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{"Title": "Petersson norms of higher-level newforms", "Definition": "For an embedded normalized newform $f$ of exact level $N>1$, this table gives its Petersson self-product $P_\\nu(f)$ in two conventions: the integral and its division by $[\\mathrm{SL}_2(\\mathbb Z):\\Gamma_0(N)]$ CITE{BK} CITE{PARI}.", "Parameters": {"form": {"type": "Symbolic", "title": "embedded normalized newform", "display": "$f$", "constraints": "a full LMFDB label $N.k.a.x.n.i$ of an embedded normalized newform with exact level $N>1$, integer weight $k\\geq2$, and $\\chi(-1)=(-1)^k$"}, "normalization": {"type": "Symbolic", "title": "Petersson convention", "display": "$\\nu$", "constraints": "integral or index-normalized"}}, "Tags": ["number theory", "modular form", "period", "integral"], "Links": {"PARI": {"title": "PARI/GP: modular forms, mfpetersson and mfsymbol", "url": "https://pari.math.u-bordeaux.fr/dochtml/html/Modular_forms.html"}, "Labels": {"title": "LMFDB: labels for classical modular forms", "url": "https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/Labels"}, "Reliability": {"title": "LMFDB: reliability of classical modular form data", "url": "https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/Reliability"}, "Eta3": {"title": "LMFDB: newform 3.6.a.a", "url": "https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/3/6/a/a/"}, "Eta5": {"title": "LMFDB: newform 5.4.a.a", "url": "https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/5/4/a/a/"}, "Eta7": {"title": "LMFDB: newform 7.3.b.a", "url": "https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/7/3/b/a/"}}, "Data properties": {"type": "R", "rigour": "heuristic (agreement-checked)", "complete": "no", "complete-note": "it holds both conventions for all $138$ embeddings in three disjoint slices: weight $4$ with trivial character and exact levels $2\\leq N\\leq30$ ($61$ embeddings), weight $6$ with trivial character and $2\\leq N\\leq15$ ($27$), and weight $3$ with primitive quadratic character $\\chi_D$ at $N=|D|$ for negative fundamental $D$ with $|D|\\leq40$ ($50$). These noncanonical bounds balance low-conductor coverage across weights, characters and coefficient fields. The shorter weight-six range preserves the common roster with the critical-value companion, which has five critical integers per form. All orbits, embeddings and both conjugates in these spaces are included; empty newspaces contribute no entries", "rigour details": "Values are recorded to 100 significant digits after fresh PARI/GP computations at 160 and 220 decimal working digits, with explicit bit precision for modular symbols and coefficient embeddings CITE{PARI}. The largest relative difference was less than $10^{-150}$. The calculation uses the Haberland period-pairing method CITE{BC}. For larger coefficient fields, the generator first pairs an exact rational basis and then applies the embedded exact coordinate vectors by sesquilinearity. These are agreement checks, not certified analytic error bounds.\n\nEvery orbit is reconstructed under an exact coefficient-field map and checked on its Fourier coefficients through the Sturm bound. Rational isolating rectangles identify all 138 embeddings. The exact dimensions of all 57 selected newspaces, including empty spaces, match the roster. The labels, coefficient-field polynomials and embedding-root data also agree with the catalogue records attached to the critical-value companion CITE{Reliability}. The catalogue's floating-point coefficients are not inputs to the Petersson computations.\n\nAn independent calculation integrates over a fundamental domain using exact eta-product expansions for $\\eta(z)^6\\eta(3z)^6$ CITE{Eta3}, $\\eta(z)^4\\eta(5z)^4$ CITE{Eta5}, and $\\eta(z)^3\\eta(7z)^3$ CITE{Eta7}. It sums the prime-level cosets, integrates the vertical coordinate using the finite incomplete-gamma formula, and applies Gauss-Legendre quadrature to the horizontal coordinate. Exact polynomial moments control the quadrature before use. The two settings use Fourier sum cutoffs $70N$ and $90N$, 140 and 180 quadrature nodes, and 150 and 190 working digits. All three norms agree with PARI to more than 150 relative decimal digits.\n\nThe same independent integral, with coefficients reconstructed from exact catalogue Hecke data by Euler recurrences, checks both embeddings of 11.4.a.a and both of 19.3.b.b. Cutoffs $23N$ and $30N$, 80 and 110 quadrature nodes, and 85 and 115 working digits agree with PARI to more than 50 relative decimal digits. These spot checks do not constitute an independent calculation of every entry.\n\nPositivity, conjugation and the conversion between the two conventions are checked throughout. The full generator, argument-selection explanation, exact roster, source hashes and numerical evidence are attached."}, "Numbers": {"3.6.a.a.1.1": {"integral": "0.00005749811764526189734273482793649287781267406259434159320157403773849887405956292102575776562139835167", "index-normalized": "0.00001437452941131547433568370698412321945316851564858539830039350943462471851489073025643944140534958792"}, "4.6.a.a.1.1": {"integral": "0.00006583658338850941609748045103133439662766537045465413083205806973494904377314600215812020602705340110", "index-normalized": "0.00001097276389808490268291340850522239943794422840910902180534301162249150729552433369302003433784223352"}, "5.4.a.a.1.1": {"integral": "0.0008708001049786912568455608132545555778639959958598938873321969914127633088917739858675692898906143957", "index-normalized": "0.0001451333508297818761409268022090925963106659993099823145553661652354605514819623309779282149817690660"}, "5.6.a.a.1.1": {"integral": 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"0.08591765283182571867061771197203132703713260484429099893709572205010694941411974102774325997090611853", "index-normalized": "0.001193300733775357203758579332944879542182397289504041651904107250695929852973885292051989721818140535"}, "40.3.e.c.19.8": {"integral": "0.08591765283182571867061771197203132703713260484429099893709572205010694941411974102774325997090611853", "index-normalized": "0.001193300733775357203758579332944879542182397289504041651904107250695929852973885292051989721818140535"}}, "Comments": {"comment-labels": "Write $f(z)=\\sum_{n\\geq1}a_n e^{2\\pi inz}$, with $a_1=1$. The weight-$k$ form belongs to $S_k^{\\mathrm{new}}(\\Gamma_0(N),\\chi)$ and is identified by its full LMFDB label $N.k.a.x.n.i$ CITE{Labels}. The labels $a$ and $x$ identify the character orbit and newform orbit, $n$ is the Conrey character index, and $i$ distinguishes embeddings with that character. The attached roster specifies an exact coefficient-field polynomial, a rational isolating rectangle for its generator and Fourier coefficients through the Sturm bound for every embedding. Software list positions do not define these labels.", "comment-characters": "The weight-three slice has primitive Kronecker character $\\chi_D(m)=(D/m)$ for a negative fundamental discriminant $D$, with $N=|D|$. Its value is zero at nonunits modulo $N$. The weight-four and weight-six slices have trivial character. All forms occur at their exact level, with every coefficient-field embedding and both members of conjugate pairs.", "comment-normalization": "The word norm here denotes the self-product, not its square root. The integral convention has no division by the hyperbolic volume of the fundamental domain. PARI uses the index-normalized convention CITE{PARI}. Both conventions describe the same self-product and determine each other by the exact positive integer $\\mu(N)$. They agree at level one, which is covered by HREF{Petersson_norms_of_level_one_cusp_forms}[the level-one Petersson table].", "comment-periods": "HREF{Critical_L-values_of_higher-level_newforms}[Critical L-values of higher-level newforms] records Mellin periods of exactly the same embedded forms, with the same labels and Fourier normalization. The Petersson self-product is a different modular period, used in normalizing pairings CITE{BK}. It is not a critical arithmetic L-value."}, "Formulas": {"formula-integral": "The unaveraged self-product is $I(f)=\\int_{\\Gamma_0(N)\\backslash\\mathbb H}|f(z)|^2 y^k\\,dx\\,dy/y^2$, where $z=x+iy$ and $k$ is the weight CITE{BK}.", "formula-index": "With $\\mu(N)=N\\prod_{p\\mid N}(1+1/p)$, the two conventions are $P_{\\mathrm{integral}}(f)=I(f)$ and $P_{\\mathrm{index\\text{-}normalized}}(f)=I(f)/\\mu(N)$. Thus $P_{\\mathrm{integral}}(f)=\\mu(N)P_{\\mathrm{index\\text{-}normalized}}(f)$.", "formula-conjugation": "If $\\bar f(z)=\\sum_{n\\geq1}\\overline{a_n}e^{2\\pi inz}$, then $I(\\bar f)=I(f)>0$."}, "Similar tables": [{"table": "HREF{Critical_L-values_of_higher-level_newforms}[Critical L-values of higher-level newforms]", "relation": "records critical Mellin periods of the same 138 embedded forms, using the same full labels and Fourier normalization"}, {"table": "HREF{Petersson_norms_of_level_one_cusp_forms}[Petersson norms of level-one cusp forms]", "relation": "records the same self-product at level one, where the subgroup index is one and these two conventions coincide"}], "References": {"BC": {"bib": "K. Belabas and H. Cohen, Modular Forms in Pari/GP (2018), sections 2, 6 and 9.2.", "arxiv": "1810.00547"}, "BK": {"bib": "J. Brown and K. Klosin, On the norms of p-stabilized elliptic newforms, with an appendix by K. Conrad (2015), section 2.", "arxiv": "1402.0900"}}, "Display properties": {"number-header": "$P_\\nu(f)$"}, "Keywords": ["Petersson inner product", "modular periods", "Petersson self-products"], "Programs": {"program-pari": {"language": "PARI/GP", "code": "\\\\ Example: the unique newform 5.4.a.a.1.1.\ndefault(realprecision, 160);\nmf = mfinit([5, 4], 0);\nf = mfeigenbasis(mf)[1];\np = mfpetersson(mfsymbol(mf, f));\n[p, 6*p] \\\\ index-normalized and integral conventions"}}}