🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 929 β€Ί 929.2.a β€Ί 929.2.a.c

Newform orbit 929.2.a.c

Trace form

The 47 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

47q+3q2+3q3+63q4+2q5+6q6+39q7+6q8+56q9+15q10+21q11βˆ’10q12+16q13+2q14+35q15+87q16+3q17βˆ’q18+46q19βˆ’10q20+17q21+25q22+12q23+O(q25)47q + 3q^{2} + 3q^{3} + 63q^{4} + 2q^{5} + 6q^{6} + 39q^{7} + 6q^{8} + 56q^{9} + 15q^{10} + 21q^{11} - 10q^{12} + 16q^{13} + 2q^{14} + 35q^{15} + 87q^{16} + 3q^{17} - q^{18} + 46q^{19} - 10q^{20} + 17q^{21} + 25q^{22} + 12q^{23} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 47. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x47βˆ’6x46βˆ’165x45+1046x44+12026x43βˆ’81513x42βˆ’511194x41+3764305x40+14101679x39βˆ’115245495x38βˆ’266016583x37+2481781385x36+3527337352x35βˆ’38966166377x34βˆ’33367256492x33+456686572553x32+227907145591x31βˆ’4059522659741x30βˆ’1158080264548x29+27653364280651x28+4830712098971x27βˆ’145138274961293x26βˆ’20137240560897x25+587205364277733x24+90844624880920x23βˆ’1823094513072909x22βˆ’369899431001629x21+4302955860054386x20+1143939832047744x19βˆ’7617412486450295x18βˆ’2501577889898149x17+9953785003797272x16+3762339838190752x15βˆ’9446528496191357x14βˆ’3822103897594547x13+6415629833745829x12+2568417251258710x11βˆ’3073214937487960x10βˆ’1103718649364606x9+1017507106759046x8+284444574106092x7βˆ’223431798842140x6βˆ’38173647931964x5+29558672724648x4+1716214031600x3βˆ’1870426745648x2+34555885272x+31739480792x^{47} - 6x^{46} - 165x^{45} + 1046x^{44} + 12026x^{43} - 81513x^{42} - 511194x^{41} + 3764305x^{40} + 14101679x^{39} - 115245495x^{38} - 266016583x^{37} + 2481781385x^{36} + 3527337352x^{35} - 38966166377x^{34} - 33367256492x^{33} + 456686572553x^{32} + 227907145591x^{31} - 4059522659741x^{30} - 1158080264548x^{29} + 27653364280651x^{28} + 4830712098971x^{27} - 145138274961293x^{26} - 20137240560897x^{25} + 587205364277733x^{24} + 90844624880920x^{23} - 1823094513072909x^{22} - 369899431001629x^{21} + 4302955860054386x^{20} + 1143939832047744x^{19} - 7617412486450295x^{18} - 2501577889898149x^{17} + 9953785003797272x^{16} + 3762339838190752x^{15} - 9446528496191357x^{14} - 3822103897594547x^{13} + 6415629833745829x^{12} + 2568417251258710x^{11} - 3073214937487960x^{10} - 1103718649364606x^{9} + 1017507106759046x^{8} + 284444574106092x^{7} - 223431798842140x^{6} - 38173647931964x^{5} + 29558672724648x^{4} + 1716214031600x^{3} - 1870426745648x^{2} + 34555885272x + 31739480792

Atkin–Lehner signs

ppwpw_p
929βˆ’1-1

For pβˆ₯Np \parallel N every form in the orbit has ap=βˆ’pk/2βˆ’1wpa_p = -p^{k/2-1} w_p. The Fricke sign is wN=βˆ’1w_N = -1, so the sign of the functional equation is (βˆ’1)k/2wN=+1(-1)^{k/2} w_N = +1: each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

pp2357111317192329313741434753596167717379838997
tr⁑ap\operatorname{tr} a_p3323921163461212812552315βˆ’52176971151862βˆ’4112

tr⁑an\operatorname{tr} a_n for every n≀1000n \le 1000 is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.291 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of TT over Z\mathbb{Z} is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and tr⁑ap\operatorname{tr} a_p for every p<1000p < 1000 equal LMFDB's.

The engine's checks
  • dim M^+ at levels [1, 929]: [0, 78] (certified at ell = 2147480161)
  • T = [(2, 1), (3, 1)]; h of degree 77 = 1 x 77 (phi(ord eps) x dim S_k^new); 8 primes, 0 rejected
Open in the notebook Β· JSON Β· stored data
Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(929, 2)[2]   # 929.2.a.c

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 929:2 1000