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Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 835 β€Ί 835.2.a β€Ί 835.2.a.c

Newform orbit 835.2.a.c

Trace form

The 16 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

16qβˆ’5q2βˆ’7q3+15q4βˆ’16q5βˆ’3q6βˆ’7q7βˆ’15q8+11q9+5q10βˆ’4q11βˆ’19q12βˆ’18q13βˆ’8q14+7q15+13q16βˆ’23q17βˆ’10q18βˆ’q19βˆ’15q20βˆ’6q21βˆ’8q22βˆ’20q23βˆ’4q24+O(q25)16q - 5q^{2} - 7q^{3} + 15q^{4} - 16q^{5} - 3q^{6} - 7q^{7} - 15q^{8} + 11q^{9} + 5q^{10} - 4q^{11} - 19q^{12} - 18q^{13} - 8q^{14} + 7q^{15} + 13q^{16} - 23q^{17} - 10q^{18} - q^{19} - 15q^{20} - 6q^{21} - 8q^{22} - 20q^{23} - 4q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 16. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x16+12x15+22x14βˆ’219x13βˆ’755x12+1318x11+6486x10βˆ’3380x9βˆ’24657x8+4481x7+45760x6βˆ’6120x5βˆ’39982x4+6958x3+13158x2βˆ’1691xβˆ’1399x^{16} + 12x^{15} + 22x^{14} - 219x^{13} - 755x^{12} + 1318x^{11} + 6486x^{10} - 3380x^{9} - 24657x^{8} + 4481x^{7} + 45760x^{6} - 6120x^{5} - 39982x^{4} + 6958x^{3} + 13158x^{2} - 1691x - 1399

Atkin–Lehner signs

ppwpw_p
5+1+1
167+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=+1w_N = +1, so the sign of the functional equation is (βˆ’1)k/2wN=βˆ’1(-1)^{k/2} w_N = -1: each L-function in the orbit has odd analytic rank.

Traces of Hecke eigenvalues

pp2357111317192329313741434753596167717379838997
tr⁑ap\operatorname{tr} a_pβˆ’5βˆ’7βˆ’16βˆ’7βˆ’4βˆ’18βˆ’23βˆ’1βˆ’20βˆ’264βˆ’16βˆ’17βˆ’12βˆ’46βˆ’45βˆ’10βˆ’10βˆ’25βˆ’8βˆ’2246βˆ’20βˆ’3βˆ’26

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 0.247 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, 5, 167, 835]: [0, 1, 15, 86] (certified at ell = 2147483237)
  • T = [(2, 1), (3, 1)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 6 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(835, 2)[2]   # 835.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 835:2 1000