🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 2 β€Ί Level 635 β€Ί 635.2.a β€Ί 635.2.a.e

Newform orbit 635.2.a.e

Trace form

The 8 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

8qβˆ’2q2βˆ’2q3+2q4βˆ’8q5βˆ’3q6βˆ’5q7βˆ’3q8+4q9+2q10+q11βˆ’3q12βˆ’17q13+4q14+2q15βˆ’6q16βˆ’5q17βˆ’11q18βˆ’8q19βˆ’2q20βˆ’11q21βˆ’14q22βˆ’8q23βˆ’4q24+O(q25)8q - 2q^{2} - 2q^{3} + 2q^{4} - 8q^{5} - 3q^{6} - 5q^{7} - 3q^{8} + 4q^{9} + 2q^{10} + q^{11} - 3q^{12} - 17q^{13} + 4q^{14} + 2q^{15} - 6q^{16} - 5q^{17} - 11q^{18} - 8q^{19} - 2q^{20} - 11q^{21} - 14q^{22} - 8q^{23} - 4q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 8. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x8+4x7βˆ’12x6βˆ’35x5+60x4+35x3βˆ’50x2+14xβˆ’1x^{8} + 4x^{7} - 12x^{6} - 35x^{5} + 60x^{4} + 35x^{3} - 50x^{2} + 14x - 1

Atkin–Lehner signs

ppwpw_p
5+1+1
127+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’2βˆ’2βˆ’8βˆ’51βˆ’17βˆ’5βˆ’8βˆ’8βˆ’4βˆ’20βˆ’459βˆ’5βˆ’8βˆ’1326βˆ’23βˆ’15βˆ’1βˆ’28βˆ’2649βˆ’18

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 0.22 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, 127, 635]: [0, 1, 11, 66] (certified at ell = 2147483269)
  • T = [(2, 1), (3, 1)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 5 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(635, 2)[4]   # 635.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 635:2 1000