🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 10 β€Ί Level 31 β€Ί 31.10.a β€Ί 31.10.a.a

Newform orbit 31.10.a.a

Trace form

The 10 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

10qβˆ’33q2βˆ’8q3+2389q4βˆ’3297q5βˆ’7680q6βˆ’495q7βˆ’64191q8+39998q9+122176q10βˆ’110700q11+126488q12βˆ’41330q13βˆ’427974q14βˆ’558898q15+388881q16βˆ’1299708q17βˆ’2735613q18βˆ’1079891q19βˆ’5556678q20βˆ’3605290q21βˆ’3059792q22βˆ’3288786q23βˆ’11419200q24+O(q25)10q - 33q^{2} - 8q^{3} + 2389q^{4} - 3297q^{5} - 7680q^{6} - 495q^{7} - 64191q^{8} + 39998q^{9} + 122176q^{10} - 110700q^{11} + 126488q^{12} - 41330q^{13} - 427974q^{14} - 558898q^{15} + 388881q^{16} - 1299708q^{17} - 2735613q^{18} - 1079891q^{19} - 5556678q^{20} - 3605290q^{21} - 3059792q^{22} - 3288786q^{23} - 11419200q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 10. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x10+41x9βˆ’113648x8βˆ’2005658x7+4382772552x6βˆ’18513936584x5βˆ’63709733855552x4+1422127897557504x3+253313193944964224x2βˆ’3379429521593077760xβˆ’261104268679451346944x^{10} + 41x^{9} - 113648x^{8} - 2005658x^{7} + 4382772552x^{6} - 18513936584x^{5} - 63709733855552x^{4} + 1422127897557504x^{3} + 253313193944964224x^{2} - 3379429521593077760x - 261104268679451346944

Atkin–Lehner signs

ppwpw_p
31+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’33βˆ’8βˆ’3297βˆ’495βˆ’110700βˆ’41330βˆ’1299708βˆ’1079891βˆ’3288786βˆ’13938312βˆ’92352108067838βˆ’15522651108971149119811641590380βˆ’216434421βˆ’112505150βˆ’763033272βˆ’744464301βˆ’253686584βˆ’723745828βˆ’27165804βˆ’1774306026βˆ’863529649

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 0.31 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 engine's checks
  • dim M^+ at levels [1, 31]: [1, 25] (certified at ell = 2147482951)
  • T = [(2, 1), (3, 1)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 7 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(31, 10)[0]   # 31.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 31:10 1000