🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 4 β€Ί Level 241 β€Ί 241.4.a β€Ί 241.4.a.a

Newform orbit 241.4.a.a

Trace form

The 27 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

27qβˆ’11q2βˆ’14q3+89q4βˆ’50q5βˆ’63q6βˆ’68q7βˆ’129q8+135q9βˆ’79q10βˆ’298q11βˆ’125q12βˆ’106q13βˆ’284q14βˆ’358q15+117q16βˆ’132q17βˆ’405q18βˆ’216q19βˆ’570q20βˆ’216q21βˆ’38q22βˆ’814q23βˆ’873q24+O(q25)27q - 11q^{2} - 14q^{3} + 89q^{4} - 50q^{5} - 63q^{6} - 68q^{7} - 129q^{8} + 135q^{9} - 79q^{10} - 298q^{11} - 125q^{12} - 106q^{13} - 284q^{14} - 358q^{15} + 117q^{16} - 132q^{17} - 405q^{18} - 216q^{19} - 570q^{20} - 216q^{21} - 38q^{22} - 814q^{23} - 873q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 27. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x27+25x26βˆ’209x25βˆ’8960x24+5093x23+1403197x22+2234726x21βˆ’127390118x20βˆ’277555725x19+7448916991x18+15063575608x17βˆ’291538954093x16βˆ’431173859379x15+7609789175720x14+6115501785968x13βˆ’127761019083425x12βˆ’26011100279290x11+1304467793438453x10βˆ’233701029100312x9βˆ’7784189928462843x8+2796666051785840x7+26026909298657596x6βˆ’10504993601392807x5βˆ’45621625046525135x4+16211128356846434x3+36185752952641940x2βˆ’8089234974528480xβˆ’8708225245120064x^{27} + 25x^{26} - 209x^{25} - 8960x^{24} + 5093x^{23} + 1403197x^{22} + 2234726x^{21} - 127390118x^{20} - 277555725x^{19} + 7448916991x^{18} + 15063575608x^{17} - 291538954093x^{16} - 431173859379x^{15} + 7609789175720x^{14} + 6115501785968x^{13} - 127761019083425x^{12} - 26011100279290x^{11} + 1304467793438453x^{10} - 233701029100312x^{9} - 7784189928462843x^{8} + 2796666051785840x^{7} + 26026909298657596x^{6} - 10504993601392807x^{5} - 45621625046525135x^{4} + 16211128356846434x^{3} + 36185752952641940x^{2} - 8089234974528480x - 8708225245120064

Atkin–Lehner signs

ppwpw_p
241βˆ’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=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’11βˆ’14βˆ’50βˆ’68βˆ’298βˆ’106βˆ’132βˆ’216βˆ’814βˆ’736βˆ’994βˆ’472βˆ’1318βˆ’974βˆ’2012βˆ’980βˆ’2590βˆ’832βˆ’132βˆ’5996932βˆ’1666878βˆ’23441234

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 0.164 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, 241]: [1, 62] (certified at ell = 2147482801)
  • T = [(2, 1), (3, 1)]; h of degree 60 = 1 x 60 (phi(ord eps) x dim S_k^new); 9 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(241, 4)[0]   # 241.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 241:4 1000