🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 4 β€Ί Level 233 β€Ί 233.4.a β€Ί 233.4.a.a

Newform orbit 233.4.a.a

Trace form

The 26 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

26qβˆ’4q2βˆ’32q3+80q4βˆ’26q5βˆ’40q6βˆ’150q7βˆ’48q8+194q9βˆ’184q10βˆ’116q11βˆ’387q12βˆ’240q13βˆ’89q14βˆ’212q15+164q16βˆ’270q17βˆ’201q18βˆ’222q19βˆ’391q20βˆ’238q21βˆ’643q22βˆ’546q23βˆ’629q24+O(q25)26q - 4q^{2} - 32q^{3} + 80q^{4} - 26q^{5} - 40q^{6} - 150q^{7} - 48q^{8} + 194q^{9} - 184q^{10} - 116q^{11} - 387q^{12} - 240q^{13} - 89q^{14} - 212q^{15} + 164q^{16} - 270q^{17} - 201q^{18} - 222q^{19} - 391q^{20} - 238q^{21} - 643q^{22} - 546q^{23} - 629q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 26. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x26+36x25+96x24βˆ’10222x23βˆ’93218x22+1119859x21+15392984x20βˆ’57995487x19βˆ’1245821503x18+1242831109x17+59081950470x16+7254372669x15βˆ’1758212761888x14βˆ’841191791792x13+33700482806663x12+13757901848292x11βˆ’413829703327832x10βˆ’39398320014245x9+3118334604050996x8βˆ’1057751632920477x7βˆ’12912266724440318x6+10524251871653906x5+21628165813806167x4βˆ’28912888193109198x3+1763555844275334x2+7496810254674912xβˆ’1360496730516642x^{26} + 36x^{25} + 96x^{24} - 10222x^{23} - 93218x^{22} + 1119859x^{21} + 15392984x^{20} - 57995487x^{19} - 1245821503x^{18} + 1242831109x^{17} + 59081950470x^{16} + 7254372669x^{15} - 1758212761888x^{14} - 841191791792x^{13} + 33700482806663x^{12} + 13757901848292x^{11} - 413829703327832x^{10} - 39398320014245x^{9} + 3118334604050996x^{8} - 1057751632920477x^{7} - 12912266724440318x^{6} + 10524251871653906x^{5} + 21628165813806167x^{4} - 28912888193109198x^{3} + 1763555844275334x^{2} + 7496810254674912x - 1360496730516642

Atkin–Lehner signs

ppwpw_p
233βˆ’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’4βˆ’32βˆ’26βˆ’150βˆ’116βˆ’240βˆ’270βˆ’222βˆ’546βˆ’122βˆ’1174βˆ’694βˆ’590βˆ’1698βˆ’1548βˆ’112248βˆ’1014βˆ’1828376βˆ’842βˆ’3258βˆ’628βˆ’380βˆ’5062

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 0.159 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, 233]: [1, 60] (certified at ell = 2147482481)
  • T = [(2, 1), (3, 1)]; h of degree 58 = 1 x 58 (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(233, 4)[0]   # 233.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 233:4 1000