🌿 Sagebrush Atlas
Atlas β€Ί Modular forms β€Ί Weight 4 β€Ί Level 211 β€Ί 211.4.a β€Ί 211.4.a.a

Newform orbit 211.4.a.a

Trace form

The 23 newforms in this orbit are conjugate over Q\mathbb{Q}; the sum of their qq-expansions is

23qβˆ’4q2βˆ’17q3+72q4βˆ’91q5βˆ’58q6βˆ’32q7βˆ’48q8+120q9βˆ’60q10βˆ’183q11βˆ’267q12βˆ’129q13βˆ’361q14βˆ’138q15+100q16βˆ’382q17βˆ’182q18βˆ’197q19βˆ’880q20βˆ’610q21βˆ’283q22βˆ’302q23βˆ’838q24+O(q25)23q - 4q^{2} - 17q^{3} + 72q^{4} - 91q^{5} - 58q^{6} - 32q^{7} - 48q^{8} + 120q^{9} - 60q^{10} - 183q^{11} - 267q^{12} - 129q^{13} - 361q^{14} - 138q^{15} + 100q^{16} - 382q^{17} - 182q^{18} - 197q^{19} - 880q^{20} - 610q^{21} - 283q^{22} - 302q^{23} - 838q^{24} + O(q^{25})

Coefficient field

The coefficients ana_n generate a number field of degree 23. It is generated by the eigenvalue Ξ±\alpha of T=T2+T3T = T_{2} + T_{3}, whose minimal polynomial is

x23+21x22βˆ’220x21βˆ’6516x20+13650x19+812988x18βˆ’30099x17βˆ’55005279x16βˆ’25235650x15+2221970164x14+956563588x13βˆ’54939911292x12βˆ’18503593424x11+837728434420x10+295352160861x9βˆ’7833843353861x8βˆ’3988848802914x7+43290763993476x6+33763135873330x5βˆ’125564892854584x4βˆ’144429469821041x3+124878229948177x2+236806934882750x+86160174317950x^{23} + 21x^{22} - 220x^{21} - 6516x^{20} + 13650x^{19} + 812988x^{18} - 30099x^{17} - 55005279x^{16} - 25235650x^{15} + 2221970164x^{14} + 956563588x^{13} - 54939911292x^{12} - 18503593424x^{11} + 837728434420x^{10} + 295352160861x^{9} - 7833843353861x^{8} - 3988848802914x^{7} + 43290763993476x^{6} + 33763135873330x^{5} - 125564892854584x^{4} - 144429469821041x^{3} + 124878229948177x^{2} + 236806934882750x + 86160174317950

Atkin–Lehner signs

ppwpw_p
211βˆ’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’17βˆ’91βˆ’32βˆ’183βˆ’129βˆ’382βˆ’197βˆ’302βˆ’1063βˆ’432βˆ’861βˆ’1026βˆ’577βˆ’1206βˆ’2790βˆ’1682βˆ’1277291βˆ’186828690364βˆ’4974βˆ’1358

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 0.141 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, 211]: [1, 54] (certified at ell = 2147482681)
  • T = [(2, 1), (3, 1)]; h of degree 52 = 1 x 52 (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(211, 4)[0]   # 211.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 211:4 1000