Atlas › Modular forms › Weight 4 › Level 211 › 211.4.a › 211.4.a.b

Newform orbit 211.4.a.b

Trace form

The 29 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$29q + 6q^{2} + 13q^{3} + 132q^{4} + 89q^{5} + 38q^{6} + 24q^{7} + 72q^{8} + 390q^{9} + 80q^{10} + 147q^{11} + 93q^{12} + 131q^{13} + 199q^{14} + 102q^{15} + 628q^{16} + 366q^{17} + 88q^{18} + 145q^{19} + 560q^{20} + 398q^{21} - 19q^{22} + 434q^{23} + 314q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 29. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{29} - 19x^{28} - 626x^{27} + 13238x^{26} + 157184x^{25} - 3873236x^{24} - 20628323x^{23} + 632487411x^{22} + 1547883114x^{21} - 64802629246x^{20} - 66792573890x^{19} + 4420530320548x^{18} + 1491022737532x^{17} - 206866592992442x^{16} - 9241598212129x^{15} + 6716206509169649x^{14} - 100154652379144x^{13} - 150802480195194600x^{12} - 9489855164796064x^{11} + 2300775891506133952x^{10} + 512360471655410173x^{9} - 22981497716222486607x^{8} - 9950476010729638608x^{7} + 140032421976899095324x^{6} + 94803787397370023880x^{5} - 450490457794146027272x^{4} - 426441598746287398256x^{3} + 517122724552261027664x^{2} + 649115010203497077360x + 97184115874110549968$

Atkin–Lehner signs

$p$	$w_p$
211	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	6	13	89	24	147	131	366	145	434	1315	436	471	2090	197	956	1400	−68	2413	−807	294	−1833	−906	954	2870	−418

$\operatorname{tr} a_n$ for every $n \le 1000$ 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 $T$ over $\mathbb{Z}$ 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 · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(211, 4)[1]   # 211.4.a.b

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

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