Atlas › Modular forms › Weight 6 › Level 97 › 97.6.a › 97.6.a.a

Newform orbit 97.6.a.a

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q - 21q^{2} - 36q^{3} + 271q^{4} - 89q^{5} - 217q^{6} - 451q^{7} - 1116q^{8} + 1025q^{9} + 34q^{10} - 1823q^{11} - 2229q^{12} - 414q^{13} - 17q^{14} - 2111q^{15} + 3295q^{16} - 3269q^{17} - 3771q^{18} - 4363q^{19} - 6409q^{20} - 1158q^{21} - 4008q^{22} - 17852q^{23} - 11417q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{19} + 57x^{18} - 1419x^{17} - 114616x^{16} + 530703x^{15} + 89842506x^{14} + 41322237x^{13} - 35779833235x^{12} - 50154130409x^{11} + 7973134290642x^{10} + 6458003318898x^{9} - 998874734142033x^{8} + 111681197774926x^{7} + 66509663868178071x^{6} - 44694438735091999x^{5} - 2156553233205876047x^{4} + 1704702184994107324x^{3} + 26701814162382725930x^{2} - 17126810926953462663x - 8843110132446779050$

Atkin–Lehner signs

$p$	$w_p$
97	$+1$

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 odd 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
$\operatorname{tr} a_p$	−21	−36	−89	−451	−1823	−414	−3269	−4363	−17852	−21325	−19824	−26906	805	−3063	−68829	−45182	−10149	58769	−24633	−78467	−59395	−20620	−130614	−79076	−178771

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.261 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, 97]: [1, 42] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 8 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(97, 6)[0]   # 97.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 97:6 1000