Atlas › Modular forms › Weight 6 › Level 29 › 29.6.a › 29.6.a.b

Newform orbit 29.6.a.b

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + 4q^{2} + 26q^{3} + 154q^{4} + 32q^{5} + 22q^{6} + 184q^{7} + 942q^{8} + 1005q^{9} + 922q^{10} + 1106q^{11} + 214q^{12} + 408q^{13} - 2008q^{14} - 614q^{15} + 242q^{16} - 874q^{17} - 5598q^{18} + 4288q^{19} - 6350q^{20} - 4200q^{21} - 6114q^{22} - 4532q^{23} - 4318q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} - 30x^{6} - 1114x^{5} + 39408x^{4} + 116505x^{3} - 9632930x^{2} + 22233300x + 350796600$

Atkin–Lehner signs

$p$	$w_p$
29	$-1$

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
$\operatorname{tr} a_p$	4	26	32	184	1106	408	−874	4288	−4532	5887	7794	5086	19826	19498	14278	−58644	12888	102866	102996	−51596	−17566	212058	−122928	−66510	−118182

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.214 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, 29]: [1, 13] (certified at ell = 2147483549)
T = [(2, 1), (3, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 3 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(29, 6)[1]   # 29.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 29:6 1000