Atlas › Modular forms › Weight 8 › Level 29 › 29.8.a › 29.8.a.a

Newform orbit 29.8.a.a

Trace form

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

$7q - 8q^{2} - 82q^{3} + 346q^{4} - 320q^{5} - 938q^{6} - 1704q^{7} + 2082q^{8} + 4061q^{9} - 14114q^{10} - 14498q^{11} - 42334q^{12} - 9024q^{13} - 45880q^{14} - 24238q^{15} - 3022q^{16} - 54882q^{17} + 17218q^{18} - 100572q^{19} - 86022q^{20} + 34304q^{21} + 9086q^{22} + 55340q^{23} - 138966q^{24} + O(q^{25})$

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

$x^{7} + 90x^{6} - 5318x^{5} - 490252x^{4} + 4031709x^{3} + 603956346x^{2} + 8841811860x + 35942621512$

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 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
$\operatorname{tr} a_p$	−8	−82	−320	−1704	−14498	−9024	−54882	−100572	55340	170723	−153330	−457050	−864230	−1590058	−204750	−2696092	1434256	−5622798	−5380324	4605140	3979266	−6355522	3077704	743498	−7924270

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.264 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, 19] (certified at ell = 2147483549)
T = [(2, 1), (3, 1)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 4 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, 8)[0]   # 29.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 29:8 1000