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

Newform orbit 29.8.a.b

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + 80q^{3} + 922q^{4} + 180q^{5} + 358q^{6} + 1040q^{7} - 4620q^{8} + 10986q^{9} + 8496q^{10} + 7384q^{11} + 49720q^{12} + 20820q^{13} + 50976q^{14} + 43516q^{15} + 122082q^{16} - 11620q^{17} + 66060q^{18} + 75068q^{19} - 42914q^{20} + 51480q^{21} - 36950q^{22} + 62040q^{23} - 205942q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} - 80x^{9} - 14687x^{8} + 1113860x^{7} + 71958463x^{6} - 5289793280x^{5} - 129086782781x^{4} + 9719420309660x^{3} + 49472203198620x^{2} - 5777530641977040x + 50106826279797696$

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
$\operatorname{tr} a_p$	0	80	180	1040	7384	20820	−11620	75068	62040	−243890	200600	−367740	932764	1443560	−286960	3953220	6712320	1905196	−2718200	3447736	−2554460	4187744	3498720	−303268	4908620

$\operatorname{tr} a_n$ 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)[1]   # 29.8.a.b

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

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