Atlas › Modular forms › Weight 10 › Level 29 › 29.10.a › 29.10.a.a

Newform orbit 29.10.a.a

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - 244q^{3} + 1194q^{4} - 738q^{5} - 2330q^{6} - 7128q^{7} - 13776q^{8} + 33017q^{9} + 37812q^{10} - 59512q^{11} - 127348q^{12} - 165758q^{13} - 406080q^{14} - 693178q^{15} - 1044958q^{16} - 394814q^{17} - 1676576q^{18} - 2256606q^{19} - 2237578q^{20} - 1750168q^{21} - 5311718q^{22} - 1699500q^{23} - 4446318q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} + 244x^{8} - 75885x^{7} - 15893670x^{6} + 2188254679x^{5} + 298460397160x^{4} - 24455930684547x^{3} - 1131583192745078x^{2} + 4771480185165784x + 43851759845977856$

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
$\operatorname{tr} a_p$	0	−244	−738	−7128	−59512	−165758	−394814	−2256606	−1699500	−6365529	−11929632	14454898	52495202	21819888	44968948	−111394302	−236142720	−241129054	−672046492	−475841956	−424813822	−170801148	−468898296	−676036598	170708754

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.308 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, 23] (certified at ell = 2147483549)
T = [(2, 1), (3, 1)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 6 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, 10)[0]   # 29.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 29:10 1000