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

Newform orbit 29.10.a.b

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + 16q^{2} + 242q^{3} + 3498q^{4} + 1762q^{5} + 5446q^{6} + 12080q^{7} + 25350q^{8} + 43026q^{9} - 46678q^{10} + 24474q^{11} - 14210q^{12} + 107722q^{13} + 677768q^{14} + 505426q^{15} + 1656882q^{16} + 982120q^{17} + 2364102q^{18} + 2084360q^{19} + 4689410q^{20} + 2911344q^{21} + 2725230q^{22} + 3004004q^{23} + 7893170q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} - 258x^{11} - 116596x^{10} + 27201108x^{9} + 5306688394x^{8} - 998994873488x^{7} - 123013940482576x^{6} + 15800217471906180x^{5} + 1514193085561310445x^{4} - 99305259061604545550x^{3} - 8618670563607879603300x^{2} + 127588494093131623077000x + 11170825094960211961980000$

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
$\operatorname{tr} a_p$	16	242	1762	12080	24474	107722	982120	2084360	3004004	8487372	17872478	452980	−69039804	5379186	−49104062	2253998	51587572	251179296	741046264	488700124	1432375020	400834638	1525085236	691159332	2494422276

$\operatorname{tr} a_n$ 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)[1]   # 29.10.a.b

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

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