Atlas › Modular forms › Weight 12 › Level 26 › 26.12.a › 26.12.a.a

Newform orbit 26.12.a.a

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 64q^{2} - 243q^{3} + 2048q^{4} - 4985q^{5} + 7776q^{6} + 10155q^{7} - 65536q^{8} + 117531q^{9} + 159520q^{10} + 690610q^{11} - 248832q^{12} + 742586q^{13} - 324960q^{14} + 3308625q^{15} + 2097152q^{16} + 4927q^{17} - 3760992q^{18} - 13606074q^{19} - 5104640q^{20} - 33128613q^{21} - 22099520q^{22} - 25659912q^{23} + 7962624q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{10921})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 5228x - 4350108$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
13	$-1$

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
$\operatorname{tr} a_p$	−64	−243	−4985	10155	690610	742586	4927	−13606074	−25659912	−89468700	−290399012	−299050245	950412634	172567923	2308446007	−1230762002	−1885996810	24364350	−14619578510	−23372029279	−8441959124	−54336993224	−23534785824	20490366052	59265724824

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.546 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, 2, 13, 26]: [2, 4, 15, 41] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 5 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(26, 12)[0]   # 26.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 26:12 1000