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

Newform orbit 26.12.a.b

Trace form

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

$2q + 64q^{2} - 243q^{3} + 2048q^{4} - 2035q^{5} - 7776q^{6} - 27601q^{7} + 65536q^{8} - 250317q^{9} - 65120q^{10} + 100336q^{11} - 248832q^{12} - 742586q^{13} - 883232q^{14} - 2358585q^{15} + 2097152q^{16} - 10853037q^{17} - 8010144q^{18} - 29292132q^{19} - 2083840q^{20} + 20552049q^{21} + 3210752q^{22} - 40320844q^{23} - 7962624q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{16545})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 2278x - 41736224$

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	−2035	−27601	100336	−742586	−10853037	−29292132	−40320844	−161271964	−16103782	−110294223	684375690	570956615	4141326907	5163836442	1880901200	4997961134	−12259964908	6206403685	−19386934280	−29776852112	13498378950	−51166597872	126097288336

$\operatorname{tr} a_n$ 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)[1]   # 26.12.a.b

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

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