Atlas › Modular forms › Weight 8 › Level 26 › 26.8.a › 26.8.a.d

Newform orbit 26.8.a.d

Trace form

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

$2q - 16q^{2} - 12q^{3} + 128q^{4} - 146q^{5} + 96q^{6} - 1780q^{7} - 1024q^{8} + 5988q^{9} + 1168q^{10} + 10904q^{11} - 768q^{12} - 4394q^{13} + 14240q^{14} + 53796q^{15} + 8192q^{16} - 14118q^{17} - 47904q^{18} + 54408q^{19} - 9344q^{20} + 50370q^{21} - 87232q^{22} - 156464q^{23} + 6144q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{105})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 158x - 187904$

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 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	−12	−146	−1780	10904	−4394	−14118	54408	−156464	−145868	45800	348558	−156240	737252	−615196	−1536024	−1762472	−4460272	1926680	2505140	3725404	3911392	8883360	−16720548	11317132

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.27 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]: [1, 3, 9, 27] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 3 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, 8)[3]   # 26.8.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 26:8 1000