Atlas › Modular forms › Weight 2 › Level 826 › 826.2.a › 826.2.a.f

Newform orbit 826.2.a.f

Trace form

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

$2q + 2q^{2} - q^{3} + 2q^{4} - 2q^{5} - q^{6} - 2q^{7} + 2q^{8} - 3q^{9} - 2q^{10} - 9q^{11} - q^{12} - 5q^{13} - 2q^{14} - 4q^{15} + 2q^{16} - 4q^{17} - 3q^{18} + 9q^{19} - 2q^{20} + q^{21} - 9q^{22} - 7q^{23} - q^{24} + O(q^{25})$

Coefficient field

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

$x^{2} + 3x + 1$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
7	$+1$
59	$-1$

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
$\operatorname{tr} a_p$	2	−1	−2	−2	−9	−5	−4	9	−7	−15	2	−4	−4	3	6	1	2	0	1	−20	1	−22	24	7	−3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.299 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 7, 14, 59, 118, 413, 826]: [0, 1, 1, 4, 6, 17, 42, 124] (certified at ell = 2147481901)
T = [(3, 1), (5, 1)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 4 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(826, 2)[5]   # 826.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 826:2 1000