Atlas › Modular forms › Weight 8 › Level 7 › 7.8.a › 7.8.a.b

Newform orbit 7.8.a.b

Trace form

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

$2q - 3q^{2} + 94q^{3} + 181q^{4} + 330q^{5} - 1006q^{6} - 686q^{7} - 1185q^{8} + 1774q^{9} - 4820q^{10} + 2844q^{11} + 11102q^{12} + 2534q^{13} + 1029q^{14} + 24160q^{15} - 35663q^{16} - 1488q^{17} - 83971q^{18} + 32810q^{19} + 42840q^{20} - 32242q^{21} + 45904q^{22} - 6576q^{23} - 27150q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{865})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} - 91x + 1854$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$

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
$\operatorname{tr} a_p$	−3	94	330	−686	2844	2534	−1488	32810	−6576	20640	−391836	367392	734664	−480476	−1089108	2858844	160170	−864646	−328648	−7500216	4301244	−6408440	11659074	9772260	10762752

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.234 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, 7]: [1, 5] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 1 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(7, 8)[1]   # 7.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 7:8 1000