Atlas › Modular forms › Weight 8 › Level 15 › 15.8.a › 15.8.a.c

Newform orbit 15.8.a.c

Trace form

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

$2q + 7q^{2} + 54q^{3} + 69q^{4} + 250q^{5} + 189q^{6} + 1304q^{7} + 1449q^{8} + 1458q^{9} + 875q^{10} + 3448q^{11} + 1863q^{12} - 8988q^{13} - 12264q^{14} + 6750q^{15} - 24495q^{16} - 5492q^{17} + 5103q^{18} - 49584q^{19} + 8625q^{20} + 35208q^{21} - 127364q^{22} + 91848q^{23} + 39123q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{601})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{7}$ , whose minimal polynomial is

$x^{2} - 1311x - 24826$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
5	$-1$

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
$\operatorname{tr} a_p$	7	54	250	1304	3448	−8988	−5492	−49584	91848	181772	304232	−502316	631172	353640	−467480	−568052	287224	−2514180	−5073832	−3748816	−1477212	−4627720	−6072936	16516356	2723428

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.255 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, 3, 5, 15]: [1, 3, 5, 16] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 2 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(15, 8)[2]   # 15.8.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 15:8 1000