Atlas › Modular forms › Weight 2 › Level 915 › 915.2.a › 915.2.a.k

Newform orbit 915.2.a.k

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{7}$ , whose minimal polynomial is

$x^{8} - 4x^{7} - 38x^{6} + 152x^{5} + 320x^{4} - 1034x^{3} - 1384x^{2} + 1258x + 1369$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
5	$-1$
61	$-1$

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
$\operatorname{tr} a_p$	2	8	8	2	10	0	2	−6	6	20	−8	18	12	2	18	6	12	8	−16	28	−16	−20	10	38	−8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.423 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, 3, 5, 15, 61, 183, 305, 915]: [0, 1, 1, 4, 5, 22, 32, 128] (certified at ell = 2147482921)
T = [(2, 1), (7, 1)]; h of degree 39 = 1 x 39 (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(915, 2)[10]   # 915.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 915:2 1000