Atlas › Modular forms › Weight 2 › Level 930 › 930.2.a › 930.2.a.q

Newform orbit 930.2.a.q

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{65})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{7} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{37} + 9T_{41} + 10T_{43} + 11T_{47} + 12T_{53}$ , whose minimal polynomial is

$x^{2} - 58x - 13784$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$-1$
5	$+1$
31	$-1$

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	2	−2	−1	5	12	−8	1	5	0	2	2	−2	−9	6	3	−2	−6	−6	−17	−9	−9	−16	−5	0

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.387 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, 3, 5, 6, 10, 15, 30, 31, 62, 93, 155, 186, 310, 465, 930]: [0, 1, 1, 1, 3, 3, 4, 10, 3, 10, 12, 18, 36, 52, 68, 200] (certified at ell = 2147482921)
T = [(7, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (37, 8), (41, 9), (43, 10), (47, 11), (53, 12)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 7 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(930, 2)[16]   # 930.2.a.q

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

cargo run --release -p sagebrush-web --example atlas -- 930:2 1000