Atlas › Modular forms › Weight 2 › Level 359 › 359.2.a › 359.2.a.d

Newform orbit 359.2.a.d

Trace form

The 24 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$24q + q^{2} + 5q^{3} + 31q^{4} + 6q^{5} + 7q^{6} + 5q^{7} + 39q^{9} + 4q^{10} + q^{11} + 8q^{12} + 19q^{13} - q^{14} - q^{15} + 37q^{16} + 5q^{17} - 7q^{18} + 25q^{19} - q^{20} + 14q^{21} - 13q^{22} - 8q^{23} - q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 24. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{24} - 16x^{23} - 432x^{22} + 7916x^{21} + 71773x^{20} - 1647787x^{19} - 5314242x^{18} + 189413891x^{17} + 79776321x^{16} - 13299677197x^{15} + 15341122200x^{14} + 597599376360x^{13} - 1254390106026x^{12} - 17542145001610x^{11} + 46730428439423x^{10} + 337385732003421x^{9} - 997367055335226x^{8} - 4179745081758668x^{7} + 12579163914229316x^{6} + 31772047024681079x^{5} - 89283636557071556x^{4} - 132128550962174365x^{3} + 305461123745837596x^{2} + 218978553474786084x - 315336482962270584$

Atkin–Lehner signs

$p$	$w_p$
359	$-1$

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
$\operatorname{tr} a_p$	1	5	6	5	1	19	5	25	−8	1	41	23	31	−4	−10	−1	7	38	3	−16	58	−1	−14	58	77

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.184 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, 359]: [0, 31] (certified at ell = 2147476963)
T = [(2, 1), (3, 3)]; h of degree 30 = 1 x 30 (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(359, 2)[3]   # 359.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 359:2 1000