Atlas › Modular forms › Weight 2 › Level 740 › 740.2.a › 740.2.a.f

Newform orbit 740.2.a.f

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 3q^{3} + 4q^{5} - 5q^{7} + 13q^{9} + 5q^{11} - 6q^{13} + 3q^{15} + 10q^{17} + 19q^{21} + 2q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17} + 11T_{19}$ , whose minimal polynomial is

$x^{4} - 61x^{3} - 6960x^{2} + 199976x + 4570078$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 740$ )
5	$-1$
37	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	0	3	4	−5	5	−6	10	0	2	2	−4	4	29	4	−9	−3	−10	8	14	−17	7	−24	31	−4	12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.395 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, 4, 5, 10, 20, 37, 74, 148, 185, 370, 740]: [0, 1, 2, 1, 3, 6, 3, 11, 22, 20, 60, 120] (certified at ell = 2147483629)
T = [(3, 1), (7, 3), (11, 5), (13, 7), (17, 9), (19, 11)]; h of degree 12 = 1 x 12 (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(740, 2)[5]   # 740.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 740:2 1000