Atlas › Modular forms › Weight 2 › Level 574 › 574.2.a › 574.2.a.m

Newform orbit 574.2.a.m

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{5} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ , whose minimal polynomial is

$x^{4} - 178x^{3} + 4924x^{2} + 393892x - 15942808$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
7	$+1$
41	$-1$

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
$\operatorname{tr} a_p$	−4	−1	3	−4	4	−6	−1	2	6	17	5	−6	4	−13	6	29	16	21	4	17	12	−27	−18	37	−3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.494 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, 7, 14, 41, 82, 287, 574]: [0, 1, 1, 4, 4, 12, 30, 88] (certified at ell = 2147482921)
T = [(3, 1), (5, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 6 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(574, 2)[12]   # 574.2.a.m

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

cargo run --release -p sagebrush-web --example atlas -- 574:2 1000