Atlas › Modular forms › Weight 4 › Level 74 › 74.4.a › 74.4.a.d

Newform orbit 74.4.a.d

Trace form

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

$4q + 8q^{2} + 4q^{3} + 16q^{4} + 21q^{5} + 8q^{6} + 23q^{7} + 32q^{8} + 88q^{9} + 42q^{10} - 66q^{11} + 16q^{12} + 53q^{13} + 46q^{14} - 60q^{15} + 64q^{16} + 12q^{17} + 176q^{18} - 34q^{19} + 84q^{20} - 439q^{21} - 132q^{22} + 45q^{23} + 32q^{24} + O(q^{25})$

Coefficient field

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

$x^{4} - 25x^{3} + 105x^{2} + 533x + 466$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
37	$-1$

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
$\operatorname{tr} a_p$	8	4	21	23	−66	53	12	−34	45	−21	17	148	−174	−514	−93	3	354	1139	965	27	2168	449	−1293	−264	842

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.102 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 engine's checks

dim M^+ at levels [1, 2, 37, 74]: [1, 2, 11, 31] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 2 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(74, 4)[3]   # 74.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 74:4 1000