Atlas › Modular forms › Weight 6 › Level 74 › 74.6.a › 74.6.a.c

Newform orbit 74.6.a.c

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 12q^{2} - 8q^{3} + 48q^{4} - 42q^{5} + 32q^{6} + 84q^{7} - 192q^{8} + 193q^{9} + 168q^{10} + 304q^{11} - 128q^{12} - 806q^{13} - 336q^{14} + 112q^{15} + 768q^{16} + 246q^{17} - 772q^{18} - 4442q^{19} - 672q^{20} - 7630q^{21} - 1216q^{22} - 2898q^{23} + 512q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 50x^{2} + 383x - 1226$

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 odd 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
$\operatorname{tr} a_p$	−12	−8	−42	84	304	−806	246	−4442	−2898	−1790	−14606	−4107	8044	−8150	19788	38328	−17062	−32890	5540	−14892	−2492	−129958	−139996	−58606	29814

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.253 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, 17, 49] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (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(74, 6)[2]   # 74.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 74:6 1000