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

Newform orbit 74.6.a.e

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 20q^{2} + 19q^{3} + 80q^{4} + 95q^{5} + 76q^{6} + 170q^{7} + 320q^{8} + 598q^{9} + 380q^{10} + 903q^{11} + 304q^{12} + 1371q^{13} + 680q^{14} - 8q^{15} + 1280q^{16} + 2070q^{17} + 2392q^{18} + 2358q^{19} + 1520q^{20} + 2832q^{21} + 3612q^{22} + 2097q^{23} + 1216q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} - 114x^{4} - 2018x^{3} + 431800x^{2} - 6144255x - 22198214$

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
$\operatorname{tr} a_p$	20	19	95	170	903	1371	2070	2358	2097	2927	−5849	−6845	−14427	−6972	−18962	−23576	−18316	−18695	−85273	8760	−10425	48425	27704	233646	251694

$\operatorname{tr} a_n$ 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)[4]   # 74.6.a.e

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

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