Atlas › Modular forms › Weight 2 › Level 754 › 754.2.a › 754.2.a.j

Newform orbit 754.2.a.j

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{31}$ , whose minimal polynomial is

$x^{6} - 41x^{5} - 6536x^{4} + 88467x^{3} + 8168261x^{2} + 32619228x - 337574068$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
13	$-1$
29	$-1$

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
$\operatorname{tr} a_p$	6	2	3	−1	7	6	8	−7	6	6	−4	22	1	4	−24	−6	−10	−3	−16	−8	1	−39	8	5	1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.506 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, 13, 26, 29, 58, 377, 754]: [0, 1, 1, 5, 3, 9, 36, 108] (certified at ell = 2147483353)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (17, 5), (19, 6), (23, 7), (31, 8)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 8 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(754, 2)[9]   # 754.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 754:2 1000