Atlas › Modular forms › Weight 2 › Level 746 › 746.2.a › 746.2.a.c

Newform orbit 746.2.a.c

Trace form

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

$6q - 6q^{2} - 3q^{3} + 6q^{4} + q^{5} + 3q^{6} - 5q^{7} - 6q^{8} + 3q^{9} - q^{10} - 8q^{11} - 3q^{12} - 8q^{13} + 5q^{14} - 6q^{15} + 6q^{16} + 8q^{17} - 3q^{18} - 26q^{19} + q^{20} - 8q^{21} + 8q^{22} + 5q^{23} + 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{6} + 2x^{5} - 18x^{4} - 5x^{3} + 83x^{2} - 84x + 20$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
373	$+1$

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
$\operatorname{tr} a_p$	−6	−3	1	−5	−8	−8	8	−26	5	3	−17	−11	−10	−11	9	15	−29	−32	−6	5	−15	−28	−13	−9	−9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.236 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, 373, 746]: [0, 1, 31, 95] (certified at ell = 2147478997)
T = [(3, 1), (5, 1)]; h of degree 32 = 1 x 32 (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(746, 2)[2]   # 746.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 746:2 1000