Atlas › Modular forms › Weight 2 › Level 942 › 942.2.a › 942.2.a.h

Newform orbit 942.2.a.h

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7}$ , whose minimal polynomial is

$x^{3} + 3x^{2} - 31x - 95$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$-1$
157	$-1$

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
$\operatorname{tr} a_p$	−3	3	−6	1	−6	−3	−9	−2	−7	−6	−4	−5	−16	−7	−14	0	−7	18	−12	8	−6	8	−4	−29	−10

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.402 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, 3, 6, 157, 314, 471, 942]: [0, 1, 1, 3, 13, 41, 54, 162] (certified at ell = 2147483053)
T = [(5, 1), (7, 3)]; h of degree 25 = 1 x 25 (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(942, 2)[7]   # 942.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 942:2 1000