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

Newform orbit 942.2.a.i

Trace form

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

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

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

$x^{3} + 19x^{2} + 59x - 17$

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	−4	−5	0	−5	−7	−12	−7	−2	−16	−7	−6	−7	−2	−2	7	−6	−12	22	−12	4	8	21	12

$\operatorname{tr} a_n$ 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)[8]   # 942.2.a.i

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

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