Atlas › Modular forms › Weight 2 › Level 938 › 938.2.a › 938.2.a.f

Newform orbit 938.2.a.f

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{4} + 194x^{3} + 11471x^{2} + 216973x + 1077119$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
7	$+1$
67	$-1$

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
$\operatorname{tr} a_p$	4	−4	−6	−4	3	−10	−13	−8	0	−9	−11	−9	−9	−5	−13	−9	12	1	4	0	−9	0	10	−9	−22

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.51 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, 7, 14, 67, 134, 469, 938]: [0, 1, 1, 4, 6, 19, 46, 140] (certified at ell = 2147483647)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 33 = 1 x 33 (phi(ord eps) x dim S_k^new); 9 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(938, 2)[5]   # 938.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 938:2 1000