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

Newform orbit 938.2.a.h

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{5} - 38x^{4} - 1271x^{3} + 42353x^{2} + 77609x - 4129288$

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 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
$\operatorname{tr} a_p$	−5	4	−2	5	3	10	−5	0	−6	7	7	15	5	15	−3	−5	4	−17	−5	−10	13	18	40	−29	36

$\operatorname{tr} a_n$ 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)[7]   # 938.2.a.h

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

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