Atlas › Modular forms › Weight 4 › Level 93 › 93.4.a › 93.4.a.c

Newform orbit 93.4.a.c

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 3q^{2} + 6q^{3} - 3q^{4} - 8q^{5} - 9q^{6} - 37q^{7} + 3q^{8} + 18q^{9} - 5q^{10} - 39q^{11} - 9q^{12} - 53q^{13} + 64q^{14} - 24q^{15} - 23q^{16} + 25q^{17} - 27q^{18} - 188q^{19} + 63q^{20} - 111q^{21} - 103q^{22} - 152q^{23} + 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} + 11x + 26$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
31	$+1$

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	6	−8	−37	−39	−53	25	−188	−152	106	−62	260	177	−288	779	138	−339	−581	−255	−30	−208	−151	−2061	−848	−882

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.108 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 engine's checks

dim M^+ at levels [1, 3, 31, 93]: [1, 2, 9, 34] (certified at ell = 2147482951)
T = [(2, 1), (5, 1)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 3 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(93, 4)[2]   # 93.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 93:4 1000