Atlas › Modular forms › Weight 6 › Level 93 › 93.6.a › 93.6.a.c

Newform orbit 93.6.a.c

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - 3q^{2} - 63q^{3} + 101q^{4} - 64q^{5} + 27q^{6} - 88q^{7} + 114q^{8} + 567q^{9} - 54q^{10} + 26q^{11} - 909q^{12} - 450q^{13} - 1795q^{14} + 576q^{15} + 25q^{16} - 2914q^{17} - 243q^{18} - 468q^{19} - 9840q^{20} + 792q^{21} - 4229q^{22} - 3732q^{23} - 1026q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{7} + 67x^{6} - 7453x^{5} - 362048x^{4} + 20876815x^{3} + 357836695x^{2} - 23583329299x + 233824598470$

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	−63	−64	−88	26	−450	−2914	−468	−3732	−16678	−6727	17902	−12242	−19060	−12864	−27254	−61688	70086	−22320	−167348	−3938	99534	−176978	−14958	−170864

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.264 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, 3, 15, 56] (certified at ell = 2147482951)
T = [(2, 1), (5, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 6 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, 6)[2]   # 93.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 93:6 1000