Atlas › Modular forms › Weight 4 › Level 97 › 97.4.a › 97.4.a.b

Newform orbit 97.4.a.b

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q + 9q^{2} + 12q^{3} + 59q^{4} + 16q^{5} + 11q^{6} + 58q^{7} + 132q^{8} + 167q^{9} + 14q^{10} + 142q^{11} + 147q^{12} + 28q^{13} + 67q^{14} + 82q^{15} + 295q^{16} + 160q^{17} + 135q^{18} + 100q^{19} + 119q^{20} + 188q^{22} + 664q^{23} + 199q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{13} - 21x^{12} - 131x^{11} + 4979x^{10} - 5182x^{9} - 421768x^{8} + 1458286x^{7} + 15168102x^{6} - 73180372x^{5} - 198084542x^{4} + 1252624925x^{3} + 198233487x^{2} - 4445148108x - 1187069816$

Atkin–Lehner signs

$p$	$w_p$
97	$+1$

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
$\operatorname{tr} a_p$	9	12	16	58	142	28	160	100	664	116	−254	50	−104	−92	1176	898	660	−748	−82	2536	994	2072	2724	1756	−1261

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.101 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, 97]: [1, 26] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 24 = 1 x 24 (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(97, 4)[1]   # 97.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 97:4 1000