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

Newform orbit 97.4.a.a

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q - 11q^{2} - 12q^{3} + 35q^{4} - 34q^{5} - 37q^{6} - 68q^{7} - 108q^{8} + 77q^{9} - 46q^{10} - 122q^{11} - 93q^{12} - 102q^{13} - 101q^{14} - 338q^{15} + 71q^{16} - 248q^{17} - 405q^{18} - 128q^{19} - 481q^{20} - 252q^{21} - 164q^{22} - 670q^{23} - 377q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{11} + 23x^{10} + 53x^{9} - 1557x^{8} - 5734x^{7} + 28560x^{6} + 93398x^{5} - 214122x^{4} - 474120x^{3} + 599054x^{2} + 676141x - 250821$

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 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
$\operatorname{tr} a_p$	−11	−12	−34	−68	−122	−102	−248	−128	−670	−346	182	−44	−468	−388	−1608	−482	−460	−428	−1058	−2274	−766	−828	−2528	−932	1067

$\operatorname{tr} a_n$ 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)[0]   # 97.4.a.a

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

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