Atlas › Modular forms › Weight 2 › Level 97 › 97.2.a › 97.2.a.a

Newform orbit 97.2.a.a

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 4q^{2} - 4q^{3} + 4q^{4} - 3q^{5} + 3q^{6} - 7q^{7} - 9q^{8} + q^{9} + 4q^{10} - 7q^{11} - 3q^{12} - 2q^{13} + 7q^{14} - 3q^{15} + 18q^{16} - 3q^{17} + 8q^{18} + 5q^{19} + 3q^{20} + 14q^{21} + 14q^{22} - 12q^{23} + 19q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 8x^{2} + 19x + 13$

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
$\operatorname{tr} a_p$	−4	−4	−3	−7	−7	−2	−3	5	−12	1	−8	−2	3	1	−17	2	19	3	−1	−23	−1	−12	2	12	−3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.16 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, 97]: [0, 8] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 1 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, 2)[0]   # 97.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 97:2 1000