Atlas › Modular forms › Weight 2 › Level 697 › 697.2.a › 697.2.a.f

Newform orbit 697.2.a.f

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} - 9x^{14} - 21x^{13} + 389x^{12} - 447x^{11} - 4990x^{10} + 12021x^{9} + 18405x^{8} - 66683x^{7} - 10249x^{6} + 113303x^{5} - 11393x^{4} - 54763x^{3} + 9996x^{2} + 6149x - 833$

Atkin–Lehner signs

$p$	$w_p$
17	$-1$
41	$+1$

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
$\operatorname{tr} a_p$	5	4	7	7	6	7	15	13	18	6	−8	−23	−15	0	68	−4	29	−4	3	24	12	−36	79	−12	−31

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.225 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, 17, 41, 697]: [0, 2, 4, 64] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 53 = 1 x 53 (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(697, 2)[5]   # 697.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 697:2 1000