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

Newform orbit 697.2.a.c

Trace form

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

$7q - 3q^{2} - 2q^{3} + 9q^{4} - 5q^{5} + 2q^{6} - 7q^{7} - 21q^{8} + 7q^{9} + q^{10} - 6q^{11} - 5q^{12} + 5q^{13} - 9q^{14} - 26q^{15} + 9q^{16} + 7q^{17} - 6q^{18} - 13q^{19} - 4q^{20} + 4q^{21} + 8q^{22} - 26q^{23} + 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} + 5x^{6} - 15x^{5} - 73x^{4} + 21x^{3} + 178x^{2} - 7x - 109$

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 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	−2	−5	−7	−6	5	7	−13	−26	10	−2	1	7	−12	−18	−6	−55	6	−13	16	−6	−12	−41	6	−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 0.225 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)[2]   # 697.2.a.c

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

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