Atlas › Modular forms › Weight 2 › Level 694 › 694.2.a › 694.2.a.c

Newform orbit 694.2.a.c

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{7} + 9x^{6} + 8x^{5} - 65x^{4} - 123x^{3} + 5x^{2} + 81x + 5$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
347	$+1$

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
$\operatorname{tr} a_p$	−7	0	−9	3	−16	−7	−8	−3	−4	−22	−3	−7	−29	4	−9	−6	−11	−30	4	−22	6	6	−33	−36	14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.222 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, 2, 347, 694]: [0, 1, 30, 89] (certified at ell = 2147481179)
T = [(3, 1), (5, 1)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 3 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(694, 2)[2]   # 694.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 694:2 1000