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

Newform orbit 694.2.a.d

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{9} - 12x^{8} + 32x^{7} + 120x^{6} - 752x^{5} + 1125x^{4} + 215x^{3} - 1817x^{2} + 1274x - 181$

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 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
$\operatorname{tr} a_p$	9	6	6	7	12	−3	4	8	2	−2	4	−16	21	6	7	−18	5	−6	−6	−5	1	−12	1	9	−20

$\operatorname{tr} a_n$ 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)[3]   # 694.2.a.d

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

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