Atlas › Modular forms › Weight 2 › Level 690 › 690.2.a › 690.2.a.l

Newform orbit 690.2.a.l

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{7} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{29} + 7T_{31} + 8T_{37} + 9T_{41} + 10T_{43} + 11T_{47} + 12T_{53} + 13T_{59} + 14T_{61} + 15T_{67} + 16T_{71}$ , whose minimal polynomial is

$x^{2} + 310x + 3200$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$+1$
5	$-1$
23	$-1$

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
$\operatorname{tr} a_p$	2	−2	2	−2	2	4	6	8	2	4	0	−6	4	0	−16	8	−12	10	−16	−4	8	−2	−2	−2	−16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.177 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, 3, 5, 6, 10, 15, 23, 30, 46, 69, 115, 138, 230, 345, 690]: [0, 1, 1, 1, 3, 3, 4, 3, 10, 8, 10, 14, 28, 40, 52, 152] (certified at ell = 2147483053)
T = [(7, 1), (11, 2), (13, 3), (17, 4), (19, 5), (29, 6), (31, 7), (37, 8), (41, 9), (43, 10), (47, 11), (53, 12), (59, 13), (61, 14), (67, 15), (71, 16)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 5 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(690, 2)[11]   # 690.2.a.l

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

cargo run --release -p sagebrush-web --example atlas -- 690:2 1000