Atlas › Modular forms › Weight 2 › Level 770 › 770.2.a › 770.2.a.k

Newform orbit 770.2.a.k

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{33})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{13} + 3T_{19} + 4T_{23} + 5T_{29} + 6T_{17} + 7T_{31} + 8T_{37} + 9T_{41} + 10T_{43} + 11T_{47} + 12T_{53}$ , whose minimal polynomial is

$x^{2} + 264x + 15312$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	$-1$
7	$-1$
11	$+1$

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	4	2	2	−2	2	−2	−2	−2	6	−2	−10	−8	8	−2	−14	−6	14	−8	−8	10	2	16	8	−22

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.872 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, 5, 7, 10, 11, 14, 22, 35, 55, 70, 77, 110, 154, 385, 770]: [0, 1, 1, 1, 3, 2, 4, 5, 6, 8, 16, 10, 22, 28, 52, 152] (certified at ell = 2147482921)
T = [(3, 1), (13, 2), (19, 3), (23, 4), (29, 5), (17, 6), (31, 7), (37, 8), (41, 9), (43, 10), (47, 11), (53, 12)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 7 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(770, 2)[10]   # 770.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 770:2 1000