Atlas › Modular forms › Weight 2 › Level 767 › 767.2.a › 767.2.a.f

Newform orbit 767.2.a.f

Trace form

The 23 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$23q + 7q^{3} + 30q^{4} + 4q^{5} - 6q^{6} + 9q^{7} + 34q^{9} + 14q^{10} - 5q^{11} + 21q^{12} + 23q^{13} - 6q^{14} - 2q^{15} + 40q^{16} + 22q^{17} - 4q^{18} + 8q^{19} - 10q^{20} - q^{21} + 30q^{22} - 4q^{23} - 37q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 23. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{23} - 7x^{22} - 59x^{21} + 473x^{20} + 1270x^{19} - 12934x^{18} - 10682x^{17} + 184572x^{16} - 11016x^{15} - 1471000x^{14} + 806348x^{13} + 6465044x^{12} - 5337305x^{11} - 14395758x^{10} + 13936888x^{9} + 13585663x^{8} - 14180819x^{7} - 4409444x^{6} + 5151475x^{5} + 689090x^{4} - 758448x^{3} - 77818x^{2} + 38320x + 5328$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$
59	$+1$

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
$\operatorname{tr} a_p$	0	7	4	9	−5	23	22	8	−4	8	−1	13	10	11	9	31	−23	37	20	12	61	22	−30	21	117

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.24 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, 13, 59, 767]: [0, 1, 6, 72] (certified at ell = 2147481901)
T = [(2, 1), (3, 1)]; h of degree 59 = 1 x 59 (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(767, 2)[5]   # 767.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 767:2 1000