Atlas › Modular forms › Weight 2 › Level 463 › 463.2.a › 463.2.a.b

Newform orbit 463.2.a.b

Trace form

The 22 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$22q + 8q^{2} + 4q^{3} + 22q^{4} + 14q^{5} + 2q^{7} + 21q^{8} + 30q^{9} + 3q^{10} + 3q^{11} + 5q^{12} + 13q^{13} + 4q^{14} - 3q^{15} + 18q^{16} + 56q^{17} + 5q^{18} + 3q^{19} + 20q^{20} + 3q^{21} - 5q^{22} - 11q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 22. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{22} - 12x^{21} - 9x^{20} + 618x^{19} - 1499x^{18} - 10987x^{17} + 45307x^{16} + 76458x^{15} - 531576x^{14} - 111388x^{13} + 3234394x^{12} - 1103715x^{11} - 11503686x^{10} + 5843232x^{9} + 25328786x^{8} - 11387828x^{7} - 34340702x^{6} + 9164589x^{5} + 26107615x^{4} - 2037034x^{3} - 9261860x^{2} - 403906x + 953995$

Atkin–Lehner signs

$p$	$w_p$
463	$-1$

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
$\operatorname{tr} a_p$	8	4	14	2	3	13	56	3	0	8	−4	−1	48	−2	24	26	21	−10	−8	7	30	−29	49	49	25

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.189 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, 463]: [0, 39] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 38 = 1 x 38 (phi(ord eps) x dim S_k^new); 4 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(463, 2)[1]   # 463.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 463:2 1000