Atlas › Modular forms › Weight 2 › Level 461 › 461.2.a › 461.2.a.d

Newform orbit 461.2.a.d

Trace form

The 26 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$26q + 3q^{2} + 8q^{3} + 39q^{4} + q^{5} - 2q^{6} + 10q^{7} + 6q^{8} + 36q^{9} + q^{10} + 22q^{11} + 5q^{12} + 6q^{13} + 6q^{14} + 15q^{15} + 61q^{16} - 7q^{17} + 3q^{18} + 50q^{19} - q^{20} + q^{21} + q^{22} - 6q^{23} - 18q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 26. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{26} - 11x^{25} - 40x^{24} + 824x^{23} - 580x^{22} - 23713x^{21} + 53252x^{20} + 344769x^{19} - 1123481x^{18} - 2780000x^{17} + 12130383x^{16} + 12327581x^{15} - 77586773x^{14} - 23992183x^{13} + 308131014x^{12} - 22145406x^{11} - 763167150x^{10} + 207951723x^{9} + 1150606318x^{8} - 396559071x^{7} - 1010330981x^{6} + 302360438x^{5} + 489321595x^{4} - 77272839x^{3} - 105860022x^{2} - 3682848x + 1540216$

Atkin–Lehner signs

$p$	$w_p$
461	$-1$

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
$\operatorname{tr} a_p$	3	8	1	10	22	6	−7	50	−6	−5	38	19	−4	43	2	−4	29	14	92	13	5	51	0	−20	−3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.195 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, 461]: [0, 39] (certified at ell = 2147478721)
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(461, 2)[3]   # 461.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 461:2 1000