Atlas › Modular forms › Weight 2 › Level 811 › 811.2.a › 811.2.a.c

Newform orbit 811.2.a.c

Trace form

The 40 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$40q + 6q^{2} + 2q^{3} + 46q^{4} + 33q^{5} + 5q^{6} + 5q^{7} + 12q^{8} + 62q^{9} + 37q^{11} - 5q^{12} + 6q^{13} + 14q^{14} + 14q^{15} + 58q^{16} + 18q^{17} - 9q^{18} - 2q^{19} + 46q^{20} + 33q^{21} - 8q^{22} + 25q^{23} - q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 40. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{40} - 8x^{39} - 127x^{38} + 1164x^{37} + 6493x^{36} - 73278x^{35} - 163964x^{34} + 2647143x^{33} + 1660959x^{32} - 61394449x^{31} + 18199710x^{30} + 969455262x^{29} - 854570328x^{28} - 10786219608x^{27} + 13755586801x^{26} + 86319425194x^{25} - 133766956119x^{24} - 502832415848x^{23} + 875961641786x^{22} + 2145003661881x^{21} - 4015308392509x^{20} - 6713584694833x^{19} + 13081748752208x^{18} + 15400191705280x^{17} - 30416756896351x^{16} - 25791164591880x^{15} + 50331633120897x^{14} + 31296173952580x^{13} - 58738036928220x^{12} - 27110201364694x^{11} + 47530721750247x^{10} + 16275710852152x^{9} - 25908161040026x^{8} - 6389832544486x^{7} + 9062203788734x^{6} + 1459725763556x^{5} - 1867948728448x^{4} - 146611677520x^{3} + 190179178848x^{2} + 884814592x - 5457135488$

Atkin–Lehner signs

$p$	$w_p$
811	$-1$

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
$\operatorname{tr} a_p$	6	2	33	5	37	6	18	−2	25	82	0	29	34	5	5	79	32	15	3	80	−5	−3	−17	59	−5

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.264 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, 811]: [0, 68] (certified at ell = 2147478481)
T = [(2, 1), (3, 1)]; h of degree 67 = 1 x 67 (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(811, 2)[2]   # 811.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 811:2 1000