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

Newform orbit 811.2.a.b

Trace form

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

$26q - 6q^{2} - 4q^{3} + 20q^{4} - 30q^{5} - 11q^{6} - 5q^{7} - 18q^{8} + 8q^{9} - 37q^{11} - 7q^{12} - 24q^{14} - 10q^{15} + 8q^{16} - 22q^{17} - 15q^{18} - 3q^{19} - 46q^{20} - 39q^{21} - 4q^{22} - 25q^{23} - 25q^{24} + O(q^{25})$

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

$x^{26} + 10x^{25} - 18x^{24} - 442x^{23} - 358x^{22} + 8148x^{21} + 13409x^{20} - 83369x^{19} - 168178x^{18} + 524468x^{17} + 1141059x^{16} - 2092609x^{15} - 4649606x^{14} + 5171728x^{13} + 11701444x^{12} - 7131687x^{11} - 18018234x^{10} + 3667494x^{9} + 15976464x^{8} + 2227308x^{7} - 6673453x^{6} - 2956654x^{5} + 432217x^{4} + 548174x^{3} + 138739x^{2} + 14858x + 588$

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 odd 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	−4	−30	−5	−37	0	−22	−3	−25	−71	−2	−26	−53	3	−17	−72	−37	−23	−2	−38	14	7	−15	−60	11

$\operatorname{tr} a_n$ 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)[1]   # 811.2.a.b

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

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