Atlas › Modular forms › Weight 2 › Level 827 › 827.2.a › 827.2.a.c

Newform orbit 827.2.a.c

Trace form

The 24 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$24q - 5q^{2} - 11q^{3} + 19q^{4} - 19q^{5} - 9q^{6} - 16q^{7} - 3q^{8} + 19q^{9} - 6q^{10} + 2q^{11} - 37q^{12} - 52q^{13} - q^{14} - 5q^{15} + q^{16} - 13q^{17} - 11q^{18} - 28q^{19} - 24q^{20} - 11q^{21} - 21q^{22} + 12q^{23} - 17q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 24. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{24} + 16x^{23} + 58x^{22} - 323x^{21} - 2478x^{20} - 184x^{19} + 31183x^{18} + 40400x^{17} - 197960x^{16} - 373240x^{15} + 751215x^{14} + 1662132x^{13} - 1863481x^{12} - 4227628x^{11} + 3164882x^{10} + 6289960x^{9} - 3612046x^{8} - 5249853x^{7} + 2496680x^{6} + 2199234x^{5} - 845287x^{4} - 371983x^{3} + 88878x^{2} + 20374x + 889$

Atkin–Lehner signs

$p$	$w_p$
827	$+1$

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
$\operatorname{tr} a_p$	−5	−11	−19	−16	2	−52	−13	−28	12	−14	−37	−58	−29	−9	−53	12	5	−67	−18	2	−71	6	−14	−10	−123

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.362 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, 827]: [0, 70] (certified at ell = 2147481883)
T = [(2, 1), (3, 1)]; h of degree 69 = 1 x 69 (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(827, 2)[2]   # 827.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 827:2 1000