Atlas › Modular forms › Weight 2 › Level 829 › 829.2.a › 829.2.a.b

Newform orbit 829.2.a.b

Trace form

The 28 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$28q - 9q^{2} - 6q^{3} + 21q^{4} - 8q^{5} - 9q^{6} - 10q^{7} - 24q^{8} + 6q^{9} - 5q^{10} - 53q^{11} - 18q^{12} - 10q^{13} - 24q^{14} - 29q^{15} - q^{16} - 26q^{17} - 19q^{18} - 32q^{19} - 30q^{20} - 11q^{21} + 12q^{22} - 22q^{23} - 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 28. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{28} + 15x^{27} + 38x^{26} - 442x^{25} - 2311x^{24} + 4117x^{23} + 41433x^{22} - 154x^{21} - 382819x^{20} - 261487x^{19} + 2102918x^{18} + 2076438x^{17} - 7280608x^{16} - 7989367x^{15} + 16175913x^{14} + 17693502x^{13} - 22774081x^{12} - 23463460x^{11} + 19524035x^{10} + 18274999x^{9} - 9445605x^{8} - 7692213x^{7} + 2282442x^{6} + 1460787x^{5} - 262310x^{4} - 113424x^{3} + 13469x^{2} + 3014x - 243$

Atkin–Lehner signs

$p$	$w_p$
829	$+1$

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
$\operatorname{tr} a_p$	−9	−6	−8	−10	−53	−10	−26	−32	−22	−34	−14	0	−22	−51	−23	−26	−71	−36	0	−46	17	−31	−53	−19	18

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.261 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, 829]: [0, 69] (certified at ell = 2147479273)
T = [(2, 1), (3, 1)]; h of degree 68 = 1 x 68 (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(829, 2)[1]   # 829.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 829:2 1000