Atlas › Modular forms › Weight 2 › Level 831 › 831.2.a › 831.2.a.e

Newform orbit 831.2.a.e

Trace form

The 20 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$20q + q^{2} + 20q^{3} + 29q^{4} + 3q^{5} + q^{6} + 11q^{7} + 20q^{9} + 6q^{10} + 6q^{11} + 29q^{12} + 16q^{13} + q^{14} + 3q^{15} + 39q^{16} + 8q^{17} + q^{18} + 21q^{19} - 9q^{20} + 11q^{21} + 20q^{22} + 4q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 20. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{20} - 10x^{19} - 673x^{18} + 7314x^{17} + 176634x^{16} - 2150838x^{15} - 22587333x^{14} + 325659710x^{13} + 1384156985x^{12} - 27179094718x^{11} - 25647808995x^{10} + 1249578364666x^{9} - 1114675498941x^{8} - 30189911724282x^{7} + 57505846032545x^{6} + 337794428499706x^{5} - 816263818980278x^{4} - 1200178044812408x^{3} + 3468285481367016x^{2} - 1907388992535700x + 92566751152991$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
277	$+1$

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
$\operatorname{tr} a_p$	1	20	3	11	6	16	8	21	4	15	19	20	1	31	−18	1	−14	14	41	−14	10	33	−14	41	44

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.27 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, 3, 277, 831]: [0, 1, 23, 94] (certified at ell = 2147483137)
T = [(2, 1), (5, 3)]; h of degree 47 = 1 x 47 (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(831, 2)[4]   # 831.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 831:2 1000