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

Newform orbit 831.2.a.c

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{6} + 10x^{5} - 39x^{4} - 378x^{3} - 19x^{2} + 1280x + 953$

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 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
$\operatorname{tr} a_p$	−1	−6	−3	3	−8	0	−10	1	−6	−19	−1	−4	−19	11	−18	−9	−6	−10	1	−10	6	−15	6	−47	0

$\operatorname{tr} a_n$ 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)[2]   # 831.2.a.c

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

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