Atlas › Modular forms › Weight 2 › Level 817 › 817.2.a › 817.2.a.c

Newform orbit 817.2.a.c

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q - 7q^{2} + 17q^{4} - 13q^{5} - 2q^{6} - 15q^{7} - 18q^{8} + 11q^{9} + 8q^{10} - 15q^{11} - 21q^{12} - 4q^{13} + 3q^{14} - 17q^{15} + 13q^{16} - 17q^{17} - 4q^{18} + 13q^{19} - 22q^{20} + 10q^{21} - 23q^{22} - 34q^{23} - 6q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ , whose minimal polynomial is

$x^{13} + 312x^{12} + 39527x^{11} + 2432288x^{10} + 50326897x^{9} - 2799766596x^{8} - 235797955345x^{7} - 7402933230100x^{6} - 103660818423194x^{5} + 53797755012828x^{4} + 21328677758768846x^{3} + 231647282333074392x^{2} + 274095075764698082x - 5470455563220129899$

Atkin–Lehner signs

$p$	$w_p$
19	$-1$
43	$-1$

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
$\operatorname{tr} a_p$	−7	0	−13	−15	−15	−4	−17	13	−34	−6	−4	2	2	13	−45	−18	−4	6	−48	−37	16	−10	−45	16	6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.412 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, 19, 43, 817]: [0, 2, 4, 74] (certified at ell = 2147483647)
T = [(2, 1), (3, 3), (5, 5), (7, 7), (11, 9)]; h of degree 63 = 1 x 63 (phi(ord eps) x dim S_k^new); 15 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(817, 2)[2]   # 817.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 817:2 1000