Atlas › Modular forms › Weight 2 › Level 794 › 794.2.a › 794.2.a.h

Newform orbit 794.2.a.h

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23}$ , whose minimal polynomial is

$x^{14} - 304x^{13} + 18246x^{12} + 2889966x^{11} - 381868768x^{10} + 3291591549x^{9} + 1395826007465x^{8} - 57029832757814x^{7} - 698095759532095x^{6} + 63320410142608730x^{5} - 179988530847046130x^{4} - 19964170861983579080x^{3} + 41245320493348547910x^{2} + 1087301063449428030905x + 1023143530380871056011$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
397	$+1$

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
$\operatorname{tr} a_p$	14	1	7	7	3	7	10	7	14	20	6	20	4	3	0	21	−30	19	−21	−11	9	22	−51	−6	−15

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.56 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, 2, 397, 794]: [0, 1, 33, 101] (certified at ell = 2147483053)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (17, 6), (19, 7), (23, 8)]; h of degree 34 = 1 x 34 (phi(ord eps) x dim S_k^new); 10 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(794, 2)[7]   # 794.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 794:2 1000