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

Newform orbit 794.2.a.g

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. 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

$x^{12} - 372x^{11} + 45863x^{10} - 990356x^{9} - 229042651x^{8} + 16883873327x^{7} - 102640865666x^{6} - 22279977144575x^{5} + 468470586676603x^{4} + 5874589987407947x^{3} - 123657782597534379x^{2} - 73401935288021843x + 624528591824845429$

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
$\operatorname{tr} a_p$	−12	7	3	7	15	7	0	21	12	−2	28	8	−10	29	6	−11	4	37	23	−15	19	50	19	4	27

$\operatorname{tr} a_n$ 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)[6]   # 794.2.a.g

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

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