Atlas › Modular forms › Weight 2 › Level 357 › 357.2.a › 357.2.a.h

Newform orbit 357.2.a.h

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{19}$ , whose minimal polynomial is

$x^{4} - 110x^{3} + 3649x^{2} - 21736x - 485392$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
7	$-1$
17	$+1$

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
$\operatorname{tr} a_p$	2	−4	−2	4	2	2	−4	10	6	−4	−4	0	−18	26	−4	20	4	−4	28	4	8	−8	28	−28	4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.331 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, 7, 17, 21, 51, 119, 357]: [0, 1, 1, 2, 4, 8, 14, 52] (certified at ell = 2147483137)
T = [(2, 1), (5, 3), (11, 5), (13, 7), (19, 9)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 4 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(357, 2)[7]   # 357.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 357:2 1000