Atlas › Modular forms › Weight 2 › Level 453 › 453.2.a › 453.2.a.g

Newform orbit 453.2.a.g

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{9} - 312x^{7} + 63x^{6} + 28690x^{5} - 30335x^{4} - 787221x^{3} + 2013114x^{2} + 720184x - 3455048$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
151	$+1$

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
$\operatorname{tr} a_p$	6	9	−2	14	2	−2	−2	8	−8	0	16	10	−8	8	−24	10	−2	14	30	−10	10	20	0	−32	30

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.295 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, 151, 453]: [0, 1, 13, 52] (certified at ell = 2147482951)
T = [(2, 1), (5, 3)]; h of degree 25 = 1 x 25 (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(453, 2)[6]   # 453.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 453:2 1000