Atlas › Modular forms › Weight 2 › Level 452 › 452.2.a › 452.2.a.b

Newform orbit 452.2.a.b

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + 3q^{3} + 3q^{5} + 12q^{9} + 4q^{11} + 16q^{13} + 3q^{15} + 14q^{17} + 8q^{19} + 3q^{21} + O(q^{25})$

Coefficient field

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

$x^{7} - 6x^{6} - 27x^{5} + 207x^{4} - 28x^{3} - 1440x^{2} + 2246x - 926$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 452$ )
113	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	0	3	3	0	4	16	14	8	0	5	−1	6	1	8	−15	−1	−15	15	−1	−24	41	−21	−14	−40	24

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.195 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, 4, 113, 226, 452]: [0, 1, 2, 10, 30, 60] (certified at ell = 2147480161)
T = [(3, 1), (5, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 2 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(452, 2)[1]   # 452.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 452:2 1000