Atlas › Modular forms › Weight 2 › Level 675 › 675.2.a › 675.2.a.p

Newform orbit 675.2.a.p

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + q^{2} + 3q^{4} - 2q^{7} + 6q^{8} + 2q^{11} - 6q^{13} + 12q^{14} - 3q^{16} + 4q^{17} + 14q^{22} + 6q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{13})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$ , whose minimal polynomial is

$x^{2} - 95x - 16025$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 675$ )
5	— ( $5^2 \mid 675$ )

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
$\operatorname{tr} a_p$	1	0	0	−2	2	−6	4	0	6	10	−4	−4	−2	6	4	4	−10	6	16	22	−18	−16	−6	−6	−16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.887 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, 5, 9, 15, 25, 27, 45, 75, 135, 225, 675]: [0, 1, 1, 2, 4, 3, 4, 8, 12, 20, 32, 92] (certified at ell = 2147482801)
T = [(2, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 8 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(675, 2)[15]   # 675.2.a.p

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

cargo run --release -p sagebrush-web --example atlas -- 675:2 1000