Atlas › Modular forms › Weight 2 › Level 825 › 825.2.a › 825.2.a.k

Newform orbit 825.2.a.k

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{3} - x^{2} - 113x - 59$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
5	— ( $5^2 \mid 825$ )
11	$-1$

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
$\operatorname{tr} a_p$	1	−3	0	0	3	2	2	8	0	−10	8	6	−14	−4	8	6	12	−6	4	8	14	12	0	−10	−22

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.45 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, 11, 15, 25, 33, 55, 75, 165, 275, 825]: [0, 1, 1, 2, 4, 3, 6, 8, 12, 28, 32, 124] (certified at ell = 2147482921)
T = [(2, 1), (7, 3)]; h of degree 32 = 1 x 32 (phi(ord eps) x dim S_k^new); 5 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(825, 2)[10]   # 825.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 825:2 1000