Atlas › Modular forms › Weight 2 › Level 275 › 275.2.a › 275.2.a.g

Newform orbit 275.2.a.g

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{2} - 10x + 5$

Atkin–Lehner signs

$p$	$w_p$
5	— ( $5^2 \mid 275$ )
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	1	2	8	1	0	3	−5	−6	16	−6	−12	6	3	10	−11	16	−6	23	5	−27	−25	1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.19 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, 5, 11, 25, 55, 275]: [0, 1, 2, 3, 8, 32] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 3 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(275, 2)[6]   # 275.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 275:2 1000