Atlas › Modular forms › Weight 2 › Level 750 › 750.2.a › 750.2.a.e

Newform orbit 750.2.a.e

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{7} + 3T_{11}$ , whose minimal polynomial is

$x^{2} + 22x + 41$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$+1$
5	— ( $5^2 \mid 750$ )

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
$\operatorname{tr} a_p$	2	−2	0	−7	−5	−9	−2	−3	−5	−2	−4	1	3	−2	−15	−3	−15	10	−2	0	16	16	−4	11	6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.375 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, 3, 5, 6, 10, 15, 25, 30, 50, 75, 125, 150, 250, 375, 750]: [0, 1, 1, 1, 3, 3, 4, 3, 10, 9, 12, 13, 34, 39, 52, 154] (certified at ell = 2147482801)
T = [(7, 1), (11, 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(750, 2)[4]   # 750.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 750:2 1000