Atlas › Modular forms › Weight 6 › Level 45 › 45.6.a › 45.6.a.f

Newform orbit 45.6.a.f

Trace form

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

$2q + 5q^{2} + 21q^{4} + 50q^{5} + 80q^{7} + 255q^{8} + 125q^{10} + 800q^{11} - 120q^{13} + 1650q^{14} - 687q^{16} + 1940q^{17} - 1512q^{19} + 525q^{20} + 550q^{22} - 1320q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{145})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{7}$ , whose minimal polynomial is

$x^{2} - 85x - 14180$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 45$ )
5	$-1$

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
$\operatorname{tr} a_p$	5	0	50	80	800	−120	1940	−1512	−1320	1300	−5824	−12560	400	25680	18920	49460	63200	−49116	6080	65200	97740	−46288	−57360	−87000	10180

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.244 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 engine's checks

dim M^+ at levels [1, 3, 5, 9, 15, 45]: [1, 3, 3, 6, 12, 32] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 9 = 1 x 9 (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(45, 6)[5]   # 45.6.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 45:6 1000