Atlas › Modular forms › Weight 2 › Level 605 › 605.2.a › 605.2.a.h

Newform orbit 605.2.a.h

Trace form

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

$3q + q^{2} + q^{3} + 9q^{4} - 3q^{5} + 5q^{6} - q^{7} - 9q^{8} + 2q^{9} - q^{10} + 9q^{12} + 6q^{13} + 5q^{14} - q^{15} + 13q^{16} + 4q^{17} + 20q^{18} + 4q^{19} - 9q^{20} - 17q^{21} - 6q^{23} + 17q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{13}$ , whose minimal polynomial is

$x^{3} - 41x^{2} - 875x + 4569$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
11	— ( $11^2 \mid 605$ )

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	1	−3	−1	0	6	4	4	−6	2	14	4	9	7	−15	−6	10	3	19	6	−12	2	18	11	−2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.268 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, 55, 121, 605]: [0, 1, 2, 8, 12, 68] (certified at ell = 2147482921)
T = [(2, 1), (3, 3), (7, 5), (13, 7)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 9 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(605, 2)[7]   # 605.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 605:2 1000