Atlas › Modular forms › Weight 2 › Level 604 › 604.2.a › 604.2.a.a

Newform orbit 604.2.a.a

Trace form

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

$3q - 3q^{3} - 3q^{5} - 3q^{7} + 12q^{9} - 9q^{11} - 3q^{13} - 6q^{15} + 12q^{17} - 9q^{19} - 15q^{21} - 6q^{23} + O(q^{25})$

Coefficient field

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

$x^{3} + 6x^{2} + 3x - 1$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 604$ )
151	$-1$

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
$\operatorname{tr} a_p$	0	−3	−3	−3	−9	−3	12	−9	−6	−21	3	−3	3	−6	−9	−6	−15	9	9	−18	3	9	−27	−24	24

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.208 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, 4, 151, 302, 604]: [0, 1, 2, 13, 40, 79] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 2 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(604, 2)[0]   # 604.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 604:2 1000