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

Newspace 604.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(604))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
604.2.a.a	3	degree 3	$3q - 3q^{3} - 3q^{5} - 3q^{7} + O(q^{8})$	−
604.2.a.b	3	degree 3	$3q - 3q^{5} + O(q^{8})$	−
604.2.a.c	6	degree 6	$6q + 3q^{3} + 6q^{5} + 5q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
604.2.a.a	$x^{3} + 6x^{2} + 3x - 1$
604.2.a.b	$x^{3} + 3x^{2} - 2x - 3$
604.2.a.c	$x^{6} - 9x^{5} + 3x^{4} + 155x^{3} - 327x^{2} - 333x + 971$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(604))$	12
Old forms	62
Cusp forms $S_{2}(\Gamma_0(604))$	74
Eisenstein series $E_{2}(\Gamma_0(604))$	5
Modular forms $M_{2}(\Gamma_0(604))$	79

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(604)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(151))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(302))^{\oplus 2}$

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 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)

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

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

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