Atlas › Modular forms › Weight 2 › Level 904 › 904.2.a

Newspace 904.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(904))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
904.2.a.a	1	$\mathbb{Q}$	$q - 4q^{7} + O(q^{8})$	−	904a1 rank 1
904.2.a.b	5	degree 5	$5q + q^{3} - 5q^{5} - 2q^{7} + O(q^{8})$	+
904.2.a.c	6	degree 6	$6q - 7q^{3} - 5q^{5} + 2q^{7} + O(q^{8})$	−
904.2.a.d	7	degree 7	$7q + 7q^{3} + 3q^{5} + 2q^{7} + O(q^{8})$	+
904.2.a.e	9	degree 9	$9q - 3q^{3} + 9q^{5} + 2q^{7} + O(q^{8})$	−

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 $T$
904.2.a.a	$x$
904.2.a.b	$x^{5} + 4x^{4} - x^{3} - 9x^{2} + 4$
904.2.a.c	$x^{6} + 12x^{5} + 35x^{4} - x^{3} - 106x^{2} - 92x - 12$
904.2.a.d	$x^{7} - 10x^{6} + 5x^{5} + 189x^{4} - 384x^{3} - 650x^{2} + 1320x + 772$
904.2.a.e	$x^{9} - 6x^{8} - 35x^{7} + 221x^{6} + 198x^{5} - 2414x^{4} + 1884x^{3} + 7452x^{2} - 14024x + 6824$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(904))$	28
Old forms	83
Cusp forms $S_{2}(\Gamma_0(904))$	111
Eisenstein series $E_{2}(\Gamma_0(904))$	7
Modular forms $M_{2}(\Gamma_0(904))$	118

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(904)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(113))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(226))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(452))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.264 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, 8, 113, 226, 452, 904]: [0, 1, 2, 3, 10, 30, 60, 118] (certified at ell = 2147480161)
T = [(3, 1), (5, 1)]; h of degree 28 = 1 x 28 (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(904, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 904:2 1000

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