Atlas › Modular forms › Weight 2 › Level 404 › 404.2.a

Newspace 404.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(404))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
404.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} + 3q^{5} + 2q^{7} + O(q^{8})$	+	404b1 rank 0
404.2.a.b	1	$\mathbb{Q}$	$q - q^{5} - 2q^{7} + O(q^{8})$	−	404a1 rank 1
404.2.a.c	7	degree 7	$7q + 2q^{3} + 2q^{7} + O(q^{8})$	+

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 $T$
404.2.a.a	$x - 1$
404.2.a.b	$x + 1$
404.2.a.c	$x^{7} - 2x^{6} - 66x^{5} + 206x^{4} + 729x^{3} - 2856x^{2} + 2253x + 24$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(404))$	9
Old forms	40
Cusp forms $S_{2}(\Gamma_0(404))$	49
Eisenstein series $E_{2}(\Gamma_0(404))$	5
Modular forms $M_{2}(\Gamma_0(404))$	54

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(404)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(101))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(202))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.191 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, 101, 202, 404]: [0, 1, 2, 9, 27, 54] (certified at ell = 2147482801)
T = [(3, 1), (5, 1)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 1 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(404, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 404:2 1000

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