Atlas › Modular forms › Weight 2 › Level 474 › 474.2.a

Newspace 474.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(474))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
474.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} - 3q^{7} + O(q^{8})$	+ + +	$-1$	474a1 rank 1
474.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} + O(q^{8})$	+ − −	$-1$	474b1 rank 1
474.2.a.c	2	$\mathbb{Q}(\sqrt{29})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - q^{5} + 2q^{6} + q^{7} + O(q^{8})$	+ + −	$+1$
474.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + 3q^{5} - 2q^{6} - q^{7} + O(q^{8})$	+ − +	$+1$
474.2.a.e	3	degree 3	$3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} + 3q^{6} + O(q^{8})$	− − −	$+1$
474.2.a.f	4	degree 4	$4q + 4q^{2} - 4q^{3} + 4q^{4} + q^{5} - 4q^{6} + 4q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
474.2.a.a	$x + 7$
474.2.a.b	$x + 5$
474.2.a.c	$x^{2} - 2x - 115$
474.2.a.d	$x^{2} - 125$
474.2.a.e	$x^{3} - 3x^{2} - 13x + 23$
474.2.a.f	$x^{4} - 13x^{3} - 67x^{2} + 1385x - 4378$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(474))$	13
Old forms	64
Cusp forms $S_{2}(\Gamma_0(474))$	77
Eisenstein series $E_{2}(\Gamma_0(474))$	7
Modular forms $M_{2}(\Gamma_0(474))$	84

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(474)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(79))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(158))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(237))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.227 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, 3, 6, 79, 158, 237, 474]: [0, 1, 1, 3, 7, 22, 28, 84] (certified at ell = 2147483053)
T = [(5, 1), (7, 3)]; h of degree 13 = 1 x 13 (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(474, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 474:2 1000

← 473.2.a · 475.2.a →