Atlas › Modular forms › Weight 2 › Level 477 › 477.2.a

Newspace 477.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(477))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
477.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - 4q^{7} + O(q^{8})$	−	477a1 rank 1
477.2.a.b	3	degree 3	$3q + q^{2} + q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	+
477.2.a.c	4	degree 4	$4q - 3q^{2} + 3q^{4} - 4q^{5} + O(q^{8})$	+
477.2.a.d	4	degree 4	$4q - 3q^{2} + 3q^{4} - 2q^{5} - 4q^{7} + O(q^{8})$	−
477.2.a.e	4	degree 4	$4q + 3q^{2} + 3q^{4} + 4q^{5} + O(q^{8})$	−
477.2.a.f	5	degree 5	$5q + 10q^{4} + 4q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
477.2.a.a	$x - 1$
477.2.a.b	$x^{3} - 3x^{2} - x + 1$
477.2.a.c	$x^{4} + 7x^{3} + 8x^{2} - 30x - 53$
477.2.a.d	$x^{4} + 5x^{3} + 4x^{2} - 8x - 9$
477.2.a.e	$x^{4} - 7x^{3} + 8x^{2} + 30x - 53$
477.2.a.f	$x^{5} - 37x^{3} - 8x^{2} + 195x + 99$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(477))$	21
Old forms	30
Cusp forms $S_{2}(\Gamma_0(477))$	51
Eisenstein series $E_{2}(\Gamma_0(477))$	7
Modular forms $M_{2}(\Gamma_0(477))$	58

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(477)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(53))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(159))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.295 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, 3, 9, 53, 159, 477]: [0, 1, 2, 5, 20, 56] (certified at ell = 2147483053)
T = [(2, 1), (5, 1)]; h of degree 21 = 1 x 21 (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(477, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 477:2 1000

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