Atlas › Modular forms › Weight 2 › Level 478 › 478.2.a

Newspace 478.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(478))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
478.2.a.a	4	degree 4	$4q - 4q^{2} - 2q^{3} + 4q^{4} - 3q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
478.2.a.b	4	degree 4	$4q + 4q^{2} - 6q^{3} + 4q^{4} - 7q^{5} - 6q^{6} - 6q^{7} + O(q^{8})$	− −	$-1$
478.2.a.c	5	degree 5	$5q + 5q^{2} + 2q^{3} + 5q^{4} + 3q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$
478.2.a.d	6	degree 6	$6q - 6q^{2} + 2q^{3} + 6q^{4} + 5q^{5} - 2q^{6} + O(q^{8})$	+ −	$+1$

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 $T$
478.2.a.a	$x^{4} + 5x^{3} - 12x - 7$
478.2.a.b	$x^{4} + 13x^{3} + 60x^{2} + 116x + 79$
478.2.a.c	$x^{5} - 5x^{4} - 6x^{3} + 48x^{2} - 25x - 30$
478.2.a.d	$x^{6} - 7x^{5} - 8x^{4} + 132x^{3} - 143x^{2} - 476x + 772$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(478))$	19
Old forms	40
Cusp forms $S_{2}(\Gamma_0(478))$	59
Eisenstein series $E_{2}(\Gamma_0(478))$	3
Modular forms $M_{2}(\Gamma_0(478))$	62

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(478)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(239))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.2 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, 239, 478]: [0, 1, 21, 62] (certified at ell = 2147482093)
T = [(3, 1), (5, 1)]; h of degree 19 = 1 x 19 (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(478, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 478:2 1000

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