Atlas › Modular forms › Weight 4 › Level 53 › 53.4.a

Newspace 53.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(53))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
53.4.a.a	1	$\mathbb{Q}$	$q + q^{3} - 8q^{4} - 18q^{5} + 2q^{7} + O(q^{8})$	−	$-1$
53.4.a.b	4	degree 4	$4q - 4q^{2} - 15q^{3} + 16q^{4} - q^{5} - 27q^{6} - 46q^{7} + O(q^{8})$	−	$-1$
53.4.a.c	8	degree 8	$8q + 2q^{2} + 16q^{3} + 44q^{4} + q^{5} + 9q^{6} + 68q^{7} + O(q^{8})$	+	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
53.4.a.a	$x - 1$
53.4.a.b	$x^{4} + 19x^{3} + 116x^{2} + 240x + 100$
53.4.a.c	$x^{8} - 18x^{7} - 50x^{6} + 2556x^{5} - 10869x^{4} - 60400x^{3} + 562168x^{2} - 1419120x + 1156176$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(53))$	13
Old forms	0
Cusp forms $S_{4}(\Gamma_0(53))$	13
Eisenstein series $E_{4}(\Gamma_0(53))$	2
Modular forms $M_{4}(\Gamma_0(53))$	15

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.096 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, 53]: [1, 15] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; 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(53, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 53:4 1000

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