Atlas › Modular forms › Weight 4 › Level 52 › 52.4.a

Newspace 52.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(52))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
52.4.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - 13q^{5} - 11q^{7} + O(q^{8})$	+
52.4.a.b	2	$\mathbb{Q}(\sqrt{217})$	$2q - 3q^{3} + 23q^{5} + 27q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
52.4.a.a	$x + 42$
52.4.a.b	$x^{2} - 66x + 872$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(52))$	3
Old forms	15
Cusp forms $S_{4}(\Gamma_0(52))$	18
Eisenstein series $E_{4}(\Gamma_0(52))$	6
Modular forms $M_{4}(\Gamma_0(52))$	24

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(52)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.113 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, 13, 26, 52]: [1, 2, 3, 5, 13, 24] (certified at ell = 2147483629)
T = [(3, 1), (5, 3)]; h of degree 3 = 1 x 3 (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(52, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 52:4 1000

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