Atlas › Modular forms › Weight 6 › Level 53 › 53.6.a

Newspace 53.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(53))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
53.6.a.a	9	degree 9	$9q - 4q^{2} - 47q^{3} + 80q^{4} - 14q^{5} - 87q^{6} - 404q^{7} + O(q^{8})$	+	$-1$
53.6.a.b	12	degree 12	$12q + 8q^{2} + 43q^{3} + 224q^{4} + 86q^{5} + 129q^{6} + 380q^{7} + O(q^{8})$	−	$+1$

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 $T$
53.6.a.a	$x^{9} + 51x^{8} - 36x^{7} - 28078x^{6} - 212939x^{5} + 4030367x^{4} + 42913558x^{3} - 92968836x^{2} - 1527177000x - 1276100320$
53.6.a.b	$x^{12} - 51x^{11} - 1101x^{10} + 77969x^{9} + 317613x^{8} - 43924901x^{7} + 27309005x^{6} + 11000703475x^{5} - 26586387006x^{4} - 1176072939536x^{3} + 4474638313036x^{2} + 43254516477864x - 211116842821728$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(53))$	21
Old forms	0
Cusp forms $S_{6}(\Gamma_0(53))$	21
Eisenstein series $E_{6}(\Gamma_0(53))$	2
Modular forms $M_{6}(\Gamma_0(53))$	23

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.228 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, 23] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 4 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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 53:6 1000

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