Atlas › Modular forms › Weight 8 › Level 53 › 53.8.a

Newspace 53.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(53))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
53.8.a.a	14	degree 14	$14q - 16q^{2} - 149q^{3} + 768q^{4} - 374q^{5} - 615q^{6} - 3076q^{7} + O(q^{8})$	−	$-1$
53.8.a.b	17	degree 17	$17q + 8q^{2} + 121q^{3} + 1344q^{4} + 126q^{5} + 681q^{6} + 2412q^{7} + O(q^{8})$	+	$+1$

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 $T$
53.8.a.a	$x^{14} + 165x^{13} - 6910x^{12} - 2028628x^{11} + 519086x^{10} + 9358400554x^{9} + 70576358220x^{8} - 20608246011552x^{7} - 125086856832391x^{6} + 23125397027566277x^{5} + 37222431383262342x^{4} - 12547592145144646792x^{3} + 41197958629672114836x^{2} + 2551728827283081067160x - 19272378287851051652800$
53.8.a.b	$x^{17} - 129x^{16} - 20539x^{15} + 2757637x^{14} + 174012168x^{13} - 23251490146x^{12} - 817438126242x^{11} + 97446799578438x^{10} + 2418180232782225x^{9} - 208219488393214277x^{8} - 4551323234399349339x^{7} + 202393624152310403885x^{6} + 4606733698182362397550x^{5} - 58555754072980933956304x^{4} - 1499972841208514027083008x^{3} + 2096489278558900188376560x^{2} + 140218548524266872776911200x + 512899227957749250911328000$

Orbit

Characteristic polynomial of

T

53.8.a.a

x^{14} + 165x^{13} - 6910x^{12} - 2028628x^{11} + 519086x^{10} + 9358400554x^{9} + 70576358220x^{8} - 20608246011552x^{7} - 125086856832391x^{6} + 23125397027566277x^{5} + 37222431383262342x^{4} - 12547592145144646792x^{3} + 41197958629672114836x^{2} + 2551728827283081067160x - 19272378287851051652800

53.8.a.b

x^{17} - 129x^{16} - 20539x^{15} + 2757637x^{14} + 174012168x^{13} - 23251490146x^{12} - 817438126242x^{11} + 97446799578438x^{10} + 2418180232782225x^{9} - 208219488393214277x^{8} - 4551323234399349339x^{7} + 202393624152310403885x^{6} + 4606733698182362397550x^{5} - 58555754072980933956304x^{4} - 1499972841208514027083008x^{3} + 2096489278558900188376560x^{2} + 140218548524266872776911200x + 512899227957749250911328000

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(53))$	31
Old forms	0
Cusp forms $S_{8}(\Gamma_0(53))$	31
Eisenstein series $E_{8}(\Gamma_0(53))$	2
Modular forms $M_{8}(\Gamma_0(53))$	33

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.287 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, 33] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 7 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, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 53:8 1000

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