Atlas › Modular forms › Weight 4 › Level 163 › 163.4.a

Newspace 163.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(163))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
163.4.a.a	19	degree 19	$19q - 10q^{2} - 15q^{3} + 62q^{4} - 71q^{5} - 36q^{6} - 48q^{7} + O(q^{8})$	−	$-1$
163.4.a.b	21	degree 21	$21q + 12q^{2} + 15q^{3} + 82q^{4} + 79q^{5} + 36q^{6} + 36q^{7} + O(q^{8})$	+	$+1$

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 $T$
163.4.a.a	$x^{19} + 25x^{18} - 70x^{17} - 5386x^{16} - 11555x^{15} + 437166x^{14} + 1329443x^{13} - 17500984x^{12} - 47366076x^{11} + 375189648x^{10} + 667335034x^{9} - 4189738957x^{8} - 2766769810x^{7} + 20728835229x^{6} - 6487780125x^{5} - 30289921070x^{4} + 24707146682x^{3} - 178333913x^{2} - 2254979968x - 12056458$
163.4.a.b	$x^{21} - 27x^{20} - 180x^{19} + 10100x^{18} - 18457x^{17} - 1505490x^{16} + 6957783x^{15} + 119283680x^{14} - 722896770x^{13} - 5583954122x^{12} + 38834768042x^{11} + 161569378985x^{10} - 1208721434654x^{9} - 2944664902583x^{8} + 22127542069019x^{7} + 34017967735010x^{6} - 226671144797564x^{5} - 242404116590783x^{4} + 1107994998598080x^{3} + 863420391687518x^{2} - 1465853232899208x - 28790063240196$

Orbit

Characteristic polynomial of

T

163.4.a.a

x^{19} + 25x^{18} - 70x^{17} - 5386x^{16} - 11555x^{15} + 437166x^{14} + 1329443x^{13} - 17500984x^{12} - 47366076x^{11} + 375189648x^{10} + 667335034x^{9} - 4189738957x^{8} - 2766769810x^{7} + 20728835229x^{6} - 6487780125x^{5} - 30289921070x^{4} + 24707146682x^{3} - 178333913x^{2} - 2254979968x - 12056458

163.4.a.b

x^{21} - 27x^{20} - 180x^{19} + 10100x^{18} - 18457x^{17} - 1505490x^{16} + 6957783x^{15} + 119283680x^{14} - 722896770x^{13} - 5583954122x^{12} + 38834768042x^{11} + 161569378985x^{10} - 1208721434654x^{9} - 2944664902583x^{8} + 22127542069019x^{7} + 34017967735010x^{6} - 226671144797564x^{5} - 242404116590783x^{4} + 1107994998598080x^{3} + 863420391687518x^{2} - 1465853232899208x - 28790063240196

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(163))$	40
Old forms	0
Cusp forms $S_{4}(\Gamma_0(163))$	40
Eisenstein series $E_{4}(\Gamma_0(163))$	2
Modular forms $M_{4}(\Gamma_0(163))$	42

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.12 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, 163]: [1, 42] (certified at ell = 2147483179)
T = [(2, 1), (3, 1)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 6 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(163, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 163:4 1000

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