Atlas › Modular forms › Weight 4 › Level 167 › 167.4.a

Newspace 167.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(167))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
167.4.a.a	15	degree 15	$15q - 7q^{2} - 13q^{3} + 41q^{4} - 26q^{5} - 20q^{6} - 85q^{7} + O(q^{8})$	−	$-1$
167.4.a.b	26	degree 26	$26q + 5q^{2} + 11q^{3} + 129q^{4} + 34q^{5} + 28q^{6} + 83q^{7} + O(q^{8})$	+	$+1$

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 $T$
167.4.a.a	$x^{15} + 20x^{14} - 77x^{13} - 3078x^{12} - 2482x^{11} + 156371x^{10} + 215309x^{9} - 3561590x^{8} - 3719741x^{7} + 37832097x^{6} + 22701406x^{5} - 185261047x^{4} - 25792078x^{3} + 354998638x^{2} - 108407162x - 74375019$
167.4.a.b	$x^{26} - 16x^{25} - 582x^{24} + 9630x^{23} + 143044x^{22} - 2434790x^{21} - 20052747x^{20} + 343764145x^{19} + 1821609445x^{18} - 30118931779x^{17} - 115021552550x^{16} + 1709155753627x^{15} + 5227183587210x^{14} - 63282460715201x^{13} - 170963052018428x^{12} + 1495393282134301x^{11} + 3881508106120314x^{10} - 21295611658780743x^{9} - 57100116623466120x^{8} + 162241488567297709x^{7} + 492320470725463832x^{6} - 480826735624486696x^{5} - 2141618186449861476x^{4} - 443666870058802386x^{3} + 3243673881937847784x^{2} + 3106979012025371639x + 782927812525957665$

Orbit

Characteristic polynomial of

T

167.4.a.a

x^{15} + 20x^{14} - 77x^{13} - 3078x^{12} - 2482x^{11} + 156371x^{10} + 215309x^{9} - 3561590x^{8} - 3719741x^{7} + 37832097x^{6} + 22701406x^{5} - 185261047x^{4} - 25792078x^{3} + 354998638x^{2} - 108407162x - 74375019

167.4.a.b

x^{26} - 16x^{25} - 582x^{24} + 9630x^{23} + 143044x^{22} - 2434790x^{21} - 20052747x^{20} + 343764145x^{19} + 1821609445x^{18} - 30118931779x^{17} - 115021552550x^{16} + 1709155753627x^{15} + 5227183587210x^{14} - 63282460715201x^{13} - 170963052018428x^{12} + 1495393282134301x^{11} + 3881508106120314x^{10} - 21295611658780743x^{9} - 57100116623466120x^{8} + 162241488567297709x^{7} + 492320470725463832x^{6} - 480826735624486696x^{5} - 2141618186449861476x^{4} - 443666870058802386x^{3} + 3243673881937847784x^{2} + 3106979012025371639x + 782927812525957665

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(167))$	41
Old forms	0
Cusp forms $S_{4}(\Gamma_0(167))$	41
Eisenstein series $E_{4}(\Gamma_0(167))$	2
Modular forms $M_{4}(\Gamma_0(167))$	43

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.124 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, 167]: [1, 43] (certified at ell = 2147483237)
T = [(2, 1), (3, 1)]; h of degree 41 = 1 x 41 (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(167, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 167:4 1000

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