Atlas › Modular forms › Weight 4 › Level 241 › 241.4.a

Newspace 241.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(241))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
241.4.a.a	27	degree 27	$27q - 11q^{2} - 14q^{3} + 89q^{4} - 50q^{5} - 63q^{6} - 68q^{7} + O(q^{8})$	−	$-1$
241.4.a.b	33	degree 33	$33q + 9q^{2} + 10q^{3} + 161q^{4} + 50q^{5} + 33q^{6} + 58q^{7} + O(q^{8})$	+	$+1$

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 $T$
241.4.a.a	$x^{27} + 25x^{26} - 209x^{25} - 8960x^{24} + 5093x^{23} + 1403197x^{22} + 2234726x^{21} - 127390118x^{20} - 277555725x^{19} + 7448916991x^{18} + 15063575608x^{17} - 291538954093x^{16} - 431173859379x^{15} + 7609789175720x^{14} + 6115501785968x^{13} - 127761019083425x^{12} - 26011100279290x^{11} + 1304467793438453x^{10} - 233701029100312x^{9} - 7784189928462843x^{8} + 2796666051785840x^{7} + 26026909298657596x^{6} - 10504993601392807x^{5} - 45621625046525135x^{4} + 16211128356846434x^{3} + 36185752952641940x^{2} - 8089234974528480x - 8708225245120064$
241.4.a.b	$x^{33} - 19x^{32} - 713x^{31} + 14888x^{30} + 206737x^{29} - 4906767x^{28} - 32410626x^{27} + 908516434x^{26} + 3117824283x^{25} - 106443625141x^{24} - 204061960084x^{23} + 8393484961463x^{22} + 10519918980897x^{21} - 460073357068844x^{20} - 502792361372960x^{19} + 17738644709032611x^{18} + 22244123433782914x^{17} - 477609633472578551x^{16} - 779122366967965080x^{15} + 8705572088486500877x^{14} + 18955452337289080172x^{13} - 99973323077140701720x^{12} - 294687612451532750995x^{11} + 602766191094844667165x^{10} + 2674786481938500317758x^{9} - 594843531330169046940x^{8} - 11985913575926843595600x^{7} - 10628640627235027452976x^{6} + 15892056079018589613856x^{5} + 30149055600762330642880x^{4} + 9731199845878694075392x^{3} - 8080600033709457220096x^{2} - 5088708972789220769792x - 412625090630475948032$

Orbit

Characteristic polynomial of

T

241.4.a.a

x^{27} + 25x^{26} - 209x^{25} - 8960x^{24} + 5093x^{23} + 1403197x^{22} + 2234726x^{21} - 127390118x^{20} - 277555725x^{19} + 7448916991x^{18} + 15063575608x^{17} - 291538954093x^{16} - 431173859379x^{15} + 7609789175720x^{14} + 6115501785968x^{13} - 127761019083425x^{12} - 26011100279290x^{11} + 1304467793438453x^{10} - 233701029100312x^{9} - 7784189928462843x^{8} + 2796666051785840x^{7} + 26026909298657596x^{6} - 10504993601392807x^{5} - 45621625046525135x^{4} + 16211128356846434x^{3} + 36185752952641940x^{2} - 8089234974528480x - 8708225245120064

241.4.a.b

x^{33} - 19x^{32} - 713x^{31} + 14888x^{30} + 206737x^{29} - 4906767x^{28} - 32410626x^{27} + 908516434x^{26} + 3117824283x^{25} - 106443625141x^{24} - 204061960084x^{23} + 8393484961463x^{22} + 10519918980897x^{21} - 460073357068844x^{20} - 502792361372960x^{19} + 17738644709032611x^{18} + 22244123433782914x^{17} - 477609633472578551x^{16} - 779122366967965080x^{15} + 8705572088486500877x^{14} + 18955452337289080172x^{13} - 99973323077140701720x^{12} - 294687612451532750995x^{11} + 602766191094844667165x^{10} + 2674786481938500317758x^{9} - 594843531330169046940x^{8} - 11985913575926843595600x^{7} - 10628640627235027452976x^{6} + 15892056079018589613856x^{5} + 30149055600762330642880x^{4} + 9731199845878694075392x^{3} - 8080600033709457220096x^{2} - 5088708972789220769792x - 412625090630475948032

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(241))$	60
Old forms	0
Cusp forms $S_{4}(\Gamma_0(241))$	60
Eisenstein series $E_{4}(\Gamma_0(241))$	2
Modular forms $M_{4}(\Gamma_0(241))$	62

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.164 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, 241]: [1, 62] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 60 = 1 x 60 (phi(ord eps) x dim S_k^new); 9 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(241, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 241:4 1000

← 240.4.a · 242.4.a →