Atlas › Modular forms › Weight 2 › Level 601 › 601.2.a

Newspace 601.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(601))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
601.2.a.a	20	degree 20	$20q - 5q^{2} - 7q^{3} + 11q^{4} - 12q^{5} - 8q^{6} - 11q^{7} + O(q^{8})$	+	$-1$
601.2.a.b	29	degree 29	$29q + 4q^{2} + 5q^{3} + 34q^{4} + 10q^{5} - 4q^{6} + 7q^{7} + O(q^{8})$	−	$+1$

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 $T$
601.2.a.a	$x^{20} + 12x^{19} + 23x^{18} - 227x^{17} - 889x^{16} + 1179x^{15} + 8477x^{14} - 10x^{13} - 35522x^{12} - 13615x^{11} + 76003x^{10} + 35031x^{9} - 89376x^{8} - 38312x^{7} + 59500x^{6} + 20887x^{5} - 22150x^{4} - 5506x^{3} + 4277x^{2} + 549x - 337$
601.2.a.b	$x^{29} - 9x^{28} - 65x^{27} + 790x^{26} + 1052x^{25} - 27646x^{24} + 15400x^{23} + 508441x^{22} - 744695x^{21} - 5484258x^{20} + 10911525x^{19} + 37059788x^{18} - 85965289x^{17} - 164374558x^{16} + 410757074x^{15} + 496184580x^{14} - 1237203075x^{13} - 1048612813x^{12} + 2344667809x^{11} + 1568175610x^{10} - 2686071131x^{9} - 1606180735x^{8} + 1682262386x^{7} + 1007931879x^{6} - 448796225x^{5} - 300480054x^{4} + 18188047x^{3} + 21033136x^{2} + 115442x - 47831$

Orbit

Characteristic polynomial of

T

601.2.a.a

x^{20} + 12x^{19} + 23x^{18} - 227x^{17} - 889x^{16} + 1179x^{15} + 8477x^{14} - 10x^{13} - 35522x^{12} - 13615x^{11} + 76003x^{10} + 35031x^{9} - 89376x^{8} - 38312x^{7} + 59500x^{6} + 20887x^{5} - 22150x^{4} - 5506x^{3} + 4277x^{2} + 549x - 337

601.2.a.b

x^{29} - 9x^{28} - 65x^{27} + 790x^{26} + 1052x^{25} - 27646x^{24} + 15400x^{23} + 508441x^{22} - 744695x^{21} - 5484258x^{20} + 10911525x^{19} + 37059788x^{18} - 85965289x^{17} - 164374558x^{16} + 410757074x^{15} + 496184580x^{14} - 1237203075x^{13} - 1048612813x^{12} + 2344667809x^{11} + 1568175610x^{10} - 2686071131x^{9} - 1606180735x^{8} + 1682262386x^{7} + 1007931879x^{6} - 448796225x^{5} - 300480054x^{4} + 18188047x^{3} + 21033136x^{2} + 115442x - 47831

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(601))$	49
Old forms	0
Cusp forms $S_{2}(\Gamma_0(601))$	49
Eisenstein series $E_{2}(\Gamma_0(601))$	1
Modular forms $M_{2}(\Gamma_0(601))$	50

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.208 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, 601]: [0, 50] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 49 = 1 x 49 (phi(ord eps) x dim S_k^new); 5 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(601, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 601:2 1000

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