Atlas › Modular forms › Weight 2 › Level 613 › 613.2.a

Newspace 613.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(613))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
613.2.a.a	5	degree 5	$5q - 4q^{2} - 5q^{3} + 8q^{4} + 4q^{6} - 2q^{7} + O(q^{8})$	+	$-1$
613.2.a.b	18	degree 18	$18q - 6q^{2} - 13q^{3} + 12q^{4} - 16q^{5} - 2q^{6} - 8q^{7} + O(q^{8})$	+	$-1$
613.2.a.c	27	degree 27	$27q + 8q^{2} + 18q^{3} + 28q^{4} + 14q^{5} + 4q^{6} + 6q^{7} + O(q^{8})$	−	$+1$

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 $T$
613.2.a.a	$x^{5} + 9x^{4} + 25x^{3} + 13x^{2} - 34x - 31$
613.2.a.b	$x^{18} + 19x^{17} + 116x^{16} + 76x^{15} - 1774x^{14} - 5085x^{13} + 5847x^{12} + 34758x^{11} + 3433x^{10} - 89357x^{9} - 26062x^{8} + 104837x^{7} + 19326x^{6} - 53732x^{5} - 4104x^{4} + 10702x^{3} + 676x^{2} - 711x - 81$
613.2.a.c	$x^{27} - 26x^{26} + 236x^{25} - 442x^{24} - 6760x^{23} + 44458x^{22} - 16365x^{21} - 723928x^{20} + 2134990x^{19} + 3184967x^{18} - 25162334x^{17} + 19399408x^{16} + 117802397x^{15} - 239428979x^{14} - 186131255x^{13} + 907940240x^{12} - 284200411x^{11} - 1560486555x^{10} + 1352634548x^{9} + 1144790962x^{8} - 1624872049x^{7} - 232710040x^{6} + 765871749x^{5} - 31404965x^{4} - 129083990x^{3} - 3369997x^{2} + 4054947x + 3577$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(613))$	50
Old forms	0
Cusp forms $S_{2}(\Gamma_0(613))$	50
Eisenstein series $E_{2}(\Gamma_0(613))$	1
Modular forms $M_{2}(\Gamma_0(613))$	51

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.212 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, 613]: [0, 51] (certified at ell = 2147478013)
T = [(2, 1), (3, 1)]; h of degree 50 = 1 x 50 (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(613, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 613:2 1000

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