Atlas › Modular forms › Weight 2 › Level 617 › 617.2.a

Newspace 617.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(617))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
617.2.a.a	23	degree 23	$23q - 6q^{2} - 13q^{3} + 18q^{4} - 6q^{5} - 5q^{6} - 37q^{7} + O(q^{8})$	+	$-1$
617.2.a.b	28	degree 28	$28q + 3q^{2} + 11q^{3} + 29q^{4} + 2q^{5} + 5q^{6} + 39q^{7} + O(q^{8})$	−	$+1$

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 $T$
617.2.a.a	$x^{23} + 19x^{22} + 108x^{21} - 87x^{20} - 2861x^{19} - 6203x^{18} + 22432x^{17} + 89110x^{16} - 60756x^{15} - 534442x^{14} - 49836x^{13} + 1784172x^{12} + 609684x^{11} - 3654933x^{10} - 1214769x^{9} + 4715758x^{8} + 766941x^{7} - 3646092x^{6} + 297069x^{5} + 1387418x^{4} - 448255x^{3} - 117350x^{2} + 70326x - 7983$
617.2.a.b	$x^{28} - 14x^{27} - 5x^{26} + 903x^{25} - 2852x^{24} - 20884x^{23} + 112859x^{22} + 187826x^{21} - 1925650x^{20} + 163328x^{19} + 17954082x^{18} - 16541212x^{17} - 99776640x^{16} + 141683001x^{15} + 345362706x^{14} - 607296711x^{13} - 768869371x^{12} + 1520905549x^{11} + 1144449363x^{10} - 2314979609x^{9} - 1193978579x^{8} + 2115923035x^{7} + 892532452x^{6} - 1076105881x^{5} - 439542455x^{4} + 238563524x^{3} + 104632508x^{2} - 2405034x - 767942$

Orbit

Characteristic polynomial of

T

617.2.a.a

x^{23} + 19x^{22} + 108x^{21} - 87x^{20} - 2861x^{19} - 6203x^{18} + 22432x^{17} + 89110x^{16} - 60756x^{15} - 534442x^{14} - 49836x^{13} + 1784172x^{12} + 609684x^{11} - 3654933x^{10} - 1214769x^{9} + 4715758x^{8} + 766941x^{7} - 3646092x^{6} + 297069x^{5} + 1387418x^{4} - 448255x^{3} - 117350x^{2} + 70326x - 7983

617.2.a.b

x^{28} - 14x^{27} - 5x^{26} + 903x^{25} - 2852x^{24} - 20884x^{23} + 112859x^{22} + 187826x^{21} - 1925650x^{20} + 163328x^{19} + 17954082x^{18} - 16541212x^{17} - 99776640x^{16} + 141683001x^{15} + 345362706x^{14} - 607296711x^{13} - 768869371x^{12} + 1520905549x^{11} + 1144449363x^{10} - 2314979609x^{9} - 1193978579x^{8} + 2115923035x^{7} + 892532452x^{6} - 1076105881x^{5} - 439542455x^{4} + 238563524x^{3} + 104632508x^{2} - 2405034x - 767942

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(617))$	51
Old forms	0
Cusp forms $S_{2}(\Gamma_0(617))$	51
Eisenstein series $E_{2}(\Gamma_0(617))$	1
Modular forms $M_{2}(\Gamma_0(617))$	52

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.209 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, 617]: [0, 52] (certified at ell = 2147481337)
T = [(2, 1), (3, 1)]; h of degree 51 = 1 x 51 (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(617, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 617:2 1000

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