Atlas › Modular forms › Weight 2 › Level 691 › 691.2.a

Newspace 691.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(691))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
691.2.a.a	24	degree 24	$24q - 8q^{2} - 5q^{3} + 18q^{4} - 23q^{5} - 11q^{6} - 6q^{7} + O(q^{8})$	+	$-1$
691.2.a.b	33	degree 33	$33q + 8q^{2} + q^{3} + 36q^{4} + 23q^{5} + 5q^{6} + 4q^{7} + O(q^{8})$	−	$+1$

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 $T$
691.2.a.a	$x^{24} + 13x^{23} + 22x^{22} - 365x^{21} - 1475x^{20} + 3076x^{19} + 21492x^{18} - 6897x^{17} - 152462x^{16} - 33583x^{15} + 630495x^{14} + 221675x^{13} - 1611236x^{12} - 451827x^{11} + 2521879x^{10} + 278620x^{9} - 2250663x^{8} + 190394x^{7} + 981585x^{6} - 261440x^{5} - 142803x^{4} + 67700x^{3} - 6581x^{2} - 289x + 16$
691.2.a.b	$x^{33} - 9x^{32} - 88x^{31} + 989x^{30} + 2658x^{29} - 45639x^{28} - 18458x^{27} + 1173792x^{26} - 809385x^{25} - 18815481x^{24} + 24706887x^{23} + 199174888x^{22} - 339009864x^{21} - 1438662908x^{20} + 2809079276x^{19} + 7213122021x^{18} - 15206721713x^{17} - 25189818947x^{16} + 55258293652x^{15} + 60441689914x^{14} - 135279780014x^{13} - 95614322796x^{12} + 220452398498x^{11} + 90485940325x^{10} - 232561988863x^{9} - 38471129804x^{8} + 149890652651x^{7} - 5873629243x^{6} - 52035311350x^{5} + 10426531406x^{4} + 7043802576x^{3} - 2185982336x^{2} - 28139248x + 18569312$

Orbit

Characteristic polynomial of

T

691.2.a.a

x^{24} + 13x^{23} + 22x^{22} - 365x^{21} - 1475x^{20} + 3076x^{19} + 21492x^{18} - 6897x^{17} - 152462x^{16} - 33583x^{15} + 630495x^{14} + 221675x^{13} - 1611236x^{12} - 451827x^{11} + 2521879x^{10} + 278620x^{9} - 2250663x^{8} + 190394x^{7} + 981585x^{6} - 261440x^{5} - 142803x^{4} + 67700x^{3} - 6581x^{2} - 289x + 16

691.2.a.b

x^{33} - 9x^{32} - 88x^{31} + 989x^{30} + 2658x^{29} - 45639x^{28} - 18458x^{27} + 1173792x^{26} - 809385x^{25} - 18815481x^{24} + 24706887x^{23} + 199174888x^{22} - 339009864x^{21} - 1438662908x^{20} + 2809079276x^{19} + 7213122021x^{18} - 15206721713x^{17} - 25189818947x^{16} + 55258293652x^{15} + 60441689914x^{14} - 135279780014x^{13} - 95614322796x^{12} + 220452398498x^{11} + 90485940325x^{10} - 232561988863x^{9} - 38471129804x^{8} + 149890652651x^{7} - 5873629243x^{6} - 52035311350x^{5} + 10426531406x^{4} + 7043802576x^{3} - 2185982336x^{2} - 28139248x + 18569312

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(691))$	57
Old forms	0
Cusp forms $S_{2}(\Gamma_0(691))$	57
Eisenstein series $E_{2}(\Gamma_0(691))$	1
Modular forms $M_{2}(\Gamma_0(691))$	58

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.229 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, 691]: [0, 58] (certified at ell = 2147480791)
T = [(2, 1), (3, 1)]; h of degree 57 = 1 x 57 (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(691, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 691:2 1000

← 690.2.a · 692.2.a →