Atlas › Modular forms › Weight 2 › Level 677 › 677.2.a

Newspace 677.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(677))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
677.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + q^{6} + q^{7} + O(q^{8})$	+	$-1$	677a1 rank 1
677.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - 3q^{3} - q^{4} + 4q^{6} - 4q^{7} + O(q^{8})$	+	$-1$
677.2.a.c	18	degree 18	$18q - 2q^{2} - 12q^{3} + 10q^{4} - 3q^{5} - 6q^{6} - 18q^{7} + O(q^{8})$	+	$-1$
677.2.a.d	35	degree 35	$35q + 4q^{2} + 18q^{3} + 46q^{4} + 3q^{5} + q^{6} + 23q^{7} + O(q^{8})$	−	$+1$

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 $T$
677.2.a.a	$x + 2$
677.2.a.b	$x^{2} + 4x - 1$
677.2.a.c	$x^{18} + 14x^{17} + 47x^{16} - 162x^{15} - 1140x^{14} - 122x^{13} + 9009x^{12} + 8290x^{11} - 34993x^{10} - 41573x^{9} + 75500x^{8} + 87692x^{7} - 93529x^{6} - 82854x^{5} + 63196x^{4} + 27921x^{3} - 18687x^{2} + 213x + 26$
677.2.a.d	$x^{35} - 22x^{34} + 108x^{33} + 1018x^{32} - 11364x^{31} + 2432x^{30} + 365666x^{29} - 1046858x^{28} - 5376124x^{27} + 28574013x^{26} + 30091547x^{25} - 385855751x^{24} + 170982348x^{23} + 3086763145x^{22} - 4030758589x^{21} - 15507323737x^{20} + 30961191168x^{19} + 49530469840x^{18} - 137162780357x^{17} - 97347061777x^{16} + 390015660321x^{15} + 101609091200x^{14} - 735125062914x^{13} - 10614957139x^{12} + 918864605755x^{11} - 108172690001x^{10} - 742235507276x^{9} + 129700042314x^{8} + 364159061001x^{7} - 62013577996x^{6} - 94777466701x^{5} + 8855084815x^{4} + 9078575714x^{3} + 1235985340x^{2} + 54282088x + 407584$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(677))$	56
Old forms	0
Cusp forms $S_{2}(\Gamma_0(677))$	56
Eisenstein series $E_{2}(\Gamma_0(677))$	1
Modular forms $M_{2}(\Gamma_0(677))$	57

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.233 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, 677]: [0, 57] (certified at ell = 2147480297)
T = [(2, 1), (3, 1)]; h of degree 56 = 1 x 56 (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(677, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 677:2 1000

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