Atlas › Modular forms › Weight 2 › Level 679 › 679.2.a

Newspace 679.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(679))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
679.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + 2q^{3} - q^{4} - 4q^{5} + 4q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$
679.2.a.b	4	degree 4	$4q - q^{2} + q^{3} + q^{4} + 2q^{5} - 9q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
679.2.a.c	4	degree 4	$4q - q^{2} + q^{3} + 3q^{4} - 7q^{5} + 4q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
679.2.a.d	8	degree 8	$8q - 2q^{2} - 4q^{3} + 4q^{4} - 9q^{5} - 3q^{6} + 8q^{7} + O(q^{8})$	− −	$-1$
679.2.a.e	14	degree 14	$14q + 2q^{2} + 4q^{3} + 16q^{4} + 19q^{5} - q^{6} + 14q^{7} + O(q^{8})$	− +	$+1$
679.2.a.f	17	degree 17	$17q + 4q^{2} + 22q^{4} + 5q^{5} - 3q^{6} - 17q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
679.2.a.a	$x^{2} - 5x - 55$
679.2.a.b	$x^{4} - 2x^{3} - 16x^{2} + 16x + 16$
679.2.a.c	$x^{4} - 2x^{3} - 56x^{2} + 99x - 43$
679.2.a.d	$x^{8} + 14x^{7} - 2x^{6} - 569x^{5} - 745x^{4} + 5598x^{3} + 2504x^{2} - 7360x - 3888$
679.2.a.e	$x^{14} - 14x^{13} - 218x^{12} + 3986x^{11} + 5835x^{10} - 338710x^{9} + 938624x^{8} + 9075647x^{7} - 46276642x^{6} - 28249739x^{5} + 405102767x^{4} - 198374080x^{3} - 721098400x^{2} - 189554200x - 7739600$
679.2.a.f	$x^{17} - 4x^{16} - 416x^{15} + 1511x^{14} + 69929x^{13} - 207648x^{12} - 6252306x^{11} + 13046898x^{10} + 324412884x^{9} - 345709103x^{8} - 9852568860x^{7} + 120765172x^{6} + 164319593677x^{5} + 156851707619x^{4} - 1223922888524x^{3} - 2139282760156x^{2} + 1564661463040x + 3477273155536$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(679))$	49
Old forms	14
Cusp forms $S_{2}(\Gamma_0(679))$	63
Eisenstein series $E_{2}(\Gamma_0(679))$	3
Modular forms $M_{2}(\Gamma_0(679))$	66

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(679)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(97))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.328 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, 7, 97, 679]: [0, 1, 8, 66] (certified at ell = 2147483137)
T = [(2, 1), (3, 3)]; h of degree 49 = 1 x 49 (phi(ord eps) x dim S_k^new); 7 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(679, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 679:2 1000

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