Atlas › Modular forms › Weight 2 › Level 674 › 674.2.a

Newspace 674.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(674))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
674.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 2q^{5} - q^{6} + O(q^{8})$	+ +	$-1$	674a1 rank 1
674.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 3q^{3} + q^{4} - 2q^{5} - 3q^{6} + 2q^{7} + O(q^{8})$	− −	$-1$	674c1 rank 1
674.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 2q^{5} - 4q^{7} + O(q^{8})$	− −	$-1$	674b1 rank 1
674.2.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} - 4q^{3} + 2q^{4} + 2q^{5} + 4q^{6} + O(q^{8})$	+ +	$-1$
674.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} + 8q^{7} + O(q^{8})$	+ −	$+1$
674.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 3q^{3} + 2q^{4} - 3q^{6} - 7q^{7} + O(q^{8})$	+ +	$-1$
674.2.a.g	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - q^{3} + 2q^{4} - 4q^{5} - q^{6} - q^{7} + O(q^{8})$	− −	$-1$
674.2.a.h	7	degree 7	$7q - 7q^{2} + 7q^{4} - 4q^{5} + q^{7} + O(q^{8})$	+ −	$+1$
674.2.a.i	11	degree 11	$11q + 11q^{2} + 4q^{3} + 11q^{4} + 8q^{5} + 4q^{6} + 5q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
674.2.a.a	$x + 1$
674.2.a.b	$x + 5$
674.2.a.c	$x + 2$
674.2.a.d	$x^{2} + 2x - 2$
674.2.a.e	$x^{2} - 8$
674.2.a.f	$x^{2} - 3x + 1$
674.2.a.g	$x^{2} + 5x + 5$
674.2.a.h	$x^{7} + 4x^{6} - 34x^{5} - 123x^{4} + 213x^{3} + 898x^{2} + 788x + 197$
674.2.a.i	$x^{11} - 12x^{10} + 18x^{9} + 297x^{8} - 1327x^{7} - 46x^{6} + 11164x^{5} - 25179x^{4} + 17384x^{3} + 5256x^{2} - 9056x + 1412$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(674))$	29
Old forms	54
Cusp forms $S_{2}(\Gamma_0(674))$	83
Eisenstein series $E_{2}(\Gamma_0(674))$	3
Modular forms $M_{2}(\Gamma_0(674))$	86

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(674)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(337))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.342 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, 2, 337, 674]: [0, 1, 28, 86] (certified at ell = 2147480161)
T = [(3, 1), (5, 1)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 4 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(674, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 674:2 1000

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