Atlas › Modular forms › Weight 2 › Level 658 › 658.2.a

Newspace 658.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(658))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
658.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$	658a1 rank 0
658.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 3q^{5} - q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$	658c1 rank 0
658.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} + 2q^{5} - 2q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$	658b1 rank 0
658.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - 3q^{3} + q^{4} - q^{5} - 3q^{6} + q^{7} + O(q^{8})$	− − +	$-1$	658f1 rank 1
658.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + O(q^{8})$	− + −	$-1$	658d1 rank 1
658.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 4q^{5} + q^{7} + O(q^{8})$	− − +	$-1$	658e1 rank 1
658.2.a.g	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} - 7q^{5} + q^{6} + 3q^{7} + O(q^{8})$	+ − −	$-1$
658.2.a.h	4	degree 4	$4q - 4q^{2} - q^{3} + 4q^{4} - q^{5} + q^{6} - 4q^{7} + O(q^{8})$	+ + +	$-1$
658.2.a.i	5	degree 5	$5q + 5q^{2} + q^{3} + 5q^{4} + 3q^{5} + q^{6} - 5q^{7} + O(q^{8})$	− + +	$+1$
658.2.a.j	5	degree 5	$5q + 5q^{2} + 3q^{3} + 5q^{4} + 5q^{5} + 3q^{6} + 5q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13}$ 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 $T$
658.2.a.a	$x - 15$
658.2.a.b	$x - 39$
658.2.a.c	$x - 12$
658.2.a.d	$x + 43$
658.2.a.e	$x + 15$
658.2.a.f	$x + 22$
658.2.a.g	$x^{3} + 9x^{2} - 3573x + 47115$
658.2.a.h	$x^{4} + 53x^{3} + 375x^{2} + 599x + 268$
658.2.a.i	$x^{5} - 87x^{4} + 1065x^{3} + 72523x^{2} - 1686638x + 4243928$
658.2.a.j	$x^{5} - 41x^{4} - 573x^{3} + 35113x^{2} - 180832x - 2294044$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(658))$	23
Old forms	70
Cusp forms $S_{2}(\Gamma_0(658))$	93
Eisenstein series $E_{2}(\Gamma_0(658))$	7
Modular forms $M_{2}(\Gamma_0(658))$	100

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(658)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(94))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(329))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.421 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, 7, 14, 47, 94, 329, 658]: [0, 1, 1, 4, 5, 14, 34, 100] (certified at ell = 2147483137)
T = [(3, 1), (5, 3), (11, 5), (13, 7)]; h of degree 23 = 1 x 23 (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(658, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 658:2 1000

← 657.2.a · 659.2.a →