Atlas › Modular forms › Weight 2 › Level 358 › 358.2.a

Newspace 358.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(358))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
358.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} - 2q^{6} - 2q^{7} + O(q^{8})$	+ −	$+1$	358a1 rank 0
358.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 2q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$	358b1 rank 0
358.2.a.c	2	$\mathbb{Q}(\sqrt{21})$	$2q - 2q^{2} + q^{3} + 2q^{4} + 6q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$
358.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 3q^{3} + 2q^{4} - 4q^{5} - 3q^{6} - 6q^{7} + O(q^{8})$	− −	$-1$
358.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} + 3q^{3} + 2q^{4} + q^{5} + 3q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$
358.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} + 3q^{3} + 2q^{4} + 2q^{5} + 3q^{6} - 4q^{7} + O(q^{8})$	− +	$+1$
358.2.a.g	4	degree 4	$4q - 4q^{2} - 2q^{3} + 4q^{4} - 7q^{5} + 2q^{6} + O(q^{8})$	+ +	$-1$

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 $T$
358.2.a.a	$x - 2$
358.2.a.b	$x + 2$
358.2.a.c	$x^{2} - 7x + 7$
358.2.a.d	$x^{2} + 7x + 11$
358.2.a.e	$x^{2} - 4x - 1$
358.2.a.f	$x^{2} - 5x + 5$
358.2.a.g	$x^{4} + 9x^{3} + 24x^{2} + 19x - 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(358))$	14
Old forms	30
Cusp forms $S_{2}(\Gamma_0(358))$	44
Eisenstein series $E_{2}(\Gamma_0(358))$	3
Modular forms $M_{2}(\Gamma_0(358))$	47

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(358)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(179))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.302 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, 179, 358]: [0, 1, 16, 47] (certified at ell = 2147483137)
T = [(3, 1), (5, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 2 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(358, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 358:2 1000

← 357.2.a · 359.2.a →