Atlas › Modular forms › Weight 2 › Level 458 › 458.2.a

Newspace 458.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(458))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
458.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 3q^{3} + q^{4} + q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$	458a1 rank 1
458.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$	458b1 rank 1
458.2.a.c	2	$\mathbb{Q}(\sqrt{13})$	$2q - 2q^{2} + 2q^{4} + q^{5} - 3q^{7} + O(q^{8})$	+ +	$-1$
458.2.a.d	7	degree 7	$7q - 7q^{2} + 4q^{3} + 7q^{4} + q^{5} - 4q^{6} + 7q^{7} + O(q^{8})$	+ −	$+1$
458.2.a.e	9	degree 9	$9q + 9q^{2} + 2q^{3} + 9q^{4} + 4q^{5} + 2q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
458.2.a.a	$x$
458.2.a.b	$x + 4$
458.2.a.c	$x^{2} - 3x - 27$
458.2.a.d	$x^{7} - 7x^{6} - 273x^{5} + 2206x^{4} + 15894x^{3} - 142679x^{2} - 69859x + 1451996$
458.2.a.e	$x^{9} - 14x^{8} - 227x^{7} + 3758x^{6} + 8467x^{5} - 280303x^{4} + 522980x^{3} + 5364982x^{2} - 21477851x + 21382184$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(458))$	20
Old forms	36
Cusp forms $S_{2}(\Gamma_0(458))$	56
Eisenstein series $E_{2}(\Gamma_0(458))$	3
Modular forms $M_{2}(\Gamma_0(458))$	59

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(458)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(229))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.202 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, 229, 458]: [0, 1, 19, 59] (certified at ell = 2147478649)
T = [(3, 1), (5, 3)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 3 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(458, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 458:2 1000

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