Atlas › Modular forms › Weight 2 › Level 258 › 258.2.a

Newspace 258.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(258))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
258.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} + 2q^{7} + O(q^{8})$	+ + −	$+1$	258b1 rank 0
258.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 5q^{7} + O(q^{8})$	+ + +	$-1$	258a1 rank 1
258.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 3q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+ − −	$-1$	258c1 rank 1
258.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + 4q^{7} + O(q^{8})$	− + +	$+1$	258d1 rank 0
258.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + 3q^{5} - q^{6} - q^{7} + O(q^{8})$	− + +	$+1$	258e1 rank 0
258.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + O(q^{8})$	− − −	$+1$	258f1 rank 0
258.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} - 2q^{7} + O(q^{8})$	− − −	$+1$	258g1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13}$ 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 $T$
258.2.a.a	$x - 18$
258.2.a.b	$x + 30$
258.2.a.c	$x + 58$
258.2.a.d	$x - 72$
258.2.a.e	$x - 2$
258.2.a.f	$x + 22$
258.2.a.g	$x + 10$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(258))$	7
Old forms	34
Cusp forms $S_{2}(\Gamma_0(258))$	41
Eisenstein series $E_{2}(\Gamma_0(258))$	7
Modular forms $M_{2}(\Gamma_0(258))$	48

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(258)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(86))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(129))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.22 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, 3, 6, 43, 86, 129, 258]: [0, 1, 1, 3, 4, 13, 16, 48] (certified at ell = 2147483647)
T = [(5, 1), (7, 3), (11, 5), (13, 7)]; h of degree 7 = 1 x 7 (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(258, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 258:2 1000

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