Atlas › Modular forms › Weight 2 › Level 130 › 130.2.a

Newspace 130.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(130))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
130.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	+ − −	$-1$	130a1 rank 1
130.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + q^{5} + O(q^{8})$	− − −	$+1$	130b1 rank 0
130.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} - q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	− + +	$+1$	130c1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{17}$ 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 $T$
130.2.a.a	$x + 86$
130.2.a.b	$x - 14$
130.2.a.c	$x + 6$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(130))$	3
Old forms	14
Cusp forms $S_{2}(\Gamma_0(130))$	17
Eisenstein series $E_{2}(\Gamma_0(130))$	7
Modular forms $M_{2}(\Gamma_0(130))$	24

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(130)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(65))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.207 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, 5, 10, 13, 26, 65, 130]: [0, 1, 1, 3, 1, 5, 8, 24] (certified at ell = 2147483629)
T = [(3, 1), (7, 3), (11, 5), (17, 7)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 1 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(130, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 130:2 1000

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