Atlas › Modular forms › Weight 4 › Level 130 › 130.4.a

Newspace 130.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(130))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
130.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 2q^{3} + 4q^{4} + 5q^{5} + 4q^{6} + 8q^{7} + O(q^{8})$	+ − −	$+1$
130.4.a.b	1	$\mathbb{Q}$	$q + 2q^{2} - 4q^{3} + 4q^{4} - 5q^{5} - 8q^{6} - 8q^{7} + O(q^{8})$	− + +	$-1$
130.4.a.c	2	$\mathbb{Q}(\sqrt{30})$	$2q - 4q^{2} - 8q^{3} + 8q^{4} + 10q^{5} + 16q^{6} - 20q^{7} + O(q^{8})$	+ − +	$-1$
130.4.a.d	2	$\mathbb{Q}(\sqrt{19})$	$2q - 4q^{2} + 2q^{3} + 8q^{4} - 10q^{5} - 4q^{6} + O(q^{8})$	+ + −	$-1$
130.4.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q - 4q^{2} + 2q^{3} + 8q^{4} - 10q^{5} - 4q^{6} + O(q^{8})$	+ + +	$+1$
130.4.a.f	2	$\mathbb{Q}(\sqrt{51})$	$2q + 4q^{2} + 2q^{3} + 8q^{4} - 10q^{5} + 4q^{6} + 16q^{7} + O(q^{8})$	− + −	$+1$
130.4.a.g	2	$\mathbb{Q}(\sqrt{10})$	$2q + 4q^{2} + 12q^{3} + 8q^{4} + 10q^{5} + 24q^{6} + 4q^{7} + O(q^{8})$	− − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7}$ 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 $T$
130.4.a.a	$x - 22$
130.4.a.b	$x + 28$
130.4.a.c	$x^{2} + 68x - 2474$
130.4.a.d	$x^{2} - 2x - 5490$
130.4.a.e	$x^{2} - 2x - 3644$
130.4.a.f	$x^{2} - 50x - 650$
130.4.a.g	$x^{2} - 24x - 1066$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(130))$	12
Old forms	48
Cusp forms $S_{4}(\Gamma_0(130))$	60
Eisenstein series $E_{4}(\Gamma_0(130))$	8
Modular forms $M_{4}(\Gamma_0(130))$	68

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(130)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(65))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.134 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]: [1, 2, 3, 7, 5, 13, 24, 68] (certified at ell = 2147483629)
T = [(3, 1), (7, 3)]; h of degree 12 = 1 x 12 (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(130, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 130:4 1000

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