Atlas › Modular forms › Weight 4 › Level 115 › 115.4.a

Newspace 115.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(115))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
115.4.a.a	1	$\mathbb{Q}$	$q + q^{2} + 4q^{3} - 7q^{4} - 5q^{5} + 4q^{6} - 32q^{7} + O(q^{8})$	+ −	$-1$
115.4.a.b	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} - 4q^{4} + 5q^{5} - 6q^{6} - 2q^{7} + O(q^{8})$	− +	$-1$
115.4.a.c	2	$\mathbb{Q}(\sqrt{109})$	$2q - 6q^{2} - 3q^{3} + 2q^{4} + 10q^{5} + 9q^{6} + q^{7} + O(q^{8})$	− +	$-1$
115.4.a.d	5	degree 5	$5q - 5q^{2} - 6q^{3} + 33q^{4} - 25q^{5} - 36q^{6} - 15q^{7} + O(q^{8})$	+ −	$-1$
115.4.a.e	5	degree 5	$5q + 6q^{2} + 4q^{3} + 22q^{4} - 25q^{5} + 19q^{6} - 3q^{7} + O(q^{8})$	+ +	$+1$
115.4.a.f	8	degree 8	$8q + 6q^{2} + 42q^{4} + 40q^{5} + 11q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
115.4.a.a	$x - 5$
115.4.a.b	$x + 1$
115.4.a.c	$x^{2} + 9x - 7$
115.4.a.d	$x^{5} + 11x^{4} - 47x^{3} - 89x^{2} + 437x - 361$
115.4.a.e	$x^{5} - 10x^{4} - 106x^{3} + 825x^{2} + 2930x - 15634$
115.4.a.f	$x^{8} - 6x^{7} - 222x^{6} + 1242x^{5} + 14956x^{4} - 75868x^{3} - 291419x^{2} + 1173478x + 1062270$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(115))$	22
Old forms	12
Cusp forms $S_{4}(\Gamma_0(115))$	34
Eisenstein series $E_{4}(\Gamma_0(115))$	4
Modular forms $M_{4}(\Gamma_0(115))$	38

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(115)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.11 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, 5, 23, 115]: [1, 3, 7, 38] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 22 = 1 x 22 (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(115, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 115:4 1000

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