Atlas › Modular forms › Weight 4 › Level 118 › 118.4.a

Newspace 118.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(118))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
118.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 5q^{3} + 4q^{4} - 5q^{5} - 10q^{6} - 33q^{7} + O(q^{8})$	+ −	$-1$
118.4.a.b	1	$\mathbb{Q}$	$q + 2q^{2} - 7q^{3} + 4q^{4} + 5q^{5} - 14q^{6} - 15q^{7} + O(q^{8})$	− +	$-1$
118.4.a.c	1	$\mathbb{Q}$	$q + 2q^{2} - q^{3} + 4q^{4} - 13q^{5} - 2q^{6} - 27q^{7} + O(q^{8})$	− +	$-1$
118.4.a.d	3	degree 3	$3q - 6q^{2} - 12q^{3} + 12q^{4} - 4q^{5} + 24q^{6} + 48q^{7} + O(q^{8})$	+ −	$-1$
118.4.a.e	4	degree 4	$4q - 8q^{2} + 2q^{3} + 16q^{4} + 16q^{5} - 4q^{6} + q^{7} + O(q^{8})$	+ +	$+1$
118.4.a.f	5	degree 5	$5q + 10q^{2} + 7q^{3} + 20q^{4} + 17q^{5} + 14q^{6} + 14q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
118.4.a.a	$x$
118.4.a.b	$x + 2$
118.4.a.c	$x + 14$
118.4.a.d	$x^{3} + 16x^{2} + 45x - 135$
118.4.a.e	$x^{4} - 18x^{3} - 300x^{2} + 4840x + 1888$
118.4.a.f	$x^{5} - 24x^{4} - 291x^{3} + 7361x^{2} - 26464x - 39680$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(118))$	15
Old forms	28
Cusp forms $S_{4}(\Gamma_0(118))$	43
Eisenstein series $E_{4}(\Gamma_0(118))$	4
Modular forms $M_{4}(\Gamma_0(118))$	47

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(118)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(59))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.113 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, 59, 118]: [1, 2, 16, 47] (certified at ell = 2147482481)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (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(118, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 118:4 1000

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