Atlas › Modular forms › Weight 2 › Level 114 › 114.2.a

Newspace 114.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(114))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
114.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{6} + 4q^{7} + O(q^{8})$	+ + −	$+1$	114b1 rank 0
114.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + 2q^{5} - q^{6} + O(q^{8})$	− + +	$+1$	114c1 rank 0
114.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{6} - 4q^{7} + O(q^{8})$	− − −	$+1$	114a1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
114.2.a.a	$x - 12$
114.2.a.b	$x - 2$
114.2.a.c	$x + 12$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(114))$	3
Old forms	14
Cusp forms $S_{2}(\Gamma_0(114))$	17
Eisenstein series $E_{2}(\Gamma_0(114))$	7
Modular forms $M_{2}(\Gamma_0(114))$	24

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(114)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(38))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(57))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.093 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, 19, 38, 57, 114]: [0, 1, 1, 3, 2, 7, 8, 24] (certified at ell = 2147483647)
T = [(5, 1), (7, 3)]; 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(114, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 114:2 1000

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