Atlas › Modular forms › Weight 4 › Level 68 › 68.4.a

Newspace 68.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(68))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
68.4.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - 8q^{5} - 12q^{7} + O(q^{8})$	+
68.4.a.b	3	degree 3	$3q + 4q^{3} + 26q^{5} + 8q^{7} + O(q^{8})$	−

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 $T$
68.4.a.a	$x + 10$
68.4.a.b	$x^{3} - 30x^{2} + 108x - 88$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(68))$	4
Old forms	20
Cusp forms $S_{4}(\Gamma_0(68))$	24
Eisenstein series $E_{4}(\Gamma_0(68))$	6
Modular forms $M_{4}(\Gamma_0(68))$	30

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(68)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.101 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, 4, 17, 34, 68]: [1, 2, 3, 6, 16, 30] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 4 = 1 x 4 (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(68, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 68:4 1000

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