Atlas › Modular forms › Weight 6 › Level 68 › 68.6.a

Newspace 68.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(68))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
68.6.a.a	2	$\mathbb{Q}(\sqrt{13})$	$2q - 8q^{3} - 20q^{5} + 128q^{7} + O(q^{8})$	−
68.6.a.b	4	degree 4	$4q + 10q^{3} - 88q^{5} + 166q^{7} + O(q^{8})$	+

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 $T$
68.6.a.a	$x^{2} + 28x - 12$
68.6.a.b	$x^{4} + 78x^{3} - 9558x^{2} - 512584x + 12125736$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(68))$	6
Old forms	36
Cusp forms $S_{6}(\Gamma_0(68))$	42
Eisenstein series $E_{6}(\Gamma_0(68))$	6
Modular forms $M_{6}(\Gamma_0(68))$	48

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(68)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.245 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, 4, 8, 24, 48] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 2 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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 68:6 1000

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