Atlas › Modular forms › Weight 6 › Level 62 › 62.6.a

Newspace 62.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(62))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
62.6.a.a	2	$\mathbb{Q}(\sqrt{94})$	$2q - 8q^{2} - 8q^{3} + 32q^{4} + 2q^{5} + 32q^{6} - 50q^{7} + O(q^{8})$	+ +	$-1$
62.6.a.b	2	$\mathbb{Q}(\sqrt{7})$	$2q + 8q^{2} - 8q^{3} + 32q^{4} - 110q^{5} - 32q^{6} - 126q^{7} + O(q^{8})$	− −	$-1$
62.6.a.c	4	degree 4	$4q - 16q^{2} + 10q^{3} + 64q^{4} + 2q^{5} - 40q^{6} + 146q^{7} + O(q^{8})$	+ −	$+1$
62.6.a.d	4	degree 4	$4q + 16q^{2} + 28q^{3} + 64q^{4} - 10q^{5} + 112q^{6} + 266q^{7} + O(q^{8})$	− +	$+1$

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 $T$
62.6.a.a	$x^{2} + 6x - 367$
62.6.a.b	$x^{2} + 118x + 3453$
62.6.a.c	$x^{4} - 12x^{3} - 13497x^{2} + 222974x + 4095090$
62.6.a.d	$x^{4} - 18x^{3} - 7387x^{2} + 83308x + 12334280$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(62))$	12
Old forms	26
Cusp forms $S_{6}(\Gamma_0(62))$	38
Eisenstein series $E_{6}(\Gamma_0(62))$	4
Modular forms $M_{6}(\Gamma_0(62))$	42

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(62)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.241 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, 31, 62]: [1, 2, 15, 42] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; h of degree 12 = 1 x 12 (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(62, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 62:6 1000

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