Atlas › Modular forms › Weight 6 › Level 64 › 64.6.a

Newspace 64.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(64))$

Newforms

Label	Dim	Coefficient field	Trace form	CM
64.6.a.a	1	$\mathbb{Q}$	$q - 20q^{3} + 74q^{5} - 24q^{7} + O(q^{8})$
64.6.a.b	1	$\mathbb{Q}$	$q - 12q^{3} - 54q^{5} + 88q^{7} + O(q^{8})$
64.6.a.c	1	$\mathbb{Q}$	$q - 8q^{3} - 14q^{5} + 208q^{7} + O(q^{8})$
64.6.a.d	1	$\mathbb{Q}$	$q + 82q^{5} + O(q^{8})$	$\mathbb{Q}(\sqrt{-4})$
64.6.a.e	1	$\mathbb{Q}$	$q + 8q^{3} - 14q^{5} - 208q^{7} + O(q^{8})$
64.6.a.f	1	$\mathbb{Q}$	$q + 12q^{3} - 54q^{5} - 88q^{7} + O(q^{8})$
64.6.a.g	1	$\mathbb{Q}$	$q + 20q^{3} + 74q^{5} + 24q^{7} + O(q^{8})$
64.6.a.h	2	$\mathbb{Q}(\sqrt{3})$	$2q - 92q^{5} + O(q^{8})$

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 $T$
64.6.a.a	$x - 54$
64.6.a.b	$x + 66$
64.6.a.c	$x + 22$
64.6.a.d	$x - 82$
64.6.a.e	$x + 6$
64.6.a.f	$x + 42$
64.6.a.g	$x - 94$
64.6.a.h	$x^{2} + 92x + 1348$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(64))$	9
Old forms	25
Cusp forms $S_{6}(\Gamma_0(64))$	34
Eisenstein series $E_{6}(\Gamma_0(64))$	12
Modular forms $M_{6}(\Gamma_0(64))$	46

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(64)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 5}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(16))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.25 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, 8, 16, 32, 64]: [1, 2, 4, 7, 12, 22, 42] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 9 = 1 x 9 (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(64, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 64:6 1000

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