Atlas › Modular forms › Weight 6 › Level 63 › 63.6.a

Newspace 63.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(63))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
63.6.a.a	1	$\mathbb{Q}$	$q - 10q^{2} + 68q^{4} + 106q^{5} - 49q^{7} + O(q^{8})$	+
63.6.a.b	1	$\mathbb{Q}$	$q - 5q^{2} - 7q^{4} - 94q^{5} - 49q^{7} + O(q^{8})$	+
63.6.a.c	1	$\mathbb{Q}$	$q - q^{2} - 31q^{4} + 34q^{5} - 49q^{7} + O(q^{8})$	+
63.6.a.d	1	$\mathbb{Q}$	$q + 6q^{2} + 4q^{4} - 78q^{5} + 49q^{7} + O(q^{8})$	−
63.6.a.e	1	$\mathbb{Q}$	$q + 10q^{2} + 68q^{4} + 56q^{5} - 49q^{7} + O(q^{8})$	+
63.6.a.f	2	$\mathbb{Q}(\sqrt{57})$	$2q - 9q^{2} + 5q^{4} + 18q^{5} + 98q^{7} + O(q^{8})$	−
63.6.a.g	2	$\mathbb{Q}(\sqrt{7})$	$2q - 8q^{4} - 98q^{7} + O(q^{8})$	+
63.6.a.h	4	degree 4	$4q + 94q^{4} + 196q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
63.6.a.a	$x - 96$
63.6.a.b	$x + 99$
63.6.a.c	$x - 33$
63.6.a.d	$x + 72$
63.6.a.e	$x - 66$
63.6.a.f	$x^{2} - 9x - 1704$
63.6.a.g	$x^{2} - 1008$
63.6.a.h	$x^{4} - 8427x^{2} + 1741824$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(63))$	13
Old forms	23
Cusp forms $S_{6}(\Gamma_0(63))$	36
Eisenstein series $E_{6}(\Gamma_0(63))$	8
Modular forms $M_{6}(\Gamma_0(63))$	44

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(63)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.252 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, 3, 7, 9, 21, 63]: [1, 3, 5, 6, 16, 42] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 13 = 1 x 13 (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(63, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 63:6 1000

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