Atlas › Modular forms › Weight 4 › Level 63 › 63.4.a

Newspace 63.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(63))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
63.4.a.a	1	$\mathbb{Q}$	$q - 4q^{2} + 8q^{4} + 4q^{5} - 7q^{7} + O(q^{8})$	+
63.4.a.b	1	$\mathbb{Q}$	$q + q^{2} - 7q^{4} - 16q^{5} - 7q^{7} + O(q^{8})$	+
63.4.a.c	1	$\mathbb{Q}$	$q + 3q^{2} + q^{4} + 18q^{5} + 7q^{7} + O(q^{8})$	−
63.4.a.d	2	$\mathbb{Q}(\sqrt{19})$	$2q + 22q^{4} - 14q^{7} + O(q^{8})$	+
63.4.a.e	2	$\mathbb{Q}(\sqrt{57})$	$2q + 3q^{2} + 17q^{4} - 6q^{5} + 14q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
63.4.a.a	$x - 8$
63.4.a.b	$x + 47$
63.4.a.c	$x - 57$
63.4.a.d	$x^{2} - 931$
63.4.a.e	$x^{2} + 15x - 300$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(63))$	7
Old forms	13
Cusp forms $S_{4}(\Gamma_0(63))$	20
Eisenstein series $E_{4}(\Gamma_0(63))$	8
Modular forms $M_{4}(\Gamma_0(63))$	28

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.107 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, 2, 3, 4, 10, 26] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 7 = 1 x 7 (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(63, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 63:4 1000

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