Atlas › Modular forms › Weight 2 › Level 632 › 632.2.a

Newspace 632.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(632))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
632.2.a.a	1	$\mathbb{Q}$	$q + q^{3} - q^{5} - 5q^{7} + O(q^{8})$	−	632a1 rank 1
632.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} - q^{5} - 4q^{7} + O(q^{8})$	+
632.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} + 4q^{5} - 4q^{7} + O(q^{8})$	+
632.2.a.d	3	degree 3	$3q - q^{3} - 6q^{5} + q^{7} + O(q^{8})$	−
632.2.a.e	6	degree 6	$6q + 2q^{3} + 9q^{5} + 6q^{7} + O(q^{8})$	+
632.2.a.f	6	degree 6	$6q + 4q^{3} - 5q^{5} + 6q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
632.2.a.a	$x + 2$
632.2.a.b	$x^{2} + 5x + 5$
632.2.a.c	$x^{2} - 10x - 20$
632.2.a.d	$x^{3} + 19x^{2} + 93x + 26$
632.2.a.e	$x^{6} - 29x^{5} + 211x^{4} + 902x^{3} - 15260x^{2} + 41232x + 1696$
632.2.a.f	$x^{6} + 11x^{5} - 161x^{4} - 1384x^{3} + 10502x^{2} + 44420x - 267224$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(632))$	20
Old forms	57
Cusp forms $S_{2}(\Gamma_0(632))$	77
Eisenstein series $E_{2}(\Gamma_0(632))$	7
Modular forms $M_{2}(\Gamma_0(632))$	84

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(632)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(79))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(158))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(316))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.333 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, 79, 158, 316, 632]: [0, 1, 2, 3, 7, 22, 43, 84] (certified at ell = 2147483053)
T = [(3, 1), (5, 3)]; h of degree 20 = 1 x 20 (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(632, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 632:2 1000

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