Atlas › Modular forms › Weight 2 › Level 468 › 468.2.a

Newspace 468.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(468))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
468.2.a.a	1	$\mathbb{Q}$	$q - 4q^{5} + 4q^{7} + O(q^{8})$	+	468b1 rank 0
468.2.a.b	1	$\mathbb{Q}$	$q - 2q^{5} - 2q^{7} + O(q^{8})$	+	468c1 rank 1
468.2.a.c	1	$\mathbb{Q}$	$q + 2q^{7} + O(q^{8})$	−	468d1 rank 0
468.2.a.d	1	$\mathbb{Q}$	$q + 4q^{5} - 2q^{7} + O(q^{8})$	−	468e1 rank 0
468.2.a.e	1	$\mathbb{Q}$	$q + 4q^{5} + 4q^{7} + O(q^{8})$	+	468a1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{17} + 9T_{19}$ 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 $T$
468.2.a.a	$x - 28$
468.2.a.b	$x + 94$
468.2.a.c	$x - 66$
468.2.a.d	$x + 14$
468.2.a.e	$x + 4$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(468))$	5
Old forms	68
Cusp forms $S_{2}(\Gamma_0(468))$	73
Eisenstein series $E_{2}(\Gamma_0(468))$	23
Modular forms $M_{2}(\Gamma_0(468))$	96

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(468)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(36))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(52))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(78))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(117))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(156))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(234))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.321 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, 3, 4, 6, 9, 12, 13, 18, 26, 36, 39, 52, 78, 117, 156, 234, 468]: [0, 1, 1, 2, 3, 2, 5, 1, 5, 5, 9, 6, 10, 18, 16, 34, 46, 90] (certified at ell = 2147483629)
T = [(5, 1), (7, 3), (11, 5), (17, 7), (19, 9)]; h of degree 5 = 1 x 5 (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(468, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 468:2 1000

← 467.2.a · 469.2.a →