Atlas › Modular forms › Weight 2 › Level 492 › 492.2.a

Newspace 492.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(492))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
492.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 2q^{7} + O(q^{8})$	+ −	492a1 rank 1
492.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} - 4q^{7} + O(q^{8})$	− +	492b1 rank 1
492.2.a.c	2	$\mathbb{Q}(\sqrt{6})$	$2q - 2q^{3} - 4q^{5} + 4q^{7} + O(q^{8})$	+ +
492.2.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + 2q^{5} + 2q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{19}$ 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 $T$
492.2.a.a	$x + 59$
492.2.a.b	$x + 83$
492.2.a.c	$x^{2} - 114x + 3243$
492.2.a.d	$x^{2} - 8x - 7187$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(492))$	6
Old forms	73
Cusp forms $S_{2}(\Gamma_0(492))$	79
Eisenstein series $E_{2}(\Gamma_0(492))$	11
Modular forms $M_{2}(\Gamma_0(492))$	90

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(492)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(82))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(123))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(164))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(246))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.301 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, 12, 41, 82, 123, 164, 246, 492]: [0, 1, 1, 2, 3, 5, 4, 12, 16, 24, 46, 90] (certified at ell = 2147482921)
T = [(5, 1), (7, 3), (13, 5), (17, 7), (19, 9)]; h of degree 6 = 1 x 6 (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(492, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 492:2 1000

← 491.2.a · 493.2.a →