Atlas › Modular forms › Weight 2 › Level 462 › 462.2.a

Newspace 462.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(462))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
462.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} + q^{7} + O(q^{8})$	+ + − −	$-1$	462c1 rank 1
462.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} + O(q^{8})$	+ + + +	$-1$	462a1 rank 1
462.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} - q^{7} + O(q^{8})$	+ + + −	$+1$	462b1 rank 0
462.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} + O(q^{8})$	+ − + +	$+1$	462d1 rank 0
462.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 4q^{5} - q^{6} + q^{7} + O(q^{8})$	− + − +	$-1$	462e1 rank 1
462.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + O(q^{8})$	− − − +	$+1$	462g1 rank 0
462.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} - q^{7} + O(q^{8})$	− − + −	$+1$	462f1 rank 0
462.2.a.h	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} + 2q^{7} + O(q^{8})$	+ − − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{13} + 3T_{17} + 4T_{19} + 5T_{23} + 6T_{29} + 7T_{31} + 8T_{37}$ 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 $T$
462.2.a.a	$x + 84$
462.2.a.b	$x + 78$
462.2.a.c	$x - 128$
462.2.a.d	$x - 130$
462.2.a.e	$x + 118$
462.2.a.f	$x - 6$
462.2.a.g	$x + 8$
462.2.a.h	$x^{2} - 12x - 552$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(462))$	9
Old forms	80
Cusp forms $S_{2}(\Gamma_0(462))$	89
Eisenstein series $E_{2}(\Gamma_0(462))$	15
Modular forms $M_{2}(\Gamma_0(462))$	104

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.608 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, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462]: [0, 1, 1, 3, 1, 2, 4, 4, 5, 6, 12, 16, 10, 28, 36, 104] (certified at ell = 2147482951)
T = [(5, 1), (13, 2), (17, 3), (19, 4), (23, 5), (29, 6), (31, 7), (37, 8)]; h of degree 9 = 1 x 9 (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(462, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 462:2 1000

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