Atlas › Modular forms › Weight 2 › Level 484 › 484.2.a

Newspace 484.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(484))$

Newforms

Label	Dim	Coefficient field	Trace form	CM	Elliptic curve
484.2.a.a	1	$\mathbb{Q}$	$q + q^{3} - 3q^{5} - 2q^{7} + O(q^{8})$		484a1 rank 1
484.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{3} - q^{5} - q^{7} + O(q^{8})$
484.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{3} - q^{5} + q^{7} + O(q^{8})$
484.2.a.d	2	$\mathbb{Q}(\sqrt{33})$	$2q + q^{3} + 3q^{5} + O(q^{8})$	$\mathbb{Q}(\sqrt{-11})$
484.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q + 4q^{3} + 6q^{5} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{13} + 9T_{17}$ 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 $T$
484.2.a.a	$x + 44$
484.2.a.b	$x^{2} - 101x + 1019$
484.2.a.c	$x^{2} + 113x + 2281$
484.2.a.d	$x^{2} - 10x - 8$
484.2.a.e	$x^{2} - 22x - 4211$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(484))$	9
Old forms	40
Cusp forms $S_{2}(\Gamma_0(484))$	49
Eisenstein series $E_{2}(\Gamma_0(484))$	35
Modular forms $M_{2}(\Gamma_0(484))$	84

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(484)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(121))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(242))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.249 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, 11, 22, 44, 121, 242, 484]: [0, 1, 2, 2, 5, 9, 12, 35, 69] (certified at ell = 2147482921)
T = [(3, 1), (5, 3), (7, 5), (13, 7), (17, 9)]; 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(484, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 484:2 1000

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