Atlas › Modular forms › Weight 2 › Level 552 › 552.2.a

Newspace 552.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(552))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
552.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 2q^{5} + 2q^{7} + O(q^{8})$	+ +	552a1 rank 1
552.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 2q^{7} + O(q^{8})$	+ −	552b1 rank 0
552.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} - 4q^{7} + O(q^{8})$	+ −	552d1 rank 1
552.2.a.d	1	$\mathbb{Q}$	$q - q^{3} + 4q^{5} + 2q^{7} + O(q^{8})$	+ −	552c1 rank 0
552.2.a.e	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} - 4q^{7} + O(q^{8})$	− +	552e1 rank 1
552.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} + 2q^{5} + 4q^{7} + O(q^{8})$	− −
552.2.a.g	3	degree 3	$3q + 3q^{3} + 2q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17} + 11T_{19}$ 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 $T$
552.2.a.a	$x + 56$
552.2.a.b	$x - 146$
552.2.a.c	$x + 62$
552.2.a.d	$x + 78$
552.2.a.e	$x + 90$
552.2.a.f	$x^{2} - 10x - 6820$
552.2.a.g	$x^{3} - 170x^{2} + 5740x - 472$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(552))$	10
Old forms	79
Cusp forms $S_{2}(\Gamma_0(552))$	89
Eisenstein series $E_{2}(\Gamma_0(552))$	15
Modular forms $M_{2}(\Gamma_0(552))$	104

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(552)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 8}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(46))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(69))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(92))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(138))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(184))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(276))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.458 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, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552]: [0, 1, 1, 2, 3, 3, 5, 3, 8, 8, 10, 15, 28, 28, 54, 104] (certified at ell = 2147483647)
T = [(5, 1), (7, 3), (11, 5), (13, 7), (17, 9), (19, 11)]; h of degree 10 = 1 x 10 (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(552, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 552:2 1000

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