Atlas › Modular forms › Weight 2 › Level 732 › 732.2.a

Newspace 732.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(732))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
732.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + q^{5} - 5q^{7} + O(q^{8})$	+ −	732b1 rank 1
732.2.a.b	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} + 2q^{7} + O(q^{8})$	+ +	732a1 rank 0
732.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} - 2q^{7} + O(q^{8})$	− +	732c1 rank 1
732.2.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + 4q^{7} + O(q^{8})$	− −
732.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + 2q^{5} + 2q^{7} + O(q^{8})$	− −
732.2.a.f	3	degree 3	$3q - 3q^{3} - 3q^{5} + 3q^{7} + O(q^{8})$	+ −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{17} + 7T_{19} + 9T_{23}$ 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 $T$
732.2.a.a	$x + 121$
732.2.a.b	$x - 80$
732.2.a.c	$x + 44$
732.2.a.d	$x^{2} + 128x + 2068$
732.2.a.e	$x^{2} - 166x + 6457$
732.2.a.f	$x^{3} + 125x^{2} + 5048x + 64972$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(732))$	10
Old forms	109
Cusp forms $S_{2}(\Gamma_0(732))$	119
Eisenstein series $E_{2}(\Gamma_0(732))$	11
Modular forms $M_{2}(\Gamma_0(732))$	130

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(732)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(61))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(122))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(183))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(244))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(366))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.405 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, 61, 122, 183, 244, 366, 732]: [0, 1, 1, 2, 3, 5, 5, 17, 22, 34, 66, 130] (certified at ell = 2147482921)
T = [(5, 1), (7, 3), (17, 5), (19, 7), (23, 9)]; 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(732, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 732:2 1000

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