Atlas › Modular forms › Weight 2 › Level 708 › 708.2.a

Newspace 708.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(708))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
708.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 2q^{5} + O(q^{8})$	+ +	708a1 rank 0
708.2.a.b	2	$\mathbb{Q}(\sqrt{13})$	$2q - 2q^{3} - 2q^{5} - q^{7} + O(q^{8})$	+ −
708.2.a.c	2	$\mathbb{Q}(\sqrt{61})$	$2q - 2q^{3} + 6q^{5} - q^{7} + O(q^{8})$	+ +
708.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} - 4q^{5} - q^{7} + O(q^{8})$	− +
708.2.a.e	3	degree 3	$3q + 3q^{3} + 2q^{5} - q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + T_{7}$ 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 $T$
708.2.a.a	$x + 2$
708.2.a.b	$x^{2} + 3x - 1$
708.2.a.c	$x^{2} - 5x - 9$
708.2.a.d	$x^{2} + 5x + 5$
708.2.a.e	$x^{3} - x^{2} - 25x + 50$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(708))$	10
Old forms	105
Cusp forms $S_{2}(\Gamma_0(708))$	115
Eisenstein series $E_{2}(\Gamma_0(708))$	11
Modular forms $M_{2}(\Gamma_0(708))$	126

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(708)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(59))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(118))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(177))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(236))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(354))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.268 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, 59, 118, 177, 236, 354, 708]: [0, 1, 1, 2, 3, 5, 6, 17, 22, 33, 64, 126] (certified at ell = 2147482481)
T = [(5, 1), (7, 1)]; h of degree 10 = 1 x 10 (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(708, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 708:2 1000

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