Atlas › Modular forms › Weight 2 › Level 310 › 310.2.a

Newspace 310.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(310))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
310.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	− + −	$-1$	310b1 rank 1
310.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} - q^{5} + 2q^{6} + O(q^{8})$	− + +	$+1$	310a1 rank 0
310.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} + O(q^{8})$	+ + +	$-1$
310.2.a.d	2	$\mathbb{Q}(\sqrt{6})$	$2q - 2q^{2} + 2q^{4} + 2q^{5} - 4q^{7} + O(q^{8})$	+ − +	$+1$
310.2.a.e	3	degree 3	$3q + 3q^{2} + 2q^{3} + 3q^{4} + 3q^{5} + 2q^{6} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{7}$ 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 $T$
310.2.a.a	$x + 6$
310.2.a.b	$x - 2$
310.2.a.c	$x^{2} + 2x - 2$
310.2.a.d	$x^{2} + 4x - 2$
310.2.a.e	$x^{3} - 2x^{2} - 8x - 4$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(310))$	9
Old forms	36
Cusp forms $S_{2}(\Gamma_0(310))$	45
Eisenstein series $E_{2}(\Gamma_0(310))$	7
Modular forms $M_{2}(\Gamma_0(310))$	52

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(310)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(62))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(155))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.295 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, 5, 10, 31, 62, 155, 310]: [0, 1, 1, 3, 3, 10, 18, 52] (certified at ell = 2147482921)
T = [(3, 1), (7, 1)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 1 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(310, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 310:2 1000

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