Atlas › Modular forms › Weight 2 › Level 312 › 312.2.a

Newspace 312.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(312))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
312.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 2q^{5} + 4q^{7} + O(q^{8})$	+ −	312d1 rank 0
312.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 4q^{7} + O(q^{8})$	+ +	312b1 rank 1
312.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + 4q^{5} + O(q^{8})$	+ +	312e1 rank 0
312.2.a.d	1	$\mathbb{Q}$	$q + q^{3} - 4q^{5} - 4q^{7} + O(q^{8})$	− +	312f1 rank 1
312.2.a.e	1	$\mathbb{Q}$	$q + q^{3} + O(q^{8})$	− +	312a1 rank 0
312.2.a.f	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} + O(q^{8})$	− −	312c1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{17} + 9T_{19} + 11T_{23}$ 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 $T$
312.2.a.a	$x - 184$
312.2.a.b	$x + 56$
312.2.a.c	$x - 124$
312.2.a.d	$x - 12$
312.2.a.e	$x - 88$
312.2.a.f	$x + 20$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(312))$	6
Old forms	43
Cusp forms $S_{2}(\Gamma_0(312))$	49
Eisenstein series $E_{2}(\Gamma_0(312))$	15
Modular forms $M_{2}(\Gamma_0(312))$	64

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(312)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(52))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(78))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(104))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(156))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.286 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, 13, 24, 26, 39, 52, 78, 104, 156, 312]: [0, 1, 1, 2, 3, 3, 5, 1, 8, 5, 6, 10, 18, 18, 34, 64] (certified at ell = 2147483629)
T = [(5, 1), (7, 3), (11, 5), (17, 7), (19, 9), (23, 11)]; h of degree 6 = 1 x 6 (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(312, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 312:2 1000

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