Atlas › Modular forms › Weight 2 › Level 304 › 304.2.a

Newspace 304.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(304))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
304.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - q^{5} + 3q^{7} + O(q^{8})$	−	304f1 rank 1
304.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 3q^{7} + O(q^{8})$	+	304c1 rank 1
304.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + q^{7} + O(q^{8})$	+	304b1 rank 0
304.2.a.d	1	$\mathbb{Q}$	$q + q^{3} - 4q^{5} - 3q^{7} + O(q^{8})$	−	304a1 rank 1
304.2.a.e	1	$\mathbb{Q}$	$q + 2q^{3} - q^{5} + 3q^{7} + O(q^{8})$	−	304d1 rank 0
304.2.a.f	1	$\mathbb{Q}$	$q + 2q^{3} + 3q^{5} + q^{7} + O(q^{8})$	+	304e1 rank 0
304.2.a.g	3	degree 3	$3q - q^{3} + q^{5} - 4q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{23} + 8T_{29}$ 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 $T$
304.2.a.a	$x + 125$
304.2.a.b	$x + 60$
304.2.a.c	$x - 120$
304.2.a.d	$x + 44$
304.2.a.e	$x - 47$
304.2.a.f	$x - 9$
304.2.a.g	$x^{3} - 9x^{2} - 2050x + 41296$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(304))$	9
Old forms	26
Cusp forms $S_{2}(\Gamma_0(304))$	35
Eisenstein series $E_{2}(\Gamma_0(304))$	11
Modular forms $M_{2}(\Gamma_0(304))$	46

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(304)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(38))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(76))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(152))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.34 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, 4, 8, 16, 19, 38, 76, 152, 304]: [0, 1, 2, 3, 4, 2, 7, 13, 24, 44] (certified at ell = 2147483629)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (17, 6), (23, 7), (29, 8)]; h of degree 9 = 1 x 9 (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(304, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 304:2 1000

← 303.2.a · 305.2.a →