Atlas › Modular forms › Weight 2 › Level 323 › 323.2.a

Newspace 323.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(323))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
323.2.a.a	1	$\mathbb{Q}$	$q + 3q^{3} - 2q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	+ −	$+1$	323a1 rank 0
323.2.a.b	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} + q^{3} + 5q^{4} + 4q^{5} + 8q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$
323.2.a.c	4	degree 4	$4q - q^{3} + 4q^{4} - 7q^{5} - 10q^{6} - 11q^{7} + O(q^{8})$	− −	$-1$
323.2.a.d	5	degree 5	$5q - 3q^{2} - 3q^{3} + 3q^{4} - 3q^{5} + 2q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
323.2.a.e	6	degree 6	$6q + 2q^{2} - 3q^{3} + 10q^{4} - q^{5} - 10q^{6} + 5q^{7} + O(q^{8})$	+ −	$+1$
323.2.a.f	7	degree 7	$7q + q^{2} + 3q^{3} + 7q^{4} + 7q^{5} - 2q^{6} + q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
323.2.a.a	$x - 3$
323.2.a.b	$x^{2} - 17$
323.2.a.c	$x^{4} + x^{3} - 4x^{2} + 1$
323.2.a.d	$x^{5} + 6x^{4} + x^{3} - 22x^{2} - 15x + 1$
323.2.a.e	$x^{6} + x^{5} - 11x^{4} - 7x^{3} + 31x^{2} + 14x - 17$
323.2.a.f	$x^{7} - 4x^{6} - 14x^{5} + 62x^{4} - 8x^{3} - 155x^{2} + 151x - 29$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(323))$	25
Old forms	4
Cusp forms $S_{2}(\Gamma_0(323))$	29
Eisenstein series $E_{2}(\Gamma_0(323))$	3
Modular forms $M_{2}(\Gamma_0(323))$	32

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(323)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.182 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, 17, 19, 323]: [0, 2, 2, 32] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 25 = 1 x 25 (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(323, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 323:2 1000

← 322.2.a · 324.2.a →