Atlas › Modular forms › Weight 2 › Level 345 › 345.2.a

Newspace 345.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(345))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
345.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} + q^{5} - 2q^{6} - 5q^{7} + O(q^{8})$	− − +	$-1$	345f1 rank 1
345.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} + 4q^{7} + O(q^{8})$	− + +	$+1$	345d1 rank 0
345.2.a.c	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} - q^{5} + q^{7} + O(q^{8})$	+ + −	$+1$	345a1 rank 0
345.2.a.d	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - q^{5} - 3q^{7} + O(q^{8})$	− + −	$-1$	345b1 rank 1
345.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} + 4q^{7} + O(q^{8})$	− + +	$+1$	345c1 rank 0
345.2.a.f	1	$\mathbb{Q}$	$q + 2q^{2} - q^{3} + 2q^{4} + q^{5} - 2q^{6} + 3q^{7} + O(q^{8})$	+ − +	$+1$	345e1 rank 0
345.2.a.g	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} - 2q^{3} + 4q^{4} + 2q^{5} + 2q^{6} - 6q^{7} + O(q^{8})$	+ − −	$-1$
345.2.a.h	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} - 2q^{5} - 2q^{7} + O(q^{8})$	+ + +	$-1$
345.2.a.i	2	$\mathbb{Q}(\sqrt{6})$	$2q + 2q^{3} + 8q^{4} - 2q^{5} - 2q^{7} + O(q^{8})$	− + +	$+1$
345.2.a.j	3	degree 3	$3q - q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - q^{6} + 6q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{19}$ 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 $T$
345.2.a.a	$x + 22$
345.2.a.b	$x - 115$
345.2.a.c	$x - 38$
345.2.a.d	$x + 102$
345.2.a.e	$x + 7$
345.2.a.f	$x - 18$
345.2.a.g	$x^{2} + 156x + 3384$
345.2.a.h	$x^{2} + 8x - 5392$
345.2.a.i	$x^{2} - 8x - 200$
345.2.a.j	$x^{3} - 41x^{2} - 216x + 2672$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(345))$	15
Old forms	30
Cusp forms $S_{2}(\Gamma_0(345))$	45
Eisenstein series $E_{2}(\Gamma_0(345))$	7
Modular forms $M_{2}(\Gamma_0(345))$	52

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(345)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(69))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(115))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.238 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, 3, 5, 15, 23, 69, 115, 345]: [0, 1, 1, 4, 3, 10, 14, 52] (certified at ell = 2147483053)
T = [(2, 1), (7, 3), (13, 5), (17, 7), (19, 9)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 4 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(345, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 345:2 1000

← 344.2.a · 346.2.a →