Atlas › Modular forms › Weight 2 › Level 684 › 684.2.a

Newspace 684.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(684))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
684.2.a.a	1	$\mathbb{Q}$	$q - 2q^{5} + O(q^{8})$	+	684b1 rank 1
684.2.a.b	1	$\mathbb{Q}$	$q + q^{5} - 3q^{7} + O(q^{8})$	+	684a1 rank 1
684.2.a.c	1	$\mathbb{Q}$	$q + 3q^{5} + q^{7} + O(q^{8})$	−	684c1 rank 0
684.2.a.d	2	$\mathbb{Q}(\sqrt{33})$	$2q - 3q^{5} + q^{7} + O(q^{8})$	−
684.2.a.e	2	$\mathbb{Q}(\sqrt{7})$	$2q + 6q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{13} + 4T_{17} + 5T_{23} + 6T_{11} + 7T_{29} + 8T_{31}$ 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 $T$
684.2.a.a	$x + 134$
684.2.a.b	$x + 29$
684.2.a.c	$x - 23$
684.2.a.d	$x^{2} - 97x - 3662$
684.2.a.e	$x^{2} - 184x + 8457$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(684))$	7
Old forms	102
Cusp forms $S_{2}(\Gamma_0(684))$	109
Eisenstein series $E_{2}(\Gamma_0(684))$	23
Modular forms $M_{2}(\Gamma_0(684))$	132

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.507 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, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, 342, 684]: [0, 1, 1, 2, 3, 2, 5, 5, 2, 9, 7, 8, 13, 24, 22, 46, 64, 126] (certified at ell = 2147483647)
T = [(5, 1), (7, 2), (13, 3), (17, 4), (23, 5), (11, 6), (29, 7), (31, 8)]; h of degree 7 = 1 x 7 (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(684, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 684:2 1000

← 683.2.a · 685.2.a →