Atlas › Modular forms › Weight 2 › Level 654 › 654.2.a

Newspace 654.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(654))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
654.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+ − −	$-1$	654a1 rank 1
654.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$	654b1 rank 1
654.2.a.c	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	+ − +	$+1$
654.2.a.d	3	degree 3	$3q + 3q^{2} - 3q^{3} + 3q^{4} + q^{5} - 3q^{6} + 5q^{7} + O(q^{8})$	− + +	$+1$
654.2.a.e	4	degree 4	$4q + 4q^{2} + 4q^{3} + 4q^{4} + 2q^{5} + 4q^{6} + 3q^{7} + O(q^{8})$	− − −	$+1$
654.2.a.f	5	degree 5	$5q - 5q^{2} - 5q^{3} + 5q^{4} - 2q^{5} + 5q^{6} + 3q^{7} + O(q^{8})$	+ + −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ 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 $T$
654.2.a.a	$x + 119$
654.2.a.b	$x + 63$
654.2.a.c	$x^{3} - 231x^{2} + 2016x + 1220991$
654.2.a.d	$x^{3} - 177x^{2} + 9120x - 113039$
654.2.a.e	$x^{4} - 48x^{3} - 6223x^{2} + 347755x - 4260189$
654.2.a.f	$x^{5} - 18x^{4} - 1901x^{3} + 59137x^{2} - 578467x + 1839856$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(654))$	17
Old forms	90
Cusp forms $S_{2}(\Gamma_0(654))$	107
Eisenstein series $E_{2}(\Gamma_0(654))$	7
Modular forms $M_{2}(\Gamma_0(654))$	114

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(654)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(109))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(218))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(327))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.463 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, 6, 109, 218, 327, 654]: [0, 1, 1, 3, 9, 29, 38, 114] (certified at ell = 2147482801)
T = [(5, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 5 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(654, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 654:2 1000

← 653.2.a · 655.2.a →