Atlas › Modular forms › Weight 2 › Level 700 › 700.2.a

Newspace 700.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(700))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
700.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - q^{7} + O(q^{8})$	+	700g1 rank 1
700.2.a.b	1	$\mathbb{Q}$	$q - 3q^{3} + q^{7} + O(q^{8})$	−	700d1 rank 1
700.2.a.c	1	$\mathbb{Q}$	$q - 2q^{3} + q^{7} + O(q^{8})$	−	700i1 rank 0
700.2.a.d	1	$\mathbb{Q}$	$q - q^{3} - q^{7} + O(q^{8})$	+	700a1 rank 0
700.2.a.e	1	$\mathbb{Q}$	$q - q^{7} + O(q^{8})$	+	700f1 rank 1
700.2.a.f	1	$\mathbb{Q}$	$q - q^{7} + O(q^{8})$	+	700e1 rank 1
700.2.a.g	1	$\mathbb{Q}$	$q + q^{7} + O(q^{8})$	−	700c1 rank 1
700.2.a.h	1	$\mathbb{Q}$	$q + q^{7} + O(q^{8})$	−	700h1 rank 0
700.2.a.i	1	$\mathbb{Q}$	$q + 2q^{3} - q^{7} + O(q^{8})$	+	700b1 rank 0
700.2.a.j	1	$\mathbb{Q}$	$q + 3q^{3} + q^{7} + O(q^{8})$	−	700j1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{11} + 3T_{19} + 4T_{23} + 5T_{29} + 6T_{13} + 7T_{17} + 8T_{31}$ 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 $T$
700.2.a.a	$x + 21$
700.2.a.b	$x + 71$
700.2.a.c	$x - 83$
700.2.a.d	$x - 81$
700.2.a.e	$x + 31$
700.2.a.f	$x + 126$
700.2.a.g	$x + 23$
700.2.a.h	$x - 42$
700.2.a.i	$x - 159$
700.2.a.j	$x + 57$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(700))$	10
Old forms	93
Cusp forms $S_{2}(\Gamma_0(700))$	103
Eisenstein series $E_{2}(\Gamma_0(700))$	35
Modular forms $M_{2}(\Gamma_0(700))$	138

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.826 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, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350, 700]: [0, 1, 2, 1, 1, 3, 4, 6, 3, 7, 6, 9, 16, 18, 30, 22, 64, 126] (certified at ell = 2147482921)
T = [(3, 1), (11, 2), (19, 3), (23, 4), (29, 5), (13, 6), (17, 7), (31, 8)]; h of degree 10 = 1 x 10 (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(700, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 700:2 1000

← 699.2.a · 701.2.a →