Atlas › Modular forms › Weight 2 › Level 400 › 400.2.a

Newspace 400.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(400))$

Newforms

Label	Dim	Coefficient field	Trace form	Elliptic curve
400.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} + 2q^{7} + O(q^{8})$	400h1 rank 1
400.2.a.b	1	$\mathbb{Q}$	$q - 2q^{3} - 2q^{7} + O(q^{8})$	400f1 rank 0
400.2.a.c	1	$\mathbb{Q}$	$q - 2q^{3} + 2q^{7} + O(q^{8})$	400e1 rank 0
400.2.a.d	1	$\mathbb{Q}$	$q - q^{3} - 2q^{7} + O(q^{8})$	400c1 rank 1
400.2.a.e	1	$\mathbb{Q}$	$q - 4q^{7} + O(q^{8})$	400a1 rank 1
400.2.a.f	1	$\mathbb{Q}$	$q + q^{3} + 2q^{7} + O(q^{8})$	400b1 rank 0
400.2.a.g	1	$\mathbb{Q}$	$q + 2q^{3} + 2q^{7} + O(q^{8})$	400d1 rank 0
400.2.a.h	1	$\mathbb{Q}$	$q + 3q^{3} - 2q^{7} + O(q^{8})$	400g1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43} + 13T_{47} + 14T_{53} + 15T_{59} + 16T_{61}$ 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 $T$
400.2.a.a	$x + 134$
400.2.a.b	$x - 252$
400.2.a.c	$x + 42$
400.2.a.d	$x + 122$
400.2.a.e	$x + 188$
400.2.a.f	$x - 42$
400.2.a.g	$x + 64$
400.2.a.h	$x + 178$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(400))$	8
Old forms	35
Cusp forms $S_{2}(\Gamma_0(400))$	43
Eisenstein series $E_{2}(\Gamma_0(400))$	35
Modular forms $M_{2}(\Gamma_0(400))$	78

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(400)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(40))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(50))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(80))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(100))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(200))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.763 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, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400]: [0, 1, 2, 1, 3, 3, 4, 6, 3, 10, 9, 16, 18, 34, 64] (certified at ell = 2147482921)
T = [(3, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12), (47, 13), (53, 14), (59, 15), (61, 16)]; h of degree 8 = 1 x 8 (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(400, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 400:2 1000

← 399.2.a · 401.2.a →