Atlas › Modular forms › Weight 2 › Level 200 › 200.2.a

Newspace 200.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(200))$

Newforms

Label	Dim	Coefficient field	Trace form	Elliptic curve
200.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} + 2q^{7} + O(q^{8})$	200a1 rank 0
200.2.a.b	1	$\mathbb{Q}$	$q - 2q^{3} - 2q^{7} + O(q^{8})$	200b1 rank 1
200.2.a.c	1	$\mathbb{Q}$	$q + 4q^{7} + O(q^{8})$	200c1 rank 0
200.2.a.d	1	$\mathbb{Q}$	$q + 2q^{3} + 2q^{7} + O(q^{8})$	200d1 rank 0
200.2.a.e	1	$\mathbb{Q}$	$q + 3q^{3} - 2q^{7} + O(q^{8})$	200e1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7}$ 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 $T$
200.2.a.a	$x - 3$
200.2.a.b	$x + 8$
200.2.a.c	$x - 12$
200.2.a.d	$x - 8$
200.2.a.e	$x + 3$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(200))$	5
Old forms	14
Cusp forms $S_{2}(\Gamma_0(200))$	19
Eisenstein series $E_{2}(\Gamma_0(200))$	23
Modular forms $M_{2}(\Gamma_0(200))$	42

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.208 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, 20, 25, 40, 50, 100, 200]: [0, 1, 2, 1, 3, 3, 6, 3, 10, 9, 18, 34] (certified at ell = 2147482921)
T = [(3, 1), (7, 3)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 1 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(200, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 200:2 1000

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