Atlas › Modular forms › Weight 6 › Level 100 › 100.6.a

Newspace 100.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(100))$

Newforms

Label	Dim	Coefficient field	Trace form
100.6.a.a	1	$\mathbb{Q}$	$q - 22q^{3} - 218q^{7} + O(q^{8})$
100.6.a.b	1	$\mathbb{Q}$	$q + 12q^{3} + 88q^{7} + O(q^{8})$
100.6.a.c	2	$\mathbb{Q}(\sqrt{409})$	$2q - 20q^{3} + 40q^{7} + O(q^{8})$
100.6.a.d	2	$\mathbb{Q}(\sqrt{31})$	$2q + O(q^{8})$
100.6.a.e	2	$\mathbb{Q}(\sqrt{409})$	$2q + 20q^{3} - 40q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{7}$ 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 $T$
100.6.a.a	$x + 240$
100.6.a.b	$x - 100$
100.6.a.c	$x^{2} - 20x - 10125$
100.6.a.d	$x^{2} - 12400$
100.6.a.e	$x^{2} + 20x - 10125$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(100))$	8
Old forms	58
Cusp forms $S_{6}(\Gamma_0(100))$	66
Eisenstein series $E_{6}(\Gamma_0(100))$	18
Modular forms $M_{6}(\Gamma_0(100))$	84

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(100)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 6}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(25))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(50))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.29 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, 10, 20, 25, 50, 100]: [1, 2, 4, 3, 9, 18, 13, 39, 78] (certified at ell = 2147482921)
T = [(3, 1), (7, 1)]; 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(100, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 100:6 1000

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