Atlas › Modular forms › Weight 6 › Level 50 › 50.6.a

Newspace 50.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(50))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
50.6.a.a	1	$\mathbb{Q}$	$q - 4q^{2} - 11q^{3} + 16q^{4} + 44q^{6} - 142q^{7} + O(q^{8})$	+
50.6.a.b	1	$\mathbb{Q}$	$q - 4q^{2} - 6q^{3} + 16q^{4} + 24q^{6} + 118q^{7} + O(q^{8})$	+
50.6.a.c	1	$\mathbb{Q}$	$q - 4q^{2} + 14q^{3} + 16q^{4} - 56q^{6} + 158q^{7} + O(q^{8})$	+
50.6.a.d	1	$\mathbb{Q}$	$q + 4q^{2} - 24q^{3} + 16q^{4} - 96q^{6} + 172q^{7} + O(q^{8})$	−
50.6.a.e	1	$\mathbb{Q}$	$q + 4q^{2} - 14q^{3} + 16q^{4} - 56q^{6} - 158q^{7} + O(q^{8})$	−
50.6.a.f	1	$\mathbb{Q}$	$q + 4q^{2} + 11q^{3} + 16q^{4} + 44q^{6} + 142q^{7} + O(q^{8})$	−
50.6.a.g	1	$\mathbb{Q}$	$q + 4q^{2} + 26q^{3} + 16q^{4} + 104q^{6} + 22q^{7} + O(q^{8})$	−

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 $T$
50.6.a.a	$x + 153$
50.6.a.b	$x - 112$
50.6.a.c	$x - 172$
50.6.a.d	$x - 148$
50.6.a.e	$x + 172$
50.6.a.f	$x - 153$
50.6.a.g	$x - 48$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(50))$	7
Old forms	24
Cusp forms $S_{6}(\Gamma_0(50))$	31
Eisenstein series $E_{6}(\Gamma_0(50))$	12
Modular forms $M_{6}(\Gamma_0(50))$	43

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.128 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, 5, 10, 25, 50]: [1, 2, 3, 9, 13, 39] (certified at ell = 2147482921)
T = [(3, 1), (7, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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(50, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 50:6 1000

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