Atlas › Modular forms › Weight 8 › Level 50 › 50.8.a

Newspace 50.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(50))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
50.8.a.a	1	$\mathbb{Q}$	$q - 8q^{2} - 43q^{3} + 64q^{4} + 344q^{6} - 974q^{7} + O(q^{8})$	+
50.8.a.b	1	$\mathbb{Q}$	$q - 8q^{2} - 28q^{3} + 64q^{4} + 224q^{6} - 104q^{7} + O(q^{8})$	+
50.8.a.c	1	$\mathbb{Q}$	$q - 8q^{2} + 57q^{3} + 64q^{4} - 456q^{6} - 1174q^{7} + O(q^{8})$	+
50.8.a.d	1	$\mathbb{Q}$	$q - 8q^{2} + 87q^{3} + 64q^{4} - 696q^{6} + 1366q^{7} + O(q^{8})$	+
50.8.a.e	1	$\mathbb{Q}$	$q + 8q^{2} - 87q^{3} + 64q^{4} - 696q^{6} - 1366q^{7} + O(q^{8})$	−
50.8.a.f	1	$\mathbb{Q}$	$q + 8q^{2} - 57q^{3} + 64q^{4} - 456q^{6} + 1174q^{7} + O(q^{8})$	−
50.8.a.g	1	$\mathbb{Q}$	$q + 8q^{2} - 12q^{3} + 64q^{4} - 96q^{6} - 1016q^{7} + O(q^{8})$	−
50.8.a.h	1	$\mathbb{Q}$	$q + 8q^{2} + 43q^{3} + 64q^{4} + 344q^{6} + 974q^{7} + O(q^{8})$	−
50.8.a.i	2	$\mathbb{Q}(\sqrt{31})$	$2q - 16q^{2} - 56q^{3} + 128q^{4} + 448q^{6} - 408q^{7} + O(q^{8})$	+
50.8.a.j	2	$\mathbb{Q}(\sqrt{31})$	$2q + 16q^{2} + 56q^{3} + 128q^{4} + 448q^{6} + 408q^{7} + O(q^{8})$	−

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 $T$
50.8.a.a	$x + 1017$
50.8.a.b	$x + 132$
50.8.a.c	$x + 1117$
50.8.a.d	$x - 1453$
50.8.a.e	$x + 1453$
50.8.a.f	$x - 1117$
50.8.a.g	$x + 1028$
50.8.a.h	$x - 1017$
50.8.a.i	$x^{2} + 464x - 57776$
50.8.a.j	$x^{2} - 464x - 57776$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(50))$	12
Old forms	35
Cusp forms $S_{8}(\Gamma_0(50))$	47
Eisenstein series $E_{8}(\Gamma_0(50))$	12
Modular forms $M_{8}(\Gamma_0(50))$	59

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(50)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 3}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(25))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.315 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, 3, 5, 13, 19, 55] (certified at ell = 2147482921)
T = [(3, 1), (7, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 5 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, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 50:8 1000

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