Atlas › Modular forms › Weight 8 › Level 42 › 42.8.a

Newspace 42.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(42))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
42.8.a.a	1	$\mathbb{Q}$	$q - 8q^{2} - 27q^{3} + 64q^{4} - 410q^{5} + 216q^{6} - 343q^{7} + O(q^{8})$	+ + +	$+1$
42.8.a.b	1	$\mathbb{Q}$	$q - 8q^{2} - 27q^{3} + 64q^{4} - 18q^{5} + 216q^{6} + 343q^{7} + O(q^{8})$	+ + −	$-1$
42.8.a.c	1	$\mathbb{Q}$	$q - 8q^{2} + 27q^{3} + 64q^{4} - 122q^{5} - 216q^{6} - 343q^{7} + O(q^{8})$	+ − +	$-1$
42.8.a.d	1	$\mathbb{Q}$	$q - 8q^{2} + 27q^{3} + 64q^{4} + 270q^{5} - 216q^{6} + 343q^{7} + O(q^{8})$	+ − −	$+1$
42.8.a.e	1	$\mathbb{Q}$	$q + 8q^{2} - 27q^{3} + 64q^{4} + 30q^{5} - 216q^{6} + 343q^{7} + O(q^{8})$	− + −	$+1$
42.8.a.f	1	$\mathbb{Q}$	$q + 8q^{2} + 27q^{3} + 64q^{4} + 470q^{5} + 216q^{6} - 343q^{7} + O(q^{8})$	− − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{11}$ 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 $T$
42.8.a.a	$x + 17054$
42.8.a.b	$x - 24498$
42.8.a.c	$x + 3158$
42.8.a.d	$x + 3942$
42.8.a.e	$x - 5394$
42.8.a.f	$x + 21334$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(42))$	6
Old forms	46
Cusp forms $S_{8}(\Gamma_0(42))$	52
Eisenstein series $E_{8}(\Gamma_0(42))$	8
Modular forms $M_{8}(\Gamma_0(42))$	60

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(42)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 4}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.326 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, 3, 6, 7, 14, 21, 42]: [1, 3, 3, 9, 5, 16, 20, 60] (certified at ell = 2147483647)
T = [(5, 1), (11, 3)]; h of degree 6 = 1 x 6 (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(42, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 42:8 1000

← 41.8.a · 43.8.a →