Atlas › Modular forms › Weight 6 › Level 42 › 42.6.a

Newspace 42.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(42))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
42.6.a.a	1	$\mathbb{Q}$	$q - 4q^{2} - 9q^{3} + 16q^{4} - 54q^{5} + 36q^{6} + 49q^{7} + O(q^{8})$	+ + −	$+1$
42.6.a.b	1	$\mathbb{Q}$	$q - 4q^{2} - 9q^{3} + 16q^{4} + 44q^{5} + 36q^{6} - 49q^{7} + O(q^{8})$	+ + +	$-1$
42.6.a.c	1	$\mathbb{Q}$	$q - 4q^{2} + 9q^{3} + 16q^{4} - 72q^{5} - 36q^{6} + 49q^{7} + O(q^{8})$	+ − −	$-1$
42.6.a.d	1	$\mathbb{Q}$	$q - 4q^{2} + 9q^{3} + 16q^{4} + 26q^{5} - 36q^{6} - 49q^{7} + O(q^{8})$	+ − +	$+1$
42.6.a.e	1	$\mathbb{Q}$	$q + 4q^{2} - 9q^{3} + 16q^{4} + 76q^{5} - 36q^{6} - 49q^{7} + O(q^{8})$	− + +	$+1$
42.6.a.f	1	$\mathbb{Q}$	$q + 4q^{2} + 9q^{3} + 16q^{4} + 24q^{5} + 36q^{6} + 49q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + T_{11}$ 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 $T$
42.6.a.a	$x - 162$
42.6.a.b	$x + 426$
42.6.a.c	$x + 486$
42.6.a.d	$x - 690$
42.6.a.e	$x - 726$
42.6.a.f	$x - 90$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(42))$	6
Old forms	30
Cusp forms $S_{6}(\Gamma_0(42))$	36
Eisenstein series $E_{6}(\Gamma_0(42))$	8
Modular forms $M_{6}(\Gamma_0(42))$	44

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.131 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, 2, 3, 7, 5, 12, 16, 44] (certified at ell = 2147483647)
T = [(5, 1), (11, 1)]; h of degree 6 = 1 x 6 (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(42, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 42:6 1000

← 41.6.a · 43.6.a →