Atlas › Modular forms › Weight 10 › Level 30 › 30.10.a

Newspace 30.10.a: $S_{10}^{\mathrm{new}}(\Gamma_0(30))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
30.10.a.a	1	$\mathbb{Q}$	$q - 16q^{2} - 81q^{3} + 256q^{4} - 625q^{5} + 1296q^{6} + 6332q^{7} + O(q^{8})$	+ + +	$-1$
30.10.a.b	1	$\mathbb{Q}$	$q - 16q^{2} - 81q^{3} + 256q^{4} + 625q^{5} + 1296q^{6} - 7168q^{7} + O(q^{8})$	+ + −	$+1$
30.10.a.c	1	$\mathbb{Q}$	$q - 16q^{2} + 81q^{3} + 256q^{4} - 625q^{5} - 1296q^{6} + 7196q^{7} + O(q^{8})$	+ − +	$+1$
30.10.a.d	1	$\mathbb{Q}$	$q + 16q^{2} - 81q^{3} + 256q^{4} - 625q^{5} - 1296q^{6} + 3164q^{7} + O(q^{8})$	− + +	$+1$
30.10.a.e	1	$\mathbb{Q}$	$q + 16q^{2} - 81q^{3} + 256q^{4} + 625q^{5} - 1296q^{6} - 10336q^{7} + O(q^{8})$	− + −	$-1$
30.10.a.f	1	$\mathbb{Q}$	$q + 16q^{2} + 81q^{3} + 256q^{4} + 625q^{5} + 1296q^{6} + 2408q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + T_{11}$ 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 $T$
30.10.a.a	$x - 14084$
30.10.a.b	$x + 90916$
30.10.a.c	$x + 38404$
30.10.a.d	$x - 50084$
30.10.a.e	$x - 17084$
30.10.a.f	$x - 17300$

Dimensions

Newforms $S_{10}^{\mathrm{new}}(\Gamma_0(30))$	6
Old forms	44
Cusp forms $S_{10}(\Gamma_0(30))$	50
Eisenstein series $E_{10}(\Gamma_0(30))$	8
Modular forms $M_{10}(\Gamma_0(30))$	58

Decomposition of the old subspace

$S_{10}^{\mathrm{old}}(\Gamma_0(30)) =$ $S_{10}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 4}$ $\oplus$ $S_{10}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 4}$ $\oplus$ $S_{10}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{10}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 2}$ $\oplus$ $S_{10}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 2}$ $\oplus$ $S_{10}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.362 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, 5, 6, 10, 15, 30]: [1, 3, 4, 5, 11, 15, 20, 58] (certified at ell = 2147483629)
T = [(7, 1), (11, 1)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 4 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(30, 10)

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

cargo run --release -p sagebrush-web --example atlas -- 30:10 1000

← 29.10.a · 31.10.a →