Atlas › Modular forms › Weight 4 › Level 34 › 34.4.a

Newspace 34.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(34))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
34.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 2q^{3} + 4q^{4} - 18q^{5} + 4q^{6} - 10q^{7} + O(q^{8})$	+ −	$-1$
34.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} - 2q^{3} + 4q^{4} + 16q^{5} + 4q^{6} + 24q^{7} + O(q^{8})$	+ +	$+1$
34.4.a.c	2	$\mathbb{Q}(\sqrt{13})$	$2q + 4q^{2} + 6q^{3} + 8q^{4} - 4q^{5} + 12q^{6} - 6q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
34.4.a.a	$x + 20$
34.4.a.b	$x - 14$
34.4.a.c	$x^{2} - 2x - 116$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(34))$	4
Old forms	8
Cusp forms $S_{4}(\Gamma_0(34))$	12
Eisenstein series $E_{4}(\Gamma_0(34))$	4
Modular forms $M_{4}(\Gamma_0(34))$	16

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(34)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.101 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, 17, 34]: [1, 2, 6, 16] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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(34, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 34:4 1000

← 33.4.a · 35.4.a →