Atlas › Modular forms › Weight 4 › Level 170 › 170.4.a

Newspace 170.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(170))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
170.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{3} + 4q^{4} - 5q^{5} - 8q^{6} - 4q^{7} + O(q^{8})$	+ + −	$-1$
170.4.a.b	1	$\mathbb{Q}$	$q + 2q^{2} + 7q^{3} + 4q^{4} + 5q^{5} + 14q^{6} - 10q^{7} + O(q^{8})$	− − +	$+1$
170.4.a.c	2	$\mathbb{Q}(\sqrt{43})$	$2q - 4q^{2} - 6q^{3} + 8q^{4} + 10q^{5} + 12q^{6} - 34q^{7} + O(q^{8})$	+ − +	$-1$
170.4.a.d	2	$\mathbb{Q}(\sqrt{145})$	$2q - 4q^{2} + 3q^{3} + 8q^{4} + 10q^{5} - 6q^{6} + 28q^{7} + O(q^{8})$	+ − −	$+1$
170.4.a.e	2	$\mathbb{Q}(\sqrt{6})$	$2q + 4q^{2} - 4q^{3} + 8q^{4} - 10q^{5} - 8q^{6} - 28q^{7} + O(q^{8})$	− + +	$-1$
170.4.a.f	2	$\mathbb{Q}(\sqrt{19})$	$2q + 4q^{2} + 2q^{3} + 8q^{4} + 10q^{5} + 4q^{6} + 46q^{7} + O(q^{8})$	− − +	$+1$
170.4.a.g	3	degree 3	$3q - 6q^{2} + 7q^{3} + 12q^{4} - 15q^{5} - 14q^{6} - 10q^{7} + O(q^{8})$	+ + +	$+1$
170.4.a.h	3	degree 3	$3q + 6q^{2} - q^{3} + 12q^{4} - 15q^{5} - 2q^{6} + 20q^{7} + O(q^{8})$	− + −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7}$ 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 $T$
170.4.a.a	$x + 8$
170.4.a.b	$x + 23$
170.4.a.c	$x^{2} + 108x + 2744$
170.4.a.d	$x^{2} - 87x + 1856$
170.4.a.e	$x^{2} + 88x - 2120$
170.4.a.f	$x^{2} - 140x + 3000$
170.4.a.g	$x^{3} + 23x^{2} - 8856x - 391000$
170.4.a.h	$x^{3} - 59x^{2} - 2884x + 128720$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(170))$	16
Old forms	62
Cusp forms $S_{4}(\Gamma_0(170))$	78
Eisenstein series $E_{4}(\Gamma_0(170))$	8
Modular forms $M_{4}(\Gamma_0(170))$	86

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(170)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(85))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.156 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, 17, 34, 85, 170]: [1, 2, 3, 7, 6, 16, 30, 86] (certified at ell = 2147483489)
T = [(3, 1), (7, 3)]; h of degree 16 = 1 x 16 (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(170, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 170:4 1000

← 169.4.a · 171.4.a →