Atlas › Modular forms › Weight 6 › Level 70 › 70.6.a

Newspace 70.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(70))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
70.6.a.a	1	$\mathbb{Q}$	$q - 4q^{2} - 23q^{3} + 16q^{4} + 25q^{5} + 92q^{6} + 49q^{7} + O(q^{8})$	+ − −	$-1$
70.6.a.b	1	$\mathbb{Q}$	$q - 4q^{2} - 9q^{3} + 16q^{4} + 25q^{5} + 36q^{6} - 49q^{7} + O(q^{8})$	+ − +	$+1$
70.6.a.c	1	$\mathbb{Q}$	$q - 4q^{2} - 3q^{3} + 16q^{4} - 25q^{5} + 12q^{6} + 49q^{7} + O(q^{8})$	+ + −	$+1$
70.6.a.d	1	$\mathbb{Q}$	$q - 4q^{2} + 11q^{3} + 16q^{4} - 25q^{5} - 44q^{6} - 49q^{7} + O(q^{8})$	+ + +	$-1$
70.6.a.e	1	$\mathbb{Q}$	$q + 4q^{2} - 17q^{3} + 16q^{4} + 25q^{5} - 68q^{6} - 49q^{7} + O(q^{8})$	− − +	$-1$
70.6.a.f	1	$\mathbb{Q}$	$q + 4q^{2} - 11q^{3} + 16q^{4} - 25q^{5} - 44q^{6} + 49q^{7} + O(q^{8})$	− + −	$-1$
70.6.a.g	2	$\mathbb{Q}(\sqrt{3369})$	$2q + 8q^{2} + 3q^{3} + 32q^{4} - 50q^{5} + 12q^{6} - 98q^{7} + O(q^{8})$	− + +	$+1$
70.6.a.h	2	$\mathbb{Q}(\sqrt{1129})$	$2q + 8q^{2} + 5q^{3} + 32q^{4} + 50q^{5} + 20q^{6} + 98q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{11}$ 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 $T$
70.6.a.a	$x - 532$
70.6.a.b	$x + 196$
70.6.a.c	$x - 402$
70.6.a.d	$x - 94$
70.6.a.e	$x + 732$
70.6.a.f	$x + 278$
70.6.a.g	$x^{2} - 962x + 227992$
70.6.a.h	$x^{2} - 420x + 3456$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(70))$	10
Old forms	46
Cusp forms $S_{6}(\Gamma_0(70))$	56
Eisenstein series $E_{6}(\Gamma_0(70))$	8
Modular forms $M_{6}(\Gamma_0(70))$	64

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(70)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.287 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, 7, 10, 14, 35, 70]: [1, 2, 3, 5, 9, 12, 22, 64] (certified at ell = 2147483629)
T = [(3, 1), (11, 1)]; h of degree 10 = 1 x 10 (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(70, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 70:6 1000

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