Atlas › Modular forms › Weight 6 › Level 105 › 105.6.a

Newspace 105.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(105))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
105.6.a.a	2	$\mathbb{Q}(\sqrt{65})$	$2q - 13q^{2} + 18q^{3} + 53q^{4} + 50q^{5} - 117q^{6} - 98q^{7} + O(q^{8})$	− − +	$-1$
105.6.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 8q^{2} + 18q^{3} + 8q^{4} - 50q^{5} - 72q^{6} - 98q^{7} + O(q^{8})$	− + +	$+1$
105.6.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 4q^{2} - 18q^{3} + 8q^{4} + 50q^{5} + 36q^{6} - 98q^{7} + O(q^{8})$	+ − +	$+1$
105.6.a.d	2	$\mathbb{Q}(\sqrt{73})$	$2q - q^{2} + 18q^{3} - 27q^{4} - 50q^{5} - 9q^{6} + 98q^{7} + O(q^{8})$	− + −	$-1$
105.6.a.e	2	$\mathbb{Q}(\sqrt{233})$	$2q + q^{2} - 18q^{3} + 53q^{4} - 50q^{5} - 9q^{6} - 98q^{7} + O(q^{8})$	+ + +	$-1$
105.6.a.f	2	$\mathbb{Q}(\sqrt{65})$	$2q + 3q^{2} - 18q^{3} - 27q^{4} + 50q^{5} - 27q^{6} + 98q^{7} + O(q^{8})$	+ − −	$-1$
105.6.a.g	4	degree 4	$4q + 8q^{2} - 36q^{3} + 114q^{4} - 100q^{5} - 72q^{6} + 196q^{7} + O(q^{8})$	+ + −	$+1$
105.6.a.h	4	degree 4	$4q + 10q^{2} + 36q^{3} + 114q^{4} + 100q^{5} + 90q^{6} + 196q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{11}$ 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 $T$
105.6.a.a	$x^{2} + 405x - 27650$
105.6.a.b	$x^{2} - 660x + 106900$
105.6.a.c	$x^{2} + 180x - 318332$
105.6.a.d	$x^{2} + 397x - 24126$
105.6.a.e	$x^{2} + 195x - 11522$
105.6.a.f	$x^{2} + 229x + 8414$
105.6.a.g	$x^{4} - 240x^{3} - 658613x^{2} + 109138580x + 83192094636$
105.6.a.h	$x^{4} - 6x^{3} - 561671x^{2} + 77853940x + 29441931204$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(105))$	20
Old forms	56
Cusp forms $S_{6}(\Gamma_0(105))$	76
Eisenstein series $E_{6}(\Gamma_0(105))$	8
Modular forms $M_{6}(\Gamma_0(105))$	84

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(105)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.337 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, 3, 5, 7, 15, 21, 35, 105]: [1, 3, 3, 5, 12, 16, 22, 84] (certified at ell = 2147483629)
T = [(2, 1), (11, 1)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 7 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(105, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 105:6 1000

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