Atlas › Modular forms › Weight 4 › Level 105 › 105.4.a

Newspace 105.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(105))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
105.4.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - 8q^{4} + 5q^{5} + 7q^{7} + O(q^{8})$	+ − −	$+1$
105.4.a.b	1	$\mathbb{Q}$	$q + 5q^{2} - 3q^{3} + 17q^{4} + 5q^{5} - 15q^{6} + 7q^{7} + O(q^{8})$	+ − −	$+1$
105.4.a.c	2	$\mathbb{Q}(\sqrt{17})$	$2q - 7q^{2} - 6q^{3} + 17q^{4} - 10q^{5} + 21q^{6} - 14q^{7} + O(q^{8})$	+ + +	$+1$
105.4.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - 4q^{2} + 6q^{3} + 2q^{4} - 10q^{5} - 12q^{6} - 14q^{7} + O(q^{8})$	− + +	$-1$
105.4.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 6q^{3} + 2q^{4} + 10q^{5} + 6q^{6} - 14q^{7} + O(q^{8})$	+ − +	$-1$
105.4.a.f	2	$\mathbb{Q}(\sqrt{65})$	$2q + q^{2} + 6q^{3} + 17q^{4} + 10q^{5} + 3q^{6} - 14q^{7} + O(q^{8})$	− − +	$+1$
105.4.a.g	2	$\mathbb{Q}(\sqrt{41})$	$2q + 3q^{2} + 6q^{3} + 9q^{4} - 10q^{5} + 9q^{6} + 14q^{7} + O(q^{8})$	− + −	$+1$

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 $T$
105.4.a.a	$x - 42$
105.4.a.b	$x - 17$
105.4.a.c	$x^{2} + 33x - 242$
105.4.a.d	$x^{2} + 96x + 2299$
105.4.a.e	$x^{2} + 18x - 2807$
105.4.a.f	$x^{2} + 21x + 94$
105.4.a.g	$x^{2} - 65x + 1046$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(105))$	12
Old forms	32
Cusp forms $S_{4}(\Gamma_0(105))$	44
Eisenstein series $E_{4}(\Gamma_0(105))$	8
Modular forms $M_{4}(\Gamma_0(105))$	52

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.124 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, 2, 3, 3, 8, 10, 14, 52] (certified at ell = 2147483629)
T = [(2, 1), (11, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 3 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 105:4 1000

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