Atlas › Modular forms › Weight 4 › Level 165 › 165.4.a

Newspace 165.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(165))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
165.4.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - 8q^{4} - 5q^{5} + 2q^{7} + O(q^{8})$	+ + +	$+1$
165.4.a.b	1	$\mathbb{Q}$	$q + q^{2} + 3q^{3} - 7q^{4} - 5q^{5} + 3q^{6} + 36q^{7} + O(q^{8})$	− + −	$+1$
165.4.a.c	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} + 6q^{3} - 7q^{4} - 10q^{5} - 3q^{6} - 4q^{7} + O(q^{8})$	− + +	$-1$
165.4.a.d	3	degree 3	$3q - 4q^{2} - 9q^{3} + 22q^{4} - 15q^{5} + 12q^{6} - 4q^{7} + O(q^{8})$	+ + −	$-1$
165.4.a.e	3	degree 3	$3q - 2q^{2} + 9q^{3} + 30q^{4} - 15q^{5} - 6q^{6} + 10q^{7} + O(q^{8})$	− + −	$+1$
165.4.a.f	3	degree 3	$3q + q^{2} - 9q^{3} - 5q^{4} + 15q^{5} - 3q^{6} - 16q^{7} + O(q^{8})$	+ − +	$-1$
165.4.a.g	3	degree 3	$3q + q^{2} - 9q^{3} + 17q^{4} + 15q^{5} - 3q^{6} + 6q^{7} + O(q^{8})$	+ − −	$+1$
165.4.a.h	4	degree 4	$4q + 4q^{2} + 12q^{3} + 26q^{4} + 20q^{5} + 12q^{6} + 34q^{7} + O(q^{8})$	− − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{7}$ 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 $T$
165.4.a.a	$x - 6$
165.4.a.b	$x - 109$
165.4.a.c	$x^{2} + 13x - 472$
165.4.a.d	$x^{3} + 16x^{2} - 3651x - 105716$
165.4.a.e	$x^{3} - 28x^{2} - 5317x + 116922$
165.4.a.f	$x^{3} + 47x^{2} - 6597x - 172951$
165.4.a.g	$x^{3} - 19x^{2} - 2846x - 20876$
165.4.a.h	$x^{4} - 106x^{3} + 1803x^{2} + 86294x - 2064924$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(165))$	20
Old forms	48
Cusp forms $S_{4}(\Gamma_0(165))$	68
Eisenstein series $E_{4}(\Gamma_0(165))$	8
Modular forms $M_{4}(\Gamma_0(165))$	76

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(165)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(55))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.15 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, 11, 15, 33, 55, 165]: [1, 2, 3, 4, 8, 14, 20, 76] (certified at ell = 2147482921)
T = [(2, 1), (7, 3)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 5 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(165, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 165:4 1000

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