Atlas › Modular forms › Weight 2 › Level 165 › 165.2.a

Newspace 165.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(165))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
165.2.a.a	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	+ + +	$-1$
165.2.a.b	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + 2q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	− + +	$+1$
165.2.a.c	3	degree 3	$3q - q^{2} + 3q^{3} + 5q^{4} + 3q^{5} - q^{6} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{7}$ 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 $T$
165.2.a.a	$x^{2} + 6x + 7$
165.2.a.b	$x^{2} - 4x + 1$
165.2.a.c	$x^{3} + x^{2} - 9x - 13$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(165))$	7
Old forms	14
Cusp forms $S_{2}(\Gamma_0(165))$	21
Eisenstein series $E_{2}(\Gamma_0(165))$	7
Modular forms $M_{2}(\Gamma_0(165))$	28

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.185 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]: [0, 1, 1, 2, 4, 6, 8, 28] (certified at ell = 2147482921)
T = [(2, 1), (7, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 1 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, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 165:2 1000

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