Atlas › Modular forms › Weight 2 › Level 161 › 161.2.a

Newspace 161.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(161))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
161.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + 2q^{5} + q^{7} + O(q^{8})$	− +	$+1$	161a1 rank 0
161.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - 2q^{3} - q^{4} - 2q^{5} + q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
161.2.a.c	3	degree 3	$3q - q^{2} + 2q^{3} + 5q^{4} + 2q^{5} + 4q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$
161.2.a.d	5	degree 5	$5q + 2q^{2} + 12q^{4} - 4q^{5} - 3q^{6} + 5q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
161.2.a.a	$x + 1$
161.2.a.b	$x^{2} + 3x + 1$
161.2.a.c	$x^{3} - x^{2} - 13x + 23$
161.2.a.d	$x^{5} - 2x^{4} - 19x^{3} + 47x^{2} - 8x - 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(161))$	11
Old forms	4
Cusp forms $S_{2}(\Gamma_0(161))$	15
Eisenstein series $E_{2}(\Gamma_0(161))$	3
Modular forms $M_{2}(\Gamma_0(161))$	18

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(161)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.172 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, 7, 23, 161]: [0, 1, 3, 18] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 2 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(161, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 161:2 1000

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