Atlas › Modular forms › Weight 2 › Level 122 › 122.2.a

Newspace 122.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(122))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
122.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + q^{5} + 2q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$	122a1 rank 1
122.2.a.b	2	$\mathbb{Q}(\sqrt{13})$	$2q - 2q^{2} + q^{3} + 2q^{4} - q^{6} + 5q^{7} + O(q^{8})$	+ −	$+1$
122.2.a.c	3	degree 3	$3q + 3q^{2} - q^{3} + 3q^{4} + q^{5} - q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
122.2.a.a	$x + 1$
122.2.a.b	$x^{2} - x - 3$
122.2.a.c	$x^{3} - 4x + 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(122))$	6
Old forms	8
Cusp forms $S_{2}(\Gamma_0(122))$	14
Eisenstein series $E_{2}(\Gamma_0(122))$	3
Modular forms $M_{2}(\Gamma_0(122))$	17

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(122)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(61))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.169 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, 2, 61, 122]: [0, 1, 5, 17] (certified at ell = 2147482921)
T = [(3, 1), (5, 1)]; h of degree 6 = 1 x 6 (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(122, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 122:2 1000

← 121.2.a · 123.2.a →