Atlas › Modular forms › Weight 2 › Level 42 › 42.2.a

Newspace 42.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(42))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
42.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} + O(q^{8})$	− + +	$+1$		42a1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + T_{11}$ 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 $T$
42.2.a.a	$x + 6$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(42))$	1
Old forms	4
Cusp forms $S_{2}(\Gamma_0(42))$	5
Eisenstein series $E_{2}(\Gamma_0(42))$	7
Modular forms $M_{2}(\Gamma_0(42))$	12

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(42)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.109 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, 3, 6, 7, 14, 21, 42]: [0, 1, 1, 3, 1, 4, 4, 12] (certified at ell = 2147483647)
T = [(5, 1), (11, 1)]; h of degree 1 = 1 x 1 (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(42, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 42:2 1000

← 41.2.a · 43.2.a →