Atlas › Modular forms › Weight 4 › Level 122 › 122.4.a

Newspace 122.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(122))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
122.4.a.a	2	$\mathbb{Q}(\sqrt{3})$	$2q + 4q^{2} - 4q^{3} + 8q^{4} - 22q^{5} - 8q^{6} - 40q^{7} + O(q^{8})$	− +	$-1$
122.4.a.b	3	degree 3	$3q - 6q^{2} - 2q^{3} + 12q^{4} - q^{5} + 4q^{6} - 31q^{7} + O(q^{8})$	+ −	$-1$
122.4.a.c	4	degree 4	$4q - 8q^{2} + 7q^{3} + 16q^{4} - 6q^{5} - 14q^{6} + 39q^{7} + O(q^{8})$	+ +	$+1$
122.4.a.d	6	degree 6	$6q + 12q^{2} + 5q^{3} + 24q^{4} + 13q^{5} + 10q^{6} + 44q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
122.4.a.a	$x^{2} + 70x + 457$
122.4.a.b	$x^{3} + 5x^{2} - 205x - 1241$
122.4.a.c	$x^{4} + 11x^{3} - 4540x^{2} - 23565x + 5132477$
122.4.a.d	$x^{6} - 44x^{5} - 4067x^{4} + 190717x^{3} - 1242320x^{2} - 10570229x + 25666806$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(122))$	15
Old forms	30
Cusp forms $S_{4}(\Gamma_0(122))$	45
Eisenstein series $E_{4}(\Gamma_0(122))$	4
Modular forms $M_{4}(\Gamma_0(122))$	49

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(122)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(61))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.115 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]: [1, 2, 17, 49] (certified at ell = 2147482921)
T = [(3, 1), (5, 3)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 4 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 122:4 1000

← 121.4.a · 123.4.a →