Atlas › Modular forms › Weight 4 › Level 121 › 121.4.a

Newspace 121.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(121))$

Newforms

Label	Dim	Coefficient field	Trace form	CM
121.4.a.a	1	$\mathbb{Q}$	$q + 8q^{3} - 8q^{4} + 18q^{5} + O(q^{8})$	$\mathbb{Q}(\sqrt{-11})$
121.4.a.b	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} - 8q^{3} + 10q^{4} - 10q^{5} + 32q^{6} - 8q^{7} + O(q^{8})$
121.4.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} - 2q^{3} - 8q^{4} + 2q^{5} + 26q^{6} - 20q^{7} + O(q^{8})$
121.4.a.d	2	$\mathbb{Q}(\sqrt{26})$	$2q - 10q^{3} + 36q^{4} + 10q^{5} + O(q^{8})$
121.4.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{2} - 8q^{3} + 10q^{4} - 10q^{5} - 32q^{6} + 8q^{7} + O(q^{8})$
121.4.a.f	4	degree 4	$4q - 4q^{2} + 6q^{3} + 14q^{4} - 11q^{5} - 43q^{6} - 25q^{7} + O(q^{8})$
121.4.a.g	4	degree 4	$4q + 4q^{2} + 6q^{3} + 14q^{4} - 11q^{5} + 43q^{6} + 25q^{7} + O(q^{8})$
121.4.a.h	6	degree 6	$6q + 8q^{3} + 20q^{4} + 14q^{5} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
121.4.a.a	$x - 24$
121.4.a.b	$x^{2} + 26x - 23$
121.4.a.c	$x^{2} + 8x - 491$
121.4.a.d	$x^{2} + 30x + 199$
121.4.a.e	$x^{2} + 22x + 73$
121.4.a.f	$x^{4} - 14x^{3} - 3x^{2} + 364x + 491$
121.4.a.g	$x^{4} - 22x^{3} - 117x^{2} + 2618x + 13421$
121.4.a.h	$x^{6} - 24x^{5} - 1150x^{4} + 24228x^{3} + 311509x^{2} - 5893140x + 12457428$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(121))$	23
Old forms	4
Cusp forms $S_{4}(\Gamma_0(121))$	27
Eisenstein series $E_{4}(\Gamma_0(121))$	12
Modular forms $M_{4}(\Gamma_0(121))$	39

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(121)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$

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, 11, 121]: [1, 4, 34] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 23 = 1 x 23 (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(121, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 121:4 1000

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