Atlas › Modular forms › Weight 12 › Level 18 › 18.12.a

Newspace 18.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(18))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
18.12.a.a	1	$\mathbb{Q}$	$q - 32q^{2} + 1024q^{4} - 3630q^{5} + 32936q^{7} + O(q^{8})$	+
18.12.a.b	1	$\mathbb{Q}$	$q - 32q^{2} + 1024q^{4} + 5280q^{5} - 49036q^{7} + O(q^{8})$	+
18.12.a.c	1	$\mathbb{Q}$	$q + 32q^{2} + 1024q^{4} - 5766q^{5} + 72464q^{7} + O(q^{8})$	−
18.12.a.d	1	$\mathbb{Q}$	$q + 32q^{2} + 1024q^{4} - 5280q^{5} - 49036q^{7} + O(q^{8})$	−
18.12.a.e	1	$\mathbb{Q}$	$q + 32q^{2} + 1024q^{4} + 11730q^{5} - 50008q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + T_{7}$ 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 $T$
18.12.a.a	$x - 29306$
18.12.a.b	$x + 43756$
18.12.a.c	$x - 66698$
18.12.a.d	$x + 54316$
18.12.a.e	$x + 38278$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(18))$	5
Old forms	24
Cusp forms $S_{12}(\Gamma_0(18))$	29
Eisenstein series $E_{12}(\Gamma_0(18))$	8
Modular forms $M_{12}(\Gamma_0(18))$	37

Decomposition of the old subspace

$S_{12}^{\mathrm{old}}(\Gamma_0(18)) =$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))^{\oplus 6}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 4}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 2}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.361 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, 9, 18]: [2, 4, 5, 13, 12, 35] (certified at ell = 2147483647)
T = [(5, 1), (7, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 3 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(18, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 18:12 1000

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