Atlas › Modular forms › Weight 4 › Level 189 › 189.4.a

Newspace 189.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(189))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
189.4.a.a	1	$\mathbb{Q}$	$q - 3q^{2} + q^{4} + 12q^{5} + 7q^{7} + O(q^{8})$	−
189.4.a.b	1	$\mathbb{Q}$	$q - 8q^{4} - 21q^{5} + 7q^{7} + O(q^{8})$	−
189.4.a.c	1	$\mathbb{Q}$	$q - 8q^{4} + 21q^{5} + 7q^{7} + O(q^{8})$	−
189.4.a.d	1	$\mathbb{Q}$	$q + 3q^{2} + q^{4} - 12q^{5} + 7q^{7} + O(q^{8})$	−
189.4.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q - 6q^{2} + 8q^{4} - 6q^{5} + 14q^{7} + O(q^{8})$	−
189.4.a.f	2	$\mathbb{Q}(\sqrt{3})$	$2q - 10q^{4} + 14q^{7} + O(q^{8})$	−
189.4.a.g	2	$\mathbb{Q}(\sqrt{7})$	$2q - 2q^{4} - 14q^{7} + O(q^{8})$	+
189.4.a.h	2	$\mathbb{Q}(\sqrt{21})$	$2q + 26q^{4} + 14q^{7} + O(q^{8})$	−
189.4.a.i	2	$\mathbb{Q}(\sqrt{3})$	$2q + 6q^{2} + 8q^{4} + 6q^{5} + 14q^{7} + O(q^{8})$	−
189.4.a.j	3	degree 3	$3q - q^{2} + 21q^{4} - 2q^{5} - 21q^{7} + O(q^{8})$	+
189.4.a.k	3	degree 3	$3q + q^{2} + 21q^{4} + 2q^{5} - 21q^{7} + O(q^{8})$	+
189.4.a.l	4	degree 4	$4q + 26q^{4} - 28q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
189.4.a.a	$x - 33$
189.4.a.b	$x + 63$
189.4.a.c	$x - 63$
189.4.a.d	$x + 33$
189.4.a.e	$x^{2} + 24x - 363$
189.4.a.f	$x^{2} - 48$
189.4.a.g	$x^{2} - 28$
189.4.a.h	$x^{2} - 336$
189.4.a.i	$x^{2} - 24x - 363$
189.4.a.j	$x^{3} + 7x^{2} - 1555x + 12911$
189.4.a.k	$x^{3} - 7x^{2} - 1555x - 12911$
189.4.a.l	$x^{4} - 4121x^{2} + 3034564$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(189))$	24
Old forms	42
Cusp forms $S_{4}(\Gamma_0(189))$	66
Eisenstein series $E_{4}(\Gamma_0(189))$	12
Modular forms $M_{4}(\Gamma_0(189))$	78

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(189)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(63))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.154 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, 3, 7, 9, 21, 27, 63, 189]: [1, 2, 3, 4, 10, 10, 26, 74] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 5 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(189, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 189:4 1000

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