# Properties

 Label 189.4.o.a Level $189$ Weight $4$ Character orbit 189.o Analytic conductor $11.151$ Analytic rank $0$ Dimension $44$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 189.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1513609911$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$22$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$44q + 6q^{2} + 78q^{4} + 5q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q + 6q^{2} + 78q^{4} + 5q^{7} + 18q^{11} - 204q^{14} - 242q^{16} - 34q^{22} + 102q^{23} - 352q^{25} + 300q^{28} - 246q^{29} - 1068q^{32} + 328q^{37} - 170q^{43} + 968q^{46} - 79q^{49} - 288q^{50} - 1212q^{56} - 538q^{58} - 808q^{64} - 4350q^{65} - 590q^{67} + 384q^{70} + 5304q^{74} + 2787q^{77} - 302q^{79} - 612q^{85} + 13692q^{86} + 1294q^{88} + 210q^{91} + 10194q^{92} - 6336q^{95} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
62.1 −4.27138 2.46608i 0 8.16312 + 14.1389i −7.65650 13.2614i 0 11.9101 14.1827i 41.0664i 0 75.5262i
62.2 −4.27138 2.46608i 0 8.16312 + 14.1389i 7.65650 + 13.2614i 0 6.32750 17.4058i 41.0664i 0 75.5262i
62.3 −3.92583 2.26658i 0 6.27474 + 10.8682i −0.0687529 0.119084i 0 2.53789 + 18.3455i 20.6235i 0 0.623335i
62.4 −3.92583 2.26658i 0 6.27474 + 10.8682i 0.0687529 + 0.119084i 0 −17.1567 + 6.97489i 20.6235i 0 0.623335i
62.5 −2.59186 1.49641i 0 0.478502 + 0.828790i −7.80147 13.5125i 0 −13.5435 12.6323i 21.0785i 0 46.6969i
62.6 −2.59186 1.49641i 0 0.478502 + 0.828790i 7.80147 + 13.5125i 0 17.7116 + 5.41283i 21.0785i 0 46.6969i
62.7 −1.93743 1.11857i 0 −1.49759 2.59390i −2.75758 4.77627i 0 13.1049 + 13.0867i 24.5978i 0 12.3382i
62.8 −1.93743 1.11857i 0 −1.49759 2.59390i 2.75758 + 4.77627i 0 −17.8859 4.80585i 24.5978i 0 12.3382i
62.9 −0.628557 0.362898i 0 −3.73661 6.47200i −5.53318 9.58374i 0 4.49570 17.9663i 11.2304i 0 8.03191i
62.10 −0.628557 0.362898i 0 −3.73661 6.47200i 5.53318 + 9.58374i 0 13.3114 12.8765i 11.2304i 0 8.03191i
62.11 0.0847887 + 0.0489528i 0 −3.99521 6.91990i −9.06347 15.6984i 0 12.7516 + 13.4312i 1.56555i 0 1.77473i
62.12 0.0847887 + 0.0489528i 0 −3.99521 6.91990i 9.06347 + 15.6984i 0 −18.0075 4.32762i 1.56555i 0 1.77473i
62.13 1.10556 + 0.638294i 0 −3.18516 5.51686i −1.59510 2.76280i 0 −13.1775 + 13.0136i 18.3450i 0 4.07258i
62.14 1.10556 + 0.638294i 0 −3.18516 5.51686i 1.59510 + 2.76280i 0 −4.68134 + 17.9188i 18.3450i 0 4.07258i
62.15 2.31807 + 1.33834i 0 −0.417710 0.723496i −0.223284 0.386739i 0 18.4159 1.96340i 23.6495i 0 1.19532i
62.16 2.31807 + 1.33834i 0 −0.417710 0.723496i 0.223284 + 0.386739i 0 −7.50759 16.9303i 23.6495i 0 1.19532i
62.17 3.28475 + 1.89645i 0 3.19307 + 5.53055i −9.97590 17.2788i 0 −13.5672 + 12.6068i 6.12125i 0 75.6753i
62.18 3.28475 + 1.89645i 0 3.19307 + 5.53055i 9.97590 + 17.2788i 0 −4.13417 + 18.0529i 6.12125i 0 75.6753i
62.19 3.38393 + 1.95371i 0 3.63397 + 6.29422i −5.82670 10.0921i 0 −2.49865 18.3509i 2.86048i 0 45.5347i
62.20 3.38393 + 1.95371i 0 3.63397 + 6.29422i 5.82670 + 10.0921i 0 17.1417 7.01157i 2.86048i 0 45.5347i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 125.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.4.o.a 44
3.b odd 2 1 63.4.o.a 44
7.b odd 2 1 inner 189.4.o.a 44
9.c even 3 1 63.4.o.a 44
9.c even 3 1 567.4.c.c 44
9.d odd 6 1 inner 189.4.o.a 44
9.d odd 6 1 567.4.c.c 44
21.c even 2 1 63.4.o.a 44
63.l odd 6 1 63.4.o.a 44
63.l odd 6 1 567.4.c.c 44
63.o even 6 1 inner 189.4.o.a 44
63.o even 6 1 567.4.c.c 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.o.a 44 3.b odd 2 1
63.4.o.a 44 9.c even 3 1
63.4.o.a 44 21.c even 2 1
63.4.o.a 44 63.l odd 6 1
189.4.o.a 44 1.a even 1 1 trivial
189.4.o.a 44 7.b odd 2 1 inner
189.4.o.a 44 9.d odd 6 1 inner
189.4.o.a 44 63.o even 6 1 inner
567.4.c.c 44 9.c even 3 1
567.4.c.c 44 9.d odd 6 1
567.4.c.c 44 63.l odd 6 1
567.4.c.c 44 63.o even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(189, [\chi])$$.