# Properties

 Label 189.3.j.b Level $189$ Weight $3$ Character orbit 189.j Analytic conductor $5.150$ Analytic rank $0$ Dimension $22$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.14987699641$$ Analytic rank: $$0$$ Dimension: $$22$$ Relative dimension: $$11$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$22q - 24q^{4} - 12q^{5} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$22q - 24q^{4} - 12q^{5} + 25q^{10} - 24q^{11} - 18q^{13} + 60q^{14} - 24q^{16} + 6q^{17} + 3q^{19} + 39q^{20} - 59q^{22} - 81q^{23} + 57q^{25} - 3q^{26} + 34q^{28} + 63q^{29} + 58q^{31} - 99q^{34} - 27q^{35} - 20q^{37} + 48q^{38} - 105q^{40} + 51q^{41} + 65q^{43} + 54q^{44} + 75q^{46} + 4q^{49} - 63q^{50} - 46q^{52} - 63q^{53} - 100q^{55} - 192q^{56} + 40q^{58} - 156q^{61} + 106q^{64} + 264q^{67} - 27q^{68} + 236q^{70} + q^{73} - 342q^{74} + 233q^{76} + 531q^{77} - 280q^{79} + 96q^{80} - 157q^{82} - 255q^{83} + 102q^{85} - 504q^{86} + 408q^{88} - 720q^{89} - 70q^{91} + 1239q^{92} - 522q^{94} + 178q^{97} + 483q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
44.1 3.69072i 0 −9.62138 −5.05096 + 2.91617i 0 3.74467 + 5.91417i 20.7469i 0 10.7628 + 18.6416i
44.2 3.22356i 0 −6.39137 4.79167 2.76647i 0 −6.99696 0.206413i 7.70873i 0 −8.91790 15.4463i
44.3 2.41355i 0 −1.82523 −5.93347 + 3.42569i 0 −6.55070 + 2.46747i 5.24892i 0 8.26808 + 14.3207i
44.4 1.46555i 0 1.85217 −0.998268 + 0.576350i 0 −4.05935 5.70277i 8.57663i 0 0.844669 + 1.46301i
44.5 1.29088i 0 2.33363 −4.18841 + 2.41818i 0 6.74202 + 1.88284i 8.17595i 0 3.12158 + 5.40673i
44.6 0.513687i 0 3.73613 6.08350 3.51231i 0 1.23968 + 6.88935i 3.97395i 0 −1.80423 3.12502i
44.7 0.0767494i 0 3.99411 3.53076 2.03848i 0 2.97142 6.33803i 0.613543i 0 0.156452 + 0.270983i
44.8 1.28539i 0 2.34778 −6.82011 + 3.93759i 0 −6.26023 3.13201i 8.15936i 0 −5.06133 8.76649i
44.9 2.15495i 0 −0.643801 −1.62693 + 0.939308i 0 6.99124 0.350157i 7.23243i 0 −2.02416 3.50595i
44.10 2.74500i 0 −3.53503 −2.32531 + 1.34252i 0 0.122457 + 6.99893i 1.27635i 0 −3.68521 6.38297i
44.11 2.87176i 0 −4.24700 6.53753 3.77444i 0 2.05575 6.69133i 0.709334i 0 10.8393 + 18.7742i
116.1 2.87176i 0 −4.24700 6.53753 + 3.77444i 0 2.05575 + 6.69133i 0.709334i 0 10.8393 18.7742i
116.2 2.74500i 0 −3.53503 −2.32531 1.34252i 0 0.122457 6.99893i 1.27635i 0 −3.68521 + 6.38297i
116.3 2.15495i 0 −0.643801 −1.62693 0.939308i 0 6.99124 + 0.350157i 7.23243i 0 −2.02416 + 3.50595i
116.4 1.28539i 0 2.34778 −6.82011 3.93759i 0 −6.26023 + 3.13201i 8.15936i 0 −5.06133 + 8.76649i
116.5 0.0767494i 0 3.99411 3.53076 + 2.03848i 0 2.97142 + 6.33803i 0.613543i 0 0.156452 0.270983i
116.6 0.513687i 0 3.73613 6.08350 + 3.51231i 0 1.23968 6.88935i 3.97395i 0 −1.80423 + 3.12502i
116.7 1.29088i 0 2.33363 −4.18841 2.41818i 0 6.74202 1.88284i 8.17595i 0 3.12158 5.40673i
116.8 1.46555i 0 1.85217 −0.998268 0.576350i 0 −4.05935 + 5.70277i 8.57663i 0 0.844669 1.46301i
116.9 2.41355i 0 −1.82523 −5.93347 3.42569i 0 −6.55070 2.46747i 5.24892i 0 8.26808 14.3207i
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 116.11 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.3.j.b 22
3.b odd 2 1 63.3.j.b 22
7.c even 3 1 189.3.n.b 22
9.c even 3 1 63.3.n.b yes 22
9.d odd 6 1 189.3.n.b 22
21.c even 2 1 441.3.j.f 22
21.g even 6 1 441.3.n.f 22
21.g even 6 1 441.3.r.f 22
21.h odd 6 1 63.3.n.b yes 22
21.h odd 6 1 441.3.r.g 22
63.g even 3 1 441.3.r.g 22
63.h even 3 1 63.3.j.b 22
63.j odd 6 1 inner 189.3.j.b 22
63.k odd 6 1 441.3.r.f 22
63.l odd 6 1 441.3.n.f 22
63.t odd 6 1 441.3.j.f 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.j.b 22 3.b odd 2 1
63.3.j.b 22 63.h even 3 1
63.3.n.b yes 22 9.c even 3 1
63.3.n.b yes 22 21.h odd 6 1
189.3.j.b 22 1.a even 1 1 trivial
189.3.j.b 22 63.j odd 6 1 inner
189.3.n.b 22 7.c even 3 1
189.3.n.b 22 9.d odd 6 1
441.3.j.f 22 21.c even 2 1
441.3.j.f 22 63.t odd 6 1
441.3.n.f 22 21.g even 6 1
441.3.n.f 22 63.l odd 6 1
441.3.r.f 22 21.g even 6 1
441.3.r.f 22 63.k odd 6 1
441.3.r.g 22 21.h odd 6 1
441.3.r.g 22 63.g even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{22} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(189, [\chi])$$.