# Properties

 Label 189.3.k.a Level $189$ Weight $3$ Character orbit 189.k Analytic conductor $5.150$ Analytic rank $0$ Dimension $28$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.14987699641$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$14$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q - q^{2} - 23q^{4} + 16q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q - q^{2} - 23q^{4} + 16q^{8} - 6q^{10} + 14q^{11} + 15q^{13} + 11q^{14} - 27q^{16} + 33q^{17} - 6q^{19} - 108q^{20} - 10q^{22} + 68q^{23} - 62q^{25} - 54q^{26} - 16q^{28} - 70q^{29} + 45q^{31} - 153q^{32} + 12q^{34} - 18q^{35} + 9q^{37} + 234q^{41} + 30q^{43} - 51q^{44} - 22q^{46} + 111q^{47} + 34q^{49} - 241q^{50} - 148q^{53} + 412q^{56} - 34q^{58} - 42q^{59} + 120q^{61} - 48q^{64} - 114q^{65} - 34q^{67} + 264q^{70} + 350q^{71} - 6q^{73} + 718q^{74} + 72q^{76} + 32q^{77} - 82q^{79} + 609q^{80} - 18q^{82} - 738q^{83} + 3q^{85} + 34q^{86} - 50q^{88} - 21q^{89} + 39q^{91} - 288q^{92} - 3q^{94} - 507q^{95} - 57q^{97} - 811q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1 −1.91801 + 3.32210i 0 −5.35756 9.27956i 0.216546i 0 6.73498 1.90790i 25.7594 0 −0.719386 0.415338i
10.2 −1.62718 + 2.81835i 0 −3.29541 5.70782i 3.39483i 0 −6.91332 1.09817i 8.43145 0 9.56782 + 5.52398i
10.3 −1.41697 + 2.45427i 0 −2.01561 3.49114i 2.39855i 0 0.0107242 + 6.99999i 0.0884848 0 5.88667 + 3.39867i
10.4 −0.902282 + 1.56280i 0 0.371774 + 0.643931i 5.75495i 0 6.44289 + 2.73664i −8.56004 0 −8.99383 5.19259i
10.5 −0.826674 + 1.43184i 0 0.633221 + 1.09677i 7.86923i 0 −5.81886 3.89113i −8.70726 0 −11.2675 6.50529i
10.6 −0.662399 + 1.14731i 0 1.12246 + 1.94415i 7.23514i 0 −3.90816 5.80744i −8.27324 0 8.30093 + 4.79254i
10.7 −0.198068 + 0.343064i 0 1.92154 + 3.32820i 2.97240i 0 2.98301 6.33259i −3.10693 0 1.01972 + 0.588737i
10.8 0.178911 0.309883i 0 1.93598 + 3.35322i 4.59004i 0 −6.01934 + 3.57317i 2.81677 0 1.42238 + 0.821210i
10.9 0.227576 0.394173i 0 1.89642 + 3.28469i 4.37081i 0 5.22047 + 4.66334i 3.54692 0 −1.72285 0.994690i
10.10 0.840995 1.45665i 0 0.585454 + 1.01404i 2.34462i 0 −3.93446 + 5.78964i 8.69742 0 3.41529 + 1.97182i
10.11 1.12025 1.94033i 0 −0.509909 0.883189i 1.93444i 0 3.87064 5.83251i 6.67708 0 −3.75345 2.16706i
10.12 1.32841 2.30087i 0 −1.52933 2.64888i 9.20400i 0 6.96620 + 0.687028i 2.50096 0 21.1772 + 12.2267i
10.13 1.67756 2.90562i 0 −3.62842 6.28461i 0.888628i 0 −3.29335 6.17688i −10.9271 0 −2.58202 1.49073i
10.14 1.67789 2.90618i 0 −3.63061 6.28839i 8.51666i 0 −2.34142 + 6.59680i −10.9439 0 −24.7510 14.2900i
19.1 −1.91801 3.32210i 0 −5.35756 + 9.27956i 0.216546i 0 6.73498 + 1.90790i 25.7594 0 −0.719386 + 0.415338i
19.2 −1.62718 2.81835i 0 −3.29541 + 5.70782i 3.39483i 0 −6.91332 + 1.09817i 8.43145 0 9.56782 5.52398i
19.3 −1.41697 2.45427i 0 −2.01561 + 3.49114i 2.39855i 0 0.0107242 6.99999i 0.0884848 0 5.88667 3.39867i
19.4 −0.902282 1.56280i 0 0.371774 0.643931i 5.75495i 0 6.44289 2.73664i −8.56004 0 −8.99383 + 5.19259i
19.5 −0.826674 1.43184i 0 0.633221 1.09677i 7.86923i 0 −5.81886 + 3.89113i −8.70726 0 −11.2675 + 6.50529i
19.6 −0.662399 1.14731i 0 1.12246 1.94415i 7.23514i 0 −3.90816 + 5.80744i −8.27324 0 8.30093 4.79254i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.14 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.k 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.k.a 28
3.b odd 2 1 63.3.k.a 28
7.d odd 6 1 189.3.t.a 28
9.c even 3 1 189.3.t.a 28
9.d odd 6 1 63.3.t.a yes 28
21.c even 2 1 441.3.k.b 28
21.g even 6 1 63.3.t.a yes 28
21.g even 6 1 441.3.l.b 28
21.h odd 6 1 441.3.l.a 28
21.h odd 6 1 441.3.t.a 28
63.i even 6 1 441.3.l.a 28
63.j odd 6 1 441.3.l.b 28
63.k odd 6 1 inner 189.3.k.a 28
63.n odd 6 1 441.3.k.b 28
63.o even 6 1 441.3.t.a 28
63.s even 6 1 63.3.k.a 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.k.a 28 3.b odd 2 1
63.3.k.a 28 63.s even 6 1
63.3.t.a yes 28 9.d odd 6 1
63.3.t.a yes 28 21.g even 6 1
189.3.k.a 28 1.a even 1 1 trivial
189.3.k.a 28 63.k odd 6 1 inner
189.3.t.a 28 7.d odd 6 1
189.3.t.a 28 9.c even 3 1
441.3.k.b 28 21.c even 2 1
441.3.k.b 28 63.n odd 6 1
441.3.l.a 28 21.h odd 6 1
441.3.l.a 28 63.i even 6 1
441.3.l.b 28 21.g even 6 1
441.3.l.b 28 63.j odd 6 1
441.3.t.a 28 21.h odd 6 1
441.3.t.a 28 63.o even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(189, [\chi])$$.