Properties

 Label 189.2.v.b Level $189$ Weight $2$ Character orbit 189.v Analytic conductor $1.509$ Analytic rank $0$ Dimension $54$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,2,Mod(22,189)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([14, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("189.22");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 189.v (of order $$9$$, degree $$6$$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.50917259820$$ Analytic rank: $$0$$ Dimension: $$54$$ Relative dimension: $$9$$ over $$\Q(\zeta_{9})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$54 q + 3 q^{3} - 3 q^{5} + 9 q^{8} + 3 q^{9}+O(q^{10})$$ 54 * q + 3 * q^3 - 3 * q^5 + 9 * q^8 + 3 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$54 q + 3 q^{3} - 3 q^{5} + 9 q^{8} + 3 q^{9} - 6 q^{11} - 60 q^{12} + 9 q^{13} - 9 q^{15} + 30 q^{17} - 3 q^{18} + 18 q^{20} + 3 q^{21} - 9 q^{22} + 36 q^{24} - 45 q^{25} - 54 q^{26} - 57 q^{27} - 54 q^{28} + 30 q^{29} + 24 q^{30} - 9 q^{31} + 51 q^{32} - 12 q^{33} - 9 q^{34} - 12 q^{35} + 48 q^{36} - 78 q^{38} - 36 q^{39} + 45 q^{40} - 51 q^{41} - 12 q^{42} - 9 q^{43} + 30 q^{44} + 51 q^{45} - 9 q^{47} + 15 q^{48} + 126 q^{50} - 12 q^{51} + 9 q^{52} - 60 q^{53} - 90 q^{54} + 9 q^{56} + 39 q^{57} - 27 q^{58} + 42 q^{59} + 135 q^{60} + 36 q^{62} + 9 q^{63} - 27 q^{64} - 18 q^{65} - 147 q^{66} - 27 q^{67} - 81 q^{68} + 48 q^{69} + 75 q^{72} + 84 q^{74} + 15 q^{75} + 54 q^{76} - 3 q^{77} - 66 q^{78} + 72 q^{79} - 222 q^{80} - 69 q^{81} - 54 q^{83} - 12 q^{84} + 18 q^{85} + 66 q^{86} + 3 q^{87} + 54 q^{88} + 90 q^{89} + 15 q^{90} - 129 q^{92} + 21 q^{93} + 36 q^{94} - 48 q^{95} + 36 q^{96} + 3 q^{98} + 51 q^{99}+O(q^{100})$$ 54 * q + 3 * q^3 - 3 * q^5 + 9 * q^8 + 3 * q^9 - 6 * q^11 - 60 * q^12 + 9 * q^13 - 9 * q^15 + 30 * q^17 - 3 * q^18 + 18 * q^20 + 3 * q^21 - 9 * q^22 + 36 * q^24 - 45 * q^25 - 54 * q^26 - 57 * q^27 - 54 * q^28 + 30 * q^29 + 24 * q^30 - 9 * q^31 + 51 * q^32 - 12 * q^33 - 9 * q^34 - 12 * q^35 + 48 * q^36 - 78 * q^38 - 36 * q^39 + 45 * q^40 - 51 * q^41 - 12 * q^42 - 9 * q^43 + 30 * q^44 + 51 * q^45 - 9 * q^47 + 15 * q^48 + 126 * q^50 - 12 * q^51 + 9 * q^52 - 60 * q^53 - 90 * q^54 + 9 * q^56 + 39 * q^57 - 27 * q^58 + 42 * q^59 + 135 * q^60 + 36 * q^62 + 9 * q^63 - 27 * q^64 - 18 * q^65 - 147 * q^66 - 27 * q^67 - 81 * q^68 + 48 * q^69 + 75 * q^72 + 84 * q^74 + 15 * q^75 + 54 * q^76 - 3 * q^77 - 66 * q^78 + 72 * q^79 - 222 * q^80 - 69 * q^81 - 54 * q^83 - 12 * q^84 + 18 * q^85 + 66 * q^86 + 3 * q^87 + 54 * q^88 + 90 * q^89 + 15 * q^90 - 129 * q^92 + 21 * q^93 + 36 * q^94 - 48 * q^95 + 36 * q^96 + 3 * q^98 + 51 * q^99

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1 −2.12375 + 1.78204i 1.53024 + 0.811393i 0.987362 5.59961i 0.371253 0.135125i −4.69579 + 1.00376i −0.173648 0.984808i 5.10945 + 8.84983i 1.68328 + 2.48326i −0.547652 + 0.948561i
22.2 −1.67142 + 1.40248i −1.72870 0.107685i 0.479372 2.71866i −2.82618 + 1.02865i 3.04040 2.24449i −0.173648 0.984808i 0.829763 + 1.43719i 2.97681 + 0.372309i 3.28107 5.68297i
22.3 −1.18344 + 0.993024i 0.993131 1.41905i 0.0671374 0.380755i 2.98981 1.08820i 0.233835 + 2.66556i −0.173648 0.984808i −1.24623 2.15853i −1.02738 2.81860i −2.45765 + 4.25678i
22.4 −0.461316 + 0.387090i 1.33551 + 1.10291i −0.284323 + 1.61247i 2.68051 0.975627i −1.04302 + 0.00817329i −0.173648 0.984808i −1.09521 1.89697i 0.567181 + 2.94590i −0.858907 + 1.48767i
22.5 −0.246848 + 0.207130i −0.915107 + 1.47057i −0.329265 + 1.86736i −1.61217 + 0.586782i −0.0787071 0.552555i −0.173648 0.984808i −0.627745 1.08729i −1.32516 2.69146i 0.276422 0.478776i
22.6 0.554554 0.465326i −1.73195 0.0184677i −0.256295 + 1.45352i 2.40653 0.875906i −0.969054 + 0.795681i −0.173648 0.984808i 1.25815 + 2.17918i 2.99932 + 0.0639705i 0.926970 1.60556i
22.7 1.19955 1.00654i 1.37519 1.05302i 0.0784988 0.445189i −0.920293 + 0.334959i 0.589706 2.64734i −0.173648 0.984808i 1.21197 + 2.09919i 0.782305 2.89620i −0.766788 + 1.32812i
22.8 1.35330 1.13555i 0.722934 + 1.57397i 0.194644 1.10388i 0.893231 0.325110i 2.76567 + 1.30912i −0.173648 0.984808i 0.776506 + 1.34495i −1.95473 + 2.27575i 0.839631 1.45428i
22.9 1.81332 1.52156i −1.25490 1.19383i 0.625702 3.54853i −0.550279 + 0.200285i −4.09202 0.255396i −0.173648 0.984808i −1.89757 3.28669i 0.149542 + 2.99627i −0.693087 + 1.20046i
43.1 −2.12375 1.78204i 1.53024 0.811393i 0.987362 + 5.59961i 0.371253 + 0.135125i −4.69579 1.00376i −0.173648 + 0.984808i 5.10945 8.84983i 1.68328 2.48326i −0.547652 0.948561i
43.2 −1.67142 1.40248i −1.72870 + 0.107685i 0.479372 + 2.71866i −2.82618 1.02865i 3.04040 + 2.24449i −0.173648 + 0.984808i 0.829763 1.43719i 2.97681 0.372309i 3.28107 + 5.68297i
43.3 −1.18344 0.993024i 0.993131 + 1.41905i 0.0671374 + 0.380755i 2.98981 + 1.08820i 0.233835 2.66556i −0.173648 + 0.984808i −1.24623 + 2.15853i −1.02738 + 2.81860i −2.45765 4.25678i
43.4 −0.461316 0.387090i 1.33551 1.10291i −0.284323 1.61247i 2.68051 + 0.975627i −1.04302 0.00817329i −0.173648 + 0.984808i −1.09521 + 1.89697i 0.567181 2.94590i −0.858907 1.48767i
43.5 −0.246848 0.207130i −0.915107 1.47057i −0.329265 1.86736i −1.61217 0.586782i −0.0787071 + 0.552555i −0.173648 + 0.984808i −0.627745 + 1.08729i −1.32516 + 2.69146i 0.276422 + 0.478776i
43.6 0.554554 + 0.465326i −1.73195 + 0.0184677i −0.256295 1.45352i 2.40653 + 0.875906i −0.969054 0.795681i −0.173648 + 0.984808i 1.25815 2.17918i 2.99932 0.0639705i 0.926970 + 1.60556i
43.7 1.19955 + 1.00654i 1.37519 + 1.05302i 0.0784988 + 0.445189i −0.920293 0.334959i 0.589706 + 2.64734i −0.173648 + 0.984808i 1.21197 2.09919i 0.782305 + 2.89620i −0.766788 1.32812i
43.8 1.35330 + 1.13555i 0.722934 1.57397i 0.194644 + 1.10388i 0.893231 + 0.325110i 2.76567 1.30912i −0.173648 + 0.984808i 0.776506 1.34495i −1.95473 2.27575i 0.839631 + 1.45428i
43.9 1.81332 + 1.52156i −1.25490 + 1.19383i 0.625702 + 3.54853i −0.550279 0.200285i −4.09202 + 0.255396i −0.173648 + 0.984808i −1.89757 + 3.28669i 0.149542 2.99627i −0.693087 1.20046i
85.1 −2.47328 0.900200i −0.969270 + 1.43545i 3.77466 + 3.16732i −0.697772 + 3.95726i 3.68947 2.67773i −0.766044 + 0.642788i −3.85257 6.67284i −1.12103 2.78268i 5.28812 9.15929i
85.2 −1.78358 0.649172i 0.189618 + 1.72164i 1.22766 + 1.03013i 0.611519 3.46809i 0.779441 3.19379i −0.766044 + 0.642788i 0.377143 + 0.653231i −2.92809 + 0.652907i −3.34208 + 5.78866i
See all 54 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 22.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.v.b 54
3.b odd 2 1 567.2.v.a 54
27.e even 9 1 inner 189.2.v.b 54
27.e even 9 1 5103.2.a.g 27
27.f odd 18 1 567.2.v.a 54
27.f odd 18 1 5103.2.a.h 27

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.v.b 54 1.a even 1 1 trivial
189.2.v.b 54 27.e even 9 1 inner
567.2.v.a 54 3.b odd 2 1
567.2.v.a 54 27.f odd 18 1
5103.2.a.g 27 27.e even 9 1
5103.2.a.h 27 27.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{54} - 7 T_{2}^{51} - 39 T_{2}^{49} + 416 T_{2}^{48} - 27 T_{2}^{47} + 156 T_{2}^{46} + \cdots + 5041$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.