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Properties

Label 189.2.w.a
Level $189$
Weight $2$
Character orbit 189.w
Analytic conductor $1.509$
Analytic rank $0$
Dimension $132$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [189,2,Mod(25,189)]
 
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([10, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.w (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: \(132\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 132 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 132 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 6 q^{8} + 3 q^{9} - 6 q^{10} + 3 q^{11} - 3 q^{12} - 12 q^{13} + 15 q^{14} - 9 q^{16} - 54 q^{17} - 3 q^{18} - 6 q^{19} - 18 q^{20} - 21 q^{21} - 12 q^{22} + 45 q^{24} - 3 q^{25} + 30 q^{26} - 12 q^{27} - 12 q^{28} - 30 q^{29} - 57 q^{30} - 3 q^{31} + 51 q^{32} + 15 q^{33} - 18 q^{34} - 12 q^{35} - 60 q^{36} + 3 q^{37} - 57 q^{38} - 66 q^{39} - 66 q^{40} + 33 q^{42} - 12 q^{43} + 3 q^{44} + 33 q^{45} + 3 q^{46} - 21 q^{47} + 90 q^{48} + 12 q^{49} - 39 q^{50} - 48 q^{51} + 9 q^{52} + 9 q^{53} - 63 q^{54} - 24 q^{55} + 57 q^{56} - 18 q^{57} - 3 q^{58} - 18 q^{59} + 81 q^{60} + 33 q^{61} + 75 q^{62} + 63 q^{63} - 30 q^{64} + 81 q^{65} + 69 q^{66} - 3 q^{67} + 6 q^{68} - 6 q^{69} - 42 q^{70} - 18 q^{71} - 105 q^{72} + 21 q^{73} - 93 q^{74} + 18 q^{75} - 24 q^{76} + 87 q^{77} - 30 q^{78} + 15 q^{79} + 102 q^{80} + 39 q^{81} - 6 q^{82} - 42 q^{83} - 36 q^{84} - 63 q^{85} + 159 q^{86} + 30 q^{87} + 57 q^{88} - 150 q^{89} - 39 q^{90} + 6 q^{91} - 66 q^{92} - 27 q^{93} + 33 q^{94} - 147 q^{95} + 81 q^{96} - 12 q^{97} + 99 q^{98} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −2.58171 + 0.939667i −1.71864 + 0.215106i 4.25019 − 3.56633i 0.335317 + 0.122045i 4.23491 − 2.17029i −0.921726 + 2.48000i −4.87420 + 8.44237i 2.90746 − 0.739379i −0.980375
25.2 −2.28724 + 0.832486i 1.36942 − 1.06052i 3.00633 − 2.52261i 4.14290 + 1.50789i −2.24932 + 3.56567i −1.33708 − 2.28303i −2.34212 + 4.05668i 0.750612 − 2.90458i −10.7311
25.3 −2.11851 + 0.771075i 1.70797 + 0.287798i 2.36145 − 1.98149i −1.51793 − 0.552482i −3.84028 + 0.707271i 2.02397 + 1.70398i −1.22040 + 2.11380i 2.83434 + 0.983104i 3.64176
25.4 −2.06527 + 0.751697i −0.131292 + 1.72707i 2.16820 − 1.81934i −1.77012 − 0.644271i −1.02708 − 3.66555i −0.610341 − 2.57439i −0.912517 + 1.58053i −2.96552 − 0.453502i 4.14007
25.5 −1.47150 + 0.535583i 0.0586596 + 1.73106i 0.346382 − 0.290649i 3.26769 + 1.18934i −1.01344 − 2.51584i −0.172863 + 2.64010i 1.21191 − 2.09908i −2.99312 + 0.203086i −5.44540
25.6 −1.36957 + 0.498483i 0.717145 − 1.57661i 0.0951481 − 0.0798388i −1.88092 − 0.684598i −0.196268 + 2.51676i −1.86025 + 1.88135i 1.36695 − 2.36763i −1.97140 − 2.26132i 2.91731
25.7 −1.30907 + 0.476464i −1.39125 − 1.03171i −0.0454330 + 0.0381228i 1.40139 + 0.510065i 2.31282 + 0.687701i −2.64009 − 0.173054i 1.43440 − 2.48445i 0.871157 + 2.87073i −2.07755
25.8 −0.928908 + 0.338095i −1.69366 + 0.362654i −0.783527 + 0.657458i −3.08847 − 1.12411i 1.45064 − 0.909490i 2.54716 + 0.715520i 1.49406 − 2.58780i 2.73696 − 1.22843i 3.24896
25.9 −0.606828 + 0.220867i 1.44289 + 0.958162i −1.21263 + 1.01752i 1.46866 + 0.534550i −1.08721 − 0.262753i 1.88474 − 1.85681i 1.15690 − 2.00380i 1.16385 + 2.76504i −1.00929
25.10 −0.332947 + 0.121183i −1.35881 + 1.07408i −1.43592 + 1.20488i 0.672463 + 0.244756i 0.322250 − 0.522275i −2.22176 − 1.43659i 0.686389 − 1.18886i 0.692708 − 2.91893i −0.253554
25.11 −0.328890 + 0.119706i 0.434559 − 1.67665i −1.43825 + 1.20683i 2.13988 + 0.778854i 0.0577835 + 0.603453i 2.62809 + 0.305198i 0.678558 − 1.17530i −2.62232 − 1.45721i −0.797020
25.12 0.0303128 − 0.0110329i 0.919170 + 1.46804i −1.53129 + 1.28491i −3.70820 − 1.34968i 0.0440593 + 0.0343591i −2.47092 + 0.945798i −0.0644996 + 0.111717i −1.31025 + 2.69875i −0.127297
25.13 0.456477 − 0.166144i 1.72709 − 0.131048i −1.35132 + 1.13389i 1.44421 + 0.525651i 0.766603 − 0.346766i −1.09247 + 2.40967i −0.914231 + 1.58349i 2.96565 − 0.452662i 0.746585
25.14 0.562011 − 0.204555i −0.692899 − 1.58742i −1.25807 + 1.05565i −2.61250 − 0.950872i −0.714132 − 0.750410i −0.229775 − 2.63575i −1.08919 + 1.88654i −2.03978 + 2.19984i −1.66276
25.15 1.00414 − 0.365478i −0.803391 + 1.53446i −0.657360 + 0.551590i 0.354797 + 0.129136i −0.245908 + 1.83444i 1.55771 + 2.13859i −1.52708 + 2.64497i −1.70913 − 2.46554i 0.403463
25.16 1.17974 − 0.429388i −1.72810 − 0.116945i −0.324689 + 0.272446i 3.35074 + 1.21957i −2.08891 + 0.604061i 1.13056 − 2.39204i −1.52151 + 2.63533i 2.97265 + 0.404186i 4.47665
25.17 1.45469 − 0.529465i 1.56527 − 0.741574i 0.303710 − 0.254843i 0.0676746 + 0.0246315i 1.88435 − 1.90752i −1.36843 − 2.26438i −1.24118 + 2.14978i 1.90014 − 2.32153i 0.111487
25.18 1.68235 − 0.612324i 1.10447 + 1.33422i 0.923259 − 0.774706i −1.38133 − 0.502765i 2.67508 + 1.56832i 2.29659 − 1.31364i −0.711445 + 1.23226i −0.560272 + 2.94722i −2.63174
25.19 1.90772 − 0.694353i 0.482660 − 1.66344i 1.62518 − 1.36369i −2.38896 − 0.869509i −0.234236 − 3.50852i 1.31108 + 2.29806i 0.123353 − 0.213653i −2.53408 − 1.60575i −5.16121
25.20 2.03199 − 0.739583i −0.528499 − 1.64945i 2.04991 − 1.72008i 3.15692 + 1.14902i −2.29381 − 2.96080i −2.22776 + 1.42726i 0.730849 − 1.26587i −2.44138 + 1.74347i 7.26462
See next 80 embeddings (of 132 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.w 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.w.a yes 132
3.b odd 2 1 567.2.w.a 132
7.c even 3 1 189.2.u.a ✓ 132
21.h odd 6 1 567.2.u.a 132
27.e even 9 1 189.2.u.a ✓ 132
27.f odd 18 1 567.2.u.a 132
189.w even 9 1 inner 189.2.w.a yes 132
189.bf odd 18 1 567.2.w.a 132
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.u.a ✓ 132 7.c even 3 1
189.2.u.a ✓ 132 27.e even 9 1
189.2.w.a yes 132 1.a even 1 1 trivial
189.2.w.a yes 132 189.w even 9 1 inner
567.2.u.a 132 21.h odd 6 1
567.2.u.a 132 27.f odd 18 1
567.2.w.a 132 3.b odd 2 1
567.2.w.a 132 189.bf odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(189, [\chi])\).