↑

Properties

Label 189.2.w.a.25.8
Level $189$
Weight $2$
Character 189.25
Analytic conductor $1.509$
Analytic rank $0$
Dimension $132$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [189,2,Mod(25,189)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.w (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content 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: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 25.8
Character \(\chi\) \(=\) 189.25
Dual form 189.2.w.a.121.8

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.928908 + 0.338095i) q^{2} +(-1.69366 + 0.362654i) q^{3} +(-0.783527 + 0.657458i) q^{4} +(-3.08847 - 1.12411i) q^{5} +(1.45064 - 0.909490i) q^{6} +(2.54716 + 0.715520i) q^{7} +(1.49406 - 2.58780i) q^{8} +(2.73696 - 1.22843i) q^{9} +3.24896 q^{10} +(3.23909 - 1.17893i) q^{11} +(1.08860 - 1.39766i) q^{12} +(-0.176997 - 1.00380i) q^{13} +(-2.60799 + 0.196530i) q^{14} +(5.63848 + 0.783815i) q^{15} +(-0.157705 + 0.894392i) q^{16} -0.671106 q^{17} +(-2.12706 + 2.06645i) q^{18} -2.12460 q^{19} +(3.15896 - 1.14977i) q^{20} +(-4.57351 - 0.288108i) q^{21} +(-2.61023 + 2.19024i) q^{22} +(-0.996522 - 5.65156i) q^{23} +(-1.59196 + 4.92467i) q^{24} +(4.44481 + 3.72964i) q^{25} +(0.503792 + 0.872594i) q^{26} +(-4.18999 + 3.07311i) q^{27} +(-2.46619 + 1.11402i) q^{28} +(0.678506 - 3.84800i) q^{29} +(-5.50263 + 1.17825i) q^{30} +(7.91115 - 6.63825i) q^{31} +(0.881871 + 5.00134i) q^{32} +(-5.05837 + 3.17138i) q^{33} +(0.623395 - 0.226897i) q^{34} +(-7.06251 - 5.07316i) q^{35} +(-1.33685 + 2.76194i) q^{36} +(4.55770 - 7.89416i) q^{37} +(1.97356 - 0.718318i) q^{38} +(0.663803 + 1.63590i) q^{39} +(-7.52335 + 6.31284i) q^{40} +(0.857967 + 4.86577i) q^{41} +(4.34578 - 1.27865i) q^{42} +(-2.16071 - 1.81305i) q^{43} +(-1.76282 + 3.05329i) q^{44} +(-9.83392 + 0.717305i) q^{45} +(2.83644 + 4.91286i) q^{46} +(-0.379189 - 0.318177i) q^{47} +(-0.0572559 - 1.57199i) q^{48} +(5.97606 + 3.64509i) q^{49} +(-5.38979 - 1.96172i) q^{50} +(1.13662 - 0.243379i) q^{51} +(0.798636 + 0.670135i) q^{52} +(-4.04846 + 7.01214i) q^{53} +(2.85311 - 4.27125i) q^{54} -11.3291 q^{55} +(5.65724 - 5.52250i) q^{56} +(3.59836 - 0.770497i) q^{57} +(0.670718 + 3.80383i) q^{58} +(-1.67822 - 9.51765i) q^{59} +(-4.93323 + 3.09292i) q^{60} +(6.71058 + 5.63085i) q^{61} +(-5.10438 + 8.84104i) q^{62} +(7.85045 - 1.17065i) q^{63} +(-3.41829 - 5.92066i) q^{64} +(-0.581732 + 3.29916i) q^{65} +(3.62653 - 4.65613i) q^{66} +(-9.15746 - 3.33304i) q^{67} +(0.525830 - 0.441223i) q^{68} +(3.73733 + 9.21042i) q^{69} +(8.27563 + 2.32470i) q^{70} +(4.72319 + 8.18081i) q^{71} +(0.910285 - 8.91805i) q^{72} +(-5.56377 - 9.63672i) q^{73} +(-1.56470 + 8.87388i) q^{74} +(-8.88056 - 4.70481i) q^{75} +(1.66469 - 1.39684i) q^{76} +(9.09404 - 0.685297i) q^{77} +(-1.16970 - 1.29517i) q^{78} +(0.194091 - 0.0706432i) q^{79} +(1.49247 - 2.58503i) q^{80} +(5.98194 - 6.72431i) q^{81} +(-2.44206 - 4.22978i) q^{82} +(2.37994 - 13.4973i) q^{83} +(3.77289 - 2.78115i) q^{84} +(2.07269 + 0.754398i) q^{85} +(2.62008 + 0.953630i) q^{86} +(0.246335 + 6.76326i) q^{87} +(1.78857 - 10.1435i) q^{88} +11.9916 q^{89} +(8.89229 - 3.99111i) q^{90} +(0.267399 - 2.68348i) q^{91} +(4.49646 + 3.77298i) q^{92} +(-10.9914 + 14.1119i) q^{93} +(0.459806 + 0.167356i) q^{94} +(6.56178 + 2.38829i) q^{95} +(-3.30735 - 8.15075i) q^{96} +(0.577101 + 0.484245i) q^{97} +(-6.78360 - 1.36548i) q^{98} +(7.41704 - 7.20568i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 6 q^{8} + 3 q^{9} - 6 q^{10} + 3 q^{11} - 3 q^{12} - 12 q^{13} + 15 q^{14} - 9 q^{16} - 54 q^{17} - 3 q^{18} - 6 q^{19} - 18 q^{20} - 21 q^{21}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.928908 + 0.338095i −0.656837 + 0.239069i −0.648870 − 0.760899i \(-0.724758\pi\)
−0.00796681 + 0.999968i \(0.502536\pi\)
\(3\) −1.69366 + 0.362654i −0.977835 + 0.209379i
\(4\) −0.783527 + 0.657458i −0.391764 + 0.328729i
\(5\) −3.08847 − 1.12411i −1.38121 − 0.502718i −0.458663 − 0.888610i \(-0.651671\pi\)
−0.922544 + 0.385892i \(0.873894\pi\)
\(6\) 1.45064 − 0.909490i 0.592222 − 0.371298i
\(7\) 2.54716 + 0.715520i 0.962736 + 0.270441i
\(8\) 1.49406 − 2.58780i 0.528232 − 0.914924i
\(9\) 2.73696 − 1.22843i 0.912321 − 0.409475i
\(10\) 3.24896 1.02741
\(11\) 3.23909 − 1.17893i 0.976622 − 0.355462i 0.196096 − 0.980585i \(-0.437173\pi\)
0.780526 + 0.625123i \(0.214951\pi\)
\(12\) 1.08860 − 1.39766i 0.314251 − 0.403469i
\(13\) −0.176997 − 1.00380i −0.0490900 − 0.278403i 0.950375 − 0.311106i \(-0.100700\pi\)
−0.999465 + 0.0327031i \(0.989588\pi\)
\(14\) −2.60799 + 0.196530i −0.697015 + 0.0525248i
\(15\) 5.63848 + 0.783815i 1.45585 + 0.202380i
\(16\) −0.157705 + 0.894392i −0.0394264 + 0.223598i
\(17\) −0.671106 −0.162767 −0.0813835 − 0.996683i \(-0.525934\pi\)
−0.0813835 + 0.996683i \(0.525934\pi\)
\(18\) −2.12706 + 2.06645i −0.501353 + 0.487066i
\(19\) −2.12460 −0.487418 −0.243709 − 0.969848i \(-0.578364\pi\)
−0.243709 + 0.969848i \(0.578364\pi\)
\(20\) 3.15896 − 1.14977i 0.706365 − 0.257096i
\(21\) −4.57351 − 0.288108i −0.998022 − 0.0628704i
\(22\) −2.61023 + 2.19024i −0.556502 + 0.466961i
\(23\) −0.996522 − 5.65156i −0.207789 − 1.17843i −0.892990 − 0.450077i \(-0.851397\pi\)
0.685201 − 0.728354i \(-0.259714\pi\)
\(24\) −1.59196 + 4.92467i −0.324958 + 1.00524i
\(25\) 4.44481 + 3.72964i 0.888962 + 0.745928i
\(26\) 0.503792 + 0.872594i 0.0988018 + 0.171130i
\(27\) −4.18999 + 3.07311i −0.806364 + 0.591420i
\(28\) −2.46619 + 1.11402i −0.466067 + 0.210530i
\(29\) 0.678506 − 3.84800i 0.125995 − 0.714555i −0.854716 − 0.519095i \(-0.826269\pi\)
0.980712 − 0.195460i \(-0.0626199\pi\)
\(30\) −5.50263 + 1.17825i −1.00464 + 0.215118i
\(31\) 7.91115 − 6.63825i 1.42089 − 1.19226i 0.470021 − 0.882655i \(-0.344246\pi\)
0.950864 − 0.309609i \(-0.100198\pi\)
\(32\) 0.881871 + 5.00134i 0.155894 + 0.884120i
\(33\) −5.05837 + 3.17138i −0.880549 + 0.552066i
\(34\) 0.623395 − 0.226897i 0.106911 − 0.0389126i
\(35\) −7.06251 − 5.07316i −1.19378 − 0.857520i
\(36\) −1.33685 + 2.76194i −0.222808 + 0.460324i
\(37\) 4.55770 − 7.89416i 0.749281 − 1.29779i −0.198887 − 0.980022i \(-0.563733\pi\)
0.948168 − 0.317770i \(-0.102934\pi\)
\(38\) 1.97356 − 0.718318i 0.320154 − 0.116527i
\(39\) 0.663803 + 1.63590i 0.106294 + 0.261954i
\(40\) −7.52335 + 6.31284i −1.18955 + 0.998148i
\(41\) 0.857967 + 4.86577i 0.133992 + 0.759906i 0.975556 + 0.219749i \(0.0705239\pi\)
−0.841564 + 0.540157i \(0.818365\pi\)
\(42\) 4.34578 − 1.27865i 0.670568 − 0.197301i
\(43\) −2.16071 − 1.81305i −0.329504 − 0.276487i 0.462993 − 0.886362i \(-0.346775\pi\)
−0.792498 + 0.609875i \(0.791220\pi\)
\(44\) −1.76282 + 3.05329i −0.265755 + 0.460301i
\(45\) −9.83392 + 0.717305i −1.46595 + 0.106930i
\(46\) 2.83644 + 4.91286i 0.418210 + 0.724361i
\(47\) −0.379189 − 0.318177i −0.0553104 − 0.0464109i 0.614713 − 0.788751i \(-0.289272\pi\)
−0.670024 + 0.742340i \(0.733716\pi\)
\(48\) −0.0572559 − 1.57199i −0.00826417 − 0.226897i
\(49\) 5.97606 + 3.64509i 0.853723 + 0.520727i
\(50\) −5.38979 − 1.96172i −0.762231 − 0.277429i
\(51\) 1.13662 − 0.243379i 0.159159 − 0.0340799i
\(52\) 0.798636 + 0.670135i 0.110751 + 0.0929310i
\(53\) −4.04846 + 7.01214i −0.556099 + 0.963191i 0.441718 + 0.897154i \(0.354369\pi\)
−0.997817 + 0.0660375i \(0.978964\pi\)
\(54\) 2.85311 − 4.27125i 0.388259 − 0.581243i
\(55\) −11.3291 −1.52761
\(56\) 5.65724 − 5.52250i 0.755981 − 0.737975i
\(57\) 3.59836 − 0.770497i 0.476614 − 0.102055i
\(58\) 0.670718 + 3.80383i 0.0880696 + 0.499468i
\(59\) −1.67822 − 9.51765i −0.218485 − 1.23909i −0.874755 − 0.484566i \(-0.838978\pi\)
0.656269 − 0.754527i \(-0.272134\pi\)
\(60\) −4.93323 + 3.09292i −0.636877 + 0.399295i
\(61\) 6.71058 + 5.63085i 0.859202 + 0.720956i 0.961796 − 0.273767i \(-0.0882698\pi\)
−0.102594 + 0.994723i \(0.532714\pi\)
\(62\) −5.10438 + 8.84104i −0.648256 + 1.12281i
\(63\) 7.85045 − 1.17065i 0.989064 − 0.147488i
\(64\) −3.41829 − 5.92066i −0.427287 − 0.740082i
\(65\) −0.581732 + 3.29916i −0.0721549 + 0.409211i
\(66\) 3.62653 − 4.65613i 0.446395 − 0.573130i
\(67\) −9.15746 − 3.33304i −1.11876 − 0.407196i −0.284562 − 0.958658i \(-0.591848\pi\)
−0.834200 + 0.551462i \(0.814070\pi\)
\(68\) 0.525830 − 0.441223i 0.0637662 − 0.0535062i
\(69\) 3.73733 + 9.21042i 0.449922 + 1.10880i
\(70\) 8.27563 + 2.32470i 0.989127 + 0.277855i
\(71\) 4.72319 + 8.18081i 0.560540 + 0.970883i 0.997449 + 0.0713778i \(0.0227396\pi\)
−0.436910 + 0.899505i \(0.643927\pi\)
\(72\) 0.910285 − 8.91805i 0.107278 − 1.05100i
\(73\) −5.56377 − 9.63672i −0.651189 − 1.12789i −0.982835 − 0.184489i \(-0.940937\pi\)
0.331645 − 0.943404i \(-0.392396\pi\)
\(74\) −1.56470 + 8.87388i −0.181893 + 1.03157i
\(75\) −8.88056 − 4.70481i −1.02544 − 0.543264i
\(76\) 1.66469 − 1.39684i 0.190953 − 0.160228i
\(77\) 9.09404 − 0.685297i 1.03636 − 0.0780968i
\(78\) −1.16970 − 1.29517i −0.132443 − 0.146650i
\(79\) 0.194091 − 0.0706432i 0.0218369 − 0.00794798i −0.331079 − 0.943603i \(-0.607413\pi\)
0.352916 + 0.935655i \(0.385190\pi\)
\(80\) 1.49247 − 2.58503i 0.166863 − 0.289015i
\(81\) 5.98194 − 6.72431i 0.664660 − 0.747146i
\(82\) −2.44206 − 4.22978i −0.269681 − 0.467101i
\(83\) 2.37994 − 13.4973i 0.261232 − 1.48152i −0.518321 − 0.855186i \(-0.673443\pi\)
0.779553 − 0.626336i \(-0.215446\pi\)
\(84\) 3.77289 − 2.78115i 0.411656 − 0.303448i
\(85\) 2.07269 + 0.754398i 0.224815 + 0.0818259i
\(86\) 2.62008 + 0.953630i 0.282530 + 0.102833i
\(87\) 0.246335 + 6.76326i 0.0264099 + 0.725097i
\(88\) 1.78857 − 10.1435i 0.190663 − 1.08130i
\(89\) 11.9916 1.27111 0.635553 − 0.772057i \(-0.280772\pi\)
0.635553 + 0.772057i \(0.280772\pi\)
\(90\) 8.89229 − 3.99111i 0.937330 − 0.420700i
\(91\) 0.267399 − 2.68348i 0.0280310 − 0.281305i
\(92\) 4.49646 + 3.77298i 0.468788 + 0.393360i
\(93\) −10.9914 + 14.1119i −1.13976 + 1.46334i
\(94\) 0.459806 + 0.167356i 0.0474253 + 0.0172614i
\(95\) 6.56178 + 2.38829i 0.673225 + 0.245034i
\(96\) −3.30735 − 8.15075i −0.337555 − 0.831882i
\(97\) 0.577101 + 0.484245i 0.0585957 + 0.0491676i 0.671615 − 0.740900i \(-0.265601\pi\)
−0.613019 + 0.790068i \(0.710045\pi\)
\(98\) −6.78360 − 1.36548i −0.685247 − 0.137934i
\(99\) 7.41704 − 7.20568i 0.745441 − 0.724198i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 189.2.w.a.25.8 yes 132
3.2 odd 2 567.2.w.a.235.15 132
7.2 even 3 189.2.u.a.79.15 yes 132
21.2 odd 6 567.2.u.a.478.8 132
27.13 even 9 189.2.u.a.67.15 ✓ 132
27.14 odd 18 567.2.u.a.172.8 132
189.121 even 9 inner 189.2.w.a.121.8 yes 132
189.149 odd 18 567.2.w.a.415.15 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.u.a.67.15 ✓ 132 27.13 even 9
189.2.u.a.79.15 yes 132 7.2 even 3
189.2.w.a.25.8 yes 132 1.1 even 1 trivial
189.2.w.a.121.8 yes 132 189.121 even 9 inner
567.2.u.a.172.8 132 27.14 odd 18
567.2.u.a.478.8 132 21.2 odd 6
567.2.w.a.235.15 132 3.2 odd 2
567.2.w.a.415.15 132 189.149 odd 18