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Properties

Label 189.2.f.a.64.2
Level $189$
Weight $2$
Character 189.64
Analytic conductor $1.509$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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(64,189)] 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("189.64"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(189, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 64.2
Root \(0.500000 + 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 189.64
Dual form 189.2.f.a.127.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.119562 - 0.207087i) q^{2} +(0.971410 - 1.68253i) q^{4} +(0.590972 - 1.02359i) q^{5} +(0.500000 + 0.866025i) q^{7} -0.942820 q^{8} -0.282630 q^{10} +(-1.85185 - 3.20750i) q^{11} +(-0.500000 + 0.866025i) q^{13} +(0.119562 - 0.207087i) q^{14} +(-1.83009 - 3.16982i) q^{16} +6.94282 q^{17} +1.94282 q^{19} +(-1.14815 - 1.98866i) q^{20} +(-0.442820 + 0.766987i) q^{22} +(-2.80150 + 4.85235i) q^{23} +(1.80150 + 3.12030i) q^{25} +0.239123 q^{26} +1.94282 q^{28} +(0.119562 + 0.207087i) q^{29} +(-0.830095 + 1.43777i) q^{31} +(-1.38044 + 2.39099i) q^{32} +(-0.830095 - 1.43777i) q^{34} +1.18194 q^{35} -9.54583 q^{37} +(-0.232287 - 0.402332i) q^{38} +(-0.557180 + 0.965064i) q^{40} +(-5.09097 + 8.81782i) q^{41} +(-1.11273 - 1.92730i) q^{43} -7.19562 q^{44} +1.33981 q^{46} +(2.91423 + 5.04759i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(0.430782 - 0.746136i) q^{50} +(0.971410 + 1.68253i) q^{52} +11.6030 q^{53} -4.37756 q^{55} +(-0.471410 - 0.816506i) q^{56} +(0.0285900 - 0.0495193i) q^{58} +(1.30150 - 2.25427i) q^{59} +(3.80150 + 6.58440i) q^{61} +0.396990 q^{62} -6.66019 q^{64} +(0.590972 + 1.02359i) q^{65} +(-1.75404 + 3.03809i) q^{67} +(6.74433 - 11.6815i) q^{68} +(-0.141315 - 0.244765i) q^{70} -8.60301 q^{71} +15.1488 q^{73} +(1.14132 + 1.97682i) q^{74} +(1.88727 - 3.26886i) q^{76} +(1.85185 - 3.20750i) q^{77} +(-3.68878 - 6.38915i) q^{79} -4.32614 q^{80} +2.43474 q^{82} +(-3.47141 - 6.01266i) q^{83} +(4.10301 - 7.10662i) q^{85} +(-0.266078 + 0.460861i) q^{86} +(1.74596 + 3.02409i) q^{88} -2.74720 q^{89} -1.00000 q^{91} +(5.44282 + 9.42724i) q^{92} +(0.696860 - 1.20700i) q^{94} +(1.14815 - 1.98866i) q^{95} +(-3.58414 - 6.20790i) q^{97} +0.239123 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - 3 q^{4} - 5 q^{5} + 3 q^{7} + 12 q^{8} - 2 q^{11} - 3 q^{13} + q^{14} - 3 q^{16} + 24 q^{17} - 6 q^{19} - 16 q^{20} + 15 q^{22} - 6 q^{25} + 2 q^{26} - 6 q^{28} + q^{29} + 3 q^{31} - 8 q^{32}+ \cdots + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.119562 − 0.207087i −0.0845428 − 0.146433i 0.820653 − 0.571426i \(-0.193610\pi\)
−0.905196 + 0.424994i \(0.860276\pi\)
\(3\) 0 0
\(4\) 0.971410 − 1.68253i 0.485705 − 0.841266i
\(5\) 0.590972 − 1.02359i 0.264291 − 0.457765i −0.703087 − 0.711104i \(-0.748196\pi\)
0.967378 + 0.253339i \(0.0815289\pi\)
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i
\(8\) −0.942820 −0.333337
\(9\) 0 0
\(10\) −0.282630 −0.0893755
\(11\) −1.85185 − 3.20750i −0.558353 − 0.967096i −0.997634 − 0.0687465i \(-0.978100\pi\)
0.439281 − 0.898350i \(-0.355233\pi\)
\(12\) 0 0
\(13\) −0.500000 + 0.866025i −0.138675 + 0.240192i −0.926995 − 0.375073i \(-0.877618\pi\)
0.788320 + 0.615265i \(0.210951\pi\)
\(14\) 0.119562 − 0.207087i 0.0319542 − 0.0553463i
\(15\) 0 0
\(16\) −1.83009 − 3.16982i −0.457524 − 0.792454i
\(17\) 6.94282 1.68388 0.841941 − 0.539570i \(-0.181413\pi\)
0.841941 + 0.539570i \(0.181413\pi\)
\(18\) 0 0
\(19\) 1.94282 0.445713 0.222857 − 0.974851i \(-0.428462\pi\)
0.222857 + 0.974851i \(0.428462\pi\)
\(20\) −1.14815 − 1.98866i −0.256735 − 0.444677i
\(21\) 0 0
\(22\) −0.442820 + 0.766987i −0.0944096 + 0.163522i
\(23\) −2.80150 + 4.85235i −0.584154 + 1.01178i 0.410826 + 0.911714i \(0.365240\pi\)
−0.994980 + 0.100071i \(0.968093\pi\)
\(24\) 0 0
\(25\) 1.80150 + 3.12030i 0.360301 + 0.624060i
\(26\) 0.239123 0.0468959
\(27\) 0 0
\(28\) 1.94282 0.367158
\(29\) 0.119562 + 0.207087i 0.0222020 + 0.0384551i 0.876913 − 0.480649i \(-0.159599\pi\)
−0.854711 + 0.519104i \(0.826266\pi\)
\(30\) 0 0
\(31\) −0.830095 + 1.43777i −0.149089 + 0.258231i −0.930891 − 0.365297i \(-0.880968\pi\)
0.781802 + 0.623527i \(0.214301\pi\)
\(32\) −1.38044 + 2.39099i −0.244029 + 0.422671i
\(33\) 0 0
\(34\) −0.830095 − 1.43777i −0.142360 − 0.246575i
\(35\) 1.18194 0.199785
\(36\) 0 0
\(37\) −9.54583 −1.56932 −0.784662 − 0.619923i \(-0.787164\pi\)
−0.784662 + 0.619923i \(0.787164\pi\)
\(38\) −0.232287 − 0.402332i −0.0376819 − 0.0652669i
\(39\) 0 0
\(40\) −0.557180 + 0.965064i −0.0880979 + 0.152590i
\(41\) −5.09097 + 8.81782i −0.795076 + 1.37711i 0.127715 + 0.991811i \(0.459236\pi\)
−0.922791 + 0.385301i \(0.874097\pi\)
\(42\) 0 0
\(43\) −1.11273 − 1.92730i −0.169689 − 0.293910i 0.768622 − 0.639704i \(-0.220943\pi\)
−0.938311 + 0.345794i \(0.887610\pi\)
\(44\) −7.19562 −1.08478
\(45\) 0 0
\(46\) 1.33981 0.197544
\(47\) 2.91423 + 5.04759i 0.425084 + 0.736267i 0.996428 − 0.0844432i \(-0.0269112\pi\)
−0.571344 + 0.820711i \(0.693578\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) 0.430782 − 0.746136i 0.0609217 − 0.105520i
\(51\) 0 0
\(52\) 0.971410 + 1.68253i 0.134710 + 0.233325i
\(53\) 11.6030 1.59380 0.796898 − 0.604114i \(-0.206473\pi\)
0.796898 + 0.604114i \(0.206473\pi\)
\(54\) 0 0
\(55\) −4.37756 −0.590270
\(56\) −0.471410 − 0.816506i −0.0629948 − 0.109110i
\(57\) 0 0
\(58\) 0.0285900 − 0.0495193i 0.00375405 − 0.00650220i
\(59\) 1.30150 − 2.25427i 0.169442 − 0.293481i −0.768782 − 0.639511i \(-0.779137\pi\)
0.938224 + 0.346029i \(0.112470\pi\)
\(60\) 0 0
\(61\) 3.80150 + 6.58440i 0.486733 + 0.843046i 0.999884 − 0.0152524i \(-0.00485519\pi\)
−0.513151 + 0.858298i \(0.671522\pi\)
\(62\) 0.396990 0.0504178
\(63\) 0 0
\(64\) −6.66019 −0.832524
\(65\) 0.590972 + 1.02359i 0.0733010 + 0.126961i
\(66\) 0 0
\(67\) −1.75404 + 3.03809i −0.214290 + 0.371161i −0.953053 − 0.302804i \(-0.902077\pi\)
0.738763 + 0.673966i \(0.235410\pi\)
\(68\) 6.74433 − 11.6815i 0.817870 − 1.41659i
\(69\) 0 0
\(70\) −0.141315 − 0.244765i −0.0168904 − 0.0292550i
\(71\) −8.60301 −1.02099 −0.510495 − 0.859881i \(-0.670538\pi\)
−0.510495 + 0.859881i \(0.670538\pi\)
\(72\) 0 0
\(73\) 15.1488 1.77304 0.886519 − 0.462693i \(-0.153117\pi\)
0.886519 + 0.462693i \(0.153117\pi\)
\(74\) 1.14132 + 1.97682i 0.132675 + 0.229800i
\(75\) 0 0
\(76\) 1.88727 − 3.26886i 0.216485 − 0.374963i
\(77\) 1.85185 − 3.20750i 0.211038 − 0.365528i
\(78\) 0 0
\(79\) −3.68878 − 6.38915i −0.415020 − 0.718836i 0.580410 − 0.814324i \(-0.302892\pi\)
−0.995431 + 0.0954881i \(0.969559\pi\)
\(80\) −4.32614 −0.483677
\(81\) 0 0
\(82\) 2.43474 0.268872
\(83\) −3.47141 − 6.01266i −0.381037 − 0.659975i 0.610174 − 0.792267i \(-0.291100\pi\)
−0.991211 + 0.132292i \(0.957766\pi\)
\(84\) 0 0
\(85\) 4.10301 − 7.10662i 0.445034 − 0.770821i
\(86\) −0.266078 + 0.460861i −0.0286920 + 0.0496960i
\(87\) 0 0
\(88\) 1.74596 + 3.02409i 0.186120 + 0.322369i
\(89\) −2.74720 −0.291203 −0.145602 − 0.989343i \(-0.546512\pi\)
−0.145602 + 0.989343i \(0.546512\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 5.44282 + 9.42724i 0.567453 + 0.982858i
\(93\) 0 0
\(94\) 0.696860 − 1.20700i 0.0718756 − 0.124492i
\(95\) 1.14815 − 1.98866i 0.117798 − 0.204032i
\(96\) 0 0
\(97\) −3.58414 − 6.20790i −0.363914 − 0.630317i 0.624687 − 0.780875i \(-0.285226\pi\)
−0.988601 + 0.150558i \(0.951893\pi\)
\(98\) 0.239123 0.0241551
\(99\) 0 0
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.f.a.64.2 6
3.2 odd 2 63.2.f.b.22.2 ✓ 6
4.3 odd 2 3024.2.r.g.1009.3 6
7.2 even 3 1323.2.g.c.361.2 6
7.3 odd 6 1323.2.h.e.226.2 6
7.4 even 3 1323.2.h.d.226.2 6
7.5 odd 6 1323.2.g.b.361.2 6
7.6 odd 2 1323.2.f.c.442.2 6
9.2 odd 6 63.2.f.b.43.2 yes 6
9.4 even 3 567.2.a.g.1.2 3
9.5 odd 6 567.2.a.d.1.2 3
9.7 even 3 inner 189.2.f.a.127.2 6
12.11 even 2 1008.2.r.k.337.2 6
21.2 odd 6 441.2.g.e.67.2 6
21.5 even 6 441.2.g.d.67.2 6
21.11 odd 6 441.2.h.c.373.2 6
21.17 even 6 441.2.h.b.373.2 6
21.20 even 2 441.2.f.d.148.2 6
36.7 odd 6 3024.2.r.g.2017.3 6
36.11 even 6 1008.2.r.k.673.2 6
36.23 even 6 9072.2.a.bq.1.3 3
36.31 odd 6 9072.2.a.cd.1.1 3
63.2 odd 6 441.2.h.c.214.2 6
63.11 odd 6 441.2.g.e.79.2 6
63.13 odd 6 3969.2.a.p.1.2 3
63.16 even 3 1323.2.h.d.802.2 6
63.20 even 6 441.2.f.d.295.2 6
63.25 even 3 1323.2.g.c.667.2 6
63.34 odd 6 1323.2.f.c.883.2 6
63.38 even 6 441.2.g.d.79.2 6
63.41 even 6 3969.2.a.m.1.2 3
63.47 even 6 441.2.h.b.214.2 6
63.52 odd 6 1323.2.g.b.667.2 6
63.61 odd 6 1323.2.h.e.802.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.2 ✓ 6 3.2 odd 2
63.2.f.b.43.2 yes 6 9.2 odd 6
189.2.f.a.64.2 6 1.1 even 1 trivial
189.2.f.a.127.2 6 9.7 even 3 inner
441.2.f.d.148.2 6 21.20 even 2
441.2.f.d.295.2 6 63.20 even 6
441.2.g.d.67.2 6 21.5 even 6
441.2.g.d.79.2 6 63.38 even 6
441.2.g.e.67.2 6 21.2 odd 6
441.2.g.e.79.2 6 63.11 odd 6
441.2.h.b.214.2 6 63.47 even 6
441.2.h.b.373.2 6 21.17 even 6
441.2.h.c.214.2 6 63.2 odd 6
441.2.h.c.373.2 6 21.11 odd 6
567.2.a.d.1.2 3 9.5 odd 6
567.2.a.g.1.2 3 9.4 even 3
1008.2.r.k.337.2 6 12.11 even 2
1008.2.r.k.673.2 6 36.11 even 6
1323.2.f.c.442.2 6 7.6 odd 2
1323.2.f.c.883.2 6 63.34 odd 6
1323.2.g.b.361.2 6 7.5 odd 6
1323.2.g.b.667.2 6 63.52 odd 6
1323.2.g.c.361.2 6 7.2 even 3
1323.2.g.c.667.2 6 63.25 even 3
1323.2.h.d.226.2 6 7.4 even 3
1323.2.h.d.802.2 6 63.16 even 3
1323.2.h.e.226.2 6 7.3 odd 6
1323.2.h.e.802.2 6 63.61 odd 6
3024.2.r.g.1009.3 6 4.3 odd 2
3024.2.r.g.2017.3 6 36.7 odd 6
3969.2.a.m.1.2 3 63.41 even 6
3969.2.a.p.1.2 3 63.13 odd 6
9072.2.a.bq.1.3 3 36.23 even 6
9072.2.a.cd.1.1 3 36.31 odd 6