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Properties

Label 189.2.f.b.64.1
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 = [6,3] 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: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) 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 64.1
Root \(0.939693 + 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 189.64
Dual form 189.2.f.b.127.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.439693 - 0.761570i) q^{2} +(0.613341 - 1.06234i) q^{4} +(0.673648 - 1.16679i) q^{5} +(-0.500000 - 0.866025i) q^{7} -2.83750 q^{8} -1.18479 q^{10} +(0.826352 + 1.43128i) q^{11} +(1.68479 - 2.91815i) q^{13} +(-0.439693 + 0.761570i) q^{14} +(0.0209445 + 0.0362770i) q^{16} -0.467911 q^{17} -3.22668 q^{19} +(-0.826352 - 1.43128i) q^{20} +(0.726682 - 1.25865i) q^{22} +(4.47178 - 7.74535i) q^{23} +(1.59240 + 2.75811i) q^{25} -2.96316 q^{26} -1.22668 q^{28} +(3.13429 + 5.42874i) q^{29} +(-4.61721 + 7.99724i) q^{31} +(-2.81908 + 4.88279i) q^{32} +(0.205737 + 0.356347i) q^{34} -1.34730 q^{35} +9.23442 q^{37} +(1.41875 + 2.45734i) q^{38} +(-1.91147 + 3.31077i) q^{40} +(1.70574 - 2.95442i) q^{41} +(2.20574 + 3.82045i) q^{43} +2.02734 q^{44} -7.86484 q^{46} +(4.67752 + 8.10170i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(1.40033 - 2.42544i) q^{50} +(-2.06670 - 3.57964i) q^{52} +0.573978 q^{53} +2.22668 q^{55} +(1.41875 + 2.45734i) q^{56} +(2.75624 - 4.77396i) q^{58} +(-5.19846 + 9.00400i) q^{59} +(-3.81908 - 6.61484i) q^{61} +8.12061 q^{62} +5.04189 q^{64} +(-2.26991 - 3.93161i) q^{65} +(-0.298133 + 0.516382i) q^{67} +(-0.286989 + 0.497079i) q^{68} +(0.592396 + 1.02606i) q^{70} +0.554378 q^{71} +2.04963 q^{73} +(-4.06031 - 7.03266i) q^{74} +(-1.97906 + 3.42782i) q^{76} +(0.826352 - 1.43128i) q^{77} +(1.20187 + 2.08169i) q^{79} +0.0564370 q^{80} -3.00000 q^{82} +(-7.52481 - 13.0334i) q^{83} +(-0.315207 + 0.545955i) q^{85} +(1.93969 - 3.35965i) q^{86} +(-2.34477 - 4.06126i) q^{88} -9.08647 q^{89} -3.36959 q^{91} +(-5.48545 - 9.50108i) q^{92} +(4.11334 - 7.12452i) q^{94} +(-2.17365 + 3.76487i) q^{95} +(0.949493 + 1.64457i) q^{97} +0.879385 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 3 q^{4} + 3 q^{5} - 3 q^{7} - 12 q^{8} + 6 q^{11} + 3 q^{13} + 3 q^{14} - 3 q^{16} - 12 q^{17} - 6 q^{19} - 6 q^{20} - 9 q^{22} + 12 q^{23} + 6 q^{25} + 6 q^{26} + 6 q^{28} + 9 q^{29} + 3 q^{31}+ \cdots - 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.439693 − 0.761570i −0.310910 − 0.538511i 0.667650 − 0.744475i \(-0.267300\pi\)
−0.978560 + 0.205964i \(0.933967\pi\)
\(3\) 0 0
\(4\) 0.613341 − 1.06234i 0.306670 − 0.531169i
\(5\) 0.673648 − 1.16679i 0.301265 − 0.521806i −0.675158 − 0.737673i \(-0.735925\pi\)
0.976423 + 0.215867i \(0.0692579\pi\)
\(6\) 0 0
\(7\) −0.500000 − 0.866025i −0.188982 − 0.327327i
\(8\) −2.83750 −1.00321
\(9\) 0 0
\(10\) −1.18479 −0.374664
\(11\) 0.826352 + 1.43128i 0.249154 + 0.431548i 0.963291 − 0.268458i \(-0.0865140\pi\)
−0.714137 + 0.700006i \(0.753181\pi\)
\(12\) 0 0
\(13\) 1.68479 − 2.91815i 0.467277 − 0.809348i −0.532024 − 0.846729i \(-0.678568\pi\)
0.999301 + 0.0373813i \(0.0119016\pi\)
\(14\) −0.439693 + 0.761570i −0.117513 + 0.203538i
\(15\) 0 0
\(16\) 0.0209445 + 0.0362770i 0.00523613 + 0.00906925i
\(17\) −0.467911 −0.113485 −0.0567426 − 0.998389i \(-0.518071\pi\)
−0.0567426 + 0.998389i \(0.518071\pi\)
\(18\) 0 0
\(19\) −3.22668 −0.740252 −0.370126 − 0.928982i \(-0.620685\pi\)
−0.370126 + 0.928982i \(0.620685\pi\)
\(20\) −0.826352 − 1.43128i −0.184778 − 0.320045i
\(21\) 0 0
\(22\) 0.726682 − 1.25865i 0.154929 − 0.268345i
\(23\) 4.47178 − 7.74535i 0.932431 − 1.61502i 0.153279 − 0.988183i \(-0.451017\pi\)
0.779152 − 0.626835i \(-0.215650\pi\)
\(24\) 0 0
\(25\) 1.59240 + 2.75811i 0.318479 + 0.551622i
\(26\) −2.96316 −0.581124
\(27\) 0 0
\(28\) −1.22668 −0.231821
\(29\) 3.13429 + 5.42874i 0.582022 + 1.00809i 0.995239 + 0.0974595i \(0.0310717\pi\)
−0.413217 + 0.910632i \(0.635595\pi\)
\(30\) 0 0
\(31\) −4.61721 + 7.99724i −0.829276 + 1.43635i 0.0693317 + 0.997594i \(0.477913\pi\)
−0.898607 + 0.438754i \(0.855420\pi\)
\(32\) −2.81908 + 4.88279i −0.498347 + 0.863163i
\(33\) 0 0
\(34\) 0.205737 + 0.356347i 0.0352836 + 0.0611130i
\(35\) −1.34730 −0.227735
\(36\) 0 0
\(37\) 9.23442 1.51813 0.759065 − 0.651015i \(-0.225657\pi\)
0.759065 + 0.651015i \(0.225657\pi\)
\(38\) 1.41875 + 2.45734i 0.230151 + 0.398634i
\(39\) 0 0
\(40\) −1.91147 + 3.31077i −0.302231 + 0.523479i
\(41\) 1.70574 − 2.95442i 0.266391 − 0.461403i −0.701536 − 0.712634i \(-0.747502\pi\)
0.967927 + 0.251231i \(0.0808353\pi\)
\(42\) 0 0
\(43\) 2.20574 + 3.82045i 0.336372 + 0.582613i 0.983747 − 0.179558i \(-0.0574668\pi\)
−0.647376 + 0.762171i \(0.724133\pi\)
\(44\) 2.02734 0.305633
\(45\) 0 0
\(46\) −7.86484 −1.15961
\(47\) 4.67752 + 8.10170i 0.682286 + 1.18175i 0.974281 + 0.225335i \(0.0723475\pi\)
−0.291995 + 0.956420i \(0.594319\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) 1.40033 − 2.42544i 0.198037 − 0.343009i
\(51\) 0 0
\(52\) −2.06670 − 3.57964i −0.286600 − 0.496406i
\(53\) 0.573978 0.0788419 0.0394210 − 0.999223i \(-0.487449\pi\)
0.0394210 + 0.999223i \(0.487449\pi\)
\(54\) 0 0
\(55\) 2.22668 0.300246
\(56\) 1.41875 + 2.45734i 0.189588 + 0.328376i
\(57\) 0 0
\(58\) 2.75624 − 4.77396i 0.361913 − 0.626851i
\(59\) −5.19846 + 9.00400i −0.676782 + 1.17222i 0.299162 + 0.954202i \(0.403293\pi\)
−0.975945 + 0.218019i \(0.930041\pi\)
\(60\) 0 0
\(61\) −3.81908 − 6.61484i −0.488983 − 0.846943i 0.510937 − 0.859618i \(-0.329299\pi\)
−0.999920 + 0.0126752i \(0.995965\pi\)
\(62\) 8.12061 1.03132
\(63\) 0 0
\(64\) 5.04189 0.630236
\(65\) −2.26991 − 3.93161i −0.281548 − 0.487656i
\(66\) 0 0
\(67\) −0.298133 + 0.516382i −0.0364228 + 0.0630861i −0.883662 − 0.468125i \(-0.844930\pi\)
0.847239 + 0.531211i \(0.178263\pi\)
\(68\) −0.286989 + 0.497079i −0.0348025 + 0.0602797i
\(69\) 0 0
\(70\) 0.592396 + 1.02606i 0.0708049 + 0.122638i
\(71\) 0.554378 0.0657925 0.0328963 − 0.999459i \(-0.489527\pi\)
0.0328963 + 0.999459i \(0.489527\pi\)
\(72\) 0 0
\(73\) 2.04963 0.239891 0.119946 − 0.992780i \(-0.461728\pi\)
0.119946 + 0.992780i \(0.461728\pi\)
\(74\) −4.06031 − 7.03266i −0.472001 − 0.817530i
\(75\) 0 0
\(76\) −1.97906 + 3.42782i −0.227013 + 0.393198i
\(77\) 0.826352 − 1.43128i 0.0941715 − 0.163110i
\(78\) 0 0
\(79\) 1.20187 + 2.08169i 0.135221 + 0.234209i 0.925682 − 0.378303i \(-0.123492\pi\)
−0.790461 + 0.612512i \(0.790159\pi\)
\(80\) 0.0564370 0.00630985
\(81\) 0 0
\(82\) −3.00000 −0.331295
\(83\) −7.52481 − 13.0334i −0.825956 − 1.43060i −0.901187 − 0.433431i \(-0.857303\pi\)
0.0752309 − 0.997166i \(-0.476031\pi\)
\(84\) 0 0
\(85\) −0.315207 + 0.545955i −0.0341891 + 0.0592172i
\(86\) 1.93969 − 3.35965i 0.209162 − 0.362280i
\(87\) 0 0
\(88\) −2.34477 − 4.06126i −0.249953 − 0.432932i
\(89\) −9.08647 −0.963164 −0.481582 − 0.876401i \(-0.659938\pi\)
−0.481582 + 0.876401i \(0.659938\pi\)
\(90\) 0 0
\(91\) −3.36959 −0.353228
\(92\) −5.48545 − 9.50108i −0.571898 − 0.990556i
\(93\) 0 0
\(94\) 4.11334 − 7.12452i 0.424259 − 0.734838i
\(95\) −2.17365 + 3.76487i −0.223012 + 0.386267i
\(96\) 0 0
\(97\) 0.949493 + 1.64457i 0.0964064 + 0.166981i 0.910195 − 0.414181i \(-0.135932\pi\)
−0.813788 + 0.581161i \(0.802598\pi\)
\(98\) 0.879385 0.0888313
\(99\) 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.b.64.1 6
3.2 odd 2 63.2.f.a.22.3 ✓ 6
4.3 odd 2 3024.2.r.k.1009.2 6
7.2 even 3 1323.2.g.d.361.1 6
7.3 odd 6 1323.2.h.b.226.3 6
7.4 even 3 1323.2.h.c.226.3 6
7.5 odd 6 1323.2.g.e.361.1 6
7.6 odd 2 1323.2.f.d.442.1 6
9.2 odd 6 63.2.f.a.43.3 yes 6
9.4 even 3 567.2.a.c.1.3 3
9.5 odd 6 567.2.a.h.1.1 3
9.7 even 3 inner 189.2.f.b.127.1 6
12.11 even 2 1008.2.r.h.337.2 6
21.2 odd 6 441.2.g.c.67.3 6
21.5 even 6 441.2.g.b.67.3 6
21.11 odd 6 441.2.h.d.373.1 6
21.17 even 6 441.2.h.e.373.1 6
21.20 even 2 441.2.f.c.148.3 6
36.7 odd 6 3024.2.r.k.2017.2 6
36.11 even 6 1008.2.r.h.673.2 6
36.23 even 6 9072.2.a.ca.1.2 3
36.31 odd 6 9072.2.a.bs.1.2 3
63.2 odd 6 441.2.h.d.214.1 6
63.11 odd 6 441.2.g.c.79.3 6
63.13 odd 6 3969.2.a.l.1.3 3
63.16 even 3 1323.2.h.c.802.3 6
63.20 even 6 441.2.f.c.295.3 6
63.25 even 3 1323.2.g.d.667.1 6
63.34 odd 6 1323.2.f.d.883.1 6
63.38 even 6 441.2.g.b.79.3 6
63.41 even 6 3969.2.a.q.1.1 3
63.47 even 6 441.2.h.e.214.1 6
63.52 odd 6 1323.2.g.e.667.1 6
63.61 odd 6 1323.2.h.b.802.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.3 ✓ 6 3.2 odd 2
63.2.f.a.43.3 yes 6 9.2 odd 6
189.2.f.b.64.1 6 1.1 even 1 trivial
189.2.f.b.127.1 6 9.7 even 3 inner
441.2.f.c.148.3 6 21.20 even 2
441.2.f.c.295.3 6 63.20 even 6
441.2.g.b.67.3 6 21.5 even 6
441.2.g.b.79.3 6 63.38 even 6
441.2.g.c.67.3 6 21.2 odd 6
441.2.g.c.79.3 6 63.11 odd 6
441.2.h.d.214.1 6 63.2 odd 6
441.2.h.d.373.1 6 21.11 odd 6
441.2.h.e.214.1 6 63.47 even 6
441.2.h.e.373.1 6 21.17 even 6
567.2.a.c.1.3 3 9.4 even 3
567.2.a.h.1.1 3 9.5 odd 6
1008.2.r.h.337.2 6 12.11 even 2
1008.2.r.h.673.2 6 36.11 even 6
1323.2.f.d.442.1 6 7.6 odd 2
1323.2.f.d.883.1 6 63.34 odd 6
1323.2.g.d.361.1 6 7.2 even 3
1323.2.g.d.667.1 6 63.25 even 3
1323.2.g.e.361.1 6 7.5 odd 6
1323.2.g.e.667.1 6 63.52 odd 6
1323.2.h.b.226.3 6 7.3 odd 6
1323.2.h.b.802.3 6 63.61 odd 6
1323.2.h.c.226.3 6 7.4 even 3
1323.2.h.c.802.3 6 63.16 even 3
3024.2.r.k.1009.2 6 4.3 odd 2
3024.2.r.k.2017.2 6 36.7 odd 6
3969.2.a.l.1.3 3 63.13 odd 6
3969.2.a.q.1.1 3 63.41 even 6
9072.2.a.bs.1.2 3 36.31 odd 6
9072.2.a.ca.1.2 3 36.23 even 6