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Properties

Label 171.2.g.c.106.6
Level $171$
Weight $2$
Character 171.106
Analytic conductor $1.365$
Analytic rank $0$
Dimension $32$
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] = [171,2,Mod(106,171)] 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("171.106"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 106.6
Character \(\chi\) \(=\) 171.106
Dual form 171.2.g.c.121.6

$q$-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+(-0.395929 - 0.685769i) q^{2} +(0.659141 + 1.60173i) q^{3} +(0.686481 - 1.18902i) q^{4} +2.59093 q^{5} +(0.837442 - 1.08619i) q^{6} +(-0.373088 + 0.646207i) q^{7} -2.67091 q^{8} +(-2.13107 + 2.11153i) q^{9} +(-1.02582 - 1.77678i) q^{10} +(-1.28837 + 2.23153i) q^{11} +(2.35697 + 0.315823i) q^{12} +(3.09365 - 5.35835i) q^{13} +0.590865 q^{14} +(1.70779 + 4.14996i) q^{15} +(-0.315472 - 0.546414i) q^{16} +(0.119999 - 0.207845i) q^{17} +(2.29177 + 0.625402i) q^{18} +(3.89399 + 1.95878i) q^{19} +(1.77862 - 3.08066i) q^{20} +(-1.28097 - 0.171643i) q^{21} +2.04042 q^{22} +(-1.93131 + 3.34513i) q^{23} +(-1.76050 - 4.27806i) q^{24} +1.71290 q^{25} -4.89946 q^{26} +(-4.78677 - 2.02159i) q^{27} +(0.512235 + 0.887218i) q^{28} -6.79737 q^{29} +(2.16975 - 2.81424i) q^{30} +(-3.77423 - 6.53716i) q^{31} +(-2.92071 + 5.05883i) q^{32} +(-4.42353 - 0.592732i) q^{33} -0.190045 q^{34} +(-0.966644 + 1.67428i) q^{35} +(1.04772 + 3.98340i) q^{36} -8.47678 q^{37} +(-0.198475 - 3.44592i) q^{38} +(10.6218 + 1.42327i) q^{39} -6.92012 q^{40} +8.15194 q^{41} +(0.389464 + 0.946405i) q^{42} +(1.44011 + 2.49434i) q^{43} +(1.76889 + 3.06380i) q^{44} +(-5.52143 + 5.47082i) q^{45} +3.05865 q^{46} -4.52565 q^{47} +(0.667266 - 0.865465i) q^{48} +(3.22161 + 5.57999i) q^{49} +(-0.678188 - 1.17466i) q^{50} +(0.412007 + 0.0552070i) q^{51} +(-4.24746 - 7.35681i) q^{52} +(-5.57774 - 9.66094i) q^{53} +(0.508878 + 4.08302i) q^{54} +(-3.33808 + 5.78173i) q^{55} +(0.996483 - 1.72596i) q^{56} +(-0.570737 + 7.52823i) q^{57} +(2.69128 + 4.66143i) q^{58} +0.344246 q^{59} +(6.10675 + 0.818275i) q^{60} +0.0790199 q^{61} +(-2.98865 + 5.17650i) q^{62} +(-0.569412 - 2.16490i) q^{63} +3.36369 q^{64} +(8.01542 - 13.8831i) q^{65} +(1.34492 + 3.26820i) q^{66} +(-4.61385 + 7.99142i) q^{67} +(-0.164754 - 0.285363i) q^{68} +(-6.63100 - 0.888523i) q^{69} +1.53089 q^{70} +(2.15288 - 3.72891i) q^{71} +(5.69187 - 5.63970i) q^{72} +(1.63071 - 2.82448i) q^{73} +(3.35620 + 5.81312i) q^{74} +(1.12905 + 2.74361i) q^{75} +(5.00218 - 3.28537i) q^{76} +(-0.961354 - 1.66511i) q^{77} +(-3.22944 - 7.84760i) q^{78} +(3.57283 + 6.18833i) q^{79} +(-0.817366 - 1.41572i) q^{80} +(0.0828771 - 8.99962i) q^{81} +(-3.22759 - 5.59035i) q^{82} +(1.78498 - 3.09167i) q^{83} +(-1.08345 + 1.40526i) q^{84} +(0.310909 - 0.538510i) q^{85} +(1.14036 - 1.97516i) q^{86} +(-4.48043 - 10.8875i) q^{87} +(3.44113 - 5.96021i) q^{88} +(5.21555 + 9.03360i) q^{89} +(5.93782 + 1.62037i) q^{90} +(2.30841 + 3.99827i) q^{91} +(2.65162 + 4.59274i) q^{92} +(7.98300 - 10.3542i) q^{93} +(1.79184 + 3.10355i) q^{94} +(10.0891 + 5.07505i) q^{95} +(-10.0280 - 1.34371i) q^{96} +(1.20626 + 2.08930i) q^{97} +(2.55106 - 4.41856i) q^{98} +(-1.96633 - 7.47598i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9} - 8 q^{10} + 7 q^{11} - 3 q^{12} - 4 q^{13} - 2 q^{14} + q^{15} - 11 q^{16} - 7 q^{17} + 6 q^{18} + 7 q^{19} - 3 q^{20}+ \cdots - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.395929 − 0.685769i −0.279964 − 0.484912i 0.691411 − 0.722461i \(-0.256989\pi\)
−0.971375 + 0.237549i \(0.923656\pi\)
\(3\) 0.659141 + 1.60173i 0.380555 + 0.924758i
\(4\) 0.686481 − 1.18902i 0.343240 − 0.594510i
\(5\) 2.59093 1.15870 0.579349 − 0.815080i \(-0.303307\pi\)
0.579349 + 0.815080i \(0.303307\pi\)
\(6\) 0.837442 − 1.08619i 0.341884 − 0.443435i
\(7\) −0.373088 + 0.646207i −0.141014 + 0.244243i −0.927879 − 0.372882i \(-0.878370\pi\)
0.786865 + 0.617126i \(0.211703\pi\)
\(8\) −2.67091 −0.944308
\(9\) −2.13107 + 2.11153i −0.710355 + 0.703844i
\(10\) −1.02582 − 1.77678i −0.324394 − 0.561866i
\(11\) −1.28837 + 2.23153i −0.388460 + 0.672832i −0.992243 − 0.124317i \(-0.960326\pi\)
0.603783 + 0.797149i \(0.293659\pi\)
\(12\) 2.35697 + 0.315823i 0.680400 + 0.0911703i
\(13\) 3.09365 − 5.35835i 0.858023 − 1.48614i −0.0157882 − 0.999875i \(-0.505026\pi\)
0.873812 − 0.486265i \(-0.161641\pi\)
\(14\) 0.590865 0.157915
\(15\) 1.70779 + 4.14996i 0.440949 + 1.07152i
\(16\) −0.315472 − 0.546414i −0.0788681 − 0.136604i
\(17\) 0.119999 − 0.207845i 0.0291041 − 0.0504097i −0.851107 − 0.524993i \(-0.824068\pi\)
0.880211 + 0.474583i \(0.157401\pi\)
\(18\) 2.29177 + 0.625402i 0.540176 + 0.147409i
\(19\) 3.89399 + 1.95878i 0.893343 + 0.449375i
\(20\) 1.77862 − 3.08066i 0.397712 − 0.688857i
\(21\) −1.28097 − 0.171643i −0.279530 − 0.0374557i
\(22\) 2.04042 0.435019
\(23\) −1.93131 + 3.34513i −0.402707 + 0.697509i −0.994052 − 0.108910i \(-0.965264\pi\)
0.591345 + 0.806419i \(0.298597\pi\)
\(24\) −1.76050 − 4.27806i −0.359362 − 0.873256i
\(25\) 1.71290 0.342581
\(26\) −4.89946 −0.960863
\(27\) −4.78677 − 2.02159i −0.921215 − 0.389055i
\(28\) 0.512235 + 0.887218i 0.0968034 + 0.167668i
\(29\) −6.79737 −1.26224 −0.631120 − 0.775685i \(-0.717405\pi\)
−0.631120 + 0.775685i \(0.717405\pi\)
\(30\) 2.16975 − 2.81424i 0.396141 − 0.513807i
\(31\) −3.77423 − 6.53716i −0.677872 − 1.17411i −0.975621 − 0.219464i \(-0.929569\pi\)
0.297749 − 0.954644i \(-0.403764\pi\)
\(32\) −2.92071 + 5.05883i −0.516314 + 0.894283i
\(33\) −4.42353 − 0.592732i −0.770037 − 0.103181i
\(34\) −0.190045 −0.0325924
\(35\) −0.966644 + 1.67428i −0.163393 + 0.283004i
\(36\) 1.04772 + 3.98340i 0.174619 + 0.663900i
\(37\) −8.47678 −1.39357 −0.696787 − 0.717278i \(-0.745388\pi\)
−0.696787 + 0.717278i \(0.745388\pi\)
\(38\) −0.198475 − 3.44592i −0.0321970 − 0.559002i
\(39\) 10.6218 + 1.42327i 1.70085 + 0.227905i
\(40\) −6.92012 −1.09417
\(41\) 8.15194 1.27312 0.636559 − 0.771228i \(-0.280357\pi\)
0.636559 + 0.771228i \(0.280357\pi\)
\(42\) 0.389464 + 0.946405i 0.0600956 + 0.146034i
\(43\) 1.44011 + 2.49434i 0.219615 + 0.380384i 0.954690 − 0.297602i \(-0.0961867\pi\)
−0.735076 + 0.677985i \(0.762853\pi\)
\(44\) 1.76889 + 3.06380i 0.266670 + 0.461886i
\(45\) −5.52143 + 5.47082i −0.823087 + 0.815542i
\(46\) 3.05865 0.450974
\(47\) −4.52565 −0.660134 −0.330067 − 0.943957i \(-0.607071\pi\)
−0.330067 + 0.943957i \(0.607071\pi\)
\(48\) 0.667266 − 0.865465i 0.0963115 − 0.124919i
\(49\) 3.22161 + 5.57999i 0.460230 + 0.797142i
\(50\) −0.678188 − 1.17466i −0.0959103 − 0.166122i
\(51\) 0.412007 + 0.0552070i 0.0576925 + 0.00773053i
\(52\) −4.24746 − 7.35681i −0.589016 − 1.02021i
\(53\) −5.57774 − 9.66094i −0.766162 − 1.32703i −0.939630 − 0.342192i \(-0.888831\pi\)
0.173468 − 0.984840i \(-0.444503\pi\)
\(54\) 0.508878 + 4.08302i 0.0692495 + 0.555629i
\(55\) −3.33808 + 5.78173i −0.450107 + 0.779609i
\(56\) 0.996483 − 1.72596i 0.133161 − 0.230641i
\(57\) −0.570737 + 7.52823i −0.0755960 + 0.997139i
\(58\) 2.69128 + 4.66143i 0.353382 + 0.612075i
\(59\) 0.344246 0.0448170 0.0224085 − 0.999749i \(-0.492867\pi\)
0.0224085 + 0.999749i \(0.492867\pi\)
\(60\) 6.10675 + 0.818275i 0.788378 + 0.105639i
\(61\) 0.0790199 0.0101175 0.00505873 − 0.999987i \(-0.498390\pi\)
0.00505873 + 0.999987i \(0.498390\pi\)
\(62\) −2.98865 + 5.17650i −0.379559 + 0.657416i
\(63\) −0.569412 − 2.16490i −0.0717391 − 0.272751i
\(64\) 3.36369 0.420462
\(65\) 8.01542 − 13.8831i 0.994190 − 1.72199i
\(66\) 1.34492 + 3.26820i 0.165549 + 0.402287i
\(67\) −4.61385 + 7.99142i −0.563671 + 0.976307i 0.433501 + 0.901153i \(0.357278\pi\)
−0.997172 + 0.0751540i \(0.976055\pi\)
\(68\) −0.164754 − 0.285363i −0.0199794 − 0.0346053i
\(69\) −6.63100 − 0.888523i −0.798279 − 0.106966i
\(70\) 1.53089 0.182976
\(71\) 2.15288 − 3.72891i 0.255500 − 0.442540i −0.709531 − 0.704674i \(-0.751093\pi\)
0.965031 + 0.262135i \(0.0844264\pi\)
\(72\) 5.69187 − 5.63970i 0.670794 − 0.664645i
\(73\) 1.63071 − 2.82448i 0.190861 − 0.330580i −0.754675 − 0.656099i \(-0.772206\pi\)
0.945536 + 0.325518i \(0.105539\pi\)
\(74\) 3.35620 + 5.81312i 0.390151 + 0.675761i
\(75\) 1.12905 + 2.74361i 0.130371 + 0.316804i
\(76\) 5.00218 − 3.28537i 0.573789 − 0.376858i
\(77\) −0.961354 − 1.66511i −0.109556 − 0.189757i
\(78\) −3.22944 − 7.84760i −0.365662 − 0.888566i
\(79\) 3.57283 + 6.18833i 0.401975 + 0.696241i 0.993964 − 0.109705i \(-0.0349905\pi\)
−0.591989 + 0.805946i \(0.701657\pi\)
\(80\) −0.817366 − 1.41572i −0.0913843 − 0.158282i
\(81\) 0.0828771 − 8.99962i 0.00920857 − 0.999958i
\(82\) −3.22759 − 5.59035i −0.356427 − 0.617350i
\(83\) 1.78498 − 3.09167i 0.195927 − 0.339355i −0.751277 − 0.659987i \(-0.770562\pi\)
0.947204 + 0.320632i \(0.103895\pi\)
\(84\) −1.08345 + 1.40526i −0.118214 + 0.153327i
\(85\) 0.310909 − 0.538510i 0.0337228 − 0.0584096i
\(86\) 1.14036 − 1.97516i 0.122968 − 0.212987i
\(87\) −4.48043 − 10.8875i −0.480353 − 1.16727i
\(88\) 3.44113 − 5.96021i 0.366825 − 0.635360i
\(89\) 5.21555 + 9.03360i 0.552848 + 0.957560i 0.998068 + 0.0621388i \(0.0197921\pi\)
−0.445220 + 0.895421i \(0.646875\pi\)
\(90\) 5.93782 + 1.62037i 0.625901 + 0.170802i
\(91\) 2.30841 + 3.99827i 0.241987 + 0.419133i
\(92\) 2.65162 + 4.59274i 0.276450 + 0.478826i
\(93\) 7.98300 − 10.3542i 0.827798 − 1.07368i
\(94\) 1.79184 + 3.10355i 0.184814 + 0.320107i
\(95\) 10.0891 + 5.07505i 1.03512 + 0.520689i
\(96\) −10.0280 − 1.34371i −1.02348 − 0.137142i
\(97\) 1.20626 + 2.08930i 0.122477 + 0.212136i 0.920744 − 0.390168i \(-0.127583\pi\)
−0.798267 + 0.602304i \(0.794250\pi\)
\(98\) 2.55106 − 4.41856i 0.257696 − 0.446342i
\(99\) −1.96633 − 7.47598i −0.197624 − 0.751364i
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 171.2.g.c.106.6 ✓ 32
3.2 odd 2 513.2.g.c.505.11 32
9.4 even 3 171.2.h.c.49.11 yes 32
9.5 odd 6 513.2.h.c.334.6 32
19.7 even 3 171.2.h.c.7.11 yes 32
57.26 odd 6 513.2.h.c.235.6 32
171.121 even 3 inner 171.2.g.c.121.6 yes 32
171.140 odd 6 513.2.g.c.64.11 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.g.c.106.6 ✓ 32 1.1 even 1 trivial
171.2.g.c.121.6 yes 32 171.121 even 3 inner
171.2.h.c.7.11 yes 32 19.7 even 3
171.2.h.c.49.11 yes 32 9.4 even 3
513.2.g.c.64.11 32 171.140 odd 6
513.2.g.c.505.11 32 3.2 odd 2
513.2.h.c.235.6 32 57.26 odd 6
513.2.h.c.334.6 32 9.5 odd 6