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Properties

Label 171.2.g.c.121.6
Level $171$
Weight $2$
Character 171.121
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 121.6
Character \(\chi\) \(=\) 171.121
Dual form 171.2.g.c.106.6

$q$-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 \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

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 −0.971375 − 0.237549i \(-0.923656\pi\)
0.691411 + 0.722461i \(0.256989\pi\)
\(3\) 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 0.786865 − 0.617126i \(-0.211703\pi\)
−0.927879 + 0.372882i \(0.878370\pi\)
\(8\) −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 0.603783 − 0.797149i \(-0.293659\pi\)
−0.992243 + 0.124317i \(0.960326\pi\)
\(12\) 2.35697 − 0.315823i 0.680400 − 0.0911703i
\(13\) 3.09365 + 5.35835i 0.858023 + 1.48614i 0.873812 + 0.486265i \(0.161641\pi\)
−0.0157882 + 0.999875i \(0.505026\pi\)
\(14\) 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 0.880211 − 0.474583i \(-0.157401\pi\)
−0.851107 + 0.524993i \(0.824068\pi\)
\(18\) 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 0.591345 − 0.806419i \(-0.298597\pi\)
−0.994052 + 0.108910i \(0.965264\pi\)
\(24\) −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 0.297749 + 0.954644i \(0.403764\pi\)
−0.975621 + 0.219464i \(0.929569\pi\)
\(32\) −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 −0.735076 − 0.677985i \(-0.762853\pi\)
0.954690 + 0.297602i \(0.0961867\pi\)
\(44\) 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 0.173468 + 0.984840i \(0.444503\pi\)
−0.939630 + 0.342192i \(0.888831\pi\)
\(54\) 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 −0.997172 − 0.0751540i \(-0.976055\pi\)
0.433501 − 0.901153i \(-0.357278\pi\)
\(68\) −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 0.965031 − 0.262135i \(-0.0844264\pi\)
−0.709531 + 0.704674i \(0.751093\pi\)
\(72\) 5.69187 + 5.63970i 0.670794 + 0.664645i
\(73\) 1.63071 + 2.82448i 0.190861 + 0.330580i 0.945536 − 0.325518i \(-0.105539\pi\)
−0.754675 + 0.656099i \(0.772206\pi\)
\(74\) 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 −0.591989 − 0.805946i \(-0.701657\pi\)
0.993964 + 0.109705i \(0.0349905\pi\)
\(80\) −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 0.947204 − 0.320632i \(-0.103895\pi\)
−0.751277 + 0.659987i \(0.770562\pi\)
\(84\) −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 −0.445220 − 0.895421i \(-0.646875\pi\)
0.998068 − 0.0621388i \(-0.0197921\pi\)
\(90\) 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 −0.798267 − 0.602304i \(-0.794250\pi\)
0.920744 + 0.390168i \(0.127583\pi\)
\(98\) 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.121.6 yes 32
3.2 odd 2 513.2.g.c.64.11 32
9.2 odd 6 513.2.h.c.235.6 32
9.7 even 3 171.2.h.c.7.11 yes 32
19.11 even 3 171.2.h.c.49.11 yes 32
57.11 odd 6 513.2.h.c.334.6 32
171.11 odd 6 513.2.g.c.505.11 32
171.106 even 3 inner 171.2.g.c.106.6 ✓ 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.g.c.106.6 ✓ 32 171.106 even 3 inner
171.2.g.c.121.6 yes 32 1.1 even 1 trivial
171.2.h.c.7.11 yes 32 9.7 even 3
171.2.h.c.49.11 yes 32 19.11 even 3
513.2.g.c.64.11 32 3.2 odd 2
513.2.g.c.505.11 32 171.11 odd 6
513.2.h.c.235.6 32 9.2 odd 6
513.2.h.c.334.6 32 57.11 odd 6