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Properties

Label 171.2.h.c.7.11
Level $171$
Weight $2$
Character 171.7
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(7,171)] 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("171.7"); 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, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,-2,1,34,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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 7.11
Character \(\chi\) \(=\) 171.7
Dual form 171.2.h.c.49.11

$q$-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+0.791858 q^{2} +(-1.71671 - 0.230031i) q^{3} -1.37296 q^{4} +(-1.29546 + 2.24381i) q^{5} +(-1.35939 - 0.182152i) q^{6} +(-0.373088 + 0.646207i) q^{7} -2.67091 q^{8} +(2.89417 + 0.789791i) q^{9} +(-1.02582 + 1.77678i) q^{10} +(-1.28837 + 2.23153i) q^{11} +(2.35697 + 0.315823i) q^{12} -6.18729 q^{13} +(-0.295433 + 0.511704i) q^{14} +(2.74008 - 3.55397i) q^{15} +0.630945 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} +(0.789131 - 1.02353i) q^{21} +(-1.02021 + 1.76705i) q^{22} +3.86263 q^{23} +(4.58516 + 0.614390i) q^{24} +(-0.856452 - 1.48342i) q^{25} -4.89946 q^{26} +(-4.78677 - 2.02159i) q^{27} +(0.512235 - 0.887218i) q^{28} +(3.39869 + 5.88670i) q^{29} +(2.16975 - 2.81424i) q^{30} +(-3.77423 - 6.53716i) q^{31} +5.84143 q^{32} +(2.72508 - 3.53452i) q^{33} +(0.0950223 + 0.164583i) q^{34} +(-0.966644 - 1.67428i) q^{35} +(-3.97359 - 1.08435i) q^{36} -8.47678 q^{37} +(3.08349 - 1.55107i) q^{38} +(10.6218 + 1.42327i) q^{39} +(3.46006 - 5.99300i) q^{40} +(-4.07597 + 7.05978i) q^{41} +(0.624879 - 0.810488i) q^{42} -2.88022 q^{43} +(1.76889 - 3.06380i) q^{44} +(-5.52143 + 5.47082i) q^{45} +3.05865 q^{46} +(2.26283 + 3.91933i) q^{47} +(-1.08315 - 0.145137i) q^{48} +(3.22161 + 5.57999i) q^{49} +(-0.678188 - 1.17466i) q^{50} +(-0.158193 - 0.384412i) q^{51} +8.49492 q^{52} +(-5.57774 + 9.66094i) q^{53} +(-3.79044 - 1.60081i) q^{54} +(-3.33808 - 5.78173i) q^{55} +(0.996483 - 1.72596i) q^{56} +(-7.13543 + 2.46691i) q^{57} +(2.69128 + 4.66143i) q^{58} +(-0.172123 + 0.298126i) q^{59} +(-3.76202 + 4.87946i) q^{60} +(-0.0395099 - 0.0684332i) q^{61} +(-2.98865 - 5.17650i) q^{62} +(-1.59015 + 1.57557i) q^{63} +3.36369 q^{64} +(8.01542 - 13.8831i) q^{65} +(2.15788 - 2.79884i) q^{66} +9.22770 q^{67} +(-0.164754 - 0.285363i) q^{68} +(-6.63100 - 0.888523i) q^{69} +(-0.765444 - 1.32579i) q^{70} +(2.15288 + 3.72891i) q^{71} +(-7.73006 - 2.10946i) q^{72} +(1.63071 + 2.82448i) q^{73} -6.71241 q^{74} +(1.12905 + 2.74361i) q^{75} +(-5.34630 + 2.68933i) q^{76} +(-0.961354 - 1.66511i) q^{77} +(8.41094 + 1.12703i) q^{78} -7.14567 q^{79} +(-0.817366 + 1.41572i) q^{80} +(7.75246 + 4.57158i) q^{81} +(-3.22759 + 5.59035i) q^{82} +(1.78498 - 3.09167i) q^{83} +(-1.08345 + 1.40526i) q^{84} -0.621818 q^{85} -2.28072 q^{86} +(-4.48043 - 10.8875i) q^{87} +(3.44113 - 5.96021i) q^{88} +(5.21555 - 9.03360i) q^{89} +(-4.37219 + 4.33211i) q^{90} +(2.30841 - 3.99827i) q^{91} -5.30324 q^{92} +(4.97550 + 12.0906i) q^{93} +(1.79184 + 3.10355i) q^{94} +(-0.649404 + 11.2749i) q^{95} +(-10.0280 - 1.34371i) q^{96} -2.41251 q^{97} +(2.55106 + 4.41856i) q^{98} +(-5.49122 + 5.44088i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 2 q^{2} + q^{3} + 34 q^{4} + 3 q^{5} - 7 q^{6} + q^{7} - 36 q^{8} + 17 q^{9} - 8 q^{10} + 7 q^{11} - 3 q^{12} + 8 q^{13} + q^{14} - 14 q^{15} + 22 q^{16} - 7 q^{17} + 6 q^{18} + 7 q^{19} - 3 q^{20}+ \cdots - 49 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(20\) \(154\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.791858 0.559928 0.279964 − 0.960010i \(-0.409677\pi\)
0.279964 + 0.960010i \(0.409677\pi\)
\(3\) −1.71671 − 0.230031i −0.991142 − 0.132808i
\(4\) −1.37296 −0.686481
\(5\) −1.29546 + 2.24381i −0.579349 + 1.00346i 0.416205 + 0.909271i \(0.363360\pi\)
−0.995554 + 0.0941911i \(0.969974\pi\)
\(6\) −1.35939 − 0.182152i −0.554968 − 0.0743631i
\(7\) −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
\(9\) 2.89417 + 0.789791i 0.964724 + 0.263264i
\(10\) −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
\(13\) −6.18729 −1.71605 −0.858023 − 0.513611i \(-0.828308\pi\)
−0.858023 + 0.513611i \(0.828308\pi\)
\(14\) −0.295433 + 0.511704i −0.0789577 + 0.136759i
\(15\) 2.74008 − 3.55397i 0.707485 − 0.917631i
\(16\) 0.630945 0.157736
\(17\) 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
\(21\) 0.789131 − 1.02353i 0.172202 − 0.223352i
\(22\) −1.02021 + 1.76705i −0.217509 + 0.376737i
\(23\) 3.86263 0.805414 0.402707 − 0.915329i \(-0.368069\pi\)
0.402707 + 0.915329i \(0.368069\pi\)
\(24\) 4.58516 + 0.614390i 0.935943 + 0.125412i
\(25\) −0.856452 − 1.48342i −0.171290 − 0.296684i
\(26\) −4.89946 −0.960863
\(27\) −4.78677 − 2.02159i −0.921215 − 0.389055i
\(28\) 0.512235 − 0.887218i 0.0968034 − 0.167668i
\(29\) 3.39869 + 5.88670i 0.631120 + 1.09313i 0.987323 + 0.158724i \(0.0507379\pi\)
−0.356203 + 0.934409i \(0.615929\pi\)
\(30\) 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 \(-0.403764\pi\)
\(32\) 5.84143 1.03263
\(33\) 2.72508 − 3.53452i 0.474376 − 0.615281i
\(34\) 0.0950223 + 0.164583i 0.0162962 + 0.0282258i
\(35\) −0.966644 − 1.67428i −0.163393 − 0.283004i
\(36\) −3.97359 − 1.08435i −0.662264 − 0.180725i
\(37\) −8.47678 −1.39357 −0.696787 − 0.717278i \(-0.745388\pi\)
−0.696787 + 0.717278i \(0.745388\pi\)
\(38\) 3.08349 − 1.55107i 0.500208 − 0.251617i
\(39\) 10.6218 + 1.42327i 1.70085 + 0.227905i
\(40\) 3.46006 − 5.99300i 0.547084 − 0.947577i
\(41\) −4.07597 + 7.05978i −0.636559 + 1.10255i 0.349623 + 0.936890i \(0.386310\pi\)
−0.986182 + 0.165663i \(0.947024\pi\)
\(42\) 0.624879 − 0.810488i 0.0964210 − 0.125061i
\(43\) −2.88022 −0.439229 −0.219615 − 0.975587i \(-0.570480\pi\)
−0.219615 + 0.975587i \(0.570480\pi\)
\(44\) 1.76889 − 3.06380i 0.266670 − 0.461886i
\(45\) −5.52143 + 5.47082i −0.823087 + 0.815542i
\(46\) 3.05865 0.450974
\(47\) 2.26283 + 3.91933i 0.330067 + 0.571693i 0.982525 − 0.186132i \(-0.0595953\pi\)
−0.652458 + 0.757825i \(0.726262\pi\)
\(48\) −1.08315 − 0.145137i −0.156339 − 0.0209487i
\(49\) 3.22161 + 5.57999i 0.460230 + 0.797142i
\(50\) −0.678188 − 1.17466i −0.0959103 − 0.166122i
\(51\) −0.158193 − 0.384412i −0.0221514 − 0.0538284i
\(52\) 8.49492 1.17803
\(53\) −5.57774 + 9.66094i −0.766162 + 1.32703i 0.173468 + 0.984840i \(0.444503\pi\)
−0.939630 + 0.342192i \(0.888831\pi\)
\(54\) −3.79044 − 1.60081i −0.515814 − 0.217843i
\(55\) −3.33808 − 5.78173i −0.450107 − 0.779609i
\(56\) 0.996483 − 1.72596i 0.133161 − 0.230641i
\(57\) −7.13543 + 2.46691i −0.945111 + 0.326750i
\(58\) 2.69128 + 4.66143i 0.353382 + 0.612075i
\(59\) −0.172123 + 0.298126i −0.0224085 + 0.0388127i −0.877012 − 0.480468i \(-0.840467\pi\)
0.854604 + 0.519281i \(0.173800\pi\)
\(60\) −3.76202 + 4.87946i −0.485675 + 0.629936i
\(61\) −0.0395099 − 0.0684332i −0.00505873 − 0.00876198i 0.863485 − 0.504375i \(-0.168277\pi\)
−0.868544 + 0.495613i \(0.834944\pi\)
\(62\) −2.98865 − 5.17650i −0.379559 − 0.657416i
\(63\) −1.59015 + 1.57557i −0.200340 + 0.198504i
\(64\) 3.36369 0.420462
\(65\) 8.01542 − 13.8831i 0.994190 − 1.72199i
\(66\) 2.15788 − 2.79884i 0.265616 − 0.344513i
\(67\) 9.22770 1.12734 0.563671 − 0.825999i \(-0.309388\pi\)
0.563671 + 0.825999i \(0.309388\pi\)
\(68\) −0.164754 − 0.285363i −0.0199794 − 0.0346053i
\(69\) −6.63100 − 0.888523i −0.798279 − 0.106966i
\(70\) −0.765444 − 1.32579i −0.0914881 − 0.158462i
\(71\) 2.15288 + 3.72891i 0.255500 + 0.442540i 0.965031 − 0.262135i \(-0.0844264\pi\)
−0.709531 + 0.704674i \(0.751093\pi\)
\(72\) −7.73006 − 2.10946i −0.910996 − 0.248602i
\(73\) 1.63071 + 2.82448i 0.190861 + 0.330580i 0.945536 − 0.325518i \(-0.105539\pi\)
−0.754675 + 0.656099i \(0.772206\pi\)
\(74\) −6.71241 −0.780302
\(75\) 1.12905 + 2.74361i 0.130371 + 0.316804i
\(76\) −5.34630 + 2.68933i −0.613263 + 0.308487i
\(77\) −0.961354 − 1.66511i −0.109556 − 0.189757i
\(78\) 8.41094 + 1.12703i 0.952351 + 0.127611i
\(79\) −7.14567 −0.803950 −0.401975 − 0.915651i \(-0.631676\pi\)
−0.401975 + 0.915651i \(0.631676\pi\)
\(80\) −0.817366 + 1.41572i −0.0913843 + 0.158282i
\(81\) 7.75246 + 4.57158i 0.861384 + 0.507954i
\(82\) −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
\(85\) −0.621818 −0.0674457
\(86\) −2.28072 −0.245937
\(87\) −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 \(-0.0197921\pi\)
\(90\) −4.37219 + 4.33211i −0.460869 + 0.456645i
\(91\) 2.30841 − 3.99827i 0.241987 − 0.419133i
\(92\) −5.30324 −0.552901
\(93\) 4.97550 + 12.0906i 0.515936 + 1.25373i
\(94\) 1.79184 + 3.10355i 0.184814 + 0.320107i
\(95\) −0.649404 + 11.2749i −0.0666274 + 1.15678i
\(96\) −10.0280 − 1.34371i −1.02348 − 0.137142i
\(97\) −2.41251 −0.244954 −0.122477 − 0.992471i \(-0.539084\pi\)
−0.122477 + 0.992471i \(0.539084\pi\)
\(98\) 2.55106 + 4.41856i 0.257696 + 0.446342i
\(99\) −5.49122 + 5.44088i −0.551888 + 0.546829i
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.h.c.7.11 yes 32
3.2 odd 2 513.2.h.c.235.6 32
9.4 even 3 171.2.g.c.121.6 yes 32
9.5 odd 6 513.2.g.c.64.11 32
19.11 even 3 171.2.g.c.106.6 ✓ 32
57.11 odd 6 513.2.g.c.505.11 32
171.49 even 3 inner 171.2.h.c.49.11 yes 32
171.68 odd 6 513.2.h.c.334.6 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.g.c.106.6 ✓ 32 19.11 even 3
171.2.g.c.121.6 yes 32 9.4 even 3
171.2.h.c.7.11 yes 32 1.1 even 1 trivial
171.2.h.c.49.11 yes 32 171.49 even 3 inner
513.2.g.c.64.11 32 9.5 odd 6
513.2.g.c.505.11 32 57.11 odd 6
513.2.h.c.235.6 32 3.2 odd 2
513.2.h.c.334.6 32 171.68 odd 6