Properties

Label 1323.2.g.g.667.3
Level 13231323
Weight 22
Character 1323.667
Analytic conductor 10.56410.564
Analytic rank 00
Dimension 1212
Inner twists 44

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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] = [1323,2,Mod(361,1323)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1323, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 4])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1323.361"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 1323=3372 1323 = 3^{3} \cdot 7^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1323.g (of order 33, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,2,0,-6,0] 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: 10.564208187410.5642081874
Analytic rank: 00
Dimension: 1212
Relative dimension: 66 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q[x]/(x12)\mathbb{Q}[x]/(x^{12} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x127x10+37x878x6+123x436x2+9 x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 35 3^{5}
Twist minimal: no (minimal twist has level 441)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

Embedding invariants

Embedding label 667.3
Root 1.29589+0.748185i1.29589 + 0.748185i of defining polynomial
Character χ\chi == 1323.667
Dual form 1323.2.g.g.361.3

qq-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)f(q) == q+(0.119562+0.207087i)q2+(0.9714101.68253i)q42.59179q5+0.942820q8+(0.3098790.536725i)q104.18194q11+(1.84155+3.18966i)q13+(1.830093.16982i)q16+(0.8555361.48183i)q17+(3.57780+6.19694i)q19+(2.51769+4.36077i)q20+(0.5000000.866025i)q22+5.12476q23+1.71737q25+(0.440358+0.762722i)q26+(1.06238+1.84010i)q29+(3.26793+5.66021i)q31+(1.380442.39099i)q32+(0.2045790.354341i)q34+(0.830095+1.43777i)q371.71107q382.44359q40+(5.10948+8.84988i)q41+(0.8300951.43777i)q43+(4.06238+7.03625i)q44+(0.612725+1.06127i)q46+(4.669128.08715i)q47+(0.205332+0.355645i)q50+7.15561q52+(5.32326+9.22015i)q53+10.8387q550.508080q58+(3.03215+5.25183i)q59+(3.992986.91605i)q611.56287q626.66019q64+(4.772928.26693i)q65+(4.13160+7.15614i)q673.32431q686.23912q71+(3.57780+6.19694i)q730.396990q74+(6.95103+12.0395i)q76+(4.91423+8.51170i)q79+(4.74322+8.21550i)q80+(1.22180+2.11621i)q82+(3.44733+5.97094i)q83+(2.21737+3.84060i)q85+0.396990q863.94282q88+(2.517694.36077i)q89+(4.978258.62258i)q92+(1.116501.93383i)q94+(9.2729216.0612i)q95+(1.531672.65294i)q97+O(q100)q+(0.119562 + 0.207087i) q^{2} +(0.971410 - 1.68253i) q^{4} -2.59179 q^{5} +0.942820 q^{8} +(-0.309879 - 0.536725i) q^{10} -4.18194 q^{11} +(1.84155 + 3.18966i) q^{13} +(-1.83009 - 3.16982i) q^{16} +(-0.855536 - 1.48183i) q^{17} +(-3.57780 + 6.19694i) q^{19} +(-2.51769 + 4.36077i) q^{20} +(-0.500000 - 0.866025i) q^{22} +5.12476 q^{23} +1.71737 q^{25} +(-0.440358 + 0.762722i) q^{26} +(-1.06238 + 1.84010i) q^{29} +(-3.26793 + 5.66021i) q^{31} +(1.38044 - 2.39099i) q^{32} +(0.204579 - 0.354341i) q^{34} +(-0.830095 + 1.43777i) q^{37} -1.71107 q^{38} -2.44359 q^{40} +(5.10948 + 8.84988i) q^{41} +(0.830095 - 1.43777i) q^{43} +(-4.06238 + 7.03625i) q^{44} +(0.612725 + 1.06127i) q^{46} +(-4.66912 - 8.08715i) q^{47} +(0.205332 + 0.355645i) q^{50} +7.15561 q^{52} +(5.32326 + 9.22015i) q^{53} +10.8387 q^{55} -0.508080 q^{58} +(-3.03215 + 5.25183i) q^{59} +(-3.99298 - 6.91605i) q^{61} -1.56287 q^{62} -6.66019 q^{64} +(-4.77292 - 8.26693i) q^{65} +(-4.13160 + 7.15614i) q^{67} -3.32431 q^{68} -6.23912 q^{71} +(3.57780 + 6.19694i) q^{73} -0.396990 q^{74} +(6.95103 + 12.0395i) q^{76} +(4.91423 + 8.51170i) q^{79} +(4.74322 + 8.21550i) q^{80} +(-1.22180 + 2.11621i) q^{82} +(-3.44733 + 5.97094i) q^{83} +(2.21737 + 3.84060i) q^{85} +0.396990 q^{86} -3.94282 q^{88} +(2.51769 - 4.36077i) q^{89} +(4.97825 - 8.62258i) q^{92} +(1.11650 - 1.93383i) q^{94} +(9.27292 - 16.0612i) q^{95} +(1.53167 - 2.65294i) q^{97} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 12q+2q26q424q816q116q166q228q23+24q25+22q29+16q32+6q376q4314q4412q46+56q50+28q53+36q5848q64++60q95+O(q100) 12 q + 2 q^{2} - 6 q^{4} - 24 q^{8} - 16 q^{11} - 6 q^{16} - 6 q^{22} - 8 q^{23} + 24 q^{25} + 22 q^{29} + 16 q^{32} + 6 q^{37} - 6 q^{43} - 14 q^{44} - 12 q^{46} + 56 q^{50} + 28 q^{53} + 36 q^{58} - 48 q^{64}+ \cdots + 60 q^{95}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1323Z)×\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times.

nn 785785 10811081
χ(n)\chi(n) e(23)e\left(\frac{2}{3}\right) e(13)e\left(\frac{1}{3}\right)

Coefficient data

For each nn we display the coefficients of the qq-expansion ana_n, the Satake parameters αp\alpha_p, and the Satake angles θp=Arg(αp)\theta_p = \textrm{Arg}(\alpha_p).



Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)
Significant digits:
nn ana_n an/n(k1)/2a_n / n^{(k-1)/2} αn \alpha_n θn \theta_n
pp apa_p ap/p(k1)/2a_p / p^{(k-1)/2} αp \alpha_p θp \theta_p
22 0.119562 + 0.207087i 0.0845428 + 0.146433i 0.905196 0.424994i 0.139724π-0.139724\pi
−0.820653 + 0.571426i 0.806390π0.806390\pi
33 0 0
44 0.971410 1.68253i 0.485705 0.841266i
55 −2.59179 −1.15908 −0.579542 0.814943i 0.696768π-0.696768\pi
−0.579542 + 0.814943i 0.696768π0.696768\pi
66 0 0
77 0 0
88 0.942820 0.333337
99 0 0
1010 −0.309879 0.536725i −0.0979922 0.169727i
1111 −4.18194 −1.26090 −0.630452 0.776228i 0.717130π-0.717130\pi
−0.630452 + 0.776228i 0.717130π0.717130\pi
1212 0 0
1313 1.84155 + 3.18966i 0.510755 + 0.884653i 0.999922 + 0.0124633i 0.00396730π0.00396730\pi
−0.489168 + 0.872190i 0.662699π0.662699\pi
1414 0 0
1515 0 0
1616 −1.83009 3.16982i −0.457524 0.792454i
1717 −0.855536 1.48183i −0.207498 0.359397i 0.743428 0.668816i 0.233199π-0.233199\pi
−0.950926 + 0.309419i 0.899865π0.899865\pi
1818 0 0
1919 −3.57780 + 6.19694i −0.820805 + 1.42168i 0.0842790 + 0.996442i 0.473141π0.473141\pi
−0.905084 + 0.425233i 0.860192π0.860192\pi
2020 −2.51769 + 4.36077i −0.562973 + 0.975097i
2121 0 0
2222 −0.500000 0.866025i −0.106600 0.184637i
2323 5.12476 1.06859 0.534294 0.845299i 0.320578π-0.320578\pi
0.534294 + 0.845299i 0.320578π0.320578\pi
2424 0 0
2525 1.71737 0.343474
2626 −0.440358 + 0.762722i −0.0863613 + 0.149582i
2727 0 0
2828 0 0
2929 −1.06238 + 1.84010i −0.197279 + 0.341698i −0.947645 0.319325i 0.896544π-0.896544\pi
0.750366 + 0.661023i 0.229877π0.229877\pi
3030 0 0
3131 −3.26793 + 5.66021i −0.586937 + 1.01660i 0.407694 + 0.913119i 0.366333π0.366333\pi
−0.994631 + 0.103486i 0.967000π0.967000\pi
3232 1.38044 2.39099i 0.244029 0.422671i
3333 0 0
3434 0.204579 0.354341i 0.0350850 0.0607689i
3535 0 0
3636 0 0
3737 −0.830095 + 1.43777i −0.136467 + 0.236367i −0.926157 0.377139i 0.876908π-0.876908\pi
0.789690 + 0.613506i 0.210241π0.210241\pi
3838 −1.71107 −0.277573
3939 0 0
4040 −2.44359 −0.386366
4141 5.10948 + 8.84988i 0.797967 + 1.38212i 0.920938 + 0.389708i 0.127424π0.127424\pi
−0.122972 + 0.992410i 0.539242π0.539242\pi
4242 0 0
4343 0.830095 1.43777i 0.126588 0.219257i −0.795764 0.605606i 0.792931π-0.792931\pi
0.922353 + 0.386349i 0.126264π0.126264\pi
4444 −4.06238 + 7.03625i −0.612427 + 1.06075i
4545 0 0
4646 0.612725 + 1.06127i 0.0903414 + 0.156476i
4747 −4.66912 8.08715i −0.681061 1.17963i −0.974657 0.223703i 0.928185π-0.928185\pi
0.293596 0.955930i 0.405148π-0.405148\pi
4848 0 0
4949 0 0
5050 0.205332 + 0.355645i 0.0290383 + 0.0502958i
5151 0 0
5252 7.15561 0.992305
5353 5.32326 + 9.22015i 0.731206 + 1.26649i 0.956368 + 0.292164i 0.0943754π0.0943754\pi
−0.225162 + 0.974321i 0.572291π0.572291\pi
5454 0 0
5555 10.8387 1.46149
5656 0 0
5757 0 0
5858 −0.508080 −0.0667142
5959 −3.03215 + 5.25183i −0.394752 + 0.683730i −0.993069 0.117529i 0.962503π-0.962503\pi
0.598318 + 0.801259i 0.295836π0.295836\pi
6060 0 0
6161 −3.99298 6.91605i −0.511249 0.885509i −0.999915 0.0130384i 0.995850π-0.995850\pi
0.488666 0.872471i 0.337484π-0.337484\pi
6262 −1.56287 −0.198485
6363 0 0
6464 −6.66019 −0.832524
6565 −4.77292 8.26693i −0.592007 1.02539i
6666 0 0
6767 −4.13160 + 7.15614i −0.504755 + 0.874262i 0.495230 + 0.868762i 0.335084π0.335084\pi
−0.999985 + 0.00549964i 0.998249π0.998249\pi
6868 −3.32431 −0.403131
6969 0 0
7070 0 0
7171 −6.23912 −0.740448 −0.370224 0.928943i 0.620719π-0.620719\pi
−0.370224 + 0.928943i 0.620719π0.620719\pi
7272 0 0
7373 3.57780 + 6.19694i 0.418750 + 0.725297i 0.995814 0.0914022i 0.0291349π-0.0291349\pi
−0.577064 + 0.816699i 0.695802π0.695802\pi
7474 −0.396990 −0.0461492
7575 0 0
7676 6.95103 + 12.0395i 0.797338 + 1.38103i
7777 0 0
7878 0 0
7979 4.91423 + 8.51170i 0.552894 + 0.957641i 0.998064 + 0.0621945i 0.0198099π0.0198099\pi
−0.445170 + 0.895446i 0.646857π0.646857\pi
8080 4.74322 + 8.21550i 0.530308 + 0.918521i
8181 0 0
8282 −1.22180 + 2.11621i −0.134925 + 0.233696i
8383 −3.44733 + 5.97094i −0.378393 + 0.655396i −0.990829 0.135124i 0.956857π-0.956857\pi
0.612436 + 0.790521i 0.290190π0.290190\pi
8484 0 0
8585 2.21737 + 3.84060i 0.240508 + 0.416571i
8686 0.396990 0.0428085
8787 0 0
8888 −3.94282 −0.420306
8989 2.51769 4.36077i 0.266875 0.462240i −0.701178 0.712986i 0.747342π-0.747342\pi
0.968053 + 0.250745i 0.0806757π0.0806757\pi
9090 0 0
9191 0 0
9292 4.97825 8.62258i 0.519018 0.898966i
9393 0 0
9494 1.11650 1.93383i 0.115158 0.199459i
9595 9.27292 16.0612i 0.951381 1.64784i
9696 0 0
9797 1.53167 2.65294i 0.155518 0.269365i −0.777730 0.628599i 0.783629π-0.783629\pi
0.933247 + 0.359234i 0.116962π0.116962\pi
9898 0 0
9999 0 0
100100 1.66827 2.88953i 0.166827 0.288953i
101101 −11.0997 −1.10446 −0.552229 0.833692i 0.686223π-0.686223\pi
−0.552229 + 0.833692i 0.686223π0.686223\pi
102102 0 0
103103 −7.98597 −0.786881 −0.393440 0.919350i 0.628715π-0.628715\pi
−0.393440 + 0.919350i 0.628715π0.628715\pi
104104 1.73625 + 3.00728i 0.170254 + 0.294888i
105105 0 0
106106 −1.27292 + 2.20475i −0.123636 + 0.214145i
107107 1.97825 3.42642i 0.191244 0.331245i −0.754419 0.656394i 0.772081π-0.772081\pi
0.945663 + 0.325149i 0.105414π0.105414\pi
108108 0 0
109109 −3.63160 6.29012i −0.347844 0.602484i 0.638022 0.770018i 0.279753π-0.279753\pi
−0.985866 + 0.167534i 0.946420π0.946420\pi
110110 1.29589 + 2.24456i 0.123559 + 0.214010i
111111 0 0
112112 0 0
113113 3.46457 + 6.00082i 0.325920 + 0.564509i 0.981698 0.190444i 0.0609928π-0.0609928\pi
−0.655778 + 0.754953i 0.727659π0.727659\pi
114114 0 0
115115 −13.2823 −1.23858
116116 2.06402 + 3.57498i 0.191639 + 0.331929i
117117 0 0
118118 −1.45011 −0.133494
119119 0 0
120120 0 0
121121 6.48865 0.589877
122122 0.954815 1.65379i 0.0864449 0.149727i
123123 0 0
124124 6.34899 + 10.9968i 0.570156 + 0.987540i
125125 8.50788 0.760968
126126 0 0
127127 9.11109 0.808479 0.404239 0.914653i 0.367536π-0.367536\pi
0.404239 + 0.914653i 0.367536π0.367536\pi
128128 −3.55718 6.16122i −0.314413 0.544580i
129129 0 0
130130 1.14132 1.97682i 0.100100 0.173378i
131131 −4.30286 −0.375943 −0.187971 0.982175i 0.560191π-0.560191\pi
−0.187971 + 0.982175i 0.560191π0.560191\pi
132132 0 0
133133 0 0
134134 −1.97592 −0.170694
135135 0 0
136136 −0.806617 1.39710i −0.0691668 0.119800i
137137 −20.5893 −1.75907 −0.879533 0.475838i 0.842145π-0.842145\pi
−0.879533 + 0.475838i 0.842145π0.842145\pi
138138 0 0
139139 −7.88067 13.6497i −0.668429 1.15775i −0.978343 0.206989i 0.933634π-0.933634\pi
0.309914 0.950765i 0.399700π-0.399700\pi
140140 0 0
141141 0 0
142142 −0.745960 1.29204i −0.0625996 0.108426i
143143 −7.70127 13.3390i −0.644012 1.11546i
144144 0 0
145145 2.75347 4.76915i 0.228663 0.396056i
146146 −0.855536 + 1.48183i −0.0708047 + 0.122637i
147147 0 0
148148 1.61273 + 2.79332i 0.132565 + 0.229610i
149149 −6.06758 −0.497076 −0.248538 0.968622i 0.579950π-0.579950\pi
−0.248538 + 0.968622i 0.579950π0.579950\pi
150150 0 0
151151 4.48865 0.365281 0.182641 0.983180i 0.441536π-0.441536\pi
0.182641 + 0.983180i 0.441536π0.441536\pi
152152 −3.37323 + 5.84260i −0.273605 + 0.473897i
153153 0 0
154154 0 0
155155 8.46978 14.6701i 0.680309 1.17833i
156156 0 0
157157 0.514457 0.891066i 0.0410582 0.0711148i −0.844766 0.535136i 0.820260π-0.820260\pi
0.885824 + 0.464021i 0.153594π0.153594\pi
158158 −1.17511 + 2.03534i −0.0934865 + 0.161923i
159159 0 0
160160 −3.57780 + 6.19694i −0.282850 + 0.489911i
161161 0 0
162162 0 0
163163 −3.41423 + 5.91362i −0.267423 + 0.463190i −0.968196 0.250194i 0.919505π-0.919505\pi
0.700772 + 0.713385i 0.252839π0.252839\pi
164164 19.8536 1.55031
165165 0 0
166166 −1.64867 −0.127962
167167 −8.99716 15.5835i −0.696221 1.20589i −0.969767 0.244032i 0.921530π-0.921530\pi
0.273546 0.961859i 0.411803π-0.411803\pi
168168 0 0
169169 −0.282630 + 0.489530i −0.0217408 + 0.0376561i
170170 −0.530225 + 0.918376i −0.0406664 + 0.0704362i
171171 0 0
172172 −1.61273 2.79332i −0.122969 0.212989i
173173 0.415178 + 0.719110i 0.0315654 + 0.0546729i 0.881377 0.472414i 0.156617π-0.156617\pi
−0.849811 + 0.527087i 0.823284π0.823284\pi
174174 0 0
175175 0 0
176176 7.65335 + 13.2560i 0.576893 + 0.999208i
177177 0 0
178178 1.20408 0.0902493
179179 3.78947 + 6.56355i 0.283238 + 0.490583i 0.972180 0.234233i 0.0752580π-0.0752580\pi
−0.688942 + 0.724816i 0.741925π0.741925\pi
180180 0 0
181181 −0.409157 −0.0304124 −0.0152062 0.999884i 0.504840π-0.504840\pi
−0.0152062 + 0.999884i 0.504840π0.504840\pi
182182 0 0
183183 0 0
184184 4.83173 0.356200
185185 2.15143 3.72639i 0.158176 0.273969i
186186 0 0
187187 3.57780 + 6.19694i 0.261635 + 0.453165i
188188 −18.1425 −1.32318
189189 0 0
190190 4.43474 0.321730
191191 8.01204 + 13.8773i 0.579731 + 1.00412i 0.995510 + 0.0946575i 0.0301756π0.0301756\pi
−0.415779 + 0.909466i 0.636491π0.636491\pi
192192 0 0
193193 6.18715 10.7164i 0.445360 0.771387i −0.552717 0.833369i 0.686409π-0.686409\pi
0.998077 + 0.0619822i 0.0197422π0.0197422\pi
194194 0.732518 0.0525917
195195 0 0
196196 0 0
197197 −23.1021 −1.64595 −0.822977 0.568075i 0.807688π-0.807688\pi
−0.822977 + 0.568075i 0.807688π0.807688\pi
198198 0 0
199199 −3.37323 5.84260i −0.239122 0.414171i 0.721341 0.692580i 0.243526π-0.243526\pi
−0.960463 + 0.278409i 0.910193π0.910193\pi
200200 1.61917 0.114493
201201 0 0
202202 −1.32710 2.29860i −0.0933741 0.161729i
203203 0 0
204204 0 0
205205 −13.2427 22.9370i −0.924910 1.60199i
206206 −0.954815 1.65379i −0.0665251 0.115225i
207207 0 0
208208 6.74043 11.6748i 0.467365 0.809500i
209209 14.9622 25.9153i 1.03496 1.79260i
210210 0 0
211211 −8.44282 14.6234i −0.581228 1.00672i −0.995334 0.0964875i 0.969239π-0.969239\pi
0.414106 0.910228i 0.364094π-0.364094\pi
212212 20.6843 1.42060
213213 0 0
214214 0.946090 0.0646734
215215 −2.15143 + 3.72639i −0.146726 + 0.254138i
216216 0 0
217217 0 0
218218 0.868400 1.50411i 0.0588155 0.101871i
219219 0 0
220220 10.5288 18.2365i 0.709854 1.22950i
221221 3.15103 5.45774i 0.211961 0.367128i
222222 0 0
223223 −2.25071 + 3.89834i −0.150719 + 0.261052i −0.931492 0.363762i 0.881492π-0.881492\pi
0.780773 + 0.624815i 0.214825π0.214825\pi
224224 0 0
225225 0 0
226226 −0.828460 + 1.43494i −0.0551084 + 0.0954505i
227227 −6.06429 −0.402501 −0.201251 0.979540i 0.564501π-0.564501\pi
−0.201251 + 0.979540i 0.564501π0.564501\pi
228228 0 0
229229 11.0493 0.730159 0.365080 0.930976i 0.381042π-0.381042\pi
0.365080 + 0.930976i 0.381042π0.381042\pi
230230 −1.58805 2.75059i −0.104713 0.181369i
231231 0 0
232232 −1.00163 + 1.73488i −0.0657605 + 0.113901i
233233 −4.06922 + 7.04809i −0.266583 + 0.461736i −0.967977 0.251038i 0.919228π-0.919228\pi
0.701394 + 0.712774i 0.252561π0.252561\pi
234234 0 0
235235 12.1014 + 20.9602i 0.789407 + 1.36729i
236236 5.89092 + 10.2034i 0.383466 + 0.664183i
237237 0 0
238238 0 0
239239 10.5813 + 18.3273i 0.684445 + 1.18549i 0.973611 + 0.228214i 0.0732886π0.0732886\pi
−0.289166 + 0.957279i 0.593378π0.593378\pi
240240 0 0
241241 −13.6915 −0.881945 −0.440972 0.897521i 0.645366π-0.645366\pi
−0.440972 + 0.897521i 0.645366π0.645366\pi
242242 0.775794 + 1.34371i 0.0498699 + 0.0863772i
243243 0 0
244244 −15.5153 −0.993265
245245 0 0
246246 0 0
247247 −26.3549 −1.67692
248248 −3.08107 + 5.33656i −0.195648 + 0.338872i
249249 0 0
250250 1.01722 + 1.76187i 0.0643344 + 0.111430i
251251 15.2040 0.959667 0.479833 0.877360i 0.340697π-0.340697\pi
0.479833 + 0.877360i 0.340697π0.340697\pi
252252 0 0
253253 −21.4315 −1.34738
254254 1.08934 + 1.88679i 0.0683511 + 0.118388i
255255 0 0
256256 −5.80959 + 10.0625i −0.363099 + 0.628906i
257257 25.6215 1.59822 0.799112 0.601182i 0.205303π-0.205303\pi
0.799112 + 0.601182i 0.205303π0.205303\pi
258258 0 0
259259 0 0
260260 −18.5458 −1.15016
261261 0 0
262262 −0.514457 0.891066i −0.0317833 0.0550502i
263263 −7.10069 −0.437847 −0.218924 0.975742i 0.570255π-0.570255\pi
−0.218924 + 0.975742i 0.570255π0.570255\pi
264264 0 0
265265 −13.7968 23.8967i −0.847528 1.46796i
266266 0 0
267267 0 0
268268 8.02696 + 13.9031i 0.490324 + 0.849267i
269269 8.21572 + 14.2301i 0.500922 + 0.867622i 0.999999 + 0.00106448i 0.000338834π0.000338834\pi
−0.499078 + 0.866557i 0.666328π0.666328\pi
270270 0 0
271271 6.34899 10.9968i 0.385674 0.668007i −0.606189 0.795321i 0.707302π-0.707302\pi
0.991862 + 0.127314i 0.0406357π0.0406357\pi
272272 −3.13143 + 5.42379i −0.189871 + 0.328865i
273273 0 0
274274 −2.46169 4.26378i −0.148716 0.257584i
275275 −7.18194 −0.433087
276276 0 0
277277 −0.828460 −0.0497773 −0.0248887 0.999690i 0.507923π-0.507923\pi
−0.0248887 + 0.999690i 0.507923π0.507923\pi
278278 1.88445 3.26396i 0.113022 0.195760i
279279 0 0
280280 0 0
281281 2.60985 4.52039i 0.155690 0.269664i −0.777620 0.628735i 0.783573π-0.783573\pi
0.933310 + 0.359071i 0.116906π0.116906\pi
282282 0 0
283283 −3.67708 + 6.36890i −0.218580 + 0.378592i −0.954374 0.298614i 0.903476π-0.903476\pi
0.735794 + 0.677205i 0.236809π0.236809\pi
284284 −6.06075 + 10.4975i −0.359639 + 0.622913i
285285 0 0
286286 1.84155 3.18966i 0.108893 0.188609i
287287 0 0
288288 0 0
289289 7.03611 12.1869i 0.413889 0.716877i
290290 1.31684 0.0773273
291291 0 0
292292 13.9021 0.813557
293293 −3.91286 6.77728i −0.228592 0.395933i 0.728799 0.684728i 0.240079π-0.240079\pi
−0.957391 + 0.288795i 0.906745π0.906745\pi
294294 0 0
295295 7.85868 13.6116i 0.457550 0.792500i
296296 −0.782630 + 1.35556i −0.0454895 + 0.0787900i
297297 0 0
298298 −0.725450 1.25652i −0.0420242 0.0727881i
299299 9.43752 + 16.3463i 0.545786 + 0.945329i
300300 0 0
301301 0 0
302302 0.536670 + 0.929540i 0.0308819 + 0.0534890i
303303 0 0
304304 26.1909 1.50215
305305 10.3490 + 17.9249i 0.592580 + 1.02638i
306306 0 0
307307 −22.6709 −1.29390 −0.646948 0.762534i 0.723955π-0.723955\pi
−0.646948 + 0.762534i 0.723955π0.723955\pi
308308 0 0
309309 0 0
310310 4.05064 0.230061
311311 16.1588 27.9879i 0.916281 1.58705i 0.111266 0.993791i 0.464509π-0.464509\pi
0.805015 0.593255i 0.202157π-0.202157\pi
312312 0 0
313313 −12.1598 21.0614i −0.687312 1.19046i −0.972704 0.232048i 0.925457π-0.925457\pi
0.285392 0.958411i 0.407876π-0.407876\pi
314314 0.246037 0.0138847
315315 0 0
316316 19.0949 1.07417
317317 2.56922 + 4.45002i 0.144302 + 0.249938i 0.929112 0.369798i 0.120573π-0.120573\pi
−0.784811 + 0.619736i 0.787240π0.787240\pi
318318 0 0
319319 4.44282 7.69519i 0.248750 0.430848i
320320 17.2618 0.964964
321321 0 0
322322 0 0
323323 12.2438 0.681262
324324 0 0
325325 3.16263 + 5.47783i 0.175431 + 0.303855i
326326 −1.63284 −0.0904349
327327 0 0
328328 4.81732 + 8.34384i 0.265992 + 0.460712i
329329 0 0
330330 0 0
331331 5.84897 + 10.1307i 0.321488 + 0.556834i 0.980795 0.195040i 0.0624835π-0.0624835\pi
−0.659307 + 0.751874i 0.729150π0.729150\pi
332332 6.69753 + 11.6005i 0.367575 + 0.636658i
333333 0 0
334334 2.15143 3.72639i 0.117721 0.203899i
335335 10.7082 18.5472i 0.585053 1.01334i
336336 0 0
337337 16.8473 + 29.1804i 0.917733 + 1.58956i 0.802850 + 0.596181i 0.203316π0.203316\pi
0.114883 + 0.993379i 0.463351π0.463351\pi
338338 −0.135167 −0.00735211
339339 0 0
340340 8.61590 0.467263
341341 13.6663 23.6707i 0.740071 1.28184i
342342 0 0
343343 0 0
344344 0.782630 1.35556i 0.0421966 0.0730866i
345345 0 0
346346 −0.0992788 + 0.171956i −0.00533726 + 0.00924441i
347347 13.6557 23.6523i 0.733075 1.26972i −0.222488 0.974936i 0.571418π-0.571418\pi
0.955563 0.294788i 0.0952490π-0.0952490\pi
348348 0 0
349349 11.4585 19.8467i 0.613358 1.06237i −0.377312 0.926086i 0.623152π-0.623152\pi
0.990670 0.136281i 0.0435150π-0.0435150\pi
350350 0 0
351351 0 0
352352 −5.77292 + 9.99898i −0.307697 + 0.532948i
353353 10.2693 0.546581 0.273290 0.961932i 0.411888π-0.411888\pi
0.273290 + 0.961932i 0.411888π0.411888\pi
354354 0 0
355355 16.1705 0.858241
356356 −4.89142 8.47218i −0.259245 0.449025i
357357 0 0
358358 −0.906150 + 1.56950i −0.0478915 + 0.0829505i
359359 5.05034 8.74745i 0.266547 0.461673i −0.701421 0.712747i 0.747451π-0.747451\pi
0.967968 + 0.251075i 0.0807839π0.0807839\pi
360360 0 0
361361 −16.1014 27.8884i −0.847441 1.46781i
362362 −0.0489195 0.0847311i −0.00257115 0.00445337i
363363 0 0
364364 0 0
365365 −9.27292 16.0612i −0.485367 0.840680i
366366 0 0
367367 7.77537 0.405871 0.202935 0.979192i 0.434952π-0.434952\pi
0.202935 + 0.979192i 0.434952π0.434952\pi
368368 −9.37880 16.2446i −0.488904 0.846806i
369369 0 0
370370 1.02891 0.0534907
371371 0 0
372372 0 0
373373 24.1111 1.24842 0.624212 0.781255i 0.285420π-0.285420\pi
0.624212 + 0.781255i 0.285420π0.285420\pi
374374 −0.855536 + 1.48183i −0.0442387 + 0.0766237i
375375 0 0
376376 −4.40214 7.62473i −0.227023 0.393215i
377377 −7.82573 −0.403045
378378 0 0
379379 −13.3581 −0.686161 −0.343081 0.939306i 0.611470π-0.611470\pi
−0.343081 + 0.939306i 0.611470π0.611470\pi
380380 −18.0156 31.2039i −0.924181 1.60073i
381381 0 0
382382 −1.91586 + 3.31838i −0.0980242 + 0.169783i
383383 −9.24040 −0.472162 −0.236081 0.971733i 0.575863π-0.575863\pi
−0.236081 + 0.971733i 0.575863π0.575863\pi
384384 0 0
385385 0 0
386386 2.95898 0.150608
387387 0 0
388388 −2.97577 5.15418i −0.151072 0.261664i
389389 −10.4484 −0.529756 −0.264878 0.964282i 0.585332π-0.585332\pi
−0.264878 + 0.964282i 0.585332π0.585332\pi
390390 0 0
391391 −4.38442 7.59404i −0.221730 0.384047i
392392 0 0
393393 0 0
394394 −2.76212 4.78413i −0.139154 0.241021i
395395 −12.7366 22.0605i −0.640850 1.10999i
396396 0 0
397397 −0.204579 + 0.354341i −0.0102675 + 0.0177838i −0.871114 0.491082i 0.836602π-0.836602\pi
0.860846 + 0.508866i 0.169935π0.169935\pi
398398 0.806617 1.39710i 0.0404321 0.0700304i
399399 0 0
400400 −3.14295 5.44375i −0.157147 0.272187i
401401 −15.2528 −0.761688 −0.380844 0.924639i 0.624367π-0.624367\pi
−0.380844 + 0.924639i 0.624367π0.624367\pi
402402 0 0
403403 −24.0722 −1.19912
404404 −10.7823 + 18.6756i −0.536441 + 0.929143i
405405 0 0
406406 0 0
407407 3.47141 6.01266i 0.172071 0.298036i
408408 0 0
409409 −3.06335 + 5.30587i −0.151473 + 0.262359i −0.931769 0.363051i 0.881735π-0.881735\pi
0.780296 + 0.625410i 0.215068π0.215068\pi
410410 3.16664 5.48477i 0.156389 0.270874i
411411 0 0
412412 −7.75765 + 13.4366i −0.382192 + 0.661976i
413413 0 0
414414 0 0
415415 8.93474 15.4754i 0.438589 0.759659i
416416 10.1686 0.498557
417417 0 0
418418 7.15561 0.349992
419419 −0.781437 1.35349i −0.0381757 0.0661223i 0.846306 0.532697i 0.178821π-0.178821\pi
−0.884482 + 0.466574i 0.845488π0.845488\pi
420420 0 0
421421 −11.6316 + 20.1465i −0.566889 + 0.981881i 0.429982 + 0.902838i 0.358520π0.358520\pi
−0.996871 + 0.0790438i 0.974813π0.974813\pi
422422 2.01887 3.49679i 0.0982773 0.170221i
423423 0 0
424424 5.01887 + 8.69295i 0.243738 + 0.422167i
425425 −1.46927 2.54485i −0.0712702 0.123444i
426426 0 0
427427 0 0
428428 −3.84338 6.65692i −0.185777 0.321775i
429429 0 0
430430 −1.02891 −0.0496187
431431 0.502879 + 0.871011i 0.0242228 + 0.0419551i 0.877883 0.478876i 0.158956π-0.158956\pi
−0.853660 + 0.520831i 0.825622π0.825622\pi
432432 0 0
433433 −13.1071 −0.629889 −0.314945 0.949110i 0.601986π-0.601986\pi
−0.314945 + 0.949110i 0.601986π0.601986\pi
434434 0 0
435435 0 0
436436 −14.1111 −0.675799
437437 −18.3354 + 31.7579i −0.877101 + 1.51918i
438438 0 0
439439 9.30704 + 16.1203i 0.444201 + 0.769378i 0.997996 0.0632744i 0.0201543π-0.0201543\pi
−0.553795 + 0.832653i 0.686821π0.686821\pi
440440 10.2190 0.487170
441441 0 0
442442 1.50697 0.0716792
443443 −0.559503 0.969088i −0.0265828 0.0460427i 0.852428 0.522845i 0.175129π-0.175129\pi
−0.879011 + 0.476802i 0.841796π0.841796\pi
444444 0 0
445445 −6.52532 + 11.3022i −0.309330 + 0.535775i
446446 −1.07639 −0.0509687
447447 0 0
448448 0 0
449449 39.4419 1.86138 0.930689 0.365813i 0.119209π-0.119209\pi
0.930689 + 0.365813i 0.119209π0.119209\pi
450450 0 0
451451 −21.3676 37.0097i −1.00616 1.74272i
452452 13.4621 0.633203
453453 0 0
454454 −0.725057 1.25584i −0.0340286 0.0589393i
455455 0 0
456456 0 0
457457 17.1202 + 29.6531i 0.800852 + 1.38712i 0.919056 + 0.394126i 0.128953π0.128953\pi
−0.118205 + 0.992989i 0.537714π0.537714\pi
458458 1.32107 + 2.28817i 0.0617297 + 0.106919i
459459 0 0
460460 −12.9026 + 22.3479i −0.601585 + 1.04198i
461461 −10.1938 + 17.6561i −0.474772 + 0.822328i −0.999583 0.0288903i 0.990803π-0.990803\pi
0.524811 + 0.851219i 0.324136π0.324136\pi
462462 0 0
463463 −3.40451 5.89679i −0.158221 0.274047i 0.776006 0.630725i 0.217243π-0.217243\pi
−0.934227 + 0.356678i 0.883909π0.883909\pi
464464 7.77704 0.361040
465465 0 0
466466 −1.94609 −0.0901509
467467 −12.3956 + 21.4698i −0.573598 + 0.993502i 0.422594 + 0.906319i 0.361120π0.361120\pi
−0.996192 + 0.0871825i 0.972214π0.972214\pi
468468 0 0
469469 0 0
470470 −2.89372 + 5.01207i −0.133477 + 0.231190i
471471 0 0
472472 −2.85877 + 4.95153i −0.131586 + 0.227913i
473473 −3.47141 + 6.01266i −0.159616 + 0.276462i
474474 0 0
475475 −6.14441 + 10.6424i −0.281925 + 0.488309i
476476 0 0
477477 0 0
478478 −2.53022 + 4.38248i −0.115730 + 0.200450i
479479 −11.0997 −0.507157 −0.253579 0.967315i 0.581608π-0.581608\pi
−0.253579 + 0.967315i 0.581608π0.581608\pi
480480 0 0
481481 −6.11465 −0.278804
482482 −1.63697 2.83532i −0.0745621 0.129145i
483483 0 0
484484 6.30314 10.9174i 0.286506 0.496244i
485485 −3.96978 + 6.87585i −0.180258 + 0.312216i
486486 0 0
487487 5.01887 + 8.69295i 0.227427 + 0.393915i 0.957045 0.289940i 0.0936354π-0.0936354\pi
−0.729618 + 0.683855i 0.760302π0.760302\pi
488488 −3.76466 6.52059i −0.170418 0.295173i
489489 0 0
490490 0 0
491491 −6.19398 10.7283i −0.279530 0.484161i 0.691738 0.722149i 0.256845π-0.256845\pi
−0.971268 + 0.237988i 0.923512π0.923512\pi
492492 0 0
493493 3.63562 0.163740
494494 −3.15103 5.45774i −0.141772 0.245556i
495495 0 0
496496 23.9225 1.07415
497497 0 0
498498 0 0
499499 10.2222 0.457608 0.228804 0.973473i 0.426519π-0.426519\pi
0.228804 + 0.973473i 0.426519π0.426519\pi
500500 8.26464 14.3148i 0.369606 0.640177i
501501 0 0
502502 1.81781 + 3.14854i 0.0811329 + 0.140526i
503503 −8.45753 −0.377102 −0.188551 0.982063i 0.560379π-0.560379\pi
−0.188551 + 0.982063i 0.560379π0.560379\pi
504504 0 0
505505 28.7680 1.28016
506506 −2.56238 4.43818i −0.113912 0.197301i
507507 0 0
508508 8.85060 15.3297i 0.392682 0.680145i
509509 −10.5657 −0.468317 −0.234159 0.972198i 0.575233π-0.575233\pi
−0.234159 + 0.972198i 0.575233π0.575233\pi
510510 0 0
511511 0 0
512512 −17.0071 −0.751616
513513 0 0
514514 3.06335 + 5.30587i 0.135118 + 0.234032i
515515 20.6979 0.912060
516516 0 0
517517 19.5260 + 33.8200i 0.858752 + 1.48740i
518518 0 0
519519 0 0
520520 −4.50000 7.79423i −0.197338 0.341800i
521521 −9.87788 17.1090i −0.432758 0.749558i 0.564352 0.825534i 0.309126π-0.309126\pi
−0.997110 + 0.0759760i 0.975793π0.975793\pi
522522 0 0
523523 −16.2641 + 28.1702i −0.711179 + 1.23180i 0.253236 + 0.967405i 0.418505π0.418505\pi
−0.964415 + 0.264394i 0.914828π0.914828\pi
524524 −4.17984 + 7.23970i −0.182597 + 0.316268i
525525 0 0
526526 −0.848970 1.47046i −0.0370168 0.0641150i
527527 11.1833 0.487153
528528 0 0
529529 3.26320 0.141878
530530 3.29913 5.71426i 0.143305 0.248211i
531531 0 0
532532 0 0
533533 −18.8187 + 32.5950i −0.815130 + 1.41185i
534534 0 0
535535 −5.12720 + 8.88057i −0.221668 + 0.383940i
536536 −3.89536 + 6.74695i −0.168254 + 0.291424i
537537 0 0
538538 −1.96457 + 3.40274i −0.0846987 + 0.146702i
539539 0 0
540540 0 0
541541 −7.61109 + 13.1828i −0.327226 + 0.566773i −0.981960 0.189087i 0.939447π-0.939447\pi
0.654734 + 0.755859i 0.272781π0.272781\pi
542542 3.03638 0.130424
543543 0 0
544544 −4.72406 −0.202542
545545 9.41234 + 16.3027i 0.403180 + 0.698329i
546546 0 0
547547 −11.6871 + 20.2427i −0.499706 + 0.865517i −1.00000 0.000339172i 0.999892π-0.999892\pi
0.500294 + 0.865856i 0.333225π0.333225\pi
548548 −20.0007 + 34.6422i −0.854387 + 1.47984i
549549 0 0
550550 −0.858685 1.48729i −0.0366144 0.0634181i
551551 −7.60199 13.1670i −0.323856 0.560934i
552552 0 0
553553 0 0
554554 −0.0990521 0.171563i −0.00420832 0.00728902i
555555 0 0
556556 −30.6214 −1.29864
557557 13.8337 + 23.9606i 0.586151 + 1.01524i 0.994731 + 0.102521i 0.0326908π0.0326908\pi
−0.408580 + 0.912723i 0.633976π0.633976\pi
558558 0 0
559559 6.11465 0.258622
560560 0 0
561561 0 0
562562 1.24815 0.0526500
563563 4.27912 7.41166i 0.180343 0.312364i −0.761654 0.647984i 0.775612π-0.775612\pi
0.941998 + 0.335620i 0.108946π0.108946\pi
564564 0 0
565565 −8.97944 15.5529i −0.377768 0.654313i
566566 −1.75855 −0.0739175
567567 0 0
568568 −5.88237 −0.246819
569569 6.86389 + 11.8886i 0.287749 + 0.498396i 0.973272 0.229655i 0.0737597π-0.0737597\pi
−0.685523 + 0.728051i 0.740426π0.740426\pi
570570 0 0
571571 −5.35868 + 9.28151i −0.224254 + 0.388419i −0.956095 0.293056i 0.905328π-0.905328\pi
0.731841 + 0.681475i 0.238661π0.238661\pi
572572 −29.9244 −1.25120
573573 0 0
574574 0 0
575575 8.80111 0.367032
576576 0 0
577577 22.8177 + 39.5214i 0.949912 + 1.64530i 0.745605 + 0.666389i 0.232161π0.232161\pi
0.204307 + 0.978907i 0.434506π0.434506\pi
578578 3.36500 0.139965
579579 0 0
580580 −5.34950 9.26560i −0.222126 0.384733i
581581 0 0
582582 0 0
583583 −22.2616 38.5582i −0.921980 1.59692i
584584 3.37323 + 5.84260i 0.139585 + 0.241768i
585585 0 0
586586 0.935657 1.62060i 0.0386516 0.0669466i
587587 −5.10948 + 8.84988i −0.210891 + 0.365274i −0.951994 0.306118i 0.900970π-0.900970\pi
0.741103 + 0.671392i 0.234303π0.234303\pi
588588 0 0
589589 −23.3840 40.5023i −0.963521 1.66887i
590590 3.75839 0.154730
591591 0 0
592592 6.07661 0.249747
593593 5.69804 9.86929i 0.233990 0.405283i −0.724988 0.688761i 0.758155π-0.758155\pi
0.958979 + 0.283478i 0.0914883π0.0914883\pi
594594 0 0
595595 0 0
596596 −5.89411 + 10.2089i −0.241432 + 0.418173i
597597 0 0
598598 −2.25673 + 3.90877i −0.0922846 + 0.159842i
599599 −17.2873 + 29.9424i −0.706339 + 1.22341i 0.259867 + 0.965644i 0.416321π0.416321\pi
−0.966206 + 0.257771i 0.917012π0.917012\pi
600600 0 0
601601 19.4207 33.6376i 0.792187 1.37211i −0.132423 0.991193i 0.542276π-0.542276\pi
0.924610 0.380915i 0.124391π-0.124391\pi
602602 0 0
603603 0 0
604604 4.36032 7.55230i 0.177419 0.307299i
605605 −16.8172 −0.683717
606606 0 0
607607 −41.3325 −1.67763 −0.838817 0.544414i 0.816752π-0.816752\pi
−0.838817 + 0.544414i 0.816752π0.816752\pi
608608 9.87788 + 17.1090i 0.400601 + 0.693861i
609609 0 0
610610 −2.47468 + 4.28627i −0.100197 + 0.173546i
611611 17.1969 29.7858i 0.695710 1.20501i
612612 0 0
613613 14.3285 + 24.8176i 0.578721 + 1.00237i 0.995626 + 0.0934244i 0.0297813π0.0297813\pi
−0.416905 + 0.908950i 0.636885π0.636885\pi
614614 −2.71057 4.69485i −0.109390 0.189469i
615615 0 0
616616 0 0
617617 −16.8518 29.1883i −0.678430 1.17508i −0.975454 0.220205i 0.929327π-0.929327\pi
0.297024 0.954870i 0.404006π-0.404006\pi
618618 0 0
619619 1.43807 0.0578010 0.0289005 0.999582i 0.490799π-0.490799\pi
0.0289005 + 0.999582i 0.490799π0.490799\pi
620620 −16.4552 28.5013i −0.660859 1.14464i
621621 0 0
622622 7.72789 0.309860
623623 0 0
624624 0 0
625625 −30.6375 −1.22550
626626 2.90769 5.03626i 0.116215 0.201290i
627627 0 0
628628 −0.999498 1.73118i −0.0398843 0.0690816i
629629 2.84071 0.113266
630630 0 0
631631 −30.7680 −1.22486 −0.612428 0.790527i 0.709807π-0.709807\pi
−0.612428 + 0.790527i 0.709807π0.709807\pi
632632 4.63323 + 8.02500i 0.184300 + 0.319217i
633633 0 0
634634 −0.614360 + 1.06410i −0.0243993 + 0.0422609i
635635 −23.6140 −0.937094
636636 0 0
637637 0 0
638638 2.12476 0.0841202
639639 0 0
640640 9.21946 + 15.9686i 0.364431 + 0.631213i
641641 −9.23912 −0.364923 −0.182462 0.983213i 0.558407π-0.558407\pi
−0.182462 + 0.983213i 0.558407π0.558407\pi
642642 0 0
643643 12.7795 + 22.1348i 0.503976 + 0.872912i 0.999989 + 0.00459728i 0.00146337π0.00146337\pi
−0.496013 + 0.868315i 0.665203π0.665203\pi
644644 0 0
645645 0 0
646646 1.46389 + 2.53552i 0.0575958 + 0.0997588i
647647 14.1556 + 24.5181i 0.556512 + 0.963908i 0.997784 + 0.0665343i 0.0211942π0.0211942\pi
−0.441272 + 0.897374i 0.645472π0.645472\pi
648648 0 0
649649 12.6803 21.9629i 0.497744 0.862118i
650650 −0.756258 + 1.30988i −0.0296629 + 0.0513776i
651651 0 0
652652 6.63323 + 11.4891i 0.259778 + 0.449948i
653653 8.35021 0.326769 0.163385 0.986562i 0.447759π-0.447759\pi
0.163385 + 0.986562i 0.447759π0.447759\pi
654654 0 0
655655 11.1521 0.435749
656656 18.7017 32.3922i 0.730177 1.26470i
657657 0 0
658658 0 0
659659 −16.7862 + 29.0745i −0.653897 + 1.13258i 0.328272 + 0.944583i 0.393534π0.393534\pi
−0.982169 + 0.188000i 0.939799π0.939799\pi
660660 0 0
661661 8.47668 14.6820i 0.329705 0.571065i −0.652748 0.757575i 0.726384π-0.726384\pi
0.982453 + 0.186509i 0.0597175π0.0597175\pi
662662 −1.39862 + 2.42249i −0.0543591 + 0.0941527i
663663 0 0
664664 −3.25021 + 5.62952i −0.126133 + 0.218468i
665665 0 0
666666 0 0
667667 −5.44445 + 9.43007i −0.210810 + 0.365134i
668668 −34.9597 −1.35263
669669 0 0
670670 5.12118 0.197848
671671 16.6984 + 28.9225i 0.644636 + 1.11654i
672672 0 0
673673 22.2157 38.4788i 0.856354 1.48325i −0.0190299 0.999819i 0.506058π-0.506058\pi
0.875384 0.483429i 0.160609π-0.160609\pi
674674 −4.02859 + 6.97772i −0.155175 + 0.268772i
675675 0 0
676676 0.549100 + 0.951068i 0.0211192 + 0.0365796i
677677 −7.18681 12.4479i −0.276212 0.478412i 0.694229 0.719755i 0.255746π-0.255746\pi
−0.970440 + 0.241342i 0.922412π0.922412\pi
678678 0 0
679679 0 0
680680 2.09058 + 3.62099i 0.0801701 + 0.138859i
681681 0 0
682682 6.53585 0.250271
683683 −16.1546 27.9806i −0.618138 1.07065i −0.989825 0.142289i 0.954554π-0.954554\pi
0.371687 0.928358i 0.378780π-0.378780\pi
684684 0 0
685685 53.3632 2.03890
686686 0 0
687687 0 0
688688 −6.07661 −0.231669
689689 −19.6061 + 33.9588i −0.746934 + 1.29373i
690690 0 0
691691 14.4981 + 25.1114i 0.551533 + 0.955283i 0.998164 + 0.0605650i 0.0192902π0.0192902\pi
−0.446631 + 0.894718i 0.647376π0.647376\pi
692692 1.61323 0.0613259
693693 0 0
694694 6.53078 0.247905
695695 20.4250 + 35.3772i 0.774765 + 1.34193i
696696 0 0
697697 8.74269 15.1428i 0.331153 0.573574i
698698 5.47997 0.207420
699699 0 0
700700 0 0
701701 26.3912 0.996783 0.498392 0.866952i 0.333924π-0.333924\pi
0.498392 + 0.866952i 0.333924π0.333924\pi
702702 0 0
703703 −5.93984 10.2881i −0.224025 0.388023i
704704 27.8525 1.04973
705705 0 0
706706 1.22782 + 2.12664i 0.0462095 + 0.0800372i
707707 0 0
708708 0 0
709709 3.94282 + 6.82916i 0.148076 + 0.256475i 0.930516 0.366251i 0.119359π-0.119359\pi
−0.782441 + 0.622725i 0.786025π0.786025\pi
710710 1.93337 + 3.34870i 0.0725581 + 0.125674i
711711 0 0
712712 2.37373 4.11142i 0.0889592 0.154082i
713713 −16.7473 + 29.0073i −0.627193 + 1.08633i
714714 0 0
715715 19.9601 + 34.5718i 0.746464 + 1.29291i
716716 14.7245 0.550281
717717 0 0
718718 2.41531 0.0901385
719719 −16.5754 + 28.7095i −0.618159 + 1.07068i 0.371663 + 0.928368i 0.378788π0.378788\pi
−0.989822 + 0.142314i 0.954546π0.954546\pi
720720 0 0
721721 0 0
722722 3.85021 6.66877i 0.143290 0.248186i
723723 0 0
724724 −0.397460 + 0.688420i −0.0147715 + 0.0255849i
725725 −1.82450 + 3.16013i −0.0677603 + 0.117364i
726726 0 0
727727 −16.5502 + 28.6658i −0.613814 + 1.06316i 0.376777 + 0.926304i 0.377032π0.377032\pi
−0.990591 + 0.136853i 0.956301π0.956301\pi
728728 0 0
729729 0 0
730730 2.21737 3.84060i 0.0820685 0.142147i
731731 −2.84071 −0.105067
732732 0 0
733733 −44.5589 −1.64582 −0.822911 0.568170i 0.807651π-0.807651\pi
−0.822911 + 0.568170i 0.807651π0.807651\pi
734734 0.929636 + 1.61018i 0.0343135 + 0.0594327i
735735 0 0
736736 7.07442 12.2533i 0.260767 0.451661i
737737 17.2781 29.9266i 0.636448 1.10236i
738738 0 0
739739 −19.9045 34.4756i −0.732199 1.26821i −0.955941 0.293558i 0.905161π-0.905161\pi
0.223742 0.974648i 0.428173π-0.428173\pi
740740 −4.17984 7.23970i −0.153654 0.266137i
741741 0 0
742742 0 0
743743 5.37072 + 9.30237i 0.197033 + 0.341271i 0.947565 0.319563i 0.103536π-0.103536\pi
−0.750532 + 0.660834i 0.770203π0.770203\pi
744744 0 0
745745 15.7259 0.576152
746746 2.88276 + 4.99309i 0.105545 + 0.182810i
747747 0 0
748748 13.9021 0.508310
749749 0 0
750750 0 0
751751 19.7141 0.719378 0.359689 0.933072i 0.382883π-0.382883\pi
0.359689 + 0.933072i 0.382883π0.382883\pi
752752 −17.0899 + 29.6005i −0.623203 + 1.07942i
753753 0 0
754754 −0.935657 1.62060i −0.0340746 0.0590189i
755755 −11.6336 −0.423391
756756 0 0
757757 35.3549 1.28499 0.642497 0.766288i 0.277898π-0.277898\pi
0.642497 + 0.766288i 0.277898π0.277898\pi
758758 −1.59712 2.76629i −0.0580100 0.100476i
759759 0 0
760760 8.74269 15.1428i 0.317131 0.549286i
761761 39.1144 1.41790 0.708948 0.705261i 0.249170π-0.249170\pi
0.708948 + 0.705261i 0.249170π0.249170\pi
762762 0 0
763763 0 0
764764 31.1319 1.12631
765765 0 0
766766 −1.10480 1.91357i −0.0399180 0.0691399i
767767 −22.3354 −0.806486
768768 0 0
769769 18.9240 + 32.7773i 0.682415 + 1.18198i 0.974242 + 0.225507i 0.0724038π0.0724038\pi
−0.291826 + 0.956471i 0.594263π0.594263\pi
770770 0 0
771771 0 0
772772 −12.0205 20.8201i −0.432628 0.749333i
773773 14.9133 + 25.8305i 0.536393 + 0.929059i 0.999095 + 0.0425453i 0.0135467π0.0135467\pi
−0.462702 + 0.886514i 0.653120π0.653120\pi
774774 0 0
775775 −5.61224 + 9.72068i −0.201598 + 0.349177i
776776 1.44409 2.50124i 0.0518399 0.0897894i
777777 0 0
778778 −1.24923 2.16373i −0.0447870 0.0775734i
779779 −73.1229 −2.61990
780780 0 0
781781 26.0917 0.933633
782782 1.04842 1.81591i 0.0374913 0.0649369i
783783 0 0
784784 0 0
785785 −1.33336 + 2.30946i −0.0475898 + 0.0824280i
786786 0 0
787787 −8.81030 + 15.2599i −0.314053 + 0.543956i −0.979236 0.202724i 0.935020π-0.935020\pi
0.665182 + 0.746681i 0.268354π0.268354\pi
788788 −22.4416 + 38.8700i −0.799448 + 1.38468i
789789 0 0
790790 3.04563 5.27518i 0.108359 0.187683i
791791 0 0
792792 0 0
793793 14.7066 25.4725i 0.522246 0.904556i
794794 −0.0978390 −0.00347218
795795 0 0
796796 −13.1071 −0.464570
797797 5.06056 + 8.76515i 0.179254 + 0.310477i 0.941625 0.336663i 0.109298π-0.109298\pi
−0.762371 + 0.647140i 0.775965π0.775965\pi
798798 0 0
799799 −7.98921 + 13.8377i −0.282638 + 0.489543i
800800 2.37072 4.10621i 0.0838177 0.145177i
801801 0 0
802802 −1.82365 3.15865i −0.0643953 0.111536i
803803 −14.9622 25.9153i −0.528004 0.914529i
804804 0 0
805805 0 0
806806 −2.87812 4.98504i −0.101377 0.175591i
807807 0 0
808808 −10.4650 −0.368157
809809 −23.5735 40.8305i −0.828799 1.43552i −0.898981 0.437988i 0.855691π-0.855691\pi
0.0701816 0.997534i 0.477642π-0.477642\pi
810810 0 0
811811 21.0577 0.739435 0.369717 0.929144i 0.379454π-0.379454\pi
0.369717 + 0.929144i 0.379454π0.379454\pi
812812 0 0
813813 0 0
814814 1.66019 0.0581896
815815 8.84896 15.3269i 0.309966 0.536876i
816816 0 0
817817 5.93984 + 10.2881i 0.207809 + 0.359935i
818818 −1.46504 −0.0512238
819819 0 0
820820 −51.4563 −1.79693
821821 5.58018 + 9.66515i 0.194750 + 0.337316i 0.946818 0.321768i 0.104277π-0.104277\pi
−0.752069 + 0.659085i 0.770944π0.770944\pi
822822 0 0
823823 −4.71737 + 8.17072i −0.164437 + 0.284814i −0.936455 0.350787i 0.885914π-0.885914\pi
0.772018 + 0.635601i 0.219247π0.219247\pi
824824 −7.52933 −0.262297
825825 0 0
826826 0 0
827827 −17.2646 −0.600348 −0.300174 0.953884i 0.597045π-0.597045\pi
−0.300174 + 0.953884i 0.597045π0.597045\pi
828828 0 0
829829 24.2263 + 41.9612i 0.841415 + 1.45737i 0.888699 + 0.458492i 0.151610π0.151610\pi
−0.0472838 + 0.998881i 0.515057π0.515057\pi
830830 4.27301 0.148318
831831 0 0
832832 −12.2651 21.2438i −0.425215 0.736495i
833833 0 0
834834 0 0
835835 23.3187 + 40.3893i 0.806978 + 1.39773i
836836 −29.0688 50.3487i −1.00537 1.74135i
837837 0 0
838838 0.186860 0.323651i 0.00645497 0.0111803i
839839 −7.43429 + 12.8766i −0.256660 + 0.444548i −0.965345 0.260977i 0.915955π-0.915955\pi
0.708685 + 0.705525i 0.249289π0.249289\pi
840840 0 0
841841 12.2427 + 21.2050i 0.422162 + 0.731206i
842842 −5.56277 −0.191706
843843 0 0
844844 −32.8058 −1.12922
845845 0.732518 1.26876i 0.0251994 0.0436466i
846846 0 0
847847 0 0
848848 19.4841 33.7475i 0.669088 1.15889i
849849 0 0
850850 0.351337 0.608534i 0.0120508 0.0208725i
851851 −4.25404 + 7.36821i −0.145827 + 0.252579i
852852 0 0
853853 −3.99900 + 6.92648i −0.136923 + 0.237158i −0.926331 0.376712i 0.877055π-0.877055\pi
0.789407 + 0.613870i 0.210388π0.210388\pi
854854 0 0
855855 0 0
856856 1.86513 3.23050i 0.0637488 0.110416i
857857 43.1322 1.47337 0.736684 0.676237i 0.236390π-0.236390\pi
0.736684 + 0.676237i 0.236390π0.236390\pi
858858 0 0
859859 −2.44359 −0.0833742 −0.0416871 0.999131i 0.513273π-0.513273\pi
−0.0416871 + 0.999131i 0.513273π0.513273\pi
860860 4.17984 + 7.23970i 0.142531 + 0.246872i
861861 0 0
862862 −0.120250 + 0.208279i −0.00409573 + 0.00709401i
863863 12.8594 22.2731i 0.437738 0.758185i −0.559777 0.828644i 0.689113π-0.689113\pi
0.997515 + 0.0704589i 0.0224464π0.0224464\pi
864864 0 0
865865 −1.07605 1.86378i −0.0365870 0.0633705i
866866 −1.56711 2.71432i −0.0532526 0.0922362i
867867 0 0
868868 0 0
869869 −20.5510 35.5954i −0.697146 1.20749i
870870 0 0
871871 −30.4342 −1.03122
872872 −3.42395 5.93045i −0.115949 0.200830i
873873 0 0
874874 −8.76884 −0.296611
875875 0 0
876876 0 0
877877 −21.9590 −0.741502 −0.370751 0.928732i 0.620900π-0.620900\pi
−0.370751 + 0.928732i 0.620900π0.620900\pi
878878 −2.22553 + 3.85473i −0.0751080 + 0.130091i
879879 0 0
880880 −19.8359 34.3567i −0.668667 1.15817i
881881 35.0576 1.18112 0.590560 0.806994i 0.298907π-0.298907\pi
0.590560 + 0.806994i 0.298907π0.298907\pi
882882 0 0
883883 26.3009 0.885097 0.442549 0.896744i 0.354074π-0.354074\pi
0.442549 + 0.896744i 0.354074π0.354074\pi
884884 −6.12188 10.6034i −0.205901 0.356631i
885885 0 0
886886 0.133790 0.231731i 0.00449477 0.00778517i
887887 −47.8180 −1.60557 −0.802785 0.596269i 0.796649π-0.796649\pi
−0.802785 + 0.596269i 0.796649π0.796649\pi
888888 0 0
889889 0 0
890890 −3.12071 −0.104607
891891 0 0
892892 4.37272 + 7.57378i 0.146410 + 0.253589i
893893 66.8208 2.23607
894894 0 0
895895 −9.82150 17.0113i −0.328296 0.568626i
896896 0 0
897897 0 0
898898 4.71574 + 8.16789i 0.157366 + 0.272566i
899899 −6.94357 12.0266i −0.231581 0.401110i
900900 0 0
901901 9.10848 15.7764i 0.303448 0.525587i
902902 5.10948 8.84988i 0.170127 0.294669i
903903 0 0
904904 3.26647 + 5.65769i 0.108641 + 0.188172i
905905 1.06045 0.0352505
906906 0 0
907907 −19.1144 −0.634682 −0.317341 0.948312i 0.602790π-0.602790\pi
−0.317341 + 0.948312i 0.602790π0.602790\pi
908908 −5.89092 + 10.2034i −0.195497 + 0.338611i
909909 0 0
910910 0 0
911911 −9.02928 + 15.6392i −0.299153 + 0.518149i −0.975942 0.218028i 0.930038π-0.930038\pi
0.676789 + 0.736177i 0.263371π0.263371\pi
912912 0 0
913913 14.4165 24.9701i 0.477117 0.826391i
914914 −4.09385 + 7.09076i −0.135413 + 0.234541i
915915 0 0
916916 10.7334 18.5908i 0.354642 0.614258i
917917 0 0
918918 0 0
919919 −8.10464 + 14.0377i −0.267348 + 0.463060i −0.968176 0.250270i 0.919481π-0.919481\pi
0.700828 + 0.713330i 0.252814π0.252814\pi
920920 −12.5228 −0.412865
921921 0 0
922922 −4.87514 −0.160554
923923 −11.4897 19.9007i −0.378187 0.655039i
924924 0 0
925925 −1.42558 + 2.46918i −0.0468728 + 0.0811860i
926926 0.814099 1.41006i 0.0267529 0.0463375i
927927 0 0
928928 2.93310 + 5.08029i 0.0962839 + 0.166769i
929929 11.3415 + 19.6440i 0.372102 + 0.644499i 0.989889 0.141846i 0.0453039π-0.0453039\pi
−0.617787 + 0.786345i 0.711971π0.711971\pi
930930 0 0
931931 0 0
932932 7.90576 + 13.6932i 0.258962 + 0.448535i
933933 0 0
934934 −5.92814 −0.193975
935935 −9.27292 16.0612i −0.303257 0.525256i
936936 0 0
937937 −51.2933 −1.67568 −0.837840 0.545915i 0.816182π-0.816182\pi
−0.837840 + 0.545915i 0.816182π0.816182\pi
938938 0 0
939939 0 0
940940 47.0216 1.53368
941941 15.9659 27.6538i 0.520474 0.901487i −0.479243 0.877682i 0.659089π-0.659089\pi
0.999717 0.0238048i 0.00757801π-0.00757801\pi
942942 0 0
943943 26.1849 + 45.3535i 0.852697 + 1.47691i
944944 22.1965 0.722433
945945 0 0
946946 −1.66019 −0.0539774
947947 −2.24665 3.89131i −0.0730063 0.126451i 0.827211 0.561891i 0.189926π-0.189926\pi
−0.900218 + 0.435440i 0.856593π0.856593\pi
948948 0 0
949949 −13.1774 + 22.8240i −0.427757 + 0.740898i
950950 −2.93854 −0.0953390
951951 0 0
952952 0 0
953953 −1.14635 −0.0371340 −0.0185670 0.999828i 0.505910π-0.505910\pi
−0.0185670 + 0.999828i 0.505910π0.505910\pi
954954 0 0
955955 −20.7655 35.9669i −0.671956 1.16386i
956956 41.1150 1.32975
957957 0 0
958958 −1.32710 2.29860i −0.0428765 0.0742643i
959959 0 0
960960 0 0
961961 −5.85868 10.1475i −0.188990 0.327340i
962962 −0.731078 1.26626i −0.0235709 0.0408260i
963963 0 0
964964 −13.3000 + 23.0363i −0.428365 + 0.741950i
965965 −16.0358 + 27.7748i −0.516210 + 0.894102i
966966 0 0
967967 −24.8080 42.9686i −0.797770 1.38178i −0.921065 0.389408i 0.872680π-0.872680\pi
0.123295 0.992370i 0.460654π-0.460654\pi
968968 6.11763 0.196628
969969 0 0
970970 −1.89853 −0.0609582
971971 2.56661 4.44550i 0.0823664 0.142663i −0.821900 0.569632i 0.807086π-0.807086\pi
0.904266 + 0.426970i 0.140419π0.140419\pi
972972 0 0
973973 0 0
974974 −1.20013 + 2.07869i −0.0384546 + 0.0666054i
975975 0 0
976976 −14.6151 + 25.3141i −0.467817 + 0.810283i
977977 15.5974 27.0155i 0.499006 0.864303i −0.500994 0.865451i 0.667032π-0.667032\pi
0.999999 + 0.00114787i 0.000365378π0.000365378\pi
978978 0 0
979979 −10.5288 + 18.2365i −0.336503 + 0.582840i
980980 0 0
981981 0 0
982982 1.48113 2.56538i 0.0472646 0.0818647i
983983 20.3401 0.648748 0.324374 0.945929i 0.394846π-0.394846\pi
0.324374 + 0.945929i 0.394846π0.394846\pi
984984 0 0
985985 59.8757 1.90780
986986 0.434681 + 0.752890i 0.0138431 + 0.0239769i
987987 0 0
988988 −25.6014 + 44.3429i −0.814488 + 1.41074i
989989 4.25404 7.36821i 0.135271 0.234296i
990990 0 0
991991 6.48276 + 11.2285i 0.205932 + 0.356684i 0.950429 0.310941i 0.100644π-0.100644\pi
−0.744498 + 0.667625i 0.767311π0.767311\pi
992992 9.02234 + 15.6272i 0.286460 + 0.496163i
993993 0 0
994994 0 0
995995 8.74269 + 15.1428i 0.277162 + 0.480059i
996996 0 0
997997 49.4816 1.56710 0.783548 0.621331i 0.213408π-0.213408\pi
0.783548 + 0.621331i 0.213408π0.213408\pi
998998 1.22218 + 2.11688i 0.0386875 + 0.0670086i
999999 0 0
Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1323.2.g.g.667.3 12
3.2 odd 2 441.2.g.g.79.4 12
7.2 even 3 1323.2.f.g.883.4 12
7.3 odd 6 1323.2.h.g.802.3 12
7.4 even 3 1323.2.h.g.802.4 12
7.5 odd 6 1323.2.f.g.883.3 12
7.6 odd 2 inner 1323.2.g.g.667.4 12
9.4 even 3 1323.2.h.g.226.4 12
9.5 odd 6 441.2.h.g.373.3 12
21.2 odd 6 441.2.f.g.295.4 yes 12
21.5 even 6 441.2.f.g.295.3 yes 12
21.11 odd 6 441.2.h.g.214.3 12
21.17 even 6 441.2.h.g.214.4 12
21.20 even 2 441.2.g.g.79.3 12
63.2 odd 6 3969.2.a.be.1.4 6
63.4 even 3 inner 1323.2.g.g.361.3 12
63.5 even 6 441.2.f.g.148.3 12
63.13 odd 6 1323.2.h.g.226.3 12
63.16 even 3 3969.2.a.bd.1.3 6
63.23 odd 6 441.2.f.g.148.4 yes 12
63.31 odd 6 inner 1323.2.g.g.361.4 12
63.32 odd 6 441.2.g.g.67.4 12
63.40 odd 6 1323.2.f.g.442.3 12
63.41 even 6 441.2.h.g.373.4 12
63.47 even 6 3969.2.a.be.1.3 6
63.58 even 3 1323.2.f.g.442.4 12
63.59 even 6 441.2.g.g.67.3 12
63.61 odd 6 3969.2.a.bd.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.2.f.g.148.3 12 63.5 even 6
441.2.f.g.148.4 yes 12 63.23 odd 6
441.2.f.g.295.3 yes 12 21.5 even 6
441.2.f.g.295.4 yes 12 21.2 odd 6
441.2.g.g.67.3 12 63.59 even 6
441.2.g.g.67.4 12 63.32 odd 6
441.2.g.g.79.3 12 21.20 even 2
441.2.g.g.79.4 12 3.2 odd 2
441.2.h.g.214.3 12 21.11 odd 6
441.2.h.g.214.4 12 21.17 even 6
441.2.h.g.373.3 12 9.5 odd 6
441.2.h.g.373.4 12 63.41 even 6
1323.2.f.g.442.3 12 63.40 odd 6
1323.2.f.g.442.4 12 63.58 even 3
1323.2.f.g.883.3 12 7.5 odd 6
1323.2.f.g.883.4 12 7.2 even 3
1323.2.g.g.361.3 12 63.4 even 3 inner
1323.2.g.g.361.4 12 63.31 odd 6 inner
1323.2.g.g.667.3 12 1.1 even 1 trivial
1323.2.g.g.667.4 12 7.6 odd 2 inner
1323.2.h.g.226.3 12 63.13 odd 6
1323.2.h.g.226.4 12 9.4 even 3
1323.2.h.g.802.3 12 7.3 odd 6
1323.2.h.g.802.4 12 7.4 even 3
3969.2.a.bd.1.3 6 63.16 even 3
3969.2.a.bd.1.4 6 63.61 odd 6
3969.2.a.be.1.3 6 63.47 even 6
3969.2.a.be.1.4 6 63.2 odd 6