Properties

Label 3971.2.a.k.1.5
Level 39713971
Weight 22
Character 3971.1
Self dual yes
Analytic conductor 31.70931.709
Analytic rank 11
Dimension 99
CM no
Inner twists 11

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3971,2,Mod(1,3971)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3971, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3971.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 3971=11192 3971 = 11 \cdot 19^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 3971.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 31.708594642731.7085946427
Analytic rank: 11
Dimension: 99
Coefficient field: Q[x]/(x9)\mathbb{Q}[x]/(x^{9} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x9x813x7+12x6+51x538x470x3+30x2+27x+3 x^{9} - x^{8} - 13x^{7} + 12x^{6} + 51x^{5} - 38x^{4} - 70x^{3} + 30x^{2} + 27x + 3 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 209)
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.5
Root 0.138860-0.138860 of defining polynomial
Character χ\chi == 3971.1

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q+0.138860q21.49679q31.98072q43.51650q50.207844q61.46902q70.552764q80.759634q90.488302q101.00000q11+2.96471q12+2.39255q130.203989q14+5.26344q15+3.88468q163.24041q170.105483q18+6.96519q20+2.19881q210.138860q227.38177q23+0.827369q24+7.36576q25+0.332231q26+5.62736q27+2.90971q28+6.40954q29+0.730884q30+0.866600q31+1.64496q32+1.49679q330.449965q34+5.16581q35+1.50462q362.78244q373.58113q39+1.94379q40+9.85358q41+0.305327q42+7.30504q43+1.98072q44+2.67125q451.02504q46+3.35902q475.81453q484.84198q49+1.02281q50+4.85020q514.73897q523.78360q53+0.781418q54+3.51650q55+0.812022q56+0.890031q58+8.40207q5910.4254q601.43767q61+0.120336q62+1.11592q637.54094q648.41340q65+0.207844q66+6.48612q67+6.41834q68+11.0489q69+0.717326q703.41568q71+0.419898q72+2.81088q730.386370q7411.0250q75+1.46902q770.497278q7817.7188q7913.6605q806.14405q81+1.36827q82+17.7129q834.35522q84+11.3949q85+1.01438q869.59370q87+0.552764q88+12.8724q89+0.370931q903.51471q91+14.6212q921.29711q93+0.466435q942.46215q9615.7259q970.672359q98+0.759634q99+O(q100)q+0.138860 q^{2} -1.49679 q^{3} -1.98072 q^{4} -3.51650 q^{5} -0.207844 q^{6} -1.46902 q^{7} -0.552764 q^{8} -0.759634 q^{9} -0.488302 q^{10} -1.00000 q^{11} +2.96471 q^{12} +2.39255 q^{13} -0.203989 q^{14} +5.26344 q^{15} +3.88468 q^{16} -3.24041 q^{17} -0.105483 q^{18} +6.96519 q^{20} +2.19881 q^{21} -0.138860 q^{22} -7.38177 q^{23} +0.827369 q^{24} +7.36576 q^{25} +0.332231 q^{26} +5.62736 q^{27} +2.90971 q^{28} +6.40954 q^{29} +0.730884 q^{30} +0.866600 q^{31} +1.64496 q^{32} +1.49679 q^{33} -0.449965 q^{34} +5.16581 q^{35} +1.50462 q^{36} -2.78244 q^{37} -3.58113 q^{39} +1.94379 q^{40} +9.85358 q^{41} +0.305327 q^{42} +7.30504 q^{43} +1.98072 q^{44} +2.67125 q^{45} -1.02504 q^{46} +3.35902 q^{47} -5.81453 q^{48} -4.84198 q^{49} +1.02281 q^{50} +4.85020 q^{51} -4.73897 q^{52} -3.78360 q^{53} +0.781418 q^{54} +3.51650 q^{55} +0.812022 q^{56} +0.890031 q^{58} +8.40207 q^{59} -10.4254 q^{60} -1.43767 q^{61} +0.120336 q^{62} +1.11592 q^{63} -7.54094 q^{64} -8.41340 q^{65} +0.207844 q^{66} +6.48612 q^{67} +6.41834 q^{68} +11.0489 q^{69} +0.717326 q^{70} -3.41568 q^{71} +0.419898 q^{72} +2.81088 q^{73} -0.386370 q^{74} -11.0250 q^{75} +1.46902 q^{77} -0.497278 q^{78} -17.7188 q^{79} -13.6605 q^{80} -6.14405 q^{81} +1.36827 q^{82} +17.7129 q^{83} -4.35522 q^{84} +11.3949 q^{85} +1.01438 q^{86} -9.59370 q^{87} +0.552764 q^{88} +12.8724 q^{89} +0.370931 q^{90} -3.51471 q^{91} +14.6212 q^{92} -1.29711 q^{93} +0.466435 q^{94} -2.46215 q^{96} -15.7259 q^{97} -0.672359 q^{98} +0.759634 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 9qq2+9q4+3q63q7+11q921q109q11+8q1211q13q147q15+13q166q174q18+2q2024q21+q22+q23+3q24+11q99+O(q100) 9 q - q^{2} + 9 q^{4} + 3 q^{6} - 3 q^{7} + 11 q^{9} - 21 q^{10} - 9 q^{11} + 8 q^{12} - 11 q^{13} - q^{14} - 7 q^{15} + 13 q^{16} - 6 q^{17} - 4 q^{18} + 2 q^{20} - 24 q^{21} + q^{22} + q^{23} + 3 q^{24}+ \cdots - 11 q^{99}+O(q^{100}) Copy content Toggle raw display

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 0.138860 0.0981891 0.0490946 0.998794i 0.484366π-0.484366\pi
0.0490946 + 0.998794i 0.484366π0.484366\pi
33 −1.49679 −0.864169 −0.432085 0.901833i 0.642222π-0.642222\pi
−0.432085 + 0.901833i 0.642222π0.642222\pi
44 −1.98072 −0.990359
55 −3.51650 −1.57263 −0.786313 0.617828i 0.788013π-0.788013\pi
−0.786313 + 0.617828i 0.788013π0.788013\pi
66 −0.207844 −0.0848520
77 −1.46902 −0.555237 −0.277619 0.960691i 0.589545π-0.589545\pi
−0.277619 + 0.960691i 0.589545π0.589545\pi
88 −0.552764 −0.195432
99 −0.759634 −0.253211
1010 −0.488302 −0.154415
1111 −1.00000 −0.301511
1212 2.96471 0.855838
1313 2.39255 0.663574 0.331787 0.943354i 0.392348π-0.392348\pi
0.331787 + 0.943354i 0.392348π0.392348\pi
1414 −0.203989 −0.0545183
1515 5.26344 1.35902
1616 3.88468 0.971170
1717 −3.24041 −0.785915 −0.392957 0.919557i 0.628548π-0.628548\pi
−0.392957 + 0.919557i 0.628548π0.628548\pi
1818 −0.105483 −0.0248626
1919 0 0
2020 6.96519 1.55746
2121 2.19881 0.479819
2222 −0.138860 −0.0296051
2323 −7.38177 −1.53921 −0.769603 0.638523i 0.779546π-0.779546\pi
−0.769603 + 0.638523i 0.779546π0.779546\pi
2424 0.827369 0.168886
2525 7.36576 1.47315
2626 0.332231 0.0651558
2727 5.62736 1.08299
2828 2.90971 0.549884
2929 6.40954 1.19022 0.595111 0.803644i 0.297108π-0.297108\pi
0.595111 + 0.803644i 0.297108π0.297108\pi
3030 0.730884 0.133441
3131 0.866600 0.155646 0.0778230 0.996967i 0.475203π-0.475203\pi
0.0778230 + 0.996967i 0.475203π0.475203\pi
3232 1.64496 0.290790
3333 1.49679 0.260557
3434 −0.449965 −0.0771683
3535 5.16581 0.873181
3636 1.50462 0.250770
3737 −2.78244 −0.457430 −0.228715 0.973493i 0.573452π-0.573452\pi
−0.228715 + 0.973493i 0.573452π0.573452\pi
3838 0 0
3939 −3.58113 −0.573440
4040 1.94379 0.307341
4141 9.85358 1.53887 0.769435 0.638725i 0.220538π-0.220538\pi
0.769435 + 0.638725i 0.220538π0.220538\pi
4242 0.305327 0.0471130
4343 7.30504 1.11401 0.557004 0.830510i 0.311951π-0.311951\pi
0.557004 + 0.830510i 0.311951π0.311951\pi
4444 1.98072 0.298604
4545 2.67125 0.398207
4646 −1.02504 −0.151133
4747 3.35902 0.489964 0.244982 0.969528i 0.421218π-0.421218\pi
0.244982 + 0.969528i 0.421218π0.421218\pi
4848 −5.81453 −0.839255
4949 −4.84198 −0.691711
5050 1.02281 0.144648
5151 4.85020 0.679164
5252 −4.73897 −0.657177
5353 −3.78360 −0.519718 −0.259859 0.965647i 0.583676π-0.583676\pi
−0.259859 + 0.965647i 0.583676π0.583676\pi
5454 0.781418 0.106338
5555 3.51650 0.474165
5656 0.812022 0.108511
5757 0 0
5858 0.890031 0.116867
5959 8.40207 1.09386 0.546928 0.837180i 0.315797π-0.315797\pi
0.546928 + 0.837180i 0.315797π0.315797\pi
6060 −10.4254 −1.34591
6161 −1.43767 −0.184075 −0.0920375 0.995756i 0.529338π-0.529338\pi
−0.0920375 + 0.995756i 0.529338π0.529338\pi
6262 0.120336 0.0152827
6363 1.11592 0.140592
6464 −7.54094 −0.942617
6565 −8.41340 −1.04355
6666 0.207844 0.0255839
6767 6.48612 0.792406 0.396203 0.918163i 0.370328π-0.370328\pi
0.396203 + 0.918163i 0.370328π0.370328\pi
6868 6.41834 0.778338
6969 11.0489 1.33013
7070 0.717326 0.0857369
7171 −3.41568 −0.405367 −0.202683 0.979244i 0.564966π-0.564966\pi
−0.202683 + 0.979244i 0.564966π0.564966\pi
7272 0.419898 0.0494855
7373 2.81088 0.328989 0.164495 0.986378i 0.447401π-0.447401\pi
0.164495 + 0.986378i 0.447401π0.447401\pi
7474 −0.386370 −0.0449146
7575 −11.0250 −1.27305
7676 0 0
7777 1.46902 0.167410
7878 −0.497278 −0.0563056
7979 −17.7188 −1.99353 −0.996763 0.0803939i 0.974382π-0.974382\pi
−0.996763 + 0.0803939i 0.974382π0.974382\pi
8080 −13.6605 −1.52729
8181 −6.14405 −0.682673
8282 1.36827 0.151100
8383 17.7129 1.94425 0.972124 0.234467i 0.0753345π-0.0753345\pi
0.972124 + 0.234467i 0.0753345π0.0753345\pi
8484 −4.35522 −0.475193
8585 11.3949 1.23595
8686 1.01438 0.109383
8787 −9.59370 −1.02855
8888 0.552764 0.0589249
8989 12.8724 1.36447 0.682236 0.731132i 0.261007π-0.261007\pi
0.682236 + 0.731132i 0.261007π0.261007\pi
9090 0.370931 0.0390996
9191 −3.51471 −0.368441
9292 14.6212 1.52437
9393 −1.29711 −0.134505
9494 0.466435 0.0481091
9595 0 0
9696 −2.46215 −0.251292
9797 −15.7259 −1.59673 −0.798364 0.602175i 0.794301π-0.794301\pi
−0.798364 + 0.602175i 0.794301π0.794301\pi
9898 −0.672359 −0.0679185
9999 0.759634 0.0763461
100100 −14.5895 −1.45895
101101 −7.83751 −0.779861 −0.389931 0.920844i 0.627501π-0.627501\pi
−0.389931 + 0.920844i 0.627501π0.627501\pi
102102 0.673501 0.0666865
103103 −7.27715 −0.717039 −0.358519 0.933522i 0.616718π-0.616718\pi
−0.358519 + 0.933522i 0.616718π0.616718\pi
104104 −1.32252 −0.129683
105105 −7.73211 −0.754576
106106 −0.525392 −0.0510306
107107 6.23745 0.602997 0.301498 0.953467i 0.402513π-0.402513\pi
0.301498 + 0.953467i 0.402513π0.402513\pi
108108 −11.1462 −1.07255
109109 2.71842 0.260377 0.130189 0.991489i 0.458442π-0.458442\pi
0.130189 + 0.991489i 0.458442π0.458442\pi
110110 0.488302 0.0465578
111111 4.16471 0.395297
112112 −5.70667 −0.539230
113113 0.486637 0.0457789 0.0228895 0.999738i 0.492713π-0.492713\pi
0.0228895 + 0.999738i 0.492713π0.492713\pi
114114 0 0
115115 25.9580 2.42060
116116 −12.6955 −1.17875
117117 −1.81746 −0.168025
118118 1.16671 0.107405
119119 4.76023 0.436369
120120 −2.90944 −0.265595
121121 1.00000 0.0909091
122122 −0.199636 −0.0180742
123123 −14.7487 −1.32984
124124 −1.71649 −0.154145
125125 −8.31921 −0.744093
126126 0.154957 0.0138046
127127 3.68254 0.326773 0.163386 0.986562i 0.447758π-0.447758\pi
0.163386 + 0.986562i 0.447758π0.447758\pi
128128 −4.33705 −0.383345
129129 −10.9341 −0.962691
130130 −1.16829 −0.102466
131131 0.0541158 0.00472812 0.00236406 0.999997i 0.499247π-0.499247\pi
0.00236406 + 0.999997i 0.499247π0.499247\pi
132132 −2.96471 −0.258045
133133 0 0
134134 0.900665 0.0778056
135135 −19.7886 −1.70313
136136 1.79118 0.153593
137137 16.9727 1.45008 0.725038 0.688709i 0.241822π-0.241822\pi
0.725038 + 0.688709i 0.241822π0.241822\pi
138138 1.53426 0.130605
139139 −18.0362 −1.52981 −0.764907 0.644141i 0.777215π-0.777215\pi
−0.764907 + 0.644141i 0.777215π0.777215\pi
140140 −10.2320 −0.864763
141141 −5.02773 −0.423412
142142 −0.474303 −0.0398026
143143 −2.39255 −0.200075
144144 −2.95093 −0.245911
145145 −22.5391 −1.87177
146146 0.390320 0.0323031
147147 7.24740 0.597756
148148 5.51122 0.453020
149149 −5.30375 −0.434501 −0.217250 0.976116i 0.569709π-0.569709\pi
−0.217250 + 0.976116i 0.569709π0.569709\pi
150150 −1.53093 −0.125000
151151 −10.8143 −0.880057 −0.440029 0.897984i 0.645032π-0.645032\pi
−0.440029 + 0.897984i 0.645032π0.645032\pi
152152 0 0
153153 2.46153 0.199003
154154 0.203989 0.0164379
155155 −3.04740 −0.244773
156156 7.09322 0.567912
157157 3.19528 0.255011 0.127506 0.991838i 0.459303π-0.459303\pi
0.127506 + 0.991838i 0.459303π0.459303\pi
158158 −2.46045 −0.195743
159159 5.66324 0.449124
160160 −5.78449 −0.457304
161161 10.8440 0.854625
162162 −0.853166 −0.0670310
163163 −14.3438 −1.12349 −0.561747 0.827309i 0.689871π-0.689871\pi
−0.561747 + 0.827309i 0.689871π0.689871\pi
164164 −19.5172 −1.52403
165165 −5.26344 −0.409759
166166 2.45963 0.190904
167167 15.2567 1.18060 0.590298 0.807186i 0.299010π-0.299010\pi
0.590298 + 0.807186i 0.299010π0.299010\pi
168168 −1.21542 −0.0937718
169169 −7.27570 −0.559669
170170 1.58230 0.121357
171171 0 0
172172 −14.4692 −1.10327
173173 −21.6749 −1.64791 −0.823954 0.566656i 0.808237π-0.808237\pi
−0.823954 + 0.566656i 0.808237π0.808237\pi
174174 −1.33219 −0.100993
175175 −10.8205 −0.817950
176176 −3.88468 −0.292819
177177 −12.5761 −0.945277
178178 1.78747 0.133976
179179 4.19425 0.313493 0.156746 0.987639i 0.449899π-0.449899\pi
0.156746 + 0.987639i 0.449899π0.449899\pi
180180 −5.29100 −0.394368
181181 −24.9278 −1.85287 −0.926436 0.376453i 0.877144π-0.877144\pi
−0.926436 + 0.376453i 0.877144π0.877144\pi
182182 −0.488053 −0.0361769
183183 2.15189 0.159072
184184 4.08038 0.300810
185185 9.78444 0.719366
186186 −0.180118 −0.0132069
187187 3.24041 0.236962
188188 −6.65327 −0.485240
189189 −8.26671 −0.601315
190190 0 0
191191 −7.74450 −0.560372 −0.280186 0.959946i 0.590396π-0.590396\pi
−0.280186 + 0.959946i 0.590396π0.590396\pi
192192 11.2872 0.814581
193193 13.9096 1.00123 0.500617 0.865669i 0.333106π-0.333106\pi
0.500617 + 0.865669i 0.333106π0.333106\pi
194194 −2.18371 −0.156781
195195 12.5931 0.901807
196196 9.59059 0.685042
197197 20.3595 1.45056 0.725279 0.688455i 0.241711π-0.241711\pi
0.725279 + 0.688455i 0.241711π0.241711\pi
198198 0.105483 0.00749636
199199 3.03426 0.215093 0.107546 0.994200i 0.465701π-0.465701\pi
0.107546 + 0.994200i 0.465701π0.465701\pi
200200 −4.07153 −0.287901
201201 −9.70833 −0.684773
202202 −1.08832 −0.0765739
203203 −9.41574 −0.660855
204204 −9.60687 −0.672616
205205 −34.6501 −2.42007
206206 −1.01051 −0.0704054
207207 5.60745 0.389744
208208 9.29429 0.644443
209209 0 0
210210 −1.07368 −0.0740912
211211 5.01758 0.345425 0.172712 0.984972i 0.444747π-0.444747\pi
0.172712 + 0.984972i 0.444747π0.444747\pi
212212 7.49425 0.514707
213213 5.11254 0.350305
214214 0.866134 0.0592077
215215 −25.6882 −1.75192
216216 −3.11061 −0.211650
217217 −1.27305 −0.0864205
218218 0.377481 0.0255662
219219 −4.20729 −0.284302
220220 −6.96519 −0.469593
221221 −7.75285 −0.521513
222222 0.578313 0.0388139
223223 −11.8616 −0.794314 −0.397157 0.917751i 0.630003π-0.630003\pi
−0.397157 + 0.917751i 0.630003π0.630003\pi
224224 −2.41647 −0.161457
225225 −5.59529 −0.373019
226226 0.0675746 0.00449499
227227 −1.40838 −0.0934777 −0.0467388 0.998907i 0.514883π-0.514883\pi
−0.0467388 + 0.998907i 0.514883π0.514883\pi
228228 0 0
229229 11.3642 0.750971 0.375485 0.926828i 0.377476π-0.377476\pi
0.375485 + 0.926828i 0.377476π0.377476\pi
230230 3.60454 0.237676
231231 −2.19881 −0.144671
232232 −3.54296 −0.232607
233233 −12.8001 −0.838563 −0.419281 0.907856i 0.637718π-0.637718\pi
−0.419281 + 0.907856i 0.637718π0.637718\pi
234234 −0.252374 −0.0164982
235235 −11.8120 −0.770530
236236 −16.6421 −1.08331
237237 26.5213 1.72274
238238 0.661007 0.0428467
239239 −3.62977 −0.234790 −0.117395 0.993085i 0.537454π-0.537454\pi
−0.117395 + 0.993085i 0.537454π0.537454\pi
240240 20.4468 1.31983
241241 19.3216 1.24461 0.622306 0.782774i 0.286196π-0.286196\pi
0.622306 + 0.782774i 0.286196π0.286196\pi
242242 0.138860 0.00892629
243243 −7.68576 −0.493042
244244 2.84762 0.182300
245245 17.0268 1.08780
246246 −2.04801 −0.130576
247247 0 0
248248 −0.479026 −0.0304182
249249 −26.5125 −1.68016
250250 −1.15521 −0.0730618
251251 18.1298 1.14434 0.572172 0.820133i 0.306101π-0.306101\pi
0.572172 + 0.820133i 0.306101π0.306101\pi
252252 −2.21032 −0.139237
253253 7.38177 0.464088
254254 0.511360 0.0320856
255255 −17.0557 −1.06807
256256 14.4796 0.904977
257257 1.55938 0.0972713 0.0486357 0.998817i 0.484513π-0.484513\pi
0.0486357 + 0.998817i 0.484513π0.484513\pi
258258 −1.51831 −0.0945258
259259 4.08746 0.253982
260260 16.6646 1.03349
261261 −4.86890 −0.301378
262262 0.00751455 0.000464250 0
263263 17.2721 1.06504 0.532522 0.846416i 0.321244π-0.321244\pi
0.532522 + 0.846416i 0.321244π0.321244\pi
264264 −0.827369 −0.0509211
265265 13.3050 0.817321
266266 0 0
267267 −19.2672 −1.17914
268268 −12.8472 −0.784766
269269 −13.9309 −0.849384 −0.424692 0.905338i 0.639618π-0.639618\pi
−0.424692 + 0.905338i 0.639618π0.639618\pi
270270 −2.74786 −0.167229
271271 6.15305 0.373771 0.186886 0.982382i 0.440161π-0.440161\pi
0.186886 + 0.982382i 0.440161π0.440161\pi
272272 −12.5880 −0.763257
273273 5.26076 0.318396
274274 2.35684 0.142382
275275 −7.36576 −0.444172
276276 −21.8848 −1.31731
277277 −18.4147 −1.10643 −0.553215 0.833039i 0.686599π-0.686599\pi
−0.553215 + 0.833039i 0.686599π0.686599\pi
278278 −2.50452 −0.150211
279279 −0.658299 −0.0394113
280280 −2.85547 −0.170647
281281 24.8330 1.48141 0.740706 0.671830i 0.234491π-0.234491\pi
0.740706 + 0.671830i 0.234491π0.234491\pi
282282 −0.698153 −0.0415744
283283 17.1761 1.02101 0.510506 0.859874i 0.329458π-0.329458\pi
0.510506 + 0.859874i 0.329458π0.329458\pi
284284 6.76550 0.401458
285285 0 0
286286 −0.332231 −0.0196452
287287 −14.4751 −0.854438
288288 −1.24956 −0.0736313
289289 −6.49974 −0.382338
290290 −3.12979 −0.183788
291291 23.5384 1.37984
292292 −5.56757 −0.325817
293293 23.1770 1.35402 0.677008 0.735976i 0.263276π-0.263276\pi
0.677008 + 0.735976i 0.263276π0.263276\pi
294294 1.00638 0.0586931
295295 −29.5459 −1.72023
296296 1.53803 0.0893963
297297 −5.62736 −0.326533
298298 −0.736482 −0.0426632
299299 −17.6613 −1.02138
300300 21.8373 1.26078
301301 −10.7312 −0.618539
302302 −1.50168 −0.0864121
303303 11.7311 0.673932
304304 0 0
305305 5.05557 0.289481
306306 0.341809 0.0195399
307307 26.3151 1.50188 0.750940 0.660370i 0.229601π-0.229601\pi
0.750940 + 0.660370i 0.229601π0.229601\pi
308308 −2.90971 −0.165796
309309 10.8923 0.619643
310310 −0.423163 −0.0240341
311311 25.2900 1.43407 0.717033 0.697039i 0.245500π-0.245500\pi
0.717033 + 0.697039i 0.245500π0.245500\pi
312312 1.97952 0.112068
313313 −27.6880 −1.56502 −0.782509 0.622640i 0.786060π-0.786060\pi
−0.782509 + 0.622640i 0.786060π0.786060\pi
314314 0.443698 0.0250393
315315 −3.92412 −0.221099
316316 35.0960 1.97431
317317 10.1236 0.568599 0.284299 0.958736i 0.408239π-0.408239\pi
0.284299 + 0.958736i 0.408239π0.408239\pi
318318 0.786400 0.0440991
319319 −6.40954 −0.358865
320320 26.5177 1.48238
321321 −9.33612 −0.521091
322322 1.50580 0.0839149
323323 0 0
324324 12.1696 0.676091
325325 17.6230 0.977546
326326 −1.99179 −0.110315
327327 −4.06889 −0.225010
328328 −5.44670 −0.300744
329329 −4.93447 −0.272046
330330 −0.730884 −0.0402338
331331 8.45572 0.464769 0.232384 0.972624i 0.425347π-0.425347\pi
0.232384 + 0.972624i 0.425347π0.425347\pi
332332 −35.0843 −1.92550
333333 2.11363 0.115826
334334 2.11855 0.115922
335335 −22.8084 −1.24616
336336 8.54166 0.465986
337337 0.0173781 0.000946646 0 0.000473323 1.00000i 0.499849π-0.499849\pi
0.000473323 1.00000i 0.499849π0.499849\pi
338338 −1.01031 −0.0549534
339339 −0.728391 −0.0395608
340340 −22.5701 −1.22403
341341 −0.866600 −0.0469290
342342 0 0
343343 17.3961 0.939302
344344 −4.03796 −0.217712
345345 −38.8535 −2.09180
346346 −3.00978 −0.161807
347347 8.63862 0.463746 0.231873 0.972746i 0.425515π-0.425515\pi
0.231873 + 0.972746i 0.425515π0.425515\pi
348348 19.0024 1.01864
349349 −3.41607 −0.182858 −0.0914289 0.995812i 0.529143π-0.529143\pi
−0.0914289 + 0.995812i 0.529143π0.529143\pi
350350 −1.50253 −0.0803138
351351 13.4638 0.718642
352352 −1.64496 −0.0876765
353353 −10.9807 −0.584443 −0.292221 0.956351i 0.594394π-0.594394\pi
−0.292221 + 0.956351i 0.594394π0.594394\pi
354354 −1.74632 −0.0928159
355355 12.0112 0.637490
356356 −25.4966 −1.35132
357357 −7.12504 −0.377097
358358 0.582415 0.0307816
359359 7.56251 0.399134 0.199567 0.979884i 0.436046π-0.436046\pi
0.199567 + 0.979884i 0.436046π0.436046\pi
360360 −1.47657 −0.0778222
361361 0 0
362362 −3.46149 −0.181932
363363 −1.49679 −0.0785609
364364 6.96164 0.364889
365365 −9.88447 −0.517377
366366 0.298812 0.0156191
367367 2.97462 0.155274 0.0776369 0.996982i 0.475263π-0.475263\pi
0.0776369 + 0.996982i 0.475263π0.475263\pi
368368 −28.6758 −1.49483
369369 −7.48511 −0.389659
370370 1.35867 0.0706339
371371 5.55819 0.288567
372372 2.56922 0.133208
373373 29.9852 1.55258 0.776288 0.630378i 0.217100π-0.217100\pi
0.776288 + 0.630378i 0.217100π0.217100\pi
374374 0.449965 0.0232671
375375 12.4521 0.643022
376376 −1.85675 −0.0957544
377377 15.3351 0.789800
378378 −1.14792 −0.0590426
379379 −30.6010 −1.57187 −0.785933 0.618312i 0.787817π-0.787817\pi
−0.785933 + 0.618312i 0.787817π0.787817\pi
380380 0 0
381381 −5.51198 −0.282387
382382 −1.07540 −0.0550225
383383 −1.69358 −0.0865379 −0.0432690 0.999063i 0.513777π-0.513777\pi
−0.0432690 + 0.999063i 0.513777π0.513777\pi
384384 6.49163 0.331275
385385 −5.16581 −0.263274
386386 1.93149 0.0983104
387387 −5.54915 −0.282079
388388 31.1487 1.58133
389389 −23.0617 −1.16928 −0.584638 0.811294i 0.698764π-0.698764\pi
−0.584638 + 0.811294i 0.698764π0.698764\pi
390390 1.74868 0.0885477
391391 23.9200 1.20969
392392 2.67647 0.135182
393393 −0.0809998 −0.00408590
394394 2.82713 0.142429
395395 62.3083 3.13507
396396 −1.50462 −0.0756100
397397 −19.0356 −0.955369 −0.477685 0.878531i 0.658524π-0.658524\pi
−0.477685 + 0.878531i 0.658524π0.658524\pi
398398 0.421338 0.0211198
399399 0 0
400400 28.6136 1.43068
401401 27.9015 1.39334 0.696668 0.717394i 0.254665π-0.254665\pi
0.696668 + 0.717394i 0.254665π0.254665\pi
402402 −1.34810 −0.0672372
403403 2.07339 0.103283
404404 15.5239 0.772343
405405 21.6056 1.07359
406406 −1.30747 −0.0648888
407407 2.78244 0.137920
408408 −2.68102 −0.132730
409409 −7.32038 −0.361970 −0.180985 0.983486i 0.557928π-0.557928\pi
−0.180985 + 0.983486i 0.557928π0.557928\pi
410410 −4.81153 −0.237624
411411 −25.4045 −1.25311
412412 14.4140 0.710126
413413 −12.3428 −0.607350
414414 0.778652 0.0382687
415415 −62.2875 −3.05758
416416 3.93564 0.192961
417417 26.9964 1.32202
418418 0 0
419419 −37.5029 −1.83214 −0.916069 0.401021i 0.868655π-0.868655\pi
−0.916069 + 0.401021i 0.868655π0.868655\pi
420420 15.3151 0.747301
421421 36.8529 1.79610 0.898050 0.439893i 0.144983π-0.144983\pi
0.898050 + 0.439893i 0.144983π0.144983\pi
422422 0.696744 0.0339170
423423 −2.55163 −0.124064
424424 2.09144 0.101569
425425 −23.8681 −1.15777
426426 0.709929 0.0343962
427427 2.11197 0.102205
428428 −12.3546 −0.597183
429429 3.58113 0.172899
430430 −3.56707 −0.172019
431431 −25.7897 −1.24225 −0.621124 0.783712i 0.713324π-0.713324\pi
−0.621124 + 0.783712i 0.713324π0.713324\pi
432432 21.8605 1.05176
433433 −31.0028 −1.48990 −0.744949 0.667121i 0.767526π-0.767526\pi
−0.744949 + 0.667121i 0.767526π0.767526\pi
434434 −0.176777 −0.00848556
435435 33.7362 1.61753
436436 −5.38442 −0.257867
437437 0 0
438438 −0.584226 −0.0279154
439439 18.5779 0.886674 0.443337 0.896355i 0.353794π-0.353794\pi
0.443337 + 0.896355i 0.353794π0.353794\pi
440440 −1.94379 −0.0926668
441441 3.67813 0.175149
442442 −1.07656 −0.0512069
443443 −26.6129 −1.26442 −0.632208 0.774799i 0.717851π-0.717851\pi
−0.632208 + 0.774799i 0.717851π0.717851\pi
444444 −8.24912 −0.391486
445445 −45.2658 −2.14581
446446 −1.64711 −0.0779930
447447 7.93858 0.375482
448448 11.0778 0.523376
449449 7.83267 0.369647 0.184823 0.982772i 0.440829π-0.440829\pi
0.184823 + 0.982772i 0.440829π0.440829\pi
450450 −0.776964 −0.0366264
451451 −9.85358 −0.463987
452452 −0.963890 −0.0453376
453453 16.1867 0.760519
454454 −0.195569 −0.00917849
455455 12.3595 0.579420
456456 0 0
457457 2.02478 0.0947150 0.0473575 0.998878i 0.484920π-0.484920\pi
0.0473575 + 0.998878i 0.484920π0.484920\pi
458458 1.57804 0.0737372
459459 −18.2350 −0.851136
460460 −51.4155 −2.39726
461461 −38.1417 −1.77644 −0.888219 0.459421i 0.848057π-0.848057\pi
−0.888219 + 0.459421i 0.848057π0.848057\pi
462462 −0.305327 −0.0142051
463463 −27.6957 −1.28713 −0.643564 0.765393i 0.722545π-0.722545\pi
−0.643564 + 0.765393i 0.722545π0.722545\pi
464464 24.8990 1.15591
465465 4.56130 0.211525
466466 −1.77743 −0.0823378
467467 −3.91063 −0.180962 −0.0904811 0.995898i 0.528840π-0.528840\pi
−0.0904811 + 0.995898i 0.528840π0.528840\pi
468468 3.59988 0.166405
469469 −9.52824 −0.439973
470470 −1.64022 −0.0756576
471471 −4.78265 −0.220373
472472 −4.64436 −0.213774
473473 −7.30504 −0.335886
474474 3.68276 0.169155
475475 0 0
476476 −9.42867 −0.432162
477477 2.87415 0.131598
478478 −0.504032 −0.0230539
479479 6.77505 0.309560 0.154780 0.987949i 0.450533π-0.450533\pi
0.154780 + 0.987949i 0.450533π0.450533\pi
480480 8.65813 0.395188
481481 −6.65712 −0.303539
482482 2.68300 0.122207
483483 −16.2311 −0.738541
484484 −1.98072 −0.0900326
485485 55.3003 2.51106
486486 −1.06725 −0.0484114
487487 15.8303 0.717340 0.358670 0.933464i 0.383230π-0.383230\pi
0.358670 + 0.933464i 0.383230π0.383230\pi
488488 0.794694 0.0359741
489489 21.4696 0.970889
490490 2.36435 0.106810
491491 −2.66404 −0.120226 −0.0601132 0.998192i 0.519146π-0.519146\pi
−0.0601132 + 0.998192i 0.519146π0.519146\pi
492492 29.2130 1.31702
493493 −20.7695 −0.935413
494494 0 0
495495 −2.67125 −0.120064
496496 3.36646 0.151159
497497 5.01770 0.225075
498498 −3.68153 −0.164973
499499 10.9165 0.488690 0.244345 0.969688i 0.421427π-0.421427\pi
0.244345 + 0.969688i 0.421427π0.421427\pi
500500 16.4780 0.736919
501501 −22.8359 −1.02023
502502 2.51752 0.112362
503503 30.6519 1.36670 0.683350 0.730091i 0.260522π-0.260522\pi
0.683350 + 0.730091i 0.260522π0.260522\pi
504504 −0.616839 −0.0274762
505505 27.5606 1.22643
506506 1.02504 0.0455684
507507 10.8902 0.483649
508508 −7.29408 −0.323622
509509 −6.69649 −0.296817 −0.148408 0.988926i 0.547415π-0.547415\pi
−0.148408 + 0.988926i 0.547415π0.547415\pi
510510 −2.36836 −0.104873
511511 −4.12924 −0.182667
512512 10.6847 0.472204
513513 0 0
514514 0.216536 0.00955099
515515 25.5901 1.12763
516516 21.6573 0.953410
517517 −3.35902 −0.147730
518518 0.567586 0.0249383
519519 32.4426 1.42407
520520 4.65063 0.203943
521521 −23.4833 −1.02882 −0.514412 0.857543i 0.671990π-0.671990\pi
−0.514412 + 0.857543i 0.671990π0.671990\pi
522522 −0.676098 −0.0295920
523523 35.0640 1.53324 0.766622 0.642099i 0.221936π-0.221936\pi
0.766622 + 0.642099i 0.221936π0.221936\pi
524524 −0.107188 −0.00468254
525525 16.1959 0.706847
526526 2.39841 0.104576
527527 −2.80814 −0.122325
528528 5.81453 0.253045
529529 31.4906 1.36916
530530 1.84754 0.0802521
531531 −6.38249 −0.276977
532532 0 0
533533 23.5752 1.02115
534534 −2.67546 −0.115778
535535 −21.9340 −0.948288
536536 −3.58529 −0.154861
537537 −6.27788 −0.270911
538538 −1.93445 −0.0834002
539539 4.84198 0.208559
540540 39.1957 1.68671
541541 31.0702 1.33581 0.667906 0.744245i 0.267191π-0.267191\pi
0.667906 + 0.744245i 0.267191π0.267191\pi
542542 0.854415 0.0367003
543543 37.3116 1.60119
544544 −5.33033 −0.228536
545545 −9.55931 −0.409476
546546 0.730511 0.0312630
547547 −19.1794 −0.820051 −0.410025 0.912074i 0.634480π-0.634480\pi
−0.410025 + 0.912074i 0.634480π0.634480\pi
548548 −33.6181 −1.43610
549549 1.09210 0.0466099
550550 −1.02281 −0.0436129
551551 0 0
552552 −6.10745 −0.259950
553553 26.0293 1.10688
554554 −2.55707 −0.108639
555555 −14.6452 −0.621654
556556 35.7247 1.51506
557557 −9.68999 −0.410578 −0.205289 0.978701i 0.565813π-0.565813\pi
−0.205289 + 0.978701i 0.565813π0.565813\pi
558558 −0.0914117 −0.00386977
559559 17.4777 0.739227
560560 20.0675 0.848007
561561 −4.85020 −0.204776
562562 3.44832 0.145458
563563 −24.4092 −1.02873 −0.514363 0.857573i 0.671972π-0.671972\pi
−0.514363 + 0.857573i 0.671972π0.671972\pi
564564 9.95852 0.419329
565565 −1.71126 −0.0719932
566566 2.38508 0.100252
567567 9.02574 0.379045
568568 1.88807 0.0792214
569569 34.3379 1.43952 0.719761 0.694222i 0.244251π-0.244251\pi
0.719761 + 0.694222i 0.244251π0.244251\pi
570570 0 0
571571 43.4564 1.81859 0.909296 0.416149i 0.136621π-0.136621\pi
0.909296 + 0.416149i 0.136621π0.136621\pi
572572 4.73897 0.198146
573573 11.5919 0.484257
574574 −2.01002 −0.0838966
575575 −54.3724 −2.26749
576576 5.72835 0.238681
577577 18.1629 0.756131 0.378066 0.925779i 0.376589π-0.376589\pi
0.378066 + 0.925779i 0.376589π0.376589\pi
578578 −0.902557 −0.0375414
579579 −20.8197 −0.865236
580580 44.6437 1.85373
581581 −26.0207 −1.07952
582582 3.26855 0.135486
583583 3.78360 0.156701
584584 −1.55376 −0.0642949
585585 6.39111 0.264240
586586 3.21837 0.132950
587587 −11.5578 −0.477040 −0.238520 0.971138i 0.576662π-0.576662\pi
−0.238520 + 0.971138i 0.576662π0.576662\pi
588588 −14.3551 −0.591993
589589 0 0
590590 −4.10275 −0.168908
591591 −30.4739 −1.25353
592592 −10.8089 −0.444242
593593 30.8182 1.26555 0.632776 0.774335i 0.281916π-0.281916\pi
0.632776 + 0.774335i 0.281916π0.281916\pi
594594 −0.781418 −0.0320620
595595 −16.7393 −0.686246
596596 10.5052 0.430311
597597 −4.54163 −0.185877
598598 −2.45245 −0.100288
599599 −22.9029 −0.935788 −0.467894 0.883785i 0.654987π-0.654987\pi
−0.467894 + 0.883785i 0.654987π0.654987\pi
600600 6.09421 0.248795
601601 −5.27555 −0.215194 −0.107597 0.994195i 0.534316π-0.534316\pi
−0.107597 + 0.994195i 0.534316π0.534316\pi
602602 −1.49015 −0.0607338
603603 −4.92708 −0.200646
604604 21.4201 0.871573
605605 −3.51650 −0.142966
606606 1.62898 0.0661728
607607 −22.3060 −0.905372 −0.452686 0.891670i 0.649534π-0.649534\pi
−0.452686 + 0.891670i 0.649534π0.649534\pi
608608 0 0
609609 14.0933 0.571091
610610 0.702019 0.0284239
611611 8.03663 0.325127
612612 −4.87559 −0.197084
613613 37.7943 1.52650 0.763249 0.646104i 0.223603π-0.223603\pi
0.763249 + 0.646104i 0.223603π0.223603\pi
614614 3.65412 0.147468
615615 51.8637 2.09135
616616 −0.812022 −0.0327173
617617 9.69316 0.390232 0.195116 0.980780i 0.437492π-0.437492\pi
0.195116 + 0.980780i 0.437492π0.437492\pi
618618 1.51251 0.0608422
619619 −18.4686 −0.742314 −0.371157 0.928570i 0.621039π-0.621039\pi
−0.371157 + 0.928570i 0.621039π0.621039\pi
620620 6.03604 0.242413
621621 −41.5399 −1.66694
622622 3.51178 0.140810
623623 −18.9098 −0.757606
624624 −13.9116 −0.556908
625625 −7.57434 −0.302973
626626 −3.84476 −0.153668
627627 0 0
628628 −6.32896 −0.252553
629629 9.01624 0.359501
630630 −0.544905 −0.0217096
631631 12.8342 0.510921 0.255460 0.966820i 0.417773π-0.417773\pi
0.255460 + 0.966820i 0.417773π0.417773\pi
632632 9.79434 0.389598
633633 −7.51025 −0.298505
634634 1.40577 0.0558302
635635 −12.9497 −0.513892
636636 −11.2173 −0.444794
637637 −11.5847 −0.459002
638638 −0.890031 −0.0352367
639639 2.59467 0.102643
640640 15.2512 0.602858
641641 41.2232 1.62822 0.814110 0.580711i 0.197225π-0.197225\pi
0.814110 + 0.580711i 0.197225π0.197225\pi
642642 −1.29642 −0.0511655
643643 −15.6032 −0.615332 −0.307666 0.951494i 0.599548π-0.599548\pi
−0.307666 + 0.951494i 0.599548π0.599548\pi
644644 −21.4789 −0.846385
645645 38.4497 1.51395
646646 0 0
647647 −12.7748 −0.502228 −0.251114 0.967958i 0.580797π-0.580797\pi
−0.251114 + 0.967958i 0.580797π0.580797\pi
648648 3.39621 0.133416
649649 −8.40207 −0.329810
650650 2.44713 0.0959844
651651 1.90549 0.0746820
652652 28.4110 1.11266
653653 −5.05990 −0.198009 −0.0990045 0.995087i 0.531566π-0.531566\pi
−0.0990045 + 0.995087i 0.531566π0.531566\pi
654654 −0.565007 −0.0220935
655655 −0.190298 −0.00743557
656656 38.2780 1.49450
657657 −2.13524 −0.0833037
658658 −0.685203 −0.0267120
659659 −29.3796 −1.14447 −0.572233 0.820091i 0.693923π-0.693923\pi
−0.572233 + 0.820091i 0.693923π0.693923\pi
660660 10.4254 0.405808
661661 −32.5399 −1.26565 −0.632827 0.774293i 0.718106π-0.718106\pi
−0.632827 + 0.774293i 0.718106π0.718106\pi
662662 1.17417 0.0456352
663663 11.6043 0.450675
664664 −9.79108 −0.379968
665665 0 0
666666 0.293500 0.0113729
667667 −47.3138 −1.83200
668668 −30.2191 −1.16921
669669 17.7543 0.686422
670670 −3.16719 −0.122359
671671 1.43767 0.0555007
672672 3.61694 0.139527
673673 −37.8116 −1.45753 −0.728765 0.684764i 0.759905π-0.759905\pi
−0.728765 + 0.684764i 0.759905π0.759905\pi
674674 0.00241313 9.29504e−5 0
675675 41.4498 1.59541
676676 14.4111 0.554273
677677 −1.66524 −0.0640004 −0.0320002 0.999488i 0.510188π-0.510188\pi
−0.0320002 + 0.999488i 0.510188π0.510188\pi
678678 −0.101145 −0.00388444
679679 23.1017 0.886563
680680 −6.29869 −0.241544
681681 2.10805 0.0807805
682682 −0.120336 −0.00460792
683683 −22.3652 −0.855779 −0.427890 0.903831i 0.640743π-0.640743\pi
−0.427890 + 0.903831i 0.640743π0.640743\pi
684684 0 0
685685 −59.6845 −2.28043
686686 2.41563 0.0922292
687687 −17.0098 −0.648966
688688 28.3777 1.08189
689689 −9.05246 −0.344871
690690 −5.39522 −0.205392
691691 22.7083 0.863865 0.431932 0.901906i 0.357832π-0.357832\pi
0.431932 + 0.901906i 0.357832π0.357832\pi
692692 42.9318 1.63202
693693 −1.11592 −0.0423902
694694 1.19956 0.0455348
695695 63.4244 2.40582
696696 5.30305 0.201012
697697 −31.9296 −1.20942
698698 −0.474356 −0.0179547
699699 19.1590 0.724660
700700 21.4323 0.810064
701701 −27.0490 −1.02163 −0.510814 0.859691i 0.670656π-0.670656\pi
−0.510814 + 0.859691i 0.670656π0.670656\pi
702702 1.86958 0.0705629
703703 0 0
704704 7.54094 0.284210
705705 17.6800 0.665868
706706 −1.52478 −0.0573860
707707 11.5135 0.433008
708708 24.9097 0.936163
709709 −31.2772 −1.17464 −0.587320 0.809355i 0.699817π-0.699817\pi
−0.587320 + 0.809355i 0.699817π0.699817\pi
710710 1.66788 0.0625946
711711 13.4598 0.504783
712712 −7.11541 −0.266661
713713 −6.39705 −0.239571
714714 −0.989386 −0.0370268
715715 8.41340 0.314643
716716 −8.30762 −0.310470
717717 5.43299 0.202899
718718 1.05013 0.0391906
719719 −10.4902 −0.391219 −0.195609 0.980682i 0.562668π-0.562668\pi
−0.195609 + 0.980682i 0.562668π0.562668\pi
720720 10.3770 0.386726
721721 10.6903 0.398127
722722 0 0
723723 −28.9203 −1.07556
724724 49.3750 1.83501
725725 47.2111 1.75338
726726 −0.207844 −0.00771382
727727 −43.4181 −1.61029 −0.805145 0.593078i 0.797913π-0.797913\pi
−0.805145 + 0.593078i 0.797913π0.797913\pi
728728 1.94280 0.0720051
729729 29.9361 1.10874
730730 −1.37256 −0.0508008
731731 −23.6713 −0.875515
732732 −4.26228 −0.157538
733733 37.7254 1.39342 0.696710 0.717353i 0.254646π-0.254646\pi
0.696710 + 0.717353i 0.254646π0.254646\pi
734734 0.413057 0.0152462
735735 −25.4855 −0.940046
736736 −12.1427 −0.447586
737737 −6.48612 −0.238919
738738 −1.03939 −0.0382603
739739 10.5904 0.389575 0.194788 0.980845i 0.437598π-0.437598\pi
0.194788 + 0.980845i 0.437598π0.437598\pi
740740 −19.3802 −0.712431
741741 0 0
742742 0.771812 0.0283341
743743 −2.32562 −0.0853187 −0.0426593 0.999090i 0.513583π-0.513583\pi
−0.0426593 + 0.999090i 0.513583π0.513583\pi
744744 0.716998 0.0262864
745745 18.6506 0.683307
746746 4.16376 0.152446
747747 −13.4554 −0.492306
748748 −6.41834 −0.234678
749749 −9.16294 −0.334806
750750 1.72910 0.0631378
751751 −7.37463 −0.269104 −0.134552 0.990907i 0.542960π-0.542960\pi
−0.134552 + 0.990907i 0.542960π0.542960\pi
752752 13.0487 0.475838
753753 −27.1365 −0.988907
754754 2.12944 0.0775498
755755 38.0286 1.38400
756756 16.3740 0.595518
757757 −36.2300 −1.31680 −0.658401 0.752667i 0.728767π-0.728767\pi
−0.658401 + 0.752667i 0.728767π0.728767\pi
758758 −4.24926 −0.154340
759759 −11.0489 −0.401051
760760 0 0
761761 8.48528 0.307591 0.153796 0.988103i 0.450850π-0.450850\pi
0.153796 + 0.988103i 0.450850π0.450850\pi
762762 −0.765395 −0.0277273
763763 −3.99341 −0.144571
764764 15.3397 0.554970
765765 −8.65595 −0.312957
766766 −0.235171 −0.00849708
767767 20.1024 0.725854
768768 −21.6729 −0.782053
769769 −8.18651 −0.295213 −0.147607 0.989046i 0.547157π-0.547157\pi
−0.147607 + 0.989046i 0.547157π0.547157\pi
770770 −0.717326 −0.0258506
771771 −2.33405 −0.0840589
772772 −27.5510 −0.991582
773773 −35.4306 −1.27435 −0.637175 0.770720i 0.719897π-0.719897\pi
−0.637175 + 0.770720i 0.719897π0.719897\pi
774774 −0.770558 −0.0276971
775775 6.38317 0.229290
776776 8.69274 0.312051
777777 −6.11804 −0.219484
778778 −3.20236 −0.114810
779779 0 0
780780 −24.9433 −0.893113
781781 3.41568 0.122223
782782 3.32154 0.118778
783783 36.0688 1.28899
784784 −18.8095 −0.671769
785785 −11.2362 −0.401038
786786 −0.0112477 −0.000401191 0
787787 −2.32432 −0.0828529 −0.0414264 0.999142i 0.513190π-0.513190\pi
−0.0414264 + 0.999142i 0.513190π0.513190\pi
788788 −40.3265 −1.43657
789789 −25.8526 −0.920378
790790 8.65216 0.307830
791791 −0.714879 −0.0254182
792792 −0.419898 −0.0149204
793793 −3.43970 −0.122147
794794 −2.64329 −0.0938069
795795 −19.9148 −0.706304
796796 −6.01000 −0.213019
797797 7.98937 0.282998 0.141499 0.989938i 0.454808π-0.454808\pi
0.141499 + 0.989938i 0.454808π0.454808\pi
798798 0 0
799799 −10.8846 −0.385070
800800 12.1164 0.428378
801801 −9.77832 −0.345500
802802 3.87442 0.136810
803803 −2.81088 −0.0991939
804804 19.2295 0.678171
805805 −38.1328 −1.34401
806806 0.287911 0.0101412
807807 20.8516 0.734011
808808 4.33229 0.152410
809809 −3.66337 −0.128797 −0.0643987 0.997924i 0.520513π-0.520513\pi
−0.0643987 + 0.997924i 0.520513π0.520513\pi
810810 3.00016 0.105415
811811 −38.8311 −1.36355 −0.681773 0.731564i 0.738791π-0.738791\pi
−0.681773 + 0.731564i 0.738791π0.738791\pi
812812 18.6499 0.654484
813813 −9.20980 −0.323002
814814 0.386370 0.0135423
815815 50.4400 1.76684
816816 18.8415 0.659583
817817 0 0
818818 −1.01651 −0.0355415
819819 2.66989 0.0932935
820820 68.6320 2.39673
821821 −21.9475 −0.765975 −0.382987 0.923754i 0.625105π-0.625105\pi
−0.382987 + 0.923754i 0.625105π0.625105\pi
822822 −3.52768 −0.123042
823823 8.08878 0.281957 0.140979 0.990013i 0.454975π-0.454975\pi
0.140979 + 0.990013i 0.454975π0.454975\pi
824824 4.02255 0.140132
825825 11.0250 0.383840
826826 −1.71393 −0.0596351
827827 28.5117 0.991449 0.495724 0.868480i 0.334903π-0.334903\pi
0.495724 + 0.868480i 0.334903π0.334903\pi
828828 −11.1068 −0.385987
829829 −52.4728 −1.82246 −0.911228 0.411901i 0.864865π-0.864865\pi
−0.911228 + 0.411901i 0.864865π0.864865\pi
830830 −8.64927 −0.300221
831831 27.5628 0.956143
832832 −18.0421 −0.625496
833833 15.6900 0.543626
834834 3.74873 0.129808
835835 −53.6500 −1.85664
836836 0 0
837837 4.87668 0.168563
838838 −5.20767 −0.179896
839839 10.9975 0.379677 0.189839 0.981815i 0.439203π-0.439203\pi
0.189839 + 0.981815i 0.439203π0.439203\pi
840840 4.27403 0.147468
841841 12.0822 0.416627
842842 5.11741 0.176358
843843 −37.1696 −1.28019
844844 −9.93842 −0.342094
845845 25.5850 0.880151
846846 −0.354320 −0.0121818
847847 −1.46902 −0.0504761
848848 −14.6981 −0.504734
849849 −25.7089 −0.882327
850850 −3.31433 −0.113681
851851 20.5393 0.704079
852852 −10.1265 −0.346928
853853 −34.0760 −1.16674 −0.583369 0.812207i 0.698266π-0.698266\pi
−0.583369 + 0.812207i 0.698266π0.698266\pi
854854 0.293269 0.0100355
855855 0 0
856856 −3.44784 −0.117845
857857 19.4670 0.664981 0.332490 0.943107i 0.392111π-0.392111\pi
0.332490 + 0.943107i 0.392111π0.392111\pi
858858 0.497278 0.0169768
859859 −41.7466 −1.42437 −0.712187 0.701990i 0.752295π-0.752295\pi
−0.712187 + 0.701990i 0.752295π0.752295\pi
860860 50.8810 1.73503
861861 21.6661 0.738379
862862 −3.58117 −0.121975
863863 20.0106 0.681169 0.340584 0.940214i 0.389375π-0.389375\pi
0.340584 + 0.940214i 0.389375π0.389375\pi
864864 9.25677 0.314922
865865 76.2196 2.59154
866866 −4.30506 −0.146292
867867 9.72871 0.330404
868868 2.52156 0.0855873
869869 17.7188 0.601071
870870 4.68463 0.158824
871871 15.5184 0.525820
872872 −1.50264 −0.0508859
873873 11.9460 0.404310
874874 0 0
875875 12.2211 0.413148
876876 8.33345 0.281561
877877 14.6756 0.495560 0.247780 0.968816i 0.420299π-0.420299\pi
0.247780 + 0.968816i 0.420299π0.420299\pi
878878 2.57973 0.0870618
879879 −34.6910 −1.17010
880880 13.6605 0.460494
881881 17.9508 0.604777 0.302388 0.953185i 0.402216π-0.402216\pi
0.302388 + 0.953185i 0.402216π0.402216\pi
882882 0.510747 0.0171977
883883 23.0794 0.776684 0.388342 0.921515i 0.373048π-0.373048\pi
0.388342 + 0.921515i 0.373048π0.373048\pi
884884 15.3562 0.516485
885885 44.2238 1.48657
886886 −3.69547 −0.124152
887887 2.24450 0.0753629 0.0376814 0.999290i 0.488003π-0.488003\pi
0.0376814 + 0.999290i 0.488003π0.488003\pi
888888 −2.30210 −0.0772535
889889 −5.40973 −0.181437
890890 −6.28563 −0.210695
891891 6.14405 0.205834
892892 23.4945 0.786656
893893 0 0
894894 1.10235 0.0368683
895895 −14.7491 −0.493007
896896 6.37121 0.212847
897897 26.4351 0.882643
898898 1.08765 0.0362953
899899 5.55451 0.185253
900900 11.0827 0.369423
901901 12.2604 0.408454
902902 −1.36827 −0.0455585
903903 16.0624 0.534522
904904 −0.268995 −0.00894665
905905 87.6587 2.91387
906906 2.24770 0.0746747
907907 −28.1759 −0.935565 −0.467783 0.883844i 0.654947π-0.654947\pi
−0.467783 + 0.883844i 0.654947π0.654947\pi
908908 2.78961 0.0925764
909909 5.95364 0.197470
910910 1.71624 0.0568928
911911 −32.9161 −1.09056 −0.545280 0.838254i 0.683577π-0.683577\pi
−0.545280 + 0.838254i 0.683577π0.683577\pi
912912 0 0
913913 −17.7129 −0.586213
914914 0.281161 0.00929998
915915 −7.56711 −0.250161
916916 −22.5094 −0.743730
917917 −0.0794973 −0.00262523
918918 −2.53212 −0.0835723
919919 54.3280 1.79211 0.896057 0.443938i 0.146419π-0.146419\pi
0.896057 + 0.443938i 0.146419π0.146419\pi
920920 −14.3486 −0.473061
921921 −39.3880 −1.29788
922922 −5.29638 −0.174427
923923 −8.17219 −0.268991
924924 4.35522 0.143276
925925 −20.4948 −0.673864
926926 −3.84583 −0.126382
927927 5.52797 0.181562
928928 10.5434 0.346104
929929 −17.8175 −0.584574 −0.292287 0.956331i 0.594416π-0.594416\pi
−0.292287 + 0.956331i 0.594416π0.594416\pi
930930 0.633384 0.0207695
931931 0 0
932932 25.3534 0.830478
933933 −37.8537 −1.23928
934934 −0.543031 −0.0177685
935935 −11.3949 −0.372653
936936 1.00463 0.0328373
937937 −52.7066 −1.72185 −0.860925 0.508732i 0.830115π-0.830115\pi
−0.860925 + 0.508732i 0.830115π0.830115\pi
938938 −1.32310 −0.0432006
939939 41.4430 1.35244
940940 23.3962 0.763101
941941 −51.4855 −1.67838 −0.839189 0.543839i 0.816970π-0.816970\pi
−0.839189 + 0.543839i 0.816970π0.816970\pi
942942 −0.664121 −0.0216382
943943 −72.7369 −2.36864
944944 32.6393 1.06232
945945 29.0699 0.945643
946946 −1.01438 −0.0329804
947947 −26.4120 −0.858274 −0.429137 0.903239i 0.641182π-0.641182\pi
−0.429137 + 0.903239i 0.641182π0.641182\pi
948948 −52.5312 −1.70614
949949 6.72518 0.218309
950950 0 0
951951 −15.1529 −0.491366
952952 −2.63128 −0.0852804
953953 −25.4872 −0.825613 −0.412806 0.910819i 0.635451π-0.635451\pi
−0.412806 + 0.910819i 0.635451π0.635451\pi
954954 0.399106 0.0129215
955955 27.2335 0.881256
956956 7.18955 0.232527
957957 9.59370 0.310120
958958 0.940787 0.0303954
959959 −24.9332 −0.805136
960960 −39.6913 −1.28103
961961 −30.2490 −0.975774
962962 −0.924411 −0.0298042
963963 −4.73818 −0.152686
964964 −38.2706 −1.23261
965965 −48.9131 −1.57457
966966 −2.25386 −0.0725167
967967 18.1540 0.583793 0.291897 0.956450i 0.405714π-0.405714\pi
0.291897 + 0.956450i 0.405714π0.405714\pi
968968 −0.552764 −0.0177665
969969 0 0
970970 7.67902 0.246558
971971 −10.1058 −0.324311 −0.162156 0.986765i 0.551845π-0.551845\pi
−0.162156 + 0.986765i 0.551845π0.551845\pi
972972 15.2233 0.488289
973973 26.4956 0.849410
974974 2.19820 0.0704350
975975 −26.3778 −0.844765
976976 −5.58490 −0.178768
977977 36.3162 1.16186 0.580929 0.813954i 0.302689π-0.302689\pi
0.580929 + 0.813954i 0.302689π0.302689\pi
978978 2.98128 0.0953308
979979 −12.8724 −0.411404
980980 −33.7253 −1.07732
981981 −2.06500 −0.0659304
982982 −0.369930 −0.0118049
983983 7.72465 0.246378 0.123189 0.992383i 0.460688π-0.460688\pi
0.123189 + 0.992383i 0.460688π0.460688\pi
984984 8.15254 0.259894
985985 −71.5943 −2.28118
986986 −2.88407 −0.0918474
987987 7.38584 0.235094
988988 0 0
989989 −53.9241 −1.71469
990990 −0.370931 −0.0117890
991991 19.1661 0.608830 0.304415 0.952539i 0.401539π-0.401539\pi
0.304415 + 0.952539i 0.401539π0.401539\pi
992992 1.42552 0.0452603
993993 −12.6564 −0.401639
994994 0.696760 0.0220999
995995 −10.6700 −0.338260
996996 52.5137 1.66396
997997 27.2322 0.862454 0.431227 0.902244i 0.358081π-0.358081\pi
0.431227 + 0.902244i 0.358081π0.358081\pi
998998 1.51587 0.0479840
999999 −15.6578 −0.495390
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 3971.2.a.k.1.5 9
19.7 even 3 209.2.e.b.144.5 yes 18
19.11 even 3 209.2.e.b.45.5 18
19.18 odd 2 3971.2.a.l.1.5 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.e.b.45.5 18 19.11 even 3
209.2.e.b.144.5 yes 18 19.7 even 3
3971.2.a.k.1.5 9 1.1 even 1 trivial
3971.2.a.l.1.5 9 19.18 odd 2