Properties

Label 735.4.a.v.1.3
Level 735735
Weight 44
Character 735.1
Self dual yes
Analytic conductor 43.36643.366
Analytic rank 00
Dimension 44
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,4,Mod(1,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 735=3572 735 = 3 \cdot 5 \cdot 7^{2}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 735.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: 43.366403854243.3664038542
Analytic rank: 00
Dimension: 44
Coefficient field: Q[x]/(x4)\mathbb{Q}[x]/(x^{4} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4x335x2+19x+196 x^{4} - x^{3} - 35x^{2} + 19x + 196 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 2 2
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.3
Root 2.927712.92771 of defining polynomial
Character χ\chi == 735.1

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+2.92771q23.00000q3+0.571487q4+5.00000q58.78313q621.7485q8+9.00000q9+14.6386q10+45.3855q111.71446q12+24.1478q1315.0000q1568.2453q16+7.89162q17+26.3494q18151.949q19+2.85744q20+132.876q22+39.3342q23+65.2456q24+25.0000q25+70.6977q2627.0000q27+200.572q2943.9157q3015.0636q3125.8142q32136.156q33+23.1044q34+5.14339q36+119.753q37444.864q3872.4434q39108.743q40+32.1480q41+358.400q43+25.9372q44+45.0000q45+115.159q46+171.051q47+204.736q48+73.1928q5023.6749q51+13.8002q52+387.811q5379.0482q54+226.927q55+455.848q57+587.218q58+504.229q598.57231q60+328.392q6144.1017q62+470.386q64+120.739q65398.627q66+519.007q67+4.50996q68118.003q69+1173.62q71195.737q72113.604q73+350.603q7475.0000q7586.8371q76212.093q78805.369q79341.227q80+81.0000q81+94.1201q821350.22q83+39.4581q85+1049.29q86601.717q87987.068q88+658.726q89+131.747q90+22.4790q92+45.1907q93+500.788q94759.747q95+77.4426q96+886.401q97+408.469q99+O(q100)q+2.92771 q^{2} -3.00000 q^{3} +0.571487 q^{4} +5.00000 q^{5} -8.78313 q^{6} -21.7485 q^{8} +9.00000 q^{9} +14.6386 q^{10} +45.3855 q^{11} -1.71446 q^{12} +24.1478 q^{13} -15.0000 q^{15} -68.2453 q^{16} +7.89162 q^{17} +26.3494 q^{18} -151.949 q^{19} +2.85744 q^{20} +132.876 q^{22} +39.3342 q^{23} +65.2456 q^{24} +25.0000 q^{25} +70.6977 q^{26} -27.0000 q^{27} +200.572 q^{29} -43.9157 q^{30} -15.0636 q^{31} -25.8142 q^{32} -136.156 q^{33} +23.1044 q^{34} +5.14339 q^{36} +119.753 q^{37} -444.864 q^{38} -72.4434 q^{39} -108.743 q^{40} +32.1480 q^{41} +358.400 q^{43} +25.9372 q^{44} +45.0000 q^{45} +115.159 q^{46} +171.051 q^{47} +204.736 q^{48} +73.1928 q^{50} -23.6749 q^{51} +13.8002 q^{52} +387.811 q^{53} -79.0482 q^{54} +226.927 q^{55} +455.848 q^{57} +587.218 q^{58} +504.229 q^{59} -8.57231 q^{60} +328.392 q^{61} -44.1017 q^{62} +470.386 q^{64} +120.739 q^{65} -398.627 q^{66} +519.007 q^{67} +4.50996 q^{68} -118.003 q^{69} +1173.62 q^{71} -195.737 q^{72} -113.604 q^{73} +350.603 q^{74} -75.0000 q^{75} -86.8371 q^{76} -212.093 q^{78} -805.369 q^{79} -341.227 q^{80} +81.0000 q^{81} +94.1201 q^{82} -1350.22 q^{83} +39.4581 q^{85} +1049.29 q^{86} -601.717 q^{87} -987.068 q^{88} +658.726 q^{89} +131.747 q^{90} +22.4790 q^{92} +45.1907 q^{93} +500.788 q^{94} -759.747 q^{95} +77.4426 q^{96} +886.401 q^{97} +408.469 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+q212q3+39q4+20q53q6+33q8+36q9+5q10+60q11117q12+14q1360q15+283q16+46q17+9q18+40q19+195q2014q22++540q99+O(q100) 4 q + q^{2} - 12 q^{3} + 39 q^{4} + 20 q^{5} - 3 q^{6} + 33 q^{8} + 36 q^{9} + 5 q^{10} + 60 q^{11} - 117 q^{12} + 14 q^{13} - 60 q^{15} + 283 q^{16} + 46 q^{17} + 9 q^{18} + 40 q^{19} + 195 q^{20} - 14 q^{22}+ \cdots + 540 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 2.92771 1.03510 0.517551 0.855652i 0.326844π-0.326844\pi
0.517551 + 0.855652i 0.326844π0.326844\pi
33 −3.00000 −0.577350
44 0.571487 0.0714359
55 5.00000 0.447214
66 −8.78313 −0.597616
77 0 0
88 −21.7485 −0.961158
99 9.00000 0.333333
1010 14.6386 0.462912
1111 45.3855 1.24402 0.622011 0.783009i 0.286316π-0.286316\pi
0.622011 + 0.783009i 0.286316π0.286316\pi
1212 −1.71446 −0.0412436
1313 24.1478 0.515184 0.257592 0.966254i 0.417071π-0.417071\pi
0.257592 + 0.966254i 0.417071π0.417071\pi
1414 0 0
1515 −15.0000 −0.258199
1616 −68.2453 −1.06633
1717 7.89162 0.112588 0.0562941 0.998414i 0.482072π-0.482072\pi
0.0562941 + 0.998414i 0.482072π0.482072\pi
1818 26.3494 0.345034
1919 −151.949 −1.83471 −0.917357 0.398066i 0.869682π-0.869682\pi
−0.917357 + 0.398066i 0.869682π0.869682\pi
2020 2.85744 0.0319471
2121 0 0
2222 132.876 1.28769
2323 39.3342 0.356598 0.178299 0.983976i 0.442941π-0.442941\pi
0.178299 + 0.983976i 0.442941π0.442941\pi
2424 65.2456 0.554925
2525 25.0000 0.200000
2626 70.6977 0.533268
2727 −27.0000 −0.192450
2828 0 0
2929 200.572 1.28432 0.642162 0.766569i 0.278038π-0.278038\pi
0.642162 + 0.766569i 0.278038π0.278038\pi
3030 −43.9157 −0.267262
3131 −15.0636 −0.0872740 −0.0436370 0.999047i 0.513895π-0.513895\pi
−0.0436370 + 0.999047i 0.513895π0.513895\pi
3232 −25.8142 −0.142605
3333 −136.156 −0.718236
3434 23.1044 0.116540
3535 0 0
3636 5.14339 0.0238120
3737 119.753 0.532089 0.266045 0.963961i 0.414283π-0.414283\pi
0.266045 + 0.963961i 0.414283π0.414283\pi
3838 −444.864 −1.89912
3939 −72.4434 −0.297442
4040 −108.743 −0.429843
4141 32.1480 0.122456 0.0612278 0.998124i 0.480498π-0.480498\pi
0.0612278 + 0.998124i 0.480498π0.480498\pi
4242 0 0
4343 358.400 1.27106 0.635530 0.772077i 0.280782π-0.280782\pi
0.635530 + 0.772077i 0.280782π0.280782\pi
4444 25.9372 0.0888678
4545 45.0000 0.149071
4646 115.159 0.369115
4747 171.051 0.530859 0.265429 0.964130i 0.414486π-0.414486\pi
0.265429 + 0.964130i 0.414486π0.414486\pi
4848 204.736 0.615648
4949 0 0
5050 73.1928 0.207020
5151 −23.6749 −0.0650028
5252 13.8002 0.0368026
5353 387.811 1.00509 0.502547 0.864550i 0.332397π-0.332397\pi
0.502547 + 0.864550i 0.332397π0.332397\pi
5454 −79.0482 −0.199205
5555 226.927 0.556343
5656 0 0
5757 455.848 1.05927
5858 587.218 1.32941
5959 504.229 1.11263 0.556314 0.830972i 0.312215π-0.312215\pi
0.556314 + 0.830972i 0.312215π0.312215\pi
6060 −8.57231 −0.0184447
6161 328.392 0.689284 0.344642 0.938734i 0.388000π-0.388000\pi
0.344642 + 0.938734i 0.388000π0.388000\pi
6262 −44.1017 −0.0903375
6363 0 0
6464 470.386 0.918722
6565 120.739 0.230397
6666 −398.627 −0.743448
6767 519.007 0.946370 0.473185 0.880963i 0.343104π-0.343104\pi
0.473185 + 0.880963i 0.343104π0.343104\pi
6868 4.50996 0.00804284
6969 −118.003 −0.205882
7070 0 0
7171 1173.62 1.96173 0.980863 0.194697i 0.0623723π-0.0623723\pi
0.980863 + 0.194697i 0.0623723π0.0623723\pi
7272 −195.737 −0.320386
7373 −113.604 −0.182142 −0.0910708 0.995844i 0.529029π-0.529029\pi
−0.0910708 + 0.995844i 0.529029π0.529029\pi
7474 350.603 0.550767
7575 −75.0000 −0.115470
7676 −86.8371 −0.131064
7777 0 0
7878 −212.093 −0.307882
7979 −805.369 −1.14698 −0.573488 0.819214i 0.694410π-0.694410\pi
−0.573488 + 0.819214i 0.694410π0.694410\pi
8080 −341.227 −0.476879
8181 81.0000 0.111111
8282 94.1201 0.126754
8383 −1350.22 −1.78561 −0.892805 0.450444i 0.851266π-0.851266\pi
−0.892805 + 0.450444i 0.851266π0.851266\pi
8484 0 0
8585 39.4581 0.0503510
8686 1049.29 1.31568
8787 −601.717 −0.741504
8888 −987.068 −1.19570
8989 658.726 0.784549 0.392274 0.919848i 0.371688π-0.371688\pi
0.392274 + 0.919848i 0.371688π0.371688\pi
9090 131.747 0.154304
9191 0 0
9292 22.4790 0.0254739
9393 45.1907 0.0503877
9494 500.788 0.549493
9595 −759.747 −0.820509
9696 77.4426 0.0823329
9797 886.401 0.927839 0.463920 0.885877i 0.346443π-0.346443\pi
0.463920 + 0.885877i 0.346443π0.346443\pi
9898 0 0
9999 408.469 0.414674
100100 14.2872 0.0142872
101101 1853.00 1.82555 0.912775 0.408464i 0.133935π-0.133935\pi
0.912775 + 0.408464i 0.133935π0.133935\pi
102102 −69.3132 −0.0672846
103103 −1673.73 −1.60114 −0.800572 0.599237i 0.795471π-0.795471\pi
−0.800572 + 0.599237i 0.795471π0.795471\pi
104104 −525.179 −0.495173
105105 0 0
106106 1135.40 1.04037
107107 −3.50571 −0.00316738 −0.00158369 0.999999i 0.500504π-0.500504\pi
−0.00158369 + 0.999999i 0.500504π0.500504\pi
108108 −15.4302 −0.0137479
109109 −1503.52 −1.32120 −0.660600 0.750738i 0.729698π-0.729698\pi
−0.660600 + 0.750738i 0.729698π0.729698\pi
110110 664.378 0.575872
111111 −359.260 −0.307202
112112 0 0
113113 211.621 0.176173 0.0880867 0.996113i 0.471925π-0.471925\pi
0.0880867 + 0.996113i 0.471925π0.471925\pi
114114 1334.59 1.09646
115115 196.671 0.159475
116116 114.625 0.0917468
117117 217.330 0.171728
118118 1476.24 1.15168
119119 0 0
120120 326.228 0.248170
121121 728.842 0.547590
122122 961.437 0.713479
123123 −96.4441 −0.0706997
124124 −8.60863 −0.00623450
125125 125.000 0.0894427
126126 0 0
127127 746.863 0.521837 0.260919 0.965361i 0.415975π-0.415975\pi
0.260919 + 0.965361i 0.415975π0.415975\pi
128128 1583.67 1.09358
129129 −1075.20 −0.733846
130130 353.489 0.238485
131131 225.367 0.150308 0.0751541 0.997172i 0.476055π-0.476055\pi
0.0751541 + 0.997172i 0.476055π0.476055\pi
132132 −77.8117 −0.0513079
133133 0 0
134134 1519.50 0.979590
135135 −135.000 −0.0860663
136136 −171.631 −0.108215
137137 −1827.22 −1.13949 −0.569743 0.821823i 0.692957π-0.692957\pi
−0.569743 + 0.821823i 0.692957π0.692957\pi
138138 −345.478 −0.213109
139139 −1656.39 −1.01074 −0.505370 0.862903i 0.668644π-0.668644\pi
−0.505370 + 0.862903i 0.668644π0.668644\pi
140140 0 0
141141 −513.153 −0.306491
142142 3436.01 2.03059
143143 1095.96 0.640900
144144 −614.208 −0.355444
145145 1002.86 0.574367
146146 −332.600 −0.188535
147147 0 0
148148 68.4375 0.0380103
149149 1116.84 0.614060 0.307030 0.951700i 0.400665π-0.400665\pi
0.307030 + 0.951700i 0.400665π0.400665\pi
150150 −219.578 −0.119523
151151 438.299 0.236214 0.118107 0.993001i 0.462317π-0.462317\pi
0.118107 + 0.993001i 0.462317π0.462317\pi
152152 3304.67 1.76345
153153 71.0246 0.0375294
154154 0 0
155155 −75.3178 −0.0390301
156156 −41.4005 −0.0212480
157157 −1136.54 −0.577743 −0.288872 0.957368i 0.593280π-0.593280\pi
−0.288872 + 0.957368i 0.593280π0.593280\pi
158158 −2357.89 −1.18724
159159 −1163.43 −0.580291
160160 −129.071 −0.0637748
161161 0 0
162162 237.145 0.115011
163163 875.223 0.420569 0.210285 0.977640i 0.432561π-0.432561\pi
0.210285 + 0.977640i 0.432561π0.432561\pi
164164 18.3722 0.00874773
165165 −680.782 −0.321205
166166 −3953.04 −1.84829
167167 2121.81 0.983178 0.491589 0.870827i 0.336416π-0.336416\pi
0.491589 + 0.870827i 0.336416π0.336416\pi
168168 0 0
169169 −1613.88 −0.734585
170170 115.522 0.0521184
171171 −1367.54 −0.611571
172172 204.821 0.0907993
173173 519.259 0.228200 0.114100 0.993469i 0.463602π-0.463602\pi
0.114100 + 0.993469i 0.463602π0.463602\pi
174174 −1761.65 −0.767533
175175 0 0
176176 −3097.35 −1.32654
177177 −1512.69 −0.642376
178178 1928.56 0.812088
179179 −432.844 −0.180739 −0.0903696 0.995908i 0.528805π-0.528805\pi
−0.0903696 + 0.995908i 0.528805π0.528805\pi
180180 25.7169 0.0106490
181181 2791.71 1.14644 0.573221 0.819401i 0.305694π-0.305694\pi
0.573221 + 0.819401i 0.305694π0.305694\pi
182182 0 0
183183 −985.177 −0.397958
184184 −855.461 −0.342747
185185 598.766 0.237958
186186 132.305 0.0521564
187187 358.165 0.140062
188188 97.7535 0.0379224
189189 0 0
190190 −2224.32 −0.849310
191191 −3221.32 −1.22035 −0.610173 0.792268i 0.708900π-0.708900\pi
−0.610173 + 0.792268i 0.708900π0.708900\pi
192192 −1411.16 −0.530425
193193 −3640.40 −1.35773 −0.678864 0.734264i 0.737527π-0.737527\pi
−0.678864 + 0.734264i 0.737527π0.737527\pi
194194 2595.13 0.960408
195195 −362.217 −0.133020
196196 0 0
197197 3178.85 1.14966 0.574832 0.818271i 0.305067π-0.305067\pi
0.574832 + 0.818271i 0.305067π0.305067\pi
198198 1195.88 0.429230
199199 3869.55 1.37842 0.689208 0.724563i 0.257959π-0.257959\pi
0.689208 + 0.724563i 0.257959π0.257959\pi
200200 −543.713 −0.192232
201201 −1557.02 −0.546387
202202 5425.05 1.88963
203203 0 0
204204 −13.5299 −0.00464354
205205 160.740 0.0547638
206206 −4900.21 −1.65735
207207 354.008 0.118866
208208 −1647.97 −0.549358
209209 −6896.29 −2.28242
210210 0 0
211211 4477.70 1.46094 0.730469 0.682946i 0.239302π-0.239302\pi
0.730469 + 0.682946i 0.239302π0.239302\pi
212212 221.629 0.0717998
213213 −3520.85 −1.13260
214214 −10.2637 −0.00327856
215215 1792.00 0.568435
216216 587.210 0.184975
217217 0 0
218218 −4401.86 −1.36758
219219 340.812 0.105160
220220 129.686 0.0397429
221221 190.565 0.0580036
222222 −1051.81 −0.317985
223223 −305.552 −0.0917546 −0.0458773 0.998947i 0.514608π-0.514608\pi
−0.0458773 + 0.998947i 0.514608π0.514608\pi
224224 0 0
225225 225.000 0.0666667
226226 619.564 0.182357
227227 −6621.96 −1.93619 −0.968094 0.250587i 0.919376π-0.919376\pi
−0.968094 + 0.250587i 0.919376π0.919376\pi
228228 260.511 0.0756701
229229 −5240.39 −1.51220 −0.756102 0.654454i 0.772899π-0.772899\pi
−0.756102 + 0.654454i 0.772899π0.772899\pi
230230 575.796 0.165073
231231 0 0
232232 −4362.16 −1.23444
233233 −2614.11 −0.735005 −0.367502 0.930023i 0.619787π-0.619787\pi
−0.367502 + 0.930023i 0.619787π0.619787\pi
234234 636.280 0.177756
235235 855.255 0.237407
236236 288.161 0.0794816
237237 2416.11 0.662207
238238 0 0
239239 −319.944 −0.0865919 −0.0432960 0.999062i 0.513786π-0.513786\pi
−0.0432960 + 0.999062i 0.513786π0.513786\pi
240240 1023.68 0.275326
241241 3429.97 0.916778 0.458389 0.888752i 0.348427π-0.348427\pi
0.458389 + 0.888752i 0.348427π0.348427\pi
242242 2133.84 0.566811
243243 −243.000 −0.0641500
244244 187.672 0.0492396
245245 0 0
246246 −282.360 −0.0731814
247247 −3669.24 −0.945215
248248 327.610 0.0838842
249249 4050.65 1.03092
250250 365.964 0.0925823
251251 −5152.46 −1.29570 −0.647849 0.761768i 0.724331π-0.724331\pi
−0.647849 + 0.761768i 0.724331π0.724331\pi
252252 0 0
253253 1785.20 0.443615
254254 2186.60 0.540155
255255 −118.374 −0.0290702
256256 873.432 0.213240
257257 1937.46 0.470254 0.235127 0.971965i 0.424449π-0.424449\pi
0.235127 + 0.971965i 0.424449π0.424449\pi
258258 −3147.88 −0.759606
259259 0 0
260260 69.0008 0.0164586
261261 1805.15 0.428108
262262 659.809 0.155584
263263 2817.53 0.660594 0.330297 0.943877i 0.392851π-0.392851\pi
0.330297 + 0.943877i 0.392851π0.392851\pi
264264 2961.20 0.690339
265265 1939.05 0.449491
266266 0 0
267267 −1976.18 −0.452959
268268 296.606 0.0676048
269269 7581.60 1.71843 0.859216 0.511612i 0.170952π-0.170952\pi
0.859216 + 0.511612i 0.170952π0.170952\pi
270270 −395.241 −0.0890874
271271 −2007.56 −0.450002 −0.225001 0.974359i 0.572239π-0.572239\pi
−0.225001 + 0.974359i 0.572239π0.572239\pi
272272 −538.566 −0.120057
273273 0 0
274274 −5349.56 −1.17948
275275 1134.64 0.248804
276276 −67.4370 −0.0147074
277277 522.535 0.113343 0.0566716 0.998393i 0.481951π-0.481951\pi
0.0566716 + 0.998393i 0.481951π0.481951\pi
278278 −4849.42 −1.04622
279279 −135.572 −0.0290913
280280 0 0
281281 2528.04 0.536691 0.268345 0.963323i 0.413523π-0.413523\pi
0.268345 + 0.963323i 0.413523π0.413523\pi
282282 −1502.36 −0.317250
283283 −2608.02 −0.547812 −0.273906 0.961756i 0.588316π-0.588316\pi
−0.273906 + 0.961756i 0.588316π0.588316\pi
284284 670.707 0.140138
285285 2279.24 0.473721
286286 3208.65 0.663397
287287 0 0
288288 −232.328 −0.0475349
289289 −4850.72 −0.987324
290290 2936.09 0.594528
291291 −2659.20 −0.535688
292292 −64.9233 −0.0130115
293293 −2830.10 −0.564287 −0.282143 0.959372i 0.591045π-0.591045\pi
−0.282143 + 0.959372i 0.591045π0.591045\pi
294294 0 0
295295 2521.15 0.497582
296296 −2604.46 −0.511422
297297 −1225.41 −0.239412
298298 3269.78 0.635614
299299 949.834 0.183713
300300 −42.8616 −0.00824871
301301 0 0
302302 1283.21 0.244505
303303 −5559.00 −1.05398
304304 10369.8 1.95642
305305 1641.96 0.308257
306306 207.939 0.0388468
307307 8752.67 1.62717 0.813585 0.581446i 0.197513π-0.197513\pi
0.813585 + 0.581446i 0.197513π0.197513\pi
308308 0 0
309309 5021.20 0.924421
310310 −220.509 −0.0404002
311311 −4245.23 −0.774034 −0.387017 0.922073i 0.626495π-0.626495\pi
−0.387017 + 0.922073i 0.626495π0.626495\pi
312312 1575.54 0.285889
313313 −5888.63 −1.06340 −0.531702 0.846932i 0.678447π-0.678447\pi
−0.531702 + 0.846932i 0.678447π0.678447\pi
314314 −3327.46 −0.598023
315315 0 0
316316 −460.258 −0.0819353
317317 −7074.82 −1.25351 −0.626753 0.779218i 0.715616π-0.715616\pi
−0.626753 + 0.779218i 0.715616π0.715616\pi
318318 −3406.19 −0.600660
319319 9103.08 1.59773
320320 2351.93 0.410865
321321 10.5171 0.00182869
322322 0 0
323323 −1199.13 −0.206567
324324 46.2905 0.00793733
325325 603.695 0.103037
326326 2562.40 0.435332
327327 4510.55 0.762795
328328 −699.172 −0.117699
329329 0 0
330330 −1993.13 −0.332480
331331 −10236.4 −1.69983 −0.849916 0.526918i 0.823347π-0.823347\pi
−0.849916 + 0.526918i 0.823347π0.823347\pi
332332 −771.632 −0.127557
333333 1077.78 0.177363
334334 6212.05 1.01769
335335 2595.04 0.423230
336336 0 0
337337 −5579.42 −0.901871 −0.450935 0.892557i 0.648909π-0.648909\pi
−0.450935 + 0.892557i 0.648909π0.648909\pi
338338 −4724.99 −0.760371
339339 −634.862 −0.101714
340340 22.5498 0.00359687
341341 −683.667 −0.108571
342342 −4003.77 −0.633039
343343 0 0
344344 −7794.68 −1.22169
345345 −590.013 −0.0920732
346346 1520.24 0.236210
347347 9526.56 1.47381 0.736906 0.675995i 0.236286π-0.236286\pi
0.736906 + 0.675995i 0.236286π0.236286\pi
348348 −343.874 −0.0529701
349349 198.263 0.0304092 0.0152046 0.999884i 0.495160π-0.495160\pi
0.0152046 + 0.999884i 0.495160π0.495160\pi
350350 0 0
351351 −651.990 −0.0991472
352352 −1171.59 −0.177403
353353 −9280.48 −1.39929 −0.699646 0.714489i 0.746659π-0.746659\pi
−0.699646 + 0.714489i 0.746659π0.746659\pi
354354 −4428.71 −0.664925
355355 5868.08 0.877311
356356 376.454 0.0560450
357357 0 0
358358 −1267.24 −0.187083
359359 −6589.47 −0.968743 −0.484372 0.874862i 0.660952π-0.660952\pi
−0.484372 + 0.874862i 0.660952π0.660952\pi
360360 −978.684 −0.143281
361361 16229.6 2.36617
362362 8173.31 1.18668
363363 −2186.53 −0.316151
364364 0 0
365365 −568.020 −0.0814562
366366 −2884.31 −0.411927
367367 10984.5 1.56236 0.781180 0.624305i 0.214618π-0.214618\pi
0.781180 + 0.624305i 0.214618π0.214618\pi
368368 −2684.37 −0.380252
369369 289.332 0.0408185
370370 1753.01 0.246310
371371 0 0
372372 25.8259 0.00359949
373373 −2190.04 −0.304010 −0.152005 0.988380i 0.548573π-0.548573\pi
−0.152005 + 0.988380i 0.548573π0.548573\pi
374374 1048.60 0.144979
375375 −375.000 −0.0516398
376376 −3720.11 −0.510239
377377 4843.38 0.661663
378378 0 0
379379 2063.29 0.279642 0.139821 0.990177i 0.455347π-0.455347\pi
0.139821 + 0.990177i 0.455347π0.455347\pi
380380 −434.186 −0.0586138
381381 −2240.59 −0.301283
382382 −9431.08 −1.26318
383383 −10582.1 −1.41180 −0.705898 0.708313i 0.749457π-0.749457\pi
−0.705898 + 0.708313i 0.749457π0.749457\pi
384384 −4751.00 −0.631376
385385 0 0
386386 −10658.0 −1.40539
387387 3225.60 0.423686
388388 506.567 0.0662811
389389 7301.27 0.951643 0.475822 0.879542i 0.342151π-0.342151\pi
0.475822 + 0.879542i 0.342151π0.342151\pi
390390 −1060.47 −0.137689
391391 310.411 0.0401487
392392 0 0
393393 −676.100 −0.0867805
394394 9306.76 1.19002
395395 −4026.84 −0.512943
396396 233.435 0.0296226
397397 7650.00 0.967110 0.483555 0.875314i 0.339345π-0.339345\pi
0.483555 + 0.875314i 0.339345π0.339345\pi
398398 11328.9 1.42680
399399 0 0
400400 −1706.13 −0.213267
401401 7383.41 0.919476 0.459738 0.888055i 0.347943π-0.347943\pi
0.459738 + 0.888055i 0.347943π0.347943\pi
402402 −4558.51 −0.565566
403403 −363.752 −0.0449622
404404 1058.97 0.130410
405405 405.000 0.0496904
406406 0 0
407407 5435.06 0.661931
408408 514.894 0.0624780
409409 13061.0 1.57903 0.789517 0.613729i 0.210331π-0.210331\pi
0.789517 + 0.613729i 0.210331π0.210331\pi
410410 470.600 0.0566861
411411 5481.65 0.657883
412412 −956.517 −0.114379
413413 0 0
414414 1036.43 0.123038
415415 −6751.08 −0.798549
416416 −623.356 −0.0734677
417417 4969.16 0.583551
418418 −20190.3 −2.36254
419419 −12329.9 −1.43760 −0.718800 0.695217i 0.755308π-0.755308\pi
−0.718800 + 0.695217i 0.755308π0.755308\pi
420420 0 0
421421 −4011.19 −0.464354 −0.232177 0.972673i 0.574585π-0.574585\pi
−0.232177 + 0.972673i 0.574585π0.574585\pi
422422 13109.4 1.51222
423423 1539.46 0.176953
424424 −8434.32 −0.966054
425425 197.291 0.0225176
426426 −10308.0 −1.17236
427427 0 0
428428 −2.00347 −0.000226265 0
429429 −3287.88 −0.370024
430430 5246.46 0.588388
431431 −12983.2 −1.45099 −0.725495 0.688227i 0.758389π-0.758389\pi
−0.725495 + 0.688227i 0.758389π0.758389\pi
432432 1842.62 0.205216
433433 10976.3 1.21822 0.609109 0.793087i 0.291527π-0.291527\pi
0.609109 + 0.793087i 0.291527π0.291527\pi
434434 0 0
435435 −3008.59 −0.331611
436436 −859.241 −0.0943812
437437 −5976.81 −0.654255
438438 997.799 0.108851
439439 17477.2 1.90009 0.950045 0.312112i 0.101036π-0.101036\pi
0.950045 + 0.312112i 0.101036π0.101036\pi
440440 −4935.34 −0.534734
441441 0 0
442442 557.920 0.0600397
443443 2083.78 0.223484 0.111742 0.993737i 0.464357π-0.464357\pi
0.111742 + 0.993737i 0.464357π0.464357\pi
444444 −205.312 −0.0219453
445445 3293.63 0.350861
446446 −894.568 −0.0949754
447447 −3350.51 −0.354527
448448 0 0
449449 −1874.17 −0.196988 −0.0984938 0.995138i 0.531402π-0.531402\pi
−0.0984938 + 0.995138i 0.531402π0.531402\pi
450450 658.735 0.0690068
451451 1459.05 0.152337
452452 120.939 0.0125851
453453 −1314.90 −0.136378
454454 −19387.2 −2.00415
455455 0 0
456456 −9914.02 −1.01813
457457 −6648.72 −0.680556 −0.340278 0.940325i 0.610521π-0.610521\pi
−0.340278 + 0.940325i 0.610521π0.610521\pi
458458 −15342.3 −1.56529
459459 −213.074 −0.0216676
460460 112.395 0.0113923
461461 15529.0 1.56888 0.784442 0.620202i 0.212949π-0.212949\pi
0.784442 + 0.620202i 0.212949π0.212949\pi
462462 0 0
463463 2365.89 0.237478 0.118739 0.992926i 0.462115π-0.462115\pi
0.118739 + 0.992926i 0.462115π0.462115\pi
464464 −13688.1 −1.36952
465465 225.953 0.0225341
466466 −7653.36 −0.760805
467467 −17310.3 −1.71525 −0.857626 0.514274i 0.828062π-0.828062\pi
−0.857626 + 0.514274i 0.828062π0.828062\pi
468468 124.201 0.0122675
469469 0 0
470470 2503.94 0.245741
471471 3409.62 0.333560
472472 −10966.2 −1.06941
473473 16266.2 1.58122
474474 7073.66 0.685452
475475 −3798.73 −0.366943
476476 0 0
477477 3490.30 0.335031
478478 −936.704 −0.0896314
479479 2363.33 0.225435 0.112717 0.993627i 0.464045π-0.464045\pi
0.112717 + 0.993627i 0.464045π0.464045\pi
480480 387.213 0.0368204
481481 2891.78 0.274124
482482 10041.9 0.948959
483483 0 0
484484 416.524 0.0391176
485485 4432.01 0.414942
486486 −711.434 −0.0664018
487487 −17815.5 −1.65770 −0.828849 0.559473i 0.811004π-0.811004\pi
−0.828849 + 0.559473i 0.811004π0.811004\pi
488488 −7142.05 −0.662511
489489 −2625.67 −0.242816
490490 0 0
491491 1968.10 0.180894 0.0904469 0.995901i 0.471170π-0.471170\pi
0.0904469 + 0.995901i 0.471170π0.471170\pi
492492 −55.1166 −0.00505050
493493 1582.84 0.144600
494494 −10742.5 −0.978394
495495 2042.35 0.185448
496496 1028.02 0.0930632
497497 0 0
498498 11859.1 1.06711
499499 17118.5 1.53573 0.767863 0.640614i 0.221320π-0.221320\pi
0.767863 + 0.640614i 0.221320π0.221320\pi
500500 71.4359 0.00638942
501501 −6365.44 −0.567638
502502 −15084.9 −1.34118
503503 −7025.91 −0.622803 −0.311401 0.950278i 0.600798π-0.600798\pi
−0.311401 + 0.950278i 0.600798π0.600798\pi
504504 0 0
505505 9265.00 0.816410
506506 5226.55 0.459187
507507 4841.65 0.424113
508508 426.823 0.0372779
509509 5155.61 0.448955 0.224478 0.974479i 0.427932π-0.427932\pi
0.224478 + 0.974479i 0.427932π0.427932\pi
510510 −346.566 −0.0300906
511511 0 0
512512 −10112.2 −0.872851
513513 4102.63 0.353091
514514 5672.32 0.486761
515515 −8368.67 −0.716053
516516 −614.464 −0.0524230
517517 7763.23 0.660400
518518 0 0
519519 −1557.78 −0.131751
520520 −2625.89 −0.221448
521521 −1673.01 −0.140683 −0.0703416 0.997523i 0.522409π-0.522409\pi
−0.0703416 + 0.997523i 0.522409π0.522409\pi
522522 5284.96 0.443135
523523 612.609 0.0512190 0.0256095 0.999672i 0.491847π-0.491847\pi
0.0256095 + 0.999672i 0.491847π0.491847\pi
524524 128.794 0.0107374
525525 0 0
526526 8248.90 0.683782
527527 −118.876 −0.00982603
528528 9292.04 0.765879
529529 −10619.8 −0.872838
530530 5676.99 0.465269
531531 4538.06 0.370876
532532 0 0
533533 776.304 0.0630871
534534 −5785.68 −0.468859
535535 −17.5285 −0.00141649
536536 −11287.6 −0.909612
537537 1298.53 0.104350
538538 22196.7 1.77875
539539 0 0
540540 −77.1508 −0.00614823
541541 12856.9 1.02174 0.510872 0.859657i 0.329323π-0.329323\pi
0.510872 + 0.859657i 0.329323π0.329323\pi
542542 −5877.55 −0.465798
543543 −8375.12 −0.661899
544544 −203.716 −0.0160556
545545 −7517.59 −0.590859
546546 0 0
547547 −15953.2 −1.24700 −0.623500 0.781824i 0.714290π-0.714290\pi
−0.623500 + 0.781824i 0.714290π0.714290\pi
548548 −1044.23 −0.0814003
549549 2955.53 0.229761
550550 3321.89 0.257538
551551 −30476.9 −2.35637
552552 2566.38 0.197885
553553 0 0
554554 1529.83 0.117322
555555 −1796.30 −0.137385
556556 −946.605 −0.0722032
557557 −6765.42 −0.514650 −0.257325 0.966325i 0.582841π-0.582841\pi
−0.257325 + 0.966325i 0.582841π0.582841\pi
558558 −396.916 −0.0301125
559559 8654.58 0.654829
560560 0 0
561561 −1074.50 −0.0808649
562562 7401.36 0.555529
563563 −23505.6 −1.75958 −0.879789 0.475364i 0.842316π-0.842316\pi
−0.879789 + 0.475364i 0.842316π0.842316\pi
564564 −293.261 −0.0218945
565565 1058.10 0.0787872
566566 −7635.53 −0.567041
567567 0 0
568568 −25524.4 −1.88553
569569 477.650 0.0351918 0.0175959 0.999845i 0.494399π-0.494399\pi
0.0175959 + 0.999845i 0.494399π0.494399\pi
570570 6672.95 0.490350
571571 14081.6 1.03204 0.516022 0.856576i 0.327412π-0.327412\pi
0.516022 + 0.856576i 0.327412π0.327412\pi
572572 626.327 0.0457833
573573 9663.95 0.704568
574574 0 0
575575 983.355 0.0713196
576576 4233.47 0.306241
577577 10405.0 0.750719 0.375359 0.926879i 0.377519π-0.377519\pi
0.375359 + 0.926879i 0.377519π0.377519\pi
578578 −14201.5 −1.02198
579579 10921.2 0.783885
580580 573.123 0.0410304
581581 0 0
582582 −7785.38 −0.554492
583583 17601.0 1.25036
584584 2470.72 0.175067
585585 1086.65 0.0767991
586586 −8285.70 −0.584094
587587 −14406.0 −1.01295 −0.506473 0.862256i 0.669051π-0.669051\pi
−0.506473 + 0.862256i 0.669051π0.669051\pi
588588 0 0
589589 2288.90 0.160123
590590 7381.19 0.515049
591591 −9536.56 −0.663759
592592 −8172.60 −0.567384
593593 −11647.6 −0.806596 −0.403298 0.915069i 0.632136π-0.632136\pi
−0.403298 + 0.915069i 0.632136π0.632136\pi
594594 −3587.64 −0.247816
595595 0 0
596596 638.258 0.0438659
597597 −11608.6 −0.795829
598598 2780.84 0.190162
599599 6289.16 0.428995 0.214498 0.976725i 0.431189π-0.431189\pi
0.214498 + 0.976725i 0.431189π0.431189\pi
600600 1631.14 0.110985
601601 10340.0 0.701796 0.350898 0.936414i 0.385876π-0.385876\pi
0.350898 + 0.936414i 0.385876π0.385876\pi
602602 0 0
603603 4671.06 0.315457
604604 250.482 0.0168741
605605 3644.21 0.244890
606606 −16275.1 −1.09098
607607 15606.3 1.04356 0.521781 0.853080i 0.325268π-0.325268\pi
0.521781 + 0.853080i 0.325268π0.325268\pi
608608 3922.45 0.261639
609609 0 0
610610 4807.19 0.319078
611611 4130.51 0.273490
612612 40.5897 0.00268095
613613 23004.4 1.51572 0.757861 0.652416i 0.226244π-0.226244\pi
0.757861 + 0.652416i 0.226244π0.226244\pi
614614 25625.3 1.68429
615615 −482.220 −0.0316179
616616 0 0
617617 21516.8 1.40394 0.701971 0.712205i 0.252304π-0.252304\pi
0.701971 + 0.712205i 0.252304π0.252304\pi
618618 14700.6 0.956870
619619 7302.79 0.474191 0.237095 0.971486i 0.423805π-0.423805\pi
0.237095 + 0.971486i 0.423805π0.423805\pi
620620 −43.0432 −0.00278815
621621 −1062.02 −0.0686273
622622 −12428.8 −0.801204
623623 0 0
624624 4943.92 0.317172
625625 625.000 0.0400000
626626 −17240.2 −1.10073
627627 20688.9 1.31776
628628 −649.518 −0.0412716
629629 945.047 0.0599070
630630 0 0
631631 28805.7 1.81733 0.908666 0.417524i 0.137102π-0.137102\pi
0.908666 + 0.417524i 0.137102π0.137102\pi
632632 17515.6 1.10243
633633 −13433.1 −0.843473
634634 −20713.0 −1.29751
635635 3734.31 0.233373
636636 −664.887 −0.0414536
637637 0 0
638638 26651.2 1.65381
639639 10562.5 0.653909
640640 7918.34 0.489062
641641 −7479.14 −0.460855 −0.230428 0.973089i 0.574013π-0.574013\pi
−0.230428 + 0.973089i 0.574013π0.574013\pi
642642 30.7911 0.00189288
643643 −25832.0 −1.58432 −0.792158 0.610316i 0.791042π-0.791042\pi
−0.792158 + 0.610316i 0.791042π0.791042\pi
644644 0 0
645645 −5376.01 −0.328186
646646 −3510.70 −0.213818
647647 14971.7 0.909736 0.454868 0.890559i 0.349686π-0.349686\pi
0.454868 + 0.890559i 0.349686π0.349686\pi
648648 −1761.63 −0.106795
649649 22884.7 1.38413
650650 1767.44 0.106654
651651 0 0
652652 500.179 0.0300438
653653 7423.31 0.444865 0.222432 0.974948i 0.428600π-0.428600\pi
0.222432 + 0.974948i 0.428600π0.428600\pi
654654 13205.6 0.789571
655655 1126.83 0.0672199
656656 −2193.95 −0.130578
657657 −1022.44 −0.0607139
658658 0 0
659659 3126.66 0.184822 0.0924109 0.995721i 0.470543π-0.470543\pi
0.0924109 + 0.995721i 0.470543π0.470543\pi
660660 −389.058 −0.0229456
661661 −7725.29 −0.454582 −0.227291 0.973827i 0.572987π-0.572987\pi
−0.227291 + 0.973827i 0.572987π0.572987\pi
662662 −29969.3 −1.75950
663663 −571.696 −0.0334884
664664 29365.2 1.71625
665665 0 0
666666 3155.42 0.183589
667667 7889.36 0.457987
668668 1212.59 0.0702343
669669 916.656 0.0529745
670670 7597.51 0.438086
671671 14904.2 0.857484
672672 0 0
673673 26556.6 1.52107 0.760536 0.649296i 0.224936π-0.224936\pi
0.760536 + 0.649296i 0.224936π0.224936\pi
674674 −16334.9 −0.933528
675675 −675.000 −0.0384900
676676 −922.315 −0.0524758
677677 −21375.9 −1.21350 −0.606752 0.794891i 0.707528π-0.707528\pi
−0.606752 + 0.794891i 0.707528π0.707528\pi
678678 −1858.69 −0.105284
679679 0 0
680680 −858.156 −0.0483953
681681 19865.9 1.11786
682682 −2001.58 −0.112382
683683 19700.2 1.10367 0.551834 0.833954i 0.313928π-0.313928\pi
0.551834 + 0.833954i 0.313928π0.313928\pi
684684 −781.534 −0.0436882
685685 −9136.08 −0.509594
686686 0 0
687687 15721.2 0.873071
688688 −24459.1 −1.35537
689689 9364.78 0.517808
690690 −1727.39 −0.0953051
691691 −4568.18 −0.251493 −0.125747 0.992062i 0.540133π-0.540133\pi
−0.125747 + 0.992062i 0.540133π0.540133\pi
692692 296.750 0.0163017
693693 0 0
694694 27891.0 1.52555
695695 −8281.94 −0.452017
696696 13086.5 0.712703
697697 253.700 0.0137871
698698 580.458 0.0314766
699699 7842.33 0.424355
700700 0 0
701701 −22203.6 −1.19632 −0.598159 0.801377i 0.704101π-0.704101\pi
−0.598159 + 0.801377i 0.704101π0.704101\pi
702702 −1908.84 −0.102627
703703 −18196.4 −0.976232
704704 21348.7 1.14291
705705 −2565.77 −0.137067
706706 −27170.6 −1.44841
707707 0 0
708708 −864.482 −0.0458887
709709 25615.6 1.35686 0.678430 0.734665i 0.262661π-0.262661\pi
0.678430 + 0.734665i 0.262661π0.262661\pi
710710 17180.0 0.908106
711711 −7248.32 −0.382325
712712 −14326.3 −0.754076
713713 −592.513 −0.0311217
714714 0 0
715715 5479.79 0.286619
716716 −247.365 −0.0129113
717717 959.832 0.0499939
718718 −19292.1 −1.00275
719719 16228.4 0.841747 0.420873 0.907119i 0.361724π-0.361724\pi
0.420873 + 0.907119i 0.361724π0.361724\pi
720720 −3071.04 −0.158960
721721 0 0
722722 47515.5 2.44923
723723 −10289.9 −0.529302
724724 1595.43 0.0818971
725725 5014.31 0.256865
726726 −6401.51 −0.327248
727727 −34994.2 −1.78523 −0.892615 0.450820i 0.851132π-0.851132\pi
−0.892615 + 0.450820i 0.851132π0.851132\pi
728728 0 0
729729 729.000 0.0370370
730730 −1663.00 −0.0843155
731731 2828.36 0.143106
732732 −563.016 −0.0284285
733733 24430.8 1.23107 0.615533 0.788111i 0.288941π-0.288941\pi
0.615533 + 0.788111i 0.288941π0.288941\pi
734734 32159.4 1.61720
735735 0 0
736736 −1015.38 −0.0508525
737737 23555.4 1.17731
738738 847.081 0.0422513
739739 −17001.5 −0.846291 −0.423146 0.906062i 0.639074π-0.639074\pi
−0.423146 + 0.906062i 0.639074π0.639074\pi
740740 342.187 0.0169987
741741 11007.7 0.545720
742742 0 0
743743 14.3426 0.000708183 0 0.000354092 1.00000i 0.499887π-0.499887\pi
0.000354092 1.00000i 0.499887π0.499887\pi
744744 −982.831 −0.0484305
745745 5584.19 0.274616
746746 −6411.80 −0.314682
747747 −12152.0 −0.595203
748748 204.687 0.0100055
749749 0 0
750750 −1097.89 −0.0534524
751751 −30724.4 −1.49287 −0.746437 0.665456i 0.768237π-0.768237\pi
−0.746437 + 0.665456i 0.768237π0.768237\pi
752752 −11673.4 −0.566072
753753 15457.4 0.748072
754754 14180.0 0.684888
755755 2191.49 0.105638
756756 0 0
757757 −27240.7 −1.30790 −0.653950 0.756538i 0.726889π-0.726889\pi
−0.653950 + 0.756538i 0.726889π0.726889\pi
758758 6040.72 0.289458
759759 −5355.61 −0.256121
760760 16523.4 0.788639
761761 10252.4 0.488370 0.244185 0.969729i 0.421480π-0.421480\pi
0.244185 + 0.969729i 0.421480π0.421480\pi
762762 −6559.79 −0.311859
763763 0 0
764764 −1840.94 −0.0871766
765765 355.123 0.0167837
766766 −30981.2 −1.46135
767767 12176.0 0.573208
768768 −2620.30 −0.123114
769769 −37870.0 −1.77585 −0.887925 0.459989i 0.847853π-0.847853\pi
−0.887925 + 0.459989i 0.847853π0.847853\pi
770770 0 0
771771 −5812.37 −0.271501
772772 −2080.44 −0.0969906
773773 16744.1 0.779098 0.389549 0.921006i 0.372631π-0.372631\pi
0.389549 + 0.921006i 0.372631π0.372631\pi
774774 9443.63 0.438559
775775 −376.589 −0.0174548
776776 −19277.9 −0.891801
777777 0 0
778778 21376.0 0.985047
779779 −4884.87 −0.224671
780780 −207.002 −0.00950240
781781 53265.1 2.44043
782782 908.793 0.0415580
783783 −5415.46 −0.247168
784784 0 0
785785 −5682.70 −0.258375
786786 −1979.43 −0.0898267
787787 −17080.1 −0.773622 −0.386811 0.922159i 0.626423π-0.626423\pi
−0.386811 + 0.922159i 0.626423π0.626423\pi
788788 1816.68 0.0821274
789789 −8452.58 −0.381394
790790 −11789.4 −0.530948
791791 0 0
792792 −8883.61 −0.398567
793793 7929.95 0.355108
794794 22397.0 1.00106
795795 −5817.16 −0.259514
796796 2211.40 0.0984684
797797 −30528.1 −1.35679 −0.678395 0.734697i 0.737324π-0.737324\pi
−0.678395 + 0.734697i 0.737324π0.737324\pi
798798 0 0
799799 1349.87 0.0597684
800800 −645.355 −0.0285209
801801 5928.54 0.261516
802802 21616.5 0.951752
803803 −5155.97 −0.226588
804804 −889.818 −0.0390317
805805 0 0
806806 −1064.96 −0.0465404
807807 −22744.8 −0.992138
808808 −40300.0 −1.75464
809809 36969.6 1.60665 0.803325 0.595541i 0.203062π-0.203062\pi
0.803325 + 0.595541i 0.203062π0.203062\pi
810810 1185.72 0.0514346
811811 35231.2 1.52544 0.762722 0.646727i 0.223863π-0.223863\pi
0.762722 + 0.646727i 0.223863π0.223863\pi
812812 0 0
813813 6022.68 0.259809
814814 15912.3 0.685166
815815 4376.12 0.188084
816816 1615.70 0.0693147
817817 −54458.7 −2.33203
818818 38238.8 1.63446
819819 0 0
820820 91.8609 0.00391210
821821 −3853.10 −0.163793 −0.0818965 0.996641i 0.526098π-0.526098\pi
−0.0818965 + 0.996641i 0.526098π0.526098\pi
822822 16048.7 0.680976
823823 −19323.3 −0.818431 −0.409216 0.912438i 0.634198π-0.634198\pi
−0.409216 + 0.912438i 0.634198π0.634198\pi
824824 36401.2 1.53895
825825 −3403.91 −0.143647
826826 0 0
827827 42985.0 1.80742 0.903710 0.428145i 0.140833π-0.140833\pi
0.903710 + 0.428145i 0.140833π0.140833\pi
828828 202.311 0.00849130
829829 7707.12 0.322894 0.161447 0.986881i 0.448384π-0.448384\pi
0.161447 + 0.986881i 0.448384π0.448384\pi
830830 −19765.2 −0.826579
831831 −1567.60 −0.0654387
832832 11358.8 0.473311
833833 0 0
834834 14548.3 0.604035
835835 10609.1 0.439691
836836 −3941.14 −0.163047
837837 406.716 0.0167959
838838 −36098.3 −1.48806
839839 15324.4 0.630579 0.315290 0.948995i 0.397898π-0.397898\pi
0.315290 + 0.948995i 0.397898π0.397898\pi
840840 0 0
841841 15840.3 0.649486
842842 −11743.6 −0.480654
843843 −7584.11 −0.309858
844844 2558.95 0.104363
845845 −8069.42 −0.328517
846846 4507.09 0.183164
847847 0 0
848848 −26466.3 −1.07176
849849 7824.06 0.316279
850850 577.610 0.0233081
851851 4710.40 0.189742
852852 −2012.12 −0.0809086
853853 21332.0 0.856266 0.428133 0.903716i 0.359172π-0.359172\pi
0.428133 + 0.903716i 0.359172π0.359172\pi
854854 0 0
855855 −6837.72 −0.273503
856856 76.2440 0.00304435
857857 32731.6 1.30466 0.652328 0.757936i 0.273792π-0.273792\pi
0.652328 + 0.757936i 0.273792π0.273792\pi
858858 −9625.95 −0.383012
859859 35989.8 1.42952 0.714759 0.699371i 0.246536π-0.246536\pi
0.714759 + 0.699371i 0.246536π0.246536\pi
860860 1024.11 0.0406067
861861 0 0
862862 −38010.9 −1.50192
863863 −40400.4 −1.59356 −0.796781 0.604268i 0.793466π-0.793466\pi
−0.796781 + 0.604268i 0.793466π0.793466\pi
864864 696.984 0.0274443
865865 2596.30 0.102054
866866 32135.5 1.26098
867867 14552.2 0.570032
868868 0 0
869869 −36552.1 −1.42686
870870 −8808.27 −0.343251
871871 12532.9 0.487555
872872 32699.3 1.26988
873873 7977.61 0.309280
874874 −17498.4 −0.677221
875875 0 0
876876 194.770 0.00751217
877877 15813.9 0.608890 0.304445 0.952530i 0.401529π-0.401529\pi
0.304445 + 0.952530i 0.401529π0.401529\pi
878878 51168.1 1.96679
879879 8490.29 0.325791
880880 −15486.7 −0.593247
881881 11465.6 0.438461 0.219231 0.975673i 0.429645π-0.429645\pi
0.219231 + 0.975673i 0.429645π0.429645\pi
882882 0 0
883883 −15044.6 −0.573376 −0.286688 0.958024i 0.592554π-0.592554\pi
−0.286688 + 0.958024i 0.592554π0.592554\pi
884884 108.906 0.00414354
885885 −7563.44 −0.287279
886886 6100.70 0.231329
887887 −4917.35 −0.186143 −0.0930714 0.995659i 0.529668π-0.529668\pi
−0.0930714 + 0.995659i 0.529668π0.529668\pi
888888 7813.37 0.295270
889889 0 0
890890 9642.80 0.363177
891891 3676.22 0.138225
892892 −174.619 −0.00655458
893893 −25991.1 −0.973974
894894 −9809.33 −0.366972
895895 −2164.22 −0.0808290
896896 0 0
897897 −2849.50 −0.106067
898898 −5487.02 −0.203902
899899 −3021.34 −0.112088
900900 128.585 0.00476240
901901 3060.46 0.113162
902902 4271.69 0.157685
903903 0 0
904904 −4602.44 −0.169331
905905 13958.5 0.512704
906906 −3849.64 −0.141165
907907 −37984.5 −1.39058 −0.695289 0.718731i 0.744723π-0.744723\pi
−0.695289 + 0.718731i 0.744723π0.744723\pi
908908 −3784.37 −0.138313
909909 16677.0 0.608516
910910 0 0
911911 5336.79 0.194090 0.0970449 0.995280i 0.469061π-0.469061\pi
0.0970449 + 0.995280i 0.469061π0.469061\pi
912912 −31109.5 −1.12954
913913 −61280.2 −2.22134
914914 −19465.5 −0.704444
915915 −4925.88 −0.177972
916916 −2994.82 −0.108026
917917 0 0
918918 −623.818 −0.0224282
919919 −37636.1 −1.35093 −0.675463 0.737394i 0.736056π-0.736056\pi
−0.675463 + 0.737394i 0.736056π0.736056\pi
920920 −4277.31 −0.153281
921921 −26258.0 −0.939447
922922 45464.3 1.62396
923923 28340.2 1.01065
924924 0 0
925925 2993.83 0.106418
926926 6926.64 0.245814
927927 −15063.6 −0.533715
928928 −5177.62 −0.183151
929929 −15972.3 −0.564084 −0.282042 0.959402i 0.591012π-0.591012\pi
−0.282042 + 0.959402i 0.591012π0.591012\pi
930930 661.526 0.0233250
931931 0 0
932932 −1493.93 −0.0525057
933933 12735.7 0.446889
934934 −50679.4 −1.77546
935935 1790.83 0.0626377
936936 −4726.61 −0.165058
937937 −47349.8 −1.65085 −0.825426 0.564510i 0.809065π-0.809065\pi
−0.825426 + 0.564510i 0.809065π0.809065\pi
938938 0 0
939939 17665.9 0.613956
940940 488.768 0.0169594
941941 30521.3 1.05735 0.528674 0.848825i 0.322689π-0.322689\pi
0.528674 + 0.848825i 0.322689π0.322689\pi
942942 9982.37 0.345269
943943 1264.52 0.0436674
944944 −34411.3 −1.18643
945945 0 0
946946 47622.6 1.63673
947947 −18950.3 −0.650267 −0.325134 0.945668i 0.605409π-0.605409\pi
−0.325134 + 0.945668i 0.605409π0.605409\pi
948948 1380.77 0.0473054
949949 −2743.29 −0.0938365
950950 −11121.6 −0.379823
951951 21224.5 0.723712
952952 0 0
953953 −34526.2 −1.17357 −0.586785 0.809743i 0.699607π-0.699607\pi
−0.586785 + 0.809743i 0.699607π0.699607\pi
954954 10218.6 0.346791
955955 −16106.6 −0.545756
956956 −182.844 −0.00618577
957957 −27309.2 −0.922447
958958 6919.14 0.233348
959959 0 0
960960 −7055.79 −0.237213
961961 −29564.1 −0.992383
962962 8466.28 0.283746
963963 −31.5514 −0.00105579
964964 1960.18 0.0654909
965965 −18202.0 −0.607195
966966 0 0
967967 −41107.3 −1.36703 −0.683517 0.729934i 0.739551π-0.739551\pi
−0.683517 + 0.729934i 0.739551π0.739551\pi
968968 −15851.2 −0.526320
969969 3597.38 0.119262
970970 12975.6 0.429508
971971 −13730.5 −0.453793 −0.226896 0.973919i 0.572858π-0.572858\pi
−0.226896 + 0.973919i 0.572858π0.572858\pi
972972 −138.871 −0.00458262
973973 0 0
974974 −52158.7 −1.71589
975975 −1811.08 −0.0594883
976976 −22411.2 −0.735006
977977 −15039.7 −0.492490 −0.246245 0.969208i 0.579197π-0.579197\pi
−0.246245 + 0.969208i 0.579197π0.579197\pi
978978 −7687.20 −0.251339
979979 29896.6 0.975996
980980 0 0
981981 −13531.7 −0.440400
982982 5762.01 0.187244
983983 −27849.6 −0.903627 −0.451814 0.892112i 0.649223π-0.649223\pi
−0.451814 + 0.892112i 0.649223π0.649223\pi
984984 2097.52 0.0679537
985985 15894.3 0.514146
986986 4634.10 0.149675
987987 0 0
988988 −2096.92 −0.0675223
989989 14097.4 0.453257
990990 5979.40 0.191957
991991 4838.08 0.155082 0.0775411 0.996989i 0.475293π-0.475293\pi
0.0775411 + 0.996989i 0.475293π0.475293\pi
992992 388.854 0.0124457
993993 30709.3 0.981399
994994 0 0
995995 19347.7 0.616446
996996 2314.90 0.0736449
997997 37173.5 1.18084 0.590419 0.807097i 0.298962π-0.298962\pi
0.590419 + 0.807097i 0.298962π0.298962\pi
998998 50117.9 1.58963
999999 −3233.34 −0.102401
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 735.4.a.v.1.3 4
3.2 odd 2 2205.4.a.bn.1.2 4
7.6 odd 2 735.4.a.w.1.3 yes 4
21.20 even 2 2205.4.a.bo.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
735.4.a.v.1.3 4 1.1 even 1 trivial
735.4.a.w.1.3 yes 4 7.6 odd 2
2205.4.a.bn.1.2 4 3.2 odd 2
2205.4.a.bo.1.2 4 21.20 even 2