Properties

Label 115.4.a.c.1.2
Level 115115
Weight 44
Character 115.1
Self dual yes
Analytic conductor 6.7856.785
Analytic rank 11
Dimension 22
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] = [115,4,Mod(1,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("115.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 115=523 115 = 5 \cdot 23
Weight: k k == 4 4
Character orbit: [χ][\chi] == 115.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: 6.785219650666.78521965066
Analytic rank: 11
Dimension: 22
Coefficient field: Q(109)\Q(\sqrt{109})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2x27 x^{2} - x - 27 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: yes
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.2
Root 4.72015-4.72015 of defining polynomial
Character χ\chi == 115.1

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) == q3.00000q2+3.72015q3+1.00000q4+5.00000q511.1605q625.6008q7+21.0000q813.1605q915.0000q10+12.6008q11+3.72015q12+8.16046q13+76.8023q14+18.6008q1571.0000q1676.0411q17+39.4814q18103.362q19+5.00000q2095.2388q2137.8023q2223.0000q23+78.1232q24+25.0000q2524.4814q26149.403q2725.6008q28267.202q2955.8023q3063.7574q31+45.0000q32+46.8768q33+228.123q34128.004q3513.1605q36+112.164q37+310.086q38+30.3582q39+105.000q40239.078q41+285.716q42+282.239q43+12.6008q4465.8023q45+69.0000q46+577.291q47264.131q48+312.399q4975.0000q50282.884q51+8.16046q522.31326q53+448.209q54+63.0038q55537.616q56384.522q57+801.605q58+272.888q59+18.6008q60+294.049q61+191.272q62+336.918q63+433.000q64+40.8023q65140.630q66+426.732q6776.0411q6885.5635q69+384.011q701020.85q71276.370q72286.650q73336.493q74+93.0038q75103.362q76322.589q7791.0745q78551.866q79355.000q80200.470q81+717.235q82+21.7021q8395.2388q84380.205q85846.716q86994.031q87+264.616q881049.63q89+197.407q90208.914q9123.0000q92237.187q931731.87q94516.810q95+167.407q96+1729.19q97937.198q98165.832q99+O(q100)q-3.00000 q^{2} +3.72015 q^{3} +1.00000 q^{4} +5.00000 q^{5} -11.1605 q^{6} -25.6008 q^{7} +21.0000 q^{8} -13.1605 q^{9} -15.0000 q^{10} +12.6008 q^{11} +3.72015 q^{12} +8.16046 q^{13} +76.8023 q^{14} +18.6008 q^{15} -71.0000 q^{16} -76.0411 q^{17} +39.4814 q^{18} -103.362 q^{19} +5.00000 q^{20} -95.2388 q^{21} -37.8023 q^{22} -23.0000 q^{23} +78.1232 q^{24} +25.0000 q^{25} -24.4814 q^{26} -149.403 q^{27} -25.6008 q^{28} -267.202 q^{29} -55.8023 q^{30} -63.7574 q^{31} +45.0000 q^{32} +46.8768 q^{33} +228.123 q^{34} -128.004 q^{35} -13.1605 q^{36} +112.164 q^{37} +310.086 q^{38} +30.3582 q^{39} +105.000 q^{40} -239.078 q^{41} +285.716 q^{42} +282.239 q^{43} +12.6008 q^{44} -65.8023 q^{45} +69.0000 q^{46} +577.291 q^{47} -264.131 q^{48} +312.399 q^{49} -75.0000 q^{50} -282.884 q^{51} +8.16046 q^{52} -2.31326 q^{53} +448.209 q^{54} +63.0038 q^{55} -537.616 q^{56} -384.522 q^{57} +801.605 q^{58} +272.888 q^{59} +18.6008 q^{60} +294.049 q^{61} +191.272 q^{62} +336.918 q^{63} +433.000 q^{64} +40.8023 q^{65} -140.630 q^{66} +426.732 q^{67} -76.0411 q^{68} -85.5635 q^{69} +384.011 q^{70} -1020.85 q^{71} -276.370 q^{72} -286.650 q^{73} -336.493 q^{74} +93.0038 q^{75} -103.362 q^{76} -322.589 q^{77} -91.0745 q^{78} -551.866 q^{79} -355.000 q^{80} -200.470 q^{81} +717.235 q^{82} +21.7021 q^{83} -95.2388 q^{84} -380.205 q^{85} -846.716 q^{86} -994.031 q^{87} +264.616 q^{88} -1049.63 q^{89} +197.407 q^{90} -208.914 q^{91} -23.0000 q^{92} -237.187 q^{93} -1731.87 q^{94} -516.810 q^{95} +167.407 q^{96} +1729.19 q^{97} -937.198 q^{98} -165.832 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q6q23q3+2q4+10q5+9q6+q7+42q8+5q930q1027q113q1215q133q1415q15142q1679q1715q1871q19+885q99+O(q100) 2 q - 6 q^{2} - 3 q^{3} + 2 q^{4} + 10 q^{5} + 9 q^{6} + q^{7} + 42 q^{8} + 5 q^{9} - 30 q^{10} - 27 q^{11} - 3 q^{12} - 15 q^{13} - 3 q^{14} - 15 q^{15} - 142 q^{16} - 79 q^{17} - 15 q^{18} - 71 q^{19}+ \cdots - 885 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 −3.00000 −1.06066 −0.530330 0.847791i 0.677932π-0.677932\pi
−0.530330 + 0.847791i 0.677932π0.677932\pi
33 3.72015 0.715944 0.357972 0.933732i 0.383468π-0.383468\pi
0.357972 + 0.933732i 0.383468π0.383468\pi
44 1.00000 0.125000
55 5.00000 0.447214
66 −11.1605 −0.759373
77 −25.6008 −1.38231 −0.691156 0.722706i 0.742898π-0.742898\pi
−0.691156 + 0.722706i 0.742898π0.742898\pi
88 21.0000 0.928078
99 −13.1605 −0.487424
1010 −15.0000 −0.474342
1111 12.6008 0.345389 0.172694 0.984975i 0.444753π-0.444753\pi
0.172694 + 0.984975i 0.444753π0.444753\pi
1212 3.72015 0.0894930
1313 8.16046 0.174100 0.0870502 0.996204i 0.472256π-0.472256\pi
0.0870502 + 0.996204i 0.472256π0.472256\pi
1414 76.8023 1.46616
1515 18.6008 0.320180
1616 −71.0000 −1.10938
1717 −76.0411 −1.08486 −0.542431 0.840100i 0.682496π-0.682496\pi
−0.542431 + 0.840100i 0.682496π0.682496\pi
1818 39.4814 0.516992
1919 −103.362 −1.24805 −0.624023 0.781406i 0.714503π-0.714503\pi
−0.624023 + 0.781406i 0.714503π0.714503\pi
2020 5.00000 0.0559017
2121 −95.2388 −0.989657
2222 −37.8023 −0.366340
2323 −23.0000 −0.208514
2424 78.1232 0.664451
2525 25.0000 0.200000
2626 −24.4814 −0.184661
2727 −149.403 −1.06491
2828 −25.6008 −0.172789
2929 −267.202 −1.71097 −0.855484 0.517829i 0.826740π-0.826740\pi
−0.855484 + 0.517829i 0.826740π0.826740\pi
3030 −55.8023 −0.339602
3131 −63.7574 −0.369392 −0.184696 0.982796i 0.559130π-0.559130\pi
−0.184696 + 0.982796i 0.559130π0.559130\pi
3232 45.0000 0.248592
3333 46.8768 0.247279
3434 228.123 1.15067
3535 −128.004 −0.618188
3636 −13.1605 −0.0609281
3737 112.164 0.498370 0.249185 0.968456i 0.419837π-0.419837\pi
0.249185 + 0.968456i 0.419837π0.419837\pi
3838 310.086 1.32375
3939 30.3582 0.124646
4040 105.000 0.415049
4141 −239.078 −0.910677 −0.455339 0.890318i 0.650482π-0.650482\pi
−0.455339 + 0.890318i 0.650482π0.650482\pi
4242 285.716 1.04969
4343 282.239 1.00095 0.500477 0.865750i 0.333158π-0.333158\pi
0.500477 + 0.865750i 0.333158π0.333158\pi
4444 12.6008 0.0431736
4545 −65.8023 −0.217983
4646 69.0000 0.221163
4747 577.291 1.79163 0.895815 0.444427i 0.146593π-0.146593\pi
0.895815 + 0.444427i 0.146593π0.146593\pi
4848 −264.131 −0.794250
4949 312.399 0.910785
5050 −75.0000 −0.212132
5151 −282.884 −0.776701
5252 8.16046 0.0217625
5353 −2.31326 −0.00599529 −0.00299764 0.999996i 0.500954π-0.500954\pi
−0.00299764 + 0.999996i 0.500954π0.500954\pi
5454 448.209 1.12951
5555 63.0038 0.154462
5656 −537.616 −1.28289
5757 −384.522 −0.893531
5858 801.605 1.81476
5959 272.888 0.602153 0.301077 0.953600i 0.402654π-0.402654\pi
0.301077 + 0.953600i 0.402654π0.402654\pi
6060 18.6008 0.0400225
6161 294.049 0.617198 0.308599 0.951192i 0.400140π-0.400140\pi
0.308599 + 0.951192i 0.400140π0.400140\pi
6262 191.272 0.391800
6363 336.918 0.673772
6464 433.000 0.845703
6565 40.8023 0.0778600
6666 −140.630 −0.262279
6767 426.732 0.778113 0.389056 0.921214i 0.372801π-0.372801\pi
0.389056 + 0.921214i 0.372801π0.372801\pi
6868 −76.0411 −0.135608
6969 −85.5635 −0.149285
7070 384.011 0.655688
7171 −1020.85 −1.70638 −0.853191 0.521598i 0.825336π-0.825336\pi
−0.853191 + 0.521598i 0.825336π0.825336\pi
7272 −276.370 −0.452368
7373 −286.650 −0.459586 −0.229793 0.973240i 0.573805π-0.573805\pi
−0.229793 + 0.973240i 0.573805π0.573805\pi
7474 −336.493 −0.528601
7575 93.0038 0.143189
7676 −103.362 −0.156006
7777 −322.589 −0.477435
7878 −91.0745 −0.132207
7979 −551.866 −0.785947 −0.392974 0.919550i 0.628554π-0.628554\pi
−0.392974 + 0.919550i 0.628554π0.628554\pi
8080 −355.000 −0.496128
8181 −200.470 −0.274993
8282 717.235 0.965919
8383 21.7021 0.0287001 0.0143501 0.999897i 0.495432π-0.495432\pi
0.0143501 + 0.999897i 0.495432π0.495432\pi
8484 −95.2388 −0.123707
8585 −380.205 −0.485165
8686 −846.716 −1.06167
8787 −994.031 −1.22496
8888 264.616 0.320547
8989 −1049.63 −1.25012 −0.625058 0.780578i 0.714925π-0.714925\pi
−0.625058 + 0.780578i 0.714925π0.714925\pi
9090 197.407 0.231206
9191 −208.914 −0.240661
9292 −23.0000 −0.0260643
9393 −237.187 −0.264464
9494 −1731.87 −1.90031
9595 −516.810 −0.558143
9696 167.407 0.177978
9797 1729.19 1.81003 0.905014 0.425381i 0.139860π-0.139860\pi
0.905014 + 0.425381i 0.139860π0.139860\pi
9898 −937.198 −0.966033
9999 −165.832 −0.168351
100100 25.0000 0.0250000
101101 855.874 0.843195 0.421597 0.906783i 0.361470π-0.361470\pi
0.421597 + 0.906783i 0.361470π0.361470\pi
102102 848.653 0.823816
103103 −632.534 −0.605101 −0.302551 0.953133i 0.597838π-0.597838\pi
−0.302551 + 0.953133i 0.597838π0.597838\pi
104104 171.370 0.161579
105105 −476.194 −0.442588
106106 6.93977 0.00635896
107107 1309.62 1.18323 0.591616 0.806220i 0.298490π-0.298490\pi
0.591616 + 0.806220i 0.298490π0.298490\pi
108108 −149.403 −0.133114
109109 1726.21 1.51689 0.758443 0.651739i 0.225960π-0.225960\pi
0.758443 + 0.651739i 0.225960π0.225960\pi
110110 −189.011 −0.163832
111111 417.268 0.356805
112112 1817.65 1.53350
113113 −1492.24 −1.24228 −0.621142 0.783698i 0.713331π-0.713331\pi
−0.621142 + 0.783698i 0.713331π0.713331\pi
114114 1153.57 0.947732
115115 −115.000 −0.0932505
116116 −267.202 −0.213871
117117 −107.395 −0.0848608
118118 −818.665 −0.638680
119119 1946.71 1.49962
120120 390.616 0.297152
121121 −1172.22 −0.880707
122122 −882.146 −0.654637
123123 −889.408 −0.651994
124124 −63.7574 −0.0461741
125125 125.000 0.0894427
126126 −1010.75 −0.714644
127127 −1368.29 −0.956034 −0.478017 0.878351i 0.658644π-0.658644\pi
−0.478017 + 0.878351i 0.658644π0.658644\pi
128128 −1659.00 −1.14560
129129 1049.97 0.716627
130130 −122.407 −0.0825830
131131 1984.24 1.32339 0.661694 0.749774i 0.269838π-0.269838\pi
0.661694 + 0.749774i 0.269838π0.269838\pi
132132 46.8768 0.0309098
133133 2646.15 1.72519
134134 −1280.19 −0.825313
135135 −747.015 −0.476243
136136 −1596.86 −1.00684
137137 273.331 0.170455 0.0852273 0.996362i 0.472838π-0.472838\pi
0.0852273 + 0.996362i 0.472838π0.472838\pi
138138 256.691 0.158340
139139 −557.889 −0.340428 −0.170214 0.985407i 0.554446π-0.554446\pi
−0.170214 + 0.985407i 0.554446π0.554446\pi
140140 −128.004 −0.0772736
141141 2147.61 1.28271
142142 3062.56 1.80989
143143 102.828 0.0601323
144144 934.393 0.540736
145145 −1336.01 −0.765168
146146 859.949 0.487465
147147 1162.17 0.652071
148148 112.164 0.0622963
149149 −1354.16 −0.744545 −0.372272 0.928124i 0.621421π-0.621421\pi
−0.372272 + 0.928124i 0.621421π0.621421\pi
150150 −279.011 −0.151875
151151 162.652 0.0876586 0.0438293 0.999039i 0.486044π-0.486044\pi
0.0438293 + 0.999039i 0.486044π0.486044\pi
152152 −2170.60 −1.15828
153153 1000.74 0.528789
154154 967.768 0.506396
155155 −318.787 −0.165197
156156 30.3582 0.0155808
157157 −2364.83 −1.20213 −0.601063 0.799201i 0.705256π-0.705256\pi
−0.601063 + 0.799201i 0.705256π0.705256\pi
158158 1655.60 0.833623
159159 −8.60567 −0.00429229
160160 225.000 0.111174
161161 588.818 0.288232
162162 601.410 0.291674
163163 471.162 0.226406 0.113203 0.993572i 0.463889π-0.463889\pi
0.113203 + 0.993572i 0.463889π0.463889\pi
164164 −239.078 −0.113835
165165 234.384 0.110586
166166 −65.1062 −0.0304411
167167 −3735.73 −1.73102 −0.865508 0.500895i 0.833004π-0.833004\pi
−0.865508 + 0.500895i 0.833004π0.833004\pi
168168 −2000.01 −0.918479
169169 −2130.41 −0.969689
170170 1140.62 0.514596
171171 1360.29 0.608328
172172 282.239 0.125119
173173 −2709.33 −1.19067 −0.595336 0.803477i 0.702981π-0.702981\pi
−0.595336 + 0.803477i 0.702981π0.702981\pi
174174 2982.09 1.29926
175175 −640.019 −0.276462
176176 −894.654 −0.383165
177177 1015.19 0.431108
178178 3148.88 1.32595
179179 3300.43 1.37813 0.689066 0.724699i 0.258021π-0.258021\pi
0.689066 + 0.724699i 0.258021π0.258021\pi
180180 −65.8023 −0.0272479
181181 −583.868 −0.239771 −0.119886 0.992788i 0.538253π-0.538253\pi
−0.119886 + 0.992788i 0.538253π0.538253\pi
182182 626.742 0.255259
183183 1093.91 0.441879
184184 −483.000 −0.193518
185185 560.821 0.222878
186186 711.562 0.280507
187187 −958.176 −0.374699
188188 577.291 0.223954
189189 3824.83 1.47204
190190 1550.43 0.592000
191191 −3058.98 −1.15885 −0.579424 0.815026i 0.696722π-0.696722\pi
−0.579424 + 0.815026i 0.696722π0.696722\pi
192192 1610.83 0.605476
193193 −1336.93 −0.498625 −0.249313 0.968423i 0.580205π-0.580205\pi
−0.249313 + 0.968423i 0.580205π0.580205\pi
194194 −5187.57 −1.91983
195195 151.791 0.0557434
196196 312.399 0.113848
197197 −2449.21 −0.885783 −0.442892 0.896575i 0.646047π-0.646047\pi
−0.442892 + 0.896575i 0.646047π0.646047\pi
198198 497.496 0.178563
199199 1138.66 0.405614 0.202807 0.979219i 0.434994π-0.434994\pi
0.202807 + 0.979219i 0.434994π0.434994\pi
200200 525.000 0.185616
201201 1587.51 0.557085
202202 −2567.62 −0.894343
203203 6840.56 2.36509
204204 −282.884 −0.0970876
205205 −1195.39 −0.407267
206206 1897.60 0.641807
207207 302.691 0.101635
208208 −579.393 −0.193143
209209 −1302.44 −0.431061
210210 1428.58 0.469436
211211 5596.81 1.82607 0.913034 0.407884i 0.133733π-0.133733\pi
0.913034 + 0.407884i 0.133733π0.133733\pi
212212 −2.31326 −0.000749411 0
213213 −3797.74 −1.22167
214214 −3928.86 −1.25501
215215 1411.19 0.447640
216216 −3137.46 −0.988321
217217 1632.24 0.510615
218218 −5178.62 −1.60890
219219 −1066.38 −0.329038
220220 63.0038 0.0193078
221221 −620.530 −0.188875
222222 −1251.81 −0.378449
223223 5580.60 1.67581 0.837903 0.545820i 0.183782π-0.183782\pi
0.837903 + 0.545820i 0.183782π0.183782\pi
224224 −1152.03 −0.343632
225225 −329.011 −0.0974849
226226 4476.72 1.31764
227227 −566.323 −0.165587 −0.0827934 0.996567i 0.526384π-0.526384\pi
−0.0827934 + 0.996567i 0.526384π0.526384\pi
228228 −384.522 −0.111691
229229 −693.702 −0.200180 −0.100090 0.994978i 0.531913π-0.531913\pi
−0.100090 + 0.994978i 0.531913π0.531913\pi
230230 345.000 0.0989071
231231 −1200.08 −0.341816
232232 −5611.23 −1.58791
233233 −2208.68 −0.621011 −0.310506 0.950572i 0.600498π-0.600498\pi
−0.310506 + 0.950572i 0.600498π0.600498\pi
234234 322.186 0.0900084
235235 2886.46 0.801241
236236 272.888 0.0752691
237237 −2053.03 −0.562694
238238 −5840.13 −1.59059
239239 −3876.79 −1.04924 −0.524621 0.851336i 0.675793π-0.675793\pi
−0.524621 + 0.851336i 0.675793π0.675793\pi
240240 −1320.65 −0.355199
241241 −2024.57 −0.541136 −0.270568 0.962701i 0.587211π-0.587211\pi
−0.270568 + 0.962701i 0.587211π0.587211\pi
242242 3516.66 0.934131
243243 3288.10 0.868033
244244 294.049 0.0771498
245245 1562.00 0.407315
246246 2668.22 0.691544
247247 −843.481 −0.217285
248248 −1338.91 −0.342825
249249 80.7350 0.0205477
250250 −375.000 −0.0948683
251251 −733.444 −0.184441 −0.0922203 0.995739i 0.529396π-0.529396\pi
−0.0922203 + 0.995739i 0.529396π0.529396\pi
252252 336.918 0.0842215
253253 −289.818 −0.0720185
254254 4104.88 1.01403
255255 −1414.42 −0.347351
256256 1513.00 0.369385
257257 2367.83 0.574713 0.287356 0.957824i 0.407224π-0.407224\pi
0.287356 + 0.957824i 0.407224π0.407224\pi
258258 −3149.91 −0.760097
259259 −2871.49 −0.688903
260260 40.8023 0.00973250
261261 3516.50 0.833968
262262 −5952.72 −1.40366
263263 6632.76 1.55511 0.777554 0.628816i 0.216460π-0.216460\pi
0.777554 + 0.628816i 0.216460π0.216460\pi
264264 984.412 0.229494
265265 −11.5663 −0.00268117
266266 −7938.44 −1.82984
267267 −3904.78 −0.895013
268268 426.732 0.0972641
269269 −6092.97 −1.38102 −0.690511 0.723322i 0.742614π-0.742614\pi
−0.690511 + 0.723322i 0.742614π0.742614\pi
270270 2241.05 0.505132
271271 −1981.30 −0.444115 −0.222058 0.975034i 0.571277π-0.571277\pi
−0.222058 + 0.975034i 0.571277π0.571277\pi
272272 5398.92 1.20352
273273 −777.192 −0.172300
274274 −819.994 −0.180794
275275 315.019 0.0690777
276276 −85.5635 −0.0186606
277277 −4097.42 −0.888773 −0.444386 0.895835i 0.646578π-0.646578\pi
−0.444386 + 0.895835i 0.646578π0.646578\pi
278278 1673.67 0.361079
279279 839.077 0.180051
280280 −2688.08 −0.573727
281281 −3866.40 −0.820819 −0.410409 0.911901i 0.634614π-0.634614\pi
−0.410409 + 0.911901i 0.634614π0.634614\pi
282282 −6442.84 −1.36052
283283 3982.00 0.836415 0.418208 0.908351i 0.362658π-0.362658\pi
0.418208 + 0.908351i 0.362658π0.362658\pi
284284 −1020.85 −0.213298
285285 −1922.61 −0.399599
286286 −308.484 −0.0637799
287287 6120.59 1.25884
288288 −592.221 −0.121170
289289 869.245 0.176927
290290 4008.02 0.811583
291291 6432.86 1.29588
292292 −286.650 −0.0574483
293293 −7491.70 −1.49375 −0.746877 0.664962i 0.768448π-0.768448\pi
−0.746877 + 0.664962i 0.768448π0.768448\pi
294294 −3486.52 −0.691626
295295 1364.44 0.269291
296296 2355.45 0.462526
297297 −1882.59 −0.367809
298298 4062.48 0.789709
299299 −187.691 −0.0363024
300300 93.0038 0.0178986
301301 −7225.53 −1.38363
302302 −487.957 −0.0929760
303303 3183.98 0.603680
304304 7338.70 1.38455
305305 1470.24 0.276019
306306 −3002.21 −0.560865
307307 −5813.01 −1.08067 −0.540335 0.841450i 0.681703π-0.681703\pi
−0.540335 + 0.841450i 0.681703π0.681703\pi
308308 −322.589 −0.0596793
309309 −2353.12 −0.433218
310310 956.361 0.175218
311311 4769.43 0.869613 0.434807 0.900524i 0.356817π-0.356817\pi
0.434807 + 0.900524i 0.356817π0.356817\pi
312312 637.521 0.115681
313313 7053.43 1.27375 0.636875 0.770967i 0.280227π-0.280227\pi
0.636875 + 0.770967i 0.280227π0.280227\pi
314314 7094.49 1.27505
315315 1684.59 0.301320
316316 −551.866 −0.0982434
317317 8845.36 1.56721 0.783604 0.621261i 0.213379π-0.213379\pi
0.783604 + 0.621261i 0.213379π0.213379\pi
318318 25.8170 0.00455266
319319 −3366.94 −0.590949
320320 2165.00 0.378210
321321 4871.99 0.847127
322322 −1766.45 −0.305716
323323 7859.76 1.35396
324324 −200.470 −0.0343741
325325 204.011 0.0348201
326326 −1413.48 −0.240140
327327 6421.75 1.08601
328328 −5020.64 −0.845179
329329 −14779.1 −2.47659
330330 −703.152 −0.117295
331331 4264.07 0.708079 0.354040 0.935230i 0.384808π-0.384808\pi
0.354040 + 0.935230i 0.384808π0.384808\pi
332332 21.7021 0.00358752
333333 −1476.13 −0.242918
334334 11207.2 1.83602
335335 2133.66 0.347983
336336 6761.95 1.09790
337337 −1290.58 −0.208613 −0.104306 0.994545i 0.533262π-0.533262\pi
−0.104306 + 0.994545i 0.533262π0.533262\pi
338338 6391.22 1.02851
339339 −5551.36 −0.889405
340340 −380.205 −0.0606457
341341 −803.392 −0.127584
342342 −4080.87 −0.645229
343343 783.403 0.123323
344344 5927.01 0.928963
345345 −427.818 −0.0667621
346346 8127.98 1.26290
347347 −1805.09 −0.279257 −0.139629 0.990204i 0.544591π-0.544591\pi
−0.139629 + 0.990204i 0.544591π0.544591\pi
348348 −994.031 −0.153120
349349 −3586.81 −0.550136 −0.275068 0.961425i 0.588700π-0.588700\pi
−0.275068 + 0.961425i 0.588700π0.588700\pi
350350 1920.06 0.293233
351351 −1219.20 −0.185402
352352 567.034 0.0858609
353353 −2292.90 −0.345719 −0.172859 0.984947i 0.555301π-0.555301\pi
−0.172859 + 0.984947i 0.555301π0.555301\pi
354354 −3045.56 −0.457259
355355 −5104.27 −0.763118
356356 −1049.63 −0.156264
357357 7242.06 1.07364
358358 −9901.28 −1.46173
359359 −3077.18 −0.452388 −0.226194 0.974082i 0.572628π-0.572628\pi
−0.226194 + 0.974082i 0.572628π0.572628\pi
360360 −1381.85 −0.202305
361361 3824.70 0.557618
362362 1751.60 0.254316
363363 −4360.84 −0.630537
364364 −208.914 −0.0300826
365365 −1433.25 −0.205533
366366 −3281.72 −0.468684
367367 −1378.56 −0.196076 −0.0980382 0.995183i 0.531257π-0.531257\pi
−0.0980382 + 0.995183i 0.531257π0.531257\pi
368368 1633.00 0.231321
369369 3146.38 0.443886
370370 −1682.46 −0.236398
371371 59.2211 0.00828736
372372 −237.187 −0.0330580
373373 11182.3 1.55227 0.776135 0.630567i 0.217178π-0.217178\pi
0.776135 + 0.630567i 0.217178π0.217178\pi
374374 2874.53 0.397429
375375 465.019 0.0640360
376376 12123.1 1.66277
377377 −2180.49 −0.297880
378378 −11474.5 −1.56133
379379 12021.9 1.62935 0.814677 0.579915i 0.196914π-0.196914\pi
0.814677 + 0.579915i 0.196914π0.196914\pi
380380 −516.810 −0.0697679
381381 −5090.26 −0.684467
382382 9176.93 1.22914
383383 −13274.6 −1.77102 −0.885512 0.464617i 0.846192π-0.846192\pi
−0.885512 + 0.464617i 0.846192π0.846192\pi
384384 −6171.73 −0.820182
385385 −1612.95 −0.213515
386386 4010.80 0.528872
387387 −3714.39 −0.487889
388388 1729.19 0.226254
389389 11016.4 1.43587 0.717934 0.696112i 0.245088π-0.245088\pi
0.717934 + 0.696112i 0.245088π0.245088\pi
390390 −455.372 −0.0591248
391391 1748.94 0.226210
392392 6560.38 0.845279
393393 7381.67 0.947471
394394 7347.64 0.939515
395395 −2759.33 −0.351486
396396 −165.832 −0.0210439
397397 −1893.18 −0.239335 −0.119668 0.992814i 0.538183π-0.538183\pi
−0.119668 + 0.992814i 0.538183π0.538183\pi
398398 −3415.97 −0.430219
399399 9844.07 1.23514
400400 −1775.00 −0.221875
401401 11281.8 1.40495 0.702475 0.711709i 0.252078π-0.252078\pi
0.702475 + 0.711709i 0.252078π0.252078\pi
402402 −4762.52 −0.590878
403403 −520.290 −0.0643113
404404 855.874 0.105399
405405 −1002.35 −0.122981
406406 −20521.7 −2.50856
407407 1413.36 0.172131
408408 −5940.57 −0.720839
409409 −14677.6 −1.77447 −0.887236 0.461316i 0.847377π-0.847377\pi
−0.887236 + 0.461316i 0.847377π0.847377\pi
410410 3586.17 0.431972
411411 1016.83 0.122036
412412 −632.534 −0.0756376
413413 −6986.15 −0.832363
414414 −908.072 −0.107800
415415 108.510 0.0128351
416416 367.221 0.0432800
417417 −2075.43 −0.243728
418418 3907.32 0.457209
419419 2196.50 0.256100 0.128050 0.991768i 0.459128π-0.459128\pi
0.128050 + 0.991768i 0.459128π0.459128\pi
420420 −476.194 −0.0553235
421421 −6571.28 −0.760723 −0.380362 0.924838i 0.624200π-0.624200\pi
−0.380362 + 0.924838i 0.624200π0.624200\pi
422422 −16790.4 −1.93684
423423 −7597.42 −0.873284
424424 −48.5784 −0.00556409
425425 −1901.03 −0.216973
426426 11393.2 1.29578
427427 −7527.87 −0.853160
428428 1309.62 0.147904
429429 382.536 0.0430513
430430 −4233.58 −0.474794
431431 −583.607 −0.0652235 −0.0326118 0.999468i 0.510382π-0.510382\pi
−0.0326118 + 0.999468i 0.510382π0.510382\pi
432432 10607.6 1.18139
433433 −2677.60 −0.297176 −0.148588 0.988899i 0.547473π-0.547473\pi
−0.148588 + 0.988899i 0.547473π0.547473\pi
434434 −4896.71 −0.541589
435435 −4970.15 −0.547817
436436 1726.21 0.189611
437437 2377.33 0.260236
438438 3199.14 0.348997
439439 −5509.18 −0.598949 −0.299475 0.954104i 0.596811π-0.596811\pi
−0.299475 + 0.954104i 0.596811π0.596811\pi
440440 1323.08 0.143353
441441 −4111.32 −0.443939
442442 1861.59 0.200332
443443 −13054.6 −1.40009 −0.700047 0.714097i 0.746838π-0.746838\pi
−0.700047 + 0.714097i 0.746838π0.746838\pi
444444 417.268 0.0446006
445445 −5248.14 −0.559069
446446 −16741.8 −1.77746
447447 −5037.68 −0.533052
448448 −11085.1 −1.16903
449449 −11819.1 −1.24227 −0.621135 0.783704i 0.713328π-0.713328\pi
−0.621135 + 0.783704i 0.713328π0.713328\pi
450450 987.034 0.103398
451451 −3012.57 −0.314537
452452 −1492.24 −0.155285
453453 605.091 0.0627587
454454 1698.97 0.175631
455455 −1044.57 −0.107627
456456 −8074.97 −0.829266
457457 −4144.87 −0.424264 −0.212132 0.977241i 0.568041π-0.568041\pi
−0.212132 + 0.977241i 0.568041π0.568041\pi
458458 2081.11 0.212323
459459 11360.8 1.15528
460460 −115.000 −0.0116563
461461 −6061.38 −0.612379 −0.306189 0.951971i 0.599054π-0.599054\pi
−0.306189 + 0.951971i 0.599054π0.599054\pi
462462 3600.24 0.362551
463463 −5729.13 −0.575066 −0.287533 0.957771i 0.592835π-0.592835\pi
−0.287533 + 0.957771i 0.592835π0.592835\pi
464464 18971.3 1.89811
465465 −1185.94 −0.118272
466466 6626.05 0.658682
467467 8958.05 0.887643 0.443821 0.896115i 0.353622π-0.353622\pi
0.443821 + 0.896115i 0.353622π0.353622\pi
468468 −107.395 −0.0106076
469469 −10924.7 −1.07559
470470 −8659.37 −0.849845
471471 −8797.53 −0.860655
472472 5730.65 0.558845
473473 3556.42 0.345718
474474 6159.08 0.596827
475475 −2584.05 −0.249609
476476 1946.71 0.187452
477477 30.4435 0.00292225
478478 11630.4 1.11289
479479 6356.60 0.606347 0.303174 0.952935i 0.401954π-0.401954\pi
0.303174 + 0.952935i 0.401954π0.401954\pi
480480 837.034 0.0795942
481481 915.312 0.0867664
482482 6073.70 0.573961
483483 2190.49 0.206358
484484 −1172.22 −0.110088
485485 8645.96 0.809469
486486 −9864.31 −0.920688
487487 13991.2 1.30185 0.650926 0.759141i 0.274381π-0.274381\pi
0.650926 + 0.759141i 0.274381π0.274381\pi
488488 6175.02 0.572808
489489 1752.79 0.162094
490490 −4685.99 −0.432023
491491 −5301.96 −0.487320 −0.243660 0.969861i 0.578348π-0.578348\pi
−0.243660 + 0.969861i 0.578348π0.578348\pi
492492 −889.408 −0.0814992
493493 20318.3 1.85617
494494 2530.44 0.230466
495495 −829.159 −0.0752888
496496 4526.78 0.409795
497497 26134.7 2.35875
498498 −242.205 −0.0217941
499499 −8013.45 −0.718900 −0.359450 0.933164i 0.617036π-0.617036\pi
−0.359450 + 0.933164i 0.617036π0.617036\pi
500500 125.000 0.0111803
501501 −13897.5 −1.23931
502502 2200.33 0.195629
503503 −11881.9 −1.05325 −0.526627 0.850097i 0.676544π-0.676544\pi
−0.526627 + 0.850097i 0.676544π0.676544\pi
504504 7075.27 0.625313
505505 4279.37 0.377088
506506 869.453 0.0763871
507507 −7925.44 −0.694243
508508 −1368.29 −0.119504
509509 −12113.8 −1.05488 −0.527441 0.849592i 0.676848π-0.676848\pi
−0.527441 + 0.849592i 0.676848π0.676848\pi
510510 4243.27 0.368422
511511 7338.45 0.635291
512512 8733.00 0.753804
513513 15442.6 1.32906
514514 −7103.49 −0.609575
515515 −3162.67 −0.270609
516516 1049.97 0.0895783
517517 7274.31 0.618808
518518 8614.48 0.730692
519519 −10079.1 −0.852454
520520 856.848 0.0722602
521521 −15327.6 −1.28890 −0.644449 0.764647i 0.722913π-0.722913\pi
−0.644449 + 0.764647i 0.722913π0.722913\pi
522522 −10549.5 −0.884556
523523 14545.8 1.21614 0.608070 0.793883i 0.291944π-0.291944\pi
0.608070 + 0.793883i 0.291944π0.291944\pi
524524 1984.24 0.165423
525525 −2380.97 −0.197931
526526 −19898.3 −1.64944
527527 4848.18 0.400740
528528 −3328.25 −0.274325
529529 529.000 0.0434783
530530 34.6989 0.00284381
531531 −3591.34 −0.293504
532532 2646.15 0.215648
533533 −1950.99 −0.158549
534534 11714.3 0.949304
535535 6548.10 0.529157
536536 8961.36 0.722149
537537 12278.1 0.986665
538538 18278.9 1.46479
539539 3936.47 0.314575
540540 −747.015 −0.0595304
541541 −5468.93 −0.434617 −0.217308 0.976103i 0.569728π-0.569728\pi
−0.217308 + 0.976103i 0.569728π0.569728\pi
542542 5943.89 0.471055
543543 −2172.08 −0.171663
544544 −3421.85 −0.269688
545545 8631.03 0.678372
546546 2331.58 0.182751
547547 −19566.3 −1.52943 −0.764713 0.644371i 0.777119π-0.777119\pi
−0.764713 + 0.644371i 0.777119π0.777119\pi
548548 273.331 0.0213068
549549 −3869.82 −0.300837
550550 −945.057 −0.0732680
551551 27618.5 2.13537
552552 −1796.83 −0.138548
553553 14128.2 1.08642
554554 12292.3 0.942686
555555 2086.34 0.159568
556556 −557.889 −0.0425536
557557 −9803.11 −0.745729 −0.372865 0.927886i 0.621624π-0.621624\pi
−0.372865 + 0.927886i 0.621624π0.621624\pi
558558 −2517.23 −0.190973
559559 2303.20 0.174266
560560 9088.27 0.685803
561561 −3564.56 −0.268264
562562 11599.2 0.870610
563563 −4909.73 −0.367532 −0.183766 0.982970i 0.558829π-0.558829\pi
−0.183766 + 0.982970i 0.558829π0.558829\pi
564564 2147.61 0.160338
565565 −7461.19 −0.555566
566566 −11946.0 −0.887153
567567 5132.18 0.380126
568568 −21438.0 −1.58366
569569 11498.2 0.847155 0.423578 0.905860i 0.360774π-0.360774\pi
0.423578 + 0.905860i 0.360774π0.360774\pi
570570 5767.84 0.423839
571571 14766.6 1.08225 0.541124 0.840943i 0.317999π-0.317999\pi
0.541124 + 0.840943i 0.317999π0.317999\pi
572572 102.828 0.00751653
573573 −11379.9 −0.829670
574574 −18361.8 −1.33520
575575 −575.000 −0.0417029
576576 −5698.48 −0.412216
577577 13364.0 0.964212 0.482106 0.876113i 0.339872π-0.339872\pi
0.482106 + 0.876113i 0.339872π0.339872\pi
578578 −2607.73 −0.187660
579579 −4973.60 −0.356988
580580 −1336.01 −0.0956460
581581 −555.590 −0.0396725
582582 −19298.6 −1.37449
583583 −29.1488 −0.00207070
584584 −6019.64 −0.426532
585585 −536.977 −0.0379509
586586 22475.1 1.58437
587587 12524.7 0.880664 0.440332 0.897835i 0.354861π-0.354861\pi
0.440332 + 0.897835i 0.354861π0.354861\pi
588588 1162.17 0.0815089
589589 6590.09 0.461019
590590 −4093.32 −0.285626
591591 −9111.45 −0.634171
592592 −7963.66 −0.552879
593593 −17938.6 −1.24224 −0.621121 0.783714i 0.713323π-0.713323\pi
−0.621121 + 0.783714i 0.713323π0.713323\pi
594594 5647.78 0.390120
595595 9733.55 0.670650
596596 −1354.16 −0.0930681
597597 4235.98 0.290397
598598 563.072 0.0385045
599599 −26735.8 −1.82370 −0.911848 0.410528i 0.865344π-0.865344\pi
−0.911848 + 0.410528i 0.865344π0.865344\pi
600600 1953.08 0.132890
601601 −21043.6 −1.42826 −0.714131 0.700012i 0.753178π-0.753178\pi
−0.714131 + 0.700012i 0.753178π0.753178\pi
602602 21676.6 1.46756
603603 −5615.98 −0.379271
604604 162.652 0.0109573
605605 −5861.10 −0.393864
606606 −9551.95 −0.640299
607607 5098.83 0.340947 0.170474 0.985362i 0.445470π-0.445470\pi
0.170474 + 0.985362i 0.445470π0.445470\pi
608608 −4651.29 −0.310254
609609 25447.9 1.69327
610610 −4410.73 −0.292763
611611 4710.96 0.311923
612612 1000.74 0.0660986
613613 −12784.8 −0.842369 −0.421185 0.906975i 0.638386π-0.638386\pi
−0.421185 + 0.906975i 0.638386π0.638386\pi
614614 17439.0 1.14622
615615 −4447.04 −0.291580
616616 −6774.37 −0.443096
617617 −8940.73 −0.583372 −0.291686 0.956514i 0.594216π-0.594216\pi
−0.291686 + 0.956514i 0.594216π0.594216\pi
618618 7059.37 0.459498
619619 1333.26 0.0865725 0.0432863 0.999063i 0.486217π-0.486217\pi
0.0432863 + 0.999063i 0.486217π0.486217\pi
620620 −318.787 −0.0206497
621621 3436.27 0.222050
622622 −14308.3 −0.922364
623623 26871.3 1.72805
624624 −2155.43 −0.138279
625625 625.000 0.0400000
626626 −21160.3 −1.35102
627627 −4845.28 −0.308615
628628 −2364.83 −0.150266
629629 −8529.09 −0.540663
630630 −5053.77 −0.319598
631631 −16667.9 −1.05157 −0.525784 0.850618i 0.676228π-0.676228\pi
−0.525784 + 0.850618i 0.676228π0.676228\pi
632632 −11589.2 −0.729420
633633 20821.0 1.30736
634634 −26536.1 −1.66228
635635 −6841.46 −0.427551
636636 −8.60567 −0.000536536 0
637637 2549.32 0.158568
638638 10100.8 0.626796
639639 13434.9 0.831733
640640 −8295.00 −0.512326
641641 27431.9 1.69032 0.845158 0.534516i 0.179506π-0.179506\pi
0.845158 + 0.534516i 0.179506π0.179506\pi
642642 −14616.0 −0.898514
643643 9618.00 0.589886 0.294943 0.955515i 0.404699π-0.404699\pi
0.294943 + 0.955515i 0.404699π0.404699\pi
644644 588.818 0.0360290
645645 5249.86 0.320485
646646 −23579.3 −1.43609
647647 −12548.6 −0.762496 −0.381248 0.924473i 0.624506π-0.624506\pi
−0.381248 + 0.924473i 0.624506π0.624506\pi
648648 −4209.87 −0.255215
649649 3438.60 0.207977
650650 −612.034 −0.0369323
651651 6072.18 0.365572
652652 471.162 0.0283008
653653 −8829.79 −0.529152 −0.264576 0.964365i 0.585232π-0.585232\pi
−0.264576 + 0.964365i 0.585232π0.585232\pi
654654 −19265.3 −1.15188
655655 9921.19 0.591837
656656 16974.6 1.01028
657657 3772.44 0.224014
658658 44337.3 2.62682
659659 5935.56 0.350860 0.175430 0.984492i 0.443868π-0.443868\pi
0.175430 + 0.984492i 0.443868π0.443868\pi
660660 234.384 0.0138233
661661 −1182.03 −0.0695549 −0.0347775 0.999395i 0.511072π-0.511072\pi
−0.0347775 + 0.999395i 0.511072π0.511072\pi
662662 −12792.2 −0.751032
663663 −2308.47 −0.135224
664664 455.743 0.0266360
665665 13230.7 0.771527
666666 4428.40 0.257653
667667 6145.64 0.356762
668668 −3735.73 −0.216377
669669 20760.7 1.19978
670670 −6400.97 −0.369091
671671 3705.24 0.213173
672672 −4285.74 −0.246021
673673 −6053.25 −0.346710 −0.173355 0.984859i 0.555461π-0.555461\pi
−0.173355 + 0.984859i 0.555461π0.555461\pi
674674 3871.74 0.221267
675675 −3735.08 −0.212982
676676 −2130.41 −0.121211
677677 16694.2 0.947728 0.473864 0.880598i 0.342859π-0.342859\pi
0.473864 + 0.880598i 0.342859π0.342859\pi
678678 16654.1 0.943357
679679 −44268.6 −2.50202
680680 −7984.31 −0.450271
681681 −2106.81 −0.118551
682682 2410.18 0.135323
683683 −4150.03 −0.232499 −0.116249 0.993220i 0.537087π-0.537087\pi
−0.116249 + 0.993220i 0.537087π0.537087\pi
684684 1360.29 0.0760410
685685 1366.66 0.0762296
686686 −2350.21 −0.130804
687687 −2580.68 −0.143317
688688 −20039.0 −1.11043
689689 −18.8772 −0.00104378
690690 1283.45 0.0708119
691691 33382.4 1.83781 0.918904 0.394481i 0.129076π-0.129076\pi
0.918904 + 0.394481i 0.129076π0.129076\pi
692692 −2709.33 −0.148834
693693 4245.42 0.232713
694694 5415.27 0.296197
695695 −2789.45 −0.152244
696696 −20874.6 −1.13686
697697 18179.8 0.987960
698698 10760.4 0.583508
699699 −8216.64 −0.444609
700700 −640.019 −0.0345578
701701 −29528.7 −1.59099 −0.795494 0.605961i 0.792789π-0.792789\pi
−0.795494 + 0.605961i 0.792789π0.792789\pi
702702 3657.59 0.196648
703703 −11593.5 −0.621989
704704 5456.13 0.292096
705705 10738.1 0.573644
706706 6878.69 0.366690
707707 −21911.0 −1.16556
708708 1015.19 0.0538885
709709 27016.3 1.43106 0.715529 0.698583i 0.246186π-0.246186\pi
0.715529 + 0.698583i 0.246186π0.246186\pi
710710 15312.8 0.809408
711711 7262.82 0.383090
712712 −22042.2 −1.16020
713713 1466.42 0.0770237
714714 −21726.2 −1.13877
715715 514.140 0.0268920
716716 3300.43 0.172266
717717 −14422.3 −0.751198
718718 9231.54 0.479830
719719 24752.9 1.28390 0.641952 0.766745i 0.278125π-0.278125\pi
0.641952 + 0.766745i 0.278125π0.278125\pi
720720 4671.96 0.241825
721721 16193.4 0.836438
722722 −11474.1 −0.591443
723723 −7531.69 −0.387423
724724 −583.868 −0.0299714
725725 −6680.04 −0.342194
726726 13082.5 0.668785
727727 −17574.5 −0.896565 −0.448283 0.893892i 0.647964π-0.647964\pi
−0.448283 + 0.893892i 0.647964π0.647964\pi
728728 −4387.19 −0.223352
729729 17644.9 0.896456
730730 4299.74 0.218001
731731 −21461.7 −1.08590
732732 1093.91 0.0552349
733733 27739.6 1.39780 0.698898 0.715221i 0.253674π-0.253674\pi
0.698898 + 0.715221i 0.253674π0.253674\pi
734734 4135.67 0.207970
735735 5810.87 0.291615
736736 −1035.00 −0.0518351
737737 5377.15 0.268751
738738 −9439.14 −0.470812
739739 −3140.78 −0.156340 −0.0781701 0.996940i 0.524908π-0.524908\pi
−0.0781701 + 0.996940i 0.524908π0.524908\pi
740740 560.821 0.0278597
741741 −3137.88 −0.155564
742742 −177.663 −0.00879007
743743 −35881.5 −1.77169 −0.885844 0.463984i 0.846420π-0.846420\pi
−0.885844 + 0.463984i 0.846420π0.846420\pi
744744 −4980.93 −0.245443
745745 −6770.80 −0.332970
746746 −33546.9 −1.64643
747747 −285.609 −0.0139891
748748 −958.176 −0.0468374
749749 −33527.3 −1.63559
750750 −1395.06 −0.0679204
751751 −7009.03 −0.340563 −0.170282 0.985395i 0.554468π-0.554468\pi
−0.170282 + 0.985395i 0.554468π0.554468\pi
752752 −40987.7 −1.98759
753753 −2728.52 −0.132049
754754 6541.46 0.315950
755755 813.261 0.0392021
756756 3824.83 0.184005
757757 29998.3 1.44030 0.720150 0.693818i 0.244073π-0.244073\pi
0.720150 + 0.693818i 0.244073π0.244073\pi
758758 −36065.8 −1.72819
759759 −1078.17 −0.0515612
760760 −10853.0 −0.518000
761761 5283.49 0.251677 0.125839 0.992051i 0.459838π-0.459838\pi
0.125839 + 0.992051i 0.459838π0.459838\pi
762762 15270.8 0.725987
763763 −44192.2 −2.09681
764764 −3058.98 −0.144856
765765 5003.68 0.236481
766766 39823.9 1.87845
767767 2226.89 0.104835
768768 5628.59 0.264459
769769 −15216.2 −0.713538 −0.356769 0.934193i 0.616122π-0.616122\pi
−0.356769 + 0.934193i 0.616122π0.616122\pi
770770 4838.84 0.226467
771771 8808.69 0.411462
772772 −1336.93 −0.0623281
773773 15651.3 0.728250 0.364125 0.931350i 0.381368π-0.381368\pi
0.364125 + 0.931350i 0.381368π0.381368\pi
774774 11143.2 0.517485
775775 −1593.93 −0.0738785
776776 36313.0 1.67985
777777 −10682.4 −0.493216
778778 −33049.1 −1.52297
779779 24711.6 1.13657
780780 151.791 0.00696793
781781 −12863.6 −0.589365
782782 −5246.83 −0.239931
783783 39920.7 1.82203
784784 −22180.3 −1.01040
785785 −11824.1 −0.537607
786786 −22145.0 −1.00494
787787 9401.09 0.425810 0.212905 0.977073i 0.431707π-0.431707\pi
0.212905 + 0.977073i 0.431707π0.431707\pi
788788 −2449.21 −0.110723
789789 24674.9 1.11337
790790 8278.00 0.372807
791791 38202.5 1.71722
792792 −3482.47 −0.156243
793793 2399.57 0.107454
794794 5679.55 0.253854
795795 −43.0283 −0.00191957
796796 1138.66 0.0507018
797797 37388.5 1.66169 0.830845 0.556504i 0.187857π-0.187857\pi
0.830845 + 0.556504i 0.187857π0.187857\pi
798798 −29532.2 −1.31006
799799 −43897.9 −1.94367
800800 1125.00 0.0497184
801801 13813.6 0.609337
802802 −33845.3 −1.49017
803803 −3612.00 −0.158736
804804 1587.51 0.0696356
805805 2944.09 0.128901
806806 1560.87 0.0682125
807807 −22666.8 −0.988734
808808 17973.4 0.782550
809809 24390.6 1.05999 0.529993 0.848002i 0.322195π-0.322195\pi
0.529993 + 0.848002i 0.322195π0.322195\pi
810810 3007.05 0.130441
811811 23750.8 1.02836 0.514181 0.857682i 0.328096π-0.328096\pi
0.514181 + 0.857682i 0.328096π0.328096\pi
812812 6840.56 0.295636
813813 −7370.73 −0.317962
814814 −4240.07 −0.182573
815815 2355.81 0.101252
816816 20084.8 0.861653
817817 −29172.8 −1.24924
818818 44032.7 1.88211
819819 2749.40 0.117304
820820 −1195.39 −0.0509084
821821 3041.59 0.129296 0.0646481 0.997908i 0.479408π-0.479408\pi
0.0646481 + 0.997908i 0.479408π0.479408\pi
822822 −3050.50 −0.129439
823823 28411.6 1.20336 0.601681 0.798737i 0.294498π-0.294498\pi
0.601681 + 0.798737i 0.294498π0.294498\pi
824824 −13283.2 −0.561581
825825 1171.92 0.0494558
826826 20958.4 0.882854
827827 −24351.5 −1.02392 −0.511962 0.859008i 0.671081π-0.671081\pi
−0.511962 + 0.859008i 0.671081π0.671081\pi
828828 302.691 0.0127044
829829 −47240.8 −1.97918 −0.989590 0.143914i 0.954031π-0.954031\pi
−0.989590 + 0.143914i 0.954031π0.954031\pi
830830 −325.531 −0.0136137
831831 −15243.0 −0.636311
832832 3533.48 0.147237
833833 −23755.2 −0.988077
834834 6226.30 0.258512
835835 −18678.7 −0.774134
836836 −1302.44 −0.0538826
837837 9525.55 0.393371
838838 −6589.49 −0.271635
839839 2646.33 0.108893 0.0544466 0.998517i 0.482661π-0.482661\pi
0.0544466 + 0.998517i 0.482661π0.482661\pi
840840 −10000.1 −0.410756
841841 47007.7 1.92741
842842 19713.8 0.806869
843843 −14383.6 −0.587660
844844 5596.81 0.228258
845845 −10652.0 −0.433658
846846 22792.3 0.926258
847847 30009.7 1.21741
848848 164.241 0.00665102
849849 14813.7 0.598827
850850 5703.08 0.230134
851851 −2579.78 −0.103917
852852 −3797.74 −0.152709
853853 −40826.4 −1.63877 −0.819385 0.573244i 0.805685π-0.805685\pi
−0.819385 + 0.573244i 0.805685π0.805685\pi
854854 22583.6 0.904913
855855 6801.46 0.272053
856856 27502.0 1.09813
857857 1516.61 0.0604510 0.0302255 0.999543i 0.490377π-0.490377\pi
0.0302255 + 0.999543i 0.490377π0.490377\pi
858858 −1147.61 −0.0456628
859859 6514.74 0.258766 0.129383 0.991595i 0.458700π-0.458700\pi
0.129383 + 0.991595i 0.458700π0.458700\pi
860860 1411.19 0.0559550
861861 22769.5 0.901258
862862 1750.82 0.0691800
863863 −7120.18 −0.280850 −0.140425 0.990091i 0.544847π-0.544847\pi
−0.140425 + 0.990091i 0.544847π0.544847\pi
864864 −6723.14 −0.264729
865865 −13546.6 −0.532485
866866 8032.79 0.315202
867867 3233.72 0.126670
868868 1632.24 0.0638269
869869 −6953.94 −0.271457
870870 14910.5 0.581048
871871 3482.33 0.135470
872872 36250.3 1.40779
873873 −22757.0 −0.882252
874874 −7131.98 −0.276021
875875 −3200.10 −0.123638
876876 −1066.38 −0.0411297
877877 −34013.4 −1.30964 −0.654818 0.755787i 0.727255π-0.727255\pi
−0.654818 + 0.755787i 0.727255π0.727255\pi
878878 16527.5 0.635282
879879 −27870.3 −1.06944
880880 −4473.27 −0.171357
881881 6704.11 0.256376 0.128188 0.991750i 0.459084π-0.459084\pi
0.128188 + 0.991750i 0.459084π0.459084\pi
882882 12334.0 0.470868
883883 47808.5 1.82207 0.911033 0.412333i 0.135286π-0.135286\pi
0.911033 + 0.412333i 0.135286π0.135286\pi
884884 −620.530 −0.0236094
885885 5075.93 0.192797
886886 39163.7 1.48502
887887 −38955.7 −1.47464 −0.737320 0.675544i 0.763909π-0.763909\pi
−0.737320 + 0.675544i 0.763909π0.763909\pi
888888 8762.64 0.331143
889889 35029.3 1.32154
890890 15744.4 0.592982
891891 −2526.07 −0.0949794
892892 5580.60 0.209476
893893 −59670.0 −2.23604
894894 15113.1 0.565387
895895 16502.1 0.616319
896896 42471.7 1.58357
897897 −698.238 −0.0259905
898898 35457.4 1.31763
899899 17036.1 0.632019
900900 −329.011 −0.0121856
901901 175.903 0.00650407
902902 9037.71 0.333617
903903 −26880.1 −0.990601
904904 −31337.0 −1.15294
905905 −2919.34 −0.107229
906906 −1815.27 −0.0665656
907907 7725.66 0.282830 0.141415 0.989950i 0.454835π-0.454835\pi
0.141415 + 0.989950i 0.454835π0.454835\pi
908908 −566.323 −0.0206983
909909 −11263.7 −0.410994
910910 3133.71 0.114155
911911 −13150.7 −0.478269 −0.239135 0.970986i 0.576864π-0.576864\pi
−0.239135 + 0.970986i 0.576864π0.576864\pi
912912 27301.1 0.991260
913913 273.463 0.00991270
914914 12434.6 0.450000
915915 5469.53 0.197614
916916 −693.702 −0.0250224
917917 −50798.0 −1.82933
918918 −34082.3 −1.22536
919919 33083.7 1.18752 0.593759 0.804643i 0.297643π-0.297643\pi
0.593759 + 0.804643i 0.297643π0.297643\pi
920920 −2415.00 −0.0865437
921921 −21625.3 −0.773699
922922 18184.1 0.649526
923923 −8330.65 −0.297082
924924 −1200.08 −0.0427270
925925 2804.11 0.0996740
926926 17187.4 0.609949
927927 8324.44 0.294941
928928 −12024.1 −0.425333
929929 −1310.12 −0.0462686 −0.0231343 0.999732i 0.507365π-0.507365\pi
−0.0231343 + 0.999732i 0.507365π0.507365\pi
930930 3557.81 0.125446
931931 −32290.2 −1.13670
932932 −2208.68 −0.0776264
933933 17743.0 0.622594
934934 −26874.1 −0.941487
935935 −4790.88 −0.167571
936936 −2255.30 −0.0787574
937937 40623.1 1.41633 0.708163 0.706049i 0.249524π-0.249524\pi
0.708163 + 0.706049i 0.249524π0.249524\pi
938938 32774.0 1.14084
939939 26239.8 0.911933
940940 2886.46 0.100155
941941 −10434.0 −0.361466 −0.180733 0.983532i 0.557847π-0.557847\pi
−0.180733 + 0.983532i 0.557847π0.557847\pi
942942 26392.6 0.912863
943943 5498.80 0.189889
944944 −19375.1 −0.668013
945945 19124.2 0.658317
946946 −10669.3 −0.366689
947947 −51333.4 −1.76147 −0.880734 0.473612i 0.842950π-0.842950\pi
−0.880734 + 0.473612i 0.842950π0.842950\pi
948948 −2053.03 −0.0703367
949949 −2339.19 −0.0800141
950950 7752.15 0.264750
951951 32906.1 1.12203
952952 40880.9 1.39176
953953 40263.6 1.36859 0.684294 0.729206i 0.260110π-0.260110\pi
0.684294 + 0.729206i 0.260110π0.260110\pi
954954 −91.3306 −0.00309951
955955 −15294.9 −0.518252
956956 −3876.79 −0.131155
957957 −12525.5 −0.423086
958958 −19069.8 −0.643128
959959 −6997.49 −0.235621
960960 8054.13 0.270777
961961 −25726.0 −0.863549
962962 −2745.94 −0.0920297
963963 −17235.2 −0.576736
964964 −2024.57 −0.0676420
965965 −6684.67 −0.222992
966966 −6571.48 −0.218876
967967 −8353.12 −0.277785 −0.138893 0.990307i 0.544354π-0.544354\pi
−0.138893 + 0.990307i 0.544354π0.544354\pi
968968 −24616.6 −0.817364
969969 29239.5 0.969358
970970 −25937.9 −0.858572
971971 −28020.1 −0.926063 −0.463032 0.886342i 0.653238π-0.653238\pi
−0.463032 + 0.886342i 0.653238π0.653238\pi
972972 3288.10 0.108504
973973 14282.4 0.470578
974974 −41973.6 −1.38082
975975 758.954 0.0249292
976976 −20877.5 −0.684704
977977 −50115.3 −1.64107 −0.820537 0.571593i 0.806326π-0.806326\pi
−0.820537 + 0.571593i 0.806326π0.806326\pi
978978 −5258.38 −0.171927
979979 −13226.1 −0.431776
980980 1562.00 0.0509144
981981 −22717.7 −0.739367
982982 15905.9 0.516881
983983 −38113.0 −1.23664 −0.618320 0.785926i 0.712186π-0.712186\pi
−0.618320 + 0.785926i 0.712186π0.712186\pi
984984 −18677.6 −0.605101
985985 −12246.1 −0.396134
986986 −60954.9 −1.96876
987987 −54980.5 −1.77310
988988 −843.481 −0.0271606
989989 −6491.49 −0.208713
990990 2487.48 0.0798558
991991 −5245.02 −0.168127 −0.0840634 0.996460i 0.526790π-0.526790\pi
−0.0840634 + 0.996460i 0.526790π0.526790\pi
992992 −2869.08 −0.0918281
993993 15863.0 0.506945
994994 −78404.0 −2.50183
995995 5693.29 0.181396
996996 80.7350 0.00256846
997997 21896.1 0.695544 0.347772 0.937579i 0.386938π-0.386938\pi
0.347772 + 0.937579i 0.386938π0.386938\pi
998998 24040.3 0.762509
999999 −16757.7 −0.530721
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 115.4.a.c.1.2 2
3.2 odd 2 1035.4.a.g.1.1 2
4.3 odd 2 1840.4.a.h.1.1 2
5.2 odd 4 575.4.b.f.24.1 4
5.3 odd 4 575.4.b.f.24.4 4
5.4 even 2 575.4.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.c.1.2 2 1.1 even 1 trivial
575.4.a.h.1.1 2 5.4 even 2
575.4.b.f.24.1 4 5.2 odd 4
575.4.b.f.24.4 4 5.3 odd 4
1035.4.a.g.1.1 2 3.2 odd 2
1840.4.a.h.1.1 2 4.3 odd 2