Properties

Label 5225.2.a.s.1.10
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $15$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 5 x^{14} - 11 x^{13} + 85 x^{12} + 6 x^{11} - 537 x^{10} + 327 x^{9} + 1556 x^{8} - 1451 x^{7} + \cdots - 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-0.324814\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.324814 q^{2} -3.24319 q^{3} -1.89450 q^{4} -1.05343 q^{6} -0.166007 q^{7} -1.26499 q^{8} +7.51826 q^{9} +O(q^{10})\) \(q+0.324814 q^{2} -3.24319 q^{3} -1.89450 q^{4} -1.05343 q^{6} -0.166007 q^{7} -1.26499 q^{8} +7.51826 q^{9} +1.00000 q^{11} +6.14420 q^{12} -6.05331 q^{13} -0.0539213 q^{14} +3.37811 q^{16} -0.875666 q^{17} +2.44204 q^{18} -1.00000 q^{19} +0.538390 q^{21} +0.324814 q^{22} -4.61502 q^{23} +4.10259 q^{24} -1.96620 q^{26} -14.6536 q^{27} +0.314499 q^{28} -1.00709 q^{29} +8.35612 q^{31} +3.62723 q^{32} -3.24319 q^{33} -0.284429 q^{34} -14.2433 q^{36} +7.03003 q^{37} -0.324814 q^{38} +19.6320 q^{39} -8.32057 q^{41} +0.174877 q^{42} -6.62275 q^{43} -1.89450 q^{44} -1.49902 q^{46} +8.43004 q^{47} -10.9558 q^{48} -6.97244 q^{49} +2.83995 q^{51} +11.4680 q^{52} +10.0757 q^{53} -4.75968 q^{54} +0.209996 q^{56} +3.24319 q^{57} -0.327117 q^{58} +4.98050 q^{59} +7.32714 q^{61} +2.71418 q^{62} -1.24808 q^{63} -5.57804 q^{64} -1.05343 q^{66} +0.360331 q^{67} +1.65895 q^{68} +14.9674 q^{69} -7.74042 q^{71} -9.51050 q^{72} +6.39400 q^{73} +2.28345 q^{74} +1.89450 q^{76} -0.166007 q^{77} +6.37676 q^{78} -0.309265 q^{79} +24.9695 q^{81} -2.70264 q^{82} +3.15788 q^{83} -1.01998 q^{84} -2.15116 q^{86} +3.26618 q^{87} -1.26499 q^{88} +13.0206 q^{89} +1.00489 q^{91} +8.74314 q^{92} -27.1005 q^{93} +2.73819 q^{94} -11.7638 q^{96} -10.6444 q^{97} -2.26475 q^{98} +7.51826 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 15 q^{7} - 15 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 15 q^{7} - 15 q^{8} + 19 q^{9} + 15 q^{11} - 9 q^{12} - 13 q^{13} + 9 q^{14} + 21 q^{16} - 11 q^{17} - 16 q^{18} - 15 q^{19} - 5 q^{22} - 10 q^{23} + 17 q^{24} - 17 q^{26} - 13 q^{27} - 30 q^{28} + 5 q^{29} + 6 q^{31} - 40 q^{32} - 4 q^{33} + 17 q^{34} + 28 q^{36} - 13 q^{37} + 5 q^{38} - 22 q^{39} - 30 q^{42} - 36 q^{43} + 17 q^{44} + 13 q^{46} + 6 q^{47} - 14 q^{48} + 16 q^{49} + 4 q^{51} - 50 q^{52} - 9 q^{53} + 9 q^{54} - 18 q^{56} + 4 q^{57} - 2 q^{58} - 7 q^{59} - 2 q^{61} - 11 q^{62} - 39 q^{63} + 17 q^{64} - q^{66} - 35 q^{67} + 18 q^{68} - 9 q^{69} + 13 q^{71} - 68 q^{72} - 2 q^{73} + 13 q^{74} - 17 q^{76} - 15 q^{77} + 10 q^{78} + 6 q^{79} + 11 q^{81} - 14 q^{82} - 30 q^{83} - 6 q^{84} - 25 q^{86} - 19 q^{87} - 15 q^{88} + 55 q^{89} + 26 q^{91} + 18 q^{92} - 14 q^{93} - 22 q^{94} - 17 q^{96} - 28 q^{97} + 22 q^{98} + 19 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.324814 0.229678 0.114839 0.993384i \(-0.463365\pi\)
0.114839 + 0.993384i \(0.463365\pi\)
\(3\) −3.24319 −1.87245 −0.936227 0.351395i \(-0.885708\pi\)
−0.936227 + 0.351395i \(0.885708\pi\)
\(4\) −1.89450 −0.947248
\(5\) 0 0
\(6\) −1.05343 −0.430062
\(7\) −0.166007 −0.0627446 −0.0313723 0.999508i \(-0.509988\pi\)
−0.0313723 + 0.999508i \(0.509988\pi\)
\(8\) −1.26499 −0.447240
\(9\) 7.51826 2.50609
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 6.14420 1.77368
\(13\) −6.05331 −1.67889 −0.839443 0.543447i \(-0.817119\pi\)
−0.839443 + 0.543447i \(0.817119\pi\)
\(14\) −0.0539213 −0.0144111
\(15\) 0 0
\(16\) 3.37811 0.844527
\(17\) −0.875666 −0.212380 −0.106190 0.994346i \(-0.533865\pi\)
−0.106190 + 0.994346i \(0.533865\pi\)
\(18\) 2.44204 0.575593
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0.538390 0.117486
\(22\) 0.324814 0.0692506
\(23\) −4.61502 −0.962299 −0.481149 0.876639i \(-0.659781\pi\)
−0.481149 + 0.876639i \(0.659781\pi\)
\(24\) 4.10259 0.837437
\(25\) 0 0
\(26\) −1.96620 −0.385604
\(27\) −14.6536 −2.82008
\(28\) 0.314499 0.0594347
\(29\) −1.00709 −0.187012 −0.0935059 0.995619i \(-0.529807\pi\)
−0.0935059 + 0.995619i \(0.529807\pi\)
\(30\) 0 0
\(31\) 8.35612 1.50080 0.750402 0.660982i \(-0.229860\pi\)
0.750402 + 0.660982i \(0.229860\pi\)
\(32\) 3.62723 0.641210
\(33\) −3.24319 −0.564566
\(34\) −0.284429 −0.0487791
\(35\) 0 0
\(36\) −14.2433 −2.37389
\(37\) 7.03003 1.15573 0.577865 0.816133i \(-0.303886\pi\)
0.577865 + 0.816133i \(0.303886\pi\)
\(38\) −0.324814 −0.0526918
\(39\) 19.6320 3.14364
\(40\) 0 0
\(41\) −8.32057 −1.29945 −0.649727 0.760167i \(-0.725117\pi\)
−0.649727 + 0.760167i \(0.725117\pi\)
\(42\) 0.174877 0.0269841
\(43\) −6.62275 −1.00996 −0.504980 0.863131i \(-0.668500\pi\)
−0.504980 + 0.863131i \(0.668500\pi\)
\(44\) −1.89450 −0.285606
\(45\) 0 0
\(46\) −1.49902 −0.221019
\(47\) 8.43004 1.22965 0.614824 0.788664i \(-0.289227\pi\)
0.614824 + 0.788664i \(0.289227\pi\)
\(48\) −10.9558 −1.58134
\(49\) −6.97244 −0.996063
\(50\) 0 0
\(51\) 2.83995 0.397673
\(52\) 11.4680 1.59032
\(53\) 10.0757 1.38401 0.692005 0.721893i \(-0.256728\pi\)
0.692005 + 0.721893i \(0.256728\pi\)
\(54\) −4.75968 −0.647711
\(55\) 0 0
\(56\) 0.209996 0.0280619
\(57\) 3.24319 0.429571
\(58\) −0.327117 −0.0429525
\(59\) 4.98050 0.648406 0.324203 0.945987i \(-0.394904\pi\)
0.324203 + 0.945987i \(0.394904\pi\)
\(60\) 0 0
\(61\) 7.32714 0.938144 0.469072 0.883160i \(-0.344588\pi\)
0.469072 + 0.883160i \(0.344588\pi\)
\(62\) 2.71418 0.344702
\(63\) −1.24808 −0.157243
\(64\) −5.57804 −0.697255
\(65\) 0 0
\(66\) −1.05343 −0.129669
\(67\) 0.360331 0.0440214 0.0220107 0.999758i \(-0.492993\pi\)
0.0220107 + 0.999758i \(0.492993\pi\)
\(68\) 1.65895 0.201177
\(69\) 14.9674 1.80186
\(70\) 0 0
\(71\) −7.74042 −0.918619 −0.459310 0.888276i \(-0.651903\pi\)
−0.459310 + 0.888276i \(0.651903\pi\)
\(72\) −9.51050 −1.12082
\(73\) 6.39400 0.748361 0.374180 0.927356i \(-0.377924\pi\)
0.374180 + 0.927356i \(0.377924\pi\)
\(74\) 2.28345 0.265446
\(75\) 0 0
\(76\) 1.89450 0.217314
\(77\) −0.166007 −0.0189182
\(78\) 6.37676 0.722025
\(79\) −0.309265 −0.0347950 −0.0173975 0.999849i \(-0.505538\pi\)
−0.0173975 + 0.999849i \(0.505538\pi\)
\(80\) 0 0
\(81\) 24.9695 2.77438
\(82\) −2.70264 −0.298456
\(83\) 3.15788 0.346623 0.173311 0.984867i \(-0.444553\pi\)
0.173311 + 0.984867i \(0.444553\pi\)
\(84\) −1.01998 −0.111289
\(85\) 0 0
\(86\) −2.15116 −0.231966
\(87\) 3.26618 0.350171
\(88\) −1.26499 −0.134848
\(89\) 13.0206 1.38018 0.690088 0.723726i \(-0.257572\pi\)
0.690088 + 0.723726i \(0.257572\pi\)
\(90\) 0 0
\(91\) 1.00489 0.105341
\(92\) 8.74314 0.911535
\(93\) −27.1005 −2.81019
\(94\) 2.73819 0.282423
\(95\) 0 0
\(96\) −11.7638 −1.20064
\(97\) −10.6444 −1.08077 −0.540385 0.841418i \(-0.681721\pi\)
−0.540385 + 0.841418i \(0.681721\pi\)
\(98\) −2.26475 −0.228774
\(99\) 7.51826 0.755614
\(100\) 0 0
\(101\) 10.8769 1.08229 0.541145 0.840929i \(-0.317991\pi\)
0.541145 + 0.840929i \(0.317991\pi\)
\(102\) 0.922455 0.0913367
\(103\) 5.78563 0.570075 0.285037 0.958516i \(-0.407994\pi\)
0.285037 + 0.958516i \(0.407994\pi\)
\(104\) 7.65736 0.750866
\(105\) 0 0
\(106\) 3.27274 0.317877
\(107\) 4.75657 0.459835 0.229918 0.973210i \(-0.426154\pi\)
0.229918 + 0.973210i \(0.426154\pi\)
\(108\) 27.7611 2.67131
\(109\) 14.0827 1.34888 0.674438 0.738332i \(-0.264386\pi\)
0.674438 + 0.738332i \(0.264386\pi\)
\(110\) 0 0
\(111\) −22.7997 −2.16405
\(112\) −0.560788 −0.0529895
\(113\) 16.5897 1.56063 0.780314 0.625387i \(-0.215059\pi\)
0.780314 + 0.625387i \(0.215059\pi\)
\(114\) 1.05343 0.0986630
\(115\) 0 0
\(116\) 1.90793 0.177147
\(117\) −45.5104 −4.20744
\(118\) 1.61774 0.148925
\(119\) 0.145366 0.0133257
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.37996 0.215471
\(123\) 26.9852 2.43317
\(124\) −15.8306 −1.42163
\(125\) 0 0
\(126\) −0.405394 −0.0361154
\(127\) −16.0352 −1.42290 −0.711448 0.702738i \(-0.751961\pi\)
−0.711448 + 0.702738i \(0.751961\pi\)
\(128\) −9.06628 −0.801354
\(129\) 21.4788 1.89110
\(130\) 0 0
\(131\) 9.50050 0.830063 0.415031 0.909807i \(-0.363771\pi\)
0.415031 + 0.909807i \(0.363771\pi\)
\(132\) 6.14420 0.534784
\(133\) 0.166007 0.0143946
\(134\) 0.117040 0.0101108
\(135\) 0 0
\(136\) 1.10771 0.0949850
\(137\) −14.1245 −1.20674 −0.603368 0.797463i \(-0.706175\pi\)
−0.603368 + 0.797463i \(0.706175\pi\)
\(138\) 4.86161 0.413848
\(139\) 8.78232 0.744907 0.372453 0.928051i \(-0.378517\pi\)
0.372453 + 0.928051i \(0.378517\pi\)
\(140\) 0 0
\(141\) −27.3402 −2.30246
\(142\) −2.51420 −0.210987
\(143\) −6.05331 −0.506203
\(144\) 25.3975 2.11646
\(145\) 0 0
\(146\) 2.07686 0.171882
\(147\) 22.6129 1.86508
\(148\) −13.3184 −1.09476
\(149\) −6.56136 −0.537528 −0.268764 0.963206i \(-0.586615\pi\)
−0.268764 + 0.963206i \(0.586615\pi\)
\(150\) 0 0
\(151\) −15.9594 −1.29876 −0.649380 0.760464i \(-0.724971\pi\)
−0.649380 + 0.760464i \(0.724971\pi\)
\(152\) 1.26499 0.102604
\(153\) −6.58349 −0.532244
\(154\) −0.0539213 −0.00434510
\(155\) 0 0
\(156\) −37.1928 −2.97781
\(157\) 13.8417 1.10469 0.552343 0.833617i \(-0.313734\pi\)
0.552343 + 0.833617i \(0.313734\pi\)
\(158\) −0.100454 −0.00799166
\(159\) −32.6775 −2.59150
\(160\) 0 0
\(161\) 0.766124 0.0603790
\(162\) 8.11043 0.637216
\(163\) 0.673322 0.0527386 0.0263693 0.999652i \(-0.491605\pi\)
0.0263693 + 0.999652i \(0.491605\pi\)
\(164\) 15.7633 1.23091
\(165\) 0 0
\(166\) 1.02572 0.0796117
\(167\) 9.05874 0.700986 0.350493 0.936565i \(-0.386014\pi\)
0.350493 + 0.936565i \(0.386014\pi\)
\(168\) −0.681057 −0.0525447
\(169\) 23.6426 1.81866
\(170\) 0 0
\(171\) −7.51826 −0.574936
\(172\) 12.5468 0.956682
\(173\) −21.7359 −1.65255 −0.826275 0.563267i \(-0.809544\pi\)
−0.826275 + 0.563267i \(0.809544\pi\)
\(174\) 1.06090 0.0804267
\(175\) 0 0
\(176\) 3.37811 0.254634
\(177\) −16.1527 −1.21411
\(178\) 4.22926 0.316996
\(179\) −1.39240 −0.104073 −0.0520363 0.998645i \(-0.516571\pi\)
−0.0520363 + 0.998645i \(0.516571\pi\)
\(180\) 0 0
\(181\) 12.5467 0.932587 0.466294 0.884630i \(-0.345589\pi\)
0.466294 + 0.884630i \(0.345589\pi\)
\(182\) 0.326402 0.0241945
\(183\) −23.7633 −1.75663
\(184\) 5.83794 0.430379
\(185\) 0 0
\(186\) −8.80261 −0.645438
\(187\) −0.875666 −0.0640351
\(188\) −15.9707 −1.16478
\(189\) 2.43259 0.176945
\(190\) 0 0
\(191\) −14.7271 −1.06561 −0.532807 0.846237i \(-0.678863\pi\)
−0.532807 + 0.846237i \(0.678863\pi\)
\(192\) 18.0906 1.30558
\(193\) −11.7487 −0.845689 −0.422845 0.906202i \(-0.638968\pi\)
−0.422845 + 0.906202i \(0.638968\pi\)
\(194\) −3.45744 −0.248229
\(195\) 0 0
\(196\) 13.2093 0.943519
\(197\) −13.0704 −0.931224 −0.465612 0.884989i \(-0.654166\pi\)
−0.465612 + 0.884989i \(0.654166\pi\)
\(198\) 2.44204 0.173548
\(199\) −16.7629 −1.18829 −0.594145 0.804358i \(-0.702509\pi\)
−0.594145 + 0.804358i \(0.702509\pi\)
\(200\) 0 0
\(201\) −1.16862 −0.0824281
\(202\) 3.53297 0.248579
\(203\) 0.167183 0.0117340
\(204\) −5.38027 −0.376694
\(205\) 0 0
\(206\) 1.87925 0.130934
\(207\) −34.6969 −2.41160
\(208\) −20.4487 −1.41786
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −19.4607 −1.33973 −0.669866 0.742482i \(-0.733648\pi\)
−0.669866 + 0.742482i \(0.733648\pi\)
\(212\) −19.0885 −1.31100
\(213\) 25.1036 1.72007
\(214\) 1.54500 0.105614
\(215\) 0 0
\(216\) 18.5366 1.26125
\(217\) −1.38717 −0.0941673
\(218\) 4.57425 0.309807
\(219\) −20.7369 −1.40127
\(220\) 0 0
\(221\) 5.30068 0.356563
\(222\) −7.40566 −0.497035
\(223\) −11.4123 −0.764224 −0.382112 0.924116i \(-0.624803\pi\)
−0.382112 + 0.924116i \(0.624803\pi\)
\(224\) −0.602144 −0.0402324
\(225\) 0 0
\(226\) 5.38857 0.358442
\(227\) −4.01475 −0.266468 −0.133234 0.991085i \(-0.542536\pi\)
−0.133234 + 0.991085i \(0.542536\pi\)
\(228\) −6.14420 −0.406910
\(229\) 26.3140 1.73888 0.869440 0.494038i \(-0.164480\pi\)
0.869440 + 0.494038i \(0.164480\pi\)
\(230\) 0 0
\(231\) 0.538390 0.0354235
\(232\) 1.27395 0.0836392
\(233\) 14.6801 0.961724 0.480862 0.876796i \(-0.340324\pi\)
0.480862 + 0.876796i \(0.340324\pi\)
\(234\) −14.7824 −0.966356
\(235\) 0 0
\(236\) −9.43554 −0.614202
\(237\) 1.00300 0.0651521
\(238\) 0.0472170 0.00306063
\(239\) −28.5560 −1.84713 −0.923567 0.383437i \(-0.874740\pi\)
−0.923567 + 0.383437i \(0.874740\pi\)
\(240\) 0 0
\(241\) −22.3426 −1.43921 −0.719607 0.694381i \(-0.755678\pi\)
−0.719607 + 0.694381i \(0.755678\pi\)
\(242\) 0.324814 0.0208798
\(243\) −37.0199 −2.37483
\(244\) −13.8812 −0.888655
\(245\) 0 0
\(246\) 8.76516 0.558846
\(247\) 6.05331 0.385163
\(248\) −10.5704 −0.671220
\(249\) −10.2416 −0.649035
\(250\) 0 0
\(251\) 11.5146 0.726795 0.363398 0.931634i \(-0.381617\pi\)
0.363398 + 0.931634i \(0.381617\pi\)
\(252\) 2.36448 0.148948
\(253\) −4.61502 −0.290144
\(254\) −5.20847 −0.326808
\(255\) 0 0
\(256\) 8.21122 0.513201
\(257\) −10.5478 −0.657955 −0.328978 0.944338i \(-0.606704\pi\)
−0.328978 + 0.944338i \(0.606704\pi\)
\(258\) 6.97662 0.434345
\(259\) −1.16703 −0.0725158
\(260\) 0 0
\(261\) −7.57156 −0.468668
\(262\) 3.08590 0.190647
\(263\) −20.3165 −1.25277 −0.626386 0.779513i \(-0.715466\pi\)
−0.626386 + 0.779513i \(0.715466\pi\)
\(264\) 4.10259 0.252497
\(265\) 0 0
\(266\) 0.0539213 0.00330612
\(267\) −42.2281 −2.58432
\(268\) −0.682645 −0.0416992
\(269\) −16.4623 −1.00373 −0.501863 0.864947i \(-0.667352\pi\)
−0.501863 + 0.864947i \(0.667352\pi\)
\(270\) 0 0
\(271\) 23.2292 1.41107 0.705537 0.708673i \(-0.250706\pi\)
0.705537 + 0.708673i \(0.250706\pi\)
\(272\) −2.95809 −0.179361
\(273\) −3.25904 −0.197246
\(274\) −4.58783 −0.277161
\(275\) 0 0
\(276\) −28.3556 −1.70681
\(277\) 17.7363 1.06567 0.532837 0.846218i \(-0.321126\pi\)
0.532837 + 0.846218i \(0.321126\pi\)
\(278\) 2.85262 0.171089
\(279\) 62.8235 3.76114
\(280\) 0 0
\(281\) −25.6914 −1.53262 −0.766310 0.642471i \(-0.777909\pi\)
−0.766310 + 0.642471i \(0.777909\pi\)
\(282\) −8.88048 −0.528825
\(283\) −5.17198 −0.307442 −0.153721 0.988114i \(-0.549126\pi\)
−0.153721 + 0.988114i \(0.549126\pi\)
\(284\) 14.6642 0.870160
\(285\) 0 0
\(286\) −1.96620 −0.116264
\(287\) 1.38127 0.0815337
\(288\) 27.2705 1.60693
\(289\) −16.2332 −0.954895
\(290\) 0 0
\(291\) 34.5216 2.02369
\(292\) −12.1134 −0.708883
\(293\) −24.8835 −1.45371 −0.726854 0.686792i \(-0.759018\pi\)
−0.726854 + 0.686792i \(0.759018\pi\)
\(294\) 7.34500 0.428369
\(295\) 0 0
\(296\) −8.89289 −0.516889
\(297\) −14.6536 −0.850286
\(298\) −2.13122 −0.123458
\(299\) 27.9362 1.61559
\(300\) 0 0
\(301\) 1.09942 0.0633695
\(302\) −5.18384 −0.298297
\(303\) −35.2758 −2.02654
\(304\) −3.37811 −0.193748
\(305\) 0 0
\(306\) −2.13841 −0.122245
\(307\) 1.37200 0.0783042 0.0391521 0.999233i \(-0.487534\pi\)
0.0391521 + 0.999233i \(0.487534\pi\)
\(308\) 0.314499 0.0179202
\(309\) −18.7639 −1.06744
\(310\) 0 0
\(311\) 1.75534 0.0995360 0.0497680 0.998761i \(-0.484152\pi\)
0.0497680 + 0.998761i \(0.484152\pi\)
\(312\) −24.8343 −1.40596
\(313\) 18.5787 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(314\) 4.49597 0.253722
\(315\) 0 0
\(316\) 0.585901 0.0329595
\(317\) −4.00886 −0.225160 −0.112580 0.993643i \(-0.535911\pi\)
−0.112580 + 0.993643i \(0.535911\pi\)
\(318\) −10.6141 −0.595210
\(319\) −1.00709 −0.0563862
\(320\) 0 0
\(321\) −15.4265 −0.861021
\(322\) 0.248848 0.0138677
\(323\) 0.875666 0.0487234
\(324\) −47.3045 −2.62803
\(325\) 0 0
\(326\) 0.218704 0.0121129
\(327\) −45.6728 −2.52571
\(328\) 10.5254 0.581169
\(329\) −1.39944 −0.0771537
\(330\) 0 0
\(331\) 6.40426 0.352010 0.176005 0.984389i \(-0.443682\pi\)
0.176005 + 0.984389i \(0.443682\pi\)
\(332\) −5.98260 −0.328338
\(333\) 52.8536 2.89636
\(334\) 2.94240 0.161001
\(335\) 0 0
\(336\) 1.81874 0.0992204
\(337\) −26.4195 −1.43916 −0.719582 0.694407i \(-0.755667\pi\)
−0.719582 + 0.694407i \(0.755667\pi\)
\(338\) 7.67945 0.417707
\(339\) −53.8035 −2.92221
\(340\) 0 0
\(341\) 8.35612 0.452509
\(342\) −2.44204 −0.132050
\(343\) 2.31952 0.125242
\(344\) 8.37769 0.451695
\(345\) 0 0
\(346\) −7.06013 −0.379555
\(347\) 6.14023 0.329625 0.164812 0.986325i \(-0.447298\pi\)
0.164812 + 0.986325i \(0.447298\pi\)
\(348\) −6.18776 −0.331699
\(349\) 17.0701 0.913742 0.456871 0.889533i \(-0.348970\pi\)
0.456871 + 0.889533i \(0.348970\pi\)
\(350\) 0 0
\(351\) 88.7026 4.73459
\(352\) 3.62723 0.193332
\(353\) 24.1803 1.28699 0.643495 0.765451i \(-0.277484\pi\)
0.643495 + 0.765451i \(0.277484\pi\)
\(354\) −5.24662 −0.278855
\(355\) 0 0
\(356\) −24.6674 −1.30737
\(357\) −0.471450 −0.0249518
\(358\) −0.452270 −0.0239032
\(359\) −27.5984 −1.45659 −0.728294 0.685265i \(-0.759687\pi\)
−0.728294 + 0.685265i \(0.759687\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 4.07534 0.214195
\(363\) −3.24319 −0.170223
\(364\) −1.90376 −0.0997841
\(365\) 0 0
\(366\) −7.71865 −0.403460
\(367\) −0.102872 −0.00536987 −0.00268494 0.999996i \(-0.500855\pi\)
−0.00268494 + 0.999996i \(0.500855\pi\)
\(368\) −15.5900 −0.812687
\(369\) −62.5562 −3.25655
\(370\) 0 0
\(371\) −1.67264 −0.0868392
\(372\) 51.3417 2.66194
\(373\) −11.1255 −0.576055 −0.288027 0.957622i \(-0.592999\pi\)
−0.288027 + 0.957622i \(0.592999\pi\)
\(374\) −0.284429 −0.0147075
\(375\) 0 0
\(376\) −10.6639 −0.549948
\(377\) 6.09623 0.313972
\(378\) 0.790138 0.0406403
\(379\) −19.2468 −0.988643 −0.494321 0.869279i \(-0.664583\pi\)
−0.494321 + 0.869279i \(0.664583\pi\)
\(380\) 0 0
\(381\) 52.0053 2.66431
\(382\) −4.78356 −0.244748
\(383\) −8.55556 −0.437169 −0.218584 0.975818i \(-0.570144\pi\)
−0.218584 + 0.975818i \(0.570144\pi\)
\(384\) 29.4037 1.50050
\(385\) 0 0
\(386\) −3.81614 −0.194236
\(387\) −49.7916 −2.53105
\(388\) 20.1657 1.02376
\(389\) 16.7276 0.848125 0.424063 0.905633i \(-0.360604\pi\)
0.424063 + 0.905633i \(0.360604\pi\)
\(390\) 0 0
\(391\) 4.04122 0.204373
\(392\) 8.82005 0.445480
\(393\) −30.8119 −1.55425
\(394\) −4.24543 −0.213882
\(395\) 0 0
\(396\) −14.2433 −0.715753
\(397\) 37.0868 1.86133 0.930667 0.365867i \(-0.119228\pi\)
0.930667 + 0.365867i \(0.119228\pi\)
\(398\) −5.44482 −0.272924
\(399\) −0.538390 −0.0269532
\(400\) 0 0
\(401\) −4.15586 −0.207534 −0.103767 0.994602i \(-0.533090\pi\)
−0.103767 + 0.994602i \(0.533090\pi\)
\(402\) −0.379584 −0.0189319
\(403\) −50.5822 −2.51968
\(404\) −20.6062 −1.02520
\(405\) 0 0
\(406\) 0.0543035 0.00269504
\(407\) 7.03003 0.348466
\(408\) −3.59250 −0.177855
\(409\) −19.6949 −0.973851 −0.486926 0.873443i \(-0.661882\pi\)
−0.486926 + 0.873443i \(0.661882\pi\)
\(410\) 0 0
\(411\) 45.8083 2.25956
\(412\) −10.9608 −0.540002
\(413\) −0.826796 −0.0406840
\(414\) −11.2701 −0.553893
\(415\) 0 0
\(416\) −21.9568 −1.07652
\(417\) −28.4827 −1.39480
\(418\) −0.324814 −0.0158872
\(419\) −4.19043 −0.204716 −0.102358 0.994748i \(-0.532639\pi\)
−0.102358 + 0.994748i \(0.532639\pi\)
\(420\) 0 0
\(421\) 22.5663 1.09981 0.549907 0.835226i \(-0.314663\pi\)
0.549907 + 0.835226i \(0.314663\pi\)
\(422\) −6.32111 −0.307707
\(423\) 63.3792 3.08160
\(424\) −12.7457 −0.618985
\(425\) 0 0
\(426\) 8.15401 0.395063
\(427\) −1.21635 −0.0588635
\(428\) −9.01131 −0.435578
\(429\) 19.6320 0.947843
\(430\) 0 0
\(431\) −18.3524 −0.884002 −0.442001 0.897014i \(-0.645731\pi\)
−0.442001 + 0.897014i \(0.645731\pi\)
\(432\) −49.5013 −2.38163
\(433\) 1.78082 0.0855810 0.0427905 0.999084i \(-0.486375\pi\)
0.0427905 + 0.999084i \(0.486375\pi\)
\(434\) −0.450572 −0.0216282
\(435\) 0 0
\(436\) −26.6796 −1.27772
\(437\) 4.61502 0.220766
\(438\) −6.73564 −0.321841
\(439\) −26.8306 −1.28055 −0.640277 0.768144i \(-0.721180\pi\)
−0.640277 + 0.768144i \(0.721180\pi\)
\(440\) 0 0
\(441\) −52.4206 −2.49622
\(442\) 1.72174 0.0818946
\(443\) 15.4460 0.733862 0.366931 0.930248i \(-0.380409\pi\)
0.366931 + 0.930248i \(0.380409\pi\)
\(444\) 43.1939 2.04989
\(445\) 0 0
\(446\) −3.70687 −0.175525
\(447\) 21.2797 1.00650
\(448\) 0.925991 0.0437490
\(449\) −14.7640 −0.696758 −0.348379 0.937354i \(-0.613268\pi\)
−0.348379 + 0.937354i \(0.613268\pi\)
\(450\) 0 0
\(451\) −8.32057 −0.391800
\(452\) −31.4291 −1.47830
\(453\) 51.7594 2.43187
\(454\) −1.30405 −0.0612020
\(455\) 0 0
\(456\) −4.10259 −0.192121
\(457\) 16.2215 0.758811 0.379406 0.925230i \(-0.376128\pi\)
0.379406 + 0.925230i \(0.376128\pi\)
\(458\) 8.54717 0.399383
\(459\) 12.8316 0.598929
\(460\) 0 0
\(461\) 13.4764 0.627659 0.313830 0.949479i \(-0.398388\pi\)
0.313830 + 0.949479i \(0.398388\pi\)
\(462\) 0.174877 0.00813600
\(463\) 28.6461 1.33130 0.665648 0.746266i \(-0.268155\pi\)
0.665648 + 0.746266i \(0.268155\pi\)
\(464\) −3.40205 −0.157936
\(465\) 0 0
\(466\) 4.76829 0.220887
\(467\) −17.0725 −0.790023 −0.395011 0.918676i \(-0.629259\pi\)
−0.395011 + 0.918676i \(0.629259\pi\)
\(468\) 86.2192 3.98549
\(469\) −0.0598173 −0.00276211
\(470\) 0 0
\(471\) −44.8911 −2.06847
\(472\) −6.30027 −0.289993
\(473\) −6.62275 −0.304514
\(474\) 0.325790 0.0149640
\(475\) 0 0
\(476\) −0.275396 −0.0126228
\(477\) 75.7521 3.46845
\(478\) −9.27539 −0.424246
\(479\) 28.3507 1.29538 0.647689 0.761905i \(-0.275736\pi\)
0.647689 + 0.761905i \(0.275736\pi\)
\(480\) 0 0
\(481\) −42.5550 −1.94034
\(482\) −7.25720 −0.330556
\(483\) −2.48468 −0.113057
\(484\) −1.89450 −0.0861134
\(485\) 0 0
\(486\) −12.0246 −0.545447
\(487\) 11.4047 0.516798 0.258399 0.966038i \(-0.416805\pi\)
0.258399 + 0.966038i \(0.416805\pi\)
\(488\) −9.26874 −0.419576
\(489\) −2.18371 −0.0987507
\(490\) 0 0
\(491\) −29.9154 −1.35006 −0.675032 0.737788i \(-0.735870\pi\)
−0.675032 + 0.737788i \(0.735870\pi\)
\(492\) −51.1233 −2.30482
\(493\) 0.881874 0.0397176
\(494\) 1.96620 0.0884636
\(495\) 0 0
\(496\) 28.2279 1.26747
\(497\) 1.28496 0.0576384
\(498\) −3.32662 −0.149069
\(499\) −37.9248 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(500\) 0 0
\(501\) −29.3792 −1.31256
\(502\) 3.74010 0.166929
\(503\) −2.74270 −0.122291 −0.0611456 0.998129i \(-0.519475\pi\)
−0.0611456 + 0.998129i \(0.519475\pi\)
\(504\) 1.57881 0.0703256
\(505\) 0 0
\(506\) −1.49902 −0.0666397
\(507\) −76.6774 −3.40536
\(508\) 30.3787 1.34784
\(509\) 0.635877 0.0281847 0.0140924 0.999901i \(-0.495514\pi\)
0.0140924 + 0.999901i \(0.495514\pi\)
\(510\) 0 0
\(511\) −1.06145 −0.0469556
\(512\) 20.7997 0.919225
\(513\) 14.6536 0.646971
\(514\) −3.42608 −0.151118
\(515\) 0 0
\(516\) −40.6915 −1.79134
\(517\) 8.43004 0.370753
\(518\) −0.379068 −0.0166553
\(519\) 70.4936 3.09433
\(520\) 0 0
\(521\) 28.8718 1.26490 0.632449 0.774602i \(-0.282050\pi\)
0.632449 + 0.774602i \(0.282050\pi\)
\(522\) −2.45935 −0.107643
\(523\) 20.2611 0.885956 0.442978 0.896532i \(-0.353922\pi\)
0.442978 + 0.896532i \(0.353922\pi\)
\(524\) −17.9987 −0.786275
\(525\) 0 0
\(526\) −6.59910 −0.287734
\(527\) −7.31717 −0.318741
\(528\) −10.9558 −0.476791
\(529\) −1.70157 −0.0739814
\(530\) 0 0
\(531\) 37.4447 1.62496
\(532\) −0.314499 −0.0136352
\(533\) 50.3670 2.18164
\(534\) −13.7163 −0.593561
\(535\) 0 0
\(536\) −0.455814 −0.0196882
\(537\) 4.51580 0.194871
\(538\) −5.34720 −0.230534
\(539\) −6.97244 −0.300324
\(540\) 0 0
\(541\) 25.4105 1.09248 0.546242 0.837628i \(-0.316058\pi\)
0.546242 + 0.837628i \(0.316058\pi\)
\(542\) 7.54518 0.324093
\(543\) −40.6912 −1.74623
\(544\) −3.17624 −0.136180
\(545\) 0 0
\(546\) −1.05858 −0.0453032
\(547\) 0.458489 0.0196036 0.00980178 0.999952i \(-0.496880\pi\)
0.00980178 + 0.999952i \(0.496880\pi\)
\(548\) 26.7588 1.14308
\(549\) 55.0874 2.35107
\(550\) 0 0
\(551\) 1.00709 0.0429034
\(552\) −18.9335 −0.805865
\(553\) 0.0513400 0.00218320
\(554\) 5.76101 0.244762
\(555\) 0 0
\(556\) −16.6381 −0.705611
\(557\) 43.5515 1.84534 0.922669 0.385593i \(-0.126003\pi\)
0.922669 + 0.385593i \(0.126003\pi\)
\(558\) 20.4059 0.863852
\(559\) 40.0896 1.69561
\(560\) 0 0
\(561\) 2.83995 0.119903
\(562\) −8.34492 −0.352009
\(563\) −46.9523 −1.97880 −0.989401 0.145208i \(-0.953615\pi\)
−0.989401 + 0.145208i \(0.953615\pi\)
\(564\) 51.7959 2.18100
\(565\) 0 0
\(566\) −1.67993 −0.0706128
\(567\) −4.14509 −0.174078
\(568\) 9.79153 0.410844
\(569\) 30.2122 1.26656 0.633282 0.773922i \(-0.281708\pi\)
0.633282 + 0.773922i \(0.281708\pi\)
\(570\) 0 0
\(571\) 23.2284 0.972077 0.486038 0.873938i \(-0.338442\pi\)
0.486038 + 0.873938i \(0.338442\pi\)
\(572\) 11.4680 0.479500
\(573\) 47.7627 1.99531
\(574\) 0.448656 0.0187265
\(575\) 0 0
\(576\) −41.9371 −1.74738
\(577\) −9.07629 −0.377851 −0.188926 0.981991i \(-0.560500\pi\)
−0.188926 + 0.981991i \(0.560500\pi\)
\(578\) −5.27277 −0.219318
\(579\) 38.1032 1.58351
\(580\) 0 0
\(581\) −0.524229 −0.0217487
\(582\) 11.2131 0.464798
\(583\) 10.0757 0.417295
\(584\) −8.08832 −0.334697
\(585\) 0 0
\(586\) −8.08250 −0.333885
\(587\) 36.7331 1.51614 0.758069 0.652174i \(-0.226143\pi\)
0.758069 + 0.652174i \(0.226143\pi\)
\(588\) −42.8401 −1.76670
\(589\) −8.35612 −0.344308
\(590\) 0 0
\(591\) 42.3896 1.74368
\(592\) 23.7482 0.976044
\(593\) 28.5431 1.17213 0.586063 0.810266i \(-0.300677\pi\)
0.586063 + 0.810266i \(0.300677\pi\)
\(594\) −4.75968 −0.195292
\(595\) 0 0
\(596\) 12.4305 0.509172
\(597\) 54.3652 2.22502
\(598\) 9.07406 0.371066
\(599\) −45.9055 −1.87565 −0.937825 0.347109i \(-0.887164\pi\)
−0.937825 + 0.347109i \(0.887164\pi\)
\(600\) 0 0
\(601\) 34.0331 1.38824 0.694120 0.719860i \(-0.255794\pi\)
0.694120 + 0.719860i \(0.255794\pi\)
\(602\) 0.357107 0.0145546
\(603\) 2.70906 0.110321
\(604\) 30.2351 1.23025
\(605\) 0 0
\(606\) −11.4581 −0.465452
\(607\) −3.84641 −0.156121 −0.0780606 0.996949i \(-0.524873\pi\)
−0.0780606 + 0.996949i \(0.524873\pi\)
\(608\) −3.62723 −0.147104
\(609\) −0.542207 −0.0219713
\(610\) 0 0
\(611\) −51.0297 −2.06444
\(612\) 12.4724 0.504167
\(613\) −17.5232 −0.707754 −0.353877 0.935292i \(-0.615137\pi\)
−0.353877 + 0.935292i \(0.615137\pi\)
\(614\) 0.445645 0.0179848
\(615\) 0 0
\(616\) 0.209996 0.00846098
\(617\) 14.0815 0.566900 0.283450 0.958987i \(-0.408521\pi\)
0.283450 + 0.958987i \(0.408521\pi\)
\(618\) −6.09477 −0.245167
\(619\) −23.1027 −0.928577 −0.464288 0.885684i \(-0.653690\pi\)
−0.464288 + 0.885684i \(0.653690\pi\)
\(620\) 0 0
\(621\) 67.6265 2.71376
\(622\) 0.570158 0.0228613
\(623\) −2.16150 −0.0865985
\(624\) 66.3191 2.65489
\(625\) 0 0
\(626\) 6.03462 0.241192
\(627\) 3.24319 0.129520
\(628\) −26.2230 −1.04641
\(629\) −6.15596 −0.245454
\(630\) 0 0
\(631\) 3.60839 0.143648 0.0718239 0.997417i \(-0.477118\pi\)
0.0718239 + 0.997417i \(0.477118\pi\)
\(632\) 0.391216 0.0155617
\(633\) 63.1148 2.50859
\(634\) −1.30213 −0.0517144
\(635\) 0 0
\(636\) 61.9074 2.45479
\(637\) 42.2064 1.67228
\(638\) −0.327117 −0.0129507
\(639\) −58.1945 −2.30214
\(640\) 0 0
\(641\) −17.5350 −0.692589 −0.346294 0.938126i \(-0.612560\pi\)
−0.346294 + 0.938126i \(0.612560\pi\)
\(642\) −5.01073 −0.197758
\(643\) −19.6051 −0.773151 −0.386575 0.922258i \(-0.626342\pi\)
−0.386575 + 0.922258i \(0.626342\pi\)
\(644\) −1.45142 −0.0571939
\(645\) 0 0
\(646\) 0.284429 0.0111907
\(647\) −41.2895 −1.62326 −0.811629 0.584173i \(-0.801419\pi\)
−0.811629 + 0.584173i \(0.801419\pi\)
\(648\) −31.5860 −1.24082
\(649\) 4.98050 0.195502
\(650\) 0 0
\(651\) 4.49885 0.176324
\(652\) −1.27561 −0.0499566
\(653\) 8.64148 0.338167 0.169084 0.985602i \(-0.445919\pi\)
0.169084 + 0.985602i \(0.445919\pi\)
\(654\) −14.8351 −0.580100
\(655\) 0 0
\(656\) −28.1078 −1.09742
\(657\) 48.0717 1.87546
\(658\) −0.454558 −0.0177205
\(659\) −2.34694 −0.0914237 −0.0457118 0.998955i \(-0.514556\pi\)
−0.0457118 + 0.998955i \(0.514556\pi\)
\(660\) 0 0
\(661\) −37.8564 −1.47244 −0.736222 0.676740i \(-0.763392\pi\)
−0.736222 + 0.676740i \(0.763392\pi\)
\(662\) 2.08019 0.0808490
\(663\) −17.1911 −0.667647
\(664\) −3.99468 −0.155024
\(665\) 0 0
\(666\) 17.1676 0.665230
\(667\) 4.64774 0.179961
\(668\) −17.1617 −0.664008
\(669\) 37.0122 1.43097
\(670\) 0 0
\(671\) 7.32714 0.282861
\(672\) 1.95287 0.0753334
\(673\) −32.1381 −1.23883 −0.619417 0.785062i \(-0.712631\pi\)
−0.619417 + 0.785062i \(0.712631\pi\)
\(674\) −8.58144 −0.330545
\(675\) 0 0
\(676\) −44.7908 −1.72272
\(677\) −46.2653 −1.77812 −0.889060 0.457791i \(-0.848641\pi\)
−0.889060 + 0.457791i \(0.848641\pi\)
\(678\) −17.4761 −0.671167
\(679\) 1.76703 0.0678125
\(680\) 0 0
\(681\) 13.0206 0.498950
\(682\) 2.71418 0.103931
\(683\) 3.51505 0.134500 0.0672498 0.997736i \(-0.478578\pi\)
0.0672498 + 0.997736i \(0.478578\pi\)
\(684\) 14.2433 0.544607
\(685\) 0 0
\(686\) 0.753412 0.0287654
\(687\) −85.3414 −3.25598
\(688\) −22.3724 −0.852938
\(689\) −60.9917 −2.32360
\(690\) 0 0
\(691\) 6.47549 0.246339 0.123170 0.992386i \(-0.460694\pi\)
0.123170 + 0.992386i \(0.460694\pi\)
\(692\) 41.1786 1.56537
\(693\) −1.24808 −0.0474107
\(694\) 1.99443 0.0757077
\(695\) 0 0
\(696\) −4.13167 −0.156611
\(697\) 7.28604 0.275979
\(698\) 5.54461 0.209867
\(699\) −47.6102 −1.80078
\(700\) 0 0
\(701\) −14.8438 −0.560643 −0.280321 0.959906i \(-0.590441\pi\)
−0.280321 + 0.959906i \(0.590441\pi\)
\(702\) 28.8118 1.08743
\(703\) −7.03003 −0.265142
\(704\) −5.57804 −0.210230
\(705\) 0 0
\(706\) 7.85411 0.295593
\(707\) −1.80564 −0.0679079
\(708\) 30.6012 1.15006
\(709\) −25.8169 −0.969572 −0.484786 0.874633i \(-0.661103\pi\)
−0.484786 + 0.874633i \(0.661103\pi\)
\(710\) 0 0
\(711\) −2.32514 −0.0871994
\(712\) −16.4708 −0.617270
\(713\) −38.5637 −1.44422
\(714\) −0.153134 −0.00573088
\(715\) 0 0
\(716\) 2.63789 0.0985826
\(717\) 92.6124 3.45868
\(718\) −8.96435 −0.334547
\(719\) 25.3406 0.945044 0.472522 0.881319i \(-0.343344\pi\)
0.472522 + 0.881319i \(0.343344\pi\)
\(720\) 0 0
\(721\) −0.960452 −0.0357691
\(722\) 0.324814 0.0120883
\(723\) 72.4613 2.69486
\(724\) −23.7696 −0.883391
\(725\) 0 0
\(726\) −1.05343 −0.0390965
\(727\) −3.83993 −0.142415 −0.0712075 0.997462i \(-0.522685\pi\)
−0.0712075 + 0.997462i \(0.522685\pi\)
\(728\) −1.27117 −0.0471128
\(729\) 45.1542 1.67238
\(730\) 0 0
\(731\) 5.79932 0.214496
\(732\) 45.0195 1.66397
\(733\) −19.8037 −0.731466 −0.365733 0.930720i \(-0.619182\pi\)
−0.365733 + 0.930720i \(0.619182\pi\)
\(734\) −0.0334142 −0.00123334
\(735\) 0 0
\(736\) −16.7397 −0.617035
\(737\) 0.360331 0.0132730
\(738\) −20.3191 −0.747958
\(739\) −43.0878 −1.58501 −0.792506 0.609864i \(-0.791224\pi\)
−0.792506 + 0.609864i \(0.791224\pi\)
\(740\) 0 0
\(741\) −19.6320 −0.721200
\(742\) −0.543297 −0.0199451
\(743\) 16.0472 0.588714 0.294357 0.955696i \(-0.404895\pi\)
0.294357 + 0.955696i \(0.404895\pi\)
\(744\) 34.2817 1.25683
\(745\) 0 0
\(746\) −3.61371 −0.132307
\(747\) 23.7418 0.868667
\(748\) 1.65895 0.0606571
\(749\) −0.789622 −0.0288522
\(750\) 0 0
\(751\) −34.8124 −1.27032 −0.635161 0.772380i \(-0.719066\pi\)
−0.635161 + 0.772380i \(0.719066\pi\)
\(752\) 28.4776 1.03847
\(753\) −37.3440 −1.36089
\(754\) 1.98014 0.0721124
\(755\) 0 0
\(756\) −4.60853 −0.167611
\(757\) 31.5794 1.14777 0.573886 0.818935i \(-0.305435\pi\)
0.573886 + 0.818935i \(0.305435\pi\)
\(758\) −6.25164 −0.227070
\(759\) 14.9674 0.543281
\(760\) 0 0
\(761\) −26.0445 −0.944113 −0.472056 0.881568i \(-0.656488\pi\)
−0.472056 + 0.881568i \(0.656488\pi\)
\(762\) 16.8920 0.611934
\(763\) −2.33782 −0.0846346
\(764\) 27.9004 1.00940
\(765\) 0 0
\(766\) −2.77897 −0.100408
\(767\) −30.1485 −1.08860
\(768\) −26.6305 −0.960946
\(769\) −1.27892 −0.0461189 −0.0230595 0.999734i \(-0.507341\pi\)
−0.0230595 + 0.999734i \(0.507341\pi\)
\(770\) 0 0
\(771\) 34.2086 1.23199
\(772\) 22.2578 0.801077
\(773\) 43.3371 1.55873 0.779364 0.626571i \(-0.215542\pi\)
0.779364 + 0.626571i \(0.215542\pi\)
\(774\) −16.1730 −0.581326
\(775\) 0 0
\(776\) 13.4650 0.483364
\(777\) 3.78490 0.135782
\(778\) 5.43337 0.194796
\(779\) 8.32057 0.298115
\(780\) 0 0
\(781\) −7.74042 −0.276974
\(782\) 1.31264 0.0469401
\(783\) 14.7574 0.527388
\(784\) −23.5537 −0.841202
\(785\) 0 0
\(786\) −10.0081 −0.356978
\(787\) −15.9911 −0.570022 −0.285011 0.958524i \(-0.591997\pi\)
−0.285011 + 0.958524i \(0.591997\pi\)
\(788\) 24.7617 0.882100
\(789\) 65.8903 2.34576
\(790\) 0 0
\(791\) −2.75400 −0.0979210
\(792\) −9.51050 −0.337941
\(793\) −44.3535 −1.57504
\(794\) 12.0463 0.427508
\(795\) 0 0
\(796\) 31.7572 1.12561
\(797\) −28.8110 −1.02054 −0.510269 0.860015i \(-0.670454\pi\)
−0.510269 + 0.860015i \(0.670454\pi\)
\(798\) −0.174877 −0.00619057
\(799\) −7.38190 −0.261153
\(800\) 0 0
\(801\) 97.8919 3.45884
\(802\) −1.34988 −0.0476659
\(803\) 6.39400 0.225639
\(804\) 2.21395 0.0780799
\(805\) 0 0
\(806\) −16.4298 −0.578715
\(807\) 53.3904 1.87943
\(808\) −13.7591 −0.484044
\(809\) 47.1962 1.65933 0.829666 0.558261i \(-0.188531\pi\)
0.829666 + 0.558261i \(0.188531\pi\)
\(810\) 0 0
\(811\) 21.4242 0.752304 0.376152 0.926558i \(-0.377247\pi\)
0.376152 + 0.926558i \(0.377247\pi\)
\(812\) −0.316728 −0.0111150
\(813\) −75.3367 −2.64217
\(814\) 2.28345 0.0800349
\(815\) 0 0
\(816\) 9.59365 0.335845
\(817\) 6.62275 0.231701
\(818\) −6.39718 −0.223672
\(819\) 7.55502 0.263994
\(820\) 0 0
\(821\) 9.70806 0.338814 0.169407 0.985546i \(-0.445815\pi\)
0.169407 + 0.985546i \(0.445815\pi\)
\(822\) 14.8792 0.518971
\(823\) −41.7049 −1.45374 −0.726871 0.686774i \(-0.759026\pi\)
−0.726871 + 0.686774i \(0.759026\pi\)
\(824\) −7.31874 −0.254960
\(825\) 0 0
\(826\) −0.268555 −0.00934422
\(827\) 19.1359 0.665420 0.332710 0.943029i \(-0.392037\pi\)
0.332710 + 0.943029i \(0.392037\pi\)
\(828\) 65.7332 2.28439
\(829\) −53.7809 −1.86789 −0.933943 0.357421i \(-0.883656\pi\)
−0.933943 + 0.357421i \(0.883656\pi\)
\(830\) 0 0
\(831\) −57.5222 −1.99542
\(832\) 33.7656 1.17061
\(833\) 6.10553 0.211544
\(834\) −9.25158 −0.320356
\(835\) 0 0
\(836\) 1.89450 0.0655225
\(837\) −122.447 −4.23238
\(838\) −1.36111 −0.0470188
\(839\) −11.9762 −0.413464 −0.206732 0.978398i \(-0.566283\pi\)
−0.206732 + 0.978398i \(0.566283\pi\)
\(840\) 0 0
\(841\) −27.9858 −0.965027
\(842\) 7.32985 0.252603
\(843\) 83.3220 2.86976
\(844\) 36.8683 1.26906
\(845\) 0 0
\(846\) 20.5865 0.707777
\(847\) −0.166007 −0.00570405
\(848\) 34.0369 1.16883
\(849\) 16.7737 0.575672
\(850\) 0 0
\(851\) −32.4437 −1.11216
\(852\) −47.5587 −1.62934
\(853\) −36.1717 −1.23850 −0.619248 0.785196i \(-0.712562\pi\)
−0.619248 + 0.785196i \(0.712562\pi\)
\(854\) −0.395089 −0.0135197
\(855\) 0 0
\(856\) −6.01700 −0.205657
\(857\) 13.2431 0.452375 0.226187 0.974084i \(-0.427374\pi\)
0.226187 + 0.974084i \(0.427374\pi\)
\(858\) 6.37676 0.217699
\(859\) −18.5582 −0.633198 −0.316599 0.948560i \(-0.602541\pi\)
−0.316599 + 0.948560i \(0.602541\pi\)
\(860\) 0 0
\(861\) −4.47971 −0.152668
\(862\) −5.96111 −0.203036
\(863\) 5.72788 0.194979 0.0974896 0.995237i \(-0.468919\pi\)
0.0974896 + 0.995237i \(0.468919\pi\)
\(864\) −53.1518 −1.80826
\(865\) 0 0
\(866\) 0.578437 0.0196561
\(867\) 52.6473 1.78800
\(868\) 2.62799 0.0891997
\(869\) −0.309265 −0.0104911
\(870\) 0 0
\(871\) −2.18120 −0.0739070
\(872\) −17.8144 −0.603272
\(873\) −80.0270 −2.70850
\(874\) 1.49902 0.0507052
\(875\) 0 0
\(876\) 39.2860 1.32735
\(877\) −1.40932 −0.0475893 −0.0237947 0.999717i \(-0.507575\pi\)
−0.0237947 + 0.999717i \(0.507575\pi\)
\(878\) −8.71495 −0.294115
\(879\) 80.7017 2.72200
\(880\) 0 0
\(881\) 21.6242 0.728538 0.364269 0.931294i \(-0.381319\pi\)
0.364269 + 0.931294i \(0.381319\pi\)
\(882\) −17.0270 −0.573327
\(883\) −45.4323 −1.52892 −0.764460 0.644671i \(-0.776994\pi\)
−0.764460 + 0.644671i \(0.776994\pi\)
\(884\) −10.0421 −0.337753
\(885\) 0 0
\(886\) 5.01708 0.168552
\(887\) −35.7575 −1.20062 −0.600310 0.799767i \(-0.704956\pi\)
−0.600310 + 0.799767i \(0.704956\pi\)
\(888\) 28.8413 0.967851
\(889\) 2.66195 0.0892791
\(890\) 0 0
\(891\) 24.9695 0.836508
\(892\) 21.6205 0.723909
\(893\) −8.43004 −0.282101
\(894\) 6.91195 0.231170
\(895\) 0 0
\(896\) 1.50506 0.0502806
\(897\) −90.6022 −3.02512
\(898\) −4.79557 −0.160030
\(899\) −8.41536 −0.280668
\(900\) 0 0
\(901\) −8.82299 −0.293937
\(902\) −2.70264 −0.0899880
\(903\) −3.56562 −0.118657
\(904\) −20.9858 −0.697976
\(905\) 0 0
\(906\) 16.8122 0.558547
\(907\) −11.5725 −0.384257 −0.192129 0.981370i \(-0.561539\pi\)
−0.192129 + 0.981370i \(0.561539\pi\)
\(908\) 7.60593 0.252412
\(909\) 81.7753 2.71232
\(910\) 0 0
\(911\) 19.3672 0.641665 0.320832 0.947136i \(-0.396037\pi\)
0.320832 + 0.947136i \(0.396037\pi\)
\(912\) 10.9558 0.362784
\(913\) 3.15788 0.104511
\(914\) 5.26898 0.174282
\(915\) 0 0
\(916\) −49.8518 −1.64715
\(917\) −1.57715 −0.0520819
\(918\) 4.16789 0.137561
\(919\) 35.9029 1.18433 0.592163 0.805818i \(-0.298274\pi\)
0.592163 + 0.805818i \(0.298274\pi\)
\(920\) 0 0
\(921\) −4.44966 −0.146621
\(922\) 4.37733 0.144160
\(923\) 46.8552 1.54226
\(924\) −1.01998 −0.0335548
\(925\) 0 0
\(926\) 9.30464 0.305769
\(927\) 43.4979 1.42866
\(928\) −3.65294 −0.119914
\(929\) −26.0456 −0.854529 −0.427265 0.904127i \(-0.640523\pi\)
−0.427265 + 0.904127i \(0.640523\pi\)
\(930\) 0 0
\(931\) 6.97244 0.228513
\(932\) −27.8113 −0.910991
\(933\) −5.69289 −0.186377
\(934\) −5.54540 −0.181451
\(935\) 0 0
\(936\) 57.5700 1.88174
\(937\) −59.8785 −1.95615 −0.978073 0.208262i \(-0.933219\pi\)
−0.978073 + 0.208262i \(0.933219\pi\)
\(938\) −0.0194295 −0.000634395 0
\(939\) −60.2542 −1.96632
\(940\) 0 0
\(941\) 17.0743 0.556606 0.278303 0.960493i \(-0.410228\pi\)
0.278303 + 0.960493i \(0.410228\pi\)
\(942\) −14.5813 −0.475083
\(943\) 38.3996 1.25046
\(944\) 16.8247 0.547596
\(945\) 0 0
\(946\) −2.15116 −0.0699403
\(947\) −9.94311 −0.323108 −0.161554 0.986864i \(-0.551651\pi\)
−0.161554 + 0.986864i \(0.551651\pi\)
\(948\) −1.90019 −0.0617152
\(949\) −38.7049 −1.25641
\(950\) 0 0
\(951\) 13.0015 0.421602
\(952\) −0.183887 −0.00595980
\(953\) −29.7554 −0.963871 −0.481935 0.876207i \(-0.660066\pi\)
−0.481935 + 0.876207i \(0.660066\pi\)
\(954\) 24.6053 0.796627
\(955\) 0 0
\(956\) 54.0992 1.74969
\(957\) 3.26618 0.105581
\(958\) 9.20872 0.297520
\(959\) 2.34476 0.0757161
\(960\) 0 0
\(961\) 38.8247 1.25241
\(962\) −13.8224 −0.445654
\(963\) 35.7612 1.15239
\(964\) 42.3280 1.36329
\(965\) 0 0
\(966\) −0.807060 −0.0259667
\(967\) −21.9469 −0.705765 −0.352883 0.935668i \(-0.614798\pi\)
−0.352883 + 0.935668i \(0.614798\pi\)
\(968\) −1.26499 −0.0406582
\(969\) −2.83995 −0.0912323
\(970\) 0 0
\(971\) 14.5115 0.465695 0.232847 0.972513i \(-0.425196\pi\)
0.232847 + 0.972513i \(0.425196\pi\)
\(972\) 70.1341 2.24955
\(973\) −1.45792 −0.0467389
\(974\) 3.70442 0.118697
\(975\) 0 0
\(976\) 24.7519 0.792288
\(977\) 50.3307 1.61022 0.805111 0.593124i \(-0.202106\pi\)
0.805111 + 0.593124i \(0.202106\pi\)
\(978\) −0.709299 −0.0226809
\(979\) 13.0206 0.416139
\(980\) 0 0
\(981\) 105.877 3.38040
\(982\) −9.71695 −0.310080
\(983\) −9.71499 −0.309860 −0.154930 0.987925i \(-0.549515\pi\)
−0.154930 + 0.987925i \(0.549515\pi\)
\(984\) −34.1359 −1.08821
\(985\) 0 0
\(986\) 0.286445 0.00912227
\(987\) 4.53865 0.144467
\(988\) −11.4680 −0.364845
\(989\) 30.5641 0.971883
\(990\) 0 0
\(991\) 24.1420 0.766897 0.383449 0.923562i \(-0.374736\pi\)
0.383449 + 0.923562i \(0.374736\pi\)
\(992\) 30.3096 0.962330
\(993\) −20.7702 −0.659123
\(994\) 0.417373 0.0132383
\(995\) 0 0
\(996\) 19.4027 0.614797
\(997\) 5.23428 0.165771 0.0828856 0.996559i \(-0.473586\pi\)
0.0828856 + 0.996559i \(0.473586\pi\)
\(998\) −12.3185 −0.389936
\(999\) −103.015 −3.25925
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5225.2.a.s.1.10 15
5.4 even 2 5225.2.a.x.1.6 yes 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.s.1.10 15 1.1 even 1 trivial
5225.2.a.x.1.6 yes 15 5.4 even 2