Properties

Label 9025.2.a.bo.1.1
Level $9025$
Weight $2$
Character 9025.1
Self dual yes
Analytic conductor $72.065$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.0649878242\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1805)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.75660\) of defining polynomial
Character \(\chi\) \(=\) 9025.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.75660 q^{2} -0.0856374 q^{3} +1.08564 q^{4} +0.150431 q^{6} +1.86144 q^{7} +1.60617 q^{8} -2.99267 q^{9} +O(q^{10})\) \(q-1.75660 q^{2} -0.0856374 q^{3} +1.08564 q^{4} +0.150431 q^{6} +1.86144 q^{7} +1.60617 q^{8} -2.99267 q^{9} -1.22420 q^{11} -0.0929712 q^{12} +2.40734 q^{13} -3.26979 q^{14} -4.99267 q^{16} -6.05910 q^{17} +5.25691 q^{18} -0.159409 q^{21} +2.15043 q^{22} +7.20500 q^{23} -0.137548 q^{24} -4.22873 q^{26} +0.513197 q^{27} +2.02084 q^{28} +1.28446 q^{29} -5.87596 q^{31} +5.55777 q^{32} +0.104837 q^{33} +10.6434 q^{34} -3.24895 q^{36} +1.59883 q^{37} -0.206159 q^{39} -9.72553 q^{41} +0.280017 q^{42} +0.697495 q^{43} -1.32904 q^{44} -12.6563 q^{46} +11.6434 q^{47} +0.427559 q^{48} -3.53506 q^{49} +0.518886 q^{51} +2.61350 q^{52} -10.2614 q^{53} -0.901480 q^{54} +2.98978 q^{56} -2.25628 q^{58} +9.40836 q^{59} +3.73575 q^{61} +10.3217 q^{62} -5.57066 q^{63} +0.222557 q^{64} -0.184157 q^{66} -7.41188 q^{67} -6.57799 q^{68} -0.617018 q^{69} -8.51320 q^{71} -4.80672 q^{72} +6.21234 q^{73} -2.80851 q^{74} -2.27877 q^{77} +0.362138 q^{78} -7.39383 q^{79} +8.93405 q^{81} +17.0839 q^{82} -0.0135105 q^{83} -0.173060 q^{84} -1.22522 q^{86} -0.109998 q^{87} -1.96627 q^{88} -1.25527 q^{89} +4.48111 q^{91} +7.82202 q^{92} +0.503202 q^{93} -20.4528 q^{94} -0.475953 q^{96} -7.97800 q^{97} +6.20967 q^{98} +3.66363 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + q^{3} + 3 q^{4} - 7 q^{6} + 11 q^{7} + 6 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + q^{3} + 3 q^{4} - 7 q^{6} + 11 q^{7} + 6 q^{8} + 5 q^{9} - 16 q^{12} - 2 q^{13} + 11 q^{14} - 3 q^{16} + 7 q^{17} + 17 q^{18} - 2 q^{21} + q^{22} + 11 q^{23} - 13 q^{24} + 9 q^{26} - 14 q^{27} + 13 q^{28} + 15 q^{29} + q^{31} + 3 q^{32} + 12 q^{33} + 22 q^{34} + 16 q^{36} - 11 q^{37} - 29 q^{39} - 22 q^{41} - 19 q^{42} + 26 q^{43} - 12 q^{44} - 10 q^{46} + 26 q^{47} - 13 q^{48} + 13 q^{49} + 11 q^{51} + 27 q^{52} - 16 q^{53} - 25 q^{54} + 8 q^{56} + 3 q^{58} + 10 q^{59} + 2 q^{61} + 31 q^{62} + 17 q^{63} + 4 q^{64} + 22 q^{66} + 3 q^{67} - 4 q^{68} - 14 q^{69} - 18 q^{71} + 29 q^{72} + 24 q^{73} - 17 q^{74} + 6 q^{77} - 15 q^{78} - 30 q^{79} - 4 q^{81} + 13 q^{82} + 12 q^{83} - 52 q^{84} + 16 q^{86} + q^{87} - 23 q^{88} - 9 q^{89} + 9 q^{91} + 25 q^{92} - 7 q^{93} + 11 q^{94} - 6 q^{96} + 19 q^{97} + 48 q^{98} - 9 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.75660 −1.24210 −0.621051 0.783770i \(-0.713294\pi\)
−0.621051 + 0.783770i \(0.713294\pi\)
\(3\) −0.0856374 −0.0494428 −0.0247214 0.999694i \(-0.507870\pi\)
−0.0247214 + 0.999694i \(0.507870\pi\)
\(4\) 1.08564 0.542819
\(5\) 0 0
\(6\) 0.150431 0.0614130
\(7\) 1.86144 0.703557 0.351778 0.936083i \(-0.385577\pi\)
0.351778 + 0.936083i \(0.385577\pi\)
\(8\) 1.60617 0.567866
\(9\) −2.99267 −0.997555
\(10\) 0 0
\(11\) −1.22420 −0.369111 −0.184555 0.982822i \(-0.559084\pi\)
−0.184555 + 0.982822i \(0.559084\pi\)
\(12\) −0.0929712 −0.0268385
\(13\) 2.40734 0.667677 0.333838 0.942630i \(-0.391656\pi\)
0.333838 + 0.942630i \(0.391656\pi\)
\(14\) −3.26979 −0.873889
\(15\) 0 0
\(16\) −4.99267 −1.24817
\(17\) −6.05910 −1.46955 −0.734774 0.678312i \(-0.762712\pi\)
−0.734774 + 0.678312i \(0.762712\pi\)
\(18\) 5.25691 1.23907
\(19\) 0 0
\(20\) 0 0
\(21\) −0.159409 −0.0347858
\(22\) 2.15043 0.458473
\(23\) 7.20500 1.50235 0.751173 0.660105i \(-0.229488\pi\)
0.751173 + 0.660105i \(0.229488\pi\)
\(24\) −0.137548 −0.0280769
\(25\) 0 0
\(26\) −4.22873 −0.829323
\(27\) 0.513197 0.0987647
\(28\) 2.02084 0.381904
\(29\) 1.28446 0.238519 0.119259 0.992863i \(-0.461948\pi\)
0.119259 + 0.992863i \(0.461948\pi\)
\(30\) 0 0
\(31\) −5.87596 −1.05535 −0.527677 0.849445i \(-0.676937\pi\)
−0.527677 + 0.849445i \(0.676937\pi\)
\(32\) 5.55777 0.982485
\(33\) 0.104837 0.0182499
\(34\) 10.6434 1.82533
\(35\) 0 0
\(36\) −3.24895 −0.541492
\(37\) 1.59883 0.262847 0.131423 0.991326i \(-0.458045\pi\)
0.131423 + 0.991326i \(0.458045\pi\)
\(38\) 0 0
\(39\) −0.206159 −0.0330118
\(40\) 0 0
\(41\) −9.72553 −1.51887 −0.759436 0.650581i \(-0.774525\pi\)
−0.759436 + 0.650581i \(0.774525\pi\)
\(42\) 0.280017 0.0432075
\(43\) 0.697495 0.106367 0.0531835 0.998585i \(-0.483063\pi\)
0.0531835 + 0.998585i \(0.483063\pi\)
\(44\) −1.32904 −0.200360
\(45\) 0 0
\(46\) −12.6563 −1.86607
\(47\) 11.6434 1.69837 0.849183 0.528099i \(-0.177095\pi\)
0.849183 + 0.528099i \(0.177095\pi\)
\(48\) 0.427559 0.0617128
\(49\) −3.53506 −0.505008
\(50\) 0 0
\(51\) 0.518886 0.0726586
\(52\) 2.61350 0.362427
\(53\) −10.2614 −1.40952 −0.704759 0.709447i \(-0.748945\pi\)
−0.704759 + 0.709447i \(0.748945\pi\)
\(54\) −0.901480 −0.122676
\(55\) 0 0
\(56\) 2.98978 0.399526
\(57\) 0 0
\(58\) −2.25628 −0.296265
\(59\) 9.40836 1.22486 0.612432 0.790523i \(-0.290191\pi\)
0.612432 + 0.790523i \(0.290191\pi\)
\(60\) 0 0
\(61\) 3.73575 0.478314 0.239157 0.970981i \(-0.423129\pi\)
0.239157 + 0.970981i \(0.423129\pi\)
\(62\) 10.3217 1.31086
\(63\) −5.57066 −0.701837
\(64\) 0.222557 0.0278197
\(65\) 0 0
\(66\) −0.184157 −0.0226682
\(67\) −7.41188 −0.905505 −0.452752 0.891636i \(-0.649558\pi\)
−0.452752 + 0.891636i \(0.649558\pi\)
\(68\) −6.57799 −0.797698
\(69\) −0.617018 −0.0742802
\(70\) 0 0
\(71\) −8.51320 −1.01033 −0.505165 0.863023i \(-0.668568\pi\)
−0.505165 + 0.863023i \(0.668568\pi\)
\(72\) −4.80672 −0.566478
\(73\) 6.21234 0.727099 0.363549 0.931575i \(-0.381565\pi\)
0.363549 + 0.931575i \(0.381565\pi\)
\(74\) −2.80851 −0.326483
\(75\) 0 0
\(76\) 0 0
\(77\) −2.27877 −0.259690
\(78\) 0.362138 0.0410040
\(79\) −7.39383 −0.831871 −0.415936 0.909394i \(-0.636546\pi\)
−0.415936 + 0.909394i \(0.636546\pi\)
\(80\) 0 0
\(81\) 8.93405 0.992672
\(82\) 17.0839 1.88660
\(83\) −0.0135105 −0.00148297 −0.000741487 1.00000i \(-0.500236\pi\)
−0.000741487 1.00000i \(0.500236\pi\)
\(84\) −0.173060 −0.0188824
\(85\) 0 0
\(86\) −1.22522 −0.132119
\(87\) −0.109998 −0.0117930
\(88\) −1.96627 −0.209605
\(89\) −1.25527 −0.133058 −0.0665291 0.997784i \(-0.521193\pi\)
−0.0665291 + 0.997784i \(0.521193\pi\)
\(90\) 0 0
\(91\) 4.48111 0.469748
\(92\) 7.82202 0.815502
\(93\) 0.503202 0.0521796
\(94\) −20.4528 −2.10954
\(95\) 0 0
\(96\) −0.475953 −0.0485768
\(97\) −7.97800 −0.810043 −0.405022 0.914307i \(-0.632736\pi\)
−0.405022 + 0.914307i \(0.632736\pi\)
\(98\) 6.20967 0.627272
\(99\) 3.66363 0.368208
\(100\) 0 0
\(101\) 18.2028 1.81125 0.905625 0.424080i \(-0.139403\pi\)
0.905625 + 0.424080i \(0.139403\pi\)
\(102\) −0.911474 −0.0902494
\(103\) −3.41352 −0.336344 −0.168172 0.985758i \(-0.553786\pi\)
−0.168172 + 0.985758i \(0.553786\pi\)
\(104\) 3.86660 0.379151
\(105\) 0 0
\(106\) 18.0252 1.75077
\(107\) −1.49515 −0.144542 −0.0722710 0.997385i \(-0.523025\pi\)
−0.0722710 + 0.997385i \(0.523025\pi\)
\(108\) 0.557145 0.0536113
\(109\) 15.5780 1.49210 0.746050 0.665889i \(-0.231948\pi\)
0.746050 + 0.665889i \(0.231948\pi\)
\(110\) 0 0
\(111\) −0.136920 −0.0129959
\(112\) −9.29353 −0.878156
\(113\) 13.6479 1.28389 0.641945 0.766750i \(-0.278128\pi\)
0.641945 + 0.766750i \(0.278128\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.39446 0.129472
\(117\) −7.20437 −0.666045
\(118\) −16.5267 −1.52141
\(119\) −11.2786 −1.03391
\(120\) 0 0
\(121\) −9.50133 −0.863757
\(122\) −6.56222 −0.594115
\(123\) 0.832870 0.0750973
\(124\) −6.37916 −0.572866
\(125\) 0 0
\(126\) 9.78540 0.871753
\(127\) −19.2055 −1.70421 −0.852106 0.523370i \(-0.824675\pi\)
−0.852106 + 0.523370i \(0.824675\pi\)
\(128\) −11.5065 −1.01704
\(129\) −0.0597316 −0.00525908
\(130\) 0 0
\(131\) −19.3097 −1.68710 −0.843548 0.537054i \(-0.819537\pi\)
−0.843548 + 0.537054i \(0.819537\pi\)
\(132\) 0.113815 0.00990637
\(133\) 0 0
\(134\) 13.0197 1.12473
\(135\) 0 0
\(136\) −9.73194 −0.834507
\(137\) 14.6242 1.24943 0.624715 0.780853i \(-0.285215\pi\)
0.624715 + 0.780853i \(0.285215\pi\)
\(138\) 1.08385 0.0922636
\(139\) 1.39485 0.118310 0.0591548 0.998249i \(-0.481159\pi\)
0.0591548 + 0.998249i \(0.481159\pi\)
\(140\) 0 0
\(141\) −0.997112 −0.0839720
\(142\) 14.9543 1.25493
\(143\) −2.94707 −0.246447
\(144\) 14.9414 1.24512
\(145\) 0 0
\(146\) −10.9126 −0.903131
\(147\) 0.302733 0.0249690
\(148\) 1.73575 0.142678
\(149\) 14.5340 1.19068 0.595338 0.803476i \(-0.297018\pi\)
0.595338 + 0.803476i \(0.297018\pi\)
\(150\) 0 0
\(151\) 23.5773 1.91870 0.959348 0.282227i \(-0.0910733\pi\)
0.959348 + 0.282227i \(0.0910733\pi\)
\(152\) 0 0
\(153\) 18.1329 1.46596
\(154\) 4.00289 0.322562
\(155\) 0 0
\(156\) −0.223814 −0.0179194
\(157\) 17.0409 1.36001 0.680007 0.733206i \(-0.261977\pi\)
0.680007 + 0.733206i \(0.261977\pi\)
\(158\) 12.9880 1.03327
\(159\) 0.878764 0.0696905
\(160\) 0 0
\(161\) 13.4116 1.05699
\(162\) −15.6935 −1.23300
\(163\) 9.98080 0.781757 0.390878 0.920442i \(-0.372171\pi\)
0.390878 + 0.920442i \(0.372171\pi\)
\(164\) −10.5584 −0.824473
\(165\) 0 0
\(166\) 0.0237326 0.00184200
\(167\) 23.1483 1.79127 0.895633 0.444794i \(-0.146723\pi\)
0.895633 + 0.444794i \(0.146723\pi\)
\(168\) −0.256037 −0.0197537
\(169\) −7.20470 −0.554208
\(170\) 0 0
\(171\) 0 0
\(172\) 0.757226 0.0577380
\(173\) −12.4017 −0.942880 −0.471440 0.881898i \(-0.656266\pi\)
−0.471440 + 0.881898i \(0.656266\pi\)
\(174\) 0.193222 0.0146482
\(175\) 0 0
\(176\) 6.11203 0.460712
\(177\) −0.805708 −0.0605607
\(178\) 2.20500 0.165272
\(179\) 22.5640 1.68651 0.843254 0.537515i \(-0.180637\pi\)
0.843254 + 0.537515i \(0.180637\pi\)
\(180\) 0 0
\(181\) 7.13669 0.530466 0.265233 0.964184i \(-0.414551\pi\)
0.265233 + 0.964184i \(0.414551\pi\)
\(182\) −7.87152 −0.583476
\(183\) −0.319920 −0.0236492
\(184\) 11.5724 0.853132
\(185\) 0 0
\(186\) −0.883924 −0.0648125
\(187\) 7.41756 0.542426
\(188\) 12.6405 0.921905
\(189\) 0.955282 0.0694866
\(190\) 0 0
\(191\) −12.6479 −0.915173 −0.457587 0.889165i \(-0.651286\pi\)
−0.457587 + 0.889165i \(0.651286\pi\)
\(192\) −0.0190592 −0.00137548
\(193\) −3.82038 −0.274997 −0.137498 0.990502i \(-0.543906\pi\)
−0.137498 + 0.990502i \(0.543906\pi\)
\(194\) 14.0141 1.00616
\(195\) 0 0
\(196\) −3.83779 −0.274128
\(197\) −5.35606 −0.381603 −0.190802 0.981629i \(-0.561109\pi\)
−0.190802 + 0.981629i \(0.561109\pi\)
\(198\) −6.43552 −0.457353
\(199\) −7.19251 −0.509863 −0.254932 0.966959i \(-0.582053\pi\)
−0.254932 + 0.966959i \(0.582053\pi\)
\(200\) 0 0
\(201\) 0.634734 0.0447707
\(202\) −31.9751 −2.24976
\(203\) 2.39094 0.167811
\(204\) 0.563322 0.0394404
\(205\) 0 0
\(206\) 5.99618 0.417774
\(207\) −21.5622 −1.49867
\(208\) −12.0191 −0.833372
\(209\) 0 0
\(210\) 0 0
\(211\) −23.3668 −1.60863 −0.804317 0.594200i \(-0.797469\pi\)
−0.804317 + 0.594200i \(0.797469\pi\)
\(212\) −11.1402 −0.765113
\(213\) 0.729048 0.0499536
\(214\) 2.62638 0.179536
\(215\) 0 0
\(216\) 0.824280 0.0560851
\(217\) −10.9377 −0.742501
\(218\) −27.3643 −1.85334
\(219\) −0.532008 −0.0359498
\(220\) 0 0
\(221\) −14.5863 −0.981183
\(222\) 0.240513 0.0161422
\(223\) −4.28166 −0.286721 −0.143361 0.989671i \(-0.545791\pi\)
−0.143361 + 0.989671i \(0.545791\pi\)
\(224\) 10.3454 0.691234
\(225\) 0 0
\(226\) −23.9740 −1.59472
\(227\) 6.60883 0.438643 0.219322 0.975653i \(-0.429616\pi\)
0.219322 + 0.975653i \(0.429616\pi\)
\(228\) 0 0
\(229\) −23.3277 −1.54154 −0.770770 0.637113i \(-0.780128\pi\)
−0.770770 + 0.637113i \(0.780128\pi\)
\(230\) 0 0
\(231\) 0.195148 0.0128398
\(232\) 2.06306 0.135447
\(233\) 17.0372 1.11615 0.558073 0.829792i \(-0.311541\pi\)
0.558073 + 0.829792i \(0.311541\pi\)
\(234\) 12.6552 0.827296
\(235\) 0 0
\(236\) 10.2141 0.664879
\(237\) 0.633189 0.0411300
\(238\) 19.8120 1.28422
\(239\) 10.4194 0.673973 0.336986 0.941509i \(-0.390592\pi\)
0.336986 + 0.941509i \(0.390592\pi\)
\(240\) 0 0
\(241\) −3.32069 −0.213904 −0.106952 0.994264i \(-0.534109\pi\)
−0.106952 + 0.994264i \(0.534109\pi\)
\(242\) 16.6900 1.07288
\(243\) −2.30468 −0.147845
\(244\) 4.05567 0.259638
\(245\) 0 0
\(246\) −1.46302 −0.0932786
\(247\) 0 0
\(248\) −9.43778 −0.599300
\(249\) 0.00115701 7.33223e−5 0
\(250\) 0 0
\(251\) −25.7802 −1.62723 −0.813616 0.581403i \(-0.802504\pi\)
−0.813616 + 0.581403i \(0.802504\pi\)
\(252\) −6.04771 −0.380970
\(253\) −8.82038 −0.554532
\(254\) 33.7363 2.11681
\(255\) 0 0
\(256\) 19.7672 1.23545
\(257\) −5.40015 −0.336852 −0.168426 0.985714i \(-0.553868\pi\)
−0.168426 + 0.985714i \(0.553868\pi\)
\(258\) 0.104925 0.00653231
\(259\) 2.97613 0.184928
\(260\) 0 0
\(261\) −3.84397 −0.237936
\(262\) 33.9194 2.09555
\(263\) −8.76997 −0.540779 −0.270390 0.962751i \(-0.587153\pi\)
−0.270390 + 0.962751i \(0.587153\pi\)
\(264\) 0.168387 0.0103635
\(265\) 0 0
\(266\) 0 0
\(267\) 0.107498 0.00657877
\(268\) −8.04661 −0.491525
\(269\) 19.3500 1.17979 0.589894 0.807481i \(-0.299169\pi\)
0.589894 + 0.807481i \(0.299169\pi\)
\(270\) 0 0
\(271\) 8.95871 0.544203 0.272101 0.962269i \(-0.412281\pi\)
0.272101 + 0.962269i \(0.412281\pi\)
\(272\) 30.2511 1.83424
\(273\) −0.383751 −0.0232257
\(274\) −25.6889 −1.55192
\(275\) 0 0
\(276\) −0.669858 −0.0403207
\(277\) 10.4586 0.628398 0.314199 0.949357i \(-0.398264\pi\)
0.314199 + 0.949357i \(0.398264\pi\)
\(278\) −2.45019 −0.146953
\(279\) 17.5848 1.05277
\(280\) 0 0
\(281\) −10.9107 −0.650878 −0.325439 0.945563i \(-0.605512\pi\)
−0.325439 + 0.945563i \(0.605512\pi\)
\(282\) 1.75152 0.104302
\(283\) 0.952763 0.0566359 0.0283179 0.999599i \(-0.490985\pi\)
0.0283179 + 0.999599i \(0.490985\pi\)
\(284\) −9.24224 −0.548426
\(285\) 0 0
\(286\) 5.17682 0.306112
\(287\) −18.1035 −1.06861
\(288\) −16.6326 −0.980083
\(289\) 19.7127 1.15957
\(290\) 0 0
\(291\) 0.683215 0.0400508
\(292\) 6.74434 0.394683
\(293\) 13.4219 0.784114 0.392057 0.919941i \(-0.371764\pi\)
0.392057 + 0.919941i \(0.371764\pi\)
\(294\) −0.531781 −0.0310141
\(295\) 0 0
\(296\) 2.56800 0.149262
\(297\) −0.628256 −0.0364551
\(298\) −25.5305 −1.47894
\(299\) 17.3449 1.00308
\(300\) 0 0
\(301\) 1.29834 0.0748352
\(302\) −41.4159 −2.38322
\(303\) −1.55884 −0.0895532
\(304\) 0 0
\(305\) 0 0
\(306\) −31.8522 −1.82087
\(307\) −18.8433 −1.07544 −0.537721 0.843123i \(-0.680714\pi\)
−0.537721 + 0.843123i \(0.680714\pi\)
\(308\) −2.47392 −0.140965
\(309\) 0.292325 0.0166298
\(310\) 0 0
\(311\) −0.912189 −0.0517255 −0.0258628 0.999666i \(-0.508233\pi\)
−0.0258628 + 0.999666i \(0.508233\pi\)
\(312\) −0.331125 −0.0187463
\(313\) 18.7665 1.06074 0.530371 0.847765i \(-0.322053\pi\)
0.530371 + 0.847765i \(0.322053\pi\)
\(314\) −29.9340 −1.68928
\(315\) 0 0
\(316\) −8.02702 −0.451555
\(317\) 8.85956 0.497603 0.248801 0.968555i \(-0.419963\pi\)
0.248801 + 0.968555i \(0.419963\pi\)
\(318\) −1.54363 −0.0865627
\(319\) −1.57244 −0.0880398
\(320\) 0 0
\(321\) 0.128041 0.00714656
\(322\) −23.5589 −1.31288
\(323\) 0 0
\(324\) 9.69914 0.538841
\(325\) 0 0
\(326\) −17.5323 −0.971022
\(327\) −1.33406 −0.0737736
\(328\) −15.6208 −0.862516
\(329\) 21.6735 1.19490
\(330\) 0 0
\(331\) −3.96044 −0.217686 −0.108843 0.994059i \(-0.534715\pi\)
−0.108843 + 0.994059i \(0.534715\pi\)
\(332\) −0.0146675 −0.000804986 0
\(333\) −4.78478 −0.262204
\(334\) −40.6622 −2.22494
\(335\) 0 0
\(336\) 0.795874 0.0434185
\(337\) −5.79148 −0.315482 −0.157741 0.987481i \(-0.550421\pi\)
−0.157741 + 0.987481i \(0.550421\pi\)
\(338\) 12.6558 0.688383
\(339\) −1.16877 −0.0634791
\(340\) 0 0
\(341\) 7.19336 0.389542
\(342\) 0 0
\(343\) −19.6103 −1.05886
\(344\) 1.12029 0.0604022
\(345\) 0 0
\(346\) 21.7847 1.17115
\(347\) −14.4126 −0.773708 −0.386854 0.922141i \(-0.626438\pi\)
−0.386854 + 0.922141i \(0.626438\pi\)
\(348\) −0.119418 −0.00640148
\(349\) 9.11150 0.487727 0.243864 0.969810i \(-0.421585\pi\)
0.243864 + 0.969810i \(0.421585\pi\)
\(350\) 0 0
\(351\) 1.23544 0.0659429
\(352\) −6.80384 −0.362646
\(353\) 3.59377 0.191277 0.0956386 0.995416i \(-0.469511\pi\)
0.0956386 + 0.995416i \(0.469511\pi\)
\(354\) 1.41530 0.0752226
\(355\) 0 0
\(356\) −1.36277 −0.0722265
\(357\) 0.965873 0.0511194
\(358\) −39.6358 −2.09482
\(359\) −10.7465 −0.567177 −0.283588 0.958946i \(-0.591525\pi\)
−0.283588 + 0.958946i \(0.591525\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) −12.5363 −0.658893
\(363\) 0.813669 0.0427066
\(364\) 4.86487 0.254988
\(365\) 0 0
\(366\) 0.561972 0.0293747
\(367\) −1.68713 −0.0880676 −0.0440338 0.999030i \(-0.514021\pi\)
−0.0440338 + 0.999030i \(0.514021\pi\)
\(368\) −35.9722 −1.87518
\(369\) 29.1053 1.51516
\(370\) 0 0
\(371\) −19.1010 −0.991676
\(372\) 0.546295 0.0283241
\(373\) 34.2844 1.77518 0.887590 0.460635i \(-0.152378\pi\)
0.887590 + 0.460635i \(0.152378\pi\)
\(374\) −13.0297 −0.673749
\(375\) 0 0
\(376\) 18.7013 0.964444
\(377\) 3.09214 0.159253
\(378\) −1.67805 −0.0863094
\(379\) −24.7637 −1.27202 −0.636012 0.771679i \(-0.719417\pi\)
−0.636012 + 0.771679i \(0.719417\pi\)
\(380\) 0 0
\(381\) 1.64471 0.0842610
\(382\) 22.2174 1.13674
\(383\) 17.2980 0.883885 0.441943 0.897043i \(-0.354290\pi\)
0.441943 + 0.897043i \(0.354290\pi\)
\(384\) 0.985386 0.0502853
\(385\) 0 0
\(386\) 6.71086 0.341574
\(387\) −2.08737 −0.106107
\(388\) −8.66121 −0.439707
\(389\) 31.5131 1.59778 0.798888 0.601480i \(-0.205422\pi\)
0.798888 + 0.601480i \(0.205422\pi\)
\(390\) 0 0
\(391\) −43.6559 −2.20777
\(392\) −5.67789 −0.286777
\(393\) 1.65363 0.0834147
\(394\) 9.40845 0.473991
\(395\) 0 0
\(396\) 3.97737 0.199870
\(397\) −27.9014 −1.40033 −0.700166 0.713980i \(-0.746891\pi\)
−0.700166 + 0.713980i \(0.746891\pi\)
\(398\) 12.6343 0.633303
\(399\) 0 0
\(400\) 0 0
\(401\) 4.73460 0.236434 0.118217 0.992988i \(-0.462282\pi\)
0.118217 + 0.992988i \(0.462282\pi\)
\(402\) −1.11497 −0.0556098
\(403\) −14.1455 −0.704635
\(404\) 19.7617 0.983180
\(405\) 0 0
\(406\) −4.19993 −0.208439
\(407\) −1.95730 −0.0970195
\(408\) 0.833418 0.0412603
\(409\) −8.34645 −0.412706 −0.206353 0.978478i \(-0.566159\pi\)
−0.206353 + 0.978478i \(0.566159\pi\)
\(410\) 0 0
\(411\) −1.25238 −0.0617753
\(412\) −3.70584 −0.182574
\(413\) 17.5131 0.861761
\(414\) 37.8761 1.86151
\(415\) 0 0
\(416\) 13.3795 0.655982
\(417\) −0.119451 −0.00584955
\(418\) 0 0
\(419\) 21.5120 1.05093 0.525466 0.850815i \(-0.323891\pi\)
0.525466 + 0.850815i \(0.323891\pi\)
\(420\) 0 0
\(421\) 30.6789 1.49520 0.747598 0.664151i \(-0.231207\pi\)
0.747598 + 0.664151i \(0.231207\pi\)
\(422\) 41.0460 1.99809
\(423\) −34.8448 −1.69421
\(424\) −16.4816 −0.800417
\(425\) 0 0
\(426\) −1.28064 −0.0620474
\(427\) 6.95387 0.336521
\(428\) −1.62319 −0.0784601
\(429\) 0.252380 0.0121850
\(430\) 0 0
\(431\) −25.5833 −1.23230 −0.616152 0.787627i \(-0.711309\pi\)
−0.616152 + 0.787627i \(0.711309\pi\)
\(432\) −2.56222 −0.123275
\(433\) −31.9328 −1.53459 −0.767295 0.641294i \(-0.778398\pi\)
−0.767295 + 0.641294i \(0.778398\pi\)
\(434\) 19.2132 0.922263
\(435\) 0 0
\(436\) 16.9120 0.809940
\(437\) 0 0
\(438\) 0.934525 0.0446533
\(439\) 9.45544 0.451283 0.225642 0.974210i \(-0.427552\pi\)
0.225642 + 0.974210i \(0.427552\pi\)
\(440\) 0 0
\(441\) 10.5792 0.503774
\(442\) 25.6223 1.21873
\(443\) −9.27713 −0.440770 −0.220385 0.975413i \(-0.570731\pi\)
−0.220385 + 0.975413i \(0.570731\pi\)
\(444\) −0.148645 −0.00705440
\(445\) 0 0
\(446\) 7.52116 0.356137
\(447\) −1.24466 −0.0588703
\(448\) 0.414276 0.0195727
\(449\) −27.6454 −1.30467 −0.652335 0.757931i \(-0.726210\pi\)
−0.652335 + 0.757931i \(0.726210\pi\)
\(450\) 0 0
\(451\) 11.9060 0.560632
\(452\) 14.8167 0.696920
\(453\) −2.01910 −0.0948656
\(454\) −11.6091 −0.544840
\(455\) 0 0
\(456\) 0 0
\(457\) 6.99174 0.327060 0.163530 0.986538i \(-0.447712\pi\)
0.163530 + 0.986538i \(0.447712\pi\)
\(458\) 40.9775 1.91475
\(459\) −3.10951 −0.145140
\(460\) 0 0
\(461\) 15.2246 0.709080 0.354540 0.935041i \(-0.384638\pi\)
0.354540 + 0.935041i \(0.384638\pi\)
\(462\) −0.342797 −0.0159484
\(463\) 27.7350 1.28895 0.644477 0.764623i \(-0.277075\pi\)
0.644477 + 0.764623i \(0.277075\pi\)
\(464\) −6.41289 −0.297711
\(465\) 0 0
\(466\) −29.9276 −1.38637
\(467\) 38.3385 1.77409 0.887046 0.461681i \(-0.152754\pi\)
0.887046 + 0.461681i \(0.152754\pi\)
\(468\) −7.82134 −0.361541
\(469\) −13.7967 −0.637074
\(470\) 0 0
\(471\) −1.45934 −0.0672429
\(472\) 15.1114 0.695559
\(473\) −0.853874 −0.0392612
\(474\) −1.11226 −0.0510877
\(475\) 0 0
\(476\) −12.2445 −0.561226
\(477\) 30.7091 1.40607
\(478\) −18.3026 −0.837144
\(479\) −10.5137 −0.480382 −0.240191 0.970726i \(-0.577210\pi\)
−0.240191 + 0.970726i \(0.577210\pi\)
\(480\) 0 0
\(481\) 3.84894 0.175497
\(482\) 5.83312 0.265691
\(483\) −1.14854 −0.0522603
\(484\) −10.3150 −0.468864
\(485\) 0 0
\(486\) 4.04839 0.183639
\(487\) 14.1296 0.640273 0.320136 0.947371i \(-0.396271\pi\)
0.320136 + 0.947371i \(0.396271\pi\)
\(488\) 6.00025 0.271618
\(489\) −0.854730 −0.0386522
\(490\) 0 0
\(491\) 18.7427 0.845847 0.422924 0.906165i \(-0.361004\pi\)
0.422924 + 0.906165i \(0.361004\pi\)
\(492\) 0.904194 0.0407642
\(493\) −7.78269 −0.350515
\(494\) 0 0
\(495\) 0 0
\(496\) 29.3367 1.31726
\(497\) −15.8468 −0.710825
\(498\) −0.00203240 −9.10739e−5 0
\(499\) 16.2646 0.728103 0.364052 0.931379i \(-0.381393\pi\)
0.364052 + 0.931379i \(0.381393\pi\)
\(500\) 0 0
\(501\) −1.98236 −0.0885652
\(502\) 45.2854 2.02119
\(503\) 26.1541 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(504\) −8.94741 −0.398549
\(505\) 0 0
\(506\) 15.4939 0.688786
\(507\) 0.616992 0.0274016
\(508\) −20.8502 −0.925078
\(509\) −17.3725 −0.770025 −0.385012 0.922911i \(-0.625803\pi\)
−0.385012 + 0.922911i \(0.625803\pi\)
\(510\) 0 0
\(511\) 11.5639 0.511555
\(512\) −11.7100 −0.517513
\(513\) 0 0
\(514\) 9.48589 0.418405
\(515\) 0 0
\(516\) −0.0648469 −0.00285473
\(517\) −14.2539 −0.626885
\(518\) −5.22786 −0.229699
\(519\) 1.06205 0.0466186
\(520\) 0 0
\(521\) −33.7615 −1.47912 −0.739559 0.673092i \(-0.764966\pi\)
−0.739559 + 0.673092i \(0.764966\pi\)
\(522\) 6.75231 0.295540
\(523\) 28.7361 1.25654 0.628271 0.777995i \(-0.283763\pi\)
0.628271 + 0.777995i \(0.283763\pi\)
\(524\) −20.9633 −0.915787
\(525\) 0 0
\(526\) 15.4053 0.671703
\(527\) 35.6031 1.55089
\(528\) −0.523419 −0.0227789
\(529\) 28.9120 1.25705
\(530\) 0 0
\(531\) −28.1561 −1.22187
\(532\) 0 0
\(533\) −23.4127 −1.01412
\(534\) −0.188831 −0.00817150
\(535\) 0 0
\(536\) −11.9047 −0.514205
\(537\) −1.93232 −0.0833857
\(538\) −33.9901 −1.46542
\(539\) 4.32762 0.186404
\(540\) 0 0
\(541\) −17.7661 −0.763823 −0.381911 0.924199i \(-0.624734\pi\)
−0.381911 + 0.924199i \(0.624734\pi\)
\(542\) −15.7369 −0.675956
\(543\) −0.611168 −0.0262277
\(544\) −33.6751 −1.44381
\(545\) 0 0
\(546\) 0.674096 0.0288487
\(547\) 10.4531 0.446941 0.223471 0.974711i \(-0.428261\pi\)
0.223471 + 0.974711i \(0.428261\pi\)
\(548\) 15.8766 0.678214
\(549\) −11.1799 −0.477145
\(550\) 0 0
\(551\) 0 0
\(552\) −0.991034 −0.0421812
\(553\) −13.7631 −0.585268
\(554\) −18.3716 −0.780535
\(555\) 0 0
\(556\) 1.51430 0.0642206
\(557\) 23.5768 0.998982 0.499491 0.866319i \(-0.333520\pi\)
0.499491 + 0.866319i \(0.333520\pi\)
\(558\) −30.8894 −1.30765
\(559\) 1.67911 0.0710187
\(560\) 0 0
\(561\) −0.635221 −0.0268191
\(562\) 19.1657 0.808457
\(563\) 27.8106 1.17208 0.586038 0.810283i \(-0.300687\pi\)
0.586038 + 0.810283i \(0.300687\pi\)
\(564\) −1.08250 −0.0455815
\(565\) 0 0
\(566\) −1.67362 −0.0703475
\(567\) 16.6302 0.698401
\(568\) −13.6736 −0.573732
\(569\) 9.72405 0.407653 0.203827 0.979007i \(-0.434662\pi\)
0.203827 + 0.979007i \(0.434662\pi\)
\(570\) 0 0
\(571\) −7.74753 −0.324224 −0.162112 0.986772i \(-0.551831\pi\)
−0.162112 + 0.986772i \(0.551831\pi\)
\(572\) −3.19945 −0.133776
\(573\) 1.08314 0.0452487
\(574\) 31.8005 1.32733
\(575\) 0 0
\(576\) −0.666040 −0.0277517
\(577\) 21.3536 0.888964 0.444482 0.895788i \(-0.353388\pi\)
0.444482 + 0.895788i \(0.353388\pi\)
\(578\) −34.6274 −1.44031
\(579\) 0.327167 0.0135966
\(580\) 0 0
\(581\) −0.0251490 −0.00104336
\(582\) −1.20013 −0.0497472
\(583\) 12.5621 0.520268
\(584\) 9.97805 0.412895
\(585\) 0 0
\(586\) −23.5768 −0.973950
\(587\) 34.7452 1.43409 0.717043 0.697029i \(-0.245495\pi\)
0.717043 + 0.697029i \(0.245495\pi\)
\(588\) 0.328658 0.0135536
\(589\) 0 0
\(590\) 0 0
\(591\) 0.458679 0.0188675
\(592\) −7.98244 −0.328076
\(593\) 0.557914 0.0229108 0.0114554 0.999934i \(-0.496354\pi\)
0.0114554 + 0.999934i \(0.496354\pi\)
\(594\) 1.10359 0.0452810
\(595\) 0 0
\(596\) 15.7787 0.646321
\(597\) 0.615948 0.0252091
\(598\) −30.4680 −1.24593
\(599\) −10.1872 −0.416237 −0.208119 0.978104i \(-0.566734\pi\)
−0.208119 + 0.978104i \(0.566734\pi\)
\(600\) 0 0
\(601\) −1.46619 −0.0598070 −0.0299035 0.999553i \(-0.509520\pi\)
−0.0299035 + 0.999553i \(0.509520\pi\)
\(602\) −2.28066 −0.0929529
\(603\) 22.1813 0.903291
\(604\) 25.5964 1.04150
\(605\) 0 0
\(606\) 2.73826 0.111234
\(607\) −37.3950 −1.51782 −0.758909 0.651197i \(-0.774267\pi\)
−0.758909 + 0.651197i \(0.774267\pi\)
\(608\) 0 0
\(609\) −0.204754 −0.00829706
\(610\) 0 0
\(611\) 28.0297 1.13396
\(612\) 19.6857 0.795748
\(613\) 29.0310 1.17255 0.586276 0.810111i \(-0.300593\pi\)
0.586276 + 0.810111i \(0.300593\pi\)
\(614\) 33.1000 1.33581
\(615\) 0 0
\(616\) −3.66009 −0.147469
\(617\) 0.259474 0.0104460 0.00522301 0.999986i \(-0.498337\pi\)
0.00522301 + 0.999986i \(0.498337\pi\)
\(618\) −0.513498 −0.0206559
\(619\) 23.2107 0.932917 0.466459 0.884543i \(-0.345530\pi\)
0.466459 + 0.884543i \(0.345530\pi\)
\(620\) 0 0
\(621\) 3.69758 0.148379
\(622\) 1.60235 0.0642484
\(623\) −2.33660 −0.0936139
\(624\) 1.02928 0.0412042
\(625\) 0 0
\(626\) −32.9651 −1.31755
\(627\) 0 0
\(628\) 18.5003 0.738241
\(629\) −9.68750 −0.386266
\(630\) 0 0
\(631\) −1.74339 −0.0694032 −0.0347016 0.999398i \(-0.511048\pi\)
−0.0347016 + 0.999398i \(0.511048\pi\)
\(632\) −11.8757 −0.472391
\(633\) 2.00107 0.0795354
\(634\) −15.5627 −0.618074
\(635\) 0 0
\(636\) 0.954019 0.0378293
\(637\) −8.51009 −0.337182
\(638\) 2.76215 0.109354
\(639\) 25.4772 1.00786
\(640\) 0 0
\(641\) 9.38800 0.370804 0.185402 0.982663i \(-0.440641\pi\)
0.185402 + 0.982663i \(0.440641\pi\)
\(642\) −0.224917 −0.00887676
\(643\) 35.4801 1.39920 0.699600 0.714535i \(-0.253362\pi\)
0.699600 + 0.714535i \(0.253362\pi\)
\(644\) 14.5602 0.573752
\(645\) 0 0
\(646\) 0 0
\(647\) 9.61436 0.377979 0.188990 0.981979i \(-0.439479\pi\)
0.188990 + 0.981979i \(0.439479\pi\)
\(648\) 14.3496 0.563705
\(649\) −11.5177 −0.452110
\(650\) 0 0
\(651\) 0.936679 0.0367113
\(652\) 10.8355 0.424352
\(653\) −10.9351 −0.427922 −0.213961 0.976842i \(-0.568637\pi\)
−0.213961 + 0.976842i \(0.568637\pi\)
\(654\) 2.34341 0.0916344
\(655\) 0 0
\(656\) 48.5563 1.89581
\(657\) −18.5914 −0.725321
\(658\) −38.0716 −1.48418
\(659\) 30.2427 1.17809 0.589045 0.808100i \(-0.299504\pi\)
0.589045 + 0.808100i \(0.299504\pi\)
\(660\) 0 0
\(661\) −17.8243 −0.693284 −0.346642 0.937997i \(-0.612678\pi\)
−0.346642 + 0.937997i \(0.612678\pi\)
\(662\) 6.95691 0.270388
\(663\) 1.24914 0.0485124
\(664\) −0.0217002 −0.000842130 0
\(665\) 0 0
\(666\) 8.40493 0.325684
\(667\) 9.25455 0.358338
\(668\) 25.1306 0.972333
\(669\) 0.366670 0.0141763
\(670\) 0 0
\(671\) −4.57332 −0.176551
\(672\) −0.885957 −0.0341765
\(673\) 16.6638 0.642341 0.321171 0.947021i \(-0.395924\pi\)
0.321171 + 0.947021i \(0.395924\pi\)
\(674\) 10.1733 0.391861
\(675\) 0 0
\(676\) −7.82169 −0.300834
\(677\) −36.5254 −1.40379 −0.701893 0.712283i \(-0.747661\pi\)
−0.701893 + 0.712283i \(0.747661\pi\)
\(678\) 2.05307 0.0788476
\(679\) −14.8505 −0.569911
\(680\) 0 0
\(681\) −0.565963 −0.0216878
\(682\) −12.6358 −0.483852
\(683\) 34.2305 1.30979 0.654896 0.755719i \(-0.272712\pi\)
0.654896 + 0.755719i \(0.272712\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 34.4475 1.31521
\(687\) 1.99773 0.0762181
\(688\) −3.48236 −0.132764
\(689\) −24.7028 −0.941102
\(690\) 0 0
\(691\) 4.08362 0.155348 0.0776742 0.996979i \(-0.475251\pi\)
0.0776742 + 0.996979i \(0.475251\pi\)
\(692\) −13.4637 −0.511813
\(693\) 6.81961 0.259055
\(694\) 25.3171 0.961025
\(695\) 0 0
\(696\) −0.176675 −0.00669686
\(697\) 58.9280 2.23206
\(698\) −16.0052 −0.605808
\(699\) −1.45903 −0.0551854
\(700\) 0 0
\(701\) 25.6862 0.970154 0.485077 0.874471i \(-0.338791\pi\)
0.485077 + 0.874471i \(0.338791\pi\)
\(702\) −2.17017 −0.0819078
\(703\) 0 0
\(704\) −0.272455 −0.0102685
\(705\) 0 0
\(706\) −6.31281 −0.237586
\(707\) 33.8834 1.27432
\(708\) −0.874706 −0.0328735
\(709\) −19.5953 −0.735918 −0.367959 0.929842i \(-0.619943\pi\)
−0.367959 + 0.929842i \(0.619943\pi\)
\(710\) 0 0
\(711\) 22.1273 0.829838
\(712\) −2.01617 −0.0755592
\(713\) −42.3363 −1.58551
\(714\) −1.69665 −0.0634956
\(715\) 0 0
\(716\) 24.4963 0.915469
\(717\) −0.892288 −0.0333231
\(718\) 18.8772 0.704492
\(719\) −1.29094 −0.0481439 −0.0240720 0.999710i \(-0.507663\pi\)
−0.0240720 + 0.999710i \(0.507663\pi\)
\(720\) 0 0
\(721\) −6.35405 −0.236637
\(722\) 0 0
\(723\) 0.284375 0.0105760
\(724\) 7.74786 0.287947
\(725\) 0 0
\(726\) −1.42929 −0.0530459
\(727\) 3.54012 0.131296 0.0656479 0.997843i \(-0.479089\pi\)
0.0656479 + 0.997843i \(0.479089\pi\)
\(728\) 7.19742 0.266754
\(729\) −26.6048 −0.985362
\(730\) 0 0
\(731\) −4.22619 −0.156311
\(732\) −0.347318 −0.0128372
\(733\) −17.2078 −0.635585 −0.317792 0.948160i \(-0.602941\pi\)
−0.317792 + 0.948160i \(0.602941\pi\)
\(734\) 2.96361 0.109389
\(735\) 0 0
\(736\) 40.0438 1.47603
\(737\) 9.07363 0.334231
\(738\) −51.1263 −1.88198
\(739\) −40.0918 −1.47480 −0.737401 0.675456i \(-0.763947\pi\)
−0.737401 + 0.675456i \(0.763947\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 33.5528 1.23176
\(743\) −20.2637 −0.743403 −0.371702 0.928352i \(-0.621226\pi\)
−0.371702 + 0.928352i \(0.621226\pi\)
\(744\) 0.808227 0.0296310
\(745\) 0 0
\(746\) −60.2239 −2.20495
\(747\) 0.0404325 0.00147935
\(748\) 8.05279 0.294439
\(749\) −2.78313 −0.101693
\(750\) 0 0
\(751\) −38.7484 −1.41395 −0.706975 0.707238i \(-0.749941\pi\)
−0.706975 + 0.707238i \(0.749941\pi\)
\(752\) −58.1317 −2.11984
\(753\) 2.20775 0.0804548
\(754\) −5.43165 −0.197809
\(755\) 0 0
\(756\) 1.03709 0.0377186
\(757\) −20.8068 −0.756237 −0.378118 0.925757i \(-0.623429\pi\)
−0.378118 + 0.925757i \(0.623429\pi\)
\(758\) 43.4998 1.57998
\(759\) 0.755354 0.0274176
\(760\) 0 0
\(761\) 48.6822 1.76473 0.882363 0.470569i \(-0.155951\pi\)
0.882363 + 0.470569i \(0.155951\pi\)
\(762\) −2.88909 −0.104661
\(763\) 28.9974 1.04978
\(764\) −13.7311 −0.496773
\(765\) 0 0
\(766\) −30.3856 −1.09788
\(767\) 22.6491 0.817813
\(768\) −1.69281 −0.0610840
\(769\) 20.0077 0.721495 0.360748 0.932663i \(-0.382522\pi\)
0.360748 + 0.932663i \(0.382522\pi\)
\(770\) 0 0
\(771\) 0.462455 0.0166549
\(772\) −4.14754 −0.149273
\(773\) −1.59040 −0.0572026 −0.0286013 0.999591i \(-0.509105\pi\)
−0.0286013 + 0.999591i \(0.509105\pi\)
\(774\) 3.66667 0.131796
\(775\) 0 0
\(776\) −12.8140 −0.459996
\(777\) −0.254868 −0.00914333
\(778\) −55.3558 −1.98460
\(779\) 0 0
\(780\) 0 0
\(781\) 10.4219 0.372924
\(782\) 76.6858 2.74228
\(783\) 0.659182 0.0235572
\(784\) 17.6494 0.630334
\(785\) 0 0
\(786\) −2.90477 −0.103610
\(787\) −2.87913 −0.102630 −0.0513150 0.998683i \(-0.516341\pi\)
−0.0513150 + 0.998683i \(0.516341\pi\)
\(788\) −5.81474 −0.207142
\(789\) 0.751037 0.0267376
\(790\) 0 0
\(791\) 25.4048 0.903290
\(792\) 5.88440 0.209093
\(793\) 8.99324 0.319359
\(794\) 49.0116 1.73936
\(795\) 0 0
\(796\) −7.80846 −0.276763
\(797\) −30.6071 −1.08416 −0.542080 0.840327i \(-0.682363\pi\)
−0.542080 + 0.840327i \(0.682363\pi\)
\(798\) 0 0
\(799\) −70.5486 −2.49583
\(800\) 0 0
\(801\) 3.75660 0.132733
\(802\) −8.31678 −0.293676
\(803\) −7.60515 −0.268380
\(804\) 0.689091 0.0243024
\(805\) 0 0
\(806\) 24.8479 0.875229
\(807\) −1.65708 −0.0583320
\(808\) 29.2368 1.02855
\(809\) 51.0221 1.79384 0.896920 0.442193i \(-0.145799\pi\)
0.896920 + 0.442193i \(0.145799\pi\)
\(810\) 0 0
\(811\) 34.3444 1.20600 0.602998 0.797743i \(-0.293973\pi\)
0.602998 + 0.797743i \(0.293973\pi\)
\(812\) 2.59570 0.0910912
\(813\) −0.767201 −0.0269069
\(814\) 3.43818 0.120508
\(815\) 0 0
\(816\) −2.59062 −0.0906900
\(817\) 0 0
\(818\) 14.6614 0.512623
\(819\) −13.4105 −0.468600
\(820\) 0 0
\(821\) 40.2464 1.40461 0.702305 0.711876i \(-0.252154\pi\)
0.702305 + 0.711876i \(0.252154\pi\)
\(822\) 2.19993 0.0767313
\(823\) 23.3478 0.813853 0.406927 0.913461i \(-0.366601\pi\)
0.406927 + 0.913461i \(0.366601\pi\)
\(824\) −5.48268 −0.190998
\(825\) 0 0
\(826\) −30.7634 −1.07040
\(827\) 32.7322 1.13821 0.569105 0.822265i \(-0.307290\pi\)
0.569105 + 0.822265i \(0.307290\pi\)
\(828\) −23.4087 −0.813508
\(829\) 4.83161 0.167809 0.0839044 0.996474i \(-0.473261\pi\)
0.0839044 + 0.996474i \(0.473261\pi\)
\(830\) 0 0
\(831\) −0.895650 −0.0310697
\(832\) 0.535772 0.0185746
\(833\) 21.4193 0.742134
\(834\) 0.209828 0.00726574
\(835\) 0 0
\(836\) 0 0
\(837\) −3.01552 −0.104232
\(838\) −37.7880 −1.30537
\(839\) 37.2171 1.28488 0.642438 0.766338i \(-0.277923\pi\)
0.642438 + 0.766338i \(0.277923\pi\)
\(840\) 0 0
\(841\) −27.3502 −0.943109
\(842\) −53.8904 −1.85719
\(843\) 0.934365 0.0321812
\(844\) −25.3678 −0.873197
\(845\) 0 0
\(846\) 61.2084 2.10439
\(847\) −17.6861 −0.607702
\(848\) 51.2320 1.75931
\(849\) −0.0815921 −0.00280024
\(850\) 0 0
\(851\) 11.5196 0.394887
\(852\) 0.791482 0.0271157
\(853\) 12.1087 0.414593 0.207297 0.978278i \(-0.433533\pi\)
0.207297 + 0.978278i \(0.433533\pi\)
\(854\) −12.2151 −0.417994
\(855\) 0 0
\(856\) −2.40147 −0.0820805
\(857\) 23.8743 0.815532 0.407766 0.913086i \(-0.366308\pi\)
0.407766 + 0.913086i \(0.366308\pi\)
\(858\) −0.443330 −0.0151350
\(859\) 35.4892 1.21088 0.605438 0.795893i \(-0.292998\pi\)
0.605438 + 0.795893i \(0.292998\pi\)
\(860\) 0 0
\(861\) 1.55033 0.0528352
\(862\) 44.9396 1.53065
\(863\) 26.9330 0.916810 0.458405 0.888743i \(-0.348421\pi\)
0.458405 + 0.888743i \(0.348421\pi\)
\(864\) 2.85223 0.0970348
\(865\) 0 0
\(866\) 56.0930 1.90612
\(867\) −1.68815 −0.0573325
\(868\) −11.8744 −0.403044
\(869\) 9.05154 0.307053
\(870\) 0 0
\(871\) −17.8429 −0.604584
\(872\) 25.0209 0.847313
\(873\) 23.8755 0.808063
\(874\) 0 0
\(875\) 0 0
\(876\) −0.577568 −0.0195142
\(877\) 24.7625 0.836170 0.418085 0.908408i \(-0.362701\pi\)
0.418085 + 0.908408i \(0.362701\pi\)
\(878\) −16.6094 −0.560540
\(879\) −1.14941 −0.0387688
\(880\) 0 0
\(881\) 53.6558 1.80771 0.903855 0.427840i \(-0.140725\pi\)
0.903855 + 0.427840i \(0.140725\pi\)
\(882\) −18.5835 −0.625738
\(883\) 0.946198 0.0318421 0.0159210 0.999873i \(-0.494932\pi\)
0.0159210 + 0.999873i \(0.494932\pi\)
\(884\) −15.8355 −0.532605
\(885\) 0 0
\(886\) 16.2962 0.547481
\(887\) −2.27312 −0.0763241 −0.0381620 0.999272i \(-0.512150\pi\)
−0.0381620 + 0.999272i \(0.512150\pi\)
\(888\) −0.219917 −0.00737992
\(889\) −35.7498 −1.19901
\(890\) 0 0
\(891\) −10.9371 −0.366406
\(892\) −4.64833 −0.155638
\(893\) 0 0
\(894\) 2.18636 0.0731229
\(895\) 0 0
\(896\) −21.4186 −0.715545
\(897\) −1.48537 −0.0495952
\(898\) 48.5619 1.62053
\(899\) −7.54745 −0.251722
\(900\) 0 0
\(901\) 62.1752 2.07135
\(902\) −20.9141 −0.696363
\(903\) −0.111187 −0.00370006
\(904\) 21.9209 0.729078
\(905\) 0 0
\(906\) 3.54675 0.117833
\(907\) 12.3800 0.411070 0.205535 0.978650i \(-0.434107\pi\)
0.205535 + 0.978650i \(0.434107\pi\)
\(908\) 7.17479 0.238104
\(909\) −54.4750 −1.80682
\(910\) 0 0
\(911\) −47.9424 −1.58840 −0.794202 0.607654i \(-0.792111\pi\)
−0.794202 + 0.607654i \(0.792111\pi\)
\(912\) 0 0
\(913\) 0.0165396 0.000547381 0
\(914\) −12.2817 −0.406242
\(915\) 0 0
\(916\) −25.3255 −0.836777
\(917\) −35.9438 −1.18697
\(918\) 5.46216 0.180278
\(919\) 43.4053 1.43181 0.715904 0.698199i \(-0.246015\pi\)
0.715904 + 0.698199i \(0.246015\pi\)
\(920\) 0 0
\(921\) 1.61369 0.0531728
\(922\) −26.7435 −0.880750
\(923\) −20.4942 −0.674574
\(924\) 0.211860 0.00696969
\(925\) 0 0
\(926\) −48.7192 −1.60101
\(927\) 10.2155 0.335522
\(928\) 7.13875 0.234341
\(929\) −37.9300 −1.24444 −0.622222 0.782841i \(-0.713770\pi\)
−0.622222 + 0.782841i \(0.713770\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 18.4963 0.605865
\(933\) 0.0781175 0.00255745
\(934\) −67.3453 −2.20360
\(935\) 0 0
\(936\) −11.5714 −0.378224
\(937\) 6.68720 0.218461 0.109231 0.994016i \(-0.465161\pi\)
0.109231 + 0.994016i \(0.465161\pi\)
\(938\) 24.2353 0.791311
\(939\) −1.60711 −0.0524461
\(940\) 0 0
\(941\) 4.36774 0.142384 0.0711921 0.997463i \(-0.477320\pi\)
0.0711921 + 0.997463i \(0.477320\pi\)
\(942\) 2.56347 0.0835225
\(943\) −70.0725 −2.28187
\(944\) −46.9728 −1.52883
\(945\) 0 0
\(946\) 1.49991 0.0487664
\(947\) 1.50804 0.0490046 0.0245023 0.999700i \(-0.492200\pi\)
0.0245023 + 0.999700i \(0.492200\pi\)
\(948\) 0.687413 0.0223261
\(949\) 14.9552 0.485467
\(950\) 0 0
\(951\) −0.758710 −0.0246029
\(952\) −18.1154 −0.587123
\(953\) −16.0555 −0.520090 −0.260045 0.965596i \(-0.583737\pi\)
−0.260045 + 0.965596i \(0.583737\pi\)
\(954\) −53.9435 −1.74649
\(955\) 0 0
\(956\) 11.3117 0.365845
\(957\) 0.134660 0.00435293
\(958\) 18.4683 0.596684
\(959\) 27.2220 0.879045
\(960\) 0 0
\(961\) 3.52693 0.113772
\(962\) −6.76104 −0.217985
\(963\) 4.47450 0.144189
\(964\) −3.60506 −0.116111
\(965\) 0 0
\(966\) 2.01752 0.0649127
\(967\) 45.0466 1.44860 0.724300 0.689485i \(-0.242163\pi\)
0.724300 + 0.689485i \(0.242163\pi\)
\(968\) −15.2607 −0.490498
\(969\) 0 0
\(970\) 0 0
\(971\) 15.2507 0.489417 0.244709 0.969597i \(-0.421308\pi\)
0.244709 + 0.969597i \(0.421308\pi\)
\(972\) −2.50205 −0.0802531
\(973\) 2.59642 0.0832374
\(974\) −24.8200 −0.795284
\(975\) 0 0
\(976\) −18.6514 −0.597016
\(977\) 26.8192 0.858023 0.429011 0.903299i \(-0.358862\pi\)
0.429011 + 0.903299i \(0.358862\pi\)
\(978\) 1.50142 0.0480100
\(979\) 1.53670 0.0491132
\(980\) 0 0
\(981\) −46.6197 −1.48845
\(982\) −32.9234 −1.05063
\(983\) −28.8049 −0.918735 −0.459367 0.888246i \(-0.651924\pi\)
−0.459367 + 0.888246i \(0.651924\pi\)
\(984\) 1.33773 0.0426452
\(985\) 0 0
\(986\) 13.6711 0.435375
\(987\) −1.85606 −0.0590790
\(988\) 0 0
\(989\) 5.02545 0.159800
\(990\) 0 0
\(991\) 10.3840 0.329859 0.164929 0.986305i \(-0.447260\pi\)
0.164929 + 0.986305i \(0.447260\pi\)
\(992\) −32.6573 −1.03687
\(993\) 0.339162 0.0107630
\(994\) 27.8364 0.882917
\(995\) 0 0
\(996\) 0.00125609 3.98007e−5 0
\(997\) −49.9100 −1.58066 −0.790332 0.612679i \(-0.790092\pi\)
−0.790332 + 0.612679i \(0.790092\pi\)
\(998\) −28.5704 −0.904379
\(999\) 0.820516 0.0259600
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 9025.2.a.bo.1.1 4
5.4 even 2 1805.2.a.j.1.4 4
19.18 odd 2 9025.2.a.bh.1.4 4
95.94 odd 2 1805.2.a.n.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.j.1.4 4 5.4 even 2
1805.2.a.n.1.1 yes 4 95.94 odd 2
9025.2.a.bh.1.4 4 19.18 odd 2
9025.2.a.bo.1.1 4 1.1 even 1 trivial