Properties

Label 8470.2.a.bj.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -3.23607 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.23607 q^{6} +1.00000 q^{7} -1.00000 q^{8} +7.47214 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -3.23607 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.23607 q^{6} +1.00000 q^{7} -1.00000 q^{8} +7.47214 q^{9} -1.00000 q^{10} -3.23607 q^{12} -4.47214 q^{13} -1.00000 q^{14} -3.23607 q^{15} +1.00000 q^{16} -2.00000 q^{17} -7.47214 q^{18} +5.23607 q^{19} +1.00000 q^{20} -3.23607 q^{21} -7.23607 q^{23} +3.23607 q^{24} +1.00000 q^{25} +4.47214 q^{26} -14.4721 q^{27} +1.00000 q^{28} +1.23607 q^{29} +3.23607 q^{30} +8.94427 q^{31} -1.00000 q^{32} +2.00000 q^{34} +1.00000 q^{35} +7.47214 q^{36} -3.23607 q^{37} -5.23607 q^{38} +14.4721 q^{39} -1.00000 q^{40} +7.70820 q^{41} +3.23607 q^{42} +8.00000 q^{43} +7.47214 q^{45} +7.23607 q^{46} +0.472136 q^{47} -3.23607 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.47214 q^{51} -4.47214 q^{52} -0.763932 q^{53} +14.4721 q^{54} -1.00000 q^{56} -16.9443 q^{57} -1.23607 q^{58} -8.00000 q^{59} -3.23607 q^{60} -14.4721 q^{61} -8.94427 q^{62} +7.47214 q^{63} +1.00000 q^{64} -4.47214 q^{65} -6.00000 q^{67} -2.00000 q^{68} +23.4164 q^{69} -1.00000 q^{70} -8.00000 q^{71} -7.47214 q^{72} +8.47214 q^{73} +3.23607 q^{74} -3.23607 q^{75} +5.23607 q^{76} -14.4721 q^{78} -5.23607 q^{79} +1.00000 q^{80} +24.4164 q^{81} -7.70820 q^{82} -8.00000 q^{83} -3.23607 q^{84} -2.00000 q^{85} -8.00000 q^{86} -4.00000 q^{87} -14.9443 q^{89} -7.47214 q^{90} -4.47214 q^{91} -7.23607 q^{92} -28.9443 q^{93} -0.472136 q^{94} +5.23607 q^{95} +3.23607 q^{96} +1.70820 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 6q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 6q^{9} - 2q^{10} - 2q^{12} - 2q^{14} - 2q^{15} + 2q^{16} - 4q^{17} - 6q^{18} + 6q^{19} + 2q^{20} - 2q^{21} - 10q^{23} + 2q^{24} + 2q^{25} - 20q^{27} + 2q^{28} - 2q^{29} + 2q^{30} - 2q^{32} + 4q^{34} + 2q^{35} + 6q^{36} - 2q^{37} - 6q^{38} + 20q^{39} - 2q^{40} + 2q^{41} + 2q^{42} + 16q^{43} + 6q^{45} + 10q^{46} - 8q^{47} - 2q^{48} + 2q^{49} - 2q^{50} + 4q^{51} - 6q^{53} + 20q^{54} - 2q^{56} - 16q^{57} + 2q^{58} - 16q^{59} - 2q^{60} - 20q^{61} + 6q^{63} + 2q^{64} - 12q^{67} - 4q^{68} + 20q^{69} - 2q^{70} - 16q^{71} - 6q^{72} + 8q^{73} + 2q^{74} - 2q^{75} + 6q^{76} - 20q^{78} - 6q^{79} + 2q^{80} + 22q^{81} - 2q^{82} - 16q^{83} - 2q^{84} - 4q^{85} - 16q^{86} - 8q^{87} - 12q^{89} - 6q^{90} - 10q^{92} - 40q^{93} + 8q^{94} + 6q^{95} + 2q^{96} - 10q^{97} - 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −3.23607 −1.86834 −0.934172 0.356822i \(-0.883860\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 3.23607 1.32112
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 7.47214 2.49071
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −3.23607 −0.934172
\(13\) −4.47214 −1.24035 −0.620174 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) −1.00000 −0.267261
\(15\) −3.23607 −0.835549
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −7.47214 −1.76120
\(19\) 5.23607 1.20124 0.600618 0.799536i \(-0.294921\pi\)
0.600618 + 0.799536i \(0.294921\pi\)
\(20\) 1.00000 0.223607
\(21\) −3.23607 −0.706168
\(22\) 0 0
\(23\) −7.23607 −1.50882 −0.754412 0.656401i \(-0.772078\pi\)
−0.754412 + 0.656401i \(0.772078\pi\)
\(24\) 3.23607 0.660560
\(25\) 1.00000 0.200000
\(26\) 4.47214 0.877058
\(27\) −14.4721 −2.78516
\(28\) 1.00000 0.188982
\(29\) 1.23607 0.229532 0.114766 0.993393i \(-0.463388\pi\)
0.114766 + 0.993393i \(0.463388\pi\)
\(30\) 3.23607 0.590822
\(31\) 8.94427 1.60644 0.803219 0.595683i \(-0.203119\pi\)
0.803219 + 0.595683i \(0.203119\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 1.00000 0.169031
\(36\) 7.47214 1.24536
\(37\) −3.23607 −0.532006 −0.266003 0.963972i \(-0.585703\pi\)
−0.266003 + 0.963972i \(0.585703\pi\)
\(38\) −5.23607 −0.849402
\(39\) 14.4721 2.31740
\(40\) −1.00000 −0.158114
\(41\) 7.70820 1.20382 0.601910 0.798564i \(-0.294407\pi\)
0.601910 + 0.798564i \(0.294407\pi\)
\(42\) 3.23607 0.499336
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 7.47214 1.11388
\(46\) 7.23607 1.06690
\(47\) 0.472136 0.0688681 0.0344341 0.999407i \(-0.489037\pi\)
0.0344341 + 0.999407i \(0.489037\pi\)
\(48\) −3.23607 −0.467086
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 6.47214 0.906280
\(52\) −4.47214 −0.620174
\(53\) −0.763932 −0.104934 −0.0524671 0.998623i \(-0.516708\pi\)
−0.0524671 + 0.998623i \(0.516708\pi\)
\(54\) 14.4721 1.96941
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −16.9443 −2.24432
\(58\) −1.23607 −0.162304
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) −3.23607 −0.417775
\(61\) −14.4721 −1.85297 −0.926484 0.376335i \(-0.877184\pi\)
−0.926484 + 0.376335i \(0.877184\pi\)
\(62\) −8.94427 −1.13592
\(63\) 7.47214 0.941401
\(64\) 1.00000 0.125000
\(65\) −4.47214 −0.554700
\(66\) 0 0
\(67\) −6.00000 −0.733017 −0.366508 0.930415i \(-0.619447\pi\)
−0.366508 + 0.930415i \(0.619447\pi\)
\(68\) −2.00000 −0.242536
\(69\) 23.4164 2.81900
\(70\) −1.00000 −0.119523
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −7.47214 −0.880600
\(73\) 8.47214 0.991589 0.495794 0.868440i \(-0.334877\pi\)
0.495794 + 0.868440i \(0.334877\pi\)
\(74\) 3.23607 0.376185
\(75\) −3.23607 −0.373669
\(76\) 5.23607 0.600618
\(77\) 0 0
\(78\) −14.4721 −1.63865
\(79\) −5.23607 −0.589104 −0.294552 0.955636i \(-0.595170\pi\)
−0.294552 + 0.955636i \(0.595170\pi\)
\(80\) 1.00000 0.111803
\(81\) 24.4164 2.71293
\(82\) −7.70820 −0.851229
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) −3.23607 −0.353084
\(85\) −2.00000 −0.216930
\(86\) −8.00000 −0.862662
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) −14.9443 −1.58409 −0.792045 0.610463i \(-0.790983\pi\)
−0.792045 + 0.610463i \(0.790983\pi\)
\(90\) −7.47214 −0.787632
\(91\) −4.47214 −0.468807
\(92\) −7.23607 −0.754412
\(93\) −28.9443 −3.00138
\(94\) −0.472136 −0.0486971
\(95\) 5.23607 0.537209
\(96\) 3.23607 0.330280
\(97\) 1.70820 0.173442 0.0867209 0.996233i \(-0.472361\pi\)
0.0867209 + 0.996233i \(0.472361\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 7.41641 0.737960 0.368980 0.929437i \(-0.379707\pi\)
0.368980 + 0.929437i \(0.379707\pi\)
\(102\) −6.47214 −0.640837
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 4.47214 0.438529
\(105\) −3.23607 −0.315808
\(106\) 0.763932 0.0741996
\(107\) 16.0000 1.54678 0.773389 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) −14.4721 −1.39258
\(109\) −15.7082 −1.50457 −0.752287 0.658836i \(-0.771049\pi\)
−0.752287 + 0.658836i \(0.771049\pi\)
\(110\) 0 0
\(111\) 10.4721 0.993971
\(112\) 1.00000 0.0944911
\(113\) 10.9443 1.02955 0.514775 0.857325i \(-0.327875\pi\)
0.514775 + 0.857325i \(0.327875\pi\)
\(114\) 16.9443 1.58698
\(115\) −7.23607 −0.674767
\(116\) 1.23607 0.114766
\(117\) −33.4164 −3.08935
\(118\) 8.00000 0.736460
\(119\) −2.00000 −0.183340
\(120\) 3.23607 0.295411
\(121\) 0 0
\(122\) 14.4721 1.31025
\(123\) −24.9443 −2.24915
\(124\) 8.94427 0.803219
\(125\) 1.00000 0.0894427
\(126\) −7.47214 −0.665671
\(127\) 5.52786 0.490519 0.245259 0.969458i \(-0.421127\pi\)
0.245259 + 0.969458i \(0.421127\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −25.8885 −2.27936
\(130\) 4.47214 0.392232
\(131\) −6.18034 −0.539979 −0.269989 0.962863i \(-0.587020\pi\)
−0.269989 + 0.962863i \(0.587020\pi\)
\(132\) 0 0
\(133\) 5.23607 0.454025
\(134\) 6.00000 0.518321
\(135\) −14.4721 −1.24556
\(136\) 2.00000 0.171499
\(137\) −19.8885 −1.69919 −0.849596 0.527433i \(-0.823154\pi\)
−0.849596 + 0.527433i \(0.823154\pi\)
\(138\) −23.4164 −1.99334
\(139\) −14.1803 −1.20276 −0.601380 0.798963i \(-0.705382\pi\)
−0.601380 + 0.798963i \(0.705382\pi\)
\(140\) 1.00000 0.0845154
\(141\) −1.52786 −0.128669
\(142\) 8.00000 0.671345
\(143\) 0 0
\(144\) 7.47214 0.622678
\(145\) 1.23607 0.102650
\(146\) −8.47214 −0.701159
\(147\) −3.23607 −0.266906
\(148\) −3.23607 −0.266003
\(149\) −7.70820 −0.631481 −0.315740 0.948846i \(-0.602253\pi\)
−0.315740 + 0.948846i \(0.602253\pi\)
\(150\) 3.23607 0.264224
\(151\) 19.7082 1.60383 0.801915 0.597438i \(-0.203814\pi\)
0.801915 + 0.597438i \(0.203814\pi\)
\(152\) −5.23607 −0.424701
\(153\) −14.9443 −1.20817
\(154\) 0 0
\(155\) 8.94427 0.718421
\(156\) 14.4721 1.15870
\(157\) −7.52786 −0.600789 −0.300394 0.953815i \(-0.597118\pi\)
−0.300394 + 0.953815i \(0.597118\pi\)
\(158\) 5.23607 0.416559
\(159\) 2.47214 0.196053
\(160\) −1.00000 −0.0790569
\(161\) −7.23607 −0.570282
\(162\) −24.4164 −1.91833
\(163\) 14.9443 1.17053 0.585263 0.810844i \(-0.300991\pi\)
0.585263 + 0.810844i \(0.300991\pi\)
\(164\) 7.70820 0.601910
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 4.94427 0.382599 0.191300 0.981532i \(-0.438730\pi\)
0.191300 + 0.981532i \(0.438730\pi\)
\(168\) 3.23607 0.249668
\(169\) 7.00000 0.538462
\(170\) 2.00000 0.153393
\(171\) 39.1246 2.99193
\(172\) 8.00000 0.609994
\(173\) −12.4721 −0.948239 −0.474119 0.880461i \(-0.657234\pi\)
−0.474119 + 0.880461i \(0.657234\pi\)
\(174\) 4.00000 0.303239
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 25.8885 1.94590
\(178\) 14.9443 1.12012
\(179\) 18.4721 1.38067 0.690336 0.723489i \(-0.257463\pi\)
0.690336 + 0.723489i \(0.257463\pi\)
\(180\) 7.47214 0.556940
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 4.47214 0.331497
\(183\) 46.8328 3.46198
\(184\) 7.23607 0.533450
\(185\) −3.23607 −0.237920
\(186\) 28.9443 2.12230
\(187\) 0 0
\(188\) 0.472136 0.0344341
\(189\) −14.4721 −1.05269
\(190\) −5.23607 −0.379864
\(191\) 6.47214 0.468307 0.234154 0.972200i \(-0.424768\pi\)
0.234154 + 0.972200i \(0.424768\pi\)
\(192\) −3.23607 −0.233543
\(193\) 26.9443 1.93949 0.969746 0.244118i \(-0.0784984\pi\)
0.969746 + 0.244118i \(0.0784984\pi\)
\(194\) −1.70820 −0.122642
\(195\) 14.4721 1.03637
\(196\) 1.00000 0.0714286
\(197\) 21.4164 1.52586 0.762928 0.646484i \(-0.223761\pi\)
0.762928 + 0.646484i \(0.223761\pi\)
\(198\) 0 0
\(199\) −5.52786 −0.391860 −0.195930 0.980618i \(-0.562773\pi\)
−0.195930 + 0.980618i \(0.562773\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 19.4164 1.36953
\(202\) −7.41641 −0.521817
\(203\) 1.23607 0.0867550
\(204\) 6.47214 0.453140
\(205\) 7.70820 0.538364
\(206\) −14.0000 −0.975426
\(207\) −54.0689 −3.75805
\(208\) −4.47214 −0.310087
\(209\) 0 0
\(210\) 3.23607 0.223310
\(211\) 13.8885 0.956127 0.478063 0.878325i \(-0.341339\pi\)
0.478063 + 0.878325i \(0.341339\pi\)
\(212\) −0.763932 −0.0524671
\(213\) 25.8885 1.77385
\(214\) −16.0000 −1.09374
\(215\) 8.00000 0.545595
\(216\) 14.4721 0.984704
\(217\) 8.94427 0.607177
\(218\) 15.7082 1.06389
\(219\) −27.4164 −1.85263
\(220\) 0 0
\(221\) 8.94427 0.601657
\(222\) −10.4721 −0.702844
\(223\) −1.05573 −0.0706968 −0.0353484 0.999375i \(-0.511254\pi\)
−0.0353484 + 0.999375i \(0.511254\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 7.47214 0.498142
\(226\) −10.9443 −0.728002
\(227\) −10.4721 −0.695060 −0.347530 0.937669i \(-0.612980\pi\)
−0.347530 + 0.937669i \(0.612980\pi\)
\(228\) −16.9443 −1.12216
\(229\) −20.4721 −1.35284 −0.676418 0.736518i \(-0.736469\pi\)
−0.676418 + 0.736518i \(0.736469\pi\)
\(230\) 7.23607 0.477132
\(231\) 0 0
\(232\) −1.23607 −0.0811518
\(233\) −14.9443 −0.979032 −0.489516 0.871994i \(-0.662826\pi\)
−0.489516 + 0.871994i \(0.662826\pi\)
\(234\) 33.4164 2.18450
\(235\) 0.472136 0.0307988
\(236\) −8.00000 −0.520756
\(237\) 16.9443 1.10065
\(238\) 2.00000 0.129641
\(239\) −2.76393 −0.178784 −0.0893920 0.995997i \(-0.528492\pi\)
−0.0893920 + 0.995997i \(0.528492\pi\)
\(240\) −3.23607 −0.208887
\(241\) −23.1246 −1.48959 −0.744794 0.667295i \(-0.767452\pi\)
−0.744794 + 0.667295i \(0.767452\pi\)
\(242\) 0 0
\(243\) −35.5967 −2.28353
\(244\) −14.4721 −0.926484
\(245\) 1.00000 0.0638877
\(246\) 24.9443 1.59039
\(247\) −23.4164 −1.48995
\(248\) −8.94427 −0.567962
\(249\) 25.8885 1.64062
\(250\) −1.00000 −0.0632456
\(251\) −14.4721 −0.913473 −0.456737 0.889602i \(-0.650982\pi\)
−0.456737 + 0.889602i \(0.650982\pi\)
\(252\) 7.47214 0.470700
\(253\) 0 0
\(254\) −5.52786 −0.346849
\(255\) 6.47214 0.405301
\(256\) 1.00000 0.0625000
\(257\) −25.1246 −1.56723 −0.783615 0.621247i \(-0.786627\pi\)
−0.783615 + 0.621247i \(0.786627\pi\)
\(258\) 25.8885 1.61175
\(259\) −3.23607 −0.201079
\(260\) −4.47214 −0.277350
\(261\) 9.23607 0.571698
\(262\) 6.18034 0.381823
\(263\) 28.3607 1.74879 0.874397 0.485211i \(-0.161257\pi\)
0.874397 + 0.485211i \(0.161257\pi\)
\(264\) 0 0
\(265\) −0.763932 −0.0469280
\(266\) −5.23607 −0.321044
\(267\) 48.3607 2.95963
\(268\) −6.00000 −0.366508
\(269\) −3.52786 −0.215098 −0.107549 0.994200i \(-0.534300\pi\)
−0.107549 + 0.994200i \(0.534300\pi\)
\(270\) 14.4721 0.880746
\(271\) 2.47214 0.150172 0.0750858 0.997177i \(-0.476077\pi\)
0.0750858 + 0.997177i \(0.476077\pi\)
\(272\) −2.00000 −0.121268
\(273\) 14.4721 0.875894
\(274\) 19.8885 1.20151
\(275\) 0 0
\(276\) 23.4164 1.40950
\(277\) 20.4721 1.23005 0.615026 0.788507i \(-0.289146\pi\)
0.615026 + 0.788507i \(0.289146\pi\)
\(278\) 14.1803 0.850480
\(279\) 66.8328 4.00118
\(280\) −1.00000 −0.0597614
\(281\) 16.3607 0.975996 0.487998 0.872845i \(-0.337727\pi\)
0.487998 + 0.872845i \(0.337727\pi\)
\(282\) 1.52786 0.0909830
\(283\) −7.41641 −0.440860 −0.220430 0.975403i \(-0.570746\pi\)
−0.220430 + 0.975403i \(0.570746\pi\)
\(284\) −8.00000 −0.474713
\(285\) −16.9443 −1.00369
\(286\) 0 0
\(287\) 7.70820 0.455001
\(288\) −7.47214 −0.440300
\(289\) −13.0000 −0.764706
\(290\) −1.23607 −0.0725844
\(291\) −5.52786 −0.324049
\(292\) 8.47214 0.495794
\(293\) 8.47214 0.494947 0.247474 0.968895i \(-0.420400\pi\)
0.247474 + 0.968895i \(0.420400\pi\)
\(294\) 3.23607 0.188731
\(295\) −8.00000 −0.465778
\(296\) 3.23607 0.188093
\(297\) 0 0
\(298\) 7.70820 0.446524
\(299\) 32.3607 1.87147
\(300\) −3.23607 −0.186834
\(301\) 8.00000 0.461112
\(302\) −19.7082 −1.13408
\(303\) −24.0000 −1.37876
\(304\) 5.23607 0.300309
\(305\) −14.4721 −0.828672
\(306\) 14.9443 0.854307
\(307\) −18.4721 −1.05426 −0.527130 0.849785i \(-0.676732\pi\)
−0.527130 + 0.849785i \(0.676732\pi\)
\(308\) 0 0
\(309\) −45.3050 −2.57731
\(310\) −8.94427 −0.508001
\(311\) −22.8328 −1.29473 −0.647365 0.762180i \(-0.724129\pi\)
−0.647365 + 0.762180i \(0.724129\pi\)
\(312\) −14.4721 −0.819323
\(313\) 10.6525 0.602114 0.301057 0.953606i \(-0.402661\pi\)
0.301057 + 0.953606i \(0.402661\pi\)
\(314\) 7.52786 0.424822
\(315\) 7.47214 0.421007
\(316\) −5.23607 −0.294552
\(317\) −20.1803 −1.13344 −0.566720 0.823910i \(-0.691788\pi\)
−0.566720 + 0.823910i \(0.691788\pi\)
\(318\) −2.47214 −0.138631
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −51.7771 −2.88992
\(322\) 7.23607 0.403250
\(323\) −10.4721 −0.582685
\(324\) 24.4164 1.35647
\(325\) −4.47214 −0.248069
\(326\) −14.9443 −0.827687
\(327\) 50.8328 2.81106
\(328\) −7.70820 −0.425614
\(329\) 0.472136 0.0260297
\(330\) 0 0
\(331\) 11.4164 0.627503 0.313751 0.949505i \(-0.398414\pi\)
0.313751 + 0.949505i \(0.398414\pi\)
\(332\) −8.00000 −0.439057
\(333\) −24.1803 −1.32507
\(334\) −4.94427 −0.270539
\(335\) −6.00000 −0.327815
\(336\) −3.23607 −0.176542
\(337\) 24.8328 1.35273 0.676365 0.736567i \(-0.263554\pi\)
0.676365 + 0.736567i \(0.263554\pi\)
\(338\) −7.00000 −0.380750
\(339\) −35.4164 −1.92356
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) −39.1246 −2.11562
\(343\) 1.00000 0.0539949
\(344\) −8.00000 −0.431331
\(345\) 23.4164 1.26070
\(346\) 12.4721 0.670506
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) −4.00000 −0.214423
\(349\) 7.05573 0.377684 0.188842 0.982007i \(-0.439527\pi\)
0.188842 + 0.982007i \(0.439527\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 64.7214 3.45457
\(352\) 0 0
\(353\) −23.2361 −1.23673 −0.618366 0.785891i \(-0.712205\pi\)
−0.618366 + 0.785891i \(0.712205\pi\)
\(354\) −25.8885 −1.37596
\(355\) −8.00000 −0.424596
\(356\) −14.9443 −0.792045
\(357\) 6.47214 0.342542
\(358\) −18.4721 −0.976283
\(359\) −10.1803 −0.537298 −0.268649 0.963238i \(-0.586577\pi\)
−0.268649 + 0.963238i \(0.586577\pi\)
\(360\) −7.47214 −0.393816
\(361\) 8.41641 0.442969
\(362\) −14.0000 −0.735824
\(363\) 0 0
\(364\) −4.47214 −0.234404
\(365\) 8.47214 0.443452
\(366\) −46.8328 −2.44799
\(367\) −24.8328 −1.29626 −0.648131 0.761529i \(-0.724449\pi\)
−0.648131 + 0.761529i \(0.724449\pi\)
\(368\) −7.23607 −0.377206
\(369\) 57.5967 2.99837
\(370\) 3.23607 0.168235
\(371\) −0.763932 −0.0396614
\(372\) −28.9443 −1.50069
\(373\) −6.36068 −0.329344 −0.164672 0.986348i \(-0.552656\pi\)
−0.164672 + 0.986348i \(0.552656\pi\)
\(374\) 0 0
\(375\) −3.23607 −0.167110
\(376\) −0.472136 −0.0243486
\(377\) −5.52786 −0.284699
\(378\) 14.4721 0.744366
\(379\) 12.9443 0.664903 0.332451 0.943120i \(-0.392124\pi\)
0.332451 + 0.943120i \(0.392124\pi\)
\(380\) 5.23607 0.268605
\(381\) −17.8885 −0.916458
\(382\) −6.47214 −0.331143
\(383\) 31.8885 1.62943 0.814714 0.579863i \(-0.196894\pi\)
0.814714 + 0.579863i \(0.196894\pi\)
\(384\) 3.23607 0.165140
\(385\) 0 0
\(386\) −26.9443 −1.37143
\(387\) 59.7771 3.03864
\(388\) 1.70820 0.0867209
\(389\) −6.36068 −0.322499 −0.161250 0.986914i \(-0.551552\pi\)
−0.161250 + 0.986914i \(0.551552\pi\)
\(390\) −14.4721 −0.732825
\(391\) 14.4721 0.731887
\(392\) −1.00000 −0.0505076
\(393\) 20.0000 1.00887
\(394\) −21.4164 −1.07894
\(395\) −5.23607 −0.263455
\(396\) 0 0
\(397\) −1.41641 −0.0710875 −0.0355437 0.999368i \(-0.511316\pi\)
−0.0355437 + 0.999368i \(0.511316\pi\)
\(398\) 5.52786 0.277087
\(399\) −16.9443 −0.848275
\(400\) 1.00000 0.0500000
\(401\) −12.4721 −0.622829 −0.311414 0.950274i \(-0.600803\pi\)
−0.311414 + 0.950274i \(0.600803\pi\)
\(402\) −19.4164 −0.968402
\(403\) −40.0000 −1.99254
\(404\) 7.41641 0.368980
\(405\) 24.4164 1.21326
\(406\) −1.23607 −0.0613450
\(407\) 0 0
\(408\) −6.47214 −0.320418
\(409\) −9.23607 −0.456694 −0.228347 0.973580i \(-0.573332\pi\)
−0.228347 + 0.973580i \(0.573332\pi\)
\(410\) −7.70820 −0.380681
\(411\) 64.3607 3.17468
\(412\) 14.0000 0.689730
\(413\) −8.00000 −0.393654
\(414\) 54.0689 2.65734
\(415\) −8.00000 −0.392705
\(416\) 4.47214 0.219265
\(417\) 45.8885 2.24717
\(418\) 0 0
\(419\) −3.41641 −0.166902 −0.0834512 0.996512i \(-0.526594\pi\)
−0.0834512 + 0.996512i \(0.526594\pi\)
\(420\) −3.23607 −0.157904
\(421\) −31.3050 −1.52571 −0.762855 0.646570i \(-0.776203\pi\)
−0.762855 + 0.646570i \(0.776203\pi\)
\(422\) −13.8885 −0.676084
\(423\) 3.52786 0.171531
\(424\) 0.763932 0.0370998
\(425\) −2.00000 −0.0970143
\(426\) −25.8885 −1.25430
\(427\) −14.4721 −0.700356
\(428\) 16.0000 0.773389
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 28.6525 1.38014 0.690071 0.723742i \(-0.257579\pi\)
0.690071 + 0.723742i \(0.257579\pi\)
\(432\) −14.4721 −0.696291
\(433\) −0.763932 −0.0367122 −0.0183561 0.999832i \(-0.505843\pi\)
−0.0183561 + 0.999832i \(0.505843\pi\)
\(434\) −8.94427 −0.429339
\(435\) −4.00000 −0.191785
\(436\) −15.7082 −0.752287
\(437\) −37.8885 −1.81245
\(438\) 27.4164 1.31001
\(439\) 7.41641 0.353966 0.176983 0.984214i \(-0.443366\pi\)
0.176983 + 0.984214i \(0.443366\pi\)
\(440\) 0 0
\(441\) 7.47214 0.355816
\(442\) −8.94427 −0.425436
\(443\) 14.9443 0.710024 0.355012 0.934862i \(-0.384477\pi\)
0.355012 + 0.934862i \(0.384477\pi\)
\(444\) 10.4721 0.496986
\(445\) −14.9443 −0.708426
\(446\) 1.05573 0.0499902
\(447\) 24.9443 1.17982
\(448\) 1.00000 0.0472456
\(449\) 26.3607 1.24404 0.622019 0.783002i \(-0.286313\pi\)
0.622019 + 0.783002i \(0.286313\pi\)
\(450\) −7.47214 −0.352240
\(451\) 0 0
\(452\) 10.9443 0.514775
\(453\) −63.7771 −2.99651
\(454\) 10.4721 0.491482
\(455\) −4.47214 −0.209657
\(456\) 16.9443 0.793488
\(457\) −30.9443 −1.44751 −0.723756 0.690056i \(-0.757586\pi\)
−0.723756 + 0.690056i \(0.757586\pi\)
\(458\) 20.4721 0.956600
\(459\) 28.9443 1.35100
\(460\) −7.23607 −0.337383
\(461\) −3.05573 −0.142319 −0.0711597 0.997465i \(-0.522670\pi\)
−0.0711597 + 0.997465i \(0.522670\pi\)
\(462\) 0 0
\(463\) −2.29180 −0.106509 −0.0532544 0.998581i \(-0.516959\pi\)
−0.0532544 + 0.998581i \(0.516959\pi\)
\(464\) 1.23607 0.0573830
\(465\) −28.9443 −1.34226
\(466\) 14.9443 0.692280
\(467\) 36.1803 1.67423 0.837113 0.547030i \(-0.184242\pi\)
0.837113 + 0.547030i \(0.184242\pi\)
\(468\) −33.4164 −1.54467
\(469\) −6.00000 −0.277054
\(470\) −0.472136 −0.0217780
\(471\) 24.3607 1.12248
\(472\) 8.00000 0.368230
\(473\) 0 0
\(474\) −16.9443 −0.778276
\(475\) 5.23607 0.240247
\(476\) −2.00000 −0.0916698
\(477\) −5.70820 −0.261361
\(478\) 2.76393 0.126419
\(479\) −29.8885 −1.36564 −0.682821 0.730586i \(-0.739247\pi\)
−0.682821 + 0.730586i \(0.739247\pi\)
\(480\) 3.23607 0.147706
\(481\) 14.4721 0.659873
\(482\) 23.1246 1.05330
\(483\) 23.4164 1.06548
\(484\) 0 0
\(485\) 1.70820 0.0775655
\(486\) 35.5967 1.61470
\(487\) −34.0689 −1.54381 −0.771904 0.635739i \(-0.780696\pi\)
−0.771904 + 0.635739i \(0.780696\pi\)
\(488\) 14.4721 0.655123
\(489\) −48.3607 −2.18695
\(490\) −1.00000 −0.0451754
\(491\) 8.00000 0.361035 0.180517 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) −24.9443 −1.12457
\(493\) −2.47214 −0.111339
\(494\) 23.4164 1.05355
\(495\) 0 0
\(496\) 8.94427 0.401610
\(497\) −8.00000 −0.358849
\(498\) −25.8885 −1.16009
\(499\) −29.3050 −1.31187 −0.655935 0.754817i \(-0.727725\pi\)
−0.655935 + 0.754817i \(0.727725\pi\)
\(500\) 1.00000 0.0447214
\(501\) −16.0000 −0.714827
\(502\) 14.4721 0.645923
\(503\) 38.8328 1.73147 0.865735 0.500503i \(-0.166852\pi\)
0.865735 + 0.500503i \(0.166852\pi\)
\(504\) −7.47214 −0.332835
\(505\) 7.41641 0.330026
\(506\) 0 0
\(507\) −22.6525 −1.00603
\(508\) 5.52786 0.245259
\(509\) −10.9443 −0.485096 −0.242548 0.970139i \(-0.577983\pi\)
−0.242548 + 0.970139i \(0.577983\pi\)
\(510\) −6.47214 −0.286591
\(511\) 8.47214 0.374785
\(512\) −1.00000 −0.0441942
\(513\) −75.7771 −3.34564
\(514\) 25.1246 1.10820
\(515\) 14.0000 0.616914
\(516\) −25.8885 −1.13968
\(517\) 0 0
\(518\) 3.23607 0.142185
\(519\) 40.3607 1.77164
\(520\) 4.47214 0.196116
\(521\) −0.472136 −0.0206847 −0.0103423 0.999947i \(-0.503292\pi\)
−0.0103423 + 0.999947i \(0.503292\pi\)
\(522\) −9.23607 −0.404252
\(523\) 21.5279 0.941348 0.470674 0.882307i \(-0.344011\pi\)
0.470674 + 0.882307i \(0.344011\pi\)
\(524\) −6.18034 −0.269989
\(525\) −3.23607 −0.141234
\(526\) −28.3607 −1.23658
\(527\) −17.8885 −0.779237
\(528\) 0 0
\(529\) 29.3607 1.27655
\(530\) 0.763932 0.0331831
\(531\) −59.7771 −2.59410
\(532\) 5.23607 0.227012
\(533\) −34.4721 −1.49315
\(534\) −48.3607 −2.09277
\(535\) 16.0000 0.691740
\(536\) 6.00000 0.259161
\(537\) −59.7771 −2.57957
\(538\) 3.52786 0.152097
\(539\) 0 0
\(540\) −14.4721 −0.622782
\(541\) 10.7639 0.462778 0.231389 0.972861i \(-0.425673\pi\)
0.231389 + 0.972861i \(0.425673\pi\)
\(542\) −2.47214 −0.106187
\(543\) −45.3050 −1.94422
\(544\) 2.00000 0.0857493
\(545\) −15.7082 −0.672866
\(546\) −14.4721 −0.619350
\(547\) 24.9443 1.06654 0.533270 0.845945i \(-0.320963\pi\)
0.533270 + 0.845945i \(0.320963\pi\)
\(548\) −19.8885 −0.849596
\(549\) −108.138 −4.61521
\(550\) 0 0
\(551\) 6.47214 0.275722
\(552\) −23.4164 −0.996669
\(553\) −5.23607 −0.222660
\(554\) −20.4721 −0.869778
\(555\) 10.4721 0.444517
\(556\) −14.1803 −0.601380
\(557\) 28.8328 1.22169 0.610843 0.791752i \(-0.290831\pi\)
0.610843 + 0.791752i \(0.290831\pi\)
\(558\) −66.8328 −2.82926
\(559\) −35.7771 −1.51321
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −16.3607 −0.690134
\(563\) 38.8328 1.63661 0.818304 0.574786i \(-0.194915\pi\)
0.818304 + 0.574786i \(0.194915\pi\)
\(564\) −1.52786 −0.0643347
\(565\) 10.9443 0.460429
\(566\) 7.41641 0.311735
\(567\) 24.4164 1.02539
\(568\) 8.00000 0.335673
\(569\) −6.83282 −0.286447 −0.143223 0.989690i \(-0.545747\pi\)
−0.143223 + 0.989690i \(0.545747\pi\)
\(570\) 16.9443 0.709717
\(571\) −0.583592 −0.0244226 −0.0122113 0.999925i \(-0.503887\pi\)
−0.0122113 + 0.999925i \(0.503887\pi\)
\(572\) 0 0
\(573\) −20.9443 −0.874960
\(574\) −7.70820 −0.321734
\(575\) −7.23607 −0.301765
\(576\) 7.47214 0.311339
\(577\) 1.70820 0.0711135 0.0355567 0.999368i \(-0.488680\pi\)
0.0355567 + 0.999368i \(0.488680\pi\)
\(578\) 13.0000 0.540729
\(579\) −87.1935 −3.62364
\(580\) 1.23607 0.0513249
\(581\) −8.00000 −0.331896
\(582\) 5.52786 0.229137
\(583\) 0 0
\(584\) −8.47214 −0.350579
\(585\) −33.4164 −1.38160
\(586\) −8.47214 −0.349981
\(587\) −13.7082 −0.565798 −0.282899 0.959150i \(-0.591296\pi\)
−0.282899 + 0.959150i \(0.591296\pi\)
\(588\) −3.23607 −0.133453
\(589\) 46.8328 1.92971
\(590\) 8.00000 0.329355
\(591\) −69.3050 −2.85082
\(592\) −3.23607 −0.133002
\(593\) −38.3607 −1.57528 −0.787642 0.616133i \(-0.788698\pi\)
−0.787642 + 0.616133i \(0.788698\pi\)
\(594\) 0 0
\(595\) −2.00000 −0.0819920
\(596\) −7.70820 −0.315740
\(597\) 17.8885 0.732129
\(598\) −32.3607 −1.32333
\(599\) 17.8885 0.730906 0.365453 0.930830i \(-0.380914\pi\)
0.365453 + 0.930830i \(0.380914\pi\)
\(600\) 3.23607 0.132112
\(601\) −31.7082 −1.29340 −0.646702 0.762743i \(-0.723852\pi\)
−0.646702 + 0.762743i \(0.723852\pi\)
\(602\) −8.00000 −0.326056
\(603\) −44.8328 −1.82573
\(604\) 19.7082 0.801915
\(605\) 0 0
\(606\) 24.0000 0.974933
\(607\) 18.8328 0.764400 0.382200 0.924080i \(-0.375166\pi\)
0.382200 + 0.924080i \(0.375166\pi\)
\(608\) −5.23607 −0.212351
\(609\) −4.00000 −0.162088
\(610\) 14.4721 0.585960
\(611\) −2.11146 −0.0854204
\(612\) −14.9443 −0.604086
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 18.4721 0.745475
\(615\) −24.9443 −1.00585
\(616\) 0 0
\(617\) −31.5279 −1.26926 −0.634632 0.772814i \(-0.718848\pi\)
−0.634632 + 0.772814i \(0.718848\pi\)
\(618\) 45.3050 1.82243
\(619\) 1.52786 0.0614100 0.0307050 0.999528i \(-0.490225\pi\)
0.0307050 + 0.999528i \(0.490225\pi\)
\(620\) 8.94427 0.359211
\(621\) 104.721 4.20232
\(622\) 22.8328 0.915513
\(623\) −14.9443 −0.598730
\(624\) 14.4721 0.579349
\(625\) 1.00000 0.0400000
\(626\) −10.6525 −0.425759
\(627\) 0 0
\(628\) −7.52786 −0.300394
\(629\) 6.47214 0.258061
\(630\) −7.47214 −0.297697
\(631\) 14.8328 0.590485 0.295243 0.955422i \(-0.404600\pi\)
0.295243 + 0.955422i \(0.404600\pi\)
\(632\) 5.23607 0.208280
\(633\) −44.9443 −1.78637
\(634\) 20.1803 0.801464
\(635\) 5.52786 0.219367
\(636\) 2.47214 0.0980266
\(637\) −4.47214 −0.177192
\(638\) 0 0
\(639\) −59.7771 −2.36475
\(640\) −1.00000 −0.0395285
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 51.7771 2.04348
\(643\) −48.1803 −1.90005 −0.950024 0.312178i \(-0.898941\pi\)
−0.950024 + 0.312178i \(0.898941\pi\)
\(644\) −7.23607 −0.285141
\(645\) −25.8885 −1.01936
\(646\) 10.4721 0.412021
\(647\) −30.0000 −1.17942 −0.589711 0.807614i \(-0.700758\pi\)
−0.589711 + 0.807614i \(0.700758\pi\)
\(648\) −24.4164 −0.959167
\(649\) 0 0
\(650\) 4.47214 0.175412
\(651\) −28.9443 −1.13442
\(652\) 14.9443 0.585263
\(653\) −15.5967 −0.610348 −0.305174 0.952297i \(-0.598715\pi\)
−0.305174 + 0.952297i \(0.598715\pi\)
\(654\) −50.8328 −1.98772
\(655\) −6.18034 −0.241486
\(656\) 7.70820 0.300955
\(657\) 63.3050 2.46976
\(658\) −0.472136 −0.0184058
\(659\) −32.3607 −1.26059 −0.630297 0.776354i \(-0.717067\pi\)
−0.630297 + 0.776354i \(0.717067\pi\)
\(660\) 0 0
\(661\) −2.94427 −0.114519 −0.0572595 0.998359i \(-0.518236\pi\)
−0.0572595 + 0.998359i \(0.518236\pi\)
\(662\) −11.4164 −0.443711
\(663\) −28.9443 −1.12410
\(664\) 8.00000 0.310460
\(665\) 5.23607 0.203046
\(666\) 24.1803 0.936969
\(667\) −8.94427 −0.346324
\(668\) 4.94427 0.191300
\(669\) 3.41641 0.132086
\(670\) 6.00000 0.231800
\(671\) 0 0
\(672\) 3.23607 0.124834
\(673\) −23.8885 −0.920836 −0.460418 0.887702i \(-0.652300\pi\)
−0.460418 + 0.887702i \(0.652300\pi\)
\(674\) −24.8328 −0.956524
\(675\) −14.4721 −0.557033
\(676\) 7.00000 0.269231
\(677\) −32.4721 −1.24801 −0.624003 0.781422i \(-0.714495\pi\)
−0.624003 + 0.781422i \(0.714495\pi\)
\(678\) 35.4164 1.36016
\(679\) 1.70820 0.0655549
\(680\) 2.00000 0.0766965
\(681\) 33.8885 1.29861
\(682\) 0 0
\(683\) −39.5279 −1.51249 −0.756246 0.654288i \(-0.772969\pi\)
−0.756246 + 0.654288i \(0.772969\pi\)
\(684\) 39.1246 1.49597
\(685\) −19.8885 −0.759902
\(686\) −1.00000 −0.0381802
\(687\) 66.2492 2.52757
\(688\) 8.00000 0.304997
\(689\) 3.41641 0.130155
\(690\) −23.4164 −0.891447
\(691\) 8.36068 0.318055 0.159028 0.987274i \(-0.449164\pi\)
0.159028 + 0.987274i \(0.449164\pi\)
\(692\) −12.4721 −0.474119
\(693\) 0 0
\(694\) 0 0
\(695\) −14.1803 −0.537891
\(696\) 4.00000 0.151620
\(697\) −15.4164 −0.583938
\(698\) −7.05573 −0.267063
\(699\) 48.3607 1.82917
\(700\) 1.00000 0.0377964
\(701\) 3.70820 0.140057 0.0700285 0.997545i \(-0.477691\pi\)
0.0700285 + 0.997545i \(0.477691\pi\)
\(702\) −64.7214 −2.44275
\(703\) −16.9443 −0.639065
\(704\) 0 0
\(705\) −1.52786 −0.0575427
\(706\) 23.2361 0.874501
\(707\) 7.41641 0.278923
\(708\) 25.8885 0.972951
\(709\) −2.94427 −0.110574 −0.0552872 0.998470i \(-0.517607\pi\)
−0.0552872 + 0.998470i \(0.517607\pi\)
\(710\) 8.00000 0.300235
\(711\) −39.1246 −1.46729
\(712\) 14.9443 0.560060
\(713\) −64.7214 −2.42383
\(714\) −6.47214 −0.242214
\(715\) 0 0
\(716\) 18.4721 0.690336
\(717\) 8.94427 0.334030
\(718\) 10.1803 0.379927
\(719\) −32.3607 −1.20685 −0.603425 0.797420i \(-0.706198\pi\)
−0.603425 + 0.797420i \(0.706198\pi\)
\(720\) 7.47214 0.278470
\(721\) 14.0000 0.521387
\(722\) −8.41641 −0.313226
\(723\) 74.8328 2.78306
\(724\) 14.0000 0.520306
\(725\) 1.23607 0.0459064
\(726\) 0 0
\(727\) −34.3607 −1.27437 −0.637184 0.770712i \(-0.719901\pi\)
−0.637184 + 0.770712i \(0.719901\pi\)
\(728\) 4.47214 0.165748
\(729\) 41.9443 1.55349
\(730\) −8.47214 −0.313568
\(731\) −16.0000 −0.591781
\(732\) 46.8328 1.73099
\(733\) 44.2492 1.63438 0.817191 0.576367i \(-0.195530\pi\)
0.817191 + 0.576367i \(0.195530\pi\)
\(734\) 24.8328 0.916596
\(735\) −3.23607 −0.119364
\(736\) 7.23607 0.266725
\(737\) 0 0
\(738\) −57.5967 −2.12017
\(739\) −44.7214 −1.64510 −0.822551 0.568691i \(-0.807450\pi\)
−0.822551 + 0.568691i \(0.807450\pi\)
\(740\) −3.23607 −0.118960
\(741\) 75.7771 2.78374
\(742\) 0.763932 0.0280448
\(743\) −29.5279 −1.08327 −0.541636 0.840613i \(-0.682195\pi\)
−0.541636 + 0.840613i \(0.682195\pi\)
\(744\) 28.9443 1.06115
\(745\) −7.70820 −0.282407
\(746\) 6.36068 0.232881
\(747\) −59.7771 −2.18713
\(748\) 0 0
\(749\) 16.0000 0.584627
\(750\) 3.23607 0.118164
\(751\) 41.8885 1.52853 0.764267 0.644900i \(-0.223101\pi\)
0.764267 + 0.644900i \(0.223101\pi\)
\(752\) 0.472136 0.0172170
\(753\) 46.8328 1.70668
\(754\) 5.52786 0.201313
\(755\) 19.7082 0.717255
\(756\) −14.4721 −0.526346
\(757\) −2.87539 −0.104508 −0.0522539 0.998634i \(-0.516641\pi\)
−0.0522539 + 0.998634i \(0.516641\pi\)
\(758\) −12.9443 −0.470157
\(759\) 0 0
\(760\) −5.23607 −0.189932
\(761\) −40.0689 −1.45250 −0.726248 0.687433i \(-0.758738\pi\)
−0.726248 + 0.687433i \(0.758738\pi\)
\(762\) 17.8885 0.648034
\(763\) −15.7082 −0.568675
\(764\) 6.47214 0.234154
\(765\) −14.9443 −0.540311
\(766\) −31.8885 −1.15218
\(767\) 35.7771 1.29184
\(768\) −3.23607 −0.116772
\(769\) 41.0132 1.47897 0.739486 0.673172i \(-0.235069\pi\)
0.739486 + 0.673172i \(0.235069\pi\)
\(770\) 0 0
\(771\) 81.3050 2.92813
\(772\) 26.9443 0.969746
\(773\) −41.7771 −1.50262 −0.751309 0.659951i \(-0.770577\pi\)
−0.751309 + 0.659951i \(0.770577\pi\)
\(774\) −59.7771 −2.14864
\(775\) 8.94427 0.321288
\(776\) −1.70820 −0.0613209
\(777\) 10.4721 0.375686
\(778\) 6.36068 0.228041
\(779\) 40.3607 1.44607
\(780\) 14.4721 0.518186
\(781\) 0 0
\(782\) −14.4721 −0.517523
\(783\) −17.8885 −0.639284
\(784\) 1.00000 0.0357143
\(785\) −7.52786 −0.268681
\(786\) −20.0000 −0.713376
\(787\) −6.83282 −0.243564 −0.121782 0.992557i \(-0.538861\pi\)
−0.121782 + 0.992557i \(0.538861\pi\)
\(788\) 21.4164 0.762928
\(789\) −91.7771 −3.26735
\(790\) 5.23607 0.186291
\(791\) 10.9443 0.389134
\(792\) 0 0
\(793\) 64.7214 2.29832
\(794\) 1.41641 0.0502664
\(795\) 2.47214 0.0876776
\(796\) −5.52786 −0.195930
\(797\) −33.4164 −1.18367 −0.591835 0.806059i \(-0.701596\pi\)
−0.591835 + 0.806059i \(0.701596\pi\)
\(798\) 16.9443 0.599821
\(799\) −0.944272 −0.0334059
\(800\) −1.00000 −0.0353553
\(801\) −111.666 −3.94551
\(802\) 12.4721 0.440406
\(803\) 0 0
\(804\) 19.4164 0.684764
\(805\) −7.23607 −0.255038
\(806\) 40.0000 1.40894
\(807\) 11.4164 0.401877
\(808\) −7.41641 −0.260908
\(809\) −24.3607 −0.856476 −0.428238 0.903666i \(-0.640865\pi\)
−0.428238 + 0.903666i \(0.640865\pi\)
\(810\) −24.4164 −0.857905
\(811\) −18.1803 −0.638398 −0.319199 0.947688i \(-0.603414\pi\)
−0.319199 + 0.947688i \(0.603414\pi\)
\(812\) 1.23607 0.0433775
\(813\) −8.00000 −0.280572
\(814\) 0 0
\(815\) 14.9443 0.523475
\(816\) 6.47214 0.226570
\(817\) 41.8885 1.46549
\(818\) 9.23607 0.322932
\(819\) −33.4164 −1.16766
\(820\) 7.70820 0.269182
\(821\) 35.7082 1.24622 0.623112 0.782132i \(-0.285868\pi\)
0.623112 + 0.782132i \(0.285868\pi\)
\(822\) −64.3607 −2.24484
\(823\) −39.0132 −1.35991 −0.679957 0.733252i \(-0.738001\pi\)
−0.679957 + 0.733252i \(0.738001\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) −31.7771 −1.10500 −0.552499 0.833514i \(-0.686326\pi\)
−0.552499 + 0.833514i \(0.686326\pi\)
\(828\) −54.0689 −1.87902
\(829\) −16.4721 −0.572101 −0.286050 0.958215i \(-0.592343\pi\)
−0.286050 + 0.958215i \(0.592343\pi\)
\(830\) 8.00000 0.277684
\(831\) −66.2492 −2.29816
\(832\) −4.47214 −0.155043
\(833\) −2.00000 −0.0692959
\(834\) −45.8885 −1.58899
\(835\) 4.94427 0.171104
\(836\) 0 0
\(837\) −129.443 −4.47419
\(838\) 3.41641 0.118018
\(839\) 4.00000 0.138095 0.0690477 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) 3.23607 0.111655
\(841\) −27.4721 −0.947315
\(842\) 31.3050 1.07884
\(843\) −52.9443 −1.82350
\(844\) 13.8885 0.478063
\(845\) 7.00000 0.240807
\(846\) −3.52786 −0.121290
\(847\) 0 0
\(848\) −0.763932 −0.0262335
\(849\) 24.0000 0.823678
\(850\) 2.00000 0.0685994
\(851\) 23.4164 0.802704
\(852\) 25.8885 0.886927
\(853\) −49.4164 −1.69199 −0.845993 0.533194i \(-0.820991\pi\)
−0.845993 + 0.533194i \(0.820991\pi\)
\(854\) 14.4721 0.495226
\(855\) 39.1246 1.33803
\(856\) −16.0000 −0.546869
\(857\) 0.472136 0.0161279 0.00806393 0.999967i \(-0.497433\pi\)
0.00806393 + 0.999967i \(0.497433\pi\)
\(858\) 0 0
\(859\) 48.3607 1.65005 0.825023 0.565100i \(-0.191162\pi\)
0.825023 + 0.565100i \(0.191162\pi\)
\(860\) 8.00000 0.272798
\(861\) −24.9443 −0.850099
\(862\) −28.6525 −0.975907
\(863\) −19.2361 −0.654803 −0.327402 0.944885i \(-0.606173\pi\)
−0.327402 + 0.944885i \(0.606173\pi\)
\(864\) 14.4721 0.492352
\(865\) −12.4721 −0.424065
\(866\) 0.763932 0.0259595
\(867\) 42.0689 1.42873
\(868\) 8.94427 0.303588
\(869\) 0 0
\(870\) 4.00000 0.135613
\(871\) 26.8328 0.909195
\(872\) 15.7082 0.531947
\(873\) 12.7639 0.431994
\(874\) 37.8885 1.28160
\(875\) 1.00000 0.0338062
\(876\) −27.4164 −0.926315
\(877\) −35.3050 −1.19216 −0.596082 0.802924i \(-0.703277\pi\)
−0.596082 + 0.802924i \(0.703277\pi\)
\(878\) −7.41641 −0.250292
\(879\) −27.4164 −0.924732
\(880\) 0 0
\(881\) 38.9443 1.31207 0.656033 0.754732i \(-0.272233\pi\)
0.656033 + 0.754732i \(0.272233\pi\)
\(882\) −7.47214 −0.251600
\(883\) −6.58359 −0.221556 −0.110778 0.993845i \(-0.535334\pi\)
−0.110778 + 0.993845i \(0.535334\pi\)
\(884\) 8.94427 0.300828
\(885\) 25.8885 0.870234
\(886\) −14.9443 −0.502063
\(887\) −6.11146 −0.205203 −0.102601 0.994723i \(-0.532717\pi\)
−0.102601 + 0.994723i \(0.532717\pi\)
\(888\) −10.4721 −0.351422
\(889\) 5.52786 0.185399
\(890\) 14.9443 0.500933
\(891\) 0 0
\(892\) −1.05573 −0.0353484
\(893\) 2.47214 0.0827269
\(894\) −24.9443 −0.834261
\(895\) 18.4721 0.617455
\(896\) −1.00000 −0.0334077
\(897\) −104.721 −3.49654
\(898\) −26.3607 −0.879667
\(899\) 11.0557 0.368729
\(900\) 7.47214 0.249071
\(901\) 1.52786 0.0509005
\(902\) 0 0
\(903\) −25.8885 −0.861517
\(904\) −10.9443 −0.364001
\(905\) 14.0000 0.465376
\(906\) 63.7771 2.11885
\(907\) 15.5279 0.515594 0.257797 0.966199i \(-0.417003\pi\)
0.257797 + 0.966199i \(0.417003\pi\)
\(908\) −10.4721 −0.347530
\(909\) 55.4164 1.83805
\(910\) 4.47214 0.148250
\(911\) 3.41641 0.113191 0.0565953 0.998397i \(-0.481976\pi\)
0.0565953 + 0.998397i \(0.481976\pi\)
\(912\) −16.9443 −0.561081
\(913\) 0 0
\(914\) 30.9443 1.02355
\(915\) 46.8328 1.54825
\(916\) −20.4721 −0.676418
\(917\) −6.18034 −0.204093
\(918\) −28.9443 −0.955303
\(919\) −38.1803 −1.25945 −0.629727 0.776817i \(-0.716833\pi\)
−0.629727 + 0.776817i \(0.716833\pi\)
\(920\) 7.23607 0.238566
\(921\) 59.7771 1.96972
\(922\) 3.05573 0.100635
\(923\) 35.7771 1.17762
\(924\) 0 0
\(925\) −3.23607 −0.106401
\(926\) 2.29180 0.0753131
\(927\) 104.610 3.43584
\(928\) −1.23607 −0.0405759
\(929\) 30.3607 0.996102 0.498051 0.867148i \(-0.334049\pi\)
0.498051 + 0.867148i \(0.334049\pi\)
\(930\) 28.9443 0.949120
\(931\) 5.23607 0.171605
\(932\) −14.9443 −0.489516
\(933\) 73.8885 2.41900
\(934\) −36.1803 −1.18386
\(935\) 0 0
\(936\) 33.4164 1.09225
\(937\) 30.9443 1.01091 0.505453 0.862854i \(-0.331326\pi\)
0.505453 + 0.862854i \(0.331326\pi\)
\(938\) 6.00000 0.195907
\(939\) −34.4721 −1.12496
\(940\) 0.472136 0.0153994
\(941\) −54.2492 −1.76847 −0.884237 0.467038i \(-0.845321\pi\)
−0.884237 + 0.467038i \(0.845321\pi\)
\(942\) −24.3607 −0.793714
\(943\) −55.7771 −1.81635
\(944\) −8.00000 −0.260378
\(945\) −14.4721 −0.470779
\(946\) 0 0
\(947\) 20.4721 0.665255 0.332628 0.943058i \(-0.392065\pi\)
0.332628 + 0.943058i \(0.392065\pi\)
\(948\) 16.9443 0.550324
\(949\) −37.8885 −1.22991
\(950\) −5.23607 −0.169880
\(951\) 65.3050 2.11766
\(952\) 2.00000 0.0648204
\(953\) 3.88854 0.125962 0.0629811 0.998015i \(-0.479939\pi\)
0.0629811 + 0.998015i \(0.479939\pi\)
\(954\) 5.70820 0.184810
\(955\) 6.47214 0.209433
\(956\) −2.76393 −0.0893920
\(957\) 0 0
\(958\) 29.8885 0.965655
\(959\) −19.8885 −0.642235
\(960\) −3.23607 −0.104444
\(961\) 49.0000 1.58065
\(962\) −14.4721 −0.466600
\(963\) 119.554 3.85258
\(964\) −23.1246 −0.744794
\(965\) 26.9443 0.867367
\(966\) −23.4164 −0.753411
\(967\) 33.3050 1.07102 0.535508 0.844530i \(-0.320120\pi\)
0.535508 + 0.844530i \(0.320120\pi\)
\(968\) 0 0
\(969\) 33.8885 1.08866
\(970\) −1.70820 −0.0548471
\(971\) −22.8328 −0.732740 −0.366370 0.930469i \(-0.619400\pi\)
−0.366370 + 0.930469i \(0.619400\pi\)
\(972\) −35.5967 −1.14177
\(973\) −14.1803 −0.454601
\(974\) 34.0689 1.09164
\(975\) 14.4721 0.463479
\(976\) −14.4721 −0.463242
\(977\) −0.832816 −0.0266441 −0.0133221 0.999911i \(-0.504241\pi\)
−0.0133221 + 0.999911i \(0.504241\pi\)
\(978\) 48.3607 1.54640
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −117.374 −3.74746
\(982\) −8.00000 −0.255290
\(983\) −13.0557 −0.416413 −0.208207 0.978085i \(-0.566763\pi\)
−0.208207 + 0.978085i \(0.566763\pi\)
\(984\) 24.9443 0.795194
\(985\) 21.4164 0.682383
\(986\) 2.47214 0.0787288
\(987\) −1.52786 −0.0486324
\(988\) −23.4164 −0.744975
\(989\) −57.8885 −1.84075
\(990\) 0 0
\(991\) 37.3050 1.18503 0.592515 0.805559i \(-0.298135\pi\)
0.592515 + 0.805559i \(0.298135\pi\)
\(992\) −8.94427 −0.283981
\(993\) −36.9443 −1.17239
\(994\) 8.00000 0.253745
\(995\) −5.52786 −0.175245
\(996\) 25.8885 0.820310
\(997\) −31.5279 −0.998497 −0.499249 0.866459i \(-0.666391\pi\)
−0.499249 + 0.866459i \(0.666391\pi\)
\(998\) 29.3050 0.927632
\(999\) 46.8328 1.48172
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 8470.2.a.bj.1.1 2
11.10 odd 2 8470.2.a.bv.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.bj.1.1 2 1.1 even 1 trivial
8470.2.a.bv.1.1 yes 2 11.10 odd 2