Properties

Label 8470.2.a.cs.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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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: \(4\)
Coefficient field: 4.4.4400.1
Defining polynomial: \(x^{4} - 7 x^{2} + 11\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.54336\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.54336 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.54336 q^{6} +1.00000 q^{7} +1.00000 q^{8} +3.46869 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.54336 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.54336 q^{6} +1.00000 q^{7} +1.00000 q^{8} +3.46869 q^{9} +1.00000 q^{10} -2.54336 q^{12} -6.42254 q^{13} +1.00000 q^{14} -2.54336 q^{15} +1.00000 q^{16} -3.95385 q^{17} +3.46869 q^{18} +2.46869 q^{19} +1.00000 q^{20} -2.54336 q^{21} +4.99442 q^{23} -2.54336 q^{24} +1.00000 q^{25} -6.42254 q^{26} -1.19205 q^{27} +1.00000 q^{28} -9.04402 q^{29} -2.54336 q^{30} +7.88910 q^{31} +1.00000 q^{32} -3.95385 q^{34} +1.00000 q^{35} +3.46869 q^{36} +11.0440 q^{37} +2.46869 q^{38} +16.3348 q^{39} +1.00000 q^{40} -6.95385 q^{41} -2.54336 q^{42} +3.02295 q^{43} +3.46869 q^{45} +4.99442 q^{46} -8.37984 q^{47} -2.54336 q^{48} +1.00000 q^{49} +1.00000 q^{50} +10.0561 q^{51} -6.42254 q^{52} +10.0616 q^{53} -1.19205 q^{54} +1.00000 q^{56} -6.27877 q^{57} -9.04402 q^{58} -4.09575 q^{59} -2.54336 q^{60} -3.59696 q^{61} +7.88910 q^{62} +3.46869 q^{63} +1.00000 q^{64} -6.42254 q^{65} -1.28222 q^{67} -3.95385 q^{68} -12.7026 q^{69} +1.00000 q^{70} -5.90025 q^{71} +3.46869 q^{72} -10.2521 q^{73} +11.0440 q^{74} -2.54336 q^{75} +2.46869 q^{76} +16.3348 q^{78} -6.21099 q^{79} +1.00000 q^{80} -7.37426 q^{81} -6.95385 q^{82} -13.3383 q^{83} -2.54336 q^{84} -3.95385 q^{85} +3.02295 q^{86} +23.0022 q^{87} +14.7432 q^{89} +3.46869 q^{90} -6.42254 q^{91} +4.99442 q^{92} -20.0648 q^{93} -8.37984 q^{94} +2.46869 q^{95} -2.54336 q^{96} -11.9603 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} + 4q^{7} + 4q^{8} + 6q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} + 4q^{7} + 4q^{8} + 6q^{9} + 4q^{10} - 4q^{12} - 14q^{13} + 4q^{14} - 4q^{15} + 4q^{16} - 12q^{17} + 6q^{18} + 2q^{19} + 4q^{20} - 4q^{21} - 4q^{24} + 4q^{25} - 14q^{26} - 22q^{27} + 4q^{28} - 10q^{29} - 4q^{30} - 18q^{31} + 4q^{32} - 12q^{34} + 4q^{35} + 6q^{36} + 18q^{37} + 2q^{38} + 30q^{39} + 4q^{40} - 24q^{41} - 4q^{42} - 10q^{43} + 6q^{45} - 8q^{47} - 4q^{48} + 4q^{49} + 4q^{50} - 14q^{52} + 20q^{53} - 22q^{54} + 4q^{56} - 30q^{57} - 10q^{58} - 14q^{59} - 4q^{60} - 14q^{61} - 18q^{62} + 6q^{63} + 4q^{64} - 14q^{65} - 12q^{68} - 4q^{69} + 4q^{70} - 14q^{71} + 6q^{72} - 30q^{73} + 18q^{74} - 4q^{75} + 2q^{76} + 30q^{78} - 8q^{79} + 4q^{80} + 16q^{81} - 24q^{82} - 8q^{83} - 4q^{84} - 12q^{85} - 10q^{86} - 2q^{87} - 4q^{89} + 6q^{90} - 14q^{91} - 2q^{93} - 8q^{94} + 2q^{95} - 4q^{96} - 10q^{97} + 4q^{98} + O(q^{100}) \)

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
\(3\) −2.54336 −1.46841 −0.734205 0.678927i \(-0.762445\pi\)
−0.734205 + 0.678927i \(0.762445\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −2.54336 −1.03832
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 3.46869 1.15623
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −2.54336 −0.734205
\(13\) −6.42254 −1.78129 −0.890646 0.454697i \(-0.849747\pi\)
−0.890646 + 0.454697i \(0.849747\pi\)
\(14\) 1.00000 0.267261
\(15\) −2.54336 −0.656693
\(16\) 1.00000 0.250000
\(17\) −3.95385 −0.958950 −0.479475 0.877556i \(-0.659173\pi\)
−0.479475 + 0.877556i \(0.659173\pi\)
\(18\) 3.46869 0.817578
\(19\) 2.46869 0.566356 0.283178 0.959067i \(-0.408611\pi\)
0.283178 + 0.959067i \(0.408611\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.54336 −0.555007
\(22\) 0 0
\(23\) 4.99442 1.04141 0.520705 0.853737i \(-0.325669\pi\)
0.520705 + 0.853737i \(0.325669\pi\)
\(24\) −2.54336 −0.519162
\(25\) 1.00000 0.200000
\(26\) −6.42254 −1.25956
\(27\) −1.19205 −0.229410
\(28\) 1.00000 0.188982
\(29\) −9.04402 −1.67943 −0.839716 0.543026i \(-0.817279\pi\)
−0.839716 + 0.543026i \(0.817279\pi\)
\(30\) −2.54336 −0.464352
\(31\) 7.88910 1.41692 0.708462 0.705749i \(-0.249389\pi\)
0.708462 + 0.705749i \(0.249389\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.95385 −0.678080
\(35\) 1.00000 0.169031
\(36\) 3.46869 0.578115
\(37\) 11.0440 1.81563 0.907813 0.419376i \(-0.137751\pi\)
0.907813 + 0.419376i \(0.137751\pi\)
\(38\) 2.46869 0.400474
\(39\) 16.3348 2.61567
\(40\) 1.00000 0.158114
\(41\) −6.95385 −1.08601 −0.543004 0.839730i \(-0.682713\pi\)
−0.543004 + 0.839730i \(0.682713\pi\)
\(42\) −2.54336 −0.392449
\(43\) 3.02295 0.460995 0.230497 0.973073i \(-0.425965\pi\)
0.230497 + 0.973073i \(0.425965\pi\)
\(44\) 0 0
\(45\) 3.46869 0.517082
\(46\) 4.99442 0.736388
\(47\) −8.37984 −1.22232 −0.611162 0.791505i \(-0.709298\pi\)
−0.611162 + 0.791505i \(0.709298\pi\)
\(48\) −2.54336 −0.367103
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 10.0561 1.40813
\(52\) −6.42254 −0.890646
\(53\) 10.0616 1.38207 0.691037 0.722820i \(-0.257154\pi\)
0.691037 + 0.722820i \(0.257154\pi\)
\(54\) −1.19205 −0.162217
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −6.27877 −0.831644
\(58\) −9.04402 −1.18754
\(59\) −4.09575 −0.533221 −0.266610 0.963804i \(-0.585904\pi\)
−0.266610 + 0.963804i \(0.585904\pi\)
\(60\) −2.54336 −0.328347
\(61\) −3.59696 −0.460544 −0.230272 0.973126i \(-0.573962\pi\)
−0.230272 + 0.973126i \(0.573962\pi\)
\(62\) 7.88910 1.00192
\(63\) 3.46869 0.437014
\(64\) 1.00000 0.125000
\(65\) −6.42254 −0.796618
\(66\) 0 0
\(67\) −1.28222 −0.156648 −0.0783239 0.996928i \(-0.524957\pi\)
−0.0783239 + 0.996928i \(0.524957\pi\)
\(68\) −3.95385 −0.479475
\(69\) −12.7026 −1.52922
\(70\) 1.00000 0.119523
\(71\) −5.90025 −0.700231 −0.350116 0.936707i \(-0.613858\pi\)
−0.350116 + 0.936707i \(0.613858\pi\)
\(72\) 3.46869 0.408789
\(73\) −10.2521 −1.19992 −0.599960 0.800030i \(-0.704817\pi\)
−0.599960 + 0.800030i \(0.704817\pi\)
\(74\) 11.0440 1.28384
\(75\) −2.54336 −0.293682
\(76\) 2.46869 0.283178
\(77\) 0 0
\(78\) 16.3348 1.84956
\(79\) −6.21099 −0.698791 −0.349396 0.936975i \(-0.613613\pi\)
−0.349396 + 0.936975i \(0.613613\pi\)
\(80\) 1.00000 0.111803
\(81\) −7.37426 −0.819362
\(82\) −6.95385 −0.767924
\(83\) −13.3383 −1.46407 −0.732034 0.681268i \(-0.761429\pi\)
−0.732034 + 0.681268i \(0.761429\pi\)
\(84\) −2.54336 −0.277504
\(85\) −3.95385 −0.428855
\(86\) 3.02295 0.325973
\(87\) 23.0022 2.46610
\(88\) 0 0
\(89\) 14.7432 1.56278 0.781388 0.624045i \(-0.214512\pi\)
0.781388 + 0.624045i \(0.214512\pi\)
\(90\) 3.46869 0.365632
\(91\) −6.42254 −0.673265
\(92\) 4.99442 0.520705
\(93\) −20.0648 −2.08063
\(94\) −8.37984 −0.864314
\(95\) 2.46869 0.253282
\(96\) −2.54336 −0.259581
\(97\) −11.9603 −1.21439 −0.607194 0.794554i \(-0.707705\pi\)
−0.607194 + 0.794554i \(0.707705\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 2.26114 0.224992 0.112496 0.993652i \(-0.464115\pi\)
0.112496 + 0.993652i \(0.464115\pi\)
\(102\) 10.0561 0.995699
\(103\) 13.4359 1.32388 0.661940 0.749557i \(-0.269733\pi\)
0.661940 + 0.749557i \(0.269733\pi\)
\(104\) −6.42254 −0.629782
\(105\) −2.54336 −0.248207
\(106\) 10.0616 0.977274
\(107\) −13.3879 −1.29426 −0.647128 0.762381i \(-0.724030\pi\)
−0.647128 + 0.762381i \(0.724030\pi\)
\(108\) −1.19205 −0.114705
\(109\) 8.67624 0.831033 0.415516 0.909586i \(-0.363601\pi\)
0.415516 + 0.909586i \(0.363601\pi\)
\(110\) 0 0
\(111\) −28.0889 −2.66608
\(112\) 1.00000 0.0944911
\(113\) −9.16671 −0.862332 −0.431166 0.902273i \(-0.641898\pi\)
−0.431166 + 0.902273i \(0.641898\pi\)
\(114\) −6.27877 −0.588061
\(115\) 4.99442 0.465732
\(116\) −9.04402 −0.839716
\(117\) −22.2778 −2.05958
\(118\) −4.09575 −0.377044
\(119\) −3.95385 −0.362449
\(120\) −2.54336 −0.232176
\(121\) 0 0
\(122\) −3.59696 −0.325653
\(123\) 17.6862 1.59471
\(124\) 7.88910 0.708462
\(125\) 1.00000 0.0894427
\(126\) 3.46869 0.309015
\(127\) −15.2797 −1.35585 −0.677926 0.735130i \(-0.737121\pi\)
−0.677926 + 0.735130i \(0.737121\pi\)
\(128\) 1.00000 0.0883883
\(129\) −7.68845 −0.676930
\(130\) −6.42254 −0.563294
\(131\) 5.65461 0.494045 0.247023 0.969010i \(-0.420548\pi\)
0.247023 + 0.969010i \(0.420548\pi\)
\(132\) 0 0
\(133\) 2.46869 0.214063
\(134\) −1.28222 −0.110767
\(135\) −1.19205 −0.102595
\(136\) −3.95385 −0.339040
\(137\) −3.95288 −0.337717 −0.168859 0.985640i \(-0.554008\pi\)
−0.168859 + 0.985640i \(0.554008\pi\)
\(138\) −12.7026 −1.08132
\(139\) 21.0254 1.78335 0.891676 0.452673i \(-0.149530\pi\)
0.891676 + 0.452673i \(0.149530\pi\)
\(140\) 1.00000 0.0845154
\(141\) 21.3130 1.79487
\(142\) −5.90025 −0.495138
\(143\) 0 0
\(144\) 3.46869 0.289057
\(145\) −9.04402 −0.751065
\(146\) −10.2521 −0.848472
\(147\) −2.54336 −0.209773
\(148\) 11.0440 0.907813
\(149\) 8.55328 0.700712 0.350356 0.936617i \(-0.386061\pi\)
0.350356 + 0.936617i \(0.386061\pi\)
\(150\) −2.54336 −0.207665
\(151\) 2.21155 0.179973 0.0899866 0.995943i \(-0.471318\pi\)
0.0899866 + 0.995943i \(0.471318\pi\)
\(152\) 2.46869 0.200237
\(153\) −13.7147 −1.10877
\(154\) 0 0
\(155\) 7.88910 0.633668
\(156\) 16.3348 1.30783
\(157\) −13.3863 −1.06834 −0.534172 0.845376i \(-0.679377\pi\)
−0.534172 + 0.845376i \(0.679377\pi\)
\(158\) −6.21099 −0.494120
\(159\) −25.5904 −2.02945
\(160\) 1.00000 0.0790569
\(161\) 4.99442 0.393616
\(162\) −7.37426 −0.579377
\(163\) 2.05304 0.160807 0.0804033 0.996762i \(-0.474379\pi\)
0.0804033 + 0.996762i \(0.474379\pi\)
\(164\) −6.95385 −0.543004
\(165\) 0 0
\(166\) −13.3383 −1.03525
\(167\) −21.2017 −1.64064 −0.820319 0.571906i \(-0.806204\pi\)
−0.820319 + 0.571906i \(0.806204\pi\)
\(168\) −2.54336 −0.196225
\(169\) 28.2490 2.17300
\(170\) −3.95385 −0.303246
\(171\) 8.56312 0.654838
\(172\) 3.02295 0.230497
\(173\) 5.91230 0.449504 0.224752 0.974416i \(-0.427843\pi\)
0.224752 + 0.974416i \(0.427843\pi\)
\(174\) 23.0022 1.74379
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 10.4170 0.782987
\(178\) 14.7432 1.10505
\(179\) −17.8879 −1.33701 −0.668504 0.743709i \(-0.733065\pi\)
−0.668504 + 0.743709i \(0.733065\pi\)
\(180\) 3.46869 0.258541
\(181\) 11.0710 0.822899 0.411449 0.911433i \(-0.365023\pi\)
0.411449 + 0.911433i \(0.365023\pi\)
\(182\) −6.42254 −0.476070
\(183\) 9.14837 0.676267
\(184\) 4.99442 0.368194
\(185\) 11.0440 0.811973
\(186\) −20.0648 −1.47123
\(187\) 0 0
\(188\) −8.37984 −0.611162
\(189\) −1.19205 −0.0867087
\(190\) 2.46869 0.179098
\(191\) −9.55976 −0.691720 −0.345860 0.938286i \(-0.612413\pi\)
−0.345860 + 0.938286i \(0.612413\pi\)
\(192\) −2.54336 −0.183551
\(193\) 16.6846 1.20098 0.600491 0.799631i \(-0.294972\pi\)
0.600491 + 0.799631i \(0.294972\pi\)
\(194\) −11.9603 −0.858701
\(195\) 16.3348 1.16976
\(196\) 1.00000 0.0714286
\(197\) −0.533440 −0.0380060 −0.0190030 0.999819i \(-0.506049\pi\)
−0.0190030 + 0.999819i \(0.506049\pi\)
\(198\) 0 0
\(199\) −2.81443 −0.199510 −0.0997548 0.995012i \(-0.531806\pi\)
−0.0997548 + 0.995012i \(0.531806\pi\)
\(200\) 1.00000 0.0707107
\(201\) 3.26114 0.230023
\(202\) 2.26114 0.159094
\(203\) −9.04402 −0.634766
\(204\) 10.0561 0.704066
\(205\) −6.95385 −0.485678
\(206\) 13.4359 0.936124
\(207\) 17.3241 1.20411
\(208\) −6.42254 −0.445323
\(209\) 0 0
\(210\) −2.54336 −0.175509
\(211\) 3.20239 0.220461 0.110231 0.993906i \(-0.464841\pi\)
0.110231 + 0.993906i \(0.464841\pi\)
\(212\) 10.0616 0.691037
\(213\) 15.0065 1.02823
\(214\) −13.3879 −0.915177
\(215\) 3.02295 0.206163
\(216\) −1.19205 −0.0811086
\(217\) 7.88910 0.535547
\(218\) 8.67624 0.587629
\(219\) 26.0749 1.76198
\(220\) 0 0
\(221\) 25.3938 1.70817
\(222\) −28.0889 −1.88521
\(223\) −0.768535 −0.0514649 −0.0257325 0.999669i \(-0.508192\pi\)
−0.0257325 + 0.999669i \(0.508192\pi\)
\(224\) 1.00000 0.0668153
\(225\) 3.46869 0.231246
\(226\) −9.16671 −0.609761
\(227\) −6.58738 −0.437220 −0.218610 0.975812i \(-0.570152\pi\)
−0.218610 + 0.975812i \(0.570152\pi\)
\(228\) −6.27877 −0.415822
\(229\) −23.2829 −1.53857 −0.769287 0.638903i \(-0.779389\pi\)
−0.769287 + 0.638903i \(0.779389\pi\)
\(230\) 4.99442 0.329323
\(231\) 0 0
\(232\) −9.04402 −0.593769
\(233\) 1.72239 0.112837 0.0564186 0.998407i \(-0.482032\pi\)
0.0564186 + 0.998407i \(0.482032\pi\)
\(234\) −22.2778 −1.45635
\(235\) −8.37984 −0.546640
\(236\) −4.09575 −0.266610
\(237\) 15.7968 1.02611
\(238\) −3.95385 −0.256290
\(239\) −11.6437 −0.753169 −0.376585 0.926382i \(-0.622902\pi\)
−0.376585 + 0.926382i \(0.622902\pi\)
\(240\) −2.54336 −0.164173
\(241\) 9.58236 0.617254 0.308627 0.951183i \(-0.400130\pi\)
0.308627 + 0.951183i \(0.400130\pi\)
\(242\) 0 0
\(243\) 22.3316 1.43257
\(244\) −3.59696 −0.230272
\(245\) 1.00000 0.0638877
\(246\) 17.6862 1.12763
\(247\) −15.8553 −1.00885
\(248\) 7.88910 0.500958
\(249\) 33.9241 2.14985
\(250\) 1.00000 0.0632456
\(251\) −8.45409 −0.533618 −0.266809 0.963749i \(-0.585969\pi\)
−0.266809 + 0.963749i \(0.585969\pi\)
\(252\) 3.46869 0.218507
\(253\) 0 0
\(254\) −15.2797 −0.958732
\(255\) 10.0561 0.629736
\(256\) 1.00000 0.0625000
\(257\) −20.3754 −1.27098 −0.635492 0.772108i \(-0.719203\pi\)
−0.635492 + 0.772108i \(0.719203\pi\)
\(258\) −7.68845 −0.478662
\(259\) 11.0440 0.686242
\(260\) −6.42254 −0.398309
\(261\) −31.3709 −1.94181
\(262\) 5.65461 0.349343
\(263\) −16.1920 −0.998444 −0.499222 0.866474i \(-0.666381\pi\)
−0.499222 + 0.866474i \(0.666381\pi\)
\(264\) 0 0
\(265\) 10.0616 0.618082
\(266\) 2.46869 0.151365
\(267\) −37.4973 −2.29480
\(268\) −1.28222 −0.0783239
\(269\) −26.8144 −1.63490 −0.817452 0.575996i \(-0.804614\pi\)
−0.817452 + 0.575996i \(0.804614\pi\)
\(270\) −1.19205 −0.0725457
\(271\) 22.4312 1.36260 0.681300 0.732004i \(-0.261415\pi\)
0.681300 + 0.732004i \(0.261415\pi\)
\(272\) −3.95385 −0.239737
\(273\) 16.3348 0.988630
\(274\) −3.95288 −0.238802
\(275\) 0 0
\(276\) −12.7026 −0.764608
\(277\) −22.1150 −1.32876 −0.664380 0.747395i \(-0.731305\pi\)
−0.664380 + 0.747395i \(0.731305\pi\)
\(278\) 21.0254 1.26102
\(279\) 27.3648 1.63829
\(280\) 1.00000 0.0597614
\(281\) 12.5703 0.749882 0.374941 0.927049i \(-0.377663\pi\)
0.374941 + 0.927049i \(0.377663\pi\)
\(282\) 21.3130 1.26917
\(283\) −5.10477 −0.303447 −0.151723 0.988423i \(-0.548482\pi\)
−0.151723 + 0.988423i \(0.548482\pi\)
\(284\) −5.90025 −0.350116
\(285\) −6.27877 −0.371922
\(286\) 0 0
\(287\) −6.95385 −0.410473
\(288\) 3.46869 0.204395
\(289\) −1.36707 −0.0804158
\(290\) −9.04402 −0.531083
\(291\) 30.4194 1.78322
\(292\) −10.2521 −0.599960
\(293\) −17.2556 −1.00808 −0.504041 0.863680i \(-0.668154\pi\)
−0.504041 + 0.863680i \(0.668154\pi\)
\(294\) −2.54336 −0.148332
\(295\) −4.09575 −0.238464
\(296\) 11.0440 0.641921
\(297\) 0 0
\(298\) 8.55328 0.495478
\(299\) −32.0769 −1.85505
\(300\) −2.54336 −0.146841
\(301\) 3.02295 0.174240
\(302\) 2.21155 0.127260
\(303\) −5.75091 −0.330381
\(304\) 2.46869 0.141589
\(305\) −3.59696 −0.205961
\(306\) −13.7147 −0.784016
\(307\) −24.8048 −1.41569 −0.707844 0.706369i \(-0.750332\pi\)
−0.707844 + 0.706369i \(0.750332\pi\)
\(308\) 0 0
\(309\) −34.1724 −1.94400
\(310\) 7.88910 0.448071
\(311\) −15.0718 −0.854645 −0.427322 0.904099i \(-0.640543\pi\)
−0.427322 + 0.904099i \(0.640543\pi\)
\(312\) 16.3348 0.924778
\(313\) −14.6737 −0.829406 −0.414703 0.909957i \(-0.636115\pi\)
−0.414703 + 0.909957i \(0.636115\pi\)
\(314\) −13.3863 −0.755433
\(315\) 3.46869 0.195439
\(316\) −6.21099 −0.349396
\(317\) −14.2689 −0.801423 −0.400712 0.916204i \(-0.631237\pi\)
−0.400712 + 0.916204i \(0.631237\pi\)
\(318\) −25.5904 −1.43504
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 34.0502 1.90050
\(322\) 4.99442 0.278328
\(323\) −9.76083 −0.543107
\(324\) −7.37426 −0.409681
\(325\) −6.42254 −0.356258
\(326\) 2.05304 0.113707
\(327\) −22.0668 −1.22030
\(328\) −6.95385 −0.383962
\(329\) −8.37984 −0.461995
\(330\) 0 0
\(331\) −12.9209 −0.710197 −0.355099 0.934829i \(-0.615553\pi\)
−0.355099 + 0.934829i \(0.615553\pi\)
\(332\) −13.3383 −0.732034
\(333\) 38.3083 2.09928
\(334\) −21.2017 −1.16011
\(335\) −1.28222 −0.0700550
\(336\) −2.54336 −0.138752
\(337\) −20.6003 −1.12217 −0.561086 0.827758i \(-0.689616\pi\)
−0.561086 + 0.827758i \(0.689616\pi\)
\(338\) 28.2490 1.53654
\(339\) 23.3143 1.26626
\(340\) −3.95385 −0.214428
\(341\) 0 0
\(342\) 8.56312 0.463040
\(343\) 1.00000 0.0539949
\(344\) 3.02295 0.162986
\(345\) −12.7026 −0.683887
\(346\) 5.91230 0.317847
\(347\) 3.81140 0.204607 0.102303 0.994753i \(-0.467379\pi\)
0.102303 + 0.994753i \(0.467379\pi\)
\(348\) 23.0022 1.23305
\(349\) 19.7782 1.05870 0.529351 0.848403i \(-0.322435\pi\)
0.529351 + 0.848403i \(0.322435\pi\)
\(350\) 1.00000 0.0534522
\(351\) 7.65598 0.408646
\(352\) 0 0
\(353\) −2.85223 −0.151809 −0.0759044 0.997115i \(-0.524184\pi\)
−0.0759044 + 0.997115i \(0.524184\pi\)
\(354\) 10.4170 0.553655
\(355\) −5.90025 −0.313153
\(356\) 14.7432 0.781388
\(357\) 10.0561 0.532224
\(358\) −17.8879 −0.945407
\(359\) −16.9374 −0.893921 −0.446960 0.894554i \(-0.647494\pi\)
−0.446960 + 0.894554i \(0.647494\pi\)
\(360\) 3.46869 0.182816
\(361\) −12.9056 −0.679241
\(362\) 11.0710 0.581877
\(363\) 0 0
\(364\) −6.42254 −0.336633
\(365\) −10.2521 −0.536621
\(366\) 9.14837 0.478193
\(367\) −20.0055 −1.04428 −0.522139 0.852860i \(-0.674866\pi\)
−0.522139 + 0.852860i \(0.674866\pi\)
\(368\) 4.99442 0.260352
\(369\) −24.1207 −1.25568
\(370\) 11.0440 0.574151
\(371\) 10.0616 0.522375
\(372\) −20.0648 −1.04031
\(373\) 23.1627 1.19932 0.599660 0.800255i \(-0.295303\pi\)
0.599660 + 0.800255i \(0.295303\pi\)
\(374\) 0 0
\(375\) −2.54336 −0.131339
\(376\) −8.37984 −0.432157
\(377\) 58.0856 2.99156
\(378\) −1.19205 −0.0613123
\(379\) 32.6068 1.67490 0.837450 0.546514i \(-0.184046\pi\)
0.837450 + 0.546514i \(0.184046\pi\)
\(380\) 2.46869 0.126641
\(381\) 38.8617 1.99095
\(382\) −9.55976 −0.489120
\(383\) 5.16926 0.264137 0.132068 0.991241i \(-0.457838\pi\)
0.132068 + 0.991241i \(0.457838\pi\)
\(384\) −2.54336 −0.129790
\(385\) 0 0
\(386\) 16.6846 0.849223
\(387\) 10.4857 0.533016
\(388\) −11.9603 −0.607194
\(389\) −0.868284 −0.0440237 −0.0220119 0.999758i \(-0.507007\pi\)
−0.0220119 + 0.999758i \(0.507007\pi\)
\(390\) 16.3348 0.827147
\(391\) −19.7472 −0.998659
\(392\) 1.00000 0.0505076
\(393\) −14.3817 −0.725461
\(394\) −0.533440 −0.0268743
\(395\) −6.21099 −0.312509
\(396\) 0 0
\(397\) −16.7823 −0.842280 −0.421140 0.906996i \(-0.638370\pi\)
−0.421140 + 0.906996i \(0.638370\pi\)
\(398\) −2.81443 −0.141075
\(399\) −6.27877 −0.314332
\(400\) 1.00000 0.0500000
\(401\) −1.66393 −0.0830925 −0.0415463 0.999137i \(-0.513228\pi\)
−0.0415463 + 0.999137i \(0.513228\pi\)
\(402\) 3.26114 0.162651
\(403\) −50.6681 −2.52396
\(404\) 2.26114 0.112496
\(405\) −7.37426 −0.366430
\(406\) −9.04402 −0.448847
\(407\) 0 0
\(408\) 10.0561 0.497850
\(409\) 20.4530 1.01133 0.505667 0.862729i \(-0.331246\pi\)
0.505667 + 0.862729i \(0.331246\pi\)
\(410\) −6.95385 −0.343426
\(411\) 10.0536 0.495907
\(412\) 13.4359 0.661940
\(413\) −4.09575 −0.201538
\(414\) 17.3241 0.851433
\(415\) −13.3383 −0.654751
\(416\) −6.42254 −0.314891
\(417\) −53.4753 −2.61869
\(418\) 0 0
\(419\) 18.0870 0.883609 0.441804 0.897111i \(-0.354339\pi\)
0.441804 + 0.897111i \(0.354339\pi\)
\(420\) −2.54336 −0.124103
\(421\) −14.6364 −0.713333 −0.356667 0.934232i \(-0.616087\pi\)
−0.356667 + 0.934232i \(0.616087\pi\)
\(422\) 3.20239 0.155890
\(423\) −29.0671 −1.41329
\(424\) 10.0616 0.488637
\(425\) −3.95385 −0.191790
\(426\) 15.0065 0.727066
\(427\) −3.59696 −0.174069
\(428\) −13.3879 −0.647128
\(429\) 0 0
\(430\) 3.02295 0.145779
\(431\) 0.629485 0.0303212 0.0151606 0.999885i \(-0.495174\pi\)
0.0151606 + 0.999885i \(0.495174\pi\)
\(432\) −1.19205 −0.0573524
\(433\) −15.1788 −0.729445 −0.364722 0.931116i \(-0.618836\pi\)
−0.364722 + 0.931116i \(0.618836\pi\)
\(434\) 7.88910 0.378689
\(435\) 23.0022 1.10287
\(436\) 8.67624 0.415516
\(437\) 12.3297 0.589809
\(438\) 26.0749 1.24590
\(439\) −12.5690 −0.599888 −0.299944 0.953957i \(-0.596968\pi\)
−0.299944 + 0.953957i \(0.596968\pi\)
\(440\) 0 0
\(441\) 3.46869 0.165176
\(442\) 25.3938 1.20786
\(443\) −2.85871 −0.135821 −0.0679106 0.997691i \(-0.521633\pi\)
−0.0679106 + 0.997691i \(0.521633\pi\)
\(444\) −28.0889 −1.33304
\(445\) 14.7432 0.698895
\(446\) −0.768535 −0.0363912
\(447\) −21.7541 −1.02893
\(448\) 1.00000 0.0472456
\(449\) 28.4736 1.34375 0.671877 0.740663i \(-0.265488\pi\)
0.671877 + 0.740663i \(0.265488\pi\)
\(450\) 3.46869 0.163516
\(451\) 0 0
\(452\) −9.16671 −0.431166
\(453\) −5.62477 −0.264275
\(454\) −6.58738 −0.309161
\(455\) −6.42254 −0.301093
\(456\) −6.27877 −0.294030
\(457\) −7.63383 −0.357096 −0.178548 0.983931i \(-0.557140\pi\)
−0.178548 + 0.983931i \(0.557140\pi\)
\(458\) −23.2829 −1.08794
\(459\) 4.71318 0.219992
\(460\) 4.99442 0.232866
\(461\) 4.72213 0.219931 0.109966 0.993935i \(-0.464926\pi\)
0.109966 + 0.993935i \(0.464926\pi\)
\(462\) 0 0
\(463\) −19.4328 −0.903117 −0.451558 0.892242i \(-0.649132\pi\)
−0.451558 + 0.892242i \(0.649132\pi\)
\(464\) −9.04402 −0.419858
\(465\) −20.0648 −0.930485
\(466\) 1.72239 0.0797880
\(467\) −13.5440 −0.626740 −0.313370 0.949631i \(-0.601458\pi\)
−0.313370 + 0.949631i \(0.601458\pi\)
\(468\) −22.2778 −1.02979
\(469\) −1.28222 −0.0592073
\(470\) −8.37984 −0.386533
\(471\) 34.0462 1.56877
\(472\) −4.09575 −0.188522
\(473\) 0 0
\(474\) 15.7968 0.725571
\(475\) 2.46869 0.113271
\(476\) −3.95385 −0.181224
\(477\) 34.9007 1.59799
\(478\) −11.6437 −0.532571
\(479\) −4.33024 −0.197854 −0.0989269 0.995095i \(-0.531541\pi\)
−0.0989269 + 0.995095i \(0.531541\pi\)
\(480\) −2.54336 −0.116088
\(481\) −70.9307 −3.23416
\(482\) 9.58236 0.436465
\(483\) −12.7026 −0.577990
\(484\) 0 0
\(485\) −11.9603 −0.543090
\(486\) 22.3316 1.01298
\(487\) −36.5760 −1.65742 −0.828708 0.559682i \(-0.810923\pi\)
−0.828708 + 0.559682i \(0.810923\pi\)
\(488\) −3.59696 −0.162827
\(489\) −5.22163 −0.236130
\(490\) 1.00000 0.0451754
\(491\) 31.7227 1.43163 0.715813 0.698292i \(-0.246056\pi\)
0.715813 + 0.698292i \(0.246056\pi\)
\(492\) 17.6862 0.797354
\(493\) 35.7587 1.61049
\(494\) −15.8553 −0.713362
\(495\) 0 0
\(496\) 7.88910 0.354231
\(497\) −5.90025 −0.264662
\(498\) 33.9241 1.52018
\(499\) 36.7983 1.64732 0.823659 0.567085i \(-0.191929\pi\)
0.823659 + 0.567085i \(0.191929\pi\)
\(500\) 1.00000 0.0447214
\(501\) 53.9236 2.40913
\(502\) −8.45409 −0.377325
\(503\) −9.16098 −0.408468 −0.204234 0.978922i \(-0.565470\pi\)
−0.204234 + 0.978922i \(0.565470\pi\)
\(504\) 3.46869 0.154508
\(505\) 2.26114 0.100620
\(506\) 0 0
\(507\) −71.8475 −3.19086
\(508\) −15.2797 −0.677926
\(509\) 14.9999 0.664860 0.332430 0.943128i \(-0.392131\pi\)
0.332430 + 0.943128i \(0.392131\pi\)
\(510\) 10.0561 0.445290
\(511\) −10.2521 −0.453527
\(512\) 1.00000 0.0441942
\(513\) −2.94280 −0.129928
\(514\) −20.3754 −0.898721
\(515\) 13.4359 0.592057
\(516\) −7.68845 −0.338465
\(517\) 0 0
\(518\) 11.0440 0.485246
\(519\) −15.0371 −0.660057
\(520\) −6.42254 −0.281647
\(521\) −16.8142 −0.736642 −0.368321 0.929699i \(-0.620067\pi\)
−0.368321 + 0.929699i \(0.620067\pi\)
\(522\) −31.3709 −1.37307
\(523\) −1.20886 −0.0528598 −0.0264299 0.999651i \(-0.508414\pi\)
−0.0264299 + 0.999651i \(0.508414\pi\)
\(524\) 5.65461 0.247023
\(525\) −2.54336 −0.111001
\(526\) −16.1920 −0.706007
\(527\) −31.1923 −1.35876
\(528\) 0 0
\(529\) 1.94427 0.0845336
\(530\) 10.0616 0.437050
\(531\) −14.2069 −0.616526
\(532\) 2.46869 0.107031
\(533\) 44.6614 1.93450
\(534\) −37.4973 −1.62267
\(535\) −13.3879 −0.578809
\(536\) −1.28222 −0.0553834
\(537\) 45.4955 1.96328
\(538\) −26.8144 −1.15605
\(539\) 0 0
\(540\) −1.19205 −0.0512976
\(541\) 4.66793 0.200690 0.100345 0.994953i \(-0.468005\pi\)
0.100345 + 0.994953i \(0.468005\pi\)
\(542\) 22.4312 0.963504
\(543\) −28.1575 −1.20835
\(544\) −3.95385 −0.169520
\(545\) 8.67624 0.371649
\(546\) 16.3348 0.699067
\(547\) −22.1545 −0.947260 −0.473630 0.880724i \(-0.657057\pi\)
−0.473630 + 0.880724i \(0.657057\pi\)
\(548\) −3.95288 −0.168859
\(549\) −12.4767 −0.532494
\(550\) 0 0
\(551\) −22.3269 −0.951157
\(552\) −12.7026 −0.540660
\(553\) −6.21099 −0.264118
\(554\) −22.1150 −0.939576
\(555\) −28.0889 −1.19231
\(556\) 21.0254 0.891676
\(557\) 37.8026 1.60175 0.800875 0.598832i \(-0.204368\pi\)
0.800875 + 0.598832i \(0.204368\pi\)
\(558\) 27.3648 1.15845
\(559\) −19.4150 −0.821167
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 12.5703 0.530247
\(563\) 9.89847 0.417171 0.208585 0.978004i \(-0.433114\pi\)
0.208585 + 0.978004i \(0.433114\pi\)
\(564\) 21.3130 0.897437
\(565\) −9.16671 −0.385647
\(566\) −5.10477 −0.214569
\(567\) −7.37426 −0.309690
\(568\) −5.90025 −0.247569
\(569\) −4.95198 −0.207598 −0.103799 0.994598i \(-0.533100\pi\)
−0.103799 + 0.994598i \(0.533100\pi\)
\(570\) −6.27877 −0.262989
\(571\) −8.53293 −0.357092 −0.178546 0.983932i \(-0.557139\pi\)
−0.178546 + 0.983932i \(0.557139\pi\)
\(572\) 0 0
\(573\) 24.3139 1.01573
\(574\) −6.95385 −0.290248
\(575\) 4.99442 0.208282
\(576\) 3.46869 0.144529
\(577\) −9.82861 −0.409170 −0.204585 0.978849i \(-0.565585\pi\)
−0.204585 + 0.978849i \(0.565585\pi\)
\(578\) −1.36707 −0.0568626
\(579\) −42.4349 −1.76354
\(580\) −9.04402 −0.375532
\(581\) −13.3383 −0.553365
\(582\) 30.4194 1.26093
\(583\) 0 0
\(584\) −10.2521 −0.424236
\(585\) −22.2778 −0.921074
\(586\) −17.2556 −0.712821
\(587\) 20.0651 0.828175 0.414088 0.910237i \(-0.364101\pi\)
0.414088 + 0.910237i \(0.364101\pi\)
\(588\) −2.54336 −0.104886
\(589\) 19.4757 0.802484
\(590\) −4.09575 −0.168619
\(591\) 1.35673 0.0558085
\(592\) 11.0440 0.453906
\(593\) −3.85286 −0.158218 −0.0791090 0.996866i \(-0.525208\pi\)
−0.0791090 + 0.996866i \(0.525208\pi\)
\(594\) 0 0
\(595\) −3.95385 −0.162092
\(596\) 8.55328 0.350356
\(597\) 7.15811 0.292962
\(598\) −32.0769 −1.31172
\(599\) 7.38576 0.301774 0.150887 0.988551i \(-0.451787\pi\)
0.150887 + 0.988551i \(0.451787\pi\)
\(600\) −2.54336 −0.103832
\(601\) −25.7196 −1.04912 −0.524562 0.851372i \(-0.675771\pi\)
−0.524562 + 0.851372i \(0.675771\pi\)
\(602\) 3.02295 0.123206
\(603\) −4.44762 −0.181121
\(604\) 2.21155 0.0899866
\(605\) 0 0
\(606\) −5.75091 −0.233615
\(607\) −3.02418 −0.122748 −0.0613738 0.998115i \(-0.519548\pi\)
−0.0613738 + 0.998115i \(0.519548\pi\)
\(608\) 2.46869 0.100119
\(609\) 23.0022 0.932097
\(610\) −3.59696 −0.145637
\(611\) 53.8198 2.17732
\(612\) −13.7147 −0.554383
\(613\) 20.7030 0.836185 0.418093 0.908404i \(-0.362699\pi\)
0.418093 + 0.908404i \(0.362699\pi\)
\(614\) −24.8048 −1.00104
\(615\) 17.6862 0.713175
\(616\) 0 0
\(617\) 16.2405 0.653817 0.326909 0.945056i \(-0.393993\pi\)
0.326909 + 0.945056i \(0.393993\pi\)
\(618\) −34.1724 −1.37461
\(619\) −35.9490 −1.44491 −0.722457 0.691416i \(-0.756987\pi\)
−0.722457 + 0.691416i \(0.756987\pi\)
\(620\) 7.88910 0.316834
\(621\) −5.95359 −0.238909
\(622\) −15.0718 −0.604325
\(623\) 14.7432 0.590674
\(624\) 16.3348 0.653917
\(625\) 1.00000 0.0400000
\(626\) −14.6737 −0.586479
\(627\) 0 0
\(628\) −13.3863 −0.534172
\(629\) −43.6664 −1.74109
\(630\) 3.46869 0.138196
\(631\) −41.4726 −1.65100 −0.825499 0.564403i \(-0.809106\pi\)
−0.825499 + 0.564403i \(0.809106\pi\)
\(632\) −6.21099 −0.247060
\(633\) −8.14483 −0.323728
\(634\) −14.2689 −0.566692
\(635\) −15.2797 −0.606355
\(636\) −25.5904 −1.01473
\(637\) −6.42254 −0.254470
\(638\) 0 0
\(639\) −20.4661 −0.809628
\(640\) 1.00000 0.0395285
\(641\) 3.17884 0.125557 0.0627783 0.998027i \(-0.480004\pi\)
0.0627783 + 0.998027i \(0.480004\pi\)
\(642\) 34.0502 1.34386
\(643\) 20.0750 0.791680 0.395840 0.918320i \(-0.370454\pi\)
0.395840 + 0.918320i \(0.370454\pi\)
\(644\) 4.99442 0.196808
\(645\) −7.68845 −0.302732
\(646\) −9.76083 −0.384035
\(647\) −38.1738 −1.50077 −0.750383 0.661003i \(-0.770131\pi\)
−0.750383 + 0.661003i \(0.770131\pi\)
\(648\) −7.37426 −0.289688
\(649\) 0 0
\(650\) −6.42254 −0.251913
\(651\) −20.0648 −0.786403
\(652\) 2.05304 0.0804033
\(653\) 44.2896 1.73319 0.866594 0.499014i \(-0.166304\pi\)
0.866594 + 0.499014i \(0.166304\pi\)
\(654\) −22.0668 −0.862880
\(655\) 5.65461 0.220944
\(656\) −6.95385 −0.271502
\(657\) −35.5614 −1.38738
\(658\) −8.37984 −0.326680
\(659\) −30.2781 −1.17947 −0.589734 0.807598i \(-0.700767\pi\)
−0.589734 + 0.807598i \(0.700767\pi\)
\(660\) 0 0
\(661\) 7.16382 0.278640 0.139320 0.990247i \(-0.455508\pi\)
0.139320 + 0.990247i \(0.455508\pi\)
\(662\) −12.9209 −0.502185
\(663\) −64.5855 −2.50829
\(664\) −13.3383 −0.517626
\(665\) 2.46869 0.0957317
\(666\) 38.3083 1.48442
\(667\) −45.1697 −1.74898
\(668\) −21.2017 −0.820319
\(669\) 1.95466 0.0755717
\(670\) −1.28222 −0.0495364
\(671\) 0 0
\(672\) −2.54336 −0.0981123
\(673\) −22.0439 −0.849730 −0.424865 0.905257i \(-0.639678\pi\)
−0.424865 + 0.905257i \(0.639678\pi\)
\(674\) −20.6003 −0.793495
\(675\) −1.19205 −0.0458819
\(676\) 28.2490 1.08650
\(677\) −34.8050 −1.33767 −0.668833 0.743413i \(-0.733206\pi\)
−0.668833 + 0.743413i \(0.733206\pi\)
\(678\) 23.3143 0.895379
\(679\) −11.9603 −0.458995
\(680\) −3.95385 −0.151623
\(681\) 16.7541 0.642018
\(682\) 0 0
\(683\) 35.0185 1.33995 0.669973 0.742385i \(-0.266306\pi\)
0.669973 + 0.742385i \(0.266306\pi\)
\(684\) 8.56312 0.327419
\(685\) −3.95288 −0.151032
\(686\) 1.00000 0.0381802
\(687\) 59.2167 2.25926
\(688\) 3.02295 0.115249
\(689\) −64.6213 −2.46188
\(690\) −12.7026 −0.483581
\(691\) −19.5535 −0.743849 −0.371925 0.928263i \(-0.621302\pi\)
−0.371925 + 0.928263i \(0.621302\pi\)
\(692\) 5.91230 0.224752
\(693\) 0 0
\(694\) 3.81140 0.144679
\(695\) 21.0254 0.797540
\(696\) 23.0022 0.871897
\(697\) 27.4945 1.04143
\(698\) 19.7782 0.748616
\(699\) −4.38065 −0.165691
\(700\) 1.00000 0.0377964
\(701\) 13.9458 0.526726 0.263363 0.964697i \(-0.415168\pi\)
0.263363 + 0.964697i \(0.415168\pi\)
\(702\) 7.65598 0.288956
\(703\) 27.2643 1.02829
\(704\) 0 0
\(705\) 21.3130 0.802692
\(706\) −2.85223 −0.107345
\(707\) 2.26114 0.0850391
\(708\) 10.4170 0.391493
\(709\) −0.549195 −0.0206255 −0.0103127 0.999947i \(-0.503283\pi\)
−0.0103127 + 0.999947i \(0.503283\pi\)
\(710\) −5.90025 −0.221433
\(711\) −21.5440 −0.807963
\(712\) 14.7432 0.552525
\(713\) 39.4015 1.47560
\(714\) 10.0561 0.376339
\(715\) 0 0
\(716\) −17.8879 −0.668504
\(717\) 29.6142 1.10596
\(718\) −16.9374 −0.632097
\(719\) −25.8914 −0.965586 −0.482793 0.875735i \(-0.660378\pi\)
−0.482793 + 0.875735i \(0.660378\pi\)
\(720\) 3.46869 0.129270
\(721\) 13.4359 0.500379
\(722\) −12.9056 −0.480296
\(723\) −24.3714 −0.906383
\(724\) 11.0710 0.411449
\(725\) −9.04402 −0.335886
\(726\) 0 0
\(727\) −41.6185 −1.54355 −0.771773 0.635898i \(-0.780630\pi\)
−0.771773 + 0.635898i \(0.780630\pi\)
\(728\) −6.42254 −0.238035
\(729\) −34.6744 −1.28424
\(730\) −10.2521 −0.379448
\(731\) −11.9523 −0.442071
\(732\) 9.14837 0.338134
\(733\) 2.16850 0.0800954 0.0400477 0.999198i \(-0.487249\pi\)
0.0400477 + 0.999198i \(0.487249\pi\)
\(734\) −20.0055 −0.738417
\(735\) −2.54336 −0.0938133
\(736\) 4.99442 0.184097
\(737\) 0 0
\(738\) −24.1207 −0.887897
\(739\) −3.06748 −0.112839 −0.0564196 0.998407i \(-0.517968\pi\)
−0.0564196 + 0.998407i \(0.517968\pi\)
\(740\) 11.0440 0.405986
\(741\) 40.3257 1.48140
\(742\) 10.0616 0.369375
\(743\) −38.9833 −1.43016 −0.715079 0.699044i \(-0.753609\pi\)
−0.715079 + 0.699044i \(0.753609\pi\)
\(744\) −20.0648 −0.735613
\(745\) 8.55328 0.313368
\(746\) 23.1627 0.848047
\(747\) −46.2664 −1.69280
\(748\) 0 0
\(749\) −13.3879 −0.489183
\(750\) −2.54336 −0.0928704
\(751\) 24.2379 0.884455 0.442228 0.896903i \(-0.354188\pi\)
0.442228 + 0.896903i \(0.354188\pi\)
\(752\) −8.37984 −0.305581
\(753\) 21.5018 0.783570
\(754\) 58.0856 2.11535
\(755\) 2.21155 0.0804865
\(756\) −1.19205 −0.0433544
\(757\) −24.2156 −0.880132 −0.440066 0.897965i \(-0.645045\pi\)
−0.440066 + 0.897965i \(0.645045\pi\)
\(758\) 32.6068 1.18433
\(759\) 0 0
\(760\) 2.46869 0.0895488
\(761\) 2.07860 0.0753493 0.0376746 0.999290i \(-0.488005\pi\)
0.0376746 + 0.999290i \(0.488005\pi\)
\(762\) 38.8617 1.40781
\(763\) 8.67624 0.314101
\(764\) −9.55976 −0.345860
\(765\) −13.7147 −0.495855
\(766\) 5.16926 0.186773
\(767\) 26.3051 0.949822
\(768\) −2.54336 −0.0917757
\(769\) −14.7508 −0.531929 −0.265964 0.963983i \(-0.585690\pi\)
−0.265964 + 0.963983i \(0.585690\pi\)
\(770\) 0 0
\(771\) 51.8221 1.86633
\(772\) 16.6846 0.600491
\(773\) 49.5573 1.78245 0.891225 0.453561i \(-0.149847\pi\)
0.891225 + 0.453561i \(0.149847\pi\)
\(774\) 10.4857 0.376899
\(775\) 7.88910 0.283385
\(776\) −11.9603 −0.429351
\(777\) −28.0889 −1.00769
\(778\) −0.868284 −0.0311295
\(779\) −17.1669 −0.615068
\(780\) 16.3348 0.584881
\(781\) 0 0
\(782\) −19.7472 −0.706159
\(783\) 10.7809 0.385278
\(784\) 1.00000 0.0357143
\(785\) −13.3863 −0.477778
\(786\) −14.3817 −0.512979
\(787\) 31.0744 1.10768 0.553842 0.832622i \(-0.313161\pi\)
0.553842 + 0.832622i \(0.313161\pi\)
\(788\) −0.533440 −0.0190030
\(789\) 41.1822 1.46613
\(790\) −6.21099 −0.220977
\(791\) −9.16671 −0.325931
\(792\) 0 0
\(793\) 23.1016 0.820363
\(794\) −16.7823 −0.595582
\(795\) −25.5904 −0.907598
\(796\) −2.81443 −0.0997548
\(797\) 18.3839 0.651191 0.325595 0.945509i \(-0.394435\pi\)
0.325595 + 0.945509i \(0.394435\pi\)
\(798\) −6.27877 −0.222266
\(799\) 33.1326 1.17215
\(800\) 1.00000 0.0353553
\(801\) 51.1396 1.80693
\(802\) −1.66393 −0.0587553
\(803\) 0 0
\(804\) 3.26114 0.115012
\(805\) 4.99442 0.176030
\(806\) −50.6681 −1.78471
\(807\) 68.1988 2.40071
\(808\) 2.26114 0.0795468
\(809\) −41.4458 −1.45716 −0.728578 0.684963i \(-0.759818\pi\)
−0.728578 + 0.684963i \(0.759818\pi\)
\(810\) −7.37426 −0.259105
\(811\) 7.70150 0.270436 0.135218 0.990816i \(-0.456827\pi\)
0.135218 + 0.990816i \(0.456827\pi\)
\(812\) −9.04402 −0.317383
\(813\) −57.0507 −2.00086
\(814\) 0 0
\(815\) 2.05304 0.0719149
\(816\) 10.0561 0.352033
\(817\) 7.46272 0.261087
\(818\) 20.4530 0.715122
\(819\) −22.2778 −0.778449
\(820\) −6.95385 −0.242839
\(821\) −48.2959 −1.68554 −0.842769 0.538275i \(-0.819076\pi\)
−0.842769 + 0.538275i \(0.819076\pi\)
\(822\) 10.0536 0.350660
\(823\) 36.7609 1.28140 0.640702 0.767789i \(-0.278643\pi\)
0.640702 + 0.767789i \(0.278643\pi\)
\(824\) 13.4359 0.468062
\(825\) 0 0
\(826\) −4.09575 −0.142509
\(827\) −29.9018 −1.03979 −0.519893 0.854231i \(-0.674028\pi\)
−0.519893 + 0.854231i \(0.674028\pi\)
\(828\) 17.3241 0.602054
\(829\) 18.8334 0.654110 0.327055 0.945005i \(-0.393944\pi\)
0.327055 + 0.945005i \(0.393944\pi\)
\(830\) −13.3383 −0.462979
\(831\) 56.2464 1.95117
\(832\) −6.42254 −0.222662
\(833\) −3.95385 −0.136993
\(834\) −53.4753 −1.85170
\(835\) −21.2017 −0.733716
\(836\) 0 0
\(837\) −9.40419 −0.325056
\(838\) 18.0870 0.624806
\(839\) 7.76812 0.268185 0.134093 0.990969i \(-0.457188\pi\)
0.134093 + 0.990969i \(0.457188\pi\)
\(840\) −2.54336 −0.0877543
\(841\) 52.7943 1.82049
\(842\) −14.6364 −0.504403
\(843\) −31.9708 −1.10113
\(844\) 3.20239 0.110231
\(845\) 28.2490 0.971796
\(846\) −29.0671 −0.999346
\(847\) 0 0
\(848\) 10.0616 0.345518
\(849\) 12.9833 0.445585
\(850\) −3.95385 −0.135616
\(851\) 55.1585 1.89081
\(852\) 15.0065 0.514113
\(853\) −24.1273 −0.826104 −0.413052 0.910707i \(-0.635537\pi\)
−0.413052 + 0.910707i \(0.635537\pi\)
\(854\) −3.59696 −0.123085
\(855\) 8.56312 0.292853
\(856\) −13.3879 −0.457589
\(857\) 22.2559 0.760246 0.380123 0.924936i \(-0.375882\pi\)
0.380123 + 0.924936i \(0.375882\pi\)
\(858\) 0 0
\(859\) 5.09623 0.173881 0.0869406 0.996214i \(-0.472291\pi\)
0.0869406 + 0.996214i \(0.472291\pi\)
\(860\) 3.02295 0.103082
\(861\) 17.6862 0.602743
\(862\) 0.629485 0.0214403
\(863\) 14.7292 0.501389 0.250694 0.968066i \(-0.419341\pi\)
0.250694 + 0.968066i \(0.419341\pi\)
\(864\) −1.19205 −0.0405543
\(865\) 5.91230 0.201024
\(866\) −15.1788 −0.515795
\(867\) 3.47695 0.118083
\(868\) 7.88910 0.267774
\(869\) 0 0
\(870\) 23.0022 0.779848
\(871\) 8.23510 0.279036
\(872\) 8.67624 0.293814
\(873\) −41.4867 −1.40411
\(874\) 12.3297 0.417058
\(875\) 1.00000 0.0338062
\(876\) 26.0749 0.880988
\(877\) 50.2137 1.69560 0.847798 0.530319i \(-0.177928\pi\)
0.847798 + 0.530319i \(0.177928\pi\)
\(878\) −12.5690 −0.424185
\(879\) 43.8872 1.48028
\(880\) 0 0
\(881\) 11.6181 0.391423 0.195711 0.980662i \(-0.437298\pi\)
0.195711 + 0.980662i \(0.437298\pi\)
\(882\) 3.46869 0.116797
\(883\) 2.75951 0.0928650 0.0464325 0.998921i \(-0.485215\pi\)
0.0464325 + 0.998921i \(0.485215\pi\)
\(884\) 25.3938 0.854085
\(885\) 10.4170 0.350162
\(886\) −2.85871 −0.0960401
\(887\) 34.8529 1.17024 0.585122 0.810945i \(-0.301046\pi\)
0.585122 + 0.810945i \(0.301046\pi\)
\(888\) −28.0889 −0.942603
\(889\) −15.2797 −0.512464
\(890\) 14.7432 0.494193
\(891\) 0 0
\(892\) −0.768535 −0.0257325
\(893\) −20.6872 −0.692271
\(894\) −21.7541 −0.727566
\(895\) −17.8879 −0.597928
\(896\) 1.00000 0.0334077
\(897\) 81.5831 2.72398
\(898\) 28.4736 0.950178
\(899\) −71.3492 −2.37963
\(900\) 3.46869 0.115623
\(901\) −39.7822 −1.32534
\(902\) 0 0
\(903\) −7.68845 −0.255855
\(904\) −9.16671 −0.304880
\(905\) 11.0710 0.368011
\(906\) −5.62477 −0.186870
\(907\) −5.25497 −0.174488 −0.0872442 0.996187i \(-0.527806\pi\)
−0.0872442 + 0.996187i \(0.527806\pi\)
\(908\) −6.58738 −0.218610
\(909\) 7.84321 0.260143
\(910\) −6.42254 −0.212905
\(911\) −11.1799 −0.370407 −0.185204 0.982700i \(-0.559294\pi\)
−0.185204 + 0.982700i \(0.559294\pi\)
\(912\) −6.27877 −0.207911
\(913\) 0 0
\(914\) −7.63383 −0.252505
\(915\) 9.14837 0.302436
\(916\) −23.2829 −0.769287
\(917\) 5.65461 0.186732
\(918\) 4.71318 0.155558
\(919\) 24.3862 0.804428 0.402214 0.915546i \(-0.368241\pi\)
0.402214 + 0.915546i \(0.368241\pi\)
\(920\) 4.99442 0.164661
\(921\) 63.0877 2.07881
\(922\) 4.72213 0.155515
\(923\) 37.8946 1.24732
\(924\) 0 0
\(925\) 11.0440 0.363125
\(926\) −19.4328 −0.638600
\(927\) 46.6050 1.53071
\(928\) −9.04402 −0.296885
\(929\) 19.9250 0.653718 0.326859 0.945073i \(-0.394010\pi\)
0.326859 + 0.945073i \(0.394010\pi\)
\(930\) −20.0648 −0.657952
\(931\) 2.46869 0.0809080
\(932\) 1.72239 0.0564186
\(933\) 38.3331 1.25497
\(934\) −13.5440 −0.443172
\(935\) 0 0
\(936\) −22.2778 −0.728173
\(937\) 47.9680 1.56705 0.783523 0.621363i \(-0.213421\pi\)
0.783523 + 0.621363i \(0.213421\pi\)
\(938\) −1.28222 −0.0418659
\(939\) 37.3205 1.21791
\(940\) −8.37984 −0.273320
\(941\) 43.8482 1.42941 0.714704 0.699427i \(-0.246561\pi\)
0.714704 + 0.699427i \(0.246561\pi\)
\(942\) 34.0462 1.10929
\(943\) −34.7305 −1.13098
\(944\) −4.09575 −0.133305
\(945\) −1.19205 −0.0387773
\(946\) 0 0
\(947\) −27.0791 −0.879951 −0.439976 0.898010i \(-0.645013\pi\)
−0.439976 + 0.898010i \(0.645013\pi\)
\(948\) 15.7968 0.513056
\(949\) 65.8447 2.13741
\(950\) 2.46869 0.0800949
\(951\) 36.2911 1.17682
\(952\) −3.95385 −0.128145
\(953\) 30.8977 1.00088 0.500438 0.865772i \(-0.333172\pi\)
0.500438 + 0.865772i \(0.333172\pi\)
\(954\) 34.9007 1.12995
\(955\) −9.55976 −0.309347
\(956\) −11.6437 −0.376585
\(957\) 0 0
\(958\) −4.33024 −0.139904
\(959\) −3.95288 −0.127645
\(960\) −2.54336 −0.0820867
\(961\) 31.2379 1.00767
\(962\) −70.9307 −2.28690
\(963\) −46.4384 −1.49646
\(964\) 9.58236 0.308627
\(965\) 16.6846 0.537096
\(966\) −12.7026 −0.408700
\(967\) −33.0146 −1.06168 −0.530839 0.847473i \(-0.678123\pi\)
−0.530839 + 0.847473i \(0.678123\pi\)
\(968\) 0 0
\(969\) 24.8253 0.797504
\(970\) −11.9603 −0.384023
\(971\) 17.7082 0.568283 0.284142 0.958782i \(-0.408291\pi\)
0.284142 + 0.958782i \(0.408291\pi\)
\(972\) 22.3316 0.716285
\(973\) 21.0254 0.674044
\(974\) −36.5760 −1.17197
\(975\) 16.3348 0.523134
\(976\) −3.59696 −0.115136
\(977\) 14.4633 0.462722 0.231361 0.972868i \(-0.425682\pi\)
0.231361 + 0.972868i \(0.425682\pi\)
\(978\) −5.22163 −0.166969
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 30.0952 0.960865
\(982\) 31.7227 1.01231
\(983\) −12.3858 −0.395047 −0.197523 0.980298i \(-0.563290\pi\)
−0.197523 + 0.980298i \(0.563290\pi\)
\(984\) 17.6862 0.563814
\(985\) −0.533440 −0.0169968
\(986\) 35.7587 1.13879
\(987\) 21.3130 0.678399
\(988\) −15.8553 −0.504423
\(989\) 15.0979 0.480085
\(990\) 0 0
\(991\) −48.6112 −1.54419 −0.772093 0.635509i \(-0.780790\pi\)
−0.772093 + 0.635509i \(0.780790\pi\)
\(992\) 7.88910 0.250479
\(993\) 32.8625 1.04286
\(994\) −5.90025 −0.187145
\(995\) −2.81443 −0.0892234
\(996\) 33.9241 1.07493
\(997\) 35.2749 1.11717 0.558583 0.829449i \(-0.311345\pi\)
0.558583 + 0.829449i \(0.311345\pi\)
\(998\) 36.7983 1.16483
\(999\) −13.1650 −0.416522
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.cs.1.2 4
11.2 odd 10 770.2.n.f.631.1 yes 8
11.6 odd 10 770.2.n.f.421.1 8
11.10 odd 2 8470.2.a.co.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.f.421.1 8 11.6 odd 10
770.2.n.f.631.1 yes 8 11.2 odd 10
8470.2.a.co.1.2 4 11.10 odd 2
8470.2.a.cs.1.2 4 1.1 even 1 trivial