Properties

Label 5070.2.a.bq.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +0.307979 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +0.307979 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +0.335126 q^{11} +1.00000 q^{12} -0.307979 q^{14} +1.00000 q^{15} +1.00000 q^{16} -6.85086 q^{17} -1.00000 q^{18} +5.80194 q^{19} +1.00000 q^{20} +0.307979 q^{21} -0.335126 q^{22} +2.35690 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +0.307979 q^{28} -5.91185 q^{29} -1.00000 q^{30} +0.0609989 q^{31} -1.00000 q^{32} +0.335126 q^{33} +6.85086 q^{34} +0.307979 q^{35} +1.00000 q^{36} +7.07606 q^{37} -5.80194 q^{38} -1.00000 q^{40} +3.40581 q^{41} -0.307979 q^{42} -0.0489173 q^{43} +0.335126 q^{44} +1.00000 q^{45} -2.35690 q^{46} +7.00000 q^{47} +1.00000 q^{48} -6.90515 q^{49} -1.00000 q^{50} -6.85086 q^{51} +12.1739 q^{53} -1.00000 q^{54} +0.335126 q^{55} -0.307979 q^{56} +5.80194 q^{57} +5.91185 q^{58} -13.1347 q^{59} +1.00000 q^{60} +2.14675 q^{61} -0.0609989 q^{62} +0.307979 q^{63} +1.00000 q^{64} -0.335126 q^{66} +11.9608 q^{67} -6.85086 q^{68} +2.35690 q^{69} -0.307979 q^{70} +9.56465 q^{71} -1.00000 q^{72} +2.31336 q^{73} -7.07606 q^{74} +1.00000 q^{75} +5.80194 q^{76} +0.103211 q^{77} -0.0760644 q^{79} +1.00000 q^{80} +1.00000 q^{81} -3.40581 q^{82} +3.84117 q^{83} +0.307979 q^{84} -6.85086 q^{85} +0.0489173 q^{86} -5.91185 q^{87} -0.335126 q^{88} -4.41119 q^{89} -1.00000 q^{90} +2.35690 q^{92} +0.0609989 q^{93} -7.00000 q^{94} +5.80194 q^{95} -1.00000 q^{96} +6.13169 q^{97} +6.90515 q^{98} +0.335126 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 6q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 6q^{7} - 3q^{8} + 3q^{9} - 3q^{10} + 3q^{12} - 6q^{14} + 3q^{15} + 3q^{16} - 7q^{17} - 3q^{18} + 13q^{19} + 3q^{20} + 6q^{21} + 3q^{23} - 3q^{24} + 3q^{25} + 3q^{27} + 6q^{28} - 14q^{29} - 3q^{30} + 10q^{31} - 3q^{32} + 7q^{34} + 6q^{35} + 3q^{36} + 6q^{37} - 13q^{38} - 3q^{40} - 3q^{41} - 6q^{42} + 9q^{43} + 3q^{45} - 3q^{46} + 21q^{47} + 3q^{48} + 5q^{49} - 3q^{50} - 7q^{51} + 3q^{53} - 3q^{54} - 6q^{56} + 13q^{57} + 14q^{58} + 7q^{59} + 3q^{60} - 21q^{61} - 10q^{62} + 6q^{63} + 3q^{64} + 23q^{67} - 7q^{68} + 3q^{69} - 6q^{70} + 7q^{71} - 3q^{72} + 9q^{73} - 6q^{74} + 3q^{75} + 13q^{76} - 21q^{77} + 15q^{79} + 3q^{80} + 3q^{81} + 3q^{82} + 20q^{83} + 6q^{84} - 7q^{85} - 9q^{86} - 14q^{87} + 3q^{89} - 3q^{90} + 3q^{92} + 10q^{93} - 21q^{94} + 13q^{95} - 3q^{96} + 16q^{97} - 5q^{98} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 0.307979 0.116405 0.0582025 0.998305i \(-0.481463\pi\)
0.0582025 + 0.998305i \(0.481463\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0.335126 0.101044 0.0505221 0.998723i \(-0.483911\pi\)
0.0505221 + 0.998723i \(0.483911\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −0.307979 −0.0823107
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −6.85086 −1.66158 −0.830788 0.556589i \(-0.812110\pi\)
−0.830788 + 0.556589i \(0.812110\pi\)
\(18\) −1.00000 −0.235702
\(19\) 5.80194 1.33106 0.665528 0.746373i \(-0.268206\pi\)
0.665528 + 0.746373i \(0.268206\pi\)
\(20\) 1.00000 0.223607
\(21\) 0.307979 0.0672064
\(22\) −0.335126 −0.0714490
\(23\) 2.35690 0.491447 0.245723 0.969340i \(-0.420975\pi\)
0.245723 + 0.969340i \(0.420975\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0.307979 0.0582025
\(29\) −5.91185 −1.09780 −0.548902 0.835887i \(-0.684954\pi\)
−0.548902 + 0.835887i \(0.684954\pi\)
\(30\) −1.00000 −0.182574
\(31\) 0.0609989 0.0109557 0.00547787 0.999985i \(-0.498256\pi\)
0.00547787 + 0.999985i \(0.498256\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.335126 0.0583379
\(34\) 6.85086 1.17491
\(35\) 0.307979 0.0520579
\(36\) 1.00000 0.166667
\(37\) 7.07606 1.16330 0.581649 0.813440i \(-0.302408\pi\)
0.581649 + 0.813440i \(0.302408\pi\)
\(38\) −5.80194 −0.941199
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 3.40581 0.531899 0.265949 0.963987i \(-0.414315\pi\)
0.265949 + 0.963987i \(0.414315\pi\)
\(42\) −0.307979 −0.0475221
\(43\) −0.0489173 −0.00745982 −0.00372991 0.999993i \(-0.501187\pi\)
−0.00372991 + 0.999993i \(0.501187\pi\)
\(44\) 0.335126 0.0505221
\(45\) 1.00000 0.149071
\(46\) −2.35690 −0.347505
\(47\) 7.00000 1.02105 0.510527 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.90515 −0.986450
\(50\) −1.00000 −0.141421
\(51\) −6.85086 −0.959312
\(52\) 0 0
\(53\) 12.1739 1.67221 0.836107 0.548567i \(-0.184826\pi\)
0.836107 + 0.548567i \(0.184826\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0.335126 0.0451883
\(56\) −0.307979 −0.0411554
\(57\) 5.80194 0.768485
\(58\) 5.91185 0.776264
\(59\) −13.1347 −1.70999 −0.854994 0.518638i \(-0.826439\pi\)
−0.854994 + 0.518638i \(0.826439\pi\)
\(60\) 1.00000 0.129099
\(61\) 2.14675 0.274863 0.137432 0.990511i \(-0.456115\pi\)
0.137432 + 0.990511i \(0.456115\pi\)
\(62\) −0.0609989 −0.00774687
\(63\) 0.307979 0.0388016
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −0.335126 −0.0412511
\(67\) 11.9608 1.46124 0.730620 0.682784i \(-0.239231\pi\)
0.730620 + 0.682784i \(0.239231\pi\)
\(68\) −6.85086 −0.830788
\(69\) 2.35690 0.283737
\(70\) −0.307979 −0.0368105
\(71\) 9.56465 1.13511 0.567557 0.823334i \(-0.307889\pi\)
0.567557 + 0.823334i \(0.307889\pi\)
\(72\) −1.00000 −0.117851
\(73\) 2.31336 0.270758 0.135379 0.990794i \(-0.456775\pi\)
0.135379 + 0.990794i \(0.456775\pi\)
\(74\) −7.07606 −0.822576
\(75\) 1.00000 0.115470
\(76\) 5.80194 0.665528
\(77\) 0.103211 0.0117620
\(78\) 0 0
\(79\) −0.0760644 −0.00855792 −0.00427896 0.999991i \(-0.501362\pi\)
−0.00427896 + 0.999991i \(0.501362\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −3.40581 −0.376109
\(83\) 3.84117 0.421623 0.210811 0.977527i \(-0.432389\pi\)
0.210811 + 0.977527i \(0.432389\pi\)
\(84\) 0.307979 0.0336032
\(85\) −6.85086 −0.743080
\(86\) 0.0489173 0.00527489
\(87\) −5.91185 −0.633817
\(88\) −0.335126 −0.0357245
\(89\) −4.41119 −0.467585 −0.233793 0.972286i \(-0.575114\pi\)
−0.233793 + 0.972286i \(0.575114\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 2.35690 0.245723
\(93\) 0.0609989 0.00632529
\(94\) −7.00000 −0.721995
\(95\) 5.80194 0.595266
\(96\) −1.00000 −0.102062
\(97\) 6.13169 0.622578 0.311289 0.950315i \(-0.399239\pi\)
0.311289 + 0.950315i \(0.399239\pi\)
\(98\) 6.90515 0.697525
\(99\) 0.335126 0.0336814
\(100\) 1.00000 0.100000
\(101\) 1.32975 0.132315 0.0661575 0.997809i \(-0.478926\pi\)
0.0661575 + 0.997809i \(0.478926\pi\)
\(102\) 6.85086 0.678336
\(103\) 8.92154 0.879066 0.439533 0.898227i \(-0.355144\pi\)
0.439533 + 0.898227i \(0.355144\pi\)
\(104\) 0 0
\(105\) 0.307979 0.0300556
\(106\) −12.1739 −1.18243
\(107\) 14.4644 1.39833 0.699164 0.714961i \(-0.253556\pi\)
0.699164 + 0.714961i \(0.253556\pi\)
\(108\) 1.00000 0.0962250
\(109\) 7.13706 0.683607 0.341803 0.939772i \(-0.388962\pi\)
0.341803 + 0.939772i \(0.388962\pi\)
\(110\) −0.335126 −0.0319530
\(111\) 7.07606 0.671630
\(112\) 0.307979 0.0291012
\(113\) −16.3424 −1.53737 −0.768683 0.639630i \(-0.779088\pi\)
−0.768683 + 0.639630i \(0.779088\pi\)
\(114\) −5.80194 −0.543401
\(115\) 2.35690 0.219782
\(116\) −5.91185 −0.548902
\(117\) 0 0
\(118\) 13.1347 1.20914
\(119\) −2.10992 −0.193416
\(120\) −1.00000 −0.0912871
\(121\) −10.8877 −0.989790
\(122\) −2.14675 −0.194358
\(123\) 3.40581 0.307092
\(124\) 0.0609989 0.00547787
\(125\) 1.00000 0.0894427
\(126\) −0.307979 −0.0274369
\(127\) −0.911854 −0.0809140 −0.0404570 0.999181i \(-0.512881\pi\)
−0.0404570 + 0.999181i \(0.512881\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.0489173 −0.00430693
\(130\) 0 0
\(131\) 13.0368 1.13903 0.569517 0.821980i \(-0.307130\pi\)
0.569517 + 0.821980i \(0.307130\pi\)
\(132\) 0.335126 0.0291689
\(133\) 1.78687 0.154941
\(134\) −11.9608 −1.03325
\(135\) 1.00000 0.0860663
\(136\) 6.85086 0.587456
\(137\) 12.8877 1.10107 0.550535 0.834812i \(-0.314424\pi\)
0.550535 + 0.834812i \(0.314424\pi\)
\(138\) −2.35690 −0.200632
\(139\) −2.01208 −0.170663 −0.0853313 0.996353i \(-0.527195\pi\)
−0.0853313 + 0.996353i \(0.527195\pi\)
\(140\) 0.307979 0.0260289
\(141\) 7.00000 0.589506
\(142\) −9.56465 −0.802647
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −5.91185 −0.490953
\(146\) −2.31336 −0.191455
\(147\) −6.90515 −0.569527
\(148\) 7.07606 0.581649
\(149\) 3.44935 0.282582 0.141291 0.989968i \(-0.454875\pi\)
0.141291 + 0.989968i \(0.454875\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 2.84117 0.231211 0.115605 0.993295i \(-0.463119\pi\)
0.115605 + 0.993295i \(0.463119\pi\)
\(152\) −5.80194 −0.470599
\(153\) −6.85086 −0.553859
\(154\) −0.103211 −0.00831702
\(155\) 0.0609989 0.00489955
\(156\) 0 0
\(157\) 16.2228 1.29472 0.647361 0.762184i \(-0.275873\pi\)
0.647361 + 0.762184i \(0.275873\pi\)
\(158\) 0.0760644 0.00605136
\(159\) 12.1739 0.965453
\(160\) −1.00000 −0.0790569
\(161\) 0.725873 0.0572068
\(162\) −1.00000 −0.0785674
\(163\) −0.0924579 −0.00724186 −0.00362093 0.999993i \(-0.501153\pi\)
−0.00362093 + 0.999993i \(0.501153\pi\)
\(164\) 3.40581 0.265949
\(165\) 0.335126 0.0260895
\(166\) −3.84117 −0.298132
\(167\) −9.64071 −0.746021 −0.373010 0.927827i \(-0.621674\pi\)
−0.373010 + 0.927827i \(0.621674\pi\)
\(168\) −0.307979 −0.0237611
\(169\) 0 0
\(170\) 6.85086 0.525437
\(171\) 5.80194 0.443685
\(172\) −0.0489173 −0.00372991
\(173\) 0.719169 0.0546774 0.0273387 0.999626i \(-0.491297\pi\)
0.0273387 + 0.999626i \(0.491297\pi\)
\(174\) 5.91185 0.448176
\(175\) 0.307979 0.0232810
\(176\) 0.335126 0.0252610
\(177\) −13.1347 −0.987262
\(178\) 4.41119 0.330633
\(179\) −2.01208 −0.150390 −0.0751950 0.997169i \(-0.523958\pi\)
−0.0751950 + 0.997169i \(0.523958\pi\)
\(180\) 1.00000 0.0745356
\(181\) −2.57135 −0.191127 −0.0955635 0.995423i \(-0.530465\pi\)
−0.0955635 + 0.995423i \(0.530465\pi\)
\(182\) 0 0
\(183\) 2.14675 0.158692
\(184\) −2.35690 −0.173753
\(185\) 7.07606 0.520243
\(186\) −0.0609989 −0.00447266
\(187\) −2.29590 −0.167893
\(188\) 7.00000 0.510527
\(189\) 0.307979 0.0224021
\(190\) −5.80194 −0.420917
\(191\) 3.80194 0.275099 0.137549 0.990495i \(-0.456077\pi\)
0.137549 + 0.990495i \(0.456077\pi\)
\(192\) 1.00000 0.0721688
\(193\) −8.86294 −0.637968 −0.318984 0.947760i \(-0.603342\pi\)
−0.318984 + 0.947760i \(0.603342\pi\)
\(194\) −6.13169 −0.440229
\(195\) 0 0
\(196\) −6.90515 −0.493225
\(197\) −22.6504 −1.61377 −0.806887 0.590706i \(-0.798849\pi\)
−0.806887 + 0.590706i \(0.798849\pi\)
\(198\) −0.335126 −0.0238163
\(199\) −5.46681 −0.387532 −0.193766 0.981048i \(-0.562070\pi\)
−0.193766 + 0.981048i \(0.562070\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 11.9608 0.843648
\(202\) −1.32975 −0.0935608
\(203\) −1.82072 −0.127790
\(204\) −6.85086 −0.479656
\(205\) 3.40581 0.237872
\(206\) −8.92154 −0.621593
\(207\) 2.35690 0.163816
\(208\) 0 0
\(209\) 1.94438 0.134495
\(210\) −0.307979 −0.0212525
\(211\) 11.0422 0.760177 0.380089 0.924950i \(-0.375893\pi\)
0.380089 + 0.924950i \(0.375893\pi\)
\(212\) 12.1739 0.836107
\(213\) 9.56465 0.655359
\(214\) −14.4644 −0.988767
\(215\) −0.0489173 −0.00333613
\(216\) −1.00000 −0.0680414
\(217\) 0.0187864 0.00127530
\(218\) −7.13706 −0.483383
\(219\) 2.31336 0.156322
\(220\) 0.335126 0.0225942
\(221\) 0 0
\(222\) −7.07606 −0.474914
\(223\) 18.3884 1.23138 0.615688 0.787990i \(-0.288878\pi\)
0.615688 + 0.787990i \(0.288878\pi\)
\(224\) −0.307979 −0.0205777
\(225\) 1.00000 0.0666667
\(226\) 16.3424 1.08708
\(227\) 12.0073 0.796952 0.398476 0.917179i \(-0.369539\pi\)
0.398476 + 0.917179i \(0.369539\pi\)
\(228\) 5.80194 0.384243
\(229\) 25.4601 1.68245 0.841226 0.540684i \(-0.181835\pi\)
0.841226 + 0.540684i \(0.181835\pi\)
\(230\) −2.35690 −0.155409
\(231\) 0.103211 0.00679082
\(232\) 5.91185 0.388132
\(233\) −2.10992 −0.138225 −0.0691126 0.997609i \(-0.522017\pi\)
−0.0691126 + 0.997609i \(0.522017\pi\)
\(234\) 0 0
\(235\) 7.00000 0.456630
\(236\) −13.1347 −0.854994
\(237\) −0.0760644 −0.00494091
\(238\) 2.10992 0.136766
\(239\) 3.95838 0.256046 0.128023 0.991771i \(-0.459137\pi\)
0.128023 + 0.991771i \(0.459137\pi\)
\(240\) 1.00000 0.0645497
\(241\) −23.1564 −1.49164 −0.745819 0.666149i \(-0.767942\pi\)
−0.745819 + 0.666149i \(0.767942\pi\)
\(242\) 10.8877 0.699887
\(243\) 1.00000 0.0641500
\(244\) 2.14675 0.137432
\(245\) −6.90515 −0.441154
\(246\) −3.40581 −0.217147
\(247\) 0 0
\(248\) −0.0609989 −0.00387344
\(249\) 3.84117 0.243424
\(250\) −1.00000 −0.0632456
\(251\) −5.54048 −0.349712 −0.174856 0.984594i \(-0.555946\pi\)
−0.174856 + 0.984594i \(0.555946\pi\)
\(252\) 0.307979 0.0194008
\(253\) 0.789856 0.0496578
\(254\) 0.911854 0.0572148
\(255\) −6.85086 −0.429017
\(256\) 1.00000 0.0625000
\(257\) 6.60819 0.412207 0.206104 0.978530i \(-0.433922\pi\)
0.206104 + 0.978530i \(0.433922\pi\)
\(258\) 0.0489173 0.00304546
\(259\) 2.17928 0.135414
\(260\) 0 0
\(261\) −5.91185 −0.365935
\(262\) −13.0368 −0.805418
\(263\) −0.660563 −0.0407320 −0.0203660 0.999793i \(-0.506483\pi\)
−0.0203660 + 0.999793i \(0.506483\pi\)
\(264\) −0.335126 −0.0206256
\(265\) 12.1739 0.747837
\(266\) −1.78687 −0.109560
\(267\) −4.41119 −0.269960
\(268\) 11.9608 0.730620
\(269\) −26.7928 −1.63359 −0.816794 0.576929i \(-0.804251\pi\)
−0.816794 + 0.576929i \(0.804251\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 14.0151 0.851355 0.425677 0.904875i \(-0.360036\pi\)
0.425677 + 0.904875i \(0.360036\pi\)
\(272\) −6.85086 −0.415394
\(273\) 0 0
\(274\) −12.8877 −0.778574
\(275\) 0.335126 0.0202088
\(276\) 2.35690 0.141868
\(277\) −23.4252 −1.40748 −0.703742 0.710456i \(-0.748489\pi\)
−0.703742 + 0.710456i \(0.748489\pi\)
\(278\) 2.01208 0.120677
\(279\) 0.0609989 0.00365191
\(280\) −0.307979 −0.0184052
\(281\) 8.40044 0.501128 0.250564 0.968100i \(-0.419384\pi\)
0.250564 + 0.968100i \(0.419384\pi\)
\(282\) −7.00000 −0.416844
\(283\) −31.6142 −1.87927 −0.939633 0.342183i \(-0.888834\pi\)
−0.939633 + 0.342183i \(0.888834\pi\)
\(284\) 9.56465 0.567557
\(285\) 5.80194 0.343677
\(286\) 0 0
\(287\) 1.04892 0.0619156
\(288\) −1.00000 −0.0589256
\(289\) 29.9342 1.76084
\(290\) 5.91185 0.347156
\(291\) 6.13169 0.359446
\(292\) 2.31336 0.135379
\(293\) −17.5254 −1.02385 −0.511923 0.859031i \(-0.671067\pi\)
−0.511923 + 0.859031i \(0.671067\pi\)
\(294\) 6.90515 0.402716
\(295\) −13.1347 −0.764730
\(296\) −7.07606 −0.411288
\(297\) 0.335126 0.0194460
\(298\) −3.44935 −0.199816
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −0.0150655 −0.000868360 0
\(302\) −2.84117 −0.163491
\(303\) 1.32975 0.0763921
\(304\) 5.80194 0.332764
\(305\) 2.14675 0.122923
\(306\) 6.85086 0.391637
\(307\) 20.6732 1.17988 0.589942 0.807446i \(-0.299151\pi\)
0.589942 + 0.807446i \(0.299151\pi\)
\(308\) 0.103211 0.00588102
\(309\) 8.92154 0.507529
\(310\) −0.0609989 −0.00346451
\(311\) −22.0248 −1.24891 −0.624455 0.781061i \(-0.714679\pi\)
−0.624455 + 0.781061i \(0.714679\pi\)
\(312\) 0 0
\(313\) 21.7192 1.22764 0.613820 0.789446i \(-0.289632\pi\)
0.613820 + 0.789446i \(0.289632\pi\)
\(314\) −16.2228 −0.915506
\(315\) 0.307979 0.0173526
\(316\) −0.0760644 −0.00427896
\(317\) −16.4644 −0.924734 −0.462367 0.886689i \(-0.653000\pi\)
−0.462367 + 0.886689i \(0.653000\pi\)
\(318\) −12.1739 −0.682678
\(319\) −1.98121 −0.110927
\(320\) 1.00000 0.0559017
\(321\) 14.4644 0.807325
\(322\) −0.725873 −0.0404513
\(323\) −39.7482 −2.21165
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 0.0924579 0.00512077
\(327\) 7.13706 0.394681
\(328\) −3.40581 −0.188055
\(329\) 2.15585 0.118856
\(330\) −0.335126 −0.0184481
\(331\) 26.4373 1.45312 0.726562 0.687101i \(-0.241117\pi\)
0.726562 + 0.687101i \(0.241117\pi\)
\(332\) 3.84117 0.210811
\(333\) 7.07606 0.387766
\(334\) 9.64071 0.527516
\(335\) 11.9608 0.653487
\(336\) 0.307979 0.0168016
\(337\) 27.4698 1.49638 0.748188 0.663487i \(-0.230924\pi\)
0.748188 + 0.663487i \(0.230924\pi\)
\(338\) 0 0
\(339\) −16.3424 −0.887598
\(340\) −6.85086 −0.371540
\(341\) 0.0204423 0.00110701
\(342\) −5.80194 −0.313733
\(343\) −4.28249 −0.231233
\(344\) 0.0489173 0.00263745
\(345\) 2.35690 0.126891
\(346\) −0.719169 −0.0386627
\(347\) −33.1323 −1.77863 −0.889317 0.457291i \(-0.848820\pi\)
−0.889317 + 0.457291i \(0.848820\pi\)
\(348\) −5.91185 −0.316909
\(349\) 26.3739 1.41176 0.705881 0.708331i \(-0.250551\pi\)
0.705881 + 0.708331i \(0.250551\pi\)
\(350\) −0.307979 −0.0164621
\(351\) 0 0
\(352\) −0.335126 −0.0178623
\(353\) 21.0291 1.11926 0.559632 0.828741i \(-0.310942\pi\)
0.559632 + 0.828741i \(0.310942\pi\)
\(354\) 13.1347 0.698100
\(355\) 9.56465 0.507639
\(356\) −4.41119 −0.233793
\(357\) −2.10992 −0.111669
\(358\) 2.01208 0.106342
\(359\) −30.8877 −1.63019 −0.815095 0.579327i \(-0.803315\pi\)
−0.815095 + 0.579327i \(0.803315\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 14.6625 0.771710
\(362\) 2.57135 0.135147
\(363\) −10.8877 −0.571456
\(364\) 0 0
\(365\) 2.31336 0.121087
\(366\) −2.14675 −0.112213
\(367\) 11.9584 0.624222 0.312111 0.950046i \(-0.398964\pi\)
0.312111 + 0.950046i \(0.398964\pi\)
\(368\) 2.35690 0.122862
\(369\) 3.40581 0.177300
\(370\) −7.07606 −0.367867
\(371\) 3.74930 0.194654
\(372\) 0.0609989 0.00316265
\(373\) 25.0315 1.29608 0.648040 0.761606i \(-0.275589\pi\)
0.648040 + 0.761606i \(0.275589\pi\)
\(374\) 2.29590 0.118718
\(375\) 1.00000 0.0516398
\(376\) −7.00000 −0.360997
\(377\) 0 0
\(378\) −0.307979 −0.0158407
\(379\) −34.4989 −1.77209 −0.886044 0.463602i \(-0.846557\pi\)
−0.886044 + 0.463602i \(0.846557\pi\)
\(380\) 5.80194 0.297633
\(381\) −0.911854 −0.0467157
\(382\) −3.80194 −0.194524
\(383\) 10.4644 0.534707 0.267353 0.963599i \(-0.413851\pi\)
0.267353 + 0.963599i \(0.413851\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0.103211 0.00526014
\(386\) 8.86294 0.451112
\(387\) −0.0489173 −0.00248661
\(388\) 6.13169 0.311289
\(389\) 25.2218 1.27879 0.639397 0.768877i \(-0.279184\pi\)
0.639397 + 0.768877i \(0.279184\pi\)
\(390\) 0 0
\(391\) −16.1468 −0.816576
\(392\) 6.90515 0.348763
\(393\) 13.0368 0.657621
\(394\) 22.6504 1.14111
\(395\) −0.0760644 −0.00382722
\(396\) 0.335126 0.0168407
\(397\) 21.3860 1.07333 0.536665 0.843795i \(-0.319684\pi\)
0.536665 + 0.843795i \(0.319684\pi\)
\(398\) 5.46681 0.274027
\(399\) 1.78687 0.0894555
\(400\) 1.00000 0.0500000
\(401\) −11.8062 −0.589576 −0.294788 0.955563i \(-0.595249\pi\)
−0.294788 + 0.955563i \(0.595249\pi\)
\(402\) −11.9608 −0.596549
\(403\) 0 0
\(404\) 1.32975 0.0661575
\(405\) 1.00000 0.0496904
\(406\) 1.82072 0.0903610
\(407\) 2.37137 0.117544
\(408\) 6.85086 0.339168
\(409\) −5.65950 −0.279844 −0.139922 0.990163i \(-0.544685\pi\)
−0.139922 + 0.990163i \(0.544685\pi\)
\(410\) −3.40581 −0.168201
\(411\) 12.8877 0.635703
\(412\) 8.92154 0.439533
\(413\) −4.04520 −0.199051
\(414\) −2.35690 −0.115835
\(415\) 3.84117 0.188555
\(416\) 0 0
\(417\) −2.01208 −0.0985321
\(418\) −1.94438 −0.0951026
\(419\) 17.0084 0.830913 0.415456 0.909613i \(-0.363622\pi\)
0.415456 + 0.909613i \(0.363622\pi\)
\(420\) 0.307979 0.0150278
\(421\) −10.2851 −0.501267 −0.250634 0.968082i \(-0.580639\pi\)
−0.250634 + 0.968082i \(0.580639\pi\)
\(422\) −11.0422 −0.537526
\(423\) 7.00000 0.340352
\(424\) −12.1739 −0.591217
\(425\) −6.85086 −0.332315
\(426\) −9.56465 −0.463409
\(427\) 0.661154 0.0319955
\(428\) 14.4644 0.699164
\(429\) 0 0
\(430\) 0.0489173 0.00235900
\(431\) −35.9657 −1.73241 −0.866203 0.499693i \(-0.833446\pi\)
−0.866203 + 0.499693i \(0.833446\pi\)
\(432\) 1.00000 0.0481125
\(433\) 7.63342 0.366839 0.183419 0.983035i \(-0.441283\pi\)
0.183419 + 0.983035i \(0.441283\pi\)
\(434\) −0.0187864 −0.000901774 0
\(435\) −5.91185 −0.283452
\(436\) 7.13706 0.341803
\(437\) 13.6746 0.654143
\(438\) −2.31336 −0.110536
\(439\) −34.4373 −1.64360 −0.821801 0.569775i \(-0.807030\pi\)
−0.821801 + 0.569775i \(0.807030\pi\)
\(440\) −0.335126 −0.0159765
\(441\) −6.90515 −0.328817
\(442\) 0 0
\(443\) 33.6431 1.59843 0.799216 0.601044i \(-0.205248\pi\)
0.799216 + 0.601044i \(0.205248\pi\)
\(444\) 7.07606 0.335815
\(445\) −4.41119 −0.209110
\(446\) −18.3884 −0.870714
\(447\) 3.44935 0.163149
\(448\) 0.307979 0.0145506
\(449\) −27.9729 −1.32012 −0.660060 0.751213i \(-0.729469\pi\)
−0.660060 + 0.751213i \(0.729469\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 1.14138 0.0537453
\(452\) −16.3424 −0.768683
\(453\) 2.84117 0.133490
\(454\) −12.0073 −0.563530
\(455\) 0 0
\(456\) −5.80194 −0.271701
\(457\) −6.18060 −0.289116 −0.144558 0.989496i \(-0.546176\pi\)
−0.144558 + 0.989496i \(0.546176\pi\)
\(458\) −25.4601 −1.18967
\(459\) −6.85086 −0.319771
\(460\) 2.35690 0.109891
\(461\) 38.1062 1.77478 0.887391 0.461017i \(-0.152515\pi\)
0.887391 + 0.461017i \(0.152515\pi\)
\(462\) −0.103211 −0.00480183
\(463\) −2.06829 −0.0961218 −0.0480609 0.998844i \(-0.515304\pi\)
−0.0480609 + 0.998844i \(0.515304\pi\)
\(464\) −5.91185 −0.274451
\(465\) 0.0609989 0.00282876
\(466\) 2.10992 0.0977400
\(467\) −4.27114 −0.197645 −0.0988225 0.995105i \(-0.531508\pi\)
−0.0988225 + 0.995105i \(0.531508\pi\)
\(468\) 0 0
\(469\) 3.68366 0.170096
\(470\) −7.00000 −0.322886
\(471\) 16.2228 0.747508
\(472\) 13.1347 0.604572
\(473\) −0.0163935 −0.000753772 0
\(474\) 0.0760644 0.00349375
\(475\) 5.80194 0.266211
\(476\) −2.10992 −0.0967079
\(477\) 12.1739 0.557405
\(478\) −3.95838 −0.181052
\(479\) −38.7560 −1.77081 −0.885404 0.464823i \(-0.846118\pi\)
−0.885404 + 0.464823i \(0.846118\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 23.1564 1.05475
\(483\) 0.725873 0.0330284
\(484\) −10.8877 −0.494895
\(485\) 6.13169 0.278426
\(486\) −1.00000 −0.0453609
\(487\) −33.4620 −1.51631 −0.758155 0.652075i \(-0.773899\pi\)
−0.758155 + 0.652075i \(0.773899\pi\)
\(488\) −2.14675 −0.0971789
\(489\) −0.0924579 −0.00418109
\(490\) 6.90515 0.311943
\(491\) 35.1487 1.58624 0.793119 0.609067i \(-0.208456\pi\)
0.793119 + 0.609067i \(0.208456\pi\)
\(492\) 3.40581 0.153546
\(493\) 40.5013 1.82408
\(494\) 0 0
\(495\) 0.335126 0.0150628
\(496\) 0.0609989 0.00273893
\(497\) 2.94571 0.132133
\(498\) −3.84117 −0.172127
\(499\) −9.61463 −0.430410 −0.215205 0.976569i \(-0.569042\pi\)
−0.215205 + 0.976569i \(0.569042\pi\)
\(500\) 1.00000 0.0447214
\(501\) −9.64071 −0.430715
\(502\) 5.54048 0.247284
\(503\) 18.5265 0.826055 0.413028 0.910719i \(-0.364471\pi\)
0.413028 + 0.910719i \(0.364471\pi\)
\(504\) −0.307979 −0.0137185
\(505\) 1.32975 0.0591730
\(506\) −0.789856 −0.0351134
\(507\) 0 0
\(508\) −0.911854 −0.0404570
\(509\) 5.43296 0.240812 0.120406 0.992725i \(-0.461580\pi\)
0.120406 + 0.992725i \(0.461580\pi\)
\(510\) 6.85086 0.303361
\(511\) 0.712464 0.0315175
\(512\) −1.00000 −0.0441942
\(513\) 5.80194 0.256162
\(514\) −6.60819 −0.291475
\(515\) 8.92154 0.393130
\(516\) −0.0489173 −0.00215347
\(517\) 2.34588 0.103172
\(518\) −2.17928 −0.0957519
\(519\) 0.719169 0.0315680
\(520\) 0 0
\(521\) −38.8001 −1.69986 −0.849932 0.526892i \(-0.823357\pi\)
−0.849932 + 0.526892i \(0.823357\pi\)
\(522\) 5.91185 0.258755
\(523\) 32.0629 1.40201 0.701007 0.713155i \(-0.252734\pi\)
0.701007 + 0.713155i \(0.252734\pi\)
\(524\) 13.0368 0.569517
\(525\) 0.307979 0.0134413
\(526\) 0.660563 0.0288019
\(527\) −0.417895 −0.0182038
\(528\) 0.335126 0.0145845
\(529\) −17.4450 −0.758480
\(530\) −12.1739 −0.528800
\(531\) −13.1347 −0.569996
\(532\) 1.78687 0.0774707
\(533\) 0 0
\(534\) 4.41119 0.190891
\(535\) 14.4644 0.625351
\(536\) −11.9608 −0.516627
\(537\) −2.01208 −0.0868277
\(538\) 26.7928 1.15512
\(539\) −2.31409 −0.0996750
\(540\) 1.00000 0.0430331
\(541\) 31.8334 1.36862 0.684312 0.729189i \(-0.260103\pi\)
0.684312 + 0.729189i \(0.260103\pi\)
\(542\) −14.0151 −0.601999
\(543\) −2.57135 −0.110347
\(544\) 6.85086 0.293728
\(545\) 7.13706 0.305718
\(546\) 0 0
\(547\) −44.4010 −1.89845 −0.949225 0.314597i \(-0.898131\pi\)
−0.949225 + 0.314597i \(0.898131\pi\)
\(548\) 12.8877 0.550535
\(549\) 2.14675 0.0916211
\(550\) −0.335126 −0.0142898
\(551\) −34.3002 −1.46124
\(552\) −2.35690 −0.100316
\(553\) −0.0234262 −0.000996184 0
\(554\) 23.4252 0.995241
\(555\) 7.07606 0.300362
\(556\) −2.01208 −0.0853313
\(557\) −7.73423 −0.327710 −0.163855 0.986484i \(-0.552393\pi\)
−0.163855 + 0.986484i \(0.552393\pi\)
\(558\) −0.0609989 −0.00258229
\(559\) 0 0
\(560\) 0.307979 0.0130145
\(561\) −2.29590 −0.0969328
\(562\) −8.40044 −0.354351
\(563\) 44.5803 1.87884 0.939418 0.342774i \(-0.111367\pi\)
0.939418 + 0.342774i \(0.111367\pi\)
\(564\) 7.00000 0.294753
\(565\) −16.3424 −0.687531
\(566\) 31.6142 1.32884
\(567\) 0.307979 0.0129339
\(568\) −9.56465 −0.401324
\(569\) 24.9065 1.04413 0.522067 0.852905i \(-0.325161\pi\)
0.522067 + 0.852905i \(0.325161\pi\)
\(570\) −5.80194 −0.243016
\(571\) −0.873690 −0.0365628 −0.0182814 0.999833i \(-0.505819\pi\)
−0.0182814 + 0.999833i \(0.505819\pi\)
\(572\) 0 0
\(573\) 3.80194 0.158828
\(574\) −1.04892 −0.0437810
\(575\) 2.35690 0.0982894
\(576\) 1.00000 0.0416667
\(577\) −20.0871 −0.836236 −0.418118 0.908393i \(-0.637310\pi\)
−0.418118 + 0.908393i \(0.637310\pi\)
\(578\) −29.9342 −1.24510
\(579\) −8.86294 −0.368331
\(580\) −5.91185 −0.245476
\(581\) 1.18300 0.0490790
\(582\) −6.13169 −0.254167
\(583\) 4.07979 0.168967
\(584\) −2.31336 −0.0957273
\(585\) 0 0
\(586\) 17.5254 0.723968
\(587\) −16.0901 −0.664108 −0.332054 0.943260i \(-0.607742\pi\)
−0.332054 + 0.943260i \(0.607742\pi\)
\(588\) −6.90515 −0.284764
\(589\) 0.353912 0.0145827
\(590\) 13.1347 0.540746
\(591\) −22.6504 −0.931713
\(592\) 7.07606 0.290824
\(593\) 4.85517 0.199378 0.0996889 0.995019i \(-0.468215\pi\)
0.0996889 + 0.995019i \(0.468215\pi\)
\(594\) −0.335126 −0.0137504
\(595\) −2.10992 −0.0864981
\(596\) 3.44935 0.141291
\(597\) −5.46681 −0.223742
\(598\) 0 0
\(599\) −34.4319 −1.40685 −0.703425 0.710770i \(-0.748347\pi\)
−0.703425 + 0.710770i \(0.748347\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −24.5244 −1.00037 −0.500185 0.865919i \(-0.666735\pi\)
−0.500185 + 0.865919i \(0.666735\pi\)
\(602\) 0.0150655 0.000614024 0
\(603\) 11.9608 0.487080
\(604\) 2.84117 0.115605
\(605\) −10.8877 −0.442648
\(606\) −1.32975 −0.0540174
\(607\) 17.6420 0.716068 0.358034 0.933708i \(-0.383447\pi\)
0.358034 + 0.933708i \(0.383447\pi\)
\(608\) −5.80194 −0.235300
\(609\) −1.82072 −0.0737795
\(610\) −2.14675 −0.0869194
\(611\) 0 0
\(612\) −6.85086 −0.276929
\(613\) 44.5018 1.79741 0.898706 0.438551i \(-0.144508\pi\)
0.898706 + 0.438551i \(0.144508\pi\)
\(614\) −20.6732 −0.834304
\(615\) 3.40581 0.137336
\(616\) −0.103211 −0.00415851
\(617\) 32.1618 1.29479 0.647393 0.762156i \(-0.275859\pi\)
0.647393 + 0.762156i \(0.275859\pi\)
\(618\) −8.92154 −0.358877
\(619\) 24.8340 0.998162 0.499081 0.866555i \(-0.333671\pi\)
0.499081 + 0.866555i \(0.333671\pi\)
\(620\) 0.0609989 0.00244978
\(621\) 2.35690 0.0945790
\(622\) 22.0248 0.883112
\(623\) −1.35855 −0.0544292
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −21.7192 −0.868073
\(627\) 1.94438 0.0776510
\(628\) 16.2228 0.647361
\(629\) −48.4771 −1.93291
\(630\) −0.307979 −0.0122702
\(631\) 10.5418 0.419663 0.209831 0.977738i \(-0.432708\pi\)
0.209831 + 0.977738i \(0.432708\pi\)
\(632\) 0.0760644 0.00302568
\(633\) 11.0422 0.438889
\(634\) 16.4644 0.653886
\(635\) −0.911854 −0.0361858
\(636\) 12.1739 0.482726
\(637\) 0 0
\(638\) 1.98121 0.0784370
\(639\) 9.56465 0.378372
\(640\) −1.00000 −0.0395285
\(641\) 33.5663 1.32579 0.662895 0.748713i \(-0.269328\pi\)
0.662895 + 0.748713i \(0.269328\pi\)
\(642\) −14.4644 −0.570865
\(643\) 8.24160 0.325017 0.162509 0.986707i \(-0.448041\pi\)
0.162509 + 0.986707i \(0.448041\pi\)
\(644\) 0.725873 0.0286034
\(645\) −0.0489173 −0.00192612
\(646\) 39.7482 1.56387
\(647\) −3.66056 −0.143912 −0.0719558 0.997408i \(-0.522924\pi\)
−0.0719558 + 0.997408i \(0.522924\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −4.40176 −0.172784
\(650\) 0 0
\(651\) 0.0187864 0.000736295 0
\(652\) −0.0924579 −0.00362093
\(653\) −44.6064 −1.74558 −0.872791 0.488093i \(-0.837693\pi\)
−0.872791 + 0.488093i \(0.837693\pi\)
\(654\) −7.13706 −0.279081
\(655\) 13.0368 0.509391
\(656\) 3.40581 0.132975
\(657\) 2.31336 0.0902526
\(658\) −2.15585 −0.0840438
\(659\) −2.51035 −0.0977895 −0.0488947 0.998804i \(-0.515570\pi\)
−0.0488947 + 0.998804i \(0.515570\pi\)
\(660\) 0.335126 0.0130447
\(661\) 40.3096 1.56786 0.783932 0.620847i \(-0.213211\pi\)
0.783932 + 0.620847i \(0.213211\pi\)
\(662\) −26.4373 −1.02751
\(663\) 0 0
\(664\) −3.84117 −0.149066
\(665\) 1.78687 0.0692919
\(666\) −7.07606 −0.274192
\(667\) −13.9336 −0.539512
\(668\) −9.64071 −0.373010
\(669\) 18.3884 0.710935
\(670\) −11.9608 −0.462085
\(671\) 0.719432 0.0277733
\(672\) −0.307979 −0.0118805
\(673\) 42.6340 1.64342 0.821710 0.569906i \(-0.193020\pi\)
0.821710 + 0.569906i \(0.193020\pi\)
\(674\) −27.4698 −1.05810
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 18.5254 0.711990 0.355995 0.934488i \(-0.384142\pi\)
0.355995 + 0.934488i \(0.384142\pi\)
\(678\) 16.3424 0.627627
\(679\) 1.88843 0.0724712
\(680\) 6.85086 0.262718
\(681\) 12.0073 0.460121
\(682\) −0.0204423 −0.000782776 0
\(683\) 48.1769 1.84344 0.921719 0.387859i \(-0.126785\pi\)
0.921719 + 0.387859i \(0.126785\pi\)
\(684\) 5.80194 0.221843
\(685\) 12.8877 0.492413
\(686\) 4.28249 0.163506
\(687\) 25.4601 0.971364
\(688\) −0.0489173 −0.00186496
\(689\) 0 0
\(690\) −2.35690 −0.0897255
\(691\) 37.0084 1.40786 0.703932 0.710267i \(-0.251426\pi\)
0.703932 + 0.710267i \(0.251426\pi\)
\(692\) 0.719169 0.0273387
\(693\) 0.103211 0.00392068
\(694\) 33.1323 1.25768
\(695\) −2.01208 −0.0763226
\(696\) 5.91185 0.224088
\(697\) −23.3327 −0.883790
\(698\) −26.3739 −0.998266
\(699\) −2.10992 −0.0798044
\(700\) 0.307979 0.0116405
\(701\) 9.25188 0.349439 0.174719 0.984618i \(-0.444098\pi\)
0.174719 + 0.984618i \(0.444098\pi\)
\(702\) 0 0
\(703\) 41.0549 1.54841
\(704\) 0.335126 0.0126305
\(705\) 7.00000 0.263635
\(706\) −21.0291 −0.791439
\(707\) 0.409534 0.0154021
\(708\) −13.1347 −0.493631
\(709\) −15.0573 −0.565488 −0.282744 0.959195i \(-0.591245\pi\)
−0.282744 + 0.959195i \(0.591245\pi\)
\(710\) −9.56465 −0.358955
\(711\) −0.0760644 −0.00285264
\(712\) 4.41119 0.165316
\(713\) 0.143768 0.00538416
\(714\) 2.10992 0.0789616
\(715\) 0 0
\(716\) −2.01208 −0.0751950
\(717\) 3.95838 0.147828
\(718\) 30.8877 1.15272
\(719\) 24.8364 0.926241 0.463120 0.886295i \(-0.346730\pi\)
0.463120 + 0.886295i \(0.346730\pi\)
\(720\) 1.00000 0.0372678
\(721\) 2.74764 0.102328
\(722\) −14.6625 −0.545681
\(723\) −23.1564 −0.861197
\(724\) −2.57135 −0.0955635
\(725\) −5.91185 −0.219561
\(726\) 10.8877 0.404080
\(727\) −24.7590 −0.918260 −0.459130 0.888369i \(-0.651839\pi\)
−0.459130 + 0.888369i \(0.651839\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −2.31336 −0.0856211
\(731\) 0.335126 0.0123951
\(732\) 2.14675 0.0793462
\(733\) −15.4359 −0.570140 −0.285070 0.958507i \(-0.592017\pi\)
−0.285070 + 0.958507i \(0.592017\pi\)
\(734\) −11.9584 −0.441392
\(735\) −6.90515 −0.254700
\(736\) −2.35690 −0.0868763
\(737\) 4.00836 0.147650
\(738\) −3.40581 −0.125370
\(739\) −20.3327 −0.747952 −0.373976 0.927438i \(-0.622006\pi\)
−0.373976 + 0.927438i \(0.622006\pi\)
\(740\) 7.07606 0.260121
\(741\) 0 0
\(742\) −3.74930 −0.137641
\(743\) 17.8243 0.653910 0.326955 0.945040i \(-0.393977\pi\)
0.326955 + 0.945040i \(0.393977\pi\)
\(744\) −0.0609989 −0.00223633
\(745\) 3.44935 0.126375
\(746\) −25.0315 −0.916467
\(747\) 3.84117 0.140541
\(748\) −2.29590 −0.0839463
\(749\) 4.45473 0.162772
\(750\) −1.00000 −0.0365148
\(751\) 48.0616 1.75379 0.876896 0.480680i \(-0.159610\pi\)
0.876896 + 0.480680i \(0.159610\pi\)
\(752\) 7.00000 0.255264
\(753\) −5.54048 −0.201906
\(754\) 0 0
\(755\) 2.84117 0.103401
\(756\) 0.307979 0.0112011
\(757\) 32.9748 1.19849 0.599244 0.800566i \(-0.295468\pi\)
0.599244 + 0.800566i \(0.295468\pi\)
\(758\) 34.4989 1.25306
\(759\) 0.789856 0.0286700
\(760\) −5.80194 −0.210458
\(761\) 11.9946 0.434805 0.217402 0.976082i \(-0.430242\pi\)
0.217402 + 0.976082i \(0.430242\pi\)
\(762\) 0.911854 0.0330330
\(763\) 2.19806 0.0795752
\(764\) 3.80194 0.137549
\(765\) −6.85086 −0.247693
\(766\) −10.4644 −0.378095
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −18.5036 −0.667259 −0.333629 0.942704i \(-0.608273\pi\)
−0.333629 + 0.942704i \(0.608273\pi\)
\(770\) −0.103211 −0.00371948
\(771\) 6.60819 0.237988
\(772\) −8.86294 −0.318984
\(773\) −23.4946 −0.845040 −0.422520 0.906354i \(-0.638854\pi\)
−0.422520 + 0.906354i \(0.638854\pi\)
\(774\) 0.0489173 0.00175830
\(775\) 0.0609989 0.00219115
\(776\) −6.13169 −0.220115
\(777\) 2.17928 0.0781811
\(778\) −25.2218 −0.904244
\(779\) 19.7603 0.707987
\(780\) 0 0
\(781\) 3.20536 0.114697
\(782\) 16.1468 0.577407
\(783\) −5.91185 −0.211272
\(784\) −6.90515 −0.246612
\(785\) 16.2228 0.579017
\(786\) −13.0368 −0.465009
\(787\) 7.23682 0.257965 0.128982 0.991647i \(-0.458829\pi\)
0.128982 + 0.991647i \(0.458829\pi\)
\(788\) −22.6504 −0.806887
\(789\) −0.660563 −0.0235166
\(790\) 0.0760644 0.00270625
\(791\) −5.03311 −0.178957
\(792\) −0.335126 −0.0119082
\(793\) 0 0
\(794\) −21.3860 −0.758959
\(795\) 12.1739 0.431764
\(796\) −5.46681 −0.193766
\(797\) −46.7079 −1.65448 −0.827240 0.561849i \(-0.810090\pi\)
−0.827240 + 0.561849i \(0.810090\pi\)
\(798\) −1.78687 −0.0632546
\(799\) −47.9560 −1.69656
\(800\) −1.00000 −0.0353553
\(801\) −4.41119 −0.155862
\(802\) 11.8062 0.416893
\(803\) 0.775265 0.0273585
\(804\) 11.9608 0.421824
\(805\) 0.725873 0.0255837
\(806\) 0 0
\(807\) −26.7928 −0.943153
\(808\) −1.32975 −0.0467804
\(809\) −9.95838 −0.350118 −0.175059 0.984558i \(-0.556012\pi\)
−0.175059 + 0.984558i \(0.556012\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −36.2452 −1.27274 −0.636370 0.771384i \(-0.719565\pi\)
−0.636370 + 0.771384i \(0.719565\pi\)
\(812\) −1.82072 −0.0638949
\(813\) 14.0151 0.491530
\(814\) −2.37137 −0.0831165
\(815\) −0.0924579 −0.00323866
\(816\) −6.85086 −0.239828
\(817\) −0.283815 −0.00992944
\(818\) 5.65950 0.197880
\(819\) 0 0
\(820\) 3.40581 0.118936
\(821\) −1.37867 −0.0481158 −0.0240579 0.999711i \(-0.507659\pi\)
−0.0240579 + 0.999711i \(0.507659\pi\)
\(822\) −12.8877 −0.449510
\(823\) −23.8713 −0.832101 −0.416051 0.909341i \(-0.636586\pi\)
−0.416051 + 0.909341i \(0.636586\pi\)
\(824\) −8.92154 −0.310797
\(825\) 0.335126 0.0116676
\(826\) 4.04520 0.140750
\(827\) −21.9571 −0.763521 −0.381761 0.924261i \(-0.624682\pi\)
−0.381761 + 0.924261i \(0.624682\pi\)
\(828\) 2.35690 0.0819078
\(829\) −27.4862 −0.954635 −0.477317 0.878731i \(-0.658391\pi\)
−0.477317 + 0.878731i \(0.658391\pi\)
\(830\) −3.84117 −0.133329
\(831\) −23.4252 −0.812611
\(832\) 0 0
\(833\) 47.3062 1.63906
\(834\) 2.01208 0.0696727
\(835\) −9.64071 −0.333631
\(836\) 1.94438 0.0672477
\(837\) 0.0609989 0.00210843
\(838\) −17.0084 −0.587544
\(839\) 20.5749 0.710325 0.355163 0.934805i \(-0.384425\pi\)
0.355163 + 0.934805i \(0.384425\pi\)
\(840\) −0.307979 −0.0106263
\(841\) 5.95002 0.205173
\(842\) 10.2851 0.354449
\(843\) 8.40044 0.289326
\(844\) 11.0422 0.380089
\(845\) 0 0
\(846\) −7.00000 −0.240665
\(847\) −3.35317 −0.115216
\(848\) 12.1739 0.418053
\(849\) −31.6142 −1.08499
\(850\) 6.85086 0.234982
\(851\) 16.6775 0.571699
\(852\) 9.56465 0.327679
\(853\) −50.3812 −1.72502 −0.862509 0.506041i \(-0.831108\pi\)
−0.862509 + 0.506041i \(0.831108\pi\)
\(854\) −0.661154 −0.0226242
\(855\) 5.80194 0.198422
\(856\) −14.4644 −0.494384
\(857\) 21.6045 0.737995 0.368997 0.929430i \(-0.379701\pi\)
0.368997 + 0.929430i \(0.379701\pi\)
\(858\) 0 0
\(859\) −38.2959 −1.30664 −0.653320 0.757082i \(-0.726624\pi\)
−0.653320 + 0.757082i \(0.726624\pi\)
\(860\) −0.0489173 −0.00166807
\(861\) 1.04892 0.0357470
\(862\) 35.9657 1.22500
\(863\) 8.61894 0.293392 0.146696 0.989182i \(-0.453136\pi\)
0.146696 + 0.989182i \(0.453136\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0.719169 0.0244525
\(866\) −7.63342 −0.259394
\(867\) 29.9342 1.01662
\(868\) 0.0187864 0.000637651 0
\(869\) −0.0254911 −0.000864727 0
\(870\) 5.91185 0.200431
\(871\) 0 0
\(872\) −7.13706 −0.241691
\(873\) 6.13169 0.207526
\(874\) −13.6746 −0.462549
\(875\) 0.307979 0.0104116
\(876\) 2.31336 0.0781610
\(877\) −33.1021 −1.11778 −0.558890 0.829242i \(-0.688773\pi\)
−0.558890 + 0.829242i \(0.688773\pi\)
\(878\) 34.4373 1.16220
\(879\) −17.5254 −0.591118
\(880\) 0.335126 0.0112971
\(881\) 20.8267 0.701669 0.350835 0.936437i \(-0.385898\pi\)
0.350835 + 0.936437i \(0.385898\pi\)
\(882\) 6.90515 0.232508
\(883\) 0.785807 0.0264445 0.0132223 0.999913i \(-0.495791\pi\)
0.0132223 + 0.999913i \(0.495791\pi\)
\(884\) 0 0
\(885\) −13.1347 −0.441517
\(886\) −33.6431 −1.13026
\(887\) −44.2392 −1.48541 −0.742704 0.669620i \(-0.766457\pi\)
−0.742704 + 0.669620i \(0.766457\pi\)
\(888\) −7.07606 −0.237457
\(889\) −0.280831 −0.00941878
\(890\) 4.41119 0.147863
\(891\) 0.335126 0.0112271
\(892\) 18.3884 0.615688
\(893\) 40.6136 1.35908
\(894\) −3.44935 −0.115364
\(895\) −2.01208 −0.0672565
\(896\) −0.307979 −0.0102888
\(897\) 0 0
\(898\) 27.9729 0.933466
\(899\) −0.360617 −0.0120272
\(900\) 1.00000 0.0333333
\(901\) −83.4016 −2.77851
\(902\) −1.14138 −0.0380036
\(903\) −0.0150655 −0.000501348 0
\(904\) 16.3424 0.543541
\(905\) −2.57135 −0.0854746
\(906\) −2.84117 −0.0943914
\(907\) 8.44371 0.280369 0.140184 0.990125i \(-0.455230\pi\)
0.140184 + 0.990125i \(0.455230\pi\)
\(908\) 12.0073 0.398476
\(909\) 1.32975 0.0441050
\(910\) 0 0
\(911\) −47.2801 −1.56646 −0.783230 0.621732i \(-0.786429\pi\)
−0.783230 + 0.621732i \(0.786429\pi\)
\(912\) 5.80194 0.192121
\(913\) 1.28727 0.0426025
\(914\) 6.18060 0.204436
\(915\) 2.14675 0.0709694
\(916\) 25.4601 0.841226
\(917\) 4.01507 0.132589
\(918\) 6.85086 0.226112
\(919\) −11.1927 −0.369213 −0.184606 0.982813i \(-0.559101\pi\)
−0.184606 + 0.982813i \(0.559101\pi\)
\(920\) −2.35690 −0.0777046
\(921\) 20.6732 0.681206
\(922\) −38.1062 −1.25496
\(923\) 0 0
\(924\) 0.103211 0.00339541
\(925\) 7.07606 0.232660
\(926\) 2.06829 0.0679684
\(927\) 8.92154 0.293022
\(928\) 5.91185 0.194066
\(929\) 1.11828 0.0366895 0.0183447 0.999832i \(-0.494160\pi\)
0.0183447 + 0.999832i \(0.494160\pi\)
\(930\) −0.0609989 −0.00200023
\(931\) −40.0632 −1.31302
\(932\) −2.10992 −0.0691126
\(933\) −22.0248 −0.721058
\(934\) 4.27114 0.139756
\(935\) −2.29590 −0.0750839
\(936\) 0 0
\(937\) −29.9051 −0.976959 −0.488479 0.872575i \(-0.662448\pi\)
−0.488479 + 0.872575i \(0.662448\pi\)
\(938\) −3.68366 −0.120276
\(939\) 21.7192 0.708778
\(940\) 7.00000 0.228315
\(941\) −32.7646 −1.06810 −0.534048 0.845454i \(-0.679330\pi\)
−0.534048 + 0.845454i \(0.679330\pi\)
\(942\) −16.2228 −0.528568
\(943\) 8.02715 0.261400
\(944\) −13.1347 −0.427497
\(945\) 0.307979 0.0100185
\(946\) 0.0163935 0.000532997 0
\(947\) −6.81535 −0.221469 −0.110735 0.993850i \(-0.535320\pi\)
−0.110735 + 0.993850i \(0.535320\pi\)
\(948\) −0.0760644 −0.00247046
\(949\) 0 0
\(950\) −5.80194 −0.188240
\(951\) −16.4644 −0.533895
\(952\) 2.10992 0.0683828
\(953\) 23.6886 0.767348 0.383674 0.923469i \(-0.374659\pi\)
0.383674 + 0.923469i \(0.374659\pi\)
\(954\) −12.1739 −0.394145
\(955\) 3.80194 0.123028
\(956\) 3.95838 0.128023
\(957\) −1.98121 −0.0640435
\(958\) 38.7560 1.25215
\(959\) 3.96913 0.128170
\(960\) 1.00000 0.0322749
\(961\) −30.9963 −0.999880
\(962\) 0 0
\(963\) 14.4644 0.466109
\(964\) −23.1564 −0.745819
\(965\) −8.86294 −0.285308
\(966\) −0.725873 −0.0233546
\(967\) −28.9075 −0.929604 −0.464802 0.885415i \(-0.653874\pi\)
−0.464802 + 0.885415i \(0.653874\pi\)
\(968\) 10.8877 0.349944
\(969\) −39.7482 −1.27690
\(970\) −6.13169 −0.196877
\(971\) −47.2717 −1.51702 −0.758511 0.651660i \(-0.774073\pi\)
−0.758511 + 0.651660i \(0.774073\pi\)
\(972\) 1.00000 0.0320750
\(973\) −0.619678 −0.0198660
\(974\) 33.4620 1.07219
\(975\) 0 0
\(976\) 2.14675 0.0687159
\(977\) −33.9527 −1.08624 −0.543122 0.839654i \(-0.682758\pi\)
−0.543122 + 0.839654i \(0.682758\pi\)
\(978\) 0.0924579 0.00295648
\(979\) −1.47830 −0.0472468
\(980\) −6.90515 −0.220577
\(981\) 7.13706 0.227869
\(982\) −35.1487 −1.12164
\(983\) −22.9366 −0.731564 −0.365782 0.930701i \(-0.619198\pi\)
−0.365782 + 0.930701i \(0.619198\pi\)
\(984\) −3.40581 −0.108573
\(985\) −22.6504 −0.721702
\(986\) −40.5013 −1.28982
\(987\) 2.15585 0.0686215
\(988\) 0 0
\(989\) −0.115293 −0.00366611
\(990\) −0.335126 −0.0106510
\(991\) −5.75781 −0.182903 −0.0914514 0.995810i \(-0.529151\pi\)
−0.0914514 + 0.995810i \(0.529151\pi\)
\(992\) −0.0609989 −0.00193672
\(993\) 26.4373 0.838961
\(994\) −2.94571 −0.0934321
\(995\) −5.46681 −0.173310
\(996\) 3.84117 0.121712
\(997\) 3.86353 0.122359 0.0611796 0.998127i \(-0.480514\pi\)
0.0611796 + 0.998127i \(0.480514\pi\)
\(998\) 9.61463 0.304346
\(999\) 7.07606 0.223877
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 5070.2.a.bq.1.1 3
13.5 odd 4 5070.2.b.y.1351.4 6
13.8 odd 4 5070.2.b.y.1351.3 6
13.12 even 2 5070.2.a.bv.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bq.1.1 3 1.1 even 1 trivial
5070.2.a.bv.1.3 yes 3 13.12 even 2
5070.2.b.y.1351.3 6 13.8 odd 4
5070.2.b.y.1351.4 6 13.5 odd 4