Properties

Label 5070.2.b.y.1351.3
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.3
Root \(-0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.y.1351.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -0.307979i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -0.307979i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -0.335126i q^{11} -1.00000 q^{12} -0.307979 q^{14} +1.00000i q^{15} +1.00000 q^{16} +6.85086 q^{17} -1.00000i q^{18} +5.80194i q^{19} -1.00000i q^{20} -0.307979i q^{21} -0.335126 q^{22} -2.35690 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +0.307979i q^{28} -5.91185 q^{29} +1.00000 q^{30} +0.0609989i q^{31} -1.00000i q^{32} -0.335126i q^{33} -6.85086i q^{34} +0.307979 q^{35} -1.00000 q^{36} -7.07606i q^{37} +5.80194 q^{38} -1.00000 q^{40} +3.40581i q^{41} -0.307979 q^{42} +0.0489173 q^{43} +0.335126i q^{44} +1.00000i q^{45} +2.35690i q^{46} -7.00000i q^{47} +1.00000 q^{48} +6.90515 q^{49} +1.00000i q^{50} +6.85086 q^{51} +12.1739 q^{53} -1.00000i q^{54} +0.335126 q^{55} +0.307979 q^{56} +5.80194i q^{57} +5.91185i q^{58} +13.1347i q^{59} -1.00000i q^{60} +2.14675 q^{61} +0.0609989 q^{62} -0.307979i q^{63} -1.00000 q^{64} -0.335126 q^{66} +11.9608i q^{67} -6.85086 q^{68} -2.35690 q^{69} -0.307979i q^{70} +9.56465i q^{71} +1.00000i q^{72} -2.31336i q^{73} -7.07606 q^{74} -1.00000 q^{75} -5.80194i q^{76} -0.103211 q^{77} -0.0760644 q^{79} +1.00000i q^{80} +1.00000 q^{81} +3.40581 q^{82} +3.84117i q^{83} +0.307979i q^{84} +6.85086i q^{85} -0.0489173i q^{86} -5.91185 q^{87} +0.335126 q^{88} +4.41119i q^{89} +1.00000 q^{90} +2.35690 q^{92} +0.0609989i q^{93} -7.00000 q^{94} -5.80194 q^{95} -1.00000i q^{96} +6.13169i q^{97} -6.90515i q^{98} -0.335126i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{3} - 6q^{4} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{3} - 6q^{4} + 6q^{9} + 6q^{10} - 6q^{12} - 12q^{14} + 6q^{16} + 14q^{17} - 6q^{23} - 6q^{25} + 6q^{27} - 28q^{29} + 6q^{30} + 12q^{35} - 6q^{36} + 26q^{38} - 6q^{40} - 12q^{42} - 18q^{43} + 6q^{48} - 10q^{49} + 14q^{51} + 6q^{53} + 12q^{56} - 42q^{61} + 20q^{62} - 6q^{64} - 14q^{68} - 6q^{69} - 12q^{74} - 6q^{75} + 42q^{77} + 30q^{79} + 6q^{81} - 6q^{82} - 28q^{87} + 6q^{90} + 6q^{92} - 42q^{94} - 26q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 1.00000i 0.447214i
\(6\) − 1.00000i − 0.408248i
\(7\) − 0.307979i − 0.116405i −0.998305 0.0582025i \(-0.981463\pi\)
0.998305 0.0582025i \(-0.0185369\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) − 0.335126i − 0.101044i −0.998723 0.0505221i \(-0.983911\pi\)
0.998723 0.0505221i \(-0.0160885\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −0.307979 −0.0823107
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) 6.85086 1.66158 0.830788 0.556589i \(-0.187890\pi\)
0.830788 + 0.556589i \(0.187890\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 5.80194i 1.33106i 0.746373 + 0.665528i \(0.231794\pi\)
−0.746373 + 0.665528i \(0.768206\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) − 0.307979i − 0.0672064i
\(22\) −0.335126 −0.0714490
\(23\) −2.35690 −0.491447 −0.245723 0.969340i \(-0.579025\pi\)
−0.245723 + 0.969340i \(0.579025\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0.307979i 0.0582025i
\(29\) −5.91185 −1.09780 −0.548902 0.835887i \(-0.684954\pi\)
−0.548902 + 0.835887i \(0.684954\pi\)
\(30\) 1.00000 0.182574
\(31\) 0.0609989i 0.0109557i 0.999985 + 0.00547787i \(0.00174367\pi\)
−0.999985 + 0.00547787i \(0.998256\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 0.335126i − 0.0583379i
\(34\) − 6.85086i − 1.17491i
\(35\) 0.307979 0.0520579
\(36\) −1.00000 −0.166667
\(37\) − 7.07606i − 1.16330i −0.813440 0.581649i \(-0.802408\pi\)
0.813440 0.581649i \(-0.197592\pi\)
\(38\) 5.80194 0.941199
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 3.40581i 0.531899i 0.963987 + 0.265949i \(0.0856854\pi\)
−0.963987 + 0.265949i \(0.914315\pi\)
\(42\) −0.307979 −0.0475221
\(43\) 0.0489173 0.00745982 0.00372991 0.999993i \(-0.498813\pi\)
0.00372991 + 0.999993i \(0.498813\pi\)
\(44\) 0.335126i 0.0505221i
\(45\) 1.00000i 0.149071i
\(46\) 2.35690i 0.347505i
\(47\) − 7.00000i − 1.02105i −0.859861 0.510527i \(-0.829450\pi\)
0.859861 0.510527i \(-0.170550\pi\)
\(48\) 1.00000 0.144338
\(49\) 6.90515 0.986450
\(50\) 1.00000i 0.141421i
\(51\) 6.85086 0.959312
\(52\) 0 0
\(53\) 12.1739 1.67221 0.836107 0.548567i \(-0.184826\pi\)
0.836107 + 0.548567i \(0.184826\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) 0.335126 0.0451883
\(56\) 0.307979 0.0411554
\(57\) 5.80194i 0.768485i
\(58\) 5.91185i 0.776264i
\(59\) 13.1347i 1.70999i 0.518638 + 0.854994i \(0.326439\pi\)
−0.518638 + 0.854994i \(0.673561\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) 2.14675 0.274863 0.137432 0.990511i \(-0.456115\pi\)
0.137432 + 0.990511i \(0.456115\pi\)
\(62\) 0.0609989 0.00774687
\(63\) − 0.307979i − 0.0388016i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −0.335126 −0.0412511
\(67\) 11.9608i 1.46124i 0.682784 + 0.730620i \(0.260769\pi\)
−0.682784 + 0.730620i \(0.739231\pi\)
\(68\) −6.85086 −0.830788
\(69\) −2.35690 −0.283737
\(70\) − 0.307979i − 0.0368105i
\(71\) 9.56465i 1.13511i 0.823334 + 0.567557i \(0.192111\pi\)
−0.823334 + 0.567557i \(0.807889\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 2.31336i − 0.270758i −0.990794 0.135379i \(-0.956775\pi\)
0.990794 0.135379i \(-0.0432252\pi\)
\(74\) −7.07606 −0.822576
\(75\) −1.00000 −0.115470
\(76\) − 5.80194i − 0.665528i
\(77\) −0.103211 −0.0117620
\(78\) 0 0
\(79\) −0.0760644 −0.00855792 −0.00427896 0.999991i \(-0.501362\pi\)
−0.00427896 + 0.999991i \(0.501362\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) 3.40581 0.376109
\(83\) 3.84117i 0.421623i 0.977527 + 0.210811i \(0.0676106\pi\)
−0.977527 + 0.210811i \(0.932389\pi\)
\(84\) 0.307979i 0.0336032i
\(85\) 6.85086i 0.743080i
\(86\) − 0.0489173i − 0.00527489i
\(87\) −5.91185 −0.633817
\(88\) 0.335126 0.0357245
\(89\) 4.41119i 0.467585i 0.972286 + 0.233793i \(0.0751137\pi\)
−0.972286 + 0.233793i \(0.924886\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 2.35690 0.245723
\(93\) 0.0609989i 0.00632529i
\(94\) −7.00000 −0.721995
\(95\) −5.80194 −0.595266
\(96\) − 1.00000i − 0.102062i
\(97\) 6.13169i 0.622578i 0.950315 + 0.311289i \(0.100761\pi\)
−0.950315 + 0.311289i \(0.899239\pi\)
\(98\) − 6.90515i − 0.697525i
\(99\) − 0.335126i − 0.0336814i
\(100\) 1.00000 0.100000
\(101\) −1.32975 −0.132315 −0.0661575 0.997809i \(-0.521074\pi\)
−0.0661575 + 0.997809i \(0.521074\pi\)
\(102\) − 6.85086i − 0.678336i
\(103\) −8.92154 −0.879066 −0.439533 0.898227i \(-0.644856\pi\)
−0.439533 + 0.898227i \(0.644856\pi\)
\(104\) 0 0
\(105\) 0.307979 0.0300556
\(106\) − 12.1739i − 1.18243i
\(107\) 14.4644 1.39833 0.699164 0.714961i \(-0.253556\pi\)
0.699164 + 0.714961i \(0.253556\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 7.13706i 0.683607i 0.939772 + 0.341803i \(0.111038\pi\)
−0.939772 + 0.341803i \(0.888962\pi\)
\(110\) − 0.335126i − 0.0319530i
\(111\) − 7.07606i − 0.671630i
\(112\) − 0.307979i − 0.0291012i
\(113\) −16.3424 −1.53737 −0.768683 0.639630i \(-0.779088\pi\)
−0.768683 + 0.639630i \(0.779088\pi\)
\(114\) 5.80194 0.543401
\(115\) − 2.35690i − 0.219782i
\(116\) 5.91185 0.548902
\(117\) 0 0
\(118\) 13.1347 1.20914
\(119\) − 2.10992i − 0.193416i
\(120\) −1.00000 −0.0912871
\(121\) 10.8877 0.989790
\(122\) − 2.14675i − 0.194358i
\(123\) 3.40581i 0.307092i
\(124\) − 0.0609989i − 0.00547787i
\(125\) − 1.00000i − 0.0894427i
\(126\) −0.307979 −0.0274369
\(127\) 0.911854 0.0809140 0.0404570 0.999181i \(-0.487119\pi\)
0.0404570 + 0.999181i \(0.487119\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0.0489173 0.00430693
\(130\) 0 0
\(131\) 13.0368 1.13903 0.569517 0.821980i \(-0.307130\pi\)
0.569517 + 0.821980i \(0.307130\pi\)
\(132\) 0.335126i 0.0291689i
\(133\) 1.78687 0.154941
\(134\) 11.9608 1.03325
\(135\) 1.00000i 0.0860663i
\(136\) 6.85086i 0.587456i
\(137\) − 12.8877i − 1.10107i −0.834812 0.550535i \(-0.814424\pi\)
0.834812 0.550535i \(-0.185576\pi\)
\(138\) 2.35690i 0.200632i
\(139\) −2.01208 −0.170663 −0.0853313 0.996353i \(-0.527195\pi\)
−0.0853313 + 0.996353i \(0.527195\pi\)
\(140\) −0.307979 −0.0260289
\(141\) − 7.00000i − 0.589506i
\(142\) 9.56465 0.802647
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 5.91185i − 0.490953i
\(146\) −2.31336 −0.191455
\(147\) 6.90515 0.569527
\(148\) 7.07606i 0.581649i
\(149\) 3.44935i 0.282582i 0.989968 + 0.141291i \(0.0451253\pi\)
−0.989968 + 0.141291i \(0.954875\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) − 2.84117i − 0.231211i −0.993295 0.115605i \(-0.963119\pi\)
0.993295 0.115605i \(-0.0368808\pi\)
\(152\) −5.80194 −0.470599
\(153\) 6.85086 0.553859
\(154\) 0.103211i 0.00831702i
\(155\) −0.0609989 −0.00489955
\(156\) 0 0
\(157\) 16.2228 1.29472 0.647361 0.762184i \(-0.275873\pi\)
0.647361 + 0.762184i \(0.275873\pi\)
\(158\) 0.0760644i 0.00605136i
\(159\) 12.1739 0.965453
\(160\) 1.00000 0.0790569
\(161\) 0.725873i 0.0572068i
\(162\) − 1.00000i − 0.0785674i
\(163\) 0.0924579i 0.00724186i 0.999993 + 0.00362093i \(0.00115258\pi\)
−0.999993 + 0.00362093i \(0.998847\pi\)
\(164\) − 3.40581i − 0.265949i
\(165\) 0.335126 0.0260895
\(166\) 3.84117 0.298132
\(167\) 9.64071i 0.746021i 0.927827 + 0.373010i \(0.121674\pi\)
−0.927827 + 0.373010i \(0.878326\pi\)
\(168\) 0.307979 0.0237611
\(169\) 0 0
\(170\) 6.85086 0.525437
\(171\) 5.80194i 0.443685i
\(172\) −0.0489173 −0.00372991
\(173\) −0.719169 −0.0546774 −0.0273387 0.999626i \(-0.508703\pi\)
−0.0273387 + 0.999626i \(0.508703\pi\)
\(174\) 5.91185i 0.448176i
\(175\) 0.307979i 0.0232810i
\(176\) − 0.335126i − 0.0252610i
\(177\) 13.1347i 0.987262i
\(178\) 4.41119 0.330633
\(179\) 2.01208 0.150390 0.0751950 0.997169i \(-0.476042\pi\)
0.0751950 + 0.997169i \(0.476042\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) 2.57135 0.191127 0.0955635 0.995423i \(-0.469535\pi\)
0.0955635 + 0.995423i \(0.469535\pi\)
\(182\) 0 0
\(183\) 2.14675 0.158692
\(184\) − 2.35690i − 0.173753i
\(185\) 7.07606 0.520243
\(186\) 0.0609989 0.00447266
\(187\) − 2.29590i − 0.167893i
\(188\) 7.00000i 0.510527i
\(189\) − 0.307979i − 0.0224021i
\(190\) 5.80194i 0.420917i
\(191\) 3.80194 0.275099 0.137549 0.990495i \(-0.456077\pi\)
0.137549 + 0.990495i \(0.456077\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 8.86294i 0.637968i 0.947760 + 0.318984i \(0.103342\pi\)
−0.947760 + 0.318984i \(0.896658\pi\)
\(194\) 6.13169 0.440229
\(195\) 0 0
\(196\) −6.90515 −0.493225
\(197\) − 22.6504i − 1.61377i −0.590706 0.806887i \(-0.701151\pi\)
0.590706 0.806887i \(-0.298849\pi\)
\(198\) −0.335126 −0.0238163
\(199\) 5.46681 0.387532 0.193766 0.981048i \(-0.437930\pi\)
0.193766 + 0.981048i \(0.437930\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) 11.9608i 0.843648i
\(202\) 1.32975i 0.0935608i
\(203\) 1.82072i 0.127790i
\(204\) −6.85086 −0.479656
\(205\) −3.40581 −0.237872
\(206\) 8.92154i 0.621593i
\(207\) −2.35690 −0.163816
\(208\) 0 0
\(209\) 1.94438 0.134495
\(210\) − 0.307979i − 0.0212525i
\(211\) 11.0422 0.760177 0.380089 0.924950i \(-0.375893\pi\)
0.380089 + 0.924950i \(0.375893\pi\)
\(212\) −12.1739 −0.836107
\(213\) 9.56465i 0.655359i
\(214\) − 14.4644i − 0.988767i
\(215\) 0.0489173i 0.00333613i
\(216\) 1.00000i 0.0680414i
\(217\) 0.0187864 0.00127530
\(218\) 7.13706 0.483383
\(219\) − 2.31336i − 0.156322i
\(220\) −0.335126 −0.0225942
\(221\) 0 0
\(222\) −7.07606 −0.474914
\(223\) 18.3884i 1.23138i 0.787990 + 0.615688i \(0.211122\pi\)
−0.787990 + 0.615688i \(0.788878\pi\)
\(224\) −0.307979 −0.0205777
\(225\) −1.00000 −0.0666667
\(226\) 16.3424i 1.08708i
\(227\) 12.0073i 0.796952i 0.917179 + 0.398476i \(0.130461\pi\)
−0.917179 + 0.398476i \(0.869539\pi\)
\(228\) − 5.80194i − 0.384243i
\(229\) − 25.4601i − 1.68245i −0.540684 0.841226i \(-0.681835\pi\)
0.540684 0.841226i \(-0.318165\pi\)
\(230\) −2.35690 −0.155409
\(231\) −0.103211 −0.00679082
\(232\) − 5.91185i − 0.388132i
\(233\) 2.10992 0.138225 0.0691126 0.997609i \(-0.477983\pi\)
0.0691126 + 0.997609i \(0.477983\pi\)
\(234\) 0 0
\(235\) 7.00000 0.456630
\(236\) − 13.1347i − 0.854994i
\(237\) −0.0760644 −0.00494091
\(238\) −2.10992 −0.136766
\(239\) 3.95838i 0.256046i 0.991771 + 0.128023i \(0.0408632\pi\)
−0.991771 + 0.128023i \(0.959137\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 23.1564i 1.49164i 0.666149 + 0.745819i \(0.267942\pi\)
−0.666149 + 0.745819i \(0.732058\pi\)
\(242\) − 10.8877i − 0.699887i
\(243\) 1.00000 0.0641500
\(244\) −2.14675 −0.137432
\(245\) 6.90515i 0.441154i
\(246\) 3.40581 0.217147
\(247\) 0 0
\(248\) −0.0609989 −0.00387344
\(249\) 3.84117i 0.243424i
\(250\) −1.00000 −0.0632456
\(251\) 5.54048 0.349712 0.174856 0.984594i \(-0.444054\pi\)
0.174856 + 0.984594i \(0.444054\pi\)
\(252\) 0.307979i 0.0194008i
\(253\) 0.789856i 0.0496578i
\(254\) − 0.911854i − 0.0572148i
\(255\) 6.85086i 0.429017i
\(256\) 1.00000 0.0625000
\(257\) −6.60819 −0.412207 −0.206104 0.978530i \(-0.566078\pi\)
−0.206104 + 0.978530i \(0.566078\pi\)
\(258\) − 0.0489173i − 0.00304546i
\(259\) −2.17928 −0.135414
\(260\) 0 0
\(261\) −5.91185 −0.365935
\(262\) − 13.0368i − 0.805418i
\(263\) −0.660563 −0.0407320 −0.0203660 0.999793i \(-0.506483\pi\)
−0.0203660 + 0.999793i \(0.506483\pi\)
\(264\) 0.335126 0.0206256
\(265\) 12.1739i 0.747837i
\(266\) − 1.78687i − 0.109560i
\(267\) 4.41119i 0.269960i
\(268\) − 11.9608i − 0.730620i
\(269\) −26.7928 −1.63359 −0.816794 0.576929i \(-0.804251\pi\)
−0.816794 + 0.576929i \(0.804251\pi\)
\(270\) 1.00000 0.0608581
\(271\) − 14.0151i − 0.851355i −0.904875 0.425677i \(-0.860036\pi\)
0.904875 0.425677i \(-0.139964\pi\)
\(272\) 6.85086 0.415394
\(273\) 0 0
\(274\) −12.8877 −0.778574
\(275\) 0.335126i 0.0202088i
\(276\) 2.35690 0.141868
\(277\) 23.4252 1.40748 0.703742 0.710456i \(-0.251511\pi\)
0.703742 + 0.710456i \(0.251511\pi\)
\(278\) 2.01208i 0.120677i
\(279\) 0.0609989i 0.00365191i
\(280\) 0.307979i 0.0184052i
\(281\) − 8.40044i − 0.501128i −0.968100 0.250564i \(-0.919384\pi\)
0.968100 0.250564i \(-0.0806161\pi\)
\(282\) −7.00000 −0.416844
\(283\) 31.6142 1.87927 0.939633 0.342183i \(-0.111166\pi\)
0.939633 + 0.342183i \(0.111166\pi\)
\(284\) − 9.56465i − 0.567557i
\(285\) −5.80194 −0.343677
\(286\) 0 0
\(287\) 1.04892 0.0619156
\(288\) − 1.00000i − 0.0589256i
\(289\) 29.9342 1.76084
\(290\) −5.91185 −0.347156
\(291\) 6.13169i 0.359446i
\(292\) 2.31336i 0.135379i
\(293\) 17.5254i 1.02385i 0.859031 + 0.511923i \(0.171067\pi\)
−0.859031 + 0.511923i \(0.828933\pi\)
\(294\) − 6.90515i − 0.402716i
\(295\) −13.1347 −0.764730
\(296\) 7.07606 0.411288
\(297\) − 0.335126i − 0.0194460i
\(298\) 3.44935 0.199816
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) − 0.0150655i 0 0.000868360i
\(302\) −2.84117 −0.163491
\(303\) −1.32975 −0.0763921
\(304\) 5.80194i 0.332764i
\(305\) 2.14675i 0.122923i
\(306\) − 6.85086i − 0.391637i
\(307\) − 20.6732i − 1.17988i −0.807446 0.589942i \(-0.799151\pi\)
0.807446 0.589942i \(-0.200849\pi\)
\(308\) 0.103211 0.00588102
\(309\) −8.92154 −0.507529
\(310\) 0.0609989i 0.00346451i
\(311\) 22.0248 1.24891 0.624455 0.781061i \(-0.285321\pi\)
0.624455 + 0.781061i \(0.285321\pi\)
\(312\) 0 0
\(313\) 21.7192 1.22764 0.613820 0.789446i \(-0.289632\pi\)
0.613820 + 0.789446i \(0.289632\pi\)
\(314\) − 16.2228i − 0.915506i
\(315\) 0.307979 0.0173526
\(316\) 0.0760644 0.00427896
\(317\) − 16.4644i − 0.924734i −0.886689 0.462367i \(-0.847000\pi\)
0.886689 0.462367i \(-0.153000\pi\)
\(318\) − 12.1739i − 0.682678i
\(319\) 1.98121i 0.110927i
\(320\) − 1.00000i − 0.0559017i
\(321\) 14.4644 0.807325
\(322\) 0.725873 0.0404513
\(323\) 39.7482i 2.21165i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 0.0924579 0.00512077
\(327\) 7.13706i 0.394681i
\(328\) −3.40581 −0.188055
\(329\) −2.15585 −0.118856
\(330\) − 0.335126i − 0.0184481i
\(331\) 26.4373i 1.45312i 0.687101 + 0.726562i \(0.258883\pi\)
−0.687101 + 0.726562i \(0.741117\pi\)
\(332\) − 3.84117i − 0.210811i
\(333\) − 7.07606i − 0.387766i
\(334\) 9.64071 0.527516
\(335\) −11.9608 −0.653487
\(336\) − 0.307979i − 0.0168016i
\(337\) −27.4698 −1.49638 −0.748188 0.663487i \(-0.769076\pi\)
−0.748188 + 0.663487i \(0.769076\pi\)
\(338\) 0 0
\(339\) −16.3424 −0.887598
\(340\) − 6.85086i − 0.371540i
\(341\) 0.0204423 0.00110701
\(342\) 5.80194 0.313733
\(343\) − 4.28249i − 0.231233i
\(344\) 0.0489173i 0.00263745i
\(345\) − 2.35690i − 0.126891i
\(346\) 0.719169i 0.0386627i
\(347\) −33.1323 −1.77863 −0.889317 0.457291i \(-0.848820\pi\)
−0.889317 + 0.457291i \(0.848820\pi\)
\(348\) 5.91185 0.316909
\(349\) − 26.3739i − 1.41176i −0.708331 0.705881i \(-0.750551\pi\)
0.708331 0.705881i \(-0.249449\pi\)
\(350\) 0.307979 0.0164621
\(351\) 0 0
\(352\) −0.335126 −0.0178623
\(353\) 21.0291i 1.11926i 0.828741 + 0.559632i \(0.189058\pi\)
−0.828741 + 0.559632i \(0.810942\pi\)
\(354\) 13.1347 0.698100
\(355\) −9.56465 −0.507639
\(356\) − 4.41119i − 0.233793i
\(357\) − 2.10992i − 0.111669i
\(358\) − 2.01208i − 0.106342i
\(359\) 30.8877i 1.63019i 0.579327 + 0.815095i \(0.303315\pi\)
−0.579327 + 0.815095i \(0.696685\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −14.6625 −0.771710
\(362\) − 2.57135i − 0.135147i
\(363\) 10.8877 0.571456
\(364\) 0 0
\(365\) 2.31336 0.121087
\(366\) − 2.14675i − 0.112213i
\(367\) 11.9584 0.624222 0.312111 0.950046i \(-0.398964\pi\)
0.312111 + 0.950046i \(0.398964\pi\)
\(368\) −2.35690 −0.122862
\(369\) 3.40581i 0.177300i
\(370\) − 7.07606i − 0.367867i
\(371\) − 3.74930i − 0.194654i
\(372\) − 0.0609989i − 0.00316265i
\(373\) 25.0315 1.29608 0.648040 0.761606i \(-0.275589\pi\)
0.648040 + 0.761606i \(0.275589\pi\)
\(374\) −2.29590 −0.118718
\(375\) − 1.00000i − 0.0516398i
\(376\) 7.00000 0.360997
\(377\) 0 0
\(378\) −0.307979 −0.0158407
\(379\) − 34.4989i − 1.77209i −0.463602 0.886044i \(-0.653443\pi\)
0.463602 0.886044i \(-0.346557\pi\)
\(380\) 5.80194 0.297633
\(381\) 0.911854 0.0467157
\(382\) − 3.80194i − 0.194524i
\(383\) 10.4644i 0.534707i 0.963599 + 0.267353i \(0.0861491\pi\)
−0.963599 + 0.267353i \(0.913851\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) − 0.103211i − 0.00526014i
\(386\) 8.86294 0.451112
\(387\) 0.0489173 0.00248661
\(388\) − 6.13169i − 0.311289i
\(389\) −25.2218 −1.27879 −0.639397 0.768877i \(-0.720816\pi\)
−0.639397 + 0.768877i \(0.720816\pi\)
\(390\) 0 0
\(391\) −16.1468 −0.816576
\(392\) 6.90515i 0.348763i
\(393\) 13.0368 0.657621
\(394\) −22.6504 −1.14111
\(395\) − 0.0760644i − 0.00382722i
\(396\) 0.335126i 0.0168407i
\(397\) − 21.3860i − 1.07333i −0.843795 0.536665i \(-0.819684\pi\)
0.843795 0.536665i \(-0.180316\pi\)
\(398\) − 5.46681i − 0.274027i
\(399\) 1.78687 0.0894555
\(400\) −1.00000 −0.0500000
\(401\) 11.8062i 0.589576i 0.955563 + 0.294788i \(0.0952490\pi\)
−0.955563 + 0.294788i \(0.904751\pi\)
\(402\) 11.9608 0.596549
\(403\) 0 0
\(404\) 1.32975 0.0661575
\(405\) 1.00000i 0.0496904i
\(406\) 1.82072 0.0903610
\(407\) −2.37137 −0.117544
\(408\) 6.85086i 0.339168i
\(409\) − 5.65950i − 0.279844i −0.990163 0.139922i \(-0.955315\pi\)
0.990163 0.139922i \(-0.0446852\pi\)
\(410\) 3.40581i 0.168201i
\(411\) − 12.8877i − 0.635703i
\(412\) 8.92154 0.439533
\(413\) 4.04520 0.199051
\(414\) 2.35690i 0.115835i
\(415\) −3.84117 −0.188555
\(416\) 0 0
\(417\) −2.01208 −0.0985321
\(418\) − 1.94438i − 0.0951026i
\(419\) 17.0084 0.830913 0.415456 0.909613i \(-0.363622\pi\)
0.415456 + 0.909613i \(0.363622\pi\)
\(420\) −0.307979 −0.0150278
\(421\) − 10.2851i − 0.501267i −0.968082 0.250634i \(-0.919361\pi\)
0.968082 0.250634i \(-0.0806389\pi\)
\(422\) − 11.0422i − 0.537526i
\(423\) − 7.00000i − 0.340352i
\(424\) 12.1739i 0.591217i
\(425\) −6.85086 −0.332315
\(426\) 9.56465 0.463409
\(427\) − 0.661154i − 0.0319955i
\(428\) −14.4644 −0.699164
\(429\) 0 0
\(430\) 0.0489173 0.00235900
\(431\) − 35.9657i − 1.73241i −0.499693 0.866203i \(-0.666554\pi\)
0.499693 0.866203i \(-0.333446\pi\)
\(432\) 1.00000 0.0481125
\(433\) −7.63342 −0.366839 −0.183419 0.983035i \(-0.558717\pi\)
−0.183419 + 0.983035i \(0.558717\pi\)
\(434\) − 0.0187864i 0 0.000901774i
\(435\) − 5.91185i − 0.283452i
\(436\) − 7.13706i − 0.341803i
\(437\) − 13.6746i − 0.654143i
\(438\) −2.31336 −0.110536
\(439\) 34.4373 1.64360 0.821801 0.569775i \(-0.192970\pi\)
0.821801 + 0.569775i \(0.192970\pi\)
\(440\) 0.335126i 0.0159765i
\(441\) 6.90515 0.328817
\(442\) 0 0
\(443\) 33.6431 1.59843 0.799216 0.601044i \(-0.205248\pi\)
0.799216 + 0.601044i \(0.205248\pi\)
\(444\) 7.07606i 0.335815i
\(445\) −4.41119 −0.209110
\(446\) 18.3884 0.870714
\(447\) 3.44935i 0.163149i
\(448\) 0.307979i 0.0145506i
\(449\) 27.9729i 1.32012i 0.751213 + 0.660060i \(0.229469\pi\)
−0.751213 + 0.660060i \(0.770531\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) 1.14138 0.0537453
\(452\) 16.3424 0.768683
\(453\) − 2.84117i − 0.133490i
\(454\) 12.0073 0.563530
\(455\) 0 0
\(456\) −5.80194 −0.271701
\(457\) − 6.18060i − 0.289116i −0.989496 0.144558i \(-0.953824\pi\)
0.989496 0.144558i \(-0.0461761\pi\)
\(458\) −25.4601 −1.18967
\(459\) 6.85086 0.319771
\(460\) 2.35690i 0.109891i
\(461\) 38.1062i 1.77478i 0.461017 + 0.887391i \(0.347485\pi\)
−0.461017 + 0.887391i \(0.652515\pi\)
\(462\) 0.103211i 0.00480183i
\(463\) 2.06829i 0.0961218i 0.998844 + 0.0480609i \(0.0153042\pi\)
−0.998844 + 0.0480609i \(0.984696\pi\)
\(464\) −5.91185 −0.274451
\(465\) −0.0609989 −0.00282876
\(466\) − 2.10992i − 0.0977400i
\(467\) 4.27114 0.197645 0.0988225 0.995105i \(-0.468492\pi\)
0.0988225 + 0.995105i \(0.468492\pi\)
\(468\) 0 0
\(469\) 3.68366 0.170096
\(470\) − 7.00000i − 0.322886i
\(471\) 16.2228 0.747508
\(472\) −13.1347 −0.604572
\(473\) − 0.0163935i 0 0.000753772i
\(474\) 0.0760644i 0.00349375i
\(475\) − 5.80194i − 0.266211i
\(476\) 2.10992i 0.0967079i
\(477\) 12.1739 0.557405
\(478\) 3.95838 0.181052
\(479\) 38.7560i 1.77081i 0.464823 + 0.885404i \(0.346118\pi\)
−0.464823 + 0.885404i \(0.653882\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 23.1564 1.05475
\(483\) 0.725873i 0.0330284i
\(484\) −10.8877 −0.494895
\(485\) −6.13169 −0.278426
\(486\) − 1.00000i − 0.0453609i
\(487\) − 33.4620i − 1.51631i −0.652075 0.758155i \(-0.726101\pi\)
0.652075 0.758155i \(-0.273899\pi\)
\(488\) 2.14675i 0.0971789i
\(489\) 0.0924579i 0.00418109i
\(490\) 6.90515 0.311943
\(491\) −35.1487 −1.58624 −0.793119 0.609067i \(-0.791544\pi\)
−0.793119 + 0.609067i \(0.791544\pi\)
\(492\) − 3.40581i − 0.153546i
\(493\) −40.5013 −1.82408
\(494\) 0 0
\(495\) 0.335126 0.0150628
\(496\) 0.0609989i 0.00273893i
\(497\) 2.94571 0.132133
\(498\) 3.84117 0.172127
\(499\) − 9.61463i − 0.430410i −0.976569 0.215205i \(-0.930958\pi\)
0.976569 0.215205i \(-0.0690420\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) 9.64071i 0.430715i
\(502\) − 5.54048i − 0.247284i
\(503\) 18.5265 0.826055 0.413028 0.910719i \(-0.364471\pi\)
0.413028 + 0.910719i \(0.364471\pi\)
\(504\) 0.307979 0.0137185
\(505\) − 1.32975i − 0.0591730i
\(506\) 0.789856 0.0351134
\(507\) 0 0
\(508\) −0.911854 −0.0404570
\(509\) 5.43296i 0.240812i 0.992725 + 0.120406i \(0.0384196\pi\)
−0.992725 + 0.120406i \(0.961580\pi\)
\(510\) 6.85086 0.303361
\(511\) −0.712464 −0.0315175
\(512\) − 1.00000i − 0.0441942i
\(513\) 5.80194i 0.256162i
\(514\) 6.60819i 0.291475i
\(515\) − 8.92154i − 0.393130i
\(516\) −0.0489173 −0.00215347
\(517\) −2.34588 −0.103172
\(518\) 2.17928i 0.0957519i
\(519\) −0.719169 −0.0315680
\(520\) 0 0
\(521\) −38.8001 −1.69986 −0.849932 0.526892i \(-0.823357\pi\)
−0.849932 + 0.526892i \(0.823357\pi\)
\(522\) 5.91185i 0.258755i
\(523\) 32.0629 1.40201 0.701007 0.713155i \(-0.252734\pi\)
0.701007 + 0.713155i \(0.252734\pi\)
\(524\) −13.0368 −0.569517
\(525\) 0.307979i 0.0134413i
\(526\) 0.660563i 0.0288019i
\(527\) 0.417895i 0.0182038i
\(528\) − 0.335126i − 0.0145845i
\(529\) −17.4450 −0.758480
\(530\) 12.1739 0.528800
\(531\) 13.1347i 0.569996i
\(532\) −1.78687 −0.0774707
\(533\) 0 0
\(534\) 4.41119 0.190891
\(535\) 14.4644i 0.625351i
\(536\) −11.9608 −0.516627
\(537\) 2.01208 0.0868277
\(538\) 26.7928i 1.15512i
\(539\) − 2.31409i − 0.0996750i
\(540\) − 1.00000i − 0.0430331i
\(541\) − 31.8334i − 1.36862i −0.729189 0.684312i \(-0.760103\pi\)
0.729189 0.684312i \(-0.239897\pi\)
\(542\) −14.0151 −0.601999
\(543\) 2.57135 0.110347
\(544\) − 6.85086i − 0.293728i
\(545\) −7.13706 −0.305718
\(546\) 0 0
\(547\) −44.4010 −1.89845 −0.949225 0.314597i \(-0.898131\pi\)
−0.949225 + 0.314597i \(0.898131\pi\)
\(548\) 12.8877i 0.550535i
\(549\) 2.14675 0.0916211
\(550\) 0.335126 0.0142898
\(551\) − 34.3002i − 1.46124i
\(552\) − 2.35690i − 0.100316i
\(553\) 0.0234262i 0 0.000996184i
\(554\) − 23.4252i − 0.995241i
\(555\) 7.07606 0.300362
\(556\) 2.01208 0.0853313
\(557\) 7.73423i 0.327710i 0.986484 + 0.163855i \(0.0523929\pi\)
−0.986484 + 0.163855i \(0.947607\pi\)
\(558\) 0.0609989 0.00258229
\(559\) 0 0
\(560\) 0.307979 0.0130145
\(561\) − 2.29590i − 0.0969328i
\(562\) −8.40044 −0.354351
\(563\) −44.5803 −1.87884 −0.939418 0.342774i \(-0.888633\pi\)
−0.939418 + 0.342774i \(0.888633\pi\)
\(564\) 7.00000i 0.294753i
\(565\) − 16.3424i − 0.687531i
\(566\) − 31.6142i − 1.32884i
\(567\) − 0.307979i − 0.0129339i
\(568\) −9.56465 −0.401324
\(569\) −24.9065 −1.04413 −0.522067 0.852905i \(-0.674839\pi\)
−0.522067 + 0.852905i \(0.674839\pi\)
\(570\) 5.80194i 0.243016i
\(571\) 0.873690 0.0365628 0.0182814 0.999833i \(-0.494181\pi\)
0.0182814 + 0.999833i \(0.494181\pi\)
\(572\) 0 0
\(573\) 3.80194 0.158828
\(574\) − 1.04892i − 0.0437810i
\(575\) 2.35690 0.0982894
\(576\) −1.00000 −0.0416667
\(577\) − 20.0871i − 0.836236i −0.908393 0.418118i \(-0.862690\pi\)
0.908393 0.418118i \(-0.137310\pi\)
\(578\) − 29.9342i − 1.24510i
\(579\) 8.86294i 0.368331i
\(580\) 5.91185i 0.245476i
\(581\) 1.18300 0.0490790
\(582\) 6.13169 0.254167
\(583\) − 4.07979i − 0.168967i
\(584\) 2.31336 0.0957273
\(585\) 0 0
\(586\) 17.5254 0.723968
\(587\) − 16.0901i − 0.664108i −0.943260 0.332054i \(-0.892258\pi\)
0.943260 0.332054i \(-0.107742\pi\)
\(588\) −6.90515 −0.284764
\(589\) −0.353912 −0.0145827
\(590\) 13.1347i 0.540746i
\(591\) − 22.6504i − 0.931713i
\(592\) − 7.07606i − 0.290824i
\(593\) − 4.85517i − 0.199378i −0.995019 0.0996889i \(-0.968215\pi\)
0.995019 0.0996889i \(-0.0317848\pi\)
\(594\) −0.335126 −0.0137504
\(595\) 2.10992 0.0864981
\(596\) − 3.44935i − 0.141291i
\(597\) 5.46681 0.223742
\(598\) 0 0
\(599\) −34.4319 −1.40685 −0.703425 0.710770i \(-0.748347\pi\)
−0.703425 + 0.710770i \(0.748347\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) −24.5244 −1.00037 −0.500185 0.865919i \(-0.666735\pi\)
−0.500185 + 0.865919i \(0.666735\pi\)
\(602\) −0.0150655 −0.000614024 0
\(603\) 11.9608i 0.487080i
\(604\) 2.84117i 0.115605i
\(605\) 10.8877i 0.442648i
\(606\) 1.32975i 0.0540174i
\(607\) 17.6420 0.716068 0.358034 0.933708i \(-0.383447\pi\)
0.358034 + 0.933708i \(0.383447\pi\)
\(608\) 5.80194 0.235300
\(609\) 1.82072i 0.0737795i
\(610\) 2.14675 0.0869194
\(611\) 0 0
\(612\) −6.85086 −0.276929
\(613\) 44.5018i 1.79741i 0.438551 + 0.898706i \(0.355492\pi\)
−0.438551 + 0.898706i \(0.644508\pi\)
\(614\) −20.6732 −0.834304
\(615\) −3.40581 −0.137336
\(616\) − 0.103211i − 0.00415851i
\(617\) 32.1618i 1.29479i 0.762156 + 0.647393i \(0.224141\pi\)
−0.762156 + 0.647393i \(0.775859\pi\)
\(618\) 8.92154i 0.358877i
\(619\) − 24.8340i − 0.998162i −0.866555 0.499081i \(-0.833671\pi\)
0.866555 0.499081i \(-0.166329\pi\)
\(620\) 0.0609989 0.00244978
\(621\) −2.35690 −0.0945790
\(622\) − 22.0248i − 0.883112i
\(623\) 1.35855 0.0544292
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 21.7192i − 0.868073i
\(627\) 1.94438 0.0776510
\(628\) −16.2228 −0.647361
\(629\) − 48.4771i − 1.93291i
\(630\) − 0.307979i − 0.0122702i
\(631\) − 10.5418i − 0.419663i −0.977738 0.209831i \(-0.932708\pi\)
0.977738 0.209831i \(-0.0672915\pi\)
\(632\) − 0.0760644i − 0.00302568i
\(633\) 11.0422 0.438889
\(634\) −16.4644 −0.653886
\(635\) 0.911854i 0.0361858i
\(636\) −12.1739 −0.482726
\(637\) 0 0
\(638\) 1.98121 0.0784370
\(639\) 9.56465i 0.378372i
\(640\) −1.00000 −0.0395285
\(641\) −33.5663 −1.32579 −0.662895 0.748713i \(-0.730672\pi\)
−0.662895 + 0.748713i \(0.730672\pi\)
\(642\) − 14.4644i − 0.570865i
\(643\) 8.24160i 0.325017i 0.986707 + 0.162509i \(0.0519585\pi\)
−0.986707 + 0.162509i \(0.948041\pi\)
\(644\) − 0.725873i − 0.0286034i
\(645\) 0.0489173i 0.00192612i
\(646\) 39.7482 1.56387
\(647\) 3.66056 0.143912 0.0719558 0.997408i \(-0.477076\pi\)
0.0719558 + 0.997408i \(0.477076\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 4.40176 0.172784
\(650\) 0 0
\(651\) 0.0187864 0.000736295 0
\(652\) − 0.0924579i − 0.00362093i
\(653\) −44.6064 −1.74558 −0.872791 0.488093i \(-0.837693\pi\)
−0.872791 + 0.488093i \(0.837693\pi\)
\(654\) 7.13706 0.279081
\(655\) 13.0368i 0.509391i
\(656\) 3.40581i 0.132975i
\(657\) − 2.31336i − 0.0902526i
\(658\) 2.15585i 0.0840438i
\(659\) −2.51035 −0.0977895 −0.0488947 0.998804i \(-0.515570\pi\)
−0.0488947 + 0.998804i \(0.515570\pi\)
\(660\) −0.335126 −0.0130447
\(661\) − 40.3096i − 1.56786i −0.620847 0.783932i \(-0.713211\pi\)
0.620847 0.783932i \(-0.286789\pi\)
\(662\) 26.4373 1.02751
\(663\) 0 0
\(664\) −3.84117 −0.149066
\(665\) 1.78687i 0.0692919i
\(666\) −7.07606 −0.274192
\(667\) 13.9336 0.539512
\(668\) − 9.64071i − 0.373010i
\(669\) 18.3884i 0.710935i
\(670\) 11.9608i 0.462085i
\(671\) − 0.719432i − 0.0277733i
\(672\) −0.307979 −0.0118805
\(673\) −42.6340 −1.64342 −0.821710 0.569906i \(-0.806980\pi\)
−0.821710 + 0.569906i \(0.806980\pi\)
\(674\) 27.4698i 1.05810i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 18.5254 0.711990 0.355995 0.934488i \(-0.384142\pi\)
0.355995 + 0.934488i \(0.384142\pi\)
\(678\) 16.3424i 0.627627i
\(679\) 1.88843 0.0724712
\(680\) −6.85086 −0.262718
\(681\) 12.0073i 0.460121i
\(682\) − 0.0204423i 0 0.000782776i
\(683\) − 48.1769i − 1.84344i −0.387859 0.921719i \(-0.626785\pi\)
0.387859 0.921719i \(-0.373215\pi\)
\(684\) − 5.80194i − 0.221843i
\(685\) 12.8877 0.492413
\(686\) −4.28249 −0.163506
\(687\) − 25.4601i − 0.971364i
\(688\) 0.0489173 0.00186496
\(689\) 0 0
\(690\) −2.35690 −0.0897255
\(691\) 37.0084i 1.40786i 0.710267 + 0.703932i \(0.248574\pi\)
−0.710267 + 0.703932i \(0.751426\pi\)
\(692\) 0.719169 0.0273387
\(693\) −0.103211 −0.00392068
\(694\) 33.1323i 1.25768i
\(695\) − 2.01208i − 0.0763226i
\(696\) − 5.91185i − 0.224088i
\(697\) 23.3327i 0.883790i
\(698\) −26.3739 −0.998266
\(699\) 2.10992 0.0798044
\(700\) − 0.307979i − 0.0116405i
\(701\) −9.25188 −0.349439 −0.174719 0.984618i \(-0.555902\pi\)
−0.174719 + 0.984618i \(0.555902\pi\)
\(702\) 0 0
\(703\) 41.0549 1.54841
\(704\) 0.335126i 0.0126305i
\(705\) 7.00000 0.263635
\(706\) 21.0291 0.791439
\(707\) 0.409534i 0.0154021i
\(708\) − 13.1347i − 0.493631i
\(709\) 15.0573i 0.565488i 0.959195 + 0.282744i \(0.0912447\pi\)
−0.959195 + 0.282744i \(0.908755\pi\)
\(710\) 9.56465i 0.358955i
\(711\) −0.0760644 −0.00285264
\(712\) −4.41119 −0.165316
\(713\) − 0.143768i − 0.00538416i
\(714\) −2.10992 −0.0789616
\(715\) 0 0
\(716\) −2.01208 −0.0751950
\(717\) 3.95838i 0.147828i
\(718\) 30.8877 1.15272
\(719\) −24.8364 −0.926241 −0.463120 0.886295i \(-0.653270\pi\)
−0.463120 + 0.886295i \(0.653270\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) 2.74764i 0.102328i
\(722\) 14.6625i 0.545681i
\(723\) 23.1564i 0.861197i
\(724\) −2.57135 −0.0955635
\(725\) 5.91185 0.219561
\(726\) − 10.8877i − 0.404080i
\(727\) 24.7590 0.918260 0.459130 0.888369i \(-0.348161\pi\)
0.459130 + 0.888369i \(0.348161\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 2.31336i − 0.0856211i
\(731\) 0.335126 0.0123951
\(732\) −2.14675 −0.0793462
\(733\) − 15.4359i − 0.570140i −0.958507 0.285070i \(-0.907983\pi\)
0.958507 0.285070i \(-0.0920168\pi\)
\(734\) − 11.9584i − 0.441392i
\(735\) 6.90515i 0.254700i
\(736\) 2.35690i 0.0868763i
\(737\) 4.00836 0.147650
\(738\) 3.40581 0.125370
\(739\) 20.3327i 0.747952i 0.927438 + 0.373976i \(0.122006\pi\)
−0.927438 + 0.373976i \(0.877994\pi\)
\(740\) −7.07606 −0.260121
\(741\) 0 0
\(742\) −3.74930 −0.137641
\(743\) 17.8243i 0.653910i 0.945040 + 0.326955i \(0.106023\pi\)
−0.945040 + 0.326955i \(0.893977\pi\)
\(744\) −0.0609989 −0.00223633
\(745\) −3.44935 −0.126375
\(746\) − 25.0315i − 0.916467i
\(747\) 3.84117i 0.140541i
\(748\) 2.29590i 0.0839463i
\(749\) − 4.45473i − 0.162772i
\(750\) −1.00000 −0.0365148
\(751\) −48.0616 −1.75379 −0.876896 0.480680i \(-0.840390\pi\)
−0.876896 + 0.480680i \(0.840390\pi\)
\(752\) − 7.00000i − 0.255264i
\(753\) 5.54048 0.201906
\(754\) 0 0
\(755\) 2.84117 0.103401
\(756\) 0.307979i 0.0112011i
\(757\) 32.9748 1.19849 0.599244 0.800566i \(-0.295468\pi\)
0.599244 + 0.800566i \(0.295468\pi\)
\(758\) −34.4989 −1.25306
\(759\) 0.789856i 0.0286700i
\(760\) − 5.80194i − 0.210458i
\(761\) − 11.9946i − 0.434805i −0.976082 0.217402i \(-0.930242\pi\)
0.976082 0.217402i \(-0.0697584\pi\)
\(762\) − 0.911854i − 0.0330330i
\(763\) 2.19806 0.0795752
\(764\) −3.80194 −0.137549
\(765\) 6.85086i 0.247693i
\(766\) 10.4644 0.378095
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) − 18.5036i − 0.667259i −0.942704 0.333629i \(-0.891727\pi\)
0.942704 0.333629i \(-0.108273\pi\)
\(770\) −0.103211 −0.00371948
\(771\) −6.60819 −0.237988
\(772\) − 8.86294i − 0.318984i
\(773\) − 23.4946i − 0.845040i −0.906354 0.422520i \(-0.861146\pi\)
0.906354 0.422520i \(-0.138854\pi\)
\(774\) − 0.0489173i − 0.00175830i
\(775\) − 0.0609989i − 0.00219115i
\(776\) −6.13169 −0.220115
\(777\) −2.17928 −0.0781811
\(778\) 25.2218i 0.904244i
\(779\) −19.7603 −0.707987
\(780\) 0 0
\(781\) 3.20536 0.114697
\(782\) 16.1468i 0.577407i
\(783\) −5.91185 −0.211272
\(784\) 6.90515 0.246612
\(785\) 16.2228i 0.579017i
\(786\) − 13.0368i − 0.465009i
\(787\) − 7.23682i − 0.257965i −0.991647 0.128982i \(-0.958829\pi\)
0.991647 0.128982i \(-0.0411711\pi\)
\(788\) 22.6504i 0.806887i
\(789\) −0.660563 −0.0235166
\(790\) −0.0760644 −0.00270625
\(791\) 5.03311i 0.178957i
\(792\) 0.335126 0.0119082
\(793\) 0 0
\(794\) −21.3860 −0.758959
\(795\) 12.1739i 0.431764i
\(796\) −5.46681 −0.193766
\(797\) 46.7079 1.65448 0.827240 0.561849i \(-0.189910\pi\)
0.827240 + 0.561849i \(0.189910\pi\)
\(798\) − 1.78687i − 0.0632546i
\(799\) − 47.9560i − 1.69656i
\(800\) 1.00000i 0.0353553i
\(801\) 4.41119i 0.155862i
\(802\) 11.8062 0.416893
\(803\) −0.775265 −0.0273585
\(804\) − 11.9608i − 0.421824i
\(805\) −0.725873 −0.0255837
\(806\) 0 0
\(807\) −26.7928 −0.943153
\(808\) − 1.32975i − 0.0467804i
\(809\) −9.95838 −0.350118 −0.175059 0.984558i \(-0.556012\pi\)
−0.175059 + 0.984558i \(0.556012\pi\)
\(810\) 1.00000 0.0351364
\(811\) − 36.2452i − 1.27274i −0.771384 0.636370i \(-0.780435\pi\)
0.771384 0.636370i \(-0.219565\pi\)
\(812\) − 1.82072i − 0.0638949i
\(813\) − 14.0151i − 0.491530i
\(814\) 2.37137i 0.0831165i
\(815\) −0.0924579 −0.00323866
\(816\) 6.85086 0.239828
\(817\) 0.283815i 0.00992944i
\(818\) −5.65950 −0.197880
\(819\) 0 0
\(820\) 3.40581 0.118936
\(821\) − 1.37867i − 0.0481158i −0.999711 0.0240579i \(-0.992341\pi\)
0.999711 0.0240579i \(-0.00765860\pi\)
\(822\) −12.8877 −0.449510
\(823\) 23.8713 0.832101 0.416051 0.909341i \(-0.363414\pi\)
0.416051 + 0.909341i \(0.363414\pi\)
\(824\) − 8.92154i − 0.310797i
\(825\) 0.335126i 0.0116676i
\(826\) − 4.04520i − 0.140750i
\(827\) 21.9571i 0.763521i 0.924261 + 0.381761i \(0.124682\pi\)
−0.924261 + 0.381761i \(0.875318\pi\)
\(828\) 2.35690 0.0819078
\(829\) 27.4862 0.954635 0.477317 0.878731i \(-0.341609\pi\)
0.477317 + 0.878731i \(0.341609\pi\)
\(830\) 3.84117i 0.133329i
\(831\) 23.4252 0.812611
\(832\) 0 0
\(833\) 47.3062 1.63906
\(834\) 2.01208i 0.0696727i
\(835\) −9.64071 −0.333631
\(836\) −1.94438 −0.0672477
\(837\) 0.0609989i 0.00210843i
\(838\) − 17.0084i − 0.587544i
\(839\) − 20.5749i − 0.710325i −0.934805 0.355163i \(-0.884425\pi\)
0.934805 0.355163i \(-0.115575\pi\)
\(840\) 0.307979i 0.0106263i
\(841\) 5.95002 0.205173
\(842\) −10.2851 −0.354449
\(843\) − 8.40044i − 0.289326i
\(844\) −11.0422 −0.380089
\(845\) 0 0
\(846\) −7.00000 −0.240665
\(847\) − 3.35317i − 0.115216i
\(848\) 12.1739 0.418053
\(849\) 31.6142 1.08499
\(850\) 6.85086i 0.234982i
\(851\) 16.6775i 0.571699i
\(852\) − 9.56465i − 0.327679i
\(853\) 50.3812i 1.72502i 0.506041 + 0.862509i \(0.331108\pi\)
−0.506041 + 0.862509i \(0.668892\pi\)
\(854\) −0.661154 −0.0226242
\(855\) −5.80194 −0.198422
\(856\) 14.4644i 0.494384i
\(857\) −21.6045 −0.737995 −0.368997 0.929430i \(-0.620299\pi\)
−0.368997 + 0.929430i \(0.620299\pi\)
\(858\) 0 0
\(859\) −38.2959 −1.30664 −0.653320 0.757082i \(-0.726624\pi\)
−0.653320 + 0.757082i \(0.726624\pi\)
\(860\) − 0.0489173i − 0.00166807i
\(861\) 1.04892 0.0357470
\(862\) −35.9657 −1.22500
\(863\) 8.61894i 0.293392i 0.989182 + 0.146696i \(0.0468639\pi\)
−0.989182 + 0.146696i \(0.953136\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) − 0.719169i − 0.0244525i
\(866\) 7.63342i 0.259394i
\(867\) 29.9342 1.01662
\(868\) −0.0187864 −0.000637651 0
\(869\) 0.0254911i 0 0.000864727i
\(870\) −5.91185 −0.200431
\(871\) 0 0
\(872\) −7.13706 −0.241691
\(873\) 6.13169i 0.207526i
\(874\) −13.6746 −0.462549
\(875\) −0.307979 −0.0104116
\(876\) 2.31336i 0.0781610i
\(877\) − 33.1021i − 1.11778i −0.829242 0.558890i \(-0.811227\pi\)
0.829242 0.558890i \(-0.188773\pi\)
\(878\) − 34.4373i − 1.16220i
\(879\) 17.5254i 0.591118i
\(880\) 0.335126 0.0112971
\(881\) −20.8267 −0.701669 −0.350835 0.936437i \(-0.614102\pi\)
−0.350835 + 0.936437i \(0.614102\pi\)
\(882\) − 6.90515i − 0.232508i
\(883\) −0.785807 −0.0264445 −0.0132223 0.999913i \(-0.504209\pi\)
−0.0132223 + 0.999913i \(0.504209\pi\)
\(884\) 0 0
\(885\) −13.1347 −0.441517
\(886\) − 33.6431i − 1.13026i
\(887\) −44.2392 −1.48541 −0.742704 0.669620i \(-0.766457\pi\)
−0.742704 + 0.669620i \(0.766457\pi\)
\(888\) 7.07606 0.237457
\(889\) − 0.280831i − 0.00941878i
\(890\) 4.41119i 0.147863i
\(891\) − 0.335126i − 0.0112271i
\(892\) − 18.3884i − 0.615688i
\(893\) 40.6136 1.35908
\(894\) 3.44935 0.115364
\(895\) 2.01208i 0.0672565i
\(896\) 0.307979 0.0102888
\(897\) 0 0
\(898\) 27.9729 0.933466
\(899\) − 0.360617i − 0.0120272i
\(900\) 1.00000 0.0333333
\(901\) 83.4016 2.77851
\(902\) − 1.14138i − 0.0380036i
\(903\) − 0.0150655i 0 0.000501348i
\(904\) − 16.3424i − 0.543541i
\(905\) 2.57135i 0.0854746i
\(906\) −2.84117 −0.0943914
\(907\) −8.44371 −0.280369 −0.140184 0.990125i \(-0.544770\pi\)
−0.140184 + 0.990125i \(0.544770\pi\)
\(908\) − 12.0073i − 0.398476i
\(909\) −1.32975 −0.0441050
\(910\) 0 0
\(911\) −47.2801 −1.56646 −0.783230 0.621732i \(-0.786429\pi\)
−0.783230 + 0.621732i \(0.786429\pi\)
\(912\) 5.80194i 0.192121i
\(913\) 1.28727 0.0426025
\(914\) −6.18060 −0.204436
\(915\) 2.14675i 0.0709694i
\(916\) 25.4601i 0.841226i
\(917\) − 4.01507i − 0.132589i
\(918\) − 6.85086i − 0.226112i
\(919\) −11.1927 −0.369213 −0.184606 0.982813i \(-0.559101\pi\)
−0.184606 + 0.982813i \(0.559101\pi\)
\(920\) 2.35690 0.0777046
\(921\) − 20.6732i − 0.681206i
\(922\) 38.1062 1.25496
\(923\) 0 0
\(924\) 0.103211 0.00339541
\(925\) 7.07606i 0.232660i
\(926\) 2.06829 0.0679684
\(927\) −8.92154 −0.293022
\(928\) 5.91185i 0.194066i
\(929\) 1.11828i 0.0366895i 0.999832 + 0.0183447i \(0.00583964\pi\)
−0.999832 + 0.0183447i \(0.994160\pi\)
\(930\) 0.0609989i 0.00200023i
\(931\) 40.0632i 1.31302i
\(932\) −2.10992 −0.0691126
\(933\) 22.0248 0.721058
\(934\) − 4.27114i − 0.139756i
\(935\) 2.29590 0.0750839
\(936\) 0 0
\(937\) −29.9051 −0.976959 −0.488479 0.872575i \(-0.662448\pi\)
−0.488479 + 0.872575i \(0.662448\pi\)
\(938\) − 3.68366i − 0.120276i
\(939\) 21.7192 0.708778
\(940\) −7.00000 −0.228315
\(941\) − 32.7646i − 1.06810i −0.845454 0.534048i \(-0.820670\pi\)
0.845454 0.534048i \(-0.179330\pi\)
\(942\) − 16.2228i − 0.528568i
\(943\) − 8.02715i − 0.261400i
\(944\) 13.1347i 0.427497i
\(945\) 0.307979 0.0100185
\(946\) −0.0163935 −0.000532997 0
\(947\) 6.81535i 0.221469i 0.993850 + 0.110735i \(0.0353203\pi\)
−0.993850 + 0.110735i \(0.964680\pi\)
\(948\) 0.0760644 0.00247046
\(949\) 0 0
\(950\) −5.80194 −0.188240
\(951\) − 16.4644i − 0.533895i
\(952\) 2.10992 0.0683828
\(953\) −23.6886 −0.767348 −0.383674 0.923469i \(-0.625341\pi\)
−0.383674 + 0.923469i \(0.625341\pi\)
\(954\) − 12.1739i − 0.394145i
\(955\) 3.80194i 0.123028i
\(956\) − 3.95838i − 0.128023i
\(957\) 1.98121i 0.0640435i
\(958\) 38.7560 1.25215
\(959\) −3.96913 −0.128170
\(960\) − 1.00000i − 0.0322749i
\(961\) 30.9963 0.999880
\(962\) 0 0
\(963\) 14.4644 0.466109
\(964\) − 23.1564i − 0.745819i
\(965\) −8.86294 −0.285308
\(966\) 0.725873 0.0233546
\(967\) − 28.9075i − 0.929604i −0.885415 0.464802i \(-0.846126\pi\)
0.885415 0.464802i \(-0.153874\pi\)
\(968\) 10.8877i 0.349944i
\(969\) 39.7482i 1.27690i
\(970\) 6.13169i 0.196877i
\(971\) −47.2717 −1.51702 −0.758511 0.651660i \(-0.774073\pi\)
−0.758511 + 0.651660i \(0.774073\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0.619678i 0.0198660i
\(974\) −33.4620 −1.07219
\(975\) 0 0
\(976\) 2.14675 0.0687159
\(977\) − 33.9527i − 1.08624i −0.839654 0.543122i \(-0.817242\pi\)
0.839654 0.543122i \(-0.182758\pi\)
\(978\) 0.0924579 0.00295648
\(979\) 1.47830 0.0472468
\(980\) − 6.90515i − 0.220577i
\(981\) 7.13706i 0.227869i
\(982\) 35.1487i 1.12164i
\(983\) 22.9366i 0.731564i 0.930701 + 0.365782i \(0.119198\pi\)
−0.930701 + 0.365782i \(0.880802\pi\)
\(984\) −3.40581 −0.108573
\(985\) 22.6504 0.721702
\(986\) 40.5013i 1.28982i
\(987\) −2.15585 −0.0686215
\(988\) 0 0
\(989\) −0.115293 −0.00366611
\(990\) − 0.335126i − 0.0106510i
\(991\) −5.75781 −0.182903 −0.0914514 0.995810i \(-0.529151\pi\)
−0.0914514 + 0.995810i \(0.529151\pi\)
\(992\) 0.0609989 0.00193672
\(993\) 26.4373i 0.838961i
\(994\) − 2.94571i − 0.0934321i
\(995\) 5.46681i 0.173310i
\(996\) − 3.84117i − 0.121712i
\(997\) 3.86353 0.122359 0.0611796 0.998127i \(-0.480514\pi\)
0.0611796 + 0.998127i \(0.480514\pi\)
\(998\) −9.61463 −0.304346
\(999\) − 7.07606i − 0.223877i
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.b.y.1351.3 6
13.5 odd 4 5070.2.a.bq.1.1 3
13.8 odd 4 5070.2.a.bv.1.3 yes 3
13.12 even 2 inner 5070.2.b.y.1351.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bq.1.1 3 13.5 odd 4
5070.2.a.bv.1.3 yes 3 13.8 odd 4
5070.2.b.y.1351.3 6 1.1 even 1 trivial
5070.2.b.y.1351.4 6 13.12 even 2 inner