Properties

Label 5070.2.b.z.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 \(1.24698i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.z.1351.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +3.69202i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +3.69202i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.04892i q^{11} -1.00000 q^{12} +3.69202 q^{14} +1.00000i q^{15} +1.00000 q^{16} +6.85086 q^{17} -1.00000i q^{18} -0.911854i q^{19} -1.00000i q^{20} +3.69202i q^{21} -3.04892 q^{22} +0.356896 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -3.69202i q^{28} +10.5036 q^{29} +1.00000 q^{30} -2.06100i q^{31} -1.00000i q^{32} -3.04892i q^{33} -6.85086i q^{34} -3.69202 q^{35} -1.00000 q^{36} -0.899772i q^{37} -0.911854 q^{38} -1.00000 q^{40} -10.2959i q^{41} +3.69202 q^{42} -9.43296 q^{43} +3.04892i q^{44} +1.00000i q^{45} -0.356896i q^{46} +11.5918i q^{47} +1.00000 q^{48} -6.63102 q^{49} +1.00000i q^{50} +6.85086 q^{51} -3.40581 q^{53} -1.00000i q^{54} +3.04892 q^{55} -3.69202 q^{56} -0.911854i q^{57} -10.5036i q^{58} +8.54288i q^{59} -1.00000i q^{60} +1.55496 q^{61} -2.06100 q^{62} +3.69202i q^{63} -1.00000 q^{64} -3.04892 q^{66} -1.14914i q^{67} -6.85086 q^{68} +0.356896 q^{69} +3.69202i q^{70} +4.13706i q^{71} +1.00000i q^{72} +11.1685i q^{73} -0.899772 q^{74} -1.00000 q^{75} +0.911854i q^{76} +11.2567 q^{77} +9.62565 q^{79} +1.00000i q^{80} +1.00000 q^{81} -10.2959 q^{82} +9.86054i q^{83} -3.69202i q^{84} +6.85086i q^{85} +9.43296i q^{86} +10.5036 q^{87} +3.04892 q^{88} -6.41119i q^{89} +1.00000 q^{90} -0.356896 q^{92} -2.06100i q^{93} +11.5918 q^{94} +0.911854 q^{95} -1.00000i q^{96} -6.97823i q^{97} +6.63102i q^{98} -3.04892i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{3} - 6q^{4} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{3} - 6q^{4} + 6q^{9} + 6q^{10} - 6q^{12} + 12q^{14} + 6q^{16} + 14q^{17} - 6q^{23} - 6q^{25} + 6q^{27} + 6q^{30} - 12q^{35} - 6q^{36} + 2q^{38} - 6q^{40} + 12q^{42} - 18q^{43} + 6q^{48} - 10q^{49} + 14q^{51} + 6q^{53} - 12q^{56} + 10q^{61} - 32q^{62} - 6q^{64} - 14q^{68} - 6q^{69} + 40q^{74} - 6q^{75} + 14q^{77} + 34q^{79} + 6q^{81} - 34q^{82} + 6q^{90} + 6q^{92} + 14q^{94} - 2q^{95} + O(q^{100}) \)

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
\(7\) 3.69202i 1.39545i 0.716364 + 0.697726i \(0.245805\pi\)
−0.716364 + 0.697726i \(0.754195\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) − 3.04892i − 0.919283i −0.888105 0.459642i \(-0.847978\pi\)
0.888105 0.459642i \(-0.152022\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 3.69202 0.986734
\(15\) 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
\(19\) − 0.911854i − 0.209194i −0.994515 0.104597i \(-0.966645\pi\)
0.994515 0.104597i \(-0.0333552\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) 3.69202i 0.805665i
\(22\) −3.04892 −0.650031
\(23\) 0.356896 0.0744179 0.0372090 0.999308i \(-0.488153\pi\)
0.0372090 + 0.999308i \(0.488153\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) − 3.69202i − 0.697726i
\(29\) 10.5036 1.95048 0.975239 0.221153i \(-0.0709819\pi\)
0.975239 + 0.221153i \(0.0709819\pi\)
\(30\) 1.00000 0.182574
\(31\) − 2.06100i − 0.370166i −0.982723 0.185083i \(-0.940745\pi\)
0.982723 0.185083i \(-0.0592555\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 3.04892i − 0.530748i
\(34\) − 6.85086i − 1.17491i
\(35\) −3.69202 −0.624066
\(36\) −1.00000 −0.166667
\(37\) − 0.899772i − 0.147922i −0.997261 0.0739608i \(-0.976436\pi\)
0.997261 0.0739608i \(-0.0235640\pi\)
\(38\) −0.911854 −0.147922
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) − 10.2959i − 1.60795i −0.594664 0.803974i \(-0.702715\pi\)
0.594664 0.803974i \(-0.297285\pi\)
\(42\) 3.69202 0.569691
\(43\) −9.43296 −1.43851 −0.719256 0.694745i \(-0.755517\pi\)
−0.719256 + 0.694745i \(0.755517\pi\)
\(44\) 3.04892i 0.459642i
\(45\) 1.00000i 0.149071i
\(46\) − 0.356896i − 0.0526214i
\(47\) 11.5918i 1.69084i 0.534105 + 0.845418i \(0.320649\pi\)
−0.534105 + 0.845418i \(0.679351\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.63102 −0.947289
\(50\) 1.00000i 0.141421i
\(51\) 6.85086 0.959312
\(52\) 0 0
\(53\) −3.40581 −0.467824 −0.233912 0.972258i \(-0.575153\pi\)
−0.233912 + 0.972258i \(0.575153\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) 3.04892 0.411116
\(56\) −3.69202 −0.493367
\(57\) − 0.911854i − 0.120778i
\(58\) − 10.5036i − 1.37920i
\(59\) 8.54288i 1.11219i 0.831119 + 0.556094i \(0.187701\pi\)
−0.831119 + 0.556094i \(0.812299\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) 1.55496 0.199092 0.0995460 0.995033i \(-0.468261\pi\)
0.0995460 + 0.995033i \(0.468261\pi\)
\(62\) −2.06100 −0.261747
\(63\) 3.69202i 0.465151i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −3.04892 −0.375296
\(67\) − 1.14914i − 0.140390i −0.997533 0.0701952i \(-0.977638\pi\)
0.997533 0.0701952i \(-0.0223622\pi\)
\(68\) −6.85086 −0.830788
\(69\) 0.356896 0.0429652
\(70\) 3.69202i 0.441281i
\(71\) 4.13706i 0.490979i 0.969399 + 0.245490i \(0.0789487\pi\)
−0.969399 + 0.245490i \(0.921051\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 11.1685i 1.30718i 0.756850 + 0.653588i \(0.226737\pi\)
−0.756850 + 0.653588i \(0.773263\pi\)
\(74\) −0.899772 −0.104596
\(75\) −1.00000 −0.115470
\(76\) 0.911854i 0.104597i
\(77\) 11.2567 1.28282
\(78\) 0 0
\(79\) 9.62565 1.08297 0.541485 0.840710i \(-0.317862\pi\)
0.541485 + 0.840710i \(0.317862\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) −10.2959 −1.13699
\(83\) 9.86054i 1.08234i 0.840915 + 0.541168i \(0.182018\pi\)
−0.840915 + 0.541168i \(0.817982\pi\)
\(84\) − 3.69202i − 0.402833i
\(85\) 6.85086i 0.743080i
\(86\) 9.43296i 1.01718i
\(87\) 10.5036 1.12611
\(88\) 3.04892 0.325016
\(89\) − 6.41119i − 0.679585i −0.940500 0.339792i \(-0.889643\pi\)
0.940500 0.339792i \(-0.110357\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −0.356896 −0.0372090
\(93\) − 2.06100i − 0.213716i
\(94\) 11.5918 1.19560
\(95\) 0.911854 0.0935542
\(96\) − 1.00000i − 0.102062i
\(97\) − 6.97823i − 0.708532i −0.935145 0.354266i \(-0.884731\pi\)
0.935145 0.354266i \(-0.115269\pi\)
\(98\) 6.63102i 0.669834i
\(99\) − 3.04892i − 0.306428i
\(100\) 1.00000 0.100000
\(101\) 4.09783 0.407750 0.203875 0.978997i \(-0.434646\pi\)
0.203875 + 0.978997i \(0.434646\pi\)
\(102\) − 6.85086i − 0.678336i
\(103\) 16.9215 1.66733 0.833665 0.552271i \(-0.186239\pi\)
0.833665 + 0.552271i \(0.186239\pi\)
\(104\) 0 0
\(105\) −3.69202 −0.360304
\(106\) 3.40581i 0.330802i
\(107\) −12.7192 −1.22961 −0.614804 0.788680i \(-0.710765\pi\)
−0.614804 + 0.788680i \(0.710765\pi\)
\(108\) −1.00000 −0.0962250
\(109\) − 19.9922i − 1.91491i −0.288585 0.957454i \(-0.593185\pi\)
0.288585 0.957454i \(-0.406815\pi\)
\(110\) − 3.04892i − 0.290703i
\(111\) − 0.899772i − 0.0854026i
\(112\) 3.69202i 0.348863i
\(113\) −4.49934 −0.423262 −0.211631 0.977350i \(-0.567877\pi\)
−0.211631 + 0.977350i \(0.567877\pi\)
\(114\) −0.911854 −0.0854030
\(115\) 0.356896i 0.0332807i
\(116\) −10.5036 −0.975239
\(117\) 0 0
\(118\) 8.54288 0.786436
\(119\) 25.2935i 2.31865i
\(120\) −1.00000 −0.0912871
\(121\) 1.70410 0.154918
\(122\) − 1.55496i − 0.140779i
\(123\) − 10.2959i − 0.928350i
\(124\) 2.06100i 0.185083i
\(125\) − 1.00000i − 0.0894427i
\(126\) 3.69202 0.328911
\(127\) 7.08815 0.628971 0.314486 0.949262i \(-0.398168\pi\)
0.314486 + 0.949262i \(0.398168\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −9.43296 −0.830526
\(130\) 0 0
\(131\) 19.7506 1.72562 0.862810 0.505528i \(-0.168702\pi\)
0.862810 + 0.505528i \(0.168702\pi\)
\(132\) 3.04892i 0.265374i
\(133\) 3.36658 0.291920
\(134\) −1.14914 −0.0992710
\(135\) 1.00000i 0.0860663i
\(136\) 6.85086i 0.587456i
\(137\) − 7.94331i − 0.678643i −0.940670 0.339322i \(-0.889803\pi\)
0.940670 0.339322i \(-0.110197\pi\)
\(138\) − 0.356896i − 0.0303810i
\(139\) −1.71379 −0.145362 −0.0726810 0.997355i \(-0.523155\pi\)
−0.0726810 + 0.997355i \(0.523155\pi\)
\(140\) 3.69202 0.312033
\(141\) 11.5918i 0.976205i
\(142\) 4.13706 0.347175
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 10.5036i 0.872280i
\(146\) 11.1685 0.924313
\(147\) −6.63102 −0.546918
\(148\) 0.899772i 0.0739608i
\(149\) 6.91185i 0.566241i 0.959084 + 0.283121i \(0.0913697\pi\)
−0.959084 + 0.283121i \(0.908630\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) 5.43296i 0.442128i 0.975259 + 0.221064i \(0.0709530\pi\)
−0.975259 + 0.221064i \(0.929047\pi\)
\(152\) 0.911854 0.0739611
\(153\) 6.85086 0.553859
\(154\) − 11.2567i − 0.907088i
\(155\) 2.06100 0.165543
\(156\) 0 0
\(157\) −9.13706 −0.729217 −0.364609 0.931161i \(-0.618797\pi\)
−0.364609 + 0.931161i \(0.618797\pi\)
\(158\) − 9.62565i − 0.765775i
\(159\) −3.40581 −0.270099
\(160\) 1.00000 0.0790569
\(161\) 1.31767i 0.103847i
\(162\) − 1.00000i − 0.0785674i
\(163\) 14.3773i 1.12612i 0.826416 + 0.563060i \(0.190376\pi\)
−0.826416 + 0.563060i \(0.809624\pi\)
\(164\) 10.2959i 0.803974i
\(165\) 3.04892 0.237358
\(166\) 9.86054 0.765327
\(167\) − 3.00538i − 0.232563i −0.993216 0.116282i \(-0.962903\pi\)
0.993216 0.116282i \(-0.0370975\pi\)
\(168\) −3.69202 −0.284846
\(169\) 0 0
\(170\) 6.85086 0.525437
\(171\) − 0.911854i − 0.0697312i
\(172\) 9.43296 0.719256
\(173\) −6.89546 −0.524252 −0.262126 0.965034i \(-0.584424\pi\)
−0.262126 + 0.965034i \(0.584424\pi\)
\(174\) − 10.5036i − 0.796279i
\(175\) − 3.69202i − 0.279091i
\(176\) − 3.04892i − 0.229821i
\(177\) 8.54288i 0.642122i
\(178\) −6.41119 −0.480539
\(179\) 25.6461 1.91688 0.958439 0.285296i \(-0.0920921\pi\)
0.958439 + 0.285296i \(0.0920921\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) 25.8092 1.91839 0.959193 0.282753i \(-0.0912478\pi\)
0.959193 + 0.282753i \(0.0912478\pi\)
\(182\) 0 0
\(183\) 1.55496 0.114946
\(184\) 0.356896i 0.0263107i
\(185\) 0.899772 0.0661526
\(186\) −2.06100 −0.151120
\(187\) − 20.8877i − 1.52746i
\(188\) − 11.5918i − 0.845418i
\(189\) 3.69202i 0.268555i
\(190\) − 0.911854i − 0.0661528i
\(191\) −14.5700 −1.05425 −0.527125 0.849788i \(-0.676730\pi\)
−0.527125 + 0.849788i \(0.676730\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 8.32544i 0.599278i 0.954053 + 0.299639i \(0.0968662\pi\)
−0.954053 + 0.299639i \(0.903134\pi\)
\(194\) −6.97823 −0.501008
\(195\) 0 0
\(196\) 6.63102 0.473644
\(197\) 26.8810i 1.91519i 0.288116 + 0.957595i \(0.406971\pi\)
−0.288116 + 0.957595i \(0.593029\pi\)
\(198\) −3.04892 −0.216677
\(199\) 18.7047 1.32594 0.662970 0.748646i \(-0.269296\pi\)
0.662970 + 0.748646i \(0.269296\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) − 1.14914i − 0.0810544i
\(202\) − 4.09783i − 0.288323i
\(203\) 38.7797i 2.72180i
\(204\) −6.85086 −0.479656
\(205\) 10.2959 0.719097
\(206\) − 16.9215i − 1.17898i
\(207\) 0.356896 0.0248060
\(208\) 0 0
\(209\) −2.78017 −0.192308
\(210\) 3.69202i 0.254774i
\(211\) 2.93362 0.201959 0.100980 0.994889i \(-0.467802\pi\)
0.100980 + 0.994889i \(0.467802\pi\)
\(212\) 3.40581 0.233912
\(213\) 4.13706i 0.283467i
\(214\) 12.7192i 0.869464i
\(215\) − 9.43296i − 0.643323i
\(216\) 1.00000i 0.0680414i
\(217\) 7.60925 0.516550
\(218\) −19.9922 −1.35404
\(219\) 11.1685i 0.754699i
\(220\) −3.04892 −0.205558
\(221\) 0 0
\(222\) −0.899772 −0.0603888
\(223\) 18.1444i 1.21504i 0.794306 + 0.607518i \(0.207835\pi\)
−0.794306 + 0.607518i \(0.792165\pi\)
\(224\) 3.69202 0.246684
\(225\) −1.00000 −0.0666667
\(226\) 4.49934i 0.299291i
\(227\) − 12.3056i − 0.816750i −0.912814 0.408375i \(-0.866096\pi\)
0.912814 0.408375i \(-0.133904\pi\)
\(228\) 0.911854i 0.0603890i
\(229\) 4.64848i 0.307180i 0.988135 + 0.153590i \(0.0490835\pi\)
−0.988135 + 0.153590i \(0.950916\pi\)
\(230\) 0.356896 0.0235330
\(231\) 11.2567 0.740634
\(232\) 10.5036i 0.689598i
\(233\) −0.658170 −0.0431181 −0.0215591 0.999768i \(-0.506863\pi\)
−0.0215591 + 0.999768i \(0.506863\pi\)
\(234\) 0 0
\(235\) −11.5918 −0.756165
\(236\) − 8.54288i − 0.556094i
\(237\) 9.62565 0.625253
\(238\) 25.2935 1.63953
\(239\) 18.7748i 1.21444i 0.794534 + 0.607220i \(0.207715\pi\)
−0.794534 + 0.607220i \(0.792285\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 18.0271i 1.16123i 0.814178 + 0.580615i \(0.197188\pi\)
−0.814178 + 0.580615i \(0.802812\pi\)
\(242\) − 1.70410i − 0.109544i
\(243\) 1.00000 0.0641500
\(244\) −1.55496 −0.0995460
\(245\) − 6.63102i − 0.423640i
\(246\) −10.2959 −0.656442
\(247\) 0 0
\(248\) 2.06100 0.130874
\(249\) 9.86054i 0.624887i
\(250\) −1.00000 −0.0632456
\(251\) 2.18060 0.137638 0.0688192 0.997629i \(-0.478077\pi\)
0.0688192 + 0.997629i \(0.478077\pi\)
\(252\) − 3.69202i − 0.232575i
\(253\) − 1.08815i − 0.0684112i
\(254\) − 7.08815i − 0.444750i
\(255\) 6.85086i 0.429017i
\(256\) 1.00000 0.0625000
\(257\) 9.58748 0.598051 0.299025 0.954245i \(-0.403338\pi\)
0.299025 + 0.954245i \(0.403338\pi\)
\(258\) 9.43296i 0.587270i
\(259\) 3.32198 0.206418
\(260\) 0 0
\(261\) 10.5036 0.650159
\(262\) − 19.7506i − 1.22020i
\(263\) −1.41789 −0.0874311 −0.0437156 0.999044i \(-0.513920\pi\)
−0.0437156 + 0.999044i \(0.513920\pi\)
\(264\) 3.04892 0.187648
\(265\) − 3.40581i − 0.209217i
\(266\) − 3.36658i − 0.206419i
\(267\) − 6.41119i − 0.392358i
\(268\) 1.14914i 0.0701952i
\(269\) −4.14675 −0.252832 −0.126416 0.991977i \(-0.540347\pi\)
−0.126416 + 0.991977i \(0.540347\pi\)
\(270\) 1.00000 0.0608581
\(271\) 6.40044i 0.388799i 0.980922 + 0.194399i \(0.0622758\pi\)
−0.980922 + 0.194399i \(0.937724\pi\)
\(272\) 6.85086 0.415394
\(273\) 0 0
\(274\) −7.94331 −0.479873
\(275\) 3.04892i 0.183857i
\(276\) −0.356896 −0.0214826
\(277\) −10.3394 −0.621237 −0.310618 0.950535i \(-0.600536\pi\)
−0.310618 + 0.950535i \(0.600536\pi\)
\(278\) 1.71379i 0.102786i
\(279\) − 2.06100i − 0.123389i
\(280\) − 3.69202i − 0.220640i
\(281\) 5.08144i 0.303133i 0.988447 + 0.151567i \(0.0484318\pi\)
−0.988447 + 0.151567i \(0.951568\pi\)
\(282\) 11.5918 0.690281
\(283\) −24.5284 −1.45806 −0.729031 0.684481i \(-0.760029\pi\)
−0.729031 + 0.684481i \(0.760029\pi\)
\(284\) − 4.13706i − 0.245490i
\(285\) 0.911854 0.0540136
\(286\) 0 0
\(287\) 38.0127 2.24382
\(288\) − 1.00000i − 0.0589256i
\(289\) 29.9342 1.76084
\(290\) 10.5036 0.616795
\(291\) − 6.97823i − 0.409071i
\(292\) − 11.1685i − 0.653588i
\(293\) 15.8538i 0.926191i 0.886308 + 0.463096i \(0.153261\pi\)
−0.886308 + 0.463096i \(0.846739\pi\)
\(294\) 6.63102i 0.386729i
\(295\) −8.54288 −0.497386
\(296\) 0.899772 0.0522982
\(297\) − 3.04892i − 0.176916i
\(298\) 6.91185 0.400393
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) − 34.8267i − 2.00738i
\(302\) 5.43296 0.312632
\(303\) 4.09783 0.235414
\(304\) − 0.911854i − 0.0522984i
\(305\) 1.55496i 0.0890366i
\(306\) − 6.85086i − 0.391637i
\(307\) − 14.2121i − 0.811125i −0.914067 0.405563i \(-0.867076\pi\)
0.914067 0.405563i \(-0.132924\pi\)
\(308\) −11.2567 −0.641408
\(309\) 16.9215 0.962633
\(310\) − 2.06100i − 0.117057i
\(311\) −15.0127 −0.851291 −0.425645 0.904890i \(-0.639953\pi\)
−0.425645 + 0.904890i \(0.639953\pi\)
\(312\) 0 0
\(313\) −13.4993 −0.763028 −0.381514 0.924363i \(-0.624597\pi\)
−0.381514 + 0.924363i \(0.624597\pi\)
\(314\) 9.13706i 0.515634i
\(315\) −3.69202 −0.208022
\(316\) −9.62565 −0.541485
\(317\) − 33.9463i − 1.90661i −0.302003 0.953307i \(-0.597655\pi\)
0.302003 0.953307i \(-0.402345\pi\)
\(318\) 3.40581i 0.190989i
\(319\) − 32.0248i − 1.79304i
\(320\) − 1.00000i − 0.0559017i
\(321\) −12.7192 −0.709915
\(322\) 1.31767 0.0734307
\(323\) − 6.24698i − 0.347591i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 14.3773 0.796287
\(327\) − 19.9922i − 1.10557i
\(328\) 10.2959 0.568496
\(329\) −42.7972 −2.35948
\(330\) − 3.04892i − 0.167837i
\(331\) 6.84415i 0.376189i 0.982151 + 0.188094i \(0.0602310\pi\)
−0.982151 + 0.188094i \(0.939769\pi\)
\(332\) − 9.86054i − 0.541168i
\(333\) − 0.899772i − 0.0493072i
\(334\) −3.00538 −0.164447
\(335\) 1.14914 0.0627845
\(336\) 3.69202i 0.201416i
\(337\) −27.7332 −1.51072 −0.755361 0.655309i \(-0.772538\pi\)
−0.755361 + 0.655309i \(0.772538\pi\)
\(338\) 0 0
\(339\) −4.49934 −0.244370
\(340\) − 6.85086i − 0.371540i
\(341\) −6.28382 −0.340288
\(342\) −0.911854 −0.0493074
\(343\) 1.36227i 0.0735558i
\(344\) − 9.43296i − 0.508591i
\(345\) 0.356896i 0.0192146i
\(346\) 6.89546i 0.370702i
\(347\) −10.4862 −0.562928 −0.281464 0.959572i \(-0.590820\pi\)
−0.281464 + 0.959572i \(0.590820\pi\)
\(348\) −10.5036 −0.563055
\(349\) − 8.13467i − 0.435439i −0.976011 0.217719i \(-0.930138\pi\)
0.976011 0.217719i \(-0.0698618\pi\)
\(350\) −3.69202 −0.197347
\(351\) 0 0
\(352\) −3.04892 −0.162508
\(353\) 25.8649i 1.37665i 0.725404 + 0.688324i \(0.241653\pi\)
−0.725404 + 0.688324i \(0.758347\pi\)
\(354\) 8.54288 0.454049
\(355\) −4.13706 −0.219573
\(356\) 6.41119i 0.339792i
\(357\) 25.2935i 1.33867i
\(358\) − 25.6461i − 1.35544i
\(359\) − 30.2959i − 1.59896i −0.600695 0.799478i \(-0.705109\pi\)
0.600695 0.799478i \(-0.294891\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 18.1685 0.956238
\(362\) − 25.8092i − 1.35650i
\(363\) 1.70410 0.0894422
\(364\) 0 0
\(365\) −11.1685 −0.584587
\(366\) − 1.55496i − 0.0812790i
\(367\) −3.41013 −0.178007 −0.0890035 0.996031i \(-0.528368\pi\)
−0.0890035 + 0.996031i \(0.528368\pi\)
\(368\) 0.356896 0.0186045
\(369\) − 10.2959i − 0.535983i
\(370\) − 0.899772i − 0.0467769i
\(371\) − 12.5743i − 0.652827i
\(372\) 2.06100i 0.106858i
\(373\) 3.35988 0.173968 0.0869840 0.996210i \(-0.472277\pi\)
0.0869840 + 0.996210i \(0.472277\pi\)
\(374\) −20.8877 −1.08008
\(375\) − 1.00000i − 0.0516398i
\(376\) −11.5918 −0.597801
\(377\) 0 0
\(378\) 3.69202 0.189897
\(379\) 31.4916i 1.61761i 0.588075 + 0.808807i \(0.299886\pi\)
−0.588075 + 0.808807i \(0.700114\pi\)
\(380\) −0.911854 −0.0467771
\(381\) 7.08815 0.363137
\(382\) 14.5700i 0.745467i
\(383\) − 21.8485i − 1.11640i −0.829705 0.558202i \(-0.811491\pi\)
0.829705 0.558202i \(-0.188509\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 11.2567i 0.573693i
\(386\) 8.32544 0.423754
\(387\) −9.43296 −0.479504
\(388\) 6.97823i 0.354266i
\(389\) 11.3787 0.576921 0.288461 0.957492i \(-0.406857\pi\)
0.288461 + 0.957492i \(0.406857\pi\)
\(390\) 0 0
\(391\) 2.44504 0.123651
\(392\) − 6.63102i − 0.334917i
\(393\) 19.7506 0.996287
\(394\) 26.8810 1.35424
\(395\) 9.62565i 0.484319i
\(396\) 3.04892i 0.153214i
\(397\) 22.0476i 1.10654i 0.833003 + 0.553268i \(0.186620\pi\)
−0.833003 + 0.553268i \(0.813380\pi\)
\(398\) − 18.7047i − 0.937582i
\(399\) 3.36658 0.168540
\(400\) −1.00000 −0.0500000
\(401\) 10.2459i 0.511657i 0.966722 + 0.255828i \(0.0823482\pi\)
−0.966722 + 0.255828i \(0.917652\pi\)
\(402\) −1.14914 −0.0573141
\(403\) 0 0
\(404\) −4.09783 −0.203875
\(405\) 1.00000i 0.0496904i
\(406\) 38.7797 1.92460
\(407\) −2.74333 −0.135982
\(408\) 6.85086i 0.339168i
\(409\) − 22.8189i − 1.12832i −0.825664 0.564162i \(-0.809200\pi\)
0.825664 0.564162i \(-0.190800\pi\)
\(410\) − 10.2959i − 0.508478i
\(411\) − 7.94331i − 0.391815i
\(412\) −16.9215 −0.833665
\(413\) −31.5405 −1.55201
\(414\) − 0.356896i − 0.0175405i
\(415\) −9.86054 −0.484035
\(416\) 0 0
\(417\) −1.71379 −0.0839247
\(418\) 2.78017i 0.135982i
\(419\) −28.5773 −1.39609 −0.698047 0.716052i \(-0.745947\pi\)
−0.698047 + 0.716052i \(0.745947\pi\)
\(420\) 3.69202 0.180152
\(421\) − 32.3937i − 1.57877i −0.613896 0.789387i \(-0.710399\pi\)
0.613896 0.789387i \(-0.289601\pi\)
\(422\) − 2.93362i − 0.142807i
\(423\) 11.5918i 0.563612i
\(424\) − 3.40581i − 0.165401i
\(425\) −6.85086 −0.332315
\(426\) 4.13706 0.200441
\(427\) 5.74094i 0.277824i
\(428\) 12.7192 0.614804
\(429\) 0 0
\(430\) −9.43296 −0.454898
\(431\) 14.6455i 0.705449i 0.935727 + 0.352724i \(0.114745\pi\)
−0.935727 + 0.352724i \(0.885255\pi\)
\(432\) 1.00000 0.0481125
\(433\) 35.8485 1.72277 0.861384 0.507955i \(-0.169598\pi\)
0.861384 + 0.507955i \(0.169598\pi\)
\(434\) − 7.60925i − 0.365256i
\(435\) 10.5036i 0.503611i
\(436\) 19.9922i 0.957454i
\(437\) − 0.325437i − 0.0155678i
\(438\) 11.1685 0.533653
\(439\) −2.55927 −0.122147 −0.0610736 0.998133i \(-0.519452\pi\)
−0.0610736 + 0.998133i \(0.519452\pi\)
\(440\) 3.04892i 0.145351i
\(441\) −6.63102 −0.315763
\(442\) 0 0
\(443\) 23.2857 1.10634 0.553169 0.833069i \(-0.313418\pi\)
0.553169 + 0.833069i \(0.313418\pi\)
\(444\) 0.899772i 0.0427013i
\(445\) 6.41119 0.303920
\(446\) 18.1444 0.859160
\(447\) 6.91185i 0.326919i
\(448\) − 3.69202i − 0.174432i
\(449\) − 27.3327i − 1.28991i −0.764220 0.644956i \(-0.776876\pi\)
0.764220 0.644956i \(-0.223124\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) −31.3913 −1.47816
\(452\) 4.49934 0.211631
\(453\) 5.43296i 0.255263i
\(454\) −12.3056 −0.577530
\(455\) 0 0
\(456\) 0.911854 0.0427015
\(457\) 7.74094i 0.362106i 0.983473 + 0.181053i \(0.0579505\pi\)
−0.983473 + 0.181053i \(0.942049\pi\)
\(458\) 4.64848 0.217209
\(459\) 6.85086 0.319771
\(460\) − 0.356896i − 0.0166404i
\(461\) − 17.1511i − 0.798805i −0.916776 0.399402i \(-0.869218\pi\)
0.916776 0.399402i \(-0.130782\pi\)
\(462\) − 11.2567i − 0.523708i
\(463\) − 34.2747i − 1.59288i −0.604717 0.796441i \(-0.706714\pi\)
0.604717 0.796441i \(-0.293286\pi\)
\(464\) 10.5036 0.487620
\(465\) 2.06100 0.0955765
\(466\) 0.658170i 0.0304891i
\(467\) −20.9474 −0.969328 −0.484664 0.874700i \(-0.661058\pi\)
−0.484664 + 0.874700i \(0.661058\pi\)
\(468\) 0 0
\(469\) 4.24267 0.195908
\(470\) 11.5918i 0.534690i
\(471\) −9.13706 −0.421014
\(472\) −8.54288 −0.393218
\(473\) 28.7603i 1.32240i
\(474\) − 9.62565i − 0.442121i
\(475\) 0.911854i 0.0418387i
\(476\) − 25.2935i − 1.15933i
\(477\) −3.40581 −0.155941
\(478\) 18.7748 0.858739
\(479\) 15.8310i 0.723337i 0.932307 + 0.361669i \(0.117793\pi\)
−0.932307 + 0.361669i \(0.882207\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 18.0271 0.821114
\(483\) 1.31767i 0.0599559i
\(484\) −1.70410 −0.0774592
\(485\) 6.97823 0.316865
\(486\) − 1.00000i − 0.0453609i
\(487\) − 33.7060i − 1.52737i −0.645592 0.763683i \(-0.723389\pi\)
0.645592 0.763683i \(-0.276611\pi\)
\(488\) 1.55496i 0.0703896i
\(489\) 14.3773i 0.650166i
\(490\) −6.63102 −0.299559
\(491\) −3.98925 −0.180032 −0.0900161 0.995940i \(-0.528692\pi\)
−0.0900161 + 0.995940i \(0.528692\pi\)
\(492\) 10.2959i 0.464175i
\(493\) 71.9590 3.24087
\(494\) 0 0
\(495\) 3.04892 0.137039
\(496\) − 2.06100i − 0.0925416i
\(497\) −15.2741 −0.685138
\(498\) 9.86054 0.441862
\(499\) − 20.4698i − 0.916354i −0.888861 0.458177i \(-0.848503\pi\)
0.888861 0.458177i \(-0.151497\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) − 3.00538i − 0.134270i
\(502\) − 2.18060i − 0.0973251i
\(503\) 22.5351 1.00479 0.502395 0.864638i \(-0.332452\pi\)
0.502395 + 0.864638i \(0.332452\pi\)
\(504\) −3.69202 −0.164456
\(505\) 4.09783i 0.182351i
\(506\) −1.08815 −0.0483740
\(507\) 0 0
\(508\) −7.08815 −0.314486
\(509\) 10.9718i 0.486316i 0.969987 + 0.243158i \(0.0781833\pi\)
−0.969987 + 0.243158i \(0.921817\pi\)
\(510\) 6.85086 0.303361
\(511\) −41.2344 −1.82410
\(512\) − 1.00000i − 0.0441942i
\(513\) − 0.911854i − 0.0402593i
\(514\) − 9.58748i − 0.422886i
\(515\) 16.9215i 0.745652i
\(516\) 9.43296 0.415263
\(517\) 35.3424 1.55436
\(518\) − 3.32198i − 0.145959i
\(519\) −6.89546 −0.302677
\(520\) 0 0
\(521\) 15.9691 0.699620 0.349810 0.936821i \(-0.386246\pi\)
0.349810 + 0.936821i \(0.386246\pi\)
\(522\) − 10.5036i − 0.459732i
\(523\) −24.7681 −1.08303 −0.541516 0.840690i \(-0.682150\pi\)
−0.541516 + 0.840690i \(0.682150\pi\)
\(524\) −19.7506 −0.862810
\(525\) − 3.69202i − 0.161133i
\(526\) 1.41789i 0.0618232i
\(527\) − 14.1196i − 0.615060i
\(528\) − 3.04892i − 0.132687i
\(529\) −22.8726 −0.994462
\(530\) −3.40581 −0.147939
\(531\) 8.54288i 0.370729i
\(532\) −3.36658 −0.145960
\(533\) 0 0
\(534\) −6.41119 −0.277439
\(535\) − 12.7192i − 0.549898i
\(536\) 1.14914 0.0496355
\(537\) 25.6461 1.10671
\(538\) 4.14675i 0.178779i
\(539\) 20.2174i 0.870827i
\(540\) − 1.00000i − 0.0430331i
\(541\) 0.460107i 0.0197816i 0.999951 + 0.00989078i \(0.00314838\pi\)
−0.999951 + 0.00989078i \(0.996852\pi\)
\(542\) 6.40044 0.274922
\(543\) 25.8092 1.10758
\(544\) − 6.85086i − 0.293728i
\(545\) 19.9922 0.856373
\(546\) 0 0
\(547\) −8.70410 −0.372161 −0.186080 0.982535i \(-0.559578\pi\)
−0.186080 + 0.982535i \(0.559578\pi\)
\(548\) 7.94331i 0.339322i
\(549\) 1.55496 0.0663640
\(550\) 3.04892 0.130006
\(551\) − 9.57779i − 0.408028i
\(552\) 0.356896i 0.0151905i
\(553\) 35.5381i 1.51123i
\(554\) 10.3394i 0.439281i
\(555\) 0.899772 0.0381932
\(556\) 1.71379 0.0726810
\(557\) − 36.7415i − 1.55679i −0.627776 0.778394i \(-0.716034\pi\)
0.627776 0.778394i \(-0.283966\pi\)
\(558\) −2.06100 −0.0872490
\(559\) 0 0
\(560\) −3.69202 −0.156016
\(561\) − 20.8877i − 0.881879i
\(562\) 5.08144 0.214348
\(563\) 27.7713 1.17042 0.585211 0.810881i \(-0.301012\pi\)
0.585211 + 0.810881i \(0.301012\pi\)
\(564\) − 11.5918i − 0.488103i
\(565\) − 4.49934i − 0.189288i
\(566\) 24.5284i 1.03101i
\(567\) 3.69202i 0.155050i
\(568\) −4.13706 −0.173587
\(569\) −38.4379 −1.61140 −0.805700 0.592324i \(-0.798210\pi\)
−0.805700 + 0.592324i \(0.798210\pi\)
\(570\) − 0.911854i − 0.0381934i
\(571\) −10.8086 −0.452328 −0.226164 0.974089i \(-0.572618\pi\)
−0.226164 + 0.974089i \(0.572618\pi\)
\(572\) 0 0
\(573\) −14.5700 −0.608671
\(574\) − 38.0127i − 1.58662i
\(575\) −0.356896 −0.0148836
\(576\) −1.00000 −0.0416667
\(577\) − 35.5991i − 1.48201i −0.671500 0.741005i \(-0.734350\pi\)
0.671500 0.741005i \(-0.265650\pi\)
\(578\) − 29.9342i − 1.24510i
\(579\) 8.32544i 0.345993i
\(580\) − 10.5036i − 0.436140i
\(581\) −36.4053 −1.51035
\(582\) −6.97823 −0.289257
\(583\) 10.3840i 0.430063i
\(584\) −11.1685 −0.462157
\(585\) 0 0
\(586\) 15.8538 0.654916
\(587\) − 11.4741i − 0.473587i −0.971560 0.236794i \(-0.923903\pi\)
0.971560 0.236794i \(-0.0760965\pi\)
\(588\) 6.63102 0.273459
\(589\) −1.87933 −0.0774364
\(590\) 8.54288i 0.351705i
\(591\) 26.8810i 1.10574i
\(592\) − 0.899772i − 0.0369804i
\(593\) 15.5147i 0.637111i 0.947904 + 0.318555i \(0.103198\pi\)
−0.947904 + 0.318555i \(0.896802\pi\)
\(594\) −3.04892 −0.125099
\(595\) −25.2935 −1.03693
\(596\) − 6.91185i − 0.283121i
\(597\) 18.7047 0.765532
\(598\) 0 0
\(599\) −14.2034 −0.580337 −0.290168 0.956976i \(-0.593711\pi\)
−0.290168 + 0.956976i \(0.593711\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) −39.5579 −1.61360 −0.806801 0.590823i \(-0.798803\pi\)
−0.806801 + 0.590823i \(0.798803\pi\)
\(602\) −34.8267 −1.41943
\(603\) − 1.14914i − 0.0467968i
\(604\) − 5.43296i − 0.221064i
\(605\) 1.70410i 0.0692816i
\(606\) − 4.09783i − 0.166463i
\(607\) 4.77346 0.193749 0.0968744 0.995297i \(-0.469115\pi\)
0.0968744 + 0.995297i \(0.469115\pi\)
\(608\) −0.911854 −0.0369806
\(609\) 38.7797i 1.57143i
\(610\) 1.55496 0.0629584
\(611\) 0 0
\(612\) −6.85086 −0.276929
\(613\) − 5.43429i − 0.219489i −0.993960 0.109744i \(-0.964997\pi\)
0.993960 0.109744i \(-0.0350032\pi\)
\(614\) −14.2121 −0.573552
\(615\) 10.2959 0.415171
\(616\) 11.2567i 0.453544i
\(617\) 20.6692i 0.832110i 0.909339 + 0.416055i \(0.136588\pi\)
−0.909339 + 0.416055i \(0.863412\pi\)
\(618\) − 16.9215i − 0.680684i
\(619\) 24.4077i 0.981030i 0.871433 + 0.490515i \(0.163191\pi\)
−0.871433 + 0.490515i \(0.836809\pi\)
\(620\) −2.06100 −0.0827717
\(621\) 0.356896 0.0143217
\(622\) 15.0127i 0.601953i
\(623\) 23.6703 0.948329
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 13.4993i 0.539542i
\(627\) −2.78017 −0.111029
\(628\) 9.13706 0.364609
\(629\) − 6.16421i − 0.245783i
\(630\) 3.69202i 0.147094i
\(631\) − 32.0538i − 1.27604i −0.770019 0.638021i \(-0.779753\pi\)
0.770019 0.638021i \(-0.220247\pi\)
\(632\) 9.62565i 0.382888i
\(633\) 2.93362 0.116601
\(634\) −33.9463 −1.34818
\(635\) 7.08815i 0.281284i
\(636\) 3.40581 0.135049
\(637\) 0 0
\(638\) −32.0248 −1.26787
\(639\) 4.13706i 0.163660i
\(640\) −1.00000 −0.0395285
\(641\) −18.5327 −0.731998 −0.365999 0.930615i \(-0.619273\pi\)
−0.365999 + 0.930615i \(0.619273\pi\)
\(642\) 12.7192i 0.501986i
\(643\) 22.3454i 0.881217i 0.897699 + 0.440608i \(0.145237\pi\)
−0.897699 + 0.440608i \(0.854763\pi\)
\(644\) − 1.31767i − 0.0519234i
\(645\) − 9.43296i − 0.371422i
\(646\) −6.24698 −0.245784
\(647\) −7.86964 −0.309388 −0.154694 0.987962i \(-0.549439\pi\)
−0.154694 + 0.987962i \(0.549439\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 26.0465 1.02242
\(650\) 0 0
\(651\) 7.60925 0.298230
\(652\) − 14.3773i − 0.563060i
\(653\) −47.2271 −1.84814 −0.924070 0.382223i \(-0.875159\pi\)
−0.924070 + 0.382223i \(0.875159\pi\)
\(654\) −19.9922 −0.781758
\(655\) 19.7506i 0.771721i
\(656\) − 10.2959i − 0.401987i
\(657\) 11.1685i 0.435726i
\(658\) 42.7972i 1.66841i
\(659\) −46.6276 −1.81635 −0.908176 0.418588i \(-0.862525\pi\)
−0.908176 + 0.418588i \(0.862525\pi\)
\(660\) −3.04892 −0.118679
\(661\) − 39.9415i − 1.55354i −0.629781 0.776772i \(-0.716856\pi\)
0.629781 0.776772i \(-0.283144\pi\)
\(662\) 6.84415 0.266005
\(663\) 0 0
\(664\) −9.86054 −0.382663
\(665\) 3.36658i 0.130551i
\(666\) −0.899772 −0.0348655
\(667\) 3.74871 0.145151
\(668\) 3.00538i 0.116282i
\(669\) 18.1444i 0.701501i
\(670\) − 1.14914i − 0.0443953i
\(671\) − 4.74094i − 0.183022i
\(672\) 3.69202 0.142423
\(673\) 35.3612 1.36307 0.681537 0.731783i \(-0.261312\pi\)
0.681537 + 0.731783i \(0.261312\pi\)
\(674\) 27.7332i 1.06824i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 19.1715 0.736821 0.368410 0.929663i \(-0.379902\pi\)
0.368410 + 0.929663i \(0.379902\pi\)
\(678\) 4.49934i 0.172796i
\(679\) 25.7638 0.988723
\(680\) −6.85086 −0.262718
\(681\) − 12.3056i − 0.471551i
\(682\) 6.28382i 0.240620i
\(683\) − 1.13946i − 0.0436001i −0.999762 0.0218000i \(-0.993060\pi\)
0.999762 0.0218000i \(-0.00693972\pi\)
\(684\) 0.911854i 0.0348656i
\(685\) 7.94331 0.303498
\(686\) 1.36227 0.0520118
\(687\) 4.64848i 0.177351i
\(688\) −9.43296 −0.359628
\(689\) 0 0
\(690\) 0.356896 0.0135868
\(691\) − 11.9071i − 0.452966i −0.974015 0.226483i \(-0.927277\pi\)
0.974015 0.226483i \(-0.0727228\pi\)
\(692\) 6.89546 0.262126
\(693\) 11.2567 0.427605
\(694\) 10.4862i 0.398050i
\(695\) − 1.71379i − 0.0650078i
\(696\) 10.5036i 0.398140i
\(697\) − 70.5357i − 2.67173i
\(698\) −8.13467 −0.307902
\(699\) −0.658170 −0.0248943
\(700\) 3.69202i 0.139545i
\(701\) −38.0575 −1.43741 −0.718707 0.695313i \(-0.755266\pi\)
−0.718707 + 0.695313i \(0.755266\pi\)
\(702\) 0 0
\(703\) −0.820461 −0.0309443
\(704\) 3.04892i 0.114910i
\(705\) −11.5918 −0.436572
\(706\) 25.8649 0.973437
\(707\) 15.1293i 0.568996i
\(708\) − 8.54288i − 0.321061i
\(709\) − 26.1215i − 0.981014i −0.871437 0.490507i \(-0.836812\pi\)
0.871437 0.490507i \(-0.163188\pi\)
\(710\) 4.13706i 0.155261i
\(711\) 9.62565 0.360990
\(712\) 6.41119 0.240270
\(713\) − 0.735562i − 0.0275470i
\(714\) 25.2935 0.946586
\(715\) 0 0
\(716\) −25.6461 −0.958439
\(717\) 18.7748i 0.701157i
\(718\) −30.2959 −1.13063
\(719\) −21.3739 −0.797111 −0.398556 0.917144i \(-0.630488\pi\)
−0.398556 + 0.917144i \(0.630488\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) 62.4747i 2.32668i
\(722\) − 18.1685i − 0.676162i
\(723\) 18.0271i 0.670437i
\(724\) −25.8092 −0.959193
\(725\) −10.5036 −0.390096
\(726\) − 1.70410i − 0.0632452i
\(727\) 17.9124 0.664336 0.332168 0.943220i \(-0.392220\pi\)
0.332168 + 0.943220i \(0.392220\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 11.1685i 0.413366i
\(731\) −64.6238 −2.39020
\(732\) −1.55496 −0.0574729
\(733\) 27.5386i 1.01716i 0.861015 + 0.508580i \(0.169829\pi\)
−0.861015 + 0.508580i \(0.830171\pi\)
\(734\) 3.41013i 0.125870i
\(735\) − 6.63102i − 0.244589i
\(736\) − 0.356896i − 0.0131554i
\(737\) −3.50365 −0.129059
\(738\) −10.2959 −0.378997
\(739\) 7.99090i 0.293950i 0.989140 + 0.146975i \(0.0469537\pi\)
−0.989140 + 0.146975i \(0.953046\pi\)
\(740\) −0.899772 −0.0330763
\(741\) 0 0
\(742\) −12.5743 −0.461618
\(743\) 19.8398i 0.727853i 0.931428 + 0.363927i \(0.118564\pi\)
−0.931428 + 0.363927i \(0.881436\pi\)
\(744\) 2.06100 0.0755599
\(745\) −6.91185 −0.253231
\(746\) − 3.35988i − 0.123014i
\(747\) 9.86054i 0.360778i
\(748\) 20.8877i 0.763730i
\(749\) − 46.9594i − 1.71586i
\(750\) −1.00000 −0.0365148
\(751\) 25.7525 0.939724 0.469862 0.882740i \(-0.344304\pi\)
0.469862 + 0.882740i \(0.344304\pi\)
\(752\) 11.5918i 0.422709i
\(753\) 2.18060 0.0794656
\(754\) 0 0
\(755\) −5.43296 −0.197726
\(756\) − 3.69202i − 0.134278i
\(757\) −51.5096 −1.87215 −0.936074 0.351802i \(-0.885569\pi\)
−0.936074 + 0.351802i \(0.885569\pi\)
\(758\) 31.4916 1.14383
\(759\) − 1.08815i − 0.0394972i
\(760\) 0.911854i 0.0330764i
\(761\) 31.4765i 1.14102i 0.821290 + 0.570511i \(0.193255\pi\)
−0.821290 + 0.570511i \(0.806745\pi\)
\(762\) − 7.08815i − 0.256776i
\(763\) 73.8117 2.67216
\(764\) 14.5700 0.527125
\(765\) 6.85086i 0.247693i
\(766\) −21.8485 −0.789417
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) − 47.1584i − 1.70057i −0.526319 0.850287i \(-0.676428\pi\)
0.526319 0.850287i \(-0.323572\pi\)
\(770\) 11.2567 0.405662
\(771\) 9.58748 0.345285
\(772\) − 8.32544i − 0.299639i
\(773\) − 3.71917i − 0.133769i −0.997761 0.0668846i \(-0.978694\pi\)
0.997761 0.0668846i \(-0.0213059\pi\)
\(774\) 9.43296i 0.339061i
\(775\) 2.06100i 0.0740333i
\(776\) 6.97823 0.250504
\(777\) 3.32198 0.119175
\(778\) − 11.3787i − 0.407945i
\(779\) −9.38835 −0.336373
\(780\) 0 0
\(781\) 12.6136 0.451349
\(782\) − 2.44504i − 0.0874345i
\(783\) 10.5036 0.375370
\(784\) −6.63102 −0.236822
\(785\) − 9.13706i − 0.326116i
\(786\) − 19.7506i − 0.704482i
\(787\) − 47.2282i − 1.68350i −0.539865 0.841752i \(-0.681525\pi\)
0.539865 0.841752i \(-0.318475\pi\)
\(788\) − 26.8810i − 0.957595i
\(789\) −1.41789 −0.0504784
\(790\) 9.62565 0.342465
\(791\) − 16.6116i − 0.590642i
\(792\) 3.04892 0.108339
\(793\) 0 0
\(794\) 22.0476 0.782440
\(795\) − 3.40581i − 0.120792i
\(796\) −18.7047 −0.662970
\(797\) −34.4359 −1.21978 −0.609892 0.792485i \(-0.708787\pi\)
−0.609892 + 0.792485i \(0.708787\pi\)
\(798\) − 3.36658i − 0.119176i
\(799\) 79.4137i 2.80945i
\(800\) 1.00000i 0.0353553i
\(801\) − 6.41119i − 0.226528i
\(802\) 10.2459 0.361796
\(803\) 34.0519 1.20167
\(804\) 1.14914i 0.0405272i
\(805\) −1.31767 −0.0464417
\(806\) 0 0
\(807\) −4.14675 −0.145973
\(808\) 4.09783i 0.144161i
\(809\) −32.3846 −1.13858 −0.569292 0.822136i \(-0.692782\pi\)
−0.569292 + 0.822136i \(0.692782\pi\)
\(810\) 1.00000 0.0351364
\(811\) − 37.8297i − 1.32838i −0.747564 0.664190i \(-0.768777\pi\)
0.747564 0.664190i \(-0.231223\pi\)
\(812\) − 38.7797i − 1.36090i
\(813\) 6.40044i 0.224473i
\(814\) 2.74333i 0.0961537i
\(815\) −14.3773 −0.503616
\(816\) 6.85086 0.239828
\(817\) 8.60148i 0.300928i
\(818\) −22.8189 −0.797845
\(819\) 0 0
\(820\) −10.2959 −0.359548
\(821\) − 43.2820i − 1.51055i −0.655406 0.755276i \(-0.727503\pi\)
0.655406 0.755276i \(-0.272497\pi\)
\(822\) −7.94331 −0.277055
\(823\) 26.7399 0.932093 0.466047 0.884760i \(-0.345678\pi\)
0.466047 + 0.884760i \(0.345678\pi\)
\(824\) 16.9215i 0.589490i
\(825\) 3.04892i 0.106150i
\(826\) 31.5405i 1.09743i
\(827\) 21.4437i 0.745671i 0.927897 + 0.372835i \(0.121614\pi\)
−0.927897 + 0.372835i \(0.878386\pi\)
\(828\) −0.356896 −0.0124030
\(829\) −22.4547 −0.779885 −0.389943 0.920839i \(-0.627505\pi\)
−0.389943 + 0.920839i \(0.627505\pi\)
\(830\) 9.86054i 0.342264i
\(831\) −10.3394 −0.358671
\(832\) 0 0
\(833\) −45.4282 −1.57399
\(834\) 1.71379i 0.0593438i
\(835\) 3.00538 0.104005
\(836\) 2.78017 0.0961541
\(837\) − 2.06100i − 0.0712385i
\(838\) 28.5773i 0.987187i
\(839\) 24.4034i 0.842500i 0.906945 + 0.421250i \(0.138408\pi\)
−0.906945 + 0.421250i \(0.861592\pi\)
\(840\) − 3.69202i − 0.127387i
\(841\) 81.3266 2.80437
\(842\) −32.3937 −1.11636
\(843\) 5.08144i 0.175014i
\(844\) −2.93362 −0.100980
\(845\) 0 0
\(846\) 11.5918 0.398534
\(847\) 6.29159i 0.216181i
\(848\) −3.40581 −0.116956
\(849\) −24.5284 −0.841813
\(850\) 6.85086i 0.234982i
\(851\) − 0.321125i − 0.0110080i
\(852\) − 4.13706i − 0.141733i
\(853\) − 7.96184i − 0.272608i −0.990667 0.136304i \(-0.956478\pi\)
0.990667 0.136304i \(-0.0435224\pi\)
\(854\) 5.74094 0.196451
\(855\) 0.911854 0.0311847
\(856\) − 12.7192i − 0.434732i
\(857\) 8.12737 0.277626 0.138813 0.990319i \(-0.455671\pi\)
0.138813 + 0.990319i \(0.455671\pi\)
\(858\) 0 0
\(859\) 17.1666 0.585717 0.292858 0.956156i \(-0.405394\pi\)
0.292858 + 0.956156i \(0.405394\pi\)
\(860\) 9.43296i 0.321661i
\(861\) 38.0127 1.29547
\(862\) 14.6455 0.498828
\(863\) 46.2857i 1.57558i 0.615941 + 0.787792i \(0.288776\pi\)
−0.615941 + 0.787792i \(0.711224\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) − 6.89546i − 0.234453i
\(866\) − 35.8485i − 1.21818i
\(867\) 29.9342 1.01662
\(868\) −7.60925 −0.258275
\(869\) − 29.3478i − 0.995556i
\(870\) 10.5036 0.356107
\(871\) 0 0
\(872\) 19.9922 0.677022
\(873\) − 6.97823i − 0.236177i
\(874\) −0.325437 −0.0110081
\(875\) 3.69202 0.124813
\(876\) − 11.1685i − 0.377349i
\(877\) − 2.72156i − 0.0919006i −0.998944 0.0459503i \(-0.985368\pi\)
0.998944 0.0459503i \(-0.0146316\pi\)
\(878\) 2.55927i 0.0863712i
\(879\) 15.8538i 0.534737i
\(880\) 3.04892 0.102779
\(881\) 16.7241 0.563448 0.281724 0.959495i \(-0.409094\pi\)
0.281724 + 0.959495i \(0.409094\pi\)
\(882\) 6.63102i 0.223278i
\(883\) 2.88364 0.0970423 0.0485211 0.998822i \(-0.484549\pi\)
0.0485211 + 0.998822i \(0.484549\pi\)
\(884\) 0 0
\(885\) −8.54288 −0.287166
\(886\) − 23.2857i − 0.782300i
\(887\) 55.8866 1.87649 0.938245 0.345973i \(-0.112451\pi\)
0.938245 + 0.345973i \(0.112451\pi\)
\(888\) 0.899772 0.0301944
\(889\) 26.1696i 0.877700i
\(890\) − 6.41119i − 0.214904i
\(891\) − 3.04892i − 0.102143i
\(892\) − 18.1444i − 0.607518i
\(893\) 10.5700 0.353712
\(894\) 6.91185 0.231167
\(895\) 25.6461i 0.857254i
\(896\) −3.69202 −0.123342
\(897\) 0 0
\(898\) −27.3327 −0.912105
\(899\) − 21.6480i − 0.722001i
\(900\) 1.00000 0.0333333
\(901\) −23.3327 −0.777326
\(902\) 31.3913i 1.04522i
\(903\) − 34.8267i − 1.15896i
\(904\) − 4.49934i − 0.149646i
\(905\) 25.8092i 0.857928i
\(906\) 5.43296 0.180498
\(907\) −49.5381 −1.64489 −0.822443 0.568848i \(-0.807389\pi\)
−0.822443 + 0.568848i \(0.807389\pi\)
\(908\) 12.3056i 0.408375i
\(909\) 4.09783 0.135917
\(910\) 0 0
\(911\) 37.0603 1.22786 0.613931 0.789360i \(-0.289587\pi\)
0.613931 + 0.789360i \(0.289587\pi\)
\(912\) − 0.911854i − 0.0301945i
\(913\) 30.0640 0.994973
\(914\) 7.74094 0.256047
\(915\) 1.55496i 0.0514053i
\(916\) − 4.64848i − 0.153590i
\(917\) 72.9197i 2.40802i
\(918\) − 6.85086i − 0.226112i
\(919\) 53.6228 1.76885 0.884426 0.466680i \(-0.154550\pi\)
0.884426 + 0.466680i \(0.154550\pi\)
\(920\) −0.356896 −0.0117665
\(921\) − 14.2121i − 0.468303i
\(922\) −17.1511 −0.564840
\(923\) 0 0
\(924\) −11.2567 −0.370317
\(925\) 0.899772i 0.0295843i
\(926\) −34.2747 −1.12634
\(927\) 16.9215 0.555776
\(928\) − 10.5036i − 0.344799i
\(929\) 10.6726i 0.350158i 0.984554 + 0.175079i \(0.0560181\pi\)
−0.984554 + 0.175079i \(0.943982\pi\)
\(930\) − 2.06100i − 0.0675828i
\(931\) 6.04652i 0.198167i
\(932\) 0.658170 0.0215591
\(933\) −15.0127 −0.491493
\(934\) 20.9474i 0.685419i
\(935\) 20.8877 0.683101
\(936\) 0 0
\(937\) 17.0538 0.557124 0.278562 0.960418i \(-0.410142\pi\)
0.278562 + 0.960418i \(0.410142\pi\)
\(938\) − 4.24267i − 0.138528i
\(939\) −13.4993 −0.440534
\(940\) 11.5918 0.378083
\(941\) 46.7706i 1.52468i 0.647178 + 0.762339i \(0.275949\pi\)
−0.647178 + 0.762339i \(0.724051\pi\)
\(942\) 9.13706i 0.297702i
\(943\) − 3.67456i − 0.119660i
\(944\) 8.54288i 0.278047i
\(945\) −3.69202 −0.120101
\(946\) 28.7603 0.935079
\(947\) − 30.0398i − 0.976163i −0.872798 0.488081i \(-0.837697\pi\)
0.872798 0.488081i \(-0.162303\pi\)
\(948\) −9.62565 −0.312626
\(949\) 0 0
\(950\) 0.911854 0.0295845
\(951\) − 33.9463i − 1.10078i
\(952\) −25.2935 −0.819767
\(953\) 29.4252 0.953175 0.476588 0.879127i \(-0.341873\pi\)
0.476588 + 0.879127i \(0.341873\pi\)
\(954\) 3.40581i 0.110267i
\(955\) − 14.5700i − 0.471475i
\(956\) − 18.7748i − 0.607220i
\(957\) − 32.0248i − 1.03521i
\(958\) 15.8310 0.511477
\(959\) 29.3269 0.947014
\(960\) − 1.00000i − 0.0322749i
\(961\) 26.7523 0.862977
\(962\) 0 0
\(963\) −12.7192 −0.409869
\(964\) − 18.0271i − 0.580615i
\(965\) −8.32544 −0.268005
\(966\) 1.31767 0.0423952
\(967\) − 12.7023i − 0.408478i −0.978921 0.204239i \(-0.934528\pi\)
0.978921 0.204239i \(-0.0654720\pi\)
\(968\) 1.70410i 0.0547719i
\(969\) − 6.24698i − 0.200682i
\(970\) − 6.97823i − 0.224057i
\(971\) 22.9105 0.735234 0.367617 0.929977i \(-0.380174\pi\)
0.367617 + 0.929977i \(0.380174\pi\)
\(972\) −1.00000 −0.0320750
\(973\) − 6.32736i − 0.202846i
\(974\) −33.7060 −1.08001
\(975\) 0 0
\(976\) 1.55496 0.0497730
\(977\) 25.1390i 0.804267i 0.915581 + 0.402134i \(0.131731\pi\)
−0.915581 + 0.402134i \(0.868269\pi\)
\(978\) 14.3773 0.459737
\(979\) −19.5472 −0.624731
\(980\) 6.63102i 0.211820i
\(981\) − 19.9922i − 0.638303i
\(982\) 3.98925i 0.127302i
\(983\) − 50.3895i − 1.60718i −0.595186 0.803588i \(-0.702921\pi\)
0.595186 0.803588i \(-0.297079\pi\)
\(984\) 10.2959 0.328221
\(985\) −26.8810 −0.856499
\(986\) − 71.9590i − 2.29164i
\(987\) −42.7972 −1.36225
\(988\) 0 0
\(989\) −3.36658 −0.107051
\(990\) − 3.04892i − 0.0969010i
\(991\) 23.3375 0.741341 0.370670 0.928764i \(-0.379128\pi\)
0.370670 + 0.928764i \(0.379128\pi\)
\(992\) −2.06100 −0.0654368
\(993\) 6.84415i 0.217193i
\(994\) 15.2741i 0.484466i
\(995\) 18.7047i 0.592979i
\(996\) − 9.86054i − 0.312443i
\(997\) 22.8127 0.722485 0.361243 0.932472i \(-0.382353\pi\)
0.361243 + 0.932472i \(0.382353\pi\)
\(998\) −20.4698 −0.647960
\(999\) − 0.899772i − 0.0284675i
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.z.1351.3 6
13.5 odd 4 5070.2.a.bp.1.1 3
13.8 odd 4 5070.2.a.bw.1.3 yes 3
13.12 even 2 inner 5070.2.b.z.1351.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bp.1.1 3 13.5 odd 4
5070.2.a.bw.1.3 yes 3 13.8 odd 4
5070.2.b.z.1351.3 6 1.1 even 1 trivial
5070.2.b.z.1351.4 6 13.12 even 2 inner