Properties

Label 5070.2.b.z.1351.6
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 1351.6
Root \(1.80194i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.z.1351.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +1.04892i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +1.04892i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.35690i q^{11} -1.00000 q^{12} -1.04892 q^{14} -1.00000i q^{15} +1.00000 q^{16} +1.08815 q^{17} +1.00000i q^{18} +2.93900i q^{19} +1.00000i q^{20} +1.04892i q^{21} +1.35690 q^{22} +0.692021 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -1.04892i q^{28} -2.37867 q^{29} +1.00000 q^{30} +9.85086i q^{31} +1.00000i q^{32} -1.35690i q^{33} +1.08815i q^{34} +1.04892 q^{35} -1.00000 q^{36} -9.26875i q^{37} -2.93900 q^{38} -1.00000 q^{40} +2.84117i q^{41} -1.04892 q^{42} +4.45473 q^{43} +1.35690i q^{44} -1.00000i q^{45} +0.692021i q^{46} +3.31767i q^{47} +1.00000 q^{48} +5.89977 q^{49} -1.00000i q^{50} +1.08815 q^{51} +0.664874 q^{53} +1.00000i q^{54} -1.35690 q^{55} +1.04892 q^{56} +2.93900i q^{57} -2.37867i q^{58} +1.96077i q^{59} +1.00000i q^{60} +3.24698 q^{61} -9.85086 q^{62} +1.04892i q^{63} -1.00000 q^{64} +1.35690 q^{66} +6.91185i q^{67} -1.08815 q^{68} +0.692021 q^{69} +1.04892i q^{70} +2.29590i q^{71} -1.00000i q^{72} -3.36227i q^{73} +9.26875 q^{74} -1.00000 q^{75} -2.93900i q^{76} +1.42327 q^{77} +12.3230 q^{79} -1.00000i q^{80} +1.00000 q^{81} -2.84117 q^{82} +2.68664i q^{83} -1.04892i q^{84} -1.08815i q^{85} +4.45473i q^{86} -2.37867 q^{87} -1.35690 q^{88} -12.8877i q^{89} +1.00000 q^{90} -0.692021 q^{92} +9.85086i q^{93} -3.31767 q^{94} +2.93900 q^{95} +1.00000i q^{96} +1.56704i q^{97} +5.89977i q^{98} -1.35690i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{3} - 6q^{4} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{3} - 6q^{4} + 6q^{9} + 6q^{10} - 6q^{12} + 12q^{14} + 6q^{16} + 14q^{17} - 6q^{23} - 6q^{25} + 6q^{27} + 6q^{30} - 12q^{35} - 6q^{36} + 2q^{38} - 6q^{40} + 12q^{42} - 18q^{43} + 6q^{48} - 10q^{49} + 14q^{51} + 6q^{53} - 12q^{56} + 10q^{61} - 32q^{62} - 6q^{64} - 14q^{68} - 6q^{69} + 40q^{74} - 6q^{75} + 14q^{77} + 34q^{79} + 6q^{81} - 34q^{82} + 6q^{90} + 6q^{92} + 14q^{94} - 2q^{95} + O(q^{100}) \)

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
\(7\) 1.04892i 0.396453i 0.980156 + 0.198227i \(0.0635182\pi\)
−0.980156 + 0.198227i \(0.936482\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) − 1.35690i − 0.409119i −0.978854 0.204560i \(-0.934424\pi\)
0.978854 0.204560i \(-0.0655763\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.04892 −0.280335
\(15\) − 1.00000i − 0.258199i
\(16\) 1.00000 0.250000
\(17\) 1.08815 0.263914 0.131957 0.991255i \(-0.457874\pi\)
0.131957 + 0.991255i \(0.457874\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 2.93900i 0.674253i 0.941459 + 0.337127i \(0.109455\pi\)
−0.941459 + 0.337127i \(0.890545\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 1.04892i 0.228893i
\(22\) 1.35690 0.289291
\(23\) 0.692021 0.144296 0.0721482 0.997394i \(-0.477015\pi\)
0.0721482 + 0.997394i \(0.477015\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) − 1.04892i − 0.198227i
\(29\) −2.37867 −0.441707 −0.220854 0.975307i \(-0.570884\pi\)
−0.220854 + 0.975307i \(0.570884\pi\)
\(30\) 1.00000 0.182574
\(31\) 9.85086i 1.76927i 0.466288 + 0.884633i \(0.345591\pi\)
−0.466288 + 0.884633i \(0.654409\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 1.35690i − 0.236205i
\(34\) 1.08815i 0.186615i
\(35\) 1.04892 0.177299
\(36\) −1.00000 −0.166667
\(37\) − 9.26875i − 1.52377i −0.647710 0.761887i \(-0.724273\pi\)
0.647710 0.761887i \(-0.275727\pi\)
\(38\) −2.93900 −0.476769
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 2.84117i 0.443716i 0.975079 + 0.221858i \(0.0712121\pi\)
−0.975079 + 0.221858i \(0.928788\pi\)
\(42\) −1.04892 −0.161851
\(43\) 4.45473 0.679340 0.339670 0.940545i \(-0.389685\pi\)
0.339670 + 0.940545i \(0.389685\pi\)
\(44\) 1.35690i 0.204560i
\(45\) − 1.00000i − 0.149071i
\(46\) 0.692021i 0.102033i
\(47\) 3.31767i 0.483931i 0.970285 + 0.241966i \(0.0777922\pi\)
−0.970285 + 0.241966i \(0.922208\pi\)
\(48\) 1.00000 0.144338
\(49\) 5.89977 0.842825
\(50\) − 1.00000i − 0.141421i
\(51\) 1.08815 0.152371
\(52\) 0 0
\(53\) 0.664874 0.0913275 0.0456638 0.998957i \(-0.485460\pi\)
0.0456638 + 0.998957i \(0.485460\pi\)
\(54\) 1.00000i 0.136083i
\(55\) −1.35690 −0.182964
\(56\) 1.04892 0.140167
\(57\) 2.93900i 0.389280i
\(58\) − 2.37867i − 0.312334i
\(59\) 1.96077i 0.255271i 0.991821 + 0.127635i \(0.0407387\pi\)
−0.991821 + 0.127635i \(0.959261\pi\)
\(60\) 1.00000i 0.129099i
\(61\) 3.24698 0.415733 0.207867 0.978157i \(-0.433348\pi\)
0.207867 + 0.978157i \(0.433348\pi\)
\(62\) −9.85086 −1.25106
\(63\) 1.04892i 0.132151i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.35690 0.167022
\(67\) 6.91185i 0.844417i 0.906499 + 0.422209i \(0.138745\pi\)
−0.906499 + 0.422209i \(0.861255\pi\)
\(68\) −1.08815 −0.131957
\(69\) 0.692021 0.0833096
\(70\) 1.04892i 0.125370i
\(71\) 2.29590i 0.272473i 0.990676 + 0.136236i \(0.0435007\pi\)
−0.990676 + 0.136236i \(0.956499\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 3.36227i − 0.393524i −0.980451 0.196762i \(-0.936957\pi\)
0.980451 0.196762i \(-0.0630427\pi\)
\(74\) 9.26875 1.07747
\(75\) −1.00000 −0.115470
\(76\) − 2.93900i − 0.337127i
\(77\) 1.42327 0.162197
\(78\) 0 0
\(79\) 12.3230 1.38645 0.693225 0.720721i \(-0.256189\pi\)
0.693225 + 0.720721i \(0.256189\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) −2.84117 −0.313754
\(83\) 2.68664i 0.294898i 0.989070 + 0.147449i \(0.0471062\pi\)
−0.989070 + 0.147449i \(0.952894\pi\)
\(84\) − 1.04892i − 0.114446i
\(85\) − 1.08815i − 0.118026i
\(86\) 4.45473i 0.480366i
\(87\) −2.37867 −0.255020
\(88\) −1.35690 −0.144646
\(89\) − 12.8877i − 1.36609i −0.730375 0.683046i \(-0.760655\pi\)
0.730375 0.683046i \(-0.239345\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −0.692021 −0.0721482
\(93\) 9.85086i 1.02149i
\(94\) −3.31767 −0.342191
\(95\) 2.93900 0.301535
\(96\) 1.00000i 0.102062i
\(97\) 1.56704i 0.159109i 0.996831 + 0.0795544i \(0.0253497\pi\)
−0.996831 + 0.0795544i \(0.974650\pi\)
\(98\) 5.89977i 0.595967i
\(99\) − 1.35690i − 0.136373i
\(100\) 1.00000 0.100000
\(101\) −4.71379 −0.469040 −0.234520 0.972111i \(-0.575352\pi\)
−0.234520 + 0.972111i \(0.575352\pi\)
\(102\) 1.08815i 0.107743i
\(103\) 12.1642 1.19858 0.599288 0.800534i \(-0.295451\pi\)
0.599288 + 0.800534i \(0.295451\pi\)
\(104\) 0 0
\(105\) 1.04892 0.102364
\(106\) 0.664874i 0.0645783i
\(107\) 1.83877 0.177761 0.0888805 0.996042i \(-0.471671\pi\)
0.0888805 + 0.996042i \(0.471671\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 16.2403i 1.55554i 0.628551 + 0.777768i \(0.283648\pi\)
−0.628551 + 0.777768i \(0.716352\pi\)
\(110\) − 1.35690i − 0.129375i
\(111\) − 9.26875i − 0.879751i
\(112\) 1.04892i 0.0991134i
\(113\) 16.8267 1.58292 0.791461 0.611220i \(-0.209321\pi\)
0.791461 + 0.611220i \(0.209321\pi\)
\(114\) −2.93900 −0.275263
\(115\) − 0.692021i − 0.0645313i
\(116\) 2.37867 0.220854
\(117\) 0 0
\(118\) −1.96077 −0.180504
\(119\) 1.14138i 0.104630i
\(120\) −1.00000 −0.0912871
\(121\) 9.15883 0.832621
\(122\) 3.24698i 0.293968i
\(123\) 2.84117i 0.256179i
\(124\) − 9.85086i − 0.884633i
\(125\) 1.00000i 0.0894427i
\(126\) −1.04892 −0.0934450
\(127\) 5.06100 0.449091 0.224546 0.974464i \(-0.427910\pi\)
0.224546 + 0.974464i \(0.427910\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 4.45473 0.392217
\(130\) 0 0
\(131\) 3.81940 0.333702 0.166851 0.985982i \(-0.446640\pi\)
0.166851 + 0.985982i \(0.446640\pi\)
\(132\) 1.35690i 0.118103i
\(133\) −3.08277 −0.267310
\(134\) −6.91185 −0.597093
\(135\) − 1.00000i − 0.0860663i
\(136\) − 1.08815i − 0.0933077i
\(137\) 8.59717i 0.734506i 0.930121 + 0.367253i \(0.119702\pi\)
−0.930121 + 0.367253i \(0.880298\pi\)
\(138\) 0.692021i 0.0589088i
\(139\) −2.38404 −0.202212 −0.101106 0.994876i \(-0.532238\pi\)
−0.101106 + 0.994876i \(0.532238\pi\)
\(140\) −1.04892 −0.0886497
\(141\) 3.31767i 0.279398i
\(142\) −2.29590 −0.192667
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 2.37867i 0.197537i
\(146\) 3.36227 0.278264
\(147\) 5.89977 0.486605
\(148\) 9.26875i 0.761887i
\(149\) − 8.93900i − 0.732312i −0.930554 0.366156i \(-0.880674\pi\)
0.930554 0.366156i \(-0.119326\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) 8.45473i 0.688036i 0.938963 + 0.344018i \(0.111788\pi\)
−0.938963 + 0.344018i \(0.888212\pi\)
\(152\) 2.93900 0.238384
\(153\) 1.08815 0.0879714
\(154\) 1.42327i 0.114690i
\(155\) 9.85086 0.791240
\(156\) 0 0
\(157\) −2.70410 −0.215811 −0.107905 0.994161i \(-0.534414\pi\)
−0.107905 + 0.994161i \(0.534414\pi\)
\(158\) 12.3230i 0.980369i
\(159\) 0.664874 0.0527280
\(160\) 1.00000 0.0790569
\(161\) 0.725873i 0.0572068i
\(162\) 1.00000i 0.0785674i
\(163\) 22.5284i 1.76456i 0.470724 + 0.882280i \(0.343993\pi\)
−0.470724 + 0.882280i \(0.656007\pi\)
\(164\) − 2.84117i − 0.221858i
\(165\) −1.35690 −0.105634
\(166\) −2.68664 −0.208524
\(167\) − 12.2228i − 0.945830i −0.881108 0.472915i \(-0.843202\pi\)
0.881108 0.472915i \(-0.156798\pi\)
\(168\) 1.04892 0.0809257
\(169\) 0 0
\(170\) 1.08815 0.0834570
\(171\) 2.93900i 0.224751i
\(172\) −4.45473 −0.339670
\(173\) 11.7168 0.890810 0.445405 0.895329i \(-0.353060\pi\)
0.445405 + 0.895329i \(0.353060\pi\)
\(174\) − 2.37867i − 0.180326i
\(175\) − 1.04892i − 0.0792907i
\(176\) − 1.35690i − 0.102280i
\(177\) 1.96077i 0.147381i
\(178\) 12.8877 0.965973
\(179\) −8.89738 −0.665021 −0.332511 0.943099i \(-0.607896\pi\)
−0.332511 + 0.943099i \(0.607896\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) −1.31229 −0.0975418 −0.0487709 0.998810i \(-0.515530\pi\)
−0.0487709 + 0.998810i \(0.515530\pi\)
\(182\) 0 0
\(183\) 3.24698 0.240024
\(184\) − 0.692021i − 0.0510165i
\(185\) −9.26875 −0.681452
\(186\) −9.85086 −0.722300
\(187\) − 1.47650i − 0.107972i
\(188\) − 3.31767i − 0.241966i
\(189\) 1.04892i 0.0762975i
\(190\) 2.93900i 0.213218i
\(191\) 5.75063 0.416101 0.208050 0.978118i \(-0.433288\pi\)
0.208050 + 0.978118i \(0.433288\pi\)
\(192\) −1.00000 −0.0721688
\(193\) − 10.0339i − 0.722252i −0.932517 0.361126i \(-0.882392\pi\)
0.932517 0.361126i \(-0.117608\pi\)
\(194\) −1.56704 −0.112507
\(195\) 0 0
\(196\) −5.89977 −0.421412
\(197\) 22.9071i 1.63206i 0.578009 + 0.816031i \(0.303830\pi\)
−0.578009 + 0.816031i \(0.696170\pi\)
\(198\) 1.35690 0.0964304
\(199\) −27.0291 −1.91604 −0.958020 0.286702i \(-0.907441\pi\)
−0.958020 + 0.286702i \(0.907441\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 6.91185i 0.487525i
\(202\) − 4.71379i − 0.331661i
\(203\) − 2.49502i − 0.175116i
\(204\) −1.08815 −0.0761855
\(205\) 2.84117 0.198436
\(206\) 12.1642i 0.847521i
\(207\) 0.692021 0.0480988
\(208\) 0 0
\(209\) 3.98792 0.275850
\(210\) 1.04892i 0.0723822i
\(211\) 10.3720 0.714035 0.357018 0.934098i \(-0.383794\pi\)
0.357018 + 0.934098i \(0.383794\pi\)
\(212\) −0.664874 −0.0456638
\(213\) 2.29590i 0.157312i
\(214\) 1.83877i 0.125696i
\(215\) − 4.45473i − 0.303810i
\(216\) − 1.00000i − 0.0680414i
\(217\) −10.3327 −0.701432
\(218\) −16.2403 −1.09993
\(219\) − 3.36227i − 0.227201i
\(220\) 1.35690 0.0914819
\(221\) 0 0
\(222\) 9.26875 0.622078
\(223\) 14.0532i 0.941074i 0.882380 + 0.470537i \(0.155940\pi\)
−0.882380 + 0.470537i \(0.844060\pi\)
\(224\) −1.04892 −0.0700837
\(225\) −1.00000 −0.0666667
\(226\) 16.8267i 1.11929i
\(227\) − 1.93362i − 0.128339i −0.997939 0.0641696i \(-0.979560\pi\)
0.997939 0.0641696i \(-0.0204399\pi\)
\(228\) − 2.93900i − 0.194640i
\(229\) 10.9148i 0.721273i 0.932706 + 0.360636i \(0.117440\pi\)
−0.932706 + 0.360636i \(0.882560\pi\)
\(230\) 0.692021 0.0456305
\(231\) 1.42327 0.0936444
\(232\) 2.37867i 0.156167i
\(233\) 21.6896 1.42093 0.710467 0.703730i \(-0.248484\pi\)
0.710467 + 0.703730i \(0.248484\pi\)
\(234\) 0 0
\(235\) 3.31767 0.216421
\(236\) − 1.96077i − 0.127635i
\(237\) 12.3230 0.800468
\(238\) −1.14138 −0.0739844
\(239\) − 27.2349i − 1.76168i −0.473415 0.880840i \(-0.656979\pi\)
0.473415 0.880840i \(-0.343021\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) − 8.21014i − 0.528862i −0.964405 0.264431i \(-0.914816\pi\)
0.964405 0.264431i \(-0.0851841\pi\)
\(242\) 9.15883i 0.588752i
\(243\) 1.00000 0.0641500
\(244\) −3.24698 −0.207867
\(245\) − 5.89977i − 0.376923i
\(246\) −2.84117 −0.181146
\(247\) 0 0
\(248\) 9.85086 0.625530
\(249\) 2.68664i 0.170259i
\(250\) −1.00000 −0.0632456
\(251\) 6.57002 0.414696 0.207348 0.978267i \(-0.433517\pi\)
0.207348 + 0.978267i \(0.433517\pi\)
\(252\) − 1.04892i − 0.0660756i
\(253\) − 0.939001i − 0.0590345i
\(254\) 5.06100i 0.317555i
\(255\) − 1.08815i − 0.0681423i
\(256\) 1.00000 0.0625000
\(257\) −13.7657 −0.858680 −0.429340 0.903143i \(-0.641254\pi\)
−0.429340 + 0.903143i \(0.641254\pi\)
\(258\) 4.45473i 0.277339i
\(259\) 9.72215 0.604105
\(260\) 0 0
\(261\) −2.37867 −0.147236
\(262\) 3.81940i 0.235963i
\(263\) −9.54288 −0.588439 −0.294219 0.955738i \(-0.595060\pi\)
−0.294219 + 0.955738i \(0.595060\pi\)
\(264\) −1.35690 −0.0835112
\(265\) − 0.664874i − 0.0408429i
\(266\) − 3.08277i − 0.189017i
\(267\) − 12.8877i − 0.788714i
\(268\) − 6.91185i − 0.422209i
\(269\) 9.07069 0.553050 0.276525 0.961007i \(-0.410817\pi\)
0.276525 + 0.961007i \(0.410817\pi\)
\(270\) 1.00000 0.0608581
\(271\) − 17.5579i − 1.06657i −0.845936 0.533285i \(-0.820958\pi\)
0.845936 0.533285i \(-0.179042\pi\)
\(272\) 1.08815 0.0659785
\(273\) 0 0
\(274\) −8.59717 −0.519374
\(275\) 1.35690i 0.0818239i
\(276\) −0.692021 −0.0416548
\(277\) −13.7071 −0.823579 −0.411790 0.911279i \(-0.635096\pi\)
−0.411790 + 0.911279i \(0.635096\pi\)
\(278\) − 2.38404i − 0.142985i
\(279\) 9.85086i 0.589755i
\(280\) − 1.04892i − 0.0626848i
\(281\) 24.3696i 1.45377i 0.686761 + 0.726883i \(0.259032\pi\)
−0.686761 + 0.726883i \(0.740968\pi\)
\(282\) −3.31767 −0.197564
\(283\) 17.1511 1.01952 0.509762 0.860315i \(-0.329733\pi\)
0.509762 + 0.860315i \(0.329733\pi\)
\(284\) − 2.29590i − 0.136236i
\(285\) 2.93900 0.174091
\(286\) 0 0
\(287\) −2.98015 −0.175913
\(288\) 1.00000i 0.0589256i
\(289\) −15.8159 −0.930349
\(290\) −2.37867 −0.139680
\(291\) 1.56704i 0.0918615i
\(292\) 3.36227i 0.196762i
\(293\) 24.1172i 1.40894i 0.709732 + 0.704471i \(0.248816\pi\)
−0.709732 + 0.704471i \(0.751184\pi\)
\(294\) 5.89977i 0.344082i
\(295\) 1.96077 0.114161
\(296\) −9.26875 −0.538735
\(297\) − 1.35690i − 0.0787351i
\(298\) 8.93900 0.517822
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 4.67264i 0.269327i
\(302\) −8.45473 −0.486515
\(303\) −4.71379 −0.270800
\(304\) 2.93900i 0.168563i
\(305\) − 3.24698i − 0.185922i
\(306\) 1.08815i 0.0622052i
\(307\) 17.2282i 0.983265i 0.870803 + 0.491632i \(0.163600\pi\)
−0.870803 + 0.491632i \(0.836400\pi\)
\(308\) −1.42327 −0.0810984
\(309\) 12.1642 0.691998
\(310\) 9.85086i 0.559491i
\(311\) 25.9801 1.47320 0.736600 0.676329i \(-0.236430\pi\)
0.736600 + 0.676329i \(0.236430\pi\)
\(312\) 0 0
\(313\) 7.82669 0.442391 0.221195 0.975230i \(-0.429004\pi\)
0.221195 + 0.975230i \(0.429004\pi\)
\(314\) − 2.70410i − 0.152601i
\(315\) 1.04892 0.0590998
\(316\) −12.3230 −0.693225
\(317\) 0.391813i 0.0220064i 0.999939 + 0.0110032i \(0.00350250\pi\)
−0.999939 + 0.0110032i \(0.996498\pi\)
\(318\) 0.664874i 0.0372843i
\(319\) 3.22760i 0.180711i
\(320\) 1.00000i 0.0559017i
\(321\) 1.83877 0.102630
\(322\) −0.725873 −0.0404513
\(323\) 3.19806i 0.177945i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −22.5284 −1.24773
\(327\) 16.2403i 0.898089i
\(328\) 2.84117 0.156877
\(329\) −3.47996 −0.191856
\(330\) − 1.35690i − 0.0746947i
\(331\) 26.3424i 1.44791i 0.689847 + 0.723955i \(0.257678\pi\)
−0.689847 + 0.723955i \(0.742322\pi\)
\(332\) − 2.68664i − 0.147449i
\(333\) − 9.26875i − 0.507924i
\(334\) 12.2228 0.668803
\(335\) 6.91185 0.377635
\(336\) 1.04892i 0.0572231i
\(337\) −14.8345 −0.808085 −0.404042 0.914740i \(-0.632395\pi\)
−0.404042 + 0.914740i \(0.632395\pi\)
\(338\) 0 0
\(339\) 16.8267 0.913900
\(340\) 1.08815i 0.0590130i
\(341\) 13.3666 0.723841
\(342\) −2.93900 −0.158923
\(343\) 13.5308i 0.730594i
\(344\) − 4.45473i − 0.240183i
\(345\) − 0.692021i − 0.0372572i
\(346\) 11.7168i 0.629898i
\(347\) −0.636399 −0.0341637 −0.0170819 0.999854i \(-0.505438\pi\)
−0.0170819 + 0.999854i \(0.505438\pi\)
\(348\) 2.37867 0.127510
\(349\) − 17.2784i − 0.924894i −0.886647 0.462447i \(-0.846972\pi\)
0.886647 0.462447i \(-0.153028\pi\)
\(350\) 1.04892 0.0560670
\(351\) 0 0
\(352\) 1.35690 0.0723228
\(353\) − 21.7614i − 1.15824i −0.815242 0.579121i \(-0.803396\pi\)
0.815242 0.579121i \(-0.196604\pi\)
\(354\) −1.96077 −0.104214
\(355\) 2.29590 0.121854
\(356\) 12.8877i 0.683046i
\(357\) 1.14138i 0.0604080i
\(358\) − 8.89738i − 0.470241i
\(359\) 22.8412i 1.20551i 0.797926 + 0.602755i \(0.205930\pi\)
−0.797926 + 0.602755i \(0.794070\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 10.3623 0.545383
\(362\) − 1.31229i − 0.0689725i
\(363\) 9.15883 0.480714
\(364\) 0 0
\(365\) −3.36227 −0.175989
\(366\) 3.24698i 0.169722i
\(367\) −7.78315 −0.406277 −0.203139 0.979150i \(-0.565114\pi\)
−0.203139 + 0.979150i \(0.565114\pi\)
\(368\) 0.692021 0.0360741
\(369\) 2.84117i 0.147905i
\(370\) − 9.26875i − 0.481859i
\(371\) 0.697398i 0.0362071i
\(372\) − 9.85086i − 0.510743i
\(373\) −30.5133 −1.57992 −0.789960 0.613158i \(-0.789899\pi\)
−0.789960 + 0.613158i \(0.789899\pi\)
\(374\) 1.47650 0.0763480
\(375\) 1.00000i 0.0516398i
\(376\) 3.31767 0.171096
\(377\) 0 0
\(378\) −1.04892 −0.0539505
\(379\) − 6.41358i − 0.329444i −0.986340 0.164722i \(-0.947327\pi\)
0.986340 0.164722i \(-0.0526726\pi\)
\(380\) −2.93900 −0.150768
\(381\) 5.06100 0.259283
\(382\) 5.75063i 0.294228i
\(383\) − 2.89440i − 0.147897i −0.997262 0.0739484i \(-0.976440\pi\)
0.997262 0.0739484i \(-0.0235600\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) − 1.42327i − 0.0725366i
\(386\) 10.0339 0.510710
\(387\) 4.45473 0.226447
\(388\) − 1.56704i − 0.0795544i
\(389\) 17.1250 0.868271 0.434136 0.900848i \(-0.357054\pi\)
0.434136 + 0.900848i \(0.357054\pi\)
\(390\) 0 0
\(391\) 0.753020 0.0380819
\(392\) − 5.89977i − 0.297984i
\(393\) 3.81940 0.192663
\(394\) −22.9071 −1.15404
\(395\) − 12.3230i − 0.620040i
\(396\) 1.35690i 0.0681866i
\(397\) 25.0103i 1.25523i 0.778524 + 0.627615i \(0.215969\pi\)
−0.778524 + 0.627615i \(0.784031\pi\)
\(398\) − 27.0291i − 1.35484i
\(399\) −3.08277 −0.154331
\(400\) −1.00000 −0.0500000
\(401\) − 30.8689i − 1.54152i −0.637126 0.770760i \(-0.719877\pi\)
0.637126 0.770760i \(-0.280123\pi\)
\(402\) −6.91185 −0.344732
\(403\) 0 0
\(404\) 4.71379 0.234520
\(405\) − 1.00000i − 0.0496904i
\(406\) 2.49502 0.123826
\(407\) −12.5767 −0.623405
\(408\) − 1.08815i − 0.0538713i
\(409\) − 11.0871i − 0.548221i −0.961698 0.274110i \(-0.911617\pi\)
0.961698 0.274110i \(-0.0883834\pi\)
\(410\) 2.84117i 0.140315i
\(411\) 8.59717i 0.424067i
\(412\) −12.1642 −0.599288
\(413\) −2.05669 −0.101203
\(414\) 0.692021i 0.0340110i
\(415\) 2.68664 0.131882
\(416\) 0 0
\(417\) −2.38404 −0.116747
\(418\) 3.98792i 0.195055i
\(419\) 17.5080 0.855320 0.427660 0.903940i \(-0.359338\pi\)
0.427660 + 0.903940i \(0.359338\pi\)
\(420\) −1.04892 −0.0511819
\(421\) 16.1274i 0.786000i 0.919538 + 0.393000i \(0.128563\pi\)
−0.919538 + 0.393000i \(0.871437\pi\)
\(422\) 10.3720i 0.504899i
\(423\) 3.31767i 0.161310i
\(424\) − 0.664874i − 0.0322892i
\(425\) −1.08815 −0.0527828
\(426\) −2.29590 −0.111237
\(427\) 3.40581i 0.164819i
\(428\) −1.83877 −0.0888805
\(429\) 0 0
\(430\) 4.45473 0.214826
\(431\) − 33.2905i − 1.60355i −0.597627 0.801774i \(-0.703890\pi\)
0.597627 0.801774i \(-0.296110\pi\)
\(432\) 1.00000 0.0481125
\(433\) 11.1056 0.533701 0.266851 0.963738i \(-0.414017\pi\)
0.266851 + 0.963738i \(0.414017\pi\)
\(434\) − 10.3327i − 0.495987i
\(435\) 2.37867i 0.114048i
\(436\) − 16.2403i − 0.777768i
\(437\) 2.03385i 0.0972923i
\(438\) 3.36227 0.160656
\(439\) −12.6950 −0.605900 −0.302950 0.953007i \(-0.597971\pi\)
−0.302950 + 0.953007i \(0.597971\pi\)
\(440\) 1.35690i 0.0646875i
\(441\) 5.89977 0.280942
\(442\) 0 0
\(443\) −6.90110 −0.327881 −0.163941 0.986470i \(-0.552421\pi\)
−0.163941 + 0.986470i \(0.552421\pi\)
\(444\) 9.26875i 0.439875i
\(445\) −12.8877 −0.610935
\(446\) −14.0532 −0.665440
\(447\) − 8.93900i − 0.422800i
\(448\) − 1.04892i − 0.0495567i
\(449\) 3.27652i 0.154629i 0.997007 + 0.0773143i \(0.0246345\pi\)
−0.997007 + 0.0773143i \(0.975366\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) 3.85517 0.181533
\(452\) −16.8267 −0.791461
\(453\) 8.45473i 0.397238i
\(454\) 1.93362 0.0907495
\(455\) 0 0
\(456\) 2.93900 0.137631
\(457\) 1.40581i 0.0657612i 0.999459 + 0.0328806i \(0.0104681\pi\)
−0.999459 + 0.0328806i \(0.989532\pi\)
\(458\) −10.9148 −0.510017
\(459\) 1.08815 0.0507903
\(460\) 0.692021i 0.0322657i
\(461\) 12.3773i 0.576470i 0.957560 + 0.288235i \(0.0930684\pi\)
−0.957560 + 0.288235i \(0.906932\pi\)
\(462\) 1.42327i 0.0662166i
\(463\) − 31.7797i − 1.47693i −0.674293 0.738464i \(-0.735552\pi\)
0.674293 0.738464i \(-0.264448\pi\)
\(464\) −2.37867 −0.110427
\(465\) 9.85086 0.456822
\(466\) 21.6896i 1.00475i
\(467\) 36.2790 1.67879 0.839397 0.543519i \(-0.182909\pi\)
0.839397 + 0.543519i \(0.182909\pi\)
\(468\) 0 0
\(469\) −7.24996 −0.334772
\(470\) 3.31767i 0.153033i
\(471\) −2.70410 −0.124598
\(472\) 1.96077 0.0902518
\(473\) − 6.04461i − 0.277931i
\(474\) 12.3230i 0.566016i
\(475\) − 2.93900i − 0.134851i
\(476\) − 1.14138i − 0.0523148i
\(477\) 0.664874 0.0304425
\(478\) 27.2349 1.24570
\(479\) 5.87933i 0.268633i 0.990938 + 0.134317i \(0.0428840\pi\)
−0.990938 + 0.134317i \(0.957116\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 8.21014 0.373962
\(483\) 0.725873i 0.0330284i
\(484\) −9.15883 −0.416311
\(485\) 1.56704 0.0711556
\(486\) 1.00000i 0.0453609i
\(487\) 30.6243i 1.38772i 0.720110 + 0.693860i \(0.244091\pi\)
−0.720110 + 0.693860i \(0.755909\pi\)
\(488\) − 3.24698i − 0.146984i
\(489\) 22.5284i 1.01877i
\(490\) 5.89977 0.266525
\(491\) −34.4456 −1.55451 −0.777255 0.629186i \(-0.783388\pi\)
−0.777255 + 0.629186i \(0.783388\pi\)
\(492\) − 2.84117i − 0.128090i
\(493\) −2.58834 −0.116573
\(494\) 0 0
\(495\) −1.35690 −0.0609879
\(496\) 9.85086i 0.442316i
\(497\) −2.40821 −0.108023
\(498\) −2.68664 −0.120391
\(499\) − 10.0194i − 0.448529i −0.974528 0.224264i \(-0.928002\pi\)
0.974528 0.224264i \(-0.0719979\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) − 12.2228i − 0.546075i
\(502\) 6.57002i 0.293235i
\(503\) 8.27950 0.369165 0.184582 0.982817i \(-0.440907\pi\)
0.184582 + 0.982817i \(0.440907\pi\)
\(504\) 1.04892 0.0467225
\(505\) 4.71379i 0.209761i
\(506\) 0.939001 0.0417437
\(507\) 0 0
\(508\) −5.06100 −0.224546
\(509\) − 44.4607i − 1.97069i −0.170585 0.985343i \(-0.554566\pi\)
0.170585 0.985343i \(-0.445434\pi\)
\(510\) 1.08815 0.0481839
\(511\) 3.52675 0.156014
\(512\) 1.00000i 0.0441942i
\(513\) 2.93900i 0.129760i
\(514\) − 13.7657i − 0.607179i
\(515\) − 12.1642i − 0.536019i
\(516\) −4.45473 −0.196109
\(517\) 4.50173 0.197986
\(518\) 9.72215i 0.427167i
\(519\) 11.7168 0.514309
\(520\) 0 0
\(521\) −35.8461 −1.57044 −0.785222 0.619214i \(-0.787451\pi\)
−0.785222 + 0.619214i \(0.787451\pi\)
\(522\) − 2.37867i − 0.104111i
\(523\) −5.80433 −0.253806 −0.126903 0.991915i \(-0.540504\pi\)
−0.126903 + 0.991915i \(0.540504\pi\)
\(524\) −3.81940 −0.166851
\(525\) − 1.04892i − 0.0457785i
\(526\) − 9.54288i − 0.416089i
\(527\) 10.7192i 0.466934i
\(528\) − 1.35690i − 0.0590513i
\(529\) −22.5211 −0.979179
\(530\) 0.664874 0.0288803
\(531\) 1.96077i 0.0850902i
\(532\) 3.08277 0.133655
\(533\) 0 0
\(534\) 12.8877 0.557705
\(535\) − 1.83877i − 0.0794971i
\(536\) 6.91185 0.298547
\(537\) −8.89738 −0.383950
\(538\) 9.07069i 0.391065i
\(539\) − 8.00538i − 0.344816i
\(540\) 1.00000i 0.0430331i
\(541\) 23.2446i 0.999363i 0.866209 + 0.499681i \(0.166550\pi\)
−0.866209 + 0.499681i \(0.833450\pi\)
\(542\) 17.5579 0.754178
\(543\) −1.31229 −0.0563158
\(544\) 1.08815i 0.0466539i
\(545\) 16.2403 0.695657
\(546\) 0 0
\(547\) −16.1588 −0.690902 −0.345451 0.938437i \(-0.612274\pi\)
−0.345451 + 0.938437i \(0.612274\pi\)
\(548\) − 8.59717i − 0.367253i
\(549\) 3.24698 0.138578
\(550\) −1.35690 −0.0578582
\(551\) − 6.99090i − 0.297822i
\(552\) − 0.692021i − 0.0294544i
\(553\) 12.9259i 0.549663i
\(554\) − 13.7071i − 0.582358i
\(555\) −9.26875 −0.393437
\(556\) 2.38404 0.101106
\(557\) − 25.5937i − 1.08444i −0.840236 0.542220i \(-0.817584\pi\)
0.840236 0.542220i \(-0.182416\pi\)
\(558\) −9.85086 −0.417020
\(559\) 0 0
\(560\) 1.04892 0.0443248
\(561\) − 1.47650i − 0.0623379i
\(562\) −24.3696 −1.02797
\(563\) 40.9232 1.72471 0.862354 0.506307i \(-0.168990\pi\)
0.862354 + 0.506307i \(0.168990\pi\)
\(564\) − 3.31767i − 0.139699i
\(565\) − 16.8267i − 0.707904i
\(566\) 17.1511i 0.720913i
\(567\) 1.04892i 0.0440504i
\(568\) 2.29590 0.0963337
\(569\) 20.1946 0.846602 0.423301 0.905989i \(-0.360871\pi\)
0.423301 + 0.905989i \(0.360871\pi\)
\(570\) 2.93900i 0.123101i
\(571\) −36.8756 −1.54320 −0.771598 0.636110i \(-0.780542\pi\)
−0.771598 + 0.636110i \(0.780542\pi\)
\(572\) 0 0
\(573\) 5.75063 0.240236
\(574\) − 2.98015i − 0.124389i
\(575\) −0.692021 −0.0288593
\(576\) −1.00000 −0.0416667
\(577\) − 5.07500i − 0.211275i −0.994405 0.105637i \(-0.966312\pi\)
0.994405 0.105637i \(-0.0336883\pi\)
\(578\) − 15.8159i − 0.657856i
\(579\) − 10.0339i − 0.416993i
\(580\) − 2.37867i − 0.0987687i
\(581\) −2.81807 −0.116913
\(582\) −1.56704 −0.0649559
\(583\) − 0.902165i − 0.0373639i
\(584\) −3.36227 −0.139132
\(585\) 0 0
\(586\) −24.1172 −0.996273
\(587\) − 10.5714i − 0.436326i −0.975912 0.218163i \(-0.929993\pi\)
0.975912 0.218163i \(-0.0700065\pi\)
\(588\) −5.89977 −0.243303
\(589\) −28.9517 −1.19293
\(590\) 1.96077i 0.0807237i
\(591\) 22.9071i 0.942271i
\(592\) − 9.26875i − 0.380943i
\(593\) − 38.4999i − 1.58100i −0.612460 0.790501i \(-0.709820\pi\)
0.612460 0.790501i \(-0.290180\pi\)
\(594\) 1.35690 0.0556741
\(595\) 1.14138 0.0467918
\(596\) 8.93900i 0.366156i
\(597\) −27.0291 −1.10623
\(598\) 0 0
\(599\) −0.332142 −0.0135709 −0.00678547 0.999977i \(-0.502160\pi\)
−0.00678547 + 0.999977i \(0.502160\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) −7.04162 −0.287234 −0.143617 0.989633i \(-0.545873\pi\)
−0.143617 + 0.989633i \(0.545873\pi\)
\(602\) −4.67264 −0.190443
\(603\) 6.91185i 0.281472i
\(604\) − 8.45473i − 0.344018i
\(605\) − 9.15883i − 0.372360i
\(606\) − 4.71379i − 0.191485i
\(607\) −29.4185 −1.19406 −0.597030 0.802219i \(-0.703653\pi\)
−0.597030 + 0.802219i \(0.703653\pi\)
\(608\) −2.93900 −0.119192
\(609\) − 2.49502i − 0.101103i
\(610\) 3.24698 0.131466
\(611\) 0 0
\(612\) −1.08815 −0.0439857
\(613\) 34.1987i 1.38127i 0.723203 + 0.690635i \(0.242669\pi\)
−0.723203 + 0.690635i \(0.757331\pi\)
\(614\) −17.2282 −0.695273
\(615\) 2.84117 0.114567
\(616\) − 1.42327i − 0.0573452i
\(617\) − 34.1890i − 1.37640i −0.725523 0.688198i \(-0.758402\pi\)
0.725523 0.688198i \(-0.241598\pi\)
\(618\) 12.1642i 0.489316i
\(619\) − 9.80061i − 0.393920i −0.980412 0.196960i \(-0.936893\pi\)
0.980412 0.196960i \(-0.0631069\pi\)
\(620\) −9.85086 −0.395620
\(621\) 0.692021 0.0277699
\(622\) 25.9801i 1.04171i
\(623\) 13.5181 0.541592
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 7.82669i 0.312818i
\(627\) 3.98792 0.159262
\(628\) 2.70410 0.107905
\(629\) − 10.0858i − 0.402145i
\(630\) 1.04892i 0.0417899i
\(631\) − 17.0968i − 0.680612i −0.940315 0.340306i \(-0.889469\pi\)
0.940315 0.340306i \(-0.110531\pi\)
\(632\) − 12.3230i − 0.490184i
\(633\) 10.3720 0.412248
\(634\) −0.391813 −0.0155609
\(635\) − 5.06100i − 0.200840i
\(636\) −0.664874 −0.0263640
\(637\) 0 0
\(638\) −3.22760 −0.127782
\(639\) 2.29590i 0.0908243i
\(640\) −1.00000 −0.0395285
\(641\) 14.7030 0.580735 0.290368 0.956915i \(-0.406222\pi\)
0.290368 + 0.956915i \(0.406222\pi\)
\(642\) 1.83877i 0.0725706i
\(643\) 42.7036i 1.68407i 0.539425 + 0.842033i \(0.318641\pi\)
−0.539425 + 0.842033i \(0.681359\pi\)
\(644\) − 0.725873i − 0.0286034i
\(645\) − 4.45473i − 0.175405i
\(646\) −3.19806 −0.125826
\(647\) −41.7265 −1.64044 −0.820218 0.572051i \(-0.806148\pi\)
−0.820218 + 0.572051i \(0.806148\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 2.66056 0.104436
\(650\) 0 0
\(651\) −10.3327 −0.404972
\(652\) − 22.5284i − 0.882280i
\(653\) −28.2306 −1.10475 −0.552374 0.833596i \(-0.686278\pi\)
−0.552374 + 0.833596i \(0.686278\pi\)
\(654\) −16.2403 −0.635045
\(655\) − 3.81940i − 0.149236i
\(656\) 2.84117i 0.110929i
\(657\) − 3.36227i − 0.131175i
\(658\) − 3.47996i − 0.135663i
\(659\) −38.7885 −1.51099 −0.755493 0.655156i \(-0.772603\pi\)
−0.755493 + 0.655156i \(0.772603\pi\)
\(660\) 1.35690 0.0528171
\(661\) − 31.5733i − 1.22806i −0.789284 0.614029i \(-0.789548\pi\)
0.789284 0.614029i \(-0.210452\pi\)
\(662\) −26.3424 −1.02383
\(663\) 0 0
\(664\) 2.68664 0.104262
\(665\) 3.08277i 0.119545i
\(666\) 9.26875 0.359157
\(667\) −1.64609 −0.0637368
\(668\) 12.2228i 0.472915i
\(669\) 14.0532i 0.543329i
\(670\) 6.91185i 0.267028i
\(671\) − 4.40581i − 0.170085i
\(672\) −1.04892 −0.0404629
\(673\) 44.1400 1.70147 0.850737 0.525592i \(-0.176156\pi\)
0.850737 + 0.525592i \(0.176156\pi\)
\(674\) − 14.8345i − 0.571402i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −22.8431 −0.877931 −0.438966 0.898504i \(-0.644655\pi\)
−0.438966 + 0.898504i \(0.644655\pi\)
\(678\) 16.8267i 0.646225i
\(679\) −1.64370 −0.0630792
\(680\) −1.08815 −0.0417285
\(681\) − 1.93362i − 0.0740966i
\(682\) 13.3666i 0.511833i
\(683\) 13.6866i 0.523705i 0.965108 + 0.261852i \(0.0843334\pi\)
−0.965108 + 0.261852i \(0.915667\pi\)
\(684\) − 2.93900i − 0.112376i
\(685\) 8.59717 0.328481
\(686\) −13.5308 −0.516608
\(687\) 10.9148i 0.416427i
\(688\) 4.45473 0.169835
\(689\) 0 0
\(690\) 0.692021 0.0263448
\(691\) − 24.0261i − 0.913995i −0.889468 0.456998i \(-0.848925\pi\)
0.889468 0.456998i \(-0.151075\pi\)
\(692\) −11.7168 −0.445405
\(693\) 1.42327 0.0540656
\(694\) − 0.636399i − 0.0241574i
\(695\) 2.38404i 0.0904319i
\(696\) 2.37867i 0.0901631i
\(697\) 3.09160i 0.117103i
\(698\) 17.2784 0.653999
\(699\) 21.6896 0.820377
\(700\) 1.04892i 0.0396453i
\(701\) −50.5392 −1.90884 −0.954419 0.298471i \(-0.903523\pi\)
−0.954419 + 0.298471i \(0.903523\pi\)
\(702\) 0 0
\(703\) 27.2409 1.02741
\(704\) 1.35690i 0.0511399i
\(705\) 3.31767 0.124951
\(706\) 21.7614 0.819000
\(707\) − 4.94438i − 0.185952i
\(708\) − 1.96077i − 0.0736903i
\(709\) 12.1847i 0.457604i 0.973473 + 0.228802i \(0.0734809\pi\)
−0.973473 + 0.228802i \(0.926519\pi\)
\(710\) 2.29590i 0.0861635i
\(711\) 12.3230 0.462150
\(712\) −12.8877 −0.482987
\(713\) 6.81700i 0.255299i
\(714\) −1.14138 −0.0427149
\(715\) 0 0
\(716\) 8.89738 0.332511
\(717\) − 27.2349i − 1.01711i
\(718\) −22.8412 −0.852425
\(719\) 10.8401 0.404268 0.202134 0.979358i \(-0.435212\pi\)
0.202134 + 0.979358i \(0.435212\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) 12.7593i 0.475179i
\(722\) 10.3623i 0.385644i
\(723\) − 8.21014i − 0.305339i
\(724\) 1.31229 0.0487709
\(725\) 2.37867 0.0883414
\(726\) 9.15883i 0.339916i
\(727\) −33.2489 −1.23313 −0.616567 0.787303i \(-0.711477\pi\)
−0.616567 + 0.787303i \(0.711477\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 3.36227i − 0.124443i
\(731\) 4.84740 0.179287
\(732\) −3.24698 −0.120012
\(733\) − 8.59120i − 0.317323i −0.987333 0.158662i \(-0.949282\pi\)
0.987333 0.158662i \(-0.0507179\pi\)
\(734\) − 7.78315i − 0.287281i
\(735\) − 5.89977i − 0.217616i
\(736\) 0.692021i 0.0255082i
\(737\) 9.37867 0.345468
\(738\) −2.84117 −0.104585
\(739\) 38.4131i 1.41305i 0.707689 + 0.706525i \(0.249738\pi\)
−0.707689 + 0.706525i \(0.750262\pi\)
\(740\) 9.26875 0.340726
\(741\) 0 0
\(742\) −0.697398 −0.0256023
\(743\) 21.7904i 0.799414i 0.916643 + 0.399707i \(0.130888\pi\)
−0.916643 + 0.399707i \(0.869112\pi\)
\(744\) 9.85086 0.361150
\(745\) −8.93900 −0.327500
\(746\) − 30.5133i − 1.11717i
\(747\) 2.68664i 0.0982992i
\(748\) 1.47650i 0.0539862i
\(749\) 1.92872i 0.0704739i
\(750\) −1.00000 −0.0365148
\(751\) −0.715120 −0.0260951 −0.0130475 0.999915i \(-0.504153\pi\)
−0.0130475 + 0.999915i \(0.504153\pi\)
\(752\) 3.31767i 0.120983i
\(753\) 6.57002 0.239425
\(754\) 0 0
\(755\) 8.45473 0.307699
\(756\) − 1.04892i − 0.0381488i
\(757\) 29.7894 1.08271 0.541357 0.840793i \(-0.317911\pi\)
0.541357 + 0.840793i \(0.317911\pi\)
\(758\) 6.41358 0.232952
\(759\) − 0.939001i − 0.0340836i
\(760\) − 2.93900i − 0.106609i
\(761\) − 28.4112i − 1.02990i −0.857219 0.514952i \(-0.827810\pi\)
0.857219 0.514952i \(-0.172190\pi\)
\(762\) 5.06100i 0.183341i
\(763\) −17.0347 −0.616698
\(764\) −5.75063 −0.208050
\(765\) − 1.08815i − 0.0393420i
\(766\) 2.89440 0.104579
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 16.6200i 0.599333i 0.954044 + 0.299666i \(0.0968754\pi\)
−0.954044 + 0.299666i \(0.903125\pi\)
\(770\) 1.42327 0.0512911
\(771\) −13.7657 −0.495759
\(772\) 10.0339i 0.361126i
\(773\) − 10.8388i − 0.389844i −0.980819 0.194922i \(-0.937555\pi\)
0.980819 0.194922i \(-0.0624453\pi\)
\(774\) 4.45473i 0.160122i
\(775\) − 9.85086i − 0.353853i
\(776\) 1.56704 0.0562534
\(777\) 9.72215 0.348780
\(778\) 17.1250i 0.613960i
\(779\) −8.35019 −0.299177
\(780\) 0 0
\(781\) 3.11529 0.111474
\(782\) 0.753020i 0.0269280i
\(783\) −2.37867 −0.0850066
\(784\) 5.89977 0.210706
\(785\) 2.70410i 0.0965136i
\(786\) 3.81940i 0.136233i
\(787\) 4.55974i 0.162537i 0.996692 + 0.0812687i \(0.0258972\pi\)
−0.996692 + 0.0812687i \(0.974103\pi\)
\(788\) − 22.9071i − 0.816031i
\(789\) −9.54288 −0.339735
\(790\) 12.3230 0.438434
\(791\) 17.6498i 0.627555i
\(792\) −1.35690 −0.0482152
\(793\) 0 0
\(794\) −25.0103 −0.887582
\(795\) − 0.664874i − 0.0235807i
\(796\) 27.0291 0.958020
\(797\) 13.6601 0.483865 0.241933 0.970293i \(-0.422219\pi\)
0.241933 + 0.970293i \(0.422219\pi\)
\(798\) − 3.08277i − 0.109129i
\(799\) 3.61011i 0.127716i
\(800\) − 1.00000i − 0.0353553i
\(801\) − 12.8877i − 0.455364i
\(802\) 30.8689 1.09002
\(803\) −4.56225 −0.160998
\(804\) − 6.91185i − 0.243762i
\(805\) 0.725873 0.0255837
\(806\) 0 0
\(807\) 9.07069 0.319303
\(808\) 4.71379i 0.165831i
\(809\) 30.2857 1.06479 0.532395 0.846496i \(-0.321292\pi\)
0.532395 + 0.846496i \(0.321292\pi\)
\(810\) 1.00000 0.0351364
\(811\) − 26.5327i − 0.931690i −0.884866 0.465845i \(-0.845751\pi\)
0.884866 0.465845i \(-0.154249\pi\)
\(812\) 2.49502i 0.0875582i
\(813\) − 17.5579i − 0.615784i
\(814\) − 12.5767i − 0.440814i
\(815\) 22.5284 0.789136
\(816\) 1.08815 0.0380927
\(817\) 13.0925i 0.458047i
\(818\) 11.0871 0.387651
\(819\) 0 0
\(820\) −2.84117 −0.0992178
\(821\) − 48.5370i − 1.69395i −0.531630 0.846977i \(-0.678420\pi\)
0.531630 0.846977i \(-0.321580\pi\)
\(822\) −8.59717 −0.299861
\(823\) 41.2650 1.43841 0.719204 0.694799i \(-0.244507\pi\)
0.719204 + 0.694799i \(0.244507\pi\)
\(824\) − 12.1642i − 0.423760i
\(825\) 1.35690i 0.0472411i
\(826\) − 2.05669i − 0.0715613i
\(827\) 22.9004i 0.796324i 0.917315 + 0.398162i \(0.130352\pi\)
−0.917315 + 0.398162i \(0.869648\pi\)
\(828\) −0.692021 −0.0240494
\(829\) −13.9782 −0.485484 −0.242742 0.970091i \(-0.578047\pi\)
−0.242742 + 0.970091i \(0.578047\pi\)
\(830\) 2.68664i 0.0932548i
\(831\) −13.7071 −0.475494
\(832\) 0 0
\(833\) 6.41981 0.222433
\(834\) − 2.38404i − 0.0825527i
\(835\) −12.2228 −0.422988
\(836\) −3.98792 −0.137925
\(837\) 9.85086i 0.340495i
\(838\) 17.5080i 0.604802i
\(839\) − 1.35258i − 0.0466964i −0.999727 0.0233482i \(-0.992567\pi\)
0.999727 0.0233482i \(-0.00743264\pi\)
\(840\) − 1.04892i − 0.0361911i
\(841\) −23.3419 −0.804895
\(842\) −16.1274 −0.555786
\(843\) 24.3696i 0.839333i
\(844\) −10.3720 −0.357018
\(845\) 0 0
\(846\) −3.31767 −0.114064
\(847\) 9.60686i 0.330096i
\(848\) 0.664874 0.0228319
\(849\) 17.1511 0.588623
\(850\) − 1.08815i − 0.0373231i
\(851\) − 6.41417i − 0.219875i
\(852\) − 2.29590i − 0.0786561i
\(853\) − 18.0887i − 0.619347i −0.950843 0.309673i \(-0.899780\pi\)
0.950843 0.309673i \(-0.100220\pi\)
\(854\) −3.40581 −0.116545
\(855\) 2.93900 0.100512
\(856\) − 1.83877i − 0.0628480i
\(857\) 8.47889 0.289633 0.144817 0.989458i \(-0.453741\pi\)
0.144817 + 0.989458i \(0.453741\pi\)
\(858\) 0 0
\(859\) 19.8968 0.678870 0.339435 0.940630i \(-0.389764\pi\)
0.339435 + 0.940630i \(0.389764\pi\)
\(860\) 4.45473i 0.151905i
\(861\) −2.98015 −0.101563
\(862\) 33.2905 1.13388
\(863\) − 16.0989i − 0.548013i −0.961728 0.274006i \(-0.911651\pi\)
0.961728 0.274006i \(-0.0883490\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) − 11.7168i − 0.398382i
\(866\) 11.1056i 0.377384i
\(867\) −15.8159 −0.537137
\(868\) 10.3327 0.350716
\(869\) − 16.7211i − 0.567224i
\(870\) −2.37867 −0.0806443
\(871\) 0 0
\(872\) 16.2403 0.549965
\(873\) 1.56704i 0.0530363i
\(874\) −2.03385 −0.0687961
\(875\) −1.04892 −0.0354599
\(876\) 3.36227i 0.113601i
\(877\) 7.14377i 0.241228i 0.992699 + 0.120614i \(0.0384863\pi\)
−0.992699 + 0.120614i \(0.961514\pi\)
\(878\) − 12.6950i − 0.428436i
\(879\) 24.1172i 0.813453i
\(880\) −1.35690 −0.0457410
\(881\) −42.5786 −1.43451 −0.717256 0.696810i \(-0.754602\pi\)
−0.717256 + 0.696810i \(0.754602\pi\)
\(882\) 5.89977i 0.198656i
\(883\) 38.3997 1.29225 0.646126 0.763230i \(-0.276388\pi\)
0.646126 + 0.763230i \(0.276388\pi\)
\(884\) 0 0
\(885\) 1.96077 0.0659106
\(886\) − 6.90110i − 0.231847i
\(887\) 57.1943 1.92040 0.960199 0.279317i \(-0.0901079\pi\)
0.960199 + 0.279317i \(0.0901079\pi\)
\(888\) −9.26875 −0.311039
\(889\) 5.30857i 0.178044i
\(890\) − 12.8877i − 0.431996i
\(891\) − 1.35690i − 0.0454577i
\(892\) − 14.0532i − 0.470537i
\(893\) −9.75063 −0.326292
\(894\) 8.93900 0.298965
\(895\) 8.89738i 0.297407i
\(896\) 1.04892 0.0350419
\(897\) 0 0
\(898\) −3.27652 −0.109339
\(899\) − 23.4319i − 0.781497i
\(900\) 1.00000 0.0333333
\(901\) 0.723480 0.0241026
\(902\) 3.85517i 0.128363i
\(903\) 4.67264i 0.155496i
\(904\) − 16.8267i − 0.559647i
\(905\) 1.31229i 0.0436220i
\(906\) −8.45473 −0.280890
\(907\) −1.07415 −0.0356664 −0.0178332 0.999841i \(-0.505677\pi\)
−0.0178332 + 0.999841i \(0.505677\pi\)
\(908\) 1.93362i 0.0641696i
\(909\) −4.71379 −0.156347
\(910\) 0 0
\(911\) −50.9904 −1.68939 −0.844694 0.535249i \(-0.820218\pi\)
−0.844694 + 0.535249i \(0.820218\pi\)
\(912\) 2.93900i 0.0973201i
\(913\) 3.64550 0.120648
\(914\) −1.40581 −0.0465002
\(915\) − 3.24698i − 0.107342i
\(916\) − 10.9148i − 0.360636i
\(917\) 4.00623i 0.132297i
\(918\) 1.08815i 0.0359142i
\(919\) 7.82344 0.258072 0.129036 0.991640i \(-0.458812\pi\)
0.129036 + 0.991640i \(0.458812\pi\)
\(920\) −0.692021 −0.0228153
\(921\) 17.2282i 0.567688i
\(922\) −12.3773 −0.407626
\(923\) 0 0
\(924\) −1.42327 −0.0468222
\(925\) 9.26875i 0.304755i
\(926\) 31.7797 1.04435
\(927\) 12.1642 0.399525
\(928\) − 2.37867i − 0.0780835i
\(929\) − 19.5007i − 0.639796i −0.947452 0.319898i \(-0.896351\pi\)
0.947452 0.319898i \(-0.103649\pi\)
\(930\) 9.85086i 0.323022i
\(931\) 17.3394i 0.568277i
\(932\) −21.6896 −0.710467
\(933\) 25.9801 0.850552
\(934\) 36.2790i 1.18709i
\(935\) −1.47650 −0.0482867
\(936\) 0 0
\(937\) −32.0968 −1.04856 −0.524278 0.851547i \(-0.675665\pi\)
−0.524278 + 0.851547i \(0.675665\pi\)
\(938\) − 7.24996i − 0.236720i
\(939\) 7.82669 0.255414
\(940\) −3.31767 −0.108210
\(941\) 35.9181i 1.17090i 0.810710 + 0.585448i \(0.199081\pi\)
−0.810710 + 0.585448i \(0.800919\pi\)
\(942\) − 2.70410i − 0.0881044i
\(943\) 1.96615i 0.0640266i
\(944\) 1.96077i 0.0638177i
\(945\) 1.04892 0.0341213
\(946\) 6.04461 0.196527
\(947\) − 20.7700i − 0.674934i −0.941337 0.337467i \(-0.890430\pi\)
0.941337 0.337467i \(-0.109570\pi\)
\(948\) −12.3230 −0.400234
\(949\) 0 0
\(950\) 2.93900 0.0953538
\(951\) 0.391813i 0.0127054i
\(952\) 1.14138 0.0369922
\(953\) 11.7855 0.381771 0.190886 0.981612i \(-0.438864\pi\)
0.190886 + 0.981612i \(0.438864\pi\)
\(954\) 0.664874i 0.0215261i
\(955\) − 5.75063i − 0.186086i
\(956\) 27.2349i 0.880840i
\(957\) 3.22760i 0.104334i
\(958\) −5.87933 −0.189953
\(959\) −9.01772 −0.291197
\(960\) 1.00000i 0.0322749i
\(961\) −66.0393 −2.13030
\(962\) 0 0
\(963\) 1.83877 0.0592536
\(964\) 8.21014i 0.264431i
\(965\) −10.0339 −0.323001
\(966\) −0.725873 −0.0233546
\(967\) − 52.0116i − 1.67258i −0.548287 0.836290i \(-0.684720\pi\)
0.548287 0.836290i \(-0.315280\pi\)
\(968\) − 9.15883i − 0.294376i
\(969\) 3.19806i 0.102737i
\(970\) 1.56704i 0.0503146i
\(971\) −17.7144 −0.568482 −0.284241 0.958753i \(-0.591742\pi\)
−0.284241 + 0.958753i \(0.591742\pi\)
\(972\) −1.00000 −0.0320750
\(973\) − 2.50066i − 0.0801676i
\(974\) −30.6243 −0.981266
\(975\) 0 0
\(976\) 3.24698 0.103933
\(977\) − 8.16959i − 0.261368i −0.991424 0.130684i \(-0.958283\pi\)
0.991424 0.130684i \(-0.0417174\pi\)
\(978\) −22.5284 −0.720379
\(979\) −17.4873 −0.558895
\(980\) 5.89977i 0.188461i
\(981\) 16.2403i 0.518512i
\(982\) − 34.4456i − 1.09920i
\(983\) − 57.0256i − 1.81883i −0.415885 0.909417i \(-0.636528\pi\)
0.415885 0.909417i \(-0.363472\pi\)
\(984\) 2.84117 0.0905731
\(985\) 22.9071 0.729880
\(986\) − 2.58834i − 0.0824294i
\(987\) −3.47996 −0.110768
\(988\) 0 0
\(989\) 3.08277 0.0980264
\(990\) − 1.35690i − 0.0431250i
\(991\) 37.2416 1.18302 0.591509 0.806298i \(-0.298532\pi\)
0.591509 + 0.806298i \(0.298532\pi\)
\(992\) −9.85086 −0.312765
\(993\) 26.3424i 0.835951i
\(994\) − 2.40821i − 0.0763837i
\(995\) 27.0291i 0.856879i
\(996\) − 2.68664i − 0.0851296i
\(997\) −9.00059 −0.285052 −0.142526 0.989791i \(-0.545522\pi\)
−0.142526 + 0.989791i \(0.545522\pi\)
\(998\) 10.0194 0.317158
\(999\) − 9.26875i − 0.293250i
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.1351.6 6
13.5 odd 4 5070.2.a.bw.1.1 yes 3
13.8 odd 4 5070.2.a.bp.1.3 3
13.12 even 2 inner 5070.2.b.z.1351.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bp.1.3 3 13.8 odd 4
5070.2.a.bw.1.1 yes 3 13.5 odd 4
5070.2.b.z.1351.1 6 13.12 even 2 inner
5070.2.b.z.1351.6 6 1.1 even 1 trivial