Properties

Label 5070.2.b.d.1351.1
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.d.1351.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.00000i q^{11} +1.00000 q^{12} +1.00000i q^{15} +1.00000 q^{16} +6.00000 q^{17} -1.00000i q^{18} -4.00000i q^{19} +1.00000i q^{20} +4.00000 q^{22} -8.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +6.00000 q^{29} +1.00000 q^{30} +8.00000i q^{31} -1.00000i q^{32} -4.00000i q^{33} -6.00000i q^{34} -1.00000 q^{36} -10.0000i q^{37} -4.00000 q^{38} +1.00000 q^{40} +6.00000i q^{41} -4.00000 q^{43} -4.00000i q^{44} -1.00000i q^{45} +8.00000i q^{46} -1.00000 q^{48} +7.00000 q^{49} +1.00000i q^{50} -6.00000 q^{51} -10.0000 q^{53} +1.00000i q^{54} +4.00000 q^{55} +4.00000i q^{57} -6.00000i q^{58} +4.00000i q^{59} -1.00000i q^{60} -2.00000 q^{61} +8.00000 q^{62} -1.00000 q^{64} -4.00000 q^{66} +12.0000i q^{67} -6.00000 q^{68} +8.00000 q^{69} -16.0000i q^{71} +1.00000i q^{72} +2.00000i q^{73} -10.0000 q^{74} +1.00000 q^{75} +4.00000i q^{76} -16.0000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +6.00000 q^{82} +12.0000i q^{83} -6.00000i q^{85} +4.00000i q^{86} -6.00000 q^{87} -4.00000 q^{88} +10.0000i q^{89} -1.00000 q^{90} +8.00000 q^{92} -8.00000i q^{93} -4.00000 q^{95} +1.00000i q^{96} +6.00000i q^{97} -7.00000i q^{98} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{4} + 2q^{9} - 2q^{10} + 2q^{12} + 2q^{16} + 12q^{17} + 8q^{22} - 16q^{23} - 2q^{25} - 2q^{27} + 12q^{29} + 2q^{30} - 2q^{36} - 8q^{38} + 2q^{40} - 8q^{43} - 2q^{48} + 14q^{49} - 12q^{51} - 20q^{53} + 8q^{55} - 4q^{61} + 16q^{62} - 2q^{64} - 8q^{66} - 12q^{68} + 16q^{69} - 20q^{74} + 2q^{75} - 32q^{79} + 2q^{81} + 12q^{82} - 12q^{87} - 8q^{88} - 2q^{90} + 16q^{92} - 8q^{95} + O(q^{100}) \)

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 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 4.00000i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 0 0
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) − 4.00000i − 0.917663i −0.888523 0.458831i \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 1.00000 0.182574
\(31\) 8.00000i 1.43684i 0.695608 + 0.718421i \(0.255135\pi\)
−0.695608 + 0.718421i \(0.744865\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 4.00000i − 0.696311i
\(34\) − 6.00000i − 1.02899i
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 6.00000i 0.937043i 0.883452 + 0.468521i \(0.155213\pi\)
−0.883452 + 0.468521i \(0.844787\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) − 4.00000i − 0.603023i
\(45\) − 1.00000i − 0.149071i
\(46\) 8.00000i 1.17954i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −1.00000 −0.144338
\(49\) 7.00000 1.00000
\(50\) 1.00000i 0.141421i
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) − 6.00000i − 0.787839i
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) −6.00000 −0.727607
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) − 16.0000i − 1.89885i −0.313993 0.949425i \(-0.601667\pi\)
0.313993 0.949425i \(-0.398333\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) −10.0000 −1.16248
\(75\) 1.00000 0.115470
\(76\) 4.00000i 0.458831i
\(77\) 0 0
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) − 6.00000i − 0.650791i
\(86\) 4.00000i 0.431331i
\(87\) −6.00000 −0.643268
\(88\) −4.00000 −0.426401
\(89\) 10.0000i 1.06000i 0.847998 + 0.529999i \(0.177808\pi\)
−0.847998 + 0.529999i \(0.822192\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 1.00000i 0.102062i
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) − 7.00000i − 0.707107i
\(99\) 4.00000i 0.402015i
\(100\) 1.00000 0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 6.00000i 0.594089i
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 10.0000i 0.971286i
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) 10.0000i 0.957826i 0.877862 + 0.478913i \(0.158969\pi\)
−0.877862 + 0.478913i \(0.841031\pi\)
\(110\) − 4.00000i − 0.381385i
\(111\) 10.0000i 0.949158i
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 4.00000 0.374634
\(115\) 8.00000i 0.746004i
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) −5.00000 −0.454545
\(122\) 2.00000i 0.181071i
\(123\) − 6.00000i − 0.541002i
\(124\) − 8.00000i − 0.718421i
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 4.00000i 0.348155i
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 1.00000i 0.0860663i
\(136\) 6.00000i 0.514496i
\(137\) 10.0000i 0.854358i 0.904167 + 0.427179i \(0.140493\pi\)
−0.904167 + 0.427179i \(0.859507\pi\)
\(138\) − 8.00000i − 0.681005i
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −16.0000 −1.34269
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 6.00000i − 0.498273i
\(146\) 2.00000 0.165521
\(147\) −7.00000 −0.577350
\(148\) 10.0000i 0.821995i
\(149\) − 6.00000i − 0.491539i −0.969328 0.245770i \(-0.920959\pi\)
0.969328 0.245770i \(-0.0790407\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 4.00000 0.324443
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 16.0000i 1.27289i
\(159\) 10.0000 0.793052
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) − 1.00000i − 0.0785674i
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) − 6.00000i − 0.468521i
\(165\) −4.00000 −0.311400
\(166\) 12.0000 0.931381
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −6.00000 −0.460179
\(171\) − 4.00000i − 0.305888i
\(172\) 4.00000 0.304997
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 6.00000i 0.454859i
\(175\) 0 0
\(176\) 4.00000i 0.301511i
\(177\) − 4.00000i − 0.300658i
\(178\) 10.0000 0.749532
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) − 8.00000i − 0.589768i
\(185\) −10.0000 −0.735215
\(186\) −8.00000 −0.586588
\(187\) 24.0000i 1.75505i
\(188\) 0 0
\(189\) 0 0
\(190\) 4.00000i 0.290191i
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) 10.0000i 0.719816i 0.932988 + 0.359908i \(0.117192\pi\)
−0.932988 + 0.359908i \(0.882808\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 4.00000 0.284268
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) − 12.0000i − 0.846415i
\(202\) − 2.00000i − 0.140720i
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 6.00000 0.419058
\(206\) 0 0
\(207\) −8.00000 −0.556038
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 10.0000 0.686803
\(213\) 16.0000i 1.09630i
\(214\) 12.0000i 0.820303i
\(215\) 4.00000i 0.272798i
\(216\) − 1.00000i − 0.0680414i
\(217\) 0 0
\(218\) 10.0000 0.677285
\(219\) − 2.00000i − 0.135147i
\(220\) −4.00000 −0.269680
\(221\) 0 0
\(222\) 10.0000 0.671156
\(223\) − 8.00000i − 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) − 10.0000i − 0.665190i
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) − 4.00000i − 0.264906i
\(229\) 14.0000i 0.925146i 0.886581 + 0.462573i \(0.153074\pi\)
−0.886581 + 0.462573i \(0.846926\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) − 4.00000i − 0.260378i
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) 8.00000i 0.517477i 0.965947 + 0.258738i \(0.0833068\pi\)
−0.965947 + 0.258738i \(0.916693\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 18.0000i 1.15948i 0.814801 + 0.579741i \(0.196846\pi\)
−0.814801 + 0.579741i \(0.803154\pi\)
\(242\) 5.00000i 0.321412i
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) − 7.00000i − 0.447214i
\(246\) −6.00000 −0.382546
\(247\) 0 0
\(248\) −8.00000 −0.508001
\(249\) − 12.0000i − 0.760469i
\(250\) 1.00000 0.0632456
\(251\) 28.0000 1.76734 0.883672 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 0 0
\(253\) − 32.0000i − 2.01182i
\(254\) 8.00000i 0.501965i
\(255\) 6.00000i 0.375735i
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) − 4.00000i − 0.249029i
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) − 12.0000i − 0.741362i
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 4.00000 0.246183
\(265\) 10.0000i 0.614295i
\(266\) 0 0
\(267\) − 10.0000i − 0.611990i
\(268\) − 12.0000i − 0.733017i
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 1.00000 0.0608581
\(271\) 24.0000i 1.45790i 0.684569 + 0.728948i \(0.259990\pi\)
−0.684569 + 0.728948i \(0.740010\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) − 4.00000i − 0.241209i
\(276\) −8.00000 −0.481543
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) − 12.0000i − 0.719712i
\(279\) 8.00000i 0.478947i
\(280\) 0 0
\(281\) 10.0000i 0.596550i 0.954480 + 0.298275i \(0.0964112\pi\)
−0.954480 + 0.298275i \(0.903589\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 16.0000i 0.949425i
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) 0 0
\(288\) − 1.00000i − 0.0589256i
\(289\) 19.0000 1.11765
\(290\) −6.00000 −0.352332
\(291\) − 6.00000i − 0.351726i
\(292\) − 2.00000i − 0.117041i
\(293\) − 26.0000i − 1.51894i −0.650545 0.759468i \(-0.725459\pi\)
0.650545 0.759468i \(-0.274541\pi\)
\(294\) 7.00000i 0.408248i
\(295\) 4.00000 0.232889
\(296\) 10.0000 0.581238
\(297\) − 4.00000i − 0.232104i
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 0 0
\(303\) −2.00000 −0.114897
\(304\) − 4.00000i − 0.229416i
\(305\) 2.00000i 0.114520i
\(306\) − 6.00000i − 0.342997i
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) − 8.00000i − 0.454369i
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 2.00000i 0.112867i
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) − 10.0000i − 0.560772i
\(319\) 24.0000i 1.34374i
\(320\) 1.00000i 0.0559017i
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) − 24.0000i − 1.33540i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) − 10.0000i − 0.553001i
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 4.00000i 0.220193i
\(331\) 20.0000i 1.09930i 0.835395 + 0.549650i \(0.185239\pi\)
−0.835395 + 0.549650i \(0.814761\pi\)
\(332\) − 12.0000i − 0.658586i
\(333\) − 10.0000i − 0.547997i
\(334\) 8.00000 0.437741
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) −10.0000 −0.543125
\(340\) 6.00000i 0.325396i
\(341\) −32.0000 −1.73290
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) − 4.00000i − 0.215666i
\(345\) − 8.00000i − 0.430706i
\(346\) − 18.0000i − 0.967686i
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 6.00000 0.321634
\(349\) 6.00000i 0.321173i 0.987022 + 0.160586i \(0.0513385\pi\)
−0.987022 + 0.160586i \(0.948662\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) 30.0000i 1.59674i 0.602168 + 0.798369i \(0.294304\pi\)
−0.602168 + 0.798369i \(0.705696\pi\)
\(354\) −4.00000 −0.212598
\(355\) −16.0000 −0.849192
\(356\) − 10.0000i − 0.529999i
\(357\) 0 0
\(358\) 12.0000i 0.634220i
\(359\) 16.0000i 0.844448i 0.906492 + 0.422224i \(0.138750\pi\)
−0.906492 + 0.422224i \(0.861250\pi\)
\(360\) 1.00000 0.0527046
\(361\) 3.00000 0.157895
\(362\) 6.00000i 0.315353i
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) − 2.00000i − 0.104542i
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) −8.00000 −0.417029
\(369\) 6.00000i 0.312348i
\(370\) 10.0000i 0.519875i
\(371\) 0 0
\(372\) 8.00000i 0.414781i
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 24.0000 1.24101
\(375\) − 1.00000i − 0.0516398i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 4.00000i 0.205466i 0.994709 + 0.102733i \(0.0327588\pi\)
−0.994709 + 0.102733i \(0.967241\pi\)
\(380\) 4.00000 0.205196
\(381\) 8.00000 0.409852
\(382\) 0 0
\(383\) − 32.0000i − 1.63512i −0.575841 0.817562i \(-0.695325\pi\)
0.575841 0.817562i \(-0.304675\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) −4.00000 −0.203331
\(388\) − 6.00000i − 0.304604i
\(389\) 34.0000 1.72387 0.861934 0.507020i \(-0.169253\pi\)
0.861934 + 0.507020i \(0.169253\pi\)
\(390\) 0 0
\(391\) −48.0000 −2.42746
\(392\) 7.00000i 0.353553i
\(393\) −12.0000 −0.605320
\(394\) −6.00000 −0.302276
\(395\) 16.0000i 0.805047i
\(396\) − 4.00000i − 0.201008i
\(397\) 30.0000i 1.50566i 0.658217 + 0.752828i \(0.271311\pi\)
−0.658217 + 0.752828i \(0.728689\pi\)
\(398\) − 8.00000i − 0.401004i
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 34.0000i 1.69788i 0.528490 + 0.848939i \(0.322758\pi\)
−0.528490 + 0.848939i \(0.677242\pi\)
\(402\) −12.0000 −0.598506
\(403\) 0 0
\(404\) −2.00000 −0.0995037
\(405\) − 1.00000i − 0.0496904i
\(406\) 0 0
\(407\) 40.0000 1.98273
\(408\) − 6.00000i − 0.297044i
\(409\) 22.0000i 1.08783i 0.839140 + 0.543915i \(0.183059\pi\)
−0.839140 + 0.543915i \(0.816941\pi\)
\(410\) − 6.00000i − 0.296319i
\(411\) − 10.0000i − 0.493264i
\(412\) 0 0
\(413\) 0 0
\(414\) 8.00000i 0.393179i
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) − 16.0000i − 0.782586i
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) 34.0000i 1.65706i 0.559946 + 0.828529i \(0.310822\pi\)
−0.559946 + 0.828529i \(0.689178\pi\)
\(422\) 12.0000i 0.584151i
\(423\) 0 0
\(424\) − 10.0000i − 0.485643i
\(425\) −6.00000 −0.291043
\(426\) 16.0000 0.775203
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 24.0000i 1.15604i 0.816023 + 0.578020i \(0.196174\pi\)
−0.816023 + 0.578020i \(0.803826\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) 6.00000i 0.287678i
\(436\) − 10.0000i − 0.478913i
\(437\) 32.0000i 1.53077i
\(438\) −2.00000 −0.0955637
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 4.00000i 0.190693i
\(441\) 7.00000 0.333333
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) − 10.0000i − 0.474579i
\(445\) 10.0000 0.474045
\(446\) −8.00000 −0.378811
\(447\) 6.00000i 0.283790i
\(448\) 0 0
\(449\) 18.0000i 0.849473i 0.905317 + 0.424736i \(0.139633\pi\)
−0.905317 + 0.424736i \(0.860367\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) −24.0000 −1.13012
\(452\) −10.0000 −0.470360
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) − 34.0000i − 1.59045i −0.606313 0.795226i \(-0.707352\pi\)
0.606313 0.795226i \(-0.292648\pi\)
\(458\) 14.0000 0.654177
\(459\) −6.00000 −0.280056
\(460\) − 8.00000i − 0.373002i
\(461\) 18.0000i 0.838344i 0.907907 + 0.419172i \(0.137680\pi\)
−0.907907 + 0.419172i \(0.862320\pi\)
\(462\) 0 0
\(463\) 8.00000i 0.371792i 0.982569 + 0.185896i \(0.0595187\pi\)
−0.982569 + 0.185896i \(0.940481\pi\)
\(464\) 6.00000 0.278543
\(465\) −8.00000 −0.370991
\(466\) − 14.0000i − 0.648537i
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) −4.00000 −0.184115
\(473\) − 16.0000i − 0.735681i
\(474\) − 16.0000i − 0.734904i
\(475\) 4.00000i 0.183533i
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 8.00000 0.365911
\(479\) − 8.00000i − 0.365529i −0.983157 0.182765i \(-0.941495\pi\)
0.983157 0.182765i \(-0.0585046\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 18.0000 0.819878
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 6.00000 0.272446
\(486\) 1.00000i 0.0453609i
\(487\) 16.0000i 0.725029i 0.931978 + 0.362515i \(0.118082\pi\)
−0.931978 + 0.362515i \(0.881918\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) − 4.00000i − 0.180886i
\(490\) −7.00000 −0.316228
\(491\) −4.00000 −0.180517 −0.0902587 0.995918i \(-0.528769\pi\)
−0.0902587 + 0.995918i \(0.528769\pi\)
\(492\) 6.00000i 0.270501i
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 8.00000i 0.359211i
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) − 20.0000i − 0.895323i −0.894203 0.447661i \(-0.852257\pi\)
0.894203 0.447661i \(-0.147743\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) − 8.00000i − 0.357414i
\(502\) − 28.0000i − 1.24970i
\(503\) 8.00000 0.356702 0.178351 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) 0 0
\(505\) − 2.00000i − 0.0889988i
\(506\) −32.0000 −1.42257
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) − 30.0000i − 1.32973i −0.746965 0.664863i \(-0.768490\pi\)
0.746965 0.664863i \(-0.231510\pi\)
\(510\) 6.00000 0.265684
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) − 6.00000i − 0.264649i
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) − 6.00000i − 0.262613i
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) − 24.0000i − 1.04645i
\(527\) 48.0000i 2.09091i
\(528\) − 4.00000i − 0.174078i
\(529\) 41.0000 1.78261
\(530\) 10.0000 0.434372
\(531\) 4.00000i 0.173585i
\(532\) 0 0
\(533\) 0 0
\(534\) −10.0000 −0.432742
\(535\) 12.0000i 0.518805i
\(536\) −12.0000 −0.518321
\(537\) 12.0000 0.517838
\(538\) 26.0000i 1.12094i
\(539\) 28.0000i 1.20605i
\(540\) − 1.00000i − 0.0430331i
\(541\) 22.0000i 0.945854i 0.881102 + 0.472927i \(0.156803\pi\)
−0.881102 + 0.472927i \(0.843197\pi\)
\(542\) 24.0000 1.03089
\(543\) 6.00000 0.257485
\(544\) − 6.00000i − 0.257248i
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) − 10.0000i − 0.427179i
\(549\) −2.00000 −0.0853579
\(550\) −4.00000 −0.170561
\(551\) − 24.0000i − 1.02243i
\(552\) 8.00000i 0.340503i
\(553\) 0 0
\(554\) − 10.0000i − 0.424859i
\(555\) 10.0000 0.424476
\(556\) −12.0000 −0.508913
\(557\) − 18.0000i − 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 8.00000 0.338667
\(559\) 0 0
\(560\) 0 0
\(561\) − 24.0000i − 1.01328i
\(562\) 10.0000 0.421825
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) − 10.0000i − 0.420703i
\(566\) 20.0000i 0.840663i
\(567\) 0 0
\(568\) 16.0000 0.671345
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) − 4.00000i − 0.167542i
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 8.00000 0.333623
\(576\) −1.00000 −0.0416667
\(577\) − 42.0000i − 1.74848i −0.485491 0.874241i \(-0.661359\pi\)
0.485491 0.874241i \(-0.338641\pi\)
\(578\) − 19.0000i − 0.790296i
\(579\) − 10.0000i − 0.415586i
\(580\) 6.00000i 0.249136i
\(581\) 0 0
\(582\) −6.00000 −0.248708
\(583\) − 40.0000i − 1.65663i
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 7.00000 0.288675
\(589\) 32.0000 1.31854
\(590\) − 4.00000i − 0.164677i
\(591\) 6.00000i 0.246807i
\(592\) − 10.0000i − 0.410997i
\(593\) 34.0000i 1.39621i 0.715994 + 0.698106i \(0.245974\pi\)
−0.715994 + 0.698106i \(0.754026\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 6.00000i 0.245770i
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) 12.0000i 0.488678i
\(604\) 0 0
\(605\) 5.00000i 0.203279i
\(606\) 2.00000i 0.0812444i
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) 10.0000i 0.403896i 0.979396 + 0.201948i \(0.0647272\pi\)
−0.979396 + 0.201948i \(0.935273\pi\)
\(614\) −12.0000 −0.484281
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) − 42.0000i − 1.69086i −0.534089 0.845428i \(-0.679345\pi\)
0.534089 0.845428i \(-0.320655\pi\)
\(618\) 0 0
\(619\) − 4.00000i − 0.160774i −0.996764 0.0803868i \(-0.974384\pi\)
0.996764 0.0803868i \(-0.0256155\pi\)
\(620\) −8.00000 −0.321288
\(621\) 8.00000 0.321029
\(622\) − 24.0000i − 0.962312i
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 22.0000i 0.879297i
\(627\) −16.0000 −0.638978
\(628\) 2.00000 0.0798087
\(629\) − 60.0000i − 2.39236i
\(630\) 0 0
\(631\) − 16.0000i − 0.636950i −0.947931 0.318475i \(-0.896829\pi\)
0.947931 0.318475i \(-0.103171\pi\)
\(632\) − 16.0000i − 0.636446i
\(633\) 12.0000 0.476957
\(634\) 2.00000 0.0794301
\(635\) 8.00000i 0.317470i
\(636\) −10.0000 −0.396526
\(637\) 0 0
\(638\) 24.0000 0.950169
\(639\) − 16.0000i − 0.632950i
\(640\) 1.00000 0.0395285
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) − 12.0000i − 0.473602i
\(643\) − 4.00000i − 0.157745i −0.996885 0.0788723i \(-0.974868\pi\)
0.996885 0.0788723i \(-0.0251319\pi\)
\(644\) 0 0
\(645\) − 4.00000i − 0.157500i
\(646\) −24.0000 −0.944267
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) − 4.00000i − 0.156652i
\(653\) −34.0000 −1.33052 −0.665261 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) −10.0000 −0.391031
\(655\) − 12.0000i − 0.468879i
\(656\) 6.00000i 0.234261i
\(657\) 2.00000i 0.0780274i
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 4.00000 0.155700
\(661\) 14.0000i 0.544537i 0.962221 + 0.272268i \(0.0877739\pi\)
−0.962221 + 0.272268i \(0.912226\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) −48.0000 −1.85857
\(668\) − 8.00000i − 0.309529i
\(669\) 8.00000i 0.309298i
\(670\) − 12.0000i − 0.463600i
\(671\) − 8.00000i − 0.308837i
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 2.00000i 0.0770371i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 10.0000i 0.384048i
\(679\) 0 0
\(680\) 6.00000 0.230089
\(681\) 4.00000i 0.153280i
\(682\) 32.0000i 1.22534i
\(683\) 28.0000i 1.07139i 0.844411 + 0.535695i \(0.179950\pi\)
−0.844411 + 0.535695i \(0.820050\pi\)
\(684\) 4.00000i 0.152944i
\(685\) 10.0000 0.382080
\(686\) 0 0
\(687\) − 14.0000i − 0.534133i
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) −8.00000 −0.304555
\(691\) 12.0000i 0.456502i 0.973602 + 0.228251i \(0.0733006\pi\)
−0.973602 + 0.228251i \(0.926699\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) − 20.0000i − 0.759190i
\(695\) − 12.0000i − 0.455186i
\(696\) − 6.00000i − 0.227429i
\(697\) 36.0000i 1.36360i
\(698\) 6.00000 0.227103
\(699\) −14.0000 −0.529529
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) −40.0000 −1.50863
\(704\) − 4.00000i − 0.150756i
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 0 0
\(708\) 4.00000i 0.150329i
\(709\) − 50.0000i − 1.87779i −0.344204 0.938895i \(-0.611851\pi\)
0.344204 0.938895i \(-0.388149\pi\)
\(710\) 16.0000i 0.600469i
\(711\) −16.0000 −0.600047
\(712\) −10.0000 −0.374766
\(713\) − 64.0000i − 2.39682i
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) − 8.00000i − 0.298765i
\(718\) 16.0000 0.597115
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) 0 0
\(722\) − 3.00000i − 0.111648i
\(723\) − 18.0000i − 0.669427i
\(724\) 6.00000 0.222988
\(725\) −6.00000 −0.222834
\(726\) − 5.00000i − 0.185567i
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 2.00000i − 0.0740233i
\(731\) −24.0000 −0.887672
\(732\) −2.00000 −0.0739221
\(733\) 50.0000i 1.84679i 0.383849 + 0.923396i \(0.374598\pi\)
−0.383849 + 0.923396i \(0.625402\pi\)
\(734\) 24.0000i 0.885856i
\(735\) 7.00000i 0.258199i
\(736\) 8.00000i 0.294884i
\(737\) −48.0000 −1.76810
\(738\) 6.00000 0.220863
\(739\) 20.0000i 0.735712i 0.929883 + 0.367856i \(0.119908\pi\)
−0.929883 + 0.367856i \(0.880092\pi\)
\(740\) 10.0000 0.367607
\(741\) 0 0
\(742\) 0 0
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 8.00000 0.293294
\(745\) −6.00000 −0.219823
\(746\) − 22.0000i − 0.805477i
\(747\) 12.0000i 0.439057i
\(748\) − 24.0000i − 0.877527i
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) −28.0000 −1.02038
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) 4.00000 0.145287
\(759\) 32.0000i 1.16153i
\(760\) − 4.00000i − 0.145095i
\(761\) 42.0000i 1.52250i 0.648459 + 0.761249i \(0.275414\pi\)
−0.648459 + 0.761249i \(0.724586\pi\)
\(762\) − 8.00000i − 0.289809i
\(763\) 0 0
\(764\) 0 0
\(765\) − 6.00000i − 0.216930i
\(766\) −32.0000 −1.15621
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) − 34.0000i − 1.22607i −0.790055 0.613036i \(-0.789948\pi\)
0.790055 0.613036i \(-0.210052\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) − 10.0000i − 0.359908i
\(773\) − 38.0000i − 1.36677i −0.730061 0.683383i \(-0.760508\pi\)
0.730061 0.683383i \(-0.239492\pi\)
\(774\) 4.00000i 0.143777i
\(775\) − 8.00000i − 0.287368i
\(776\) −6.00000 −0.215387
\(777\) 0 0
\(778\) − 34.0000i − 1.21896i
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 64.0000 2.29010
\(782\) 48.0000i 1.71648i
\(783\) −6.00000 −0.214423
\(784\) 7.00000 0.250000
\(785\) 2.00000i 0.0713831i
\(786\) 12.0000i 0.428026i
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) 6.00000i 0.213741i
\(789\) −24.0000 −0.854423
\(790\) 16.0000 0.569254
\(791\) 0 0
\(792\) −4.00000 −0.142134
\(793\) 0 0
\(794\) 30.0000 1.06466
\(795\) − 10.0000i − 0.354663i
\(796\) −8.00000 −0.283552
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000i 0.0353553i
\(801\) 10.0000i 0.353333i
\(802\) 34.0000 1.20058
\(803\) −8.00000 −0.282314
\(804\) 12.0000i 0.423207i
\(805\) 0 0
\(806\) 0 0
\(807\) 26.0000 0.915243
\(808\) 2.00000i 0.0703598i
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) −1.00000 −0.0351364
\(811\) − 12.0000i − 0.421377i −0.977553 0.210688i \(-0.932429\pi\)
0.977553 0.210688i \(-0.0675706\pi\)
\(812\) 0 0
\(813\) − 24.0000i − 0.841717i
\(814\) − 40.0000i − 1.40200i
\(815\) 4.00000 0.140114
\(816\) −6.00000 −0.210042
\(817\) 16.0000i 0.559769i
\(818\) 22.0000 0.769212
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) − 6.00000i − 0.209401i −0.994504 0.104701i \(-0.966612\pi\)
0.994504 0.104701i \(-0.0333885\pi\)
\(822\) −10.0000 −0.348790
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 0 0
\(825\) 4.00000i 0.139262i
\(826\) 0 0
\(827\) − 36.0000i − 1.25184i −0.779886 0.625921i \(-0.784723\pi\)
0.779886 0.625921i \(-0.215277\pi\)
\(828\) 8.00000 0.278019
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) − 12.0000i − 0.416526i
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) 42.0000 1.45521
\(834\) 12.0000i 0.415526i
\(835\) 8.00000 0.276851
\(836\) −16.0000 −0.553372
\(837\) − 8.00000i − 0.276520i
\(838\) 4.00000i 0.138178i
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 34.0000 1.17172
\(843\) − 10.0000i − 0.344418i
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −10.0000 −0.343401
\(849\) 20.0000 0.686398
\(850\) 6.00000i 0.205798i
\(851\) 80.0000i 2.74236i
\(852\) − 16.0000i − 0.548151i
\(853\) − 26.0000i − 0.890223i −0.895475 0.445112i \(-0.853164\pi\)
0.895475 0.445112i \(-0.146836\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) − 12.0000i − 0.410152i
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) − 4.00000i − 0.136399i
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) − 18.0000i − 0.612018i
\(866\) 34.0000i 1.15537i
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) − 64.0000i − 2.17105i
\(870\) 6.00000 0.203419
\(871\) 0 0
\(872\) −10.0000 −0.338643
\(873\) 6.00000i 0.203069i
\(874\) 32.0000 1.08242
\(875\) 0 0
\(876\) 2.00000i 0.0675737i
\(877\) − 14.0000i − 0.472746i −0.971662 0.236373i \(-0.924041\pi\)
0.971662 0.236373i \(-0.0759588\pi\)
\(878\) − 40.0000i − 1.34993i
\(879\) 26.0000i 0.876958i
\(880\) 4.00000 0.134840
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) − 7.00000i − 0.235702i
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) −4.00000 −0.134459
\(886\) − 36.0000i − 1.20944i
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) −10.0000 −0.335578
\(889\) 0 0
\(890\) − 10.0000i − 0.335201i
\(891\) 4.00000i 0.134005i
\(892\) 8.00000i 0.267860i
\(893\) 0 0
\(894\) 6.00000 0.200670
\(895\) 12.0000i 0.401116i
\(896\) 0 0
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) 48.0000i 1.60089i
\(900\) 1.00000 0.0333333
\(901\) −60.0000 −1.99889
\(902\) 24.0000i 0.799113i
\(903\) 0 0
\(904\) 10.0000i 0.332595i
\(905\) 6.00000i 0.199447i
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 4.00000i 0.132745i
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 4.00000i 0.132453i
\(913\) −48.0000 −1.58857
\(914\) −34.0000 −1.12462
\(915\) − 2.00000i − 0.0661180i
\(916\) − 14.0000i − 0.462573i
\(917\) 0 0
\(918\) 6.00000i 0.198030i
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) −8.00000 −0.263752
\(921\) 12.0000i 0.395413i
\(922\) 18.0000 0.592798
\(923\) 0 0
\(924\) 0 0
\(925\) 10.0000i 0.328798i
\(926\) 8.00000 0.262896
\(927\) 0 0
\(928\) − 6.00000i − 0.196960i
\(929\) 14.0000i 0.459325i 0.973270 + 0.229663i \(0.0737623\pi\)
−0.973270 + 0.229663i \(0.926238\pi\)
\(930\) 8.00000i 0.262330i
\(931\) − 28.0000i − 0.917663i
\(932\) −14.0000 −0.458585
\(933\) −24.0000 −0.785725
\(934\) 12.0000i 0.392652i
\(935\) 24.0000 0.784884
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) 22.0000 0.717943
\(940\) 0 0
\(941\) − 30.0000i − 0.977972i −0.872292 0.488986i \(-0.837367\pi\)
0.872292 0.488986i \(-0.162633\pi\)
\(942\) − 2.00000i − 0.0651635i
\(943\) − 48.0000i − 1.56310i
\(944\) 4.00000i 0.130189i
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) 52.0000i 1.68977i 0.534946 + 0.844886i \(0.320332\pi\)
−0.534946 + 0.844886i \(0.679668\pi\)
\(948\) −16.0000 −0.519656
\(949\) 0 0
\(950\) 4.00000 0.129777
\(951\) − 2.00000i − 0.0648544i
\(952\) 0 0
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) 10.0000i 0.323762i
\(955\) 0 0
\(956\) − 8.00000i − 0.258738i
\(957\) − 24.0000i − 0.775810i
\(958\) −8.00000 −0.258468
\(959\) 0 0
\(960\) − 1.00000i − 0.0322749i
\(961\) −33.0000 −1.06452
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) − 18.0000i − 0.579741i
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) − 16.0000i − 0.514525i −0.966342 0.257263i \(-0.917179\pi\)
0.966342 0.257263i \(-0.0828206\pi\)
\(968\) − 5.00000i − 0.160706i
\(969\) 24.0000i 0.770991i
\(970\) − 6.00000i − 0.192648i
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 30.0000i 0.959785i 0.877327 + 0.479893i \(0.159324\pi\)
−0.877327 + 0.479893i \(0.840676\pi\)
\(978\) −4.00000 −0.127906
\(979\) −40.0000 −1.27841
\(980\) 7.00000i 0.223607i
\(981\) 10.0000i 0.319275i
\(982\) 4.00000i 0.127645i
\(983\) 8.00000i 0.255160i 0.991828 + 0.127580i \(0.0407210\pi\)
−0.991828 + 0.127580i \(0.959279\pi\)
\(984\) 6.00000 0.191273
\(985\) −6.00000 −0.191176
\(986\) − 36.0000i − 1.14647i
\(987\) 0 0
\(988\) 0 0
\(989\) 32.0000 1.01754
\(990\) − 4.00000i − 0.127128i
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 8.00000 0.254000
\(993\) − 20.0000i − 0.634681i
\(994\) 0 0
\(995\) − 8.00000i − 0.253617i
\(996\) 12.0000i 0.380235i
\(997\) −58.0000 −1.83688 −0.918439 0.395562i \(-0.870550\pi\)
−0.918439 + 0.395562i \(0.870550\pi\)
\(998\) −20.0000 −0.633089
\(999\) 10.0000i 0.316386i
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.d.1351.1 2
13.5 odd 4 5070.2.a.a.1.1 1
13.8 odd 4 390.2.a.f.1.1 1
13.12 even 2 inner 5070.2.b.d.1351.2 2
39.8 even 4 1170.2.a.a.1.1 1
52.47 even 4 3120.2.a.w.1.1 1
65.8 even 4 1950.2.e.g.1249.1 2
65.34 odd 4 1950.2.a.k.1.1 1
65.47 even 4 1950.2.e.g.1249.2 2
156.47 odd 4 9360.2.a.p.1.1 1
195.8 odd 4 5850.2.e.e.5149.2 2
195.47 odd 4 5850.2.e.e.5149.1 2
195.164 even 4 5850.2.a.bo.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.f.1.1 1 13.8 odd 4
1170.2.a.a.1.1 1 39.8 even 4
1950.2.a.k.1.1 1 65.34 odd 4
1950.2.e.g.1249.1 2 65.8 even 4
1950.2.e.g.1249.2 2 65.47 even 4
3120.2.a.w.1.1 1 52.47 even 4
5070.2.a.a.1.1 1 13.5 odd 4
5070.2.b.d.1351.1 2 1.1 even 1 trivial
5070.2.b.d.1351.2 2 13.12 even 2 inner
5850.2.a.bo.1.1 1 195.164 even 4
5850.2.e.e.5149.1 2 195.47 odd 4
5850.2.e.e.5149.2 2 195.8 odd 4
9360.2.a.p.1.1 1 156.47 odd 4