Properties

Label 3450.2.d.m.2899.1
Level $3450$
Weight $2$
Character 3450.2899
Analytic conductor $27.548$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.5483886973\)
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 690)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2899.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3450.2899
Dual form 3450.2.d.m.2899.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{12} +4.00000i q^{13} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +8.00000 q^{19} +2.00000i q^{22} -1.00000i q^{23} -1.00000 q^{24} +4.00000 q^{26} -1.00000i q^{27} -4.00000 q^{29} -1.00000i q^{32} -2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} -8.00000i q^{38} -4.00000 q^{39} -2.00000 q^{41} +2.00000i q^{43} +2.00000 q^{44} -1.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +6.00000 q^{51} -4.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +8.00000i q^{57} +4.00000i q^{58} -8.00000 q^{59} +2.00000 q^{61} -1.00000 q^{64} -2.00000 q^{66} +6.00000i q^{67} +6.00000i q^{68} +1.00000 q^{69} +10.0000 q^{71} -1.00000i q^{72} -2.00000i q^{73} +2.00000 q^{74} -8.00000 q^{76} +4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +2.00000i q^{82} +8.00000i q^{83} +2.00000 q^{86} -4.00000i q^{87} -2.00000i q^{88} +12.0000 q^{89} +1.00000i q^{92} +12.0000 q^{94} +1.00000 q^{96} +16.0000i q^{97} -7.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} - 4q^{11} + 2q^{16} + 16q^{19} - 2q^{24} + 8q^{26} - 8q^{29} - 12q^{34} + 2q^{36} - 8q^{39} - 4q^{41} + 4q^{44} - 2q^{46} + 14q^{49} + 12q^{51} - 2q^{54} - 16q^{59} + 4q^{61} - 2q^{64} - 4q^{66} + 2q^{69} + 20q^{71} + 4q^{74} - 16q^{76} + 16q^{79} + 2q^{81} + 4q^{86} + 24q^{89} + 24q^{94} + 2q^{96} + 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.00000i 0.426401i
\(23\) − 1.00000i − 0.208514i
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 2.00000i − 0.348155i
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) − 8.00000i − 1.29777i
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) − 4.00000i − 0.554700i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000i 1.05963i
\(58\) 4.00000i 0.525226i
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) 6.00000i 0.733017i 0.930415 + 0.366508i \(0.119447\pi\)
−0.930415 + 0.366508i \(0.880553\pi\)
\(68\) 6.00000i 0.727607i
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) 4.00000i 0.452911i
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000i 0.220863i
\(83\) 8.00000i 0.878114i 0.898459 + 0.439057i \(0.144687\pi\)
−0.898459 + 0.439057i \(0.855313\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) − 4.00000i − 0.428845i
\(88\) − 2.00000i − 0.213201i
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000i 0.104257i
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 16.0000i 1.62455i 0.583272 + 0.812277i \(0.301772\pi\)
−0.583272 + 0.812277i \(0.698228\pi\)
\(98\) − 7.00000i − 0.707107i
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) − 6.00000i − 0.594089i
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 18.0000i 1.69330i 0.532152 + 0.846649i \(0.321383\pi\)
−0.532152 + 0.846649i \(0.678617\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) − 4.00000i − 0.369800i
\(118\) 8.00000i 0.736460i
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) − 2.00000i − 0.181071i
\(123\) − 2.00000i − 0.180334i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 2.00000i − 0.177471i −0.996055 0.0887357i \(-0.971717\pi\)
0.996055 0.0887357i \(-0.0282826\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 2.00000i 0.174078i
\(133\) 0 0
\(134\) 6.00000 0.518321
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) − 2.00000i − 0.170872i −0.996344 0.0854358i \(-0.972772\pi\)
0.996344 0.0854358i \(-0.0272282\pi\)
\(138\) − 1.00000i − 0.0851257i
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) − 10.0000i − 0.839181i
\(143\) − 8.00000i − 0.668994i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 7.00000i 0.577350i
\(148\) − 2.00000i − 0.164399i
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 8.00000i 0.648886i
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) − 8.00000i − 0.636446i
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) − 1.00000i − 0.0785674i
\(163\) − 16.0000i − 1.25322i −0.779334 0.626608i \(-0.784443\pi\)
0.779334 0.626608i \(-0.215557\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 16.0000i 1.23812i 0.785345 + 0.619059i \(0.212486\pi\)
−0.785345 + 0.619059i \(0.787514\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) − 2.00000i − 0.152499i
\(173\) − 10.0000i − 0.760286i −0.924928 0.380143i \(-0.875875\pi\)
0.924928 0.380143i \(-0.124125\pi\)
\(174\) −4.00000 −0.303239
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) − 8.00000i − 0.601317i
\(178\) − 12.0000i − 0.899438i
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 2.00000i 0.147844i
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 12.0000i 0.877527i
\(188\) − 12.0000i − 0.875190i
\(189\) 0 0
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 26.0000i 1.87152i 0.352636 + 0.935760i \(0.385285\pi\)
−0.352636 + 0.935760i \(0.614715\pi\)
\(194\) 16.0000 1.14873
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) − 2.00000i − 0.142134i
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −6.00000 −0.423207
\(202\) − 12.0000i − 0.844317i
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 0 0
\(207\) 1.00000i 0.0695048i
\(208\) 4.00000i 0.277350i
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 10.0000i 0.685189i
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) − 10.0000i − 0.677285i
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) 2.00000i 0.134231i
\(223\) − 2.00000i − 0.133930i −0.997755 0.0669650i \(-0.978668\pi\)
0.997755 0.0669650i \(-0.0213316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) − 28.0000i − 1.85843i −0.369546 0.929213i \(-0.620487\pi\)
0.369546 0.929213i \(-0.379513\pi\)
\(228\) − 8.00000i − 0.529813i
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 4.00000i − 0.262613i
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) 8.00000 0.520756
\(237\) 8.00000i 0.519656i
\(238\) 0 0
\(239\) 14.0000 0.905585 0.452792 0.891616i \(-0.350428\pi\)
0.452792 + 0.891616i \(0.350428\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 7.00000i 0.449977i
\(243\) 1.00000i 0.0641500i
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) 32.0000i 2.03611i
\(248\) 0 0
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 0 0
\(253\) 2.00000i 0.125739i
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 22.0000i − 1.37232i −0.727450 0.686161i \(-0.759294\pi\)
0.727450 0.686161i \(-0.240706\pi\)
\(258\) 2.00000i 0.124515i
\(259\) 0 0
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) 4.00000i 0.247121i
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000i 0.734388i
\(268\) − 6.00000i − 0.366508i
\(269\) −8.00000 −0.487769 −0.243884 0.969804i \(-0.578422\pi\)
−0.243884 + 0.969804i \(0.578422\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 28.0000i 1.68236i 0.540758 + 0.841178i \(0.318138\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 4.00000i 0.239904i
\(279\) 0 0
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 12.0000i 0.714590i
\(283\) 14.0000i 0.832214i 0.909316 + 0.416107i \(0.136606\pi\)
−0.909316 + 0.416107i \(0.863394\pi\)
\(284\) −10.0000 −0.593391
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) 2.00000i 0.117041i
\(293\) − 14.0000i − 0.817889i −0.912559 0.408944i \(-0.865897\pi\)
0.912559 0.408944i \(-0.134103\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 2.00000i 0.116052i
\(298\) 14.0000i 0.810998i
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 0 0
\(302\) 4.00000i 0.230174i
\(303\) 12.0000i 0.689382i
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) − 28.0000i − 1.59804i −0.601302 0.799022i \(-0.705351\pi\)
0.601302 0.799022i \(-0.294649\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 26.0000 1.47432 0.737162 0.675716i \(-0.236165\pi\)
0.737162 + 0.675716i \(0.236165\pi\)
\(312\) − 4.00000i − 0.226455i
\(313\) 28.0000i 1.58265i 0.611393 + 0.791327i \(0.290609\pi\)
−0.611393 + 0.791327i \(0.709391\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) − 48.0000i − 2.67079i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) 10.0000i 0.553001i
\(328\) − 2.00000i − 0.110432i
\(329\) 0 0
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) − 8.00000i − 0.439057i
\(333\) − 2.00000i − 0.109599i
\(334\) 16.0000 0.875481
\(335\) 0 0
\(336\) 0 0
\(337\) 28.0000i 1.52526i 0.646837 + 0.762629i \(0.276092\pi\)
−0.646837 + 0.762629i \(0.723908\pi\)
\(338\) 3.00000i 0.163178i
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) 8.00000i 0.432590i
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) 4.00000i 0.214423i
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 2.00000i 0.106600i
\(353\) 10.0000i 0.532246i 0.963939 + 0.266123i \(0.0857428\pi\)
−0.963939 + 0.266123i \(0.914257\pi\)
\(354\) −8.00000 −0.425195
\(355\) 0 0
\(356\) −12.0000 −0.635999
\(357\) 0 0
\(358\) 16.0000i 0.845626i
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) − 10.0000i − 0.525588i
\(363\) − 7.00000i − 0.367405i
\(364\) 0 0
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) − 16.0000i − 0.835193i −0.908633 0.417597i \(-0.862873\pi\)
0.908633 0.417597i \(-0.137127\pi\)
\(368\) − 1.00000i − 0.0521286i
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 6.00000i 0.310668i 0.987862 + 0.155334i \(0.0496454\pi\)
−0.987862 + 0.155334i \(0.950355\pi\)
\(374\) 12.0000 0.620505
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) − 16.0000i − 0.824042i
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) − 24.0000i − 1.22795i
\(383\) − 32.0000i − 1.63512i −0.575841 0.817562i \(-0.695325\pi\)
0.575841 0.817562i \(-0.304675\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 26.0000 1.32337
\(387\) − 2.00000i − 0.101666i
\(388\) − 16.0000i − 0.812277i
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 7.00000i 0.353553i
\(393\) − 4.00000i − 0.201773i
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) − 28.0000i − 1.40528i −0.711546 0.702640i \(-0.752005\pi\)
0.711546 0.702640i \(-0.247995\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) 0 0
\(400\) 0 0
\(401\) −8.00000 −0.399501 −0.199750 0.979847i \(-0.564013\pi\)
−0.199750 + 0.979847i \(0.564013\pi\)
\(402\) 6.00000i 0.299253i
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) 0 0
\(407\) − 4.00000i − 0.198273i
\(408\) 6.00000i 0.297044i
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 0 0
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) − 4.00000i − 0.195881i
\(418\) 16.0000i 0.782586i
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) − 4.00000i − 0.194717i
\(423\) − 12.0000i − 0.583460i
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 10.0000 0.484502
\(427\) 0 0
\(428\) − 12.0000i − 0.580042i
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 16.0000i 0.768911i 0.923144 + 0.384455i \(0.125611\pi\)
−0.923144 + 0.384455i \(0.874389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) − 8.00000i − 0.382692i
\(438\) − 2.00000i − 0.0955637i
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) − 24.0000i − 1.14156i
\(443\) − 4.00000i − 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) − 14.0000i − 0.662177i
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) − 18.0000i − 0.846649i
\(453\) − 4.00000i − 0.187936i
\(454\) −28.0000 −1.31411
\(455\) 0 0
\(456\) −8.00000 −0.374634
\(457\) 8.00000i 0.374224i 0.982339 + 0.187112i \(0.0599128\pi\)
−0.982339 + 0.187112i \(0.940087\pi\)
\(458\) − 10.0000i − 0.467269i
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) − 26.0000i − 1.20832i −0.796862 0.604161i \(-0.793508\pi\)
0.796862 0.604161i \(-0.206492\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 24.0000i 1.11059i 0.831654 + 0.555294i \(0.187394\pi\)
−0.831654 + 0.555294i \(0.812606\pi\)
\(468\) 4.00000i 0.184900i
\(469\) 0 0
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) − 8.00000i − 0.368230i
\(473\) − 4.00000i − 0.183920i
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000i 0.274721i
\(478\) − 14.0000i − 0.640345i
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 6.00000i 0.273293i
\(483\) 0 0
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 26.0000i 1.17817i 0.808070 + 0.589086i \(0.200512\pi\)
−0.808070 + 0.589086i \(0.799488\pi\)
\(488\) 2.00000i 0.0905357i
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 2.00000i 0.0901670i
\(493\) 24.0000i 1.08091i
\(494\) 32.0000 1.43975
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 8.00000i 0.358489i
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) − 2.00000i − 0.0892644i
\(503\) − 16.0000i − 0.713405i −0.934218 0.356702i \(-0.883901\pi\)
0.934218 0.356702i \(-0.116099\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) − 3.00000i − 0.133235i
\(508\) 2.00000i 0.0887357i
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) − 8.00000i − 0.353209i
\(514\) −22.0000 −0.970378
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) − 24.0000i − 1.05552i
\(518\) 0 0
\(519\) 10.0000 0.438951
\(520\) 0 0
\(521\) −20.0000 −0.876216 −0.438108 0.898922i \(-0.644351\pi\)
−0.438108 + 0.898922i \(0.644351\pi\)
\(522\) − 4.00000i − 0.175075i
\(523\) 30.0000i 1.31181i 0.754844 + 0.655904i \(0.227712\pi\)
−0.754844 + 0.655904i \(0.772288\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) − 2.00000i − 0.0870388i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) − 8.00000i − 0.346518i
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) −6.00000 −0.259161
\(537\) − 16.0000i − 0.690451i
\(538\) 8.00000i 0.344904i
\(539\) −14.0000 −0.603023
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) − 24.0000i − 1.03089i
\(543\) 10.0000i 0.429141i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) − 4.00000i − 0.171028i −0.996337 0.0855138i \(-0.972747\pi\)
0.996337 0.0855138i \(-0.0272532\pi\)
\(548\) 2.00000i 0.0854358i
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −32.0000 −1.36325
\(552\) 1.00000i 0.0425628i
\(553\) 0 0
\(554\) 28.0000 1.18961
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) − 38.0000i − 1.61011i −0.593199 0.805056i \(-0.702135\pi\)
0.593199 0.805056i \(-0.297865\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) − 12.0000i − 0.506189i
\(563\) − 24.0000i − 1.01148i −0.862686 0.505740i \(-0.831220\pi\)
0.862686 0.505740i \(-0.168780\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) 14.0000 0.588464
\(567\) 0 0
\(568\) 10.0000i 0.419591i
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 8.00000i 0.334497i
\(573\) 24.0000i 1.00261i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 38.0000i − 1.58196i −0.611842 0.790980i \(-0.709571\pi\)
0.611842 0.790980i \(-0.290429\pi\)
\(578\) 19.0000i 0.790296i
\(579\) −26.0000 −1.08052
\(580\) 0 0
\(581\) 0 0
\(582\) 16.0000i 0.663221i
\(583\) 12.0000i 0.496989i
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) − 4.00000i − 0.165098i −0.996587 0.0825488i \(-0.973694\pi\)
0.996587 0.0825488i \(-0.0263060\pi\)
\(588\) − 7.00000i − 0.288675i
\(589\) 0 0
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 2.00000i 0.0821995i
\(593\) 6.00000i 0.246390i 0.992382 + 0.123195i \(0.0393141\pi\)
−0.992382 + 0.123195i \(0.960686\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 14.0000 0.573462
\(597\) 16.0000i 0.654836i
\(598\) − 4.00000i − 0.163572i
\(599\) 34.0000 1.38920 0.694601 0.719395i \(-0.255581\pi\)
0.694601 + 0.719395i \(0.255581\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) − 6.00000i − 0.244339i
\(604\) 4.00000 0.162758
\(605\) 0 0
\(606\) 12.0000 0.487467
\(607\) 10.0000i 0.405887i 0.979190 + 0.202944i \(0.0650509\pi\)
−0.979190 + 0.202944i \(0.934949\pi\)
\(608\) − 8.00000i − 0.324443i
\(609\) 0 0
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) − 6.00000i − 0.242536i
\(613\) − 30.0000i − 1.21169i −0.795583 0.605844i \(-0.792835\pi\)
0.795583 0.605844i \(-0.207165\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 0 0
\(619\) −32.0000 −1.28619 −0.643094 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) − 26.0000i − 1.04251i
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) 28.0000 1.11911
\(627\) − 16.0000i − 0.638978i
\(628\) − 10.0000i − 0.399043i
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 8.00000i 0.318223i
\(633\) 4.00000i 0.158986i
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 28.0000i 1.10940i
\(638\) − 8.00000i − 0.316723i
\(639\) −10.0000 −0.395594
\(640\) 0 0
\(641\) −4.00000 −0.157991 −0.0789953 0.996875i \(-0.525171\pi\)
−0.0789953 + 0.996875i \(0.525171\pi\)
\(642\) 12.0000i 0.473602i
\(643\) − 10.0000i − 0.394362i −0.980367 0.197181i \(-0.936821\pi\)
0.980367 0.197181i \(-0.0631786\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −48.0000 −1.88853
\(647\) 28.0000i 1.10079i 0.834903 + 0.550397i \(0.185524\pi\)
−0.834903 + 0.550397i \(0.814476\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 16.0000i 0.626608i
\(653\) 2.00000i 0.0782660i 0.999234 + 0.0391330i \(0.0124596\pi\)
−0.999234 + 0.0391330i \(0.987540\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 2.00000i 0.0780274i
\(658\) 0 0
\(659\) 26.0000 1.01282 0.506408 0.862294i \(-0.330973\pi\)
0.506408 + 0.862294i \(0.330973\pi\)
\(660\) 0 0
\(661\) −50.0000 −1.94477 −0.972387 0.233373i \(-0.925024\pi\)
−0.972387 + 0.233373i \(0.925024\pi\)
\(662\) − 28.0000i − 1.08825i
\(663\) 24.0000i 0.932083i
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 4.00000i 0.154881i
\(668\) − 16.0000i − 0.619059i
\(669\) 2.00000 0.0773245
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) − 10.0000i − 0.385472i −0.981251 0.192736i \(-0.938264\pi\)
0.981251 0.192736i \(-0.0617360\pi\)
\(674\) 28.0000 1.07852
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 10.0000i 0.384331i 0.981363 + 0.192166i \(0.0615511\pi\)
−0.981363 + 0.192166i \(0.938449\pi\)
\(678\) 18.0000i 0.691286i
\(679\) 0 0
\(680\) 0 0
\(681\) 28.0000 1.07296
\(682\) 0 0
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) 8.00000 0.305888
\(685\) 0 0
\(686\) 0 0
\(687\) 10.0000i 0.381524i
\(688\) 2.00000i 0.0762493i
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 10.0000i 0.380143i
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 4.00000 0.151620
\(697\) 12.0000i 0.454532i
\(698\) 26.0000i 0.984115i
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −14.0000 −0.528773 −0.264386 0.964417i \(-0.585169\pi\)
−0.264386 + 0.964417i \(0.585169\pi\)
\(702\) − 4.00000i − 0.150970i
\(703\) 16.0000i 0.603451i
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) 0 0
\(708\) 8.00000i 0.300658i
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 12.0000i 0.449719i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 16.0000 0.597948
\(717\) 14.0000i 0.522840i
\(718\) − 16.0000i − 0.597115i
\(719\) 2.00000 0.0745874 0.0372937 0.999304i \(-0.488126\pi\)
0.0372937 + 0.999304i \(0.488126\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 45.0000i − 1.67473i
\(723\) − 6.00000i − 0.223142i
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) − 28.0000i − 1.03846i −0.854634 0.519231i \(-0.826218\pi\)
0.854634 0.519231i \(-0.173782\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) − 2.00000i − 0.0739221i
\(733\) − 42.0000i − 1.55131i −0.631160 0.775653i \(-0.717421\pi\)
0.631160 0.775653i \(-0.282579\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) − 12.0000i − 0.442026i
\(738\) − 2.00000i − 0.0736210i
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 0 0
\(741\) −32.0000 −1.17555
\(742\) 0 0
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 6.00000 0.219676
\(747\) − 8.00000i − 0.292705i
\(748\) − 12.0000i − 0.438763i
\(749\) 0 0
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 12.0000i 0.437595i
\(753\) 2.00000i 0.0728841i
\(754\) −16.0000 −0.582686
\(755\) 0 0
\(756\) 0 0
\(757\) 6.00000i 0.218074i 0.994038 + 0.109037i \(0.0347767\pi\)
−0.994038 + 0.109037i \(0.965223\pi\)
\(758\) 12.0000i 0.435860i
\(759\) −2.00000 −0.0725954
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) − 2.00000i − 0.0724524i
\(763\) 0 0
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) −32.0000 −1.15621
\(767\) − 32.0000i − 1.15545i
\(768\) 1.00000i 0.0360844i
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 22.0000 0.792311
\(772\) − 26.0000i − 0.935760i
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 0 0
\(776\) −16.0000 −0.574367
\(777\) 0 0
\(778\) 26.0000i 0.932145i
\(779\) −16.0000 −0.573259
\(780\) 0 0
\(781\) −20.0000 −0.715656
\(782\) 6.00000i 0.214560i
\(783\) 4.00000i 0.142948i
\(784\) 7.00000 0.250000
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) − 22.0000i − 0.784215i −0.919919 0.392108i \(-0.871746\pi\)
0.919919 0.392108i \(-0.128254\pi\)
\(788\) − 18.0000i − 0.641223i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 2.00000i 0.0710669i
\(793\) 8.00000i 0.284088i
\(794\) −28.0000 −0.993683
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) − 14.0000i − 0.495905i −0.968772 0.247953i \(-0.920242\pi\)
0.968772 0.247953i \(-0.0797578\pi\)
\(798\) 0 0
\(799\) 72.0000 2.54718
\(800\) 0 0
\(801\) −12.0000 −0.423999
\(802\) 8.00000i 0.282490i
\(803\) 4.00000i 0.141157i
\(804\) 6.00000 0.211604
\(805\) 0 0
\(806\) 0 0
\(807\) − 8.00000i − 0.281613i
\(808\) 12.0000i 0.422159i
\(809\) 46.0000 1.61727 0.808637 0.588308i \(-0.200206\pi\)
0.808637 + 0.588308i \(0.200206\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) 24.0000i 0.841717i
\(814\) −4.00000 −0.140200
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 16.0000i 0.559769i
\(818\) − 10.0000i − 0.349642i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) − 2.00000i − 0.0697580i
\(823\) − 22.0000i − 0.766872i −0.923567 0.383436i \(-0.874741\pi\)
0.923567 0.383436i \(-0.125259\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 12.0000i − 0.417281i −0.977992 0.208640i \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) − 1.00000i − 0.0347524i
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) −28.0000 −0.971309
\(832\) − 4.00000i − 0.138675i
\(833\) − 42.0000i − 1.45521i
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) 30.0000i 1.03633i
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 14.0000i 0.482472i
\(843\) 12.0000i 0.413302i
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) − 6.00000i − 0.206041i
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) − 10.0000i − 0.342594i
\(853\) 36.0000i 1.23262i 0.787505 + 0.616308i \(0.211372\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) − 18.0000i − 0.614868i −0.951569 0.307434i \(-0.900530\pi\)
0.951569 0.307434i \(-0.0994704\pi\)
\(858\) − 8.00000i − 0.273115i
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 8.00000i 0.272481i
\(863\) − 20.0000i − 0.680808i −0.940279 0.340404i \(-0.889436\pi\)
0.940279 0.340404i \(-0.110564\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 16.0000 0.543702
\(867\) − 19.0000i − 0.645274i
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 10.0000i 0.338643i
\(873\) − 16.0000i − 0.541518i
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) 20.0000i 0.675352i 0.941262 + 0.337676i \(0.109641\pi\)
−0.941262 + 0.337676i \(0.890359\pi\)
\(878\) 28.0000i 0.944954i
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) 4.00000 0.134763 0.0673817 0.997727i \(-0.478535\pi\)
0.0673817 + 0.997727i \(0.478535\pi\)
\(882\) 7.00000i 0.235702i
\(883\) 32.0000i 1.07689i 0.842662 + 0.538443i \(0.180987\pi\)
−0.842662 + 0.538443i \(0.819013\pi\)
\(884\) −24.0000 −0.807207
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) − 36.0000i − 1.20876i −0.796696 0.604381i \(-0.793421\pi\)
0.796696 0.604381i \(-0.206579\pi\)
\(888\) − 2.00000i − 0.0671156i
\(889\) 0 0
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 2.00000i 0.0669650i
\(893\) 96.0000i 3.21252i
\(894\) −14.0000 −0.468230
\(895\) 0 0
\(896\) 0 0
\(897\) 4.00000i 0.133556i
\(898\) 30.0000i 1.00111i
\(899\) 0 0
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) − 4.00000i − 0.133185i
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) −4.00000 −0.132891
\(907\) 22.0000i 0.730498i 0.930910 + 0.365249i \(0.119016\pi\)
−0.930910 + 0.365249i \(0.880984\pi\)
\(908\) 28.0000i 0.929213i
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −4.00000 −0.132526 −0.0662630 0.997802i \(-0.521108\pi\)
−0.0662630 + 0.997802i \(0.521108\pi\)
\(912\) 8.00000i 0.264906i
\(913\) − 16.0000i − 0.529523i
\(914\) 8.00000 0.264616
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) 6.00000i 0.198030i
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) 12.0000i 0.395199i
\(923\) 40.0000i 1.31662i
\(924\) 0 0
\(925\) 0 0
\(926\) −26.0000 −0.854413
\(927\) 0 0
\(928\) 4.00000i 0.131306i
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 56.0000 1.83533
\(932\) − 6.00000i − 0.196537i
\(933\) 26.0000i 0.851202i
\(934\) 24.0000 0.785304
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) − 32.0000i − 1.04539i −0.852518 0.522697i \(-0.824926\pi\)
0.852518 0.522697i \(-0.175074\pi\)
\(938\) 0 0
\(939\) −28.0000 −0.913745
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 10.0000i 0.325818i
\(943\) 2.00000i 0.0651290i
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) − 4.00000i − 0.129983i −0.997886 0.0649913i \(-0.979298\pi\)
0.997886 0.0649913i \(-0.0207020\pi\)
\(948\) − 8.00000i − 0.259828i
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 26.0000i 0.842223i 0.907009 + 0.421111i \(0.138360\pi\)
−0.907009 + 0.421111i \(0.861640\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) −14.0000 −0.452792
\(957\) 8.00000i 0.258603i
\(958\) − 16.0000i − 0.516937i
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 8.00000i 0.257930i
\(963\) − 12.0000i − 0.386695i
\(964\) 6.00000 0.193247
\(965\) 0 0
\(966\) 0 0
\(967\) 42.0000i 1.35063i 0.737530 + 0.675314i \(0.235992\pi\)
−0.737530 + 0.675314i \(0.764008\pi\)
\(968\) − 7.00000i − 0.224989i
\(969\) 48.0000 1.54198
\(970\) 0 0
\(971\) −14.0000 −0.449281 −0.224641 0.974442i \(-0.572121\pi\)
−0.224641 + 0.974442i \(0.572121\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 0 0
\(974\) 26.0000 0.833094
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) − 16.0000i − 0.511624i
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) − 20.0000i − 0.638226i
\(983\) 48.0000i 1.53096i 0.643458 + 0.765481i \(0.277499\pi\)
−0.643458 + 0.765481i \(0.722501\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) 24.0000 0.764316
\(987\) 0 0
\(988\) − 32.0000i − 1.01806i
\(989\) 2.00000 0.0635963
\(990\) 0 0
\(991\) 12.0000 0.381193 0.190596 0.981669i \(-0.438958\pi\)
0.190596 + 0.981669i \(0.438958\pi\)
\(992\) 0 0
\(993\) 28.0000i 0.888553i
\(994\) 0 0
\(995\) 0 0
\(996\) 8.00000 0.253490
\(997\) 4.00000i 0.126681i 0.997992 + 0.0633406i \(0.0201755\pi\)
−0.997992 + 0.0633406i \(0.979825\pi\)
\(998\) − 4.00000i − 0.126618i
\(999\) 2.00000 0.0632772
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 3450.2.d.m.2899.1 2
5.2 odd 4 690.2.a.j.1.1 1
5.3 odd 4 3450.2.a.b.1.1 1
5.4 even 2 inner 3450.2.d.m.2899.2 2
15.2 even 4 2070.2.a.c.1.1 1
20.7 even 4 5520.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.j.1.1 1 5.2 odd 4
2070.2.a.c.1.1 1 15.2 even 4
3450.2.a.b.1.1 1 5.3 odd 4
3450.2.d.m.2899.1 2 1.1 even 1 trivial
3450.2.d.m.2899.2 2 5.4 even 2 inner
5520.2.a.l.1.1 1 20.7 even 4