Properties

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

Related objects

Downloads

Learn more

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.t.2899.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} -4.00000i q^{22} -1.00000i q^{23} -1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +2.00000 q^{29} -1.00000i q^{32} +4.00000i q^{33} +6.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} +2.00000 q^{39} +10.0000 q^{41} -4.00000i q^{43} -4.00000 q^{44} -1.00000 q^{46} +1.00000i q^{48} +7.00000 q^{49} -6.00000 q^{51} +2.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} -4.00000i q^{57} -2.00000i q^{58} +4.00000 q^{59} -10.0000 q^{61} -1.00000 q^{64} +4.00000 q^{66} +12.0000i q^{67} -6.00000i q^{68} +1.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} +10.0000i q^{73} +2.00000 q^{74} +4.00000 q^{76} -2.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} -4.00000i q^{83} -4.00000 q^{86} +2.00000i q^{87} +4.00000i q^{88} -18.0000 q^{89} +1.00000i q^{92} +1.00000 q^{96} -2.00000i q^{97} -7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + 8q^{11} + 2q^{16} - 8q^{19} - 2q^{24} - 4q^{26} + 4q^{29} + 12q^{34} + 2q^{36} + 4q^{39} + 20q^{41} - 8q^{44} - 2q^{46} + 14q^{49} - 12q^{51} - 2q^{54} + 8q^{59} - 20q^{61} - 2q^{64} + 8q^{66} + 2q^{69} - 16q^{71} + 4q^{74} + 8q^{76} + 16q^{79} + 2q^{81} - 8q^{86} - 36q^{89} + 2q^{96} - 8q^{99} + O(q^{100}) \)

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
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 4.00000i − 0.852803i
\(23\) − 1.00000i − 0.208514i
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 4.00000i 0.696311i
\(34\) 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 \(0.947432\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 2.00000i 0.277350i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.00000i − 0.529813i
\(58\) − 2.00000i − 0.262613i
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 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 \(0.738112\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) − 2.00000i − 0.226455i
\(79\) 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
\(82\) − 10.0000i − 1.10432i
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 2.00000i 0.214423i
\(88\) 4.00000i 0.426401i
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000i 0.104257i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) − 7.00000i − 0.707107i
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 6.00000i 0.594089i
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 2.00000 0.196116
\(105\) 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
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 2.00000i 0.184900i
\(118\) − 4.00000i − 0.368230i
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000i 0.905357i
\(123\) 10.0000i 0.901670i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) − 4.00000i − 0.348155i
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 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
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000i 0.671345i
\(143\) − 8.00000i − 0.668994i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) 7.00000i 0.577350i
\(148\) − 2.00000i − 0.164399i
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) − 4.00000i − 0.324443i
\(153\) − 6.00000i − 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 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
\(163\) 20.0000i 1.56652i 0.621694 + 0.783260i \(0.286445\pi\)
−0.621694 + 0.783260i \(0.713555\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 4.00000i 0.304997i
\(173\) 14.0000i 1.06440i 0.846619 + 0.532200i \(0.178635\pi\)
−0.846619 + 0.532200i \(0.821365\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 4.00000i 0.300658i
\(178\) 18.0000i 1.34916i
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) − 10.0000i − 0.739221i
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 24.0000i 1.75505i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 4.00000i 0.284268i
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) − 6.00000i − 0.422159i
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) 0 0
\(207\) 1.00000i 0.0695048i
\(208\) − 2.00000i − 0.138675i
\(209\) −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
\(213\) − 8.00000i − 0.548151i
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) − 10.0000i − 0.677285i
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 2.00000i 0.134231i
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 4.00000i 0.264906i
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.00000i 0.131306i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) 8.00000i 0.519656i
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) 1.00000i 0.0641500i
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) 8.00000i 0.509028i
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) − 4.00000i − 0.251478i
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.0000i 0.873296i 0.899632 + 0.436648i \(0.143834\pi\)
−0.899632 + 0.436648i \(0.856166\pi\)
\(258\) − 4.00000i − 0.249029i
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) − 20.0000i − 1.23560i
\(263\) − 24.0000i − 1.47990i −0.672660 0.739952i \(-0.734848\pi\)
0.672660 0.739952i \(-0.265152\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) − 18.0000i − 1.10158i
\(268\) − 12.0000i − 0.733017i
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 6.00000i 0.363803i
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 10.0000i 0.600842i 0.953807 + 0.300421i \(0.0971271\pi\)
−0.953807 + 0.300421i \(0.902873\pi\)
\(278\) − 20.0000i − 1.19952i
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) − 10.0000i − 0.585206i
\(293\) 22.0000i 1.28525i 0.766179 + 0.642627i \(0.222155\pi\)
−0.766179 + 0.642627i \(0.777845\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) − 4.00000i − 0.232104i
\(298\) − 10.0000i − 0.579284i
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) 0 0
\(302\) − 8.00000i − 0.460348i
\(303\) 6.00000i 0.344691i
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) − 4.00000i − 0.228292i −0.993464 0.114146i \(-0.963587\pi\)
0.993464 0.114146i \(-0.0364132\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 2.00000i 0.113228i
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) − 30.0000i − 1.68497i −0.538721 0.842484i \(-0.681092\pi\)
0.538721 0.842484i \(-0.318908\pi\)
\(318\) 6.00000i 0.336463i
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) − 24.0000i − 1.33540i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 20.0000 1.10770
\(327\) 10.0000i 0.553001i
\(328\) 10.0000i 0.552158i
\(329\) 0 0
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 4.00000i 0.219529i
\(333\) − 2.00000i − 0.109599i
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) 0 0
\(337\) − 2.00000i − 0.108947i −0.998515 0.0544735i \(-0.982652\pi\)
0.998515 0.0544735i \(-0.0173480\pi\)
\(338\) − 9.00000i − 0.489535i
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) − 4.00000i − 0.216295i
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) − 2.00000i − 0.107211i
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) − 4.00000i − 0.213201i
\(353\) 34.0000i 1.80964i 0.425797 + 0.904819i \(0.359994\pi\)
−0.425797 + 0.904819i \(0.640006\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) 4.00000i 0.211407i
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 2.00000i 0.105118i
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) − 1.00000i − 0.0521286i
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 18.0000i − 0.932005i −0.884783 0.466002i \(-0.845694\pi\)
0.884783 0.466002i \(-0.154306\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) 0 0
\(377\) − 4.00000i − 0.206010i
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) − 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
\(386\) 2.00000 0.101797
\(387\) 4.00000i 0.203331i
\(388\) 2.00000i 0.101535i
\(389\) −38.0000 −1.92668 −0.963338 0.268290i \(-0.913542\pi\)
−0.963338 + 0.268290i \(0.913542\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 7.00000i 0.353553i
\(393\) 20.0000i 1.00887i
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) 0 0
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 12.0000i 0.598506i
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) 8.00000i 0.396545i
\(408\) − 6.00000i − 0.297044i
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 0 0
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 20.0000i 0.979404i
\(418\) 16.0000i 0.782586i
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) − 4.00000i − 0.194717i
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) − 12.0000i − 0.580042i
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 34.0000i 1.63394i 0.576683 + 0.816968i \(0.304347\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 4.00000i 0.191346i
\(438\) 10.0000i 0.477818i
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) − 12.0000i − 0.570782i
\(443\) − 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
\(446\) 16.0000 0.757622
\(447\) 10.0000i 0.472984i
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 40.0000 1.88353
\(452\) 6.00000i 0.282216i
\(453\) 8.00000i 0.375873i
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) 14.0000i 0.654177i
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i 0.928316 + 0.371792i \(0.121256\pi\)
−0.928316 + 0.371792i \(0.878744\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) − 12.0000i − 0.555294i −0.960683 0.277647i \(-0.910445\pi\)
0.960683 0.277647i \(-0.0895545\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) 0 0
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 4.00000i 0.184115i
\(473\) − 16.0000i − 0.735681i
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 0 0
\(477\) − 6.00000i − 0.274721i
\(478\) 16.0000i 0.731823i
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) − 18.0000i − 0.819878i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 40.0000i − 1.81257i −0.422664 0.906287i \(-0.638905\pi\)
0.422664 0.906287i \(-0.361095\pi\)
\(488\) − 10.0000i − 0.452679i
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) −4.00000 −0.180517 −0.0902587 0.995918i \(-0.528769\pi\)
−0.0902587 + 0.995918i \(0.528769\pi\)
\(492\) − 10.0000i − 0.450835i
\(493\) 12.0000i 0.540453i
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) − 4.00000i − 0.179244i
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) − 20.0000i − 0.892644i
\(503\) − 40.0000i − 1.78351i −0.452517 0.891756i \(-0.649474\pi\)
0.452517 0.891756i \(-0.350526\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −4.00000 −0.177822
\(507\) 9.00000i 0.399704i
\(508\) − 16.0000i − 0.709885i
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) 14.0000 0.617514
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 2.00000i 0.0875376i
\(523\) − 36.0000i − 1.57417i −0.616844 0.787085i \(-0.711589\pi\)
0.616844 0.787085i \(-0.288411\pi\)
\(524\) −20.0000 −0.873704
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) 4.00000i 0.174078i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) − 20.0000i − 0.866296i
\(534\) −18.0000 −0.778936
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) − 4.00000i − 0.172613i
\(538\) 14.0000i 0.603583i
\(539\) 28.0000 1.20605
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 0 0
\(543\) − 2.00000i − 0.0858282i
\(544\) 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
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 1.00000i 0.0425628i
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) − 14.0000i − 0.593199i −0.955002 0.296600i \(-0.904147\pi\)
0.955002 0.296600i \(-0.0958526\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) − 18.0000i − 0.759284i
\(563\) − 36.0000i − 1.51722i −0.651546 0.758610i \(-0.725879\pi\)
0.651546 0.758610i \(-0.274121\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) − 8.00000i − 0.335673i
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 8.00000i 0.334497i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 19.0000i 0.790296i
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) 0 0
\(582\) − 2.00000i − 0.0829027i
\(583\) 24.0000i 0.993978i
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) 22.0000 0.908812
\(587\) − 28.0000i − 1.15568i −0.816149 0.577842i \(-0.803895\pi\)
0.816149 0.577842i \(-0.196105\pi\)
\(588\) − 7.00000i − 0.288675i
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 2.00000i 0.0821995i
\(593\) − 30.0000i − 1.23195i −0.787765 0.615976i \(-0.788762\pi\)
0.787765 0.615976i \(-0.211238\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 16.0000i 0.654836i
\(598\) 2.00000i 0.0817861i
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 0 0
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 0 0
\(603\) − 12.0000i − 0.488678i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) 16.0000i 0.649420i 0.945814 + 0.324710i \(0.105267\pi\)
−0.945814 + 0.324710i \(0.894733\pi\)
\(608\) 4.00000i 0.162221i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 6.00000i 0.242536i
\(613\) − 18.0000i − 0.727013i −0.931592 0.363507i \(-0.881579\pi\)
0.931592 0.363507i \(-0.118421\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) − 8.00000i − 0.320771i
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) − 16.0000i − 0.638978i
\(628\) − 10.0000i − 0.399043i
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 8.00000i 0.318223i
\(633\) 4.00000i 0.158986i
\(634\) −30.0000 −1.19145
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) − 14.0000i − 0.554700i
\(638\) − 8.00000i − 0.316723i
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 20.0000i 0.788723i 0.918955 + 0.394362i \(0.129034\pi\)
−0.918955 + 0.394362i \(0.870966\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) − 8.00000i − 0.314512i −0.987558 0.157256i \(-0.949735\pi\)
0.987558 0.157256i \(-0.0502649\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) − 20.0000i − 0.783260i
\(653\) 14.0000i 0.547862i 0.961749 + 0.273931i \(0.0883240\pi\)
−0.961749 + 0.273931i \(0.911676\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) − 10.0000i − 0.390137i
\(658\) 0 0
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 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
\(663\) 12.0000i 0.466041i
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) − 2.00000i − 0.0774403i
\(668\) 8.00000i 0.309529i
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) 2.00000i 0.0770943i 0.999257 + 0.0385472i \(0.0122730\pi\)
−0.999257 + 0.0385472i \(0.987727\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 38.0000i − 1.46046i −0.683202 0.730229i \(-0.739413\pi\)
0.683202 0.730229i \(-0.260587\pi\)
\(678\) − 6.00000i − 0.230429i
\(679\) 0 0
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) − 14.0000i − 0.534133i
\(688\) − 4.00000i − 0.152499i
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) − 14.0000i − 0.532200i
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) 60.0000i 2.27266i
\(698\) 14.0000i 0.529908i
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −50.0000 −1.88847 −0.944237 0.329267i \(-0.893198\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) 2.00000i 0.0754851i
\(703\) − 8.00000i − 0.301726i
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 34.0000 1.27961
\(707\) 0 0
\(708\) − 4.00000i − 0.150329i
\(709\) 34.0000 1.27690 0.638448 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) − 18.0000i − 0.674579i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) − 16.0000i − 0.597531i
\(718\) 8.00000i 0.298557i
\(719\) 32.0000 1.19340 0.596699 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.00000i 0.111648i
\(723\) 18.0000i 0.669427i
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) − 16.0000i − 0.593407i −0.954970 0.296704i \(-0.904113\pi\)
0.954970 0.296704i \(-0.0958873\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 10.0000i 0.369611i
\(733\) 6.00000i 0.221615i 0.993842 + 0.110808i \(0.0353437\pi\)
−0.993842 + 0.110808i \(0.964656\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 48.0000i 1.76810i
\(738\) 10.0000i 0.368105i
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 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
\(746\) −18.0000 −0.659027
\(747\) 4.00000i 0.146352i
\(748\) − 24.0000i − 0.877527i
\(749\) 0 0
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) 20.0000i 0.728841i
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) 0 0
\(757\) 18.0000i 0.654221i 0.944986 + 0.327111i \(0.106075\pi\)
−0.944986 + 0.327111i \(0.893925\pi\)
\(758\) 12.0000i 0.435860i
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 16.0000i 0.579619i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −32.0000 −1.15621
\(767\) − 8.00000i − 0.288863i
\(768\) 1.00000i 0.0360844i
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) − 2.00000i − 0.0719816i
\(773\) 54.0000i 1.94225i 0.238581 + 0.971123i \(0.423318\pi\)
−0.238581 + 0.971123i \(0.576682\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 38.0000i 1.36237i
\(779\) −40.0000 −1.43315
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) − 6.00000i − 0.214560i
\(783\) − 2.00000i − 0.0714742i
\(784\) 7.00000 0.250000
\(785\) 0 0
\(786\) 20.0000 0.713376
\(787\) − 4.00000i − 0.142585i −0.997455 0.0712923i \(-0.977288\pi\)
0.997455 0.0712923i \(-0.0227123\pi\)
\(788\) 6.00000i 0.213741i
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 0 0
\(792\) − 4.00000i − 0.142134i
\(793\) 20.0000i 0.710221i
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 34.0000i 1.20434i 0.798367 + 0.602171i \(0.205697\pi\)
−0.798367 + 0.602171i \(0.794303\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) − 10.0000i − 0.353112i
\(803\) 40.0000i 1.41157i
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) 0 0
\(807\) − 14.0000i − 0.492823i
\(808\) 6.00000i 0.211079i
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) 16.0000i 0.559769i
\(818\) 26.0000i 0.909069i
\(819\) 0 0
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) − 2.00000i − 0.0697580i
\(823\) − 40.0000i − 1.39431i −0.716919 0.697156i \(-0.754448\pi\)
0.716919 0.697156i \(-0.245552\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) − 1.00000i − 0.0347524i
\(829\) −46.0000 −1.59765 −0.798823 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 2.00000i 0.0693375i
\(833\) 42.0000i 1.45521i
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) 12.0000i 0.414533i
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 2.00000i 0.0689246i
\(843\) 18.0000i 0.619953i
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000i 0.206041i
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) 8.00000i 0.274075i
\(853\) 6.00000i 0.205436i 0.994711 + 0.102718i \(0.0327539\pi\)
−0.994711 + 0.102718i \(0.967246\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) − 42.0000i − 1.43469i −0.696717 0.717346i \(-0.745357\pi\)
0.696717 0.717346i \(-0.254643\pi\)
\(858\) − 8.00000i − 0.273115i
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 32.0000i 1.08992i
\(863\) − 32.0000i − 1.08929i −0.838666 0.544646i \(-0.816664\pi\)
0.838666 0.544646i \(-0.183336\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 34.0000 1.15537
\(867\) − 19.0000i − 0.645274i
\(868\) 0 0
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 10.0000i 0.338643i
\(873\) 2.00000i 0.0676897i
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) 50.0000i 1.68838i 0.536044 + 0.844190i \(0.319918\pi\)
−0.536044 + 0.844190i \(0.680082\pi\)
\(878\) − 8.00000i − 0.269987i
\(879\) −22.0000 −0.742042
\(880\) 0 0
\(881\) 10.0000 0.336909 0.168454 0.985709i \(-0.446122\pi\)
0.168454 + 0.985709i \(0.446122\pi\)
\(882\) 7.00000i 0.235702i
\(883\) 20.0000i 0.673054i 0.941674 + 0.336527i \(0.109252\pi\)
−0.941674 + 0.336527i \(0.890748\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 24.0000i 0.805841i 0.915235 + 0.402921i \(0.132005\pi\)
−0.915235 + 0.402921i \(0.867995\pi\)
\(888\) − 2.00000i − 0.0671156i
\(889\) 0 0
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) − 16.0000i − 0.535720i
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) 0 0
\(897\) − 2.00000i − 0.0667781i
\(898\) − 30.0000i − 1.00111i
\(899\) 0 0
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) − 40.0000i − 1.33185i
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) − 44.0000i − 1.46100i −0.682915 0.730498i \(-0.739288\pi\)
0.682915 0.730498i \(-0.260712\pi\)
\(908\) − 20.0000i − 0.663723i
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) − 16.0000i − 0.529523i
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) − 6.00000i − 0.198030i
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 18.0000i 0.592798i
\(923\) 16.0000i 0.526646i
\(924\) 0 0
\(925\) 0 0
\(926\) 16.0000 0.525793
\(927\) 0 0
\(928\) − 2.00000i − 0.0656532i
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −28.0000 −0.917663
\(932\) 6.00000i 0.196537i
\(933\) 8.00000i 0.261908i
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 22.0000i 0.718709i 0.933201 + 0.359354i \(0.117003\pi\)
−0.933201 + 0.359354i \(0.882997\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) 10.0000i 0.325818i
\(943\) − 10.0000i − 0.325645i
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) − 4.00000i − 0.129983i −0.997886 0.0649913i \(-0.979298\pi\)
0.997886 0.0649913i \(-0.0207020\pi\)
\(948\) − 8.00000i − 0.259828i
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) 2.00000i 0.0647864i 0.999475 + 0.0323932i \(0.0103129\pi\)
−0.999475 + 0.0323932i \(0.989687\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 8.00000i 0.258603i
\(958\) 32.0000i 1.03387i
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) − 4.00000i − 0.128965i
\(963\) − 12.0000i − 0.386695i
\(964\) −18.0000 −0.579741
\(965\) 0 0
\(966\) 0 0
\(967\) 24.0000i 0.771788i 0.922543 + 0.385894i \(0.126107\pi\)
−0.922543 + 0.385894i \(0.873893\pi\)
\(968\) 5.00000i 0.160706i
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) 52.0000 1.66876 0.834380 0.551190i \(-0.185826\pi\)
0.834380 + 0.551190i \(0.185826\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 0 0
\(974\) −40.0000 −1.28168
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) − 42.0000i − 1.34370i −0.740688 0.671850i \(-0.765500\pi\)
0.740688 0.671850i \(-0.234500\pi\)
\(978\) 20.0000i 0.639529i
\(979\) −72.0000 −2.30113
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 4.00000i 0.127645i
\(983\) 24.0000i 0.765481i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) −10.0000 −0.318788
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) − 8.00000i − 0.254514i
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 28.0000i 0.888553i
\(994\) 0 0
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) − 38.0000i − 1.20347i −0.798695 0.601736i \(-0.794476\pi\)
0.798695 0.601736i \(-0.205524\pi\)
\(998\) 20.0000i 0.633089i
\(999\) 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.t.2899.1 2
5.2 odd 4 690.2.a.k.1.1 1
5.3 odd 4 3450.2.a.d.1.1 1
5.4 even 2 inner 3450.2.d.t.2899.2 2
15.2 even 4 2070.2.a.b.1.1 1
20.7 even 4 5520.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.k.1.1 1 5.2 odd 4
2070.2.a.b.1.1 1 15.2 even 4
3450.2.a.d.1.1 1 5.3 odd 4
3450.2.d.t.2899.1 2 1.1 even 1 trivial
3450.2.d.t.2899.2 2 5.4 even 2 inner
5520.2.a.i.1.1 1 20.7 even 4