Properties

Label 3450.2.d.b.2899.1
Level $3450$
Weight $2$
Character 3450.2899
Analytic conductor $27.548$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.b.2899.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} +1.00000i q^{12} -3.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} +4.00000i q^{17} +1.00000i q^{18} +3.00000 q^{19} -3.00000 q^{21} +3.00000i q^{22} -1.00000i q^{23} +1.00000 q^{24} -3.00000 q^{26} +1.00000i q^{27} +3.00000i q^{28} -3.00000 q^{29} -10.0000 q^{31} -1.00000i q^{32} +3.00000i q^{33} +4.00000 q^{34} +1.00000 q^{36} +8.00000i q^{37} -3.00000i q^{38} -3.00000 q^{39} -9.00000 q^{41} +3.00000i q^{42} +1.00000i q^{43} +3.00000 q^{44} -1.00000 q^{46} -2.00000i q^{47} -1.00000i q^{48} -2.00000 q^{49} +4.00000 q^{51} +3.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} +3.00000 q^{56} -3.00000i q^{57} +3.00000i q^{58} -8.00000 q^{59} +12.0000 q^{61} +10.0000i q^{62} +3.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} -8.00000i q^{67} -4.00000i q^{68} -1.00000 q^{69} -14.0000 q^{71} -1.00000i q^{72} +13.0000i q^{73} +8.00000 q^{74} -3.00000 q^{76} +9.00000i q^{77} +3.00000i q^{78} +17.0000 q^{79} +1.00000 q^{81} +9.00000i q^{82} +1.00000i q^{83} +3.00000 q^{84} +1.00000 q^{86} +3.00000i q^{87} -3.00000i q^{88} +6.00000 q^{89} -9.00000 q^{91} +1.00000i q^{92} +10.0000i q^{93} -2.00000 q^{94} -1.00000 q^{96} -6.00000i q^{97} +2.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} - 6q^{11} - 6q^{14} + 2q^{16} + 6q^{19} - 6q^{21} + 2q^{24} - 6q^{26} - 6q^{29} - 20q^{31} + 8q^{34} + 2q^{36} - 6q^{39} - 18q^{41} + 6q^{44} - 2q^{46} - 4q^{49} + 8q^{51} + 2q^{54} + 6q^{56} - 16q^{59} + 24q^{61} - 2q^{64} + 6q^{66} - 2q^{69} - 28q^{71} + 16q^{74} - 6q^{76} + 34q^{79} + 2q^{81} + 6q^{84} + 2q^{86} + 12q^{89} - 18q^{91} - 4q^{94} - 2q^{96} + 6q^{99} + O(q^{100}) \)

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
\(7\) − 3.00000i − 1.13389i −0.823754 0.566947i \(-0.808125\pi\)
0.823754 0.566947i \(-0.191875\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 3.00000i − 0.832050i −0.909353 0.416025i \(-0.863423\pi\)
0.909353 0.416025i \(-0.136577\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 3.00000i 0.639602i
\(23\) − 1.00000i − 0.208514i
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −3.00000 −0.588348
\(27\) 1.00000i 0.192450i
\(28\) 3.00000i 0.566947i
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 3.00000i 0.522233i
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) − 3.00000i − 0.486664i
\(39\) −3.00000 −0.480384
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 3.00000i 0.462910i
\(43\) 1.00000i 0.152499i 0.997089 + 0.0762493i \(0.0242945\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) − 2.00000i − 0.291730i −0.989305 0.145865i \(-0.953403\pi\)
0.989305 0.145865i \(-0.0465965\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 3.00000i 0.416025i
\(53\) − 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
\(56\) 3.00000 0.400892
\(57\) − 3.00000i − 0.397360i
\(58\) 3.00000i 0.393919i
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 10.0000i 1.27000i
\(63\) 3.00000i 0.377964i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 13.0000i 1.52153i 0.649025 + 0.760767i \(0.275177\pi\)
−0.649025 + 0.760767i \(0.724823\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −3.00000 −0.344124
\(77\) 9.00000i 1.02565i
\(78\) 3.00000i 0.339683i
\(79\) 17.0000 1.91265 0.956325 0.292306i \(-0.0944227\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 9.00000i 0.993884i
\(83\) 1.00000i 0.109764i 0.998493 + 0.0548821i \(0.0174783\pi\)
−0.998493 + 0.0548821i \(0.982522\pi\)
\(84\) 3.00000 0.327327
\(85\) 0 0
\(86\) 1.00000 0.107833
\(87\) 3.00000i 0.321634i
\(88\) − 3.00000i − 0.319801i
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −9.00000 −0.943456
\(92\) 1.00000i 0.104257i
\(93\) 10.0000i 1.03695i
\(94\) −2.00000 −0.206284
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 6.00000i − 0.609208i −0.952479 0.304604i \(-0.901476\pi\)
0.952479 0.304604i \(-0.0985241\pi\)
\(98\) 2.00000i 0.202031i
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) − 4.00000i − 0.396059i
\(103\) 13.0000i 1.28093i 0.767988 + 0.640464i \(0.221258\pi\)
−0.767988 + 0.640464i \(0.778742\pi\)
\(104\) 3.00000 0.294174
\(105\) 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
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) − 3.00000i − 0.283473i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) −3.00000 −0.280976
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) 3.00000i 0.277350i
\(118\) 8.00000i 0.736460i
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) − 12.0000i − 1.08643i
\(123\) 9.00000i 0.811503i
\(124\) 10.0000 0.898027
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) − 16.0000i − 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) − 3.00000i − 0.261116i
\(133\) − 9.00000i − 0.780399i
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 18.0000i 1.53784i 0.639343 + 0.768922i \(0.279207\pi\)
−0.639343 + 0.768922i \(0.720793\pi\)
\(138\) 1.00000i 0.0851257i
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) 14.0000i 1.17485i
\(143\) 9.00000i 0.752618i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 13.0000 1.07589
\(147\) 2.00000i 0.164957i
\(148\) − 8.00000i − 0.657596i
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −22.0000 −1.79033 −0.895167 0.445730i \(-0.852944\pi\)
−0.895167 + 0.445730i \(0.852944\pi\)
\(152\) 3.00000i 0.243332i
\(153\) − 4.00000i − 0.323381i
\(154\) 9.00000 0.725241
\(155\) 0 0
\(156\) 3.00000 0.240192
\(157\) − 6.00000i − 0.478852i −0.970915 0.239426i \(-0.923041\pi\)
0.970915 0.239426i \(-0.0769593\pi\)
\(158\) − 17.0000i − 1.35245i
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) − 1.00000i − 0.0785674i
\(163\) 6.00000i 0.469956i 0.972001 + 0.234978i \(0.0755019\pi\)
−0.972001 + 0.234978i \(0.924498\pi\)
\(164\) 9.00000 0.702782
\(165\) 0 0
\(166\) 1.00000 0.0776151
\(167\) − 2.00000i − 0.154765i −0.997001 0.0773823i \(-0.975344\pi\)
0.997001 0.0773823i \(-0.0246562\pi\)
\(168\) − 3.00000i − 0.231455i
\(169\) 4.00000 0.307692
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) − 1.00000i − 0.0762493i
\(173\) 15.0000i 1.14043i 0.821496 + 0.570214i \(0.193140\pi\)
−0.821496 + 0.570214i \(0.806860\pi\)
\(174\) 3.00000 0.227429
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 8.00000i 0.601317i
\(178\) − 6.00000i − 0.449719i
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 9.00000i 0.667124i
\(183\) − 12.0000i − 0.887066i
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 10.0000 0.733236
\(187\) − 12.0000i − 0.877527i
\(188\) 2.00000i 0.145865i
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) −15.0000 −1.08536 −0.542681 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 6.00000i − 0.431889i −0.976406 0.215945i \(-0.930717\pi\)
0.976406 0.215945i \(-0.0692831\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 17.0000i 1.21120i 0.795769 + 0.605600i \(0.207067\pi\)
−0.795769 + 0.605600i \(0.792933\pi\)
\(198\) − 3.00000i − 0.213201i
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 6.00000i 0.422159i
\(203\) 9.00000i 0.631676i
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) 13.0000 0.905753
\(207\) 1.00000i 0.0695048i
\(208\) − 3.00000i − 0.208013i
\(209\) −9.00000 −0.622543
\(210\) 0 0
\(211\) 22.0000 1.51454 0.757271 0.653101i \(-0.226532\pi\)
0.757271 + 0.653101i \(0.226532\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 14.0000i 0.959264i
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 30.0000i 2.03653i
\(218\) 0 0
\(219\) 13.0000 0.878459
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) − 8.00000i − 0.536925i
\(223\) − 6.00000i − 0.401790i −0.979613 0.200895i \(-0.935615\pi\)
0.979613 0.200895i \(-0.0643850\pi\)
\(224\) −3.00000 −0.200446
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 3.00000i 0.198680i
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 9.00000 0.592157
\(232\) − 3.00000i − 0.196960i
\(233\) − 1.00000i − 0.0655122i −0.999463 0.0327561i \(-0.989572\pi\)
0.999463 0.0327561i \(-0.0104285\pi\)
\(234\) 3.00000 0.196116
\(235\) 0 0
\(236\) 8.00000 0.520756
\(237\) − 17.0000i − 1.10427i
\(238\) − 12.0000i − 0.777844i
\(239\) 10.0000 0.646846 0.323423 0.946254i \(-0.395166\pi\)
0.323423 + 0.946254i \(0.395166\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 2.00000i 0.128565i
\(243\) − 1.00000i − 0.0641500i
\(244\) −12.0000 −0.768221
\(245\) 0 0
\(246\) 9.00000 0.573819
\(247\) − 9.00000i − 0.572656i
\(248\) − 10.0000i − 0.635001i
\(249\) 1.00000 0.0633724
\(250\) 0 0
\(251\) −16.0000 −1.00991 −0.504956 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) − 3.00000i − 0.188982i
\(253\) 3.00000i 0.188608i
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) − 1.00000i − 0.0622573i
\(259\) 24.0000 1.49129
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) − 10.0000i − 0.617802i
\(263\) − 8.00000i − 0.493301i −0.969104 0.246651i \(-0.920670\pi\)
0.969104 0.246651i \(-0.0793300\pi\)
\(264\) −3.00000 −0.184637
\(265\) 0 0
\(266\) −9.00000 −0.551825
\(267\) − 6.00000i − 0.367194i
\(268\) 8.00000i 0.488678i
\(269\) 5.00000 0.304855 0.152428 0.988315i \(-0.451291\pi\)
0.152428 + 0.988315i \(0.451291\pi\)
\(270\) 0 0
\(271\) 22.0000 1.33640 0.668202 0.743980i \(-0.267064\pi\)
0.668202 + 0.743980i \(0.267064\pi\)
\(272\) 4.00000i 0.242536i
\(273\) 9.00000i 0.544705i
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) 13.0000i 0.781094i 0.920583 + 0.390547i \(0.127714\pi\)
−0.920583 + 0.390547i \(0.872286\pi\)
\(278\) 8.00000i 0.479808i
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 2.00000i 0.119098i
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 14.0000 0.830747
\(285\) 0 0
\(286\) 9.00000 0.532181
\(287\) 27.0000i 1.59376i
\(288\) 1.00000i 0.0589256i
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) − 13.0000i − 0.760767i
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) − 3.00000i − 0.174078i
\(298\) − 6.00000i − 0.347571i
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) 3.00000 0.172917
\(302\) 22.0000i 1.26596i
\(303\) 6.00000i 0.344691i
\(304\) 3.00000 0.172062
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) − 8.00000i − 0.456584i −0.973593 0.228292i \(-0.926686\pi\)
0.973593 0.228292i \(-0.0733141\pi\)
\(308\) − 9.00000i − 0.512823i
\(309\) 13.0000 0.739544
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) − 3.00000i − 0.169842i
\(313\) 24.0000i 1.35656i 0.734803 + 0.678280i \(0.237274\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) −17.0000 −0.956325
\(317\) − 31.0000i − 1.74113i −0.492050 0.870567i \(-0.663752\pi\)
0.492050 0.870567i \(-0.336248\pi\)
\(318\) 6.00000i 0.336463i
\(319\) 9.00000 0.503903
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 3.00000i 0.167183i
\(323\) 12.0000i 0.667698i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 6.00000 0.332309
\(327\) 0 0
\(328\) − 9.00000i − 0.496942i
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −26.0000 −1.42909 −0.714545 0.699590i \(-0.753366\pi\)
−0.714545 + 0.699590i \(0.753366\pi\)
\(332\) − 1.00000i − 0.0548821i
\(333\) − 8.00000i − 0.438397i
\(334\) −2.00000 −0.109435
\(335\) 0 0
\(336\) −3.00000 −0.163663
\(337\) − 14.0000i − 0.762629i −0.924445 0.381314i \(-0.875472\pi\)
0.924445 0.381314i \(-0.124528\pi\)
\(338\) − 4.00000i − 0.217571i
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 30.0000 1.62459
\(342\) 3.00000i 0.162221i
\(343\) − 15.0000i − 0.809924i
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) 15.0000 0.806405
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) − 3.00000i − 0.160817i
\(349\) 5.00000 0.267644 0.133822 0.991005i \(-0.457275\pi\)
0.133822 + 0.991005i \(0.457275\pi\)
\(350\) 0 0
\(351\) 3.00000 0.160128
\(352\) 3.00000i 0.159901i
\(353\) 9.00000i 0.479022i 0.970894 + 0.239511i \(0.0769871\pi\)
−0.970894 + 0.239511i \(0.923013\pi\)
\(354\) 8.00000 0.425195
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) − 12.0000i − 0.635107i
\(358\) − 10.0000i − 0.528516i
\(359\) −31.0000 −1.63612 −0.818059 0.575135i \(-0.804950\pi\)
−0.818059 + 0.575135i \(0.804950\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 16.0000i 0.840941i
\(363\) 2.00000i 0.104973i
\(364\) 9.00000 0.471728
\(365\) 0 0
\(366\) −12.0000 −0.627250
\(367\) − 27.0000i − 1.40939i −0.709511 0.704694i \(-0.751084\pi\)
0.709511 0.704694i \(-0.248916\pi\)
\(368\) − 1.00000i − 0.0521286i
\(369\) 9.00000 0.468521
\(370\) 0 0
\(371\) −18.0000 −0.934513
\(372\) − 10.0000i − 0.518476i
\(373\) − 16.0000i − 0.828449i −0.910175 0.414224i \(-0.864053\pi\)
0.910175 0.414224i \(-0.135947\pi\)
\(374\) −12.0000 −0.620505
\(375\) 0 0
\(376\) 2.00000 0.103142
\(377\) 9.00000i 0.463524i
\(378\) − 3.00000i − 0.154303i
\(379\) −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
\(382\) 15.0000i 0.767467i
\(383\) − 29.0000i − 1.48183i −0.671598 0.740915i \(-0.734392\pi\)
0.671598 0.740915i \(-0.265608\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) − 1.00000i − 0.0508329i
\(388\) 6.00000i 0.304604i
\(389\) −38.0000 −1.92668 −0.963338 0.268290i \(-0.913542\pi\)
−0.963338 + 0.268290i \(0.913542\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) − 2.00000i − 0.101015i
\(393\) − 10.0000i − 0.504433i
\(394\) 17.0000 0.856448
\(395\) 0 0
\(396\) −3.00000 −0.150756
\(397\) − 30.0000i − 1.50566i −0.658217 0.752828i \(-0.728689\pi\)
0.658217 0.752828i \(-0.271311\pi\)
\(398\) 11.0000i 0.551380i
\(399\) −9.00000 −0.450564
\(400\) 0 0
\(401\) −28.0000 −1.39825 −0.699127 0.714998i \(-0.746428\pi\)
−0.699127 + 0.714998i \(0.746428\pi\)
\(402\) 8.00000i 0.399004i
\(403\) 30.0000i 1.49441i
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 9.00000 0.446663
\(407\) − 24.0000i − 1.18964i
\(408\) 4.00000i 0.198030i
\(409\) 23.0000 1.13728 0.568638 0.822588i \(-0.307470\pi\)
0.568638 + 0.822588i \(0.307470\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) − 13.0000i − 0.640464i
\(413\) 24.0000i 1.18096i
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) −3.00000 −0.147087
\(417\) 8.00000i 0.391762i
\(418\) 9.00000i 0.440204i
\(419\) −19.0000 −0.928211 −0.464105 0.885780i \(-0.653624\pi\)
−0.464105 + 0.885780i \(0.653624\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) − 22.0000i − 1.07094i
\(423\) 2.00000i 0.0972433i
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 14.0000 0.678302
\(427\) − 36.0000i − 1.74216i
\(428\) − 12.0000i − 0.580042i
\(429\) 9.00000 0.434524
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 4.00000i 0.192228i 0.995370 + 0.0961139i \(0.0306413\pi\)
−0.995370 + 0.0961139i \(0.969359\pi\)
\(434\) 30.0000 1.44005
\(435\) 0 0
\(436\) 0 0
\(437\) − 3.00000i − 0.143509i
\(438\) − 13.0000i − 0.621164i
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) − 12.0000i − 0.570782i
\(443\) 26.0000i 1.23530i 0.786454 + 0.617649i \(0.211915\pi\)
−0.786454 + 0.617649i \(0.788085\pi\)
\(444\) −8.00000 −0.379663
\(445\) 0 0
\(446\) −6.00000 −0.284108
\(447\) − 6.00000i − 0.283790i
\(448\) 3.00000i 0.141737i
\(449\) 26.0000 1.22702 0.613508 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) 0 0
\(451\) 27.0000 1.27138
\(452\) − 6.00000i − 0.282216i
\(453\) 22.0000i 1.03365i
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) − 22.0000i − 1.02912i −0.857455 0.514558i \(-0.827956\pi\)
0.857455 0.514558i \(-0.172044\pi\)
\(458\) 6.00000i 0.280362i
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −9.00000 −0.419172 −0.209586 0.977790i \(-0.567212\pi\)
−0.209586 + 0.977790i \(0.567212\pi\)
\(462\) − 9.00000i − 0.418718i
\(463\) − 4.00000i − 0.185896i −0.995671 0.0929479i \(-0.970371\pi\)
0.995671 0.0929479i \(-0.0296290\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) −1.00000 −0.0463241
\(467\) − 11.0000i − 0.509019i −0.967070 0.254510i \(-0.918086\pi\)
0.967070 0.254510i \(-0.0819141\pi\)
\(468\) − 3.00000i − 0.138675i
\(469\) −24.0000 −1.10822
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) − 8.00000i − 0.368230i
\(473\) − 3.00000i − 0.137940i
\(474\) −17.0000 −0.780836
\(475\) 0 0
\(476\) −12.0000 −0.550019
\(477\) 6.00000i 0.274721i
\(478\) − 10.0000i − 0.457389i
\(479\) −33.0000 −1.50781 −0.753904 0.656984i \(-0.771832\pi\)
−0.753904 + 0.656984i \(0.771832\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) 10.0000i 0.455488i
\(483\) 3.00000i 0.136505i
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) 12.0000i 0.543214i
\(489\) 6.00000 0.271329
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) − 9.00000i − 0.405751i
\(493\) − 12.0000i − 0.540453i
\(494\) −9.00000 −0.404929
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) 42.0000i 1.88396i
\(498\) − 1.00000i − 0.0448111i
\(499\) −34.0000 −1.52205 −0.761025 0.648723i \(-0.775303\pi\)
−0.761025 + 0.648723i \(0.775303\pi\)
\(500\) 0 0
\(501\) −2.00000 −0.0893534
\(502\) 16.0000i 0.714115i
\(503\) − 35.0000i − 1.56057i −0.625422 0.780286i \(-0.715073\pi\)
0.625422 0.780286i \(-0.284927\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) 3.00000 0.133366
\(507\) − 4.00000i − 0.177646i
\(508\) 16.0000i 0.709885i
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) 39.0000 1.72526
\(512\) − 1.00000i − 0.0441942i
\(513\) 3.00000i 0.132453i
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) −1.00000 −0.0440225
\(517\) 6.00000i 0.263880i
\(518\) − 24.0000i − 1.05450i
\(519\) 15.0000 0.658427
\(520\) 0 0
\(521\) 4.00000 0.175243 0.0876216 0.996154i \(-0.472073\pi\)
0.0876216 + 0.996154i \(0.472073\pi\)
\(522\) − 3.00000i − 0.131306i
\(523\) 11.0000i 0.480996i 0.970650 + 0.240498i \(0.0773108\pi\)
−0.970650 + 0.240498i \(0.922689\pi\)
\(524\) −10.0000 −0.436852
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) − 40.0000i − 1.74243i
\(528\) 3.00000i 0.130558i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 9.00000i 0.390199i
\(533\) 27.0000i 1.16950i
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) − 10.0000i − 0.431532i
\(538\) − 5.00000i − 0.215565i
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −13.0000 −0.558914 −0.279457 0.960158i \(-0.590154\pi\)
−0.279457 + 0.960158i \(0.590154\pi\)
\(542\) − 22.0000i − 0.944981i
\(543\) 16.0000i 0.686626i
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) 9.00000 0.385164
\(547\) − 22.0000i − 0.940652i −0.882493 0.470326i \(-0.844136\pi\)
0.882493 0.470326i \(-0.155864\pi\)
\(548\) − 18.0000i − 0.768922i
\(549\) −12.0000 −0.512148
\(550\) 0 0
\(551\) −9.00000 −0.383413
\(552\) − 1.00000i − 0.0425628i
\(553\) − 51.0000i − 2.16874i
\(554\) 13.0000 0.552317
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 32.0000i 1.35588i 0.735116 + 0.677942i \(0.237128\pi\)
−0.735116 + 0.677942i \(0.762872\pi\)
\(558\) − 10.0000i − 0.423334i
\(559\) 3.00000 0.126886
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) 10.0000i 0.421825i
\(563\) 5.00000i 0.210725i 0.994434 + 0.105362i \(0.0336003\pi\)
−0.994434 + 0.105362i \(0.966400\pi\)
\(564\) 2.00000 0.0842152
\(565\) 0 0
\(566\) 0 0
\(567\) − 3.00000i − 0.125988i
\(568\) − 14.0000i − 0.587427i
\(569\) 20.0000 0.838444 0.419222 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) − 9.00000i − 0.376309i
\(573\) 15.0000i 0.626634i
\(574\) 27.0000 1.12696
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 7.00000i 0.291414i 0.989328 + 0.145707i \(0.0465456\pi\)
−0.989328 + 0.145707i \(0.953454\pi\)
\(578\) − 1.00000i − 0.0415945i
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) 3.00000 0.124461
\(582\) 6.00000i 0.248708i
\(583\) 18.0000i 0.745484i
\(584\) −13.0000 −0.537944
\(585\) 0 0
\(586\) 0 0
\(587\) − 24.0000i − 0.990586i −0.868726 0.495293i \(-0.835061\pi\)
0.868726 0.495293i \(-0.164939\pi\)
\(588\) − 2.00000i − 0.0824786i
\(589\) −30.0000 −1.23613
\(590\) 0 0
\(591\) 17.0000 0.699287
\(592\) 8.00000i 0.328798i
\(593\) − 21.0000i − 0.862367i −0.902264 0.431183i \(-0.858096\pi\)
0.902264 0.431183i \(-0.141904\pi\)
\(594\) −3.00000 −0.123091
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 11.0000i 0.450200i
\(598\) 3.00000i 0.122679i
\(599\) −6.00000 −0.245153 −0.122577 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) − 3.00000i − 0.122271i
\(603\) 8.00000i 0.325785i
\(604\) 22.0000 0.895167
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) − 2.00000i − 0.0811775i −0.999176 0.0405887i \(-0.987077\pi\)
0.999176 0.0405887i \(-0.0129233\pi\)
\(608\) − 3.00000i − 0.121666i
\(609\) 9.00000 0.364698
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 4.00000i 0.161690i
\(613\) 20.0000i 0.807792i 0.914805 + 0.403896i \(0.132344\pi\)
−0.914805 + 0.403896i \(0.867656\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) −9.00000 −0.362620
\(617\) 16.0000i 0.644136i 0.946717 + 0.322068i \(0.104378\pi\)
−0.946717 + 0.322068i \(0.895622\pi\)
\(618\) − 13.0000i − 0.522937i
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) − 24.0000i − 0.962312i
\(623\) − 18.0000i − 0.721155i
\(624\) −3.00000 −0.120096
\(625\) 0 0
\(626\) 24.0000 0.959233
\(627\) 9.00000i 0.359425i
\(628\) 6.00000i 0.239426i
\(629\) −32.0000 −1.27592
\(630\) 0 0
\(631\) 5.00000 0.199047 0.0995234 0.995035i \(-0.468268\pi\)
0.0995234 + 0.995035i \(0.468268\pi\)
\(632\) 17.0000i 0.676224i
\(633\) − 22.0000i − 0.874421i
\(634\) −31.0000 −1.23117
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 6.00000i 0.237729i
\(638\) − 9.00000i − 0.356313i
\(639\) 14.0000 0.553831
\(640\) 0 0
\(641\) −16.0000 −0.631962 −0.315981 0.948766i \(-0.602334\pi\)
−0.315981 + 0.948766i \(0.602334\pi\)
\(642\) − 12.0000i − 0.473602i
\(643\) − 31.0000i − 1.22252i −0.791430 0.611260i \(-0.790663\pi\)
0.791430 0.611260i \(-0.209337\pi\)
\(644\) 3.00000 0.118217
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 18.0000i 0.707653i 0.935311 + 0.353827i \(0.115120\pi\)
−0.935311 + 0.353827i \(0.884880\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) 30.0000 1.17579
\(652\) − 6.00000i − 0.234978i
\(653\) − 3.00000i − 0.117399i −0.998276 0.0586995i \(-0.981305\pi\)
0.998276 0.0586995i \(-0.0186954\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −9.00000 −0.351391
\(657\) − 13.0000i − 0.507178i
\(658\) 6.00000i 0.233904i
\(659\) 33.0000 1.28550 0.642749 0.766077i \(-0.277794\pi\)
0.642749 + 0.766077i \(0.277794\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 26.0000i 1.01052i
\(663\) − 12.0000i − 0.466041i
\(664\) −1.00000 −0.0388075
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) 3.00000i 0.116160i
\(668\) 2.00000i 0.0773823i
\(669\) −6.00000 −0.231973
\(670\) 0 0
\(671\) −36.0000 −1.38976
\(672\) 3.00000i 0.115728i
\(673\) 3.00000i 0.115642i 0.998327 + 0.0578208i \(0.0184152\pi\)
−0.998327 + 0.0578208i \(0.981585\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) − 6.00000i − 0.230429i
\(679\) −18.0000 −0.690777
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) − 30.0000i − 1.14876i
\(683\) 38.0000i 1.45403i 0.686622 + 0.727015i \(0.259093\pi\)
−0.686622 + 0.727015i \(0.740907\pi\)
\(684\) 3.00000 0.114708
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) 6.00000i 0.228914i
\(688\) 1.00000i 0.0381246i
\(689\) −18.0000 −0.685745
\(690\) 0 0
\(691\) −48.0000 −1.82601 −0.913003 0.407953i \(-0.866243\pi\)
−0.913003 + 0.407953i \(0.866243\pi\)
\(692\) − 15.0000i − 0.570214i
\(693\) − 9.00000i − 0.341882i
\(694\) 0 0
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) − 36.0000i − 1.36360i
\(698\) − 5.00000i − 0.189253i
\(699\) −1.00000 −0.0378235
\(700\) 0 0
\(701\) −36.0000 −1.35970 −0.679851 0.733351i \(-0.737955\pi\)
−0.679851 + 0.733351i \(0.737955\pi\)
\(702\) − 3.00000i − 0.113228i
\(703\) 24.0000i 0.905177i
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) 9.00000 0.338719
\(707\) 18.0000i 0.676960i
\(708\) − 8.00000i − 0.300658i
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) −17.0000 −0.637550
\(712\) 6.00000i 0.224860i
\(713\) 10.0000i 0.374503i
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) −10.0000 −0.373718
\(717\) − 10.0000i − 0.373457i
\(718\) 31.0000i 1.15691i
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 0 0
\(721\) 39.0000 1.45244
\(722\) 10.0000i 0.372161i
\(723\) 10.0000i 0.371904i
\(724\) 16.0000 0.594635
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) 16.0000i 0.593407i 0.954970 + 0.296704i \(0.0958873\pi\)
−0.954970 + 0.296704i \(0.904113\pi\)
\(728\) − 9.00000i − 0.333562i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −4.00000 −0.147945
\(732\) 12.0000i 0.443533i
\(733\) 14.0000i 0.517102i 0.965998 + 0.258551i \(0.0832450\pi\)
−0.965998 + 0.258551i \(0.916755\pi\)
\(734\) −27.0000 −0.996588
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 24.0000i 0.884051i
\(738\) − 9.00000i − 0.331295i
\(739\) −50.0000 −1.83928 −0.919640 0.392763i \(-0.871519\pi\)
−0.919640 + 0.392763i \(0.871519\pi\)
\(740\) 0 0
\(741\) −9.00000 −0.330623
\(742\) 18.0000i 0.660801i
\(743\) − 29.0000i − 1.06391i −0.846774 0.531953i \(-0.821458\pi\)
0.846774 0.531953i \(-0.178542\pi\)
\(744\) −10.0000 −0.366618
\(745\) 0 0
\(746\) −16.0000 −0.585802
\(747\) − 1.00000i − 0.0365881i
\(748\) 12.0000i 0.438763i
\(749\) 36.0000 1.31541
\(750\) 0 0
\(751\) 45.0000 1.64207 0.821037 0.570875i \(-0.193396\pi\)
0.821037 + 0.570875i \(0.193396\pi\)
\(752\) − 2.00000i − 0.0729325i
\(753\) 16.0000i 0.583072i
\(754\) 9.00000 0.327761
\(755\) 0 0
\(756\) −3.00000 −0.109109
\(757\) 20.0000i 0.726912i 0.931611 + 0.363456i \(0.118403\pi\)
−0.931611 + 0.363456i \(0.881597\pi\)
\(758\) 12.0000i 0.435860i
\(759\) 3.00000 0.108893
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 16.0000i 0.579619i
\(763\) 0 0
\(764\) 15.0000 0.542681
\(765\) 0 0
\(766\) −29.0000 −1.04781
\(767\) 24.0000i 0.866590i
\(768\) − 1.00000i − 0.0360844i
\(769\) −36.0000 −1.29819 −0.649097 0.760706i \(-0.724853\pi\)
−0.649097 + 0.760706i \(0.724853\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 6.00000i 0.215945i
\(773\) − 40.0000i − 1.43870i −0.694648 0.719350i \(-0.744440\pi\)
0.694648 0.719350i \(-0.255560\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) − 24.0000i − 0.860995i
\(778\) 38.0000i 1.36237i
\(779\) −27.0000 −0.967375
\(780\) 0 0
\(781\) 42.0000 1.50288
\(782\) − 4.00000i − 0.143040i
\(783\) − 3.00000i − 0.107211i
\(784\) −2.00000 −0.0714286
\(785\) 0 0
\(786\) −10.0000 −0.356688
\(787\) 5.00000i 0.178231i 0.996021 + 0.0891154i \(0.0284040\pi\)
−0.996021 + 0.0891154i \(0.971596\pi\)
\(788\) − 17.0000i − 0.605600i
\(789\) −8.00000 −0.284808
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 3.00000i 0.106600i
\(793\) − 36.0000i − 1.27840i
\(794\) −30.0000 −1.06466
\(795\) 0 0
\(796\) 11.0000 0.389885
\(797\) − 24.0000i − 0.850124i −0.905164 0.425062i \(-0.860252\pi\)
0.905164 0.425062i \(-0.139748\pi\)
\(798\) 9.00000i 0.318597i
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 28.0000i 0.988714i
\(803\) − 39.0000i − 1.37628i
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 30.0000 1.05670
\(807\) − 5.00000i − 0.176008i
\(808\) − 6.00000i − 0.211079i
\(809\) −21.0000 −0.738321 −0.369160 0.929366i \(-0.620355\pi\)
−0.369160 + 0.929366i \(0.620355\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) − 9.00000i − 0.315838i
\(813\) − 22.0000i − 0.771574i
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) 3.00000i 0.104957i
\(818\) − 23.0000i − 0.804176i
\(819\) 9.00000 0.314485
\(820\) 0 0
\(821\) −55.0000 −1.91951 −0.959757 0.280833i \(-0.909389\pi\)
−0.959757 + 0.280833i \(0.909389\pi\)
\(822\) − 18.0000i − 0.627822i
\(823\) − 16.0000i − 0.557725i −0.960331 0.278862i \(-0.910043\pi\)
0.960331 0.278862i \(-0.0899574\pi\)
\(824\) −13.0000 −0.452876
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) 9.00000i 0.312961i 0.987681 + 0.156480i \(0.0500148\pi\)
−0.987681 + 0.156480i \(0.949985\pi\)
\(828\) − 1.00000i − 0.0347524i
\(829\) −11.0000 −0.382046 −0.191023 0.981586i \(-0.561180\pi\)
−0.191023 + 0.981586i \(0.561180\pi\)
\(830\) 0 0
\(831\) 13.0000 0.450965
\(832\) 3.00000i 0.104006i
\(833\) − 8.00000i − 0.277184i
\(834\) 8.00000 0.277017
\(835\) 0 0
\(836\) 9.00000 0.311272
\(837\) − 10.0000i − 0.345651i
\(838\) 19.0000i 0.656344i
\(839\) −33.0000 −1.13929 −0.569643 0.821892i \(-0.692919\pi\)
−0.569643 + 0.821892i \(0.692919\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 10.0000i 0.344623i
\(843\) 10.0000i 0.344418i
\(844\) −22.0000 −0.757271
\(845\) 0 0
\(846\) 2.00000 0.0687614
\(847\) 6.00000i 0.206162i
\(848\) − 6.00000i − 0.206041i
\(849\) 0 0
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) − 14.0000i − 0.479632i
\(853\) − 23.0000i − 0.787505i −0.919216 0.393753i \(-0.871177\pi\)
0.919216 0.393753i \(-0.128823\pi\)
\(854\) −36.0000 −1.23189
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) − 22.0000i − 0.751506i −0.926720 0.375753i \(-0.877384\pi\)
0.926720 0.375753i \(-0.122616\pi\)
\(858\) − 9.00000i − 0.307255i
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 27.0000 0.920158
\(862\) − 16.0000i − 0.544962i
\(863\) 54.0000i 1.83818i 0.394046 + 0.919091i \(0.371075\pi\)
−0.394046 + 0.919091i \(0.628925\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 4.00000 0.135926
\(867\) − 1.00000i − 0.0339618i
\(868\) − 30.0000i − 1.01827i
\(869\) −51.0000 −1.73006
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 0 0
\(873\) 6.00000i 0.203069i
\(874\) −3.00000 −0.101477
\(875\) 0 0
\(876\) −13.0000 −0.439229
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) − 4.00000i − 0.134993i
\(879\) 0 0
\(880\) 0 0
\(881\) −40.0000 −1.34763 −0.673817 0.738898i \(-0.735346\pi\)
−0.673817 + 0.738898i \(0.735346\pi\)
\(882\) − 2.00000i − 0.0673435i
\(883\) − 48.0000i − 1.61533i −0.589643 0.807664i \(-0.700731\pi\)
0.589643 0.807664i \(-0.299269\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 26.0000 0.873487
\(887\) 8.00000i 0.268614i 0.990940 + 0.134307i \(0.0428808\pi\)
−0.990940 + 0.134307i \(0.957119\pi\)
\(888\) 8.00000i 0.268462i
\(889\) −48.0000 −1.60987
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 6.00000i 0.200895i
\(893\) − 6.00000i − 0.200782i
\(894\) −6.00000 −0.200670
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) 3.00000i 0.100167i
\(898\) − 26.0000i − 0.867631i
\(899\) 30.0000 1.00056
\(900\) 0 0
\(901\) 24.0000 0.799556
\(902\) − 27.0000i − 0.899002i
\(903\) − 3.00000i − 0.0998337i
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 22.0000 0.730901
\(907\) 1.00000i 0.0332045i 0.999862 + 0.0166022i \(0.00528490\pi\)
−0.999862 + 0.0166022i \(0.994715\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 55.0000 1.82223 0.911116 0.412151i \(-0.135222\pi\)
0.911116 + 0.412151i \(0.135222\pi\)
\(912\) − 3.00000i − 0.0993399i
\(913\) − 3.00000i − 0.0992855i
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) − 30.0000i − 0.990687i
\(918\) 4.00000i 0.132020i
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) −8.00000 −0.263609
\(922\) 9.00000i 0.296399i
\(923\) 42.0000i 1.38245i
\(924\) −9.00000 −0.296078
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) − 13.0000i − 0.426976i
\(928\) 3.00000i 0.0984798i
\(929\) 41.0000 1.34517 0.672583 0.740022i \(-0.265185\pi\)
0.672583 + 0.740022i \(0.265185\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 1.00000i 0.0327561i
\(933\) − 24.0000i − 0.785725i
\(934\) −11.0000 −0.359931
\(935\) 0 0
\(936\) −3.00000 −0.0980581
\(937\) 22.0000i 0.718709i 0.933201 + 0.359354i \(0.117003\pi\)
−0.933201 + 0.359354i \(0.882997\pi\)
\(938\) 24.0000i 0.783628i
\(939\) 24.0000 0.783210
\(940\) 0 0
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 6.00000i 0.195491i
\(943\) 9.00000i 0.293080i
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) −3.00000 −0.0975384
\(947\) − 2.00000i − 0.0649913i −0.999472 0.0324956i \(-0.989654\pi\)
0.999472 0.0324956i \(-0.0103455\pi\)
\(948\) 17.0000i 0.552134i
\(949\) 39.0000 1.26599
\(950\) 0 0
\(951\) −31.0000 −1.00524
\(952\) 12.0000i 0.388922i
\(953\) − 24.0000i − 0.777436i −0.921357 0.388718i \(-0.872918\pi\)
0.921357 0.388718i \(-0.127082\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) −10.0000 −0.323423
\(957\) − 9.00000i − 0.290929i
\(958\) 33.0000i 1.06618i
\(959\) 54.0000 1.74375
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) − 24.0000i − 0.773791i
\(963\) − 12.0000i − 0.386695i
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 3.00000 0.0965234
\(967\) 44.0000i 1.41494i 0.706741 + 0.707472i \(0.250165\pi\)
−0.706741 + 0.707472i \(0.749835\pi\)
\(968\) − 2.00000i − 0.0642824i
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) 43.0000 1.37994 0.689968 0.723840i \(-0.257625\pi\)
0.689968 + 0.723840i \(0.257625\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 24.0000i 0.769405i
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) 12.0000 0.384111
\(977\) − 38.0000i − 1.21573i −0.794041 0.607864i \(-0.792027\pi\)
0.794041 0.607864i \(-0.207973\pi\)
\(978\) − 6.00000i − 0.191859i
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) 0 0
\(982\) − 12.0000i − 0.382935i
\(983\) − 39.0000i − 1.24391i −0.783054 0.621953i \(-0.786339\pi\)
0.783054 0.621953i \(-0.213661\pi\)
\(984\) −9.00000 −0.286910
\(985\) 0 0
\(986\) −12.0000 −0.382158
\(987\) 6.00000i 0.190982i
\(988\) 9.00000i 0.286328i
\(989\) 1.00000 0.0317982
\(990\) 0 0
\(991\) −26.0000 −0.825917 −0.412959 0.910750i \(-0.635505\pi\)
−0.412959 + 0.910750i \(0.635505\pi\)
\(992\) 10.0000i 0.317500i
\(993\) 26.0000i 0.825085i
\(994\) 42.0000 1.33216
\(995\) 0 0
\(996\) −1.00000 −0.0316862
\(997\) 49.0000i 1.55185i 0.630828 + 0.775923i \(0.282715\pi\)
−0.630828 + 0.775923i \(0.717285\pi\)
\(998\) 34.0000i 1.07625i
\(999\) −8.00000 −0.253109
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.b.2899.1 2
5.2 odd 4 3450.2.a.s.1.1 yes 1
5.3 odd 4 3450.2.a.h.1.1 1
5.4 even 2 inner 3450.2.d.b.2899.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.h.1.1 1 5.3 odd 4
3450.2.a.s.1.1 yes 1 5.2 odd 4
3450.2.d.b.2899.1 2 1.1 even 1 trivial
3450.2.d.b.2899.2 2 5.4 even 2 inner