Properties

Label 3450.2.a.bi.1.2
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +5.12311 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +5.12311 q^{7} -1.00000 q^{8} +1.00000 q^{9} +5.12311 q^{11} +1.00000 q^{12} -2.00000 q^{13} -5.12311 q^{14} +1.00000 q^{16} -7.12311 q^{17} -1.00000 q^{18} +4.00000 q^{19} +5.12311 q^{21} -5.12311 q^{22} -1.00000 q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} +5.12311 q^{28} +2.00000 q^{29} -1.00000 q^{32} +5.12311 q^{33} +7.12311 q^{34} +1.00000 q^{36} +7.12311 q^{37} -4.00000 q^{38} -2.00000 q^{39} +2.00000 q^{41} -5.12311 q^{42} +5.12311 q^{44} +1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +19.2462 q^{49} -7.12311 q^{51} -2.00000 q^{52} +4.24621 q^{53} -1.00000 q^{54} -5.12311 q^{56} +4.00000 q^{57} -2.00000 q^{58} -14.2462 q^{59} +0.876894 q^{61} +5.12311 q^{63} +1.00000 q^{64} -5.12311 q^{66} +8.00000 q^{67} -7.12311 q^{68} -1.00000 q^{69} +6.24621 q^{71} -1.00000 q^{72} -12.2462 q^{73} -7.12311 q^{74} +4.00000 q^{76} +26.2462 q^{77} +2.00000 q^{78} -5.12311 q^{79} +1.00000 q^{81} -2.00000 q^{82} -11.3693 q^{83} +5.12311 q^{84} +2.00000 q^{87} -5.12311 q^{88} +3.12311 q^{89} -10.2462 q^{91} -1.00000 q^{92} -8.00000 q^{94} -1.00000 q^{96} -0.246211 q^{97} -19.2462 q^{98} +5.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + 2q^{11} + 2q^{12} - 4q^{13} - 2q^{14} + 2q^{16} - 6q^{17} - 2q^{18} + 8q^{19} + 2q^{21} - 2q^{22} - 2q^{23} - 2q^{24} + 4q^{26} + 2q^{27} + 2q^{28} + 4q^{29} - 2q^{32} + 2q^{33} + 6q^{34} + 2q^{36} + 6q^{37} - 8q^{38} - 4q^{39} + 4q^{41} - 2q^{42} + 2q^{44} + 2q^{46} + 16q^{47} + 2q^{48} + 22q^{49} - 6q^{51} - 4q^{52} - 8q^{53} - 2q^{54} - 2q^{56} + 8q^{57} - 4q^{58} - 12q^{59} + 10q^{61} + 2q^{63} + 2q^{64} - 2q^{66} + 16q^{67} - 6q^{68} - 2q^{69} - 4q^{71} - 2q^{72} - 8q^{73} - 6q^{74} + 8q^{76} + 36q^{77} + 4q^{78} - 2q^{79} + 2q^{81} - 4q^{82} + 2q^{83} + 2q^{84} + 4q^{87} - 2q^{88} - 2q^{89} - 4q^{91} - 2q^{92} - 16q^{94} - 2q^{96} + 16q^{97} - 22q^{98} + 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 5.12311 1.93635 0.968176 0.250270i \(-0.0805195\pi\)
0.968176 + 0.250270i \(0.0805195\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.12311 1.54467 0.772337 0.635213i \(-0.219088\pi\)
0.772337 + 0.635213i \(0.219088\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −5.12311 −1.36921
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −7.12311 −1.72761 −0.863803 0.503829i \(-0.831924\pi\)
−0.863803 + 0.503829i \(0.831924\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 5.12311 1.11795
\(22\) −5.12311 −1.09225
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) 5.12311 0.968176
\(29\) 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 \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.12311 0.891818
\(34\) 7.12311 1.22160
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 7.12311 1.17103 0.585516 0.810661i \(-0.300892\pi\)
0.585516 + 0.810661i \(0.300892\pi\)
\(38\) −4.00000 −0.648886
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) −5.12311 −0.790512
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 5.12311 0.772337
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) 19.2462 2.74946
\(50\) 0 0
\(51\) −7.12311 −0.997434
\(52\) −2.00000 −0.277350
\(53\) 4.24621 0.583262 0.291631 0.956531i \(-0.405802\pi\)
0.291631 + 0.956531i \(0.405802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −5.12311 −0.684604
\(57\) 4.00000 0.529813
\(58\) −2.00000 −0.262613
\(59\) −14.2462 −1.85470 −0.927349 0.374197i \(-0.877918\pi\)
−0.927349 + 0.374197i \(0.877918\pi\)
\(60\) 0 0
\(61\) 0.876894 0.112275 0.0561374 0.998423i \(-0.482122\pi\)
0.0561374 + 0.998423i \(0.482122\pi\)
\(62\) 0 0
\(63\) 5.12311 0.645451
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −5.12311 −0.630611
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −7.12311 −0.863803
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 6.24621 0.741289 0.370644 0.928775i \(-0.379137\pi\)
0.370644 + 0.928775i \(0.379137\pi\)
\(72\) −1.00000 −0.117851
\(73\) −12.2462 −1.43331 −0.716655 0.697428i \(-0.754328\pi\)
−0.716655 + 0.697428i \(0.754328\pi\)
\(74\) −7.12311 −0.828044
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 26.2462 2.99103
\(78\) 2.00000 0.226455
\(79\) −5.12311 −0.576394 −0.288197 0.957571i \(-0.593056\pi\)
−0.288197 + 0.957571i \(0.593056\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −11.3693 −1.24794 −0.623972 0.781446i \(-0.714482\pi\)
−0.623972 + 0.781446i \(0.714482\pi\)
\(84\) 5.12311 0.558977
\(85\) 0 0
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) −5.12311 −0.546125
\(89\) 3.12311 0.331049 0.165524 0.986206i \(-0.447068\pi\)
0.165524 + 0.986206i \(0.447068\pi\)
\(90\) 0 0
\(91\) −10.2462 −1.07409
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −0.246211 −0.0249990 −0.0124995 0.999922i \(-0.503979\pi\)
−0.0124995 + 0.999922i \(0.503979\pi\)
\(98\) −19.2462 −1.94416
\(99\) 5.12311 0.514891
\(100\) 0 0
\(101\) −16.2462 −1.61656 −0.808279 0.588799i \(-0.799601\pi\)
−0.808279 + 0.588799i \(0.799601\pi\)
\(102\) 7.12311 0.705293
\(103\) −15.3693 −1.51438 −0.757192 0.653192i \(-0.773429\pi\)
−0.757192 + 0.653192i \(0.773429\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −4.24621 −0.412428
\(107\) −11.3693 −1.09911 −0.549557 0.835456i \(-0.685203\pi\)
−0.549557 + 0.835456i \(0.685203\pi\)
\(108\) 1.00000 0.0962250
\(109\) 11.1231 1.06540 0.532700 0.846304i \(-0.321177\pi\)
0.532700 + 0.846304i \(0.321177\pi\)
\(110\) 0 0
\(111\) 7.12311 0.676095
\(112\) 5.12311 0.484088
\(113\) 0.876894 0.0824913 0.0412456 0.999149i \(-0.486867\pi\)
0.0412456 + 0.999149i \(0.486867\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) −2.00000 −0.184900
\(118\) 14.2462 1.31147
\(119\) −36.4924 −3.34525
\(120\) 0 0
\(121\) 15.2462 1.38602
\(122\) −0.876894 −0.0793903
\(123\) 2.00000 0.180334
\(124\) 0 0
\(125\) 0 0
\(126\) −5.12311 −0.456403
\(127\) 22.2462 1.97403 0.987016 0.160622i \(-0.0513500\pi\)
0.987016 + 0.160622i \(0.0513500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 5.12311 0.445909
\(133\) 20.4924 1.77692
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 7.12311 0.610801
\(137\) 19.1231 1.63380 0.816899 0.576781i \(-0.195692\pi\)
0.816899 + 0.576781i \(0.195692\pi\)
\(138\) 1.00000 0.0851257
\(139\) −16.4924 −1.39887 −0.699435 0.714697i \(-0.746565\pi\)
−0.699435 + 0.714697i \(0.746565\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) −6.24621 −0.524170
\(143\) −10.2462 −0.856831
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 12.2462 1.01350
\(147\) 19.2462 1.58740
\(148\) 7.12311 0.585516
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −10.2462 −0.833825 −0.416912 0.908947i \(-0.636888\pi\)
−0.416912 + 0.908947i \(0.636888\pi\)
\(152\) −4.00000 −0.324443
\(153\) −7.12311 −0.575869
\(154\) −26.2462 −2.11498
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) −0.876894 −0.0699838 −0.0349919 0.999388i \(-0.511141\pi\)
−0.0349919 + 0.999388i \(0.511141\pi\)
\(158\) 5.12311 0.407572
\(159\) 4.24621 0.336746
\(160\) 0 0
\(161\) −5.12311 −0.403757
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 11.3693 0.882430
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) −5.12311 −0.395256
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) −3.75379 −0.285395 −0.142698 0.989766i \(-0.545578\pi\)
−0.142698 + 0.989766i \(0.545578\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) 5.12311 0.386169
\(177\) −14.2462 −1.07081
\(178\) −3.12311 −0.234087
\(179\) 24.4924 1.83065 0.915325 0.402716i \(-0.131934\pi\)
0.915325 + 0.402716i \(0.131934\pi\)
\(180\) 0 0
\(181\) 19.1231 1.42141 0.710705 0.703491i \(-0.248376\pi\)
0.710705 + 0.703491i \(0.248376\pi\)
\(182\) 10.2462 0.759500
\(183\) 0.876894 0.0648219
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) −36.4924 −2.66859
\(188\) 8.00000 0.583460
\(189\) 5.12311 0.372651
\(190\) 0 0
\(191\) −20.4924 −1.48278 −0.741390 0.671075i \(-0.765833\pi\)
−0.741390 + 0.671075i \(0.765833\pi\)
\(192\) 1.00000 0.0721688
\(193\) 8.24621 0.593575 0.296788 0.954944i \(-0.404085\pi\)
0.296788 + 0.954944i \(0.404085\pi\)
\(194\) 0.246211 0.0176769
\(195\) 0 0
\(196\) 19.2462 1.37473
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) −5.12311 −0.364083
\(199\) 10.8769 0.771043 0.385521 0.922699i \(-0.374022\pi\)
0.385521 + 0.922699i \(0.374022\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 16.2462 1.14308
\(203\) 10.2462 0.719143
\(204\) −7.12311 −0.498717
\(205\) 0 0
\(206\) 15.3693 1.07083
\(207\) −1.00000 −0.0695048
\(208\) −2.00000 −0.138675
\(209\) 20.4924 1.41749
\(210\) 0 0
\(211\) 16.4924 1.13539 0.567693 0.823241i \(-0.307836\pi\)
0.567693 + 0.823241i \(0.307836\pi\)
\(212\) 4.24621 0.291631
\(213\) 6.24621 0.427983
\(214\) 11.3693 0.777191
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −11.1231 −0.753352
\(219\) −12.2462 −0.827522
\(220\) 0 0
\(221\) 14.2462 0.958304
\(222\) −7.12311 −0.478072
\(223\) −6.24621 −0.418277 −0.209139 0.977886i \(-0.567066\pi\)
−0.209139 + 0.977886i \(0.567066\pi\)
\(224\) −5.12311 −0.342302
\(225\) 0 0
\(226\) −0.876894 −0.0583301
\(227\) −19.3693 −1.28559 −0.642793 0.766040i \(-0.722225\pi\)
−0.642793 + 0.766040i \(0.722225\pi\)
\(228\) 4.00000 0.264906
\(229\) −1.36932 −0.0904870 −0.0452435 0.998976i \(-0.514406\pi\)
−0.0452435 + 0.998976i \(0.514406\pi\)
\(230\) 0 0
\(231\) 26.2462 1.72687
\(232\) −2.00000 −0.131306
\(233\) 20.7386 1.35863 0.679317 0.733845i \(-0.262276\pi\)
0.679317 + 0.733845i \(0.262276\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −14.2462 −0.927349
\(237\) −5.12311 −0.332781
\(238\) 36.4924 2.36545
\(239\) −1.75379 −0.113443 −0.0567216 0.998390i \(-0.518065\pi\)
−0.0567216 + 0.998390i \(0.518065\pi\)
\(240\) 0 0
\(241\) −2.49242 −0.160551 −0.0802755 0.996773i \(-0.525580\pi\)
−0.0802755 + 0.996773i \(0.525580\pi\)
\(242\) −15.2462 −0.980064
\(243\) 1.00000 0.0641500
\(244\) 0.876894 0.0561374
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) −11.3693 −0.720501
\(250\) 0 0
\(251\) −7.36932 −0.465147 −0.232574 0.972579i \(-0.574715\pi\)
−0.232574 + 0.972579i \(0.574715\pi\)
\(252\) 5.12311 0.322725
\(253\) −5.12311 −0.322087
\(254\) −22.2462 −1.39585
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.24621 0.514385 0.257192 0.966360i \(-0.417203\pi\)
0.257192 + 0.966360i \(0.417203\pi\)
\(258\) 0 0
\(259\) 36.4924 2.26753
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 4.00000 0.247121
\(263\) −10.2462 −0.631808 −0.315904 0.948791i \(-0.602308\pi\)
−0.315904 + 0.948791i \(0.602308\pi\)
\(264\) −5.12311 −0.315305
\(265\) 0 0
\(266\) −20.4924 −1.25647
\(267\) 3.12311 0.191131
\(268\) 8.00000 0.488678
\(269\) −16.2462 −0.990549 −0.495274 0.868737i \(-0.664933\pi\)
−0.495274 + 0.868737i \(0.664933\pi\)
\(270\) 0 0
\(271\) 10.2462 0.622413 0.311207 0.950342i \(-0.399267\pi\)
0.311207 + 0.950342i \(0.399267\pi\)
\(272\) −7.12311 −0.431902
\(273\) −10.2462 −0.620129
\(274\) −19.1231 −1.15527
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −20.2462 −1.21648 −0.608238 0.793754i \(-0.708124\pi\)
−0.608238 + 0.793754i \(0.708124\pi\)
\(278\) 16.4924 0.989150
\(279\) 0 0
\(280\) 0 0
\(281\) −12.8769 −0.768171 −0.384086 0.923298i \(-0.625483\pi\)
−0.384086 + 0.923298i \(0.625483\pi\)
\(282\) −8.00000 −0.476393
\(283\) 2.24621 0.133523 0.0667617 0.997769i \(-0.478733\pi\)
0.0667617 + 0.997769i \(0.478733\pi\)
\(284\) 6.24621 0.370644
\(285\) 0 0
\(286\) 10.2462 0.605871
\(287\) 10.2462 0.604815
\(288\) −1.00000 −0.0589256
\(289\) 33.7386 1.98463
\(290\) 0 0
\(291\) −0.246211 −0.0144332
\(292\) −12.2462 −0.716655
\(293\) −11.7538 −0.686664 −0.343332 0.939214i \(-0.611556\pi\)
−0.343332 + 0.939214i \(0.611556\pi\)
\(294\) −19.2462 −1.12246
\(295\) 0 0
\(296\) −7.12311 −0.414022
\(297\) 5.12311 0.297273
\(298\) 10.0000 0.579284
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) 0 0
\(302\) 10.2462 0.589603
\(303\) −16.2462 −0.933320
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 7.12311 0.407201
\(307\) −0.492423 −0.0281040 −0.0140520 0.999901i \(-0.504473\pi\)
−0.0140520 + 0.999901i \(0.504473\pi\)
\(308\) 26.2462 1.49552
\(309\) −15.3693 −0.874330
\(310\) 0 0
\(311\) 26.7386 1.51621 0.758104 0.652133i \(-0.226126\pi\)
0.758104 + 0.652133i \(0.226126\pi\)
\(312\) 2.00000 0.113228
\(313\) −10.4924 −0.593067 −0.296533 0.955022i \(-0.595831\pi\)
−0.296533 + 0.955022i \(0.595831\pi\)
\(314\) 0.876894 0.0494860
\(315\) 0 0
\(316\) −5.12311 −0.288197
\(317\) −20.7386 −1.16480 −0.582399 0.812903i \(-0.697886\pi\)
−0.582399 + 0.812903i \(0.697886\pi\)
\(318\) −4.24621 −0.238116
\(319\) 10.2462 0.573678
\(320\) 0 0
\(321\) −11.3693 −0.634573
\(322\) 5.12311 0.285500
\(323\) −28.4924 −1.58536
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) 11.1231 0.615109
\(328\) −2.00000 −0.110432
\(329\) 40.9848 2.25957
\(330\) 0 0
\(331\) −32.4924 −1.78595 −0.892973 0.450111i \(-0.851384\pi\)
−0.892973 + 0.450111i \(0.851384\pi\)
\(332\) −11.3693 −0.623972
\(333\) 7.12311 0.390344
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) 5.12311 0.279488
\(337\) 6.49242 0.353665 0.176832 0.984241i \(-0.443415\pi\)
0.176832 + 0.984241i \(0.443415\pi\)
\(338\) 9.00000 0.489535
\(339\) 0.876894 0.0476264
\(340\) 0 0
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 62.7386 3.38757
\(344\) 0 0
\(345\) 0 0
\(346\) 3.75379 0.201805
\(347\) 34.7386 1.86487 0.932434 0.361341i \(-0.117681\pi\)
0.932434 + 0.361341i \(0.117681\pi\)
\(348\) 2.00000 0.107211
\(349\) −24.7386 −1.32423 −0.662114 0.749403i \(-0.730341\pi\)
−0.662114 + 0.749403i \(0.730341\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −5.12311 −0.273062
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 14.2462 0.757178
\(355\) 0 0
\(356\) 3.12311 0.165524
\(357\) −36.4924 −1.93138
\(358\) −24.4924 −1.29446
\(359\) −12.4924 −0.659325 −0.329662 0.944099i \(-0.606935\pi\)
−0.329662 + 0.944099i \(0.606935\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −19.1231 −1.00509
\(363\) 15.2462 0.800219
\(364\) −10.2462 −0.537047
\(365\) 0 0
\(366\) −0.876894 −0.0458360
\(367\) −13.1231 −0.685021 −0.342510 0.939514i \(-0.611277\pi\)
−0.342510 + 0.939514i \(0.611277\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 21.7538 1.12940
\(372\) 0 0
\(373\) −13.3693 −0.692237 −0.346118 0.938191i \(-0.612500\pi\)
−0.346118 + 0.938191i \(0.612500\pi\)
\(374\) 36.4924 1.88698
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) −4.00000 −0.206010
\(378\) −5.12311 −0.263504
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 22.2462 1.13971
\(382\) 20.4924 1.04848
\(383\) −13.7538 −0.702786 −0.351393 0.936228i \(-0.614292\pi\)
−0.351393 + 0.936228i \(0.614292\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −8.24621 −0.419721
\(387\) 0 0
\(388\) −0.246211 −0.0124995
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 7.12311 0.360231
\(392\) −19.2462 −0.972080
\(393\) −4.00000 −0.201773
\(394\) −10.0000 −0.503793
\(395\) 0 0
\(396\) 5.12311 0.257446
\(397\) 34.4924 1.73113 0.865563 0.500801i \(-0.166961\pi\)
0.865563 + 0.500801i \(0.166961\pi\)
\(398\) −10.8769 −0.545209
\(399\) 20.4924 1.02590
\(400\) 0 0
\(401\) −17.3693 −0.867382 −0.433691 0.901062i \(-0.642789\pi\)
−0.433691 + 0.901062i \(0.642789\pi\)
\(402\) −8.00000 −0.399004
\(403\) 0 0
\(404\) −16.2462 −0.808279
\(405\) 0 0
\(406\) −10.2462 −0.508511
\(407\) 36.4924 1.80886
\(408\) 7.12311 0.352646
\(409\) −16.2462 −0.803323 −0.401662 0.915788i \(-0.631567\pi\)
−0.401662 + 0.915788i \(0.631567\pi\)
\(410\) 0 0
\(411\) 19.1231 0.943273
\(412\) −15.3693 −0.757192
\(413\) −72.9848 −3.59135
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −16.4924 −0.807637
\(418\) −20.4924 −1.00232
\(419\) −1.61553 −0.0789237 −0.0394619 0.999221i \(-0.512564\pi\)
−0.0394619 + 0.999221i \(0.512564\pi\)
\(420\) 0 0
\(421\) 13.3693 0.651581 0.325790 0.945442i \(-0.394370\pi\)
0.325790 + 0.945442i \(0.394370\pi\)
\(422\) −16.4924 −0.802839
\(423\) 8.00000 0.388973
\(424\) −4.24621 −0.206214
\(425\) 0 0
\(426\) −6.24621 −0.302630
\(427\) 4.49242 0.217404
\(428\) −11.3693 −0.549557
\(429\) −10.2462 −0.494692
\(430\) 0 0
\(431\) 26.2462 1.26424 0.632118 0.774872i \(-0.282186\pi\)
0.632118 + 0.774872i \(0.282186\pi\)
\(432\) 1.00000 0.0481125
\(433\) −10.4924 −0.504234 −0.252117 0.967697i \(-0.581127\pi\)
−0.252117 + 0.967697i \(0.581127\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 11.1231 0.532700
\(437\) −4.00000 −0.191346
\(438\) 12.2462 0.585147
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 19.2462 0.916486
\(442\) −14.2462 −0.677623
\(443\) 24.4924 1.16367 0.581835 0.813307i \(-0.302335\pi\)
0.581835 + 0.813307i \(0.302335\pi\)
\(444\) 7.12311 0.338048
\(445\) 0 0
\(446\) 6.24621 0.295767
\(447\) −10.0000 −0.472984
\(448\) 5.12311 0.242044
\(449\) −10.4924 −0.495168 −0.247584 0.968866i \(-0.579637\pi\)
−0.247584 + 0.968866i \(0.579637\pi\)
\(450\) 0 0
\(451\) 10.2462 0.482475
\(452\) 0.876894 0.0412456
\(453\) −10.2462 −0.481409
\(454\) 19.3693 0.909047
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 32.7386 1.53145 0.765724 0.643169i \(-0.222381\pi\)
0.765724 + 0.643169i \(0.222381\pi\)
\(458\) 1.36932 0.0639840
\(459\) −7.12311 −0.332478
\(460\) 0 0
\(461\) 32.7386 1.52479 0.762395 0.647112i \(-0.224023\pi\)
0.762395 + 0.647112i \(0.224023\pi\)
\(462\) −26.2462 −1.22108
\(463\) 36.0000 1.67306 0.836531 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −20.7386 −0.960699
\(467\) 3.36932 0.155913 0.0779567 0.996957i \(-0.475160\pi\)
0.0779567 + 0.996957i \(0.475160\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 40.9848 1.89250
\(470\) 0 0
\(471\) −0.876894 −0.0404052
\(472\) 14.2462 0.655735
\(473\) 0 0
\(474\) 5.12311 0.235312
\(475\) 0 0
\(476\) −36.4924 −1.67263
\(477\) 4.24621 0.194421
\(478\) 1.75379 0.0802164
\(479\) −20.4924 −0.936323 −0.468161 0.883643i \(-0.655083\pi\)
−0.468161 + 0.883643i \(0.655083\pi\)
\(480\) 0 0
\(481\) −14.2462 −0.649571
\(482\) 2.49242 0.113527
\(483\) −5.12311 −0.233109
\(484\) 15.2462 0.693010
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −40.4924 −1.83489 −0.917443 0.397866i \(-0.869751\pi\)
−0.917443 + 0.397866i \(0.869751\pi\)
\(488\) −0.876894 −0.0396951
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 2.00000 0.0901670
\(493\) −14.2462 −0.641617
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 32.0000 1.43540
\(498\) 11.3693 0.509471
\(499\) −28.9848 −1.29754 −0.648770 0.760985i \(-0.724716\pi\)
−0.648770 + 0.760985i \(0.724716\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) 7.36932 0.328909
\(503\) 6.73863 0.300461 0.150230 0.988651i \(-0.451998\pi\)
0.150230 + 0.988651i \(0.451998\pi\)
\(504\) −5.12311 −0.228201
\(505\) 0 0
\(506\) 5.12311 0.227750
\(507\) −9.00000 −0.399704
\(508\) 22.2462 0.987016
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) 0 0
\(511\) −62.7386 −2.77539
\(512\) −1.00000 −0.0441942
\(513\) 4.00000 0.176604
\(514\) −8.24621 −0.363725
\(515\) 0 0
\(516\) 0 0
\(517\) 40.9848 1.80251
\(518\) −36.4924 −1.60338
\(519\) −3.75379 −0.164773
\(520\) 0 0
\(521\) 8.87689 0.388904 0.194452 0.980912i \(-0.437707\pi\)
0.194452 + 0.980912i \(0.437707\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 28.4924 1.24589 0.622943 0.782267i \(-0.285937\pi\)
0.622943 + 0.782267i \(0.285937\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 10.2462 0.446756
\(527\) 0 0
\(528\) 5.12311 0.222955
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −14.2462 −0.618233
\(532\) 20.4924 0.888459
\(533\) −4.00000 −0.173259
\(534\) −3.12311 −0.135150
\(535\) 0 0
\(536\) −8.00000 −0.345547
\(537\) 24.4924 1.05693
\(538\) 16.2462 0.700424
\(539\) 98.6004 4.24702
\(540\) 0 0
\(541\) −15.7538 −0.677308 −0.338654 0.940911i \(-0.609972\pi\)
−0.338654 + 0.940911i \(0.609972\pi\)
\(542\) −10.2462 −0.440112
\(543\) 19.1231 0.820651
\(544\) 7.12311 0.305401
\(545\) 0 0
\(546\) 10.2462 0.438497
\(547\) −0.492423 −0.0210545 −0.0105272 0.999945i \(-0.503351\pi\)
−0.0105272 + 0.999945i \(0.503351\pi\)
\(548\) 19.1231 0.816899
\(549\) 0.876894 0.0374249
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 1.00000 0.0425628
\(553\) −26.2462 −1.11610
\(554\) 20.2462 0.860179
\(555\) 0 0
\(556\) −16.4924 −0.699435
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −36.4924 −1.54071
\(562\) 12.8769 0.543179
\(563\) −11.3693 −0.479160 −0.239580 0.970877i \(-0.577010\pi\)
−0.239580 + 0.970877i \(0.577010\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −2.24621 −0.0944153
\(567\) 5.12311 0.215150
\(568\) −6.24621 −0.262085
\(569\) 27.1231 1.13706 0.568530 0.822663i \(-0.307512\pi\)
0.568530 + 0.822663i \(0.307512\pi\)
\(570\) 0 0
\(571\) 34.7386 1.45377 0.726883 0.686761i \(-0.240968\pi\)
0.726883 + 0.686761i \(0.240968\pi\)
\(572\) −10.2462 −0.428416
\(573\) −20.4924 −0.856083
\(574\) −10.2462 −0.427669
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −14.4924 −0.603327 −0.301664 0.953414i \(-0.597542\pi\)
−0.301664 + 0.953414i \(0.597542\pi\)
\(578\) −33.7386 −1.40334
\(579\) 8.24621 0.342701
\(580\) 0 0
\(581\) −58.2462 −2.41646
\(582\) 0.246211 0.0102058
\(583\) 21.7538 0.900950
\(584\) 12.2462 0.506752
\(585\) 0 0
\(586\) 11.7538 0.485545
\(587\) 9.75379 0.402582 0.201291 0.979531i \(-0.435486\pi\)
0.201291 + 0.979531i \(0.435486\pi\)
\(588\) 19.2462 0.793700
\(589\) 0 0
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) 7.12311 0.292758
\(593\) −7.75379 −0.318410 −0.159205 0.987246i \(-0.550893\pi\)
−0.159205 + 0.987246i \(0.550893\pi\)
\(594\) −5.12311 −0.210204
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 10.8769 0.445162
\(598\) −2.00000 −0.0817861
\(599\) −6.24621 −0.255213 −0.127607 0.991825i \(-0.540730\pi\)
−0.127607 + 0.991825i \(0.540730\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) −10.2462 −0.416912
\(605\) 0 0
\(606\) 16.2462 0.659957
\(607\) −26.7386 −1.08529 −0.542644 0.839963i \(-0.682577\pi\)
−0.542644 + 0.839963i \(0.682577\pi\)
\(608\) −4.00000 −0.162221
\(609\) 10.2462 0.415197
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) −7.12311 −0.287934
\(613\) −43.1231 −1.74173 −0.870863 0.491526i \(-0.836439\pi\)
−0.870863 + 0.491526i \(0.836439\pi\)
\(614\) 0.492423 0.0198726
\(615\) 0 0
\(616\) −26.2462 −1.05749
\(617\) −40.1080 −1.61469 −0.807343 0.590083i \(-0.799095\pi\)
−0.807343 + 0.590083i \(0.799095\pi\)
\(618\) 15.3693 0.618245
\(619\) −36.0000 −1.44696 −0.723481 0.690344i \(-0.757459\pi\)
−0.723481 + 0.690344i \(0.757459\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) −26.7386 −1.07212
\(623\) 16.0000 0.641026
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) 10.4924 0.419362
\(627\) 20.4924 0.818389
\(628\) −0.876894 −0.0349919
\(629\) −50.7386 −2.02308
\(630\) 0 0
\(631\) −39.3693 −1.56727 −0.783634 0.621223i \(-0.786636\pi\)
−0.783634 + 0.621223i \(0.786636\pi\)
\(632\) 5.12311 0.203786
\(633\) 16.4924 0.655515
\(634\) 20.7386 0.823636
\(635\) 0 0
\(636\) 4.24621 0.168373
\(637\) −38.4924 −1.52513
\(638\) −10.2462 −0.405651
\(639\) 6.24621 0.247096
\(640\) 0 0
\(641\) −9.36932 −0.370066 −0.185033 0.982732i \(-0.559239\pi\)
−0.185033 + 0.982732i \(0.559239\pi\)
\(642\) 11.3693 0.448711
\(643\) −36.4924 −1.43912 −0.719560 0.694430i \(-0.755657\pi\)
−0.719560 + 0.694430i \(0.755657\pi\)
\(644\) −5.12311 −0.201879
\(645\) 0 0
\(646\) 28.4924 1.12102
\(647\) −11.5076 −0.452410 −0.226205 0.974080i \(-0.572632\pi\)
−0.226205 + 0.974080i \(0.572632\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −72.9848 −2.86491
\(650\) 0 0
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) −10.4924 −0.410600 −0.205300 0.978699i \(-0.565817\pi\)
−0.205300 + 0.978699i \(0.565817\pi\)
\(654\) −11.1231 −0.434948
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) −12.2462 −0.477770
\(658\) −40.9848 −1.59776
\(659\) −7.36932 −0.287068 −0.143534 0.989645i \(-0.545847\pi\)
−0.143534 + 0.989645i \(0.545847\pi\)
\(660\) 0 0
\(661\) 33.8617 1.31707 0.658535 0.752551i \(-0.271177\pi\)
0.658535 + 0.752551i \(0.271177\pi\)
\(662\) 32.4924 1.26285
\(663\) 14.2462 0.553277
\(664\) 11.3693 0.441215
\(665\) 0 0
\(666\) −7.12311 −0.276015
\(667\) −2.00000 −0.0774403
\(668\) 8.00000 0.309529
\(669\) −6.24621 −0.241492
\(670\) 0 0
\(671\) 4.49242 0.173428
\(672\) −5.12311 −0.197628
\(673\) 46.9848 1.81113 0.905566 0.424205i \(-0.139446\pi\)
0.905566 + 0.424205i \(0.139446\pi\)
\(674\) −6.49242 −0.250079
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 46.4924 1.78685 0.893424 0.449213i \(-0.148296\pi\)
0.893424 + 0.449213i \(0.148296\pi\)
\(678\) −0.876894 −0.0336769
\(679\) −1.26137 −0.0484068
\(680\) 0 0
\(681\) −19.3693 −0.742234
\(682\) 0 0
\(683\) −17.7538 −0.679330 −0.339665 0.940547i \(-0.610314\pi\)
−0.339665 + 0.940547i \(0.610314\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) −62.7386 −2.39537
\(687\) −1.36932 −0.0522427
\(688\) 0 0
\(689\) −8.49242 −0.323536
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −3.75379 −0.142698
\(693\) 26.2462 0.997011
\(694\) −34.7386 −1.31866
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) −14.2462 −0.539614
\(698\) 24.7386 0.936371
\(699\) 20.7386 0.784407
\(700\) 0 0
\(701\) −36.2462 −1.36900 −0.684500 0.729013i \(-0.739980\pi\)
−0.684500 + 0.729013i \(0.739980\pi\)
\(702\) 2.00000 0.0754851
\(703\) 28.4924 1.07461
\(704\) 5.12311 0.193084
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) −83.2311 −3.13023
\(708\) −14.2462 −0.535405
\(709\) 13.3693 0.502095 0.251048 0.967975i \(-0.419225\pi\)
0.251048 + 0.967975i \(0.419225\pi\)
\(710\) 0 0
\(711\) −5.12311 −0.192131
\(712\) −3.12311 −0.117043
\(713\) 0 0
\(714\) 36.4924 1.36569
\(715\) 0 0
\(716\) 24.4924 0.915325
\(717\) −1.75379 −0.0654964
\(718\) 12.4924 0.466213
\(719\) −9.75379 −0.363755 −0.181877 0.983321i \(-0.558217\pi\)
−0.181877 + 0.983321i \(0.558217\pi\)
\(720\) 0 0
\(721\) −78.7386 −2.93238
\(722\) 3.00000 0.111648
\(723\) −2.49242 −0.0926942
\(724\) 19.1231 0.710705
\(725\) 0 0
\(726\) −15.2462 −0.565840
\(727\) −27.8617 −1.03333 −0.516667 0.856186i \(-0.672828\pi\)
−0.516667 + 0.856186i \(0.672828\pi\)
\(728\) 10.2462 0.379750
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0.876894 0.0324109
\(733\) 13.8617 0.511995 0.255998 0.966677i \(-0.417596\pi\)
0.255998 + 0.966677i \(0.417596\pi\)
\(734\) 13.1231 0.484383
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 40.9848 1.50970
\(738\) −2.00000 −0.0736210
\(739\) 40.4924 1.48954 0.744769 0.667322i \(-0.232560\pi\)
0.744769 + 0.667322i \(0.232560\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) −21.7538 −0.798607
\(743\) 14.7386 0.540708 0.270354 0.962761i \(-0.412859\pi\)
0.270354 + 0.962761i \(0.412859\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 13.3693 0.489485
\(747\) −11.3693 −0.415982
\(748\) −36.4924 −1.33430
\(749\) −58.2462 −2.12827
\(750\) 0 0
\(751\) 43.8617 1.60054 0.800269 0.599641i \(-0.204690\pi\)
0.800269 + 0.599641i \(0.204690\pi\)
\(752\) 8.00000 0.291730
\(753\) −7.36932 −0.268553
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) 5.12311 0.186326
\(757\) −9.86174 −0.358431 −0.179216 0.983810i \(-0.557356\pi\)
−0.179216 + 0.983810i \(0.557356\pi\)
\(758\) −4.00000 −0.145287
\(759\) −5.12311 −0.185957
\(760\) 0 0
\(761\) −48.2462 −1.74892 −0.874462 0.485094i \(-0.838785\pi\)
−0.874462 + 0.485094i \(0.838785\pi\)
\(762\) −22.2462 −0.805895
\(763\) 56.9848 2.06299
\(764\) −20.4924 −0.741390
\(765\) 0 0
\(766\) 13.7538 0.496945
\(767\) 28.4924 1.02880
\(768\) 1.00000 0.0360844
\(769\) −20.7386 −0.747854 −0.373927 0.927458i \(-0.621989\pi\)
−0.373927 + 0.927458i \(0.621989\pi\)
\(770\) 0 0
\(771\) 8.24621 0.296980
\(772\) 8.24621 0.296788
\(773\) 36.2462 1.30369 0.651843 0.758354i \(-0.273996\pi\)
0.651843 + 0.758354i \(0.273996\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0.246211 0.00883847
\(777\) 36.4924 1.30916
\(778\) 18.0000 0.645331
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) −7.12311 −0.254722
\(783\) 2.00000 0.0714742
\(784\) 19.2462 0.687365
\(785\) 0 0
\(786\) 4.00000 0.142675
\(787\) −48.0000 −1.71102 −0.855508 0.517790i \(-0.826755\pi\)
−0.855508 + 0.517790i \(0.826755\pi\)
\(788\) 10.0000 0.356235
\(789\) −10.2462 −0.364775
\(790\) 0 0
\(791\) 4.49242 0.159732
\(792\) −5.12311 −0.182042
\(793\) −1.75379 −0.0622789
\(794\) −34.4924 −1.22409
\(795\) 0 0
\(796\) 10.8769 0.385521
\(797\) 22.4924 0.796722 0.398361 0.917229i \(-0.369579\pi\)
0.398361 + 0.917229i \(0.369579\pi\)
\(798\) −20.4924 −0.725424
\(799\) −56.9848 −2.01598
\(800\) 0 0
\(801\) 3.12311 0.110350
\(802\) 17.3693 0.613332
\(803\) −62.7386 −2.21400
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 0 0
\(807\) −16.2462 −0.571894
\(808\) 16.2462 0.571540
\(809\) 4.24621 0.149289 0.0746444 0.997210i \(-0.476218\pi\)
0.0746444 + 0.997210i \(0.476218\pi\)
\(810\) 0 0
\(811\) −36.0000 −1.26413 −0.632065 0.774915i \(-0.717793\pi\)
−0.632065 + 0.774915i \(0.717793\pi\)
\(812\) 10.2462 0.359572
\(813\) 10.2462 0.359350
\(814\) −36.4924 −1.27906
\(815\) 0 0
\(816\) −7.12311 −0.249359
\(817\) 0 0
\(818\) 16.2462 0.568035
\(819\) −10.2462 −0.358032
\(820\) 0 0
\(821\) −19.7538 −0.689412 −0.344706 0.938711i \(-0.612021\pi\)
−0.344706 + 0.938711i \(0.612021\pi\)
\(822\) −19.1231 −0.666995
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 15.3693 0.535416
\(825\) 0 0
\(826\) 72.9848 2.53947
\(827\) −17.1231 −0.595429 −0.297714 0.954655i \(-0.596224\pi\)
−0.297714 + 0.954655i \(0.596224\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 32.2462 1.11996 0.559979 0.828507i \(-0.310809\pi\)
0.559979 + 0.828507i \(0.310809\pi\)
\(830\) 0 0
\(831\) −20.2462 −0.702333
\(832\) −2.00000 −0.0693375
\(833\) −137.093 −4.74998
\(834\) 16.4924 0.571086
\(835\) 0 0
\(836\) 20.4924 0.708745
\(837\) 0 0
\(838\) 1.61553 0.0558075
\(839\) −26.2462 −0.906120 −0.453060 0.891480i \(-0.649668\pi\)
−0.453060 + 0.891480i \(0.649668\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −13.3693 −0.460737
\(843\) −12.8769 −0.443504
\(844\) 16.4924 0.567693
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) 78.1080 2.68382
\(848\) 4.24621 0.145815
\(849\) 2.24621 0.0770898
\(850\) 0 0
\(851\) −7.12311 −0.244177
\(852\) 6.24621 0.213992
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) −4.49242 −0.153728
\(855\) 0 0
\(856\) 11.3693 0.388595
\(857\) −24.7386 −0.845056 −0.422528 0.906350i \(-0.638857\pi\)
−0.422528 + 0.906350i \(0.638857\pi\)
\(858\) 10.2462 0.349800
\(859\) 48.4924 1.65454 0.827270 0.561804i \(-0.189893\pi\)
0.827270 + 0.561804i \(0.189893\pi\)
\(860\) 0 0
\(861\) 10.2462 0.349190
\(862\) −26.2462 −0.893950
\(863\) −48.9848 −1.66746 −0.833732 0.552170i \(-0.813800\pi\)
−0.833732 + 0.552170i \(0.813800\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 10.4924 0.356547
\(867\) 33.7386 1.14582
\(868\) 0 0
\(869\) −26.2462 −0.890342
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) −11.1231 −0.376676
\(873\) −0.246211 −0.00833299
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) −12.2462 −0.413761
\(877\) −3.26137 −0.110129 −0.0550643 0.998483i \(-0.517536\pi\)
−0.0550643 + 0.998483i \(0.517536\pi\)
\(878\) 0 0
\(879\) −11.7538 −0.396445
\(880\) 0 0
\(881\) 31.6155 1.06515 0.532577 0.846381i \(-0.321224\pi\)
0.532577 + 0.846381i \(0.321224\pi\)
\(882\) −19.2462 −0.648054
\(883\) 20.9848 0.706196 0.353098 0.935586i \(-0.385128\pi\)
0.353098 + 0.935586i \(0.385128\pi\)
\(884\) 14.2462 0.479152
\(885\) 0 0
\(886\) −24.4924 −0.822839
\(887\) −52.4924 −1.76252 −0.881262 0.472629i \(-0.843305\pi\)
−0.881262 + 0.472629i \(0.843305\pi\)
\(888\) −7.12311 −0.239036
\(889\) 113.970 3.82242
\(890\) 0 0
\(891\) 5.12311 0.171630
\(892\) −6.24621 −0.209139
\(893\) 32.0000 1.07084
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) −5.12311 −0.171151
\(897\) 2.00000 0.0667781
\(898\) 10.4924 0.350137
\(899\) 0 0
\(900\) 0 0
\(901\) −30.2462 −1.00765
\(902\) −10.2462 −0.341162
\(903\) 0 0
\(904\) −0.876894 −0.0291651
\(905\) 0 0
\(906\) 10.2462 0.340408
\(907\) −37.7538 −1.25359 −0.626797 0.779183i \(-0.715634\pi\)
−0.626797 + 0.779183i \(0.715634\pi\)
\(908\) −19.3693 −0.642793
\(909\) −16.2462 −0.538853
\(910\) 0 0
\(911\) −13.7538 −0.455683 −0.227842 0.973698i \(-0.573167\pi\)
−0.227842 + 0.973698i \(0.573167\pi\)
\(912\) 4.00000 0.132453
\(913\) −58.2462 −1.92767
\(914\) −32.7386 −1.08290
\(915\) 0 0
\(916\) −1.36932 −0.0452435
\(917\) −20.4924 −0.676719
\(918\) 7.12311 0.235098
\(919\) 10.8769 0.358796 0.179398 0.983777i \(-0.442585\pi\)
0.179398 + 0.983777i \(0.442585\pi\)
\(920\) 0 0
\(921\) −0.492423 −0.0162259
\(922\) −32.7386 −1.07819
\(923\) −12.4924 −0.411193
\(924\) 26.2462 0.863437
\(925\) 0 0
\(926\) −36.0000 −1.18303
\(927\) −15.3693 −0.504795
\(928\) −2.00000 −0.0656532
\(929\) −30.9848 −1.01658 −0.508290 0.861186i \(-0.669722\pi\)
−0.508290 + 0.861186i \(0.669722\pi\)
\(930\) 0 0
\(931\) 76.9848 2.52308
\(932\) 20.7386 0.679317
\(933\) 26.7386 0.875384
\(934\) −3.36932 −0.110247
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 16.7386 0.546827 0.273414 0.961897i \(-0.411847\pi\)
0.273414 + 0.961897i \(0.411847\pi\)
\(938\) −40.9848 −1.33820
\(939\) −10.4924 −0.342407
\(940\) 0 0
\(941\) −6.49242 −0.211647 −0.105823 0.994385i \(-0.533748\pi\)
−0.105823 + 0.994385i \(0.533748\pi\)
\(942\) 0.876894 0.0285708
\(943\) −2.00000 −0.0651290
\(944\) −14.2462 −0.463675
\(945\) 0 0
\(946\) 0 0
\(947\) −22.2462 −0.722905 −0.361452 0.932391i \(-0.617719\pi\)
−0.361452 + 0.932391i \(0.617719\pi\)
\(948\) −5.12311 −0.166391
\(949\) 24.4924 0.795058
\(950\) 0 0
\(951\) −20.7386 −0.672496
\(952\) 36.4924 1.18273
\(953\) 33.8617 1.09689 0.548445 0.836187i \(-0.315220\pi\)
0.548445 + 0.836187i \(0.315220\pi\)
\(954\) −4.24621 −0.137476
\(955\) 0 0
\(956\) −1.75379 −0.0567216
\(957\) 10.2462 0.331213
\(958\) 20.4924 0.662080
\(959\) 97.9697 3.16361
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 14.2462 0.459316
\(963\) −11.3693 −0.366371
\(964\) −2.49242 −0.0802755
\(965\) 0 0
\(966\) 5.12311 0.164833
\(967\) 4.98485 0.160302 0.0801509 0.996783i \(-0.474460\pi\)
0.0801509 + 0.996783i \(0.474460\pi\)
\(968\) −15.2462 −0.490032
\(969\) −28.4924 −0.915308
\(970\) 0 0
\(971\) 34.8769 1.11925 0.559626 0.828745i \(-0.310945\pi\)
0.559626 + 0.828745i \(0.310945\pi\)
\(972\) 1.00000 0.0320750
\(973\) −84.4924 −2.70870
\(974\) 40.4924 1.29746
\(975\) 0 0
\(976\) 0.876894 0.0280687
\(977\) −21.8617 −0.699419 −0.349710 0.936858i \(-0.613720\pi\)
−0.349710 + 0.936858i \(0.613720\pi\)
\(978\) −12.0000 −0.383718
\(979\) 16.0000 0.511362
\(980\) 0 0
\(981\) 11.1231 0.355133
\(982\) 12.0000 0.382935
\(983\) 26.2462 0.837124 0.418562 0.908188i \(-0.362534\pi\)
0.418562 + 0.908188i \(0.362534\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 0 0
\(986\) 14.2462 0.453692
\(987\) 40.9848 1.30456
\(988\) −8.00000 −0.254514
\(989\) 0 0
\(990\) 0 0
\(991\) −37.4773 −1.19050 −0.595252 0.803539i \(-0.702948\pi\)
−0.595252 + 0.803539i \(0.702948\pi\)
\(992\) 0 0
\(993\) −32.4924 −1.03112
\(994\) −32.0000 −1.01498
\(995\) 0 0
\(996\) −11.3693 −0.360251
\(997\) −20.2462 −0.641204 −0.320602 0.947214i \(-0.603885\pi\)
−0.320602 + 0.947214i \(0.603885\pi\)
\(998\) 28.9848 0.917499
\(999\) 7.12311 0.225365
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.a.bi.1.2 2
5.2 odd 4 3450.2.d.v.2899.2 4
5.3 odd 4 3450.2.d.v.2899.3 4
5.4 even 2 690.2.a.l.1.1 2
15.14 odd 2 2070.2.a.t.1.1 2
20.19 odd 2 5520.2.a.bs.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.l.1.1 2 5.4 even 2
2070.2.a.t.1.1 2 15.14 odd 2
3450.2.a.bi.1.2 2 1.1 even 1 trivial
3450.2.d.v.2899.2 4 5.2 odd 4
3450.2.d.v.2899.3 4 5.3 odd 4
5520.2.a.bs.1.2 2 20.19 odd 2