Properties

Label 3450.2.a.bj.1.1
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: \(\Q(\sqrt{73}) \)
Defining polynomial: \(x^{2} - x - 18\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.77200\) of defining polynomial
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.77200 q^{11} -1.00000 q^{12} -4.77200 q^{13} +3.00000 q^{14} +1.00000 q^{16} +7.77200 q^{17} +1.00000 q^{18} -0.772002 q^{19} -3.00000 q^{21} -2.77200 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.77200 q^{26} -1.00000 q^{27} +3.00000 q^{28} +3.00000 q^{29} +7.54400 q^{31} +1.00000 q^{32} +2.77200 q^{33} +7.77200 q^{34} +1.00000 q^{36} +1.77200 q^{37} -0.772002 q^{38} +4.77200 q^{39} +2.77200 q^{41} -3.00000 q^{42} +0.772002 q^{43} -2.77200 q^{44} -1.00000 q^{46} +0.227998 q^{47} -1.00000 q^{48} +2.00000 q^{49} -7.77200 q^{51} -4.77200 q^{52} -4.00000 q^{53} -1.00000 q^{54} +3.00000 q^{56} +0.772002 q^{57} +3.00000 q^{58} +6.00000 q^{59} +7.54400 q^{61} +7.54400 q^{62} +3.00000 q^{63} +1.00000 q^{64} +2.77200 q^{66} -7.54400 q^{67} +7.77200 q^{68} +1.00000 q^{69} +3.77200 q^{71} +1.00000 q^{72} +10.5440 q^{73} +1.77200 q^{74} -0.772002 q^{76} -8.31601 q^{77} +4.77200 q^{78} +10.7720 q^{79} +1.00000 q^{81} +2.77200 q^{82} +1.00000 q^{83} -3.00000 q^{84} +0.772002 q^{86} -3.00000 q^{87} -2.77200 q^{88} +8.22800 q^{89} -14.3160 q^{91} -1.00000 q^{92} -7.54400 q^{93} +0.227998 q^{94} -1.00000 q^{96} -0.455996 q^{97} +2.00000 q^{98} -2.77200 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 6q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 6q^{7} + 2q^{8} + 2q^{9} + 3q^{11} - 2q^{12} - q^{13} + 6q^{14} + 2q^{16} + 7q^{17} + 2q^{18} + 7q^{19} - 6q^{21} + 3q^{22} - 2q^{23} - 2q^{24} - q^{26} - 2q^{27} + 6q^{28} + 6q^{29} - 2q^{31} + 2q^{32} - 3q^{33} + 7q^{34} + 2q^{36} - 5q^{37} + 7q^{38} + q^{39} - 3q^{41} - 6q^{42} - 7q^{43} + 3q^{44} - 2q^{46} + 9q^{47} - 2q^{48} + 4q^{49} - 7q^{51} - q^{52} - 8q^{53} - 2q^{54} + 6q^{56} - 7q^{57} + 6q^{58} + 12q^{59} - 2q^{61} - 2q^{62} + 6q^{63} + 2q^{64} - 3q^{66} + 2q^{67} + 7q^{68} + 2q^{69} - q^{71} + 2q^{72} + 4q^{73} - 5q^{74} + 7q^{76} + 9q^{77} + q^{78} + 13q^{79} + 2q^{81} - 3q^{82} + 2q^{83} - 6q^{84} - 7q^{86} - 6q^{87} + 3q^{88} + 25q^{89} - 3q^{91} - 2q^{92} + 2q^{93} + 9q^{94} - 2q^{96} - 18q^{97} + 4q^{98} + 3q^{99} + O(q^{100}) \)

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
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.77200 −0.835790 −0.417895 0.908495i \(-0.637232\pi\)
−0.417895 + 0.908495i \(0.637232\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.77200 −1.32352 −0.661758 0.749718i \(-0.730189\pi\)
−0.661758 + 0.749718i \(0.730189\pi\)
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.77200 1.88499 0.942494 0.334224i \(-0.108474\pi\)
0.942494 + 0.334224i \(0.108474\pi\)
\(18\) 1.00000 0.235702
\(19\) −0.772002 −0.177109 −0.0885547 0.996071i \(-0.528225\pi\)
−0.0885547 + 0.996071i \(0.528225\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) −2.77200 −0.590993
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −4.77200 −0.935867
\(27\) −1.00000 −0.192450
\(28\) 3.00000 0.566947
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 7.54400 1.35494 0.677472 0.735549i \(-0.263076\pi\)
0.677472 + 0.735549i \(0.263076\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.77200 0.482544
\(34\) 7.77200 1.33289
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.77200 0.291315 0.145658 0.989335i \(-0.453470\pi\)
0.145658 + 0.989335i \(0.453470\pi\)
\(38\) −0.772002 −0.125235
\(39\) 4.77200 0.764132
\(40\) 0 0
\(41\) 2.77200 0.432914 0.216457 0.976292i \(-0.430550\pi\)
0.216457 + 0.976292i \(0.430550\pi\)
\(42\) −3.00000 −0.462910
\(43\) 0.772002 0.117729 0.0588646 0.998266i \(-0.481252\pi\)
0.0588646 + 0.998266i \(0.481252\pi\)
\(44\) −2.77200 −0.417895
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 0.227998 0.0332569 0.0166285 0.999862i \(-0.494707\pi\)
0.0166285 + 0.999862i \(0.494707\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −7.77200 −1.08830
\(52\) −4.77200 −0.661758
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) 0.772002 0.102254
\(58\) 3.00000 0.393919
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 7.54400 0.965911 0.482955 0.875645i \(-0.339563\pi\)
0.482955 + 0.875645i \(0.339563\pi\)
\(62\) 7.54400 0.958089
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.77200 0.341210
\(67\) −7.54400 −0.921647 −0.460823 0.887492i \(-0.652446\pi\)
−0.460823 + 0.887492i \(0.652446\pi\)
\(68\) 7.77200 0.942494
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 3.77200 0.447654 0.223827 0.974629i \(-0.428145\pi\)
0.223827 + 0.974629i \(0.428145\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.5440 1.23408 0.617041 0.786931i \(-0.288331\pi\)
0.617041 + 0.786931i \(0.288331\pi\)
\(74\) 1.77200 0.205991
\(75\) 0 0
\(76\) −0.772002 −0.0885547
\(77\) −8.31601 −0.947697
\(78\) 4.77200 0.540323
\(79\) 10.7720 1.21194 0.605972 0.795486i \(-0.292784\pi\)
0.605972 + 0.795486i \(0.292784\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.77200 0.306116
\(83\) 1.00000 0.109764 0.0548821 0.998493i \(-0.482522\pi\)
0.0548821 + 0.998493i \(0.482522\pi\)
\(84\) −3.00000 −0.327327
\(85\) 0 0
\(86\) 0.772002 0.0832471
\(87\) −3.00000 −0.321634
\(88\) −2.77200 −0.295496
\(89\) 8.22800 0.872166 0.436083 0.899906i \(-0.356365\pi\)
0.436083 + 0.899906i \(0.356365\pi\)
\(90\) 0 0
\(91\) −14.3160 −1.50073
\(92\) −1.00000 −0.104257
\(93\) −7.54400 −0.782277
\(94\) 0.227998 0.0235162
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −0.455996 −0.0462994 −0.0231497 0.999732i \(-0.507369\pi\)
−0.0231497 + 0.999732i \(0.507369\pi\)
\(98\) 2.00000 0.202031
\(99\) −2.77200 −0.278597
\(100\) 0 0
\(101\) −11.3160 −1.12598 −0.562992 0.826462i \(-0.690350\pi\)
−0.562992 + 0.826462i \(0.690350\pi\)
\(102\) −7.77200 −0.769543
\(103\) −11.0000 −1.08386 −0.541931 0.840423i \(-0.682307\pi\)
−0.541931 + 0.840423i \(0.682307\pi\)
\(104\) −4.77200 −0.467933
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.22800 0.596534 0.298267 0.954482i \(-0.403591\pi\)
0.298267 + 0.954482i \(0.403591\pi\)
\(110\) 0 0
\(111\) −1.77200 −0.168191
\(112\) 3.00000 0.283473
\(113\) 3.77200 0.354840 0.177420 0.984135i \(-0.443225\pi\)
0.177420 + 0.984135i \(0.443225\pi\)
\(114\) 0.772002 0.0723046
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) −4.77200 −0.441172
\(118\) 6.00000 0.552345
\(119\) 23.3160 2.13737
\(120\) 0 0
\(121\) −3.31601 −0.301455
\(122\) 7.54400 0.683002
\(123\) −2.77200 −0.249943
\(124\) 7.54400 0.677472
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) −21.5440 −1.91172 −0.955861 0.293821i \(-0.905073\pi\)
−0.955861 + 0.293821i \(0.905073\pi\)
\(128\) 1.00000 0.0883883
\(129\) −0.772002 −0.0679710
\(130\) 0 0
\(131\) 15.0880 1.31825 0.659123 0.752035i \(-0.270928\pi\)
0.659123 + 0.752035i \(0.270928\pi\)
\(132\) 2.77200 0.241272
\(133\) −2.31601 −0.200823
\(134\) −7.54400 −0.651703
\(135\) 0 0
\(136\) 7.77200 0.666444
\(137\) −6.22800 −0.532094 −0.266047 0.963960i \(-0.585718\pi\)
−0.266047 + 0.963960i \(0.585718\pi\)
\(138\) 1.00000 0.0851257
\(139\) 7.77200 0.659213 0.329606 0.944118i \(-0.393084\pi\)
0.329606 + 0.944118i \(0.393084\pi\)
\(140\) 0 0
\(141\) −0.227998 −0.0192009
\(142\) 3.77200 0.316539
\(143\) 13.2280 1.10618
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 10.5440 0.872628
\(147\) −2.00000 −0.164957
\(148\) 1.77200 0.145658
\(149\) 23.0880 1.89144 0.945722 0.324978i \(-0.105357\pi\)
0.945722 + 0.324978i \(0.105357\pi\)
\(150\) 0 0
\(151\) 6.45600 0.525382 0.262691 0.964880i \(-0.415390\pi\)
0.262691 + 0.964880i \(0.415390\pi\)
\(152\) −0.772002 −0.0626176
\(153\) 7.77200 0.628329
\(154\) −8.31601 −0.670123
\(155\) 0 0
\(156\) 4.77200 0.382066
\(157\) 3.54400 0.282842 0.141421 0.989950i \(-0.454833\pi\)
0.141421 + 0.989950i \(0.454833\pi\)
\(158\) 10.7720 0.856974
\(159\) 4.00000 0.317221
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 1.00000 0.0785674
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 2.77200 0.216457
\(165\) 0 0
\(166\) 1.00000 0.0776151
\(167\) 9.31601 0.720894 0.360447 0.932780i \(-0.382624\pi\)
0.360447 + 0.932780i \(0.382624\pi\)
\(168\) −3.00000 −0.231455
\(169\) 9.77200 0.751692
\(170\) 0 0
\(171\) −0.772002 −0.0590365
\(172\) 0.772002 0.0588646
\(173\) 7.22800 0.549535 0.274767 0.961511i \(-0.411399\pi\)
0.274767 + 0.961511i \(0.411399\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) −2.77200 −0.208948
\(177\) −6.00000 −0.450988
\(178\) 8.22800 0.616715
\(179\) −9.54400 −0.713352 −0.356676 0.934228i \(-0.616090\pi\)
−0.356676 + 0.934228i \(0.616090\pi\)
\(180\) 0 0
\(181\) −11.3160 −0.841112 −0.420556 0.907267i \(-0.638165\pi\)
−0.420556 + 0.907267i \(0.638165\pi\)
\(182\) −14.3160 −1.06117
\(183\) −7.54400 −0.557669
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −7.54400 −0.553153
\(187\) −21.5440 −1.57545
\(188\) 0.227998 0.0166285
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) 18.3160 1.32530 0.662650 0.748929i \(-0.269432\pi\)
0.662650 + 0.748929i \(0.269432\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 3.77200 0.271515 0.135757 0.990742i \(-0.456653\pi\)
0.135757 + 0.990742i \(0.456653\pi\)
\(194\) −0.455996 −0.0327386
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 13.0000 0.926212 0.463106 0.886303i \(-0.346735\pi\)
0.463106 + 0.886303i \(0.346735\pi\)
\(198\) −2.77200 −0.196998
\(199\) −22.0880 −1.56578 −0.782889 0.622162i \(-0.786255\pi\)
−0.782889 + 0.622162i \(0.786255\pi\)
\(200\) 0 0
\(201\) 7.54400 0.532113
\(202\) −11.3160 −0.796191
\(203\) 9.00000 0.631676
\(204\) −7.77200 −0.544149
\(205\) 0 0
\(206\) −11.0000 −0.766406
\(207\) −1.00000 −0.0695048
\(208\) −4.77200 −0.330879
\(209\) 2.13999 0.148026
\(210\) 0 0
\(211\) 15.3160 1.05440 0.527199 0.849742i \(-0.323242\pi\)
0.527199 + 0.849742i \(0.323242\pi\)
\(212\) −4.00000 −0.274721
\(213\) −3.77200 −0.258453
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 22.6320 1.53636
\(218\) 6.22800 0.421813
\(219\) −10.5440 −0.712498
\(220\) 0 0
\(221\) −37.0880 −2.49481
\(222\) −1.77200 −0.118929
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 3.00000 0.200446
\(225\) 0 0
\(226\) 3.77200 0.250910
\(227\) 2.22800 0.147877 0.0739387 0.997263i \(-0.476443\pi\)
0.0739387 + 0.997263i \(0.476443\pi\)
\(228\) 0.772002 0.0511271
\(229\) −23.5440 −1.55583 −0.777916 0.628369i \(-0.783723\pi\)
−0.777916 + 0.628369i \(0.783723\pi\)
\(230\) 0 0
\(231\) 8.31601 0.547153
\(232\) 3.00000 0.196960
\(233\) 28.3160 1.85504 0.927522 0.373770i \(-0.121935\pi\)
0.927522 + 0.373770i \(0.121935\pi\)
\(234\) −4.77200 −0.311956
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) −10.7720 −0.699717
\(238\) 23.3160 1.51135
\(239\) 14.2280 0.920333 0.460166 0.887833i \(-0.347790\pi\)
0.460166 + 0.887833i \(0.347790\pi\)
\(240\) 0 0
\(241\) 24.6320 1.58669 0.793344 0.608774i \(-0.208338\pi\)
0.793344 + 0.608774i \(0.208338\pi\)
\(242\) −3.31601 −0.213161
\(243\) −1.00000 −0.0641500
\(244\) 7.54400 0.482955
\(245\) 0 0
\(246\) −2.77200 −0.176736
\(247\) 3.68399 0.234407
\(248\) 7.54400 0.479045
\(249\) −1.00000 −0.0633724
\(250\) 0 0
\(251\) −29.3160 −1.85041 −0.925205 0.379468i \(-0.876107\pi\)
−0.925205 + 0.379468i \(0.876107\pi\)
\(252\) 3.00000 0.188982
\(253\) 2.77200 0.174274
\(254\) −21.5440 −1.35179
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −25.0880 −1.56495 −0.782473 0.622684i \(-0.786042\pi\)
−0.782473 + 0.622684i \(0.786042\pi\)
\(258\) −0.772002 −0.0480627
\(259\) 5.31601 0.330321
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 15.0880 0.932140
\(263\) 23.5440 1.45179 0.725893 0.687808i \(-0.241427\pi\)
0.725893 + 0.687808i \(0.241427\pi\)
\(264\) 2.77200 0.170605
\(265\) 0 0
\(266\) −2.31601 −0.142003
\(267\) −8.22800 −0.503545
\(268\) −7.54400 −0.460823
\(269\) 15.8600 0.967002 0.483501 0.875344i \(-0.339365\pi\)
0.483501 + 0.875344i \(0.339365\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 7.77200 0.471247
\(273\) 14.3160 0.866444
\(274\) −6.22800 −0.376247
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) 16.7720 1.00773 0.503866 0.863782i \(-0.331911\pi\)
0.503866 + 0.863782i \(0.331911\pi\)
\(278\) 7.77200 0.466134
\(279\) 7.54400 0.451648
\(280\) 0 0
\(281\) −26.8600 −1.60233 −0.801167 0.598441i \(-0.795787\pi\)
−0.801167 + 0.598441i \(0.795787\pi\)
\(282\) −0.227998 −0.0135771
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 3.77200 0.223827
\(285\) 0 0
\(286\) 13.2280 0.782188
\(287\) 8.31601 0.490878
\(288\) 1.00000 0.0589256
\(289\) 43.4040 2.55318
\(290\) 0 0
\(291\) 0.455996 0.0267310
\(292\) 10.5440 0.617041
\(293\) −7.54400 −0.440725 −0.220363 0.975418i \(-0.570724\pi\)
−0.220363 + 0.975418i \(0.570724\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) 1.77200 0.102996
\(297\) 2.77200 0.160848
\(298\) 23.0880 1.33745
\(299\) 4.77200 0.275972
\(300\) 0 0
\(301\) 2.31601 0.133492
\(302\) 6.45600 0.371501
\(303\) 11.3160 0.650088
\(304\) −0.772002 −0.0442773
\(305\) 0 0
\(306\) 7.77200 0.444296
\(307\) −5.31601 −0.303400 −0.151700 0.988427i \(-0.548475\pi\)
−0.151700 + 0.988427i \(0.548475\pi\)
\(308\) −8.31601 −0.473848
\(309\) 11.0000 0.625768
\(310\) 0 0
\(311\) 24.8600 1.40968 0.704841 0.709365i \(-0.251018\pi\)
0.704841 + 0.709365i \(0.251018\pi\)
\(312\) 4.77200 0.270161
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 3.54400 0.200000
\(315\) 0 0
\(316\) 10.7720 0.605972
\(317\) 9.00000 0.505490 0.252745 0.967533i \(-0.418667\pi\)
0.252745 + 0.967533i \(0.418667\pi\)
\(318\) 4.00000 0.224309
\(319\) −8.31601 −0.465607
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) −3.00000 −0.167183
\(323\) −6.00000 −0.333849
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) −6.22800 −0.344409
\(328\) 2.77200 0.153058
\(329\) 0.683994 0.0377098
\(330\) 0 0
\(331\) 26.2280 1.44162 0.720811 0.693132i \(-0.243770\pi\)
0.720811 + 0.693132i \(0.243770\pi\)
\(332\) 1.00000 0.0548821
\(333\) 1.77200 0.0971051
\(334\) 9.31601 0.509749
\(335\) 0 0
\(336\) −3.00000 −0.163663
\(337\) −4.00000 −0.217894 −0.108947 0.994048i \(-0.534748\pi\)
−0.108947 + 0.994048i \(0.534748\pi\)
\(338\) 9.77200 0.531527
\(339\) −3.77200 −0.204867
\(340\) 0 0
\(341\) −20.9120 −1.13245
\(342\) −0.772002 −0.0417451
\(343\) −15.0000 −0.809924
\(344\) 0.772002 0.0416236
\(345\) 0 0
\(346\) 7.22800 0.388580
\(347\) 7.54400 0.404983 0.202492 0.979284i \(-0.435096\pi\)
0.202492 + 0.979284i \(0.435096\pi\)
\(348\) −3.00000 −0.160817
\(349\) −8.77200 −0.469554 −0.234777 0.972049i \(-0.575436\pi\)
−0.234777 + 0.972049i \(0.575436\pi\)
\(350\) 0 0
\(351\) 4.77200 0.254711
\(352\) −2.77200 −0.147748
\(353\) −23.8600 −1.26994 −0.634970 0.772537i \(-0.718988\pi\)
−0.634970 + 0.772537i \(0.718988\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) 8.22800 0.436083
\(357\) −23.3160 −1.23401
\(358\) −9.54400 −0.504416
\(359\) −18.7720 −0.990748 −0.495374 0.868680i \(-0.664969\pi\)
−0.495374 + 0.868680i \(0.664969\pi\)
\(360\) 0 0
\(361\) −18.4040 −0.968632
\(362\) −11.3160 −0.594756
\(363\) 3.31601 0.174045
\(364\) −14.3160 −0.750363
\(365\) 0 0
\(366\) −7.54400 −0.394331
\(367\) −7.68399 −0.401101 −0.200551 0.979683i \(-0.564273\pi\)
−0.200551 + 0.979683i \(0.564273\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 2.77200 0.144305
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) −7.54400 −0.391138
\(373\) 4.22800 0.218917 0.109459 0.993991i \(-0.465088\pi\)
0.109459 + 0.993991i \(0.465088\pi\)
\(374\) −21.5440 −1.11401
\(375\) 0 0
\(376\) 0.227998 0.0117581
\(377\) −14.3160 −0.737312
\(378\) −3.00000 −0.154303
\(379\) 11.0880 0.569553 0.284776 0.958594i \(-0.408081\pi\)
0.284776 + 0.958594i \(0.408081\pi\)
\(380\) 0 0
\(381\) 21.5440 1.10373
\(382\) 18.3160 0.937128
\(383\) −5.22800 −0.267138 −0.133569 0.991040i \(-0.542644\pi\)
−0.133569 + 0.991040i \(0.542644\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 3.77200 0.191990
\(387\) 0.772002 0.0392431
\(388\) −0.455996 −0.0231497
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) −7.77200 −0.393047
\(392\) 2.00000 0.101015
\(393\) −15.0880 −0.761089
\(394\) 13.0000 0.654931
\(395\) 0 0
\(396\) −2.77200 −0.139298
\(397\) −36.6320 −1.83851 −0.919254 0.393665i \(-0.871207\pi\)
−0.919254 + 0.393665i \(0.871207\pi\)
\(398\) −22.0880 −1.10717
\(399\) 2.31601 0.115945
\(400\) 0 0
\(401\) 22.6320 1.13019 0.565094 0.825026i \(-0.308840\pi\)
0.565094 + 0.825026i \(0.308840\pi\)
\(402\) 7.54400 0.376261
\(403\) −36.0000 −1.79329
\(404\) −11.3160 −0.562992
\(405\) 0 0
\(406\) 9.00000 0.446663
\(407\) −4.91199 −0.243478
\(408\) −7.77200 −0.384771
\(409\) −23.0000 −1.13728 −0.568638 0.822588i \(-0.692530\pi\)
−0.568638 + 0.822588i \(0.692530\pi\)
\(410\) 0 0
\(411\) 6.22800 0.307204
\(412\) −11.0000 −0.541931
\(413\) 18.0000 0.885722
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) −4.77200 −0.233967
\(417\) −7.77200 −0.380597
\(418\) 2.13999 0.104670
\(419\) −25.6320 −1.25221 −0.626103 0.779740i \(-0.715351\pi\)
−0.626103 + 0.779740i \(0.715351\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 15.3160 0.745571
\(423\) 0.227998 0.0110856
\(424\) −4.00000 −0.194257
\(425\) 0 0
\(426\) −3.77200 −0.182754
\(427\) 22.6320 1.09524
\(428\) 4.00000 0.193347
\(429\) −13.2280 −0.638654
\(430\) 0 0
\(431\) −26.6320 −1.28282 −0.641409 0.767199i \(-0.721650\pi\)
−0.641409 + 0.767199i \(0.721650\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −23.5440 −1.13145 −0.565726 0.824593i \(-0.691404\pi\)
−0.565726 + 0.824593i \(0.691404\pi\)
\(434\) 22.6320 1.08637
\(435\) 0 0
\(436\) 6.22800 0.298267
\(437\) 0.772002 0.0369299
\(438\) −10.5440 −0.503812
\(439\) −12.0000 −0.572729 −0.286364 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) −37.0880 −1.76410
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) −1.77200 −0.0840955
\(445\) 0 0
\(446\) −10.0000 −0.473514
\(447\) −23.0880 −1.09203
\(448\) 3.00000 0.141737
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) −7.68399 −0.361825
\(452\) 3.77200 0.177420
\(453\) −6.45600 −0.303329
\(454\) 2.22800 0.104565
\(455\) 0 0
\(456\) 0.772002 0.0361523
\(457\) −20.6320 −0.965125 −0.482562 0.875862i \(-0.660294\pi\)
−0.482562 + 0.875862i \(0.660294\pi\)
\(458\) −23.5440 −1.10014
\(459\) −7.77200 −0.362766
\(460\) 0 0
\(461\) −4.54400 −0.211635 −0.105818 0.994386i \(-0.533746\pi\)
−0.105818 + 0.994386i \(0.533746\pi\)
\(462\) 8.31601 0.386896
\(463\) −35.0880 −1.63068 −0.815339 0.578984i \(-0.803449\pi\)
−0.815339 + 0.578984i \(0.803449\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) 28.3160 1.31171
\(467\) −1.45600 −0.0673755 −0.0336877 0.999432i \(-0.510725\pi\)
−0.0336877 + 0.999432i \(0.510725\pi\)
\(468\) −4.77200 −0.220586
\(469\) −22.6320 −1.04505
\(470\) 0 0
\(471\) −3.54400 −0.163299
\(472\) 6.00000 0.276172
\(473\) −2.13999 −0.0983969
\(474\) −10.7720 −0.494774
\(475\) 0 0
\(476\) 23.3160 1.06869
\(477\) −4.00000 −0.183147
\(478\) 14.2280 0.650773
\(479\) −9.68399 −0.442473 −0.221236 0.975220i \(-0.571009\pi\)
−0.221236 + 0.975220i \(0.571009\pi\)
\(480\) 0 0
\(481\) −8.45600 −0.385560
\(482\) 24.6320 1.12196
\(483\) 3.00000 0.136505
\(484\) −3.31601 −0.150728
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −29.5440 −1.33877 −0.669383 0.742917i \(-0.733442\pi\)
−0.669383 + 0.742917i \(0.733442\pi\)
\(488\) 7.54400 0.341501
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) −27.5440 −1.24304 −0.621522 0.783397i \(-0.713485\pi\)
−0.621522 + 0.783397i \(0.713485\pi\)
\(492\) −2.77200 −0.124971
\(493\) 23.3160 1.05010
\(494\) 3.68399 0.165751
\(495\) 0 0
\(496\) 7.54400 0.338736
\(497\) 11.3160 0.507592
\(498\) −1.00000 −0.0448111
\(499\) −35.3160 −1.58096 −0.790481 0.612487i \(-0.790169\pi\)
−0.790481 + 0.612487i \(0.790169\pi\)
\(500\) 0 0
\(501\) −9.31601 −0.416208
\(502\) −29.3160 −1.30844
\(503\) −36.9480 −1.64743 −0.823715 0.567004i \(-0.808103\pi\)
−0.823715 + 0.567004i \(0.808103\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) 2.77200 0.123231
\(507\) −9.77200 −0.433990
\(508\) −21.5440 −0.955861
\(509\) −14.8600 −0.658658 −0.329329 0.944215i \(-0.606823\pi\)
−0.329329 + 0.944215i \(0.606823\pi\)
\(510\) 0 0
\(511\) 31.6320 1.39932
\(512\) 1.00000 0.0441942
\(513\) 0.772002 0.0340847
\(514\) −25.0880 −1.10658
\(515\) 0 0
\(516\) −0.772002 −0.0339855
\(517\) −0.632011 −0.0277958
\(518\) 5.31601 0.233572
\(519\) −7.22800 −0.317274
\(520\) 0 0
\(521\) −1.77200 −0.0776328 −0.0388164 0.999246i \(-0.512359\pi\)
−0.0388164 + 0.999246i \(0.512359\pi\)
\(522\) 3.00000 0.131306
\(523\) −12.3160 −0.538541 −0.269271 0.963065i \(-0.586783\pi\)
−0.269271 + 0.963065i \(0.586783\pi\)
\(524\) 15.0880 0.659123
\(525\) 0 0
\(526\) 23.5440 1.02657
\(527\) 58.6320 2.55405
\(528\) 2.77200 0.120636
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) −2.31601 −0.100412
\(533\) −13.2280 −0.572968
\(534\) −8.22800 −0.356060
\(535\) 0 0
\(536\) −7.54400 −0.325851
\(537\) 9.54400 0.411854
\(538\) 15.8600 0.683774
\(539\) −5.54400 −0.238797
\(540\) 0 0
\(541\) −36.3160 −1.56135 −0.780674 0.624939i \(-0.785124\pi\)
−0.780674 + 0.624939i \(0.785124\pi\)
\(542\) 16.0000 0.687259
\(543\) 11.3160 0.485616
\(544\) 7.77200 0.333222
\(545\) 0 0
\(546\) 14.3160 0.612668
\(547\) 30.8600 1.31948 0.659739 0.751494i \(-0.270667\pi\)
0.659739 + 0.751494i \(0.270667\pi\)
\(548\) −6.22800 −0.266047
\(549\) 7.54400 0.321970
\(550\) 0 0
\(551\) −2.31601 −0.0986652
\(552\) 1.00000 0.0425628
\(553\) 32.3160 1.37422
\(554\) 16.7720 0.712574
\(555\) 0 0
\(556\) 7.77200 0.329606
\(557\) −13.0880 −0.554557 −0.277278 0.960790i \(-0.589432\pi\)
−0.277278 + 0.960790i \(0.589432\pi\)
\(558\) 7.54400 0.319363
\(559\) −3.68399 −0.155816
\(560\) 0 0
\(561\) 21.5440 0.909589
\(562\) −26.8600 −1.13302
\(563\) 32.9480 1.38859 0.694297 0.719689i \(-0.255716\pi\)
0.694297 + 0.719689i \(0.255716\pi\)
\(564\) −0.227998 −0.00960045
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) 3.00000 0.125988
\(568\) 3.77200 0.158270
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −38.6320 −1.61670 −0.808350 0.588703i \(-0.799639\pi\)
−0.808350 + 0.588703i \(0.799639\pi\)
\(572\) 13.2280 0.553090
\(573\) −18.3160 −0.765162
\(574\) 8.31601 0.347103
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 5.68399 0.236628 0.118314 0.992976i \(-0.462251\pi\)
0.118314 + 0.992976i \(0.462251\pi\)
\(578\) 43.4040 1.80537
\(579\) −3.77200 −0.156759
\(580\) 0 0
\(581\) 3.00000 0.124461
\(582\) 0.455996 0.0189017
\(583\) 11.0880 0.459218
\(584\) 10.5440 0.436314
\(585\) 0 0
\(586\) −7.54400 −0.311640
\(587\) −41.5440 −1.71470 −0.857352 0.514730i \(-0.827892\pi\)
−0.857352 + 0.514730i \(0.827892\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −5.82399 −0.239973
\(590\) 0 0
\(591\) −13.0000 −0.534749
\(592\) 1.77200 0.0728288
\(593\) −15.6840 −0.644064 −0.322032 0.946729i \(-0.604366\pi\)
−0.322032 + 0.946729i \(0.604366\pi\)
\(594\) 2.77200 0.113737
\(595\) 0 0
\(596\) 23.0880 0.945722
\(597\) 22.0880 0.904002
\(598\) 4.77200 0.195142
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) −24.2280 −0.988281 −0.494140 0.869382i \(-0.664517\pi\)
−0.494140 + 0.869382i \(0.664517\pi\)
\(602\) 2.31601 0.0943933
\(603\) −7.54400 −0.307216
\(604\) 6.45600 0.262691
\(605\) 0 0
\(606\) 11.3160 0.459681
\(607\) 29.0880 1.18065 0.590323 0.807167i \(-0.299000\pi\)
0.590323 + 0.807167i \(0.299000\pi\)
\(608\) −0.772002 −0.0313088
\(609\) −9.00000 −0.364698
\(610\) 0 0
\(611\) −1.08801 −0.0440161
\(612\) 7.77200 0.314165
\(613\) −12.2280 −0.493884 −0.246942 0.969030i \(-0.579426\pi\)
−0.246942 + 0.969030i \(0.579426\pi\)
\(614\) −5.31601 −0.214537
\(615\) 0 0
\(616\) −8.31601 −0.335061
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 11.0000 0.442485
\(619\) −35.0880 −1.41031 −0.705153 0.709055i \(-0.749122\pi\)
−0.705153 + 0.709055i \(0.749122\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 24.8600 0.996796
\(623\) 24.6840 0.988943
\(624\) 4.77200 0.191033
\(625\) 0 0
\(626\) 8.00000 0.319744
\(627\) −2.13999 −0.0854630
\(628\) 3.54400 0.141421
\(629\) 13.7720 0.549126
\(630\) 0 0
\(631\) −20.0880 −0.799691 −0.399845 0.916583i \(-0.630936\pi\)
−0.399845 + 0.916583i \(0.630936\pi\)
\(632\) 10.7720 0.428487
\(633\) −15.3160 −0.608757
\(634\) 9.00000 0.357436
\(635\) 0 0
\(636\) 4.00000 0.158610
\(637\) −9.54400 −0.378147
\(638\) −8.31601 −0.329234
\(639\) 3.77200 0.149218
\(640\) 0 0
\(641\) 24.8600 0.981911 0.490956 0.871185i \(-0.336648\pi\)
0.490956 + 0.871185i \(0.336648\pi\)
\(642\) −4.00000 −0.157867
\(643\) −6.31601 −0.249079 −0.124539 0.992215i \(-0.539745\pi\)
−0.124539 + 0.992215i \(0.539745\pi\)
\(644\) −3.00000 −0.118217
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) 22.2280 0.873873 0.436936 0.899492i \(-0.356063\pi\)
0.436936 + 0.899492i \(0.356063\pi\)
\(648\) 1.00000 0.0392837
\(649\) −16.6320 −0.652864
\(650\) 0 0
\(651\) −22.6320 −0.887018
\(652\) −16.0000 −0.626608
\(653\) −40.5440 −1.58661 −0.793305 0.608825i \(-0.791641\pi\)
−0.793305 + 0.608825i \(0.791641\pi\)
\(654\) −6.22800 −0.243534
\(655\) 0 0
\(656\) 2.77200 0.108228
\(657\) 10.5440 0.411361
\(658\) 0.683994 0.0266649
\(659\) 7.63201 0.297301 0.148650 0.988890i \(-0.452507\pi\)
0.148650 + 0.988890i \(0.452507\pi\)
\(660\) 0 0
\(661\) 24.8600 0.966942 0.483471 0.875360i \(-0.339376\pi\)
0.483471 + 0.875360i \(0.339376\pi\)
\(662\) 26.2280 1.01938
\(663\) 37.0880 1.44038
\(664\) 1.00000 0.0388075
\(665\) 0 0
\(666\) 1.77200 0.0686637
\(667\) −3.00000 −0.116160
\(668\) 9.31601 0.360447
\(669\) 10.0000 0.386622
\(670\) 0 0
\(671\) −20.9120 −0.807299
\(672\) −3.00000 −0.115728
\(673\) 9.45600 0.364502 0.182251 0.983252i \(-0.441662\pi\)
0.182251 + 0.983252i \(0.441662\pi\)
\(674\) −4.00000 −0.154074
\(675\) 0 0
\(676\) 9.77200 0.375846
\(677\) 48.0000 1.84479 0.922395 0.386248i \(-0.126229\pi\)
0.922395 + 0.386248i \(0.126229\pi\)
\(678\) −3.77200 −0.144863
\(679\) −1.36799 −0.0524986
\(680\) 0 0
\(681\) −2.22800 −0.0853771
\(682\) −20.9120 −0.800762
\(683\) −24.6320 −0.942518 −0.471259 0.881995i \(-0.656200\pi\)
−0.471259 + 0.881995i \(0.656200\pi\)
\(684\) −0.772002 −0.0295182
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) 23.5440 0.898260
\(688\) 0.772002 0.0294323
\(689\) 19.0880 0.727195
\(690\) 0 0
\(691\) −47.7720 −1.81733 −0.908666 0.417523i \(-0.862898\pi\)
−0.908666 + 0.417523i \(0.862898\pi\)
\(692\) 7.22800 0.274767
\(693\) −8.31601 −0.315899
\(694\) 7.54400 0.286366
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) 21.5440 0.816037
\(698\) −8.77200 −0.332025
\(699\) −28.3160 −1.07101
\(700\) 0 0
\(701\) −15.5440 −0.587089 −0.293544 0.955945i \(-0.594835\pi\)
−0.293544 + 0.955945i \(0.594835\pi\)
\(702\) 4.77200 0.180108
\(703\) −1.36799 −0.0515947
\(704\) −2.77200 −0.104474
\(705\) 0 0
\(706\) −23.8600 −0.897983
\(707\) −33.9480 −1.27675
\(708\) −6.00000 −0.225494
\(709\) −1.31601 −0.0494236 −0.0247118 0.999695i \(-0.507867\pi\)
−0.0247118 + 0.999695i \(0.507867\pi\)
\(710\) 0 0
\(711\) 10.7720 0.403982
\(712\) 8.22800 0.308357
\(713\) −7.54400 −0.282525
\(714\) −23.3160 −0.872580
\(715\) 0 0
\(716\) −9.54400 −0.356676
\(717\) −14.2280 −0.531354
\(718\) −18.7720 −0.700565
\(719\) −4.68399 −0.174684 −0.0873418 0.996178i \(-0.527837\pi\)
−0.0873418 + 0.996178i \(0.527837\pi\)
\(720\) 0 0
\(721\) −33.0000 −1.22898
\(722\) −18.4040 −0.684926
\(723\) −24.6320 −0.916074
\(724\) −11.3160 −0.420556
\(725\) 0 0
\(726\) 3.31601 0.123069
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) −14.3160 −0.530586
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.00000 0.221918
\(732\) −7.54400 −0.278834
\(733\) −25.7720 −0.951911 −0.475955 0.879469i \(-0.657898\pi\)
−0.475955 + 0.879469i \(0.657898\pi\)
\(734\) −7.68399 −0.283621
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 20.9120 0.770303
\(738\) 2.77200 0.102039
\(739\) 25.7720 0.948038 0.474019 0.880515i \(-0.342803\pi\)
0.474019 + 0.880515i \(0.342803\pi\)
\(740\) 0 0
\(741\) −3.68399 −0.135335
\(742\) −12.0000 −0.440534
\(743\) −28.7720 −1.05554 −0.527771 0.849387i \(-0.676972\pi\)
−0.527771 + 0.849387i \(0.676972\pi\)
\(744\) −7.54400 −0.276577
\(745\) 0 0
\(746\) 4.22800 0.154798
\(747\) 1.00000 0.0365881
\(748\) −21.5440 −0.787727
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −9.45600 −0.345054 −0.172527 0.985005i \(-0.555193\pi\)
−0.172527 + 0.985005i \(0.555193\pi\)
\(752\) 0.227998 0.00831424
\(753\) 29.3160 1.06833
\(754\) −14.3160 −0.521358
\(755\) 0 0
\(756\) −3.00000 −0.109109
\(757\) 49.7720 1.80899 0.904497 0.426480i \(-0.140246\pi\)
0.904497 + 0.426480i \(0.140246\pi\)
\(758\) 11.0880 0.402735
\(759\) −2.77200 −0.100617
\(760\) 0 0
\(761\) 27.8600 1.00992 0.504962 0.863141i \(-0.331506\pi\)
0.504962 + 0.863141i \(0.331506\pi\)
\(762\) 21.5440 0.780457
\(763\) 18.6840 0.676406
\(764\) 18.3160 0.662650
\(765\) 0 0
\(766\) −5.22800 −0.188895
\(767\) −28.6320 −1.03384
\(768\) −1.00000 −0.0360844
\(769\) 25.0880 0.904697 0.452348 0.891841i \(-0.350586\pi\)
0.452348 + 0.891841i \(0.350586\pi\)
\(770\) 0 0
\(771\) 25.0880 0.903523
\(772\) 3.77200 0.135757
\(773\) 31.5440 1.13456 0.567279 0.823525i \(-0.307996\pi\)
0.567279 + 0.823525i \(0.307996\pi\)
\(774\) 0.772002 0.0277490
\(775\) 0 0
\(776\) −0.455996 −0.0163693
\(777\) −5.31601 −0.190711
\(778\) 18.0000 0.645331
\(779\) −2.13999 −0.0766731
\(780\) 0 0
\(781\) −10.4560 −0.374145
\(782\) −7.77200 −0.277926
\(783\) −3.00000 −0.107211
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) −15.0880 −0.538171
\(787\) −13.2280 −0.471527 −0.235764 0.971810i \(-0.575759\pi\)
−0.235764 + 0.971810i \(0.575759\pi\)
\(788\) 13.0000 0.463106
\(789\) −23.5440 −0.838189
\(790\) 0 0
\(791\) 11.3160 0.402351
\(792\) −2.77200 −0.0984988
\(793\) −36.0000 −1.27840
\(794\) −36.6320 −1.30002
\(795\) 0 0
\(796\) −22.0880 −0.782889
\(797\) 37.0880 1.31372 0.656862 0.754011i \(-0.271883\pi\)
0.656862 + 0.754011i \(0.271883\pi\)
\(798\) 2.31601 0.0819857
\(799\) 1.77200 0.0626889
\(800\) 0 0
\(801\) 8.22800 0.290722
\(802\) 22.6320 0.799164
\(803\) −29.2280 −1.03143
\(804\) 7.54400 0.266056
\(805\) 0 0
\(806\) −36.0000 −1.26805
\(807\) −15.8600 −0.558299
\(808\) −11.3160 −0.398096
\(809\) 21.6840 0.762369 0.381184 0.924499i \(-0.375516\pi\)
0.381184 + 0.924499i \(0.375516\pi\)
\(810\) 0 0
\(811\) 12.6320 0.443570 0.221785 0.975096i \(-0.428812\pi\)
0.221785 + 0.975096i \(0.428812\pi\)
\(812\) 9.00000 0.315838
\(813\) −16.0000 −0.561144
\(814\) −4.91199 −0.172165
\(815\) 0 0
\(816\) −7.77200 −0.272074
\(817\) −0.595987 −0.0208509
\(818\) −23.0000 −0.804176
\(819\) −14.3160 −0.500242
\(820\) 0 0
\(821\) −20.7720 −0.724948 −0.362474 0.931994i \(-0.618068\pi\)
−0.362474 + 0.931994i \(0.618068\pi\)
\(822\) 6.22800 0.217226
\(823\) 6.45600 0.225042 0.112521 0.993649i \(-0.464107\pi\)
0.112521 + 0.993649i \(0.464107\pi\)
\(824\) −11.0000 −0.383203
\(825\) 0 0
\(826\) 18.0000 0.626300
\(827\) 0.367989 0.0127962 0.00639811 0.999980i \(-0.497963\pi\)
0.00639811 + 0.999980i \(0.497963\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 21.4040 0.743392 0.371696 0.928354i \(-0.378776\pi\)
0.371696 + 0.928354i \(0.378776\pi\)
\(830\) 0 0
\(831\) −16.7720 −0.581814
\(832\) −4.77200 −0.165439
\(833\) 15.5440 0.538568
\(834\) −7.77200 −0.269122
\(835\) 0 0
\(836\) 2.13999 0.0740131
\(837\) −7.54400 −0.260759
\(838\) −25.6320 −0.885443
\(839\) −27.4040 −0.946092 −0.473046 0.881038i \(-0.656845\pi\)
−0.473046 + 0.881038i \(0.656845\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 22.0000 0.758170
\(843\) 26.8600 0.925108
\(844\) 15.3160 0.527199
\(845\) 0 0
\(846\) 0.227998 0.00783874
\(847\) −9.94802 −0.341818
\(848\) −4.00000 −0.137361
\(849\) 28.0000 0.960958
\(850\) 0 0
\(851\) −1.77200 −0.0607434
\(852\) −3.77200 −0.129227
\(853\) 33.8600 1.15934 0.579672 0.814850i \(-0.303181\pi\)
0.579672 + 0.814850i \(0.303181\pi\)
\(854\) 22.6320 0.774451
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) −13.2280 −0.451596
\(859\) −41.5440 −1.41746 −0.708732 0.705478i \(-0.750732\pi\)
−0.708732 + 0.705478i \(0.750732\pi\)
\(860\) 0 0
\(861\) −8.31601 −0.283409
\(862\) −26.6320 −0.907090
\(863\) 23.3160 0.793686 0.396843 0.917886i \(-0.370106\pi\)
0.396843 + 0.917886i \(0.370106\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −23.5440 −0.800058
\(867\) −43.4040 −1.47408
\(868\) 22.6320 0.768181
\(869\) −29.8600 −1.01293
\(870\) 0 0
\(871\) 36.0000 1.21981
\(872\) 6.22800 0.210907
\(873\) −0.455996 −0.0154331
\(874\) 0.772002 0.0261134
\(875\) 0 0
\(876\) −10.5440 −0.356249
\(877\) 55.2640 1.86613 0.933067 0.359703i \(-0.117122\pi\)
0.933067 + 0.359703i \(0.117122\pi\)
\(878\) −12.0000 −0.404980
\(879\) 7.54400 0.254453
\(880\) 0 0
\(881\) 27.5440 0.927981 0.463991 0.885840i \(-0.346417\pi\)
0.463991 + 0.885840i \(0.346417\pi\)
\(882\) 2.00000 0.0673435
\(883\) 2.22800 0.0749781 0.0374891 0.999297i \(-0.488064\pi\)
0.0374891 + 0.999297i \(0.488064\pi\)
\(884\) −37.0880 −1.24740
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) −29.9480 −1.00556 −0.502778 0.864416i \(-0.667689\pi\)
−0.502778 + 0.864416i \(0.667689\pi\)
\(888\) −1.77200 −0.0594645
\(889\) −64.6320 −2.16769
\(890\) 0 0
\(891\) −2.77200 −0.0928656
\(892\) −10.0000 −0.334825
\(893\) −0.176015 −0.00589012
\(894\) −23.0880 −0.772178
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) −4.77200 −0.159333
\(898\) −2.00000 −0.0667409
\(899\) 22.6320 0.754820
\(900\) 0 0
\(901\) −31.0880 −1.03569
\(902\) −7.68399 −0.255849
\(903\) −2.31601 −0.0770718
\(904\) 3.77200 0.125455
\(905\) 0 0
\(906\) −6.45600 −0.214486
\(907\) 25.4040 0.843526 0.421763 0.906706i \(-0.361411\pi\)
0.421763 + 0.906706i \(0.361411\pi\)
\(908\) 2.22800 0.0739387
\(909\) −11.3160 −0.375328
\(910\) 0 0
\(911\) −2.77200 −0.0918405 −0.0459203 0.998945i \(-0.514622\pi\)
−0.0459203 + 0.998945i \(0.514622\pi\)
\(912\) 0.772002 0.0255635
\(913\) −2.77200 −0.0917399
\(914\) −20.6320 −0.682446
\(915\) 0 0
\(916\) −23.5440 −0.777916
\(917\) 45.2640 1.49475
\(918\) −7.77200 −0.256514
\(919\) −5.31601 −0.175359 −0.0876794 0.996149i \(-0.527945\pi\)
−0.0876794 + 0.996149i \(0.527945\pi\)
\(920\) 0 0
\(921\) 5.31601 0.175168
\(922\) −4.54400 −0.149649
\(923\) −18.0000 −0.592477
\(924\) 8.31601 0.273576
\(925\) 0 0
\(926\) −35.0880 −1.15306
\(927\) −11.0000 −0.361287
\(928\) 3.00000 0.0984798
\(929\) 50.3160 1.65081 0.825407 0.564538i \(-0.190946\pi\)
0.825407 + 0.564538i \(0.190946\pi\)
\(930\) 0 0
\(931\) −1.54400 −0.0506027
\(932\) 28.3160 0.927522
\(933\) −24.8600 −0.813880
\(934\) −1.45600 −0.0476417
\(935\) 0 0
\(936\) −4.77200 −0.155978
\(937\) 42.6320 1.39273 0.696364 0.717689i \(-0.254800\pi\)
0.696364 + 0.717689i \(0.254800\pi\)
\(938\) −22.6320 −0.738961
\(939\) −8.00000 −0.261070
\(940\) 0 0
\(941\) −30.6320 −0.998575 −0.499288 0.866436i \(-0.666405\pi\)
−0.499288 + 0.866436i \(0.666405\pi\)
\(942\) −3.54400 −0.115470
\(943\) −2.77200 −0.0902688
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) −2.13999 −0.0695771
\(947\) 3.08801 0.100347 0.0501734 0.998741i \(-0.484023\pi\)
0.0501734 + 0.998741i \(0.484023\pi\)
\(948\) −10.7720 −0.349858
\(949\) −50.3160 −1.63333
\(950\) 0 0
\(951\) −9.00000 −0.291845
\(952\) 23.3160 0.755676
\(953\) 38.2280 1.23833 0.619163 0.785262i \(-0.287472\pi\)
0.619163 + 0.785262i \(0.287472\pi\)
\(954\) −4.00000 −0.129505
\(955\) 0 0
\(956\) 14.2280 0.460166
\(957\) 8.31601 0.268818
\(958\) −9.68399 −0.312876
\(959\) −18.6840 −0.603338
\(960\) 0 0
\(961\) 25.9120 0.835871
\(962\) −8.45600 −0.272632
\(963\) 4.00000 0.128898
\(964\) 24.6320 0.793344
\(965\) 0 0
\(966\) 3.00000 0.0965234
\(967\) 42.6320 1.37095 0.685477 0.728095i \(-0.259594\pi\)
0.685477 + 0.728095i \(0.259594\pi\)
\(968\) −3.31601 −0.106580
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) 19.6320 0.630021 0.315011 0.949088i \(-0.397992\pi\)
0.315011 + 0.949088i \(0.397992\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 23.3160 0.747477
\(974\) −29.5440 −0.946651
\(975\) 0 0
\(976\) 7.54400 0.241478
\(977\) 34.6840 1.10964 0.554820 0.831971i \(-0.312787\pi\)
0.554820 + 0.831971i \(0.312787\pi\)
\(978\) 16.0000 0.511624
\(979\) −22.8080 −0.728948
\(980\) 0 0
\(981\) 6.22800 0.198845
\(982\) −27.5440 −0.878964
\(983\) 51.4040 1.63953 0.819767 0.572698i \(-0.194103\pi\)
0.819767 + 0.572698i \(0.194103\pi\)
\(984\) −2.77200 −0.0883682
\(985\) 0 0
\(986\) 23.3160 0.742533
\(987\) −0.683994 −0.0217718
\(988\) 3.68399 0.117203
\(989\) −0.772002 −0.0245482
\(990\) 0 0
\(991\) −38.1760 −1.21270 −0.606351 0.795197i \(-0.707367\pi\)
−0.606351 + 0.795197i \(0.707367\pi\)
\(992\) 7.54400 0.239522
\(993\) −26.2280 −0.832320
\(994\) 11.3160 0.358922
\(995\) 0 0
\(996\) −1.00000 −0.0316862
\(997\) 8.77200 0.277812 0.138906 0.990306i \(-0.455641\pi\)
0.138906 + 0.990306i \(0.455641\pi\)
\(998\) −35.3160 −1.11791
\(999\) −1.77200 −0.0560637
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.bj.1.1 yes 2
5.2 odd 4 3450.2.d.w.2899.3 4
5.3 odd 4 3450.2.d.w.2899.1 4
5.4 even 2 3450.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.bh.1.1 2 5.4 even 2
3450.2.a.bj.1.1 yes 2 1.1 even 1 trivial
3450.2.d.w.2899.1 4 5.3 odd 4
3450.2.d.w.2899.3 4 5.2 odd 4