Properties

Label 3450.2.a.bj.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: \(\Q(\sqrt{73}) \)
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 1.2
Root \(-3.77200\) of defining polynomial
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +5.77200 q^{11} -1.00000 q^{12} +3.77200 q^{13} +3.00000 q^{14} +1.00000 q^{16} -0.772002 q^{17} +1.00000 q^{18} +7.77200 q^{19} -3.00000 q^{21} +5.77200 q^{22} -1.00000 q^{23} -1.00000 q^{24} +3.77200 q^{26} -1.00000 q^{27} +3.00000 q^{28} +3.00000 q^{29} -9.54400 q^{31} +1.00000 q^{32} -5.77200 q^{33} -0.772002 q^{34} +1.00000 q^{36} -6.77200 q^{37} +7.77200 q^{38} -3.77200 q^{39} -5.77200 q^{41} -3.00000 q^{42} -7.77200 q^{43} +5.77200 q^{44} -1.00000 q^{46} +8.77200 q^{47} -1.00000 q^{48} +2.00000 q^{49} +0.772002 q^{51} +3.77200 q^{52} -4.00000 q^{53} -1.00000 q^{54} +3.00000 q^{56} -7.77200 q^{57} +3.00000 q^{58} +6.00000 q^{59} -9.54400 q^{61} -9.54400 q^{62} +3.00000 q^{63} +1.00000 q^{64} -5.77200 q^{66} +9.54400 q^{67} -0.772002 q^{68} +1.00000 q^{69} -4.77200 q^{71} +1.00000 q^{72} -6.54400 q^{73} -6.77200 q^{74} +7.77200 q^{76} +17.3160 q^{77} -3.77200 q^{78} +2.22800 q^{79} +1.00000 q^{81} -5.77200 q^{82} +1.00000 q^{83} -3.00000 q^{84} -7.77200 q^{86} -3.00000 q^{87} +5.77200 q^{88} +16.7720 q^{89} +11.3160 q^{91} -1.00000 q^{92} +9.54400 q^{93} +8.77200 q^{94} -1.00000 q^{96} -17.5440 q^{97} +2.00000 q^{98} +5.77200 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 6q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 6q^{7} + 2q^{8} + 2q^{9} + 3q^{11} - 2q^{12} - q^{13} + 6q^{14} + 2q^{16} + 7q^{17} + 2q^{18} + 7q^{19} - 6q^{21} + 3q^{22} - 2q^{23} - 2q^{24} - q^{26} - 2q^{27} + 6q^{28} + 6q^{29} - 2q^{31} + 2q^{32} - 3q^{33} + 7q^{34} + 2q^{36} - 5q^{37} + 7q^{38} + q^{39} - 3q^{41} - 6q^{42} - 7q^{43} + 3q^{44} - 2q^{46} + 9q^{47} - 2q^{48} + 4q^{49} - 7q^{51} - q^{52} - 8q^{53} - 2q^{54} + 6q^{56} - 7q^{57} + 6q^{58} + 12q^{59} - 2q^{61} - 2q^{62} + 6q^{63} + 2q^{64} - 3q^{66} + 2q^{67} + 7q^{68} + 2q^{69} - q^{71} + 2q^{72} + 4q^{73} - 5q^{74} + 7q^{76} + 9q^{77} + q^{78} + 13q^{79} + 2q^{81} - 3q^{82} + 2q^{83} - 6q^{84} - 7q^{86} - 6q^{87} + 3q^{88} + 25q^{89} - 3q^{91} - 2q^{92} + 2q^{93} + 9q^{94} - 2q^{96} - 18q^{97} + 4q^{98} + 3q^{99} + O(q^{100}) \)

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
\(11\) 5.77200 1.74032 0.870162 0.492766i \(-0.164014\pi\)
0.870162 + 0.492766i \(0.164014\pi\)
\(12\) −1.00000 −0.288675
\(13\) 3.77200 1.04617 0.523083 0.852282i \(-0.324782\pi\)
0.523083 + 0.852282i \(0.324782\pi\)
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.772002 −0.187238 −0.0936190 0.995608i \(-0.529844\pi\)
−0.0936190 + 0.995608i \(0.529844\pi\)
\(18\) 1.00000 0.235702
\(19\) 7.77200 1.78302 0.891510 0.453001i \(-0.149647\pi\)
0.891510 + 0.453001i \(0.149647\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 5.77200 1.23059
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 3.77200 0.739750
\(27\) −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
\(31\) −9.54400 −1.71415 −0.857077 0.515189i \(-0.827722\pi\)
−0.857077 + 0.515189i \(0.827722\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.77200 −1.00478
\(34\) −0.772002 −0.132397
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −6.77200 −1.11331 −0.556655 0.830744i \(-0.687916\pi\)
−0.556655 + 0.830744i \(0.687916\pi\)
\(38\) 7.77200 1.26079
\(39\) −3.77200 −0.604004
\(40\) 0 0
\(41\) −5.77200 −0.901435 −0.450718 0.892667i \(-0.648832\pi\)
−0.450718 + 0.892667i \(0.648832\pi\)
\(42\) −3.00000 −0.462910
\(43\) −7.77200 −1.18522 −0.592610 0.805490i \(-0.701902\pi\)
−0.592610 + 0.805490i \(0.701902\pi\)
\(44\) 5.77200 0.870162
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 8.77200 1.27953 0.639764 0.768571i \(-0.279032\pi\)
0.639764 + 0.768571i \(0.279032\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 0.772002 0.108102
\(52\) 3.77200 0.523083
\(53\) −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
\(57\) −7.77200 −1.02943
\(58\) 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
\(61\) −9.54400 −1.22198 −0.610992 0.791637i \(-0.709229\pi\)
−0.610992 + 0.791637i \(0.709229\pi\)
\(62\) −9.54400 −1.21209
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −5.77200 −0.710484
\(67\) 9.54400 1.16599 0.582993 0.812477i \(-0.301882\pi\)
0.582993 + 0.812477i \(0.301882\pi\)
\(68\) −0.772002 −0.0936190
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −4.77200 −0.566332 −0.283166 0.959071i \(-0.591385\pi\)
−0.283166 + 0.959071i \(0.591385\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.54400 −0.765918 −0.382959 0.923765i \(-0.625095\pi\)
−0.382959 + 0.923765i \(0.625095\pi\)
\(74\) −6.77200 −0.787229
\(75\) 0 0
\(76\) 7.77200 0.891510
\(77\) 17.3160 1.97334
\(78\) −3.77200 −0.427095
\(79\) 2.22800 0.250669 0.125335 0.992115i \(-0.460000\pi\)
0.125335 + 0.992115i \(0.460000\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −5.77200 −0.637411
\(83\) 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
\(86\) −7.77200 −0.838077
\(87\) −3.00000 −0.321634
\(88\) 5.77200 0.615297
\(89\) 16.7720 1.77783 0.888914 0.458073i \(-0.151460\pi\)
0.888914 + 0.458073i \(0.151460\pi\)
\(90\) 0 0
\(91\) 11.3160 1.18624
\(92\) −1.00000 −0.104257
\(93\) 9.54400 0.989667
\(94\) 8.77200 0.904763
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −17.5440 −1.78132 −0.890662 0.454666i \(-0.849759\pi\)
−0.890662 + 0.454666i \(0.849759\pi\)
\(98\) 2.00000 0.202031
\(99\) 5.77200 0.580108
\(100\) 0 0
\(101\) 14.3160 1.42450 0.712248 0.701928i \(-0.247677\pi\)
0.712248 + 0.701928i \(0.247677\pi\)
\(102\) 0.772002 0.0764396
\(103\) −11.0000 −1.08386 −0.541931 0.840423i \(-0.682307\pi\)
−0.541931 + 0.840423i \(0.682307\pi\)
\(104\) 3.77200 0.369875
\(105\) 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
\(109\) 14.7720 1.41490 0.707451 0.706763i \(-0.249845\pi\)
0.707451 + 0.706763i \(0.249845\pi\)
\(110\) 0 0
\(111\) 6.77200 0.642770
\(112\) 3.00000 0.283473
\(113\) −4.77200 −0.448912 −0.224456 0.974484i \(-0.572061\pi\)
−0.224456 + 0.974484i \(0.572061\pi\)
\(114\) −7.77200 −0.727915
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) 3.77200 0.348722
\(118\) 6.00000 0.552345
\(119\) −2.31601 −0.212308
\(120\) 0 0
\(121\) 22.3160 2.02873
\(122\) −9.54400 −0.864073
\(123\) 5.77200 0.520444
\(124\) −9.54400 −0.857077
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) −4.45600 −0.395406 −0.197703 0.980262i \(-0.563348\pi\)
−0.197703 + 0.980262i \(0.563348\pi\)
\(128\) 1.00000 0.0883883
\(129\) 7.77200 0.684287
\(130\) 0 0
\(131\) −19.0880 −1.66773 −0.833863 0.551971i \(-0.813876\pi\)
−0.833863 + 0.551971i \(0.813876\pi\)
\(132\) −5.77200 −0.502388
\(133\) 23.3160 2.02175
\(134\) 9.54400 0.824476
\(135\) 0 0
\(136\) −0.772002 −0.0661986
\(137\) −14.7720 −1.26206 −0.631029 0.775760i \(-0.717367\pi\)
−0.631029 + 0.775760i \(0.717367\pi\)
\(138\) 1.00000 0.0851257
\(139\) −0.772002 −0.0654803 −0.0327402 0.999464i \(-0.510423\pi\)
−0.0327402 + 0.999464i \(0.510423\pi\)
\(140\) 0 0
\(141\) −8.77200 −0.738736
\(142\) −4.77200 −0.400458
\(143\) 21.7720 1.82067
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −6.54400 −0.541586
\(147\) −2.00000 −0.164957
\(148\) −6.77200 −0.556655
\(149\) −11.0880 −0.908365 −0.454182 0.890909i \(-0.650069\pi\)
−0.454182 + 0.890909i \(0.650069\pi\)
\(150\) 0 0
\(151\) 23.5440 1.91598 0.957992 0.286795i \(-0.0925899\pi\)
0.957992 + 0.286795i \(0.0925899\pi\)
\(152\) 7.77200 0.630393
\(153\) −0.772002 −0.0624127
\(154\) 17.3160 1.39536
\(155\) 0 0
\(156\) −3.77200 −0.302002
\(157\) −13.5440 −1.08093 −0.540465 0.841367i \(-0.681752\pi\)
−0.540465 + 0.841367i \(0.681752\pi\)
\(158\) 2.22800 0.177250
\(159\) 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 \(0.715557\pi\)
\(164\) −5.77200 −0.450718
\(165\) 0 0
\(166\) 1.00000 0.0776151
\(167\) −16.3160 −1.26257 −0.631285 0.775551i \(-0.717472\pi\)
−0.631285 + 0.775551i \(0.717472\pi\)
\(168\) −3.00000 −0.231455
\(169\) 1.22800 0.0944614
\(170\) 0 0
\(171\) 7.77200 0.594340
\(172\) −7.77200 −0.592610
\(173\) 15.7720 1.19912 0.599562 0.800329i \(-0.295342\pi\)
0.599562 + 0.800329i \(0.295342\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) 5.77200 0.435081
\(177\) −6.00000 −0.450988
\(178\) 16.7720 1.25711
\(179\) 7.54400 0.563865 0.281933 0.959434i \(-0.409025\pi\)
0.281933 + 0.959434i \(0.409025\pi\)
\(180\) 0 0
\(181\) 14.3160 1.06410 0.532050 0.846713i \(-0.321422\pi\)
0.532050 + 0.846713i \(0.321422\pi\)
\(182\) 11.3160 0.838798
\(183\) 9.54400 0.705513
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 9.54400 0.699800
\(187\) −4.45600 −0.325855
\(188\) 8.77200 0.639764
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) −7.31601 −0.529368 −0.264684 0.964335i \(-0.585268\pi\)
−0.264684 + 0.964335i \(0.585268\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −4.77200 −0.343496 −0.171748 0.985141i \(-0.554942\pi\)
−0.171748 + 0.985141i \(0.554942\pi\)
\(194\) −17.5440 −1.25959
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 13.0000 0.926212 0.463106 0.886303i \(-0.346735\pi\)
0.463106 + 0.886303i \(0.346735\pi\)
\(198\) 5.77200 0.410198
\(199\) 12.0880 0.856896 0.428448 0.903566i \(-0.359060\pi\)
0.428448 + 0.903566i \(0.359060\pi\)
\(200\) 0 0
\(201\) −9.54400 −0.673182
\(202\) 14.3160 1.00727
\(203\) 9.00000 0.631676
\(204\) 0.772002 0.0540509
\(205\) 0 0
\(206\) −11.0000 −0.766406
\(207\) −1.00000 −0.0695048
\(208\) 3.77200 0.261541
\(209\) 44.8600 3.10303
\(210\) 0 0
\(211\) −10.3160 −0.710183 −0.355092 0.934832i \(-0.615550\pi\)
−0.355092 + 0.934832i \(0.615550\pi\)
\(212\) −4.00000 −0.274721
\(213\) 4.77200 0.326972
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −28.6320 −1.94367
\(218\) 14.7720 1.00049
\(219\) 6.54400 0.442203
\(220\) 0 0
\(221\) −2.91199 −0.195882
\(222\) 6.77200 0.454507
\(223\) −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
\(226\) −4.77200 −0.317429
\(227\) 10.7720 0.714963 0.357481 0.933920i \(-0.383636\pi\)
0.357481 + 0.933920i \(0.383636\pi\)
\(228\) −7.77200 −0.514713
\(229\) −6.45600 −0.426624 −0.213312 0.976984i \(-0.568425\pi\)
−0.213312 + 0.976984i \(0.568425\pi\)
\(230\) 0 0
\(231\) −17.3160 −1.13931
\(232\) 3.00000 0.196960
\(233\) 2.68399 0.175834 0.0879172 0.996128i \(-0.471979\pi\)
0.0879172 + 0.996128i \(0.471979\pi\)
\(234\) 3.77200 0.246583
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) −2.22800 −0.144724
\(238\) −2.31601 −0.150124
\(239\) 22.7720 1.47300 0.736499 0.676438i \(-0.236478\pi\)
0.736499 + 0.676438i \(0.236478\pi\)
\(240\) 0 0
\(241\) −26.6320 −1.71552 −0.857759 0.514051i \(-0.828144\pi\)
−0.857759 + 0.514051i \(0.828144\pi\)
\(242\) 22.3160 1.43453
\(243\) −1.00000 −0.0641500
\(244\) −9.54400 −0.610992
\(245\) 0 0
\(246\) 5.77200 0.368009
\(247\) 29.3160 1.86533
\(248\) −9.54400 −0.606045
\(249\) −1.00000 −0.0633724
\(250\) 0 0
\(251\) −3.68399 −0.232532 −0.116266 0.993218i \(-0.537092\pi\)
−0.116266 + 0.993218i \(0.537092\pi\)
\(252\) 3.00000 0.188982
\(253\) −5.77200 −0.362883
\(254\) −4.45600 −0.279594
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 9.08801 0.566894 0.283447 0.958988i \(-0.408522\pi\)
0.283447 + 0.958988i \(0.408522\pi\)
\(258\) 7.77200 0.483864
\(259\) −20.3160 −1.26238
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) −19.0880 −1.17926
\(263\) 6.45600 0.398094 0.199047 0.979990i \(-0.436215\pi\)
0.199047 + 0.979990i \(0.436215\pi\)
\(264\) −5.77200 −0.355242
\(265\) 0 0
\(266\) 23.3160 1.42960
\(267\) −16.7720 −1.02643
\(268\) 9.54400 0.582993
\(269\) −26.8600 −1.63768 −0.818842 0.574019i \(-0.805383\pi\)
−0.818842 + 0.574019i \(0.805383\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −0.772002 −0.0468095
\(273\) −11.3160 −0.684876
\(274\) −14.7720 −0.892409
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) 8.22800 0.494372 0.247186 0.968968i \(-0.420494\pi\)
0.247186 + 0.968968i \(0.420494\pi\)
\(278\) −0.772002 −0.0463016
\(279\) −9.54400 −0.571385
\(280\) 0 0
\(281\) 15.8600 0.946129 0.473064 0.881028i \(-0.343148\pi\)
0.473064 + 0.881028i \(0.343148\pi\)
\(282\) −8.77200 −0.522365
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) −4.77200 −0.283166
\(285\) 0 0
\(286\) 21.7720 1.28741
\(287\) −17.3160 −1.02213
\(288\) 1.00000 0.0589256
\(289\) −16.4040 −0.964942
\(290\) 0 0
\(291\) 17.5440 1.02845
\(292\) −6.54400 −0.382959
\(293\) 9.54400 0.557567 0.278783 0.960354i \(-0.410069\pi\)
0.278783 + 0.960354i \(0.410069\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) −6.77200 −0.393615
\(297\) −5.77200 −0.334926
\(298\) −11.0880 −0.642311
\(299\) −3.77200 −0.218141
\(300\) 0 0
\(301\) −23.3160 −1.34391
\(302\) 23.5440 1.35481
\(303\) −14.3160 −0.822433
\(304\) 7.77200 0.445755
\(305\) 0 0
\(306\) −0.772002 −0.0441324
\(307\) 20.3160 1.15950 0.579748 0.814796i \(-0.303151\pi\)
0.579748 + 0.814796i \(0.303151\pi\)
\(308\) 17.3160 0.986671
\(309\) 11.0000 0.625768
\(310\) 0 0
\(311\) −17.8600 −1.01275 −0.506374 0.862314i \(-0.669015\pi\)
−0.506374 + 0.862314i \(0.669015\pi\)
\(312\) −3.77200 −0.213548
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) −13.5440 −0.764332
\(315\) 0 0
\(316\) 2.22800 0.125335
\(317\) 9.00000 0.505490 0.252745 0.967533i \(-0.418667\pi\)
0.252745 + 0.967533i \(0.418667\pi\)
\(318\) 4.00000 0.224309
\(319\) 17.3160 0.969510
\(320\) 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
\(327\) −14.7720 −0.816894
\(328\) −5.77200 −0.318705
\(329\) 26.3160 1.45085
\(330\) 0 0
\(331\) 34.7720 1.91124 0.955621 0.294599i \(-0.0951860\pi\)
0.955621 + 0.294599i \(0.0951860\pi\)
\(332\) 1.00000 0.0548821
\(333\) −6.77200 −0.371103
\(334\) −16.3160 −0.892772
\(335\) 0 0
\(336\) −3.00000 −0.163663
\(337\) −4.00000 −0.217894 −0.108947 0.994048i \(-0.534748\pi\)
−0.108947 + 0.994048i \(0.534748\pi\)
\(338\) 1.22800 0.0667943
\(339\) 4.77200 0.259180
\(340\) 0 0
\(341\) −55.0880 −2.98318
\(342\) 7.77200 0.420262
\(343\) −15.0000 −0.809924
\(344\) −7.77200 −0.419038
\(345\) 0 0
\(346\) 15.7720 0.847908
\(347\) −9.54400 −0.512349 −0.256174 0.966631i \(-0.582462\pi\)
−0.256174 + 0.966631i \(0.582462\pi\)
\(348\) −3.00000 −0.160817
\(349\) −0.227998 −0.0122045 −0.00610223 0.999981i \(-0.501942\pi\)
−0.00610223 + 0.999981i \(0.501942\pi\)
\(350\) 0 0
\(351\) −3.77200 −0.201335
\(352\) 5.77200 0.307649
\(353\) 18.8600 1.00382 0.501909 0.864921i \(-0.332631\pi\)
0.501909 + 0.864921i \(0.332631\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) 16.7720 0.888914
\(357\) 2.31601 0.122576
\(358\) 7.54400 0.398713
\(359\) −10.2280 −0.539813 −0.269907 0.962887i \(-0.586993\pi\)
−0.269907 + 0.962887i \(0.586993\pi\)
\(360\) 0 0
\(361\) 41.4040 2.17916
\(362\) 14.3160 0.752433
\(363\) −22.3160 −1.17129
\(364\) 11.3160 0.593120
\(365\) 0 0
\(366\) 9.54400 0.498873
\(367\) −33.3160 −1.73908 −0.869541 0.493861i \(-0.835585\pi\)
−0.869541 + 0.493861i \(0.835585\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −5.77200 −0.300478
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 9.54400 0.494834
\(373\) 12.7720 0.661309 0.330655 0.943752i \(-0.392730\pi\)
0.330655 + 0.943752i \(0.392730\pi\)
\(374\) −4.45600 −0.230414
\(375\) 0 0
\(376\) 8.77200 0.452381
\(377\) 11.3160 0.582804
\(378\) −3.00000 −0.154303
\(379\) −23.0880 −1.18595 −0.592976 0.805220i \(-0.702047\pi\)
−0.592976 + 0.805220i \(0.702047\pi\)
\(380\) 0 0
\(381\) 4.45600 0.228288
\(382\) −7.31601 −0.374319
\(383\) −13.7720 −0.703716 −0.351858 0.936053i \(-0.614450\pi\)
−0.351858 + 0.936053i \(0.614450\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −4.77200 −0.242889
\(387\) −7.77200 −0.395073
\(388\) −17.5440 −0.890662
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 0.772002 0.0390418
\(392\) 2.00000 0.101015
\(393\) 19.0880 0.962863
\(394\) 13.0000 0.654931
\(395\) 0 0
\(396\) 5.77200 0.290054
\(397\) 14.6320 0.734360 0.367180 0.930150i \(-0.380323\pi\)
0.367180 + 0.930150i \(0.380323\pi\)
\(398\) 12.0880 0.605917
\(399\) −23.3160 −1.16726
\(400\) 0 0
\(401\) −28.6320 −1.42981 −0.714907 0.699219i \(-0.753531\pi\)
−0.714907 + 0.699219i \(0.753531\pi\)
\(402\) −9.54400 −0.476012
\(403\) −36.0000 −1.79329
\(404\) 14.3160 0.712248
\(405\) 0 0
\(406\) 9.00000 0.446663
\(407\) −39.0880 −1.93752
\(408\) 0.772002 0.0382198
\(409\) −23.0000 −1.13728 −0.568638 0.822588i \(-0.692530\pi\)
−0.568638 + 0.822588i \(0.692530\pi\)
\(410\) 0 0
\(411\) 14.7720 0.728649
\(412\) −11.0000 −0.541931
\(413\) 18.0000 0.885722
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 3.77200 0.184938
\(417\) 0.772002 0.0378051
\(418\) 44.8600 2.19417
\(419\) 25.6320 1.25221 0.626103 0.779740i \(-0.284649\pi\)
0.626103 + 0.779740i \(0.284649\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −10.3160 −0.502175
\(423\) 8.77200 0.426509
\(424\) −4.00000 −0.194257
\(425\) 0 0
\(426\) 4.77200 0.231204
\(427\) −28.6320 −1.38560
\(428\) 4.00000 0.193347
\(429\) −21.7720 −1.05116
\(430\) 0 0
\(431\) 24.6320 1.18648 0.593241 0.805025i \(-0.297848\pi\)
0.593241 + 0.805025i \(0.297848\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −6.45600 −0.310255 −0.155128 0.987894i \(-0.549579\pi\)
−0.155128 + 0.987894i \(0.549579\pi\)
\(434\) −28.6320 −1.37438
\(435\) 0 0
\(436\) 14.7720 0.707451
\(437\) −7.77200 −0.371785
\(438\) 6.54400 0.312685
\(439\) −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
\(442\) −2.91199 −0.138509
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 6.77200 0.321385
\(445\) 0 0
\(446\) −10.0000 −0.473514
\(447\) 11.0880 0.524445
\(448\) 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
\(451\) −33.3160 −1.56879
\(452\) −4.77200 −0.224456
\(453\) −23.5440 −1.10619
\(454\) 10.7720 0.505555
\(455\) 0 0
\(456\) −7.77200 −0.363957
\(457\) 30.6320 1.43291 0.716453 0.697636i \(-0.245765\pi\)
0.716453 + 0.697636i \(0.245765\pi\)
\(458\) −6.45600 −0.301669
\(459\) 0.772002 0.0360340
\(460\) 0 0
\(461\) 12.5440 0.584232 0.292116 0.956383i \(-0.405641\pi\)
0.292116 + 0.956383i \(0.405641\pi\)
\(462\) −17.3160 −0.805613
\(463\) −0.911993 −0.0423839 −0.0211919 0.999775i \(-0.506746\pi\)
−0.0211919 + 0.999775i \(0.506746\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) 2.68399 0.124334
\(467\) −18.5440 −0.858114 −0.429057 0.903277i \(-0.641154\pi\)
−0.429057 + 0.903277i \(0.641154\pi\)
\(468\) 3.77200 0.174361
\(469\) 28.6320 1.32210
\(470\) 0 0
\(471\) 13.5440 0.624075
\(472\) 6.00000 0.276172
\(473\) −44.8600 −2.06267
\(474\) −2.22800 −0.102335
\(475\) 0 0
\(476\) −2.31601 −0.106154
\(477\) −4.00000 −0.183147
\(478\) 22.7720 1.04157
\(479\) −35.3160 −1.61363 −0.806815 0.590805i \(-0.798810\pi\)
−0.806815 + 0.590805i \(0.798810\pi\)
\(480\) 0 0
\(481\) −25.5440 −1.16471
\(482\) −26.6320 −1.21305
\(483\) 3.00000 0.136505
\(484\) 22.3160 1.01436
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −12.4560 −0.564435 −0.282218 0.959350i \(-0.591070\pi\)
−0.282218 + 0.959350i \(0.591070\pi\)
\(488\) −9.54400 −0.432037
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) −10.4560 −0.471873 −0.235936 0.971769i \(-0.575816\pi\)
−0.235936 + 0.971769i \(0.575816\pi\)
\(492\) 5.77200 0.260222
\(493\) −2.31601 −0.104308
\(494\) 29.3160 1.31899
\(495\) 0 0
\(496\) −9.54400 −0.428538
\(497\) −14.3160 −0.642161
\(498\) −1.00000 −0.0448111
\(499\) −9.68399 −0.433515 −0.216758 0.976225i \(-0.569548\pi\)
−0.216758 + 0.976225i \(0.569548\pi\)
\(500\) 0 0
\(501\) 16.3160 0.728945
\(502\) −3.68399 −0.164425
\(503\) 39.9480 1.78119 0.890597 0.454793i \(-0.150287\pi\)
0.890597 + 0.454793i \(0.150287\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) −5.77200 −0.256597
\(507\) −1.22800 −0.0545373
\(508\) −4.45600 −0.197703
\(509\) 27.8600 1.23487 0.617437 0.786621i \(-0.288171\pi\)
0.617437 + 0.786621i \(0.288171\pi\)
\(510\) 0 0
\(511\) −19.6320 −0.868469
\(512\) 1.00000 0.0441942
\(513\) −7.77200 −0.343142
\(514\) 9.08801 0.400855
\(515\) 0 0
\(516\) 7.77200 0.342143
\(517\) 50.6320 2.22679
\(518\) −20.3160 −0.892634
\(519\) −15.7720 −0.692314
\(520\) 0 0
\(521\) 6.77200 0.296687 0.148343 0.988936i \(-0.452606\pi\)
0.148343 + 0.988936i \(0.452606\pi\)
\(522\) 3.00000 0.131306
\(523\) 13.3160 0.582268 0.291134 0.956682i \(-0.405967\pi\)
0.291134 + 0.956682i \(0.405967\pi\)
\(524\) −19.0880 −0.833863
\(525\) 0 0
\(526\) 6.45600 0.281495
\(527\) 7.36799 0.320955
\(528\) −5.77200 −0.251194
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 23.3160 1.01088
\(533\) −21.7720 −0.943050
\(534\) −16.7720 −0.725796
\(535\) 0 0
\(536\) 9.54400 0.412238
\(537\) −7.54400 −0.325548
\(538\) −26.8600 −1.15802
\(539\) 11.5440 0.497235
\(540\) 0 0
\(541\) −10.6840 −0.459341 −0.229670 0.973268i \(-0.573765\pi\)
−0.229670 + 0.973268i \(0.573765\pi\)
\(542\) 16.0000 0.687259
\(543\) −14.3160 −0.614359
\(544\) −0.772002 −0.0330993
\(545\) 0 0
\(546\) −11.3160 −0.484280
\(547\) −11.8600 −0.507097 −0.253549 0.967323i \(-0.581598\pi\)
−0.253549 + 0.967323i \(0.581598\pi\)
\(548\) −14.7720 −0.631029
\(549\) −9.54400 −0.407328
\(550\) 0 0
\(551\) 23.3160 0.993295
\(552\) 1.00000 0.0425628
\(553\) 6.68399 0.284232
\(554\) 8.22800 0.349574
\(555\) 0 0
\(556\) −0.772002 −0.0327402
\(557\) 21.0880 0.893528 0.446764 0.894652i \(-0.352576\pi\)
0.446764 + 0.894652i \(0.352576\pi\)
\(558\) −9.54400 −0.404030
\(559\) −29.3160 −1.23993
\(560\) 0 0
\(561\) 4.45600 0.188132
\(562\) 15.8600 0.669014
\(563\) −43.9480 −1.85219 −0.926094 0.377293i \(-0.876855\pi\)
−0.926094 + 0.377293i \(0.876855\pi\)
\(564\) −8.77200 −0.369368
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) 3.00000 0.125988
\(568\) −4.77200 −0.200229
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 12.6320 0.528633 0.264317 0.964436i \(-0.414854\pi\)
0.264317 + 0.964436i \(0.414854\pi\)
\(572\) 21.7720 0.910333
\(573\) 7.31601 0.305631
\(574\) −17.3160 −0.722756
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 31.3160 1.30370 0.651851 0.758347i \(-0.273993\pi\)
0.651851 + 0.758347i \(0.273993\pi\)
\(578\) −16.4040 −0.682317
\(579\) 4.77200 0.198318
\(580\) 0 0
\(581\) 3.00000 0.124461
\(582\) 17.5440 0.727222
\(583\) −23.0880 −0.956208
\(584\) −6.54400 −0.270793
\(585\) 0 0
\(586\) 9.54400 0.394259
\(587\) −24.4560 −1.00941 −0.504703 0.863293i \(-0.668398\pi\)
−0.504703 + 0.863293i \(0.668398\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −74.1760 −3.05637
\(590\) 0 0
\(591\) −13.0000 −0.534749
\(592\) −6.77200 −0.278328
\(593\) −41.3160 −1.69664 −0.848322 0.529480i \(-0.822387\pi\)
−0.848322 + 0.529480i \(0.822387\pi\)
\(594\) −5.77200 −0.236828
\(595\) 0 0
\(596\) −11.0880 −0.454182
\(597\) −12.0880 −0.494729
\(598\) −3.77200 −0.154249
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) −32.7720 −1.33680 −0.668399 0.743803i \(-0.733020\pi\)
−0.668399 + 0.743803i \(0.733020\pi\)
\(602\) −23.3160 −0.950289
\(603\) 9.54400 0.388662
\(604\) 23.5440 0.957992
\(605\) 0 0
\(606\) −14.3160 −0.581548
\(607\) −5.08801 −0.206516 −0.103258 0.994655i \(-0.532927\pi\)
−0.103258 + 0.994655i \(0.532927\pi\)
\(608\) 7.77200 0.315196
\(609\) −9.00000 −0.364698
\(610\) 0 0
\(611\) 33.0880 1.33860
\(612\) −0.772002 −0.0312063
\(613\) −20.7720 −0.838973 −0.419487 0.907762i \(-0.637790\pi\)
−0.419487 + 0.907762i \(0.637790\pi\)
\(614\) 20.3160 0.819887
\(615\) 0 0
\(616\) 17.3160 0.697682
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 11.0000 0.442485
\(619\) −0.911993 −0.0366561 −0.0183280 0.999832i \(-0.505834\pi\)
−0.0183280 + 0.999832i \(0.505834\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) −17.8600 −0.716121
\(623\) 50.3160 2.01587
\(624\) −3.77200 −0.151001
\(625\) 0 0
\(626\) 8.00000 0.319744
\(627\) −44.8600 −1.79154
\(628\) −13.5440 −0.540465
\(629\) 5.22800 0.208454
\(630\) 0 0
\(631\) 14.0880 0.560835 0.280417 0.959878i \(-0.409527\pi\)
0.280417 + 0.959878i \(0.409527\pi\)
\(632\) 2.22800 0.0886250
\(633\) 10.3160 0.410024
\(634\) 9.00000 0.357436
\(635\) 0 0
\(636\) 4.00000 0.158610
\(637\) 7.54400 0.298904
\(638\) 17.3160 0.685547
\(639\) −4.77200 −0.188777
\(640\) 0 0
\(641\) −17.8600 −0.705428 −0.352714 0.935731i \(-0.614741\pi\)
−0.352714 + 0.935731i \(0.614741\pi\)
\(642\) −4.00000 −0.157867
\(643\) 19.3160 0.761749 0.380874 0.924627i \(-0.375623\pi\)
0.380874 + 0.924627i \(0.375623\pi\)
\(644\) −3.00000 −0.118217
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) 30.7720 1.20977 0.604886 0.796312i \(-0.293219\pi\)
0.604886 + 0.796312i \(0.293219\pi\)
\(648\) 1.00000 0.0392837
\(649\) 34.6320 1.35943
\(650\) 0 0
\(651\) 28.6320 1.12218
\(652\) −16.0000 −0.626608
\(653\) −23.4560 −0.917904 −0.458952 0.888461i \(-0.651775\pi\)
−0.458952 + 0.888461i \(0.651775\pi\)
\(654\) −14.7720 −0.577631
\(655\) 0 0
\(656\) −5.77200 −0.225359
\(657\) −6.54400 −0.255306
\(658\) 26.3160 1.02590
\(659\) −43.6320 −1.69966 −0.849831 0.527055i \(-0.823296\pi\)
−0.849831 + 0.527055i \(0.823296\pi\)
\(660\) 0 0
\(661\) −17.8600 −0.694674 −0.347337 0.937740i \(-0.612914\pi\)
−0.347337 + 0.937740i \(0.612914\pi\)
\(662\) 34.7720 1.35145
\(663\) 2.91199 0.113092
\(664\) 1.00000 0.0388075
\(665\) 0 0
\(666\) −6.77200 −0.262410
\(667\) −3.00000 −0.116160
\(668\) −16.3160 −0.631285
\(669\) 10.0000 0.386622
\(670\) 0 0
\(671\) −55.0880 −2.12665
\(672\) −3.00000 −0.115728
\(673\) 26.5440 1.02320 0.511598 0.859225i \(-0.329054\pi\)
0.511598 + 0.859225i \(0.329054\pi\)
\(674\) −4.00000 −0.154074
\(675\) 0 0
\(676\) 1.22800 0.0472307
\(677\) 48.0000 1.84479 0.922395 0.386248i \(-0.126229\pi\)
0.922395 + 0.386248i \(0.126229\pi\)
\(678\) 4.77200 0.183268
\(679\) −52.6320 −2.01983
\(680\) 0 0
\(681\) −10.7720 −0.412784
\(682\) −55.0880 −2.10943
\(683\) 26.6320 1.01905 0.509523 0.860457i \(-0.329822\pi\)
0.509523 + 0.860457i \(0.329822\pi\)
\(684\) 7.77200 0.297170
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) 6.45600 0.246312
\(688\) −7.77200 −0.296305
\(689\) −15.0880 −0.574807
\(690\) 0 0
\(691\) −39.2280 −1.49230 −0.746152 0.665776i \(-0.768101\pi\)
−0.746152 + 0.665776i \(0.768101\pi\)
\(692\) 15.7720 0.599562
\(693\) 17.3160 0.657781
\(694\) −9.54400 −0.362285
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) 4.45600 0.168783
\(698\) −0.227998 −0.00862986
\(699\) −2.68399 −0.101518
\(700\) 0 0
\(701\) 1.54400 0.0583162 0.0291581 0.999575i \(-0.490717\pi\)
0.0291581 + 0.999575i \(0.490717\pi\)
\(702\) −3.77200 −0.142365
\(703\) −52.6320 −1.98505
\(704\) 5.77200 0.217541
\(705\) 0 0
\(706\) 18.8600 0.709806
\(707\) 42.9480 1.61523
\(708\) −6.00000 −0.225494
\(709\) 24.3160 0.913207 0.456603 0.889670i \(-0.349066\pi\)
0.456603 + 0.889670i \(0.349066\pi\)
\(710\) 0 0
\(711\) 2.22800 0.0835565
\(712\) 16.7720 0.628557
\(713\) 9.54400 0.357426
\(714\) 2.31601 0.0866743
\(715\) 0 0
\(716\) 7.54400 0.281933
\(717\) −22.7720 −0.850436
\(718\) −10.2280 −0.381705
\(719\) −30.3160 −1.13060 −0.565298 0.824887i \(-0.691239\pi\)
−0.565298 + 0.824887i \(0.691239\pi\)
\(720\) 0 0
\(721\) −33.0000 −1.22898
\(722\) 41.4040 1.54090
\(723\) 26.6320 0.990455
\(724\) 14.3160 0.532050
\(725\) 0 0
\(726\) −22.3160 −0.828225
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 11.3160 0.419399
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.00000 0.221918
\(732\) 9.54400 0.352757
\(733\) −17.2280 −0.636331 −0.318165 0.948035i \(-0.603067\pi\)
−0.318165 + 0.948035i \(0.603067\pi\)
\(734\) −33.3160 −1.22972
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 55.0880 2.02919
\(738\) −5.77200 −0.212470
\(739\) 17.2280 0.633742 0.316871 0.948469i \(-0.397368\pi\)
0.316871 + 0.948469i \(0.397368\pi\)
\(740\) 0 0
\(741\) −29.3160 −1.07695
\(742\) −12.0000 −0.440534
\(743\) −20.2280 −0.742093 −0.371047 0.928614i \(-0.621001\pi\)
−0.371047 + 0.928614i \(0.621001\pi\)
\(744\) 9.54400 0.349900
\(745\) 0 0
\(746\) 12.7720 0.467616
\(747\) 1.00000 0.0365881
\(748\) −4.45600 −0.162927
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −26.5440 −0.968604 −0.484302 0.874901i \(-0.660926\pi\)
−0.484302 + 0.874901i \(0.660926\pi\)
\(752\) 8.77200 0.319882
\(753\) 3.68399 0.134252
\(754\) 11.3160 0.412105
\(755\) 0 0
\(756\) −3.00000 −0.109109
\(757\) 41.2280 1.49846 0.749229 0.662312i \(-0.230424\pi\)
0.749229 + 0.662312i \(0.230424\pi\)
\(758\) −23.0880 −0.838594
\(759\) 5.77200 0.209510
\(760\) 0 0
\(761\) −14.8600 −0.538675 −0.269337 0.963046i \(-0.586805\pi\)
−0.269337 + 0.963046i \(0.586805\pi\)
\(762\) 4.45600 0.161424
\(763\) 44.3160 1.60435
\(764\) −7.31601 −0.264684
\(765\) 0 0
\(766\) −13.7720 −0.497603
\(767\) 22.6320 0.817195
\(768\) −1.00000 −0.0360844
\(769\) −9.08801 −0.327722 −0.163861 0.986483i \(-0.552395\pi\)
−0.163861 + 0.986483i \(0.552395\pi\)
\(770\) 0 0
\(771\) −9.08801 −0.327297
\(772\) −4.77200 −0.171748
\(773\) 14.4560 0.519946 0.259973 0.965616i \(-0.416286\pi\)
0.259973 + 0.965616i \(0.416286\pi\)
\(774\) −7.77200 −0.279359
\(775\) 0 0
\(776\) −17.5440 −0.629793
\(777\) 20.3160 0.728833
\(778\) 18.0000 0.645331
\(779\) −44.8600 −1.60728
\(780\) 0 0
\(781\) −27.5440 −0.985602
\(782\) 0.772002 0.0276067
\(783\) −3.00000 −0.107211
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) 19.0880 0.680847
\(787\) −21.7720 −0.776088 −0.388044 0.921641i \(-0.626849\pi\)
−0.388044 + 0.921641i \(0.626849\pi\)
\(788\) 13.0000 0.463106
\(789\) −6.45600 −0.229840
\(790\) 0 0
\(791\) −14.3160 −0.509019
\(792\) 5.77200 0.205099
\(793\) −36.0000 −1.27840
\(794\) 14.6320 0.519271
\(795\) 0 0
\(796\) 12.0880 0.428448
\(797\) 2.91199 0.103148 0.0515740 0.998669i \(-0.483576\pi\)
0.0515740 + 0.998669i \(0.483576\pi\)
\(798\) −23.3160 −0.825378
\(799\) −6.77200 −0.239576
\(800\) 0 0
\(801\) 16.7720 0.592610
\(802\) −28.6320 −1.01103
\(803\) −37.7720 −1.33294
\(804\) −9.54400 −0.336591
\(805\) 0 0
\(806\) −36.0000 −1.26805
\(807\) 26.8600 0.945517
\(808\) 14.3160 0.503635
\(809\) 47.3160 1.66354 0.831771 0.555119i \(-0.187327\pi\)
0.831771 + 0.555119i \(0.187327\pi\)
\(810\) 0 0
\(811\) −38.6320 −1.35655 −0.678277 0.734807i \(-0.737273\pi\)
−0.678277 + 0.734807i \(0.737273\pi\)
\(812\) 9.00000 0.315838
\(813\) −16.0000 −0.561144
\(814\) −39.0880 −1.37003
\(815\) 0 0
\(816\) 0.772002 0.0270255
\(817\) −60.4040 −2.11327
\(818\) −23.0000 −0.804176
\(819\) 11.3160 0.395413
\(820\) 0 0
\(821\) −12.2280 −0.426760 −0.213380 0.976969i \(-0.568447\pi\)
−0.213380 + 0.976969i \(0.568447\pi\)
\(822\) 14.7720 0.515233
\(823\) 23.5440 0.820692 0.410346 0.911930i \(-0.365408\pi\)
0.410346 + 0.911930i \(0.365408\pi\)
\(824\) −11.0000 −0.383203
\(825\) 0 0
\(826\) 18.0000 0.626300
\(827\) 51.6320 1.79542 0.897710 0.440586i \(-0.145229\pi\)
0.897710 + 0.440586i \(0.145229\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −38.4040 −1.33383 −0.666913 0.745135i \(-0.732385\pi\)
−0.666913 + 0.745135i \(0.732385\pi\)
\(830\) 0 0
\(831\) −8.22800 −0.285426
\(832\) 3.77200 0.130771
\(833\) −1.54400 −0.0534966
\(834\) 0.772002 0.0267322
\(835\) 0 0
\(836\) 44.8600 1.55152
\(837\) 9.54400 0.329889
\(838\) 25.6320 0.885443
\(839\) 32.4040 1.11871 0.559355 0.828928i \(-0.311049\pi\)
0.559355 + 0.828928i \(0.311049\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 22.0000 0.758170
\(843\) −15.8600 −0.546248
\(844\) −10.3160 −0.355092
\(845\) 0 0
\(846\) 8.77200 0.301588
\(847\) 66.9480 2.30036
\(848\) −4.00000 −0.137361
\(849\) 28.0000 0.960958
\(850\) 0 0
\(851\) 6.77200 0.232141
\(852\) 4.77200 0.163486
\(853\) −8.86001 −0.303361 −0.151680 0.988430i \(-0.548468\pi\)
−0.151680 + 0.988430i \(0.548468\pi\)
\(854\) −28.6320 −0.979767
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) −21.7720 −0.743284
\(859\) −24.4560 −0.834428 −0.417214 0.908808i \(-0.636993\pi\)
−0.417214 + 0.908808i \(0.636993\pi\)
\(860\) 0 0
\(861\) 17.3160 0.590128
\(862\) 24.6320 0.838970
\(863\) −2.31601 −0.0788377 −0.0394189 0.999223i \(-0.512551\pi\)
−0.0394189 + 0.999223i \(0.512551\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −6.45600 −0.219384
\(867\) 16.4040 0.557109
\(868\) −28.6320 −0.971834
\(869\) 12.8600 0.436246
\(870\) 0 0
\(871\) 36.0000 1.21981
\(872\) 14.7720 0.500243
\(873\) −17.5440 −0.593775
\(874\) −7.77200 −0.262892
\(875\) 0 0
\(876\) 6.54400 0.221101
\(877\) −47.2640 −1.59599 −0.797996 0.602662i \(-0.794107\pi\)
−0.797996 + 0.602662i \(0.794107\pi\)
\(878\) −12.0000 −0.404980
\(879\) −9.54400 −0.321911
\(880\) 0 0
\(881\) 10.4560 0.352271 0.176136 0.984366i \(-0.443640\pi\)
0.176136 + 0.984366i \(0.443640\pi\)
\(882\) 2.00000 0.0673435
\(883\) 10.7720 0.362507 0.181253 0.983436i \(-0.441985\pi\)
0.181253 + 0.983436i \(0.441985\pi\)
\(884\) −2.91199 −0.0979409
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 46.9480 1.57636 0.788180 0.615445i \(-0.211024\pi\)
0.788180 + 0.615445i \(0.211024\pi\)
\(888\) 6.77200 0.227254
\(889\) −13.3680 −0.448348
\(890\) 0 0
\(891\) 5.77200 0.193369
\(892\) −10.0000 −0.334825
\(893\) 68.1760 2.28142
\(894\) 11.0880 0.370838
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) 3.77200 0.125943
\(898\) −2.00000 −0.0667409
\(899\) −28.6320 −0.954931
\(900\) 0 0
\(901\) 3.08801 0.102876
\(902\) −33.3160 −1.10930
\(903\) 23.3160 0.775908
\(904\) −4.77200 −0.158714
\(905\) 0 0
\(906\) −23.5440 −0.782197
\(907\) −34.4040 −1.14237 −0.571183 0.820823i \(-0.693515\pi\)
−0.571183 + 0.820823i \(0.693515\pi\)
\(908\) 10.7720 0.357481
\(909\) 14.3160 0.474832
\(910\) 0 0
\(911\) 5.77200 0.191235 0.0956175 0.995418i \(-0.469517\pi\)
0.0956175 + 0.995418i \(0.469517\pi\)
\(912\) −7.77200 −0.257357
\(913\) 5.77200 0.191025
\(914\) 30.6320 1.01322
\(915\) 0 0
\(916\) −6.45600 −0.213312
\(917\) −57.2640 −1.89102
\(918\) 0.772002 0.0254799
\(919\) 20.3160 0.670163 0.335082 0.942189i \(-0.391236\pi\)
0.335082 + 0.942189i \(0.391236\pi\)
\(920\) 0 0
\(921\) −20.3160 −0.669435
\(922\) 12.5440 0.413115
\(923\) −18.0000 −0.592477
\(924\) −17.3160 −0.569655
\(925\) 0 0
\(926\) −0.911993 −0.0299699
\(927\) −11.0000 −0.361287
\(928\) 3.00000 0.0984798
\(929\) 24.6840 0.809856 0.404928 0.914349i \(-0.367297\pi\)
0.404928 + 0.914349i \(0.367297\pi\)
\(930\) 0 0
\(931\) 15.5440 0.509434
\(932\) 2.68399 0.0879172
\(933\) 17.8600 0.584710
\(934\) −18.5440 −0.606778
\(935\) 0 0
\(936\) 3.77200 0.123292
\(937\) −8.63201 −0.281996 −0.140998 0.990010i \(-0.545031\pi\)
−0.140998 + 0.990010i \(0.545031\pi\)
\(938\) 28.6320 0.934868
\(939\) −8.00000 −0.261070
\(940\) 0 0
\(941\) 20.6320 0.672584 0.336292 0.941758i \(-0.390827\pi\)
0.336292 + 0.941758i \(0.390827\pi\)
\(942\) 13.5440 0.441287
\(943\) 5.77200 0.187962
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) −44.8600 −1.45852
\(947\) −31.0880 −1.01022 −0.505112 0.863054i \(-0.668549\pi\)
−0.505112 + 0.863054i \(0.668549\pi\)
\(948\) −2.22800 −0.0723620
\(949\) −24.6840 −0.801276
\(950\) 0 0
\(951\) −9.00000 −0.291845
\(952\) −2.31601 −0.0750622
\(953\) 46.7720 1.51509 0.757547 0.652781i \(-0.226398\pi\)
0.757547 + 0.652781i \(0.226398\pi\)
\(954\) −4.00000 −0.129505
\(955\) 0 0
\(956\) 22.7720 0.736499
\(957\) −17.3160 −0.559747
\(958\) −35.3160 −1.14101
\(959\) −44.3160 −1.43104
\(960\) 0 0
\(961\) 60.0880 1.93832
\(962\) −25.5440 −0.823572
\(963\) 4.00000 0.128898
\(964\) −26.6320 −0.857759
\(965\) 0 0
\(966\) 3.00000 0.0965234
\(967\) −8.63201 −0.277587 −0.138793 0.990321i \(-0.544322\pi\)
−0.138793 + 0.990321i \(0.544322\pi\)
\(968\) 22.3160 0.717264
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) −31.6320 −1.01512 −0.507560 0.861617i \(-0.669452\pi\)
−0.507560 + 0.861617i \(0.669452\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −2.31601 −0.0742477
\(974\) −12.4560 −0.399116
\(975\) 0 0
\(976\) −9.54400 −0.305496
\(977\) 60.3160 1.92968 0.964840 0.262838i \(-0.0846585\pi\)
0.964840 + 0.262838i \(0.0846585\pi\)
\(978\) 16.0000 0.511624
\(979\) 96.8080 3.09400
\(980\) 0 0
\(981\) 14.7720 0.471634
\(982\) −10.4560 −0.333664
\(983\) −8.40401 −0.268046 −0.134023 0.990978i \(-0.542790\pi\)
−0.134023 + 0.990978i \(0.542790\pi\)
\(984\) 5.77200 0.184005
\(985\) 0 0
\(986\) −2.31601 −0.0737566
\(987\) −26.3160 −0.837648
\(988\) 29.3160 0.932666
\(989\) 7.77200 0.247135
\(990\) 0 0
\(991\) 30.1760 0.958573 0.479286 0.877659i \(-0.340896\pi\)
0.479286 + 0.877659i \(0.340896\pi\)
\(992\) −9.54400 −0.303022
\(993\) −34.7720 −1.10346
\(994\) −14.3160 −0.454076
\(995\) 0 0
\(996\) −1.00000 −0.0316862
\(997\) 0.227998 0.00722077 0.00361039 0.999993i \(-0.498851\pi\)
0.00361039 + 0.999993i \(0.498851\pi\)
\(998\) −9.68399 −0.306541
\(999\) 6.77200 0.214257
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.1.2 yes 2
5.2 odd 4 3450.2.d.w.2899.4 4
5.3 odd 4 3450.2.d.w.2899.2 4
5.4 even 2 3450.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.bh.1.2 2 5.4 even 2
3450.2.a.bj.1.2 yes 2 1.1 even 1 trivial
3450.2.d.w.2899.2 4 5.3 odd 4
3450.2.d.w.2899.4 4 5.2 odd 4