Properties

Label 3525.2.a.be.1.6
Level $3525$
Weight $2$
Character 3525.1
Self dual yes
Analytic conductor $28.147$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1472667125\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 3 x^{7} - 7 x^{6} + 24 x^{5} + 8 x^{4} - 47 x^{3} + 8 x^{2} + 13 x + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(1.60965\) of defining polynomial
Character \(\chi\) \(=\) 3525.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.60965 q^{2} -1.00000 q^{3} +0.590961 q^{4} -1.60965 q^{6} +2.89141 q^{7} -2.26805 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.60965 q^{2} -1.00000 q^{3} +0.590961 q^{4} -1.60965 q^{6} +2.89141 q^{7} -2.26805 q^{8} +1.00000 q^{9} -2.05110 q^{11} -0.590961 q^{12} +4.52660 q^{13} +4.65415 q^{14} -4.83269 q^{16} -4.14437 q^{17} +1.60965 q^{18} +3.21947 q^{19} -2.89141 q^{21} -3.30155 q^{22} +2.15638 q^{23} +2.26805 q^{24} +7.28623 q^{26} -1.00000 q^{27} +1.70871 q^{28} +9.10114 q^{29} +0.697696 q^{31} -3.24281 q^{32} +2.05110 q^{33} -6.67097 q^{34} +0.590961 q^{36} -8.61642 q^{37} +5.18220 q^{38} -4.52660 q^{39} +3.28361 q^{41} -4.65415 q^{42} +4.46014 q^{43} -1.21212 q^{44} +3.47101 q^{46} +1.00000 q^{47} +4.83269 q^{48} +1.36026 q^{49} +4.14437 q^{51} +2.67505 q^{52} +2.71638 q^{53} -1.60965 q^{54} -6.55788 q^{56} -3.21947 q^{57} +14.6496 q^{58} -8.56992 q^{59} +8.48391 q^{61} +1.12304 q^{62} +2.89141 q^{63} +4.44560 q^{64} +3.30155 q^{66} +13.9564 q^{67} -2.44916 q^{68} -2.15638 q^{69} -15.7695 q^{71} -2.26805 q^{72} +8.88127 q^{73} -13.8694 q^{74} +1.90258 q^{76} -5.93059 q^{77} -7.28623 q^{78} +15.6685 q^{79} +1.00000 q^{81} +5.28545 q^{82} +5.17115 q^{83} -1.70871 q^{84} +7.17925 q^{86} -9.10114 q^{87} +4.65202 q^{88} +14.5447 q^{89} +13.0883 q^{91} +1.27434 q^{92} -0.697696 q^{93} +1.60965 q^{94} +3.24281 q^{96} -0.136874 q^{97} +2.18954 q^{98} -2.05110 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 3q^{2} - 8q^{3} + 7q^{4} - 3q^{6} + 8q^{7} + 6q^{8} + 8q^{9} + O(q^{10}) \) \( 8q + 3q^{2} - 8q^{3} + 7q^{4} - 3q^{6} + 8q^{7} + 6q^{8} + 8q^{9} - 8q^{11} - 7q^{12} + 10q^{13} + q^{14} + 5q^{16} + 6q^{17} + 3q^{18} - 2q^{19} - 8q^{21} + 10q^{23} - 6q^{24} - 14q^{26} - 8q^{27} + 44q^{28} - 13q^{29} + 10q^{32} + 8q^{33} + 28q^{34} + 7q^{36} + 3q^{37} + 36q^{38} - 10q^{39} - 16q^{41} - q^{42} + 25q^{43} - 17q^{44} - 5q^{46} + 8q^{47} - 5q^{48} + 16q^{49} - 6q^{51} - 17q^{52} + 4q^{53} - 3q^{54} + 37q^{56} + 2q^{57} + 15q^{58} - 8q^{59} + 15q^{61} + 6q^{62} + 8q^{63} - 14q^{64} + 27q^{67} + 14q^{68} - 10q^{69} + 14q^{71} + 6q^{72} + 28q^{73} - 21q^{74} + 6q^{76} + 4q^{77} + 14q^{78} + 7q^{79} + 8q^{81} - 53q^{82} + 60q^{83} - 44q^{84} - 3q^{86} + 13q^{87} + 54q^{88} - 34q^{89} + 23q^{91} - 43q^{92} + 3q^{94} - 10q^{96} + 7q^{97} + 40q^{98} - 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.60965 1.13819 0.569096 0.822271i \(-0.307293\pi\)
0.569096 + 0.822271i \(0.307293\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.590961 0.295481
\(5\) 0 0
\(6\) −1.60965 −0.657135
\(7\) 2.89141 1.09285 0.546425 0.837508i \(-0.315988\pi\)
0.546425 + 0.837508i \(0.315988\pi\)
\(8\) −2.26805 −0.801878
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.05110 −0.618431 −0.309216 0.950992i \(-0.600066\pi\)
−0.309216 + 0.950992i \(0.600066\pi\)
\(12\) −0.590961 −0.170596
\(13\) 4.52660 1.25545 0.627727 0.778434i \(-0.283985\pi\)
0.627727 + 0.778434i \(0.283985\pi\)
\(14\) 4.65415 1.24387
\(15\) 0 0
\(16\) −4.83269 −1.20817
\(17\) −4.14437 −1.00516 −0.502579 0.864532i \(-0.667615\pi\)
−0.502579 + 0.864532i \(0.667615\pi\)
\(18\) 1.60965 0.379397
\(19\) 3.21947 0.738597 0.369298 0.929311i \(-0.379598\pi\)
0.369298 + 0.929311i \(0.379598\pi\)
\(20\) 0 0
\(21\) −2.89141 −0.630958
\(22\) −3.30155 −0.703893
\(23\) 2.15638 0.449637 0.224818 0.974401i \(-0.427821\pi\)
0.224818 + 0.974401i \(0.427821\pi\)
\(24\) 2.26805 0.462965
\(25\) 0 0
\(26\) 7.28623 1.42895
\(27\) −1.00000 −0.192450
\(28\) 1.70871 0.322916
\(29\) 9.10114 1.69004 0.845020 0.534735i \(-0.179589\pi\)
0.845020 + 0.534735i \(0.179589\pi\)
\(30\) 0 0
\(31\) 0.697696 0.125310 0.0626550 0.998035i \(-0.480043\pi\)
0.0626550 + 0.998035i \(0.480043\pi\)
\(32\) −3.24281 −0.573253
\(33\) 2.05110 0.357052
\(34\) −6.67097 −1.14406
\(35\) 0 0
\(36\) 0.590961 0.0984935
\(37\) −8.61642 −1.41653 −0.708265 0.705946i \(-0.750522\pi\)
−0.708265 + 0.705946i \(0.750522\pi\)
\(38\) 5.18220 0.840665
\(39\) −4.52660 −0.724837
\(40\) 0 0
\(41\) 3.28361 0.512813 0.256407 0.966569i \(-0.417461\pi\)
0.256407 + 0.966569i \(0.417461\pi\)
\(42\) −4.65415 −0.718151
\(43\) 4.46014 0.680166 0.340083 0.940396i \(-0.389545\pi\)
0.340083 + 0.940396i \(0.389545\pi\)
\(44\) −1.21212 −0.182734
\(45\) 0 0
\(46\) 3.47101 0.511773
\(47\) 1.00000 0.145865
\(48\) 4.83269 0.697538
\(49\) 1.36026 0.194323
\(50\) 0 0
\(51\) 4.14437 0.580328
\(52\) 2.67505 0.370962
\(53\) 2.71638 0.373124 0.186562 0.982443i \(-0.440266\pi\)
0.186562 + 0.982443i \(0.440266\pi\)
\(54\) −1.60965 −0.219045
\(55\) 0 0
\(56\) −6.55788 −0.876333
\(57\) −3.21947 −0.426429
\(58\) 14.6496 1.92359
\(59\) −8.56992 −1.11571 −0.557854 0.829939i \(-0.688375\pi\)
−0.557854 + 0.829939i \(0.688375\pi\)
\(60\) 0 0
\(61\) 8.48391 1.08625 0.543127 0.839651i \(-0.317240\pi\)
0.543127 + 0.839651i \(0.317240\pi\)
\(62\) 1.12304 0.142627
\(63\) 2.89141 0.364284
\(64\) 4.44560 0.555700
\(65\) 0 0
\(66\) 3.30155 0.406393
\(67\) 13.9564 1.70505 0.852524 0.522687i \(-0.175071\pi\)
0.852524 + 0.522687i \(0.175071\pi\)
\(68\) −2.44916 −0.297004
\(69\) −2.15638 −0.259598
\(70\) 0 0
\(71\) −15.7695 −1.87149 −0.935746 0.352674i \(-0.885273\pi\)
−0.935746 + 0.352674i \(0.885273\pi\)
\(72\) −2.26805 −0.267293
\(73\) 8.88127 1.03947 0.519737 0.854326i \(-0.326030\pi\)
0.519737 + 0.854326i \(0.326030\pi\)
\(74\) −13.8694 −1.61228
\(75\) 0 0
\(76\) 1.90258 0.218241
\(77\) −5.93059 −0.675853
\(78\) −7.28623 −0.825003
\(79\) 15.6685 1.76284 0.881421 0.472332i \(-0.156588\pi\)
0.881421 + 0.472332i \(0.156588\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 5.28545 0.583680
\(83\) 5.17115 0.567607 0.283804 0.958882i \(-0.408404\pi\)
0.283804 + 0.958882i \(0.408404\pi\)
\(84\) −1.70871 −0.186436
\(85\) 0 0
\(86\) 7.17925 0.774159
\(87\) −9.10114 −0.975745
\(88\) 4.65202 0.495907
\(89\) 14.5447 1.54174 0.770868 0.636994i \(-0.219823\pi\)
0.770868 + 0.636994i \(0.219823\pi\)
\(90\) 0 0
\(91\) 13.0883 1.37202
\(92\) 1.27434 0.132859
\(93\) −0.697696 −0.0723477
\(94\) 1.60965 0.166022
\(95\) 0 0
\(96\) 3.24281 0.330968
\(97\) −0.136874 −0.0138975 −0.00694874 0.999976i \(-0.502212\pi\)
−0.00694874 + 0.999976i \(0.502212\pi\)
\(98\) 2.18954 0.221177
\(99\) −2.05110 −0.206144
\(100\) 0 0
\(101\) −7.06652 −0.703145 −0.351572 0.936161i \(-0.614353\pi\)
−0.351572 + 0.936161i \(0.614353\pi\)
\(102\) 6.67097 0.660524
\(103\) 8.01574 0.789814 0.394907 0.918721i \(-0.370777\pi\)
0.394907 + 0.918721i \(0.370777\pi\)
\(104\) −10.2666 −1.00672
\(105\) 0 0
\(106\) 4.37241 0.424686
\(107\) 19.7935 1.91351 0.956753 0.290900i \(-0.0939547\pi\)
0.956753 + 0.290900i \(0.0939547\pi\)
\(108\) −0.590961 −0.0568653
\(109\) −1.61314 −0.154511 −0.0772554 0.997011i \(-0.524616\pi\)
−0.0772554 + 0.997011i \(0.524616\pi\)
\(110\) 0 0
\(111\) 8.61642 0.817835
\(112\) −13.9733 −1.32035
\(113\) −6.17689 −0.581073 −0.290536 0.956864i \(-0.593834\pi\)
−0.290536 + 0.956864i \(0.593834\pi\)
\(114\) −5.18220 −0.485358
\(115\) 0 0
\(116\) 5.37842 0.499374
\(117\) 4.52660 0.418485
\(118\) −13.7945 −1.26989
\(119\) −11.9831 −1.09849
\(120\) 0 0
\(121\) −6.79297 −0.617543
\(122\) 13.6561 1.23636
\(123\) −3.28361 −0.296073
\(124\) 0.412311 0.0370266
\(125\) 0 0
\(126\) 4.65415 0.414625
\(127\) 7.60694 0.675006 0.337503 0.941324i \(-0.390418\pi\)
0.337503 + 0.941324i \(0.390418\pi\)
\(128\) 13.6415 1.20575
\(129\) −4.46014 −0.392694
\(130\) 0 0
\(131\) −16.1974 −1.41517 −0.707585 0.706628i \(-0.750215\pi\)
−0.707585 + 0.706628i \(0.750215\pi\)
\(132\) 1.21212 0.105502
\(133\) 9.30881 0.807176
\(134\) 22.4649 1.94067
\(135\) 0 0
\(136\) 9.39965 0.806014
\(137\) 13.7313 1.17315 0.586573 0.809896i \(-0.300477\pi\)
0.586573 + 0.809896i \(0.300477\pi\)
\(138\) −3.47101 −0.295472
\(139\) 0.415128 0.0352107 0.0176053 0.999845i \(-0.494396\pi\)
0.0176053 + 0.999845i \(0.494396\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) −25.3833 −2.13012
\(143\) −9.28454 −0.776412
\(144\) −4.83269 −0.402724
\(145\) 0 0
\(146\) 14.2957 1.18312
\(147\) −1.36026 −0.112192
\(148\) −5.09197 −0.418557
\(149\) −7.34727 −0.601911 −0.300956 0.953638i \(-0.597306\pi\)
−0.300956 + 0.953638i \(0.597306\pi\)
\(150\) 0 0
\(151\) 5.58738 0.454695 0.227347 0.973814i \(-0.426995\pi\)
0.227347 + 0.973814i \(0.426995\pi\)
\(152\) −7.30193 −0.592265
\(153\) −4.14437 −0.335052
\(154\) −9.54615 −0.769251
\(155\) 0 0
\(156\) −2.67505 −0.214175
\(157\) −13.6795 −1.09175 −0.545873 0.837868i \(-0.683802\pi\)
−0.545873 + 0.837868i \(0.683802\pi\)
\(158\) 25.2207 2.00645
\(159\) −2.71638 −0.215423
\(160\) 0 0
\(161\) 6.23499 0.491386
\(162\) 1.60965 0.126466
\(163\) −17.5811 −1.37706 −0.688530 0.725208i \(-0.741744\pi\)
−0.688530 + 0.725208i \(0.741744\pi\)
\(164\) 1.94048 0.151526
\(165\) 0 0
\(166\) 8.32372 0.646046
\(167\) 1.61778 0.125188 0.0625938 0.998039i \(-0.480063\pi\)
0.0625938 + 0.998039i \(0.480063\pi\)
\(168\) 6.55788 0.505951
\(169\) 7.49013 0.576164
\(170\) 0 0
\(171\) 3.21947 0.246199
\(172\) 2.63577 0.200976
\(173\) 11.9840 0.911129 0.455565 0.890203i \(-0.349437\pi\)
0.455565 + 0.890203i \(0.349437\pi\)
\(174\) −14.6496 −1.11058
\(175\) 0 0
\(176\) 9.91235 0.747171
\(177\) 8.56992 0.644154
\(178\) 23.4118 1.75479
\(179\) 13.5550 1.01315 0.506574 0.862197i \(-0.330912\pi\)
0.506574 + 0.862197i \(0.330912\pi\)
\(180\) 0 0
\(181\) −5.72613 −0.425620 −0.212810 0.977094i \(-0.568262\pi\)
−0.212810 + 0.977094i \(0.568262\pi\)
\(182\) 21.0675 1.56163
\(183\) −8.48391 −0.627149
\(184\) −4.89079 −0.360554
\(185\) 0 0
\(186\) −1.12304 −0.0823456
\(187\) 8.50054 0.621621
\(188\) 0.590961 0.0431003
\(189\) −2.89141 −0.210319
\(190\) 0 0
\(191\) 8.20512 0.593701 0.296851 0.954924i \(-0.404064\pi\)
0.296851 + 0.954924i \(0.404064\pi\)
\(192\) −4.44560 −0.320834
\(193\) 21.2576 1.53016 0.765079 0.643936i \(-0.222700\pi\)
0.765079 + 0.643936i \(0.222700\pi\)
\(194\) −0.220319 −0.0158180
\(195\) 0 0
\(196\) 0.803861 0.0574186
\(197\) 3.62958 0.258597 0.129298 0.991606i \(-0.458727\pi\)
0.129298 + 0.991606i \(0.458727\pi\)
\(198\) −3.30155 −0.234631
\(199\) 3.79766 0.269209 0.134605 0.990899i \(-0.457024\pi\)
0.134605 + 0.990899i \(0.457024\pi\)
\(200\) 0 0
\(201\) −13.9564 −0.984410
\(202\) −11.3746 −0.800314
\(203\) 26.3151 1.84696
\(204\) 2.44916 0.171476
\(205\) 0 0
\(206\) 12.9025 0.898960
\(207\) 2.15638 0.149879
\(208\) −21.8757 −1.51680
\(209\) −6.60347 −0.456771
\(210\) 0 0
\(211\) 14.6034 1.00534 0.502669 0.864479i \(-0.332352\pi\)
0.502669 + 0.864479i \(0.332352\pi\)
\(212\) 1.60528 0.110251
\(213\) 15.7695 1.08051
\(214\) 31.8605 2.17794
\(215\) 0 0
\(216\) 2.26805 0.154322
\(217\) 2.01733 0.136945
\(218\) −2.59658 −0.175863
\(219\) −8.88127 −0.600140
\(220\) 0 0
\(221\) −18.7599 −1.26193
\(222\) 13.8694 0.930853
\(223\) 23.7647 1.59140 0.795701 0.605689i \(-0.207102\pi\)
0.795701 + 0.605689i \(0.207102\pi\)
\(224\) −9.37629 −0.626480
\(225\) 0 0
\(226\) −9.94261 −0.661372
\(227\) −6.10001 −0.404872 −0.202436 0.979295i \(-0.564886\pi\)
−0.202436 + 0.979295i \(0.564886\pi\)
\(228\) −1.90258 −0.126001
\(229\) −22.3505 −1.47696 −0.738482 0.674274i \(-0.764457\pi\)
−0.738482 + 0.674274i \(0.764457\pi\)
\(230\) 0 0
\(231\) 5.93059 0.390204
\(232\) −20.6419 −1.35521
\(233\) 6.55140 0.429197 0.214598 0.976702i \(-0.431156\pi\)
0.214598 + 0.976702i \(0.431156\pi\)
\(234\) 7.28623 0.476316
\(235\) 0 0
\(236\) −5.06449 −0.329670
\(237\) −15.6685 −1.01778
\(238\) −19.2885 −1.25029
\(239\) −26.9521 −1.74339 −0.871693 0.490053i \(-0.836978\pi\)
−0.871693 + 0.490053i \(0.836978\pi\)
\(240\) 0 0
\(241\) 16.0041 1.03091 0.515457 0.856915i \(-0.327622\pi\)
0.515457 + 0.856915i \(0.327622\pi\)
\(242\) −10.9343 −0.702882
\(243\) −1.00000 −0.0641500
\(244\) 5.01366 0.320967
\(245\) 0 0
\(246\) −5.28545 −0.336988
\(247\) 14.5733 0.927274
\(248\) −1.58241 −0.100483
\(249\) −5.17115 −0.327708
\(250\) 0 0
\(251\) −26.5909 −1.67840 −0.839201 0.543821i \(-0.816977\pi\)
−0.839201 + 0.543821i \(0.816977\pi\)
\(252\) 1.70871 0.107639
\(253\) −4.42297 −0.278070
\(254\) 12.2445 0.768287
\(255\) 0 0
\(256\) 13.0667 0.816670
\(257\) −2.59505 −0.161875 −0.0809373 0.996719i \(-0.525791\pi\)
−0.0809373 + 0.996719i \(0.525791\pi\)
\(258\) −7.17925 −0.446961
\(259\) −24.9136 −1.54806
\(260\) 0 0
\(261\) 9.10114 0.563346
\(262\) −26.0720 −1.61074
\(263\) −2.09258 −0.129034 −0.0645170 0.997917i \(-0.520551\pi\)
−0.0645170 + 0.997917i \(0.520551\pi\)
\(264\) −4.65202 −0.286312
\(265\) 0 0
\(266\) 14.9839 0.918721
\(267\) −14.5447 −0.890122
\(268\) 8.24771 0.503809
\(269\) 11.9754 0.730153 0.365077 0.930977i \(-0.381043\pi\)
0.365077 + 0.930977i \(0.381043\pi\)
\(270\) 0 0
\(271\) 13.4698 0.818230 0.409115 0.912483i \(-0.365837\pi\)
0.409115 + 0.912483i \(0.365837\pi\)
\(272\) 20.0284 1.21440
\(273\) −13.0883 −0.792138
\(274\) 22.1026 1.33527
\(275\) 0 0
\(276\) −1.27434 −0.0767062
\(277\) 11.6830 0.701965 0.350982 0.936382i \(-0.385848\pi\)
0.350982 + 0.936382i \(0.385848\pi\)
\(278\) 0.668209 0.0400765
\(279\) 0.697696 0.0417700
\(280\) 0 0
\(281\) −26.0065 −1.55142 −0.775710 0.631090i \(-0.782608\pi\)
−0.775710 + 0.631090i \(0.782608\pi\)
\(282\) −1.60965 −0.0958530
\(283\) −14.4211 −0.857246 −0.428623 0.903484i \(-0.641001\pi\)
−0.428623 + 0.903484i \(0.641001\pi\)
\(284\) −9.31915 −0.552990
\(285\) 0 0
\(286\) −14.9448 −0.883706
\(287\) 9.49426 0.560428
\(288\) −3.24281 −0.191084
\(289\) 0.175799 0.0103411
\(290\) 0 0
\(291\) 0.136874 0.00802372
\(292\) 5.24848 0.307144
\(293\) −5.61335 −0.327936 −0.163968 0.986466i \(-0.552429\pi\)
−0.163968 + 0.986466i \(0.552429\pi\)
\(294\) −2.18954 −0.127696
\(295\) 0 0
\(296\) 19.5425 1.13589
\(297\) 2.05110 0.119017
\(298\) −11.8265 −0.685091
\(299\) 9.76109 0.564498
\(300\) 0 0
\(301\) 12.8961 0.743319
\(302\) 8.99371 0.517530
\(303\) 7.06652 0.405961
\(304\) −15.5587 −0.892352
\(305\) 0 0
\(306\) −6.67097 −0.381354
\(307\) −13.5331 −0.772375 −0.386187 0.922420i \(-0.626208\pi\)
−0.386187 + 0.922420i \(0.626208\pi\)
\(308\) −3.50475 −0.199701
\(309\) −8.01574 −0.455999
\(310\) 0 0
\(311\) −0.849986 −0.0481983 −0.0240991 0.999710i \(-0.507672\pi\)
−0.0240991 + 0.999710i \(0.507672\pi\)
\(312\) 10.2666 0.581231
\(313\) 2.68904 0.151993 0.0759966 0.997108i \(-0.475786\pi\)
0.0759966 + 0.997108i \(0.475786\pi\)
\(314\) −22.0192 −1.24262
\(315\) 0 0
\(316\) 9.25946 0.520885
\(317\) 4.65829 0.261635 0.130818 0.991406i \(-0.458240\pi\)
0.130818 + 0.991406i \(0.458240\pi\)
\(318\) −4.37241 −0.245193
\(319\) −18.6674 −1.04517
\(320\) 0 0
\(321\) −19.7935 −1.10476
\(322\) 10.0361 0.559292
\(323\) −13.3427 −0.742406
\(324\) 0.590961 0.0328312
\(325\) 0 0
\(326\) −28.2994 −1.56736
\(327\) 1.61314 0.0892068
\(328\) −7.44740 −0.411214
\(329\) 2.89141 0.159409
\(330\) 0 0
\(331\) −17.7902 −0.977837 −0.488918 0.872330i \(-0.662608\pi\)
−0.488918 + 0.872330i \(0.662608\pi\)
\(332\) 3.05595 0.167717
\(333\) −8.61642 −0.472177
\(334\) 2.60405 0.142487
\(335\) 0 0
\(336\) 13.9733 0.762305
\(337\) −34.0593 −1.85533 −0.927663 0.373419i \(-0.878185\pi\)
−0.927663 + 0.373419i \(0.878185\pi\)
\(338\) 12.0565 0.655785
\(339\) 6.17689 0.335483
\(340\) 0 0
\(341\) −1.43105 −0.0774956
\(342\) 5.18220 0.280222
\(343\) −16.3068 −0.880485
\(344\) −10.1158 −0.545410
\(345\) 0 0
\(346\) 19.2901 1.03704
\(347\) 22.5182 1.20884 0.604419 0.796667i \(-0.293405\pi\)
0.604419 + 0.796667i \(0.293405\pi\)
\(348\) −5.37842 −0.288314
\(349\) 26.7475 1.43176 0.715879 0.698224i \(-0.246026\pi\)
0.715879 + 0.698224i \(0.246026\pi\)
\(350\) 0 0
\(351\) −4.52660 −0.241612
\(352\) 6.65134 0.354518
\(353\) −13.9790 −0.744025 −0.372013 0.928228i \(-0.621332\pi\)
−0.372013 + 0.928228i \(0.621332\pi\)
\(354\) 13.7945 0.733171
\(355\) 0 0
\(356\) 8.59536 0.455553
\(357\) 11.9831 0.634212
\(358\) 21.8187 1.15316
\(359\) 8.06400 0.425602 0.212801 0.977096i \(-0.431741\pi\)
0.212801 + 0.977096i \(0.431741\pi\)
\(360\) 0 0
\(361\) −8.63503 −0.454475
\(362\) −9.21705 −0.484437
\(363\) 6.79297 0.356538
\(364\) 7.73466 0.405406
\(365\) 0 0
\(366\) −13.6561 −0.713816
\(367\) 4.15872 0.217083 0.108542 0.994092i \(-0.465382\pi\)
0.108542 + 0.994092i \(0.465382\pi\)
\(368\) −10.4211 −0.543239
\(369\) 3.28361 0.170938
\(370\) 0 0
\(371\) 7.85418 0.407768
\(372\) −0.412311 −0.0213773
\(373\) −3.98023 −0.206089 −0.103044 0.994677i \(-0.532858\pi\)
−0.103044 + 0.994677i \(0.532858\pi\)
\(374\) 13.6829 0.707524
\(375\) 0 0
\(376\) −2.26805 −0.116966
\(377\) 41.1972 2.12177
\(378\) −4.65415 −0.239384
\(379\) 4.75448 0.244221 0.122111 0.992516i \(-0.461034\pi\)
0.122111 + 0.992516i \(0.461034\pi\)
\(380\) 0 0
\(381\) −7.60694 −0.389715
\(382\) 13.2073 0.675746
\(383\) 7.55458 0.386021 0.193010 0.981197i \(-0.438175\pi\)
0.193010 + 0.981197i \(0.438175\pi\)
\(384\) −13.6415 −0.696138
\(385\) 0 0
\(386\) 34.2173 1.74161
\(387\) 4.46014 0.226722
\(388\) −0.0808874 −0.00410644
\(389\) 12.0545 0.611189 0.305594 0.952162i \(-0.401145\pi\)
0.305594 + 0.952162i \(0.401145\pi\)
\(390\) 0 0
\(391\) −8.93685 −0.451956
\(392\) −3.08514 −0.155823
\(393\) 16.1974 0.817049
\(394\) 5.84234 0.294333
\(395\) 0 0
\(396\) −1.21212 −0.0609115
\(397\) 5.70537 0.286344 0.143172 0.989698i \(-0.454270\pi\)
0.143172 + 0.989698i \(0.454270\pi\)
\(398\) 6.11289 0.306412
\(399\) −9.30881 −0.466023
\(400\) 0 0
\(401\) −25.6860 −1.28270 −0.641349 0.767249i \(-0.721625\pi\)
−0.641349 + 0.767249i \(0.721625\pi\)
\(402\) −22.4649 −1.12045
\(403\) 3.15819 0.157321
\(404\) −4.17604 −0.207766
\(405\) 0 0
\(406\) 42.3581 2.10220
\(407\) 17.6732 0.876027
\(408\) −9.39965 −0.465352
\(409\) −37.8814 −1.87311 −0.936557 0.350515i \(-0.886006\pi\)
−0.936557 + 0.350515i \(0.886006\pi\)
\(410\) 0 0
\(411\) −13.7313 −0.677316
\(412\) 4.73699 0.233375
\(413\) −24.7792 −1.21930
\(414\) 3.47101 0.170591
\(415\) 0 0
\(416\) −14.6789 −0.719693
\(417\) −0.415128 −0.0203289
\(418\) −10.6292 −0.519893
\(419\) −39.3561 −1.92267 −0.961336 0.275377i \(-0.911197\pi\)
−0.961336 + 0.275377i \(0.911197\pi\)
\(420\) 0 0
\(421\) −12.9116 −0.629273 −0.314636 0.949212i \(-0.601883\pi\)
−0.314636 + 0.949212i \(0.601883\pi\)
\(422\) 23.5063 1.14427
\(423\) 1.00000 0.0486217
\(424\) −6.16090 −0.299200
\(425\) 0 0
\(426\) 25.3833 1.22982
\(427\) 24.5305 1.18711
\(428\) 11.6972 0.565404
\(429\) 9.28454 0.448262
\(430\) 0 0
\(431\) −3.38916 −0.163250 −0.0816250 0.996663i \(-0.526011\pi\)
−0.0816250 + 0.996663i \(0.526011\pi\)
\(432\) 4.83269 0.232513
\(433\) 34.1171 1.63956 0.819782 0.572676i \(-0.194095\pi\)
0.819782 + 0.572676i \(0.194095\pi\)
\(434\) 3.24718 0.155870
\(435\) 0 0
\(436\) −0.953303 −0.0456549
\(437\) 6.94241 0.332100
\(438\) −14.2957 −0.683075
\(439\) −13.2170 −0.630811 −0.315406 0.948957i \(-0.602141\pi\)
−0.315406 + 0.948957i \(0.602141\pi\)
\(440\) 0 0
\(441\) 1.36026 0.0647743
\(442\) −30.1968 −1.43632
\(443\) −28.3665 −1.34773 −0.673867 0.738853i \(-0.735368\pi\)
−0.673867 + 0.738853i \(0.735368\pi\)
\(444\) 5.09197 0.241654
\(445\) 0 0
\(446\) 38.2528 1.81132
\(447\) 7.34727 0.347514
\(448\) 12.8541 0.607297
\(449\) −5.94981 −0.280789 −0.140394 0.990096i \(-0.544837\pi\)
−0.140394 + 0.990096i \(0.544837\pi\)
\(450\) 0 0
\(451\) −6.73502 −0.317140
\(452\) −3.65030 −0.171696
\(453\) −5.58738 −0.262518
\(454\) −9.81886 −0.460822
\(455\) 0 0
\(456\) 7.30193 0.341944
\(457\) −6.58351 −0.307963 −0.153982 0.988074i \(-0.549210\pi\)
−0.153982 + 0.988074i \(0.549210\pi\)
\(458\) −35.9764 −1.68107
\(459\) 4.14437 0.193443
\(460\) 0 0
\(461\) −35.2196 −1.64034 −0.820171 0.572118i \(-0.806122\pi\)
−0.820171 + 0.572118i \(0.806122\pi\)
\(462\) 9.54615 0.444127
\(463\) 13.7978 0.641239 0.320619 0.947208i \(-0.396109\pi\)
0.320619 + 0.947208i \(0.396109\pi\)
\(464\) −43.9830 −2.04186
\(465\) 0 0
\(466\) 10.5454 0.488508
\(467\) 8.02180 0.371205 0.185602 0.982625i \(-0.440576\pi\)
0.185602 + 0.982625i \(0.440576\pi\)
\(468\) 2.67505 0.123654
\(469\) 40.3538 1.86336
\(470\) 0 0
\(471\) 13.6795 0.630320
\(472\) 19.4370 0.894662
\(473\) −9.14822 −0.420636
\(474\) −25.2207 −1.15843
\(475\) 0 0
\(476\) −7.08153 −0.324582
\(477\) 2.71638 0.124375
\(478\) −43.3833 −1.98431
\(479\) −0.339149 −0.0154961 −0.00774806 0.999970i \(-0.502466\pi\)
−0.00774806 + 0.999970i \(0.502466\pi\)
\(480\) 0 0
\(481\) −39.0031 −1.77839
\(482\) 25.7609 1.17338
\(483\) −6.23499 −0.283702
\(484\) −4.01438 −0.182472
\(485\) 0 0
\(486\) −1.60965 −0.0730150
\(487\) −32.9320 −1.49229 −0.746146 0.665782i \(-0.768098\pi\)
−0.746146 + 0.665782i \(0.768098\pi\)
\(488\) −19.2420 −0.871043
\(489\) 17.5811 0.795046
\(490\) 0 0
\(491\) 2.35165 0.106129 0.0530643 0.998591i \(-0.483101\pi\)
0.0530643 + 0.998591i \(0.483101\pi\)
\(492\) −1.94048 −0.0874838
\(493\) −37.7185 −1.69876
\(494\) 23.4578 1.05542
\(495\) 0 0
\(496\) −3.37175 −0.151396
\(497\) −45.5960 −2.04526
\(498\) −8.32372 −0.372995
\(499\) −3.77188 −0.168853 −0.0844263 0.996430i \(-0.526906\pi\)
−0.0844263 + 0.996430i \(0.526906\pi\)
\(500\) 0 0
\(501\) −1.61778 −0.0722771
\(502\) −42.8019 −1.91034
\(503\) −0.270322 −0.0120530 −0.00602652 0.999982i \(-0.501918\pi\)
−0.00602652 + 0.999982i \(0.501918\pi\)
\(504\) −6.55788 −0.292111
\(505\) 0 0
\(506\) −7.11941 −0.316497
\(507\) −7.49013 −0.332648
\(508\) 4.49540 0.199451
\(509\) −0.465277 −0.0206230 −0.0103115 0.999947i \(-0.503282\pi\)
−0.0103115 + 0.999947i \(0.503282\pi\)
\(510\) 0 0
\(511\) 25.6794 1.13599
\(512\) −6.25011 −0.276219
\(513\) −3.21947 −0.142143
\(514\) −4.17711 −0.184244
\(515\) 0 0
\(516\) −2.63577 −0.116033
\(517\) −2.05110 −0.0902075
\(518\) −40.1021 −1.76199
\(519\) −11.9840 −0.526041
\(520\) 0 0
\(521\) −20.3235 −0.890387 −0.445193 0.895434i \(-0.646865\pi\)
−0.445193 + 0.895434i \(0.646865\pi\)
\(522\) 14.6496 0.641196
\(523\) 43.0920 1.88428 0.942140 0.335219i \(-0.108810\pi\)
0.942140 + 0.335219i \(0.108810\pi\)
\(524\) −9.57201 −0.418155
\(525\) 0 0
\(526\) −3.36831 −0.146865
\(527\) −2.89151 −0.125956
\(528\) −9.91235 −0.431380
\(529\) −18.3500 −0.797827
\(530\) 0 0
\(531\) −8.56992 −0.371903
\(532\) 5.50114 0.238505
\(533\) 14.8636 0.643813
\(534\) −23.4118 −1.01313
\(535\) 0 0
\(536\) −31.6539 −1.36724
\(537\) −13.5550 −0.584941
\(538\) 19.2762 0.831054
\(539\) −2.79003 −0.120175
\(540\) 0 0
\(541\) −27.1465 −1.16712 −0.583560 0.812070i \(-0.698341\pi\)
−0.583560 + 0.812070i \(0.698341\pi\)
\(542\) 21.6816 0.931303
\(543\) 5.72613 0.245732
\(544\) 13.4394 0.576209
\(545\) 0 0
\(546\) −21.0675 −0.901605
\(547\) 6.65030 0.284346 0.142173 0.989842i \(-0.454591\pi\)
0.142173 + 0.989842i \(0.454591\pi\)
\(548\) 8.11468 0.346642
\(549\) 8.48391 0.362085
\(550\) 0 0
\(551\) 29.3008 1.24826
\(552\) 4.89079 0.208166
\(553\) 45.3040 1.92652
\(554\) 18.8055 0.798970
\(555\) 0 0
\(556\) 0.245324 0.0104041
\(557\) 7.78871 0.330018 0.165009 0.986292i \(-0.447235\pi\)
0.165009 + 0.986292i \(0.447235\pi\)
\(558\) 1.12304 0.0475422
\(559\) 20.1893 0.853916
\(560\) 0 0
\(561\) −8.50054 −0.358893
\(562\) −41.8613 −1.76581
\(563\) −17.8053 −0.750405 −0.375203 0.926943i \(-0.622427\pi\)
−0.375203 + 0.926943i \(0.622427\pi\)
\(564\) −0.590961 −0.0248840
\(565\) 0 0
\(566\) −23.2129 −0.975710
\(567\) 2.89141 0.121428
\(568\) 35.7660 1.50071
\(569\) −1.15450 −0.0483993 −0.0241996 0.999707i \(-0.507704\pi\)
−0.0241996 + 0.999707i \(0.507704\pi\)
\(570\) 0 0
\(571\) −45.8755 −1.91983 −0.959914 0.280293i \(-0.909568\pi\)
−0.959914 + 0.280293i \(0.909568\pi\)
\(572\) −5.48680 −0.229415
\(573\) −8.20512 −0.342774
\(574\) 15.2824 0.637875
\(575\) 0 0
\(576\) 4.44560 0.185233
\(577\) −17.1596 −0.714362 −0.357181 0.934035i \(-0.616262\pi\)
−0.357181 + 0.934035i \(0.616262\pi\)
\(578\) 0.282975 0.0117702
\(579\) −21.2576 −0.883437
\(580\) 0 0
\(581\) 14.9519 0.620310
\(582\) 0.220319 0.00913253
\(583\) −5.57158 −0.230751
\(584\) −20.1432 −0.833531
\(585\) 0 0
\(586\) −9.03551 −0.373254
\(587\) 36.9440 1.52484 0.762422 0.647080i \(-0.224010\pi\)
0.762422 + 0.647080i \(0.224010\pi\)
\(588\) −0.803861 −0.0331507
\(589\) 2.24621 0.0925535
\(590\) 0 0
\(591\) −3.62958 −0.149301
\(592\) 41.6405 1.71141
\(593\) −13.9078 −0.571127 −0.285563 0.958360i \(-0.592181\pi\)
−0.285563 + 0.958360i \(0.592181\pi\)
\(594\) 3.30155 0.135464
\(595\) 0 0
\(596\) −4.34195 −0.177853
\(597\) −3.79766 −0.155428
\(598\) 15.7119 0.642507
\(599\) 39.5466 1.61583 0.807916 0.589298i \(-0.200596\pi\)
0.807916 + 0.589298i \(0.200596\pi\)
\(600\) 0 0
\(601\) 8.97526 0.366109 0.183054 0.983103i \(-0.441402\pi\)
0.183054 + 0.983103i \(0.441402\pi\)
\(602\) 20.7582 0.846040
\(603\) 13.9564 0.568350
\(604\) 3.30192 0.134353
\(605\) 0 0
\(606\) 11.3746 0.462061
\(607\) −0.157215 −0.00638115 −0.00319058 0.999995i \(-0.501016\pi\)
−0.00319058 + 0.999995i \(0.501016\pi\)
\(608\) −10.4401 −0.423403
\(609\) −26.3151 −1.06634
\(610\) 0 0
\(611\) 4.52660 0.183127
\(612\) −2.44916 −0.0990015
\(613\) −9.37271 −0.378560 −0.189280 0.981923i \(-0.560615\pi\)
−0.189280 + 0.981923i \(0.560615\pi\)
\(614\) −21.7835 −0.879111
\(615\) 0 0
\(616\) 13.4509 0.541952
\(617\) −46.0532 −1.85403 −0.927017 0.375020i \(-0.877636\pi\)
−0.927017 + 0.375020i \(0.877636\pi\)
\(618\) −12.9025 −0.519015
\(619\) 32.5780 1.30942 0.654710 0.755880i \(-0.272791\pi\)
0.654710 + 0.755880i \(0.272791\pi\)
\(620\) 0 0
\(621\) −2.15638 −0.0865327
\(622\) −1.36818 −0.0548589
\(623\) 42.0548 1.68489
\(624\) 21.8757 0.875727
\(625\) 0 0
\(626\) 4.32840 0.172997
\(627\) 6.60347 0.263717
\(628\) −8.08407 −0.322590
\(629\) 35.7096 1.42384
\(630\) 0 0
\(631\) −31.6591 −1.26033 −0.630163 0.776462i \(-0.717012\pi\)
−0.630163 + 0.776462i \(0.717012\pi\)
\(632\) −35.5370 −1.41358
\(633\) −14.6034 −0.580432
\(634\) 7.49819 0.297791
\(635\) 0 0
\(636\) −1.60528 −0.0636533
\(637\) 6.15736 0.243963
\(638\) −30.0479 −1.18961
\(639\) −15.7695 −0.623831
\(640\) 0 0
\(641\) −13.6035 −0.537306 −0.268653 0.963237i \(-0.586579\pi\)
−0.268653 + 0.963237i \(0.586579\pi\)
\(642\) −31.8605 −1.25743
\(643\) −4.19435 −0.165409 −0.0827045 0.996574i \(-0.526356\pi\)
−0.0827045 + 0.996574i \(0.526356\pi\)
\(644\) 3.68464 0.145195
\(645\) 0 0
\(646\) −21.4770 −0.845000
\(647\) −21.9387 −0.862499 −0.431249 0.902233i \(-0.641927\pi\)
−0.431249 + 0.902233i \(0.641927\pi\)
\(648\) −2.26805 −0.0890976
\(649\) 17.5778 0.689989
\(650\) 0 0
\(651\) −2.01733 −0.0790652
\(652\) −10.3898 −0.406895
\(653\) 3.23345 0.126534 0.0632672 0.997997i \(-0.479848\pi\)
0.0632672 + 0.997997i \(0.479848\pi\)
\(654\) 2.59658 0.101534
\(655\) 0 0
\(656\) −15.8686 −0.619567
\(657\) 8.88127 0.346491
\(658\) 4.65415 0.181438
\(659\) −15.4658 −0.602461 −0.301230 0.953551i \(-0.597397\pi\)
−0.301230 + 0.953551i \(0.597397\pi\)
\(660\) 0 0
\(661\) 23.2375 0.903835 0.451917 0.892060i \(-0.350740\pi\)
0.451917 + 0.892060i \(0.350740\pi\)
\(662\) −28.6359 −1.11297
\(663\) 18.7599 0.728575
\(664\) −11.7284 −0.455152
\(665\) 0 0
\(666\) −13.8694 −0.537428
\(667\) 19.6255 0.759904
\(668\) 0.956045 0.0369905
\(669\) −23.7647 −0.918797
\(670\) 0 0
\(671\) −17.4014 −0.671773
\(672\) 9.37629 0.361698
\(673\) −44.2939 −1.70741 −0.853703 0.520760i \(-0.825649\pi\)
−0.853703 + 0.520760i \(0.825649\pi\)
\(674\) −54.8233 −2.11172
\(675\) 0 0
\(676\) 4.42638 0.170245
\(677\) −3.26240 −0.125384 −0.0626921 0.998033i \(-0.519969\pi\)
−0.0626921 + 0.998033i \(0.519969\pi\)
\(678\) 9.94261 0.381844
\(679\) −0.395760 −0.0151879
\(680\) 0 0
\(681\) 6.10001 0.233753
\(682\) −2.30348 −0.0882048
\(683\) 46.2534 1.76984 0.884918 0.465747i \(-0.154214\pi\)
0.884918 + 0.465747i \(0.154214\pi\)
\(684\) 1.90258 0.0727470
\(685\) 0 0
\(686\) −26.2482 −1.00216
\(687\) 22.3505 0.852725
\(688\) −21.5545 −0.821757
\(689\) 12.2960 0.468439
\(690\) 0 0
\(691\) 18.4048 0.700153 0.350077 0.936721i \(-0.386156\pi\)
0.350077 + 0.936721i \(0.386156\pi\)
\(692\) 7.08210 0.269221
\(693\) −5.93059 −0.225284
\(694\) 36.2463 1.37589
\(695\) 0 0
\(696\) 20.6419 0.782428
\(697\) −13.6085 −0.515458
\(698\) 43.0540 1.62962
\(699\) −6.55140 −0.247797
\(700\) 0 0
\(701\) −18.0583 −0.682051 −0.341026 0.940054i \(-0.610774\pi\)
−0.341026 + 0.940054i \(0.610774\pi\)
\(702\) −7.28623 −0.275001
\(703\) −27.7403 −1.04624
\(704\) −9.11839 −0.343662
\(705\) 0 0
\(706\) −22.5012 −0.846844
\(707\) −20.4322 −0.768432
\(708\) 5.06449 0.190335
\(709\) 40.7703 1.53116 0.765580 0.643340i \(-0.222452\pi\)
0.765580 + 0.643340i \(0.222452\pi\)
\(710\) 0 0
\(711\) 15.6685 0.587614
\(712\) −32.9882 −1.23629
\(713\) 1.50450 0.0563440
\(714\) 19.2885 0.721855
\(715\) 0 0
\(716\) 8.01047 0.299365
\(717\) 26.9521 1.00654
\(718\) 12.9802 0.484416
\(719\) 7.63475 0.284728 0.142364 0.989814i \(-0.454530\pi\)
0.142364 + 0.989814i \(0.454530\pi\)
\(720\) 0 0
\(721\) 23.1768 0.863149
\(722\) −13.8993 −0.517280
\(723\) −16.0041 −0.595199
\(724\) −3.38392 −0.125762
\(725\) 0 0
\(726\) 10.9343 0.405809
\(727\) −9.38310 −0.348000 −0.174000 0.984746i \(-0.555669\pi\)
−0.174000 + 0.984746i \(0.555669\pi\)
\(728\) −29.6849 −1.10020
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −18.4845 −0.683673
\(732\) −5.01366 −0.185310
\(733\) 8.20997 0.303242 0.151621 0.988439i \(-0.451551\pi\)
0.151621 + 0.988439i \(0.451551\pi\)
\(734\) 6.69407 0.247083
\(735\) 0 0
\(736\) −6.99274 −0.257756
\(737\) −28.6261 −1.05446
\(738\) 5.28545 0.194560
\(739\) 53.7810 1.97837 0.989184 0.146683i \(-0.0468596\pi\)
0.989184 + 0.146683i \(0.0468596\pi\)
\(740\) 0 0
\(741\) −14.5733 −0.535362
\(742\) 12.6424 0.464119
\(743\) 6.71423 0.246321 0.123161 0.992387i \(-0.460697\pi\)
0.123161 + 0.992387i \(0.460697\pi\)
\(744\) 1.58241 0.0580141
\(745\) 0 0
\(746\) −6.40676 −0.234568
\(747\) 5.17115 0.189202
\(748\) 5.02349 0.183677
\(749\) 57.2311 2.09118
\(750\) 0 0
\(751\) 15.4143 0.562476 0.281238 0.959638i \(-0.409255\pi\)
0.281238 + 0.959638i \(0.409255\pi\)
\(752\) −4.83269 −0.176230
\(753\) 26.5909 0.969026
\(754\) 66.3130 2.41498
\(755\) 0 0
\(756\) −1.70871 −0.0621452
\(757\) −21.9633 −0.798269 −0.399135 0.916892i \(-0.630689\pi\)
−0.399135 + 0.916892i \(0.630689\pi\)
\(758\) 7.65303 0.277971
\(759\) 4.42297 0.160544
\(760\) 0 0
\(761\) 23.5171 0.852494 0.426247 0.904607i \(-0.359835\pi\)
0.426247 + 0.904607i \(0.359835\pi\)
\(762\) −12.2445 −0.443571
\(763\) −4.66425 −0.168857
\(764\) 4.84890 0.175427
\(765\) 0 0
\(766\) 12.1602 0.439366
\(767\) −38.7926 −1.40072
\(768\) −13.0667 −0.471505
\(769\) −49.0593 −1.76912 −0.884562 0.466422i \(-0.845543\pi\)
−0.884562 + 0.466422i \(0.845543\pi\)
\(770\) 0 0
\(771\) 2.59505 0.0934584
\(772\) 12.5624 0.452132
\(773\) −15.5967 −0.560973 −0.280487 0.959858i \(-0.590496\pi\)
−0.280487 + 0.959858i \(0.590496\pi\)
\(774\) 7.17925 0.258053
\(775\) 0 0
\(776\) 0.310439 0.0111441
\(777\) 24.9136 0.893771
\(778\) 19.4035 0.695650
\(779\) 10.5715 0.378762
\(780\) 0 0
\(781\) 32.3448 1.15739
\(782\) −14.3852 −0.514412
\(783\) −9.10114 −0.325248
\(784\) −6.57371 −0.234775
\(785\) 0 0
\(786\) 26.0720 0.929958
\(787\) 7.26119 0.258833 0.129417 0.991590i \(-0.458690\pi\)
0.129417 + 0.991590i \(0.458690\pi\)
\(788\) 2.14494 0.0764104
\(789\) 2.09258 0.0744978
\(790\) 0 0
\(791\) −17.8599 −0.635026
\(792\) 4.65202 0.165302
\(793\) 38.4033 1.36374
\(794\) 9.18362 0.325914
\(795\) 0 0
\(796\) 2.24427 0.0795461
\(797\) −15.4601 −0.547624 −0.273812 0.961783i \(-0.588285\pi\)
−0.273812 + 0.961783i \(0.588285\pi\)
\(798\) −14.9839 −0.530424
\(799\) −4.14437 −0.146617
\(800\) 0 0
\(801\) 14.5447 0.513912
\(802\) −41.3454 −1.45996
\(803\) −18.2164 −0.642843
\(804\) −8.24771 −0.290874
\(805\) 0 0
\(806\) 5.08357 0.179061
\(807\) −11.9754 −0.421554
\(808\) 16.0272 0.563837
\(809\) −50.3702 −1.77092 −0.885461 0.464713i \(-0.846158\pi\)
−0.885461 + 0.464713i \(0.846158\pi\)
\(810\) 0 0
\(811\) −39.0167 −1.37006 −0.685032 0.728513i \(-0.740212\pi\)
−0.685032 + 0.728513i \(0.740212\pi\)
\(812\) 15.5512 0.545741
\(813\) −13.4698 −0.472405
\(814\) 28.4476 0.997087
\(815\) 0 0
\(816\) −20.0284 −0.701136
\(817\) 14.3593 0.502368
\(818\) −60.9756 −2.13196
\(819\) 13.0883 0.457341
\(820\) 0 0
\(821\) 17.5530 0.612604 0.306302 0.951934i \(-0.400908\pi\)
0.306302 + 0.951934i \(0.400908\pi\)
\(822\) −22.1026 −0.770916
\(823\) −11.6225 −0.405134 −0.202567 0.979268i \(-0.564928\pi\)
−0.202567 + 0.979268i \(0.564928\pi\)
\(824\) −18.1801 −0.633335
\(825\) 0 0
\(826\) −39.8857 −1.38780
\(827\) 4.12817 0.143550 0.0717752 0.997421i \(-0.477134\pi\)
0.0717752 + 0.997421i \(0.477134\pi\)
\(828\) 1.27434 0.0442863
\(829\) 35.8722 1.24589 0.622946 0.782265i \(-0.285936\pi\)
0.622946 + 0.782265i \(0.285936\pi\)
\(830\) 0 0
\(831\) −11.6830 −0.405279
\(832\) 20.1235 0.697656
\(833\) −5.63742 −0.195325
\(834\) −0.668209 −0.0231382
\(835\) 0 0
\(836\) −3.90239 −0.134967
\(837\) −0.697696 −0.0241159
\(838\) −63.3494 −2.18837
\(839\) 36.0129 1.24330 0.621652 0.783294i \(-0.286462\pi\)
0.621652 + 0.783294i \(0.286462\pi\)
\(840\) 0 0
\(841\) 53.8307 1.85623
\(842\) −20.7831 −0.716233
\(843\) 26.0065 0.895713
\(844\) 8.63002 0.297058
\(845\) 0 0
\(846\) 1.60965 0.0553408
\(847\) −19.6413 −0.674882
\(848\) −13.1274 −0.450797
\(849\) 14.4211 0.494931
\(850\) 0 0
\(851\) −18.5803 −0.636925
\(852\) 9.31915 0.319269
\(853\) −12.6467 −0.433014 −0.216507 0.976281i \(-0.569466\pi\)
−0.216507 + 0.976281i \(0.569466\pi\)
\(854\) 39.4854 1.35116
\(855\) 0 0
\(856\) −44.8927 −1.53440
\(857\) −7.19169 −0.245664 −0.122832 0.992428i \(-0.539198\pi\)
−0.122832 + 0.992428i \(0.539198\pi\)
\(858\) 14.9448 0.510208
\(859\) 35.7583 1.22006 0.610028 0.792379i \(-0.291158\pi\)
0.610028 + 0.792379i \(0.291158\pi\)
\(860\) 0 0
\(861\) −9.49426 −0.323564
\(862\) −5.45535 −0.185810
\(863\) 26.9381 0.916984 0.458492 0.888699i \(-0.348390\pi\)
0.458492 + 0.888699i \(0.348390\pi\)
\(864\) 3.24281 0.110323
\(865\) 0 0
\(866\) 54.9165 1.86614
\(867\) −0.175799 −0.00597046
\(868\) 1.19216 0.0404646
\(869\) −32.1377 −1.09020
\(870\) 0 0
\(871\) 63.1752 2.14061
\(872\) 3.65869 0.123899
\(873\) −0.136874 −0.00463250
\(874\) 11.1748 0.377994
\(875\) 0 0
\(876\) −5.24848 −0.177330
\(877\) 45.4016 1.53310 0.766552 0.642182i \(-0.221971\pi\)
0.766552 + 0.642182i \(0.221971\pi\)
\(878\) −21.2746 −0.717984
\(879\) 5.61335 0.189334
\(880\) 0 0
\(881\) 13.6687 0.460511 0.230255 0.973130i \(-0.426044\pi\)
0.230255 + 0.973130i \(0.426044\pi\)
\(882\) 2.18954 0.0737255
\(883\) −7.44286 −0.250472 −0.125236 0.992127i \(-0.539969\pi\)
−0.125236 + 0.992127i \(0.539969\pi\)
\(884\) −11.0864 −0.372875
\(885\) 0 0
\(886\) −45.6601 −1.53398
\(887\) 10.6253 0.356762 0.178381 0.983962i \(-0.442914\pi\)
0.178381 + 0.983962i \(0.442914\pi\)
\(888\) −19.5425 −0.655804
\(889\) 21.9948 0.737681
\(890\) 0 0
\(891\) −2.05110 −0.0687146
\(892\) 14.0440 0.470229
\(893\) 3.21947 0.107735
\(894\) 11.8265 0.395537
\(895\) 0 0
\(896\) 39.4431 1.31770
\(897\) −9.76109 −0.325913
\(898\) −9.57709 −0.319592
\(899\) 6.34983 0.211779
\(900\) 0 0
\(901\) −11.2577 −0.375048
\(902\) −10.8410 −0.360966
\(903\) −12.8961 −0.429156
\(904\) 14.0095 0.465950
\(905\) 0 0
\(906\) −8.99371 −0.298796
\(907\) 2.96687 0.0985133 0.0492566 0.998786i \(-0.484315\pi\)
0.0492566 + 0.998786i \(0.484315\pi\)
\(908\) −3.60487 −0.119632
\(909\) −7.06652 −0.234382
\(910\) 0 0
\(911\) −2.44851 −0.0811227 −0.0405614 0.999177i \(-0.512915\pi\)
−0.0405614 + 0.999177i \(0.512915\pi\)
\(912\) 15.5587 0.515199
\(913\) −10.6066 −0.351026
\(914\) −10.5971 −0.350521
\(915\) 0 0
\(916\) −13.2083 −0.436414
\(917\) −46.8332 −1.54657
\(918\) 6.67097 0.220175
\(919\) −2.21230 −0.0729769 −0.0364885 0.999334i \(-0.511617\pi\)
−0.0364885 + 0.999334i \(0.511617\pi\)
\(920\) 0 0
\(921\) 13.5331 0.445931
\(922\) −56.6912 −1.86702
\(923\) −71.3822 −2.34957
\(924\) 3.50475 0.115298
\(925\) 0 0
\(926\) 22.2096 0.729853
\(927\) 8.01574 0.263271
\(928\) −29.5133 −0.968820
\(929\) 14.9520 0.490558 0.245279 0.969453i \(-0.421120\pi\)
0.245279 + 0.969453i \(0.421120\pi\)
\(930\) 0 0
\(931\) 4.37931 0.143526
\(932\) 3.87162 0.126819
\(933\) 0.849986 0.0278273
\(934\) 12.9123 0.422502
\(935\) 0 0
\(936\) −10.2666 −0.335574
\(937\) 0.266191 0.00869607 0.00434804 0.999991i \(-0.498616\pi\)
0.00434804 + 0.999991i \(0.498616\pi\)
\(938\) 64.9553 2.12087
\(939\) −2.68904 −0.0877533
\(940\) 0 0
\(941\) 51.4216 1.67630 0.838149 0.545442i \(-0.183638\pi\)
0.838149 + 0.545442i \(0.183638\pi\)
\(942\) 22.0192 0.717424
\(943\) 7.08072 0.230580
\(944\) 41.4157 1.34797
\(945\) 0 0
\(946\) −14.7254 −0.478764
\(947\) −27.9541 −0.908385 −0.454192 0.890904i \(-0.650072\pi\)
−0.454192 + 0.890904i \(0.650072\pi\)
\(948\) −9.25946 −0.300733
\(949\) 40.2020 1.30501
\(950\) 0 0
\(951\) −4.65829 −0.151055
\(952\) 27.1783 0.880853
\(953\) 3.48949 0.113036 0.0565178 0.998402i \(-0.482000\pi\)
0.0565178 + 0.998402i \(0.482000\pi\)
\(954\) 4.37241 0.141562
\(955\) 0 0
\(956\) −15.9276 −0.515137
\(957\) 18.6674 0.603431
\(958\) −0.545910 −0.0176376
\(959\) 39.7029 1.28207
\(960\) 0 0
\(961\) −30.5132 −0.984297
\(962\) −62.7812 −2.02415
\(963\) 19.7935 0.637836
\(964\) 9.45780 0.304615
\(965\) 0 0
\(966\) −10.0361 −0.322907
\(967\) −15.8726 −0.510430 −0.255215 0.966884i \(-0.582146\pi\)
−0.255215 + 0.966884i \(0.582146\pi\)
\(968\) 15.4068 0.495194
\(969\) 13.3427 0.428628
\(970\) 0 0
\(971\) 44.2670 1.42060 0.710298 0.703901i \(-0.248560\pi\)
0.710298 + 0.703901i \(0.248560\pi\)
\(972\) −0.590961 −0.0189551
\(973\) 1.20031 0.0384800
\(974\) −53.0089 −1.69851
\(975\) 0 0
\(976\) −41.0001 −1.31238
\(977\) −0.648897 −0.0207601 −0.0103800 0.999946i \(-0.503304\pi\)
−0.0103800 + 0.999946i \(0.503304\pi\)
\(978\) 28.2994 0.904915
\(979\) −29.8327 −0.953458
\(980\) 0 0
\(981\) −1.61314 −0.0515036
\(982\) 3.78533 0.120795
\(983\) 28.4868 0.908586 0.454293 0.890852i \(-0.349892\pi\)
0.454293 + 0.890852i \(0.349892\pi\)
\(984\) 7.44740 0.237414
\(985\) 0 0
\(986\) −60.7134 −1.93351
\(987\) −2.89141 −0.0920346
\(988\) 8.61223 0.273991
\(989\) 9.61778 0.305828
\(990\) 0 0
\(991\) 45.7840 1.45438 0.727189 0.686437i \(-0.240826\pi\)
0.727189 + 0.686437i \(0.240826\pi\)
\(992\) −2.26249 −0.0718343
\(993\) 17.7902 0.564554
\(994\) −73.3935 −2.32790
\(995\) 0 0
\(996\) −3.05595 −0.0968314
\(997\) 44.8082 1.41909 0.709546 0.704659i \(-0.248900\pi\)
0.709546 + 0.704659i \(0.248900\pi\)
\(998\) −6.07140 −0.192187
\(999\) 8.61642 0.272612
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 3525.2.a.be.1.6 yes 8
5.4 even 2 3525.2.a.bd.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3525.2.a.bd.1.3 8 5.4 even 2
3525.2.a.be.1.6 yes 8 1.1 even 1 trivial