Properties

Label 8470.2.a.da.1.3
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.38498000.1
Defining polynomial: \(x^{6} - x^{5} - 12 x^{4} + 5 x^{3} + 38 x^{2} - x - 19\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.824369\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.82437 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.82437 q^{6} +1.00000 q^{7} +1.00000 q^{8} +0.328323 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.82437 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.82437 q^{6} +1.00000 q^{7} +1.00000 q^{8} +0.328323 q^{9} -1.00000 q^{10} -1.82437 q^{12} -2.21432 q^{13} +1.00000 q^{14} +1.82437 q^{15} +1.00000 q^{16} -7.96638 q^{17} +0.328323 q^{18} -0.915653 q^{19} -1.00000 q^{20} -1.82437 q^{21} +6.07737 q^{23} -1.82437 q^{24} +1.00000 q^{25} -2.21432 q^{26} +4.87412 q^{27} +1.00000 q^{28} +5.06597 q^{29} +1.82437 q^{30} +9.33892 q^{31} +1.00000 q^{32} -7.96638 q^{34} -1.00000 q^{35} +0.328323 q^{36} +4.32937 q^{37} -0.915653 q^{38} +4.03973 q^{39} -1.00000 q^{40} -8.99953 q^{41} -1.82437 q^{42} +2.76947 q^{43} -0.328323 q^{45} +6.07737 q^{46} +4.22816 q^{47} -1.82437 q^{48} +1.00000 q^{49} +1.00000 q^{50} +14.5336 q^{51} -2.21432 q^{52} -9.33444 q^{53} +4.87412 q^{54} +1.00000 q^{56} +1.67049 q^{57} +5.06597 q^{58} +7.19487 q^{59} +1.82437 q^{60} -12.6895 q^{61} +9.33892 q^{62} +0.328323 q^{63} +1.00000 q^{64} +2.21432 q^{65} -1.29018 q^{67} -7.96638 q^{68} -11.0874 q^{69} -1.00000 q^{70} +0.0202370 q^{71} +0.328323 q^{72} +11.8463 q^{73} +4.32937 q^{74} -1.82437 q^{75} -0.915653 q^{76} +4.03973 q^{78} -9.39226 q^{79} -1.00000 q^{80} -9.87717 q^{81} -8.99953 q^{82} +0.747575 q^{83} -1.82437 q^{84} +7.96638 q^{85} +2.76947 q^{86} -9.24220 q^{87} -14.2639 q^{89} -0.328323 q^{90} -2.21432 q^{91} +6.07737 q^{92} -17.0376 q^{93} +4.22816 q^{94} +0.915653 q^{95} -1.82437 q^{96} -7.27431 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} - 5q^{3} + 6q^{4} - 6q^{5} - 5q^{6} + 6q^{7} + 6q^{8} + 11q^{9} + O(q^{10}) \) \( 6q + 6q^{2} - 5q^{3} + 6q^{4} - 6q^{5} - 5q^{6} + 6q^{7} + 6q^{8} + 11q^{9} - 6q^{10} - 5q^{12} + 6q^{14} + 5q^{15} + 6q^{16} - 15q^{17} + 11q^{18} - 13q^{19} - 6q^{20} - 5q^{21} - 2q^{23} - 5q^{24} + 6q^{25} - 26q^{27} + 6q^{28} - 4q^{29} + 5q^{30} - 2q^{31} + 6q^{32} - 15q^{34} - 6q^{35} + 11q^{36} - 13q^{38} - 30q^{39} - 6q^{40} - 17q^{41} - 5q^{42} + 15q^{43} - 11q^{45} - 2q^{46} - 18q^{47} - 5q^{48} + 6q^{49} + 6q^{50} + 6q^{51} + 22q^{53} - 26q^{54} + 6q^{56} - 2q^{57} - 4q^{58} - 7q^{59} + 5q^{60} - 20q^{61} - 2q^{62} + 11q^{63} + 6q^{64} - 29q^{67} - 15q^{68} + 12q^{69} - 6q^{70} - 2q^{71} + 11q^{72} + q^{73} - 5q^{75} - 13q^{76} - 30q^{78} - 12q^{79} - 6q^{80} + 22q^{81} - 17q^{82} - 35q^{83} - 5q^{84} + 15q^{85} + 15q^{86} - 29q^{89} - 11q^{90} - 2q^{92} + 8q^{93} - 18q^{94} + 13q^{95} - 5q^{96} - 19q^{97} + 6q^{98} + O(q^{100}) \)

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
\(3\) −1.82437 −1.05330 −0.526650 0.850082i \(-0.676552\pi\)
−0.526650 + 0.850082i \(0.676552\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.82437 −0.744796
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0.328323 0.109441
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −1.82437 −0.526650
\(13\) −2.21432 −0.614141 −0.307071 0.951687i \(-0.599349\pi\)
−0.307071 + 0.951687i \(0.599349\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.82437 0.471050
\(16\) 1.00000 0.250000
\(17\) −7.96638 −1.93213 −0.966066 0.258297i \(-0.916839\pi\)
−0.966066 + 0.258297i \(0.916839\pi\)
\(18\) 0.328323 0.0773866
\(19\) −0.915653 −0.210065 −0.105033 0.994469i \(-0.533495\pi\)
−0.105033 + 0.994469i \(0.533495\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.82437 −0.398110
\(22\) 0 0
\(23\) 6.07737 1.26722 0.633610 0.773653i \(-0.281572\pi\)
0.633610 + 0.773653i \(0.281572\pi\)
\(24\) −1.82437 −0.372398
\(25\) 1.00000 0.200000
\(26\) −2.21432 −0.434263
\(27\) 4.87412 0.938026
\(28\) 1.00000 0.188982
\(29\) 5.06597 0.940727 0.470364 0.882473i \(-0.344123\pi\)
0.470364 + 0.882473i \(0.344123\pi\)
\(30\) 1.82437 0.333083
\(31\) 9.33892 1.67732 0.838660 0.544655i \(-0.183339\pi\)
0.838660 + 0.544655i \(0.183339\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −7.96638 −1.36622
\(35\) −1.00000 −0.169031
\(36\) 0.328323 0.0547206
\(37\) 4.32937 0.711744 0.355872 0.934535i \(-0.384184\pi\)
0.355872 + 0.934535i \(0.384184\pi\)
\(38\) −0.915653 −0.148538
\(39\) 4.03973 0.646875
\(40\) −1.00000 −0.158114
\(41\) −8.99953 −1.40549 −0.702745 0.711441i \(-0.748043\pi\)
−0.702745 + 0.711441i \(0.748043\pi\)
\(42\) −1.82437 −0.281506
\(43\) 2.76947 0.422340 0.211170 0.977449i \(-0.432273\pi\)
0.211170 + 0.977449i \(0.432273\pi\)
\(44\) 0 0
\(45\) −0.328323 −0.0489436
\(46\) 6.07737 0.896060
\(47\) 4.22816 0.616740 0.308370 0.951266i \(-0.400216\pi\)
0.308370 + 0.951266i \(0.400216\pi\)
\(48\) −1.82437 −0.263325
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 14.5336 2.03511
\(52\) −2.21432 −0.307071
\(53\) −9.33444 −1.28218 −0.641092 0.767464i \(-0.721518\pi\)
−0.641092 + 0.767464i \(0.721518\pi\)
\(54\) 4.87412 0.663284
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 1.67049 0.221262
\(58\) 5.06597 0.665195
\(59\) 7.19487 0.936693 0.468346 0.883545i \(-0.344850\pi\)
0.468346 + 0.883545i \(0.344850\pi\)
\(60\) 1.82437 0.235525
\(61\) −12.6895 −1.62473 −0.812365 0.583150i \(-0.801820\pi\)
−0.812365 + 0.583150i \(0.801820\pi\)
\(62\) 9.33892 1.18604
\(63\) 0.328323 0.0413649
\(64\) 1.00000 0.125000
\(65\) 2.21432 0.274652
\(66\) 0 0
\(67\) −1.29018 −0.157620 −0.0788101 0.996890i \(-0.525112\pi\)
−0.0788101 + 0.996890i \(0.525112\pi\)
\(68\) −7.96638 −0.966066
\(69\) −11.0874 −1.33476
\(70\) −1.00000 −0.119523
\(71\) 0.0202370 0.00240169 0.00120084 0.999999i \(-0.499618\pi\)
0.00120084 + 0.999999i \(0.499618\pi\)
\(72\) 0.328323 0.0386933
\(73\) 11.8463 1.38650 0.693250 0.720698i \(-0.256178\pi\)
0.693250 + 0.720698i \(0.256178\pi\)
\(74\) 4.32937 0.503279
\(75\) −1.82437 −0.210660
\(76\) −0.915653 −0.105033
\(77\) 0 0
\(78\) 4.03973 0.457410
\(79\) −9.39226 −1.05671 −0.528356 0.849023i \(-0.677191\pi\)
−0.528356 + 0.849023i \(0.677191\pi\)
\(80\) −1.00000 −0.111803
\(81\) −9.87717 −1.09746
\(82\) −8.99953 −0.993832
\(83\) 0.747575 0.0820570 0.0410285 0.999158i \(-0.486937\pi\)
0.0410285 + 0.999158i \(0.486937\pi\)
\(84\) −1.82437 −0.199055
\(85\) 7.96638 0.864075
\(86\) 2.76947 0.298639
\(87\) −9.24220 −0.990868
\(88\) 0 0
\(89\) −14.2639 −1.51197 −0.755983 0.654591i \(-0.772841\pi\)
−0.755983 + 0.654591i \(0.772841\pi\)
\(90\) −0.328323 −0.0346083
\(91\) −2.21432 −0.232124
\(92\) 6.07737 0.633610
\(93\) −17.0376 −1.76672
\(94\) 4.22816 0.436101
\(95\) 0.915653 0.0939440
\(96\) −1.82437 −0.186199
\(97\) −7.27431 −0.738594 −0.369297 0.929311i \(-0.620402\pi\)
−0.369297 + 0.929311i \(0.620402\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −5.35283 −0.532627 −0.266313 0.963886i \(-0.585806\pi\)
−0.266313 + 0.963886i \(0.585806\pi\)
\(102\) 14.5336 1.43904
\(103\) −14.3938 −1.41826 −0.709130 0.705078i \(-0.750912\pi\)
−0.709130 + 0.705078i \(0.750912\pi\)
\(104\) −2.21432 −0.217132
\(105\) 1.82437 0.178040
\(106\) −9.33444 −0.906641
\(107\) −4.27627 −0.413402 −0.206701 0.978404i \(-0.566273\pi\)
−0.206701 + 0.978404i \(0.566273\pi\)
\(108\) 4.87412 0.469013
\(109\) −5.04265 −0.482998 −0.241499 0.970401i \(-0.577639\pi\)
−0.241499 + 0.970401i \(0.577639\pi\)
\(110\) 0 0
\(111\) −7.89837 −0.749680
\(112\) 1.00000 0.0944911
\(113\) 13.8169 1.29979 0.649894 0.760025i \(-0.274813\pi\)
0.649894 + 0.760025i \(0.274813\pi\)
\(114\) 1.67049 0.156456
\(115\) −6.07737 −0.566718
\(116\) 5.06597 0.470364
\(117\) −0.727012 −0.0672123
\(118\) 7.19487 0.662342
\(119\) −7.96638 −0.730277
\(120\) 1.82437 0.166541
\(121\) 0 0
\(122\) −12.6895 −1.14886
\(123\) 16.4185 1.48040
\(124\) 9.33892 0.838660
\(125\) −1.00000 −0.0894427
\(126\) 0.328323 0.0292494
\(127\) 14.6820 1.30281 0.651407 0.758729i \(-0.274179\pi\)
0.651407 + 0.758729i \(0.274179\pi\)
\(128\) 1.00000 0.0883883
\(129\) −5.05253 −0.444850
\(130\) 2.21432 0.194209
\(131\) −7.62582 −0.666271 −0.333135 0.942879i \(-0.608107\pi\)
−0.333135 + 0.942879i \(0.608107\pi\)
\(132\) 0 0
\(133\) −0.915653 −0.0793972
\(134\) −1.29018 −0.111454
\(135\) −4.87412 −0.419498
\(136\) −7.96638 −0.683111
\(137\) −4.93059 −0.421248 −0.210624 0.977567i \(-0.567550\pi\)
−0.210624 + 0.977567i \(0.567550\pi\)
\(138\) −11.0874 −0.943820
\(139\) −4.17718 −0.354304 −0.177152 0.984184i \(-0.556688\pi\)
−0.177152 + 0.984184i \(0.556688\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −7.71373 −0.649613
\(142\) 0.0202370 0.00169825
\(143\) 0 0
\(144\) 0.328323 0.0273603
\(145\) −5.06597 −0.420706
\(146\) 11.8463 0.980403
\(147\) −1.82437 −0.150471
\(148\) 4.32937 0.355872
\(149\) −13.4785 −1.10420 −0.552102 0.833776i \(-0.686174\pi\)
−0.552102 + 0.833776i \(0.686174\pi\)
\(150\) −1.82437 −0.148959
\(151\) 17.6102 1.43309 0.716547 0.697538i \(-0.245721\pi\)
0.716547 + 0.697538i \(0.245721\pi\)
\(152\) −0.915653 −0.0742692
\(153\) −2.61555 −0.211455
\(154\) 0 0
\(155\) −9.33892 −0.750121
\(156\) 4.03973 0.323438
\(157\) 11.4836 0.916491 0.458245 0.888826i \(-0.348478\pi\)
0.458245 + 0.888826i \(0.348478\pi\)
\(158\) −9.39226 −0.747208
\(159\) 17.0295 1.35052
\(160\) −1.00000 −0.0790569
\(161\) 6.07737 0.478964
\(162\) −9.87717 −0.776024
\(163\) −0.139250 −0.0109069 −0.00545347 0.999985i \(-0.501736\pi\)
−0.00545347 + 0.999985i \(0.501736\pi\)
\(164\) −8.99953 −0.702745
\(165\) 0 0
\(166\) 0.747575 0.0580230
\(167\) 7.04367 0.545056 0.272528 0.962148i \(-0.412140\pi\)
0.272528 + 0.962148i \(0.412140\pi\)
\(168\) −1.82437 −0.140753
\(169\) −8.09680 −0.622830
\(170\) 7.96638 0.610993
\(171\) −0.300630 −0.0229898
\(172\) 2.76947 0.211170
\(173\) −14.0443 −1.06777 −0.533883 0.845558i \(-0.679268\pi\)
−0.533883 + 0.845558i \(0.679268\pi\)
\(174\) −9.24220 −0.700650
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −13.1261 −0.986619
\(178\) −14.2639 −1.06912
\(179\) 12.8339 0.959254 0.479627 0.877473i \(-0.340772\pi\)
0.479627 + 0.877473i \(0.340772\pi\)
\(180\) −0.328323 −0.0244718
\(181\) 19.2871 1.43360 0.716798 0.697281i \(-0.245607\pi\)
0.716798 + 0.697281i \(0.245607\pi\)
\(182\) −2.21432 −0.164136
\(183\) 23.1504 1.71133
\(184\) 6.07737 0.448030
\(185\) −4.32937 −0.318302
\(186\) −17.0376 −1.24926
\(187\) 0 0
\(188\) 4.22816 0.308370
\(189\) 4.87412 0.354540
\(190\) 0.915653 0.0664284
\(191\) −12.0109 −0.869076 −0.434538 0.900653i \(-0.643088\pi\)
−0.434538 + 0.900653i \(0.643088\pi\)
\(192\) −1.82437 −0.131663
\(193\) −17.2998 −1.24527 −0.622635 0.782512i \(-0.713938\pi\)
−0.622635 + 0.782512i \(0.713938\pi\)
\(194\) −7.27431 −0.522265
\(195\) −4.03973 −0.289291
\(196\) 1.00000 0.0714286
\(197\) −13.1724 −0.938497 −0.469248 0.883066i \(-0.655475\pi\)
−0.469248 + 0.883066i \(0.655475\pi\)
\(198\) 0 0
\(199\) −22.9779 −1.62886 −0.814431 0.580260i \(-0.802951\pi\)
−0.814431 + 0.580260i \(0.802951\pi\)
\(200\) 1.00000 0.0707107
\(201\) 2.35376 0.166021
\(202\) −5.35283 −0.376624
\(203\) 5.06597 0.355561
\(204\) 14.5336 1.01756
\(205\) 8.99953 0.628555
\(206\) −14.3938 −1.00286
\(207\) 1.99534 0.138686
\(208\) −2.21432 −0.153535
\(209\) 0 0
\(210\) 1.82437 0.125893
\(211\) −16.5609 −1.14010 −0.570050 0.821610i \(-0.693076\pi\)
−0.570050 + 0.821610i \(0.693076\pi\)
\(212\) −9.33444 −0.641092
\(213\) −0.0369197 −0.00252970
\(214\) −4.27627 −0.292320
\(215\) −2.76947 −0.188876
\(216\) 4.87412 0.331642
\(217\) 9.33892 0.633968
\(218\) −5.04265 −0.341531
\(219\) −21.6119 −1.46040
\(220\) 0 0
\(221\) 17.6401 1.18660
\(222\) −7.89837 −0.530104
\(223\) −19.1235 −1.28060 −0.640301 0.768124i \(-0.721190\pi\)
−0.640301 + 0.768124i \(0.721190\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0.328323 0.0218882
\(226\) 13.8169 0.919088
\(227\) 19.6989 1.30746 0.653732 0.756726i \(-0.273202\pi\)
0.653732 + 0.756726i \(0.273202\pi\)
\(228\) 1.67049 0.110631
\(229\) 27.5813 1.82263 0.911313 0.411714i \(-0.135070\pi\)
0.911313 + 0.411714i \(0.135070\pi\)
\(230\) −6.07737 −0.400730
\(231\) 0 0
\(232\) 5.06597 0.332597
\(233\) −12.9463 −0.848141 −0.424071 0.905629i \(-0.639399\pi\)
−0.424071 + 0.905629i \(0.639399\pi\)
\(234\) −0.727012 −0.0475263
\(235\) −4.22816 −0.275815
\(236\) 7.19487 0.468346
\(237\) 17.1349 1.11303
\(238\) −7.96638 −0.516384
\(239\) −5.77626 −0.373635 −0.186818 0.982395i \(-0.559817\pi\)
−0.186818 + 0.982395i \(0.559817\pi\)
\(240\) 1.82437 0.117763
\(241\) −25.0323 −1.61247 −0.806237 0.591593i \(-0.798499\pi\)
−0.806237 + 0.591593i \(0.798499\pi\)
\(242\) 0 0
\(243\) 3.39724 0.217933
\(244\) −12.6895 −0.812365
\(245\) −1.00000 −0.0638877
\(246\) 16.4185 1.04680
\(247\) 2.02755 0.129010
\(248\) 9.33892 0.593022
\(249\) −1.36385 −0.0864306
\(250\) −1.00000 −0.0632456
\(251\) −6.66594 −0.420750 −0.210375 0.977621i \(-0.567469\pi\)
−0.210375 + 0.977621i \(0.567469\pi\)
\(252\) 0.328323 0.0206824
\(253\) 0 0
\(254\) 14.6820 0.921228
\(255\) −14.5336 −0.910131
\(256\) 1.00000 0.0625000
\(257\) −22.7153 −1.41694 −0.708472 0.705739i \(-0.750615\pi\)
−0.708472 + 0.705739i \(0.750615\pi\)
\(258\) −5.05253 −0.314557
\(259\) 4.32937 0.269014
\(260\) 2.21432 0.137326
\(261\) 1.66328 0.102954
\(262\) −7.62582 −0.471124
\(263\) 14.5015 0.894200 0.447100 0.894484i \(-0.352457\pi\)
0.447100 + 0.894484i \(0.352457\pi\)
\(264\) 0 0
\(265\) 9.33444 0.573410
\(266\) −0.915653 −0.0561423
\(267\) 26.0225 1.59255
\(268\) −1.29018 −0.0788101
\(269\) −21.2753 −1.29718 −0.648590 0.761138i \(-0.724641\pi\)
−0.648590 + 0.761138i \(0.724641\pi\)
\(270\) −4.87412 −0.296630
\(271\) 8.06611 0.489981 0.244991 0.969525i \(-0.421215\pi\)
0.244991 + 0.969525i \(0.421215\pi\)
\(272\) −7.96638 −0.483033
\(273\) 4.03973 0.244496
\(274\) −4.93059 −0.297868
\(275\) 0 0
\(276\) −11.0874 −0.667382
\(277\) 17.4563 1.04885 0.524423 0.851458i \(-0.324281\pi\)
0.524423 + 0.851458i \(0.324281\pi\)
\(278\) −4.17718 −0.250531
\(279\) 3.06619 0.183568
\(280\) −1.00000 −0.0597614
\(281\) −20.4688 −1.22106 −0.610532 0.791992i \(-0.709044\pi\)
−0.610532 + 0.791992i \(0.709044\pi\)
\(282\) −7.71373 −0.459346
\(283\) 5.27282 0.313437 0.156718 0.987643i \(-0.449909\pi\)
0.156718 + 0.987643i \(0.449909\pi\)
\(284\) 0.0202370 0.00120084
\(285\) −1.67049 −0.0989512
\(286\) 0 0
\(287\) −8.99953 −0.531226
\(288\) 0.328323 0.0193466
\(289\) 46.4632 2.73313
\(290\) −5.06597 −0.297484
\(291\) 13.2710 0.777961
\(292\) 11.8463 0.693250
\(293\) 6.40613 0.374250 0.187125 0.982336i \(-0.440083\pi\)
0.187125 + 0.982336i \(0.440083\pi\)
\(294\) −1.82437 −0.106399
\(295\) −7.19487 −0.418902
\(296\) 4.32937 0.251640
\(297\) 0 0
\(298\) −13.4785 −0.780790
\(299\) −13.4572 −0.778252
\(300\) −1.82437 −0.105330
\(301\) 2.76947 0.159629
\(302\) 17.6102 1.01335
\(303\) 9.76555 0.561016
\(304\) −0.915653 −0.0525163
\(305\) 12.6895 0.726601
\(306\) −2.61555 −0.149521
\(307\) 5.35970 0.305894 0.152947 0.988234i \(-0.451124\pi\)
0.152947 + 0.988234i \(0.451124\pi\)
\(308\) 0 0
\(309\) 26.2595 1.49385
\(310\) −9.33892 −0.530415
\(311\) −26.9819 −1.53001 −0.765003 0.644027i \(-0.777262\pi\)
−0.765003 + 0.644027i \(0.777262\pi\)
\(312\) 4.03973 0.228705
\(313\) −28.5633 −1.61449 −0.807245 0.590216i \(-0.799043\pi\)
−0.807245 + 0.590216i \(0.799043\pi\)
\(314\) 11.4836 0.648057
\(315\) −0.328323 −0.0184989
\(316\) −9.39226 −0.528356
\(317\) 21.0077 1.17991 0.589954 0.807437i \(-0.299146\pi\)
0.589954 + 0.807437i \(0.299146\pi\)
\(318\) 17.0295 0.954965
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 7.80149 0.435437
\(322\) 6.07737 0.338679
\(323\) 7.29444 0.405873
\(324\) −9.87717 −0.548732
\(325\) −2.21432 −0.122828
\(326\) −0.139250 −0.00771236
\(327\) 9.19965 0.508742
\(328\) −8.99953 −0.496916
\(329\) 4.22816 0.233106
\(330\) 0 0
\(331\) 11.6022 0.637717 0.318858 0.947802i \(-0.396701\pi\)
0.318858 + 0.947802i \(0.396701\pi\)
\(332\) 0.747575 0.0410285
\(333\) 1.42143 0.0778941
\(334\) 7.04367 0.385413
\(335\) 1.29018 0.0704899
\(336\) −1.82437 −0.0995275
\(337\) 2.34874 0.127944 0.0639720 0.997952i \(-0.479623\pi\)
0.0639720 + 0.997952i \(0.479623\pi\)
\(338\) −8.09680 −0.440408
\(339\) −25.2072 −1.36907
\(340\) 7.96638 0.432038
\(341\) 0 0
\(342\) −0.300630 −0.0162562
\(343\) 1.00000 0.0539949
\(344\) 2.76947 0.149320
\(345\) 11.0874 0.596924
\(346\) −14.0443 −0.755025
\(347\) −31.4811 −1.68999 −0.844997 0.534771i \(-0.820398\pi\)
−0.844997 + 0.534771i \(0.820398\pi\)
\(348\) −9.24220 −0.495434
\(349\) 2.60777 0.139591 0.0697955 0.997561i \(-0.477765\pi\)
0.0697955 + 0.997561i \(0.477765\pi\)
\(350\) 1.00000 0.0534522
\(351\) −10.7929 −0.576080
\(352\) 0 0
\(353\) −0.416007 −0.0221418 −0.0110709 0.999939i \(-0.503524\pi\)
−0.0110709 + 0.999939i \(0.503524\pi\)
\(354\) −13.1261 −0.697645
\(355\) −0.0202370 −0.00107407
\(356\) −14.2639 −0.755983
\(357\) 14.5336 0.769201
\(358\) 12.8339 0.678295
\(359\) −11.6150 −0.613017 −0.306508 0.951868i \(-0.599161\pi\)
−0.306508 + 0.951868i \(0.599161\pi\)
\(360\) −0.328323 −0.0173042
\(361\) −18.1616 −0.955873
\(362\) 19.2871 1.01371
\(363\) 0 0
\(364\) −2.21432 −0.116062
\(365\) −11.8463 −0.620061
\(366\) 23.1504 1.21009
\(367\) −8.00709 −0.417966 −0.208983 0.977919i \(-0.567015\pi\)
−0.208983 + 0.977919i \(0.567015\pi\)
\(368\) 6.07737 0.316805
\(369\) −2.95476 −0.153819
\(370\) −4.32937 −0.225073
\(371\) −9.33444 −0.484620
\(372\) −17.0376 −0.883361
\(373\) −16.6329 −0.861217 −0.430609 0.902539i \(-0.641701\pi\)
−0.430609 + 0.902539i \(0.641701\pi\)
\(374\) 0 0
\(375\) 1.82437 0.0942100
\(376\) 4.22816 0.218051
\(377\) −11.2177 −0.577739
\(378\) 4.87412 0.250698
\(379\) −0.0831686 −0.00427208 −0.00213604 0.999998i \(-0.500680\pi\)
−0.00213604 + 0.999998i \(0.500680\pi\)
\(380\) 0.915653 0.0469720
\(381\) −26.7853 −1.37225
\(382\) −12.0109 −0.614530
\(383\) 6.28284 0.321038 0.160519 0.987033i \(-0.448683\pi\)
0.160519 + 0.987033i \(0.448683\pi\)
\(384\) −1.82437 −0.0930995
\(385\) 0 0
\(386\) −17.2998 −0.880539
\(387\) 0.909280 0.0462213
\(388\) −7.27431 −0.369297
\(389\) −3.67862 −0.186513 −0.0932567 0.995642i \(-0.529728\pi\)
−0.0932567 + 0.995642i \(0.529728\pi\)
\(390\) −4.03973 −0.204560
\(391\) −48.4147 −2.44844
\(392\) 1.00000 0.0505076
\(393\) 13.9123 0.701783
\(394\) −13.1724 −0.663618
\(395\) 9.39226 0.472576
\(396\) 0 0
\(397\) 7.12152 0.357419 0.178709 0.983902i \(-0.442808\pi\)
0.178709 + 0.983902i \(0.442808\pi\)
\(398\) −22.9779 −1.15178
\(399\) 1.67049 0.0836290
\(400\) 1.00000 0.0500000
\(401\) 5.70426 0.284857 0.142429 0.989805i \(-0.454509\pi\)
0.142429 + 0.989805i \(0.454509\pi\)
\(402\) 2.35376 0.117395
\(403\) −20.6793 −1.03011
\(404\) −5.35283 −0.266313
\(405\) 9.87717 0.490801
\(406\) 5.06597 0.251420
\(407\) 0 0
\(408\) 14.5336 0.719521
\(409\) 23.7324 1.17349 0.586745 0.809772i \(-0.300409\pi\)
0.586745 + 0.809772i \(0.300409\pi\)
\(410\) 8.99953 0.444455
\(411\) 8.99521 0.443701
\(412\) −14.3938 −0.709130
\(413\) 7.19487 0.354037
\(414\) 1.99534 0.0980658
\(415\) −0.747575 −0.0366970
\(416\) −2.21432 −0.108566
\(417\) 7.62072 0.373188
\(418\) 0 0
\(419\) 26.3585 1.28770 0.643848 0.765154i \(-0.277337\pi\)
0.643848 + 0.765154i \(0.277337\pi\)
\(420\) 1.82437 0.0890201
\(421\) −37.5160 −1.82842 −0.914209 0.405243i \(-0.867187\pi\)
−0.914209 + 0.405243i \(0.867187\pi\)
\(422\) −16.5609 −0.806172
\(423\) 1.38820 0.0674968
\(424\) −9.33444 −0.453320
\(425\) −7.96638 −0.386426
\(426\) −0.0369197 −0.00178877
\(427\) −12.6895 −0.614090
\(428\) −4.27627 −0.206701
\(429\) 0 0
\(430\) −2.76947 −0.133556
\(431\) −6.43804 −0.310110 −0.155055 0.987906i \(-0.549555\pi\)
−0.155055 + 0.987906i \(0.549555\pi\)
\(432\) 4.87412 0.234506
\(433\) 14.6478 0.703930 0.351965 0.936013i \(-0.385514\pi\)
0.351965 + 0.936013i \(0.385514\pi\)
\(434\) 9.33892 0.448283
\(435\) 9.24220 0.443130
\(436\) −5.04265 −0.241499
\(437\) −5.56476 −0.266199
\(438\) −21.6119 −1.03266
\(439\) −31.4583 −1.50143 −0.750713 0.660629i \(-0.770290\pi\)
−0.750713 + 0.660629i \(0.770290\pi\)
\(440\) 0 0
\(441\) 0.328323 0.0156344
\(442\) 17.6401 0.839054
\(443\) −31.0007 −1.47289 −0.736444 0.676499i \(-0.763496\pi\)
−0.736444 + 0.676499i \(0.763496\pi\)
\(444\) −7.89837 −0.374840
\(445\) 14.2639 0.676172
\(446\) −19.1235 −0.905523
\(447\) 24.5898 1.16306
\(448\) 1.00000 0.0472456
\(449\) 17.5963 0.830419 0.415209 0.909726i \(-0.363708\pi\)
0.415209 + 0.909726i \(0.363708\pi\)
\(450\) 0.328323 0.0154773
\(451\) 0 0
\(452\) 13.8169 0.649894
\(453\) −32.1274 −1.50948
\(454\) 19.6989 0.924517
\(455\) 2.21432 0.103809
\(456\) 1.67049 0.0782278
\(457\) 34.3749 1.60799 0.803996 0.594635i \(-0.202703\pi\)
0.803996 + 0.594635i \(0.202703\pi\)
\(458\) 27.5813 1.28879
\(459\) −38.8291 −1.81239
\(460\) −6.07737 −0.283359
\(461\) −10.0757 −0.469271 −0.234635 0.972083i \(-0.575390\pi\)
−0.234635 + 0.972083i \(0.575390\pi\)
\(462\) 0 0
\(463\) −30.0439 −1.39626 −0.698128 0.715973i \(-0.745983\pi\)
−0.698128 + 0.715973i \(0.745983\pi\)
\(464\) 5.06597 0.235182
\(465\) 17.0376 0.790102
\(466\) −12.9463 −0.599726
\(467\) −19.7362 −0.913280 −0.456640 0.889652i \(-0.650947\pi\)
−0.456640 + 0.889652i \(0.650947\pi\)
\(468\) −0.727012 −0.0336062
\(469\) −1.29018 −0.0595749
\(470\) −4.22816 −0.195030
\(471\) −20.9503 −0.965340
\(472\) 7.19487 0.331171
\(473\) 0 0
\(474\) 17.1349 0.787034
\(475\) −0.915653 −0.0420130
\(476\) −7.96638 −0.365138
\(477\) −3.06471 −0.140324
\(478\) −5.77626 −0.264200
\(479\) 5.39968 0.246718 0.123359 0.992362i \(-0.460633\pi\)
0.123359 + 0.992362i \(0.460633\pi\)
\(480\) 1.82437 0.0832707
\(481\) −9.58660 −0.437111
\(482\) −25.0323 −1.14019
\(483\) −11.0874 −0.504493
\(484\) 0 0
\(485\) 7.27431 0.330309
\(486\) 3.39724 0.154102
\(487\) −12.6656 −0.573933 −0.286967 0.957941i \(-0.592647\pi\)
−0.286967 + 0.957941i \(0.592647\pi\)
\(488\) −12.6895 −0.574428
\(489\) 0.254044 0.0114883
\(490\) −1.00000 −0.0451754
\(491\) −28.7791 −1.29878 −0.649392 0.760454i \(-0.724977\pi\)
−0.649392 + 0.760454i \(0.724977\pi\)
\(492\) 16.4185 0.740202
\(493\) −40.3575 −1.81761
\(494\) 2.02755 0.0912236
\(495\) 0 0
\(496\) 9.33892 0.419330
\(497\) 0.0202370 0.000907752 0
\(498\) −1.36385 −0.0611157
\(499\) −19.3858 −0.867826 −0.433913 0.900955i \(-0.642867\pi\)
−0.433913 + 0.900955i \(0.642867\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −12.8503 −0.574107
\(502\) −6.66594 −0.297516
\(503\) 1.14878 0.0512218 0.0256109 0.999672i \(-0.491847\pi\)
0.0256109 + 0.999672i \(0.491847\pi\)
\(504\) 0.328323 0.0146247
\(505\) 5.35283 0.238198
\(506\) 0 0
\(507\) 14.7715 0.656027
\(508\) 14.6820 0.651407
\(509\) −1.32413 −0.0586909 −0.0293454 0.999569i \(-0.509342\pi\)
−0.0293454 + 0.999569i \(0.509342\pi\)
\(510\) −14.5336 −0.643559
\(511\) 11.8463 0.524047
\(512\) 1.00000 0.0441942
\(513\) −4.46301 −0.197047
\(514\) −22.7153 −1.00193
\(515\) 14.3938 0.634265
\(516\) −5.05253 −0.222425
\(517\) 0 0
\(518\) 4.32937 0.190222
\(519\) 25.6220 1.12468
\(520\) 2.21432 0.0971043
\(521\) −27.2341 −1.19315 −0.596573 0.802559i \(-0.703471\pi\)
−0.596573 + 0.802559i \(0.703471\pi\)
\(522\) 1.66328 0.0727997
\(523\) 23.8982 1.04499 0.522497 0.852641i \(-0.325001\pi\)
0.522497 + 0.852641i \(0.325001\pi\)
\(524\) −7.62582 −0.333135
\(525\) −1.82437 −0.0796220
\(526\) 14.5015 0.632295
\(527\) −74.3974 −3.24080
\(528\) 0 0
\(529\) 13.9345 0.605847
\(530\) 9.33444 0.405462
\(531\) 2.36225 0.102513
\(532\) −0.915653 −0.0396986
\(533\) 19.9278 0.863170
\(534\) 26.0225 1.12611
\(535\) 4.27627 0.184879
\(536\) −1.29018 −0.0557272
\(537\) −23.4138 −1.01038
\(538\) −21.2753 −0.917244
\(539\) 0 0
\(540\) −4.87412 −0.209749
\(541\) −33.8027 −1.45329 −0.726645 0.687013i \(-0.758921\pi\)
−0.726645 + 0.687013i \(0.758921\pi\)
\(542\) 8.06611 0.346469
\(543\) −35.1867 −1.51001
\(544\) −7.96638 −0.341556
\(545\) 5.04265 0.216003
\(546\) 4.03973 0.172885
\(547\) −23.6009 −1.00910 −0.504550 0.863382i \(-0.668342\pi\)
−0.504550 + 0.863382i \(0.668342\pi\)
\(548\) −4.93059 −0.210624
\(549\) −4.16627 −0.177812
\(550\) 0 0
\(551\) −4.63867 −0.197614
\(552\) −11.0874 −0.471910
\(553\) −9.39226 −0.399399
\(554\) 17.4563 0.741646
\(555\) 7.89837 0.335267
\(556\) −4.17718 −0.177152
\(557\) 6.01368 0.254808 0.127404 0.991851i \(-0.459336\pi\)
0.127404 + 0.991851i \(0.459336\pi\)
\(558\) 3.06619 0.129802
\(559\) −6.13248 −0.259376
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −20.4688 −0.863423
\(563\) 6.87116 0.289585 0.144793 0.989462i \(-0.453749\pi\)
0.144793 + 0.989462i \(0.453749\pi\)
\(564\) −7.71373 −0.324806
\(565\) −13.8169 −0.581283
\(566\) 5.27282 0.221633
\(567\) −9.87717 −0.414802
\(568\) 0.0202370 0.000849124 0
\(569\) 18.9976 0.796420 0.398210 0.917294i \(-0.369632\pi\)
0.398210 + 0.917294i \(0.369632\pi\)
\(570\) −1.67049 −0.0699691
\(571\) 0.393382 0.0164625 0.00823127 0.999966i \(-0.497380\pi\)
0.00823127 + 0.999966i \(0.497380\pi\)
\(572\) 0 0
\(573\) 21.9123 0.915398
\(574\) −8.99953 −0.375633
\(575\) 6.07737 0.253444
\(576\) 0.328323 0.0136801
\(577\) 8.43804 0.351280 0.175640 0.984454i \(-0.443800\pi\)
0.175640 + 0.984454i \(0.443800\pi\)
\(578\) 46.4632 1.93261
\(579\) 31.5613 1.31164
\(580\) −5.06597 −0.210353
\(581\) 0.747575 0.0310146
\(582\) 13.2710 0.550102
\(583\) 0 0
\(584\) 11.8463 0.490201
\(585\) 0.727012 0.0300583
\(586\) 6.40613 0.264635
\(587\) 46.3445 1.91284 0.956422 0.291988i \(-0.0943166\pi\)
0.956422 + 0.291988i \(0.0943166\pi\)
\(588\) −1.82437 −0.0752357
\(589\) −8.55121 −0.352347
\(590\) −7.19487 −0.296208
\(591\) 24.0314 0.988519
\(592\) 4.32937 0.177936
\(593\) −25.9525 −1.06574 −0.532871 0.846197i \(-0.678887\pi\)
−0.532871 + 0.846197i \(0.678887\pi\)
\(594\) 0 0
\(595\) 7.96638 0.326590
\(596\) −13.4785 −0.552102
\(597\) 41.9202 1.71568
\(598\) −13.4572 −0.550307
\(599\) 4.80045 0.196141 0.0980705 0.995179i \(-0.468733\pi\)
0.0980705 + 0.995179i \(0.468733\pi\)
\(600\) −1.82437 −0.0744796
\(601\) −30.5112 −1.24458 −0.622289 0.782787i \(-0.713797\pi\)
−0.622289 + 0.782787i \(0.713797\pi\)
\(602\) 2.76947 0.112875
\(603\) −0.423595 −0.0172501
\(604\) 17.6102 0.716547
\(605\) 0 0
\(606\) 9.76555 0.396698
\(607\) 16.5488 0.671697 0.335848 0.941916i \(-0.390977\pi\)
0.335848 + 0.941916i \(0.390977\pi\)
\(608\) −0.915653 −0.0371346
\(609\) −9.24220 −0.374513
\(610\) 12.6895 0.513784
\(611\) −9.36249 −0.378766
\(612\) −2.61555 −0.105727
\(613\) −42.0921 −1.70008 −0.850041 0.526716i \(-0.823423\pi\)
−0.850041 + 0.526716i \(0.823423\pi\)
\(614\) 5.35970 0.216300
\(615\) −16.4185 −0.662057
\(616\) 0 0
\(617\) 7.32983 0.295088 0.147544 0.989056i \(-0.452863\pi\)
0.147544 + 0.989056i \(0.452863\pi\)
\(618\) 26.2595 1.05631
\(619\) −11.9509 −0.480347 −0.240173 0.970730i \(-0.577204\pi\)
−0.240173 + 0.970730i \(0.577204\pi\)
\(620\) −9.33892 −0.375060
\(621\) 29.6219 1.18869
\(622\) −26.9819 −1.08188
\(623\) −14.2639 −0.571469
\(624\) 4.03973 0.161719
\(625\) 1.00000 0.0400000
\(626\) −28.5633 −1.14162
\(627\) 0 0
\(628\) 11.4836 0.458245
\(629\) −34.4894 −1.37518
\(630\) −0.328323 −0.0130807
\(631\) 43.7266 1.74073 0.870363 0.492410i \(-0.163884\pi\)
0.870363 + 0.492410i \(0.163884\pi\)
\(632\) −9.39226 −0.373604
\(633\) 30.2132 1.20087
\(634\) 21.0077 0.834321
\(635\) −14.6820 −0.582636
\(636\) 17.0295 0.675262
\(637\) −2.21432 −0.0877345
\(638\) 0 0
\(639\) 0.00664427 0.000262843 0
\(640\) −1.00000 −0.0395285
\(641\) 37.6723 1.48796 0.743982 0.668199i \(-0.232935\pi\)
0.743982 + 0.668199i \(0.232935\pi\)
\(642\) 7.80149 0.307900
\(643\) 24.7735 0.976973 0.488486 0.872572i \(-0.337549\pi\)
0.488486 + 0.872572i \(0.337549\pi\)
\(644\) 6.07737 0.239482
\(645\) 5.05253 0.198943
\(646\) 7.29444 0.286996
\(647\) 36.2066 1.42343 0.711714 0.702470i \(-0.247919\pi\)
0.711714 + 0.702470i \(0.247919\pi\)
\(648\) −9.87717 −0.388012
\(649\) 0 0
\(650\) −2.21432 −0.0868527
\(651\) −17.0376 −0.667758
\(652\) −0.139250 −0.00545347
\(653\) −16.7136 −0.654055 −0.327027 0.945015i \(-0.606047\pi\)
−0.327027 + 0.945015i \(0.606047\pi\)
\(654\) 9.19965 0.359735
\(655\) 7.62582 0.297965
\(656\) −8.99953 −0.351373
\(657\) 3.88940 0.151740
\(658\) 4.22816 0.164831
\(659\) 34.0997 1.32834 0.664168 0.747584i \(-0.268786\pi\)
0.664168 + 0.747584i \(0.268786\pi\)
\(660\) 0 0
\(661\) −29.1326 −1.13313 −0.566563 0.824018i \(-0.691727\pi\)
−0.566563 + 0.824018i \(0.691727\pi\)
\(662\) 11.6022 0.450934
\(663\) −32.1821 −1.24985
\(664\) 0.747575 0.0290115
\(665\) 0.915653 0.0355075
\(666\) 1.42143 0.0550794
\(667\) 30.7878 1.19211
\(668\) 7.04367 0.272528
\(669\) 34.8883 1.34886
\(670\) 1.29018 0.0498439
\(671\) 0 0
\(672\) −1.82437 −0.0703766
\(673\) 27.1266 1.04565 0.522826 0.852439i \(-0.324878\pi\)
0.522826 + 0.852439i \(0.324878\pi\)
\(674\) 2.34874 0.0904700
\(675\) 4.87412 0.187605
\(676\) −8.09680 −0.311415
\(677\) −45.5790 −1.75174 −0.875872 0.482543i \(-0.839713\pi\)
−0.875872 + 0.482543i \(0.839713\pi\)
\(678\) −25.2072 −0.968076
\(679\) −7.27431 −0.279162
\(680\) 7.96638 0.305497
\(681\) −35.9381 −1.37715
\(682\) 0 0
\(683\) −6.68497 −0.255793 −0.127897 0.991787i \(-0.540823\pi\)
−0.127897 + 0.991787i \(0.540823\pi\)
\(684\) −0.300630 −0.0114949
\(685\) 4.93059 0.188388
\(686\) 1.00000 0.0381802
\(687\) −50.3185 −1.91977
\(688\) 2.76947 0.105585
\(689\) 20.6694 0.787442
\(690\) 11.0874 0.422089
\(691\) 2.37244 0.0902518 0.0451259 0.998981i \(-0.485631\pi\)
0.0451259 + 0.998981i \(0.485631\pi\)
\(692\) −14.0443 −0.533883
\(693\) 0 0
\(694\) −31.4811 −1.19501
\(695\) 4.17718 0.158450
\(696\) −9.24220 −0.350325
\(697\) 71.6937 2.71559
\(698\) 2.60777 0.0987057
\(699\) 23.6189 0.893347
\(700\) 1.00000 0.0377964
\(701\) −30.5560 −1.15408 −0.577042 0.816714i \(-0.695793\pi\)
−0.577042 + 0.816714i \(0.695793\pi\)
\(702\) −10.7929 −0.407350
\(703\) −3.96420 −0.149513
\(704\) 0 0
\(705\) 7.71373 0.290516
\(706\) −0.416007 −0.0156566
\(707\) −5.35283 −0.201314
\(708\) −13.1261 −0.493309
\(709\) 10.5970 0.397980 0.198990 0.980002i \(-0.436234\pi\)
0.198990 + 0.980002i \(0.436234\pi\)
\(710\) −0.0202370 −0.000759480 0
\(711\) −3.08370 −0.115648
\(712\) −14.2639 −0.534561
\(713\) 56.7561 2.12553
\(714\) 14.5336 0.543907
\(715\) 0 0
\(716\) 12.8339 0.479627
\(717\) 10.5380 0.393550
\(718\) −11.6150 −0.433468
\(719\) 11.8939 0.443568 0.221784 0.975096i \(-0.428812\pi\)
0.221784 + 0.975096i \(0.428812\pi\)
\(720\) −0.328323 −0.0122359
\(721\) −14.3938 −0.536052
\(722\) −18.1616 −0.675904
\(723\) 45.6682 1.69842
\(724\) 19.2871 0.716798
\(725\) 5.06597 0.188145
\(726\) 0 0
\(727\) −23.4180 −0.868527 −0.434264 0.900786i \(-0.642991\pi\)
−0.434264 + 0.900786i \(0.642991\pi\)
\(728\) −2.21432 −0.0820681
\(729\) 23.4337 0.867915
\(730\) −11.8463 −0.438450
\(731\) −22.0626 −0.816016
\(732\) 23.1504 0.855664
\(733\) 12.0816 0.446242 0.223121 0.974791i \(-0.428375\pi\)
0.223121 + 0.974791i \(0.428375\pi\)
\(734\) −8.00709 −0.295547
\(735\) 1.82437 0.0672929
\(736\) 6.07737 0.224015
\(737\) 0 0
\(738\) −2.95476 −0.108766
\(739\) 35.0133 1.28798 0.643992 0.765032i \(-0.277277\pi\)
0.643992 + 0.765032i \(0.277277\pi\)
\(740\) −4.32937 −0.159151
\(741\) −3.69899 −0.135886
\(742\) −9.33444 −0.342678
\(743\) −48.5288 −1.78035 −0.890174 0.455620i \(-0.849417\pi\)
−0.890174 + 0.455620i \(0.849417\pi\)
\(744\) −17.0376 −0.624630
\(745\) 13.4785 0.493815
\(746\) −16.6329 −0.608973
\(747\) 0.245446 0.00898041
\(748\) 0 0
\(749\) −4.27627 −0.156251
\(750\) 1.82437 0.0666165
\(751\) −0.0449682 −0.00164091 −0.000820457 1.00000i \(-0.500261\pi\)
−0.000820457 1.00000i \(0.500261\pi\)
\(752\) 4.22816 0.154185
\(753\) 12.1611 0.443177
\(754\) −11.2177 −0.408523
\(755\) −17.6102 −0.640899
\(756\) 4.87412 0.177270
\(757\) 28.8381 1.04814 0.524069 0.851676i \(-0.324413\pi\)
0.524069 + 0.851676i \(0.324413\pi\)
\(758\) −0.0831686 −0.00302082
\(759\) 0 0
\(760\) 0.915653 0.0332142
\(761\) −18.3746 −0.666078 −0.333039 0.942913i \(-0.608074\pi\)
−0.333039 + 0.942913i \(0.608074\pi\)
\(762\) −26.7853 −0.970330
\(763\) −5.04265 −0.182556
\(764\) −12.0109 −0.434538
\(765\) 2.61555 0.0945654
\(766\) 6.28284 0.227008
\(767\) −15.9317 −0.575262
\(768\) −1.82437 −0.0658313
\(769\) 20.2436 0.730003 0.365002 0.931007i \(-0.381068\pi\)
0.365002 + 0.931007i \(0.381068\pi\)
\(770\) 0 0
\(771\) 41.4412 1.49247
\(772\) −17.2998 −0.622635
\(773\) −27.2469 −0.980003 −0.490001 0.871722i \(-0.663004\pi\)
−0.490001 + 0.871722i \(0.663004\pi\)
\(774\) 0.909280 0.0326834
\(775\) 9.33892 0.335464
\(776\) −7.27431 −0.261132
\(777\) −7.89837 −0.283352
\(778\) −3.67862 −0.131885
\(779\) 8.24045 0.295245
\(780\) −4.03973 −0.144646
\(781\) 0 0
\(782\) −48.4147 −1.73131
\(783\) 24.6922 0.882426
\(784\) 1.00000 0.0357143
\(785\) −11.4836 −0.409867
\(786\) 13.9123 0.496235
\(787\) −26.8288 −0.956342 −0.478171 0.878267i \(-0.658700\pi\)
−0.478171 + 0.878267i \(0.658700\pi\)
\(788\) −13.1724 −0.469248
\(789\) −26.4561 −0.941861
\(790\) 9.39226 0.334161
\(791\) 13.8169 0.491273
\(792\) 0 0
\(793\) 28.0987 0.997813
\(794\) 7.12152 0.252733
\(795\) −17.0295 −0.603973
\(796\) −22.9779 −0.814431
\(797\) 36.4937 1.29267 0.646337 0.763052i \(-0.276300\pi\)
0.646337 + 0.763052i \(0.276300\pi\)
\(798\) 1.67049 0.0591347
\(799\) −33.6831 −1.19162
\(800\) 1.00000 0.0353553
\(801\) −4.68316 −0.165471
\(802\) 5.70426 0.201424
\(803\) 0 0
\(804\) 2.35376 0.0830107
\(805\) −6.07737 −0.214199
\(806\) −20.6793 −0.728399
\(807\) 38.8141 1.36632
\(808\) −5.35283 −0.188312
\(809\) −8.52614 −0.299763 −0.149882 0.988704i \(-0.547889\pi\)
−0.149882 + 0.988704i \(0.547889\pi\)
\(810\) 9.87717 0.347049
\(811\) 43.5605 1.52962 0.764808 0.644258i \(-0.222834\pi\)
0.764808 + 0.644258i \(0.222834\pi\)
\(812\) 5.06597 0.177781
\(813\) −14.7156 −0.516097
\(814\) 0 0
\(815\) 0.139250 0.00487773
\(816\) 14.5336 0.508778
\(817\) −2.53587 −0.0887188
\(818\) 23.7324 0.829783
\(819\) −0.727012 −0.0254039
\(820\) 8.99953 0.314277
\(821\) −22.0579 −0.769825 −0.384913 0.922953i \(-0.625768\pi\)
−0.384913 + 0.922953i \(0.625768\pi\)
\(822\) 8.99521 0.313744
\(823\) 3.37693 0.117712 0.0588562 0.998266i \(-0.481255\pi\)
0.0588562 + 0.998266i \(0.481255\pi\)
\(824\) −14.3938 −0.501431
\(825\) 0 0
\(826\) 7.19487 0.250342
\(827\) 17.2810 0.600919 0.300460 0.953795i \(-0.402860\pi\)
0.300460 + 0.953795i \(0.402860\pi\)
\(828\) 1.99534 0.0693430
\(829\) −23.8021 −0.826682 −0.413341 0.910576i \(-0.635638\pi\)
−0.413341 + 0.910576i \(0.635638\pi\)
\(830\) −0.747575 −0.0259487
\(831\) −31.8467 −1.10475
\(832\) −2.21432 −0.0767677
\(833\) −7.96638 −0.276019
\(834\) 7.62072 0.263884
\(835\) −7.04367 −0.243756
\(836\) 0 0
\(837\) 45.5191 1.57337
\(838\) 26.3585 0.910538
\(839\) 2.14765 0.0741451 0.0370726 0.999313i \(-0.488197\pi\)
0.0370726 + 0.999313i \(0.488197\pi\)
\(840\) 1.82437 0.0629467
\(841\) −3.33593 −0.115032
\(842\) −37.5160 −1.29289
\(843\) 37.3426 1.28615
\(844\) −16.5609 −0.570050
\(845\) 8.09680 0.278538
\(846\) 1.38820 0.0477274
\(847\) 0 0
\(848\) −9.33444 −0.320546
\(849\) −9.61957 −0.330143
\(850\) −7.96638 −0.273245
\(851\) 26.3112 0.901936
\(852\) −0.0369197 −0.00126485
\(853\) 10.0469 0.343998 0.171999 0.985097i \(-0.444977\pi\)
0.171999 + 0.985097i \(0.444977\pi\)
\(854\) −12.6895 −0.434227
\(855\) 0.300630 0.0102813
\(856\) −4.27627 −0.146160
\(857\) −27.9578 −0.955019 −0.477510 0.878626i \(-0.658460\pi\)
−0.477510 + 0.878626i \(0.658460\pi\)
\(858\) 0 0
\(859\) −20.7449 −0.707805 −0.353903 0.935282i \(-0.615146\pi\)
−0.353903 + 0.935282i \(0.615146\pi\)
\(860\) −2.76947 −0.0944380
\(861\) 16.4185 0.559540
\(862\) −6.43804 −0.219281
\(863\) 2.03460 0.0692587 0.0346293 0.999400i \(-0.488975\pi\)
0.0346293 + 0.999400i \(0.488975\pi\)
\(864\) 4.87412 0.165821
\(865\) 14.0443 0.477520
\(866\) 14.6478 0.497754
\(867\) −84.7661 −2.87881
\(868\) 9.33892 0.316984
\(869\) 0 0
\(870\) 9.24220 0.313340
\(871\) 2.85686 0.0968011
\(872\) −5.04265 −0.170766
\(873\) −2.38833 −0.0808326
\(874\) −5.56476 −0.188231
\(875\) −1.00000 −0.0338062
\(876\) −21.6119 −0.730200
\(877\) 32.2740 1.08982 0.544908 0.838496i \(-0.316565\pi\)
0.544908 + 0.838496i \(0.316565\pi\)
\(878\) −31.4583 −1.06167
\(879\) −11.6872 −0.394198
\(880\) 0 0
\(881\) −3.99069 −0.134450 −0.0672248 0.997738i \(-0.521414\pi\)
−0.0672248 + 0.997738i \(0.521414\pi\)
\(882\) 0.328323 0.0110552
\(883\) 16.2835 0.547982 0.273991 0.961732i \(-0.411656\pi\)
0.273991 + 0.961732i \(0.411656\pi\)
\(884\) 17.6401 0.593301
\(885\) 13.1261 0.441229
\(886\) −31.0007 −1.04149
\(887\) −37.4984 −1.25907 −0.629536 0.776971i \(-0.716755\pi\)
−0.629536 + 0.776971i \(0.716755\pi\)
\(888\) −7.89837 −0.265052
\(889\) 14.6820 0.492417
\(890\) 14.2639 0.478126
\(891\) 0 0
\(892\) −19.1235 −0.640301
\(893\) −3.87153 −0.129556
\(894\) 24.5898 0.822407
\(895\) −12.8339 −0.428991
\(896\) 1.00000 0.0334077
\(897\) 24.5510 0.819733
\(898\) 17.5963 0.587195
\(899\) 47.3107 1.57790
\(900\) 0.328323 0.0109441
\(901\) 74.3617 2.47735
\(902\) 0 0
\(903\) −5.05253 −0.168138
\(904\) 13.8169 0.459544
\(905\) −19.2871 −0.641124
\(906\) −32.1274 −1.06736
\(907\) 4.16415 0.138268 0.0691341 0.997607i \(-0.477976\pi\)
0.0691341 + 0.997607i \(0.477976\pi\)
\(908\) 19.6989 0.653732
\(909\) −1.75746 −0.0582913
\(910\) 2.21432 0.0734039
\(911\) −45.0657 −1.49309 −0.746546 0.665333i \(-0.768289\pi\)
−0.746546 + 0.665333i \(0.768289\pi\)
\(912\) 1.67049 0.0553154
\(913\) 0 0
\(914\) 34.3749 1.13702
\(915\) −23.1504 −0.765329
\(916\) 27.5813 0.911313
\(917\) −7.62582 −0.251827
\(918\) −38.8291 −1.28155
\(919\) 32.0466 1.05712 0.528561 0.848896i \(-0.322732\pi\)
0.528561 + 0.848896i \(0.322732\pi\)
\(920\) −6.07737 −0.200365
\(921\) −9.77808 −0.322199
\(922\) −10.0757 −0.331824
\(923\) −0.0448111 −0.00147497
\(924\) 0 0
\(925\) 4.32937 0.142349
\(926\) −30.0439 −0.987302
\(927\) −4.72581 −0.155216
\(928\) 5.06597 0.166299
\(929\) 51.0738 1.67568 0.837839 0.545918i \(-0.183819\pi\)
0.837839 + 0.545918i \(0.183819\pi\)
\(930\) 17.0376 0.558686
\(931\) −0.915653 −0.0300093
\(932\) −12.9463 −0.424071
\(933\) 49.2250 1.61156
\(934\) −19.7362 −0.645787
\(935\) 0 0
\(936\) −0.727012 −0.0237631
\(937\) −5.41242 −0.176816 −0.0884080 0.996084i \(-0.528178\pi\)
−0.0884080 + 0.996084i \(0.528178\pi\)
\(938\) −1.29018 −0.0421258
\(939\) 52.1099 1.70054
\(940\) −4.22816 −0.137907
\(941\) 4.74009 0.154523 0.0772613 0.997011i \(-0.475382\pi\)
0.0772613 + 0.997011i \(0.475382\pi\)
\(942\) −20.9503 −0.682598
\(943\) −54.6935 −1.78107
\(944\) 7.19487 0.234173
\(945\) −4.87412 −0.158555
\(946\) 0 0
\(947\) 16.4424 0.534306 0.267153 0.963654i \(-0.413917\pi\)
0.267153 + 0.963654i \(0.413917\pi\)
\(948\) 17.1349 0.556517
\(949\) −26.2314 −0.851506
\(950\) −0.915653 −0.0297077
\(951\) −38.3258 −1.24280
\(952\) −7.96638 −0.258192
\(953\) 45.6007 1.47715 0.738575 0.674171i \(-0.235499\pi\)
0.738575 + 0.674171i \(0.235499\pi\)
\(954\) −3.06471 −0.0992238
\(955\) 12.0109 0.388663
\(956\) −5.77626 −0.186818
\(957\) 0 0
\(958\) 5.39968 0.174456
\(959\) −4.93059 −0.159217
\(960\) 1.82437 0.0588813
\(961\) 56.2155 1.81340
\(962\) −9.58660 −0.309084
\(963\) −1.40400 −0.0452432
\(964\) −25.0323 −0.806237
\(965\) 17.2998 0.556902
\(966\) −11.0874 −0.356730
\(967\) 28.9513 0.931012 0.465506 0.885045i \(-0.345872\pi\)
0.465506 + 0.885045i \(0.345872\pi\)
\(968\) 0 0
\(969\) −13.3077 −0.427506
\(970\) 7.27431 0.233564
\(971\) 7.09959 0.227837 0.113918 0.993490i \(-0.463660\pi\)
0.113918 + 0.993490i \(0.463660\pi\)
\(972\) 3.39724 0.108966
\(973\) −4.17718 −0.133914
\(974\) −12.6656 −0.405832
\(975\) 4.03973 0.129375
\(976\) −12.6895 −0.406182
\(977\) −0.0166861 −0.000533837 0 −0.000266918 1.00000i \(-0.500085\pi\)
−0.000266918 1.00000i \(0.500085\pi\)
\(978\) 0.254044 0.00812343
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −1.65562 −0.0528599
\(982\) −28.7791 −0.918379
\(983\) −30.3585 −0.968286 −0.484143 0.874989i \(-0.660869\pi\)
−0.484143 + 0.874989i \(0.660869\pi\)
\(984\) 16.4185 0.523402
\(985\) 13.1724 0.419709
\(986\) −40.3575 −1.28524
\(987\) −7.71373 −0.245531
\(988\) 2.02755 0.0645048
\(989\) 16.8311 0.535197
\(990\) 0 0
\(991\) −55.9611 −1.77766 −0.888831 0.458234i \(-0.848482\pi\)
−0.888831 + 0.458234i \(0.848482\pi\)
\(992\) 9.33892 0.296511
\(993\) −21.1668 −0.671707
\(994\) 0.0202370 0.000641878 0
\(995\) 22.9779 0.728449
\(996\) −1.36385 −0.0432153
\(997\) 46.4336 1.47057 0.735284 0.677759i \(-0.237049\pi\)
0.735284 + 0.677759i \(0.237049\pi\)
\(998\) −19.3858 −0.613645
\(999\) 21.1019 0.667634
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 8470.2.a.da.1.3 6
11.5 even 5 770.2.n.h.421.2 12
11.9 even 5 770.2.n.h.631.2 yes 12
11.10 odd 2 8470.2.a.cu.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.h.421.2 12 11.5 even 5
770.2.n.h.631.2 yes 12 11.9 even 5
8470.2.a.cu.1.3 6 11.10 odd 2
8470.2.a.da.1.3 6 1.1 even 1 trivial