Properties

Label 1200.4.a.bt.1.1
Level $1200$
Weight $4$
Character 1200.1
Self dual yes
Analytic conductor $70.802$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.8022920069\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.70156\) of defining polynomial
Character \(\chi\) \(=\) 1200.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} -16.2094 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -16.2094 q^{7} +9.00000 q^{9} +40.2094 q^{11} -19.7906 q^{13} -83.0156 q^{17} +48.8375 q^{19} -48.6281 q^{21} +1.61250 q^{23} +27.0000 q^{27} -24.5344 q^{29} +12.4187 q^{31} +120.628 q^{33} -325.884 q^{37} -59.3719 q^{39} -242.419 q^{41} +367.350 q^{43} -204.544 q^{47} -80.2562 q^{49} -249.047 q^{51} +61.5281 q^{53} +146.512 q^{57} +112.209 q^{59} +477.350 q^{61} -145.884 q^{63} +558.094 q^{67} +4.83749 q^{69} -558.281 q^{71} -1011.77 q^{73} -651.769 q^{77} -1150.47 q^{79} +81.0000 q^{81} -1157.92 q^{83} -73.6032 q^{87} +96.9751 q^{89} +320.794 q^{91} +37.2562 q^{93} +1152.37 q^{97} +361.884 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 6 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} + 6 q^{7} + 18 q^{9} + 42 q^{11} - 78 q^{13} - 102 q^{17} - 56 q^{19} + 18 q^{21} - 48 q^{23} + 54 q^{27} - 318 q^{29} - 52 q^{31} + 126 q^{33} - 306 q^{37} - 234 q^{39} - 408 q^{41} + 120 q^{43} + 180 q^{47} + 70 q^{49} - 306 q^{51} - 402 q^{53} - 168 q^{57} + 186 q^{59} + 340 q^{61} + 54 q^{63} + 732 q^{67} - 144 q^{69} + 36 q^{71} - 1332 q^{73} - 612 q^{77} - 380 q^{79} + 162 q^{81} - 984 q^{83} - 954 q^{87} + 1116 q^{89} - 972 q^{91} - 156 q^{93} + 768 q^{97} + 378 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −16.2094 −0.875224 −0.437612 0.899164i \(-0.644176\pi\)
−0.437612 + 0.899164i \(0.644176\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 40.2094 1.10214 0.551072 0.834458i \(-0.314219\pi\)
0.551072 + 0.834458i \(0.314219\pi\)
\(12\) 0 0
\(13\) −19.7906 −0.422226 −0.211113 0.977462i \(-0.567709\pi\)
−0.211113 + 0.977462i \(0.567709\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −83.0156 −1.18437 −0.592184 0.805803i \(-0.701734\pi\)
−0.592184 + 0.805803i \(0.701734\pi\)
\(18\) 0 0
\(19\) 48.8375 0.589689 0.294844 0.955545i \(-0.404732\pi\)
0.294844 + 0.955545i \(0.404732\pi\)
\(20\) 0 0
\(21\) −48.6281 −0.505311
\(22\) 0 0
\(23\) 1.61250 0.0146186 0.00730932 0.999973i \(-0.497673\pi\)
0.00730932 + 0.999973i \(0.497673\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −24.5344 −0.157101 −0.0785504 0.996910i \(-0.525029\pi\)
−0.0785504 + 0.996910i \(0.525029\pi\)
\(30\) 0 0
\(31\) 12.4187 0.0719507 0.0359754 0.999353i \(-0.488546\pi\)
0.0359754 + 0.999353i \(0.488546\pi\)
\(32\) 0 0
\(33\) 120.628 0.636323
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −325.884 −1.44797 −0.723987 0.689813i \(-0.757693\pi\)
−0.723987 + 0.689813i \(0.757693\pi\)
\(38\) 0 0
\(39\) −59.3719 −0.243772
\(40\) 0 0
\(41\) −242.419 −0.923401 −0.461701 0.887036i \(-0.652761\pi\)
−0.461701 + 0.887036i \(0.652761\pi\)
\(42\) 0 0
\(43\) 367.350 1.30280 0.651399 0.758735i \(-0.274182\pi\)
0.651399 + 0.758735i \(0.274182\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −204.544 −0.634804 −0.317402 0.948291i \(-0.602810\pi\)
−0.317402 + 0.948291i \(0.602810\pi\)
\(48\) 0 0
\(49\) −80.2562 −0.233983
\(50\) 0 0
\(51\) −249.047 −0.683795
\(52\) 0 0
\(53\) 61.5281 0.159463 0.0797314 0.996816i \(-0.474594\pi\)
0.0797314 + 0.996816i \(0.474594\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 146.512 0.340457
\(58\) 0 0
\(59\) 112.209 0.247600 0.123800 0.992307i \(-0.460492\pi\)
0.123800 + 0.992307i \(0.460492\pi\)
\(60\) 0 0
\(61\) 477.350 1.00194 0.500970 0.865464i \(-0.332977\pi\)
0.500970 + 0.865464i \(0.332977\pi\)
\(62\) 0 0
\(63\) −145.884 −0.291741
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 558.094 1.01764 0.508821 0.860872i \(-0.330082\pi\)
0.508821 + 0.860872i \(0.330082\pi\)
\(68\) 0 0
\(69\) 4.83749 0.00844008
\(70\) 0 0
\(71\) −558.281 −0.933180 −0.466590 0.884474i \(-0.654518\pi\)
−0.466590 + 0.884474i \(0.654518\pi\)
\(72\) 0 0
\(73\) −1011.77 −1.62217 −0.811086 0.584927i \(-0.801123\pi\)
−0.811086 + 0.584927i \(0.801123\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −651.769 −0.964623
\(78\) 0 0
\(79\) −1150.47 −1.63845 −0.819227 0.573470i \(-0.805597\pi\)
−0.819227 + 0.573470i \(0.805597\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −1157.92 −1.53131 −0.765655 0.643251i \(-0.777585\pi\)
−0.765655 + 0.643251i \(0.777585\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −73.6032 −0.0907022
\(88\) 0 0
\(89\) 96.9751 0.115498 0.0577491 0.998331i \(-0.481608\pi\)
0.0577491 + 0.998331i \(0.481608\pi\)
\(90\) 0 0
\(91\) 320.794 0.369542
\(92\) 0 0
\(93\) 37.2562 0.0415408
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1152.37 1.20625 0.603123 0.797648i \(-0.293923\pi\)
0.603123 + 0.797648i \(0.293923\pi\)
\(98\) 0 0
\(99\) 361.884 0.367381
\(100\) 0 0
\(101\) −1156.49 −1.13936 −0.569679 0.821867i \(-0.692932\pi\)
−0.569679 + 0.821867i \(0.692932\pi\)
\(102\) 0 0
\(103\) −1333.70 −1.27585 −0.637927 0.770096i \(-0.720208\pi\)
−0.637927 + 0.770096i \(0.720208\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −798.263 −0.721224 −0.360612 0.932716i \(-0.617432\pi\)
−0.360612 + 0.932716i \(0.617432\pi\)
\(108\) 0 0
\(109\) −985.119 −0.865663 −0.432831 0.901475i \(-0.642485\pi\)
−0.432831 + 0.901475i \(0.642485\pi\)
\(110\) 0 0
\(111\) −977.653 −0.835988
\(112\) 0 0
\(113\) −1888.25 −1.57196 −0.785979 0.618253i \(-0.787841\pi\)
−0.785979 + 0.618253i \(0.787841\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −178.116 −0.140742
\(118\) 0 0
\(119\) 1345.63 1.03659
\(120\) 0 0
\(121\) 285.794 0.214721
\(122\) 0 0
\(123\) −727.256 −0.533126
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −620.859 −0.433798 −0.216899 0.976194i \(-0.569594\pi\)
−0.216899 + 0.976194i \(0.569594\pi\)
\(128\) 0 0
\(129\) 1102.05 0.752171
\(130\) 0 0
\(131\) 2588.35 1.72630 0.863151 0.504947i \(-0.168488\pi\)
0.863151 + 0.504947i \(0.168488\pi\)
\(132\) 0 0
\(133\) −791.625 −0.516110
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1656.29 1.03289 0.516447 0.856319i \(-0.327254\pi\)
0.516447 + 0.856319i \(0.327254\pi\)
\(138\) 0 0
\(139\) 153.256 0.0935182 0.0467591 0.998906i \(-0.485111\pi\)
0.0467591 + 0.998906i \(0.485111\pi\)
\(140\) 0 0
\(141\) −613.631 −0.366504
\(142\) 0 0
\(143\) −795.769 −0.465353
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −240.769 −0.135090
\(148\) 0 0
\(149\) 1483.38 0.815591 0.407795 0.913073i \(-0.366298\pi\)
0.407795 + 0.913073i \(0.366298\pi\)
\(150\) 0 0
\(151\) 394.281 0.212491 0.106246 0.994340i \(-0.466117\pi\)
0.106246 + 0.994340i \(0.466117\pi\)
\(152\) 0 0
\(153\) −747.141 −0.394789
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1727.05 −0.877922 −0.438961 0.898506i \(-0.644653\pi\)
−0.438961 + 0.898506i \(0.644653\pi\)
\(158\) 0 0
\(159\) 184.584 0.0920659
\(160\) 0 0
\(161\) −26.1376 −0.0127946
\(162\) 0 0
\(163\) −2034.28 −0.977529 −0.488764 0.872416i \(-0.662552\pi\)
−0.488764 + 0.872416i \(0.662552\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 192.900 0.0893835 0.0446918 0.999001i \(-0.485769\pi\)
0.0446918 + 0.999001i \(0.485769\pi\)
\(168\) 0 0
\(169\) −1805.33 −0.821726
\(170\) 0 0
\(171\) 439.537 0.196563
\(172\) 0 0
\(173\) 1239.91 0.544905 0.272452 0.962169i \(-0.412165\pi\)
0.272452 + 0.962169i \(0.412165\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 336.628 0.142952
\(178\) 0 0
\(179\) 2636.86 1.10105 0.550525 0.834818i \(-0.314427\pi\)
0.550525 + 0.834818i \(0.314427\pi\)
\(180\) 0 0
\(181\) 3317.58 1.36240 0.681199 0.732099i \(-0.261459\pi\)
0.681199 + 0.732099i \(0.261459\pi\)
\(182\) 0 0
\(183\) 1432.05 0.578471
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3338.01 −1.30534
\(188\) 0 0
\(189\) −437.653 −0.168437
\(190\) 0 0
\(191\) 624.506 0.236585 0.118292 0.992979i \(-0.462258\pi\)
0.118292 + 0.992979i \(0.462258\pi\)
\(192\) 0 0
\(193\) 436.144 0.162665 0.0813324 0.996687i \(-0.474082\pi\)
0.0813324 + 0.996687i \(0.474082\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3355.81 −1.21366 −0.606831 0.794831i \(-0.707560\pi\)
−0.606831 + 0.794831i \(0.707560\pi\)
\(198\) 0 0
\(199\) −3799.77 −1.35356 −0.676780 0.736185i \(-0.736625\pi\)
−0.676780 + 0.736185i \(0.736625\pi\)
\(200\) 0 0
\(201\) 1674.28 0.587536
\(202\) 0 0
\(203\) 397.687 0.137498
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 14.5125 0.00487288
\(208\) 0 0
\(209\) 1963.72 0.649922
\(210\) 0 0
\(211\) −2365.27 −0.771715 −0.385857 0.922558i \(-0.626094\pi\)
−0.385857 + 0.922558i \(0.626094\pi\)
\(212\) 0 0
\(213\) −1674.84 −0.538772
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −201.300 −0.0629730
\(218\) 0 0
\(219\) −3035.31 −0.936562
\(220\) 0 0
\(221\) 1642.93 0.500070
\(222\) 0 0
\(223\) −3328.58 −0.999545 −0.499772 0.866157i \(-0.666583\pi\)
−0.499772 + 0.866157i \(0.666583\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −527.100 −0.154118 −0.0770592 0.997027i \(-0.524553\pi\)
−0.0770592 + 0.997027i \(0.524553\pi\)
\(228\) 0 0
\(229\) 2566.06 0.740479 0.370240 0.928936i \(-0.379276\pi\)
0.370240 + 0.928936i \(0.379276\pi\)
\(230\) 0 0
\(231\) −1955.31 −0.556925
\(232\) 0 0
\(233\) −5534.99 −1.55626 −0.778132 0.628101i \(-0.783832\pi\)
−0.778132 + 0.628101i \(0.783832\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −3451.41 −0.945962
\(238\) 0 0
\(239\) −1010.01 −0.273355 −0.136678 0.990616i \(-0.543642\pi\)
−0.136678 + 0.990616i \(0.543642\pi\)
\(240\) 0 0
\(241\) −4074.29 −1.08900 −0.544498 0.838762i \(-0.683280\pi\)
−0.544498 + 0.838762i \(0.683280\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −966.525 −0.248982
\(248\) 0 0
\(249\) −3473.77 −0.884103
\(250\) 0 0
\(251\) −1773.98 −0.446107 −0.223054 0.974806i \(-0.571602\pi\)
−0.223054 + 0.974806i \(0.571602\pi\)
\(252\) 0 0
\(253\) 64.8375 0.0161119
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −662.784 −0.160869 −0.0804345 0.996760i \(-0.525631\pi\)
−0.0804345 + 0.996760i \(0.525631\pi\)
\(258\) 0 0
\(259\) 5282.38 1.26730
\(260\) 0 0
\(261\) −220.810 −0.0523669
\(262\) 0 0
\(263\) 712.312 0.167008 0.0835039 0.996507i \(-0.473389\pi\)
0.0835039 + 0.996507i \(0.473389\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 290.925 0.0666829
\(268\) 0 0
\(269\) −3136.41 −0.710894 −0.355447 0.934696i \(-0.615671\pi\)
−0.355447 + 0.934696i \(0.615671\pi\)
\(270\) 0 0
\(271\) 2275.69 0.510105 0.255053 0.966927i \(-0.417907\pi\)
0.255053 + 0.966927i \(0.417907\pi\)
\(272\) 0 0
\(273\) 962.381 0.213355
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −5171.00 −1.12164 −0.560821 0.827937i \(-0.689515\pi\)
−0.560821 + 0.827937i \(0.689515\pi\)
\(278\) 0 0
\(279\) 111.769 0.0239836
\(280\) 0 0
\(281\) 2240.14 0.475571 0.237785 0.971318i \(-0.423578\pi\)
0.237785 + 0.971318i \(0.423578\pi\)
\(282\) 0 0
\(283\) 225.244 0.0473123 0.0236561 0.999720i \(-0.492469\pi\)
0.0236561 + 0.999720i \(0.492469\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3929.46 0.808183
\(288\) 0 0
\(289\) 1978.59 0.402726
\(290\) 0 0
\(291\) 3457.12 0.696427
\(292\) 0 0
\(293\) −1139.86 −0.227274 −0.113637 0.993522i \(-0.536250\pi\)
−0.113637 + 0.993522i \(0.536250\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1085.65 0.212108
\(298\) 0 0
\(299\) −31.9123 −0.00617237
\(300\) 0 0
\(301\) −5954.51 −1.14024
\(302\) 0 0
\(303\) −3469.47 −0.657808
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5244.86 0.975049 0.487525 0.873109i \(-0.337900\pi\)
0.487525 + 0.873109i \(0.337900\pi\)
\(308\) 0 0
\(309\) −4001.09 −0.736615
\(310\) 0 0
\(311\) 5188.26 0.945977 0.472989 0.881068i \(-0.343175\pi\)
0.472989 + 0.881068i \(0.343175\pi\)
\(312\) 0 0
\(313\) −486.656 −0.0878832 −0.0439416 0.999034i \(-0.513992\pi\)
−0.0439416 + 0.999034i \(0.513992\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4218.87 0.747493 0.373747 0.927531i \(-0.378073\pi\)
0.373747 + 0.927531i \(0.378073\pi\)
\(318\) 0 0
\(319\) −986.512 −0.173148
\(320\) 0 0
\(321\) −2394.79 −0.416399
\(322\) 0 0
\(323\) −4054.27 −0.698408
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2955.36 −0.499791
\(328\) 0 0
\(329\) 3315.53 0.555595
\(330\) 0 0
\(331\) −7439.94 −1.23546 −0.617728 0.786392i \(-0.711947\pi\)
−0.617728 + 0.786392i \(0.711947\pi\)
\(332\) 0 0
\(333\) −2932.96 −0.482658
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 6555.39 1.05963 0.529815 0.848113i \(-0.322261\pi\)
0.529815 + 0.848113i \(0.322261\pi\)
\(338\) 0 0
\(339\) −5664.74 −0.907571
\(340\) 0 0
\(341\) 499.350 0.0793000
\(342\) 0 0
\(343\) 6860.72 1.08001
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1950.56 0.301763 0.150881 0.988552i \(-0.451789\pi\)
0.150881 + 0.988552i \(0.451789\pi\)
\(348\) 0 0
\(349\) −1426.74 −0.218830 −0.109415 0.993996i \(-0.534898\pi\)
−0.109415 + 0.993996i \(0.534898\pi\)
\(350\) 0 0
\(351\) −534.347 −0.0812573
\(352\) 0 0
\(353\) 7078.96 1.06735 0.533676 0.845689i \(-0.320810\pi\)
0.533676 + 0.845689i \(0.320810\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 4036.89 0.598474
\(358\) 0 0
\(359\) 5409.79 0.795314 0.397657 0.917534i \(-0.369823\pi\)
0.397657 + 0.917534i \(0.369823\pi\)
\(360\) 0 0
\(361\) −4473.90 −0.652267
\(362\) 0 0
\(363\) 857.381 0.123969
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4940.09 0.702645 0.351322 0.936255i \(-0.385732\pi\)
0.351322 + 0.936255i \(0.385732\pi\)
\(368\) 0 0
\(369\) −2181.77 −0.307800
\(370\) 0 0
\(371\) −997.332 −0.139566
\(372\) 0 0
\(373\) −12891.9 −1.78959 −0.894797 0.446473i \(-0.852680\pi\)
−0.894797 + 0.446473i \(0.852680\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 485.551 0.0663320
\(378\) 0 0
\(379\) 9475.15 1.28418 0.642092 0.766627i \(-0.278067\pi\)
0.642092 + 0.766627i \(0.278067\pi\)
\(380\) 0 0
\(381\) −1862.58 −0.250453
\(382\) 0 0
\(383\) −5800.97 −0.773931 −0.386966 0.922094i \(-0.626477\pi\)
−0.386966 + 0.922094i \(0.626477\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3306.15 0.434266
\(388\) 0 0
\(389\) 13779.7 1.79603 0.898016 0.439962i \(-0.145008\pi\)
0.898016 + 0.439962i \(0.145008\pi\)
\(390\) 0 0
\(391\) −133.862 −0.0173138
\(392\) 0 0
\(393\) 7765.06 0.996680
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 2816.46 0.356056 0.178028 0.984025i \(-0.443028\pi\)
0.178028 + 0.984025i \(0.443028\pi\)
\(398\) 0 0
\(399\) −2374.88 −0.297976
\(400\) 0 0
\(401\) 11986.4 1.49270 0.746352 0.665551i \(-0.231804\pi\)
0.746352 + 0.665551i \(0.231804\pi\)
\(402\) 0 0
\(403\) −245.775 −0.0303794
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −13103.6 −1.59588
\(408\) 0 0
\(409\) −3339.07 −0.403683 −0.201841 0.979418i \(-0.564693\pi\)
−0.201841 + 0.979418i \(0.564693\pi\)
\(410\) 0 0
\(411\) 4968.87 0.596342
\(412\) 0 0
\(413\) −1818.84 −0.216706
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 459.769 0.0539927
\(418\) 0 0
\(419\) −1688.52 −0.196873 −0.0984363 0.995143i \(-0.531384\pi\)
−0.0984363 + 0.995143i \(0.531384\pi\)
\(420\) 0 0
\(421\) −2664.27 −0.308429 −0.154214 0.988037i \(-0.549285\pi\)
−0.154214 + 0.988037i \(0.549285\pi\)
\(422\) 0 0
\(423\) −1840.89 −0.211601
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −7737.54 −0.876923
\(428\) 0 0
\(429\) −2387.31 −0.268672
\(430\) 0 0
\(431\) −12266.0 −1.37084 −0.685420 0.728148i \(-0.740381\pi\)
−0.685420 + 0.728148i \(0.740381\pi\)
\(432\) 0 0
\(433\) 15647.3 1.73664 0.868318 0.496008i \(-0.165201\pi\)
0.868318 + 0.496008i \(0.165201\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 78.7503 0.00862045
\(438\) 0 0
\(439\) −16131.0 −1.75373 −0.876867 0.480733i \(-0.840371\pi\)
−0.876867 + 0.480733i \(0.840371\pi\)
\(440\) 0 0
\(441\) −722.306 −0.0779944
\(442\) 0 0
\(443\) 10053.7 1.07825 0.539127 0.842225i \(-0.318754\pi\)
0.539127 + 0.842225i \(0.318754\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4450.13 0.470882
\(448\) 0 0
\(449\) 7477.71 0.785957 0.392979 0.919548i \(-0.371445\pi\)
0.392979 + 0.919548i \(0.371445\pi\)
\(450\) 0 0
\(451\) −9747.51 −1.01772
\(452\) 0 0
\(453\) 1182.84 0.122682
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1363.46 −0.139562 −0.0697812 0.997562i \(-0.522230\pi\)
−0.0697812 + 0.997562i \(0.522230\pi\)
\(458\) 0 0
\(459\) −2241.42 −0.227932
\(460\) 0 0
\(461\) 5276.77 0.533109 0.266555 0.963820i \(-0.414115\pi\)
0.266555 + 0.963820i \(0.414115\pi\)
\(462\) 0 0
\(463\) 5740.02 0.576159 0.288079 0.957607i \(-0.406983\pi\)
0.288079 + 0.957607i \(0.406983\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6233.36 0.617657 0.308828 0.951118i \(-0.400063\pi\)
0.308828 + 0.951118i \(0.400063\pi\)
\(468\) 0 0
\(469\) −9046.35 −0.890664
\(470\) 0 0
\(471\) −5181.16 −0.506869
\(472\) 0 0
\(473\) 14770.9 1.43587
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 553.753 0.0531543
\(478\) 0 0
\(479\) 19688.2 1.87803 0.939013 0.343881i \(-0.111742\pi\)
0.939013 + 0.343881i \(0.111742\pi\)
\(480\) 0 0
\(481\) 6449.46 0.611372
\(482\) 0 0
\(483\) −78.4127 −0.00738696
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −3955.08 −0.368012 −0.184006 0.982925i \(-0.558907\pi\)
−0.184006 + 0.982925i \(0.558907\pi\)
\(488\) 0 0
\(489\) −6102.84 −0.564377
\(490\) 0 0
\(491\) 13893.5 1.27699 0.638497 0.769624i \(-0.279557\pi\)
0.638497 + 0.769624i \(0.279557\pi\)
\(492\) 0 0
\(493\) 2036.74 0.186065
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9049.39 0.816741
\(498\) 0 0
\(499\) −13523.7 −1.21324 −0.606618 0.794993i \(-0.707474\pi\)
−0.606618 + 0.794993i \(0.707474\pi\)
\(500\) 0 0
\(501\) 578.700 0.0516056
\(502\) 0 0
\(503\) 13135.4 1.16437 0.582184 0.813057i \(-0.302198\pi\)
0.582184 + 0.813057i \(0.302198\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −5415.99 −0.474423
\(508\) 0 0
\(509\) −2222.71 −0.193556 −0.0967778 0.995306i \(-0.530854\pi\)
−0.0967778 + 0.995306i \(0.530854\pi\)
\(510\) 0 0
\(511\) 16400.1 1.41976
\(512\) 0 0
\(513\) 1318.61 0.113486
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −8224.57 −0.699645
\(518\) 0 0
\(519\) 3719.73 0.314601
\(520\) 0 0
\(521\) −4916.42 −0.413421 −0.206710 0.978402i \(-0.566276\pi\)
−0.206710 + 0.978402i \(0.566276\pi\)
\(522\) 0 0
\(523\) 17743.4 1.48349 0.741746 0.670681i \(-0.233998\pi\)
0.741746 + 0.670681i \(0.233998\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1030.95 −0.0852161
\(528\) 0 0
\(529\) −12164.4 −0.999786
\(530\) 0 0
\(531\) 1009.88 0.0825334
\(532\) 0 0
\(533\) 4797.62 0.389884
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 7910.58 0.635692
\(538\) 0 0
\(539\) −3227.05 −0.257883
\(540\) 0 0
\(541\) −12671.3 −1.00699 −0.503495 0.863998i \(-0.667953\pi\)
−0.503495 + 0.863998i \(0.667953\pi\)
\(542\) 0 0
\(543\) 9952.74 0.786580
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 5250.90 0.410443 0.205221 0.978716i \(-0.434209\pi\)
0.205221 + 0.978716i \(0.434209\pi\)
\(548\) 0 0
\(549\) 4296.15 0.333980
\(550\) 0 0
\(551\) −1198.20 −0.0926406
\(552\) 0 0
\(553\) 18648.4 1.43401
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −25830.2 −1.96492 −0.982462 0.186465i \(-0.940297\pi\)
−0.982462 + 0.186465i \(0.940297\pi\)
\(558\) 0 0
\(559\) −7270.09 −0.550075
\(560\) 0 0
\(561\) −10014.0 −0.753640
\(562\) 0 0
\(563\) 2021.14 0.151298 0.0756490 0.997135i \(-0.475897\pi\)
0.0756490 + 0.997135i \(0.475897\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1312.96 −0.0972471
\(568\) 0 0
\(569\) 8706.51 0.641469 0.320734 0.947169i \(-0.396070\pi\)
0.320734 + 0.947169i \(0.396070\pi\)
\(570\) 0 0
\(571\) 12194.5 0.893740 0.446870 0.894599i \(-0.352539\pi\)
0.446870 + 0.894599i \(0.352539\pi\)
\(572\) 0 0
\(573\) 1873.52 0.136592
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 15264.0 1.10130 0.550649 0.834737i \(-0.314380\pi\)
0.550649 + 0.834737i \(0.314380\pi\)
\(578\) 0 0
\(579\) 1308.43 0.0939146
\(580\) 0 0
\(581\) 18769.2 1.34024
\(582\) 0 0
\(583\) 2474.01 0.175751
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −8456.89 −0.594639 −0.297319 0.954778i \(-0.596093\pi\)
−0.297319 + 0.954778i \(0.596093\pi\)
\(588\) 0 0
\(589\) 606.500 0.0424285
\(590\) 0 0
\(591\) −10067.4 −0.700708
\(592\) 0 0
\(593\) 1225.23 0.0848467 0.0424234 0.999100i \(-0.486492\pi\)
0.0424234 + 0.999100i \(0.486492\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −11399.3 −0.781478
\(598\) 0 0
\(599\) 16060.0 1.09548 0.547741 0.836648i \(-0.315488\pi\)
0.547741 + 0.836648i \(0.315488\pi\)
\(600\) 0 0
\(601\) 9699.93 0.658350 0.329175 0.944269i \(-0.393229\pi\)
0.329175 + 0.944269i \(0.393229\pi\)
\(602\) 0 0
\(603\) 5022.84 0.339214
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 23661.2 1.58217 0.791087 0.611703i \(-0.209515\pi\)
0.791087 + 0.611703i \(0.209515\pi\)
\(608\) 0 0
\(609\) 1193.06 0.0793847
\(610\) 0 0
\(611\) 4048.05 0.268030
\(612\) 0 0
\(613\) 8085.63 0.532749 0.266375 0.963870i \(-0.414174\pi\)
0.266375 + 0.963870i \(0.414174\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 11035.1 0.720029 0.360014 0.932947i \(-0.382772\pi\)
0.360014 + 0.932947i \(0.382772\pi\)
\(618\) 0 0
\(619\) 16826.3 1.09258 0.546290 0.837596i \(-0.316040\pi\)
0.546290 + 0.837596i \(0.316040\pi\)
\(620\) 0 0
\(621\) 43.5374 0.00281336
\(622\) 0 0
\(623\) −1571.90 −0.101087
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 5891.17 0.375233
\(628\) 0 0
\(629\) 27053.5 1.71493
\(630\) 0 0
\(631\) 3705.91 0.233803 0.116902 0.993144i \(-0.462704\pi\)
0.116902 + 0.993144i \(0.462704\pi\)
\(632\) 0 0
\(633\) −7095.81 −0.445550
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1588.32 0.0987937
\(638\) 0 0
\(639\) −5024.53 −0.311060
\(640\) 0 0
\(641\) −24597.4 −1.51566 −0.757829 0.652453i \(-0.773740\pi\)
−0.757829 + 0.652453i \(0.773740\pi\)
\(642\) 0 0
\(643\) −21479.5 −1.31737 −0.658685 0.752419i \(-0.728887\pi\)
−0.658685 + 0.752419i \(0.728887\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 27119.7 1.64789 0.823946 0.566668i \(-0.191768\pi\)
0.823946 + 0.566668i \(0.191768\pi\)
\(648\) 0 0
\(649\) 4511.87 0.272891
\(650\) 0 0
\(651\) −603.900 −0.0363575
\(652\) 0 0
\(653\) −18476.4 −1.10725 −0.553627 0.832765i \(-0.686757\pi\)
−0.553627 + 0.832765i \(0.686757\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −9105.92 −0.540724
\(658\) 0 0
\(659\) 19273.5 1.13928 0.569641 0.821894i \(-0.307082\pi\)
0.569641 + 0.821894i \(0.307082\pi\)
\(660\) 0 0
\(661\) 25605.3 1.50670 0.753352 0.657618i \(-0.228436\pi\)
0.753352 + 0.657618i \(0.228436\pi\)
\(662\) 0 0
\(663\) 4928.79 0.288716
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −39.5616 −0.00229660
\(668\) 0 0
\(669\) −9985.75 −0.577087
\(670\) 0 0
\(671\) 19193.9 1.10428
\(672\) 0 0
\(673\) −7855.52 −0.449938 −0.224969 0.974366i \(-0.572228\pi\)
−0.224969 + 0.974366i \(0.572228\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6763.09 0.383939 0.191970 0.981401i \(-0.438512\pi\)
0.191970 + 0.981401i \(0.438512\pi\)
\(678\) 0 0
\(679\) −18679.3 −1.05574
\(680\) 0 0
\(681\) −1581.30 −0.0889803
\(682\) 0 0
\(683\) 15608.6 0.874447 0.437224 0.899353i \(-0.355962\pi\)
0.437224 + 0.899353i \(0.355962\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 7698.17 0.427516
\(688\) 0 0
\(689\) −1217.68 −0.0673293
\(690\) 0 0
\(691\) −6203.15 −0.341504 −0.170752 0.985314i \(-0.554620\pi\)
−0.170752 + 0.985314i \(0.554620\pi\)
\(692\) 0 0
\(693\) −5865.92 −0.321541
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 20124.5 1.09365
\(698\) 0 0
\(699\) −16605.0 −0.898509
\(700\) 0 0
\(701\) −16507.9 −0.889435 −0.444718 0.895671i \(-0.646696\pi\)
−0.444718 + 0.895671i \(0.646696\pi\)
\(702\) 0 0
\(703\) −15915.4 −0.853854
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18746.0 0.997193
\(708\) 0 0
\(709\) −25539.6 −1.35283 −0.676416 0.736520i \(-0.736468\pi\)
−0.676416 + 0.736520i \(0.736468\pi\)
\(710\) 0 0
\(711\) −10354.2 −0.546151
\(712\) 0 0
\(713\) 20.0252 0.00105182
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −3030.02 −0.157822
\(718\) 0 0
\(719\) 7353.45 0.381415 0.190708 0.981647i \(-0.438922\pi\)
0.190708 + 0.981647i \(0.438922\pi\)
\(720\) 0 0
\(721\) 21618.4 1.11666
\(722\) 0 0
\(723\) −12222.9 −0.628732
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −21696.5 −1.10685 −0.553424 0.832900i \(-0.686679\pi\)
−0.553424 + 0.832900i \(0.686679\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −30495.8 −1.54299
\(732\) 0 0
\(733\) −90.2714 −0.00454877 −0.00227439 0.999997i \(-0.500724\pi\)
−0.00227439 + 0.999997i \(0.500724\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 22440.6 1.12159
\(738\) 0 0
\(739\) −14273.1 −0.710479 −0.355239 0.934775i \(-0.615601\pi\)
−0.355239 + 0.934775i \(0.615601\pi\)
\(740\) 0 0
\(741\) −2899.57 −0.143750
\(742\) 0 0
\(743\) 15866.6 0.783429 0.391715 0.920087i \(-0.371882\pi\)
0.391715 + 0.920087i \(0.371882\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −10421.3 −0.510437
\(748\) 0 0
\(749\) 12939.3 0.631232
\(750\) 0 0
\(751\) −26776.9 −1.30107 −0.650534 0.759477i \(-0.725455\pi\)
−0.650534 + 0.759477i \(0.725455\pi\)
\(752\) 0 0
\(753\) −5321.95 −0.257560
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −30478.0 −1.46333 −0.731666 0.681663i \(-0.761257\pi\)
−0.731666 + 0.681663i \(0.761257\pi\)
\(758\) 0 0
\(759\) 194.512 0.00930218
\(760\) 0 0
\(761\) 29104.7 1.38639 0.693195 0.720750i \(-0.256202\pi\)
0.693195 + 0.720750i \(0.256202\pi\)
\(762\) 0 0
\(763\) 15968.2 0.757649
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2220.69 −0.104543
\(768\) 0 0
\(769\) 4170.65 0.195575 0.0977876 0.995207i \(-0.468823\pi\)
0.0977876 + 0.995207i \(0.468823\pi\)
\(770\) 0 0
\(771\) −1988.35 −0.0928778
\(772\) 0 0
\(773\) 17738.5 0.825367 0.412684 0.910874i \(-0.364591\pi\)
0.412684 + 0.910874i \(0.364591\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 15847.1 0.731677
\(778\) 0 0
\(779\) −11839.1 −0.544519
\(780\) 0 0
\(781\) −22448.1 −1.02850
\(782\) 0 0
\(783\) −662.429 −0.0302341
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −3807.92 −0.172475 −0.0862374 0.996275i \(-0.527484\pi\)
−0.0862374 + 0.996275i \(0.527484\pi\)
\(788\) 0 0
\(789\) 2136.94 0.0964220
\(790\) 0 0
\(791\) 30607.3 1.37582
\(792\) 0 0
\(793\) −9447.06 −0.423045
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −23840.3 −1.05956 −0.529779 0.848136i \(-0.677725\pi\)
−0.529779 + 0.848136i \(0.677725\pi\)
\(798\) 0 0
\(799\) 16980.3 0.751841
\(800\) 0 0
\(801\) 872.775 0.0384994
\(802\) 0 0
\(803\) −40682.6 −1.78787
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −9409.24 −0.410435
\(808\) 0 0
\(809\) −1984.22 −0.0862316 −0.0431158 0.999070i \(-0.513728\pi\)
−0.0431158 + 0.999070i \(0.513728\pi\)
\(810\) 0 0
\(811\) 9713.78 0.420588 0.210294 0.977638i \(-0.432558\pi\)
0.210294 + 0.977638i \(0.432558\pi\)
\(812\) 0 0
\(813\) 6827.08 0.294509
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 17940.5 0.768246
\(818\) 0 0
\(819\) 2887.14 0.123181
\(820\) 0 0
\(821\) −19235.4 −0.817686 −0.408843 0.912605i \(-0.634068\pi\)
−0.408843 + 0.912605i \(0.634068\pi\)
\(822\) 0 0
\(823\) −12717.6 −0.538650 −0.269325 0.963049i \(-0.586801\pi\)
−0.269325 + 0.963049i \(0.586801\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −6744.75 −0.283601 −0.141800 0.989895i \(-0.545289\pi\)
−0.141800 + 0.989895i \(0.545289\pi\)
\(828\) 0 0
\(829\) 3404.22 0.142622 0.0713108 0.997454i \(-0.477282\pi\)
0.0713108 + 0.997454i \(0.477282\pi\)
\(830\) 0 0
\(831\) −15513.0 −0.647581
\(832\) 0 0
\(833\) 6662.52 0.277122
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 335.306 0.0138469
\(838\) 0 0
\(839\) 21361.9 0.879015 0.439508 0.898239i \(-0.355153\pi\)
0.439508 + 0.898239i \(0.355153\pi\)
\(840\) 0 0
\(841\) −23787.1 −0.975319
\(842\) 0 0
\(843\) 6720.41 0.274571
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −4632.54 −0.187929
\(848\) 0 0
\(849\) 675.732 0.0273158
\(850\) 0 0
\(851\) −525.488 −0.0211674
\(852\) 0 0
\(853\) 10728.9 0.430657 0.215328 0.976542i \(-0.430918\pi\)
0.215328 + 0.976542i \(0.430918\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 42895.2 1.70977 0.854885 0.518817i \(-0.173627\pi\)
0.854885 + 0.518817i \(0.173627\pi\)
\(858\) 0 0
\(859\) −35530.5 −1.41127 −0.705637 0.708574i \(-0.749339\pi\)
−0.705637 + 0.708574i \(0.749339\pi\)
\(860\) 0 0
\(861\) 11788.4 0.466605
\(862\) 0 0
\(863\) 5704.35 0.225004 0.112502 0.993652i \(-0.464114\pi\)
0.112502 + 0.993652i \(0.464114\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 5935.78 0.232514
\(868\) 0 0
\(869\) −46259.6 −1.80581
\(870\) 0 0
\(871\) −11045.0 −0.429674
\(872\) 0 0
\(873\) 10371.4 0.402082
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −50249.0 −1.93476 −0.967382 0.253324i \(-0.918476\pi\)
−0.967382 + 0.253324i \(0.918476\pi\)
\(878\) 0 0
\(879\) −3419.58 −0.131217
\(880\) 0 0
\(881\) −26864.5 −1.02734 −0.513672 0.857987i \(-0.671715\pi\)
−0.513672 + 0.857987i \(0.671715\pi\)
\(882\) 0 0
\(883\) 18942.1 0.721918 0.360959 0.932582i \(-0.382449\pi\)
0.360959 + 0.932582i \(0.382449\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −25344.8 −0.959409 −0.479705 0.877430i \(-0.659256\pi\)
−0.479705 + 0.877430i \(0.659256\pi\)
\(888\) 0 0
\(889\) 10063.7 0.379670
\(890\) 0 0
\(891\) 3256.96 0.122460
\(892\) 0 0
\(893\) −9989.40 −0.374337
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −95.7370 −0.00356362
\(898\) 0 0
\(899\) −304.686 −0.0113035
\(900\) 0 0
\(901\) −5107.79 −0.188863
\(902\) 0 0
\(903\) −17863.5 −0.658318
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4800.11 0.175728 0.0878639 0.996132i \(-0.471996\pi\)
0.0878639 + 0.996132i \(0.471996\pi\)
\(908\) 0 0
\(909\) −10408.4 −0.379786
\(910\) 0 0
\(911\) 25731.7 0.935819 0.467909 0.883776i \(-0.345007\pi\)
0.467909 + 0.883776i \(0.345007\pi\)
\(912\) 0 0
\(913\) −46559.4 −1.68772
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −41955.6 −1.51090
\(918\) 0 0
\(919\) 12751.9 0.457722 0.228861 0.973459i \(-0.426500\pi\)
0.228861 + 0.973459i \(0.426500\pi\)
\(920\) 0 0
\(921\) 15734.6 0.562945
\(922\) 0 0
\(923\) 11048.7 0.394012
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −12003.3 −0.425285
\(928\) 0 0
\(929\) 15557.8 0.549444 0.274722 0.961524i \(-0.411414\pi\)
0.274722 + 0.961524i \(0.411414\pi\)
\(930\) 0 0
\(931\) −3919.51 −0.137977
\(932\) 0 0
\(933\) 15564.8 0.546160
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 23858.0 0.831811 0.415905 0.909408i \(-0.363465\pi\)
0.415905 + 0.909408i \(0.363465\pi\)
\(938\) 0 0
\(939\) −1459.97 −0.0507394
\(940\) 0 0
\(941\) 9748.00 0.337700 0.168850 0.985642i \(-0.445995\pi\)
0.168850 + 0.985642i \(0.445995\pi\)
\(942\) 0 0
\(943\) −390.899 −0.0134989
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −51537.0 −1.76845 −0.884227 0.467057i \(-0.845314\pi\)
−0.884227 + 0.467057i \(0.845314\pi\)
\(948\) 0 0
\(949\) 20023.5 0.684923
\(950\) 0 0
\(951\) 12656.6 0.431566
\(952\) 0 0
\(953\) 5631.36 0.191414 0.0957071 0.995410i \(-0.469489\pi\)
0.0957071 + 0.995410i \(0.469489\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −2959.54 −0.0999668
\(958\) 0 0
\(959\) −26847.4 −0.904013
\(960\) 0 0
\(961\) −29636.8 −0.994823
\(962\) 0 0
\(963\) −7184.36 −0.240408
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −43360.9 −1.44198 −0.720989 0.692946i \(-0.756312\pi\)
−0.720989 + 0.692946i \(0.756312\pi\)
\(968\) 0 0
\(969\) −12162.8 −0.403226
\(970\) 0 0
\(971\) −12920.0 −0.427007 −0.213503 0.976942i \(-0.568487\pi\)
−0.213503 + 0.976942i \(0.568487\pi\)
\(972\) 0 0
\(973\) −2484.19 −0.0818493
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 10650.4 0.348759 0.174379 0.984679i \(-0.444208\pi\)
0.174379 + 0.984679i \(0.444208\pi\)
\(978\) 0 0
\(979\) 3899.31 0.127296
\(980\) 0 0
\(981\) −8866.07 −0.288554
\(982\) 0 0
\(983\) −49450.3 −1.60450 −0.802248 0.596991i \(-0.796363\pi\)
−0.802248 + 0.596991i \(0.796363\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 9946.58 0.320773
\(988\) 0 0
\(989\) 592.351 0.0190452
\(990\) 0 0
\(991\) −9410.47 −0.301648 −0.150824 0.988561i \(-0.548193\pi\)
−0.150824 + 0.988561i \(0.548193\pi\)
\(992\) 0 0
\(993\) −22319.8 −0.713291
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −532.117 −0.0169030 −0.00845151 0.999964i \(-0.502690\pi\)
−0.00845151 + 0.999964i \(0.502690\pi\)
\(998\) 0 0
\(999\) −8798.88 −0.278663
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 1200.4.a.bt.1.1 2
4.3 odd 2 75.4.a.c.1.1 2
5.2 odd 4 240.4.f.f.49.2 4
5.3 odd 4 240.4.f.f.49.4 4
5.4 even 2 1200.4.a.bn.1.2 2
12.11 even 2 225.4.a.o.1.2 2
15.2 even 4 720.4.f.j.289.2 4
15.8 even 4 720.4.f.j.289.1 4
20.3 even 4 15.4.b.a.4.4 yes 4
20.7 even 4 15.4.b.a.4.1 4
20.19 odd 2 75.4.a.f.1.2 2
40.3 even 4 960.4.f.q.769.3 4
40.13 odd 4 960.4.f.p.769.1 4
40.27 even 4 960.4.f.q.769.1 4
40.37 odd 4 960.4.f.p.769.3 4
60.23 odd 4 45.4.b.b.19.1 4
60.47 odd 4 45.4.b.b.19.4 4
60.59 even 2 225.4.a.i.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.b.a.4.1 4 20.7 even 4
15.4.b.a.4.4 yes 4 20.3 even 4
45.4.b.b.19.1 4 60.23 odd 4
45.4.b.b.19.4 4 60.47 odd 4
75.4.a.c.1.1 2 4.3 odd 2
75.4.a.f.1.2 2 20.19 odd 2
225.4.a.i.1.1 2 60.59 even 2
225.4.a.o.1.2 2 12.11 even 2
240.4.f.f.49.2 4 5.2 odd 4
240.4.f.f.49.4 4 5.3 odd 4
720.4.f.j.289.1 4 15.8 even 4
720.4.f.j.289.2 4 15.2 even 4
960.4.f.p.769.1 4 40.13 odd 4
960.4.f.p.769.3 4 40.37 odd 4
960.4.f.q.769.1 4 40.27 even 4
960.4.f.q.769.3 4 40.3 even 4
1200.4.a.bn.1.2 2 5.4 even 2
1200.4.a.bt.1.1 2 1.1 even 1 trivial