Properties

Label 2312.4.a.k.1.2
Level $2312$
Weight $4$
Character 2312.1
Self dual yes
Analytic conductor $136.412$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2312.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(136.412415933\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 95x^{6} + 756x^{4} - 1780x^{2} + 1152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: no (minimal twist has level 136)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.03229\) of defining polynomial
Character \(\chi\) \(=\) 2312.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.52350 q^{3} -18.2701 q^{5} +13.8757 q^{7} +45.6501 q^{9} +O(q^{10})\) \(q-8.52350 q^{3} -18.2701 q^{5} +13.8757 q^{7} +45.6501 q^{9} -60.8512 q^{11} +61.8450 q^{13} +155.725 q^{15} +40.0349 q^{19} -118.270 q^{21} -4.88830 q^{23} +208.795 q^{25} -158.964 q^{27} -113.141 q^{29} -95.1610 q^{31} +518.665 q^{33} -253.510 q^{35} +273.445 q^{37} -527.136 q^{39} +446.056 q^{41} +274.325 q^{43} -834.029 q^{45} -27.6551 q^{47} -150.464 q^{49} -488.605 q^{53} +1111.76 q^{55} -341.237 q^{57} -266.325 q^{59} -502.818 q^{61} +633.428 q^{63} -1129.91 q^{65} -1008.84 q^{67} +41.6654 q^{69} +724.061 q^{71} -188.092 q^{73} -1779.67 q^{75} -844.355 q^{77} +48.3382 q^{79} +122.376 q^{81} -1384.89 q^{83} +964.357 q^{87} -50.3364 q^{89} +858.144 q^{91} +811.105 q^{93} -731.440 q^{95} -1285.79 q^{97} -2777.86 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 132 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 132 q^{9} + 44 q^{13} - 24 q^{15} - 48 q^{19} + 308 q^{21} + 520 q^{25} + 812 q^{33} - 1064 q^{35} - 8 q^{43} + 312 q^{47} + 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 180 q^{69} - 1660 q^{77} + 3156 q^{81} - 2472 q^{83} + 6664 q^{87} + 68 q^{89} + 4036 q^{93}+O(q^{100}) \) 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
\(3\) −8.52350 −1.64035 −0.820174 0.572114i \(-0.806124\pi\)
−0.820174 + 0.572114i \(0.806124\pi\)
\(4\) 0 0
\(5\) −18.2701 −1.63412 −0.817062 0.576550i \(-0.804399\pi\)
−0.817062 + 0.576550i \(0.804399\pi\)
\(6\) 0 0
\(7\) 13.8757 0.749219 0.374609 0.927183i \(-0.377777\pi\)
0.374609 + 0.927183i \(0.377777\pi\)
\(8\) 0 0
\(9\) 45.6501 1.69074
\(10\) 0 0
\(11\) −60.8512 −1.66794 −0.833969 0.551811i \(-0.813937\pi\)
−0.833969 + 0.551811i \(0.813937\pi\)
\(12\) 0 0
\(13\) 61.8450 1.31944 0.659720 0.751512i \(-0.270675\pi\)
0.659720 + 0.751512i \(0.270675\pi\)
\(14\) 0 0
\(15\) 155.725 2.68053
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) 40.0349 0.483402 0.241701 0.970351i \(-0.422295\pi\)
0.241701 + 0.970351i \(0.422295\pi\)
\(20\) 0 0
\(21\) −118.270 −1.22898
\(22\) 0 0
\(23\) −4.88830 −0.0443165 −0.0221583 0.999754i \(-0.507054\pi\)
−0.0221583 + 0.999754i \(0.507054\pi\)
\(24\) 0 0
\(25\) 208.795 1.67036
\(26\) 0 0
\(27\) −158.964 −1.13306
\(28\) 0 0
\(29\) −113.141 −0.724474 −0.362237 0.932086i \(-0.617987\pi\)
−0.362237 + 0.932086i \(0.617987\pi\)
\(30\) 0 0
\(31\) −95.1610 −0.551336 −0.275668 0.961253i \(-0.588899\pi\)
−0.275668 + 0.961253i \(0.588899\pi\)
\(32\) 0 0
\(33\) 518.665 2.73600
\(34\) 0 0
\(35\) −253.510 −1.22432
\(36\) 0 0
\(37\) 273.445 1.21497 0.607487 0.794330i \(-0.292178\pi\)
0.607487 + 0.794330i \(0.292178\pi\)
\(38\) 0 0
\(39\) −527.136 −2.16434
\(40\) 0 0
\(41\) 446.056 1.69908 0.849540 0.527525i \(-0.176880\pi\)
0.849540 + 0.527525i \(0.176880\pi\)
\(42\) 0 0
\(43\) 274.325 0.972888 0.486444 0.873712i \(-0.338294\pi\)
0.486444 + 0.873712i \(0.338294\pi\)
\(44\) 0 0
\(45\) −834.029 −2.76288
\(46\) 0 0
\(47\) −27.6551 −0.0858280 −0.0429140 0.999079i \(-0.513664\pi\)
−0.0429140 + 0.999079i \(0.513664\pi\)
\(48\) 0 0
\(49\) −150.464 −0.438671
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −488.605 −1.26632 −0.633161 0.774020i \(-0.718243\pi\)
−0.633161 + 0.774020i \(0.718243\pi\)
\(54\) 0 0
\(55\) 1111.76 2.72562
\(56\) 0 0
\(57\) −341.237 −0.792947
\(58\) 0 0
\(59\) −266.325 −0.587671 −0.293835 0.955856i \(-0.594932\pi\)
−0.293835 + 0.955856i \(0.594932\pi\)
\(60\) 0 0
\(61\) −502.818 −1.05540 −0.527699 0.849432i \(-0.676945\pi\)
−0.527699 + 0.849432i \(0.676945\pi\)
\(62\) 0 0
\(63\) 633.428 1.26674
\(64\) 0 0
\(65\) −1129.91 −2.15613
\(66\) 0 0
\(67\) −1008.84 −1.83955 −0.919776 0.392443i \(-0.871630\pi\)
−0.919776 + 0.392443i \(0.871630\pi\)
\(68\) 0 0
\(69\) 41.6654 0.0726946
\(70\) 0 0
\(71\) 724.061 1.21029 0.605143 0.796117i \(-0.293116\pi\)
0.605143 + 0.796117i \(0.293116\pi\)
\(72\) 0 0
\(73\) −188.092 −0.301569 −0.150785 0.988567i \(-0.548180\pi\)
−0.150785 + 0.988567i \(0.548180\pi\)
\(74\) 0 0
\(75\) −1779.67 −2.73997
\(76\) 0 0
\(77\) −844.355 −1.24965
\(78\) 0 0
\(79\) 48.3382 0.0688415 0.0344207 0.999407i \(-0.489041\pi\)
0.0344207 + 0.999407i \(0.489041\pi\)
\(80\) 0 0
\(81\) 122.376 0.167868
\(82\) 0 0
\(83\) −1384.89 −1.83146 −0.915729 0.401797i \(-0.868386\pi\)
−0.915729 + 0.401797i \(0.868386\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 964.357 1.18839
\(88\) 0 0
\(89\) −50.3364 −0.0599511 −0.0299756 0.999551i \(-0.509543\pi\)
−0.0299756 + 0.999551i \(0.509543\pi\)
\(90\) 0 0
\(91\) 858.144 0.988549
\(92\) 0 0
\(93\) 811.105 0.904384
\(94\) 0 0
\(95\) −731.440 −0.789938
\(96\) 0 0
\(97\) −1285.79 −1.34589 −0.672947 0.739691i \(-0.734972\pi\)
−0.672947 + 0.739691i \(0.734972\pi\)
\(98\) 0 0
\(99\) −2777.86 −2.82005
\(100\) 0 0
\(101\) −932.025 −0.918217 −0.459109 0.888380i \(-0.651831\pi\)
−0.459109 + 0.888380i \(0.651831\pi\)
\(102\) 0 0
\(103\) −283.119 −0.270840 −0.135420 0.990788i \(-0.543238\pi\)
−0.135420 + 0.990788i \(0.543238\pi\)
\(104\) 0 0
\(105\) 2160.80 2.00831
\(106\) 0 0
\(107\) −273.739 −0.247321 −0.123661 0.992325i \(-0.539463\pi\)
−0.123661 + 0.992325i \(0.539463\pi\)
\(108\) 0 0
\(109\) 834.584 0.733382 0.366691 0.930343i \(-0.380491\pi\)
0.366691 + 0.930343i \(0.380491\pi\)
\(110\) 0 0
\(111\) −2330.71 −1.99298
\(112\) 0 0
\(113\) −193.136 −0.160785 −0.0803927 0.996763i \(-0.525617\pi\)
−0.0803927 + 0.996763i \(0.525617\pi\)
\(114\) 0 0
\(115\) 89.3095 0.0724187
\(116\) 0 0
\(117\) 2823.23 2.23083
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2371.87 1.78202
\(122\) 0 0
\(123\) −3801.96 −2.78708
\(124\) 0 0
\(125\) −1530.94 −1.09545
\(126\) 0 0
\(127\) −1114.62 −0.778793 −0.389396 0.921070i \(-0.627316\pi\)
−0.389396 + 0.921070i \(0.627316\pi\)
\(128\) 0 0
\(129\) −2338.21 −1.59588
\(130\) 0 0
\(131\) 673.762 0.449365 0.224683 0.974432i \(-0.427865\pi\)
0.224683 + 0.974432i \(0.427865\pi\)
\(132\) 0 0
\(133\) 555.513 0.362174
\(134\) 0 0
\(135\) 2904.28 1.85156
\(136\) 0 0
\(137\) 1213.64 0.756849 0.378425 0.925632i \(-0.376466\pi\)
0.378425 + 0.925632i \(0.376466\pi\)
\(138\) 0 0
\(139\) 818.573 0.499500 0.249750 0.968310i \(-0.419652\pi\)
0.249750 + 0.968310i \(0.419652\pi\)
\(140\) 0 0
\(141\) 235.718 0.140788
\(142\) 0 0
\(143\) −3763.34 −2.20074
\(144\) 0 0
\(145\) 2067.09 1.18388
\(146\) 0 0
\(147\) 1282.48 0.719573
\(148\) 0 0
\(149\) 2397.27 1.31807 0.659033 0.752114i \(-0.270966\pi\)
0.659033 + 0.752114i \(0.270966\pi\)
\(150\) 0 0
\(151\) 1197.87 0.645573 0.322787 0.946472i \(-0.395380\pi\)
0.322787 + 0.946472i \(0.395380\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1738.60 0.900952
\(156\) 0 0
\(157\) 1926.67 0.979393 0.489696 0.871893i \(-0.337108\pi\)
0.489696 + 0.871893i \(0.337108\pi\)
\(158\) 0 0
\(159\) 4164.63 2.07721
\(160\) 0 0
\(161\) −67.8287 −0.0332028
\(162\) 0 0
\(163\) −1307.62 −0.628347 −0.314174 0.949366i \(-0.601727\pi\)
−0.314174 + 0.949366i \(0.601727\pi\)
\(164\) 0 0
\(165\) −9476.04 −4.47096
\(166\) 0 0
\(167\) −2548.82 −1.18104 −0.590520 0.807023i \(-0.701077\pi\)
−0.590520 + 0.807023i \(0.701077\pi\)
\(168\) 0 0
\(169\) 1627.80 0.740921
\(170\) 0 0
\(171\) 1827.59 0.817308
\(172\) 0 0
\(173\) −661.688 −0.290793 −0.145397 0.989373i \(-0.546446\pi\)
−0.145397 + 0.989373i \(0.546446\pi\)
\(174\) 0 0
\(175\) 2897.18 1.25147
\(176\) 0 0
\(177\) 2270.02 0.963985
\(178\) 0 0
\(179\) −2927.34 −1.22235 −0.611173 0.791497i \(-0.709302\pi\)
−0.611173 + 0.791497i \(0.709302\pi\)
\(180\) 0 0
\(181\) −209.349 −0.0859711 −0.0429856 0.999076i \(-0.513687\pi\)
−0.0429856 + 0.999076i \(0.513687\pi\)
\(182\) 0 0
\(183\) 4285.77 1.73122
\(184\) 0 0
\(185\) −4995.85 −1.98542
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −2205.74 −0.848909
\(190\) 0 0
\(191\) 4259.39 1.61361 0.806803 0.590820i \(-0.201196\pi\)
0.806803 + 0.590820i \(0.201196\pi\)
\(192\) 0 0
\(193\) 72.8650 0.0271758 0.0135879 0.999908i \(-0.495675\pi\)
0.0135879 + 0.999908i \(0.495675\pi\)
\(194\) 0 0
\(195\) 9630.80 3.53680
\(196\) 0 0
\(197\) −4200.16 −1.51903 −0.759516 0.650489i \(-0.774564\pi\)
−0.759516 + 0.650489i \(0.774564\pi\)
\(198\) 0 0
\(199\) −2013.74 −0.717337 −0.358669 0.933465i \(-0.616769\pi\)
−0.358669 + 0.933465i \(0.616769\pi\)
\(200\) 0 0
\(201\) 8598.89 3.01751
\(202\) 0 0
\(203\) −1569.91 −0.542790
\(204\) 0 0
\(205\) −8149.47 −2.77651
\(206\) 0 0
\(207\) −223.151 −0.0749279
\(208\) 0 0
\(209\) −2436.17 −0.806284
\(210\) 0 0
\(211\) −1431.34 −0.467001 −0.233500 0.972357i \(-0.575018\pi\)
−0.233500 + 0.972357i \(0.575018\pi\)
\(212\) 0 0
\(213\) −6171.54 −1.98529
\(214\) 0 0
\(215\) −5011.94 −1.58982
\(216\) 0 0
\(217\) −1320.43 −0.413072
\(218\) 0 0
\(219\) 1603.21 0.494679
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1220.63 −0.366544 −0.183272 0.983062i \(-0.558669\pi\)
−0.183272 + 0.983062i \(0.558669\pi\)
\(224\) 0 0
\(225\) 9531.51 2.82415
\(226\) 0 0
\(227\) 4338.39 1.26850 0.634250 0.773128i \(-0.281309\pi\)
0.634250 + 0.773128i \(0.281309\pi\)
\(228\) 0 0
\(229\) 1887.81 0.544760 0.272380 0.962190i \(-0.412189\pi\)
0.272380 + 0.962190i \(0.412189\pi\)
\(230\) 0 0
\(231\) 7196.86 2.04986
\(232\) 0 0
\(233\) 4991.46 1.40344 0.701720 0.712453i \(-0.252416\pi\)
0.701720 + 0.712453i \(0.252416\pi\)
\(234\) 0 0
\(235\) 505.261 0.140254
\(236\) 0 0
\(237\) −412.011 −0.112924
\(238\) 0 0
\(239\) −2220.06 −0.600853 −0.300427 0.953805i \(-0.597129\pi\)
−0.300427 + 0.953805i \(0.597129\pi\)
\(240\) 0 0
\(241\) 4989.64 1.33366 0.666828 0.745212i \(-0.267652\pi\)
0.666828 + 0.745212i \(0.267652\pi\)
\(242\) 0 0
\(243\) 3248.95 0.857697
\(244\) 0 0
\(245\) 2748.99 0.716843
\(246\) 0 0
\(247\) 2475.96 0.637819
\(248\) 0 0
\(249\) 11804.1 3.00423
\(250\) 0 0
\(251\) −1077.08 −0.270855 −0.135427 0.990787i \(-0.543241\pi\)
−0.135427 + 0.990787i \(0.543241\pi\)
\(252\) 0 0
\(253\) 297.459 0.0739173
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6521.35 1.58284 0.791421 0.611271i \(-0.209342\pi\)
0.791421 + 0.611271i \(0.209342\pi\)
\(258\) 0 0
\(259\) 3794.24 0.910281
\(260\) 0 0
\(261\) −5164.89 −1.22490
\(262\) 0 0
\(263\) −7879.02 −1.84730 −0.923652 0.383231i \(-0.874811\pi\)
−0.923652 + 0.383231i \(0.874811\pi\)
\(264\) 0 0
\(265\) 8926.85 2.06933
\(266\) 0 0
\(267\) 429.043 0.0983408
\(268\) 0 0
\(269\) 2539.91 0.575691 0.287845 0.957677i \(-0.407061\pi\)
0.287845 + 0.957677i \(0.407061\pi\)
\(270\) 0 0
\(271\) −919.402 −0.206087 −0.103044 0.994677i \(-0.532858\pi\)
−0.103044 + 0.994677i \(0.532858\pi\)
\(272\) 0 0
\(273\) −7314.39 −1.62156
\(274\) 0 0
\(275\) −12705.4 −2.78606
\(276\) 0 0
\(277\) 6999.15 1.51819 0.759094 0.650981i \(-0.225642\pi\)
0.759094 + 0.650981i \(0.225642\pi\)
\(278\) 0 0
\(279\) −4344.11 −0.932168
\(280\) 0 0
\(281\) −1310.63 −0.278242 −0.139121 0.990275i \(-0.544428\pi\)
−0.139121 + 0.990275i \(0.544428\pi\)
\(282\) 0 0
\(283\) 2119.86 0.445275 0.222638 0.974901i \(-0.428533\pi\)
0.222638 + 0.974901i \(0.428533\pi\)
\(284\) 0 0
\(285\) 6234.42 1.29577
\(286\) 0 0
\(287\) 6189.35 1.27298
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 10959.4 2.20773
\(292\) 0 0
\(293\) −4192.22 −0.835877 −0.417939 0.908475i \(-0.637247\pi\)
−0.417939 + 0.908475i \(0.637247\pi\)
\(294\) 0 0
\(295\) 4865.78 0.960327
\(296\) 0 0
\(297\) 9673.13 1.88987
\(298\) 0 0
\(299\) −302.317 −0.0584730
\(300\) 0 0
\(301\) 3806.46 0.728906
\(302\) 0 0
\(303\) 7944.11 1.50620
\(304\) 0 0
\(305\) 9186.51 1.72465
\(306\) 0 0
\(307\) 899.329 0.167190 0.0835952 0.996500i \(-0.473360\pi\)
0.0835952 + 0.996500i \(0.473360\pi\)
\(308\) 0 0
\(309\) 2413.17 0.444273
\(310\) 0 0
\(311\) −8053.67 −1.46843 −0.734215 0.678917i \(-0.762450\pi\)
−0.734215 + 0.678917i \(0.762450\pi\)
\(312\) 0 0
\(313\) 3940.63 0.711622 0.355811 0.934558i \(-0.384205\pi\)
0.355811 + 0.934558i \(0.384205\pi\)
\(314\) 0 0
\(315\) −11572.8 −2.07000
\(316\) 0 0
\(317\) −5790.24 −1.02591 −0.512953 0.858416i \(-0.671449\pi\)
−0.512953 + 0.858416i \(0.671449\pi\)
\(318\) 0 0
\(319\) 6884.76 1.20838
\(320\) 0 0
\(321\) 2333.22 0.405693
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 12912.9 2.20394
\(326\) 0 0
\(327\) −7113.57 −1.20300
\(328\) 0 0
\(329\) −383.735 −0.0643039
\(330\) 0 0
\(331\) 6184.41 1.02697 0.513484 0.858099i \(-0.328355\pi\)
0.513484 + 0.858099i \(0.328355\pi\)
\(332\) 0 0
\(333\) 12482.8 2.05421
\(334\) 0 0
\(335\) 18431.7 3.00606
\(336\) 0 0
\(337\) −2895.51 −0.468038 −0.234019 0.972232i \(-0.575188\pi\)
−0.234019 + 0.972232i \(0.575188\pi\)
\(338\) 0 0
\(339\) 1646.20 0.263744
\(340\) 0 0
\(341\) 5790.66 0.919595
\(342\) 0 0
\(343\) −6847.17 −1.07788
\(344\) 0 0
\(345\) −761.230 −0.118792
\(346\) 0 0
\(347\) 1555.79 0.240690 0.120345 0.992732i \(-0.461600\pi\)
0.120345 + 0.992732i \(0.461600\pi\)
\(348\) 0 0
\(349\) −1103.98 −0.169325 −0.0846627 0.996410i \(-0.526981\pi\)
−0.0846627 + 0.996410i \(0.526981\pi\)
\(350\) 0 0
\(351\) −9831.11 −1.49500
\(352\) 0 0
\(353\) 9622.07 1.45080 0.725398 0.688330i \(-0.241656\pi\)
0.725398 + 0.688330i \(0.241656\pi\)
\(354\) 0 0
\(355\) −13228.6 −1.97776
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3095.00 0.455008 0.227504 0.973777i \(-0.426944\pi\)
0.227504 + 0.973777i \(0.426944\pi\)
\(360\) 0 0
\(361\) −5256.21 −0.766323
\(362\) 0 0
\(363\) −20216.6 −2.92313
\(364\) 0 0
\(365\) 3436.46 0.492802
\(366\) 0 0
\(367\) 1805.83 0.256849 0.128425 0.991719i \(-0.459008\pi\)
0.128425 + 0.991719i \(0.459008\pi\)
\(368\) 0 0
\(369\) 20362.5 2.87271
\(370\) 0 0
\(371\) −6779.75 −0.948752
\(372\) 0 0
\(373\) 6265.65 0.869767 0.434884 0.900487i \(-0.356789\pi\)
0.434884 + 0.900487i \(0.356789\pi\)
\(374\) 0 0
\(375\) 13049.0 1.79693
\(376\) 0 0
\(377\) −6997.20 −0.955900
\(378\) 0 0
\(379\) −1997.53 −0.270729 −0.135364 0.990796i \(-0.543221\pi\)
−0.135364 + 0.990796i \(0.543221\pi\)
\(380\) 0 0
\(381\) 9500.48 1.27749
\(382\) 0 0
\(383\) −13879.9 −1.85178 −0.925889 0.377797i \(-0.876682\pi\)
−0.925889 + 0.377797i \(0.876682\pi\)
\(384\) 0 0
\(385\) 15426.4 2.04208
\(386\) 0 0
\(387\) 12523.0 1.64490
\(388\) 0 0
\(389\) 7261.69 0.946483 0.473242 0.880933i \(-0.343084\pi\)
0.473242 + 0.880933i \(0.343084\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −5742.81 −0.737115
\(394\) 0 0
\(395\) −883.143 −0.112496
\(396\) 0 0
\(397\) −7324.04 −0.925902 −0.462951 0.886384i \(-0.653209\pi\)
−0.462951 + 0.886384i \(0.653209\pi\)
\(398\) 0 0
\(399\) −4734.91 −0.594091
\(400\) 0 0
\(401\) 2017.73 0.251274 0.125637 0.992076i \(-0.459903\pi\)
0.125637 + 0.992076i \(0.459903\pi\)
\(402\) 0 0
\(403\) −5885.23 −0.727455
\(404\) 0 0
\(405\) −2235.81 −0.274317
\(406\) 0 0
\(407\) −16639.4 −2.02650
\(408\) 0 0
\(409\) −837.375 −0.101236 −0.0506180 0.998718i \(-0.516119\pi\)
−0.0506180 + 0.998718i \(0.516119\pi\)
\(410\) 0 0
\(411\) −10344.5 −1.24150
\(412\) 0 0
\(413\) −3695.45 −0.440294
\(414\) 0 0
\(415\) 25301.9 2.99283
\(416\) 0 0
\(417\) −6977.11 −0.819353
\(418\) 0 0
\(419\) −6292.34 −0.733653 −0.366827 0.930289i \(-0.619556\pi\)
−0.366827 + 0.930289i \(0.619556\pi\)
\(420\) 0 0
\(421\) −3436.25 −0.397797 −0.198899 0.980020i \(-0.563736\pi\)
−0.198899 + 0.980020i \(0.563736\pi\)
\(422\) 0 0
\(423\) −1262.46 −0.145113
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −6976.96 −0.790724
\(428\) 0 0
\(429\) 32076.8 3.60999
\(430\) 0 0
\(431\) 1198.92 0.133991 0.0669955 0.997753i \(-0.478659\pi\)
0.0669955 + 0.997753i \(0.478659\pi\)
\(432\) 0 0
\(433\) 1921.84 0.213297 0.106649 0.994297i \(-0.465988\pi\)
0.106649 + 0.994297i \(0.465988\pi\)
\(434\) 0 0
\(435\) −17618.9 −1.94198
\(436\) 0 0
\(437\) −195.702 −0.0214227
\(438\) 0 0
\(439\) 17813.5 1.93666 0.968328 0.249681i \(-0.0803258\pi\)
0.968328 + 0.249681i \(0.0803258\pi\)
\(440\) 0 0
\(441\) −6868.70 −0.741680
\(442\) 0 0
\(443\) −10707.1 −1.14833 −0.574165 0.818739i \(-0.694673\pi\)
−0.574165 + 0.818739i \(0.694673\pi\)
\(444\) 0 0
\(445\) 919.650 0.0979676
\(446\) 0 0
\(447\) −20433.1 −2.16209
\(448\) 0 0
\(449\) −9542.98 −1.00303 −0.501516 0.865149i \(-0.667224\pi\)
−0.501516 + 0.865149i \(0.667224\pi\)
\(450\) 0 0
\(451\) −27143.0 −2.83396
\(452\) 0 0
\(453\) −10210.1 −1.05896
\(454\) 0 0
\(455\) −15678.4 −1.61541
\(456\) 0 0
\(457\) 17234.9 1.76415 0.882075 0.471109i \(-0.156146\pi\)
0.882075 + 0.471109i \(0.156146\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −11358.1 −1.14751 −0.573755 0.819027i \(-0.694514\pi\)
−0.573755 + 0.819027i \(0.694514\pi\)
\(462\) 0 0
\(463\) 5407.62 0.542793 0.271397 0.962468i \(-0.412514\pi\)
0.271397 + 0.962468i \(0.412514\pi\)
\(464\) 0 0
\(465\) −14818.9 −1.47787
\(466\) 0 0
\(467\) 5059.61 0.501350 0.250675 0.968071i \(-0.419347\pi\)
0.250675 + 0.968071i \(0.419347\pi\)
\(468\) 0 0
\(469\) −13998.5 −1.37823
\(470\) 0 0
\(471\) −16421.9 −1.60655
\(472\) 0 0
\(473\) −16693.0 −1.62272
\(474\) 0 0
\(475\) 8359.09 0.807455
\(476\) 0 0
\(477\) −22304.8 −2.14102
\(478\) 0 0
\(479\) −2151.09 −0.205190 −0.102595 0.994723i \(-0.532715\pi\)
−0.102595 + 0.994723i \(0.532715\pi\)
\(480\) 0 0
\(481\) 16911.2 1.60308
\(482\) 0 0
\(483\) 578.138 0.0544641
\(484\) 0 0
\(485\) 23491.4 2.19936
\(486\) 0 0
\(487\) −12035.6 −1.11988 −0.559942 0.828532i \(-0.689177\pi\)
−0.559942 + 0.828532i \(0.689177\pi\)
\(488\) 0 0
\(489\) 11145.5 1.03071
\(490\) 0 0
\(491\) −16672.1 −1.53238 −0.766191 0.642613i \(-0.777850\pi\)
−0.766191 + 0.642613i \(0.777850\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 50751.7 4.60832
\(496\) 0 0
\(497\) 10046.9 0.906769
\(498\) 0 0
\(499\) −6097.31 −0.547000 −0.273500 0.961872i \(-0.588181\pi\)
−0.273500 + 0.961872i \(0.588181\pi\)
\(500\) 0 0
\(501\) 21724.9 1.93732
\(502\) 0 0
\(503\) 4307.43 0.381827 0.190913 0.981607i \(-0.438855\pi\)
0.190913 + 0.981607i \(0.438855\pi\)
\(504\) 0 0
\(505\) 17028.2 1.50048
\(506\) 0 0
\(507\) −13874.6 −1.21537
\(508\) 0 0
\(509\) 12433.5 1.08272 0.541362 0.840790i \(-0.317909\pi\)
0.541362 + 0.840790i \(0.317909\pi\)
\(510\) 0 0
\(511\) −2609.92 −0.225941
\(512\) 0 0
\(513\) −6364.09 −0.547722
\(514\) 0 0
\(515\) 5172.60 0.442587
\(516\) 0 0
\(517\) 1682.85 0.143156
\(518\) 0 0
\(519\) 5639.90 0.477002
\(520\) 0 0
\(521\) −628.628 −0.0528612 −0.0264306 0.999651i \(-0.508414\pi\)
−0.0264306 + 0.999651i \(0.508414\pi\)
\(522\) 0 0
\(523\) −10333.5 −0.863963 −0.431981 0.901883i \(-0.642185\pi\)
−0.431981 + 0.901883i \(0.642185\pi\)
\(524\) 0 0
\(525\) −24694.2 −2.05284
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −12143.1 −0.998036
\(530\) 0 0
\(531\) −12157.8 −0.993600
\(532\) 0 0
\(533\) 27586.3 2.24183
\(534\) 0 0
\(535\) 5001.24 0.404154
\(536\) 0 0
\(537\) 24951.2 2.00507
\(538\) 0 0
\(539\) 9155.92 0.731676
\(540\) 0 0
\(541\) −4122.30 −0.327599 −0.163800 0.986494i \(-0.552375\pi\)
−0.163800 + 0.986494i \(0.552375\pi\)
\(542\) 0 0
\(543\) 1784.38 0.141023
\(544\) 0 0
\(545\) −15247.9 −1.19844
\(546\) 0 0
\(547\) 21268.5 1.66248 0.831241 0.555913i \(-0.187631\pi\)
0.831241 + 0.555913i \(0.187631\pi\)
\(548\) 0 0
\(549\) −22953.7 −1.78441
\(550\) 0 0
\(551\) −4529.58 −0.350212
\(552\) 0 0
\(553\) 670.728 0.0515773
\(554\) 0 0
\(555\) 42582.1 3.25678
\(556\) 0 0
\(557\) 8773.96 0.667441 0.333721 0.942672i \(-0.391696\pi\)
0.333721 + 0.942672i \(0.391696\pi\)
\(558\) 0 0
\(559\) 16965.6 1.28367
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5507.46 0.412276 0.206138 0.978523i \(-0.433910\pi\)
0.206138 + 0.978523i \(0.433910\pi\)
\(564\) 0 0
\(565\) 3528.62 0.262743
\(566\) 0 0
\(567\) 1698.05 0.125770
\(568\) 0 0
\(569\) 13023.4 0.959528 0.479764 0.877398i \(-0.340722\pi\)
0.479764 + 0.877398i \(0.340722\pi\)
\(570\) 0 0
\(571\) −13591.0 −0.996089 −0.498044 0.867152i \(-0.665948\pi\)
−0.498044 + 0.867152i \(0.665948\pi\)
\(572\) 0 0
\(573\) −36304.9 −2.64688
\(574\) 0 0
\(575\) −1020.65 −0.0740246
\(576\) 0 0
\(577\) 2171.91 0.156703 0.0783517 0.996926i \(-0.475034\pi\)
0.0783517 + 0.996926i \(0.475034\pi\)
\(578\) 0 0
\(579\) −621.065 −0.0445778
\(580\) 0 0
\(581\) −19216.3 −1.37216
\(582\) 0 0
\(583\) 29732.2 2.11215
\(584\) 0 0
\(585\) −51580.5 −3.64546
\(586\) 0 0
\(587\) 25902.3 1.82130 0.910651 0.413176i \(-0.135581\pi\)
0.910651 + 0.413176i \(0.135581\pi\)
\(588\) 0 0
\(589\) −3809.76 −0.266517
\(590\) 0 0
\(591\) 35800.1 2.49174
\(592\) 0 0
\(593\) 24167.0 1.67356 0.836780 0.547539i \(-0.184435\pi\)
0.836780 + 0.547539i \(0.184435\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 17164.1 1.17668
\(598\) 0 0
\(599\) −14936.3 −1.01883 −0.509415 0.860521i \(-0.670138\pi\)
−0.509415 + 0.860521i \(0.670138\pi\)
\(600\) 0 0
\(601\) 15669.9 1.06354 0.531770 0.846889i \(-0.321527\pi\)
0.531770 + 0.846889i \(0.321527\pi\)
\(602\) 0 0
\(603\) −46053.8 −3.11021
\(604\) 0 0
\(605\) −43334.2 −2.91204
\(606\) 0 0
\(607\) −16641.8 −1.11280 −0.556401 0.830914i \(-0.687818\pi\)
−0.556401 + 0.830914i \(0.687818\pi\)
\(608\) 0 0
\(609\) 13381.2 0.890364
\(610\) 0 0
\(611\) −1710.33 −0.113245
\(612\) 0 0
\(613\) 17114.6 1.12766 0.563828 0.825892i \(-0.309328\pi\)
0.563828 + 0.825892i \(0.309328\pi\)
\(614\) 0 0
\(615\) 69462.0 4.55444
\(616\) 0 0
\(617\) 21652.6 1.41280 0.706402 0.707811i \(-0.250317\pi\)
0.706402 + 0.707811i \(0.250317\pi\)
\(618\) 0 0
\(619\) 17883.9 1.16125 0.580625 0.814171i \(-0.302808\pi\)
0.580625 + 0.814171i \(0.302808\pi\)
\(620\) 0 0
\(621\) 777.062 0.0502132
\(622\) 0 0
\(623\) −698.455 −0.0449165
\(624\) 0 0
\(625\) 1871.02 0.119745
\(626\) 0 0
\(627\) 20764.7 1.32259
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −18222.4 −1.14964 −0.574820 0.818280i \(-0.694928\pi\)
−0.574820 + 0.818280i \(0.694928\pi\)
\(632\) 0 0
\(633\) 12200.0 0.766044
\(634\) 0 0
\(635\) 20364.2 1.27264
\(636\) 0 0
\(637\) −9305.46 −0.578800
\(638\) 0 0
\(639\) 33053.4 2.04628
\(640\) 0 0
\(641\) −17105.1 −1.05399 −0.526996 0.849868i \(-0.676682\pi\)
−0.526996 + 0.849868i \(0.676682\pi\)
\(642\) 0 0
\(643\) −155.386 −0.00953008 −0.00476504 0.999989i \(-0.501517\pi\)
−0.00476504 + 0.999989i \(0.501517\pi\)
\(644\) 0 0
\(645\) 42719.2 2.60786
\(646\) 0 0
\(647\) 17013.9 1.03383 0.516914 0.856037i \(-0.327081\pi\)
0.516914 + 0.856037i \(0.327081\pi\)
\(648\) 0 0
\(649\) 16206.2 0.980199
\(650\) 0 0
\(651\) 11254.7 0.677581
\(652\) 0 0
\(653\) −31766.8 −1.90373 −0.951863 0.306524i \(-0.900834\pi\)
−0.951863 + 0.306524i \(0.900834\pi\)
\(654\) 0 0
\(655\) −12309.7 −0.734318
\(656\) 0 0
\(657\) −8586.43 −0.509876
\(658\) 0 0
\(659\) 5545.66 0.327812 0.163906 0.986476i \(-0.447591\pi\)
0.163906 + 0.986476i \(0.447591\pi\)
\(660\) 0 0
\(661\) −12808.2 −0.753676 −0.376838 0.926279i \(-0.622989\pi\)
−0.376838 + 0.926279i \(0.622989\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −10149.3 −0.591836
\(666\) 0 0
\(667\) 553.067 0.0321062
\(668\) 0 0
\(669\) 10404.0 0.601260
\(670\) 0 0
\(671\) 30597.1 1.76034
\(672\) 0 0
\(673\) 13921.9 0.797399 0.398699 0.917082i \(-0.369462\pi\)
0.398699 + 0.917082i \(0.369462\pi\)
\(674\) 0 0
\(675\) −33190.9 −1.89262
\(676\) 0 0
\(677\) 15570.9 0.883954 0.441977 0.897026i \(-0.354277\pi\)
0.441977 + 0.897026i \(0.354277\pi\)
\(678\) 0 0
\(679\) −17841.2 −1.00837
\(680\) 0 0
\(681\) −36978.3 −2.08078
\(682\) 0 0
\(683\) −6547.76 −0.366827 −0.183414 0.983036i \(-0.558715\pi\)
−0.183414 + 0.983036i \(0.558715\pi\)
\(684\) 0 0
\(685\) −22173.3 −1.23679
\(686\) 0 0
\(687\) −16090.8 −0.893597
\(688\) 0 0
\(689\) −30217.8 −1.67084
\(690\) 0 0
\(691\) 12241.5 0.673936 0.336968 0.941516i \(-0.390599\pi\)
0.336968 + 0.941516i \(0.390599\pi\)
\(692\) 0 0
\(693\) −38544.8 −2.11284
\(694\) 0 0
\(695\) −14955.4 −0.816244
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −42544.7 −2.30213
\(700\) 0 0
\(701\) −20002.0 −1.07770 −0.538849 0.842403i \(-0.681141\pi\)
−0.538849 + 0.842403i \(0.681141\pi\)
\(702\) 0 0
\(703\) 10947.3 0.587320
\(704\) 0 0
\(705\) −4306.59 −0.230065
\(706\) 0 0
\(707\) −12932.5 −0.687946
\(708\) 0 0
\(709\) −781.933 −0.0414191 −0.0207095 0.999786i \(-0.506593\pi\)
−0.0207095 + 0.999786i \(0.506593\pi\)
\(710\) 0 0
\(711\) 2206.64 0.116393
\(712\) 0 0
\(713\) 465.175 0.0244333
\(714\) 0 0
\(715\) 68756.5 3.59629
\(716\) 0 0
\(717\) 18922.7 0.985609
\(718\) 0 0
\(719\) −3489.63 −0.181003 −0.0905017 0.995896i \(-0.528847\pi\)
−0.0905017 + 0.995896i \(0.528847\pi\)
\(720\) 0 0
\(721\) −3928.48 −0.202919
\(722\) 0 0
\(723\) −42529.2 −2.18766
\(724\) 0 0
\(725\) −23623.3 −1.21013
\(726\) 0 0
\(727\) −4894.00 −0.249667 −0.124834 0.992178i \(-0.539840\pi\)
−0.124834 + 0.992178i \(0.539840\pi\)
\(728\) 0 0
\(729\) −30996.6 −1.57479
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 5187.08 0.261377 0.130688 0.991423i \(-0.458281\pi\)
0.130688 + 0.991423i \(0.458281\pi\)
\(734\) 0 0
\(735\) −23431.0 −1.17587
\(736\) 0 0
\(737\) 61389.4 3.06826
\(738\) 0 0
\(739\) −10603.0 −0.527793 −0.263896 0.964551i \(-0.585008\pi\)
−0.263896 + 0.964551i \(0.585008\pi\)
\(740\) 0 0
\(741\) −21103.8 −1.04625
\(742\) 0 0
\(743\) 4784.54 0.236242 0.118121 0.992999i \(-0.462313\pi\)
0.118121 + 0.992999i \(0.462313\pi\)
\(744\) 0 0
\(745\) −43798.2 −2.15388
\(746\) 0 0
\(747\) −63220.1 −3.09652
\(748\) 0 0
\(749\) −3798.33 −0.185298
\(750\) 0 0
\(751\) 6361.82 0.309116 0.154558 0.987984i \(-0.450605\pi\)
0.154558 + 0.987984i \(0.450605\pi\)
\(752\) 0 0
\(753\) 9180.47 0.444296
\(754\) 0 0
\(755\) −21885.2 −1.05495
\(756\) 0 0
\(757\) 28528.8 1.36975 0.684873 0.728662i \(-0.259858\pi\)
0.684873 + 0.728662i \(0.259858\pi\)
\(758\) 0 0
\(759\) −2535.39 −0.121250
\(760\) 0 0
\(761\) −18097.5 −0.862069 −0.431035 0.902335i \(-0.641851\pi\)
−0.431035 + 0.902335i \(0.641851\pi\)
\(762\) 0 0
\(763\) 11580.5 0.549463
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −16470.9 −0.775396
\(768\) 0 0
\(769\) 7189.93 0.337159 0.168580 0.985688i \(-0.446082\pi\)
0.168580 + 0.985688i \(0.446082\pi\)
\(770\) 0 0
\(771\) −55584.7 −2.59641
\(772\) 0 0
\(773\) 5316.50 0.247375 0.123688 0.992321i \(-0.460528\pi\)
0.123688 + 0.992321i \(0.460528\pi\)
\(774\) 0 0
\(775\) −19869.2 −0.920931
\(776\) 0 0
\(777\) −32340.2 −1.49318
\(778\) 0 0
\(779\) 17857.8 0.821338
\(780\) 0 0
\(781\) −44060.0 −2.01868
\(782\) 0 0
\(783\) 17985.3 0.820871
\(784\) 0 0
\(785\) −35200.3 −1.60045
\(786\) 0 0
\(787\) 8498.14 0.384912 0.192456 0.981306i \(-0.438355\pi\)
0.192456 + 0.981306i \(0.438355\pi\)
\(788\) 0 0
\(789\) 67156.8 3.03022
\(790\) 0 0
\(791\) −2679.91 −0.120463
\(792\) 0 0
\(793\) −31096.8 −1.39253
\(794\) 0 0
\(795\) −76088.0 −3.39442
\(796\) 0 0
\(797\) 19009.1 0.844841 0.422420 0.906400i \(-0.361181\pi\)
0.422420 + 0.906400i \(0.361181\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −2297.86 −0.101362
\(802\) 0 0
\(803\) 11445.7 0.502999
\(804\) 0 0
\(805\) 1239.23 0.0542575
\(806\) 0 0
\(807\) −21648.9 −0.944334
\(808\) 0 0
\(809\) −16963.6 −0.737215 −0.368608 0.929585i \(-0.620165\pi\)
−0.368608 + 0.929585i \(0.620165\pi\)
\(810\) 0 0
\(811\) −24943.1 −1.07999 −0.539994 0.841669i \(-0.681574\pi\)
−0.539994 + 0.841669i \(0.681574\pi\)
\(812\) 0 0
\(813\) 7836.52 0.338055
\(814\) 0 0
\(815\) 23890.3 1.02680
\(816\) 0 0
\(817\) 10982.6 0.470296
\(818\) 0 0
\(819\) 39174.3 1.67138
\(820\) 0 0
\(821\) −24317.6 −1.03373 −0.516863 0.856068i \(-0.672900\pi\)
−0.516863 + 0.856068i \(0.672900\pi\)
\(822\) 0 0
\(823\) 4969.93 0.210499 0.105250 0.994446i \(-0.466436\pi\)
0.105250 + 0.994446i \(0.466436\pi\)
\(824\) 0 0
\(825\) 108295. 4.57011
\(826\) 0 0
\(827\) 13711.0 0.576516 0.288258 0.957553i \(-0.406924\pi\)
0.288258 + 0.957553i \(0.406924\pi\)
\(828\) 0 0
\(829\) 35645.2 1.49338 0.746689 0.665173i \(-0.231642\pi\)
0.746689 + 0.665173i \(0.231642\pi\)
\(830\) 0 0
\(831\) −59657.3 −2.49036
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 46567.1 1.92996
\(836\) 0 0
\(837\) 15127.1 0.624696
\(838\) 0 0
\(839\) 39704.8 1.63380 0.816902 0.576777i \(-0.195690\pi\)
0.816902 + 0.576777i \(0.195690\pi\)
\(840\) 0 0
\(841\) −11588.1 −0.475137
\(842\) 0 0
\(843\) 11171.2 0.456413
\(844\) 0 0
\(845\) −29740.1 −1.21076
\(846\) 0 0
\(847\) 32911.4 1.33512
\(848\) 0 0
\(849\) −18068.7 −0.730406
\(850\) 0 0
\(851\) −1336.68 −0.0538434
\(852\) 0 0
\(853\) 4378.77 0.175763 0.0878817 0.996131i \(-0.471990\pi\)
0.0878817 + 0.996131i \(0.471990\pi\)
\(854\) 0 0
\(855\) −33390.3 −1.33558
\(856\) 0 0
\(857\) −14706.1 −0.586173 −0.293086 0.956086i \(-0.594682\pi\)
−0.293086 + 0.956086i \(0.594682\pi\)
\(858\) 0 0
\(859\) 22056.6 0.876092 0.438046 0.898953i \(-0.355671\pi\)
0.438046 + 0.898953i \(0.355671\pi\)
\(860\) 0 0
\(861\) −52755.0 −2.08813
\(862\) 0 0
\(863\) 13451.2 0.530571 0.265286 0.964170i \(-0.414534\pi\)
0.265286 + 0.964170i \(0.414534\pi\)
\(864\) 0 0
\(865\) 12089.1 0.475192
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2941.44 −0.114823
\(870\) 0 0
\(871\) −62392.0 −2.42718
\(872\) 0 0
\(873\) −58696.2 −2.27556
\(874\) 0 0
\(875\) −21242.9 −0.820734
\(876\) 0 0
\(877\) −36821.0 −1.41774 −0.708870 0.705339i \(-0.750795\pi\)
−0.708870 + 0.705339i \(0.750795\pi\)
\(878\) 0 0
\(879\) 35732.4 1.37113
\(880\) 0 0
\(881\) −20929.4 −0.800372 −0.400186 0.916434i \(-0.631055\pi\)
−0.400186 + 0.916434i \(0.631055\pi\)
\(882\) 0 0
\(883\) 14362.9 0.547394 0.273697 0.961816i \(-0.411753\pi\)
0.273697 + 0.961816i \(0.411753\pi\)
\(884\) 0 0
\(885\) −41473.4 −1.57527
\(886\) 0 0
\(887\) −15832.4 −0.599323 −0.299662 0.954046i \(-0.596874\pi\)
−0.299662 + 0.954046i \(0.596874\pi\)
\(888\) 0 0
\(889\) −15466.2 −0.583486
\(890\) 0 0
\(891\) −7446.71 −0.279993
\(892\) 0 0
\(893\) −1107.17 −0.0414894
\(894\) 0 0
\(895\) 53482.7 1.99746
\(896\) 0 0
\(897\) 2576.80 0.0959161
\(898\) 0 0
\(899\) 10766.6 0.399429
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −32444.4 −1.19566
\(904\) 0 0
\(905\) 3824.82 0.140487
\(906\) 0 0
\(907\) −34058.7 −1.24686 −0.623429 0.781880i \(-0.714261\pi\)
−0.623429 + 0.781880i \(0.714261\pi\)
\(908\) 0 0
\(909\) −42547.0 −1.55247
\(910\) 0 0
\(911\) 19317.1 0.702531 0.351265 0.936276i \(-0.385752\pi\)
0.351265 + 0.936276i \(0.385752\pi\)
\(912\) 0 0
\(913\) 84271.9 3.05476
\(914\) 0 0
\(915\) −78301.2 −2.82903
\(916\) 0 0
\(917\) 9348.93 0.336673
\(918\) 0 0
\(919\) −19221.2 −0.689934 −0.344967 0.938615i \(-0.612110\pi\)
−0.344967 + 0.938615i \(0.612110\pi\)
\(920\) 0 0
\(921\) −7665.43 −0.274250
\(922\) 0 0
\(923\) 44779.6 1.59690
\(924\) 0 0
\(925\) 57093.9 2.02945
\(926\) 0 0
\(927\) −12924.4 −0.457921
\(928\) 0 0
\(929\) 34753.9 1.22738 0.613692 0.789546i \(-0.289684\pi\)
0.613692 + 0.789546i \(0.289684\pi\)
\(930\) 0 0
\(931\) −6023.81 −0.212054
\(932\) 0 0
\(933\) 68645.5 2.40874
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −649.069 −0.0226298 −0.0113149 0.999936i \(-0.503602\pi\)
−0.0113149 + 0.999936i \(0.503602\pi\)
\(938\) 0 0
\(939\) −33588.0 −1.16731
\(940\) 0 0
\(941\) 13716.6 0.475183 0.237591 0.971365i \(-0.423642\pi\)
0.237591 + 0.971365i \(0.423642\pi\)
\(942\) 0 0
\(943\) −2180.46 −0.0752973
\(944\) 0 0
\(945\) 40299.0 1.38722
\(946\) 0 0
\(947\) 43131.8 1.48004 0.740019 0.672586i \(-0.234816\pi\)
0.740019 + 0.672586i \(0.234816\pi\)
\(948\) 0 0
\(949\) −11632.6 −0.397903
\(950\) 0 0
\(951\) 49353.1 1.68284
\(952\) 0 0
\(953\) −4974.39 −0.169083 −0.0845415 0.996420i \(-0.526943\pi\)
−0.0845415 + 0.996420i \(0.526943\pi\)
\(954\) 0 0
\(955\) −77819.4 −2.63683
\(956\) 0 0
\(957\) −58682.3 −1.98216
\(958\) 0 0
\(959\) 16840.2 0.567046
\(960\) 0 0
\(961\) −20735.4 −0.696028
\(962\) 0 0
\(963\) −12496.2 −0.418157
\(964\) 0 0
\(965\) −1331.25 −0.0444087
\(966\) 0 0
\(967\) 83.3000 0.00277016 0.00138508 0.999999i \(-0.499559\pi\)
0.00138508 + 0.999999i \(0.499559\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1577.72 0.0521438 0.0260719 0.999660i \(-0.491700\pi\)
0.0260719 + 0.999660i \(0.491700\pi\)
\(972\) 0 0
\(973\) 11358.3 0.374235
\(974\) 0 0
\(975\) −110063. −3.61523
\(976\) 0 0
\(977\) −30161.6 −0.987672 −0.493836 0.869555i \(-0.664406\pi\)
−0.493836 + 0.869555i \(0.664406\pi\)
\(978\) 0 0
\(979\) 3063.03 0.0999948
\(980\) 0 0
\(981\) 38098.8 1.23996
\(982\) 0 0
\(983\) 3072.86 0.0997040 0.0498520 0.998757i \(-0.484125\pi\)
0.0498520 + 0.998757i \(0.484125\pi\)
\(984\) 0 0
\(985\) 76737.3 2.48229
\(986\) 0 0
\(987\) 3270.77 0.105481
\(988\) 0 0
\(989\) −1340.98 −0.0431150
\(990\) 0 0
\(991\) 22582.4 0.723869 0.361934 0.932204i \(-0.382116\pi\)
0.361934 + 0.932204i \(0.382116\pi\)
\(992\) 0 0
\(993\) −52712.8 −1.68458
\(994\) 0 0
\(995\) 36791.1 1.17222
\(996\) 0 0
\(997\) −17226.0 −0.547195 −0.273597 0.961844i \(-0.588214\pi\)
−0.273597 + 0.961844i \(0.588214\pi\)
\(998\) 0 0
\(999\) −43467.8 −1.37664
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 2312.4.a.k.1.2 8
17.4 even 4 136.4.b.b.33.7 yes 8
17.13 even 4 136.4.b.b.33.2 8
17.16 even 2 inner 2312.4.a.k.1.7 8
51.38 odd 4 1224.4.c.e.577.1 8
51.47 odd 4 1224.4.c.e.577.8 8
68.47 odd 4 272.4.b.f.33.7 8
68.55 odd 4 272.4.b.f.33.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.4.b.b.33.2 8 17.13 even 4
136.4.b.b.33.7 yes 8 17.4 even 4
272.4.b.f.33.2 8 68.55 odd 4
272.4.b.f.33.7 8 68.47 odd 4
1224.4.c.e.577.1 8 51.38 odd 4
1224.4.c.e.577.8 8 51.47 odd 4
2312.4.a.k.1.2 8 1.1 even 1 trivial
2312.4.a.k.1.7 8 17.16 even 2 inner