Properties

Label 2352.4.a.bn.1.2
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 294)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} +3.89949 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +3.89949 q^{5} +9.00000 q^{9} +61.3970 q^{11} -53.6985 q^{13} -11.6985 q^{15} -32.1005 q^{17} -55.7990 q^{19} +94.6030 q^{23} -109.794 q^{25} -27.0000 q^{27} +138.191 q^{29} +132.603 q^{31} -184.191 q^{33} +149.206 q^{37} +161.095 q^{39} -427.497 q^{41} -437.588 q^{43} +35.0955 q^{45} +57.0051 q^{47} +96.3015 q^{51} -263.588 q^{53} +239.417 q^{55} +167.397 q^{57} +451.799 q^{59} -579.307 q^{61} -209.397 q^{65} -309.588 q^{67} -283.809 q^{69} +1058.98 q^{71} -1193.66 q^{73} +329.382 q^{75} -1319.56 q^{79} +81.0000 q^{81} +1190.33 q^{83} -125.176 q^{85} -414.573 q^{87} +233.085 q^{89} -397.809 q^{93} -217.588 q^{95} +1609.44 q^{97} +552.573 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} - 12q^{5} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} - 12q^{5} + 18q^{9} + 4q^{11} - 48q^{13} + 36q^{15} - 84q^{17} - 72q^{19} + 308q^{23} + 18q^{25} - 54q^{27} - 80q^{29} + 384q^{31} - 12q^{33} + 536q^{37} + 144q^{39} - 756q^{41} - 400q^{43} - 108q^{45} + 312q^{47} + 252q^{51} - 52q^{53} + 1152q^{55} + 216q^{57} + 864q^{59} - 1416q^{61} - 300q^{65} - 144q^{67} - 924q^{69} + 1524q^{71} - 744q^{73} - 54q^{75} - 976q^{79} + 162q^{81} - 312q^{83} + 700q^{85} + 240q^{87} - 108q^{89} - 1152q^{93} + 40q^{95} + 744q^{97} + 36q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 3.89949 0.348781 0.174391 0.984677i \(-0.444204\pi\)
0.174391 + 0.984677i \(0.444204\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 61.3970 1.68290 0.841449 0.540336i \(-0.181703\pi\)
0.841449 + 0.540336i \(0.181703\pi\)
\(12\) 0 0
\(13\) −53.6985 −1.14564 −0.572818 0.819682i \(-0.694150\pi\)
−0.572818 + 0.819682i \(0.694150\pi\)
\(14\) 0 0
\(15\) −11.6985 −0.201369
\(16\) 0 0
\(17\) −32.1005 −0.457972 −0.228986 0.973430i \(-0.573541\pi\)
−0.228986 + 0.973430i \(0.573541\pi\)
\(18\) 0 0
\(19\) −55.7990 −0.673746 −0.336873 0.941550i \(-0.609369\pi\)
−0.336873 + 0.941550i \(0.609369\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 94.6030 0.857656 0.428828 0.903386i \(-0.358927\pi\)
0.428828 + 0.903386i \(0.358927\pi\)
\(24\) 0 0
\(25\) −109.794 −0.878352
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 138.191 0.884876 0.442438 0.896799i \(-0.354114\pi\)
0.442438 + 0.896799i \(0.354114\pi\)
\(30\) 0 0
\(31\) 132.603 0.768265 0.384132 0.923278i \(-0.374501\pi\)
0.384132 + 0.923278i \(0.374501\pi\)
\(32\) 0 0
\(33\) −184.191 −0.971622
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 149.206 0.662955 0.331477 0.943463i \(-0.392453\pi\)
0.331477 + 0.943463i \(0.392453\pi\)
\(38\) 0 0
\(39\) 161.095 0.661434
\(40\) 0 0
\(41\) −427.497 −1.62839 −0.814194 0.580593i \(-0.802821\pi\)
−0.814194 + 0.580593i \(0.802821\pi\)
\(42\) 0 0
\(43\) −437.588 −1.55190 −0.775948 0.630797i \(-0.782728\pi\)
−0.775948 + 0.630797i \(0.782728\pi\)
\(44\) 0 0
\(45\) 35.0955 0.116260
\(46\) 0 0
\(47\) 57.0051 0.176916 0.0884579 0.996080i \(-0.471806\pi\)
0.0884579 + 0.996080i \(0.471806\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 96.3015 0.264410
\(52\) 0 0
\(53\) −263.588 −0.683143 −0.341572 0.939856i \(-0.610959\pi\)
−0.341572 + 0.939856i \(0.610959\pi\)
\(54\) 0 0
\(55\) 239.417 0.586964
\(56\) 0 0
\(57\) 167.397 0.388987
\(58\) 0 0
\(59\) 451.799 0.996936 0.498468 0.866908i \(-0.333896\pi\)
0.498468 + 0.866908i \(0.333896\pi\)
\(60\) 0 0
\(61\) −579.307 −1.21594 −0.607972 0.793958i \(-0.708017\pi\)
−0.607972 + 0.793958i \(0.708017\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −209.397 −0.399577
\(66\) 0 0
\(67\) −309.588 −0.564510 −0.282255 0.959339i \(-0.591082\pi\)
−0.282255 + 0.959339i \(0.591082\pi\)
\(68\) 0 0
\(69\) −283.809 −0.495168
\(70\) 0 0
\(71\) 1058.98 1.77012 0.885059 0.465479i \(-0.154118\pi\)
0.885059 + 0.465479i \(0.154118\pi\)
\(72\) 0 0
\(73\) −1193.66 −1.91380 −0.956898 0.290424i \(-0.906204\pi\)
−0.956898 + 0.290424i \(0.906204\pi\)
\(74\) 0 0
\(75\) 329.382 0.507116
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1319.56 −1.87926 −0.939632 0.342187i \(-0.888832\pi\)
−0.939632 + 0.342187i \(0.888832\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1190.33 1.57417 0.787083 0.616847i \(-0.211590\pi\)
0.787083 + 0.616847i \(0.211590\pi\)
\(84\) 0 0
\(85\) −125.176 −0.159732
\(86\) 0 0
\(87\) −414.573 −0.510883
\(88\) 0 0
\(89\) 233.085 0.277607 0.138803 0.990320i \(-0.455674\pi\)
0.138803 + 0.990320i \(0.455674\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −397.809 −0.443558
\(94\) 0 0
\(95\) −217.588 −0.234990
\(96\) 0 0
\(97\) 1609.44 1.68468 0.842338 0.538950i \(-0.181179\pi\)
0.842338 + 0.538950i \(0.181179\pi\)
\(98\) 0 0
\(99\) 552.573 0.560966
\(100\) 0 0
\(101\) 1479.26 1.45734 0.728671 0.684864i \(-0.240139\pi\)
0.728671 + 0.684864i \(0.240139\pi\)
\(102\) 0 0
\(103\) −1145.35 −1.09567 −0.547837 0.836585i \(-0.684548\pi\)
−0.547837 + 0.836585i \(0.684548\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −436.955 −0.394785 −0.197392 0.980325i \(-0.563247\pi\)
−0.197392 + 0.980325i \(0.563247\pi\)
\(108\) 0 0
\(109\) −166.352 −0.146180 −0.0730898 0.997325i \(-0.523286\pi\)
−0.0730898 + 0.997325i \(0.523286\pi\)
\(110\) 0 0
\(111\) −447.618 −0.382757
\(112\) 0 0
\(113\) 490.824 0.408609 0.204305 0.978907i \(-0.434507\pi\)
0.204305 + 0.978907i \(0.434507\pi\)
\(114\) 0 0
\(115\) 368.904 0.299135
\(116\) 0 0
\(117\) −483.286 −0.381879
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2438.59 1.83215
\(122\) 0 0
\(123\) 1282.49 0.940150
\(124\) 0 0
\(125\) −915.578 −0.655134
\(126\) 0 0
\(127\) 2616.70 1.82831 0.914153 0.405369i \(-0.132857\pi\)
0.914153 + 0.405369i \(0.132857\pi\)
\(128\) 0 0
\(129\) 1312.76 0.895988
\(130\) 0 0
\(131\) 177.588 0.118442 0.0592211 0.998245i \(-0.481138\pi\)
0.0592211 + 0.998245i \(0.481138\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −105.286 −0.0671230
\(136\) 0 0
\(137\) 27.0152 0.0168472 0.00842358 0.999965i \(-0.497319\pi\)
0.00842358 + 0.999965i \(0.497319\pi\)
\(138\) 0 0
\(139\) −922.754 −0.563071 −0.281536 0.959551i \(-0.590844\pi\)
−0.281536 + 0.959551i \(0.590844\pi\)
\(140\) 0 0
\(141\) −171.015 −0.102142
\(142\) 0 0
\(143\) −3296.92 −1.92799
\(144\) 0 0
\(145\) 538.875 0.308628
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 746.703 0.410552 0.205276 0.978704i \(-0.434191\pi\)
0.205276 + 0.978704i \(0.434191\pi\)
\(150\) 0 0
\(151\) −2073.15 −1.11729 −0.558643 0.829408i \(-0.688678\pi\)
−0.558643 + 0.829408i \(0.688678\pi\)
\(152\) 0 0
\(153\) −288.905 −0.152657
\(154\) 0 0
\(155\) 517.085 0.267956
\(156\) 0 0
\(157\) −1566.22 −0.796166 −0.398083 0.917349i \(-0.630324\pi\)
−0.398083 + 0.917349i \(0.630324\pi\)
\(158\) 0 0
\(159\) 790.764 0.394413
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −98.7333 −0.0474441 −0.0237221 0.999719i \(-0.507552\pi\)
−0.0237221 + 0.999719i \(0.507552\pi\)
\(164\) 0 0
\(165\) −718.252 −0.338884
\(166\) 0 0
\(167\) −2231.36 −1.03394 −0.516969 0.856004i \(-0.672940\pi\)
−0.516969 + 0.856004i \(0.672940\pi\)
\(168\) 0 0
\(169\) 686.527 0.312484
\(170\) 0 0
\(171\) −502.191 −0.224582
\(172\) 0 0
\(173\) −2100.84 −0.923261 −0.461631 0.887072i \(-0.652735\pi\)
−0.461631 + 0.887072i \(0.652735\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1355.40 −0.575581
\(178\) 0 0
\(179\) −1722.54 −0.719267 −0.359634 0.933094i \(-0.617098\pi\)
−0.359634 + 0.933094i \(0.617098\pi\)
\(180\) 0 0
\(181\) −1655.00 −0.679644 −0.339822 0.940490i \(-0.610367\pi\)
−0.339822 + 0.940490i \(0.610367\pi\)
\(182\) 0 0
\(183\) 1737.92 0.702026
\(184\) 0 0
\(185\) 581.828 0.231226
\(186\) 0 0
\(187\) −1970.87 −0.770720
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1007.69 0.381747 0.190874 0.981615i \(-0.438868\pi\)
0.190874 + 0.981615i \(0.438868\pi\)
\(192\) 0 0
\(193\) −7.64849 −0.00285259 −0.00142630 0.999999i \(-0.500454\pi\)
−0.00142630 + 0.999999i \(0.500454\pi\)
\(194\) 0 0
\(195\) 628.191 0.230696
\(196\) 0 0
\(197\) 2689.88 0.972822 0.486411 0.873730i \(-0.338306\pi\)
0.486411 + 0.873730i \(0.338306\pi\)
\(198\) 0 0
\(199\) 867.497 0.309021 0.154511 0.987991i \(-0.450620\pi\)
0.154511 + 0.987991i \(0.450620\pi\)
\(200\) 0 0
\(201\) 928.764 0.325920
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1667.02 −0.567951
\(206\) 0 0
\(207\) 851.427 0.285885
\(208\) 0 0
\(209\) −3425.89 −1.13385
\(210\) 0 0
\(211\) −162.030 −0.0528655 −0.0264328 0.999651i \(-0.508415\pi\)
−0.0264328 + 0.999651i \(0.508415\pi\)
\(212\) 0 0
\(213\) −3176.95 −1.02198
\(214\) 0 0
\(215\) −1706.37 −0.541272
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 3580.97 1.10493
\(220\) 0 0
\(221\) 1723.75 0.524669
\(222\) 0 0
\(223\) −4577.85 −1.37469 −0.687344 0.726332i \(-0.741223\pi\)
−0.687344 + 0.726332i \(0.741223\pi\)
\(224\) 0 0
\(225\) −988.145 −0.292784
\(226\) 0 0
\(227\) −2218.19 −0.648575 −0.324287 0.945959i \(-0.605125\pi\)
−0.324287 + 0.945959i \(0.605125\pi\)
\(228\) 0 0
\(229\) −785.217 −0.226588 −0.113294 0.993562i \(-0.536140\pi\)
−0.113294 + 0.993562i \(0.536140\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5369.12 −1.50963 −0.754813 0.655940i \(-0.772273\pi\)
−0.754813 + 0.655940i \(0.772273\pi\)
\(234\) 0 0
\(235\) 222.291 0.0617049
\(236\) 0 0
\(237\) 3958.67 1.08499
\(238\) 0 0
\(239\) −3713.28 −1.00499 −0.502493 0.864581i \(-0.667584\pi\)
−0.502493 + 0.864581i \(0.667584\pi\)
\(240\) 0 0
\(241\) −6998.62 −1.87063 −0.935313 0.353821i \(-0.884882\pi\)
−0.935313 + 0.353821i \(0.884882\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2996.32 0.771868
\(248\) 0 0
\(249\) −3570.99 −0.908846
\(250\) 0 0
\(251\) 3722.75 0.936168 0.468084 0.883684i \(-0.344945\pi\)
0.468084 + 0.883684i \(0.344945\pi\)
\(252\) 0 0
\(253\) 5808.34 1.44335
\(254\) 0 0
\(255\) 375.527 0.0922213
\(256\) 0 0
\(257\) −1230.15 −0.298578 −0.149289 0.988794i \(-0.547699\pi\)
−0.149289 + 0.988794i \(0.547699\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1243.72 0.294959
\(262\) 0 0
\(263\) −2388.63 −0.560036 −0.280018 0.959995i \(-0.590340\pi\)
−0.280018 + 0.959995i \(0.590340\pi\)
\(264\) 0 0
\(265\) −1027.86 −0.238268
\(266\) 0 0
\(267\) −699.256 −0.160276
\(268\) 0 0
\(269\) −6702.31 −1.51913 −0.759567 0.650429i \(-0.774589\pi\)
−0.759567 + 0.650429i \(0.774589\pi\)
\(270\) 0 0
\(271\) 4950.37 1.10964 0.554822 0.831969i \(-0.312786\pi\)
0.554822 + 0.831969i \(0.312786\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6741.02 −1.47818
\(276\) 0 0
\(277\) −3705.18 −0.803691 −0.401846 0.915707i \(-0.631631\pi\)
−0.401846 + 0.915707i \(0.631631\pi\)
\(278\) 0 0
\(279\) 1193.43 0.256088
\(280\) 0 0
\(281\) 9324.74 1.97960 0.989800 0.142465i \(-0.0455029\pi\)
0.989800 + 0.142465i \(0.0455029\pi\)
\(282\) 0 0
\(283\) −5569.49 −1.16986 −0.584932 0.811082i \(-0.698879\pi\)
−0.584932 + 0.811082i \(0.698879\pi\)
\(284\) 0 0
\(285\) 652.764 0.135672
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −3882.56 −0.790262
\(290\) 0 0
\(291\) −4828.31 −0.972648
\(292\) 0 0
\(293\) 1665.31 0.332042 0.166021 0.986122i \(-0.446908\pi\)
0.166021 + 0.986122i \(0.446908\pi\)
\(294\) 0 0
\(295\) 1761.79 0.347713
\(296\) 0 0
\(297\) −1657.72 −0.323874
\(298\) 0 0
\(299\) −5080.04 −0.982563
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −4437.77 −0.841396
\(304\) 0 0
\(305\) −2259.00 −0.424099
\(306\) 0 0
\(307\) −5303.32 −0.985916 −0.492958 0.870053i \(-0.664084\pi\)
−0.492958 + 0.870053i \(0.664084\pi\)
\(308\) 0 0
\(309\) 3436.04 0.632587
\(310\) 0 0
\(311\) 1125.99 0.205302 0.102651 0.994717i \(-0.467267\pi\)
0.102651 + 0.994717i \(0.467267\pi\)
\(312\) 0 0
\(313\) 8299.51 1.49877 0.749386 0.662133i \(-0.230349\pi\)
0.749386 + 0.662133i \(0.230349\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4278.76 −0.758105 −0.379053 0.925375i \(-0.623750\pi\)
−0.379053 + 0.925375i \(0.623750\pi\)
\(318\) 0 0
\(319\) 8484.50 1.48916
\(320\) 0 0
\(321\) 1310.86 0.227929
\(322\) 0 0
\(323\) 1791.18 0.308556
\(324\) 0 0
\(325\) 5895.77 1.00627
\(326\) 0 0
\(327\) 499.055 0.0843969
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1707.12 0.283479 0.141739 0.989904i \(-0.454730\pi\)
0.141739 + 0.989904i \(0.454730\pi\)
\(332\) 0 0
\(333\) 1342.85 0.220985
\(334\) 0 0
\(335\) −1207.24 −0.196891
\(336\) 0 0
\(337\) 1710.67 0.276517 0.138259 0.990396i \(-0.455849\pi\)
0.138259 + 0.990396i \(0.455849\pi\)
\(338\) 0 0
\(339\) −1472.47 −0.235911
\(340\) 0 0
\(341\) 8141.42 1.29291
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −1106.71 −0.172705
\(346\) 0 0
\(347\) −8910.30 −1.37847 −0.689236 0.724537i \(-0.742054\pi\)
−0.689236 + 0.724537i \(0.742054\pi\)
\(348\) 0 0
\(349\) −5378.68 −0.824969 −0.412485 0.910965i \(-0.635339\pi\)
−0.412485 + 0.910965i \(0.635339\pi\)
\(350\) 0 0
\(351\) 1449.86 0.220478
\(352\) 0 0
\(353\) 4252.56 0.641193 0.320596 0.947216i \(-0.396117\pi\)
0.320596 + 0.947216i \(0.396117\pi\)
\(354\) 0 0
\(355\) 4129.51 0.617384
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4903.89 −0.720940 −0.360470 0.932771i \(-0.617384\pi\)
−0.360470 + 0.932771i \(0.617384\pi\)
\(360\) 0 0
\(361\) −3745.47 −0.546067
\(362\) 0 0
\(363\) −7315.76 −1.05779
\(364\) 0 0
\(365\) −4654.66 −0.667497
\(366\) 0 0
\(367\) −4041.57 −0.574845 −0.287423 0.957804i \(-0.592798\pi\)
−0.287423 + 0.957804i \(0.592798\pi\)
\(368\) 0 0
\(369\) −3847.48 −0.542796
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −7451.35 −1.03436 −0.517180 0.855877i \(-0.673018\pi\)
−0.517180 + 0.855877i \(0.673018\pi\)
\(374\) 0 0
\(375\) 2746.73 0.378242
\(376\) 0 0
\(377\) −7420.64 −1.01375
\(378\) 0 0
\(379\) 12564.4 1.70288 0.851438 0.524456i \(-0.175731\pi\)
0.851438 + 0.524456i \(0.175731\pi\)
\(380\) 0 0
\(381\) −7850.11 −1.05557
\(382\) 0 0
\(383\) −4289.93 −0.572337 −0.286169 0.958179i \(-0.592382\pi\)
−0.286169 + 0.958179i \(0.592382\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3938.29 −0.517299
\(388\) 0 0
\(389\) 5662.77 0.738082 0.369041 0.929413i \(-0.379686\pi\)
0.369041 + 0.929413i \(0.379686\pi\)
\(390\) 0 0
\(391\) −3036.81 −0.392782
\(392\) 0 0
\(393\) −532.764 −0.0683826
\(394\) 0 0
\(395\) −5145.61 −0.655452
\(396\) 0 0
\(397\) −14561.4 −1.84084 −0.920421 0.390928i \(-0.872154\pi\)
−0.920421 + 0.390928i \(0.872154\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3742.19 −0.466025 −0.233013 0.972474i \(-0.574858\pi\)
−0.233013 + 0.972474i \(0.574858\pi\)
\(402\) 0 0
\(403\) −7120.58 −0.880152
\(404\) 0 0
\(405\) 315.859 0.0387535
\(406\) 0 0
\(407\) 9160.80 1.11569
\(408\) 0 0
\(409\) 3517.17 0.425215 0.212608 0.977138i \(-0.431804\pi\)
0.212608 + 0.977138i \(0.431804\pi\)
\(410\) 0 0
\(411\) −81.0455 −0.00972671
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4641.69 0.549040
\(416\) 0 0
\(417\) 2768.26 0.325089
\(418\) 0 0
\(419\) −7579.52 −0.883732 −0.441866 0.897081i \(-0.645683\pi\)
−0.441866 + 0.897081i \(0.645683\pi\)
\(420\) 0 0
\(421\) −4980.87 −0.576610 −0.288305 0.957539i \(-0.593092\pi\)
−0.288305 + 0.957539i \(0.593092\pi\)
\(422\) 0 0
\(423\) 513.045 0.0589719
\(424\) 0 0
\(425\) 3524.44 0.402260
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 9890.77 1.11313
\(430\) 0 0
\(431\) −14203.3 −1.58736 −0.793678 0.608339i \(-0.791836\pi\)
−0.793678 + 0.608339i \(0.791836\pi\)
\(432\) 0 0
\(433\) −3874.82 −0.430051 −0.215026 0.976608i \(-0.568983\pi\)
−0.215026 + 0.976608i \(0.568983\pi\)
\(434\) 0 0
\(435\) −1616.62 −0.178187
\(436\) 0 0
\(437\) −5278.75 −0.577842
\(438\) 0 0
\(439\) 7763.82 0.844070 0.422035 0.906579i \(-0.361316\pi\)
0.422035 + 0.906579i \(0.361316\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −9662.24 −1.03627 −0.518134 0.855299i \(-0.673373\pi\)
−0.518134 + 0.855299i \(0.673373\pi\)
\(444\) 0 0
\(445\) 908.915 0.0968241
\(446\) 0 0
\(447\) −2240.11 −0.237032
\(448\) 0 0
\(449\) −10942.6 −1.15014 −0.575069 0.818105i \(-0.695025\pi\)
−0.575069 + 0.818105i \(0.695025\pi\)
\(450\) 0 0
\(451\) −26247.0 −2.74041
\(452\) 0 0
\(453\) 6219.44 0.645065
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 13618.3 1.39396 0.696979 0.717091i \(-0.254527\pi\)
0.696979 + 0.717091i \(0.254527\pi\)
\(458\) 0 0
\(459\) 866.714 0.0881367
\(460\) 0 0
\(461\) −11955.8 −1.20789 −0.603947 0.797025i \(-0.706406\pi\)
−0.603947 + 0.797025i \(0.706406\pi\)
\(462\) 0 0
\(463\) −648.503 −0.0650939 −0.0325470 0.999470i \(-0.510362\pi\)
−0.0325470 + 0.999470i \(0.510362\pi\)
\(464\) 0 0
\(465\) −1551.25 −0.154705
\(466\) 0 0
\(467\) 2784.74 0.275937 0.137969 0.990437i \(-0.455943\pi\)
0.137969 + 0.990437i \(0.455943\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 4698.66 0.459667
\(472\) 0 0
\(473\) −26866.6 −2.61168
\(474\) 0 0
\(475\) 6126.39 0.591785
\(476\) 0 0
\(477\) −2372.29 −0.227714
\(478\) 0 0
\(479\) 11113.4 1.06009 0.530046 0.847969i \(-0.322175\pi\)
0.530046 + 0.847969i \(0.322175\pi\)
\(480\) 0 0
\(481\) −8012.14 −0.759505
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6275.99 0.587584
\(486\) 0 0
\(487\) 3786.27 0.352305 0.176152 0.984363i \(-0.443635\pi\)
0.176152 + 0.984363i \(0.443635\pi\)
\(488\) 0 0
\(489\) 296.200 0.0273919
\(490\) 0 0
\(491\) 9582.12 0.880723 0.440361 0.897821i \(-0.354850\pi\)
0.440361 + 0.897821i \(0.354850\pi\)
\(492\) 0 0
\(493\) −4436.00 −0.405248
\(494\) 0 0
\(495\) 2154.75 0.195655
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −5581.55 −0.500730 −0.250365 0.968152i \(-0.580551\pi\)
−0.250365 + 0.968152i \(0.580551\pi\)
\(500\) 0 0
\(501\) 6694.07 0.596944
\(502\) 0 0
\(503\) −14116.3 −1.25132 −0.625661 0.780095i \(-0.715171\pi\)
−0.625661 + 0.780095i \(0.715171\pi\)
\(504\) 0 0
\(505\) 5768.35 0.508294
\(506\) 0 0
\(507\) −2059.58 −0.180413
\(508\) 0 0
\(509\) −16787.6 −1.46188 −0.730941 0.682441i \(-0.760919\pi\)
−0.730941 + 0.682441i \(0.760919\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 1506.57 0.129662
\(514\) 0 0
\(515\) −4466.27 −0.382150
\(516\) 0 0
\(517\) 3499.94 0.297731
\(518\) 0 0
\(519\) 6302.53 0.533045
\(520\) 0 0
\(521\) 5598.61 0.470786 0.235393 0.971900i \(-0.424362\pi\)
0.235393 + 0.971900i \(0.424362\pi\)
\(522\) 0 0
\(523\) −13270.7 −1.10954 −0.554769 0.832004i \(-0.687193\pi\)
−0.554769 + 0.832004i \(0.687193\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4256.62 −0.351843
\(528\) 0 0
\(529\) −3217.27 −0.264426
\(530\) 0 0
\(531\) 4066.19 0.332312
\(532\) 0 0
\(533\) 22956.0 1.86554
\(534\) 0 0
\(535\) −1703.90 −0.137694
\(536\) 0 0
\(537\) 5167.63 0.415269
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −23017.5 −1.82921 −0.914603 0.404353i \(-0.867497\pi\)
−0.914603 + 0.404353i \(0.867497\pi\)
\(542\) 0 0
\(543\) 4965.01 0.392393
\(544\) 0 0
\(545\) −648.687 −0.0509848
\(546\) 0 0
\(547\) −4475.84 −0.349859 −0.174930 0.984581i \(-0.555970\pi\)
−0.174930 + 0.984581i \(0.555970\pi\)
\(548\) 0 0
\(549\) −5213.76 −0.405315
\(550\) 0 0
\(551\) −7710.91 −0.596181
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1745.48 −0.133499
\(556\) 0 0
\(557\) 1541.70 0.117278 0.0586390 0.998279i \(-0.481324\pi\)
0.0586390 + 0.998279i \(0.481324\pi\)
\(558\) 0 0
\(559\) 23497.8 1.77791
\(560\) 0 0
\(561\) 5912.62 0.444975
\(562\) 0 0
\(563\) 13079.7 0.979115 0.489557 0.871971i \(-0.337158\pi\)
0.489557 + 0.871971i \(0.337158\pi\)
\(564\) 0 0
\(565\) 1913.97 0.142515
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 11411.2 0.840740 0.420370 0.907353i \(-0.361900\pi\)
0.420370 + 0.907353i \(0.361900\pi\)
\(570\) 0 0
\(571\) −2311.74 −0.169428 −0.0847139 0.996405i \(-0.526998\pi\)
−0.0847139 + 0.996405i \(0.526998\pi\)
\(572\) 0 0
\(573\) −3023.06 −0.220402
\(574\) 0 0
\(575\) −10386.8 −0.753324
\(576\) 0 0
\(577\) −25097.0 −1.81075 −0.905373 0.424617i \(-0.860409\pi\)
−0.905373 + 0.424617i \(0.860409\pi\)
\(578\) 0 0
\(579\) 22.9455 0.00164694
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −16183.5 −1.14966
\(584\) 0 0
\(585\) −1884.57 −0.133192
\(586\) 0 0
\(587\) 19789.1 1.39145 0.695726 0.718307i \(-0.255083\pi\)
0.695726 + 0.718307i \(0.255083\pi\)
\(588\) 0 0
\(589\) −7399.12 −0.517615
\(590\) 0 0
\(591\) −8069.64 −0.561659
\(592\) 0 0
\(593\) 2774.13 0.192108 0.0960540 0.995376i \(-0.469378\pi\)
0.0960540 + 0.995376i \(0.469378\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2602.49 −0.178414
\(598\) 0 0
\(599\) 26190.3 1.78649 0.893245 0.449570i \(-0.148423\pi\)
0.893245 + 0.449570i \(0.148423\pi\)
\(600\) 0 0
\(601\) 11038.6 0.749211 0.374605 0.927184i \(-0.377778\pi\)
0.374605 + 0.927184i \(0.377778\pi\)
\(602\) 0 0
\(603\) −2786.29 −0.188170
\(604\) 0 0
\(605\) 9509.26 0.639019
\(606\) 0 0
\(607\) −1854.39 −0.123999 −0.0619996 0.998076i \(-0.519748\pi\)
−0.0619996 + 0.998076i \(0.519748\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3061.08 −0.202681
\(612\) 0 0
\(613\) −14856.8 −0.978894 −0.489447 0.872033i \(-0.662801\pi\)
−0.489447 + 0.872033i \(0.662801\pi\)
\(614\) 0 0
\(615\) 5001.07 0.327907
\(616\) 0 0
\(617\) 7600.00 0.495890 0.247945 0.968774i \(-0.420245\pi\)
0.247945 + 0.968774i \(0.420245\pi\)
\(618\) 0 0
\(619\) 22685.3 1.47302 0.736509 0.676427i \(-0.236473\pi\)
0.736509 + 0.676427i \(0.236473\pi\)
\(620\) 0 0
\(621\) −2554.28 −0.165056
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 10154.0 0.649853
\(626\) 0 0
\(627\) 10277.7 0.654626
\(628\) 0 0
\(629\) −4789.59 −0.303614
\(630\) 0 0
\(631\) 12024.1 0.758595 0.379297 0.925275i \(-0.376166\pi\)
0.379297 + 0.925275i \(0.376166\pi\)
\(632\) 0 0
\(633\) 486.091 0.0305219
\(634\) 0 0
\(635\) 10203.8 0.637679
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 9530.86 0.590039
\(640\) 0 0
\(641\) 11320.6 0.697559 0.348779 0.937205i \(-0.386596\pi\)
0.348779 + 0.937205i \(0.386596\pi\)
\(642\) 0 0
\(643\) 16843.6 1.03304 0.516521 0.856275i \(-0.327227\pi\)
0.516521 + 0.856275i \(0.327227\pi\)
\(644\) 0 0
\(645\) 5119.12 0.312504
\(646\) 0 0
\(647\) 7719.32 0.469054 0.234527 0.972110i \(-0.424646\pi\)
0.234527 + 0.972110i \(0.424646\pi\)
\(648\) 0 0
\(649\) 27739.1 1.67774
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −30803.2 −1.84597 −0.922987 0.384832i \(-0.874259\pi\)
−0.922987 + 0.384832i \(0.874259\pi\)
\(654\) 0 0
\(655\) 692.503 0.0413104
\(656\) 0 0
\(657\) −10742.9 −0.637932
\(658\) 0 0
\(659\) −9760.68 −0.576968 −0.288484 0.957485i \(-0.593151\pi\)
−0.288484 + 0.957485i \(0.593151\pi\)
\(660\) 0 0
\(661\) −18071.1 −1.06336 −0.531682 0.846944i \(-0.678440\pi\)
−0.531682 + 0.846944i \(0.678440\pi\)
\(662\) 0 0
\(663\) −5171.25 −0.302918
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 13073.3 0.758920
\(668\) 0 0
\(669\) 13733.5 0.793676
\(670\) 0 0
\(671\) −35567.7 −2.04631
\(672\) 0 0
\(673\) 26591.1 1.52305 0.761524 0.648137i \(-0.224451\pi\)
0.761524 + 0.648137i \(0.224451\pi\)
\(674\) 0 0
\(675\) 2964.44 0.169039
\(676\) 0 0
\(677\) −33080.3 −1.87796 −0.938979 0.343975i \(-0.888227\pi\)
−0.938979 + 0.343975i \(0.888227\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 6654.57 0.374455
\(682\) 0 0
\(683\) 10466.1 0.586344 0.293172 0.956060i \(-0.405289\pi\)
0.293172 + 0.956060i \(0.405289\pi\)
\(684\) 0 0
\(685\) 105.345 0.00587597
\(686\) 0 0
\(687\) 2355.65 0.130820
\(688\) 0 0
\(689\) 14154.3 0.782634
\(690\) 0 0
\(691\) 13269.9 0.730551 0.365275 0.930900i \(-0.380975\pi\)
0.365275 + 0.930900i \(0.380975\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3598.27 −0.196389
\(696\) 0 0
\(697\) 13722.9 0.745755
\(698\) 0 0
\(699\) 16107.4 0.871583
\(700\) 0 0
\(701\) −15169.9 −0.817344 −0.408672 0.912681i \(-0.634008\pi\)
−0.408672 + 0.912681i \(0.634008\pi\)
\(702\) 0 0
\(703\) −8325.55 −0.446663
\(704\) 0 0
\(705\) −666.873 −0.0356254
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 26317.7 1.39405 0.697026 0.717046i \(-0.254506\pi\)
0.697026 + 0.717046i \(0.254506\pi\)
\(710\) 0 0
\(711\) −11876.0 −0.626421
\(712\) 0 0
\(713\) 12544.6 0.658907
\(714\) 0 0
\(715\) −12856.3 −0.672447
\(716\) 0 0
\(717\) 11139.8 0.580229
\(718\) 0 0
\(719\) 23013.1 1.19366 0.596831 0.802367i \(-0.296426\pi\)
0.596831 + 0.802367i \(0.296426\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 20995.9 1.08001
\(724\) 0 0
\(725\) −15172.5 −0.777232
\(726\) 0 0
\(727\) −16265.4 −0.829780 −0.414890 0.909872i \(-0.636180\pi\)
−0.414890 + 0.909872i \(0.636180\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 14046.8 0.710724
\(732\) 0 0
\(733\) 866.444 0.0436601 0.0218300 0.999762i \(-0.493051\pi\)
0.0218300 + 0.999762i \(0.493051\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −19007.8 −0.950013
\(738\) 0 0
\(739\) −10990.1 −0.547058 −0.273529 0.961864i \(-0.588191\pi\)
−0.273529 + 0.961864i \(0.588191\pi\)
\(740\) 0 0
\(741\) −8988.96 −0.445638
\(742\) 0 0
\(743\) 22416.4 1.10683 0.553417 0.832905i \(-0.313324\pi\)
0.553417 + 0.832905i \(0.313324\pi\)
\(744\) 0 0
\(745\) 2911.76 0.143193
\(746\) 0 0
\(747\) 10713.0 0.524722
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 19717.9 0.958080 0.479040 0.877793i \(-0.340985\pi\)
0.479040 + 0.877793i \(0.340985\pi\)
\(752\) 0 0
\(753\) −11168.3 −0.540497
\(754\) 0 0
\(755\) −8084.22 −0.389689
\(756\) 0 0
\(757\) 839.321 0.0402981 0.0201490 0.999797i \(-0.493586\pi\)
0.0201490 + 0.999797i \(0.493586\pi\)
\(758\) 0 0
\(759\) −17425.0 −0.833318
\(760\) 0 0
\(761\) −18364.5 −0.874786 −0.437393 0.899270i \(-0.644098\pi\)
−0.437393 + 0.899270i \(0.644098\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1126.58 −0.0532440
\(766\) 0 0
\(767\) −24260.9 −1.14213
\(768\) 0 0
\(769\) 14890.8 0.698277 0.349138 0.937071i \(-0.386474\pi\)
0.349138 + 0.937071i \(0.386474\pi\)
\(770\) 0 0
\(771\) 3690.45 0.172384
\(772\) 0 0
\(773\) 16606.2 0.772683 0.386341 0.922356i \(-0.373739\pi\)
0.386341 + 0.922356i \(0.373739\pi\)
\(774\) 0 0
\(775\) −14559.0 −0.674807
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 23853.9 1.09712
\(780\) 0 0
\(781\) 65018.5 2.97893
\(782\) 0 0
\(783\) −3731.15 −0.170294
\(784\) 0 0
\(785\) −6107.47 −0.277688
\(786\) 0 0
\(787\) 3156.20 0.142956 0.0714781 0.997442i \(-0.477228\pi\)
0.0714781 + 0.997442i \(0.477228\pi\)
\(788\) 0 0
\(789\) 7165.90 0.323337
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 31107.9 1.39303
\(794\) 0 0
\(795\) 3083.58 0.137564
\(796\) 0 0
\(797\) 13514.8 0.600652 0.300326 0.953837i \(-0.402904\pi\)
0.300326 + 0.953837i \(0.402904\pi\)
\(798\) 0 0
\(799\) −1829.89 −0.0810224
\(800\) 0 0
\(801\) 2097.77 0.0925356
\(802\) 0 0
\(803\) −73287.0 −3.22072
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 20106.9 0.877073
\(808\) 0 0
\(809\) −26758.1 −1.16287 −0.581437 0.813591i \(-0.697509\pi\)
−0.581437 + 0.813591i \(0.697509\pi\)
\(810\) 0 0
\(811\) −15920.7 −0.689338 −0.344669 0.938724i \(-0.612009\pi\)
−0.344669 + 0.938724i \(0.612009\pi\)
\(812\) 0 0
\(813\) −14851.1 −0.640653
\(814\) 0 0
\(815\) −385.010 −0.0165476
\(816\) 0 0
\(817\) 24417.0 1.04558
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −19306.2 −0.820696 −0.410348 0.911929i \(-0.634593\pi\)
−0.410348 + 0.911929i \(0.634593\pi\)
\(822\) 0 0
\(823\) −791.000 −0.0335025 −0.0167512 0.999860i \(-0.505332\pi\)
−0.0167512 + 0.999860i \(0.505332\pi\)
\(824\) 0 0
\(825\) 20223.0 0.853426
\(826\) 0 0
\(827\) −29537.6 −1.24199 −0.620993 0.783816i \(-0.713270\pi\)
−0.620993 + 0.783816i \(0.713270\pi\)
\(828\) 0 0
\(829\) 5766.52 0.241592 0.120796 0.992677i \(-0.461455\pi\)
0.120796 + 0.992677i \(0.461455\pi\)
\(830\) 0 0
\(831\) 11115.5 0.464011
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −8701.16 −0.360618
\(836\) 0 0
\(837\) −3580.28 −0.147853
\(838\) 0 0
\(839\) 29726.4 1.22321 0.611603 0.791165i \(-0.290525\pi\)
0.611603 + 0.791165i \(0.290525\pi\)
\(840\) 0 0
\(841\) −5292.27 −0.216994
\(842\) 0 0
\(843\) −27974.2 −1.14292
\(844\) 0 0
\(845\) 2677.11 0.108989
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 16708.5 0.675421
\(850\) 0 0
\(851\) 14115.3 0.568587
\(852\) 0 0
\(853\) 17829.6 0.715677 0.357838 0.933784i \(-0.383514\pi\)
0.357838 + 0.933784i \(0.383514\pi\)
\(854\) 0 0
\(855\) −1958.29 −0.0783300
\(856\) 0 0
\(857\) 39682.4 1.58171 0.790856 0.612003i \(-0.209636\pi\)
0.790856 + 0.612003i \(0.209636\pi\)
\(858\) 0 0
\(859\) −2195.13 −0.0871909 −0.0435955 0.999049i \(-0.513881\pi\)
−0.0435955 + 0.999049i \(0.513881\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −31917.1 −1.25894 −0.629472 0.777023i \(-0.716729\pi\)
−0.629472 + 0.777023i \(0.716729\pi\)
\(864\) 0 0
\(865\) −8192.23 −0.322016
\(866\) 0 0
\(867\) 11647.7 0.456258
\(868\) 0 0
\(869\) −81016.8 −3.16261
\(870\) 0 0
\(871\) 16624.4 0.646724
\(872\) 0 0
\(873\) 14484.9 0.561559
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 28842.9 1.11055 0.555276 0.831666i \(-0.312613\pi\)
0.555276 + 0.831666i \(0.312613\pi\)
\(878\) 0 0
\(879\) −4995.92 −0.191704
\(880\) 0 0
\(881\) 15350.2 0.587015 0.293508 0.955957i \(-0.405177\pi\)
0.293508 + 0.955957i \(0.405177\pi\)
\(882\) 0 0
\(883\) −6089.64 −0.232087 −0.116043 0.993244i \(-0.537021\pi\)
−0.116043 + 0.993244i \(0.537021\pi\)
\(884\) 0 0
\(885\) −5285.36 −0.200752
\(886\) 0 0
\(887\) −384.441 −0.0145527 −0.00727637 0.999974i \(-0.502316\pi\)
−0.00727637 + 0.999974i \(0.502316\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 4973.15 0.186989
\(892\) 0 0
\(893\) −3180.82 −0.119196
\(894\) 0 0
\(895\) −6717.05 −0.250867
\(896\) 0 0
\(897\) 15240.1 0.567283
\(898\) 0 0
\(899\) 18324.5 0.679819
\(900\) 0 0
\(901\) 8461.30 0.312860
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −6453.68 −0.237047
\(906\) 0 0
\(907\) 7267.93 0.266073 0.133036 0.991111i \(-0.457527\pi\)
0.133036 + 0.991111i \(0.457527\pi\)
\(908\) 0 0
\(909\) 13313.3 0.485780
\(910\) 0 0
\(911\) 8535.12 0.310408 0.155204 0.987882i \(-0.450397\pi\)
0.155204 + 0.987882i \(0.450397\pi\)
\(912\) 0 0
\(913\) 73082.7 2.64916
\(914\) 0 0
\(915\) 6777.01 0.244854
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 10851.7 0.389516 0.194758 0.980851i \(-0.437608\pi\)
0.194758 + 0.980851i \(0.437608\pi\)
\(920\) 0 0
\(921\) 15909.9 0.569219
\(922\) 0 0
\(923\) −56865.9 −2.02791
\(924\) 0 0
\(925\) −16381.9 −0.582307
\(926\) 0 0
\(927\) −10308.1 −0.365224
\(928\) 0 0
\(929\) 1560.64 0.0551161 0.0275581 0.999620i \(-0.491227\pi\)
0.0275581 + 0.999620i \(0.491227\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −3377.97 −0.118531
\(934\) 0 0
\(935\) −7685.41 −0.268813
\(936\) 0 0
\(937\) −11978.4 −0.417627 −0.208813 0.977956i \(-0.566960\pi\)
−0.208813 + 0.977956i \(0.566960\pi\)
\(938\) 0 0
\(939\) −24898.5 −0.865317
\(940\) 0 0
\(941\) −24597.7 −0.852137 −0.426068 0.904691i \(-0.640102\pi\)
−0.426068 + 0.904691i \(0.640102\pi\)
\(942\) 0 0
\(943\) −40442.6 −1.39660
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 10834.6 0.371783 0.185891 0.982570i \(-0.440483\pi\)
0.185891 + 0.982570i \(0.440483\pi\)
\(948\) 0 0
\(949\) 64097.6 2.19252
\(950\) 0 0
\(951\) 12836.3 0.437692
\(952\) 0 0
\(953\) 701.418 0.0238417 0.0119209 0.999929i \(-0.496205\pi\)
0.0119209 + 0.999929i \(0.496205\pi\)
\(954\) 0 0
\(955\) 3929.47 0.133146
\(956\) 0 0
\(957\) −25453.5 −0.859765
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −12207.4 −0.409769
\(962\) 0 0
\(963\) −3932.59 −0.131595
\(964\) 0 0
\(965\) −29.8252 −0.000994931 0
\(966\) 0 0
\(967\) −42402.5 −1.41011 −0.705053 0.709154i \(-0.749077\pi\)
−0.705053 + 0.709154i \(0.749077\pi\)
\(968\) 0 0
\(969\) −5373.53 −0.178145
\(970\) 0 0
\(971\) 15463.4 0.511064 0.255532 0.966801i \(-0.417749\pi\)
0.255532 + 0.966801i \(0.417749\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −17687.3 −0.580971
\(976\) 0 0
\(977\) −34484.1 −1.12922 −0.564609 0.825359i \(-0.690973\pi\)
−0.564609 + 0.825359i \(0.690973\pi\)
\(978\) 0 0
\(979\) 14310.7 0.467184
\(980\) 0 0
\(981\) −1497.16 −0.0487266
\(982\) 0 0
\(983\) 7335.36 0.238008 0.119004 0.992894i \(-0.462030\pi\)
0.119004 + 0.992894i \(0.462030\pi\)
\(984\) 0 0
\(985\) 10489.2 0.339302
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −41397.1 −1.33099
\(990\) 0 0
\(991\) 10123.0 0.324487 0.162243 0.986751i \(-0.448127\pi\)
0.162243 + 0.986751i \(0.448127\pi\)
\(992\) 0 0
\(993\) −5121.35 −0.163667
\(994\) 0 0
\(995\) 3382.80 0.107781
\(996\) 0 0
\(997\) −56669.2 −1.80013 −0.900066 0.435755i \(-0.856482\pi\)
−0.900066 + 0.435755i \(0.856482\pi\)
\(998\) 0 0
\(999\) −4028.56 −0.127586
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 2352.4.a.bn.1.2 2
4.3 odd 2 294.4.a.k.1.2 yes 2
7.6 odd 2 2352.4.a.cd.1.1 2
12.11 even 2 882.4.a.bi.1.1 2
28.3 even 6 294.4.e.o.79.2 4
28.11 odd 6 294.4.e.n.79.1 4
28.19 even 6 294.4.e.o.67.2 4
28.23 odd 6 294.4.e.n.67.1 4
28.27 even 2 294.4.a.j.1.1 2
84.11 even 6 882.4.g.y.667.2 4
84.23 even 6 882.4.g.y.361.2 4
84.47 odd 6 882.4.g.bd.361.1 4
84.59 odd 6 882.4.g.bd.667.1 4
84.83 odd 2 882.4.a.bc.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.4.a.j.1.1 2 28.27 even 2
294.4.a.k.1.2 yes 2 4.3 odd 2
294.4.e.n.67.1 4 28.23 odd 6
294.4.e.n.79.1 4 28.11 odd 6
294.4.e.o.67.2 4 28.19 even 6
294.4.e.o.79.2 4 28.3 even 6
882.4.a.bc.1.2 2 84.83 odd 2
882.4.a.bi.1.1 2 12.11 even 2
882.4.g.y.361.2 4 84.23 even 6
882.4.g.y.667.2 4 84.11 even 6
882.4.g.bd.361.1 4 84.47 odd 6
882.4.g.bd.667.1 4 84.59 odd 6
2352.4.a.bn.1.2 2 1.1 even 1 trivial
2352.4.a.cd.1.1 2 7.6 odd 2