Properties

Label 4100.2.d.c.1149.6
Level $4100$
Weight $2$
Character 4100.1149
Analytic conductor $32.739$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4100.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.7386648287\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} + 2 x^{6} + 4 x^{5} + 14 x^{4} - 14 x^{3} + 8 x^{2} + 4 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 164)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1149.6
Root \(0.626295 + 0.626295i\) of defining polynomial
Character \(\chi\) \(=\) 4100.1149
Dual form 4100.2.d.c.1149.3

$q$-expansion

\(f(q)\) \(=\) \(q+2.21551i q^{3} +5.06479i q^{7} -1.90849 q^{9} +O(q^{10})\) \(q+2.21551i q^{3} +5.06479i q^{7} -1.90849 q^{9} -2.55961 q^{11} -4.93620i q^{13} -2.68821i q^{17} -4.72069 q^{19} -11.2211 q^{21} -1.49482i q^{23} +2.41826i q^{27} -2.43102 q^{29} +3.19339 q^{31} -5.67085i q^{33} -2.90849i q^{37} +10.9362 q^{39} -1.00000 q^{41} -7.62441i q^{43} -5.15171i q^{47} -18.6521 q^{49} +5.95575 q^{51} -11.1192i q^{53} -10.4587i q^{57} -1.49482 q^{59} +1.06380 q^{61} -9.66608i q^{63} -10.5596i q^{67} +3.31179 q^{69} -12.6023 q^{71} +8.28490i q^{73} -12.9639i q^{77} +4.28967 q^{79} -11.0831 q^{81} +10.5606i q^{83} -5.38595i q^{87} -9.36722 q^{89} +25.0008 q^{91} +7.07498i q^{93} +3.37641i q^{97} +4.88499 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 24q^{9} + O(q^{10}) \) \( 8q - 24q^{9} + 8q^{11} - 12q^{19} + 8q^{29} - 16q^{31} + 48q^{39} - 8q^{41} - 32q^{49} - 8q^{51} - 24q^{59} + 48q^{61} + 56q^{69} - 4q^{71} + 36q^{79} + 56q^{81} - 8q^{89} + 72q^{91} - 116q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4100\mathbb{Z}\right)^\times\).

\(n\) \(1477\) \(2051\) \(3901\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 2.21551i 1.27913i 0.768739 + 0.639563i \(0.220885\pi\)
−0.768739 + 0.639563i \(0.779115\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 5.06479i 1.91431i 0.289574 + 0.957156i \(0.406486\pi\)
−0.289574 + 0.957156i \(0.593514\pi\)
\(8\) 0 0
\(9\) −1.90849 −0.636162
\(10\) 0 0
\(11\) −2.55961 −0.771753 −0.385876 0.922551i \(-0.626101\pi\)
−0.385876 + 0.922551i \(0.626101\pi\)
\(12\) 0 0
\(13\) − 4.93620i − 1.36906i −0.728987 0.684528i \(-0.760009\pi\)
0.728987 0.684528i \(-0.239991\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.68821i − 0.651986i −0.945372 0.325993i \(-0.894301\pi\)
0.945372 0.325993i \(-0.105699\pi\)
\(18\) 0 0
\(19\) −4.72069 −1.08300 −0.541500 0.840701i \(-0.682143\pi\)
−0.541500 + 0.840701i \(0.682143\pi\)
\(20\) 0 0
\(21\) −11.2211 −2.44864
\(22\) 0 0
\(23\) − 1.49482i − 0.311692i −0.987781 0.155846i \(-0.950190\pi\)
0.987781 0.155846i \(-0.0498103\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.41826i 0.465395i
\(28\) 0 0
\(29\) −2.43102 −0.451429 −0.225715 0.974193i \(-0.572472\pi\)
−0.225715 + 0.974193i \(0.572472\pi\)
\(30\) 0 0
\(31\) 3.19339 0.573549 0.286774 0.957998i \(-0.407417\pi\)
0.286774 + 0.957998i \(0.407417\pi\)
\(32\) 0 0
\(33\) − 5.67085i − 0.987168i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.90849i − 0.478152i −0.971001 0.239076i \(-0.923155\pi\)
0.971001 0.239076i \(-0.0768445\pi\)
\(38\) 0 0
\(39\) 10.9362 1.75119
\(40\) 0 0
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) − 7.62441i − 1.16271i −0.813650 0.581356i \(-0.802523\pi\)
0.813650 0.581356i \(-0.197477\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 5.15171i − 0.751454i −0.926730 0.375727i \(-0.877393\pi\)
0.926730 0.375727i \(-0.122607\pi\)
\(48\) 0 0
\(49\) −18.6521 −2.66459
\(50\) 0 0
\(51\) 5.95575 0.833972
\(52\) 0 0
\(53\) − 11.1192i − 1.52734i −0.645605 0.763672i \(-0.723395\pi\)
0.645605 0.763672i \(-0.276605\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 10.4587i − 1.38529i
\(58\) 0 0
\(59\) −1.49482 −0.194609 −0.0973046 0.995255i \(-0.531022\pi\)
−0.0973046 + 0.995255i \(0.531022\pi\)
\(60\) 0 0
\(61\) 1.06380 0.136206 0.0681029 0.997678i \(-0.478305\pi\)
0.0681029 + 0.997678i \(0.478305\pi\)
\(62\) 0 0
\(63\) − 9.66608i − 1.21781i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 10.5596i − 1.29006i −0.764156 0.645031i \(-0.776844\pi\)
0.764156 0.645031i \(-0.223156\pi\)
\(68\) 0 0
\(69\) 3.31179 0.398693
\(70\) 0 0
\(71\) −12.6023 −1.49562 −0.747808 0.663915i \(-0.768894\pi\)
−0.747808 + 0.663915i \(0.768894\pi\)
\(72\) 0 0
\(73\) 8.28490i 0.969674i 0.874604 + 0.484837i \(0.161121\pi\)
−0.874604 + 0.484837i \(0.838879\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 12.9639i − 1.47737i
\(78\) 0 0
\(79\) 4.28967 0.482625 0.241313 0.970447i \(-0.422422\pi\)
0.241313 + 0.970447i \(0.422422\pi\)
\(80\) 0 0
\(81\) −11.0831 −1.23146
\(82\) 0 0
\(83\) 10.5606i 1.15918i 0.814909 + 0.579588i \(0.196787\pi\)
−0.814909 + 0.579588i \(0.803213\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 5.38595i − 0.577435i
\(88\) 0 0
\(89\) −9.36722 −0.992923 −0.496462 0.868059i \(-0.665368\pi\)
−0.496462 + 0.868059i \(0.665368\pi\)
\(90\) 0 0
\(91\) 25.0008 2.62080
\(92\) 0 0
\(93\) 7.07498i 0.733641i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.37641i 0.342823i 0.985200 + 0.171411i \(0.0548328\pi\)
−0.985200 + 0.171411i \(0.945167\pi\)
\(98\) 0 0
\(99\) 4.88499 0.490960
\(100\) 0 0
\(101\) −4.24799 −0.422691 −0.211345 0.977411i \(-0.567785\pi\)
−0.211345 + 0.977411i \(0.567785\pi\)
\(102\) 0 0
\(103\) 1.11923i 0.110281i 0.998479 + 0.0551404i \(0.0175606\pi\)
−0.998479 + 0.0551404i \(0.982439\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.322149i 0.0311434i 0.999879 + 0.0155717i \(0.00495682\pi\)
−0.999879 + 0.0155717i \(0.995043\pi\)
\(108\) 0 0
\(109\) −3.23763 −0.310109 −0.155055 0.987906i \(-0.549555\pi\)
−0.155055 + 0.987906i \(0.549555\pi\)
\(110\) 0 0
\(111\) 6.44378 0.611616
\(112\) 0 0
\(113\) 2.10187i 0.197727i 0.995101 + 0.0988637i \(0.0315208\pi\)
−0.995101 + 0.0988637i \(0.968479\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 9.42066i 0.870941i
\(118\) 0 0
\(119\) 13.6152 1.24810
\(120\) 0 0
\(121\) −4.44838 −0.404398
\(122\) 0 0
\(123\) − 2.21551i − 0.199766i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 13.0658i 1.15940i 0.814830 + 0.579700i \(0.196830\pi\)
−0.814830 + 0.579700i \(0.803170\pi\)
\(128\) 0 0
\(129\) 16.8919 1.48725
\(130\) 0 0
\(131\) 8.43102 0.736622 0.368311 0.929703i \(-0.379936\pi\)
0.368311 + 0.929703i \(0.379936\pi\)
\(132\) 0 0
\(133\) − 23.9093i − 2.07320i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.5606i 1.41487i 0.706779 + 0.707434i \(0.250147\pi\)
−0.706779 + 0.707434i \(0.749853\pi\)
\(138\) 0 0
\(139\) 2.18303 0.185162 0.0925810 0.995705i \(-0.470488\pi\)
0.0925810 + 0.995705i \(0.470488\pi\)
\(140\) 0 0
\(141\) 11.4137 0.961204
\(142\) 0 0
\(143\) 12.6348i 1.05657i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 41.3240i − 3.40834i
\(148\) 0 0
\(149\) −2.43102 −0.199157 −0.0995785 0.995030i \(-0.531749\pi\)
−0.0995785 + 0.995030i \(0.531749\pi\)
\(150\) 0 0
\(151\) −0.343113 −0.0279221 −0.0139611 0.999903i \(-0.504444\pi\)
−0.0139611 + 0.999903i \(0.504444\pi\)
\(152\) 0 0
\(153\) 5.13040i 0.414769i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.6348i 0.848746i 0.905487 + 0.424373i \(0.139506\pi\)
−0.905487 + 0.424373i \(0.860494\pi\)
\(158\) 0 0
\(159\) 24.6348 1.95366
\(160\) 0 0
\(161\) 7.57096 0.596675
\(162\) 0 0
\(163\) − 0.173834i − 0.0136157i −0.999977 0.00680785i \(-0.997833\pi\)
0.999977 0.00680785i \(-0.00216702\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 14.5596i − 1.12666i −0.826233 0.563328i \(-0.809521\pi\)
0.826233 0.563328i \(-0.190479\pi\)
\(168\) 0 0
\(169\) −11.3661 −0.874312
\(170\) 0 0
\(171\) 9.00937 0.688963
\(172\) 0 0
\(173\) − 22.2592i − 1.69233i −0.532918 0.846167i \(-0.678905\pi\)
0.532918 0.846167i \(-0.321095\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 3.31179i − 0.248930i
\(178\) 0 0
\(179\) 17.9268 1.33991 0.669957 0.742400i \(-0.266312\pi\)
0.669957 + 0.742400i \(0.266312\pi\)
\(180\) 0 0
\(181\) 20.9916 1.56030 0.780148 0.625595i \(-0.215144\pi\)
0.780148 + 0.625595i \(0.215144\pi\)
\(182\) 0 0
\(183\) 2.35686i 0.174224i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.88077i 0.503172i
\(188\) 0 0
\(189\) −12.2480 −0.890910
\(190\) 0 0
\(191\) 6.45074 0.466759 0.233380 0.972386i \(-0.425022\pi\)
0.233380 + 0.972386i \(0.425022\pi\)
\(192\) 0 0
\(193\) 6.20374i 0.446555i 0.974755 + 0.223278i \(0.0716756\pi\)
−0.974755 + 0.223278i \(0.928324\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 14.1850i − 1.01064i −0.862932 0.505320i \(-0.831374\pi\)
0.862932 0.505320i \(-0.168626\pi\)
\(198\) 0 0
\(199\) −18.7099 −1.32631 −0.663155 0.748482i \(-0.730783\pi\)
−0.663155 + 0.748482i \(0.730783\pi\)
\(200\) 0 0
\(201\) 23.3949 1.65015
\(202\) 0 0
\(203\) − 12.3126i − 0.864176i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.85285i 0.198286i
\(208\) 0 0
\(209\) 12.0831 0.835808
\(210\) 0 0
\(211\) 14.6984 1.01188 0.505940 0.862569i \(-0.331146\pi\)
0.505940 + 0.862569i \(0.331146\pi\)
\(212\) 0 0
\(213\) − 27.9205i − 1.91308i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 16.1738i 1.09795i
\(218\) 0 0
\(219\) −18.3553 −1.24033
\(220\) 0 0
\(221\) −13.2695 −0.892605
\(222\) 0 0
\(223\) 20.1109i 1.34672i 0.739314 + 0.673361i \(0.235150\pi\)
−0.739314 + 0.673361i \(0.764850\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.71149i 0.445457i 0.974880 + 0.222729i \(0.0714964\pi\)
−0.974880 + 0.222729i \(0.928504\pi\)
\(228\) 0 0
\(229\) −12.7624 −0.843361 −0.421680 0.906745i \(-0.638560\pi\)
−0.421680 + 0.906745i \(0.638560\pi\)
\(230\) 0 0
\(231\) 28.7217 1.88975
\(232\) 0 0
\(233\) − 29.0750i − 1.90477i −0.304906 0.952383i \(-0.598625\pi\)
0.304906 0.952383i \(-0.401375\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 9.50380i 0.617338i
\(238\) 0 0
\(239\) −26.0971 −1.68808 −0.844041 0.536279i \(-0.819829\pi\)
−0.844041 + 0.536279i \(0.819829\pi\)
\(240\) 0 0
\(241\) 1.06380 0.0685255 0.0342627 0.999413i \(-0.489092\pi\)
0.0342627 + 0.999413i \(0.489092\pi\)
\(242\) 0 0
\(243\) − 17.3000i − 1.10980i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 23.3023i 1.48269i
\(248\) 0 0
\(249\) −23.3971 −1.48273
\(250\) 0 0
\(251\) −7.82815 −0.494108 −0.247054 0.969002i \(-0.579463\pi\)
−0.247054 + 0.969002i \(0.579463\pi\)
\(252\) 0 0
\(253\) 3.82617i 0.240549i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 7.49482i − 0.467514i −0.972295 0.233757i \(-0.924898\pi\)
0.972295 0.233757i \(-0.0751020\pi\)
\(258\) 0 0
\(259\) 14.7309 0.915332
\(260\) 0 0
\(261\) 4.63957 0.287182
\(262\) 0 0
\(263\) 15.5919i 0.961439i 0.876874 + 0.480720i \(0.159625\pi\)
−0.876874 + 0.480720i \(0.840375\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 20.7532i − 1.27007i
\(268\) 0 0
\(269\) −8.80860 −0.537070 −0.268535 0.963270i \(-0.586539\pi\)
−0.268535 + 0.963270i \(0.586539\pi\)
\(270\) 0 0
\(271\) −19.8816 −1.20772 −0.603860 0.797090i \(-0.706372\pi\)
−0.603860 + 0.797090i \(0.706372\pi\)
\(272\) 0 0
\(273\) 55.3896i 3.35233i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.08232i 0.425535i 0.977103 + 0.212768i \(0.0682477\pi\)
−0.977103 + 0.212768i \(0.931752\pi\)
\(278\) 0 0
\(279\) −6.09453 −0.364870
\(280\) 0 0
\(281\) −31.2695 −1.86538 −0.932692 0.360675i \(-0.882547\pi\)
−0.932692 + 0.360675i \(0.882547\pi\)
\(282\) 0 0
\(283\) − 2.18303i − 0.129768i −0.997893 0.0648838i \(-0.979332\pi\)
0.997893 0.0648838i \(-0.0206677\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 5.06479i − 0.298965i
\(288\) 0 0
\(289\) 9.77354 0.574914
\(290\) 0 0
\(291\) −7.48048 −0.438514
\(292\) 0 0
\(293\) − 33.6798i − 1.96760i −0.179278 0.983798i \(-0.557376\pi\)
0.179278 0.983798i \(-0.442624\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 6.18982i − 0.359170i
\(298\) 0 0
\(299\) −7.37874 −0.426723
\(300\) 0 0
\(301\) 38.6160 2.22579
\(302\) 0 0
\(303\) − 9.41147i − 0.540675i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.7532i 0.727862i 0.931426 + 0.363931i \(0.118566\pi\)
−0.931426 + 0.363931i \(0.881434\pi\)
\(308\) 0 0
\(309\) −2.47966 −0.141063
\(310\) 0 0
\(311\) 2.18204 0.123732 0.0618660 0.998084i \(-0.480295\pi\)
0.0618660 + 0.998084i \(0.480295\pi\)
\(312\) 0 0
\(313\) − 29.3042i − 1.65637i −0.560452 0.828187i \(-0.689373\pi\)
0.560452 0.828187i \(-0.310627\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 28.5511i 1.60359i 0.597601 + 0.801794i \(0.296121\pi\)
−0.597601 + 0.801794i \(0.703879\pi\)
\(318\) 0 0
\(319\) 6.22247 0.348392
\(320\) 0 0
\(321\) −0.713725 −0.0398363
\(322\) 0 0
\(323\) 12.6902i 0.706101i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 7.17301i − 0.396669i
\(328\) 0 0
\(329\) 26.0923 1.43852
\(330\) 0 0
\(331\) −4.80761 −0.264250 −0.132125 0.991233i \(-0.542180\pi\)
−0.132125 + 0.991233i \(0.542180\pi\)
\(332\) 0 0
\(333\) 5.55080i 0.304182i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 16.3148i − 0.888724i −0.895847 0.444362i \(-0.853430\pi\)
0.895847 0.444362i \(-0.146570\pi\)
\(338\) 0 0
\(339\) −4.65672 −0.252918
\(340\) 0 0
\(341\) −8.17383 −0.442638
\(342\) 0 0
\(343\) − 59.0156i − 3.18654i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 31.6034i − 1.69656i −0.529547 0.848281i \(-0.677638\pi\)
0.529547 0.848281i \(-0.322362\pi\)
\(348\) 0 0
\(349\) −4.97311 −0.266204 −0.133102 0.991102i \(-0.542494\pi\)
−0.133102 + 0.991102i \(0.542494\pi\)
\(350\) 0 0
\(351\) 11.9370 0.637151
\(352\) 0 0
\(353\) 1.72430i 0.0917750i 0.998947 + 0.0458875i \(0.0146116\pi\)
−0.998947 + 0.0458875i \(0.985388\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 30.1646i 1.59648i
\(358\) 0 0
\(359\) 7.13796 0.376727 0.188364 0.982099i \(-0.439682\pi\)
0.188364 + 0.982099i \(0.439682\pi\)
\(360\) 0 0
\(361\) 3.28490 0.172889
\(362\) 0 0
\(363\) − 9.85542i − 0.517276i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 9.38595i 0.489943i 0.969530 + 0.244971i \(0.0787785\pi\)
−0.969530 + 0.244971i \(0.921221\pi\)
\(368\) 0 0
\(369\) 1.90849 0.0993518
\(370\) 0 0
\(371\) 56.3166 2.92381
\(372\) 0 0
\(373\) 23.1212i 1.19717i 0.801059 + 0.598585i \(0.204270\pi\)
−0.801059 + 0.598585i \(0.795730\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) −5.99080 −0.307727 −0.153863 0.988092i \(-0.549172\pi\)
−0.153863 + 0.988092i \(0.549172\pi\)
\(380\) 0 0
\(381\) −28.9474 −1.48302
\(382\) 0 0
\(383\) − 29.5811i − 1.51153i −0.654845 0.755763i \(-0.727266\pi\)
0.654845 0.755763i \(-0.272734\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 14.5511i 0.739672i
\(388\) 0 0
\(389\) −22.8620 −1.15915 −0.579576 0.814918i \(-0.696782\pi\)
−0.579576 + 0.814918i \(0.696782\pi\)
\(390\) 0 0
\(391\) −4.01839 −0.203219
\(392\) 0 0
\(393\) 18.6790i 0.942232i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.588531i 0.0295375i 0.999891 + 0.0147688i \(0.00470122\pi\)
−0.999891 + 0.0147688i \(0.995299\pi\)
\(398\) 0 0
\(399\) 52.9713 2.65188
\(400\) 0 0
\(401\) −0.0926758 −0.00462801 −0.00231400 0.999997i \(-0.500737\pi\)
−0.00231400 + 0.999997i \(0.500737\pi\)
\(402\) 0 0
\(403\) − 15.7632i − 0.785220i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.44460i 0.369015i
\(408\) 0 0
\(409\) 1.29526 0.0640463 0.0320232 0.999487i \(-0.489805\pi\)
0.0320232 + 0.999487i \(0.489805\pi\)
\(410\) 0 0
\(411\) −36.6902 −1.80979
\(412\) 0 0
\(413\) − 7.57096i − 0.372543i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.83652i 0.236846i
\(418\) 0 0
\(419\) −1.75201 −0.0855912 −0.0427956 0.999084i \(-0.513626\pi\)
−0.0427956 + 0.999084i \(0.513626\pi\)
\(420\) 0 0
\(421\) 36.7364 1.79042 0.895212 0.445641i \(-0.147024\pi\)
0.895212 + 0.445641i \(0.147024\pi\)
\(422\) 0 0
\(423\) 9.83196i 0.478046i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 5.38793i 0.260740i
\(428\) 0 0
\(429\) −27.9924 −1.35149
\(430\) 0 0
\(431\) 12.3126 0.593078 0.296539 0.955021i \(-0.404168\pi\)
0.296539 + 0.955021i \(0.404168\pi\)
\(432\) 0 0
\(433\) − 17.7632i − 0.853644i −0.904336 0.426822i \(-0.859633\pi\)
0.904336 0.426822i \(-0.140367\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.05659i 0.337562i
\(438\) 0 0
\(439\) −3.22230 −0.153792 −0.0768961 0.997039i \(-0.524501\pi\)
−0.0768961 + 0.997039i \(0.524501\pi\)
\(440\) 0 0
\(441\) 35.5973 1.69511
\(442\) 0 0
\(443\) − 35.8537i − 1.70346i −0.523982 0.851730i \(-0.675554\pi\)
0.523982 0.851730i \(-0.324446\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 5.38595i − 0.254747i
\(448\) 0 0
\(449\) 15.3437 0.724113 0.362057 0.932156i \(-0.382075\pi\)
0.362057 + 0.932156i \(0.382075\pi\)
\(450\) 0 0
\(451\) 2.55961 0.120528
\(452\) 0 0
\(453\) − 0.760170i − 0.0357159i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 27.5917i − 1.29068i −0.763894 0.645342i \(-0.776715\pi\)
0.763894 0.645342i \(-0.223285\pi\)
\(458\) 0 0
\(459\) 6.50079 0.303431
\(460\) 0 0
\(461\) −8.08116 −0.376377 −0.188189 0.982133i \(-0.560262\pi\)
−0.188189 + 0.982133i \(0.560262\pi\)
\(462\) 0 0
\(463\) − 8.23622i − 0.382770i −0.981515 0.191385i \(-0.938702\pi\)
0.981515 0.191385i \(-0.0612979\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 23.7333i 1.09825i 0.835742 + 0.549123i \(0.185038\pi\)
−0.835742 + 0.549123i \(0.814962\pi\)
\(468\) 0 0
\(469\) 53.4822 2.46958
\(470\) 0 0
\(471\) −23.5614 −1.08565
\(472\) 0 0
\(473\) 19.5155i 0.897325i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 21.2209i 0.971638i
\(478\) 0 0
\(479\) 16.6987 0.762985 0.381492 0.924372i \(-0.375410\pi\)
0.381492 + 0.924372i \(0.375410\pi\)
\(480\) 0 0
\(481\) −14.3569 −0.654617
\(482\) 0 0
\(483\) 16.7735i 0.763223i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 0.927003i − 0.0420065i −0.999779 0.0210033i \(-0.993314\pi\)
0.999779 0.0210033i \(-0.00668604\pi\)
\(488\) 0 0
\(489\) 0.385130 0.0174162
\(490\) 0 0
\(491\) −26.8752 −1.21286 −0.606430 0.795137i \(-0.707399\pi\)
−0.606430 + 0.795137i \(0.707399\pi\)
\(492\) 0 0
\(493\) 6.53509i 0.294326i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 63.8279i − 2.86307i
\(498\) 0 0
\(499\) −18.2155 −0.815438 −0.407719 0.913107i \(-0.633676\pi\)
−0.407719 + 0.913107i \(0.633676\pi\)
\(500\) 0 0
\(501\) 32.2570 1.44114
\(502\) 0 0
\(503\) 18.9591i 0.845346i 0.906282 + 0.422673i \(0.138908\pi\)
−0.906282 + 0.422673i \(0.861092\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 25.1816i − 1.11835i
\(508\) 0 0
\(509\) −12.4218 −0.550588 −0.275294 0.961360i \(-0.588775\pi\)
−0.275294 + 0.961360i \(0.588775\pi\)
\(510\) 0 0
\(511\) −41.9613 −1.85626
\(512\) 0 0
\(513\) − 11.4159i − 0.504022i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 13.1864i 0.579937i
\(518\) 0 0
\(519\) 49.3154 2.16471
\(520\) 0 0
\(521\) −7.17267 −0.314240 −0.157120 0.987579i \(-0.550221\pi\)
−0.157120 + 0.987579i \(0.550221\pi\)
\(522\) 0 0
\(523\) 35.6563i 1.55914i 0.626315 + 0.779570i \(0.284563\pi\)
−0.626315 + 0.779570i \(0.715437\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 8.58448i − 0.373946i
\(528\) 0 0
\(529\) 20.7655 0.902848
\(530\) 0 0
\(531\) 2.85285 0.123803
\(532\) 0 0
\(533\) 4.93620i 0.213811i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 39.7171i 1.71392i
\(538\) 0 0
\(539\) 47.7422 2.05640
\(540\) 0 0
\(541\) 28.9731 1.24565 0.622826 0.782361i \(-0.285985\pi\)
0.622826 + 0.782361i \(0.285985\pi\)
\(542\) 0 0
\(543\) 46.5072i 1.99581i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 16.1178i − 0.689148i −0.938759 0.344574i \(-0.888023\pi\)
0.938759 0.344574i \(-0.111977\pi\)
\(548\) 0 0
\(549\) −2.03025 −0.0866489
\(550\) 0 0
\(551\) 11.4761 0.488898
\(552\) 0 0
\(553\) 21.7263i 0.923895i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 27.2675i 1.15536i 0.816262 + 0.577681i \(0.196042\pi\)
−0.816262 + 0.577681i \(0.803958\pi\)
\(558\) 0 0
\(559\) −37.6356 −1.59182
\(560\) 0 0
\(561\) −15.2444 −0.643620
\(562\) 0 0
\(563\) 41.9173i 1.76660i 0.468805 + 0.883302i \(0.344685\pi\)
−0.468805 + 0.883302i \(0.655315\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 56.1338i − 2.35740i
\(568\) 0 0
\(569\) −37.3762 −1.56689 −0.783446 0.621460i \(-0.786540\pi\)
−0.783446 + 0.621460i \(0.786540\pi\)
\(570\) 0 0
\(571\) −26.3228 −1.10157 −0.550787 0.834646i \(-0.685673\pi\)
−0.550787 + 0.834646i \(0.685673\pi\)
\(572\) 0 0
\(573\) 14.2917i 0.597044i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 13.8262i 0.575591i 0.957692 + 0.287795i \(0.0929223\pi\)
−0.957692 + 0.287795i \(0.907078\pi\)
\(578\) 0 0
\(579\) −13.7445 −0.571200
\(580\) 0 0
\(581\) −53.4873 −2.21903
\(582\) 0 0
\(583\) 28.4609i 1.17873i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 22.7004i − 0.936945i −0.883478 0.468472i \(-0.844805\pi\)
0.883478 0.468472i \(-0.155195\pi\)
\(588\) 0 0
\(589\) −15.0750 −0.621154
\(590\) 0 0
\(591\) 31.4270 1.29274
\(592\) 0 0
\(593\) 16.4167i 0.674152i 0.941477 + 0.337076i \(0.109438\pi\)
−0.941477 + 0.337076i \(0.890562\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 41.4520i − 1.69652i
\(598\) 0 0
\(599\) 6.54909 0.267588 0.133794 0.991009i \(-0.457284\pi\)
0.133794 + 0.991009i \(0.457284\pi\)
\(600\) 0 0
\(601\) −24.7783 −1.01073 −0.505365 0.862906i \(-0.668642\pi\)
−0.505365 + 0.862906i \(0.668642\pi\)
\(602\) 0 0
\(603\) 20.1529i 0.820688i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 15.7982i 0.641231i 0.947209 + 0.320615i \(0.103890\pi\)
−0.947209 + 0.320615i \(0.896110\pi\)
\(608\) 0 0
\(609\) 27.2787 1.10539
\(610\) 0 0
\(611\) −25.4299 −1.02878
\(612\) 0 0
\(613\) 39.3415i 1.58899i 0.607272 + 0.794494i \(0.292264\pi\)
−0.607272 + 0.794494i \(0.707736\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.4075i 1.06313i 0.847019 + 0.531563i \(0.178395\pi\)
−0.847019 + 0.531563i \(0.821605\pi\)
\(618\) 0 0
\(619\) −33.9278 −1.36367 −0.681837 0.731504i \(-0.738819\pi\)
−0.681837 + 0.731504i \(0.738819\pi\)
\(620\) 0 0
\(621\) 3.61487 0.145060
\(622\) 0 0
\(623\) − 47.4430i − 1.90076i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 26.7703i 1.06910i
\(628\) 0 0
\(629\) −7.81861 −0.311748
\(630\) 0 0
\(631\) −5.68017 −0.226124 −0.113062 0.993588i \(-0.536066\pi\)
−0.113062 + 0.993588i \(0.536066\pi\)
\(632\) 0 0
\(633\) 32.5644i 1.29432i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 92.0706i 3.64797i
\(638\) 0 0
\(639\) 24.0513 0.951454
\(640\) 0 0
\(641\) 15.5802 0.615379 0.307690 0.951487i \(-0.400444\pi\)
0.307690 + 0.951487i \(0.400444\pi\)
\(642\) 0 0
\(643\) 22.9372i 0.904554i 0.891877 + 0.452277i \(0.149388\pi\)
−0.891877 + 0.452277i \(0.850612\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 15.0546i − 0.591858i −0.955210 0.295929i \(-0.904371\pi\)
0.955210 0.295929i \(-0.0956291\pi\)
\(648\) 0 0
\(649\) 3.82617 0.150190
\(650\) 0 0
\(651\) −35.8333 −1.40442
\(652\) 0 0
\(653\) − 45.8652i − 1.79484i −0.441174 0.897422i \(-0.645438\pi\)
0.441174 0.897422i \(-0.354562\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 15.8116i − 0.616870i
\(658\) 0 0
\(659\) 4.37302 0.170349 0.0851744 0.996366i \(-0.472855\pi\)
0.0851744 + 0.996366i \(0.472855\pi\)
\(660\) 0 0
\(661\) −25.4946 −0.991625 −0.495813 0.868430i \(-0.665130\pi\)
−0.495813 + 0.868430i \(0.665130\pi\)
\(662\) 0 0
\(663\) − 29.3988i − 1.14175i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.63394i 0.140707i
\(668\) 0 0
\(669\) −44.5558 −1.72263
\(670\) 0 0
\(671\) −2.72292 −0.105117
\(672\) 0 0
\(673\) 31.7540i 1.22403i 0.790848 + 0.612013i \(0.209640\pi\)
−0.790848 + 0.612013i \(0.790360\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 27.8998i 1.07228i 0.844131 + 0.536138i \(0.180117\pi\)
−0.844131 + 0.536138i \(0.819883\pi\)
\(678\) 0 0
\(679\) −17.1008 −0.656270
\(680\) 0 0
\(681\) −14.8694 −0.569796
\(682\) 0 0
\(683\) 8.11782i 0.310620i 0.987866 + 0.155310i \(0.0496376\pi\)
−0.987866 + 0.155310i \(0.950362\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 28.2752i − 1.07876i
\(688\) 0 0
\(689\) −54.8867 −2.09102
\(690\) 0 0
\(691\) −26.6390 −1.01339 −0.506697 0.862124i \(-0.669134\pi\)
−0.506697 + 0.862124i \(0.669134\pi\)
\(692\) 0 0
\(693\) 24.7414i 0.939850i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.68821i 0.101823i
\(698\) 0 0
\(699\) 64.4159 2.43643
\(700\) 0 0
\(701\) 43.1302 1.62900 0.814502 0.580160i \(-0.197010\pi\)
0.814502 + 0.580160i \(0.197010\pi\)
\(702\) 0 0
\(703\) 13.7301i 0.517839i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 21.5152i − 0.809162i
\(708\) 0 0
\(709\) −49.5175 −1.85967 −0.929835 0.367978i \(-0.880050\pi\)
−0.929835 + 0.367978i \(0.880050\pi\)
\(710\) 0 0
\(711\) −8.18677 −0.307028
\(712\) 0 0
\(713\) − 4.77354i − 0.178771i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 57.8184i − 2.15927i
\(718\) 0 0
\(719\) 16.4839 0.614745 0.307372 0.951589i \(-0.400550\pi\)
0.307372 + 0.951589i \(0.400550\pi\)
\(720\) 0 0
\(721\) −5.66866 −0.211112
\(722\) 0 0
\(723\) 2.35686i 0.0876527i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 48.5947i − 1.80228i −0.433530 0.901139i \(-0.642732\pi\)
0.433530 0.901139i \(-0.357268\pi\)
\(728\) 0 0
\(729\) 5.07896 0.188110
\(730\) 0 0
\(731\) −20.4960 −0.758071
\(732\) 0 0
\(733\) 11.8166i 0.436457i 0.975898 + 0.218229i \(0.0700278\pi\)
−0.975898 + 0.218229i \(0.929972\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 27.0285i 0.995609i
\(738\) 0 0
\(739\) 5.68017 0.208949 0.104474 0.994528i \(-0.466684\pi\)
0.104474 + 0.994528i \(0.466684\pi\)
\(740\) 0 0
\(741\) −51.6264 −1.89654
\(742\) 0 0
\(743\) − 24.4980i − 0.898743i −0.893345 0.449372i \(-0.851648\pi\)
0.893345 0.449372i \(-0.148352\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 20.1548i − 0.737424i
\(748\) 0 0
\(749\) −1.63162 −0.0596181
\(750\) 0 0
\(751\) −35.7897 −1.30598 −0.652992 0.757365i \(-0.726487\pi\)
−0.652992 + 0.757365i \(0.726487\pi\)
\(752\) 0 0
\(753\) − 17.3433i − 0.632027i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 37.7169i − 1.37084i −0.728147 0.685421i \(-0.759618\pi\)
0.728147 0.685421i \(-0.240382\pi\)
\(758\) 0 0
\(759\) −8.47691 −0.307692
\(760\) 0 0
\(761\) −22.6533 −0.821181 −0.410590 0.911820i \(-0.634677\pi\)
−0.410590 + 0.911820i \(0.634677\pi\)
\(762\) 0 0
\(763\) − 16.3979i − 0.593646i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7.37874i 0.266431i
\(768\) 0 0
\(769\) 42.6292 1.53725 0.768624 0.639701i \(-0.220942\pi\)
0.768624 + 0.639701i \(0.220942\pi\)
\(770\) 0 0
\(771\) 16.6049 0.598009
\(772\) 0 0
\(773\) 11.7635i 0.423105i 0.977367 + 0.211552i \(0.0678519\pi\)
−0.977367 + 0.211552i \(0.932148\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 32.6364i 1.17082i
\(778\) 0 0
\(779\) 4.72069 0.169136
\(780\) 0 0
\(781\) 32.2570 1.15425
\(782\) 0 0
\(783\) − 5.87884i − 0.210093i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 34.4517i − 1.22807i −0.789278 0.614036i \(-0.789545\pi\)
0.789278 0.614036i \(-0.210455\pi\)
\(788\) 0 0
\(789\) −34.5441 −1.22980
\(790\) 0 0
\(791\) −10.6455 −0.378512
\(792\) 0 0
\(793\) − 5.25113i − 0.186473i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 41.2161i − 1.45995i −0.683475 0.729974i \(-0.739532\pi\)
0.683475 0.729974i \(-0.260468\pi\)
\(798\) 0 0
\(799\) −13.8489 −0.489937
\(800\) 0 0
\(801\) 17.8772 0.631660
\(802\) 0 0
\(803\) − 21.2061i − 0.748349i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 19.5155i − 0.686979i
\(808\) 0 0
\(809\) −12.4123 −0.436393 −0.218196 0.975905i \(-0.570017\pi\)
−0.218196 + 0.975905i \(0.570017\pi\)
\(810\) 0 0
\(811\) 3.91665 0.137532 0.0687660 0.997633i \(-0.478094\pi\)
0.0687660 + 0.997633i \(0.478094\pi\)
\(812\) 0 0
\(813\) − 44.0479i − 1.54483i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 35.9924i 1.25922i
\(818\) 0 0
\(819\) −47.7137 −1.66725
\(820\) 0 0
\(821\) 28.6326 0.999283 0.499642 0.866232i \(-0.333465\pi\)
0.499642 + 0.866232i \(0.333465\pi\)
\(822\) 0 0
\(823\) 43.8212i 1.52751i 0.645506 + 0.763755i \(0.276647\pi\)
−0.645506 + 0.763755i \(0.723353\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 11.1924i − 0.389198i −0.980883 0.194599i \(-0.937659\pi\)
0.980883 0.194599i \(-0.0623405\pi\)
\(828\) 0 0
\(829\) −26.6278 −0.924820 −0.462410 0.886666i \(-0.653015\pi\)
−0.462410 + 0.886666i \(0.653015\pi\)
\(830\) 0 0
\(831\) −15.6910 −0.544313
\(832\) 0 0
\(833\) 50.1408i 1.73727i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 7.72244i 0.266927i
\(838\) 0 0
\(839\) −9.43085 −0.325589 −0.162795 0.986660i \(-0.552051\pi\)
−0.162795 + 0.986660i \(0.552051\pi\)
\(840\) 0 0
\(841\) −23.0901 −0.796212
\(842\) 0 0
\(843\) − 69.2780i − 2.38606i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 22.5301i − 0.774144i
\(848\) 0 0
\(849\) 4.83652 0.165989
\(850\) 0 0
\(851\) −4.34767 −0.149036
\(852\) 0 0
\(853\) 32.6045i 1.11636i 0.829721 + 0.558179i \(0.188500\pi\)
−0.829721 + 0.558179i \(0.811500\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 4.25499i 0.145348i 0.997356 + 0.0726739i \(0.0231532\pi\)
−0.997356 + 0.0726739i \(0.976847\pi\)
\(858\) 0 0
\(859\) −24.5164 −0.836487 −0.418244 0.908335i \(-0.637354\pi\)
−0.418244 + 0.908335i \(0.637354\pi\)
\(860\) 0 0
\(861\) 11.2211 0.382414
\(862\) 0 0
\(863\) − 4.11643i − 0.140125i −0.997543 0.0700624i \(-0.977680\pi\)
0.997543 0.0700624i \(-0.0223198\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 21.6534i 0.735387i
\(868\) 0 0
\(869\) −10.9799 −0.372467
\(870\) 0 0
\(871\) −52.1244 −1.76617
\(872\) 0 0
\(873\) − 6.44384i − 0.218091i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3.17301i 0.107145i 0.998564 + 0.0535725i \(0.0170608\pi\)
−0.998564 + 0.0535725i \(0.982939\pi\)
\(878\) 0 0
\(879\) 74.6180 2.51680
\(880\) 0 0
\(881\) 16.0391 0.540371 0.270186 0.962808i \(-0.412915\pi\)
0.270186 + 0.962808i \(0.412915\pi\)
\(882\) 0 0
\(883\) − 36.5648i − 1.23050i −0.788330 0.615252i \(-0.789054\pi\)
0.788330 0.615252i \(-0.210946\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 44.6498i 1.49919i 0.661896 + 0.749596i \(0.269752\pi\)
−0.661896 + 0.749596i \(0.730248\pi\)
\(888\) 0 0
\(889\) −66.1755 −2.21945
\(890\) 0 0
\(891\) 28.3686 0.950382
\(892\) 0 0
\(893\) 24.3196i 0.813825i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 16.3477i − 0.545833i
\(898\) 0 0
\(899\) −7.76319 −0.258917
\(900\) 0 0
\(901\) −29.8908 −0.995807
\(902\) 0 0
\(903\) 85.5542i 2.84707i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 43.2692i 1.43673i 0.695667 + 0.718365i \(0.255109\pi\)
−0.695667 + 0.718365i \(0.744891\pi\)
\(908\) 0 0
\(909\) 8.10723 0.268900
\(910\) 0 0
\(911\) −1.64116 −0.0543739 −0.0271870 0.999630i \(-0.508655\pi\)
−0.0271870 + 0.999630i \(0.508655\pi\)
\(912\) 0 0
\(913\) − 27.0311i − 0.894598i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 42.7014i 1.41012i
\(918\) 0 0
\(919\) −6.67802 −0.220288 −0.110144 0.993916i \(-0.535131\pi\)
−0.110144 + 0.993916i \(0.535131\pi\)
\(920\) 0 0
\(921\) −28.2548 −0.931027
\(922\) 0 0
\(923\) 62.2074i 2.04758i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 2.13603i − 0.0701564i
\(928\) 0 0
\(929\) 23.3505 0.766104 0.383052 0.923727i \(-0.374873\pi\)
0.383052 + 0.923727i \(0.374873\pi\)
\(930\) 0 0
\(931\) 88.0508 2.88575
\(932\) 0 0
\(933\) 4.83433i 0.158269i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 45.9295i − 1.50045i −0.661182 0.750225i \(-0.729945\pi\)
0.661182 0.750225i \(-0.270055\pi\)
\(938\) 0 0
\(939\) 64.9238 2.11871
\(940\) 0 0
\(941\) −13.2488 −0.431899 −0.215949 0.976405i \(-0.569285\pi\)
−0.215949 + 0.976405i \(0.569285\pi\)
\(942\) 0 0
\(943\) 1.49482i 0.0486781i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 22.1017i 0.718207i 0.933298 + 0.359104i \(0.116918\pi\)
−0.933298 + 0.359104i \(0.883082\pi\)
\(948\) 0 0
\(949\) 40.8959 1.32754
\(950\) 0 0
\(951\) −63.2552 −2.05119
\(952\) 0 0
\(953\) − 18.2642i − 0.591635i −0.955244 0.295818i \(-0.904408\pi\)
0.955244 0.295818i \(-0.0955920\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 13.7860i 0.445637i
\(958\) 0 0
\(959\) −83.8760 −2.70850
\(960\) 0 0
\(961\) −20.8023 −0.671042
\(962\) 0 0
\(963\) − 0.614817i − 0.0198122i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 28.0843i 0.903132i 0.892238 + 0.451566i \(0.149134\pi\)
−0.892238 + 0.451566i \(0.850866\pi\)
\(968\) 0 0
\(969\) −28.1152 −0.903192
\(970\) 0 0
\(971\) −34.1505 −1.09594 −0.547972 0.836497i \(-0.684600\pi\)
−0.547972 + 0.836497i \(0.684600\pi\)
\(972\) 0 0
\(973\) 11.0566i 0.354458i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 41.4597i − 1.32641i −0.748437 0.663206i \(-0.769195\pi\)
0.748437 0.663206i \(-0.230805\pi\)
\(978\) 0 0
\(979\) 23.9765 0.766291
\(980\) 0 0
\(981\) 6.17898 0.197280
\(982\) 0 0
\(983\) − 34.1240i − 1.08839i −0.838960 0.544193i \(-0.816836\pi\)
0.838960 0.544193i \(-0.183164\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 57.8078i 1.84004i
\(988\) 0 0
\(989\) −11.3971 −0.362408
\(990\) 0 0
\(991\) −21.3535 −0.678315 −0.339158 0.940730i \(-0.610142\pi\)
−0.339158 + 0.940730i \(0.610142\pi\)
\(992\) 0 0
\(993\) − 10.6513i − 0.338009i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 19.3560i 0.613012i 0.951869 + 0.306506i \(0.0991600\pi\)
−0.951869 + 0.306506i \(0.900840\pi\)
\(998\) 0 0
\(999\) 7.03348 0.222529
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 4100.2.d.c.1149.6 8
5.2 odd 4 164.2.a.a.1.3 4
5.3 odd 4 4100.2.a.c.1.2 4
5.4 even 2 inner 4100.2.d.c.1149.3 8
15.2 even 4 1476.2.a.g.1.1 4
20.7 even 4 656.2.a.i.1.2 4
35.27 even 4 8036.2.a.i.1.2 4
40.27 even 4 2624.2.a.y.1.3 4
40.37 odd 4 2624.2.a.v.1.2 4
60.47 odd 4 5904.2.a.bp.1.1 4
205.122 odd 4 6724.2.a.c.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.2.a.a.1.3 4 5.2 odd 4
656.2.a.i.1.2 4 20.7 even 4
1476.2.a.g.1.1 4 15.2 even 4
2624.2.a.v.1.2 4 40.37 odd 4
2624.2.a.y.1.3 4 40.27 even 4
4100.2.a.c.1.2 4 5.3 odd 4
4100.2.d.c.1149.3 8 5.4 even 2 inner
4100.2.d.c.1149.6 8 1.1 even 1 trivial
5904.2.a.bp.1.1 4 60.47 odd 4
6724.2.a.c.1.2 4 205.122 odd 4
8036.2.a.i.1.2 4 35.27 even 4