Properties

Label 4100.2.d.c.1149.5
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 1149.5
Root \(-1.20518 - 1.20518i\) of defining polynomial
Character \(\chi\) \(=\) 4100.1149
Dual form 4100.2.d.c.1149.4

$q$-expansion

\(f(q)\) \(=\) \(q+0.0950939i q^{3} -3.14501i q^{7} +2.99096 q^{9} +O(q^{10})\) \(q+0.0950939i q^{3} -3.14501i q^{7} +2.99096 q^{9} -1.67570 q^{11} +6.63052i q^{13} -5.16120i q^{17} +4.72562 q^{19} +0.299072 q^{21} -8.82071i q^{23} +0.569704i q^{27} +1.80981 q^{29} -1.65951 q^{31} -0.159349i q^{33} +1.99096i q^{37} -0.630522 q^{39} -1.00000 q^{41} +1.46932i q^{43} +8.53543i q^{47} -2.89112 q^{49} +0.490799 q^{51} -9.35139i q^{53} +0.449377i q^{57} -8.82071 q^{59} +12.6305 q^{61} -9.40660i q^{63} -9.67570i q^{67} +0.838796 q^{69} -0.776081 q^{71} +8.33145i q^{73} +5.27009i q^{77} -0.915804 q^{79} +8.91870 q^{81} -10.0998i q^{83} +0.172102i q^{87} +6.44033 q^{89} +20.8531 q^{91} -0.157809i q^{93} +8.32241i q^{97} -5.01193 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 24q^{9} + O(q^{10}) \) \( 8q - 24q^{9} + 8q^{11} - 12q^{19} + 8q^{29} - 16q^{31} + 48q^{39} - 8q^{41} - 32q^{49} - 8q^{51} - 24q^{59} + 48q^{61} + 56q^{69} - 4q^{71} + 36q^{79} + 56q^{81} - 8q^{89} + 72q^{91} - 116q^{99} + O(q^{100}) \)

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
\(3\) 0.0950939i 0.0549025i 0.999623 + 0.0274513i \(0.00873910\pi\)
−0.999623 + 0.0274513i \(0.991261\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 3.14501i − 1.18870i −0.804205 0.594352i \(-0.797409\pi\)
0.804205 0.594352i \(-0.202591\pi\)
\(8\) 0 0
\(9\) 2.99096 0.996986
\(10\) 0 0
\(11\) −1.67570 −0.505241 −0.252621 0.967565i \(-0.581292\pi\)
−0.252621 + 0.967565i \(0.581292\pi\)
\(12\) 0 0
\(13\) 6.63052i 1.83898i 0.393118 + 0.919488i \(0.371396\pi\)
−0.393118 + 0.919488i \(0.628604\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 5.16120i − 1.25178i −0.779913 0.625888i \(-0.784737\pi\)
0.779913 0.625888i \(-0.215263\pi\)
\(18\) 0 0
\(19\) 4.72562 1.08413 0.542065 0.840336i \(-0.317643\pi\)
0.542065 + 0.840336i \(0.317643\pi\)
\(20\) 0 0
\(21\) 0.299072 0.0652628
\(22\) 0 0
\(23\) − 8.82071i − 1.83925i −0.392803 0.919623i \(-0.628495\pi\)
0.392803 0.919623i \(-0.371505\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0.569704i 0.109640i
\(28\) 0 0
\(29\) 1.80981 0.336074 0.168037 0.985781i \(-0.446257\pi\)
0.168037 + 0.985781i \(0.446257\pi\)
\(30\) 0 0
\(31\) −1.65951 −0.298056 −0.149028 0.988833i \(-0.547614\pi\)
−0.149028 + 0.988833i \(0.547614\pi\)
\(32\) 0 0
\(33\) − 0.159349i − 0.0277390i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.99096i 0.327311i 0.986518 + 0.163656i \(0.0523286\pi\)
−0.986518 + 0.163656i \(0.947671\pi\)
\(38\) 0 0
\(39\) −0.630522 −0.100964
\(40\) 0 0
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) 1.46932i 0.224069i 0.993704 + 0.112034i \(0.0357367\pi\)
−0.993704 + 0.112034i \(0.964263\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.53543i 1.24502i 0.782612 + 0.622510i \(0.213887\pi\)
−0.782612 + 0.622510i \(0.786113\pi\)
\(48\) 0 0
\(49\) −2.89112 −0.413017
\(50\) 0 0
\(51\) 0.490799 0.0687256
\(52\) 0 0
\(53\) − 9.35139i − 1.28451i −0.766490 0.642256i \(-0.777999\pi\)
0.766490 0.642256i \(-0.222001\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.449377i 0.0595215i
\(58\) 0 0
\(59\) −8.82071 −1.14836 −0.574179 0.818730i \(-0.694679\pi\)
−0.574179 + 0.818730i \(0.694679\pi\)
\(60\) 0 0
\(61\) 12.6305 1.61717 0.808586 0.588378i \(-0.200233\pi\)
0.808586 + 0.588378i \(0.200233\pi\)
\(62\) 0 0
\(63\) − 9.40660i − 1.18512i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 9.67570i − 1.18207i −0.806644 0.591037i \(-0.798719\pi\)
0.806644 0.591037i \(-0.201281\pi\)
\(68\) 0 0
\(69\) 0.838796 0.100979
\(70\) 0 0
\(71\) −0.776081 −0.0921039 −0.0460519 0.998939i \(-0.514664\pi\)
−0.0460519 + 0.998939i \(0.514664\pi\)
\(72\) 0 0
\(73\) 8.33145i 0.975123i 0.873089 + 0.487561i \(0.162113\pi\)
−0.873089 + 0.487561i \(0.837887\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.27009i 0.600582i
\(78\) 0 0
\(79\) −0.915804 −0.103036 −0.0515180 0.998672i \(-0.516406\pi\)
−0.0515180 + 0.998672i \(0.516406\pi\)
\(80\) 0 0
\(81\) 8.91870 0.990966
\(82\) 0 0
\(83\) − 10.0998i − 1.10860i −0.832316 0.554301i \(-0.812986\pi\)
0.832316 0.554301i \(-0.187014\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.172102i 0.0184513i
\(88\) 0 0
\(89\) 6.44033 0.682674 0.341337 0.939941i \(-0.389120\pi\)
0.341337 + 0.939941i \(0.389120\pi\)
\(90\) 0 0
\(91\) 20.8531 2.18600
\(92\) 0 0
\(93\) − 0.157809i − 0.0163640i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.32241i 0.845012i 0.906360 + 0.422506i \(0.138850\pi\)
−0.906360 + 0.422506i \(0.861150\pi\)
\(98\) 0 0
\(99\) −5.01193 −0.503718
\(100\) 0 0
\(101\) 9.79173 0.974313 0.487157 0.873315i \(-0.338034\pi\)
0.487157 + 0.873315i \(0.338034\pi\)
\(102\) 0 0
\(103\) − 0.648608i − 0.0639093i −0.999489 0.0319546i \(-0.989827\pi\)
0.999489 0.0319546i \(-0.0101732\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 16.8026i − 1.62437i −0.583399 0.812186i \(-0.698278\pi\)
0.583399 0.812186i \(-0.301722\pi\)
\(108\) 0 0
\(109\) −3.84969 −0.368734 −0.184367 0.982857i \(-0.559023\pi\)
−0.184367 + 0.982857i \(0.559023\pi\)
\(110\) 0 0
\(111\) −0.189328 −0.0179702
\(112\) 0 0
\(113\) − 7.65046i − 0.719695i −0.933011 0.359848i \(-0.882829\pi\)
0.933011 0.359848i \(-0.117171\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 19.8316i 1.83343i
\(118\) 0 0
\(119\) −16.2321 −1.48799
\(120\) 0 0
\(121\) −8.19204 −0.744731
\(122\) 0 0
\(123\) − 0.0950939i − 0.00857433i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 14.9206i − 1.32398i −0.749511 0.661992i \(-0.769711\pi\)
0.749511 0.661992i \(-0.230289\pi\)
\(128\) 0 0
\(129\) −0.139723 −0.0123019
\(130\) 0 0
\(131\) 4.19019 0.366098 0.183049 0.983104i \(-0.441403\pi\)
0.183049 + 0.983104i \(0.441403\pi\)
\(132\) 0 0
\(133\) − 14.8621i − 1.28871i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 4.09984i − 0.350273i −0.984544 0.175137i \(-0.943963\pi\)
0.984544 0.175137i \(-0.0560367\pi\)
\(138\) 0 0
\(139\) 11.9819 1.01629 0.508146 0.861271i \(-0.330331\pi\)
0.508146 + 0.861271i \(0.330331\pi\)
\(140\) 0 0
\(141\) −0.811668 −0.0683547
\(142\) 0 0
\(143\) − 11.1107i − 0.929127i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 0.274928i − 0.0226756i
\(148\) 0 0
\(149\) 1.80981 0.148266 0.0741328 0.997248i \(-0.476381\pi\)
0.0741328 + 0.997248i \(0.476381\pi\)
\(150\) 0 0
\(151\) −21.3561 −1.73794 −0.868969 0.494867i \(-0.835217\pi\)
−0.868969 + 0.494867i \(0.835217\pi\)
\(152\) 0 0
\(153\) − 15.4369i − 1.24800i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 13.1107i − 1.04635i −0.852225 0.523175i \(-0.824747\pi\)
0.852225 0.523175i \(-0.175253\pi\)
\(158\) 0 0
\(159\) 0.889261 0.0705230
\(160\) 0 0
\(161\) −27.7413 −2.18632
\(162\) 0 0
\(163\) 10.7808i 0.844420i 0.906498 + 0.422210i \(0.138745\pi\)
−0.906498 + 0.422210i \(0.861255\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 13.6757i − 1.05826i −0.848542 0.529129i \(-0.822519\pi\)
0.848542 0.529129i \(-0.177481\pi\)
\(168\) 0 0
\(169\) −30.9638 −2.38183
\(170\) 0 0
\(171\) 14.1341 1.08086
\(172\) 0 0
\(173\) 10.5801i 0.804387i 0.915555 + 0.402193i \(0.131752\pi\)
−0.915555 + 0.402193i \(0.868248\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 0.838796i − 0.0630478i
\(178\) 0 0
\(179\) 1.23536 0.0923352 0.0461676 0.998934i \(-0.485299\pi\)
0.0461676 + 0.998934i \(0.485299\pi\)
\(180\) 0 0
\(181\) −3.90965 −0.290602 −0.145301 0.989387i \(-0.546415\pi\)
−0.145301 + 0.989387i \(0.546415\pi\)
\(182\) 0 0
\(183\) 1.20109i 0.0887868i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 8.64861i 0.632449i
\(188\) 0 0
\(189\) 1.79173 0.130329
\(190\) 0 0
\(191\) −7.31712 −0.529448 −0.264724 0.964324i \(-0.585281\pi\)
−0.264724 + 0.964324i \(0.585281\pi\)
\(192\) 0 0
\(193\) − 13.3009i − 0.957422i −0.877973 0.478711i \(-0.841104\pi\)
0.877973 0.478711i \(-0.158896\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.5692i 1.10926i 0.832098 + 0.554628i \(0.187140\pi\)
−0.832098 + 0.554628i \(0.812860\pi\)
\(198\) 0 0
\(199\) 27.8972 1.97758 0.988789 0.149319i \(-0.0477080\pi\)
0.988789 + 0.149319i \(0.0477080\pi\)
\(200\) 0 0
\(201\) 0.920100 0.0648989
\(202\) 0 0
\(203\) − 5.69189i − 0.399492i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 26.3824i − 1.83370i
\(208\) 0 0
\(209\) −7.91870 −0.547748
\(210\) 0 0
\(211\) 18.1484 1.24939 0.624694 0.780870i \(-0.285224\pi\)
0.624694 + 0.780870i \(0.285224\pi\)
\(212\) 0 0
\(213\) − 0.0738006i − 0.00505673i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.21917i 0.354300i
\(218\) 0 0
\(219\) −0.792270 −0.0535367
\(220\) 0 0
\(221\) 34.2215 2.30199
\(222\) 0 0
\(223\) − 6.55826i − 0.439174i −0.975593 0.219587i \(-0.929529\pi\)
0.975593 0.219587i \(-0.0704709\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 23.4884i − 1.55898i −0.626416 0.779489i \(-0.715479\pi\)
0.626416 0.779489i \(-0.284521\pi\)
\(228\) 0 0
\(229\) −12.1503 −0.802915 −0.401457 0.915878i \(-0.631496\pi\)
−0.401457 + 0.915878i \(0.631496\pi\)
\(230\) 0 0
\(231\) −0.501153 −0.0329735
\(232\) 0 0
\(233\) − 21.8422i − 1.43093i −0.698649 0.715465i \(-0.746215\pi\)
0.698649 0.715465i \(-0.253785\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 0.0870874i − 0.00565694i
\(238\) 0 0
\(239\) −21.5968 −1.39698 −0.698490 0.715620i \(-0.746144\pi\)
−0.698490 + 0.715620i \(0.746144\pi\)
\(240\) 0 0
\(241\) 12.6305 0.813603 0.406802 0.913516i \(-0.366644\pi\)
0.406802 + 0.913516i \(0.366644\pi\)
\(242\) 0 0
\(243\) 2.55723i 0.164046i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 31.3333i 1.99369i
\(248\) 0 0
\(249\) 0.960434 0.0608650
\(250\) 0 0
\(251\) 20.7702 1.31101 0.655503 0.755192i \(-0.272457\pi\)
0.655503 + 0.755192i \(0.272457\pi\)
\(252\) 0 0
\(253\) 14.7808i 0.929263i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 14.8207i − 0.924491i −0.886752 0.462245i \(-0.847044\pi\)
0.886752 0.462245i \(-0.152956\pi\)
\(258\) 0 0
\(259\) 6.26159 0.389076
\(260\) 0 0
\(261\) 5.41307 0.335061
\(262\) 0 0
\(263\) 18.4175i 1.13567i 0.823142 + 0.567836i \(0.192219\pi\)
−0.823142 + 0.567836i \(0.807781\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0.612437i 0.0374805i
\(268\) 0 0
\(269\) 25.8916 1.57864 0.789318 0.613984i \(-0.210434\pi\)
0.789318 + 0.613984i \(0.210434\pi\)
\(270\) 0 0
\(271\) −17.5017 −1.06315 −0.531576 0.847010i \(-0.678400\pi\)
−0.531576 + 0.847010i \(0.678400\pi\)
\(272\) 0 0
\(273\) 1.98300i 0.120017i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 8.77179i − 0.527045i −0.964653 0.263523i \(-0.915116\pi\)
0.964653 0.263523i \(-0.0848844\pi\)
\(278\) 0 0
\(279\) −4.96351 −0.297158
\(280\) 0 0
\(281\) 16.2215 0.967692 0.483846 0.875153i \(-0.339239\pi\)
0.483846 + 0.875153i \(0.339239\pi\)
\(282\) 0 0
\(283\) − 11.9819i − 0.712251i −0.934438 0.356125i \(-0.884098\pi\)
0.934438 0.356125i \(-0.115902\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.14501i 0.185644i
\(288\) 0 0
\(289\) −9.63803 −0.566943
\(290\) 0 0
\(291\) −0.791411 −0.0463933
\(292\) 0 0
\(293\) − 11.2516i − 0.657323i −0.944448 0.328661i \(-0.893403\pi\)
0.944448 0.328661i \(-0.106597\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 0.954650i − 0.0553944i
\(298\) 0 0
\(299\) 58.4859 3.38233
\(300\) 0 0
\(301\) 4.62103 0.266352
\(302\) 0 0
\(303\) 0.931134i 0.0534922i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 8.61244i − 0.491538i −0.969328 0.245769i \(-0.920960\pi\)
0.969328 0.245769i \(-0.0790404\pi\)
\(308\) 0 0
\(309\) 0.0616787 0.00350878
\(310\) 0 0
\(311\) 31.7575 1.80080 0.900400 0.435063i \(-0.143274\pi\)
0.900400 + 0.435063i \(0.143274\pi\)
\(312\) 0 0
\(313\) 2.21777i 0.125356i 0.998034 + 0.0626778i \(0.0199641\pi\)
−0.998034 + 0.0626778i \(0.980036\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.3947i 1.03315i 0.856243 + 0.516574i \(0.172793\pi\)
−0.856243 + 0.516574i \(0.827207\pi\)
\(318\) 0 0
\(319\) −3.03269 −0.169798
\(320\) 0 0
\(321\) 1.59783 0.0891820
\(322\) 0 0
\(323\) − 24.3899i − 1.35709i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 0.366083i − 0.0202444i
\(328\) 0 0
\(329\) 26.8440 1.47996
\(330\) 0 0
\(331\) 10.1160 0.556027 0.278014 0.960577i \(-0.410324\pi\)
0.278014 + 0.960577i \(0.410324\pi\)
\(332\) 0 0
\(333\) 5.95487i 0.326325i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 7.81135i − 0.425511i −0.977105 0.212756i \(-0.931756\pi\)
0.977105 0.212756i \(-0.0682439\pi\)
\(338\) 0 0
\(339\) 0.727513 0.0395131
\(340\) 0 0
\(341\) 2.78083 0.150590
\(342\) 0 0
\(343\) − 12.9225i − 0.697749i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.6281i 0.838959i 0.907765 + 0.419480i \(0.137787\pi\)
−0.907765 + 0.419480i \(0.862213\pi\)
\(348\) 0 0
\(349\) −7.49265 −0.401073 −0.200536 0.979686i \(-0.564268\pi\)
−0.200536 + 0.979686i \(0.564268\pi\)
\(350\) 0 0
\(351\) −3.77743 −0.201624
\(352\) 0 0
\(353\) 22.4313i 1.19390i 0.802279 + 0.596949i \(0.203620\pi\)
−0.802279 + 0.596949i \(0.796380\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 1.54357i − 0.0816944i
\(358\) 0 0
\(359\) 15.6196 0.824372 0.412186 0.911100i \(-0.364765\pi\)
0.412186 + 0.911100i \(0.364765\pi\)
\(360\) 0 0
\(361\) 3.33145 0.175340
\(362\) 0 0
\(363\) − 0.779014i − 0.0408876i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3.82790i 0.199815i 0.994997 + 0.0999073i \(0.0318546\pi\)
−0.994997 + 0.0999073i \(0.968145\pi\)
\(368\) 0 0
\(369\) −2.99096 −0.155703
\(370\) 0 0
\(371\) −29.4103 −1.52690
\(372\) 0 0
\(373\) − 18.1997i − 0.942344i −0.882041 0.471172i \(-0.843831\pi\)
0.882041 0.471172i \(-0.156169\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) 14.7627 0.758311 0.379156 0.925333i \(-0.376215\pi\)
0.379156 + 0.925333i \(0.376215\pi\)
\(380\) 0 0
\(381\) 1.41885 0.0726901
\(382\) 0 0
\(383\) 4.75406i 0.242921i 0.992596 + 0.121460i \(0.0387578\pi\)
−0.992596 + 0.121460i \(0.961242\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.39467i 0.223394i
\(388\) 0 0
\(389\) −14.3804 −0.729114 −0.364557 0.931181i \(-0.618780\pi\)
−0.364557 + 0.931181i \(0.618780\pi\)
\(390\) 0 0
\(391\) −45.5255 −2.30232
\(392\) 0 0
\(393\) 0.398461i 0.0200997i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 10.9311i 0.548618i 0.961642 + 0.274309i \(0.0884491\pi\)
−0.961642 + 0.274309i \(0.911551\pi\)
\(398\) 0 0
\(399\) 1.41330 0.0707534
\(400\) 0 0
\(401\) 30.4132 1.51876 0.759382 0.650646i \(-0.225502\pi\)
0.759382 + 0.650646i \(0.225502\pi\)
\(402\) 0 0
\(403\) − 11.0034i − 0.548118i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 3.33624i − 0.165371i
\(408\) 0 0
\(409\) −13.3100 −0.658136 −0.329068 0.944306i \(-0.606734\pi\)
−0.329068 + 0.944306i \(0.606734\pi\)
\(410\) 0 0
\(411\) 0.389870 0.0192309
\(412\) 0 0
\(413\) 27.7413i 1.36506i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.13941i 0.0557970i
\(418\) 0 0
\(419\) −15.7917 −0.771476 −0.385738 0.922608i \(-0.626053\pi\)
−0.385738 + 0.922608i \(0.626053\pi\)
\(420\) 0 0
\(421\) −34.4317 −1.67810 −0.839050 0.544054i \(-0.816889\pi\)
−0.839050 + 0.544054i \(0.816889\pi\)
\(422\) 0 0
\(423\) 25.5291i 1.24127i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 39.7232i − 1.92234i
\(428\) 0 0
\(429\) 1.05656 0.0510114
\(430\) 0 0
\(431\) 5.69189 0.274168 0.137084 0.990559i \(-0.456227\pi\)
0.137084 + 0.990559i \(0.456227\pi\)
\(432\) 0 0
\(433\) − 13.0034i − 0.624903i −0.949934 0.312452i \(-0.898850\pi\)
0.949934 0.312452i \(-0.101150\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 41.6833i − 1.99398i
\(438\) 0 0
\(439\) −9.59593 −0.457989 −0.228994 0.973428i \(-0.573544\pi\)
−0.228994 + 0.973428i \(0.573544\pi\)
\(440\) 0 0
\(441\) −8.64720 −0.411772
\(442\) 0 0
\(443\) − 2.47072i − 0.117388i −0.998276 0.0586938i \(-0.981306\pi\)
0.998276 0.0586938i \(-0.0186936\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0.172102i 0.00814015i
\(448\) 0 0
\(449\) −35.2324 −1.66272 −0.831359 0.555735i \(-0.812437\pi\)
−0.831359 + 0.555735i \(0.812437\pi\)
\(450\) 0 0
\(451\) 1.67570 0.0789054
\(452\) 0 0
\(453\) − 2.03084i − 0.0954172i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 37.0241i 1.73191i 0.500118 + 0.865957i \(0.333290\pi\)
−0.500118 + 0.865957i \(0.666710\pi\)
\(458\) 0 0
\(459\) 2.94036 0.137244
\(460\) 0 0
\(461\) −27.6324 −1.28697 −0.643484 0.765460i \(-0.722512\pi\)
−0.643484 + 0.765460i \(0.722512\pi\)
\(462\) 0 0
\(463\) 23.1877i 1.07763i 0.842425 + 0.538813i \(0.181127\pi\)
−0.842425 + 0.538813i \(0.818873\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 27.5235i 1.27364i 0.771014 + 0.636818i \(0.219750\pi\)
−0.771014 + 0.636818i \(0.780250\pi\)
\(468\) 0 0
\(469\) −30.4302 −1.40514
\(470\) 0 0
\(471\) 1.24675 0.0574473
\(472\) 0 0
\(473\) − 2.46213i − 0.113209i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 27.9696i − 1.28064i
\(478\) 0 0
\(479\) −11.1088 −0.507576 −0.253788 0.967260i \(-0.581677\pi\)
−0.253788 + 0.967260i \(0.581677\pi\)
\(480\) 0 0
\(481\) −13.2011 −0.601918
\(482\) 0 0
\(483\) − 2.63803i − 0.120034i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 31.3933i 1.42256i 0.702906 + 0.711282i \(0.251885\pi\)
−0.702906 + 0.711282i \(0.748115\pi\)
\(488\) 0 0
\(489\) −1.02519 −0.0463608
\(490\) 0 0
\(491\) 39.9590 1.80333 0.901663 0.432440i \(-0.142347\pi\)
0.901663 + 0.432440i \(0.142347\pi\)
\(492\) 0 0
\(493\) − 9.34081i − 0.420689i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.44079i 0.109484i
\(498\) 0 0
\(499\) −16.0951 −0.720515 −0.360258 0.932853i \(-0.617311\pi\)
−0.360258 + 0.932853i \(0.617311\pi\)
\(500\) 0 0
\(501\) 1.30048 0.0581010
\(502\) 0 0
\(503\) 5.97717i 0.266509i 0.991082 + 0.133254i \(0.0425427\pi\)
−0.991082 + 0.133254i \(0.957457\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 2.94447i − 0.130769i
\(508\) 0 0
\(509\) 12.5726 0.557269 0.278634 0.960397i \(-0.410118\pi\)
0.278634 + 0.960397i \(0.410118\pi\)
\(510\) 0 0
\(511\) 26.2025 1.15913
\(512\) 0 0
\(513\) 2.69220i 0.118864i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 14.3028i − 0.629036i
\(518\) 0 0
\(519\) −1.00610 −0.0441629
\(520\) 0 0
\(521\) −31.6233 −1.38544 −0.692722 0.721205i \(-0.743589\pi\)
−0.692722 + 0.721205i \(0.743589\pi\)
\(522\) 0 0
\(523\) − 21.5405i − 0.941900i −0.882160 0.470950i \(-0.843911\pi\)
0.882160 0.470950i \(-0.156089\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.56505i 0.373099i
\(528\) 0 0
\(529\) −54.8049 −2.38282
\(530\) 0 0
\(531\) −26.3824 −1.14490
\(532\) 0 0
\(533\) − 6.63052i − 0.287200i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0.117475i 0.00506944i
\(538\) 0 0
\(539\) 4.84463 0.208673
\(540\) 0 0
\(541\) 31.4927 1.35397 0.676987 0.735995i \(-0.263285\pi\)
0.676987 + 0.735995i \(0.263285\pi\)
\(542\) 0 0
\(543\) − 0.371784i − 0.0159548i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 17.6861i 0.756201i 0.925765 + 0.378100i \(0.123423\pi\)
−0.925765 + 0.378100i \(0.876577\pi\)
\(548\) 0 0
\(549\) 37.7774 1.61230
\(550\) 0 0
\(551\) 8.55248 0.364348
\(552\) 0 0
\(553\) 2.88022i 0.122479i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 19.3296i 0.819021i 0.912305 + 0.409511i \(0.134301\pi\)
−0.912305 + 0.409511i \(0.865699\pi\)
\(558\) 0 0
\(559\) −9.74235 −0.412057
\(560\) 0 0
\(561\) −0.822430 −0.0347230
\(562\) 0 0
\(563\) 35.7299i 1.50583i 0.658115 + 0.752917i \(0.271354\pi\)
−0.658115 + 0.752917i \(0.728646\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 28.0494i − 1.17797i
\(568\) 0 0
\(569\) −4.65187 −0.195016 −0.0975082 0.995235i \(-0.531087\pi\)
−0.0975082 + 0.995235i \(0.531087\pi\)
\(570\) 0 0
\(571\) −20.6791 −0.865393 −0.432697 0.901540i \(-0.642438\pi\)
−0.432697 + 0.901540i \(0.642438\pi\)
\(572\) 0 0
\(573\) − 0.695813i − 0.0290680i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 24.7808i 1.03164i 0.856697 + 0.515820i \(0.172513\pi\)
−0.856697 + 0.515820i \(0.827487\pi\)
\(578\) 0 0
\(579\) 1.26484 0.0525649
\(580\) 0 0
\(581\) −31.7641 −1.31780
\(582\) 0 0
\(583\) 15.6701i 0.648989i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.4027i 0.553187i 0.960987 + 0.276594i \(0.0892056\pi\)
−0.960987 + 0.276594i \(0.910794\pi\)
\(588\) 0 0
\(589\) −7.84219 −0.323132
\(590\) 0 0
\(591\) −1.48053 −0.0609010
\(592\) 0 0
\(593\) − 1.83911i − 0.0755233i −0.999287 0.0377616i \(-0.987977\pi\)
0.999287 0.0377616i \(-0.0120228\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 2.65285i 0.108574i
\(598\) 0 0
\(599\) 35.9457 1.46870 0.734352 0.678769i \(-0.237486\pi\)
0.734352 + 0.678769i \(0.237486\pi\)
\(600\) 0 0
\(601\) −29.8858 −1.21907 −0.609533 0.792760i \(-0.708643\pi\)
−0.609533 + 0.792760i \(0.708643\pi\)
\(602\) 0 0
\(603\) − 28.9396i − 1.17851i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 4.25015i − 0.172508i −0.996273 0.0862541i \(-0.972510\pi\)
0.996273 0.0862541i \(-0.0274897\pi\)
\(608\) 0 0
\(609\) 0.541264 0.0219331
\(610\) 0 0
\(611\) −56.5944 −2.28956
\(612\) 0 0
\(613\) − 9.35184i − 0.377717i −0.982004 0.188859i \(-0.939521\pi\)
0.982004 0.188859i \(-0.0604788\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 12.6019i − 0.507332i −0.967292 0.253666i \(-0.918364\pi\)
0.967292 0.253666i \(-0.0816363\pi\)
\(618\) 0 0
\(619\) 2.54018 0.102098 0.0510491 0.998696i \(-0.483743\pi\)
0.0510491 + 0.998696i \(0.483743\pi\)
\(620\) 0 0
\(621\) 5.02519 0.201654
\(622\) 0 0
\(623\) − 20.2549i − 0.811497i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 0.753020i − 0.0300727i
\(628\) 0 0
\(629\) 10.2757 0.409720
\(630\) 0 0
\(631\) 48.0057 1.91108 0.955538 0.294867i \(-0.0952752\pi\)
0.955538 + 0.294867i \(0.0952752\pi\)
\(632\) 0 0
\(633\) 1.72580i 0.0685945i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 19.1696i − 0.759528i
\(638\) 0 0
\(639\) −2.32123 −0.0918262
\(640\) 0 0
\(641\) 1.02148 0.0403461 0.0201730 0.999797i \(-0.493578\pi\)
0.0201730 + 0.999797i \(0.493578\pi\)
\(642\) 0 0
\(643\) − 8.40606i − 0.331503i −0.986168 0.165751i \(-0.946995\pi\)
0.986168 0.165751i \(-0.0530049\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 5.86778i − 0.230686i −0.993326 0.115343i \(-0.963203\pi\)
0.993326 0.115343i \(-0.0367968\pi\)
\(648\) 0 0
\(649\) 14.7808 0.580198
\(650\) 0 0
\(651\) −0.496312 −0.0194520
\(652\) 0 0
\(653\) 37.5749i 1.47042i 0.677841 + 0.735209i \(0.262916\pi\)
−0.677841 + 0.735209i \(0.737084\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 24.9190i 0.972183i
\(658\) 0 0
\(659\) 16.8360 0.655839 0.327919 0.944706i \(-0.393653\pi\)
0.327919 + 0.944706i \(0.393653\pi\)
\(660\) 0 0
\(661\) 4.84983 0.188637 0.0943183 0.995542i \(-0.469933\pi\)
0.0943183 + 0.995542i \(0.469933\pi\)
\(662\) 0 0
\(663\) 3.25426i 0.126385i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 15.9638i − 0.618122i
\(668\) 0 0
\(669\) 0.623651 0.0241117
\(670\) 0 0
\(671\) −21.1649 −0.817062
\(672\) 0 0
\(673\) 6.24065i 0.240559i 0.992740 + 0.120280i \(0.0383792\pi\)
−0.992740 + 0.120280i \(0.961621\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 29.3566i 1.12827i 0.825683 + 0.564134i \(0.190790\pi\)
−0.825683 + 0.564134i \(0.809210\pi\)
\(678\) 0 0
\(679\) 26.1741 1.00447
\(680\) 0 0
\(681\) 2.23360 0.0855918
\(682\) 0 0
\(683\) − 25.6861i − 0.982849i −0.870920 0.491425i \(-0.836476\pi\)
0.870920 0.491425i \(-0.163524\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 1.15542i − 0.0440820i
\(688\) 0 0
\(689\) 62.0046 2.36219
\(690\) 0 0
\(691\) 8.77128 0.333675 0.166838 0.985984i \(-0.446644\pi\)
0.166838 + 0.985984i \(0.446644\pi\)
\(692\) 0 0
\(693\) 15.7626i 0.598772i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 5.16120i 0.195495i
\(698\) 0 0
\(699\) 2.07706 0.0785616
\(700\) 0 0
\(701\) −15.1075 −0.570602 −0.285301 0.958438i \(-0.592094\pi\)
−0.285301 + 0.958438i \(0.592094\pi\)
\(702\) 0 0
\(703\) 9.40850i 0.354848i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 30.7951i − 1.15817i
\(708\) 0 0
\(709\) 12.0132 0.451165 0.225583 0.974224i \(-0.427571\pi\)
0.225583 + 0.974224i \(0.427571\pi\)
\(710\) 0 0
\(711\) −2.73913 −0.102725
\(712\) 0 0
\(713\) 14.6380i 0.548198i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 2.05372i − 0.0766977i
\(718\) 0 0
\(719\) 2.27778 0.0849468 0.0424734 0.999098i \(-0.486476\pi\)
0.0424734 + 0.999098i \(0.486476\pi\)
\(720\) 0 0
\(721\) −2.03988 −0.0759692
\(722\) 0 0
\(723\) 1.20109i 0.0446689i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 7.71952i − 0.286301i −0.989701 0.143151i \(-0.954277\pi\)
0.989701 0.143151i \(-0.0457233\pi\)
\(728\) 0 0
\(729\) 26.5129 0.981960
\(730\) 0 0
\(731\) 7.58345 0.280484
\(732\) 0 0
\(733\) 33.2753i 1.22905i 0.788896 + 0.614526i \(0.210653\pi\)
−0.788896 + 0.614526i \(0.789347\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.2135i 0.597233i
\(738\) 0 0
\(739\) −48.0057 −1.76592 −0.882959 0.469450i \(-0.844452\pi\)
−0.882959 + 0.469450i \(0.844452\pi\)
\(740\) 0 0
\(741\) −2.97961 −0.109459
\(742\) 0 0
\(743\) 43.1345i 1.58245i 0.611524 + 0.791226i \(0.290557\pi\)
−0.611524 + 0.791226i \(0.709443\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 30.2082i − 1.10526i
\(748\) 0 0
\(749\) −52.8445 −1.93090
\(750\) 0 0
\(751\) −6.46883 −0.236051 −0.118025 0.993011i \(-0.537656\pi\)
−0.118025 + 0.993011i \(0.537656\pi\)
\(752\) 0 0
\(753\) 1.97512i 0.0719775i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 39.5531i 1.43758i 0.695227 + 0.718790i \(0.255304\pi\)
−0.695227 + 0.718790i \(0.744696\pi\)
\(758\) 0 0
\(759\) −1.40557 −0.0510189
\(760\) 0 0
\(761\) 28.5130 1.03360 0.516799 0.856107i \(-0.327124\pi\)
0.516799 + 0.856107i \(0.327124\pi\)
\(762\) 0 0
\(763\) 12.1073i 0.438315i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 58.4859i − 2.11180i
\(768\) 0 0
\(769\) −49.7184 −1.79289 −0.896445 0.443154i \(-0.853859\pi\)
−0.896445 + 0.443154i \(0.853859\pi\)
\(770\) 0 0
\(771\) 1.40936 0.0507569
\(772\) 0 0
\(773\) − 24.2539i − 0.872351i −0.899862 0.436175i \(-0.856333\pi\)
0.899862 0.436175i \(-0.143667\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0.595439i 0.0213613i
\(778\) 0 0
\(779\) −4.72562 −0.169313
\(780\) 0 0
\(781\) 1.30048 0.0465347
\(782\) 0 0
\(783\) 1.03106i 0.0368470i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 0.907346i − 0.0323434i −0.999869 0.0161717i \(-0.994852\pi\)
0.999869 0.0161717i \(-0.00514784\pi\)
\(788\) 0 0
\(789\) −1.75139 −0.0623512
\(790\) 0 0
\(791\) −24.0608 −0.855504
\(792\) 0 0
\(793\) 83.7470i 2.97394i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 32.4934i 1.15098i 0.817810 + 0.575488i \(0.195188\pi\)
−0.817810 + 0.575488i \(0.804812\pi\)
\(798\) 0 0
\(799\) 44.0531 1.55849
\(800\) 0 0
\(801\) 19.2628 0.680616
\(802\) 0 0
\(803\) − 13.9610i − 0.492672i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 2.46213i 0.0866711i
\(808\) 0 0
\(809\) 2.07804 0.0730602 0.0365301 0.999333i \(-0.488370\pi\)
0.0365301 + 0.999333i \(0.488370\pi\)
\(810\) 0 0
\(811\) −13.7518 −0.482893 −0.241446 0.970414i \(-0.577622\pi\)
−0.241446 + 0.970414i \(0.577622\pi\)
\(812\) 0 0
\(813\) − 1.66431i − 0.0583697i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 6.94344i 0.242920i
\(818\) 0 0
\(819\) 62.3707 2.17941
\(820\) 0 0
\(821\) 6.76979 0.236267 0.118134 0.992998i \(-0.462309\pi\)
0.118134 + 0.992998i \(0.462309\pi\)
\(822\) 0 0
\(823\) 22.3575i 0.779335i 0.920956 + 0.389667i \(0.127410\pi\)
−0.920956 + 0.389667i \(0.872590\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 26.1160i − 0.908143i −0.890965 0.454072i \(-0.849971\pi\)
0.890965 0.454072i \(-0.150029\pi\)
\(828\) 0 0
\(829\) 19.7540 0.686085 0.343043 0.939320i \(-0.388542\pi\)
0.343043 + 0.939320i \(0.388542\pi\)
\(830\) 0 0
\(831\) 0.834144 0.0289361
\(832\) 0 0
\(833\) 14.9216i 0.517004i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 0.945427i − 0.0326787i
\(838\) 0 0
\(839\) −20.8188 −0.718745 −0.359373 0.933194i \(-0.617009\pi\)
−0.359373 + 0.933194i \(0.617009\pi\)
\(840\) 0 0
\(841\) −25.7246 −0.887054
\(842\) 0 0
\(843\) 1.54256i 0.0531287i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 25.7641i 0.885265i
\(848\) 0 0
\(849\) 1.13941 0.0391044
\(850\) 0 0
\(851\) 17.5617 0.602006
\(852\) 0 0
\(853\) 48.6666i 1.66631i 0.553037 + 0.833157i \(0.313469\pi\)
−0.553037 + 0.833157i \(0.686531\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 12.8515i 0.439001i 0.975612 + 0.219500i \(0.0704427\pi\)
−0.975612 + 0.219500i \(0.929557\pi\)
\(858\) 0 0
\(859\) 1.60904 0.0548998 0.0274499 0.999623i \(-0.491261\pi\)
0.0274499 + 0.999623i \(0.491261\pi\)
\(860\) 0 0
\(861\) −0.299072 −0.0101923
\(862\) 0 0
\(863\) − 46.0494i − 1.56754i −0.621052 0.783770i \(-0.713294\pi\)
0.621052 0.783770i \(-0.286706\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 0.916518i − 0.0311266i
\(868\) 0 0
\(869\) 1.53461 0.0520581
\(870\) 0 0
\(871\) 64.1549 2.17381
\(872\) 0 0
\(873\) 24.8920i 0.842465i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 3.63392i − 0.122709i −0.998116 0.0613543i \(-0.980458\pi\)
0.998116 0.0613543i \(-0.0195420\pi\)
\(878\) 0 0
\(879\) 1.06995 0.0360887
\(880\) 0 0
\(881\) 28.2426 0.951519 0.475759 0.879575i \(-0.342173\pi\)
0.475759 + 0.879575i \(0.342173\pi\)
\(882\) 0 0
\(883\) − 4.23962i − 0.142674i −0.997452 0.0713372i \(-0.977273\pi\)
0.997452 0.0713372i \(-0.0227266\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 46.4003i 1.55797i 0.627043 + 0.778984i \(0.284265\pi\)
−0.627043 + 0.778984i \(0.715735\pi\)
\(888\) 0 0
\(889\) −46.9254 −1.57383
\(890\) 0 0
\(891\) −14.9450 −0.500677
\(892\) 0 0
\(893\) 40.3352i 1.34976i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 5.56166i 0.185698i
\(898\) 0 0
\(899\) −3.00339 −0.100169
\(900\) 0 0
\(901\) −48.2644 −1.60792
\(902\) 0 0
\(903\) 0.439432i 0.0146234i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 27.0358i 0.897708i 0.893605 + 0.448854i \(0.148168\pi\)
−0.893605 + 0.448854i \(0.851832\pi\)
\(908\) 0 0
\(909\) 29.2866 0.971376
\(910\) 0 0
\(911\) −42.3500 −1.40312 −0.701559 0.712612i \(-0.747512\pi\)
−0.701559 + 0.712612i \(0.747512\pi\)
\(912\) 0 0
\(913\) 16.9243i 0.560111i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 13.1782i − 0.435183i
\(918\) 0 0
\(919\) −8.17400 −0.269635 −0.134818 0.990870i \(-0.543045\pi\)
−0.134818 + 0.990870i \(0.543045\pi\)
\(920\) 0 0
\(921\) 0.818991 0.0269867
\(922\) 0 0
\(923\) − 5.14582i − 0.169377i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 1.93996i − 0.0637166i
\(928\) 0 0
\(929\) −42.2596 −1.38649 −0.693247 0.720700i \(-0.743820\pi\)
−0.693247 + 0.720700i \(0.743820\pi\)
\(930\) 0 0
\(931\) −13.6623 −0.447764
\(932\) 0 0
\(933\) 3.01994i 0.0988684i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 1.16600i − 0.0380917i −0.999819 0.0190458i \(-0.993937\pi\)
0.999819 0.0190458i \(-0.00606284\pi\)
\(938\) 0 0
\(939\) −0.210896 −0.00688234
\(940\) 0 0
\(941\) 4.93864 0.160995 0.0804975 0.996755i \(-0.474349\pi\)
0.0804975 + 0.996755i \(0.474349\pi\)
\(942\) 0 0
\(943\) 8.82071i 0.287242i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 25.3210i − 0.822822i −0.911450 0.411411i \(-0.865036\pi\)
0.911450 0.411411i \(-0.134964\pi\)
\(948\) 0 0
\(949\) −55.2419 −1.79323
\(950\) 0 0
\(951\) −1.74922 −0.0567224
\(952\) 0 0
\(953\) − 47.6143i − 1.54238i −0.636606 0.771189i \(-0.719662\pi\)
0.636606 0.771189i \(-0.280338\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 0.288391i − 0.00932235i
\(958\) 0 0
\(959\) −12.8941 −0.416371
\(960\) 0 0
\(961\) −28.2460 −0.911163
\(962\) 0 0
\(963\) − 50.2559i − 1.61947i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 25.9763i 0.835342i 0.908598 + 0.417671i \(0.137153\pi\)
−0.908598 + 0.417671i \(0.862847\pi\)
\(968\) 0 0
\(969\) 2.31933 0.0745076
\(970\) 0 0
\(971\) −55.8687 −1.79291 −0.896457 0.443132i \(-0.853867\pi\)
−0.896457 + 0.443132i \(0.853867\pi\)
\(972\) 0 0
\(973\) − 37.6833i − 1.20807i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 42.5241i 1.36047i 0.732996 + 0.680233i \(0.238121\pi\)
−0.732996 + 0.680233i \(0.761879\pi\)
\(978\) 0 0
\(979\) −10.7920 −0.344915
\(980\) 0 0
\(981\) −11.5143 −0.367622
\(982\) 0 0
\(983\) 50.8977i 1.62338i 0.584086 + 0.811692i \(0.301453\pi\)
−0.584086 + 0.811692i \(0.698547\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 2.55271i 0.0812535i
\(988\) 0 0
\(989\) 12.9604 0.412118
\(990\) 0 0
\(991\) −27.7147 −0.880387 −0.440194 0.897903i \(-0.645090\pi\)
−0.440194 + 0.897903i \(0.645090\pi\)
\(992\) 0 0
\(993\) 0.961973i 0.0305273i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 22.3480i 0.707768i 0.935289 + 0.353884i \(0.115139\pi\)
−0.935289 + 0.353884i \(0.884861\pi\)
\(998\) 0 0
\(999\) −1.13426 −0.0358863
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.1149.5 8
5.2 odd 4 164.2.a.a.1.2 4
5.3 odd 4 4100.2.a.c.1.3 4
5.4 even 2 inner 4100.2.d.c.1149.4 8
15.2 even 4 1476.2.a.g.1.3 4
20.7 even 4 656.2.a.i.1.3 4
35.27 even 4 8036.2.a.i.1.3 4
40.27 even 4 2624.2.a.y.1.2 4
40.37 odd 4 2624.2.a.v.1.3 4
60.47 odd 4 5904.2.a.bp.1.3 4
205.122 odd 4 6724.2.a.c.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.2.a.a.1.2 4 5.2 odd 4
656.2.a.i.1.3 4 20.7 even 4
1476.2.a.g.1.3 4 15.2 even 4
2624.2.a.v.1.3 4 40.37 odd 4
2624.2.a.y.1.2 4 40.27 even 4
4100.2.a.c.1.3 4 5.3 odd 4
4100.2.d.c.1149.4 8 5.4 even 2 inner
4100.2.d.c.1149.5 8 1.1 even 1 trivial
5904.2.a.bp.1.3 4 60.47 odd 4
6724.2.a.c.1.3 4 205.122 odd 4
8036.2.a.i.1.3 4 35.27 even 4