Properties

Label 4800.2.f.z.3649.2
Level 4800
Weight 2
Character 4800.3649
Analytic conductor 38.328
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(38.3281929702\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 600)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4800.3649
Dual form 4800.2.f.z.3649.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} +3.00000i q^{13} -6.00000i q^{17} +7.00000 q^{19} -3.00000 q^{21} -6.00000i q^{23} -1.00000i q^{27} -2.00000 q^{29} +5.00000 q^{31} +2.00000i q^{33} +10.0000i q^{37} -3.00000 q^{39} +12.0000 q^{41} +3.00000i q^{43} -10.0000i q^{47} -2.00000 q^{49} +6.00000 q^{51} +7.00000i q^{57} +6.00000 q^{59} +13.0000 q^{61} -3.00000i q^{63} -7.00000i q^{67} +6.00000 q^{69} +4.00000 q^{71} -6.00000i q^{73} +6.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{83} -2.00000i q^{87} -16.0000 q^{89} -9.00000 q^{91} +5.00000i q^{93} +7.00000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 4q^{11} + 14q^{19} - 6q^{21} - 4q^{29} + 10q^{31} - 6q^{39} + 24q^{41} - 4q^{49} + 12q^{51} + 12q^{59} + 26q^{61} + 12q^{69} + 8q^{71} - 16q^{79} + 2q^{81} - 32q^{89} - 18q^{91} - 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(4351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 3.00000i 0.832050i 0.909353 + 0.416025i \(0.136577\pi\)
−0.909353 + 0.416025i \(0.863423\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) − 6.00000i − 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 0 0
\(33\) 2.00000i 0.348155i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 0 0
\(39\) −3.00000 −0.480384
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 0 0
\(43\) 3.00000i 0.457496i 0.973486 + 0.228748i \(0.0734631\pi\)
−0.973486 + 0.228748i \(0.926537\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 10.0000i − 1.45865i −0.684167 0.729325i \(-0.739834\pi\)
0.684167 0.729325i \(-0.260166\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 7.00000i 0.927173i
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 0 0
\(63\) − 3.00000i − 0.377964i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 7.00000i − 0.855186i −0.903971 0.427593i \(-0.859362\pi\)
0.903971 0.427593i \(-0.140638\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000i 0.683763i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 2.00000i − 0.214423i
\(88\) 0 0
\(89\) −16.0000 −1.69600 −0.847998 0.529999i \(-0.822192\pi\)
−0.847998 + 0.529999i \(0.822192\pi\)
\(90\) 0 0
\(91\) −9.00000 −0.943456
\(92\) 0 0
\(93\) 5.00000i 0.518476i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.00000i 0.710742i 0.934725 + 0.355371i \(0.115646\pi\)
−0.934725 + 0.355371i \(0.884354\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 12.0000i 1.18240i 0.806527 + 0.591198i \(0.201345\pi\)
−0.806527 + 0.591198i \(0.798655\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.0000i 1.54678i 0.633932 + 0.773389i \(0.281440\pi\)
−0.633932 + 0.773389i \(0.718560\pi\)
\(108\) 0 0
\(109\) 9.00000 0.862044 0.431022 0.902342i \(-0.358153\pi\)
0.431022 + 0.902342i \(0.358153\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) 12.0000i 1.12887i 0.825479 + 0.564433i \(0.190905\pi\)
−0.825479 + 0.564433i \(0.809095\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 3.00000i − 0.277350i
\(118\) 0 0
\(119\) 18.0000 1.65006
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 12.0000i 1.08200i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) 0 0
\(129\) −3.00000 −0.264135
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 21.0000i 1.82093i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.0000i 0.854358i 0.904167 + 0.427179i \(0.140493\pi\)
−0.904167 + 0.427179i \(0.859507\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 10.0000 0.842152
\(142\) 0 0
\(143\) 6.00000i 0.501745i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 2.00000i − 0.164957i
\(148\) 0 0
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) −1.00000 −0.0813788 −0.0406894 0.999172i \(-0.512955\pi\)
−0.0406894 + 0.999172i \(0.512955\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 9.00000i − 0.718278i −0.933284 0.359139i \(-0.883070\pi\)
0.933284 0.359139i \(-0.116930\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 18.0000 1.41860
\(162\) 0 0
\(163\) 1.00000i 0.0783260i 0.999233 + 0.0391630i \(0.0124692\pi\)
−0.999233 + 0.0391630i \(0.987531\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) 0 0
\(169\) 4.00000 0.307692
\(170\) 0 0
\(171\) −7.00000 −0.535303
\(172\) 0 0
\(173\) − 2.00000i − 0.152057i −0.997106 0.0760286i \(-0.975776\pi\)
0.997106 0.0760286i \(-0.0242240\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.00000i 0.450988i
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 19.0000 1.41226 0.706129 0.708083i \(-0.250440\pi\)
0.706129 + 0.708083i \(0.250440\pi\)
\(182\) 0 0
\(183\) 13.0000i 0.960988i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 12.0000i − 0.877527i
\(188\) 0 0
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) 19.0000i 1.36765i 0.729646 + 0.683825i \(0.239685\pi\)
−0.729646 + 0.683825i \(0.760315\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.0000i 0.997459i 0.866758 + 0.498729i \(0.166200\pi\)
−0.866758 + 0.498729i \(0.833800\pi\)
\(198\) 0 0
\(199\) −3.00000 −0.212664 −0.106332 0.994331i \(-0.533911\pi\)
−0.106332 + 0.994331i \(0.533911\pi\)
\(200\) 0 0
\(201\) 7.00000 0.493742
\(202\) 0 0
\(203\) − 6.00000i − 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.00000i 0.417029i
\(208\) 0 0
\(209\) 14.0000 0.968400
\(210\) 0 0
\(211\) 9.00000 0.619586 0.309793 0.950804i \(-0.399740\pi\)
0.309793 + 0.950804i \(0.399740\pi\)
\(212\) 0 0
\(213\) 4.00000i 0.274075i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 15.0000i 1.01827i
\(218\) 0 0
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 18.0000 1.21081
\(222\) 0 0
\(223\) − 11.0000i − 0.736614i −0.929704 0.368307i \(-0.879937\pi\)
0.929704 0.368307i \(-0.120063\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) 0 0
\(229\) −19.0000 −1.25556 −0.627778 0.778393i \(-0.716035\pi\)
−0.627778 + 0.778393i \(0.716035\pi\)
\(230\) 0 0
\(231\) −6.00000 −0.394771
\(232\) 0 0
\(233\) − 4.00000i − 0.262049i −0.991379 0.131024i \(-0.958173\pi\)
0.991379 0.131024i \(-0.0418266\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 8.00000i − 0.519656i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 25.0000 1.61039 0.805196 0.593009i \(-0.202060\pi\)
0.805196 + 0.593009i \(0.202060\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 21.0000i 1.33620i
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) 28.0000 1.76734 0.883672 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 0 0
\(253\) − 12.0000i − 0.754434i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 4.00000i − 0.249513i −0.992187 0.124757i \(-0.960185\pi\)
0.992187 0.124757i \(-0.0398150\pi\)
\(258\) 0 0
\(259\) −30.0000 −1.86411
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) − 12.0000i − 0.739952i −0.929041 0.369976i \(-0.879366\pi\)
0.929041 0.369976i \(-0.120634\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 16.0000i − 0.979184i
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 0 0
\(273\) − 9.00000i − 0.544705i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.00000i 0.0600842i 0.999549 + 0.0300421i \(0.00956413\pi\)
−0.999549 + 0.0300421i \(0.990436\pi\)
\(278\) 0 0
\(279\) −5.00000 −0.299342
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) 5.00000i 0.297219i 0.988896 + 0.148610i \(0.0474798\pi\)
−0.988896 + 0.148610i \(0.952520\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 36.0000i 2.12501i
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −7.00000 −0.410347
\(292\) 0 0
\(293\) 2.00000i 0.116841i 0.998292 + 0.0584206i \(0.0186065\pi\)
−0.998292 + 0.0584206i \(0.981394\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 2.00000i − 0.116052i
\(298\) 0 0
\(299\) 18.0000 1.04097
\(300\) 0 0
\(301\) −9.00000 −0.518751
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 5.00000i − 0.285365i −0.989769 0.142683i \(-0.954427\pi\)
0.989769 0.142683i \(-0.0455728\pi\)
\(308\) 0 0
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) −2.00000 −0.113410 −0.0567048 0.998391i \(-0.518059\pi\)
−0.0567048 + 0.998391i \(0.518059\pi\)
\(312\) 0 0
\(313\) 19.0000i 1.07394i 0.843600 + 0.536972i \(0.180432\pi\)
−0.843600 + 0.536972i \(0.819568\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 32.0000i 1.79730i 0.438667 + 0.898650i \(0.355451\pi\)
−0.438667 + 0.898650i \(0.644549\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) −16.0000 −0.893033
\(322\) 0 0
\(323\) − 42.0000i − 2.33694i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 9.00000i 0.497701i
\(328\) 0 0
\(329\) 30.0000 1.65395
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) − 10.0000i − 0.547997i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 7.00000i 0.381314i 0.981657 + 0.190657i \(0.0610619\pi\)
−0.981657 + 0.190657i \(0.938938\pi\)
\(338\) 0 0
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) 10.0000 0.541530
\(342\) 0 0
\(343\) 15.0000i 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.0000i 0.966291i 0.875540 + 0.483145i \(0.160506\pi\)
−0.875540 + 0.483145i \(0.839494\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) 3.00000 0.160128
\(352\) 0 0
\(353\) 26.0000i 1.38384i 0.721974 + 0.691920i \(0.243235\pi\)
−0.721974 + 0.691920i \(0.756765\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 18.0000i 0.952661i
\(358\) 0 0
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) − 7.00000i − 0.367405i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 11.0000i − 0.574195i −0.957901 0.287098i \(-0.907310\pi\)
0.957901 0.287098i \(-0.0926904\pi\)
\(368\) 0 0
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 7.00000i − 0.362446i −0.983442 0.181223i \(-0.941994\pi\)
0.983442 0.181223i \(-0.0580056\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 6.00000i − 0.309016i
\(378\) 0 0
\(379\) 29.0000 1.48963 0.744815 0.667271i \(-0.232538\pi\)
0.744815 + 0.667271i \(0.232538\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 0 0
\(383\) − 36.0000i − 1.83951i −0.392488 0.919757i \(-0.628386\pi\)
0.392488 0.919757i \(-0.371614\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 3.00000i − 0.152499i
\(388\) 0 0
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) 0 0
\(393\) − 8.00000i − 0.403547i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 11.0000i 0.552074i 0.961147 + 0.276037i \(0.0890213\pi\)
−0.961147 + 0.276037i \(0.910979\pi\)
\(398\) 0 0
\(399\) −21.0000 −1.05131
\(400\) 0 0
\(401\) −8.00000 −0.399501 −0.199750 0.979847i \(-0.564013\pi\)
−0.199750 + 0.979847i \(0.564013\pi\)
\(402\) 0 0
\(403\) 15.0000i 0.747203i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 20.0000i 0.991363i
\(408\) 0 0
\(409\) −5.00000 −0.247234 −0.123617 0.992330i \(-0.539449\pi\)
−0.123617 + 0.992330i \(0.539449\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) 0 0
\(413\) 18.0000i 0.885722i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 4.00000i − 0.195881i
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) 0 0
\(423\) 10.0000i 0.486217i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 39.0000i 1.88734i
\(428\) 0 0
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) 34.0000 1.63772 0.818861 0.573992i \(-0.194606\pi\)
0.818861 + 0.573992i \(0.194606\pi\)
\(432\) 0 0
\(433\) 3.00000i 0.144171i 0.997398 + 0.0720854i \(0.0229654\pi\)
−0.997398 + 0.0720854i \(0.977035\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 42.0000i − 2.00913i
\(438\) 0 0
\(439\) 19.0000 0.906821 0.453410 0.891302i \(-0.350207\pi\)
0.453410 + 0.891302i \(0.350207\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) − 16.0000i − 0.760183i −0.924949 0.380091i \(-0.875893\pi\)
0.924949 0.380091i \(-0.124107\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 22.0000i − 1.04056i
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 0 0
\(453\) − 1.00000i − 0.0469841i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000i 0.467780i 0.972263 + 0.233890i \(0.0751456\pi\)
−0.972263 + 0.233890i \(0.924854\pi\)
\(458\) 0 0
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −8.00000 −0.372597 −0.186299 0.982493i \(-0.559649\pi\)
−0.186299 + 0.982493i \(0.559649\pi\)
\(462\) 0 0
\(463\) − 8.00000i − 0.371792i −0.982569 0.185896i \(-0.940481\pi\)
0.982569 0.185896i \(-0.0595187\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 26.0000i − 1.20314i −0.798821 0.601568i \(-0.794543\pi\)
0.798821 0.601568i \(-0.205457\pi\)
\(468\) 0 0
\(469\) 21.0000 0.969690
\(470\) 0 0
\(471\) 9.00000 0.414698
\(472\) 0 0
\(473\) 6.00000i 0.275880i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −38.0000 −1.73626 −0.868132 0.496333i \(-0.834679\pi\)
−0.868132 + 0.496333i \(0.834679\pi\)
\(480\) 0 0
\(481\) −30.0000 −1.36788
\(482\) 0 0
\(483\) 18.0000i 0.819028i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 11.0000i − 0.498458i −0.968445 0.249229i \(-0.919823\pi\)
0.968445 0.249229i \(-0.0801771\pi\)
\(488\) 0 0
\(489\) −1.00000 −0.0452216
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 0 0
\(493\) 12.0000i 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000i 0.538274i
\(498\) 0 0
\(499\) −25.0000 −1.11915 −0.559577 0.828778i \(-0.689036\pi\)
−0.559577 + 0.828778i \(0.689036\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) 0 0
\(503\) 4.00000i 0.178351i 0.996016 + 0.0891756i \(0.0284232\pi\)
−0.996016 + 0.0891756i \(0.971577\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 4.00000i 0.177646i
\(508\) 0 0
\(509\) 22.0000 0.975133 0.487566 0.873086i \(-0.337885\pi\)
0.487566 + 0.873086i \(0.337885\pi\)
\(510\) 0 0
\(511\) 18.0000 0.796273
\(512\) 0 0
\(513\) − 7.00000i − 0.309058i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 20.0000i − 0.879599i
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 29.0000i 1.26808i 0.773300 + 0.634041i \(0.218605\pi\)
−0.773300 + 0.634041i \(0.781395\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 30.0000i − 1.30682i
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) 36.0000i 1.55933i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 18.0000i − 0.776757i
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 15.0000 0.644900 0.322450 0.946586i \(-0.395494\pi\)
0.322450 + 0.946586i \(0.395494\pi\)
\(542\) 0 0
\(543\) 19.0000i 0.815368i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) −13.0000 −0.554826
\(550\) 0 0
\(551\) −14.0000 −0.596420
\(552\) 0 0
\(553\) − 24.0000i − 1.02058i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 0 0
\(559\) −9.00000 −0.380659
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) − 26.0000i − 1.09577i −0.836554 0.547885i \(-0.815433\pi\)
0.836554 0.547885i \(-0.184567\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.00000i 0.125988i
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −39.0000 −1.63210 −0.816050 0.577982i \(-0.803840\pi\)
−0.816050 + 0.577982i \(0.803840\pi\)
\(572\) 0 0
\(573\) 18.0000i 0.751961i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 11.0000i − 0.457936i −0.973434 0.228968i \(-0.926465\pi\)
0.973434 0.228968i \(-0.0735351\pi\)
\(578\) 0 0
\(579\) −19.0000 −0.789613
\(580\) 0 0
\(581\) 18.0000 0.746766
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.0000i 0.660391i 0.943913 + 0.330195i \(0.107115\pi\)
−0.943913 + 0.330195i \(0.892885\pi\)
\(588\) 0 0
\(589\) 35.0000 1.44215
\(590\) 0 0
\(591\) −14.0000 −0.575883
\(592\) 0 0
\(593\) − 36.0000i − 1.47834i −0.673517 0.739171i \(-0.735217\pi\)
0.673517 0.739171i \(-0.264783\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 3.00000i − 0.122782i
\(598\) 0 0
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 0 0
\(601\) 35.0000 1.42768 0.713840 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) 0 0
\(603\) 7.00000i 0.285062i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 32.0000i − 1.29884i −0.760430 0.649420i \(-0.775012\pi\)
0.760430 0.649420i \(-0.224988\pi\)
\(608\) 0 0
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) 30.0000 1.21367
\(612\) 0 0
\(613\) − 34.0000i − 1.37325i −0.727013 0.686624i \(-0.759092\pi\)
0.727013 0.686624i \(-0.240908\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 40.0000i 1.61034i 0.593045 + 0.805170i \(0.297926\pi\)
−0.593045 + 0.805170i \(0.702074\pi\)
\(618\) 0 0
\(619\) −5.00000 −0.200967 −0.100483 0.994939i \(-0.532039\pi\)
−0.100483 + 0.994939i \(0.532039\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) 0 0
\(623\) − 48.0000i − 1.92308i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 14.0000i 0.559106i
\(628\) 0 0
\(629\) 60.0000 2.39236
\(630\) 0 0
\(631\) 25.0000 0.995234 0.497617 0.867397i \(-0.334208\pi\)
0.497617 + 0.867397i \(0.334208\pi\)
\(632\) 0 0
\(633\) 9.00000i 0.357718i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 6.00000i − 0.237729i
\(638\) 0 0
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 0 0
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.0000i 0.471769i 0.971781 + 0.235884i \(0.0757987\pi\)
−0.971781 + 0.235884i \(0.924201\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) −15.0000 −0.587896
\(652\) 0 0
\(653\) − 26.0000i − 1.01746i −0.860927 0.508729i \(-0.830115\pi\)
0.860927 0.508729i \(-0.169885\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) −42.0000 −1.63361 −0.816805 0.576913i \(-0.804257\pi\)
−0.816805 + 0.576913i \(0.804257\pi\)
\(662\) 0 0
\(663\) 18.0000i 0.699062i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12.0000i 0.464642i
\(668\) 0 0
\(669\) 11.0000 0.425285
\(670\) 0 0
\(671\) 26.0000 1.00372
\(672\) 0 0
\(673\) − 50.0000i − 1.92736i −0.267063 0.963679i \(-0.586053\pi\)
0.267063 0.963679i \(-0.413947\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 10.0000i − 0.384331i −0.981363 0.192166i \(-0.938449\pi\)
0.981363 0.192166i \(-0.0615511\pi\)
\(678\) 0 0
\(679\) −21.0000 −0.805906
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) 40.0000i 1.53056i 0.643699 + 0.765279i \(0.277399\pi\)
−0.643699 + 0.765279i \(0.722601\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 19.0000i − 0.724895i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 0 0
\(693\) − 6.00000i − 0.227921i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 72.0000i − 2.72719i
\(698\) 0 0
\(699\) 4.00000 0.151294
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) 70.0000i 2.64010i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −19.0000 −0.713560 −0.356780 0.934188i \(-0.616125\pi\)
−0.356780 + 0.934188i \(0.616125\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) − 30.0000i − 1.12351i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) −36.0000 −1.34071
\(722\) 0 0
\(723\) 25.0000i 0.929760i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 5.00000i − 0.185440i −0.995692 0.0927199i \(-0.970444\pi\)
0.995692 0.0927199i \(-0.0295561\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 18.0000 0.665754
\(732\) 0 0
\(733\) 2.00000i 0.0738717i 0.999318 + 0.0369358i \(0.0117597\pi\)
−0.999318 + 0.0369358i \(0.988240\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 14.0000i − 0.515697i
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −21.0000 −0.771454
\(742\) 0 0
\(743\) − 20.0000i − 0.733729i −0.930274 0.366864i \(-0.880431\pi\)
0.930274 0.366864i \(-0.119569\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 6.00000i 0.219529i
\(748\) 0 0
\(749\) −48.0000 −1.75388
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 0 0
\(753\) 28.0000i 1.02038i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 13.0000i 0.472493i 0.971693 + 0.236247i \(0.0759173\pi\)
−0.971693 + 0.236247i \(0.924083\pi\)
\(758\) 0 0
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) −52.0000 −1.88500 −0.942499 0.334208i \(-0.891531\pi\)
−0.942499 + 0.334208i \(0.891531\pi\)
\(762\) 0 0
\(763\) 27.0000i 0.977466i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.0000i 0.649942i
\(768\) 0 0
\(769\) 21.0000 0.757279 0.378640 0.925544i \(-0.376392\pi\)
0.378640 + 0.925544i \(0.376392\pi\)
\(770\) 0 0
\(771\) 4.00000 0.144056
\(772\) 0 0
\(773\) 12.0000i 0.431610i 0.976436 + 0.215805i \(0.0692376\pi\)
−0.976436 + 0.215805i \(0.930762\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 30.0000i − 1.07624i
\(778\) 0 0
\(779\) 84.0000 3.00961
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) 2.00000i 0.0714742i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 37.0000i − 1.31891i −0.751745 0.659454i \(-0.770788\pi\)
0.751745 0.659454i \(-0.229212\pi\)
\(788\) 0 0
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) −36.0000 −1.28001
\(792\) 0 0
\(793\) 39.0000i 1.38493i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 20.0000i − 0.708436i −0.935163 0.354218i \(-0.884747\pi\)
0.935163 0.354218i \(-0.115253\pi\)
\(798\) 0 0
\(799\) −60.0000 −2.12265
\(800\) 0 0
\(801\) 16.0000 0.565332
\(802\) 0 0
\(803\) − 12.0000i − 0.423471i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 6.00000i − 0.211210i
\(808\) 0 0
\(809\) −48.0000 −1.68759 −0.843795 0.536666i \(-0.819684\pi\)
−0.843795 + 0.536666i \(0.819684\pi\)
\(810\) 0 0
\(811\) 17.0000 0.596951 0.298475 0.954417i \(-0.403522\pi\)
0.298475 + 0.954417i \(0.403522\pi\)
\(812\) 0 0
\(813\) − 24.0000i − 0.841717i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 21.0000i 0.734697i
\(818\) 0 0
\(819\) 9.00000 0.314485
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) 9.00000i 0.313720i 0.987621 + 0.156860i \(0.0501372\pi\)
−0.987621 + 0.156860i \(0.949863\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 24.0000i − 0.834562i −0.908778 0.417281i \(-0.862983\pi\)
0.908778 0.417281i \(-0.137017\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) −1.00000 −0.0346896
\(832\) 0 0
\(833\) 12.0000i 0.415775i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 5.00000i − 0.172825i
\(838\) 0 0
\(839\) 34.0000 1.17381 0.586905 0.809656i \(-0.300346\pi\)
0.586905 + 0.809656i \(0.300346\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 2.00000i 0.0688837i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 21.0000i − 0.721569i
\(848\) 0 0
\(849\) −5.00000 −0.171600
\(850\) 0 0
\(851\) 60.0000 2.05677
\(852\) 0 0
\(853\) 25.0000i 0.855984i 0.903783 + 0.427992i \(0.140779\pi\)
−0.903783 + 0.427992i \(0.859221\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 4.00000i 0.136637i 0.997664 + 0.0683187i \(0.0217635\pi\)
−0.997664 + 0.0683187i \(0.978237\pi\)
\(858\) 0 0
\(859\) 32.0000 1.09183 0.545913 0.837842i \(-0.316183\pi\)
0.545913 + 0.837842i \(0.316183\pi\)
\(860\) 0 0
\(861\) −36.0000 −1.22688
\(862\) 0 0
\(863\) − 8.00000i − 0.272323i −0.990687 0.136162i \(-0.956523\pi\)
0.990687 0.136162i \(-0.0434766\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 19.0000i − 0.645274i
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 21.0000 0.711558
\(872\) 0 0
\(873\) − 7.00000i − 0.236914i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 31.0000i 1.04680i 0.852088 + 0.523398i \(0.175336\pi\)
−0.852088 + 0.523398i \(0.824664\pi\)
\(878\) 0 0
\(879\) −2.00000 −0.0674583
\(880\) 0 0
\(881\) 8.00000 0.269527 0.134763 0.990878i \(-0.456973\pi\)
0.134763 + 0.990878i \(0.456973\pi\)
\(882\) 0 0
\(883\) − 53.0000i − 1.78359i −0.452438 0.891796i \(-0.649446\pi\)
0.452438 0.891796i \(-0.350554\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 42.0000i 1.41022i 0.709097 + 0.705111i \(0.249103\pi\)
−0.709097 + 0.705111i \(0.750897\pi\)
\(888\) 0 0
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 0 0
\(893\) − 70.0000i − 2.34246i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 18.0000i 0.601003i
\(898\) 0 0
\(899\) −10.0000 −0.333519
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) − 9.00000i − 0.299501i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 52.0000i − 1.72663i −0.504664 0.863316i \(-0.668384\pi\)
0.504664 0.863316i \(-0.331616\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −18.0000 −0.596367 −0.298183 0.954509i \(-0.596381\pi\)
−0.298183 + 0.954509i \(0.596381\pi\)
\(912\) 0 0
\(913\) − 12.0000i − 0.397142i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 24.0000i − 0.792550i
\(918\) 0 0
\(919\) 41.0000 1.35247 0.676233 0.736688i \(-0.263611\pi\)
0.676233 + 0.736688i \(0.263611\pi\)
\(920\) 0 0
\(921\) 5.00000 0.164756
\(922\) 0 0
\(923\) 12.0000i 0.394985i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 12.0000i − 0.394132i
\(928\) 0 0
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −14.0000 −0.458831
\(932\) 0 0
\(933\) − 2.00000i − 0.0654771i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 39.0000i − 1.27407i −0.770833 0.637037i \(-0.780160\pi\)
0.770833 0.637037i \(-0.219840\pi\)
\(938\) 0 0
\(939\) −19.0000 −0.620042
\(940\) 0 0
\(941\) −46.0000 −1.49956 −0.749779 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) 0 0
\(943\) − 72.0000i − 2.34464i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 6.00000i − 0.194974i −0.995237 0.0974869i \(-0.968920\pi\)
0.995237 0.0974869i \(-0.0310804\pi\)
\(948\) 0 0
\(949\) 18.0000 0.584305
\(950\) 0 0
\(951\) −32.0000 −1.03767
\(952\) 0 0
\(953\) − 8.00000i − 0.259145i −0.991570 0.129573i \(-0.958639\pi\)
0.991570 0.129573i \(-0.0413606\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 4.00000i − 0.129302i
\(958\) 0 0
\(959\) −30.0000 −0.968751
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) − 16.0000i − 0.515593i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 24.0000i 0.771788i 0.922543 + 0.385894i \(0.126107\pi\)
−0.922543 + 0.385894i \(0.873893\pi\)
\(968\) 0 0
\(969\) 42.0000 1.34923
\(970\) 0 0
\(971\) 30.0000 0.962746 0.481373 0.876516i \(-0.340138\pi\)
0.481373 + 0.876516i \(0.340138\pi\)
\(972\) 0 0
\(973\) − 12.0000i − 0.384702i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 58.0000i − 1.85558i −0.373097 0.927792i \(-0.621704\pi\)
0.373097 0.927792i \(-0.378296\pi\)
\(978\) 0 0
\(979\) −32.0000 −1.02272
\(980\) 0 0
\(981\) −9.00000 −0.287348
\(982\) 0 0
\(983\) 4.00000i 0.127580i 0.997963 + 0.0637901i \(0.0203188\pi\)
−0.997963 + 0.0637901i \(0.979681\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 30.0000i 0.954911i
\(988\) 0 0
\(989\) 18.0000 0.572367
\(990\) 0 0
\(991\) 17.0000 0.540023 0.270011 0.962857i \(-0.412973\pi\)
0.270011 + 0.962857i \(0.412973\pi\)
\(992\) 0 0
\(993\) − 4.00000i − 0.126936i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 42.0000i 1.33015i 0.746775 + 0.665077i \(0.231601\pi\)
−0.746775 + 0.665077i \(0.768399\pi\)
\(998\) 0 0
\(999\) 10.0000 0.316386
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 4800.2.f.z.3649.2 2
4.3 odd 2 4800.2.f.k.3649.1 2
5.2 odd 4 4800.2.a.bs.1.1 1
5.3 odd 4 4800.2.a.bd.1.1 1
5.4 even 2 inner 4800.2.f.z.3649.1 2
8.3 odd 2 600.2.f.d.49.2 2
8.5 even 2 1200.2.f.c.49.1 2
20.3 even 4 4800.2.a.bp.1.1 1
20.7 even 4 4800.2.a.bc.1.1 1
20.19 odd 2 4800.2.f.k.3649.2 2
24.5 odd 2 3600.2.f.o.2449.2 2
24.11 even 2 1800.2.f.e.649.1 2
40.3 even 4 600.2.a.b.1.1 1
40.13 odd 4 1200.2.a.q.1.1 1
40.19 odd 2 600.2.f.d.49.1 2
40.27 even 4 600.2.a.i.1.1 yes 1
40.29 even 2 1200.2.f.c.49.2 2
40.37 odd 4 1200.2.a.b.1.1 1
120.29 odd 2 3600.2.f.o.2449.1 2
120.53 even 4 3600.2.a.bl.1.1 1
120.59 even 2 1800.2.f.e.649.2 2
120.77 even 4 3600.2.a.i.1.1 1
120.83 odd 4 1800.2.a.e.1.1 1
120.107 odd 4 1800.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.2.a.b.1.1 1 40.3 even 4
600.2.a.i.1.1 yes 1 40.27 even 4
600.2.f.d.49.1 2 40.19 odd 2
600.2.f.d.49.2 2 8.3 odd 2
1200.2.a.b.1.1 1 40.37 odd 4
1200.2.a.q.1.1 1 40.13 odd 4
1200.2.f.c.49.1 2 8.5 even 2
1200.2.f.c.49.2 2 40.29 even 2
1800.2.a.e.1.1 1 120.83 odd 4
1800.2.a.t.1.1 1 120.107 odd 4
1800.2.f.e.649.1 2 24.11 even 2
1800.2.f.e.649.2 2 120.59 even 2
3600.2.a.i.1.1 1 120.77 even 4
3600.2.a.bl.1.1 1 120.53 even 4
3600.2.f.o.2449.1 2 120.29 odd 2
3600.2.f.o.2449.2 2 24.5 odd 2
4800.2.a.bc.1.1 1 20.7 even 4
4800.2.a.bd.1.1 1 5.3 odd 4
4800.2.a.bp.1.1 1 20.3 even 4
4800.2.a.bs.1.1 1 5.2 odd 4
4800.2.f.k.3649.1 2 4.3 odd 2
4800.2.f.k.3649.2 2 20.19 odd 2
4800.2.f.z.3649.1 2 5.4 even 2 inner
4800.2.f.z.3649.2 2 1.1 even 1 trivial