Properties

Label 1425.2.c.b.799.1
Level $1425$
Weight $2$
Character 1425.799
Analytic conductor $11.379$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.3786822880\)
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 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1425.799
Dual form 1425.2.c.b.799.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} -5.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} -5.00000i q^{7} -1.00000 q^{9} +1.00000 q^{11} -2.00000i q^{12} -2.00000i q^{13} -10.0000 q^{14} -4.00000 q^{16} -1.00000i q^{17} +2.00000i q^{18} +1.00000 q^{19} +5.00000 q^{21} -2.00000i q^{22} +4.00000i q^{23} -4.00000 q^{26} -1.00000i q^{27} +10.0000i q^{28} +2.00000 q^{29} -6.00000 q^{31} +8.00000i q^{32} +1.00000i q^{33} -2.00000 q^{34} +2.00000 q^{36} -2.00000i q^{38} +2.00000 q^{39} -10.0000i q^{42} +1.00000i q^{43} -2.00000 q^{44} +8.00000 q^{46} -9.00000i q^{47} -4.00000i q^{48} -18.0000 q^{49} +1.00000 q^{51} +4.00000i q^{52} -10.0000i q^{53} -2.00000 q^{54} +1.00000i q^{57} -4.00000i q^{58} +8.00000 q^{59} -1.00000 q^{61} +12.0000i q^{62} +5.00000i q^{63} +8.00000 q^{64} +2.00000 q^{66} +8.00000i q^{67} +2.00000i q^{68} -4.00000 q^{69} -12.0000 q^{71} +11.0000i q^{73} -2.00000 q^{76} -5.00000i q^{77} -4.00000i q^{78} -16.0000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -10.0000 q^{84} +2.00000 q^{86} +2.00000i q^{87} +6.00000 q^{89} -10.0000 q^{91} -8.00000i q^{92} -6.00000i q^{93} -18.0000 q^{94} -8.00000 q^{96} -10.0000i q^{97} +36.0000i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} + 4q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 4q^{4} + 4q^{6} - 2q^{9} + 2q^{11} - 20q^{14} - 8q^{16} + 2q^{19} + 10q^{21} - 8q^{26} + 4q^{29} - 12q^{31} - 4q^{34} + 4q^{36} + 4q^{39} - 4q^{44} + 16q^{46} - 36q^{49} + 2q^{51} - 4q^{54} + 16q^{59} - 2q^{61} + 16q^{64} + 4q^{66} - 8q^{69} - 24q^{71} - 4q^{76} - 32q^{79} + 2q^{81} - 20q^{84} + 4q^{86} + 12q^{89} - 20q^{91} - 36q^{94} - 16q^{96} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) − 5.00000i − 1.88982i −0.327327 0.944911i \(-0.606148\pi\)
0.327327 0.944911i \(-0.393852\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) − 2.00000i − 0.577350i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) −10.0000 −2.67261
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) − 1.00000i − 0.242536i −0.992620 0.121268i \(-0.961304\pi\)
0.992620 0.121268i \(-0.0386960\pi\)
\(18\) 2.00000i 0.471405i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 5.00000 1.09109
\(22\) − 2.00000i − 0.426401i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) − 1.00000i − 0.192450i
\(28\) 10.0000i 1.88982i
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 8.00000i 1.41421i
\(33\) 1.00000i 0.174078i
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) − 2.00000i − 0.324443i
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) − 10.0000i − 1.54303i
\(43\) 1.00000i 0.152499i 0.997089 + 0.0762493i \(0.0242945\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) − 9.00000i − 1.31278i −0.754420 0.656392i \(-0.772082\pi\)
0.754420 0.656392i \(-0.227918\pi\)
\(48\) − 4.00000i − 0.577350i
\(49\) −18.0000 −2.57143
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 4.00000i 0.554700i
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) −2.00000 −0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000i 0.132453i
\(58\) − 4.00000i − 0.525226i
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 12.0000i 1.52400i
\(63\) 5.00000i 0.629941i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 2.00000i 0.242536i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 11.0000i 1.28745i 0.765256 + 0.643726i \(0.222612\pi\)
−0.765256 + 0.643726i \(0.777388\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) − 5.00000i − 0.569803i
\(78\) − 4.00000i − 0.452911i
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) −10.0000 −1.09109
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) 2.00000i 0.214423i
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −10.0000 −1.04828
\(92\) − 8.00000i − 0.834058i
\(93\) − 6.00000i − 0.622171i
\(94\) −18.0000 −1.85656
\(95\) 0 0
\(96\) −8.00000 −0.816497
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 36.0000i 3.63655i
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) − 2.00000i − 0.198030i
\(103\) 2.00000i 0.197066i 0.995134 + 0.0985329i \(0.0314150\pi\)
−0.995134 + 0.0985329i \(0.968585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −20.0000 −1.94257
\(107\) 6.00000i 0.580042i 0.957020 + 0.290021i \(0.0936623\pi\)
−0.957020 + 0.290021i \(0.906338\pi\)
\(108\) 2.00000i 0.192450i
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 20.0000i 1.88982i
\(113\) − 2.00000i − 0.188144i −0.995565 0.0940721i \(-0.970012\pi\)
0.995565 0.0940721i \(-0.0299884\pi\)
\(114\) 2.00000 0.187317
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 2.00000i 0.184900i
\(118\) − 16.0000i − 1.47292i
\(119\) −5.00000 −0.458349
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 2.00000i 0.181071i
\(123\) 0 0
\(124\) 12.0000 1.07763
\(125\) 0 0
\(126\) 10.0000 0.890871
\(127\) − 2.00000i − 0.177471i −0.996055 0.0887357i \(-0.971717\pi\)
0.996055 0.0887357i \(-0.0282826\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) 7.00000 0.611593 0.305796 0.952097i \(-0.401077\pi\)
0.305796 + 0.952097i \(0.401077\pi\)
\(132\) − 2.00000i − 0.174078i
\(133\) − 5.00000i − 0.433555i
\(134\) 16.0000 1.38219
\(135\) 0 0
\(136\) 0 0
\(137\) − 9.00000i − 0.768922i −0.923141 0.384461i \(-0.874387\pi\)
0.923141 0.384461i \(-0.125613\pi\)
\(138\) 8.00000i 0.681005i
\(139\) 13.0000 1.10265 0.551323 0.834292i \(-0.314123\pi\)
0.551323 + 0.834292i \(0.314123\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) 24.0000i 2.01404i
\(143\) − 2.00000i − 0.167248i
\(144\) 4.00000 0.333333
\(145\) 0 0
\(146\) 22.0000 1.82073
\(147\) − 18.0000i − 1.48461i
\(148\) 0 0
\(149\) 21.0000 1.72039 0.860194 0.509968i \(-0.170343\pi\)
0.860194 + 0.509968i \(0.170343\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 1.00000i 0.0808452i
\(154\) −10.0000 −0.805823
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) − 18.0000i − 1.43656i −0.695756 0.718278i \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) 32.0000i 2.54578i
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 20.0000 1.57622
\(162\) − 2.00000i − 0.157135i
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −24.0000 −1.86276
\(167\) 10.0000i 0.773823i 0.922117 + 0.386912i \(0.126458\pi\)
−0.922117 + 0.386912i \(0.873542\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) − 2.00000i − 0.152499i
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 8.00000i 0.601317i
\(178\) − 12.0000i − 0.899438i
\(179\) 18.0000 1.34538 0.672692 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 20.0000i 1.48250i
\(183\) − 1.00000i − 0.0739221i
\(184\) 0 0
\(185\) 0 0
\(186\) −12.0000 −0.879883
\(187\) − 1.00000i − 0.0731272i
\(188\) 18.0000i 1.31278i
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) 9.00000 0.651217 0.325609 0.945505i \(-0.394431\pi\)
0.325609 + 0.945505i \(0.394431\pi\)
\(192\) 8.00000i 0.577350i
\(193\) − 4.00000i − 0.287926i −0.989583 0.143963i \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) −20.0000 −1.43592
\(195\) 0 0
\(196\) 36.0000 2.57143
\(197\) − 2.00000i − 0.142494i −0.997459 0.0712470i \(-0.977302\pi\)
0.997459 0.0712470i \(-0.0226979\pi\)
\(198\) 2.00000i 0.142134i
\(199\) 21.0000 1.48865 0.744325 0.667817i \(-0.232771\pi\)
0.744325 + 0.667817i \(0.232771\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) − 4.00000i − 0.281439i
\(203\) − 10.0000i − 0.701862i
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) − 4.00000i − 0.278019i
\(208\) 8.00000i 0.554700i
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 20.0000i 1.37361i
\(213\) − 12.0000i − 0.822226i
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 30.0000i 2.03653i
\(218\) 8.00000i 0.541828i
\(219\) −11.0000 −0.743311
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) − 12.0000i − 0.803579i −0.915732 0.401790i \(-0.868388\pi\)
0.915732 0.401790i \(-0.131612\pi\)
\(224\) 40.0000 2.67261
\(225\) 0 0
\(226\) −4.00000 −0.266076
\(227\) 18.0000i 1.19470i 0.801980 + 0.597351i \(0.203780\pi\)
−0.801980 + 0.597351i \(0.796220\pi\)
\(228\) − 2.00000i − 0.132453i
\(229\) −25.0000 −1.65205 −0.826023 0.563636i \(-0.809402\pi\)
−0.826023 + 0.563636i \(0.809402\pi\)
\(230\) 0 0
\(231\) 5.00000 0.328976
\(232\) 0 0
\(233\) − 9.00000i − 0.589610i −0.955557 0.294805i \(-0.904745\pi\)
0.955557 0.294805i \(-0.0952546\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) −16.0000 −1.04151
\(237\) − 16.0000i − 1.03931i
\(238\) 10.0000i 0.648204i
\(239\) 3.00000 0.194054 0.0970269 0.995282i \(-0.469067\pi\)
0.0970269 + 0.995282i \(0.469067\pi\)
\(240\) 0 0
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 20.0000i 1.28565i
\(243\) 1.00000i 0.0641500i
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) − 2.00000i − 0.127257i
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 7.00000 0.441836 0.220918 0.975292i \(-0.429095\pi\)
0.220918 + 0.975292i \(0.429095\pi\)
\(252\) − 10.0000i − 0.629941i
\(253\) 4.00000i 0.251478i
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) − 8.00000i − 0.499026i −0.968371 0.249513i \(-0.919729\pi\)
0.968371 0.249513i \(-0.0802706\pi\)
\(258\) 2.00000i 0.124515i
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) − 14.0000i − 0.864923i
\(263\) − 23.0000i − 1.41824i −0.705087 0.709120i \(-0.749092\pi\)
0.705087 0.709120i \(-0.250908\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −10.0000 −0.613139
\(267\) 6.00000i 0.367194i
\(268\) − 16.0000i − 0.977356i
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 4.00000i 0.242536i
\(273\) − 10.0000i − 0.605228i
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) − 11.0000i − 0.660926i −0.943819 0.330463i \(-0.892795\pi\)
0.943819 0.330463i \(-0.107205\pi\)
\(278\) − 26.0000i − 1.55938i
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) − 18.0000i − 1.07188i
\(283\) 13.0000i 0.772770i 0.922338 + 0.386385i \(0.126276\pi\)
−0.922338 + 0.386385i \(0.873724\pi\)
\(284\) 24.0000 1.42414
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) − 8.00000i − 0.471405i
\(289\) 16.0000 0.941176
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) − 22.0000i − 1.28745i
\(293\) 28.0000i 1.63578i 0.575376 + 0.817889i \(0.304856\pi\)
−0.575376 + 0.817889i \(0.695144\pi\)
\(294\) −36.0000 −2.09956
\(295\) 0 0
\(296\) 0 0
\(297\) − 1.00000i − 0.0580259i
\(298\) − 42.0000i − 2.43299i
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 5.00000 0.288195
\(302\) 0 0
\(303\) 2.00000i 0.114897i
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 10.0000i 0.569803i
\(309\) −2.00000 −0.113776
\(310\) 0 0
\(311\) −21.0000 −1.19080 −0.595400 0.803429i \(-0.703007\pi\)
−0.595400 + 0.803429i \(0.703007\pi\)
\(312\) 0 0
\(313\) 2.00000i 0.113047i 0.998401 + 0.0565233i \(0.0180015\pi\)
−0.998401 + 0.0565233i \(0.981998\pi\)
\(314\) −36.0000 −2.03160
\(315\) 0 0
\(316\) 32.0000 1.80014
\(317\) − 4.00000i − 0.224662i −0.993671 0.112331i \(-0.964168\pi\)
0.993671 0.112331i \(-0.0358318\pi\)
\(318\) − 20.0000i − 1.12154i
\(319\) 2.00000 0.111979
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) − 40.0000i − 2.22911i
\(323\) − 1.00000i − 0.0556415i
\(324\) −2.00000 −0.111111
\(325\) 0 0
\(326\) 0 0
\(327\) − 4.00000i − 0.221201i
\(328\) 0 0
\(329\) −45.0000 −2.48093
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 24.0000i 1.31717i
\(333\) 0 0
\(334\) 20.0000 1.09435
\(335\) 0 0
\(336\) −20.0000 −1.09109
\(337\) − 14.0000i − 0.762629i −0.924445 0.381314i \(-0.875472\pi\)
0.924445 0.381314i \(-0.124528\pi\)
\(338\) − 18.0000i − 0.979071i
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 2.00000i 0.108148i
\(343\) 55.0000i 2.96972i
\(344\) 0 0
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) − 25.0000i − 1.34207i −0.741426 0.671035i \(-0.765850\pi\)
0.741426 0.671035i \(-0.234150\pi\)
\(348\) − 4.00000i − 0.214423i
\(349\) −9.00000 −0.481759 −0.240879 0.970555i \(-0.577436\pi\)
−0.240879 + 0.970555i \(0.577436\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 8.00000i 0.426401i
\(353\) 2.00000i 0.106449i 0.998583 + 0.0532246i \(0.0169499\pi\)
−0.998583 + 0.0532246i \(0.983050\pi\)
\(354\) 16.0000 0.850390
\(355\) 0 0
\(356\) −12.0000 −0.635999
\(357\) − 5.00000i − 0.264628i
\(358\) − 36.0000i − 1.90266i
\(359\) −37.0000 −1.95279 −0.976393 0.216003i \(-0.930698\pi\)
−0.976393 + 0.216003i \(0.930698\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 28.0000i 1.47165i
\(363\) − 10.0000i − 0.524864i
\(364\) 20.0000 1.04828
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) − 8.00000i − 0.417597i −0.977959 0.208798i \(-0.933045\pi\)
0.977959 0.208798i \(-0.0669552\pi\)
\(368\) − 16.0000i − 0.834058i
\(369\) 0 0
\(370\) 0 0
\(371\) −50.0000 −2.59587
\(372\) 12.0000i 0.622171i
\(373\) − 16.0000i − 0.828449i −0.910175 0.414224i \(-0.864053\pi\)
0.910175 0.414224i \(-0.135947\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) 0 0
\(377\) − 4.00000i − 0.206010i
\(378\) 10.0000i 0.514344i
\(379\) −34.0000 −1.74646 −0.873231 0.487306i \(-0.837980\pi\)
−0.873231 + 0.487306i \(0.837980\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) − 18.0000i − 0.920960i
\(383\) 34.0000i 1.73732i 0.495410 + 0.868659i \(0.335018\pi\)
−0.495410 + 0.868659i \(0.664982\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −8.00000 −0.407189
\(387\) − 1.00000i − 0.0508329i
\(388\) 20.0000i 1.01535i
\(389\) 27.0000 1.36895 0.684477 0.729034i \(-0.260031\pi\)
0.684477 + 0.729034i \(0.260031\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 7.00000i 0.353103i
\(394\) −4.00000 −0.201517
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 25.0000i 1.25471i 0.778732 + 0.627357i \(0.215863\pi\)
−0.778732 + 0.627357i \(0.784137\pi\)
\(398\) − 42.0000i − 2.10527i
\(399\) 5.00000 0.250313
\(400\) 0 0
\(401\) 36.0000 1.79775 0.898877 0.438201i \(-0.144384\pi\)
0.898877 + 0.438201i \(0.144384\pi\)
\(402\) 16.0000i 0.798007i
\(403\) 12.0000i 0.597763i
\(404\) −4.00000 −0.199007
\(405\) 0 0
\(406\) −20.0000 −0.992583
\(407\) 0 0
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 9.00000 0.443937
\(412\) − 4.00000i − 0.197066i
\(413\) − 40.0000i − 1.96827i
\(414\) −8.00000 −0.393179
\(415\) 0 0
\(416\) 16.0000 0.784465
\(417\) 13.0000i 0.636613i
\(418\) − 2.00000i − 0.0978232i
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) − 24.0000i − 1.16830i
\(423\) 9.00000i 0.437595i
\(424\) 0 0
\(425\) 0 0
\(426\) −24.0000 −1.16280
\(427\) 5.00000i 0.241967i
\(428\) − 12.0000i − 0.580042i
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) −34.0000 −1.63772 −0.818861 0.573992i \(-0.805394\pi\)
−0.818861 + 0.573992i \(0.805394\pi\)
\(432\) 4.00000i 0.192450i
\(433\) − 6.00000i − 0.288342i −0.989553 0.144171i \(-0.953949\pi\)
0.989553 0.144171i \(-0.0460515\pi\)
\(434\) 60.0000 2.88009
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) 4.00000i 0.191346i
\(438\) 22.0000i 1.05120i
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 18.0000 0.857143
\(442\) 4.00000i 0.190261i
\(443\) 5.00000i 0.237557i 0.992921 + 0.118779i \(0.0378979\pi\)
−0.992921 + 0.118779i \(0.962102\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −24.0000 −1.13643
\(447\) 21.0000i 0.993266i
\(448\) − 40.0000i − 1.88982i
\(449\) 36.0000 1.69895 0.849473 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 4.00000i 0.188144i
\(453\) 0 0
\(454\) 36.0000 1.68956
\(455\) 0 0
\(456\) 0 0
\(457\) − 29.0000i − 1.35656i −0.734802 0.678281i \(-0.762725\pi\)
0.734802 0.678281i \(-0.237275\pi\)
\(458\) 50.0000i 2.33635i
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) 27.0000 1.25752 0.628758 0.777601i \(-0.283564\pi\)
0.628758 + 0.777601i \(0.283564\pi\)
\(462\) − 10.0000i − 0.465242i
\(463\) − 17.0000i − 0.790057i −0.918669 0.395029i \(-0.870735\pi\)
0.918669 0.395029i \(-0.129265\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) − 5.00000i − 0.231372i −0.993286 0.115686i \(-0.963093\pi\)
0.993286 0.115686i \(-0.0369067\pi\)
\(468\) − 4.00000i − 0.184900i
\(469\) 40.0000 1.84703
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 1.00000i 0.0459800i
\(474\) −32.0000 −1.46981
\(475\) 0 0
\(476\) 10.0000 0.458349
\(477\) 10.0000i 0.457869i
\(478\) − 6.00000i − 0.274434i
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) − 40.0000i − 1.82195i
\(483\) 20.0000i 0.910032i
\(484\) 20.0000 0.909091
\(485\) 0 0
\(486\) 2.00000 0.0907218
\(487\) − 16.0000i − 0.725029i −0.931978 0.362515i \(-0.881918\pi\)
0.931978 0.362515i \(-0.118082\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) − 2.00000i − 0.0900755i
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 24.0000 1.07763
\(497\) 60.0000i 2.69137i
\(498\) − 24.0000i − 1.07547i
\(499\) −5.00000 −0.223831 −0.111915 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) 0 0
\(501\) −10.0000 −0.446767
\(502\) − 14.0000i − 0.624851i
\(503\) − 16.0000i − 0.713405i −0.934218 0.356702i \(-0.883901\pi\)
0.934218 0.356702i \(-0.116099\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 8.00000 0.355643
\(507\) 9.00000i 0.399704i
\(508\) 4.00000i 0.177471i
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 55.0000 2.43306
\(512\) − 32.0000i − 1.41421i
\(513\) − 1.00000i − 0.0441511i
\(514\) −16.0000 −0.705730
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) − 9.00000i − 0.395820i
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 36.0000 1.57719 0.788594 0.614914i \(-0.210809\pi\)
0.788594 + 0.614914i \(0.210809\pi\)
\(522\) 4.00000i 0.175075i
\(523\) 34.0000i 1.48672i 0.668894 + 0.743358i \(0.266768\pi\)
−0.668894 + 0.743358i \(0.733232\pi\)
\(524\) −14.0000 −0.611593
\(525\) 0 0
\(526\) −46.0000 −2.00570
\(527\) 6.00000i 0.261364i
\(528\) − 4.00000i − 0.174078i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 10.0000i 0.433555i
\(533\) 0 0
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) 0 0
\(537\) 18.0000i 0.776757i
\(538\) − 28.0000i − 1.20717i
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) 3.00000 0.128980 0.0644900 0.997918i \(-0.479458\pi\)
0.0644900 + 0.997918i \(0.479458\pi\)
\(542\) − 24.0000i − 1.03089i
\(543\) − 14.0000i − 0.600798i
\(544\) 8.00000 0.342997
\(545\) 0 0
\(546\) −20.0000 −0.855921
\(547\) − 26.0000i − 1.11168i −0.831289 0.555840i \(-0.812397\pi\)
0.831289 0.555840i \(-0.187603\pi\)
\(548\) 18.0000i 0.768922i
\(549\) 1.00000 0.0426790
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) 0 0
\(553\) 80.0000i 3.40195i
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −26.0000 −1.10265
\(557\) − 41.0000i − 1.73723i −0.495491 0.868613i \(-0.665012\pi\)
0.495491 0.868613i \(-0.334988\pi\)
\(558\) − 12.0000i − 0.508001i
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 1.00000 0.0422200
\(562\) − 20.0000i − 0.843649i
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) −18.0000 −0.757937
\(565\) 0 0
\(566\) 26.0000 1.09286
\(567\) − 5.00000i − 0.209980i
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 4.00000i 0.167248i
\(573\) 9.00000i 0.375980i
\(574\) 0 0
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) 27.0000i 1.12402i 0.827129 + 0.562012i \(0.189973\pi\)
−0.827129 + 0.562012i \(0.810027\pi\)
\(578\) − 32.0000i − 1.33102i
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) −60.0000 −2.48922
\(582\) − 20.0000i − 0.829027i
\(583\) − 10.0000i − 0.414158i
\(584\) 0 0
\(585\) 0 0
\(586\) 56.0000 2.31334
\(587\) 7.00000i 0.288921i 0.989511 + 0.144460i \(0.0461446\pi\)
−0.989511 + 0.144460i \(0.953855\pi\)
\(588\) 36.0000i 1.48461i
\(589\) −6.00000 −0.247226
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) 0 0
\(593\) 6.00000i 0.246390i 0.992382 + 0.123195i \(0.0393141\pi\)
−0.992382 + 0.123195i \(0.960686\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −42.0000 −1.72039
\(597\) 21.0000i 0.859473i
\(598\) − 16.0000i − 0.654289i
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) − 10.0000i − 0.407570i
\(603\) − 8.00000i − 0.325785i
\(604\) 0 0
\(605\) 0 0
\(606\) 4.00000 0.162489
\(607\) 26.0000i 1.05531i 0.849460 + 0.527654i \(0.176928\pi\)
−0.849460 + 0.527654i \(0.823072\pi\)
\(608\) 8.00000i 0.324443i
\(609\) 10.0000 0.405220
\(610\) 0 0
\(611\) −18.0000 −0.728202
\(612\) − 2.00000i − 0.0808452i
\(613\) − 33.0000i − 1.33286i −0.745569 0.666429i \(-0.767822\pi\)
0.745569 0.666429i \(-0.232178\pi\)
\(614\) −24.0000 −0.968561
\(615\) 0 0
\(616\) 0 0
\(617\) 27.0000i 1.08698i 0.839416 + 0.543490i \(0.182897\pi\)
−0.839416 + 0.543490i \(0.817103\pi\)
\(618\) 4.00000i 0.160904i
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 42.0000i 1.68405i
\(623\) − 30.0000i − 1.20192i
\(624\) −8.00000 −0.320256
\(625\) 0 0
\(626\) 4.00000 0.159872
\(627\) 1.00000i 0.0399362i
\(628\) 36.0000i 1.43656i
\(629\) 0 0
\(630\) 0 0
\(631\) 15.0000 0.597141 0.298570 0.954388i \(-0.403490\pi\)
0.298570 + 0.954388i \(0.403490\pi\)
\(632\) 0 0
\(633\) 12.0000i 0.476957i
\(634\) −8.00000 −0.317721
\(635\) 0 0
\(636\) −20.0000 −0.793052
\(637\) 36.0000i 1.42637i
\(638\) − 4.00000i − 0.158362i
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 1.00000i 0.0394362i 0.999806 + 0.0197181i \(0.00627687\pi\)
−0.999806 + 0.0197181i \(0.993723\pi\)
\(644\) −40.0000 −1.57622
\(645\) 0 0
\(646\) −2.00000 −0.0786889
\(647\) − 39.0000i − 1.53325i −0.642096 0.766624i \(-0.721935\pi\)
0.642096 0.766624i \(-0.278065\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) −30.0000 −1.17579
\(652\) 0 0
\(653\) − 3.00000i − 0.117399i −0.998276 0.0586995i \(-0.981305\pi\)
0.998276 0.0586995i \(-0.0186954\pi\)
\(654\) −8.00000 −0.312825
\(655\) 0 0
\(656\) 0 0
\(657\) − 11.0000i − 0.429151i
\(658\) 90.0000i 3.50857i
\(659\) 14.0000 0.545363 0.272681 0.962104i \(-0.412090\pi\)
0.272681 + 0.962104i \(0.412090\pi\)
\(660\) 0 0
\(661\) 12.0000 0.466746 0.233373 0.972387i \(-0.425024\pi\)
0.233373 + 0.972387i \(0.425024\pi\)
\(662\) 8.00000i 0.310929i
\(663\) − 2.00000i − 0.0776736i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 8.00000i 0.309761i
\(668\) − 20.0000i − 0.773823i
\(669\) 12.0000 0.463947
\(670\) 0 0
\(671\) −1.00000 −0.0386046
\(672\) 40.0000i 1.54303i
\(673\) 24.0000i 0.925132i 0.886585 + 0.462566i \(0.153071\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) −28.0000 −1.07852
\(675\) 0 0
\(676\) −18.0000 −0.692308
\(677\) 34.0000i 1.30673i 0.757045 + 0.653363i \(0.226642\pi\)
−0.757045 + 0.653363i \(0.773358\pi\)
\(678\) − 4.00000i − 0.153619i
\(679\) −50.0000 −1.91882
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 12.0000i 0.459504i
\(683\) 6.00000i 0.229584i 0.993390 + 0.114792i \(0.0366201\pi\)
−0.993390 + 0.114792i \(0.963380\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) 110.000 4.19982
\(687\) − 25.0000i − 0.953809i
\(688\) − 4.00000i − 0.152499i
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) −31.0000 −1.17930 −0.589648 0.807661i \(-0.700733\pi\)
−0.589648 + 0.807661i \(0.700733\pi\)
\(692\) 12.0000i 0.456172i
\(693\) 5.00000i 0.189934i
\(694\) −50.0000 −1.89797
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 18.0000i 0.681310i
\(699\) 9.00000 0.340411
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 4.00000i 0.150970i
\(703\) 0 0
\(704\) 8.00000 0.301511
\(705\) 0 0
\(706\) 4.00000 0.150542
\(707\) − 10.0000i − 0.376089i
\(708\) − 16.0000i − 0.601317i
\(709\) 42.0000 1.57734 0.788672 0.614815i \(-0.210769\pi\)
0.788672 + 0.614815i \(0.210769\pi\)
\(710\) 0 0
\(711\) 16.0000 0.600047
\(712\) 0 0
\(713\) − 24.0000i − 0.898807i
\(714\) −10.0000 −0.374241
\(715\) 0 0
\(716\) −36.0000 −1.34538
\(717\) 3.00000i 0.112037i
\(718\) 74.0000i 2.76166i
\(719\) −33.0000 −1.23069 −0.615346 0.788257i \(-0.710984\pi\)
−0.615346 + 0.788257i \(0.710984\pi\)
\(720\) 0 0
\(721\) 10.0000 0.372419
\(722\) − 2.00000i − 0.0744323i
\(723\) 20.0000i 0.743808i
\(724\) 28.0000 1.04061
\(725\) 0 0
\(726\) −20.0000 −0.742270
\(727\) − 23.0000i − 0.853023i −0.904482 0.426511i \(-0.859742\pi\)
0.904482 0.426511i \(-0.140258\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 1.00000 0.0369863
\(732\) 2.00000i 0.0739221i
\(733\) 14.0000i 0.517102i 0.965998 + 0.258551i \(0.0832450\pi\)
−0.965998 + 0.258551i \(0.916755\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) −32.0000 −1.17954
\(737\) 8.00000i 0.294684i
\(738\) 0 0
\(739\) 5.00000 0.183928 0.0919640 0.995762i \(-0.470686\pi\)
0.0919640 + 0.995762i \(0.470686\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 100.000i 3.67112i
\(743\) 8.00000i 0.293492i 0.989174 + 0.146746i \(0.0468799\pi\)
−0.989174 + 0.146746i \(0.953120\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −32.0000 −1.17160
\(747\) 12.0000i 0.439057i
\(748\) 2.00000i 0.0731272i
\(749\) 30.0000 1.09618
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 36.0000i 1.31278i
\(753\) 7.00000i 0.255094i
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) 10.0000 0.363696
\(757\) − 17.0000i − 0.617876i −0.951082 0.308938i \(-0.900027\pi\)
0.951082 0.308938i \(-0.0999735\pi\)
\(758\) 68.0000i 2.46987i
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) 15.0000 0.543750 0.271875 0.962333i \(-0.412356\pi\)
0.271875 + 0.962333i \(0.412356\pi\)
\(762\) − 4.00000i − 0.144905i
\(763\) 20.0000i 0.724049i
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) 68.0000 2.45694
\(767\) − 16.0000i − 0.577727i
\(768\) 16.0000i 0.577350i
\(769\) −11.0000 −0.396670 −0.198335 0.980134i \(-0.563553\pi\)
−0.198335 + 0.980134i \(0.563553\pi\)
\(770\) 0 0
\(771\) 8.00000 0.288113
\(772\) 8.00000i 0.287926i
\(773\) − 20.0000i − 0.719350i −0.933078 0.359675i \(-0.882888\pi\)
0.933078 0.359675i \(-0.117112\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) − 54.0000i − 1.93599i
\(779\) 0 0
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) − 8.00000i − 0.286079i
\(783\) − 2.00000i − 0.0714742i
\(784\) 72.0000 2.57143
\(785\) 0 0
\(786\) 14.0000 0.499363
\(787\) 40.0000i 1.42585i 0.701242 + 0.712923i \(0.252629\pi\)
−0.701242 + 0.712923i \(0.747371\pi\)
\(788\) 4.00000i 0.142494i
\(789\) 23.0000 0.818822
\(790\) 0 0
\(791\) −10.0000 −0.355559
\(792\) 0 0
\(793\) 2.00000i 0.0710221i
\(794\) 50.0000 1.77443
\(795\) 0 0
\(796\) −42.0000 −1.48865
\(797\) − 44.0000i − 1.55856i −0.626676 0.779280i \(-0.715585\pi\)
0.626676 0.779280i \(-0.284415\pi\)
\(798\) − 10.0000i − 0.353996i
\(799\) −9.00000 −0.318397
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) − 72.0000i − 2.54241i
\(803\) 11.0000i 0.388182i
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) 24.0000 0.845364
\(807\) 14.0000i 0.492823i
\(808\) 0 0
\(809\) 55.0000 1.93370 0.966849 0.255351i \(-0.0821909\pi\)
0.966849 + 0.255351i \(0.0821909\pi\)
\(810\) 0 0
\(811\) −38.0000 −1.33436 −0.667180 0.744896i \(-0.732499\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) 20.0000i 0.701862i
\(813\) 12.0000i 0.420858i
\(814\) 0 0
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) 1.00000i 0.0349856i
\(818\) − 28.0000i − 0.978997i
\(819\) 10.0000 0.349428
\(820\) 0 0
\(821\) −45.0000 −1.57051 −0.785255 0.619172i \(-0.787468\pi\)
−0.785255 + 0.619172i \(0.787468\pi\)
\(822\) − 18.0000i − 0.627822i
\(823\) − 43.0000i − 1.49889i −0.662069 0.749443i \(-0.730321\pi\)
0.662069 0.749443i \(-0.269679\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −80.0000 −2.78356
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 8.00000i 0.278019i
\(829\) −52.0000 −1.80603 −0.903017 0.429604i \(-0.858653\pi\)
−0.903017 + 0.429604i \(0.858653\pi\)
\(830\) 0 0
\(831\) 11.0000 0.381586
\(832\) − 16.0000i − 0.554700i
\(833\) 18.0000i 0.623663i
\(834\) 26.0000 0.900306
\(835\) 0 0
\(836\) −2.00000 −0.0691714
\(837\) 6.00000i 0.207390i
\(838\) 56.0000i 1.93449i
\(839\) −54.0000 −1.86429 −0.932144 0.362089i \(-0.882064\pi\)
−0.932144 + 0.362089i \(0.882064\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) − 52.0000i − 1.79204i
\(843\) 10.0000i 0.344418i
\(844\) −24.0000 −0.826114
\(845\) 0 0
\(846\) 18.0000 0.618853
\(847\) 50.0000i 1.71802i
\(848\) 40.0000i 1.37361i
\(849\) −13.0000 −0.446159
\(850\) 0 0
\(851\) 0 0
\(852\) 24.0000i 0.822226i
\(853\) 14.0000i 0.479351i 0.970853 + 0.239675i \(0.0770410\pi\)
−0.970853 + 0.239675i \(0.922959\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) 0 0
\(857\) − 8.00000i − 0.273275i −0.990621 0.136637i \(-0.956370\pi\)
0.990621 0.136637i \(-0.0436295\pi\)
\(858\) − 4.00000i − 0.136558i
\(859\) −27.0000 −0.921228 −0.460614 0.887601i \(-0.652371\pi\)
−0.460614 + 0.887601i \(0.652371\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 68.0000i 2.31609i
\(863\) 44.0000i 1.49778i 0.662696 + 0.748889i \(0.269412\pi\)
−0.662696 + 0.748889i \(0.730588\pi\)
\(864\) 8.00000 0.272166
\(865\) 0 0
\(866\) −12.0000 −0.407777
\(867\) 16.0000i 0.543388i
\(868\) − 60.0000i − 2.03653i
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 0 0
\(873\) 10.0000i 0.338449i
\(874\) 8.00000 0.270604
\(875\) 0 0
\(876\) 22.0000 0.743311
\(877\) − 6.00000i − 0.202606i −0.994856 0.101303i \(-0.967699\pi\)
0.994856 0.101303i \(-0.0323011\pi\)
\(878\) 52.0000i 1.75491i
\(879\) −28.0000 −0.944417
\(880\) 0 0
\(881\) −37.0000 −1.24656 −0.623281 0.781998i \(-0.714201\pi\)
−0.623281 + 0.781998i \(0.714201\pi\)
\(882\) − 36.0000i − 1.21218i
\(883\) − 35.0000i − 1.17784i −0.808190 0.588922i \(-0.799553\pi\)
0.808190 0.588922i \(-0.200447\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) 10.0000 0.335957
\(887\) − 12.0000i − 0.402921i −0.979497 0.201460i \(-0.935431\pi\)
0.979497 0.201460i \(-0.0645687\pi\)
\(888\) 0 0
\(889\) −10.0000 −0.335389
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 24.0000i 0.803579i
\(893\) − 9.00000i − 0.301174i
\(894\) 42.0000 1.40469
\(895\) 0 0
\(896\) 0 0
\(897\) 8.00000i 0.267112i
\(898\) − 72.0000i − 2.40267i
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) −10.0000 −0.333148
\(902\) 0 0
\(903\) 5.00000i 0.166390i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 26.0000i − 0.863316i −0.902037 0.431658i \(-0.857929\pi\)
0.902037 0.431658i \(-0.142071\pi\)
\(908\) − 36.0000i − 1.19470i
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) −6.00000 −0.198789 −0.0993944 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) − 12.0000i − 0.397142i
\(914\) −58.0000 −1.91847
\(915\) 0 0
\(916\) 50.0000 1.65205
\(917\) − 35.0000i − 1.15580i
\(918\) 2.00000i 0.0660098i
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) − 54.0000i − 1.77840i
\(923\) 24.0000i 0.789970i
\(924\) −10.0000 −0.328976
\(925\) 0 0
\(926\) −34.0000 −1.11731
\(927\) − 2.00000i − 0.0656886i
\(928\) 16.0000i 0.525226i
\(929\) 2.00000 0.0656179 0.0328089 0.999462i \(-0.489555\pi\)
0.0328089 + 0.999462i \(0.489555\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) 18.0000i 0.589610i
\(933\) − 21.0000i − 0.687509i
\(934\) −10.0000 −0.327210
\(935\) 0 0
\(936\) 0 0
\(937\) 21.0000i 0.686040i 0.939328 + 0.343020i \(0.111450\pi\)
−0.939328 + 0.343020i \(0.888550\pi\)
\(938\) − 80.0000i − 2.61209i
\(939\) −2.00000 −0.0652675
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) − 36.0000i − 1.17294i
\(943\) 0 0
\(944\) −32.0000 −1.04151
\(945\) 0 0
\(946\) 2.00000 0.0650256
\(947\) − 28.0000i − 0.909878i −0.890523 0.454939i \(-0.849661\pi\)
0.890523 0.454939i \(-0.150339\pi\)
\(948\) 32.0000i 1.03931i
\(949\) 22.0000 0.714150
\(950\) 0 0
\(951\) 4.00000 0.129709
\(952\) 0 0
\(953\) 32.0000i 1.03658i 0.855204 + 0.518291i \(0.173432\pi\)
−0.855204 + 0.518291i \(0.826568\pi\)
\(954\) 20.0000 0.647524
\(955\) 0 0
\(956\) −6.00000 −0.194054
\(957\) 2.00000i 0.0646508i
\(958\) − 32.0000i − 1.03387i
\(959\) −45.0000 −1.45313
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) − 6.00000i − 0.193347i
\(964\) −40.0000 −1.28831
\(965\) 0 0
\(966\) 40.0000 1.28698
\(967\) − 32.0000i − 1.02905i −0.857475 0.514525i \(-0.827968\pi\)
0.857475 0.514525i \(-0.172032\pi\)
\(968\) 0 0
\(969\) 1.00000 0.0321246
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) − 2.00000i − 0.0641500i
\(973\) − 65.0000i − 2.08380i
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) 54.0000i 1.72761i 0.503824 + 0.863807i \(0.331926\pi\)
−0.503824 + 0.863807i \(0.668074\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 0 0
\(983\) 44.0000i 1.40338i 0.712481 + 0.701691i \(0.247571\pi\)
−0.712481 + 0.701691i \(0.752429\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −4.00000 −0.127386
\(987\) − 45.0000i − 1.43237i
\(988\) 4.00000i 0.127257i
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) − 48.0000i − 1.52400i
\(993\) − 4.00000i − 0.126936i
\(994\) 120.000 3.80617
\(995\) 0 0
\(996\) −24.0000 −0.760469
\(997\) − 47.0000i − 1.48850i −0.667898 0.744252i \(-0.732806\pi\)
0.667898 0.744252i \(-0.267194\pi\)
\(998\) 10.0000i 0.316544i
\(999\) 0 0
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 1425.2.c.b.799.1 2
5.2 odd 4 1425.2.a.j.1.1 1
5.3 odd 4 57.2.a.a.1.1 1
5.4 even 2 inner 1425.2.c.b.799.2 2
15.2 even 4 4275.2.a.b.1.1 1
15.8 even 4 171.2.a.d.1.1 1
20.3 even 4 912.2.a.g.1.1 1
35.13 even 4 2793.2.a.b.1.1 1
40.3 even 4 3648.2.a.r.1.1 1
40.13 odd 4 3648.2.a.bh.1.1 1
55.43 even 4 6897.2.a.f.1.1 1
60.23 odd 4 2736.2.a.v.1.1 1
65.38 odd 4 9633.2.a.o.1.1 1
95.18 even 4 1083.2.a.e.1.1 1
105.83 odd 4 8379.2.a.p.1.1 1
285.113 odd 4 3249.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.a.1.1 1 5.3 odd 4
171.2.a.d.1.1 1 15.8 even 4
912.2.a.g.1.1 1 20.3 even 4
1083.2.a.e.1.1 1 95.18 even 4
1425.2.a.j.1.1 1 5.2 odd 4
1425.2.c.b.799.1 2 1.1 even 1 trivial
1425.2.c.b.799.2 2 5.4 even 2 inner
2736.2.a.v.1.1 1 60.23 odd 4
2793.2.a.b.1.1 1 35.13 even 4
3249.2.a.b.1.1 1 285.113 odd 4
3648.2.a.r.1.1 1 40.3 even 4
3648.2.a.bh.1.1 1 40.13 odd 4
4275.2.a.b.1.1 1 15.2 even 4
6897.2.a.f.1.1 1 55.43 even 4
8379.2.a.p.1.1 1 105.83 odd 4
9633.2.a.o.1.1 1 65.38 odd 4