Properties

Label 3234.2.a.z.1.1
Level $3234$
Weight $2$
Character 3234.1
Self dual yes
Analytic conductor $25.824$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.8236200137\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3234.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{12} -1.41421 q^{13} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -5.41421 q^{19} -1.00000 q^{22} +7.65685 q^{23} -1.00000 q^{24} -5.00000 q^{25} +1.41421 q^{26} +1.00000 q^{27} -5.65685 q^{29} +3.07107 q^{31} -1.00000 q^{32} +1.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} +3.65685 q^{37} +5.41421 q^{38} -1.41421 q^{39} -9.65685 q^{41} -10.0000 q^{43} +1.00000 q^{44} -7.65685 q^{46} +1.41421 q^{47} +1.00000 q^{48} +5.00000 q^{50} -4.00000 q^{51} -1.41421 q^{52} -3.65685 q^{53} -1.00000 q^{54} -5.41421 q^{57} +5.65685 q^{58} -6.82843 q^{59} +1.41421 q^{61} -3.07107 q^{62} +1.00000 q^{64} -1.00000 q^{66} -11.3137 q^{67} -4.00000 q^{68} +7.65685 q^{69} +13.6569 q^{71} -1.00000 q^{72} -4.00000 q^{73} -3.65685 q^{74} -5.00000 q^{75} -5.41421 q^{76} +1.41421 q^{78} +1.65685 q^{79} +1.00000 q^{81} +9.65685 q^{82} +15.0711 q^{83} +10.0000 q^{86} -5.65685 q^{87} -1.00000 q^{88} -2.58579 q^{89} +7.65685 q^{92} +3.07107 q^{93} -1.41421 q^{94} -1.00000 q^{96} -12.7279 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} + 2 q^{12} + 2 q^{16} - 8 q^{17} - 2 q^{18} - 8 q^{19} - 2 q^{22} + 4 q^{23} - 2 q^{24} - 10 q^{25} + 2 q^{27} - 8 q^{31} - 2 q^{32} + 2 q^{33} + 8 q^{34} + 2 q^{36} - 4 q^{37} + 8 q^{38} - 8 q^{41} - 20 q^{43} + 2 q^{44} - 4 q^{46} + 2 q^{48} + 10 q^{50} - 8 q^{51} + 4 q^{53} - 2 q^{54} - 8 q^{57} - 8 q^{59} + 8 q^{62} + 2 q^{64} - 2 q^{66} - 8 q^{68} + 4 q^{69} + 16 q^{71} - 2 q^{72} - 8 q^{73} + 4 q^{74} - 10 q^{75} - 8 q^{76} - 8 q^{79} + 2 q^{81} + 8 q^{82} + 16 q^{83} + 20 q^{86} - 2 q^{88} - 8 q^{89} + 4 q^{92} - 8 q^{93} - 2 q^{96} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 1.00000 0.288675
\(13\) −1.41421 −0.392232 −0.196116 0.980581i \(-0.562833\pi\)
−0.196116 + 0.980581i \(0.562833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) −1.00000 −0.235702
\(19\) −5.41421 −1.24211 −0.621053 0.783769i \(-0.713295\pi\)
−0.621053 + 0.783769i \(0.713295\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 7.65685 1.59656 0.798282 0.602284i \(-0.205742\pi\)
0.798282 + 0.602284i \(0.205742\pi\)
\(24\) −1.00000 −0.204124
\(25\) −5.00000 −1.00000
\(26\) 1.41421 0.277350
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −5.65685 −1.05045 −0.525226 0.850963i \(-0.676019\pi\)
−0.525226 + 0.850963i \(0.676019\pi\)
\(30\) 0 0
\(31\) 3.07107 0.551580 0.275790 0.961218i \(-0.411061\pi\)
0.275790 + 0.961218i \(0.411061\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 3.65685 0.601183 0.300592 0.953753i \(-0.402816\pi\)
0.300592 + 0.953753i \(0.402816\pi\)
\(38\) 5.41421 0.878301
\(39\) −1.41421 −0.226455
\(40\) 0 0
\(41\) −9.65685 −1.50815 −0.754074 0.656790i \(-0.771914\pi\)
−0.754074 + 0.656790i \(0.771914\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −7.65685 −1.12894
\(47\) 1.41421 0.206284 0.103142 0.994667i \(-0.467110\pi\)
0.103142 + 0.994667i \(0.467110\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 5.00000 0.707107
\(51\) −4.00000 −0.560112
\(52\) −1.41421 −0.196116
\(53\) −3.65685 −0.502308 −0.251154 0.967947i \(-0.580810\pi\)
−0.251154 + 0.967947i \(0.580810\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −5.41421 −0.717130
\(58\) 5.65685 0.742781
\(59\) −6.82843 −0.888985 −0.444493 0.895782i \(-0.646616\pi\)
−0.444493 + 0.895782i \(0.646616\pi\)
\(60\) 0 0
\(61\) 1.41421 0.181071 0.0905357 0.995893i \(-0.471142\pi\)
0.0905357 + 0.995893i \(0.471142\pi\)
\(62\) −3.07107 −0.390026
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) −11.3137 −1.38219 −0.691095 0.722764i \(-0.742871\pi\)
−0.691095 + 0.722764i \(0.742871\pi\)
\(68\) −4.00000 −0.485071
\(69\) 7.65685 0.921777
\(70\) 0 0
\(71\) 13.6569 1.62077 0.810385 0.585897i \(-0.199258\pi\)
0.810385 + 0.585897i \(0.199258\pi\)
\(72\) −1.00000 −0.117851
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −3.65685 −0.425101
\(75\) −5.00000 −0.577350
\(76\) −5.41421 −0.621053
\(77\) 0 0
\(78\) 1.41421 0.160128
\(79\) 1.65685 0.186411 0.0932053 0.995647i \(-0.470289\pi\)
0.0932053 + 0.995647i \(0.470289\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 9.65685 1.06642
\(83\) 15.0711 1.65426 0.827132 0.562007i \(-0.189971\pi\)
0.827132 + 0.562007i \(0.189971\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.0000 1.07833
\(87\) −5.65685 −0.606478
\(88\) −1.00000 −0.106600
\(89\) −2.58579 −0.274093 −0.137046 0.990565i \(-0.543761\pi\)
−0.137046 + 0.990565i \(0.543761\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 7.65685 0.798282
\(93\) 3.07107 0.318455
\(94\) −1.41421 −0.145865
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −12.7279 −1.29232 −0.646162 0.763200i \(-0.723627\pi\)
−0.646162 + 0.763200i \(0.723627\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) −5.00000 −0.500000
\(101\) 0.242641 0.0241437 0.0120718 0.999927i \(-0.496157\pi\)
0.0120718 + 0.999927i \(0.496157\pi\)
\(102\) 4.00000 0.396059
\(103\) −13.8995 −1.36956 −0.684779 0.728751i \(-0.740101\pi\)
−0.684779 + 0.728751i \(0.740101\pi\)
\(104\) 1.41421 0.138675
\(105\) 0 0
\(106\) 3.65685 0.355185
\(107\) 17.3137 1.67378 0.836890 0.547372i \(-0.184372\pi\)
0.836890 + 0.547372i \(0.184372\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 3.65685 0.347093
\(112\) 0 0
\(113\) −5.65685 −0.532152 −0.266076 0.963952i \(-0.585727\pi\)
−0.266076 + 0.963952i \(0.585727\pi\)
\(114\) 5.41421 0.507088
\(115\) 0 0
\(116\) −5.65685 −0.525226
\(117\) −1.41421 −0.130744
\(118\) 6.82843 0.628608
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −1.41421 −0.128037
\(123\) −9.65685 −0.870729
\(124\) 3.07107 0.275790
\(125\) 0 0
\(126\) 0 0
\(127\) −15.3137 −1.35887 −0.679436 0.733735i \(-0.737775\pi\)
−0.679436 + 0.733735i \(0.737775\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) −3.75736 −0.328282 −0.164141 0.986437i \(-0.552485\pi\)
−0.164141 + 0.986437i \(0.552485\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) 11.3137 0.977356
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) −9.31371 −0.795724 −0.397862 0.917445i \(-0.630248\pi\)
−0.397862 + 0.917445i \(0.630248\pi\)
\(138\) −7.65685 −0.651795
\(139\) −2.10051 −0.178163 −0.0890813 0.996024i \(-0.528393\pi\)
−0.0890813 + 0.996024i \(0.528393\pi\)
\(140\) 0 0
\(141\) 1.41421 0.119098
\(142\) −13.6569 −1.14606
\(143\) −1.41421 −0.118262
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) 3.65685 0.300592
\(149\) 12.9706 1.06259 0.531295 0.847187i \(-0.321706\pi\)
0.531295 + 0.847187i \(0.321706\pi\)
\(150\) 5.00000 0.408248
\(151\) −1.65685 −0.134833 −0.0674164 0.997725i \(-0.521476\pi\)
−0.0674164 + 0.997725i \(0.521476\pi\)
\(152\) 5.41421 0.439151
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) −1.41421 −0.113228
\(157\) 2.82843 0.225733 0.112867 0.993610i \(-0.463997\pi\)
0.112867 + 0.993610i \(0.463997\pi\)
\(158\) −1.65685 −0.131812
\(159\) −3.65685 −0.290007
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 5.65685 0.443079 0.221540 0.975151i \(-0.428892\pi\)
0.221540 + 0.975151i \(0.428892\pi\)
\(164\) −9.65685 −0.754074
\(165\) 0 0
\(166\) −15.0711 −1.16974
\(167\) −10.8284 −0.837929 −0.418964 0.908003i \(-0.637607\pi\)
−0.418964 + 0.908003i \(0.637607\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) −5.41421 −0.414035
\(172\) −10.0000 −0.762493
\(173\) −4.92893 −0.374740 −0.187370 0.982289i \(-0.559996\pi\)
−0.187370 + 0.982289i \(0.559996\pi\)
\(174\) 5.65685 0.428845
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −6.82843 −0.513256
\(178\) 2.58579 0.193813
\(179\) 2.34315 0.175135 0.0875675 0.996159i \(-0.472091\pi\)
0.0875675 + 0.996159i \(0.472091\pi\)
\(180\) 0 0
\(181\) 21.6569 1.60974 0.804871 0.593450i \(-0.202235\pi\)
0.804871 + 0.593450i \(0.202235\pi\)
\(182\) 0 0
\(183\) 1.41421 0.104542
\(184\) −7.65685 −0.564471
\(185\) 0 0
\(186\) −3.07107 −0.225182
\(187\) −4.00000 −0.292509
\(188\) 1.41421 0.103142
\(189\) 0 0
\(190\) 0 0
\(191\) −10.9706 −0.793802 −0.396901 0.917861i \(-0.629914\pi\)
−0.396901 + 0.917861i \(0.629914\pi\)
\(192\) 1.00000 0.0721688
\(193\) 7.65685 0.551152 0.275576 0.961279i \(-0.411131\pi\)
0.275576 + 0.961279i \(0.411131\pi\)
\(194\) 12.7279 0.913812
\(195\) 0 0
\(196\) 0 0
\(197\) 21.3137 1.51854 0.759269 0.650776i \(-0.225556\pi\)
0.759269 + 0.650776i \(0.225556\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −10.5858 −0.750407 −0.375203 0.926943i \(-0.622427\pi\)
−0.375203 + 0.926943i \(0.622427\pi\)
\(200\) 5.00000 0.353553
\(201\) −11.3137 −0.798007
\(202\) −0.242641 −0.0170721
\(203\) 0 0
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) 13.8995 0.968424
\(207\) 7.65685 0.532188
\(208\) −1.41421 −0.0980581
\(209\) −5.41421 −0.374509
\(210\) 0 0
\(211\) 17.3137 1.19192 0.595962 0.803012i \(-0.296771\pi\)
0.595962 + 0.803012i \(0.296771\pi\)
\(212\) −3.65685 −0.251154
\(213\) 13.6569 0.935752
\(214\) −17.3137 −1.18354
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 10.0000 0.677285
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) 5.65685 0.380521
\(222\) −3.65685 −0.245432
\(223\) −19.5563 −1.30959 −0.654795 0.755807i \(-0.727245\pi\)
−0.654795 + 0.755807i \(0.727245\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 5.65685 0.376288
\(227\) −23.5563 −1.56349 −0.781745 0.623598i \(-0.785670\pi\)
−0.781745 + 0.623598i \(0.785670\pi\)
\(228\) −5.41421 −0.358565
\(229\) −5.65685 −0.373815 −0.186908 0.982377i \(-0.559847\pi\)
−0.186908 + 0.982377i \(0.559847\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 5.65685 0.371391
\(233\) 12.3431 0.808626 0.404313 0.914621i \(-0.367511\pi\)
0.404313 + 0.914621i \(0.367511\pi\)
\(234\) 1.41421 0.0924500
\(235\) 0 0
\(236\) −6.82843 −0.444493
\(237\) 1.65685 0.107624
\(238\) 0 0
\(239\) 14.3431 0.927781 0.463890 0.885893i \(-0.346453\pi\)
0.463890 + 0.885893i \(0.346453\pi\)
\(240\) 0 0
\(241\) 7.31371 0.471117 0.235559 0.971860i \(-0.424308\pi\)
0.235559 + 0.971860i \(0.424308\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 1.00000 0.0641500
\(244\) 1.41421 0.0905357
\(245\) 0 0
\(246\) 9.65685 0.615699
\(247\) 7.65685 0.487194
\(248\) −3.07107 −0.195013
\(249\) 15.0711 0.955090
\(250\) 0 0
\(251\) −14.8284 −0.935962 −0.467981 0.883739i \(-0.655018\pi\)
−0.467981 + 0.883739i \(0.655018\pi\)
\(252\) 0 0
\(253\) 7.65685 0.481382
\(254\) 15.3137 0.960868
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 16.7279 1.04346 0.521730 0.853111i \(-0.325287\pi\)
0.521730 + 0.853111i \(0.325287\pi\)
\(258\) 10.0000 0.622573
\(259\) 0 0
\(260\) 0 0
\(261\) −5.65685 −0.350150
\(262\) 3.75736 0.232130
\(263\) 13.6569 0.842118 0.421059 0.907033i \(-0.361659\pi\)
0.421059 + 0.907033i \(0.361659\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 0 0
\(266\) 0 0
\(267\) −2.58579 −0.158248
\(268\) −11.3137 −0.691095
\(269\) −22.6274 −1.37962 −0.689809 0.723991i \(-0.742306\pi\)
−0.689809 + 0.723991i \(0.742306\pi\)
\(270\) 0 0
\(271\) −21.1716 −1.28608 −0.643041 0.765832i \(-0.722327\pi\)
−0.643041 + 0.765832i \(0.722327\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 9.31371 0.562662
\(275\) −5.00000 −0.301511
\(276\) 7.65685 0.460888
\(277\) 4.00000 0.240337 0.120168 0.992754i \(-0.461657\pi\)
0.120168 + 0.992754i \(0.461657\pi\)
\(278\) 2.10051 0.125980
\(279\) 3.07107 0.183860
\(280\) 0 0
\(281\) −18.9706 −1.13169 −0.565844 0.824512i \(-0.691450\pi\)
−0.565844 + 0.824512i \(0.691450\pi\)
\(282\) −1.41421 −0.0842152
\(283\) −15.7574 −0.936678 −0.468339 0.883549i \(-0.655147\pi\)
−0.468339 + 0.883549i \(0.655147\pi\)
\(284\) 13.6569 0.810385
\(285\) 0 0
\(286\) 1.41421 0.0836242
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −12.7279 −0.746124
\(292\) −4.00000 −0.234082
\(293\) 3.07107 0.179414 0.0897068 0.995968i \(-0.471407\pi\)
0.0897068 + 0.995968i \(0.471407\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −3.65685 −0.212550
\(297\) 1.00000 0.0580259
\(298\) −12.9706 −0.751365
\(299\) −10.8284 −0.626224
\(300\) −5.00000 −0.288675
\(301\) 0 0
\(302\) 1.65685 0.0953412
\(303\) 0.242641 0.0139393
\(304\) −5.41421 −0.310526
\(305\) 0 0
\(306\) 4.00000 0.228665
\(307\) 0.727922 0.0415447 0.0207724 0.999784i \(-0.493387\pi\)
0.0207724 + 0.999784i \(0.493387\pi\)
\(308\) 0 0
\(309\) −13.8995 −0.790715
\(310\) 0 0
\(311\) −20.2426 −1.14785 −0.573927 0.818906i \(-0.694581\pi\)
−0.573927 + 0.818906i \(0.694581\pi\)
\(312\) 1.41421 0.0800641
\(313\) 12.2426 0.691995 0.345997 0.938235i \(-0.387541\pi\)
0.345997 + 0.938235i \(0.387541\pi\)
\(314\) −2.82843 −0.159617
\(315\) 0 0
\(316\) 1.65685 0.0932053
\(317\) −5.31371 −0.298448 −0.149224 0.988803i \(-0.547677\pi\)
−0.149224 + 0.988803i \(0.547677\pi\)
\(318\) 3.65685 0.205066
\(319\) −5.65685 −0.316723
\(320\) 0 0
\(321\) 17.3137 0.966357
\(322\) 0 0
\(323\) 21.6569 1.20502
\(324\) 1.00000 0.0555556
\(325\) 7.07107 0.392232
\(326\) −5.65685 −0.313304
\(327\) −10.0000 −0.553001
\(328\) 9.65685 0.533211
\(329\) 0 0
\(330\) 0 0
\(331\) 28.9706 1.59237 0.796183 0.605056i \(-0.206849\pi\)
0.796183 + 0.605056i \(0.206849\pi\)
\(332\) 15.0711 0.827132
\(333\) 3.65685 0.200394
\(334\) 10.8284 0.592505
\(335\) 0 0
\(336\) 0 0
\(337\) −17.3137 −0.943138 −0.471569 0.881829i \(-0.656312\pi\)
−0.471569 + 0.881829i \(0.656312\pi\)
\(338\) 11.0000 0.598321
\(339\) −5.65685 −0.307238
\(340\) 0 0
\(341\) 3.07107 0.166308
\(342\) 5.41421 0.292767
\(343\) 0 0
\(344\) 10.0000 0.539164
\(345\) 0 0
\(346\) 4.92893 0.264981
\(347\) −2.00000 −0.107366 −0.0536828 0.998558i \(-0.517096\pi\)
−0.0536828 + 0.998558i \(0.517096\pi\)
\(348\) −5.65685 −0.303239
\(349\) −22.5858 −1.20899 −0.604495 0.796609i \(-0.706625\pi\)
−0.604495 + 0.796609i \(0.706625\pi\)
\(350\) 0 0
\(351\) −1.41421 −0.0754851
\(352\) −1.00000 −0.0533002
\(353\) 13.8995 0.739795 0.369898 0.929072i \(-0.379393\pi\)
0.369898 + 0.929072i \(0.379393\pi\)
\(354\) 6.82843 0.362927
\(355\) 0 0
\(356\) −2.58579 −0.137046
\(357\) 0 0
\(358\) −2.34315 −0.123839
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 0 0
\(361\) 10.3137 0.542827
\(362\) −21.6569 −1.13826
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 0 0
\(366\) −1.41421 −0.0739221
\(367\) 30.8701 1.61140 0.805702 0.592321i \(-0.201788\pi\)
0.805702 + 0.592321i \(0.201788\pi\)
\(368\) 7.65685 0.399141
\(369\) −9.65685 −0.502716
\(370\) 0 0
\(371\) 0 0
\(372\) 3.07107 0.159227
\(373\) −36.6274 −1.89650 −0.948248 0.317531i \(-0.897146\pi\)
−0.948248 + 0.317531i \(0.897146\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) −1.41421 −0.0729325
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) −17.6569 −0.906972 −0.453486 0.891263i \(-0.649820\pi\)
−0.453486 + 0.891263i \(0.649820\pi\)
\(380\) 0 0
\(381\) −15.3137 −0.784545
\(382\) 10.9706 0.561303
\(383\) 7.55635 0.386111 0.193056 0.981188i \(-0.438160\pi\)
0.193056 + 0.981188i \(0.438160\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −7.65685 −0.389724
\(387\) −10.0000 −0.508329
\(388\) −12.7279 −0.646162
\(389\) 12.6274 0.640235 0.320118 0.947378i \(-0.396278\pi\)
0.320118 + 0.947378i \(0.396278\pi\)
\(390\) 0 0
\(391\) −30.6274 −1.54890
\(392\) 0 0
\(393\) −3.75736 −0.189534
\(394\) −21.3137 −1.07377
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) −11.7990 −0.592174 −0.296087 0.955161i \(-0.595682\pi\)
−0.296087 + 0.955161i \(0.595682\pi\)
\(398\) 10.5858 0.530618
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 11.3137 0.564276
\(403\) −4.34315 −0.216347
\(404\) 0.242641 0.0120718
\(405\) 0 0
\(406\) 0 0
\(407\) 3.65685 0.181264
\(408\) 4.00000 0.198030
\(409\) 12.4853 0.617357 0.308679 0.951166i \(-0.400113\pi\)
0.308679 + 0.951166i \(0.400113\pi\)
\(410\) 0 0
\(411\) −9.31371 −0.459411
\(412\) −13.8995 −0.684779
\(413\) 0 0
\(414\) −7.65685 −0.376314
\(415\) 0 0
\(416\) 1.41421 0.0693375
\(417\) −2.10051 −0.102862
\(418\) 5.41421 0.264818
\(419\) −16.2843 −0.795539 −0.397769 0.917485i \(-0.630216\pi\)
−0.397769 + 0.917485i \(0.630216\pi\)
\(420\) 0 0
\(421\) −14.9706 −0.729621 −0.364810 0.931082i \(-0.618866\pi\)
−0.364810 + 0.931082i \(0.618866\pi\)
\(422\) −17.3137 −0.842818
\(423\) 1.41421 0.0687614
\(424\) 3.65685 0.177593
\(425\) 20.0000 0.970143
\(426\) −13.6569 −0.661677
\(427\) 0 0
\(428\) 17.3137 0.836890
\(429\) −1.41421 −0.0682789
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 1.00000 0.0481125
\(433\) −37.2132 −1.78835 −0.894176 0.447715i \(-0.852238\pi\)
−0.894176 + 0.447715i \(0.852238\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) −41.4558 −1.98310
\(438\) 4.00000 0.191127
\(439\) −35.7990 −1.70859 −0.854296 0.519786i \(-0.826012\pi\)
−0.854296 + 0.519786i \(0.826012\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −5.65685 −0.269069
\(443\) 28.2843 1.34383 0.671913 0.740630i \(-0.265473\pi\)
0.671913 + 0.740630i \(0.265473\pi\)
\(444\) 3.65685 0.173547
\(445\) 0 0
\(446\) 19.5563 0.926020
\(447\) 12.9706 0.613487
\(448\) 0 0
\(449\) 10.3431 0.488123 0.244062 0.969760i \(-0.421520\pi\)
0.244062 + 0.969760i \(0.421520\pi\)
\(450\) 5.00000 0.235702
\(451\) −9.65685 −0.454724
\(452\) −5.65685 −0.266076
\(453\) −1.65685 −0.0778458
\(454\) 23.5563 1.10555
\(455\) 0 0
\(456\) 5.41421 0.253544
\(457\) 19.6569 0.919509 0.459754 0.888046i \(-0.347937\pi\)
0.459754 + 0.888046i \(0.347937\pi\)
\(458\) 5.65685 0.264327
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −37.8995 −1.76516 −0.882578 0.470167i \(-0.844194\pi\)
−0.882578 + 0.470167i \(0.844194\pi\)
\(462\) 0 0
\(463\) −27.3137 −1.26938 −0.634688 0.772769i \(-0.718871\pi\)
−0.634688 + 0.772769i \(0.718871\pi\)
\(464\) −5.65685 −0.262613
\(465\) 0 0
\(466\) −12.3431 −0.571785
\(467\) 37.9411 1.75571 0.877853 0.478930i \(-0.158975\pi\)
0.877853 + 0.478930i \(0.158975\pi\)
\(468\) −1.41421 −0.0653720
\(469\) 0 0
\(470\) 0 0
\(471\) 2.82843 0.130327
\(472\) 6.82843 0.314304
\(473\) −10.0000 −0.459800
\(474\) −1.65685 −0.0761018
\(475\) 27.0711 1.24211
\(476\) 0 0
\(477\) −3.65685 −0.167436
\(478\) −14.3431 −0.656040
\(479\) −32.9706 −1.50646 −0.753232 0.657755i \(-0.771506\pi\)
−0.753232 + 0.657755i \(0.771506\pi\)
\(480\) 0 0
\(481\) −5.17157 −0.235803
\(482\) −7.31371 −0.333130
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −0.343146 −0.0155494 −0.00777471 0.999970i \(-0.502475\pi\)
−0.00777471 + 0.999970i \(0.502475\pi\)
\(488\) −1.41421 −0.0640184
\(489\) 5.65685 0.255812
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) −9.65685 −0.435365
\(493\) 22.6274 1.01909
\(494\) −7.65685 −0.344498
\(495\) 0 0
\(496\) 3.07107 0.137895
\(497\) 0 0
\(498\) −15.0711 −0.675351
\(499\) −10.6274 −0.475749 −0.237874 0.971296i \(-0.576451\pi\)
−0.237874 + 0.971296i \(0.576451\pi\)
\(500\) 0 0
\(501\) −10.8284 −0.483778
\(502\) 14.8284 0.661825
\(503\) 0.485281 0.0216376 0.0108188 0.999941i \(-0.496556\pi\)
0.0108188 + 0.999941i \(0.496556\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −7.65685 −0.340389
\(507\) −11.0000 −0.488527
\(508\) −15.3137 −0.679436
\(509\) 41.9411 1.85901 0.929504 0.368812i \(-0.120236\pi\)
0.929504 + 0.368812i \(0.120236\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −5.41421 −0.239043
\(514\) −16.7279 −0.737837
\(515\) 0 0
\(516\) −10.0000 −0.440225
\(517\) 1.41421 0.0621970
\(518\) 0 0
\(519\) −4.92893 −0.216356
\(520\) 0 0
\(521\) 7.27208 0.318596 0.159298 0.987231i \(-0.449077\pi\)
0.159298 + 0.987231i \(0.449077\pi\)
\(522\) 5.65685 0.247594
\(523\) 31.7574 1.38865 0.694326 0.719660i \(-0.255703\pi\)
0.694326 + 0.719660i \(0.255703\pi\)
\(524\) −3.75736 −0.164141
\(525\) 0 0
\(526\) −13.6569 −0.595467
\(527\) −12.2843 −0.535111
\(528\) 1.00000 0.0435194
\(529\) 35.6274 1.54902
\(530\) 0 0
\(531\) −6.82843 −0.296328
\(532\) 0 0
\(533\) 13.6569 0.591544
\(534\) 2.58579 0.111898
\(535\) 0 0
\(536\) 11.3137 0.488678
\(537\) 2.34315 0.101114
\(538\) 22.6274 0.975537
\(539\) 0 0
\(540\) 0 0
\(541\) −8.62742 −0.370922 −0.185461 0.982652i \(-0.559378\pi\)
−0.185461 + 0.982652i \(0.559378\pi\)
\(542\) 21.1716 0.909397
\(543\) 21.6569 0.929385
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) 0 0
\(547\) −1.65685 −0.0708420 −0.0354210 0.999372i \(-0.511277\pi\)
−0.0354210 + 0.999372i \(0.511277\pi\)
\(548\) −9.31371 −0.397862
\(549\) 1.41421 0.0603572
\(550\) 5.00000 0.213201
\(551\) 30.6274 1.30477
\(552\) −7.65685 −0.325897
\(553\) 0 0
\(554\) −4.00000 −0.169944
\(555\) 0 0
\(556\) −2.10051 −0.0890813
\(557\) 31.9411 1.35339 0.676694 0.736264i \(-0.263412\pi\)
0.676694 + 0.736264i \(0.263412\pi\)
\(558\) −3.07107 −0.130009
\(559\) 14.1421 0.598149
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 18.9706 0.800225
\(563\) 22.5858 0.951877 0.475939 0.879478i \(-0.342108\pi\)
0.475939 + 0.879478i \(0.342108\pi\)
\(564\) 1.41421 0.0595491
\(565\) 0 0
\(566\) 15.7574 0.662331
\(567\) 0 0
\(568\) −13.6569 −0.573029
\(569\) −6.68629 −0.280304 −0.140152 0.990130i \(-0.544759\pi\)
−0.140152 + 0.990130i \(0.544759\pi\)
\(570\) 0 0
\(571\) 40.2843 1.68584 0.842922 0.538036i \(-0.180833\pi\)
0.842922 + 0.538036i \(0.180833\pi\)
\(572\) −1.41421 −0.0591312
\(573\) −10.9706 −0.458302
\(574\) 0 0
\(575\) −38.2843 −1.59656
\(576\) 1.00000 0.0416667
\(577\) 27.7574 1.15555 0.577777 0.816195i \(-0.303920\pi\)
0.577777 + 0.816195i \(0.303920\pi\)
\(578\) 1.00000 0.0415945
\(579\) 7.65685 0.318208
\(580\) 0 0
\(581\) 0 0
\(582\) 12.7279 0.527589
\(583\) −3.65685 −0.151451
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −3.07107 −0.126865
\(587\) −3.51472 −0.145068 −0.0725340 0.997366i \(-0.523109\pi\)
−0.0725340 + 0.997366i \(0.523109\pi\)
\(588\) 0 0
\(589\) −16.6274 −0.685121
\(590\) 0 0
\(591\) 21.3137 0.876729
\(592\) 3.65685 0.150296
\(593\) 23.3137 0.957379 0.478690 0.877984i \(-0.341112\pi\)
0.478690 + 0.877984i \(0.341112\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) 12.9706 0.531295
\(597\) −10.5858 −0.433247
\(598\) 10.8284 0.442807
\(599\) −32.9706 −1.34714 −0.673570 0.739123i \(-0.735240\pi\)
−0.673570 + 0.739123i \(0.735240\pi\)
\(600\) 5.00000 0.204124
\(601\) −13.4558 −0.548875 −0.274438 0.961605i \(-0.588492\pi\)
−0.274438 + 0.961605i \(0.588492\pi\)
\(602\) 0 0
\(603\) −11.3137 −0.460730
\(604\) −1.65685 −0.0674164
\(605\) 0 0
\(606\) −0.242641 −0.00985660
\(607\) −3.31371 −0.134499 −0.0672496 0.997736i \(-0.521422\pi\)
−0.0672496 + 0.997736i \(0.521422\pi\)
\(608\) 5.41421 0.219575
\(609\) 0 0
\(610\) 0 0
\(611\) −2.00000 −0.0809113
\(612\) −4.00000 −0.161690
\(613\) 12.0000 0.484675 0.242338 0.970192i \(-0.422086\pi\)
0.242338 + 0.970192i \(0.422086\pi\)
\(614\) −0.727922 −0.0293765
\(615\) 0 0
\(616\) 0 0
\(617\) 14.3431 0.577433 0.288717 0.957415i \(-0.406771\pi\)
0.288717 + 0.957415i \(0.406771\pi\)
\(618\) 13.8995 0.559120
\(619\) 7.31371 0.293963 0.146981 0.989139i \(-0.453044\pi\)
0.146981 + 0.989139i \(0.453044\pi\)
\(620\) 0 0
\(621\) 7.65685 0.307259
\(622\) 20.2426 0.811656
\(623\) 0 0
\(624\) −1.41421 −0.0566139
\(625\) 25.0000 1.00000
\(626\) −12.2426 −0.489314
\(627\) −5.41421 −0.216223
\(628\) 2.82843 0.112867
\(629\) −14.6274 −0.583233
\(630\) 0 0
\(631\) 37.5980 1.49675 0.748376 0.663275i \(-0.230834\pi\)
0.748376 + 0.663275i \(0.230834\pi\)
\(632\) −1.65685 −0.0659061
\(633\) 17.3137 0.688158
\(634\) 5.31371 0.211034
\(635\) 0 0
\(636\) −3.65685 −0.145004
\(637\) 0 0
\(638\) 5.65685 0.223957
\(639\) 13.6569 0.540257
\(640\) 0 0
\(641\) 8.28427 0.327209 0.163605 0.986526i \(-0.447688\pi\)
0.163605 + 0.986526i \(0.447688\pi\)
\(642\) −17.3137 −0.683318
\(643\) 46.8284 1.84673 0.923366 0.383920i \(-0.125426\pi\)
0.923366 + 0.383920i \(0.125426\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −21.6569 −0.852078
\(647\) −17.8995 −0.703702 −0.351851 0.936056i \(-0.614448\pi\)
−0.351851 + 0.936056i \(0.614448\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −6.82843 −0.268039
\(650\) −7.07107 −0.277350
\(651\) 0 0
\(652\) 5.65685 0.221540
\(653\) −32.6274 −1.27681 −0.638405 0.769701i \(-0.720405\pi\)
−0.638405 + 0.769701i \(0.720405\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) −9.65685 −0.377037
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −25.3137 −0.986082 −0.493041 0.870006i \(-0.664115\pi\)
−0.493041 + 0.870006i \(0.664115\pi\)
\(660\) 0 0
\(661\) 24.9706 0.971242 0.485621 0.874169i \(-0.338593\pi\)
0.485621 + 0.874169i \(0.338593\pi\)
\(662\) −28.9706 −1.12597
\(663\) 5.65685 0.219694
\(664\) −15.0711 −0.584871
\(665\) 0 0
\(666\) −3.65685 −0.141700
\(667\) −43.3137 −1.67711
\(668\) −10.8284 −0.418964
\(669\) −19.5563 −0.756092
\(670\) 0 0
\(671\) 1.41421 0.0545951
\(672\) 0 0
\(673\) 19.6569 0.757716 0.378858 0.925455i \(-0.376317\pi\)
0.378858 + 0.925455i \(0.376317\pi\)
\(674\) 17.3137 0.666899
\(675\) −5.00000 −0.192450
\(676\) −11.0000 −0.423077
\(677\) 18.5858 0.714310 0.357155 0.934045i \(-0.383747\pi\)
0.357155 + 0.934045i \(0.383747\pi\)
\(678\) 5.65685 0.217250
\(679\) 0 0
\(680\) 0 0
\(681\) −23.5563 −0.902681
\(682\) −3.07107 −0.117597
\(683\) −48.2843 −1.84755 −0.923773 0.382940i \(-0.874912\pi\)
−0.923773 + 0.382940i \(0.874912\pi\)
\(684\) −5.41421 −0.207018
\(685\) 0 0
\(686\) 0 0
\(687\) −5.65685 −0.215822
\(688\) −10.0000 −0.381246
\(689\) 5.17157 0.197021
\(690\) 0 0
\(691\) −4.48528 −0.170628 −0.0853141 0.996354i \(-0.527189\pi\)
−0.0853141 + 0.996354i \(0.527189\pi\)
\(692\) −4.92893 −0.187370
\(693\) 0 0
\(694\) 2.00000 0.0759190
\(695\) 0 0
\(696\) 5.65685 0.214423
\(697\) 38.6274 1.46312
\(698\) 22.5858 0.854885
\(699\) 12.3431 0.466861
\(700\) 0 0
\(701\) −41.3137 −1.56040 −0.780199 0.625532i \(-0.784882\pi\)
−0.780199 + 0.625532i \(0.784882\pi\)
\(702\) 1.41421 0.0533761
\(703\) −19.7990 −0.746733
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −13.8995 −0.523114
\(707\) 0 0
\(708\) −6.82843 −0.256628
\(709\) 0.627417 0.0235631 0.0117816 0.999931i \(-0.496250\pi\)
0.0117816 + 0.999931i \(0.496250\pi\)
\(710\) 0 0
\(711\) 1.65685 0.0621369
\(712\) 2.58579 0.0969064
\(713\) 23.5147 0.880633
\(714\) 0 0
\(715\) 0 0
\(716\) 2.34315 0.0875675
\(717\) 14.3431 0.535655
\(718\) 4.00000 0.149279
\(719\) −14.5858 −0.543958 −0.271979 0.962303i \(-0.587678\pi\)
−0.271979 + 0.962303i \(0.587678\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −10.3137 −0.383836
\(723\) 7.31371 0.272000
\(724\) 21.6569 0.804871
\(725\) 28.2843 1.05045
\(726\) −1.00000 −0.0371135
\(727\) 39.3553 1.45961 0.729804 0.683656i \(-0.239611\pi\)
0.729804 + 0.683656i \(0.239611\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 40.0000 1.47945
\(732\) 1.41421 0.0522708
\(733\) 35.7574 1.32073 0.660364 0.750946i \(-0.270402\pi\)
0.660364 + 0.750946i \(0.270402\pi\)
\(734\) −30.8701 −1.13943
\(735\) 0 0
\(736\) −7.65685 −0.282235
\(737\) −11.3137 −0.416746
\(738\) 9.65685 0.355474
\(739\) 6.34315 0.233336 0.116668 0.993171i \(-0.462779\pi\)
0.116668 + 0.993171i \(0.462779\pi\)
\(740\) 0 0
\(741\) 7.65685 0.281282
\(742\) 0 0
\(743\) −4.97056 −0.182352 −0.0911761 0.995835i \(-0.529063\pi\)
−0.0911761 + 0.995835i \(0.529063\pi\)
\(744\) −3.07107 −0.112591
\(745\) 0 0
\(746\) 36.6274 1.34103
\(747\) 15.0711 0.551422
\(748\) −4.00000 −0.146254
\(749\) 0 0
\(750\) 0 0
\(751\) 26.9706 0.984170 0.492085 0.870547i \(-0.336235\pi\)
0.492085 + 0.870547i \(0.336235\pi\)
\(752\) 1.41421 0.0515711
\(753\) −14.8284 −0.540378
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) 0 0
\(757\) −6.68629 −0.243017 −0.121509 0.992590i \(-0.538773\pi\)
−0.121509 + 0.992590i \(0.538773\pi\)
\(758\) 17.6569 0.641326
\(759\) 7.65685 0.277926
\(760\) 0 0
\(761\) 13.8579 0.502347 0.251174 0.967942i \(-0.419183\pi\)
0.251174 + 0.967942i \(0.419183\pi\)
\(762\) 15.3137 0.554757
\(763\) 0 0
\(764\) −10.9706 −0.396901
\(765\) 0 0
\(766\) −7.55635 −0.273022
\(767\) 9.65685 0.348689
\(768\) 1.00000 0.0360844
\(769\) −39.7990 −1.43519 −0.717594 0.696462i \(-0.754757\pi\)
−0.717594 + 0.696462i \(0.754757\pi\)
\(770\) 0 0
\(771\) 16.7279 0.602441
\(772\) 7.65685 0.275576
\(773\) 14.1421 0.508657 0.254329 0.967118i \(-0.418146\pi\)
0.254329 + 0.967118i \(0.418146\pi\)
\(774\) 10.0000 0.359443
\(775\) −15.3553 −0.551580
\(776\) 12.7279 0.456906
\(777\) 0 0
\(778\) −12.6274 −0.452715
\(779\) 52.2843 1.87328
\(780\) 0 0
\(781\) 13.6569 0.488681
\(782\) 30.6274 1.09523
\(783\) −5.65685 −0.202159
\(784\) 0 0
\(785\) 0 0
\(786\) 3.75736 0.134021
\(787\) 38.3848 1.36827 0.684135 0.729356i \(-0.260180\pi\)
0.684135 + 0.729356i \(0.260180\pi\)
\(788\) 21.3137 0.759269
\(789\) 13.6569 0.486197
\(790\) 0 0
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) −2.00000 −0.0710221
\(794\) 11.7990 0.418730
\(795\) 0 0
\(796\) −10.5858 −0.375203
\(797\) 50.8284 1.80044 0.900218 0.435440i \(-0.143407\pi\)
0.900218 + 0.435440i \(0.143407\pi\)
\(798\) 0 0
\(799\) −5.65685 −0.200125
\(800\) 5.00000 0.176777
\(801\) −2.58579 −0.0913643
\(802\) 6.00000 0.211867
\(803\) −4.00000 −0.141157
\(804\) −11.3137 −0.399004
\(805\) 0 0
\(806\) 4.34315 0.152981
\(807\) −22.6274 −0.796523
\(808\) −0.242641 −0.00853607
\(809\) 18.9706 0.666969 0.333485 0.942755i \(-0.391775\pi\)
0.333485 + 0.942755i \(0.391775\pi\)
\(810\) 0 0
\(811\) −1.12994 −0.0396776 −0.0198388 0.999803i \(-0.506315\pi\)
−0.0198388 + 0.999803i \(0.506315\pi\)
\(812\) 0 0
\(813\) −21.1716 −0.742519
\(814\) −3.65685 −0.128173
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) 54.1421 1.89419
\(818\) −12.4853 −0.436538
\(819\) 0 0
\(820\) 0 0
\(821\) 3.02944 0.105728 0.0528640 0.998602i \(-0.483165\pi\)
0.0528640 + 0.998602i \(0.483165\pi\)
\(822\) 9.31371 0.324853
\(823\) −53.5980 −1.86831 −0.934154 0.356870i \(-0.883844\pi\)
−0.934154 + 0.356870i \(0.883844\pi\)
\(824\) 13.8995 0.484212
\(825\) −5.00000 −0.174078
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 7.65685 0.266094
\(829\) −19.3137 −0.670793 −0.335396 0.942077i \(-0.608870\pi\)
−0.335396 + 0.942077i \(0.608870\pi\)
\(830\) 0 0
\(831\) 4.00000 0.138758
\(832\) −1.41421 −0.0490290
\(833\) 0 0
\(834\) 2.10051 0.0727345
\(835\) 0 0
\(836\) −5.41421 −0.187254
\(837\) 3.07107 0.106152
\(838\) 16.2843 0.562531
\(839\) −12.7279 −0.439417 −0.219708 0.975566i \(-0.570511\pi\)
−0.219708 + 0.975566i \(0.570511\pi\)
\(840\) 0 0
\(841\) 3.00000 0.103448
\(842\) 14.9706 0.515920
\(843\) −18.9706 −0.653381
\(844\) 17.3137 0.595962
\(845\) 0 0
\(846\) −1.41421 −0.0486217
\(847\) 0 0
\(848\) −3.65685 −0.125577
\(849\) −15.7574 −0.540791
\(850\) −20.0000 −0.685994
\(851\) 28.0000 0.959828
\(852\) 13.6569 0.467876
\(853\) 52.7279 1.80537 0.902685 0.430302i \(-0.141593\pi\)
0.902685 + 0.430302i \(0.141593\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −17.3137 −0.591770
\(857\) 14.8284 0.506529 0.253265 0.967397i \(-0.418496\pi\)
0.253265 + 0.967397i \(0.418496\pi\)
\(858\) 1.41421 0.0482805
\(859\) −46.4264 −1.58405 −0.792024 0.610490i \(-0.790973\pi\)
−0.792024 + 0.610490i \(0.790973\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −16.0000 −0.544962
\(863\) 49.5980 1.68833 0.844167 0.536080i \(-0.180095\pi\)
0.844167 + 0.536080i \(0.180095\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 37.2132 1.26456
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 1.65685 0.0562049
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 10.0000 0.338643
\(873\) −12.7279 −0.430775
\(874\) 41.4558 1.40226
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) 53.3137 1.80028 0.900138 0.435605i \(-0.143465\pi\)
0.900138 + 0.435605i \(0.143465\pi\)
\(878\) 35.7990 1.20816
\(879\) 3.07107 0.103585
\(880\) 0 0
\(881\) 21.8995 0.737813 0.368906 0.929467i \(-0.379732\pi\)
0.368906 + 0.929467i \(0.379732\pi\)
\(882\) 0 0
\(883\) −26.6274 −0.896084 −0.448042 0.894013i \(-0.647878\pi\)
−0.448042 + 0.894013i \(0.647878\pi\)
\(884\) 5.65685 0.190261
\(885\) 0 0
\(886\) −28.2843 −0.950229
\(887\) 4.28427 0.143852 0.0719259 0.997410i \(-0.477085\pi\)
0.0719259 + 0.997410i \(0.477085\pi\)
\(888\) −3.65685 −0.122716
\(889\) 0 0
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) −19.5563 −0.654795
\(893\) −7.65685 −0.256227
\(894\) −12.9706 −0.433801
\(895\) 0 0
\(896\) 0 0
\(897\) −10.8284 −0.361551
\(898\) −10.3431 −0.345155
\(899\) −17.3726 −0.579408
\(900\) −5.00000 −0.166667
\(901\) 14.6274 0.487310
\(902\) 9.65685 0.321538
\(903\) 0 0
\(904\) 5.65685 0.188144
\(905\) 0 0
\(906\) 1.65685 0.0550453
\(907\) −33.9411 −1.12700 −0.563498 0.826117i \(-0.690545\pi\)
−0.563498 + 0.826117i \(0.690545\pi\)
\(908\) −23.5563 −0.781745
\(909\) 0.242641 0.00804788
\(910\) 0 0
\(911\) 41.5980 1.37820 0.689101 0.724665i \(-0.258006\pi\)
0.689101 + 0.724665i \(0.258006\pi\)
\(912\) −5.41421 −0.179283
\(913\) 15.0711 0.498780
\(914\) −19.6569 −0.650191
\(915\) 0 0
\(916\) −5.65685 −0.186908
\(917\) 0 0
\(918\) 4.00000 0.132020
\(919\) 14.6274 0.482514 0.241257 0.970461i \(-0.422440\pi\)
0.241257 + 0.970461i \(0.422440\pi\)
\(920\) 0 0
\(921\) 0.727922 0.0239858
\(922\) 37.8995 1.24815
\(923\) −19.3137 −0.635718
\(924\) 0 0
\(925\) −18.2843 −0.601183
\(926\) 27.3137 0.897584
\(927\) −13.8995 −0.456519
\(928\) 5.65685 0.185695
\(929\) 35.0711 1.15064 0.575322 0.817927i \(-0.304877\pi\)
0.575322 + 0.817927i \(0.304877\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 12.3431 0.404313
\(933\) −20.2426 −0.662714
\(934\) −37.9411 −1.24147
\(935\) 0 0
\(936\) 1.41421 0.0462250
\(937\) 50.6274 1.65393 0.826963 0.562257i \(-0.190067\pi\)
0.826963 + 0.562257i \(0.190067\pi\)
\(938\) 0 0
\(939\) 12.2426 0.399523
\(940\) 0 0
\(941\) −1.69848 −0.0553690 −0.0276845 0.999617i \(-0.508813\pi\)
−0.0276845 + 0.999617i \(0.508813\pi\)
\(942\) −2.82843 −0.0921551
\(943\) −73.9411 −2.40785
\(944\) −6.82843 −0.222246
\(945\) 0 0
\(946\) 10.0000 0.325128
\(947\) −24.6863 −0.802197 −0.401098 0.916035i \(-0.631371\pi\)
−0.401098 + 0.916035i \(0.631371\pi\)
\(948\) 1.65685 0.0538121
\(949\) 5.65685 0.183629
\(950\) −27.0711 −0.878301
\(951\) −5.31371 −0.172309
\(952\) 0 0
\(953\) −26.0000 −0.842223 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 3.65685 0.118395
\(955\) 0 0
\(956\) 14.3431 0.463890
\(957\) −5.65685 −0.182860
\(958\) 32.9706 1.06523
\(959\) 0 0
\(960\) 0 0
\(961\) −21.5685 −0.695759
\(962\) 5.17157 0.166738
\(963\) 17.3137 0.557926
\(964\) 7.31371 0.235559
\(965\) 0 0
\(966\) 0 0
\(967\) 44.2843 1.42409 0.712043 0.702136i \(-0.247770\pi\)
0.712043 + 0.702136i \(0.247770\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 21.6569 0.695718
\(970\) 0 0
\(971\) −46.9117 −1.50547 −0.752734 0.658325i \(-0.771265\pi\)
−0.752734 + 0.658325i \(0.771265\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 0.343146 0.0109951
\(975\) 7.07107 0.226455
\(976\) 1.41421 0.0452679
\(977\) 26.3431 0.842792 0.421396 0.906877i \(-0.361540\pi\)
0.421396 + 0.906877i \(0.361540\pi\)
\(978\) −5.65685 −0.180886
\(979\) −2.58579 −0.0826421
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) −20.0000 −0.638226
\(983\) −41.0122 −1.30809 −0.654043 0.756457i \(-0.726928\pi\)
−0.654043 + 0.756457i \(0.726928\pi\)
\(984\) 9.65685 0.307849
\(985\) 0 0
\(986\) −22.6274 −0.720604
\(987\) 0 0
\(988\) 7.65685 0.243597
\(989\) −76.5685 −2.43474
\(990\) 0 0
\(991\) 43.3137 1.37591 0.687953 0.725756i \(-0.258510\pi\)
0.687953 + 0.725756i \(0.258510\pi\)
\(992\) −3.07107 −0.0975065
\(993\) 28.9706 0.919353
\(994\) 0 0
\(995\) 0 0
\(996\) 15.0711 0.477545
\(997\) −9.89949 −0.313520 −0.156760 0.987637i \(-0.550105\pi\)
−0.156760 + 0.987637i \(0.550105\pi\)
\(998\) 10.6274 0.336405
\(999\) 3.65685 0.115698
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 3234.2.a.z.1.1 yes 2
3.2 odd 2 9702.2.a.dl.1.1 2
7.6 odd 2 3234.2.a.y.1.2 2
21.20 even 2 9702.2.a.de.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.y.1.2 2 7.6 odd 2
3234.2.a.z.1.1 yes 2 1.1 even 1 trivial
9702.2.a.de.1.2 2 21.20 even 2
9702.2.a.dl.1.1 2 3.2 odd 2