Properties

Label 2646.2.a.bi.1.2
Level $2646$
Weight $2$
Character 2646.1
Self dual yes
Analytic conductor $21.128$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.1284163748\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 378)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 2646.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +3.64575 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +3.64575 q^{5} -1.00000 q^{8} -3.64575 q^{10} -3.64575 q^{11} -4.64575 q^{13} +1.00000 q^{16} -3.64575 q^{17} +2.00000 q^{19} +3.64575 q^{20} +3.64575 q^{22} +1.29150 q^{23} +8.29150 q^{25} +4.64575 q^{26} -2.35425 q^{29} -4.64575 q^{31} -1.00000 q^{32} +3.64575 q^{34} -11.9373 q^{37} -2.00000 q^{38} -3.64575 q^{40} -10.9373 q^{41} +5.00000 q^{43} -3.64575 q^{44} -1.29150 q^{46} -4.93725 q^{47} -8.29150 q^{50} -4.64575 q^{52} -6.00000 q^{53} -13.2915 q^{55} +2.35425 q^{58} -8.35425 q^{59} +7.35425 q^{61} +4.64575 q^{62} +1.00000 q^{64} -16.9373 q^{65} -2.29150 q^{67} -3.64575 q^{68} -15.6458 q^{71} +10.5830 q^{73} +11.9373 q^{74} +2.00000 q^{76} +11.2288 q^{79} +3.64575 q^{80} +10.9373 q^{82} +13.2915 q^{83} -13.2915 q^{85} -5.00000 q^{86} +3.64575 q^{88} -4.93725 q^{89} +1.29150 q^{92} +4.93725 q^{94} +7.29150 q^{95} +13.5830 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} - 2 q^{10} - 2 q^{11} - 4 q^{13} + 2 q^{16} - 2 q^{17} + 4 q^{19} + 2 q^{20} + 2 q^{22} - 8 q^{23} + 6 q^{25} + 4 q^{26} - 10 q^{29} - 4 q^{31} - 2 q^{32} + 2 q^{34} - 8 q^{37} - 4 q^{38} - 2 q^{40} - 6 q^{41} + 10 q^{43} - 2 q^{44} + 8 q^{46} + 6 q^{47} - 6 q^{50} - 4 q^{52} - 12 q^{53} - 16 q^{55} + 10 q^{58} - 22 q^{59} + 20 q^{61} + 4 q^{62} + 2 q^{64} - 18 q^{65} + 6 q^{67} - 2 q^{68} - 26 q^{71} + 8 q^{74} + 4 q^{76} - 4 q^{79} + 2 q^{80} + 6 q^{82} + 16 q^{83} - 16 q^{85} - 10 q^{86} + 2 q^{88} + 6 q^{89} - 8 q^{92} - 6 q^{94} + 4 q^{95} + 6 q^{97} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 3.64575 1.63043 0.815215 0.579159i \(-0.196619\pi\)
0.815215 + 0.579159i \(0.196619\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −3.64575 −1.15289
\(11\) −3.64575 −1.09924 −0.549618 0.835416i \(-0.685227\pi\)
−0.549618 + 0.835416i \(0.685227\pi\)
\(12\) 0 0
\(13\) −4.64575 −1.28850 −0.644250 0.764815i \(-0.722830\pi\)
−0.644250 + 0.764815i \(0.722830\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.64575 −0.884225 −0.442112 0.896960i \(-0.645771\pi\)
−0.442112 + 0.896960i \(0.645771\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 3.64575 0.815215
\(21\) 0 0
\(22\) 3.64575 0.777277
\(23\) 1.29150 0.269297 0.134648 0.990893i \(-0.457009\pi\)
0.134648 + 0.990893i \(0.457009\pi\)
\(24\) 0 0
\(25\) 8.29150 1.65830
\(26\) 4.64575 0.911107
\(27\) 0 0
\(28\) 0 0
\(29\) −2.35425 −0.437173 −0.218587 0.975818i \(-0.570145\pi\)
−0.218587 + 0.975818i \(0.570145\pi\)
\(30\) 0 0
\(31\) −4.64575 −0.834402 −0.417201 0.908814i \(-0.636989\pi\)
−0.417201 + 0.908814i \(0.636989\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.64575 0.625241
\(35\) 0 0
\(36\) 0 0
\(37\) −11.9373 −1.96247 −0.981236 0.192809i \(-0.938240\pi\)
−0.981236 + 0.192809i \(0.938240\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) −3.64575 −0.576444
\(41\) −10.9373 −1.70811 −0.854056 0.520181i \(-0.825864\pi\)
−0.854056 + 0.520181i \(0.825864\pi\)
\(42\) 0 0
\(43\) 5.00000 0.762493 0.381246 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) −3.64575 −0.549618
\(45\) 0 0
\(46\) −1.29150 −0.190422
\(47\) −4.93725 −0.720173 −0.360086 0.932919i \(-0.617253\pi\)
−0.360086 + 0.932919i \(0.617253\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −8.29150 −1.17260
\(51\) 0 0
\(52\) −4.64575 −0.644250
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −13.2915 −1.79223
\(56\) 0 0
\(57\) 0 0
\(58\) 2.35425 0.309128
\(59\) −8.35425 −1.08763 −0.543815 0.839205i \(-0.683021\pi\)
−0.543815 + 0.839205i \(0.683021\pi\)
\(60\) 0 0
\(61\) 7.35425 0.941615 0.470808 0.882236i \(-0.343963\pi\)
0.470808 + 0.882236i \(0.343963\pi\)
\(62\) 4.64575 0.590011
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −16.9373 −2.10081
\(66\) 0 0
\(67\) −2.29150 −0.279952 −0.139976 0.990155i \(-0.544702\pi\)
−0.139976 + 0.990155i \(0.544702\pi\)
\(68\) −3.64575 −0.442112
\(69\) 0 0
\(70\) 0 0
\(71\) −15.6458 −1.85681 −0.928405 0.371571i \(-0.878819\pi\)
−0.928405 + 0.371571i \(0.878819\pi\)
\(72\) 0 0
\(73\) 10.5830 1.23865 0.619324 0.785136i \(-0.287407\pi\)
0.619324 + 0.785136i \(0.287407\pi\)
\(74\) 11.9373 1.38768
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 0 0
\(79\) 11.2288 1.26333 0.631667 0.775240i \(-0.282371\pi\)
0.631667 + 0.775240i \(0.282371\pi\)
\(80\) 3.64575 0.407607
\(81\) 0 0
\(82\) 10.9373 1.20782
\(83\) 13.2915 1.45893 0.729466 0.684017i \(-0.239769\pi\)
0.729466 + 0.684017i \(0.239769\pi\)
\(84\) 0 0
\(85\) −13.2915 −1.44167
\(86\) −5.00000 −0.539164
\(87\) 0 0
\(88\) 3.64575 0.388638
\(89\) −4.93725 −0.523348 −0.261674 0.965156i \(-0.584275\pi\)
−0.261674 + 0.965156i \(0.584275\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.29150 0.134648
\(93\) 0 0
\(94\) 4.93725 0.509239
\(95\) 7.29150 0.748092
\(96\) 0 0
\(97\) 13.5830 1.37915 0.689573 0.724217i \(-0.257798\pi\)
0.689573 + 0.724217i \(0.257798\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 8.29150 0.829150
\(101\) −8.35425 −0.831279 −0.415639 0.909529i \(-0.636442\pi\)
−0.415639 + 0.909529i \(0.636442\pi\)
\(102\) 0 0
\(103\) 3.93725 0.387949 0.193975 0.981007i \(-0.437862\pi\)
0.193975 + 0.981007i \(0.437862\pi\)
\(104\) 4.64575 0.455553
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) −3.35425 −0.321279 −0.160639 0.987013i \(-0.551356\pi\)
−0.160639 + 0.987013i \(0.551356\pi\)
\(110\) 13.2915 1.26730
\(111\) 0 0
\(112\) 0 0
\(113\) −2.35425 −0.221469 −0.110735 0.993850i \(-0.535320\pi\)
−0.110735 + 0.993850i \(0.535320\pi\)
\(114\) 0 0
\(115\) 4.70850 0.439070
\(116\) −2.35425 −0.218587
\(117\) 0 0
\(118\) 8.35425 0.769071
\(119\) 0 0
\(120\) 0 0
\(121\) 2.29150 0.208318
\(122\) −7.35425 −0.665822
\(123\) 0 0
\(124\) −4.64575 −0.417201
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 1.35425 0.120170 0.0600851 0.998193i \(-0.480863\pi\)
0.0600851 + 0.998193i \(0.480863\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 16.9373 1.48550
\(131\) 14.5830 1.27412 0.637062 0.770813i \(-0.280150\pi\)
0.637062 + 0.770813i \(0.280150\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 2.29150 0.197956
\(135\) 0 0
\(136\) 3.64575 0.312621
\(137\) −1.29150 −0.110341 −0.0551703 0.998477i \(-0.517570\pi\)
−0.0551703 + 0.998477i \(0.517570\pi\)
\(138\) 0 0
\(139\) 1.58301 0.134269 0.0671344 0.997744i \(-0.478614\pi\)
0.0671344 + 0.997744i \(0.478614\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 15.6458 1.31296
\(143\) 16.9373 1.41636
\(144\) 0 0
\(145\) −8.58301 −0.712780
\(146\) −10.5830 −0.875856
\(147\) 0 0
\(148\) −11.9373 −0.981236
\(149\) −22.9373 −1.87909 −0.939547 0.342421i \(-0.888753\pi\)
−0.939547 + 0.342421i \(0.888753\pi\)
\(150\) 0 0
\(151\) −19.2288 −1.56481 −0.782407 0.622767i \(-0.786008\pi\)
−0.782407 + 0.622767i \(0.786008\pi\)
\(152\) −2.00000 −0.162221
\(153\) 0 0
\(154\) 0 0
\(155\) −16.9373 −1.36043
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) −11.2288 −0.893312
\(159\) 0 0
\(160\) −3.64575 −0.288222
\(161\) 0 0
\(162\) 0 0
\(163\) −1.00000 −0.0783260 −0.0391630 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) −10.9373 −0.854056
\(165\) 0 0
\(166\) −13.2915 −1.03162
\(167\) 8.58301 0.664173 0.332086 0.943249i \(-0.392247\pi\)
0.332086 + 0.943249i \(0.392247\pi\)
\(168\) 0 0
\(169\) 8.58301 0.660231
\(170\) 13.2915 1.01941
\(171\) 0 0
\(172\) 5.00000 0.381246
\(173\) 14.5830 1.10873 0.554363 0.832275i \(-0.312962\pi\)
0.554363 + 0.832275i \(0.312962\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.64575 −0.274809
\(177\) 0 0
\(178\) 4.93725 0.370063
\(179\) −16.9373 −1.26595 −0.632975 0.774172i \(-0.718166\pi\)
−0.632975 + 0.774172i \(0.718166\pi\)
\(180\) 0 0
\(181\) −2.70850 −0.201321 −0.100661 0.994921i \(-0.532096\pi\)
−0.100661 + 0.994921i \(0.532096\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1.29150 −0.0952108
\(185\) −43.5203 −3.19967
\(186\) 0 0
\(187\) 13.2915 0.971971
\(188\) −4.93725 −0.360086
\(189\) 0 0
\(190\) −7.29150 −0.528981
\(191\) 20.8118 1.50589 0.752943 0.658086i \(-0.228634\pi\)
0.752943 + 0.658086i \(0.228634\pi\)
\(192\) 0 0
\(193\) −7.00000 −0.503871 −0.251936 0.967744i \(-0.581067\pi\)
−0.251936 + 0.967744i \(0.581067\pi\)
\(194\) −13.5830 −0.975203
\(195\) 0 0
\(196\) 0 0
\(197\) −4.70850 −0.335467 −0.167733 0.985832i \(-0.553645\pi\)
−0.167733 + 0.985832i \(0.553645\pi\)
\(198\) 0 0
\(199\) 6.06275 0.429777 0.214888 0.976639i \(-0.431061\pi\)
0.214888 + 0.976639i \(0.431061\pi\)
\(200\) −8.29150 −0.586298
\(201\) 0 0
\(202\) 8.35425 0.587803
\(203\) 0 0
\(204\) 0 0
\(205\) −39.8745 −2.78496
\(206\) −3.93725 −0.274321
\(207\) 0 0
\(208\) −4.64575 −0.322125
\(209\) −7.29150 −0.504364
\(210\) 0 0
\(211\) 14.8745 1.02400 0.512002 0.858984i \(-0.328904\pi\)
0.512002 + 0.858984i \(0.328904\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −6.00000 −0.410152
\(215\) 18.2288 1.24319
\(216\) 0 0
\(217\) 0 0
\(218\) 3.35425 0.227178
\(219\) 0 0
\(220\) −13.2915 −0.896113
\(221\) 16.9373 1.13932
\(222\) 0 0
\(223\) −13.8745 −0.929106 −0.464553 0.885545i \(-0.653785\pi\)
−0.464553 + 0.885545i \(0.653785\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2.35425 0.156602
\(227\) −6.00000 −0.398234 −0.199117 0.979976i \(-0.563807\pi\)
−0.199117 + 0.979976i \(0.563807\pi\)
\(228\) 0 0
\(229\) −9.35425 −0.618146 −0.309073 0.951038i \(-0.600019\pi\)
−0.309073 + 0.951038i \(0.600019\pi\)
\(230\) −4.70850 −0.310469
\(231\) 0 0
\(232\) 2.35425 0.154564
\(233\) 19.2915 1.26383 0.631914 0.775038i \(-0.282270\pi\)
0.631914 + 0.775038i \(0.282270\pi\)
\(234\) 0 0
\(235\) −18.0000 −1.17419
\(236\) −8.35425 −0.543815
\(237\) 0 0
\(238\) 0 0
\(239\) 10.9373 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(240\) 0 0
\(241\) 5.00000 0.322078 0.161039 0.986948i \(-0.448515\pi\)
0.161039 + 0.986948i \(0.448515\pi\)
\(242\) −2.29150 −0.147303
\(243\) 0 0
\(244\) 7.35425 0.470808
\(245\) 0 0
\(246\) 0 0
\(247\) −9.29150 −0.591204
\(248\) 4.64575 0.295006
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) −4.70850 −0.296021
\(254\) −1.35425 −0.0849731
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −15.8745 −0.990225 −0.495112 0.868829i \(-0.664873\pi\)
−0.495112 + 0.868829i \(0.664873\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −16.9373 −1.05040
\(261\) 0 0
\(262\) −14.5830 −0.900941
\(263\) 4.93725 0.304444 0.152222 0.988346i \(-0.451357\pi\)
0.152222 + 0.988346i \(0.451357\pi\)
\(264\) 0 0
\(265\) −21.8745 −1.34374
\(266\) 0 0
\(267\) 0 0
\(268\) −2.29150 −0.139976
\(269\) 16.7085 1.01874 0.509368 0.860549i \(-0.329879\pi\)
0.509368 + 0.860549i \(0.329879\pi\)
\(270\) 0 0
\(271\) 5.22876 0.317624 0.158812 0.987309i \(-0.449234\pi\)
0.158812 + 0.987309i \(0.449234\pi\)
\(272\) −3.64575 −0.221056
\(273\) 0 0
\(274\) 1.29150 0.0780225
\(275\) −30.2288 −1.82286
\(276\) 0 0
\(277\) −2.06275 −0.123938 −0.0619692 0.998078i \(-0.519738\pi\)
−0.0619692 + 0.998078i \(0.519738\pi\)
\(278\) −1.58301 −0.0949423
\(279\) 0 0
\(280\) 0 0
\(281\) −19.5203 −1.16448 −0.582241 0.813017i \(-0.697824\pi\)
−0.582241 + 0.813017i \(0.697824\pi\)
\(282\) 0 0
\(283\) −2.29150 −0.136216 −0.0681078 0.997678i \(-0.521696\pi\)
−0.0681078 + 0.997678i \(0.521696\pi\)
\(284\) −15.6458 −0.928405
\(285\) 0 0
\(286\) −16.9373 −1.00152
\(287\) 0 0
\(288\) 0 0
\(289\) −3.70850 −0.218147
\(290\) 8.58301 0.504011
\(291\) 0 0
\(292\) 10.5830 0.619324
\(293\) −7.06275 −0.412610 −0.206305 0.978488i \(-0.566144\pi\)
−0.206305 + 0.978488i \(0.566144\pi\)
\(294\) 0 0
\(295\) −30.4575 −1.77330
\(296\) 11.9373 0.693839
\(297\) 0 0
\(298\) 22.9373 1.32872
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) 19.2288 1.10649
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 26.8118 1.53524
\(306\) 0 0
\(307\) 7.58301 0.432785 0.216392 0.976306i \(-0.430571\pi\)
0.216392 + 0.976306i \(0.430571\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 16.9373 0.961971
\(311\) 13.5203 0.766664 0.383332 0.923611i \(-0.374777\pi\)
0.383332 + 0.923611i \(0.374777\pi\)
\(312\) 0 0
\(313\) 12.7085 0.718327 0.359163 0.933275i \(-0.383062\pi\)
0.359163 + 0.933275i \(0.383062\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) 11.2288 0.631667
\(317\) −26.8118 −1.50590 −0.752949 0.658079i \(-0.771369\pi\)
−0.752949 + 0.658079i \(0.771369\pi\)
\(318\) 0 0
\(319\) 8.58301 0.480556
\(320\) 3.64575 0.203804
\(321\) 0 0
\(322\) 0 0
\(323\) −7.29150 −0.405710
\(324\) 0 0
\(325\) −38.5203 −2.13672
\(326\) 1.00000 0.0553849
\(327\) 0 0
\(328\) 10.9373 0.603909
\(329\) 0 0
\(330\) 0 0
\(331\) −31.8745 −1.75198 −0.875991 0.482328i \(-0.839791\pi\)
−0.875991 + 0.482328i \(0.839791\pi\)
\(332\) 13.2915 0.729466
\(333\) 0 0
\(334\) −8.58301 −0.469641
\(335\) −8.35425 −0.456441
\(336\) 0 0
\(337\) −30.5830 −1.66596 −0.832981 0.553301i \(-0.813368\pi\)
−0.832981 + 0.553301i \(0.813368\pi\)
\(338\) −8.58301 −0.466854
\(339\) 0 0
\(340\) −13.2915 −0.720833
\(341\) 16.9373 0.917204
\(342\) 0 0
\(343\) 0 0
\(344\) −5.00000 −0.269582
\(345\) 0 0
\(346\) −14.5830 −0.783987
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −1.22876 −0.0657738 −0.0328869 0.999459i \(-0.510470\pi\)
−0.0328869 + 0.999459i \(0.510470\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.64575 0.194319
\(353\) 12.0000 0.638696 0.319348 0.947638i \(-0.396536\pi\)
0.319348 + 0.947638i \(0.396536\pi\)
\(354\) 0 0
\(355\) −57.0405 −3.02740
\(356\) −4.93725 −0.261674
\(357\) 0 0
\(358\) 16.9373 0.895162
\(359\) −11.1660 −0.589319 −0.294660 0.955602i \(-0.595206\pi\)
−0.294660 + 0.955602i \(0.595206\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 2.70850 0.142355
\(363\) 0 0
\(364\) 0 0
\(365\) 38.5830 2.01953
\(366\) 0 0
\(367\) 29.8745 1.55944 0.779718 0.626130i \(-0.215362\pi\)
0.779718 + 0.626130i \(0.215362\pi\)
\(368\) 1.29150 0.0673242
\(369\) 0 0
\(370\) 43.5203 2.26251
\(371\) 0 0
\(372\) 0 0
\(373\) 4.58301 0.237299 0.118650 0.992936i \(-0.462144\pi\)
0.118650 + 0.992936i \(0.462144\pi\)
\(374\) −13.2915 −0.687287
\(375\) 0 0
\(376\) 4.93725 0.254619
\(377\) 10.9373 0.563297
\(378\) 0 0
\(379\) 25.5830 1.31411 0.657055 0.753842i \(-0.271802\pi\)
0.657055 + 0.753842i \(0.271802\pi\)
\(380\) 7.29150 0.374046
\(381\) 0 0
\(382\) −20.8118 −1.06482
\(383\) −20.5830 −1.05174 −0.525871 0.850564i \(-0.676261\pi\)
−0.525871 + 0.850564i \(0.676261\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 7.00000 0.356291
\(387\) 0 0
\(388\) 13.5830 0.689573
\(389\) −19.2915 −0.978118 −0.489059 0.872251i \(-0.662660\pi\)
−0.489059 + 0.872251i \(0.662660\pi\)
\(390\) 0 0
\(391\) −4.70850 −0.238119
\(392\) 0 0
\(393\) 0 0
\(394\) 4.70850 0.237211
\(395\) 40.9373 2.05978
\(396\) 0 0
\(397\) −16.6458 −0.835426 −0.417713 0.908579i \(-0.637168\pi\)
−0.417713 + 0.908579i \(0.637168\pi\)
\(398\) −6.06275 −0.303898
\(399\) 0 0
\(400\) 8.29150 0.414575
\(401\) 20.8118 1.03929 0.519645 0.854382i \(-0.326064\pi\)
0.519645 + 0.854382i \(0.326064\pi\)
\(402\) 0 0
\(403\) 21.5830 1.07513
\(404\) −8.35425 −0.415639
\(405\) 0 0
\(406\) 0 0
\(407\) 43.5203 2.15722
\(408\) 0 0
\(409\) 14.8745 0.735497 0.367749 0.929925i \(-0.380129\pi\)
0.367749 + 0.929925i \(0.380129\pi\)
\(410\) 39.8745 1.96926
\(411\) 0 0
\(412\) 3.93725 0.193975
\(413\) 0 0
\(414\) 0 0
\(415\) 48.4575 2.37869
\(416\) 4.64575 0.227777
\(417\) 0 0
\(418\) 7.29150 0.356639
\(419\) −28.9373 −1.41368 −0.706839 0.707375i \(-0.749879\pi\)
−0.706839 + 0.707375i \(0.749879\pi\)
\(420\) 0 0
\(421\) −14.7085 −0.716848 −0.358424 0.933559i \(-0.616686\pi\)
−0.358424 + 0.933559i \(0.616686\pi\)
\(422\) −14.8745 −0.724080
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −30.2288 −1.46631
\(426\) 0 0
\(427\) 0 0
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) −18.2288 −0.879069
\(431\) −38.8118 −1.86950 −0.934748 0.355310i \(-0.884375\pi\)
−0.934748 + 0.355310i \(0.884375\pi\)
\(432\) 0 0
\(433\) −19.0000 −0.913082 −0.456541 0.889702i \(-0.650912\pi\)
−0.456541 + 0.889702i \(0.650912\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.35425 −0.160639
\(437\) 2.58301 0.123562
\(438\) 0 0
\(439\) 25.1660 1.20111 0.600554 0.799584i \(-0.294947\pi\)
0.600554 + 0.799584i \(0.294947\pi\)
\(440\) 13.2915 0.633648
\(441\) 0 0
\(442\) −16.9373 −0.805623
\(443\) 19.2915 0.916567 0.458283 0.888806i \(-0.348464\pi\)
0.458283 + 0.888806i \(0.348464\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 13.8745 0.656977
\(447\) 0 0
\(448\) 0 0
\(449\) −2.35425 −0.111104 −0.0555519 0.998456i \(-0.517692\pi\)
−0.0555519 + 0.998456i \(0.517692\pi\)
\(450\) 0 0
\(451\) 39.8745 1.87762
\(452\) −2.35425 −0.110735
\(453\) 0 0
\(454\) 6.00000 0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) −8.29150 −0.387860 −0.193930 0.981015i \(-0.562123\pi\)
−0.193930 + 0.981015i \(0.562123\pi\)
\(458\) 9.35425 0.437095
\(459\) 0 0
\(460\) 4.70850 0.219535
\(461\) −24.2288 −1.12845 −0.564223 0.825623i \(-0.690824\pi\)
−0.564223 + 0.825623i \(0.690824\pi\)
\(462\) 0 0
\(463\) 18.7085 0.869458 0.434729 0.900561i \(-0.356844\pi\)
0.434729 + 0.900561i \(0.356844\pi\)
\(464\) −2.35425 −0.109293
\(465\) 0 0
\(466\) −19.2915 −0.893662
\(467\) −0.228757 −0.0105856 −0.00529280 0.999986i \(-0.501685\pi\)
−0.00529280 + 0.999986i \(0.501685\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 18.0000 0.830278
\(471\) 0 0
\(472\) 8.35425 0.384535
\(473\) −18.2288 −0.838159
\(474\) 0 0
\(475\) 16.5830 0.760880
\(476\) 0 0
\(477\) 0 0
\(478\) −10.9373 −0.500258
\(479\) −3.64575 −0.166579 −0.0832893 0.996525i \(-0.526543\pi\)
−0.0832893 + 0.996525i \(0.526543\pi\)
\(480\) 0 0
\(481\) 55.4575 2.52864
\(482\) −5.00000 −0.227744
\(483\) 0 0
\(484\) 2.29150 0.104159
\(485\) 49.5203 2.24860
\(486\) 0 0
\(487\) 23.8745 1.08186 0.540929 0.841069i \(-0.318073\pi\)
0.540929 + 0.841069i \(0.318073\pi\)
\(488\) −7.35425 −0.332911
\(489\) 0 0
\(490\) 0 0
\(491\) −25.7490 −1.16204 −0.581018 0.813890i \(-0.697346\pi\)
−0.581018 + 0.813890i \(0.697346\pi\)
\(492\) 0 0
\(493\) 8.58301 0.386559
\(494\) 9.29150 0.418044
\(495\) 0 0
\(496\) −4.64575 −0.208600
\(497\) 0 0
\(498\) 0 0
\(499\) −6.16601 −0.276029 −0.138014 0.990430i \(-0.544072\pi\)
−0.138014 + 0.990430i \(0.544072\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) −18.0000 −0.803379
\(503\) −3.87451 −0.172756 −0.0863779 0.996262i \(-0.527529\pi\)
−0.0863779 + 0.996262i \(0.527529\pi\)
\(504\) 0 0
\(505\) −30.4575 −1.35534
\(506\) 4.70850 0.209318
\(507\) 0 0
\(508\) 1.35425 0.0600851
\(509\) −8.12549 −0.360156 −0.180078 0.983652i \(-0.557635\pi\)
−0.180078 + 0.983652i \(0.557635\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 15.8745 0.700195
\(515\) 14.3542 0.632524
\(516\) 0 0
\(517\) 18.0000 0.791639
\(518\) 0 0
\(519\) 0 0
\(520\) 16.9373 0.742748
\(521\) 8.12549 0.355984 0.177992 0.984032i \(-0.443040\pi\)
0.177992 + 0.984032i \(0.443040\pi\)
\(522\) 0 0
\(523\) −1.00000 −0.0437269 −0.0218635 0.999761i \(-0.506960\pi\)
−0.0218635 + 0.999761i \(0.506960\pi\)
\(524\) 14.5830 0.637062
\(525\) 0 0
\(526\) −4.93725 −0.215275
\(527\) 16.9373 0.737798
\(528\) 0 0
\(529\) −21.3320 −0.927479
\(530\) 21.8745 0.950168
\(531\) 0 0
\(532\) 0 0
\(533\) 50.8118 2.20090
\(534\) 0 0
\(535\) 21.8745 0.945717
\(536\) 2.29150 0.0989778
\(537\) 0 0
\(538\) −16.7085 −0.720354
\(539\) 0 0
\(540\) 0 0
\(541\) 1.16601 0.0501307 0.0250654 0.999686i \(-0.492021\pi\)
0.0250654 + 0.999686i \(0.492021\pi\)
\(542\) −5.22876 −0.224594
\(543\) 0 0
\(544\) 3.64575 0.156310
\(545\) −12.2288 −0.523822
\(546\) 0 0
\(547\) 18.2915 0.782088 0.391044 0.920372i \(-0.372114\pi\)
0.391044 + 0.920372i \(0.372114\pi\)
\(548\) −1.29150 −0.0551703
\(549\) 0 0
\(550\) 30.2288 1.28896
\(551\) −4.70850 −0.200589
\(552\) 0 0
\(553\) 0 0
\(554\) 2.06275 0.0876377
\(555\) 0 0
\(556\) 1.58301 0.0671344
\(557\) 5.77124 0.244535 0.122268 0.992497i \(-0.460983\pi\)
0.122268 + 0.992497i \(0.460983\pi\)
\(558\) 0 0
\(559\) −23.2288 −0.982472
\(560\) 0 0
\(561\) 0 0
\(562\) 19.5203 0.823412
\(563\) −18.4575 −0.777891 −0.388946 0.921261i \(-0.627161\pi\)
−0.388946 + 0.921261i \(0.627161\pi\)
\(564\) 0 0
\(565\) −8.58301 −0.361090
\(566\) 2.29150 0.0963190
\(567\) 0 0
\(568\) 15.6458 0.656481
\(569\) 16.9373 0.710047 0.355023 0.934857i \(-0.384473\pi\)
0.355023 + 0.934857i \(0.384473\pi\)
\(570\) 0 0
\(571\) 9.29150 0.388837 0.194419 0.980919i \(-0.437718\pi\)
0.194419 + 0.980919i \(0.437718\pi\)
\(572\) 16.9373 0.708182
\(573\) 0 0
\(574\) 0 0
\(575\) 10.7085 0.446575
\(576\) 0 0
\(577\) −32.2915 −1.34431 −0.672156 0.740409i \(-0.734632\pi\)
−0.672156 + 0.740409i \(0.734632\pi\)
\(578\) 3.70850 0.154253
\(579\) 0 0
\(580\) −8.58301 −0.356390
\(581\) 0 0
\(582\) 0 0
\(583\) 21.8745 0.905950
\(584\) −10.5830 −0.437928
\(585\) 0 0
\(586\) 7.06275 0.291759
\(587\) −11.7712 −0.485851 −0.242926 0.970045i \(-0.578107\pi\)
−0.242926 + 0.970045i \(0.578107\pi\)
\(588\) 0 0
\(589\) −9.29150 −0.382850
\(590\) 30.4575 1.25392
\(591\) 0 0
\(592\) −11.9373 −0.490618
\(593\) −40.9373 −1.68109 −0.840546 0.541741i \(-0.817766\pi\)
−0.840546 + 0.541741i \(0.817766\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −22.9373 −0.939547
\(597\) 0 0
\(598\) 6.00000 0.245358
\(599\) −9.87451 −0.403461 −0.201731 0.979441i \(-0.564657\pi\)
−0.201731 + 0.979441i \(0.564657\pi\)
\(600\) 0 0
\(601\) 26.8745 1.09623 0.548117 0.836402i \(-0.315345\pi\)
0.548117 + 0.836402i \(0.315345\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −19.2288 −0.782407
\(605\) 8.35425 0.339649
\(606\) 0 0
\(607\) 5.41699 0.219869 0.109935 0.993939i \(-0.464936\pi\)
0.109935 + 0.993939i \(0.464936\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) −26.8118 −1.08558
\(611\) 22.9373 0.927942
\(612\) 0 0
\(613\) −42.3948 −1.71231 −0.856154 0.516720i \(-0.827153\pi\)
−0.856154 + 0.516720i \(0.827153\pi\)
\(614\) −7.58301 −0.306025
\(615\) 0 0
\(616\) 0 0
\(617\) −1.52026 −0.0612033 −0.0306017 0.999532i \(-0.509742\pi\)
−0.0306017 + 0.999532i \(0.509742\pi\)
\(618\) 0 0
\(619\) −8.29150 −0.333264 −0.166632 0.986019i \(-0.553289\pi\)
−0.166632 + 0.986019i \(0.553289\pi\)
\(620\) −16.9373 −0.680216
\(621\) 0 0
\(622\) −13.5203 −0.542113
\(623\) 0 0
\(624\) 0 0
\(625\) 2.29150 0.0916601
\(626\) −12.7085 −0.507934
\(627\) 0 0
\(628\) −4.00000 −0.159617
\(629\) 43.5203 1.73527
\(630\) 0 0
\(631\) 6.06275 0.241354 0.120677 0.992692i \(-0.461493\pi\)
0.120677 + 0.992692i \(0.461493\pi\)
\(632\) −11.2288 −0.446656
\(633\) 0 0
\(634\) 26.8118 1.06483
\(635\) 4.93725 0.195929
\(636\) 0 0
\(637\) 0 0
\(638\) −8.58301 −0.339804
\(639\) 0 0
\(640\) −3.64575 −0.144111
\(641\) 18.2288 0.719993 0.359996 0.932954i \(-0.382778\pi\)
0.359996 + 0.932954i \(0.382778\pi\)
\(642\) 0 0
\(643\) −3.12549 −0.123257 −0.0616287 0.998099i \(-0.519629\pi\)
−0.0616287 + 0.998099i \(0.519629\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 7.29150 0.286880
\(647\) −9.87451 −0.388207 −0.194103 0.980981i \(-0.562180\pi\)
−0.194103 + 0.980981i \(0.562180\pi\)
\(648\) 0 0
\(649\) 30.4575 1.19556
\(650\) 38.5203 1.51089
\(651\) 0 0
\(652\) −1.00000 −0.0391630
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) 53.1660 2.07737
\(656\) −10.9373 −0.427028
\(657\) 0 0
\(658\) 0 0
\(659\) 24.2288 0.943818 0.471909 0.881647i \(-0.343565\pi\)
0.471909 + 0.881647i \(0.343565\pi\)
\(660\) 0 0
\(661\) −3.16601 −0.123144 −0.0615718 0.998103i \(-0.519611\pi\)
−0.0615718 + 0.998103i \(0.519611\pi\)
\(662\) 31.8745 1.23884
\(663\) 0 0
\(664\) −13.2915 −0.515810
\(665\) 0 0
\(666\) 0 0
\(667\) −3.04052 −0.117729
\(668\) 8.58301 0.332086
\(669\) 0 0
\(670\) 8.35425 0.322753
\(671\) −26.8118 −1.03506
\(672\) 0 0
\(673\) 15.7490 0.607080 0.303540 0.952819i \(-0.401831\pi\)
0.303540 + 0.952819i \(0.401831\pi\)
\(674\) 30.5830 1.17801
\(675\) 0 0
\(676\) 8.58301 0.330116
\(677\) 45.8745 1.76310 0.881550 0.472090i \(-0.156500\pi\)
0.881550 + 0.472090i \(0.156500\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 13.2915 0.509706
\(681\) 0 0
\(682\) −16.9373 −0.648561
\(683\) 26.8118 1.02592 0.512962 0.858411i \(-0.328548\pi\)
0.512962 + 0.858411i \(0.328548\pi\)
\(684\) 0 0
\(685\) −4.70850 −0.179902
\(686\) 0 0
\(687\) 0 0
\(688\) 5.00000 0.190623
\(689\) 27.8745 1.06193
\(690\) 0 0
\(691\) −38.7490 −1.47408 −0.737041 0.675848i \(-0.763778\pi\)
−0.737041 + 0.675848i \(0.763778\pi\)
\(692\) 14.5830 0.554363
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 5.77124 0.218916
\(696\) 0 0
\(697\) 39.8745 1.51035
\(698\) 1.22876 0.0465091
\(699\) 0 0
\(700\) 0 0
\(701\) 26.5830 1.00403 0.502013 0.864860i \(-0.332593\pi\)
0.502013 + 0.864860i \(0.332593\pi\)
\(702\) 0 0
\(703\) −23.8745 −0.900444
\(704\) −3.64575 −0.137404
\(705\) 0 0
\(706\) −12.0000 −0.451626
\(707\) 0 0
\(708\) 0 0
\(709\) 37.8118 1.42005 0.710025 0.704176i \(-0.248683\pi\)
0.710025 + 0.704176i \(0.248683\pi\)
\(710\) 57.0405 2.14069
\(711\) 0 0
\(712\) 4.93725 0.185031
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) 61.7490 2.30928
\(716\) −16.9373 −0.632975
\(717\) 0 0
\(718\) 11.1660 0.416712
\(719\) −22.9373 −0.855415 −0.427708 0.903917i \(-0.640679\pi\)
−0.427708 + 0.903917i \(0.640679\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 15.0000 0.558242
\(723\) 0 0
\(724\) −2.70850 −0.100661
\(725\) −19.5203 −0.724964
\(726\) 0 0
\(727\) −1.22876 −0.0455721 −0.0227860 0.999740i \(-0.507254\pi\)
−0.0227860 + 0.999740i \(0.507254\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −38.5830 −1.42802
\(731\) −18.2288 −0.674215
\(732\) 0 0
\(733\) 35.2288 1.30120 0.650602 0.759419i \(-0.274517\pi\)
0.650602 + 0.759419i \(0.274517\pi\)
\(734\) −29.8745 −1.10269
\(735\) 0 0
\(736\) −1.29150 −0.0476054
\(737\) 8.35425 0.307733
\(738\) 0 0
\(739\) −31.4575 −1.15718 −0.578592 0.815617i \(-0.696397\pi\)
−0.578592 + 0.815617i \(0.696397\pi\)
\(740\) −43.5203 −1.59984
\(741\) 0 0
\(742\) 0 0
\(743\) 28.9373 1.06160 0.530802 0.847496i \(-0.321891\pi\)
0.530802 + 0.847496i \(0.321891\pi\)
\(744\) 0 0
\(745\) −83.6235 −3.06373
\(746\) −4.58301 −0.167796
\(747\) 0 0
\(748\) 13.2915 0.485985
\(749\) 0 0
\(750\) 0 0
\(751\) −52.4575 −1.91420 −0.957101 0.289755i \(-0.906426\pi\)
−0.957101 + 0.289755i \(0.906426\pi\)
\(752\) −4.93725 −0.180043
\(753\) 0 0
\(754\) −10.9373 −0.398311
\(755\) −70.1033 −2.55132
\(756\) 0 0
\(757\) 48.9778 1.78013 0.890064 0.455836i \(-0.150660\pi\)
0.890064 + 0.455836i \(0.150660\pi\)
\(758\) −25.5830 −0.929217
\(759\) 0 0
\(760\) −7.29150 −0.264491
\(761\) −2.58301 −0.0936339 −0.0468169 0.998903i \(-0.514908\pi\)
−0.0468169 + 0.998903i \(0.514908\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 20.8118 0.752943
\(765\) 0 0
\(766\) 20.5830 0.743694
\(767\) 38.8118 1.40141
\(768\) 0 0
\(769\) 40.5830 1.46346 0.731730 0.681594i \(-0.238713\pi\)
0.731730 + 0.681594i \(0.238713\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.00000 −0.251936
\(773\) 46.3320 1.66645 0.833223 0.552936i \(-0.186493\pi\)
0.833223 + 0.552936i \(0.186493\pi\)
\(774\) 0 0
\(775\) −38.5203 −1.38369
\(776\) −13.5830 −0.487601
\(777\) 0 0
\(778\) 19.2915 0.691634
\(779\) −21.8745 −0.783736
\(780\) 0 0
\(781\) 57.0405 2.04107
\(782\) 4.70850 0.168376
\(783\) 0 0
\(784\) 0 0
\(785\) −14.5830 −0.520490
\(786\) 0 0
\(787\) −11.7085 −0.417363 −0.208681 0.977984i \(-0.566917\pi\)
−0.208681 + 0.977984i \(0.566917\pi\)
\(788\) −4.70850 −0.167733
\(789\) 0 0
\(790\) −40.9373 −1.45648
\(791\) 0 0
\(792\) 0 0
\(793\) −34.1660 −1.21327
\(794\) 16.6458 0.590736
\(795\) 0 0
\(796\) 6.06275 0.214888
\(797\) 14.8118 0.524660 0.262330 0.964978i \(-0.415509\pi\)
0.262330 + 0.964978i \(0.415509\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) −8.29150 −0.293149
\(801\) 0 0
\(802\) −20.8118 −0.734889
\(803\) −38.5830 −1.36156
\(804\) 0 0
\(805\) 0 0
\(806\) −21.5830 −0.760229
\(807\) 0 0
\(808\) 8.35425 0.293901
\(809\) −15.4170 −0.542033 −0.271016 0.962575i \(-0.587360\pi\)
−0.271016 + 0.962575i \(0.587360\pi\)
\(810\) 0 0
\(811\) −29.2915 −1.02856 −0.514282 0.857621i \(-0.671941\pi\)
−0.514282 + 0.857621i \(0.671941\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −43.5203 −1.52538
\(815\) −3.64575 −0.127705
\(816\) 0 0
\(817\) 10.0000 0.349856
\(818\) −14.8745 −0.520075
\(819\) 0 0
\(820\) −39.8745 −1.39248
\(821\) −20.5830 −0.718352 −0.359176 0.933270i \(-0.616942\pi\)
−0.359176 + 0.933270i \(0.616942\pi\)
\(822\) 0 0
\(823\) −35.1033 −1.22362 −0.611811 0.791004i \(-0.709559\pi\)
−0.611811 + 0.791004i \(0.709559\pi\)
\(824\) −3.93725 −0.137161
\(825\) 0 0
\(826\) 0 0
\(827\) 43.7490 1.52130 0.760651 0.649161i \(-0.224880\pi\)
0.760651 + 0.649161i \(0.224880\pi\)
\(828\) 0 0
\(829\) −29.2915 −1.01734 −0.508668 0.860963i \(-0.669862\pi\)
−0.508668 + 0.860963i \(0.669862\pi\)
\(830\) −48.4575 −1.68198
\(831\) 0 0
\(832\) −4.64575 −0.161062
\(833\) 0 0
\(834\) 0 0
\(835\) 31.2915 1.08289
\(836\) −7.29150 −0.252182
\(837\) 0 0
\(838\) 28.9373 0.999621
\(839\) 49.2915 1.70173 0.850866 0.525383i \(-0.176078\pi\)
0.850866 + 0.525383i \(0.176078\pi\)
\(840\) 0 0
\(841\) −23.4575 −0.808880
\(842\) 14.7085 0.506888
\(843\) 0 0
\(844\) 14.8745 0.512002
\(845\) 31.2915 1.07646
\(846\) 0 0
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 30.2288 1.03684
\(851\) −15.4170 −0.528488
\(852\) 0 0
\(853\) 47.8745 1.63919 0.819596 0.572942i \(-0.194198\pi\)
0.819596 + 0.572942i \(0.194198\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −6.00000 −0.205076
\(857\) 26.8118 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(858\) 0 0
\(859\) −7.00000 −0.238837 −0.119418 0.992844i \(-0.538103\pi\)
−0.119418 + 0.992844i \(0.538103\pi\)
\(860\) 18.2288 0.621595
\(861\) 0 0
\(862\) 38.8118 1.32193
\(863\) −29.1660 −0.992823 −0.496411 0.868087i \(-0.665349\pi\)
−0.496411 + 0.868087i \(0.665349\pi\)
\(864\) 0 0
\(865\) 53.1660 1.80770
\(866\) 19.0000 0.645646
\(867\) 0 0
\(868\) 0 0
\(869\) −40.9373 −1.38870
\(870\) 0 0
\(871\) 10.6458 0.360718
\(872\) 3.35425 0.113589
\(873\) 0 0
\(874\) −2.58301 −0.0873715
\(875\) 0 0
\(876\) 0 0
\(877\) 26.6458 0.899763 0.449882 0.893088i \(-0.351466\pi\)
0.449882 + 0.893088i \(0.351466\pi\)
\(878\) −25.1660 −0.849312
\(879\) 0 0
\(880\) −13.2915 −0.448056
\(881\) 3.87451 0.130535 0.0652677 0.997868i \(-0.479210\pi\)
0.0652677 + 0.997868i \(0.479210\pi\)
\(882\) 0 0
\(883\) −19.8745 −0.668830 −0.334415 0.942426i \(-0.608539\pi\)
−0.334415 + 0.942426i \(0.608539\pi\)
\(884\) 16.9373 0.569661
\(885\) 0 0
\(886\) −19.2915 −0.648111
\(887\) −15.8745 −0.533014 −0.266507 0.963833i \(-0.585870\pi\)
−0.266507 + 0.963833i \(0.585870\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 18.0000 0.603361
\(891\) 0 0
\(892\) −13.8745 −0.464553
\(893\) −9.87451 −0.330438
\(894\) 0 0
\(895\) −61.7490 −2.06404
\(896\) 0 0
\(897\) 0 0
\(898\) 2.35425 0.0785623
\(899\) 10.9373 0.364778
\(900\) 0 0
\(901\) 21.8745 0.728746
\(902\) −39.8745 −1.32768
\(903\) 0 0
\(904\) 2.35425 0.0783011
\(905\) −9.87451 −0.328240
\(906\) 0 0
\(907\) 36.2915 1.20504 0.602520 0.798104i \(-0.294163\pi\)
0.602520 + 0.798104i \(0.294163\pi\)
\(908\) −6.00000 −0.199117
\(909\) 0 0
\(910\) 0 0
\(911\) −28.9373 −0.958734 −0.479367 0.877615i \(-0.659134\pi\)
−0.479367 + 0.877615i \(0.659134\pi\)
\(912\) 0 0
\(913\) −48.4575 −1.60371
\(914\) 8.29150 0.274259
\(915\) 0 0
\(916\) −9.35425 −0.309073
\(917\) 0 0
\(918\) 0 0
\(919\) 28.7712 0.949076 0.474538 0.880235i \(-0.342615\pi\)
0.474538 + 0.880235i \(0.342615\pi\)
\(920\) −4.70850 −0.155235
\(921\) 0 0
\(922\) 24.2288 0.797932
\(923\) 72.6863 2.39250
\(924\) 0 0
\(925\) −98.9778 −3.25437
\(926\) −18.7085 −0.614799
\(927\) 0 0
\(928\) 2.35425 0.0772820
\(929\) 16.9373 0.555693 0.277847 0.960625i \(-0.410379\pi\)
0.277847 + 0.960625i \(0.410379\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 19.2915 0.631914
\(933\) 0 0
\(934\) 0.228757 0.00748514
\(935\) 48.4575 1.58473
\(936\) 0 0
\(937\) −44.7490 −1.46189 −0.730943 0.682438i \(-0.760920\pi\)
−0.730943 + 0.682438i \(0.760920\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −18.0000 −0.587095
\(941\) 53.3948 1.74062 0.870310 0.492505i \(-0.163919\pi\)
0.870310 + 0.492505i \(0.163919\pi\)
\(942\) 0 0
\(943\) −14.1255 −0.459989
\(944\) −8.35425 −0.271908
\(945\) 0 0
\(946\) 18.2288 0.592668
\(947\) −10.4797 −0.340546 −0.170273 0.985397i \(-0.554465\pi\)
−0.170273 + 0.985397i \(0.554465\pi\)
\(948\) 0 0
\(949\) −49.1660 −1.59600
\(950\) −16.5830 −0.538024
\(951\) 0 0
\(952\) 0 0
\(953\) −52.3320 −1.69520 −0.847600 0.530635i \(-0.821953\pi\)
−0.847600 + 0.530635i \(0.821953\pi\)
\(954\) 0 0
\(955\) 75.8745 2.45524
\(956\) 10.9373 0.353736
\(957\) 0 0
\(958\) 3.64575 0.117789
\(959\) 0 0
\(960\) 0 0
\(961\) −9.41699 −0.303774
\(962\) −55.4575 −1.78802
\(963\) 0 0
\(964\) 5.00000 0.161039
\(965\) −25.5203 −0.821526
\(966\) 0 0
\(967\) 36.0627 1.15970 0.579850 0.814723i \(-0.303111\pi\)
0.579850 + 0.814723i \(0.303111\pi\)
\(968\) −2.29150 −0.0736517
\(969\) 0 0
\(970\) −49.5203 −1.59000
\(971\) 9.87451 0.316888 0.158444 0.987368i \(-0.449352\pi\)
0.158444 + 0.987368i \(0.449352\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −23.8745 −0.764989
\(975\) 0 0
\(976\) 7.35425 0.235404
\(977\) 51.8745 1.65961 0.829806 0.558052i \(-0.188451\pi\)
0.829806 + 0.558052i \(0.188451\pi\)
\(978\) 0 0
\(979\) 18.0000 0.575282
\(980\) 0 0
\(981\) 0 0
\(982\) 25.7490 0.821684
\(983\) −20.8118 −0.663792 −0.331896 0.943316i \(-0.607688\pi\)
−0.331896 + 0.943316i \(0.607688\pi\)
\(984\) 0 0
\(985\) −17.1660 −0.546955
\(986\) −8.58301 −0.273339
\(987\) 0 0
\(988\) −9.29150 −0.295602
\(989\) 6.45751 0.205337
\(990\) 0 0
\(991\) −26.0627 −0.827910 −0.413955 0.910297i \(-0.635853\pi\)
−0.413955 + 0.910297i \(0.635853\pi\)
\(992\) 4.64575 0.147503
\(993\) 0 0
\(994\) 0 0
\(995\) 22.1033 0.700721
\(996\) 0 0
\(997\) 23.6863 0.750152 0.375076 0.926994i \(-0.377617\pi\)
0.375076 + 0.926994i \(0.377617\pi\)
\(998\) 6.16601 0.195182
\(999\) 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 2646.2.a.bi.1.2 2
3.2 odd 2 2646.2.a.bl.1.1 2
7.2 even 3 378.2.g.h.109.1 yes 4
7.4 even 3 378.2.g.h.163.1 yes 4
7.6 odd 2 2646.2.a.bf.1.1 2
21.2 odd 6 378.2.g.g.109.2 4
21.11 odd 6 378.2.g.g.163.2 yes 4
21.20 even 2 2646.2.a.bo.1.2 2
63.2 odd 6 1134.2.h.q.109.1 4
63.4 even 3 1134.2.h.t.541.2 4
63.11 odd 6 1134.2.e.t.919.2 4
63.16 even 3 1134.2.h.t.109.2 4
63.23 odd 6 1134.2.e.t.865.2 4
63.25 even 3 1134.2.e.q.919.1 4
63.32 odd 6 1134.2.h.q.541.1 4
63.58 even 3 1134.2.e.q.865.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.g.g.109.2 4 21.2 odd 6
378.2.g.g.163.2 yes 4 21.11 odd 6
378.2.g.h.109.1 yes 4 7.2 even 3
378.2.g.h.163.1 yes 4 7.4 even 3
1134.2.e.q.865.1 4 63.58 even 3
1134.2.e.q.919.1 4 63.25 even 3
1134.2.e.t.865.2 4 63.23 odd 6
1134.2.e.t.919.2 4 63.11 odd 6
1134.2.h.q.109.1 4 63.2 odd 6
1134.2.h.q.541.1 4 63.32 odd 6
1134.2.h.t.109.2 4 63.16 even 3
1134.2.h.t.541.2 4 63.4 even 3
2646.2.a.bf.1.1 2 7.6 odd 2
2646.2.a.bi.1.2 2 1.1 even 1 trivial
2646.2.a.bl.1.1 2 3.2 odd 2
2646.2.a.bo.1.2 2 21.20 even 2