Properties

Label 3450.2.a.bc.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} -6.47214 q^{13} +4.00000 q^{14} +1.00000 q^{16} +6.47214 q^{17} -1.00000 q^{18} +2.00000 q^{19} +4.00000 q^{21} +1.00000 q^{23} +1.00000 q^{24} +6.47214 q^{26} -1.00000 q^{27} -4.00000 q^{28} +8.47214 q^{29} -1.00000 q^{32} -6.47214 q^{34} +1.00000 q^{36} +8.47214 q^{37} -2.00000 q^{38} +6.47214 q^{39} -6.94427 q^{41} -4.00000 q^{42} -1.00000 q^{46} +12.9443 q^{47} -1.00000 q^{48} +9.00000 q^{49} -6.47214 q^{51} -6.47214 q^{52} -8.94427 q^{53} +1.00000 q^{54} +4.00000 q^{56} -2.00000 q^{57} -8.47214 q^{58} -6.00000 q^{59} +8.47214 q^{61} -4.00000 q^{63} +1.00000 q^{64} -12.9443 q^{67} +6.47214 q^{68} -1.00000 q^{69} -16.4721 q^{71} -1.00000 q^{72} +12.9443 q^{73} -8.47214 q^{74} +2.00000 q^{76} -6.47214 q^{78} +3.52786 q^{79} +1.00000 q^{81} +6.94427 q^{82} -10.4721 q^{83} +4.00000 q^{84} -8.47214 q^{87} -7.52786 q^{89} +25.8885 q^{91} +1.00000 q^{92} -12.9443 q^{94} +1.00000 q^{96} -13.4164 q^{97} -9.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 8q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 8q^{7} - 2q^{8} + 2q^{9} - 2q^{12} - 4q^{13} + 8q^{14} + 2q^{16} + 4q^{17} - 2q^{18} + 4q^{19} + 8q^{21} + 2q^{23} + 2q^{24} + 4q^{26} - 2q^{27} - 8q^{28} + 8q^{29} - 2q^{32} - 4q^{34} + 2q^{36} + 8q^{37} - 4q^{38} + 4q^{39} + 4q^{41} - 8q^{42} - 2q^{46} + 8q^{47} - 2q^{48} + 18q^{49} - 4q^{51} - 4q^{52} + 2q^{54} + 8q^{56} - 4q^{57} - 8q^{58} - 12q^{59} + 8q^{61} - 8q^{63} + 2q^{64} - 8q^{67} + 4q^{68} - 2q^{69} - 24q^{71} - 2q^{72} + 8q^{73} - 8q^{74} + 4q^{76} - 4q^{78} + 16q^{79} + 2q^{81} - 4q^{82} - 12q^{83} + 8q^{84} - 8q^{87} - 24q^{89} + 16q^{91} + 2q^{92} - 8q^{94} + 2q^{96} - 18q^{98} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) −6.47214 −1.79505 −0.897524 0.440966i \(-0.854636\pi\)
−0.897524 + 0.440966i \(0.854636\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.47214 1.56972 0.784862 0.619671i \(-0.212734\pi\)
0.784862 + 0.619671i \(0.212734\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 6.47214 1.26929
\(27\) −1.00000 −0.192450
\(28\) −4.00000 −0.755929
\(29\) 8.47214 1.57324 0.786618 0.617440i \(-0.211830\pi\)
0.786618 + 0.617440i \(0.211830\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −6.47214 −1.10996
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 8.47214 1.39281 0.696405 0.717649i \(-0.254782\pi\)
0.696405 + 0.717649i \(0.254782\pi\)
\(38\) −2.00000 −0.324443
\(39\) 6.47214 1.03637
\(40\) 0 0
\(41\) −6.94427 −1.08451 −0.542257 0.840213i \(-0.682430\pi\)
−0.542257 + 0.840213i \(0.682430\pi\)
\(42\) −4.00000 −0.617213
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 12.9443 1.88812 0.944058 0.329779i \(-0.106974\pi\)
0.944058 + 0.329779i \(0.106974\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −6.47214 −0.906280
\(52\) −6.47214 −0.897524
\(53\) −8.94427 −1.22859 −0.614295 0.789076i \(-0.710560\pi\)
−0.614295 + 0.789076i \(0.710560\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) −2.00000 −0.264906
\(58\) −8.47214 −1.11245
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 8.47214 1.08475 0.542373 0.840138i \(-0.317526\pi\)
0.542373 + 0.840138i \(0.317526\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −12.9443 −1.58139 −0.790697 0.612207i \(-0.790282\pi\)
−0.790697 + 0.612207i \(0.790282\pi\)
\(68\) 6.47214 0.784862
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −16.4721 −1.95488 −0.977441 0.211207i \(-0.932261\pi\)
−0.977441 + 0.211207i \(0.932261\pi\)
\(72\) −1.00000 −0.117851
\(73\) 12.9443 1.51501 0.757506 0.652828i \(-0.226418\pi\)
0.757506 + 0.652828i \(0.226418\pi\)
\(74\) −8.47214 −0.984866
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) −6.47214 −0.732825
\(79\) 3.52786 0.396916 0.198458 0.980109i \(-0.436407\pi\)
0.198458 + 0.980109i \(0.436407\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.94427 0.766867
\(83\) −10.4721 −1.14947 −0.574733 0.818341i \(-0.694894\pi\)
−0.574733 + 0.818341i \(0.694894\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 0 0
\(87\) −8.47214 −0.908308
\(88\) 0 0
\(89\) −7.52786 −0.797952 −0.398976 0.916961i \(-0.630634\pi\)
−0.398976 + 0.916961i \(0.630634\pi\)
\(90\) 0 0
\(91\) 25.8885 2.71386
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −12.9443 −1.33510
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −13.4164 −1.36223 −0.681115 0.732177i \(-0.738505\pi\)
−0.681115 + 0.732177i \(0.738505\pi\)
\(98\) −9.00000 −0.909137
\(99\) 0 0
\(100\) 0 0
\(101\) −8.47214 −0.843009 −0.421505 0.906826i \(-0.638498\pi\)
−0.421505 + 0.906826i \(0.638498\pi\)
\(102\) 6.47214 0.640837
\(103\) 0.944272 0.0930419 0.0465209 0.998917i \(-0.485187\pi\)
0.0465209 + 0.998917i \(0.485187\pi\)
\(104\) 6.47214 0.634645
\(105\) 0 0
\(106\) 8.94427 0.868744
\(107\) −6.47214 −0.625685 −0.312842 0.949805i \(-0.601281\pi\)
−0.312842 + 0.949805i \(0.601281\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −8.47214 −0.811483 −0.405742 0.913988i \(-0.632987\pi\)
−0.405742 + 0.913988i \(0.632987\pi\)
\(110\) 0 0
\(111\) −8.47214 −0.804140
\(112\) −4.00000 −0.377964
\(113\) 7.41641 0.697677 0.348838 0.937183i \(-0.386576\pi\)
0.348838 + 0.937183i \(0.386576\pi\)
\(114\) 2.00000 0.187317
\(115\) 0 0
\(116\) 8.47214 0.786618
\(117\) −6.47214 −0.598349
\(118\) 6.00000 0.552345
\(119\) −25.8885 −2.37320
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −8.47214 −0.767031
\(123\) 6.94427 0.626144
\(124\) 0 0
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) 15.4164 1.36798 0.683992 0.729489i \(-0.260242\pi\)
0.683992 + 0.729489i \(0.260242\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 10.9443 0.956205 0.478103 0.878304i \(-0.341325\pi\)
0.478103 + 0.878304i \(0.341325\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 12.9443 1.11821
\(135\) 0 0
\(136\) −6.47214 −0.554981
\(137\) −7.41641 −0.633626 −0.316813 0.948488i \(-0.602613\pi\)
−0.316813 + 0.948488i \(0.602613\pi\)
\(138\) 1.00000 0.0851257
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −12.9443 −1.09010
\(142\) 16.4721 1.38231
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −12.9443 −1.07128
\(147\) −9.00000 −0.742307
\(148\) 8.47214 0.696405
\(149\) 13.4164 1.09911 0.549557 0.835456i \(-0.314796\pi\)
0.549557 + 0.835456i \(0.314796\pi\)
\(150\) 0 0
\(151\) −0.944272 −0.0768438 −0.0384219 0.999262i \(-0.512233\pi\)
−0.0384219 + 0.999262i \(0.512233\pi\)
\(152\) −2.00000 −0.162221
\(153\) 6.47214 0.523241
\(154\) 0 0
\(155\) 0 0
\(156\) 6.47214 0.518186
\(157\) 20.4721 1.63385 0.816927 0.576741i \(-0.195676\pi\)
0.816927 + 0.576741i \(0.195676\pi\)
\(158\) −3.52786 −0.280662
\(159\) 8.94427 0.709327
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) −1.00000 −0.0785674
\(163\) −12.9443 −1.01387 −0.506937 0.861983i \(-0.669222\pi\)
−0.506937 + 0.861983i \(0.669222\pi\)
\(164\) −6.94427 −0.542257
\(165\) 0 0
\(166\) 10.4721 0.812795
\(167\) −20.9443 −1.62072 −0.810358 0.585935i \(-0.800727\pi\)
−0.810358 + 0.585935i \(0.800727\pi\)
\(168\) −4.00000 −0.308607
\(169\) 28.8885 2.22220
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) −2.94427 −0.223849 −0.111924 0.993717i \(-0.535701\pi\)
−0.111924 + 0.993717i \(0.535701\pi\)
\(174\) 8.47214 0.642271
\(175\) 0 0
\(176\) 0 0
\(177\) 6.00000 0.450988
\(178\) 7.52786 0.564237
\(179\) −1.05573 −0.0789088 −0.0394544 0.999221i \(-0.512562\pi\)
−0.0394544 + 0.999221i \(0.512562\pi\)
\(180\) 0 0
\(181\) −0.472136 −0.0350936 −0.0175468 0.999846i \(-0.505586\pi\)
−0.0175468 + 0.999846i \(0.505586\pi\)
\(182\) −25.8885 −1.91899
\(183\) −8.47214 −0.626278
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 12.9443 0.944058
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 9.88854 0.715510 0.357755 0.933816i \(-0.383542\pi\)
0.357755 + 0.933816i \(0.383542\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −20.0000 −1.43963 −0.719816 0.694165i \(-0.755774\pi\)
−0.719816 + 0.694165i \(0.755774\pi\)
\(194\) 13.4164 0.963242
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −1.05573 −0.0752175 −0.0376088 0.999293i \(-0.511974\pi\)
−0.0376088 + 0.999293i \(0.511974\pi\)
\(198\) 0 0
\(199\) −9.41641 −0.667511 −0.333756 0.942660i \(-0.608316\pi\)
−0.333756 + 0.942660i \(0.608316\pi\)
\(200\) 0 0
\(201\) 12.9443 0.913019
\(202\) 8.47214 0.596097
\(203\) −33.8885 −2.37851
\(204\) −6.47214 −0.453140
\(205\) 0 0
\(206\) −0.944272 −0.0657905
\(207\) 1.00000 0.0695048
\(208\) −6.47214 −0.448762
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −8.94427 −0.614295
\(213\) 16.4721 1.12865
\(214\) 6.47214 0.442426
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 8.47214 0.573805
\(219\) −12.9443 −0.874693
\(220\) 0 0
\(221\) −41.8885 −2.81773
\(222\) 8.47214 0.568613
\(223\) 15.4164 1.03236 0.516180 0.856480i \(-0.327354\pi\)
0.516180 + 0.856480i \(0.327354\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) −7.41641 −0.493332
\(227\) −9.52786 −0.632387 −0.316193 0.948695i \(-0.602405\pi\)
−0.316193 + 0.948695i \(0.602405\pi\)
\(228\) −2.00000 −0.132453
\(229\) −1.41641 −0.0935989 −0.0467994 0.998904i \(-0.514902\pi\)
−0.0467994 + 0.998904i \(0.514902\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −8.47214 −0.556223
\(233\) 2.94427 0.192886 0.0964428 0.995339i \(-0.469254\pi\)
0.0964428 + 0.995339i \(0.469254\pi\)
\(234\) 6.47214 0.423097
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) −3.52786 −0.229159
\(238\) 25.8885 1.67811
\(239\) −3.52786 −0.228199 −0.114099 0.993469i \(-0.536398\pi\)
−0.114099 + 0.993469i \(0.536398\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 11.0000 0.707107
\(243\) −1.00000 −0.0641500
\(244\) 8.47214 0.542373
\(245\) 0 0
\(246\) −6.94427 −0.442751
\(247\) −12.9443 −0.823624
\(248\) 0 0
\(249\) 10.4721 0.663645
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −4.00000 −0.251976
\(253\) 0 0
\(254\) −15.4164 −0.967311
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) −33.8885 −2.10573
\(260\) 0 0
\(261\) 8.47214 0.524412
\(262\) −10.9443 −0.676139
\(263\) −3.05573 −0.188424 −0.0942121 0.995552i \(-0.530033\pi\)
−0.0942121 + 0.995552i \(0.530033\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 8.00000 0.490511
\(267\) 7.52786 0.460698
\(268\) −12.9443 −0.790697
\(269\) −8.47214 −0.516555 −0.258278 0.966071i \(-0.583155\pi\)
−0.258278 + 0.966071i \(0.583155\pi\)
\(270\) 0 0
\(271\) −16.9443 −1.02929 −0.514646 0.857403i \(-0.672076\pi\)
−0.514646 + 0.857403i \(0.672076\pi\)
\(272\) 6.47214 0.392431
\(273\) −25.8885 −1.56685
\(274\) 7.41641 0.448042
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 0.583592 0.0350647 0.0175323 0.999846i \(-0.494419\pi\)
0.0175323 + 0.999846i \(0.494419\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.4721 0.744025 0.372013 0.928228i \(-0.378668\pi\)
0.372013 + 0.928228i \(0.378668\pi\)
\(282\) 12.9443 0.770820
\(283\) −24.0000 −1.42665 −0.713326 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) −16.4721 −0.977441
\(285\) 0 0
\(286\) 0 0
\(287\) 27.7771 1.63963
\(288\) −1.00000 −0.0589256
\(289\) 24.8885 1.46403
\(290\) 0 0
\(291\) 13.4164 0.786484
\(292\) 12.9443 0.757506
\(293\) −16.9443 −0.989895 −0.494947 0.868923i \(-0.664813\pi\)
−0.494947 + 0.868923i \(0.664813\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) −8.47214 −0.492433
\(297\) 0 0
\(298\) −13.4164 −0.777192
\(299\) −6.47214 −0.374293
\(300\) 0 0
\(301\) 0 0
\(302\) 0.944272 0.0543367
\(303\) 8.47214 0.486711
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) −6.47214 −0.369987
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) −0.944272 −0.0537178
\(310\) 0 0
\(311\) −15.5279 −0.880504 −0.440252 0.897874i \(-0.645111\pi\)
−0.440252 + 0.897874i \(0.645111\pi\)
\(312\) −6.47214 −0.366413
\(313\) −1.41641 −0.0800601 −0.0400301 0.999198i \(-0.512745\pi\)
−0.0400301 + 0.999198i \(0.512745\pi\)
\(314\) −20.4721 −1.15531
\(315\) 0 0
\(316\) 3.52786 0.198458
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −8.94427 −0.501570
\(319\) 0 0
\(320\) 0 0
\(321\) 6.47214 0.361239
\(322\) 4.00000 0.222911
\(323\) 12.9443 0.720239
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 12.9443 0.716917
\(327\) 8.47214 0.468510
\(328\) 6.94427 0.383433
\(329\) −51.7771 −2.85456
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) −10.4721 −0.574733
\(333\) 8.47214 0.464270
\(334\) 20.9443 1.14602
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) 1.41641 0.0771567 0.0385783 0.999256i \(-0.487717\pi\)
0.0385783 + 0.999256i \(0.487717\pi\)
\(338\) −28.8885 −1.57133
\(339\) −7.41641 −0.402804
\(340\) 0 0
\(341\) 0 0
\(342\) −2.00000 −0.108148
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 2.94427 0.158285
\(347\) 8.94427 0.480154 0.240077 0.970754i \(-0.422827\pi\)
0.240077 + 0.970754i \(0.422827\pi\)
\(348\) −8.47214 −0.454154
\(349\) 7.88854 0.422264 0.211132 0.977458i \(-0.432285\pi\)
0.211132 + 0.977458i \(0.432285\pi\)
\(350\) 0 0
\(351\) 6.47214 0.345457
\(352\) 0 0
\(353\) 14.9443 0.795403 0.397702 0.917515i \(-0.369808\pi\)
0.397702 + 0.917515i \(0.369808\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) −7.52786 −0.398976
\(357\) 25.8885 1.37017
\(358\) 1.05573 0.0557970
\(359\) −15.0557 −0.794611 −0.397305 0.917686i \(-0.630055\pi\)
−0.397305 + 0.917686i \(0.630055\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0.472136 0.0248149
\(363\) 11.0000 0.577350
\(364\) 25.8885 1.35693
\(365\) 0 0
\(366\) 8.47214 0.442846
\(367\) 12.0000 0.626395 0.313197 0.949688i \(-0.398600\pi\)
0.313197 + 0.949688i \(0.398600\pi\)
\(368\) 1.00000 0.0521286
\(369\) −6.94427 −0.361504
\(370\) 0 0
\(371\) 35.7771 1.85745
\(372\) 0 0
\(373\) −5.41641 −0.280451 −0.140225 0.990120i \(-0.544783\pi\)
−0.140225 + 0.990120i \(0.544783\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.9443 −0.667550
\(377\) −54.8328 −2.82403
\(378\) −4.00000 −0.205738
\(379\) −36.8328 −1.89197 −0.945987 0.324204i \(-0.894904\pi\)
−0.945987 + 0.324204i \(0.894904\pi\)
\(380\) 0 0
\(381\) −15.4164 −0.789807
\(382\) −9.88854 −0.505942
\(383\) −21.8885 −1.11845 −0.559226 0.829015i \(-0.688902\pi\)
−0.559226 + 0.829015i \(0.688902\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 20.0000 1.01797
\(387\) 0 0
\(388\) −13.4164 −0.681115
\(389\) −4.47214 −0.226746 −0.113373 0.993552i \(-0.536166\pi\)
−0.113373 + 0.993552i \(0.536166\pi\)
\(390\) 0 0
\(391\) 6.47214 0.327310
\(392\) −9.00000 −0.454569
\(393\) −10.9443 −0.552065
\(394\) 1.05573 0.0531868
\(395\) 0 0
\(396\) 0 0
\(397\) 33.3050 1.67153 0.835764 0.549089i \(-0.185025\pi\)
0.835764 + 0.549089i \(0.185025\pi\)
\(398\) 9.41641 0.472002
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) −1.41641 −0.0707320 −0.0353660 0.999374i \(-0.511260\pi\)
−0.0353660 + 0.999374i \(0.511260\pi\)
\(402\) −12.9443 −0.645602
\(403\) 0 0
\(404\) −8.47214 −0.421505
\(405\) 0 0
\(406\) 33.8885 1.68186
\(407\) 0 0
\(408\) 6.47214 0.320418
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) 7.41641 0.365824
\(412\) 0.944272 0.0465209
\(413\) 24.0000 1.18096
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 6.47214 0.317323
\(417\) 0 0
\(418\) 0 0
\(419\) −2.11146 −0.103151 −0.0515757 0.998669i \(-0.516424\pi\)
−0.0515757 + 0.998669i \(0.516424\pi\)
\(420\) 0 0
\(421\) −10.5836 −0.515813 −0.257906 0.966170i \(-0.583033\pi\)
−0.257906 + 0.966170i \(0.583033\pi\)
\(422\) 12.0000 0.584151
\(423\) 12.9443 0.629372
\(424\) 8.94427 0.434372
\(425\) 0 0
\(426\) −16.4721 −0.798078
\(427\) −33.8885 −1.63998
\(428\) −6.47214 −0.312842
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −5.41641 −0.260296 −0.130148 0.991495i \(-0.541545\pi\)
−0.130148 + 0.991495i \(0.541545\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −8.47214 −0.405742
\(437\) 2.00000 0.0956730
\(438\) 12.9443 0.618501
\(439\) 26.8328 1.28066 0.640330 0.768100i \(-0.278798\pi\)
0.640330 + 0.768100i \(0.278798\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 41.8885 1.99243
\(443\) −8.94427 −0.424955 −0.212478 0.977166i \(-0.568153\pi\)
−0.212478 + 0.977166i \(0.568153\pi\)
\(444\) −8.47214 −0.402070
\(445\) 0 0
\(446\) −15.4164 −0.729988
\(447\) −13.4164 −0.634574
\(448\) −4.00000 −0.188982
\(449\) −17.0557 −0.804910 −0.402455 0.915440i \(-0.631843\pi\)
−0.402455 + 0.915440i \(0.631843\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 7.41641 0.348838
\(453\) 0.944272 0.0443658
\(454\) 9.52786 0.447165
\(455\) 0 0
\(456\) 2.00000 0.0936586
\(457\) 1.41641 0.0662568 0.0331284 0.999451i \(-0.489453\pi\)
0.0331284 + 0.999451i \(0.489453\pi\)
\(458\) 1.41641 0.0661844
\(459\) −6.47214 −0.302093
\(460\) 0 0
\(461\) 30.3607 1.41404 0.707019 0.707195i \(-0.250040\pi\)
0.707019 + 0.707195i \(0.250040\pi\)
\(462\) 0 0
\(463\) −11.4164 −0.530565 −0.265283 0.964171i \(-0.585465\pi\)
−0.265283 + 0.964171i \(0.585465\pi\)
\(464\) 8.47214 0.393309
\(465\) 0 0
\(466\) −2.94427 −0.136391
\(467\) −17.5279 −0.811093 −0.405546 0.914074i \(-0.632919\pi\)
−0.405546 + 0.914074i \(0.632919\pi\)
\(468\) −6.47214 −0.299175
\(469\) 51.7771 2.39084
\(470\) 0 0
\(471\) −20.4721 −0.943306
\(472\) 6.00000 0.276172
\(473\) 0 0
\(474\) 3.52786 0.162040
\(475\) 0 0
\(476\) −25.8885 −1.18660
\(477\) −8.94427 −0.409530
\(478\) 3.52786 0.161361
\(479\) 0.944272 0.0431449 0.0215724 0.999767i \(-0.493133\pi\)
0.0215724 + 0.999767i \(0.493133\pi\)
\(480\) 0 0
\(481\) −54.8328 −2.50016
\(482\) 2.00000 0.0910975
\(483\) 4.00000 0.182006
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 4.58359 0.207702 0.103851 0.994593i \(-0.466883\pi\)
0.103851 + 0.994593i \(0.466883\pi\)
\(488\) −8.47214 −0.383516
\(489\) 12.9443 0.585360
\(490\) 0 0
\(491\) −7.88854 −0.356005 −0.178002 0.984030i \(-0.556964\pi\)
−0.178002 + 0.984030i \(0.556964\pi\)
\(492\) 6.94427 0.313072
\(493\) 54.8328 2.46955
\(494\) 12.9443 0.582390
\(495\) 0 0
\(496\) 0 0
\(497\) 65.8885 2.95551
\(498\) −10.4721 −0.469268
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 0 0
\(501\) 20.9443 0.935721
\(502\) 12.0000 0.535586
\(503\) 21.8885 0.975962 0.487981 0.872854i \(-0.337734\pi\)
0.487981 + 0.872854i \(0.337734\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 0 0
\(507\) −28.8885 −1.28299
\(508\) 15.4164 0.683992
\(509\) −25.4164 −1.12656 −0.563281 0.826265i \(-0.690461\pi\)
−0.563281 + 0.826265i \(0.690461\pi\)
\(510\) 0 0
\(511\) −51.7771 −2.29048
\(512\) −1.00000 −0.0441942
\(513\) −2.00000 −0.0883022
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 33.8885 1.48898
\(519\) 2.94427 0.129239
\(520\) 0 0
\(521\) −28.4721 −1.24739 −0.623693 0.781669i \(-0.714369\pi\)
−0.623693 + 0.781669i \(0.714369\pi\)
\(522\) −8.47214 −0.370815
\(523\) 12.9443 0.566013 0.283007 0.959118i \(-0.408668\pi\)
0.283007 + 0.959118i \(0.408668\pi\)
\(524\) 10.9443 0.478103
\(525\) 0 0
\(526\) 3.05573 0.133236
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) −8.00000 −0.346844
\(533\) 44.9443 1.94675
\(534\) −7.52786 −0.325763
\(535\) 0 0
\(536\) 12.9443 0.559107
\(537\) 1.05573 0.0455580
\(538\) 8.47214 0.365260
\(539\) 0 0
\(540\) 0 0
\(541\) −12.8328 −0.551726 −0.275863 0.961197i \(-0.588964\pi\)
−0.275863 + 0.961197i \(0.588964\pi\)
\(542\) 16.9443 0.727819
\(543\) 0.472136 0.0202613
\(544\) −6.47214 −0.277491
\(545\) 0 0
\(546\) 25.8885 1.10793
\(547\) −30.8328 −1.31832 −0.659158 0.752004i \(-0.729087\pi\)
−0.659158 + 0.752004i \(0.729087\pi\)
\(548\) −7.41641 −0.316813
\(549\) 8.47214 0.361582
\(550\) 0 0
\(551\) 16.9443 0.721850
\(552\) 1.00000 0.0425628
\(553\) −14.1115 −0.600080
\(554\) −0.583592 −0.0247945
\(555\) 0 0
\(556\) 0 0
\(557\) −5.88854 −0.249506 −0.124753 0.992188i \(-0.539814\pi\)
−0.124753 + 0.992188i \(0.539814\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −12.4721 −0.526105
\(563\) −25.3050 −1.06648 −0.533238 0.845965i \(-0.679025\pi\)
−0.533238 + 0.845965i \(0.679025\pi\)
\(564\) −12.9443 −0.545052
\(565\) 0 0
\(566\) 24.0000 1.00880
\(567\) −4.00000 −0.167984
\(568\) 16.4721 0.691155
\(569\) −23.3050 −0.976994 −0.488497 0.872565i \(-0.662455\pi\)
−0.488497 + 0.872565i \(0.662455\pi\)
\(570\) 0 0
\(571\) −20.8328 −0.871826 −0.435913 0.899989i \(-0.643575\pi\)
−0.435913 + 0.899989i \(0.643575\pi\)
\(572\) 0 0
\(573\) −9.88854 −0.413100
\(574\) −27.7771 −1.15939
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 0.944272 0.0393106 0.0196553 0.999807i \(-0.493743\pi\)
0.0196553 + 0.999807i \(0.493743\pi\)
\(578\) −24.8885 −1.03523
\(579\) 20.0000 0.831172
\(580\) 0 0
\(581\) 41.8885 1.73783
\(582\) −13.4164 −0.556128
\(583\) 0 0
\(584\) −12.9443 −0.535638
\(585\) 0 0
\(586\) 16.9443 0.699961
\(587\) −5.88854 −0.243046 −0.121523 0.992589i \(-0.538778\pi\)
−0.121523 + 0.992589i \(0.538778\pi\)
\(588\) −9.00000 −0.371154
\(589\) 0 0
\(590\) 0 0
\(591\) 1.05573 0.0434269
\(592\) 8.47214 0.348203
\(593\) 11.8885 0.488204 0.244102 0.969750i \(-0.421507\pi\)
0.244102 + 0.969750i \(0.421507\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 13.4164 0.549557
\(597\) 9.41641 0.385388
\(598\) 6.47214 0.264665
\(599\) −18.3607 −0.750197 −0.375099 0.926985i \(-0.622391\pi\)
−0.375099 + 0.926985i \(0.622391\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) −12.9443 −0.527132
\(604\) −0.944272 −0.0384219
\(605\) 0 0
\(606\) −8.47214 −0.344157
\(607\) 19.4164 0.788088 0.394044 0.919092i \(-0.371076\pi\)
0.394044 + 0.919092i \(0.371076\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 33.8885 1.37323
\(610\) 0 0
\(611\) −83.7771 −3.38926
\(612\) 6.47214 0.261621
\(613\) 17.4164 0.703442 0.351721 0.936105i \(-0.385597\pi\)
0.351721 + 0.936105i \(0.385597\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5.52786 −0.222543 −0.111272 0.993790i \(-0.535492\pi\)
−0.111272 + 0.993790i \(0.535492\pi\)
\(618\) 0.944272 0.0379842
\(619\) −12.8328 −0.515794 −0.257897 0.966172i \(-0.583030\pi\)
−0.257897 + 0.966172i \(0.583030\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 15.5279 0.622611
\(623\) 30.1115 1.20639
\(624\) 6.47214 0.259093
\(625\) 0 0
\(626\) 1.41641 0.0566110
\(627\) 0 0
\(628\) 20.4721 0.816927
\(629\) 54.8328 2.18633
\(630\) 0 0
\(631\) 42.3607 1.68635 0.843176 0.537638i \(-0.180683\pi\)
0.843176 + 0.537638i \(0.180683\pi\)
\(632\) −3.52786 −0.140331
\(633\) 12.0000 0.476957
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 8.94427 0.354663
\(637\) −58.2492 −2.30792
\(638\) 0 0
\(639\) −16.4721 −0.651628
\(640\) 0 0
\(641\) −20.4721 −0.808601 −0.404300 0.914626i \(-0.632485\pi\)
−0.404300 + 0.914626i \(0.632485\pi\)
\(642\) −6.47214 −0.255435
\(643\) 4.94427 0.194983 0.0974915 0.995236i \(-0.468918\pi\)
0.0974915 + 0.995236i \(0.468918\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) −12.9443 −0.509286
\(647\) 6.11146 0.240266 0.120133 0.992758i \(-0.461668\pi\)
0.120133 + 0.992758i \(0.461668\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −12.9443 −0.506937
\(653\) 32.8328 1.28485 0.642424 0.766350i \(-0.277929\pi\)
0.642424 + 0.766350i \(0.277929\pi\)
\(654\) −8.47214 −0.331287
\(655\) 0 0
\(656\) −6.94427 −0.271128
\(657\) 12.9443 0.505004
\(658\) 51.7771 2.01848
\(659\) −17.8885 −0.696839 −0.348419 0.937339i \(-0.613281\pi\)
−0.348419 + 0.937339i \(0.613281\pi\)
\(660\) 0 0
\(661\) 44.2492 1.72110 0.860548 0.509370i \(-0.170121\pi\)
0.860548 + 0.509370i \(0.170121\pi\)
\(662\) 0 0
\(663\) 41.8885 1.62682
\(664\) 10.4721 0.406398
\(665\) 0 0
\(666\) −8.47214 −0.328289
\(667\) 8.47214 0.328042
\(668\) −20.9443 −0.810358
\(669\) −15.4164 −0.596033
\(670\) 0 0
\(671\) 0 0
\(672\) −4.00000 −0.154303
\(673\) −14.8328 −0.571763 −0.285882 0.958265i \(-0.592286\pi\)
−0.285882 + 0.958265i \(0.592286\pi\)
\(674\) −1.41641 −0.0545580
\(675\) 0 0
\(676\) 28.8885 1.11110
\(677\) −9.88854 −0.380048 −0.190024 0.981779i \(-0.560857\pi\)
−0.190024 + 0.981779i \(0.560857\pi\)
\(678\) 7.41641 0.284825
\(679\) 53.6656 2.05950
\(680\) 0 0
\(681\) 9.52786 0.365109
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) 1.41641 0.0540393
\(688\) 0 0
\(689\) 57.8885 2.20538
\(690\) 0 0
\(691\) 31.7771 1.20886 0.604429 0.796659i \(-0.293401\pi\)
0.604429 + 0.796659i \(0.293401\pi\)
\(692\) −2.94427 −0.111924
\(693\) 0 0
\(694\) −8.94427 −0.339520
\(695\) 0 0
\(696\) 8.47214 0.321135
\(697\) −44.9443 −1.70239
\(698\) −7.88854 −0.298586
\(699\) −2.94427 −0.111363
\(700\) 0 0
\(701\) 10.5836 0.399737 0.199868 0.979823i \(-0.435949\pi\)
0.199868 + 0.979823i \(0.435949\pi\)
\(702\) −6.47214 −0.244275
\(703\) 16.9443 0.639065
\(704\) 0 0
\(705\) 0 0
\(706\) −14.9443 −0.562435
\(707\) 33.8885 1.27451
\(708\) 6.00000 0.225494
\(709\) 11.5279 0.432938 0.216469 0.976289i \(-0.430546\pi\)
0.216469 + 0.976289i \(0.430546\pi\)
\(710\) 0 0
\(711\) 3.52786 0.132305
\(712\) 7.52786 0.282119
\(713\) 0 0
\(714\) −25.8885 −0.968854
\(715\) 0 0
\(716\) −1.05573 −0.0394544
\(717\) 3.52786 0.131750
\(718\) 15.0557 0.561875
\(719\) −44.4721 −1.65853 −0.829265 0.558855i \(-0.811241\pi\)
−0.829265 + 0.558855i \(0.811241\pi\)
\(720\) 0 0
\(721\) −3.77709 −0.140666
\(722\) 15.0000 0.558242
\(723\) 2.00000 0.0743808
\(724\) −0.472136 −0.0175468
\(725\) 0 0
\(726\) −11.0000 −0.408248
\(727\) 16.9443 0.628428 0.314214 0.949352i \(-0.398259\pi\)
0.314214 + 0.949352i \(0.398259\pi\)
\(728\) −25.8885 −0.959493
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) −8.47214 −0.313139
\(733\) −8.47214 −0.312925 −0.156463 0.987684i \(-0.550009\pi\)
−0.156463 + 0.987684i \(0.550009\pi\)
\(734\) −12.0000 −0.442928
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) 6.94427 0.255622
\(739\) −33.8885 −1.24661 −0.623305 0.781979i \(-0.714211\pi\)
−0.623305 + 0.781979i \(0.714211\pi\)
\(740\) 0 0
\(741\) 12.9443 0.475520
\(742\) −35.7771 −1.31342
\(743\) −37.8885 −1.39000 −0.694998 0.719012i \(-0.744595\pi\)
−0.694998 + 0.719012i \(0.744595\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 5.41641 0.198309
\(747\) −10.4721 −0.383155
\(748\) 0 0
\(749\) 25.8885 0.945947
\(750\) 0 0
\(751\) 27.5279 1.00451 0.502253 0.864721i \(-0.332505\pi\)
0.502253 + 0.864721i \(0.332505\pi\)
\(752\) 12.9443 0.472029
\(753\) 12.0000 0.437304
\(754\) 54.8328 1.99689
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) −18.5836 −0.675432 −0.337716 0.941248i \(-0.609654\pi\)
−0.337716 + 0.941248i \(0.609654\pi\)
\(758\) 36.8328 1.33783
\(759\) 0 0
\(760\) 0 0
\(761\) 53.7771 1.94942 0.974709 0.223478i \(-0.0717411\pi\)
0.974709 + 0.223478i \(0.0717411\pi\)
\(762\) 15.4164 0.558478
\(763\) 33.8885 1.22685
\(764\) 9.88854 0.357755
\(765\) 0 0
\(766\) 21.8885 0.790865
\(767\) 38.8328 1.40217
\(768\) −1.00000 −0.0360844
\(769\) 17.0557 0.615045 0.307523 0.951541i \(-0.400500\pi\)
0.307523 + 0.951541i \(0.400500\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) −20.0000 −0.719816
\(773\) 20.9443 0.753313 0.376657 0.926353i \(-0.377074\pi\)
0.376657 + 0.926353i \(0.377074\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 13.4164 0.481621
\(777\) 33.8885 1.21574
\(778\) 4.47214 0.160334
\(779\) −13.8885 −0.497609
\(780\) 0 0
\(781\) 0 0
\(782\) −6.47214 −0.231443
\(783\) −8.47214 −0.302769
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 10.9443 0.390369
\(787\) 54.8328 1.95458 0.977289 0.211909i \(-0.0679681\pi\)
0.977289 + 0.211909i \(0.0679681\pi\)
\(788\) −1.05573 −0.0376088
\(789\) 3.05573 0.108787
\(790\) 0 0
\(791\) −29.6656 −1.05479
\(792\) 0 0
\(793\) −54.8328 −1.94717
\(794\) −33.3050 −1.18195
\(795\) 0 0
\(796\) −9.41641 −0.333756
\(797\) 2.11146 0.0747916 0.0373958 0.999301i \(-0.488094\pi\)
0.0373958 + 0.999301i \(0.488094\pi\)
\(798\) −8.00000 −0.283197
\(799\) 83.7771 2.96382
\(800\) 0 0
\(801\) −7.52786 −0.265984
\(802\) 1.41641 0.0500151
\(803\) 0 0
\(804\) 12.9443 0.456509
\(805\) 0 0
\(806\) 0 0
\(807\) 8.47214 0.298233
\(808\) 8.47214 0.298049
\(809\) 31.8885 1.12114 0.560571 0.828107i \(-0.310582\pi\)
0.560571 + 0.828107i \(0.310582\pi\)
\(810\) 0 0
\(811\) −14.1115 −0.495520 −0.247760 0.968821i \(-0.579694\pi\)
−0.247760 + 0.968821i \(0.579694\pi\)
\(812\) −33.8885 −1.18925
\(813\) 16.9443 0.594262
\(814\) 0 0
\(815\) 0 0
\(816\) −6.47214 −0.226570
\(817\) 0 0
\(818\) 18.0000 0.629355
\(819\) 25.8885 0.904619
\(820\) 0 0
\(821\) −7.52786 −0.262724 −0.131362 0.991334i \(-0.541935\pi\)
−0.131362 + 0.991334i \(0.541935\pi\)
\(822\) −7.41641 −0.258677
\(823\) 24.3607 0.849160 0.424580 0.905390i \(-0.360422\pi\)
0.424580 + 0.905390i \(0.360422\pi\)
\(824\) −0.944272 −0.0328953
\(825\) 0 0
\(826\) −24.0000 −0.835067
\(827\) 33.5279 1.16588 0.582939 0.812516i \(-0.301903\pi\)
0.582939 + 0.812516i \(0.301903\pi\)
\(828\) 1.00000 0.0347524
\(829\) 13.0557 0.453444 0.226722 0.973959i \(-0.427199\pi\)
0.226722 + 0.973959i \(0.427199\pi\)
\(830\) 0 0
\(831\) −0.583592 −0.0202446
\(832\) −6.47214 −0.224381
\(833\) 58.2492 2.01822
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 2.11146 0.0729390
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 42.7771 1.47507
\(842\) 10.5836 0.364735
\(843\) −12.4721 −0.429563
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) −12.9443 −0.445033
\(847\) 44.0000 1.51186
\(848\) −8.94427 −0.307148
\(849\) 24.0000 0.823678
\(850\) 0 0
\(851\) 8.47214 0.290421
\(852\) 16.4721 0.564326
\(853\) −18.4721 −0.632474 −0.316237 0.948680i \(-0.602419\pi\)
−0.316237 + 0.948680i \(0.602419\pi\)
\(854\) 33.8885 1.15964
\(855\) 0 0
\(856\) 6.47214 0.221213
\(857\) −11.8885 −0.406105 −0.203052 0.979168i \(-0.565086\pi\)
−0.203052 + 0.979168i \(0.565086\pi\)
\(858\) 0 0
\(859\) 31.7771 1.08422 0.542110 0.840307i \(-0.317626\pi\)
0.542110 + 0.840307i \(0.317626\pi\)
\(860\) 0 0
\(861\) −27.7771 −0.946641
\(862\) 0 0
\(863\) −20.9443 −0.712951 −0.356476 0.934305i \(-0.616022\pi\)
−0.356476 + 0.934305i \(0.616022\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 5.41641 0.184057
\(867\) −24.8885 −0.845259
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 83.7771 2.83868
\(872\) 8.47214 0.286903
\(873\) −13.4164 −0.454077
\(874\) −2.00000 −0.0676510
\(875\) 0 0
\(876\) −12.9443 −0.437346
\(877\) 24.3607 0.822602 0.411301 0.911500i \(-0.365075\pi\)
0.411301 + 0.911500i \(0.365075\pi\)
\(878\) −26.8328 −0.905564
\(879\) 16.9443 0.571516
\(880\) 0 0
\(881\) 4.47214 0.150670 0.0753350 0.997158i \(-0.475997\pi\)
0.0753350 + 0.997158i \(0.475997\pi\)
\(882\) −9.00000 −0.303046
\(883\) −51.7771 −1.74244 −0.871219 0.490895i \(-0.836670\pi\)
−0.871219 + 0.490895i \(0.836670\pi\)
\(884\) −41.8885 −1.40886
\(885\) 0 0
\(886\) 8.94427 0.300489
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 8.47214 0.284306
\(889\) −61.6656 −2.06820
\(890\) 0 0
\(891\) 0 0
\(892\) 15.4164 0.516180
\(893\) 25.8885 0.866327
\(894\) 13.4164 0.448712
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) 6.47214 0.216098
\(898\) 17.0557 0.569157
\(899\) 0 0
\(900\) 0 0
\(901\) −57.8885 −1.92855
\(902\) 0 0
\(903\) 0 0
\(904\) −7.41641 −0.246666
\(905\) 0 0
\(906\) −0.944272 −0.0313713
\(907\) 38.8328 1.28942 0.644711 0.764426i \(-0.276978\pi\)
0.644711 + 0.764426i \(0.276978\pi\)
\(908\) −9.52786 −0.316193
\(909\) −8.47214 −0.281003
\(910\) 0 0
\(911\) 7.05573 0.233767 0.116883 0.993146i \(-0.462710\pi\)
0.116883 + 0.993146i \(0.462710\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 0 0
\(914\) −1.41641 −0.0468506
\(915\) 0 0
\(916\) −1.41641 −0.0467994
\(917\) −43.7771 −1.44565
\(918\) 6.47214 0.213612
\(919\) 17.4164 0.574514 0.287257 0.957854i \(-0.407257\pi\)
0.287257 + 0.957854i \(0.407257\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −30.3607 −0.999876
\(923\) 106.610 3.50911
\(924\) 0 0
\(925\) 0 0
\(926\) 11.4164 0.375166
\(927\) 0.944272 0.0310140
\(928\) −8.47214 −0.278111
\(929\) 6.94427 0.227834 0.113917 0.993490i \(-0.463660\pi\)
0.113917 + 0.993490i \(0.463660\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) 2.94427 0.0964428
\(933\) 15.5279 0.508359
\(934\) 17.5279 0.573529
\(935\) 0 0
\(936\) 6.47214 0.211548
\(937\) −33.4164 −1.09167 −0.545833 0.837894i \(-0.683787\pi\)
−0.545833 + 0.837894i \(0.683787\pi\)
\(938\) −51.7771 −1.69058
\(939\) 1.41641 0.0462227
\(940\) 0 0
\(941\) 27.5279 0.897383 0.448691 0.893687i \(-0.351890\pi\)
0.448691 + 0.893687i \(0.351890\pi\)
\(942\) 20.4721 0.667018
\(943\) −6.94427 −0.226137
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −3.52786 −0.114580
\(949\) −83.7771 −2.71952
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 25.8885 0.839053
\(953\) −32.3607 −1.04827 −0.524133 0.851637i \(-0.675610\pi\)
−0.524133 + 0.851637i \(0.675610\pi\)
\(954\) 8.94427 0.289581
\(955\) 0 0
\(956\) −3.52786 −0.114099
\(957\) 0 0
\(958\) −0.944272 −0.0305080
\(959\) 29.6656 0.957953
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 54.8328 1.76788
\(963\) −6.47214 −0.208562
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) −4.00000 −0.128698
\(967\) −28.5836 −0.919186 −0.459593 0.888130i \(-0.652005\pi\)
−0.459593 + 0.888130i \(0.652005\pi\)
\(968\) 11.0000 0.353553
\(969\) −12.9443 −0.415830
\(970\) 0 0
\(971\) −53.8885 −1.72937 −0.864683 0.502318i \(-0.832481\pi\)
−0.864683 + 0.502318i \(0.832481\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −4.58359 −0.146868
\(975\) 0 0
\(976\) 8.47214 0.271186
\(977\) −10.4721 −0.335033 −0.167517 0.985869i \(-0.553575\pi\)
−0.167517 + 0.985869i \(0.553575\pi\)
\(978\) −12.9443 −0.413912
\(979\) 0 0
\(980\) 0 0
\(981\) −8.47214 −0.270494
\(982\) 7.88854 0.251734
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −6.94427 −0.221375
\(985\) 0 0
\(986\) −54.8328 −1.74623
\(987\) 51.7771 1.64808
\(988\) −12.9443 −0.411812
\(989\) 0 0
\(990\) 0 0
\(991\) −53.6656 −1.70474 −0.852372 0.522935i \(-0.824837\pi\)
−0.852372 + 0.522935i \(0.824837\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −65.8885 −2.08986
\(995\) 0 0
\(996\) 10.4721 0.331822
\(997\) −19.4164 −0.614924 −0.307462 0.951560i \(-0.599480\pi\)
−0.307462 + 0.951560i \(0.599480\pi\)
\(998\) 24.0000 0.759707
\(999\) −8.47214 −0.268047
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 3450.2.a.bc.1.1 2
5.2 odd 4 690.2.d.a.139.1 4
5.3 odd 4 690.2.d.a.139.3 yes 4
5.4 even 2 3450.2.a.bn.1.2 2
15.2 even 4 2070.2.d.b.829.4 4
15.8 even 4 2070.2.d.b.829.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.d.a.139.1 4 5.2 odd 4
690.2.d.a.139.3 yes 4 5.3 odd 4
2070.2.d.b.829.2 4 15.8 even 4
2070.2.d.b.829.4 4 15.2 even 4
3450.2.a.bc.1.1 2 1.1 even 1 trivial
3450.2.a.bn.1.2 2 5.4 even 2