Properties

Label 3450.2.a.bo.1.2
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 1.2
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -0.622216 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -0.622216 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.42864 q^{11} +1.00000 q^{12} -1.37778 q^{13} +0.622216 q^{14} +1.00000 q^{16} -6.42864 q^{17} -1.00000 q^{18} -1.37778 q^{19} -0.622216 q^{21} -4.42864 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.37778 q^{26} +1.00000 q^{27} -0.622216 q^{28} -4.23506 q^{29} -4.00000 q^{31} -1.00000 q^{32} +4.42864 q^{33} +6.42864 q^{34} +1.00000 q^{36} -11.8064 q^{37} +1.37778 q^{38} -1.37778 q^{39} -10.8573 q^{41} +0.622216 q^{42} +1.05086 q^{43} +4.42864 q^{44} +1.00000 q^{46} +11.6128 q^{47} +1.00000 q^{48} -6.61285 q^{49} -6.42864 q^{51} -1.37778 q^{52} -3.37778 q^{53} -1.00000 q^{54} +0.622216 q^{56} -1.37778 q^{57} +4.23506 q^{58} +9.61285 q^{59} +8.66370 q^{61} +4.00000 q^{62} -0.622216 q^{63} +1.00000 q^{64} -4.42864 q^{66} -11.8064 q^{67} -6.42864 q^{68} -1.00000 q^{69} -6.99063 q^{71} -1.00000 q^{72} -2.75557 q^{73} +11.8064 q^{74} -1.37778 q^{76} -2.75557 q^{77} +1.37778 q^{78} +12.0415 q^{79} +1.00000 q^{81} +10.8573 q^{82} -2.19358 q^{83} -0.622216 q^{84} -1.05086 q^{86} -4.23506 q^{87} -4.42864 q^{88} +2.75557 q^{89} +0.857279 q^{91} -1.00000 q^{92} -4.00000 q^{93} -11.6128 q^{94} -1.00000 q^{96} +2.13335 q^{97} +6.61285 q^{98} +4.42864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{6} - 2q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{6} - 2q^{7} - 3q^{8} + 3q^{9} + 3q^{12} - 4q^{13} + 2q^{14} + 3q^{16} - 6q^{17} - 3q^{18} - 4q^{19} - 2q^{21} - 3q^{23} - 3q^{24} + 4q^{26} + 3q^{27} - 2q^{28} + 14q^{29} - 12q^{31} - 3q^{32} + 6q^{34} + 3q^{36} - 22q^{37} + 4q^{38} - 4q^{39} - 6q^{41} + 2q^{42} - 10q^{43} + 3q^{46} + 8q^{47} + 3q^{48} + 7q^{49} - 6q^{51} - 4q^{52} - 10q^{53} - 3q^{54} + 2q^{56} - 4q^{57} - 14q^{58} + 2q^{59} - 14q^{61} + 12q^{62} - 2q^{63} + 3q^{64} - 22q^{67} - 6q^{68} - 3q^{69} + 6q^{71} - 3q^{72} - 8q^{73} + 22q^{74} - 4q^{76} - 8q^{77} + 4q^{78} - 4q^{79} + 3q^{81} + 6q^{82} - 20q^{83} - 2q^{84} + 10q^{86} + 14q^{87} + 8q^{89} - 24q^{91} - 3q^{92} - 12q^{93} - 8q^{94} - 3q^{96} + 6q^{97} - 7q^{98} + O(q^{100}) \)

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
\(7\) −0.622216 −0.235175 −0.117588 0.993063i \(-0.537516\pi\)
−0.117588 + 0.993063i \(0.537516\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.42864 1.33529 0.667643 0.744482i \(-0.267303\pi\)
0.667643 + 0.744482i \(0.267303\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.37778 −0.382129 −0.191064 0.981578i \(-0.561194\pi\)
−0.191064 + 0.981578i \(0.561194\pi\)
\(14\) 0.622216 0.166294
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.42864 −1.55917 −0.779587 0.626294i \(-0.784571\pi\)
−0.779587 + 0.626294i \(0.784571\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.37778 −0.316085 −0.158043 0.987432i \(-0.550518\pi\)
−0.158043 + 0.987432i \(0.550518\pi\)
\(20\) 0 0
\(21\) −0.622216 −0.135779
\(22\) −4.42864 −0.944189
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 1.37778 0.270206
\(27\) 1.00000 0.192450
\(28\) −0.622216 −0.117588
\(29\) −4.23506 −0.786432 −0.393216 0.919446i \(-0.628637\pi\)
−0.393216 + 0.919446i \(0.628637\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.42864 0.770927
\(34\) 6.42864 1.10250
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −11.8064 −1.94096 −0.970482 0.241173i \(-0.922468\pi\)
−0.970482 + 0.241173i \(0.922468\pi\)
\(38\) 1.37778 0.223506
\(39\) −1.37778 −0.220622
\(40\) 0 0
\(41\) −10.8573 −1.69562 −0.847811 0.530298i \(-0.822080\pi\)
−0.847811 + 0.530298i \(0.822080\pi\)
\(42\) 0.622216 0.0960100
\(43\) 1.05086 0.160254 0.0801270 0.996785i \(-0.474467\pi\)
0.0801270 + 0.996785i \(0.474467\pi\)
\(44\) 4.42864 0.667643
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 11.6128 1.69391 0.846954 0.531666i \(-0.178434\pi\)
0.846954 + 0.531666i \(0.178434\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.61285 −0.944693
\(50\) 0 0
\(51\) −6.42864 −0.900190
\(52\) −1.37778 −0.191064
\(53\) −3.37778 −0.463974 −0.231987 0.972719i \(-0.574523\pi\)
−0.231987 + 0.972719i \(0.574523\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0.622216 0.0831471
\(57\) −1.37778 −0.182492
\(58\) 4.23506 0.556091
\(59\) 9.61285 1.25149 0.625743 0.780029i \(-0.284796\pi\)
0.625743 + 0.780029i \(0.284796\pi\)
\(60\) 0 0
\(61\) 8.66370 1.10927 0.554637 0.832093i \(-0.312857\pi\)
0.554637 + 0.832093i \(0.312857\pi\)
\(62\) 4.00000 0.508001
\(63\) −0.622216 −0.0783918
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.42864 −0.545128
\(67\) −11.8064 −1.44238 −0.721192 0.692735i \(-0.756406\pi\)
−0.721192 + 0.692735i \(0.756406\pi\)
\(68\) −6.42864 −0.779587
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −6.99063 −0.829635 −0.414818 0.909905i \(-0.636155\pi\)
−0.414818 + 0.909905i \(0.636155\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.75557 −0.322515 −0.161257 0.986912i \(-0.551555\pi\)
−0.161257 + 0.986912i \(0.551555\pi\)
\(74\) 11.8064 1.37247
\(75\) 0 0
\(76\) −1.37778 −0.158043
\(77\) −2.75557 −0.314026
\(78\) 1.37778 0.156003
\(79\) 12.0415 1.35477 0.677387 0.735627i \(-0.263112\pi\)
0.677387 + 0.735627i \(0.263112\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 10.8573 1.19899
\(83\) −2.19358 −0.240776 −0.120388 0.992727i \(-0.538414\pi\)
−0.120388 + 0.992727i \(0.538414\pi\)
\(84\) −0.622216 −0.0678893
\(85\) 0 0
\(86\) −1.05086 −0.113317
\(87\) −4.23506 −0.454046
\(88\) −4.42864 −0.472095
\(89\) 2.75557 0.292090 0.146045 0.989278i \(-0.453346\pi\)
0.146045 + 0.989278i \(0.453346\pi\)
\(90\) 0 0
\(91\) 0.857279 0.0898673
\(92\) −1.00000 −0.104257
\(93\) −4.00000 −0.414781
\(94\) −11.6128 −1.19777
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 2.13335 0.216609 0.108305 0.994118i \(-0.465458\pi\)
0.108305 + 0.994118i \(0.465458\pi\)
\(98\) 6.61285 0.667998
\(99\) 4.42864 0.445095
\(100\) 0 0
\(101\) 16.2351 1.61545 0.807725 0.589560i \(-0.200699\pi\)
0.807725 + 0.589560i \(0.200699\pi\)
\(102\) 6.42864 0.636530
\(103\) −12.2351 −1.20556 −0.602778 0.797909i \(-0.705940\pi\)
−0.602778 + 0.797909i \(0.705940\pi\)
\(104\) 1.37778 0.135103
\(105\) 0 0
\(106\) 3.37778 0.328079
\(107\) 10.6637 1.03090 0.515450 0.856920i \(-0.327625\pi\)
0.515450 + 0.856920i \(0.327625\pi\)
\(108\) 1.00000 0.0962250
\(109\) −15.1526 −1.45135 −0.725676 0.688036i \(-0.758473\pi\)
−0.725676 + 0.688036i \(0.758473\pi\)
\(110\) 0 0
\(111\) −11.8064 −1.12062
\(112\) −0.622216 −0.0587939
\(113\) −14.4286 −1.35733 −0.678666 0.734447i \(-0.737442\pi\)
−0.678666 + 0.734447i \(0.737442\pi\)
\(114\) 1.37778 0.129041
\(115\) 0 0
\(116\) −4.23506 −0.393216
\(117\) −1.37778 −0.127376
\(118\) −9.61285 −0.884934
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 8.61285 0.782986
\(122\) −8.66370 −0.784375
\(123\) −10.8573 −0.978968
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 0.622216 0.0554314
\(127\) 4.99063 0.442847 0.221423 0.975178i \(-0.428930\pi\)
0.221423 + 0.975178i \(0.428930\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.05086 0.0925226
\(130\) 0 0
\(131\) 0.755569 0.0660143 0.0330072 0.999455i \(-0.489492\pi\)
0.0330072 + 0.999455i \(0.489492\pi\)
\(132\) 4.42864 0.385464
\(133\) 0.857279 0.0743355
\(134\) 11.8064 1.01992
\(135\) 0 0
\(136\) 6.42864 0.551251
\(137\) −12.9175 −1.10362 −0.551808 0.833971i \(-0.686062\pi\)
−0.551808 + 0.833971i \(0.686062\pi\)
\(138\) 1.00000 0.0851257
\(139\) −14.1017 −1.19609 −0.598046 0.801462i \(-0.704056\pi\)
−0.598046 + 0.801462i \(0.704056\pi\)
\(140\) 0 0
\(141\) 11.6128 0.977978
\(142\) 6.99063 0.586641
\(143\) −6.10171 −0.510251
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 2.75557 0.228052
\(147\) −6.61285 −0.545418
\(148\) −11.8064 −0.970482
\(149\) 12.4286 1.01819 0.509097 0.860709i \(-0.329979\pi\)
0.509097 + 0.860709i \(0.329979\pi\)
\(150\) 0 0
\(151\) −4.85728 −0.395280 −0.197640 0.980275i \(-0.563328\pi\)
−0.197640 + 0.980275i \(0.563328\pi\)
\(152\) 1.37778 0.111753
\(153\) −6.42864 −0.519725
\(154\) 2.75557 0.222050
\(155\) 0 0
\(156\) −1.37778 −0.110311
\(157\) 9.90813 0.790755 0.395378 0.918519i \(-0.370614\pi\)
0.395378 + 0.918519i \(0.370614\pi\)
\(158\) −12.0415 −0.957969
\(159\) −3.37778 −0.267876
\(160\) 0 0
\(161\) 0.622216 0.0490375
\(162\) −1.00000 −0.0785674
\(163\) −10.1017 −0.791227 −0.395614 0.918417i \(-0.629468\pi\)
−0.395614 + 0.918417i \(0.629468\pi\)
\(164\) −10.8573 −0.847811
\(165\) 0 0
\(166\) 2.19358 0.170255
\(167\) −1.89829 −0.146894 −0.0734470 0.997299i \(-0.523400\pi\)
−0.0734470 + 0.997299i \(0.523400\pi\)
\(168\) 0.622216 0.0480050
\(169\) −11.1017 −0.853978
\(170\) 0 0
\(171\) −1.37778 −0.105362
\(172\) 1.05086 0.0801270
\(173\) −21.2257 −1.61376 −0.806880 0.590716i \(-0.798846\pi\)
−0.806880 + 0.590716i \(0.798846\pi\)
\(174\) 4.23506 0.321059
\(175\) 0 0
\(176\) 4.42864 0.333821
\(177\) 9.61285 0.722546
\(178\) −2.75557 −0.206539
\(179\) 0.488863 0.0365393 0.0182697 0.999833i \(-0.494184\pi\)
0.0182697 + 0.999833i \(0.494184\pi\)
\(180\) 0 0
\(181\) −20.9304 −1.55575 −0.777873 0.628422i \(-0.783701\pi\)
−0.777873 + 0.628422i \(0.783701\pi\)
\(182\) −0.857279 −0.0635457
\(183\) 8.66370 0.640439
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) −28.4701 −2.08194
\(188\) 11.6128 0.846954
\(189\) −0.622216 −0.0452595
\(190\) 0 0
\(191\) 18.9590 1.37182 0.685912 0.727684i \(-0.259403\pi\)
0.685912 + 0.727684i \(0.259403\pi\)
\(192\) 1.00000 0.0721688
\(193\) 1.24443 0.0895761 0.0447881 0.998997i \(-0.485739\pi\)
0.0447881 + 0.998997i \(0.485739\pi\)
\(194\) −2.13335 −0.153166
\(195\) 0 0
\(196\) −6.61285 −0.472346
\(197\) 15.2444 1.08612 0.543060 0.839694i \(-0.317265\pi\)
0.543060 + 0.839694i \(0.317265\pi\)
\(198\) −4.42864 −0.314730
\(199\) −14.5303 −1.03003 −0.515015 0.857181i \(-0.672214\pi\)
−0.515015 + 0.857181i \(0.672214\pi\)
\(200\) 0 0
\(201\) −11.8064 −0.832761
\(202\) −16.2351 −1.14230
\(203\) 2.63512 0.184949
\(204\) −6.42864 −0.450095
\(205\) 0 0
\(206\) 12.2351 0.852457
\(207\) −1.00000 −0.0695048
\(208\) −1.37778 −0.0955322
\(209\) −6.10171 −0.422064
\(210\) 0 0
\(211\) 0.266706 0.0183608 0.00918041 0.999958i \(-0.497078\pi\)
0.00918041 + 0.999958i \(0.497078\pi\)
\(212\) −3.37778 −0.231987
\(213\) −6.99063 −0.478990
\(214\) −10.6637 −0.728956
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 2.48886 0.168955
\(218\) 15.1526 1.02626
\(219\) −2.75557 −0.186204
\(220\) 0 0
\(221\) 8.85728 0.595805
\(222\) 11.8064 0.792395
\(223\) 0.990632 0.0663376 0.0331688 0.999450i \(-0.489440\pi\)
0.0331688 + 0.999450i \(0.489440\pi\)
\(224\) 0.622216 0.0415735
\(225\) 0 0
\(226\) 14.4286 0.959779
\(227\) 17.1526 1.13846 0.569228 0.822180i \(-0.307242\pi\)
0.569228 + 0.822180i \(0.307242\pi\)
\(228\) −1.37778 −0.0912460
\(229\) 7.15257 0.472655 0.236327 0.971673i \(-0.424056\pi\)
0.236327 + 0.971673i \(0.424056\pi\)
\(230\) 0 0
\(231\) −2.75557 −0.181303
\(232\) 4.23506 0.278046
\(233\) 3.71456 0.243349 0.121674 0.992570i \(-0.461174\pi\)
0.121674 + 0.992570i \(0.461174\pi\)
\(234\) 1.37778 0.0900686
\(235\) 0 0
\(236\) 9.61285 0.625743
\(237\) 12.0415 0.782179
\(238\) −4.00000 −0.259281
\(239\) 27.8479 1.80133 0.900666 0.434512i \(-0.143079\pi\)
0.900666 + 0.434512i \(0.143079\pi\)
\(240\) 0 0
\(241\) −7.12399 −0.458896 −0.229448 0.973321i \(-0.573692\pi\)
−0.229448 + 0.973321i \(0.573692\pi\)
\(242\) −8.61285 −0.553655
\(243\) 1.00000 0.0641500
\(244\) 8.66370 0.554637
\(245\) 0 0
\(246\) 10.8573 0.692235
\(247\) 1.89829 0.120785
\(248\) 4.00000 0.254000
\(249\) −2.19358 −0.139012
\(250\) 0 0
\(251\) −22.4099 −1.41450 −0.707250 0.706963i \(-0.750065\pi\)
−0.707250 + 0.706963i \(0.750065\pi\)
\(252\) −0.622216 −0.0391959
\(253\) −4.42864 −0.278426
\(254\) −4.99063 −0.313140
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −11.7146 −0.730734 −0.365367 0.930864i \(-0.619057\pi\)
−0.365367 + 0.930864i \(0.619057\pi\)
\(258\) −1.05086 −0.0654234
\(259\) 7.34614 0.456467
\(260\) 0 0
\(261\) −4.23506 −0.262144
\(262\) −0.755569 −0.0466792
\(263\) −8.47013 −0.522290 −0.261145 0.965300i \(-0.584100\pi\)
−0.261145 + 0.965300i \(0.584100\pi\)
\(264\) −4.42864 −0.272564
\(265\) 0 0
\(266\) −0.857279 −0.0525631
\(267\) 2.75557 0.168638
\(268\) −11.8064 −0.721192
\(269\) 28.8256 1.75753 0.878765 0.477255i \(-0.158368\pi\)
0.878765 + 0.477255i \(0.158368\pi\)
\(270\) 0 0
\(271\) −13.8350 −0.840417 −0.420208 0.907428i \(-0.638043\pi\)
−0.420208 + 0.907428i \(0.638043\pi\)
\(272\) −6.42864 −0.389794
\(273\) 0.857279 0.0518849
\(274\) 12.9175 0.780375
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 2.88892 0.173578 0.0867892 0.996227i \(-0.472339\pi\)
0.0867892 + 0.996227i \(0.472339\pi\)
\(278\) 14.1017 0.845764
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 5.51114 0.328767 0.164383 0.986397i \(-0.447437\pi\)
0.164383 + 0.986397i \(0.447437\pi\)
\(282\) −11.6128 −0.691535
\(283\) 8.19358 0.487058 0.243529 0.969894i \(-0.421695\pi\)
0.243529 + 0.969894i \(0.421695\pi\)
\(284\) −6.99063 −0.414818
\(285\) 0 0
\(286\) 6.10171 0.360802
\(287\) 6.75557 0.398769
\(288\) −1.00000 −0.0589256
\(289\) 24.3274 1.43102
\(290\) 0 0
\(291\) 2.13335 0.125059
\(292\) −2.75557 −0.161257
\(293\) 12.6222 0.737398 0.368699 0.929549i \(-0.379803\pi\)
0.368699 + 0.929549i \(0.379803\pi\)
\(294\) 6.61285 0.385669
\(295\) 0 0
\(296\) 11.8064 0.686234
\(297\) 4.42864 0.256976
\(298\) −12.4286 −0.719972
\(299\) 1.37778 0.0796793
\(300\) 0 0
\(301\) −0.653858 −0.0376878
\(302\) 4.85728 0.279505
\(303\) 16.2351 0.932680
\(304\) −1.37778 −0.0790214
\(305\) 0 0
\(306\) 6.42864 0.367501
\(307\) −0.653858 −0.0373177 −0.0186588 0.999826i \(-0.505940\pi\)
−0.0186588 + 0.999826i \(0.505940\pi\)
\(308\) −2.75557 −0.157013
\(309\) −12.2351 −0.696028
\(310\) 0 0
\(311\) −13.0923 −0.742399 −0.371199 0.928553i \(-0.621053\pi\)
−0.371199 + 0.928553i \(0.621053\pi\)
\(312\) 1.37778 0.0780017
\(313\) 11.2573 0.636302 0.318151 0.948040i \(-0.396938\pi\)
0.318151 + 0.948040i \(0.396938\pi\)
\(314\) −9.90813 −0.559148
\(315\) 0 0
\(316\) 12.0415 0.677387
\(317\) −22.4701 −1.26205 −0.631024 0.775763i \(-0.717365\pi\)
−0.631024 + 0.775763i \(0.717365\pi\)
\(318\) 3.37778 0.189417
\(319\) −18.7556 −1.05011
\(320\) 0 0
\(321\) 10.6637 0.595190
\(322\) −0.622216 −0.0346747
\(323\) 8.85728 0.492832
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 10.1017 0.559482
\(327\) −15.1526 −0.837939
\(328\) 10.8573 0.599493
\(329\) −7.22570 −0.398365
\(330\) 0 0
\(331\) −29.4479 −1.61860 −0.809300 0.587395i \(-0.800153\pi\)
−0.809300 + 0.587395i \(0.800153\pi\)
\(332\) −2.19358 −0.120388
\(333\) −11.8064 −0.646988
\(334\) 1.89829 0.103870
\(335\) 0 0
\(336\) −0.622216 −0.0339446
\(337\) −24.6222 −1.34126 −0.670629 0.741793i \(-0.733976\pi\)
−0.670629 + 0.741793i \(0.733976\pi\)
\(338\) 11.1017 0.603853
\(339\) −14.4286 −0.783656
\(340\) 0 0
\(341\) −17.7146 −0.959297
\(342\) 1.37778 0.0745020
\(343\) 8.47013 0.457344
\(344\) −1.05086 −0.0566583
\(345\) 0 0
\(346\) 21.2257 1.14110
\(347\) 26.1017 1.40121 0.700607 0.713548i \(-0.252913\pi\)
0.700607 + 0.713548i \(0.252913\pi\)
\(348\) −4.23506 −0.227023
\(349\) −21.2257 −1.13619 −0.568093 0.822965i \(-0.692319\pi\)
−0.568093 + 0.822965i \(0.692319\pi\)
\(350\) 0 0
\(351\) −1.37778 −0.0735407
\(352\) −4.42864 −0.236047
\(353\) −4.28544 −0.228091 −0.114046 0.993476i \(-0.536381\pi\)
−0.114046 + 0.993476i \(0.536381\pi\)
\(354\) −9.61285 −0.510917
\(355\) 0 0
\(356\) 2.75557 0.146045
\(357\) 4.00000 0.211702
\(358\) −0.488863 −0.0258372
\(359\) −4.65386 −0.245621 −0.122811 0.992430i \(-0.539191\pi\)
−0.122811 + 0.992430i \(0.539191\pi\)
\(360\) 0 0
\(361\) −17.1017 −0.900090
\(362\) 20.9304 1.10008
\(363\) 8.61285 0.452057
\(364\) 0.857279 0.0449336
\(365\) 0 0
\(366\) −8.66370 −0.452859
\(367\) 4.88892 0.255200 0.127600 0.991826i \(-0.459273\pi\)
0.127600 + 0.991826i \(0.459273\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −10.8573 −0.565207
\(370\) 0 0
\(371\) 2.10171 0.109115
\(372\) −4.00000 −0.207390
\(373\) 23.4193 1.21260 0.606302 0.795235i \(-0.292652\pi\)
0.606302 + 0.795235i \(0.292652\pi\)
\(374\) 28.4701 1.47216
\(375\) 0 0
\(376\) −11.6128 −0.598887
\(377\) 5.83500 0.300518
\(378\) 0.622216 0.0320033
\(379\) −4.90766 −0.252089 −0.126045 0.992025i \(-0.540228\pi\)
−0.126045 + 0.992025i \(0.540228\pi\)
\(380\) 0 0
\(381\) 4.99063 0.255678
\(382\) −18.9590 −0.970026
\(383\) −19.8796 −1.01580 −0.507899 0.861417i \(-0.669578\pi\)
−0.507899 + 0.861417i \(0.669578\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −1.24443 −0.0633399
\(387\) 1.05086 0.0534180
\(388\) 2.13335 0.108305
\(389\) −0.161933 −0.00821034 −0.00410517 0.999992i \(-0.501307\pi\)
−0.00410517 + 0.999992i \(0.501307\pi\)
\(390\) 0 0
\(391\) 6.42864 0.325110
\(392\) 6.61285 0.333999
\(393\) 0.755569 0.0381134
\(394\) −15.2444 −0.768003
\(395\) 0 0
\(396\) 4.42864 0.222548
\(397\) 1.76494 0.0885796 0.0442898 0.999019i \(-0.485898\pi\)
0.0442898 + 0.999019i \(0.485898\pi\)
\(398\) 14.5303 0.728341
\(399\) 0.857279 0.0429176
\(400\) 0 0
\(401\) −31.3461 −1.56535 −0.782676 0.622430i \(-0.786146\pi\)
−0.782676 + 0.622430i \(0.786146\pi\)
\(402\) 11.8064 0.588851
\(403\) 5.51114 0.274529
\(404\) 16.2351 0.807725
\(405\) 0 0
\(406\) −2.63512 −0.130779
\(407\) −52.2864 −2.59174
\(408\) 6.42864 0.318265
\(409\) −3.51114 −0.173615 −0.0868073 0.996225i \(-0.527666\pi\)
−0.0868073 + 0.996225i \(0.527666\pi\)
\(410\) 0 0
\(411\) −12.9175 −0.637173
\(412\) −12.2351 −0.602778
\(413\) −5.98126 −0.294319
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 1.37778 0.0675514
\(417\) −14.1017 −0.690564
\(418\) 6.10171 0.298444
\(419\) −19.7748 −0.966061 −0.483031 0.875603i \(-0.660464\pi\)
−0.483031 + 0.875603i \(0.660464\pi\)
\(420\) 0 0
\(421\) 11.8064 0.575410 0.287705 0.957719i \(-0.407108\pi\)
0.287705 + 0.957719i \(0.407108\pi\)
\(422\) −0.266706 −0.0129831
\(423\) 11.6128 0.564636
\(424\) 3.37778 0.164040
\(425\) 0 0
\(426\) 6.99063 0.338697
\(427\) −5.39069 −0.260874
\(428\) 10.6637 0.515450
\(429\) −6.10171 −0.294593
\(430\) 0 0
\(431\) 24.9403 1.20133 0.600665 0.799501i \(-0.294903\pi\)
0.600665 + 0.799501i \(0.294903\pi\)
\(432\) 1.00000 0.0481125
\(433\) −32.0513 −1.54029 −0.770144 0.637870i \(-0.779816\pi\)
−0.770144 + 0.637870i \(0.779816\pi\)
\(434\) −2.48886 −0.119469
\(435\) 0 0
\(436\) −15.1526 −0.725676
\(437\) 1.37778 0.0659084
\(438\) 2.75557 0.131666
\(439\) 0.470127 0.0224379 0.0112190 0.999937i \(-0.496429\pi\)
0.0112190 + 0.999937i \(0.496429\pi\)
\(440\) 0 0
\(441\) −6.61285 −0.314898
\(442\) −8.85728 −0.421298
\(443\) −0.653858 −0.0310658 −0.0155329 0.999879i \(-0.504944\pi\)
−0.0155329 + 0.999879i \(0.504944\pi\)
\(444\) −11.8064 −0.560308
\(445\) 0 0
\(446\) −0.990632 −0.0469078
\(447\) 12.4286 0.587854
\(448\) −0.622216 −0.0293969
\(449\) 17.1427 0.809015 0.404508 0.914535i \(-0.367443\pi\)
0.404508 + 0.914535i \(0.367443\pi\)
\(450\) 0 0
\(451\) −48.0830 −2.26414
\(452\) −14.4286 −0.678666
\(453\) −4.85728 −0.228215
\(454\) −17.1526 −0.805010
\(455\) 0 0
\(456\) 1.37778 0.0645207
\(457\) 1.00937 0.0472162 0.0236081 0.999721i \(-0.492485\pi\)
0.0236081 + 0.999721i \(0.492485\pi\)
\(458\) −7.15257 −0.334217
\(459\) −6.42864 −0.300063
\(460\) 0 0
\(461\) 20.3180 0.946305 0.473153 0.880980i \(-0.343116\pi\)
0.473153 + 0.880980i \(0.343116\pi\)
\(462\) 2.75557 0.128201
\(463\) 12.4572 0.578936 0.289468 0.957188i \(-0.406522\pi\)
0.289468 + 0.957188i \(0.406522\pi\)
\(464\) −4.23506 −0.196608
\(465\) 0 0
\(466\) −3.71456 −0.172074
\(467\) −15.7877 −0.730567 −0.365284 0.930896i \(-0.619028\pi\)
−0.365284 + 0.930896i \(0.619028\pi\)
\(468\) −1.37778 −0.0636881
\(469\) 7.34614 0.339213
\(470\) 0 0
\(471\) 9.90813 0.456543
\(472\) −9.61285 −0.442467
\(473\) 4.65386 0.213985
\(474\) −12.0415 −0.553084
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) −3.37778 −0.154658
\(478\) −27.8479 −1.27373
\(479\) 5.12399 0.234121 0.117060 0.993125i \(-0.462653\pi\)
0.117060 + 0.993125i \(0.462653\pi\)
\(480\) 0 0
\(481\) 16.2667 0.741698
\(482\) 7.12399 0.324489
\(483\) 0.622216 0.0283118
\(484\) 8.61285 0.391493
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −23.2128 −1.05187 −0.525936 0.850524i \(-0.676285\pi\)
−0.525936 + 0.850524i \(0.676285\pi\)
\(488\) −8.66370 −0.392187
\(489\) −10.1017 −0.456815
\(490\) 0 0
\(491\) −40.1847 −1.81351 −0.906755 0.421658i \(-0.861448\pi\)
−0.906755 + 0.421658i \(0.861448\pi\)
\(492\) −10.8573 −0.489484
\(493\) 27.2257 1.22618
\(494\) −1.89829 −0.0854081
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 4.34968 0.195110
\(498\) 2.19358 0.0982965
\(499\) 30.5718 1.36858 0.684292 0.729208i \(-0.260112\pi\)
0.684292 + 0.729208i \(0.260112\pi\)
\(500\) 0 0
\(501\) −1.89829 −0.0848093
\(502\) 22.4099 1.00020
\(503\) −23.4291 −1.04465 −0.522326 0.852746i \(-0.674936\pi\)
−0.522326 + 0.852746i \(0.674936\pi\)
\(504\) 0.622216 0.0277157
\(505\) 0 0
\(506\) 4.42864 0.196877
\(507\) −11.1017 −0.493044
\(508\) 4.99063 0.221423
\(509\) −17.2128 −0.762943 −0.381472 0.924381i \(-0.624583\pi\)
−0.381472 + 0.924381i \(0.624583\pi\)
\(510\) 0 0
\(511\) 1.71456 0.0758476
\(512\) −1.00000 −0.0441942
\(513\) −1.37778 −0.0608307
\(514\) 11.7146 0.516707
\(515\) 0 0
\(516\) 1.05086 0.0462613
\(517\) 51.4291 2.26185
\(518\) −7.34614 −0.322771
\(519\) −21.2257 −0.931705
\(520\) 0 0
\(521\) 22.3684 0.979978 0.489989 0.871729i \(-0.337001\pi\)
0.489989 + 0.871729i \(0.337001\pi\)
\(522\) 4.23506 0.185364
\(523\) 30.2953 1.32472 0.662360 0.749186i \(-0.269555\pi\)
0.662360 + 0.749186i \(0.269555\pi\)
\(524\) 0.755569 0.0330072
\(525\) 0 0
\(526\) 8.47013 0.369315
\(527\) 25.7146 1.12014
\(528\) 4.42864 0.192732
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 9.61285 0.417162
\(532\) 0.857279 0.0371678
\(533\) 14.9590 0.647946
\(534\) −2.75557 −0.119245
\(535\) 0 0
\(536\) 11.8064 0.509960
\(537\) 0.488863 0.0210960
\(538\) −28.8256 −1.24276
\(539\) −29.2859 −1.26143
\(540\) 0 0
\(541\) 31.4479 1.35205 0.676024 0.736879i \(-0.263701\pi\)
0.676024 + 0.736879i \(0.263701\pi\)
\(542\) 13.8350 0.594264
\(543\) −20.9304 −0.898210
\(544\) 6.42864 0.275626
\(545\) 0 0
\(546\) −0.857279 −0.0366882
\(547\) 20.8573 0.891793 0.445896 0.895085i \(-0.352885\pi\)
0.445896 + 0.895085i \(0.352885\pi\)
\(548\) −12.9175 −0.551808
\(549\) 8.66370 0.369758
\(550\) 0 0
\(551\) 5.83500 0.248580
\(552\) 1.00000 0.0425628
\(553\) −7.49240 −0.318609
\(554\) −2.88892 −0.122739
\(555\) 0 0
\(556\) −14.1017 −0.598046
\(557\) 36.3180 1.53884 0.769422 0.638740i \(-0.220544\pi\)
0.769422 + 0.638740i \(0.220544\pi\)
\(558\) 4.00000 0.169334
\(559\) −1.44785 −0.0612376
\(560\) 0 0
\(561\) −28.4701 −1.20201
\(562\) −5.51114 −0.232473
\(563\) 2.58073 0.108765 0.0543824 0.998520i \(-0.482681\pi\)
0.0543824 + 0.998520i \(0.482681\pi\)
\(564\) 11.6128 0.488989
\(565\) 0 0
\(566\) −8.19358 −0.344402
\(567\) −0.622216 −0.0261306
\(568\) 6.99063 0.293320
\(569\) −36.3497 −1.52386 −0.761929 0.647661i \(-0.775747\pi\)
−0.761929 + 0.647661i \(0.775747\pi\)
\(570\) 0 0
\(571\) −45.9309 −1.92215 −0.961074 0.276292i \(-0.910894\pi\)
−0.961074 + 0.276292i \(0.910894\pi\)
\(572\) −6.10171 −0.255125
\(573\) 18.9590 0.792023
\(574\) −6.75557 −0.281972
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) −24.3274 −1.01189
\(579\) 1.24443 0.0517168
\(580\) 0 0
\(581\) 1.36488 0.0566247
\(582\) −2.13335 −0.0884303
\(583\) −14.9590 −0.619538
\(584\) 2.75557 0.114026
\(585\) 0 0
\(586\) −12.6222 −0.521419
\(587\) 42.1847 1.74115 0.870574 0.492037i \(-0.163748\pi\)
0.870574 + 0.492037i \(0.163748\pi\)
\(588\) −6.61285 −0.272709
\(589\) 5.51114 0.227082
\(590\) 0 0
\(591\) 15.2444 0.627072
\(592\) −11.8064 −0.485241
\(593\) 18.7368 0.769430 0.384715 0.923036i \(-0.374300\pi\)
0.384715 + 0.923036i \(0.374300\pi\)
\(594\) −4.42864 −0.181709
\(595\) 0 0
\(596\) 12.4286 0.509097
\(597\) −14.5303 −0.594688
\(598\) −1.37778 −0.0563418
\(599\) −41.5625 −1.69820 −0.849098 0.528235i \(-0.822854\pi\)
−0.849098 + 0.528235i \(0.822854\pi\)
\(600\) 0 0
\(601\) 23.7146 0.967337 0.483668 0.875251i \(-0.339304\pi\)
0.483668 + 0.875251i \(0.339304\pi\)
\(602\) 0.653858 0.0266493
\(603\) −11.8064 −0.480795
\(604\) −4.85728 −0.197640
\(605\) 0 0
\(606\) −16.2351 −0.659504
\(607\) 2.74266 0.111321 0.0556606 0.998450i \(-0.482274\pi\)
0.0556606 + 0.998450i \(0.482274\pi\)
\(608\) 1.37778 0.0558765
\(609\) 2.63512 0.106781
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) −6.42864 −0.259862
\(613\) −24.0731 −0.972305 −0.486152 0.873874i \(-0.661600\pi\)
−0.486152 + 0.873874i \(0.661600\pi\)
\(614\) 0.653858 0.0263876
\(615\) 0 0
\(616\) 2.75557 0.111025
\(617\) −39.4893 −1.58978 −0.794890 0.606753i \(-0.792472\pi\)
−0.794890 + 0.606753i \(0.792472\pi\)
\(618\) 12.2351 0.492166
\(619\) −22.6222 −0.909264 −0.454632 0.890679i \(-0.650229\pi\)
−0.454632 + 0.890679i \(0.650229\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 13.0923 0.524955
\(623\) −1.71456 −0.0686923
\(624\) −1.37778 −0.0551555
\(625\) 0 0
\(626\) −11.2573 −0.449934
\(627\) −6.10171 −0.243679
\(628\) 9.90813 0.395378
\(629\) 75.8992 3.02630
\(630\) 0 0
\(631\) 5.93978 0.236459 0.118229 0.992986i \(-0.462278\pi\)
0.118229 + 0.992986i \(0.462278\pi\)
\(632\) −12.0415 −0.478985
\(633\) 0.266706 0.0106006
\(634\) 22.4701 0.892403
\(635\) 0 0
\(636\) −3.37778 −0.133938
\(637\) 9.11108 0.360994
\(638\) 18.7556 0.742540
\(639\) −6.99063 −0.276545
\(640\) 0 0
\(641\) 19.8163 0.782696 0.391348 0.920243i \(-0.372009\pi\)
0.391348 + 0.920243i \(0.372009\pi\)
\(642\) −10.6637 −0.420863
\(643\) 44.7467 1.76464 0.882318 0.470653i \(-0.155982\pi\)
0.882318 + 0.470653i \(0.155982\pi\)
\(644\) 0.622216 0.0245187
\(645\) 0 0
\(646\) −8.85728 −0.348485
\(647\) 32.9403 1.29501 0.647507 0.762059i \(-0.275811\pi\)
0.647507 + 0.762059i \(0.275811\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 42.5718 1.67109
\(650\) 0 0
\(651\) 2.48886 0.0975462
\(652\) −10.1017 −0.395614
\(653\) 37.2257 1.45675 0.728377 0.685177i \(-0.240275\pi\)
0.728377 + 0.685177i \(0.240275\pi\)
\(654\) 15.1526 0.592512
\(655\) 0 0
\(656\) −10.8573 −0.423906
\(657\) −2.75557 −0.107505
\(658\) 7.22570 0.281687
\(659\) 24.6953 0.961994 0.480997 0.876722i \(-0.340275\pi\)
0.480997 + 0.876722i \(0.340275\pi\)
\(660\) 0 0
\(661\) −14.8287 −0.576770 −0.288385 0.957515i \(-0.593118\pi\)
−0.288385 + 0.957515i \(0.593118\pi\)
\(662\) 29.4479 1.14452
\(663\) 8.85728 0.343988
\(664\) 2.19358 0.0851273
\(665\) 0 0
\(666\) 11.8064 0.457490
\(667\) 4.23506 0.163982
\(668\) −1.89829 −0.0734470
\(669\) 0.990632 0.0383000
\(670\) 0 0
\(671\) 38.3684 1.48120
\(672\) 0.622216 0.0240025
\(673\) 8.77430 0.338225 0.169112 0.985597i \(-0.445910\pi\)
0.169112 + 0.985597i \(0.445910\pi\)
\(674\) 24.6222 0.948412
\(675\) 0 0
\(676\) −11.1017 −0.426989
\(677\) −12.8256 −0.492929 −0.246465 0.969152i \(-0.579269\pi\)
−0.246465 + 0.969152i \(0.579269\pi\)
\(678\) 14.4286 0.554129
\(679\) −1.32741 −0.0509412
\(680\) 0 0
\(681\) 17.1526 0.657288
\(682\) 17.7146 0.678325
\(683\) 8.47013 0.324100 0.162050 0.986783i \(-0.448189\pi\)
0.162050 + 0.986783i \(0.448189\pi\)
\(684\) −1.37778 −0.0526809
\(685\) 0 0
\(686\) −8.47013 −0.323391
\(687\) 7.15257 0.272887
\(688\) 1.05086 0.0400635
\(689\) 4.65386 0.177298
\(690\) 0 0
\(691\) 25.7975 0.981384 0.490692 0.871333i \(-0.336744\pi\)
0.490692 + 0.871333i \(0.336744\pi\)
\(692\) −21.2257 −0.806880
\(693\) −2.75557 −0.104675
\(694\) −26.1017 −0.990807
\(695\) 0 0
\(696\) 4.23506 0.160530
\(697\) 69.7975 2.64377
\(698\) 21.2257 0.803404
\(699\) 3.71456 0.140497
\(700\) 0 0
\(701\) 7.45091 0.281417 0.140709 0.990051i \(-0.455062\pi\)
0.140709 + 0.990051i \(0.455062\pi\)
\(702\) 1.37778 0.0520011
\(703\) 16.2667 0.613510
\(704\) 4.42864 0.166911
\(705\) 0 0
\(706\) 4.28544 0.161285
\(707\) −10.1017 −0.379914
\(708\) 9.61285 0.361273
\(709\) 13.2543 0.497775 0.248887 0.968532i \(-0.419935\pi\)
0.248887 + 0.968532i \(0.419935\pi\)
\(710\) 0 0
\(711\) 12.0415 0.451591
\(712\) −2.75557 −0.103269
\(713\) 4.00000 0.149801
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) 0.488863 0.0182697
\(717\) 27.8479 1.04000
\(718\) 4.65386 0.173680
\(719\) −2.33677 −0.0871469 −0.0435735 0.999050i \(-0.513874\pi\)
−0.0435735 + 0.999050i \(0.513874\pi\)
\(720\) 0 0
\(721\) 7.61285 0.283517
\(722\) 17.1017 0.636460
\(723\) −7.12399 −0.264944
\(724\) −20.9304 −0.777873
\(725\) 0 0
\(726\) −8.61285 −0.319653
\(727\) −3.37778 −0.125275 −0.0626375 0.998036i \(-0.519951\pi\)
−0.0626375 + 0.998036i \(0.519951\pi\)
\(728\) −0.857279 −0.0317729
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.75557 −0.249864
\(732\) 8.66370 0.320220
\(733\) 21.0509 0.777531 0.388766 0.921337i \(-0.372902\pi\)
0.388766 + 0.921337i \(0.372902\pi\)
\(734\) −4.88892 −0.180453
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −52.2864 −1.92599
\(738\) 10.8573 0.399662
\(739\) −22.6351 −0.832646 −0.416323 0.909217i \(-0.636682\pi\)
−0.416323 + 0.909217i \(0.636682\pi\)
\(740\) 0 0
\(741\) 1.89829 0.0697354
\(742\) −2.10171 −0.0771562
\(743\) −3.87955 −0.142327 −0.0711635 0.997465i \(-0.522671\pi\)
−0.0711635 + 0.997465i \(0.522671\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) −23.4193 −0.857440
\(747\) −2.19358 −0.0802588
\(748\) −28.4701 −1.04097
\(749\) −6.63512 −0.242442
\(750\) 0 0
\(751\) 19.7748 0.721592 0.360796 0.932645i \(-0.382505\pi\)
0.360796 + 0.932645i \(0.382505\pi\)
\(752\) 11.6128 0.423477
\(753\) −22.4099 −0.816662
\(754\) −5.83500 −0.212498
\(755\) 0 0
\(756\) −0.622216 −0.0226298
\(757\) 8.19358 0.297801 0.148900 0.988852i \(-0.452427\pi\)
0.148900 + 0.988852i \(0.452427\pi\)
\(758\) 4.90766 0.178254
\(759\) −4.42864 −0.160749
\(760\) 0 0
\(761\) −45.1624 −1.63714 −0.818568 0.574410i \(-0.805232\pi\)
−0.818568 + 0.574410i \(0.805232\pi\)
\(762\) −4.99063 −0.180792
\(763\) 9.42816 0.341322
\(764\) 18.9590 0.685912
\(765\) 0 0
\(766\) 19.8796 0.718277
\(767\) −13.2444 −0.478229
\(768\) 1.00000 0.0360844
\(769\) 29.4924 1.06352 0.531762 0.846894i \(-0.321530\pi\)
0.531762 + 0.846894i \(0.321530\pi\)
\(770\) 0 0
\(771\) −11.7146 −0.421889
\(772\) 1.24443 0.0447881
\(773\) −41.0923 −1.47799 −0.738994 0.673712i \(-0.764699\pi\)
−0.738994 + 0.673712i \(0.764699\pi\)
\(774\) −1.05086 −0.0377722
\(775\) 0 0
\(776\) −2.13335 −0.0765829
\(777\) 7.34614 0.263541
\(778\) 0.161933 0.00580559
\(779\) 14.9590 0.535961
\(780\) 0 0
\(781\) −30.9590 −1.10780
\(782\) −6.42864 −0.229888
\(783\) −4.23506 −0.151349
\(784\) −6.61285 −0.236173
\(785\) 0 0
\(786\) −0.755569 −0.0269502
\(787\) 10.0919 0.359736 0.179868 0.983691i \(-0.442433\pi\)
0.179868 + 0.983691i \(0.442433\pi\)
\(788\) 15.2444 0.543060
\(789\) −8.47013 −0.301544
\(790\) 0 0
\(791\) 8.97773 0.319211
\(792\) −4.42864 −0.157365
\(793\) −11.9367 −0.423885
\(794\) −1.76494 −0.0626353
\(795\) 0 0
\(796\) −14.5303 −0.515015
\(797\) −25.2958 −0.896022 −0.448011 0.894028i \(-0.647867\pi\)
−0.448011 + 0.894028i \(0.647867\pi\)
\(798\) −0.857279 −0.0303473
\(799\) −74.6548 −2.64110
\(800\) 0 0
\(801\) 2.75557 0.0973632
\(802\) 31.3461 1.10687
\(803\) −12.2034 −0.430649
\(804\) −11.8064 −0.416380
\(805\) 0 0
\(806\) −5.51114 −0.194122
\(807\) 28.8256 1.01471
\(808\) −16.2351 −0.571148
\(809\) 43.4479 1.52755 0.763773 0.645485i \(-0.223345\pi\)
0.763773 + 0.645485i \(0.223345\pi\)
\(810\) 0 0
\(811\) −8.65386 −0.303878 −0.151939 0.988390i \(-0.548552\pi\)
−0.151939 + 0.988390i \(0.548552\pi\)
\(812\) 2.63512 0.0924747
\(813\) −13.8350 −0.485215
\(814\) 52.2864 1.83264
\(815\) 0 0
\(816\) −6.42864 −0.225047
\(817\) −1.44785 −0.0506539
\(818\) 3.51114 0.122764
\(819\) 0.857279 0.0299558
\(820\) 0 0
\(821\) 2.78721 0.0972744 0.0486372 0.998817i \(-0.484512\pi\)
0.0486372 + 0.998817i \(0.484512\pi\)
\(822\) 12.9175 0.450550
\(823\) 29.9684 1.04463 0.522316 0.852752i \(-0.325068\pi\)
0.522316 + 0.852752i \(0.325068\pi\)
\(824\) 12.2351 0.426229
\(825\) 0 0
\(826\) 5.98126 0.208115
\(827\) 47.2543 1.64319 0.821596 0.570070i \(-0.193084\pi\)
0.821596 + 0.570070i \(0.193084\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −53.1624 −1.84641 −0.923203 0.384312i \(-0.874439\pi\)
−0.923203 + 0.384312i \(0.874439\pi\)
\(830\) 0 0
\(831\) 2.88892 0.100216
\(832\) −1.37778 −0.0477661
\(833\) 42.5116 1.47294
\(834\) 14.1017 0.488302
\(835\) 0 0
\(836\) −6.10171 −0.211032
\(837\) −4.00000 −0.138260
\(838\) 19.7748 0.683108
\(839\) −17.3274 −0.598208 −0.299104 0.954220i \(-0.596688\pi\)
−0.299104 + 0.954220i \(0.596688\pi\)
\(840\) 0 0
\(841\) −11.0642 −0.381525
\(842\) −11.8064 −0.406876
\(843\) 5.51114 0.189814
\(844\) 0.266706 0.00918041
\(845\) 0 0
\(846\) −11.6128 −0.399258
\(847\) −5.35905 −0.184139
\(848\) −3.37778 −0.115994
\(849\) 8.19358 0.281203
\(850\) 0 0
\(851\) 11.8064 0.404719
\(852\) −6.99063 −0.239495
\(853\) 17.7649 0.608260 0.304130 0.952631i \(-0.401634\pi\)
0.304130 + 0.952631i \(0.401634\pi\)
\(854\) 5.39069 0.184466
\(855\) 0 0
\(856\) −10.6637 −0.364478
\(857\) 10.0830 0.344428 0.172214 0.985060i \(-0.444908\pi\)
0.172214 + 0.985060i \(0.444908\pi\)
\(858\) 6.10171 0.208309
\(859\) 5.12399 0.174828 0.0874141 0.996172i \(-0.472140\pi\)
0.0874141 + 0.996172i \(0.472140\pi\)
\(860\) 0 0
\(861\) 6.75557 0.230229
\(862\) −24.9403 −0.849468
\(863\) 49.2070 1.67502 0.837512 0.546419i \(-0.184009\pi\)
0.837512 + 0.546419i \(0.184009\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 32.0513 1.08915
\(867\) 24.3274 0.826202
\(868\) 2.48886 0.0844775
\(869\) 53.3274 1.80901
\(870\) 0 0
\(871\) 16.2667 0.551176
\(872\) 15.1526 0.513131
\(873\) 2.13335 0.0722031
\(874\) −1.37778 −0.0466043
\(875\) 0 0
\(876\) −2.75557 −0.0931020
\(877\) −9.11108 −0.307659 −0.153830 0.988097i \(-0.549161\pi\)
−0.153830 + 0.988097i \(0.549161\pi\)
\(878\) −0.470127 −0.0158660
\(879\) 12.6222 0.425737
\(880\) 0 0
\(881\) −9.71456 −0.327292 −0.163646 0.986519i \(-0.552325\pi\)
−0.163646 + 0.986519i \(0.552325\pi\)
\(882\) 6.61285 0.222666
\(883\) 33.3274 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(884\) 8.85728 0.297903
\(885\) 0 0
\(886\) 0.653858 0.0219668
\(887\) 53.5941 1.79951 0.899757 0.436391i \(-0.143744\pi\)
0.899757 + 0.436391i \(0.143744\pi\)
\(888\) 11.8064 0.396198
\(889\) −3.10525 −0.104147
\(890\) 0 0
\(891\) 4.42864 0.148365
\(892\) 0.990632 0.0331688
\(893\) −16.0000 −0.535420
\(894\) −12.4286 −0.415676
\(895\) 0 0
\(896\) 0.622216 0.0207868
\(897\) 1.37778 0.0460029
\(898\) −17.1427 −0.572060
\(899\) 16.9403 0.564989
\(900\) 0 0
\(901\) 21.7146 0.723417
\(902\) 48.0830 1.60099
\(903\) −0.653858 −0.0217590
\(904\) 14.4286 0.479889
\(905\) 0 0
\(906\) 4.85728 0.161372
\(907\) −2.56199 −0.0850696 −0.0425348 0.999095i \(-0.513543\pi\)
−0.0425348 + 0.999095i \(0.513543\pi\)
\(908\) 17.1526 0.569228
\(909\) 16.2351 0.538483
\(910\) 0 0
\(911\) 8.47013 0.280628 0.140314 0.990107i \(-0.455189\pi\)
0.140314 + 0.990107i \(0.455189\pi\)
\(912\) −1.37778 −0.0456230
\(913\) −9.71456 −0.321505
\(914\) −1.00937 −0.0333869
\(915\) 0 0
\(916\) 7.15257 0.236327
\(917\) −0.470127 −0.0155250
\(918\) 6.42864 0.212177
\(919\) −36.8988 −1.21718 −0.608589 0.793486i \(-0.708264\pi\)
−0.608589 + 0.793486i \(0.708264\pi\)
\(920\) 0 0
\(921\) −0.653858 −0.0215454
\(922\) −20.3180 −0.669139
\(923\) 9.63158 0.317027
\(924\) −2.75557 −0.0906516
\(925\) 0 0
\(926\) −12.4572 −0.409370
\(927\) −12.2351 −0.401852
\(928\) 4.23506 0.139023
\(929\) −0.164996 −0.00541334 −0.00270667 0.999996i \(-0.500862\pi\)
−0.00270667 + 0.999996i \(0.500862\pi\)
\(930\) 0 0
\(931\) 9.11108 0.298604
\(932\) 3.71456 0.121674
\(933\) −13.0923 −0.428624
\(934\) 15.7877 0.516589
\(935\) 0 0
\(936\) 1.37778 0.0450343
\(937\) 40.3180 1.31713 0.658566 0.752523i \(-0.271163\pi\)
0.658566 + 0.752523i \(0.271163\pi\)
\(938\) −7.34614 −0.239860
\(939\) 11.2573 0.367369
\(940\) 0 0
\(941\) 53.3689 1.73978 0.869888 0.493249i \(-0.164191\pi\)
0.869888 + 0.493249i \(0.164191\pi\)
\(942\) −9.90813 −0.322824
\(943\) 10.8573 0.353562
\(944\) 9.61285 0.312872
\(945\) 0 0
\(946\) −4.65386 −0.151310
\(947\) −44.6735 −1.45170 −0.725848 0.687856i \(-0.758552\pi\)
−0.725848 + 0.687856i \(0.758552\pi\)
\(948\) 12.0415 0.391089
\(949\) 3.79658 0.123242
\(950\) 0 0
\(951\) −22.4701 −0.728644
\(952\) −4.00000 −0.129641
\(953\) 21.2672 0.688912 0.344456 0.938803i \(-0.388063\pi\)
0.344456 + 0.938803i \(0.388063\pi\)
\(954\) 3.37778 0.109360
\(955\) 0 0
\(956\) 27.8479 0.900666
\(957\) −18.7556 −0.606281
\(958\) −5.12399 −0.165548
\(959\) 8.03747 0.259543
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −16.2667 −0.524460
\(963\) 10.6637 0.343633
\(964\) −7.12399 −0.229448
\(965\) 0 0
\(966\) −0.622216 −0.0200195
\(967\) −10.9461 −0.352002 −0.176001 0.984390i \(-0.556316\pi\)
−0.176001 + 0.984390i \(0.556316\pi\)
\(968\) −8.61285 −0.276827
\(969\) 8.85728 0.284537
\(970\) 0 0
\(971\) −16.8988 −0.542307 −0.271154 0.962536i \(-0.587405\pi\)
−0.271154 + 0.962536i \(0.587405\pi\)
\(972\) 1.00000 0.0320750
\(973\) 8.77430 0.281291
\(974\) 23.2128 0.743786
\(975\) 0 0
\(976\) 8.66370 0.277318
\(977\) −59.2484 −1.89553 −0.947763 0.318976i \(-0.896661\pi\)
−0.947763 + 0.318976i \(0.896661\pi\)
\(978\) 10.1017 0.323017
\(979\) 12.2034 0.390023
\(980\) 0 0
\(981\) −15.1526 −0.483784
\(982\) 40.1847 1.28234
\(983\) 23.8163 0.759621 0.379810 0.925064i \(-0.375989\pi\)
0.379810 + 0.925064i \(0.375989\pi\)
\(984\) 10.8573 0.346117
\(985\) 0 0
\(986\) −27.2257 −0.867043
\(987\) −7.22570 −0.229996
\(988\) 1.89829 0.0603926
\(989\) −1.05086 −0.0334152
\(990\) 0 0
\(991\) 29.7146 0.943914 0.471957 0.881622i \(-0.343548\pi\)
0.471957 + 0.881622i \(0.343548\pi\)
\(992\) 4.00000 0.127000
\(993\) −29.4479 −0.934499
\(994\) −4.34968 −0.137963
\(995\) 0 0
\(996\) −2.19358 −0.0695061
\(997\) −22.7052 −0.719081 −0.359540 0.933130i \(-0.617066\pi\)
−0.359540 + 0.933130i \(0.617066\pi\)
\(998\) −30.5718 −0.967735
\(999\) −11.8064 −0.373539
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.bo.1.2 3
5.2 odd 4 690.2.d.c.139.3 6
5.3 odd 4 690.2.d.c.139.6 yes 6
5.4 even 2 3450.2.a.bt.1.2 3
15.2 even 4 2070.2.d.e.829.4 6
15.8 even 4 2070.2.d.e.829.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.d.c.139.3 6 5.2 odd 4
690.2.d.c.139.6 yes 6 5.3 odd 4
2070.2.d.e.829.1 6 15.8 even 4
2070.2.d.e.829.4 6 15.2 even 4
3450.2.a.bo.1.2 3 1.1 even 1 trivial
3450.2.a.bt.1.2 3 5.4 even 2