Properties

Label 350.2.a.h.1.2
Level $350$
Weight $2$
Character 350.1
Self dual yes
Analytic conductor $2.795$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 350.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} +1.00000 q^{7} +1.00000 q^{8} +3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} +1.00000 q^{7} +1.00000 q^{8} +3.00000 q^{9} -4.89898 q^{11} +2.44949 q^{12} -4.44949 q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +3.00000 q^{18} +1.55051 q^{19} +2.44949 q^{21} -4.89898 q^{22} -2.89898 q^{23} +2.44949 q^{24} -4.44949 q^{26} +1.00000 q^{28} +6.89898 q^{29} +8.89898 q^{31} +1.00000 q^{32} -12.0000 q^{33} -2.00000 q^{34} +3.00000 q^{36} -2.00000 q^{37} +1.55051 q^{38} -10.8990 q^{39} -1.10102 q^{41} +2.44949 q^{42} +0.898979 q^{43} -4.89898 q^{44} -2.89898 q^{46} -8.89898 q^{47} +2.44949 q^{48} +1.00000 q^{49} -4.89898 q^{51} -4.44949 q^{52} +10.8990 q^{53} +1.00000 q^{56} +3.79796 q^{57} +6.89898 q^{58} -1.55051 q^{59} +3.55051 q^{61} +8.89898 q^{62} +3.00000 q^{63} +1.00000 q^{64} -12.0000 q^{66} +8.00000 q^{67} -2.00000 q^{68} -7.10102 q^{69} -1.10102 q^{71} +3.00000 q^{72} -2.89898 q^{73} -2.00000 q^{74} +1.55051 q^{76} -4.89898 q^{77} -10.8990 q^{78} +6.89898 q^{79} -9.00000 q^{81} -1.10102 q^{82} +2.44949 q^{83} +2.44949 q^{84} +0.898979 q^{86} +16.8990 q^{87} -4.89898 q^{88} -10.0000 q^{89} -4.44949 q^{91} -2.89898 q^{92} +21.7980 q^{93} -8.89898 q^{94} +2.44949 q^{96} -15.7980 q^{97} +1.00000 q^{98} -14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 6 q^{9} - 4 q^{13} + 2 q^{14} + 2 q^{16} - 4 q^{17} + 6 q^{18} + 8 q^{19} + 4 q^{23} - 4 q^{26} + 2 q^{28} + 4 q^{29} + 8 q^{31} + 2 q^{32} - 24 q^{33} - 4 q^{34} + 6 q^{36} - 4 q^{37} + 8 q^{38} - 12 q^{39} - 12 q^{41} - 8 q^{43} + 4 q^{46} - 8 q^{47} + 2 q^{49} - 4 q^{52} + 12 q^{53} + 2 q^{56} - 12 q^{57} + 4 q^{58} - 8 q^{59} + 12 q^{61} + 8 q^{62} + 6 q^{63} + 2 q^{64} - 24 q^{66} + 16 q^{67} - 4 q^{68} - 24 q^{69} - 12 q^{71} + 6 q^{72} + 4 q^{73} - 4 q^{74} + 8 q^{76} - 12 q^{78} + 4 q^{79} - 18 q^{81} - 12 q^{82} - 8 q^{86} + 24 q^{87} - 20 q^{89} - 4 q^{91} + 4 q^{92} + 24 q^{93} - 8 q^{94} - 12 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.44949 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.44949 1.00000
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) −4.89898 −1.47710 −0.738549 0.674200i \(-0.764489\pi\)
−0.738549 + 0.674200i \(0.764489\pi\)
\(12\) 2.44949 0.707107
\(13\) −4.44949 −1.23407 −0.617033 0.786937i \(-0.711666\pi\)
−0.617033 + 0.786937i \(0.711666\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 3.00000 0.707107
\(19\) 1.55051 0.355711 0.177856 0.984057i \(-0.443084\pi\)
0.177856 + 0.984057i \(0.443084\pi\)
\(20\) 0 0
\(21\) 2.44949 0.534522
\(22\) −4.89898 −1.04447
\(23\) −2.89898 −0.604479 −0.302240 0.953232i \(-0.597734\pi\)
−0.302240 + 0.953232i \(0.597734\pi\)
\(24\) 2.44949 0.500000
\(25\) 0 0
\(26\) −4.44949 −0.872617
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 6.89898 1.28111 0.640554 0.767913i \(-0.278705\pi\)
0.640554 + 0.767913i \(0.278705\pi\)
\(30\) 0 0
\(31\) 8.89898 1.59830 0.799152 0.601129i \(-0.205282\pi\)
0.799152 + 0.601129i \(0.205282\pi\)
\(32\) 1.00000 0.176777
\(33\) −12.0000 −2.08893
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 1.55051 0.251526
\(39\) −10.8990 −1.74523
\(40\) 0 0
\(41\) −1.10102 −0.171951 −0.0859753 0.996297i \(-0.527401\pi\)
−0.0859753 + 0.996297i \(0.527401\pi\)
\(42\) 2.44949 0.377964
\(43\) 0.898979 0.137093 0.0685465 0.997648i \(-0.478164\pi\)
0.0685465 + 0.997648i \(0.478164\pi\)
\(44\) −4.89898 −0.738549
\(45\) 0 0
\(46\) −2.89898 −0.427431
\(47\) −8.89898 −1.29805 −0.649025 0.760767i \(-0.724823\pi\)
−0.649025 + 0.760767i \(0.724823\pi\)
\(48\) 2.44949 0.353553
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.89898 −0.685994
\(52\) −4.44949 −0.617033
\(53\) 10.8990 1.49709 0.748545 0.663084i \(-0.230753\pi\)
0.748545 + 0.663084i \(0.230753\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 3.79796 0.503052
\(58\) 6.89898 0.905880
\(59\) −1.55051 −0.201859 −0.100930 0.994894i \(-0.532182\pi\)
−0.100930 + 0.994894i \(0.532182\pi\)
\(60\) 0 0
\(61\) 3.55051 0.454596 0.227298 0.973825i \(-0.427011\pi\)
0.227298 + 0.973825i \(0.427011\pi\)
\(62\) 8.89898 1.13017
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −12.0000 −1.47710
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −2.00000 −0.242536
\(69\) −7.10102 −0.854862
\(70\) 0 0
\(71\) −1.10102 −0.130667 −0.0653335 0.997863i \(-0.520811\pi\)
−0.0653335 + 0.997863i \(0.520811\pi\)
\(72\) 3.00000 0.353553
\(73\) −2.89898 −0.339300 −0.169650 0.985504i \(-0.554264\pi\)
−0.169650 + 0.985504i \(0.554264\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 1.55051 0.177856
\(77\) −4.89898 −0.558291
\(78\) −10.8990 −1.23407
\(79\) 6.89898 0.776196 0.388098 0.921618i \(-0.373132\pi\)
0.388098 + 0.921618i \(0.373132\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) −1.10102 −0.121587
\(83\) 2.44949 0.268866 0.134433 0.990923i \(-0.457079\pi\)
0.134433 + 0.990923i \(0.457079\pi\)
\(84\) 2.44949 0.267261
\(85\) 0 0
\(86\) 0.898979 0.0969395
\(87\) 16.8990 1.81176
\(88\) −4.89898 −0.522233
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −4.44949 −0.466433
\(92\) −2.89898 −0.302240
\(93\) 21.7980 2.26034
\(94\) −8.89898 −0.917860
\(95\) 0 0
\(96\) 2.44949 0.250000
\(97\) −15.7980 −1.60404 −0.802020 0.597297i \(-0.796241\pi\)
−0.802020 + 0.597297i \(0.796241\pi\)
\(98\) 1.00000 0.101015
\(99\) −14.6969 −1.47710
\(100\) 0 0
\(101\) 3.55051 0.353289 0.176644 0.984275i \(-0.443476\pi\)
0.176644 + 0.984275i \(0.443476\pi\)
\(102\) −4.89898 −0.485071
\(103\) −12.8990 −1.27097 −0.635487 0.772111i \(-0.719201\pi\)
−0.635487 + 0.772111i \(0.719201\pi\)
\(104\) −4.44949 −0.436308
\(105\) 0 0
\(106\) 10.8990 1.05860
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 6.89898 0.660802 0.330401 0.943841i \(-0.392816\pi\)
0.330401 + 0.943841i \(0.392816\pi\)
\(110\) 0 0
\(111\) −4.89898 −0.464991
\(112\) 1.00000 0.0944911
\(113\) −19.7980 −1.86244 −0.931218 0.364464i \(-0.881252\pi\)
−0.931218 + 0.364464i \(0.881252\pi\)
\(114\) 3.79796 0.355711
\(115\) 0 0
\(116\) 6.89898 0.640554
\(117\) −13.3485 −1.23407
\(118\) −1.55051 −0.142736
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 3.55051 0.321448
\(123\) −2.69694 −0.243175
\(124\) 8.89898 0.799152
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) 14.8990 1.32207 0.661035 0.750355i \(-0.270117\pi\)
0.661035 + 0.750355i \(0.270117\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.20204 0.193879
\(130\) 0 0
\(131\) −6.44949 −0.563495 −0.281747 0.959489i \(-0.590914\pi\)
−0.281747 + 0.959489i \(0.590914\pi\)
\(132\) −12.0000 −1.04447
\(133\) 1.55051 0.134446
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 1.79796 0.153610 0.0768050 0.997046i \(-0.475528\pi\)
0.0768050 + 0.997046i \(0.475528\pi\)
\(138\) −7.10102 −0.604479
\(139\) 1.55051 0.131513 0.0657563 0.997836i \(-0.479054\pi\)
0.0657563 + 0.997836i \(0.479054\pi\)
\(140\) 0 0
\(141\) −21.7980 −1.83572
\(142\) −1.10102 −0.0923956
\(143\) 21.7980 1.82284
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) −2.89898 −0.239921
\(147\) 2.44949 0.202031
\(148\) −2.00000 −0.164399
\(149\) −3.79796 −0.311141 −0.155570 0.987825i \(-0.549722\pi\)
−0.155570 + 0.987825i \(0.549722\pi\)
\(150\) 0 0
\(151\) 19.5959 1.59469 0.797347 0.603522i \(-0.206236\pi\)
0.797347 + 0.603522i \(0.206236\pi\)
\(152\) 1.55051 0.125763
\(153\) −6.00000 −0.485071
\(154\) −4.89898 −0.394771
\(155\) 0 0
\(156\) −10.8990 −0.872617
\(157\) −3.55051 −0.283362 −0.141681 0.989912i \(-0.545251\pi\)
−0.141681 + 0.989912i \(0.545251\pi\)
\(158\) 6.89898 0.548853
\(159\) 26.6969 2.11720
\(160\) 0 0
\(161\) −2.89898 −0.228472
\(162\) −9.00000 −0.707107
\(163\) 7.10102 0.556195 0.278097 0.960553i \(-0.410296\pi\)
0.278097 + 0.960553i \(0.410296\pi\)
\(164\) −1.10102 −0.0859753
\(165\) 0 0
\(166\) 2.44949 0.190117
\(167\) 4.89898 0.379094 0.189547 0.981872i \(-0.439298\pi\)
0.189547 + 0.981872i \(0.439298\pi\)
\(168\) 2.44949 0.188982
\(169\) 6.79796 0.522920
\(170\) 0 0
\(171\) 4.65153 0.355711
\(172\) 0.898979 0.0685465
\(173\) 6.24745 0.474985 0.237492 0.971389i \(-0.423675\pi\)
0.237492 + 0.971389i \(0.423675\pi\)
\(174\) 16.8990 1.28111
\(175\) 0 0
\(176\) −4.89898 −0.369274
\(177\) −3.79796 −0.285472
\(178\) −10.0000 −0.749532
\(179\) 13.7980 1.03131 0.515654 0.856797i \(-0.327549\pi\)
0.515654 + 0.856797i \(0.327549\pi\)
\(180\) 0 0
\(181\) −10.2474 −0.761687 −0.380843 0.924640i \(-0.624366\pi\)
−0.380843 + 0.924640i \(0.624366\pi\)
\(182\) −4.44949 −0.329818
\(183\) 8.69694 0.642896
\(184\) −2.89898 −0.213716
\(185\) 0 0
\(186\) 21.7980 1.59830
\(187\) 9.79796 0.716498
\(188\) −8.89898 −0.649025
\(189\) 0 0
\(190\) 0 0
\(191\) 12.6969 0.918718 0.459359 0.888251i \(-0.348079\pi\)
0.459359 + 0.888251i \(0.348079\pi\)
\(192\) 2.44949 0.176777
\(193\) 21.5959 1.55451 0.777254 0.629187i \(-0.216612\pi\)
0.777254 + 0.629187i \(0.216612\pi\)
\(194\) −15.7980 −1.13423
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −18.8990 −1.34650 −0.673248 0.739417i \(-0.735101\pi\)
−0.673248 + 0.739417i \(0.735101\pi\)
\(198\) −14.6969 −1.04447
\(199\) 16.8990 1.19794 0.598968 0.800773i \(-0.295577\pi\)
0.598968 + 0.800773i \(0.295577\pi\)
\(200\) 0 0
\(201\) 19.5959 1.38219
\(202\) 3.55051 0.249813
\(203\) 6.89898 0.484213
\(204\) −4.89898 −0.342997
\(205\) 0 0
\(206\) −12.8990 −0.898714
\(207\) −8.69694 −0.604479
\(208\) −4.44949 −0.308517
\(209\) −7.59592 −0.525421
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 10.8990 0.748545
\(213\) −2.69694 −0.184791
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) 0 0
\(217\) 8.89898 0.604102
\(218\) 6.89898 0.467258
\(219\) −7.10102 −0.479842
\(220\) 0 0
\(221\) 8.89898 0.598610
\(222\) −4.89898 −0.328798
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −19.7980 −1.31694
\(227\) −7.34847 −0.487735 −0.243868 0.969809i \(-0.578416\pi\)
−0.243868 + 0.969809i \(0.578416\pi\)
\(228\) 3.79796 0.251526
\(229\) −19.1464 −1.26523 −0.632616 0.774466i \(-0.718019\pi\)
−0.632616 + 0.774466i \(0.718019\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 6.89898 0.452940
\(233\) −29.7980 −1.95213 −0.976065 0.217481i \(-0.930216\pi\)
−0.976065 + 0.217481i \(0.930216\pi\)
\(234\) −13.3485 −0.872617
\(235\) 0 0
\(236\) −1.55051 −0.100930
\(237\) 16.8990 1.09771
\(238\) −2.00000 −0.129641
\(239\) −6.20204 −0.401177 −0.200588 0.979676i \(-0.564285\pi\)
−0.200588 + 0.979676i \(0.564285\pi\)
\(240\) 0 0
\(241\) −8.69694 −0.560219 −0.280110 0.959968i \(-0.590371\pi\)
−0.280110 + 0.959968i \(0.590371\pi\)
\(242\) 13.0000 0.835672
\(243\) −22.0454 −1.41421
\(244\) 3.55051 0.227298
\(245\) 0 0
\(246\) −2.69694 −0.171951
\(247\) −6.89898 −0.438972
\(248\) 8.89898 0.565086
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) −6.44949 −0.407088 −0.203544 0.979066i \(-0.565246\pi\)
−0.203544 + 0.979066i \(0.565246\pi\)
\(252\) 3.00000 0.188982
\(253\) 14.2020 0.892875
\(254\) 14.8990 0.934845
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.69694 0.542500 0.271250 0.962509i \(-0.412563\pi\)
0.271250 + 0.962509i \(0.412563\pi\)
\(258\) 2.20204 0.137093
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 20.6969 1.28111
\(262\) −6.44949 −0.398451
\(263\) −9.79796 −0.604168 −0.302084 0.953281i \(-0.597682\pi\)
−0.302084 + 0.953281i \(0.597682\pi\)
\(264\) −12.0000 −0.738549
\(265\) 0 0
\(266\) 1.55051 0.0950679
\(267\) −24.4949 −1.49906
\(268\) 8.00000 0.488678
\(269\) 19.1464 1.16738 0.583689 0.811977i \(-0.301609\pi\)
0.583689 + 0.811977i \(0.301609\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) −2.00000 −0.121268
\(273\) −10.8990 −0.659636
\(274\) 1.79796 0.108619
\(275\) 0 0
\(276\) −7.10102 −0.427431
\(277\) 14.8990 0.895193 0.447596 0.894236i \(-0.352280\pi\)
0.447596 + 0.894236i \(0.352280\pi\)
\(278\) 1.55051 0.0929934
\(279\) 26.6969 1.59830
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −21.7980 −1.29805
\(283\) −3.75255 −0.223066 −0.111533 0.993761i \(-0.535576\pi\)
−0.111533 + 0.993761i \(0.535576\pi\)
\(284\) −1.10102 −0.0653335
\(285\) 0 0
\(286\) 21.7980 1.28894
\(287\) −1.10102 −0.0649912
\(288\) 3.00000 0.176777
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −38.6969 −2.26845
\(292\) −2.89898 −0.169650
\(293\) −18.2474 −1.06603 −0.533014 0.846107i \(-0.678941\pi\)
−0.533014 + 0.846107i \(0.678941\pi\)
\(294\) 2.44949 0.142857
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −3.79796 −0.220010
\(299\) 12.8990 0.745967
\(300\) 0 0
\(301\) 0.898979 0.0518163
\(302\) 19.5959 1.12762
\(303\) 8.69694 0.499626
\(304\) 1.55051 0.0889279
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) 20.2474 1.15558 0.577791 0.816184i \(-0.303915\pi\)
0.577791 + 0.816184i \(0.303915\pi\)
\(308\) −4.89898 −0.279145
\(309\) −31.5959 −1.79743
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) −10.8990 −0.617033
\(313\) 21.5959 1.22067 0.610337 0.792142i \(-0.291034\pi\)
0.610337 + 0.792142i \(0.291034\pi\)
\(314\) −3.55051 −0.200367
\(315\) 0 0
\(316\) 6.89898 0.388098
\(317\) 22.4949 1.26344 0.631720 0.775197i \(-0.282349\pi\)
0.631720 + 0.775197i \(0.282349\pi\)
\(318\) 26.6969 1.49709
\(319\) −33.7980 −1.89232
\(320\) 0 0
\(321\) 19.5959 1.09374
\(322\) −2.89898 −0.161554
\(323\) −3.10102 −0.172545
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) 7.10102 0.393289
\(327\) 16.8990 0.934516
\(328\) −1.10102 −0.0607937
\(329\) −8.89898 −0.490617
\(330\) 0 0
\(331\) −18.6969 −1.02768 −0.513838 0.857887i \(-0.671777\pi\)
−0.513838 + 0.857887i \(0.671777\pi\)
\(332\) 2.44949 0.134433
\(333\) −6.00000 −0.328798
\(334\) 4.89898 0.268060
\(335\) 0 0
\(336\) 2.44949 0.133631
\(337\) −9.59592 −0.522723 −0.261361 0.965241i \(-0.584171\pi\)
−0.261361 + 0.965241i \(0.584171\pi\)
\(338\) 6.79796 0.369760
\(339\) −48.4949 −2.63388
\(340\) 0 0
\(341\) −43.5959 −2.36085
\(342\) 4.65153 0.251526
\(343\) 1.00000 0.0539949
\(344\) 0.898979 0.0484697
\(345\) 0 0
\(346\) 6.24745 0.335865
\(347\) −28.8990 −1.55138 −0.775689 0.631115i \(-0.782598\pi\)
−0.775689 + 0.631115i \(0.782598\pi\)
\(348\) 16.8990 0.905880
\(349\) −8.44949 −0.452291 −0.226145 0.974094i \(-0.572612\pi\)
−0.226145 + 0.974094i \(0.572612\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.89898 −0.261116
\(353\) −22.8990 −1.21879 −0.609395 0.792867i \(-0.708588\pi\)
−0.609395 + 0.792867i \(0.708588\pi\)
\(354\) −3.79796 −0.201859
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) −4.89898 −0.259281
\(358\) 13.7980 0.729245
\(359\) −27.5959 −1.45646 −0.728228 0.685334i \(-0.759656\pi\)
−0.728228 + 0.685334i \(0.759656\pi\)
\(360\) 0 0
\(361\) −16.5959 −0.873469
\(362\) −10.2474 −0.538594
\(363\) 31.8434 1.67134
\(364\) −4.44949 −0.233217
\(365\) 0 0
\(366\) 8.69694 0.454596
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) −2.89898 −0.151120
\(369\) −3.30306 −0.171951
\(370\) 0 0
\(371\) 10.8990 0.565847
\(372\) 21.7980 1.13017
\(373\) 4.69694 0.243198 0.121599 0.992579i \(-0.461198\pi\)
0.121599 + 0.992579i \(0.461198\pi\)
\(374\) 9.79796 0.506640
\(375\) 0 0
\(376\) −8.89898 −0.458930
\(377\) −30.6969 −1.58097
\(378\) 0 0
\(379\) 30.6969 1.57680 0.788398 0.615166i \(-0.210911\pi\)
0.788398 + 0.615166i \(0.210911\pi\)
\(380\) 0 0
\(381\) 36.4949 1.86969
\(382\) 12.6969 0.649632
\(383\) 7.10102 0.362845 0.181423 0.983405i \(-0.441930\pi\)
0.181423 + 0.983405i \(0.441930\pi\)
\(384\) 2.44949 0.125000
\(385\) 0 0
\(386\) 21.5959 1.09920
\(387\) 2.69694 0.137093
\(388\) −15.7980 −0.802020
\(389\) 13.1010 0.664248 0.332124 0.943236i \(-0.392235\pi\)
0.332124 + 0.943236i \(0.392235\pi\)
\(390\) 0 0
\(391\) 5.79796 0.293215
\(392\) 1.00000 0.0505076
\(393\) −15.7980 −0.796902
\(394\) −18.8990 −0.952117
\(395\) 0 0
\(396\) −14.6969 −0.738549
\(397\) 2.65153 0.133077 0.0665383 0.997784i \(-0.478805\pi\)
0.0665383 + 0.997784i \(0.478805\pi\)
\(398\) 16.8990 0.847069
\(399\) 3.79796 0.190136
\(400\) 0 0
\(401\) −29.3939 −1.46786 −0.733930 0.679225i \(-0.762316\pi\)
−0.733930 + 0.679225i \(0.762316\pi\)
\(402\) 19.5959 0.977356
\(403\) −39.5959 −1.97241
\(404\) 3.55051 0.176644
\(405\) 0 0
\(406\) 6.89898 0.342391
\(407\) 9.79796 0.485667
\(408\) −4.89898 −0.242536
\(409\) 34.4949 1.70566 0.852831 0.522186i \(-0.174883\pi\)
0.852831 + 0.522186i \(0.174883\pi\)
\(410\) 0 0
\(411\) 4.40408 0.217237
\(412\) −12.8990 −0.635487
\(413\) −1.55051 −0.0762956
\(414\) −8.69694 −0.427431
\(415\) 0 0
\(416\) −4.44949 −0.218154
\(417\) 3.79796 0.185987
\(418\) −7.59592 −0.371528
\(419\) −1.55051 −0.0757474 −0.0378737 0.999283i \(-0.512058\pi\)
−0.0378737 + 0.999283i \(0.512058\pi\)
\(420\) 0 0
\(421\) −4.20204 −0.204795 −0.102397 0.994744i \(-0.532651\pi\)
−0.102397 + 0.994744i \(0.532651\pi\)
\(422\) 12.0000 0.584151
\(423\) −26.6969 −1.29805
\(424\) 10.8990 0.529301
\(425\) 0 0
\(426\) −2.69694 −0.130667
\(427\) 3.55051 0.171821
\(428\) 8.00000 0.386695
\(429\) 53.3939 2.57788
\(430\) 0 0
\(431\) −1.79796 −0.0866046 −0.0433023 0.999062i \(-0.513788\pi\)
−0.0433023 + 0.999062i \(0.513788\pi\)
\(432\) 0 0
\(433\) 0.202041 0.00970947 0.00485474 0.999988i \(-0.498455\pi\)
0.00485474 + 0.999988i \(0.498455\pi\)
\(434\) 8.89898 0.427165
\(435\) 0 0
\(436\) 6.89898 0.330401
\(437\) −4.49490 −0.215020
\(438\) −7.10102 −0.339300
\(439\) −21.3939 −1.02107 −0.510537 0.859856i \(-0.670553\pi\)
−0.510537 + 0.859856i \(0.670553\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 8.89898 0.423281
\(443\) −9.79796 −0.465515 −0.232758 0.972535i \(-0.574775\pi\)
−0.232758 + 0.972535i \(0.574775\pi\)
\(444\) −4.89898 −0.232495
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) −9.30306 −0.440020
\(448\) 1.00000 0.0472456
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) 5.39388 0.253988
\(452\) −19.7980 −0.931218
\(453\) 48.0000 2.25524
\(454\) −7.34847 −0.344881
\(455\) 0 0
\(456\) 3.79796 0.177856
\(457\) −29.5959 −1.38444 −0.692219 0.721687i \(-0.743367\pi\)
−0.692219 + 0.721687i \(0.743367\pi\)
\(458\) −19.1464 −0.894654
\(459\) 0 0
\(460\) 0 0
\(461\) 17.3485 0.807999 0.403999 0.914759i \(-0.367620\pi\)
0.403999 + 0.914759i \(0.367620\pi\)
\(462\) −12.0000 −0.558291
\(463\) −3.59592 −0.167116 −0.0835582 0.996503i \(-0.526628\pi\)
−0.0835582 + 0.996503i \(0.526628\pi\)
\(464\) 6.89898 0.320277
\(465\) 0 0
\(466\) −29.7980 −1.38036
\(467\) −10.4495 −0.483545 −0.241772 0.970333i \(-0.577729\pi\)
−0.241772 + 0.970333i \(0.577729\pi\)
\(468\) −13.3485 −0.617033
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) −8.69694 −0.400734
\(472\) −1.55051 −0.0713680
\(473\) −4.40408 −0.202500
\(474\) 16.8990 0.776196
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) 32.6969 1.49709
\(478\) −6.20204 −0.283675
\(479\) 9.30306 0.425068 0.212534 0.977154i \(-0.431828\pi\)
0.212534 + 0.977154i \(0.431828\pi\)
\(480\) 0 0
\(481\) 8.89898 0.405759
\(482\) −8.69694 −0.396135
\(483\) −7.10102 −0.323108
\(484\) 13.0000 0.590909
\(485\) 0 0
\(486\) −22.0454 −1.00000
\(487\) 7.30306 0.330933 0.165467 0.986215i \(-0.447087\pi\)
0.165467 + 0.986215i \(0.447087\pi\)
\(488\) 3.55051 0.160724
\(489\) 17.3939 0.786578
\(490\) 0 0
\(491\) 19.5959 0.884351 0.442176 0.896928i \(-0.354207\pi\)
0.442176 + 0.896928i \(0.354207\pi\)
\(492\) −2.69694 −0.121587
\(493\) −13.7980 −0.621429
\(494\) −6.89898 −0.310400
\(495\) 0 0
\(496\) 8.89898 0.399576
\(497\) −1.10102 −0.0493875
\(498\) 6.00000 0.268866
\(499\) 6.20204 0.277641 0.138821 0.990318i \(-0.455669\pi\)
0.138821 + 0.990318i \(0.455669\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) −6.44949 −0.287855
\(503\) 4.00000 0.178351 0.0891756 0.996016i \(-0.471577\pi\)
0.0891756 + 0.996016i \(0.471577\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) 14.2020 0.631358
\(507\) 16.6515 0.739520
\(508\) 14.8990 0.661035
\(509\) −31.5505 −1.39845 −0.699226 0.714901i \(-0.746472\pi\)
−0.699226 + 0.714901i \(0.746472\pi\)
\(510\) 0 0
\(511\) −2.89898 −0.128243
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 8.69694 0.383606
\(515\) 0 0
\(516\) 2.20204 0.0969395
\(517\) 43.5959 1.91735
\(518\) −2.00000 −0.0878750
\(519\) 15.3031 0.671730
\(520\) 0 0
\(521\) 32.6969 1.43248 0.716239 0.697855i \(-0.245862\pi\)
0.716239 + 0.697855i \(0.245862\pi\)
\(522\) 20.6969 0.905880
\(523\) 33.1464 1.44939 0.724696 0.689069i \(-0.241980\pi\)
0.724696 + 0.689069i \(0.241980\pi\)
\(524\) −6.44949 −0.281747
\(525\) 0 0
\(526\) −9.79796 −0.427211
\(527\) −17.7980 −0.775291
\(528\) −12.0000 −0.522233
\(529\) −14.5959 −0.634605
\(530\) 0 0
\(531\) −4.65153 −0.201859
\(532\) 1.55051 0.0672231
\(533\) 4.89898 0.212198
\(534\) −24.4949 −1.06000
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 33.7980 1.45849
\(538\) 19.1464 0.825461
\(539\) −4.89898 −0.211014
\(540\) 0 0
\(541\) 9.59592 0.412561 0.206280 0.978493i \(-0.433864\pi\)
0.206280 + 0.978493i \(0.433864\pi\)
\(542\) 12.0000 0.515444
\(543\) −25.1010 −1.07719
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) −10.8990 −0.466433
\(547\) 18.6969 0.799423 0.399712 0.916641i \(-0.369110\pi\)
0.399712 + 0.916641i \(0.369110\pi\)
\(548\) 1.79796 0.0768050
\(549\) 10.6515 0.454596
\(550\) 0 0
\(551\) 10.6969 0.455705
\(552\) −7.10102 −0.302240
\(553\) 6.89898 0.293374
\(554\) 14.8990 0.632997
\(555\) 0 0
\(556\) 1.55051 0.0657563
\(557\) −12.6969 −0.537987 −0.268993 0.963142i \(-0.586691\pi\)
−0.268993 + 0.963142i \(0.586691\pi\)
\(558\) 26.6969 1.13017
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) −18.0000 −0.759284
\(563\) 30.0454 1.26626 0.633131 0.774044i \(-0.281769\pi\)
0.633131 + 0.774044i \(0.281769\pi\)
\(564\) −21.7980 −0.917860
\(565\) 0 0
\(566\) −3.75255 −0.157731
\(567\) −9.00000 −0.377964
\(568\) −1.10102 −0.0461978
\(569\) 33.7980 1.41688 0.708442 0.705769i \(-0.249398\pi\)
0.708442 + 0.705769i \(0.249398\pi\)
\(570\) 0 0
\(571\) −11.1010 −0.464563 −0.232282 0.972649i \(-0.574619\pi\)
−0.232282 + 0.972649i \(0.574619\pi\)
\(572\) 21.7980 0.911418
\(573\) 31.1010 1.29926
\(574\) −1.10102 −0.0459557
\(575\) 0 0
\(576\) 3.00000 0.125000
\(577\) 2.49490 0.103864 0.0519320 0.998651i \(-0.483462\pi\)
0.0519320 + 0.998651i \(0.483462\pi\)
\(578\) −13.0000 −0.540729
\(579\) 52.8990 2.19841
\(580\) 0 0
\(581\) 2.44949 0.101622
\(582\) −38.6969 −1.60404
\(583\) −53.3939 −2.21135
\(584\) −2.89898 −0.119961
\(585\) 0 0
\(586\) −18.2474 −0.753795
\(587\) −1.14643 −0.0473182 −0.0236591 0.999720i \(-0.507532\pi\)
−0.0236591 + 0.999720i \(0.507532\pi\)
\(588\) 2.44949 0.101015
\(589\) 13.7980 0.568535
\(590\) 0 0
\(591\) −46.2929 −1.90423
\(592\) −2.00000 −0.0821995
\(593\) 10.8990 0.447567 0.223784 0.974639i \(-0.428159\pi\)
0.223784 + 0.974639i \(0.428159\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.79796 −0.155570
\(597\) 41.3939 1.69414
\(598\) 12.8990 0.527478
\(599\) −13.1010 −0.535293 −0.267647 0.963517i \(-0.586246\pi\)
−0.267647 + 0.963517i \(0.586246\pi\)
\(600\) 0 0
\(601\) −39.3939 −1.60691 −0.803455 0.595366i \(-0.797007\pi\)
−0.803455 + 0.595366i \(0.797007\pi\)
\(602\) 0.898979 0.0366397
\(603\) 24.0000 0.977356
\(604\) 19.5959 0.797347
\(605\) 0 0
\(606\) 8.69694 0.353289
\(607\) −33.3939 −1.35542 −0.677708 0.735331i \(-0.737027\pi\)
−0.677708 + 0.735331i \(0.737027\pi\)
\(608\) 1.55051 0.0628815
\(609\) 16.8990 0.684781
\(610\) 0 0
\(611\) 39.5959 1.60188
\(612\) −6.00000 −0.242536
\(613\) 27.7980 1.12275 0.561374 0.827562i \(-0.310273\pi\)
0.561374 + 0.827562i \(0.310273\pi\)
\(614\) 20.2474 0.817121
\(615\) 0 0
\(616\) −4.89898 −0.197386
\(617\) −29.5959 −1.19149 −0.595743 0.803175i \(-0.703142\pi\)
−0.595743 + 0.803175i \(0.703142\pi\)
\(618\) −31.5959 −1.27097
\(619\) −41.5505 −1.67006 −0.835028 0.550207i \(-0.814549\pi\)
−0.835028 + 0.550207i \(0.814549\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) −10.0000 −0.400642
\(624\) −10.8990 −0.436308
\(625\) 0 0
\(626\) 21.5959 0.863146
\(627\) −18.6061 −0.743057
\(628\) −3.55051 −0.141681
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) −42.4949 −1.69170 −0.845848 0.533425i \(-0.820905\pi\)
−0.845848 + 0.533425i \(0.820905\pi\)
\(632\) 6.89898 0.274427
\(633\) 29.3939 1.16830
\(634\) 22.4949 0.893387
\(635\) 0 0
\(636\) 26.6969 1.05860
\(637\) −4.44949 −0.176295
\(638\) −33.7980 −1.33807
\(639\) −3.30306 −0.130667
\(640\) 0 0
\(641\) 25.7980 1.01896 0.509479 0.860483i \(-0.329838\pi\)
0.509479 + 0.860483i \(0.329838\pi\)
\(642\) 19.5959 0.773389
\(643\) −25.1464 −0.991678 −0.495839 0.868414i \(-0.665139\pi\)
−0.495839 + 0.868414i \(0.665139\pi\)
\(644\) −2.89898 −0.114236
\(645\) 0 0
\(646\) −3.10102 −0.122008
\(647\) 46.2929 1.81996 0.909980 0.414652i \(-0.136097\pi\)
0.909980 + 0.414652i \(0.136097\pi\)
\(648\) −9.00000 −0.353553
\(649\) 7.59592 0.298166
\(650\) 0 0
\(651\) 21.7980 0.854329
\(652\) 7.10102 0.278097
\(653\) 20.2020 0.790567 0.395283 0.918559i \(-0.370646\pi\)
0.395283 + 0.918559i \(0.370646\pi\)
\(654\) 16.8990 0.660802
\(655\) 0 0
\(656\) −1.10102 −0.0429876
\(657\) −8.69694 −0.339300
\(658\) −8.89898 −0.346918
\(659\) 16.8990 0.658291 0.329145 0.944279i \(-0.393239\pi\)
0.329145 + 0.944279i \(0.393239\pi\)
\(660\) 0 0
\(661\) −40.9444 −1.59255 −0.796276 0.604933i \(-0.793200\pi\)
−0.796276 + 0.604933i \(0.793200\pi\)
\(662\) −18.6969 −0.726677
\(663\) 21.7980 0.846563
\(664\) 2.44949 0.0950586
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) −20.0000 −0.774403
\(668\) 4.89898 0.189547
\(669\) 9.79796 0.378811
\(670\) 0 0
\(671\) −17.3939 −0.671483
\(672\) 2.44949 0.0944911
\(673\) 17.7980 0.686061 0.343030 0.939324i \(-0.388547\pi\)
0.343030 + 0.939324i \(0.388547\pi\)
\(674\) −9.59592 −0.369621
\(675\) 0 0
\(676\) 6.79796 0.261460
\(677\) 36.4495 1.40087 0.700434 0.713717i \(-0.252990\pi\)
0.700434 + 0.713717i \(0.252990\pi\)
\(678\) −48.4949 −1.86244
\(679\) −15.7980 −0.606270
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) −43.5959 −1.66937
\(683\) −3.59592 −0.137594 −0.0687970 0.997631i \(-0.521916\pi\)
−0.0687970 + 0.997631i \(0.521916\pi\)
\(684\) 4.65153 0.177856
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) −46.8990 −1.78931
\(688\) 0.898979 0.0342733
\(689\) −48.4949 −1.84751
\(690\) 0 0
\(691\) 21.1464 0.804448 0.402224 0.915541i \(-0.368237\pi\)
0.402224 + 0.915541i \(0.368237\pi\)
\(692\) 6.24745 0.237492
\(693\) −14.6969 −0.558291
\(694\) −28.8990 −1.09699
\(695\) 0 0
\(696\) 16.8990 0.640554
\(697\) 2.20204 0.0834083
\(698\) −8.44949 −0.319818
\(699\) −72.9898 −2.76073
\(700\) 0 0
\(701\) 11.3031 0.426911 0.213455 0.976953i \(-0.431528\pi\)
0.213455 + 0.976953i \(0.431528\pi\)
\(702\) 0 0
\(703\) −3.10102 −0.116957
\(704\) −4.89898 −0.184637
\(705\) 0 0
\(706\) −22.8990 −0.861814
\(707\) 3.55051 0.133531
\(708\) −3.79796 −0.142736
\(709\) −28.2929 −1.06256 −0.531280 0.847196i \(-0.678289\pi\)
−0.531280 + 0.847196i \(0.678289\pi\)
\(710\) 0 0
\(711\) 20.6969 0.776196
\(712\) −10.0000 −0.374766
\(713\) −25.7980 −0.966141
\(714\) −4.89898 −0.183340
\(715\) 0 0
\(716\) 13.7980 0.515654
\(717\) −15.1918 −0.567350
\(718\) −27.5959 −1.02987
\(719\) −4.49490 −0.167631 −0.0838157 0.996481i \(-0.526711\pi\)
−0.0838157 + 0.996481i \(0.526711\pi\)
\(720\) 0 0
\(721\) −12.8990 −0.480383
\(722\) −16.5959 −0.617636
\(723\) −21.3031 −0.792269
\(724\) −10.2474 −0.380843
\(725\) 0 0
\(726\) 31.8434 1.18182
\(727\) −22.6969 −0.841783 −0.420891 0.907111i \(-0.638283\pi\)
−0.420891 + 0.907111i \(0.638283\pi\)
\(728\) −4.44949 −0.164909
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −1.79796 −0.0664999
\(732\) 8.69694 0.321448
\(733\) −39.6413 −1.46419 −0.732093 0.681205i \(-0.761456\pi\)
−0.732093 + 0.681205i \(0.761456\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) −2.89898 −0.106858
\(737\) −39.1918 −1.44365
\(738\) −3.30306 −0.121587
\(739\) 4.49490 0.165347 0.0826737 0.996577i \(-0.473654\pi\)
0.0826737 + 0.996577i \(0.473654\pi\)
\(740\) 0 0
\(741\) −16.8990 −0.620800
\(742\) 10.8990 0.400114
\(743\) 44.6969 1.63977 0.819886 0.572527i \(-0.194037\pi\)
0.819886 + 0.572527i \(0.194037\pi\)
\(744\) 21.7980 0.799152
\(745\) 0 0
\(746\) 4.69694 0.171967
\(747\) 7.34847 0.268866
\(748\) 9.79796 0.358249
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) −41.7980 −1.52523 −0.762615 0.646853i \(-0.776085\pi\)
−0.762615 + 0.646853i \(0.776085\pi\)
\(752\) −8.89898 −0.324512
\(753\) −15.7980 −0.575710
\(754\) −30.6969 −1.11792
\(755\) 0 0
\(756\) 0 0
\(757\) 51.7980 1.88263 0.941314 0.337531i \(-0.109592\pi\)
0.941314 + 0.337531i \(0.109592\pi\)
\(758\) 30.6969 1.11496
\(759\) 34.7878 1.26272
\(760\) 0 0
\(761\) −21.1010 −0.764911 −0.382456 0.923974i \(-0.624922\pi\)
−0.382456 + 0.923974i \(0.624922\pi\)
\(762\) 36.4949 1.32207
\(763\) 6.89898 0.249760
\(764\) 12.6969 0.459359
\(765\) 0 0
\(766\) 7.10102 0.256570
\(767\) 6.89898 0.249108
\(768\) 2.44949 0.0883883
\(769\) −40.6969 −1.46757 −0.733785 0.679382i \(-0.762248\pi\)
−0.733785 + 0.679382i \(0.762248\pi\)
\(770\) 0 0
\(771\) 21.3031 0.767211
\(772\) 21.5959 0.777254
\(773\) −1.34847 −0.0485011 −0.0242505 0.999706i \(-0.507720\pi\)
−0.0242505 + 0.999706i \(0.507720\pi\)
\(774\) 2.69694 0.0969395
\(775\) 0 0
\(776\) −15.7980 −0.567114
\(777\) −4.89898 −0.175750
\(778\) 13.1010 0.469694
\(779\) −1.70714 −0.0611648
\(780\) 0 0
\(781\) 5.39388 0.193008
\(782\) 5.79796 0.207335
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) −15.7980 −0.563495
\(787\) −50.4495 −1.79833 −0.899165 0.437610i \(-0.855825\pi\)
−0.899165 + 0.437610i \(0.855825\pi\)
\(788\) −18.8990 −0.673248
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −19.7980 −0.703934
\(792\) −14.6969 −0.522233
\(793\) −15.7980 −0.561002
\(794\) 2.65153 0.0940993
\(795\) 0 0
\(796\) 16.8990 0.598968
\(797\) 0.944387 0.0334519 0.0167260 0.999860i \(-0.494676\pi\)
0.0167260 + 0.999860i \(0.494676\pi\)
\(798\) 3.79796 0.134446
\(799\) 17.7980 0.629647
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) −29.3939 −1.03793
\(803\) 14.2020 0.501179
\(804\) 19.5959 0.691095
\(805\) 0 0
\(806\) −39.5959 −1.39471
\(807\) 46.8990 1.65092
\(808\) 3.55051 0.124907
\(809\) 47.5959 1.67338 0.836692 0.547674i \(-0.184487\pi\)
0.836692 + 0.547674i \(0.184487\pi\)
\(810\) 0 0
\(811\) 14.9444 0.524768 0.262384 0.964963i \(-0.415491\pi\)
0.262384 + 0.964963i \(0.415491\pi\)
\(812\) 6.89898 0.242107
\(813\) 29.3939 1.03089
\(814\) 9.79796 0.343418
\(815\) 0 0
\(816\) −4.89898 −0.171499
\(817\) 1.39388 0.0487656
\(818\) 34.4949 1.20609
\(819\) −13.3485 −0.466433
\(820\) 0 0
\(821\) 8.20204 0.286253 0.143127 0.989704i \(-0.454284\pi\)
0.143127 + 0.989704i \(0.454284\pi\)
\(822\) 4.40408 0.153610
\(823\) 39.1918 1.36614 0.683071 0.730352i \(-0.260644\pi\)
0.683071 + 0.730352i \(0.260644\pi\)
\(824\) −12.8990 −0.449357
\(825\) 0 0
\(826\) −1.55051 −0.0539492
\(827\) 15.5959 0.542323 0.271162 0.962534i \(-0.412592\pi\)
0.271162 + 0.962534i \(0.412592\pi\)
\(828\) −8.69694 −0.302240
\(829\) −43.6413 −1.51573 −0.757863 0.652414i \(-0.773756\pi\)
−0.757863 + 0.652414i \(0.773756\pi\)
\(830\) 0 0
\(831\) 36.4949 1.26599
\(832\) −4.44949 −0.154258
\(833\) −2.00000 −0.0692959
\(834\) 3.79796 0.131513
\(835\) 0 0
\(836\) −7.59592 −0.262710
\(837\) 0 0
\(838\) −1.55051 −0.0535615
\(839\) 36.8990 1.27389 0.636947 0.770907i \(-0.280197\pi\)
0.636947 + 0.770907i \(0.280197\pi\)
\(840\) 0 0
\(841\) 18.5959 0.641239
\(842\) −4.20204 −0.144812
\(843\) −44.0908 −1.51857
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) −26.6969 −0.917860
\(847\) 13.0000 0.446685
\(848\) 10.8990 0.374272
\(849\) −9.19184 −0.315463
\(850\) 0 0
\(851\) 5.79796 0.198751
\(852\) −2.69694 −0.0923956
\(853\) 33.8434 1.15877 0.579387 0.815052i \(-0.303292\pi\)
0.579387 + 0.815052i \(0.303292\pi\)
\(854\) 3.55051 0.121496
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) 53.1918 1.81700 0.908499 0.417886i \(-0.137229\pi\)
0.908499 + 0.417886i \(0.137229\pi\)
\(858\) 53.3939 1.82284
\(859\) 53.6413 1.83022 0.915109 0.403206i \(-0.132104\pi\)
0.915109 + 0.403206i \(0.132104\pi\)
\(860\) 0 0
\(861\) −2.69694 −0.0919114
\(862\) −1.79796 −0.0612387
\(863\) 45.3939 1.54523 0.772613 0.634878i \(-0.218949\pi\)
0.772613 + 0.634878i \(0.218949\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0.202041 0.00686563
\(867\) −31.8434 −1.08146
\(868\) 8.89898 0.302051
\(869\) −33.7980 −1.14652
\(870\) 0 0
\(871\) −35.5959 −1.20612
\(872\) 6.89898 0.233629
\(873\) −47.3939 −1.60404
\(874\) −4.49490 −0.152042
\(875\) 0 0
\(876\) −7.10102 −0.239921
\(877\) 39.3939 1.33024 0.665118 0.746738i \(-0.268381\pi\)
0.665118 + 0.746738i \(0.268381\pi\)
\(878\) −21.3939 −0.722008
\(879\) −44.6969 −1.50759
\(880\) 0 0
\(881\) 8.20204 0.276334 0.138167 0.990409i \(-0.455879\pi\)
0.138167 + 0.990409i \(0.455879\pi\)
\(882\) 3.00000 0.101015
\(883\) −22.2020 −0.747158 −0.373579 0.927598i \(-0.621870\pi\)
−0.373579 + 0.927598i \(0.621870\pi\)
\(884\) 8.89898 0.299305
\(885\) 0 0
\(886\) −9.79796 −0.329169
\(887\) −2.69694 −0.0905543 −0.0452772 0.998974i \(-0.514417\pi\)
−0.0452772 + 0.998974i \(0.514417\pi\)
\(888\) −4.89898 −0.164399
\(889\) 14.8990 0.499696
\(890\) 0 0
\(891\) 44.0908 1.47710
\(892\) 4.00000 0.133930
\(893\) −13.7980 −0.461731
\(894\) −9.30306 −0.311141
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 31.5959 1.05496
\(898\) −10.0000 −0.333704
\(899\) 61.3939 2.04760
\(900\) 0 0
\(901\) −21.7980 −0.726195
\(902\) 5.39388 0.179596
\(903\) 2.20204 0.0732793
\(904\) −19.7980 −0.658470
\(905\) 0 0
\(906\) 48.0000 1.59469
\(907\) 41.7980 1.38788 0.693939 0.720034i \(-0.255874\pi\)
0.693939 + 0.720034i \(0.255874\pi\)
\(908\) −7.34847 −0.243868
\(909\) 10.6515 0.353289
\(910\) 0 0
\(911\) −35.5959 −1.17935 −0.589673 0.807642i \(-0.700743\pi\)
−0.589673 + 0.807642i \(0.700743\pi\)
\(912\) 3.79796 0.125763
\(913\) −12.0000 −0.397142
\(914\) −29.5959 −0.978946
\(915\) 0 0
\(916\) −19.1464 −0.632616
\(917\) −6.44949 −0.212981
\(918\) 0 0
\(919\) −26.8990 −0.887315 −0.443658 0.896196i \(-0.646319\pi\)
−0.443658 + 0.896196i \(0.646319\pi\)
\(920\) 0 0
\(921\) 49.5959 1.63424
\(922\) 17.3485 0.571341
\(923\) 4.89898 0.161252
\(924\) −12.0000 −0.394771
\(925\) 0 0
\(926\) −3.59592 −0.118169
\(927\) −38.6969 −1.27097
\(928\) 6.89898 0.226470
\(929\) −28.2929 −0.928259 −0.464129 0.885767i \(-0.653633\pi\)
−0.464129 + 0.885767i \(0.653633\pi\)
\(930\) 0 0
\(931\) 1.55051 0.0508159
\(932\) −29.7980 −0.976065
\(933\) 29.3939 0.962312
\(934\) −10.4495 −0.341918
\(935\) 0 0
\(936\) −13.3485 −0.436308
\(937\) 41.1010 1.34271 0.671356 0.741135i \(-0.265712\pi\)
0.671356 + 0.741135i \(0.265712\pi\)
\(938\) 8.00000 0.261209
\(939\) 52.8990 1.72629
\(940\) 0 0
\(941\) −19.5505 −0.637328 −0.318664 0.947868i \(-0.603234\pi\)
−0.318664 + 0.947868i \(0.603234\pi\)
\(942\) −8.69694 −0.283362
\(943\) 3.19184 0.103940
\(944\) −1.55051 −0.0504648
\(945\) 0 0
\(946\) −4.40408 −0.143189
\(947\) −44.0908 −1.43276 −0.716379 0.697711i \(-0.754202\pi\)
−0.716379 + 0.697711i \(0.754202\pi\)
\(948\) 16.8990 0.548853
\(949\) 12.8990 0.418719
\(950\) 0 0
\(951\) 55.1010 1.78677
\(952\) −2.00000 −0.0648204
\(953\) −2.20204 −0.0713311 −0.0356656 0.999364i \(-0.511355\pi\)
−0.0356656 + 0.999364i \(0.511355\pi\)
\(954\) 32.6969 1.05860
\(955\) 0 0
\(956\) −6.20204 −0.200588
\(957\) −82.7878 −2.67615
\(958\) 9.30306 0.300568
\(959\) 1.79796 0.0580591
\(960\) 0 0
\(961\) 48.1918 1.55458
\(962\) 8.89898 0.286915
\(963\) 24.0000 0.773389
\(964\) −8.69694 −0.280110
\(965\) 0 0
\(966\) −7.10102 −0.228472
\(967\) 36.2929 1.16710 0.583550 0.812077i \(-0.301663\pi\)
0.583550 + 0.812077i \(0.301663\pi\)
\(968\) 13.0000 0.417836
\(969\) −7.59592 −0.244016
\(970\) 0 0
\(971\) −9.55051 −0.306490 −0.153245 0.988188i \(-0.548972\pi\)
−0.153245 + 0.988188i \(0.548972\pi\)
\(972\) −22.0454 −0.707107
\(973\) 1.55051 0.0497071
\(974\) 7.30306 0.234005
\(975\) 0 0
\(976\) 3.55051 0.113649
\(977\) 29.3939 0.940393 0.470197 0.882562i \(-0.344183\pi\)
0.470197 + 0.882562i \(0.344183\pi\)
\(978\) 17.3939 0.556195
\(979\) 48.9898 1.56572
\(980\) 0 0
\(981\) 20.6969 0.660802
\(982\) 19.5959 0.625331
\(983\) 13.3031 0.424302 0.212151 0.977237i \(-0.431953\pi\)
0.212151 + 0.977237i \(0.431953\pi\)
\(984\) −2.69694 −0.0859753
\(985\) 0 0
\(986\) −13.7980 −0.439417
\(987\) −21.7980 −0.693837
\(988\) −6.89898 −0.219486
\(989\) −2.60612 −0.0828699
\(990\) 0 0
\(991\) 31.3031 0.994375 0.497187 0.867643i \(-0.334366\pi\)
0.497187 + 0.867643i \(0.334366\pi\)
\(992\) 8.89898 0.282543
\(993\) −45.7980 −1.45335
\(994\) −1.10102 −0.0349223
\(995\) 0 0
\(996\) 6.00000 0.190117
\(997\) −57.3485 −1.81624 −0.908122 0.418705i \(-0.862484\pi\)
−0.908122 + 0.418705i \(0.862484\pi\)
\(998\) 6.20204 0.196322
\(999\) 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 350.2.a.h.1.2 2
3.2 odd 2 3150.2.a.bs.1.2 2
4.3 odd 2 2800.2.a.bl.1.1 2
5.2 odd 4 70.2.c.a.29.3 yes 4
5.3 odd 4 70.2.c.a.29.2 4
5.4 even 2 350.2.a.g.1.1 2
7.6 odd 2 2450.2.a.bq.1.1 2
15.2 even 4 630.2.g.g.379.1 4
15.8 even 4 630.2.g.g.379.3 4
15.14 odd 2 3150.2.a.bt.1.2 2
20.3 even 4 560.2.g.e.449.2 4
20.7 even 4 560.2.g.e.449.4 4
20.19 odd 2 2800.2.a.bm.1.2 2
35.2 odd 12 490.2.i.c.459.4 8
35.3 even 12 490.2.i.f.79.3 8
35.12 even 12 490.2.i.f.459.3 8
35.13 even 4 490.2.c.e.99.1 4
35.17 even 12 490.2.i.f.79.2 8
35.18 odd 12 490.2.i.c.79.4 8
35.23 odd 12 490.2.i.c.459.1 8
35.27 even 4 490.2.c.e.99.4 4
35.32 odd 12 490.2.i.c.79.1 8
35.33 even 12 490.2.i.f.459.2 8
35.34 odd 2 2450.2.a.bl.1.2 2
40.3 even 4 2240.2.g.i.449.3 4
40.13 odd 4 2240.2.g.j.449.1 4
40.27 even 4 2240.2.g.i.449.1 4
40.37 odd 4 2240.2.g.j.449.3 4
60.23 odd 4 5040.2.t.t.1009.2 4
60.47 odd 4 5040.2.t.t.1009.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.c.a.29.2 4 5.3 odd 4
70.2.c.a.29.3 yes 4 5.2 odd 4
350.2.a.g.1.1 2 5.4 even 2
350.2.a.h.1.2 2 1.1 even 1 trivial
490.2.c.e.99.1 4 35.13 even 4
490.2.c.e.99.4 4 35.27 even 4
490.2.i.c.79.1 8 35.32 odd 12
490.2.i.c.79.4 8 35.18 odd 12
490.2.i.c.459.1 8 35.23 odd 12
490.2.i.c.459.4 8 35.2 odd 12
490.2.i.f.79.2 8 35.17 even 12
490.2.i.f.79.3 8 35.3 even 12
490.2.i.f.459.2 8 35.33 even 12
490.2.i.f.459.3 8 35.12 even 12
560.2.g.e.449.2 4 20.3 even 4
560.2.g.e.449.4 4 20.7 even 4
630.2.g.g.379.1 4 15.2 even 4
630.2.g.g.379.3 4 15.8 even 4
2240.2.g.i.449.1 4 40.27 even 4
2240.2.g.i.449.3 4 40.3 even 4
2240.2.g.j.449.1 4 40.13 odd 4
2240.2.g.j.449.3 4 40.37 odd 4
2450.2.a.bl.1.2 2 35.34 odd 2
2450.2.a.bq.1.1 2 7.6 odd 2
2800.2.a.bl.1.1 2 4.3 odd 2
2800.2.a.bm.1.2 2 20.19 odd 2
3150.2.a.bs.1.2 2 3.2 odd 2
3150.2.a.bt.1.2 2 15.14 odd 2
5040.2.t.t.1009.1 4 60.47 odd 4
5040.2.t.t.1009.2 4 60.23 odd 4