Properties

Label 5070.2.a.ca.1.3
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.131472.2
Defining polynomial: \(x^{4} - 2 x^{3} - 19 x^{2} + 20 x + 52\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.32258\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.32258 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.32258 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +5.34541 q^{11} +1.00000 q^{12} +2.32258 q^{14} -1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} -4.02283 q^{19} -1.00000 q^{20} +2.32258 q^{21} +5.34541 q^{22} -4.93593 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +2.32258 q^{28} +4.29078 q^{29} -1.00000 q^{30} +3.47183 q^{31} +1.00000 q^{32} +5.34541 q^{33} +4.00000 q^{34} -2.32258 q^{35} +1.00000 q^{36} -3.14925 q^{37} -4.02283 q^{38} -1.00000 q^{40} -2.64516 q^{41} +2.32258 q^{42} +12.2508 q^{43} +5.34541 q^{44} -1.00000 q^{45} -4.93593 q^{46} +1.81894 q^{47} +1.00000 q^{48} -1.60562 q^{49} +1.00000 q^{50} +4.00000 q^{51} -5.48693 q^{53} +1.00000 q^{54} -5.34541 q^{55} +2.32258 q^{56} -4.02283 q^{57} +4.29078 q^{58} -6.78668 q^{59} -1.00000 q^{60} +0.535898 q^{61} +3.47183 q^{62} +2.32258 q^{63} +1.00000 q^{64} +5.34541 q^{66} -4.10926 q^{67} +4.00000 q^{68} -4.93593 q^{69} -2.32258 q^{70} +15.8719 q^{71} +1.00000 q^{72} +13.5734 q^{73} -3.14925 q^{74} +1.00000 q^{75} -4.02283 q^{76} +12.4151 q^{77} -7.96774 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.64516 q^{82} -11.3360 q^{83} +2.32258 q^{84} -4.00000 q^{85} +12.2508 q^{86} +4.29078 q^{87} +5.34541 q^{88} -1.73978 q^{89} -1.00000 q^{90} -4.93593 q^{92} +3.47183 q^{93} +1.81894 q^{94} +4.02283 q^{95} +1.00000 q^{96} -16.1093 q^{97} -1.60562 q^{98} +5.34541 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + 2q^{7} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + 2q^{7} + 4q^{8} + 4q^{9} - 4q^{10} - 2q^{11} + 4q^{12} + 2q^{14} - 4q^{15} + 4q^{16} + 16q^{17} + 4q^{18} - 4q^{20} + 2q^{21} - 2q^{22} + 4q^{23} + 4q^{24} + 4q^{25} + 4q^{27} + 2q^{28} + 8q^{29} - 4q^{30} + 4q^{31} + 4q^{32} - 2q^{33} + 16q^{34} - 2q^{35} + 4q^{36} - 10q^{37} - 4q^{40} + 4q^{41} + 2q^{42} + 14q^{43} - 2q^{44} - 4q^{45} + 4q^{46} + 8q^{47} + 4q^{48} + 14q^{49} + 4q^{50} + 16q^{51} + 8q^{53} + 4q^{54} + 2q^{55} + 2q^{56} + 8q^{58} - 6q^{59} - 4q^{60} + 16q^{61} + 4q^{62} + 2q^{63} + 4q^{64} - 2q^{66} + 12q^{67} + 16q^{68} + 4q^{69} - 2q^{70} + 16q^{71} + 4q^{72} + 12q^{73} - 10q^{74} + 4q^{75} - 8q^{77} - 10q^{79} - 4q^{80} + 4q^{81} + 4q^{82} + 16q^{83} + 2q^{84} - 16q^{85} + 14q^{86} + 8q^{87} - 2q^{88} - 4q^{89} - 4q^{90} + 4q^{92} + 4q^{93} + 8q^{94} + 4q^{96} - 36q^{97} + 14q^{98} - 2q^{99} + O(q^{100}) \)

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
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) 2.32258 0.877853 0.438926 0.898523i \(-0.355359\pi\)
0.438926 + 0.898523i \(0.355359\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 5.34541 1.61170 0.805850 0.592119i \(-0.201709\pi\)
0.805850 + 0.592119i \(0.201709\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 2.32258 0.620736
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.02283 −0.922900 −0.461450 0.887166i \(-0.652671\pi\)
−0.461450 + 0.887166i \(0.652671\pi\)
\(20\) −1.00000 −0.223607
\(21\) 2.32258 0.506828
\(22\) 5.34541 1.13964
\(23\) −4.93593 −1.02921 −0.514607 0.857426i \(-0.672062\pi\)
−0.514607 + 0.857426i \(0.672062\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 2.32258 0.438926
\(29\) 4.29078 0.796777 0.398388 0.917217i \(-0.369570\pi\)
0.398388 + 0.917217i \(0.369570\pi\)
\(30\) −1.00000 −0.182574
\(31\) 3.47183 0.623560 0.311780 0.950154i \(-0.399075\pi\)
0.311780 + 0.950154i \(0.399075\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.34541 0.930516
\(34\) 4.00000 0.685994
\(35\) −2.32258 −0.392588
\(36\) 1.00000 0.166667
\(37\) −3.14925 −0.517734 −0.258867 0.965913i \(-0.583349\pi\)
−0.258867 + 0.965913i \(0.583349\pi\)
\(38\) −4.02283 −0.652589
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −2.64516 −0.413104 −0.206552 0.978436i \(-0.566224\pi\)
−0.206552 + 0.978436i \(0.566224\pi\)
\(42\) 2.32258 0.358382
\(43\) 12.2508 1.86823 0.934113 0.356976i \(-0.116192\pi\)
0.934113 + 0.356976i \(0.116192\pi\)
\(44\) 5.34541 0.805850
\(45\) −1.00000 −0.149071
\(46\) −4.93593 −0.727764
\(47\) 1.81894 0.265320 0.132660 0.991162i \(-0.457648\pi\)
0.132660 + 0.991162i \(0.457648\pi\)
\(48\) 1.00000 0.144338
\(49\) −1.60562 −0.229375
\(50\) 1.00000 0.141421
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) −5.48693 −0.753687 −0.376844 0.926277i \(-0.622991\pi\)
−0.376844 + 0.926277i \(0.622991\pi\)
\(54\) 1.00000 0.136083
\(55\) −5.34541 −0.720774
\(56\) 2.32258 0.310368
\(57\) −4.02283 −0.532836
\(58\) 4.29078 0.563406
\(59\) −6.78668 −0.883551 −0.441775 0.897126i \(-0.645651\pi\)
−0.441775 + 0.897126i \(0.645651\pi\)
\(60\) −1.00000 −0.129099
\(61\) 0.535898 0.0686148 0.0343074 0.999411i \(-0.489077\pi\)
0.0343074 + 0.999411i \(0.489077\pi\)
\(62\) 3.47183 0.440923
\(63\) 2.32258 0.292618
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 5.34541 0.657974
\(67\) −4.10926 −0.502026 −0.251013 0.967984i \(-0.580764\pi\)
−0.251013 + 0.967984i \(0.580764\pi\)
\(68\) 4.00000 0.485071
\(69\) −4.93593 −0.594217
\(70\) −2.32258 −0.277601
\(71\) 15.8719 1.88364 0.941822 0.336112i \(-0.109112\pi\)
0.941822 + 0.336112i \(0.109112\pi\)
\(72\) 1.00000 0.117851
\(73\) 13.5734 1.58864 0.794321 0.607498i \(-0.207827\pi\)
0.794321 + 0.607498i \(0.207827\pi\)
\(74\) −3.14925 −0.366093
\(75\) 1.00000 0.115470
\(76\) −4.02283 −0.461450
\(77\) 12.4151 1.41484
\(78\) 0 0
\(79\) −7.96774 −0.896441 −0.448220 0.893923i \(-0.647942\pi\)
−0.448220 + 0.893923i \(0.647942\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −2.64516 −0.292109
\(83\) −11.3360 −1.24428 −0.622142 0.782904i \(-0.713737\pi\)
−0.622142 + 0.782904i \(0.713737\pi\)
\(84\) 2.32258 0.253414
\(85\) −4.00000 −0.433861
\(86\) 12.2508 1.32104
\(87\) 4.29078 0.460019
\(88\) 5.34541 0.569822
\(89\) −1.73978 −0.184417 −0.0922083 0.995740i \(-0.529393\pi\)
−0.0922083 + 0.995740i \(0.529393\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −4.93593 −0.514607
\(93\) 3.47183 0.360012
\(94\) 1.81894 0.187610
\(95\) 4.02283 0.412733
\(96\) 1.00000 0.102062
\(97\) −16.1093 −1.63565 −0.817824 0.575469i \(-0.804820\pi\)
−0.817824 + 0.575469i \(0.804820\pi\)
\(98\) −1.60562 −0.162192
\(99\) 5.34541 0.537233
\(100\) 1.00000 0.100000
\(101\) 12.6908 1.26278 0.631391 0.775464i \(-0.282484\pi\)
0.631391 + 0.775464i \(0.282484\pi\)
\(102\) 4.00000 0.396059
\(103\) 4.79612 0.472575 0.236288 0.971683i \(-0.424069\pi\)
0.236288 + 0.971683i \(0.424069\pi\)
\(104\) 0 0
\(105\) −2.32258 −0.226661
\(106\) −5.48693 −0.532938
\(107\) 3.07180 0.296962 0.148481 0.988915i \(-0.452562\pi\)
0.148481 + 0.988915i \(0.452562\pi\)
\(108\) 1.00000 0.0962250
\(109\) −6.69081 −0.640864 −0.320432 0.947272i \(-0.603828\pi\)
−0.320432 + 0.947272i \(0.603828\pi\)
\(110\) −5.34541 −0.509664
\(111\) −3.14925 −0.298914
\(112\) 2.32258 0.219463
\(113\) −7.10972 −0.668826 −0.334413 0.942427i \(-0.608538\pi\)
−0.334413 + 0.942427i \(0.608538\pi\)
\(114\) −4.02283 −0.376772
\(115\) 4.93593 0.460278
\(116\) 4.29078 0.398388
\(117\) 0 0
\(118\) −6.78668 −0.624765
\(119\) 9.29032 0.851642
\(120\) −1.00000 −0.0912871
\(121\) 17.5734 1.59758
\(122\) 0.535898 0.0485180
\(123\) −2.64516 −0.238506
\(124\) 3.47183 0.311780
\(125\) −1.00000 −0.0894427
\(126\) 2.32258 0.206912
\(127\) −2.13209 −0.189192 −0.0945961 0.995516i \(-0.530156\pi\)
−0.0945961 + 0.995516i \(0.530156\pi\)
\(128\) 1.00000 0.0883883
\(129\) 12.2508 1.07862
\(130\) 0 0
\(131\) 1.25851 0.109957 0.0549785 0.998488i \(-0.482491\pi\)
0.0549785 + 0.998488i \(0.482491\pi\)
\(132\) 5.34541 0.465258
\(133\) −9.34333 −0.810170
\(134\) −4.10926 −0.354986
\(135\) −1.00000 −0.0860663
\(136\) 4.00000 0.342997
\(137\) −19.7149 −1.68436 −0.842178 0.539199i \(-0.818727\pi\)
−0.842178 + 0.539199i \(0.818727\pi\)
\(138\) −4.93593 −0.420175
\(139\) 5.67742 0.481553 0.240776 0.970581i \(-0.422598\pi\)
0.240776 + 0.970581i \(0.422598\pi\)
\(140\) −2.32258 −0.196294
\(141\) 1.81894 0.153183
\(142\) 15.8719 1.33194
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −4.29078 −0.356329
\(146\) 13.5734 1.12334
\(147\) −1.60562 −0.132430
\(148\) −3.14925 −0.258867
\(149\) −19.4775 −1.59566 −0.797829 0.602884i \(-0.794018\pi\)
−0.797829 + 0.602884i \(0.794018\pi\)
\(150\) 1.00000 0.0816497
\(151\) −14.5170 −1.18138 −0.590690 0.806899i \(-0.701144\pi\)
−0.590690 + 0.806899i \(0.701144\pi\)
\(152\) −4.02283 −0.326294
\(153\) 4.00000 0.323381
\(154\) 12.4151 1.00044
\(155\) −3.47183 −0.278864
\(156\) 0 0
\(157\) 24.3829 1.94596 0.972982 0.230879i \(-0.0741602\pi\)
0.972982 + 0.230879i \(0.0741602\pi\)
\(158\) −7.96774 −0.633879
\(159\) −5.48693 −0.435142
\(160\) −1.00000 −0.0790569
\(161\) −11.4641 −0.903498
\(162\) 1.00000 0.0785674
\(163\) −23.6267 −1.85059 −0.925295 0.379249i \(-0.876182\pi\)
−0.925295 + 0.379249i \(0.876182\pi\)
\(164\) −2.64516 −0.206552
\(165\) −5.34541 −0.416139
\(166\) −11.3360 −0.879842
\(167\) 16.7471 1.29593 0.647967 0.761669i \(-0.275620\pi\)
0.647967 + 0.761669i \(0.275620\pi\)
\(168\) 2.32258 0.179191
\(169\) 0 0
\(170\) −4.00000 −0.306786
\(171\) −4.02283 −0.307633
\(172\) 12.2508 0.934113
\(173\) 16.1321 1.22650 0.613250 0.789889i \(-0.289862\pi\)
0.613250 + 0.789889i \(0.289862\pi\)
\(174\) 4.29078 0.325283
\(175\) 2.32258 0.175571
\(176\) 5.34541 0.402925
\(177\) −6.78668 −0.510118
\(178\) −1.73978 −0.130402
\(179\) 7.32824 0.547738 0.273869 0.961767i \(-0.411696\pi\)
0.273869 + 0.961767i \(0.411696\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 19.5734 1.45488 0.727438 0.686173i \(-0.240711\pi\)
0.727438 + 0.686173i \(0.240711\pi\)
\(182\) 0 0
\(183\) 0.535898 0.0396147
\(184\) −4.93593 −0.363882
\(185\) 3.14925 0.231538
\(186\) 3.47183 0.254567
\(187\) 21.3816 1.56358
\(188\) 1.81894 0.132660
\(189\) 2.32258 0.168943
\(190\) 4.02283 0.291846
\(191\) −17.0375 −1.23279 −0.616394 0.787438i \(-0.711407\pi\)
−0.616394 + 0.787438i \(0.711407\pi\)
\(192\) 1.00000 0.0721688
\(193\) 6.15491 0.443040 0.221520 0.975156i \(-0.428898\pi\)
0.221520 + 0.975156i \(0.428898\pi\)
\(194\) −16.1093 −1.15658
\(195\) 0 0
\(196\) −1.60562 −0.114687
\(197\) 13.7867 0.982260 0.491130 0.871086i \(-0.336584\pi\)
0.491130 + 0.871086i \(0.336584\pi\)
\(198\) 5.34541 0.379881
\(199\) 2.28304 0.161841 0.0809203 0.996721i \(-0.474214\pi\)
0.0809203 + 0.996721i \(0.474214\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.10926 −0.289845
\(202\) 12.6908 0.892922
\(203\) 9.96567 0.699453
\(204\) 4.00000 0.280056
\(205\) 2.64516 0.184746
\(206\) 4.79612 0.334161
\(207\) −4.93593 −0.343071
\(208\) 0 0
\(209\) −21.5036 −1.48744
\(210\) −2.32258 −0.160273
\(211\) 22.4775 1.54741 0.773707 0.633543i \(-0.218400\pi\)
0.773707 + 0.633543i \(0.218400\pi\)
\(212\) −5.48693 −0.376844
\(213\) 15.8719 1.08752
\(214\) 3.07180 0.209984
\(215\) −12.2508 −0.835496
\(216\) 1.00000 0.0680414
\(217\) 8.06361 0.547393
\(218\) −6.69081 −0.453159
\(219\) 13.5734 0.917203
\(220\) −5.34541 −0.360387
\(221\) 0 0
\(222\) −3.14925 −0.211364
\(223\) −1.37891 −0.0923389 −0.0461694 0.998934i \(-0.514701\pi\)
−0.0461694 + 0.998934i \(0.514701\pi\)
\(224\) 2.32258 0.155184
\(225\) 1.00000 0.0666667
\(226\) −7.10972 −0.472931
\(227\) −19.3205 −1.28235 −0.641174 0.767396i \(-0.721552\pi\)
−0.641174 + 0.767396i \(0.721552\pi\)
\(228\) −4.02283 −0.266418
\(229\) 15.7626 1.04162 0.520811 0.853672i \(-0.325630\pi\)
0.520811 + 0.853672i \(0.325630\pi\)
\(230\) 4.93593 0.325466
\(231\) 12.4151 0.816856
\(232\) 4.29078 0.281703
\(233\) 16.5549 1.08455 0.542275 0.840201i \(-0.317563\pi\)
0.542275 + 0.840201i \(0.317563\pi\)
\(234\) 0 0
\(235\) −1.81894 −0.118655
\(236\) −6.78668 −0.441775
\(237\) −7.96774 −0.517560
\(238\) 9.29032 0.602202
\(239\) 26.2006 1.69477 0.847387 0.530976i \(-0.178175\pi\)
0.847387 + 0.530976i \(0.178175\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 15.6496 1.00808 0.504039 0.863681i \(-0.331847\pi\)
0.504039 + 0.863681i \(0.331847\pi\)
\(242\) 17.5734 1.12966
\(243\) 1.00000 0.0641500
\(244\) 0.535898 0.0343074
\(245\) 1.60562 0.102580
\(246\) −2.64516 −0.168649
\(247\) 0 0
\(248\) 3.47183 0.220462
\(249\) −11.3360 −0.718388
\(250\) −1.00000 −0.0632456
\(251\) −28.3416 −1.78891 −0.894454 0.447160i \(-0.852435\pi\)
−0.894454 + 0.447160i \(0.852435\pi\)
\(252\) 2.32258 0.146309
\(253\) −26.3846 −1.65878
\(254\) −2.13209 −0.133779
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) −12.5093 −0.780309 −0.390154 0.920750i \(-0.627578\pi\)
−0.390154 + 0.920750i \(0.627578\pi\)
\(258\) 12.2508 0.762700
\(259\) −7.31439 −0.454494
\(260\) 0 0
\(261\) 4.29078 0.265592
\(262\) 1.25851 0.0777513
\(263\) −10.7059 −0.660154 −0.330077 0.943954i \(-0.607075\pi\)
−0.330077 + 0.943954i \(0.607075\pi\)
\(264\) 5.34541 0.328987
\(265\) 5.48693 0.337059
\(266\) −9.34333 −0.572877
\(267\) −1.73978 −0.106473
\(268\) −4.10926 −0.251013
\(269\) 7.52522 0.458821 0.229410 0.973330i \(-0.426320\pi\)
0.229410 + 0.973330i \(0.426320\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 9.84868 0.598264 0.299132 0.954212i \(-0.403303\pi\)
0.299132 + 0.954212i \(0.403303\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −19.7149 −1.19102
\(275\) 5.34541 0.322340
\(276\) −4.93593 −0.297108
\(277\) 0.953101 0.0572663 0.0286331 0.999590i \(-0.490885\pi\)
0.0286331 + 0.999590i \(0.490885\pi\)
\(278\) 5.67742 0.340509
\(279\) 3.47183 0.207853
\(280\) −2.32258 −0.138801
\(281\) 5.57336 0.332479 0.166239 0.986085i \(-0.446838\pi\)
0.166239 + 0.986085i \(0.446838\pi\)
\(282\) 1.81894 0.108316
\(283\) 22.0134 1.30856 0.654280 0.756252i \(-0.272972\pi\)
0.654280 + 0.756252i \(0.272972\pi\)
\(284\) 15.8719 0.941822
\(285\) 4.02283 0.238292
\(286\) 0 0
\(287\) −6.14359 −0.362645
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) −4.29078 −0.251963
\(291\) −16.1093 −0.944342
\(292\) 13.5734 0.794321
\(293\) −6.84302 −0.399773 −0.199887 0.979819i \(-0.564057\pi\)
−0.199887 + 0.979819i \(0.564057\pi\)
\(294\) −1.60562 −0.0936419
\(295\) 6.78668 0.395136
\(296\) −3.14925 −0.183047
\(297\) 5.34541 0.310172
\(298\) −19.4775 −1.12830
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 28.4534 1.64003
\(302\) −14.5170 −0.835361
\(303\) 12.6908 0.729068
\(304\) −4.02283 −0.230725
\(305\) −0.535898 −0.0306855
\(306\) 4.00000 0.228665
\(307\) 4.75442 0.271349 0.135675 0.990753i \(-0.456680\pi\)
0.135675 + 0.990753i \(0.456680\pi\)
\(308\) 12.4151 0.707418
\(309\) 4.79612 0.272842
\(310\) −3.47183 −0.197187
\(311\) −1.93639 −0.109803 −0.0549013 0.998492i \(-0.517484\pi\)
−0.0549013 + 0.998492i \(0.517484\pi\)
\(312\) 0 0
\(313\) 25.5545 1.44443 0.722213 0.691671i \(-0.243125\pi\)
0.722213 + 0.691671i \(0.243125\pi\)
\(314\) 24.3829 1.37600
\(315\) −2.32258 −0.130863
\(316\) −7.96774 −0.448220
\(317\) −12.6667 −0.711435 −0.355717 0.934594i \(-0.615763\pi\)
−0.355717 + 0.934594i \(0.615763\pi\)
\(318\) −5.48693 −0.307692
\(319\) 22.9359 1.28417
\(320\) −1.00000 −0.0559017
\(321\) 3.07180 0.171451
\(322\) −11.4641 −0.638869
\(323\) −16.0913 −0.895344
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −23.6267 −1.30856
\(327\) −6.69081 −0.370003
\(328\) −2.64516 −0.146054
\(329\) 4.22464 0.232912
\(330\) −5.34541 −0.294255
\(331\) −23.6981 −1.30256 −0.651282 0.758836i \(-0.725769\pi\)
−0.651282 + 0.758836i \(0.725769\pi\)
\(332\) −11.3360 −0.622142
\(333\) −3.14925 −0.172578
\(334\) 16.7471 0.916363
\(335\) 4.10926 0.224513
\(336\) 2.32258 0.126707
\(337\) −19.5554 −1.06525 −0.532625 0.846351i \(-0.678795\pi\)
−0.532625 + 0.846351i \(0.678795\pi\)
\(338\) 0 0
\(339\) −7.10972 −0.386147
\(340\) −4.00000 −0.216930
\(341\) 18.5584 1.00499
\(342\) −4.02283 −0.217530
\(343\) −19.9872 −1.07921
\(344\) 12.2508 0.660518
\(345\) 4.93593 0.265742
\(346\) 16.1321 0.867266
\(347\) −23.0375 −1.23672 −0.618358 0.785897i \(-0.712202\pi\)
−0.618358 + 0.785897i \(0.712202\pi\)
\(348\) 4.29078 0.230010
\(349\) 15.3205 0.820088 0.410044 0.912066i \(-0.365513\pi\)
0.410044 + 0.912066i \(0.365513\pi\)
\(350\) 2.32258 0.124147
\(351\) 0 0
\(352\) 5.34541 0.284911
\(353\) −28.4078 −1.51199 −0.755996 0.654576i \(-0.772847\pi\)
−0.755996 + 0.654576i \(0.772847\pi\)
\(354\) −6.78668 −0.360708
\(355\) −15.8719 −0.842391
\(356\) −1.73978 −0.0922083
\(357\) 9.29032 0.491696
\(358\) 7.32824 0.387309
\(359\) −23.5734 −1.24415 −0.622077 0.782956i \(-0.713711\pi\)
−0.622077 + 0.782956i \(0.713711\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −2.81687 −0.148256
\(362\) 19.5734 1.02875
\(363\) 17.5734 0.922362
\(364\) 0 0
\(365\) −13.5734 −0.710462
\(366\) 0.535898 0.0280119
\(367\) 27.4909 1.43501 0.717506 0.696552i \(-0.245283\pi\)
0.717506 + 0.696552i \(0.245283\pi\)
\(368\) −4.93593 −0.257303
\(369\) −2.64516 −0.137701
\(370\) 3.14925 0.163722
\(371\) −12.7438 −0.661627
\(372\) 3.47183 0.180006
\(373\) 26.3421 1.36394 0.681971 0.731379i \(-0.261123\pi\)
0.681971 + 0.731379i \(0.261123\pi\)
\(374\) 21.3816 1.10562
\(375\) −1.00000 −0.0516398
\(376\) 1.81894 0.0938048
\(377\) 0 0
\(378\) 2.32258 0.119461
\(379\) −0.448551 −0.0230405 −0.0115202 0.999934i \(-0.503667\pi\)
−0.0115202 + 0.999934i \(0.503667\pi\)
\(380\) 4.02283 0.206367
\(381\) −2.13209 −0.109230
\(382\) −17.0375 −0.871713
\(383\) 12.1227 0.619439 0.309719 0.950828i \(-0.399765\pi\)
0.309719 + 0.950828i \(0.399765\pi\)
\(384\) 1.00000 0.0510310
\(385\) −12.4151 −0.632734
\(386\) 6.15491 0.313277
\(387\) 12.2508 0.622742
\(388\) −16.1093 −0.817824
\(389\) −18.6195 −0.944045 −0.472022 0.881587i \(-0.656476\pi\)
−0.472022 + 0.881587i \(0.656476\pi\)
\(390\) 0 0
\(391\) −19.7437 −0.998484
\(392\) −1.60562 −0.0810962
\(393\) 1.25851 0.0634837
\(394\) 13.7867 0.694563
\(395\) 7.96774 0.400900
\(396\) 5.34541 0.268617
\(397\) −11.3042 −0.567340 −0.283670 0.958922i \(-0.591552\pi\)
−0.283670 + 0.958922i \(0.591552\pi\)
\(398\) 2.28304 0.114439
\(399\) −9.34333 −0.467752
\(400\) 1.00000 0.0500000
\(401\) −1.50580 −0.0751959 −0.0375980 0.999293i \(-0.511971\pi\)
−0.0375980 + 0.999293i \(0.511971\pi\)
\(402\) −4.10926 −0.204951
\(403\) 0 0
\(404\) 12.6908 0.631391
\(405\) −1.00000 −0.0496904
\(406\) 9.96567 0.494588
\(407\) −16.8340 −0.834432
\(408\) 4.00000 0.198030
\(409\) 11.1394 0.550806 0.275403 0.961329i \(-0.411189\pi\)
0.275403 + 0.961329i \(0.411189\pi\)
\(410\) 2.64516 0.130635
\(411\) −19.7149 −0.972464
\(412\) 4.79612 0.236288
\(413\) −15.7626 −0.775627
\(414\) −4.93593 −0.242588
\(415\) 11.3360 0.556461
\(416\) 0 0
\(417\) 5.67742 0.278024
\(418\) −21.5036 −1.05178
\(419\) −15.5098 −0.757701 −0.378851 0.925458i \(-0.623681\pi\)
−0.378851 + 0.925458i \(0.623681\pi\)
\(420\) −2.32258 −0.113330
\(421\) −39.4452 −1.92244 −0.961221 0.275778i \(-0.911065\pi\)
−0.961221 + 0.275778i \(0.911065\pi\)
\(422\) 22.4775 1.09419
\(423\) 1.81894 0.0884400
\(424\) −5.48693 −0.266469
\(425\) 4.00000 0.194029
\(426\) 15.8719 0.768995
\(427\) 1.24467 0.0602336
\(428\) 3.07180 0.148481
\(429\) 0 0
\(430\) −12.2508 −0.590785
\(431\) −2.15491 −0.103799 −0.0518993 0.998652i \(-0.516527\pi\)
−0.0518993 + 0.998652i \(0.516527\pi\)
\(432\) 1.00000 0.0481125
\(433\) 1.33938 0.0643664 0.0321832 0.999482i \(-0.489754\pi\)
0.0321832 + 0.999482i \(0.489754\pi\)
\(434\) 8.06361 0.387066
\(435\) −4.29078 −0.205727
\(436\) −6.69081 −0.320432
\(437\) 19.8564 0.949861
\(438\) 13.5734 0.648560
\(439\) −31.6981 −1.51287 −0.756434 0.654071i \(-0.773060\pi\)
−0.756434 + 0.654071i \(0.773060\pi\)
\(440\) −5.34541 −0.254832
\(441\) −1.60562 −0.0764583
\(442\) 0 0
\(443\) −28.0904 −1.33461 −0.667307 0.744782i \(-0.732553\pi\)
−0.667307 + 0.744782i \(0.732553\pi\)
\(444\) −3.14925 −0.149457
\(445\) 1.73978 0.0824736
\(446\) −1.37891 −0.0652935
\(447\) −19.4775 −0.921254
\(448\) 2.32258 0.109732
\(449\) −28.9167 −1.36466 −0.682332 0.731043i \(-0.739034\pi\)
−0.682332 + 0.731043i \(0.739034\pi\)
\(450\) 1.00000 0.0471405
\(451\) −14.1395 −0.665801
\(452\) −7.10972 −0.334413
\(453\) −14.5170 −0.682070
\(454\) −19.3205 −0.906756
\(455\) 0 0
\(456\) −4.02283 −0.188386
\(457\) −9.69900 −0.453700 −0.226850 0.973930i \(-0.572843\pi\)
−0.226850 + 0.973930i \(0.572843\pi\)
\(458\) 15.7626 0.736538
\(459\) 4.00000 0.186704
\(460\) 4.93593 0.230139
\(461\) 11.9979 0.558799 0.279400 0.960175i \(-0.409865\pi\)
0.279400 + 0.960175i \(0.409865\pi\)
\(462\) 12.4151 0.577604
\(463\) −6.42664 −0.298671 −0.149336 0.988787i \(-0.547713\pi\)
−0.149336 + 0.988787i \(0.547713\pi\)
\(464\) 4.29078 0.199194
\(465\) −3.47183 −0.161002
\(466\) 16.5549 0.766893
\(467\) −26.2642 −1.21536 −0.607681 0.794182i \(-0.707900\pi\)
−0.607681 + 0.794182i \(0.707900\pi\)
\(468\) 0 0
\(469\) −9.54409 −0.440705
\(470\) −1.81894 −0.0839015
\(471\) 24.3829 1.12350
\(472\) −6.78668 −0.312382
\(473\) 65.4854 3.01102
\(474\) −7.96774 −0.365970
\(475\) −4.02283 −0.184580
\(476\) 9.29032 0.425821
\(477\) −5.48693 −0.251229
\(478\) 26.2006 1.19839
\(479\) −25.6826 −1.17347 −0.586735 0.809779i \(-0.699587\pi\)
−0.586735 + 0.809779i \(0.699587\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 15.6496 0.712818
\(483\) −11.4641 −0.521635
\(484\) 17.5734 0.798789
\(485\) 16.1093 0.731484
\(486\) 1.00000 0.0453609
\(487\) −9.37891 −0.424999 −0.212500 0.977161i \(-0.568160\pi\)
−0.212500 + 0.977161i \(0.568160\pi\)
\(488\) 0.535898 0.0242590
\(489\) −23.6267 −1.06844
\(490\) 1.60562 0.0725347
\(491\) −7.73854 −0.349235 −0.174618 0.984636i \(-0.555869\pi\)
−0.174618 + 0.984636i \(0.555869\pi\)
\(492\) −2.64516 −0.119253
\(493\) 17.1631 0.772987
\(494\) 0 0
\(495\) −5.34541 −0.240258
\(496\) 3.47183 0.155890
\(497\) 36.8637 1.65356
\(498\) −11.3360 −0.507977
\(499\) 3.18106 0.142404 0.0712019 0.997462i \(-0.477317\pi\)
0.0712019 + 0.997462i \(0.477317\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 16.7471 0.748207
\(502\) −28.3416 −1.26495
\(503\) −35.0758 −1.56395 −0.781975 0.623309i \(-0.785788\pi\)
−0.781975 + 0.623309i \(0.785788\pi\)
\(504\) 2.32258 0.103456
\(505\) −12.6908 −0.564734
\(506\) −26.3846 −1.17294
\(507\) 0 0
\(508\) −2.13209 −0.0945961
\(509\) 18.7867 0.832705 0.416353 0.909203i \(-0.363308\pi\)
0.416353 + 0.909203i \(0.363308\pi\)
\(510\) −4.00000 −0.177123
\(511\) 31.5252 1.39459
\(512\) 1.00000 0.0441942
\(513\) −4.02283 −0.177612
\(514\) −12.5093 −0.551761
\(515\) −4.79612 −0.211342
\(516\) 12.2508 0.539311
\(517\) 9.72298 0.427616
\(518\) −7.31439 −0.321376
\(519\) 16.1321 0.708120
\(520\) 0 0
\(521\) 5.28512 0.231545 0.115773 0.993276i \(-0.463066\pi\)
0.115773 + 0.993276i \(0.463066\pi\)
\(522\) 4.29078 0.187802
\(523\) −20.7523 −0.907437 −0.453718 0.891145i \(-0.649903\pi\)
−0.453718 + 0.891145i \(0.649903\pi\)
\(524\) 1.25851 0.0549785
\(525\) 2.32258 0.101366
\(526\) −10.7059 −0.466800
\(527\) 13.8873 0.604942
\(528\) 5.34541 0.232629
\(529\) 1.36345 0.0592805
\(530\) 5.48693 0.238337
\(531\) −6.78668 −0.294517
\(532\) −9.34333 −0.405085
\(533\) 0 0
\(534\) −1.73978 −0.0752877
\(535\) −3.07180 −0.132805
\(536\) −4.10926 −0.177493
\(537\) 7.32824 0.316237
\(538\) 7.52522 0.324435
\(539\) −8.58271 −0.369683
\(540\) −1.00000 −0.0430331
\(541\) 28.7365 1.23548 0.617739 0.786384i \(-0.288049\pi\)
0.617739 + 0.786384i \(0.288049\pi\)
\(542\) 9.84868 0.423037
\(543\) 19.5734 0.839973
\(544\) 4.00000 0.171499
\(545\) 6.69081 0.286603
\(546\) 0 0
\(547\) 18.3768 0.785737 0.392869 0.919595i \(-0.371483\pi\)
0.392869 + 0.919595i \(0.371483\pi\)
\(548\) −19.7149 −0.842178
\(549\) 0.535898 0.0228716
\(550\) 5.34541 0.227929
\(551\) −17.2610 −0.735345
\(552\) −4.93593 −0.210087
\(553\) −18.5057 −0.786943
\(554\) 0.953101 0.0404934
\(555\) 3.14925 0.133678
\(556\) 5.67742 0.240776
\(557\) 26.5386 1.12448 0.562238 0.826975i \(-0.309940\pi\)
0.562238 + 0.826975i \(0.309940\pi\)
\(558\) 3.47183 0.146974
\(559\) 0 0
\(560\) −2.32258 −0.0981469
\(561\) 21.3816 0.902733
\(562\) 5.57336 0.235098
\(563\) −6.06111 −0.255446 −0.127723 0.991810i \(-0.540767\pi\)
−0.127723 + 0.991810i \(0.540767\pi\)
\(564\) 1.81894 0.0765913
\(565\) 7.10972 0.299108
\(566\) 22.0134 0.925292
\(567\) 2.32258 0.0975392
\(568\) 15.8719 0.665969
\(569\) −14.4964 −0.607719 −0.303860 0.952717i \(-0.598275\pi\)
−0.303860 + 0.952717i \(0.598275\pi\)
\(570\) 4.02283 0.168498
\(571\) 2.90413 0.121534 0.0607670 0.998152i \(-0.480645\pi\)
0.0607670 + 0.998152i \(0.480645\pi\)
\(572\) 0 0
\(573\) −17.0375 −0.711750
\(574\) −6.14359 −0.256429
\(575\) −4.93593 −0.205843
\(576\) 1.00000 0.0416667
\(577\) −18.8180 −0.783405 −0.391702 0.920092i \(-0.628114\pi\)
−0.391702 + 0.920092i \(0.628114\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 6.15491 0.255789
\(580\) −4.29078 −0.178165
\(581\) −26.3287 −1.09230
\(582\) −16.1093 −0.667750
\(583\) −29.3299 −1.21472
\(584\) 13.5734 0.561670
\(585\) 0 0
\(586\) −6.84302 −0.282682
\(587\) 3.93639 0.162472 0.0812361 0.996695i \(-0.474113\pi\)
0.0812361 + 0.996695i \(0.474113\pi\)
\(588\) −1.60562 −0.0662148
\(589\) −13.9666 −0.575483
\(590\) 6.78668 0.279403
\(591\) 13.7867 0.567108
\(592\) −3.14925 −0.129434
\(593\) −20.1227 −0.826338 −0.413169 0.910654i \(-0.635578\pi\)
−0.413169 + 0.910654i \(0.635578\pi\)
\(594\) 5.34541 0.219325
\(595\) −9.29032 −0.380866
\(596\) −19.4775 −0.797829
\(597\) 2.28304 0.0934388
\(598\) 0 0
\(599\) 48.1172 1.96601 0.983007 0.183567i \(-0.0587644\pi\)
0.983007 + 0.183567i \(0.0587644\pi\)
\(600\) 1.00000 0.0408248
\(601\) −4.24217 −0.173042 −0.0865209 0.996250i \(-0.527575\pi\)
−0.0865209 + 0.996250i \(0.527575\pi\)
\(602\) 28.4534 1.15967
\(603\) −4.10926 −0.167342
\(604\) −14.5170 −0.590690
\(605\) −17.5734 −0.714459
\(606\) 12.6908 0.515529
\(607\) 12.0685 0.489844 0.244922 0.969543i \(-0.421238\pi\)
0.244922 + 0.969543i \(0.421238\pi\)
\(608\) −4.02283 −0.163147
\(609\) 9.96567 0.403829
\(610\) −0.535898 −0.0216979
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) −41.7421 −1.68595 −0.842974 0.537954i \(-0.819197\pi\)
−0.842974 + 0.537954i \(0.819197\pi\)
\(614\) 4.75442 0.191873
\(615\) 2.64516 0.106663
\(616\) 12.4151 0.500220
\(617\) −33.4586 −1.34699 −0.673497 0.739190i \(-0.735208\pi\)
−0.673497 + 0.739190i \(0.735208\pi\)
\(618\) 4.79612 0.192928
\(619\) −38.0978 −1.53128 −0.765639 0.643270i \(-0.777577\pi\)
−0.765639 + 0.643270i \(0.777577\pi\)
\(620\) −3.47183 −0.139432
\(621\) −4.93593 −0.198072
\(622\) −1.93639 −0.0776422
\(623\) −4.04078 −0.161891
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 25.5545 1.02136
\(627\) −21.5036 −0.858773
\(628\) 24.3829 0.972982
\(629\) −12.5970 −0.502276
\(630\) −2.32258 −0.0925338
\(631\) 24.8001 0.987275 0.493638 0.869668i \(-0.335667\pi\)
0.493638 + 0.869668i \(0.335667\pi\)
\(632\) −7.96774 −0.316940
\(633\) 22.4775 0.893400
\(634\) −12.6667 −0.503060
\(635\) 2.13209 0.0846093
\(636\) −5.48693 −0.217571
\(637\) 0 0
\(638\) 22.9359 0.908042
\(639\) 15.8719 0.627881
\(640\) −1.00000 −0.0395285
\(641\) 4.08519 0.161355 0.0806776 0.996740i \(-0.474292\pi\)
0.0806776 + 0.996740i \(0.474292\pi\)
\(642\) 3.07180 0.121234
\(643\) −36.5816 −1.44264 −0.721318 0.692604i \(-0.756463\pi\)
−0.721318 + 0.692604i \(0.756463\pi\)
\(644\) −11.4641 −0.451749
\(645\) −12.2508 −0.482374
\(646\) −16.0913 −0.633104
\(647\) 17.2341 0.677541 0.338771 0.940869i \(-0.389989\pi\)
0.338771 + 0.940869i \(0.389989\pi\)
\(648\) 1.00000 0.0392837
\(649\) −36.2776 −1.42402
\(650\) 0 0
\(651\) 8.06361 0.316038
\(652\) −23.6267 −0.925295
\(653\) −21.9274 −0.858085 −0.429042 0.903284i \(-0.641149\pi\)
−0.429042 + 0.903284i \(0.641149\pi\)
\(654\) −6.69081 −0.261631
\(655\) −1.25851 −0.0491742
\(656\) −2.64516 −0.103276
\(657\) 13.5734 0.529547
\(658\) 4.22464 0.164694
\(659\) −48.4759 −1.88835 −0.944176 0.329441i \(-0.893140\pi\)
−0.944176 + 0.329441i \(0.893140\pi\)
\(660\) −5.34541 −0.208070
\(661\) −29.8221 −1.15994 −0.579972 0.814636i \(-0.696937\pi\)
−0.579972 + 0.814636i \(0.696937\pi\)
\(662\) −23.6981 −0.921052
\(663\) 0 0
\(664\) −11.3360 −0.439921
\(665\) 9.34333 0.362319
\(666\) −3.14925 −0.122031
\(667\) −21.1790 −0.820054
\(668\) 16.7471 0.647967
\(669\) −1.37891 −0.0533119
\(670\) 4.10926 0.158755
\(671\) 2.86459 0.110586
\(672\) 2.32258 0.0895955
\(673\) −30.6719 −1.18232 −0.591158 0.806556i \(-0.701329\pi\)
−0.591158 + 0.806556i \(0.701329\pi\)
\(674\) −19.5554 −0.753246
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 21.0831 0.810290 0.405145 0.914252i \(-0.367221\pi\)
0.405145 + 0.914252i \(0.367221\pi\)
\(678\) −7.10972 −0.273047
\(679\) −37.4150 −1.43586
\(680\) −4.00000 −0.153393
\(681\) −19.3205 −0.740363
\(682\) 18.5584 0.710636
\(683\) 13.3360 0.510287 0.255143 0.966903i \(-0.417877\pi\)
0.255143 + 0.966903i \(0.417877\pi\)
\(684\) −4.02283 −0.153817
\(685\) 19.7149 0.753267
\(686\) −19.9872 −0.763117
\(687\) 15.7626 0.601381
\(688\) 12.2508 0.467057
\(689\) 0 0
\(690\) 4.93593 0.187908
\(691\) 34.9054 1.32786 0.663932 0.747793i \(-0.268887\pi\)
0.663932 + 0.747793i \(0.268887\pi\)
\(692\) 16.1321 0.613250
\(693\) 12.4151 0.471612
\(694\) −23.0375 −0.874490
\(695\) −5.67742 −0.215357
\(696\) 4.29078 0.162641
\(697\) −10.5806 −0.400770
\(698\) 15.3205 0.579890
\(699\) 16.5549 0.626166
\(700\) 2.32258 0.0877853
\(701\) −39.6715 −1.49837 −0.749186 0.662360i \(-0.769555\pi\)
−0.749186 + 0.662360i \(0.769555\pi\)
\(702\) 0 0
\(703\) 12.6689 0.477817
\(704\) 5.34541 0.201463
\(705\) −1.81894 −0.0685053
\(706\) −28.4078 −1.06914
\(707\) 29.4754 1.10854
\(708\) −6.78668 −0.255059
\(709\) −2.83441 −0.106448 −0.0532242 0.998583i \(-0.516950\pi\)
−0.0532242 + 0.998583i \(0.516950\pi\)
\(710\) −15.8719 −0.595661
\(711\) −7.96774 −0.298814
\(712\) −1.73978 −0.0652011
\(713\) −17.1367 −0.641776
\(714\) 9.29032 0.347681
\(715\) 0 0
\(716\) 7.32824 0.273869
\(717\) 26.2006 0.978478
\(718\) −23.5734 −0.879750
\(719\) −43.7128 −1.63021 −0.815106 0.579311i \(-0.803322\pi\)
−0.815106 + 0.579311i \(0.803322\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 11.1394 0.414852
\(722\) −2.81687 −0.104833
\(723\) 15.6496 0.582014
\(724\) 19.5734 0.727438
\(725\) 4.29078 0.159355
\(726\) 17.5734 0.652209
\(727\) 16.2568 0.602932 0.301466 0.953477i \(-0.402524\pi\)
0.301466 + 0.953477i \(0.402524\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −13.5734 −0.502373
\(731\) 49.0031 1.81245
\(732\) 0.535898 0.0198074
\(733\) 4.96774 0.183488 0.0917438 0.995783i \(-0.470756\pi\)
0.0917438 + 0.995783i \(0.470756\pi\)
\(734\) 27.4909 1.01471
\(735\) 1.60562 0.0592243
\(736\) −4.93593 −0.181941
\(737\) −21.9657 −0.809116
\(738\) −2.64516 −0.0973697
\(739\) 49.6875 1.82778 0.913892 0.405957i \(-0.133062\pi\)
0.913892 + 0.405957i \(0.133062\pi\)
\(740\) 3.14925 0.115769
\(741\) 0 0
\(742\) −12.7438 −0.467841
\(743\) −21.5420 −0.790300 −0.395150 0.918617i \(-0.629307\pi\)
−0.395150 + 0.918617i \(0.629307\pi\)
\(744\) 3.47183 0.127284
\(745\) 19.4775 0.713600
\(746\) 26.3421 0.964452
\(747\) −11.3360 −0.414761
\(748\) 21.3816 0.781790
\(749\) 7.13449 0.260689
\(750\) −1.00000 −0.0365148
\(751\) −6.34872 −0.231668 −0.115834 0.993269i \(-0.536954\pi\)
−0.115834 + 0.993269i \(0.536954\pi\)
\(752\) 1.81894 0.0663300
\(753\) −28.3416 −1.03283
\(754\) 0 0
\(755\) 14.5170 0.528329
\(756\) 2.32258 0.0844714
\(757\) −48.1107 −1.74861 −0.874307 0.485373i \(-0.838684\pi\)
−0.874307 + 0.485373i \(0.838684\pi\)
\(758\) −0.448551 −0.0162921
\(759\) −26.3846 −0.957699
\(760\) 4.02283 0.145923
\(761\) −33.6866 −1.22114 −0.610569 0.791963i \(-0.709059\pi\)
−0.610569 + 0.791963i \(0.709059\pi\)
\(762\) −2.13209 −0.0772374
\(763\) −15.5399 −0.562584
\(764\) −17.0375 −0.616394
\(765\) −4.00000 −0.144620
\(766\) 12.1227 0.438009
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 5.98886 0.215964 0.107982 0.994153i \(-0.465561\pi\)
0.107982 + 0.994153i \(0.465561\pi\)
\(770\) −12.4151 −0.447410
\(771\) −12.5093 −0.450511
\(772\) 6.15491 0.221520
\(773\) 13.4882 0.485136 0.242568 0.970134i \(-0.422010\pi\)
0.242568 + 0.970134i \(0.422010\pi\)
\(774\) 12.2508 0.440345
\(775\) 3.47183 0.124712
\(776\) −16.1093 −0.578289
\(777\) −7.31439 −0.262402
\(778\) −18.6195 −0.667540
\(779\) 10.6410 0.381254
\(780\) 0 0
\(781\) 84.8416 3.03587
\(782\) −19.7437 −0.706035
\(783\) 4.29078 0.153340
\(784\) −1.60562 −0.0573437
\(785\) −24.3829 −0.870262
\(786\) 1.25851 0.0448897
\(787\) 54.6487 1.94802 0.974009 0.226510i \(-0.0727317\pi\)
0.974009 + 0.226510i \(0.0727317\pi\)
\(788\) 13.7867 0.491130
\(789\) −10.7059 −0.381140
\(790\) 7.96774 0.283479
\(791\) −16.5129 −0.587131
\(792\) 5.34541 0.189941
\(793\) 0 0
\(794\) −11.3042 −0.401170
\(795\) 5.48693 0.194601
\(796\) 2.28304 0.0809203
\(797\) −38.3244 −1.35752 −0.678760 0.734361i \(-0.737482\pi\)
−0.678760 + 0.734361i \(0.737482\pi\)
\(798\) −9.34333 −0.330750
\(799\) 7.27577 0.257398
\(800\) 1.00000 0.0353553
\(801\) −1.73978 −0.0614722
\(802\) −1.50580 −0.0531716
\(803\) 72.5551 2.56041
\(804\) −4.10926 −0.144922
\(805\) 11.4641 0.404056
\(806\) 0 0
\(807\) 7.52522 0.264900
\(808\) 12.6908 0.446461
\(809\) 11.3816 0.400157 0.200078 0.979780i \(-0.435880\pi\)
0.200078 + 0.979780i \(0.435880\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −10.8899 −0.382397 −0.191198 0.981551i \(-0.561237\pi\)
−0.191198 + 0.981551i \(0.561237\pi\)
\(812\) 9.96567 0.349726
\(813\) 9.84868 0.345408
\(814\) −16.8340 −0.590033
\(815\) 23.6267 0.827609
\(816\) 4.00000 0.140028
\(817\) −49.2828 −1.72419
\(818\) 11.1394 0.389479
\(819\) 0 0
\(820\) 2.64516 0.0923730
\(821\) 1.68810 0.0589152 0.0294576 0.999566i \(-0.490622\pi\)
0.0294576 + 0.999566i \(0.490622\pi\)
\(822\) −19.7149 −0.687636
\(823\) 6.77816 0.236272 0.118136 0.992997i \(-0.462308\pi\)
0.118136 + 0.992997i \(0.462308\pi\)
\(824\) 4.79612 0.167081
\(825\) 5.34541 0.186103
\(826\) −15.7626 −0.548451
\(827\) 30.2298 1.05119 0.525597 0.850733i \(-0.323842\pi\)
0.525597 + 0.850733i \(0.323842\pi\)
\(828\) −4.93593 −0.171536
\(829\) −10.4405 −0.362612 −0.181306 0.983427i \(-0.558032\pi\)
−0.181306 + 0.983427i \(0.558032\pi\)
\(830\) 11.3360 0.393477
\(831\) 0.953101 0.0330627
\(832\) 0 0
\(833\) −6.42249 −0.222526
\(834\) 5.67742 0.196593
\(835\) −16.7471 −0.579559
\(836\) −21.5036 −0.743719
\(837\) 3.47183 0.120004
\(838\) −15.5098 −0.535776
\(839\) −11.1965 −0.386547 −0.193273 0.981145i \(-0.561910\pi\)
−0.193273 + 0.981145i \(0.561910\pi\)
\(840\) −2.32258 −0.0801366
\(841\) −10.5892 −0.365146
\(842\) −39.4452 −1.35937
\(843\) 5.57336 0.191957
\(844\) 22.4775 0.773707
\(845\) 0 0
\(846\) 1.81894 0.0625365
\(847\) 40.8155 1.40244
\(848\) −5.48693 −0.188422
\(849\) 22.0134 0.755498
\(850\) 4.00000 0.137199
\(851\) 15.5445 0.532859
\(852\) 15.8719 0.543761
\(853\) −21.8185 −0.747051 −0.373525 0.927620i \(-0.621851\pi\)
−0.373525 + 0.927620i \(0.621851\pi\)
\(854\) 1.24467 0.0425916
\(855\) 4.02283 0.137578
\(856\) 3.07180 0.104992
\(857\) 9.42369 0.321907 0.160954 0.986962i \(-0.448543\pi\)
0.160954 + 0.986962i \(0.448543\pi\)
\(858\) 0 0
\(859\) 24.4775 0.835161 0.417581 0.908640i \(-0.362878\pi\)
0.417581 + 0.908640i \(0.362878\pi\)
\(860\) −12.2508 −0.417748
\(861\) −6.14359 −0.209373
\(862\) −2.15491 −0.0733966
\(863\) −29.7873 −1.01397 −0.506986 0.861954i \(-0.669240\pi\)
−0.506986 + 0.861954i \(0.669240\pi\)
\(864\) 1.00000 0.0340207
\(865\) −16.1321 −0.548507
\(866\) 1.33938 0.0455139
\(867\) −1.00000 −0.0339618
\(868\) 8.06361 0.273697
\(869\) −42.5908 −1.44479
\(870\) −4.29078 −0.145471
\(871\) 0 0
\(872\) −6.69081 −0.226579
\(873\) −16.1093 −0.545216
\(874\) 19.8564 0.671653
\(875\) −2.32258 −0.0785175
\(876\) 13.5734 0.458601
\(877\) 1.90979 0.0644890 0.0322445 0.999480i \(-0.489734\pi\)
0.0322445 + 0.999480i \(0.489734\pi\)
\(878\) −31.6981 −1.06976
\(879\) −6.84302 −0.230809
\(880\) −5.34541 −0.180194
\(881\) 6.36823 0.214551 0.107276 0.994229i \(-0.465787\pi\)
0.107276 + 0.994229i \(0.465787\pi\)
\(882\) −1.60562 −0.0540642
\(883\) 34.9897 1.17750 0.588749 0.808316i \(-0.299621\pi\)
0.588749 + 0.808316i \(0.299621\pi\)
\(884\) 0 0
\(885\) 6.78668 0.228132
\(886\) −28.0904 −0.943715
\(887\) 21.4001 0.718546 0.359273 0.933233i \(-0.383025\pi\)
0.359273 + 0.933233i \(0.383025\pi\)
\(888\) −3.14925 −0.105682
\(889\) −4.95194 −0.166083
\(890\) 1.73978 0.0583176
\(891\) 5.34541 0.179078
\(892\) −1.37891 −0.0461694
\(893\) −7.31729 −0.244864
\(894\) −19.4775 −0.651425
\(895\) −7.32824 −0.244956
\(896\) 2.32258 0.0775919
\(897\) 0 0
\(898\) −28.9167 −0.964963
\(899\) 14.8969 0.496838
\(900\) 1.00000 0.0333333
\(901\) −21.9477 −0.731184
\(902\) −14.1395 −0.470792
\(903\) 28.4534 0.946871
\(904\) −7.10972 −0.236466
\(905\) −19.5734 −0.650641
\(906\) −14.5170 −0.482296
\(907\) 32.4693 1.07813 0.539063 0.842266i \(-0.318779\pi\)
0.539063 + 0.842266i \(0.318779\pi\)
\(908\) −19.3205 −0.641174
\(909\) 12.6908 0.420928
\(910\) 0 0
\(911\) 34.4380 1.14098 0.570490 0.821304i \(-0.306753\pi\)
0.570490 + 0.821304i \(0.306753\pi\)
\(912\) −4.02283 −0.133209
\(913\) −60.5954 −2.00541
\(914\) −9.69900 −0.320814
\(915\) −0.535898 −0.0177163
\(916\) 15.7626 0.520811
\(917\) 2.92300 0.0965260
\(918\) 4.00000 0.132020
\(919\) 36.4827 1.20345 0.601727 0.798702i \(-0.294480\pi\)
0.601727 + 0.798702i \(0.294480\pi\)
\(920\) 4.93593 0.162733
\(921\) 4.75442 0.156663
\(922\) 11.9979 0.395131
\(923\) 0 0
\(924\) 12.4151 0.408428
\(925\) −3.14925 −0.103547
\(926\) −6.42664 −0.211192
\(927\) 4.79612 0.157525
\(928\) 4.29078 0.140852
\(929\) 49.9100 1.63749 0.818747 0.574155i \(-0.194669\pi\)
0.818747 + 0.574155i \(0.194669\pi\)
\(930\) −3.47183 −0.113846
\(931\) 6.45914 0.211690
\(932\) 16.5549 0.542275
\(933\) −1.93639 −0.0633946
\(934\) −26.2642 −0.859390
\(935\) −21.3816 −0.699254
\(936\) 0 0
\(937\) −15.8873 −0.519017 −0.259508 0.965741i \(-0.583560\pi\)
−0.259508 + 0.965741i \(0.583560\pi\)
\(938\) −9.54409 −0.311625
\(939\) 25.5545 0.833939
\(940\) −1.81894 −0.0593274
\(941\) 6.99273 0.227956 0.113978 0.993483i \(-0.463641\pi\)
0.113978 + 0.993483i \(0.463641\pi\)
\(942\) 24.3829 0.794437
\(943\) 13.0563 0.425173
\(944\) −6.78668 −0.220888
\(945\) −2.32258 −0.0755535
\(946\) 65.4854 2.12911
\(947\) −52.3064 −1.69973 −0.849865 0.527000i \(-0.823317\pi\)
−0.849865 + 0.527000i \(0.823317\pi\)
\(948\) −7.96774 −0.258780
\(949\) 0 0
\(950\) −4.02283 −0.130518
\(951\) −12.6667 −0.410747
\(952\) 9.29032 0.301101
\(953\) 28.8827 0.935603 0.467802 0.883833i \(-0.345046\pi\)
0.467802 + 0.883833i \(0.345046\pi\)
\(954\) −5.48693 −0.177646
\(955\) 17.0375 0.551319
\(956\) 26.2006 0.847387
\(957\) 22.9359 0.741413
\(958\) −25.6826 −0.829768
\(959\) −45.7894 −1.47862
\(960\) −1.00000 −0.0322749
\(961\) −18.9464 −0.611173
\(962\) 0 0
\(963\) 3.07180 0.0989873
\(964\) 15.6496 0.504039
\(965\) −6.15491 −0.198134
\(966\) −11.4641 −0.368851
\(967\) 10.8610 0.349265 0.174633 0.984634i \(-0.444126\pi\)
0.174633 + 0.984634i \(0.444126\pi\)
\(968\) 17.5734 0.564829
\(969\) −16.0913 −0.516927
\(970\) 16.1093 0.517237
\(971\) −24.9979 −0.802222 −0.401111 0.916030i \(-0.631376\pi\)
−0.401111 + 0.916030i \(0.631376\pi\)
\(972\) 1.00000 0.0320750
\(973\) 13.1863 0.422732
\(974\) −9.37891 −0.300520
\(975\) 0 0
\(976\) 0.535898 0.0171537
\(977\) 42.7653 1.36818 0.684092 0.729396i \(-0.260199\pi\)
0.684092 + 0.729396i \(0.260199\pi\)
\(978\) −23.6267 −0.755500
\(979\) −9.29984 −0.297224
\(980\) 1.60562 0.0512898
\(981\) −6.69081 −0.213621
\(982\) −7.73854 −0.246947
\(983\) 10.1288 0.323058 0.161529 0.986868i \(-0.448358\pi\)
0.161529 + 0.986868i \(0.448358\pi\)
\(984\) −2.64516 −0.0843246
\(985\) −13.7867 −0.439280
\(986\) 17.1631 0.546584
\(987\) 4.22464 0.134472
\(988\) 0 0
\(989\) −60.4691 −1.92280
\(990\) −5.34541 −0.169888
\(991\) 20.2483 0.643208 0.321604 0.946874i \(-0.395778\pi\)
0.321604 + 0.946874i \(0.395778\pi\)
\(992\) 3.47183 0.110231
\(993\) −23.6981 −0.752036
\(994\) 36.8637 1.16924
\(995\) −2.28304 −0.0723774
\(996\) −11.3360 −0.359194
\(997\) −20.7961 −0.658620 −0.329310 0.944222i \(-0.606816\pi\)
−0.329310 + 0.944222i \(0.606816\pi\)
\(998\) 3.18106 0.100695
\(999\) −3.14925 −0.0996380
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 5070.2.a.ca.1.3 4
13.5 odd 4 5070.2.b.ba.1351.3 8
13.6 odd 12 390.2.bb.c.361.1 yes 8
13.8 odd 4 5070.2.b.ba.1351.6 8
13.11 odd 12 390.2.bb.c.121.1 8
13.12 even 2 5070.2.a.bz.1.2 4
39.11 even 12 1170.2.bs.f.901.3 8
39.32 even 12 1170.2.bs.f.361.3 8
65.19 odd 12 1950.2.bc.g.751.4 8
65.24 odd 12 1950.2.bc.g.901.4 8
65.32 even 12 1950.2.y.j.49.2 8
65.37 even 12 1950.2.y.k.199.3 8
65.58 even 12 1950.2.y.k.49.3 8
65.63 even 12 1950.2.y.j.199.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.c.121.1 8 13.11 odd 12
390.2.bb.c.361.1 yes 8 13.6 odd 12
1170.2.bs.f.361.3 8 39.32 even 12
1170.2.bs.f.901.3 8 39.11 even 12
1950.2.y.j.49.2 8 65.32 even 12
1950.2.y.j.199.2 8 65.63 even 12
1950.2.y.k.49.3 8 65.58 even 12
1950.2.y.k.199.3 8 65.37 even 12
1950.2.bc.g.751.4 8 65.19 odd 12
1950.2.bc.g.901.4 8 65.24 odd 12
5070.2.a.bz.1.2 4 13.12 even 2
5070.2.a.ca.1.3 4 1.1 even 1 trivial
5070.2.b.ba.1351.3 8 13.5 odd 4
5070.2.b.ba.1351.6 8 13.8 odd 4