Properties

Label 169.4.b.b.168.1
Level $169$
Weight $4$
Character 169.168
Analytic conductor $9.971$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 168.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 169.168
Dual form 169.4.b.b.168.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.46410i q^{2} -7.00000 q^{3} -4.00000 q^{4} -13.8564i q^{5} +24.2487i q^{6} -22.5167i q^{7} -13.8564i q^{8} +22.0000 q^{9} +O(q^{10})\) \(q-3.46410i q^{2} -7.00000 q^{3} -4.00000 q^{4} -13.8564i q^{5} +24.2487i q^{6} -22.5167i q^{7} -13.8564i q^{8} +22.0000 q^{9} -48.0000 q^{10} -22.5167i q^{11} +28.0000 q^{12} -78.0000 q^{14} +96.9948i q^{15} -80.0000 q^{16} +27.0000 q^{17} -76.2102i q^{18} +88.3346i q^{19} +55.4256i q^{20} +157.617i q^{21} -78.0000 q^{22} +57.0000 q^{23} +96.9948i q^{24} -67.0000 q^{25} +35.0000 q^{27} +90.0666i q^{28} -69.0000 q^{29} +336.000 q^{30} +72.7461i q^{31} +166.277i q^{32} +157.617i q^{33} -93.5307i q^{34} -312.000 q^{35} -88.0000 q^{36} -39.8372i q^{37} +306.000 q^{38} -192.000 q^{40} -393.176i q^{41} +546.000 q^{42} -85.0000 q^{43} +90.0666i q^{44} -304.841i q^{45} -197.454i q^{46} +342.946i q^{47} +560.000 q^{48} -164.000 q^{49} +232.095i q^{50} -189.000 q^{51} +426.000 q^{53} -121.244i q^{54} -312.000 q^{55} -312.000 q^{56} -618.342i q^{57} +239.023i q^{58} +19.0526i q^{59} -387.979i q^{60} -17.0000 q^{61} +252.000 q^{62} -495.367i q^{63} -64.0000 q^{64} +546.000 q^{66} +164.545i q^{67} -108.000 q^{68} -399.000 q^{69} +1080.80i q^{70} -583.701i q^{71} -304.841i q^{72} -1004.59i q^{73} -138.000 q^{74} +469.000 q^{75} -353.338i q^{76} -507.000 q^{77} -1244.00 q^{79} +1108.51i q^{80} -839.000 q^{81} -1362.00 q^{82} +426.084i q^{83} -630.466i q^{84} -374.123i q^{85} +294.449i q^{86} +483.000 q^{87} -312.000 q^{88} +306.573i q^{89} -1056.00 q^{90} -228.000 q^{92} -509.223i q^{93} +1188.00 q^{94} +1224.00 q^{95} -1163.94i q^{96} -1234.95i q^{97} +568.113i q^{98} -495.367i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{3} - 8 q^{4} + 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{3} - 8 q^{4} + 44 q^{9} - 96 q^{10} + 56 q^{12} - 156 q^{14} - 160 q^{16} + 54 q^{17} - 156 q^{22} + 114 q^{23} - 134 q^{25} + 70 q^{27} - 138 q^{29} + 672 q^{30} - 624 q^{35} - 176 q^{36} + 612 q^{38} - 384 q^{40} + 1092 q^{42} - 170 q^{43} + 1120 q^{48} - 328 q^{49} - 378 q^{51} + 852 q^{53} - 624 q^{55} - 624 q^{56} - 34 q^{61} + 504 q^{62} - 128 q^{64} + 1092 q^{66} - 216 q^{68} - 798 q^{69} - 276 q^{74} + 938 q^{75} - 1014 q^{77} - 2488 q^{79} - 1678 q^{81} - 2724 q^{82} + 966 q^{87} - 624 q^{88} - 2112 q^{90} - 456 q^{92} + 2376 q^{94} + 2448 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.46410i − 1.22474i −0.790569 0.612372i \(-0.790215\pi\)
0.790569 0.612372i \(-0.209785\pi\)
\(3\) −7.00000 −1.34715 −0.673575 0.739119i \(-0.735242\pi\)
−0.673575 + 0.739119i \(0.735242\pi\)
\(4\) −4.00000 −0.500000
\(5\) − 13.8564i − 1.23935i −0.784857 0.619677i \(-0.787263\pi\)
0.784857 0.619677i \(-0.212737\pi\)
\(6\) 24.2487i 1.64992i
\(7\) − 22.5167i − 1.21579i −0.794019 0.607893i \(-0.792015\pi\)
0.794019 0.607893i \(-0.207985\pi\)
\(8\) − 13.8564i − 0.612372i
\(9\) 22.0000 0.814815
\(10\) −48.0000 −1.51789
\(11\) − 22.5167i − 0.617184i −0.951194 0.308592i \(-0.900142\pi\)
0.951194 0.308592i \(-0.0998578\pi\)
\(12\) 28.0000 0.673575
\(13\) 0 0
\(14\) −78.0000 −1.48903
\(15\) 96.9948i 1.66960i
\(16\) −80.0000 −1.25000
\(17\) 27.0000 0.385204 0.192602 0.981277i \(-0.438307\pi\)
0.192602 + 0.981277i \(0.438307\pi\)
\(18\) − 76.2102i − 0.997940i
\(19\) 88.3346i 1.06660i 0.845927 + 0.533299i \(0.179048\pi\)
−0.845927 + 0.533299i \(0.820952\pi\)
\(20\) 55.4256i 0.619677i
\(21\) 157.617i 1.63785i
\(22\) −78.0000 −0.755893
\(23\) 57.0000 0.516753 0.258377 0.966044i \(-0.416812\pi\)
0.258377 + 0.966044i \(0.416812\pi\)
\(24\) 96.9948i 0.824958i
\(25\) −67.0000 −0.536000
\(26\) 0 0
\(27\) 35.0000 0.249472
\(28\) 90.0666i 0.607893i
\(29\) −69.0000 −0.441827 −0.220913 0.975293i \(-0.570904\pi\)
−0.220913 + 0.975293i \(0.570904\pi\)
\(30\) 336.000 2.04483
\(31\) 72.7461i 0.421471i 0.977543 + 0.210735i \(0.0675858\pi\)
−0.977543 + 0.210735i \(0.932414\pi\)
\(32\) 166.277i 0.918559i
\(33\) 157.617i 0.831440i
\(34\) − 93.5307i − 0.471776i
\(35\) −312.000 −1.50679
\(36\) −88.0000 −0.407407
\(37\) − 39.8372i − 0.177005i −0.996076 0.0885026i \(-0.971792\pi\)
0.996076 0.0885026i \(-0.0282081\pi\)
\(38\) 306.000 1.30631
\(39\) 0 0
\(40\) −192.000 −0.758947
\(41\) − 393.176i − 1.49765i −0.662767 0.748826i \(-0.730618\pi\)
0.662767 0.748826i \(-0.269382\pi\)
\(42\) 546.000 2.00594
\(43\) −85.0000 −0.301451 −0.150725 0.988576i \(-0.548161\pi\)
−0.150725 + 0.988576i \(0.548161\pi\)
\(44\) 90.0666i 0.308592i
\(45\) − 304.841i − 1.00984i
\(46\) − 197.454i − 0.632891i
\(47\) 342.946i 1.06434i 0.846639 + 0.532168i \(0.178623\pi\)
−0.846639 + 0.532168i \(0.821377\pi\)
\(48\) 560.000 1.68394
\(49\) −164.000 −0.478134
\(50\) 232.095i 0.656463i
\(51\) −189.000 −0.518927
\(52\) 0 0
\(53\) 426.000 1.10407 0.552034 0.833822i \(-0.313852\pi\)
0.552034 + 0.833822i \(0.313852\pi\)
\(54\) − 121.244i − 0.305540i
\(55\) −312.000 −0.764910
\(56\) −312.000 −0.744513
\(57\) − 618.342i − 1.43687i
\(58\) 239.023i 0.541125i
\(59\) 19.0526i 0.0420412i 0.999779 + 0.0210206i \(0.00669156\pi\)
−0.999779 + 0.0210206i \(0.993308\pi\)
\(60\) − 387.979i − 0.834799i
\(61\) −17.0000 −0.0356824 −0.0178412 0.999841i \(-0.505679\pi\)
−0.0178412 + 0.999841i \(0.505679\pi\)
\(62\) 252.000 0.516194
\(63\) − 495.367i − 0.990640i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 546.000 1.01830
\(67\) 164.545i 0.300035i 0.988683 + 0.150018i \(0.0479330\pi\)
−0.988683 + 0.150018i \(0.952067\pi\)
\(68\) −108.000 −0.192602
\(69\) −399.000 −0.696144
\(70\) 1080.80i 1.84543i
\(71\) − 583.701i − 0.975670i −0.872936 0.487835i \(-0.837787\pi\)
0.872936 0.487835i \(-0.162213\pi\)
\(72\) − 304.841i − 0.498970i
\(73\) − 1004.59i − 1.61066i −0.592826 0.805331i \(-0.701988\pi\)
0.592826 0.805331i \(-0.298012\pi\)
\(74\) −138.000 −0.216786
\(75\) 469.000 0.722073
\(76\) − 353.338i − 0.533299i
\(77\) −507.000 −0.750364
\(78\) 0 0
\(79\) −1244.00 −1.77166 −0.885829 0.464012i \(-0.846409\pi\)
−0.885829 + 0.464012i \(0.846409\pi\)
\(80\) 1108.51i 1.54919i
\(81\) −839.000 −1.15089
\(82\) −1362.00 −1.83424
\(83\) 426.084i 0.563480i 0.959491 + 0.281740i \(0.0909116\pi\)
−0.959491 + 0.281740i \(0.909088\pi\)
\(84\) − 630.466i − 0.818923i
\(85\) − 374.123i − 0.477404i
\(86\) 294.449i 0.369200i
\(87\) 483.000 0.595207
\(88\) −312.000 −0.377947
\(89\) 306.573i 0.365131i 0.983194 + 0.182566i \(0.0584402\pi\)
−0.983194 + 0.182566i \(0.941560\pi\)
\(90\) −1056.00 −1.23680
\(91\) 0 0
\(92\) −228.000 −0.258377
\(93\) − 509.223i − 0.567785i
\(94\) 1188.00 1.30354
\(95\) 1224.00 1.32189
\(96\) − 1163.94i − 1.23744i
\(97\) − 1234.95i − 1.29268i −0.763048 0.646342i \(-0.776298\pi\)
0.763048 0.646342i \(-0.223702\pi\)
\(98\) 568.113i 0.585592i
\(99\) − 495.367i − 0.502891i
\(100\) 268.000 0.268000
\(101\) 1959.00 1.92998 0.964989 0.262290i \(-0.0844778\pi\)
0.964989 + 0.262290i \(0.0844778\pi\)
\(102\) 654.715i 0.635554i
\(103\) 1856.00 1.77551 0.887753 0.460320i \(-0.152265\pi\)
0.887753 + 0.460320i \(0.152265\pi\)
\(104\) 0 0
\(105\) 2184.00 2.02987
\(106\) − 1475.71i − 1.35220i
\(107\) −255.000 −0.230390 −0.115195 0.993343i \(-0.536749\pi\)
−0.115195 + 0.993343i \(0.536749\pi\)
\(108\) −140.000 −0.124736
\(109\) 609.682i 0.535752i 0.963453 + 0.267876i \(0.0863217\pi\)
−0.963453 + 0.267876i \(0.913678\pi\)
\(110\) 1080.80i 0.936820i
\(111\) 278.860i 0.238453i
\(112\) 1801.33i 1.51973i
\(113\) 411.000 0.342156 0.171078 0.985257i \(-0.445275\pi\)
0.171078 + 0.985257i \(0.445275\pi\)
\(114\) −2142.00 −1.75980
\(115\) − 789.815i − 0.640440i
\(116\) 276.000 0.220913
\(117\) 0 0
\(118\) 66.0000 0.0514898
\(119\) − 607.950i − 0.468325i
\(120\) 1344.00 1.02242
\(121\) 824.000 0.619083
\(122\) 58.8897i 0.0437018i
\(123\) 2752.23i 2.01756i
\(124\) − 290.985i − 0.210735i
\(125\) − 803.672i − 0.575061i
\(126\) −1716.00 −1.21328
\(127\) −2243.00 −1.56720 −0.783599 0.621267i \(-0.786618\pi\)
−0.783599 + 0.621267i \(0.786618\pi\)
\(128\) 1551.92i 1.07165i
\(129\) 595.000 0.406099
\(130\) 0 0
\(131\) −372.000 −0.248105 −0.124053 0.992276i \(-0.539589\pi\)
−0.124053 + 0.992276i \(0.539589\pi\)
\(132\) − 630.466i − 0.415720i
\(133\) 1989.00 1.29675
\(134\) 570.000 0.367466
\(135\) − 484.974i − 0.309185i
\(136\) − 374.123i − 0.235888i
\(137\) 1189.92i 0.742056i 0.928622 + 0.371028i \(0.120995\pi\)
−0.928622 + 0.371028i \(0.879005\pi\)
\(138\) 1382.18i 0.852599i
\(139\) −2545.00 −1.55298 −0.776490 0.630130i \(-0.783002\pi\)
−0.776490 + 0.630130i \(0.783002\pi\)
\(140\) 1248.00 0.753395
\(141\) − 2400.62i − 1.43382i
\(142\) −2022.00 −1.19495
\(143\) 0 0
\(144\) −1760.00 −1.01852
\(145\) 956.092i 0.547580i
\(146\) −3480.00 −1.97265
\(147\) 1148.00 0.644119
\(148\) 159.349i 0.0885026i
\(149\) − 1304.23i − 0.717094i −0.933512 0.358547i \(-0.883272\pi\)
0.933512 0.358547i \(-0.116728\pi\)
\(150\) − 1624.66i − 0.884355i
\(151\) 86.6025i 0.0466729i 0.999728 + 0.0233365i \(0.00742890\pi\)
−0.999728 + 0.0233365i \(0.992571\pi\)
\(152\) 1224.00 0.653155
\(153\) 594.000 0.313870
\(154\) 1756.30i 0.919004i
\(155\) 1008.00 0.522352
\(156\) 0 0
\(157\) −1534.00 −0.779787 −0.389893 0.920860i \(-0.627488\pi\)
−0.389893 + 0.920860i \(0.627488\pi\)
\(158\) 4309.34i 2.16983i
\(159\) −2982.00 −1.48735
\(160\) 2304.00 1.13842
\(161\) − 1283.45i − 0.628261i
\(162\) 2906.38i 1.40955i
\(163\) 1633.32i 0.784858i 0.919782 + 0.392429i \(0.128365\pi\)
−0.919782 + 0.392429i \(0.871635\pi\)
\(164\) 1572.70i 0.748826i
\(165\) 2184.00 1.03045
\(166\) 1476.00 0.690119
\(167\) 1626.40i 0.753618i 0.926291 + 0.376809i \(0.122979\pi\)
−0.926291 + 0.376809i \(0.877021\pi\)
\(168\) 2184.00 1.00297
\(169\) 0 0
\(170\) −1296.00 −0.584698
\(171\) 1943.36i 0.869079i
\(172\) 340.000 0.150725
\(173\) −873.000 −0.383659 −0.191829 0.981428i \(-0.561442\pi\)
−0.191829 + 0.981428i \(0.561442\pi\)
\(174\) − 1673.16i − 0.728977i
\(175\) 1508.62i 0.651661i
\(176\) 1801.33i 0.771481i
\(177\) − 133.368i − 0.0566359i
\(178\) 1062.00 0.447193
\(179\) −1287.00 −0.537402 −0.268701 0.963224i \(-0.586594\pi\)
−0.268701 + 0.963224i \(0.586594\pi\)
\(180\) 1219.36i 0.504922i
\(181\) 2.00000 0.000821319 0 0.000410660 1.00000i \(-0.499869\pi\)
0.000410660 1.00000i \(0.499869\pi\)
\(182\) 0 0
\(183\) 119.000 0.0480696
\(184\) − 789.815i − 0.316445i
\(185\) −552.000 −0.219372
\(186\) −1764.00 −0.695391
\(187\) − 607.950i − 0.237742i
\(188\) − 1371.78i − 0.532168i
\(189\) − 788.083i − 0.303305i
\(190\) − 4240.06i − 1.61898i
\(191\) −2841.00 −1.07627 −0.538135 0.842859i \(-0.680871\pi\)
−0.538135 + 0.842859i \(0.680871\pi\)
\(192\) 448.000 0.168394
\(193\) − 4245.26i − 1.58332i −0.610964 0.791659i \(-0.709218\pi\)
0.610964 0.791659i \(-0.290782\pi\)
\(194\) −4278.00 −1.58321
\(195\) 0 0
\(196\) 656.000 0.239067
\(197\) 2752.23i 0.995371i 0.867357 + 0.497686i \(0.165817\pi\)
−0.867357 + 0.497686i \(0.834183\pi\)
\(198\) −1716.00 −0.615913
\(199\) −1685.00 −0.600234 −0.300117 0.953902i \(-0.597026\pi\)
−0.300117 + 0.953902i \(0.597026\pi\)
\(200\) 928.379i 0.328232i
\(201\) − 1151.81i − 0.404192i
\(202\) − 6786.18i − 2.36373i
\(203\) 1553.65i 0.537167i
\(204\) 756.000 0.259464
\(205\) −5448.00 −1.85612
\(206\) − 6429.37i − 2.17454i
\(207\) 1254.00 0.421058
\(208\) 0 0
\(209\) 1989.00 0.658287
\(210\) − 7565.60i − 2.48608i
\(211\) 1681.00 0.548459 0.274229 0.961664i \(-0.411577\pi\)
0.274229 + 0.961664i \(0.411577\pi\)
\(212\) −1704.00 −0.552034
\(213\) 4085.91i 1.31437i
\(214\) 883.346i 0.282170i
\(215\) 1177.79i 0.373604i
\(216\) − 484.974i − 0.152770i
\(217\) 1638.00 0.512418
\(218\) 2112.00 0.656159
\(219\) 7032.13i 2.16980i
\(220\) 1248.00 0.382455
\(221\) 0 0
\(222\) 966.000 0.292044
\(223\) − 4096.30i − 1.23008i −0.788495 0.615042i \(-0.789139\pi\)
0.788495 0.615042i \(-0.210861\pi\)
\(224\) 3744.00 1.11677
\(225\) −1474.00 −0.436741
\(226\) − 1423.75i − 0.419054i
\(227\) − 438.209i − 0.128128i −0.997946 0.0640638i \(-0.979594\pi\)
0.997946 0.0640638i \(-0.0204061\pi\)
\(228\) 2473.37i 0.718433i
\(229\) − 180.133i − 0.0519805i −0.999662 0.0259903i \(-0.991726\pi\)
0.999662 0.0259903i \(-0.00827389\pi\)
\(230\) −2736.00 −0.784376
\(231\) 3549.00 1.01085
\(232\) 956.092i 0.270563i
\(233\) −5778.00 −1.62459 −0.812295 0.583247i \(-0.801782\pi\)
−0.812295 + 0.583247i \(0.801782\pi\)
\(234\) 0 0
\(235\) 4752.00 1.31909
\(236\) − 76.2102i − 0.0210206i
\(237\) 8708.00 2.38669
\(238\) −2106.00 −0.573579
\(239\) 1860.22i 0.503464i 0.967797 + 0.251732i \(0.0810001\pi\)
−0.967797 + 0.251732i \(0.919000\pi\)
\(240\) − 7759.59i − 2.08700i
\(241\) − 2059.41i − 0.550449i −0.961380 0.275224i \(-0.911248\pi\)
0.961380 0.275224i \(-0.0887521\pi\)
\(242\) − 2854.42i − 0.758219i
\(243\) 4928.00 1.30095
\(244\) 68.0000 0.0178412
\(245\) 2272.45i 0.592578i
\(246\) 9534.00 2.47100
\(247\) 0 0
\(248\) 1008.00 0.258097
\(249\) − 2982.59i − 0.759093i
\(250\) −2784.00 −0.704302
\(251\) 4491.00 1.12936 0.564680 0.825310i \(-0.309000\pi\)
0.564680 + 0.825310i \(0.309000\pi\)
\(252\) 1981.47i 0.495320i
\(253\) − 1283.45i − 0.318932i
\(254\) 7769.98i 1.91942i
\(255\) 2618.86i 0.643135i
\(256\) 4864.00 1.18750
\(257\) 5451.00 1.32305 0.661525 0.749923i \(-0.269909\pi\)
0.661525 + 0.749923i \(0.269909\pi\)
\(258\) − 2061.14i − 0.497368i
\(259\) −897.000 −0.215200
\(260\) 0 0
\(261\) −1518.00 −0.360007
\(262\) 1288.65i 0.303866i
\(263\) −783.000 −0.183581 −0.0917906 0.995778i \(-0.529259\pi\)
−0.0917906 + 0.995778i \(0.529259\pi\)
\(264\) 2184.00 0.509151
\(265\) − 5902.83i − 1.36833i
\(266\) − 6890.10i − 1.58819i
\(267\) − 2146.01i − 0.491887i
\(268\) − 658.179i − 0.150018i
\(269\) −5085.00 −1.15256 −0.576279 0.817253i \(-0.695496\pi\)
−0.576279 + 0.817253i \(0.695496\pi\)
\(270\) −1680.00 −0.378672
\(271\) − 1325.02i − 0.297008i −0.988912 0.148504i \(-0.952554\pi\)
0.988912 0.148504i \(-0.0474458\pi\)
\(272\) −2160.00 −0.481505
\(273\) 0 0
\(274\) 4122.00 0.908829
\(275\) 1508.62i 0.330811i
\(276\) 1596.00 0.348072
\(277\) −3421.00 −0.742050 −0.371025 0.928623i \(-0.620994\pi\)
−0.371025 + 0.928623i \(0.620994\pi\)
\(278\) 8816.14i 1.90200i
\(279\) 1600.41i 0.343421i
\(280\) 4323.20i 0.922716i
\(281\) − 810.600i − 0.172087i −0.996291 0.0860433i \(-0.972578\pi\)
0.996291 0.0860433i \(-0.0274223\pi\)
\(282\) −8316.00 −1.75607
\(283\) 7177.00 1.50752 0.753760 0.657149i \(-0.228238\pi\)
0.753760 + 0.657149i \(0.228238\pi\)
\(284\) 2334.80i 0.487835i
\(285\) −8568.00 −1.78079
\(286\) 0 0
\(287\) −8853.00 −1.82082
\(288\) 3658.09i 0.748455i
\(289\) −4184.00 −0.851618
\(290\) 3312.00 0.670646
\(291\) 8644.67i 1.74144i
\(292\) 4018.36i 0.805331i
\(293\) − 9313.24i − 1.85695i −0.371400 0.928473i \(-0.621122\pi\)
0.371400 0.928473i \(-0.378878\pi\)
\(294\) − 3976.79i − 0.788881i
\(295\) 264.000 0.0521040
\(296\) −552.000 −0.108393
\(297\) − 788.083i − 0.153970i
\(298\) −4518.00 −0.878257
\(299\) 0 0
\(300\) −1876.00 −0.361036
\(301\) 1913.92i 0.366499i
\(302\) 300.000 0.0571625
\(303\) −13713.0 −2.59997
\(304\) − 7066.77i − 1.33325i
\(305\) 235.559i 0.0442232i
\(306\) − 2057.68i − 0.384410i
\(307\) 4777.00i 0.888070i 0.896009 + 0.444035i \(0.146453\pi\)
−0.896009 + 0.444035i \(0.853547\pi\)
\(308\) 2028.00 0.375182
\(309\) −12992.0 −2.39187
\(310\) − 3491.81i − 0.639748i
\(311\) 6192.00 1.12899 0.564495 0.825436i \(-0.309071\pi\)
0.564495 + 0.825436i \(0.309071\pi\)
\(312\) 0 0
\(313\) −770.000 −0.139051 −0.0695255 0.997580i \(-0.522149\pi\)
−0.0695255 + 0.997580i \(0.522149\pi\)
\(314\) 5313.93i 0.955040i
\(315\) −6864.00 −1.22775
\(316\) 4976.00 0.885829
\(317\) − 8057.50i − 1.42762i −0.700341 0.713808i \(-0.746969\pi\)
0.700341 0.713808i \(-0.253031\pi\)
\(318\) 10330.0i 1.82162i
\(319\) 1553.65i 0.272689i
\(320\) 886.810i 0.154919i
\(321\) 1785.00 0.310371
\(322\) −4446.00 −0.769459
\(323\) 2385.03i 0.410857i
\(324\) 3356.00 0.575446
\(325\) 0 0
\(326\) 5658.00 0.961250
\(327\) − 4267.77i − 0.721738i
\(328\) −5448.00 −0.917120
\(329\) 7722.00 1.29400
\(330\) − 7565.60i − 1.26204i
\(331\) 5277.56i 0.876377i 0.898883 + 0.438189i \(0.144380\pi\)
−0.898883 + 0.438189i \(0.855620\pi\)
\(332\) − 1704.34i − 0.281740i
\(333\) − 876.418i − 0.144226i
\(334\) 5634.00 0.922990
\(335\) 2280.00 0.371850
\(336\) − 12609.3i − 2.04731i
\(337\) 8278.00 1.33808 0.669038 0.743228i \(-0.266706\pi\)
0.669038 + 0.743228i \(0.266706\pi\)
\(338\) 0 0
\(339\) −2877.00 −0.460936
\(340\) 1496.49i 0.238702i
\(341\) 1638.00 0.260125
\(342\) 6732.00 1.06440
\(343\) − 4030.48i − 0.634477i
\(344\) 1177.79i 0.184600i
\(345\) 5528.71i 0.862770i
\(346\) 3024.16i 0.469884i
\(347\) 6867.00 1.06236 0.531181 0.847258i \(-0.321748\pi\)
0.531181 + 0.847258i \(0.321748\pi\)
\(348\) −1932.00 −0.297604
\(349\) 12153.8i 1.86412i 0.362303 + 0.932060i \(0.381990\pi\)
−0.362303 + 0.932060i \(0.618010\pi\)
\(350\) 5226.00 0.798118
\(351\) 0 0
\(352\) 3744.00 0.566920
\(353\) 5807.57i 0.875653i 0.899059 + 0.437827i \(0.144252\pi\)
−0.899059 + 0.437827i \(0.855748\pi\)
\(354\) −462.000 −0.0693645
\(355\) −8088.00 −1.20920
\(356\) − 1226.29i − 0.182566i
\(357\) 4255.65i 0.630904i
\(358\) 4458.30i 0.658180i
\(359\) − 1340.61i − 0.197088i −0.995133 0.0985439i \(-0.968581\pi\)
0.995133 0.0985439i \(-0.0314185\pi\)
\(360\) −4224.00 −0.618401
\(361\) −944.000 −0.137629
\(362\) − 6.92820i − 0.00100591i
\(363\) −5768.00 −0.833999
\(364\) 0 0
\(365\) −13920.0 −1.99618
\(366\) − 412.228i − 0.0588730i
\(367\) 3665.00 0.521285 0.260642 0.965435i \(-0.416066\pi\)
0.260642 + 0.965435i \(0.416066\pi\)
\(368\) −4560.00 −0.645941
\(369\) − 8649.86i − 1.22031i
\(370\) 1912.18i 0.268675i
\(371\) − 9592.10i − 1.34231i
\(372\) 2036.89i 0.283892i
\(373\) 5371.00 0.745576 0.372788 0.927917i \(-0.378402\pi\)
0.372788 + 0.927917i \(0.378402\pi\)
\(374\) −2106.00 −0.291173
\(375\) 5625.70i 0.774693i
\(376\) 4752.00 0.651770
\(377\) 0 0
\(378\) −2730.00 −0.371471
\(379\) − 11509.5i − 1.55990i −0.625842 0.779950i \(-0.715244\pi\)
0.625842 0.779950i \(-0.284756\pi\)
\(380\) −4896.00 −0.660946
\(381\) 15701.0 2.11125
\(382\) 9841.51i 1.31816i
\(383\) − 2419.67i − 0.322819i −0.986888 0.161409i \(-0.948396\pi\)
0.986888 0.161409i \(-0.0516040\pi\)
\(384\) − 10863.4i − 1.44368i
\(385\) 7025.20i 0.929967i
\(386\) −14706.0 −1.93916
\(387\) −1870.00 −0.245626
\(388\) 4939.81i 0.646342i
\(389\) −9858.00 −1.28489 −0.642443 0.766334i \(-0.722079\pi\)
−0.642443 + 0.766334i \(0.722079\pi\)
\(390\) 0 0
\(391\) 1539.00 0.199055
\(392\) 2272.45i 0.292796i
\(393\) 2604.00 0.334235
\(394\) 9534.00 1.21908
\(395\) 17237.4i 2.19571i
\(396\) 1981.47i 0.251446i
\(397\) 8720.88i 1.10249i 0.834344 + 0.551245i \(0.185847\pi\)
−0.834344 + 0.551245i \(0.814153\pi\)
\(398\) 5837.01i 0.735133i
\(399\) −13923.0 −1.74692
\(400\) 5360.00 0.670000
\(401\) − 7584.65i − 0.944537i −0.881455 0.472269i \(-0.843435\pi\)
0.881455 0.472269i \(-0.156565\pi\)
\(402\) −3990.00 −0.495033
\(403\) 0 0
\(404\) −7836.00 −0.964989
\(405\) 11625.5i 1.42636i
\(406\) 5382.00 0.657892
\(407\) −897.000 −0.109245
\(408\) 2618.86i 0.317777i
\(409\) − 4304.15i − 0.520358i −0.965560 0.260179i \(-0.916218\pi\)
0.965560 0.260179i \(-0.0837815\pi\)
\(410\) 18872.4i 2.27327i
\(411\) − 8329.43i − 0.999661i
\(412\) −7424.00 −0.887753
\(413\) 429.000 0.0511131
\(414\) − 4343.98i − 0.515689i
\(415\) 5904.00 0.698352
\(416\) 0 0
\(417\) 17815.0 2.09210
\(418\) − 6890.10i − 0.806234i
\(419\) 5397.00 0.629262 0.314631 0.949214i \(-0.398119\pi\)
0.314631 + 0.949214i \(0.398119\pi\)
\(420\) −8736.00 −1.01494
\(421\) − 7260.76i − 0.840541i −0.907399 0.420270i \(-0.861935\pi\)
0.907399 0.420270i \(-0.138065\pi\)
\(422\) − 5823.15i − 0.671722i
\(423\) 7544.81i 0.867237i
\(424\) − 5902.83i − 0.676101i
\(425\) −1809.00 −0.206469
\(426\) 14154.0 1.60977
\(427\) 382.783i 0.0433822i
\(428\) 1020.00 0.115195
\(429\) 0 0
\(430\) 4080.00 0.457570
\(431\) − 486.706i − 0.0543940i −0.999630 0.0271970i \(-0.991342\pi\)
0.999630 0.0271970i \(-0.00865814\pi\)
\(432\) −2800.00 −0.311840
\(433\) 12139.0 1.34726 0.673629 0.739069i \(-0.264734\pi\)
0.673629 + 0.739069i \(0.264734\pi\)
\(434\) − 5674.20i − 0.627581i
\(435\) − 6692.64i − 0.737673i
\(436\) − 2438.73i − 0.267876i
\(437\) 5035.07i 0.551167i
\(438\) 24360.0 2.65746
\(439\) 461.000 0.0501192 0.0250596 0.999686i \(-0.492022\pi\)
0.0250596 + 0.999686i \(0.492022\pi\)
\(440\) 4323.20i 0.468410i
\(441\) −3608.00 −0.389591
\(442\) 0 0
\(443\) 12156.0 1.30372 0.651861 0.758338i \(-0.273988\pi\)
0.651861 + 0.758338i \(0.273988\pi\)
\(444\) − 1115.44i − 0.119226i
\(445\) 4248.00 0.452527
\(446\) −14190.0 −1.50654
\(447\) 9129.64i 0.966034i
\(448\) 1441.07i 0.151973i
\(449\) 296.181i 0.0311306i 0.999879 + 0.0155653i \(0.00495479\pi\)
−0.999879 + 0.0155653i \(0.995045\pi\)
\(450\) 5106.09i 0.534896i
\(451\) −8853.00 −0.924327
\(452\) −1644.00 −0.171078
\(453\) − 606.218i − 0.0628755i
\(454\) −1518.00 −0.156924
\(455\) 0 0
\(456\) −8568.00 −0.879898
\(457\) 611.414i 0.0625837i 0.999510 + 0.0312918i \(0.00996213\pi\)
−0.999510 + 0.0312918i \(0.990038\pi\)
\(458\) −624.000 −0.0636629
\(459\) 945.000 0.0960977
\(460\) 3159.26i 0.320220i
\(461\) 13127.2i 1.32624i 0.748514 + 0.663119i \(0.230767\pi\)
−0.748514 + 0.663119i \(0.769233\pi\)
\(462\) − 12294.1i − 1.23804i
\(463\) 834.848i 0.0837985i 0.999122 + 0.0418992i \(0.0133408\pi\)
−0.999122 + 0.0418992i \(0.986659\pi\)
\(464\) 5520.00 0.552284
\(465\) −7056.00 −0.703686
\(466\) 20015.6i 1.98971i
\(467\) 14496.0 1.43639 0.718196 0.695841i \(-0.244968\pi\)
0.718196 + 0.695841i \(0.244968\pi\)
\(468\) 0 0
\(469\) 3705.00 0.364778
\(470\) − 16461.4i − 1.61555i
\(471\) 10738.0 1.05049
\(472\) 264.000 0.0257449
\(473\) 1913.92i 0.186051i
\(474\) − 30165.4i − 2.92309i
\(475\) − 5918.42i − 0.571696i
\(476\) 2431.80i 0.234162i
\(477\) 9372.00 0.899611
\(478\) 6444.00 0.616614
\(479\) − 8897.54i − 0.848725i −0.905492 0.424362i \(-0.860498\pi\)
0.905492 0.424362i \(-0.139502\pi\)
\(480\) −16128.0 −1.53362
\(481\) 0 0
\(482\) −7134.00 −0.674159
\(483\) 8984.15i 0.846362i
\(484\) −3296.00 −0.309542
\(485\) −17112.0 −1.60209
\(486\) − 17071.1i − 1.59333i
\(487\) − 4754.48i − 0.442394i −0.975229 0.221197i \(-0.929004\pi\)
0.975229 0.221197i \(-0.0709964\pi\)
\(488\) 235.559i 0.0218509i
\(489\) − 11433.3i − 1.05732i
\(490\) 7872.00 0.725757
\(491\) −1635.00 −0.150278 −0.0751390 0.997173i \(-0.523940\pi\)
−0.0751390 + 0.997173i \(0.523940\pi\)
\(492\) − 11008.9i − 1.00878i
\(493\) −1863.00 −0.170193
\(494\) 0 0
\(495\) −6864.00 −0.623260
\(496\) − 5819.69i − 0.526838i
\(497\) −13143.0 −1.18621
\(498\) −10332.0 −0.929695
\(499\) 14434.9i 1.29498i 0.762074 + 0.647490i \(0.224181\pi\)
−0.762074 + 0.647490i \(0.775819\pi\)
\(500\) 3214.69i 0.287530i
\(501\) − 11384.8i − 1.01524i
\(502\) − 15557.3i − 1.38318i
\(503\) 12687.0 1.12462 0.562312 0.826925i \(-0.309912\pi\)
0.562312 + 0.826925i \(0.309912\pi\)
\(504\) −6864.00 −0.606641
\(505\) − 27144.7i − 2.39193i
\(506\) −4446.00 −0.390610
\(507\) 0 0
\(508\) 8972.00 0.783599
\(509\) − 5748.68i − 0.500600i −0.968168 0.250300i \(-0.919471\pi\)
0.968168 0.250300i \(-0.0805293\pi\)
\(510\) 9072.00 0.787676
\(511\) −22620.0 −1.95822
\(512\) − 4434.05i − 0.382733i
\(513\) 3091.71i 0.266086i
\(514\) − 18882.8i − 1.62040i
\(515\) − 25717.5i − 2.20048i
\(516\) −2380.00 −0.203050
\(517\) 7722.00 0.656892
\(518\) 3107.30i 0.263565i
\(519\) 6111.00 0.516846
\(520\) 0 0
\(521\) 6054.00 0.509080 0.254540 0.967062i \(-0.418076\pi\)
0.254540 + 0.967062i \(0.418076\pi\)
\(522\) 5258.51i 0.440917i
\(523\) −14803.0 −1.23765 −0.618824 0.785530i \(-0.712391\pi\)
−0.618824 + 0.785530i \(0.712391\pi\)
\(524\) 1488.00 0.124053
\(525\) − 10560.3i − 0.877885i
\(526\) 2712.39i 0.224840i
\(527\) 1964.15i 0.162352i
\(528\) − 12609.3i − 1.03930i
\(529\) −8918.00 −0.732966
\(530\) −20448.0 −1.67586
\(531\) 419.156i 0.0342558i
\(532\) −7956.00 −0.648377
\(533\) 0 0
\(534\) −7434.00 −0.602436
\(535\) 3533.38i 0.285536i
\(536\) 2280.00 0.183733
\(537\) 9009.00 0.723961
\(538\) 17615.0i 1.41159i
\(539\) 3692.73i 0.295097i
\(540\) 1939.90i 0.154592i
\(541\) 21470.5i 1.70626i 0.521695 + 0.853132i \(0.325300\pi\)
−0.521695 + 0.853132i \(0.674700\pi\)
\(542\) −4590.00 −0.363759
\(543\) −14.0000 −0.00110644
\(544\) 4489.48i 0.353832i
\(545\) 8448.00 0.663986
\(546\) 0 0
\(547\) −13516.0 −1.05649 −0.528247 0.849091i \(-0.677151\pi\)
−0.528247 + 0.849091i \(0.677151\pi\)
\(548\) − 4759.68i − 0.371028i
\(549\) −374.000 −0.0290746
\(550\) 5226.00 0.405159
\(551\) − 6095.09i − 0.471251i
\(552\) 5528.71i 0.426300i
\(553\) 28010.7i 2.15396i
\(554\) 11850.7i 0.908822i
\(555\) 3864.00 0.295527
\(556\) 10180.0 0.776490
\(557\) 2890.79i 0.219905i 0.993937 + 0.109952i \(0.0350698\pi\)
−0.993937 + 0.109952i \(0.964930\pi\)
\(558\) 5544.00 0.420603
\(559\) 0 0
\(560\) 24960.0 1.88349
\(561\) 4255.65i 0.320274i
\(562\) −2808.00 −0.210762
\(563\) 11583.0 0.867079 0.433539 0.901135i \(-0.357265\pi\)
0.433539 + 0.901135i \(0.357265\pi\)
\(564\) 9602.49i 0.716911i
\(565\) − 5694.98i − 0.424053i
\(566\) − 24861.9i − 1.84633i
\(567\) 18891.5i 1.39924i
\(568\) −8088.00 −0.597473
\(569\) 12879.0 0.948885 0.474443 0.880286i \(-0.342650\pi\)
0.474443 + 0.880286i \(0.342650\pi\)
\(570\) 29680.4i 2.18101i
\(571\) −11636.0 −0.852805 −0.426402 0.904534i \(-0.640219\pi\)
−0.426402 + 0.904534i \(0.640219\pi\)
\(572\) 0 0
\(573\) 19887.0 1.44990
\(574\) 30667.7i 2.23004i
\(575\) −3819.00 −0.276980
\(576\) −1408.00 −0.101852
\(577\) − 12311.4i − 0.888269i −0.895960 0.444134i \(-0.853511\pi\)
0.895960 0.444134i \(-0.146489\pi\)
\(578\) 14493.8i 1.04301i
\(579\) 29716.8i 2.13297i
\(580\) − 3824.37i − 0.273790i
\(581\) 9594.00 0.685071
\(582\) 29946.0 2.13282
\(583\) − 9592.10i − 0.681414i
\(584\) −13920.0 −0.986325
\(585\) 0 0
\(586\) −32262.0 −2.27428
\(587\) − 15645.6i − 1.10011i −0.835129 0.550054i \(-0.814607\pi\)
0.835129 0.550054i \(-0.185393\pi\)
\(588\) −4592.00 −0.322059
\(589\) −6426.00 −0.449539
\(590\) − 914.523i − 0.0638141i
\(591\) − 19265.6i − 1.34092i
\(592\) 3186.97i 0.221256i
\(593\) − 25821.4i − 1.78813i −0.447942 0.894063i \(-0.647843\pi\)
0.447942 0.894063i \(-0.352157\pi\)
\(594\) −2730.00 −0.188575
\(595\) −8424.00 −0.580421
\(596\) 5216.94i 0.358547i
\(597\) 11795.0 0.808605
\(598\) 0 0
\(599\) 1668.00 0.113777 0.0568887 0.998381i \(-0.481882\pi\)
0.0568887 + 0.998381i \(0.481882\pi\)
\(600\) − 6498.65i − 0.442177i
\(601\) 13699.0 0.929773 0.464887 0.885370i \(-0.346095\pi\)
0.464887 + 0.885370i \(0.346095\pi\)
\(602\) 6630.00 0.448868
\(603\) 3619.99i 0.244473i
\(604\) − 346.410i − 0.0233365i
\(605\) − 11417.7i − 0.767264i
\(606\) 47503.2i 3.18430i
\(607\) −23173.0 −1.54953 −0.774764 0.632251i \(-0.782131\pi\)
−0.774764 + 0.632251i \(0.782131\pi\)
\(608\) −14688.0 −0.979732
\(609\) − 10875.5i − 0.723644i
\(610\) 816.000 0.0541621
\(611\) 0 0
\(612\) −2376.00 −0.156935
\(613\) 16615.6i 1.09477i 0.836880 + 0.547387i \(0.184377\pi\)
−0.836880 + 0.547387i \(0.815623\pi\)
\(614\) 16548.0 1.08766
\(615\) 38136.0 2.50047
\(616\) 7025.20i 0.459502i
\(617\) − 28393.5i − 1.85264i −0.376736 0.926321i \(-0.622954\pi\)
0.376736 0.926321i \(-0.377046\pi\)
\(618\) 45005.6i 2.92944i
\(619\) 6245.78i 0.405556i 0.979225 + 0.202778i \(0.0649969\pi\)
−0.979225 + 0.202778i \(0.935003\pi\)
\(620\) −4032.00 −0.261176
\(621\) 1995.00 0.128916
\(622\) − 21449.7i − 1.38273i
\(623\) 6903.00 0.443921
\(624\) 0 0
\(625\) −19511.0 −1.24870
\(626\) 2667.36i 0.170302i
\(627\) −13923.0 −0.886812
\(628\) 6136.00 0.389893
\(629\) − 1075.60i − 0.0681830i
\(630\) 23777.6i 1.50369i
\(631\) − 22379.8i − 1.41193i −0.708247 0.705964i \(-0.750514\pi\)
0.708247 0.705964i \(-0.249486\pi\)
\(632\) 17237.4i 1.08491i
\(633\) −11767.0 −0.738857
\(634\) −27912.0 −1.74847
\(635\) 31079.9i 1.94231i
\(636\) 11928.0 0.743673
\(637\) 0 0
\(638\) 5382.00 0.333974
\(639\) − 12841.4i − 0.794990i
\(640\) 21504.0 1.32816
\(641\) 19827.0 1.22172 0.610858 0.791740i \(-0.290825\pi\)
0.610858 + 0.791740i \(0.290825\pi\)
\(642\) − 6183.42i − 0.380125i
\(643\) 8450.68i 0.518293i 0.965838 + 0.259146i \(0.0834412\pi\)
−0.965838 + 0.259146i \(0.916559\pi\)
\(644\) 5133.80i 0.314130i
\(645\) − 8244.56i − 0.503301i
\(646\) 8262.00 0.503195
\(647\) 2949.00 0.179192 0.0895959 0.995978i \(-0.471442\pi\)
0.0895959 + 0.995978i \(0.471442\pi\)
\(648\) 11625.5i 0.704774i
\(649\) 429.000 0.0259472
\(650\) 0 0
\(651\) −11466.0 −0.690304
\(652\) − 6533.30i − 0.392429i
\(653\) 12039.0 0.721474 0.360737 0.932668i \(-0.382525\pi\)
0.360737 + 0.932668i \(0.382525\pi\)
\(654\) −14784.0 −0.883945
\(655\) 5154.58i 0.307490i
\(656\) 31454.0i 1.87206i
\(657\) − 22101.0i − 1.31239i
\(658\) − 26749.8i − 1.58483i
\(659\) 3363.00 0.198792 0.0993960 0.995048i \(-0.468309\pi\)
0.0993960 + 0.995048i \(0.468309\pi\)
\(660\) −8736.00 −0.515225
\(661\) 10158.5i 0.597759i 0.954291 + 0.298880i \(0.0966129\pi\)
−0.954291 + 0.298880i \(0.903387\pi\)
\(662\) 18282.0 1.07334
\(663\) 0 0
\(664\) 5904.00 0.345060
\(665\) − 27560.4i − 1.60714i
\(666\) −3036.00 −0.176641
\(667\) −3933.00 −0.228315
\(668\) − 6505.58i − 0.376809i
\(669\) 28674.1i 1.65711i
\(670\) − 7898.15i − 0.455421i
\(671\) 382.783i 0.0220226i
\(672\) −26208.0 −1.50446
\(673\) −18169.0 −1.04066 −0.520329 0.853966i \(-0.674191\pi\)
−0.520329 + 0.853966i \(0.674191\pi\)
\(674\) − 28675.8i − 1.63880i
\(675\) −2345.00 −0.133717
\(676\) 0 0
\(677\) 9042.00 0.513312 0.256656 0.966503i \(-0.417379\pi\)
0.256656 + 0.966503i \(0.417379\pi\)
\(678\) 9966.22i 0.564529i
\(679\) −27807.0 −1.57163
\(680\) −5184.00 −0.292349
\(681\) 3067.46i 0.172607i
\(682\) − 5674.20i − 0.318587i
\(683\) 12462.1i 0.698169i 0.937091 + 0.349084i \(0.113507\pi\)
−0.937091 + 0.349084i \(0.886493\pi\)
\(684\) − 7773.44i − 0.434540i
\(685\) 16488.0 0.919670
\(686\) −13962.0 −0.777072
\(687\) 1260.93i 0.0700256i
\(688\) 6800.00 0.376813
\(689\) 0 0
\(690\) 19152.0 1.05667
\(691\) − 4318.00i − 0.237720i −0.992911 0.118860i \(-0.962076\pi\)
0.992911 0.118860i \(-0.0379240\pi\)
\(692\) 3492.00 0.191829
\(693\) −11154.0 −0.611408
\(694\) − 23788.0i − 1.30112i
\(695\) 35264.6i 1.92469i
\(696\) − 6692.64i − 0.364489i
\(697\) − 10615.7i − 0.576901i
\(698\) 42102.0 2.28307
\(699\) 40446.0 2.18857
\(700\) − 6034.47i − 0.325830i
\(701\) −18270.0 −0.984377 −0.492189 0.870489i \(-0.663803\pi\)
−0.492189 + 0.870489i \(0.663803\pi\)
\(702\) 0 0
\(703\) 3519.00 0.188793
\(704\) 1441.07i 0.0771481i
\(705\) −33264.0 −1.77701
\(706\) 20118.0 1.07245
\(707\) − 44110.1i − 2.34644i
\(708\) 533.472i 0.0283179i
\(709\) − 1629.86i − 0.0863338i −0.999068 0.0431669i \(-0.986255\pi\)
0.999068 0.0431669i \(-0.0137447\pi\)
\(710\) 28017.7i 1.48096i
\(711\) −27368.0 −1.44357
\(712\) 4248.00 0.223596
\(713\) 4146.53i 0.217796i
\(714\) 14742.0 0.772697
\(715\) 0 0
\(716\) 5148.00 0.268701
\(717\) − 13021.6i − 0.678241i
\(718\) −4644.00 −0.241382
\(719\) 9831.00 0.509923 0.254961 0.966951i \(-0.417937\pi\)
0.254961 + 0.966951i \(0.417937\pi\)
\(720\) 24387.3i 1.26231i
\(721\) − 41790.9i − 2.15863i
\(722\) 3270.11i 0.168561i
\(723\) 14415.9i 0.741537i
\(724\) −8.00000 −0.000410660 0
\(725\) 4623.00 0.236819
\(726\) 19980.9i 1.02144i
\(727\) −15464.0 −0.788897 −0.394448 0.918918i \(-0.629064\pi\)
−0.394448 + 0.918918i \(0.629064\pi\)
\(728\) 0 0
\(729\) −11843.0 −0.601687
\(730\) 48220.3i 2.44481i
\(731\) −2295.00 −0.116120
\(732\) −476.000 −0.0240348
\(733\) 12616.3i 0.635733i 0.948136 + 0.317866i \(0.102966\pi\)
−0.948136 + 0.317866i \(0.897034\pi\)
\(734\) − 12695.9i − 0.638441i
\(735\) − 15907.2i − 0.798291i
\(736\) 9477.78i 0.474668i
\(737\) 3705.00 0.185177
\(738\) −29964.0 −1.49457
\(739\) − 16283.0i − 0.810528i −0.914200 0.405264i \(-0.867180\pi\)
0.914200 0.405264i \(-0.132820\pi\)
\(740\) 2208.00 0.109686
\(741\) 0 0
\(742\) −33228.0 −1.64399
\(743\) 10806.3i 0.533571i 0.963756 + 0.266786i \(0.0859616\pi\)
−0.963756 + 0.266786i \(0.914038\pi\)
\(744\) −7056.00 −0.347696
\(745\) −18072.0 −0.888734
\(746\) − 18605.7i − 0.913140i
\(747\) 9373.86i 0.459132i
\(748\) 2431.80i 0.118871i
\(749\) 5741.75i 0.280105i
\(750\) 19488.0 0.948802
\(751\) −13615.0 −0.661542 −0.330771 0.943711i \(-0.607309\pi\)
−0.330771 + 0.943711i \(0.607309\pi\)
\(752\) − 27435.7i − 1.33042i
\(753\) −31437.0 −1.52142
\(754\) 0 0
\(755\) 1200.00 0.0578443
\(756\) 3152.33i 0.151652i
\(757\) 5551.00 0.266519 0.133259 0.991081i \(-0.457456\pi\)
0.133259 + 0.991081i \(0.457456\pi\)
\(758\) −39870.0 −1.91048
\(759\) 8984.15i 0.429649i
\(760\) − 16960.2i − 0.809490i
\(761\) − 10082.3i − 0.480265i −0.970740 0.240133i \(-0.922809\pi\)
0.970740 0.240133i \(-0.0771909\pi\)
\(762\) − 54389.9i − 2.58574i
\(763\) 13728.0 0.651359
\(764\) 11364.0 0.538135
\(765\) − 8230.71i − 0.388996i
\(766\) −8382.00 −0.395371
\(767\) 0 0
\(768\) −34048.0 −1.59974
\(769\) 29758.4i 1.39547i 0.716357 + 0.697733i \(0.245808\pi\)
−0.716357 + 0.697733i \(0.754192\pi\)
\(770\) 24336.0 1.13897
\(771\) −38157.0 −1.78235
\(772\) 16981.0i 0.791659i
\(773\) − 27735.3i − 1.29052i −0.763964 0.645259i \(-0.776749\pi\)
0.763964 0.645259i \(-0.223251\pi\)
\(774\) 6477.87i 0.300830i
\(775\) − 4873.99i − 0.225908i
\(776\) −17112.0 −0.791604
\(777\) 6279.00 0.289907
\(778\) 34149.1i 1.57366i
\(779\) 34731.0 1.59739
\(780\) 0 0
\(781\) −13143.0 −0.602168
\(782\) − 5331.25i − 0.243792i
\(783\) −2415.00 −0.110224
\(784\) 13120.0 0.597668
\(785\) 21255.7i 0.966432i
\(786\) − 9020.52i − 0.409353i
\(787\) 31549.3i 1.42899i 0.699643 + 0.714493i \(0.253342\pi\)
−0.699643 + 0.714493i \(0.746658\pi\)
\(788\) − 11008.9i − 0.497686i
\(789\) 5481.00 0.247311
\(790\) 59712.0 2.68919
\(791\) − 9254.35i − 0.415988i
\(792\) −6864.00 −0.307957
\(793\) 0 0
\(794\) 30210.0 1.35027
\(795\) 41319.8i 1.84335i
\(796\) 6740.00 0.300117
\(797\) 1455.00 0.0646659 0.0323330 0.999477i \(-0.489706\pi\)
0.0323330 + 0.999477i \(0.489706\pi\)
\(798\) 48230.7i 2.13953i
\(799\) 9259.54i 0.409986i
\(800\) − 11140.6i − 0.492347i
\(801\) 6744.61i 0.297514i
\(802\) −26274.0 −1.15682
\(803\) −22620.0 −0.994075
\(804\) 4607.26i 0.202096i
\(805\) −17784.0 −0.778638
\(806\) 0 0
\(807\) 35595.0 1.55267
\(808\) − 27144.7i − 1.18187i
\(809\) 1659.00 0.0720981 0.0360490 0.999350i \(-0.488523\pi\)
0.0360490 + 0.999350i \(0.488523\pi\)
\(810\) 40272.0 1.74693
\(811\) − 4402.87i − 0.190636i −0.995447 0.0953180i \(-0.969613\pi\)
0.995447 0.0953180i \(-0.0303868\pi\)
\(812\) − 6214.60i − 0.268583i
\(813\) 9275.13i 0.400114i
\(814\) 3107.30i 0.133797i
\(815\) 22632.0 0.972717
\(816\) 15120.0 0.648659
\(817\) − 7508.44i − 0.321526i
\(818\) −14910.0 −0.637306
\(819\) 0 0
\(820\) 21792.0 0.928061
\(821\) − 28701.8i − 1.22010i −0.792364 0.610049i \(-0.791150\pi\)
0.792364 0.610049i \(-0.208850\pi\)
\(822\) −28854.0 −1.22433
\(823\) 15779.0 0.668313 0.334156 0.942518i \(-0.391549\pi\)
0.334156 + 0.942518i \(0.391549\pi\)
\(824\) − 25717.5i − 1.08727i
\(825\) − 10560.3i − 0.445652i
\(826\) − 1486.10i − 0.0626005i
\(827\) 7354.29i 0.309231i 0.987975 + 0.154615i \(0.0494138\pi\)
−0.987975 + 0.154615i \(0.950586\pi\)
\(828\) −5016.00 −0.210529
\(829\) 17371.0 0.727768 0.363884 0.931444i \(-0.381450\pi\)
0.363884 + 0.931444i \(0.381450\pi\)
\(830\) − 20452.1i − 0.855303i
\(831\) 23947.0 0.999654
\(832\) 0 0
\(833\) −4428.00 −0.184179
\(834\) − 61713.0i − 2.56228i
\(835\) 22536.0 0.934001
\(836\) −7956.00 −0.329144
\(837\) 2546.11i 0.105145i
\(838\) − 18695.8i − 0.770685i
\(839\) − 29474.3i − 1.21283i −0.795148 0.606416i \(-0.792607\pi\)
0.795148 0.606416i \(-0.207393\pi\)
\(840\) − 30262.4i − 1.24304i
\(841\) −19628.0 −0.804789
\(842\) −25152.0 −1.02945
\(843\) 5674.20i 0.231827i
\(844\) −6724.00 −0.274229
\(845\) 0 0
\(846\) 26136.0 1.06214
\(847\) − 18553.7i − 0.752673i
\(848\) −34080.0 −1.38008
\(849\) −50239.0 −2.03086
\(850\) 6266.56i 0.252872i
\(851\) − 2270.72i − 0.0914680i
\(852\) − 16343.6i − 0.657187i
\(853\) − 2909.85i − 0.116801i −0.998293 0.0584005i \(-0.981400\pi\)
0.998293 0.0584005i \(-0.0186000\pi\)
\(854\) 1326.00 0.0531321
\(855\) 26928.0 1.07710
\(856\) 3533.38i 0.141085i
\(857\) −5346.00 −0.213087 −0.106544 0.994308i \(-0.533978\pi\)
−0.106544 + 0.994308i \(0.533978\pi\)
\(858\) 0 0
\(859\) 24244.0 0.962974 0.481487 0.876453i \(-0.340097\pi\)
0.481487 + 0.876453i \(0.340097\pi\)
\(860\) − 4711.18i − 0.186802i
\(861\) 61971.0 2.45292
\(862\) −1686.00 −0.0666188
\(863\) − 32780.8i − 1.29301i −0.762908 0.646507i \(-0.776229\pi\)
0.762908 0.646507i \(-0.223771\pi\)
\(864\) 5819.69i 0.229155i
\(865\) 12096.6i 0.475489i
\(866\) − 42050.7i − 1.65005i
\(867\) 29288.0 1.14726
\(868\) −6552.00 −0.256209
\(869\) 28010.7i 1.09344i
\(870\) −23184.0 −0.903461
\(871\) 0 0
\(872\) 8448.00 0.328080
\(873\) − 27168.9i − 1.05330i
\(874\) 17442.0 0.675039
\(875\) −18096.0 −0.699150
\(876\) − 28128.5i − 1.08490i
\(877\) 4543.17i 0.174928i 0.996168 + 0.0874640i \(0.0278763\pi\)
−0.996168 + 0.0874640i \(0.972124\pi\)
\(878\) − 1596.95i − 0.0613832i
\(879\) 65192.7i 2.50159i
\(880\) 24960.0 0.956138
\(881\) −20517.0 −0.784603 −0.392302 0.919837i \(-0.628321\pi\)
−0.392302 + 0.919837i \(0.628321\pi\)
\(882\) 12498.5i 0.477149i
\(883\) −23852.0 −0.909042 −0.454521 0.890736i \(-0.650189\pi\)
−0.454521 + 0.890736i \(0.650189\pi\)
\(884\) 0 0
\(885\) −1848.00 −0.0701919
\(886\) − 42109.6i − 1.59673i
\(887\) 38757.0 1.46712 0.733558 0.679626i \(-0.237858\pi\)
0.733558 + 0.679626i \(0.237858\pi\)
\(888\) 3864.00 0.146022
\(889\) 50504.9i 1.90538i
\(890\) − 14715.5i − 0.554230i
\(891\) 18891.5i 0.710312i
\(892\) 16385.2i 0.615042i
\(893\) −30294.0 −1.13522
\(894\) 31626.0 1.18315
\(895\) 17833.2i 0.666031i
\(896\) 34944.0 1.30290
\(897\) 0 0
\(898\) 1026.00 0.0381270
\(899\) − 5019.48i − 0.186217i
\(900\) 5896.00 0.218370
\(901\) 11502.0 0.425291
\(902\) 30667.7i 1.13206i
\(903\) − 13397.4i − 0.493730i
\(904\) − 5694.98i − 0.209527i
\(905\) − 27.7128i − 0.00101791i
\(906\) −2100.00 −0.0770064
\(907\) −39071.0 −1.43035 −0.715177 0.698943i \(-0.753654\pi\)
−0.715177 + 0.698943i \(0.753654\pi\)
\(908\) 1752.84i 0.0640638i
\(909\) 43098.0 1.57257
\(910\) 0 0
\(911\) −53040.0 −1.92897 −0.964486 0.264134i \(-0.914914\pi\)
−0.964486 + 0.264134i \(0.914914\pi\)
\(912\) 49467.4i 1.79608i
\(913\) 9594.00 0.347771
\(914\) 2118.00 0.0766490
\(915\) − 1648.91i − 0.0595753i
\(916\) 720.533i 0.0259903i
\(917\) 8376.20i 0.301643i
\(918\) − 3273.58i − 0.117695i
\(919\) 367.000 0.0131732 0.00658662 0.999978i \(-0.497903\pi\)
0.00658662 + 0.999978i \(0.497903\pi\)
\(920\) −10944.0 −0.392188
\(921\) − 33439.0i − 1.19636i
\(922\) 45474.0 1.62430
\(923\) 0 0
\(924\) −14196.0 −0.505427
\(925\) 2669.09i 0.0948748i
\(926\) 2892.00 0.102632
\(927\) 40832.0 1.44671
\(928\) − 11473.1i − 0.405844i
\(929\) 29935.0i 1.05720i 0.848872 + 0.528599i \(0.177282\pi\)
−0.848872 + 0.528599i \(0.822718\pi\)
\(930\) 24442.7i 0.861836i
\(931\) − 14486.9i − 0.509976i
\(932\) 23112.0 0.812295
\(933\) −43344.0 −1.52092
\(934\) − 50215.6i − 1.75921i
\(935\) −8424.00 −0.294646
\(936\) 0 0
\(937\) 42166.0 1.47012 0.735060 0.678002i \(-0.237154\pi\)
0.735060 + 0.678002i \(0.237154\pi\)
\(938\) − 12834.5i − 0.446760i
\(939\) 5390.00 0.187323
\(940\) −19008.0 −0.659545
\(941\) 35022.1i 1.21327i 0.794981 + 0.606635i \(0.207481\pi\)
−0.794981 + 0.606635i \(0.792519\pi\)
\(942\) − 37197.5i − 1.28658i
\(943\) − 22411.0i − 0.773916i
\(944\) − 1524.20i − 0.0525515i
\(945\) −10920.0 −0.375902
\(946\) 6630.00 0.227865
\(947\) − 2599.81i − 0.0892106i −0.999005 0.0446053i \(-0.985797\pi\)
0.999005 0.0446053i \(-0.0142030\pi\)
\(948\) −34832.0 −1.19334
\(949\) 0 0
\(950\) −20502.0 −0.700182
\(951\) 56402.5i 1.92321i
\(952\) −8424.00 −0.286789
\(953\) 10623.0 0.361084 0.180542 0.983567i \(-0.442215\pi\)
0.180542 + 0.983567i \(0.442215\pi\)
\(954\) − 32465.6i − 1.10179i
\(955\) 39366.1i 1.33388i
\(956\) − 7440.89i − 0.251732i
\(957\) − 10875.5i − 0.367353i
\(958\) −30822.0 −1.03947
\(959\) 26793.0 0.902180
\(960\) − 6207.67i − 0.208700i
\(961\) 24499.0 0.822362
\(962\) 0 0
\(963\) −5610.00 −0.187726
\(964\) 8237.63i 0.275224i
\(965\) −58824.0 −1.96229
\(966\) 31122.0 1.03658
\(967\) − 20199.2i − 0.671729i −0.941910 0.335864i \(-0.890972\pi\)
0.941910 0.335864i \(-0.109028\pi\)
\(968\) − 11417.7i − 0.379110i
\(969\) − 16695.2i − 0.553486i
\(970\) 59277.7i 1.96216i
\(971\) −2325.00 −0.0768412 −0.0384206 0.999262i \(-0.512233\pi\)
−0.0384206 + 0.999262i \(0.512233\pi\)
\(972\) −19712.0 −0.650476
\(973\) 57304.9i 1.88809i
\(974\) −16470.0 −0.541820
\(975\) 0 0
\(976\) 1360.00 0.0446030
\(977\) 32938.4i 1.07860i 0.842113 + 0.539300i \(0.181311\pi\)
−0.842113 + 0.539300i \(0.818689\pi\)
\(978\) −39606.0 −1.29495
\(979\) 6903.00 0.225353
\(980\) − 9089.80i − 0.296289i
\(981\) 13413.0i 0.436538i
\(982\) 5663.81i 0.184052i
\(983\) − 42702.0i − 1.38554i −0.721161 0.692768i \(-0.756391\pi\)
0.721161 0.692768i \(-0.243609\pi\)
\(984\) 38136.0 1.23550
\(985\) 38136.0 1.23362
\(986\) 6453.62i 0.208443i
\(987\) −54054.0 −1.74322
\(988\) 0 0
\(989\) −4845.00 −0.155776
\(990\) 23777.6i 0.763335i
\(991\) −4843.00 −0.155240 −0.0776201 0.996983i \(-0.524732\pi\)
−0.0776201 + 0.996983i \(0.524732\pi\)
\(992\) −12096.0 −0.387146
\(993\) − 36942.9i − 1.18061i
\(994\) 45528.7i 1.45280i
\(995\) 23348.0i 0.743902i
\(996\) 11930.4i 0.379546i
\(997\) 10943.0 0.347611 0.173806 0.984780i \(-0.444394\pi\)
0.173806 + 0.984780i \(0.444394\pi\)
\(998\) 50004.0 1.58602
\(999\) − 1394.30i − 0.0441579i
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 169.4.b.b.168.1 2
13.2 odd 12 169.4.c.i.22.2 4
13.3 even 3 169.4.e.b.147.1 2
13.4 even 6 169.4.e.b.23.1 2
13.5 odd 4 169.4.a.h.1.1 2
13.6 odd 12 169.4.c.i.146.2 4
13.7 odd 12 169.4.c.i.146.1 4
13.8 odd 4 169.4.a.h.1.2 2
13.9 even 3 13.4.e.a.10.1 yes 2
13.10 even 6 13.4.e.a.4.1 2
13.11 odd 12 169.4.c.i.22.1 4
13.12 even 2 inner 169.4.b.b.168.2 2
39.5 even 4 1521.4.a.q.1.2 2
39.8 even 4 1521.4.a.q.1.1 2
39.23 odd 6 117.4.q.c.82.1 2
39.35 odd 6 117.4.q.c.10.1 2
52.23 odd 6 208.4.w.a.17.1 2
52.35 odd 6 208.4.w.a.49.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.e.a.4.1 2 13.10 even 6
13.4.e.a.10.1 yes 2 13.9 even 3
117.4.q.c.10.1 2 39.35 odd 6
117.4.q.c.82.1 2 39.23 odd 6
169.4.a.h.1.1 2 13.5 odd 4
169.4.a.h.1.2 2 13.8 odd 4
169.4.b.b.168.1 2 1.1 even 1 trivial
169.4.b.b.168.2 2 13.12 even 2 inner
169.4.c.i.22.1 4 13.11 odd 12
169.4.c.i.22.2 4 13.2 odd 12
169.4.c.i.146.1 4 13.7 odd 12
169.4.c.i.146.2 4 13.6 odd 12
169.4.e.b.23.1 2 13.4 even 6
169.4.e.b.147.1 2 13.3 even 3
208.4.w.a.17.1 2 52.23 odd 6
208.4.w.a.49.1 2 52.35 odd 6
1521.4.a.q.1.1 2 39.8 even 4
1521.4.a.q.1.2 2 39.5 even 4