Properties

Label 5070.2.b.e.1351.2
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.4841538248\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.e.1351.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} -8.00000 q^{17} +1.00000i q^{18} -6.00000i q^{19} +1.00000i q^{20} +2.00000i q^{21} +4.00000 q^{22} -6.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +2.00000i q^{28} -4.00000 q^{29} -1.00000 q^{30} +1.00000i q^{32} +4.00000i q^{33} -8.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} +2.00000i q^{37} +6.00000 q^{38} -1.00000 q^{40} -2.00000i q^{41} -2.00000 q^{42} +4.00000 q^{43} +4.00000i q^{44} -1.00000i q^{45} -6.00000i q^{46} -1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} +8.00000 q^{51} -10.0000 q^{53} -1.00000i q^{54} -4.00000 q^{55} -2.00000 q^{56} +6.00000i q^{57} -4.00000i q^{58} -4.00000i q^{59} -1.00000i q^{60} -10.0000 q^{61} -2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} +12.0000i q^{67} +8.00000 q^{68} +6.00000 q^{69} -2.00000i q^{70} -8.00000i q^{71} -1.00000i q^{72} +8.00000i q^{73} -2.00000 q^{74} +1.00000 q^{75} +6.00000i q^{76} -8.00000 q^{77} +8.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +2.00000 q^{82} +12.0000i q^{83} -2.00000i q^{84} +8.00000i q^{85} +4.00000i q^{86} +4.00000 q^{87} -4.00000 q^{88} +14.0000i q^{89} +1.00000 q^{90} +6.00000 q^{92} -6.00000 q^{95} -1.00000i q^{96} -16.0000i q^{97} +3.00000i q^{98} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{4} + 2q^{9} + 2q^{10} + 2q^{12} + 4q^{14} + 2q^{16} - 16q^{17} + 8q^{22} - 12q^{23} - 2q^{25} - 2q^{27} - 8q^{29} - 2q^{30} - 4q^{35} - 2q^{36} + 12q^{38} - 2q^{40} - 4q^{42} + 8q^{43} - 2q^{48} + 6q^{49} + 16q^{51} - 20q^{53} - 8q^{55} - 4q^{56} - 20q^{61} - 2q^{64} - 8q^{66} + 16q^{68} + 12q^{69} - 4q^{74} + 2q^{75} - 16q^{77} + 16q^{79} + 2q^{81} + 4q^{82} + 8q^{87} - 8q^{88} + 2q^{90} + 12q^{92} - 12q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) − 1.00000i − 0.447214i
\(6\) − 1.00000i − 0.408248i
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) − 4.00000i − 1.20605i −0.797724 0.603023i \(-0.793963\pi\)
0.797724 0.603023i \(-0.206037\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) −8.00000 −1.94029 −0.970143 0.242536i \(-0.922021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 1.00000i 0.235702i
\(19\) − 6.00000i − 1.37649i −0.725476 0.688247i \(-0.758380\pi\)
0.725476 0.688247i \(-0.241620\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 2.00000i 0.436436i
\(22\) 4.00000 0.852803
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.00000i 0.377964i
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) −1.00000 −0.182574
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 4.00000i 0.696311i
\(34\) − 8.00000i − 1.37199i
\(35\) −2.00000 −0.338062
\(36\) −1.00000 −0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) − 2.00000i − 0.312348i −0.987730 0.156174i \(-0.950084\pi\)
0.987730 0.156174i \(-0.0499160\pi\)
\(42\) −2.00000 −0.308607
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 4.00000i 0.603023i
\(45\) − 1.00000i − 0.149071i
\(46\) − 6.00000i − 0.884652i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −1.00000 −0.144338
\(49\) 3.00000 0.428571
\(50\) − 1.00000i − 0.141421i
\(51\) 8.00000 1.12022
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) −4.00000 −0.539360
\(56\) −2.00000 −0.267261
\(57\) 6.00000i 0.794719i
\(58\) − 4.00000i − 0.525226i
\(59\) − 4.00000i − 0.520756i −0.965507 0.260378i \(-0.916153\pi\)
0.965507 0.260378i \(-0.0838471\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 8.00000 0.970143
\(69\) 6.00000 0.722315
\(70\) − 2.00000i − 0.239046i
\(71\) − 8.00000i − 0.949425i −0.880141 0.474713i \(-0.842552\pi\)
0.880141 0.474713i \(-0.157448\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 8.00000i 0.936329i 0.883641 + 0.468165i \(0.155085\pi\)
−0.883641 + 0.468165i \(0.844915\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) 6.00000i 0.688247i
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) − 2.00000i − 0.218218i
\(85\) 8.00000i 0.867722i
\(86\) 4.00000i 0.431331i
\(87\) 4.00000 0.428845
\(88\) −4.00000 −0.426401
\(89\) 14.0000i 1.48400i 0.670402 + 0.741999i \(0.266122\pi\)
−0.670402 + 0.741999i \(0.733878\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) − 1.00000i − 0.102062i
\(97\) − 16.0000i − 1.62455i −0.583272 0.812277i \(-0.698228\pi\)
0.583272 0.812277i \(-0.301772\pi\)
\(98\) 3.00000i 0.303046i
\(99\) − 4.00000i − 0.402015i
\(100\) 1.00000 0.100000
\(101\) 16.0000 1.59206 0.796030 0.605257i \(-0.206930\pi\)
0.796030 + 0.605257i \(0.206930\pi\)
\(102\) 8.00000i 0.792118i
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) − 10.0000i − 0.971286i
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) 12.0000i 1.14939i 0.818367 + 0.574696i \(0.194880\pi\)
−0.818367 + 0.574696i \(0.805120\pi\)
\(110\) − 4.00000i − 0.381385i
\(111\) − 2.00000i − 0.189832i
\(112\) − 2.00000i − 0.188982i
\(113\) 20.0000 1.88144 0.940721 0.339182i \(-0.110150\pi\)
0.940721 + 0.339182i \(0.110150\pi\)
\(114\) −6.00000 −0.561951
\(115\) 6.00000i 0.559503i
\(116\) 4.00000 0.371391
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) 16.0000i 1.46672i
\(120\) 1.00000 0.0912871
\(121\) −5.00000 −0.454545
\(122\) − 10.0000i − 0.905357i
\(123\) 2.00000i 0.180334i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 2.00000 0.178174
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) − 4.00000i − 0.348155i
\(133\) −12.0000 −1.04053
\(134\) −12.0000 −1.03664
\(135\) 1.00000i 0.0860663i
\(136\) 8.00000i 0.685994i
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 6.00000i 0.510754i
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 4.00000i 0.332182i
\(146\) −8.00000 −0.662085
\(147\) −3.00000 −0.247436
\(148\) − 2.00000i − 0.164399i
\(149\) − 10.0000i − 0.819232i −0.912258 0.409616i \(-0.865663\pi\)
0.912258 0.409616i \(-0.134337\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) −6.00000 −0.486664
\(153\) −8.00000 −0.646762
\(154\) − 8.00000i − 0.644658i
\(155\) 0 0
\(156\) 0 0
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 8.00000i 0.636446i
\(159\) 10.0000 0.793052
\(160\) 1.00000 0.0790569
\(161\) 12.0000i 0.945732i
\(162\) 1.00000i 0.0785674i
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) 2.00000i 0.156174i
\(165\) 4.00000 0.311400
\(166\) −12.0000 −0.931381
\(167\) 4.00000i 0.309529i 0.987951 + 0.154765i \(0.0494619\pi\)
−0.987951 + 0.154765i \(0.950538\pi\)
\(168\) 2.00000 0.154303
\(169\) 0 0
\(170\) −8.00000 −0.613572
\(171\) − 6.00000i − 0.458831i
\(172\) −4.00000 −0.304997
\(173\) −22.0000 −1.67263 −0.836315 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) 4.00000i 0.303239i
\(175\) 2.00000i 0.151186i
\(176\) − 4.00000i − 0.301511i
\(177\) 4.00000i 0.300658i
\(178\) −14.0000 −1.04934
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 6.00000i 0.442326i
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) 32.0000i 2.34007i
\(188\) 0 0
\(189\) 2.00000i 0.145479i
\(190\) − 6.00000i − 0.435286i
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) 4.00000i 0.287926i 0.989583 + 0.143963i \(0.0459847\pi\)
−0.989583 + 0.143963i \(0.954015\pi\)
\(194\) 16.0000 1.14873
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 4.00000 0.284268
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) − 12.0000i − 0.846415i
\(202\) 16.0000i 1.12576i
\(203\) 8.00000i 0.561490i
\(204\) −8.00000 −0.560112
\(205\) −2.00000 −0.139686
\(206\) 12.0000i 0.836080i
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) −24.0000 −1.66011
\(210\) 2.00000i 0.138013i
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 10.0000 0.686803
\(213\) 8.00000i 0.548151i
\(214\) 12.0000i 0.820303i
\(215\) − 4.00000i − 0.272798i
\(216\) 1.00000i 0.0680414i
\(217\) 0 0
\(218\) −12.0000 −0.812743
\(219\) − 8.00000i − 0.540590i
\(220\) 4.00000 0.269680
\(221\) 0 0
\(222\) 2.00000 0.134231
\(223\) − 2.00000i − 0.133930i −0.997755 0.0669650i \(-0.978668\pi\)
0.997755 0.0669650i \(-0.0213316\pi\)
\(224\) 2.00000 0.133631
\(225\) −1.00000 −0.0666667
\(226\) 20.0000i 1.33038i
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) − 6.00000i − 0.397360i
\(229\) − 4.00000i − 0.264327i −0.991228 0.132164i \(-0.957808\pi\)
0.991228 0.132164i \(-0.0421925\pi\)
\(230\) −6.00000 −0.395628
\(231\) 8.00000 0.526361
\(232\) 4.00000i 0.262613i
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 4.00000i 0.260378i
\(237\) −8.00000 −0.519656
\(238\) −16.0000 −1.03713
\(239\) 16.0000i 1.03495i 0.855697 + 0.517477i \(0.173129\pi\)
−0.855697 + 0.517477i \(0.826871\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) − 2.00000i − 0.128831i −0.997923 0.0644157i \(-0.979482\pi\)
0.997923 0.0644157i \(-0.0205183\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) − 3.00000i − 0.191663i
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) 0 0
\(249\) − 12.0000i − 0.760469i
\(250\) −1.00000 −0.0632456
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 2.00000i 0.125988i
\(253\) 24.0000i 1.50887i
\(254\) − 4.00000i − 0.250982i
\(255\) − 8.00000i − 0.500979i
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) − 4.00000i − 0.249029i
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) 10.0000i 0.617802i
\(263\) 2.00000 0.123325 0.0616626 0.998097i \(-0.480360\pi\)
0.0616626 + 0.998097i \(0.480360\pi\)
\(264\) 4.00000 0.246183
\(265\) 10.0000i 0.614295i
\(266\) − 12.0000i − 0.735767i
\(267\) − 14.0000i − 0.856786i
\(268\) − 12.0000i − 0.733017i
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) −1.00000 −0.0608581
\(271\) − 16.0000i − 0.971931i −0.873978 0.485965i \(-0.838468\pi\)
0.873978 0.485965i \(-0.161532\pi\)
\(272\) −8.00000 −0.485071
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 4.00000i 0.241209i
\(276\) −6.00000 −0.361158
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) − 8.00000i − 0.479808i
\(279\) 0 0
\(280\) 2.00000i 0.119523i
\(281\) 6.00000i 0.357930i 0.983855 + 0.178965i \(0.0572749\pi\)
−0.983855 + 0.178965i \(0.942725\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 8.00000i 0.474713i
\(285\) 6.00000 0.355409
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 1.00000i 0.0589256i
\(289\) 47.0000 2.76471
\(290\) −4.00000 −0.234888
\(291\) 16.0000i 0.937937i
\(292\) − 8.00000i − 0.468165i
\(293\) 26.0000i 1.51894i 0.650545 + 0.759468i \(0.274541\pi\)
−0.650545 + 0.759468i \(0.725459\pi\)
\(294\) − 3.00000i − 0.174964i
\(295\) −4.00000 −0.232889
\(296\) 2.00000 0.116248
\(297\) 4.00000i 0.232104i
\(298\) 10.0000 0.579284
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) − 8.00000i − 0.461112i
\(302\) 0 0
\(303\) −16.0000 −0.919176
\(304\) − 6.00000i − 0.344124i
\(305\) 10.0000i 0.572598i
\(306\) − 8.00000i − 0.457330i
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 8.00000 0.455842
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 22.0000i 1.24153i
\(315\) −2.00000 −0.112687
\(316\) −8.00000 −0.450035
\(317\) − 30.0000i − 1.68497i −0.538721 0.842484i \(-0.681092\pi\)
0.538721 0.842484i \(-0.318908\pi\)
\(318\) 10.0000i 0.560772i
\(319\) 16.0000i 0.895828i
\(320\) 1.00000i 0.0559017i
\(321\) −12.0000 −0.669775
\(322\) −12.0000 −0.668734
\(323\) 48.0000i 2.67079i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) − 12.0000i − 0.663602i
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) 4.00000i 0.220193i
\(331\) 10.0000i 0.549650i 0.961494 + 0.274825i \(0.0886199\pi\)
−0.961494 + 0.274825i \(0.911380\pi\)
\(332\) − 12.0000i − 0.658586i
\(333\) 2.00000i 0.109599i
\(334\) −4.00000 −0.218870
\(335\) 12.0000 0.655630
\(336\) 2.00000i 0.109109i
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) −20.0000 −1.08625
\(340\) − 8.00000i − 0.433861i
\(341\) 0 0
\(342\) 6.00000 0.324443
\(343\) − 20.0000i − 1.07990i
\(344\) − 4.00000i − 0.215666i
\(345\) − 6.00000i − 0.323029i
\(346\) − 22.0000i − 1.18273i
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) −4.00000 −0.214423
\(349\) 28.0000i 1.49881i 0.662114 + 0.749403i \(0.269659\pi\)
−0.662114 + 0.749403i \(0.730341\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) − 14.0000i − 0.745145i −0.928003 0.372572i \(-0.878476\pi\)
0.928003 0.372572i \(-0.121524\pi\)
\(354\) −4.00000 −0.212598
\(355\) −8.00000 −0.424596
\(356\) − 14.0000i − 0.741999i
\(357\) − 16.0000i − 0.846810i
\(358\) 10.0000i 0.528516i
\(359\) 24.0000i 1.26667i 0.773877 + 0.633336i \(0.218315\pi\)
−0.773877 + 0.633336i \(0.781685\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −17.0000 −0.894737
\(362\) − 10.0000i − 0.525588i
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 8.00000 0.418739
\(366\) 10.0000i 0.522708i
\(367\) −36.0000 −1.87918 −0.939592 0.342296i \(-0.888796\pi\)
−0.939592 + 0.342296i \(0.888796\pi\)
\(368\) −6.00000 −0.312772
\(369\) − 2.00000i − 0.104116i
\(370\) 2.00000i 0.103975i
\(371\) 20.0000i 1.03835i
\(372\) 0 0
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) −32.0000 −1.65468
\(375\) − 1.00000i − 0.0516398i
\(376\) 0 0
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) − 18.0000i − 0.924598i −0.886724 0.462299i \(-0.847025\pi\)
0.886724 0.462299i \(-0.152975\pi\)
\(380\) 6.00000 0.307794
\(381\) 4.00000 0.204926
\(382\) 0 0
\(383\) − 12.0000i − 0.613171i −0.951843 0.306586i \(-0.900813\pi\)
0.951843 0.306586i \(-0.0991866\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 8.00000i 0.407718i
\(386\) −4.00000 −0.203595
\(387\) 4.00000 0.203331
\(388\) 16.0000i 0.812277i
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) 0 0
\(391\) 48.0000 2.42746
\(392\) − 3.00000i − 0.151523i
\(393\) −10.0000 −0.504433
\(394\) 18.0000 0.906827
\(395\) − 8.00000i − 0.402524i
\(396\) 4.00000i 0.201008i
\(397\) − 14.0000i − 0.702640i −0.936255 0.351320i \(-0.885733\pi\)
0.936255 0.351320i \(-0.114267\pi\)
\(398\) 0 0
\(399\) 12.0000 0.600751
\(400\) −1.00000 −0.0500000
\(401\) − 30.0000i − 1.49813i −0.662497 0.749064i \(-0.730503\pi\)
0.662497 0.749064i \(-0.269497\pi\)
\(402\) 12.0000 0.598506
\(403\) 0 0
\(404\) −16.0000 −0.796030
\(405\) − 1.00000i − 0.0496904i
\(406\) −8.00000 −0.397033
\(407\) 8.00000 0.396545
\(408\) − 8.00000i − 0.396059i
\(409\) − 26.0000i − 1.28562i −0.766027 0.642809i \(-0.777769\pi\)
0.766027 0.642809i \(-0.222231\pi\)
\(410\) − 2.00000i − 0.0987730i
\(411\) − 6.00000i − 0.295958i
\(412\) −12.0000 −0.591198
\(413\) −8.00000 −0.393654
\(414\) − 6.00000i − 0.294884i
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) 8.00000 0.391762
\(418\) − 24.0000i − 1.17388i
\(419\) 10.0000 0.488532 0.244266 0.969708i \(-0.421453\pi\)
0.244266 + 0.969708i \(0.421453\pi\)
\(420\) −2.00000 −0.0975900
\(421\) − 8.00000i − 0.389896i −0.980814 0.194948i \(-0.937546\pi\)
0.980814 0.194948i \(-0.0624538\pi\)
\(422\) − 20.0000i − 0.973585i
\(423\) 0 0
\(424\) 10.0000i 0.485643i
\(425\) 8.00000 0.388057
\(426\) −8.00000 −0.387601
\(427\) 20.0000i 0.967868i
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) − 4.00000i − 0.191785i
\(436\) − 12.0000i − 0.574696i
\(437\) 36.0000i 1.72211i
\(438\) 8.00000 0.382255
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 4.00000i 0.190693i
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 16.0000 0.760183 0.380091 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 2.00000i 0.0949158i
\(445\) 14.0000 0.663664
\(446\) 2.00000 0.0947027
\(447\) 10.0000i 0.472984i
\(448\) 2.00000i 0.0944911i
\(449\) 6.00000i 0.283158i 0.989927 + 0.141579i \(0.0452178\pi\)
−0.989927 + 0.141579i \(0.954782\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) −8.00000 −0.376705
\(452\) −20.0000 −0.940721
\(453\) 0 0
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 6.00000 0.280976
\(457\) 8.00000i 0.374224i 0.982339 + 0.187112i \(0.0599128\pi\)
−0.982339 + 0.187112i \(0.940087\pi\)
\(458\) 4.00000 0.186908
\(459\) 8.00000 0.373408
\(460\) − 6.00000i − 0.279751i
\(461\) − 6.00000i − 0.279448i −0.990190 0.139724i \(-0.955378\pi\)
0.990190 0.139724i \(-0.0446215\pi\)
\(462\) 8.00000i 0.372194i
\(463\) 26.0000i 1.20832i 0.796862 + 0.604161i \(0.206492\pi\)
−0.796862 + 0.604161i \(0.793508\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) − 24.0000i − 1.11178i
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) 24.0000 1.10822
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) −4.00000 −0.184115
\(473\) − 16.0000i − 0.735681i
\(474\) − 8.00000i − 0.367452i
\(475\) 6.00000i 0.275299i
\(476\) − 16.0000i − 0.733359i
\(477\) −10.0000 −0.457869
\(478\) −16.0000 −0.731823
\(479\) − 32.0000i − 1.46212i −0.682315 0.731059i \(-0.739027\pi\)
0.682315 0.731059i \(-0.260973\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 2.00000 0.0910975
\(483\) − 12.0000i − 0.546019i
\(484\) 5.00000 0.227273
\(485\) −16.0000 −0.726523
\(486\) − 1.00000i − 0.0453609i
\(487\) 26.0000i 1.17817i 0.808070 + 0.589086i \(0.200512\pi\)
−0.808070 + 0.589086i \(0.799488\pi\)
\(488\) 10.0000i 0.452679i
\(489\) − 16.0000i − 0.723545i
\(490\) 3.00000 0.135526
\(491\) −42.0000 −1.89543 −0.947717 0.319113i \(-0.896615\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) − 2.00000i − 0.0901670i
\(493\) 32.0000 1.44121
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) 0 0
\(497\) −16.0000 −0.717698
\(498\) 12.0000 0.537733
\(499\) 38.0000i 1.70111i 0.525883 + 0.850557i \(0.323735\pi\)
−0.525883 + 0.850557i \(0.676265\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) − 4.00000i − 0.178707i
\(502\) − 6.00000i − 0.267793i
\(503\) 10.0000 0.445878 0.222939 0.974832i \(-0.428435\pi\)
0.222939 + 0.974832i \(0.428435\pi\)
\(504\) −2.00000 −0.0890871
\(505\) − 16.0000i − 0.711991i
\(506\) −24.0000 −1.06693
\(507\) 0 0
\(508\) 4.00000 0.177471
\(509\) 18.0000i 0.797836i 0.916987 + 0.398918i \(0.130614\pi\)
−0.916987 + 0.398918i \(0.869386\pi\)
\(510\) 8.00000 0.354246
\(511\) 16.0000 0.707798
\(512\) 1.00000i 0.0441942i
\(513\) 6.00000i 0.264906i
\(514\) − 12.0000i − 0.529297i
\(515\) − 12.0000i − 0.528783i
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 4.00000i 0.175750i
\(519\) 22.0000 0.965693
\(520\) 0 0
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) − 4.00000i − 0.175075i
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) −10.0000 −0.436852
\(525\) − 2.00000i − 0.0872872i
\(526\) 2.00000i 0.0872041i
\(527\) 0 0
\(528\) 4.00000i 0.174078i
\(529\) 13.0000 0.565217
\(530\) −10.0000 −0.434372
\(531\) − 4.00000i − 0.173585i
\(532\) 12.0000 0.520266
\(533\) 0 0
\(534\) 14.0000 0.605839
\(535\) − 12.0000i − 0.518805i
\(536\) 12.0000 0.518321
\(537\) −10.0000 −0.431532
\(538\) − 24.0000i − 1.03471i
\(539\) − 12.0000i − 0.516877i
\(540\) − 1.00000i − 0.0430331i
\(541\) 8.00000i 0.343947i 0.985102 + 0.171973i \(0.0550143\pi\)
−0.985102 + 0.171973i \(0.944986\pi\)
\(542\) 16.0000 0.687259
\(543\) 10.0000 0.429141
\(544\) − 8.00000i − 0.342997i
\(545\) 12.0000 0.514024
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) −10.0000 −0.426790
\(550\) −4.00000 −0.170561
\(551\) 24.0000i 1.02243i
\(552\) − 6.00000i − 0.255377i
\(553\) − 16.0000i − 0.680389i
\(554\) 10.0000i 0.424859i
\(555\) −2.00000 −0.0848953
\(556\) 8.00000 0.339276
\(557\) − 2.00000i − 0.0847427i −0.999102 0.0423714i \(-0.986509\pi\)
0.999102 0.0423714i \(-0.0134913\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −2.00000 −0.0845154
\(561\) − 32.0000i − 1.35104i
\(562\) −6.00000 −0.253095
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) − 20.0000i − 0.841406i
\(566\) 4.00000i 0.168133i
\(567\) − 2.00000i − 0.0839921i
\(568\) −8.00000 −0.335673
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 6.00000i 0.251312i
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 4.00000i − 0.166957i
\(575\) 6.00000 0.250217
\(576\) −1.00000 −0.0416667
\(577\) 12.0000i 0.499567i 0.968302 + 0.249783i \(0.0803594\pi\)
−0.968302 + 0.249783i \(0.919641\pi\)
\(578\) 47.0000i 1.95494i
\(579\) − 4.00000i − 0.166234i
\(580\) − 4.00000i − 0.166091i
\(581\) 24.0000 0.995688
\(582\) −16.0000 −0.663221
\(583\) 40.0000i 1.65663i
\(584\) 8.00000 0.331042
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) 36.0000i 1.48588i 0.669359 + 0.742940i \(0.266569\pi\)
−0.669359 + 0.742940i \(0.733431\pi\)
\(588\) 3.00000 0.123718
\(589\) 0 0
\(590\) − 4.00000i − 0.164677i
\(591\) 18.0000i 0.740421i
\(592\) 2.00000i 0.0821995i
\(593\) − 22.0000i − 0.903432i −0.892162 0.451716i \(-0.850812\pi\)
0.892162 0.451716i \(-0.149188\pi\)
\(594\) −4.00000 −0.164122
\(595\) 16.0000 0.655936
\(596\) 10.0000i 0.409616i
\(597\) 0 0
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 8.00000 0.326056
\(603\) 12.0000i 0.488678i
\(604\) 0 0
\(605\) 5.00000i 0.203279i
\(606\) − 16.0000i − 0.649956i
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 6.00000 0.243332
\(609\) − 8.00000i − 0.324176i
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) 8.00000 0.323381
\(613\) − 10.0000i − 0.403896i −0.979396 0.201948i \(-0.935273\pi\)
0.979396 0.201948i \(-0.0647272\pi\)
\(614\) 12.0000 0.484281
\(615\) 2.00000 0.0806478
\(616\) 8.00000i 0.322329i
\(617\) − 6.00000i − 0.241551i −0.992680 0.120775i \(-0.961462\pi\)
0.992680 0.120775i \(-0.0385381\pi\)
\(618\) − 12.0000i − 0.482711i
\(619\) 10.0000i 0.401934i 0.979598 + 0.200967i \(0.0644084\pi\)
−0.979598 + 0.200967i \(0.935592\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) − 24.0000i − 0.962312i
\(623\) 28.0000 1.12180
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 6.00000i 0.239808i
\(627\) 24.0000 0.958468
\(628\) −22.0000 −0.877896
\(629\) − 16.0000i − 0.637962i
\(630\) − 2.00000i − 0.0796819i
\(631\) − 12.0000i − 0.477712i −0.971055 0.238856i \(-0.923228\pi\)
0.971055 0.238856i \(-0.0767725\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) 20.0000 0.794929
\(634\) 30.0000 1.19145
\(635\) 4.00000i 0.158735i
\(636\) −10.0000 −0.396526
\(637\) 0 0
\(638\) −16.0000 −0.633446
\(639\) − 8.00000i − 0.316475i
\(640\) −1.00000 −0.0395285
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) − 12.0000i − 0.473602i
\(643\) 44.0000i 1.73519i 0.497271 + 0.867595i \(0.334335\pi\)
−0.497271 + 0.867595i \(0.665665\pi\)
\(644\) − 12.0000i − 0.472866i
\(645\) 4.00000i 0.157500i
\(646\) −48.0000 −1.88853
\(647\) 30.0000 1.17942 0.589711 0.807614i \(-0.299242\pi\)
0.589711 + 0.807614i \(0.299242\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) − 16.0000i − 0.626608i
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 12.0000 0.469237
\(655\) − 10.0000i − 0.390732i
\(656\) − 2.00000i − 0.0780869i
\(657\) 8.00000i 0.312110i
\(658\) 0 0
\(659\) 2.00000 0.0779089 0.0389545 0.999241i \(-0.487597\pi\)
0.0389545 + 0.999241i \(0.487597\pi\)
\(660\) −4.00000 −0.155700
\(661\) − 48.0000i − 1.86698i −0.358599 0.933492i \(-0.616745\pi\)
0.358599 0.933492i \(-0.383255\pi\)
\(662\) −10.0000 −0.388661
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 12.0000i 0.465340i
\(666\) −2.00000 −0.0774984
\(667\) 24.0000 0.929284
\(668\) − 4.00000i − 0.154765i
\(669\) 2.00000i 0.0773245i
\(670\) 12.0000i 0.463600i
\(671\) 40.0000i 1.54418i
\(672\) −2.00000 −0.0771517
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) − 14.0000i − 0.539260i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) − 20.0000i − 0.768095i
\(679\) −32.0000 −1.22805
\(680\) 8.00000 0.306786
\(681\) 4.00000i 0.153280i
\(682\) 0 0
\(683\) 44.0000i 1.68361i 0.539779 + 0.841807i \(0.318508\pi\)
−0.539779 + 0.841807i \(0.681492\pi\)
\(684\) 6.00000i 0.229416i
\(685\) 6.00000 0.229248
\(686\) 20.0000 0.763604
\(687\) 4.00000i 0.152610i
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) 6.00000 0.228416
\(691\) − 14.0000i − 0.532585i −0.963892 0.266293i \(-0.914201\pi\)
0.963892 0.266293i \(-0.0857987\pi\)
\(692\) 22.0000 0.836315
\(693\) −8.00000 −0.303895
\(694\) − 24.0000i − 0.911028i
\(695\) 8.00000i 0.303457i
\(696\) − 4.00000i − 0.151620i
\(697\) 16.0000i 0.606043i
\(698\) −28.0000 −1.05982
\(699\) 24.0000 0.907763
\(700\) − 2.00000i − 0.0755929i
\(701\) −32.0000 −1.20862 −0.604312 0.796748i \(-0.706552\pi\)
−0.604312 + 0.796748i \(0.706552\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) 4.00000i 0.150756i
\(705\) 0 0
\(706\) 14.0000 0.526897
\(707\) − 32.0000i − 1.20348i
\(708\) − 4.00000i − 0.150329i
\(709\) 4.00000i 0.150223i 0.997175 + 0.0751116i \(0.0239313\pi\)
−0.997175 + 0.0751116i \(0.976069\pi\)
\(710\) − 8.00000i − 0.300235i
\(711\) 8.00000 0.300023
\(712\) 14.0000 0.524672
\(713\) 0 0
\(714\) 16.0000 0.598785
\(715\) 0 0
\(716\) −10.0000 −0.373718
\(717\) − 16.0000i − 0.597531i
\(718\) −24.0000 −0.895672
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) − 24.0000i − 0.893807i
\(722\) − 17.0000i − 0.632674i
\(723\) 2.00000i 0.0743808i
\(724\) 10.0000 0.371647
\(725\) 4.00000 0.148556
\(726\) 5.00000i 0.185567i
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 8.00000i 0.296093i
\(731\) −32.0000 −1.18356
\(732\) −10.0000 −0.369611
\(733\) 38.0000i 1.40356i 0.712393 + 0.701781i \(0.247612\pi\)
−0.712393 + 0.701781i \(0.752388\pi\)
\(734\) − 36.0000i − 1.32878i
\(735\) 3.00000i 0.110657i
\(736\) − 6.00000i − 0.221163i
\(737\) 48.0000 1.76810
\(738\) 2.00000 0.0736210
\(739\) − 30.0000i − 1.10357i −0.833987 0.551784i \(-0.813947\pi\)
0.833987 0.551784i \(-0.186053\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) −20.0000 −0.734223
\(743\) 12.0000i 0.440237i 0.975473 + 0.220119i \(0.0706445\pi\)
−0.975473 + 0.220119i \(0.929356\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) − 2.00000i − 0.0732252i
\(747\) 12.0000i 0.439057i
\(748\) − 32.0000i − 1.17004i
\(749\) − 24.0000i − 0.876941i
\(750\) 1.00000 0.0365148
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 0 0
\(753\) 6.00000 0.218652
\(754\) 0 0
\(755\) 0 0
\(756\) − 2.00000i − 0.0727393i
\(757\) −6.00000 −0.218074 −0.109037 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 18.0000 0.653789
\(759\) − 24.0000i − 0.871145i
\(760\) 6.00000i 0.217643i
\(761\) − 22.0000i − 0.797499i −0.917060 0.398750i \(-0.869444\pi\)
0.917060 0.398750i \(-0.130556\pi\)
\(762\) 4.00000i 0.144905i
\(763\) 24.0000 0.868858
\(764\) 0 0
\(765\) 8.00000i 0.289241i
\(766\) 12.0000 0.433578
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 10.0000i 0.360609i 0.983611 + 0.180305i \(0.0577084\pi\)
−0.983611 + 0.180305i \(0.942292\pi\)
\(770\) −8.00000 −0.288300
\(771\) 12.0000 0.432169
\(772\) − 4.00000i − 0.143963i
\(773\) 38.0000i 1.36677i 0.730061 + 0.683383i \(0.239492\pi\)
−0.730061 + 0.683383i \(0.760508\pi\)
\(774\) 4.00000i 0.143777i
\(775\) 0 0
\(776\) −16.0000 −0.574367
\(777\) −4.00000 −0.143499
\(778\) − 8.00000i − 0.286814i
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 48.0000i 1.71648i
\(783\) 4.00000 0.142948
\(784\) 3.00000 0.107143
\(785\) − 22.0000i − 0.785214i
\(786\) − 10.0000i − 0.356688i
\(787\) 32.0000i 1.14068i 0.821410 + 0.570338i \(0.193188\pi\)
−0.821410 + 0.570338i \(0.806812\pi\)
\(788\) 18.0000i 0.641223i
\(789\) −2.00000 −0.0712019
\(790\) 8.00000 0.284627
\(791\) − 40.0000i − 1.42224i
\(792\) −4.00000 −0.142134
\(793\) 0 0
\(794\) 14.0000 0.496841
\(795\) − 10.0000i − 0.354663i
\(796\) 0 0
\(797\) 10.0000 0.354218 0.177109 0.984191i \(-0.443325\pi\)
0.177109 + 0.984191i \(0.443325\pi\)
\(798\) 12.0000i 0.424795i
\(799\) 0 0
\(800\) − 1.00000i − 0.0353553i
\(801\) 14.0000i 0.494666i
\(802\) 30.0000 1.05934
\(803\) 32.0000 1.12926
\(804\) 12.0000i 0.423207i
\(805\) 12.0000 0.422944
\(806\) 0 0
\(807\) 24.0000 0.844840
\(808\) − 16.0000i − 0.562878i
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 1.00000 0.0351364
\(811\) − 38.0000i − 1.33436i −0.744896 0.667180i \(-0.767501\pi\)
0.744896 0.667180i \(-0.232499\pi\)
\(812\) − 8.00000i − 0.280745i
\(813\) 16.0000i 0.561144i
\(814\) 8.00000i 0.280400i
\(815\) 16.0000 0.560456
\(816\) 8.00000 0.280056
\(817\) − 24.0000i − 0.839654i
\(818\) 26.0000 0.909069
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) − 10.0000i − 0.349002i −0.984657 0.174501i \(-0.944169\pi\)
0.984657 0.174501i \(-0.0558313\pi\)
\(822\) 6.00000 0.209274
\(823\) 52.0000 1.81261 0.906303 0.422628i \(-0.138892\pi\)
0.906303 + 0.422628i \(0.138892\pi\)
\(824\) − 12.0000i − 0.418040i
\(825\) − 4.00000i − 0.139262i
\(826\) − 8.00000i − 0.278356i
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 6.00000 0.208514
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 12.0000i 0.416526i
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) −24.0000 −0.831551
\(834\) 8.00000i 0.277017i
\(835\) 4.00000 0.138426
\(836\) 24.0000 0.830057
\(837\) 0 0
\(838\) 10.0000i 0.345444i
\(839\) 40.0000i 1.38095i 0.723355 + 0.690477i \(0.242599\pi\)
−0.723355 + 0.690477i \(0.757401\pi\)
\(840\) − 2.00000i − 0.0690066i
\(841\) −13.0000 −0.448276
\(842\) 8.00000 0.275698
\(843\) − 6.00000i − 0.206651i
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) 10.0000i 0.343604i
\(848\) −10.0000 −0.343401
\(849\) −4.00000 −0.137280
\(850\) 8.00000i 0.274398i
\(851\) − 12.0000i − 0.411355i
\(852\) − 8.00000i − 0.274075i
\(853\) − 46.0000i − 1.57501i −0.616308 0.787505i \(-0.711372\pi\)
0.616308 0.787505i \(-0.288628\pi\)
\(854\) −20.0000 −0.684386
\(855\) −6.00000 −0.205196
\(856\) − 12.0000i − 0.410152i
\(857\) −4.00000 −0.136637 −0.0683187 0.997664i \(-0.521763\pi\)
−0.0683187 + 0.997664i \(0.521763\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 4.00000i 0.136399i
\(861\) 4.00000 0.136320
\(862\) 0 0
\(863\) − 12.0000i − 0.408485i −0.978920 0.204242i \(-0.934527\pi\)
0.978920 0.204242i \(-0.0654731\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) 22.0000i 0.748022i
\(866\) 2.00000i 0.0679628i
\(867\) −47.0000 −1.59620
\(868\) 0 0
\(869\) − 32.0000i − 1.08553i
\(870\) 4.00000 0.135613
\(871\) 0 0
\(872\) 12.0000 0.406371
\(873\) − 16.0000i − 0.541518i
\(874\) −36.0000 −1.21772
\(875\) 2.00000 0.0676123
\(876\) 8.00000i 0.270295i
\(877\) − 14.0000i − 0.472746i −0.971662 0.236373i \(-0.924041\pi\)
0.971662 0.236373i \(-0.0759588\pi\)
\(878\) 32.0000i 1.07995i
\(879\) − 26.0000i − 0.876958i
\(880\) −4.00000 −0.134840
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 3.00000i 0.101015i
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 0 0
\(885\) 4.00000 0.134459
\(886\) 16.0000i 0.537531i
\(887\) −2.00000 −0.0671534 −0.0335767 0.999436i \(-0.510690\pi\)
−0.0335767 + 0.999436i \(0.510690\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 8.00000i 0.268311i
\(890\) 14.0000i 0.469281i
\(891\) − 4.00000i − 0.134005i
\(892\) 2.00000i 0.0669650i
\(893\) 0 0
\(894\) −10.0000 −0.334450
\(895\) − 10.0000i − 0.334263i
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 80.0000 2.66519
\(902\) − 8.00000i − 0.266371i
\(903\) 8.00000i 0.266223i
\(904\) − 20.0000i − 0.665190i
\(905\) 10.0000i 0.332411i
\(906\) 0 0
\(907\) −52.0000 −1.72663 −0.863316 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) 4.00000i 0.132745i
\(909\) 16.0000 0.530687
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 6.00000i 0.198680i
\(913\) 48.0000 1.58857
\(914\) −8.00000 −0.264616
\(915\) − 10.0000i − 0.330590i
\(916\) 4.00000i 0.132164i
\(917\) − 20.0000i − 0.660458i
\(918\) 8.00000i 0.264039i
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 6.00000 0.197814
\(921\) 12.0000i 0.395413i
\(922\) 6.00000 0.197599
\(923\) 0 0
\(924\) −8.00000 −0.263181
\(925\) − 2.00000i − 0.0657596i
\(926\) −26.0000 −0.854413
\(927\) 12.0000 0.394132
\(928\) − 4.00000i − 0.131306i
\(929\) − 30.0000i − 0.984268i −0.870519 0.492134i \(-0.836217\pi\)
0.870519 0.492134i \(-0.163783\pi\)
\(930\) 0 0
\(931\) − 18.0000i − 0.589926i
\(932\) 24.0000 0.786146
\(933\) 24.0000 0.785725
\(934\) − 28.0000i − 0.916188i
\(935\) 32.0000 1.04651
\(936\) 0 0
\(937\) 18.0000 0.588034 0.294017 0.955800i \(-0.405008\pi\)
0.294017 + 0.955800i \(0.405008\pi\)
\(938\) 24.0000i 0.783628i
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) − 34.0000i − 1.10837i −0.832394 0.554184i \(-0.813030\pi\)
0.832394 0.554184i \(-0.186970\pi\)
\(942\) − 22.0000i − 0.716799i
\(943\) 12.0000i 0.390774i
\(944\) − 4.00000i − 0.130189i
\(945\) 2.00000 0.0650600
\(946\) 16.0000 0.520205
\(947\) − 4.00000i − 0.129983i −0.997886 0.0649913i \(-0.979298\pi\)
0.997886 0.0649913i \(-0.0207020\pi\)
\(948\) 8.00000 0.259828
\(949\) 0 0
\(950\) −6.00000 −0.194666
\(951\) 30.0000i 0.972817i
\(952\) 16.0000 0.518563
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) − 10.0000i − 0.323762i
\(955\) 0 0
\(956\) − 16.0000i − 0.517477i
\(957\) − 16.0000i − 0.517207i
\(958\) 32.0000 1.03387
\(959\) 12.0000 0.387500
\(960\) − 1.00000i − 0.0322749i
\(961\) 31.0000 1.00000
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 2.00000i 0.0644157i
\(965\) 4.00000 0.128765
\(966\) 12.0000 0.386094
\(967\) − 14.0000i − 0.450210i −0.974335 0.225105i \(-0.927728\pi\)
0.974335 0.225105i \(-0.0722725\pi\)
\(968\) 5.00000i 0.160706i
\(969\) − 48.0000i − 1.54198i
\(970\) − 16.0000i − 0.513729i
\(971\) 38.0000 1.21948 0.609739 0.792602i \(-0.291274\pi\)
0.609739 + 0.792602i \(0.291274\pi\)
\(972\) 1.00000 0.0320750
\(973\) 16.0000i 0.512936i
\(974\) −26.0000 −0.833094
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 30.0000i 0.959785i 0.877327 + 0.479893i \(0.159324\pi\)
−0.877327 + 0.479893i \(0.840676\pi\)
\(978\) 16.0000 0.511624
\(979\) 56.0000 1.78977
\(980\) 3.00000i 0.0958315i
\(981\) 12.0000i 0.383131i
\(982\) − 42.0000i − 1.34027i
\(983\) 52.0000i 1.65854i 0.558846 + 0.829271i \(0.311244\pi\)
−0.558846 + 0.829271i \(0.688756\pi\)
\(984\) 2.00000 0.0637577
\(985\) −18.0000 −0.573528
\(986\) 32.0000i 1.01909i
\(987\) 0 0
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) − 4.00000i − 0.127128i
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) − 10.0000i − 0.317340i
\(994\) − 16.0000i − 0.507489i
\(995\) 0 0
\(996\) 12.0000i 0.380235i
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) −38.0000 −1.20287
\(999\) − 2.00000i − 0.0632772i
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.b.e.1351.2 2
13.5 odd 4 390.2.a.e.1.1 1
13.8 odd 4 5070.2.a.e.1.1 1
13.12 even 2 inner 5070.2.b.e.1351.1 2
39.5 even 4 1170.2.a.e.1.1 1
52.31 even 4 3120.2.a.o.1.1 1
65.18 even 4 1950.2.e.f.1249.1 2
65.44 odd 4 1950.2.a.h.1.1 1
65.57 even 4 1950.2.e.f.1249.2 2
156.83 odd 4 9360.2.a.bh.1.1 1
195.44 even 4 5850.2.a.bi.1.1 1
195.83 odd 4 5850.2.e.i.5149.2 2
195.122 odd 4 5850.2.e.i.5149.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.e.1.1 1 13.5 odd 4
1170.2.a.e.1.1 1 39.5 even 4
1950.2.a.h.1.1 1 65.44 odd 4
1950.2.e.f.1249.1 2 65.18 even 4
1950.2.e.f.1249.2 2 65.57 even 4
3120.2.a.o.1.1 1 52.31 even 4
5070.2.a.e.1.1 1 13.8 odd 4
5070.2.b.e.1351.1 2 13.12 even 2 inner
5070.2.b.e.1351.2 2 1.1 even 1 trivial
5850.2.a.bi.1.1 1 195.44 even 4
5850.2.e.i.5149.1 2 195.122 odd 4
5850.2.e.i.5149.2 2 195.83 odd 4
9360.2.a.bh.1.1 1 156.83 odd 4