Properties

Label 975.2.c.i.274.3
Level $975$
Weight $2$
Character 975.274
Analytic conductor $7.785$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.3
Root \(1.40680 + 0.144584i\) of defining polynomial
Character \(\chi\) \(=\) 975.274
Dual form 975.2.c.i.274.4

$q$-expansion

\(f(q)\) \(=\) \(q-0.289169i q^{2} +1.00000i q^{3} +1.91638 q^{4} +0.289169 q^{6} +4.91638i q^{7} -1.13249i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-0.289169i q^{2} +1.00000i q^{3} +1.91638 q^{4} +0.289169 q^{6} +4.91638i q^{7} -1.13249i q^{8} -1.00000 q^{9} +4.91638 q^{11} +1.91638i q^{12} -1.00000i q^{13} +1.42166 q^{14} +3.50528 q^{16} -4.33804i q^{17} +0.289169i q^{18} -2.57834 q^{19} -4.91638 q^{21} -1.42166i q^{22} +6.33804i q^{23} +1.13249 q^{24} -0.289169 q^{26} -1.00000i q^{27} +9.42166i q^{28} -6.00000 q^{29} +1.42166 q^{31} -3.27861i q^{32} +4.91638i q^{33} -1.25443 q^{34} -1.91638 q^{36} +9.49472i q^{37} +0.745574i q^{38} +1.00000 q^{39} +4.33804 q^{41} +1.42166i q^{42} +1.15667i q^{43} +9.42166 q^{44} +1.83276 q^{46} -5.42166i q^{47} +3.50528i q^{48} -17.1708 q^{49} +4.33804 q^{51} -1.91638i q^{52} +0.338044i q^{53} -0.289169 q^{54} +5.56777 q^{56} -2.57834i q^{57} +1.73501i q^{58} +11.2544 q^{59} -10.1708 q^{61} -0.411100i q^{62} -4.91638i q^{63} +6.06249 q^{64} +1.42166 q^{66} -7.25443i q^{67} -8.31335i q^{68} -6.33804 q^{69} +0.916382 q^{71} +1.13249i q^{72} +3.15667i q^{73} +2.74557 q^{74} -4.94108 q^{76} +24.1708i q^{77} -0.289169i q^{78} +3.49472 q^{79} +1.00000 q^{81} -1.25443i q^{82} -11.2544i q^{83} -9.42166 q^{84} +0.334474 q^{86} -6.00000i q^{87} -5.56777i q^{88} +0.338044 q^{89} +4.91638 q^{91} +12.1461i q^{92} +1.42166i q^{93} -1.56777 q^{94} +3.27861 q^{96} -12.3380i q^{97} +4.96526i q^{98} -4.91638 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 16 q^{4} - 6 q^{9} + 2 q^{11} + 12 q^{14} + 52 q^{16} - 12 q^{19} - 2 q^{21} + 12 q^{24} - 36 q^{29} + 12 q^{31} + 44 q^{34} + 16 q^{36} + 6 q^{39} + 2 q^{41} + 60 q^{44} - 44 q^{46} - 24 q^{49} + 2 q^{51} - 32 q^{56} + 16 q^{59} + 18 q^{61} - 60 q^{64} + 12 q^{66} - 14 q^{69} - 22 q^{71} + 68 q^{74} + 8 q^{76} - 10 q^{79} + 6 q^{81} - 60 q^{84} + 112 q^{86} - 22 q^{89} + 2 q^{91} + 56 q^{94} - 44 q^{96} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.289169i − 0.204473i −0.994760 0.102237i \(-0.967400\pi\)
0.994760 0.102237i \(-0.0325999\pi\)
\(3\) 1.00000i 0.577350i
\(4\) 1.91638 0.958191
\(5\) 0 0
\(6\) 0.289169 0.118053
\(7\) 4.91638i 1.85822i 0.369807 + 0.929109i \(0.379424\pi\)
−0.369807 + 0.929109i \(0.620576\pi\)
\(8\) − 1.13249i − 0.400397i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.91638 1.48234 0.741172 0.671315i \(-0.234270\pi\)
0.741172 + 0.671315i \(0.234270\pi\)
\(12\) 1.91638i 0.553212i
\(13\) − 1.00000i − 0.277350i
\(14\) 1.42166 0.379955
\(15\) 0 0
\(16\) 3.50528 0.876320
\(17\) − 4.33804i − 1.05213i −0.850444 0.526065i \(-0.823667\pi\)
0.850444 0.526065i \(-0.176333\pi\)
\(18\) 0.289169i 0.0681577i
\(19\) −2.57834 −0.591511 −0.295756 0.955264i \(-0.595571\pi\)
−0.295756 + 0.955264i \(0.595571\pi\)
\(20\) 0 0
\(21\) −4.91638 −1.07284
\(22\) − 1.42166i − 0.303100i
\(23\) 6.33804i 1.32157i 0.750574 + 0.660787i \(0.229777\pi\)
−0.750574 + 0.660787i \(0.770223\pi\)
\(24\) 1.13249 0.231169
\(25\) 0 0
\(26\) −0.289169 −0.0567106
\(27\) − 1.00000i − 0.192450i
\(28\) 9.42166i 1.78053i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 1.42166 0.255338 0.127669 0.991817i \(-0.459250\pi\)
0.127669 + 0.991817i \(0.459250\pi\)
\(32\) − 3.27861i − 0.579581i
\(33\) 4.91638i 0.855832i
\(34\) −1.25443 −0.215132
\(35\) 0 0
\(36\) −1.91638 −0.319397
\(37\) 9.49472i 1.56092i 0.625204 + 0.780461i \(0.285016\pi\)
−0.625204 + 0.780461i \(0.714984\pi\)
\(38\) 0.745574i 0.120948i
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 4.33804 0.677489 0.338744 0.940878i \(-0.389998\pi\)
0.338744 + 0.940878i \(0.389998\pi\)
\(42\) 1.42166i 0.219367i
\(43\) 1.15667i 0.176391i 0.996103 + 0.0881956i \(0.0281100\pi\)
−0.996103 + 0.0881956i \(0.971890\pi\)
\(44\) 9.42166 1.42037
\(45\) 0 0
\(46\) 1.83276 0.270226
\(47\) − 5.42166i − 0.790831i −0.918502 0.395415i \(-0.870601\pi\)
0.918502 0.395415i \(-0.129399\pi\)
\(48\) 3.50528i 0.505944i
\(49\) −17.1708 −2.45297
\(50\) 0 0
\(51\) 4.33804 0.607448
\(52\) − 1.91638i − 0.265754i
\(53\) 0.338044i 0.0464340i 0.999730 + 0.0232170i \(0.00739086\pi\)
−0.999730 + 0.0232170i \(0.992609\pi\)
\(54\) −0.289169 −0.0393509
\(55\) 0 0
\(56\) 5.56777 0.744025
\(57\) − 2.57834i − 0.341509i
\(58\) 1.73501i 0.227818i
\(59\) 11.2544 1.46520 0.732601 0.680659i \(-0.238306\pi\)
0.732601 + 0.680659i \(0.238306\pi\)
\(60\) 0 0
\(61\) −10.1708 −1.30224 −0.651119 0.758975i \(-0.725700\pi\)
−0.651119 + 0.758975i \(0.725700\pi\)
\(62\) − 0.411100i − 0.0522098i
\(63\) − 4.91638i − 0.619406i
\(64\) 6.06249 0.757812
\(65\) 0 0
\(66\) 1.42166 0.174995
\(67\) − 7.25443i − 0.886269i −0.896455 0.443135i \(-0.853866\pi\)
0.896455 0.443135i \(-0.146134\pi\)
\(68\) − 8.31335i − 1.00814i
\(69\) −6.33804 −0.763011
\(70\) 0 0
\(71\) 0.916382 0.108754 0.0543772 0.998520i \(-0.482683\pi\)
0.0543772 + 0.998520i \(0.482683\pi\)
\(72\) 1.13249i 0.133466i
\(73\) 3.15667i 0.369461i 0.982789 + 0.184730i \(0.0591412\pi\)
−0.982789 + 0.184730i \(0.940859\pi\)
\(74\) 2.74557 0.319166
\(75\) 0 0
\(76\) −4.94108 −0.566780
\(77\) 24.1708i 2.75452i
\(78\) − 0.289169i − 0.0327419i
\(79\) 3.49472 0.393187 0.196593 0.980485i \(-0.437012\pi\)
0.196593 + 0.980485i \(0.437012\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 1.25443i − 0.138528i
\(83\) − 11.2544i − 1.23533i −0.786440 0.617667i \(-0.788078\pi\)
0.786440 0.617667i \(-0.211922\pi\)
\(84\) −9.42166 −1.02799
\(85\) 0 0
\(86\) 0.334474 0.0360672
\(87\) − 6.00000i − 0.643268i
\(88\) − 5.56777i − 0.593527i
\(89\) 0.338044 0.0358326 0.0179163 0.999839i \(-0.494297\pi\)
0.0179163 + 0.999839i \(0.494297\pi\)
\(90\) 0 0
\(91\) 4.91638 0.515377
\(92\) 12.1461i 1.26632i
\(93\) 1.42166i 0.147420i
\(94\) −1.56777 −0.161704
\(95\) 0 0
\(96\) 3.27861 0.334621
\(97\) − 12.3380i − 1.25274i −0.779526 0.626369i \(-0.784540\pi\)
0.779526 0.626369i \(-0.215460\pi\)
\(98\) 4.96526i 0.501567i
\(99\) −4.91638 −0.494115
\(100\) 0 0
\(101\) 10.6761 1.06231 0.531155 0.847274i \(-0.321758\pi\)
0.531155 + 0.847274i \(0.321758\pi\)
\(102\) − 1.25443i − 0.124207i
\(103\) 14.5089i 1.42960i 0.699329 + 0.714800i \(0.253482\pi\)
−0.699329 + 0.714800i \(0.746518\pi\)
\(104\) −1.13249 −0.111050
\(105\) 0 0
\(106\) 0.0977518 0.00949450
\(107\) 4.17081i 0.403207i 0.979467 + 0.201604i \(0.0646153\pi\)
−0.979467 + 0.201604i \(0.935385\pi\)
\(108\) − 1.91638i − 0.184404i
\(109\) 3.83276 0.367112 0.183556 0.983009i \(-0.441239\pi\)
0.183556 + 0.983009i \(0.441239\pi\)
\(110\) 0 0
\(111\) −9.49472 −0.901199
\(112\) 17.2333i 1.62839i
\(113\) 0.843326i 0.0793334i 0.999213 + 0.0396667i \(0.0126296\pi\)
−0.999213 + 0.0396667i \(0.987370\pi\)
\(114\) −0.745574 −0.0698294
\(115\) 0 0
\(116\) −11.4983 −1.06759
\(117\) 1.00000i 0.0924500i
\(118\) − 3.25443i − 0.299594i
\(119\) 21.3275 1.95509
\(120\) 0 0
\(121\) 13.1708 1.19735
\(122\) 2.94108i 0.266273i
\(123\) 4.33804i 0.391148i
\(124\) 2.72445 0.244663
\(125\) 0 0
\(126\) −1.42166 −0.126652
\(127\) 1.83276i 0.162631i 0.996688 + 0.0813157i \(0.0259122\pi\)
−0.996688 + 0.0813157i \(0.974088\pi\)
\(128\) − 8.31029i − 0.734533i
\(129\) −1.15667 −0.101839
\(130\) 0 0
\(131\) 5.83276 0.509611 0.254805 0.966992i \(-0.417989\pi\)
0.254805 + 0.966992i \(0.417989\pi\)
\(132\) 9.42166i 0.820050i
\(133\) − 12.6761i − 1.09916i
\(134\) −2.09775 −0.181218
\(135\) 0 0
\(136\) −4.91281 −0.421270
\(137\) − 16.5089i − 1.41045i −0.708985 0.705223i \(-0.750847\pi\)
0.708985 0.705223i \(-0.249153\pi\)
\(138\) 1.83276i 0.156015i
\(139\) −7.49472 −0.635694 −0.317847 0.948142i \(-0.602960\pi\)
−0.317847 + 0.948142i \(0.602960\pi\)
\(140\) 0 0
\(141\) 5.42166 0.456586
\(142\) − 0.264989i − 0.0222374i
\(143\) − 4.91638i − 0.411128i
\(144\) −3.50528 −0.292107
\(145\) 0 0
\(146\) 0.912811 0.0755448
\(147\) − 17.1708i − 1.41622i
\(148\) 18.1955i 1.49566i
\(149\) −20.4842 −1.67813 −0.839064 0.544033i \(-0.816897\pi\)
−0.839064 + 0.544033i \(0.816897\pi\)
\(150\) 0 0
\(151\) −16.4111 −1.33552 −0.667758 0.744378i \(-0.732746\pi\)
−0.667758 + 0.744378i \(0.732746\pi\)
\(152\) 2.91995i 0.236839i
\(153\) 4.33804i 0.350710i
\(154\) 6.98944 0.563225
\(155\) 0 0
\(156\) 1.91638 0.153433
\(157\) − 21.6655i − 1.72910i −0.502549 0.864549i \(-0.667604\pi\)
0.502549 0.864549i \(-0.332396\pi\)
\(158\) − 1.01056i − 0.0803961i
\(159\) −0.338044 −0.0268087
\(160\) 0 0
\(161\) −31.1602 −2.45577
\(162\) − 0.289169i − 0.0227192i
\(163\) − 6.07306i − 0.475678i −0.971305 0.237839i \(-0.923561\pi\)
0.971305 0.237839i \(-0.0764391\pi\)
\(164\) 8.31335 0.649163
\(165\) 0 0
\(166\) −3.25443 −0.252592
\(167\) − 0.745574i − 0.0576942i −0.999584 0.0288471i \(-0.990816\pi\)
0.999584 0.0288471i \(-0.00918360\pi\)
\(168\) 5.56777i 0.429563i
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 2.57834 0.197170
\(172\) 2.21663i 0.169016i
\(173\) − 0.843326i − 0.0641169i −0.999486 0.0320584i \(-0.989794\pi\)
0.999486 0.0320584i \(-0.0102063\pi\)
\(174\) −1.73501 −0.131531
\(175\) 0 0
\(176\) 17.2333 1.29901
\(177\) 11.2544i 0.845934i
\(178\) − 0.0977518i − 0.00732681i
\(179\) 18.9894 1.41934 0.709669 0.704536i \(-0.248845\pi\)
0.709669 + 0.704536i \(0.248845\pi\)
\(180\) 0 0
\(181\) 17.4947 1.30037 0.650186 0.759775i \(-0.274691\pi\)
0.650186 + 0.759775i \(0.274691\pi\)
\(182\) − 1.42166i − 0.105381i
\(183\) − 10.1708i − 0.751848i
\(184\) 7.17780 0.529154
\(185\) 0 0
\(186\) 0.411100 0.0301433
\(187\) − 21.3275i − 1.55962i
\(188\) − 10.3900i − 0.757767i
\(189\) 4.91638 0.357614
\(190\) 0 0
\(191\) −22.5089 −1.62868 −0.814342 0.580386i \(-0.802902\pi\)
−0.814342 + 0.580386i \(0.802902\pi\)
\(192\) 6.06249i 0.437523i
\(193\) − 2.65139i − 0.190851i −0.995437 0.0954257i \(-0.969579\pi\)
0.995437 0.0954257i \(-0.0304212\pi\)
\(194\) −3.56777 −0.256151
\(195\) 0 0
\(196\) −32.9058 −2.35042
\(197\) − 12.9894i − 0.925459i −0.886500 0.462730i \(-0.846870\pi\)
0.886500 0.462730i \(-0.153130\pi\)
\(198\) 1.42166i 0.101033i
\(199\) 2.84333 0.201558 0.100779 0.994909i \(-0.467866\pi\)
0.100779 + 0.994909i \(0.467866\pi\)
\(200\) 0 0
\(201\) 7.25443 0.511688
\(202\) − 3.08719i − 0.217214i
\(203\) − 29.4983i − 2.07037i
\(204\) 8.31335 0.582051
\(205\) 0 0
\(206\) 4.19550 0.292315
\(207\) − 6.33804i − 0.440525i
\(208\) − 3.50528i − 0.243048i
\(209\) −12.6761 −0.876823
\(210\) 0 0
\(211\) 6.31335 0.434629 0.217314 0.976102i \(-0.430270\pi\)
0.217314 + 0.976102i \(0.430270\pi\)
\(212\) 0.647822i 0.0444926i
\(213\) 0.916382i 0.0627894i
\(214\) 1.20607 0.0824450
\(215\) 0 0
\(216\) −1.13249 −0.0770565
\(217\) 6.98944i 0.474474i
\(218\) − 1.10831i − 0.0750645i
\(219\) −3.15667 −0.213308
\(220\) 0 0
\(221\) −4.33804 −0.291808
\(222\) 2.74557i 0.184271i
\(223\) − 19.2544i − 1.28937i −0.764448 0.644686i \(-0.776988\pi\)
0.764448 0.644686i \(-0.223012\pi\)
\(224\) 16.1189 1.07699
\(225\) 0 0
\(226\) 0.243863 0.0162215
\(227\) − 13.0872i − 0.868627i −0.900762 0.434314i \(-0.856991\pi\)
0.900762 0.434314i \(-0.143009\pi\)
\(228\) − 4.94108i − 0.327231i
\(229\) 24.5089 1.61959 0.809795 0.586713i \(-0.199578\pi\)
0.809795 + 0.586713i \(0.199578\pi\)
\(230\) 0 0
\(231\) −24.1708 −1.59032
\(232\) 6.79497i 0.446111i
\(233\) − 8.33804i − 0.546243i −0.961979 0.273122i \(-0.911944\pi\)
0.961979 0.273122i \(-0.0880562\pi\)
\(234\) 0.289169 0.0189035
\(235\) 0 0
\(236\) 21.5678 1.40394
\(237\) 3.49472i 0.227006i
\(238\) − 6.16724i − 0.399763i
\(239\) −8.91638 −0.576753 −0.288376 0.957517i \(-0.593115\pi\)
−0.288376 + 0.957517i \(0.593115\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) − 3.80858i − 0.244825i
\(243\) 1.00000i 0.0641500i
\(244\) −19.4911 −1.24779
\(245\) 0 0
\(246\) 1.25443 0.0799793
\(247\) 2.57834i 0.164056i
\(248\) − 1.61003i − 0.102237i
\(249\) 11.2544 0.713220
\(250\) 0 0
\(251\) −6.31335 −0.398495 −0.199248 0.979949i \(-0.563850\pi\)
−0.199248 + 0.979949i \(0.563850\pi\)
\(252\) − 9.42166i − 0.593509i
\(253\) 31.1602i 1.95903i
\(254\) 0.529977 0.0332537
\(255\) 0 0
\(256\) 9.72191 0.607619
\(257\) − 11.1567i − 0.695934i −0.937507 0.347967i \(-0.886872\pi\)
0.937507 0.347967i \(-0.113128\pi\)
\(258\) 0.334474i 0.0208234i
\(259\) −46.6797 −2.90053
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) − 1.68665i − 0.104202i
\(263\) − 8.00000i − 0.493301i −0.969104 0.246651i \(-0.920670\pi\)
0.969104 0.246651i \(-0.0793300\pi\)
\(264\) 5.56777 0.342673
\(265\) 0 0
\(266\) −3.66553 −0.224748
\(267\) 0.338044i 0.0206880i
\(268\) − 13.9022i − 0.849215i
\(269\) −18.6761 −1.13870 −0.569351 0.822095i \(-0.692805\pi\)
−0.569351 + 0.822095i \(0.692805\pi\)
\(270\) 0 0
\(271\) 6.57834 0.399606 0.199803 0.979836i \(-0.435970\pi\)
0.199803 + 0.979836i \(0.435970\pi\)
\(272\) − 15.2061i − 0.922003i
\(273\) 4.91638i 0.297553i
\(274\) −4.77384 −0.288398
\(275\) 0 0
\(276\) −12.1461 −0.731110
\(277\) 25.6655i 1.54209i 0.636779 + 0.771046i \(0.280266\pi\)
−0.636779 + 0.771046i \(0.719734\pi\)
\(278\) 2.16724i 0.129982i
\(279\) −1.42166 −0.0851127
\(280\) 0 0
\(281\) 3.15667 0.188311 0.0941557 0.995557i \(-0.469985\pi\)
0.0941557 + 0.995557i \(0.469985\pi\)
\(282\) − 1.56777i − 0.0933596i
\(283\) 3.47002i 0.206271i 0.994667 + 0.103136i \(0.0328876\pi\)
−0.994667 + 0.103136i \(0.967112\pi\)
\(284\) 1.75614 0.104208
\(285\) 0 0
\(286\) −1.42166 −0.0840647
\(287\) 21.3275i 1.25892i
\(288\) 3.27861i 0.193194i
\(289\) −1.81863 −0.106978
\(290\) 0 0
\(291\) 12.3380 0.723269
\(292\) 6.04939i 0.354014i
\(293\) − 28.6550i − 1.67404i −0.547172 0.837020i \(-0.684296\pi\)
0.547172 0.837020i \(-0.315704\pi\)
\(294\) −4.96526 −0.289580
\(295\) 0 0
\(296\) 10.7527 0.624989
\(297\) − 4.91638i − 0.285277i
\(298\) 5.92337i 0.343132i
\(299\) 6.33804 0.366539
\(300\) 0 0
\(301\) −5.68665 −0.327773
\(302\) 4.74557i 0.273077i
\(303\) 10.6761i 0.613325i
\(304\) −9.03780 −0.518353
\(305\) 0 0
\(306\) 1.25443 0.0717108
\(307\) − 1.92694i − 0.109977i −0.998487 0.0549883i \(-0.982488\pi\)
0.998487 0.0549883i \(-0.0175121\pi\)
\(308\) 46.3205i 2.63935i
\(309\) −14.5089 −0.825380
\(310\) 0 0
\(311\) −29.9789 −1.69995 −0.849973 0.526826i \(-0.823382\pi\)
−0.849973 + 0.526826i \(0.823382\pi\)
\(312\) − 1.13249i − 0.0641149i
\(313\) 16.3133i 0.922085i 0.887378 + 0.461042i \(0.152524\pi\)
−0.887378 + 0.461042i \(0.847476\pi\)
\(314\) −6.26499 −0.353554
\(315\) 0 0
\(316\) 6.69721 0.376748
\(317\) 30.6761i 1.72294i 0.507808 + 0.861470i \(0.330456\pi\)
−0.507808 + 0.861470i \(0.669544\pi\)
\(318\) 0.0977518i 0.00548165i
\(319\) −29.4983 −1.65159
\(320\) 0 0
\(321\) −4.17081 −0.232792
\(322\) 9.01056i 0.502139i
\(323\) 11.1849i 0.622347i
\(324\) 1.91638 0.106466
\(325\) 0 0
\(326\) −1.75614 −0.0972634
\(327\) 3.83276i 0.211952i
\(328\) − 4.91281i − 0.271265i
\(329\) 26.6550 1.46954
\(330\) 0 0
\(331\) −10.0978 −0.555023 −0.277511 0.960722i \(-0.589510\pi\)
−0.277511 + 0.960722i \(0.589510\pi\)
\(332\) − 21.5678i − 1.18369i
\(333\) − 9.49472i − 0.520307i
\(334\) −0.215597 −0.0117969
\(335\) 0 0
\(336\) −17.2333 −0.940154
\(337\) − 1.32391i − 0.0721180i −0.999350 0.0360590i \(-0.988520\pi\)
0.999350 0.0360590i \(-0.0114804\pi\)
\(338\) 0.289169i 0.0157287i
\(339\) −0.843326 −0.0458032
\(340\) 0 0
\(341\) 6.98944 0.378499
\(342\) − 0.745574i − 0.0403160i
\(343\) − 50.0036i − 2.69994i
\(344\) 1.30993 0.0706265
\(345\) 0 0
\(346\) −0.243863 −0.0131102
\(347\) − 7.49472i − 0.402338i −0.979557 0.201169i \(-0.935526\pi\)
0.979557 0.201169i \(-0.0644740\pi\)
\(348\) − 11.4983i − 0.616373i
\(349\) 22.1461 1.18545 0.592727 0.805403i \(-0.298051\pi\)
0.592727 + 0.805403i \(0.298051\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) − 16.1189i − 0.859139i
\(353\) − 4.50885i − 0.239982i −0.992775 0.119991i \(-0.961713\pi\)
0.992775 0.119991i \(-0.0382866\pi\)
\(354\) 3.25443 0.172971
\(355\) 0 0
\(356\) 0.647822 0.0343345
\(357\) 21.3275i 1.12877i
\(358\) − 5.49115i − 0.290216i
\(359\) 20.4111 1.07726 0.538628 0.842543i \(-0.318943\pi\)
0.538628 + 0.842543i \(0.318943\pi\)
\(360\) 0 0
\(361\) −12.3522 −0.650115
\(362\) − 5.05892i − 0.265891i
\(363\) 13.1708i 0.691288i
\(364\) 9.42166 0.493829
\(365\) 0 0
\(366\) −2.94108 −0.153733
\(367\) − 10.3133i − 0.538352i −0.963091 0.269176i \(-0.913249\pi\)
0.963091 0.269176i \(-0.0867514\pi\)
\(368\) 22.2166i 1.15812i
\(369\) −4.33804 −0.225830
\(370\) 0 0
\(371\) −1.66196 −0.0862844
\(372\) 2.72445i 0.141256i
\(373\) − 18.6761i − 0.967011i −0.875341 0.483506i \(-0.839363\pi\)
0.875341 0.483506i \(-0.160637\pi\)
\(374\) −6.16724 −0.318900
\(375\) 0 0
\(376\) −6.14000 −0.316646
\(377\) 6.00000i 0.309016i
\(378\) − 1.42166i − 0.0731224i
\(379\) 28.7527 1.47693 0.738464 0.674293i \(-0.235552\pi\)
0.738464 + 0.674293i \(0.235552\pi\)
\(380\) 0 0
\(381\) −1.83276 −0.0938953
\(382\) 6.50885i 0.333022i
\(383\) 14.2439i 0.727827i 0.931433 + 0.363914i \(0.118560\pi\)
−0.931433 + 0.363914i \(0.881440\pi\)
\(384\) 8.31029 0.424083
\(385\) 0 0
\(386\) −0.766699 −0.0390240
\(387\) − 1.15667i − 0.0587971i
\(388\) − 23.6444i − 1.20036i
\(389\) −34.6761 −1.75815 −0.879074 0.476686i \(-0.841838\pi\)
−0.879074 + 0.476686i \(0.841838\pi\)
\(390\) 0 0
\(391\) 27.4947 1.39047
\(392\) 19.4458i 0.982163i
\(393\) 5.83276i 0.294224i
\(394\) −3.75614 −0.189231
\(395\) 0 0
\(396\) −9.42166 −0.473456
\(397\) 7.18137i 0.360423i 0.983628 + 0.180211i \(0.0576782\pi\)
−0.983628 + 0.180211i \(0.942322\pi\)
\(398\) − 0.822200i − 0.0412132i
\(399\) 12.6761 0.634598
\(400\) 0 0
\(401\) 37.8610 1.89069 0.945345 0.326072i \(-0.105725\pi\)
0.945345 + 0.326072i \(0.105725\pi\)
\(402\) − 2.09775i − 0.104626i
\(403\) − 1.42166i − 0.0708181i
\(404\) 20.4595 1.01790
\(405\) 0 0
\(406\) −8.52998 −0.423336
\(407\) 46.6797i 2.31382i
\(408\) − 4.91281i − 0.243220i
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 16.5089 0.814322
\(412\) 27.8045i 1.36983i
\(413\) 55.3311i 2.72266i
\(414\) −1.83276 −0.0900754
\(415\) 0 0
\(416\) −3.27861 −0.160747
\(417\) − 7.49472i − 0.367018i
\(418\) 3.66553i 0.179287i
\(419\) 33.4983 1.63650 0.818249 0.574864i \(-0.194945\pi\)
0.818249 + 0.574864i \(0.194945\pi\)
\(420\) 0 0
\(421\) 13.5194 0.658896 0.329448 0.944174i \(-0.393137\pi\)
0.329448 + 0.944174i \(0.393137\pi\)
\(422\) − 1.82562i − 0.0888699i
\(423\) 5.42166i 0.263610i
\(424\) 0.382833 0.0185920
\(425\) 0 0
\(426\) 0.264989 0.0128387
\(427\) − 50.0036i − 2.41984i
\(428\) 7.99286i 0.386349i
\(429\) 4.91638 0.237365
\(430\) 0 0
\(431\) 12.4111 0.597822 0.298911 0.954281i \(-0.403377\pi\)
0.298911 + 0.954281i \(0.403377\pi\)
\(432\) − 3.50528i − 0.168648i
\(433\) 17.3239i 0.832534i 0.909242 + 0.416267i \(0.136662\pi\)
−0.909242 + 0.416267i \(0.863338\pi\)
\(434\) 2.02113 0.0970171
\(435\) 0 0
\(436\) 7.34504 0.351763
\(437\) − 16.3416i − 0.781725i
\(438\) 0.912811i 0.0436158i
\(439\) −0.651393 −0.0310893 −0.0155446 0.999879i \(-0.504948\pi\)
−0.0155446 + 0.999879i \(0.504948\pi\)
\(440\) 0 0
\(441\) 17.1708 0.817658
\(442\) 1.25443i 0.0596670i
\(443\) 8.84690i 0.420329i 0.977666 + 0.210164i \(0.0673999\pi\)
−0.977666 + 0.210164i \(0.932600\pi\)
\(444\) −18.1955 −0.863520
\(445\) 0 0
\(446\) −5.56777 −0.263642
\(447\) − 20.4842i − 0.968867i
\(448\) 29.8055i 1.40818i
\(449\) −4.33804 −0.204725 −0.102362 0.994747i \(-0.532640\pi\)
−0.102362 + 0.994747i \(0.532640\pi\)
\(450\) 0 0
\(451\) 21.3275 1.00427
\(452\) 1.61613i 0.0760166i
\(453\) − 16.4111i − 0.771061i
\(454\) −3.78440 −0.177611
\(455\) 0 0
\(456\) −2.91995 −0.136739
\(457\) 15.3275i 0.716989i 0.933532 + 0.358495i \(0.116710\pi\)
−0.933532 + 0.358495i \(0.883290\pi\)
\(458\) − 7.08719i − 0.331163i
\(459\) −4.33804 −0.202483
\(460\) 0 0
\(461\) −11.8575 −0.552257 −0.276128 0.961121i \(-0.589052\pi\)
−0.276128 + 0.961121i \(0.589052\pi\)
\(462\) 6.98944i 0.325178i
\(463\) 26.4147i 1.22759i 0.789464 + 0.613797i \(0.210359\pi\)
−0.789464 + 0.613797i \(0.789641\pi\)
\(464\) −21.0317 −0.976372
\(465\) 0 0
\(466\) −2.41110 −0.111692
\(467\) − 33.6691i − 1.55802i −0.627012 0.779010i \(-0.715722\pi\)
0.627012 0.779010i \(-0.284278\pi\)
\(468\) 1.91638i 0.0885848i
\(469\) 35.6655 1.64688
\(470\) 0 0
\(471\) 21.6655 0.998295
\(472\) − 12.7456i − 0.586663i
\(473\) 5.68665i 0.261473i
\(474\) 1.01056 0.0464167
\(475\) 0 0
\(476\) 40.8716 1.87335
\(477\) − 0.338044i − 0.0154780i
\(478\) 2.57834i 0.117930i
\(479\) −10.7491 −0.491141 −0.245570 0.969379i \(-0.578975\pi\)
−0.245570 + 0.969379i \(0.578975\pi\)
\(480\) 0 0
\(481\) 9.49472 0.432922
\(482\) 1.73501i 0.0790276i
\(483\) − 31.1602i − 1.41784i
\(484\) 25.2403 1.14729
\(485\) 0 0
\(486\) 0.289169 0.0131170
\(487\) 22.7491i 1.03086i 0.856931 + 0.515431i \(0.172368\pi\)
−0.856931 + 0.515431i \(0.827632\pi\)
\(488\) 11.5184i 0.521413i
\(489\) 6.07306 0.274633
\(490\) 0 0
\(491\) −17.6867 −0.798187 −0.399094 0.916910i \(-0.630675\pi\)
−0.399094 + 0.916910i \(0.630675\pi\)
\(492\) 8.31335i 0.374795i
\(493\) 26.0283i 1.17225i
\(494\) 0.745574 0.0335450
\(495\) 0 0
\(496\) 4.98333 0.223758
\(497\) 4.50528i 0.202089i
\(498\) − 3.25443i − 0.145834i
\(499\) −19.9305 −0.892212 −0.446106 0.894980i \(-0.647190\pi\)
−0.446106 + 0.894980i \(0.647190\pi\)
\(500\) 0 0
\(501\) 0.745574 0.0333098
\(502\) 1.82562i 0.0814815i
\(503\) − 33.3522i − 1.48710i −0.668680 0.743550i \(-0.733140\pi\)
0.668680 0.743550i \(-0.266860\pi\)
\(504\) −5.56777 −0.248008
\(505\) 0 0
\(506\) 9.01056 0.400568
\(507\) − 1.00000i − 0.0444116i
\(508\) 3.51227i 0.155832i
\(509\) −13.8363 −0.613285 −0.306642 0.951825i \(-0.599206\pi\)
−0.306642 + 0.951825i \(0.599206\pi\)
\(510\) 0 0
\(511\) −15.5194 −0.686538
\(512\) − 19.4319i − 0.858775i
\(513\) 2.57834i 0.113836i
\(514\) −3.22616 −0.142300
\(515\) 0 0
\(516\) −2.21663 −0.0975817
\(517\) − 26.6550i − 1.17228i
\(518\) 13.4983i 0.593081i
\(519\) 0.843326 0.0370179
\(520\) 0 0
\(521\) −23.3522 −1.02308 −0.511539 0.859260i \(-0.670924\pi\)
−0.511539 + 0.859260i \(0.670924\pi\)
\(522\) − 1.73501i − 0.0759394i
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) 11.1778 0.488304
\(525\) 0 0
\(526\) −2.31335 −0.100867
\(527\) − 6.16724i − 0.268649i
\(528\) 17.2333i 0.749983i
\(529\) −17.1708 −0.746557
\(530\) 0 0
\(531\) −11.2544 −0.488400
\(532\) − 24.2922i − 1.05320i
\(533\) − 4.33804i − 0.187902i
\(534\) 0.0977518 0.00423014
\(535\) 0 0
\(536\) −8.21560 −0.354860
\(537\) 18.9894i 0.819455i
\(538\) 5.40054i 0.232834i
\(539\) −84.4182 −3.63615
\(540\) 0 0
\(541\) 16.1744 0.695391 0.347695 0.937608i \(-0.386964\pi\)
0.347695 + 0.937608i \(0.386964\pi\)
\(542\) − 1.90225i − 0.0817086i
\(543\) 17.4947i 0.750770i
\(544\) −14.2227 −0.609795
\(545\) 0 0
\(546\) 1.42166 0.0608416
\(547\) 9.68665i 0.414171i 0.978323 + 0.207086i \(0.0663979\pi\)
−0.978323 + 0.207086i \(0.933602\pi\)
\(548\) − 31.6373i − 1.35148i
\(549\) 10.1708 0.434079
\(550\) 0 0
\(551\) 15.4700 0.659045
\(552\) 7.17780i 0.305507i
\(553\) 17.1814i 0.730626i
\(554\) 7.42166 0.315316
\(555\) 0 0
\(556\) −14.3627 −0.609116
\(557\) 0.647822i 0.0274491i 0.999906 + 0.0137246i \(0.00436880\pi\)
−0.999906 + 0.0137246i \(0.995631\pi\)
\(558\) 0.411100i 0.0174033i
\(559\) 1.15667 0.0489221
\(560\) 0 0
\(561\) 21.3275 0.900447
\(562\) − 0.912811i − 0.0385046i
\(563\) 16.3169i 0.687676i 0.939029 + 0.343838i \(0.111727\pi\)
−0.939029 + 0.343838i \(0.888273\pi\)
\(564\) 10.3900 0.437497
\(565\) 0 0
\(566\) 1.00342 0.0421769
\(567\) 4.91638i 0.206469i
\(568\) − 1.03780i − 0.0435450i
\(569\) −44.6550 −1.87203 −0.936017 0.351956i \(-0.885517\pi\)
−0.936017 + 0.351956i \(0.885517\pi\)
\(570\) 0 0
\(571\) −6.67252 −0.279236 −0.139618 0.990205i \(-0.544587\pi\)
−0.139618 + 0.990205i \(0.544587\pi\)
\(572\) − 9.42166i − 0.393940i
\(573\) − 22.5089i − 0.940321i
\(574\) 6.16724 0.257415
\(575\) 0 0
\(576\) −6.06249 −0.252604
\(577\) 15.3275i 0.638091i 0.947739 + 0.319046i \(0.103362\pi\)
−0.947739 + 0.319046i \(0.896638\pi\)
\(578\) 0.525891i 0.0218742i
\(579\) 2.65139 0.110188
\(580\) 0 0
\(581\) 55.3311 2.29552
\(582\) − 3.56777i − 0.147889i
\(583\) 1.66196i 0.0688312i
\(584\) 3.57492 0.147931
\(585\) 0 0
\(586\) −8.28611 −0.342296
\(587\) 12.2650i 0.506230i 0.967436 + 0.253115i \(0.0814551\pi\)
−0.967436 + 0.253115i \(0.918545\pi\)
\(588\) − 32.9058i − 1.35701i
\(589\) −3.66553 −0.151035
\(590\) 0 0
\(591\) 12.9894 0.534314
\(592\) 33.2817i 1.36787i
\(593\) 5.85389i 0.240390i 0.992750 + 0.120195i \(0.0383520\pi\)
−0.992750 + 0.120195i \(0.961648\pi\)
\(594\) −1.42166 −0.0583315
\(595\) 0 0
\(596\) −39.2555 −1.60797
\(597\) 2.84333i 0.116370i
\(598\) − 1.83276i − 0.0749473i
\(599\) −27.1355 −1.10873 −0.554364 0.832274i \(-0.687039\pi\)
−0.554364 + 0.832274i \(0.687039\pi\)
\(600\) 0 0
\(601\) 34.1708 1.39386 0.696928 0.717141i \(-0.254550\pi\)
0.696928 + 0.717141i \(0.254550\pi\)
\(602\) 1.64440i 0.0670208i
\(603\) 7.25443i 0.295423i
\(604\) −31.4499 −1.27968
\(605\) 0 0
\(606\) 3.08719 0.125408
\(607\) 10.3133i 0.418606i 0.977851 + 0.209303i \(0.0671195\pi\)
−0.977851 + 0.209303i \(0.932881\pi\)
\(608\) 8.45335i 0.342829i
\(609\) 29.4983 1.19533
\(610\) 0 0
\(611\) −5.42166 −0.219337
\(612\) 8.31335i 0.336047i
\(613\) − 0.484156i − 0.0195549i −0.999952 0.00977744i \(-0.996888\pi\)
0.999952 0.00977744i \(-0.00311230\pi\)
\(614\) −0.557212 −0.0224872
\(615\) 0 0
\(616\) 27.3733 1.10290
\(617\) − 7.15667i − 0.288117i −0.989569 0.144058i \(-0.953985\pi\)
0.989569 0.144058i \(-0.0460153\pi\)
\(618\) 4.19550i 0.168768i
\(619\) −5.42166 −0.217915 −0.108958 0.994046i \(-0.534751\pi\)
−0.108958 + 0.994046i \(0.534751\pi\)
\(620\) 0 0
\(621\) 6.33804 0.254337
\(622\) 8.66895i 0.347593i
\(623\) 1.66196i 0.0665848i
\(624\) 3.50528 0.140324
\(625\) 0 0
\(626\) 4.71731 0.188542
\(627\) − 12.6761i − 0.506234i
\(628\) − 41.5194i − 1.65681i
\(629\) 41.1885 1.64229
\(630\) 0 0
\(631\) −10.7244 −0.426934 −0.213467 0.976950i \(-0.568476\pi\)
−0.213467 + 0.976950i \(0.568476\pi\)
\(632\) − 3.95775i − 0.157431i
\(633\) 6.31335i 0.250933i
\(634\) 8.87056 0.352295
\(635\) 0 0
\(636\) −0.647822 −0.0256878
\(637\) 17.1708i 0.680332i
\(638\) 8.52998i 0.337705i
\(639\) −0.916382 −0.0362515
\(640\) 0 0
\(641\) −0.362741 −0.0143274 −0.00716370 0.999974i \(-0.502280\pi\)
−0.00716370 + 0.999974i \(0.502280\pi\)
\(642\) 1.20607i 0.0475996i
\(643\) 9.39697i 0.370580i 0.982684 + 0.185290i \(0.0593225\pi\)
−0.982684 + 0.185290i \(0.940678\pi\)
\(644\) −59.7149 −2.35310
\(645\) 0 0
\(646\) 3.23433 0.127253
\(647\) 18.0036i 0.707793i 0.935285 + 0.353897i \(0.115144\pi\)
−0.935285 + 0.353897i \(0.884856\pi\)
\(648\) − 1.13249i − 0.0444886i
\(649\) 55.3311 2.17193
\(650\) 0 0
\(651\) −6.98944 −0.273938
\(652\) − 11.6383i − 0.455791i
\(653\) 34.8222i 1.36270i 0.731959 + 0.681349i \(0.238606\pi\)
−0.731959 + 0.681349i \(0.761394\pi\)
\(654\) 1.10831 0.0433385
\(655\) 0 0
\(656\) 15.2061 0.593697
\(657\) − 3.15667i − 0.123154i
\(658\) − 7.70778i − 0.300480i
\(659\) −11.4700 −0.446809 −0.223404 0.974726i \(-0.571717\pi\)
−0.223404 + 0.974726i \(0.571717\pi\)
\(660\) 0 0
\(661\) 12.1672 0.473251 0.236625 0.971601i \(-0.423959\pi\)
0.236625 + 0.971601i \(0.423959\pi\)
\(662\) 2.91995i 0.113487i
\(663\) − 4.33804i − 0.168476i
\(664\) −12.7456 −0.494624
\(665\) 0 0
\(666\) −2.74557 −0.106389
\(667\) − 38.0283i − 1.47246i
\(668\) − 1.42880i − 0.0552821i
\(669\) 19.2544 0.744419
\(670\) 0 0
\(671\) −50.0036 −1.93037
\(672\) 16.1189i 0.621799i
\(673\) 27.9789i 1.07851i 0.842144 + 0.539253i \(0.181293\pi\)
−0.842144 + 0.539253i \(0.818707\pi\)
\(674\) −0.382833 −0.0147462
\(675\) 0 0
\(676\) −1.91638 −0.0737070
\(677\) 22.9930i 0.883693i 0.897091 + 0.441847i \(0.145676\pi\)
−0.897091 + 0.441847i \(0.854324\pi\)
\(678\) 0.243863i 0.00936551i
\(679\) 60.6585 2.32786
\(680\) 0 0
\(681\) 13.0872 0.501502
\(682\) − 2.02113i − 0.0773929i
\(683\) 28.6066i 1.09460i 0.836936 + 0.547301i \(0.184345\pi\)
−0.836936 + 0.547301i \(0.815655\pi\)
\(684\) 4.94108 0.188927
\(685\) 0 0
\(686\) −14.4595 −0.552065
\(687\) 24.5089i 0.935071i
\(688\) 4.05447i 0.154575i
\(689\) 0.338044 0.0128785
\(690\) 0 0
\(691\) −19.4005 −0.738031 −0.369016 0.929423i \(-0.620305\pi\)
−0.369016 + 0.929423i \(0.620305\pi\)
\(692\) − 1.61613i − 0.0614362i
\(693\) − 24.1708i − 0.918173i
\(694\) −2.16724 −0.0822672
\(695\) 0 0
\(696\) −6.79497 −0.257563
\(697\) − 18.8186i − 0.712806i
\(698\) − 6.40396i − 0.242393i
\(699\) 8.33804 0.315374
\(700\) 0 0
\(701\) −38.9683 −1.47181 −0.735906 0.677083i \(-0.763244\pi\)
−0.735906 + 0.677083i \(0.763244\pi\)
\(702\) 0.289169i 0.0109140i
\(703\) − 24.4806i − 0.923303i
\(704\) 29.8055 1.12334
\(705\) 0 0
\(706\) −1.30382 −0.0490698
\(707\) 52.4877i 1.97400i
\(708\) 21.5678i 0.810567i
\(709\) 17.5194 0.657955 0.328978 0.944338i \(-0.393296\pi\)
0.328978 + 0.944338i \(0.393296\pi\)
\(710\) 0 0
\(711\) −3.49472 −0.131062
\(712\) − 0.382833i − 0.0143473i
\(713\) 9.01056i 0.337448i
\(714\) 6.16724 0.230803
\(715\) 0 0
\(716\) 36.3910 1.36000
\(717\) − 8.91638i − 0.332988i
\(718\) − 5.90225i − 0.220270i
\(719\) 4.33447 0.161649 0.0808243 0.996728i \(-0.474245\pi\)
0.0808243 + 0.996728i \(0.474245\pi\)
\(720\) 0 0
\(721\) −71.3311 −2.65651
\(722\) 3.57186i 0.132931i
\(723\) − 6.00000i − 0.223142i
\(724\) 33.5266 1.24600
\(725\) 0 0
\(726\) 3.80858 0.141350
\(727\) 22.1672i 0.822137i 0.911604 + 0.411069i \(0.134844\pi\)
−0.911604 + 0.411069i \(0.865156\pi\)
\(728\) − 5.56777i − 0.206355i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 5.01770 0.185586
\(732\) − 19.4911i − 0.720414i
\(733\) 2.83976i 0.104889i 0.998624 + 0.0524444i \(0.0167012\pi\)
−0.998624 + 0.0524444i \(0.983299\pi\)
\(734\) −2.98230 −0.110079
\(735\) 0 0
\(736\) 20.7799 0.765959
\(737\) − 35.6655i − 1.31376i
\(738\) 1.25443i 0.0461761i
\(739\) 43.9305 1.61601 0.808005 0.589176i \(-0.200547\pi\)
0.808005 + 0.589176i \(0.200547\pi\)
\(740\) 0 0
\(741\) −2.57834 −0.0947176
\(742\) 0.480585i 0.0176428i
\(743\) − 41.0872i − 1.50734i −0.657251 0.753671i \(-0.728281\pi\)
0.657251 0.753671i \(-0.271719\pi\)
\(744\) 1.61003 0.0590264
\(745\) 0 0
\(746\) −5.40054 −0.197728
\(747\) 11.2544i 0.411778i
\(748\) − 40.8716i − 1.49441i
\(749\) −20.5053 −0.749247
\(750\) 0 0
\(751\) 23.6902 0.864468 0.432234 0.901761i \(-0.357725\pi\)
0.432234 + 0.901761i \(0.357725\pi\)
\(752\) − 19.0045i − 0.693021i
\(753\) − 6.31335i − 0.230071i
\(754\) 1.73501 0.0631854
\(755\) 0 0
\(756\) 9.42166 0.342663
\(757\) 9.32391i 0.338883i 0.985540 + 0.169442i \(0.0541964\pi\)
−0.985540 + 0.169442i \(0.945804\pi\)
\(758\) − 8.31438i − 0.301992i
\(759\) −31.1602 −1.13105
\(760\) 0 0
\(761\) −42.8222 −1.55230 −0.776152 0.630546i \(-0.782831\pi\)
−0.776152 + 0.630546i \(0.782831\pi\)
\(762\) 0.529977i 0.0191991i
\(763\) 18.8433i 0.682174i
\(764\) −43.1355 −1.56059
\(765\) 0 0
\(766\) 4.11888 0.148821
\(767\) − 11.2544i − 0.406374i
\(768\) 9.72191i 0.350809i
\(769\) 17.3239 0.624716 0.312358 0.949964i \(-0.398881\pi\)
0.312358 + 0.949964i \(0.398881\pi\)
\(770\) 0 0
\(771\) 11.1567 0.401798
\(772\) − 5.08108i − 0.182872i
\(773\) 11.6373i 0.418563i 0.977855 + 0.209282i \(0.0671125\pi\)
−0.977855 + 0.209282i \(0.932887\pi\)
\(774\) −0.334474 −0.0120224
\(775\) 0 0
\(776\) −13.9728 −0.501593
\(777\) − 46.6797i − 1.67462i
\(778\) 10.0272i 0.359494i
\(779\) −11.1849 −0.400742
\(780\) 0 0
\(781\) 4.50528 0.161212
\(782\) − 7.95061i − 0.284313i
\(783\) 6.00000i 0.214423i
\(784\) −60.1885 −2.14959
\(785\) 0 0
\(786\) 1.68665 0.0601609
\(787\) 20.9411i 0.746469i 0.927737 + 0.373234i \(0.121751\pi\)
−0.927737 + 0.373234i \(0.878249\pi\)
\(788\) − 24.8927i − 0.886766i
\(789\) 8.00000 0.284808
\(790\) 0 0
\(791\) −4.14611 −0.147419
\(792\) 5.56777i 0.197842i
\(793\) 10.1708i 0.361176i
\(794\) 2.07663 0.0736967
\(795\) 0 0
\(796\) 5.44890 0.193131
\(797\) 20.3380i 0.720410i 0.932873 + 0.360205i \(0.117293\pi\)
−0.932873 + 0.360205i \(0.882707\pi\)
\(798\) − 3.66553i − 0.129758i
\(799\) −23.5194 −0.832057
\(800\) 0 0
\(801\) −0.338044 −0.0119442
\(802\) − 10.9482i − 0.386595i
\(803\) 15.5194i 0.547668i
\(804\) 13.9022 0.490294
\(805\) 0 0
\(806\) −0.411100 −0.0144804
\(807\) − 18.6761i − 0.657429i
\(808\) − 12.0906i − 0.425346i
\(809\) −7.68665 −0.270248 −0.135124 0.990829i \(-0.543143\pi\)
−0.135124 + 0.990829i \(0.543143\pi\)
\(810\) 0 0
\(811\) −44.4111 −1.55948 −0.779742 0.626101i \(-0.784650\pi\)
−0.779742 + 0.626101i \(0.784650\pi\)
\(812\) − 56.5300i − 1.98381i
\(813\) 6.57834i 0.230712i
\(814\) 13.4983 0.473115
\(815\) 0 0
\(816\) 15.2061 0.532319
\(817\) − 2.98230i − 0.104337i
\(818\) − 4.04836i − 0.141548i
\(819\) −4.91638 −0.171792
\(820\) 0 0
\(821\) −46.4630 −1.62157 −0.810785 0.585343i \(-0.800960\pi\)
−0.810785 + 0.585343i \(0.800960\pi\)
\(822\) − 4.77384i − 0.166507i
\(823\) − 46.5089i − 1.62120i −0.585603 0.810598i \(-0.699142\pi\)
0.585603 0.810598i \(-0.300858\pi\)
\(824\) 16.4312 0.572408
\(825\) 0 0
\(826\) 16.0000 0.556711
\(827\) 39.4005i 1.37009i 0.728500 + 0.685045i \(0.240218\pi\)
−0.728500 + 0.685045i \(0.759782\pi\)
\(828\) − 12.1461i − 0.422107i
\(829\) −47.6444 −1.65476 −0.827379 0.561644i \(-0.810169\pi\)
−0.827379 + 0.561644i \(0.810169\pi\)
\(830\) 0 0
\(831\) −25.6655 −0.890327
\(832\) − 6.06249i − 0.210179i
\(833\) 74.4877i 2.58085i
\(834\) −2.16724 −0.0750453
\(835\) 0 0
\(836\) −24.2922 −0.840164
\(837\) − 1.42166i − 0.0491399i
\(838\) − 9.68665i − 0.334620i
\(839\) 39.9058 1.37770 0.688851 0.724903i \(-0.258115\pi\)
0.688851 + 0.724903i \(0.258115\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) − 3.90939i − 0.134726i
\(843\) 3.15667i 0.108722i
\(844\) 12.0988 0.416457
\(845\) 0 0
\(846\) 1.56777 0.0539012
\(847\) 64.7527i 2.22493i
\(848\) 1.18494i 0.0406910i
\(849\) −3.47002 −0.119091
\(850\) 0 0
\(851\) −60.1779 −2.06287
\(852\) 1.75614i 0.0601643i
\(853\) − 29.5019i − 1.01012i −0.863083 0.505062i \(-0.831470\pi\)
0.863083 0.505062i \(-0.168530\pi\)
\(854\) −14.4595 −0.494793
\(855\) 0 0
\(856\) 4.72342 0.161443
\(857\) 8.33804i 0.284822i 0.989808 + 0.142411i \(0.0454855\pi\)
−0.989808 + 0.142411i \(0.954515\pi\)
\(858\) − 1.42166i − 0.0485348i
\(859\) 4.17081 0.142306 0.0711531 0.997465i \(-0.477332\pi\)
0.0711531 + 0.997465i \(0.477332\pi\)
\(860\) 0 0
\(861\) −21.3275 −0.726839
\(862\) − 3.58890i − 0.122238i
\(863\) 3.93051i 0.133796i 0.997760 + 0.0668981i \(0.0213103\pi\)
−0.997760 + 0.0668981i \(0.978690\pi\)
\(864\) −3.27861 −0.111540
\(865\) 0 0
\(866\) 5.00953 0.170231
\(867\) − 1.81863i − 0.0617639i
\(868\) 13.3944i 0.454637i
\(869\) 17.1814 0.582838
\(870\) 0 0
\(871\) −7.25443 −0.245807
\(872\) − 4.34058i − 0.146991i
\(873\) 12.3380i 0.417580i
\(874\) −4.72548 −0.159842
\(875\) 0 0
\(876\) −6.04939 −0.204390
\(877\) − 23.0177i − 0.777253i −0.921395 0.388626i \(-0.872950\pi\)
0.921395 0.388626i \(-0.127050\pi\)
\(878\) 0.188362i 0.00635692i
\(879\) 28.6550 0.966508
\(880\) 0 0
\(881\) −15.3522 −0.517228 −0.258614 0.965981i \(-0.583266\pi\)
−0.258614 + 0.965981i \(0.583266\pi\)
\(882\) − 4.96526i − 0.167189i
\(883\) − 42.8011i − 1.44037i −0.693782 0.720185i \(-0.744057\pi\)
0.693782 0.720185i \(-0.255943\pi\)
\(884\) −8.31335 −0.279608
\(885\) 0 0
\(886\) 2.55824 0.0859459
\(887\) − 53.1885i − 1.78590i −0.450160 0.892948i \(-0.648633\pi\)
0.450160 0.892948i \(-0.351367\pi\)
\(888\) 10.7527i 0.360837i
\(889\) −9.01056 −0.302205
\(890\) 0 0
\(891\) 4.91638 0.164705
\(892\) − 36.8988i − 1.23546i
\(893\) 13.9789i 0.467785i
\(894\) −5.92337 −0.198107
\(895\) 0 0
\(896\) 40.8566 1.36492
\(897\) 6.33804i 0.211621i
\(898\) 1.25443i 0.0418607i
\(899\) −8.52998 −0.284491
\(900\) 0 0
\(901\) 1.46645 0.0488546
\(902\) − 6.16724i − 0.205347i
\(903\) − 5.68665i − 0.189240i
\(904\) 0.955062 0.0317649
\(905\) 0 0
\(906\) −4.74557 −0.157661
\(907\) 11.8116i 0.392199i 0.980584 + 0.196099i \(0.0628276\pi\)
−0.980584 + 0.196099i \(0.937172\pi\)
\(908\) − 25.0800i − 0.832311i
\(909\) −10.6761 −0.354104
\(910\) 0 0
\(911\) 44.1955 1.46426 0.732131 0.681164i \(-0.238526\pi\)
0.732131 + 0.681164i \(0.238526\pi\)
\(912\) − 9.03780i − 0.299271i
\(913\) − 55.3311i − 1.83119i
\(914\) 4.43223 0.146605
\(915\) 0 0
\(916\) 46.9683 1.55188
\(917\) 28.6761i 0.946968i
\(918\) 1.25443i 0.0414022i
\(919\) −55.2096 −1.82120 −0.910599 0.413291i \(-0.864379\pi\)
−0.910599 + 0.413291i \(0.864379\pi\)
\(920\) 0 0
\(921\) 1.92694 0.0634950
\(922\) 3.42880i 0.112922i
\(923\) − 0.916382i − 0.0301631i
\(924\) −46.3205 −1.52383
\(925\) 0 0
\(926\) 7.63829 0.251010
\(927\) − 14.5089i − 0.476533i
\(928\) 19.6716i 0.645753i
\(929\) −22.9930 −0.754376 −0.377188 0.926137i \(-0.623109\pi\)
−0.377188 + 0.926137i \(0.623109\pi\)
\(930\) 0 0
\(931\) 44.2721 1.45096
\(932\) − 15.9789i − 0.523405i
\(933\) − 29.9789i − 0.981464i
\(934\) −9.73604 −0.318573
\(935\) 0 0
\(936\) 1.13249 0.0370167
\(937\) 7.97887i 0.260658i 0.991471 + 0.130329i \(0.0416034\pi\)
−0.991471 + 0.130329i \(0.958397\pi\)
\(938\) − 10.3133i − 0.336743i
\(939\) −16.3133 −0.532366
\(940\) 0 0
\(941\) 41.5019 1.35292 0.676461 0.736478i \(-0.263513\pi\)
0.676461 + 0.736478i \(0.263513\pi\)
\(942\) − 6.26499i − 0.204124i
\(943\) 27.4947i 0.895351i
\(944\) 39.4499 1.28399
\(945\) 0 0
\(946\) 1.64440 0.0534641
\(947\) 47.4499i 1.54192i 0.636886 + 0.770958i \(0.280222\pi\)
−0.636886 + 0.770958i \(0.719778\pi\)
\(948\) 6.69721i 0.217515i
\(949\) 3.15667 0.102470
\(950\) 0 0
\(951\) −30.6761 −0.994740
\(952\) − 24.1533i − 0.782811i
\(953\) − 30.3663i − 0.983661i −0.870691 0.491831i \(-0.836328\pi\)
0.870691 0.491831i \(-0.163672\pi\)
\(954\) −0.0977518 −0.00316483
\(955\) 0 0
\(956\) −17.0872 −0.552639
\(957\) − 29.4983i − 0.953544i
\(958\) 3.10831i 0.100425i
\(959\) 81.1638 2.62092
\(960\) 0 0
\(961\) −28.9789 −0.934802
\(962\) − 2.74557i − 0.0885209i
\(963\) − 4.17081i − 0.134402i
\(964\) −11.4983 −0.370335
\(965\) 0 0
\(966\) −9.01056 −0.289910
\(967\) − 41.0943i − 1.32150i −0.750604 0.660752i \(-0.770237\pi\)
0.750604 0.660752i \(-0.229763\pi\)
\(968\) − 14.9159i − 0.479414i
\(969\) −11.1849 −0.359312
\(970\) 0 0
\(971\) 38.1744 1.22507 0.612537 0.790442i \(-0.290149\pi\)
0.612537 + 0.790442i \(0.290149\pi\)
\(972\) 1.91638i 0.0614680i
\(973\) − 36.8469i − 1.18126i
\(974\) 6.57834 0.210784
\(975\) 0 0
\(976\) −35.6515 −1.14118
\(977\) − 10.4806i − 0.335304i −0.985846 0.167652i \(-0.946382\pi\)
0.985846 0.167652i \(-0.0536184\pi\)
\(978\) − 1.75614i − 0.0561551i
\(979\) 1.66196 0.0531163
\(980\) 0 0
\(981\) −3.83276 −0.122371
\(982\) 5.11442i 0.163208i
\(983\) 0.0766264i 0.00244400i 0.999999 + 0.00122200i \(0.000388975\pi\)
−0.999999 + 0.00122200i \(0.999611\pi\)
\(984\) 4.91281 0.156615
\(985\) 0 0
\(986\) 7.52656 0.239694
\(987\) 26.6550i 0.848437i
\(988\) 4.94108i 0.157197i
\(989\) −7.33105 −0.233114
\(990\) 0 0
\(991\) −13.8575 −0.440197 −0.220098 0.975478i \(-0.570638\pi\)
−0.220098 + 0.975478i \(0.570638\pi\)
\(992\) − 4.66107i − 0.147989i
\(993\) − 10.0978i − 0.320442i
\(994\) 1.30279 0.0413219
\(995\) 0 0
\(996\) 21.5678 0.683401
\(997\) − 10.3416i − 0.327522i −0.986500 0.163761i \(-0.947637\pi\)
0.986500 0.163761i \(-0.0523626\pi\)
\(998\) 5.76328i 0.182433i
\(999\) 9.49472 0.300400
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 975.2.c.i.274.3 6
3.2 odd 2 2925.2.c.w.2224.4 6
5.2 odd 4 975.2.a.o.1.2 3
5.3 odd 4 195.2.a.e.1.2 3
5.4 even 2 inner 975.2.c.i.274.4 6
15.2 even 4 2925.2.a.bh.1.2 3
15.8 even 4 585.2.a.n.1.2 3
15.14 odd 2 2925.2.c.w.2224.3 6
20.3 even 4 3120.2.a.bj.1.1 3
35.13 even 4 9555.2.a.bq.1.2 3
60.23 odd 4 9360.2.a.dd.1.1 3
65.38 odd 4 2535.2.a.bc.1.2 3
195.38 even 4 7605.2.a.bx.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.e.1.2 3 5.3 odd 4
585.2.a.n.1.2 3 15.8 even 4
975.2.a.o.1.2 3 5.2 odd 4
975.2.c.i.274.3 6 1.1 even 1 trivial
975.2.c.i.274.4 6 5.4 even 2 inner
2535.2.a.bc.1.2 3 65.38 odd 4
2925.2.a.bh.1.2 3 15.2 even 4
2925.2.c.w.2224.3 6 15.14 odd 2
2925.2.c.w.2224.4 6 3.2 odd 2
3120.2.a.bj.1.1 3 20.3 even 4
7605.2.a.bx.1.2 3 195.38 even 4
9360.2.a.dd.1.1 3 60.23 odd 4
9555.2.a.bq.1.2 3 35.13 even 4