Properties

Label 8034.2.a.bc.1.3
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - x^{14} - 48 x^{13} + 44 x^{12} + 872 x^{11} - 707 x^{10} - 7580 x^{9} + 5112 x^{8} + 33191 x^{7} - 16428 x^{6} - 71361 x^{5} + 21747 x^{4} + 65434 x^{3} - 11840 x^{2} - 17600 x + 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.75355\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.75355 q^{5} -1.00000 q^{6} +0.824838 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.75355 q^{5} -1.00000 q^{6} +0.824838 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.75355 q^{10} -4.01662 q^{11} -1.00000 q^{12} -1.00000 q^{13} +0.824838 q^{14} +2.75355 q^{15} +1.00000 q^{16} -4.88498 q^{17} +1.00000 q^{18} +5.85873 q^{19} -2.75355 q^{20} -0.824838 q^{21} -4.01662 q^{22} +6.04631 q^{23} -1.00000 q^{24} +2.58202 q^{25} -1.00000 q^{26} -1.00000 q^{27} +0.824838 q^{28} -6.86586 q^{29} +2.75355 q^{30} +0.898245 q^{31} +1.00000 q^{32} +4.01662 q^{33} -4.88498 q^{34} -2.27123 q^{35} +1.00000 q^{36} +4.98543 q^{37} +5.85873 q^{38} +1.00000 q^{39} -2.75355 q^{40} -8.56990 q^{41} -0.824838 q^{42} +12.5165 q^{43} -4.01662 q^{44} -2.75355 q^{45} +6.04631 q^{46} -11.9912 q^{47} -1.00000 q^{48} -6.31964 q^{49} +2.58202 q^{50} +4.88498 q^{51} -1.00000 q^{52} -6.44142 q^{53} -1.00000 q^{54} +11.0599 q^{55} +0.824838 q^{56} -5.85873 q^{57} -6.86586 q^{58} -2.07171 q^{59} +2.75355 q^{60} -12.3179 q^{61} +0.898245 q^{62} +0.824838 q^{63} +1.00000 q^{64} +2.75355 q^{65} +4.01662 q^{66} +9.95884 q^{67} -4.88498 q^{68} -6.04631 q^{69} -2.27123 q^{70} +5.23569 q^{71} +1.00000 q^{72} -2.85779 q^{73} +4.98543 q^{74} -2.58202 q^{75} +5.85873 q^{76} -3.31306 q^{77} +1.00000 q^{78} +8.39757 q^{79} -2.75355 q^{80} +1.00000 q^{81} -8.56990 q^{82} -5.21802 q^{83} -0.824838 q^{84} +13.4510 q^{85} +12.5165 q^{86} +6.86586 q^{87} -4.01662 q^{88} +15.1576 q^{89} -2.75355 q^{90} -0.824838 q^{91} +6.04631 q^{92} -0.898245 q^{93} -11.9912 q^{94} -16.1323 q^{95} -1.00000 q^{96} +6.12659 q^{97} -6.31964 q^{98} -4.01662 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} - q^{10} + 3q^{11} - 15q^{12} - 15q^{13} + 5q^{14} + q^{15} + 15q^{16} - 2q^{17} + 15q^{18} + 8q^{19} - q^{20} - 5q^{21} + 3q^{22} + 3q^{23} - 15q^{24} + 22q^{25} - 15q^{26} - 15q^{27} + 5q^{28} + 26q^{29} + q^{30} + 15q^{32} - 3q^{33} - 2q^{34} - 8q^{35} + 15q^{36} + 25q^{37} + 8q^{38} + 15q^{39} - q^{40} - q^{41} - 5q^{42} + 10q^{43} + 3q^{44} - q^{45} + 3q^{46} - 3q^{47} - 15q^{48} + 32q^{49} + 22q^{50} + 2q^{51} - 15q^{52} + 13q^{53} - 15q^{54} - 2q^{55} + 5q^{56} - 8q^{57} + 26q^{58} + 28q^{59} + q^{60} + 22q^{61} + 5q^{63} + 15q^{64} + q^{65} - 3q^{66} + 29q^{67} - 2q^{68} - 3q^{69} - 8q^{70} + 18q^{71} + 15q^{72} + 23q^{73} + 25q^{74} - 22q^{75} + 8q^{76} + 17q^{77} + 15q^{78} + 27q^{79} - q^{80} + 15q^{81} - q^{82} + 7q^{83} - 5q^{84} + 43q^{85} + 10q^{86} - 26q^{87} + 3q^{88} + 35q^{89} - q^{90} - 5q^{91} + 3q^{92} - 3q^{94} + 6q^{95} - 15q^{96} + 19q^{97} + 32q^{98} + 3q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −2.75355 −1.23142 −0.615712 0.787971i \(-0.711131\pi\)
−0.615712 + 0.787971i \(0.711131\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0.824838 0.311760 0.155880 0.987776i \(-0.450179\pi\)
0.155880 + 0.987776i \(0.450179\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.75355 −0.870748
\(11\) −4.01662 −1.21106 −0.605528 0.795824i \(-0.707038\pi\)
−0.605528 + 0.795824i \(0.707038\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 0.824838 0.220447
\(15\) 2.75355 0.710963
\(16\) 1.00000 0.250000
\(17\) −4.88498 −1.18478 −0.592391 0.805650i \(-0.701816\pi\)
−0.592391 + 0.805650i \(0.701816\pi\)
\(18\) 1.00000 0.235702
\(19\) 5.85873 1.34408 0.672042 0.740513i \(-0.265417\pi\)
0.672042 + 0.740513i \(0.265417\pi\)
\(20\) −2.75355 −0.615712
\(21\) −0.824838 −0.179994
\(22\) −4.01662 −0.856346
\(23\) 6.04631 1.26074 0.630372 0.776294i \(-0.282903\pi\)
0.630372 + 0.776294i \(0.282903\pi\)
\(24\) −1.00000 −0.204124
\(25\) 2.58202 0.516404
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 0.824838 0.155880
\(29\) −6.86586 −1.27496 −0.637479 0.770468i \(-0.720023\pi\)
−0.637479 + 0.770468i \(0.720023\pi\)
\(30\) 2.75355 0.502727
\(31\) 0.898245 0.161330 0.0806648 0.996741i \(-0.474296\pi\)
0.0806648 + 0.996741i \(0.474296\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.01662 0.699204
\(34\) −4.88498 −0.837768
\(35\) −2.27123 −0.383908
\(36\) 1.00000 0.166667
\(37\) 4.98543 0.819599 0.409800 0.912176i \(-0.365599\pi\)
0.409800 + 0.912176i \(0.365599\pi\)
\(38\) 5.85873 0.950411
\(39\) 1.00000 0.160128
\(40\) −2.75355 −0.435374
\(41\) −8.56990 −1.33839 −0.669197 0.743085i \(-0.733362\pi\)
−0.669197 + 0.743085i \(0.733362\pi\)
\(42\) −0.824838 −0.127275
\(43\) 12.5165 1.90875 0.954373 0.298617i \(-0.0965252\pi\)
0.954373 + 0.298617i \(0.0965252\pi\)
\(44\) −4.01662 −0.605528
\(45\) −2.75355 −0.410474
\(46\) 6.04631 0.891480
\(47\) −11.9912 −1.74910 −0.874549 0.484938i \(-0.838842\pi\)
−0.874549 + 0.484938i \(0.838842\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.31964 −0.902806
\(50\) 2.58202 0.365153
\(51\) 4.88498 0.684035
\(52\) −1.00000 −0.138675
\(53\) −6.44142 −0.884797 −0.442399 0.896818i \(-0.645872\pi\)
−0.442399 + 0.896818i \(0.645872\pi\)
\(54\) −1.00000 −0.136083
\(55\) 11.0599 1.49132
\(56\) 0.824838 0.110224
\(57\) −5.85873 −0.776008
\(58\) −6.86586 −0.901532
\(59\) −2.07171 −0.269713 −0.134857 0.990865i \(-0.543057\pi\)
−0.134857 + 0.990865i \(0.543057\pi\)
\(60\) 2.75355 0.355481
\(61\) −12.3179 −1.57714 −0.788570 0.614945i \(-0.789178\pi\)
−0.788570 + 0.614945i \(0.789178\pi\)
\(62\) 0.898245 0.114077
\(63\) 0.824838 0.103920
\(64\) 1.00000 0.125000
\(65\) 2.75355 0.341535
\(66\) 4.01662 0.494412
\(67\) 9.95884 1.21667 0.608333 0.793682i \(-0.291838\pi\)
0.608333 + 0.793682i \(0.291838\pi\)
\(68\) −4.88498 −0.592391
\(69\) −6.04631 −0.727890
\(70\) −2.27123 −0.271464
\(71\) 5.23569 0.621363 0.310681 0.950514i \(-0.399443\pi\)
0.310681 + 0.950514i \(0.399443\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.85779 −0.334479 −0.167240 0.985916i \(-0.553485\pi\)
−0.167240 + 0.985916i \(0.553485\pi\)
\(74\) 4.98543 0.579544
\(75\) −2.58202 −0.298146
\(76\) 5.85873 0.672042
\(77\) −3.31306 −0.377559
\(78\) 1.00000 0.113228
\(79\) 8.39757 0.944800 0.472400 0.881384i \(-0.343388\pi\)
0.472400 + 0.881384i \(0.343388\pi\)
\(80\) −2.75355 −0.307856
\(81\) 1.00000 0.111111
\(82\) −8.56990 −0.946388
\(83\) −5.21802 −0.572752 −0.286376 0.958117i \(-0.592451\pi\)
−0.286376 + 0.958117i \(0.592451\pi\)
\(84\) −0.824838 −0.0899972
\(85\) 13.4510 1.45897
\(86\) 12.5165 1.34969
\(87\) 6.86586 0.736097
\(88\) −4.01662 −0.428173
\(89\) 15.1576 1.60670 0.803349 0.595508i \(-0.203049\pi\)
0.803349 + 0.595508i \(0.203049\pi\)
\(90\) −2.75355 −0.290249
\(91\) −0.824838 −0.0864666
\(92\) 6.04631 0.630372
\(93\) −0.898245 −0.0931437
\(94\) −11.9912 −1.23680
\(95\) −16.1323 −1.65514
\(96\) −1.00000 −0.102062
\(97\) 6.12659 0.622061 0.311030 0.950400i \(-0.399326\pi\)
0.311030 + 0.950400i \(0.399326\pi\)
\(98\) −6.31964 −0.638380
\(99\) −4.01662 −0.403685
\(100\) 2.58202 0.258202
\(101\) −17.5626 −1.74754 −0.873771 0.486338i \(-0.838332\pi\)
−0.873771 + 0.486338i \(0.838332\pi\)
\(102\) 4.88498 0.483685
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) 2.27123 0.221649
\(106\) −6.44142 −0.625646
\(107\) 20.3015 1.96262 0.981310 0.192434i \(-0.0616381\pi\)
0.981310 + 0.192434i \(0.0616381\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −3.09481 −0.296429 −0.148215 0.988955i \(-0.547353\pi\)
−0.148215 + 0.988955i \(0.547353\pi\)
\(110\) 11.0599 1.05452
\(111\) −4.98543 −0.473196
\(112\) 0.824838 0.0779399
\(113\) 14.6484 1.37801 0.689004 0.724758i \(-0.258048\pi\)
0.689004 + 0.724758i \(0.258048\pi\)
\(114\) −5.85873 −0.548720
\(115\) −16.6488 −1.55251
\(116\) −6.86586 −0.637479
\(117\) −1.00000 −0.0924500
\(118\) −2.07171 −0.190716
\(119\) −4.02932 −0.369367
\(120\) 2.75355 0.251363
\(121\) 5.13324 0.466658
\(122\) −12.3179 −1.11521
\(123\) 8.56990 0.772722
\(124\) 0.898245 0.0806648
\(125\) 6.65802 0.595512
\(126\) 0.824838 0.0734824
\(127\) 14.5076 1.28735 0.643673 0.765301i \(-0.277410\pi\)
0.643673 + 0.765301i \(0.277410\pi\)
\(128\) 1.00000 0.0883883
\(129\) −12.5165 −1.10202
\(130\) 2.75355 0.241502
\(131\) 13.3775 1.16880 0.584399 0.811467i \(-0.301330\pi\)
0.584399 + 0.811467i \(0.301330\pi\)
\(132\) 4.01662 0.349602
\(133\) 4.83251 0.419031
\(134\) 9.95884 0.860313
\(135\) 2.75355 0.236988
\(136\) −4.88498 −0.418884
\(137\) −3.65086 −0.311914 −0.155957 0.987764i \(-0.549846\pi\)
−0.155957 + 0.987764i \(0.549846\pi\)
\(138\) −6.04631 −0.514696
\(139\) 14.9078 1.26446 0.632232 0.774779i \(-0.282139\pi\)
0.632232 + 0.774779i \(0.282139\pi\)
\(140\) −2.27123 −0.191954
\(141\) 11.9912 1.00984
\(142\) 5.23569 0.439370
\(143\) 4.01662 0.335887
\(144\) 1.00000 0.0833333
\(145\) 18.9055 1.57001
\(146\) −2.85779 −0.236513
\(147\) 6.31964 0.521235
\(148\) 4.98543 0.409800
\(149\) 2.99240 0.245147 0.122573 0.992459i \(-0.460885\pi\)
0.122573 + 0.992459i \(0.460885\pi\)
\(150\) −2.58202 −0.210821
\(151\) −19.8765 −1.61753 −0.808765 0.588132i \(-0.799863\pi\)
−0.808765 + 0.588132i \(0.799863\pi\)
\(152\) 5.85873 0.475206
\(153\) −4.88498 −0.394928
\(154\) −3.31306 −0.266974
\(155\) −2.47336 −0.198665
\(156\) 1.00000 0.0800641
\(157\) −5.48855 −0.438034 −0.219017 0.975721i \(-0.570285\pi\)
−0.219017 + 0.975721i \(0.570285\pi\)
\(158\) 8.39757 0.668075
\(159\) 6.44142 0.510838
\(160\) −2.75355 −0.217687
\(161\) 4.98723 0.393049
\(162\) 1.00000 0.0785674
\(163\) 7.06211 0.553147 0.276574 0.960993i \(-0.410801\pi\)
0.276574 + 0.960993i \(0.410801\pi\)
\(164\) −8.56990 −0.669197
\(165\) −11.0599 −0.861016
\(166\) −5.21802 −0.404997
\(167\) 16.9953 1.31513 0.657567 0.753396i \(-0.271586\pi\)
0.657567 + 0.753396i \(0.271586\pi\)
\(168\) −0.824838 −0.0636377
\(169\) 1.00000 0.0769231
\(170\) 13.4510 1.03165
\(171\) 5.85873 0.448028
\(172\) 12.5165 0.954373
\(173\) 4.34169 0.330093 0.165046 0.986286i \(-0.447223\pi\)
0.165046 + 0.986286i \(0.447223\pi\)
\(174\) 6.86586 0.520499
\(175\) 2.12975 0.160994
\(176\) −4.01662 −0.302764
\(177\) 2.07171 0.155719
\(178\) 15.1576 1.13611
\(179\) 17.0414 1.27373 0.636865 0.770975i \(-0.280231\pi\)
0.636865 + 0.770975i \(0.280231\pi\)
\(180\) −2.75355 −0.205237
\(181\) 17.2522 1.28234 0.641172 0.767397i \(-0.278448\pi\)
0.641172 + 0.767397i \(0.278448\pi\)
\(182\) −0.824838 −0.0611411
\(183\) 12.3179 0.910562
\(184\) 6.04631 0.445740
\(185\) −13.7276 −1.00927
\(186\) −0.898245 −0.0658626
\(187\) 19.6211 1.43484
\(188\) −11.9912 −0.874549
\(189\) −0.824838 −0.0599982
\(190\) −16.1323 −1.17036
\(191\) −16.2534 −1.17606 −0.588029 0.808840i \(-0.700096\pi\)
−0.588029 + 0.808840i \(0.700096\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.9122 −1.07341 −0.536703 0.843771i \(-0.680330\pi\)
−0.536703 + 0.843771i \(0.680330\pi\)
\(194\) 6.12659 0.439863
\(195\) −2.75355 −0.197186
\(196\) −6.31964 −0.451403
\(197\) 17.5228 1.24845 0.624224 0.781245i \(-0.285415\pi\)
0.624224 + 0.781245i \(0.285415\pi\)
\(198\) −4.01662 −0.285449
\(199\) −8.66641 −0.614346 −0.307173 0.951654i \(-0.599383\pi\)
−0.307173 + 0.951654i \(0.599383\pi\)
\(200\) 2.58202 0.182576
\(201\) −9.95884 −0.702443
\(202\) −17.5626 −1.23570
\(203\) −5.66322 −0.397480
\(204\) 4.88498 0.342017
\(205\) 23.5976 1.64813
\(206\) −1.00000 −0.0696733
\(207\) 6.04631 0.420248
\(208\) −1.00000 −0.0693375
\(209\) −23.5323 −1.62776
\(210\) 2.27123 0.156730
\(211\) −7.82353 −0.538594 −0.269297 0.963057i \(-0.586791\pi\)
−0.269297 + 0.963057i \(0.586791\pi\)
\(212\) −6.44142 −0.442399
\(213\) −5.23569 −0.358744
\(214\) 20.3015 1.38778
\(215\) −34.4647 −2.35047
\(216\) −1.00000 −0.0680414
\(217\) 0.740907 0.0502961
\(218\) −3.09481 −0.209607
\(219\) 2.85779 0.193112
\(220\) 11.0599 0.745662
\(221\) 4.88498 0.328600
\(222\) −4.98543 −0.334600
\(223\) −0.973709 −0.0652044 −0.0326022 0.999468i \(-0.510379\pi\)
−0.0326022 + 0.999468i \(0.510379\pi\)
\(224\) 0.824838 0.0551118
\(225\) 2.58202 0.172135
\(226\) 14.6484 0.974398
\(227\) 25.0223 1.66079 0.830395 0.557175i \(-0.188115\pi\)
0.830395 + 0.557175i \(0.188115\pi\)
\(228\) −5.85873 −0.388004
\(229\) 20.2628 1.33900 0.669500 0.742812i \(-0.266508\pi\)
0.669500 + 0.742812i \(0.266508\pi\)
\(230\) −16.6488 −1.09779
\(231\) 3.31306 0.217984
\(232\) −6.86586 −0.450766
\(233\) −10.4040 −0.681589 −0.340795 0.940138i \(-0.610696\pi\)
−0.340795 + 0.940138i \(0.610696\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 33.0183 2.15388
\(236\) −2.07171 −0.134857
\(237\) −8.39757 −0.545481
\(238\) −4.02932 −0.261182
\(239\) 26.4940 1.71375 0.856876 0.515523i \(-0.172402\pi\)
0.856876 + 0.515523i \(0.172402\pi\)
\(240\) 2.75355 0.177741
\(241\) 1.31680 0.0848228 0.0424114 0.999100i \(-0.486496\pi\)
0.0424114 + 0.999100i \(0.486496\pi\)
\(242\) 5.13324 0.329977
\(243\) −1.00000 −0.0641500
\(244\) −12.3179 −0.788570
\(245\) 17.4014 1.11174
\(246\) 8.56990 0.546397
\(247\) −5.85873 −0.372782
\(248\) 0.898245 0.0570386
\(249\) 5.21802 0.330679
\(250\) 6.65802 0.421090
\(251\) −13.8996 −0.877337 −0.438669 0.898649i \(-0.644550\pi\)
−0.438669 + 0.898649i \(0.644550\pi\)
\(252\) 0.824838 0.0519599
\(253\) −24.2857 −1.52683
\(254\) 14.5076 0.910291
\(255\) −13.4510 −0.842336
\(256\) 1.00000 0.0625000
\(257\) 22.3217 1.39239 0.696193 0.717854i \(-0.254876\pi\)
0.696193 + 0.717854i \(0.254876\pi\)
\(258\) −12.5165 −0.779242
\(259\) 4.11217 0.255518
\(260\) 2.75355 0.170768
\(261\) −6.86586 −0.424986
\(262\) 13.3775 0.826465
\(263\) −16.1592 −0.996420 −0.498210 0.867056i \(-0.666009\pi\)
−0.498210 + 0.867056i \(0.666009\pi\)
\(264\) 4.01662 0.247206
\(265\) 17.7368 1.08956
\(266\) 4.83251 0.296300
\(267\) −15.1576 −0.927628
\(268\) 9.95884 0.608333
\(269\) 6.48600 0.395459 0.197729 0.980257i \(-0.436643\pi\)
0.197729 + 0.980257i \(0.436643\pi\)
\(270\) 2.75355 0.167576
\(271\) −25.7060 −1.56153 −0.780763 0.624827i \(-0.785170\pi\)
−0.780763 + 0.624827i \(0.785170\pi\)
\(272\) −4.88498 −0.296196
\(273\) 0.824838 0.0499215
\(274\) −3.65086 −0.220556
\(275\) −10.3710 −0.625394
\(276\) −6.04631 −0.363945
\(277\) −16.0917 −0.966856 −0.483428 0.875384i \(-0.660608\pi\)
−0.483428 + 0.875384i \(0.660608\pi\)
\(278\) 14.9078 0.894111
\(279\) 0.898245 0.0537765
\(280\) −2.27123 −0.135732
\(281\) 0.0961083 0.00573334 0.00286667 0.999996i \(-0.499088\pi\)
0.00286667 + 0.999996i \(0.499088\pi\)
\(282\) 11.9912 0.714066
\(283\) 5.39853 0.320909 0.160455 0.987043i \(-0.448704\pi\)
0.160455 + 0.987043i \(0.448704\pi\)
\(284\) 5.23569 0.310681
\(285\) 16.1323 0.955594
\(286\) 4.01662 0.237508
\(287\) −7.06879 −0.417257
\(288\) 1.00000 0.0589256
\(289\) 6.86306 0.403710
\(290\) 18.9055 1.11017
\(291\) −6.12659 −0.359147
\(292\) −2.85779 −0.167240
\(293\) 22.2615 1.30053 0.650266 0.759706i \(-0.274657\pi\)
0.650266 + 0.759706i \(0.274657\pi\)
\(294\) 6.31964 0.368569
\(295\) 5.70455 0.332131
\(296\) 4.98543 0.289772
\(297\) 4.01662 0.233068
\(298\) 2.99240 0.173345
\(299\) −6.04631 −0.349667
\(300\) −2.58202 −0.149073
\(301\) 10.3241 0.595070
\(302\) −19.8765 −1.14377
\(303\) 17.5626 1.00894
\(304\) 5.85873 0.336021
\(305\) 33.9178 1.94213
\(306\) −4.88498 −0.279256
\(307\) −2.18512 −0.124711 −0.0623557 0.998054i \(-0.519861\pi\)
−0.0623557 + 0.998054i \(0.519861\pi\)
\(308\) −3.31306 −0.188779
\(309\) 1.00000 0.0568880
\(310\) −2.47336 −0.140477
\(311\) −0.0801534 −0.00454508 −0.00227254 0.999997i \(-0.500723\pi\)
−0.00227254 + 0.999997i \(0.500723\pi\)
\(312\) 1.00000 0.0566139
\(313\) 9.05156 0.511624 0.255812 0.966726i \(-0.417657\pi\)
0.255812 + 0.966726i \(0.417657\pi\)
\(314\) −5.48855 −0.309737
\(315\) −2.27123 −0.127969
\(316\) 8.39757 0.472400
\(317\) 4.18981 0.235323 0.117662 0.993054i \(-0.462460\pi\)
0.117662 + 0.993054i \(0.462460\pi\)
\(318\) 6.44142 0.361217
\(319\) 27.5775 1.54405
\(320\) −2.75355 −0.153928
\(321\) −20.3015 −1.13312
\(322\) 4.98723 0.277927
\(323\) −28.6198 −1.59245
\(324\) 1.00000 0.0555556
\(325\) −2.58202 −0.143225
\(326\) 7.06211 0.391134
\(327\) 3.09481 0.171144
\(328\) −8.56990 −0.473194
\(329\) −9.89081 −0.545298
\(330\) −11.0599 −0.608830
\(331\) 18.0911 0.994377 0.497188 0.867643i \(-0.334366\pi\)
0.497188 + 0.867643i \(0.334366\pi\)
\(332\) −5.21802 −0.286376
\(333\) 4.98543 0.273200
\(334\) 16.9953 0.929940
\(335\) −27.4221 −1.49823
\(336\) −0.824838 −0.0449986
\(337\) −2.00641 −0.109296 −0.0546480 0.998506i \(-0.517404\pi\)
−0.0546480 + 0.998506i \(0.517404\pi\)
\(338\) 1.00000 0.0543928
\(339\) −14.6484 −0.795593
\(340\) 13.4510 0.729484
\(341\) −3.60791 −0.195379
\(342\) 5.85873 0.316804
\(343\) −10.9866 −0.593218
\(344\) 12.5165 0.674844
\(345\) 16.6488 0.896341
\(346\) 4.34169 0.233411
\(347\) −3.06551 −0.164565 −0.0822826 0.996609i \(-0.526221\pi\)
−0.0822826 + 0.996609i \(0.526221\pi\)
\(348\) 6.86586 0.368049
\(349\) 11.4089 0.610703 0.305351 0.952240i \(-0.401226\pi\)
0.305351 + 0.952240i \(0.401226\pi\)
\(350\) 2.12975 0.113840
\(351\) 1.00000 0.0533761
\(352\) −4.01662 −0.214087
\(353\) −27.4910 −1.46320 −0.731599 0.681735i \(-0.761226\pi\)
−0.731599 + 0.681735i \(0.761226\pi\)
\(354\) 2.07171 0.110110
\(355\) −14.4167 −0.765160
\(356\) 15.1576 0.803349
\(357\) 4.02932 0.213254
\(358\) 17.0414 0.900664
\(359\) −12.5262 −0.661106 −0.330553 0.943787i \(-0.607235\pi\)
−0.330553 + 0.943787i \(0.607235\pi\)
\(360\) −2.75355 −0.145125
\(361\) 15.3247 0.806564
\(362\) 17.2522 0.906754
\(363\) −5.13324 −0.269425
\(364\) −0.824838 −0.0432333
\(365\) 7.86906 0.411886
\(366\) 12.3179 0.643865
\(367\) 21.5573 1.12528 0.562642 0.826701i \(-0.309785\pi\)
0.562642 + 0.826701i \(0.309785\pi\)
\(368\) 6.04631 0.315186
\(369\) −8.56990 −0.446131
\(370\) −13.7276 −0.713664
\(371\) −5.31313 −0.275844
\(372\) −0.898245 −0.0465719
\(373\) 34.6777 1.79554 0.897772 0.440460i \(-0.145185\pi\)
0.897772 + 0.440460i \(0.145185\pi\)
\(374\) 19.6211 1.01458
\(375\) −6.65802 −0.343819
\(376\) −11.9912 −0.618399
\(377\) 6.86586 0.353610
\(378\) −0.824838 −0.0424251
\(379\) −10.8080 −0.555172 −0.277586 0.960701i \(-0.589534\pi\)
−0.277586 + 0.960701i \(0.589534\pi\)
\(380\) −16.1323 −0.827569
\(381\) −14.5076 −0.743249
\(382\) −16.2534 −0.831598
\(383\) 2.54440 0.130013 0.0650063 0.997885i \(-0.479293\pi\)
0.0650063 + 0.997885i \(0.479293\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 9.12267 0.464934
\(386\) −14.9122 −0.759013
\(387\) 12.5165 0.636249
\(388\) 6.12659 0.311030
\(389\) 18.6438 0.945277 0.472639 0.881256i \(-0.343302\pi\)
0.472639 + 0.881256i \(0.343302\pi\)
\(390\) −2.75355 −0.139431
\(391\) −29.5361 −1.49371
\(392\) −6.31964 −0.319190
\(393\) −13.3775 −0.674806
\(394\) 17.5228 0.882787
\(395\) −23.1231 −1.16345
\(396\) −4.01662 −0.201843
\(397\) −12.1568 −0.610133 −0.305067 0.952331i \(-0.598679\pi\)
−0.305067 + 0.952331i \(0.598679\pi\)
\(398\) −8.66641 −0.434408
\(399\) −4.83251 −0.241928
\(400\) 2.58202 0.129101
\(401\) 17.1721 0.857536 0.428768 0.903415i \(-0.358948\pi\)
0.428768 + 0.903415i \(0.358948\pi\)
\(402\) −9.95884 −0.496702
\(403\) −0.898245 −0.0447448
\(404\) −17.5626 −0.873771
\(405\) −2.75355 −0.136825
\(406\) −5.66322 −0.281061
\(407\) −20.0246 −0.992581
\(408\) 4.88498 0.241843
\(409\) −2.80065 −0.138483 −0.0692415 0.997600i \(-0.522058\pi\)
−0.0692415 + 0.997600i \(0.522058\pi\)
\(410\) 23.5976 1.16540
\(411\) 3.65086 0.180084
\(412\) −1.00000 −0.0492665
\(413\) −1.70882 −0.0840858
\(414\) 6.04631 0.297160
\(415\) 14.3681 0.705300
\(416\) −1.00000 −0.0490290
\(417\) −14.9078 −0.730038
\(418\) −23.5323 −1.15100
\(419\) 40.3556 1.97150 0.985751 0.168210i \(-0.0537988\pi\)
0.985751 + 0.168210i \(0.0537988\pi\)
\(420\) 2.27123 0.110825
\(421\) −15.1113 −0.736481 −0.368240 0.929731i \(-0.620040\pi\)
−0.368240 + 0.929731i \(0.620040\pi\)
\(422\) −7.82353 −0.380843
\(423\) −11.9912 −0.583032
\(424\) −6.44142 −0.312823
\(425\) −12.6131 −0.611826
\(426\) −5.23569 −0.253670
\(427\) −10.1602 −0.491689
\(428\) 20.3015 0.981310
\(429\) −4.01662 −0.193924
\(430\) −34.4647 −1.66204
\(431\) 23.8797 1.15024 0.575122 0.818068i \(-0.304955\pi\)
0.575122 + 0.818068i \(0.304955\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −22.2116 −1.06742 −0.533712 0.845667i \(-0.679203\pi\)
−0.533712 + 0.845667i \(0.679203\pi\)
\(434\) 0.740907 0.0355647
\(435\) −18.9055 −0.906448
\(436\) −3.09481 −0.148215
\(437\) 35.4237 1.69455
\(438\) 2.85779 0.136551
\(439\) 13.5722 0.647765 0.323882 0.946097i \(-0.395012\pi\)
0.323882 + 0.946097i \(0.395012\pi\)
\(440\) 11.0599 0.527262
\(441\) −6.31964 −0.300935
\(442\) 4.88498 0.232355
\(443\) 3.04113 0.144488 0.0722441 0.997387i \(-0.476984\pi\)
0.0722441 + 0.997387i \(0.476984\pi\)
\(444\) −4.98543 −0.236598
\(445\) −41.7370 −1.97853
\(446\) −0.973709 −0.0461064
\(447\) −2.99240 −0.141536
\(448\) 0.824838 0.0389700
\(449\) −14.6556 −0.691641 −0.345821 0.938301i \(-0.612399\pi\)
−0.345821 + 0.938301i \(0.612399\pi\)
\(450\) 2.58202 0.121718
\(451\) 34.4220 1.62087
\(452\) 14.6484 0.689004
\(453\) 19.8765 0.933881
\(454\) 25.0223 1.17436
\(455\) 2.27123 0.106477
\(456\) −5.85873 −0.274360
\(457\) 19.5597 0.914966 0.457483 0.889218i \(-0.348751\pi\)
0.457483 + 0.889218i \(0.348751\pi\)
\(458\) 20.2628 0.946817
\(459\) 4.88498 0.228012
\(460\) −16.6488 −0.776254
\(461\) 6.48694 0.302127 0.151063 0.988524i \(-0.451730\pi\)
0.151063 + 0.988524i \(0.451730\pi\)
\(462\) 3.31306 0.154138
\(463\) 30.7177 1.42757 0.713787 0.700362i \(-0.246978\pi\)
0.713787 + 0.700362i \(0.246978\pi\)
\(464\) −6.86586 −0.318740
\(465\) 2.47336 0.114699
\(466\) −10.4040 −0.481956
\(467\) 22.1129 1.02327 0.511633 0.859204i \(-0.329041\pi\)
0.511633 + 0.859204i \(0.329041\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 8.21444 0.379307
\(470\) 33.0183 1.52302
\(471\) 5.48855 0.252899
\(472\) −2.07171 −0.0953581
\(473\) −50.2740 −2.31160
\(474\) −8.39757 −0.385713
\(475\) 15.1273 0.694090
\(476\) −4.02932 −0.184684
\(477\) −6.44142 −0.294932
\(478\) 26.4940 1.21181
\(479\) 28.0901 1.28347 0.641734 0.766927i \(-0.278215\pi\)
0.641734 + 0.766927i \(0.278215\pi\)
\(480\) 2.75355 0.125682
\(481\) −4.98543 −0.227316
\(482\) 1.31680 0.0599788
\(483\) −4.98723 −0.226927
\(484\) 5.13324 0.233329
\(485\) −16.8698 −0.766020
\(486\) −1.00000 −0.0453609
\(487\) −43.1659 −1.95603 −0.978016 0.208529i \(-0.933133\pi\)
−0.978016 + 0.208529i \(0.933133\pi\)
\(488\) −12.3179 −0.557603
\(489\) −7.06211 −0.319360
\(490\) 17.4014 0.786116
\(491\) 4.59882 0.207542 0.103771 0.994601i \(-0.466909\pi\)
0.103771 + 0.994601i \(0.466909\pi\)
\(492\) 8.56990 0.386361
\(493\) 33.5396 1.51055
\(494\) −5.85873 −0.263597
\(495\) 11.0599 0.497108
\(496\) 0.898245 0.0403324
\(497\) 4.31860 0.193716
\(498\) 5.21802 0.233825
\(499\) 14.7370 0.659717 0.329858 0.944030i \(-0.392999\pi\)
0.329858 + 0.944030i \(0.392999\pi\)
\(500\) 6.65802 0.297756
\(501\) −16.9953 −0.759293
\(502\) −13.8996 −0.620371
\(503\) 34.2400 1.52669 0.763343 0.645994i \(-0.223557\pi\)
0.763343 + 0.645994i \(0.223557\pi\)
\(504\) 0.824838 0.0367412
\(505\) 48.3594 2.15196
\(506\) −24.2857 −1.07963
\(507\) −1.00000 −0.0444116
\(508\) 14.5076 0.643673
\(509\) −13.8444 −0.613640 −0.306820 0.951767i \(-0.599265\pi\)
−0.306820 + 0.951767i \(0.599265\pi\)
\(510\) −13.4510 −0.595622
\(511\) −2.35722 −0.104277
\(512\) 1.00000 0.0441942
\(513\) −5.85873 −0.258669
\(514\) 22.3217 0.984566
\(515\) 2.75355 0.121336
\(516\) −12.5165 −0.551008
\(517\) 48.1641 2.11826
\(518\) 4.11217 0.180679
\(519\) −4.34169 −0.190579
\(520\) 2.75355 0.120751
\(521\) −37.9825 −1.66404 −0.832021 0.554744i \(-0.812816\pi\)
−0.832021 + 0.554744i \(0.812816\pi\)
\(522\) −6.86586 −0.300511
\(523\) −44.5784 −1.94928 −0.974639 0.223784i \(-0.928159\pi\)
−0.974639 + 0.223784i \(0.928159\pi\)
\(524\) 13.3775 0.584399
\(525\) −2.12975 −0.0929498
\(526\) −16.1592 −0.704576
\(527\) −4.38791 −0.191141
\(528\) 4.01662 0.174801
\(529\) 13.5579 0.589473
\(530\) 17.7368 0.770435
\(531\) −2.07171 −0.0899045
\(532\) 4.83251 0.209516
\(533\) 8.56990 0.371204
\(534\) −15.1576 −0.655932
\(535\) −55.9011 −2.41682
\(536\) 9.95884 0.430156
\(537\) −17.0414 −0.735389
\(538\) 6.48600 0.279631
\(539\) 25.3836 1.09335
\(540\) 2.75355 0.118494
\(541\) 31.0027 1.33291 0.666455 0.745545i \(-0.267811\pi\)
0.666455 + 0.745545i \(0.267811\pi\)
\(542\) −25.7060 −1.10417
\(543\) −17.2522 −0.740362
\(544\) −4.88498 −0.209442
\(545\) 8.52171 0.365030
\(546\) 0.824838 0.0352998
\(547\) −40.7446 −1.74211 −0.871056 0.491184i \(-0.836564\pi\)
−0.871056 + 0.491184i \(0.836564\pi\)
\(548\) −3.65086 −0.155957
\(549\) −12.3179 −0.525713
\(550\) −10.3710 −0.442220
\(551\) −40.2252 −1.71365
\(552\) −6.04631 −0.257348
\(553\) 6.92664 0.294551
\(554\) −16.0917 −0.683670
\(555\) 13.7276 0.582705
\(556\) 14.9078 0.632232
\(557\) −13.2768 −0.562555 −0.281277 0.959627i \(-0.590758\pi\)
−0.281277 + 0.959627i \(0.590758\pi\)
\(558\) 0.898245 0.0380258
\(559\) −12.5165 −0.529391
\(560\) −2.27123 −0.0959770
\(561\) −19.6211 −0.828404
\(562\) 0.0961083 0.00405408
\(563\) −16.6142 −0.700207 −0.350103 0.936711i \(-0.613854\pi\)
−0.350103 + 0.936711i \(0.613854\pi\)
\(564\) 11.9912 0.504921
\(565\) −40.3351 −1.69691
\(566\) 5.39853 0.226917
\(567\) 0.824838 0.0346400
\(568\) 5.23569 0.219685
\(569\) −12.6190 −0.529018 −0.264509 0.964383i \(-0.585210\pi\)
−0.264509 + 0.964383i \(0.585210\pi\)
\(570\) 16.1323 0.675707
\(571\) 1.19760 0.0501178 0.0250589 0.999686i \(-0.492023\pi\)
0.0250589 + 0.999686i \(0.492023\pi\)
\(572\) 4.01662 0.167943
\(573\) 16.2534 0.678997
\(574\) −7.06879 −0.295045
\(575\) 15.6117 0.651052
\(576\) 1.00000 0.0416667
\(577\) −0.301179 −0.0125382 −0.00626912 0.999980i \(-0.501996\pi\)
−0.00626912 + 0.999980i \(0.501996\pi\)
\(578\) 6.86306 0.285466
\(579\) 14.9122 0.619731
\(580\) 18.9055 0.785007
\(581\) −4.30402 −0.178561
\(582\) −6.12659 −0.253955
\(583\) 25.8727 1.07154
\(584\) −2.85779 −0.118256
\(585\) 2.75355 0.113845
\(586\) 22.2615 0.919616
\(587\) 21.8124 0.900294 0.450147 0.892954i \(-0.351372\pi\)
0.450147 + 0.892954i \(0.351372\pi\)
\(588\) 6.31964 0.260618
\(589\) 5.26258 0.216841
\(590\) 5.70455 0.234852
\(591\) −17.5228 −0.720792
\(592\) 4.98543 0.204900
\(593\) −5.08535 −0.208830 −0.104415 0.994534i \(-0.533297\pi\)
−0.104415 + 0.994534i \(0.533297\pi\)
\(594\) 4.01662 0.164804
\(595\) 11.0949 0.454848
\(596\) 2.99240 0.122573
\(597\) 8.66641 0.354693
\(598\) −6.04631 −0.247252
\(599\) 37.2656 1.52263 0.761315 0.648382i \(-0.224554\pi\)
0.761315 + 0.648382i \(0.224554\pi\)
\(600\) −2.58202 −0.105410
\(601\) 39.5448 1.61306 0.806532 0.591190i \(-0.201342\pi\)
0.806532 + 0.591190i \(0.201342\pi\)
\(602\) 10.3241 0.420778
\(603\) 9.95884 0.405555
\(604\) −19.8765 −0.808765
\(605\) −14.1346 −0.574653
\(606\) 17.5626 0.713431
\(607\) 2.47845 0.100597 0.0502986 0.998734i \(-0.483983\pi\)
0.0502986 + 0.998734i \(0.483983\pi\)
\(608\) 5.85873 0.237603
\(609\) 5.66322 0.229485
\(610\) 33.9178 1.37329
\(611\) 11.9912 0.485112
\(612\) −4.88498 −0.197464
\(613\) 17.3008 0.698774 0.349387 0.936978i \(-0.386390\pi\)
0.349387 + 0.936978i \(0.386390\pi\)
\(614\) −2.18512 −0.0881842
\(615\) −23.5976 −0.951548
\(616\) −3.31306 −0.133487
\(617\) 26.0657 1.04937 0.524683 0.851298i \(-0.324184\pi\)
0.524683 + 0.851298i \(0.324184\pi\)
\(618\) 1.00000 0.0402259
\(619\) −20.0939 −0.807641 −0.403821 0.914838i \(-0.632318\pi\)
−0.403821 + 0.914838i \(0.632318\pi\)
\(620\) −2.47336 −0.0993326
\(621\) −6.04631 −0.242630
\(622\) −0.0801534 −0.00321386
\(623\) 12.5025 0.500904
\(624\) 1.00000 0.0400320
\(625\) −31.2433 −1.24973
\(626\) 9.05156 0.361773
\(627\) 23.5323 0.939789
\(628\) −5.48855 −0.219017
\(629\) −24.3537 −0.971047
\(630\) −2.27123 −0.0904880
\(631\) 45.0779 1.79452 0.897261 0.441501i \(-0.145554\pi\)
0.897261 + 0.441501i \(0.145554\pi\)
\(632\) 8.39757 0.334037
\(633\) 7.82353 0.310957
\(634\) 4.18981 0.166399
\(635\) −39.9475 −1.58527
\(636\) 6.44142 0.255419
\(637\) 6.31964 0.250393
\(638\) 27.5775 1.09181
\(639\) 5.23569 0.207121
\(640\) −2.75355 −0.108843
\(641\) 5.02160 0.198341 0.0991707 0.995070i \(-0.468381\pi\)
0.0991707 + 0.995070i \(0.468381\pi\)
\(642\) −20.3015 −0.801236
\(643\) 25.4848 1.00502 0.502511 0.864571i \(-0.332410\pi\)
0.502511 + 0.864571i \(0.332410\pi\)
\(644\) 4.98723 0.196524
\(645\) 34.4647 1.35705
\(646\) −28.6198 −1.12603
\(647\) 15.6812 0.616492 0.308246 0.951307i \(-0.400258\pi\)
0.308246 + 0.951307i \(0.400258\pi\)
\(648\) 1.00000 0.0392837
\(649\) 8.32127 0.326638
\(650\) −2.58202 −0.101275
\(651\) −0.740907 −0.0290384
\(652\) 7.06211 0.276574
\(653\) −19.0441 −0.745255 −0.372627 0.927981i \(-0.621543\pi\)
−0.372627 + 0.927981i \(0.621543\pi\)
\(654\) 3.09481 0.121017
\(655\) −36.8356 −1.43928
\(656\) −8.56990 −0.334599
\(657\) −2.85779 −0.111493
\(658\) −9.89081 −0.385584
\(659\) 28.4790 1.10938 0.554692 0.832056i \(-0.312836\pi\)
0.554692 + 0.832056i \(0.312836\pi\)
\(660\) −11.0599 −0.430508
\(661\) −36.2555 −1.41017 −0.705087 0.709121i \(-0.749092\pi\)
−0.705087 + 0.709121i \(0.749092\pi\)
\(662\) 18.0911 0.703131
\(663\) −4.88498 −0.189717
\(664\) −5.21802 −0.202498
\(665\) −13.3065 −0.516005
\(666\) 4.98543 0.193181
\(667\) −41.5131 −1.60739
\(668\) 16.9953 0.657567
\(669\) 0.973709 0.0376458
\(670\) −27.4221 −1.05941
\(671\) 49.4761 1.91001
\(672\) −0.824838 −0.0318188
\(673\) 19.2441 0.741804 0.370902 0.928672i \(-0.379049\pi\)
0.370902 + 0.928672i \(0.379049\pi\)
\(674\) −2.00641 −0.0772839
\(675\) −2.58202 −0.0993819
\(676\) 1.00000 0.0384615
\(677\) −19.9685 −0.767453 −0.383726 0.923447i \(-0.625359\pi\)
−0.383726 + 0.923447i \(0.625359\pi\)
\(678\) −14.6484 −0.562569
\(679\) 5.05345 0.193933
\(680\) 13.4510 0.515823
\(681\) −25.0223 −0.958858
\(682\) −3.60791 −0.138154
\(683\) −34.1733 −1.30761 −0.653803 0.756665i \(-0.726828\pi\)
−0.653803 + 0.756665i \(0.726828\pi\)
\(684\) 5.85873 0.224014
\(685\) 10.0528 0.384098
\(686\) −10.9866 −0.419469
\(687\) −20.2628 −0.773072
\(688\) 12.5165 0.477187
\(689\) 6.44142 0.245399
\(690\) 16.6488 0.633809
\(691\) −1.84750 −0.0702824 −0.0351412 0.999382i \(-0.511188\pi\)
−0.0351412 + 0.999382i \(0.511188\pi\)
\(692\) 4.34169 0.165046
\(693\) −3.31306 −0.125853
\(694\) −3.06551 −0.116365
\(695\) −41.0493 −1.55709
\(696\) 6.86586 0.260250
\(697\) 41.8638 1.58571
\(698\) 11.4089 0.431832
\(699\) 10.4040 0.393516
\(700\) 2.12975 0.0804969
\(701\) 10.1157 0.382065 0.191033 0.981584i \(-0.438816\pi\)
0.191033 + 0.981584i \(0.438816\pi\)
\(702\) 1.00000 0.0377426
\(703\) 29.2083 1.10161
\(704\) −4.01662 −0.151382
\(705\) −33.0183 −1.24354
\(706\) −27.4910 −1.03464
\(707\) −14.4863 −0.544813
\(708\) 2.07171 0.0778596
\(709\) −10.8613 −0.407906 −0.203953 0.978981i \(-0.565379\pi\)
−0.203953 + 0.978981i \(0.565379\pi\)
\(710\) −14.4167 −0.541050
\(711\) 8.39757 0.314933
\(712\) 15.1576 0.568054
\(713\) 5.43107 0.203395
\(714\) 4.02932 0.150794
\(715\) −11.0599 −0.413619
\(716\) 17.0414 0.636865
\(717\) −26.4940 −0.989435
\(718\) −12.5262 −0.467473
\(719\) −3.69888 −0.137945 −0.0689725 0.997619i \(-0.521972\pi\)
−0.0689725 + 0.997619i \(0.521972\pi\)
\(720\) −2.75355 −0.102619
\(721\) −0.824838 −0.0307186
\(722\) 15.3247 0.570327
\(723\) −1.31680 −0.0489725
\(724\) 17.2522 0.641172
\(725\) −17.7278 −0.658393
\(726\) −5.13324 −0.190512
\(727\) 47.4382 1.75938 0.879692 0.475543i \(-0.157749\pi\)
0.879692 + 0.475543i \(0.157749\pi\)
\(728\) −0.824838 −0.0305705
\(729\) 1.00000 0.0370370
\(730\) 7.86906 0.291247
\(731\) −61.1428 −2.26145
\(732\) 12.3179 0.455281
\(733\) 42.9033 1.58467 0.792335 0.610086i \(-0.208865\pi\)
0.792335 + 0.610086i \(0.208865\pi\)
\(734\) 21.5573 0.795696
\(735\) −17.4014 −0.641861
\(736\) 6.04631 0.222870
\(737\) −40.0009 −1.47345
\(738\) −8.56990 −0.315463
\(739\) −18.5493 −0.682348 −0.341174 0.940000i \(-0.610825\pi\)
−0.341174 + 0.940000i \(0.610825\pi\)
\(740\) −13.7276 −0.504637
\(741\) 5.85873 0.215226
\(742\) −5.31313 −0.195051
\(743\) 1.24993 0.0458556 0.0229278 0.999737i \(-0.492701\pi\)
0.0229278 + 0.999737i \(0.492701\pi\)
\(744\) −0.898245 −0.0329313
\(745\) −8.23971 −0.301879
\(746\) 34.6777 1.26964
\(747\) −5.21802 −0.190917
\(748\) 19.6211 0.717419
\(749\) 16.7455 0.611866
\(750\) −6.65802 −0.243117
\(751\) 6.97426 0.254494 0.127247 0.991871i \(-0.459386\pi\)
0.127247 + 0.991871i \(0.459386\pi\)
\(752\) −11.9912 −0.437274
\(753\) 13.8996 0.506531
\(754\) 6.86586 0.250040
\(755\) 54.7310 1.99186
\(756\) −0.824838 −0.0299991
\(757\) 2.68607 0.0976270 0.0488135 0.998808i \(-0.484456\pi\)
0.0488135 + 0.998808i \(0.484456\pi\)
\(758\) −10.8080 −0.392566
\(759\) 24.2857 0.881516
\(760\) −16.1323 −0.585179
\(761\) −31.4888 −1.14147 −0.570734 0.821135i \(-0.693341\pi\)
−0.570734 + 0.821135i \(0.693341\pi\)
\(762\) −14.5076 −0.525557
\(763\) −2.55272 −0.0924147
\(764\) −16.2534 −0.588029
\(765\) 13.4510 0.486323
\(766\) 2.54440 0.0919327
\(767\) 2.07171 0.0748051
\(768\) −1.00000 −0.0360844
\(769\) −39.1019 −1.41005 −0.705025 0.709182i \(-0.749064\pi\)
−0.705025 + 0.709182i \(0.749064\pi\)
\(770\) 9.12267 0.328758
\(771\) −22.3217 −0.803895
\(772\) −14.9122 −0.536703
\(773\) 29.1520 1.04852 0.524262 0.851557i \(-0.324341\pi\)
0.524262 + 0.851557i \(0.324341\pi\)
\(774\) 12.5165 0.449896
\(775\) 2.31929 0.0833112
\(776\) 6.12659 0.219932
\(777\) −4.11217 −0.147523
\(778\) 18.6438 0.668412
\(779\) −50.2088 −1.79892
\(780\) −2.75355 −0.0985928
\(781\) −21.0298 −0.752505
\(782\) −29.5361 −1.05621
\(783\) 6.86586 0.245366
\(784\) −6.31964 −0.225701
\(785\) 15.1130 0.539405
\(786\) −13.3775 −0.477160
\(787\) 37.0525 1.32078 0.660389 0.750924i \(-0.270391\pi\)
0.660389 + 0.750924i \(0.270391\pi\)
\(788\) 17.5228 0.624224
\(789\) 16.1592 0.575284
\(790\) −23.1231 −0.822683
\(791\) 12.0826 0.429607
\(792\) −4.01662 −0.142724
\(793\) 12.3179 0.437420
\(794\) −12.1568 −0.431429
\(795\) −17.7368 −0.629058
\(796\) −8.66641 −0.307173
\(797\) −28.2779 −1.00166 −0.500828 0.865547i \(-0.666971\pi\)
−0.500828 + 0.865547i \(0.666971\pi\)
\(798\) −4.83251 −0.171069
\(799\) 58.5768 2.07230
\(800\) 2.58202 0.0912881
\(801\) 15.1576 0.535566
\(802\) 17.1721 0.606370
\(803\) 11.4787 0.405073
\(804\) −9.95884 −0.351221
\(805\) −13.7326 −0.484009
\(806\) −0.898245 −0.0316393
\(807\) −6.48600 −0.228318
\(808\) −17.5626 −0.617849
\(809\) 15.9799 0.561822 0.280911 0.959734i \(-0.409363\pi\)
0.280911 + 0.959734i \(0.409363\pi\)
\(810\) −2.75355 −0.0967498
\(811\) −37.0996 −1.30274 −0.651372 0.758758i \(-0.725806\pi\)
−0.651372 + 0.758758i \(0.725806\pi\)
\(812\) −5.66322 −0.198740
\(813\) 25.7060 0.901548
\(814\) −20.0246 −0.701861
\(815\) −19.4459 −0.681158
\(816\) 4.88498 0.171009
\(817\) 73.3307 2.56552
\(818\) −2.80065 −0.0979223
\(819\) −0.824838 −0.0288222
\(820\) 23.5976 0.824065
\(821\) −3.11002 −0.108541 −0.0542703 0.998526i \(-0.517283\pi\)
−0.0542703 + 0.998526i \(0.517283\pi\)
\(822\) 3.65086 0.127338
\(823\) −26.3115 −0.917162 −0.458581 0.888653i \(-0.651642\pi\)
−0.458581 + 0.888653i \(0.651642\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 10.3710 0.361071
\(826\) −1.70882 −0.0594576
\(827\) −36.0829 −1.25472 −0.627362 0.778727i \(-0.715865\pi\)
−0.627362 + 0.778727i \(0.715865\pi\)
\(828\) 6.04631 0.210124
\(829\) −0.0737644 −0.00256194 −0.00128097 0.999999i \(-0.500408\pi\)
−0.00128097 + 0.999999i \(0.500408\pi\)
\(830\) 14.3681 0.498723
\(831\) 16.0917 0.558214
\(832\) −1.00000 −0.0346688
\(833\) 30.8713 1.06963
\(834\) −14.9078 −0.516215
\(835\) −46.7973 −1.61949
\(836\) −23.5323 −0.813881
\(837\) −0.898245 −0.0310479
\(838\) 40.3556 1.39406
\(839\) −15.1825 −0.524160 −0.262080 0.965046i \(-0.584408\pi\)
−0.262080 + 0.965046i \(0.584408\pi\)
\(840\) 2.27123 0.0783649
\(841\) 18.1400 0.625518
\(842\) −15.1113 −0.520771
\(843\) −0.0961083 −0.00331014
\(844\) −7.82353 −0.269297
\(845\) −2.75355 −0.0947249
\(846\) −11.9912 −0.412266
\(847\) 4.23409 0.145485
\(848\) −6.44142 −0.221199
\(849\) −5.39853 −0.185277
\(850\) −12.6131 −0.432626
\(851\) 30.1435 1.03330
\(852\) −5.23569 −0.179372
\(853\) −44.9003 −1.53736 −0.768679 0.639634i \(-0.779086\pi\)
−0.768679 + 0.639634i \(0.779086\pi\)
\(854\) −10.1602 −0.347676
\(855\) −16.1323 −0.551712
\(856\) 20.3015 0.693891
\(857\) 27.5661 0.941641 0.470821 0.882229i \(-0.343958\pi\)
0.470821 + 0.882229i \(0.343958\pi\)
\(858\) −4.01662 −0.137125
\(859\) −17.7050 −0.604087 −0.302044 0.953294i \(-0.597669\pi\)
−0.302044 + 0.953294i \(0.597669\pi\)
\(860\) −34.4647 −1.17524
\(861\) 7.06879 0.240904
\(862\) 23.8797 0.813345
\(863\) −28.7685 −0.979291 −0.489645 0.871922i \(-0.662874\pi\)
−0.489645 + 0.871922i \(0.662874\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −11.9551 −0.406484
\(866\) −22.2116 −0.754782
\(867\) −6.86306 −0.233082
\(868\) 0.740907 0.0251480
\(869\) −33.7298 −1.14421
\(870\) −18.9055 −0.640955
\(871\) −9.95884 −0.337443
\(872\) −3.09481 −0.104804
\(873\) 6.12659 0.207354
\(874\) 35.4237 1.19822
\(875\) 5.49179 0.185657
\(876\) 2.85779 0.0965558
\(877\) 0.0975824 0.00329512 0.00164756 0.999999i \(-0.499476\pi\)
0.00164756 + 0.999999i \(0.499476\pi\)
\(878\) 13.5722 0.458039
\(879\) −22.2615 −0.750863
\(880\) 11.0599 0.372831
\(881\) 27.4355 0.924325 0.462163 0.886795i \(-0.347074\pi\)
0.462163 + 0.886795i \(0.347074\pi\)
\(882\) −6.31964 −0.212793
\(883\) 6.74837 0.227101 0.113550 0.993532i \(-0.463778\pi\)
0.113550 + 0.993532i \(0.463778\pi\)
\(884\) 4.88498 0.164300
\(885\) −5.70455 −0.191756
\(886\) 3.04113 0.102169
\(887\) −53.6082 −1.79999 −0.899993 0.435904i \(-0.856429\pi\)
−0.899993 + 0.435904i \(0.856429\pi\)
\(888\) −4.98543 −0.167300
\(889\) 11.9665 0.401342
\(890\) −41.7370 −1.39903
\(891\) −4.01662 −0.134562
\(892\) −0.973709 −0.0326022
\(893\) −70.2532 −2.35094
\(894\) −2.99240 −0.100081
\(895\) −46.9242 −1.56850
\(896\) 0.824838 0.0275559
\(897\) 6.04631 0.201880
\(898\) −14.6556 −0.489064
\(899\) −6.16723 −0.205689
\(900\) 2.58202 0.0860673
\(901\) 31.4662 1.04829
\(902\) 34.4220 1.14613
\(903\) −10.3241 −0.343564
\(904\) 14.6484 0.487199
\(905\) −47.5047 −1.57911
\(906\) 19.8765 0.660354
\(907\) 14.6663 0.486987 0.243494 0.969903i \(-0.421707\pi\)
0.243494 + 0.969903i \(0.421707\pi\)
\(908\) 25.0223 0.830395
\(909\) −17.5626 −0.582514
\(910\) 2.27123 0.0752906
\(911\) −15.7552 −0.521992 −0.260996 0.965340i \(-0.584051\pi\)
−0.260996 + 0.965340i \(0.584051\pi\)
\(912\) −5.85873 −0.194002
\(913\) 20.9588 0.693635
\(914\) 19.5597 0.646979
\(915\) −33.9178 −1.12129
\(916\) 20.2628 0.669500
\(917\) 11.0343 0.364384
\(918\) 4.88498 0.161228
\(919\) −12.0221 −0.396574 −0.198287 0.980144i \(-0.563538\pi\)
−0.198287 + 0.980144i \(0.563538\pi\)
\(920\) −16.6488 −0.548895
\(921\) 2.18512 0.0720021
\(922\) 6.48694 0.213636
\(923\) −5.23569 −0.172335
\(924\) 3.31306 0.108992
\(925\) 12.8725 0.423244
\(926\) 30.7177 1.00945
\(927\) −1.00000 −0.0328443
\(928\) −6.86586 −0.225383
\(929\) 23.3913 0.767444 0.383722 0.923449i \(-0.374642\pi\)
0.383722 + 0.923449i \(0.374642\pi\)
\(930\) 2.47336 0.0811047
\(931\) −37.0251 −1.21345
\(932\) −10.4040 −0.340795
\(933\) 0.0801534 0.00262411
\(934\) 22.1129 0.723558
\(935\) −54.0277 −1.76689
\(936\) −1.00000 −0.0326860
\(937\) −56.1198 −1.83336 −0.916678 0.399627i \(-0.869140\pi\)
−0.916678 + 0.399627i \(0.869140\pi\)
\(938\) 8.21444 0.268211
\(939\) −9.05156 −0.295387
\(940\) 33.0183 1.07694
\(941\) 3.75419 0.122383 0.0611915 0.998126i \(-0.480510\pi\)
0.0611915 + 0.998126i \(0.480510\pi\)
\(942\) 5.48855 0.178827
\(943\) −51.8163 −1.68737
\(944\) −2.07171 −0.0674284
\(945\) 2.27123 0.0738831
\(946\) −50.2740 −1.63455
\(947\) 48.9388 1.59030 0.795149 0.606414i \(-0.207392\pi\)
0.795149 + 0.606414i \(0.207392\pi\)
\(948\) −8.39757 −0.272740
\(949\) 2.85779 0.0927678
\(950\) 15.1273 0.490796
\(951\) −4.18981 −0.135864
\(952\) −4.02932 −0.130591
\(953\) −56.1519 −1.81894 −0.909469 0.415772i \(-0.863512\pi\)
−0.909469 + 0.415772i \(0.863512\pi\)
\(954\) −6.44142 −0.208549
\(955\) 44.7546 1.44823
\(956\) 26.4940 0.856876
\(957\) −27.5775 −0.891456
\(958\) 28.0901 0.907549
\(959\) −3.01137 −0.0972421
\(960\) 2.75355 0.0888703
\(961\) −30.1932 −0.973973
\(962\) −4.98543 −0.160737
\(963\) 20.3015 0.654207
\(964\) 1.31680 0.0424114
\(965\) 41.0615 1.32182
\(966\) −4.98723 −0.160461
\(967\) −26.4792 −0.851514 −0.425757 0.904837i \(-0.639992\pi\)
−0.425757 + 0.904837i \(0.639992\pi\)
\(968\) 5.13324 0.164988
\(969\) 28.6198 0.919400
\(970\) −16.8698 −0.541658
\(971\) −32.1713 −1.03243 −0.516213 0.856460i \(-0.672659\pi\)
−0.516213 + 0.856460i \(0.672659\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 12.2965 0.394209
\(974\) −43.1659 −1.38312
\(975\) 2.58202 0.0826908
\(976\) −12.3179 −0.394285
\(977\) −27.8028 −0.889490 −0.444745 0.895657i \(-0.646706\pi\)
−0.444745 + 0.895657i \(0.646706\pi\)
\(978\) −7.06211 −0.225821
\(979\) −60.8822 −1.94580
\(980\) 17.4014 0.555868
\(981\) −3.09481 −0.0988098
\(982\) 4.59882 0.146754
\(983\) −13.9383 −0.444561 −0.222281 0.974983i \(-0.571350\pi\)
−0.222281 + 0.974983i \(0.571350\pi\)
\(984\) 8.56990 0.273199
\(985\) −48.2499 −1.53737
\(986\) 33.5396 1.06812
\(987\) 9.89081 0.314828
\(988\) −5.85873 −0.186391
\(989\) 75.6786 2.40644
\(990\) 11.0599 0.351508
\(991\) −16.6732 −0.529642 −0.264821 0.964298i \(-0.585313\pi\)
−0.264821 + 0.964298i \(0.585313\pi\)
\(992\) 0.898245 0.0285193
\(993\) −18.0911 −0.574104
\(994\) 4.31860 0.136978
\(995\) 23.8634 0.756520
\(996\) 5.21802 0.165339
\(997\) −30.0597 −0.952000 −0.476000 0.879445i \(-0.657914\pi\)
−0.476000 + 0.879445i \(0.657914\pi\)
\(998\) 14.7370 0.466490
\(999\) −4.98543 −0.157732
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 8034.2.a.bc.1.3 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bc.1.3 15 1.1 even 1 trivial