Properties

Label 3381.2.a.bc.1.3
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.7997584.1
Defining polynomial: \(x^{6} - x^{5} - 7 x^{4} + 5 x^{3} + 12 x^{2} - 4 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.593528\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.593528 q^{2} -1.00000 q^{3} -1.64772 q^{4} -3.15120 q^{5} +0.593528 q^{6} +2.16503 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.593528 q^{2} -1.00000 q^{3} -1.64772 q^{4} -3.15120 q^{5} +0.593528 q^{6} +2.16503 q^{8} +1.00000 q^{9} +1.87033 q^{10} -6.54534 q^{11} +1.64772 q^{12} -0.692965 q^{13} +3.15120 q^{15} +2.01044 q^{16} +6.84417 q^{17} -0.593528 q^{18} -2.91662 q^{19} +5.19231 q^{20} +3.88485 q^{22} -1.00000 q^{23} -2.16503 q^{24} +4.93009 q^{25} +0.411295 q^{26} -1.00000 q^{27} +2.94619 q^{29} -1.87033 q^{30} -0.539333 q^{31} -5.52331 q^{32} +6.54534 q^{33} -4.06221 q^{34} -1.64772 q^{36} -2.89613 q^{37} +1.73110 q^{38} +0.692965 q^{39} -6.82244 q^{40} +9.26871 q^{41} -4.27043 q^{43} +10.7849 q^{44} -3.15120 q^{45} +0.593528 q^{46} +10.4674 q^{47} -2.01044 q^{48} -2.92615 q^{50} -6.84417 q^{51} +1.14182 q^{52} +12.5749 q^{53} +0.593528 q^{54} +20.6257 q^{55} +2.91662 q^{57} -1.74865 q^{58} +6.70326 q^{59} -5.19231 q^{60} +5.30183 q^{61} +0.320109 q^{62} -0.742642 q^{64} +2.18368 q^{65} -3.88485 q^{66} -2.73628 q^{67} -11.2773 q^{68} +1.00000 q^{69} -0.587217 q^{71} +2.16503 q^{72} +10.5266 q^{73} +1.71893 q^{74} -4.93009 q^{75} +4.80579 q^{76} -0.411295 q^{78} -13.3870 q^{79} -6.33531 q^{80} +1.00000 q^{81} -5.50124 q^{82} -9.77501 q^{83} -21.5674 q^{85} +2.53462 q^{86} -2.94619 q^{87} -14.1708 q^{88} +2.92939 q^{89} +1.87033 q^{90} +1.64772 q^{92} +0.539333 q^{93} -6.21272 q^{94} +9.19088 q^{95} +5.52331 q^{96} -8.14170 q^{97} -6.54534 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{2} - 6q^{3} + 3q^{4} + 3q^{5} + q^{6} - 3q^{8} + 6q^{9} + O(q^{10}) \) \( 6q - q^{2} - 6q^{3} + 3q^{4} + 3q^{5} + q^{6} - 3q^{8} + 6q^{9} - 3q^{10} - 14q^{11} - 3q^{12} - 3q^{15} - 7q^{16} + 15q^{17} - q^{18} - q^{19} + 17q^{20} - 6q^{22} - 6q^{23} + 3q^{24} + 9q^{25} - 15q^{26} - 6q^{27} - 6q^{29} + 3q^{30} - 11q^{31} + 3q^{32} + 14q^{33} + 15q^{34} + 3q^{36} - 5q^{37} + 14q^{38} - 17q^{40} + 18q^{41} - 37q^{43} - 10q^{44} + 3q^{45} + q^{46} + 3q^{47} + 7q^{48} - 30q^{50} - 15q^{51} - 7q^{52} - 15q^{53} + q^{54} - 2q^{55} + q^{57} + 4q^{58} - 2q^{59} - 17q^{60} - 12q^{61} + 36q^{62} - 23q^{64} - 17q^{65} + 6q^{66} - 10q^{67} - q^{68} + 6q^{69} - 21q^{71} - 3q^{72} - 8q^{73} - 16q^{74} - 9q^{75} + 18q^{76} + 15q^{78} - 17q^{79} + 3q^{80} + 6q^{81} - 48q^{82} + 12q^{83} - 13q^{85} + 22q^{86} + 6q^{87} - 2q^{88} + 18q^{89} - 3q^{90} - 3q^{92} + 11q^{93} + 3q^{94} - 16q^{95} - 3q^{96} - 2q^{97} - 14q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.593528 −0.419688 −0.209844 0.977735i \(-0.567296\pi\)
−0.209844 + 0.977735i \(0.567296\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.64772 −0.823862
\(5\) −3.15120 −1.40926 −0.704631 0.709574i \(-0.748887\pi\)
−0.704631 + 0.709574i \(0.748887\pi\)
\(6\) 0.593528 0.242307
\(7\) 0 0
\(8\) 2.16503 0.765453
\(9\) 1.00000 0.333333
\(10\) 1.87033 0.591450
\(11\) −6.54534 −1.97349 −0.986747 0.162266i \(-0.948120\pi\)
−0.986747 + 0.162266i \(0.948120\pi\)
\(12\) 1.64772 0.475657
\(13\) −0.692965 −0.192194 −0.0960970 0.995372i \(-0.530636\pi\)
−0.0960970 + 0.995372i \(0.530636\pi\)
\(14\) 0 0
\(15\) 3.15120 0.813637
\(16\) 2.01044 0.502611
\(17\) 6.84417 1.65995 0.829977 0.557797i \(-0.188353\pi\)
0.829977 + 0.557797i \(0.188353\pi\)
\(18\) −0.593528 −0.139896
\(19\) −2.91662 −0.669120 −0.334560 0.942375i \(-0.608588\pi\)
−0.334560 + 0.942375i \(0.608588\pi\)
\(20\) 5.19231 1.16104
\(21\) 0 0
\(22\) 3.88485 0.828252
\(23\) −1.00000 −0.208514
\(24\) −2.16503 −0.441934
\(25\) 4.93009 0.986017
\(26\) 0.411295 0.0806615
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.94619 0.547094 0.273547 0.961859i \(-0.411803\pi\)
0.273547 + 0.961859i \(0.411803\pi\)
\(30\) −1.87033 −0.341474
\(31\) −0.539333 −0.0968671 −0.0484335 0.998826i \(-0.515423\pi\)
−0.0484335 + 0.998826i \(0.515423\pi\)
\(32\) −5.52331 −0.976393
\(33\) 6.54534 1.13940
\(34\) −4.06221 −0.696663
\(35\) 0 0
\(36\) −1.64772 −0.274621
\(37\) −2.89613 −0.476120 −0.238060 0.971250i \(-0.576512\pi\)
−0.238060 + 0.971250i \(0.576512\pi\)
\(38\) 1.73110 0.280821
\(39\) 0.692965 0.110963
\(40\) −6.82244 −1.07872
\(41\) 9.26871 1.44753 0.723765 0.690047i \(-0.242410\pi\)
0.723765 + 0.690047i \(0.242410\pi\)
\(42\) 0 0
\(43\) −4.27043 −0.651235 −0.325617 0.945502i \(-0.605572\pi\)
−0.325617 + 0.945502i \(0.605572\pi\)
\(44\) 10.7849 1.62589
\(45\) −3.15120 −0.469754
\(46\) 0.593528 0.0875110
\(47\) 10.4674 1.52683 0.763416 0.645907i \(-0.223521\pi\)
0.763416 + 0.645907i \(0.223521\pi\)
\(48\) −2.01044 −0.290182
\(49\) 0 0
\(50\) −2.92615 −0.413820
\(51\) −6.84417 −0.958375
\(52\) 1.14182 0.158341
\(53\) 12.5749 1.72729 0.863647 0.504097i \(-0.168175\pi\)
0.863647 + 0.504097i \(0.168175\pi\)
\(54\) 0.593528 0.0807690
\(55\) 20.6257 2.78117
\(56\) 0 0
\(57\) 2.91662 0.386316
\(58\) −1.74865 −0.229609
\(59\) 6.70326 0.872690 0.436345 0.899779i \(-0.356273\pi\)
0.436345 + 0.899779i \(0.356273\pi\)
\(60\) −5.19231 −0.670325
\(61\) 5.30183 0.678830 0.339415 0.940637i \(-0.389771\pi\)
0.339415 + 0.940637i \(0.389771\pi\)
\(62\) 0.320109 0.0406539
\(63\) 0 0
\(64\) −0.742642 −0.0928303
\(65\) 2.18368 0.270852
\(66\) −3.88485 −0.478191
\(67\) −2.73628 −0.334290 −0.167145 0.985932i \(-0.553455\pi\)
−0.167145 + 0.985932i \(0.553455\pi\)
\(68\) −11.2773 −1.36757
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −0.587217 −0.0696899 −0.0348449 0.999393i \(-0.511094\pi\)
−0.0348449 + 0.999393i \(0.511094\pi\)
\(72\) 2.16503 0.255151
\(73\) 10.5266 1.23205 0.616023 0.787728i \(-0.288743\pi\)
0.616023 + 0.787728i \(0.288743\pi\)
\(74\) 1.71893 0.199822
\(75\) −4.93009 −0.569277
\(76\) 4.80579 0.551262
\(77\) 0 0
\(78\) −0.411295 −0.0465699
\(79\) −13.3870 −1.50616 −0.753078 0.657931i \(-0.771432\pi\)
−0.753078 + 0.657931i \(0.771432\pi\)
\(80\) −6.33531 −0.708310
\(81\) 1.00000 0.111111
\(82\) −5.50124 −0.607511
\(83\) −9.77501 −1.07295 −0.536473 0.843917i \(-0.680244\pi\)
−0.536473 + 0.843917i \(0.680244\pi\)
\(84\) 0 0
\(85\) −21.5674 −2.33931
\(86\) 2.53462 0.273315
\(87\) −2.94619 −0.315865
\(88\) −14.1708 −1.51062
\(89\) 2.92939 0.310515 0.155258 0.987874i \(-0.450379\pi\)
0.155258 + 0.987874i \(0.450379\pi\)
\(90\) 1.87033 0.197150
\(91\) 0 0
\(92\) 1.64772 0.171787
\(93\) 0.539333 0.0559262
\(94\) −6.21272 −0.640793
\(95\) 9.19088 0.942964
\(96\) 5.52331 0.563721
\(97\) −8.14170 −0.826664 −0.413332 0.910580i \(-0.635635\pi\)
−0.413332 + 0.910580i \(0.635635\pi\)
\(98\) 0 0
\(99\) −6.54534 −0.657831
\(100\) −8.12342 −0.812342
\(101\) −0.133249 −0.0132588 −0.00662939 0.999978i \(-0.502110\pi\)
−0.00662939 + 0.999978i \(0.502110\pi\)
\(102\) 4.06221 0.402219
\(103\) −11.0107 −1.08491 −0.542456 0.840084i \(-0.682506\pi\)
−0.542456 + 0.840084i \(0.682506\pi\)
\(104\) −1.50029 −0.147115
\(105\) 0 0
\(106\) −7.46355 −0.724924
\(107\) −8.57510 −0.828986 −0.414493 0.910053i \(-0.636041\pi\)
−0.414493 + 0.910053i \(0.636041\pi\)
\(108\) 1.64772 0.158552
\(109\) 11.2823 1.08064 0.540322 0.841458i \(-0.318302\pi\)
0.540322 + 0.841458i \(0.318302\pi\)
\(110\) −12.2419 −1.16722
\(111\) 2.89613 0.274888
\(112\) 0 0
\(113\) 3.14165 0.295541 0.147771 0.989022i \(-0.452790\pi\)
0.147771 + 0.989022i \(0.452790\pi\)
\(114\) −1.73110 −0.162132
\(115\) 3.15120 0.293851
\(116\) −4.85451 −0.450730
\(117\) −0.692965 −0.0640647
\(118\) −3.97858 −0.366258
\(119\) 0 0
\(120\) 6.82244 0.622801
\(121\) 31.8415 2.89468
\(122\) −3.14679 −0.284897
\(123\) −9.26871 −0.835731
\(124\) 0.888672 0.0798051
\(125\) 0.220311 0.0197052
\(126\) 0 0
\(127\) −12.9337 −1.14768 −0.573841 0.818967i \(-0.694547\pi\)
−0.573841 + 0.818967i \(0.694547\pi\)
\(128\) 11.4874 1.01535
\(129\) 4.27043 0.375991
\(130\) −1.29607 −0.113673
\(131\) 6.14667 0.537037 0.268518 0.963275i \(-0.413466\pi\)
0.268518 + 0.963275i \(0.413466\pi\)
\(132\) −10.7849 −0.938706
\(133\) 0 0
\(134\) 1.62406 0.140298
\(135\) 3.15120 0.271212
\(136\) 14.8178 1.27062
\(137\) −19.6122 −1.67558 −0.837792 0.545989i \(-0.816154\pi\)
−0.837792 + 0.545989i \(0.816154\pi\)
\(138\) −0.593528 −0.0505245
\(139\) −11.8453 −1.00470 −0.502352 0.864663i \(-0.667532\pi\)
−0.502352 + 0.864663i \(0.667532\pi\)
\(140\) 0 0
\(141\) −10.4674 −0.881517
\(142\) 0.348530 0.0292480
\(143\) 4.53569 0.379294
\(144\) 2.01044 0.167537
\(145\) −9.28405 −0.770999
\(146\) −6.24784 −0.517075
\(147\) 0 0
\(148\) 4.77202 0.392258
\(149\) −5.17374 −0.423849 −0.211925 0.977286i \(-0.567973\pi\)
−0.211925 + 0.977286i \(0.567973\pi\)
\(150\) 2.92615 0.238919
\(151\) −5.05014 −0.410974 −0.205487 0.978660i \(-0.565878\pi\)
−0.205487 + 0.978660i \(0.565878\pi\)
\(152\) −6.31457 −0.512180
\(153\) 6.84417 0.553318
\(154\) 0 0
\(155\) 1.69955 0.136511
\(156\) −1.14182 −0.0914184
\(157\) −9.64814 −0.770005 −0.385003 0.922915i \(-0.625799\pi\)
−0.385003 + 0.922915i \(0.625799\pi\)
\(158\) 7.94557 0.632116
\(159\) −12.5749 −0.997253
\(160\) 17.4051 1.37599
\(161\) 0 0
\(162\) −0.593528 −0.0466320
\(163\) 23.9325 1.87454 0.937271 0.348602i \(-0.113344\pi\)
0.937271 + 0.348602i \(0.113344\pi\)
\(164\) −15.2723 −1.19256
\(165\) −20.6257 −1.60571
\(166\) 5.80174 0.450303
\(167\) 12.0368 0.931435 0.465718 0.884933i \(-0.345796\pi\)
0.465718 + 0.884933i \(0.345796\pi\)
\(168\) 0 0
\(169\) −12.5198 −0.963061
\(170\) 12.8009 0.981780
\(171\) −2.91662 −0.223040
\(172\) 7.03649 0.536528
\(173\) −16.6038 −1.26236 −0.631182 0.775634i \(-0.717430\pi\)
−0.631182 + 0.775634i \(0.717430\pi\)
\(174\) 1.74865 0.132565
\(175\) 0 0
\(176\) −13.1590 −0.991899
\(177\) −6.70326 −0.503848
\(178\) −1.73868 −0.130320
\(179\) 3.24538 0.242571 0.121285 0.992618i \(-0.461298\pi\)
0.121285 + 0.992618i \(0.461298\pi\)
\(180\) 5.19231 0.387012
\(181\) 14.3797 1.06883 0.534417 0.845221i \(-0.320531\pi\)
0.534417 + 0.845221i \(0.320531\pi\)
\(182\) 0 0
\(183\) −5.30183 −0.391923
\(184\) −2.16503 −0.159608
\(185\) 9.12629 0.670978
\(186\) −0.320109 −0.0234716
\(187\) −44.7974 −3.27591
\(188\) −17.2474 −1.25790
\(189\) 0 0
\(190\) −5.45505 −0.395751
\(191\) −9.88912 −0.715552 −0.357776 0.933808i \(-0.616465\pi\)
−0.357776 + 0.933808i \(0.616465\pi\)
\(192\) 0.742642 0.0535956
\(193\) −3.59051 −0.258450 −0.129225 0.991615i \(-0.541249\pi\)
−0.129225 + 0.991615i \(0.541249\pi\)
\(194\) 4.83233 0.346941
\(195\) −2.18368 −0.156376
\(196\) 0 0
\(197\) 8.21279 0.585137 0.292569 0.956245i \(-0.405490\pi\)
0.292569 + 0.956245i \(0.405490\pi\)
\(198\) 3.88485 0.276084
\(199\) 2.24507 0.159149 0.0795743 0.996829i \(-0.474644\pi\)
0.0795743 + 0.996829i \(0.474644\pi\)
\(200\) 10.6738 0.754750
\(201\) 2.73628 0.193003
\(202\) 0.0790871 0.00556455
\(203\) 0 0
\(204\) 11.2773 0.789569
\(205\) −29.2076 −2.03995
\(206\) 6.53514 0.455325
\(207\) −1.00000 −0.0695048
\(208\) −1.39317 −0.0965987
\(209\) 19.0903 1.32050
\(210\) 0 0
\(211\) −23.0784 −1.58878 −0.794392 0.607406i \(-0.792210\pi\)
−0.794392 + 0.607406i \(0.792210\pi\)
\(212\) −20.7199 −1.42305
\(213\) 0.587217 0.0402355
\(214\) 5.08956 0.347915
\(215\) 13.4570 0.917760
\(216\) −2.16503 −0.147311
\(217\) 0 0
\(218\) −6.69634 −0.453533
\(219\) −10.5266 −0.711322
\(220\) −33.9855 −2.29130
\(221\) −4.74277 −0.319033
\(222\) −1.71893 −0.115367
\(223\) 4.28573 0.286993 0.143497 0.989651i \(-0.454165\pi\)
0.143497 + 0.989651i \(0.454165\pi\)
\(224\) 0 0
\(225\) 4.93009 0.328672
\(226\) −1.86466 −0.124035
\(227\) 10.9334 0.725673 0.362837 0.931853i \(-0.381808\pi\)
0.362837 + 0.931853i \(0.381808\pi\)
\(228\) −4.80579 −0.318271
\(229\) 5.57751 0.368572 0.184286 0.982873i \(-0.441003\pi\)
0.184286 + 0.982873i \(0.441003\pi\)
\(230\) −1.87033 −0.123326
\(231\) 0 0
\(232\) 6.37859 0.418775
\(233\) −22.2706 −1.45899 −0.729497 0.683984i \(-0.760246\pi\)
−0.729497 + 0.683984i \(0.760246\pi\)
\(234\) 0.411295 0.0268872
\(235\) −32.9850 −2.15171
\(236\) −11.0451 −0.718976
\(237\) 13.3870 0.869580
\(238\) 0 0
\(239\) 13.3901 0.866134 0.433067 0.901362i \(-0.357431\pi\)
0.433067 + 0.901362i \(0.357431\pi\)
\(240\) 6.33531 0.408943
\(241\) −3.28645 −0.211699 −0.105849 0.994382i \(-0.533756\pi\)
−0.105849 + 0.994382i \(0.533756\pi\)
\(242\) −18.8988 −1.21486
\(243\) −1.00000 −0.0641500
\(244\) −8.73595 −0.559262
\(245\) 0 0
\(246\) 5.50124 0.350746
\(247\) 2.02112 0.128601
\(248\) −1.16767 −0.0741472
\(249\) 9.77501 0.619466
\(250\) −0.130761 −0.00827004
\(251\) 10.2769 0.648672 0.324336 0.945942i \(-0.394859\pi\)
0.324336 + 0.945942i \(0.394859\pi\)
\(252\) 0 0
\(253\) 6.54534 0.411502
\(254\) 7.67653 0.481668
\(255\) 21.5674 1.35060
\(256\) −5.33281 −0.333301
\(257\) 23.4070 1.46009 0.730046 0.683398i \(-0.239499\pi\)
0.730046 + 0.683398i \(0.239499\pi\)
\(258\) −2.53462 −0.157799
\(259\) 0 0
\(260\) −3.59809 −0.223144
\(261\) 2.94619 0.182365
\(262\) −3.64822 −0.225388
\(263\) −27.3656 −1.68744 −0.843719 0.536785i \(-0.819638\pi\)
−0.843719 + 0.536785i \(0.819638\pi\)
\(264\) 14.1708 0.872155
\(265\) −39.6260 −2.43421
\(266\) 0 0
\(267\) −2.92939 −0.179276
\(268\) 4.50864 0.275409
\(269\) −26.3771 −1.60824 −0.804119 0.594469i \(-0.797362\pi\)
−0.804119 + 0.594469i \(0.797362\pi\)
\(270\) −1.87033 −0.113825
\(271\) 4.30436 0.261471 0.130736 0.991417i \(-0.458266\pi\)
0.130736 + 0.991417i \(0.458266\pi\)
\(272\) 13.7598 0.834311
\(273\) 0 0
\(274\) 11.6404 0.703223
\(275\) −32.2691 −1.94590
\(276\) −1.64772 −0.0991813
\(277\) −14.6493 −0.880192 −0.440096 0.897951i \(-0.645056\pi\)
−0.440096 + 0.897951i \(0.645056\pi\)
\(278\) 7.03052 0.421663
\(279\) −0.539333 −0.0322890
\(280\) 0 0
\(281\) 17.0380 1.01640 0.508202 0.861238i \(-0.330310\pi\)
0.508202 + 0.861238i \(0.330310\pi\)
\(282\) 6.21272 0.369962
\(283\) −17.9128 −1.06481 −0.532403 0.846491i \(-0.678711\pi\)
−0.532403 + 0.846491i \(0.678711\pi\)
\(284\) 0.967572 0.0574148
\(285\) −9.19088 −0.544421
\(286\) −2.69206 −0.159185
\(287\) 0 0
\(288\) −5.52331 −0.325464
\(289\) 29.8427 1.75545
\(290\) 5.51035 0.323579
\(291\) 8.14170 0.477275
\(292\) −17.3449 −1.01504
\(293\) −18.7699 −1.09655 −0.548273 0.836299i \(-0.684715\pi\)
−0.548273 + 0.836299i \(0.684715\pi\)
\(294\) 0 0
\(295\) −21.1233 −1.22985
\(296\) −6.27020 −0.364448
\(297\) 6.54534 0.379799
\(298\) 3.07076 0.177884
\(299\) 0.692965 0.0400752
\(300\) 8.12342 0.469006
\(301\) 0 0
\(302\) 2.99740 0.172481
\(303\) 0.133249 0.00765496
\(304\) −5.86370 −0.336307
\(305\) −16.7071 −0.956648
\(306\) −4.06221 −0.232221
\(307\) −1.42373 −0.0812566 −0.0406283 0.999174i \(-0.512936\pi\)
−0.0406283 + 0.999174i \(0.512936\pi\)
\(308\) 0 0
\(309\) 11.0107 0.626374
\(310\) −1.00873 −0.0572920
\(311\) −4.42471 −0.250902 −0.125451 0.992100i \(-0.540038\pi\)
−0.125451 + 0.992100i \(0.540038\pi\)
\(312\) 1.50029 0.0849372
\(313\) 21.6865 1.22580 0.612898 0.790162i \(-0.290004\pi\)
0.612898 + 0.790162i \(0.290004\pi\)
\(314\) 5.72644 0.323162
\(315\) 0 0
\(316\) 22.0581 1.24086
\(317\) −11.6270 −0.653037 −0.326518 0.945191i \(-0.605875\pi\)
−0.326518 + 0.945191i \(0.605875\pi\)
\(318\) 7.46355 0.418535
\(319\) −19.2838 −1.07969
\(320\) 2.34022 0.130822
\(321\) 8.57510 0.478615
\(322\) 0 0
\(323\) −19.9619 −1.11071
\(324\) −1.64772 −0.0915402
\(325\) −3.41638 −0.189507
\(326\) −14.2046 −0.786723
\(327\) −11.2823 −0.623910
\(328\) 20.0670 1.10802
\(329\) 0 0
\(330\) 12.2419 0.673897
\(331\) 7.89484 0.433940 0.216970 0.976178i \(-0.430383\pi\)
0.216970 + 0.976178i \(0.430383\pi\)
\(332\) 16.1065 0.883960
\(333\) −2.89613 −0.158707
\(334\) −7.14418 −0.390912
\(335\) 8.62259 0.471102
\(336\) 0 0
\(337\) 12.1596 0.662378 0.331189 0.943565i \(-0.392550\pi\)
0.331189 + 0.943565i \(0.392550\pi\)
\(338\) 7.43086 0.404185
\(339\) −3.14165 −0.170631
\(340\) 35.5371 1.92727
\(341\) 3.53012 0.191167
\(342\) 1.73110 0.0936071
\(343\) 0 0
\(344\) −9.24561 −0.498490
\(345\) −3.15120 −0.169655
\(346\) 9.85484 0.529799
\(347\) 14.0192 0.752588 0.376294 0.926500i \(-0.377198\pi\)
0.376294 + 0.926500i \(0.377198\pi\)
\(348\) 4.85451 0.260229
\(349\) 2.94132 0.157445 0.0787226 0.996897i \(-0.474916\pi\)
0.0787226 + 0.996897i \(0.474916\pi\)
\(350\) 0 0
\(351\) 0.692965 0.0369878
\(352\) 36.1519 1.92691
\(353\) −10.0734 −0.536153 −0.268076 0.963398i \(-0.586388\pi\)
−0.268076 + 0.963398i \(0.586388\pi\)
\(354\) 3.97858 0.211459
\(355\) 1.85044 0.0982112
\(356\) −4.82683 −0.255822
\(357\) 0 0
\(358\) −1.92622 −0.101804
\(359\) −32.4956 −1.71505 −0.857526 0.514441i \(-0.827999\pi\)
−0.857526 + 0.514441i \(0.827999\pi\)
\(360\) −6.82244 −0.359574
\(361\) −10.4933 −0.552279
\(362\) −8.53476 −0.448577
\(363\) −31.8415 −1.67124
\(364\) 0 0
\(365\) −33.1715 −1.73627
\(366\) 3.14679 0.164485
\(367\) 24.2742 1.26710 0.633552 0.773701i \(-0.281596\pi\)
0.633552 + 0.773701i \(0.281596\pi\)
\(368\) −2.01044 −0.104802
\(369\) 9.26871 0.482510
\(370\) −5.41671 −0.281601
\(371\) 0 0
\(372\) −0.888672 −0.0460755
\(373\) −22.0627 −1.14236 −0.571181 0.820824i \(-0.693515\pi\)
−0.571181 + 0.820824i \(0.693515\pi\)
\(374\) 26.5885 1.37486
\(375\) −0.220311 −0.0113768
\(376\) 22.6623 1.16872
\(377\) −2.04161 −0.105148
\(378\) 0 0
\(379\) 0.844591 0.0433837 0.0216919 0.999765i \(-0.493095\pi\)
0.0216919 + 0.999765i \(0.493095\pi\)
\(380\) −15.1440 −0.776872
\(381\) 12.9337 0.662615
\(382\) 5.86947 0.300308
\(383\) 33.0352 1.68802 0.844009 0.536329i \(-0.180189\pi\)
0.844009 + 0.536329i \(0.180189\pi\)
\(384\) −11.4874 −0.586214
\(385\) 0 0
\(386\) 2.13107 0.108469
\(387\) −4.27043 −0.217078
\(388\) 13.4153 0.681057
\(389\) 0.772812 0.0391831 0.0195916 0.999808i \(-0.493763\pi\)
0.0195916 + 0.999808i \(0.493763\pi\)
\(390\) 1.29607 0.0656292
\(391\) −6.84417 −0.346125
\(392\) 0 0
\(393\) −6.14667 −0.310058
\(394\) −4.87453 −0.245575
\(395\) 42.1852 2.12257
\(396\) 10.7849 0.541962
\(397\) −21.1380 −1.06089 −0.530444 0.847720i \(-0.677975\pi\)
−0.530444 + 0.847720i \(0.677975\pi\)
\(398\) −1.33251 −0.0667928
\(399\) 0 0
\(400\) 9.91165 0.495583
\(401\) 28.3143 1.41395 0.706975 0.707238i \(-0.250059\pi\)
0.706975 + 0.707238i \(0.250059\pi\)
\(402\) −1.62406 −0.0810009
\(403\) 0.373739 0.0186173
\(404\) 0.219558 0.0109234
\(405\) −3.15120 −0.156585
\(406\) 0 0
\(407\) 18.9561 0.939621
\(408\) −14.8178 −0.733591
\(409\) −22.3144 −1.10338 −0.551688 0.834051i \(-0.686016\pi\)
−0.551688 + 0.834051i \(0.686016\pi\)
\(410\) 17.3355 0.856141
\(411\) 19.6122 0.967399
\(412\) 18.1425 0.893818
\(413\) 0 0
\(414\) 0.593528 0.0291703
\(415\) 30.8030 1.51206
\(416\) 3.82746 0.187657
\(417\) 11.8453 0.580067
\(418\) −11.3306 −0.554199
\(419\) −5.99719 −0.292982 −0.146491 0.989212i \(-0.546798\pi\)
−0.146491 + 0.989212i \(0.546798\pi\)
\(420\) 0 0
\(421\) 12.4472 0.606638 0.303319 0.952889i \(-0.401905\pi\)
0.303319 + 0.952889i \(0.401905\pi\)
\(422\) 13.6977 0.666793
\(423\) 10.4674 0.508944
\(424\) 27.2250 1.32216
\(425\) 33.7423 1.63674
\(426\) −0.348530 −0.0168863
\(427\) 0 0
\(428\) 14.1294 0.682970
\(429\) −4.53569 −0.218985
\(430\) −7.98712 −0.385173
\(431\) 5.38995 0.259625 0.129812 0.991539i \(-0.458563\pi\)
0.129812 + 0.991539i \(0.458563\pi\)
\(432\) −2.01044 −0.0967274
\(433\) −32.6738 −1.57020 −0.785102 0.619366i \(-0.787390\pi\)
−0.785102 + 0.619366i \(0.787390\pi\)
\(434\) 0 0
\(435\) 9.28405 0.445136
\(436\) −18.5900 −0.890302
\(437\) 2.91662 0.139521
\(438\) 6.24784 0.298533
\(439\) 0.0288317 0.00137606 0.000688030 1.00000i \(-0.499781\pi\)
0.000688030 1.00000i \(0.499781\pi\)
\(440\) 44.6552 2.12885
\(441\) 0 0
\(442\) 2.81497 0.133894
\(443\) 16.9327 0.804499 0.402249 0.915530i \(-0.368229\pi\)
0.402249 + 0.915530i \(0.368229\pi\)
\(444\) −4.77202 −0.226470
\(445\) −9.23112 −0.437597
\(446\) −2.54370 −0.120448
\(447\) 5.17374 0.244709
\(448\) 0 0
\(449\) 20.8257 0.982825 0.491412 0.870927i \(-0.336481\pi\)
0.491412 + 0.870927i \(0.336481\pi\)
\(450\) −2.92615 −0.137940
\(451\) −60.6669 −2.85669
\(452\) −5.17656 −0.243485
\(453\) 5.05014 0.237276
\(454\) −6.48927 −0.304556
\(455\) 0 0
\(456\) 6.31457 0.295707
\(457\) −14.3513 −0.671326 −0.335663 0.941982i \(-0.608960\pi\)
−0.335663 + 0.941982i \(0.608960\pi\)
\(458\) −3.31041 −0.154685
\(459\) −6.84417 −0.319458
\(460\) −5.19231 −0.242093
\(461\) −3.54822 −0.165257 −0.0826286 0.996580i \(-0.526332\pi\)
−0.0826286 + 0.996580i \(0.526332\pi\)
\(462\) 0 0
\(463\) 12.2267 0.568223 0.284111 0.958791i \(-0.408301\pi\)
0.284111 + 0.958791i \(0.408301\pi\)
\(464\) 5.92315 0.274975
\(465\) −1.69955 −0.0788147
\(466\) 13.2182 0.612323
\(467\) −28.5860 −1.32280 −0.661401 0.750032i \(-0.730038\pi\)
−0.661401 + 0.750032i \(0.730038\pi\)
\(468\) 1.14182 0.0527804
\(469\) 0 0
\(470\) 19.5776 0.903045
\(471\) 9.64814 0.444563
\(472\) 14.5127 0.668003
\(473\) 27.9514 1.28521
\(474\) −7.94557 −0.364952
\(475\) −14.3792 −0.659763
\(476\) 0 0
\(477\) 12.5749 0.575765
\(478\) −7.94741 −0.363506
\(479\) 30.2691 1.38303 0.691516 0.722361i \(-0.256943\pi\)
0.691516 + 0.722361i \(0.256943\pi\)
\(480\) −17.4051 −0.794430
\(481\) 2.00692 0.0915075
\(482\) 1.95060 0.0888474
\(483\) 0 0
\(484\) −52.4660 −2.38482
\(485\) 25.6561 1.16499
\(486\) 0.593528 0.0269230
\(487\) 28.9369 1.31125 0.655627 0.755085i \(-0.272404\pi\)
0.655627 + 0.755085i \(0.272404\pi\)
\(488\) 11.4786 0.519612
\(489\) −23.9325 −1.08227
\(490\) 0 0
\(491\) −11.9185 −0.537876 −0.268938 0.963158i \(-0.586673\pi\)
−0.268938 + 0.963158i \(0.586673\pi\)
\(492\) 15.2723 0.688527
\(493\) 20.1642 0.908152
\(494\) −1.19959 −0.0539722
\(495\) 20.6257 0.927056
\(496\) −1.08430 −0.0486864
\(497\) 0 0
\(498\) −5.80174 −0.259982
\(499\) 10.3738 0.464394 0.232197 0.972669i \(-0.425409\pi\)
0.232197 + 0.972669i \(0.425409\pi\)
\(500\) −0.363012 −0.0162344
\(501\) −12.0368 −0.537765
\(502\) −6.09963 −0.272240
\(503\) 22.4429 1.00068 0.500340 0.865829i \(-0.333209\pi\)
0.500340 + 0.865829i \(0.333209\pi\)
\(504\) 0 0
\(505\) 0.419895 0.0186851
\(506\) −3.88485 −0.172702
\(507\) 12.5198 0.556024
\(508\) 21.3112 0.945532
\(509\) −5.96381 −0.264341 −0.132171 0.991227i \(-0.542195\pi\)
−0.132171 + 0.991227i \(0.542195\pi\)
\(510\) −12.8009 −0.566831
\(511\) 0 0
\(512\) −19.8096 −0.875470
\(513\) 2.91662 0.128772
\(514\) −13.8927 −0.612783
\(515\) 34.6968 1.52892
\(516\) −7.03649 −0.309764
\(517\) −68.5129 −3.01319
\(518\) 0 0
\(519\) 16.6038 0.728827
\(520\) 4.72772 0.207324
\(521\) 1.02187 0.0447691 0.0223845 0.999749i \(-0.492874\pi\)
0.0223845 + 0.999749i \(0.492874\pi\)
\(522\) −1.74865 −0.0765363
\(523\) 4.98369 0.217922 0.108961 0.994046i \(-0.465248\pi\)
0.108961 + 0.994046i \(0.465248\pi\)
\(524\) −10.1280 −0.442444
\(525\) 0 0
\(526\) 16.2423 0.708197
\(527\) −3.69129 −0.160795
\(528\) 13.1590 0.572673
\(529\) 1.00000 0.0434783
\(530\) 23.5192 1.02161
\(531\) 6.70326 0.290897
\(532\) 0 0
\(533\) −6.42289 −0.278206
\(534\) 1.73868 0.0752400
\(535\) 27.0219 1.16826
\(536\) −5.92413 −0.255884
\(537\) −3.24538 −0.140048
\(538\) 15.6555 0.674958
\(539\) 0 0
\(540\) −5.19231 −0.223442
\(541\) 15.3340 0.659260 0.329630 0.944110i \(-0.393076\pi\)
0.329630 + 0.944110i \(0.393076\pi\)
\(542\) −2.55476 −0.109736
\(543\) −14.3797 −0.617092
\(544\) −37.8025 −1.62077
\(545\) −35.5527 −1.52291
\(546\) 0 0
\(547\) −6.89796 −0.294935 −0.147468 0.989067i \(-0.547112\pi\)
−0.147468 + 0.989067i \(0.547112\pi\)
\(548\) 32.3155 1.38045
\(549\) 5.30183 0.226277
\(550\) 19.1526 0.816671
\(551\) −8.59294 −0.366071
\(552\) 2.16503 0.0921497
\(553\) 0 0
\(554\) 8.69479 0.369406
\(555\) −9.12629 −0.387389
\(556\) 19.5178 0.827738
\(557\) 29.4889 1.24949 0.624743 0.780830i \(-0.285204\pi\)
0.624743 + 0.780830i \(0.285204\pi\)
\(558\) 0.320109 0.0135513
\(559\) 2.95926 0.125163
\(560\) 0 0
\(561\) 44.7974 1.89135
\(562\) −10.1126 −0.426573
\(563\) −7.22423 −0.304465 −0.152232 0.988345i \(-0.548646\pi\)
−0.152232 + 0.988345i \(0.548646\pi\)
\(564\) 17.2474 0.726248
\(565\) −9.89996 −0.416495
\(566\) 10.6318 0.446887
\(567\) 0 0
\(568\) −1.27134 −0.0533443
\(569\) −3.61375 −0.151496 −0.0757482 0.997127i \(-0.524135\pi\)
−0.0757482 + 0.997127i \(0.524135\pi\)
\(570\) 5.45505 0.228487
\(571\) −2.53390 −0.106041 −0.0530203 0.998593i \(-0.516885\pi\)
−0.0530203 + 0.998593i \(0.516885\pi\)
\(572\) −7.47357 −0.312486
\(573\) 9.88912 0.413124
\(574\) 0 0
\(575\) −4.93009 −0.205599
\(576\) −0.742642 −0.0309434
\(577\) −2.10686 −0.0877097 −0.0438548 0.999038i \(-0.513964\pi\)
−0.0438548 + 0.999038i \(0.513964\pi\)
\(578\) −17.7125 −0.736741
\(579\) 3.59051 0.149216
\(580\) 15.2976 0.635196
\(581\) 0 0
\(582\) −4.83233 −0.200306
\(583\) −82.3069 −3.40880
\(584\) 22.7904 0.943073
\(585\) 2.18368 0.0902839
\(586\) 11.1404 0.460208
\(587\) 3.01620 0.124492 0.0622460 0.998061i \(-0.480174\pi\)
0.0622460 + 0.998061i \(0.480174\pi\)
\(588\) 0 0
\(589\) 1.57303 0.0648156
\(590\) 12.5373 0.516153
\(591\) −8.21279 −0.337829
\(592\) −5.82250 −0.239303
\(593\) 10.4536 0.429278 0.214639 0.976693i \(-0.431143\pi\)
0.214639 + 0.976693i \(0.431143\pi\)
\(594\) −3.88485 −0.159397
\(595\) 0 0
\(596\) 8.52489 0.349193
\(597\) −2.24507 −0.0918845
\(598\) −0.411295 −0.0168191
\(599\) −1.77010 −0.0723245 −0.0361623 0.999346i \(-0.511513\pi\)
−0.0361623 + 0.999346i \(0.511513\pi\)
\(600\) −10.6738 −0.435755
\(601\) −37.1024 −1.51344 −0.756720 0.653739i \(-0.773199\pi\)
−0.756720 + 0.653739i \(0.773199\pi\)
\(602\) 0 0
\(603\) −2.73628 −0.111430
\(604\) 8.32123 0.338586
\(605\) −100.339 −4.07936
\(606\) −0.0790871 −0.00321269
\(607\) −9.90037 −0.401844 −0.200922 0.979607i \(-0.564394\pi\)
−0.200922 + 0.979607i \(0.564394\pi\)
\(608\) 16.1094 0.653323
\(609\) 0 0
\(610\) 9.91617 0.401494
\(611\) −7.25357 −0.293448
\(612\) −11.2773 −0.455858
\(613\) −33.3962 −1.34886 −0.674430 0.738338i \(-0.735611\pi\)
−0.674430 + 0.738338i \(0.735611\pi\)
\(614\) 0.845025 0.0341024
\(615\) 29.2076 1.17776
\(616\) 0 0
\(617\) −19.5997 −0.789054 −0.394527 0.918884i \(-0.629092\pi\)
−0.394527 + 0.918884i \(0.629092\pi\)
\(618\) −6.53514 −0.262882
\(619\) −20.4682 −0.822685 −0.411342 0.911481i \(-0.634940\pi\)
−0.411342 + 0.911481i \(0.634940\pi\)
\(620\) −2.80039 −0.112466
\(621\) 1.00000 0.0401286
\(622\) 2.62619 0.105301
\(623\) 0 0
\(624\) 1.39317 0.0557713
\(625\) −25.3447 −1.01379
\(626\) −12.8716 −0.514452
\(627\) −19.0903 −0.762393
\(628\) 15.8975 0.634378
\(629\) −19.8216 −0.790338
\(630\) 0 0
\(631\) −16.9942 −0.676529 −0.338265 0.941051i \(-0.609840\pi\)
−0.338265 + 0.941051i \(0.609840\pi\)
\(632\) −28.9833 −1.15289
\(633\) 23.0784 0.917285
\(634\) 6.90095 0.274072
\(635\) 40.7568 1.61738
\(636\) 20.7199 0.821599
\(637\) 0 0
\(638\) 11.4455 0.453132
\(639\) −0.587217 −0.0232300
\(640\) −36.1991 −1.43090
\(641\) 10.9633 0.433023 0.216512 0.976280i \(-0.430532\pi\)
0.216512 + 0.976280i \(0.430532\pi\)
\(642\) −5.08956 −0.200869
\(643\) 11.7658 0.463996 0.231998 0.972716i \(-0.425474\pi\)
0.231998 + 0.972716i \(0.425474\pi\)
\(644\) 0 0
\(645\) −13.4570 −0.529869
\(646\) 11.8479 0.466151
\(647\) 1.76940 0.0695624 0.0347812 0.999395i \(-0.488927\pi\)
0.0347812 + 0.999395i \(0.488927\pi\)
\(648\) 2.16503 0.0850503
\(649\) −43.8751 −1.72225
\(650\) 2.02772 0.0795337
\(651\) 0 0
\(652\) −39.4342 −1.54436
\(653\) 10.5678 0.413551 0.206776 0.978388i \(-0.433703\pi\)
0.206776 + 0.978388i \(0.433703\pi\)
\(654\) 6.69634 0.261848
\(655\) −19.3694 −0.756825
\(656\) 18.6342 0.727543
\(657\) 10.5266 0.410682
\(658\) 0 0
\(659\) −10.3840 −0.404504 −0.202252 0.979334i \(-0.564826\pi\)
−0.202252 + 0.979334i \(0.564826\pi\)
\(660\) 33.9855 1.32288
\(661\) −33.3468 −1.29704 −0.648521 0.761197i \(-0.724612\pi\)
−0.648521 + 0.761197i \(0.724612\pi\)
\(662\) −4.68581 −0.182119
\(663\) 4.74277 0.184194
\(664\) −21.1632 −0.821290
\(665\) 0 0
\(666\) 1.71893 0.0666073
\(667\) −2.94619 −0.114077
\(668\) −19.8333 −0.767374
\(669\) −4.28573 −0.165696
\(670\) −5.11775 −0.197716
\(671\) −34.7023 −1.33967
\(672\) 0 0
\(673\) 11.0030 0.424134 0.212067 0.977255i \(-0.431980\pi\)
0.212067 + 0.977255i \(0.431980\pi\)
\(674\) −7.21709 −0.277992
\(675\) −4.93009 −0.189759
\(676\) 20.6292 0.793430
\(677\) 33.6967 1.29507 0.647536 0.762035i \(-0.275800\pi\)
0.647536 + 0.762035i \(0.275800\pi\)
\(678\) 1.86466 0.0716117
\(679\) 0 0
\(680\) −46.6940 −1.79063
\(681\) −10.9334 −0.418968
\(682\) −2.09523 −0.0802303
\(683\) −47.1617 −1.80459 −0.902297 0.431115i \(-0.858120\pi\)
−0.902297 + 0.431115i \(0.858120\pi\)
\(684\) 4.80579 0.183754
\(685\) 61.8021 2.36134
\(686\) 0 0
\(687\) −5.57751 −0.212795
\(688\) −8.58546 −0.327318
\(689\) −8.71396 −0.331975
\(690\) 1.87033 0.0712022
\(691\) 23.8501 0.907300 0.453650 0.891180i \(-0.350122\pi\)
0.453650 + 0.891180i \(0.350122\pi\)
\(692\) 27.3585 1.04001
\(693\) 0 0
\(694\) −8.32077 −0.315852
\(695\) 37.3269 1.41589
\(696\) −6.37859 −0.241780
\(697\) 63.4366 2.40283
\(698\) −1.74576 −0.0660779
\(699\) 22.2706 0.842351
\(700\) 0 0
\(701\) 6.20352 0.234304 0.117152 0.993114i \(-0.462624\pi\)
0.117152 + 0.993114i \(0.462624\pi\)
\(702\) −0.411295 −0.0155233
\(703\) 8.44692 0.318581
\(704\) 4.86085 0.183200
\(705\) 32.9850 1.24229
\(706\) 5.97885 0.225017
\(707\) 0 0
\(708\) 11.0451 0.415101
\(709\) 16.2491 0.610249 0.305124 0.952312i \(-0.401302\pi\)
0.305124 + 0.952312i \(0.401302\pi\)
\(710\) −1.09829 −0.0412181
\(711\) −13.3870 −0.502052
\(712\) 6.34222 0.237685
\(713\) 0.539333 0.0201982
\(714\) 0 0
\(715\) −14.2929 −0.534524
\(716\) −5.34749 −0.199845
\(717\) −13.3901 −0.500063
\(718\) 19.2871 0.719786
\(719\) 40.0502 1.49362 0.746811 0.665036i \(-0.231584\pi\)
0.746811 + 0.665036i \(0.231584\pi\)
\(720\) −6.33531 −0.236103
\(721\) 0 0
\(722\) 6.22807 0.231785
\(723\) 3.28645 0.122224
\(724\) −23.6938 −0.880572
\(725\) 14.5250 0.539444
\(726\) 18.8988 0.701401
\(727\) 40.4544 1.50037 0.750185 0.661228i \(-0.229964\pi\)
0.750185 + 0.661228i \(0.229964\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 19.6882 0.728693
\(731\) −29.2276 −1.08102
\(732\) 8.73595 0.322890
\(733\) −53.0168 −1.95822 −0.979110 0.203332i \(-0.934823\pi\)
−0.979110 + 0.203332i \(0.934823\pi\)
\(734\) −14.4074 −0.531788
\(735\) 0 0
\(736\) 5.52331 0.203592
\(737\) 17.9099 0.659720
\(738\) −5.50124 −0.202504
\(739\) 30.5517 1.12386 0.561930 0.827185i \(-0.310059\pi\)
0.561930 + 0.827185i \(0.310059\pi\)
\(740\) −15.0376 −0.552793
\(741\) −2.02112 −0.0742477
\(742\) 0 0
\(743\) −49.1733 −1.80399 −0.901997 0.431743i \(-0.857899\pi\)
−0.901997 + 0.431743i \(0.857899\pi\)
\(744\) 1.16767 0.0428089
\(745\) 16.3035 0.597314
\(746\) 13.0948 0.479436
\(747\) −9.77501 −0.357649
\(748\) 73.8138 2.69890
\(749\) 0 0
\(750\) 0.130761 0.00477471
\(751\) 41.5221 1.51516 0.757582 0.652740i \(-0.226381\pi\)
0.757582 + 0.652740i \(0.226381\pi\)
\(752\) 21.0442 0.767402
\(753\) −10.2769 −0.374511
\(754\) 1.21175 0.0441294
\(755\) 15.9140 0.579170
\(756\) 0 0
\(757\) −26.7837 −0.973470 −0.486735 0.873550i \(-0.661812\pi\)
−0.486735 + 0.873550i \(0.661812\pi\)
\(758\) −0.501289 −0.0182076
\(759\) −6.54534 −0.237581
\(760\) 19.8985 0.721795
\(761\) −23.6683 −0.857976 −0.428988 0.903310i \(-0.641130\pi\)
−0.428988 + 0.903310i \(0.641130\pi\)
\(762\) −7.67653 −0.278091
\(763\) 0 0
\(764\) 16.2945 0.589516
\(765\) −21.5674 −0.779770
\(766\) −19.6073 −0.708441
\(767\) −4.64513 −0.167726
\(768\) 5.33281 0.192431
\(769\) −16.2072 −0.584448 −0.292224 0.956350i \(-0.594395\pi\)
−0.292224 + 0.956350i \(0.594395\pi\)
\(770\) 0 0
\(771\) −23.4070 −0.842984
\(772\) 5.91617 0.212927
\(773\) 43.8899 1.57861 0.789305 0.614002i \(-0.210441\pi\)
0.789305 + 0.614002i \(0.210441\pi\)
\(774\) 2.53462 0.0911052
\(775\) −2.65896 −0.0955126
\(776\) −17.6270 −0.632773
\(777\) 0 0
\(778\) −0.458686 −0.0164447
\(779\) −27.0333 −0.968570
\(780\) 3.59809 0.128832
\(781\) 3.84354 0.137533
\(782\) 4.06221 0.145264
\(783\) −2.94619 −0.105288
\(784\) 0 0
\(785\) 30.4033 1.08514
\(786\) 3.64822 0.130128
\(787\) −25.7349 −0.917351 −0.458675 0.888604i \(-0.651676\pi\)
−0.458675 + 0.888604i \(0.651676\pi\)
\(788\) −13.5324 −0.482072
\(789\) 27.3656 0.974243
\(790\) −25.0381 −0.890816
\(791\) 0 0
\(792\) −14.1708 −0.503539
\(793\) −3.67398 −0.130467
\(794\) 12.5460 0.445242
\(795\) 39.6260 1.40539
\(796\) −3.69925 −0.131117
\(797\) −0.541418 −0.0191780 −0.00958901 0.999954i \(-0.503052\pi\)
−0.00958901 + 0.999954i \(0.503052\pi\)
\(798\) 0 0
\(799\) 71.6409 2.53447
\(800\) −27.2304 −0.962740
\(801\) 2.92939 0.103505
\(802\) −16.8054 −0.593418
\(803\) −68.9002 −2.43143
\(804\) −4.50864 −0.159008
\(805\) 0 0
\(806\) −0.221825 −0.00781344
\(807\) 26.3771 0.928516
\(808\) −0.288488 −0.0101490
\(809\) −3.89720 −0.137018 −0.0685091 0.997650i \(-0.521824\pi\)
−0.0685091 + 0.997650i \(0.521824\pi\)
\(810\) 1.87033 0.0657167
\(811\) 17.8242 0.625892 0.312946 0.949771i \(-0.398684\pi\)
0.312946 + 0.949771i \(0.398684\pi\)
\(812\) 0 0
\(813\) −4.30436 −0.150960
\(814\) −11.2510 −0.394348
\(815\) −75.4163 −2.64172
\(816\) −13.7598 −0.481690
\(817\) 12.4552 0.435754
\(818\) 13.2442 0.463073
\(819\) 0 0
\(820\) 48.1261 1.68063
\(821\) −18.6898 −0.652278 −0.326139 0.945322i \(-0.605748\pi\)
−0.326139 + 0.945322i \(0.605748\pi\)
\(822\) −11.6404 −0.406006
\(823\) 3.44929 0.120235 0.0601174 0.998191i \(-0.480853\pi\)
0.0601174 + 0.998191i \(0.480853\pi\)
\(824\) −23.8384 −0.830449
\(825\) 32.2691 1.12347
\(826\) 0 0
\(827\) 55.4418 1.92790 0.963950 0.266085i \(-0.0857301\pi\)
0.963950 + 0.266085i \(0.0857301\pi\)
\(828\) 1.64772 0.0572624
\(829\) 36.7658 1.27693 0.638465 0.769651i \(-0.279570\pi\)
0.638465 + 0.769651i \(0.279570\pi\)
\(830\) −18.2825 −0.634594
\(831\) 14.6493 0.508179
\(832\) 0.514625 0.0178414
\(833\) 0 0
\(834\) −7.03052 −0.243447
\(835\) −37.9304 −1.31264
\(836\) −31.4555 −1.08791
\(837\) 0.539333 0.0186421
\(838\) 3.55950 0.122961
\(839\) −25.3306 −0.874508 −0.437254 0.899338i \(-0.644049\pi\)
−0.437254 + 0.899338i \(0.644049\pi\)
\(840\) 0 0
\(841\) −20.3200 −0.700688
\(842\) −7.38775 −0.254599
\(843\) −17.0380 −0.586821
\(844\) 38.0269 1.30894
\(845\) 39.4524 1.35721
\(846\) −6.21272 −0.213598
\(847\) 0 0
\(848\) 25.2811 0.868156
\(849\) 17.9128 0.614766
\(850\) −20.0270 −0.686922
\(851\) 2.89613 0.0992780
\(852\) −0.967572 −0.0331485
\(853\) −51.3694 −1.75885 −0.879427 0.476034i \(-0.842074\pi\)
−0.879427 + 0.476034i \(0.842074\pi\)
\(854\) 0 0
\(855\) 9.19088 0.314321
\(856\) −18.5653 −0.634550
\(857\) 9.55532 0.326403 0.163202 0.986593i \(-0.447818\pi\)
0.163202 + 0.986593i \(0.447818\pi\)
\(858\) 2.69206 0.0919055
\(859\) −3.63424 −0.123999 −0.0619994 0.998076i \(-0.519748\pi\)
−0.0619994 + 0.998076i \(0.519748\pi\)
\(860\) −22.1734 −0.756108
\(861\) 0 0
\(862\) −3.19909 −0.108961
\(863\) 8.57788 0.291995 0.145997 0.989285i \(-0.453361\pi\)
0.145997 + 0.989285i \(0.453361\pi\)
\(864\) 5.52331 0.187907
\(865\) 52.3220 1.77900
\(866\) 19.3929 0.658996
\(867\) −29.8427 −1.01351
\(868\) 0 0
\(869\) 87.6225 2.97239
\(870\) −5.51035 −0.186818
\(871\) 1.89615 0.0642486
\(872\) 24.4264 0.827182
\(873\) −8.14170 −0.275555
\(874\) −1.73110 −0.0585553
\(875\) 0 0
\(876\) 17.3449 0.586031
\(877\) 45.6311 1.54085 0.770427 0.637528i \(-0.220043\pi\)
0.770427 + 0.637528i \(0.220043\pi\)
\(878\) −0.0171124 −0.000577516 0
\(879\) 18.7699 0.633092
\(880\) 41.4668 1.39784
\(881\) −17.3176 −0.583446 −0.291723 0.956503i \(-0.594229\pi\)
−0.291723 + 0.956503i \(0.594229\pi\)
\(882\) 0 0
\(883\) 14.3872 0.484166 0.242083 0.970256i \(-0.422169\pi\)
0.242083 + 0.970256i \(0.422169\pi\)
\(884\) 7.81478 0.262839
\(885\) 21.1233 0.710053
\(886\) −10.0501 −0.337638
\(887\) 2.17198 0.0729278 0.0364639 0.999335i \(-0.488391\pi\)
0.0364639 + 0.999335i \(0.488391\pi\)
\(888\) 6.27020 0.210414
\(889\) 0 0
\(890\) 5.47893 0.183654
\(891\) −6.54534 −0.219277
\(892\) −7.06169 −0.236443
\(893\) −30.5296 −1.02163
\(894\) −3.07076 −0.102702
\(895\) −10.2268 −0.341846
\(896\) 0 0
\(897\) −0.692965 −0.0231374
\(898\) −12.3606 −0.412480
\(899\) −1.58898 −0.0529954
\(900\) −8.12342 −0.270781
\(901\) 86.0647 2.86723
\(902\) 36.0075 1.19892
\(903\) 0 0
\(904\) 6.80175 0.226223
\(905\) −45.3134 −1.50627
\(906\) −2.99740 −0.0995820
\(907\) −12.8687 −0.427297 −0.213648 0.976911i \(-0.568535\pi\)
−0.213648 + 0.976911i \(0.568535\pi\)
\(908\) −18.0152 −0.597855
\(909\) −0.133249 −0.00441959
\(910\) 0 0
\(911\) −4.61820 −0.153008 −0.0765039 0.997069i \(-0.524376\pi\)
−0.0765039 + 0.997069i \(0.524376\pi\)
\(912\) 5.86370 0.194167
\(913\) 63.9807 2.11745
\(914\) 8.51792 0.281748
\(915\) 16.7071 0.552321
\(916\) −9.19019 −0.303652
\(917\) 0 0
\(918\) 4.06221 0.134073
\(919\) −0.395383 −0.0130425 −0.00652124 0.999979i \(-0.502076\pi\)
−0.00652124 + 0.999979i \(0.502076\pi\)
\(920\) 6.82244 0.224929
\(921\) 1.42373 0.0469135
\(922\) 2.10597 0.0693564
\(923\) 0.406921 0.0133940
\(924\) 0 0
\(925\) −14.2782 −0.469463
\(926\) −7.25689 −0.238476
\(927\) −11.0107 −0.361637
\(928\) −16.2727 −0.534179
\(929\) −36.6487 −1.20240 −0.601202 0.799097i \(-0.705311\pi\)
−0.601202 + 0.799097i \(0.705311\pi\)
\(930\) 1.00873 0.0330776
\(931\) 0 0
\(932\) 36.6958 1.20201
\(933\) 4.42471 0.144859
\(934\) 16.9666 0.555164
\(935\) 141.166 4.61661
\(936\) −1.50029 −0.0490385
\(937\) −37.1650 −1.21413 −0.607063 0.794654i \(-0.707652\pi\)
−0.607063 + 0.794654i \(0.707652\pi\)
\(938\) 0 0
\(939\) −21.6865 −0.707714
\(940\) 54.3502 1.77271
\(941\) 21.3747 0.696796 0.348398 0.937347i \(-0.386726\pi\)
0.348398 + 0.937347i \(0.386726\pi\)
\(942\) −5.72644 −0.186578
\(943\) −9.26871 −0.301831
\(944\) 13.4765 0.438623
\(945\) 0 0
\(946\) −16.5900 −0.539386
\(947\) 58.0483 1.88632 0.943158 0.332345i \(-0.107840\pi\)
0.943158 + 0.332345i \(0.107840\pi\)
\(948\) −22.0581 −0.716414
\(949\) −7.29457 −0.236792
\(950\) 8.53447 0.276895
\(951\) 11.6270 0.377031
\(952\) 0 0
\(953\) 0.853059 0.0276333 0.0138166 0.999905i \(-0.495602\pi\)
0.0138166 + 0.999905i \(0.495602\pi\)
\(954\) −7.46355 −0.241641
\(955\) 31.1626 1.00840
\(956\) −22.0632 −0.713575
\(957\) 19.2838 0.623358
\(958\) −17.9656 −0.580442
\(959\) 0 0
\(960\) −2.34022 −0.0755302
\(961\) −30.7091 −0.990617
\(962\) −1.19116 −0.0384046
\(963\) −8.57510 −0.276329
\(964\) 5.41516 0.174410
\(965\) 11.3144 0.364224
\(966\) 0 0
\(967\) 38.6397 1.24257 0.621284 0.783585i \(-0.286611\pi\)
0.621284 + 0.783585i \(0.286611\pi\)
\(968\) 68.9377 2.21574
\(969\) 19.9619 0.641268
\(970\) −15.2277 −0.488931
\(971\) −41.1365 −1.32013 −0.660066 0.751208i \(-0.729472\pi\)
−0.660066 + 0.751208i \(0.729472\pi\)
\(972\) 1.64772 0.0528508
\(973\) 0 0
\(974\) −17.1748 −0.550318
\(975\) 3.41638 0.109412
\(976\) 10.6590 0.341187
\(977\) 7.59884 0.243108 0.121554 0.992585i \(-0.461212\pi\)
0.121554 + 0.992585i \(0.461212\pi\)
\(978\) 14.2046 0.454214
\(979\) −19.1739 −0.612800
\(980\) 0 0
\(981\) 11.2823 0.360215
\(982\) 7.07399 0.225740
\(983\) 39.0631 1.24592 0.622959 0.782255i \(-0.285930\pi\)
0.622959 + 0.782255i \(0.285930\pi\)
\(984\) −20.0670 −0.639713
\(985\) −25.8802 −0.824611
\(986\) −11.9680 −0.381140
\(987\) 0 0
\(988\) −3.33025 −0.105949
\(989\) 4.27043 0.135792
\(990\) −12.2419 −0.389074
\(991\) 1.80629 0.0573786 0.0286893 0.999588i \(-0.490867\pi\)
0.0286893 + 0.999588i \(0.490867\pi\)
\(992\) 2.97890 0.0945803
\(993\) −7.89484 −0.250535
\(994\) 0 0
\(995\) −7.07467 −0.224282
\(996\) −16.1065 −0.510354
\(997\) −39.5007 −1.25100 −0.625499 0.780225i \(-0.715105\pi\)
−0.625499 + 0.780225i \(0.715105\pi\)
\(998\) −6.15714 −0.194901
\(999\) 2.89613 0.0916294
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 3381.2.a.bc.1.3 6
7.2 even 3 483.2.i.f.277.4 12
7.4 even 3 483.2.i.f.415.4 yes 12
7.6 odd 2 3381.2.a.bd.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.f.277.4 12 7.2 even 3
483.2.i.f.415.4 yes 12 7.4 even 3
3381.2.a.bc.1.3 6 1.1 even 1 trivial
3381.2.a.bd.1.3 6 7.6 odd 2