Properties

Label 3630.2.a.bs.1.4
Level $3630$
Weight $2$
Character 3630.1
Self dual yes
Analytic conductor $28.986$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9856959337\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.52625.1
Defining polynomial: \(x^{4} - x^{3} - 24 x^{2} + 19 x + 121\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 330)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.77556\) of defining polynomial
Character \(\chi\) \(=\) 3630.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +4.49096 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +4.49096 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +2.33343 q^{13} +4.49096 q^{14} -1.00000 q^{15} +1.00000 q^{16} +5.15753 q^{17} +1.00000 q^{18} +6.01163 q^{19} -1.00000 q^{20} +4.49096 q^{21} -8.88456 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.33343 q^{26} +1.00000 q^{27} +4.49096 q^{28} -7.58113 q^{29} -1.00000 q^{30} -1.10899 q^{31} +1.00000 q^{32} +5.15753 q^{34} -4.49096 q^{35} +1.00000 q^{36} +7.93310 q^{37} +6.01163 q^{38} +2.33343 q^{39} -1.00000 q^{40} -9.56995 q^{41} +4.49096 q^{42} +0.0785371 q^{43} -1.00000 q^{45} -8.88456 q^{46} +2.90264 q^{47} +1.00000 q^{48} +13.1687 q^{49} +1.00000 q^{50} +5.15753 q^{51} +2.33343 q^{52} +0.745110 q^{53} +1.00000 q^{54} +4.49096 q^{56} +6.01163 q^{57} -7.58113 q^{58} -5.32624 q^{59} -1.00000 q^{60} -15.4541 q^{61} -1.10899 q^{62} +4.49096 q^{63} +1.00000 q^{64} -2.33343 q^{65} -1.67821 q^{67} +5.15753 q^{68} -8.88456 q^{69} -4.49096 q^{70} +9.55113 q^{71} +1.00000 q^{72} -10.9443 q^{73} +7.93310 q^{74} +1.00000 q^{75} +6.01163 q^{76} +2.33343 q^{78} +5.47259 q^{79} -1.00000 q^{80} +1.00000 q^{81} -9.56995 q^{82} +8.00000 q^{83} +4.49096 q^{84} -5.15753 q^{85} +0.0785371 q^{86} -7.58113 q^{87} -7.41196 q^{89} -1.00000 q^{90} +10.4793 q^{91} -8.88456 q^{92} -1.10899 q^{93} +2.90264 q^{94} -6.01163 q^{95} +1.00000 q^{96} +5.74585 q^{97} +13.1687 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + q^{7} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + q^{7} + 4q^{8} + 4q^{9} - 4q^{10} + 4q^{12} + 2q^{13} + q^{14} - 4q^{15} + 4q^{16} + 11q^{17} + 4q^{18} + q^{19} - 4q^{20} + q^{21} + 4q^{24} + 4q^{25} + 2q^{26} + 4q^{27} + q^{28} + 9q^{29} - 4q^{30} + 17q^{31} + 4q^{32} + 11q^{34} - q^{35} + 4q^{36} + 8q^{37} + q^{38} + 2q^{39} - 4q^{40} - 11q^{41} + q^{42} + q^{43} - 4q^{45} + 10q^{47} + 4q^{48} + 31q^{49} + 4q^{50} + 11q^{51} + 2q^{52} + 11q^{53} + 4q^{54} + q^{56} + q^{57} + 9q^{58} + 10q^{59} - 4q^{60} - 10q^{61} + 17q^{62} + q^{63} + 4q^{64} - 2q^{65} + 9q^{67} + 11q^{68} - q^{70} + 10q^{71} + 4q^{72} - 8q^{73} + 8q^{74} + 4q^{75} + q^{76} + 2q^{78} - 7q^{79} - 4q^{80} + 4q^{81} - 11q^{82} + 32q^{83} + q^{84} - 11q^{85} + q^{86} + 9q^{87} - 23q^{89} - 4q^{90} + 48q^{91} + 17q^{93} + 10q^{94} - q^{95} + 4q^{96} - 2q^{97} + 31q^{98} + O(q^{100}) \)

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
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) 4.49096 1.69742 0.848711 0.528856i \(-0.177379\pi\)
0.848711 + 0.528856i \(0.177379\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) 2.33343 0.647176 0.323588 0.946198i \(-0.395111\pi\)
0.323588 + 0.946198i \(0.395111\pi\)
\(14\) 4.49096 1.20026
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 5.15753 1.25088 0.625442 0.780270i \(-0.284918\pi\)
0.625442 + 0.780270i \(0.284918\pi\)
\(18\) 1.00000 0.235702
\(19\) 6.01163 1.37916 0.689582 0.724208i \(-0.257794\pi\)
0.689582 + 0.724208i \(0.257794\pi\)
\(20\) −1.00000 −0.223607
\(21\) 4.49096 0.980007
\(22\) 0 0
\(23\) −8.88456 −1.85256 −0.926279 0.376838i \(-0.877011\pi\)
−0.926279 + 0.376838i \(0.877011\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 2.33343 0.457623
\(27\) 1.00000 0.192450
\(28\) 4.49096 0.848711
\(29\) −7.58113 −1.40778 −0.703890 0.710309i \(-0.748555\pi\)
−0.703890 + 0.710309i \(0.748555\pi\)
\(30\) −1.00000 −0.182574
\(31\) −1.10899 −0.199181 −0.0995905 0.995029i \(-0.531753\pi\)
−0.0995905 + 0.995029i \(0.531753\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 5.15753 0.884509
\(35\) −4.49096 −0.759111
\(36\) 1.00000 0.166667
\(37\) 7.93310 1.30419 0.652096 0.758136i \(-0.273890\pi\)
0.652096 + 0.758136i \(0.273890\pi\)
\(38\) 6.01163 0.975216
\(39\) 2.33343 0.373647
\(40\) −1.00000 −0.158114
\(41\) −9.56995 −1.49458 −0.747288 0.664501i \(-0.768644\pi\)
−0.747288 + 0.664501i \(0.768644\pi\)
\(42\) 4.49096 0.692970
\(43\) 0.0785371 0.0119768 0.00598839 0.999982i \(-0.498094\pi\)
0.00598839 + 0.999982i \(0.498094\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) −8.88456 −1.30996
\(47\) 2.90264 0.423394 0.211697 0.977335i \(-0.432101\pi\)
0.211697 + 0.977335i \(0.432101\pi\)
\(48\) 1.00000 0.144338
\(49\) 13.1687 1.88124
\(50\) 1.00000 0.141421
\(51\) 5.15753 0.722199
\(52\) 2.33343 0.323588
\(53\) 0.745110 0.102349 0.0511743 0.998690i \(-0.483704\pi\)
0.0511743 + 0.998690i \(0.483704\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 4.49096 0.600130
\(57\) 6.01163 0.796260
\(58\) −7.58113 −0.995451
\(59\) −5.32624 −0.693417 −0.346709 0.937973i \(-0.612701\pi\)
−0.346709 + 0.937973i \(0.612701\pi\)
\(60\) −1.00000 −0.129099
\(61\) −15.4541 −1.97869 −0.989344 0.145595i \(-0.953491\pi\)
−0.989344 + 0.145595i \(0.953491\pi\)
\(62\) −1.10899 −0.140842
\(63\) 4.49096 0.565808
\(64\) 1.00000 0.125000
\(65\) −2.33343 −0.289426
\(66\) 0 0
\(67\) −1.67821 −0.205025 −0.102513 0.994732i \(-0.532688\pi\)
−0.102513 + 0.994732i \(0.532688\pi\)
\(68\) 5.15753 0.625442
\(69\) −8.88456 −1.06958
\(70\) −4.49096 −0.536772
\(71\) 9.55113 1.13351 0.566755 0.823886i \(-0.308198\pi\)
0.566755 + 0.823886i \(0.308198\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.9443 −1.28093 −0.640465 0.767987i \(-0.721258\pi\)
−0.640465 + 0.767987i \(0.721258\pi\)
\(74\) 7.93310 0.922204
\(75\) 1.00000 0.115470
\(76\) 6.01163 0.689582
\(77\) 0 0
\(78\) 2.33343 0.264209
\(79\) 5.47259 0.615715 0.307857 0.951433i \(-0.400388\pi\)
0.307857 + 0.951433i \(0.400388\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −9.56995 −1.05682
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 4.49096 0.490004
\(85\) −5.15753 −0.559413
\(86\) 0.0785371 0.00846887
\(87\) −7.58113 −0.812782
\(88\) 0 0
\(89\) −7.41196 −0.785667 −0.392833 0.919610i \(-0.628505\pi\)
−0.392833 + 0.919610i \(0.628505\pi\)
\(90\) −1.00000 −0.105409
\(91\) 10.4793 1.09853
\(92\) −8.88456 −0.926279
\(93\) −1.10899 −0.114997
\(94\) 2.90264 0.299385
\(95\) −6.01163 −0.616781
\(96\) 1.00000 0.102062
\(97\) 5.74585 0.583403 0.291701 0.956509i \(-0.405779\pi\)
0.291701 + 0.956509i \(0.405779\pi\)
\(98\) 13.1687 1.33024
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 16.5739 1.64917 0.824584 0.565739i \(-0.191409\pi\)
0.824584 + 0.565739i \(0.191409\pi\)
\(102\) 5.15753 0.510672
\(103\) −1.09063 −0.107463 −0.0537313 0.998555i \(-0.517111\pi\)
−0.0537313 + 0.998555i \(0.517111\pi\)
\(104\) 2.33343 0.228811
\(105\) −4.49096 −0.438273
\(106\) 0.745110 0.0723714
\(107\) −5.68494 −0.549584 −0.274792 0.961504i \(-0.588609\pi\)
−0.274792 + 0.961504i \(0.588609\pi\)
\(108\) 1.00000 0.0962250
\(109\) 8.94427 0.856706 0.428353 0.903612i \(-0.359094\pi\)
0.428353 + 0.903612i \(0.359094\pi\)
\(110\) 0 0
\(111\) 7.93310 0.752976
\(112\) 4.49096 0.424356
\(113\) 3.87292 0.364334 0.182167 0.983268i \(-0.441689\pi\)
0.182167 + 0.983268i \(0.441689\pi\)
\(114\) 6.01163 0.563041
\(115\) 8.88456 0.828489
\(116\) −7.58113 −0.703890
\(117\) 2.33343 0.215725
\(118\) −5.32624 −0.490320
\(119\) 23.1623 2.12328
\(120\) −1.00000 −0.0912871
\(121\) 0 0
\(122\) −15.4541 −1.39914
\(123\) −9.56995 −0.862894
\(124\) −1.10899 −0.0995905
\(125\) −1.00000 −0.0894427
\(126\) 4.49096 0.400086
\(127\) 8.36343 0.742134 0.371067 0.928606i \(-0.378992\pi\)
0.371067 + 0.928606i \(0.378992\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0.0785371 0.00691480
\(130\) −2.33343 −0.204655
\(131\) −7.40079 −0.646610 −0.323305 0.946295i \(-0.604794\pi\)
−0.323305 + 0.946295i \(0.604794\pi\)
\(132\) 0 0
\(133\) 26.9980 2.34102
\(134\) −1.67821 −0.144975
\(135\) −1.00000 −0.0860663
\(136\) 5.15753 0.442255
\(137\) −4.44214 −0.379517 −0.189759 0.981831i \(-0.560771\pi\)
−0.189759 + 0.981831i \(0.560771\pi\)
\(138\) −8.88456 −0.756304
\(139\) −19.9074 −1.68852 −0.844260 0.535933i \(-0.819960\pi\)
−0.844260 + 0.535933i \(0.819960\pi\)
\(140\) −4.49096 −0.379555
\(141\) 2.90264 0.244446
\(142\) 9.55113 0.801513
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 7.58113 0.629578
\(146\) −10.9443 −0.905754
\(147\) 13.1687 1.08614
\(148\) 7.93310 0.652096
\(149\) 9.31461 0.763082 0.381541 0.924352i \(-0.375393\pi\)
0.381541 + 0.924352i \(0.375393\pi\)
\(150\) 1.00000 0.0816497
\(151\) −13.9029 −1.13140 −0.565702 0.824610i \(-0.691395\pi\)
−0.565702 + 0.824610i \(0.691395\pi\)
\(152\) 6.01163 0.487608
\(153\) 5.15753 0.416962
\(154\) 0 0
\(155\) 1.10899 0.0890764
\(156\) 2.33343 0.186824
\(157\) 9.89175 0.789447 0.394724 0.918800i \(-0.370840\pi\)
0.394724 + 0.918800i \(0.370840\pi\)
\(158\) 5.47259 0.435376
\(159\) 0.745110 0.0590910
\(160\) −1.00000 −0.0790569
\(161\) −39.9002 −3.14457
\(162\) 1.00000 0.0785674
\(163\) 6.43124 0.503734 0.251867 0.967762i \(-0.418955\pi\)
0.251867 + 0.967762i \(0.418955\pi\)
\(164\) −9.56995 −0.747288
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 6.04163 0.467515 0.233758 0.972295i \(-0.424898\pi\)
0.233758 + 0.972295i \(0.424898\pi\)
\(168\) 4.49096 0.346485
\(169\) −7.55512 −0.581163
\(170\) −5.15753 −0.395565
\(171\) 6.01163 0.459721
\(172\) 0.0785371 0.00598839
\(173\) −0.648032 −0.0492690 −0.0246345 0.999697i \(-0.507842\pi\)
−0.0246345 + 0.999697i \(0.507842\pi\)
\(174\) −7.58113 −0.574724
\(175\) 4.49096 0.339485
\(176\) 0 0
\(177\) −5.32624 −0.400345
\(178\) −7.41196 −0.555550
\(179\) 1.56231 0.116772 0.0583861 0.998294i \(-0.481405\pi\)
0.0583861 + 0.998294i \(0.481405\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 18.4721 1.37302 0.686512 0.727119i \(-0.259141\pi\)
0.686512 + 0.727119i \(0.259141\pi\)
\(182\) 10.4793 0.776779
\(183\) −15.4541 −1.14240
\(184\) −8.88456 −0.654978
\(185\) −7.93310 −0.583253
\(186\) −1.10899 −0.0813153
\(187\) 0 0
\(188\) 2.90264 0.211697
\(189\) 4.49096 0.326669
\(190\) −6.01163 −0.436130
\(191\) −17.6720 −1.27870 −0.639352 0.768914i \(-0.720797\pi\)
−0.639352 + 0.768914i \(0.720797\pi\)
\(192\) 1.00000 0.0721688
\(193\) 10.4954 0.755476 0.377738 0.925913i \(-0.376702\pi\)
0.377738 + 0.925913i \(0.376702\pi\)
\(194\) 5.74585 0.412528
\(195\) −2.33343 −0.167100
\(196\) 13.1687 0.940622
\(197\) 10.0788 0.718086 0.359043 0.933321i \(-0.383103\pi\)
0.359043 + 0.933321i \(0.383103\pi\)
\(198\) 0 0
\(199\) −13.6906 −0.970499 −0.485250 0.874376i \(-0.661271\pi\)
−0.485250 + 0.874376i \(0.661271\pi\)
\(200\) 1.00000 0.0707107
\(201\) −1.67821 −0.118371
\(202\) 16.5739 1.16614
\(203\) −34.0465 −2.38960
\(204\) 5.15753 0.361099
\(205\) 9.56995 0.668394
\(206\) −1.09063 −0.0759876
\(207\) −8.88456 −0.617519
\(208\) 2.33343 0.161794
\(209\) 0 0
\(210\) −4.49096 −0.309906
\(211\) −22.8481 −1.57293 −0.786464 0.617636i \(-0.788091\pi\)
−0.786464 + 0.617636i \(0.788091\pi\)
\(212\) 0.745110 0.0511743
\(213\) 9.55113 0.654433
\(214\) −5.68494 −0.388614
\(215\) −0.0785371 −0.00535618
\(216\) 1.00000 0.0680414
\(217\) −4.98044 −0.338094
\(218\) 8.94427 0.605783
\(219\) −10.9443 −0.739545
\(220\) 0 0
\(221\) 12.0347 0.809543
\(222\) 7.93310 0.532435
\(223\) −0.490958 −0.0328770 −0.0164385 0.999865i \(-0.505233\pi\)
−0.0164385 + 0.999865i \(0.505233\pi\)
\(224\) 4.49096 0.300065
\(225\) 1.00000 0.0666667
\(226\) 3.87292 0.257623
\(227\) −1.49022 −0.0989093 −0.0494547 0.998776i \(-0.515748\pi\)
−0.0494547 + 0.998776i \(0.515748\pi\)
\(228\) 6.01163 0.398130
\(229\) −8.21798 −0.543060 −0.271530 0.962430i \(-0.587530\pi\)
−0.271530 + 0.962430i \(0.587530\pi\)
\(230\) 8.88456 0.585830
\(231\) 0 0
\(232\) −7.58113 −0.497725
\(233\) −6.63686 −0.434795 −0.217397 0.976083i \(-0.569757\pi\)
−0.217397 + 0.976083i \(0.569757\pi\)
\(234\) 2.33343 0.152541
\(235\) −2.90264 −0.189347
\(236\) −5.32624 −0.346709
\(237\) 5.47259 0.355483
\(238\) 23.1623 1.50139
\(239\) 6.09764 0.394424 0.197212 0.980361i \(-0.436811\pi\)
0.197212 + 0.980361i \(0.436811\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −5.70376 −0.367412 −0.183706 0.982981i \(-0.558809\pi\)
−0.183706 + 0.982981i \(0.558809\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −15.4541 −0.989344
\(245\) −13.1687 −0.841318
\(246\) −9.56995 −0.610158
\(247\) 14.0277 0.892562
\(248\) −1.10899 −0.0704211
\(249\) 8.00000 0.506979
\(250\) −1.00000 −0.0632456
\(251\) −2.60714 −0.164561 −0.0822806 0.996609i \(-0.526220\pi\)
−0.0822806 + 0.996609i \(0.526220\pi\)
\(252\) 4.49096 0.282904
\(253\) 0 0
\(254\) 8.36343 0.524768
\(255\) −5.15753 −0.322977
\(256\) 1.00000 0.0625000
\(257\) 18.4360 1.15000 0.575002 0.818152i \(-0.305001\pi\)
0.575002 + 0.818152i \(0.305001\pi\)
\(258\) 0.0785371 0.00488950
\(259\) 35.6272 2.21377
\(260\) −2.33343 −0.144713
\(261\) −7.58113 −0.469260
\(262\) −7.40079 −0.457222
\(263\) 24.4285 1.50633 0.753163 0.657834i \(-0.228527\pi\)
0.753163 + 0.657834i \(0.228527\pi\)
\(264\) 0 0
\(265\) −0.745110 −0.0457717
\(266\) 26.9980 1.65535
\(267\) −7.41196 −0.453605
\(268\) −1.67821 −0.102513
\(269\) −11.0119 −0.671408 −0.335704 0.941967i \(-0.608974\pi\)
−0.335704 + 0.941967i \(0.608974\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 0.647750 0.0393480 0.0196740 0.999806i \(-0.493737\pi\)
0.0196740 + 0.999806i \(0.493737\pi\)
\(272\) 5.15753 0.312721
\(273\) 10.4793 0.634238
\(274\) −4.44214 −0.268359
\(275\) 0 0
\(276\) −8.88456 −0.534788
\(277\) −24.6304 −1.47990 −0.739949 0.672663i \(-0.765150\pi\)
−0.739949 + 0.672663i \(0.765150\pi\)
\(278\) −19.9074 −1.19396
\(279\) −1.10899 −0.0663936
\(280\) −4.49096 −0.268386
\(281\) 9.49170 0.566227 0.283114 0.959086i \(-0.408633\pi\)
0.283114 + 0.959086i \(0.408633\pi\)
\(282\) 2.90264 0.172850
\(283\) −20.2113 −1.20143 −0.600717 0.799462i \(-0.705118\pi\)
−0.600717 + 0.799462i \(0.705118\pi\)
\(284\) 9.55113 0.566755
\(285\) −6.01163 −0.356098
\(286\) 0 0
\(287\) −42.9783 −2.53693
\(288\) 1.00000 0.0589256
\(289\) 9.60013 0.564713
\(290\) 7.58113 0.445179
\(291\) 5.74585 0.336828
\(292\) −10.9443 −0.640465
\(293\) −1.19398 −0.0697530 −0.0348765 0.999392i \(-0.511104\pi\)
−0.0348765 + 0.999392i \(0.511104\pi\)
\(294\) 13.1687 0.768015
\(295\) 5.32624 0.310106
\(296\) 7.93310 0.461102
\(297\) 0 0
\(298\) 9.31461 0.539581
\(299\) −20.7315 −1.19893
\(300\) 1.00000 0.0577350
\(301\) 0.352707 0.0203297
\(302\) −13.9029 −0.800023
\(303\) 16.5739 0.952148
\(304\) 6.01163 0.344791
\(305\) 15.4541 0.884896
\(306\) 5.15753 0.294836
\(307\) −11.3502 −0.647793 −0.323896 0.946093i \(-0.604993\pi\)
−0.323896 + 0.946093i \(0.604993\pi\)
\(308\) 0 0
\(309\) −1.09063 −0.0620436
\(310\) 1.10899 0.0629865
\(311\) −15.1766 −0.860588 −0.430294 0.902689i \(-0.641590\pi\)
−0.430294 + 0.902689i \(0.641590\pi\)
\(312\) 2.33343 0.132104
\(313\) 23.3570 1.32021 0.660107 0.751171i \(-0.270511\pi\)
0.660107 + 0.751171i \(0.270511\pi\)
\(314\) 9.89175 0.558224
\(315\) −4.49096 −0.253037
\(316\) 5.47259 0.307857
\(317\) 10.6641 0.598956 0.299478 0.954103i \(-0.403187\pi\)
0.299478 + 0.954103i \(0.403187\pi\)
\(318\) 0.745110 0.0417837
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −5.68494 −0.317302
\(322\) −39.9002 −2.22355
\(323\) 31.0052 1.72517
\(324\) 1.00000 0.0555556
\(325\) 2.33343 0.129435
\(326\) 6.43124 0.356194
\(327\) 8.94427 0.494619
\(328\) −9.56995 −0.528412
\(329\) 13.0356 0.718678
\(330\) 0 0
\(331\) −2.42434 −0.133254 −0.0666268 0.997778i \(-0.521224\pi\)
−0.0666268 + 0.997778i \(0.521224\pi\)
\(332\) 8.00000 0.439057
\(333\) 7.93310 0.434731
\(334\) 6.04163 0.330583
\(335\) 1.67821 0.0916902
\(336\) 4.49096 0.245002
\(337\) −1.17664 −0.0640954 −0.0320477 0.999486i \(-0.510203\pi\)
−0.0320477 + 0.999486i \(0.510203\pi\)
\(338\) −7.55512 −0.410944
\(339\) 3.87292 0.210348
\(340\) −5.15753 −0.279706
\(341\) 0 0
\(342\) 6.01163 0.325072
\(343\) 27.7034 1.49584
\(344\) 0.0785371 0.00423443
\(345\) 8.88456 0.478328
\(346\) −0.648032 −0.0348384
\(347\) −23.3570 −1.25387 −0.626934 0.779072i \(-0.715690\pi\)
−0.626934 + 0.779072i \(0.715690\pi\)
\(348\) −7.58113 −0.406391
\(349\) −0.375059 −0.0200764 −0.0100382 0.999950i \(-0.503195\pi\)
−0.0100382 + 0.999950i \(0.503195\pi\)
\(350\) 4.49096 0.240052
\(351\) 2.33343 0.124549
\(352\) 0 0
\(353\) 2.07854 0.110629 0.0553147 0.998469i \(-0.482384\pi\)
0.0553147 + 0.998469i \(0.482384\pi\)
\(354\) −5.32624 −0.283086
\(355\) −9.55113 −0.506921
\(356\) −7.41196 −0.392833
\(357\) 23.1623 1.22588
\(358\) 1.56231 0.0825704
\(359\) 0.180340 0.00951798 0.00475899 0.999989i \(-0.498485\pi\)
0.00475899 + 0.999989i \(0.498485\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 17.1397 0.902091
\(362\) 18.4721 0.970874
\(363\) 0 0
\(364\) 10.4793 0.549266
\(365\) 10.9443 0.572849
\(366\) −15.4541 −0.807796
\(367\) 11.6706 0.609198 0.304599 0.952481i \(-0.401478\pi\)
0.304599 + 0.952481i \(0.401478\pi\)
\(368\) −8.88456 −0.463140
\(369\) −9.56995 −0.498192
\(370\) −7.93310 −0.412422
\(371\) 3.34626 0.173729
\(372\) −1.10899 −0.0574986
\(373\) −10.1687 −0.526516 −0.263258 0.964726i \(-0.584797\pi\)
−0.263258 + 0.964726i \(0.584797\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 2.90264 0.149692
\(377\) −17.6900 −0.911082
\(378\) 4.49096 0.230990
\(379\) 11.0825 0.569271 0.284636 0.958636i \(-0.408127\pi\)
0.284636 + 0.958636i \(0.408127\pi\)
\(380\) −6.01163 −0.308390
\(381\) 8.36343 0.428471
\(382\) −17.6720 −0.904180
\(383\) 21.3082 1.08880 0.544398 0.838827i \(-0.316758\pi\)
0.544398 + 0.838827i \(0.316758\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 10.4954 0.534202
\(387\) 0.0785371 0.00399226
\(388\) 5.74585 0.291701
\(389\) −37.7362 −1.91330 −0.956651 0.291238i \(-0.905933\pi\)
−0.956651 + 0.291238i \(0.905933\pi\)
\(390\) −2.33343 −0.118158
\(391\) −45.8224 −2.31734
\(392\) 13.1687 0.665120
\(393\) −7.40079 −0.373320
\(394\) 10.0788 0.507764
\(395\) −5.47259 −0.275356
\(396\) 0 0
\(397\) −22.4348 −1.12597 −0.562985 0.826467i \(-0.690347\pi\)
−0.562985 + 0.826467i \(0.690347\pi\)
\(398\) −13.6906 −0.686247
\(399\) 26.9980 1.35159
\(400\) 1.00000 0.0500000
\(401\) −25.9002 −1.29339 −0.646697 0.762747i \(-0.723850\pi\)
−0.646697 + 0.762747i \(0.723850\pi\)
\(402\) −1.67821 −0.0837013
\(403\) −2.58775 −0.128905
\(404\) 16.5739 0.824584
\(405\) −1.00000 −0.0496904
\(406\) −34.0465 −1.68970
\(407\) 0 0
\(408\) 5.15753 0.255336
\(409\) −24.7921 −1.22589 −0.612945 0.790125i \(-0.710015\pi\)
−0.612945 + 0.790125i \(0.710015\pi\)
\(410\) 9.56995 0.472626
\(411\) −4.44214 −0.219114
\(412\) −1.09063 −0.0537313
\(413\) −23.9199 −1.17702
\(414\) −8.88456 −0.436652
\(415\) −8.00000 −0.392705
\(416\) 2.33343 0.114406
\(417\) −19.9074 −0.974868
\(418\) 0 0
\(419\) 27.8969 1.36285 0.681427 0.731886i \(-0.261360\pi\)
0.681427 + 0.731886i \(0.261360\pi\)
\(420\) −4.49096 −0.219136
\(421\) 25.0052 1.21868 0.609339 0.792910i \(-0.291435\pi\)
0.609339 + 0.792910i \(0.291435\pi\)
\(422\) −22.8481 −1.11223
\(423\) 2.90264 0.141131
\(424\) 0.745110 0.0361857
\(425\) 5.15753 0.250177
\(426\) 9.55113 0.462754
\(427\) −69.4035 −3.35867
\(428\) −5.68494 −0.274792
\(429\) 0 0
\(430\) −0.0785371 −0.00378739
\(431\) −17.9624 −0.865216 −0.432608 0.901582i \(-0.642407\pi\)
−0.432608 + 0.901582i \(0.642407\pi\)
\(432\) 1.00000 0.0481125
\(433\) −26.9458 −1.29493 −0.647465 0.762095i \(-0.724171\pi\)
−0.647465 + 0.762095i \(0.724171\pi\)
\(434\) −4.98044 −0.239069
\(435\) 7.58113 0.363487
\(436\) 8.94427 0.428353
\(437\) −53.4107 −2.55498
\(438\) −10.9443 −0.522938
\(439\) −4.79393 −0.228802 −0.114401 0.993435i \(-0.536495\pi\)
−0.114401 + 0.993435i \(0.536495\pi\)
\(440\) 0 0
\(441\) 13.1687 0.627081
\(442\) 12.0347 0.572433
\(443\) 10.0600 0.477965 0.238982 0.971024i \(-0.423186\pi\)
0.238982 + 0.971024i \(0.423186\pi\)
\(444\) 7.93310 0.376488
\(445\) 7.41196 0.351361
\(446\) −0.490958 −0.0232476
\(447\) 9.31461 0.440566
\(448\) 4.49096 0.212178
\(449\) −0.498148 −0.0235091 −0.0117545 0.999931i \(-0.503742\pi\)
−0.0117545 + 0.999931i \(0.503742\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 3.87292 0.182167
\(453\) −13.9029 −0.653216
\(454\) −1.49022 −0.0699395
\(455\) −10.4793 −0.491278
\(456\) 6.01163 0.281521
\(457\) −1.85730 −0.0868811 −0.0434405 0.999056i \(-0.513832\pi\)
−0.0434405 + 0.999056i \(0.513832\pi\)
\(458\) −8.21798 −0.384001
\(459\) 5.15753 0.240733
\(460\) 8.88456 0.414245
\(461\) −22.1394 −1.03114 −0.515568 0.856848i \(-0.672419\pi\)
−0.515568 + 0.856848i \(0.672419\pi\)
\(462\) 0 0
\(463\) −22.1920 −1.03135 −0.515674 0.856785i \(-0.672459\pi\)
−0.515674 + 0.856785i \(0.672459\pi\)
\(464\) −7.58113 −0.351945
\(465\) 1.10899 0.0514283
\(466\) −6.63686 −0.307446
\(467\) 31.6953 1.46668 0.733342 0.679860i \(-0.237959\pi\)
0.733342 + 0.679860i \(0.237959\pi\)
\(468\) 2.33343 0.107863
\(469\) −7.53675 −0.348015
\(470\) −2.90264 −0.133889
\(471\) 9.89175 0.455788
\(472\) −5.32624 −0.245160
\(473\) 0 0
\(474\) 5.47259 0.251365
\(475\) 6.01163 0.275833
\(476\) 23.1623 1.06164
\(477\) 0.745110 0.0341162
\(478\) 6.09764 0.278900
\(479\) 23.2217 1.06103 0.530513 0.847677i \(-0.321999\pi\)
0.530513 + 0.847677i \(0.321999\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 18.5113 0.844043
\(482\) −5.70376 −0.259799
\(483\) −39.9002 −1.81552
\(484\) 0 0
\(485\) −5.74585 −0.260906
\(486\) 1.00000 0.0453609
\(487\) −38.9081 −1.76309 −0.881547 0.472096i \(-0.843498\pi\)
−0.881547 + 0.472096i \(0.843498\pi\)
\(488\) −15.4541 −0.699572
\(489\) 6.43124 0.290831
\(490\) −13.1687 −0.594902
\(491\) −4.66139 −0.210366 −0.105183 0.994453i \(-0.533543\pi\)
−0.105183 + 0.994453i \(0.533543\pi\)
\(492\) −9.56995 −0.431447
\(493\) −39.0999 −1.76097
\(494\) 14.0277 0.631136
\(495\) 0 0
\(496\) −1.10899 −0.0497952
\(497\) 42.8937 1.92405
\(498\) 8.00000 0.358489
\(499\) 16.3933 0.733866 0.366933 0.930247i \(-0.380408\pi\)
0.366933 + 0.930247i \(0.380408\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 6.04163 0.269920
\(502\) −2.60714 −0.116362
\(503\) 0.684656 0.0305273 0.0152636 0.999884i \(-0.495141\pi\)
0.0152636 + 0.999884i \(0.495141\pi\)
\(504\) 4.49096 0.200043
\(505\) −16.5739 −0.737531
\(506\) 0 0
\(507\) −7.55512 −0.335534
\(508\) 8.36343 0.371067
\(509\) 5.38550 0.238708 0.119354 0.992852i \(-0.461918\pi\)
0.119354 + 0.992852i \(0.461918\pi\)
\(510\) −5.15753 −0.228379
\(511\) −49.1503 −2.17428
\(512\) 1.00000 0.0441942
\(513\) 6.01163 0.265420
\(514\) 18.4360 0.813176
\(515\) 1.09063 0.0480588
\(516\) 0.0785371 0.00345740
\(517\) 0 0
\(518\) 35.6272 1.56537
\(519\) −0.648032 −0.0284455
\(520\) −2.33343 −0.102328
\(521\) 7.60668 0.333255 0.166627 0.986020i \(-0.446712\pi\)
0.166627 + 0.986020i \(0.446712\pi\)
\(522\) −7.58113 −0.331817
\(523\) 5.61204 0.245397 0.122699 0.992444i \(-0.460845\pi\)
0.122699 + 0.992444i \(0.460845\pi\)
\(524\) −7.40079 −0.323305
\(525\) 4.49096 0.196001
\(526\) 24.4285 1.06513
\(527\) −5.71966 −0.249152
\(528\) 0 0
\(529\) 55.9354 2.43197
\(530\) −0.745110 −0.0323655
\(531\) −5.32624 −0.231139
\(532\) 26.9980 1.17051
\(533\) −22.3308 −0.967254
\(534\) −7.41196 −0.320747
\(535\) 5.68494 0.245781
\(536\) −1.67821 −0.0724874
\(537\) 1.56231 0.0674185
\(538\) −11.0119 −0.474757
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) 44.8337 1.92755 0.963776 0.266712i \(-0.0859372\pi\)
0.963776 + 0.266712i \(0.0859372\pi\)
\(542\) 0.647750 0.0278233
\(543\) 18.4721 0.792715
\(544\) 5.15753 0.221127
\(545\) −8.94427 −0.383131
\(546\) 10.4793 0.448474
\(547\) −10.7578 −0.459969 −0.229984 0.973194i \(-0.573868\pi\)
−0.229984 + 0.973194i \(0.573868\pi\)
\(548\) −4.44214 −0.189759
\(549\) −15.4541 −0.659563
\(550\) 0 0
\(551\) −45.5750 −1.94156
\(552\) −8.88456 −0.378152
\(553\) 24.5772 1.04513
\(554\) −24.6304 −1.04645
\(555\) −7.93310 −0.336741
\(556\) −19.9074 −0.844260
\(557\) 21.0762 0.893029 0.446515 0.894776i \(-0.352665\pi\)
0.446515 + 0.894776i \(0.352665\pi\)
\(558\) −1.10899 −0.0469474
\(559\) 0.183261 0.00775109
\(560\) −4.49096 −0.189778
\(561\) 0 0
\(562\) 9.49170 0.400383
\(563\) −26.4216 −1.11354 −0.556769 0.830668i \(-0.687959\pi\)
−0.556769 + 0.830668i \(0.687959\pi\)
\(564\) 2.90264 0.122223
\(565\) −3.87292 −0.162935
\(566\) −20.2113 −0.849542
\(567\) 4.49096 0.188603
\(568\) 9.55113 0.400757
\(569\) −32.0892 −1.34525 −0.672624 0.739984i \(-0.734833\pi\)
−0.672624 + 0.739984i \(0.734833\pi\)
\(570\) −6.01163 −0.251800
\(571\) −0.842187 −0.0352444 −0.0176222 0.999845i \(-0.505610\pi\)
−0.0176222 + 0.999845i \(0.505610\pi\)
\(572\) 0 0
\(573\) −17.6720 −0.738260
\(574\) −42.9783 −1.79388
\(575\) −8.88456 −0.370512
\(576\) 1.00000 0.0416667
\(577\) −15.5655 −0.648001 −0.324000 0.946057i \(-0.605028\pi\)
−0.324000 + 0.946057i \(0.605028\pi\)
\(578\) 9.60013 0.399313
\(579\) 10.4954 0.436174
\(580\) 7.58113 0.314789
\(581\) 35.9277 1.49053
\(582\) 5.74585 0.238173
\(583\) 0 0
\(584\) −10.9443 −0.452877
\(585\) −2.33343 −0.0964754
\(586\) −1.19398 −0.0493229
\(587\) 19.2593 0.794918 0.397459 0.917620i \(-0.369892\pi\)
0.397459 + 0.917620i \(0.369892\pi\)
\(588\) 13.1687 0.543068
\(589\) −6.66685 −0.274703
\(590\) 5.32624 0.219278
\(591\) 10.0788 0.414587
\(592\) 7.93310 0.326048
\(593\) −21.1332 −0.867835 −0.433918 0.900953i \(-0.642869\pi\)
−0.433918 + 0.900953i \(0.642869\pi\)
\(594\) 0 0
\(595\) −23.1623 −0.949560
\(596\) 9.31461 0.381541
\(597\) −13.6906 −0.560318
\(598\) −20.7315 −0.847773
\(599\) −13.2217 −0.540224 −0.270112 0.962829i \(-0.587061\pi\)
−0.270112 + 0.962829i \(0.587061\pi\)
\(600\) 1.00000 0.0408248
\(601\) −39.2115 −1.59947 −0.799736 0.600352i \(-0.795027\pi\)
−0.799736 + 0.600352i \(0.795027\pi\)
\(602\) 0.352707 0.0143752
\(603\) −1.67821 −0.0683418
\(604\) −13.9029 −0.565702
\(605\) 0 0
\(606\) 16.5739 0.673270
\(607\) −8.33889 −0.338465 −0.169233 0.985576i \(-0.554129\pi\)
−0.169233 + 0.985576i \(0.554129\pi\)
\(608\) 6.01163 0.243804
\(609\) −34.0465 −1.37964
\(610\) 15.4541 0.625716
\(611\) 6.77310 0.274010
\(612\) 5.15753 0.208481
\(613\) 46.7966 1.89010 0.945049 0.326927i \(-0.106013\pi\)
0.945049 + 0.326927i \(0.106013\pi\)
\(614\) −11.3502 −0.458059
\(615\) 9.56995 0.385898
\(616\) 0 0
\(617\) −18.5692 −0.747568 −0.373784 0.927516i \(-0.621940\pi\)
−0.373784 + 0.927516i \(0.621940\pi\)
\(618\) −1.09063 −0.0438714
\(619\) −9.05390 −0.363907 −0.181953 0.983307i \(-0.558242\pi\)
−0.181953 + 0.983307i \(0.558242\pi\)
\(620\) 1.10899 0.0445382
\(621\) −8.88456 −0.356525
\(622\) −15.1766 −0.608528
\(623\) −33.2868 −1.33361
\(624\) 2.33343 0.0934119
\(625\) 1.00000 0.0400000
\(626\) 23.3570 0.933532
\(627\) 0 0
\(628\) 9.89175 0.394724
\(629\) 40.9152 1.63140
\(630\) −4.49096 −0.178924
\(631\) 37.0223 1.47383 0.736917 0.675983i \(-0.236281\pi\)
0.736917 + 0.675983i \(0.236281\pi\)
\(632\) 5.47259 0.217688
\(633\) −22.8481 −0.908131
\(634\) 10.6641 0.423526
\(635\) −8.36343 −0.331892
\(636\) 0.745110 0.0295455
\(637\) 30.7282 1.21750
\(638\) 0 0
\(639\) 9.55113 0.377837
\(640\) −1.00000 −0.0395285
\(641\) −10.8499 −0.428547 −0.214273 0.976774i \(-0.568738\pi\)
−0.214273 + 0.976774i \(0.568738\pi\)
\(642\) −5.68494 −0.224367
\(643\) 29.4482 1.16132 0.580662 0.814144i \(-0.302794\pi\)
0.580662 + 0.814144i \(0.302794\pi\)
\(644\) −39.9002 −1.57229
\(645\) −0.0785371 −0.00309239
\(646\) 31.0052 1.21988
\(647\) −33.9447 −1.33450 −0.667252 0.744832i \(-0.732530\pi\)
−0.667252 + 0.744832i \(0.732530\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 2.33343 0.0915246
\(651\) −4.98044 −0.195199
\(652\) 6.43124 0.251867
\(653\) −19.0906 −0.747074 −0.373537 0.927615i \(-0.621855\pi\)
−0.373537 + 0.927615i \(0.621855\pi\)
\(654\) 8.94427 0.349749
\(655\) 7.40079 0.289173
\(656\) −9.56995 −0.373644
\(657\) −10.9443 −0.426977
\(658\) 13.0356 0.508182
\(659\) 9.56904 0.372757 0.186378 0.982478i \(-0.440325\pi\)
0.186378 + 0.982478i \(0.440325\pi\)
\(660\) 0 0
\(661\) 16.7878 0.652968 0.326484 0.945203i \(-0.394136\pi\)
0.326484 + 0.945203i \(0.394136\pi\)
\(662\) −2.42434 −0.0942245
\(663\) 12.0347 0.467390
\(664\) 8.00000 0.310460
\(665\) −26.9980 −1.04694
\(666\) 7.93310 0.307401
\(667\) 67.3550 2.60799
\(668\) 6.04163 0.233758
\(669\) −0.490958 −0.0189815
\(670\) 1.67821 0.0648347
\(671\) 0 0
\(672\) 4.49096 0.173242
\(673\) −46.2878 −1.78426 −0.892131 0.451776i \(-0.850791\pi\)
−0.892131 + 0.451776i \(0.850791\pi\)
\(674\) −1.17664 −0.0453223
\(675\) 1.00000 0.0384900
\(676\) −7.55512 −0.290581
\(677\) −27.2450 −1.04711 −0.523554 0.851992i \(-0.675394\pi\)
−0.523554 + 0.851992i \(0.675394\pi\)
\(678\) 3.87292 0.148739
\(679\) 25.8044 0.990281
\(680\) −5.15753 −0.197782
\(681\) −1.49022 −0.0571053
\(682\) 0 0
\(683\) 3.62551 0.138726 0.0693631 0.997591i \(-0.477903\pi\)
0.0693631 + 0.997591i \(0.477903\pi\)
\(684\) 6.01163 0.229861
\(685\) 4.44214 0.169725
\(686\) 27.7034 1.05772
\(687\) −8.21798 −0.313536
\(688\) 0.0785371 0.00299420
\(689\) 1.73866 0.0662376
\(690\) 8.88456 0.338229
\(691\) −50.3666 −1.91604 −0.958018 0.286708i \(-0.907439\pi\)
−0.958018 + 0.286708i \(0.907439\pi\)
\(692\) −0.648032 −0.0246345
\(693\) 0 0
\(694\) −23.3570 −0.886619
\(695\) 19.9074 0.755129
\(696\) −7.58113 −0.287362
\(697\) −49.3573 −1.86954
\(698\) −0.375059 −0.0141962
\(699\) −6.63686 −0.251029
\(700\) 4.49096 0.169742
\(701\) 17.1399 0.647365 0.323683 0.946166i \(-0.395079\pi\)
0.323683 + 0.946166i \(0.395079\pi\)
\(702\) 2.33343 0.0880695
\(703\) 47.6909 1.79869
\(704\) 0 0
\(705\) −2.90264 −0.109320
\(706\) 2.07854 0.0782268
\(707\) 74.4329 2.79934
\(708\) −5.32624 −0.200172
\(709\) 29.1399 1.09437 0.547186 0.837011i \(-0.315699\pi\)
0.547186 + 0.837011i \(0.315699\pi\)
\(710\) −9.55113 −0.358448
\(711\) 5.47259 0.205238
\(712\) −7.41196 −0.277775
\(713\) 9.85291 0.368994
\(714\) 23.1623 0.866826
\(715\) 0 0
\(716\) 1.56231 0.0583861
\(717\) 6.09764 0.227721
\(718\) 0.180340 0.00673022
\(719\) 30.8625 1.15098 0.575488 0.817810i \(-0.304812\pi\)
0.575488 + 0.817810i \(0.304812\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −4.89796 −0.182410
\(722\) 17.1397 0.637875
\(723\) −5.70376 −0.212125
\(724\) 18.4721 0.686512
\(725\) −7.58113 −0.281556
\(726\) 0 0
\(727\) −3.64376 −0.135140 −0.0675699 0.997715i \(-0.521525\pi\)
−0.0675699 + 0.997715i \(0.521525\pi\)
\(728\) 10.4793 0.388390
\(729\) 1.00000 0.0370370
\(730\) 10.9443 0.405066
\(731\) 0.405057 0.0149816
\(732\) −15.4541 −0.571198
\(733\) 7.78765 0.287644 0.143822 0.989604i \(-0.454061\pi\)
0.143822 + 0.989604i \(0.454061\pi\)
\(734\) 11.6706 0.430768
\(735\) −13.1687 −0.485735
\(736\) −8.88456 −0.327489
\(737\) 0 0
\(738\) −9.56995 −0.352275
\(739\) 9.07455 0.333813 0.166906 0.985973i \(-0.446622\pi\)
0.166906 + 0.985973i \(0.446622\pi\)
\(740\) −7.93310 −0.291626
\(741\) 14.0277 0.515321
\(742\) 3.34626 0.122845
\(743\) −9.48702 −0.348045 −0.174022 0.984742i \(-0.555677\pi\)
−0.174022 + 0.984742i \(0.555677\pi\)
\(744\) −1.10899 −0.0406576
\(745\) −9.31461 −0.341261
\(746\) −10.1687 −0.372303
\(747\) 8.00000 0.292705
\(748\) 0 0
\(749\) −25.5308 −0.932876
\(750\) −1.00000 −0.0365148
\(751\) 49.4211 1.80340 0.901701 0.432359i \(-0.142319\pi\)
0.901701 + 0.432359i \(0.142319\pi\)
\(752\) 2.90264 0.105848
\(753\) −2.60714 −0.0950095
\(754\) −17.6900 −0.644232
\(755\) 13.9029 0.505979
\(756\) 4.49096 0.163335
\(757\) −25.2771 −0.918713 −0.459357 0.888252i \(-0.651920\pi\)
−0.459357 + 0.888252i \(0.651920\pi\)
\(758\) 11.0825 0.402535
\(759\) 0 0
\(760\) −6.01163 −0.218065
\(761\) −22.6301 −0.820341 −0.410171 0.912009i \(-0.634531\pi\)
−0.410171 + 0.912009i \(0.634531\pi\)
\(762\) 8.36343 0.302975
\(763\) 40.1684 1.45419
\(764\) −17.6720 −0.639352
\(765\) −5.15753 −0.186471
\(766\) 21.3082 0.769895
\(767\) −12.4284 −0.448763
\(768\) 1.00000 0.0360844
\(769\) −11.4015 −0.411150 −0.205575 0.978641i \(-0.565906\pi\)
−0.205575 + 0.978641i \(0.565906\pi\)
\(770\) 0 0
\(771\) 18.4360 0.663955
\(772\) 10.4954 0.377738
\(773\) 18.8896 0.679413 0.339706 0.940532i \(-0.389672\pi\)
0.339706 + 0.940532i \(0.389672\pi\)
\(774\) 0.0785371 0.00282296
\(775\) −1.10899 −0.0398362
\(776\) 5.74585 0.206264
\(777\) 35.6272 1.27812
\(778\) −37.7362 −1.35291
\(779\) −57.5310 −2.06126
\(780\) −2.33343 −0.0835501
\(781\) 0 0
\(782\) −45.8224 −1.63860
\(783\) −7.58113 −0.270927
\(784\) 13.1687 0.470311
\(785\) −9.89175 −0.353052
\(786\) −7.40079 −0.263977
\(787\) 30.9628 1.10371 0.551853 0.833942i \(-0.313921\pi\)
0.551853 + 0.833942i \(0.313921\pi\)
\(788\) 10.0788 0.359043
\(789\) 24.4285 0.869678
\(790\) −5.47259 −0.194706
\(791\) 17.3931 0.618429
\(792\) 0 0
\(793\) −36.0609 −1.28056
\(794\) −22.4348 −0.796180
\(795\) −0.745110 −0.0264263
\(796\) −13.6906 −0.485250
\(797\) −7.93634 −0.281120 −0.140560 0.990072i \(-0.544890\pi\)
−0.140560 + 0.990072i \(0.544890\pi\)
\(798\) 26.9980 0.955719
\(799\) 14.9705 0.529617
\(800\) 1.00000 0.0353553
\(801\) −7.41196 −0.261889
\(802\) −25.9002 −0.914567
\(803\) 0 0
\(804\) −1.67821 −0.0591857
\(805\) 39.9002 1.40630
\(806\) −2.58775 −0.0911497
\(807\) −11.0119 −0.387638
\(808\) 16.5739 0.583069
\(809\) 5.56276 0.195576 0.0977882 0.995207i \(-0.468823\pi\)
0.0977882 + 0.995207i \(0.468823\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 32.1021 1.12726 0.563628 0.826029i \(-0.309405\pi\)
0.563628 + 0.826029i \(0.309405\pi\)
\(812\) −34.0465 −1.19480
\(813\) 0.647750 0.0227176
\(814\) 0 0
\(815\) −6.43124 −0.225277
\(816\) 5.15753 0.180550
\(817\) 0.472136 0.0165179
\(818\) −24.7921 −0.866835
\(819\) 10.4793 0.366177
\(820\) 9.56995 0.334197
\(821\) −21.1255 −0.737286 −0.368643 0.929571i \(-0.620177\pi\)
−0.368643 + 0.929571i \(0.620177\pi\)
\(822\) −4.44214 −0.154937
\(823\) 4.94702 0.172442 0.0862211 0.996276i \(-0.472521\pi\)
0.0862211 + 0.996276i \(0.472521\pi\)
\(824\) −1.09063 −0.0379938
\(825\) 0 0
\(826\) −23.9199 −0.832280
\(827\) 44.5554 1.54934 0.774672 0.632364i \(-0.217915\pi\)
0.774672 + 0.632364i \(0.217915\pi\)
\(828\) −8.88456 −0.308760
\(829\) −21.6706 −0.752650 −0.376325 0.926488i \(-0.622812\pi\)
−0.376325 + 0.926488i \(0.622812\pi\)
\(830\) −8.00000 −0.277684
\(831\) −24.6304 −0.854419
\(832\) 2.33343 0.0808970
\(833\) 67.9180 2.35322
\(834\) −19.9074 −0.689336
\(835\) −6.04163 −0.209079
\(836\) 0 0
\(837\) −1.10899 −0.0383324
\(838\) 27.8969 0.963683
\(839\) 45.7186 1.57838 0.789190 0.614149i \(-0.210501\pi\)
0.789190 + 0.614149i \(0.210501\pi\)
\(840\) −4.49096 −0.154953
\(841\) 28.4735 0.981845
\(842\) 25.0052 0.861735
\(843\) 9.49170 0.326911
\(844\) −22.8481 −0.786464
\(845\) 7.55512 0.259904
\(846\) 2.90264 0.0997948
\(847\) 0 0
\(848\) 0.745110 0.0255872
\(849\) −20.2113 −0.693648
\(850\) 5.15753 0.176902
\(851\) −70.4820 −2.41609
\(852\) 9.55113 0.327216
\(853\) 3.27816 0.112242 0.0561210 0.998424i \(-0.482127\pi\)
0.0561210 + 0.998424i \(0.482127\pi\)
\(854\) −69.4035 −2.37494
\(855\) −6.01163 −0.205594
\(856\) −5.68494 −0.194307
\(857\) 0.297435 0.0101602 0.00508010 0.999987i \(-0.498383\pi\)
0.00508010 + 0.999987i \(0.498383\pi\)
\(858\) 0 0
\(859\) 39.7745 1.35709 0.678544 0.734560i \(-0.262611\pi\)
0.678544 + 0.734560i \(0.262611\pi\)
\(860\) −0.0785371 −0.00267809
\(861\) −42.9783 −1.46470
\(862\) −17.9624 −0.611800
\(863\) −15.0483 −0.512249 −0.256124 0.966644i \(-0.582446\pi\)
−0.256124 + 0.966644i \(0.582446\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0.648032 0.0220338
\(866\) −26.9458 −0.915654
\(867\) 9.60013 0.326037
\(868\) −4.98044 −0.169047
\(869\) 0 0
\(870\) 7.58113 0.257024
\(871\) −3.91597 −0.132688
\(872\) 8.94427 0.302891
\(873\) 5.74585 0.194468
\(874\) −53.4107 −1.80664
\(875\) −4.49096 −0.151822
\(876\) −10.9443 −0.369773
\(877\) −38.5856 −1.30294 −0.651471 0.758673i \(-0.725848\pi\)
−0.651471 + 0.758673i \(0.725848\pi\)
\(878\) −4.79393 −0.161787
\(879\) −1.19398 −0.0402719
\(880\) 0 0
\(881\) −14.2135 −0.478866 −0.239433 0.970913i \(-0.576962\pi\)
−0.239433 + 0.970913i \(0.576962\pi\)
\(882\) 13.1687 0.443413
\(883\) −2.27417 −0.0765319 −0.0382660 0.999268i \(-0.512183\pi\)
−0.0382660 + 0.999268i \(0.512183\pi\)
\(884\) 12.0347 0.404772
\(885\) 5.32624 0.179040
\(886\) 10.0600 0.337972
\(887\) −52.2657 −1.75491 −0.877455 0.479658i \(-0.840761\pi\)
−0.877455 + 0.479658i \(0.840761\pi\)
\(888\) 7.93310 0.266217
\(889\) 37.5598 1.25972
\(890\) 7.41196 0.248450
\(891\) 0 0
\(892\) −0.490958 −0.0164385
\(893\) 17.4496 0.583929
\(894\) 9.31461 0.311527
\(895\) −1.56231 −0.0522221
\(896\) 4.49096 0.150032
\(897\) −20.7315 −0.692204
\(898\) −0.498148 −0.0166234
\(899\) 8.40741 0.280403
\(900\) 1.00000 0.0333333
\(901\) 3.84293 0.128026
\(902\) 0 0
\(903\) 0.352707 0.0117373
\(904\) 3.87292 0.128812
\(905\) −18.4721 −0.614035
\(906\) −13.9029 −0.461894
\(907\) −53.0605 −1.76184 −0.880922 0.473262i \(-0.843076\pi\)
−0.880922 + 0.473262i \(0.843076\pi\)
\(908\) −1.49022 −0.0494547
\(909\) 16.5739 0.549723
\(910\) −10.4793 −0.347386
\(911\) −39.3570 −1.30395 −0.651977 0.758238i \(-0.726060\pi\)
−0.651977 + 0.758238i \(0.726060\pi\)
\(912\) 6.01163 0.199065
\(913\) 0 0
\(914\) −1.85730 −0.0614342
\(915\) 15.4541 0.510895
\(916\) −8.21798 −0.271530
\(917\) −33.2366 −1.09757
\(918\) 5.15753 0.170224
\(919\) 7.44841 0.245700 0.122850 0.992425i \(-0.460797\pi\)
0.122850 + 0.992425i \(0.460797\pi\)
\(920\) 8.88456 0.292915
\(921\) −11.3502 −0.374003
\(922\) −22.1394 −0.729124
\(923\) 22.2869 0.733581
\(924\) 0 0
\(925\) 7.93310 0.260839
\(926\) −22.1920 −0.729273
\(927\) −1.09063 −0.0358209
\(928\) −7.58113 −0.248863
\(929\) 7.70376 0.252752 0.126376 0.991982i \(-0.459665\pi\)
0.126376 + 0.991982i \(0.459665\pi\)
\(930\) 1.10899 0.0363653
\(931\) 79.1654 2.59454
\(932\) −6.63686 −0.217397
\(933\) −15.1766 −0.496861
\(934\) 31.6953 1.03710
\(935\) 0 0
\(936\) 2.33343 0.0762705
\(937\) −23.8429 −0.778914 −0.389457 0.921045i \(-0.627337\pi\)
−0.389457 + 0.921045i \(0.627337\pi\)
\(938\) −7.53675 −0.246084
\(939\) 23.3570 0.762226
\(940\) −2.90264 −0.0946737
\(941\) 20.4887 0.667912 0.333956 0.942589i \(-0.391616\pi\)
0.333956 + 0.942589i \(0.391616\pi\)
\(942\) 9.89175 0.322291
\(943\) 85.0248 2.76879
\(944\) −5.32624 −0.173354
\(945\) −4.49096 −0.146091
\(946\) 0 0
\(947\) 18.3389 0.595934 0.297967 0.954576i \(-0.403691\pi\)
0.297967 + 0.954576i \(0.403691\pi\)
\(948\) 5.47259 0.177742
\(949\) −25.5377 −0.828988
\(950\) 6.01163 0.195043
\(951\) 10.6641 0.345807
\(952\) 23.1623 0.750693
\(953\) 43.6768 1.41483 0.707415 0.706799i \(-0.249861\pi\)
0.707415 + 0.706799i \(0.249861\pi\)
\(954\) 0.745110 0.0241238
\(955\) 17.6720 0.571854
\(956\) 6.09764 0.197212
\(957\) 0 0
\(958\) 23.2217 0.750259
\(959\) −19.9495 −0.644201
\(960\) −1.00000 −0.0322749
\(961\) −29.7701 −0.960327
\(962\) 18.5113 0.596828
\(963\) −5.68494 −0.183195
\(964\) −5.70376 −0.183706
\(965\) −10.4954 −0.337859
\(966\) −39.9002 −1.28377
\(967\) 44.2296 1.42233 0.711164 0.703026i \(-0.248168\pi\)
0.711164 + 0.703026i \(0.248168\pi\)
\(968\) 0 0
\(969\) 31.0052 0.996030
\(970\) −5.74585 −0.184488
\(971\) 19.8847 0.638129 0.319065 0.947733i \(-0.396631\pi\)
0.319065 + 0.947733i \(0.396631\pi\)
\(972\) 1.00000 0.0320750
\(973\) −89.4032 −2.86613
\(974\) −38.9081 −1.24670
\(975\) 2.33343 0.0747295
\(976\) −15.4541 −0.494672
\(977\) −33.8058 −1.08154 −0.540772 0.841169i \(-0.681868\pi\)
−0.540772 + 0.841169i \(0.681868\pi\)
\(978\) 6.43124 0.205648
\(979\) 0 0
\(980\) −13.1687 −0.420659
\(981\) 8.94427 0.285569
\(982\) −4.66139 −0.148751
\(983\) −10.0792 −0.321476 −0.160738 0.986997i \(-0.551387\pi\)
−0.160738 + 0.986997i \(0.551387\pi\)
\(984\) −9.56995 −0.305079
\(985\) −10.0788 −0.321138
\(986\) −39.0999 −1.24519
\(987\) 13.0356 0.414929
\(988\) 14.0277 0.446281
\(989\) −0.697767 −0.0221877
\(990\) 0 0
\(991\) 25.6906 0.816088 0.408044 0.912962i \(-0.366211\pi\)
0.408044 + 0.912962i \(0.366211\pi\)
\(992\) −1.10899 −0.0352105
\(993\) −2.42434 −0.0769340
\(994\) 42.8937 1.36051
\(995\) 13.6906 0.434020
\(996\) 8.00000 0.253490
\(997\) −42.8994 −1.35864 −0.679318 0.733844i \(-0.737724\pi\)
−0.679318 + 0.733844i \(0.737724\pi\)
\(998\) 16.3933 0.518921
\(999\) 7.93310 0.250992
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 3630.2.a.bs.1.4 4
11.2 odd 10 330.2.m.f.301.1 yes 8
11.6 odd 10 330.2.m.f.91.1 8
11.10 odd 2 3630.2.a.bq.1.1 4
33.2 even 10 990.2.n.i.631.1 8
33.17 even 10 990.2.n.i.91.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
330.2.m.f.91.1 8 11.6 odd 10
330.2.m.f.301.1 yes 8 11.2 odd 10
990.2.n.i.91.1 8 33.17 even 10
990.2.n.i.631.1 8 33.2 even 10
3630.2.a.bq.1.1 4 11.10 odd 2
3630.2.a.bs.1.4 4 1.1 even 1 trivial