Properties

Label 8470.2.a.dd.1.3
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.745749504.1
Defining polynomial: \(x^{6} - 18 x^{4} - 4 x^{3} + 81 x^{2} + 36 x - 44\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.567932\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.567932 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.567932 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.67745 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.567932 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.567932 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.67745 q^{9} -1.00000 q^{10} -0.567932 q^{12} -5.26170 q^{13} -1.00000 q^{14} +0.567932 q^{15} +1.00000 q^{16} -5.00401 q^{17} -2.67745 q^{18} +2.96172 q^{19} -1.00000 q^{20} +0.567932 q^{21} +0.983687 q^{23} -0.567932 q^{24} +1.00000 q^{25} -5.26170 q^{26} +3.22441 q^{27} -1.00000 q^{28} +1.43608 q^{29} +0.567932 q^{30} -9.68146 q^{31} +1.00000 q^{32} -5.00401 q^{34} +1.00000 q^{35} -2.67745 q^{36} +7.43608 q^{37} +2.96172 q^{38} +2.98829 q^{39} -1.00000 q^{40} -9.74211 q^{41} +0.567932 q^{42} +5.89617 q^{43} +2.67745 q^{45} +0.983687 q^{46} -5.70949 q^{47} -0.567932 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.84194 q^{51} -5.26170 q^{52} +10.1844 q^{53} +3.22441 q^{54} -1.00000 q^{56} -1.68205 q^{57} +1.43608 q^{58} +7.39355 q^{59} +0.567932 q^{60} -7.04229 q^{61} -9.68146 q^{62} +2.67745 q^{63} +1.00000 q^{64} +5.26170 q^{65} -2.77639 q^{67} -5.00401 q^{68} -0.558667 q^{69} +1.00000 q^{70} -2.71008 q^{71} -2.67745 q^{72} +9.81960 q^{73} +7.43608 q^{74} -0.567932 q^{75} +2.96172 q^{76} +2.98829 q^{78} -0.874198 q^{79} -1.00000 q^{80} +6.20112 q^{81} -9.74211 q^{82} -2.65545 q^{83} +0.567932 q^{84} +5.00401 q^{85} +5.89617 q^{86} -0.815593 q^{87} -4.84535 q^{89} +2.67745 q^{90} +5.26170 q^{91} +0.983687 q^{92} +5.49841 q^{93} -5.70949 q^{94} -2.96172 q^{95} -0.567932 q^{96} +17.2128 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} + 6q^{4} - 6q^{5} - 6q^{7} + 6q^{8} + 18q^{9} + O(q^{10}) \) \( 6q + 6q^{2} + 6q^{4} - 6q^{5} - 6q^{7} + 6q^{8} + 18q^{9} - 6q^{10} - 6q^{14} + 6q^{16} - 6q^{17} + 18q^{18} - 6q^{20} + 6q^{25} - 12q^{27} - 6q^{28} - 12q^{29} + 6q^{32} - 6q^{34} + 6q^{35} + 18q^{36} + 24q^{37} + 24q^{39} - 6q^{40} - 12q^{41} + 18q^{43} - 18q^{45} + 24q^{47} + 6q^{49} + 6q^{50} + 12q^{51} + 36q^{53} - 12q^{54} - 6q^{56} + 12q^{57} - 12q^{58} + 30q^{59} - 36q^{61} - 18q^{63} + 6q^{64} - 12q^{67} - 6q^{68} + 6q^{70} + 6q^{71} + 18q^{72} + 6q^{73} + 24q^{74} + 24q^{78} + 24q^{79} - 6q^{80} + 54q^{81} - 12q^{82} - 24q^{83} + 6q^{85} + 18q^{86} + 24q^{87} + 36q^{89} - 18q^{90} + 24q^{94} + 6q^{98} + O(q^{100}) \)

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
\(3\) −0.567932 −0.327896 −0.163948 0.986469i \(-0.552423\pi\)
−0.163948 + 0.986469i \(0.552423\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −0.567932 −0.231857
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.67745 −0.892484
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −0.567932 −0.163948
\(13\) −5.26170 −1.45933 −0.729666 0.683803i \(-0.760325\pi\)
−0.729666 + 0.683803i \(0.760325\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0.567932 0.146639
\(16\) 1.00000 0.250000
\(17\) −5.00401 −1.21365 −0.606825 0.794835i \(-0.707557\pi\)
−0.606825 + 0.794835i \(0.707557\pi\)
\(18\) −2.67745 −0.631082
\(19\) 2.96172 0.679464 0.339732 0.940522i \(-0.389664\pi\)
0.339732 + 0.940522i \(0.389664\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0.567932 0.123933
\(22\) 0 0
\(23\) 0.983687 0.205113 0.102556 0.994727i \(-0.467298\pi\)
0.102556 + 0.994727i \(0.467298\pi\)
\(24\) −0.567932 −0.115929
\(25\) 1.00000 0.200000
\(26\) −5.26170 −1.03190
\(27\) 3.22441 0.620537
\(28\) −1.00000 −0.188982
\(29\) 1.43608 0.266672 0.133336 0.991071i \(-0.457431\pi\)
0.133336 + 0.991071i \(0.457431\pi\)
\(30\) 0.567932 0.103690
\(31\) −9.68146 −1.73884 −0.869421 0.494072i \(-0.835508\pi\)
−0.869421 + 0.494072i \(0.835508\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −5.00401 −0.858180
\(35\) 1.00000 0.169031
\(36\) −2.67745 −0.446242
\(37\) 7.43608 1.22248 0.611242 0.791444i \(-0.290670\pi\)
0.611242 + 0.791444i \(0.290670\pi\)
\(38\) 2.96172 0.480454
\(39\) 2.98829 0.478509
\(40\) −1.00000 −0.158114
\(41\) −9.74211 −1.52146 −0.760731 0.649067i \(-0.775159\pi\)
−0.760731 + 0.649067i \(0.775159\pi\)
\(42\) 0.567932 0.0876338
\(43\) 5.89617 0.899157 0.449579 0.893241i \(-0.351574\pi\)
0.449579 + 0.893241i \(0.351574\pi\)
\(44\) 0 0
\(45\) 2.67745 0.399131
\(46\) 0.983687 0.145037
\(47\) −5.70949 −0.832814 −0.416407 0.909178i \(-0.636711\pi\)
−0.416407 + 0.909178i \(0.636711\pi\)
\(48\) −0.567932 −0.0819739
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 2.84194 0.397951
\(52\) −5.26170 −0.729666
\(53\) 10.1844 1.39894 0.699470 0.714662i \(-0.253419\pi\)
0.699470 + 0.714662i \(0.253419\pi\)
\(54\) 3.22441 0.438786
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −1.68205 −0.222793
\(58\) 1.43608 0.188566
\(59\) 7.39355 0.962559 0.481279 0.876567i \(-0.340172\pi\)
0.481279 + 0.876567i \(0.340172\pi\)
\(60\) 0.567932 0.0733197
\(61\) −7.04229 −0.901673 −0.450837 0.892607i \(-0.648874\pi\)
−0.450837 + 0.892607i \(0.648874\pi\)
\(62\) −9.68146 −1.22955
\(63\) 2.67745 0.337327
\(64\) 1.00000 0.125000
\(65\) 5.26170 0.652633
\(66\) 0 0
\(67\) −2.77639 −0.339190 −0.169595 0.985514i \(-0.554246\pi\)
−0.169595 + 0.985514i \(0.554246\pi\)
\(68\) −5.00401 −0.606825
\(69\) −0.558667 −0.0672556
\(70\) 1.00000 0.119523
\(71\) −2.71008 −0.321627 −0.160814 0.986985i \(-0.551412\pi\)
−0.160814 + 0.986985i \(0.551412\pi\)
\(72\) −2.67745 −0.315541
\(73\) 9.81960 1.14930 0.574649 0.818400i \(-0.305139\pi\)
0.574649 + 0.818400i \(0.305139\pi\)
\(74\) 7.43608 0.864426
\(75\) −0.567932 −0.0655791
\(76\) 2.96172 0.339732
\(77\) 0 0
\(78\) 2.98829 0.338357
\(79\) −0.874198 −0.0983550 −0.0491775 0.998790i \(-0.515660\pi\)
−0.0491775 + 0.998790i \(0.515660\pi\)
\(80\) −1.00000 −0.111803
\(81\) 6.20112 0.689013
\(82\) −9.74211 −1.07584
\(83\) −2.65545 −0.291473 −0.145737 0.989323i \(-0.546555\pi\)
−0.145737 + 0.989323i \(0.546555\pi\)
\(84\) 0.567932 0.0619665
\(85\) 5.00401 0.542761
\(86\) 5.89617 0.635800
\(87\) −0.815593 −0.0874408
\(88\) 0 0
\(89\) −4.84535 −0.513606 −0.256803 0.966464i \(-0.582669\pi\)
−0.256803 + 0.966464i \(0.582669\pi\)
\(90\) 2.67745 0.282228
\(91\) 5.26170 0.551576
\(92\) 0.983687 0.102556
\(93\) 5.49841 0.570159
\(94\) −5.70949 −0.588889
\(95\) −2.96172 −0.303866
\(96\) −0.567932 −0.0579643
\(97\) 17.2128 1.74769 0.873847 0.486201i \(-0.161618\pi\)
0.873847 + 0.486201i \(0.161618\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 15.0324 1.49578 0.747888 0.663825i \(-0.231068\pi\)
0.747888 + 0.663825i \(0.231068\pi\)
\(102\) 2.84194 0.281394
\(103\) −7.80766 −0.769311 −0.384656 0.923060i \(-0.625680\pi\)
−0.384656 + 0.923060i \(0.625680\pi\)
\(104\) −5.26170 −0.515952
\(105\) −0.567932 −0.0554245
\(106\) 10.1844 0.989200
\(107\) 4.64943 0.449477 0.224739 0.974419i \(-0.427847\pi\)
0.224739 + 0.974419i \(0.427847\pi\)
\(108\) 3.22441 0.310269
\(109\) −4.78588 −0.458404 −0.229202 0.973379i \(-0.573612\pi\)
−0.229202 + 0.973379i \(0.573612\pi\)
\(110\) 0 0
\(111\) −4.22319 −0.400847
\(112\) −1.00000 −0.0944911
\(113\) −0.418079 −0.0393295 −0.0196648 0.999807i \(-0.506260\pi\)
−0.0196648 + 0.999807i \(0.506260\pi\)
\(114\) −1.68205 −0.157539
\(115\) −0.983687 −0.0917293
\(116\) 1.43608 0.133336
\(117\) 14.0880 1.30243
\(118\) 7.39355 0.680632
\(119\) 5.00401 0.458717
\(120\) 0.567932 0.0518449
\(121\) 0 0
\(122\) −7.04229 −0.637579
\(123\) 5.53286 0.498881
\(124\) −9.68146 −0.869421
\(125\) −1.00000 −0.0894427
\(126\) 2.67745 0.238526
\(127\) −18.7745 −1.66597 −0.832983 0.553299i \(-0.813369\pi\)
−0.832983 + 0.553299i \(0.813369\pi\)
\(128\) 1.00000 0.0883883
\(129\) −3.34862 −0.294830
\(130\) 5.26170 0.461481
\(131\) 17.7838 1.55378 0.776891 0.629635i \(-0.216796\pi\)
0.776891 + 0.629635i \(0.216796\pi\)
\(132\) 0 0
\(133\) −2.96172 −0.256813
\(134\) −2.77639 −0.239844
\(135\) −3.22441 −0.277513
\(136\) −5.00401 −0.429090
\(137\) 12.9900 1.10981 0.554904 0.831914i \(-0.312755\pi\)
0.554904 + 0.831914i \(0.312755\pi\)
\(138\) −0.558667 −0.0475569
\(139\) 13.2415 1.12313 0.561563 0.827434i \(-0.310200\pi\)
0.561563 + 0.827434i \(0.310200\pi\)
\(140\) 1.00000 0.0845154
\(141\) 3.24260 0.273076
\(142\) −2.71008 −0.227425
\(143\) 0 0
\(144\) −2.67745 −0.223121
\(145\) −1.43608 −0.119260
\(146\) 9.81960 0.812676
\(147\) −0.567932 −0.0468422
\(148\) 7.43608 0.611242
\(149\) 9.63123 0.789021 0.394511 0.918891i \(-0.370914\pi\)
0.394511 + 0.918891i \(0.370914\pi\)
\(150\) −0.567932 −0.0463715
\(151\) 12.9025 1.04999 0.524996 0.851105i \(-0.324067\pi\)
0.524996 + 0.851105i \(0.324067\pi\)
\(152\) 2.96172 0.240227
\(153\) 13.3980 1.08316
\(154\) 0 0
\(155\) 9.68146 0.777634
\(156\) 2.98829 0.239254
\(157\) −4.31464 −0.344346 −0.172173 0.985067i \(-0.555079\pi\)
−0.172173 + 0.985067i \(0.555079\pi\)
\(158\) −0.874198 −0.0695475
\(159\) −5.78407 −0.458706
\(160\) −1.00000 −0.0790569
\(161\) −0.983687 −0.0775254
\(162\) 6.20112 0.487206
\(163\) −11.3379 −0.888053 −0.444027 0.896014i \(-0.646450\pi\)
−0.444027 + 0.896014i \(0.646450\pi\)
\(164\) −9.74211 −0.760731
\(165\) 0 0
\(166\) −2.65545 −0.206103
\(167\) 5.03663 0.389746 0.194873 0.980828i \(-0.437570\pi\)
0.194873 + 0.980828i \(0.437570\pi\)
\(168\) 0.567932 0.0438169
\(169\) 14.6855 1.12965
\(170\) 5.00401 0.383790
\(171\) −7.92985 −0.606411
\(172\) 5.89617 0.449579
\(173\) −19.9314 −1.51536 −0.757680 0.652626i \(-0.773667\pi\)
−0.757680 + 0.652626i \(0.773667\pi\)
\(174\) −0.815593 −0.0618300
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −4.19904 −0.315619
\(178\) −4.84535 −0.363174
\(179\) −3.70949 −0.277260 −0.138630 0.990344i \(-0.544270\pi\)
−0.138630 + 0.990344i \(0.544270\pi\)
\(180\) 2.67745 0.199566
\(181\) 15.6470 1.16303 0.581514 0.813536i \(-0.302460\pi\)
0.581514 + 0.813536i \(0.302460\pi\)
\(182\) 5.26170 0.390023
\(183\) 3.99954 0.295655
\(184\) 0.983687 0.0725184
\(185\) −7.43608 −0.546711
\(186\) 5.49841 0.403163
\(187\) 0 0
\(188\) −5.70949 −0.416407
\(189\) −3.22441 −0.234541
\(190\) −2.96172 −0.214865
\(191\) 7.13186 0.516043 0.258022 0.966139i \(-0.416929\pi\)
0.258022 + 0.966139i \(0.416929\pi\)
\(192\) −0.567932 −0.0409870
\(193\) −12.9824 −0.934496 −0.467248 0.884126i \(-0.654755\pi\)
−0.467248 + 0.884126i \(0.654755\pi\)
\(194\) 17.2128 1.23581
\(195\) −2.98829 −0.213996
\(196\) 1.00000 0.0714286
\(197\) 8.92115 0.635606 0.317803 0.948157i \(-0.397055\pi\)
0.317803 + 0.948157i \(0.397055\pi\)
\(198\) 0 0
\(199\) −2.90018 −0.205588 −0.102794 0.994703i \(-0.532778\pi\)
−0.102794 + 0.994703i \(0.532778\pi\)
\(200\) 1.00000 0.0707107
\(201\) 1.57680 0.111219
\(202\) 15.0324 1.05767
\(203\) −1.43608 −0.100793
\(204\) 2.84194 0.198975
\(205\) 9.74211 0.680419
\(206\) −7.80766 −0.543985
\(207\) −2.63378 −0.183060
\(208\) −5.26170 −0.364833
\(209\) 0 0
\(210\) −0.567932 −0.0391910
\(211\) 23.6547 1.62845 0.814227 0.580547i \(-0.197161\pi\)
0.814227 + 0.580547i \(0.197161\pi\)
\(212\) 10.1844 0.699470
\(213\) 1.53914 0.105460
\(214\) 4.64943 0.317828
\(215\) −5.89617 −0.402115
\(216\) 3.22441 0.219393
\(217\) 9.68146 0.657220
\(218\) −4.78588 −0.324141
\(219\) −5.57687 −0.376850
\(220\) 0 0
\(221\) 26.3296 1.77112
\(222\) −4.22319 −0.283442
\(223\) 0.387500 0.0259489 0.0129745 0.999916i \(-0.495870\pi\)
0.0129745 + 0.999916i \(0.495870\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −2.67745 −0.178497
\(226\) −0.418079 −0.0278102
\(227\) −15.0347 −0.997887 −0.498943 0.866635i \(-0.666278\pi\)
−0.498943 + 0.866635i \(0.666278\pi\)
\(228\) −1.68205 −0.111397
\(229\) 8.17418 0.540165 0.270083 0.962837i \(-0.412949\pi\)
0.270083 + 0.962837i \(0.412949\pi\)
\(230\) −0.983687 −0.0648624
\(231\) 0 0
\(232\) 1.43608 0.0942830
\(233\) 17.6289 1.15491 0.577454 0.816423i \(-0.304046\pi\)
0.577454 + 0.816423i \(0.304046\pi\)
\(234\) 14.0880 0.920958
\(235\) 5.70949 0.372446
\(236\) 7.39355 0.481279
\(237\) 0.496485 0.0322502
\(238\) 5.00401 0.324362
\(239\) −28.2746 −1.82893 −0.914467 0.404660i \(-0.867390\pi\)
−0.914467 + 0.404660i \(0.867390\pi\)
\(240\) 0.567932 0.0366599
\(241\) −7.13172 −0.459395 −0.229697 0.973262i \(-0.573774\pi\)
−0.229697 + 0.973262i \(0.573774\pi\)
\(242\) 0 0
\(243\) −13.1950 −0.846462
\(244\) −7.04229 −0.450837
\(245\) −1.00000 −0.0638877
\(246\) 5.53286 0.352762
\(247\) −15.5837 −0.991564
\(248\) −9.68146 −0.614773
\(249\) 1.50811 0.0955729
\(250\) −1.00000 −0.0632456
\(251\) 26.8166 1.69265 0.846323 0.532670i \(-0.178811\pi\)
0.846323 + 0.532670i \(0.178811\pi\)
\(252\) 2.67745 0.168664
\(253\) 0 0
\(254\) −18.7745 −1.17802
\(255\) −2.84194 −0.177969
\(256\) 1.00000 0.0625000
\(257\) 20.6513 1.28819 0.644096 0.764945i \(-0.277234\pi\)
0.644096 + 0.764945i \(0.277234\pi\)
\(258\) −3.34862 −0.208476
\(259\) −7.43608 −0.462055
\(260\) 5.26170 0.326317
\(261\) −3.84502 −0.238001
\(262\) 17.7838 1.09869
\(263\) −18.9482 −1.16840 −0.584198 0.811611i \(-0.698591\pi\)
−0.584198 + 0.811611i \(0.698591\pi\)
\(264\) 0 0
\(265\) −10.1844 −0.625625
\(266\) −2.96172 −0.181594
\(267\) 2.75183 0.168409
\(268\) −2.77639 −0.169595
\(269\) 26.3034 1.60375 0.801874 0.597493i \(-0.203836\pi\)
0.801874 + 0.597493i \(0.203836\pi\)
\(270\) −3.22441 −0.196231
\(271\) 6.40481 0.389064 0.194532 0.980896i \(-0.437681\pi\)
0.194532 + 0.980896i \(0.437681\pi\)
\(272\) −5.00401 −0.303413
\(273\) −2.98829 −0.180859
\(274\) 12.9900 0.784753
\(275\) 0 0
\(276\) −0.558667 −0.0336278
\(277\) 3.41388 0.205120 0.102560 0.994727i \(-0.467297\pi\)
0.102560 + 0.994727i \(0.467297\pi\)
\(278\) 13.2415 0.794170
\(279\) 25.9217 1.55189
\(280\) 1.00000 0.0597614
\(281\) −9.93058 −0.592409 −0.296204 0.955125i \(-0.595721\pi\)
−0.296204 + 0.955125i \(0.595721\pi\)
\(282\) 3.24260 0.193094
\(283\) 14.3247 0.851516 0.425758 0.904837i \(-0.360008\pi\)
0.425758 + 0.904837i \(0.360008\pi\)
\(284\) −2.71008 −0.160814
\(285\) 1.68205 0.0996362
\(286\) 0 0
\(287\) 9.74211 0.575059
\(288\) −2.67745 −0.157770
\(289\) 8.04009 0.472946
\(290\) −1.43608 −0.0843292
\(291\) −9.77569 −0.573061
\(292\) 9.81960 0.574649
\(293\) 13.0495 0.762359 0.381180 0.924501i \(-0.375518\pi\)
0.381180 + 0.924501i \(0.375518\pi\)
\(294\) −0.567932 −0.0331225
\(295\) −7.39355 −0.430469
\(296\) 7.43608 0.432213
\(297\) 0 0
\(298\) 9.63123 0.557922
\(299\) −5.17586 −0.299328
\(300\) −0.567932 −0.0327896
\(301\) −5.89617 −0.339850
\(302\) 12.9025 0.742456
\(303\) −8.53736 −0.490458
\(304\) 2.96172 0.169866
\(305\) 7.04229 0.403240
\(306\) 13.3980 0.765912
\(307\) 12.2190 0.697378 0.348689 0.937239i \(-0.386627\pi\)
0.348689 + 0.937239i \(0.386627\pi\)
\(308\) 0 0
\(309\) 4.43422 0.252254
\(310\) 9.68146 0.549870
\(311\) 32.3753 1.83583 0.917917 0.396772i \(-0.129870\pi\)
0.917917 + 0.396772i \(0.129870\pi\)
\(312\) 2.98829 0.169178
\(313\) −33.0363 −1.86732 −0.933660 0.358159i \(-0.883404\pi\)
−0.933660 + 0.358159i \(0.883404\pi\)
\(314\) −4.31464 −0.243489
\(315\) −2.67745 −0.150857
\(316\) −0.874198 −0.0491775
\(317\) −18.7597 −1.05365 −0.526826 0.849973i \(-0.676618\pi\)
−0.526826 + 0.849973i \(0.676618\pi\)
\(318\) −5.78407 −0.324354
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −2.64056 −0.147382
\(322\) −0.983687 −0.0548187
\(323\) −14.8204 −0.824632
\(324\) 6.20112 0.344506
\(325\) −5.26170 −0.291867
\(326\) −11.3379 −0.627948
\(327\) 2.71806 0.150309
\(328\) −9.74211 −0.537918
\(329\) 5.70949 0.314774
\(330\) 0 0
\(331\) 17.5094 0.962403 0.481201 0.876610i \(-0.340201\pi\)
0.481201 + 0.876610i \(0.340201\pi\)
\(332\) −2.65545 −0.145737
\(333\) −19.9097 −1.09105
\(334\) 5.03663 0.275592
\(335\) 2.77639 0.151690
\(336\) 0.567932 0.0309832
\(337\) −19.9652 −1.08757 −0.543786 0.839224i \(-0.683010\pi\)
−0.543786 + 0.839224i \(0.683010\pi\)
\(338\) 14.6855 0.798784
\(339\) 0.237440 0.0128960
\(340\) 5.00401 0.271380
\(341\) 0 0
\(342\) −7.92985 −0.428797
\(343\) −1.00000 −0.0539949
\(344\) 5.89617 0.317900
\(345\) 0.558667 0.0300776
\(346\) −19.9314 −1.07152
\(347\) −35.7762 −1.92057 −0.960284 0.279026i \(-0.909989\pi\)
−0.960284 + 0.279026i \(0.909989\pi\)
\(348\) −0.815593 −0.0437204
\(349\) 3.27265 0.175181 0.0875904 0.996157i \(-0.472083\pi\)
0.0875904 + 0.996157i \(0.472083\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −16.9659 −0.905571
\(352\) 0 0
\(353\) −3.21106 −0.170907 −0.0854537 0.996342i \(-0.527234\pi\)
−0.0854537 + 0.996342i \(0.527234\pi\)
\(354\) −4.19904 −0.223176
\(355\) 2.71008 0.143836
\(356\) −4.84535 −0.256803
\(357\) −2.84194 −0.150411
\(358\) −3.70949 −0.196052
\(359\) −5.16418 −0.272555 −0.136278 0.990671i \(-0.543514\pi\)
−0.136278 + 0.990671i \(0.543514\pi\)
\(360\) 2.67745 0.141114
\(361\) −10.2282 −0.538329
\(362\) 15.6470 0.822386
\(363\) 0 0
\(364\) 5.26170 0.275788
\(365\) −9.81960 −0.513981
\(366\) 3.99954 0.209059
\(367\) −21.0921 −1.10100 −0.550500 0.834835i \(-0.685563\pi\)
−0.550500 + 0.834835i \(0.685563\pi\)
\(368\) 0.983687 0.0512782
\(369\) 26.0841 1.35788
\(370\) −7.43608 −0.386583
\(371\) −10.1844 −0.528750
\(372\) 5.49841 0.285079
\(373\) 20.7449 1.07413 0.537066 0.843540i \(-0.319533\pi\)
0.537066 + 0.843540i \(0.319533\pi\)
\(374\) 0 0
\(375\) 0.567932 0.0293279
\(376\) −5.70949 −0.294444
\(377\) −7.55620 −0.389164
\(378\) −3.22441 −0.165846
\(379\) 31.1753 1.60137 0.800683 0.599088i \(-0.204470\pi\)
0.800683 + 0.599088i \(0.204470\pi\)
\(380\) −2.96172 −0.151933
\(381\) 10.6626 0.546263
\(382\) 7.13186 0.364898
\(383\) −23.5582 −1.20377 −0.601883 0.798584i \(-0.705583\pi\)
−0.601883 + 0.798584i \(0.705583\pi\)
\(384\) −0.567932 −0.0289822
\(385\) 0 0
\(386\) −12.9824 −0.660789
\(387\) −15.7867 −0.802484
\(388\) 17.2128 0.873847
\(389\) −7.39973 −0.375181 −0.187591 0.982247i \(-0.560068\pi\)
−0.187591 + 0.982247i \(0.560068\pi\)
\(390\) −2.98829 −0.151318
\(391\) −4.92238 −0.248935
\(392\) 1.00000 0.0505076
\(393\) −10.1000 −0.509478
\(394\) 8.92115 0.449441
\(395\) 0.874198 0.0439857
\(396\) 0 0
\(397\) −15.2670 −0.766227 −0.383113 0.923701i \(-0.625148\pi\)
−0.383113 + 0.923701i \(0.625148\pi\)
\(398\) −2.90018 −0.145373
\(399\) 1.68205 0.0842080
\(400\) 1.00000 0.0500000
\(401\) 35.4584 1.77071 0.885355 0.464916i \(-0.153916\pi\)
0.885355 + 0.464916i \(0.153916\pi\)
\(402\) 1.57680 0.0786437
\(403\) 50.9409 2.53755
\(404\) 15.0324 0.747888
\(405\) −6.20112 −0.308136
\(406\) −1.43608 −0.0712712
\(407\) 0 0
\(408\) 2.84194 0.140697
\(409\) −17.2938 −0.855123 −0.427562 0.903986i \(-0.640627\pi\)
−0.427562 + 0.903986i \(0.640627\pi\)
\(410\) 9.74211 0.481129
\(411\) −7.37742 −0.363901
\(412\) −7.80766 −0.384656
\(413\) −7.39355 −0.363813
\(414\) −2.63378 −0.129443
\(415\) 2.65545 0.130351
\(416\) −5.26170 −0.257976
\(417\) −7.52025 −0.368268
\(418\) 0 0
\(419\) −3.77494 −0.184418 −0.0922088 0.995740i \(-0.529393\pi\)
−0.0922088 + 0.995740i \(0.529393\pi\)
\(420\) −0.567932 −0.0277122
\(421\) 3.56528 0.173761 0.0868806 0.996219i \(-0.472310\pi\)
0.0868806 + 0.996219i \(0.472310\pi\)
\(422\) 23.6547 1.15149
\(423\) 15.2869 0.743274
\(424\) 10.1844 0.494600
\(425\) −5.00401 −0.242730
\(426\) 1.53914 0.0745716
\(427\) 7.04229 0.340800
\(428\) 4.64943 0.224739
\(429\) 0 0
\(430\) −5.89617 −0.284339
\(431\) −2.54773 −0.122720 −0.0613600 0.998116i \(-0.519544\pi\)
−0.0613600 + 0.998116i \(0.519544\pi\)
\(432\) 3.22441 0.155134
\(433\) −2.45692 −0.118072 −0.0590360 0.998256i \(-0.518803\pi\)
−0.0590360 + 0.998256i \(0.518803\pi\)
\(434\) 9.68146 0.464725
\(435\) 0.815593 0.0391047
\(436\) −4.78588 −0.229202
\(437\) 2.91340 0.139367
\(438\) −5.57687 −0.266473
\(439\) −5.99413 −0.286084 −0.143042 0.989717i \(-0.545688\pi\)
−0.143042 + 0.989717i \(0.545688\pi\)
\(440\) 0 0
\(441\) −2.67745 −0.127498
\(442\) 26.3296 1.25237
\(443\) 18.2630 0.867701 0.433850 0.900985i \(-0.357155\pi\)
0.433850 + 0.900985i \(0.357155\pi\)
\(444\) −4.22319 −0.200423
\(445\) 4.84535 0.229692
\(446\) 0.387500 0.0183487
\(447\) −5.46989 −0.258717
\(448\) −1.00000 −0.0472456
\(449\) −4.60321 −0.217239 −0.108619 0.994083i \(-0.534643\pi\)
−0.108619 + 0.994083i \(0.534643\pi\)
\(450\) −2.67745 −0.126216
\(451\) 0 0
\(452\) −0.418079 −0.0196648
\(453\) −7.32775 −0.344288
\(454\) −15.0347 −0.705613
\(455\) −5.26170 −0.246672
\(456\) −1.68205 −0.0787693
\(457\) 20.3512 0.951990 0.475995 0.879448i \(-0.342088\pi\)
0.475995 + 0.879448i \(0.342088\pi\)
\(458\) 8.17418 0.381954
\(459\) −16.1350 −0.753115
\(460\) −0.983687 −0.0458646
\(461\) −22.1050 −1.02953 −0.514765 0.857331i \(-0.672121\pi\)
−0.514765 + 0.857331i \(0.672121\pi\)
\(462\) 0 0
\(463\) 30.5074 1.41780 0.708899 0.705310i \(-0.249192\pi\)
0.708899 + 0.705310i \(0.249192\pi\)
\(464\) 1.43608 0.0666681
\(465\) −5.49841 −0.254983
\(466\) 17.6289 0.816644
\(467\) 10.5885 0.489978 0.244989 0.969526i \(-0.421216\pi\)
0.244989 + 0.969526i \(0.421216\pi\)
\(468\) 14.0880 0.651216
\(469\) 2.77639 0.128202
\(470\) 5.70949 0.263359
\(471\) 2.45042 0.112910
\(472\) 7.39355 0.340316
\(473\) 0 0
\(474\) 0.496485 0.0228043
\(475\) 2.96172 0.135893
\(476\) 5.00401 0.229358
\(477\) −27.2684 −1.24853
\(478\) −28.2746 −1.29325
\(479\) −39.9165 −1.82383 −0.911915 0.410378i \(-0.865397\pi\)
−0.911915 + 0.410378i \(0.865397\pi\)
\(480\) 0.567932 0.0259224
\(481\) −39.1264 −1.78401
\(482\) −7.13172 −0.324841
\(483\) 0.558667 0.0254202
\(484\) 0 0
\(485\) −17.2128 −0.781593
\(486\) −13.1950 −0.598539
\(487\) 38.0184 1.72278 0.861390 0.507944i \(-0.169594\pi\)
0.861390 + 0.507944i \(0.169594\pi\)
\(488\) −7.04229 −0.318790
\(489\) 6.43916 0.291189
\(490\) −1.00000 −0.0451754
\(491\) 19.3973 0.875387 0.437693 0.899124i \(-0.355796\pi\)
0.437693 + 0.899124i \(0.355796\pi\)
\(492\) 5.53286 0.249440
\(493\) −7.18613 −0.323647
\(494\) −15.5837 −0.701142
\(495\) 0 0
\(496\) −9.68146 −0.434710
\(497\) 2.71008 0.121564
\(498\) 1.50811 0.0675802
\(499\) 2.76107 0.123603 0.0618013 0.998088i \(-0.480316\pi\)
0.0618013 + 0.998088i \(0.480316\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −2.86047 −0.127796
\(502\) 26.8166 1.19688
\(503\) 8.34764 0.372203 0.186102 0.982531i \(-0.440415\pi\)
0.186102 + 0.982531i \(0.440415\pi\)
\(504\) 2.67745 0.119263
\(505\) −15.0324 −0.668931
\(506\) 0 0
\(507\) −8.34035 −0.370408
\(508\) −18.7745 −0.832983
\(509\) 13.2459 0.587115 0.293557 0.955941i \(-0.405161\pi\)
0.293557 + 0.955941i \(0.405161\pi\)
\(510\) −2.84194 −0.125843
\(511\) −9.81960 −0.434394
\(512\) 1.00000 0.0441942
\(513\) 9.54978 0.421633
\(514\) 20.6513 0.910889
\(515\) 7.80766 0.344047
\(516\) −3.34862 −0.147415
\(517\) 0 0
\(518\) −7.43608 −0.326722
\(519\) 11.3197 0.496880
\(520\) 5.26170 0.230741
\(521\) 31.8038 1.39335 0.696674 0.717388i \(-0.254662\pi\)
0.696674 + 0.717388i \(0.254662\pi\)
\(522\) −3.84502 −0.168292
\(523\) −19.7709 −0.864519 −0.432260 0.901749i \(-0.642284\pi\)
−0.432260 + 0.901749i \(0.642284\pi\)
\(524\) 17.7838 0.776891
\(525\) 0.567932 0.0247866
\(526\) −18.9482 −0.826180
\(527\) 48.4461 2.11035
\(528\) 0 0
\(529\) −22.0324 −0.957929
\(530\) −10.1844 −0.442384
\(531\) −19.7959 −0.859069
\(532\) −2.96172 −0.128407
\(533\) 51.2601 2.22032
\(534\) 2.75183 0.119083
\(535\) −4.64943 −0.201012
\(536\) −2.77639 −0.119922
\(537\) 2.10674 0.0909124
\(538\) 26.3034 1.13402
\(539\) 0 0
\(540\) −3.22441 −0.138756
\(541\) 14.0034 0.602052 0.301026 0.953616i \(-0.402671\pi\)
0.301026 + 0.953616i \(0.402671\pi\)
\(542\) 6.40481 0.275110
\(543\) −8.88641 −0.381352
\(544\) −5.00401 −0.214545
\(545\) 4.78588 0.205005
\(546\) −2.98829 −0.127887
\(547\) −31.9169 −1.36467 −0.682334 0.731041i \(-0.739035\pi\)
−0.682334 + 0.731041i \(0.739035\pi\)
\(548\) 12.9900 0.554904
\(549\) 18.8554 0.804729
\(550\) 0 0
\(551\) 4.25325 0.181194
\(552\) −0.558667 −0.0237785
\(553\) 0.874198 0.0371747
\(554\) 3.41388 0.145042
\(555\) 4.22319 0.179264
\(556\) 13.2415 0.561563
\(557\) 34.1317 1.44621 0.723103 0.690740i \(-0.242715\pi\)
0.723103 + 0.690740i \(0.242715\pi\)
\(558\) 25.9217 1.09735
\(559\) −31.0239 −1.31217
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −9.93058 −0.418896
\(563\) −17.9194 −0.755215 −0.377607 0.925966i \(-0.623253\pi\)
−0.377607 + 0.925966i \(0.623253\pi\)
\(564\) 3.24260 0.136538
\(565\) 0.418079 0.0175887
\(566\) 14.3247 0.602113
\(567\) −6.20112 −0.260422
\(568\) −2.71008 −0.113712
\(569\) −28.7622 −1.20578 −0.602888 0.797826i \(-0.705983\pi\)
−0.602888 + 0.797826i \(0.705983\pi\)
\(570\) 1.68205 0.0704534
\(571\) −3.85772 −0.161441 −0.0807203 0.996737i \(-0.525722\pi\)
−0.0807203 + 0.996737i \(0.525722\pi\)
\(572\) 0 0
\(573\) −4.05041 −0.169208
\(574\) 9.74211 0.406628
\(575\) 0.983687 0.0410226
\(576\) −2.67745 −0.111561
\(577\) −11.3881 −0.474094 −0.237047 0.971498i \(-0.576180\pi\)
−0.237047 + 0.971498i \(0.576180\pi\)
\(578\) 8.04009 0.334424
\(579\) 7.37314 0.306417
\(580\) −1.43608 −0.0596298
\(581\) 2.65545 0.110167
\(582\) −9.77569 −0.405216
\(583\) 0 0
\(584\) 9.81960 0.406338
\(585\) −14.0880 −0.582465
\(586\) 13.0495 0.539069
\(587\) 5.98730 0.247122 0.123561 0.992337i \(-0.460568\pi\)
0.123561 + 0.992337i \(0.460568\pi\)
\(588\) −0.567932 −0.0234211
\(589\) −28.6737 −1.18148
\(590\) −7.39355 −0.304388
\(591\) −5.06661 −0.208412
\(592\) 7.43608 0.305621
\(593\) 7.70485 0.316400 0.158200 0.987407i \(-0.449431\pi\)
0.158200 + 0.987407i \(0.449431\pi\)
\(594\) 0 0
\(595\) −5.00401 −0.205144
\(596\) 9.63123 0.394511
\(597\) 1.64710 0.0674114
\(598\) −5.17586 −0.211657
\(599\) −18.3628 −0.750285 −0.375142 0.926967i \(-0.622406\pi\)
−0.375142 + 0.926967i \(0.622406\pi\)
\(600\) −0.567932 −0.0231857
\(601\) 17.7182 0.722740 0.361370 0.932423i \(-0.382309\pi\)
0.361370 + 0.932423i \(0.382309\pi\)
\(602\) −5.89617 −0.240310
\(603\) 7.43365 0.302722
\(604\) 12.9025 0.524996
\(605\) 0 0
\(606\) −8.53736 −0.346806
\(607\) 7.88194 0.319918 0.159959 0.987124i \(-0.448864\pi\)
0.159959 + 0.987124i \(0.448864\pi\)
\(608\) 2.96172 0.120113
\(609\) 0.815593 0.0330495
\(610\) 7.04229 0.285134
\(611\) 30.0416 1.21535
\(612\) 13.3980 0.541582
\(613\) 37.3291 1.50771 0.753855 0.657041i \(-0.228192\pi\)
0.753855 + 0.657041i \(0.228192\pi\)
\(614\) 12.2190 0.493120
\(615\) −5.53286 −0.223106
\(616\) 0 0
\(617\) 7.79365 0.313761 0.156880 0.987618i \(-0.449856\pi\)
0.156880 + 0.987618i \(0.449856\pi\)
\(618\) 4.43422 0.178370
\(619\) −9.04031 −0.363361 −0.181680 0.983358i \(-0.558154\pi\)
−0.181680 + 0.983358i \(0.558154\pi\)
\(620\) 9.68146 0.388817
\(621\) 3.17181 0.127280
\(622\) 32.3753 1.29813
\(623\) 4.84535 0.194125
\(624\) 2.98829 0.119627
\(625\) 1.00000 0.0400000
\(626\) −33.0363 −1.32040
\(627\) 0 0
\(628\) −4.31464 −0.172173
\(629\) −37.2102 −1.48367
\(630\) −2.67745 −0.106672
\(631\) 44.5917 1.77517 0.887585 0.460644i \(-0.152382\pi\)
0.887585 + 0.460644i \(0.152382\pi\)
\(632\) −0.874198 −0.0347737
\(633\) −13.4342 −0.533963
\(634\) −18.7597 −0.745045
\(635\) 18.7745 0.745042
\(636\) −5.78407 −0.229353
\(637\) −5.26170 −0.208476
\(638\) 0 0
\(639\) 7.25611 0.287047
\(640\) −1.00000 −0.0395285
\(641\) 5.36041 0.211723 0.105862 0.994381i \(-0.466240\pi\)
0.105862 + 0.994381i \(0.466240\pi\)
\(642\) −2.64056 −0.104215
\(643\) 11.9636 0.471799 0.235899 0.971777i \(-0.424196\pi\)
0.235899 + 0.971777i \(0.424196\pi\)
\(644\) −0.983687 −0.0387627
\(645\) 3.34862 0.131852
\(646\) −14.8204 −0.583103
\(647\) 6.61351 0.260004 0.130002 0.991514i \(-0.458502\pi\)
0.130002 + 0.991514i \(0.458502\pi\)
\(648\) 6.20112 0.243603
\(649\) 0 0
\(650\) −5.26170 −0.206381
\(651\) −5.49841 −0.215500
\(652\) −11.3379 −0.444027
\(653\) 21.7705 0.851947 0.425974 0.904736i \(-0.359932\pi\)
0.425974 + 0.904736i \(0.359932\pi\)
\(654\) 2.71806 0.106284
\(655\) −17.7838 −0.694872
\(656\) −9.74211 −0.380366
\(657\) −26.2915 −1.02573
\(658\) 5.70949 0.222579
\(659\) −43.1812 −1.68210 −0.841051 0.540957i \(-0.818062\pi\)
−0.841051 + 0.540957i \(0.818062\pi\)
\(660\) 0 0
\(661\) 33.9413 1.32016 0.660081 0.751195i \(-0.270522\pi\)
0.660081 + 0.751195i \(0.270522\pi\)
\(662\) 17.5094 0.680522
\(663\) −14.9534 −0.580742
\(664\) −2.65545 −0.103051
\(665\) 2.96172 0.114850
\(666\) −19.9097 −0.771487
\(667\) 1.41265 0.0546980
\(668\) 5.03663 0.194873
\(669\) −0.220074 −0.00850854
\(670\) 2.77639 0.107261
\(671\) 0 0
\(672\) 0.567932 0.0219085
\(673\) −25.9366 −0.999784 −0.499892 0.866088i \(-0.666627\pi\)
−0.499892 + 0.866088i \(0.666627\pi\)
\(674\) −19.9652 −0.769029
\(675\) 3.22441 0.124107
\(676\) 14.6855 0.564826
\(677\) −34.3810 −1.32137 −0.660684 0.750664i \(-0.729734\pi\)
−0.660684 + 0.750664i \(0.729734\pi\)
\(678\) 0.237440 0.00911884
\(679\) −17.2128 −0.660566
\(680\) 5.00401 0.191895
\(681\) 8.53868 0.327203
\(682\) 0 0
\(683\) 36.4734 1.39562 0.697808 0.716284i \(-0.254159\pi\)
0.697808 + 0.716284i \(0.254159\pi\)
\(684\) −7.92985 −0.303206
\(685\) −12.9900 −0.496321
\(686\) −1.00000 −0.0381802
\(687\) −4.64238 −0.177118
\(688\) 5.89617 0.224789
\(689\) −53.5874 −2.04152
\(690\) 0.558667 0.0212681
\(691\) 18.5975 0.707483 0.353741 0.935343i \(-0.384909\pi\)
0.353741 + 0.935343i \(0.384909\pi\)
\(692\) −19.9314 −0.757680
\(693\) 0 0
\(694\) −35.7762 −1.35805
\(695\) −13.2415 −0.502277
\(696\) −0.815593 −0.0309150
\(697\) 48.7496 1.84652
\(698\) 3.27265 0.123871
\(699\) −10.0120 −0.378689
\(700\) −1.00000 −0.0377964
\(701\) 49.2961 1.86189 0.930943 0.365164i \(-0.118987\pi\)
0.930943 + 0.365164i \(0.118987\pi\)
\(702\) −16.9659 −0.640335
\(703\) 22.0235 0.830634
\(704\) 0 0
\(705\) −3.24260 −0.122123
\(706\) −3.21106 −0.120850
\(707\) −15.0324 −0.565350
\(708\) −4.19904 −0.157809
\(709\) 35.1026 1.31831 0.659153 0.752009i \(-0.270915\pi\)
0.659153 + 0.752009i \(0.270915\pi\)
\(710\) 2.71008 0.101707
\(711\) 2.34062 0.0877803
\(712\) −4.84535 −0.181587
\(713\) −9.52353 −0.356659
\(714\) −2.84194 −0.106357
\(715\) 0 0
\(716\) −3.70949 −0.138630
\(717\) 16.0581 0.599700
\(718\) −5.16418 −0.192726
\(719\) 13.5610 0.505741 0.252871 0.967500i \(-0.418625\pi\)
0.252871 + 0.967500i \(0.418625\pi\)
\(720\) 2.67745 0.0997828
\(721\) 7.80766 0.290772
\(722\) −10.2282 −0.380656
\(723\) 4.05033 0.150633
\(724\) 15.6470 0.581514
\(725\) 1.43608 0.0533345
\(726\) 0 0
\(727\) 22.6155 0.838762 0.419381 0.907810i \(-0.362247\pi\)
0.419381 + 0.907810i \(0.362247\pi\)
\(728\) 5.26170 0.195012
\(729\) −11.1095 −0.411462
\(730\) −9.81960 −0.363440
\(731\) −29.5045 −1.09126
\(732\) 3.99954 0.147827
\(733\) −20.1667 −0.744873 −0.372436 0.928058i \(-0.621478\pi\)
−0.372436 + 0.928058i \(0.621478\pi\)
\(734\) −21.0921 −0.778525
\(735\) 0.567932 0.0209485
\(736\) 0.983687 0.0362592
\(737\) 0 0
\(738\) 26.0841 0.960167
\(739\) −6.91436 −0.254349 −0.127174 0.991880i \(-0.540591\pi\)
−0.127174 + 0.991880i \(0.540591\pi\)
\(740\) −7.43608 −0.273356
\(741\) 8.85045 0.325130
\(742\) −10.1844 −0.373883
\(743\) 25.7485 0.944622 0.472311 0.881432i \(-0.343420\pi\)
0.472311 + 0.881432i \(0.343420\pi\)
\(744\) 5.49841 0.201582
\(745\) −9.63123 −0.352861
\(746\) 20.7449 0.759526
\(747\) 7.10984 0.260136
\(748\) 0 0
\(749\) −4.64943 −0.169886
\(750\) 0.567932 0.0207379
\(751\) −25.4150 −0.927405 −0.463703 0.885991i \(-0.653479\pi\)
−0.463703 + 0.885991i \(0.653479\pi\)
\(752\) −5.70949 −0.208204
\(753\) −15.2300 −0.555011
\(754\) −7.55620 −0.275180
\(755\) −12.9025 −0.469571
\(756\) −3.22441 −0.117271
\(757\) −17.5209 −0.636809 −0.318404 0.947955i \(-0.603147\pi\)
−0.318404 + 0.947955i \(0.603147\pi\)
\(758\) 31.1753 1.13234
\(759\) 0 0
\(760\) −2.96172 −0.107433
\(761\) −11.5164 −0.417470 −0.208735 0.977972i \(-0.566935\pi\)
−0.208735 + 0.977972i \(0.566935\pi\)
\(762\) 10.6626 0.386266
\(763\) 4.78588 0.173261
\(764\) 7.13186 0.258022
\(765\) −13.3980 −0.484406
\(766\) −23.5582 −0.851191
\(767\) −38.9027 −1.40469
\(768\) −0.567932 −0.0204935
\(769\) 24.9065 0.898153 0.449077 0.893493i \(-0.351753\pi\)
0.449077 + 0.893493i \(0.351753\pi\)
\(770\) 0 0
\(771\) −11.7285 −0.422392
\(772\) −12.9824 −0.467248
\(773\) −12.8193 −0.461078 −0.230539 0.973063i \(-0.574049\pi\)
−0.230539 + 0.973063i \(0.574049\pi\)
\(774\) −15.7867 −0.567442
\(775\) −9.68146 −0.347768
\(776\) 17.2128 0.617903
\(777\) 4.22319 0.151506
\(778\) −7.39973 −0.265293
\(779\) −28.8534 −1.03378
\(780\) −2.98829 −0.106998
\(781\) 0 0
\(782\) −4.92238 −0.176024
\(783\) 4.63049 0.165480
\(784\) 1.00000 0.0357143
\(785\) 4.31464 0.153996
\(786\) −10.1000 −0.360256
\(787\) 12.4754 0.444699 0.222349 0.974967i \(-0.428627\pi\)
0.222349 + 0.974967i \(0.428627\pi\)
\(788\) 8.92115 0.317803
\(789\) 10.7613 0.383112
\(790\) 0.874198 0.0311026
\(791\) 0.418079 0.0148652
\(792\) 0 0
\(793\) 37.0544 1.31584
\(794\) −15.2670 −0.541804
\(795\) 5.78407 0.205140
\(796\) −2.90018 −0.102794
\(797\) −5.21655 −0.184780 −0.0923898 0.995723i \(-0.529451\pi\)
−0.0923898 + 0.995723i \(0.529451\pi\)
\(798\) 1.68205 0.0595440
\(799\) 28.5703 1.01075
\(800\) 1.00000 0.0353553
\(801\) 12.9732 0.458385
\(802\) 35.4584 1.25208
\(803\) 0 0
\(804\) 1.57680 0.0556095
\(805\) 0.983687 0.0346704
\(806\) 50.9409 1.79432
\(807\) −14.9386 −0.525862
\(808\) 15.0324 0.528837
\(809\) 6.12370 0.215298 0.107649 0.994189i \(-0.465668\pi\)
0.107649 + 0.994189i \(0.465668\pi\)
\(810\) −6.20112 −0.217885
\(811\) −7.24103 −0.254267 −0.127133 0.991886i \(-0.540578\pi\)
−0.127133 + 0.991886i \(0.540578\pi\)
\(812\) −1.43608 −0.0503964
\(813\) −3.63749 −0.127573
\(814\) 0 0
\(815\) 11.3379 0.397149
\(816\) 2.84194 0.0994877
\(817\) 17.4628 0.610945
\(818\) −17.2938 −0.604663
\(819\) −14.0880 −0.492273
\(820\) 9.74211 0.340209
\(821\) −25.8519 −0.902236 −0.451118 0.892464i \(-0.648975\pi\)
−0.451118 + 0.892464i \(0.648975\pi\)
\(822\) −7.37742 −0.257317
\(823\) −49.9541 −1.74129 −0.870646 0.491910i \(-0.836299\pi\)
−0.870646 + 0.491910i \(0.836299\pi\)
\(824\) −7.80766 −0.271993
\(825\) 0 0
\(826\) −7.39355 −0.257255
\(827\) 36.9973 1.28652 0.643261 0.765647i \(-0.277581\pi\)
0.643261 + 0.765647i \(0.277581\pi\)
\(828\) −2.63378 −0.0915300
\(829\) 20.0676 0.696978 0.348489 0.937313i \(-0.386695\pi\)
0.348489 + 0.937313i \(0.386695\pi\)
\(830\) 2.65545 0.0921720
\(831\) −1.93885 −0.0672579
\(832\) −5.26170 −0.182417
\(833\) −5.00401 −0.173379
\(834\) −7.52025 −0.260405
\(835\) −5.03663 −0.174300
\(836\) 0 0
\(837\) −31.2170 −1.07902
\(838\) −3.77494 −0.130403
\(839\) 31.8473 1.09949 0.549745 0.835332i \(-0.314725\pi\)
0.549745 + 0.835332i \(0.314725\pi\)
\(840\) −0.567932 −0.0195955
\(841\) −26.9377 −0.928886
\(842\) 3.56528 0.122868
\(843\) 5.63989 0.194248
\(844\) 23.6547 0.814227
\(845\) −14.6855 −0.505195
\(846\) 15.2869 0.525574
\(847\) 0 0
\(848\) 10.1844 0.349735
\(849\) −8.13546 −0.279208
\(850\) −5.00401 −0.171636
\(851\) 7.31477 0.250747
\(852\) 1.53914 0.0527301
\(853\) −54.3232 −1.85999 −0.929996 0.367569i \(-0.880190\pi\)
−0.929996 + 0.367569i \(0.880190\pi\)
\(854\) 7.04229 0.240982
\(855\) 7.92985 0.271195
\(856\) 4.64943 0.158914
\(857\) −50.4050 −1.72180 −0.860902 0.508772i \(-0.830100\pi\)
−0.860902 + 0.508772i \(0.830100\pi\)
\(858\) 0 0
\(859\) 32.4311 1.10654 0.553268 0.833003i \(-0.313380\pi\)
0.553268 + 0.833003i \(0.313380\pi\)
\(860\) −5.89617 −0.201058
\(861\) −5.53286 −0.188559
\(862\) −2.54773 −0.0867761
\(863\) 5.57666 0.189832 0.0949159 0.995485i \(-0.469742\pi\)
0.0949159 + 0.995485i \(0.469742\pi\)
\(864\) 3.22441 0.109697
\(865\) 19.9314 0.677689
\(866\) −2.45692 −0.0834895
\(867\) −4.56622 −0.155077
\(868\) 9.68146 0.328610
\(869\) 0 0
\(870\) 0.815593 0.0276512
\(871\) 14.6085 0.494991
\(872\) −4.78588 −0.162070
\(873\) −46.0864 −1.55979
\(874\) 2.91340 0.0985473
\(875\) 1.00000 0.0338062
\(876\) −5.57687 −0.188425
\(877\) −55.2215 −1.86470 −0.932349 0.361559i \(-0.882245\pi\)
−0.932349 + 0.361559i \(0.882245\pi\)
\(878\) −5.99413 −0.202292
\(879\) −7.41122 −0.249974
\(880\) 0 0
\(881\) −29.3724 −0.989582 −0.494791 0.869012i \(-0.664755\pi\)
−0.494791 + 0.869012i \(0.664755\pi\)
\(882\) −2.67745 −0.0901545
\(883\) 53.4760 1.79961 0.899805 0.436292i \(-0.143708\pi\)
0.899805 + 0.436292i \(0.143708\pi\)
\(884\) 26.3296 0.885559
\(885\) 4.19904 0.141149
\(886\) 18.2630 0.613557
\(887\) 29.0365 0.974950 0.487475 0.873137i \(-0.337918\pi\)
0.487475 + 0.873137i \(0.337918\pi\)
\(888\) −4.22319 −0.141721
\(889\) 18.7745 0.629676
\(890\) 4.84535 0.162417
\(891\) 0 0
\(892\) 0.387500 0.0129745
\(893\) −16.9099 −0.565867
\(894\) −5.46989 −0.182940
\(895\) 3.70949 0.123994
\(896\) −1.00000 −0.0334077
\(897\) 2.93954 0.0981483
\(898\) −4.60321 −0.153611
\(899\) −13.9033 −0.463701
\(900\) −2.67745 −0.0892484
\(901\) −50.9630 −1.69782
\(902\) 0 0
\(903\) 3.34862 0.111435
\(904\) −0.418079 −0.0139051
\(905\) −15.6470 −0.520122
\(906\) −7.32775 −0.243448
\(907\) −36.4927 −1.21172 −0.605860 0.795571i \(-0.707171\pi\)
−0.605860 + 0.795571i \(0.707171\pi\)
\(908\) −15.0347 −0.498943
\(909\) −40.2484 −1.33496
\(910\) −5.26170 −0.174424
\(911\) 14.9704 0.495993 0.247996 0.968761i \(-0.420228\pi\)
0.247996 + 0.968761i \(0.420228\pi\)
\(912\) −1.68205 −0.0556983
\(913\) 0 0
\(914\) 20.3512 0.673159
\(915\) −3.99954 −0.132221
\(916\) 8.17418 0.270083
\(917\) −17.7838 −0.587274
\(918\) −16.1350 −0.532533
\(919\) 46.3079 1.52756 0.763778 0.645479i \(-0.223342\pi\)
0.763778 + 0.645479i \(0.223342\pi\)
\(920\) −0.983687 −0.0324312
\(921\) −6.93959 −0.228667
\(922\) −22.1050 −0.727988
\(923\) 14.2596 0.469361
\(924\) 0 0
\(925\) 7.43608 0.244497
\(926\) 30.5074 1.00253
\(927\) 20.9046 0.686598
\(928\) 1.43608 0.0471415
\(929\) −51.7132 −1.69666 −0.848328 0.529471i \(-0.822390\pi\)
−0.848328 + 0.529471i \(0.822390\pi\)
\(930\) −5.49841 −0.180300
\(931\) 2.96172 0.0970663
\(932\) 17.6289 0.577454
\(933\) −18.3870 −0.601962
\(934\) 10.5885 0.346467
\(935\) 0 0
\(936\) 14.0880 0.460479
\(937\) −10.4357 −0.340921 −0.170460 0.985365i \(-0.554526\pi\)
−0.170460 + 0.985365i \(0.554526\pi\)
\(938\) 2.77639 0.0906523
\(939\) 18.7624 0.612286
\(940\) 5.70949 0.186223
\(941\) −47.0321 −1.53320 −0.766601 0.642124i \(-0.778054\pi\)
−0.766601 + 0.642124i \(0.778054\pi\)
\(942\) 2.45042 0.0798391
\(943\) −9.58319 −0.312072
\(944\) 7.39355 0.240640
\(945\) 3.22441 0.104890
\(946\) 0 0
\(947\) 30.3259 0.985461 0.492730 0.870182i \(-0.335999\pi\)
0.492730 + 0.870182i \(0.335999\pi\)
\(948\) 0.496485 0.0161251
\(949\) −51.6678 −1.67721
\(950\) 2.96172 0.0960907
\(951\) 10.6543 0.345488
\(952\) 5.00401 0.162181
\(953\) −0.609007 −0.0197277 −0.00986384 0.999951i \(-0.503140\pi\)
−0.00986384 + 0.999951i \(0.503140\pi\)
\(954\) −27.2684 −0.882846
\(955\) −7.13186 −0.230781
\(956\) −28.2746 −0.914467
\(957\) 0 0
\(958\) −39.9165 −1.28964
\(959\) −12.9900 −0.419468
\(960\) 0.567932 0.0183299
\(961\) 62.7307 2.02357
\(962\) −39.1264 −1.26149
\(963\) −12.4486 −0.401151
\(964\) −7.13172 −0.229697
\(965\) 12.9824 0.417919
\(966\) 0.558667 0.0179748
\(967\) −14.2048 −0.456796 −0.228398 0.973568i \(-0.573349\pi\)
−0.228398 + 0.973568i \(0.573349\pi\)
\(968\) 0 0
\(969\) 8.41701 0.270393
\(970\) −17.2128 −0.552669
\(971\) 49.1969 1.57880 0.789402 0.613877i \(-0.210391\pi\)
0.789402 + 0.613877i \(0.210391\pi\)
\(972\) −13.1950 −0.423231
\(973\) −13.2415 −0.424502
\(974\) 38.0184 1.21819
\(975\) 2.98829 0.0957018
\(976\) −7.04229 −0.225418
\(977\) 19.6319 0.628080 0.314040 0.949410i \(-0.398317\pi\)
0.314040 + 0.949410i \(0.398317\pi\)
\(978\) 6.43916 0.205902
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 12.8140 0.409119
\(982\) 19.3973 0.618992
\(983\) −21.3138 −0.679804 −0.339902 0.940461i \(-0.610394\pi\)
−0.339902 + 0.940461i \(0.610394\pi\)
\(984\) 5.53286 0.176381
\(985\) −8.92115 −0.284252
\(986\) −7.18613 −0.228853
\(987\) −3.24260 −0.103213
\(988\) −15.5837 −0.495782
\(989\) 5.79999 0.184429
\(990\) 0 0
\(991\) −27.8962 −0.886152 −0.443076 0.896484i \(-0.646113\pi\)
−0.443076 + 0.896484i \(0.646113\pi\)
\(992\) −9.68146 −0.307387
\(993\) −9.94414 −0.315568
\(994\) 2.71008 0.0859585
\(995\) 2.90018 0.0919418
\(996\) 1.50811 0.0477864
\(997\) −36.3507 −1.15124 −0.575619 0.817718i \(-0.695239\pi\)
−0.575619 + 0.817718i \(0.695239\pi\)
\(998\) 2.76107 0.0874002
\(999\) 23.9769 0.758597
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 8470.2.a.dd.1.3 yes 6
11.10 odd 2 8470.2.a.cx.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cx.1.3 6 11.10 odd 2
8470.2.a.dd.1.3 yes 6 1.1 even 1 trivial