Properties

Label 7098.2.a.bw.1.2
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.561553 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.561553 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.561553 q^{10} -1.56155 q^{11} +1.00000 q^{12} -1.00000 q^{14} +0.561553 q^{15} +1.00000 q^{16} -5.00000 q^{17} +1.00000 q^{18} -4.68466 q^{19} +0.561553 q^{20} -1.00000 q^{21} -1.56155 q^{22} -0.876894 q^{23} +1.00000 q^{24} -4.68466 q^{25} +1.00000 q^{27} -1.00000 q^{28} -1.00000 q^{29} +0.561553 q^{30} -3.12311 q^{31} +1.00000 q^{32} -1.56155 q^{33} -5.00000 q^{34} -0.561553 q^{35} +1.00000 q^{36} +10.8078 q^{37} -4.68466 q^{38} +0.561553 q^{40} -4.12311 q^{41} -1.00000 q^{42} -11.1231 q^{43} -1.56155 q^{44} +0.561553 q^{45} -0.876894 q^{46} +4.68466 q^{47} +1.00000 q^{48} +1.00000 q^{49} -4.68466 q^{50} -5.00000 q^{51} -12.1231 q^{53} +1.00000 q^{54} -0.876894 q^{55} -1.00000 q^{56} -4.68466 q^{57} -1.00000 q^{58} +4.87689 q^{59} +0.561553 q^{60} -1.00000 q^{61} -3.12311 q^{62} -1.00000 q^{63} +1.00000 q^{64} -1.56155 q^{66} -7.12311 q^{67} -5.00000 q^{68} -0.876894 q^{69} -0.561553 q^{70} -4.00000 q^{71} +1.00000 q^{72} -6.56155 q^{73} +10.8078 q^{74} -4.68466 q^{75} -4.68466 q^{76} +1.56155 q^{77} +5.56155 q^{79} +0.561553 q^{80} +1.00000 q^{81} -4.12311 q^{82} +6.24621 q^{83} -1.00000 q^{84} -2.80776 q^{85} -11.1231 q^{86} -1.00000 q^{87} -1.56155 q^{88} +2.68466 q^{89} +0.561553 q^{90} -0.876894 q^{92} -3.12311 q^{93} +4.68466 q^{94} -2.63068 q^{95} +1.00000 q^{96} -4.24621 q^{97} +1.00000 q^{98} -1.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} - 3q^{10} + q^{11} + 2q^{12} - 2q^{14} - 3q^{15} + 2q^{16} - 10q^{17} + 2q^{18} + 3q^{19} - 3q^{20} - 2q^{21} + q^{22} - 10q^{23} + 2q^{24} + 3q^{25} + 2q^{27} - 2q^{28} - 2q^{29} - 3q^{30} + 2q^{31} + 2q^{32} + q^{33} - 10q^{34} + 3q^{35} + 2q^{36} + q^{37} + 3q^{38} - 3q^{40} - 2q^{42} - 14q^{43} + q^{44} - 3q^{45} - 10q^{46} - 3q^{47} + 2q^{48} + 2q^{49} + 3q^{50} - 10q^{51} - 16q^{53} + 2q^{54} - 10q^{55} - 2q^{56} + 3q^{57} - 2q^{58} + 18q^{59} - 3q^{60} - 2q^{61} + 2q^{62} - 2q^{63} + 2q^{64} + q^{66} - 6q^{67} - 10q^{68} - 10q^{69} + 3q^{70} - 8q^{71} + 2q^{72} - 9q^{73} + q^{74} + 3q^{75} + 3q^{76} - q^{77} + 7q^{79} - 3q^{80} + 2q^{81} - 4q^{83} - 2q^{84} + 15q^{85} - 14q^{86} - 2q^{87} + q^{88} - 7q^{89} - 3q^{90} - 10q^{92} + 2q^{93} - 3q^{94} - 30q^{95} + 2q^{96} + 8q^{97} + 2q^{98} + q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0.561553 0.251134 0.125567 0.992085i \(-0.459925\pi\)
0.125567 + 0.992085i \(0.459925\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0.561553 0.177579
\(11\) −1.56155 −0.470826 −0.235413 0.971895i \(-0.575644\pi\)
−0.235413 + 0.971895i \(0.575644\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) 0.561553 0.144992
\(16\) 1.00000 0.250000
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.68466 −1.07473 −0.537367 0.843348i \(-0.680581\pi\)
−0.537367 + 0.843348i \(0.680581\pi\)
\(20\) 0.561553 0.125567
\(21\) −1.00000 −0.218218
\(22\) −1.56155 −0.332924
\(23\) −0.876894 −0.182845 −0.0914226 0.995812i \(-0.529141\pi\)
−0.0914226 + 0.995812i \(0.529141\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.68466 −0.936932
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0.561553 0.102525
\(31\) −3.12311 −0.560926 −0.280463 0.959865i \(-0.590488\pi\)
−0.280463 + 0.959865i \(0.590488\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.56155 −0.271831
\(34\) −5.00000 −0.857493
\(35\) −0.561553 −0.0949197
\(36\) 1.00000 0.166667
\(37\) 10.8078 1.77679 0.888393 0.459084i \(-0.151822\pi\)
0.888393 + 0.459084i \(0.151822\pi\)
\(38\) −4.68466 −0.759952
\(39\) 0 0
\(40\) 0.561553 0.0887893
\(41\) −4.12311 −0.643921 −0.321960 0.946753i \(-0.604342\pi\)
−0.321960 + 0.946753i \(0.604342\pi\)
\(42\) −1.00000 −0.154303
\(43\) −11.1231 −1.69626 −0.848129 0.529790i \(-0.822271\pi\)
−0.848129 + 0.529790i \(0.822271\pi\)
\(44\) −1.56155 −0.235413
\(45\) 0.561553 0.0837114
\(46\) −0.876894 −0.129291
\(47\) 4.68466 0.683328 0.341664 0.939822i \(-0.389010\pi\)
0.341664 + 0.939822i \(0.389010\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −4.68466 −0.662511
\(51\) −5.00000 −0.700140
\(52\) 0 0
\(53\) −12.1231 −1.66524 −0.832618 0.553847i \(-0.813159\pi\)
−0.832618 + 0.553847i \(0.813159\pi\)
\(54\) 1.00000 0.136083
\(55\) −0.876894 −0.118240
\(56\) −1.00000 −0.133631
\(57\) −4.68466 −0.620498
\(58\) −1.00000 −0.131306
\(59\) 4.87689 0.634918 0.317459 0.948272i \(-0.397170\pi\)
0.317459 + 0.948272i \(0.397170\pi\)
\(60\) 0.561553 0.0724962
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) −3.12311 −0.396635
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.56155 −0.192214
\(67\) −7.12311 −0.870226 −0.435113 0.900376i \(-0.643292\pi\)
−0.435113 + 0.900376i \(0.643292\pi\)
\(68\) −5.00000 −0.606339
\(69\) −0.876894 −0.105566
\(70\) −0.561553 −0.0671184
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.56155 −0.767972 −0.383986 0.923339i \(-0.625449\pi\)
−0.383986 + 0.923339i \(0.625449\pi\)
\(74\) 10.8078 1.25638
\(75\) −4.68466 −0.540938
\(76\) −4.68466 −0.537367
\(77\) 1.56155 0.177955
\(78\) 0 0
\(79\) 5.56155 0.625724 0.312862 0.949799i \(-0.398712\pi\)
0.312862 + 0.949799i \(0.398712\pi\)
\(80\) 0.561553 0.0627835
\(81\) 1.00000 0.111111
\(82\) −4.12311 −0.455321
\(83\) 6.24621 0.685611 0.342805 0.939406i \(-0.388623\pi\)
0.342805 + 0.939406i \(0.388623\pi\)
\(84\) −1.00000 −0.109109
\(85\) −2.80776 −0.304545
\(86\) −11.1231 −1.19944
\(87\) −1.00000 −0.107211
\(88\) −1.56155 −0.166462
\(89\) 2.68466 0.284573 0.142287 0.989825i \(-0.454555\pi\)
0.142287 + 0.989825i \(0.454555\pi\)
\(90\) 0.561553 0.0591929
\(91\) 0 0
\(92\) −0.876894 −0.0914226
\(93\) −3.12311 −0.323851
\(94\) 4.68466 0.483186
\(95\) −2.63068 −0.269902
\(96\) 1.00000 0.102062
\(97\) −4.24621 −0.431137 −0.215569 0.976489i \(-0.569161\pi\)
−0.215569 + 0.976489i \(0.569161\pi\)
\(98\) 1.00000 0.101015
\(99\) −1.56155 −0.156942
\(100\) −4.68466 −0.468466
\(101\) −15.4384 −1.53618 −0.768091 0.640340i \(-0.778793\pi\)
−0.768091 + 0.640340i \(0.778793\pi\)
\(102\) −5.00000 −0.495074
\(103\) 13.3693 1.31732 0.658659 0.752442i \(-0.271124\pi\)
0.658659 + 0.752442i \(0.271124\pi\)
\(104\) 0 0
\(105\) −0.561553 −0.0548019
\(106\) −12.1231 −1.17750
\(107\) −13.5616 −1.31104 −0.655522 0.755176i \(-0.727552\pi\)
−0.655522 + 0.755176i \(0.727552\pi\)
\(108\) 1.00000 0.0962250
\(109\) 9.12311 0.873835 0.436918 0.899502i \(-0.356070\pi\)
0.436918 + 0.899502i \(0.356070\pi\)
\(110\) −0.876894 −0.0836086
\(111\) 10.8078 1.02583
\(112\) −1.00000 −0.0944911
\(113\) −11.9309 −1.12236 −0.561181 0.827693i \(-0.689653\pi\)
−0.561181 + 0.827693i \(0.689653\pi\)
\(114\) −4.68466 −0.438758
\(115\) −0.492423 −0.0459186
\(116\) −1.00000 −0.0928477
\(117\) 0 0
\(118\) 4.87689 0.448955
\(119\) 5.00000 0.458349
\(120\) 0.561553 0.0512625
\(121\) −8.56155 −0.778323
\(122\) −1.00000 −0.0905357
\(123\) −4.12311 −0.371768
\(124\) −3.12311 −0.280463
\(125\) −5.43845 −0.486430
\(126\) −1.00000 −0.0890871
\(127\) −14.2462 −1.26415 −0.632073 0.774909i \(-0.717796\pi\)
−0.632073 + 0.774909i \(0.717796\pi\)
\(128\) 1.00000 0.0883883
\(129\) −11.1231 −0.979335
\(130\) 0 0
\(131\) 16.4924 1.44095 0.720475 0.693481i \(-0.243924\pi\)
0.720475 + 0.693481i \(0.243924\pi\)
\(132\) −1.56155 −0.135916
\(133\) 4.68466 0.406211
\(134\) −7.12311 −0.615343
\(135\) 0.561553 0.0483308
\(136\) −5.00000 −0.428746
\(137\) 14.8078 1.26511 0.632556 0.774514i \(-0.282006\pi\)
0.632556 + 0.774514i \(0.282006\pi\)
\(138\) −0.876894 −0.0746462
\(139\) 4.68466 0.397348 0.198674 0.980066i \(-0.436337\pi\)
0.198674 + 0.980066i \(0.436337\pi\)
\(140\) −0.561553 −0.0474599
\(141\) 4.68466 0.394519
\(142\) −4.00000 −0.335673
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −0.561553 −0.0466344
\(146\) −6.56155 −0.543038
\(147\) 1.00000 0.0824786
\(148\) 10.8078 0.888393
\(149\) 2.80776 0.230021 0.115010 0.993364i \(-0.463310\pi\)
0.115010 + 0.993364i \(0.463310\pi\)
\(150\) −4.68466 −0.382501
\(151\) −21.5616 −1.75465 −0.877327 0.479893i \(-0.840676\pi\)
−0.877327 + 0.479893i \(0.840676\pi\)
\(152\) −4.68466 −0.379976
\(153\) −5.00000 −0.404226
\(154\) 1.56155 0.125834
\(155\) −1.75379 −0.140868
\(156\) 0 0
\(157\) 14.8078 1.18179 0.590894 0.806749i \(-0.298775\pi\)
0.590894 + 0.806749i \(0.298775\pi\)
\(158\) 5.56155 0.442453
\(159\) −12.1231 −0.961425
\(160\) 0.561553 0.0443946
\(161\) 0.876894 0.0691090
\(162\) 1.00000 0.0785674
\(163\) 7.12311 0.557925 0.278962 0.960302i \(-0.410010\pi\)
0.278962 + 0.960302i \(0.410010\pi\)
\(164\) −4.12311 −0.321960
\(165\) −0.876894 −0.0682661
\(166\) 6.24621 0.484800
\(167\) −12.4924 −0.966693 −0.483346 0.875429i \(-0.660579\pi\)
−0.483346 + 0.875429i \(0.660579\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 0 0
\(170\) −2.80776 −0.215346
\(171\) −4.68466 −0.358245
\(172\) −11.1231 −0.848129
\(173\) −13.1231 −0.997731 −0.498866 0.866679i \(-0.666250\pi\)
−0.498866 + 0.866679i \(0.666250\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 4.68466 0.354127
\(176\) −1.56155 −0.117706
\(177\) 4.87689 0.366570
\(178\) 2.68466 0.201224
\(179\) 16.4924 1.23270 0.616351 0.787472i \(-0.288610\pi\)
0.616351 + 0.787472i \(0.288610\pi\)
\(180\) 0.561553 0.0418557
\(181\) −3.24621 −0.241289 −0.120644 0.992696i \(-0.538496\pi\)
−0.120644 + 0.992696i \(0.538496\pi\)
\(182\) 0 0
\(183\) −1.00000 −0.0739221
\(184\) −0.876894 −0.0646455
\(185\) 6.06913 0.446211
\(186\) −3.12311 −0.228997
\(187\) 7.80776 0.570960
\(188\) 4.68466 0.341664
\(189\) −1.00000 −0.0727393
\(190\) −2.63068 −0.190850
\(191\) 22.2462 1.60968 0.804840 0.593492i \(-0.202251\pi\)
0.804840 + 0.593492i \(0.202251\pi\)
\(192\) 1.00000 0.0721688
\(193\) 4.36932 0.314510 0.157255 0.987558i \(-0.449735\pi\)
0.157255 + 0.987558i \(0.449735\pi\)
\(194\) −4.24621 −0.304860
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 3.56155 0.253750 0.126875 0.991919i \(-0.459505\pi\)
0.126875 + 0.991919i \(0.459505\pi\)
\(198\) −1.56155 −0.110975
\(199\) 23.6155 1.67406 0.837030 0.547157i \(-0.184290\pi\)
0.837030 + 0.547157i \(0.184290\pi\)
\(200\) −4.68466 −0.331255
\(201\) −7.12311 −0.502425
\(202\) −15.4384 −1.08625
\(203\) 1.00000 0.0701862
\(204\) −5.00000 −0.350070
\(205\) −2.31534 −0.161710
\(206\) 13.3693 0.931484
\(207\) −0.876894 −0.0609484
\(208\) 0 0
\(209\) 7.31534 0.506013
\(210\) −0.561553 −0.0387508
\(211\) −11.1231 −0.765746 −0.382873 0.923801i \(-0.625065\pi\)
−0.382873 + 0.923801i \(0.625065\pi\)
\(212\) −12.1231 −0.832618
\(213\) −4.00000 −0.274075
\(214\) −13.5616 −0.927049
\(215\) −6.24621 −0.425988
\(216\) 1.00000 0.0680414
\(217\) 3.12311 0.212010
\(218\) 9.12311 0.617895
\(219\) −6.56155 −0.443389
\(220\) −0.876894 −0.0591202
\(221\) 0 0
\(222\) 10.8078 0.725370
\(223\) 16.4924 1.10441 0.552207 0.833707i \(-0.313786\pi\)
0.552207 + 0.833707i \(0.313786\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −4.68466 −0.312311
\(226\) −11.9309 −0.793630
\(227\) 4.87689 0.323691 0.161845 0.986816i \(-0.448255\pi\)
0.161845 + 0.986816i \(0.448255\pi\)
\(228\) −4.68466 −0.310249
\(229\) 2.19224 0.144867 0.0724335 0.997373i \(-0.476923\pi\)
0.0724335 + 0.997373i \(0.476923\pi\)
\(230\) −0.492423 −0.0324694
\(231\) 1.56155 0.102743
\(232\) −1.00000 −0.0656532
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 2.63068 0.171607
\(236\) 4.87689 0.317459
\(237\) 5.56155 0.361262
\(238\) 5.00000 0.324102
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0.561553 0.0362481
\(241\) 23.6847 1.52566 0.762831 0.646597i \(-0.223809\pi\)
0.762831 + 0.646597i \(0.223809\pi\)
\(242\) −8.56155 −0.550357
\(243\) 1.00000 0.0641500
\(244\) −1.00000 −0.0640184
\(245\) 0.561553 0.0358763
\(246\) −4.12311 −0.262880
\(247\) 0 0
\(248\) −3.12311 −0.198317
\(249\) 6.24621 0.395838
\(250\) −5.43845 −0.343958
\(251\) −24.4924 −1.54595 −0.772974 0.634438i \(-0.781232\pi\)
−0.772974 + 0.634438i \(0.781232\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 1.36932 0.0860882
\(254\) −14.2462 −0.893887
\(255\) −2.80776 −0.175829
\(256\) 1.00000 0.0625000
\(257\) −31.2462 −1.94909 −0.974543 0.224203i \(-0.928022\pi\)
−0.974543 + 0.224203i \(0.928022\pi\)
\(258\) −11.1231 −0.692494
\(259\) −10.8078 −0.671562
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 16.4924 1.01891
\(263\) 12.4924 0.770316 0.385158 0.922851i \(-0.374147\pi\)
0.385158 + 0.922851i \(0.374147\pi\)
\(264\) −1.56155 −0.0961069
\(265\) −6.80776 −0.418198
\(266\) 4.68466 0.287235
\(267\) 2.68466 0.164298
\(268\) −7.12311 −0.435113
\(269\) −13.1231 −0.800130 −0.400065 0.916487i \(-0.631012\pi\)
−0.400065 + 0.916487i \(0.631012\pi\)
\(270\) 0.561553 0.0341750
\(271\) 4.87689 0.296250 0.148125 0.988969i \(-0.452676\pi\)
0.148125 + 0.988969i \(0.452676\pi\)
\(272\) −5.00000 −0.303170
\(273\) 0 0
\(274\) 14.8078 0.894570
\(275\) 7.31534 0.441132
\(276\) −0.876894 −0.0527828
\(277\) 0.561553 0.0337404 0.0168702 0.999858i \(-0.494630\pi\)
0.0168702 + 0.999858i \(0.494630\pi\)
\(278\) 4.68466 0.280967
\(279\) −3.12311 −0.186975
\(280\) −0.561553 −0.0335592
\(281\) −12.8078 −0.764047 −0.382024 0.924153i \(-0.624773\pi\)
−0.382024 + 0.924153i \(0.624773\pi\)
\(282\) 4.68466 0.278967
\(283\) −24.4924 −1.45592 −0.727962 0.685618i \(-0.759532\pi\)
−0.727962 + 0.685618i \(0.759532\pi\)
\(284\) −4.00000 −0.237356
\(285\) −2.63068 −0.155828
\(286\) 0 0
\(287\) 4.12311 0.243379
\(288\) 1.00000 0.0589256
\(289\) 8.00000 0.470588
\(290\) −0.561553 −0.0329755
\(291\) −4.24621 −0.248917
\(292\) −6.56155 −0.383986
\(293\) 7.68466 0.448943 0.224471 0.974481i \(-0.427935\pi\)
0.224471 + 0.974481i \(0.427935\pi\)
\(294\) 1.00000 0.0583212
\(295\) 2.73863 0.159449
\(296\) 10.8078 0.628189
\(297\) −1.56155 −0.0906105
\(298\) 2.80776 0.162649
\(299\) 0 0
\(300\) −4.68466 −0.270469
\(301\) 11.1231 0.641125
\(302\) −21.5616 −1.24073
\(303\) −15.4384 −0.886916
\(304\) −4.68466 −0.268684
\(305\) −0.561553 −0.0321544
\(306\) −5.00000 −0.285831
\(307\) −5.06913 −0.289311 −0.144655 0.989482i \(-0.546207\pi\)
−0.144655 + 0.989482i \(0.546207\pi\)
\(308\) 1.56155 0.0889777
\(309\) 13.3693 0.760554
\(310\) −1.75379 −0.0996085
\(311\) 18.4384 1.04555 0.522774 0.852471i \(-0.324897\pi\)
0.522774 + 0.852471i \(0.324897\pi\)
\(312\) 0 0
\(313\) −13.6155 −0.769595 −0.384798 0.923001i \(-0.625729\pi\)
−0.384798 + 0.923001i \(0.625729\pi\)
\(314\) 14.8078 0.835650
\(315\) −0.561553 −0.0316399
\(316\) 5.56155 0.312862
\(317\) −7.43845 −0.417785 −0.208892 0.977939i \(-0.566986\pi\)
−0.208892 + 0.977939i \(0.566986\pi\)
\(318\) −12.1231 −0.679830
\(319\) 1.56155 0.0874302
\(320\) 0.561553 0.0313918
\(321\) −13.5616 −0.756932
\(322\) 0.876894 0.0488674
\(323\) 23.4233 1.30331
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 7.12311 0.394512
\(327\) 9.12311 0.504509
\(328\) −4.12311 −0.227660
\(329\) −4.68466 −0.258274
\(330\) −0.876894 −0.0482714
\(331\) −10.6307 −0.584315 −0.292158 0.956370i \(-0.594373\pi\)
−0.292158 + 0.956370i \(0.594373\pi\)
\(332\) 6.24621 0.342805
\(333\) 10.8078 0.592262
\(334\) −12.4924 −0.683555
\(335\) −4.00000 −0.218543
\(336\) −1.00000 −0.0545545
\(337\) 3.49242 0.190244 0.0951222 0.995466i \(-0.469676\pi\)
0.0951222 + 0.995466i \(0.469676\pi\)
\(338\) 0 0
\(339\) −11.9309 −0.647996
\(340\) −2.80776 −0.152272
\(341\) 4.87689 0.264099
\(342\) −4.68466 −0.253317
\(343\) −1.00000 −0.0539949
\(344\) −11.1231 −0.599718
\(345\) −0.492423 −0.0265111
\(346\) −13.1231 −0.705503
\(347\) −12.6847 −0.680948 −0.340474 0.940254i \(-0.610588\pi\)
−0.340474 + 0.940254i \(0.610588\pi\)
\(348\) −1.00000 −0.0536056
\(349\) 26.4924 1.41811 0.709053 0.705155i \(-0.249122\pi\)
0.709053 + 0.705155i \(0.249122\pi\)
\(350\) 4.68466 0.250406
\(351\) 0 0
\(352\) −1.56155 −0.0832310
\(353\) 10.8078 0.575239 0.287620 0.957745i \(-0.407136\pi\)
0.287620 + 0.957745i \(0.407136\pi\)
\(354\) 4.87689 0.259204
\(355\) −2.24621 −0.119217
\(356\) 2.68466 0.142287
\(357\) 5.00000 0.264628
\(358\) 16.4924 0.871652
\(359\) −29.3693 −1.55005 −0.775027 0.631929i \(-0.782264\pi\)
−0.775027 + 0.631929i \(0.782264\pi\)
\(360\) 0.561553 0.0295964
\(361\) 2.94602 0.155054
\(362\) −3.24621 −0.170617
\(363\) −8.56155 −0.449365
\(364\) 0 0
\(365\) −3.68466 −0.192864
\(366\) −1.00000 −0.0522708
\(367\) −13.3693 −0.697873 −0.348936 0.937146i \(-0.613457\pi\)
−0.348936 + 0.937146i \(0.613457\pi\)
\(368\) −0.876894 −0.0457113
\(369\) −4.12311 −0.214640
\(370\) 6.06913 0.315519
\(371\) 12.1231 0.629400
\(372\) −3.12311 −0.161925
\(373\) 17.4384 0.902929 0.451464 0.892289i \(-0.350902\pi\)
0.451464 + 0.892289i \(0.350902\pi\)
\(374\) 7.80776 0.403730
\(375\) −5.43845 −0.280840
\(376\) 4.68466 0.241593
\(377\) 0 0
\(378\) −1.00000 −0.0514344
\(379\) −32.4924 −1.66902 −0.834512 0.550990i \(-0.814250\pi\)
−0.834512 + 0.550990i \(0.814250\pi\)
\(380\) −2.63068 −0.134951
\(381\) −14.2462 −0.729856
\(382\) 22.2462 1.13822
\(383\) 20.6847 1.05694 0.528468 0.848953i \(-0.322767\pi\)
0.528468 + 0.848953i \(0.322767\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0.876894 0.0446907
\(386\) 4.36932 0.222392
\(387\) −11.1231 −0.565419
\(388\) −4.24621 −0.215569
\(389\) 6.31534 0.320201 0.160100 0.987101i \(-0.448818\pi\)
0.160100 + 0.987101i \(0.448818\pi\)
\(390\) 0 0
\(391\) 4.38447 0.221732
\(392\) 1.00000 0.0505076
\(393\) 16.4924 0.831933
\(394\) 3.56155 0.179428
\(395\) 3.12311 0.157140
\(396\) −1.56155 −0.0784710
\(397\) −10.6847 −0.536248 −0.268124 0.963384i \(-0.586404\pi\)
−0.268124 + 0.963384i \(0.586404\pi\)
\(398\) 23.6155 1.18374
\(399\) 4.68466 0.234526
\(400\) −4.68466 −0.234233
\(401\) −30.1771 −1.50697 −0.753486 0.657464i \(-0.771629\pi\)
−0.753486 + 0.657464i \(0.771629\pi\)
\(402\) −7.12311 −0.355268
\(403\) 0 0
\(404\) −15.4384 −0.768091
\(405\) 0.561553 0.0279038
\(406\) 1.00000 0.0496292
\(407\) −16.8769 −0.836557
\(408\) −5.00000 −0.247537
\(409\) −34.5616 −1.70896 −0.854479 0.519485i \(-0.826124\pi\)
−0.854479 + 0.519485i \(0.826124\pi\)
\(410\) −2.31534 −0.114347
\(411\) 14.8078 0.730413
\(412\) 13.3693 0.658659
\(413\) −4.87689 −0.239976
\(414\) −0.876894 −0.0430970
\(415\) 3.50758 0.172180
\(416\) 0 0
\(417\) 4.68466 0.229409
\(418\) 7.31534 0.357805
\(419\) −26.7386 −1.30627 −0.653134 0.757242i \(-0.726546\pi\)
−0.653134 + 0.757242i \(0.726546\pi\)
\(420\) −0.561553 −0.0274010
\(421\) 12.5616 0.612213 0.306106 0.951997i \(-0.400974\pi\)
0.306106 + 0.951997i \(0.400974\pi\)
\(422\) −11.1231 −0.541464
\(423\) 4.68466 0.227776
\(424\) −12.1231 −0.588750
\(425\) 23.4233 1.13620
\(426\) −4.00000 −0.193801
\(427\) 1.00000 0.0483934
\(428\) −13.5616 −0.655522
\(429\) 0 0
\(430\) −6.24621 −0.301219
\(431\) 30.7386 1.48063 0.740314 0.672261i \(-0.234677\pi\)
0.740314 + 0.672261i \(0.234677\pi\)
\(432\) 1.00000 0.0481125
\(433\) −17.6847 −0.849870 −0.424935 0.905224i \(-0.639703\pi\)
−0.424935 + 0.905224i \(0.639703\pi\)
\(434\) 3.12311 0.149914
\(435\) −0.561553 −0.0269244
\(436\) 9.12311 0.436918
\(437\) 4.10795 0.196510
\(438\) −6.56155 −0.313523
\(439\) 4.49242 0.214412 0.107206 0.994237i \(-0.465810\pi\)
0.107206 + 0.994237i \(0.465810\pi\)
\(440\) −0.876894 −0.0418043
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 27.4233 1.30292 0.651460 0.758683i \(-0.274157\pi\)
0.651460 + 0.758683i \(0.274157\pi\)
\(444\) 10.8078 0.512914
\(445\) 1.50758 0.0714660
\(446\) 16.4924 0.780939
\(447\) 2.80776 0.132803
\(448\) −1.00000 −0.0472456
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) −4.68466 −0.220837
\(451\) 6.43845 0.303175
\(452\) −11.9309 −0.561181
\(453\) −21.5616 −1.01305
\(454\) 4.87689 0.228884
\(455\) 0 0
\(456\) −4.68466 −0.219379
\(457\) −25.6847 −1.20148 −0.600739 0.799445i \(-0.705127\pi\)
−0.600739 + 0.799445i \(0.705127\pi\)
\(458\) 2.19224 0.102436
\(459\) −5.00000 −0.233380
\(460\) −0.492423 −0.0229593
\(461\) −7.05398 −0.328536 −0.164268 0.986416i \(-0.552526\pi\)
−0.164268 + 0.986416i \(0.552526\pi\)
\(462\) 1.56155 0.0726500
\(463\) 0.684658 0.0318188 0.0159094 0.999873i \(-0.494936\pi\)
0.0159094 + 0.999873i \(0.494936\pi\)
\(464\) −1.00000 −0.0464238
\(465\) −1.75379 −0.0813300
\(466\) −6.00000 −0.277945
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 0 0
\(469\) 7.12311 0.328914
\(470\) 2.63068 0.121344
\(471\) 14.8078 0.682306
\(472\) 4.87689 0.224477
\(473\) 17.3693 0.798642
\(474\) 5.56155 0.255451
\(475\) 21.9460 1.00695
\(476\) 5.00000 0.229175
\(477\) −12.1231 −0.555079
\(478\) 4.00000 0.182956
\(479\) 1.56155 0.0713492 0.0356746 0.999363i \(-0.488642\pi\)
0.0356746 + 0.999363i \(0.488642\pi\)
\(480\) 0.561553 0.0256313
\(481\) 0 0
\(482\) 23.6847 1.07881
\(483\) 0.876894 0.0399001
\(484\) −8.56155 −0.389161
\(485\) −2.38447 −0.108273
\(486\) 1.00000 0.0453609
\(487\) 11.3153 0.512747 0.256374 0.966578i \(-0.417472\pi\)
0.256374 + 0.966578i \(0.417472\pi\)
\(488\) −1.00000 −0.0452679
\(489\) 7.12311 0.322118
\(490\) 0.561553 0.0253684
\(491\) −18.2462 −0.823440 −0.411720 0.911310i \(-0.635072\pi\)
−0.411720 + 0.911310i \(0.635072\pi\)
\(492\) −4.12311 −0.185884
\(493\) 5.00000 0.225189
\(494\) 0 0
\(495\) −0.876894 −0.0394135
\(496\) −3.12311 −0.140232
\(497\) 4.00000 0.179425
\(498\) 6.24621 0.279899
\(499\) −2.63068 −0.117766 −0.0588828 0.998265i \(-0.518754\pi\)
−0.0588828 + 0.998265i \(0.518754\pi\)
\(500\) −5.43845 −0.243215
\(501\) −12.4924 −0.558120
\(502\) −24.4924 −1.09315
\(503\) −12.4924 −0.557010 −0.278505 0.960435i \(-0.589839\pi\)
−0.278505 + 0.960435i \(0.589839\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −8.66950 −0.385788
\(506\) 1.36932 0.0608736
\(507\) 0 0
\(508\) −14.2462 −0.632073
\(509\) 39.6847 1.75899 0.879496 0.475907i \(-0.157880\pi\)
0.879496 + 0.475907i \(0.157880\pi\)
\(510\) −2.80776 −0.124330
\(511\) 6.56155 0.290266
\(512\) 1.00000 0.0441942
\(513\) −4.68466 −0.206833
\(514\) −31.2462 −1.37821
\(515\) 7.50758 0.330823
\(516\) −11.1231 −0.489667
\(517\) −7.31534 −0.321728
\(518\) −10.8078 −0.474866
\(519\) −13.1231 −0.576040
\(520\) 0 0
\(521\) −17.0000 −0.744784 −0.372392 0.928076i \(-0.621462\pi\)
−0.372392 + 0.928076i \(0.621462\pi\)
\(522\) −1.00000 −0.0437688
\(523\) 35.8078 1.56576 0.782882 0.622170i \(-0.213749\pi\)
0.782882 + 0.622170i \(0.213749\pi\)
\(524\) 16.4924 0.720475
\(525\) 4.68466 0.204455
\(526\) 12.4924 0.544696
\(527\) 15.6155 0.680223
\(528\) −1.56155 −0.0679579
\(529\) −22.2311 −0.966568
\(530\) −6.80776 −0.295710
\(531\) 4.87689 0.211639
\(532\) 4.68466 0.203106
\(533\) 0 0
\(534\) 2.68466 0.116177
\(535\) −7.61553 −0.329248
\(536\) −7.12311 −0.307671
\(537\) 16.4924 0.711701
\(538\) −13.1231 −0.565777
\(539\) −1.56155 −0.0672608
\(540\) 0.561553 0.0241654
\(541\) −2.56155 −0.110130 −0.0550649 0.998483i \(-0.517537\pi\)
−0.0550649 + 0.998483i \(0.517537\pi\)
\(542\) 4.87689 0.209481
\(543\) −3.24621 −0.139308
\(544\) −5.00000 −0.214373
\(545\) 5.12311 0.219450
\(546\) 0 0
\(547\) 26.7386 1.14326 0.571631 0.820511i \(-0.306311\pi\)
0.571631 + 0.820511i \(0.306311\pi\)
\(548\) 14.8078 0.632556
\(549\) −1.00000 −0.0426790
\(550\) 7.31534 0.311927
\(551\) 4.68466 0.199573
\(552\) −0.876894 −0.0373231
\(553\) −5.56155 −0.236501
\(554\) 0.561553 0.0238581
\(555\) 6.06913 0.257620
\(556\) 4.68466 0.198674
\(557\) −27.7386 −1.17532 −0.587662 0.809107i \(-0.699951\pi\)
−0.587662 + 0.809107i \(0.699951\pi\)
\(558\) −3.12311 −0.132212
\(559\) 0 0
\(560\) −0.561553 −0.0237299
\(561\) 7.80776 0.329644
\(562\) −12.8078 −0.540263
\(563\) −25.3693 −1.06919 −0.534595 0.845109i \(-0.679536\pi\)
−0.534595 + 0.845109i \(0.679536\pi\)
\(564\) 4.68466 0.197260
\(565\) −6.69981 −0.281863
\(566\) −24.4924 −1.02949
\(567\) −1.00000 −0.0419961
\(568\) −4.00000 −0.167836
\(569\) −29.6155 −1.24155 −0.620774 0.783990i \(-0.713181\pi\)
−0.620774 + 0.783990i \(0.713181\pi\)
\(570\) −2.63068 −0.110187
\(571\) 30.2462 1.26576 0.632882 0.774248i \(-0.281872\pi\)
0.632882 + 0.774248i \(0.281872\pi\)
\(572\) 0 0
\(573\) 22.2462 0.929349
\(574\) 4.12311 0.172095
\(575\) 4.10795 0.171313
\(576\) 1.00000 0.0416667
\(577\) 43.6847 1.81862 0.909308 0.416124i \(-0.136612\pi\)
0.909308 + 0.416124i \(0.136612\pi\)
\(578\) 8.00000 0.332756
\(579\) 4.36932 0.181583
\(580\) −0.561553 −0.0233172
\(581\) −6.24621 −0.259137
\(582\) −4.24621 −0.176011
\(583\) 18.9309 0.784037
\(584\) −6.56155 −0.271519
\(585\) 0 0
\(586\) 7.68466 0.317450
\(587\) 6.63068 0.273678 0.136839 0.990593i \(-0.456306\pi\)
0.136839 + 0.990593i \(0.456306\pi\)
\(588\) 1.00000 0.0412393
\(589\) 14.6307 0.602847
\(590\) 2.73863 0.112748
\(591\) 3.56155 0.146503
\(592\) 10.8078 0.444196
\(593\) −8.61553 −0.353797 −0.176899 0.984229i \(-0.556606\pi\)
−0.176899 + 0.984229i \(0.556606\pi\)
\(594\) −1.56155 −0.0640713
\(595\) 2.80776 0.115107
\(596\) 2.80776 0.115010
\(597\) 23.6155 0.966519
\(598\) 0 0
\(599\) −31.1231 −1.27166 −0.635828 0.771831i \(-0.719341\pi\)
−0.635828 + 0.771831i \(0.719341\pi\)
\(600\) −4.68466 −0.191250
\(601\) 8.06913 0.329147 0.164573 0.986365i \(-0.447375\pi\)
0.164573 + 0.986365i \(0.447375\pi\)
\(602\) 11.1231 0.453344
\(603\) −7.12311 −0.290075
\(604\) −21.5616 −0.877327
\(605\) −4.80776 −0.195463
\(606\) −15.4384 −0.627144
\(607\) 12.8769 0.522657 0.261329 0.965250i \(-0.415839\pi\)
0.261329 + 0.965250i \(0.415839\pi\)
\(608\) −4.68466 −0.189988
\(609\) 1.00000 0.0405220
\(610\) −0.561553 −0.0227366
\(611\) 0 0
\(612\) −5.00000 −0.202113
\(613\) −17.3002 −0.698748 −0.349374 0.936983i \(-0.613606\pi\)
−0.349374 + 0.936983i \(0.613606\pi\)
\(614\) −5.06913 −0.204573
\(615\) −2.31534 −0.0933636
\(616\) 1.56155 0.0629168
\(617\) 34.4233 1.38583 0.692915 0.721019i \(-0.256326\pi\)
0.692915 + 0.721019i \(0.256326\pi\)
\(618\) 13.3693 0.537793
\(619\) 16.6847 0.670613 0.335307 0.942109i \(-0.391160\pi\)
0.335307 + 0.942109i \(0.391160\pi\)
\(620\) −1.75379 −0.0704339
\(621\) −0.876894 −0.0351886
\(622\) 18.4384 0.739314
\(623\) −2.68466 −0.107559
\(624\) 0 0
\(625\) 20.3693 0.814773
\(626\) −13.6155 −0.544186
\(627\) 7.31534 0.292147
\(628\) 14.8078 0.590894
\(629\) −54.0388 −2.15467
\(630\) −0.561553 −0.0223728
\(631\) 43.8078 1.74396 0.871980 0.489542i \(-0.162836\pi\)
0.871980 + 0.489542i \(0.162836\pi\)
\(632\) 5.56155 0.221227
\(633\) −11.1231 −0.442104
\(634\) −7.43845 −0.295418
\(635\) −8.00000 −0.317470
\(636\) −12.1231 −0.480712
\(637\) 0 0
\(638\) 1.56155 0.0618225
\(639\) −4.00000 −0.158238
\(640\) 0.561553 0.0221973
\(641\) 8.06913 0.318711 0.159356 0.987221i \(-0.449058\pi\)
0.159356 + 0.987221i \(0.449058\pi\)
\(642\) −13.5616 −0.535232
\(643\) −22.4384 −0.884886 −0.442443 0.896797i \(-0.645888\pi\)
−0.442443 + 0.896797i \(0.645888\pi\)
\(644\) 0.876894 0.0345545
\(645\) −6.24621 −0.245944
\(646\) 23.4233 0.921577
\(647\) −0.192236 −0.00755757 −0.00377879 0.999993i \(-0.501203\pi\)
−0.00377879 + 0.999993i \(0.501203\pi\)
\(648\) 1.00000 0.0392837
\(649\) −7.61553 −0.298936
\(650\) 0 0
\(651\) 3.12311 0.122404
\(652\) 7.12311 0.278962
\(653\) 1.80776 0.0707433 0.0353716 0.999374i \(-0.488739\pi\)
0.0353716 + 0.999374i \(0.488739\pi\)
\(654\) 9.12311 0.356742
\(655\) 9.26137 0.361872
\(656\) −4.12311 −0.160980
\(657\) −6.56155 −0.255991
\(658\) −4.68466 −0.182627
\(659\) −40.6847 −1.58485 −0.792425 0.609970i \(-0.791182\pi\)
−0.792425 + 0.609970i \(0.791182\pi\)
\(660\) −0.876894 −0.0341331
\(661\) 10.8078 0.420373 0.210187 0.977661i \(-0.432593\pi\)
0.210187 + 0.977661i \(0.432593\pi\)
\(662\) −10.6307 −0.413173
\(663\) 0 0
\(664\) 6.24621 0.242400
\(665\) 2.63068 0.102014
\(666\) 10.8078 0.418792
\(667\) 0.876894 0.0339535
\(668\) −12.4924 −0.483346
\(669\) 16.4924 0.637634
\(670\) −4.00000 −0.154533
\(671\) 1.56155 0.0602831
\(672\) −1.00000 −0.0385758
\(673\) 25.2462 0.973170 0.486585 0.873633i \(-0.338242\pi\)
0.486585 + 0.873633i \(0.338242\pi\)
\(674\) 3.49242 0.134523
\(675\) −4.68466 −0.180313
\(676\) 0 0
\(677\) −49.6155 −1.90688 −0.953440 0.301583i \(-0.902485\pi\)
−0.953440 + 0.301583i \(0.902485\pi\)
\(678\) −11.9309 −0.458202
\(679\) 4.24621 0.162955
\(680\) −2.80776 −0.107673
\(681\) 4.87689 0.186883
\(682\) 4.87689 0.186746
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) −4.68466 −0.179122
\(685\) 8.31534 0.317713
\(686\) −1.00000 −0.0381802
\(687\) 2.19224 0.0836390
\(688\) −11.1231 −0.424064
\(689\) 0 0
\(690\) −0.492423 −0.0187462
\(691\) 4.49242 0.170900 0.0854499 0.996342i \(-0.472767\pi\)
0.0854499 + 0.996342i \(0.472767\pi\)
\(692\) −13.1231 −0.498866
\(693\) 1.56155 0.0593185
\(694\) −12.6847 −0.481503
\(695\) 2.63068 0.0997875
\(696\) −1.00000 −0.0379049
\(697\) 20.6155 0.780869
\(698\) 26.4924 1.00275
\(699\) −6.00000 −0.226941
\(700\) 4.68466 0.177063
\(701\) −28.0540 −1.05958 −0.529792 0.848128i \(-0.677730\pi\)
−0.529792 + 0.848128i \(0.677730\pi\)
\(702\) 0 0
\(703\) −50.6307 −1.90957
\(704\) −1.56155 −0.0588532
\(705\) 2.63068 0.0990773
\(706\) 10.8078 0.406756
\(707\) 15.4384 0.580623
\(708\) 4.87689 0.183285
\(709\) −33.3002 −1.25061 −0.625307 0.780379i \(-0.715026\pi\)
−0.625307 + 0.780379i \(0.715026\pi\)
\(710\) −2.24621 −0.0842988
\(711\) 5.56155 0.208575
\(712\) 2.68466 0.100612
\(713\) 2.73863 0.102563
\(714\) 5.00000 0.187120
\(715\) 0 0
\(716\) 16.4924 0.616351
\(717\) 4.00000 0.149383
\(718\) −29.3693 −1.09605
\(719\) −6.93087 −0.258478 −0.129239 0.991613i \(-0.541253\pi\)
−0.129239 + 0.991613i \(0.541253\pi\)
\(720\) 0.561553 0.0209278
\(721\) −13.3693 −0.497899
\(722\) 2.94602 0.109640
\(723\) 23.6847 0.880842
\(724\) −3.24621 −0.120644
\(725\) 4.68466 0.173984
\(726\) −8.56155 −0.317749
\(727\) 33.7538 1.25186 0.625929 0.779880i \(-0.284720\pi\)
0.625929 + 0.779880i \(0.284720\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −3.68466 −0.136375
\(731\) 55.6155 2.05701
\(732\) −1.00000 −0.0369611
\(733\) −8.61553 −0.318222 −0.159111 0.987261i \(-0.550863\pi\)
−0.159111 + 0.987261i \(0.550863\pi\)
\(734\) −13.3693 −0.493470
\(735\) 0.561553 0.0207132
\(736\) −0.876894 −0.0323228
\(737\) 11.1231 0.409725
\(738\) −4.12311 −0.151774
\(739\) 22.6307 0.832483 0.416242 0.909254i \(-0.363347\pi\)
0.416242 + 0.909254i \(0.363347\pi\)
\(740\) 6.06913 0.223106
\(741\) 0 0
\(742\) 12.1231 0.445053
\(743\) 33.3693 1.22420 0.612101 0.790780i \(-0.290325\pi\)
0.612101 + 0.790780i \(0.290325\pi\)
\(744\) −3.12311 −0.114499
\(745\) 1.57671 0.0577661
\(746\) 17.4384 0.638467
\(747\) 6.24621 0.228537
\(748\) 7.80776 0.285480
\(749\) 13.5616 0.495528
\(750\) −5.43845 −0.198584
\(751\) 19.8078 0.722796 0.361398 0.932412i \(-0.382300\pi\)
0.361398 + 0.932412i \(0.382300\pi\)
\(752\) 4.68466 0.170832
\(753\) −24.4924 −0.892553
\(754\) 0 0
\(755\) −12.1080 −0.440653
\(756\) −1.00000 −0.0363696
\(757\) 1.12311 0.0408200 0.0204100 0.999792i \(-0.493503\pi\)
0.0204100 + 0.999792i \(0.493503\pi\)
\(758\) −32.4924 −1.18018
\(759\) 1.36932 0.0497031
\(760\) −2.63068 −0.0954249
\(761\) −9.50758 −0.344649 −0.172325 0.985040i \(-0.555128\pi\)
−0.172325 + 0.985040i \(0.555128\pi\)
\(762\) −14.2462 −0.516086
\(763\) −9.12311 −0.330279
\(764\) 22.2462 0.804840
\(765\) −2.80776 −0.101515
\(766\) 20.6847 0.747367
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 42.9848 1.55007 0.775037 0.631916i \(-0.217731\pi\)
0.775037 + 0.631916i \(0.217731\pi\)
\(770\) 0.876894 0.0316011
\(771\) −31.2462 −1.12530
\(772\) 4.36932 0.157255
\(773\) −6.49242 −0.233516 −0.116758 0.993160i \(-0.537250\pi\)
−0.116758 + 0.993160i \(0.537250\pi\)
\(774\) −11.1231 −0.399812
\(775\) 14.6307 0.525550
\(776\) −4.24621 −0.152430
\(777\) −10.8078 −0.387726
\(778\) 6.31534 0.226416
\(779\) 19.3153 0.692044
\(780\) 0 0
\(781\) 6.24621 0.223507
\(782\) 4.38447 0.156788
\(783\) −1.00000 −0.0357371
\(784\) 1.00000 0.0357143
\(785\) 8.31534 0.296787
\(786\) 16.4924 0.588265
\(787\) −19.8078 −0.706071 −0.353035 0.935610i \(-0.614850\pi\)
−0.353035 + 0.935610i \(0.614850\pi\)
\(788\) 3.56155 0.126875
\(789\) 12.4924 0.444742
\(790\) 3.12311 0.111115
\(791\) 11.9309 0.424213
\(792\) −1.56155 −0.0554874
\(793\) 0 0
\(794\) −10.6847 −0.379184
\(795\) −6.80776 −0.241447
\(796\) 23.6155 0.837030
\(797\) 50.1080 1.77491 0.887457 0.460890i \(-0.152470\pi\)
0.887457 + 0.460890i \(0.152470\pi\)
\(798\) 4.68466 0.165835
\(799\) −23.4233 −0.828657
\(800\) −4.68466 −0.165628
\(801\) 2.68466 0.0948577
\(802\) −30.1771 −1.06559
\(803\) 10.2462 0.361581
\(804\) −7.12311 −0.251213
\(805\) 0.492423 0.0173556
\(806\) 0 0
\(807\) −13.1231 −0.461955
\(808\) −15.4384 −0.543123
\(809\) 17.9309 0.630416 0.315208 0.949023i \(-0.397926\pi\)
0.315208 + 0.949023i \(0.397926\pi\)
\(810\) 0.561553 0.0197310
\(811\) 22.2462 0.781170 0.390585 0.920567i \(-0.372273\pi\)
0.390585 + 0.920567i \(0.372273\pi\)
\(812\) 1.00000 0.0350931
\(813\) 4.87689 0.171040
\(814\) −16.8769 −0.591535
\(815\) 4.00000 0.140114
\(816\) −5.00000 −0.175035
\(817\) 52.1080 1.82303
\(818\) −34.5616 −1.20842
\(819\) 0 0
\(820\) −2.31534 −0.0808552
\(821\) −40.9309 −1.42850 −0.714249 0.699892i \(-0.753231\pi\)
−0.714249 + 0.699892i \(0.753231\pi\)
\(822\) 14.8078 0.516480
\(823\) −2.24621 −0.0782980 −0.0391490 0.999233i \(-0.512465\pi\)
−0.0391490 + 0.999233i \(0.512465\pi\)
\(824\) 13.3693 0.465742
\(825\) 7.31534 0.254688
\(826\) −4.87689 −0.169689
\(827\) −44.4924 −1.54715 −0.773577 0.633703i \(-0.781534\pi\)
−0.773577 + 0.633703i \(0.781534\pi\)
\(828\) −0.876894 −0.0304742
\(829\) −45.1080 −1.56666 −0.783332 0.621604i \(-0.786481\pi\)
−0.783332 + 0.621604i \(0.786481\pi\)
\(830\) 3.50758 0.121750
\(831\) 0.561553 0.0194801
\(832\) 0 0
\(833\) −5.00000 −0.173240
\(834\) 4.68466 0.162216
\(835\) −7.01515 −0.242769
\(836\) 7.31534 0.253006
\(837\) −3.12311 −0.107950
\(838\) −26.7386 −0.923671
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) −0.561553 −0.0193754
\(841\) −28.0000 −0.965517
\(842\) 12.5616 0.432900
\(843\) −12.8078 −0.441123
\(844\) −11.1231 −0.382873
\(845\) 0 0
\(846\) 4.68466 0.161062
\(847\) 8.56155 0.294178
\(848\) −12.1231 −0.416309
\(849\) −24.4924 −0.840578
\(850\) 23.4233 0.803412
\(851\) −9.47727 −0.324877
\(852\) −4.00000 −0.137038
\(853\) 24.3693 0.834390 0.417195 0.908817i \(-0.363013\pi\)
0.417195 + 0.908817i \(0.363013\pi\)
\(854\) 1.00000 0.0342193
\(855\) −2.63068 −0.0899675
\(856\) −13.5616 −0.463524
\(857\) 31.3002 1.06919 0.534597 0.845107i \(-0.320463\pi\)
0.534597 + 0.845107i \(0.320463\pi\)
\(858\) 0 0
\(859\) −16.1922 −0.552472 −0.276236 0.961090i \(-0.589087\pi\)
−0.276236 + 0.961090i \(0.589087\pi\)
\(860\) −6.24621 −0.212994
\(861\) 4.12311 0.140515
\(862\) 30.7386 1.04696
\(863\) −48.1080 −1.63761 −0.818807 0.574069i \(-0.805364\pi\)
−0.818807 + 0.574069i \(0.805364\pi\)
\(864\) 1.00000 0.0340207
\(865\) −7.36932 −0.250564
\(866\) −17.6847 −0.600949
\(867\) 8.00000 0.271694
\(868\) 3.12311 0.106005
\(869\) −8.68466 −0.294607
\(870\) −0.561553 −0.0190384
\(871\) 0 0
\(872\) 9.12311 0.308947
\(873\) −4.24621 −0.143712
\(874\) 4.10795 0.138954
\(875\) 5.43845 0.183853
\(876\) −6.56155 −0.221694
\(877\) −45.3002 −1.52968 −0.764839 0.644221i \(-0.777182\pi\)
−0.764839 + 0.644221i \(0.777182\pi\)
\(878\) 4.49242 0.151612
\(879\) 7.68466 0.259197
\(880\) −0.876894 −0.0295601
\(881\) 1.82292 0.0614157 0.0307079 0.999528i \(-0.490224\pi\)
0.0307079 + 0.999528i \(0.490224\pi\)
\(882\) 1.00000 0.0336718
\(883\) 8.87689 0.298731 0.149366 0.988782i \(-0.452277\pi\)
0.149366 + 0.988782i \(0.452277\pi\)
\(884\) 0 0
\(885\) 2.73863 0.0920582
\(886\) 27.4233 0.921304
\(887\) 9.56155 0.321046 0.160523 0.987032i \(-0.448682\pi\)
0.160523 + 0.987032i \(0.448682\pi\)
\(888\) 10.8078 0.362685
\(889\) 14.2462 0.477803
\(890\) 1.50758 0.0505341
\(891\) −1.56155 −0.0523140
\(892\) 16.4924 0.552207
\(893\) −21.9460 −0.734396
\(894\) 2.80776 0.0939057
\(895\) 9.26137 0.309573
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) 3.12311 0.104161
\(900\) −4.68466 −0.156155
\(901\) 60.6155 2.01940
\(902\) 6.43845 0.214377
\(903\) 11.1231 0.370154
\(904\) −11.9309 −0.396815
\(905\) −1.82292 −0.0605959
\(906\) −21.5616 −0.716335
\(907\) 31.1231 1.03343 0.516713 0.856159i \(-0.327155\pi\)
0.516713 + 0.856159i \(0.327155\pi\)
\(908\) 4.87689 0.161845
\(909\) −15.4384 −0.512061
\(910\) 0 0
\(911\) −54.7386 −1.81357 −0.906786 0.421591i \(-0.861472\pi\)
−0.906786 + 0.421591i \(0.861472\pi\)
\(912\) −4.68466 −0.155125
\(913\) −9.75379 −0.322803
\(914\) −25.6847 −0.849573
\(915\) −0.561553 −0.0185644
\(916\) 2.19224 0.0724335
\(917\) −16.4924 −0.544628
\(918\) −5.00000 −0.165025
\(919\) 2.43845 0.0804370 0.0402185 0.999191i \(-0.487195\pi\)
0.0402185 + 0.999191i \(0.487195\pi\)
\(920\) −0.492423 −0.0162347
\(921\) −5.06913 −0.167034
\(922\) −7.05398 −0.232310
\(923\) 0 0
\(924\) 1.56155 0.0513713
\(925\) −50.6307 −1.66473
\(926\) 0.684658 0.0224993
\(927\) 13.3693 0.439106
\(928\) −1.00000 −0.0328266
\(929\) 34.6155 1.13570 0.567849 0.823133i \(-0.307776\pi\)
0.567849 + 0.823133i \(0.307776\pi\)
\(930\) −1.75379 −0.0575090
\(931\) −4.68466 −0.153533
\(932\) −6.00000 −0.196537
\(933\) 18.4384 0.603648
\(934\) 28.0000 0.916188
\(935\) 4.38447 0.143388
\(936\) 0 0
\(937\) 9.43845 0.308341 0.154170 0.988044i \(-0.450730\pi\)
0.154170 + 0.988044i \(0.450730\pi\)
\(938\) 7.12311 0.232578
\(939\) −13.6155 −0.444326
\(940\) 2.63068 0.0858034
\(941\) 28.6307 0.933334 0.466667 0.884433i \(-0.345455\pi\)
0.466667 + 0.884433i \(0.345455\pi\)
\(942\) 14.8078 0.482463
\(943\) 3.61553 0.117738
\(944\) 4.87689 0.158729
\(945\) −0.561553 −0.0182673
\(946\) 17.3693 0.564725
\(947\) −12.6847 −0.412196 −0.206098 0.978531i \(-0.566077\pi\)
−0.206098 + 0.978531i \(0.566077\pi\)
\(948\) 5.56155 0.180631
\(949\) 0 0
\(950\) 21.9460 0.712023
\(951\) −7.43845 −0.241208
\(952\) 5.00000 0.162051
\(953\) −3.26137 −0.105646 −0.0528230 0.998604i \(-0.516822\pi\)
−0.0528230 + 0.998604i \(0.516822\pi\)
\(954\) −12.1231 −0.392500
\(955\) 12.4924 0.404245
\(956\) 4.00000 0.129369
\(957\) 1.56155 0.0504778
\(958\) 1.56155 0.0504515
\(959\) −14.8078 −0.478168
\(960\) 0.561553 0.0181240
\(961\) −21.2462 −0.685362
\(962\) 0 0
\(963\) −13.5616 −0.437015
\(964\) 23.6847 0.762831
\(965\) 2.45360 0.0789842
\(966\) 0.876894 0.0282136
\(967\) 28.4924 0.916255 0.458127 0.888887i \(-0.348520\pi\)
0.458127 + 0.888887i \(0.348520\pi\)
\(968\) −8.56155 −0.275179
\(969\) 23.4233 0.752465
\(970\) −2.38447 −0.0765608
\(971\) −19.1231 −0.613690 −0.306845 0.951760i \(-0.599273\pi\)
−0.306845 + 0.951760i \(0.599273\pi\)
\(972\) 1.00000 0.0320750
\(973\) −4.68466 −0.150183
\(974\) 11.3153 0.362567
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) −35.0540 −1.12148 −0.560738 0.827993i \(-0.689483\pi\)
−0.560738 + 0.827993i \(0.689483\pi\)
\(978\) 7.12311 0.227772
\(979\) −4.19224 −0.133984
\(980\) 0.561553 0.0179381
\(981\) 9.12311 0.291278
\(982\) −18.2462 −0.582260
\(983\) −27.2311 −0.868536 −0.434268 0.900784i \(-0.642993\pi\)
−0.434268 + 0.900784i \(0.642993\pi\)
\(984\) −4.12311 −0.131440
\(985\) 2.00000 0.0637253
\(986\) 5.00000 0.159232
\(987\) −4.68466 −0.149114
\(988\) 0 0
\(989\) 9.75379 0.310152
\(990\) −0.876894 −0.0278695
\(991\) −5.56155 −0.176669 −0.0883343 0.996091i \(-0.528154\pi\)
−0.0883343 + 0.996091i \(0.528154\pi\)
\(992\) −3.12311 −0.0991587
\(993\) −10.6307 −0.337355
\(994\) 4.00000 0.126872
\(995\) 13.2614 0.420414
\(996\) 6.24621 0.197919
\(997\) −17.0000 −0.538395 −0.269198 0.963085i \(-0.586759\pi\)
−0.269198 + 0.963085i \(0.586759\pi\)
\(998\) −2.63068 −0.0832728
\(999\) 10.8078 0.341943
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 7098.2.a.bw.1.2 2
13.3 even 3 546.2.l.i.295.2 yes 4
13.9 even 3 546.2.l.i.211.2 4
13.12 even 2 7098.2.a.bq.1.1 2
39.29 odd 6 1638.2.r.z.1387.1 4
39.35 odd 6 1638.2.r.z.757.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.l.i.211.2 4 13.9 even 3
546.2.l.i.295.2 yes 4 13.3 even 3
1638.2.r.z.757.1 4 39.35 odd 6
1638.2.r.z.1387.1 4 39.29 odd 6
7098.2.a.bq.1.1 2 13.12 even 2
7098.2.a.bw.1.2 2 1.1 even 1 trivial