Properties

Label 7098.2.a.ck.1.3
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.04892 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.04892 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.04892 q^{10} -5.74094 q^{11} -1.00000 q^{12} -1.00000 q^{14} -4.04892 q^{15} +1.00000 q^{16} +1.40581 q^{17} +1.00000 q^{18} +6.26875 q^{19} +4.04892 q^{20} +1.00000 q^{21} -5.74094 q^{22} +1.86294 q^{23} -1.00000 q^{24} +11.3937 q^{25} -1.00000 q^{27} -1.00000 q^{28} -3.20775 q^{29} -4.04892 q^{30} +8.63102 q^{31} +1.00000 q^{32} +5.74094 q^{33} +1.40581 q^{34} -4.04892 q^{35} +1.00000 q^{36} +3.36227 q^{37} +6.26875 q^{38} +4.04892 q^{40} -2.59419 q^{41} +1.00000 q^{42} -5.70171 q^{43} -5.74094 q^{44} +4.04892 q^{45} +1.86294 q^{46} -8.27413 q^{47} -1.00000 q^{48} +1.00000 q^{49} +11.3937 q^{50} -1.40581 q^{51} -3.16421 q^{53} -1.00000 q^{54} -23.2446 q^{55} -1.00000 q^{56} -6.26875 q^{57} -3.20775 q^{58} +7.50604 q^{59} -4.04892 q^{60} +13.5797 q^{61} +8.63102 q^{62} -1.00000 q^{63} +1.00000 q^{64} +5.74094 q^{66} +9.20775 q^{67} +1.40581 q^{68} -1.86294 q^{69} -4.04892 q^{70} -16.8334 q^{71} +1.00000 q^{72} -0.317667 q^{73} +3.36227 q^{74} -11.3937 q^{75} +6.26875 q^{76} +5.74094 q^{77} +1.10992 q^{79} +4.04892 q^{80} +1.00000 q^{81} -2.59419 q^{82} +7.10992 q^{83} +1.00000 q^{84} +5.69202 q^{85} -5.70171 q^{86} +3.20775 q^{87} -5.74094 q^{88} -8.02177 q^{89} +4.04892 q^{90} +1.86294 q^{92} -8.63102 q^{93} -8.27413 q^{94} +25.3817 q^{95} -1.00000 q^{96} +18.0978 q^{97} +1.00000 q^{98} -5.74094 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + 3q^{10} - 3q^{11} - 3q^{12} - 3q^{14} - 3q^{15} + 3q^{16} - 9q^{17} + 3q^{18} + 11q^{19} + 3q^{20} + 3q^{21} - 3q^{22} + 11q^{23} - 3q^{24} + 2q^{25} - 3q^{27} - 3q^{28} + 8q^{29} - 3q^{30} + 11q^{31} + 3q^{32} + 3q^{33} - 9q^{34} - 3q^{35} + 3q^{36} + 3q^{37} + 11q^{38} + 3q^{40} - 21q^{41} + 3q^{42} + 10q^{43} - 3q^{44} + 3q^{45} + 11q^{46} - 14q^{47} - 3q^{48} + 3q^{49} + 2q^{50} + 9q^{51} + 2q^{53} - 3q^{54} - 24q^{55} - 3q^{56} - 11q^{57} + 8q^{58} + 32q^{59} - 3q^{60} - 6q^{61} + 11q^{62} - 3q^{63} + 3q^{64} + 3q^{66} + 10q^{67} - 9q^{68} - 11q^{69} - 3q^{70} - 21q^{71} + 3q^{72} + 16q^{73} + 3q^{74} - 2q^{75} + 11q^{76} + 3q^{77} + 4q^{79} + 3q^{80} + 3q^{81} - 21q^{82} + 22q^{83} + 3q^{84} + 12q^{85} + 10q^{86} - 8q^{87} - 3q^{88} - 21q^{89} + 3q^{90} + 11q^{92} - 11q^{93} - 14q^{94} + 25q^{95} - 3q^{96} + 36q^{97} + 3q^{98} - 3q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 4.04892 1.81073 0.905365 0.424633i \(-0.139597\pi\)
0.905365 + 0.424633i \(0.139597\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 4.04892 1.28038
\(11\) −5.74094 −1.73096 −0.865479 0.500945i \(-0.832986\pi\)
−0.865479 + 0.500945i \(0.832986\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) −4.04892 −1.04543
\(16\) 1.00000 0.250000
\(17\) 1.40581 0.340960 0.170480 0.985361i \(-0.445468\pi\)
0.170480 + 0.985361i \(0.445468\pi\)
\(18\) 1.00000 0.235702
\(19\) 6.26875 1.43815 0.719075 0.694933i \(-0.244566\pi\)
0.719075 + 0.694933i \(0.244566\pi\)
\(20\) 4.04892 0.905365
\(21\) 1.00000 0.218218
\(22\) −5.74094 −1.22397
\(23\) 1.86294 0.388449 0.194225 0.980957i \(-0.437781\pi\)
0.194225 + 0.980957i \(0.437781\pi\)
\(24\) −1.00000 −0.204124
\(25\) 11.3937 2.27875
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) −3.20775 −0.595664 −0.297832 0.954618i \(-0.596264\pi\)
−0.297832 + 0.954618i \(0.596264\pi\)
\(30\) −4.04892 −0.739228
\(31\) 8.63102 1.55018 0.775089 0.631852i \(-0.217705\pi\)
0.775089 + 0.631852i \(0.217705\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.74094 0.999369
\(34\) 1.40581 0.241095
\(35\) −4.04892 −0.684392
\(36\) 1.00000 0.166667
\(37\) 3.36227 0.552754 0.276377 0.961049i \(-0.410866\pi\)
0.276377 + 0.961049i \(0.410866\pi\)
\(38\) 6.26875 1.01693
\(39\) 0 0
\(40\) 4.04892 0.640190
\(41\) −2.59419 −0.405144 −0.202572 0.979267i \(-0.564930\pi\)
−0.202572 + 0.979267i \(0.564930\pi\)
\(42\) 1.00000 0.154303
\(43\) −5.70171 −0.869503 −0.434751 0.900551i \(-0.643164\pi\)
−0.434751 + 0.900551i \(0.643164\pi\)
\(44\) −5.74094 −0.865479
\(45\) 4.04892 0.603577
\(46\) 1.86294 0.274675
\(47\) −8.27413 −1.20691 −0.603453 0.797399i \(-0.706209\pi\)
−0.603453 + 0.797399i \(0.706209\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 11.3937 1.61132
\(51\) −1.40581 −0.196853
\(52\) 0 0
\(53\) −3.16421 −0.434638 −0.217319 0.976101i \(-0.569731\pi\)
−0.217319 + 0.976101i \(0.569731\pi\)
\(54\) −1.00000 −0.136083
\(55\) −23.2446 −3.13430
\(56\) −1.00000 −0.133631
\(57\) −6.26875 −0.830316
\(58\) −3.20775 −0.421198
\(59\) 7.50604 0.977203 0.488602 0.872507i \(-0.337507\pi\)
0.488602 + 0.872507i \(0.337507\pi\)
\(60\) −4.04892 −0.522713
\(61\) 13.5797 1.73870 0.869352 0.494193i \(-0.164537\pi\)
0.869352 + 0.494193i \(0.164537\pi\)
\(62\) 8.63102 1.09614
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 5.74094 0.706661
\(67\) 9.20775 1.12491 0.562453 0.826829i \(-0.309858\pi\)
0.562453 + 0.826829i \(0.309858\pi\)
\(68\) 1.40581 0.170480
\(69\) −1.86294 −0.224271
\(70\) −4.04892 −0.483938
\(71\) −16.8334 −1.99776 −0.998878 0.0473508i \(-0.984922\pi\)
−0.998878 + 0.0473508i \(0.984922\pi\)
\(72\) 1.00000 0.117851
\(73\) −0.317667 −0.0371801 −0.0185901 0.999827i \(-0.505918\pi\)
−0.0185901 + 0.999827i \(0.505918\pi\)
\(74\) 3.36227 0.390856
\(75\) −11.3937 −1.31563
\(76\) 6.26875 0.719075
\(77\) 5.74094 0.654241
\(78\) 0 0
\(79\) 1.10992 0.124875 0.0624377 0.998049i \(-0.480113\pi\)
0.0624377 + 0.998049i \(0.480113\pi\)
\(80\) 4.04892 0.452683
\(81\) 1.00000 0.111111
\(82\) −2.59419 −0.286480
\(83\) 7.10992 0.780415 0.390207 0.920727i \(-0.372403\pi\)
0.390207 + 0.920727i \(0.372403\pi\)
\(84\) 1.00000 0.109109
\(85\) 5.69202 0.617386
\(86\) −5.70171 −0.614831
\(87\) 3.20775 0.343907
\(88\) −5.74094 −0.611986
\(89\) −8.02177 −0.850306 −0.425153 0.905122i \(-0.639780\pi\)
−0.425153 + 0.905122i \(0.639780\pi\)
\(90\) 4.04892 0.426793
\(91\) 0 0
\(92\) 1.86294 0.194225
\(93\) −8.63102 −0.894995
\(94\) −8.27413 −0.853411
\(95\) 25.3817 2.60410
\(96\) −1.00000 −0.102062
\(97\) 18.0978 1.83756 0.918778 0.394774i \(-0.129177\pi\)
0.918778 + 0.394774i \(0.129177\pi\)
\(98\) 1.00000 0.101015
\(99\) −5.74094 −0.576986
\(100\) 11.3937 1.13937
\(101\) 11.6093 1.15516 0.577582 0.816333i \(-0.303996\pi\)
0.577582 + 0.816333i \(0.303996\pi\)
\(102\) −1.40581 −0.139196
\(103\) 17.5405 1.72832 0.864158 0.503221i \(-0.167852\pi\)
0.864158 + 0.503221i \(0.167852\pi\)
\(104\) 0 0
\(105\) 4.04892 0.395134
\(106\) −3.16421 −0.307335
\(107\) 4.04892 0.391424 0.195712 0.980661i \(-0.437298\pi\)
0.195712 + 0.980661i \(0.437298\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.4179 0.997853 0.498927 0.866644i \(-0.333728\pi\)
0.498927 + 0.866644i \(0.333728\pi\)
\(110\) −23.2446 −2.21628
\(111\) −3.36227 −0.319133
\(112\) −1.00000 −0.0944911
\(113\) −2.31767 −0.218028 −0.109014 0.994040i \(-0.534769\pi\)
−0.109014 + 0.994040i \(0.534769\pi\)
\(114\) −6.26875 −0.587122
\(115\) 7.54288 0.703377
\(116\) −3.20775 −0.297832
\(117\) 0 0
\(118\) 7.50604 0.690987
\(119\) −1.40581 −0.128871
\(120\) −4.04892 −0.369614
\(121\) 21.9584 1.99622
\(122\) 13.5797 1.22945
\(123\) 2.59419 0.233910
\(124\) 8.63102 0.775089
\(125\) 25.8877 2.31547
\(126\) −1.00000 −0.0890871
\(127\) −12.1957 −1.08219 −0.541095 0.840961i \(-0.681990\pi\)
−0.541095 + 0.840961i \(0.681990\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.70171 0.502008
\(130\) 0 0
\(131\) −11.2078 −0.979226 −0.489613 0.871940i \(-0.662862\pi\)
−0.489613 + 0.871940i \(0.662862\pi\)
\(132\) 5.74094 0.499685
\(133\) −6.26875 −0.543570
\(134\) 9.20775 0.795429
\(135\) −4.04892 −0.348475
\(136\) 1.40581 0.120547
\(137\) 0.768086 0.0656220 0.0328110 0.999462i \(-0.489554\pi\)
0.0328110 + 0.999462i \(0.489554\pi\)
\(138\) −1.86294 −0.158584
\(139\) 13.0271 1.10495 0.552474 0.833530i \(-0.313684\pi\)
0.552474 + 0.833530i \(0.313684\pi\)
\(140\) −4.04892 −0.342196
\(141\) 8.27413 0.696807
\(142\) −16.8334 −1.41263
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −12.9879 −1.07859
\(146\) −0.317667 −0.0262903
\(147\) −1.00000 −0.0824786
\(148\) 3.36227 0.276377
\(149\) 1.54958 0.126947 0.0634733 0.997984i \(-0.479782\pi\)
0.0634733 + 0.997984i \(0.479782\pi\)
\(150\) −11.3937 −0.930294
\(151\) −15.9323 −1.29655 −0.648276 0.761405i \(-0.724510\pi\)
−0.648276 + 0.761405i \(0.724510\pi\)
\(152\) 6.26875 0.508463
\(153\) 1.40581 0.113653
\(154\) 5.74094 0.462618
\(155\) 34.9463 2.80695
\(156\) 0 0
\(157\) 15.4276 1.23126 0.615628 0.788037i \(-0.288903\pi\)
0.615628 + 0.788037i \(0.288903\pi\)
\(158\) 1.10992 0.0883002
\(159\) 3.16421 0.250938
\(160\) 4.04892 0.320095
\(161\) −1.86294 −0.146820
\(162\) 1.00000 0.0785674
\(163\) −14.5133 −1.13677 −0.568386 0.822762i \(-0.692432\pi\)
−0.568386 + 0.822762i \(0.692432\pi\)
\(164\) −2.59419 −0.202572
\(165\) 23.2446 1.80959
\(166\) 7.10992 0.551837
\(167\) 0.846543 0.0655075 0.0327537 0.999463i \(-0.489572\pi\)
0.0327537 + 0.999463i \(0.489572\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) 5.69202 0.436558
\(171\) 6.26875 0.479383
\(172\) −5.70171 −0.434751
\(173\) 7.06100 0.536838 0.268419 0.963302i \(-0.413499\pi\)
0.268419 + 0.963302i \(0.413499\pi\)
\(174\) 3.20775 0.243179
\(175\) −11.3937 −0.861285
\(176\) −5.74094 −0.432740
\(177\) −7.50604 −0.564189
\(178\) −8.02177 −0.601257
\(179\) −3.20775 −0.239759 −0.119879 0.992788i \(-0.538251\pi\)
−0.119879 + 0.992788i \(0.538251\pi\)
\(180\) 4.04892 0.301788
\(181\) −25.4577 −1.89226 −0.946129 0.323791i \(-0.895042\pi\)
−0.946129 + 0.323791i \(0.895042\pi\)
\(182\) 0 0
\(183\) −13.5797 −1.00384
\(184\) 1.86294 0.137338
\(185\) 13.6136 1.00089
\(186\) −8.63102 −0.632857
\(187\) −8.07069 −0.590187
\(188\) −8.27413 −0.603453
\(189\) 1.00000 0.0727393
\(190\) 25.3817 1.84138
\(191\) 21.1782 1.53240 0.766201 0.642601i \(-0.222145\pi\)
0.766201 + 0.642601i \(0.222145\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −10.4306 −0.750809 −0.375404 0.926861i \(-0.622496\pi\)
−0.375404 + 0.926861i \(0.622496\pi\)
\(194\) 18.0978 1.29935
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 12.0435 0.858067 0.429033 0.903289i \(-0.358854\pi\)
0.429033 + 0.903289i \(0.358854\pi\)
\(198\) −5.74094 −0.407991
\(199\) −24.7899 −1.75731 −0.878653 0.477461i \(-0.841557\pi\)
−0.878653 + 0.477461i \(0.841557\pi\)
\(200\) 11.3937 0.805658
\(201\) −9.20775 −0.649465
\(202\) 11.6093 0.816824
\(203\) 3.20775 0.225140
\(204\) −1.40581 −0.0984266
\(205\) −10.5036 −0.733607
\(206\) 17.5405 1.22210
\(207\) 1.86294 0.129483
\(208\) 0 0
\(209\) −35.9885 −2.48938
\(210\) 4.04892 0.279402
\(211\) 5.20775 0.358516 0.179258 0.983802i \(-0.442630\pi\)
0.179258 + 0.983802i \(0.442630\pi\)
\(212\) −3.16421 −0.217319
\(213\) 16.8334 1.15341
\(214\) 4.04892 0.276778
\(215\) −23.0858 −1.57444
\(216\) −1.00000 −0.0680414
\(217\) −8.63102 −0.585912
\(218\) 10.4179 0.705589
\(219\) 0.317667 0.0214660
\(220\) −23.2446 −1.56715
\(221\) 0 0
\(222\) −3.36227 −0.225661
\(223\) 24.2784 1.62580 0.812902 0.582400i \(-0.197886\pi\)
0.812902 + 0.582400i \(0.197886\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 11.3937 0.759582
\(226\) −2.31767 −0.154169
\(227\) −19.7995 −1.31414 −0.657071 0.753829i \(-0.728205\pi\)
−0.657071 + 0.753829i \(0.728205\pi\)
\(228\) −6.26875 −0.415158
\(229\) 18.3177 1.21047 0.605233 0.796049i \(-0.293080\pi\)
0.605233 + 0.796049i \(0.293080\pi\)
\(230\) 7.54288 0.497363
\(231\) −5.74094 −0.377726
\(232\) −3.20775 −0.210599
\(233\) 12.2392 0.801817 0.400909 0.916118i \(-0.368694\pi\)
0.400909 + 0.916118i \(0.368694\pi\)
\(234\) 0 0
\(235\) −33.5013 −2.18538
\(236\) 7.50604 0.488602
\(237\) −1.10992 −0.0720968
\(238\) −1.40581 −0.0911253
\(239\) −18.8713 −1.22068 −0.610341 0.792139i \(-0.708968\pi\)
−0.610341 + 0.792139i \(0.708968\pi\)
\(240\) −4.04892 −0.261356
\(241\) 10.8659 0.699935 0.349968 0.936762i \(-0.386193\pi\)
0.349968 + 0.936762i \(0.386193\pi\)
\(242\) 21.9584 1.41154
\(243\) −1.00000 −0.0641500
\(244\) 13.5797 0.869352
\(245\) 4.04892 0.258676
\(246\) 2.59419 0.165399
\(247\) 0 0
\(248\) 8.63102 0.548070
\(249\) −7.10992 −0.450573
\(250\) 25.8877 1.63728
\(251\) −13.0664 −0.824742 −0.412371 0.911016i \(-0.635299\pi\)
−0.412371 + 0.911016i \(0.635299\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −10.6950 −0.672389
\(254\) −12.1957 −0.765224
\(255\) −5.69202 −0.356448
\(256\) 1.00000 0.0625000
\(257\) −16.4155 −1.02397 −0.511985 0.858994i \(-0.671090\pi\)
−0.511985 + 0.858994i \(0.671090\pi\)
\(258\) 5.70171 0.354973
\(259\) −3.36227 −0.208921
\(260\) 0 0
\(261\) −3.20775 −0.198555
\(262\) −11.2078 −0.692417
\(263\) 15.1250 0.932646 0.466323 0.884614i \(-0.345578\pi\)
0.466323 + 0.884614i \(0.345578\pi\)
\(264\) 5.74094 0.353330
\(265\) −12.8116 −0.787012
\(266\) −6.26875 −0.384362
\(267\) 8.02177 0.490924
\(268\) 9.20775 0.562453
\(269\) 5.58211 0.340347 0.170173 0.985414i \(-0.445567\pi\)
0.170173 + 0.985414i \(0.445567\pi\)
\(270\) −4.04892 −0.246409
\(271\) 25.5308 1.55089 0.775443 0.631418i \(-0.217527\pi\)
0.775443 + 0.631418i \(0.217527\pi\)
\(272\) 1.40581 0.0852399
\(273\) 0 0
\(274\) 0.768086 0.0464018
\(275\) −65.4107 −3.94441
\(276\) −1.86294 −0.112136
\(277\) −23.0019 −1.38205 −0.691026 0.722830i \(-0.742841\pi\)
−0.691026 + 0.722830i \(0.742841\pi\)
\(278\) 13.0271 0.781316
\(279\) 8.63102 0.516726
\(280\) −4.04892 −0.241969
\(281\) 2.89008 0.172408 0.0862040 0.996278i \(-0.472526\pi\)
0.0862040 + 0.996278i \(0.472526\pi\)
\(282\) 8.27413 0.492717
\(283\) −6.25906 −0.372063 −0.186031 0.982544i \(-0.559563\pi\)
−0.186031 + 0.982544i \(0.559563\pi\)
\(284\) −16.8334 −0.998878
\(285\) −25.3817 −1.50348
\(286\) 0 0
\(287\) 2.59419 0.153130
\(288\) 1.00000 0.0589256
\(289\) −15.0237 −0.883746
\(290\) −12.9879 −0.762677
\(291\) −18.0978 −1.06091
\(292\) −0.317667 −0.0185901
\(293\) 23.8019 1.39052 0.695262 0.718756i \(-0.255288\pi\)
0.695262 + 0.718756i \(0.255288\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 30.3913 1.76945
\(296\) 3.36227 0.195428
\(297\) 5.74094 0.333123
\(298\) 1.54958 0.0897648
\(299\) 0 0
\(300\) −11.3937 −0.657817
\(301\) 5.70171 0.328641
\(302\) −15.9323 −0.916801
\(303\) −11.6093 −0.666934
\(304\) 6.26875 0.359537
\(305\) 54.9831 3.14833
\(306\) 1.40581 0.0803650
\(307\) 8.13036 0.464024 0.232012 0.972713i \(-0.425469\pi\)
0.232012 + 0.972713i \(0.425469\pi\)
\(308\) 5.74094 0.327120
\(309\) −17.5405 −0.997843
\(310\) 34.9463 1.98482
\(311\) −5.87800 −0.333311 −0.166655 0.986015i \(-0.553297\pi\)
−0.166655 + 0.986015i \(0.553297\pi\)
\(312\) 0 0
\(313\) 6.13275 0.346644 0.173322 0.984865i \(-0.444550\pi\)
0.173322 + 0.984865i \(0.444550\pi\)
\(314\) 15.4276 0.870629
\(315\) −4.04892 −0.228131
\(316\) 1.10992 0.0624377
\(317\) 1.60388 0.0900826 0.0450413 0.998985i \(-0.485658\pi\)
0.0450413 + 0.998985i \(0.485658\pi\)
\(318\) 3.16421 0.177440
\(319\) 18.4155 1.03107
\(320\) 4.04892 0.226341
\(321\) −4.04892 −0.225989
\(322\) −1.86294 −0.103817
\(323\) 8.81269 0.490351
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −14.5133 −0.803819
\(327\) −10.4179 −0.576111
\(328\) −2.59419 −0.143240
\(329\) 8.27413 0.456167
\(330\) 23.2446 1.27957
\(331\) −20.2500 −1.11304 −0.556519 0.830835i \(-0.687863\pi\)
−0.556519 + 0.830835i \(0.687863\pi\)
\(332\) 7.10992 0.390207
\(333\) 3.36227 0.184251
\(334\) 0.846543 0.0463208
\(335\) 37.2814 2.03690
\(336\) 1.00000 0.0545545
\(337\) 3.11529 0.169701 0.0848504 0.996394i \(-0.472959\pi\)
0.0848504 + 0.996394i \(0.472959\pi\)
\(338\) 0 0
\(339\) 2.31767 0.125878
\(340\) 5.69202 0.308693
\(341\) −49.5502 −2.68329
\(342\) 6.26875 0.338975
\(343\) −1.00000 −0.0539949
\(344\) −5.70171 −0.307416
\(345\) −7.54288 −0.406095
\(346\) 7.06100 0.379602
\(347\) −3.64742 −0.195804 −0.0979018 0.995196i \(-0.531213\pi\)
−0.0979018 + 0.995196i \(0.531213\pi\)
\(348\) 3.20775 0.171953
\(349\) −10.4397 −0.558822 −0.279411 0.960172i \(-0.590139\pi\)
−0.279411 + 0.960172i \(0.590139\pi\)
\(350\) −11.3937 −0.609021
\(351\) 0 0
\(352\) −5.74094 −0.305993
\(353\) 14.7530 0.785224 0.392612 0.919704i \(-0.371572\pi\)
0.392612 + 0.919704i \(0.371572\pi\)
\(354\) −7.50604 −0.398942
\(355\) −68.1570 −3.61740
\(356\) −8.02177 −0.425153
\(357\) 1.40581 0.0744035
\(358\) −3.20775 −0.169535
\(359\) 31.3207 1.65304 0.826520 0.562907i \(-0.190317\pi\)
0.826520 + 0.562907i \(0.190317\pi\)
\(360\) 4.04892 0.213397
\(361\) 20.2972 1.06827
\(362\) −25.4577 −1.33803
\(363\) −21.9584 −1.15252
\(364\) 0 0
\(365\) −1.28621 −0.0673232
\(366\) −13.5797 −0.709823
\(367\) −30.5080 −1.59250 −0.796251 0.604966i \(-0.793187\pi\)
−0.796251 + 0.604966i \(0.793187\pi\)
\(368\) 1.86294 0.0971123
\(369\) −2.59419 −0.135048
\(370\) 13.6136 0.707735
\(371\) 3.16421 0.164278
\(372\) −8.63102 −0.447498
\(373\) 2.57002 0.133071 0.0665354 0.997784i \(-0.478805\pi\)
0.0665354 + 0.997784i \(0.478805\pi\)
\(374\) −8.07069 −0.417325
\(375\) −25.8877 −1.33683
\(376\) −8.27413 −0.426706
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) 30.3913 1.56110 0.780549 0.625094i \(-0.214940\pi\)
0.780549 + 0.625094i \(0.214940\pi\)
\(380\) 25.3817 1.30205
\(381\) 12.1957 0.624803
\(382\) 21.1782 1.08357
\(383\) −2.98792 −0.152675 −0.0763377 0.997082i \(-0.524323\pi\)
−0.0763377 + 0.997082i \(0.524323\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 23.2446 1.18465
\(386\) −10.4306 −0.530902
\(387\) −5.70171 −0.289834
\(388\) 18.0978 0.918778
\(389\) 17.2078 0.872468 0.436234 0.899833i \(-0.356312\pi\)
0.436234 + 0.899833i \(0.356312\pi\)
\(390\) 0 0
\(391\) 2.61894 0.132446
\(392\) 1.00000 0.0505076
\(393\) 11.2078 0.565356
\(394\) 12.0435 0.606745
\(395\) 4.49396 0.226116
\(396\) −5.74094 −0.288493
\(397\) −28.0737 −1.40898 −0.704489 0.709715i \(-0.748824\pi\)
−0.704489 + 0.709715i \(0.748824\pi\)
\(398\) −24.7899 −1.24260
\(399\) 6.26875 0.313830
\(400\) 11.3937 0.569687
\(401\) 27.9866 1.39758 0.698792 0.715325i \(-0.253721\pi\)
0.698792 + 0.715325i \(0.253721\pi\)
\(402\) −9.20775 −0.459241
\(403\) 0 0
\(404\) 11.6093 0.577582
\(405\) 4.04892 0.201192
\(406\) 3.20775 0.159198
\(407\) −19.3026 −0.956794
\(408\) −1.40581 −0.0695981
\(409\) −1.89738 −0.0938194 −0.0469097 0.998899i \(-0.514937\pi\)
−0.0469097 + 0.998899i \(0.514937\pi\)
\(410\) −10.5036 −0.518738
\(411\) −0.768086 −0.0378869
\(412\) 17.5405 0.864158
\(413\) −7.50604 −0.369348
\(414\) 1.86294 0.0915583
\(415\) 28.7875 1.41312
\(416\) 0 0
\(417\) −13.0271 −0.637942
\(418\) −35.9885 −1.76026
\(419\) −4.48321 −0.219019 −0.109509 0.993986i \(-0.534928\pi\)
−0.109509 + 0.993986i \(0.534928\pi\)
\(420\) 4.04892 0.197567
\(421\) −24.4403 −1.19115 −0.595573 0.803301i \(-0.703075\pi\)
−0.595573 + 0.803301i \(0.703075\pi\)
\(422\) 5.20775 0.253509
\(423\) −8.27413 −0.402302
\(424\) −3.16421 −0.153668
\(425\) 16.0175 0.776961
\(426\) 16.8334 0.815581
\(427\) −13.5797 −0.657168
\(428\) 4.04892 0.195712
\(429\) 0 0
\(430\) −23.0858 −1.11329
\(431\) −9.43296 −0.454370 −0.227185 0.973852i \(-0.572952\pi\)
−0.227185 + 0.973852i \(0.572952\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 0.781495 0.0375563 0.0187781 0.999824i \(-0.494022\pi\)
0.0187781 + 0.999824i \(0.494022\pi\)
\(434\) −8.63102 −0.414302
\(435\) 12.9879 0.622723
\(436\) 10.4179 0.498927
\(437\) 11.6783 0.558648
\(438\) 0.317667 0.0151787
\(439\) −26.7735 −1.27783 −0.638914 0.769278i \(-0.720616\pi\)
−0.638914 + 0.769278i \(0.720616\pi\)
\(440\) −23.2446 −1.10814
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 4.50498 0.214038 0.107019 0.994257i \(-0.465869\pi\)
0.107019 + 0.994257i \(0.465869\pi\)
\(444\) −3.36227 −0.159566
\(445\) −32.4795 −1.53968
\(446\) 24.2784 1.14962
\(447\) −1.54958 −0.0732927
\(448\) −1.00000 −0.0472456
\(449\) 5.87800 0.277400 0.138700 0.990334i \(-0.455708\pi\)
0.138700 + 0.990334i \(0.455708\pi\)
\(450\) 11.3937 0.537106
\(451\) 14.8931 0.701287
\(452\) −2.31767 −0.109014
\(453\) 15.9323 0.748565
\(454\) −19.7995 −0.929239
\(455\) 0 0
\(456\) −6.26875 −0.293561
\(457\) 39.5066 1.84804 0.924021 0.382341i \(-0.124882\pi\)
0.924021 + 0.382341i \(0.124882\pi\)
\(458\) 18.3177 0.855928
\(459\) −1.40581 −0.0656177
\(460\) 7.54288 0.351688
\(461\) 7.34481 0.342082 0.171041 0.985264i \(-0.445287\pi\)
0.171041 + 0.985264i \(0.445287\pi\)
\(462\) −5.74094 −0.267093
\(463\) 2.87071 0.133413 0.0667065 0.997773i \(-0.478751\pi\)
0.0667065 + 0.997773i \(0.478751\pi\)
\(464\) −3.20775 −0.148916
\(465\) −34.9463 −1.62060
\(466\) 12.2392 0.566970
\(467\) −33.5905 −1.55438 −0.777191 0.629265i \(-0.783356\pi\)
−0.777191 + 0.629265i \(0.783356\pi\)
\(468\) 0 0
\(469\) −9.20775 −0.425174
\(470\) −33.5013 −1.54530
\(471\) −15.4276 −0.710866
\(472\) 7.50604 0.345494
\(473\) 32.7332 1.50507
\(474\) −1.10992 −0.0509801
\(475\) 71.4245 3.27718
\(476\) −1.40581 −0.0644353
\(477\) −3.16421 −0.144879
\(478\) −18.8713 −0.863153
\(479\) −22.5133 −1.02866 −0.514330 0.857592i \(-0.671959\pi\)
−0.514330 + 0.857592i \(0.671959\pi\)
\(480\) −4.04892 −0.184807
\(481\) 0 0
\(482\) 10.8659 0.494929
\(483\) 1.86294 0.0847666
\(484\) 21.9584 0.998108
\(485\) 73.2766 3.32732
\(486\) −1.00000 −0.0453609
\(487\) −5.22713 −0.236864 −0.118432 0.992962i \(-0.537787\pi\)
−0.118432 + 0.992962i \(0.537787\pi\)
\(488\) 13.5797 0.614725
\(489\) 14.5133 0.656316
\(490\) 4.04892 0.182911
\(491\) −42.7375 −1.92872 −0.964358 0.264602i \(-0.914759\pi\)
−0.964358 + 0.264602i \(0.914759\pi\)
\(492\) 2.59419 0.116955
\(493\) −4.50950 −0.203098
\(494\) 0 0
\(495\) −23.2446 −1.04477
\(496\) 8.63102 0.387544
\(497\) 16.8334 0.755081
\(498\) −7.10992 −0.318603
\(499\) −20.5870 −0.921601 −0.460801 0.887504i \(-0.652438\pi\)
−0.460801 + 0.887504i \(0.652438\pi\)
\(500\) 25.8877 1.15773
\(501\) −0.846543 −0.0378208
\(502\) −13.0664 −0.583181
\(503\) 31.1051 1.38691 0.693455 0.720500i \(-0.256088\pi\)
0.693455 + 0.720500i \(0.256088\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 47.0049 2.09169
\(506\) −10.6950 −0.475451
\(507\) 0 0
\(508\) −12.1957 −0.541095
\(509\) −16.7724 −0.743423 −0.371712 0.928348i \(-0.621229\pi\)
−0.371712 + 0.928348i \(0.621229\pi\)
\(510\) −5.69202 −0.252047
\(511\) 0.317667 0.0140528
\(512\) 1.00000 0.0441942
\(513\) −6.26875 −0.276772
\(514\) −16.4155 −0.724057
\(515\) 71.0200 3.12951
\(516\) 5.70171 0.251004
\(517\) 47.5013 2.08910
\(518\) −3.36227 −0.147730
\(519\) −7.06100 −0.309943
\(520\) 0 0
\(521\) −0.777184 −0.0340490 −0.0170245 0.999855i \(-0.505419\pi\)
−0.0170245 + 0.999855i \(0.505419\pi\)
\(522\) −3.20775 −0.140399
\(523\) 21.1347 0.924155 0.462077 0.886840i \(-0.347104\pi\)
0.462077 + 0.886840i \(0.347104\pi\)
\(524\) −11.2078 −0.489613
\(525\) 11.3937 0.497263
\(526\) 15.1250 0.659481
\(527\) 12.1336 0.528548
\(528\) 5.74094 0.249842
\(529\) −19.5295 −0.849107
\(530\) −12.8116 −0.556501
\(531\) 7.50604 0.325734
\(532\) −6.26875 −0.271785
\(533\) 0 0
\(534\) 8.02177 0.347136
\(535\) 16.3937 0.708763
\(536\) 9.20775 0.397714
\(537\) 3.20775 0.138425
\(538\) 5.58211 0.240662
\(539\) −5.74094 −0.247280
\(540\) −4.04892 −0.174238
\(541\) 1.57434 0.0676860 0.0338430 0.999427i \(-0.489225\pi\)
0.0338430 + 0.999427i \(0.489225\pi\)
\(542\) 25.5308 1.09664
\(543\) 25.4577 1.09250
\(544\) 1.40581 0.0602737
\(545\) 42.1812 1.80684
\(546\) 0 0
\(547\) 11.8043 0.504717 0.252358 0.967634i \(-0.418794\pi\)
0.252358 + 0.967634i \(0.418794\pi\)
\(548\) 0.768086 0.0328110
\(549\) 13.5797 0.579568
\(550\) −65.4107 −2.78912
\(551\) −20.1086 −0.856655
\(552\) −1.86294 −0.0792918
\(553\) −1.10992 −0.0471984
\(554\) −23.0019 −0.977258
\(555\) −13.6136 −0.577864
\(556\) 13.0271 0.552474
\(557\) −17.2078 −0.729116 −0.364558 0.931181i \(-0.618780\pi\)
−0.364558 + 0.931181i \(0.618780\pi\)
\(558\) 8.63102 0.365380
\(559\) 0 0
\(560\) −4.04892 −0.171098
\(561\) 8.07069 0.340745
\(562\) 2.89008 0.121911
\(563\) −24.0629 −1.01413 −0.507066 0.861908i \(-0.669270\pi\)
−0.507066 + 0.861908i \(0.669270\pi\)
\(564\) 8.27413 0.348404
\(565\) −9.38404 −0.394790
\(566\) −6.25906 −0.263088
\(567\) −1.00000 −0.0419961
\(568\) −16.8334 −0.706314
\(569\) −16.8009 −0.704329 −0.352165 0.935938i \(-0.614554\pi\)
−0.352165 + 0.935938i \(0.614554\pi\)
\(570\) −25.3817 −1.06312
\(571\) 25.3793 1.06209 0.531044 0.847344i \(-0.321800\pi\)
0.531044 + 0.847344i \(0.321800\pi\)
\(572\) 0 0
\(573\) −21.1782 −0.884732
\(574\) 2.59419 0.108279
\(575\) 21.2258 0.885177
\(576\) 1.00000 0.0416667
\(577\) 11.0121 0.458439 0.229219 0.973375i \(-0.426383\pi\)
0.229219 + 0.973375i \(0.426383\pi\)
\(578\) −15.0237 −0.624903
\(579\) 10.4306 0.433480
\(580\) −12.9879 −0.539294
\(581\) −7.10992 −0.294969
\(582\) −18.0978 −0.750179
\(583\) 18.1655 0.752340
\(584\) −0.317667 −0.0131452
\(585\) 0 0
\(586\) 23.8019 0.983249
\(587\) 3.17762 0.131154 0.0655772 0.997847i \(-0.479111\pi\)
0.0655772 + 0.997847i \(0.479111\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 54.1057 2.22939
\(590\) 30.3913 1.25119
\(591\) −12.0435 −0.495405
\(592\) 3.36227 0.138189
\(593\) −3.58450 −0.147198 −0.0735988 0.997288i \(-0.523448\pi\)
−0.0735988 + 0.997288i \(0.523448\pi\)
\(594\) 5.74094 0.235554
\(595\) −5.69202 −0.233350
\(596\) 1.54958 0.0634733
\(597\) 24.7899 1.01458
\(598\) 0 0
\(599\) 28.3913 1.16004 0.580019 0.814603i \(-0.303045\pi\)
0.580019 + 0.814603i \(0.303045\pi\)
\(600\) −11.3937 −0.465147
\(601\) 17.1642 0.700143 0.350071 0.936723i \(-0.386157\pi\)
0.350071 + 0.936723i \(0.386157\pi\)
\(602\) 5.70171 0.232384
\(603\) 9.20775 0.374969
\(604\) −15.9323 −0.648276
\(605\) 88.9077 3.61461
\(606\) −11.6093 −0.471594
\(607\) −31.7584 −1.28903 −0.644517 0.764590i \(-0.722941\pi\)
−0.644517 + 0.764590i \(0.722941\pi\)
\(608\) 6.26875 0.254231
\(609\) −3.20775 −0.129985
\(610\) 54.9831 2.22620
\(611\) 0 0
\(612\) 1.40581 0.0568266
\(613\) 2.95348 0.119290 0.0596449 0.998220i \(-0.481003\pi\)
0.0596449 + 0.998220i \(0.481003\pi\)
\(614\) 8.13036 0.328115
\(615\) 10.5036 0.423548
\(616\) 5.74094 0.231309
\(617\) −14.3526 −0.577813 −0.288907 0.957357i \(-0.593292\pi\)
−0.288907 + 0.957357i \(0.593292\pi\)
\(618\) −17.5405 −0.705582
\(619\) −12.0000 −0.482321 −0.241160 0.970485i \(-0.577528\pi\)
−0.241160 + 0.970485i \(0.577528\pi\)
\(620\) 34.9463 1.40348
\(621\) −1.86294 −0.0747571
\(622\) −5.87800 −0.235686
\(623\) 8.02177 0.321385
\(624\) 0 0
\(625\) 47.8485 1.91394
\(626\) 6.13275 0.245114
\(627\) 35.9885 1.43724
\(628\) 15.4276 0.615628
\(629\) 4.72673 0.188467
\(630\) −4.04892 −0.161313
\(631\) −27.3793 −1.08995 −0.544976 0.838452i \(-0.683461\pi\)
−0.544976 + 0.838452i \(0.683461\pi\)
\(632\) 1.10992 0.0441501
\(633\) −5.20775 −0.206990
\(634\) 1.60388 0.0636980
\(635\) −49.3793 −1.95956
\(636\) 3.16421 0.125469
\(637\) 0 0
\(638\) 18.4155 0.729077
\(639\) −16.8334 −0.665919
\(640\) 4.04892 0.160048
\(641\) 6.51334 0.257261 0.128631 0.991693i \(-0.458942\pi\)
0.128631 + 0.991693i \(0.458942\pi\)
\(642\) −4.04892 −0.159798
\(643\) −35.0331 −1.38157 −0.690786 0.723060i \(-0.742735\pi\)
−0.690786 + 0.723060i \(0.742735\pi\)
\(644\) −1.86294 −0.0734100
\(645\) 23.0858 0.909001
\(646\) 8.81269 0.346731
\(647\) −39.6340 −1.55817 −0.779087 0.626916i \(-0.784317\pi\)
−0.779087 + 0.626916i \(0.784317\pi\)
\(648\) 1.00000 0.0392837
\(649\) −43.0917 −1.69150
\(650\) 0 0
\(651\) 8.63102 0.338276
\(652\) −14.5133 −0.568386
\(653\) −50.2103 −1.96488 −0.982440 0.186581i \(-0.940259\pi\)
−0.982440 + 0.186581i \(0.940259\pi\)
\(654\) −10.4179 −0.407372
\(655\) −45.3793 −1.77311
\(656\) −2.59419 −0.101286
\(657\) −0.317667 −0.0123934
\(658\) 8.27413 0.322559
\(659\) −49.9667 −1.94643 −0.973214 0.229901i \(-0.926160\pi\)
−0.973214 + 0.229901i \(0.926160\pi\)
\(660\) 23.2446 0.904794
\(661\) −8.73795 −0.339867 −0.169934 0.985456i \(-0.554355\pi\)
−0.169934 + 0.985456i \(0.554355\pi\)
\(662\) −20.2500 −0.787037
\(663\) 0 0
\(664\) 7.10992 0.275918
\(665\) −25.3817 −0.984258
\(666\) 3.36227 0.130285
\(667\) −5.97584 −0.231385
\(668\) 0.846543 0.0327537
\(669\) −24.2784 −0.938659
\(670\) 37.2814 1.44031
\(671\) −77.9603 −3.00962
\(672\) 1.00000 0.0385758
\(673\) −30.0790 −1.15946 −0.579731 0.814808i \(-0.696842\pi\)
−0.579731 + 0.814808i \(0.696842\pi\)
\(674\) 3.11529 0.119997
\(675\) −11.3937 −0.438545
\(676\) 0 0
\(677\) 29.5991 1.13759 0.568793 0.822481i \(-0.307411\pi\)
0.568793 + 0.822481i \(0.307411\pi\)
\(678\) 2.31767 0.0890095
\(679\) −18.0978 −0.694531
\(680\) 5.69202 0.218279
\(681\) 19.7995 0.758720
\(682\) −49.5502 −1.89737
\(683\) 47.7294 1.82632 0.913158 0.407605i \(-0.133636\pi\)
0.913158 + 0.407605i \(0.133636\pi\)
\(684\) 6.26875 0.239692
\(685\) 3.10992 0.118824
\(686\) −1.00000 −0.0381802
\(687\) −18.3177 −0.698863
\(688\) −5.70171 −0.217376
\(689\) 0 0
\(690\) −7.54288 −0.287152
\(691\) 3.15990 0.120208 0.0601041 0.998192i \(-0.480857\pi\)
0.0601041 + 0.998192i \(0.480857\pi\)
\(692\) 7.06100 0.268419
\(693\) 5.74094 0.218080
\(694\) −3.64742 −0.138454
\(695\) 52.7458 2.00076
\(696\) 3.20775 0.121589
\(697\) −3.64694 −0.138138
\(698\) −10.4397 −0.395147
\(699\) −12.2392 −0.462929
\(700\) −11.3937 −0.430643
\(701\) 46.2452 1.74666 0.873328 0.487132i \(-0.161957\pi\)
0.873328 + 0.487132i \(0.161957\pi\)
\(702\) 0 0
\(703\) 21.0772 0.794943
\(704\) −5.74094 −0.216370
\(705\) 33.5013 1.26173
\(706\) 14.7530 0.555237
\(707\) −11.6093 −0.436611
\(708\) −7.50604 −0.282094
\(709\) 18.7084 0.702609 0.351305 0.936261i \(-0.385738\pi\)
0.351305 + 0.936261i \(0.385738\pi\)
\(710\) −68.1570 −2.55789
\(711\) 1.10992 0.0416251
\(712\) −8.02177 −0.300629
\(713\) 16.0790 0.602165
\(714\) 1.40581 0.0526112
\(715\) 0 0
\(716\) −3.20775 −0.119879
\(717\) 18.8713 0.704762
\(718\) 31.3207 1.16888
\(719\) −11.2078 −0.417979 −0.208989 0.977918i \(-0.567017\pi\)
−0.208989 + 0.977918i \(0.567017\pi\)
\(720\) 4.04892 0.150894
\(721\) −17.5405 −0.653242
\(722\) 20.2972 0.755384
\(723\) −10.8659 −0.404108
\(724\) −25.4577 −0.946129
\(725\) −36.5483 −1.35737
\(726\) −21.9584 −0.814952
\(727\) −12.3472 −0.457933 −0.228966 0.973434i \(-0.573535\pi\)
−0.228966 + 0.973434i \(0.573535\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −1.28621 −0.0476047
\(731\) −8.01554 −0.296465
\(732\) −13.5797 −0.501921
\(733\) −37.1487 −1.37212 −0.686059 0.727546i \(-0.740661\pi\)
−0.686059 + 0.727546i \(0.740661\pi\)
\(734\) −30.5080 −1.12607
\(735\) −4.04892 −0.149347
\(736\) 1.86294 0.0686688
\(737\) −52.8611 −1.94716
\(738\) −2.59419 −0.0954933
\(739\) 15.9323 0.586079 0.293039 0.956100i \(-0.405333\pi\)
0.293039 + 0.956100i \(0.405333\pi\)
\(740\) 13.6136 0.500445
\(741\) 0 0
\(742\) 3.16421 0.116162
\(743\) 10.8595 0.398396 0.199198 0.979959i \(-0.436166\pi\)
0.199198 + 0.979959i \(0.436166\pi\)
\(744\) −8.63102 −0.316429
\(745\) 6.27413 0.229866
\(746\) 2.57002 0.0940953
\(747\) 7.10992 0.260138
\(748\) −8.07069 −0.295094
\(749\) −4.04892 −0.147944
\(750\) −25.8877 −0.945285
\(751\) 0.0435405 0.00158882 0.000794408 1.00000i \(-0.499747\pi\)
0.000794408 1.00000i \(0.499747\pi\)
\(752\) −8.27413 −0.301726
\(753\) 13.0664 0.476165
\(754\) 0 0
\(755\) −64.5086 −2.34771
\(756\) 1.00000 0.0363696
\(757\) −24.5023 −0.890552 −0.445276 0.895393i \(-0.646894\pi\)
−0.445276 + 0.895393i \(0.646894\pi\)
\(758\) 30.3913 1.10386
\(759\) 10.6950 0.388204
\(760\) 25.3817 0.920689
\(761\) −25.8726 −0.937882 −0.468941 0.883230i \(-0.655364\pi\)
−0.468941 + 0.883230i \(0.655364\pi\)
\(762\) 12.1957 0.441802
\(763\) −10.4179 −0.377153
\(764\) 21.1782 0.766201
\(765\) 5.69202 0.205795
\(766\) −2.98792 −0.107958
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −13.8237 −0.498496 −0.249248 0.968440i \(-0.580183\pi\)
−0.249248 + 0.968440i \(0.580183\pi\)
\(770\) 23.2446 0.837677
\(771\) 16.4155 0.591190
\(772\) −10.4306 −0.375404
\(773\) 14.3043 0.514488 0.257244 0.966346i \(-0.417186\pi\)
0.257244 + 0.966346i \(0.417186\pi\)
\(774\) −5.70171 −0.204944
\(775\) 98.3396 3.53246
\(776\) 18.0978 0.649674
\(777\) 3.36227 0.120621
\(778\) 17.2078 0.616928
\(779\) −16.2623 −0.582658
\(780\) 0 0
\(781\) 96.6395 3.45803
\(782\) 2.61894 0.0936531
\(783\) 3.20775 0.114636
\(784\) 1.00000 0.0357143
\(785\) 62.4650 2.22947
\(786\) 11.2078 0.399767
\(787\) 33.7496 1.20304 0.601521 0.798857i \(-0.294562\pi\)
0.601521 + 0.798857i \(0.294562\pi\)
\(788\) 12.0435 0.429033
\(789\) −15.1250 −0.538464
\(790\) 4.49396 0.159888
\(791\) 2.31767 0.0824068
\(792\) −5.74094 −0.203995
\(793\) 0 0
\(794\) −28.0737 −0.996297
\(795\) 12.8116 0.454382
\(796\) −24.7899 −0.878653
\(797\) −28.5459 −1.01115 −0.505573 0.862784i \(-0.668719\pi\)
−0.505573 + 0.862784i \(0.668719\pi\)
\(798\) 6.26875 0.221911
\(799\) −11.6319 −0.411506
\(800\) 11.3937 0.402829
\(801\) −8.02177 −0.283435
\(802\) 27.9866 0.988241
\(803\) 1.82371 0.0643573
\(804\) −9.20775 −0.324732
\(805\) −7.54288 −0.265851
\(806\) 0 0
\(807\) −5.58211 −0.196499
\(808\) 11.6093 0.408412
\(809\) −42.8659 −1.50709 −0.753543 0.657399i \(-0.771657\pi\)
−0.753543 + 0.657399i \(0.771657\pi\)
\(810\) 4.04892 0.142264
\(811\) 20.4131 0.716801 0.358401 0.933568i \(-0.383322\pi\)
0.358401 + 0.933568i \(0.383322\pi\)
\(812\) 3.20775 0.112570
\(813\) −25.5308 −0.895404
\(814\) −19.3026 −0.676556
\(815\) −58.7633 −2.05839
\(816\) −1.40581 −0.0492133
\(817\) −35.7426 −1.25048
\(818\) −1.89738 −0.0663403
\(819\) 0 0
\(820\) −10.5036 −0.366803
\(821\) −14.8465 −0.518148 −0.259074 0.965857i \(-0.583417\pi\)
−0.259074 + 0.965857i \(0.583417\pi\)
\(822\) −0.768086 −0.0267901
\(823\) 0.328421 0.0114480 0.00572402 0.999984i \(-0.498178\pi\)
0.00572402 + 0.999984i \(0.498178\pi\)
\(824\) 17.5405 0.611052
\(825\) 65.4107 2.27731
\(826\) −7.50604 −0.261169
\(827\) −18.5894 −0.646417 −0.323208 0.946328i \(-0.604761\pi\)
−0.323208 + 0.946328i \(0.604761\pi\)
\(828\) 1.86294 0.0647415
\(829\) −21.2814 −0.739134 −0.369567 0.929204i \(-0.620494\pi\)
−0.369567 + 0.929204i \(0.620494\pi\)
\(830\) 28.7875 0.999227
\(831\) 23.0019 0.797928
\(832\) 0 0
\(833\) 1.40581 0.0487085
\(834\) −13.0271 −0.451093
\(835\) 3.42758 0.118616
\(836\) −35.9885 −1.24469
\(837\) −8.63102 −0.298332
\(838\) −4.48321 −0.154870
\(839\) −0.396125 −0.0136757 −0.00683787 0.999977i \(-0.502177\pi\)
−0.00683787 + 0.999977i \(0.502177\pi\)
\(840\) 4.04892 0.139701
\(841\) −18.7103 −0.645184
\(842\) −24.4403 −0.842267
\(843\) −2.89008 −0.0995398
\(844\) 5.20775 0.179258
\(845\) 0 0
\(846\) −8.27413 −0.284470
\(847\) −21.9584 −0.754499
\(848\) −3.16421 −0.108659
\(849\) 6.25906 0.214810
\(850\) 16.0175 0.549394
\(851\) 6.26370 0.214717
\(852\) 16.8334 0.576703
\(853\) 48.4940 1.66040 0.830201 0.557465i \(-0.188226\pi\)
0.830201 + 0.557465i \(0.188226\pi\)
\(854\) −13.5797 −0.464688
\(855\) 25.3817 0.868034
\(856\) 4.04892 0.138389
\(857\) −43.9463 −1.50118 −0.750588 0.660770i \(-0.770230\pi\)
−0.750588 + 0.660770i \(0.770230\pi\)
\(858\) 0 0
\(859\) −9.87741 −0.337013 −0.168506 0.985701i \(-0.553894\pi\)
−0.168506 + 0.985701i \(0.553894\pi\)
\(860\) −23.0858 −0.787218
\(861\) −2.59419 −0.0884096
\(862\) −9.43296 −0.321288
\(863\) 13.7342 0.467519 0.233759 0.972294i \(-0.424897\pi\)
0.233759 + 0.972294i \(0.424897\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 28.5894 0.972069
\(866\) 0.781495 0.0265563
\(867\) 15.0237 0.510231
\(868\) −8.63102 −0.292956
\(869\) −6.37196 −0.216154
\(870\) 12.9879 0.440332
\(871\) 0 0
\(872\) 10.4179 0.352794
\(873\) 18.0978 0.612519
\(874\) 11.6783 0.395024
\(875\) −25.8877 −0.875164
\(876\) 0.317667 0.0107330
\(877\) 2.70543 0.0913559 0.0456780 0.998956i \(-0.485455\pi\)
0.0456780 + 0.998956i \(0.485455\pi\)
\(878\) −26.7735 −0.903561
\(879\) −23.8019 −0.802819
\(880\) −23.2446 −0.783575
\(881\) 15.3110 0.515839 0.257920 0.966166i \(-0.416963\pi\)
0.257920 + 0.966166i \(0.416963\pi\)
\(882\) 1.00000 0.0336718
\(883\) −43.0858 −1.44995 −0.724976 0.688775i \(-0.758149\pi\)
−0.724976 + 0.688775i \(0.758149\pi\)
\(884\) 0 0
\(885\) −30.3913 −1.02159
\(886\) 4.50498 0.151348
\(887\) −17.5845 −0.590430 −0.295215 0.955431i \(-0.595391\pi\)
−0.295215 + 0.955431i \(0.595391\pi\)
\(888\) −3.36227 −0.112830
\(889\) 12.1957 0.409030
\(890\) −32.4795 −1.08871
\(891\) −5.74094 −0.192329
\(892\) 24.2784 0.812902
\(893\) −51.8684 −1.73571
\(894\) −1.54958 −0.0518258
\(895\) −12.9879 −0.434138
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 5.87800 0.196151
\(899\) −27.6862 −0.923385
\(900\) 11.3937 0.379791
\(901\) −4.44829 −0.148194
\(902\) 14.8931 0.495885
\(903\) −5.70171 −0.189741
\(904\) −2.31767 −0.0770845
\(905\) −103.076 −3.42637
\(906\) 15.9323 0.529315
\(907\) 22.5870 0.749989 0.374995 0.927027i \(-0.377645\pi\)
0.374995 + 0.927027i \(0.377645\pi\)
\(908\) −19.7995 −0.657071
\(909\) 11.6093 0.385055
\(910\) 0 0
\(911\) −24.5429 −0.813142 −0.406571 0.913619i \(-0.633276\pi\)
−0.406571 + 0.913619i \(0.633276\pi\)
\(912\) −6.26875 −0.207579
\(913\) −40.8176 −1.35087
\(914\) 39.5066 1.30676
\(915\) −54.9831 −1.81769
\(916\) 18.3177 0.605233
\(917\) 11.2078 0.370113
\(918\) −1.40581 −0.0463987
\(919\) 60.0340 1.98034 0.990169 0.139877i \(-0.0446706\pi\)
0.990169 + 0.139877i \(0.0446706\pi\)
\(920\) 7.54288 0.248681
\(921\) −8.13036 −0.267904
\(922\) 7.34481 0.241889
\(923\) 0 0
\(924\) −5.74094 −0.188863
\(925\) 38.3088 1.25959
\(926\) 2.87071 0.0943373
\(927\) 17.5405 0.576105
\(928\) −3.20775 −0.105300
\(929\) −15.2319 −0.499743 −0.249871 0.968279i \(-0.580388\pi\)
−0.249871 + 0.968279i \(0.580388\pi\)
\(930\) −34.9463 −1.14593
\(931\) 6.26875 0.205450
\(932\) 12.2392 0.400909
\(933\) 5.87800 0.192437
\(934\) −33.5905 −1.09911
\(935\) −32.6775 −1.06867
\(936\) 0 0
\(937\) −13.1487 −0.429548 −0.214774 0.976664i \(-0.568902\pi\)
−0.214774 + 0.976664i \(0.568902\pi\)
\(938\) −9.20775 −0.300644
\(939\) −6.13275 −0.200135
\(940\) −33.5013 −1.09269
\(941\) 14.1817 0.462309 0.231155 0.972917i \(-0.425750\pi\)
0.231155 + 0.972917i \(0.425750\pi\)
\(942\) −15.4276 −0.502658
\(943\) −4.83281 −0.157378
\(944\) 7.50604 0.244301
\(945\) 4.04892 0.131711
\(946\) 32.7332 1.06425
\(947\) 28.4510 0.924534 0.462267 0.886741i \(-0.347036\pi\)
0.462267 + 0.886741i \(0.347036\pi\)
\(948\) −1.10992 −0.0360484
\(949\) 0 0
\(950\) 71.4245 2.31732
\(951\) −1.60388 −0.0520092
\(952\) −1.40581 −0.0455627
\(953\) −35.0702 −1.13604 −0.568018 0.823016i \(-0.692290\pi\)
−0.568018 + 0.823016i \(0.692290\pi\)
\(954\) −3.16421 −0.102445
\(955\) 85.7488 2.77477
\(956\) −18.8713 −0.610341
\(957\) −18.4155 −0.595289
\(958\) −22.5133 −0.727373
\(959\) −0.768086 −0.0248028
\(960\) −4.04892 −0.130678
\(961\) 43.4946 1.40305
\(962\) 0 0
\(963\) 4.04892 0.130475
\(964\) 10.8659 0.349968
\(965\) −42.2325 −1.35951
\(966\) 1.86294 0.0599390
\(967\) 60.1581 1.93455 0.967277 0.253723i \(-0.0816551\pi\)
0.967277 + 0.253723i \(0.0816551\pi\)
\(968\) 21.9584 0.705769
\(969\) −8.81269 −0.283104
\(970\) 73.2766 2.35277
\(971\) −6.42626 −0.206228 −0.103114 0.994670i \(-0.532881\pi\)
−0.103114 + 0.994670i \(0.532881\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −13.0271 −0.417631
\(974\) −5.22713 −0.167488
\(975\) 0 0
\(976\) 13.5797 0.434676
\(977\) 55.1896 1.76567 0.882835 0.469683i \(-0.155632\pi\)
0.882835 + 0.469683i \(0.155632\pi\)
\(978\) 14.5133 0.464085
\(979\) 46.0525 1.47184
\(980\) 4.04892 0.129338
\(981\) 10.4179 0.332618
\(982\) −42.7375 −1.36381
\(983\) −38.2452 −1.21983 −0.609916 0.792466i \(-0.708797\pi\)
−0.609916 + 0.792466i \(0.708797\pi\)
\(984\) 2.59419 0.0826997
\(985\) 48.7633 1.55373
\(986\) −4.50950 −0.143612
\(987\) −8.27413 −0.263368
\(988\) 0 0
\(989\) −10.6219 −0.337758
\(990\) −23.2446 −0.738761
\(991\) 54.9396 1.74521 0.872607 0.488423i \(-0.162428\pi\)
0.872607 + 0.488423i \(0.162428\pi\)
\(992\) 8.63102 0.274035
\(993\) 20.2500 0.642613
\(994\) 16.8334 0.533923
\(995\) −100.372 −3.18201
\(996\) −7.10992 −0.225286
\(997\) −60.8745 −1.92792 −0.963958 0.266054i \(-0.914280\pi\)
−0.963958 + 0.266054i \(0.914280\pi\)
\(998\) −20.5870 −0.651670
\(999\) −3.36227 −0.106378
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.ck.1.3 yes 3
13.12 even 2 7098.2.a.cb.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7098.2.a.cb.1.1 3 13.12 even 2
7098.2.a.ck.1.3 yes 3 1.1 even 1 trivial