Properties

Label 7098.2.a.bw.1.1
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 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.56155 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.56155 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.56155 q^{10} +2.56155 q^{11} +1.00000 q^{12} -1.00000 q^{14} -3.56155 q^{15} +1.00000 q^{16} -5.00000 q^{17} +1.00000 q^{18} +7.68466 q^{19} -3.56155 q^{20} -1.00000 q^{21} +2.56155 q^{22} -9.12311 q^{23} +1.00000 q^{24} +7.68466 q^{25} +1.00000 q^{27} -1.00000 q^{28} -1.00000 q^{29} -3.56155 q^{30} +5.12311 q^{31} +1.00000 q^{32} +2.56155 q^{33} -5.00000 q^{34} +3.56155 q^{35} +1.00000 q^{36} -9.80776 q^{37} +7.68466 q^{38} -3.56155 q^{40} +4.12311 q^{41} -1.00000 q^{42} -2.87689 q^{43} +2.56155 q^{44} -3.56155 q^{45} -9.12311 q^{46} -7.68466 q^{47} +1.00000 q^{48} +1.00000 q^{49} +7.68466 q^{50} -5.00000 q^{51} -3.87689 q^{53} +1.00000 q^{54} -9.12311 q^{55} -1.00000 q^{56} +7.68466 q^{57} -1.00000 q^{58} +13.1231 q^{59} -3.56155 q^{60} -1.00000 q^{61} +5.12311 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.56155 q^{66} +1.12311 q^{67} -5.00000 q^{68} -9.12311 q^{69} +3.56155 q^{70} -4.00000 q^{71} +1.00000 q^{72} -2.43845 q^{73} -9.80776 q^{74} +7.68466 q^{75} +7.68466 q^{76} -2.56155 q^{77} +1.43845 q^{79} -3.56155 q^{80} +1.00000 q^{81} +4.12311 q^{82} -10.2462 q^{83} -1.00000 q^{84} +17.8078 q^{85} -2.87689 q^{86} -1.00000 q^{87} +2.56155 q^{88} -9.68466 q^{89} -3.56155 q^{90} -9.12311 q^{92} +5.12311 q^{93} -7.68466 q^{94} -27.3693 q^{95} +1.00000 q^{96} +12.2462 q^{97} +1.00000 q^{98} +2.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} - 3q^{10} + q^{11} + 2q^{12} - 2q^{14} - 3q^{15} + 2q^{16} - 10q^{17} + 2q^{18} + 3q^{19} - 3q^{20} - 2q^{21} + q^{22} - 10q^{23} + 2q^{24} + 3q^{25} + 2q^{27} - 2q^{28} - 2q^{29} - 3q^{30} + 2q^{31} + 2q^{32} + q^{33} - 10q^{34} + 3q^{35} + 2q^{36} + q^{37} + 3q^{38} - 3q^{40} - 2q^{42} - 14q^{43} + q^{44} - 3q^{45} - 10q^{46} - 3q^{47} + 2q^{48} + 2q^{49} + 3q^{50} - 10q^{51} - 16q^{53} + 2q^{54} - 10q^{55} - 2q^{56} + 3q^{57} - 2q^{58} + 18q^{59} - 3q^{60} - 2q^{61} + 2q^{62} - 2q^{63} + 2q^{64} + q^{66} - 6q^{67} - 10q^{68} - 10q^{69} + 3q^{70} - 8q^{71} + 2q^{72} - 9q^{73} + q^{74} + 3q^{75} + 3q^{76} - q^{77} + 7q^{79} - 3q^{80} + 2q^{81} - 4q^{83} - 2q^{84} + 15q^{85} - 14q^{86} - 2q^{87} + q^{88} - 7q^{89} - 3q^{90} - 10q^{92} + 2q^{93} - 3q^{94} - 30q^{95} + 2q^{96} + 8q^{97} + 2q^{98} + q^{99} + O(q^{100}) \)

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
\(5\) −3.56155 −1.59277 −0.796387 0.604787i \(-0.793258\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.56155 −1.12626
\(11\) 2.56155 0.772337 0.386169 0.922428i \(-0.373798\pi\)
0.386169 + 0.922428i \(0.373798\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) −3.56155 −0.919589
\(16\) 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
\(19\) 7.68466 1.76298 0.881491 0.472201i \(-0.156540\pi\)
0.881491 + 0.472201i \(0.156540\pi\)
\(20\) −3.56155 −0.796387
\(21\) −1.00000 −0.218218
\(22\) 2.56155 0.546125
\(23\) −9.12311 −1.90230 −0.951150 0.308731i \(-0.900096\pi\)
−0.951150 + 0.308731i \(0.900096\pi\)
\(24\) 1.00000 0.204124
\(25\) 7.68466 1.53693
\(26\) 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 \(0.529597\pi\)
\(30\) −3.56155 −0.650248
\(31\) 5.12311 0.920137 0.460068 0.887883i \(-0.347825\pi\)
0.460068 + 0.887883i \(0.347825\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.56155 0.445909
\(34\) −5.00000 −0.857493
\(35\) 3.56155 0.602012
\(36\) 1.00000 0.166667
\(37\) −9.80776 −1.61239 −0.806193 0.591652i \(-0.798476\pi\)
−0.806193 + 0.591652i \(0.798476\pi\)
\(38\) 7.68466 1.24662
\(39\) 0 0
\(40\) −3.56155 −0.563131
\(41\) 4.12311 0.643921 0.321960 0.946753i \(-0.395658\pi\)
0.321960 + 0.946753i \(0.395658\pi\)
\(42\) −1.00000 −0.154303
\(43\) −2.87689 −0.438722 −0.219361 0.975644i \(-0.570397\pi\)
−0.219361 + 0.975644i \(0.570397\pi\)
\(44\) 2.56155 0.386169
\(45\) −3.56155 −0.530925
\(46\) −9.12311 −1.34513
\(47\) −7.68466 −1.12092 −0.560461 0.828181i \(-0.689376\pi\)
−0.560461 + 0.828181i \(0.689376\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 7.68466 1.08677
\(51\) −5.00000 −0.700140
\(52\) 0 0
\(53\) −3.87689 −0.532532 −0.266266 0.963900i \(-0.585790\pi\)
−0.266266 + 0.963900i \(0.585790\pi\)
\(54\) 1.00000 0.136083
\(55\) −9.12311 −1.23016
\(56\) −1.00000 −0.133631
\(57\) 7.68466 1.01786
\(58\) −1.00000 −0.131306
\(59\) 13.1231 1.70848 0.854241 0.519877i \(-0.174022\pi\)
0.854241 + 0.519877i \(0.174022\pi\)
\(60\) −3.56155 −0.459794
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 5.12311 0.650635
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.56155 0.315305
\(67\) 1.12311 0.137209 0.0686046 0.997644i \(-0.478145\pi\)
0.0686046 + 0.997644i \(0.478145\pi\)
\(68\) −5.00000 −0.606339
\(69\) −9.12311 −1.09829
\(70\) 3.56155 0.425687
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.43845 −0.285399 −0.142699 0.989766i \(-0.545578\pi\)
−0.142699 + 0.989766i \(0.545578\pi\)
\(74\) −9.80776 −1.14013
\(75\) 7.68466 0.887348
\(76\) 7.68466 0.881491
\(77\) −2.56155 −0.291916
\(78\) 0 0
\(79\) 1.43845 0.161838 0.0809190 0.996721i \(-0.474214\pi\)
0.0809190 + 0.996721i \(0.474214\pi\)
\(80\) −3.56155 −0.398194
\(81\) 1.00000 0.111111
\(82\) 4.12311 0.455321
\(83\) −10.2462 −1.12467 −0.562334 0.826910i \(-0.690096\pi\)
−0.562334 + 0.826910i \(0.690096\pi\)
\(84\) −1.00000 −0.109109
\(85\) 17.8078 1.93152
\(86\) −2.87689 −0.310223
\(87\) −1.00000 −0.107211
\(88\) 2.56155 0.273062
\(89\) −9.68466 −1.02657 −0.513286 0.858218i \(-0.671572\pi\)
−0.513286 + 0.858218i \(0.671572\pi\)
\(90\) −3.56155 −0.375421
\(91\) 0 0
\(92\) −9.12311 −0.951150
\(93\) 5.12311 0.531241
\(94\) −7.68466 −0.792612
\(95\) −27.3693 −2.80803
\(96\) 1.00000 0.102062
\(97\) 12.2462 1.24341 0.621707 0.783250i \(-0.286439\pi\)
0.621707 + 0.783250i \(0.286439\pi\)
\(98\) 1.00000 0.101015
\(99\) 2.56155 0.257446
\(100\) 7.68466 0.768466
\(101\) −19.5616 −1.94645 −0.973224 0.229860i \(-0.926173\pi\)
−0.973224 + 0.229860i \(0.926173\pi\)
\(102\) −5.00000 −0.495074
\(103\) −11.3693 −1.12025 −0.560126 0.828407i \(-0.689247\pi\)
−0.560126 + 0.828407i \(0.689247\pi\)
\(104\) 0 0
\(105\) 3.56155 0.347572
\(106\) −3.87689 −0.376557
\(107\) −9.43845 −0.912449 −0.456225 0.889865i \(-0.650799\pi\)
−0.456225 + 0.889865i \(0.650799\pi\)
\(108\) 1.00000 0.0962250
\(109\) 0.876894 0.0839912 0.0419956 0.999118i \(-0.486628\pi\)
0.0419956 + 0.999118i \(0.486628\pi\)
\(110\) −9.12311 −0.869854
\(111\) −9.80776 −0.930912
\(112\) −1.00000 −0.0944911
\(113\) 16.9309 1.59272 0.796361 0.604821i \(-0.206756\pi\)
0.796361 + 0.604821i \(0.206756\pi\)
\(114\) 7.68466 0.719734
\(115\) 32.4924 3.02993
\(116\) −1.00000 −0.0928477
\(117\) 0 0
\(118\) 13.1231 1.20808
\(119\) 5.00000 0.458349
\(120\) −3.56155 −0.325124
\(121\) −4.43845 −0.403495
\(122\) −1.00000 −0.0905357
\(123\) 4.12311 0.371768
\(124\) 5.12311 0.460068
\(125\) −9.56155 −0.855211
\(126\) −1.00000 −0.0890871
\(127\) 2.24621 0.199319 0.0996595 0.995022i \(-0.468225\pi\)
0.0996595 + 0.995022i \(0.468225\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.87689 −0.253296
\(130\) 0 0
\(131\) −16.4924 −1.44095 −0.720475 0.693481i \(-0.756076\pi\)
−0.720475 + 0.693481i \(0.756076\pi\)
\(132\) 2.56155 0.222955
\(133\) −7.68466 −0.666344
\(134\) 1.12311 0.0970215
\(135\) −3.56155 −0.306530
\(136\) −5.00000 −0.428746
\(137\) −5.80776 −0.496191 −0.248095 0.968736i \(-0.579805\pi\)
−0.248095 + 0.968736i \(0.579805\pi\)
\(138\) −9.12311 −0.776610
\(139\) −7.68466 −0.651804 −0.325902 0.945404i \(-0.605668\pi\)
−0.325902 + 0.945404i \(0.605668\pi\)
\(140\) 3.56155 0.301006
\(141\) −7.68466 −0.647165
\(142\) −4.00000 −0.335673
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 3.56155 0.295771
\(146\) −2.43845 −0.201807
\(147\) 1.00000 0.0824786
\(148\) −9.80776 −0.806193
\(149\) −17.8078 −1.45887 −0.729434 0.684051i \(-0.760217\pi\)
−0.729434 + 0.684051i \(0.760217\pi\)
\(150\) 7.68466 0.627450
\(151\) −17.4384 −1.41912 −0.709560 0.704645i \(-0.751106\pi\)
−0.709560 + 0.704645i \(0.751106\pi\)
\(152\) 7.68466 0.623308
\(153\) −5.00000 −0.404226
\(154\) −2.56155 −0.206416
\(155\) −18.2462 −1.46557
\(156\) 0 0
\(157\) −5.80776 −0.463510 −0.231755 0.972774i \(-0.574447\pi\)
−0.231755 + 0.972774i \(0.574447\pi\)
\(158\) 1.43845 0.114437
\(159\) −3.87689 −0.307458
\(160\) −3.56155 −0.281565
\(161\) 9.12311 0.719001
\(162\) 1.00000 0.0785674
\(163\) −1.12311 −0.0879684 −0.0439842 0.999032i \(-0.514005\pi\)
−0.0439842 + 0.999032i \(0.514005\pi\)
\(164\) 4.12311 0.321960
\(165\) −9.12311 −0.710233
\(166\) −10.2462 −0.795260
\(167\) 20.4924 1.58575 0.792876 0.609383i \(-0.208583\pi\)
0.792876 + 0.609383i \(0.208583\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 0 0
\(170\) 17.8078 1.36579
\(171\) 7.68466 0.587661
\(172\) −2.87689 −0.219361
\(173\) −4.87689 −0.370783 −0.185392 0.982665i \(-0.559355\pi\)
−0.185392 + 0.982665i \(0.559355\pi\)
\(174\) −1.00000 −0.0758098
\(175\) −7.68466 −0.580906
\(176\) 2.56155 0.193084
\(177\) 13.1231 0.986393
\(178\) −9.68466 −0.725896
\(179\) −16.4924 −1.23270 −0.616351 0.787472i \(-0.711390\pi\)
−0.616351 + 0.787472i \(0.711390\pi\)
\(180\) −3.56155 −0.265462
\(181\) 13.2462 0.984583 0.492292 0.870430i \(-0.336159\pi\)
0.492292 + 0.870430i \(0.336159\pi\)
\(182\) 0 0
\(183\) −1.00000 −0.0739221
\(184\) −9.12311 −0.672564
\(185\) 34.9309 2.56817
\(186\) 5.12311 0.375644
\(187\) −12.8078 −0.936596
\(188\) −7.68466 −0.560461
\(189\) −1.00000 −0.0727393
\(190\) −27.3693 −1.98558
\(191\) 5.75379 0.416330 0.208165 0.978094i \(-0.433251\pi\)
0.208165 + 0.978094i \(0.433251\pi\)
\(192\) 1.00000 0.0721688
\(193\) −20.3693 −1.46622 −0.733108 0.680113i \(-0.761931\pi\)
−0.733108 + 0.680113i \(0.761931\pi\)
\(194\) 12.2462 0.879227
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −0.561553 −0.0400090 −0.0200045 0.999800i \(-0.506368\pi\)
−0.0200045 + 0.999800i \(0.506368\pi\)
\(198\) 2.56155 0.182042
\(199\) −17.6155 −1.24873 −0.624366 0.781132i \(-0.714643\pi\)
−0.624366 + 0.781132i \(0.714643\pi\)
\(200\) 7.68466 0.543387
\(201\) 1.12311 0.0792178
\(202\) −19.5616 −1.37635
\(203\) 1.00000 0.0701862
\(204\) −5.00000 −0.350070
\(205\) −14.6847 −1.02562
\(206\) −11.3693 −0.792138
\(207\) −9.12311 −0.634100
\(208\) 0 0
\(209\) 19.6847 1.36162
\(210\) 3.56155 0.245770
\(211\) −2.87689 −0.198054 −0.0990268 0.995085i \(-0.531573\pi\)
−0.0990268 + 0.995085i \(0.531573\pi\)
\(212\) −3.87689 −0.266266
\(213\) −4.00000 −0.274075
\(214\) −9.43845 −0.645199
\(215\) 10.2462 0.698786
\(216\) 1.00000 0.0680414
\(217\) −5.12311 −0.347779
\(218\) 0.876894 0.0593908
\(219\) −2.43845 −0.164775
\(220\) −9.12311 −0.615080
\(221\) 0 0
\(222\) −9.80776 −0.658254
\(223\) −16.4924 −1.10441 −0.552207 0.833707i \(-0.686214\pi\)
−0.552207 + 0.833707i \(0.686214\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 7.68466 0.512311
\(226\) 16.9309 1.12622
\(227\) 13.1231 0.871011 0.435506 0.900186i \(-0.356570\pi\)
0.435506 + 0.900186i \(0.356570\pi\)
\(228\) 7.68466 0.508929
\(229\) 22.8078 1.50718 0.753590 0.657345i \(-0.228321\pi\)
0.753590 + 0.657345i \(0.228321\pi\)
\(230\) 32.4924 2.14249
\(231\) −2.56155 −0.168538
\(232\) −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
\(235\) 27.3693 1.78538
\(236\) 13.1231 0.854241
\(237\) 1.43845 0.0934372
\(238\) 5.00000 0.324102
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) −3.56155 −0.229897
\(241\) 11.3153 0.728885 0.364443 0.931226i \(-0.381260\pi\)
0.364443 + 0.931226i \(0.381260\pi\)
\(242\) −4.43845 −0.285314
\(243\) 1.00000 0.0641500
\(244\) −1.00000 −0.0640184
\(245\) −3.56155 −0.227539
\(246\) 4.12311 0.262880
\(247\) 0 0
\(248\) 5.12311 0.325318
\(249\) −10.2462 −0.649327
\(250\) −9.56155 −0.604726
\(251\) 8.49242 0.536037 0.268018 0.963414i \(-0.413631\pi\)
0.268018 + 0.963414i \(0.413631\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −23.3693 −1.46922
\(254\) 2.24621 0.140940
\(255\) 17.8078 1.11517
\(256\) 1.00000 0.0625000
\(257\) −14.7538 −0.920316 −0.460158 0.887837i \(-0.652207\pi\)
−0.460158 + 0.887837i \(0.652207\pi\)
\(258\) −2.87689 −0.179108
\(259\) 9.80776 0.609425
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) −16.4924 −1.01891
\(263\) −20.4924 −1.26362 −0.631808 0.775125i \(-0.717687\pi\)
−0.631808 + 0.775125i \(0.717687\pi\)
\(264\) 2.56155 0.157653
\(265\) 13.8078 0.848204
\(266\) −7.68466 −0.471177
\(267\) −9.68466 −0.592691
\(268\) 1.12311 0.0686046
\(269\) −4.87689 −0.297349 −0.148675 0.988886i \(-0.547501\pi\)
−0.148675 + 0.988886i \(0.547501\pi\)
\(270\) −3.56155 −0.216749
\(271\) 13.1231 0.797172 0.398586 0.917131i \(-0.369501\pi\)
0.398586 + 0.917131i \(0.369501\pi\)
\(272\) −5.00000 −0.303170
\(273\) 0 0
\(274\) −5.80776 −0.350860
\(275\) 19.6847 1.18703
\(276\) −9.12311 −0.549146
\(277\) −3.56155 −0.213993 −0.106996 0.994259i \(-0.534123\pi\)
−0.106996 + 0.994259i \(0.534123\pi\)
\(278\) −7.68466 −0.460895
\(279\) 5.12311 0.306712
\(280\) 3.56155 0.212843
\(281\) 7.80776 0.465772 0.232886 0.972504i \(-0.425183\pi\)
0.232886 + 0.972504i \(0.425183\pi\)
\(282\) −7.68466 −0.457615
\(283\) 8.49242 0.504822 0.252411 0.967620i \(-0.418776\pi\)
0.252411 + 0.967620i \(0.418776\pi\)
\(284\) −4.00000 −0.237356
\(285\) −27.3693 −1.62122
\(286\) 0 0
\(287\) −4.12311 −0.243379
\(288\) 1.00000 0.0589256
\(289\) 8.00000 0.470588
\(290\) 3.56155 0.209142
\(291\) 12.2462 0.717886
\(292\) −2.43845 −0.142699
\(293\) −4.68466 −0.273681 −0.136840 0.990593i \(-0.543695\pi\)
−0.136840 + 0.990593i \(0.543695\pi\)
\(294\) 1.00000 0.0583212
\(295\) −46.7386 −2.72123
\(296\) −9.80776 −0.570065
\(297\) 2.56155 0.148636
\(298\) −17.8078 −1.03158
\(299\) 0 0
\(300\) 7.68466 0.443674
\(301\) 2.87689 0.165821
\(302\) −17.4384 −1.00347
\(303\) −19.5616 −1.12378
\(304\) 7.68466 0.440745
\(305\) 3.56155 0.203934
\(306\) −5.00000 −0.285831
\(307\) −33.9309 −1.93654 −0.968269 0.249912i \(-0.919598\pi\)
−0.968269 + 0.249912i \(0.919598\pi\)
\(308\) −2.56155 −0.145958
\(309\) −11.3693 −0.646778
\(310\) −18.2462 −1.03632
\(311\) 22.5616 1.27935 0.639674 0.768646i \(-0.279069\pi\)
0.639674 + 0.768646i \(0.279069\pi\)
\(312\) 0 0
\(313\) 27.6155 1.56092 0.780461 0.625205i \(-0.214984\pi\)
0.780461 + 0.625205i \(0.214984\pi\)
\(314\) −5.80776 −0.327751
\(315\) 3.56155 0.200671
\(316\) 1.43845 0.0809190
\(317\) −11.5616 −0.649362 −0.324681 0.945824i \(-0.605257\pi\)
−0.324681 + 0.945824i \(0.605257\pi\)
\(318\) −3.87689 −0.217405
\(319\) −2.56155 −0.143419
\(320\) −3.56155 −0.199097
\(321\) −9.43845 −0.526803
\(322\) 9.12311 0.508411
\(323\) −38.4233 −2.13793
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −1.12311 −0.0622031
\(327\) 0.876894 0.0484924
\(328\) 4.12311 0.227660
\(329\) 7.68466 0.423669
\(330\) −9.12311 −0.502210
\(331\) −35.3693 −1.94407 −0.972037 0.234829i \(-0.924547\pi\)
−0.972037 + 0.234829i \(0.924547\pi\)
\(332\) −10.2462 −0.562334
\(333\) −9.80776 −0.537462
\(334\) 20.4924 1.12130
\(335\) −4.00000 −0.218543
\(336\) −1.00000 −0.0545545
\(337\) −29.4924 −1.60655 −0.803277 0.595605i \(-0.796912\pi\)
−0.803277 + 0.595605i \(0.796912\pi\)
\(338\) 0 0
\(339\) 16.9309 0.919559
\(340\) 17.8078 0.965762
\(341\) 13.1231 0.710656
\(342\) 7.68466 0.415539
\(343\) −1.00000 −0.0539949
\(344\) −2.87689 −0.155112
\(345\) 32.4924 1.74933
\(346\) −4.87689 −0.262183
\(347\) −0.315342 −0.0169284 −0.00846421 0.999964i \(-0.502694\pi\)
−0.00846421 + 0.999964i \(0.502694\pi\)
\(348\) −1.00000 −0.0536056
\(349\) −6.49242 −0.347531 −0.173766 0.984787i \(-0.555594\pi\)
−0.173766 + 0.984787i \(0.555594\pi\)
\(350\) −7.68466 −0.410762
\(351\) 0 0
\(352\) 2.56155 0.136531
\(353\) −9.80776 −0.522015 −0.261007 0.965337i \(-0.584055\pi\)
−0.261007 + 0.965337i \(0.584055\pi\)
\(354\) 13.1231 0.697485
\(355\) 14.2462 0.756110
\(356\) −9.68466 −0.513286
\(357\) 5.00000 0.264628
\(358\) −16.4924 −0.871652
\(359\) −4.63068 −0.244398 −0.122199 0.992506i \(-0.538995\pi\)
−0.122199 + 0.992506i \(0.538995\pi\)
\(360\) −3.56155 −0.187710
\(361\) 40.0540 2.10810
\(362\) 13.2462 0.696205
\(363\) −4.43845 −0.232958
\(364\) 0 0
\(365\) 8.68466 0.454576
\(366\) −1.00000 −0.0522708
\(367\) 11.3693 0.593474 0.296737 0.954959i \(-0.404102\pi\)
0.296737 + 0.954959i \(0.404102\pi\)
\(368\) −9.12311 −0.475575
\(369\) 4.12311 0.214640
\(370\) 34.9309 1.81597
\(371\) 3.87689 0.201278
\(372\) 5.12311 0.265621
\(373\) 21.5616 1.11641 0.558207 0.829701i \(-0.311489\pi\)
0.558207 + 0.829701i \(0.311489\pi\)
\(374\) −12.8078 −0.662274
\(375\) −9.56155 −0.493756
\(376\) −7.68466 −0.396306
\(377\) 0 0
\(378\) −1.00000 −0.0514344
\(379\) 0.492423 0.0252940 0.0126470 0.999920i \(-0.495974\pi\)
0.0126470 + 0.999920i \(0.495974\pi\)
\(380\) −27.3693 −1.40402
\(381\) 2.24621 0.115077
\(382\) 5.75379 0.294389
\(383\) 8.31534 0.424894 0.212447 0.977173i \(-0.431857\pi\)
0.212447 + 0.977173i \(0.431857\pi\)
\(384\) 1.00000 0.0510310
\(385\) 9.12311 0.464957
\(386\) −20.3693 −1.03677
\(387\) −2.87689 −0.146241
\(388\) 12.2462 0.621707
\(389\) 18.6847 0.947350 0.473675 0.880700i \(-0.342927\pi\)
0.473675 + 0.880700i \(0.342927\pi\)
\(390\) 0 0
\(391\) 45.6155 2.30688
\(392\) 1.00000 0.0505076
\(393\) −16.4924 −0.831933
\(394\) −0.561553 −0.0282906
\(395\) −5.12311 −0.257771
\(396\) 2.56155 0.128723
\(397\) 1.68466 0.0845506 0.0422753 0.999106i \(-0.486539\pi\)
0.0422753 + 0.999106i \(0.486539\pi\)
\(398\) −17.6155 −0.882987
\(399\) −7.68466 −0.384714
\(400\) 7.68466 0.384233
\(401\) 15.1771 0.757907 0.378954 0.925416i \(-0.376284\pi\)
0.378954 + 0.925416i \(0.376284\pi\)
\(402\) 1.12311 0.0560154
\(403\) 0 0
\(404\) −19.5616 −0.973224
\(405\) −3.56155 −0.176975
\(406\) 1.00000 0.0496292
\(407\) −25.1231 −1.24531
\(408\) −5.00000 −0.247537
\(409\) −30.4384 −1.50508 −0.752542 0.658544i \(-0.771173\pi\)
−0.752542 + 0.658544i \(0.771173\pi\)
\(410\) −14.6847 −0.725224
\(411\) −5.80776 −0.286476
\(412\) −11.3693 −0.560126
\(413\) −13.1231 −0.645746
\(414\) −9.12311 −0.448376
\(415\) 36.4924 1.79134
\(416\) 0 0
\(417\) −7.68466 −0.376319
\(418\) 19.6847 0.962808
\(419\) 22.7386 1.11085 0.555427 0.831565i \(-0.312555\pi\)
0.555427 + 0.831565i \(0.312555\pi\)
\(420\) 3.56155 0.173786
\(421\) 8.43845 0.411265 0.205632 0.978629i \(-0.434075\pi\)
0.205632 + 0.978629i \(0.434075\pi\)
\(422\) −2.87689 −0.140045
\(423\) −7.68466 −0.373641
\(424\) −3.87689 −0.188279
\(425\) −38.4233 −1.86380
\(426\) −4.00000 −0.193801
\(427\) 1.00000 0.0483934
\(428\) −9.43845 −0.456225
\(429\) 0 0
\(430\) 10.2462 0.494116
\(431\) −18.7386 −0.902608 −0.451304 0.892370i \(-0.649041\pi\)
−0.451304 + 0.892370i \(0.649041\pi\)
\(432\) 1.00000 0.0481125
\(433\) −5.31534 −0.255439 −0.127720 0.991810i \(-0.540766\pi\)
−0.127720 + 0.991810i \(0.540766\pi\)
\(434\) −5.12311 −0.245917
\(435\) 3.56155 0.170763
\(436\) 0.876894 0.0419956
\(437\) −70.1080 −3.35372
\(438\) −2.43845 −0.116514
\(439\) −28.4924 −1.35987 −0.679935 0.733273i \(-0.737992\pi\)
−0.679935 + 0.733273i \(0.737992\pi\)
\(440\) −9.12311 −0.434927
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −34.4233 −1.63550 −0.817750 0.575574i \(-0.804779\pi\)
−0.817750 + 0.575574i \(0.804779\pi\)
\(444\) −9.80776 −0.465456
\(445\) 34.4924 1.63510
\(446\) −16.4924 −0.780939
\(447\) −17.8078 −0.842278
\(448\) −1.00000 −0.0472456
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 7.68466 0.362258
\(451\) 10.5616 0.497324
\(452\) 16.9309 0.796361
\(453\) −17.4384 −0.819330
\(454\) 13.1231 0.615898
\(455\) 0 0
\(456\) 7.68466 0.359867
\(457\) −13.3153 −0.622865 −0.311433 0.950268i \(-0.600809\pi\)
−0.311433 + 0.950268i \(0.600809\pi\)
\(458\) 22.8078 1.06574
\(459\) −5.00000 −0.233380
\(460\) 32.4924 1.51497
\(461\) 30.0540 1.39975 0.699877 0.714264i \(-0.253238\pi\)
0.699877 + 0.714264i \(0.253238\pi\)
\(462\) −2.56155 −0.119174
\(463\) −11.6847 −0.543032 −0.271516 0.962434i \(-0.587525\pi\)
−0.271516 + 0.962434i \(0.587525\pi\)
\(464\) −1.00000 −0.0464238
\(465\) −18.2462 −0.846148
\(466\) −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
\(469\) −1.12311 −0.0518602
\(470\) 27.3693 1.26245
\(471\) −5.80776 −0.267608
\(472\) 13.1231 0.604040
\(473\) −7.36932 −0.338842
\(474\) 1.43845 0.0660701
\(475\) 59.0540 2.70958
\(476\) 5.00000 0.229175
\(477\) −3.87689 −0.177511
\(478\) 4.00000 0.182956
\(479\) −2.56155 −0.117040 −0.0585202 0.998286i \(-0.518638\pi\)
−0.0585202 + 0.998286i \(0.518638\pi\)
\(480\) −3.56155 −0.162562
\(481\) 0 0
\(482\) 11.3153 0.515400
\(483\) 9.12311 0.415116
\(484\) −4.43845 −0.201748
\(485\) −43.6155 −1.98048
\(486\) 1.00000 0.0453609
\(487\) 23.6847 1.07325 0.536627 0.843819i \(-0.319698\pi\)
0.536627 + 0.843819i \(0.319698\pi\)
\(488\) −1.00000 −0.0452679
\(489\) −1.12311 −0.0507886
\(490\) −3.56155 −0.160895
\(491\) −1.75379 −0.0791474 −0.0395737 0.999217i \(-0.512600\pi\)
−0.0395737 + 0.999217i \(0.512600\pi\)
\(492\) 4.12311 0.185884
\(493\) 5.00000 0.225189
\(494\) 0 0
\(495\) −9.12311 −0.410053
\(496\) 5.12311 0.230034
\(497\) 4.00000 0.179425
\(498\) −10.2462 −0.459144
\(499\) −27.3693 −1.22522 −0.612609 0.790386i \(-0.709880\pi\)
−0.612609 + 0.790386i \(0.709880\pi\)
\(500\) −9.56155 −0.427606
\(501\) 20.4924 0.915534
\(502\) 8.49242 0.379035
\(503\) 20.4924 0.913712 0.456856 0.889541i \(-0.348975\pi\)
0.456856 + 0.889541i \(0.348975\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 69.6695 3.10025
\(506\) −23.3693 −1.03889
\(507\) 0 0
\(508\) 2.24621 0.0996595
\(509\) 27.3153 1.21073 0.605366 0.795948i \(-0.293027\pi\)
0.605366 + 0.795948i \(0.293027\pi\)
\(510\) 17.8078 0.788541
\(511\) 2.43845 0.107871
\(512\) 1.00000 0.0441942
\(513\) 7.68466 0.339286
\(514\) −14.7538 −0.650762
\(515\) 40.4924 1.78431
\(516\) −2.87689 −0.126648
\(517\) −19.6847 −0.865730
\(518\) 9.80776 0.430928
\(519\) −4.87689 −0.214072
\(520\) 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
\(523\) 15.1922 0.664310 0.332155 0.943225i \(-0.392224\pi\)
0.332155 + 0.943225i \(0.392224\pi\)
\(524\) −16.4924 −0.720475
\(525\) −7.68466 −0.335386
\(526\) −20.4924 −0.893512
\(527\) −25.6155 −1.11583
\(528\) 2.56155 0.111477
\(529\) 60.2311 2.61874
\(530\) 13.8078 0.599771
\(531\) 13.1231 0.569494
\(532\) −7.68466 −0.333172
\(533\) 0 0
\(534\) −9.68466 −0.419096
\(535\) 33.6155 1.45333
\(536\) 1.12311 0.0485108
\(537\) −16.4924 −0.711701
\(538\) −4.87689 −0.210258
\(539\) 2.56155 0.110334
\(540\) −3.56155 −0.153265
\(541\) 1.56155 0.0671364 0.0335682 0.999436i \(-0.489313\pi\)
0.0335682 + 0.999436i \(0.489313\pi\)
\(542\) 13.1231 0.563686
\(543\) 13.2462 0.568449
\(544\) −5.00000 −0.214373
\(545\) −3.12311 −0.133779
\(546\) 0 0
\(547\) −22.7386 −0.972234 −0.486117 0.873894i \(-0.661587\pi\)
−0.486117 + 0.873894i \(0.661587\pi\)
\(548\) −5.80776 −0.248095
\(549\) −1.00000 −0.0426790
\(550\) 19.6847 0.839357
\(551\) −7.68466 −0.327377
\(552\) −9.12311 −0.388305
\(553\) −1.43845 −0.0611690
\(554\) −3.56155 −0.151316
\(555\) 34.9309 1.48273
\(556\) −7.68466 −0.325902
\(557\) 21.7386 0.921095 0.460548 0.887635i \(-0.347653\pi\)
0.460548 + 0.887635i \(0.347653\pi\)
\(558\) 5.12311 0.216878
\(559\) 0 0
\(560\) 3.56155 0.150503
\(561\) −12.8078 −0.540744
\(562\) 7.80776 0.329351
\(563\) −0.630683 −0.0265801 −0.0132901 0.999912i \(-0.504230\pi\)
−0.0132901 + 0.999912i \(0.504230\pi\)
\(564\) −7.68466 −0.323582
\(565\) −60.3002 −2.53685
\(566\) 8.49242 0.356963
\(567\) −1.00000 −0.0419961
\(568\) −4.00000 −0.167836
\(569\) 11.6155 0.486948 0.243474 0.969907i \(-0.421713\pi\)
0.243474 + 0.969907i \(0.421713\pi\)
\(570\) −27.3693 −1.14637
\(571\) 13.7538 0.575578 0.287789 0.957694i \(-0.407080\pi\)
0.287789 + 0.957694i \(0.407080\pi\)
\(572\) 0 0
\(573\) 5.75379 0.240368
\(574\) −4.12311 −0.172095
\(575\) −70.1080 −2.92370
\(576\) 1.00000 0.0416667
\(577\) 31.3153 1.30367 0.651837 0.758359i \(-0.273998\pi\)
0.651837 + 0.758359i \(0.273998\pi\)
\(578\) 8.00000 0.332756
\(579\) −20.3693 −0.846520
\(580\) 3.56155 0.147885
\(581\) 10.2462 0.425084
\(582\) 12.2462 0.507622
\(583\) −9.93087 −0.411295
\(584\) −2.43845 −0.100904
\(585\) 0 0
\(586\) −4.68466 −0.193521
\(587\) 31.3693 1.29475 0.647375 0.762172i \(-0.275867\pi\)
0.647375 + 0.762172i \(0.275867\pi\)
\(588\) 1.00000 0.0412393
\(589\) 39.3693 1.62218
\(590\) −46.7386 −1.92420
\(591\) −0.561553 −0.0230992
\(592\) −9.80776 −0.403097
\(593\) 32.6155 1.33936 0.669680 0.742650i \(-0.266431\pi\)
0.669680 + 0.742650i \(0.266431\pi\)
\(594\) 2.56155 0.105102
\(595\) −17.8078 −0.730047
\(596\) −17.8078 −0.729434
\(597\) −17.6155 −0.720956
\(598\) 0 0
\(599\) −22.8769 −0.934725 −0.467362 0.884066i \(-0.654796\pi\)
−0.467362 + 0.884066i \(0.654796\pi\)
\(600\) 7.68466 0.313725
\(601\) 36.9309 1.50644 0.753221 0.657768i \(-0.228499\pi\)
0.753221 + 0.657768i \(0.228499\pi\)
\(602\) 2.87689 0.117253
\(603\) 1.12311 0.0457364
\(604\) −17.4384 −0.709560
\(605\) 15.8078 0.642677
\(606\) −19.5616 −0.794634
\(607\) 21.1231 0.857360 0.428680 0.903456i \(-0.358979\pi\)
0.428680 + 0.903456i \(0.358979\pi\)
\(608\) 7.68466 0.311654
\(609\) 1.00000 0.0405220
\(610\) 3.56155 0.144203
\(611\) 0 0
\(612\) −5.00000 −0.202113
\(613\) 36.3002 1.46615 0.733075 0.680147i \(-0.238084\pi\)
0.733075 + 0.680147i \(0.238084\pi\)
\(614\) −33.9309 −1.36934
\(615\) −14.6847 −0.592143
\(616\) −2.56155 −0.103208
\(617\) −27.4233 −1.10402 −0.552010 0.833837i \(-0.686139\pi\)
−0.552010 + 0.833837i \(0.686139\pi\)
\(618\) −11.3693 −0.457341
\(619\) 4.31534 0.173448 0.0867241 0.996232i \(-0.472360\pi\)
0.0867241 + 0.996232i \(0.472360\pi\)
\(620\) −18.2462 −0.732785
\(621\) −9.12311 −0.366098
\(622\) 22.5616 0.904636
\(623\) 9.68466 0.388008
\(624\) 0 0
\(625\) −4.36932 −0.174773
\(626\) 27.6155 1.10374
\(627\) 19.6847 0.786130
\(628\) −5.80776 −0.231755
\(629\) 49.0388 1.95531
\(630\) 3.56155 0.141896
\(631\) 23.1922 0.923268 0.461634 0.887070i \(-0.347263\pi\)
0.461634 + 0.887070i \(0.347263\pi\)
\(632\) 1.43845 0.0572184
\(633\) −2.87689 −0.114346
\(634\) −11.5616 −0.459168
\(635\) −8.00000 −0.317470
\(636\) −3.87689 −0.153729
\(637\) 0 0
\(638\) −2.56155 −0.101413
\(639\) −4.00000 −0.158238
\(640\) −3.56155 −0.140783
\(641\) 36.9309 1.45868 0.729341 0.684151i \(-0.239827\pi\)
0.729341 + 0.684151i \(0.239827\pi\)
\(642\) −9.43845 −0.372506
\(643\) −26.5616 −1.04749 −0.523743 0.851877i \(-0.675465\pi\)
−0.523743 + 0.851877i \(0.675465\pi\)
\(644\) 9.12311 0.359501
\(645\) 10.2462 0.403444
\(646\) −38.4233 −1.51174
\(647\) −20.8078 −0.818038 −0.409019 0.912526i \(-0.634129\pi\)
−0.409019 + 0.912526i \(0.634129\pi\)
\(648\) 1.00000 0.0392837
\(649\) 33.6155 1.31952
\(650\) 0 0
\(651\) −5.12311 −0.200790
\(652\) −1.12311 −0.0439842
\(653\) −18.8078 −0.736005 −0.368002 0.929825i \(-0.619958\pi\)
−0.368002 + 0.929825i \(0.619958\pi\)
\(654\) 0.876894 0.0342893
\(655\) 58.7386 2.29511
\(656\) 4.12311 0.160980
\(657\) −2.43845 −0.0951329
\(658\) 7.68466 0.299579
\(659\) −28.3153 −1.10301 −0.551505 0.834172i \(-0.685946\pi\)
−0.551505 + 0.834172i \(0.685946\pi\)
\(660\) −9.12311 −0.355116
\(661\) −9.80776 −0.381478 −0.190739 0.981641i \(-0.561088\pi\)
−0.190739 + 0.981641i \(0.561088\pi\)
\(662\) −35.3693 −1.37467
\(663\) 0 0
\(664\) −10.2462 −0.397630
\(665\) 27.3693 1.06134
\(666\) −9.80776 −0.380043
\(667\) 9.12311 0.353248
\(668\) 20.4924 0.792876
\(669\) −16.4924 −0.637634
\(670\) −4.00000 −0.154533
\(671\) −2.56155 −0.0988876
\(672\) −1.00000 −0.0385758
\(673\) 8.75379 0.337434 0.168717 0.985665i \(-0.446038\pi\)
0.168717 + 0.985665i \(0.446038\pi\)
\(674\) −29.4924 −1.13601
\(675\) 7.68466 0.295783
\(676\) 0 0
\(677\) −8.38447 −0.322241 −0.161121 0.986935i \(-0.551511\pi\)
−0.161121 + 0.986935i \(0.551511\pi\)
\(678\) 16.9309 0.650226
\(679\) −12.2462 −0.469966
\(680\) 17.8078 0.682897
\(681\) 13.1231 0.502879
\(682\) 13.1231 0.502510
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 7.68466 0.293830
\(685\) 20.6847 0.790320
\(686\) −1.00000 −0.0381802
\(687\) 22.8078 0.870170
\(688\) −2.87689 −0.109681
\(689\) 0 0
\(690\) 32.4924 1.23697
\(691\) −28.4924 −1.08390 −0.541951 0.840410i \(-0.682314\pi\)
−0.541951 + 0.840410i \(0.682314\pi\)
\(692\) −4.87689 −0.185392
\(693\) −2.56155 −0.0973053
\(694\) −0.315342 −0.0119702
\(695\) 27.3693 1.03818
\(696\) −1.00000 −0.0379049
\(697\) −20.6155 −0.780869
\(698\) −6.49242 −0.245742
\(699\) −6.00000 −0.226941
\(700\) −7.68466 −0.290453
\(701\) 9.05398 0.341964 0.170982 0.985274i \(-0.445306\pi\)
0.170982 + 0.985274i \(0.445306\pi\)
\(702\) 0 0
\(703\) −75.3693 −2.84261
\(704\) 2.56155 0.0965422
\(705\) 27.3693 1.03079
\(706\) −9.80776 −0.369120
\(707\) 19.5616 0.735688
\(708\) 13.1231 0.493197
\(709\) 20.3002 0.762390 0.381195 0.924495i \(-0.375513\pi\)
0.381195 + 0.924495i \(0.375513\pi\)
\(710\) 14.2462 0.534651
\(711\) 1.43845 0.0539460
\(712\) −9.68466 −0.362948
\(713\) −46.7386 −1.75038
\(714\) 5.00000 0.187120
\(715\) 0 0
\(716\) −16.4924 −0.616351
\(717\) 4.00000 0.149383
\(718\) −4.63068 −0.172816
\(719\) 21.9309 0.817883 0.408942 0.912561i \(-0.365898\pi\)
0.408942 + 0.912561i \(0.365898\pi\)
\(720\) −3.56155 −0.132731
\(721\) 11.3693 0.423415
\(722\) 40.0540 1.49065
\(723\) 11.3153 0.420822
\(724\) 13.2462 0.492292
\(725\) −7.68466 −0.285401
\(726\) −4.43845 −0.164726
\(727\) 50.2462 1.86353 0.931764 0.363063i \(-0.118269\pi\)
0.931764 + 0.363063i \(0.118269\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 8.68466 0.321434
\(731\) 14.3845 0.532029
\(732\) −1.00000 −0.0369611
\(733\) 32.6155 1.20468 0.602341 0.798239i \(-0.294235\pi\)
0.602341 + 0.798239i \(0.294235\pi\)
\(734\) 11.3693 0.419649
\(735\) −3.56155 −0.131370
\(736\) −9.12311 −0.336282
\(737\) 2.87689 0.105972
\(738\) 4.12311 0.151774
\(739\) 47.3693 1.74251 0.871254 0.490832i \(-0.163307\pi\)
0.871254 + 0.490832i \(0.163307\pi\)
\(740\) 34.9309 1.28408
\(741\) 0 0
\(742\) 3.87689 0.142325
\(743\) 8.63068 0.316629 0.158315 0.987389i \(-0.449394\pi\)
0.158315 + 0.987389i \(0.449394\pi\)
\(744\) 5.12311 0.187822
\(745\) 63.4233 2.32365
\(746\) 21.5616 0.789425
\(747\) −10.2462 −0.374889
\(748\) −12.8078 −0.468298
\(749\) 9.43845 0.344873
\(750\) −9.56155 −0.349139
\(751\) −0.807764 −0.0294757 −0.0147379 0.999891i \(-0.504691\pi\)
−0.0147379 + 0.999891i \(0.504691\pi\)
\(752\) −7.68466 −0.280231
\(753\) 8.49242 0.309481
\(754\) 0 0
\(755\) 62.1080 2.26034
\(756\) −1.00000 −0.0363696
\(757\) −7.12311 −0.258894 −0.129447 0.991586i \(-0.541320\pi\)
−0.129447 + 0.991586i \(0.541320\pi\)
\(758\) 0.492423 0.0178856
\(759\) −23.3693 −0.848252
\(760\) −27.3693 −0.992789
\(761\) −42.4924 −1.54035 −0.770175 0.637833i \(-0.779831\pi\)
−0.770175 + 0.637833i \(0.779831\pi\)
\(762\) 2.24621 0.0813716
\(763\) −0.876894 −0.0317457
\(764\) 5.75379 0.208165
\(765\) 17.8078 0.643841
\(766\) 8.31534 0.300446
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −22.9848 −0.828855 −0.414427 0.910082i \(-0.636018\pi\)
−0.414427 + 0.910082i \(0.636018\pi\)
\(770\) 9.12311 0.328774
\(771\) −14.7538 −0.531345
\(772\) −20.3693 −0.733108
\(773\) 26.4924 0.952866 0.476433 0.879211i \(-0.341929\pi\)
0.476433 + 0.879211i \(0.341929\pi\)
\(774\) −2.87689 −0.103408
\(775\) 39.3693 1.41419
\(776\) 12.2462 0.439613
\(777\) 9.80776 0.351852
\(778\) 18.6847 0.669877
\(779\) 31.6847 1.13522
\(780\) 0 0
\(781\) −10.2462 −0.366638
\(782\) 45.6155 1.63121
\(783\) −1.00000 −0.0357371
\(784\) 1.00000 0.0357143
\(785\) 20.6847 0.738267
\(786\) −16.4924 −0.588265
\(787\) 0.807764 0.0287937 0.0143968 0.999896i \(-0.495417\pi\)
0.0143968 + 0.999896i \(0.495417\pi\)
\(788\) −0.561553 −0.0200045
\(789\) −20.4924 −0.729550
\(790\) −5.12311 −0.182272
\(791\) −16.9309 −0.601992
\(792\) 2.56155 0.0910208
\(793\) 0 0
\(794\) 1.68466 0.0597863
\(795\) 13.8078 0.489711
\(796\) −17.6155 −0.624366
\(797\) −24.1080 −0.853947 −0.426974 0.904264i \(-0.640420\pi\)
−0.426974 + 0.904264i \(0.640420\pi\)
\(798\) −7.68466 −0.272034
\(799\) 38.4233 1.35932
\(800\) 7.68466 0.271694
\(801\) −9.68466 −0.342191
\(802\) 15.1771 0.535921
\(803\) −6.24621 −0.220424
\(804\) 1.12311 0.0396089
\(805\) −32.4924 −1.14521
\(806\) 0 0
\(807\) −4.87689 −0.171675
\(808\) −19.5616 −0.688173
\(809\) −10.9309 −0.384309 −0.192154 0.981365i \(-0.561547\pi\)
−0.192154 + 0.981365i \(0.561547\pi\)
\(810\) −3.56155 −0.125140
\(811\) 5.75379 0.202043 0.101021 0.994884i \(-0.467789\pi\)
0.101021 + 0.994884i \(0.467789\pi\)
\(812\) 1.00000 0.0350931
\(813\) 13.1231 0.460247
\(814\) −25.1231 −0.880564
\(815\) 4.00000 0.140114
\(816\) −5.00000 −0.175035
\(817\) −22.1080 −0.773459
\(818\) −30.4384 −1.06426
\(819\) 0 0
\(820\) −14.6847 −0.512811
\(821\) −12.0691 −0.421216 −0.210608 0.977571i \(-0.567544\pi\)
−0.210608 + 0.977571i \(0.567544\pi\)
\(822\) −5.80776 −0.202569
\(823\) 14.2462 0.496592 0.248296 0.968684i \(-0.420129\pi\)
0.248296 + 0.968684i \(0.420129\pi\)
\(824\) −11.3693 −0.396069
\(825\) 19.6847 0.685332
\(826\) −13.1231 −0.456611
\(827\) −11.5076 −0.400158 −0.200079 0.979780i \(-0.564120\pi\)
−0.200079 + 0.979780i \(0.564120\pi\)
\(828\) −9.12311 −0.317050
\(829\) 29.1080 1.01096 0.505480 0.862838i \(-0.331315\pi\)
0.505480 + 0.862838i \(0.331315\pi\)
\(830\) 36.4924 1.26667
\(831\) −3.56155 −0.123549
\(832\) 0 0
\(833\) −5.00000 −0.173240
\(834\) −7.68466 −0.266098
\(835\) −72.9848 −2.52574
\(836\) 19.6847 0.680808
\(837\) 5.12311 0.177080
\(838\) 22.7386 0.785493
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) 3.56155 0.122885
\(841\) −28.0000 −0.965517
\(842\) 8.43845 0.290808
\(843\) 7.80776 0.268914
\(844\) −2.87689 −0.0990268
\(845\) 0 0
\(846\) −7.68466 −0.264204
\(847\) 4.43845 0.152507
\(848\) −3.87689 −0.133133
\(849\) 8.49242 0.291459
\(850\) −38.4233 −1.31791
\(851\) 89.4773 3.06724
\(852\) −4.00000 −0.137038
\(853\) −0.369317 −0.0126452 −0.00632258 0.999980i \(-0.502013\pi\)
−0.00632258 + 0.999980i \(0.502013\pi\)
\(854\) 1.00000 0.0342193
\(855\) −27.3693 −0.936011
\(856\) −9.43845 −0.322599
\(857\) −22.3002 −0.761760 −0.380880 0.924625i \(-0.624379\pi\)
−0.380880 + 0.924625i \(0.624379\pi\)
\(858\) 0 0
\(859\) −36.8078 −1.25586 −0.627932 0.778268i \(-0.716099\pi\)
−0.627932 + 0.778268i \(0.716099\pi\)
\(860\) 10.2462 0.349393
\(861\) −4.12311 −0.140515
\(862\) −18.7386 −0.638240
\(863\) 26.1080 0.888725 0.444362 0.895847i \(-0.353430\pi\)
0.444362 + 0.895847i \(0.353430\pi\)
\(864\) 1.00000 0.0340207
\(865\) 17.3693 0.590574
\(866\) −5.31534 −0.180623
\(867\) 8.00000 0.271694
\(868\) −5.12311 −0.173890
\(869\) 3.68466 0.124993
\(870\) 3.56155 0.120748
\(871\) 0 0
\(872\) 0.876894 0.0296954
\(873\) 12.2462 0.414471
\(874\) −70.1080 −2.37144
\(875\) 9.56155 0.323239
\(876\) −2.43845 −0.0823875
\(877\) 8.30019 0.280277 0.140139 0.990132i \(-0.455245\pi\)
0.140139 + 0.990132i \(0.455245\pi\)
\(878\) −28.4924 −0.961573
\(879\) −4.68466 −0.158010
\(880\) −9.12311 −0.307540
\(881\) 47.1771 1.58944 0.794718 0.606979i \(-0.207619\pi\)
0.794718 + 0.606979i \(0.207619\pi\)
\(882\) 1.00000 0.0336718
\(883\) 17.1231 0.576238 0.288119 0.957595i \(-0.406970\pi\)
0.288119 + 0.957595i \(0.406970\pi\)
\(884\) 0 0
\(885\) −46.7386 −1.57110
\(886\) −34.4233 −1.15647
\(887\) 5.43845 0.182605 0.0913026 0.995823i \(-0.470897\pi\)
0.0913026 + 0.995823i \(0.470897\pi\)
\(888\) −9.80776 −0.329127
\(889\) −2.24621 −0.0753355
\(890\) 34.4924 1.15619
\(891\) 2.56155 0.0858152
\(892\) −16.4924 −0.552207
\(893\) −59.0540 −1.97617
\(894\) −17.8078 −0.595581
\(895\) 58.7386 1.96342
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) −5.12311 −0.170865
\(900\) 7.68466 0.256155
\(901\) 19.3845 0.645790
\(902\) 10.5616 0.351661
\(903\) 2.87689 0.0957371
\(904\) 16.9309 0.563112
\(905\) −47.1771 −1.56822
\(906\) −17.4384 −0.579354
\(907\) 22.8769 0.759615 0.379807 0.925066i \(-0.375990\pi\)
0.379807 + 0.925066i \(0.375990\pi\)
\(908\) 13.1231 0.435506
\(909\) −19.5616 −0.648816
\(910\) 0 0
\(911\) −5.26137 −0.174317 −0.0871584 0.996194i \(-0.527779\pi\)
−0.0871584 + 0.996194i \(0.527779\pi\)
\(912\) 7.68466 0.254464
\(913\) −26.2462 −0.868623
\(914\) −13.3153 −0.440432
\(915\) 3.56155 0.117741
\(916\) 22.8078 0.753590
\(917\) 16.4924 0.544628
\(918\) −5.00000 −0.165025
\(919\) 6.56155 0.216446 0.108223 0.994127i \(-0.465484\pi\)
0.108223 + 0.994127i \(0.465484\pi\)
\(920\) 32.4924 1.07124
\(921\) −33.9309 −1.11806
\(922\) 30.0540 0.989775
\(923\) 0 0
\(924\) −2.56155 −0.0842689
\(925\) −75.3693 −2.47813
\(926\) −11.6847 −0.383982
\(927\) −11.3693 −0.373417
\(928\) −1.00000 −0.0328266
\(929\) −6.61553 −0.217048 −0.108524 0.994094i \(-0.534613\pi\)
−0.108524 + 0.994094i \(0.534613\pi\)
\(930\) −18.2462 −0.598317
\(931\) 7.68466 0.251855
\(932\) −6.00000 −0.196537
\(933\) 22.5616 0.738632
\(934\) 28.0000 0.916188
\(935\) 45.6155 1.49179
\(936\) 0 0
\(937\) 13.5616 0.443037 0.221518 0.975156i \(-0.428899\pi\)
0.221518 + 0.975156i \(0.428899\pi\)
\(938\) −1.12311 −0.0366707
\(939\) 27.6155 0.901199
\(940\) 27.3693 0.892689
\(941\) 53.3693 1.73979 0.869895 0.493237i \(-0.164186\pi\)
0.869895 + 0.493237i \(0.164186\pi\)
\(942\) −5.80776 −0.189227
\(943\) −37.6155 −1.22493
\(944\) 13.1231 0.427121
\(945\) 3.56155 0.115857
\(946\) −7.36932 −0.239597
\(947\) −0.315342 −0.0102472 −0.00512361 0.999987i \(-0.501631\pi\)
−0.00512361 + 0.999987i \(0.501631\pi\)
\(948\) 1.43845 0.0467186
\(949\) 0 0
\(950\) 59.0540 1.91596
\(951\) −11.5616 −0.374909
\(952\) 5.00000 0.162051
\(953\) −52.7386 −1.70837 −0.854186 0.519968i \(-0.825944\pi\)
−0.854186 + 0.519968i \(0.825944\pi\)
\(954\) −3.87689 −0.125519
\(955\) −20.4924 −0.663119
\(956\) 4.00000 0.129369
\(957\) −2.56155 −0.0828032
\(958\) −2.56155 −0.0827600
\(959\) 5.80776 0.187542
\(960\) −3.56155 −0.114949
\(961\) −4.75379 −0.153348
\(962\) 0 0
\(963\) −9.43845 −0.304150
\(964\) 11.3153 0.364443
\(965\) 72.5464 2.33535
\(966\) 9.12311 0.293531
\(967\) −4.49242 −0.144467 −0.0722333 0.997388i \(-0.523013\pi\)
−0.0722333 + 0.997388i \(0.523013\pi\)
\(968\) −4.43845 −0.142657
\(969\) −38.4233 −1.23433
\(970\) −43.6155 −1.40041
\(971\) −10.8769 −0.349056 −0.174528 0.984652i \(-0.555840\pi\)
−0.174528 + 0.984652i \(0.555840\pi\)
\(972\) 1.00000 0.0320750
\(973\) 7.68466 0.246359
\(974\) 23.6847 0.758905
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) 2.05398 0.0657125 0.0328562 0.999460i \(-0.489540\pi\)
0.0328562 + 0.999460i \(0.489540\pi\)
\(978\) −1.12311 −0.0359130
\(979\) −24.8078 −0.792860
\(980\) −3.56155 −0.113770
\(981\) 0.876894 0.0279971
\(982\) −1.75379 −0.0559656
\(983\) 55.2311 1.76160 0.880799 0.473491i \(-0.157006\pi\)
0.880799 + 0.473491i \(0.157006\pi\)
\(984\) 4.12311 0.131440
\(985\) 2.00000 0.0637253
\(986\) 5.00000 0.159232
\(987\) 7.68466 0.244605
\(988\) 0 0
\(989\) 26.2462 0.834581
\(990\) −9.12311 −0.289951
\(991\) −1.43845 −0.0456938 −0.0228469 0.999739i \(-0.507273\pi\)
−0.0228469 + 0.999739i \(0.507273\pi\)
\(992\) 5.12311 0.162659
\(993\) −35.3693 −1.12241
\(994\) 4.00000 0.126872
\(995\) 62.7386 1.98895
\(996\) −10.2462 −0.324664
\(997\) −17.0000 −0.538395 −0.269198 0.963085i \(-0.586759\pi\)
−0.269198 + 0.963085i \(0.586759\pi\)
\(998\) −27.3693 −0.866361
\(999\) −9.80776 −0.310304
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.1.1 2
13.3 even 3 546.2.l.i.295.1 yes 4
13.9 even 3 546.2.l.i.211.1 4
13.12 even 2 7098.2.a.bq.1.2 2
39.29 odd 6 1638.2.r.z.1387.2 4
39.35 odd 6 1638.2.r.z.757.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.l.i.211.1 4 13.9 even 3
546.2.l.i.295.1 yes 4 13.3 even 3
1638.2.r.z.757.2 4 39.35 odd 6
1638.2.r.z.1387.2 4 39.29 odd 6
7098.2.a.bq.1.2 2 13.12 even 2
7098.2.a.bw.1.1 2 1.1 even 1 trivial