Properties

Label 7098.2.a.cr.1.4
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.48406561.1
Defining polynomial: \(x^{6} - 3 x^{5} - 17 x^{4} + 39 x^{3} + 111 x^{2} - 131 x - 281\)
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 1.4
Root \(-1.83411\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.83411 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.83411 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.83411 q^{10} -0.198062 q^{11} +1.00000 q^{12} +1.00000 q^{14} +1.83411 q^{15} +1.00000 q^{16} +0.445042 q^{17} -1.00000 q^{18} -1.57281 q^{19} +1.83411 q^{20} -1.00000 q^{21} +0.198062 q^{22} +9.19592 q^{23} -1.00000 q^{24} -1.63605 q^{25} +1.00000 q^{27} -1.00000 q^{28} -9.88794 q^{29} -1.83411 q^{30} -6.88303 q^{31} -1.00000 q^{32} -0.198062 q^{33} -0.445042 q^{34} -1.83411 q^{35} +1.00000 q^{36} -5.13552 q^{37} +1.57281 q^{38} -1.83411 q^{40} -12.1234 q^{41} +1.00000 q^{42} -7.51027 q^{43} -0.198062 q^{44} +1.83411 q^{45} -9.19592 q^{46} +8.51673 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.63605 q^{50} +0.445042 q^{51} -1.38956 q^{53} -1.00000 q^{54} -0.363268 q^{55} +1.00000 q^{56} -1.57281 q^{57} +9.88794 q^{58} -11.7846 q^{59} +1.83411 q^{60} -10.1849 q^{61} +6.88303 q^{62} -1.00000 q^{63} +1.00000 q^{64} +0.198062 q^{66} -0.307727 q^{67} +0.445042 q^{68} +9.19592 q^{69} +1.83411 q^{70} -13.0597 q^{71} -1.00000 q^{72} +7.90088 q^{73} +5.13552 q^{74} -1.63605 q^{75} -1.57281 q^{76} +0.198062 q^{77} +6.98527 q^{79} +1.83411 q^{80} +1.00000 q^{81} +12.1234 q^{82} -3.19757 q^{83} -1.00000 q^{84} +0.816255 q^{85} +7.51027 q^{86} -9.88794 q^{87} +0.198062 q^{88} +5.55962 q^{89} -1.83411 q^{90} +9.19592 q^{92} -6.88303 q^{93} -8.51673 q^{94} -2.88471 q^{95} -1.00000 q^{96} -4.95178 q^{97} -1.00000 q^{98} -0.198062 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} + 6q^{3} + 6q^{4} - 3q^{5} - 6q^{6} - 6q^{7} - 6q^{8} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{2} + 6q^{3} + 6q^{4} - 3q^{5} - 6q^{6} - 6q^{7} - 6q^{8} + 6q^{9} + 3q^{10} - 10q^{11} + 6q^{12} + 6q^{14} - 3q^{15} + 6q^{16} + 2q^{17} - 6q^{18} - 2q^{19} - 3q^{20} - 6q^{21} + 10q^{22} + 4q^{23} - 6q^{24} + 13q^{25} + 6q^{27} - 6q^{28} + 2q^{29} + 3q^{30} - 9q^{31} - 6q^{32} - 10q^{33} - 2q^{34} + 3q^{35} + 6q^{36} - 7q^{37} + 2q^{38} + 3q^{40} - 11q^{41} + 6q^{42} - 5q^{43} - 10q^{44} - 3q^{45} - 4q^{46} - 5q^{47} + 6q^{48} + 6q^{49} - 13q^{50} + 2q^{51} - 6q^{53} - 6q^{54} + 5q^{55} + 6q^{56} - 2q^{57} - 2q^{58} - 28q^{59} - 3q^{60} + 23q^{61} + 9q^{62} - 6q^{63} + 6q^{64} + 10q^{66} + 10q^{67} + 2q^{68} + 4q^{69} - 3q^{70} - 21q^{71} - 6q^{72} + 7q^{73} + 7q^{74} + 13q^{75} - 2q^{76} + 10q^{77} - 14q^{79} - 3q^{80} + 6q^{81} + 11q^{82} - 17q^{83} - 6q^{84} - q^{85} + 5q^{86} + 2q^{87} + 10q^{88} - 17q^{89} + 3q^{90} + 4q^{92} - 9q^{93} + 5q^{94} - 22q^{95} - 6q^{96} - 6q^{98} - 10q^{99} + O(q^{100}) \)

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
\(5\) 1.83411 0.820238 0.410119 0.912032i \(-0.365487\pi\)
0.410119 + 0.912032i \(0.365487\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.83411 −0.579996
\(11\) −0.198062 −0.0597180 −0.0298590 0.999554i \(-0.509506\pi\)
−0.0298590 + 0.999554i \(0.509506\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) 1.83411 0.473565
\(16\) 1.00000 0.250000
\(17\) 0.445042 0.107939 0.0539693 0.998543i \(-0.482813\pi\)
0.0539693 + 0.998543i \(0.482813\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.57281 −0.360828 −0.180414 0.983591i \(-0.557744\pi\)
−0.180414 + 0.983591i \(0.557744\pi\)
\(20\) 1.83411 0.410119
\(21\) −1.00000 −0.218218
\(22\) 0.198062 0.0422270
\(23\) 9.19592 1.91748 0.958741 0.284281i \(-0.0917549\pi\)
0.958741 + 0.284281i \(0.0917549\pi\)
\(24\) −1.00000 −0.204124
\(25\) −1.63605 −0.327209
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −9.88794 −1.83614 −0.918072 0.396413i \(-0.870255\pi\)
−0.918072 + 0.396413i \(0.870255\pi\)
\(30\) −1.83411 −0.334861
\(31\) −6.88303 −1.23623 −0.618114 0.786088i \(-0.712103\pi\)
−0.618114 + 0.786088i \(0.712103\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.198062 −0.0344782
\(34\) −0.445042 −0.0763241
\(35\) −1.83411 −0.310021
\(36\) 1.00000 0.166667
\(37\) −5.13552 −0.844275 −0.422137 0.906532i \(-0.638720\pi\)
−0.422137 + 0.906532i \(0.638720\pi\)
\(38\) 1.57281 0.255144
\(39\) 0 0
\(40\) −1.83411 −0.289998
\(41\) −12.1234 −1.89336 −0.946682 0.322171i \(-0.895588\pi\)
−0.946682 + 0.322171i \(0.895588\pi\)
\(42\) 1.00000 0.154303
\(43\) −7.51027 −1.14531 −0.572653 0.819798i \(-0.694086\pi\)
−0.572653 + 0.819798i \(0.694086\pi\)
\(44\) −0.198062 −0.0298590
\(45\) 1.83411 0.273413
\(46\) −9.19592 −1.35586
\(47\) 8.51673 1.24229 0.621146 0.783695i \(-0.286667\pi\)
0.621146 + 0.783695i \(0.286667\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.63605 0.231372
\(51\) 0.445042 0.0623183
\(52\) 0 0
\(53\) −1.38956 −0.190871 −0.0954353 0.995436i \(-0.530424\pi\)
−0.0954353 + 0.995436i \(0.530424\pi\)
\(54\) −1.00000 −0.136083
\(55\) −0.363268 −0.0489830
\(56\) 1.00000 0.133631
\(57\) −1.57281 −0.208324
\(58\) 9.88794 1.29835
\(59\) −11.7846 −1.53422 −0.767112 0.641514i \(-0.778307\pi\)
−0.767112 + 0.641514i \(0.778307\pi\)
\(60\) 1.83411 0.236782
\(61\) −10.1849 −1.30405 −0.652024 0.758198i \(-0.726080\pi\)
−0.652024 + 0.758198i \(0.726080\pi\)
\(62\) 6.88303 0.874145
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0.198062 0.0243798
\(67\) −0.307727 −0.0375949 −0.0187974 0.999823i \(-0.505984\pi\)
−0.0187974 + 0.999823i \(0.505984\pi\)
\(68\) 0.445042 0.0539693
\(69\) 9.19592 1.10706
\(70\) 1.83411 0.219218
\(71\) −13.0597 −1.54990 −0.774951 0.632021i \(-0.782226\pi\)
−0.774951 + 0.632021i \(0.782226\pi\)
\(72\) −1.00000 −0.117851
\(73\) 7.90088 0.924728 0.462364 0.886690i \(-0.347001\pi\)
0.462364 + 0.886690i \(0.347001\pi\)
\(74\) 5.13552 0.596992
\(75\) −1.63605 −0.188914
\(76\) −1.57281 −0.180414
\(77\) 0.198062 0.0225713
\(78\) 0 0
\(79\) 6.98527 0.785904 0.392952 0.919559i \(-0.371454\pi\)
0.392952 + 0.919559i \(0.371454\pi\)
\(80\) 1.83411 0.205060
\(81\) 1.00000 0.111111
\(82\) 12.1234 1.33881
\(83\) −3.19757 −0.350979 −0.175489 0.984481i \(-0.556151\pi\)
−0.175489 + 0.984481i \(0.556151\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0.816255 0.0885353
\(86\) 7.51027 0.809853
\(87\) −9.88794 −1.06010
\(88\) 0.198062 0.0211135
\(89\) 5.55962 0.589319 0.294659 0.955602i \(-0.404794\pi\)
0.294659 + 0.955602i \(0.404794\pi\)
\(90\) −1.83411 −0.193332
\(91\) 0 0
\(92\) 9.19592 0.958741
\(93\) −6.88303 −0.713737
\(94\) −8.51673 −0.878434
\(95\) −2.88471 −0.295965
\(96\) −1.00000 −0.102062
\(97\) −4.95178 −0.502777 −0.251388 0.967886i \(-0.580887\pi\)
−0.251388 + 0.967886i \(0.580887\pi\)
\(98\) −1.00000 −0.101015
\(99\) −0.198062 −0.0199060
\(100\) −1.63605 −0.163605
\(101\) −0.0853103 −0.00848869 −0.00424435 0.999991i \(-0.501351\pi\)
−0.00424435 + 0.999991i \(0.501351\pi\)
\(102\) −0.445042 −0.0440657
\(103\) 7.20409 0.709840 0.354920 0.934897i \(-0.384508\pi\)
0.354920 + 0.934897i \(0.384508\pi\)
\(104\) 0 0
\(105\) −1.83411 −0.178991
\(106\) 1.38956 0.134966
\(107\) 8.94825 0.865060 0.432530 0.901619i \(-0.357621\pi\)
0.432530 + 0.901619i \(0.357621\pi\)
\(108\) 1.00000 0.0962250
\(109\) −0.615818 −0.0589846 −0.0294923 0.999565i \(-0.509389\pi\)
−0.0294923 + 0.999565i \(0.509389\pi\)
\(110\) 0.363268 0.0346362
\(111\) −5.13552 −0.487442
\(112\) −1.00000 −0.0944911
\(113\) 8.18052 0.769559 0.384779 0.923009i \(-0.374278\pi\)
0.384779 + 0.923009i \(0.374278\pi\)
\(114\) 1.57281 0.147307
\(115\) 16.8663 1.57279
\(116\) −9.88794 −0.918072
\(117\) 0 0
\(118\) 11.7846 1.08486
\(119\) −0.445042 −0.0407969
\(120\) −1.83411 −0.167430
\(121\) −10.9608 −0.996434
\(122\) 10.1849 0.922102
\(123\) −12.1234 −1.09313
\(124\) −6.88303 −0.618114
\(125\) −12.1712 −1.08863
\(126\) 1.00000 0.0890871
\(127\) 5.91677 0.525028 0.262514 0.964928i \(-0.415448\pi\)
0.262514 + 0.964928i \(0.415448\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −7.51027 −0.661242
\(130\) 0 0
\(131\) 4.83450 0.422393 0.211196 0.977444i \(-0.432264\pi\)
0.211196 + 0.977444i \(0.432264\pi\)
\(132\) −0.198062 −0.0172391
\(133\) 1.57281 0.136380
\(134\) 0.307727 0.0265836
\(135\) 1.83411 0.157855
\(136\) −0.445042 −0.0381620
\(137\) 6.26783 0.535497 0.267748 0.963489i \(-0.413720\pi\)
0.267748 + 0.963489i \(0.413720\pi\)
\(138\) −9.19592 −0.782809
\(139\) 13.6770 1.16007 0.580036 0.814591i \(-0.303038\pi\)
0.580036 + 0.814591i \(0.303038\pi\)
\(140\) −1.83411 −0.155010
\(141\) 8.51673 0.717238
\(142\) 13.0597 1.09595
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −18.1356 −1.50608
\(146\) −7.90088 −0.653881
\(147\) 1.00000 0.0824786
\(148\) −5.13552 −0.422137
\(149\) −13.3470 −1.09343 −0.546715 0.837319i \(-0.684122\pi\)
−0.546715 + 0.837319i \(0.684122\pi\)
\(150\) 1.63605 0.133583
\(151\) −1.21548 −0.0989144 −0.0494572 0.998776i \(-0.515749\pi\)
−0.0494572 + 0.998776i \(0.515749\pi\)
\(152\) 1.57281 0.127572
\(153\) 0.445042 0.0359795
\(154\) −0.198062 −0.0159603
\(155\) −12.6242 −1.01400
\(156\) 0 0
\(157\) 7.68582 0.613395 0.306698 0.951807i \(-0.400776\pi\)
0.306698 + 0.951807i \(0.400776\pi\)
\(158\) −6.98527 −0.555718
\(159\) −1.38956 −0.110199
\(160\) −1.83411 −0.144999
\(161\) −9.19592 −0.724740
\(162\) −1.00000 −0.0785674
\(163\) −12.1104 −0.948562 −0.474281 0.880374i \(-0.657292\pi\)
−0.474281 + 0.880374i \(0.657292\pi\)
\(164\) −12.1234 −0.946682
\(165\) −0.363268 −0.0282803
\(166\) 3.19757 0.248180
\(167\) −14.8734 −1.15094 −0.575469 0.817823i \(-0.695181\pi\)
−0.575469 + 0.817823i \(0.695181\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) −0.816255 −0.0626039
\(171\) −1.57281 −0.120276
\(172\) −7.51027 −0.572653
\(173\) −1.87323 −0.142419 −0.0712093 0.997461i \(-0.522686\pi\)
−0.0712093 + 0.997461i \(0.522686\pi\)
\(174\) 9.88794 0.749603
\(175\) 1.63605 0.123673
\(176\) −0.198062 −0.0149295
\(177\) −11.7846 −0.885784
\(178\) −5.55962 −0.416711
\(179\) −8.91285 −0.666178 −0.333089 0.942895i \(-0.608091\pi\)
−0.333089 + 0.942895i \(0.608091\pi\)
\(180\) 1.83411 0.136706
\(181\) −13.2448 −0.984478 −0.492239 0.870460i \(-0.663821\pi\)
−0.492239 + 0.870460i \(0.663821\pi\)
\(182\) 0 0
\(183\) −10.1849 −0.752893
\(184\) −9.19592 −0.677932
\(185\) −9.41910 −0.692506
\(186\) 6.88303 0.504688
\(187\) −0.0881460 −0.00644587
\(188\) 8.51673 0.621146
\(189\) −1.00000 −0.0727393
\(190\) 2.88471 0.209279
\(191\) −6.56414 −0.474965 −0.237482 0.971392i \(-0.576322\pi\)
−0.237482 + 0.971392i \(0.576322\pi\)
\(192\) 1.00000 0.0721688
\(193\) 21.6428 1.55788 0.778940 0.627099i \(-0.215758\pi\)
0.778940 + 0.627099i \(0.215758\pi\)
\(194\) 4.95178 0.355517
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −0.437589 −0.0311769 −0.0155885 0.999878i \(-0.504962\pi\)
−0.0155885 + 0.999878i \(0.504962\pi\)
\(198\) 0.198062 0.0140757
\(199\) 24.1946 1.71511 0.857553 0.514395i \(-0.171984\pi\)
0.857553 + 0.514395i \(0.171984\pi\)
\(200\) 1.63605 0.115686
\(201\) −0.307727 −0.0217054
\(202\) 0.0853103 0.00600241
\(203\) 9.88794 0.693997
\(204\) 0.445042 0.0311592
\(205\) −22.2357 −1.55301
\(206\) −7.20409 −0.501932
\(207\) 9.19592 0.639161
\(208\) 0 0
\(209\) 0.311515 0.0215479
\(210\) 1.83411 0.126566
\(211\) −11.6272 −0.800452 −0.400226 0.916416i \(-0.631068\pi\)
−0.400226 + 0.916416i \(0.631068\pi\)
\(212\) −1.38956 −0.0954353
\(213\) −13.0597 −0.894837
\(214\) −8.94825 −0.611690
\(215\) −13.7747 −0.939423
\(216\) −1.00000 −0.0680414
\(217\) 6.88303 0.467250
\(218\) 0.615818 0.0417084
\(219\) 7.90088 0.533892
\(220\) −0.363268 −0.0244915
\(221\) 0 0
\(222\) 5.13552 0.344674
\(223\) −4.17401 −0.279512 −0.139756 0.990186i \(-0.544632\pi\)
−0.139756 + 0.990186i \(0.544632\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.63605 −0.109070
\(226\) −8.18052 −0.544160
\(227\) −26.8243 −1.78039 −0.890194 0.455581i \(-0.849432\pi\)
−0.890194 + 0.455581i \(0.849432\pi\)
\(228\) −1.57281 −0.104162
\(229\) 0.0599178 0.00395948 0.00197974 0.999998i \(-0.499370\pi\)
0.00197974 + 0.999998i \(0.499370\pi\)
\(230\) −16.8663 −1.11213
\(231\) 0.198062 0.0130315
\(232\) 9.88794 0.649175
\(233\) 14.7018 0.963149 0.481574 0.876405i \(-0.340065\pi\)
0.481574 + 0.876405i \(0.340065\pi\)
\(234\) 0 0
\(235\) 15.6206 1.01898
\(236\) −11.7846 −0.767112
\(237\) 6.98527 0.453742
\(238\) 0.445042 0.0288478
\(239\) −8.06112 −0.521431 −0.260715 0.965416i \(-0.583958\pi\)
−0.260715 + 0.965416i \(0.583958\pi\)
\(240\) 1.83411 0.118391
\(241\) 1.03927 0.0669456 0.0334728 0.999440i \(-0.489343\pi\)
0.0334728 + 0.999440i \(0.489343\pi\)
\(242\) 10.9608 0.704585
\(243\) 1.00000 0.0641500
\(244\) −10.1849 −0.652024
\(245\) 1.83411 0.117177
\(246\) 12.1234 0.772962
\(247\) 0 0
\(248\) 6.88303 0.437073
\(249\) −3.19757 −0.202638
\(250\) 12.1712 0.769776
\(251\) −18.6029 −1.17421 −0.587103 0.809512i \(-0.699732\pi\)
−0.587103 + 0.809512i \(0.699732\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −1.82136 −0.114508
\(254\) −5.91677 −0.371251
\(255\) 0.816255 0.0511159
\(256\) 1.00000 0.0625000
\(257\) 22.7150 1.41692 0.708462 0.705749i \(-0.249389\pi\)
0.708462 + 0.705749i \(0.249389\pi\)
\(258\) 7.51027 0.467569
\(259\) 5.13552 0.319106
\(260\) 0 0
\(261\) −9.88794 −0.612048
\(262\) −4.83450 −0.298677
\(263\) −16.2280 −1.00066 −0.500329 0.865835i \(-0.666788\pi\)
−0.500329 + 0.865835i \(0.666788\pi\)
\(264\) 0.198062 0.0121899
\(265\) −2.54860 −0.156559
\(266\) −1.57281 −0.0964353
\(267\) 5.55962 0.340243
\(268\) −0.307727 −0.0187974
\(269\) 3.99465 0.243558 0.121779 0.992557i \(-0.461140\pi\)
0.121779 + 0.992557i \(0.461140\pi\)
\(270\) −1.83411 −0.111620
\(271\) 25.3380 1.53917 0.769587 0.638542i \(-0.220462\pi\)
0.769587 + 0.638542i \(0.220462\pi\)
\(272\) 0.445042 0.0269846
\(273\) 0 0
\(274\) −6.26783 −0.378653
\(275\) 0.324039 0.0195403
\(276\) 9.19592 0.553529
\(277\) −18.8962 −1.13536 −0.567680 0.823249i \(-0.692159\pi\)
−0.567680 + 0.823249i \(0.692159\pi\)
\(278\) −13.6770 −0.820295
\(279\) −6.88303 −0.412076
\(280\) 1.83411 0.109609
\(281\) 15.7447 0.939251 0.469625 0.882866i \(-0.344389\pi\)
0.469625 + 0.882866i \(0.344389\pi\)
\(282\) −8.51673 −0.507164
\(283\) −6.46520 −0.384316 −0.192158 0.981364i \(-0.561549\pi\)
−0.192158 + 0.981364i \(0.561549\pi\)
\(284\) −13.0597 −0.774951
\(285\) −2.88471 −0.170875
\(286\) 0 0
\(287\) 12.1234 0.715624
\(288\) −1.00000 −0.0589256
\(289\) −16.8019 −0.988349
\(290\) 18.1356 1.06496
\(291\) −4.95178 −0.290278
\(292\) 7.90088 0.462364
\(293\) 5.37856 0.314219 0.157109 0.987581i \(-0.449782\pi\)
0.157109 + 0.987581i \(0.449782\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −21.6142 −1.25843
\(296\) 5.13552 0.298496
\(297\) −0.198062 −0.0114927
\(298\) 13.3470 0.773171
\(299\) 0 0
\(300\) −1.63605 −0.0944572
\(301\) 7.51027 0.432885
\(302\) 1.21548 0.0699431
\(303\) −0.0853103 −0.00490095
\(304\) −1.57281 −0.0902069
\(305\) −18.6803 −1.06963
\(306\) −0.445042 −0.0254414
\(307\) −26.8264 −1.53106 −0.765532 0.643398i \(-0.777524\pi\)
−0.765532 + 0.643398i \(0.777524\pi\)
\(308\) 0.198062 0.0112856
\(309\) 7.20409 0.409826
\(310\) 12.6242 0.717007
\(311\) 22.2579 1.26213 0.631066 0.775729i \(-0.282618\pi\)
0.631066 + 0.775729i \(0.282618\pi\)
\(312\) 0 0
\(313\) 8.24762 0.466183 0.233091 0.972455i \(-0.425116\pi\)
0.233091 + 0.972455i \(0.425116\pi\)
\(314\) −7.68582 −0.433736
\(315\) −1.83411 −0.103340
\(316\) 6.98527 0.392952
\(317\) 13.7293 0.771112 0.385556 0.922685i \(-0.374010\pi\)
0.385556 + 0.922685i \(0.374010\pi\)
\(318\) 1.38956 0.0779226
\(319\) 1.95843 0.109651
\(320\) 1.83411 0.102530
\(321\) 8.94825 0.499443
\(322\) 9.19592 0.512469
\(323\) −0.699967 −0.0389472
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 12.1104 0.670734
\(327\) −0.615818 −0.0340548
\(328\) 12.1234 0.669405
\(329\) −8.51673 −0.469542
\(330\) 0.363268 0.0199972
\(331\) 17.9130 0.984586 0.492293 0.870430i \(-0.336159\pi\)
0.492293 + 0.870430i \(0.336159\pi\)
\(332\) −3.19757 −0.175489
\(333\) −5.13552 −0.281425
\(334\) 14.8734 0.813837
\(335\) −0.564405 −0.0308368
\(336\) −1.00000 −0.0545545
\(337\) 35.7827 1.94921 0.974604 0.223936i \(-0.0718906\pi\)
0.974604 + 0.223936i \(0.0718906\pi\)
\(338\) 0 0
\(339\) 8.18052 0.444305
\(340\) 0.816255 0.0442676
\(341\) 1.36327 0.0738251
\(342\) 1.57281 0.0850479
\(343\) −1.00000 −0.0539949
\(344\) 7.51027 0.404927
\(345\) 16.8663 0.908052
\(346\) 1.87323 0.100705
\(347\) −20.8470 −1.11913 −0.559563 0.828788i \(-0.689031\pi\)
−0.559563 + 0.828788i \(0.689031\pi\)
\(348\) −9.88794 −0.530049
\(349\) 31.6493 1.69415 0.847073 0.531476i \(-0.178362\pi\)
0.847073 + 0.531476i \(0.178362\pi\)
\(350\) −1.63605 −0.0874503
\(351\) 0 0
\(352\) 0.198062 0.0105568
\(353\) 6.19014 0.329468 0.164734 0.986338i \(-0.447323\pi\)
0.164734 + 0.986338i \(0.447323\pi\)
\(354\) 11.7846 0.626344
\(355\) −23.9529 −1.27129
\(356\) 5.55962 0.294659
\(357\) −0.445042 −0.0235541
\(358\) 8.91285 0.471059
\(359\) −27.9172 −1.47341 −0.736706 0.676213i \(-0.763620\pi\)
−0.736706 + 0.676213i \(0.763620\pi\)
\(360\) −1.83411 −0.0966660
\(361\) −16.5263 −0.869803
\(362\) 13.2448 0.696131
\(363\) −10.9608 −0.575291
\(364\) 0 0
\(365\) 14.4911 0.758497
\(366\) 10.1849 0.532376
\(367\) 2.49317 0.130142 0.0650712 0.997881i \(-0.479273\pi\)
0.0650712 + 0.997881i \(0.479273\pi\)
\(368\) 9.19592 0.479370
\(369\) −12.1234 −0.631121
\(370\) 9.41910 0.489676
\(371\) 1.38956 0.0721423
\(372\) −6.88303 −0.356868
\(373\) −31.8691 −1.65012 −0.825061 0.565044i \(-0.808859\pi\)
−0.825061 + 0.565044i \(0.808859\pi\)
\(374\) 0.0881460 0.00455792
\(375\) −12.1712 −0.628520
\(376\) −8.51673 −0.439217
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) −0.810157 −0.0416149 −0.0208075 0.999784i \(-0.506624\pi\)
−0.0208075 + 0.999784i \(0.506624\pi\)
\(380\) −2.88471 −0.147982
\(381\) 5.91677 0.303125
\(382\) 6.56414 0.335851
\(383\) −25.2437 −1.28989 −0.644946 0.764228i \(-0.723120\pi\)
−0.644946 + 0.764228i \(0.723120\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0.363268 0.0185138
\(386\) −21.6428 −1.10159
\(387\) −7.51027 −0.381769
\(388\) −4.95178 −0.251388
\(389\) 3.03323 0.153791 0.0768954 0.997039i \(-0.475499\pi\)
0.0768954 + 0.997039i \(0.475499\pi\)
\(390\) 0 0
\(391\) 4.09257 0.206970
\(392\) −1.00000 −0.0505076
\(393\) 4.83450 0.243868
\(394\) 0.437589 0.0220454
\(395\) 12.8117 0.644629
\(396\) −0.198062 −0.00995300
\(397\) −14.1365 −0.709490 −0.354745 0.934963i \(-0.615432\pi\)
−0.354745 + 0.934963i \(0.615432\pi\)
\(398\) −24.1946 −1.21276
\(399\) 1.57281 0.0787391
\(400\) −1.63605 −0.0818023
\(401\) 17.6596 0.881878 0.440939 0.897537i \(-0.354645\pi\)
0.440939 + 0.897537i \(0.354645\pi\)
\(402\) 0.307727 0.0153480
\(403\) 0 0
\(404\) −0.0853103 −0.00424435
\(405\) 1.83411 0.0911376
\(406\) −9.88794 −0.490730
\(407\) 1.01715 0.0504184
\(408\) −0.445042 −0.0220329
\(409\) 11.6727 0.577180 0.288590 0.957453i \(-0.406814\pi\)
0.288590 + 0.957453i \(0.406814\pi\)
\(410\) 22.2357 1.09814
\(411\) 6.26783 0.309169
\(412\) 7.20409 0.354920
\(413\) 11.7846 0.579882
\(414\) −9.19592 −0.451955
\(415\) −5.86469 −0.287886
\(416\) 0 0
\(417\) 13.6770 0.669768
\(418\) −0.311515 −0.0152367
\(419\) −31.2971 −1.52896 −0.764482 0.644645i \(-0.777005\pi\)
−0.764482 + 0.644645i \(0.777005\pi\)
\(420\) −1.83411 −0.0894953
\(421\) 32.8881 1.60287 0.801433 0.598085i \(-0.204071\pi\)
0.801433 + 0.598085i \(0.204071\pi\)
\(422\) 11.6272 0.566005
\(423\) 8.51673 0.414098
\(424\) 1.38956 0.0674830
\(425\) −0.728109 −0.0353185
\(426\) 13.0597 0.632745
\(427\) 10.1849 0.492884
\(428\) 8.94825 0.432530
\(429\) 0 0
\(430\) 13.7747 0.664273
\(431\) −37.9406 −1.82754 −0.913768 0.406237i \(-0.866841\pi\)
−0.913768 + 0.406237i \(0.866841\pi\)
\(432\) 1.00000 0.0481125
\(433\) 11.0673 0.531861 0.265931 0.963992i \(-0.414321\pi\)
0.265931 + 0.963992i \(0.414321\pi\)
\(434\) −6.88303 −0.330396
\(435\) −18.1356 −0.869533
\(436\) −0.615818 −0.0294923
\(437\) −14.4634 −0.691881
\(438\) −7.90088 −0.377519
\(439\) 0.955444 0.0456009 0.0228004 0.999740i \(-0.492742\pi\)
0.0228004 + 0.999740i \(0.492742\pi\)
\(440\) 0.363268 0.0173181
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −26.1946 −1.24454 −0.622272 0.782801i \(-0.713790\pi\)
−0.622272 + 0.782801i \(0.713790\pi\)
\(444\) −5.13552 −0.243721
\(445\) 10.1970 0.483382
\(446\) 4.17401 0.197645
\(447\) −13.3470 −0.631292
\(448\) −1.00000 −0.0472456
\(449\) 24.1431 1.13938 0.569691 0.821859i \(-0.307063\pi\)
0.569691 + 0.821859i \(0.307063\pi\)
\(450\) 1.63605 0.0771240
\(451\) 2.40120 0.113068
\(452\) 8.18052 0.384779
\(453\) −1.21548 −0.0571083
\(454\) 26.8243 1.25893
\(455\) 0 0
\(456\) 1.57281 0.0736536
\(457\) 10.6583 0.498574 0.249287 0.968430i \(-0.419804\pi\)
0.249287 + 0.968430i \(0.419804\pi\)
\(458\) −0.0599178 −0.00279977
\(459\) 0.445042 0.0207728
\(460\) 16.8663 0.786396
\(461\) −26.8656 −1.25126 −0.625628 0.780122i \(-0.715157\pi\)
−0.625628 + 0.780122i \(0.715157\pi\)
\(462\) −0.198062 −0.00921469
\(463\) 17.0698 0.793301 0.396650 0.917970i \(-0.370173\pi\)
0.396650 + 0.917970i \(0.370173\pi\)
\(464\) −9.88794 −0.459036
\(465\) −12.6242 −0.585434
\(466\) −14.7018 −0.681049
\(467\) −32.1453 −1.48751 −0.743754 0.668454i \(-0.766956\pi\)
−0.743754 + 0.668454i \(0.766956\pi\)
\(468\) 0 0
\(469\) 0.307727 0.0142095
\(470\) −15.6206 −0.720525
\(471\) 7.68582 0.354144
\(472\) 11.7846 0.542430
\(473\) 1.48750 0.0683954
\(474\) −6.98527 −0.320844
\(475\) 2.57319 0.118066
\(476\) −0.445042 −0.0203985
\(477\) −1.38956 −0.0636235
\(478\) 8.06112 0.368707
\(479\) −22.1790 −1.01338 −0.506692 0.862127i \(-0.669132\pi\)
−0.506692 + 0.862127i \(0.669132\pi\)
\(480\) −1.83411 −0.0837152
\(481\) 0 0
\(482\) −1.03927 −0.0473377
\(483\) −9.19592 −0.418429
\(484\) −10.9608 −0.498217
\(485\) −9.08210 −0.412397
\(486\) −1.00000 −0.0453609
\(487\) −13.8814 −0.629024 −0.314512 0.949253i \(-0.601841\pi\)
−0.314512 + 0.949253i \(0.601841\pi\)
\(488\) 10.1849 0.461051
\(489\) −12.1104 −0.547652
\(490\) −1.83411 −0.0828566
\(491\) 12.7715 0.576369 0.288185 0.957575i \(-0.406948\pi\)
0.288185 + 0.957575i \(0.406948\pi\)
\(492\) −12.1234 −0.546567
\(493\) −4.40055 −0.198191
\(494\) 0 0
\(495\) −0.363268 −0.0163277
\(496\) −6.88303 −0.309057
\(497\) 13.0597 0.585808
\(498\) 3.19757 0.143287
\(499\) −17.0569 −0.763573 −0.381787 0.924251i \(-0.624691\pi\)
−0.381787 + 0.924251i \(0.624691\pi\)
\(500\) −12.1712 −0.544314
\(501\) −14.8734 −0.664495
\(502\) 18.6029 0.830289
\(503\) 14.4879 0.645985 0.322992 0.946402i \(-0.395311\pi\)
0.322992 + 0.946402i \(0.395311\pi\)
\(504\) 1.00000 0.0445435
\(505\) −0.156468 −0.00696275
\(506\) 1.82136 0.0809695
\(507\) 0 0
\(508\) 5.91677 0.262514
\(509\) 3.77985 0.167539 0.0837695 0.996485i \(-0.473304\pi\)
0.0837695 + 0.996485i \(0.473304\pi\)
\(510\) −0.816255 −0.0361444
\(511\) −7.90088 −0.349514
\(512\) −1.00000 −0.0441942
\(513\) −1.57281 −0.0694413
\(514\) −22.7150 −1.00192
\(515\) 13.2131 0.582238
\(516\) −7.51027 −0.330621
\(517\) −1.68684 −0.0741873
\(518\) −5.13552 −0.225642
\(519\) −1.87323 −0.0822255
\(520\) 0 0
\(521\) 6.59714 0.289026 0.144513 0.989503i \(-0.453838\pi\)
0.144513 + 0.989503i \(0.453838\pi\)
\(522\) 9.88794 0.432783
\(523\) −24.8111 −1.08491 −0.542456 0.840084i \(-0.682505\pi\)
−0.542456 + 0.840084i \(0.682505\pi\)
\(524\) 4.83450 0.211196
\(525\) 1.63605 0.0714029
\(526\) 16.2280 0.707572
\(527\) −3.06323 −0.133437
\(528\) −0.198062 −0.00861955
\(529\) 61.5650 2.67674
\(530\) 2.54860 0.110704
\(531\) −11.7846 −0.511408
\(532\) 1.57281 0.0681900
\(533\) 0 0
\(534\) −5.55962 −0.240588
\(535\) 16.4121 0.709556
\(536\) 0.307727 0.0132918
\(537\) −8.91285 −0.384618
\(538\) −3.99465 −0.172222
\(539\) −0.198062 −0.00853115
\(540\) 1.83411 0.0789275
\(541\) −30.4570 −1.30945 −0.654724 0.755868i \(-0.727215\pi\)
−0.654724 + 0.755868i \(0.727215\pi\)
\(542\) −25.3380 −1.08836
\(543\) −13.2448 −0.568389
\(544\) −0.445042 −0.0190810
\(545\) −1.12948 −0.0483815
\(546\) 0 0
\(547\) 32.5691 1.39255 0.696277 0.717773i \(-0.254838\pi\)
0.696277 + 0.717773i \(0.254838\pi\)
\(548\) 6.26783 0.267748
\(549\) −10.1849 −0.434683
\(550\) −0.324039 −0.0138171
\(551\) 15.5519 0.662532
\(552\) −9.19592 −0.391404
\(553\) −6.98527 −0.297044
\(554\) 18.8962 0.802821
\(555\) −9.41910 −0.399819
\(556\) 13.6770 0.580036
\(557\) 20.8644 0.884052 0.442026 0.897002i \(-0.354260\pi\)
0.442026 + 0.897002i \(0.354260\pi\)
\(558\) 6.88303 0.291382
\(559\) 0 0
\(560\) −1.83411 −0.0775052
\(561\) −0.0881460 −0.00372153
\(562\) −15.7447 −0.664150
\(563\) 11.9854 0.505124 0.252562 0.967581i \(-0.418727\pi\)
0.252562 + 0.967581i \(0.418727\pi\)
\(564\) 8.51673 0.358619
\(565\) 15.0040 0.631222
\(566\) 6.46520 0.271753
\(567\) −1.00000 −0.0419961
\(568\) 13.0597 0.547973
\(569\) 5.07479 0.212746 0.106373 0.994326i \(-0.466076\pi\)
0.106373 + 0.994326i \(0.466076\pi\)
\(570\) 2.88471 0.120827
\(571\) 5.93197 0.248245 0.124123 0.992267i \(-0.460388\pi\)
0.124123 + 0.992267i \(0.460388\pi\)
\(572\) 0 0
\(573\) −6.56414 −0.274221
\(574\) −12.1234 −0.506023
\(575\) −15.0450 −0.627418
\(576\) 1.00000 0.0416667
\(577\) 10.1542 0.422724 0.211362 0.977408i \(-0.432210\pi\)
0.211362 + 0.977408i \(0.432210\pi\)
\(578\) 16.8019 0.698868
\(579\) 21.6428 0.899442
\(580\) −18.1356 −0.753038
\(581\) 3.19757 0.132658
\(582\) 4.95178 0.205258
\(583\) 0.275219 0.0113984
\(584\) −7.90088 −0.326941
\(585\) 0 0
\(586\) −5.37856 −0.222186
\(587\) 20.6043 0.850432 0.425216 0.905092i \(-0.360198\pi\)
0.425216 + 0.905092i \(0.360198\pi\)
\(588\) 1.00000 0.0412393
\(589\) 10.8257 0.446065
\(590\) 21.6142 0.889843
\(591\) −0.437589 −0.0180000
\(592\) −5.13552 −0.211069
\(593\) −14.2590 −0.585548 −0.292774 0.956182i \(-0.594578\pi\)
−0.292774 + 0.956182i \(0.594578\pi\)
\(594\) 0.198062 0.00812659
\(595\) −0.816255 −0.0334632
\(596\) −13.3470 −0.546715
\(597\) 24.1946 0.990217
\(598\) 0 0
\(599\) −5.47672 −0.223773 −0.111886 0.993721i \(-0.535689\pi\)
−0.111886 + 0.993721i \(0.535689\pi\)
\(600\) 1.63605 0.0667913
\(601\) −30.9403 −1.26208 −0.631041 0.775750i \(-0.717372\pi\)
−0.631041 + 0.775750i \(0.717372\pi\)
\(602\) −7.51027 −0.306096
\(603\) −0.307727 −0.0125316
\(604\) −1.21548 −0.0494572
\(605\) −20.1032 −0.817313
\(606\) 0.0853103 0.00346549
\(607\) 6.80757 0.276311 0.138155 0.990411i \(-0.455883\pi\)
0.138155 + 0.990411i \(0.455883\pi\)
\(608\) 1.57281 0.0637859
\(609\) 9.88794 0.400680
\(610\) 18.6803 0.756343
\(611\) 0 0
\(612\) 0.445042 0.0179898
\(613\) −24.2320 −0.978720 −0.489360 0.872082i \(-0.662770\pi\)
−0.489360 + 0.872082i \(0.662770\pi\)
\(614\) 26.8264 1.08263
\(615\) −22.2357 −0.896630
\(616\) −0.198062 −0.00798016
\(617\) 34.1377 1.37433 0.687165 0.726501i \(-0.258855\pi\)
0.687165 + 0.726501i \(0.258855\pi\)
\(618\) −7.20409 −0.289791
\(619\) −12.5873 −0.505925 −0.252963 0.967476i \(-0.581405\pi\)
−0.252963 + 0.967476i \(0.581405\pi\)
\(620\) −12.6242 −0.507001
\(621\) 9.19592 0.369020
\(622\) −22.2579 −0.892462
\(623\) −5.55962 −0.222742
\(624\) 0 0
\(625\) −14.1431 −0.565725
\(626\) −8.24762 −0.329641
\(627\) 0.311515 0.0124407
\(628\) 7.68582 0.306698
\(629\) −2.28552 −0.0911297
\(630\) 1.83411 0.0730726
\(631\) 39.8142 1.58498 0.792489 0.609887i \(-0.208785\pi\)
0.792489 + 0.609887i \(0.208785\pi\)
\(632\) −6.98527 −0.277859
\(633\) −11.6272 −0.462141
\(634\) −13.7293 −0.545258
\(635\) 10.8520 0.430648
\(636\) −1.38956 −0.0550996
\(637\) 0 0
\(638\) −1.95843 −0.0775349
\(639\) −13.0597 −0.516634
\(640\) −1.83411 −0.0724995
\(641\) −4.70878 −0.185986 −0.0929928 0.995667i \(-0.529643\pi\)
−0.0929928 + 0.995667i \(0.529643\pi\)
\(642\) −8.94825 −0.353159
\(643\) 36.1977 1.42750 0.713749 0.700402i \(-0.246996\pi\)
0.713749 + 0.700402i \(0.246996\pi\)
\(644\) −9.19592 −0.362370
\(645\) −13.7747 −0.542376
\(646\) 0.699967 0.0275398
\(647\) 0.587951 0.0231147 0.0115574 0.999933i \(-0.496321\pi\)
0.0115574 + 0.999933i \(0.496321\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 2.33408 0.0916208
\(650\) 0 0
\(651\) 6.88303 0.269767
\(652\) −12.1104 −0.474281
\(653\) −15.0839 −0.590279 −0.295139 0.955454i \(-0.595366\pi\)
−0.295139 + 0.955454i \(0.595366\pi\)
\(654\) 0.615818 0.0240804
\(655\) 8.86700 0.346463
\(656\) −12.1234 −0.473341
\(657\) 7.90088 0.308243
\(658\) 8.51673 0.332017
\(659\) −26.3996 −1.02838 −0.514191 0.857676i \(-0.671908\pi\)
−0.514191 + 0.857676i \(0.671908\pi\)
\(660\) −0.363268 −0.0141402
\(661\) 49.5192 1.92607 0.963037 0.269369i \(-0.0868151\pi\)
0.963037 + 0.269369i \(0.0868151\pi\)
\(662\) −17.9130 −0.696207
\(663\) 0 0
\(664\) 3.19757 0.124090
\(665\) 2.88471 0.111864
\(666\) 5.13552 0.198997
\(667\) −90.9287 −3.52077
\(668\) −14.8734 −0.575469
\(669\) −4.17401 −0.161376
\(670\) 0.564405 0.0218049
\(671\) 2.01725 0.0778752
\(672\) 1.00000 0.0385758
\(673\) −20.4206 −0.787156 −0.393578 0.919291i \(-0.628763\pi\)
−0.393578 + 0.919291i \(0.628763\pi\)
\(674\) −35.7827 −1.37830
\(675\) −1.63605 −0.0629714
\(676\) 0 0
\(677\) −35.0603 −1.34748 −0.673738 0.738970i \(-0.735312\pi\)
−0.673738 + 0.738970i \(0.735312\pi\)
\(678\) −8.18052 −0.314171
\(679\) 4.95178 0.190032
\(680\) −0.816255 −0.0313020
\(681\) −26.8243 −1.02791
\(682\) −1.36327 −0.0522022
\(683\) −24.0064 −0.918579 −0.459289 0.888287i \(-0.651896\pi\)
−0.459289 + 0.888287i \(0.651896\pi\)
\(684\) −1.57281 −0.0601380
\(685\) 11.4959 0.439235
\(686\) 1.00000 0.0381802
\(687\) 0.0599178 0.00228601
\(688\) −7.51027 −0.286326
\(689\) 0 0
\(690\) −16.8663 −0.642090
\(691\) 5.29898 0.201583 0.100791 0.994908i \(-0.467863\pi\)
0.100791 + 0.994908i \(0.467863\pi\)
\(692\) −1.87323 −0.0712093
\(693\) 0.198062 0.00752376
\(694\) 20.8470 0.791341
\(695\) 25.0852 0.951536
\(696\) 9.88794 0.374801
\(697\) −5.39544 −0.204367
\(698\) −31.6493 −1.19794
\(699\) 14.7018 0.556074
\(700\) 1.63605 0.0618367
\(701\) 47.1066 1.77919 0.889596 0.456749i \(-0.150986\pi\)
0.889596 + 0.456749i \(0.150986\pi\)
\(702\) 0 0
\(703\) 8.07721 0.304638
\(704\) −0.198062 −0.00746475
\(705\) 15.6206 0.588306
\(706\) −6.19014 −0.232969
\(707\) 0.0853103 0.00320842
\(708\) −11.7846 −0.442892
\(709\) 48.5978 1.82513 0.912565 0.408932i \(-0.134099\pi\)
0.912565 + 0.408932i \(0.134099\pi\)
\(710\) 23.9529 0.898937
\(711\) 6.98527 0.261968
\(712\) −5.55962 −0.208356
\(713\) −63.2958 −2.37044
\(714\) 0.445042 0.0166553
\(715\) 0 0
\(716\) −8.91285 −0.333089
\(717\) −8.06112 −0.301048
\(718\) 27.9172 1.04186
\(719\) 1.43382 0.0534725 0.0267363 0.999643i \(-0.491489\pi\)
0.0267363 + 0.999643i \(0.491489\pi\)
\(720\) 1.83411 0.0683532
\(721\) −7.20409 −0.268294
\(722\) 16.5263 0.615044
\(723\) 1.03927 0.0386510
\(724\) −13.2448 −0.492239
\(725\) 16.1771 0.600804
\(726\) 10.9608 0.406792
\(727\) −10.0393 −0.372338 −0.186169 0.982518i \(-0.559607\pi\)
−0.186169 + 0.982518i \(0.559607\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −14.4911 −0.536339
\(731\) −3.34238 −0.123623
\(732\) −10.1849 −0.376446
\(733\) 11.2968 0.417257 0.208628 0.977995i \(-0.433100\pi\)
0.208628 + 0.977995i \(0.433100\pi\)
\(734\) −2.49317 −0.0920246
\(735\) 1.83411 0.0676521
\(736\) −9.19592 −0.338966
\(737\) 0.0609492 0.00224509
\(738\) 12.1234 0.446270
\(739\) −22.9571 −0.844489 −0.422245 0.906482i \(-0.638758\pi\)
−0.422245 + 0.906482i \(0.638758\pi\)
\(740\) −9.41910 −0.346253
\(741\) 0 0
\(742\) −1.38956 −0.0510123
\(743\) 47.1372 1.72930 0.864649 0.502377i \(-0.167541\pi\)
0.864649 + 0.502377i \(0.167541\pi\)
\(744\) 6.88303 0.252344
\(745\) −24.4798 −0.896872
\(746\) 31.8691 1.16681
\(747\) −3.19757 −0.116993
\(748\) −0.0881460 −0.00322294
\(749\) −8.94825 −0.326962
\(750\) 12.1712 0.444430
\(751\) 4.33483 0.158180 0.0790901 0.996867i \(-0.474799\pi\)
0.0790901 + 0.996867i \(0.474799\pi\)
\(752\) 8.51673 0.310573
\(753\) −18.6029 −0.677928
\(754\) 0 0
\(755\) −2.22932 −0.0811334
\(756\) −1.00000 −0.0363696
\(757\) −51.5048 −1.87197 −0.935987 0.352034i \(-0.885490\pi\)
−0.935987 + 0.352034i \(0.885490\pi\)
\(758\) 0.810157 0.0294262
\(759\) −1.82136 −0.0661114
\(760\) 2.88471 0.104639
\(761\) −15.3697 −0.557153 −0.278576 0.960414i \(-0.589863\pi\)
−0.278576 + 0.960414i \(0.589863\pi\)
\(762\) −5.91677 −0.214342
\(763\) 0.615818 0.0222941
\(764\) −6.56414 −0.237482
\(765\) 0.816255 0.0295118
\(766\) 25.2437 0.912092
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −22.3499 −0.805958 −0.402979 0.915209i \(-0.632025\pi\)
−0.402979 + 0.915209i \(0.632025\pi\)
\(770\) −0.363268 −0.0130913
\(771\) 22.7150 0.818062
\(772\) 21.6428 0.778940
\(773\) −4.13659 −0.148783 −0.0743913 0.997229i \(-0.523701\pi\)
−0.0743913 + 0.997229i \(0.523701\pi\)
\(774\) 7.51027 0.269951
\(775\) 11.2609 0.404505
\(776\) 4.95178 0.177758
\(777\) 5.13552 0.184236
\(778\) −3.03323 −0.108746
\(779\) 19.0679 0.683178
\(780\) 0 0
\(781\) 2.58664 0.0925571
\(782\) −4.09257 −0.146350
\(783\) −9.88794 −0.353366
\(784\) 1.00000 0.0357143
\(785\) 14.0966 0.503130
\(786\) −4.83450 −0.172441
\(787\) −18.9440 −0.675279 −0.337640 0.941275i \(-0.609628\pi\)
−0.337640 + 0.941275i \(0.609628\pi\)
\(788\) −0.437589 −0.0155885
\(789\) −16.2280 −0.577730
\(790\) −12.8117 −0.455821
\(791\) −8.18052 −0.290866
\(792\) 0.198062 0.00703784
\(793\) 0 0
\(794\) 14.1365 0.501685
\(795\) −2.54860 −0.0903896
\(796\) 24.1946 0.857553
\(797\) −16.7957 −0.594934 −0.297467 0.954732i \(-0.596142\pi\)
−0.297467 + 0.954732i \(0.596142\pi\)
\(798\) −1.57281 −0.0556769
\(799\) 3.79030 0.134091
\(800\) 1.63605 0.0578430
\(801\) 5.55962 0.196440
\(802\) −17.6596 −0.623582
\(803\) −1.56487 −0.0552229
\(804\) −0.307727 −0.0108527
\(805\) −16.8663 −0.594460
\(806\) 0 0
\(807\) 3.99465 0.140618
\(808\) 0.0853103 0.00300121
\(809\) −14.9738 −0.526450 −0.263225 0.964734i \(-0.584786\pi\)
−0.263225 + 0.964734i \(0.584786\pi\)
\(810\) −1.83411 −0.0644440
\(811\) 41.5904 1.46044 0.730218 0.683214i \(-0.239418\pi\)
0.730218 + 0.683214i \(0.239418\pi\)
\(812\) 9.88794 0.346999
\(813\) 25.3380 0.888643
\(814\) −1.01715 −0.0356512
\(815\) −22.2118 −0.778046
\(816\) 0.445042 0.0155796
\(817\) 11.8122 0.413258
\(818\) −11.6727 −0.408128
\(819\) 0 0
\(820\) −22.2357 −0.776504
\(821\) −12.6711 −0.442225 −0.221112 0.975248i \(-0.570969\pi\)
−0.221112 + 0.975248i \(0.570969\pi\)
\(822\) −6.26783 −0.218616
\(823\) 19.9118 0.694082 0.347041 0.937850i \(-0.387186\pi\)
0.347041 + 0.937850i \(0.387186\pi\)
\(824\) −7.20409 −0.250966
\(825\) 0.324039 0.0112816
\(826\) −11.7846 −0.410038
\(827\) 23.9099 0.831430 0.415715 0.909495i \(-0.363531\pi\)
0.415715 + 0.909495i \(0.363531\pi\)
\(828\) 9.19592 0.319580
\(829\) −23.7810 −0.825949 −0.412975 0.910743i \(-0.635510\pi\)
−0.412975 + 0.910743i \(0.635510\pi\)
\(830\) 5.86469 0.203566
\(831\) −18.8962 −0.655501
\(832\) 0 0
\(833\) 0.445042 0.0154198
\(834\) −13.6770 −0.473598
\(835\) −27.2794 −0.944044
\(836\) 0.311515 0.0107740
\(837\) −6.88303 −0.237912
\(838\) 31.2971 1.08114
\(839\) −52.1839 −1.80159 −0.900794 0.434247i \(-0.857015\pi\)
−0.900794 + 0.434247i \(0.857015\pi\)
\(840\) 1.83411 0.0632828
\(841\) 68.7714 2.37143
\(842\) −32.8881 −1.13340
\(843\) 15.7447 0.542277
\(844\) −11.6272 −0.400226
\(845\) 0 0
\(846\) −8.51673 −0.292811
\(847\) 10.9608 0.376617
\(848\) −1.38956 −0.0477177
\(849\) −6.46520 −0.221885
\(850\) 0.728109 0.0249739
\(851\) −47.2258 −1.61888
\(852\) −13.0597 −0.447418
\(853\) −8.03941 −0.275264 −0.137632 0.990483i \(-0.543949\pi\)
−0.137632 + 0.990483i \(0.543949\pi\)
\(854\) −10.1849 −0.348522
\(855\) −2.88471 −0.0986549
\(856\) −8.94825 −0.305845
\(857\) −46.1043 −1.57489 −0.787447 0.616382i \(-0.788598\pi\)
−0.787447 + 0.616382i \(0.788598\pi\)
\(858\) 0 0
\(859\) 52.6147 1.79519 0.897595 0.440821i \(-0.145313\pi\)
0.897595 + 0.440821i \(0.145313\pi\)
\(860\) −13.7747 −0.469712
\(861\) 12.1234 0.413166
\(862\) 37.9406 1.29226
\(863\) −12.4191 −0.422752 −0.211376 0.977405i \(-0.567794\pi\)
−0.211376 + 0.977405i \(0.567794\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −3.43570 −0.116817
\(866\) −11.0673 −0.376083
\(867\) −16.8019 −0.570624
\(868\) 6.88303 0.233625
\(869\) −1.38352 −0.0469326
\(870\) 18.1356 0.614853
\(871\) 0 0
\(872\) 0.615818 0.0208542
\(873\) −4.95178 −0.167592
\(874\) 14.4634 0.489233
\(875\) 12.1712 0.411463
\(876\) 7.90088 0.266946
\(877\) 40.2554 1.35933 0.679665 0.733523i \(-0.262125\pi\)
0.679665 + 0.733523i \(0.262125\pi\)
\(878\) −0.955444 −0.0322447
\(879\) 5.37856 0.181414
\(880\) −0.363268 −0.0122458
\(881\) 19.3338 0.651372 0.325686 0.945478i \(-0.394405\pi\)
0.325686 + 0.945478i \(0.394405\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −47.3204 −1.59246 −0.796230 0.604994i \(-0.793175\pi\)
−0.796230 + 0.604994i \(0.793175\pi\)
\(884\) 0 0
\(885\) −21.6142 −0.726554
\(886\) 26.1946 0.880025
\(887\) −30.3048 −1.01754 −0.508768 0.860904i \(-0.669899\pi\)
−0.508768 + 0.860904i \(0.669899\pi\)
\(888\) 5.13552 0.172337
\(889\) −5.91677 −0.198442
\(890\) −10.1970 −0.341803
\(891\) −0.198062 −0.00663534
\(892\) −4.17401 −0.139756
\(893\) −13.3952 −0.448254
\(894\) 13.3470 0.446391
\(895\) −16.3471 −0.546425
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −24.1431 −0.805665
\(899\) 68.0590 2.26989
\(900\) −1.63605 −0.0545349
\(901\) −0.618412 −0.0206023
\(902\) −2.40120 −0.0799511
\(903\) 7.51027 0.249926
\(904\) −8.18052 −0.272080
\(905\) −24.2924 −0.807507
\(906\) 1.21548 0.0403816
\(907\) −23.5100 −0.780636 −0.390318 0.920680i \(-0.627635\pi\)
−0.390318 + 0.920680i \(0.627635\pi\)
\(908\) −26.8243 −0.890194
\(909\) −0.0853103 −0.00282956
\(910\) 0 0
\(911\) −51.9982 −1.72278 −0.861389 0.507946i \(-0.830405\pi\)
−0.861389 + 0.507946i \(0.830405\pi\)
\(912\) −1.57281 −0.0520810
\(913\) 0.633318 0.0209598
\(914\) −10.6583 −0.352545
\(915\) −18.6803 −0.617552
\(916\) 0.0599178 0.00197974
\(917\) −4.83450 −0.159649
\(918\) −0.445042 −0.0146886
\(919\) 23.4363 0.773091 0.386546 0.922270i \(-0.373668\pi\)
0.386546 + 0.922270i \(0.373668\pi\)
\(920\) −16.8663 −0.556066
\(921\) −26.8264 −0.883960
\(922\) 26.8656 0.884771
\(923\) 0 0
\(924\) 0.198062 0.00651577
\(925\) 8.40195 0.276254
\(926\) −17.0698 −0.560948
\(927\) 7.20409 0.236613
\(928\) 9.88794 0.324588
\(929\) 0.112654 0.00369607 0.00184804 0.999998i \(-0.499412\pi\)
0.00184804 + 0.999998i \(0.499412\pi\)
\(930\) 12.6242 0.413964
\(931\) −1.57281 −0.0515468
\(932\) 14.7018 0.481574
\(933\) 22.2579 0.728692
\(934\) 32.1453 1.05183
\(935\) −0.161669 −0.00528715
\(936\) 0 0
\(937\) −48.1961 −1.57450 −0.787249 0.616636i \(-0.788495\pi\)
−0.787249 + 0.616636i \(0.788495\pi\)
\(938\) −0.307727 −0.0100477
\(939\) 8.24762 0.269151
\(940\) 15.6206 0.509488
\(941\) −49.5239 −1.61443 −0.807217 0.590255i \(-0.799027\pi\)
−0.807217 + 0.590255i \(0.799027\pi\)
\(942\) −7.68582 −0.250418
\(943\) −111.486 −3.63049
\(944\) −11.7846 −0.383556
\(945\) −1.83411 −0.0596636
\(946\) −1.48750 −0.0483628
\(947\) −28.1767 −0.915620 −0.457810 0.889050i \(-0.651366\pi\)
−0.457810 + 0.889050i \(0.651366\pi\)
\(948\) 6.98527 0.226871
\(949\) 0 0
\(950\) −2.57319 −0.0834854
\(951\) 13.7293 0.445202
\(952\) 0.445042 0.0144239
\(953\) −55.7822 −1.80696 −0.903481 0.428627i \(-0.858997\pi\)
−0.903481 + 0.428627i \(0.858997\pi\)
\(954\) 1.38956 0.0449886
\(955\) −12.0393 −0.389584
\(956\) −8.06112 −0.260715
\(957\) 1.95843 0.0633070
\(958\) 22.1790 0.716571
\(959\) −6.26783 −0.202399
\(960\) 1.83411 0.0591956
\(961\) 16.3760 0.528259
\(962\) 0 0
\(963\) 8.94825 0.288353
\(964\) 1.03927 0.0334728
\(965\) 39.6952 1.27783
\(966\) 9.19592 0.295874
\(967\) −19.7898 −0.636396 −0.318198 0.948024i \(-0.603078\pi\)
−0.318198 + 0.948024i \(0.603078\pi\)
\(968\) 10.9608 0.352293
\(969\) −0.699967 −0.0224862
\(970\) 9.08210 0.291609
\(971\) 1.89350 0.0607653 0.0303826 0.999538i \(-0.490327\pi\)
0.0303826 + 0.999538i \(0.490327\pi\)
\(972\) 1.00000 0.0320750
\(973\) −13.6770 −0.438466
\(974\) 13.8814 0.444787
\(975\) 0 0
\(976\) −10.1849 −0.326012
\(977\) −3.77878 −0.120894 −0.0604470 0.998171i \(-0.519253\pi\)
−0.0604470 + 0.998171i \(0.519253\pi\)
\(978\) 12.1104 0.387249
\(979\) −1.10115 −0.0351930
\(980\) 1.83411 0.0585884
\(981\) −0.615818 −0.0196615
\(982\) −12.7715 −0.407555
\(983\) −6.78621 −0.216447 −0.108223 0.994127i \(-0.534516\pi\)
−0.108223 + 0.994127i \(0.534516\pi\)
\(984\) 12.1234 0.386481
\(985\) −0.802586 −0.0255725
\(986\) 4.40055 0.140142
\(987\) −8.51673 −0.271090
\(988\) 0 0
\(989\) −69.0639 −2.19610
\(990\) 0.363268 0.0115454
\(991\) −22.6239 −0.718672 −0.359336 0.933208i \(-0.616997\pi\)
−0.359336 + 0.933208i \(0.616997\pi\)
\(992\) 6.88303 0.218536
\(993\) 17.9130 0.568451
\(994\) −13.0597 −0.414229
\(995\) 44.3754 1.40680
\(996\) −3.19757 −0.101319
\(997\) 37.3318 1.18231 0.591155 0.806558i \(-0.298672\pi\)
0.591155 + 0.806558i \(0.298672\pi\)
\(998\) 17.0569 0.539928
\(999\) −5.13552 −0.162481
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.cr.1.4 6
13.12 even 2 7098.2.a.ct.1.3 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7098.2.a.cr.1.4 6 1.1 even 1 trivial
7098.2.a.ct.1.3 yes 6 13.12 even 2