Properties

Label 847.2.a.g.1.2
Level $847$
Weight $2$
Character 847.1
Self dual yes
Analytic conductor $6.763$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 847.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.30278 q^{2} -2.30278 q^{3} +3.30278 q^{4} +3.60555 q^{5} -5.30278 q^{6} -1.00000 q^{7} +3.00000 q^{8} +2.30278 q^{9} +O(q^{10})\) \(q+2.30278 q^{2} -2.30278 q^{3} +3.30278 q^{4} +3.60555 q^{5} -5.30278 q^{6} -1.00000 q^{7} +3.00000 q^{8} +2.30278 q^{9} +8.30278 q^{10} -7.60555 q^{12} +6.60555 q^{13} -2.30278 q^{14} -8.30278 q^{15} +0.302776 q^{16} +2.69722 q^{17} +5.30278 q^{18} -3.00000 q^{19} +11.9083 q^{20} +2.30278 q^{21} -2.69722 q^{23} -6.90833 q^{24} +8.00000 q^{25} +15.2111 q^{26} +1.60555 q^{27} -3.30278 q^{28} +4.69722 q^{29} -19.1194 q^{30} +1.00000 q^{31} -5.30278 q^{32} +6.21110 q^{34} -3.60555 q^{35} +7.60555 q^{36} -5.21110 q^{37} -6.90833 q^{38} -15.2111 q^{39} +10.8167 q^{40} +7.00000 q^{41} +5.30278 q^{42} +1.69722 q^{43} +8.30278 q^{45} -6.21110 q^{46} -1.90833 q^{47} -0.697224 q^{48} +1.00000 q^{49} +18.4222 q^{50} -6.21110 q^{51} +21.8167 q^{52} -12.9083 q^{53} +3.69722 q^{54} -3.00000 q^{56} +6.90833 q^{57} +10.8167 q^{58} +6.69722 q^{59} -27.4222 q^{60} -4.30278 q^{61} +2.30278 q^{62} -2.30278 q^{63} -12.8167 q^{64} +23.8167 q^{65} +8.51388 q^{67} +8.90833 q^{68} +6.21110 q^{69} -8.30278 q^{70} -4.30278 q^{71} +6.90833 q^{72} -5.00000 q^{73} -12.0000 q^{74} -18.4222 q^{75} -9.90833 q^{76} -35.0278 q^{78} -8.30278 q^{79} +1.09167 q^{80} -10.6056 q^{81} +16.1194 q^{82} +3.00000 q^{83} +7.60555 q^{84} +9.72498 q^{85} +3.90833 q^{86} -10.8167 q^{87} -14.7250 q^{89} +19.1194 q^{90} -6.60555 q^{91} -8.90833 q^{92} -2.30278 q^{93} -4.39445 q^{94} -10.8167 q^{95} +12.2111 q^{96} -3.60555 q^{97} +2.30278 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} + 3 q^{4} - 7 q^{6} - 2 q^{7} + 6 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{3} + 3 q^{4} - 7 q^{6} - 2 q^{7} + 6 q^{8} + q^{9} + 13 q^{10} - 8 q^{12} + 6 q^{13} - q^{14} - 13 q^{15} - 3 q^{16} + 9 q^{17} + 7 q^{18} - 6 q^{19} + 13 q^{20} + q^{21} - 9 q^{23} - 3 q^{24} + 16 q^{25} + 16 q^{26} - 4 q^{27} - 3 q^{28} + 13 q^{29} - 13 q^{30} + 2 q^{31} - 7 q^{32} - 2 q^{34} + 8 q^{36} + 4 q^{37} - 3 q^{38} - 16 q^{39} + 14 q^{41} + 7 q^{42} + 7 q^{43} + 13 q^{45} + 2 q^{46} + 7 q^{47} - 5 q^{48} + 2 q^{49} + 8 q^{50} + 2 q^{51} + 22 q^{52} - 15 q^{53} + 11 q^{54} - 6 q^{56} + 3 q^{57} + 17 q^{59} - 26 q^{60} - 5 q^{61} + q^{62} - q^{63} - 4 q^{64} + 26 q^{65} - q^{67} + 7 q^{68} - 2 q^{69} - 13 q^{70} - 5 q^{71} + 3 q^{72} - 10 q^{73} - 24 q^{74} - 8 q^{75} - 9 q^{76} - 34 q^{78} - 13 q^{79} + 13 q^{80} - 14 q^{81} + 7 q^{82} + 6 q^{83} + 8 q^{84} - 13 q^{85} - 3 q^{86} + 3 q^{89} + 13 q^{90} - 6 q^{91} - 7 q^{92} - q^{93} - 16 q^{94} + 10 q^{96} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.30278 1.62831 0.814154 0.580649i \(-0.197201\pi\)
0.814154 + 0.580649i \(0.197201\pi\)
\(3\) −2.30278 −1.32951 −0.664754 0.747062i \(-0.731464\pi\)
−0.664754 + 0.747062i \(0.731464\pi\)
\(4\) 3.30278 1.65139
\(5\) 3.60555 1.61245 0.806226 0.591608i \(-0.201507\pi\)
0.806226 + 0.591608i \(0.201507\pi\)
\(6\) −5.30278 −2.16485
\(7\) −1.00000 −0.377964
\(8\) 3.00000 1.06066
\(9\) 2.30278 0.767592
\(10\) 8.30278 2.62557
\(11\) 0 0
\(12\) −7.60555 −2.19553
\(13\) 6.60555 1.83205 0.916025 0.401121i \(-0.131379\pi\)
0.916025 + 0.401121i \(0.131379\pi\)
\(14\) −2.30278 −0.615443
\(15\) −8.30278 −2.14377
\(16\) 0.302776 0.0756939
\(17\) 2.69722 0.654173 0.327086 0.944994i \(-0.393933\pi\)
0.327086 + 0.944994i \(0.393933\pi\)
\(18\) 5.30278 1.24988
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 11.9083 2.66278
\(21\) 2.30278 0.502507
\(22\) 0 0
\(23\) −2.69722 −0.562410 −0.281205 0.959648i \(-0.590734\pi\)
−0.281205 + 0.959648i \(0.590734\pi\)
\(24\) −6.90833 −1.41016
\(25\) 8.00000 1.60000
\(26\) 15.2111 2.98314
\(27\) 1.60555 0.308988
\(28\) −3.30278 −0.624166
\(29\) 4.69722 0.872253 0.436126 0.899885i \(-0.356350\pi\)
0.436126 + 0.899885i \(0.356350\pi\)
\(30\) −19.1194 −3.49071
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) −5.30278 −0.937407
\(33\) 0 0
\(34\) 6.21110 1.06520
\(35\) −3.60555 −0.609449
\(36\) 7.60555 1.26759
\(37\) −5.21110 −0.856700 −0.428350 0.903613i \(-0.640905\pi\)
−0.428350 + 0.903613i \(0.640905\pi\)
\(38\) −6.90833 −1.12068
\(39\) −15.2111 −2.43573
\(40\) 10.8167 1.71026
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 5.30278 0.818236
\(43\) 1.69722 0.258824 0.129412 0.991591i \(-0.458691\pi\)
0.129412 + 0.991591i \(0.458691\pi\)
\(44\) 0 0
\(45\) 8.30278 1.23770
\(46\) −6.21110 −0.915777
\(47\) −1.90833 −0.278358 −0.139179 0.990267i \(-0.544446\pi\)
−0.139179 + 0.990267i \(0.544446\pi\)
\(48\) −0.697224 −0.100636
\(49\) 1.00000 0.142857
\(50\) 18.4222 2.60529
\(51\) −6.21110 −0.869728
\(52\) 21.8167 3.02543
\(53\) −12.9083 −1.77310 −0.886548 0.462638i \(-0.846903\pi\)
−0.886548 + 0.462638i \(0.846903\pi\)
\(54\) 3.69722 0.503129
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) 6.90833 0.915030
\(58\) 10.8167 1.42030
\(59\) 6.69722 0.871904 0.435952 0.899970i \(-0.356412\pi\)
0.435952 + 0.899970i \(0.356412\pi\)
\(60\) −27.4222 −3.54019
\(61\) −4.30278 −0.550914 −0.275457 0.961313i \(-0.588829\pi\)
−0.275457 + 0.961313i \(0.588829\pi\)
\(62\) 2.30278 0.292453
\(63\) −2.30278 −0.290122
\(64\) −12.8167 −1.60208
\(65\) 23.8167 2.95409
\(66\) 0 0
\(67\) 8.51388 1.04014 0.520068 0.854125i \(-0.325907\pi\)
0.520068 + 0.854125i \(0.325907\pi\)
\(68\) 8.90833 1.08029
\(69\) 6.21110 0.747729
\(70\) −8.30278 −0.992371
\(71\) −4.30278 −0.510646 −0.255323 0.966856i \(-0.582182\pi\)
−0.255323 + 0.966856i \(0.582182\pi\)
\(72\) 6.90833 0.814154
\(73\) −5.00000 −0.585206 −0.292603 0.956234i \(-0.594521\pi\)
−0.292603 + 0.956234i \(0.594521\pi\)
\(74\) −12.0000 −1.39497
\(75\) −18.4222 −2.12721
\(76\) −9.90833 −1.13656
\(77\) 0 0
\(78\) −35.0278 −3.96611
\(79\) −8.30278 −0.934135 −0.467068 0.884222i \(-0.654690\pi\)
−0.467068 + 0.884222i \(0.654690\pi\)
\(80\) 1.09167 0.122053
\(81\) −10.6056 −1.17839
\(82\) 16.1194 1.78009
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 7.60555 0.829834
\(85\) 9.72498 1.05482
\(86\) 3.90833 0.421446
\(87\) −10.8167 −1.15967
\(88\) 0 0
\(89\) −14.7250 −1.56084 −0.780422 0.625253i \(-0.784996\pi\)
−0.780422 + 0.625253i \(0.784996\pi\)
\(90\) 19.1194 2.01536
\(91\) −6.60555 −0.692450
\(92\) −8.90833 −0.928757
\(93\) −2.30278 −0.238787
\(94\) −4.39445 −0.453253
\(95\) −10.8167 −1.10977
\(96\) 12.2111 1.24629
\(97\) −3.60555 −0.366088 −0.183044 0.983105i \(-0.558595\pi\)
−0.183044 + 0.983105i \(0.558595\pi\)
\(98\) 2.30278 0.232615
\(99\) 0 0
\(100\) 26.4222 2.64222
\(101\) −5.30278 −0.527646 −0.263823 0.964571i \(-0.584983\pi\)
−0.263823 + 0.964571i \(0.584983\pi\)
\(102\) −14.3028 −1.41619
\(103\) −14.3028 −1.40929 −0.704647 0.709558i \(-0.748895\pi\)
−0.704647 + 0.709558i \(0.748895\pi\)
\(104\) 19.8167 1.94318
\(105\) 8.30278 0.810268
\(106\) −29.7250 −2.88715
\(107\) 12.3944 1.19822 0.599108 0.800668i \(-0.295522\pi\)
0.599108 + 0.800668i \(0.295522\pi\)
\(108\) 5.30278 0.510260
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 0 0
\(111\) 12.0000 1.13899
\(112\) −0.302776 −0.0286096
\(113\) −10.6972 −1.00631 −0.503155 0.864196i \(-0.667828\pi\)
−0.503155 + 0.864196i \(0.667828\pi\)
\(114\) 15.9083 1.48995
\(115\) −9.72498 −0.906859
\(116\) 15.5139 1.44043
\(117\) 15.2111 1.40627
\(118\) 15.4222 1.41973
\(119\) −2.69722 −0.247254
\(120\) −24.9083 −2.27381
\(121\) 0 0
\(122\) −9.90833 −0.897058
\(123\) −16.1194 −1.45344
\(124\) 3.30278 0.296598
\(125\) 10.8167 0.967471
\(126\) −5.30278 −0.472409
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −18.9083 −1.67128
\(129\) −3.90833 −0.344109
\(130\) 54.8444 4.81017
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 3.00000 0.260133
\(134\) 19.6056 1.69366
\(135\) 5.78890 0.498229
\(136\) 8.09167 0.693855
\(137\) −10.1194 −0.864561 −0.432281 0.901739i \(-0.642291\pi\)
−0.432281 + 0.901739i \(0.642291\pi\)
\(138\) 14.3028 1.21753
\(139\) 21.6056 1.83256 0.916279 0.400540i \(-0.131177\pi\)
0.916279 + 0.400540i \(0.131177\pi\)
\(140\) −11.9083 −1.00644
\(141\) 4.39445 0.370079
\(142\) −9.90833 −0.831488
\(143\) 0 0
\(144\) 0.697224 0.0581020
\(145\) 16.9361 1.40647
\(146\) −11.5139 −0.952895
\(147\) −2.30278 −0.189930
\(148\) −17.2111 −1.41474
\(149\) −4.90833 −0.402106 −0.201053 0.979580i \(-0.564436\pi\)
−0.201053 + 0.979580i \(0.564436\pi\)
\(150\) −42.4222 −3.46376
\(151\) −0.211103 −0.0171793 −0.00858964 0.999963i \(-0.502734\pi\)
−0.00858964 + 0.999963i \(0.502734\pi\)
\(152\) −9.00000 −0.729996
\(153\) 6.21110 0.502138
\(154\) 0 0
\(155\) 3.60555 0.289605
\(156\) −50.2389 −4.02233
\(157\) 7.21110 0.575509 0.287754 0.957704i \(-0.407091\pi\)
0.287754 + 0.957704i \(0.407091\pi\)
\(158\) −19.1194 −1.52106
\(159\) 29.7250 2.35734
\(160\) −19.1194 −1.51152
\(161\) 2.69722 0.212571
\(162\) −24.4222 −1.91879
\(163\) 2.81665 0.220617 0.110309 0.993897i \(-0.464816\pi\)
0.110309 + 0.993897i \(0.464816\pi\)
\(164\) 23.1194 1.80532
\(165\) 0 0
\(166\) 6.90833 0.536190
\(167\) −5.21110 −0.403247 −0.201624 0.979463i \(-0.564622\pi\)
−0.201624 + 0.979463i \(0.564622\pi\)
\(168\) 6.90833 0.532989
\(169\) 30.6333 2.35641
\(170\) 22.3944 1.71758
\(171\) −6.90833 −0.528293
\(172\) 5.60555 0.427419
\(173\) −1.30278 −0.0990482 −0.0495241 0.998773i \(-0.515770\pi\)
−0.0495241 + 0.998773i \(0.515770\pi\)
\(174\) −24.9083 −1.88830
\(175\) −8.00000 −0.604743
\(176\) 0 0
\(177\) −15.4222 −1.15920
\(178\) −33.9083 −2.54154
\(179\) −6.39445 −0.477944 −0.238972 0.971027i \(-0.576810\pi\)
−0.238972 + 0.971027i \(0.576810\pi\)
\(180\) 27.4222 2.04393
\(181\) −25.2111 −1.87393 −0.936963 0.349428i \(-0.886376\pi\)
−0.936963 + 0.349428i \(0.886376\pi\)
\(182\) −15.2111 −1.12752
\(183\) 9.90833 0.732445
\(184\) −8.09167 −0.596526
\(185\) −18.7889 −1.38139
\(186\) −5.30278 −0.388818
\(187\) 0 0
\(188\) −6.30278 −0.459677
\(189\) −1.60555 −0.116787
\(190\) −24.9083 −1.80704
\(191\) −26.8167 −1.94038 −0.970192 0.242336i \(-0.922086\pi\)
−0.970192 + 0.242336i \(0.922086\pi\)
\(192\) 29.5139 2.12998
\(193\) −2.11943 −0.152560 −0.0762799 0.997086i \(-0.524304\pi\)
−0.0762799 + 0.997086i \(0.524304\pi\)
\(194\) −8.30278 −0.596105
\(195\) −54.8444 −3.92749
\(196\) 3.30278 0.235913
\(197\) −10.6056 −0.755614 −0.377807 0.925884i \(-0.623322\pi\)
−0.377807 + 0.925884i \(0.623322\pi\)
\(198\) 0 0
\(199\) 19.4222 1.37680 0.688402 0.725330i \(-0.258313\pi\)
0.688402 + 0.725330i \(0.258313\pi\)
\(200\) 24.0000 1.69706
\(201\) −19.6056 −1.38287
\(202\) −12.2111 −0.859170
\(203\) −4.69722 −0.329681
\(204\) −20.5139 −1.43626
\(205\) 25.2389 1.76276
\(206\) −32.9361 −2.29477
\(207\) −6.21110 −0.431701
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 19.1194 1.31937
\(211\) 15.5139 1.06802 0.534010 0.845478i \(-0.320685\pi\)
0.534010 + 0.845478i \(0.320685\pi\)
\(212\) −42.6333 −2.92807
\(213\) 9.90833 0.678907
\(214\) 28.5416 1.95107
\(215\) 6.11943 0.417342
\(216\) 4.81665 0.327732
\(217\) −1.00000 −0.0678844
\(218\) −18.4222 −1.24771
\(219\) 11.5139 0.778036
\(220\) 0 0
\(221\) 17.8167 1.19848
\(222\) 27.6333 1.85463
\(223\) 13.9083 0.931370 0.465685 0.884950i \(-0.345808\pi\)
0.465685 + 0.884950i \(0.345808\pi\)
\(224\) 5.30278 0.354307
\(225\) 18.4222 1.22815
\(226\) −24.6333 −1.63858
\(227\) 6.51388 0.432341 0.216171 0.976356i \(-0.430643\pi\)
0.216171 + 0.976356i \(0.430643\pi\)
\(228\) 22.8167 1.51107
\(229\) −2.60555 −0.172180 −0.0860898 0.996287i \(-0.527437\pi\)
−0.0860898 + 0.996287i \(0.527437\pi\)
\(230\) −22.3944 −1.47665
\(231\) 0 0
\(232\) 14.0917 0.925164
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 35.0278 2.28984
\(235\) −6.88057 −0.448839
\(236\) 22.1194 1.43985
\(237\) 19.1194 1.24194
\(238\) −6.21110 −0.402606
\(239\) 26.3305 1.70318 0.851590 0.524208i \(-0.175639\pi\)
0.851590 + 0.524208i \(0.175639\pi\)
\(240\) −2.51388 −0.162270
\(241\) 0.486122 0.0313139 0.0156569 0.999877i \(-0.495016\pi\)
0.0156569 + 0.999877i \(0.495016\pi\)
\(242\) 0 0
\(243\) 19.6056 1.25770
\(244\) −14.2111 −0.909773
\(245\) 3.60555 0.230350
\(246\) −37.1194 −2.36665
\(247\) −19.8167 −1.26090
\(248\) 3.00000 0.190500
\(249\) −6.90833 −0.437797
\(250\) 24.9083 1.57534
\(251\) 4.39445 0.277375 0.138688 0.990336i \(-0.455712\pi\)
0.138688 + 0.990336i \(0.455712\pi\)
\(252\) −7.60555 −0.479105
\(253\) 0 0
\(254\) 4.60555 0.288978
\(255\) −22.3944 −1.40239
\(256\) −17.9083 −1.11927
\(257\) 6.48612 0.404593 0.202297 0.979324i \(-0.435160\pi\)
0.202297 + 0.979324i \(0.435160\pi\)
\(258\) −9.00000 −0.560316
\(259\) 5.21110 0.323802
\(260\) 78.6611 4.87835
\(261\) 10.8167 0.669534
\(262\) −13.8167 −0.853596
\(263\) 20.2389 1.24798 0.623991 0.781432i \(-0.285510\pi\)
0.623991 + 0.781432i \(0.285510\pi\)
\(264\) 0 0
\(265\) −46.5416 −2.85903
\(266\) 6.90833 0.423577
\(267\) 33.9083 2.07516
\(268\) 28.1194 1.71767
\(269\) 16.1194 0.982819 0.491409 0.870929i \(-0.336482\pi\)
0.491409 + 0.870929i \(0.336482\pi\)
\(270\) 13.3305 0.811270
\(271\) −28.5139 −1.73209 −0.866047 0.499962i \(-0.833347\pi\)
−0.866047 + 0.499962i \(0.833347\pi\)
\(272\) 0.816654 0.0495169
\(273\) 15.2111 0.920618
\(274\) −23.3028 −1.40777
\(275\) 0 0
\(276\) 20.5139 1.23479
\(277\) 29.9361 1.79868 0.899342 0.437245i \(-0.144046\pi\)
0.899342 + 0.437245i \(0.144046\pi\)
\(278\) 49.7527 2.98397
\(279\) 2.30278 0.137864
\(280\) −10.8167 −0.646419
\(281\) 6.39445 0.381461 0.190730 0.981642i \(-0.438914\pi\)
0.190730 + 0.981642i \(0.438914\pi\)
\(282\) 10.1194 0.602603
\(283\) 4.39445 0.261223 0.130611 0.991434i \(-0.458306\pi\)
0.130611 + 0.991434i \(0.458306\pi\)
\(284\) −14.2111 −0.843274
\(285\) 24.9083 1.47544
\(286\) 0 0
\(287\) −7.00000 −0.413197
\(288\) −12.2111 −0.719546
\(289\) −9.72498 −0.572058
\(290\) 39.0000 2.29016
\(291\) 8.30278 0.486717
\(292\) −16.5139 −0.966402
\(293\) 4.81665 0.281392 0.140696 0.990053i \(-0.455066\pi\)
0.140696 + 0.990053i \(0.455066\pi\)
\(294\) −5.30278 −0.309264
\(295\) 24.1472 1.40590
\(296\) −15.6333 −0.908668
\(297\) 0 0
\(298\) −11.3028 −0.654752
\(299\) −17.8167 −1.03036
\(300\) −60.8444 −3.51285
\(301\) −1.69722 −0.0978264
\(302\) −0.486122 −0.0279732
\(303\) 12.2111 0.701510
\(304\) −0.908327 −0.0520961
\(305\) −15.5139 −0.888322
\(306\) 14.3028 0.817635
\(307\) −16.6333 −0.949313 −0.474657 0.880171i \(-0.657428\pi\)
−0.474657 + 0.880171i \(0.657428\pi\)
\(308\) 0 0
\(309\) 32.9361 1.87367
\(310\) 8.30278 0.471566
\(311\) 6.39445 0.362596 0.181298 0.983428i \(-0.441970\pi\)
0.181298 + 0.983428i \(0.441970\pi\)
\(312\) −45.6333 −2.58348
\(313\) 15.3028 0.864964 0.432482 0.901643i \(-0.357638\pi\)
0.432482 + 0.901643i \(0.357638\pi\)
\(314\) 16.6056 0.937105
\(315\) −8.30278 −0.467808
\(316\) −27.4222 −1.54262
\(317\) 17.3028 0.971821 0.485910 0.874009i \(-0.338488\pi\)
0.485910 + 0.874009i \(0.338488\pi\)
\(318\) 68.4500 3.83848
\(319\) 0 0
\(320\) −46.2111 −2.58328
\(321\) −28.5416 −1.59304
\(322\) 6.21110 0.346131
\(323\) −8.09167 −0.450233
\(324\) −35.0278 −1.94599
\(325\) 52.8444 2.93128
\(326\) 6.48612 0.359233
\(327\) 18.4222 1.01875
\(328\) 21.0000 1.15953
\(329\) 1.90833 0.105209
\(330\) 0 0
\(331\) 23.8167 1.30908 0.654541 0.756027i \(-0.272862\pi\)
0.654541 + 0.756027i \(0.272862\pi\)
\(332\) 9.90833 0.543790
\(333\) −12.0000 −0.657596
\(334\) −12.0000 −0.656611
\(335\) 30.6972 1.67717
\(336\) 0.697224 0.0380367
\(337\) 11.7889 0.642182 0.321091 0.947048i \(-0.395950\pi\)
0.321091 + 0.947048i \(0.395950\pi\)
\(338\) 70.5416 3.83696
\(339\) 24.6333 1.33790
\(340\) 32.1194 1.74192
\(341\) 0 0
\(342\) −15.9083 −0.860224
\(343\) −1.00000 −0.0539949
\(344\) 5.09167 0.274525
\(345\) 22.3944 1.20568
\(346\) −3.00000 −0.161281
\(347\) 9.60555 0.515653 0.257827 0.966191i \(-0.416994\pi\)
0.257827 + 0.966191i \(0.416994\pi\)
\(348\) −35.7250 −1.91506
\(349\) −4.69722 −0.251437 −0.125718 0.992066i \(-0.540124\pi\)
−0.125718 + 0.992066i \(0.540124\pi\)
\(350\) −18.4222 −0.984708
\(351\) 10.6056 0.566082
\(352\) 0 0
\(353\) 5.09167 0.271002 0.135501 0.990777i \(-0.456736\pi\)
0.135501 + 0.990777i \(0.456736\pi\)
\(354\) −35.5139 −1.88754
\(355\) −15.5139 −0.823391
\(356\) −48.6333 −2.57756
\(357\) 6.21110 0.328726
\(358\) −14.7250 −0.778239
\(359\) −33.9361 −1.79108 −0.895539 0.444983i \(-0.853210\pi\)
−0.895539 + 0.444983i \(0.853210\pi\)
\(360\) 24.9083 1.31278
\(361\) −10.0000 −0.526316
\(362\) −58.0555 −3.05133
\(363\) 0 0
\(364\) −21.8167 −1.14350
\(365\) −18.0278 −0.943616
\(366\) 22.8167 1.19265
\(367\) −17.6333 −0.920451 −0.460226 0.887802i \(-0.652231\pi\)
−0.460226 + 0.887802i \(0.652231\pi\)
\(368\) −0.816654 −0.0425710
\(369\) 16.1194 0.839144
\(370\) −43.2666 −2.24932
\(371\) 12.9083 0.670167
\(372\) −7.60555 −0.394329
\(373\) −20.1194 −1.04174 −0.520872 0.853635i \(-0.674393\pi\)
−0.520872 + 0.853635i \(0.674393\pi\)
\(374\) 0 0
\(375\) −24.9083 −1.28626
\(376\) −5.72498 −0.295243
\(377\) 31.0278 1.59801
\(378\) −3.69722 −0.190165
\(379\) −15.8167 −0.812447 −0.406223 0.913774i \(-0.633155\pi\)
−0.406223 + 0.913774i \(0.633155\pi\)
\(380\) −35.7250 −1.83265
\(381\) −4.60555 −0.235950
\(382\) −61.7527 −3.15954
\(383\) 38.6333 1.97407 0.987035 0.160506i \(-0.0513125\pi\)
0.987035 + 0.160506i \(0.0513125\pi\)
\(384\) 43.5416 2.22197
\(385\) 0 0
\(386\) −4.88057 −0.248414
\(387\) 3.90833 0.198671
\(388\) −11.9083 −0.604554
\(389\) −7.81665 −0.396320 −0.198160 0.980170i \(-0.563497\pi\)
−0.198160 + 0.980170i \(0.563497\pi\)
\(390\) −126.294 −6.39516
\(391\) −7.27502 −0.367914
\(392\) 3.00000 0.151523
\(393\) 13.8167 0.696958
\(394\) −24.4222 −1.23037
\(395\) −29.9361 −1.50625
\(396\) 0 0
\(397\) 30.2111 1.51625 0.758126 0.652108i \(-0.226115\pi\)
0.758126 + 0.652108i \(0.226115\pi\)
\(398\) 44.7250 2.24186
\(399\) −6.90833 −0.345849
\(400\) 2.42221 0.121110
\(401\) −21.2111 −1.05923 −0.529616 0.848238i \(-0.677664\pi\)
−0.529616 + 0.848238i \(0.677664\pi\)
\(402\) −45.1472 −2.25174
\(403\) 6.60555 0.329046
\(404\) −17.5139 −0.871348
\(405\) −38.2389 −1.90010
\(406\) −10.8167 −0.536822
\(407\) 0 0
\(408\) −18.6333 −0.922486
\(409\) 20.6056 1.01888 0.509439 0.860506i \(-0.329853\pi\)
0.509439 + 0.860506i \(0.329853\pi\)
\(410\) 58.1194 2.87031
\(411\) 23.3028 1.14944
\(412\) −47.2389 −2.32729
\(413\) −6.69722 −0.329549
\(414\) −14.3028 −0.702943
\(415\) 10.8167 0.530969
\(416\) −35.0278 −1.71738
\(417\) −49.7527 −2.43640
\(418\) 0 0
\(419\) −38.4222 −1.87705 −0.938524 0.345215i \(-0.887806\pi\)
−0.938524 + 0.345215i \(0.887806\pi\)
\(420\) 27.4222 1.33807
\(421\) −21.8167 −1.06328 −0.531639 0.846971i \(-0.678424\pi\)
−0.531639 + 0.846971i \(0.678424\pi\)
\(422\) 35.7250 1.73906
\(423\) −4.39445 −0.213665
\(424\) −38.7250 −1.88065
\(425\) 21.5778 1.04668
\(426\) 22.8167 1.10547
\(427\) 4.30278 0.208226
\(428\) 40.9361 1.97872
\(429\) 0 0
\(430\) 14.0917 0.679561
\(431\) −3.11943 −0.150258 −0.0751288 0.997174i \(-0.523937\pi\)
−0.0751288 + 0.997174i \(0.523937\pi\)
\(432\) 0.486122 0.0233885
\(433\) 29.1472 1.40072 0.700362 0.713788i \(-0.253022\pi\)
0.700362 + 0.713788i \(0.253022\pi\)
\(434\) −2.30278 −0.110537
\(435\) −39.0000 −1.86991
\(436\) −26.4222 −1.26539
\(437\) 8.09167 0.387077
\(438\) 26.5139 1.26688
\(439\) 27.7250 1.32324 0.661621 0.749839i \(-0.269869\pi\)
0.661621 + 0.749839i \(0.269869\pi\)
\(440\) 0 0
\(441\) 2.30278 0.109656
\(442\) 41.0278 1.95149
\(443\) 26.9361 1.27977 0.639886 0.768470i \(-0.278982\pi\)
0.639886 + 0.768470i \(0.278982\pi\)
\(444\) 39.6333 1.88091
\(445\) −53.0917 −2.51679
\(446\) 32.0278 1.51656
\(447\) 11.3028 0.534603
\(448\) 12.8167 0.605530
\(449\) 40.3305 1.90332 0.951658 0.307160i \(-0.0993788\pi\)
0.951658 + 0.307160i \(0.0993788\pi\)
\(450\) 42.4222 1.99980
\(451\) 0 0
\(452\) −35.3305 −1.66181
\(453\) 0.486122 0.0228400
\(454\) 15.0000 0.703985
\(455\) −23.8167 −1.11654
\(456\) 20.7250 0.970536
\(457\) 21.1194 0.987925 0.493963 0.869483i \(-0.335548\pi\)
0.493963 + 0.869483i \(0.335548\pi\)
\(458\) −6.00000 −0.280362
\(459\) 4.33053 0.202132
\(460\) −32.1194 −1.49758
\(461\) 8.09167 0.376867 0.188433 0.982086i \(-0.439659\pi\)
0.188433 + 0.982086i \(0.439659\pi\)
\(462\) 0 0
\(463\) −30.8167 −1.43217 −0.716086 0.698012i \(-0.754068\pi\)
−0.716086 + 0.698012i \(0.754068\pi\)
\(464\) 1.42221 0.0660242
\(465\) −8.30278 −0.385032
\(466\) 41.4500 1.92013
\(467\) −13.5416 −0.626632 −0.313316 0.949649i \(-0.601440\pi\)
−0.313316 + 0.949649i \(0.601440\pi\)
\(468\) 50.2389 2.32229
\(469\) −8.51388 −0.393134
\(470\) −15.8444 −0.730848
\(471\) −16.6056 −0.765143
\(472\) 20.0917 0.924794
\(473\) 0 0
\(474\) 44.0278 2.02226
\(475\) −24.0000 −1.10120
\(476\) −8.90833 −0.408312
\(477\) −29.7250 −1.36101
\(478\) 60.6333 2.77330
\(479\) −28.6333 −1.30829 −0.654145 0.756370i \(-0.726971\pi\)
−0.654145 + 0.756370i \(0.726971\pi\)
\(480\) 44.0278 2.00958
\(481\) −34.4222 −1.56952
\(482\) 1.11943 0.0509886
\(483\) −6.21110 −0.282615
\(484\) 0 0
\(485\) −13.0000 −0.590300
\(486\) 45.1472 2.04792
\(487\) −2.81665 −0.127635 −0.0638174 0.997962i \(-0.520328\pi\)
−0.0638174 + 0.997962i \(0.520328\pi\)
\(488\) −12.9083 −0.584333
\(489\) −6.48612 −0.293313
\(490\) 8.30278 0.375081
\(491\) −12.6972 −0.573018 −0.286509 0.958078i \(-0.592495\pi\)
−0.286509 + 0.958078i \(0.592495\pi\)
\(492\) −53.2389 −2.40019
\(493\) 12.6695 0.570604
\(494\) −45.6333 −2.05314
\(495\) 0 0
\(496\) 0.302776 0.0135950
\(497\) 4.30278 0.193006
\(498\) −15.9083 −0.712869
\(499\) 2.90833 0.130195 0.0650973 0.997879i \(-0.479264\pi\)
0.0650973 + 0.997879i \(0.479264\pi\)
\(500\) 35.7250 1.59767
\(501\) 12.0000 0.536120
\(502\) 10.1194 0.451652
\(503\) 5.57779 0.248702 0.124351 0.992238i \(-0.460315\pi\)
0.124351 + 0.992238i \(0.460315\pi\)
\(504\) −6.90833 −0.307721
\(505\) −19.1194 −0.850803
\(506\) 0 0
\(507\) −70.5416 −3.13286
\(508\) 6.60555 0.293074
\(509\) 22.3028 0.988553 0.494277 0.869305i \(-0.335433\pi\)
0.494277 + 0.869305i \(0.335433\pi\)
\(510\) −51.5694 −2.28353
\(511\) 5.00000 0.221187
\(512\) −3.42221 −0.151242
\(513\) −4.81665 −0.212660
\(514\) 14.9361 0.658802
\(515\) −51.5694 −2.27242
\(516\) −12.9083 −0.568257
\(517\) 0 0
\(518\) 12.0000 0.527250
\(519\) 3.00000 0.131685
\(520\) 71.4500 3.13329
\(521\) 7.42221 0.325173 0.162586 0.986694i \(-0.448016\pi\)
0.162586 + 0.986694i \(0.448016\pi\)
\(522\) 24.9083 1.09021
\(523\) −44.5416 −1.94767 −0.973835 0.227257i \(-0.927024\pi\)
−0.973835 + 0.227257i \(0.927024\pi\)
\(524\) −19.8167 −0.865695
\(525\) 18.4222 0.804011
\(526\) 46.6056 2.03210
\(527\) 2.69722 0.117493
\(528\) 0 0
\(529\) −15.7250 −0.683695
\(530\) −107.175 −4.65538
\(531\) 15.4222 0.669267
\(532\) 9.90833 0.429580
\(533\) 46.2389 2.00283
\(534\) 78.0833 3.37899
\(535\) 44.6888 1.93207
\(536\) 25.5416 1.10323
\(537\) 14.7250 0.635430
\(538\) 37.1194 1.60033
\(539\) 0 0
\(540\) 19.1194 0.822769
\(541\) −32.3944 −1.39275 −0.696373 0.717680i \(-0.745204\pi\)
−0.696373 + 0.717680i \(0.745204\pi\)
\(542\) −65.6611 −2.82038
\(543\) 58.0555 2.49140
\(544\) −14.3028 −0.613226
\(545\) −28.8444 −1.23556
\(546\) 35.0278 1.49905
\(547\) −29.9361 −1.27997 −0.639987 0.768386i \(-0.721060\pi\)
−0.639987 + 0.768386i \(0.721060\pi\)
\(548\) −33.4222 −1.42773
\(549\) −9.90833 −0.422877
\(550\) 0 0
\(551\) −14.0917 −0.600325
\(552\) 18.6333 0.793086
\(553\) 8.30278 0.353070
\(554\) 68.9361 2.92881
\(555\) 43.2666 1.83657
\(556\) 71.3583 3.02627
\(557\) 7.23886 0.306720 0.153360 0.988170i \(-0.450991\pi\)
0.153360 + 0.988170i \(0.450991\pi\)
\(558\) 5.30278 0.224484
\(559\) 11.2111 0.474179
\(560\) −1.09167 −0.0461316
\(561\) 0 0
\(562\) 14.7250 0.621136
\(563\) −0.302776 −0.0127605 −0.00638024 0.999980i \(-0.502031\pi\)
−0.00638024 + 0.999980i \(0.502031\pi\)
\(564\) 14.5139 0.611145
\(565\) −38.5694 −1.62263
\(566\) 10.1194 0.425351
\(567\) 10.6056 0.445391
\(568\) −12.9083 −0.541621
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 57.3583 2.40247
\(571\) 25.8444 1.08155 0.540777 0.841166i \(-0.318130\pi\)
0.540777 + 0.841166i \(0.318130\pi\)
\(572\) 0 0
\(573\) 61.7527 2.57976
\(574\) −16.1194 −0.672812
\(575\) −21.5778 −0.899856
\(576\) −29.5139 −1.22974
\(577\) 2.21110 0.0920494 0.0460247 0.998940i \(-0.485345\pi\)
0.0460247 + 0.998940i \(0.485345\pi\)
\(578\) −22.3944 −0.931486
\(579\) 4.88057 0.202830
\(580\) 55.9361 2.32262
\(581\) −3.00000 −0.124461
\(582\) 19.1194 0.792526
\(583\) 0 0
\(584\) −15.0000 −0.620704
\(585\) 54.8444 2.26754
\(586\) 11.0917 0.458193
\(587\) −13.6056 −0.561561 −0.280781 0.959772i \(-0.590593\pi\)
−0.280781 + 0.959772i \(0.590593\pi\)
\(588\) −7.60555 −0.313648
\(589\) −3.00000 −0.123613
\(590\) 55.6056 2.28924
\(591\) 24.4222 1.00460
\(592\) −1.57779 −0.0648470
\(593\) 22.6056 0.928299 0.464149 0.885757i \(-0.346360\pi\)
0.464149 + 0.885757i \(0.346360\pi\)
\(594\) 0 0
\(595\) −9.72498 −0.398685
\(596\) −16.2111 −0.664033
\(597\) −44.7250 −1.83047
\(598\) −41.0278 −1.67775
\(599\) −14.7889 −0.604258 −0.302129 0.953267i \(-0.597697\pi\)
−0.302129 + 0.953267i \(0.597697\pi\)
\(600\) −55.2666 −2.25625
\(601\) −5.81665 −0.237266 −0.118633 0.992938i \(-0.537851\pi\)
−0.118633 + 0.992938i \(0.537851\pi\)
\(602\) −3.90833 −0.159292
\(603\) 19.6056 0.798400
\(604\) −0.697224 −0.0283697
\(605\) 0 0
\(606\) 28.1194 1.14227
\(607\) −20.5416 −0.833759 −0.416880 0.908962i \(-0.636876\pi\)
−0.416880 + 0.908962i \(0.636876\pi\)
\(608\) 15.9083 0.645168
\(609\) 10.8167 0.438313
\(610\) −35.7250 −1.44646
\(611\) −12.6056 −0.509966
\(612\) 20.5139 0.829224
\(613\) −33.0555 −1.33510 −0.667550 0.744565i \(-0.732657\pi\)
−0.667550 + 0.744565i \(0.732657\pi\)
\(614\) −38.3028 −1.54577
\(615\) −58.1194 −2.34360
\(616\) 0 0
\(617\) 21.6333 0.870924 0.435462 0.900207i \(-0.356585\pi\)
0.435462 + 0.900207i \(0.356585\pi\)
\(618\) 75.8444 3.05091
\(619\) 39.8167 1.60037 0.800183 0.599756i \(-0.204736\pi\)
0.800183 + 0.599756i \(0.204736\pi\)
\(620\) 11.9083 0.478250
\(621\) −4.33053 −0.173778
\(622\) 14.7250 0.590418
\(623\) 14.7250 0.589944
\(624\) −4.60555 −0.184370
\(625\) −1.00000 −0.0400000
\(626\) 35.2389 1.40843
\(627\) 0 0
\(628\) 23.8167 0.950388
\(629\) −14.0555 −0.560430
\(630\) −19.1194 −0.761736
\(631\) −31.0555 −1.23630 −0.618150 0.786060i \(-0.712118\pi\)
−0.618150 + 0.786060i \(0.712118\pi\)
\(632\) −24.9083 −0.990800
\(633\) −35.7250 −1.41994
\(634\) 39.8444 1.58242
\(635\) 7.21110 0.286164
\(636\) 98.1749 3.89289
\(637\) 6.60555 0.261721
\(638\) 0 0
\(639\) −9.90833 −0.391967
\(640\) −68.1749 −2.69485
\(641\) −28.8167 −1.13819 −0.569095 0.822272i \(-0.692706\pi\)
−0.569095 + 0.822272i \(0.692706\pi\)
\(642\) −65.7250 −2.59396
\(643\) −1.23886 −0.0488558 −0.0244279 0.999702i \(-0.507776\pi\)
−0.0244279 + 0.999702i \(0.507776\pi\)
\(644\) 8.90833 0.351037
\(645\) −14.0917 −0.554859
\(646\) −18.6333 −0.733118
\(647\) 10.6056 0.416947 0.208474 0.978028i \(-0.433150\pi\)
0.208474 + 0.978028i \(0.433150\pi\)
\(648\) −31.8167 −1.24988
\(649\) 0 0
\(650\) 121.689 4.77303
\(651\) 2.30278 0.0902529
\(652\) 9.30278 0.364325
\(653\) −6.81665 −0.266756 −0.133378 0.991065i \(-0.542582\pi\)
−0.133378 + 0.991065i \(0.542582\pi\)
\(654\) 42.4222 1.65884
\(655\) −21.6333 −0.845283
\(656\) 2.11943 0.0827498
\(657\) −11.5139 −0.449199
\(658\) 4.39445 0.171313
\(659\) 4.63331 0.180488 0.0902440 0.995920i \(-0.471235\pi\)
0.0902440 + 0.995920i \(0.471235\pi\)
\(660\) 0 0
\(661\) −18.3028 −0.711895 −0.355948 0.934506i \(-0.615842\pi\)
−0.355948 + 0.934506i \(0.615842\pi\)
\(662\) 54.8444 2.13159
\(663\) −41.0278 −1.59339
\(664\) 9.00000 0.349268
\(665\) 10.8167 0.419452
\(666\) −27.6333 −1.07077
\(667\) −12.6695 −0.490564
\(668\) −17.2111 −0.665918
\(669\) −32.0278 −1.23826
\(670\) 70.6888 2.73095
\(671\) 0 0
\(672\) −12.2111 −0.471054
\(673\) −8.42221 −0.324652 −0.162326 0.986737i \(-0.551900\pi\)
−0.162326 + 0.986737i \(0.551900\pi\)
\(674\) 27.1472 1.04567
\(675\) 12.8444 0.494382
\(676\) 101.175 3.89134
\(677\) 10.3028 0.395968 0.197984 0.980205i \(-0.436561\pi\)
0.197984 + 0.980205i \(0.436561\pi\)
\(678\) 56.7250 2.17851
\(679\) 3.60555 0.138368
\(680\) 29.1749 1.11881
\(681\) −15.0000 −0.574801
\(682\) 0 0
\(683\) 15.6972 0.600638 0.300319 0.953839i \(-0.402907\pi\)
0.300319 + 0.953839i \(0.402907\pi\)
\(684\) −22.8167 −0.872417
\(685\) −36.4861 −1.39406
\(686\) −2.30278 −0.0879204
\(687\) 6.00000 0.228914
\(688\) 0.513878 0.0195914
\(689\) −85.2666 −3.24840
\(690\) 51.5694 1.96321
\(691\) −17.0278 −0.647766 −0.323883 0.946097i \(-0.604988\pi\)
−0.323883 + 0.946097i \(0.604988\pi\)
\(692\) −4.30278 −0.163567
\(693\) 0 0
\(694\) 22.1194 0.839642
\(695\) 77.8999 2.95491
\(696\) −32.4500 −1.23001
\(697\) 18.8806 0.715153
\(698\) −10.8167 −0.409416
\(699\) −41.4500 −1.56778
\(700\) −26.4222 −0.998665
\(701\) −31.1194 −1.17536 −0.587682 0.809092i \(-0.699960\pi\)
−0.587682 + 0.809092i \(0.699960\pi\)
\(702\) 24.4222 0.921757
\(703\) 15.6333 0.589621
\(704\) 0 0
\(705\) 15.8444 0.596735
\(706\) 11.7250 0.441275
\(707\) 5.30278 0.199431
\(708\) −50.9361 −1.91430
\(709\) 25.3305 0.951308 0.475654 0.879632i \(-0.342211\pi\)
0.475654 + 0.879632i \(0.342211\pi\)
\(710\) −35.7250 −1.34073
\(711\) −19.1194 −0.717035
\(712\) −44.1749 −1.65553
\(713\) −2.69722 −0.101012
\(714\) 14.3028 0.535268
\(715\) 0 0
\(716\) −21.1194 −0.789270
\(717\) −60.6333 −2.26439
\(718\) −78.1472 −2.91643
\(719\) 21.2389 0.792076 0.396038 0.918234i \(-0.370385\pi\)
0.396038 + 0.918234i \(0.370385\pi\)
\(720\) 2.51388 0.0936867
\(721\) 14.3028 0.532663
\(722\) −23.0278 −0.857004
\(723\) −1.11943 −0.0416320
\(724\) −83.2666 −3.09458
\(725\) 37.5778 1.39560
\(726\) 0 0
\(727\) 24.1194 0.894540 0.447270 0.894399i \(-0.352396\pi\)
0.447270 + 0.894399i \(0.352396\pi\)
\(728\) −19.8167 −0.734454
\(729\) −13.3305 −0.493723
\(730\) −41.5139 −1.53650
\(731\) 4.57779 0.169316
\(732\) 32.7250 1.20955
\(733\) −6.78890 −0.250754 −0.125377 0.992109i \(-0.540014\pi\)
−0.125377 + 0.992109i \(0.540014\pi\)
\(734\) −40.6056 −1.49878
\(735\) −8.30278 −0.306252
\(736\) 14.3028 0.527207
\(737\) 0 0
\(738\) 37.1194 1.36639
\(739\) −43.1194 −1.58617 −0.793087 0.609108i \(-0.791527\pi\)
−0.793087 + 0.609108i \(0.791527\pi\)
\(740\) −62.0555 −2.28121
\(741\) 45.6333 1.67638
\(742\) 29.7250 1.09124
\(743\) 44.9361 1.64855 0.824273 0.566193i \(-0.191584\pi\)
0.824273 + 0.566193i \(0.191584\pi\)
\(744\) −6.90833 −0.253272
\(745\) −17.6972 −0.648376
\(746\) −46.3305 −1.69628
\(747\) 6.90833 0.252762
\(748\) 0 0
\(749\) −12.3944 −0.452883
\(750\) −57.3583 −2.09443
\(751\) 7.63331 0.278543 0.139272 0.990254i \(-0.455524\pi\)
0.139272 + 0.990254i \(0.455524\pi\)
\(752\) −0.577795 −0.0210700
\(753\) −10.1194 −0.368773
\(754\) 71.4500 2.60205
\(755\) −0.761141 −0.0277008
\(756\) −5.30278 −0.192860
\(757\) −23.8167 −0.865631 −0.432816 0.901483i \(-0.642480\pi\)
−0.432816 + 0.901483i \(0.642480\pi\)
\(758\) −36.4222 −1.32291
\(759\) 0 0
\(760\) −32.4500 −1.17708
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) −10.6056 −0.384199
\(763\) 8.00000 0.289619
\(764\) −88.5694 −3.20433
\(765\) 22.3944 0.809673
\(766\) 88.9638 3.21439
\(767\) 44.2389 1.59737
\(768\) 41.2389 1.48808
\(769\) −39.3305 −1.41830 −0.709148 0.705060i \(-0.750920\pi\)
−0.709148 + 0.705060i \(0.750920\pi\)
\(770\) 0 0
\(771\) −14.9361 −0.537910
\(772\) −7.00000 −0.251936
\(773\) 30.6333 1.10180 0.550902 0.834570i \(-0.314284\pi\)
0.550902 + 0.834570i \(0.314284\pi\)
\(774\) 9.00000 0.323498
\(775\) 8.00000 0.287368
\(776\) −10.8167 −0.388295
\(777\) −12.0000 −0.430498
\(778\) −18.0000 −0.645331
\(779\) −21.0000 −0.752403
\(780\) −181.139 −6.48581
\(781\) 0 0
\(782\) −16.7527 −0.599077
\(783\) 7.54163 0.269516
\(784\) 0.302776 0.0108134
\(785\) 26.0000 0.927980
\(786\) 31.8167 1.13486
\(787\) −22.4861 −0.801544 −0.400772 0.916178i \(-0.631258\pi\)
−0.400772 + 0.916178i \(0.631258\pi\)
\(788\) −35.0278 −1.24781
\(789\) −46.6056 −1.65920
\(790\) −68.9361 −2.45264
\(791\) 10.6972 0.380350
\(792\) 0 0
\(793\) −28.4222 −1.00930
\(794\) 69.5694 2.46893
\(795\) 107.175 3.80110
\(796\) 64.1472 2.27364
\(797\) −28.6333 −1.01424 −0.507122 0.861874i \(-0.669291\pi\)
−0.507122 + 0.861874i \(0.669291\pi\)
\(798\) −15.9083 −0.563149
\(799\) −5.14719 −0.182094
\(800\) −42.4222 −1.49985
\(801\) −33.9083 −1.19809
\(802\) −48.8444 −1.72476
\(803\) 0 0
\(804\) −64.7527 −2.28365
\(805\) 9.72498 0.342761
\(806\) 15.2111 0.535788
\(807\) −37.1194 −1.30667
\(808\) −15.9083 −0.559653
\(809\) 8.09167 0.284488 0.142244 0.989832i \(-0.454568\pi\)
0.142244 + 0.989832i \(0.454568\pi\)
\(810\) −88.0555 −3.09396
\(811\) −40.8444 −1.43424 −0.717121 0.696949i \(-0.754540\pi\)
−0.717121 + 0.696949i \(0.754540\pi\)
\(812\) −15.5139 −0.544430
\(813\) 65.6611 2.30283
\(814\) 0 0
\(815\) 10.1556 0.355735
\(816\) −1.88057 −0.0658331
\(817\) −5.09167 −0.178135
\(818\) 47.4500 1.65905
\(819\) −15.2111 −0.531519
\(820\) 83.3583 2.91100
\(821\) −27.4222 −0.957042 −0.478521 0.878076i \(-0.658827\pi\)
−0.478521 + 0.878076i \(0.658827\pi\)
\(822\) 53.6611 1.87164
\(823\) 35.2111 1.22738 0.613691 0.789546i \(-0.289684\pi\)
0.613691 + 0.789546i \(0.289684\pi\)
\(824\) −42.9083 −1.49478
\(825\) 0 0
\(826\) −15.4222 −0.536607
\(827\) −14.1833 −0.493203 −0.246602 0.969117i \(-0.579314\pi\)
−0.246602 + 0.969117i \(0.579314\pi\)
\(828\) −20.5139 −0.712907
\(829\) 13.2750 0.461060 0.230530 0.973065i \(-0.425954\pi\)
0.230530 + 0.973065i \(0.425954\pi\)
\(830\) 24.9083 0.864581
\(831\) −68.9361 −2.39137
\(832\) −84.6611 −2.93509
\(833\) 2.69722 0.0934533
\(834\) −114.569 −3.96721
\(835\) −18.7889 −0.650217
\(836\) 0 0
\(837\) 1.60555 0.0554960
\(838\) −88.4777 −3.05641
\(839\) 42.2111 1.45729 0.728645 0.684892i \(-0.240151\pi\)
0.728645 + 0.684892i \(0.240151\pi\)
\(840\) 24.9083 0.859419
\(841\) −6.93608 −0.239175
\(842\) −50.2389 −1.73135
\(843\) −14.7250 −0.507155
\(844\) 51.2389 1.76371
\(845\) 110.450 3.79959
\(846\) −10.1194 −0.347913
\(847\) 0 0
\(848\) −3.90833 −0.134212
\(849\) −10.1194 −0.347298
\(850\) 49.6888 1.70431
\(851\) 14.0555 0.481817
\(852\) 32.7250 1.12114
\(853\) −45.7250 −1.56559 −0.782797 0.622277i \(-0.786208\pi\)
−0.782797 + 0.622277i \(0.786208\pi\)
\(854\) 9.90833 0.339056
\(855\) −24.9083 −0.851847
\(856\) 37.1833 1.27090
\(857\) 35.3583 1.20782 0.603908 0.797054i \(-0.293609\pi\)
0.603908 + 0.797054i \(0.293609\pi\)
\(858\) 0 0
\(859\) −25.3028 −0.863320 −0.431660 0.902036i \(-0.642072\pi\)
−0.431660 + 0.902036i \(0.642072\pi\)
\(860\) 20.2111 0.689193
\(861\) 16.1194 0.549349
\(862\) −7.18335 −0.244666
\(863\) 31.4222 1.06962 0.534812 0.844971i \(-0.320382\pi\)
0.534812 + 0.844971i \(0.320382\pi\)
\(864\) −8.51388 −0.289648
\(865\) −4.69722 −0.159710
\(866\) 67.1194 2.28081
\(867\) 22.3944 0.760555
\(868\) −3.30278 −0.112104
\(869\) 0 0
\(870\) −89.8082 −3.04478
\(871\) 56.2389 1.90558
\(872\) −24.0000 −0.812743
\(873\) −8.30278 −0.281006
\(874\) 18.6333 0.630281
\(875\) −10.8167 −0.365670
\(876\) 38.0278 1.28484
\(877\) 47.0555 1.58895 0.794476 0.607296i \(-0.207746\pi\)
0.794476 + 0.607296i \(0.207746\pi\)
\(878\) 63.8444 2.15464
\(879\) −11.0917 −0.374113
\(880\) 0 0
\(881\) 34.3305 1.15663 0.578313 0.815815i \(-0.303711\pi\)
0.578313 + 0.815815i \(0.303711\pi\)
\(882\) 5.30278 0.178554
\(883\) −7.72498 −0.259966 −0.129983 0.991516i \(-0.541492\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(884\) 58.8444 1.97915
\(885\) −55.6056 −1.86916
\(886\) 62.0278 2.08386
\(887\) 44.1194 1.48139 0.740693 0.671844i \(-0.234497\pi\)
0.740693 + 0.671844i \(0.234497\pi\)
\(888\) 36.0000 1.20808
\(889\) −2.00000 −0.0670778
\(890\) −122.258 −4.09810
\(891\) 0 0
\(892\) 45.9361 1.53805
\(893\) 5.72498 0.191579
\(894\) 26.0278 0.870498
\(895\) −23.0555 −0.770661
\(896\) 18.9083 0.631683
\(897\) 41.0278 1.36988
\(898\) 92.8722 3.09918
\(899\) 4.69722 0.156661
\(900\) 60.8444 2.02815
\(901\) −34.8167 −1.15991
\(902\) 0 0
\(903\) 3.90833 0.130061
\(904\) −32.0917 −1.06735
\(905\) −90.8999 −3.02162
\(906\) 1.11943 0.0371906
\(907\) 0.394449 0.0130975 0.00654873 0.999979i \(-0.497915\pi\)
0.00654873 + 0.999979i \(0.497915\pi\)
\(908\) 21.5139 0.713963
\(909\) −12.2111 −0.405017
\(910\) −54.8444 −1.81807
\(911\) −50.4500 −1.67148 −0.835741 0.549124i \(-0.814961\pi\)
−0.835741 + 0.549124i \(0.814961\pi\)
\(912\) 2.09167 0.0692622
\(913\) 0 0
\(914\) 48.6333 1.60865
\(915\) 35.7250 1.18103
\(916\) −8.60555 −0.284335
\(917\) 6.00000 0.198137
\(918\) 9.97224 0.329133
\(919\) 55.4777 1.83004 0.915021 0.403407i \(-0.132174\pi\)
0.915021 + 0.403407i \(0.132174\pi\)
\(920\) −29.1749 −0.961869
\(921\) 38.3028 1.26212
\(922\) 18.6333 0.613655
\(923\) −28.4222 −0.935528
\(924\) 0 0
\(925\) −41.6888 −1.37072
\(926\) −70.9638 −2.33202
\(927\) −32.9361 −1.08176
\(928\) −24.9083 −0.817656
\(929\) −27.6333 −0.906619 −0.453310 0.891353i \(-0.649757\pi\)
−0.453310 + 0.891353i \(0.649757\pi\)
\(930\) −19.1194 −0.626951
\(931\) −3.00000 −0.0983210
\(932\) 59.4500 1.94735
\(933\) −14.7250 −0.482074
\(934\) −31.1833 −1.02035
\(935\) 0 0
\(936\) 45.6333 1.49157
\(937\) 31.6056 1.03251 0.516254 0.856435i \(-0.327326\pi\)
0.516254 + 0.856435i \(0.327326\pi\)
\(938\) −19.6056 −0.640144
\(939\) −35.2389 −1.14998
\(940\) −22.7250 −0.741207
\(941\) 37.4222 1.21993 0.609965 0.792429i \(-0.291184\pi\)
0.609965 + 0.792429i \(0.291184\pi\)
\(942\) −38.2389 −1.24589
\(943\) −18.8806 −0.614836
\(944\) 2.02776 0.0659978
\(945\) −5.78890 −0.188313
\(946\) 0 0
\(947\) 46.6611 1.51628 0.758140 0.652091i \(-0.226108\pi\)
0.758140 + 0.652091i \(0.226108\pi\)
\(948\) 63.1472 2.05093
\(949\) −33.0278 −1.07213
\(950\) −55.2666 −1.79309
\(951\) −39.8444 −1.29204
\(952\) −8.09167 −0.262253
\(953\) −6.02776 −0.195258 −0.0976291 0.995223i \(-0.531126\pi\)
−0.0976291 + 0.995223i \(0.531126\pi\)
\(954\) −68.4500 −2.21615
\(955\) −96.6888 −3.12878
\(956\) 86.9638 2.81261
\(957\) 0 0
\(958\) −65.9361 −2.13030
\(959\) 10.1194 0.326773
\(960\) 106.414 3.43449
\(961\) −30.0000 −0.967742
\(962\) −79.2666 −2.55566
\(963\) 28.5416 0.919741
\(964\) 1.60555 0.0517113
\(965\) −7.64171 −0.245995
\(966\) −14.3028 −0.460184
\(967\) 40.5139 1.30284 0.651419 0.758718i \(-0.274174\pi\)
0.651419 + 0.758718i \(0.274174\pi\)
\(968\) 0 0
\(969\) 18.6333 0.598588
\(970\) −29.9361 −0.961190
\(971\) −6.36669 −0.204317 −0.102158 0.994768i \(-0.532575\pi\)
−0.102158 + 0.994768i \(0.532575\pi\)
\(972\) 64.7527 2.07695
\(973\) −21.6056 −0.692642
\(974\) −6.48612 −0.207829
\(975\) −121.689 −3.89716
\(976\) −1.30278 −0.0417008
\(977\) 40.0278 1.28060 0.640301 0.768124i \(-0.278810\pi\)
0.640301 + 0.768124i \(0.278810\pi\)
\(978\) −14.9361 −0.477603
\(979\) 0 0
\(980\) 11.9083 0.380398
\(981\) −18.4222 −0.588176
\(982\) −29.2389 −0.933049
\(983\) −9.27502 −0.295827 −0.147914 0.989000i \(-0.547256\pi\)
−0.147914 + 0.989000i \(0.547256\pi\)
\(984\) −48.3583 −1.54161
\(985\) −38.2389 −1.21839
\(986\) 29.1749 0.929119
\(987\) −4.39445 −0.139877
\(988\) −65.4500 −2.08224
\(989\) −4.57779 −0.145565
\(990\) 0 0
\(991\) −43.7250 −1.38897 −0.694485 0.719507i \(-0.744368\pi\)
−0.694485 + 0.719507i \(0.744368\pi\)
\(992\) −5.30278 −0.168363
\(993\) −54.8444 −1.74043
\(994\) 9.90833 0.314273
\(995\) 70.0278 2.22003
\(996\) −22.8167 −0.722973
\(997\) 45.6972 1.44725 0.723623 0.690196i \(-0.242476\pi\)
0.723623 + 0.690196i \(0.242476\pi\)
\(998\) 6.69722 0.211997
\(999\) −8.36669 −0.264710
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 847.2.a.g.1.2 yes 2
3.2 odd 2 7623.2.a.bc.1.1 2
7.6 odd 2 5929.2.a.p.1.2 2
11.2 odd 10 847.2.f.r.323.2 8
11.3 even 5 847.2.f.o.372.2 8
11.4 even 5 847.2.f.o.148.2 8
11.5 even 5 847.2.f.o.729.1 8
11.6 odd 10 847.2.f.r.729.2 8
11.7 odd 10 847.2.f.r.148.1 8
11.8 odd 10 847.2.f.r.372.1 8
11.9 even 5 847.2.f.o.323.1 8
11.10 odd 2 847.2.a.e.1.1 2
33.32 even 2 7623.2.a.bs.1.2 2
77.76 even 2 5929.2.a.k.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
847.2.a.e.1.1 2 11.10 odd 2
847.2.a.g.1.2 yes 2 1.1 even 1 trivial
847.2.f.o.148.2 8 11.4 even 5
847.2.f.o.323.1 8 11.9 even 5
847.2.f.o.372.2 8 11.3 even 5
847.2.f.o.729.1 8 11.5 even 5
847.2.f.r.148.1 8 11.7 odd 10
847.2.f.r.323.2 8 11.2 odd 10
847.2.f.r.372.1 8 11.8 odd 10
847.2.f.r.729.2 8 11.6 odd 10
5929.2.a.k.1.1 2 77.76 even 2
5929.2.a.p.1.2 2 7.6 odd 2
7623.2.a.bc.1.1 2 3.2 odd 2
7623.2.a.bs.1.2 2 33.32 even 2