Properties

Label 7098.2.a.bn.1.2
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.46410 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.46410 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.46410 q^{10} +4.00000 q^{11} +1.00000 q^{12} -1.00000 q^{14} +3.46410 q^{15} +1.00000 q^{16} -0.464102 q^{17} -1.00000 q^{18} +0.535898 q^{19} +3.46410 q^{20} +1.00000 q^{21} -4.00000 q^{22} +2.26795 q^{23} -1.00000 q^{24} +7.00000 q^{25} +1.00000 q^{27} +1.00000 q^{28} +8.00000 q^{29} -3.46410 q^{30} +0.464102 q^{31} -1.00000 q^{32} +4.00000 q^{33} +0.464102 q^{34} +3.46410 q^{35} +1.00000 q^{36} +8.92820 q^{37} -0.535898 q^{38} -3.46410 q^{40} -6.92820 q^{41} -1.00000 q^{42} -2.46410 q^{43} +4.00000 q^{44} +3.46410 q^{45} -2.26795 q^{46} -7.46410 q^{47} +1.00000 q^{48} +1.00000 q^{49} -7.00000 q^{50} -0.464102 q^{51} -11.7321 q^{53} -1.00000 q^{54} +13.8564 q^{55} -1.00000 q^{56} +0.535898 q^{57} -8.00000 q^{58} +9.73205 q^{59} +3.46410 q^{60} +5.19615 q^{61} -0.464102 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} +3.19615 q^{67} -0.464102 q^{68} +2.26795 q^{69} -3.46410 q^{70} -11.9282 q^{71} -1.00000 q^{72} +2.92820 q^{73} -8.92820 q^{74} +7.00000 q^{75} +0.535898 q^{76} +4.00000 q^{77} -2.53590 q^{79} +3.46410 q^{80} +1.00000 q^{81} +6.92820 q^{82} -1.73205 q^{83} +1.00000 q^{84} -1.60770 q^{85} +2.46410 q^{86} +8.00000 q^{87} -4.00000 q^{88} -8.26795 q^{89} -3.46410 q^{90} +2.26795 q^{92} +0.464102 q^{93} +7.46410 q^{94} +1.85641 q^{95} -1.00000 q^{96} -8.39230 q^{97} -1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + 8q^{11} + 2q^{12} - 2q^{14} + 2q^{16} + 6q^{17} - 2q^{18} + 8q^{19} + 2q^{21} - 8q^{22} + 8q^{23} - 2q^{24} + 14q^{25} + 2q^{27} + 2q^{28} + 16q^{29} - 6q^{31} - 2q^{32} + 8q^{33} - 6q^{34} + 2q^{36} + 4q^{37} - 8q^{38} - 2q^{42} + 2q^{43} + 8q^{44} - 8q^{46} - 8q^{47} + 2q^{48} + 2q^{49} - 14q^{50} + 6q^{51} - 20q^{53} - 2q^{54} - 2q^{56} + 8q^{57} - 16q^{58} + 16q^{59} + 6q^{62} + 2q^{63} + 2q^{64} - 8q^{66} - 4q^{67} + 6q^{68} + 8q^{69} - 10q^{71} - 2q^{72} - 8q^{73} - 4q^{74} + 14q^{75} + 8q^{76} + 8q^{77} - 12q^{79} + 2q^{81} + 2q^{84} - 24q^{85} - 2q^{86} + 16q^{87} - 8q^{88} - 20q^{89} + 8q^{92} - 6q^{93} + 8q^{94} - 24q^{95} - 2q^{96} + 4q^{97} - 2q^{98} + 8q^{99} + O(q^{100}) \)

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
\(5\) 3.46410 1.54919 0.774597 0.632456i \(-0.217953\pi\)
0.774597 + 0.632456i \(0.217953\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.46410 −1.09545
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) 3.46410 0.894427
\(16\) 1.00000 0.250000
\(17\) −0.464102 −0.112561 −0.0562806 0.998415i \(-0.517924\pi\)
−0.0562806 + 0.998415i \(0.517924\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0.535898 0.122944 0.0614718 0.998109i \(-0.480421\pi\)
0.0614718 + 0.998109i \(0.480421\pi\)
\(20\) 3.46410 0.774597
\(21\) 1.00000 0.218218
\(22\) −4.00000 −0.852803
\(23\) 2.26795 0.472900 0.236450 0.971644i \(-0.424016\pi\)
0.236450 + 0.971644i \(0.424016\pi\)
\(24\) −1.00000 −0.204124
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) −3.46410 −0.632456
\(31\) 0.464102 0.0833551 0.0416776 0.999131i \(-0.486730\pi\)
0.0416776 + 0.999131i \(0.486730\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) 0.464102 0.0795928
\(35\) 3.46410 0.585540
\(36\) 1.00000 0.166667
\(37\) 8.92820 1.46779 0.733894 0.679264i \(-0.237701\pi\)
0.733894 + 0.679264i \(0.237701\pi\)
\(38\) −0.535898 −0.0869342
\(39\) 0 0
\(40\) −3.46410 −0.547723
\(41\) −6.92820 −1.08200 −0.541002 0.841021i \(-0.681955\pi\)
−0.541002 + 0.841021i \(0.681955\pi\)
\(42\) −1.00000 −0.154303
\(43\) −2.46410 −0.375772 −0.187886 0.982191i \(-0.560164\pi\)
−0.187886 + 0.982191i \(0.560164\pi\)
\(44\) 4.00000 0.603023
\(45\) 3.46410 0.516398
\(46\) −2.26795 −0.334391
\(47\) −7.46410 −1.08875 −0.544376 0.838842i \(-0.683233\pi\)
−0.544376 + 0.838842i \(0.683233\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −7.00000 −0.989949
\(51\) −0.464102 −0.0649872
\(52\) 0 0
\(53\) −11.7321 −1.61152 −0.805761 0.592241i \(-0.798243\pi\)
−0.805761 + 0.592241i \(0.798243\pi\)
\(54\) −1.00000 −0.136083
\(55\) 13.8564 1.86840
\(56\) −1.00000 −0.133631
\(57\) 0.535898 0.0709815
\(58\) −8.00000 −1.05045
\(59\) 9.73205 1.26701 0.633503 0.773741i \(-0.281617\pi\)
0.633503 + 0.773741i \(0.281617\pi\)
\(60\) 3.46410 0.447214
\(61\) 5.19615 0.665299 0.332650 0.943051i \(-0.392057\pi\)
0.332650 + 0.943051i \(0.392057\pi\)
\(62\) −0.464102 −0.0589410
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) 3.19615 0.390472 0.195236 0.980756i \(-0.437453\pi\)
0.195236 + 0.980756i \(0.437453\pi\)
\(68\) −0.464102 −0.0562806
\(69\) 2.26795 0.273029
\(70\) −3.46410 −0.414039
\(71\) −11.9282 −1.41562 −0.707809 0.706404i \(-0.750316\pi\)
−0.707809 + 0.706404i \(0.750316\pi\)
\(72\) −1.00000 −0.117851
\(73\) 2.92820 0.342720 0.171360 0.985208i \(-0.445184\pi\)
0.171360 + 0.985208i \(0.445184\pi\)
\(74\) −8.92820 −1.03788
\(75\) 7.00000 0.808290
\(76\) 0.535898 0.0614718
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −2.53590 −0.285311 −0.142655 0.989772i \(-0.545564\pi\)
−0.142655 + 0.989772i \(0.545564\pi\)
\(80\) 3.46410 0.387298
\(81\) 1.00000 0.111111
\(82\) 6.92820 0.765092
\(83\) −1.73205 −0.190117 −0.0950586 0.995472i \(-0.530304\pi\)
−0.0950586 + 0.995472i \(0.530304\pi\)
\(84\) 1.00000 0.109109
\(85\) −1.60770 −0.174379
\(86\) 2.46410 0.265711
\(87\) 8.00000 0.857690
\(88\) −4.00000 −0.426401
\(89\) −8.26795 −0.876401 −0.438200 0.898877i \(-0.644384\pi\)
−0.438200 + 0.898877i \(0.644384\pi\)
\(90\) −3.46410 −0.365148
\(91\) 0 0
\(92\) 2.26795 0.236450
\(93\) 0.464102 0.0481251
\(94\) 7.46410 0.769863
\(95\) 1.85641 0.190463
\(96\) −1.00000 −0.102062
\(97\) −8.39230 −0.852109 −0.426055 0.904697i \(-0.640097\pi\)
−0.426055 + 0.904697i \(0.640097\pi\)
\(98\) −1.00000 −0.101015
\(99\) 4.00000 0.402015
\(100\) 7.00000 0.700000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0.464102 0.0459529
\(103\) −6.66025 −0.656254 −0.328127 0.944634i \(-0.606417\pi\)
−0.328127 + 0.944634i \(0.606417\pi\)
\(104\) 0 0
\(105\) 3.46410 0.338062
\(106\) 11.7321 1.13952
\(107\) 15.4641 1.49497 0.747486 0.664278i \(-0.231261\pi\)
0.747486 + 0.664278i \(0.231261\pi\)
\(108\) 1.00000 0.0962250
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) −13.8564 −1.32116
\(111\) 8.92820 0.847428
\(112\) 1.00000 0.0944911
\(113\) 16.3923 1.54206 0.771029 0.636800i \(-0.219742\pi\)
0.771029 + 0.636800i \(0.219742\pi\)
\(114\) −0.535898 −0.0501915
\(115\) 7.85641 0.732614
\(116\) 8.00000 0.742781
\(117\) 0 0
\(118\) −9.73205 −0.895908
\(119\) −0.464102 −0.0425441
\(120\) −3.46410 −0.316228
\(121\) 5.00000 0.454545
\(122\) −5.19615 −0.470438
\(123\) −6.92820 −0.624695
\(124\) 0.464102 0.0416776
\(125\) 6.92820 0.619677
\(126\) −1.00000 −0.0890871
\(127\) 18.9282 1.67961 0.839803 0.542891i \(-0.182670\pi\)
0.839803 + 0.542891i \(0.182670\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.46410 −0.216952
\(130\) 0 0
\(131\) 19.0000 1.66004 0.830019 0.557735i \(-0.188330\pi\)
0.830019 + 0.557735i \(0.188330\pi\)
\(132\) 4.00000 0.348155
\(133\) 0.535898 0.0464683
\(134\) −3.19615 −0.276106
\(135\) 3.46410 0.298142
\(136\) 0.464102 0.0397964
\(137\) −13.8564 −1.18383 −0.591916 0.805999i \(-0.701628\pi\)
−0.591916 + 0.805999i \(0.701628\pi\)
\(138\) −2.26795 −0.193061
\(139\) −10.9282 −0.926918 −0.463459 0.886118i \(-0.653392\pi\)
−0.463459 + 0.886118i \(0.653392\pi\)
\(140\) 3.46410 0.292770
\(141\) −7.46410 −0.628591
\(142\) 11.9282 1.00099
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 27.7128 2.30142
\(146\) −2.92820 −0.242340
\(147\) 1.00000 0.0824786
\(148\) 8.92820 0.733894
\(149\) −18.4641 −1.51264 −0.756319 0.654203i \(-0.773004\pi\)
−0.756319 + 0.654203i \(0.773004\pi\)
\(150\) −7.00000 −0.571548
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −0.535898 −0.0434671
\(153\) −0.464102 −0.0375204
\(154\) −4.00000 −0.322329
\(155\) 1.60770 0.129133
\(156\) 0 0
\(157\) 10.9282 0.872166 0.436083 0.899907i \(-0.356365\pi\)
0.436083 + 0.899907i \(0.356365\pi\)
\(158\) 2.53590 0.201745
\(159\) −11.7321 −0.930412
\(160\) −3.46410 −0.273861
\(161\) 2.26795 0.178739
\(162\) −1.00000 −0.0785674
\(163\) −18.6603 −1.46158 −0.730792 0.682600i \(-0.760849\pi\)
−0.730792 + 0.682600i \(0.760849\pi\)
\(164\) −6.92820 −0.541002
\(165\) 13.8564 1.07872
\(166\) 1.73205 0.134433
\(167\) 20.0000 1.54765 0.773823 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 0 0
\(170\) 1.60770 0.123305
\(171\) 0.535898 0.0409812
\(172\) −2.46410 −0.187886
\(173\) 14.9282 1.13497 0.567485 0.823384i \(-0.307916\pi\)
0.567485 + 0.823384i \(0.307916\pi\)
\(174\) −8.00000 −0.606478
\(175\) 7.00000 0.529150
\(176\) 4.00000 0.301511
\(177\) 9.73205 0.731506
\(178\) 8.26795 0.619709
\(179\) −3.07180 −0.229597 −0.114798 0.993389i \(-0.536622\pi\)
−0.114798 + 0.993389i \(0.536622\pi\)
\(180\) 3.46410 0.258199
\(181\) −16.7846 −1.24759 −0.623795 0.781588i \(-0.714410\pi\)
−0.623795 + 0.781588i \(0.714410\pi\)
\(182\) 0 0
\(183\) 5.19615 0.384111
\(184\) −2.26795 −0.167195
\(185\) 30.9282 2.27389
\(186\) −0.464102 −0.0340296
\(187\) −1.85641 −0.135754
\(188\) −7.46410 −0.544376
\(189\) 1.00000 0.0727393
\(190\) −1.85641 −0.134678
\(191\) 18.5167 1.33982 0.669909 0.742443i \(-0.266333\pi\)
0.669909 + 0.742443i \(0.266333\pi\)
\(192\) 1.00000 0.0721688
\(193\) −9.85641 −0.709480 −0.354740 0.934965i \(-0.615431\pi\)
−0.354740 + 0.934965i \(0.615431\pi\)
\(194\) 8.39230 0.602532
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −23.3923 −1.66663 −0.833316 0.552796i \(-0.813561\pi\)
−0.833316 + 0.552796i \(0.813561\pi\)
\(198\) −4.00000 −0.284268
\(199\) 2.12436 0.150592 0.0752958 0.997161i \(-0.476010\pi\)
0.0752958 + 0.997161i \(0.476010\pi\)
\(200\) −7.00000 −0.494975
\(201\) 3.19615 0.225439
\(202\) 0 0
\(203\) 8.00000 0.561490
\(204\) −0.464102 −0.0324936
\(205\) −24.0000 −1.67623
\(206\) 6.66025 0.464042
\(207\) 2.26795 0.157633
\(208\) 0 0
\(209\) 2.14359 0.148275
\(210\) −3.46410 −0.239046
\(211\) −22.9282 −1.57844 −0.789221 0.614109i \(-0.789516\pi\)
−0.789221 + 0.614109i \(0.789516\pi\)
\(212\) −11.7321 −0.805761
\(213\) −11.9282 −0.817307
\(214\) −15.4641 −1.05710
\(215\) −8.53590 −0.582143
\(216\) −1.00000 −0.0680414
\(217\) 0.464102 0.0315053
\(218\) 12.0000 0.812743
\(219\) 2.92820 0.197870
\(220\) 13.8564 0.934199
\(221\) 0 0
\(222\) −8.92820 −0.599222
\(223\) 14.4641 0.968588 0.484294 0.874905i \(-0.339077\pi\)
0.484294 + 0.874905i \(0.339077\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 7.00000 0.466667
\(226\) −16.3923 −1.09040
\(227\) 21.3205 1.41509 0.707546 0.706667i \(-0.249802\pi\)
0.707546 + 0.706667i \(0.249802\pi\)
\(228\) 0.535898 0.0354907
\(229\) 6.07180 0.401236 0.200618 0.979670i \(-0.435705\pi\)
0.200618 + 0.979670i \(0.435705\pi\)
\(230\) −7.85641 −0.518036
\(231\) 4.00000 0.263181
\(232\) −8.00000 −0.525226
\(233\) 6.53590 0.428181 0.214090 0.976814i \(-0.431321\pi\)
0.214090 + 0.976814i \(0.431321\pi\)
\(234\) 0 0
\(235\) −25.8564 −1.68669
\(236\) 9.73205 0.633503
\(237\) −2.53590 −0.164724
\(238\) 0.464102 0.0300832
\(239\) −14.8564 −0.960981 −0.480491 0.877000i \(-0.659541\pi\)
−0.480491 + 0.877000i \(0.659541\pi\)
\(240\) 3.46410 0.223607
\(241\) 11.8564 0.763738 0.381869 0.924216i \(-0.375281\pi\)
0.381869 + 0.924216i \(0.375281\pi\)
\(242\) −5.00000 −0.321412
\(243\) 1.00000 0.0641500
\(244\) 5.19615 0.332650
\(245\) 3.46410 0.221313
\(246\) 6.92820 0.441726
\(247\) 0 0
\(248\) −0.464102 −0.0294705
\(249\) −1.73205 −0.109764
\(250\) −6.92820 −0.438178
\(251\) −23.9282 −1.51033 −0.755167 0.655532i \(-0.772444\pi\)
−0.755167 + 0.655532i \(0.772444\pi\)
\(252\) 1.00000 0.0629941
\(253\) 9.07180 0.570339
\(254\) −18.9282 −1.18766
\(255\) −1.60770 −0.100678
\(256\) 1.00000 0.0625000
\(257\) −6.46410 −0.403220 −0.201610 0.979466i \(-0.564617\pi\)
−0.201610 + 0.979466i \(0.564617\pi\)
\(258\) 2.46410 0.153408
\(259\) 8.92820 0.554772
\(260\) 0 0
\(261\) 8.00000 0.495188
\(262\) −19.0000 −1.17382
\(263\) 17.3205 1.06803 0.534014 0.845476i \(-0.320683\pi\)
0.534014 + 0.845476i \(0.320683\pi\)
\(264\) −4.00000 −0.246183
\(265\) −40.6410 −2.49656
\(266\) −0.535898 −0.0328580
\(267\) −8.26795 −0.505990
\(268\) 3.19615 0.195236
\(269\) 1.60770 0.0980229 0.0490115 0.998798i \(-0.484393\pi\)
0.0490115 + 0.998798i \(0.484393\pi\)
\(270\) −3.46410 −0.210819
\(271\) 5.53590 0.336282 0.168141 0.985763i \(-0.446224\pi\)
0.168141 + 0.985763i \(0.446224\pi\)
\(272\) −0.464102 −0.0281403
\(273\) 0 0
\(274\) 13.8564 0.837096
\(275\) 28.0000 1.68846
\(276\) 2.26795 0.136514
\(277\) −27.3205 −1.64153 −0.820765 0.571266i \(-0.806453\pi\)
−0.820765 + 0.571266i \(0.806453\pi\)
\(278\) 10.9282 0.655430
\(279\) 0.464102 0.0277850
\(280\) −3.46410 −0.207020
\(281\) 18.3923 1.09719 0.548596 0.836087i \(-0.315162\pi\)
0.548596 + 0.836087i \(0.315162\pi\)
\(282\) 7.46410 0.444481
\(283\) −4.53590 −0.269631 −0.134816 0.990871i \(-0.543044\pi\)
−0.134816 + 0.990871i \(0.543044\pi\)
\(284\) −11.9282 −0.707809
\(285\) 1.85641 0.109964
\(286\) 0 0
\(287\) −6.92820 −0.408959
\(288\) −1.00000 −0.0589256
\(289\) −16.7846 −0.987330
\(290\) −27.7128 −1.62735
\(291\) −8.39230 −0.491966
\(292\) 2.92820 0.171360
\(293\) 26.5359 1.55024 0.775122 0.631812i \(-0.217688\pi\)
0.775122 + 0.631812i \(0.217688\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 33.7128 1.96284
\(296\) −8.92820 −0.518941
\(297\) 4.00000 0.232104
\(298\) 18.4641 1.06960
\(299\) 0 0
\(300\) 7.00000 0.404145
\(301\) −2.46410 −0.142028
\(302\) 12.0000 0.690522
\(303\) 0 0
\(304\) 0.535898 0.0307359
\(305\) 18.0000 1.03068
\(306\) 0.464102 0.0265309
\(307\) −22.0000 −1.25561 −0.627803 0.778372i \(-0.716046\pi\)
−0.627803 + 0.778372i \(0.716046\pi\)
\(308\) 4.00000 0.227921
\(309\) −6.66025 −0.378889
\(310\) −1.60770 −0.0913109
\(311\) −21.3205 −1.20898 −0.604488 0.796615i \(-0.706622\pi\)
−0.604488 + 0.796615i \(0.706622\pi\)
\(312\) 0 0
\(313\) 35.3205 1.99643 0.998217 0.0596964i \(-0.0190133\pi\)
0.998217 + 0.0596964i \(0.0190133\pi\)
\(314\) −10.9282 −0.616714
\(315\) 3.46410 0.195180
\(316\) −2.53590 −0.142655
\(317\) 9.53590 0.535589 0.267795 0.963476i \(-0.413705\pi\)
0.267795 + 0.963476i \(0.413705\pi\)
\(318\) 11.7321 0.657901
\(319\) 32.0000 1.79166
\(320\) 3.46410 0.193649
\(321\) 15.4641 0.863122
\(322\) −2.26795 −0.126388
\(323\) −0.248711 −0.0138387
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 18.6603 1.03350
\(327\) −12.0000 −0.663602
\(328\) 6.92820 0.382546
\(329\) −7.46410 −0.411509
\(330\) −13.8564 −0.762770
\(331\) −13.3205 −0.732161 −0.366081 0.930583i \(-0.619301\pi\)
−0.366081 + 0.930583i \(0.619301\pi\)
\(332\) −1.73205 −0.0950586
\(333\) 8.92820 0.489263
\(334\) −20.0000 −1.09435
\(335\) 11.0718 0.604917
\(336\) 1.00000 0.0545545
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 0 0
\(339\) 16.3923 0.890308
\(340\) −1.60770 −0.0871895
\(341\) 1.85641 0.100530
\(342\) −0.535898 −0.0289781
\(343\) 1.00000 0.0539949
\(344\) 2.46410 0.132855
\(345\) 7.85641 0.422975
\(346\) −14.9282 −0.802545
\(347\) −29.7128 −1.59507 −0.797534 0.603274i \(-0.793862\pi\)
−0.797534 + 0.603274i \(0.793862\pi\)
\(348\) 8.00000 0.428845
\(349\) 33.6410 1.80076 0.900381 0.435102i \(-0.143288\pi\)
0.900381 + 0.435102i \(0.143288\pi\)
\(350\) −7.00000 −0.374166
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) −26.6603 −1.41898 −0.709491 0.704714i \(-0.751075\pi\)
−0.709491 + 0.704714i \(0.751075\pi\)
\(354\) −9.73205 −0.517253
\(355\) −41.3205 −2.19306
\(356\) −8.26795 −0.438200
\(357\) −0.464102 −0.0245629
\(358\) 3.07180 0.162350
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) −3.46410 −0.182574
\(361\) −18.7128 −0.984885
\(362\) 16.7846 0.882179
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 10.1436 0.530940
\(366\) −5.19615 −0.271607
\(367\) 5.05256 0.263741 0.131871 0.991267i \(-0.457902\pi\)
0.131871 + 0.991267i \(0.457902\pi\)
\(368\) 2.26795 0.118225
\(369\) −6.92820 −0.360668
\(370\) −30.9282 −1.60788
\(371\) −11.7321 −0.609098
\(372\) 0.464102 0.0240625
\(373\) 22.2487 1.15199 0.575997 0.817452i \(-0.304614\pi\)
0.575997 + 0.817452i \(0.304614\pi\)
\(374\) 1.85641 0.0959925
\(375\) 6.92820 0.357771
\(376\) 7.46410 0.384932
\(377\) 0 0
\(378\) −1.00000 −0.0514344
\(379\) 30.3923 1.56115 0.780574 0.625063i \(-0.214927\pi\)
0.780574 + 0.625063i \(0.214927\pi\)
\(380\) 1.85641 0.0952316
\(381\) 18.9282 0.969721
\(382\) −18.5167 −0.947395
\(383\) −27.3205 −1.39601 −0.698006 0.716092i \(-0.745929\pi\)
−0.698006 + 0.716092i \(0.745929\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 13.8564 0.706188
\(386\) 9.85641 0.501678
\(387\) −2.46410 −0.125257
\(388\) −8.39230 −0.426055
\(389\) 2.66025 0.134880 0.0674401 0.997723i \(-0.478517\pi\)
0.0674401 + 0.997723i \(0.478517\pi\)
\(390\) 0 0
\(391\) −1.05256 −0.0532302
\(392\) −1.00000 −0.0505076
\(393\) 19.0000 0.958423
\(394\) 23.3923 1.17849
\(395\) −8.78461 −0.442002
\(396\) 4.00000 0.201008
\(397\) −11.7846 −0.591453 −0.295726 0.955273i \(-0.595562\pi\)
−0.295726 + 0.955273i \(0.595562\pi\)
\(398\) −2.12436 −0.106484
\(399\) 0.535898 0.0268285
\(400\) 7.00000 0.350000
\(401\) 4.39230 0.219341 0.109671 0.993968i \(-0.465020\pi\)
0.109671 + 0.993968i \(0.465020\pi\)
\(402\) −3.19615 −0.159410
\(403\) 0 0
\(404\) 0 0
\(405\) 3.46410 0.172133
\(406\) −8.00000 −0.397033
\(407\) 35.7128 1.77022
\(408\) 0.464102 0.0229765
\(409\) −14.3923 −0.711654 −0.355827 0.934552i \(-0.615801\pi\)
−0.355827 + 0.934552i \(0.615801\pi\)
\(410\) 24.0000 1.18528
\(411\) −13.8564 −0.683486
\(412\) −6.66025 −0.328127
\(413\) 9.73205 0.478883
\(414\) −2.26795 −0.111464
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) −10.9282 −0.535156
\(418\) −2.14359 −0.104847
\(419\) −5.00000 −0.244266 −0.122133 0.992514i \(-0.538973\pi\)
−0.122133 + 0.992514i \(0.538973\pi\)
\(420\) 3.46410 0.169031
\(421\) −22.3923 −1.09133 −0.545667 0.838002i \(-0.683724\pi\)
−0.545667 + 0.838002i \(0.683724\pi\)
\(422\) 22.9282 1.11613
\(423\) −7.46410 −0.362917
\(424\) 11.7321 0.569759
\(425\) −3.24871 −0.157586
\(426\) 11.9282 0.577923
\(427\) 5.19615 0.251459
\(428\) 15.4641 0.747486
\(429\) 0 0
\(430\) 8.53590 0.411638
\(431\) 22.8564 1.10095 0.550477 0.834850i \(-0.314446\pi\)
0.550477 + 0.834850i \(0.314446\pi\)
\(432\) 1.00000 0.0481125
\(433\) −17.8564 −0.858124 −0.429062 0.903275i \(-0.641156\pi\)
−0.429062 + 0.903275i \(0.641156\pi\)
\(434\) −0.464102 −0.0222776
\(435\) 27.7128 1.32873
\(436\) −12.0000 −0.574696
\(437\) 1.21539 0.0581400
\(438\) −2.92820 −0.139915
\(439\) −34.3923 −1.64146 −0.820728 0.571320i \(-0.806432\pi\)
−0.820728 + 0.571320i \(0.806432\pi\)
\(440\) −13.8564 −0.660578
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −1.60770 −0.0763839 −0.0381920 0.999270i \(-0.512160\pi\)
−0.0381920 + 0.999270i \(0.512160\pi\)
\(444\) 8.92820 0.423714
\(445\) −28.6410 −1.35771
\(446\) −14.4641 −0.684895
\(447\) −18.4641 −0.873322
\(448\) 1.00000 0.0472456
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) −7.00000 −0.329983
\(451\) −27.7128 −1.30495
\(452\) 16.3923 0.771029
\(453\) −12.0000 −0.563809
\(454\) −21.3205 −1.00062
\(455\) 0 0
\(456\) −0.535898 −0.0250957
\(457\) 9.19615 0.430178 0.215089 0.976594i \(-0.430996\pi\)
0.215089 + 0.976594i \(0.430996\pi\)
\(458\) −6.07180 −0.283716
\(459\) −0.464102 −0.0216624
\(460\) 7.85641 0.366307
\(461\) −18.9282 −0.881574 −0.440787 0.897612i \(-0.645301\pi\)
−0.440787 + 0.897612i \(0.645301\pi\)
\(462\) −4.00000 −0.186097
\(463\) −40.2487 −1.87052 −0.935258 0.353966i \(-0.884833\pi\)
−0.935258 + 0.353966i \(0.884833\pi\)
\(464\) 8.00000 0.371391
\(465\) 1.60770 0.0745551
\(466\) −6.53590 −0.302770
\(467\) 29.6410 1.37162 0.685811 0.727779i \(-0.259448\pi\)
0.685811 + 0.727779i \(0.259448\pi\)
\(468\) 0 0
\(469\) 3.19615 0.147585
\(470\) 25.8564 1.19267
\(471\) 10.9282 0.503545
\(472\) −9.73205 −0.447954
\(473\) −9.85641 −0.453198
\(474\) 2.53590 0.116478
\(475\) 3.75129 0.172121
\(476\) −0.464102 −0.0212721
\(477\) −11.7321 −0.537174
\(478\) 14.8564 0.679516
\(479\) 16.3923 0.748984 0.374492 0.927230i \(-0.377817\pi\)
0.374492 + 0.927230i \(0.377817\pi\)
\(480\) −3.46410 −0.158114
\(481\) 0 0
\(482\) −11.8564 −0.540045
\(483\) 2.26795 0.103195
\(484\) 5.00000 0.227273
\(485\) −29.0718 −1.32008
\(486\) −1.00000 −0.0453609
\(487\) −20.3923 −0.924064 −0.462032 0.886863i \(-0.652879\pi\)
−0.462032 + 0.886863i \(0.652879\pi\)
\(488\) −5.19615 −0.235219
\(489\) −18.6603 −0.843846
\(490\) −3.46410 −0.156492
\(491\) 17.8564 0.805848 0.402924 0.915233i \(-0.367994\pi\)
0.402924 + 0.915233i \(0.367994\pi\)
\(492\) −6.92820 −0.312348
\(493\) −3.71281 −0.167217
\(494\) 0 0
\(495\) 13.8564 0.622799
\(496\) 0.464102 0.0208388
\(497\) −11.9282 −0.535053
\(498\) 1.73205 0.0776151
\(499\) −3.73205 −0.167070 −0.0835348 0.996505i \(-0.526621\pi\)
−0.0835348 + 0.996505i \(0.526621\pi\)
\(500\) 6.92820 0.309839
\(501\) 20.0000 0.893534
\(502\) 23.9282 1.06797
\(503\) 6.92820 0.308913 0.154457 0.988000i \(-0.450637\pi\)
0.154457 + 0.988000i \(0.450637\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) −9.07180 −0.403291
\(507\) 0 0
\(508\) 18.9282 0.839803
\(509\) −14.2487 −0.631563 −0.315782 0.948832i \(-0.602267\pi\)
−0.315782 + 0.948832i \(0.602267\pi\)
\(510\) 1.60770 0.0711899
\(511\) 2.92820 0.129536
\(512\) −1.00000 −0.0441942
\(513\) 0.535898 0.0236605
\(514\) 6.46410 0.285119
\(515\) −23.0718 −1.01666
\(516\) −2.46410 −0.108476
\(517\) −29.8564 −1.31308
\(518\) −8.92820 −0.392283
\(519\) 14.9282 0.655275
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −8.00000 −0.350150
\(523\) 1.07180 0.0468664 0.0234332 0.999725i \(-0.492540\pi\)
0.0234332 + 0.999725i \(0.492540\pi\)
\(524\) 19.0000 0.830019
\(525\) 7.00000 0.305505
\(526\) −17.3205 −0.755210
\(527\) −0.215390 −0.00938255
\(528\) 4.00000 0.174078
\(529\) −17.8564 −0.776365
\(530\) 40.6410 1.76533
\(531\) 9.73205 0.422335
\(532\) 0.535898 0.0232341
\(533\) 0 0
\(534\) 8.26795 0.357789
\(535\) 53.5692 2.31600
\(536\) −3.19615 −0.138053
\(537\) −3.07180 −0.132558
\(538\) −1.60770 −0.0693127
\(539\) 4.00000 0.172292
\(540\) 3.46410 0.149071
\(541\) 17.3205 0.744667 0.372333 0.928099i \(-0.378558\pi\)
0.372333 + 0.928099i \(0.378558\pi\)
\(542\) −5.53590 −0.237787
\(543\) −16.7846 −0.720297
\(544\) 0.464102 0.0198982
\(545\) −41.5692 −1.78063
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −13.8564 −0.591916
\(549\) 5.19615 0.221766
\(550\) −28.0000 −1.19392
\(551\) 4.28719 0.182640
\(552\) −2.26795 −0.0965303
\(553\) −2.53590 −0.107837
\(554\) 27.3205 1.16074
\(555\) 30.9282 1.31283
\(556\) −10.9282 −0.463459
\(557\) 45.3923 1.92333 0.961667 0.274221i \(-0.0884198\pi\)
0.961667 + 0.274221i \(0.0884198\pi\)
\(558\) −0.464102 −0.0196470
\(559\) 0 0
\(560\) 3.46410 0.146385
\(561\) −1.85641 −0.0783775
\(562\) −18.3923 −0.775833
\(563\) 13.8564 0.583978 0.291989 0.956422i \(-0.405683\pi\)
0.291989 + 0.956422i \(0.405683\pi\)
\(564\) −7.46410 −0.314295
\(565\) 56.7846 2.38895
\(566\) 4.53590 0.190658
\(567\) 1.00000 0.0419961
\(568\) 11.9282 0.500496
\(569\) 8.53590 0.357843 0.178922 0.983863i \(-0.442739\pi\)
0.178922 + 0.983863i \(0.442739\pi\)
\(570\) −1.85641 −0.0777563
\(571\) 17.3923 0.727845 0.363923 0.931429i \(-0.381437\pi\)
0.363923 + 0.931429i \(0.381437\pi\)
\(572\) 0 0
\(573\) 18.5167 0.773545
\(574\) 6.92820 0.289178
\(575\) 15.8756 0.662060
\(576\) 1.00000 0.0416667
\(577\) 42.7846 1.78115 0.890573 0.454840i \(-0.150303\pi\)
0.890573 + 0.454840i \(0.150303\pi\)
\(578\) 16.7846 0.698148
\(579\) −9.85641 −0.409618
\(580\) 27.7128 1.15071
\(581\) −1.73205 −0.0718576
\(582\) 8.39230 0.347872
\(583\) −46.9282 −1.94357
\(584\) −2.92820 −0.121170
\(585\) 0 0
\(586\) −26.5359 −1.09619
\(587\) 16.1244 0.665523 0.332762 0.943011i \(-0.392020\pi\)
0.332762 + 0.943011i \(0.392020\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0.248711 0.0102480
\(590\) −33.7128 −1.38793
\(591\) −23.3923 −0.962231
\(592\) 8.92820 0.366947
\(593\) 7.73205 0.317517 0.158759 0.987317i \(-0.449251\pi\)
0.158759 + 0.987317i \(0.449251\pi\)
\(594\) −4.00000 −0.164122
\(595\) −1.60770 −0.0659091
\(596\) −18.4641 −0.756319
\(597\) 2.12436 0.0869441
\(598\) 0 0
\(599\) 19.0526 0.778466 0.389233 0.921139i \(-0.372740\pi\)
0.389233 + 0.921139i \(0.372740\pi\)
\(600\) −7.00000 −0.285774
\(601\) 24.2487 0.989126 0.494563 0.869142i \(-0.335328\pi\)
0.494563 + 0.869142i \(0.335328\pi\)
\(602\) 2.46410 0.100429
\(603\) 3.19615 0.130157
\(604\) −12.0000 −0.488273
\(605\) 17.3205 0.704179
\(606\) 0 0
\(607\) −4.80385 −0.194982 −0.0974911 0.995236i \(-0.531082\pi\)
−0.0974911 + 0.995236i \(0.531082\pi\)
\(608\) −0.535898 −0.0217335
\(609\) 8.00000 0.324176
\(610\) −18.0000 −0.728799
\(611\) 0 0
\(612\) −0.464102 −0.0187602
\(613\) −13.6077 −0.549610 −0.274805 0.961500i \(-0.588613\pi\)
−0.274805 + 0.961500i \(0.588613\pi\)
\(614\) 22.0000 0.887848
\(615\) −24.0000 −0.967773
\(616\) −4.00000 −0.161165
\(617\) 21.8564 0.879906 0.439953 0.898021i \(-0.354995\pi\)
0.439953 + 0.898021i \(0.354995\pi\)
\(618\) 6.66025 0.267915
\(619\) −10.7846 −0.433470 −0.216735 0.976230i \(-0.569541\pi\)
−0.216735 + 0.976230i \(0.569541\pi\)
\(620\) 1.60770 0.0645666
\(621\) 2.26795 0.0910097
\(622\) 21.3205 0.854874
\(623\) −8.26795 −0.331248
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) −35.3205 −1.41169
\(627\) 2.14359 0.0856069
\(628\) 10.9282 0.436083
\(629\) −4.14359 −0.165216
\(630\) −3.46410 −0.138013
\(631\) −21.0718 −0.838855 −0.419427 0.907789i \(-0.637769\pi\)
−0.419427 + 0.907789i \(0.637769\pi\)
\(632\) 2.53590 0.100873
\(633\) −22.9282 −0.911314
\(634\) −9.53590 −0.378719
\(635\) 65.5692 2.60204
\(636\) −11.7321 −0.465206
\(637\) 0 0
\(638\) −32.0000 −1.26689
\(639\) −11.9282 −0.471872
\(640\) −3.46410 −0.136931
\(641\) 42.6410 1.68422 0.842109 0.539307i \(-0.181314\pi\)
0.842109 + 0.539307i \(0.181314\pi\)
\(642\) −15.4641 −0.610319
\(643\) 28.2487 1.11402 0.557010 0.830506i \(-0.311948\pi\)
0.557010 + 0.830506i \(0.311948\pi\)
\(644\) 2.26795 0.0893697
\(645\) −8.53590 −0.336101
\(646\) 0.248711 0.00978542
\(647\) 2.67949 0.105342 0.0526708 0.998612i \(-0.483227\pi\)
0.0526708 + 0.998612i \(0.483227\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 38.9282 1.52807
\(650\) 0 0
\(651\) 0.464102 0.0181896
\(652\) −18.6603 −0.730792
\(653\) −6.41154 −0.250903 −0.125452 0.992100i \(-0.540038\pi\)
−0.125452 + 0.992100i \(0.540038\pi\)
\(654\) 12.0000 0.469237
\(655\) 65.8179 2.57172
\(656\) −6.92820 −0.270501
\(657\) 2.92820 0.114240
\(658\) 7.46410 0.290981
\(659\) 22.3923 0.872280 0.436140 0.899879i \(-0.356345\pi\)
0.436140 + 0.899879i \(0.356345\pi\)
\(660\) 13.8564 0.539360
\(661\) −4.71281 −0.183307 −0.0916536 0.995791i \(-0.529215\pi\)
−0.0916536 + 0.995791i \(0.529215\pi\)
\(662\) 13.3205 0.517716
\(663\) 0 0
\(664\) 1.73205 0.0672166
\(665\) 1.85641 0.0719884
\(666\) −8.92820 −0.345961
\(667\) 18.1436 0.702523
\(668\) 20.0000 0.773823
\(669\) 14.4641 0.559214
\(670\) −11.0718 −0.427741
\(671\) 20.7846 0.802381
\(672\) −1.00000 −0.0385758
\(673\) −29.6410 −1.14258 −0.571289 0.820749i \(-0.693556\pi\)
−0.571289 + 0.820749i \(0.693556\pi\)
\(674\) −6.00000 −0.231111
\(675\) 7.00000 0.269430
\(676\) 0 0
\(677\) 14.9282 0.573737 0.286869 0.957970i \(-0.407386\pi\)
0.286869 + 0.957970i \(0.407386\pi\)
\(678\) −16.3923 −0.629543
\(679\) −8.39230 −0.322067
\(680\) 1.60770 0.0616523
\(681\) 21.3205 0.817004
\(682\) −1.85641 −0.0710855
\(683\) 11.1769 0.427673 0.213836 0.976869i \(-0.431404\pi\)
0.213836 + 0.976869i \(0.431404\pi\)
\(684\) 0.535898 0.0204906
\(685\) −48.0000 −1.83399
\(686\) −1.00000 −0.0381802
\(687\) 6.07180 0.231653
\(688\) −2.46410 −0.0939430
\(689\) 0 0
\(690\) −7.85641 −0.299088
\(691\) −21.1769 −0.805608 −0.402804 0.915286i \(-0.631964\pi\)
−0.402804 + 0.915286i \(0.631964\pi\)
\(692\) 14.9282 0.567485
\(693\) 4.00000 0.151947
\(694\) 29.7128 1.12788
\(695\) −37.8564 −1.43598
\(696\) −8.00000 −0.303239
\(697\) 3.21539 0.121792
\(698\) −33.6410 −1.27333
\(699\) 6.53590 0.247210
\(700\) 7.00000 0.264575
\(701\) 22.1244 0.835625 0.417813 0.908533i \(-0.362797\pi\)
0.417813 + 0.908533i \(0.362797\pi\)
\(702\) 0 0
\(703\) 4.78461 0.180455
\(704\) 4.00000 0.150756
\(705\) −25.8564 −0.973809
\(706\) 26.6603 1.00337
\(707\) 0 0
\(708\) 9.73205 0.365753
\(709\) −47.7128 −1.79189 −0.895946 0.444163i \(-0.853501\pi\)
−0.895946 + 0.444163i \(0.853501\pi\)
\(710\) 41.3205 1.55073
\(711\) −2.53590 −0.0951036
\(712\) 8.26795 0.309854
\(713\) 1.05256 0.0394186
\(714\) 0.464102 0.0173686
\(715\) 0 0
\(716\) −3.07180 −0.114798
\(717\) −14.8564 −0.554823
\(718\) 8.00000 0.298557
\(719\) 41.4641 1.54635 0.773175 0.634193i \(-0.218667\pi\)
0.773175 + 0.634193i \(0.218667\pi\)
\(720\) 3.46410 0.129099
\(721\) −6.66025 −0.248041
\(722\) 18.7128 0.696419
\(723\) 11.8564 0.440945
\(724\) −16.7846 −0.623795
\(725\) 56.0000 2.07979
\(726\) −5.00000 −0.185567
\(727\) 42.9090 1.59141 0.795703 0.605687i \(-0.207102\pi\)
0.795703 + 0.605687i \(0.207102\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −10.1436 −0.375431
\(731\) 1.14359 0.0422973
\(732\) 5.19615 0.192055
\(733\) −22.8564 −0.844221 −0.422110 0.906544i \(-0.638711\pi\)
−0.422110 + 0.906544i \(0.638711\pi\)
\(734\) −5.05256 −0.186493
\(735\) 3.46410 0.127775
\(736\) −2.26795 −0.0835977
\(737\) 12.7846 0.470927
\(738\) 6.92820 0.255031
\(739\) 4.26795 0.156999 0.0784995 0.996914i \(-0.474987\pi\)
0.0784995 + 0.996914i \(0.474987\pi\)
\(740\) 30.9282 1.13694
\(741\) 0 0
\(742\) 11.7321 0.430697
\(743\) 39.7846 1.45956 0.729778 0.683684i \(-0.239623\pi\)
0.729778 + 0.683684i \(0.239623\pi\)
\(744\) −0.464102 −0.0170148
\(745\) −63.9615 −2.34337
\(746\) −22.2487 −0.814583
\(747\) −1.73205 −0.0633724
\(748\) −1.85641 −0.0678769
\(749\) 15.4641 0.565046
\(750\) −6.92820 −0.252982
\(751\) −5.32051 −0.194148 −0.0970740 0.995277i \(-0.530948\pi\)
−0.0970740 + 0.995277i \(0.530948\pi\)
\(752\) −7.46410 −0.272188
\(753\) −23.9282 −0.871992
\(754\) 0 0
\(755\) −41.5692 −1.51286
\(756\) 1.00000 0.0363696
\(757\) −7.85641 −0.285546 −0.142773 0.989755i \(-0.545602\pi\)
−0.142773 + 0.989755i \(0.545602\pi\)
\(758\) −30.3923 −1.10390
\(759\) 9.07180 0.329285
\(760\) −1.85641 −0.0673389
\(761\) 6.92820 0.251147 0.125574 0.992084i \(-0.459923\pi\)
0.125574 + 0.992084i \(0.459923\pi\)
\(762\) −18.9282 −0.685696
\(763\) −12.0000 −0.434429
\(764\) 18.5167 0.669909
\(765\) −1.60770 −0.0581263
\(766\) 27.3205 0.987130
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 26.6410 0.960700 0.480350 0.877077i \(-0.340510\pi\)
0.480350 + 0.877077i \(0.340510\pi\)
\(770\) −13.8564 −0.499350
\(771\) −6.46410 −0.232799
\(772\) −9.85641 −0.354740
\(773\) −0.928203 −0.0333851 −0.0166926 0.999861i \(-0.505314\pi\)
−0.0166926 + 0.999861i \(0.505314\pi\)
\(774\) 2.46410 0.0885703
\(775\) 3.24871 0.116697
\(776\) 8.39230 0.301266
\(777\) 8.92820 0.320298
\(778\) −2.66025 −0.0953747
\(779\) −3.71281 −0.133025
\(780\) 0 0
\(781\) −47.7128 −1.70730
\(782\) 1.05256 0.0376394
\(783\) 8.00000 0.285897
\(784\) 1.00000 0.0357143
\(785\) 37.8564 1.35115
\(786\) −19.0000 −0.677708
\(787\) −12.3923 −0.441738 −0.220869 0.975303i \(-0.570889\pi\)
−0.220869 + 0.975303i \(0.570889\pi\)
\(788\) −23.3923 −0.833316
\(789\) 17.3205 0.616626
\(790\) 8.78461 0.312542
\(791\) 16.3923 0.582843
\(792\) −4.00000 −0.142134
\(793\) 0 0
\(794\) 11.7846 0.418220
\(795\) −40.6410 −1.44139
\(796\) 2.12436 0.0752958
\(797\) −6.24871 −0.221341 −0.110670 0.993857i \(-0.535300\pi\)
−0.110670 + 0.993857i \(0.535300\pi\)
\(798\) −0.535898 −0.0189706
\(799\) 3.46410 0.122551
\(800\) −7.00000 −0.247487
\(801\) −8.26795 −0.292134
\(802\) −4.39230 −0.155098
\(803\) 11.7128 0.413336
\(804\) 3.19615 0.112720
\(805\) 7.85641 0.276902
\(806\) 0 0
\(807\) 1.60770 0.0565935
\(808\) 0 0
\(809\) −39.7128 −1.39623 −0.698114 0.715987i \(-0.745977\pi\)
−0.698114 + 0.715987i \(0.745977\pi\)
\(810\) −3.46410 −0.121716
\(811\) 29.8564 1.04840 0.524200 0.851595i \(-0.324364\pi\)
0.524200 + 0.851595i \(0.324364\pi\)
\(812\) 8.00000 0.280745
\(813\) 5.53590 0.194152
\(814\) −35.7128 −1.25173
\(815\) −64.6410 −2.26428
\(816\) −0.464102 −0.0162468
\(817\) −1.32051 −0.0461987
\(818\) 14.3923 0.503215
\(819\) 0 0
\(820\) −24.0000 −0.838116
\(821\) 17.2487 0.601984 0.300992 0.953627i \(-0.402682\pi\)
0.300992 + 0.953627i \(0.402682\pi\)
\(822\) 13.8564 0.483298
\(823\) 44.3923 1.54742 0.773709 0.633541i \(-0.218399\pi\)
0.773709 + 0.633541i \(0.218399\pi\)
\(824\) 6.66025 0.232021
\(825\) 28.0000 0.974835
\(826\) −9.73205 −0.338621
\(827\) 15.6077 0.542733 0.271366 0.962476i \(-0.412525\pi\)
0.271366 + 0.962476i \(0.412525\pi\)
\(828\) 2.26795 0.0788167
\(829\) 48.7846 1.69436 0.847180 0.531306i \(-0.178299\pi\)
0.847180 + 0.531306i \(0.178299\pi\)
\(830\) 6.00000 0.208263
\(831\) −27.3205 −0.947738
\(832\) 0 0
\(833\) −0.464102 −0.0160802
\(834\) 10.9282 0.378413
\(835\) 69.2820 2.39760
\(836\) 2.14359 0.0741377
\(837\) 0.464102 0.0160417
\(838\) 5.00000 0.172722
\(839\) 22.3923 0.773068 0.386534 0.922275i \(-0.373672\pi\)
0.386534 + 0.922275i \(0.373672\pi\)
\(840\) −3.46410 −0.119523
\(841\) 35.0000 1.20690
\(842\) 22.3923 0.771690
\(843\) 18.3923 0.633465
\(844\) −22.9282 −0.789221
\(845\) 0 0
\(846\) 7.46410 0.256621
\(847\) 5.00000 0.171802
\(848\) −11.7321 −0.402880
\(849\) −4.53590 −0.155672
\(850\) 3.24871 0.111430
\(851\) 20.2487 0.694117
\(852\) −11.9282 −0.408654
\(853\) −21.6410 −0.740974 −0.370487 0.928838i \(-0.620809\pi\)
−0.370487 + 0.928838i \(0.620809\pi\)
\(854\) −5.19615 −0.177809
\(855\) 1.85641 0.0634878
\(856\) −15.4641 −0.528552
\(857\) 21.7128 0.741696 0.370848 0.928694i \(-0.379067\pi\)
0.370848 + 0.928694i \(0.379067\pi\)
\(858\) 0 0
\(859\) 11.8564 0.404535 0.202268 0.979330i \(-0.435169\pi\)
0.202268 + 0.979330i \(0.435169\pi\)
\(860\) −8.53590 −0.291072
\(861\) −6.92820 −0.236113
\(862\) −22.8564 −0.778492
\(863\) 29.0718 0.989615 0.494808 0.869002i \(-0.335238\pi\)
0.494808 + 0.869002i \(0.335238\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 51.7128 1.75829
\(866\) 17.8564 0.606785
\(867\) −16.7846 −0.570035
\(868\) 0.464102 0.0157526
\(869\) −10.1436 −0.344098
\(870\) −27.7128 −0.939552
\(871\) 0 0
\(872\) 12.0000 0.406371
\(873\) −8.39230 −0.284036
\(874\) −1.21539 −0.0411112
\(875\) 6.92820 0.234216
\(876\) 2.92820 0.0989348
\(877\) −29.7128 −1.00333 −0.501665 0.865062i \(-0.667279\pi\)
−0.501665 + 0.865062i \(0.667279\pi\)
\(878\) 34.3923 1.16068
\(879\) 26.5359 0.895034
\(880\) 13.8564 0.467099
\(881\) −58.0333 −1.95519 −0.977596 0.210489i \(-0.932494\pi\)
−0.977596 + 0.210489i \(0.932494\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 34.3205 1.15498 0.577489 0.816399i \(-0.304033\pi\)
0.577489 + 0.816399i \(0.304033\pi\)
\(884\) 0 0
\(885\) 33.7128 1.13324
\(886\) 1.60770 0.0540116
\(887\) −15.8564 −0.532406 −0.266203 0.963917i \(-0.585769\pi\)
−0.266203 + 0.963917i \(0.585769\pi\)
\(888\) −8.92820 −0.299611
\(889\) 18.9282 0.634832
\(890\) 28.6410 0.960049
\(891\) 4.00000 0.134005
\(892\) 14.4641 0.484294
\(893\) −4.00000 −0.133855
\(894\) 18.4641 0.617532
\(895\) −10.6410 −0.355690
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) 3.71281 0.123829
\(900\) 7.00000 0.233333
\(901\) 5.44486 0.181395
\(902\) 27.7128 0.922736
\(903\) −2.46410 −0.0820002
\(904\) −16.3923 −0.545200
\(905\) −58.1436 −1.93276
\(906\) 12.0000 0.398673
\(907\) −11.5359 −0.383043 −0.191522 0.981488i \(-0.561342\pi\)
−0.191522 + 0.981488i \(0.561342\pi\)
\(908\) 21.3205 0.707546
\(909\) 0 0
\(910\) 0 0
\(911\) −35.1769 −1.16546 −0.582732 0.812665i \(-0.698016\pi\)
−0.582732 + 0.812665i \(0.698016\pi\)
\(912\) 0.535898 0.0177454
\(913\) −6.92820 −0.229290
\(914\) −9.19615 −0.304182
\(915\) 18.0000 0.595062
\(916\) 6.07180 0.200618
\(917\) 19.0000 0.627435
\(918\) 0.464102 0.0153176
\(919\) −26.0000 −0.857661 −0.428830 0.903385i \(-0.641074\pi\)
−0.428830 + 0.903385i \(0.641074\pi\)
\(920\) −7.85641 −0.259018
\(921\) −22.0000 −0.724925
\(922\) 18.9282 0.623367
\(923\) 0 0
\(924\) 4.00000 0.131590
\(925\) 62.4974 2.05490
\(926\) 40.2487 1.32265
\(927\) −6.66025 −0.218751
\(928\) −8.00000 −0.262613
\(929\) 30.1244 0.988348 0.494174 0.869363i \(-0.335471\pi\)
0.494174 + 0.869363i \(0.335471\pi\)
\(930\) −1.60770 −0.0527184
\(931\) 0.535898 0.0175634
\(932\) 6.53590 0.214090
\(933\) −21.3205 −0.698002
\(934\) −29.6410 −0.969884
\(935\) −6.43078 −0.210309
\(936\) 0 0
\(937\) −31.0718 −1.01507 −0.507536 0.861631i \(-0.669443\pi\)
−0.507536 + 0.861631i \(0.669443\pi\)
\(938\) −3.19615 −0.104358
\(939\) 35.3205 1.15264
\(940\) −25.8564 −0.843343
\(941\) −40.3923 −1.31675 −0.658376 0.752689i \(-0.728756\pi\)
−0.658376 + 0.752689i \(0.728756\pi\)
\(942\) −10.9282 −0.356060
\(943\) −15.7128 −0.511680
\(944\) 9.73205 0.316751
\(945\) 3.46410 0.112687
\(946\) 9.85641 0.320459
\(947\) −18.2487 −0.593003 −0.296502 0.955032i \(-0.595820\pi\)
−0.296502 + 0.955032i \(0.595820\pi\)
\(948\) −2.53590 −0.0823622
\(949\) 0 0
\(950\) −3.75129 −0.121708
\(951\) 9.53590 0.309223
\(952\) 0.464102 0.0150416
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 11.7321 0.379839
\(955\) 64.1436 2.07564
\(956\) −14.8564 −0.480491
\(957\) 32.0000 1.03441
\(958\) −16.3923 −0.529611
\(959\) −13.8564 −0.447447
\(960\) 3.46410 0.111803
\(961\) −30.7846 −0.993052
\(962\) 0 0
\(963\) 15.4641 0.498324
\(964\) 11.8564 0.381869
\(965\) −34.1436 −1.09912
\(966\) −2.26795 −0.0729701
\(967\) 23.4641 0.754555 0.377277 0.926100i \(-0.376860\pi\)
0.377277 + 0.926100i \(0.376860\pi\)
\(968\) −5.00000 −0.160706
\(969\) −0.248711 −0.00798976
\(970\) 29.0718 0.933439
\(971\) 10.0718 0.323219 0.161610 0.986855i \(-0.448331\pi\)
0.161610 + 0.986855i \(0.448331\pi\)
\(972\) 1.00000 0.0320750
\(973\) −10.9282 −0.350342
\(974\) 20.3923 0.653412
\(975\) 0 0
\(976\) 5.19615 0.166325
\(977\) −1.85641 −0.0593917 −0.0296959 0.999559i \(-0.509454\pi\)
−0.0296959 + 0.999559i \(0.509454\pi\)
\(978\) 18.6603 0.596689
\(979\) −33.0718 −1.05698
\(980\) 3.46410 0.110657
\(981\) −12.0000 −0.383131
\(982\) −17.8564 −0.569821
\(983\) 32.9282 1.05025 0.525123 0.851026i \(-0.324019\pi\)
0.525123 + 0.851026i \(0.324019\pi\)
\(984\) 6.92820 0.220863
\(985\) −81.0333 −2.58194
\(986\) 3.71281 0.118240
\(987\) −7.46410 −0.237585
\(988\) 0 0
\(989\) −5.58846 −0.177703
\(990\) −13.8564 −0.440386
\(991\) 27.8564 0.884888 0.442444 0.896796i \(-0.354112\pi\)
0.442444 + 0.896796i \(0.354112\pi\)
\(992\) −0.464102 −0.0147352
\(993\) −13.3205 −0.422714
\(994\) 11.9282 0.378340
\(995\) 7.35898 0.233295
\(996\) −1.73205 −0.0548821
\(997\) −17.9808 −0.569456 −0.284728 0.958608i \(-0.591903\pi\)
−0.284728 + 0.958608i \(0.591903\pi\)
\(998\) 3.73205 0.118136
\(999\) 8.92820 0.282476
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.bn.1.2 2
13.2 odd 12 546.2.s.c.43.1 4
13.7 odd 12 546.2.s.c.127.1 yes 4
13.12 even 2 7098.2.a.bz.1.1 2
39.2 even 12 1638.2.bj.e.1135.2 4
39.20 even 12 1638.2.bj.e.127.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.s.c.43.1 4 13.2 odd 12
546.2.s.c.127.1 yes 4 13.7 odd 12
1638.2.bj.e.127.2 4 39.20 even 12
1638.2.bj.e.1135.2 4 39.2 even 12
7098.2.a.bn.1.2 2 1.1 even 1 trivial
7098.2.a.bz.1.1 2 13.12 even 2