Properties

Label 4235.2.a.q.1.2
Level $4235$
Weight $2$
Character 4235.1
Self dual yes
Analytic conductor $33.817$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.8166452560\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 4235.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.53919 q^{2} +1.17009 q^{3} +0.369102 q^{4} -1.00000 q^{5} +1.80098 q^{6} +1.00000 q^{7} -2.51026 q^{8} -1.63090 q^{9} +O(q^{10})\) \(q+1.53919 q^{2} +1.17009 q^{3} +0.369102 q^{4} -1.00000 q^{5} +1.80098 q^{6} +1.00000 q^{7} -2.51026 q^{8} -1.63090 q^{9} -1.53919 q^{10} +0.431882 q^{12} +0.0917087 q^{13} +1.53919 q^{14} -1.17009 q^{15} -4.60197 q^{16} +5.51026 q^{17} -2.51026 q^{18} -0.921622 q^{19} -0.369102 q^{20} +1.17009 q^{21} -5.70928 q^{23} -2.93722 q^{24} +1.00000 q^{25} +0.141157 q^{26} -5.41855 q^{27} +0.369102 q^{28} -1.41855 q^{29} -1.80098 q^{30} +0.879362 q^{31} -2.06278 q^{32} +8.48133 q^{34} -1.00000 q^{35} -0.601968 q^{36} -8.78765 q^{37} -1.41855 q^{38} +0.107307 q^{39} +2.51026 q^{40} +1.61757 q^{41} +1.80098 q^{42} -3.86603 q^{43} +1.63090 q^{45} -8.78765 q^{46} -5.90829 q^{47} -5.38470 q^{48} +1.00000 q^{49} +1.53919 q^{50} +6.44748 q^{51} +0.0338499 q^{52} -10.0494 q^{53} -8.34017 q^{54} -2.51026 q^{56} -1.07838 q^{57} -2.18342 q^{58} -2.14116 q^{59} -0.431882 q^{60} +3.03612 q^{61} +1.35350 q^{62} -1.63090 q^{63} +6.02893 q^{64} -0.0917087 q^{65} -1.52586 q^{67} +2.03385 q^{68} -6.68035 q^{69} -1.53919 q^{70} +4.09890 q^{71} +4.09398 q^{72} -14.1906 q^{73} -13.5259 q^{74} +1.17009 q^{75} -0.340173 q^{76} +0.165166 q^{78} -14.5464 q^{79} +4.60197 q^{80} -1.44748 q^{81} +2.48974 q^{82} +8.52359 q^{83} +0.431882 q^{84} -5.51026 q^{85} -5.95055 q^{86} -1.65983 q^{87} -2.83710 q^{89} +2.51026 q^{90} +0.0917087 q^{91} -2.10731 q^{92} +1.02893 q^{93} -9.09398 q^{94} +0.921622 q^{95} -2.41363 q^{96} +14.2557 q^{97} +1.53919 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} - 4 q^{6} + 3 q^{7} + 9 q^{8} - q^{9} + O(q^{10}) \) \( 3 q + 3 q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} - 4 q^{6} + 3 q^{7} + 9 q^{8} - q^{9} - 3 q^{10} - 12 q^{12} - 2 q^{13} + 3 q^{14} + 2 q^{15} + 5 q^{16} + 9 q^{18} - 6 q^{19} - 5 q^{20} - 2 q^{21} - 10 q^{23} - 26 q^{24} + 3 q^{25} - 20 q^{26} - 2 q^{27} + 5 q^{28} + 10 q^{29} + 4 q^{30} - 10 q^{31} + 11 q^{32} - 6 q^{34} - 3 q^{35} + 17 q^{36} - 16 q^{37} + 10 q^{38} + 12 q^{39} - 9 q^{40} - 4 q^{42} + 2 q^{43} + q^{45} - 16 q^{46} - 20 q^{47} - 34 q^{48} + 3 q^{49} + 3 q^{50} + 20 q^{51} - 32 q^{52} - 12 q^{53} - 14 q^{54} + 9 q^{56} - 2 q^{58} + 14 q^{59} + 12 q^{60} - 10 q^{61} - 6 q^{62} - q^{63} + 33 q^{64} + 2 q^{65} - 2 q^{67} - 26 q^{68} + 2 q^{69} - 3 q^{70} - 24 q^{71} + 23 q^{72} - 4 q^{73} - 38 q^{74} - 2 q^{75} + 10 q^{76} + 42 q^{78} - 8 q^{79} - 5 q^{80} - 5 q^{81} + 24 q^{82} + 10 q^{83} - 12 q^{84} - 36 q^{86} - 16 q^{87} + 20 q^{89} - 9 q^{90} - 2 q^{91} - 18 q^{92} + 18 q^{93} - 38 q^{94} + 6 q^{95} - 40 q^{96} + 3 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.53919 1.08837 0.544185 0.838965i \(-0.316839\pi\)
0.544185 + 0.838965i \(0.316839\pi\)
\(3\) 1.17009 0.675550 0.337775 0.941227i \(-0.390326\pi\)
0.337775 + 0.941227i \(0.390326\pi\)
\(4\) 0.369102 0.184551
\(5\) −1.00000 −0.447214
\(6\) 1.80098 0.735249
\(7\) 1.00000 0.377964
\(8\) −2.51026 −0.887511
\(9\) −1.63090 −0.543633
\(10\) −1.53919 −0.486734
\(11\) 0 0
\(12\) 0.431882 0.124674
\(13\) 0.0917087 0.0254354 0.0127177 0.999919i \(-0.495952\pi\)
0.0127177 + 0.999919i \(0.495952\pi\)
\(14\) 1.53919 0.411366
\(15\) −1.17009 −0.302115
\(16\) −4.60197 −1.15049
\(17\) 5.51026 1.33643 0.668217 0.743966i \(-0.267058\pi\)
0.668217 + 0.743966i \(0.267058\pi\)
\(18\) −2.51026 −0.591674
\(19\) −0.921622 −0.211435 −0.105717 0.994396i \(-0.533714\pi\)
−0.105717 + 0.994396i \(0.533714\pi\)
\(20\) −0.369102 −0.0825338
\(21\) 1.17009 0.255334
\(22\) 0 0
\(23\) −5.70928 −1.19047 −0.595233 0.803553i \(-0.702940\pi\)
−0.595233 + 0.803553i \(0.702940\pi\)
\(24\) −2.93722 −0.599558
\(25\) 1.00000 0.200000
\(26\) 0.141157 0.0276832
\(27\) −5.41855 −1.04280
\(28\) 0.369102 0.0697538
\(29\) −1.41855 −0.263418 −0.131709 0.991288i \(-0.542046\pi\)
−0.131709 + 0.991288i \(0.542046\pi\)
\(30\) −1.80098 −0.328813
\(31\) 0.879362 0.157938 0.0789690 0.996877i \(-0.474837\pi\)
0.0789690 + 0.996877i \(0.474837\pi\)
\(32\) −2.06278 −0.364651
\(33\) 0 0
\(34\) 8.48133 1.45454
\(35\) −1.00000 −0.169031
\(36\) −0.601968 −0.100328
\(37\) −8.78765 −1.44468 −0.722341 0.691537i \(-0.756934\pi\)
−0.722341 + 0.691537i \(0.756934\pi\)
\(38\) −1.41855 −0.230119
\(39\) 0.107307 0.0171829
\(40\) 2.51026 0.396907
\(41\) 1.61757 0.252621 0.126311 0.991991i \(-0.459686\pi\)
0.126311 + 0.991991i \(0.459686\pi\)
\(42\) 1.80098 0.277898
\(43\) −3.86603 −0.589564 −0.294782 0.955565i \(-0.595247\pi\)
−0.294782 + 0.955565i \(0.595247\pi\)
\(44\) 0 0
\(45\) 1.63090 0.243120
\(46\) −8.78765 −1.29567
\(47\) −5.90829 −0.861813 −0.430906 0.902397i \(-0.641806\pi\)
−0.430906 + 0.902397i \(0.641806\pi\)
\(48\) −5.38470 −0.777215
\(49\) 1.00000 0.142857
\(50\) 1.53919 0.217674
\(51\) 6.44748 0.902828
\(52\) 0.0338499 0.00469414
\(53\) −10.0494 −1.38040 −0.690199 0.723620i \(-0.742477\pi\)
−0.690199 + 0.723620i \(0.742477\pi\)
\(54\) −8.34017 −1.13495
\(55\) 0 0
\(56\) −2.51026 −0.335448
\(57\) −1.07838 −0.142835
\(58\) −2.18342 −0.286697
\(59\) −2.14116 −0.278755 −0.139377 0.990239i \(-0.544510\pi\)
−0.139377 + 0.990239i \(0.544510\pi\)
\(60\) −0.431882 −0.0557557
\(61\) 3.03612 0.388735 0.194367 0.980929i \(-0.437735\pi\)
0.194367 + 0.980929i \(0.437735\pi\)
\(62\) 1.35350 0.171895
\(63\) −1.63090 −0.205474
\(64\) 6.02893 0.753616
\(65\) −0.0917087 −0.0113751
\(66\) 0 0
\(67\) −1.52586 −0.186413 −0.0932066 0.995647i \(-0.529712\pi\)
−0.0932066 + 0.995647i \(0.529712\pi\)
\(68\) 2.03385 0.246641
\(69\) −6.68035 −0.804219
\(70\) −1.53919 −0.183968
\(71\) 4.09890 0.486450 0.243225 0.969970i \(-0.421795\pi\)
0.243225 + 0.969970i \(0.421795\pi\)
\(72\) 4.09398 0.482480
\(73\) −14.1906 −1.66088 −0.830442 0.557105i \(-0.811912\pi\)
−0.830442 + 0.557105i \(0.811912\pi\)
\(74\) −13.5259 −1.57235
\(75\) 1.17009 0.135110
\(76\) −0.340173 −0.0390205
\(77\) 0 0
\(78\) 0.165166 0.0187014
\(79\) −14.5464 −1.63660 −0.818298 0.574795i \(-0.805082\pi\)
−0.818298 + 0.574795i \(0.805082\pi\)
\(80\) 4.60197 0.514516
\(81\) −1.44748 −0.160831
\(82\) 2.48974 0.274946
\(83\) 8.52359 0.935586 0.467793 0.883838i \(-0.345049\pi\)
0.467793 + 0.883838i \(0.345049\pi\)
\(84\) 0.431882 0.0471222
\(85\) −5.51026 −0.597672
\(86\) −5.95055 −0.641664
\(87\) −1.65983 −0.177952
\(88\) 0 0
\(89\) −2.83710 −0.300732 −0.150366 0.988630i \(-0.548045\pi\)
−0.150366 + 0.988630i \(0.548045\pi\)
\(90\) 2.51026 0.264605
\(91\) 0.0917087 0.00961369
\(92\) −2.10731 −0.219702
\(93\) 1.02893 0.106695
\(94\) −9.09398 −0.937972
\(95\) 0.921622 0.0945564
\(96\) −2.41363 −0.246340
\(97\) 14.2557 1.44744 0.723721 0.690093i \(-0.242430\pi\)
0.723721 + 0.690093i \(0.242430\pi\)
\(98\) 1.53919 0.155482
\(99\) 0 0
\(100\) 0.369102 0.0369102
\(101\) −9.03612 −0.899127 −0.449564 0.893248i \(-0.648421\pi\)
−0.449564 + 0.893248i \(0.648421\pi\)
\(102\) 9.92389 0.982611
\(103\) 3.32684 0.327803 0.163902 0.986477i \(-0.447592\pi\)
0.163902 + 0.986477i \(0.447592\pi\)
\(104\) −0.230213 −0.0225742
\(105\) −1.17009 −0.114189
\(106\) −15.4680 −1.50238
\(107\) −8.09890 −0.782950 −0.391475 0.920189i \(-0.628035\pi\)
−0.391475 + 0.920189i \(0.628035\pi\)
\(108\) −2.00000 −0.192450
\(109\) −15.1773 −1.45372 −0.726860 0.686786i \(-0.759021\pi\)
−0.726860 + 0.686786i \(0.759021\pi\)
\(110\) 0 0
\(111\) −10.2823 −0.975954
\(112\) −4.60197 −0.434845
\(113\) −7.07838 −0.665878 −0.332939 0.942948i \(-0.608040\pi\)
−0.332939 + 0.942948i \(0.608040\pi\)
\(114\) −1.65983 −0.155457
\(115\) 5.70928 0.532393
\(116\) −0.523590 −0.0486142
\(117\) −0.149568 −0.0138275
\(118\) −3.29565 −0.303389
\(119\) 5.51026 0.505125
\(120\) 2.93722 0.268130
\(121\) 0 0
\(122\) 4.67316 0.423088
\(123\) 1.89269 0.170658
\(124\) 0.324575 0.0291477
\(125\) −1.00000 −0.0894427
\(126\) −2.51026 −0.223632
\(127\) 9.65983 0.857171 0.428586 0.903501i \(-0.359012\pi\)
0.428586 + 0.903501i \(0.359012\pi\)
\(128\) 13.4052 1.18487
\(129\) −4.52359 −0.398280
\(130\) −0.141157 −0.0123803
\(131\) −4.68035 −0.408924 −0.204462 0.978875i \(-0.565544\pi\)
−0.204462 + 0.978875i \(0.565544\pi\)
\(132\) 0 0
\(133\) −0.921622 −0.0799148
\(134\) −2.34858 −0.202887
\(135\) 5.41855 0.466355
\(136\) −13.8322 −1.18610
\(137\) 8.88655 0.759229 0.379615 0.925145i \(-0.376057\pi\)
0.379615 + 0.925145i \(0.376057\pi\)
\(138\) −10.2823 −0.875289
\(139\) 15.0205 1.27402 0.637012 0.770854i \(-0.280170\pi\)
0.637012 + 0.770854i \(0.280170\pi\)
\(140\) −0.369102 −0.0311948
\(141\) −6.91321 −0.582197
\(142\) 6.30898 0.529438
\(143\) 0 0
\(144\) 7.50534 0.625445
\(145\) 1.41855 0.117804
\(146\) −21.8420 −1.80766
\(147\) 1.17009 0.0965071
\(148\) −3.24354 −0.266618
\(149\) 13.7009 1.12242 0.561209 0.827674i \(-0.310336\pi\)
0.561209 + 0.827674i \(0.310336\pi\)
\(150\) 1.80098 0.147050
\(151\) −1.05559 −0.0859028 −0.0429514 0.999077i \(-0.513676\pi\)
−0.0429514 + 0.999077i \(0.513676\pi\)
\(152\) 2.31351 0.187651
\(153\) −8.98667 −0.726529
\(154\) 0 0
\(155\) −0.879362 −0.0706320
\(156\) 0.0396073 0.00317112
\(157\) −17.7587 −1.41730 −0.708650 0.705560i \(-0.750696\pi\)
−0.708650 + 0.705560i \(0.750696\pi\)
\(158\) −22.3896 −1.78122
\(159\) −11.7587 −0.932527
\(160\) 2.06278 0.163077
\(161\) −5.70928 −0.449954
\(162\) −2.22795 −0.175044
\(163\) −11.4680 −0.898243 −0.449122 0.893471i \(-0.648263\pi\)
−0.449122 + 0.893471i \(0.648263\pi\)
\(164\) 0.597048 0.0466216
\(165\) 0 0
\(166\) 13.1194 1.01826
\(167\) 5.60197 0.433493 0.216747 0.976228i \(-0.430455\pi\)
0.216747 + 0.976228i \(0.430455\pi\)
\(168\) −2.93722 −0.226611
\(169\) −12.9916 −0.999353
\(170\) −8.48133 −0.650488
\(171\) 1.50307 0.114943
\(172\) −1.42696 −0.108805
\(173\) −21.6092 −1.64291 −0.821457 0.570271i \(-0.806838\pi\)
−0.821457 + 0.570271i \(0.806838\pi\)
\(174\) −2.55479 −0.193678
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −2.50534 −0.188313
\(178\) −4.36683 −0.327308
\(179\) −2.05786 −0.153812 −0.0769058 0.997038i \(-0.524504\pi\)
−0.0769058 + 0.997038i \(0.524504\pi\)
\(180\) 0.601968 0.0448681
\(181\) 20.2823 1.50757 0.753786 0.657120i \(-0.228225\pi\)
0.753786 + 0.657120i \(0.228225\pi\)
\(182\) 0.141157 0.0104633
\(183\) 3.55252 0.262610
\(184\) 14.3318 1.05655
\(185\) 8.78765 0.646081
\(186\) 1.58372 0.116124
\(187\) 0 0
\(188\) −2.18076 −0.159049
\(189\) −5.41855 −0.394142
\(190\) 1.41855 0.102912
\(191\) −20.2823 −1.46758 −0.733788 0.679378i \(-0.762250\pi\)
−0.733788 + 0.679378i \(0.762250\pi\)
\(192\) 7.05437 0.509105
\(193\) 24.3051 1.74952 0.874760 0.484557i \(-0.161019\pi\)
0.874760 + 0.484557i \(0.161019\pi\)
\(194\) 21.9421 1.57535
\(195\) −0.107307 −0.00768443
\(196\) 0.369102 0.0263645
\(197\) 14.1483 1.00803 0.504014 0.863696i \(-0.331856\pi\)
0.504014 + 0.863696i \(0.331856\pi\)
\(198\) 0 0
\(199\) 10.4813 0.743002 0.371501 0.928433i \(-0.378843\pi\)
0.371501 + 0.928433i \(0.378843\pi\)
\(200\) −2.51026 −0.177502
\(201\) −1.78539 −0.125931
\(202\) −13.9083 −0.978584
\(203\) −1.41855 −0.0995627
\(204\) 2.37978 0.166618
\(205\) −1.61757 −0.112976
\(206\) 5.12064 0.356772
\(207\) 9.31124 0.647176
\(208\) −0.422041 −0.0292633
\(209\) 0 0
\(210\) −1.80098 −0.124280
\(211\) 2.65368 0.182687 0.0913436 0.995819i \(-0.470884\pi\)
0.0913436 + 0.995819i \(0.470884\pi\)
\(212\) −3.70928 −0.254754
\(213\) 4.79606 0.328621
\(214\) −12.4657 −0.852140
\(215\) 3.86603 0.263661
\(216\) 13.6020 0.925497
\(217\) 0.879362 0.0596950
\(218\) −23.3607 −1.58219
\(219\) −16.6042 −1.12201
\(220\) 0 0
\(221\) 0.505339 0.0339928
\(222\) −15.8264 −1.06220
\(223\) −8.67316 −0.580798 −0.290399 0.956906i \(-0.593788\pi\)
−0.290399 + 0.956906i \(0.593788\pi\)
\(224\) −2.06278 −0.137825
\(225\) −1.63090 −0.108727
\(226\) −10.8950 −0.724722
\(227\) −9.67420 −0.642099 −0.321050 0.947062i \(-0.604036\pi\)
−0.321050 + 0.947062i \(0.604036\pi\)
\(228\) −0.398032 −0.0263603
\(229\) −13.5486 −0.895320 −0.447660 0.894204i \(-0.647742\pi\)
−0.447660 + 0.894204i \(0.647742\pi\)
\(230\) 8.78765 0.579441
\(231\) 0 0
\(232\) 3.56093 0.233787
\(233\) −8.38962 −0.549622 −0.274811 0.961498i \(-0.588615\pi\)
−0.274811 + 0.961498i \(0.588615\pi\)
\(234\) −0.230213 −0.0150495
\(235\) 5.90829 0.385414
\(236\) −0.790306 −0.0514446
\(237\) −17.0205 −1.10560
\(238\) 8.48133 0.549763
\(239\) 29.4908 1.90760 0.953800 0.300442i \(-0.0971341\pi\)
0.953800 + 0.300442i \(0.0971341\pi\)
\(240\) 5.38470 0.347581
\(241\) −3.64423 −0.234745 −0.117373 0.993088i \(-0.537447\pi\)
−0.117373 + 0.993088i \(0.537447\pi\)
\(242\) 0 0
\(243\) 14.5620 0.934151
\(244\) 1.12064 0.0717415
\(245\) −1.00000 −0.0638877
\(246\) 2.91321 0.185740
\(247\) −0.0845208 −0.00537793
\(248\) −2.20743 −0.140172
\(249\) 9.97334 0.632035
\(250\) −1.53919 −0.0973469
\(251\) 23.1350 1.46027 0.730135 0.683303i \(-0.239457\pi\)
0.730135 + 0.683303i \(0.239457\pi\)
\(252\) −0.601968 −0.0379204
\(253\) 0 0
\(254\) 14.8683 0.932920
\(255\) −6.44748 −0.403757
\(256\) 8.57531 0.535957
\(257\) 20.8104 1.29812 0.649060 0.760737i \(-0.275162\pi\)
0.649060 + 0.760737i \(0.275162\pi\)
\(258\) −6.96266 −0.433476
\(259\) −8.78765 −0.546038
\(260\) −0.0338499 −0.00209928
\(261\) 2.31351 0.143203
\(262\) −7.20394 −0.445061
\(263\) 23.7009 1.46146 0.730729 0.682668i \(-0.239180\pi\)
0.730729 + 0.682668i \(0.239180\pi\)
\(264\) 0 0
\(265\) 10.0494 0.617333
\(266\) −1.41855 −0.0869769
\(267\) −3.31965 −0.203160
\(268\) −0.563198 −0.0344028
\(269\) −3.50307 −0.213586 −0.106793 0.994281i \(-0.534058\pi\)
−0.106793 + 0.994281i \(0.534058\pi\)
\(270\) 8.34017 0.507567
\(271\) 8.49693 0.516152 0.258076 0.966125i \(-0.416912\pi\)
0.258076 + 0.966125i \(0.416912\pi\)
\(272\) −25.3580 −1.53756
\(273\) 0.107307 0.00649452
\(274\) 13.6781 0.826323
\(275\) 0 0
\(276\) −2.46573 −0.148420
\(277\) 25.9649 1.56008 0.780041 0.625729i \(-0.215198\pi\)
0.780041 + 0.625729i \(0.215198\pi\)
\(278\) 23.1194 1.38661
\(279\) −1.43415 −0.0858603
\(280\) 2.51026 0.150017
\(281\) −11.6742 −0.696425 −0.348212 0.937416i \(-0.613211\pi\)
−0.348212 + 0.937416i \(0.613211\pi\)
\(282\) −10.6407 −0.633647
\(283\) 14.2557 0.847411 0.423705 0.905800i \(-0.360729\pi\)
0.423705 + 0.905800i \(0.360729\pi\)
\(284\) 1.51291 0.0897748
\(285\) 1.07838 0.0638776
\(286\) 0 0
\(287\) 1.61757 0.0954819
\(288\) 3.36418 0.198236
\(289\) 13.3630 0.786056
\(290\) 2.18342 0.128215
\(291\) 16.6803 0.977819
\(292\) −5.23779 −0.306518
\(293\) −25.1122 −1.46707 −0.733536 0.679651i \(-0.762131\pi\)
−0.733536 + 0.679651i \(0.762131\pi\)
\(294\) 1.80098 0.105036
\(295\) 2.14116 0.124663
\(296\) 22.0593 1.28217
\(297\) 0 0
\(298\) 21.0882 1.22161
\(299\) −0.523590 −0.0302800
\(300\) 0.431882 0.0249347
\(301\) −3.86603 −0.222834
\(302\) −1.62475 −0.0934941
\(303\) −10.5730 −0.607405
\(304\) 4.24128 0.243254
\(305\) −3.03612 −0.173848
\(306\) −13.8322 −0.790733
\(307\) −8.02666 −0.458106 −0.229053 0.973414i \(-0.573563\pi\)
−0.229053 + 0.973414i \(0.573563\pi\)
\(308\) 0 0
\(309\) 3.89269 0.221448
\(310\) −1.35350 −0.0768739
\(311\) −26.3968 −1.49683 −0.748413 0.663233i \(-0.769184\pi\)
−0.748413 + 0.663233i \(0.769184\pi\)
\(312\) −0.269369 −0.0152500
\(313\) 25.7321 1.45446 0.727231 0.686393i \(-0.240807\pi\)
0.727231 + 0.686393i \(0.240807\pi\)
\(314\) −27.3340 −1.54255
\(315\) 1.63090 0.0918907
\(316\) −5.36910 −0.302036
\(317\) 6.31351 0.354602 0.177301 0.984157i \(-0.443263\pi\)
0.177301 + 0.984157i \(0.443263\pi\)
\(318\) −18.0989 −1.01494
\(319\) 0 0
\(320\) −6.02893 −0.337027
\(321\) −9.47641 −0.528922
\(322\) −8.78765 −0.489717
\(323\) −5.07838 −0.282568
\(324\) −0.534268 −0.0296816
\(325\) 0.0917087 0.00508709
\(326\) −17.6514 −0.977622
\(327\) −17.7587 −0.982060
\(328\) −4.06051 −0.224204
\(329\) −5.90829 −0.325735
\(330\) 0 0
\(331\) 3.50307 0.192546 0.0962731 0.995355i \(-0.469308\pi\)
0.0962731 + 0.995355i \(0.469308\pi\)
\(332\) 3.14608 0.172663
\(333\) 14.3318 0.785376
\(334\) 8.62249 0.471802
\(335\) 1.52586 0.0833665
\(336\) −5.38470 −0.293760
\(337\) −7.57918 −0.412864 −0.206432 0.978461i \(-0.566185\pi\)
−0.206432 + 0.978461i \(0.566185\pi\)
\(338\) −19.9965 −1.08767
\(339\) −8.28231 −0.449834
\(340\) −2.03385 −0.110301
\(341\) 0 0
\(342\) 2.31351 0.125100
\(343\) 1.00000 0.0539949
\(344\) 9.70474 0.523245
\(345\) 6.68035 0.359658
\(346\) −33.2606 −1.78810
\(347\) 35.4824 1.90479 0.952397 0.304861i \(-0.0986100\pi\)
0.952397 + 0.304861i \(0.0986100\pi\)
\(348\) −0.612646 −0.0328413
\(349\) 13.6586 0.731128 0.365564 0.930786i \(-0.380876\pi\)
0.365564 + 0.930786i \(0.380876\pi\)
\(350\) 1.53919 0.0822731
\(351\) −0.496928 −0.0265241
\(352\) 0 0
\(353\) 26.4657 1.40863 0.704314 0.709888i \(-0.251255\pi\)
0.704314 + 0.709888i \(0.251255\pi\)
\(354\) −3.85619 −0.204954
\(355\) −4.09890 −0.217547
\(356\) −1.04718 −0.0555005
\(357\) 6.44748 0.341237
\(358\) −3.16743 −0.167404
\(359\) 15.3958 0.812557 0.406279 0.913749i \(-0.366826\pi\)
0.406279 + 0.913749i \(0.366826\pi\)
\(360\) −4.09398 −0.215771
\(361\) −18.1506 −0.955295
\(362\) 31.2183 1.64080
\(363\) 0 0
\(364\) 0.0338499 0.00177422
\(365\) 14.1906 0.742770
\(366\) 5.46800 0.285817
\(367\) −34.6875 −1.81067 −0.905337 0.424693i \(-0.860382\pi\)
−0.905337 + 0.424693i \(0.860382\pi\)
\(368\) 26.2739 1.36962
\(369\) −2.63809 −0.137333
\(370\) 13.5259 0.703176
\(371\) −10.0494 −0.521741
\(372\) 0.379780 0.0196907
\(373\) −36.3584 −1.88257 −0.941284 0.337616i \(-0.890379\pi\)
−0.941284 + 0.337616i \(0.890379\pi\)
\(374\) 0 0
\(375\) −1.17009 −0.0604230
\(376\) 14.8313 0.764868
\(377\) −0.130094 −0.00670016
\(378\) −8.34017 −0.428972
\(379\) −33.1461 −1.70260 −0.851300 0.524680i \(-0.824185\pi\)
−0.851300 + 0.524680i \(0.824185\pi\)
\(380\) 0.340173 0.0174505
\(381\) 11.3028 0.579062
\(382\) −31.2183 −1.59727
\(383\) −34.2628 −1.75075 −0.875375 0.483445i \(-0.839385\pi\)
−0.875375 + 0.483445i \(0.839385\pi\)
\(384\) 15.6853 0.800435
\(385\) 0 0
\(386\) 37.4101 1.90413
\(387\) 6.30510 0.320506
\(388\) 5.26180 0.267127
\(389\) 23.2762 1.18015 0.590074 0.807349i \(-0.299098\pi\)
0.590074 + 0.807349i \(0.299098\pi\)
\(390\) −0.165166 −0.00836350
\(391\) −31.4596 −1.59098
\(392\) −2.51026 −0.126787
\(393\) −5.47641 −0.276248
\(394\) 21.7770 1.09711
\(395\) 14.5464 0.731908
\(396\) 0 0
\(397\) −29.8576 −1.49851 −0.749256 0.662281i \(-0.769588\pi\)
−0.749256 + 0.662281i \(0.769588\pi\)
\(398\) 16.1327 0.808662
\(399\) −1.07838 −0.0539864
\(400\) −4.60197 −0.230098
\(401\) −5.51745 −0.275528 −0.137764 0.990465i \(-0.543992\pi\)
−0.137764 + 0.990465i \(0.543992\pi\)
\(402\) −2.74805 −0.137060
\(403\) 0.0806452 0.00401722
\(404\) −3.33525 −0.165935
\(405\) 1.44748 0.0719259
\(406\) −2.18342 −0.108361
\(407\) 0 0
\(408\) −16.1848 −0.801269
\(409\) 3.43415 0.169808 0.0849039 0.996389i \(-0.472942\pi\)
0.0849039 + 0.996389i \(0.472942\pi\)
\(410\) −2.48974 −0.122960
\(411\) 10.3980 0.512897
\(412\) 1.22795 0.0604965
\(413\) −2.14116 −0.105359
\(414\) 14.3318 0.704368
\(415\) −8.52359 −0.418407
\(416\) −0.189175 −0.00927506
\(417\) 17.5753 0.860666
\(418\) 0 0
\(419\) 18.2134 0.889782 0.444891 0.895585i \(-0.353242\pi\)
0.444891 + 0.895585i \(0.353242\pi\)
\(420\) −0.431882 −0.0210737
\(421\) 10.6576 0.519418 0.259709 0.965687i \(-0.416373\pi\)
0.259709 + 0.965687i \(0.416373\pi\)
\(422\) 4.08452 0.198831
\(423\) 9.63582 0.468510
\(424\) 25.2267 1.22512
\(425\) 5.51026 0.267287
\(426\) 7.38205 0.357661
\(427\) 3.03612 0.146928
\(428\) −2.98932 −0.144494
\(429\) 0 0
\(430\) 5.95055 0.286961
\(431\) 7.61038 0.366579 0.183290 0.983059i \(-0.441325\pi\)
0.183290 + 0.983059i \(0.441325\pi\)
\(432\) 24.9360 1.19973
\(433\) 33.5318 1.61144 0.805718 0.592299i \(-0.201780\pi\)
0.805718 + 0.592299i \(0.201780\pi\)
\(434\) 1.35350 0.0649703
\(435\) 1.65983 0.0795826
\(436\) −5.60197 −0.268286
\(437\) 5.26180 0.251706
\(438\) −25.5571 −1.22116
\(439\) 13.7587 0.656668 0.328334 0.944562i \(-0.393513\pi\)
0.328334 + 0.944562i \(0.393513\pi\)
\(440\) 0 0
\(441\) −1.63090 −0.0776618
\(442\) 0.777812 0.0369967
\(443\) −11.1689 −0.530649 −0.265324 0.964159i \(-0.585479\pi\)
−0.265324 + 0.964159i \(0.585479\pi\)
\(444\) −3.79523 −0.180113
\(445\) 2.83710 0.134492
\(446\) −13.3496 −0.632123
\(447\) 16.0312 0.758250
\(448\) 6.02893 0.284840
\(449\) −19.0700 −0.899967 −0.449984 0.893037i \(-0.648570\pi\)
−0.449984 + 0.893037i \(0.648570\pi\)
\(450\) −2.51026 −0.118335
\(451\) 0 0
\(452\) −2.61265 −0.122889
\(453\) −1.23513 −0.0580316
\(454\) −14.8904 −0.698842
\(455\) −0.0917087 −0.00429937
\(456\) 2.70701 0.126767
\(457\) 0.787653 0.0368449 0.0184224 0.999830i \(-0.494136\pi\)
0.0184224 + 0.999830i \(0.494136\pi\)
\(458\) −20.8539 −0.974440
\(459\) −29.8576 −1.39363
\(460\) 2.10731 0.0982537
\(461\) −25.5864 −1.19168 −0.595838 0.803105i \(-0.703180\pi\)
−0.595838 + 0.803105i \(0.703180\pi\)
\(462\) 0 0
\(463\) −14.2595 −0.662696 −0.331348 0.943509i \(-0.607503\pi\)
−0.331348 + 0.943509i \(0.607503\pi\)
\(464\) 6.52813 0.303061
\(465\) −1.02893 −0.0477155
\(466\) −12.9132 −0.598193
\(467\) 18.3474 0.849015 0.424507 0.905425i \(-0.360447\pi\)
0.424507 + 0.905425i \(0.360447\pi\)
\(468\) −0.0552057 −0.00255189
\(469\) −1.52586 −0.0704576
\(470\) 9.09398 0.419474
\(471\) −20.7792 −0.957457
\(472\) 5.37486 0.247398
\(473\) 0 0
\(474\) −26.1978 −1.20330
\(475\) −0.921622 −0.0422869
\(476\) 2.03385 0.0932214
\(477\) 16.3896 0.750429
\(478\) 45.3919 2.07618
\(479\) −12.1711 −0.556113 −0.278057 0.960565i \(-0.589690\pi\)
−0.278057 + 0.960565i \(0.589690\pi\)
\(480\) 2.41363 0.110167
\(481\) −0.805905 −0.0367461
\(482\) −5.60916 −0.255490
\(483\) −6.68035 −0.303966
\(484\) 0 0
\(485\) −14.2557 −0.647316
\(486\) 22.4136 1.01670
\(487\) −4.23287 −0.191809 −0.0959047 0.995391i \(-0.530574\pi\)
−0.0959047 + 0.995391i \(0.530574\pi\)
\(488\) −7.62144 −0.345006
\(489\) −13.4186 −0.606808
\(490\) −1.53919 −0.0695335
\(491\) −22.0183 −0.993670 −0.496835 0.867845i \(-0.665505\pi\)
−0.496835 + 0.867845i \(0.665505\pi\)
\(492\) 0.698597 0.0314952
\(493\) −7.81658 −0.352041
\(494\) −0.130094 −0.00585318
\(495\) 0 0
\(496\) −4.04680 −0.181706
\(497\) 4.09890 0.183861
\(498\) 15.3509 0.687888
\(499\) 32.9939 1.47701 0.738504 0.674249i \(-0.235533\pi\)
0.738504 + 0.674249i \(0.235533\pi\)
\(500\) −0.369102 −0.0165068
\(501\) 6.55479 0.292846
\(502\) 35.6092 1.58931
\(503\) 29.0349 1.29460 0.647301 0.762235i \(-0.275898\pi\)
0.647301 + 0.762235i \(0.275898\pi\)
\(504\) 4.09398 0.182360
\(505\) 9.03612 0.402102
\(506\) 0 0
\(507\) −15.2013 −0.675113
\(508\) 3.56547 0.158192
\(509\) −21.5031 −0.953107 −0.476553 0.879146i \(-0.658114\pi\)
−0.476553 + 0.879146i \(0.658114\pi\)
\(510\) −9.92389 −0.439437
\(511\) −14.1906 −0.627755
\(512\) −13.6114 −0.601546
\(513\) 4.99386 0.220484
\(514\) 32.0312 1.41284
\(515\) −3.32684 −0.146598
\(516\) −1.66967 −0.0735030
\(517\) 0 0
\(518\) −13.5259 −0.594292
\(519\) −25.2846 −1.10987
\(520\) 0.230213 0.0100955
\(521\) 24.5958 1.07756 0.538781 0.842446i \(-0.318885\pi\)
0.538781 + 0.842446i \(0.318885\pi\)
\(522\) 3.56093 0.155858
\(523\) −38.0677 −1.66458 −0.832292 0.554337i \(-0.812972\pi\)
−0.832292 + 0.554337i \(0.812972\pi\)
\(524\) −1.72753 −0.0754674
\(525\) 1.17009 0.0510668
\(526\) 36.4801 1.59061
\(527\) 4.84551 0.211074
\(528\) 0 0
\(529\) 9.59583 0.417210
\(530\) 15.4680 0.671887
\(531\) 3.49201 0.151540
\(532\) −0.340173 −0.0147484
\(533\) 0.148345 0.00642554
\(534\) −5.10957 −0.221113
\(535\) 8.09890 0.350146
\(536\) 3.83030 0.165444
\(537\) −2.40787 −0.103907
\(538\) −5.39189 −0.232461
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) −2.13009 −0.0915799 −0.0457899 0.998951i \(-0.514580\pi\)
−0.0457899 + 0.998951i \(0.514580\pi\)
\(542\) 13.0784 0.561764
\(543\) 23.7321 1.01844
\(544\) −11.3664 −0.487332
\(545\) 15.1773 0.650123
\(546\) 0.165166 0.00706845
\(547\) 9.13170 0.390443 0.195222 0.980759i \(-0.437457\pi\)
0.195222 + 0.980759i \(0.437457\pi\)
\(548\) 3.28005 0.140117
\(549\) −4.95160 −0.211329
\(550\) 0 0
\(551\) 1.30737 0.0556957
\(552\) 16.7694 0.713753
\(553\) −14.5464 −0.618575
\(554\) 39.9649 1.69795
\(555\) 10.2823 0.436460
\(556\) 5.54411 0.235123
\(557\) 11.7093 0.496138 0.248069 0.968742i \(-0.420204\pi\)
0.248069 + 0.968742i \(0.420204\pi\)
\(558\) −2.20743 −0.0934478
\(559\) −0.354549 −0.0149958
\(560\) 4.60197 0.194469
\(561\) 0 0
\(562\) −17.9688 −0.757968
\(563\) −12.5958 −0.530851 −0.265425 0.964131i \(-0.585512\pi\)
−0.265425 + 0.964131i \(0.585512\pi\)
\(564\) −2.55168 −0.107445
\(565\) 7.07838 0.297790
\(566\) 21.9421 0.922297
\(567\) −1.44748 −0.0607885
\(568\) −10.2893 −0.431729
\(569\) −7.54411 −0.316266 −0.158133 0.987418i \(-0.550547\pi\)
−0.158133 + 0.987418i \(0.550547\pi\)
\(570\) 1.65983 0.0695225
\(571\) −36.8104 −1.54047 −0.770234 0.637761i \(-0.779861\pi\)
−0.770234 + 0.637761i \(0.779861\pi\)
\(572\) 0 0
\(573\) −23.7321 −0.991421
\(574\) 2.48974 0.103920
\(575\) −5.70928 −0.238093
\(576\) −9.83257 −0.409690
\(577\) −39.5174 −1.64513 −0.822566 0.568669i \(-0.807459\pi\)
−0.822566 + 0.568669i \(0.807459\pi\)
\(578\) 20.5681 0.855521
\(579\) 28.4391 1.18189
\(580\) 0.523590 0.0217409
\(581\) 8.52359 0.353618
\(582\) 25.6742 1.06423
\(583\) 0 0
\(584\) 35.6221 1.47405
\(585\) 0.149568 0.00618386
\(586\) −38.6525 −1.59672
\(587\) 1.56812 0.0647232 0.0323616 0.999476i \(-0.489697\pi\)
0.0323616 + 0.999476i \(0.489697\pi\)
\(588\) 0.431882 0.0178105
\(589\) −0.810439 −0.0333936
\(590\) 3.29565 0.135680
\(591\) 16.5548 0.680973
\(592\) 40.4405 1.66209
\(593\) −4.95547 −0.203497 −0.101748 0.994810i \(-0.532444\pi\)
−0.101748 + 0.994810i \(0.532444\pi\)
\(594\) 0 0
\(595\) −5.51026 −0.225899
\(596\) 5.05702 0.207144
\(597\) 12.2641 0.501935
\(598\) −0.805905 −0.0329559
\(599\) −12.3668 −0.505295 −0.252648 0.967558i \(-0.581301\pi\)
−0.252648 + 0.967558i \(0.581301\pi\)
\(600\) −2.93722 −0.119912
\(601\) −24.9516 −1.01780 −0.508898 0.860827i \(-0.669947\pi\)
−0.508898 + 0.860827i \(0.669947\pi\)
\(602\) −5.95055 −0.242526
\(603\) 2.48852 0.101340
\(604\) −0.389621 −0.0158535
\(605\) 0 0
\(606\) −16.2739 −0.661082
\(607\) 19.2762 0.782396 0.391198 0.920307i \(-0.372061\pi\)
0.391198 + 0.920307i \(0.372061\pi\)
\(608\) 1.90110 0.0770999
\(609\) −1.65983 −0.0672596
\(610\) −4.67316 −0.189211
\(611\) −0.541842 −0.0219206
\(612\) −3.31700 −0.134082
\(613\) 42.6986 1.72458 0.862290 0.506415i \(-0.169029\pi\)
0.862290 + 0.506415i \(0.169029\pi\)
\(614\) −12.3545 −0.498589
\(615\) −1.89269 −0.0763207
\(616\) 0 0
\(617\) 31.7770 1.27929 0.639646 0.768669i \(-0.279081\pi\)
0.639646 + 0.768669i \(0.279081\pi\)
\(618\) 5.99159 0.241017
\(619\) −19.8420 −0.797518 −0.398759 0.917056i \(-0.630559\pi\)
−0.398759 + 0.917056i \(0.630559\pi\)
\(620\) −0.324575 −0.0130352
\(621\) 30.9360 1.24142
\(622\) −40.6297 −1.62910
\(623\) −2.83710 −0.113666
\(624\) −0.493824 −0.0197688
\(625\) 1.00000 0.0400000
\(626\) 39.6065 1.58299
\(627\) 0 0
\(628\) −6.55479 −0.261564
\(629\) −48.4222 −1.93072
\(630\) 2.51026 0.100011
\(631\) −3.63317 −0.144634 −0.0723170 0.997382i \(-0.523039\pi\)
−0.0723170 + 0.997382i \(0.523039\pi\)
\(632\) 36.5152 1.45250
\(633\) 3.10504 0.123414
\(634\) 9.71769 0.385939
\(635\) −9.65983 −0.383339
\(636\) −4.34017 −0.172099
\(637\) 0.0917087 0.00363363
\(638\) 0 0
\(639\) −6.68488 −0.264450
\(640\) −13.4052 −0.529888
\(641\) 37.5402 1.48275 0.741375 0.671091i \(-0.234174\pi\)
0.741375 + 0.671091i \(0.234174\pi\)
\(642\) −14.5860 −0.575663
\(643\) −34.8710 −1.37518 −0.687588 0.726101i \(-0.741330\pi\)
−0.687588 + 0.726101i \(0.741330\pi\)
\(644\) −2.10731 −0.0830395
\(645\) 4.52359 0.178116
\(646\) −7.81658 −0.307539
\(647\) 30.6342 1.20436 0.602178 0.798362i \(-0.294300\pi\)
0.602178 + 0.798362i \(0.294300\pi\)
\(648\) 3.63355 0.142739
\(649\) 0 0
\(650\) 0.141157 0.00553664
\(651\) 1.02893 0.0403269
\(652\) −4.23287 −0.165772
\(653\) 5.40417 0.211482 0.105741 0.994394i \(-0.466279\pi\)
0.105741 + 0.994394i \(0.466279\pi\)
\(654\) −27.3340 −1.06885
\(655\) 4.68035 0.182876
\(656\) −7.44399 −0.290639
\(657\) 23.1434 0.902911
\(658\) −9.09398 −0.354520
\(659\) 19.4101 0.756112 0.378056 0.925783i \(-0.376593\pi\)
0.378056 + 0.925783i \(0.376593\pi\)
\(660\) 0 0
\(661\) −26.3090 −1.02330 −0.511650 0.859194i \(-0.670966\pi\)
−0.511650 + 0.859194i \(0.670966\pi\)
\(662\) 5.39189 0.209562
\(663\) 0.591290 0.0229638
\(664\) −21.3964 −0.830342
\(665\) 0.921622 0.0357390
\(666\) 22.0593 0.854780
\(667\) 8.09890 0.313591
\(668\) 2.06770 0.0800017
\(669\) −10.1483 −0.392358
\(670\) 2.34858 0.0907337
\(671\) 0 0
\(672\) −2.41363 −0.0931078
\(673\) 15.7938 0.608806 0.304403 0.952543i \(-0.401543\pi\)
0.304403 + 0.952543i \(0.401543\pi\)
\(674\) −11.6658 −0.449350
\(675\) −5.41855 −0.208560
\(676\) −4.79523 −0.184432
\(677\) 15.7081 0.603710 0.301855 0.953354i \(-0.402394\pi\)
0.301855 + 0.953354i \(0.402394\pi\)
\(678\) −12.7480 −0.489586
\(679\) 14.2557 0.547082
\(680\) 13.8322 0.530440
\(681\) −11.3197 −0.433770
\(682\) 0 0
\(683\) −44.0326 −1.68486 −0.842431 0.538805i \(-0.818876\pi\)
−0.842431 + 0.538805i \(0.818876\pi\)
\(684\) 0.554787 0.0212128
\(685\) −8.88655 −0.339538
\(686\) 1.53919 0.0587665
\(687\) −15.8531 −0.604833
\(688\) 17.7914 0.678289
\(689\) −0.921622 −0.0351110
\(690\) 10.2823 0.391441
\(691\) 4.63809 0.176441 0.0882205 0.996101i \(-0.471882\pi\)
0.0882205 + 0.996101i \(0.471882\pi\)
\(692\) −7.97599 −0.303202
\(693\) 0 0
\(694\) 54.6141 2.07312
\(695\) −15.0205 −0.569761
\(696\) 4.16660 0.157934
\(697\) 8.91321 0.337612
\(698\) 21.0232 0.795739
\(699\) −9.81658 −0.371297
\(700\) 0.369102 0.0139508
\(701\) 14.6491 0.553291 0.276645 0.960972i \(-0.410777\pi\)
0.276645 + 0.960972i \(0.410777\pi\)
\(702\) −0.764867 −0.0288680
\(703\) 8.09890 0.305456
\(704\) 0 0
\(705\) 6.91321 0.260367
\(706\) 40.7358 1.53311
\(707\) −9.03612 −0.339838
\(708\) −0.924727 −0.0347534
\(709\) −25.5174 −0.958328 −0.479164 0.877725i \(-0.659060\pi\)
−0.479164 + 0.877725i \(0.659060\pi\)
\(710\) −6.30898 −0.236772
\(711\) 23.7237 0.889706
\(712\) 7.12186 0.266903
\(713\) −5.02052 −0.188020
\(714\) 9.92389 0.371392
\(715\) 0 0
\(716\) −0.759561 −0.0283861
\(717\) 34.5068 1.28868
\(718\) 23.6970 0.884364
\(719\) 39.2918 1.46534 0.732668 0.680586i \(-0.238275\pi\)
0.732668 + 0.680586i \(0.238275\pi\)
\(720\) −7.50534 −0.279707
\(721\) 3.32684 0.123898
\(722\) −27.9372 −1.03972
\(723\) −4.26406 −0.158582
\(724\) 7.48625 0.278224
\(725\) −1.41855 −0.0526837
\(726\) 0 0
\(727\) −37.7081 −1.39851 −0.699257 0.714870i \(-0.746486\pi\)
−0.699257 + 0.714870i \(0.746486\pi\)
\(728\) −0.230213 −0.00853225
\(729\) 21.3812 0.791897
\(730\) 21.8420 0.808410
\(731\) −21.3028 −0.787914
\(732\) 1.31124 0.0484650
\(733\) 4.34736 0.160573 0.0802867 0.996772i \(-0.474416\pi\)
0.0802867 + 0.996772i \(0.474416\pi\)
\(734\) −53.3907 −1.97069
\(735\) −1.17009 −0.0431593
\(736\) 11.7770 0.434105
\(737\) 0 0
\(738\) −4.06051 −0.149470
\(739\) −38.1568 −1.40362 −0.701809 0.712365i \(-0.747624\pi\)
−0.701809 + 0.712365i \(0.747624\pi\)
\(740\) 3.24354 0.119235
\(741\) −0.0988967 −0.00363306
\(742\) −15.4680 −0.567848
\(743\) 29.2618 1.07351 0.536756 0.843738i \(-0.319650\pi\)
0.536756 + 0.843738i \(0.319650\pi\)
\(744\) −2.58288 −0.0946930
\(745\) −13.7009 −0.501961
\(746\) −55.9625 −2.04893
\(747\) −13.9011 −0.508615
\(748\) 0 0
\(749\) −8.09890 −0.295927
\(750\) −1.80098 −0.0657626
\(751\) 41.6886 1.52124 0.760619 0.649199i \(-0.224896\pi\)
0.760619 + 0.649199i \(0.224896\pi\)
\(752\) 27.1898 0.991509
\(753\) 27.0700 0.986484
\(754\) −0.200238 −0.00729225
\(755\) 1.05559 0.0384169
\(756\) −2.00000 −0.0727393
\(757\) 39.7419 1.44444 0.722222 0.691661i \(-0.243121\pi\)
0.722222 + 0.691661i \(0.243121\pi\)
\(758\) −51.0181 −1.85306
\(759\) 0 0
\(760\) −2.31351 −0.0839199
\(761\) 36.4112 1.31990 0.659952 0.751308i \(-0.270577\pi\)
0.659952 + 0.751308i \(0.270577\pi\)
\(762\) 17.3972 0.630234
\(763\) −15.1773 −0.549454
\(764\) −7.48625 −0.270843
\(765\) 8.98667 0.324914
\(766\) −52.7370 −1.90546
\(767\) −0.196363 −0.00709025
\(768\) 10.0338 0.362065
\(769\) 10.5347 0.379889 0.189945 0.981795i \(-0.439169\pi\)
0.189945 + 0.981795i \(0.439169\pi\)
\(770\) 0 0
\(771\) 24.3500 0.876944
\(772\) 8.97107 0.322876
\(773\) −1.52198 −0.0547419 −0.0273709 0.999625i \(-0.508714\pi\)
−0.0273709 + 0.999625i \(0.508714\pi\)
\(774\) 9.70474 0.348830
\(775\) 0.879362 0.0315876
\(776\) −35.7854 −1.28462
\(777\) −10.2823 −0.368876
\(778\) 35.8264 1.28444
\(779\) −1.49079 −0.0534129
\(780\) −0.0396073 −0.00141817
\(781\) 0 0
\(782\) −48.4222 −1.73158
\(783\) 7.68649 0.274693
\(784\) −4.60197 −0.164356
\(785\) 17.7587 0.633836
\(786\) −8.42923 −0.300661
\(787\) 15.6020 0.556150 0.278075 0.960559i \(-0.410304\pi\)
0.278075 + 0.960559i \(0.410304\pi\)
\(788\) 5.22219 0.186033
\(789\) 27.7321 0.987288
\(790\) 22.3896 0.796587
\(791\) −7.07838 −0.251678
\(792\) 0 0
\(793\) 0.278438 0.00988764
\(794\) −45.9565 −1.63094
\(795\) 11.7587 0.417039
\(796\) 3.86868 0.137122
\(797\) 12.6491 0.448056 0.224028 0.974583i \(-0.428079\pi\)
0.224028 + 0.974583i \(0.428079\pi\)
\(798\) −1.65983 −0.0587572
\(799\) −32.5562 −1.15176
\(800\) −2.06278 −0.0729303
\(801\) 4.62702 0.163488
\(802\) −8.49239 −0.299877
\(803\) 0 0
\(804\) −0.658990 −0.0232408
\(805\) 5.70928 0.201226
\(806\) 0.124128 0.00437223
\(807\) −4.09890 −0.144288
\(808\) 22.6830 0.797985
\(809\) 49.9299 1.75544 0.877720 0.479174i \(-0.159064\pi\)
0.877720 + 0.479174i \(0.159064\pi\)
\(810\) 2.22795 0.0782820
\(811\) −7.95896 −0.279477 −0.139738 0.990188i \(-0.544626\pi\)
−0.139738 + 0.990188i \(0.544626\pi\)
\(812\) −0.523590 −0.0183744
\(813\) 9.94214 0.348686
\(814\) 0 0
\(815\) 11.4680 0.401706
\(816\) −29.6711 −1.03870
\(817\) 3.56302 0.124654
\(818\) 5.28580 0.184814
\(819\) −0.149568 −0.00522631
\(820\) −0.597048 −0.0208498
\(821\) −4.92162 −0.171766 −0.0858829 0.996305i \(-0.527371\pi\)
−0.0858829 + 0.996305i \(0.527371\pi\)
\(822\) 16.0045 0.558222
\(823\) −4.04945 −0.141155 −0.0705774 0.997506i \(-0.522484\pi\)
−0.0705774 + 0.997506i \(0.522484\pi\)
\(824\) −8.35124 −0.290929
\(825\) 0 0
\(826\) −3.29565 −0.114670
\(827\) −33.3256 −1.15885 −0.579423 0.815027i \(-0.696722\pi\)
−0.579423 + 0.815027i \(0.696722\pi\)
\(828\) 3.43680 0.119437
\(829\) −0.156755 −0.00544434 −0.00272217 0.999996i \(-0.500866\pi\)
−0.00272217 + 0.999996i \(0.500866\pi\)
\(830\) −13.1194 −0.455382
\(831\) 30.3812 1.05391
\(832\) 0.552906 0.0191686
\(833\) 5.51026 0.190919
\(834\) 27.0517 0.936724
\(835\) −5.60197 −0.193864
\(836\) 0 0
\(837\) −4.76487 −0.164698
\(838\) 28.0338 0.968413
\(839\) −49.6775 −1.71506 −0.857529 0.514435i \(-0.828002\pi\)
−0.857529 + 0.514435i \(0.828002\pi\)
\(840\) 2.93722 0.101344
\(841\) −26.9877 −0.930611
\(842\) 16.4040 0.565319
\(843\) −13.6598 −0.470469
\(844\) 0.979481 0.0337151
\(845\) 12.9916 0.446924
\(846\) 14.8313 0.509912
\(847\) 0 0
\(848\) 46.2472 1.58814
\(849\) 16.6803 0.572468
\(850\) 8.48133 0.290907
\(851\) 50.1711 1.71984
\(852\) 1.77024 0.0606474
\(853\) −39.4257 −1.34991 −0.674956 0.737858i \(-0.735837\pi\)
−0.674956 + 0.737858i \(0.735837\pi\)
\(854\) 4.67316 0.159912
\(855\) −1.50307 −0.0514040
\(856\) 20.3303 0.694876
\(857\) −33.2423 −1.13554 −0.567768 0.823189i \(-0.692193\pi\)
−0.567768 + 0.823189i \(0.692193\pi\)
\(858\) 0 0
\(859\) 7.71646 0.263282 0.131641 0.991297i \(-0.457975\pi\)
0.131641 + 0.991297i \(0.457975\pi\)
\(860\) 1.42696 0.0486590
\(861\) 1.89269 0.0645028
\(862\) 11.7138 0.398974
\(863\) −24.6453 −0.838935 −0.419467 0.907770i \(-0.637783\pi\)
−0.419467 + 0.907770i \(0.637783\pi\)
\(864\) 11.1773 0.380259
\(865\) 21.6092 0.734733
\(866\) 51.6118 1.75384
\(867\) 15.6358 0.531020
\(868\) 0.324575 0.0110168
\(869\) 0 0
\(870\) 2.55479 0.0866154
\(871\) −0.139935 −0.00474150
\(872\) 38.0989 1.29019
\(873\) −23.2495 −0.786877
\(874\) 8.09890 0.273949
\(875\) −1.00000 −0.0338062
\(876\) −6.12866 −0.207068
\(877\) −6.17954 −0.208668 −0.104334 0.994542i \(-0.533271\pi\)
−0.104334 + 0.994542i \(0.533271\pi\)
\(878\) 21.1773 0.714698
\(879\) −29.3835 −0.991080
\(880\) 0 0
\(881\) −49.3295 −1.66195 −0.830976 0.556308i \(-0.812218\pi\)
−0.830976 + 0.556308i \(0.812218\pi\)
\(882\) −2.51026 −0.0845248
\(883\) −22.1529 −0.745504 −0.372752 0.927931i \(-0.621586\pi\)
−0.372752 + 0.927931i \(0.621586\pi\)
\(884\) 0.186522 0.00627341
\(885\) 2.50534 0.0842160
\(886\) −17.1910 −0.577543
\(887\) −38.7358 −1.30062 −0.650310 0.759669i \(-0.725361\pi\)
−0.650310 + 0.759669i \(0.725361\pi\)
\(888\) 25.8113 0.866170
\(889\) 9.65983 0.323980
\(890\) 4.36683 0.146377
\(891\) 0 0
\(892\) −3.20128 −0.107187
\(893\) 5.44521 0.182217
\(894\) 24.6750 0.825257
\(895\) 2.05786 0.0687866
\(896\) 13.4052 0.447837
\(897\) −0.612646 −0.0204557
\(898\) −29.3523 −0.979498
\(899\) −1.24742 −0.0416038
\(900\) −0.601968 −0.0200656
\(901\) −55.3751 −1.84481
\(902\) 0 0
\(903\) −4.52359 −0.150536
\(904\) 17.7686 0.590974
\(905\) −20.2823 −0.674207
\(906\) −1.90110 −0.0631599
\(907\) 18.4352 0.612131 0.306065 0.952011i \(-0.400987\pi\)
0.306065 + 0.952011i \(0.400987\pi\)
\(908\) −3.57077 −0.118500
\(909\) 14.7370 0.488795
\(910\) −0.141157 −0.00467931
\(911\) −11.9011 −0.394301 −0.197151 0.980373i \(-0.563169\pi\)
−0.197151 + 0.980373i \(0.563169\pi\)
\(912\) 4.96266 0.164330
\(913\) 0 0
\(914\) 1.21235 0.0401009
\(915\) −3.55252 −0.117443
\(916\) −5.00084 −0.165232
\(917\) −4.68035 −0.154559
\(918\) −45.9565 −1.51679
\(919\) −56.3812 −1.85984 −0.929922 0.367756i \(-0.880126\pi\)
−0.929922 + 0.367756i \(0.880126\pi\)
\(920\) −14.3318 −0.472504
\(921\) −9.39189 −0.309473
\(922\) −39.3823 −1.29699
\(923\) 0.375905 0.0123731
\(924\) 0 0
\(925\) −8.78765 −0.288936
\(926\) −21.9481 −0.721260
\(927\) −5.42574 −0.178205
\(928\) 2.92616 0.0960558
\(929\) 19.2351 0.631084 0.315542 0.948912i \(-0.397814\pi\)
0.315542 + 0.948912i \(0.397814\pi\)
\(930\) −1.58372 −0.0519321
\(931\) −0.921622 −0.0302049
\(932\) −3.09663 −0.101433
\(933\) −30.8865 −1.01118
\(934\) 28.2401 0.924043
\(935\) 0 0
\(936\) 0.375453 0.0122721
\(937\) −17.6358 −0.576137 −0.288069 0.957610i \(-0.593013\pi\)
−0.288069 + 0.957610i \(0.593013\pi\)
\(938\) −2.34858 −0.0766840
\(939\) 30.1087 0.982562
\(940\) 2.18076 0.0711287
\(941\) 20.1990 0.658469 0.329235 0.944248i \(-0.393209\pi\)
0.329235 + 0.944248i \(0.393209\pi\)
\(942\) −31.9832 −1.04207
\(943\) −9.23513 −0.300737
\(944\) 9.85354 0.320705
\(945\) 5.41855 0.176265
\(946\) 0 0
\(947\) 17.6925 0.574928 0.287464 0.957792i \(-0.407188\pi\)
0.287464 + 0.957792i \(0.407188\pi\)
\(948\) −6.28231 −0.204040
\(949\) −1.30140 −0.0422453
\(950\) −1.41855 −0.0460239
\(951\) 7.38735 0.239551
\(952\) −13.8322 −0.448304
\(953\) −39.7093 −1.28631 −0.643155 0.765736i \(-0.722375\pi\)
−0.643155 + 0.765736i \(0.722375\pi\)
\(954\) 25.2267 0.816745
\(955\) 20.2823 0.656320
\(956\) 10.8851 0.352050
\(957\) 0 0
\(958\) −18.7337 −0.605257
\(959\) 8.88655 0.286962
\(960\) −7.05437 −0.227679
\(961\) −30.2267 −0.975056
\(962\) −1.24044 −0.0399934
\(963\) 13.2085 0.425637
\(964\) −1.34509 −0.0433225
\(965\) −24.3051 −0.782409
\(966\) −10.2823 −0.330828
\(967\) −3.01664 −0.0970087 −0.0485044 0.998823i \(-0.515445\pi\)
−0.0485044 + 0.998823i \(0.515445\pi\)
\(968\) 0 0
\(969\) −5.94214 −0.190889
\(970\) −21.9421 −0.704520
\(971\) −18.2134 −0.584496 −0.292248 0.956343i \(-0.594403\pi\)
−0.292248 + 0.956343i \(0.594403\pi\)
\(972\) 5.37486 0.172399
\(973\) 15.0205 0.481536
\(974\) −6.51518 −0.208760
\(975\) 0.107307 0.00343658
\(976\) −13.9721 −0.447237
\(977\) −30.0845 −0.962489 −0.481245 0.876586i \(-0.659815\pi\)
−0.481245 + 0.876586i \(0.659815\pi\)
\(978\) −20.6537 −0.660432
\(979\) 0 0
\(980\) −0.369102 −0.0117905
\(981\) 24.7526 0.790289
\(982\) −33.8902 −1.08148
\(983\) 5.92267 0.188904 0.0944519 0.995529i \(-0.469890\pi\)
0.0944519 + 0.995529i \(0.469890\pi\)
\(984\) −4.75115 −0.151461
\(985\) −14.1483 −0.450804
\(986\) −12.0312 −0.383151
\(987\) −6.91321 −0.220050
\(988\) −0.0311968 −0.000992504 0
\(989\) 22.0722 0.701856
\(990\) 0 0
\(991\) 9.24742 0.293754 0.146877 0.989155i \(-0.453078\pi\)
0.146877 + 0.989155i \(0.453078\pi\)
\(992\) −1.81393 −0.0575923
\(993\) 4.09890 0.130075
\(994\) 6.30898 0.200109
\(995\) −10.4813 −0.332281
\(996\) 3.68118 0.116643
\(997\) −32.1496 −1.01819 −0.509094 0.860711i \(-0.670019\pi\)
−0.509094 + 0.860711i \(0.670019\pi\)
\(998\) 50.7838 1.60753
\(999\) 47.6163 1.50651
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 4235.2.a.q.1.2 3
11.10 odd 2 385.2.a.f.1.2 3
33.32 even 2 3465.2.a.bh.1.2 3
44.43 even 2 6160.2.a.bn.1.1 3
55.32 even 4 1925.2.b.n.1849.2 6
55.43 even 4 1925.2.b.n.1849.5 6
55.54 odd 2 1925.2.a.v.1.2 3
77.76 even 2 2695.2.a.g.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.a.f.1.2 3 11.10 odd 2
1925.2.a.v.1.2 3 55.54 odd 2
1925.2.b.n.1849.2 6 55.32 even 4
1925.2.b.n.1849.5 6 55.43 even 4
2695.2.a.g.1.2 3 77.76 even 2
3465.2.a.bh.1.2 3 33.32 even 2
4235.2.a.q.1.2 3 1.1 even 1 trivial
6160.2.a.bn.1.1 3 44.43 even 2