Properties

Label 5054.2.a.bl.1.2
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 3 x^{11} - 24 x^{10} + 68 x^{9} + 213 x^{8} - 555 x^{7} - 814 x^{6} + 1911 x^{5} + 1005 x^{4} - 2326 x^{3} + 570 x^{2} + 105 x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 266)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.98161\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.98161 q^{3} +1.00000 q^{4} -4.31337 q^{5} +2.98161 q^{6} +1.00000 q^{7} -1.00000 q^{8} +5.89001 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.98161 q^{3} +1.00000 q^{4} -4.31337 q^{5} +2.98161 q^{6} +1.00000 q^{7} -1.00000 q^{8} +5.89001 q^{9} +4.31337 q^{10} +0.362266 q^{11} -2.98161 q^{12} -5.24373 q^{13} -1.00000 q^{14} +12.8608 q^{15} +1.00000 q^{16} -4.63000 q^{17} -5.89001 q^{18} -4.31337 q^{20} -2.98161 q^{21} -0.362266 q^{22} +1.78046 q^{23} +2.98161 q^{24} +13.6052 q^{25} +5.24373 q^{26} -8.61690 q^{27} +1.00000 q^{28} -2.08498 q^{29} -12.8608 q^{30} +2.02090 q^{31} -1.00000 q^{32} -1.08014 q^{33} +4.63000 q^{34} -4.31337 q^{35} +5.89001 q^{36} -7.44807 q^{37} +15.6348 q^{39} +4.31337 q^{40} +2.71367 q^{41} +2.98161 q^{42} +3.97133 q^{43} +0.362266 q^{44} -25.4058 q^{45} -1.78046 q^{46} -3.27349 q^{47} -2.98161 q^{48} +1.00000 q^{49} -13.6052 q^{50} +13.8049 q^{51} -5.24373 q^{52} -11.7483 q^{53} +8.61690 q^{54} -1.56259 q^{55} -1.00000 q^{56} +2.08498 q^{58} -0.777397 q^{59} +12.8608 q^{60} -10.7422 q^{61} -2.02090 q^{62} +5.89001 q^{63} +1.00000 q^{64} +22.6181 q^{65} +1.08014 q^{66} +2.61230 q^{67} -4.63000 q^{68} -5.30866 q^{69} +4.31337 q^{70} -14.5430 q^{71} -5.89001 q^{72} +2.61980 q^{73} +7.44807 q^{74} -40.5654 q^{75} +0.362266 q^{77} -15.6348 q^{78} -0.554543 q^{79} -4.31337 q^{80} +8.02223 q^{81} -2.71367 q^{82} -5.06311 q^{83} -2.98161 q^{84} +19.9709 q^{85} -3.97133 q^{86} +6.21661 q^{87} -0.362266 q^{88} -7.02669 q^{89} +25.4058 q^{90} -5.24373 q^{91} +1.78046 q^{92} -6.02554 q^{93} +3.27349 q^{94} +2.98161 q^{96} -1.38858 q^{97} -1.00000 q^{98} +2.13375 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} - 3 q^{3} + 12 q^{4} + 3 q^{5} + 3 q^{6} + 12 q^{7} - 12 q^{8} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} - 3 q^{3} + 12 q^{4} + 3 q^{5} + 3 q^{6} + 12 q^{7} - 12 q^{8} + 21 q^{9} - 3 q^{10} + 3 q^{11} - 3 q^{12} + 6 q^{13} - 12 q^{14} + 6 q^{15} + 12 q^{16} + 3 q^{17} - 21 q^{18} + 3 q^{20} - 3 q^{21} - 3 q^{22} + 12 q^{23} + 3 q^{24} + 33 q^{25} - 6 q^{26} - 21 q^{27} + 12 q^{28} - 6 q^{29} - 6 q^{30} - 3 q^{31} - 12 q^{32} + 6 q^{33} - 3 q^{34} + 3 q^{35} + 21 q^{36} - 27 q^{37} + 27 q^{39} - 3 q^{40} + 27 q^{41} + 3 q^{42} + 30 q^{43} + 3 q^{44} + 30 q^{45} - 12 q^{46} + 21 q^{47} - 3 q^{48} + 12 q^{49} - 33 q^{50} - 3 q^{51} + 6 q^{52} - 6 q^{53} + 21 q^{54} + 48 q^{55} - 12 q^{56} + 6 q^{58} + 27 q^{59} + 6 q^{60} + 18 q^{61} + 3 q^{62} + 21 q^{63} + 12 q^{64} + 9 q^{65} - 6 q^{66} - 9 q^{67} + 3 q^{68} - 18 q^{69} - 3 q^{70} - 27 q^{71} - 21 q^{72} - 9 q^{73} + 27 q^{74} - 33 q^{75} + 3 q^{77} - 27 q^{78} + 12 q^{79} + 3 q^{80} + 24 q^{81} - 27 q^{82} + 9 q^{83} - 3 q^{84} + 78 q^{85} - 30 q^{86} - 45 q^{87} - 3 q^{88} + 24 q^{89} - 30 q^{90} + 6 q^{91} + 12 q^{92} + 3 q^{93} - 21 q^{94} + 3 q^{96} + 18 q^{97} - 12 q^{98} + 48 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.98161 −1.72143 −0.860717 0.509083i \(-0.829985\pi\)
−0.860717 + 0.509083i \(0.829985\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.31337 −1.92900 −0.964499 0.264085i \(-0.914930\pi\)
−0.964499 + 0.264085i \(0.914930\pi\)
\(6\) 2.98161 1.21724
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 5.89001 1.96334
\(10\) 4.31337 1.36401
\(11\) 0.362266 0.109227 0.0546137 0.998508i \(-0.482607\pi\)
0.0546137 + 0.998508i \(0.482607\pi\)
\(12\) −2.98161 −0.860717
\(13\) −5.24373 −1.45435 −0.727174 0.686453i \(-0.759167\pi\)
−0.727174 + 0.686453i \(0.759167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 12.8608 3.32065
\(16\) 1.00000 0.250000
\(17\) −4.63000 −1.12294 −0.561470 0.827497i \(-0.689764\pi\)
−0.561470 + 0.827497i \(0.689764\pi\)
\(18\) −5.89001 −1.38829
\(19\) 0 0
\(20\) −4.31337 −0.964499
\(21\) −2.98161 −0.650641
\(22\) −0.362266 −0.0772354
\(23\) 1.78046 0.371253 0.185626 0.982620i \(-0.440569\pi\)
0.185626 + 0.982620i \(0.440569\pi\)
\(24\) 2.98161 0.608619
\(25\) 13.6052 2.72104
\(26\) 5.24373 1.02838
\(27\) −8.61690 −1.65832
\(28\) 1.00000 0.188982
\(29\) −2.08498 −0.387171 −0.193586 0.981083i \(-0.562012\pi\)
−0.193586 + 0.981083i \(0.562012\pi\)
\(30\) −12.8608 −2.34805
\(31\) 2.02090 0.362965 0.181482 0.983394i \(-0.441910\pi\)
0.181482 + 0.983394i \(0.441910\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.08014 −0.188028
\(34\) 4.63000 0.794038
\(35\) −4.31337 −0.729093
\(36\) 5.89001 0.981669
\(37\) −7.44807 −1.22446 −0.612228 0.790682i \(-0.709726\pi\)
−0.612228 + 0.790682i \(0.709726\pi\)
\(38\) 0 0
\(39\) 15.6348 2.50357
\(40\) 4.31337 0.682004
\(41\) 2.71367 0.423804 0.211902 0.977291i \(-0.432034\pi\)
0.211902 + 0.977291i \(0.432034\pi\)
\(42\) 2.98161 0.460073
\(43\) 3.97133 0.605623 0.302811 0.953051i \(-0.402075\pi\)
0.302811 + 0.953051i \(0.402075\pi\)
\(44\) 0.362266 0.0546137
\(45\) −25.4058 −3.78728
\(46\) −1.78046 −0.262515
\(47\) −3.27349 −0.477487 −0.238744 0.971083i \(-0.576736\pi\)
−0.238744 + 0.971083i \(0.576736\pi\)
\(48\) −2.98161 −0.430359
\(49\) 1.00000 0.142857
\(50\) −13.6052 −1.92406
\(51\) 13.8049 1.93307
\(52\) −5.24373 −0.727174
\(53\) −11.7483 −1.61375 −0.806874 0.590724i \(-0.798842\pi\)
−0.806874 + 0.590724i \(0.798842\pi\)
\(54\) 8.61690 1.17261
\(55\) −1.56259 −0.210699
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 2.08498 0.273772
\(59\) −0.777397 −0.101208 −0.0506042 0.998719i \(-0.516115\pi\)
−0.0506042 + 0.998719i \(0.516115\pi\)
\(60\) 12.8608 1.66032
\(61\) −10.7422 −1.37540 −0.687698 0.725997i \(-0.741379\pi\)
−0.687698 + 0.725997i \(0.741379\pi\)
\(62\) −2.02090 −0.256655
\(63\) 5.89001 0.742072
\(64\) 1.00000 0.125000
\(65\) 22.6181 2.80544
\(66\) 1.08014 0.132956
\(67\) 2.61230 0.319143 0.159572 0.987186i \(-0.448989\pi\)
0.159572 + 0.987186i \(0.448989\pi\)
\(68\) −4.63000 −0.561470
\(69\) −5.30866 −0.639087
\(70\) 4.31337 0.515547
\(71\) −14.5430 −1.72594 −0.862968 0.505258i \(-0.831397\pi\)
−0.862968 + 0.505258i \(0.831397\pi\)
\(72\) −5.89001 −0.694145
\(73\) 2.61980 0.306624 0.153312 0.988178i \(-0.451006\pi\)
0.153312 + 0.988178i \(0.451006\pi\)
\(74\) 7.44807 0.865821
\(75\) −40.5654 −4.68409
\(76\) 0 0
\(77\) 0.362266 0.0412841
\(78\) −15.6348 −1.77029
\(79\) −0.554543 −0.0623910 −0.0311955 0.999513i \(-0.509931\pi\)
−0.0311955 + 0.999513i \(0.509931\pi\)
\(80\) −4.31337 −0.482250
\(81\) 8.02223 0.891358
\(82\) −2.71367 −0.299675
\(83\) −5.06311 −0.555748 −0.277874 0.960617i \(-0.589630\pi\)
−0.277874 + 0.960617i \(0.589630\pi\)
\(84\) −2.98161 −0.325321
\(85\) 19.9709 2.16615
\(86\) −3.97133 −0.428240
\(87\) 6.21661 0.666490
\(88\) −0.362266 −0.0386177
\(89\) −7.02669 −0.744827 −0.372414 0.928067i \(-0.621470\pi\)
−0.372414 + 0.928067i \(0.621470\pi\)
\(90\) 25.4058 2.67801
\(91\) −5.24373 −0.549692
\(92\) 1.78046 0.185626
\(93\) −6.02554 −0.624820
\(94\) 3.27349 0.337634
\(95\) 0 0
\(96\) 2.98161 0.304310
\(97\) −1.38858 −0.140989 −0.0704944 0.997512i \(-0.522458\pi\)
−0.0704944 + 0.997512i \(0.522458\pi\)
\(98\) −1.00000 −0.101015
\(99\) 2.13375 0.214450
\(100\) 13.6052 1.36052
\(101\) −0.278655 −0.0277272 −0.0138636 0.999904i \(-0.504413\pi\)
−0.0138636 + 0.999904i \(0.504413\pi\)
\(102\) −13.8049 −1.36688
\(103\) −10.9855 −1.08244 −0.541218 0.840882i \(-0.682037\pi\)
−0.541218 + 0.840882i \(0.682037\pi\)
\(104\) 5.24373 0.514190
\(105\) 12.8608 1.25509
\(106\) 11.7483 1.14109
\(107\) −3.59786 −0.347819 −0.173909 0.984762i \(-0.555640\pi\)
−0.173909 + 0.984762i \(0.555640\pi\)
\(108\) −8.61690 −0.829162
\(109\) −10.9045 −1.04446 −0.522231 0.852804i \(-0.674900\pi\)
−0.522231 + 0.852804i \(0.674900\pi\)
\(110\) 1.56259 0.148987
\(111\) 22.2073 2.10782
\(112\) 1.00000 0.0944911
\(113\) −4.29382 −0.403928 −0.201964 0.979393i \(-0.564732\pi\)
−0.201964 + 0.979393i \(0.564732\pi\)
\(114\) 0 0
\(115\) −7.67981 −0.716146
\(116\) −2.08498 −0.193586
\(117\) −30.8856 −2.85538
\(118\) 0.777397 0.0715652
\(119\) −4.63000 −0.424431
\(120\) −12.8608 −1.17403
\(121\) −10.8688 −0.988069
\(122\) 10.7422 0.972552
\(123\) −8.09111 −0.729551
\(124\) 2.02090 0.181482
\(125\) −37.1173 −3.31988
\(126\) −5.89001 −0.524724
\(127\) −11.0424 −0.979851 −0.489926 0.871764i \(-0.662976\pi\)
−0.489926 + 0.871764i \(0.662976\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −11.8410 −1.04254
\(130\) −22.6181 −1.98374
\(131\) 3.90396 0.341091 0.170546 0.985350i \(-0.445447\pi\)
0.170546 + 0.985350i \(0.445447\pi\)
\(132\) −1.08014 −0.0940139
\(133\) 0 0
\(134\) −2.61230 −0.225668
\(135\) 37.1679 3.19890
\(136\) 4.63000 0.397019
\(137\) −10.8789 −0.929448 −0.464724 0.885456i \(-0.653846\pi\)
−0.464724 + 0.885456i \(0.653846\pi\)
\(138\) 5.30866 0.451903
\(139\) 12.0869 1.02520 0.512601 0.858627i \(-0.328682\pi\)
0.512601 + 0.858627i \(0.328682\pi\)
\(140\) −4.31337 −0.364546
\(141\) 9.76027 0.821963
\(142\) 14.5430 1.22042
\(143\) −1.89962 −0.158855
\(144\) 5.89001 0.490835
\(145\) 8.99330 0.746853
\(146\) −2.61980 −0.216816
\(147\) −2.98161 −0.245919
\(148\) −7.44807 −0.612228
\(149\) −19.8129 −1.62313 −0.811566 0.584260i \(-0.801385\pi\)
−0.811566 + 0.584260i \(0.801385\pi\)
\(150\) 40.5654 3.31215
\(151\) 16.3282 1.32877 0.664386 0.747390i \(-0.268693\pi\)
0.664386 + 0.747390i \(0.268693\pi\)
\(152\) 0 0
\(153\) −27.2708 −2.20471
\(154\) −0.362266 −0.0291922
\(155\) −8.71690 −0.700158
\(156\) 15.6348 1.25178
\(157\) −5.37439 −0.428923 −0.214462 0.976732i \(-0.568800\pi\)
−0.214462 + 0.976732i \(0.568800\pi\)
\(158\) 0.554543 0.0441171
\(159\) 35.0288 2.77796
\(160\) 4.31337 0.341002
\(161\) 1.78046 0.140320
\(162\) −8.02223 −0.630286
\(163\) −4.35537 −0.341139 −0.170569 0.985346i \(-0.554561\pi\)
−0.170569 + 0.985346i \(0.554561\pi\)
\(164\) 2.71367 0.211902
\(165\) 4.65903 0.362705
\(166\) 5.06311 0.392973
\(167\) −23.9411 −1.85262 −0.926309 0.376766i \(-0.877036\pi\)
−0.926309 + 0.376766i \(0.877036\pi\)
\(168\) 2.98161 0.230036
\(169\) 14.4967 1.11513
\(170\) −19.9709 −1.53170
\(171\) 0 0
\(172\) 3.97133 0.302811
\(173\) −5.35580 −0.407194 −0.203597 0.979055i \(-0.565263\pi\)
−0.203597 + 0.979055i \(0.565263\pi\)
\(174\) −6.21661 −0.471280
\(175\) 13.6052 1.02845
\(176\) 0.362266 0.0273068
\(177\) 2.31790 0.174224
\(178\) 7.02669 0.526672
\(179\) 19.3211 1.44413 0.722064 0.691826i \(-0.243193\pi\)
0.722064 + 0.691826i \(0.243193\pi\)
\(180\) −25.4058 −1.89364
\(181\) −9.50541 −0.706531 −0.353266 0.935523i \(-0.614929\pi\)
−0.353266 + 0.935523i \(0.614929\pi\)
\(182\) 5.24373 0.388691
\(183\) 32.0290 2.36765
\(184\) −1.78046 −0.131258
\(185\) 32.1263 2.36197
\(186\) 6.02554 0.441814
\(187\) −1.67729 −0.122656
\(188\) −3.27349 −0.238744
\(189\) −8.61690 −0.626787
\(190\) 0 0
\(191\) −16.3533 −1.18329 −0.591643 0.806200i \(-0.701521\pi\)
−0.591643 + 0.806200i \(0.701521\pi\)
\(192\) −2.98161 −0.215179
\(193\) 0.911504 0.0656115 0.0328057 0.999462i \(-0.489556\pi\)
0.0328057 + 0.999462i \(0.489556\pi\)
\(194\) 1.38858 0.0996941
\(195\) −67.4385 −4.82937
\(196\) 1.00000 0.0714286
\(197\) −27.7280 −1.97554 −0.987770 0.155920i \(-0.950166\pi\)
−0.987770 + 0.155920i \(0.950166\pi\)
\(198\) −2.13375 −0.151639
\(199\) −6.58815 −0.467022 −0.233511 0.972354i \(-0.575021\pi\)
−0.233511 + 0.972354i \(0.575021\pi\)
\(200\) −13.6052 −0.962031
\(201\) −7.78887 −0.549385
\(202\) 0.278655 0.0196061
\(203\) −2.08498 −0.146337
\(204\) 13.8049 0.966534
\(205\) −11.7051 −0.817518
\(206\) 10.9855 0.765398
\(207\) 10.4870 0.728894
\(208\) −5.24373 −0.363587
\(209\) 0 0
\(210\) −12.8608 −0.887480
\(211\) 3.92985 0.270542 0.135271 0.990809i \(-0.456810\pi\)
0.135271 + 0.990809i \(0.456810\pi\)
\(212\) −11.7483 −0.806874
\(213\) 43.3616 2.97109
\(214\) 3.59786 0.245945
\(215\) −17.1298 −1.16825
\(216\) 8.61690 0.586306
\(217\) 2.02090 0.137188
\(218\) 10.9045 0.738547
\(219\) −7.81121 −0.527833
\(220\) −1.56259 −0.105350
\(221\) 24.2784 1.63314
\(222\) −22.2073 −1.49045
\(223\) −15.3681 −1.02912 −0.514562 0.857453i \(-0.672046\pi\)
−0.514562 + 0.857453i \(0.672046\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 80.1347 5.34231
\(226\) 4.29382 0.285621
\(227\) 15.6050 1.03574 0.517870 0.855459i \(-0.326725\pi\)
0.517870 + 0.855459i \(0.326725\pi\)
\(228\) 0 0
\(229\) 8.96359 0.592331 0.296165 0.955137i \(-0.404292\pi\)
0.296165 + 0.955137i \(0.404292\pi\)
\(230\) 7.67981 0.506391
\(231\) −1.08014 −0.0710678
\(232\) 2.08498 0.136886
\(233\) 15.2270 0.997553 0.498776 0.866731i \(-0.333783\pi\)
0.498776 + 0.866731i \(0.333783\pi\)
\(234\) 30.8856 2.01906
\(235\) 14.1198 0.921072
\(236\) −0.777397 −0.0506042
\(237\) 1.65343 0.107402
\(238\) 4.63000 0.300118
\(239\) 18.9069 1.22298 0.611492 0.791250i \(-0.290570\pi\)
0.611492 + 0.791250i \(0.290570\pi\)
\(240\) 12.8608 0.830161
\(241\) 18.6898 1.20391 0.601957 0.798529i \(-0.294388\pi\)
0.601957 + 0.798529i \(0.294388\pi\)
\(242\) 10.8688 0.698671
\(243\) 1.93154 0.123908
\(244\) −10.7422 −0.687698
\(245\) −4.31337 −0.275571
\(246\) 8.09111 0.515871
\(247\) 0 0
\(248\) −2.02090 −0.128327
\(249\) 15.0962 0.956685
\(250\) 37.1173 2.34751
\(251\) −17.5364 −1.10689 −0.553445 0.832885i \(-0.686687\pi\)
−0.553445 + 0.832885i \(0.686687\pi\)
\(252\) 5.89001 0.371036
\(253\) 0.645002 0.0405509
\(254\) 11.0424 0.692860
\(255\) −59.5455 −3.72888
\(256\) 1.00000 0.0625000
\(257\) 11.5288 0.719145 0.359572 0.933117i \(-0.382923\pi\)
0.359572 + 0.933117i \(0.382923\pi\)
\(258\) 11.8410 0.737187
\(259\) −7.44807 −0.462801
\(260\) 22.6181 1.40272
\(261\) −12.2806 −0.760148
\(262\) −3.90396 −0.241188
\(263\) −5.34136 −0.329362 −0.164681 0.986347i \(-0.552660\pi\)
−0.164681 + 0.986347i \(0.552660\pi\)
\(264\) 1.08014 0.0664779
\(265\) 50.6746 3.11292
\(266\) 0 0
\(267\) 20.9509 1.28217
\(268\) 2.61230 0.159572
\(269\) 26.1618 1.59511 0.797557 0.603244i \(-0.206125\pi\)
0.797557 + 0.603244i \(0.206125\pi\)
\(270\) −37.1679 −2.26197
\(271\) −20.2805 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(272\) −4.63000 −0.280735
\(273\) 15.6348 0.946259
\(274\) 10.8789 0.657219
\(275\) 4.92870 0.297212
\(276\) −5.30866 −0.319544
\(277\) 31.1729 1.87300 0.936500 0.350666i \(-0.114045\pi\)
0.936500 + 0.350666i \(0.114045\pi\)
\(278\) −12.0869 −0.724927
\(279\) 11.9031 0.712622
\(280\) 4.31337 0.257773
\(281\) −6.00910 −0.358473 −0.179236 0.983806i \(-0.557363\pi\)
−0.179236 + 0.983806i \(0.557363\pi\)
\(282\) −9.76027 −0.581216
\(283\) 12.2967 0.730963 0.365481 0.930819i \(-0.380904\pi\)
0.365481 + 0.930819i \(0.380904\pi\)
\(284\) −14.5430 −0.862968
\(285\) 0 0
\(286\) 1.89962 0.112327
\(287\) 2.71367 0.160183
\(288\) −5.89001 −0.347072
\(289\) 4.43688 0.260993
\(290\) −8.99330 −0.528105
\(291\) 4.14020 0.242703
\(292\) 2.61980 0.153312
\(293\) 4.02655 0.235234 0.117617 0.993059i \(-0.462475\pi\)
0.117617 + 0.993059i \(0.462475\pi\)
\(294\) 2.98161 0.173891
\(295\) 3.35320 0.195231
\(296\) 7.44807 0.432910
\(297\) −3.12161 −0.181134
\(298\) 19.8129 1.14773
\(299\) −9.33627 −0.539930
\(300\) −40.5654 −2.34204
\(301\) 3.97133 0.228904
\(302\) −16.3282 −0.939584
\(303\) 0.830842 0.0477306
\(304\) 0 0
\(305\) 46.3350 2.65314
\(306\) 27.2708 1.55897
\(307\) −22.6810 −1.29447 −0.647237 0.762289i \(-0.724075\pi\)
−0.647237 + 0.762289i \(0.724075\pi\)
\(308\) 0.362266 0.0206420
\(309\) 32.7546 1.86334
\(310\) 8.71690 0.495087
\(311\) 16.9035 0.958512 0.479256 0.877675i \(-0.340906\pi\)
0.479256 + 0.877675i \(0.340906\pi\)
\(312\) −15.6348 −0.885144
\(313\) −1.39972 −0.0791169 −0.0395585 0.999217i \(-0.512595\pi\)
−0.0395585 + 0.999217i \(0.512595\pi\)
\(314\) 5.37439 0.303295
\(315\) −25.4058 −1.43146
\(316\) −0.554543 −0.0311955
\(317\) 10.9472 0.614856 0.307428 0.951571i \(-0.400532\pi\)
0.307428 + 0.951571i \(0.400532\pi\)
\(318\) −35.0288 −1.96432
\(319\) −0.755318 −0.0422897
\(320\) −4.31337 −0.241125
\(321\) 10.7274 0.598747
\(322\) −1.78046 −0.0992214
\(323\) 0 0
\(324\) 8.02223 0.445679
\(325\) −71.3418 −3.95733
\(326\) 4.35537 0.241222
\(327\) 32.5130 1.79797
\(328\) −2.71367 −0.149837
\(329\) −3.27349 −0.180473
\(330\) −4.65903 −0.256471
\(331\) 3.12383 0.171701 0.0858506 0.996308i \(-0.472639\pi\)
0.0858506 + 0.996308i \(0.472639\pi\)
\(332\) −5.06311 −0.277874
\(333\) −43.8692 −2.40402
\(334\) 23.9411 1.31000
\(335\) −11.2678 −0.615627
\(336\) −2.98161 −0.162660
\(337\) −3.66902 −0.199864 −0.0999322 0.994994i \(-0.531863\pi\)
−0.0999322 + 0.994994i \(0.531863\pi\)
\(338\) −14.4967 −0.788515
\(339\) 12.8025 0.695337
\(340\) 19.9709 1.08307
\(341\) 0.732104 0.0396457
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −3.97133 −0.214120
\(345\) 22.8982 1.23280
\(346\) 5.35580 0.287930
\(347\) −14.2657 −0.765825 −0.382913 0.923785i \(-0.625079\pi\)
−0.382913 + 0.923785i \(0.625079\pi\)
\(348\) 6.21661 0.333245
\(349\) −3.19923 −0.171251 −0.0856254 0.996327i \(-0.527289\pi\)
−0.0856254 + 0.996327i \(0.527289\pi\)
\(350\) −13.6052 −0.727227
\(351\) 45.1847 2.41178
\(352\) −0.362266 −0.0193089
\(353\) −2.96498 −0.157810 −0.0789051 0.996882i \(-0.525142\pi\)
−0.0789051 + 0.996882i \(0.525142\pi\)
\(354\) −2.31790 −0.123195
\(355\) 62.7294 3.32933
\(356\) −7.02669 −0.372414
\(357\) 13.8049 0.730631
\(358\) −19.3211 −1.02115
\(359\) −20.5387 −1.08399 −0.541996 0.840381i \(-0.682331\pi\)
−0.541996 + 0.840381i \(0.682331\pi\)
\(360\) 25.4058 1.33900
\(361\) 0 0
\(362\) 9.50541 0.499593
\(363\) 32.4064 1.70090
\(364\) −5.24373 −0.274846
\(365\) −11.3002 −0.591477
\(366\) −32.0290 −1.67418
\(367\) −9.22982 −0.481792 −0.240896 0.970551i \(-0.577441\pi\)
−0.240896 + 0.970551i \(0.577441\pi\)
\(368\) 1.78046 0.0928131
\(369\) 15.9836 0.832071
\(370\) −32.1263 −1.67017
\(371\) −11.7483 −0.609939
\(372\) −6.02554 −0.312410
\(373\) −18.5739 −0.961722 −0.480861 0.876797i \(-0.659676\pi\)
−0.480861 + 0.876797i \(0.659676\pi\)
\(374\) 1.67729 0.0867307
\(375\) 110.670 5.71495
\(376\) 3.27349 0.168817
\(377\) 10.9331 0.563082
\(378\) 8.61690 0.443206
\(379\) 7.87312 0.404415 0.202207 0.979343i \(-0.435188\pi\)
0.202207 + 0.979343i \(0.435188\pi\)
\(380\) 0 0
\(381\) 32.9241 1.68675
\(382\) 16.3533 0.836710
\(383\) 14.6552 0.748846 0.374423 0.927258i \(-0.377841\pi\)
0.374423 + 0.927258i \(0.377841\pi\)
\(384\) 2.98161 0.152155
\(385\) −1.56259 −0.0796369
\(386\) −0.911504 −0.0463943
\(387\) 23.3912 1.18904
\(388\) −1.38858 −0.0704944
\(389\) 17.0207 0.862986 0.431493 0.902116i \(-0.357987\pi\)
0.431493 + 0.902116i \(0.357987\pi\)
\(390\) 67.4385 3.41488
\(391\) −8.24355 −0.416894
\(392\) −1.00000 −0.0505076
\(393\) −11.6401 −0.587166
\(394\) 27.7280 1.39692
\(395\) 2.39195 0.120352
\(396\) 2.13375 0.107225
\(397\) 10.4213 0.523032 0.261516 0.965199i \(-0.415778\pi\)
0.261516 + 0.965199i \(0.415778\pi\)
\(398\) 6.58815 0.330234
\(399\) 0 0
\(400\) 13.6052 0.680259
\(401\) 30.0758 1.50191 0.750956 0.660353i \(-0.229593\pi\)
0.750956 + 0.660353i \(0.229593\pi\)
\(402\) 7.78887 0.388474
\(403\) −10.5971 −0.527877
\(404\) −0.278655 −0.0138636
\(405\) −34.6028 −1.71943
\(406\) 2.08498 0.103476
\(407\) −2.69818 −0.133744
\(408\) −13.8049 −0.683442
\(409\) −11.0533 −0.546553 −0.273276 0.961936i \(-0.588107\pi\)
−0.273276 + 0.961936i \(0.588107\pi\)
\(410\) 11.7051 0.578072
\(411\) 32.4367 1.59998
\(412\) −10.9855 −0.541218
\(413\) −0.777397 −0.0382532
\(414\) −10.4870 −0.515406
\(415\) 21.8391 1.07204
\(416\) 5.24373 0.257095
\(417\) −36.0386 −1.76482
\(418\) 0 0
\(419\) −14.4315 −0.705023 −0.352511 0.935808i \(-0.614672\pi\)
−0.352511 + 0.935808i \(0.614672\pi\)
\(420\) 12.8608 0.627543
\(421\) 26.6057 1.29668 0.648341 0.761350i \(-0.275463\pi\)
0.648341 + 0.761350i \(0.275463\pi\)
\(422\) −3.92985 −0.191302
\(423\) −19.2809 −0.937469
\(424\) 11.7483 0.570546
\(425\) −62.9919 −3.05556
\(426\) −43.3616 −2.10088
\(427\) −10.7422 −0.519851
\(428\) −3.59786 −0.173909
\(429\) 5.66395 0.273458
\(430\) 17.1298 0.826074
\(431\) −20.2109 −0.973524 −0.486762 0.873534i \(-0.661822\pi\)
−0.486762 + 0.873534i \(0.661822\pi\)
\(432\) −8.61690 −0.414581
\(433\) 36.1393 1.73674 0.868372 0.495913i \(-0.165167\pi\)
0.868372 + 0.495913i \(0.165167\pi\)
\(434\) −2.02090 −0.0970064
\(435\) −26.8145 −1.28566
\(436\) −10.9045 −0.522231
\(437\) 0 0
\(438\) 7.81121 0.373234
\(439\) 18.9829 0.906004 0.453002 0.891509i \(-0.350353\pi\)
0.453002 + 0.891509i \(0.350353\pi\)
\(440\) 1.56259 0.0744935
\(441\) 5.89001 0.280477
\(442\) −24.2784 −1.15481
\(443\) −32.2823 −1.53378 −0.766888 0.641781i \(-0.778196\pi\)
−0.766888 + 0.641781i \(0.778196\pi\)
\(444\) 22.2073 1.05391
\(445\) 30.3087 1.43677
\(446\) 15.3681 0.727701
\(447\) 59.0743 2.79412
\(448\) 1.00000 0.0472456
\(449\) −11.0150 −0.519829 −0.259914 0.965632i \(-0.583694\pi\)
−0.259914 + 0.965632i \(0.583694\pi\)
\(450\) −80.1347 −3.77759
\(451\) 0.983071 0.0462910
\(452\) −4.29382 −0.201964
\(453\) −48.6844 −2.28739
\(454\) −15.6050 −0.732379
\(455\) 22.6181 1.06035
\(456\) 0 0
\(457\) −1.92274 −0.0899421 −0.0449710 0.998988i \(-0.514320\pi\)
−0.0449710 + 0.998988i \(0.514320\pi\)
\(458\) −8.96359 −0.418841
\(459\) 39.8962 1.86220
\(460\) −7.67981 −0.358073
\(461\) −3.81073 −0.177483 −0.0887416 0.996055i \(-0.528285\pi\)
−0.0887416 + 0.996055i \(0.528285\pi\)
\(462\) 1.08014 0.0502525
\(463\) −34.9106 −1.62243 −0.811217 0.584745i \(-0.801195\pi\)
−0.811217 + 0.584745i \(0.801195\pi\)
\(464\) −2.08498 −0.0967929
\(465\) 25.9904 1.20528
\(466\) −15.2270 −0.705376
\(467\) −33.2523 −1.53874 −0.769368 0.638806i \(-0.779428\pi\)
−0.769368 + 0.638806i \(0.779428\pi\)
\(468\) −30.8856 −1.42769
\(469\) 2.61230 0.120625
\(470\) −14.1198 −0.651296
\(471\) 16.0244 0.738363
\(472\) 0.777397 0.0357826
\(473\) 1.43868 0.0661506
\(474\) −1.65343 −0.0759447
\(475\) 0 0
\(476\) −4.63000 −0.212216
\(477\) −69.1974 −3.16833
\(478\) −18.9069 −0.864780
\(479\) 3.31481 0.151458 0.0757288 0.997128i \(-0.475872\pi\)
0.0757288 + 0.997128i \(0.475872\pi\)
\(480\) −12.8608 −0.587013
\(481\) 39.0556 1.78078
\(482\) −18.6898 −0.851295
\(483\) −5.30866 −0.241552
\(484\) −10.8688 −0.494035
\(485\) 5.98945 0.271967
\(486\) −1.93154 −0.0876165
\(487\) 22.3772 1.01401 0.507005 0.861943i \(-0.330753\pi\)
0.507005 + 0.861943i \(0.330753\pi\)
\(488\) 10.7422 0.486276
\(489\) 12.9860 0.587249
\(490\) 4.31337 0.194858
\(491\) −4.94993 −0.223387 −0.111693 0.993743i \(-0.535627\pi\)
−0.111693 + 0.993743i \(0.535627\pi\)
\(492\) −8.09111 −0.364776
\(493\) 9.65346 0.434770
\(494\) 0 0
\(495\) −9.20367 −0.413674
\(496\) 2.02090 0.0907411
\(497\) −14.5430 −0.652343
\(498\) −15.0962 −0.676478
\(499\) −20.2919 −0.908389 −0.454194 0.890903i \(-0.650073\pi\)
−0.454194 + 0.890903i \(0.650073\pi\)
\(500\) −37.1173 −1.65994
\(501\) 71.3831 3.18916
\(502\) 17.5364 0.782690
\(503\) 23.3232 1.03993 0.519965 0.854187i \(-0.325945\pi\)
0.519965 + 0.854187i \(0.325945\pi\)
\(504\) −5.89001 −0.262362
\(505\) 1.20194 0.0534858
\(506\) −0.645002 −0.0286738
\(507\) −43.2234 −1.91962
\(508\) −11.0424 −0.489926
\(509\) −27.1210 −1.20212 −0.601058 0.799205i \(-0.705254\pi\)
−0.601058 + 0.799205i \(0.705254\pi\)
\(510\) 59.5455 2.63672
\(511\) 2.61980 0.115893
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −11.5288 −0.508512
\(515\) 47.3847 2.08802
\(516\) −11.8410 −0.521270
\(517\) −1.18587 −0.0521547
\(518\) 7.44807 0.327249
\(519\) 15.9689 0.700958
\(520\) −22.6181 −0.991871
\(521\) 16.1295 0.706648 0.353324 0.935501i \(-0.385051\pi\)
0.353324 + 0.935501i \(0.385051\pi\)
\(522\) 12.2806 0.537506
\(523\) 24.9933 1.09288 0.546440 0.837498i \(-0.315983\pi\)
0.546440 + 0.837498i \(0.315983\pi\)
\(524\) 3.90396 0.170546
\(525\) −40.5654 −1.77042
\(526\) 5.34136 0.232894
\(527\) −9.35677 −0.407587
\(528\) −1.08014 −0.0470069
\(529\) −19.8299 −0.862172
\(530\) −50.6746 −2.20116
\(531\) −4.57888 −0.198706
\(532\) 0 0
\(533\) −14.2297 −0.616359
\(534\) −20.9509 −0.906632
\(535\) 15.5189 0.670942
\(536\) −2.61230 −0.112834
\(537\) −57.6081 −2.48597
\(538\) −26.1618 −1.12792
\(539\) 0.362266 0.0156039
\(540\) 37.1679 1.59945
\(541\) −11.3911 −0.489741 −0.244871 0.969556i \(-0.578745\pi\)
−0.244871 + 0.969556i \(0.578745\pi\)
\(542\) 20.2805 0.871122
\(543\) 28.3414 1.21625
\(544\) 4.63000 0.198510
\(545\) 47.0352 2.01477
\(546\) −15.6348 −0.669106
\(547\) −24.8812 −1.06384 −0.531921 0.846794i \(-0.678530\pi\)
−0.531921 + 0.846794i \(0.678530\pi\)
\(548\) −10.8789 −0.464724
\(549\) −63.2716 −2.70037
\(550\) −4.92870 −0.210160
\(551\) 0 0
\(552\) 5.30866 0.225951
\(553\) −0.554543 −0.0235816
\(554\) −31.1729 −1.32441
\(555\) −95.7882 −4.06598
\(556\) 12.0869 0.512601
\(557\) 11.1581 0.472784 0.236392 0.971658i \(-0.424035\pi\)
0.236392 + 0.971658i \(0.424035\pi\)
\(558\) −11.9031 −0.503900
\(559\) −20.8246 −0.880786
\(560\) −4.31337 −0.182273
\(561\) 5.00103 0.211144
\(562\) 6.00910 0.253479
\(563\) −18.8504 −0.794449 −0.397224 0.917722i \(-0.630026\pi\)
−0.397224 + 0.917722i \(0.630026\pi\)
\(564\) 9.76027 0.410981
\(565\) 18.5208 0.779177
\(566\) −12.2967 −0.516869
\(567\) 8.02223 0.336902
\(568\) 14.5430 0.610211
\(569\) −37.7923 −1.58434 −0.792168 0.610303i \(-0.791048\pi\)
−0.792168 + 0.610303i \(0.791048\pi\)
\(570\) 0 0
\(571\) −10.6904 −0.447380 −0.223690 0.974660i \(-0.571810\pi\)
−0.223690 + 0.974660i \(0.571810\pi\)
\(572\) −1.89962 −0.0794273
\(573\) 48.7593 2.03695
\(574\) −2.71367 −0.113266
\(575\) 24.2235 1.01019
\(576\) 5.89001 0.245417
\(577\) 20.6689 0.860459 0.430229 0.902720i \(-0.358433\pi\)
0.430229 + 0.902720i \(0.358433\pi\)
\(578\) −4.43688 −0.184550
\(579\) −2.71775 −0.112946
\(580\) 8.99330 0.373427
\(581\) −5.06311 −0.210053
\(582\) −4.14020 −0.171617
\(583\) −4.25600 −0.176265
\(584\) −2.61980 −0.108408
\(585\) 133.221 5.50802
\(586\) −4.02655 −0.166335
\(587\) 9.31019 0.384273 0.192136 0.981368i \(-0.438458\pi\)
0.192136 + 0.981368i \(0.438458\pi\)
\(588\) −2.98161 −0.122960
\(589\) 0 0
\(590\) −3.35320 −0.138049
\(591\) 82.6742 3.40076
\(592\) −7.44807 −0.306114
\(593\) −42.1221 −1.72975 −0.864873 0.501990i \(-0.832601\pi\)
−0.864873 + 0.501990i \(0.832601\pi\)
\(594\) 3.12161 0.128081
\(595\) 19.9709 0.818727
\(596\) −19.8129 −0.811566
\(597\) 19.6433 0.803948
\(598\) 9.33627 0.381788
\(599\) −21.5720 −0.881406 −0.440703 0.897653i \(-0.645271\pi\)
−0.440703 + 0.897653i \(0.645271\pi\)
\(600\) 40.5654 1.65607
\(601\) 17.4784 0.712959 0.356479 0.934303i \(-0.383977\pi\)
0.356479 + 0.934303i \(0.383977\pi\)
\(602\) −3.97133 −0.161859
\(603\) 15.3865 0.626586
\(604\) 16.3282 0.664386
\(605\) 46.8810 1.90598
\(606\) −0.830842 −0.0337506
\(607\) −13.8157 −0.560763 −0.280381 0.959889i \(-0.590461\pi\)
−0.280381 + 0.959889i \(0.590461\pi\)
\(608\) 0 0
\(609\) 6.21661 0.251910
\(610\) −46.3350 −1.87605
\(611\) 17.1653 0.694432
\(612\) −27.2708 −1.10235
\(613\) 35.7849 1.44534 0.722669 0.691194i \(-0.242915\pi\)
0.722669 + 0.691194i \(0.242915\pi\)
\(614\) 22.6810 0.915331
\(615\) 34.9000 1.40730
\(616\) −0.362266 −0.0145961
\(617\) −28.2075 −1.13559 −0.567795 0.823170i \(-0.692204\pi\)
−0.567795 + 0.823170i \(0.692204\pi\)
\(618\) −32.7546 −1.31758
\(619\) −35.4746 −1.42584 −0.712922 0.701243i \(-0.752629\pi\)
−0.712922 + 0.701243i \(0.752629\pi\)
\(620\) −8.71690 −0.350079
\(621\) −15.3421 −0.615657
\(622\) −16.9035 −0.677771
\(623\) −7.02669 −0.281518
\(624\) 15.6348 0.625891
\(625\) 92.0750 3.68300
\(626\) 1.39972 0.0559441
\(627\) 0 0
\(628\) −5.37439 −0.214462
\(629\) 34.4845 1.37499
\(630\) 25.4058 1.01219
\(631\) 43.6368 1.73716 0.868578 0.495553i \(-0.165035\pi\)
0.868578 + 0.495553i \(0.165035\pi\)
\(632\) 0.554543 0.0220586
\(633\) −11.7173 −0.465720
\(634\) −10.9472 −0.434769
\(635\) 47.6298 1.89013
\(636\) 35.0288 1.38898
\(637\) −5.24373 −0.207764
\(638\) 0.755318 0.0299033
\(639\) −85.6585 −3.38860
\(640\) 4.31337 0.170501
\(641\) −4.51479 −0.178323 −0.0891616 0.996017i \(-0.528419\pi\)
−0.0891616 + 0.996017i \(0.528419\pi\)
\(642\) −10.7274 −0.423378
\(643\) −4.32950 −0.170739 −0.0853693 0.996349i \(-0.527207\pi\)
−0.0853693 + 0.996349i \(0.527207\pi\)
\(644\) 1.78046 0.0701601
\(645\) 51.0745 2.01106
\(646\) 0 0
\(647\) 21.4445 0.843071 0.421535 0.906812i \(-0.361491\pi\)
0.421535 + 0.906812i \(0.361491\pi\)
\(648\) −8.02223 −0.315143
\(649\) −0.281625 −0.0110547
\(650\) 71.3418 2.79826
\(651\) −6.02554 −0.236160
\(652\) −4.35537 −0.170569
\(653\) 36.7156 1.43679 0.718397 0.695634i \(-0.244876\pi\)
0.718397 + 0.695634i \(0.244876\pi\)
\(654\) −32.5130 −1.27136
\(655\) −16.8393 −0.657964
\(656\) 2.71367 0.105951
\(657\) 15.4306 0.602006
\(658\) 3.27349 0.127614
\(659\) 20.9897 0.817641 0.408821 0.912615i \(-0.365940\pi\)
0.408821 + 0.912615i \(0.365940\pi\)
\(660\) 4.65903 0.181353
\(661\) −1.01862 −0.0396198 −0.0198099 0.999804i \(-0.506306\pi\)
−0.0198099 + 0.999804i \(0.506306\pi\)
\(662\) −3.12383 −0.121411
\(663\) −72.3889 −2.81135
\(664\) 5.06311 0.196487
\(665\) 0 0
\(666\) 43.8692 1.69990
\(667\) −3.71224 −0.143738
\(668\) −23.9411 −0.926309
\(669\) 45.8218 1.77157
\(670\) 11.2678 0.435314
\(671\) −3.89153 −0.150231
\(672\) 2.98161 0.115018
\(673\) −45.4641 −1.75251 −0.876256 0.481846i \(-0.839966\pi\)
−0.876256 + 0.481846i \(0.839966\pi\)
\(674\) 3.66902 0.141325
\(675\) −117.235 −4.51236
\(676\) 14.4967 0.557564
\(677\) 33.9501 1.30481 0.652405 0.757871i \(-0.273760\pi\)
0.652405 + 0.757871i \(0.273760\pi\)
\(678\) −12.8025 −0.491677
\(679\) −1.38858 −0.0532887
\(680\) −19.9709 −0.765849
\(681\) −46.5281 −1.78296
\(682\) −0.732104 −0.0280337
\(683\) −1.09066 −0.0417330 −0.0208665 0.999782i \(-0.506643\pi\)
−0.0208665 + 0.999782i \(0.506643\pi\)
\(684\) 0 0
\(685\) 46.9248 1.79290
\(686\) −1.00000 −0.0381802
\(687\) −26.7259 −1.01966
\(688\) 3.97133 0.151406
\(689\) 61.6047 2.34695
\(690\) −22.8982 −0.871720
\(691\) −28.6243 −1.08892 −0.544459 0.838787i \(-0.683265\pi\)
−0.544459 + 0.838787i \(0.683265\pi\)
\(692\) −5.35580 −0.203597
\(693\) 2.13375 0.0810546
\(694\) 14.2657 0.541520
\(695\) −52.1355 −1.97761
\(696\) −6.21661 −0.235640
\(697\) −12.5643 −0.475906
\(698\) 3.19923 0.121093
\(699\) −45.4010 −1.71722
\(700\) 13.6052 0.514227
\(701\) −31.4941 −1.18952 −0.594759 0.803904i \(-0.702752\pi\)
−0.594759 + 0.803904i \(0.702752\pi\)
\(702\) −45.1847 −1.70539
\(703\) 0 0
\(704\) 0.362266 0.0136534
\(705\) −42.0997 −1.58557
\(706\) 2.96498 0.111589
\(707\) −0.278655 −0.0104799
\(708\) 2.31790 0.0871119
\(709\) −2.98911 −0.112258 −0.0561292 0.998424i \(-0.517876\pi\)
−0.0561292 + 0.998424i \(0.517876\pi\)
\(710\) −62.7294 −2.35419
\(711\) −3.26627 −0.122495
\(712\) 7.02669 0.263336
\(713\) 3.59814 0.134752
\(714\) −13.8049 −0.516634
\(715\) 8.19379 0.306430
\(716\) 19.3211 0.722064
\(717\) −56.3730 −2.10529
\(718\) 20.5387 0.766498
\(719\) −50.7556 −1.89286 −0.946431 0.322905i \(-0.895341\pi\)
−0.946431 + 0.322905i \(0.895341\pi\)
\(720\) −25.4058 −0.946819
\(721\) −10.9855 −0.409122
\(722\) 0 0
\(723\) −55.7256 −2.07246
\(724\) −9.50541 −0.353266
\(725\) −28.3665 −1.05351
\(726\) −32.4064 −1.20272
\(727\) 44.4894 1.65002 0.825011 0.565117i \(-0.191169\pi\)
0.825011 + 0.565117i \(0.191169\pi\)
\(728\) 5.24373 0.194345
\(729\) −29.8258 −1.10466
\(730\) 11.3002 0.418237
\(731\) −18.3873 −0.680077
\(732\) 32.0290 1.18383
\(733\) −24.6274 −0.909635 −0.454818 0.890585i \(-0.650296\pi\)
−0.454818 + 0.890585i \(0.650296\pi\)
\(734\) 9.22982 0.340679
\(735\) 12.8608 0.474378
\(736\) −1.78046 −0.0656288
\(737\) 0.946349 0.0348592
\(738\) −15.9836 −0.588363
\(739\) −29.9616 −1.10216 −0.551078 0.834453i \(-0.685784\pi\)
−0.551078 + 0.834453i \(0.685784\pi\)
\(740\) 32.1263 1.18099
\(741\) 0 0
\(742\) 11.7483 0.431292
\(743\) 9.44786 0.346608 0.173304 0.984868i \(-0.444556\pi\)
0.173304 + 0.984868i \(0.444556\pi\)
\(744\) 6.02554 0.220907
\(745\) 85.4602 3.13102
\(746\) 18.5739 0.680040
\(747\) −29.8218 −1.09112
\(748\) −1.67729 −0.0613279
\(749\) −3.59786 −0.131463
\(750\) −110.670 −4.04108
\(751\) −8.84575 −0.322786 −0.161393 0.986890i \(-0.551599\pi\)
−0.161393 + 0.986890i \(0.551599\pi\)
\(752\) −3.27349 −0.119372
\(753\) 52.2869 1.90544
\(754\) −10.9331 −0.398159
\(755\) −70.4297 −2.56320
\(756\) −8.61690 −0.313394
\(757\) 22.9044 0.832475 0.416238 0.909256i \(-0.363348\pi\)
0.416238 + 0.909256i \(0.363348\pi\)
\(758\) −7.87312 −0.285964
\(759\) −1.92315 −0.0698058
\(760\) 0 0
\(761\) 20.6868 0.749894 0.374947 0.927046i \(-0.377661\pi\)
0.374947 + 0.927046i \(0.377661\pi\)
\(762\) −32.9241 −1.19271
\(763\) −10.9045 −0.394770
\(764\) −16.3533 −0.591643
\(765\) 117.629 4.25288
\(766\) −14.6552 −0.529514
\(767\) 4.07646 0.147192
\(768\) −2.98161 −0.107590
\(769\) 3.55285 0.128119 0.0640596 0.997946i \(-0.479595\pi\)
0.0640596 + 0.997946i \(0.479595\pi\)
\(770\) 1.56259 0.0563118
\(771\) −34.3743 −1.23796
\(772\) 0.911504 0.0328057
\(773\) 29.9640 1.07773 0.538865 0.842392i \(-0.318853\pi\)
0.538865 + 0.842392i \(0.318853\pi\)
\(774\) −23.3912 −0.840780
\(775\) 27.4947 0.987639
\(776\) 1.38858 0.0498471
\(777\) 22.2073 0.796681
\(778\) −17.0207 −0.610223
\(779\) 0 0
\(780\) −67.4385 −2.41469
\(781\) −5.26844 −0.188519
\(782\) 8.24355 0.294789
\(783\) 17.9661 0.642056
\(784\) 1.00000 0.0357143
\(785\) 23.1818 0.827392
\(786\) 11.6401 0.415189
\(787\) 35.8277 1.27712 0.638559 0.769573i \(-0.279531\pi\)
0.638559 + 0.769573i \(0.279531\pi\)
\(788\) −27.7280 −0.987770
\(789\) 15.9259 0.566976
\(790\) −2.39195 −0.0851018
\(791\) −4.29382 −0.152671
\(792\) −2.13375 −0.0758196
\(793\) 56.3291 2.00030
\(794\) −10.4213 −0.369839
\(795\) −151.092 −5.35868
\(796\) −6.58815 −0.233511
\(797\) −31.7865 −1.12594 −0.562968 0.826479i \(-0.690341\pi\)
−0.562968 + 0.826479i \(0.690341\pi\)
\(798\) 0 0
\(799\) 15.1562 0.536189
\(800\) −13.6052 −0.481016
\(801\) −41.3873 −1.46235
\(802\) −30.0758 −1.06201
\(803\) 0.949063 0.0334917
\(804\) −7.78887 −0.274692
\(805\) −7.67981 −0.270678
\(806\) 10.5971 0.373265
\(807\) −78.0044 −2.74588
\(808\) 0.278655 0.00980306
\(809\) −12.2092 −0.429252 −0.214626 0.976696i \(-0.568853\pi\)
−0.214626 + 0.976696i \(0.568853\pi\)
\(810\) 34.6028 1.21582
\(811\) 47.0483 1.65209 0.826044 0.563606i \(-0.190586\pi\)
0.826044 + 0.563606i \(0.190586\pi\)
\(812\) −2.08498 −0.0731685
\(813\) 60.4686 2.12072
\(814\) 2.69818 0.0945713
\(815\) 18.7863 0.658057
\(816\) 13.8049 0.483267
\(817\) 0 0
\(818\) 11.0533 0.386471
\(819\) −30.8856 −1.07923
\(820\) −11.7051 −0.408759
\(821\) 26.1942 0.914183 0.457092 0.889420i \(-0.348891\pi\)
0.457092 + 0.889420i \(0.348891\pi\)
\(822\) −32.4367 −1.13136
\(823\) 36.6952 1.27911 0.639557 0.768744i \(-0.279118\pi\)
0.639557 + 0.768744i \(0.279118\pi\)
\(824\) 10.9855 0.382699
\(825\) −14.6955 −0.511630
\(826\) 0.777397 0.0270491
\(827\) 36.9160 1.28369 0.641847 0.766832i \(-0.278168\pi\)
0.641847 + 0.766832i \(0.278168\pi\)
\(828\) 10.4870 0.364447
\(829\) −38.1788 −1.32600 −0.663002 0.748618i \(-0.730718\pi\)
−0.663002 + 0.748618i \(0.730718\pi\)
\(830\) −21.8391 −0.758045
\(831\) −92.9457 −3.22425
\(832\) −5.24373 −0.181794
\(833\) −4.63000 −0.160420
\(834\) 36.0386 1.24791
\(835\) 103.267 3.57370
\(836\) 0 0
\(837\) −17.4139 −0.601913
\(838\) 14.4315 0.498526
\(839\) 5.08377 0.175511 0.0877556 0.996142i \(-0.472031\pi\)
0.0877556 + 0.996142i \(0.472031\pi\)
\(840\) −12.8608 −0.443740
\(841\) −24.6529 −0.850098
\(842\) −26.6057 −0.916893
\(843\) 17.9168 0.617088
\(844\) 3.92985 0.135271
\(845\) −62.5295 −2.15108
\(846\) 19.2809 0.662890
\(847\) −10.8688 −0.373455
\(848\) −11.7483 −0.403437
\(849\) −36.6640 −1.25830
\(850\) 62.9919 2.16061
\(851\) −13.2610 −0.454582
\(852\) 43.3616 1.48554
\(853\) −10.1943 −0.349046 −0.174523 0.984653i \(-0.555838\pi\)
−0.174523 + 0.984653i \(0.555838\pi\)
\(854\) 10.7422 0.367590
\(855\) 0 0
\(856\) 3.59786 0.122972
\(857\) 8.02856 0.274250 0.137125 0.990554i \(-0.456214\pi\)
0.137125 + 0.990554i \(0.456214\pi\)
\(858\) −5.66395 −0.193364
\(859\) 41.2373 1.40700 0.703499 0.710696i \(-0.251620\pi\)
0.703499 + 0.710696i \(0.251620\pi\)
\(860\) −17.1298 −0.584123
\(861\) −8.09111 −0.275744
\(862\) 20.2109 0.688386
\(863\) 0.552239 0.0187985 0.00939923 0.999956i \(-0.497008\pi\)
0.00939923 + 0.999956i \(0.497008\pi\)
\(864\) 8.61690 0.293153
\(865\) 23.1016 0.785477
\(866\) −36.1393 −1.22806
\(867\) −13.2291 −0.449282
\(868\) 2.02090 0.0685939
\(869\) −0.200892 −0.00681481
\(870\) 26.8145 0.909098
\(871\) −13.6982 −0.464146
\(872\) 10.9045 0.369273
\(873\) −8.17875 −0.276809
\(874\) 0 0
\(875\) −37.1173 −1.25479
\(876\) −7.81121 −0.263916
\(877\) 40.4583 1.36618 0.683090 0.730334i \(-0.260636\pi\)
0.683090 + 0.730334i \(0.260636\pi\)
\(878\) −18.9829 −0.640642
\(879\) −12.0056 −0.404939
\(880\) −1.56259 −0.0526749
\(881\) 25.8951 0.872427 0.436213 0.899843i \(-0.356319\pi\)
0.436213 + 0.899843i \(0.356319\pi\)
\(882\) −5.89001 −0.198327
\(883\) 51.1233 1.72044 0.860218 0.509926i \(-0.170327\pi\)
0.860218 + 0.509926i \(0.170327\pi\)
\(884\) 24.2784 0.816572
\(885\) −9.99795 −0.336077
\(886\) 32.2823 1.08454
\(887\) −31.1733 −1.04670 −0.523349 0.852119i \(-0.675318\pi\)
−0.523349 + 0.852119i \(0.675318\pi\)
\(888\) −22.2073 −0.745227
\(889\) −11.0424 −0.370349
\(890\) −30.3087 −1.01595
\(891\) 2.90618 0.0973607
\(892\) −15.3681 −0.514562
\(893\) 0 0
\(894\) −59.0743 −1.97574
\(895\) −83.3392 −2.78572
\(896\) −1.00000 −0.0334077
\(897\) 27.8371 0.929455
\(898\) 11.0150 0.367574
\(899\) −4.21354 −0.140530
\(900\) 80.1347 2.67116
\(901\) 54.3944 1.81214
\(902\) −0.983071 −0.0327327
\(903\) −11.8410 −0.394043
\(904\) 4.29382 0.142810
\(905\) 41.0004 1.36290
\(906\) 48.6844 1.61743
\(907\) −7.77664 −0.258219 −0.129110 0.991630i \(-0.541212\pi\)
−0.129110 + 0.991630i \(0.541212\pi\)
\(908\) 15.6050 0.517870
\(909\) −1.64128 −0.0544379
\(910\) −22.6181 −0.749784
\(911\) −13.8287 −0.458166 −0.229083 0.973407i \(-0.573573\pi\)
−0.229083 + 0.973407i \(0.573573\pi\)
\(912\) 0 0
\(913\) −1.83419 −0.0607029
\(914\) 1.92274 0.0635986
\(915\) −138.153 −4.56720
\(916\) 8.96359 0.296165
\(917\) 3.90396 0.128920
\(918\) −39.8962 −1.31677
\(919\) −52.4671 −1.73073 −0.865365 0.501143i \(-0.832913\pi\)
−0.865365 + 0.501143i \(0.832913\pi\)
\(920\) 7.67981 0.253196
\(921\) 67.6260 2.22835
\(922\) 3.81073 0.125500
\(923\) 76.2595 2.51011
\(924\) −1.08014 −0.0355339
\(925\) −101.332 −3.33179
\(926\) 34.9106 1.14723
\(927\) −64.7049 −2.12519
\(928\) 2.08498 0.0684429
\(929\) 44.7025 1.46664 0.733320 0.679883i \(-0.237970\pi\)
0.733320 + 0.679883i \(0.237970\pi\)
\(930\) −25.9904 −0.852259
\(931\) 0 0
\(932\) 15.2270 0.498776
\(933\) −50.3998 −1.65002
\(934\) 33.2523 1.08805
\(935\) 7.23478 0.236603
\(936\) 30.8856 1.00953
\(937\) −33.9615 −1.10947 −0.554737 0.832026i \(-0.687181\pi\)
−0.554737 + 0.832026i \(0.687181\pi\)
\(938\) −2.61230 −0.0852947
\(939\) 4.17343 0.136195
\(940\) 14.1198 0.460536
\(941\) 3.18819 0.103932 0.0519659 0.998649i \(-0.483451\pi\)
0.0519659 + 0.998649i \(0.483451\pi\)
\(942\) −16.0244 −0.522102
\(943\) 4.83159 0.157338
\(944\) −0.777397 −0.0253021
\(945\) 37.1679 1.20907
\(946\) −1.43868 −0.0467755
\(947\) −5.45182 −0.177160 −0.0885801 0.996069i \(-0.528233\pi\)
−0.0885801 + 0.996069i \(0.528233\pi\)
\(948\) 1.65343 0.0537010
\(949\) −13.7375 −0.445938
\(950\) 0 0
\(951\) −32.6403 −1.05843
\(952\) 4.63000 0.150059
\(953\) −11.1272 −0.360446 −0.180223 0.983626i \(-0.557682\pi\)
−0.180223 + 0.983626i \(0.557682\pi\)
\(954\) 69.1974 2.24035
\(955\) 70.5381 2.28256
\(956\) 18.9069 0.611492
\(957\) 2.25207 0.0727990
\(958\) −3.31481 −0.107097
\(959\) −10.8789 −0.351298
\(960\) 12.8608 0.415081
\(961\) −26.9160 −0.868257
\(962\) −39.0556 −1.25920
\(963\) −21.1915 −0.682886
\(964\) 18.6898 0.601957
\(965\) −3.93166 −0.126564
\(966\) 5.30866 0.170803
\(967\) −49.5193 −1.59243 −0.796217 0.605012i \(-0.793168\pi\)
−0.796217 + 0.605012i \(0.793168\pi\)
\(968\) 10.8688 0.349335
\(969\) 0 0
\(970\) −5.98945 −0.192310
\(971\) 11.6443 0.373683 0.186842 0.982390i \(-0.440175\pi\)
0.186842 + 0.982390i \(0.440175\pi\)
\(972\) 1.93154 0.0619542
\(973\) 12.0869 0.387490
\(974\) −22.3772 −0.717013
\(975\) 212.714 6.81229
\(976\) −10.7422 −0.343849
\(977\) 36.1804 1.15751 0.578757 0.815500i \(-0.303538\pi\)
0.578757 + 0.815500i \(0.303538\pi\)
\(978\) −12.9860 −0.415247
\(979\) −2.54553 −0.0813555
\(980\) −4.31337 −0.137786
\(981\) −64.2277 −2.05063
\(982\) 4.94993 0.157958
\(983\) −9.63945 −0.307451 −0.153725 0.988114i \(-0.549127\pi\)
−0.153725 + 0.988114i \(0.549127\pi\)
\(984\) 8.09111 0.257935
\(985\) 119.601 3.81081
\(986\) −9.65346 −0.307429
\(987\) 9.76027 0.310673
\(988\) 0 0
\(989\) 7.07082 0.224839
\(990\) 9.20367 0.292512
\(991\) 50.0192 1.58891 0.794456 0.607321i \(-0.207756\pi\)
0.794456 + 0.607321i \(0.207756\pi\)
\(992\) −2.02090 −0.0641637
\(993\) −9.31405 −0.295572
\(994\) 14.5430 0.461276
\(995\) 28.4172 0.900884
\(996\) 15.0962 0.478342
\(997\) 27.5717 0.873203 0.436602 0.899655i \(-0.356182\pi\)
0.436602 + 0.899655i \(0.356182\pi\)
\(998\) 20.2919 0.642328
\(999\) 64.1793 2.03054
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 5054.2.a.bl.1.2 12
19.4 even 9 266.2.u.d.225.4 24
19.5 even 9 266.2.u.d.253.4 yes 24
19.18 odd 2 5054.2.a.bm.1.11 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.u.d.225.4 24 19.4 even 9
266.2.u.d.253.4 yes 24 19.5 even 9
5054.2.a.bl.1.2 12 1.1 even 1 trivial
5054.2.a.bm.1.11 12 19.18 odd 2