Properties

Label 5054.2.a.v.1.2
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.17557\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.17557 q^{3} +1.00000 q^{4} +3.52015 q^{5} +2.17557 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.73311 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.17557 q^{3} +1.00000 q^{4} +3.52015 q^{5} +2.17557 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.73311 q^{9} -3.52015 q^{10} -2.80423 q^{11} -2.17557 q^{12} -1.34458 q^{13} -1.00000 q^{14} -7.65833 q^{15} +1.00000 q^{16} -3.69572 q^{17} -1.73311 q^{18} +3.52015 q^{20} -2.17557 q^{21} +2.80423 q^{22} +4.80423 q^{23} +2.17557 q^{24} +7.39144 q^{25} +1.34458 q^{26} +2.75621 q^{27} +1.00000 q^{28} -5.35114 q^{29} +7.65833 q^{30} +3.32437 q^{31} -1.00000 q^{32} +6.10079 q^{33} +3.69572 q^{34} +3.52015 q^{35} +1.73311 q^{36} -9.91930 q^{37} +2.92522 q^{39} -3.52015 q^{40} +6.54795 q^{41} +2.17557 q^{42} +10.6583 q^{43} -2.80423 q^{44} +6.10079 q^{45} -4.80423 q^{46} -9.44537 q^{47} -2.17557 q^{48} +1.00000 q^{49} -7.39144 q^{50} +8.04029 q^{51} -1.34458 q^{52} +2.46621 q^{53} -2.75621 q^{54} -9.87129 q^{55} -1.00000 q^{56} +5.35114 q^{58} -0.501096 q^{59} -7.65833 q^{60} -9.22835 q^{61} -3.32437 q^{62} +1.73311 q^{63} +1.00000 q^{64} -4.73311 q^{65} -6.10079 q^{66} -11.9882 q^{67} -3.69572 q^{68} -10.4519 q^{69} -3.52015 q^{70} +13.3167 q^{71} -1.73311 q^{72} -1.92586 q^{73} +9.91930 q^{74} -16.0806 q^{75} -2.80423 q^{77} -2.92522 q^{78} -17.5028 q^{79} +3.52015 q^{80} -11.1957 q^{81} -6.54795 q^{82} +10.8576 q^{83} -2.17557 q^{84} -13.0095 q^{85} -10.6583 q^{86} +11.6418 q^{87} +2.80423 q^{88} -16.5402 q^{89} -6.10079 q^{90} -1.34458 q^{91} +4.80423 q^{92} -7.23241 q^{93} +9.44537 q^{94} +2.17557 q^{96} +10.2012 q^{97} -1.00000 q^{98} -4.86002 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 4 q^{7} - 4 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 4 q^{7} - 4 q^{8} + 2 q^{9} - 2 q^{10} + 4 q^{11} - 4 q^{12} + 2 q^{13} - 4 q^{14} - 2 q^{15} + 4 q^{16} + 2 q^{17} - 2 q^{18} + 2 q^{20} - 4 q^{21} - 4 q^{22} + 4 q^{23} + 4 q^{24} - 4 q^{25} - 2 q^{26} - 10 q^{27} + 4 q^{28} - 12 q^{29} + 2 q^{30} - 14 q^{31} - 4 q^{32} - 4 q^{33} - 2 q^{34} + 2 q^{35} + 2 q^{36} - 24 q^{37} - 12 q^{39} - 2 q^{40} + 4 q^{42} + 14 q^{43} + 4 q^{44} - 4 q^{45} - 4 q^{46} - 2 q^{47} - 4 q^{48} + 4 q^{49} + 4 q^{50} + 8 q^{51} + 2 q^{52} + 10 q^{54} - 18 q^{55} - 4 q^{56} + 12 q^{58} - 2 q^{59} - 2 q^{60} + 2 q^{61} + 14 q^{62} + 2 q^{63} + 4 q^{64} - 14 q^{65} + 4 q^{66} - 22 q^{67} + 2 q^{68} - 4 q^{69} - 2 q^{70} - 4 q^{71} - 2 q^{72} + 10 q^{73} + 24 q^{74} - 16 q^{75} + 4 q^{77} + 12 q^{78} - 2 q^{79} + 2 q^{80} + 4 q^{81} - 4 q^{84} - 14 q^{85} - 14 q^{86} + 32 q^{87} - 4 q^{88} - 10 q^{89} + 4 q^{90} + 2 q^{91} + 4 q^{92} + 14 q^{93} + 2 q^{94} + 4 q^{96} + 22 q^{97} - 4 q^{98} + 2 q^{99}+O(q^{100}) \) 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
\(3\) −2.17557 −1.25607 −0.628033 0.778187i \(-0.716140\pi\)
−0.628033 + 0.778187i \(0.716140\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.52015 1.57426 0.787129 0.616789i \(-0.211567\pi\)
0.787129 + 0.616789i \(0.211567\pi\)
\(6\) 2.17557 0.888173
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.73311 0.577702
\(10\) −3.52015 −1.11317
\(11\) −2.80423 −0.845506 −0.422753 0.906245i \(-0.638936\pi\)
−0.422753 + 0.906245i \(0.638936\pi\)
\(12\) −2.17557 −0.628033
\(13\) −1.34458 −0.372918 −0.186459 0.982463i \(-0.559701\pi\)
−0.186459 + 0.982463i \(0.559701\pi\)
\(14\) −1.00000 −0.267261
\(15\) −7.65833 −1.97737
\(16\) 1.00000 0.250000
\(17\) −3.69572 −0.896343 −0.448172 0.893948i \(-0.647925\pi\)
−0.448172 + 0.893948i \(0.647925\pi\)
\(18\) −1.73311 −0.408497
\(19\) 0 0
\(20\) 3.52015 0.787129
\(21\) −2.17557 −0.474748
\(22\) 2.80423 0.597863
\(23\) 4.80423 1.00175 0.500875 0.865520i \(-0.333012\pi\)
0.500875 + 0.865520i \(0.333012\pi\)
\(24\) 2.17557 0.444086
\(25\) 7.39144 1.47829
\(26\) 1.34458 0.263693
\(27\) 2.75621 0.530434
\(28\) 1.00000 0.188982
\(29\) −5.35114 −0.993682 −0.496841 0.867842i \(-0.665507\pi\)
−0.496841 + 0.867842i \(0.665507\pi\)
\(30\) 7.65833 1.39821
\(31\) 3.32437 0.597075 0.298538 0.954398i \(-0.403501\pi\)
0.298538 + 0.954398i \(0.403501\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.10079 1.06201
\(34\) 3.69572 0.633810
\(35\) 3.52015 0.595013
\(36\) 1.73311 0.288851
\(37\) −9.91930 −1.63072 −0.815361 0.578952i \(-0.803462\pi\)
−0.815361 + 0.578952i \(0.803462\pi\)
\(38\) 0 0
\(39\) 2.92522 0.468410
\(40\) −3.52015 −0.556584
\(41\) 6.54795 1.02262 0.511309 0.859397i \(-0.329161\pi\)
0.511309 + 0.859397i \(0.329161\pi\)
\(42\) 2.17557 0.335698
\(43\) 10.6583 1.62538 0.812690 0.582696i \(-0.198002\pi\)
0.812690 + 0.582696i \(0.198002\pi\)
\(44\) −2.80423 −0.422753
\(45\) 6.10079 0.909452
\(46\) −4.80423 −0.708344
\(47\) −9.44537 −1.37775 −0.688874 0.724881i \(-0.741895\pi\)
−0.688874 + 0.724881i \(0.741895\pi\)
\(48\) −2.17557 −0.314017
\(49\) 1.00000 0.142857
\(50\) −7.39144 −1.04531
\(51\) 8.04029 1.12587
\(52\) −1.34458 −0.186459
\(53\) 2.46621 0.338761 0.169380 0.985551i \(-0.445823\pi\)
0.169380 + 0.985551i \(0.445823\pi\)
\(54\) −2.75621 −0.375073
\(55\) −9.87129 −1.33104
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 5.35114 0.702639
\(59\) −0.501096 −0.0652372 −0.0326186 0.999468i \(-0.510385\pi\)
−0.0326186 + 0.999468i \(0.510385\pi\)
\(60\) −7.65833 −0.988686
\(61\) −9.22835 −1.18157 −0.590785 0.806829i \(-0.701182\pi\)
−0.590785 + 0.806829i \(0.701182\pi\)
\(62\) −3.32437 −0.422196
\(63\) 1.73311 0.218351
\(64\) 1.00000 0.125000
\(65\) −4.73311 −0.587070
\(66\) −6.10079 −0.750956
\(67\) −11.9882 −1.46459 −0.732297 0.680985i \(-0.761552\pi\)
−0.732297 + 0.680985i \(0.761552\pi\)
\(68\) −3.69572 −0.448172
\(69\) −10.4519 −1.25826
\(70\) −3.52015 −0.420738
\(71\) 13.3167 1.58040 0.790198 0.612851i \(-0.209978\pi\)
0.790198 + 0.612851i \(0.209978\pi\)
\(72\) −1.73311 −0.204249
\(73\) −1.92586 −0.225405 −0.112703 0.993629i \(-0.535951\pi\)
−0.112703 + 0.993629i \(0.535951\pi\)
\(74\) 9.91930 1.15310
\(75\) −16.0806 −1.85683
\(76\) 0 0
\(77\) −2.80423 −0.319571
\(78\) −2.92522 −0.331216
\(79\) −17.5028 −1.96922 −0.984612 0.174754i \(-0.944087\pi\)
−0.984612 + 0.174754i \(0.944087\pi\)
\(80\) 3.52015 0.393564
\(81\) −11.1957 −1.24396
\(82\) −6.54795 −0.723101
\(83\) 10.8576 1.19178 0.595891 0.803065i \(-0.296799\pi\)
0.595891 + 0.803065i \(0.296799\pi\)
\(84\) −2.17557 −0.237374
\(85\) −13.0095 −1.41108
\(86\) −10.6583 −1.14932
\(87\) 11.6418 1.24813
\(88\) 2.80423 0.298932
\(89\) −16.5402 −1.75326 −0.876631 0.481164i \(-0.840214\pi\)
−0.876631 + 0.481164i \(0.840214\pi\)
\(90\) −6.10079 −0.643080
\(91\) −1.34458 −0.140950
\(92\) 4.80423 0.500875
\(93\) −7.23241 −0.749966
\(94\) 9.44537 0.974215
\(95\) 0 0
\(96\) 2.17557 0.222043
\(97\) 10.2012 1.03577 0.517887 0.855449i \(-0.326719\pi\)
0.517887 + 0.855449i \(0.326719\pi\)
\(98\) −1.00000 −0.101015
\(99\) −4.86002 −0.488451
\(100\) 7.39144 0.739144
\(101\) 1.16025 0.115449 0.0577246 0.998333i \(-0.481615\pi\)
0.0577246 + 0.998333i \(0.481615\pi\)
\(102\) −8.04029 −0.796108
\(103\) −10.5480 −1.03932 −0.519660 0.854373i \(-0.673942\pi\)
−0.519660 + 0.854373i \(0.673942\pi\)
\(104\) 1.34458 0.131847
\(105\) −7.65833 −0.747376
\(106\) −2.46621 −0.239540
\(107\) 9.16850 0.886352 0.443176 0.896435i \(-0.353852\pi\)
0.443176 + 0.896435i \(0.353852\pi\)
\(108\) 2.75621 0.265217
\(109\) 5.46255 0.523218 0.261609 0.965174i \(-0.415747\pi\)
0.261609 + 0.965174i \(0.415747\pi\)
\(110\) 9.87129 0.941190
\(111\) 21.5801 2.04830
\(112\) 1.00000 0.0944911
\(113\) 2.00947 0.189035 0.0945175 0.995523i \(-0.469869\pi\)
0.0945175 + 0.995523i \(0.469869\pi\)
\(114\) 0 0
\(115\) 16.9116 1.57701
\(116\) −5.35114 −0.496841
\(117\) −2.33030 −0.215436
\(118\) 0.501096 0.0461296
\(119\) −3.69572 −0.338786
\(120\) 7.65833 0.699107
\(121\) −3.13632 −0.285120
\(122\) 9.22835 0.835496
\(123\) −14.2455 −1.28448
\(124\) 3.32437 0.298538
\(125\) 8.41820 0.752947
\(126\) −1.73311 −0.154397
\(127\) 2.30719 0.204730 0.102365 0.994747i \(-0.467359\pi\)
0.102365 + 0.994747i \(0.467359\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −23.1879 −2.04158
\(130\) 4.73311 0.415121
\(131\) −21.9508 −1.91785 −0.958927 0.283654i \(-0.908453\pi\)
−0.958927 + 0.283654i \(0.908453\pi\)
\(132\) 6.10079 0.531006
\(133\) 0 0
\(134\) 11.9882 1.03562
\(135\) 9.70228 0.835039
\(136\) 3.69572 0.316905
\(137\) −22.7616 −1.94466 −0.972329 0.233618i \(-0.924944\pi\)
−0.972329 + 0.233618i \(0.924944\pi\)
\(138\) 10.4519 0.889728
\(139\) −11.6162 −0.985271 −0.492635 0.870236i \(-0.663966\pi\)
−0.492635 + 0.870236i \(0.663966\pi\)
\(140\) 3.52015 0.297507
\(141\) 20.5491 1.73054
\(142\) −13.3167 −1.11751
\(143\) 3.77050 0.315305
\(144\) 1.73311 0.144426
\(145\) −18.8368 −1.56431
\(146\) 1.92586 0.159386
\(147\) −2.17557 −0.179438
\(148\) −9.91930 −0.815361
\(149\) 11.0770 0.907467 0.453733 0.891138i \(-0.350092\pi\)
0.453733 + 0.891138i \(0.350092\pi\)
\(150\) 16.0806 1.31297
\(151\) 1.25512 0.102140 0.0510701 0.998695i \(-0.483737\pi\)
0.0510701 + 0.998695i \(0.483737\pi\)
\(152\) 0 0
\(153\) −6.40507 −0.517820
\(154\) 2.80423 0.225971
\(155\) 11.7023 0.939950
\(156\) 2.92522 0.234205
\(157\) −0.350101 −0.0279411 −0.0139706 0.999902i \(-0.504447\pi\)
−0.0139706 + 0.999902i \(0.504447\pi\)
\(158\) 17.5028 1.39245
\(159\) −5.36542 −0.425506
\(160\) −3.52015 −0.278292
\(161\) 4.80423 0.378626
\(162\) 11.1957 0.879614
\(163\) 16.4934 1.29186 0.645931 0.763396i \(-0.276470\pi\)
0.645931 + 0.763396i \(0.276470\pi\)
\(164\) 6.54795 0.511309
\(165\) 21.4757 1.67188
\(166\) −10.8576 −0.842717
\(167\) 3.14945 0.243711 0.121856 0.992548i \(-0.461115\pi\)
0.121856 + 0.992548i \(0.461115\pi\)
\(168\) 2.17557 0.167849
\(169\) −11.1921 −0.860932
\(170\) 13.0095 0.997781
\(171\) 0 0
\(172\) 10.6583 0.812690
\(173\) 12.4263 0.944755 0.472378 0.881396i \(-0.343396\pi\)
0.472378 + 0.881396i \(0.343396\pi\)
\(174\) −11.6418 −0.882561
\(175\) 7.39144 0.558740
\(176\) −2.80423 −0.211376
\(177\) 1.09017 0.0819422
\(178\) 16.5402 1.23974
\(179\) 10.5505 0.788579 0.394289 0.918986i \(-0.370991\pi\)
0.394289 + 0.918986i \(0.370991\pi\)
\(180\) 6.10079 0.454726
\(181\) 22.4122 1.66588 0.832942 0.553361i \(-0.186655\pi\)
0.832942 + 0.553361i \(0.186655\pi\)
\(182\) 1.34458 0.0996666
\(183\) 20.0769 1.48413
\(184\) −4.80423 −0.354172
\(185\) −34.9174 −2.56718
\(186\) 7.23241 0.530306
\(187\) 10.3636 0.757863
\(188\) −9.44537 −0.688874
\(189\) 2.75621 0.200485
\(190\) 0 0
\(191\) 5.98468 0.433036 0.216518 0.976279i \(-0.430530\pi\)
0.216518 + 0.976279i \(0.430530\pi\)
\(192\) −2.17557 −0.157008
\(193\) −15.4318 −1.11081 −0.555404 0.831581i \(-0.687436\pi\)
−0.555404 + 0.831581i \(0.687436\pi\)
\(194\) −10.2012 −0.732402
\(195\) 10.2972 0.737398
\(196\) 1.00000 0.0714286
\(197\) −4.97729 −0.354617 −0.177309 0.984155i \(-0.556739\pi\)
−0.177309 + 0.984155i \(0.556739\pi\)
\(198\) 4.86002 0.345387
\(199\) 0.473176 0.0335425 0.0167713 0.999859i \(-0.494661\pi\)
0.0167713 + 0.999859i \(0.494661\pi\)
\(200\) −7.39144 −0.522653
\(201\) 26.0812 1.83963
\(202\) −1.16025 −0.0816349
\(203\) −5.35114 −0.375576
\(204\) 8.04029 0.562933
\(205\) 23.0498 1.60987
\(206\) 10.5480 0.734911
\(207\) 8.32624 0.578714
\(208\) −1.34458 −0.0932296
\(209\) 0 0
\(210\) 7.65833 0.528475
\(211\) 20.0211 1.37831 0.689156 0.724613i \(-0.257982\pi\)
0.689156 + 0.724613i \(0.257982\pi\)
\(212\) 2.46621 0.169380
\(213\) −28.9713 −1.98508
\(214\) −9.16850 −0.626745
\(215\) 37.5189 2.55877
\(216\) −2.75621 −0.187537
\(217\) 3.32437 0.225673
\(218\) −5.46255 −0.369971
\(219\) 4.18985 0.283124
\(220\) −9.87129 −0.665522
\(221\) 4.96917 0.334263
\(222\) −21.5801 −1.44836
\(223\) 2.11456 0.141602 0.0708008 0.997490i \(-0.477445\pi\)
0.0708008 + 0.997490i \(0.477445\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 12.8101 0.854010
\(226\) −2.00947 −0.133668
\(227\) −4.53547 −0.301030 −0.150515 0.988608i \(-0.548093\pi\)
−0.150515 + 0.988608i \(0.548093\pi\)
\(228\) 0 0
\(229\) −22.8914 −1.51270 −0.756352 0.654164i \(-0.773020\pi\)
−0.756352 + 0.654164i \(0.773020\pi\)
\(230\) −16.9116 −1.11512
\(231\) 6.10079 0.401403
\(232\) 5.35114 0.351320
\(233\) −15.2646 −1.00002 −0.500008 0.866021i \(-0.666670\pi\)
−0.500008 + 0.866021i \(0.666670\pi\)
\(234\) 2.33030 0.152336
\(235\) −33.2491 −2.16893
\(236\) −0.501096 −0.0326186
\(237\) 38.0787 2.47348
\(238\) 3.69572 0.239558
\(239\) −16.1839 −1.04685 −0.523424 0.852072i \(-0.675346\pi\)
−0.523424 + 0.852072i \(0.675346\pi\)
\(240\) −7.65833 −0.494343
\(241\) −3.37895 −0.217657 −0.108829 0.994061i \(-0.534710\pi\)
−0.108829 + 0.994061i \(0.534710\pi\)
\(242\) 3.13632 0.201610
\(243\) 16.0883 1.03207
\(244\) −9.22835 −0.590785
\(245\) 3.52015 0.224894
\(246\) 14.2455 0.908262
\(247\) 0 0
\(248\) −3.32437 −0.211098
\(249\) −23.6216 −1.49696
\(250\) −8.41820 −0.532414
\(251\) −22.2609 −1.40510 −0.702549 0.711636i \(-0.747955\pi\)
−0.702549 + 0.711636i \(0.747955\pi\)
\(252\) 1.73311 0.109175
\(253\) −13.4721 −0.846986
\(254\) −2.30719 −0.144766
\(255\) 28.3030 1.77240
\(256\) 1.00000 0.0625000
\(257\) −14.1227 −0.880948 −0.440474 0.897765i \(-0.645190\pi\)
−0.440474 + 0.897765i \(0.645190\pi\)
\(258\) 23.1879 1.44362
\(259\) −9.91930 −0.616355
\(260\) −4.73311 −0.293535
\(261\) −9.27410 −0.574052
\(262\) 21.9508 1.35613
\(263\) 1.58732 0.0978785 0.0489392 0.998802i \(-0.484416\pi\)
0.0489392 + 0.998802i \(0.484416\pi\)
\(264\) −6.10079 −0.375478
\(265\) 8.68144 0.533296
\(266\) 0 0
\(267\) 35.9845 2.20221
\(268\) −11.9882 −0.732297
\(269\) 21.7180 1.32417 0.662085 0.749429i \(-0.269672\pi\)
0.662085 + 0.749429i \(0.269672\pi\)
\(270\) −9.70228 −0.590462
\(271\) −16.0267 −0.973550 −0.486775 0.873527i \(-0.661827\pi\)
−0.486775 + 0.873527i \(0.661827\pi\)
\(272\) −3.69572 −0.224086
\(273\) 2.92522 0.177042
\(274\) 22.7616 1.37508
\(275\) −20.7273 −1.24990
\(276\) −10.4519 −0.629132
\(277\) −18.4186 −1.10667 −0.553333 0.832960i \(-0.686644\pi\)
−0.553333 + 0.832960i \(0.686644\pi\)
\(278\) 11.6162 0.696692
\(279\) 5.76149 0.344932
\(280\) −3.52015 −0.210369
\(281\) −31.4082 −1.87365 −0.936827 0.349792i \(-0.886252\pi\)
−0.936827 + 0.349792i \(0.886252\pi\)
\(282\) −20.5491 −1.22368
\(283\) 24.7246 1.46972 0.734861 0.678217i \(-0.237247\pi\)
0.734861 + 0.678217i \(0.237247\pi\)
\(284\) 13.3167 0.790198
\(285\) 0 0
\(286\) −3.77050 −0.222954
\(287\) 6.54795 0.386514
\(288\) −1.73311 −0.102124
\(289\) −3.34167 −0.196569
\(290\) 18.8368 1.10614
\(291\) −22.1934 −1.30100
\(292\) −1.92586 −0.112703
\(293\) 4.10747 0.239961 0.119980 0.992776i \(-0.461717\pi\)
0.119980 + 0.992776i \(0.461717\pi\)
\(294\) 2.17557 0.126882
\(295\) −1.76393 −0.102700
\(296\) 9.91930 0.576548
\(297\) −7.72905 −0.448485
\(298\) −11.0770 −0.641676
\(299\) −6.45965 −0.373571
\(300\) −16.0806 −0.928413
\(301\) 10.6583 0.614336
\(302\) −1.25512 −0.0722240
\(303\) −2.52420 −0.145012
\(304\) 0 0
\(305\) −32.4852 −1.86009
\(306\) 6.40507 0.366154
\(307\) −14.3380 −0.818314 −0.409157 0.912464i \(-0.634177\pi\)
−0.409157 + 0.912464i \(0.634177\pi\)
\(308\) −2.80423 −0.159786
\(309\) 22.9478 1.30546
\(310\) −11.7023 −0.664645
\(311\) −13.2034 −0.748695 −0.374347 0.927289i \(-0.622133\pi\)
−0.374347 + 0.927289i \(0.622133\pi\)
\(312\) −2.92522 −0.165608
\(313\) 12.8412 0.725826 0.362913 0.931823i \(-0.381782\pi\)
0.362913 + 0.931823i \(0.381782\pi\)
\(314\) 0.350101 0.0197574
\(315\) 6.10079 0.343741
\(316\) −17.5028 −0.984612
\(317\) −7.23015 −0.406085 −0.203043 0.979170i \(-0.565083\pi\)
−0.203043 + 0.979170i \(0.565083\pi\)
\(318\) 5.36542 0.300878
\(319\) 15.0058 0.840164
\(320\) 3.52015 0.196782
\(321\) −19.9467 −1.11332
\(322\) −4.80423 −0.267729
\(323\) 0 0
\(324\) −11.1957 −0.621981
\(325\) −9.93835 −0.551280
\(326\) −16.4934 −0.913484
\(327\) −11.8842 −0.657196
\(328\) −6.54795 −0.361550
\(329\) −9.44537 −0.520740
\(330\) −21.4757 −1.18220
\(331\) −18.5184 −1.01786 −0.508930 0.860808i \(-0.669959\pi\)
−0.508930 + 0.860808i \(0.669959\pi\)
\(332\) 10.8576 0.595891
\(333\) −17.1912 −0.942072
\(334\) −3.14945 −0.172330
\(335\) −42.2003 −2.30565
\(336\) −2.17557 −0.118687
\(337\) −2.30938 −0.125800 −0.0629000 0.998020i \(-0.520035\pi\)
−0.0629000 + 0.998020i \(0.520035\pi\)
\(338\) 11.1921 0.608771
\(339\) −4.37174 −0.237440
\(340\) −13.0095 −0.705538
\(341\) −9.32229 −0.504831
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −10.6583 −0.574659
\(345\) −36.7923 −1.98083
\(346\) −12.4263 −0.668043
\(347\) 28.7044 1.54093 0.770465 0.637482i \(-0.220024\pi\)
0.770465 + 0.637482i \(0.220024\pi\)
\(348\) 11.6418 0.624065
\(349\) 1.87129 0.100168 0.0500839 0.998745i \(-0.484051\pi\)
0.0500839 + 0.998745i \(0.484051\pi\)
\(350\) −7.39144 −0.395089
\(351\) −3.70594 −0.197809
\(352\) 2.80423 0.149466
\(353\) −23.4056 −1.24575 −0.622877 0.782320i \(-0.714036\pi\)
−0.622877 + 0.782320i \(0.714036\pi\)
\(354\) −1.09017 −0.0579419
\(355\) 46.8766 2.48795
\(356\) −16.5402 −0.876631
\(357\) 8.04029 0.425537
\(358\) −10.5505 −0.557609
\(359\) 8.35333 0.440872 0.220436 0.975401i \(-0.429252\pi\)
0.220436 + 0.975401i \(0.429252\pi\)
\(360\) −6.10079 −0.321540
\(361\) 0 0
\(362\) −22.4122 −1.17796
\(363\) 6.82328 0.358129
\(364\) −1.34458 −0.0704750
\(365\) −6.77932 −0.354846
\(366\) −20.0769 −1.04944
\(367\) 9.73550 0.508189 0.254095 0.967179i \(-0.418223\pi\)
0.254095 + 0.967179i \(0.418223\pi\)
\(368\) 4.80423 0.250438
\(369\) 11.3483 0.590769
\(370\) 34.9174 1.81527
\(371\) 2.46621 0.128039
\(372\) −7.23241 −0.374983
\(373\) −27.4991 −1.42385 −0.711925 0.702255i \(-0.752177\pi\)
−0.711925 + 0.702255i \(0.752177\pi\)
\(374\) −10.3636 −0.535890
\(375\) −18.3144 −0.945751
\(376\) 9.44537 0.487108
\(377\) 7.19502 0.370562
\(378\) −2.75621 −0.141764
\(379\) −14.0748 −0.722976 −0.361488 0.932377i \(-0.617731\pi\)
−0.361488 + 0.932377i \(0.617731\pi\)
\(380\) 0 0
\(381\) −5.01945 −0.257154
\(382\) −5.98468 −0.306203
\(383\) 29.2284 1.49350 0.746749 0.665105i \(-0.231613\pi\)
0.746749 + 0.665105i \(0.231613\pi\)
\(384\) 2.17557 0.111022
\(385\) −9.87129 −0.503087
\(386\) 15.4318 0.785460
\(387\) 18.4720 0.938986
\(388\) 10.2012 0.517887
\(389\) −13.8753 −0.703508 −0.351754 0.936092i \(-0.614415\pi\)
−0.351754 + 0.936092i \(0.614415\pi\)
\(390\) −10.2972 −0.521419
\(391\) −17.7551 −0.897912
\(392\) −1.00000 −0.0505076
\(393\) 47.7556 2.40895
\(394\) 4.97729 0.250752
\(395\) −61.6126 −3.10007
\(396\) −4.86002 −0.244225
\(397\) −36.4442 −1.82908 −0.914541 0.404492i \(-0.867448\pi\)
−0.914541 + 0.404492i \(0.867448\pi\)
\(398\) −0.473176 −0.0237182
\(399\) 0 0
\(400\) 7.39144 0.369572
\(401\) −5.55284 −0.277295 −0.138648 0.990342i \(-0.544276\pi\)
−0.138648 + 0.990342i \(0.544276\pi\)
\(402\) −26.0812 −1.30081
\(403\) −4.46987 −0.222660
\(404\) 1.16025 0.0577246
\(405\) −39.4104 −1.95832
\(406\) 5.35114 0.265573
\(407\) 27.8160 1.37879
\(408\) −8.04029 −0.398054
\(409\) 25.8973 1.28054 0.640270 0.768150i \(-0.278823\pi\)
0.640270 + 0.768150i \(0.278823\pi\)
\(410\) −23.0498 −1.13835
\(411\) 49.5195 2.44262
\(412\) −10.5480 −0.519660
\(413\) −0.501096 −0.0246573
\(414\) −8.32624 −0.409212
\(415\) 38.2205 1.87617
\(416\) 1.34458 0.0659233
\(417\) 25.2718 1.23757
\(418\) 0 0
\(419\) 23.7413 1.15984 0.579919 0.814674i \(-0.303084\pi\)
0.579919 + 0.814674i \(0.303084\pi\)
\(420\) −7.65833 −0.373688
\(421\) −38.0474 −1.85432 −0.927158 0.374670i \(-0.877756\pi\)
−0.927158 + 0.374670i \(0.877756\pi\)
\(422\) −20.0211 −0.974613
\(423\) −16.3698 −0.795929
\(424\) −2.46621 −0.119770
\(425\) −27.3167 −1.32505
\(426\) 28.9713 1.40367
\(427\) −9.22835 −0.446591
\(428\) 9.16850 0.443176
\(429\) −8.20298 −0.396044
\(430\) −37.5189 −1.80932
\(431\) −4.34741 −0.209407 −0.104704 0.994503i \(-0.533389\pi\)
−0.104704 + 0.994503i \(0.533389\pi\)
\(432\) 2.75621 0.132608
\(433\) 4.40984 0.211924 0.105962 0.994370i \(-0.466208\pi\)
0.105962 + 0.994370i \(0.466208\pi\)
\(434\) −3.32437 −0.159575
\(435\) 40.9808 1.96488
\(436\) 5.46255 0.261609
\(437\) 0 0
\(438\) −4.18985 −0.200199
\(439\) −27.8784 −1.33056 −0.665281 0.746593i \(-0.731688\pi\)
−0.665281 + 0.746593i \(0.731688\pi\)
\(440\) 9.87129 0.470595
\(441\) 1.73311 0.0825289
\(442\) −4.96917 −0.236360
\(443\) −12.3334 −0.585980 −0.292990 0.956116i \(-0.594650\pi\)
−0.292990 + 0.956116i \(0.594650\pi\)
\(444\) 21.5801 1.02415
\(445\) −58.2241 −2.76009
\(446\) −2.11456 −0.100127
\(447\) −24.0989 −1.13984
\(448\) 1.00000 0.0472456
\(449\) 16.7235 0.789232 0.394616 0.918846i \(-0.370878\pi\)
0.394616 + 0.918846i \(0.370878\pi\)
\(450\) −12.8101 −0.603876
\(451\) −18.3619 −0.864630
\(452\) 2.00947 0.0945175
\(453\) −2.73060 −0.128295
\(454\) 4.53547 0.212860
\(455\) −4.73311 −0.221891
\(456\) 0 0
\(457\) 10.6854 0.499843 0.249922 0.968266i \(-0.419595\pi\)
0.249922 + 0.968266i \(0.419595\pi\)
\(458\) 22.8914 1.06964
\(459\) −10.1862 −0.475451
\(460\) 16.9116 0.788507
\(461\) −26.4779 −1.23320 −0.616600 0.787276i \(-0.711491\pi\)
−0.616600 + 0.787276i \(0.711491\pi\)
\(462\) −6.10079 −0.283835
\(463\) 15.6061 0.725276 0.362638 0.931930i \(-0.381876\pi\)
0.362638 + 0.931930i \(0.381876\pi\)
\(464\) −5.35114 −0.248420
\(465\) −25.4591 −1.18064
\(466\) 15.2646 0.707118
\(467\) −4.57730 −0.211812 −0.105906 0.994376i \(-0.533774\pi\)
−0.105906 + 0.994376i \(0.533774\pi\)
\(468\) −2.33030 −0.107718
\(469\) −11.9882 −0.553565
\(470\) 33.2491 1.53367
\(471\) 0.761670 0.0350959
\(472\) 0.501096 0.0230648
\(473\) −29.8884 −1.37427
\(474\) −38.0787 −1.74901
\(475\) 0 0
\(476\) −3.69572 −0.169393
\(477\) 4.27421 0.195703
\(478\) 16.1839 0.740234
\(479\) 11.9098 0.544172 0.272086 0.962273i \(-0.412286\pi\)
0.272086 + 0.962273i \(0.412286\pi\)
\(480\) 7.65833 0.349553
\(481\) 13.3373 0.608127
\(482\) 3.37895 0.153907
\(483\) −10.4519 −0.475579
\(484\) −3.13632 −0.142560
\(485\) 35.9097 1.63057
\(486\) −16.0883 −0.729780
\(487\) 11.8435 0.536680 0.268340 0.963324i \(-0.413525\pi\)
0.268340 + 0.963324i \(0.413525\pi\)
\(488\) 9.22835 0.417748
\(489\) −35.8825 −1.62266
\(490\) −3.52015 −0.159024
\(491\) 5.95019 0.268528 0.134264 0.990946i \(-0.457133\pi\)
0.134264 + 0.990946i \(0.457133\pi\)
\(492\) −14.2455 −0.642238
\(493\) 19.7763 0.890680
\(494\) 0 0
\(495\) −17.1080 −0.768947
\(496\) 3.32437 0.149269
\(497\) 13.3167 0.597334
\(498\) 23.6216 1.05851
\(499\) 15.8576 0.709886 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(500\) 8.41820 0.376473
\(501\) −6.85184 −0.306118
\(502\) 22.2609 0.993554
\(503\) 10.6100 0.473076 0.236538 0.971622i \(-0.423987\pi\)
0.236538 + 0.971622i \(0.423987\pi\)
\(504\) −1.73311 −0.0771987
\(505\) 4.08425 0.181747
\(506\) 13.4721 0.598909
\(507\) 24.3492 1.08139
\(508\) 2.30719 0.102365
\(509\) −11.3454 −0.502874 −0.251437 0.967874i \(-0.580903\pi\)
−0.251437 + 0.967874i \(0.580903\pi\)
\(510\) −28.3030 −1.25328
\(511\) −1.92586 −0.0851952
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 14.1227 0.622925
\(515\) −37.1304 −1.63616
\(516\) −23.1879 −1.02079
\(517\) 26.4869 1.16489
\(518\) 9.91930 0.435829
\(519\) −27.0343 −1.18668
\(520\) 4.73311 0.207560
\(521\) −19.1051 −0.837007 −0.418504 0.908215i \(-0.637445\pi\)
−0.418504 + 0.908215i \(0.637445\pi\)
\(522\) 9.27410 0.405916
\(523\) −23.0875 −1.00955 −0.504774 0.863252i \(-0.668424\pi\)
−0.504774 + 0.863252i \(0.668424\pi\)
\(524\) −21.9508 −0.958927
\(525\) −16.0806 −0.701814
\(526\) −1.58732 −0.0692105
\(527\) −12.2859 −0.535184
\(528\) 6.10079 0.265503
\(529\) 0.0805881 0.00350383
\(530\) −8.68144 −0.377097
\(531\) −0.868453 −0.0376877
\(532\) 0 0
\(533\) −8.80423 −0.381353
\(534\) −35.9845 −1.55720
\(535\) 32.2745 1.39535
\(536\) 11.9882 0.517812
\(537\) −22.9533 −0.990507
\(538\) −21.7180 −0.936329
\(539\) −2.80423 −0.120787
\(540\) 9.70228 0.417520
\(541\) −15.6929 −0.674690 −0.337345 0.941381i \(-0.609529\pi\)
−0.337345 + 0.941381i \(0.609529\pi\)
\(542\) 16.0267 0.688404
\(543\) −48.7593 −2.09246
\(544\) 3.69572 0.158453
\(545\) 19.2290 0.823680
\(546\) −2.92522 −0.125188
\(547\) −12.8444 −0.549187 −0.274594 0.961560i \(-0.588543\pi\)
−0.274594 + 0.961560i \(0.588543\pi\)
\(548\) −22.7616 −0.972329
\(549\) −15.9937 −0.682595
\(550\) 20.7273 0.883813
\(551\) 0 0
\(552\) 10.4519 0.444864
\(553\) −17.5028 −0.744297
\(554\) 18.4186 0.782531
\(555\) 75.9652 3.22455
\(556\) −11.6162 −0.492635
\(557\) 17.6165 0.746433 0.373217 0.927744i \(-0.378255\pi\)
0.373217 + 0.927744i \(0.378255\pi\)
\(558\) −5.76149 −0.243904
\(559\) −14.3309 −0.606134
\(560\) 3.52015 0.148753
\(561\) −22.5468 −0.951927
\(562\) 31.4082 1.32487
\(563\) −33.2321 −1.40057 −0.700283 0.713865i \(-0.746943\pi\)
−0.700283 + 0.713865i \(0.746943\pi\)
\(564\) 20.5491 0.865272
\(565\) 7.07363 0.297590
\(566\) −24.7246 −1.03925
\(567\) −11.1957 −0.470174
\(568\) −13.3167 −0.558754
\(569\) 8.98091 0.376499 0.188250 0.982121i \(-0.439719\pi\)
0.188250 + 0.982121i \(0.439719\pi\)
\(570\) 0 0
\(571\) −23.0380 −0.964110 −0.482055 0.876141i \(-0.660109\pi\)
−0.482055 + 0.876141i \(0.660109\pi\)
\(572\) 3.77050 0.157652
\(573\) −13.0201 −0.543922
\(574\) −6.54795 −0.273306
\(575\) 35.5101 1.48087
\(576\) 1.73311 0.0722128
\(577\) 21.3662 0.889487 0.444743 0.895658i \(-0.353295\pi\)
0.444743 + 0.895658i \(0.353295\pi\)
\(578\) 3.34167 0.138995
\(579\) 33.5731 1.39525
\(580\) −18.8368 −0.782156
\(581\) 10.8576 0.450451
\(582\) 22.1934 0.919946
\(583\) −6.91582 −0.286424
\(584\) 1.92586 0.0796928
\(585\) −8.20298 −0.339152
\(586\) −4.10747 −0.169678
\(587\) 6.45630 0.266480 0.133240 0.991084i \(-0.457462\pi\)
0.133240 + 0.991084i \(0.457462\pi\)
\(588\) −2.17557 −0.0897190
\(589\) 0 0
\(590\) 1.76393 0.0726199
\(591\) 10.8284 0.445423
\(592\) −9.91930 −0.407681
\(593\) 43.6349 1.79187 0.895935 0.444186i \(-0.146507\pi\)
0.895935 + 0.444186i \(0.146507\pi\)
\(594\) 7.72905 0.317127
\(595\) −13.0095 −0.533336
\(596\) 11.0770 0.453733
\(597\) −1.02943 −0.0421316
\(598\) 6.45965 0.264155
\(599\) −1.41049 −0.0576309 −0.0288154 0.999585i \(-0.509174\pi\)
−0.0288154 + 0.999585i \(0.509174\pi\)
\(600\) 16.0806 0.656487
\(601\) −47.7648 −1.94837 −0.974184 0.225754i \(-0.927515\pi\)
−0.974184 + 0.225754i \(0.927515\pi\)
\(602\) −10.6583 −0.434401
\(603\) −20.7769 −0.846100
\(604\) 1.25512 0.0510701
\(605\) −11.0403 −0.448852
\(606\) 2.52420 0.102539
\(607\) −19.5543 −0.793683 −0.396841 0.917887i \(-0.629894\pi\)
−0.396841 + 0.917887i \(0.629894\pi\)
\(608\) 0 0
\(609\) 11.6418 0.471749
\(610\) 32.4852 1.31529
\(611\) 12.7000 0.513788
\(612\) −6.40507 −0.258910
\(613\) −38.2184 −1.54363 −0.771814 0.635848i \(-0.780650\pi\)
−0.771814 + 0.635848i \(0.780650\pi\)
\(614\) 14.3380 0.578635
\(615\) −50.1464 −2.02210
\(616\) 2.80423 0.112985
\(617\) 39.4623 1.58869 0.794347 0.607465i \(-0.207813\pi\)
0.794347 + 0.607465i \(0.207813\pi\)
\(618\) −22.9478 −0.923097
\(619\) 20.0816 0.807149 0.403574 0.914947i \(-0.367768\pi\)
0.403574 + 0.914947i \(0.367768\pi\)
\(620\) 11.7023 0.469975
\(621\) 13.2415 0.531362
\(622\) 13.2034 0.529407
\(623\) −16.5402 −0.662671
\(624\) 2.92522 0.117103
\(625\) −7.32386 −0.292955
\(626\) −12.8412 −0.513237
\(627\) 0 0
\(628\) −0.350101 −0.0139706
\(629\) 36.6589 1.46169
\(630\) −6.10079 −0.243061
\(631\) −33.1030 −1.31781 −0.658905 0.752226i \(-0.728980\pi\)
−0.658905 + 0.752226i \(0.728980\pi\)
\(632\) 17.5028 0.696226
\(633\) −43.5574 −1.73125
\(634\) 7.23015 0.287146
\(635\) 8.12164 0.322297
\(636\) −5.36542 −0.212753
\(637\) −1.34458 −0.0532741
\(638\) −15.0058 −0.594086
\(639\) 23.0792 0.912999
\(640\) −3.52015 −0.139146
\(641\) −9.88036 −0.390251 −0.195125 0.980778i \(-0.562511\pi\)
−0.195125 + 0.980778i \(0.562511\pi\)
\(642\) 19.9467 0.787234
\(643\) −26.7710 −1.05575 −0.527873 0.849323i \(-0.677010\pi\)
−0.527873 + 0.849323i \(0.677010\pi\)
\(644\) 4.80423 0.189313
\(645\) −81.6250 −3.21398
\(646\) 0 0
\(647\) 14.3101 0.562588 0.281294 0.959622i \(-0.409236\pi\)
0.281294 + 0.959622i \(0.409236\pi\)
\(648\) 11.1957 0.439807
\(649\) 1.40519 0.0551584
\(650\) 9.93835 0.389814
\(651\) −7.23241 −0.283460
\(652\) 16.4934 0.645931
\(653\) 30.2338 1.18314 0.591571 0.806253i \(-0.298508\pi\)
0.591571 + 0.806253i \(0.298508\pi\)
\(654\) 11.8842 0.464708
\(655\) −77.2702 −3.01920
\(656\) 6.54795 0.255655
\(657\) −3.33773 −0.130217
\(658\) 9.44537 0.368219
\(659\) −24.2975 −0.946496 −0.473248 0.880929i \(-0.656919\pi\)
−0.473248 + 0.880929i \(0.656919\pi\)
\(660\) 21.4757 0.835940
\(661\) 40.1393 1.56124 0.780619 0.625007i \(-0.214904\pi\)
0.780619 + 0.625007i \(0.214904\pi\)
\(662\) 18.5184 0.719736
\(663\) −10.8108 −0.419856
\(664\) −10.8576 −0.421359
\(665\) 0 0
\(666\) 17.1912 0.666146
\(667\) −25.7081 −0.995421
\(668\) 3.14945 0.121856
\(669\) −4.60038 −0.177861
\(670\) 42.2003 1.63034
\(671\) 25.8784 0.999024
\(672\) 2.17557 0.0839245
\(673\) −14.8482 −0.572355 −0.286178 0.958177i \(-0.592385\pi\)
−0.286178 + 0.958177i \(0.592385\pi\)
\(674\) 2.30938 0.0889540
\(675\) 20.3724 0.784133
\(676\) −11.1921 −0.430466
\(677\) −25.9381 −0.996883 −0.498442 0.866923i \(-0.666094\pi\)
−0.498442 + 0.866923i \(0.666094\pi\)
\(678\) 4.37174 0.167896
\(679\) 10.2012 0.391486
\(680\) 13.0095 0.498890
\(681\) 9.86723 0.378113
\(682\) 9.32229 0.356969
\(683\) 22.0346 0.843129 0.421564 0.906798i \(-0.361481\pi\)
0.421564 + 0.906798i \(0.361481\pi\)
\(684\) 0 0
\(685\) −80.1243 −3.06139
\(686\) −1.00000 −0.0381802
\(687\) 49.8018 1.90006
\(688\) 10.6583 0.406345
\(689\) −3.31601 −0.126330
\(690\) 36.7923 1.40066
\(691\) −18.1301 −0.689703 −0.344851 0.938657i \(-0.612071\pi\)
−0.344851 + 0.938657i \(0.612071\pi\)
\(692\) 12.4263 0.472378
\(693\) −4.86002 −0.184617
\(694\) −28.7044 −1.08960
\(695\) −40.8906 −1.55107
\(696\) −11.6418 −0.441281
\(697\) −24.1994 −0.916617
\(698\) −1.87129 −0.0708293
\(699\) 33.2092 1.25609
\(700\) 7.39144 0.279370
\(701\) 35.8742 1.35495 0.677475 0.735546i \(-0.263074\pi\)
0.677475 + 0.735546i \(0.263074\pi\)
\(702\) 3.70594 0.139872
\(703\) 0 0
\(704\) −2.80423 −0.105688
\(705\) 72.3357 2.72432
\(706\) 23.4056 0.880882
\(707\) 1.16025 0.0436357
\(708\) 1.09017 0.0409711
\(709\) 1.44343 0.0542093 0.0271046 0.999633i \(-0.491371\pi\)
0.0271046 + 0.999633i \(0.491371\pi\)
\(710\) −46.8766 −1.75925
\(711\) −30.3343 −1.13763
\(712\) 16.5402 0.619872
\(713\) 15.9710 0.598120
\(714\) −8.04029 −0.300900
\(715\) 13.2727 0.496371
\(716\) 10.5505 0.394289
\(717\) 35.2092 1.31491
\(718\) −8.35333 −0.311744
\(719\) 22.9686 0.856585 0.428293 0.903640i \(-0.359115\pi\)
0.428293 + 0.903640i \(0.359115\pi\)
\(720\) 6.10079 0.227363
\(721\) −10.5480 −0.392826
\(722\) 0 0
\(723\) 7.35114 0.273392
\(724\) 22.4122 0.832942
\(725\) −39.5526 −1.46895
\(726\) −6.82328 −0.253236
\(727\) −23.4373 −0.869240 −0.434620 0.900614i \(-0.643117\pi\)
−0.434620 + 0.900614i \(0.643117\pi\)
\(728\) 1.34458 0.0498333
\(729\) −1.41426 −0.0523800
\(730\) 6.77932 0.250914
\(731\) −39.3902 −1.45690
\(732\) 20.0769 0.742065
\(733\) 39.9842 1.47685 0.738424 0.674336i \(-0.235570\pi\)
0.738424 + 0.674336i \(0.235570\pi\)
\(734\) −9.73550 −0.359344
\(735\) −7.65833 −0.282482
\(736\) −4.80423 −0.177086
\(737\) 33.6177 1.23832
\(738\) −11.3483 −0.417737
\(739\) 2.93035 0.107795 0.0538973 0.998546i \(-0.482836\pi\)
0.0538973 + 0.998546i \(0.482836\pi\)
\(740\) −34.9174 −1.28359
\(741\) 0 0
\(742\) −2.46621 −0.0905376
\(743\) −16.7593 −0.614837 −0.307419 0.951574i \(-0.599465\pi\)
−0.307419 + 0.951574i \(0.599465\pi\)
\(744\) 7.23241 0.265153
\(745\) 38.9928 1.42859
\(746\) 27.4991 1.00681
\(747\) 18.8175 0.688495
\(748\) 10.3636 0.378932
\(749\) 9.16850 0.335010
\(750\) 18.3144 0.668747
\(751\) −10.2383 −0.373599 −0.186800 0.982398i \(-0.559812\pi\)
−0.186800 + 0.982398i \(0.559812\pi\)
\(752\) −9.44537 −0.344437
\(753\) 48.4302 1.76490
\(754\) −7.19502 −0.262027
\(755\) 4.41820 0.160795
\(756\) 2.75621 0.100243
\(757\) −51.8313 −1.88384 −0.941920 0.335838i \(-0.890981\pi\)
−0.941920 + 0.335838i \(0.890981\pi\)
\(758\) 14.0748 0.511222
\(759\) 29.3096 1.06387
\(760\) 0 0
\(761\) 1.45246 0.0526517 0.0263259 0.999653i \(-0.491619\pi\)
0.0263259 + 0.999653i \(0.491619\pi\)
\(762\) 5.01945 0.181835
\(763\) 5.46255 0.197758
\(764\) 5.98468 0.216518
\(765\) −22.5468 −0.815181
\(766\) −29.2284 −1.05606
\(767\) 0.673762 0.0243281
\(768\) −2.17557 −0.0785041
\(769\) −0.848905 −0.0306123 −0.0153061 0.999883i \(-0.504872\pi\)
−0.0153061 + 0.999883i \(0.504872\pi\)
\(770\) 9.87129 0.355737
\(771\) 30.7249 1.10653
\(772\) −15.4318 −0.555404
\(773\) −13.5539 −0.487499 −0.243750 0.969838i \(-0.578377\pi\)
−0.243750 + 0.969838i \(0.578377\pi\)
\(774\) −18.4720 −0.663963
\(775\) 24.5719 0.882648
\(776\) −10.2012 −0.366201
\(777\) 21.5801 0.774183
\(778\) 13.8753 0.497455
\(779\) 0 0
\(780\) 10.2972 0.368699
\(781\) −37.3429 −1.33623
\(782\) 17.7551 0.634920
\(783\) −14.7489 −0.527083
\(784\) 1.00000 0.0357143
\(785\) −1.23241 −0.0439865
\(786\) −47.7556 −1.70339
\(787\) 3.86358 0.137722 0.0688609 0.997626i \(-0.478064\pi\)
0.0688609 + 0.997626i \(0.478064\pi\)
\(788\) −4.97729 −0.177309
\(789\) −3.45333 −0.122942
\(790\) 61.6126 2.19208
\(791\) 2.00947 0.0714485
\(792\) 4.86002 0.172693
\(793\) 12.4082 0.440629
\(794\) 36.4442 1.29336
\(795\) −18.8871 −0.669856
\(796\) 0.473176 0.0167713
\(797\) −15.1020 −0.534939 −0.267470 0.963566i \(-0.586187\pi\)
−0.267470 + 0.963566i \(0.586187\pi\)
\(798\) 0 0
\(799\) 34.9074 1.23494
\(800\) −7.39144 −0.261327
\(801\) −28.6660 −1.01286
\(802\) 5.55284 0.196077
\(803\) 5.40056 0.190582
\(804\) 26.0812 0.919814
\(805\) 16.9116 0.596055
\(806\) 4.46987 0.157445
\(807\) −47.2490 −1.66325
\(808\) −1.16025 −0.0408174
\(809\) 9.73661 0.342321 0.171161 0.985243i \(-0.445248\pi\)
0.171161 + 0.985243i \(0.445248\pi\)
\(810\) 39.4104 1.38474
\(811\) 0.160117 0.00562246 0.00281123 0.999996i \(-0.499105\pi\)
0.00281123 + 0.999996i \(0.499105\pi\)
\(812\) −5.35114 −0.187788
\(813\) 34.8671 1.22284
\(814\) −27.8160 −0.974949
\(815\) 58.0591 2.03372
\(816\) 8.04029 0.281467
\(817\) 0 0
\(818\) −25.8973 −0.905478
\(819\) −2.33030 −0.0814271
\(820\) 23.0498 0.804933
\(821\) −36.7541 −1.28273 −0.641363 0.767238i \(-0.721631\pi\)
−0.641363 + 0.767238i \(0.721631\pi\)
\(822\) −49.5195 −1.72719
\(823\) 5.24097 0.182689 0.0913444 0.995819i \(-0.470884\pi\)
0.0913444 + 0.995819i \(0.470884\pi\)
\(824\) 10.5480 0.367455
\(825\) 45.0936 1.56996
\(826\) 0.501096 0.0174354
\(827\) 39.6523 1.37885 0.689423 0.724359i \(-0.257864\pi\)
0.689423 + 0.724359i \(0.257864\pi\)
\(828\) 8.32624 0.289357
\(829\) 37.8829 1.31573 0.657864 0.753137i \(-0.271460\pi\)
0.657864 + 0.753137i \(0.271460\pi\)
\(830\) −38.2205 −1.32665
\(831\) 40.0710 1.39005
\(832\) −1.34458 −0.0466148
\(833\) −3.69572 −0.128049
\(834\) −25.2718 −0.875091
\(835\) 11.0865 0.383664
\(836\) 0 0
\(837\) 9.16269 0.316709
\(838\) −23.7413 −0.820130
\(839\) 21.7871 0.752175 0.376087 0.926584i \(-0.377269\pi\)
0.376087 + 0.926584i \(0.377269\pi\)
\(840\) 7.65833 0.264237
\(841\) −0.365290 −0.0125962
\(842\) 38.0474 1.31120
\(843\) 68.3307 2.35343
\(844\) 20.0211 0.689156
\(845\) −39.3979 −1.35533
\(846\) 16.3698 0.562806
\(847\) −3.13632 −0.107765
\(848\) 2.46621 0.0846901
\(849\) −53.7900 −1.84607
\(850\) 27.3167 0.936954
\(851\) −47.6546 −1.63358
\(852\) −28.9713 −0.992541
\(853\) 45.5176 1.55849 0.779247 0.626717i \(-0.215602\pi\)
0.779247 + 0.626717i \(0.215602\pi\)
\(854\) 9.22835 0.315788
\(855\) 0 0
\(856\) −9.16850 −0.313373
\(857\) −14.8920 −0.508701 −0.254351 0.967112i \(-0.581862\pi\)
−0.254351 + 0.967112i \(0.581862\pi\)
\(858\) 8.20298 0.280045
\(859\) −5.73987 −0.195842 −0.0979210 0.995194i \(-0.531219\pi\)
−0.0979210 + 0.995194i \(0.531219\pi\)
\(860\) 37.5189 1.27938
\(861\) −14.2455 −0.485487
\(862\) 4.34741 0.148073
\(863\) −37.9478 −1.29176 −0.645879 0.763440i \(-0.723509\pi\)
−0.645879 + 0.763440i \(0.723509\pi\)
\(864\) −2.75621 −0.0937683
\(865\) 43.7425 1.48729
\(866\) −4.40984 −0.149853
\(867\) 7.27004 0.246904
\(868\) 3.32437 0.112837
\(869\) 49.0819 1.66499
\(870\) −40.9808 −1.38938
\(871\) 16.1191 0.546174
\(872\) −5.46255 −0.184985
\(873\) 17.6797 0.598369
\(874\) 0 0
\(875\) 8.41820 0.284587
\(876\) 4.18985 0.141562
\(877\) 12.6048 0.425634 0.212817 0.977092i \(-0.431736\pi\)
0.212817 + 0.977092i \(0.431736\pi\)
\(878\) 27.8784 0.940850
\(879\) −8.93609 −0.301407
\(880\) −9.87129 −0.332761
\(881\) 47.5368 1.60156 0.800778 0.598962i \(-0.204420\pi\)
0.800778 + 0.598962i \(0.204420\pi\)
\(882\) −1.73311 −0.0583567
\(883\) 29.6583 0.998082 0.499041 0.866578i \(-0.333686\pi\)
0.499041 + 0.866578i \(0.333686\pi\)
\(884\) 4.96917 0.167131
\(885\) 3.83756 0.128998
\(886\) 12.3334 0.414350
\(887\) 4.97155 0.166928 0.0834642 0.996511i \(-0.473402\pi\)
0.0834642 + 0.996511i \(0.473402\pi\)
\(888\) −21.5801 −0.724182
\(889\) 2.30719 0.0773806
\(890\) 58.2241 1.95168
\(891\) 31.3952 1.05178
\(892\) 2.11456 0.0708008
\(893\) 0 0
\(894\) 24.0989 0.805987
\(895\) 37.1392 1.24143
\(896\) −1.00000 −0.0334077
\(897\) 14.0534 0.469230
\(898\) −16.7235 −0.558071
\(899\) −17.7892 −0.593303
\(900\) 12.8101 0.427005
\(901\) −9.11443 −0.303646
\(902\) 18.3619 0.611386
\(903\) −23.1879 −0.771646
\(904\) −2.00947 −0.0668339
\(905\) 78.8941 2.62253
\(906\) 2.73060 0.0907181
\(907\) 1.83004 0.0607654 0.0303827 0.999538i \(-0.490327\pi\)
0.0303827 + 0.999538i \(0.490327\pi\)
\(908\) −4.53547 −0.150515
\(909\) 2.01084 0.0666952
\(910\) 4.73311 0.156901
\(911\) −54.8341 −1.81673 −0.908367 0.418175i \(-0.862670\pi\)
−0.908367 + 0.418175i \(0.862670\pi\)
\(912\) 0 0
\(913\) −30.4473 −1.00766
\(914\) −10.6854 −0.353442
\(915\) 70.6737 2.33640
\(916\) −22.8914 −0.756352
\(917\) −21.9508 −0.724880
\(918\) 10.1862 0.336194
\(919\) −10.5422 −0.347756 −0.173878 0.984767i \(-0.555630\pi\)
−0.173878 + 0.984767i \(0.555630\pi\)
\(920\) −16.9116 −0.557558
\(921\) 31.1934 1.02786
\(922\) 26.4779 0.872005
\(923\) −17.9053 −0.589359
\(924\) 6.10079 0.200701
\(925\) −73.3179 −2.41068
\(926\) −15.6061 −0.512848
\(927\) −18.2807 −0.600418
\(928\) 5.35114 0.175660
\(929\) −49.4703 −1.62307 −0.811535 0.584304i \(-0.801367\pi\)
−0.811535 + 0.584304i \(0.801367\pi\)
\(930\) 25.4591 0.834838
\(931\) 0 0
\(932\) −15.2646 −0.500008
\(933\) 28.7249 0.940410
\(934\) 4.57730 0.149774
\(935\) 36.4815 1.19307
\(936\) 2.33030 0.0761681
\(937\) −9.69970 −0.316876 −0.158438 0.987369i \(-0.550646\pi\)
−0.158438 + 0.987369i \(0.550646\pi\)
\(938\) 11.9882 0.391429
\(939\) −27.9369 −0.911686
\(940\) −33.2491 −1.08447
\(941\) −25.9990 −0.847543 −0.423771 0.905769i \(-0.639294\pi\)
−0.423771 + 0.905769i \(0.639294\pi\)
\(942\) −0.761670 −0.0248166
\(943\) 31.4579 1.02441
\(944\) −0.501096 −0.0163093
\(945\) 9.70228 0.315615
\(946\) 29.8884 0.971754
\(947\) 47.0532 1.52902 0.764512 0.644610i \(-0.222980\pi\)
0.764512 + 0.644610i \(0.222980\pi\)
\(948\) 38.0787 1.23674
\(949\) 2.58947 0.0840578
\(950\) 0 0
\(951\) 15.7297 0.510070
\(952\) 3.69572 0.119779
\(953\) 14.0830 0.456193 0.228097 0.973638i \(-0.426750\pi\)
0.228097 + 0.973638i \(0.426750\pi\)
\(954\) −4.27421 −0.138383
\(955\) 21.0669 0.681710
\(956\) −16.1839 −0.523424
\(957\) −32.6462 −1.05530
\(958\) −11.9098 −0.384788
\(959\) −22.7616 −0.735011
\(960\) −7.65833 −0.247171
\(961\) −19.9485 −0.643501
\(962\) −13.3373 −0.430010
\(963\) 15.8900 0.512048
\(964\) −3.37895 −0.108829
\(965\) −54.3224 −1.74870
\(966\) 10.4519 0.336285
\(967\) −21.5394 −0.692661 −0.346330 0.938113i \(-0.612572\pi\)
−0.346330 + 0.938113i \(0.612572\pi\)
\(968\) 3.13632 0.100805
\(969\) 0 0
\(970\) −35.9097 −1.15299
\(971\) 29.4006 0.943509 0.471755 0.881730i \(-0.343621\pi\)
0.471755 + 0.881730i \(0.343621\pi\)
\(972\) 16.0883 0.516033
\(973\) −11.6162 −0.372397
\(974\) −11.8435 −0.379490
\(975\) 21.6216 0.692445
\(976\) −9.22835 −0.295392
\(977\) −18.1047 −0.579219 −0.289610 0.957145i \(-0.593525\pi\)
−0.289610 + 0.957145i \(0.593525\pi\)
\(978\) 35.8825 1.14740
\(979\) 46.3826 1.48239
\(980\) 3.52015 0.112447
\(981\) 9.46719 0.302264
\(982\) −5.95019 −0.189878
\(983\) −21.3333 −0.680426 −0.340213 0.940348i \(-0.610499\pi\)
−0.340213 + 0.940348i \(0.610499\pi\)
\(984\) 14.2455 0.454131
\(985\) −17.5208 −0.558259
\(986\) −19.7763 −0.629806
\(987\) 20.5491 0.654084
\(988\) 0 0
\(989\) 51.2050 1.62822
\(990\) 17.1080 0.543728
\(991\) −17.4816 −0.555323 −0.277661 0.960679i \(-0.589559\pi\)
−0.277661 + 0.960679i \(0.589559\pi\)
\(992\) −3.32437 −0.105549
\(993\) 40.2880 1.27850
\(994\) −13.3167 −0.422379
\(995\) 1.66565 0.0528046
\(996\) −23.6216 −0.748478
\(997\) 24.4311 0.773740 0.386870 0.922134i \(-0.373556\pi\)
0.386870 + 0.922134i \(0.373556\pi\)
\(998\) −15.8576 −0.501965
\(999\) −27.3397 −0.864990
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.v.1.2 4
19.18 odd 2 5054.2.a.y.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5054.2.a.v.1.2 4 1.1 even 1 trivial
5054.2.a.y.1.3 yes 4 19.18 odd 2