Properties

Label 4006.2.a.i.1.10
Level $4006$
Weight $2$
Character 4006.1
Self dual yes
Analytic conductor $31.988$
Analytic rank $0$
Dimension $46$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4006,2,Mod(1,4006)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4006, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4006.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4006 = 2 \cdot 2003 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4006.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.9880710497\)
Analytic rank: \(0\)
Dimension: \(46\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Character \(\chi\) \(=\) 4006.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.64474 q^{3} +1.00000 q^{4} -0.535091 q^{5} -1.64474 q^{6} -1.74380 q^{7} +1.00000 q^{8} -0.294845 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.64474 q^{3} +1.00000 q^{4} -0.535091 q^{5} -1.64474 q^{6} -1.74380 q^{7} +1.00000 q^{8} -0.294845 q^{9} -0.535091 q^{10} -1.93819 q^{11} -1.64474 q^{12} +3.27734 q^{13} -1.74380 q^{14} +0.880083 q^{15} +1.00000 q^{16} -1.55193 q^{17} -0.294845 q^{18} +0.183349 q^{19} -0.535091 q^{20} +2.86809 q^{21} -1.93819 q^{22} +8.78022 q^{23} -1.64474 q^{24} -4.71368 q^{25} +3.27734 q^{26} +5.41915 q^{27} -1.74380 q^{28} -3.77491 q^{29} +0.880083 q^{30} -8.42757 q^{31} +1.00000 q^{32} +3.18781 q^{33} -1.55193 q^{34} +0.933092 q^{35} -0.294845 q^{36} -0.166018 q^{37} +0.183349 q^{38} -5.39035 q^{39} -0.535091 q^{40} -2.70104 q^{41} +2.86809 q^{42} +4.36327 q^{43} -1.93819 q^{44} +0.157769 q^{45} +8.78022 q^{46} -12.8366 q^{47} -1.64474 q^{48} -3.95916 q^{49} -4.71368 q^{50} +2.55252 q^{51} +3.27734 q^{52} +8.19730 q^{53} +5.41915 q^{54} +1.03711 q^{55} -1.74380 q^{56} -0.301560 q^{57} -3.77491 q^{58} +12.9123 q^{59} +0.880083 q^{60} +8.27695 q^{61} -8.42757 q^{62} +0.514150 q^{63} +1.00000 q^{64} -1.75367 q^{65} +3.18781 q^{66} +1.83260 q^{67} -1.55193 q^{68} -14.4411 q^{69} +0.933092 q^{70} -2.99104 q^{71} -0.294845 q^{72} +7.79312 q^{73} -0.166018 q^{74} +7.75275 q^{75} +0.183349 q^{76} +3.37982 q^{77} -5.39035 q^{78} +7.03516 q^{79} -0.535091 q^{80} -8.02853 q^{81} -2.70104 q^{82} +11.9340 q^{83} +2.86809 q^{84} +0.830425 q^{85} +4.36327 q^{86} +6.20872 q^{87} -1.93819 q^{88} -1.25449 q^{89} +0.157769 q^{90} -5.71503 q^{91} +8.78022 q^{92} +13.8611 q^{93} -12.8366 q^{94} -0.0981081 q^{95} -1.64474 q^{96} +14.8335 q^{97} -3.95916 q^{98} +0.571465 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46 q + 46 q^{2} + 21 q^{3} + 46 q^{4} + 23 q^{5} + 21 q^{6} + 26 q^{7} + 46 q^{8} + 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 46 q + 46 q^{2} + 21 q^{3} + 46 q^{4} + 23 q^{5} + 21 q^{6} + 26 q^{7} + 46 q^{8} + 59 q^{9} + 23 q^{10} + 39 q^{11} + 21 q^{12} + 8 q^{13} + 26 q^{14} + 14 q^{15} + 46 q^{16} + 36 q^{17} + 59 q^{18} + 37 q^{19} + 23 q^{20} + 20 q^{21} + 39 q^{22} + 38 q^{23} + 21 q^{24} + 57 q^{25} + 8 q^{26} + 63 q^{27} + 26 q^{28} + 23 q^{29} + 14 q^{30} + 44 q^{31} + 46 q^{32} + 25 q^{33} + 36 q^{34} + 26 q^{35} + 59 q^{36} + 9 q^{37} + 37 q^{38} - 2 q^{39} + 23 q^{40} + 50 q^{41} + 20 q^{42} + 46 q^{43} + 39 q^{44} + 30 q^{45} + 38 q^{46} + 57 q^{47} + 21 q^{48} + 62 q^{49} + 57 q^{50} + 5 q^{51} + 8 q^{52} + 21 q^{53} + 63 q^{54} + 40 q^{55} + 26 q^{56} + 3 q^{57} + 23 q^{58} + 68 q^{59} + 14 q^{60} - q^{61} + 44 q^{62} + 40 q^{63} + 46 q^{64} + 18 q^{65} + 25 q^{66} + 42 q^{67} + 36 q^{68} - 7 q^{69} + 26 q^{70} + 67 q^{71} + 59 q^{72} + 48 q^{73} + 9 q^{74} + 71 q^{75} + 37 q^{76} - 5 q^{77} - 2 q^{78} + 40 q^{79} + 23 q^{80} + 82 q^{81} + 50 q^{82} + 48 q^{83} + 20 q^{84} - 68 q^{85} + 46 q^{86} + 18 q^{87} + 39 q^{88} + 90 q^{89} + 30 q^{90} + 9 q^{91} + 38 q^{92} - 42 q^{93} + 57 q^{94} + 6 q^{95} + 21 q^{96} + 46 q^{97} + 62 q^{98} + 61 q^{99}+O(q^{100}) \) 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
\(3\) −1.64474 −0.949589 −0.474794 0.880097i \(-0.657478\pi\)
−0.474794 + 0.880097i \(0.657478\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.535091 −0.239300 −0.119650 0.992816i \(-0.538177\pi\)
−0.119650 + 0.992816i \(0.538177\pi\)
\(6\) −1.64474 −0.671461
\(7\) −1.74380 −0.659095 −0.329547 0.944139i \(-0.606896\pi\)
−0.329547 + 0.944139i \(0.606896\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.294845 −0.0982815
\(10\) −0.535091 −0.169211
\(11\) −1.93819 −0.584386 −0.292193 0.956359i \(-0.594385\pi\)
−0.292193 + 0.956359i \(0.594385\pi\)
\(12\) −1.64474 −0.474794
\(13\) 3.27734 0.908970 0.454485 0.890754i \(-0.349823\pi\)
0.454485 + 0.890754i \(0.349823\pi\)
\(14\) −1.74380 −0.466051
\(15\) 0.880083 0.227236
\(16\) 1.00000 0.250000
\(17\) −1.55193 −0.376399 −0.188200 0.982131i \(-0.560265\pi\)
−0.188200 + 0.982131i \(0.560265\pi\)
\(18\) −0.294845 −0.0694955
\(19\) 0.183349 0.0420630 0.0210315 0.999779i \(-0.493305\pi\)
0.0210315 + 0.999779i \(0.493305\pi\)
\(20\) −0.535091 −0.119650
\(21\) 2.86809 0.625869
\(22\) −1.93819 −0.413224
\(23\) 8.78022 1.83080 0.915401 0.402543i \(-0.131874\pi\)
0.915401 + 0.402543i \(0.131874\pi\)
\(24\) −1.64474 −0.335730
\(25\) −4.71368 −0.942736
\(26\) 3.27734 0.642739
\(27\) 5.41915 1.04292
\(28\) −1.74380 −0.329547
\(29\) −3.77491 −0.700982 −0.350491 0.936566i \(-0.613985\pi\)
−0.350491 + 0.936566i \(0.613985\pi\)
\(30\) 0.880083 0.160680
\(31\) −8.42757 −1.51364 −0.756818 0.653625i \(-0.773247\pi\)
−0.756818 + 0.653625i \(0.773247\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.18781 0.554927
\(34\) −1.55193 −0.266154
\(35\) 0.933092 0.157721
\(36\) −0.294845 −0.0491408
\(37\) −0.166018 −0.0272931 −0.0136466 0.999907i \(-0.504344\pi\)
−0.0136466 + 0.999907i \(0.504344\pi\)
\(38\) 0.183349 0.0297431
\(39\) −5.39035 −0.863147
\(40\) −0.535091 −0.0846053
\(41\) −2.70104 −0.421832 −0.210916 0.977504i \(-0.567645\pi\)
−0.210916 + 0.977504i \(0.567645\pi\)
\(42\) 2.86809 0.442556
\(43\) 4.36327 0.665393 0.332696 0.943034i \(-0.392042\pi\)
0.332696 + 0.943034i \(0.392042\pi\)
\(44\) −1.93819 −0.292193
\(45\) 0.157769 0.0235188
\(46\) 8.78022 1.29457
\(47\) −12.8366 −1.87241 −0.936205 0.351454i \(-0.885688\pi\)
−0.936205 + 0.351454i \(0.885688\pi\)
\(48\) −1.64474 −0.237397
\(49\) −3.95916 −0.565594
\(50\) −4.71368 −0.666615
\(51\) 2.55252 0.357424
\(52\) 3.27734 0.454485
\(53\) 8.19730 1.12599 0.562993 0.826462i \(-0.309650\pi\)
0.562993 + 0.826462i \(0.309650\pi\)
\(54\) 5.41915 0.737453
\(55\) 1.03711 0.139844
\(56\) −1.74380 −0.233025
\(57\) −0.301560 −0.0399426
\(58\) −3.77491 −0.495669
\(59\) 12.9123 1.68103 0.840517 0.541784i \(-0.182251\pi\)
0.840517 + 0.541784i \(0.182251\pi\)
\(60\) 0.880083 0.113618
\(61\) 8.27695 1.05975 0.529877 0.848074i \(-0.322238\pi\)
0.529877 + 0.848074i \(0.322238\pi\)
\(62\) −8.42757 −1.07030
\(63\) 0.514150 0.0647769
\(64\) 1.00000 0.125000
\(65\) −1.75367 −0.217516
\(66\) 3.18781 0.392392
\(67\) 1.83260 0.223888 0.111944 0.993715i \(-0.464292\pi\)
0.111944 + 0.993715i \(0.464292\pi\)
\(68\) −1.55193 −0.188200
\(69\) −14.4411 −1.73851
\(70\) 0.933092 0.111526
\(71\) −2.99104 −0.354972 −0.177486 0.984123i \(-0.556796\pi\)
−0.177486 + 0.984123i \(0.556796\pi\)
\(72\) −0.294845 −0.0347478
\(73\) 7.79312 0.912115 0.456058 0.889950i \(-0.349261\pi\)
0.456058 + 0.889950i \(0.349261\pi\)
\(74\) −0.166018 −0.0192992
\(75\) 7.75275 0.895211
\(76\) 0.183349 0.0210315
\(77\) 3.37982 0.385166
\(78\) −5.39035 −0.610337
\(79\) 7.03516 0.791517 0.395758 0.918355i \(-0.370482\pi\)
0.395758 + 0.918355i \(0.370482\pi\)
\(80\) −0.535091 −0.0598250
\(81\) −8.02853 −0.892059
\(82\) −2.70104 −0.298280
\(83\) 11.9340 1.30993 0.654963 0.755661i \(-0.272684\pi\)
0.654963 + 0.755661i \(0.272684\pi\)
\(84\) 2.86809 0.312935
\(85\) 0.830425 0.0900723
\(86\) 4.36327 0.470504
\(87\) 6.20872 0.665645
\(88\) −1.93819 −0.206612
\(89\) −1.25449 −0.132976 −0.0664879 0.997787i \(-0.521179\pi\)
−0.0664879 + 0.997787i \(0.521179\pi\)
\(90\) 0.157769 0.0166303
\(91\) −5.71503 −0.599098
\(92\) 8.78022 0.915401
\(93\) 13.8611 1.43733
\(94\) −12.8366 −1.32399
\(95\) −0.0981081 −0.0100657
\(96\) −1.64474 −0.167865
\(97\) 14.8335 1.50612 0.753059 0.657953i \(-0.228577\pi\)
0.753059 + 0.657953i \(0.228577\pi\)
\(98\) −3.95916 −0.399935
\(99\) 0.571465 0.0574344
\(100\) −4.71368 −0.471368
\(101\) 10.3409 1.02896 0.514481 0.857502i \(-0.327985\pi\)
0.514481 + 0.857502i \(0.327985\pi\)
\(102\) 2.55252 0.252737
\(103\) 13.0167 1.28257 0.641285 0.767303i \(-0.278402\pi\)
0.641285 + 0.767303i \(0.278402\pi\)
\(104\) 3.27734 0.321369
\(105\) −1.53469 −0.149770
\(106\) 8.19730 0.796192
\(107\) −1.60704 −0.155359 −0.0776795 0.996978i \(-0.524751\pi\)
−0.0776795 + 0.996978i \(0.524751\pi\)
\(108\) 5.41915 0.521458
\(109\) 12.3196 1.18000 0.590002 0.807402i \(-0.299127\pi\)
0.590002 + 0.807402i \(0.299127\pi\)
\(110\) 1.03711 0.0988844
\(111\) 0.273055 0.0259173
\(112\) −1.74380 −0.164774
\(113\) −7.06350 −0.664478 −0.332239 0.943195i \(-0.607804\pi\)
−0.332239 + 0.943195i \(0.607804\pi\)
\(114\) −0.301560 −0.0282437
\(115\) −4.69821 −0.438111
\(116\) −3.77491 −0.350491
\(117\) −0.966305 −0.0893350
\(118\) 12.9123 1.18867
\(119\) 2.70626 0.248083
\(120\) 0.880083 0.0803402
\(121\) −7.24342 −0.658492
\(122\) 8.27695 0.749360
\(123\) 4.44250 0.400567
\(124\) −8.42757 −0.756818
\(125\) 5.19770 0.464896
\(126\) 0.514150 0.0458042
\(127\) −0.412758 −0.0366263 −0.0183132 0.999832i \(-0.505830\pi\)
−0.0183132 + 0.999832i \(0.505830\pi\)
\(128\) 1.00000 0.0883883
\(129\) −7.17643 −0.631849
\(130\) −1.75367 −0.153807
\(131\) 15.3237 1.33884 0.669419 0.742885i \(-0.266543\pi\)
0.669419 + 0.742885i \(0.266543\pi\)
\(132\) 3.18781 0.277463
\(133\) −0.319723 −0.0277235
\(134\) 1.83260 0.158313
\(135\) −2.89974 −0.249570
\(136\) −1.55193 −0.133077
\(137\) −0.194785 −0.0166416 −0.00832080 0.999965i \(-0.502649\pi\)
−0.00832080 + 0.999965i \(0.502649\pi\)
\(138\) −14.4411 −1.22931
\(139\) 11.1840 0.948613 0.474306 0.880360i \(-0.342699\pi\)
0.474306 + 0.880360i \(0.342699\pi\)
\(140\) 0.933092 0.0788607
\(141\) 21.1128 1.77802
\(142\) −2.99104 −0.251003
\(143\) −6.35210 −0.531190
\(144\) −0.294845 −0.0245704
\(145\) 2.01992 0.167745
\(146\) 7.79312 0.644963
\(147\) 6.51177 0.537081
\(148\) −0.166018 −0.0136466
\(149\) −11.9277 −0.977153 −0.488577 0.872521i \(-0.662484\pi\)
−0.488577 + 0.872521i \(0.662484\pi\)
\(150\) 7.75275 0.633010
\(151\) 10.4538 0.850720 0.425360 0.905024i \(-0.360147\pi\)
0.425360 + 0.905024i \(0.360147\pi\)
\(152\) 0.183349 0.0148715
\(153\) 0.457579 0.0369931
\(154\) 3.37982 0.272354
\(155\) 4.50952 0.362213
\(156\) −5.39035 −0.431574
\(157\) −15.6334 −1.24768 −0.623841 0.781551i \(-0.714429\pi\)
−0.623841 + 0.781551i \(0.714429\pi\)
\(158\) 7.03516 0.559687
\(159\) −13.4824 −1.06922
\(160\) −0.535091 −0.0423026
\(161\) −15.3110 −1.20667
\(162\) −8.02853 −0.630781
\(163\) 3.44930 0.270170 0.135085 0.990834i \(-0.456869\pi\)
0.135085 + 0.990834i \(0.456869\pi\)
\(164\) −2.70104 −0.210916
\(165\) −1.70577 −0.132794
\(166\) 11.9340 0.926257
\(167\) 3.38984 0.262313 0.131157 0.991362i \(-0.458131\pi\)
0.131157 + 0.991362i \(0.458131\pi\)
\(168\) 2.86809 0.221278
\(169\) −2.25906 −0.173774
\(170\) 0.830425 0.0636907
\(171\) −0.0540593 −0.00413402
\(172\) 4.36327 0.332696
\(173\) 14.0345 1.06702 0.533511 0.845793i \(-0.320872\pi\)
0.533511 + 0.845793i \(0.320872\pi\)
\(174\) 6.20872 0.470682
\(175\) 8.21972 0.621352
\(176\) −1.93819 −0.146097
\(177\) −21.2373 −1.59629
\(178\) −1.25449 −0.0940280
\(179\) −8.40342 −0.628101 −0.314051 0.949406i \(-0.601686\pi\)
−0.314051 + 0.949406i \(0.601686\pi\)
\(180\) 0.157769 0.0117594
\(181\) −24.6257 −1.83042 −0.915209 0.402980i \(-0.867974\pi\)
−0.915209 + 0.402980i \(0.867974\pi\)
\(182\) −5.71503 −0.423626
\(183\) −13.6134 −1.00633
\(184\) 8.78022 0.647286
\(185\) 0.0888345 0.00653125
\(186\) 13.8611 1.01635
\(187\) 3.00794 0.219963
\(188\) −12.8366 −0.936205
\(189\) −9.44992 −0.687380
\(190\) −0.0981081 −0.00711751
\(191\) −9.53239 −0.689740 −0.344870 0.938651i \(-0.612077\pi\)
−0.344870 + 0.938651i \(0.612077\pi\)
\(192\) −1.64474 −0.118699
\(193\) 16.9460 1.21980 0.609900 0.792478i \(-0.291209\pi\)
0.609900 + 0.792478i \(0.291209\pi\)
\(194\) 14.8335 1.06499
\(195\) 2.88433 0.206551
\(196\) −3.95916 −0.282797
\(197\) 22.0413 1.57038 0.785188 0.619258i \(-0.212566\pi\)
0.785188 + 0.619258i \(0.212566\pi\)
\(198\) 0.571465 0.0406123
\(199\) 27.4022 1.94249 0.971244 0.238088i \(-0.0765208\pi\)
0.971244 + 0.238088i \(0.0765208\pi\)
\(200\) −4.71368 −0.333307
\(201\) −3.01414 −0.212601
\(202\) 10.3409 0.727586
\(203\) 6.58269 0.462014
\(204\) 2.55252 0.178712
\(205\) 1.44530 0.100944
\(206\) 13.0167 0.906914
\(207\) −2.58880 −0.179934
\(208\) 3.27734 0.227242
\(209\) −0.355364 −0.0245811
\(210\) −1.53469 −0.105904
\(211\) 14.5885 1.00431 0.502155 0.864777i \(-0.332541\pi\)
0.502155 + 0.864777i \(0.332541\pi\)
\(212\) 8.19730 0.562993
\(213\) 4.91948 0.337077
\(214\) −1.60704 −0.109855
\(215\) −2.33475 −0.159228
\(216\) 5.41915 0.368726
\(217\) 14.6960 0.997630
\(218\) 12.3196 0.834389
\(219\) −12.8176 −0.866134
\(220\) 1.03711 0.0699218
\(221\) −5.08621 −0.342136
\(222\) 0.273055 0.0183263
\(223\) −1.37324 −0.0919592 −0.0459796 0.998942i \(-0.514641\pi\)
−0.0459796 + 0.998942i \(0.514641\pi\)
\(224\) −1.74380 −0.116513
\(225\) 1.38980 0.0926535
\(226\) −7.06350 −0.469857
\(227\) −24.7682 −1.64392 −0.821962 0.569543i \(-0.807120\pi\)
−0.821962 + 0.569543i \(0.807120\pi\)
\(228\) −0.301560 −0.0199713
\(229\) −29.1508 −1.92634 −0.963169 0.268898i \(-0.913341\pi\)
−0.963169 + 0.268898i \(0.913341\pi\)
\(230\) −4.69821 −0.309791
\(231\) −5.55891 −0.365749
\(232\) −3.77491 −0.247835
\(233\) 16.9061 1.10756 0.553779 0.832664i \(-0.313185\pi\)
0.553779 + 0.832664i \(0.313185\pi\)
\(234\) −0.966305 −0.0631694
\(235\) 6.86875 0.448068
\(236\) 12.9123 0.840517
\(237\) −11.5710 −0.751615
\(238\) 2.70626 0.175421
\(239\) 20.5110 1.32675 0.663374 0.748288i \(-0.269124\pi\)
0.663374 + 0.748288i \(0.269124\pi\)
\(240\) 0.880083 0.0568091
\(241\) −14.6757 −0.945344 −0.472672 0.881239i \(-0.656710\pi\)
−0.472672 + 0.881239i \(0.656710\pi\)
\(242\) −7.24342 −0.465625
\(243\) −3.05263 −0.195826
\(244\) 8.27695 0.529877
\(245\) 2.11851 0.135347
\(246\) 4.44250 0.283243
\(247\) 0.600895 0.0382340
\(248\) −8.42757 −0.535151
\(249\) −19.6283 −1.24389
\(250\) 5.19770 0.328731
\(251\) 10.7043 0.675647 0.337823 0.941209i \(-0.390309\pi\)
0.337823 + 0.941209i \(0.390309\pi\)
\(252\) 0.514150 0.0323884
\(253\) −17.0177 −1.06990
\(254\) −0.412758 −0.0258987
\(255\) −1.36583 −0.0855316
\(256\) 1.00000 0.0625000
\(257\) 22.0649 1.37637 0.688186 0.725534i \(-0.258407\pi\)
0.688186 + 0.725534i \(0.258407\pi\)
\(258\) −7.17643 −0.446785
\(259\) 0.289502 0.0179888
\(260\) −1.75367 −0.108758
\(261\) 1.11301 0.0688936
\(262\) 15.3237 0.946701
\(263\) 16.7320 1.03174 0.515870 0.856667i \(-0.327469\pi\)
0.515870 + 0.856667i \(0.327469\pi\)
\(264\) 3.18781 0.196196
\(265\) −4.38630 −0.269448
\(266\) −0.319723 −0.0196035
\(267\) 2.06331 0.126272
\(268\) 1.83260 0.111944
\(269\) −9.70358 −0.591638 −0.295819 0.955244i \(-0.595592\pi\)
−0.295819 + 0.955244i \(0.595592\pi\)
\(270\) −2.89974 −0.176472
\(271\) 20.9871 1.27487 0.637437 0.770502i \(-0.279995\pi\)
0.637437 + 0.770502i \(0.279995\pi\)
\(272\) −1.55193 −0.0940998
\(273\) 9.39971 0.568896
\(274\) −0.194785 −0.0117674
\(275\) 9.13601 0.550922
\(276\) −14.4411 −0.869254
\(277\) −29.8619 −1.79423 −0.897115 0.441797i \(-0.854341\pi\)
−0.897115 + 0.441797i \(0.854341\pi\)
\(278\) 11.1840 0.670770
\(279\) 2.48482 0.148763
\(280\) 0.933092 0.0557629
\(281\) 30.3835 1.81253 0.906265 0.422711i \(-0.138921\pi\)
0.906265 + 0.422711i \(0.138921\pi\)
\(282\) 21.1128 1.25725
\(283\) −25.3062 −1.50430 −0.752148 0.658994i \(-0.770982\pi\)
−0.752148 + 0.658994i \(0.770982\pi\)
\(284\) −2.99104 −0.177486
\(285\) 0.161362 0.00955826
\(286\) −6.35210 −0.375608
\(287\) 4.71008 0.278027
\(288\) −0.294845 −0.0173739
\(289\) −14.5915 −0.858324
\(290\) 2.01992 0.118614
\(291\) −24.3973 −1.43019
\(292\) 7.79312 0.456058
\(293\) 15.4568 0.902994 0.451497 0.892273i \(-0.350890\pi\)
0.451497 + 0.892273i \(0.350890\pi\)
\(294\) 6.51177 0.379774
\(295\) −6.90924 −0.402271
\(296\) −0.166018 −0.00964958
\(297\) −10.5033 −0.609466
\(298\) −11.9277 −0.690952
\(299\) 28.7757 1.66414
\(300\) 7.75275 0.447605
\(301\) −7.60868 −0.438557
\(302\) 10.4538 0.601550
\(303\) −17.0081 −0.977091
\(304\) 0.183349 0.0105158
\(305\) −4.42892 −0.253599
\(306\) 0.457579 0.0261581
\(307\) 15.0136 0.856873 0.428436 0.903572i \(-0.359065\pi\)
0.428436 + 0.903572i \(0.359065\pi\)
\(308\) 3.37982 0.192583
\(309\) −21.4090 −1.21791
\(310\) 4.50952 0.256123
\(311\) −15.4281 −0.874849 −0.437425 0.899255i \(-0.644109\pi\)
−0.437425 + 0.899255i \(0.644109\pi\)
\(312\) −5.39035 −0.305169
\(313\) 13.7594 0.777728 0.388864 0.921295i \(-0.372868\pi\)
0.388864 + 0.921295i \(0.372868\pi\)
\(314\) −15.6334 −0.882244
\(315\) −0.275117 −0.0155011
\(316\) 7.03516 0.395758
\(317\) 9.64703 0.541831 0.270915 0.962603i \(-0.412674\pi\)
0.270915 + 0.962603i \(0.412674\pi\)
\(318\) −13.4824 −0.756055
\(319\) 7.31649 0.409645
\(320\) −0.535091 −0.0299125
\(321\) 2.64316 0.147527
\(322\) −15.3110 −0.853246
\(323\) −0.284545 −0.0158325
\(324\) −8.02853 −0.446030
\(325\) −15.4483 −0.856918
\(326\) 3.44930 0.191039
\(327\) −20.2625 −1.12052
\(328\) −2.70104 −0.149140
\(329\) 22.3845 1.23410
\(330\) −1.70577 −0.0938995
\(331\) 8.02708 0.441208 0.220604 0.975363i \(-0.429197\pi\)
0.220604 + 0.975363i \(0.429197\pi\)
\(332\) 11.9340 0.654963
\(333\) 0.0489494 0.00268241
\(334\) 3.38984 0.185484
\(335\) −0.980608 −0.0535763
\(336\) 2.86809 0.156467
\(337\) −22.2909 −1.21426 −0.607131 0.794602i \(-0.707680\pi\)
−0.607131 + 0.794602i \(0.707680\pi\)
\(338\) −2.25906 −0.122877
\(339\) 11.6176 0.630981
\(340\) 0.830425 0.0450361
\(341\) 16.3342 0.884549
\(342\) −0.0540593 −0.00292319
\(343\) 19.1106 1.03188
\(344\) 4.36327 0.235252
\(345\) 7.72732 0.416025
\(346\) 14.0345 0.754499
\(347\) −5.97388 −0.320695 −0.160347 0.987061i \(-0.551261\pi\)
−0.160347 + 0.987061i \(0.551261\pi\)
\(348\) 6.20872 0.332822
\(349\) −2.97557 −0.159279 −0.0796394 0.996824i \(-0.525377\pi\)
−0.0796394 + 0.996824i \(0.525377\pi\)
\(350\) 8.21972 0.439362
\(351\) 17.7604 0.947979
\(352\) −1.93819 −0.103306
\(353\) −12.1499 −0.646674 −0.323337 0.946284i \(-0.604805\pi\)
−0.323337 + 0.946284i \(0.604805\pi\)
\(354\) −21.2373 −1.12875
\(355\) 1.60048 0.0849447
\(356\) −1.25449 −0.0664879
\(357\) −4.45109 −0.235577
\(358\) −8.40342 −0.444135
\(359\) 9.90806 0.522927 0.261464 0.965213i \(-0.415795\pi\)
0.261464 + 0.965213i \(0.415795\pi\)
\(360\) 0.157769 0.00831514
\(361\) −18.9664 −0.998231
\(362\) −24.6257 −1.29430
\(363\) 11.9135 0.625297
\(364\) −5.71503 −0.299549
\(365\) −4.17003 −0.218269
\(366\) −13.6134 −0.711584
\(367\) 31.9933 1.67004 0.835018 0.550223i \(-0.185457\pi\)
0.835018 + 0.550223i \(0.185457\pi\)
\(368\) 8.78022 0.457700
\(369\) 0.796387 0.0414583
\(370\) 0.0888345 0.00461829
\(371\) −14.2945 −0.742132
\(372\) 13.8611 0.718666
\(373\) 8.48716 0.439449 0.219724 0.975562i \(-0.429484\pi\)
0.219724 + 0.975562i \(0.429484\pi\)
\(374\) 3.00794 0.155537
\(375\) −8.54884 −0.441460
\(376\) −12.8366 −0.661997
\(377\) −12.3716 −0.637172
\(378\) −9.44992 −0.486051
\(379\) 3.11422 0.159967 0.0799835 0.996796i \(-0.474513\pi\)
0.0799835 + 0.996796i \(0.474513\pi\)
\(380\) −0.0981081 −0.00503284
\(381\) 0.678877 0.0347799
\(382\) −9.53239 −0.487720
\(383\) −6.12086 −0.312761 −0.156381 0.987697i \(-0.549983\pi\)
−0.156381 + 0.987697i \(0.549983\pi\)
\(384\) −1.64474 −0.0839326
\(385\) −1.80851 −0.0921702
\(386\) 16.9460 0.862529
\(387\) −1.28649 −0.0653958
\(388\) 14.8335 0.753059
\(389\) 0.488904 0.0247884 0.0123942 0.999923i \(-0.496055\pi\)
0.0123942 + 0.999923i \(0.496055\pi\)
\(390\) 2.88433 0.146054
\(391\) −13.6263 −0.689112
\(392\) −3.95916 −0.199968
\(393\) −25.2034 −1.27135
\(394\) 22.0413 1.11042
\(395\) −3.76445 −0.189410
\(396\) 0.571465 0.0287172
\(397\) 20.3480 1.02124 0.510618 0.859807i \(-0.329416\pi\)
0.510618 + 0.859807i \(0.329416\pi\)
\(398\) 27.4022 1.37355
\(399\) 0.525861 0.0263260
\(400\) −4.71368 −0.235684
\(401\) −32.4212 −1.61904 −0.809519 0.587094i \(-0.800272\pi\)
−0.809519 + 0.587094i \(0.800272\pi\)
\(402\) −3.01414 −0.150332
\(403\) −27.6200 −1.37585
\(404\) 10.3409 0.514481
\(405\) 4.29599 0.213470
\(406\) 6.58269 0.326693
\(407\) 0.321774 0.0159497
\(408\) 2.55252 0.126369
\(409\) −18.4173 −0.910675 −0.455338 0.890319i \(-0.650481\pi\)
−0.455338 + 0.890319i \(0.650481\pi\)
\(410\) 1.44530 0.0713784
\(411\) 0.320370 0.0158027
\(412\) 13.0167 0.641285
\(413\) −22.5164 −1.10796
\(414\) −2.58880 −0.127233
\(415\) −6.38577 −0.313465
\(416\) 3.27734 0.160685
\(417\) −18.3947 −0.900792
\(418\) −0.355364 −0.0173814
\(419\) −36.7509 −1.79540 −0.897700 0.440607i \(-0.854763\pi\)
−0.897700 + 0.440607i \(0.854763\pi\)
\(420\) −1.53469 −0.0748852
\(421\) −24.9153 −1.21430 −0.607148 0.794588i \(-0.707687\pi\)
−0.607148 + 0.794588i \(0.707687\pi\)
\(422\) 14.5885 0.710155
\(423\) 3.78480 0.184023
\(424\) 8.19730 0.398096
\(425\) 7.31531 0.354845
\(426\) 4.91948 0.238349
\(427\) −14.4334 −0.698479
\(428\) −1.60704 −0.0776795
\(429\) 10.4475 0.504412
\(430\) −2.33475 −0.112591
\(431\) 9.14506 0.440502 0.220251 0.975443i \(-0.429312\pi\)
0.220251 + 0.975443i \(0.429312\pi\)
\(432\) 5.41915 0.260729
\(433\) 3.73613 0.179547 0.0897735 0.995962i \(-0.471386\pi\)
0.0897735 + 0.995962i \(0.471386\pi\)
\(434\) 14.6960 0.705431
\(435\) −3.32223 −0.159289
\(436\) 12.3196 0.590002
\(437\) 1.60984 0.0770091
\(438\) −12.8176 −0.612449
\(439\) 39.0826 1.86531 0.932656 0.360768i \(-0.117485\pi\)
0.932656 + 0.360768i \(0.117485\pi\)
\(440\) 1.03711 0.0494422
\(441\) 1.16734 0.0555874
\(442\) −5.08621 −0.241926
\(443\) 32.9595 1.56595 0.782977 0.622051i \(-0.213700\pi\)
0.782977 + 0.622051i \(0.213700\pi\)
\(444\) 0.273055 0.0129586
\(445\) 0.671266 0.0318211
\(446\) −1.37324 −0.0650250
\(447\) 19.6179 0.927894
\(448\) −1.74380 −0.0823869
\(449\) −36.3922 −1.71745 −0.858726 0.512434i \(-0.828744\pi\)
−0.858726 + 0.512434i \(0.828744\pi\)
\(450\) 1.38980 0.0655159
\(451\) 5.23513 0.246513
\(452\) −7.06350 −0.332239
\(453\) −17.1938 −0.807834
\(454\) −24.7682 −1.16243
\(455\) 3.05806 0.143364
\(456\) −0.301560 −0.0141218
\(457\) −22.8886 −1.07068 −0.535342 0.844635i \(-0.679817\pi\)
−0.535342 + 0.844635i \(0.679817\pi\)
\(458\) −29.1508 −1.36213
\(459\) −8.41016 −0.392553
\(460\) −4.69821 −0.219055
\(461\) 5.22632 0.243414 0.121707 0.992566i \(-0.461163\pi\)
0.121707 + 0.992566i \(0.461163\pi\)
\(462\) −5.55891 −0.258624
\(463\) −33.9321 −1.57696 −0.788480 0.615060i \(-0.789132\pi\)
−0.788480 + 0.615060i \(0.789132\pi\)
\(464\) −3.77491 −0.175246
\(465\) −7.41696 −0.343953
\(466\) 16.9061 0.783161
\(467\) −2.48794 −0.115128 −0.0575641 0.998342i \(-0.518333\pi\)
−0.0575641 + 0.998342i \(0.518333\pi\)
\(468\) −0.966305 −0.0446675
\(469\) −3.19569 −0.147563
\(470\) 6.86875 0.316832
\(471\) 25.7128 1.18478
\(472\) 12.9123 0.594336
\(473\) −8.45685 −0.388846
\(474\) −11.5710 −0.531472
\(475\) −0.864246 −0.0396543
\(476\) 2.70626 0.124041
\(477\) −2.41693 −0.110664
\(478\) 20.5110 0.938152
\(479\) 8.97749 0.410192 0.205096 0.978742i \(-0.434249\pi\)
0.205096 + 0.978742i \(0.434249\pi\)
\(480\) 0.880083 0.0401701
\(481\) −0.544096 −0.0248086
\(482\) −14.6757 −0.668459
\(483\) 25.1825 1.14584
\(484\) −7.24342 −0.329246
\(485\) −7.93730 −0.360414
\(486\) −3.05263 −0.138470
\(487\) −9.40078 −0.425990 −0.212995 0.977053i \(-0.568322\pi\)
−0.212995 + 0.977053i \(0.568322\pi\)
\(488\) 8.27695 0.374680
\(489\) −5.67318 −0.256550
\(490\) 2.11851 0.0957044
\(491\) 11.4028 0.514601 0.257300 0.966332i \(-0.417167\pi\)
0.257300 + 0.966332i \(0.417167\pi\)
\(492\) 4.44250 0.200283
\(493\) 5.85840 0.263849
\(494\) 0.600895 0.0270355
\(495\) −0.305786 −0.0137440
\(496\) −8.42757 −0.378409
\(497\) 5.21579 0.233960
\(498\) −19.6283 −0.879564
\(499\) −8.69850 −0.389398 −0.194699 0.980863i \(-0.562373\pi\)
−0.194699 + 0.980863i \(0.562373\pi\)
\(500\) 5.19770 0.232448
\(501\) −5.57538 −0.249090
\(502\) 10.7043 0.477755
\(503\) 41.0139 1.82872 0.914360 0.404902i \(-0.132694\pi\)
0.914360 + 0.404902i \(0.132694\pi\)
\(504\) 0.514150 0.0229021
\(505\) −5.53334 −0.246231
\(506\) −17.0177 −0.756531
\(507\) 3.71555 0.165013
\(508\) −0.412758 −0.0183132
\(509\) 30.3228 1.34404 0.672018 0.740535i \(-0.265428\pi\)
0.672018 + 0.740535i \(0.265428\pi\)
\(510\) −1.36583 −0.0604800
\(511\) −13.5896 −0.601171
\(512\) 1.00000 0.0441942
\(513\) 0.993593 0.0438682
\(514\) 22.0649 0.973242
\(515\) −6.96510 −0.306919
\(516\) −7.17643 −0.315925
\(517\) 24.8798 1.09421
\(518\) 0.289502 0.0127200
\(519\) −23.0830 −1.01323
\(520\) −1.75367 −0.0769037
\(521\) −6.35833 −0.278564 −0.139282 0.990253i \(-0.544479\pi\)
−0.139282 + 0.990253i \(0.544479\pi\)
\(522\) 1.11301 0.0487152
\(523\) −34.6341 −1.51444 −0.757221 0.653158i \(-0.773444\pi\)
−0.757221 + 0.653158i \(0.773444\pi\)
\(524\) 15.3237 0.669419
\(525\) −13.5193 −0.590029
\(526\) 16.7320 0.729551
\(527\) 13.0790 0.569731
\(528\) 3.18781 0.138732
\(529\) 54.0922 2.35184
\(530\) −4.38630 −0.190529
\(531\) −3.80711 −0.165215
\(532\) −0.319723 −0.0138618
\(533\) −8.85222 −0.383432
\(534\) 2.06331 0.0892879
\(535\) 0.859915 0.0371774
\(536\) 1.83260 0.0791563
\(537\) 13.8214 0.596438
\(538\) −9.70358 −0.418351
\(539\) 7.67360 0.330525
\(540\) −2.89974 −0.124785
\(541\) 0.985205 0.0423573 0.0211786 0.999776i \(-0.493258\pi\)
0.0211786 + 0.999776i \(0.493258\pi\)
\(542\) 20.9871 0.901472
\(543\) 40.5028 1.73814
\(544\) −1.55193 −0.0665386
\(545\) −6.59211 −0.282375
\(546\) 9.39971 0.402270
\(547\) 4.56775 0.195303 0.0976513 0.995221i \(-0.468867\pi\)
0.0976513 + 0.995221i \(0.468867\pi\)
\(548\) −0.194785 −0.00832080
\(549\) −2.44041 −0.104154
\(550\) 9.13601 0.389561
\(551\) −0.692123 −0.0294855
\(552\) −14.4411 −0.614656
\(553\) −12.2679 −0.521685
\(554\) −29.8619 −1.26871
\(555\) −0.146109 −0.00620200
\(556\) 11.1840 0.474306
\(557\) 21.4967 0.910844 0.455422 0.890276i \(-0.349488\pi\)
0.455422 + 0.890276i \(0.349488\pi\)
\(558\) 2.48482 0.105191
\(559\) 14.2999 0.604822
\(560\) 0.933092 0.0394303
\(561\) −4.94727 −0.208874
\(562\) 30.3835 1.28165
\(563\) −18.5685 −0.782569 −0.391285 0.920270i \(-0.627969\pi\)
−0.391285 + 0.920270i \(0.627969\pi\)
\(564\) 21.1128 0.889010
\(565\) 3.77961 0.159010
\(566\) −25.3062 −1.06370
\(567\) 14.0002 0.587952
\(568\) −2.99104 −0.125501
\(569\) −23.9375 −1.00351 −0.501755 0.865010i \(-0.667312\pi\)
−0.501755 + 0.865010i \(0.667312\pi\)
\(570\) 0.161362 0.00675871
\(571\) −41.9227 −1.75441 −0.877206 0.480114i \(-0.840595\pi\)
−0.877206 + 0.480114i \(0.840595\pi\)
\(572\) −6.35210 −0.265595
\(573\) 15.6783 0.654969
\(574\) 4.71008 0.196595
\(575\) −41.3871 −1.72596
\(576\) −0.294845 −0.0122852
\(577\) −40.5422 −1.68779 −0.843897 0.536505i \(-0.819744\pi\)
−0.843897 + 0.536505i \(0.819744\pi\)
\(578\) −14.5915 −0.606926
\(579\) −27.8717 −1.15831
\(580\) 2.01992 0.0838725
\(581\) −20.8105 −0.863366
\(582\) −24.3973 −1.01130
\(583\) −15.8879 −0.658011
\(584\) 7.79312 0.322481
\(585\) 0.517061 0.0213778
\(586\) 15.4568 0.638513
\(587\) 17.5449 0.724155 0.362078 0.932148i \(-0.382068\pi\)
0.362078 + 0.932148i \(0.382068\pi\)
\(588\) 6.51177 0.268541
\(589\) −1.54518 −0.0636681
\(590\) −6.90924 −0.284449
\(591\) −36.2521 −1.49121
\(592\) −0.166018 −0.00682329
\(593\) −19.0693 −0.783083 −0.391542 0.920160i \(-0.628058\pi\)
−0.391542 + 0.920160i \(0.628058\pi\)
\(594\) −10.5033 −0.430957
\(595\) −1.44810 −0.0593662
\(596\) −11.9277 −0.488577
\(597\) −45.0693 −1.84456
\(598\) 28.7757 1.17673
\(599\) −15.1197 −0.617772 −0.308886 0.951099i \(-0.599956\pi\)
−0.308886 + 0.951099i \(0.599956\pi\)
\(600\) 7.75275 0.316505
\(601\) 26.3651 1.07546 0.537728 0.843118i \(-0.319283\pi\)
0.537728 + 0.843118i \(0.319283\pi\)
\(602\) −7.60868 −0.310107
\(603\) −0.540333 −0.0220040
\(604\) 10.4538 0.425360
\(605\) 3.87589 0.157577
\(606\) −17.0081 −0.690907
\(607\) 11.2277 0.455719 0.227860 0.973694i \(-0.426827\pi\)
0.227860 + 0.973694i \(0.426827\pi\)
\(608\) 0.183349 0.00743577
\(609\) −10.8268 −0.438723
\(610\) −4.42892 −0.179322
\(611\) −42.0699 −1.70196
\(612\) 0.457579 0.0184965
\(613\) −1.25138 −0.0505426 −0.0252713 0.999681i \(-0.508045\pi\)
−0.0252713 + 0.999681i \(0.508045\pi\)
\(614\) 15.0136 0.605901
\(615\) −2.37714 −0.0958555
\(616\) 3.37982 0.136177
\(617\) 33.9446 1.36656 0.683278 0.730158i \(-0.260554\pi\)
0.683278 + 0.730158i \(0.260554\pi\)
\(618\) −21.4090 −0.861195
\(619\) 11.5161 0.462872 0.231436 0.972850i \(-0.425658\pi\)
0.231436 + 0.972850i \(0.425658\pi\)
\(620\) 4.50952 0.181106
\(621\) 47.5813 1.90937
\(622\) −15.4281 −0.618612
\(623\) 2.18758 0.0876436
\(624\) −5.39035 −0.215787
\(625\) 20.7871 0.831486
\(626\) 13.7594 0.549937
\(627\) 0.584481 0.0233419
\(628\) −15.6334 −0.623841
\(629\) 0.257648 0.0102731
\(630\) −0.275117 −0.0109609
\(631\) 4.51047 0.179559 0.0897794 0.995962i \(-0.471384\pi\)
0.0897794 + 0.995962i \(0.471384\pi\)
\(632\) 7.03516 0.279843
\(633\) −23.9942 −0.953682
\(634\) 9.64703 0.383132
\(635\) 0.220863 0.00876468
\(636\) −13.4824 −0.534612
\(637\) −12.9755 −0.514108
\(638\) 7.31649 0.289662
\(639\) 0.881893 0.0348872
\(640\) −0.535091 −0.0211513
\(641\) 32.2052 1.27203 0.636015 0.771677i \(-0.280582\pi\)
0.636015 + 0.771677i \(0.280582\pi\)
\(642\) 2.64316 0.104317
\(643\) −5.22458 −0.206037 −0.103019 0.994679i \(-0.532850\pi\)
−0.103019 + 0.994679i \(0.532850\pi\)
\(644\) −15.3110 −0.603336
\(645\) 3.84004 0.151201
\(646\) −0.284545 −0.0111953
\(647\) −25.7603 −1.01274 −0.506371 0.862316i \(-0.669013\pi\)
−0.506371 + 0.862316i \(0.669013\pi\)
\(648\) −8.02853 −0.315391
\(649\) −25.0264 −0.982374
\(650\) −15.4483 −0.605933
\(651\) −24.1711 −0.947338
\(652\) 3.44930 0.135085
\(653\) −48.7147 −1.90636 −0.953178 0.302411i \(-0.902209\pi\)
−0.953178 + 0.302411i \(0.902209\pi\)
\(654\) −20.2625 −0.792326
\(655\) −8.19957 −0.320384
\(656\) −2.70104 −0.105458
\(657\) −2.29776 −0.0896441
\(658\) 22.3845 0.872638
\(659\) 9.10766 0.354784 0.177392 0.984140i \(-0.443234\pi\)
0.177392 + 0.984140i \(0.443234\pi\)
\(660\) −1.70577 −0.0663969
\(661\) −4.35433 −0.169364 −0.0846819 0.996408i \(-0.526987\pi\)
−0.0846819 + 0.996408i \(0.526987\pi\)
\(662\) 8.02708 0.311981
\(663\) 8.36547 0.324888
\(664\) 11.9340 0.463129
\(665\) 0.171081 0.00663424
\(666\) 0.0489494 0.00189675
\(667\) −33.1445 −1.28336
\(668\) 3.38984 0.131157
\(669\) 2.25862 0.0873234
\(670\) −0.980608 −0.0378842
\(671\) −16.0423 −0.619306
\(672\) 2.86809 0.110639
\(673\) 34.8823 1.34461 0.672307 0.740272i \(-0.265303\pi\)
0.672307 + 0.740272i \(0.265303\pi\)
\(674\) −22.2909 −0.858613
\(675\) −25.5441 −0.983194
\(676\) −2.25906 −0.0868868
\(677\) 1.43170 0.0550245 0.0275123 0.999621i \(-0.491241\pi\)
0.0275123 + 0.999621i \(0.491241\pi\)
\(678\) 11.6176 0.446171
\(679\) −25.8668 −0.992675
\(680\) 0.830425 0.0318454
\(681\) 40.7372 1.56105
\(682\) 16.3342 0.625470
\(683\) −36.8512 −1.41007 −0.705036 0.709172i \(-0.749069\pi\)
−0.705036 + 0.709172i \(0.749069\pi\)
\(684\) −0.0540593 −0.00206701
\(685\) 0.104228 0.00398233
\(686\) 19.1106 0.729646
\(687\) 47.9453 1.82923
\(688\) 4.36327 0.166348
\(689\) 26.8653 1.02349
\(690\) 7.72732 0.294174
\(691\) 22.6630 0.862140 0.431070 0.902318i \(-0.358136\pi\)
0.431070 + 0.902318i \(0.358136\pi\)
\(692\) 14.0345 0.533511
\(693\) −0.996522 −0.0378547
\(694\) −5.97388 −0.226765
\(695\) −5.98444 −0.227003
\(696\) 6.20872 0.235341
\(697\) 4.19184 0.158777
\(698\) −2.97557 −0.112627
\(699\) −27.8061 −1.05172
\(700\) 8.21972 0.310676
\(701\) 3.18887 0.120442 0.0602210 0.998185i \(-0.480819\pi\)
0.0602210 + 0.998185i \(0.480819\pi\)
\(702\) 17.7604 0.670322
\(703\) −0.0304391 −0.00114803
\(704\) −1.93819 −0.0730483
\(705\) −11.2973 −0.425480
\(706\) −12.1499 −0.457268
\(707\) −18.0325 −0.678184
\(708\) −21.2373 −0.798146
\(709\) 5.34912 0.200891 0.100445 0.994943i \(-0.467973\pi\)
0.100445 + 0.994943i \(0.467973\pi\)
\(710\) 1.60048 0.0600650
\(711\) −2.07428 −0.0777915
\(712\) −1.25449 −0.0470140
\(713\) −73.9959 −2.77117
\(714\) −4.45109 −0.166578
\(715\) 3.39895 0.127114
\(716\) −8.40342 −0.314051
\(717\) −33.7352 −1.25986
\(718\) 9.90806 0.369765
\(719\) 6.60652 0.246382 0.123191 0.992383i \(-0.460687\pi\)
0.123191 + 0.992383i \(0.460687\pi\)
\(720\) 0.157769 0.00587969
\(721\) −22.6985 −0.845336
\(722\) −18.9664 −0.705856
\(723\) 24.1376 0.897687
\(724\) −24.6257 −0.915209
\(725\) 17.7937 0.660841
\(726\) 11.9135 0.442152
\(727\) 9.17929 0.340441 0.170221 0.985406i \(-0.445552\pi\)
0.170221 + 0.985406i \(0.445552\pi\)
\(728\) −5.71503 −0.211813
\(729\) 29.1064 1.07801
\(730\) −4.17003 −0.154340
\(731\) −6.77151 −0.250453
\(732\) −13.6134 −0.503166
\(733\) 50.6904 1.87229 0.936145 0.351613i \(-0.114367\pi\)
0.936145 + 0.351613i \(0.114367\pi\)
\(734\) 31.9933 1.18089
\(735\) −3.48439 −0.128524
\(736\) 8.78022 0.323643
\(737\) −3.55193 −0.130837
\(738\) 0.796387 0.0293154
\(739\) 15.5996 0.573840 0.286920 0.957955i \(-0.407369\pi\)
0.286920 + 0.957955i \(0.407369\pi\)
\(740\) 0.0888345 0.00326562
\(741\) −0.988314 −0.0363066
\(742\) −14.2945 −0.524766
\(743\) −45.9533 −1.68586 −0.842932 0.538021i \(-0.819172\pi\)
−0.842932 + 0.538021i \(0.819172\pi\)
\(744\) 13.8611 0.508174
\(745\) 6.38239 0.233833
\(746\) 8.48716 0.310737
\(747\) −3.51867 −0.128742
\(748\) 3.00794 0.109981
\(749\) 2.80237 0.102396
\(750\) −8.54884 −0.312160
\(751\) 24.3091 0.887050 0.443525 0.896262i \(-0.353728\pi\)
0.443525 + 0.896262i \(0.353728\pi\)
\(752\) −12.8366 −0.468103
\(753\) −17.6057 −0.641587
\(754\) −12.3716 −0.450549
\(755\) −5.59374 −0.203577
\(756\) −9.44992 −0.343690
\(757\) −25.3950 −0.922996 −0.461498 0.887141i \(-0.652688\pi\)
−0.461498 + 0.887141i \(0.652688\pi\)
\(758\) 3.11422 0.113114
\(759\) 27.9897 1.01596
\(760\) −0.0981081 −0.00355876
\(761\) 32.8269 1.18998 0.594988 0.803735i \(-0.297157\pi\)
0.594988 + 0.803735i \(0.297157\pi\)
\(762\) 0.678877 0.0245931
\(763\) −21.4829 −0.777735
\(764\) −9.53239 −0.344870
\(765\) −0.244846 −0.00885244
\(766\) −6.12086 −0.221155
\(767\) 42.3179 1.52801
\(768\) −1.64474 −0.0593493
\(769\) 25.0222 0.902324 0.451162 0.892442i \(-0.351010\pi\)
0.451162 + 0.892442i \(0.351010\pi\)
\(770\) −1.80851 −0.0651742
\(771\) −36.2910 −1.30699
\(772\) 16.9460 0.609900
\(773\) −7.83331 −0.281744 −0.140872 0.990028i \(-0.544991\pi\)
−0.140872 + 0.990028i \(0.544991\pi\)
\(774\) −1.28649 −0.0462418
\(775\) 39.7249 1.42696
\(776\) 14.8335 0.532493
\(777\) −0.476154 −0.0170819
\(778\) 0.488904 0.0175280
\(779\) −0.495232 −0.0177435
\(780\) 2.88433 0.103276
\(781\) 5.79721 0.207441
\(782\) −13.6263 −0.487276
\(783\) −20.4568 −0.731065
\(784\) −3.95916 −0.141398
\(785\) 8.36529 0.298570
\(786\) −25.2034 −0.898977
\(787\) 0.717460 0.0255747 0.0127874 0.999918i \(-0.495930\pi\)
0.0127874 + 0.999918i \(0.495930\pi\)
\(788\) 22.0413 0.785188
\(789\) −27.5198 −0.979729
\(790\) −3.76445 −0.133933
\(791\) 12.3173 0.437954
\(792\) 0.571465 0.0203061
\(793\) 27.1264 0.963285
\(794\) 20.3480 0.722123
\(795\) 7.21430 0.255865
\(796\) 27.4022 0.971244
\(797\) −23.6900 −0.839144 −0.419572 0.907722i \(-0.637820\pi\)
−0.419572 + 0.907722i \(0.637820\pi\)
\(798\) 0.525861 0.0186153
\(799\) 19.9216 0.704774
\(800\) −4.71368 −0.166654
\(801\) 0.369880 0.0130691
\(802\) −32.4212 −1.14483
\(803\) −15.1045 −0.533028
\(804\) −3.01414 −0.106301
\(805\) 8.19275 0.288757
\(806\) −27.6200 −0.972873
\(807\) 15.9598 0.561812
\(808\) 10.3409 0.363793
\(809\) −15.1652 −0.533180 −0.266590 0.963810i \(-0.585897\pi\)
−0.266590 + 0.963810i \(0.585897\pi\)
\(810\) 4.29599 0.150946
\(811\) −1.82428 −0.0640593 −0.0320296 0.999487i \(-0.510197\pi\)
−0.0320296 + 0.999487i \(0.510197\pi\)
\(812\) 6.58269 0.231007
\(813\) −34.5182 −1.21061
\(814\) 0.321774 0.0112782
\(815\) −1.84569 −0.0646516
\(816\) 2.55252 0.0893561
\(817\) 0.799999 0.0279884
\(818\) −18.4173 −0.643945
\(819\) 1.68504 0.0588802
\(820\) 1.44530 0.0504721
\(821\) 5.00871 0.174805 0.0874026 0.996173i \(-0.472143\pi\)
0.0874026 + 0.996173i \(0.472143\pi\)
\(822\) 0.320370 0.0111742
\(823\) −39.4462 −1.37501 −0.687504 0.726180i \(-0.741294\pi\)
−0.687504 + 0.726180i \(0.741294\pi\)
\(824\) 13.0167 0.453457
\(825\) −15.0263 −0.523149
\(826\) −22.5164 −0.783447
\(827\) 5.90966 0.205499 0.102749 0.994707i \(-0.467236\pi\)
0.102749 + 0.994707i \(0.467236\pi\)
\(828\) −2.58880 −0.0899670
\(829\) −50.6004 −1.75742 −0.878712 0.477353i \(-0.841596\pi\)
−0.878712 + 0.477353i \(0.841596\pi\)
\(830\) −6.38577 −0.221653
\(831\) 49.1150 1.70378
\(832\) 3.27734 0.113621
\(833\) 6.14435 0.212889
\(834\) −18.3947 −0.636956
\(835\) −1.81387 −0.0627715
\(836\) −0.355364 −0.0122905
\(837\) −45.6703 −1.57859
\(838\) −36.7509 −1.26954
\(839\) −25.2327 −0.871128 −0.435564 0.900158i \(-0.643451\pi\)
−0.435564 + 0.900158i \(0.643451\pi\)
\(840\) −1.53469 −0.0529518
\(841\) −14.7501 −0.508624
\(842\) −24.9153 −0.858638
\(843\) −49.9729 −1.72116
\(844\) 14.5885 0.502155
\(845\) 1.20880 0.0415840
\(846\) 3.78480 0.130124
\(847\) 12.6311 0.434009
\(848\) 8.19730 0.281496
\(849\) 41.6220 1.42846
\(850\) 7.31531 0.250913
\(851\) −1.45767 −0.0499683
\(852\) 4.91948 0.168539
\(853\) 0.945622 0.0323775 0.0161887 0.999869i \(-0.494847\pi\)
0.0161887 + 0.999869i \(0.494847\pi\)
\(854\) −14.4334 −0.493899
\(855\) 0.0289267 0.000989271 0
\(856\) −1.60704 −0.0549277
\(857\) 18.3184 0.625743 0.312872 0.949795i \(-0.398709\pi\)
0.312872 + 0.949795i \(0.398709\pi\)
\(858\) 10.4475 0.356673
\(859\) −21.1262 −0.720817 −0.360408 0.932795i \(-0.617363\pi\)
−0.360408 + 0.932795i \(0.617363\pi\)
\(860\) −2.33475 −0.0796142
\(861\) −7.74684 −0.264011
\(862\) 9.14506 0.311482
\(863\) 34.5413 1.17580 0.587899 0.808934i \(-0.299955\pi\)
0.587899 + 0.808934i \(0.299955\pi\)
\(864\) 5.41915 0.184363
\(865\) −7.50973 −0.255338
\(866\) 3.73613 0.126959
\(867\) 23.9992 0.815054
\(868\) 14.6960 0.498815
\(869\) −13.6355 −0.462552
\(870\) −3.32223 −0.112634
\(871\) 6.00605 0.203507
\(872\) 12.3196 0.417194
\(873\) −4.37359 −0.148024
\(874\) 1.60984 0.0544537
\(875\) −9.06376 −0.306411
\(876\) −12.8176 −0.433067
\(877\) 35.2280 1.18957 0.594783 0.803886i \(-0.297238\pi\)
0.594783 + 0.803886i \(0.297238\pi\)
\(878\) 39.0826 1.31897
\(879\) −25.4223 −0.857473
\(880\) 1.03711 0.0349609
\(881\) 23.0409 0.776267 0.388133 0.921603i \(-0.373120\pi\)
0.388133 + 0.921603i \(0.373120\pi\)
\(882\) 1.16734 0.0393062
\(883\) −4.67656 −0.157379 −0.0786894 0.996899i \(-0.525074\pi\)
−0.0786894 + 0.996899i \(0.525074\pi\)
\(884\) −5.08621 −0.171068
\(885\) 11.3639 0.381992
\(886\) 32.9595 1.10730
\(887\) 23.4326 0.786789 0.393395 0.919370i \(-0.371301\pi\)
0.393395 + 0.919370i \(0.371301\pi\)
\(888\) 0.273055 0.00916313
\(889\) 0.719768 0.0241402
\(890\) 0.671266 0.0225009
\(891\) 15.5608 0.521307
\(892\) −1.37324 −0.0459796
\(893\) −2.35357 −0.0787593
\(894\) 19.6179 0.656120
\(895\) 4.49659 0.150305
\(896\) −1.74380 −0.0582563
\(897\) −47.3285 −1.58025
\(898\) −36.3922 −1.21442
\(899\) 31.8133 1.06103
\(900\) 1.38980 0.0463268
\(901\) −12.7217 −0.423820
\(902\) 5.23513 0.174311
\(903\) 12.5143 0.416449
\(904\) −7.06350 −0.234928
\(905\) 13.1770 0.438019
\(906\) −17.1938 −0.571225
\(907\) 34.5699 1.14788 0.573938 0.818899i \(-0.305415\pi\)
0.573938 + 0.818899i \(0.305415\pi\)
\(908\) −24.7682 −0.821962
\(909\) −3.04897 −0.101128
\(910\) 3.05806 0.101374
\(911\) 24.8411 0.823021 0.411510 0.911405i \(-0.365001\pi\)
0.411510 + 0.911405i \(0.365001\pi\)
\(912\) −0.301560 −0.00998565
\(913\) −23.1304 −0.765503
\(914\) −22.8886 −0.757088
\(915\) 7.28440 0.240815
\(916\) −29.1508 −0.963169
\(917\) −26.7215 −0.882421
\(918\) −8.41016 −0.277577
\(919\) 17.9118 0.590857 0.295429 0.955365i \(-0.404538\pi\)
0.295429 + 0.955365i \(0.404538\pi\)
\(920\) −4.69821 −0.154896
\(921\) −24.6934 −0.813677
\(922\) 5.22632 0.172120
\(923\) −9.80266 −0.322659
\(924\) −5.55891 −0.182875
\(925\) 0.782554 0.0257302
\(926\) −33.9321 −1.11508
\(927\) −3.83789 −0.126053
\(928\) −3.77491 −0.123917
\(929\) 58.6445 1.92406 0.962032 0.272937i \(-0.0879950\pi\)
0.962032 + 0.272937i \(0.0879950\pi\)
\(930\) −7.41696 −0.243212
\(931\) −0.725906 −0.0237906
\(932\) 16.9061 0.553779
\(933\) 25.3752 0.830747
\(934\) −2.48794 −0.0814079
\(935\) −1.60952 −0.0526370
\(936\) −0.966305 −0.0315847
\(937\) −58.5693 −1.91338 −0.956688 0.291114i \(-0.905974\pi\)
−0.956688 + 0.291114i \(0.905974\pi\)
\(938\) −3.19569 −0.104343
\(939\) −22.6306 −0.738522
\(940\) 6.86875 0.224034
\(941\) 41.5663 1.35502 0.677512 0.735512i \(-0.263058\pi\)
0.677512 + 0.735512i \(0.263058\pi\)
\(942\) 25.7128 0.837769
\(943\) −23.7157 −0.772290
\(944\) 12.9123 0.420259
\(945\) 5.05656 0.164490
\(946\) −8.45685 −0.274956
\(947\) −39.8619 −1.29534 −0.647668 0.761922i \(-0.724256\pi\)
−0.647668 + 0.761922i \(0.724256\pi\)
\(948\) −11.5710 −0.375808
\(949\) 25.5407 0.829085
\(950\) −0.864246 −0.0280398
\(951\) −15.8668 −0.514517
\(952\) 2.70626 0.0877105
\(953\) −30.5586 −0.989890 −0.494945 0.868924i \(-0.664812\pi\)
−0.494945 + 0.868924i \(0.664812\pi\)
\(954\) −2.41693 −0.0782510
\(955\) 5.10069 0.165055
\(956\) 20.5110 0.663374
\(957\) −12.0337 −0.388994
\(958\) 8.97749 0.290049
\(959\) 0.339666 0.0109684
\(960\) 0.880083 0.0284046
\(961\) 40.0239 1.29109
\(962\) −0.544096 −0.0175424
\(963\) 0.473829 0.0152689
\(964\) −14.6757 −0.472672
\(965\) −9.06765 −0.291898
\(966\) 25.1825 0.810233
\(967\) −30.7685 −0.989449 −0.494725 0.869050i \(-0.664731\pi\)
−0.494725 + 0.869050i \(0.664731\pi\)
\(968\) −7.24342 −0.232812
\(969\) 0.468001 0.0150344
\(970\) −7.93730 −0.254851
\(971\) 18.7378 0.601323 0.300662 0.953731i \(-0.402793\pi\)
0.300662 + 0.953731i \(0.402793\pi\)
\(972\) −3.05263 −0.0979132
\(973\) −19.5026 −0.625226
\(974\) −9.40078 −0.301220
\(975\) 25.4084 0.813720
\(976\) 8.27695 0.264939
\(977\) −37.0943 −1.18675 −0.593375 0.804926i \(-0.702205\pi\)
−0.593375 + 0.804926i \(0.702205\pi\)
\(978\) −5.67318 −0.181408
\(979\) 2.43144 0.0777092
\(980\) 2.11851 0.0676733
\(981\) −3.63237 −0.115973
\(982\) 11.4028 0.363878
\(983\) −31.7506 −1.01269 −0.506344 0.862332i \(-0.669003\pi\)
−0.506344 + 0.862332i \(0.669003\pi\)
\(984\) 4.44250 0.141622
\(985\) −11.7941 −0.375791
\(986\) 5.85840 0.186570
\(987\) −36.8166 −1.17188
\(988\) 0.600895 0.0191170
\(989\) 38.3105 1.21820
\(990\) −0.305786 −0.00971851
\(991\) −42.7285 −1.35731 −0.678657 0.734455i \(-0.737438\pi\)
−0.678657 + 0.734455i \(0.737438\pi\)
\(992\) −8.42757 −0.267576
\(993\) −13.2024 −0.418966
\(994\) 5.21579 0.165435
\(995\) −14.6626 −0.464837
\(996\) −19.6283 −0.621945
\(997\) −11.1318 −0.352547 −0.176274 0.984341i \(-0.556404\pi\)
−0.176274 + 0.984341i \(0.556404\pi\)
\(998\) −8.69850 −0.275346
\(999\) −0.899675 −0.0284644
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 4006.2.a.i.1.10 46
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4006.2.a.i.1.10 46 1.1 even 1 trivial