Properties

Label 4021.2.a.b.1.16
Level $4021$
Weight $2$
Character 4021.1
Self dual yes
Analytic conductor $32.108$
Analytic rank $1$
Dimension $151$
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] = [4021,2,Mod(1,4021)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4021, 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("4021.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4021 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4021.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: \(32.1078466528\)
Analytic rank: \(1\)
Dimension: \(151\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Character \(\chi\) \(=\) 4021.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.37064 q^{2} -0.181119 q^{3} +3.61993 q^{4} +1.77155 q^{5} +0.429369 q^{6} -0.953276 q^{7} -3.84026 q^{8} -2.96720 q^{9} +O(q^{10})\) \(q-2.37064 q^{2} -0.181119 q^{3} +3.61993 q^{4} +1.77155 q^{5} +0.429369 q^{6} -0.953276 q^{7} -3.84026 q^{8} -2.96720 q^{9} -4.19971 q^{10} +3.10092 q^{11} -0.655639 q^{12} -2.76366 q^{13} +2.25987 q^{14} -0.320862 q^{15} +1.86402 q^{16} -1.69260 q^{17} +7.03415 q^{18} -2.33313 q^{19} +6.41289 q^{20} +0.172657 q^{21} -7.35116 q^{22} +2.41355 q^{23} +0.695546 q^{24} -1.86161 q^{25} +6.55164 q^{26} +1.08078 q^{27} -3.45079 q^{28} +6.33499 q^{29} +0.760649 q^{30} -2.67778 q^{31} +3.26161 q^{32} -0.561637 q^{33} +4.01254 q^{34} -1.68878 q^{35} -10.7410 q^{36} +7.83933 q^{37} +5.53101 q^{38} +0.500553 q^{39} -6.80322 q^{40} -1.91690 q^{41} -0.409307 q^{42} +7.68521 q^{43} +11.2251 q^{44} -5.25654 q^{45} -5.72165 q^{46} +3.92709 q^{47} -0.337610 q^{48} -6.09127 q^{49} +4.41320 q^{50} +0.306562 q^{51} -10.0043 q^{52} -8.91148 q^{53} -2.56213 q^{54} +5.49344 q^{55} +3.66083 q^{56} +0.422575 q^{57} -15.0180 q^{58} +8.29463 q^{59} -1.16150 q^{60} -1.58369 q^{61} +6.34806 q^{62} +2.82856 q^{63} -11.4601 q^{64} -4.89597 q^{65} +1.33144 q^{66} -11.6908 q^{67} -6.12708 q^{68} -0.437141 q^{69} +4.00348 q^{70} +0.457924 q^{71} +11.3948 q^{72} +9.82549 q^{73} -18.5842 q^{74} +0.337173 q^{75} -8.44576 q^{76} -2.95603 q^{77} -1.18663 q^{78} +6.39444 q^{79} +3.30221 q^{80} +8.70584 q^{81} +4.54428 q^{82} -4.34921 q^{83} +0.625005 q^{84} -2.99852 q^{85} -18.2189 q^{86} -1.14739 q^{87} -11.9083 q^{88} -5.61886 q^{89} +12.4614 q^{90} +2.63453 q^{91} +8.73687 q^{92} +0.484999 q^{93} -9.30971 q^{94} -4.13326 q^{95} -0.590741 q^{96} -6.65009 q^{97} +14.4402 q^{98} -9.20103 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 151 q - 16 q^{2} - 29 q^{3} + 120 q^{4} - 27 q^{5} - 18 q^{6} - 18 q^{7} - 48 q^{8} + 98 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 151 q - 16 q^{2} - 29 q^{3} + 120 q^{4} - 27 q^{5} - 18 q^{6} - 18 q^{7} - 48 q^{8} + 98 q^{9} - 7 q^{10} - 126 q^{11} - 47 q^{12} - 10 q^{13} - 58 q^{14} - 50 q^{15} + 74 q^{16} - 39 q^{17} - 47 q^{18} - 48 q^{19} - 64 q^{20} - 50 q^{21} - 7 q^{22} - 91 q^{23} - 44 q^{24} + 102 q^{25} - 94 q^{26} - 92 q^{27} - 27 q^{28} - 79 q^{29} - 12 q^{30} - 33 q^{31} - 102 q^{32} - 7 q^{33} - 17 q^{34} - 183 q^{35} - 6 q^{36} - 24 q^{37} - 77 q^{38} - 81 q^{39} - 37 q^{40} - 62 q^{41} + 4 q^{42} - 74 q^{43} - 209 q^{44} - 35 q^{45} - 37 q^{46} - 127 q^{47} - 59 q^{48} + 75 q^{49} - 79 q^{50} - 138 q^{51} - 12 q^{52} - 104 q^{53} - 43 q^{54} - 49 q^{55} - 167 q^{56} - 19 q^{57} - 32 q^{58} - 207 q^{59} - 94 q^{60} - 57 q^{61} - 69 q^{62} - 49 q^{63} - 4 q^{64} - 78 q^{65} - 51 q^{66} - 92 q^{67} - 80 q^{68} - 84 q^{69} + 5 q^{70} - 217 q^{71} - 79 q^{72} + 2 q^{73} - 113 q^{74} - 173 q^{75} - 70 q^{76} - 85 q^{77} - 46 q^{78} - 70 q^{79} - 95 q^{80} - 49 q^{81} - 12 q^{82} - 109 q^{83} - 114 q^{84} - 25 q^{85} - 136 q^{86} - 60 q^{87} - 2 q^{88} - 144 q^{89} - 55 q^{90} - 62 q^{91} - 175 q^{92} - 42 q^{93} - 15 q^{94} - 189 q^{95} - 64 q^{96} - 24 q^{97} - 37 q^{98} - 293 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.37064 −1.67629 −0.838147 0.545444i \(-0.816361\pi\)
−0.838147 + 0.545444i \(0.816361\pi\)
\(3\) −0.181119 −0.104569 −0.0522847 0.998632i \(-0.516650\pi\)
−0.0522847 + 0.998632i \(0.516650\pi\)
\(4\) 3.61993 1.80996
\(5\) 1.77155 0.792262 0.396131 0.918194i \(-0.370353\pi\)
0.396131 + 0.918194i \(0.370353\pi\)
\(6\) 0.429369 0.175289
\(7\) −0.953276 −0.360304 −0.180152 0.983639i \(-0.557659\pi\)
−0.180152 + 0.983639i \(0.557659\pi\)
\(8\) −3.84026 −1.35774
\(9\) −2.96720 −0.989065
\(10\) −4.19971 −1.32806
\(11\) 3.10092 0.934962 0.467481 0.884003i \(-0.345162\pi\)
0.467481 + 0.884003i \(0.345162\pi\)
\(12\) −0.655639 −0.189267
\(13\) −2.76366 −0.766502 −0.383251 0.923644i \(-0.625196\pi\)
−0.383251 + 0.923644i \(0.625196\pi\)
\(14\) 2.25987 0.603976
\(15\) −0.320862 −0.0828463
\(16\) 1.86402 0.466005
\(17\) −1.69260 −0.410515 −0.205258 0.978708i \(-0.565803\pi\)
−0.205258 + 0.978708i \(0.565803\pi\)
\(18\) 7.03415 1.65796
\(19\) −2.33313 −0.535257 −0.267628 0.963522i \(-0.586240\pi\)
−0.267628 + 0.963522i \(0.586240\pi\)
\(20\) 6.41289 1.43397
\(21\) 0.172657 0.0376768
\(22\) −7.35116 −1.56727
\(23\) 2.41355 0.503260 0.251630 0.967824i \(-0.419033\pi\)
0.251630 + 0.967824i \(0.419033\pi\)
\(24\) 0.695546 0.141978
\(25\) −1.86161 −0.372321
\(26\) 6.55164 1.28488
\(27\) 1.08078 0.207995
\(28\) −3.45079 −0.652138
\(29\) 6.33499 1.17638 0.588189 0.808724i \(-0.299841\pi\)
0.588189 + 0.808724i \(0.299841\pi\)
\(30\) 0.760649 0.138875
\(31\) −2.67778 −0.480944 −0.240472 0.970656i \(-0.577302\pi\)
−0.240472 + 0.970656i \(0.577302\pi\)
\(32\) 3.26161 0.576576
\(33\) −0.561637 −0.0977684
\(34\) 4.01254 0.688145
\(35\) −1.68878 −0.285455
\(36\) −10.7410 −1.79017
\(37\) 7.83933 1.28878 0.644389 0.764698i \(-0.277112\pi\)
0.644389 + 0.764698i \(0.277112\pi\)
\(38\) 5.53101 0.897248
\(39\) 0.500553 0.0801526
\(40\) −6.80322 −1.07568
\(41\) −1.91690 −0.299370 −0.149685 0.988734i \(-0.547826\pi\)
−0.149685 + 0.988734i \(0.547826\pi\)
\(42\) −0.409307 −0.0631574
\(43\) 7.68521 1.17198 0.585992 0.810317i \(-0.300705\pi\)
0.585992 + 0.810317i \(0.300705\pi\)
\(44\) 11.2251 1.69225
\(45\) −5.25654 −0.783599
\(46\) −5.72165 −0.843612
\(47\) 3.92709 0.572825 0.286412 0.958106i \(-0.407537\pi\)
0.286412 + 0.958106i \(0.407537\pi\)
\(48\) −0.337610 −0.0487298
\(49\) −6.09127 −0.870181
\(50\) 4.41320 0.624120
\(51\) 0.306562 0.0429273
\(52\) −10.0043 −1.38734
\(53\) −8.91148 −1.22409 −0.612043 0.790824i \(-0.709652\pi\)
−0.612043 + 0.790824i \(0.709652\pi\)
\(54\) −2.56213 −0.348661
\(55\) 5.49344 0.740735
\(56\) 3.66083 0.489199
\(57\) 0.422575 0.0559714
\(58\) −15.0180 −1.97196
\(59\) 8.29463 1.07987 0.539935 0.841707i \(-0.318449\pi\)
0.539935 + 0.841707i \(0.318449\pi\)
\(60\) −1.16150 −0.149949
\(61\) −1.58369 −0.202770 −0.101385 0.994847i \(-0.532327\pi\)
−0.101385 + 0.994847i \(0.532327\pi\)
\(62\) 6.34806 0.806204
\(63\) 2.82856 0.356365
\(64\) −11.4601 −1.43252
\(65\) −4.89597 −0.607270
\(66\) 1.33144 0.163889
\(67\) −11.6908 −1.42826 −0.714130 0.700013i \(-0.753177\pi\)
−0.714130 + 0.700013i \(0.753177\pi\)
\(68\) −6.12708 −0.743018
\(69\) −0.437141 −0.0526256
\(70\) 4.00348 0.478507
\(71\) 0.457924 0.0543456 0.0271728 0.999631i \(-0.491350\pi\)
0.0271728 + 0.999631i \(0.491350\pi\)
\(72\) 11.3948 1.34289
\(73\) 9.82549 1.14999 0.574994 0.818158i \(-0.305005\pi\)
0.574994 + 0.818158i \(0.305005\pi\)
\(74\) −18.5842 −2.16037
\(75\) 0.337173 0.0389334
\(76\) −8.44576 −0.968795
\(77\) −2.95603 −0.336871
\(78\) −1.18663 −0.134359
\(79\) 6.39444 0.719431 0.359715 0.933062i \(-0.382874\pi\)
0.359715 + 0.933062i \(0.382874\pi\)
\(80\) 3.30221 0.369198
\(81\) 8.70584 0.967315
\(82\) 4.54428 0.501832
\(83\) −4.34921 −0.477388 −0.238694 0.971095i \(-0.576719\pi\)
−0.238694 + 0.971095i \(0.576719\pi\)
\(84\) 0.625005 0.0681936
\(85\) −2.99852 −0.325236
\(86\) −18.2189 −1.96459
\(87\) −1.14739 −0.123013
\(88\) −11.9083 −1.26943
\(89\) −5.61886 −0.595598 −0.297799 0.954629i \(-0.596253\pi\)
−0.297799 + 0.954629i \(0.596253\pi\)
\(90\) 12.4614 1.31354
\(91\) 2.63453 0.276174
\(92\) 8.73687 0.910882
\(93\) 0.484999 0.0502920
\(94\) −9.30971 −0.960223
\(95\) −4.13326 −0.424063
\(96\) −0.590741 −0.0602922
\(97\) −6.65009 −0.675214 −0.337607 0.941287i \(-0.609618\pi\)
−0.337607 + 0.941287i \(0.609618\pi\)
\(98\) 14.4402 1.45868
\(99\) −9.20103 −0.924739
\(100\) −6.73888 −0.673888
\(101\) −19.2858 −1.91901 −0.959505 0.281691i \(-0.909105\pi\)
−0.959505 + 0.281691i \(0.909105\pi\)
\(102\) −0.726749 −0.0719588
\(103\) 12.2732 1.20931 0.604656 0.796487i \(-0.293311\pi\)
0.604656 + 0.796487i \(0.293311\pi\)
\(104\) 10.6132 1.04071
\(105\) 0.305870 0.0298499
\(106\) 21.1259 2.05193
\(107\) 0.337182 0.0325967 0.0162983 0.999867i \(-0.494812\pi\)
0.0162983 + 0.999867i \(0.494812\pi\)
\(108\) 3.91233 0.376464
\(109\) −13.7850 −1.32036 −0.660182 0.751106i \(-0.729521\pi\)
−0.660182 + 0.751106i \(0.729521\pi\)
\(110\) −13.0230 −1.24169
\(111\) −1.41985 −0.134767
\(112\) −1.77692 −0.167904
\(113\) −4.75007 −0.446849 −0.223424 0.974721i \(-0.571724\pi\)
−0.223424 + 0.974721i \(0.571724\pi\)
\(114\) −1.00177 −0.0938246
\(115\) 4.27573 0.398714
\(116\) 22.9322 2.12920
\(117\) 8.20032 0.758120
\(118\) −19.6636 −1.81018
\(119\) 1.61351 0.147910
\(120\) 1.23220 0.112484
\(121\) −1.38430 −0.125846
\(122\) 3.75435 0.339903
\(123\) 0.347188 0.0313049
\(124\) −9.69338 −0.870492
\(125\) −12.1557 −1.08724
\(126\) −6.70548 −0.597372
\(127\) −12.8518 −1.14041 −0.570206 0.821502i \(-0.693136\pi\)
−0.570206 + 0.821502i \(0.693136\pi\)
\(128\) 20.6446 1.82474
\(129\) −1.39194 −0.122554
\(130\) 11.6066 1.01796
\(131\) −19.5908 −1.71165 −0.855827 0.517262i \(-0.826951\pi\)
−0.855827 + 0.517262i \(0.826951\pi\)
\(132\) −2.03308 −0.176957
\(133\) 2.22412 0.192855
\(134\) 27.7147 2.39418
\(135\) 1.91465 0.164787
\(136\) 6.50002 0.557372
\(137\) −8.10114 −0.692127 −0.346064 0.938211i \(-0.612482\pi\)
−0.346064 + 0.938211i \(0.612482\pi\)
\(138\) 1.03630 0.0882160
\(139\) 7.33058 0.621771 0.310886 0.950447i \(-0.399374\pi\)
0.310886 + 0.950447i \(0.399374\pi\)
\(140\) −6.11325 −0.516664
\(141\) −0.711272 −0.0598999
\(142\) −1.08557 −0.0910993
\(143\) −8.56989 −0.716650
\(144\) −5.53091 −0.460909
\(145\) 11.2228 0.931999
\(146\) −23.2927 −1.92772
\(147\) 1.10325 0.0909942
\(148\) 28.3778 2.33264
\(149\) −5.17227 −0.423728 −0.211864 0.977299i \(-0.567953\pi\)
−0.211864 + 0.977299i \(0.567953\pi\)
\(150\) −0.799316 −0.0652639
\(151\) −9.32418 −0.758791 −0.379396 0.925235i \(-0.623868\pi\)
−0.379396 + 0.925235i \(0.623868\pi\)
\(152\) 8.95983 0.726738
\(153\) 5.02227 0.406026
\(154\) 7.00768 0.564695
\(155\) −4.74383 −0.381034
\(156\) 1.81196 0.145073
\(157\) 22.9694 1.83316 0.916580 0.399851i \(-0.130938\pi\)
0.916580 + 0.399851i \(0.130938\pi\)
\(158\) −15.1589 −1.20598
\(159\) 1.61404 0.128002
\(160\) 5.77811 0.456799
\(161\) −2.30078 −0.181327
\(162\) −20.6384 −1.62151
\(163\) 1.05812 0.0828786 0.0414393 0.999141i \(-0.486806\pi\)
0.0414393 + 0.999141i \(0.486806\pi\)
\(164\) −6.93905 −0.541849
\(165\) −0.994968 −0.0774582
\(166\) 10.3104 0.800243
\(167\) −16.9648 −1.31277 −0.656387 0.754424i \(-0.727916\pi\)
−0.656387 + 0.754424i \(0.727916\pi\)
\(168\) −0.663047 −0.0511552
\(169\) −5.36218 −0.412475
\(170\) 7.10842 0.545191
\(171\) 6.92285 0.529404
\(172\) 27.8199 2.12125
\(173\) 3.64897 0.277426 0.138713 0.990333i \(-0.455703\pi\)
0.138713 + 0.990333i \(0.455703\pi\)
\(174\) 2.72005 0.206206
\(175\) 1.77462 0.134149
\(176\) 5.78017 0.435697
\(177\) −1.50232 −0.112921
\(178\) 13.3203 0.998398
\(179\) −3.69440 −0.276133 −0.138066 0.990423i \(-0.544089\pi\)
−0.138066 + 0.990423i \(0.544089\pi\)
\(180\) −19.0283 −1.41828
\(181\) −0.875066 −0.0650431 −0.0325216 0.999471i \(-0.510354\pi\)
−0.0325216 + 0.999471i \(0.510354\pi\)
\(182\) −6.24552 −0.462949
\(183\) 0.286836 0.0212036
\(184\) −9.26866 −0.683295
\(185\) 13.8878 1.02105
\(186\) −1.14976 −0.0843043
\(187\) −5.24861 −0.383816
\(188\) 14.2158 1.03679
\(189\) −1.03028 −0.0749416
\(190\) 9.79846 0.710855
\(191\) −2.00117 −0.144799 −0.0723996 0.997376i \(-0.523066\pi\)
−0.0723996 + 0.997376i \(0.523066\pi\)
\(192\) 2.07565 0.149797
\(193\) 24.7896 1.78439 0.892197 0.451646i \(-0.149163\pi\)
0.892197 + 0.451646i \(0.149163\pi\)
\(194\) 15.7650 1.13186
\(195\) 0.886755 0.0635018
\(196\) −22.0499 −1.57500
\(197\) 17.3859 1.23869 0.619346 0.785118i \(-0.287398\pi\)
0.619346 + 0.785118i \(0.287398\pi\)
\(198\) 21.8123 1.55013
\(199\) −12.9673 −0.919227 −0.459613 0.888119i \(-0.652012\pi\)
−0.459613 + 0.888119i \(0.652012\pi\)
\(200\) 7.14906 0.505515
\(201\) 2.11743 0.149352
\(202\) 45.7197 3.21683
\(203\) −6.03899 −0.423854
\(204\) 1.10973 0.0776969
\(205\) −3.39589 −0.237179
\(206\) −29.0952 −2.02716
\(207\) −7.16147 −0.497757
\(208\) −5.15152 −0.357194
\(209\) −7.23485 −0.500445
\(210\) −0.725108 −0.0500372
\(211\) 19.4139 1.33651 0.668255 0.743933i \(-0.267042\pi\)
0.668255 + 0.743933i \(0.267042\pi\)
\(212\) −32.2589 −2.21555
\(213\) −0.0829390 −0.00568289
\(214\) −0.799338 −0.0546416
\(215\) 13.6147 0.928518
\(216\) −4.15046 −0.282403
\(217\) 2.55267 0.173286
\(218\) 32.6793 2.21332
\(219\) −1.77959 −0.120253
\(220\) 19.8858 1.34070
\(221\) 4.67777 0.314661
\(222\) 3.36596 0.225909
\(223\) −20.2865 −1.35849 −0.679243 0.733913i \(-0.737692\pi\)
−0.679243 + 0.733913i \(0.737692\pi\)
\(224\) −3.10921 −0.207743
\(225\) 5.52375 0.368250
\(226\) 11.2607 0.749050
\(227\) 2.27392 0.150926 0.0754628 0.997149i \(-0.475957\pi\)
0.0754628 + 0.997149i \(0.475957\pi\)
\(228\) 1.52969 0.101306
\(229\) 4.53366 0.299593 0.149796 0.988717i \(-0.452138\pi\)
0.149796 + 0.988717i \(0.452138\pi\)
\(230\) −10.1362 −0.668361
\(231\) 0.535395 0.0352264
\(232\) −24.3280 −1.59721
\(233\) 24.5448 1.60798 0.803992 0.594640i \(-0.202705\pi\)
0.803992 + 0.594640i \(0.202705\pi\)
\(234\) −19.4400 −1.27083
\(235\) 6.95704 0.453827
\(236\) 30.0260 1.95452
\(237\) −1.15816 −0.0752304
\(238\) −3.82506 −0.247942
\(239\) 6.47840 0.419053 0.209526 0.977803i \(-0.432808\pi\)
0.209526 + 0.977803i \(0.432808\pi\)
\(240\) −0.598094 −0.0386068
\(241\) −7.62545 −0.491198 −0.245599 0.969371i \(-0.578985\pi\)
−0.245599 + 0.969371i \(0.578985\pi\)
\(242\) 3.28168 0.210955
\(243\) −4.81912 −0.309147
\(244\) −5.73283 −0.367007
\(245\) −10.7910 −0.689411
\(246\) −0.823058 −0.0524763
\(247\) 6.44798 0.410275
\(248\) 10.2834 0.652996
\(249\) 0.787727 0.0499202
\(250\) 28.8167 1.82253
\(251\) −1.23544 −0.0779805 −0.0389903 0.999240i \(-0.512414\pi\)
−0.0389903 + 0.999240i \(0.512414\pi\)
\(252\) 10.2392 0.645007
\(253\) 7.48422 0.470529
\(254\) 30.4669 1.91167
\(255\) 0.543091 0.0340097
\(256\) −26.0207 −1.62629
\(257\) −0.607426 −0.0378902 −0.0189451 0.999821i \(-0.506031\pi\)
−0.0189451 + 0.999821i \(0.506031\pi\)
\(258\) 3.29979 0.205436
\(259\) −7.47304 −0.464352
\(260\) −17.7230 −1.09914
\(261\) −18.7971 −1.16351
\(262\) 46.4426 2.86924
\(263\) −25.7861 −1.59004 −0.795019 0.606584i \(-0.792539\pi\)
−0.795019 + 0.606584i \(0.792539\pi\)
\(264\) 2.15683 0.132744
\(265\) −15.7871 −0.969797
\(266\) −5.27258 −0.323282
\(267\) 1.01769 0.0622813
\(268\) −42.3199 −2.58510
\(269\) 22.8107 1.39080 0.695398 0.718625i \(-0.255228\pi\)
0.695398 + 0.718625i \(0.255228\pi\)
\(270\) −4.53894 −0.276231
\(271\) 4.17703 0.253736 0.126868 0.991920i \(-0.459507\pi\)
0.126868 + 0.991920i \(0.459507\pi\)
\(272\) −3.15504 −0.191302
\(273\) −0.477165 −0.0288793
\(274\) 19.2049 1.16021
\(275\) −5.77269 −0.348106
\(276\) −1.58242 −0.0952504
\(277\) 9.06109 0.544428 0.272214 0.962237i \(-0.412244\pi\)
0.272214 + 0.962237i \(0.412244\pi\)
\(278\) −17.3781 −1.04227
\(279\) 7.94551 0.475685
\(280\) 6.48535 0.387574
\(281\) −30.0583 −1.79313 −0.896565 0.442912i \(-0.853945\pi\)
−0.896565 + 0.442912i \(0.853945\pi\)
\(282\) 1.68617 0.100410
\(283\) −0.613366 −0.0364608 −0.0182304 0.999834i \(-0.505803\pi\)
−0.0182304 + 0.999834i \(0.505803\pi\)
\(284\) 1.65765 0.0983636
\(285\) 0.748614 0.0443440
\(286\) 20.3161 1.20132
\(287\) 1.82734 0.107864
\(288\) −9.67783 −0.570272
\(289\) −14.1351 −0.831477
\(290\) −26.6051 −1.56230
\(291\) 1.20446 0.0706067
\(292\) 35.5676 2.08143
\(293\) −10.8221 −0.632236 −0.316118 0.948720i \(-0.602379\pi\)
−0.316118 + 0.948720i \(0.602379\pi\)
\(294\) −2.61540 −0.152533
\(295\) 14.6944 0.855539
\(296\) −30.1051 −1.74982
\(297\) 3.35140 0.194468
\(298\) 12.2616 0.710294
\(299\) −6.67023 −0.385750
\(300\) 1.22054 0.0704680
\(301\) −7.32613 −0.422271
\(302\) 22.1043 1.27196
\(303\) 3.49304 0.200670
\(304\) −4.34900 −0.249432
\(305\) −2.80558 −0.160647
\(306\) −11.9060 −0.680620
\(307\) −12.8336 −0.732455 −0.366227 0.930525i \(-0.619351\pi\)
−0.366227 + 0.930525i \(0.619351\pi\)
\(308\) −10.7006 −0.609724
\(309\) −2.22291 −0.126457
\(310\) 11.2459 0.638725
\(311\) −4.87295 −0.276320 −0.138160 0.990410i \(-0.544119\pi\)
−0.138160 + 0.990410i \(0.544119\pi\)
\(312\) −1.92225 −0.108826
\(313\) −27.4070 −1.54914 −0.774569 0.632489i \(-0.782033\pi\)
−0.774569 + 0.632489i \(0.782033\pi\)
\(314\) −54.4522 −3.07292
\(315\) 5.01093 0.282334
\(316\) 23.1474 1.30214
\(317\) −5.47432 −0.307469 −0.153734 0.988112i \(-0.549130\pi\)
−0.153734 + 0.988112i \(0.549130\pi\)
\(318\) −3.82631 −0.214569
\(319\) 19.6443 1.09987
\(320\) −20.3022 −1.13493
\(321\) −0.0610703 −0.00340861
\(322\) 5.45431 0.303957
\(323\) 3.94905 0.219731
\(324\) 31.5145 1.75081
\(325\) 5.14485 0.285385
\(326\) −2.50843 −0.138929
\(327\) 2.49673 0.138070
\(328\) 7.36141 0.406466
\(329\) −3.74360 −0.206391
\(330\) 2.35871 0.129843
\(331\) 19.9632 1.09728 0.548638 0.836060i \(-0.315146\pi\)
0.548638 + 0.836060i \(0.315146\pi\)
\(332\) −15.7438 −0.864055
\(333\) −23.2608 −1.27469
\(334\) 40.2174 2.20060
\(335\) −20.7109 −1.13156
\(336\) 0.321836 0.0175576
\(337\) −0.257107 −0.0140055 −0.00700276 0.999975i \(-0.502229\pi\)
−0.00700276 + 0.999975i \(0.502229\pi\)
\(338\) 12.7118 0.691430
\(339\) 0.860330 0.0467267
\(340\) −10.8544 −0.588665
\(341\) −8.30359 −0.449665
\(342\) −16.4116 −0.887437
\(343\) 12.4796 0.673834
\(344\) −29.5132 −1.59125
\(345\) −0.774417 −0.0416932
\(346\) −8.65039 −0.465048
\(347\) −9.47879 −0.508848 −0.254424 0.967093i \(-0.581886\pi\)
−0.254424 + 0.967093i \(0.581886\pi\)
\(348\) −4.15347 −0.222649
\(349\) −31.9053 −1.70785 −0.853927 0.520394i \(-0.825785\pi\)
−0.853927 + 0.520394i \(0.825785\pi\)
\(350\) −4.20699 −0.224873
\(351\) −2.98690 −0.159429
\(352\) 10.1140 0.539077
\(353\) 24.9653 1.32877 0.664384 0.747391i \(-0.268694\pi\)
0.664384 + 0.747391i \(0.268694\pi\)
\(354\) 3.56146 0.189289
\(355\) 0.811236 0.0430560
\(356\) −20.3399 −1.07801
\(357\) −0.292239 −0.0154669
\(358\) 8.75809 0.462880
\(359\) −30.5887 −1.61441 −0.807204 0.590273i \(-0.799020\pi\)
−0.807204 + 0.590273i \(0.799020\pi\)
\(360\) 20.1865 1.06392
\(361\) −13.5565 −0.713500
\(362\) 2.07447 0.109031
\(363\) 0.250724 0.0131596
\(364\) 9.53681 0.499865
\(365\) 17.4064 0.911091
\(366\) −0.679986 −0.0355434
\(367\) −19.9193 −1.03978 −0.519890 0.854233i \(-0.674027\pi\)
−0.519890 + 0.854233i \(0.674027\pi\)
\(368\) 4.49890 0.234522
\(369\) 5.68782 0.296096
\(370\) −32.9229 −1.71158
\(371\) 8.49510 0.441044
\(372\) 1.75566 0.0910267
\(373\) −34.7446 −1.79901 −0.899503 0.436914i \(-0.856072\pi\)
−0.899503 + 0.436914i \(0.856072\pi\)
\(374\) 12.4426 0.643389
\(375\) 2.20163 0.113692
\(376\) −15.0811 −0.777746
\(377\) −17.5078 −0.901695
\(378\) 2.44241 0.125624
\(379\) 16.6281 0.854128 0.427064 0.904222i \(-0.359548\pi\)
0.427064 + 0.904222i \(0.359548\pi\)
\(380\) −14.9621 −0.767539
\(381\) 2.32771 0.119252
\(382\) 4.74404 0.242726
\(383\) −34.9864 −1.78772 −0.893860 0.448345i \(-0.852014\pi\)
−0.893860 + 0.448345i \(0.852014\pi\)
\(384\) −3.73914 −0.190812
\(385\) −5.23676 −0.266890
\(386\) −58.7672 −2.99117
\(387\) −22.8035 −1.15917
\(388\) −24.0728 −1.22211
\(389\) 15.8376 0.803000 0.401500 0.915859i \(-0.368489\pi\)
0.401500 + 0.915859i \(0.368489\pi\)
\(390\) −2.10218 −0.106448
\(391\) −4.08517 −0.206596
\(392\) 23.3921 1.18148
\(393\) 3.54827 0.178987
\(394\) −41.2156 −2.07641
\(395\) 11.3281 0.569977
\(396\) −33.3071 −1.67374
\(397\) −2.34761 −0.117823 −0.0589116 0.998263i \(-0.518763\pi\)
−0.0589116 + 0.998263i \(0.518763\pi\)
\(398\) 30.7408 1.54090
\(399\) −0.402831 −0.0201668
\(400\) −3.47007 −0.173504
\(401\) −25.4144 −1.26913 −0.634566 0.772868i \(-0.718821\pi\)
−0.634566 + 0.772868i \(0.718821\pi\)
\(402\) −5.01967 −0.250358
\(403\) 7.40049 0.368645
\(404\) −69.8133 −3.47334
\(405\) 15.4228 0.766367
\(406\) 14.3163 0.710504
\(407\) 24.3091 1.20496
\(408\) −1.17728 −0.0582841
\(409\) 13.4957 0.667319 0.333659 0.942694i \(-0.391717\pi\)
0.333659 + 0.942694i \(0.391717\pi\)
\(410\) 8.05043 0.397582
\(411\) 1.46727 0.0723753
\(412\) 44.4280 2.18881
\(413\) −7.90707 −0.389082
\(414\) 16.9773 0.834387
\(415\) −7.70485 −0.378216
\(416\) −9.01398 −0.441947
\(417\) −1.32771 −0.0650182
\(418\) 17.1512 0.838893
\(419\) −5.67009 −0.277002 −0.138501 0.990362i \(-0.544228\pi\)
−0.138501 + 0.990362i \(0.544228\pi\)
\(420\) 1.10723 0.0540272
\(421\) −33.8168 −1.64813 −0.824064 0.566496i \(-0.808299\pi\)
−0.824064 + 0.566496i \(0.808299\pi\)
\(422\) −46.0234 −2.24038
\(423\) −11.6524 −0.566561
\(424\) 34.2224 1.66199
\(425\) 3.15095 0.152844
\(426\) 0.196618 0.00952619
\(427\) 1.50969 0.0730590
\(428\) 1.22058 0.0589988
\(429\) 1.55217 0.0749396
\(430\) −32.2756 −1.55647
\(431\) −20.9507 −1.00916 −0.504581 0.863364i \(-0.668353\pi\)
−0.504581 + 0.863364i \(0.668353\pi\)
\(432\) 2.01459 0.0969268
\(433\) 27.2153 1.30789 0.653943 0.756544i \(-0.273114\pi\)
0.653943 + 0.756544i \(0.273114\pi\)
\(434\) −6.05145 −0.290479
\(435\) −2.03266 −0.0974585
\(436\) −49.9007 −2.38981
\(437\) −5.63112 −0.269373
\(438\) 4.21876 0.201580
\(439\) −10.3784 −0.495333 −0.247667 0.968845i \(-0.579664\pi\)
−0.247667 + 0.968845i \(0.579664\pi\)
\(440\) −21.0962 −1.00572
\(441\) 18.0740 0.860666
\(442\) −11.0893 −0.527464
\(443\) −31.8810 −1.51471 −0.757356 0.653002i \(-0.773509\pi\)
−0.757356 + 0.653002i \(0.773509\pi\)
\(444\) −5.13977 −0.243923
\(445\) −9.95410 −0.471870
\(446\) 48.0920 2.27722
\(447\) 0.936798 0.0443090
\(448\) 10.9247 0.516142
\(449\) −26.7452 −1.26218 −0.631092 0.775708i \(-0.717393\pi\)
−0.631092 + 0.775708i \(0.717393\pi\)
\(450\) −13.0948 −0.617296
\(451\) −5.94416 −0.279899
\(452\) −17.1949 −0.808780
\(453\) 1.68879 0.0793463
\(454\) −5.39065 −0.252996
\(455\) 4.66721 0.218802
\(456\) −1.62280 −0.0759945
\(457\) 21.0590 0.985101 0.492550 0.870284i \(-0.336065\pi\)
0.492550 + 0.870284i \(0.336065\pi\)
\(458\) −10.7477 −0.502206
\(459\) −1.82932 −0.0853852
\(460\) 15.4778 0.721657
\(461\) 16.3333 0.760719 0.380359 0.924839i \(-0.375800\pi\)
0.380359 + 0.924839i \(0.375800\pi\)
\(462\) −1.26923 −0.0590498
\(463\) −23.1742 −1.07700 −0.538498 0.842627i \(-0.681008\pi\)
−0.538498 + 0.842627i \(0.681008\pi\)
\(464\) 11.8085 0.548198
\(465\) 0.859200 0.0398445
\(466\) −58.1869 −2.69546
\(467\) −22.9556 −1.06226 −0.531130 0.847290i \(-0.678233\pi\)
−0.531130 + 0.847290i \(0.678233\pi\)
\(468\) 29.6846 1.37217
\(469\) 11.1446 0.514608
\(470\) −16.4926 −0.760748
\(471\) −4.16021 −0.191692
\(472\) −31.8536 −1.46618
\(473\) 23.8312 1.09576
\(474\) 2.74557 0.126108
\(475\) 4.34337 0.199287
\(476\) 5.84080 0.267713
\(477\) 26.4421 1.21070
\(478\) −15.3579 −0.702456
\(479\) −0.0771714 −0.00352605 −0.00176303 0.999998i \(-0.500561\pi\)
−0.00176303 + 0.999998i \(0.500561\pi\)
\(480\) −1.04653 −0.0477672
\(481\) −21.6652 −0.987850
\(482\) 18.0772 0.823393
\(483\) 0.416716 0.0189612
\(484\) −5.01108 −0.227776
\(485\) −11.7810 −0.534947
\(486\) 11.4244 0.518221
\(487\) 30.5714 1.38532 0.692661 0.721263i \(-0.256438\pi\)
0.692661 + 0.721263i \(0.256438\pi\)
\(488\) 6.08177 0.275309
\(489\) −0.191647 −0.00866656
\(490\) 25.5815 1.15566
\(491\) 23.3870 1.05544 0.527719 0.849419i \(-0.323047\pi\)
0.527719 + 0.849419i \(0.323047\pi\)
\(492\) 1.25680 0.0566608
\(493\) −10.7226 −0.482921
\(494\) −15.2858 −0.687742
\(495\) −16.3001 −0.732635
\(496\) −4.99144 −0.224122
\(497\) −0.436528 −0.0195810
\(498\) −1.86742 −0.0836809
\(499\) −35.8812 −1.60626 −0.803132 0.595802i \(-0.796834\pi\)
−0.803132 + 0.595802i \(0.796834\pi\)
\(500\) −44.0027 −1.96786
\(501\) 3.07265 0.137276
\(502\) 2.92879 0.130718
\(503\) −9.34711 −0.416767 −0.208384 0.978047i \(-0.566820\pi\)
−0.208384 + 0.978047i \(0.566820\pi\)
\(504\) −10.8624 −0.483850
\(505\) −34.1658 −1.52036
\(506\) −17.7424 −0.788745
\(507\) 0.971194 0.0431323
\(508\) −46.5225 −2.06410
\(509\) 1.20262 0.0533052 0.0266526 0.999645i \(-0.491515\pi\)
0.0266526 + 0.999645i \(0.491515\pi\)
\(510\) −1.28747 −0.0570102
\(511\) −9.36641 −0.414345
\(512\) 20.3963 0.901400
\(513\) −2.52159 −0.111331
\(514\) 1.43999 0.0635152
\(515\) 21.7425 0.958091
\(516\) −5.03873 −0.221818
\(517\) 12.1776 0.535570
\(518\) 17.7159 0.778391
\(519\) −0.660899 −0.0290103
\(520\) 18.8018 0.824513
\(521\) −0.253487 −0.0111055 −0.00555273 0.999985i \(-0.501767\pi\)
−0.00555273 + 0.999985i \(0.501767\pi\)
\(522\) 44.5612 1.95039
\(523\) 22.7182 0.993399 0.496700 0.867923i \(-0.334545\pi\)
0.496700 + 0.867923i \(0.334545\pi\)
\(524\) −70.9172 −3.09803
\(525\) −0.321419 −0.0140279
\(526\) 61.1295 2.66537
\(527\) 4.53241 0.197435
\(528\) −1.04690 −0.0455605
\(529\) −17.1748 −0.746729
\(530\) 37.4256 1.62567
\(531\) −24.6118 −1.06806
\(532\) 8.05114 0.349061
\(533\) 5.29767 0.229468
\(534\) −2.41256 −0.104402
\(535\) 0.597336 0.0258251
\(536\) 44.8958 1.93920
\(537\) 0.669128 0.0288750
\(538\) −54.0760 −2.33138
\(539\) −18.8885 −0.813586
\(540\) 6.93089 0.298258
\(541\) −16.7625 −0.720677 −0.360339 0.932822i \(-0.617339\pi\)
−0.360339 + 0.932822i \(0.617339\pi\)
\(542\) −9.90222 −0.425337
\(543\) 0.158491 0.00680152
\(544\) −5.52059 −0.236693
\(545\) −24.4208 −1.04607
\(546\) 1.13119 0.0484103
\(547\) 13.2498 0.566522 0.283261 0.959043i \(-0.408584\pi\)
0.283261 + 0.959043i \(0.408584\pi\)
\(548\) −29.3255 −1.25273
\(549\) 4.69911 0.200553
\(550\) 13.6850 0.583529
\(551\) −14.7803 −0.629664
\(552\) 1.67874 0.0714517
\(553\) −6.09566 −0.259214
\(554\) −21.4806 −0.912622
\(555\) −2.51535 −0.106770
\(556\) 26.5362 1.12538
\(557\) 1.88308 0.0797888 0.0398944 0.999204i \(-0.487298\pi\)
0.0398944 + 0.999204i \(0.487298\pi\)
\(558\) −18.8359 −0.797389
\(559\) −21.2393 −0.898328
\(560\) −3.14791 −0.133024
\(561\) 0.950625 0.0401354
\(562\) 71.2574 3.00581
\(563\) 37.2453 1.56970 0.784851 0.619684i \(-0.212739\pi\)
0.784851 + 0.619684i \(0.212739\pi\)
\(564\) −2.57475 −0.108417
\(565\) −8.41499 −0.354021
\(566\) 1.45407 0.0611190
\(567\) −8.29907 −0.348528
\(568\) −1.75855 −0.0737871
\(569\) −17.6538 −0.740085 −0.370043 0.929015i \(-0.620657\pi\)
−0.370043 + 0.929015i \(0.620657\pi\)
\(570\) −1.77469 −0.0743337
\(571\) 40.7896 1.70699 0.853495 0.521101i \(-0.174479\pi\)
0.853495 + 0.521101i \(0.174479\pi\)
\(572\) −31.0224 −1.29711
\(573\) 0.362450 0.0151416
\(574\) −4.33196 −0.180812
\(575\) −4.49308 −0.187374
\(576\) 34.0045 1.41685
\(577\) 28.5791 1.18976 0.594881 0.803814i \(-0.297199\pi\)
0.594881 + 0.803814i \(0.297199\pi\)
\(578\) 33.5092 1.39380
\(579\) −4.48988 −0.186593
\(580\) 40.6255 1.68688
\(581\) 4.14600 0.172005
\(582\) −2.85534 −0.118358
\(583\) −27.6338 −1.14447
\(584\) −37.7325 −1.56138
\(585\) 14.5273 0.600630
\(586\) 25.6554 1.05981
\(587\) −28.6256 −1.18151 −0.590753 0.806852i \(-0.701169\pi\)
−0.590753 + 0.806852i \(0.701169\pi\)
\(588\) 3.99367 0.164696
\(589\) 6.24762 0.257429
\(590\) −34.8350 −1.43414
\(591\) −3.14892 −0.129529
\(592\) 14.6127 0.600577
\(593\) 11.3825 0.467423 0.233712 0.972306i \(-0.424913\pi\)
0.233712 + 0.972306i \(0.424913\pi\)
\(594\) −7.94495 −0.325985
\(595\) 2.85842 0.117184
\(596\) −18.7232 −0.766933
\(597\) 2.34863 0.0961230
\(598\) 15.8127 0.646630
\(599\) −25.6843 −1.04943 −0.524717 0.851277i \(-0.675829\pi\)
−0.524717 + 0.851277i \(0.675829\pi\)
\(600\) −1.29483 −0.0528613
\(601\) −4.70524 −0.191931 −0.0959654 0.995385i \(-0.530594\pi\)
−0.0959654 + 0.995385i \(0.530594\pi\)
\(602\) 17.3676 0.707851
\(603\) 34.6889 1.41264
\(604\) −33.7529 −1.37338
\(605\) −2.45237 −0.0997028
\(606\) −8.28073 −0.336382
\(607\) 17.6769 0.717481 0.358741 0.933437i \(-0.383206\pi\)
0.358741 + 0.933437i \(0.383206\pi\)
\(608\) −7.60975 −0.308616
\(609\) 1.09378 0.0443221
\(610\) 6.65102 0.269292
\(611\) −10.8531 −0.439071
\(612\) 18.1803 0.734893
\(613\) 45.3400 1.83127 0.915633 0.402015i \(-0.131690\pi\)
0.915633 + 0.402015i \(0.131690\pi\)
\(614\) 30.4239 1.22781
\(615\) 0.615062 0.0248017
\(616\) 11.3519 0.457382
\(617\) −29.9150 −1.20433 −0.602166 0.798371i \(-0.705696\pi\)
−0.602166 + 0.798371i \(0.705696\pi\)
\(618\) 5.26971 0.211979
\(619\) 44.0915 1.77219 0.886093 0.463508i \(-0.153410\pi\)
0.886093 + 0.463508i \(0.153410\pi\)
\(620\) −17.1723 −0.689657
\(621\) 2.60850 0.104676
\(622\) 11.5520 0.463193
\(623\) 5.35633 0.214597
\(624\) 0.933040 0.0373515
\(625\) −12.2264 −0.489056
\(626\) 64.9722 2.59681
\(627\) 1.31037 0.0523312
\(628\) 83.1477 3.31795
\(629\) −13.2688 −0.529063
\(630\) −11.8791 −0.473275
\(631\) −26.5892 −1.05850 −0.529249 0.848466i \(-0.677526\pi\)
−0.529249 + 0.848466i \(0.677526\pi\)
\(632\) −24.5563 −0.976798
\(633\) −3.51624 −0.139758
\(634\) 12.9776 0.515408
\(635\) −22.7676 −0.903504
\(636\) 5.84272 0.231679
\(637\) 16.8342 0.666995
\(638\) −46.5695 −1.84370
\(639\) −1.35875 −0.0537514
\(640\) 36.5730 1.44567
\(641\) −9.93314 −0.392336 −0.196168 0.980570i \(-0.562850\pi\)
−0.196168 + 0.980570i \(0.562850\pi\)
\(642\) 0.144776 0.00571384
\(643\) −0.926771 −0.0365483 −0.0182741 0.999833i \(-0.505817\pi\)
−0.0182741 + 0.999833i \(0.505817\pi\)
\(644\) −8.32865 −0.328195
\(645\) −2.46590 −0.0970945
\(646\) −9.36177 −0.368334
\(647\) −2.57708 −0.101315 −0.0506577 0.998716i \(-0.516132\pi\)
−0.0506577 + 0.998716i \(0.516132\pi\)
\(648\) −33.4327 −1.31336
\(649\) 25.7210 1.00964
\(650\) −12.1966 −0.478389
\(651\) −0.462338 −0.0181204
\(652\) 3.83033 0.150007
\(653\) −30.4848 −1.19296 −0.596482 0.802626i \(-0.703435\pi\)
−0.596482 + 0.802626i \(0.703435\pi\)
\(654\) −5.91885 −0.231445
\(655\) −34.7061 −1.35608
\(656\) −3.57314 −0.139508
\(657\) −29.1542 −1.13741
\(658\) 8.87472 0.345973
\(659\) 37.1021 1.44529 0.722647 0.691218i \(-0.242925\pi\)
0.722647 + 0.691218i \(0.242925\pi\)
\(660\) −3.60171 −0.140196
\(661\) −2.33331 −0.0907551 −0.0453775 0.998970i \(-0.514449\pi\)
−0.0453775 + 0.998970i \(0.514449\pi\)
\(662\) −47.3255 −1.83936
\(663\) −0.847235 −0.0329039
\(664\) 16.7021 0.648168
\(665\) 3.94014 0.152792
\(666\) 55.1430 2.13675
\(667\) 15.2898 0.592024
\(668\) −61.4113 −2.37607
\(669\) 3.67429 0.142056
\(670\) 49.0980 1.89682
\(671\) −4.91088 −0.189583
\(672\) 0.563139 0.0217236
\(673\) −35.8433 −1.38166 −0.690829 0.723018i \(-0.742754\pi\)
−0.690829 + 0.723018i \(0.742754\pi\)
\(674\) 0.609508 0.0234774
\(675\) −2.01198 −0.0774411
\(676\) −19.4107 −0.746565
\(677\) 42.8400 1.64647 0.823237 0.567697i \(-0.192166\pi\)
0.823237 + 0.567697i \(0.192166\pi\)
\(678\) −2.03953 −0.0783277
\(679\) 6.33937 0.243283
\(680\) 11.5151 0.441585
\(681\) −0.411852 −0.0157822
\(682\) 19.6848 0.753770
\(683\) 2.24546 0.0859200 0.0429600 0.999077i \(-0.486321\pi\)
0.0429600 + 0.999077i \(0.486321\pi\)
\(684\) 25.0602 0.958202
\(685\) −14.3516 −0.548346
\(686\) −29.5846 −1.12954
\(687\) −0.821135 −0.0313282
\(688\) 14.3254 0.546150
\(689\) 24.6283 0.938264
\(690\) 1.83586 0.0698901
\(691\) −12.4776 −0.474671 −0.237336 0.971428i \(-0.576274\pi\)
−0.237336 + 0.971428i \(0.576274\pi\)
\(692\) 13.2090 0.502131
\(693\) 8.77112 0.333187
\(694\) 22.4708 0.852980
\(695\) 12.9865 0.492606
\(696\) 4.40628 0.167019
\(697\) 3.24455 0.122896
\(698\) 75.6360 2.86286
\(699\) −4.44554 −0.168146
\(700\) 6.42401 0.242805
\(701\) −19.7583 −0.746262 −0.373131 0.927779i \(-0.621716\pi\)
−0.373131 + 0.927779i \(0.621716\pi\)
\(702\) 7.08085 0.267250
\(703\) −18.2902 −0.689827
\(704\) −35.5369 −1.33935
\(705\) −1.26006 −0.0474564
\(706\) −59.1837 −2.22741
\(707\) 18.3847 0.691428
\(708\) −5.43829 −0.204383
\(709\) 17.5081 0.657531 0.328765 0.944412i \(-0.393367\pi\)
0.328765 + 0.944412i \(0.393367\pi\)
\(710\) −1.92315 −0.0721745
\(711\) −18.9736 −0.711564
\(712\) 21.5779 0.808666
\(713\) −6.46296 −0.242040
\(714\) 0.692792 0.0259271
\(715\) −15.1820 −0.567774
\(716\) −13.3735 −0.499790
\(717\) −1.17336 −0.0438201
\(718\) 72.5147 2.70622
\(719\) 30.8017 1.14871 0.574355 0.818607i \(-0.305253\pi\)
0.574355 + 0.818607i \(0.305253\pi\)
\(720\) −9.79829 −0.365161
\(721\) −11.6997 −0.435720
\(722\) 32.1376 1.19604
\(723\) 1.38112 0.0513643
\(724\) −3.16768 −0.117726
\(725\) −11.7933 −0.437990
\(726\) −0.594377 −0.0220594
\(727\) 1.43952 0.0533888 0.0266944 0.999644i \(-0.491502\pi\)
0.0266944 + 0.999644i \(0.491502\pi\)
\(728\) −10.1173 −0.374972
\(729\) −25.2447 −0.934988
\(730\) −41.2642 −1.52726
\(731\) −13.0080 −0.481117
\(732\) 1.03833 0.0383777
\(733\) 46.0606 1.70129 0.850643 0.525743i \(-0.176213\pi\)
0.850643 + 0.525743i \(0.176213\pi\)
\(734\) 47.2215 1.74298
\(735\) 1.95446 0.0720913
\(736\) 7.87205 0.290168
\(737\) −36.2523 −1.33537
\(738\) −13.4838 −0.496345
\(739\) −53.2450 −1.95865 −0.979325 0.202294i \(-0.935160\pi\)
−0.979325 + 0.202294i \(0.935160\pi\)
\(740\) 50.2727 1.84806
\(741\) −1.16785 −0.0429022
\(742\) −20.1388 −0.739319
\(743\) −22.3241 −0.818991 −0.409496 0.912312i \(-0.634295\pi\)
−0.409496 + 0.912312i \(0.634295\pi\)
\(744\) −1.86252 −0.0682834
\(745\) −9.16293 −0.335704
\(746\) 82.3669 3.01567
\(747\) 12.9050 0.472168
\(748\) −18.9996 −0.694694
\(749\) −0.321428 −0.0117447
\(750\) −5.21927 −0.190581
\(751\) 9.25668 0.337781 0.168891 0.985635i \(-0.445982\pi\)
0.168891 + 0.985635i \(0.445982\pi\)
\(752\) 7.32017 0.266939
\(753\) 0.223763 0.00815437
\(754\) 41.5046 1.51151
\(755\) −16.5183 −0.601161
\(756\) −3.72953 −0.135642
\(757\) 13.2985 0.483343 0.241672 0.970358i \(-0.422304\pi\)
0.241672 + 0.970358i \(0.422304\pi\)
\(758\) −39.4192 −1.43177
\(759\) −1.35554 −0.0492029
\(760\) 15.8728 0.575767
\(761\) −19.0709 −0.691321 −0.345660 0.938360i \(-0.612345\pi\)
−0.345660 + 0.938360i \(0.612345\pi\)
\(762\) −5.51815 −0.199902
\(763\) 13.1409 0.475733
\(764\) −7.24408 −0.262081
\(765\) 8.89721 0.321679
\(766\) 82.9401 2.99675
\(767\) −22.9236 −0.827722
\(768\) 4.71285 0.170060
\(769\) 12.0504 0.434550 0.217275 0.976110i \(-0.430283\pi\)
0.217275 + 0.976110i \(0.430283\pi\)
\(770\) 12.4145 0.447386
\(771\) 0.110017 0.00396216
\(772\) 89.7366 3.22969
\(773\) 39.9603 1.43727 0.718635 0.695387i \(-0.244767\pi\)
0.718635 + 0.695387i \(0.244767\pi\)
\(774\) 54.0589 1.94311
\(775\) 4.98498 0.179066
\(776\) 25.5381 0.916764
\(777\) 1.35351 0.0485570
\(778\) −37.5453 −1.34606
\(779\) 4.47238 0.160240
\(780\) 3.20999 0.114936
\(781\) 1.41999 0.0508111
\(782\) 9.68446 0.346316
\(783\) 6.84670 0.244681
\(784\) −11.3542 −0.405508
\(785\) 40.6915 1.45234
\(786\) −8.41166 −0.300034
\(787\) −22.4812 −0.801367 −0.400683 0.916217i \(-0.631227\pi\)
−0.400683 + 0.916217i \(0.631227\pi\)
\(788\) 62.9356 2.24199
\(789\) 4.67036 0.166269
\(790\) −26.8548 −0.955450
\(791\) 4.52813 0.161002
\(792\) 35.3344 1.25555
\(793\) 4.37677 0.155424
\(794\) 5.56533 0.197506
\(795\) 2.85936 0.101411
\(796\) −46.9407 −1.66377
\(797\) −2.66718 −0.0944763 −0.0472382 0.998884i \(-0.515042\pi\)
−0.0472382 + 0.998884i \(0.515042\pi\)
\(798\) 0.954966 0.0338054
\(799\) −6.64698 −0.235153
\(800\) −6.07183 −0.214672
\(801\) 16.6723 0.589086
\(802\) 60.2483 2.12744
\(803\) 30.4681 1.07519
\(804\) 7.66496 0.270322
\(805\) −4.07595 −0.143658
\(806\) −17.5439 −0.617957
\(807\) −4.13147 −0.145435
\(808\) 74.0626 2.60551
\(809\) 2.22276 0.0781481 0.0390741 0.999236i \(-0.487559\pi\)
0.0390741 + 0.999236i \(0.487559\pi\)
\(810\) −36.5620 −1.28466
\(811\) 27.9050 0.979878 0.489939 0.871757i \(-0.337019\pi\)
0.489939 + 0.871757i \(0.337019\pi\)
\(812\) −21.8607 −0.767160
\(813\) −0.756541 −0.0265330
\(814\) −57.6281 −2.01986
\(815\) 1.87452 0.0656615
\(816\) 0.571438 0.0200043
\(817\) −17.9306 −0.627312
\(818\) −31.9934 −1.11862
\(819\) −7.81717 −0.273154
\(820\) −12.2929 −0.429286
\(821\) 7.67411 0.267828 0.133914 0.990993i \(-0.457245\pi\)
0.133914 + 0.990993i \(0.457245\pi\)
\(822\) −3.47838 −0.121322
\(823\) 47.7600 1.66481 0.832405 0.554168i \(-0.186964\pi\)
0.832405 + 0.554168i \(0.186964\pi\)
\(824\) −47.1322 −1.64193
\(825\) 1.04555 0.0364013
\(826\) 18.7448 0.652216
\(827\) 18.6901 0.649917 0.324958 0.945728i \(-0.394650\pi\)
0.324958 + 0.945728i \(0.394650\pi\)
\(828\) −25.9240 −0.900922
\(829\) −0.843871 −0.0293088 −0.0146544 0.999893i \(-0.504665\pi\)
−0.0146544 + 0.999893i \(0.504665\pi\)
\(830\) 18.2654 0.634002
\(831\) −1.64114 −0.0569305
\(832\) 31.6719 1.09803
\(833\) 10.3101 0.357223
\(834\) 3.14752 0.108990
\(835\) −30.0540 −1.04006
\(836\) −26.1896 −0.905787
\(837\) −2.89408 −0.100034
\(838\) 13.4417 0.464337
\(839\) 53.8824 1.86023 0.930113 0.367274i \(-0.119709\pi\)
0.930113 + 0.367274i \(0.119709\pi\)
\(840\) −1.17462 −0.0405283
\(841\) 11.1321 0.383864
\(842\) 80.1674 2.76275
\(843\) 5.44415 0.187506
\(844\) 70.2770 2.41903
\(845\) −9.49937 −0.326788
\(846\) 27.6237 0.949724
\(847\) 1.31962 0.0453428
\(848\) −16.6112 −0.570430
\(849\) 0.111092 0.00381268
\(850\) −7.46977 −0.256211
\(851\) 18.9206 0.648590
\(852\) −0.300233 −0.0102858
\(853\) 4.28689 0.146780 0.0733902 0.997303i \(-0.476618\pi\)
0.0733902 + 0.997303i \(0.476618\pi\)
\(854\) −3.57893 −0.122468
\(855\) 12.2642 0.419426
\(856\) −1.29487 −0.0442577
\(857\) 35.5916 1.21579 0.607894 0.794019i \(-0.292015\pi\)
0.607894 + 0.794019i \(0.292015\pi\)
\(858\) −3.67964 −0.125621
\(859\) −38.4568 −1.31213 −0.656064 0.754705i \(-0.727780\pi\)
−0.656064 + 0.754705i \(0.727780\pi\)
\(860\) 49.2844 1.68058
\(861\) −0.330966 −0.0112793
\(862\) 49.6666 1.69165
\(863\) −49.1038 −1.67151 −0.835757 0.549100i \(-0.814971\pi\)
−0.835757 + 0.549100i \(0.814971\pi\)
\(864\) 3.52507 0.119925
\(865\) 6.46434 0.219794
\(866\) −64.5177 −2.19240
\(867\) 2.56014 0.0869470
\(868\) 9.24047 0.313642
\(869\) 19.8286 0.672640
\(870\) 4.81870 0.163369
\(871\) 32.3094 1.09476
\(872\) 52.9381 1.79271
\(873\) 19.7321 0.667831
\(874\) 13.3494 0.451549
\(875\) 11.5877 0.391737
\(876\) −6.44198 −0.217654
\(877\) −26.6763 −0.900794 −0.450397 0.892828i \(-0.648718\pi\)
−0.450397 + 0.892828i \(0.648718\pi\)
\(878\) 24.6034 0.830324
\(879\) 1.96010 0.0661125
\(880\) 10.2399 0.345186
\(881\) −1.98755 −0.0669624 −0.0334812 0.999439i \(-0.510659\pi\)
−0.0334812 + 0.999439i \(0.510659\pi\)
\(882\) −42.8469 −1.44273
\(883\) −28.4545 −0.957571 −0.478785 0.877932i \(-0.658923\pi\)
−0.478785 + 0.877932i \(0.658923\pi\)
\(884\) 16.9332 0.569524
\(885\) −2.66144 −0.0894632
\(886\) 75.5783 2.53910
\(887\) 23.3548 0.784178 0.392089 0.919927i \(-0.371752\pi\)
0.392089 + 0.919927i \(0.371752\pi\)
\(888\) 5.45261 0.182978
\(889\) 12.2513 0.410895
\(890\) 23.5976 0.790993
\(891\) 26.9961 0.904403
\(892\) −73.4358 −2.45881
\(893\) −9.16241 −0.306608
\(894\) −2.22081 −0.0742750
\(895\) −6.54482 −0.218769
\(896\) −19.6800 −0.657463
\(897\) 1.20811 0.0403376
\(898\) 63.4032 2.11579
\(899\) −16.9637 −0.565772
\(900\) 19.9956 0.666519
\(901\) 15.0836 0.502506
\(902\) 14.0915 0.469194
\(903\) 1.32690 0.0441566
\(904\) 18.2415 0.606704
\(905\) −1.55022 −0.0515312
\(906\) −4.00351 −0.133008
\(907\) 4.13188 0.137197 0.0685983 0.997644i \(-0.478147\pi\)
0.0685983 + 0.997644i \(0.478147\pi\)
\(908\) 8.23144 0.273170
\(909\) 57.2248 1.89803
\(910\) −11.0643 −0.366777
\(911\) −35.7070 −1.18303 −0.591513 0.806296i \(-0.701469\pi\)
−0.591513 + 0.806296i \(0.701469\pi\)
\(912\) 0.787688 0.0260830
\(913\) −13.4866 −0.446340
\(914\) −49.9234 −1.65132
\(915\) 0.508146 0.0167988
\(916\) 16.4115 0.542252
\(917\) 18.6754 0.616716
\(918\) 4.33665 0.143131
\(919\) −44.2652 −1.46017 −0.730087 0.683355i \(-0.760520\pi\)
−0.730087 + 0.683355i \(0.760520\pi\)
\(920\) −16.4199 −0.541348
\(921\) 2.32442 0.0765923
\(922\) −38.7204 −1.27519
\(923\) −1.26555 −0.0416560
\(924\) 1.93809 0.0637585
\(925\) −14.5937 −0.479839
\(926\) 54.9377 1.80536
\(927\) −36.4169 −1.19609
\(928\) 20.6622 0.678271
\(929\) 16.5049 0.541507 0.270754 0.962649i \(-0.412727\pi\)
0.270754 + 0.962649i \(0.412727\pi\)
\(930\) −2.03685 −0.0667910
\(931\) 14.2117 0.465770
\(932\) 88.8505 2.91039
\(933\) 0.882586 0.0288946
\(934\) 54.4195 1.78066
\(935\) −9.29818 −0.304083
\(936\) −31.4914 −1.02933
\(937\) −32.4305 −1.05946 −0.529729 0.848167i \(-0.677706\pi\)
−0.529729 + 0.848167i \(0.677706\pi\)
\(938\) −26.4197 −0.862635
\(939\) 4.96395 0.161992
\(940\) 25.1840 0.821411
\(941\) −49.3511 −1.60880 −0.804400 0.594088i \(-0.797513\pi\)
−0.804400 + 0.594088i \(0.797513\pi\)
\(942\) 9.86236 0.321333
\(943\) −4.62654 −0.150661
\(944\) 15.4614 0.503224
\(945\) −1.82519 −0.0593734
\(946\) −56.4952 −1.83682
\(947\) −30.6757 −0.996825 −0.498413 0.866940i \(-0.666084\pi\)
−0.498413 + 0.866940i \(0.666084\pi\)
\(948\) −4.19244 −0.136164
\(949\) −27.1543 −0.881467
\(950\) −10.2966 −0.334064
\(951\) 0.991506 0.0321518
\(952\) −6.19631 −0.200824
\(953\) −3.63547 −0.117764 −0.0588822 0.998265i \(-0.518754\pi\)
−0.0588822 + 0.998265i \(0.518754\pi\)
\(954\) −62.6847 −2.02949
\(955\) −3.54517 −0.114719
\(956\) 23.4513 0.758470
\(957\) −3.55796 −0.115013
\(958\) 0.182946 0.00591070
\(959\) 7.72262 0.249377
\(960\) 3.67713 0.118679
\(961\) −23.8295 −0.768693
\(962\) 51.3605 1.65593
\(963\) −1.00049 −0.0322402
\(964\) −27.6036 −0.889051
\(965\) 43.9160 1.41371
\(966\) −0.987882 −0.0317846
\(967\) 20.2103 0.649920 0.324960 0.945728i \(-0.394649\pi\)
0.324960 + 0.945728i \(0.394649\pi\)
\(968\) 5.31609 0.170866
\(969\) −0.715250 −0.0229771
\(970\) 27.9284 0.896728
\(971\) 27.4648 0.881388 0.440694 0.897657i \(-0.354732\pi\)
0.440694 + 0.897657i \(0.354732\pi\)
\(972\) −17.4449 −0.559545
\(973\) −6.98806 −0.224027
\(974\) −72.4737 −2.32221
\(975\) −0.931832 −0.0298425
\(976\) −2.95202 −0.0944920
\(977\) −27.1733 −0.869351 −0.434676 0.900587i \(-0.643137\pi\)
−0.434676 + 0.900587i \(0.643137\pi\)
\(978\) 0.454325 0.0145277
\(979\) −17.4236 −0.556862
\(980\) −39.0626 −1.24781
\(981\) 40.9028 1.30593
\(982\) −55.4420 −1.76923
\(983\) 40.0402 1.27708 0.638542 0.769587i \(-0.279538\pi\)
0.638542 + 0.769587i \(0.279538\pi\)
\(984\) −1.33329 −0.0425039
\(985\) 30.7999 0.981368
\(986\) 25.4194 0.809518
\(987\) 0.678039 0.0215822
\(988\) 23.3412 0.742583
\(989\) 18.5486 0.589812
\(990\) 38.6416 1.22811
\(991\) −29.3816 −0.933339 −0.466669 0.884432i \(-0.654546\pi\)
−0.466669 + 0.884432i \(0.654546\pi\)
\(992\) −8.73388 −0.277301
\(993\) −3.61572 −0.114741
\(994\) 1.03485 0.0328235
\(995\) −22.9722 −0.728268
\(996\) 2.85151 0.0903537
\(997\) −18.5995 −0.589052 −0.294526 0.955643i \(-0.595162\pi\)
−0.294526 + 0.955643i \(0.595162\pi\)
\(998\) 85.0614 2.69257
\(999\) 8.47255 0.268060
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 4021.2.a.b.1.16 151
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4021.2.a.b.1.16 151 1.1 even 1 trivial