Properties

Label 6031.2.a.c.1.16
Level $6031$
Weight $2$
Character 6031.1
Self dual yes
Analytic conductor $48.158$
Analytic rank $1$
Dimension $110$
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] = [6031,2,Mod(1,6031)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6031, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6031.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6031 = 37 \cdot 163 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6031.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: \(48.1577774590\)
Analytic rank: \(1\)
Dimension: \(110\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Character \(\chi\) \(=\) 6031.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.17795 q^{2} +2.23759 q^{3} +2.74348 q^{4} +1.66421 q^{5} -4.87337 q^{6} -3.06154 q^{7} -1.61926 q^{8} +2.00681 q^{9} +O(q^{10})\) \(q-2.17795 q^{2} +2.23759 q^{3} +2.74348 q^{4} +1.66421 q^{5} -4.87337 q^{6} -3.06154 q^{7} -1.61926 q^{8} +2.00681 q^{9} -3.62457 q^{10} -1.57786 q^{11} +6.13878 q^{12} -0.211680 q^{13} +6.66790 q^{14} +3.72382 q^{15} -1.96028 q^{16} +2.77554 q^{17} -4.37074 q^{18} -6.50524 q^{19} +4.56572 q^{20} -6.85048 q^{21} +3.43650 q^{22} +3.87991 q^{23} -3.62325 q^{24} -2.23041 q^{25} +0.461028 q^{26} -2.22235 q^{27} -8.39928 q^{28} +2.19193 q^{29} -8.11030 q^{30} +6.91264 q^{31} +7.50792 q^{32} -3.53060 q^{33} -6.04499 q^{34} -5.09504 q^{35} +5.50564 q^{36} -1.00000 q^{37} +14.1681 q^{38} -0.473652 q^{39} -2.69479 q^{40} +1.14950 q^{41} +14.9200 q^{42} +6.19271 q^{43} -4.32882 q^{44} +3.33975 q^{45} -8.45026 q^{46} -0.352658 q^{47} -4.38630 q^{48} +2.37304 q^{49} +4.85773 q^{50} +6.21051 q^{51} -0.580739 q^{52} -13.5084 q^{53} +4.84018 q^{54} -2.62588 q^{55} +4.95744 q^{56} -14.5561 q^{57} -4.77392 q^{58} +3.36630 q^{59} +10.2162 q^{60} +1.43260 q^{61} -15.0554 q^{62} -6.14394 q^{63} -12.4313 q^{64} -0.352279 q^{65} +7.68948 q^{66} -1.92342 q^{67} +7.61463 q^{68} +8.68165 q^{69} +11.0968 q^{70} +6.00523 q^{71} -3.24955 q^{72} -8.32780 q^{73} +2.17795 q^{74} -4.99074 q^{75} -17.8470 q^{76} +4.83068 q^{77} +1.03159 q^{78} -7.66645 q^{79} -3.26231 q^{80} -10.9931 q^{81} -2.50356 q^{82} +2.04666 q^{83} -18.7941 q^{84} +4.61907 q^{85} -13.4874 q^{86} +4.90464 q^{87} +2.55497 q^{88} +13.6855 q^{89} -7.27382 q^{90} +0.648066 q^{91} +10.6445 q^{92} +15.4677 q^{93} +0.768073 q^{94} -10.8261 q^{95} +16.7997 q^{96} -4.63698 q^{97} -5.16837 q^{98} -3.16646 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 110 q - 9 q^{2} + 97 q^{4} - 26 q^{5} - 26 q^{6} - 4 q^{7} - 27 q^{8} + 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 110 q - 9 q^{2} + 97 q^{4} - 26 q^{5} - 26 q^{6} - 4 q^{7} - 27 q^{8} + 62 q^{9} - 17 q^{10} - 9 q^{11} - 21 q^{13} - 29 q^{14} - 23 q^{15} + 79 q^{16} - 76 q^{17} - 31 q^{18} - 27 q^{19} - 67 q^{20} - 30 q^{21} - 28 q^{22} - 32 q^{23} - 63 q^{24} + 66 q^{25} - 55 q^{26} - 4 q^{28} - 81 q^{29} - 48 q^{30} - 30 q^{31} - 73 q^{32} - 53 q^{33} - 23 q^{34} - 78 q^{35} + 7 q^{36} - 110 q^{37} - 50 q^{38} - 64 q^{39} - 37 q^{40} - 123 q^{41} - 63 q^{42} - 40 q^{43} - 31 q^{44} - 73 q^{45} + 16 q^{46} - 37 q^{47} - 29 q^{48} + 46 q^{49} - 58 q^{50} - 73 q^{51} - 39 q^{52} - 16 q^{53} - 53 q^{54} - 59 q^{55} - 113 q^{56} - 39 q^{57} + 11 q^{58} - 93 q^{59} - 18 q^{60} - 66 q^{61} - 40 q^{62} - 21 q^{63} + 23 q^{64} - 92 q^{65} - 31 q^{66} + q^{67} - 121 q^{68} - 80 q^{69} - 3 q^{70} - 75 q^{71} - 114 q^{72} - 39 q^{73} + 9 q^{74} - 25 q^{75} - 58 q^{76} - 31 q^{77} + 68 q^{78} - 36 q^{79} - 82 q^{80} - 50 q^{81} - 18 q^{82} - 57 q^{83} - 9 q^{84} - 14 q^{85} - 58 q^{86} - 58 q^{87} - 15 q^{88} - 181 q^{89} + 8 q^{90} - 55 q^{91} - 116 q^{92} - 86 q^{93} - 39 q^{94} - 70 q^{95} - 127 q^{96} - 91 q^{97} - 19 q^{98} - 21 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.17795 −1.54005 −0.770023 0.638017i \(-0.779755\pi\)
−0.770023 + 0.638017i \(0.779755\pi\)
\(3\) 2.23759 1.29187 0.645937 0.763391i \(-0.276467\pi\)
0.645937 + 0.763391i \(0.276467\pi\)
\(4\) 2.74348 1.37174
\(5\) 1.66421 0.744257 0.372128 0.928181i \(-0.378628\pi\)
0.372128 + 0.928181i \(0.378628\pi\)
\(6\) −4.87337 −1.98954
\(7\) −3.06154 −1.15715 −0.578577 0.815628i \(-0.696392\pi\)
−0.578577 + 0.815628i \(0.696392\pi\)
\(8\) −1.61926 −0.572496
\(9\) 2.00681 0.668937
\(10\) −3.62457 −1.14619
\(11\) −1.57786 −0.475742 −0.237871 0.971297i \(-0.576450\pi\)
−0.237871 + 0.971297i \(0.576450\pi\)
\(12\) 6.13878 1.77211
\(13\) −0.211680 −0.0587094 −0.0293547 0.999569i \(-0.509345\pi\)
−0.0293547 + 0.999569i \(0.509345\pi\)
\(14\) 6.66790 1.78207
\(15\) 3.72382 0.961485
\(16\) −1.96028 −0.490070
\(17\) 2.77554 0.673167 0.336583 0.941654i \(-0.390729\pi\)
0.336583 + 0.941654i \(0.390729\pi\)
\(18\) −4.37074 −1.03019
\(19\) −6.50524 −1.49240 −0.746202 0.665719i \(-0.768125\pi\)
−0.746202 + 0.665719i \(0.768125\pi\)
\(20\) 4.56572 1.02093
\(21\) −6.85048 −1.49490
\(22\) 3.43650 0.732664
\(23\) 3.87991 0.809017 0.404509 0.914534i \(-0.367443\pi\)
0.404509 + 0.914534i \(0.367443\pi\)
\(24\) −3.62325 −0.739592
\(25\) −2.23041 −0.446082
\(26\) 0.461028 0.0904151
\(27\) −2.22235 −0.427692
\(28\) −8.39928 −1.58731
\(29\) 2.19193 0.407031 0.203516 0.979072i \(-0.434763\pi\)
0.203516 + 0.979072i \(0.434763\pi\)
\(30\) −8.11030 −1.48073
\(31\) 6.91264 1.24155 0.620774 0.783990i \(-0.286819\pi\)
0.620774 + 0.783990i \(0.286819\pi\)
\(32\) 7.50792 1.32723
\(33\) −3.53060 −0.614598
\(34\) −6.04499 −1.03671
\(35\) −5.09504 −0.861220
\(36\) 5.50564 0.917607
\(37\) −1.00000 −0.164399
\(38\) 14.1681 2.29837
\(39\) −0.473652 −0.0758451
\(40\) −2.69479 −0.426084
\(41\) 1.14950 0.179522 0.0897610 0.995963i \(-0.471390\pi\)
0.0897610 + 0.995963i \(0.471390\pi\)
\(42\) 14.9200 2.30221
\(43\) 6.19271 0.944379 0.472189 0.881497i \(-0.343464\pi\)
0.472189 + 0.881497i \(0.343464\pi\)
\(44\) −4.32882 −0.652594
\(45\) 3.33975 0.497861
\(46\) −8.45026 −1.24592
\(47\) −0.352658 −0.0514405 −0.0257202 0.999669i \(-0.508188\pi\)
−0.0257202 + 0.999669i \(0.508188\pi\)
\(48\) −4.38630 −0.633108
\(49\) 2.37304 0.339006
\(50\) 4.85773 0.686986
\(51\) 6.21051 0.869646
\(52\) −0.580739 −0.0805339
\(53\) −13.5084 −1.85552 −0.927761 0.373175i \(-0.878269\pi\)
−0.927761 + 0.373175i \(0.878269\pi\)
\(54\) 4.84018 0.658664
\(55\) −2.62588 −0.354074
\(56\) 4.95744 0.662466
\(57\) −14.5561 −1.92800
\(58\) −4.77392 −0.626846
\(59\) 3.36630 0.438255 0.219128 0.975696i \(-0.429679\pi\)
0.219128 + 0.975696i \(0.429679\pi\)
\(60\) 10.2162 1.31891
\(61\) 1.43260 0.183426 0.0917131 0.995785i \(-0.470766\pi\)
0.0917131 + 0.995785i \(0.470766\pi\)
\(62\) −15.0554 −1.91204
\(63\) −6.14394 −0.774063
\(64\) −12.4313 −1.55392
\(65\) −0.352279 −0.0436948
\(66\) 7.68948 0.946509
\(67\) −1.92342 −0.234983 −0.117492 0.993074i \(-0.537485\pi\)
−0.117492 + 0.993074i \(0.537485\pi\)
\(68\) 7.61463 0.923409
\(69\) 8.68165 1.04515
\(70\) 11.0968 1.32632
\(71\) 6.00523 0.712689 0.356345 0.934355i \(-0.384023\pi\)
0.356345 + 0.934355i \(0.384023\pi\)
\(72\) −3.24955 −0.382964
\(73\) −8.32780 −0.974696 −0.487348 0.873208i \(-0.662036\pi\)
−0.487348 + 0.873208i \(0.662036\pi\)
\(74\) 2.17795 0.253182
\(75\) −4.99074 −0.576281
\(76\) −17.8470 −2.04719
\(77\) 4.83068 0.550507
\(78\) 1.03159 0.116805
\(79\) −7.66645 −0.862543 −0.431271 0.902222i \(-0.641935\pi\)
−0.431271 + 0.902222i \(0.641935\pi\)
\(80\) −3.26231 −0.364738
\(81\) −10.9931 −1.22146
\(82\) −2.50356 −0.276472
\(83\) 2.04666 0.224651 0.112325 0.993671i \(-0.464170\pi\)
0.112325 + 0.993671i \(0.464170\pi\)
\(84\) −18.7941 −2.05061
\(85\) 4.61907 0.501009
\(86\) −13.4874 −1.45439
\(87\) 4.90464 0.525833
\(88\) 2.55497 0.272360
\(89\) 13.6855 1.45066 0.725331 0.688400i \(-0.241687\pi\)
0.725331 + 0.688400i \(0.241687\pi\)
\(90\) −7.27382 −0.766728
\(91\) 0.648066 0.0679358
\(92\) 10.6445 1.10976
\(93\) 15.4677 1.60392
\(94\) 0.768073 0.0792207
\(95\) −10.8261 −1.11073
\(96\) 16.7997 1.71461
\(97\) −4.63698 −0.470814 −0.235407 0.971897i \(-0.575642\pi\)
−0.235407 + 0.971897i \(0.575642\pi\)
\(98\) −5.16837 −0.522085
\(99\) −3.16646 −0.318241
\(100\) −6.11908 −0.611908
\(101\) 4.50301 0.448066 0.224033 0.974582i \(-0.428078\pi\)
0.224033 + 0.974582i \(0.428078\pi\)
\(102\) −13.5262 −1.33929
\(103\) −1.91684 −0.188872 −0.0944358 0.995531i \(-0.530105\pi\)
−0.0944358 + 0.995531i \(0.530105\pi\)
\(104\) 0.342765 0.0336109
\(105\) −11.4006 −1.11259
\(106\) 29.4207 2.85759
\(107\) 4.59889 0.444592 0.222296 0.974979i \(-0.428645\pi\)
0.222296 + 0.974979i \(0.428645\pi\)
\(108\) −6.09697 −0.586681
\(109\) −12.5070 −1.19796 −0.598978 0.800765i \(-0.704426\pi\)
−0.598978 + 0.800765i \(0.704426\pi\)
\(110\) 5.71905 0.545290
\(111\) −2.23759 −0.212383
\(112\) 6.00148 0.567087
\(113\) 10.7719 1.01334 0.506668 0.862141i \(-0.330877\pi\)
0.506668 + 0.862141i \(0.330877\pi\)
\(114\) 31.7024 2.96920
\(115\) 6.45698 0.602116
\(116\) 6.01351 0.558341
\(117\) −0.424801 −0.0392729
\(118\) −7.33165 −0.674933
\(119\) −8.49742 −0.778958
\(120\) −6.02984 −0.550446
\(121\) −8.51037 −0.773670
\(122\) −3.12015 −0.282485
\(123\) 2.57211 0.231920
\(124\) 18.9647 1.70308
\(125\) −12.0329 −1.07626
\(126\) 13.3812 1.19209
\(127\) −17.9311 −1.59113 −0.795566 0.605867i \(-0.792826\pi\)
−0.795566 + 0.605867i \(0.792826\pi\)
\(128\) 12.0590 1.06588
\(129\) 13.8567 1.22002
\(130\) 0.767247 0.0672920
\(131\) −10.8428 −0.947336 −0.473668 0.880703i \(-0.657070\pi\)
−0.473668 + 0.880703i \(0.657070\pi\)
\(132\) −9.68612 −0.843069
\(133\) 19.9161 1.72694
\(134\) 4.18912 0.361885
\(135\) −3.69845 −0.318312
\(136\) −4.49432 −0.385385
\(137\) 17.1676 1.46673 0.733364 0.679836i \(-0.237949\pi\)
0.733364 + 0.679836i \(0.237949\pi\)
\(138\) −18.9082 −1.60958
\(139\) −17.2896 −1.46648 −0.733242 0.679967i \(-0.761994\pi\)
−0.733242 + 0.679967i \(0.761994\pi\)
\(140\) −13.9782 −1.18137
\(141\) −0.789104 −0.0664546
\(142\) −13.0791 −1.09757
\(143\) 0.334000 0.0279305
\(144\) −3.93391 −0.327826
\(145\) 3.64783 0.302936
\(146\) 18.1376 1.50108
\(147\) 5.30990 0.437953
\(148\) −2.74348 −0.225513
\(149\) −11.4017 −0.934065 −0.467032 0.884240i \(-0.654677\pi\)
−0.467032 + 0.884240i \(0.654677\pi\)
\(150\) 10.8696 0.887500
\(151\) 1.31952 0.107381 0.0536906 0.998558i \(-0.482902\pi\)
0.0536906 + 0.998558i \(0.482902\pi\)
\(152\) 10.5337 0.854396
\(153\) 5.56998 0.450306
\(154\) −10.5210 −0.847805
\(155\) 11.5041 0.924030
\(156\) −1.29946 −0.104040
\(157\) −16.2682 −1.29834 −0.649172 0.760641i \(-0.724885\pi\)
−0.649172 + 0.760641i \(0.724885\pi\)
\(158\) 16.6972 1.32835
\(159\) −30.2263 −2.39710
\(160\) 12.4948 0.987797
\(161\) −11.8785 −0.936158
\(162\) 23.9425 1.88110
\(163\) −1.00000 −0.0783260
\(164\) 3.15363 0.246257
\(165\) −5.87565 −0.457419
\(166\) −4.45754 −0.345972
\(167\) −11.9549 −0.925101 −0.462551 0.886593i \(-0.653066\pi\)
−0.462551 + 0.886593i \(0.653066\pi\)
\(168\) 11.0927 0.855822
\(169\) −12.9552 −0.996553
\(170\) −10.0601 −0.771576
\(171\) −13.0548 −0.998325
\(172\) 16.9896 1.29544
\(173\) −17.6997 −1.34569 −0.672843 0.739786i \(-0.734927\pi\)
−0.672843 + 0.739786i \(0.734927\pi\)
\(174\) −10.6821 −0.809806
\(175\) 6.82849 0.516186
\(176\) 3.09304 0.233147
\(177\) 7.53240 0.566170
\(178\) −29.8064 −2.23408
\(179\) −14.9414 −1.11678 −0.558388 0.829580i \(-0.688580\pi\)
−0.558388 + 0.829580i \(0.688580\pi\)
\(180\) 9.16254 0.682935
\(181\) 5.94801 0.442112 0.221056 0.975261i \(-0.429050\pi\)
0.221056 + 0.975261i \(0.429050\pi\)
\(182\) −1.41146 −0.104624
\(183\) 3.20558 0.236963
\(184\) −6.28259 −0.463159
\(185\) −1.66421 −0.122355
\(186\) −33.6878 −2.47011
\(187\) −4.37940 −0.320253
\(188\) −0.967510 −0.0705629
\(189\) 6.80382 0.494905
\(190\) 23.5787 1.71058
\(191\) −16.2068 −1.17268 −0.586340 0.810065i \(-0.699432\pi\)
−0.586340 + 0.810065i \(0.699432\pi\)
\(192\) −27.8163 −2.00747
\(193\) −13.3399 −0.960227 −0.480114 0.877206i \(-0.659405\pi\)
−0.480114 + 0.877206i \(0.659405\pi\)
\(194\) 10.0991 0.725076
\(195\) −0.788256 −0.0564482
\(196\) 6.51039 0.465028
\(197\) −13.6324 −0.971267 −0.485634 0.874162i \(-0.661411\pi\)
−0.485634 + 0.874162i \(0.661411\pi\)
\(198\) 6.89640 0.490106
\(199\) 18.0998 1.28306 0.641532 0.767096i \(-0.278299\pi\)
0.641532 + 0.767096i \(0.278299\pi\)
\(200\) 3.61162 0.255380
\(201\) −4.30383 −0.303569
\(202\) −9.80733 −0.690042
\(203\) −6.71069 −0.470998
\(204\) 17.0384 1.19293
\(205\) 1.91301 0.133610
\(206\) 4.17478 0.290871
\(207\) 7.78625 0.541181
\(208\) 0.414951 0.0287717
\(209\) 10.2643 0.709999
\(210\) 24.8300 1.71343
\(211\) 2.46892 0.169967 0.0849837 0.996382i \(-0.472916\pi\)
0.0849837 + 0.996382i \(0.472916\pi\)
\(212\) −37.0600 −2.54529
\(213\) 13.4372 0.920704
\(214\) −10.0162 −0.684691
\(215\) 10.3060 0.702860
\(216\) 3.59857 0.244852
\(217\) −21.1634 −1.43666
\(218\) 27.2397 1.84491
\(219\) −18.6342 −1.25918
\(220\) −7.20406 −0.485697
\(221\) −0.587525 −0.0395212
\(222\) 4.87337 0.327079
\(223\) 9.70937 0.650187 0.325094 0.945682i \(-0.394604\pi\)
0.325094 + 0.945682i \(0.394604\pi\)
\(224\) −22.9858 −1.53581
\(225\) −4.47601 −0.298401
\(226\) −23.4607 −1.56058
\(227\) −24.5150 −1.62712 −0.813559 0.581482i \(-0.802473\pi\)
−0.813559 + 0.581482i \(0.802473\pi\)
\(228\) −39.9343 −2.64471
\(229\) −25.7628 −1.70246 −0.851228 0.524797i \(-0.824141\pi\)
−0.851228 + 0.524797i \(0.824141\pi\)
\(230\) −14.0630 −0.927287
\(231\) 10.8091 0.711185
\(232\) −3.54931 −0.233024
\(233\) 3.31972 0.217482 0.108741 0.994070i \(-0.465318\pi\)
0.108741 + 0.994070i \(0.465318\pi\)
\(234\) 0.925196 0.0604820
\(235\) −0.586897 −0.0382849
\(236\) 9.23538 0.601172
\(237\) −17.1544 −1.11430
\(238\) 18.5070 1.19963
\(239\) −27.8008 −1.79828 −0.899142 0.437658i \(-0.855808\pi\)
−0.899142 + 0.437658i \(0.855808\pi\)
\(240\) −7.29972 −0.471195
\(241\) 11.6947 0.753325 0.376662 0.926351i \(-0.377072\pi\)
0.376662 + 0.926351i \(0.377072\pi\)
\(242\) 18.5352 1.19149
\(243\) −17.9311 −1.15028
\(244\) 3.93032 0.251613
\(245\) 3.94924 0.252307
\(246\) −5.60194 −0.357167
\(247\) 1.37703 0.0876181
\(248\) −11.1934 −0.710781
\(249\) 4.57960 0.290220
\(250\) 26.2071 1.65748
\(251\) 5.45539 0.344341 0.172171 0.985067i \(-0.444922\pi\)
0.172171 + 0.985067i \(0.444922\pi\)
\(252\) −16.8558 −1.06181
\(253\) −6.12194 −0.384883
\(254\) 39.0532 2.45042
\(255\) 10.3356 0.647240
\(256\) −1.40133 −0.0875829
\(257\) −21.7313 −1.35556 −0.677780 0.735265i \(-0.737058\pi\)
−0.677780 + 0.735265i \(0.737058\pi\)
\(258\) −30.1793 −1.87888
\(259\) 3.06154 0.190235
\(260\) −0.966470 −0.0599379
\(261\) 4.39879 0.272278
\(262\) 23.6150 1.45894
\(263\) −13.2484 −0.816930 −0.408465 0.912774i \(-0.633936\pi\)
−0.408465 + 0.912774i \(0.633936\pi\)
\(264\) 5.71697 0.351855
\(265\) −22.4808 −1.38098
\(266\) −43.3763 −2.65957
\(267\) 30.6226 1.87407
\(268\) −5.27687 −0.322336
\(269\) 19.6607 1.19874 0.599368 0.800474i \(-0.295419\pi\)
0.599368 + 0.800474i \(0.295419\pi\)
\(270\) 8.05506 0.490215
\(271\) 22.2540 1.35184 0.675918 0.736976i \(-0.263747\pi\)
0.675918 + 0.736976i \(0.263747\pi\)
\(272\) −5.44083 −0.329899
\(273\) 1.45011 0.0877644
\(274\) −37.3903 −2.25883
\(275\) 3.51927 0.212220
\(276\) 23.8179 1.43367
\(277\) 2.32516 0.139705 0.0698527 0.997557i \(-0.477747\pi\)
0.0698527 + 0.997557i \(0.477747\pi\)
\(278\) 37.6559 2.25845
\(279\) 13.8724 0.830517
\(280\) 8.25022 0.493045
\(281\) 18.3593 1.09522 0.547612 0.836733i \(-0.315537\pi\)
0.547612 + 0.836733i \(0.315537\pi\)
\(282\) 1.71863 0.102343
\(283\) −5.84818 −0.347638 −0.173819 0.984778i \(-0.555611\pi\)
−0.173819 + 0.984778i \(0.555611\pi\)
\(284\) 16.4752 0.977624
\(285\) −24.2243 −1.43493
\(286\) −0.727437 −0.0430142
\(287\) −3.51925 −0.207735
\(288\) 15.0670 0.887830
\(289\) −9.29639 −0.546847
\(290\) −7.94480 −0.466535
\(291\) −10.3757 −0.608233
\(292\) −22.8472 −1.33703
\(293\) −20.4865 −1.19683 −0.598416 0.801186i \(-0.704203\pi\)
−0.598416 + 0.801186i \(0.704203\pi\)
\(294\) −11.5647 −0.674467
\(295\) 5.60223 0.326174
\(296\) 1.61926 0.0941177
\(297\) 3.50655 0.203471
\(298\) 24.8324 1.43850
\(299\) −0.821298 −0.0474969
\(300\) −13.6920 −0.790508
\(301\) −18.9592 −1.09279
\(302\) −2.87386 −0.165372
\(303\) 10.0759 0.578844
\(304\) 12.7521 0.731383
\(305\) 2.38415 0.136516
\(306\) −12.1311 −0.693492
\(307\) −13.3681 −0.762957 −0.381478 0.924378i \(-0.624585\pi\)
−0.381478 + 0.924378i \(0.624585\pi\)
\(308\) 13.2529 0.755152
\(309\) −4.28910 −0.243998
\(310\) −25.0553 −1.42305
\(311\) 14.0947 0.799236 0.399618 0.916682i \(-0.369143\pi\)
0.399618 + 0.916682i \(0.369143\pi\)
\(312\) 0.766967 0.0434210
\(313\) 16.6098 0.938839 0.469419 0.882975i \(-0.344463\pi\)
0.469419 + 0.882975i \(0.344463\pi\)
\(314\) 35.4314 1.99951
\(315\) −10.2248 −0.576102
\(316\) −21.0327 −1.18318
\(317\) −0.563916 −0.0316727 −0.0158363 0.999875i \(-0.505041\pi\)
−0.0158363 + 0.999875i \(0.505041\pi\)
\(318\) 65.8314 3.69164
\(319\) −3.45855 −0.193642
\(320\) −20.6884 −1.15651
\(321\) 10.2904 0.574356
\(322\) 25.8708 1.44173
\(323\) −18.0555 −1.00464
\(324\) −30.1595 −1.67553
\(325\) 0.472132 0.0261892
\(326\) 2.17795 0.120626
\(327\) −27.9856 −1.54761
\(328\) −1.86135 −0.102776
\(329\) 1.07968 0.0595246
\(330\) 12.7969 0.704446
\(331\) 24.1833 1.32924 0.664618 0.747183i \(-0.268594\pi\)
0.664618 + 0.747183i \(0.268594\pi\)
\(332\) 5.61498 0.308162
\(333\) −2.00681 −0.109973
\(334\) 26.0373 1.42470
\(335\) −3.20097 −0.174888
\(336\) 13.4289 0.732604
\(337\) 13.8672 0.755394 0.377697 0.925929i \(-0.376716\pi\)
0.377697 + 0.925929i \(0.376716\pi\)
\(338\) 28.2158 1.53474
\(339\) 24.1031 1.30910
\(340\) 12.6723 0.687254
\(341\) −10.9072 −0.590656
\(342\) 28.4327 1.53747
\(343\) 14.1656 0.764872
\(344\) −10.0276 −0.540653
\(345\) 14.4481 0.777858
\(346\) 38.5492 2.07242
\(347\) 8.73119 0.468715 0.234357 0.972150i \(-0.424701\pi\)
0.234357 + 0.972150i \(0.424701\pi\)
\(348\) 13.4558 0.721306
\(349\) 25.6201 1.37141 0.685706 0.727879i \(-0.259494\pi\)
0.685706 + 0.727879i \(0.259494\pi\)
\(350\) −14.8721 −0.794949
\(351\) 0.470426 0.0251095
\(352\) −11.8464 −0.631417
\(353\) 15.3174 0.815260 0.407630 0.913147i \(-0.366355\pi\)
0.407630 + 0.913147i \(0.366355\pi\)
\(354\) −16.4052 −0.871928
\(355\) 9.99395 0.530424
\(356\) 37.5459 1.98993
\(357\) −19.0138 −1.00631
\(358\) 32.5418 1.71989
\(359\) −10.2809 −0.542604 −0.271302 0.962494i \(-0.587454\pi\)
−0.271302 + 0.962494i \(0.587454\pi\)
\(360\) −5.40794 −0.285023
\(361\) 23.3182 1.22727
\(362\) −12.9545 −0.680873
\(363\) −19.0427 −0.999483
\(364\) 1.77796 0.0931902
\(365\) −13.8592 −0.725424
\(366\) −6.98161 −0.364934
\(367\) −7.14373 −0.372900 −0.186450 0.982464i \(-0.559698\pi\)
−0.186450 + 0.982464i \(0.559698\pi\)
\(368\) −7.60571 −0.396475
\(369\) 2.30683 0.120089
\(370\) 3.62457 0.188432
\(371\) 41.3565 2.14712
\(372\) 42.4352 2.20016
\(373\) −3.14111 −0.162640 −0.0813202 0.996688i \(-0.525914\pi\)
−0.0813202 + 0.996688i \(0.525914\pi\)
\(374\) 9.53813 0.493205
\(375\) −26.9247 −1.39039
\(376\) 0.571046 0.0294495
\(377\) −0.463987 −0.0238965
\(378\) −14.8184 −0.762176
\(379\) 8.36914 0.429894 0.214947 0.976626i \(-0.431042\pi\)
0.214947 + 0.976626i \(0.431042\pi\)
\(380\) −29.7011 −1.52364
\(381\) −40.1226 −2.05554
\(382\) 35.2976 1.80598
\(383\) 13.2778 0.678462 0.339231 0.940703i \(-0.389833\pi\)
0.339231 + 0.940703i \(0.389833\pi\)
\(384\) 26.9832 1.37698
\(385\) 8.03925 0.409718
\(386\) 29.0537 1.47879
\(387\) 12.4276 0.631730
\(388\) −12.7215 −0.645835
\(389\) −32.2691 −1.63611 −0.818053 0.575143i \(-0.804947\pi\)
−0.818053 + 0.575143i \(0.804947\pi\)
\(390\) 1.71678 0.0869328
\(391\) 10.7688 0.544603
\(392\) −3.84258 −0.194080
\(393\) −24.2617 −1.22384
\(394\) 29.6907 1.49580
\(395\) −12.7586 −0.641953
\(396\) −8.68712 −0.436544
\(397\) 33.3729 1.67494 0.837468 0.546486i \(-0.184035\pi\)
0.837468 + 0.546486i \(0.184035\pi\)
\(398\) −39.4206 −1.97598
\(399\) 44.5640 2.23099
\(400\) 4.37223 0.218611
\(401\) −19.7770 −0.987615 −0.493808 0.869571i \(-0.664395\pi\)
−0.493808 + 0.869571i \(0.664395\pi\)
\(402\) 9.37354 0.467510
\(403\) −1.46327 −0.0728905
\(404\) 12.3539 0.614630
\(405\) −18.2949 −0.909080
\(406\) 14.6156 0.725358
\(407\) 1.57786 0.0782115
\(408\) −10.0565 −0.497869
\(409\) −35.8166 −1.77102 −0.885508 0.464625i \(-0.846189\pi\)
−0.885508 + 0.464625i \(0.846189\pi\)
\(410\) −4.16645 −0.205766
\(411\) 38.4141 1.89483
\(412\) −5.25881 −0.259083
\(413\) −10.3061 −0.507129
\(414\) −16.9581 −0.833444
\(415\) 3.40608 0.167198
\(416\) −1.58927 −0.0779206
\(417\) −38.6870 −1.89451
\(418\) −22.3553 −1.09343
\(419\) 5.47375 0.267410 0.133705 0.991021i \(-0.457313\pi\)
0.133705 + 0.991021i \(0.457313\pi\)
\(420\) −31.2774 −1.52618
\(421\) −1.44517 −0.0704333 −0.0352167 0.999380i \(-0.511212\pi\)
−0.0352167 + 0.999380i \(0.511212\pi\)
\(422\) −5.37719 −0.261757
\(423\) −0.707718 −0.0344104
\(424\) 21.8737 1.06228
\(425\) −6.19059 −0.300287
\(426\) −29.2657 −1.41793
\(427\) −4.38598 −0.212252
\(428\) 12.6170 0.609864
\(429\) 0.747356 0.0360827
\(430\) −22.4459 −1.08244
\(431\) 2.66243 0.128245 0.0641224 0.997942i \(-0.479575\pi\)
0.0641224 + 0.997942i \(0.479575\pi\)
\(432\) 4.35643 0.209599
\(433\) 21.1460 1.01621 0.508106 0.861295i \(-0.330346\pi\)
0.508106 + 0.861295i \(0.330346\pi\)
\(434\) 46.0928 2.21252
\(435\) 8.16235 0.391355
\(436\) −34.3128 −1.64328
\(437\) −25.2398 −1.20738
\(438\) 40.5844 1.93920
\(439\) −0.970109 −0.0463008 −0.0231504 0.999732i \(-0.507370\pi\)
−0.0231504 + 0.999732i \(0.507370\pi\)
\(440\) 4.25200 0.202706
\(441\) 4.76225 0.226774
\(442\) 1.27960 0.0608644
\(443\) 6.05898 0.287871 0.143935 0.989587i \(-0.454024\pi\)
0.143935 + 0.989587i \(0.454024\pi\)
\(444\) −6.13878 −0.291334
\(445\) 22.7756 1.07966
\(446\) −21.1465 −1.00132
\(447\) −25.5124 −1.20669
\(448\) 38.0591 1.79812
\(449\) −20.2042 −0.953496 −0.476748 0.879040i \(-0.658185\pi\)
−0.476748 + 0.879040i \(0.658185\pi\)
\(450\) 9.74854 0.459551
\(451\) −1.81375 −0.0854061
\(452\) 29.5525 1.39003
\(453\) 2.95255 0.138723
\(454\) 53.3925 2.50584
\(455\) 1.07852 0.0505617
\(456\) 23.5701 1.10377
\(457\) −34.6527 −1.62098 −0.810492 0.585750i \(-0.800800\pi\)
−0.810492 + 0.585750i \(0.800800\pi\)
\(458\) 56.1102 2.62186
\(459\) −6.16822 −0.287908
\(460\) 17.7146 0.825947
\(461\) 28.2054 1.31365 0.656827 0.754041i \(-0.271898\pi\)
0.656827 + 0.754041i \(0.271898\pi\)
\(462\) −23.5417 −1.09526
\(463\) −11.8701 −0.551650 −0.275825 0.961208i \(-0.588951\pi\)
−0.275825 + 0.961208i \(0.588951\pi\)
\(464\) −4.29680 −0.199474
\(465\) 25.7414 1.19373
\(466\) −7.23019 −0.334932
\(467\) −16.8305 −0.778822 −0.389411 0.921064i \(-0.627321\pi\)
−0.389411 + 0.921064i \(0.627321\pi\)
\(468\) −1.16543 −0.0538721
\(469\) 5.88864 0.271912
\(470\) 1.27823 0.0589605
\(471\) −36.4016 −1.67730
\(472\) −5.45093 −0.250899
\(473\) −9.77121 −0.449281
\(474\) 37.3614 1.71607
\(475\) 14.5094 0.665735
\(476\) −23.3125 −1.06853
\(477\) −27.1088 −1.24123
\(478\) 60.5488 2.76944
\(479\) 21.0644 0.962459 0.481229 0.876595i \(-0.340190\pi\)
0.481229 + 0.876595i \(0.340190\pi\)
\(480\) 27.9581 1.27611
\(481\) 0.211680 0.00965176
\(482\) −25.4706 −1.16015
\(483\) −26.5792 −1.20940
\(484\) −23.3480 −1.06127
\(485\) −7.71691 −0.350407
\(486\) 39.0531 1.77148
\(487\) 4.03781 0.182971 0.0914853 0.995806i \(-0.470839\pi\)
0.0914853 + 0.995806i \(0.470839\pi\)
\(488\) −2.31976 −0.105011
\(489\) −2.23759 −0.101187
\(490\) −8.60125 −0.388565
\(491\) −9.68684 −0.437161 −0.218580 0.975819i \(-0.570143\pi\)
−0.218580 + 0.975819i \(0.570143\pi\)
\(492\) 7.05654 0.318133
\(493\) 6.08378 0.274000
\(494\) −2.99910 −0.134936
\(495\) −5.26965 −0.236853
\(496\) −13.5507 −0.608445
\(497\) −18.3853 −0.824692
\(498\) −9.97415 −0.446952
\(499\) 15.0909 0.675562 0.337781 0.941225i \(-0.390324\pi\)
0.337781 + 0.941225i \(0.390324\pi\)
\(500\) −33.0120 −1.47634
\(501\) −26.7503 −1.19511
\(502\) −11.8816 −0.530301
\(503\) 18.0593 0.805223 0.402611 0.915371i \(-0.368103\pi\)
0.402611 + 0.915371i \(0.368103\pi\)
\(504\) 9.94865 0.443148
\(505\) 7.49394 0.333476
\(506\) 13.3333 0.592738
\(507\) −28.9884 −1.28742
\(508\) −49.1937 −2.18262
\(509\) 16.0031 0.709327 0.354663 0.934994i \(-0.384596\pi\)
0.354663 + 0.934994i \(0.384596\pi\)
\(510\) −22.5104 −0.996779
\(511\) 25.4959 1.12787
\(512\) −21.0660 −0.930997
\(513\) 14.4569 0.638289
\(514\) 47.3297 2.08762
\(515\) −3.19002 −0.140569
\(516\) 38.0157 1.67355
\(517\) 0.556444 0.0244724
\(518\) −6.66790 −0.292971
\(519\) −39.6047 −1.73846
\(520\) 0.570432 0.0250151
\(521\) −29.4987 −1.29236 −0.646180 0.763185i \(-0.723635\pi\)
−0.646180 + 0.763185i \(0.723635\pi\)
\(522\) −9.58035 −0.419321
\(523\) 20.1316 0.880295 0.440148 0.897925i \(-0.354926\pi\)
0.440148 + 0.897925i \(0.354926\pi\)
\(524\) −29.7469 −1.29950
\(525\) 15.2794 0.666847
\(526\) 28.8543 1.25811
\(527\) 19.1863 0.835768
\(528\) 6.92096 0.301196
\(529\) −7.94630 −0.345491
\(530\) 48.9621 2.12678
\(531\) 6.75553 0.293165
\(532\) 54.6393 2.36892
\(533\) −0.243326 −0.0105396
\(534\) −66.6945 −2.88615
\(535\) 7.65351 0.330890
\(536\) 3.11452 0.134527
\(537\) −33.4328 −1.44273
\(538\) −42.8201 −1.84611
\(539\) −3.74432 −0.161279
\(540\) −10.1466 −0.436642
\(541\) 20.4381 0.878704 0.439352 0.898315i \(-0.355208\pi\)
0.439352 + 0.898315i \(0.355208\pi\)
\(542\) −48.4683 −2.08189
\(543\) 13.3092 0.571153
\(544\) 20.8385 0.893444
\(545\) −20.8143 −0.891587
\(546\) −3.15826 −0.135161
\(547\) 44.5050 1.90290 0.951448 0.307810i \(-0.0995960\pi\)
0.951448 + 0.307810i \(0.0995960\pi\)
\(548\) 47.0990 2.01197
\(549\) 2.87497 0.122701
\(550\) −7.66480 −0.326828
\(551\) −14.2590 −0.607455
\(552\) −14.0579 −0.598343
\(553\) 23.4711 0.998095
\(554\) −5.06410 −0.215153
\(555\) −3.72382 −0.158067
\(556\) −47.4337 −2.01164
\(557\) 18.7315 0.793679 0.396839 0.917888i \(-0.370107\pi\)
0.396839 + 0.917888i \(0.370107\pi\)
\(558\) −30.2134 −1.27903
\(559\) −1.31087 −0.0554439
\(560\) 9.98771 0.422058
\(561\) −9.79930 −0.413727
\(562\) −39.9857 −1.68669
\(563\) 1.82548 0.0769348 0.0384674 0.999260i \(-0.487752\pi\)
0.0384674 + 0.999260i \(0.487752\pi\)
\(564\) −2.16489 −0.0911584
\(565\) 17.9267 0.754183
\(566\) 12.7371 0.535379
\(567\) 33.6560 1.41342
\(568\) −9.72404 −0.408012
\(569\) −27.0464 −1.13384 −0.566922 0.823772i \(-0.691866\pi\)
−0.566922 + 0.823772i \(0.691866\pi\)
\(570\) 52.7595 2.20985
\(571\) 30.8653 1.29167 0.645835 0.763477i \(-0.276509\pi\)
0.645835 + 0.763477i \(0.276509\pi\)
\(572\) 0.916322 0.0383134
\(573\) −36.2641 −1.51495
\(574\) 7.66476 0.319921
\(575\) −8.65379 −0.360888
\(576\) −24.9474 −1.03947
\(577\) 30.3918 1.26523 0.632614 0.774467i \(-0.281982\pi\)
0.632614 + 0.774467i \(0.281982\pi\)
\(578\) 20.2471 0.842169
\(579\) −29.8492 −1.24049
\(580\) 10.0077 0.415549
\(581\) −6.26595 −0.259955
\(582\) 22.5977 0.936706
\(583\) 21.3143 0.882749
\(584\) 13.4849 0.558009
\(585\) −0.706957 −0.0292291
\(586\) 44.6185 1.84317
\(587\) 44.9641 1.85587 0.927934 0.372745i \(-0.121584\pi\)
0.927934 + 0.372745i \(0.121584\pi\)
\(588\) 14.5676 0.600757
\(589\) −44.9684 −1.85289
\(590\) −12.2014 −0.502323
\(591\) −30.5037 −1.25475
\(592\) 1.96028 0.0805670
\(593\) 41.0025 1.68377 0.841885 0.539657i \(-0.181446\pi\)
0.841885 + 0.539657i \(0.181446\pi\)
\(594\) −7.63711 −0.313354
\(595\) −14.1415 −0.579744
\(596\) −31.2804 −1.28129
\(597\) 40.5001 1.65756
\(598\) 1.78875 0.0731473
\(599\) 5.24433 0.214278 0.107139 0.994244i \(-0.465831\pi\)
0.107139 + 0.994244i \(0.465831\pi\)
\(600\) 8.08133 0.329919
\(601\) −23.9038 −0.975057 −0.487528 0.873107i \(-0.662101\pi\)
−0.487528 + 0.873107i \(0.662101\pi\)
\(602\) 41.2923 1.68295
\(603\) −3.85994 −0.157189
\(604\) 3.62008 0.147299
\(605\) −14.1630 −0.575809
\(606\) −21.9448 −0.891446
\(607\) −47.8565 −1.94244 −0.971218 0.238193i \(-0.923445\pi\)
−0.971218 + 0.238193i \(0.923445\pi\)
\(608\) −48.8409 −1.98076
\(609\) −15.0158 −0.608470
\(610\) −5.19257 −0.210241
\(611\) 0.0746505 0.00302004
\(612\) 15.2811 0.617703
\(613\) −35.4248 −1.43080 −0.715398 0.698717i \(-0.753754\pi\)
−0.715398 + 0.698717i \(0.753754\pi\)
\(614\) 29.1151 1.17499
\(615\) 4.28053 0.172608
\(616\) −7.82214 −0.315163
\(617\) −47.0432 −1.89389 −0.946944 0.321400i \(-0.895847\pi\)
−0.946944 + 0.321400i \(0.895847\pi\)
\(618\) 9.34145 0.375768
\(619\) 11.8891 0.477862 0.238931 0.971037i \(-0.423203\pi\)
0.238931 + 0.971037i \(0.423203\pi\)
\(620\) 31.5612 1.26753
\(621\) −8.62252 −0.346010
\(622\) −30.6976 −1.23086
\(623\) −41.8988 −1.67864
\(624\) 0.928491 0.0371694
\(625\) −8.87322 −0.354929
\(626\) −36.1753 −1.44585
\(627\) 22.9674 0.917229
\(628\) −44.6315 −1.78099
\(629\) −2.77554 −0.110668
\(630\) 22.2691 0.887223
\(631\) 20.8728 0.830932 0.415466 0.909609i \(-0.363618\pi\)
0.415466 + 0.909609i \(0.363618\pi\)
\(632\) 12.4140 0.493802
\(633\) 5.52443 0.219576
\(634\) 1.22818 0.0487774
\(635\) −29.8412 −1.18421
\(636\) −82.9251 −3.28820
\(637\) −0.502324 −0.0199028
\(638\) 7.53256 0.298217
\(639\) 12.0514 0.476744
\(640\) 20.0688 0.793287
\(641\) 6.55250 0.258808 0.129404 0.991592i \(-0.458694\pi\)
0.129404 + 0.991592i \(0.458694\pi\)
\(642\) −22.4121 −0.884534
\(643\) 0.298843 0.0117852 0.00589260 0.999983i \(-0.498124\pi\)
0.00589260 + 0.999983i \(0.498124\pi\)
\(644\) −32.5884 −1.28416
\(645\) 23.0605 0.908007
\(646\) 39.3241 1.54719
\(647\) −27.4302 −1.07839 −0.539196 0.842180i \(-0.681272\pi\)
−0.539196 + 0.842180i \(0.681272\pi\)
\(648\) 17.8008 0.699281
\(649\) −5.31154 −0.208496
\(650\) −1.02828 −0.0403325
\(651\) −47.3549 −1.85599
\(652\) −2.74348 −0.107443
\(653\) 16.8664 0.660032 0.330016 0.943975i \(-0.392946\pi\)
0.330016 + 0.943975i \(0.392946\pi\)
\(654\) 60.9514 2.38339
\(655\) −18.0446 −0.705061
\(656\) −2.25335 −0.0879784
\(657\) −16.7123 −0.652010
\(658\) −2.35149 −0.0916705
\(659\) 14.9053 0.580629 0.290314 0.956931i \(-0.406240\pi\)
0.290314 + 0.956931i \(0.406240\pi\)
\(660\) −16.1197 −0.627460
\(661\) 28.9641 1.12657 0.563287 0.826261i \(-0.309536\pi\)
0.563287 + 0.826261i \(0.309536\pi\)
\(662\) −52.6702 −2.04708
\(663\) −1.31464 −0.0510564
\(664\) −3.31409 −0.128612
\(665\) 33.1445 1.28529
\(666\) 4.37074 0.169363
\(667\) 8.50449 0.329295
\(668\) −32.7981 −1.26900
\(669\) 21.7256 0.839960
\(670\) 6.97157 0.269335
\(671\) −2.26045 −0.0872635
\(672\) −51.4329 −1.98407
\(673\) −22.1710 −0.854629 −0.427314 0.904103i \(-0.640540\pi\)
−0.427314 + 0.904103i \(0.640540\pi\)
\(674\) −30.2021 −1.16334
\(675\) 4.95675 0.190786
\(676\) −35.5423 −1.36701
\(677\) −7.24291 −0.278368 −0.139184 0.990267i \(-0.544448\pi\)
−0.139184 + 0.990267i \(0.544448\pi\)
\(678\) −52.4955 −2.01608
\(679\) 14.1963 0.544805
\(680\) −7.47949 −0.286825
\(681\) −54.8545 −2.10203
\(682\) 23.7553 0.909637
\(683\) −30.9066 −1.18261 −0.591305 0.806448i \(-0.701387\pi\)
−0.591305 + 0.806448i \(0.701387\pi\)
\(684\) −35.8155 −1.36944
\(685\) 28.5705 1.09162
\(686\) −30.8521 −1.17794
\(687\) −57.6466 −2.19936
\(688\) −12.1394 −0.462812
\(689\) 2.85945 0.108936
\(690\) −31.4672 −1.19794
\(691\) −25.6614 −0.976206 −0.488103 0.872786i \(-0.662311\pi\)
−0.488103 + 0.872786i \(0.662311\pi\)
\(692\) −48.5588 −1.84593
\(693\) 9.69425 0.368254
\(694\) −19.0161 −0.721842
\(695\) −28.7735 −1.09144
\(696\) −7.94190 −0.301037
\(697\) 3.19048 0.120848
\(698\) −55.7994 −2.11204
\(699\) 7.42817 0.280959
\(700\) 18.7338 0.708072
\(701\) 2.09900 0.0792781 0.0396390 0.999214i \(-0.487379\pi\)
0.0396390 + 0.999214i \(0.487379\pi\)
\(702\) −1.02457 −0.0386698
\(703\) 6.50524 0.245350
\(704\) 19.6149 0.739264
\(705\) −1.31323 −0.0494593
\(706\) −33.3605 −1.25554
\(707\) −13.7861 −0.518481
\(708\) 20.6650 0.776638
\(709\) −5.27148 −0.197974 −0.0989872 0.995089i \(-0.531560\pi\)
−0.0989872 + 0.995089i \(0.531560\pi\)
\(710\) −21.7664 −0.816877
\(711\) −15.3851 −0.576987
\(712\) −22.1604 −0.830498
\(713\) 26.8204 1.00443
\(714\) 41.4111 1.54977
\(715\) 0.555846 0.0207875
\(716\) −40.9915 −1.53193
\(717\) −62.2068 −2.32315
\(718\) 22.3913 0.835634
\(719\) 19.7331 0.735920 0.367960 0.929842i \(-0.380056\pi\)
0.367960 + 0.929842i \(0.380056\pi\)
\(720\) −6.54685 −0.243987
\(721\) 5.86848 0.218554
\(722\) −50.7859 −1.89006
\(723\) 26.1680 0.973200
\(724\) 16.3183 0.606463
\(725\) −4.88890 −0.181569
\(726\) 41.4741 1.53925
\(727\) 11.9747 0.444118 0.222059 0.975033i \(-0.428722\pi\)
0.222059 + 0.975033i \(0.428722\pi\)
\(728\) −1.04939 −0.0388930
\(729\) −7.14303 −0.264557
\(730\) 30.1847 1.11719
\(731\) 17.1881 0.635724
\(732\) 8.79445 0.325052
\(733\) 44.7076 1.65131 0.825656 0.564174i \(-0.190805\pi\)
0.825656 + 0.564174i \(0.190805\pi\)
\(734\) 15.5587 0.574282
\(735\) 8.83677 0.325949
\(736\) 29.1301 1.07375
\(737\) 3.03488 0.111791
\(738\) −5.02417 −0.184942
\(739\) 37.9602 1.39639 0.698194 0.715908i \(-0.253987\pi\)
0.698194 + 0.715908i \(0.253987\pi\)
\(740\) −4.56572 −0.167839
\(741\) 3.08122 0.113192
\(742\) −90.0726 −3.30667
\(743\) −5.70078 −0.209141 −0.104571 0.994517i \(-0.533347\pi\)
−0.104571 + 0.994517i \(0.533347\pi\)
\(744\) −25.0462 −0.918239
\(745\) −18.9748 −0.695184
\(746\) 6.84118 0.250473
\(747\) 4.10727 0.150277
\(748\) −12.0148 −0.439304
\(749\) −14.0797 −0.514461
\(750\) 58.6408 2.14126
\(751\) −20.6289 −0.752760 −0.376380 0.926465i \(-0.622831\pi\)
−0.376380 + 0.926465i \(0.622831\pi\)
\(752\) 0.691309 0.0252094
\(753\) 12.2069 0.444846
\(754\) 1.01054 0.0368017
\(755\) 2.19596 0.0799192
\(756\) 18.6661 0.678881
\(757\) 18.2338 0.662720 0.331360 0.943504i \(-0.392493\pi\)
0.331360 + 0.943504i \(0.392493\pi\)
\(758\) −18.2276 −0.662056
\(759\) −13.6984 −0.497221
\(760\) 17.5303 0.635890
\(761\) −27.4534 −0.995184 −0.497592 0.867411i \(-0.665782\pi\)
−0.497592 + 0.867411i \(0.665782\pi\)
\(762\) 87.3851 3.16563
\(763\) 38.2908 1.38622
\(764\) −44.4629 −1.60861
\(765\) 9.26960 0.335143
\(766\) −28.9183 −1.04486
\(767\) −0.712577 −0.0257297
\(768\) −3.13560 −0.113146
\(769\) −31.0418 −1.11940 −0.559699 0.828696i \(-0.689083\pi\)
−0.559699 + 0.828696i \(0.689083\pi\)
\(770\) −17.5091 −0.630985
\(771\) −48.6257 −1.75121
\(772\) −36.5978 −1.31718
\(773\) 21.1360 0.760210 0.380105 0.924943i \(-0.375888\pi\)
0.380105 + 0.924943i \(0.375888\pi\)
\(774\) −27.0667 −0.972893
\(775\) −15.4180 −0.553832
\(776\) 7.50850 0.269539
\(777\) 6.85048 0.245760
\(778\) 70.2805 2.51968
\(779\) −7.47779 −0.267920
\(780\) −2.16256 −0.0774322
\(781\) −9.47539 −0.339056
\(782\) −23.4540 −0.838714
\(783\) −4.87124 −0.174084
\(784\) −4.65183 −0.166137
\(785\) −27.0737 −0.966302
\(786\) 52.8407 1.88477
\(787\) 39.9800 1.42513 0.712567 0.701604i \(-0.247532\pi\)
0.712567 + 0.701604i \(0.247532\pi\)
\(788\) −37.4002 −1.33233
\(789\) −29.6444 −1.05537
\(790\) 27.7876 0.988637
\(791\) −32.9787 −1.17259
\(792\) 5.12733 0.182192
\(793\) −0.303253 −0.0107688
\(794\) −72.6845 −2.57948
\(795\) −50.3028 −1.78406
\(796\) 49.6566 1.76003
\(797\) −50.5254 −1.78970 −0.894850 0.446367i \(-0.852718\pi\)
−0.894850 + 0.446367i \(0.852718\pi\)
\(798\) −97.0583 −3.43583
\(799\) −0.978816 −0.0346280
\(800\) −16.7457 −0.592052
\(801\) 27.4642 0.970401
\(802\) 43.0733 1.52097
\(803\) 13.1401 0.463704
\(804\) −11.8075 −0.416417
\(805\) −19.7683 −0.696742
\(806\) 3.18692 0.112255
\(807\) 43.9926 1.54861
\(808\) −7.29155 −0.256516
\(809\) −14.8578 −0.522371 −0.261186 0.965289i \(-0.584113\pi\)
−0.261186 + 0.965289i \(0.584113\pi\)
\(810\) 39.8454 1.40002
\(811\) −9.81834 −0.344769 −0.172384 0.985030i \(-0.555147\pi\)
−0.172384 + 0.985030i \(0.555147\pi\)
\(812\) −18.4106 −0.646086
\(813\) 49.7954 1.74640
\(814\) −3.43650 −0.120449
\(815\) −1.66421 −0.0582947
\(816\) −12.1743 −0.426187
\(817\) −40.2851 −1.40940
\(818\) 78.0068 2.72744
\(819\) 1.30055 0.0454448
\(820\) 5.24831 0.183279
\(821\) −46.2810 −1.61522 −0.807609 0.589719i \(-0.799239\pi\)
−0.807609 + 0.589719i \(0.799239\pi\)
\(822\) −83.6641 −2.91812
\(823\) −37.3876 −1.30325 −0.651624 0.758542i \(-0.725912\pi\)
−0.651624 + 0.758542i \(0.725912\pi\)
\(824\) 3.10386 0.108128
\(825\) 7.87468 0.274161
\(826\) 22.4461 0.781001
\(827\) 20.1688 0.701340 0.350670 0.936499i \(-0.385954\pi\)
0.350670 + 0.936499i \(0.385954\pi\)
\(828\) 21.3614 0.742360
\(829\) −23.9360 −0.831331 −0.415665 0.909518i \(-0.636451\pi\)
−0.415665 + 0.909518i \(0.636451\pi\)
\(830\) −7.41827 −0.257492
\(831\) 5.20276 0.180482
\(832\) 2.63146 0.0912295
\(833\) 6.58646 0.228207
\(834\) 84.2586 2.91764
\(835\) −19.8955 −0.688513
\(836\) 28.1600 0.973934
\(837\) −15.3623 −0.530999
\(838\) −11.9216 −0.411824
\(839\) −23.9225 −0.825895 −0.412948 0.910755i \(-0.635501\pi\)
−0.412948 + 0.910755i \(0.635501\pi\)
\(840\) 18.4606 0.636951
\(841\) −24.1954 −0.834326
\(842\) 3.14752 0.108471
\(843\) 41.0806 1.41489
\(844\) 6.77343 0.233151
\(845\) −21.5601 −0.741691
\(846\) 1.54138 0.0529936
\(847\) 26.0548 0.895255
\(848\) 26.4802 0.909335
\(849\) −13.0858 −0.449105
\(850\) 13.4828 0.462456
\(851\) −3.87991 −0.133002
\(852\) 36.8648 1.26297
\(853\) 36.5859 1.25268 0.626339 0.779551i \(-0.284553\pi\)
0.626339 + 0.779551i \(0.284553\pi\)
\(854\) 9.55246 0.326878
\(855\) −21.7259 −0.743010
\(856\) −7.44681 −0.254527
\(857\) −2.64142 −0.0902292 −0.0451146 0.998982i \(-0.514365\pi\)
−0.0451146 + 0.998982i \(0.514365\pi\)
\(858\) −1.62771 −0.0555689
\(859\) −25.5410 −0.871447 −0.435723 0.900081i \(-0.643507\pi\)
−0.435723 + 0.900081i \(0.643507\pi\)
\(860\) 28.2742 0.964141
\(861\) −7.87464 −0.268367
\(862\) −5.79865 −0.197503
\(863\) 16.9554 0.577168 0.288584 0.957455i \(-0.406816\pi\)
0.288584 + 0.957455i \(0.406816\pi\)
\(864\) −16.6852 −0.567643
\(865\) −29.4560 −1.00154
\(866\) −46.0550 −1.56501
\(867\) −20.8015 −0.706457
\(868\) −58.0612 −1.97073
\(869\) 12.0966 0.410348
\(870\) −17.7772 −0.602704
\(871\) 0.407149 0.0137957
\(872\) 20.2522 0.685825
\(873\) −9.30555 −0.314945
\(874\) 54.9710 1.85942
\(875\) 36.8393 1.24539
\(876\) −51.1226 −1.72727
\(877\) 25.8789 0.873868 0.436934 0.899494i \(-0.356064\pi\)
0.436934 + 0.899494i \(0.356064\pi\)
\(878\) 2.11285 0.0713053
\(879\) −45.8403 −1.54615
\(880\) 5.14747 0.173521
\(881\) −21.4890 −0.723984 −0.361992 0.932181i \(-0.617903\pi\)
−0.361992 + 0.932181i \(0.617903\pi\)
\(882\) −10.3719 −0.349242
\(883\) −27.4683 −0.924383 −0.462192 0.886780i \(-0.652937\pi\)
−0.462192 + 0.886780i \(0.652937\pi\)
\(884\) −1.61186 −0.0542128
\(885\) 12.5355 0.421376
\(886\) −13.1962 −0.443334
\(887\) −23.2596 −0.780980 −0.390490 0.920607i \(-0.627694\pi\)
−0.390490 + 0.920607i \(0.627694\pi\)
\(888\) 3.62325 0.121588
\(889\) 54.8970 1.84119
\(890\) −49.6041 −1.66273
\(891\) 17.3456 0.581100
\(892\) 26.6375 0.891888
\(893\) 2.29413 0.0767700
\(894\) 55.5647 1.85836
\(895\) −24.8657 −0.831168
\(896\) −36.9193 −1.23339
\(897\) −1.83773 −0.0613600
\(898\) 44.0039 1.46843
\(899\) 15.1520 0.505349
\(900\) −12.2798 −0.409328
\(901\) −37.4931 −1.24908
\(902\) 3.95026 0.131529
\(903\) −42.4230 −1.41175
\(904\) −17.4426 −0.580131
\(905\) 9.89873 0.329045
\(906\) −6.43052 −0.213640
\(907\) 20.6327 0.685099 0.342550 0.939500i \(-0.388710\pi\)
0.342550 + 0.939500i \(0.388710\pi\)
\(908\) −67.2564 −2.23198
\(909\) 9.03668 0.299728
\(910\) −2.34896 −0.0778672
\(911\) 8.48694 0.281185 0.140592 0.990068i \(-0.455099\pi\)
0.140592 + 0.990068i \(0.455099\pi\)
\(912\) 28.5340 0.944854
\(913\) −3.22934 −0.106876
\(914\) 75.4719 2.49639
\(915\) 5.33476 0.176362
\(916\) −70.6798 −2.33533
\(917\) 33.1956 1.09621
\(918\) 13.4341 0.443391
\(919\) −45.0377 −1.48566 −0.742829 0.669481i \(-0.766516\pi\)
−0.742829 + 0.669481i \(0.766516\pi\)
\(920\) −10.4555 −0.344709
\(921\) −29.9123 −0.985644
\(922\) −61.4299 −2.02309
\(923\) −1.27118 −0.0418415
\(924\) 29.6545 0.975561
\(925\) 2.23041 0.0733354
\(926\) 25.8525 0.849566
\(927\) −3.84673 −0.126343
\(928\) 16.4568 0.540222
\(929\) 18.5910 0.609950 0.304975 0.952360i \(-0.401352\pi\)
0.304975 + 0.952360i \(0.401352\pi\)
\(930\) −56.0636 −1.83840
\(931\) −15.4372 −0.505934
\(932\) 9.10757 0.298329
\(933\) 31.5381 1.03251
\(934\) 36.6560 1.19942
\(935\) −7.28824 −0.238351
\(936\) 0.687864 0.0224835
\(937\) −0.172970 −0.00565069 −0.00282534 0.999996i \(-0.500899\pi\)
−0.00282534 + 0.999996i \(0.500899\pi\)
\(938\) −12.8252 −0.418757
\(939\) 37.1658 1.21286
\(940\) −1.61014 −0.0525169
\(941\) −36.8477 −1.20120 −0.600601 0.799549i \(-0.705072\pi\)
−0.600601 + 0.799549i \(0.705072\pi\)
\(942\) 79.2810 2.58311
\(943\) 4.45996 0.145236
\(944\) −6.59889 −0.214776
\(945\) 11.3230 0.368336
\(946\) 21.2812 0.691912
\(947\) 57.3352 1.86314 0.931572 0.363558i \(-0.118438\pi\)
0.931572 + 0.363558i \(0.118438\pi\)
\(948\) −47.0626 −1.52852
\(949\) 1.76283 0.0572238
\(950\) −31.6007 −1.02526
\(951\) −1.26181 −0.0409171
\(952\) 13.7596 0.445950
\(953\) −12.9136 −0.418312 −0.209156 0.977882i \(-0.567072\pi\)
−0.209156 + 0.977882i \(0.567072\pi\)
\(954\) 59.0417 1.91155
\(955\) −26.9714 −0.872775
\(956\) −76.2709 −2.46678
\(957\) −7.73882 −0.250161
\(958\) −45.8774 −1.48223
\(959\) −52.5594 −1.69723
\(960\) −46.2921 −1.49407
\(961\) 16.7846 0.541440
\(962\) −0.461028 −0.0148641
\(963\) 9.22910 0.297404
\(964\) 32.0843 1.03337
\(965\) −22.2004 −0.714656
\(966\) 57.8883 1.86253
\(967\) −18.3592 −0.590392 −0.295196 0.955437i \(-0.595385\pi\)
−0.295196 + 0.955437i \(0.595385\pi\)
\(968\) 13.7805 0.442923
\(969\) −40.4009 −1.29786
\(970\) 16.8071 0.539642
\(971\) −41.6272 −1.33588 −0.667941 0.744214i \(-0.732824\pi\)
−0.667941 + 0.744214i \(0.732824\pi\)
\(972\) −49.1936 −1.57789
\(973\) 52.9328 1.69695
\(974\) −8.79416 −0.281783
\(975\) 1.05644 0.0338331
\(976\) −2.80831 −0.0898917
\(977\) 31.8934 1.02036 0.510180 0.860068i \(-0.329579\pi\)
0.510180 + 0.860068i \(0.329579\pi\)
\(978\) 4.87337 0.155833
\(979\) −21.5938 −0.690140
\(980\) 10.8346 0.346100
\(981\) −25.0992 −0.801357
\(982\) 21.0975 0.673247
\(983\) 41.1876 1.31368 0.656840 0.754030i \(-0.271893\pi\)
0.656840 + 0.754030i \(0.271893\pi\)
\(984\) −4.16493 −0.132773
\(985\) −22.6871 −0.722872
\(986\) −13.2502 −0.421972
\(987\) 2.41588 0.0768982
\(988\) 3.77785 0.120189
\(989\) 24.0271 0.764019
\(990\) 11.4771 0.364765
\(991\) −16.4554 −0.522723 −0.261362 0.965241i \(-0.584171\pi\)
−0.261362 + 0.965241i \(0.584171\pi\)
\(992\) 51.8996 1.64781
\(993\) 54.1124 1.71720
\(994\) 40.0422 1.27006
\(995\) 30.1219 0.954929
\(996\) 12.5640 0.398106
\(997\) −55.5583 −1.75955 −0.879775 0.475391i \(-0.842307\pi\)
−0.879775 + 0.475391i \(0.842307\pi\)
\(998\) −32.8673 −1.04040
\(999\) 2.22235 0.0703121
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 6031.2.a.c.1.16 110
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6031.2.a.c.1.16 110 1.1 even 1 trivial