Properties

Label 6027.2.a.bb.1.3
Level $6027$
Weight $2$
Character 6027.1
Self dual yes
Analytic conductor $48.126$
Analytic rank $1$
Dimension $8$
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] = [6027,2,Mod(1,6027)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6027, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6027.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6027.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: \(48.1258372982\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.7457527933.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 6x^{6} + 23x^{5} - 4x^{4} - 27x^{3} + 8x^{2} + 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 861)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.21768\) of defining polynomial
Character \(\chi\) \(=\) 6027.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.51387 q^{2} -1.00000 q^{3} +0.291794 q^{4} +3.80505 q^{5} +1.51387 q^{6} +2.58600 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.51387 q^{2} -1.00000 q^{3} +0.291794 q^{4} +3.80505 q^{5} +1.51387 q^{6} +2.58600 q^{8} +1.00000 q^{9} -5.76034 q^{10} +1.15660 q^{11} -0.291794 q^{12} -4.56081 q^{13} -3.80505 q^{15} -4.49844 q^{16} -6.92527 q^{17} -1.51387 q^{18} +2.40606 q^{19} +1.11029 q^{20} -1.75093 q^{22} +3.28488 q^{23} -2.58600 q^{24} +9.47840 q^{25} +6.90445 q^{26} -1.00000 q^{27} +1.28140 q^{29} +5.76034 q^{30} -0.334671 q^{31} +1.63805 q^{32} -1.15660 q^{33} +10.4839 q^{34} +0.291794 q^{36} +1.13227 q^{37} -3.64245 q^{38} +4.56081 q^{39} +9.83985 q^{40} -1.00000 q^{41} +2.12116 q^{43} +0.337487 q^{44} +3.80505 q^{45} -4.97287 q^{46} -7.39779 q^{47} +4.49844 q^{48} -14.3490 q^{50} +6.92527 q^{51} -1.33082 q^{52} -2.25189 q^{53} +1.51387 q^{54} +4.40090 q^{55} -2.40606 q^{57} -1.93987 q^{58} -12.9132 q^{59} -1.11029 q^{60} +2.57480 q^{61} +0.506648 q^{62} +6.51709 q^{64} -17.3541 q^{65} +1.75093 q^{66} -6.21525 q^{67} -2.02075 q^{68} -3.28488 q^{69} +2.30854 q^{71} +2.58600 q^{72} +11.9079 q^{73} -1.71411 q^{74} -9.47840 q^{75} +0.702072 q^{76} -6.90445 q^{78} -12.7517 q^{79} -17.1168 q^{80} +1.00000 q^{81} +1.51387 q^{82} +2.58930 q^{83} -26.3510 q^{85} -3.21115 q^{86} -1.28140 q^{87} +2.99095 q^{88} -9.21039 q^{89} -5.76034 q^{90} +0.958507 q^{92} +0.334671 q^{93} +11.1993 q^{94} +9.15516 q^{95} -1.63805 q^{96} -12.8974 q^{97} +1.15660 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 8 q^{3} + 13 q^{4} - 7 q^{5} - q^{6} + 6 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} - 8 q^{3} + 13 q^{4} - 7 q^{5} - q^{6} + 6 q^{8} + 8 q^{9} - 8 q^{10} + 11 q^{11} - 13 q^{12} - 10 q^{13} + 7 q^{15} - 17 q^{16} - 3 q^{17} + q^{18} - 6 q^{19} - 11 q^{20} + 15 q^{22} + 14 q^{23} - 6 q^{24} + 25 q^{25} - 24 q^{26} - 8 q^{27} + 2 q^{29} + 8 q^{30} - 16 q^{31} + 3 q^{32} - 11 q^{33} + 4 q^{34} + 13 q^{36} - 20 q^{37} - 10 q^{38} + 10 q^{39} + 3 q^{40} - 8 q^{41} + 7 q^{43} - 7 q^{45} - 5 q^{46} - 14 q^{47} + 17 q^{48} - 5 q^{50} + 3 q^{51} - 23 q^{52} + 7 q^{53} - q^{54} - 48 q^{55} + 6 q^{57} - 20 q^{58} - 22 q^{59} + 11 q^{60} + 33 q^{62} - 10 q^{64} - 14 q^{65} - 15 q^{66} + 12 q^{67} + 27 q^{68} - 14 q^{69} - 5 q^{71} + 6 q^{72} - 2 q^{73} + 6 q^{74} - 25 q^{75} - 43 q^{76} + 24 q^{78} - 15 q^{79} + 7 q^{80} + 8 q^{81} - q^{82} - 15 q^{83} - 43 q^{85} + 31 q^{86} - 2 q^{87} + 17 q^{88} - 29 q^{89} - 8 q^{90} + 19 q^{92} + 16 q^{93} - 20 q^{94} + 14 q^{95} - 3 q^{96} - 19 q^{97} + 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.51387 −1.07047 −0.535233 0.844705i \(-0.679776\pi\)
−0.535233 + 0.844705i \(0.679776\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.291794 0.145897
\(5\) 3.80505 1.70167 0.850835 0.525433i \(-0.176097\pi\)
0.850835 + 0.525433i \(0.176097\pi\)
\(6\) 1.51387 0.618034
\(7\) 0 0
\(8\) 2.58600 0.914288
\(9\) 1.00000 0.333333
\(10\) −5.76034 −1.82158
\(11\) 1.15660 0.348727 0.174363 0.984681i \(-0.444213\pi\)
0.174363 + 0.984681i \(0.444213\pi\)
\(12\) −0.291794 −0.0842336
\(13\) −4.56081 −1.26494 −0.632470 0.774585i \(-0.717959\pi\)
−0.632470 + 0.774585i \(0.717959\pi\)
\(14\) 0 0
\(15\) −3.80505 −0.982460
\(16\) −4.49844 −1.12461
\(17\) −6.92527 −1.67963 −0.839813 0.542876i \(-0.817335\pi\)
−0.839813 + 0.542876i \(0.817335\pi\)
\(18\) −1.51387 −0.356822
\(19\) 2.40606 0.551987 0.275994 0.961159i \(-0.410993\pi\)
0.275994 + 0.961159i \(0.410993\pi\)
\(20\) 1.11029 0.248268
\(21\) 0 0
\(22\) −1.75093 −0.373300
\(23\) 3.28488 0.684944 0.342472 0.939528i \(-0.388736\pi\)
0.342472 + 0.939528i \(0.388736\pi\)
\(24\) −2.58600 −0.527864
\(25\) 9.47840 1.89568
\(26\) 6.90445 1.35407
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.28140 0.237950 0.118975 0.992897i \(-0.462039\pi\)
0.118975 + 0.992897i \(0.462039\pi\)
\(30\) 5.76034 1.05169
\(31\) −0.334671 −0.0601087 −0.0300544 0.999548i \(-0.509568\pi\)
−0.0300544 + 0.999548i \(0.509568\pi\)
\(32\) 1.63805 0.289569
\(33\) −1.15660 −0.201337
\(34\) 10.4839 1.79798
\(35\) 0 0
\(36\) 0.291794 0.0486323
\(37\) 1.13227 0.186145 0.0930724 0.995659i \(-0.470331\pi\)
0.0930724 + 0.995659i \(0.470331\pi\)
\(38\) −3.64245 −0.590883
\(39\) 4.56081 0.730313
\(40\) 9.83985 1.55582
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) 2.12116 0.323474 0.161737 0.986834i \(-0.448290\pi\)
0.161737 + 0.986834i \(0.448290\pi\)
\(44\) 0.337487 0.0508781
\(45\) 3.80505 0.567223
\(46\) −4.97287 −0.733209
\(47\) −7.39779 −1.07908 −0.539540 0.841960i \(-0.681402\pi\)
−0.539540 + 0.841960i \(0.681402\pi\)
\(48\) 4.49844 0.649294
\(49\) 0 0
\(50\) −14.3490 −2.02926
\(51\) 6.92527 0.969732
\(52\) −1.33082 −0.184551
\(53\) −2.25189 −0.309322 −0.154661 0.987968i \(-0.549428\pi\)
−0.154661 + 0.987968i \(0.549428\pi\)
\(54\) 1.51387 0.206011
\(55\) 4.40090 0.593418
\(56\) 0 0
\(57\) −2.40606 −0.318690
\(58\) −1.93987 −0.254717
\(59\) −12.9132 −1.68115 −0.840575 0.541696i \(-0.817782\pi\)
−0.840575 + 0.541696i \(0.817782\pi\)
\(60\) −1.11029 −0.143338
\(61\) 2.57480 0.329669 0.164834 0.986321i \(-0.447291\pi\)
0.164834 + 0.986321i \(0.447291\pi\)
\(62\) 0.506648 0.0643443
\(63\) 0 0
\(64\) 6.51709 0.814637
\(65\) −17.3541 −2.15251
\(66\) 1.75093 0.215525
\(67\) −6.21525 −0.759313 −0.379657 0.925127i \(-0.623958\pi\)
−0.379657 + 0.925127i \(0.623958\pi\)
\(68\) −2.02075 −0.245052
\(69\) −3.28488 −0.395453
\(70\) 0 0
\(71\) 2.30854 0.273973 0.136986 0.990573i \(-0.456258\pi\)
0.136986 + 0.990573i \(0.456258\pi\)
\(72\) 2.58600 0.304763
\(73\) 11.9079 1.39371 0.696855 0.717212i \(-0.254582\pi\)
0.696855 + 0.717212i \(0.254582\pi\)
\(74\) −1.71411 −0.199262
\(75\) −9.47840 −1.09447
\(76\) 0.702072 0.0805332
\(77\) 0 0
\(78\) −6.90445 −0.781775
\(79\) −12.7517 −1.43467 −0.717337 0.696727i \(-0.754639\pi\)
−0.717337 + 0.696727i \(0.754639\pi\)
\(80\) −17.1168 −1.91372
\(81\) 1.00000 0.111111
\(82\) 1.51387 0.167179
\(83\) 2.58930 0.284213 0.142106 0.989851i \(-0.454612\pi\)
0.142106 + 0.989851i \(0.454612\pi\)
\(84\) 0 0
\(85\) −26.3510 −2.85817
\(86\) −3.21115 −0.346267
\(87\) −1.28140 −0.137380
\(88\) 2.99095 0.318837
\(89\) −9.21039 −0.976299 −0.488150 0.872760i \(-0.662328\pi\)
−0.488150 + 0.872760i \(0.662328\pi\)
\(90\) −5.76034 −0.607193
\(91\) 0 0
\(92\) 0.958507 0.0999312
\(93\) 0.334671 0.0347038
\(94\) 11.1993 1.15512
\(95\) 9.15516 0.939300
\(96\) −1.63805 −0.167183
\(97\) −12.8974 −1.30953 −0.654766 0.755832i \(-0.727233\pi\)
−0.654766 + 0.755832i \(0.727233\pi\)
\(98\) 0 0
\(99\) 1.15660 0.116242
\(100\) 2.76574 0.276574
\(101\) 0.177513 0.0176632 0.00883162 0.999961i \(-0.497189\pi\)
0.00883162 + 0.999961i \(0.497189\pi\)
\(102\) −10.4839 −1.03807
\(103\) −14.5341 −1.43209 −0.716046 0.698053i \(-0.754050\pi\)
−0.716046 + 0.698053i \(0.754050\pi\)
\(104\) −11.7942 −1.15652
\(105\) 0 0
\(106\) 3.40907 0.331118
\(107\) −12.0732 −1.16716 −0.583579 0.812056i \(-0.698348\pi\)
−0.583579 + 0.812056i \(0.698348\pi\)
\(108\) −0.291794 −0.0280779
\(109\) 17.0080 1.62907 0.814537 0.580111i \(-0.196991\pi\)
0.814537 + 0.580111i \(0.196991\pi\)
\(110\) −6.66238 −0.635233
\(111\) −1.13227 −0.107471
\(112\) 0 0
\(113\) 1.63814 0.154103 0.0770516 0.997027i \(-0.475449\pi\)
0.0770516 + 0.997027i \(0.475449\pi\)
\(114\) 3.64245 0.341147
\(115\) 12.4991 1.16555
\(116\) 0.373905 0.0347162
\(117\) −4.56081 −0.421647
\(118\) 19.5488 1.79961
\(119\) 0 0
\(120\) −9.83985 −0.898251
\(121\) −9.66229 −0.878390
\(122\) −3.89790 −0.352899
\(123\) 1.00000 0.0901670
\(124\) −0.0976550 −0.00876968
\(125\) 17.0405 1.52415
\(126\) 0 0
\(127\) −20.7696 −1.84300 −0.921502 0.388374i \(-0.873037\pi\)
−0.921502 + 0.388374i \(0.873037\pi\)
\(128\) −13.1421 −1.16161
\(129\) −2.12116 −0.186758
\(130\) 26.2718 2.30419
\(131\) 12.0351 1.05151 0.525756 0.850636i \(-0.323783\pi\)
0.525756 + 0.850636i \(0.323783\pi\)
\(132\) −0.337487 −0.0293745
\(133\) 0 0
\(134\) 9.40906 0.812819
\(135\) −3.80505 −0.327487
\(136\) −17.9087 −1.53566
\(137\) −6.50063 −0.555386 −0.277693 0.960670i \(-0.589570\pi\)
−0.277693 + 0.960670i \(0.589570\pi\)
\(138\) 4.97287 0.423318
\(139\) 23.1305 1.96191 0.980954 0.194243i \(-0.0622249\pi\)
0.980954 + 0.194243i \(0.0622249\pi\)
\(140\) 0 0
\(141\) 7.39779 0.623007
\(142\) −3.49482 −0.293279
\(143\) −5.27501 −0.441118
\(144\) −4.49844 −0.374870
\(145\) 4.87579 0.404912
\(146\) −18.0269 −1.49192
\(147\) 0 0
\(148\) 0.330391 0.0271580
\(149\) −5.20043 −0.426036 −0.213018 0.977048i \(-0.568329\pi\)
−0.213018 + 0.977048i \(0.568329\pi\)
\(150\) 14.3490 1.17159
\(151\) −10.2898 −0.837370 −0.418685 0.908131i \(-0.637509\pi\)
−0.418685 + 0.908131i \(0.637509\pi\)
\(152\) 6.22205 0.504675
\(153\) −6.92527 −0.559875
\(154\) 0 0
\(155\) −1.27344 −0.102285
\(156\) 1.33082 0.106550
\(157\) 4.03587 0.322097 0.161049 0.986946i \(-0.448512\pi\)
0.161049 + 0.986946i \(0.448512\pi\)
\(158\) 19.3043 1.53577
\(159\) 2.25189 0.178587
\(160\) 6.23287 0.492752
\(161\) 0 0
\(162\) −1.51387 −0.118941
\(163\) 6.84917 0.536468 0.268234 0.963354i \(-0.413560\pi\)
0.268234 + 0.963354i \(0.413560\pi\)
\(164\) −0.291794 −0.0227853
\(165\) −4.40090 −0.342610
\(166\) −3.91986 −0.304240
\(167\) 2.72776 0.211080 0.105540 0.994415i \(-0.466343\pi\)
0.105540 + 0.994415i \(0.466343\pi\)
\(168\) 0 0
\(169\) 7.80095 0.600073
\(170\) 39.8919 3.05957
\(171\) 2.40606 0.183996
\(172\) 0.618941 0.0471938
\(173\) −3.57517 −0.271815 −0.135908 0.990722i \(-0.543395\pi\)
−0.135908 + 0.990722i \(0.543395\pi\)
\(174\) 1.93987 0.147061
\(175\) 0 0
\(176\) −5.20288 −0.392182
\(177\) 12.9132 0.970612
\(178\) 13.9433 1.04510
\(179\) 13.3401 0.997084 0.498542 0.866865i \(-0.333869\pi\)
0.498542 + 0.866865i \(0.333869\pi\)
\(180\) 1.11029 0.0827561
\(181\) −6.15975 −0.457850 −0.228925 0.973444i \(-0.573521\pi\)
−0.228925 + 0.973444i \(0.573521\pi\)
\(182\) 0 0
\(183\) −2.57480 −0.190334
\(184\) 8.49468 0.626236
\(185\) 4.30836 0.316757
\(186\) −0.506648 −0.0371492
\(187\) −8.00974 −0.585730
\(188\) −2.15863 −0.157434
\(189\) 0 0
\(190\) −13.8597 −1.00549
\(191\) 23.8691 1.72711 0.863554 0.504256i \(-0.168233\pi\)
0.863554 + 0.504256i \(0.168233\pi\)
\(192\) −6.51709 −0.470331
\(193\) −21.6807 −1.56061 −0.780304 0.625400i \(-0.784936\pi\)
−0.780304 + 0.625400i \(0.784936\pi\)
\(194\) 19.5249 1.40181
\(195\) 17.3541 1.24275
\(196\) 0 0
\(197\) 22.1707 1.57959 0.789797 0.613368i \(-0.210186\pi\)
0.789797 + 0.613368i \(0.210186\pi\)
\(198\) −1.75093 −0.124433
\(199\) −7.79167 −0.552337 −0.276168 0.961109i \(-0.589065\pi\)
−0.276168 + 0.961109i \(0.589065\pi\)
\(200\) 24.5111 1.73320
\(201\) 6.21525 0.438390
\(202\) −0.268732 −0.0189079
\(203\) 0 0
\(204\) 2.02075 0.141481
\(205\) −3.80505 −0.265756
\(206\) 22.0028 1.53301
\(207\) 3.28488 0.228315
\(208\) 20.5165 1.42257
\(209\) 2.78283 0.192493
\(210\) 0 0
\(211\) 7.81577 0.538060 0.269030 0.963132i \(-0.413297\pi\)
0.269030 + 0.963132i \(0.413297\pi\)
\(212\) −0.657089 −0.0451291
\(213\) −2.30854 −0.158178
\(214\) 18.2772 1.24940
\(215\) 8.07111 0.550445
\(216\) −2.58600 −0.175955
\(217\) 0 0
\(218\) −25.7479 −1.74387
\(219\) −11.9079 −0.804658
\(220\) 1.28416 0.0865778
\(221\) 31.5848 2.12463
\(222\) 1.71411 0.115044
\(223\) 22.6963 1.51986 0.759929 0.650006i \(-0.225234\pi\)
0.759929 + 0.650006i \(0.225234\pi\)
\(224\) 0 0
\(225\) 9.47840 0.631893
\(226\) −2.47992 −0.164962
\(227\) −1.16540 −0.0773504 −0.0386752 0.999252i \(-0.512314\pi\)
−0.0386752 + 0.999252i \(0.512314\pi\)
\(228\) −0.702072 −0.0464959
\(229\) −24.8081 −1.63936 −0.819682 0.572819i \(-0.805850\pi\)
−0.819682 + 0.572819i \(0.805850\pi\)
\(230\) −18.9220 −1.24768
\(231\) 0 0
\(232\) 3.31370 0.217555
\(233\) −18.9135 −1.23906 −0.619531 0.784972i \(-0.712677\pi\)
−0.619531 + 0.784972i \(0.712677\pi\)
\(234\) 6.90445 0.451358
\(235\) −28.1490 −1.83624
\(236\) −3.76798 −0.245275
\(237\) 12.7517 0.828309
\(238\) 0 0
\(239\) 5.69526 0.368396 0.184198 0.982889i \(-0.441031\pi\)
0.184198 + 0.982889i \(0.441031\pi\)
\(240\) 17.1168 1.10488
\(241\) 12.4796 0.803884 0.401942 0.915665i \(-0.368335\pi\)
0.401942 + 0.915665i \(0.368335\pi\)
\(242\) 14.6274 0.940286
\(243\) −1.00000 −0.0641500
\(244\) 0.751310 0.0480977
\(245\) 0 0
\(246\) −1.51387 −0.0965206
\(247\) −10.9736 −0.698230
\(248\) −0.865459 −0.0549567
\(249\) −2.58930 −0.164090
\(250\) −25.7971 −1.63155
\(251\) −29.7508 −1.87785 −0.938926 0.344120i \(-0.888177\pi\)
−0.938926 + 0.344120i \(0.888177\pi\)
\(252\) 0 0
\(253\) 3.79927 0.238858
\(254\) 31.4424 1.97287
\(255\) 26.3510 1.65016
\(256\) 6.86124 0.428827
\(257\) 4.66090 0.290739 0.145369 0.989377i \(-0.453563\pi\)
0.145369 + 0.989377i \(0.453563\pi\)
\(258\) 3.21115 0.199918
\(259\) 0 0
\(260\) −5.06382 −0.314045
\(261\) 1.28140 0.0793166
\(262\) −18.2195 −1.12561
\(263\) 24.2173 1.49330 0.746650 0.665217i \(-0.231661\pi\)
0.746650 + 0.665217i \(0.231661\pi\)
\(264\) −2.99095 −0.184080
\(265\) −8.56857 −0.526363
\(266\) 0 0
\(267\) 9.21039 0.563667
\(268\) −1.81357 −0.110782
\(269\) −26.7174 −1.62899 −0.814495 0.580170i \(-0.802986\pi\)
−0.814495 + 0.580170i \(0.802986\pi\)
\(270\) 5.76034 0.350563
\(271\) −21.2293 −1.28959 −0.644795 0.764356i \(-0.723057\pi\)
−0.644795 + 0.764356i \(0.723057\pi\)
\(272\) 31.1530 1.88893
\(273\) 0 0
\(274\) 9.84108 0.594522
\(275\) 10.9627 0.661074
\(276\) −0.958507 −0.0576953
\(277\) −8.30807 −0.499184 −0.249592 0.968351i \(-0.580296\pi\)
−0.249592 + 0.968351i \(0.580296\pi\)
\(278\) −35.0166 −2.10015
\(279\) −0.334671 −0.0200362
\(280\) 0 0
\(281\) −19.0193 −1.13460 −0.567300 0.823511i \(-0.692012\pi\)
−0.567300 + 0.823511i \(0.692012\pi\)
\(282\) −11.1993 −0.666907
\(283\) 8.80867 0.523621 0.261811 0.965119i \(-0.415680\pi\)
0.261811 + 0.965119i \(0.415680\pi\)
\(284\) 0.673617 0.0399718
\(285\) −9.15516 −0.542305
\(286\) 7.98566 0.472202
\(287\) 0 0
\(288\) 1.63805 0.0965232
\(289\) 30.9594 1.82114
\(290\) −7.38130 −0.433445
\(291\) 12.8974 0.756058
\(292\) 3.47464 0.203338
\(293\) −30.0047 −1.75289 −0.876446 0.481500i \(-0.840092\pi\)
−0.876446 + 0.481500i \(0.840092\pi\)
\(294\) 0 0
\(295\) −49.1352 −2.86076
\(296\) 2.92806 0.170190
\(297\) −1.15660 −0.0671125
\(298\) 7.87276 0.456056
\(299\) −14.9817 −0.866413
\(300\) −2.76574 −0.159680
\(301\) 0 0
\(302\) 15.5774 0.896376
\(303\) −0.177513 −0.0101979
\(304\) −10.8235 −0.620771
\(305\) 9.79723 0.560988
\(306\) 10.4839 0.599327
\(307\) −26.3241 −1.50240 −0.751199 0.660076i \(-0.770524\pi\)
−0.751199 + 0.660076i \(0.770524\pi\)
\(308\) 0 0
\(309\) 14.5341 0.826819
\(310\) 1.92782 0.109493
\(311\) 21.2895 1.20721 0.603607 0.797282i \(-0.293730\pi\)
0.603607 + 0.797282i \(0.293730\pi\)
\(312\) 11.7942 0.667717
\(313\) −14.8092 −0.837065 −0.418532 0.908202i \(-0.637455\pi\)
−0.418532 + 0.908202i \(0.637455\pi\)
\(314\) −6.10977 −0.344794
\(315\) 0 0
\(316\) −3.72085 −0.209314
\(317\) 3.95277 0.222010 0.111005 0.993820i \(-0.464593\pi\)
0.111005 + 0.993820i \(0.464593\pi\)
\(318\) −3.40907 −0.191171
\(319\) 1.48206 0.0829795
\(320\) 24.7979 1.38624
\(321\) 12.0732 0.673859
\(322\) 0 0
\(323\) −16.6626 −0.927132
\(324\) 0.291794 0.0162108
\(325\) −43.2291 −2.39792
\(326\) −10.3687 −0.574271
\(327\) −17.0080 −0.940547
\(328\) −2.58600 −0.142788
\(329\) 0 0
\(330\) 6.66238 0.366752
\(331\) 30.9459 1.70094 0.850470 0.526023i \(-0.176317\pi\)
0.850470 + 0.526023i \(0.176317\pi\)
\(332\) 0.755542 0.0414658
\(333\) 1.13227 0.0620483
\(334\) −4.12946 −0.225954
\(335\) −23.6493 −1.29210
\(336\) 0 0
\(337\) 32.5965 1.77564 0.887822 0.460188i \(-0.152218\pi\)
0.887822 + 0.460188i \(0.152218\pi\)
\(338\) −11.8096 −0.642357
\(339\) −1.63814 −0.0889715
\(340\) −7.68906 −0.416998
\(341\) −0.387079 −0.0209615
\(342\) −3.64245 −0.196961
\(343\) 0 0
\(344\) 5.48531 0.295748
\(345\) −12.4991 −0.672930
\(346\) 5.41234 0.290969
\(347\) −24.1324 −1.29549 −0.647747 0.761855i \(-0.724289\pi\)
−0.647747 + 0.761855i \(0.724289\pi\)
\(348\) −0.373905 −0.0200434
\(349\) −24.9455 −1.33530 −0.667651 0.744474i \(-0.732700\pi\)
−0.667651 + 0.744474i \(0.732700\pi\)
\(350\) 0 0
\(351\) 4.56081 0.243438
\(352\) 1.89456 0.100981
\(353\) −35.1182 −1.86916 −0.934578 0.355759i \(-0.884222\pi\)
−0.934578 + 0.355759i \(0.884222\pi\)
\(354\) −19.5488 −1.03901
\(355\) 8.78410 0.466211
\(356\) −2.68754 −0.142439
\(357\) 0 0
\(358\) −20.1951 −1.06734
\(359\) −2.09790 −0.110723 −0.0553614 0.998466i \(-0.517631\pi\)
−0.0553614 + 0.998466i \(0.517631\pi\)
\(360\) 9.83985 0.518605
\(361\) −13.2109 −0.695310
\(362\) 9.32504 0.490113
\(363\) 9.66229 0.507139
\(364\) 0 0
\(365\) 45.3100 2.37163
\(366\) 3.89790 0.203746
\(367\) 15.7507 0.822180 0.411090 0.911595i \(-0.365148\pi\)
0.411090 + 0.911595i \(0.365148\pi\)
\(368\) −14.7768 −0.770295
\(369\) −1.00000 −0.0520579
\(370\) −6.52229 −0.339077
\(371\) 0 0
\(372\) 0.0976550 0.00506318
\(373\) −10.4347 −0.540289 −0.270144 0.962820i \(-0.587071\pi\)
−0.270144 + 0.962820i \(0.587071\pi\)
\(374\) 12.1257 0.627004
\(375\) −17.0405 −0.879969
\(376\) −19.1307 −0.986589
\(377\) −5.84421 −0.300992
\(378\) 0 0
\(379\) −13.0415 −0.669895 −0.334947 0.942237i \(-0.608719\pi\)
−0.334947 + 0.942237i \(0.608719\pi\)
\(380\) 2.67142 0.137041
\(381\) 20.7696 1.06406
\(382\) −36.1347 −1.84881
\(383\) 13.8502 0.707714 0.353857 0.935299i \(-0.384870\pi\)
0.353857 + 0.935299i \(0.384870\pi\)
\(384\) 13.1421 0.670656
\(385\) 0 0
\(386\) 32.8216 1.67058
\(387\) 2.12116 0.107825
\(388\) −3.76338 −0.191057
\(389\) 29.4776 1.49458 0.747288 0.664500i \(-0.231356\pi\)
0.747288 + 0.664500i \(0.231356\pi\)
\(390\) −26.2718 −1.33032
\(391\) −22.7487 −1.15045
\(392\) 0 0
\(393\) −12.0351 −0.607090
\(394\) −33.5634 −1.69090
\(395\) −48.5207 −2.44134
\(396\) 0.337487 0.0169594
\(397\) −25.5717 −1.28341 −0.641703 0.766954i \(-0.721772\pi\)
−0.641703 + 0.766954i \(0.721772\pi\)
\(398\) 11.7955 0.591257
\(399\) 0 0
\(400\) −42.6381 −2.13190
\(401\) 6.69303 0.334234 0.167117 0.985937i \(-0.446554\pi\)
0.167117 + 0.985937i \(0.446554\pi\)
\(402\) −9.40906 −0.469281
\(403\) 1.52637 0.0760339
\(404\) 0.0517973 0.00257701
\(405\) 3.80505 0.189074
\(406\) 0 0
\(407\) 1.30958 0.0649136
\(408\) 17.9087 0.886615
\(409\) −18.9785 −0.938425 −0.469212 0.883085i \(-0.655462\pi\)
−0.469212 + 0.883085i \(0.655462\pi\)
\(410\) 5.76034 0.284483
\(411\) 6.50063 0.320652
\(412\) −4.24097 −0.208938
\(413\) 0 0
\(414\) −4.97287 −0.244403
\(415\) 9.85242 0.483636
\(416\) −7.47084 −0.366288
\(417\) −23.1305 −1.13271
\(418\) −4.21284 −0.206057
\(419\) 38.3019 1.87117 0.935586 0.353099i \(-0.114872\pi\)
0.935586 + 0.353099i \(0.114872\pi\)
\(420\) 0 0
\(421\) 13.7293 0.669124 0.334562 0.942374i \(-0.391412\pi\)
0.334562 + 0.942374i \(0.391412\pi\)
\(422\) −11.8320 −0.575974
\(423\) −7.39779 −0.359693
\(424\) −5.82339 −0.282809
\(425\) −65.6405 −3.18403
\(426\) 3.49482 0.169325
\(427\) 0 0
\(428\) −3.52288 −0.170285
\(429\) 5.27501 0.254680
\(430\) −12.2186 −0.589233
\(431\) −1.39769 −0.0673244 −0.0336622 0.999433i \(-0.510717\pi\)
−0.0336622 + 0.999433i \(0.510717\pi\)
\(432\) 4.49844 0.216431
\(433\) 19.3022 0.927605 0.463802 0.885939i \(-0.346485\pi\)
0.463802 + 0.885939i \(0.346485\pi\)
\(434\) 0 0
\(435\) −4.87579 −0.233776
\(436\) 4.96284 0.237677
\(437\) 7.90359 0.378080
\(438\) 18.0269 0.861359
\(439\) −2.98535 −0.142483 −0.0712416 0.997459i \(-0.522696\pi\)
−0.0712416 + 0.997459i \(0.522696\pi\)
\(440\) 11.3807 0.542555
\(441\) 0 0
\(442\) −47.8152 −2.27434
\(443\) 35.3757 1.68075 0.840374 0.542007i \(-0.182335\pi\)
0.840374 + 0.542007i \(0.182335\pi\)
\(444\) −0.330391 −0.0156797
\(445\) −35.0460 −1.66134
\(446\) −34.3592 −1.62696
\(447\) 5.20043 0.245972
\(448\) 0 0
\(449\) −24.7278 −1.16698 −0.583489 0.812121i \(-0.698313\pi\)
−0.583489 + 0.812121i \(0.698313\pi\)
\(450\) −14.3490 −0.676420
\(451\) −1.15660 −0.0544619
\(452\) 0.477999 0.0224832
\(453\) 10.2898 0.483456
\(454\) 1.76426 0.0828010
\(455\) 0 0
\(456\) −6.22205 −0.291374
\(457\) −9.46161 −0.442595 −0.221298 0.975206i \(-0.571029\pi\)
−0.221298 + 0.975206i \(0.571029\pi\)
\(458\) 37.5561 1.75488
\(459\) 6.92527 0.323244
\(460\) 3.64716 0.170050
\(461\) −20.5811 −0.958557 −0.479279 0.877663i \(-0.659102\pi\)
−0.479279 + 0.877663i \(0.659102\pi\)
\(462\) 0 0
\(463\) −27.1195 −1.26035 −0.630175 0.776453i \(-0.717017\pi\)
−0.630175 + 0.776453i \(0.717017\pi\)
\(464\) −5.76430 −0.267601
\(465\) 1.27344 0.0590544
\(466\) 28.6325 1.32637
\(467\) 6.72518 0.311204 0.155602 0.987820i \(-0.450268\pi\)
0.155602 + 0.987820i \(0.450268\pi\)
\(468\) −1.33082 −0.0615169
\(469\) 0 0
\(470\) 42.6138 1.96563
\(471\) −4.03587 −0.185963
\(472\) −33.3934 −1.53705
\(473\) 2.45332 0.112804
\(474\) −19.3043 −0.886676
\(475\) 22.8056 1.04639
\(476\) 0 0
\(477\) −2.25189 −0.103107
\(478\) −8.62186 −0.394355
\(479\) 18.3927 0.840382 0.420191 0.907436i \(-0.361963\pi\)
0.420191 + 0.907436i \(0.361963\pi\)
\(480\) −6.23287 −0.284490
\(481\) −5.16408 −0.235462
\(482\) −18.8925 −0.860531
\(483\) 0 0
\(484\) −2.81940 −0.128154
\(485\) −49.0752 −2.22839
\(486\) 1.51387 0.0686704
\(487\) 8.31877 0.376960 0.188480 0.982077i \(-0.439644\pi\)
0.188480 + 0.982077i \(0.439644\pi\)
\(488\) 6.65842 0.301412
\(489\) −6.84917 −0.309730
\(490\) 0 0
\(491\) −0.703140 −0.0317323 −0.0158661 0.999874i \(-0.505051\pi\)
−0.0158661 + 0.999874i \(0.505051\pi\)
\(492\) 0.291794 0.0131551
\(493\) −8.87404 −0.399667
\(494\) 16.6125 0.747432
\(495\) 4.40090 0.197806
\(496\) 1.50550 0.0675989
\(497\) 0 0
\(498\) 3.91986 0.175653
\(499\) −25.1798 −1.12720 −0.563601 0.826047i \(-0.690584\pi\)
−0.563601 + 0.826047i \(0.690584\pi\)
\(500\) 4.97232 0.222369
\(501\) −2.72776 −0.121867
\(502\) 45.0387 2.01018
\(503\) −9.10380 −0.405918 −0.202959 0.979187i \(-0.565056\pi\)
−0.202959 + 0.979187i \(0.565056\pi\)
\(504\) 0 0
\(505\) 0.675447 0.0300570
\(506\) −5.75159 −0.255689
\(507\) −7.80095 −0.346452
\(508\) −6.06044 −0.268889
\(509\) 8.52853 0.378021 0.189010 0.981975i \(-0.439472\pi\)
0.189010 + 0.981975i \(0.439472\pi\)
\(510\) −39.8919 −1.76644
\(511\) 0 0
\(512\) 15.8972 0.702565
\(513\) −2.40606 −0.106230
\(514\) −7.05598 −0.311226
\(515\) −55.3031 −2.43695
\(516\) −0.618941 −0.0272474
\(517\) −8.55625 −0.376304
\(518\) 0 0
\(519\) 3.57517 0.156933
\(520\) −44.8776 −1.96801
\(521\) 29.7200 1.30206 0.651029 0.759053i \(-0.274338\pi\)
0.651029 + 0.759053i \(0.274338\pi\)
\(522\) −1.93987 −0.0849058
\(523\) 23.3081 1.01919 0.509597 0.860413i \(-0.329795\pi\)
0.509597 + 0.860413i \(0.329795\pi\)
\(524\) 3.51177 0.153412
\(525\) 0 0
\(526\) −36.6617 −1.59853
\(527\) 2.31769 0.100960
\(528\) 5.20288 0.226426
\(529\) −12.2096 −0.530852
\(530\) 12.9717 0.563454
\(531\) −12.9132 −0.560383
\(532\) 0 0
\(533\) 4.56081 0.197550
\(534\) −13.9433 −0.603386
\(535\) −45.9390 −1.98612
\(536\) −16.0726 −0.694231
\(537\) −13.3401 −0.575667
\(538\) 40.4466 1.74378
\(539\) 0 0
\(540\) −1.11029 −0.0477793
\(541\) 22.9665 0.987407 0.493704 0.869630i \(-0.335643\pi\)
0.493704 + 0.869630i \(0.335643\pi\)
\(542\) 32.1384 1.38046
\(543\) 6.15975 0.264340
\(544\) −11.3440 −0.486368
\(545\) 64.7164 2.77215
\(546\) 0 0
\(547\) −19.0779 −0.815711 −0.407856 0.913046i \(-0.633723\pi\)
−0.407856 + 0.913046i \(0.633723\pi\)
\(548\) −1.89684 −0.0810291
\(549\) 2.57480 0.109890
\(550\) −16.5960 −0.707657
\(551\) 3.08312 0.131345
\(552\) −8.49468 −0.361558
\(553\) 0 0
\(554\) 12.5773 0.534359
\(555\) −4.30836 −0.182880
\(556\) 6.74935 0.286236
\(557\) −1.60892 −0.0681723 −0.0340861 0.999419i \(-0.510852\pi\)
−0.0340861 + 0.999419i \(0.510852\pi\)
\(558\) 0.506648 0.0214481
\(559\) −9.67419 −0.409175
\(560\) 0 0
\(561\) 8.00974 0.338171
\(562\) 28.7928 1.21455
\(563\) 40.0650 1.68854 0.844268 0.535921i \(-0.180035\pi\)
0.844268 + 0.535921i \(0.180035\pi\)
\(564\) 2.15863 0.0908948
\(565\) 6.23320 0.262233
\(566\) −13.3352 −0.560519
\(567\) 0 0
\(568\) 5.96987 0.250490
\(569\) 0.910623 0.0381753 0.0190876 0.999818i \(-0.493924\pi\)
0.0190876 + 0.999818i \(0.493924\pi\)
\(570\) 13.8597 0.580519
\(571\) 0.733915 0.0307134 0.0153567 0.999882i \(-0.495112\pi\)
0.0153567 + 0.999882i \(0.495112\pi\)
\(572\) −1.53921 −0.0643578
\(573\) −23.8691 −0.997146
\(574\) 0 0
\(575\) 31.1354 1.29843
\(576\) 6.51709 0.271546
\(577\) −31.6908 −1.31930 −0.659652 0.751571i \(-0.729296\pi\)
−0.659652 + 0.751571i \(0.729296\pi\)
\(578\) −46.8684 −1.94947
\(579\) 21.6807 0.901017
\(580\) 1.42273 0.0590754
\(581\) 0 0
\(582\) −19.5249 −0.809334
\(583\) −2.60453 −0.107869
\(584\) 30.7937 1.27425
\(585\) −17.3541 −0.717503
\(586\) 45.4231 1.87641
\(587\) −20.9045 −0.862823 −0.431411 0.902155i \(-0.641984\pi\)
−0.431411 + 0.902155i \(0.641984\pi\)
\(588\) 0 0
\(589\) −0.805238 −0.0331792
\(590\) 74.3841 3.06235
\(591\) −22.1707 −0.911979
\(592\) −5.09347 −0.209340
\(593\) 1.44402 0.0592987 0.0296494 0.999560i \(-0.490561\pi\)
0.0296494 + 0.999560i \(0.490561\pi\)
\(594\) 1.75093 0.0718416
\(595\) 0 0
\(596\) −1.51745 −0.0621573
\(597\) 7.79167 0.318892
\(598\) 22.6803 0.927465
\(599\) 10.3458 0.422719 0.211360 0.977408i \(-0.432211\pi\)
0.211360 + 0.977408i \(0.432211\pi\)
\(600\) −24.5111 −1.00066
\(601\) 8.35641 0.340865 0.170432 0.985369i \(-0.445484\pi\)
0.170432 + 0.985369i \(0.445484\pi\)
\(602\) 0 0
\(603\) −6.21525 −0.253104
\(604\) −3.00249 −0.122170
\(605\) −36.7655 −1.49473
\(606\) 0.268732 0.0109165
\(607\) −0.600544 −0.0243753 −0.0121877 0.999926i \(-0.503880\pi\)
−0.0121877 + 0.999926i \(0.503880\pi\)
\(608\) 3.94125 0.159839
\(609\) 0 0
\(610\) −14.8317 −0.600518
\(611\) 33.7399 1.36497
\(612\) −2.02075 −0.0816841
\(613\) −20.2750 −0.818899 −0.409449 0.912333i \(-0.634279\pi\)
−0.409449 + 0.912333i \(0.634279\pi\)
\(614\) 39.8512 1.60827
\(615\) 3.80505 0.153434
\(616\) 0 0
\(617\) −23.4448 −0.943854 −0.471927 0.881638i \(-0.656441\pi\)
−0.471927 + 0.881638i \(0.656441\pi\)
\(618\) −22.0028 −0.885081
\(619\) 10.3180 0.414716 0.207358 0.978265i \(-0.433513\pi\)
0.207358 + 0.978265i \(0.433513\pi\)
\(620\) −0.371582 −0.0149231
\(621\) −3.28488 −0.131818
\(622\) −32.2294 −1.29228
\(623\) 0 0
\(624\) −20.5165 −0.821318
\(625\) 17.4481 0.697923
\(626\) 22.4191 0.896049
\(627\) −2.78283 −0.111136
\(628\) 1.17764 0.0469930
\(629\) −7.84131 −0.312654
\(630\) 0 0
\(631\) 5.95019 0.236873 0.118437 0.992962i \(-0.462212\pi\)
0.118437 + 0.992962i \(0.462212\pi\)
\(632\) −32.9757 −1.31170
\(633\) −7.81577 −0.310649
\(634\) −5.98397 −0.237654
\(635\) −79.0293 −3.13618
\(636\) 0.657089 0.0260553
\(637\) 0 0
\(638\) −2.24364 −0.0888267
\(639\) 2.30854 0.0913243
\(640\) −50.0064 −1.97668
\(641\) −18.7783 −0.741697 −0.370848 0.928693i \(-0.620933\pi\)
−0.370848 + 0.928693i \(0.620933\pi\)
\(642\) −18.2772 −0.721343
\(643\) −26.6022 −1.04909 −0.524543 0.851384i \(-0.675764\pi\)
−0.524543 + 0.851384i \(0.675764\pi\)
\(644\) 0 0
\(645\) −8.07111 −0.317800
\(646\) 25.2250 0.992463
\(647\) −23.5755 −0.926847 −0.463424 0.886137i \(-0.653379\pi\)
−0.463424 + 0.886137i \(0.653379\pi\)
\(648\) 2.58600 0.101588
\(649\) −14.9353 −0.586261
\(650\) 65.4432 2.56689
\(651\) 0 0
\(652\) 1.99855 0.0782691
\(653\) −17.2768 −0.676093 −0.338046 0.941129i \(-0.609766\pi\)
−0.338046 + 0.941129i \(0.609766\pi\)
\(654\) 25.7479 1.00682
\(655\) 45.7941 1.78933
\(656\) 4.49844 0.175635
\(657\) 11.9079 0.464570
\(658\) 0 0
\(659\) 32.3358 1.25963 0.629813 0.776747i \(-0.283132\pi\)
0.629813 + 0.776747i \(0.283132\pi\)
\(660\) −1.28416 −0.0499857
\(661\) −41.8828 −1.62905 −0.814525 0.580128i \(-0.803002\pi\)
−0.814525 + 0.580128i \(0.803002\pi\)
\(662\) −46.8480 −1.82080
\(663\) −31.5848 −1.22665
\(664\) 6.69593 0.259852
\(665\) 0 0
\(666\) −1.71411 −0.0664205
\(667\) 4.20924 0.162982
\(668\) 0.795943 0.0307960
\(669\) −22.6963 −0.877490
\(670\) 35.8019 1.38315
\(671\) 2.97800 0.114964
\(672\) 0 0
\(673\) −32.0621 −1.23590 −0.617951 0.786217i \(-0.712037\pi\)
−0.617951 + 0.786217i \(0.712037\pi\)
\(674\) −49.3467 −1.90077
\(675\) −9.47840 −0.364824
\(676\) 2.27627 0.0875488
\(677\) 39.2517 1.50857 0.754283 0.656549i \(-0.227984\pi\)
0.754283 + 0.656549i \(0.227984\pi\)
\(678\) 2.47992 0.0952409
\(679\) 0 0
\(680\) −68.1436 −2.61319
\(681\) 1.16540 0.0446583
\(682\) 0.585987 0.0224386
\(683\) −15.5806 −0.596176 −0.298088 0.954538i \(-0.596349\pi\)
−0.298088 + 0.954538i \(0.596349\pi\)
\(684\) 0.702072 0.0268444
\(685\) −24.7352 −0.945084
\(686\) 0 0
\(687\) 24.8081 0.946487
\(688\) −9.54191 −0.363782
\(689\) 10.2705 0.391273
\(690\) 18.9220 0.720348
\(691\) −18.0215 −0.685569 −0.342784 0.939414i \(-0.611370\pi\)
−0.342784 + 0.939414i \(0.611370\pi\)
\(692\) −1.04321 −0.0396570
\(693\) 0 0
\(694\) 36.5332 1.38678
\(695\) 88.0129 3.33852
\(696\) −3.31370 −0.125605
\(697\) 6.92527 0.262313
\(698\) 37.7642 1.42940
\(699\) 18.9135 0.715373
\(700\) 0 0
\(701\) −5.16472 −0.195069 −0.0975344 0.995232i \(-0.531096\pi\)
−0.0975344 + 0.995232i \(0.531096\pi\)
\(702\) −6.90445 −0.260592
\(703\) 2.72432 0.102750
\(704\) 7.53764 0.284086
\(705\) 28.1490 1.06015
\(706\) 53.1643 2.00087
\(707\) 0 0
\(708\) 3.76798 0.141609
\(709\) 11.3643 0.426796 0.213398 0.976965i \(-0.431547\pi\)
0.213398 + 0.976965i \(0.431547\pi\)
\(710\) −13.2980 −0.499063
\(711\) −12.7517 −0.478224
\(712\) −23.8180 −0.892619
\(713\) −1.09935 −0.0411711
\(714\) 0 0
\(715\) −20.0717 −0.750637
\(716\) 3.89255 0.145472
\(717\) −5.69526 −0.212693
\(718\) 3.17594 0.118525
\(719\) −30.5843 −1.14060 −0.570302 0.821435i \(-0.693174\pi\)
−0.570302 + 0.821435i \(0.693174\pi\)
\(720\) −17.1168 −0.637906
\(721\) 0 0
\(722\) 19.9995 0.744306
\(723\) −12.4796 −0.464123
\(724\) −1.79738 −0.0667990
\(725\) 12.1456 0.451077
\(726\) −14.6274 −0.542874
\(727\) −18.0711 −0.670221 −0.335111 0.942179i \(-0.608774\pi\)
−0.335111 + 0.942179i \(0.608774\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −68.5933 −2.53875
\(731\) −14.6896 −0.543315
\(732\) −0.751310 −0.0277692
\(733\) 49.3640 1.82330 0.911651 0.410966i \(-0.134808\pi\)
0.911651 + 0.410966i \(0.134808\pi\)
\(734\) −23.8445 −0.880115
\(735\) 0 0
\(736\) 5.38080 0.198339
\(737\) −7.18853 −0.264793
\(738\) 1.51387 0.0557262
\(739\) −9.21630 −0.339027 −0.169514 0.985528i \(-0.554220\pi\)
−0.169514 + 0.985528i \(0.554220\pi\)
\(740\) 1.25715 0.0462139
\(741\) 10.9736 0.403124
\(742\) 0 0
\(743\) −47.8911 −1.75696 −0.878478 0.477783i \(-0.841440\pi\)
−0.878478 + 0.477783i \(0.841440\pi\)
\(744\) 0.865459 0.0317293
\(745\) −19.7879 −0.724972
\(746\) 15.7968 0.578361
\(747\) 2.58930 0.0947376
\(748\) −2.33719 −0.0854562
\(749\) 0 0
\(750\) 25.7971 0.941977
\(751\) −0.992350 −0.0362114 −0.0181057 0.999836i \(-0.505764\pi\)
−0.0181057 + 0.999836i \(0.505764\pi\)
\(752\) 33.2786 1.21354
\(753\) 29.7508 1.08418
\(754\) 8.84736 0.322202
\(755\) −39.1531 −1.42493
\(756\) 0 0
\(757\) 17.2370 0.626490 0.313245 0.949672i \(-0.398584\pi\)
0.313245 + 0.949672i \(0.398584\pi\)
\(758\) 19.7430 0.717100
\(759\) −3.79927 −0.137905
\(760\) 23.6752 0.858791
\(761\) 25.0256 0.907177 0.453589 0.891211i \(-0.350144\pi\)
0.453589 + 0.891211i \(0.350144\pi\)
\(762\) −31.4424 −1.13904
\(763\) 0 0
\(764\) 6.96486 0.251980
\(765\) −26.3510 −0.952723
\(766\) −20.9674 −0.757584
\(767\) 58.8944 2.12655
\(768\) −6.86124 −0.247584
\(769\) −12.7073 −0.458239 −0.229119 0.973398i \(-0.573585\pi\)
−0.229119 + 0.973398i \(0.573585\pi\)
\(770\) 0 0
\(771\) −4.66090 −0.167858
\(772\) −6.32628 −0.227688
\(773\) −47.3022 −1.70134 −0.850671 0.525699i \(-0.823804\pi\)
−0.850671 + 0.525699i \(0.823804\pi\)
\(774\) −3.21115 −0.115422
\(775\) −3.17215 −0.113947
\(776\) −33.3526 −1.19729
\(777\) 0 0
\(778\) −44.6252 −1.59989
\(779\) −2.40606 −0.0862059
\(780\) 5.06382 0.181314
\(781\) 2.67004 0.0955417
\(782\) 34.4385 1.23152
\(783\) −1.28140 −0.0457935
\(784\) 0 0
\(785\) 15.3567 0.548103
\(786\) 18.2195 0.649869
\(787\) −41.5945 −1.48269 −0.741343 0.671127i \(-0.765811\pi\)
−0.741343 + 0.671127i \(0.765811\pi\)
\(788\) 6.46927 0.230458
\(789\) −24.2173 −0.862158
\(790\) 73.4539 2.61337
\(791\) 0 0
\(792\) 2.99095 0.106279
\(793\) −11.7431 −0.417011
\(794\) 38.7121 1.37384
\(795\) 8.56857 0.303896
\(796\) −2.27356 −0.0805842
\(797\) −50.4009 −1.78529 −0.892645 0.450760i \(-0.851153\pi\)
−0.892645 + 0.450760i \(0.851153\pi\)
\(798\) 0 0
\(799\) 51.2317 1.81245
\(800\) 15.5261 0.548931
\(801\) −9.21039 −0.325433
\(802\) −10.1324 −0.357786
\(803\) 13.7726 0.486024
\(804\) 1.81357 0.0639597
\(805\) 0 0
\(806\) −2.31072 −0.0813917
\(807\) 26.7174 0.940498
\(808\) 0.459049 0.0161493
\(809\) 18.2063 0.640098 0.320049 0.947401i \(-0.396301\pi\)
0.320049 + 0.947401i \(0.396301\pi\)
\(810\) −5.76034 −0.202398
\(811\) −47.0365 −1.65168 −0.825838 0.563908i \(-0.809297\pi\)
−0.825838 + 0.563908i \(0.809297\pi\)
\(812\) 0 0
\(813\) 21.2293 0.744545
\(814\) −1.98254 −0.0694878
\(815\) 26.0614 0.912892
\(816\) −31.1530 −1.09057
\(817\) 5.10363 0.178553
\(818\) 28.7309 1.00455
\(819\) 0 0
\(820\) −1.11029 −0.0387730
\(821\) −52.1811 −1.82113 −0.910566 0.413364i \(-0.864354\pi\)
−0.910566 + 0.413364i \(0.864354\pi\)
\(822\) −9.84108 −0.343247
\(823\) −4.29085 −0.149570 −0.0747848 0.997200i \(-0.523827\pi\)
−0.0747848 + 0.997200i \(0.523827\pi\)
\(824\) −37.5853 −1.30934
\(825\) −10.9627 −0.381671
\(826\) 0 0
\(827\) 46.0862 1.60257 0.801287 0.598280i \(-0.204149\pi\)
0.801287 + 0.598280i \(0.204149\pi\)
\(828\) 0.958507 0.0333104
\(829\) −12.1577 −0.422256 −0.211128 0.977458i \(-0.567714\pi\)
−0.211128 + 0.977458i \(0.567714\pi\)
\(830\) −14.9153 −0.517716
\(831\) 8.30807 0.288204
\(832\) −29.7232 −1.03047
\(833\) 0 0
\(834\) 35.0166 1.21252
\(835\) 10.3793 0.359189
\(836\) 0.812014 0.0280841
\(837\) 0.334671 0.0115679
\(838\) −57.9840 −2.00303
\(839\) −4.79517 −0.165548 −0.0827739 0.996568i \(-0.526378\pi\)
−0.0827739 + 0.996568i \(0.526378\pi\)
\(840\) 0 0
\(841\) −27.3580 −0.943380
\(842\) −20.7843 −0.716274
\(843\) 19.0193 0.655061
\(844\) 2.28059 0.0785012
\(845\) 29.6830 1.02113
\(846\) 11.1993 0.385039
\(847\) 0 0
\(848\) 10.1300 0.347866
\(849\) −8.80867 −0.302313
\(850\) 99.3710 3.40840
\(851\) 3.71938 0.127499
\(852\) −0.673617 −0.0230777
\(853\) −5.26286 −0.180197 −0.0900984 0.995933i \(-0.528718\pi\)
−0.0900984 + 0.995933i \(0.528718\pi\)
\(854\) 0 0
\(855\) 9.15516 0.313100
\(856\) −31.2212 −1.06712
\(857\) −0.907168 −0.0309883 −0.0154941 0.999880i \(-0.504932\pi\)
−0.0154941 + 0.999880i \(0.504932\pi\)
\(858\) −7.98566 −0.272626
\(859\) 28.1280 0.959715 0.479858 0.877346i \(-0.340688\pi\)
0.479858 + 0.877346i \(0.340688\pi\)
\(860\) 2.35510 0.0803083
\(861\) 0 0
\(862\) 2.11592 0.0720685
\(863\) 4.62233 0.157346 0.0786730 0.996900i \(-0.474932\pi\)
0.0786730 + 0.996900i \(0.474932\pi\)
\(864\) −1.63805 −0.0557277
\(865\) −13.6037 −0.462540
\(866\) −29.2210 −0.992969
\(867\) −30.9594 −1.05144
\(868\) 0 0
\(869\) −14.7485 −0.500309
\(870\) 7.38130 0.250249
\(871\) 28.3465 0.960486
\(872\) 43.9827 1.48944
\(873\) −12.8974 −0.436510
\(874\) −11.9650 −0.404722
\(875\) 0 0
\(876\) −3.47464 −0.117397
\(877\) −33.5121 −1.13162 −0.565812 0.824534i \(-0.691437\pi\)
−0.565812 + 0.824534i \(0.691437\pi\)
\(878\) 4.51943 0.152523
\(879\) 30.0047 1.01203
\(880\) −19.7972 −0.667364
\(881\) 5.55431 0.187129 0.0935647 0.995613i \(-0.470174\pi\)
0.0935647 + 0.995613i \(0.470174\pi\)
\(882\) 0 0
\(883\) 33.3288 1.12160 0.560802 0.827950i \(-0.310493\pi\)
0.560802 + 0.827950i \(0.310493\pi\)
\(884\) 9.21626 0.309976
\(885\) 49.1352 1.65166
\(886\) −53.5541 −1.79918
\(887\) −31.4901 −1.05733 −0.528667 0.848829i \(-0.677308\pi\)
−0.528667 + 0.848829i \(0.677308\pi\)
\(888\) −2.92806 −0.0982592
\(889\) 0 0
\(890\) 53.0550 1.77841
\(891\) 1.15660 0.0387474
\(892\) 6.62265 0.221743
\(893\) −17.7995 −0.595638
\(894\) −7.87276 −0.263304
\(895\) 50.7597 1.69671
\(896\) 0 0
\(897\) 14.9817 0.500224
\(898\) 37.4346 1.24921
\(899\) −0.428848 −0.0143029
\(900\) 2.76574 0.0921913
\(901\) 15.5950 0.519544
\(902\) 1.75093 0.0582996
\(903\) 0 0
\(904\) 4.23622 0.140895
\(905\) −23.4381 −0.779110
\(906\) −15.5774 −0.517523
\(907\) −24.1540 −0.802021 −0.401010 0.916074i \(-0.631341\pi\)
−0.401010 + 0.916074i \(0.631341\pi\)
\(908\) −0.340057 −0.0112852
\(909\) 0.177513 0.00588775
\(910\) 0 0
\(911\) 13.5342 0.448407 0.224203 0.974542i \(-0.428022\pi\)
0.224203 + 0.974542i \(0.428022\pi\)
\(912\) 10.8235 0.358402
\(913\) 2.99477 0.0991126
\(914\) 14.3236 0.473783
\(915\) −9.79723 −0.323886
\(916\) −7.23884 −0.239178
\(917\) 0 0
\(918\) −10.4839 −0.346022
\(919\) −6.10358 −0.201339 −0.100669 0.994920i \(-0.532098\pi\)
−0.100669 + 0.994920i \(0.532098\pi\)
\(920\) 32.3227 1.06565
\(921\) 26.3241 0.867410
\(922\) 31.1571 1.02610
\(923\) −10.5288 −0.346559
\(924\) 0 0
\(925\) 10.7322 0.352871
\(926\) 41.0553 1.34916
\(927\) −14.5341 −0.477364
\(928\) 2.09900 0.0689030
\(929\) −34.5625 −1.13396 −0.566979 0.823732i \(-0.691888\pi\)
−0.566979 + 0.823732i \(0.691888\pi\)
\(930\) −1.92782 −0.0632157
\(931\) 0 0
\(932\) −5.51883 −0.180775
\(933\) −21.2895 −0.696985
\(934\) −10.1810 −0.333133
\(935\) −30.4775 −0.996719
\(936\) −11.7942 −0.385506
\(937\) 3.45162 0.112759 0.0563797 0.998409i \(-0.482044\pi\)
0.0563797 + 0.998409i \(0.482044\pi\)
\(938\) 0 0
\(939\) 14.8092 0.483280
\(940\) −8.21370 −0.267901
\(941\) 33.2147 1.08277 0.541384 0.840775i \(-0.317900\pi\)
0.541384 + 0.840775i \(0.317900\pi\)
\(942\) 6.10977 0.199067
\(943\) −3.28488 −0.106970
\(944\) 58.0891 1.89064
\(945\) 0 0
\(946\) −3.71400 −0.120753
\(947\) 29.1330 0.946694 0.473347 0.880876i \(-0.343046\pi\)
0.473347 + 0.880876i \(0.343046\pi\)
\(948\) 3.72085 0.120848
\(949\) −54.3094 −1.76296
\(950\) −34.5246 −1.12013
\(951\) −3.95277 −0.128177
\(952\) 0 0
\(953\) 37.7997 1.22445 0.612226 0.790683i \(-0.290274\pi\)
0.612226 + 0.790683i \(0.290274\pi\)
\(954\) 3.40907 0.110373
\(955\) 90.8231 2.93897
\(956\) 1.66184 0.0537478
\(957\) −1.48206 −0.0479082
\(958\) −27.8440 −0.899600
\(959\) 0 0
\(960\) −24.7979 −0.800348
\(961\) −30.8880 −0.996387
\(962\) 7.81774 0.252054
\(963\) −12.0732 −0.389053
\(964\) 3.64148 0.117284
\(965\) −82.4960 −2.65564
\(966\) 0 0
\(967\) 17.1460 0.551379 0.275690 0.961247i \(-0.411094\pi\)
0.275690 + 0.961247i \(0.411094\pi\)
\(968\) −24.9866 −0.803101
\(969\) 16.6626 0.535280
\(970\) 74.2933 2.38541
\(971\) −10.5979 −0.340104 −0.170052 0.985435i \(-0.554394\pi\)
−0.170052 + 0.985435i \(0.554394\pi\)
\(972\) −0.291794 −0.00935929
\(973\) 0 0
\(974\) −12.5935 −0.403522
\(975\) 43.2291 1.38444
\(976\) −11.5826 −0.370749
\(977\) 26.8876 0.860209 0.430105 0.902779i \(-0.358477\pi\)
0.430105 + 0.902779i \(0.358477\pi\)
\(978\) 10.3687 0.331555
\(979\) −10.6527 −0.340462
\(980\) 0 0
\(981\) 17.0080 0.543025
\(982\) 1.06446 0.0339683
\(983\) −27.7596 −0.885392 −0.442696 0.896672i \(-0.645978\pi\)
−0.442696 + 0.896672i \(0.645978\pi\)
\(984\) 2.58600 0.0824386
\(985\) 84.3605 2.68795
\(986\) 13.4341 0.427830
\(987\) 0 0
\(988\) −3.20202 −0.101870
\(989\) 6.96774 0.221561
\(990\) −6.66238 −0.211744
\(991\) 12.1510 0.385988 0.192994 0.981200i \(-0.438180\pi\)
0.192994 + 0.981200i \(0.438180\pi\)
\(992\) −0.548209 −0.0174057
\(993\) −30.9459 −0.982039
\(994\) 0 0
\(995\) −29.6477 −0.939894
\(996\) −0.755542 −0.0239403
\(997\) 15.7997 0.500382 0.250191 0.968197i \(-0.419507\pi\)
0.250191 + 0.968197i \(0.419507\pi\)
\(998\) 38.1188 1.20663
\(999\) −1.13227 −0.0358236
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 6027.2.a.bb.1.3 8
7.2 even 3 861.2.i.d.739.6 yes 16
7.4 even 3 861.2.i.d.247.6 16
7.6 odd 2 6027.2.a.bc.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
861.2.i.d.247.6 16 7.4 even 3
861.2.i.d.739.6 yes 16 7.2 even 3
6027.2.a.bb.1.3 8 1.1 even 1 trivial
6027.2.a.bc.1.3 8 7.6 odd 2