Properties

Label 6003.2.a.v.1.14
Level $6003$
Weight $2$
Character 6003.1
Self dual yes
Analytic conductor $47.934$
Analytic rank $0$
Dimension $30$
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] = [6003,2,Mod(1,6003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, 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("6003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.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: \(47.9341963334\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Character \(\chi\) \(=\) 6003.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.473233 q^{2} -1.77605 q^{4} +1.04483 q^{5} +3.83278 q^{7} +1.78695 q^{8} +O(q^{10})\) \(q-0.473233 q^{2} -1.77605 q^{4} +1.04483 q^{5} +3.83278 q^{7} +1.78695 q^{8} -0.494446 q^{10} -2.56396 q^{11} +2.32837 q^{13} -1.81380 q^{14} +2.70646 q^{16} -6.35999 q^{17} +7.47559 q^{19} -1.85566 q^{20} +1.21335 q^{22} +1.00000 q^{23} -3.90834 q^{25} -1.10186 q^{26} -6.80721 q^{28} -1.00000 q^{29} +8.99906 q^{31} -4.85469 q^{32} +3.00976 q^{34} +4.00459 q^{35} -5.17971 q^{37} -3.53770 q^{38} +1.86705 q^{40} -0.118500 q^{41} -2.97677 q^{43} +4.55372 q^{44} -0.473233 q^{46} +10.7617 q^{47} +7.69021 q^{49} +1.84956 q^{50} -4.13531 q^{52} +5.29192 q^{53} -2.67889 q^{55} +6.84899 q^{56} +0.473233 q^{58} +9.96740 q^{59} +3.51664 q^{61} -4.25865 q^{62} -3.11551 q^{64} +2.43274 q^{65} +14.1207 q^{67} +11.2957 q^{68} -1.89510 q^{70} -10.5253 q^{71} -3.05985 q^{73} +2.45121 q^{74} -13.2770 q^{76} -9.82709 q^{77} -0.286819 q^{79} +2.82777 q^{80} +0.0560781 q^{82} -4.82285 q^{83} -6.64508 q^{85} +1.40870 q^{86} -4.58167 q^{88} +1.13200 q^{89} +8.92414 q^{91} -1.77605 q^{92} -5.09277 q^{94} +7.81069 q^{95} -2.76909 q^{97} -3.63926 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8} + 8 q^{10} + 36 q^{13} - 7 q^{14} + 47 q^{16} - 18 q^{17} + 16 q^{19} + 25 q^{22} + 30 q^{23} + 56 q^{25} - 11 q^{26} + 27 q^{28} - 30 q^{29} + 14 q^{31} + 7 q^{32} + 3 q^{34} + 22 q^{35} + 40 q^{37} - 6 q^{38} + 30 q^{40} - 14 q^{41} + 34 q^{43} - 5 q^{44} - q^{46} + 2 q^{47} + 74 q^{49} + 21 q^{50} + 71 q^{52} - 16 q^{53} + 22 q^{55} - 14 q^{56} + q^{58} + 32 q^{59} + 46 q^{61} - 20 q^{62} + 68 q^{64} - 12 q^{65} + 14 q^{67} - 27 q^{68} + 32 q^{71} + 50 q^{73} + 26 q^{74} + 56 q^{76} - 34 q^{77} + 16 q^{79} - 2 q^{80} + 38 q^{82} + 14 q^{83} + 38 q^{85} - 10 q^{86} + 40 q^{88} + 2 q^{89} + 32 q^{91} + 37 q^{92} + 29 q^{94} + 28 q^{95} + 56 q^{97} - 8 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.473233 −0.334626 −0.167313 0.985904i \(-0.553509\pi\)
−0.167313 + 0.985904i \(0.553509\pi\)
\(3\) 0 0
\(4\) −1.77605 −0.888025
\(5\) 1.04483 0.467260 0.233630 0.972326i \(-0.424940\pi\)
0.233630 + 0.972326i \(0.424940\pi\)
\(6\) 0 0
\(7\) 3.83278 1.44866 0.724328 0.689456i \(-0.242150\pi\)
0.724328 + 0.689456i \(0.242150\pi\)
\(8\) 1.78695 0.631783
\(9\) 0 0
\(10\) −0.494446 −0.156357
\(11\) −2.56396 −0.773063 −0.386531 0.922276i \(-0.626327\pi\)
−0.386531 + 0.922276i \(0.626327\pi\)
\(12\) 0 0
\(13\) 2.32837 0.645774 0.322887 0.946437i \(-0.395347\pi\)
0.322887 + 0.946437i \(0.395347\pi\)
\(14\) −1.81380 −0.484758
\(15\) 0 0
\(16\) 2.70646 0.676614
\(17\) −6.35999 −1.54253 −0.771263 0.636517i \(-0.780374\pi\)
−0.771263 + 0.636517i \(0.780374\pi\)
\(18\) 0 0
\(19\) 7.47559 1.71502 0.857509 0.514469i \(-0.172011\pi\)
0.857509 + 0.514469i \(0.172011\pi\)
\(20\) −1.85566 −0.414939
\(21\) 0 0
\(22\) 1.21335 0.258687
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −3.90834 −0.781668
\(26\) −1.10186 −0.216093
\(27\) 0 0
\(28\) −6.80721 −1.28644
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 8.99906 1.61628 0.808139 0.588991i \(-0.200475\pi\)
0.808139 + 0.588991i \(0.200475\pi\)
\(32\) −4.85469 −0.858196
\(33\) 0 0
\(34\) 3.00976 0.516169
\(35\) 4.00459 0.676899
\(36\) 0 0
\(37\) −5.17971 −0.851539 −0.425769 0.904832i \(-0.639996\pi\)
−0.425769 + 0.904832i \(0.639996\pi\)
\(38\) −3.53770 −0.573890
\(39\) 0 0
\(40\) 1.86705 0.295207
\(41\) −0.118500 −0.0185066 −0.00925330 0.999957i \(-0.502945\pi\)
−0.00925330 + 0.999957i \(0.502945\pi\)
\(42\) 0 0
\(43\) −2.97677 −0.453953 −0.226976 0.973900i \(-0.572884\pi\)
−0.226976 + 0.973900i \(0.572884\pi\)
\(44\) 4.55372 0.686499
\(45\) 0 0
\(46\) −0.473233 −0.0697744
\(47\) 10.7617 1.56975 0.784875 0.619654i \(-0.212727\pi\)
0.784875 + 0.619654i \(0.212727\pi\)
\(48\) 0 0
\(49\) 7.69021 1.09860
\(50\) 1.84956 0.261567
\(51\) 0 0
\(52\) −4.13531 −0.573464
\(53\) 5.29192 0.726901 0.363451 0.931613i \(-0.381599\pi\)
0.363451 + 0.931613i \(0.381599\pi\)
\(54\) 0 0
\(55\) −2.67889 −0.361221
\(56\) 6.84899 0.915235
\(57\) 0 0
\(58\) 0.473233 0.0621385
\(59\) 9.96740 1.29764 0.648822 0.760940i \(-0.275262\pi\)
0.648822 + 0.760940i \(0.275262\pi\)
\(60\) 0 0
\(61\) 3.51664 0.450260 0.225130 0.974329i \(-0.427719\pi\)
0.225130 + 0.974329i \(0.427719\pi\)
\(62\) −4.25865 −0.540849
\(63\) 0 0
\(64\) −3.11551 −0.389439
\(65\) 2.43274 0.301745
\(66\) 0 0
\(67\) 14.1207 1.72512 0.862561 0.505952i \(-0.168859\pi\)
0.862561 + 0.505952i \(0.168859\pi\)
\(68\) 11.2957 1.36980
\(69\) 0 0
\(70\) −1.89510 −0.226508
\(71\) −10.5253 −1.24912 −0.624560 0.780977i \(-0.714722\pi\)
−0.624560 + 0.780977i \(0.714722\pi\)
\(72\) 0 0
\(73\) −3.05985 −0.358129 −0.179064 0.983837i \(-0.557307\pi\)
−0.179064 + 0.983837i \(0.557307\pi\)
\(74\) 2.45121 0.284947
\(75\) 0 0
\(76\) −13.2770 −1.52298
\(77\) −9.82709 −1.11990
\(78\) 0 0
\(79\) −0.286819 −0.0322696 −0.0161348 0.999870i \(-0.505136\pi\)
−0.0161348 + 0.999870i \(0.505136\pi\)
\(80\) 2.82777 0.316155
\(81\) 0 0
\(82\) 0.0560781 0.00619279
\(83\) −4.82285 −0.529376 −0.264688 0.964334i \(-0.585269\pi\)
−0.264688 + 0.964334i \(0.585269\pi\)
\(84\) 0 0
\(85\) −6.64508 −0.720760
\(86\) 1.40870 0.151904
\(87\) 0 0
\(88\) −4.58167 −0.488408
\(89\) 1.13200 0.119992 0.0599961 0.998199i \(-0.480891\pi\)
0.0599961 + 0.998199i \(0.480891\pi\)
\(90\) 0 0
\(91\) 8.92414 0.935504
\(92\) −1.77605 −0.185166
\(93\) 0 0
\(94\) −5.09277 −0.525279
\(95\) 7.81069 0.801360
\(96\) 0 0
\(97\) −2.76909 −0.281158 −0.140579 0.990069i \(-0.544896\pi\)
−0.140579 + 0.990069i \(0.544896\pi\)
\(98\) −3.63926 −0.367621
\(99\) 0 0
\(100\) 6.94141 0.694141
\(101\) 7.62918 0.759132 0.379566 0.925165i \(-0.376073\pi\)
0.379566 + 0.925165i \(0.376073\pi\)
\(102\) 0 0
\(103\) −1.31505 −0.129575 −0.0647877 0.997899i \(-0.520637\pi\)
−0.0647877 + 0.997899i \(0.520637\pi\)
\(104\) 4.16069 0.407989
\(105\) 0 0
\(106\) −2.50431 −0.243240
\(107\) 3.09134 0.298852 0.149426 0.988773i \(-0.452258\pi\)
0.149426 + 0.988773i \(0.452258\pi\)
\(108\) 0 0
\(109\) −10.6005 −1.01534 −0.507672 0.861551i \(-0.669494\pi\)
−0.507672 + 0.861551i \(0.669494\pi\)
\(110\) 1.26774 0.120874
\(111\) 0 0
\(112\) 10.3733 0.980181
\(113\) −2.36263 −0.222257 −0.111129 0.993806i \(-0.535447\pi\)
−0.111129 + 0.993806i \(0.535447\pi\)
\(114\) 0 0
\(115\) 1.04483 0.0974304
\(116\) 1.77605 0.164902
\(117\) 0 0
\(118\) −4.71690 −0.434226
\(119\) −24.3765 −2.23459
\(120\) 0 0
\(121\) −4.42611 −0.402374
\(122\) −1.66419 −0.150669
\(123\) 0 0
\(124\) −15.9828 −1.43530
\(125\) −9.30766 −0.832502
\(126\) 0 0
\(127\) 14.1438 1.25506 0.627528 0.778594i \(-0.284067\pi\)
0.627528 + 0.778594i \(0.284067\pi\)
\(128\) 11.1837 0.988512
\(129\) 0 0
\(130\) −1.15125 −0.100972
\(131\) −17.7401 −1.54996 −0.774979 0.631986i \(-0.782240\pi\)
−0.774979 + 0.631986i \(0.782240\pi\)
\(132\) 0 0
\(133\) 28.6523 2.48447
\(134\) −6.68240 −0.577271
\(135\) 0 0
\(136\) −11.3650 −0.974541
\(137\) −8.12457 −0.694129 −0.347064 0.937841i \(-0.612821\pi\)
−0.347064 + 0.937841i \(0.612821\pi\)
\(138\) 0 0
\(139\) 3.00074 0.254519 0.127260 0.991869i \(-0.459382\pi\)
0.127260 + 0.991869i \(0.459382\pi\)
\(140\) −7.11235 −0.601103
\(141\) 0 0
\(142\) 4.98091 0.417989
\(143\) −5.96985 −0.499224
\(144\) 0 0
\(145\) −1.04483 −0.0867680
\(146\) 1.44802 0.119839
\(147\) 0 0
\(148\) 9.19942 0.756188
\(149\) −15.2424 −1.24870 −0.624352 0.781143i \(-0.714637\pi\)
−0.624352 + 0.781143i \(0.714637\pi\)
\(150\) 0 0
\(151\) 7.08563 0.576620 0.288310 0.957537i \(-0.406907\pi\)
0.288310 + 0.957537i \(0.406907\pi\)
\(152\) 13.3585 1.08352
\(153\) 0 0
\(154\) 4.65051 0.374748
\(155\) 9.40244 0.755222
\(156\) 0 0
\(157\) −1.33644 −0.106659 −0.0533297 0.998577i \(-0.516983\pi\)
−0.0533297 + 0.998577i \(0.516983\pi\)
\(158\) 0.135732 0.0107983
\(159\) 0 0
\(160\) −5.07230 −0.401001
\(161\) 3.83278 0.302065
\(162\) 0 0
\(163\) 9.55986 0.748786 0.374393 0.927270i \(-0.377851\pi\)
0.374393 + 0.927270i \(0.377851\pi\)
\(164\) 0.210462 0.0164343
\(165\) 0 0
\(166\) 2.28233 0.177143
\(167\) 12.1300 0.938648 0.469324 0.883026i \(-0.344498\pi\)
0.469324 + 0.883026i \(0.344498\pi\)
\(168\) 0 0
\(169\) −7.57868 −0.582975
\(170\) 3.14467 0.241185
\(171\) 0 0
\(172\) 5.28689 0.403121
\(173\) 23.8506 1.81333 0.906664 0.421853i \(-0.138620\pi\)
0.906664 + 0.421853i \(0.138620\pi\)
\(174\) 0 0
\(175\) −14.9798 −1.13237
\(176\) −6.93924 −0.523065
\(177\) 0 0
\(178\) −0.535702 −0.0401525
\(179\) 20.3159 1.51848 0.759240 0.650811i \(-0.225571\pi\)
0.759240 + 0.650811i \(0.225571\pi\)
\(180\) 0 0
\(181\) 15.6061 1.15999 0.579997 0.814619i \(-0.303054\pi\)
0.579997 + 0.814619i \(0.303054\pi\)
\(182\) −4.22320 −0.313044
\(183\) 0 0
\(184\) 1.78695 0.131736
\(185\) −5.41189 −0.397890
\(186\) 0 0
\(187\) 16.3068 1.19247
\(188\) −19.1133 −1.39398
\(189\) 0 0
\(190\) −3.69627 −0.268156
\(191\) 19.5710 1.41611 0.708055 0.706157i \(-0.249573\pi\)
0.708055 + 0.706157i \(0.249573\pi\)
\(192\) 0 0
\(193\) 5.09645 0.366850 0.183425 0.983034i \(-0.441282\pi\)
0.183425 + 0.983034i \(0.441282\pi\)
\(194\) 1.31042 0.0940829
\(195\) 0 0
\(196\) −13.6582 −0.975586
\(197\) −23.2485 −1.65639 −0.828195 0.560440i \(-0.810632\pi\)
−0.828195 + 0.560440i \(0.810632\pi\)
\(198\) 0 0
\(199\) 24.2684 1.72034 0.860170 0.510007i \(-0.170357\pi\)
0.860170 + 0.510007i \(0.170357\pi\)
\(200\) −6.98402 −0.493844
\(201\) 0 0
\(202\) −3.61038 −0.254025
\(203\) −3.83278 −0.269009
\(204\) 0 0
\(205\) −0.123812 −0.00864739
\(206\) 0.622324 0.0433594
\(207\) 0 0
\(208\) 6.30164 0.436940
\(209\) −19.1671 −1.32582
\(210\) 0 0
\(211\) −13.0565 −0.898847 −0.449423 0.893319i \(-0.648371\pi\)
−0.449423 + 0.893319i \(0.648371\pi\)
\(212\) −9.39872 −0.645507
\(213\) 0 0
\(214\) −1.46293 −0.100004
\(215\) −3.11020 −0.212114
\(216\) 0 0
\(217\) 34.4914 2.34143
\(218\) 5.01651 0.339761
\(219\) 0 0
\(220\) 4.75784 0.320774
\(221\) −14.8084 −0.996123
\(222\) 0 0
\(223\) 21.0983 1.41285 0.706424 0.707789i \(-0.250307\pi\)
0.706424 + 0.707789i \(0.250307\pi\)
\(224\) −18.6070 −1.24323
\(225\) 0 0
\(226\) 1.11807 0.0743731
\(227\) −28.6355 −1.90060 −0.950301 0.311333i \(-0.899225\pi\)
−0.950301 + 0.311333i \(0.899225\pi\)
\(228\) 0 0
\(229\) −26.5257 −1.75287 −0.876433 0.481524i \(-0.840084\pi\)
−0.876433 + 0.481524i \(0.840084\pi\)
\(230\) −0.494446 −0.0326028
\(231\) 0 0
\(232\) −1.78695 −0.117319
\(233\) 2.34122 0.153378 0.0766892 0.997055i \(-0.475565\pi\)
0.0766892 + 0.997055i \(0.475565\pi\)
\(234\) 0 0
\(235\) 11.2441 0.733481
\(236\) −17.7026 −1.15234
\(237\) 0 0
\(238\) 11.5357 0.747751
\(239\) −3.07650 −0.199002 −0.0995011 0.995037i \(-0.531725\pi\)
−0.0995011 + 0.995037i \(0.531725\pi\)
\(240\) 0 0
\(241\) −19.8645 −1.27958 −0.639791 0.768549i \(-0.720979\pi\)
−0.639791 + 0.768549i \(0.720979\pi\)
\(242\) 2.09458 0.134645
\(243\) 0 0
\(244\) −6.24573 −0.399842
\(245\) 8.03493 0.513333
\(246\) 0 0
\(247\) 17.4060 1.10751
\(248\) 16.0809 1.02114
\(249\) 0 0
\(250\) 4.40469 0.278577
\(251\) 12.3435 0.779116 0.389558 0.921002i \(-0.372628\pi\)
0.389558 + 0.921002i \(0.372628\pi\)
\(252\) 0 0
\(253\) −2.56396 −0.161195
\(254\) −6.69329 −0.419975
\(255\) 0 0
\(256\) 0.938515 0.0586572
\(257\) −0.764367 −0.0476799 −0.0238399 0.999716i \(-0.507589\pi\)
−0.0238399 + 0.999716i \(0.507589\pi\)
\(258\) 0 0
\(259\) −19.8527 −1.23359
\(260\) −4.32067 −0.267957
\(261\) 0 0
\(262\) 8.39520 0.518657
\(263\) 12.4337 0.766693 0.383346 0.923605i \(-0.374772\pi\)
0.383346 + 0.923605i \(0.374772\pi\)
\(264\) 0 0
\(265\) 5.52913 0.339652
\(266\) −13.5592 −0.831369
\(267\) 0 0
\(268\) −25.0791 −1.53195
\(269\) −17.2895 −1.05416 −0.527080 0.849816i \(-0.676713\pi\)
−0.527080 + 0.849816i \(0.676713\pi\)
\(270\) 0 0
\(271\) 12.2849 0.746254 0.373127 0.927780i \(-0.378286\pi\)
0.373127 + 0.927780i \(0.378286\pi\)
\(272\) −17.2130 −1.04369
\(273\) 0 0
\(274\) 3.84481 0.232274
\(275\) 10.0208 0.604279
\(276\) 0 0
\(277\) 24.0877 1.44729 0.723645 0.690172i \(-0.242465\pi\)
0.723645 + 0.690172i \(0.242465\pi\)
\(278\) −1.42005 −0.0851688
\(279\) 0 0
\(280\) 7.15600 0.427653
\(281\) 25.8168 1.54010 0.770050 0.637983i \(-0.220231\pi\)
0.770050 + 0.637983i \(0.220231\pi\)
\(282\) 0 0
\(283\) −1.67500 −0.0995687 −0.0497843 0.998760i \(-0.515853\pi\)
−0.0497843 + 0.998760i \(0.515853\pi\)
\(284\) 18.6934 1.10925
\(285\) 0 0
\(286\) 2.82513 0.167053
\(287\) −0.454185 −0.0268097
\(288\) 0 0
\(289\) 23.4495 1.37938
\(290\) 0.494446 0.0290349
\(291\) 0 0
\(292\) 5.43445 0.318027
\(293\) 3.20666 0.187335 0.0936674 0.995604i \(-0.470141\pi\)
0.0936674 + 0.995604i \(0.470141\pi\)
\(294\) 0 0
\(295\) 10.4142 0.606337
\(296\) −9.25589 −0.537987
\(297\) 0 0
\(298\) 7.21319 0.417849
\(299\) 2.32837 0.134653
\(300\) 0 0
\(301\) −11.4093 −0.657621
\(302\) −3.35315 −0.192952
\(303\) 0 0
\(304\) 20.2324 1.16041
\(305\) 3.67427 0.210388
\(306\) 0 0
\(307\) −10.2790 −0.586655 −0.293328 0.956012i \(-0.594763\pi\)
−0.293328 + 0.956012i \(0.594763\pi\)
\(308\) 17.4534 0.994501
\(309\) 0 0
\(310\) −4.44955 −0.252717
\(311\) 8.33747 0.472775 0.236387 0.971659i \(-0.424037\pi\)
0.236387 + 0.971659i \(0.424037\pi\)
\(312\) 0 0
\(313\) −0.141232 −0.00798291 −0.00399146 0.999992i \(-0.501271\pi\)
−0.00399146 + 0.999992i \(0.501271\pi\)
\(314\) 0.632447 0.0356911
\(315\) 0 0
\(316\) 0.509404 0.0286562
\(317\) 7.06189 0.396635 0.198318 0.980138i \(-0.436452\pi\)
0.198318 + 0.980138i \(0.436452\pi\)
\(318\) 0 0
\(319\) 2.56396 0.143554
\(320\) −3.25517 −0.181969
\(321\) 0 0
\(322\) −1.81380 −0.101079
\(323\) −47.5447 −2.64546
\(324\) 0 0
\(325\) −9.10007 −0.504781
\(326\) −4.52404 −0.250563
\(327\) 0 0
\(328\) −0.211754 −0.0116922
\(329\) 41.2471 2.27403
\(330\) 0 0
\(331\) 25.4463 1.39866 0.699329 0.714800i \(-0.253482\pi\)
0.699329 + 0.714800i \(0.253482\pi\)
\(332\) 8.56562 0.470099
\(333\) 0 0
\(334\) −5.74032 −0.314096
\(335\) 14.7537 0.806081
\(336\) 0 0
\(337\) −15.8723 −0.864617 −0.432308 0.901726i \(-0.642301\pi\)
−0.432308 + 0.901726i \(0.642301\pi\)
\(338\) 3.58648 0.195079
\(339\) 0 0
\(340\) 11.8020 0.640053
\(341\) −23.0732 −1.24948
\(342\) 0 0
\(343\) 2.64543 0.142840
\(344\) −5.31934 −0.286800
\(345\) 0 0
\(346\) −11.2869 −0.606787
\(347\) 3.77366 0.202581 0.101290 0.994857i \(-0.467703\pi\)
0.101290 + 0.994857i \(0.467703\pi\)
\(348\) 0 0
\(349\) −8.83978 −0.473183 −0.236591 0.971609i \(-0.576030\pi\)
−0.236591 + 0.971609i \(0.576030\pi\)
\(350\) 7.08894 0.378920
\(351\) 0 0
\(352\) 12.4472 0.663439
\(353\) 26.2225 1.39568 0.697840 0.716253i \(-0.254144\pi\)
0.697840 + 0.716253i \(0.254144\pi\)
\(354\) 0 0
\(355\) −10.9971 −0.583664
\(356\) −2.01050 −0.106556
\(357\) 0 0
\(358\) −9.61414 −0.508123
\(359\) 17.3443 0.915398 0.457699 0.889107i \(-0.348674\pi\)
0.457699 + 0.889107i \(0.348674\pi\)
\(360\) 0 0
\(361\) 36.8845 1.94129
\(362\) −7.38533 −0.388164
\(363\) 0 0
\(364\) −15.8497 −0.830751
\(365\) −3.19701 −0.167339
\(366\) 0 0
\(367\) −10.0965 −0.527032 −0.263516 0.964655i \(-0.584882\pi\)
−0.263516 + 0.964655i \(0.584882\pi\)
\(368\) 2.70646 0.141084
\(369\) 0 0
\(370\) 2.56108 0.133144
\(371\) 20.2828 1.05303
\(372\) 0 0
\(373\) −13.0445 −0.675421 −0.337710 0.941250i \(-0.609652\pi\)
−0.337710 + 0.941250i \(0.609652\pi\)
\(374\) −7.71690 −0.399031
\(375\) 0 0
\(376\) 19.2306 0.991741
\(377\) −2.32837 −0.119917
\(378\) 0 0
\(379\) 7.81802 0.401585 0.200792 0.979634i \(-0.435648\pi\)
0.200792 + 0.979634i \(0.435648\pi\)
\(380\) −13.8722 −0.711628
\(381\) 0 0
\(382\) −9.26166 −0.473868
\(383\) 3.73452 0.190825 0.0954126 0.995438i \(-0.469583\pi\)
0.0954126 + 0.995438i \(0.469583\pi\)
\(384\) 0 0
\(385\) −10.2676 −0.523285
\(386\) −2.41181 −0.122758
\(387\) 0 0
\(388\) 4.91804 0.249675
\(389\) 22.7578 1.15387 0.576933 0.816792i \(-0.304249\pi\)
0.576933 + 0.816792i \(0.304249\pi\)
\(390\) 0 0
\(391\) −6.35999 −0.321639
\(392\) 13.7420 0.694078
\(393\) 0 0
\(394\) 11.0020 0.554272
\(395\) −0.299675 −0.0150783
\(396\) 0 0
\(397\) 14.6267 0.734093 0.367046 0.930203i \(-0.380369\pi\)
0.367046 + 0.930203i \(0.380369\pi\)
\(398\) −11.4846 −0.575671
\(399\) 0 0
\(400\) −10.5778 −0.528888
\(401\) −28.2873 −1.41260 −0.706301 0.707912i \(-0.749637\pi\)
−0.706301 + 0.707912i \(0.749637\pi\)
\(402\) 0 0
\(403\) 20.9532 1.04375
\(404\) −13.5498 −0.674128
\(405\) 0 0
\(406\) 1.81380 0.0900173
\(407\) 13.2806 0.658293
\(408\) 0 0
\(409\) −15.9885 −0.790581 −0.395291 0.918556i \(-0.629356\pi\)
−0.395291 + 0.918556i \(0.629356\pi\)
\(410\) 0.0585918 0.00289364
\(411\) 0 0
\(412\) 2.33559 0.115066
\(413\) 38.2028 1.87984
\(414\) 0 0
\(415\) −5.03903 −0.247356
\(416\) −11.3035 −0.554201
\(417\) 0 0
\(418\) 9.07051 0.443653
\(419\) 20.1167 0.982766 0.491383 0.870943i \(-0.336491\pi\)
0.491383 + 0.870943i \(0.336491\pi\)
\(420\) 0 0
\(421\) 6.27858 0.305999 0.153000 0.988226i \(-0.451107\pi\)
0.153000 + 0.988226i \(0.451107\pi\)
\(422\) 6.17877 0.300778
\(423\) 0 0
\(424\) 9.45641 0.459244
\(425\) 24.8570 1.20574
\(426\) 0 0
\(427\) 13.4785 0.652271
\(428\) −5.49038 −0.265388
\(429\) 0 0
\(430\) 1.47185 0.0709789
\(431\) −12.1897 −0.587159 −0.293579 0.955935i \(-0.594847\pi\)
−0.293579 + 0.955935i \(0.594847\pi\)
\(432\) 0 0
\(433\) 25.3379 1.21766 0.608830 0.793301i \(-0.291639\pi\)
0.608830 + 0.793301i \(0.291639\pi\)
\(434\) −16.3225 −0.783504
\(435\) 0 0
\(436\) 18.8270 0.901651
\(437\) 7.47559 0.357606
\(438\) 0 0
\(439\) 3.21937 0.153652 0.0768260 0.997045i \(-0.475521\pi\)
0.0768260 + 0.997045i \(0.475521\pi\)
\(440\) −4.78705 −0.228213
\(441\) 0 0
\(442\) 7.00784 0.333329
\(443\) −20.6170 −0.979541 −0.489771 0.871851i \(-0.662919\pi\)
−0.489771 + 0.871851i \(0.662919\pi\)
\(444\) 0 0
\(445\) 1.18275 0.0560675
\(446\) −9.98442 −0.472776
\(447\) 0 0
\(448\) −11.9411 −0.564163
\(449\) 32.3690 1.52759 0.763793 0.645461i \(-0.223335\pi\)
0.763793 + 0.645461i \(0.223335\pi\)
\(450\) 0 0
\(451\) 0.303829 0.0143068
\(452\) 4.19614 0.197370
\(453\) 0 0
\(454\) 13.5512 0.635991
\(455\) 9.32417 0.437124
\(456\) 0 0
\(457\) 0.711685 0.0332912 0.0166456 0.999861i \(-0.494701\pi\)
0.0166456 + 0.999861i \(0.494701\pi\)
\(458\) 12.5528 0.586555
\(459\) 0 0
\(460\) −1.85566 −0.0865207
\(461\) −13.5927 −0.633076 −0.316538 0.948580i \(-0.602520\pi\)
−0.316538 + 0.948580i \(0.602520\pi\)
\(462\) 0 0
\(463\) 39.6700 1.84362 0.921811 0.387640i \(-0.126710\pi\)
0.921811 + 0.387640i \(0.126710\pi\)
\(464\) −2.70646 −0.125644
\(465\) 0 0
\(466\) −1.10794 −0.0513244
\(467\) −3.81127 −0.176365 −0.0881824 0.996104i \(-0.528106\pi\)
−0.0881824 + 0.996104i \(0.528106\pi\)
\(468\) 0 0
\(469\) 54.1217 2.49911
\(470\) −5.32106 −0.245442
\(471\) 0 0
\(472\) 17.8113 0.819829
\(473\) 7.63231 0.350934
\(474\) 0 0
\(475\) −29.2172 −1.34058
\(476\) 43.2938 1.98437
\(477\) 0 0
\(478\) 1.45590 0.0665914
\(479\) −24.6634 −1.12690 −0.563450 0.826150i \(-0.690526\pi\)
−0.563450 + 0.826150i \(0.690526\pi\)
\(480\) 0 0
\(481\) −12.0603 −0.549902
\(482\) 9.40051 0.428182
\(483\) 0 0
\(484\) 7.86100 0.357318
\(485\) −2.89321 −0.131374
\(486\) 0 0
\(487\) 29.2115 1.32370 0.661850 0.749636i \(-0.269771\pi\)
0.661850 + 0.749636i \(0.269771\pi\)
\(488\) 6.28407 0.284466
\(489\) 0 0
\(490\) −3.80239 −0.171775
\(491\) −7.93069 −0.357907 −0.178953 0.983858i \(-0.557271\pi\)
−0.178953 + 0.983858i \(0.557271\pi\)
\(492\) 0 0
\(493\) 6.35999 0.286440
\(494\) −8.23708 −0.370604
\(495\) 0 0
\(496\) 24.3556 1.09360
\(497\) −40.3411 −1.80954
\(498\) 0 0
\(499\) 36.4840 1.63325 0.816623 0.577171i \(-0.195843\pi\)
0.816623 + 0.577171i \(0.195843\pi\)
\(500\) 16.5309 0.739283
\(501\) 0 0
\(502\) −5.84136 −0.260713
\(503\) −26.4478 −1.17925 −0.589624 0.807678i \(-0.700724\pi\)
−0.589624 + 0.807678i \(0.700724\pi\)
\(504\) 0 0
\(505\) 7.97116 0.354712
\(506\) 1.21335 0.0539400
\(507\) 0 0
\(508\) −25.1200 −1.11452
\(509\) 34.7697 1.54114 0.770571 0.637354i \(-0.219971\pi\)
0.770571 + 0.637354i \(0.219971\pi\)
\(510\) 0 0
\(511\) −11.7278 −0.518805
\(512\) −22.8116 −1.00814
\(513\) 0 0
\(514\) 0.361724 0.0159549
\(515\) −1.37399 −0.0605454
\(516\) 0 0
\(517\) −27.5925 −1.21352
\(518\) 9.39495 0.412790
\(519\) 0 0
\(520\) 4.34719 0.190637
\(521\) 18.9963 0.832244 0.416122 0.909309i \(-0.363389\pi\)
0.416122 + 0.909309i \(0.363389\pi\)
\(522\) 0 0
\(523\) 4.84802 0.211989 0.105995 0.994367i \(-0.466197\pi\)
0.105995 + 0.994367i \(0.466197\pi\)
\(524\) 31.5073 1.37640
\(525\) 0 0
\(526\) −5.88402 −0.256556
\(527\) −57.2340 −2.49315
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −2.61657 −0.113656
\(531\) 0 0
\(532\) −50.8880 −2.20627
\(533\) −0.275912 −0.0119511
\(534\) 0 0
\(535\) 3.22991 0.139641
\(536\) 25.2331 1.08990
\(537\) 0 0
\(538\) 8.18197 0.352750
\(539\) −19.7174 −0.849288
\(540\) 0 0
\(541\) 8.52437 0.366491 0.183246 0.983067i \(-0.441340\pi\)
0.183246 + 0.983067i \(0.441340\pi\)
\(542\) −5.81361 −0.249716
\(543\) 0 0
\(544\) 30.8758 1.32379
\(545\) −11.0757 −0.474429
\(546\) 0 0
\(547\) 40.4348 1.72887 0.864433 0.502748i \(-0.167678\pi\)
0.864433 + 0.502748i \(0.167678\pi\)
\(548\) 14.4296 0.616404
\(549\) 0 0
\(550\) −4.74219 −0.202207
\(551\) −7.47559 −0.318471
\(552\) 0 0
\(553\) −1.09931 −0.0467475
\(554\) −11.3991 −0.484301
\(555\) 0 0
\(556\) −5.32946 −0.226019
\(557\) 0.156519 0.00663191 0.00331595 0.999995i \(-0.498944\pi\)
0.00331595 + 0.999995i \(0.498944\pi\)
\(558\) 0 0
\(559\) −6.93102 −0.293151
\(560\) 10.8382 0.457999
\(561\) 0 0
\(562\) −12.2174 −0.515358
\(563\) 9.14136 0.385262 0.192631 0.981271i \(-0.438298\pi\)
0.192631 + 0.981271i \(0.438298\pi\)
\(564\) 0 0
\(565\) −2.46853 −0.103852
\(566\) 0.792667 0.0333183
\(567\) 0 0
\(568\) −18.8082 −0.789173
\(569\) 43.2851 1.81461 0.907304 0.420476i \(-0.138137\pi\)
0.907304 + 0.420476i \(0.138137\pi\)
\(570\) 0 0
\(571\) 26.9946 1.12969 0.564844 0.825198i \(-0.308937\pi\)
0.564844 + 0.825198i \(0.308937\pi\)
\(572\) 10.6028 0.443324
\(573\) 0 0
\(574\) 0.214935 0.00897122
\(575\) −3.90834 −0.162989
\(576\) 0 0
\(577\) 15.2683 0.635626 0.317813 0.948153i \(-0.397052\pi\)
0.317813 + 0.948153i \(0.397052\pi\)
\(578\) −11.0971 −0.461578
\(579\) 0 0
\(580\) 1.85566 0.0770522
\(581\) −18.4849 −0.766884
\(582\) 0 0
\(583\) −13.5683 −0.561940
\(584\) −5.46781 −0.226260
\(585\) 0 0
\(586\) −1.51750 −0.0626872
\(587\) 13.8956 0.573534 0.286767 0.958000i \(-0.407419\pi\)
0.286767 + 0.958000i \(0.407419\pi\)
\(588\) 0 0
\(589\) 67.2733 2.77195
\(590\) −4.92834 −0.202896
\(591\) 0 0
\(592\) −14.0187 −0.576163
\(593\) −37.4682 −1.53864 −0.769318 0.638866i \(-0.779404\pi\)
−0.769318 + 0.638866i \(0.779404\pi\)
\(594\) 0 0
\(595\) −25.4691 −1.04413
\(596\) 27.0712 1.10888
\(597\) 0 0
\(598\) −1.10186 −0.0450585
\(599\) −38.3625 −1.56745 −0.783725 0.621108i \(-0.786683\pi\)
−0.783725 + 0.621108i \(0.786683\pi\)
\(600\) 0 0
\(601\) −40.5272 −1.65314 −0.826570 0.562834i \(-0.809711\pi\)
−0.826570 + 0.562834i \(0.809711\pi\)
\(602\) 5.39926 0.220057
\(603\) 0 0
\(604\) −12.5844 −0.512054
\(605\) −4.62451 −0.188013
\(606\) 0 0
\(607\) 0.348810 0.0141578 0.00707888 0.999975i \(-0.497747\pi\)
0.00707888 + 0.999975i \(0.497747\pi\)
\(608\) −36.2917 −1.47182
\(609\) 0 0
\(610\) −1.73879 −0.0704015
\(611\) 25.0572 1.01370
\(612\) 0 0
\(613\) −11.9753 −0.483679 −0.241839 0.970316i \(-0.577751\pi\)
−0.241839 + 0.970316i \(0.577751\pi\)
\(614\) 4.86438 0.196310
\(615\) 0 0
\(616\) −17.5605 −0.707534
\(617\) 4.83441 0.194626 0.0973131 0.995254i \(-0.468975\pi\)
0.0973131 + 0.995254i \(0.468975\pi\)
\(618\) 0 0
\(619\) −10.0325 −0.403239 −0.201620 0.979464i \(-0.564621\pi\)
−0.201620 + 0.979464i \(0.564621\pi\)
\(620\) −16.6992 −0.670657
\(621\) 0 0
\(622\) −3.94557 −0.158203
\(623\) 4.33872 0.173827
\(624\) 0 0
\(625\) 9.81683 0.392673
\(626\) 0.0668357 0.00267129
\(627\) 0 0
\(628\) 2.37358 0.0947163
\(629\) 32.9429 1.31352
\(630\) 0 0
\(631\) 11.9638 0.476271 0.238136 0.971232i \(-0.423464\pi\)
0.238136 + 0.971232i \(0.423464\pi\)
\(632\) −0.512531 −0.0203874
\(633\) 0 0
\(634\) −3.34192 −0.132724
\(635\) 14.7778 0.586437
\(636\) 0 0
\(637\) 17.9057 0.709449
\(638\) −1.21335 −0.0480370
\(639\) 0 0
\(640\) 11.6851 0.461892
\(641\) 34.0018 1.34299 0.671494 0.741010i \(-0.265653\pi\)
0.671494 + 0.741010i \(0.265653\pi\)
\(642\) 0 0
\(643\) −4.25288 −0.167717 −0.0838587 0.996478i \(-0.526724\pi\)
−0.0838587 + 0.996478i \(0.526724\pi\)
\(644\) −6.80721 −0.268242
\(645\) 0 0
\(646\) 22.4997 0.885240
\(647\) 46.7915 1.83956 0.919781 0.392431i \(-0.128366\pi\)
0.919781 + 0.392431i \(0.128366\pi\)
\(648\) 0 0
\(649\) −25.5560 −1.00316
\(650\) 4.30645 0.168913
\(651\) 0 0
\(652\) −16.9788 −0.664941
\(653\) −36.0850 −1.41212 −0.706058 0.708154i \(-0.749528\pi\)
−0.706058 + 0.708154i \(0.749528\pi\)
\(654\) 0 0
\(655\) −18.5353 −0.724234
\(656\) −0.320715 −0.0125218
\(657\) 0 0
\(658\) −19.5195 −0.760949
\(659\) 2.53082 0.0985867 0.0492934 0.998784i \(-0.484303\pi\)
0.0492934 + 0.998784i \(0.484303\pi\)
\(660\) 0 0
\(661\) −13.0031 −0.505763 −0.252882 0.967497i \(-0.581378\pi\)
−0.252882 + 0.967497i \(0.581378\pi\)
\(662\) −12.0421 −0.468028
\(663\) 0 0
\(664\) −8.61819 −0.334451
\(665\) 29.9367 1.16089
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) −21.5435 −0.833543
\(669\) 0 0
\(670\) −6.98194 −0.269736
\(671\) −9.01652 −0.348079
\(672\) 0 0
\(673\) −36.1647 −1.39405 −0.697023 0.717049i \(-0.745493\pi\)
−0.697023 + 0.717049i \(0.745493\pi\)
\(674\) 7.51128 0.289324
\(675\) 0 0
\(676\) 13.4601 0.517697
\(677\) 15.7179 0.604089 0.302044 0.953294i \(-0.402331\pi\)
0.302044 + 0.953294i \(0.402331\pi\)
\(678\) 0 0
\(679\) −10.6133 −0.407301
\(680\) −11.8744 −0.455364
\(681\) 0 0
\(682\) 10.9190 0.418110
\(683\) 39.2207 1.50074 0.750370 0.661018i \(-0.229875\pi\)
0.750370 + 0.661018i \(0.229875\pi\)
\(684\) 0 0
\(685\) −8.48875 −0.324339
\(686\) −1.25191 −0.0477980
\(687\) 0 0
\(688\) −8.05649 −0.307151
\(689\) 12.3216 0.469414
\(690\) 0 0
\(691\) −48.3185 −1.83812 −0.919060 0.394117i \(-0.871050\pi\)
−0.919060 + 0.394117i \(0.871050\pi\)
\(692\) −42.3599 −1.61028
\(693\) 0 0
\(694\) −1.78582 −0.0677888
\(695\) 3.13524 0.118927
\(696\) 0 0
\(697\) 0.753660 0.0285469
\(698\) 4.18328 0.158339
\(699\) 0 0
\(700\) 26.6049 1.00557
\(701\) −25.3649 −0.958019 −0.479010 0.877810i \(-0.659004\pi\)
−0.479010 + 0.877810i \(0.659004\pi\)
\(702\) 0 0
\(703\) −38.7214 −1.46040
\(704\) 7.98805 0.301061
\(705\) 0 0
\(706\) −12.4093 −0.467031
\(707\) 29.2410 1.09972
\(708\) 0 0
\(709\) 21.1340 0.793703 0.396852 0.917883i \(-0.370103\pi\)
0.396852 + 0.917883i \(0.370103\pi\)
\(710\) 5.20418 0.195309
\(711\) 0 0
\(712\) 2.02284 0.0758090
\(713\) 8.99906 0.337017
\(714\) 0 0
\(715\) −6.23745 −0.233267
\(716\) −36.0820 −1.34845
\(717\) 0 0
\(718\) −8.20790 −0.306316
\(719\) −41.6848 −1.55458 −0.777291 0.629142i \(-0.783407\pi\)
−0.777291 + 0.629142i \(0.783407\pi\)
\(720\) 0 0
\(721\) −5.04029 −0.187710
\(722\) −17.4550 −0.649606
\(723\) 0 0
\(724\) −27.7173 −1.03010
\(725\) 3.90834 0.145152
\(726\) 0 0
\(727\) 10.1346 0.375870 0.187935 0.982181i \(-0.439821\pi\)
0.187935 + 0.982181i \(0.439821\pi\)
\(728\) 15.9470 0.591036
\(729\) 0 0
\(730\) 1.51293 0.0559961
\(731\) 18.9322 0.700233
\(732\) 0 0
\(733\) −41.3568 −1.52755 −0.763775 0.645483i \(-0.776656\pi\)
−0.763775 + 0.645483i \(0.776656\pi\)
\(734\) 4.77798 0.176359
\(735\) 0 0
\(736\) −4.85469 −0.178946
\(737\) −36.2050 −1.33363
\(738\) 0 0
\(739\) 31.4713 1.15769 0.578845 0.815438i \(-0.303504\pi\)
0.578845 + 0.815438i \(0.303504\pi\)
\(740\) 9.61179 0.353336
\(741\) 0 0
\(742\) −9.59848 −0.352371
\(743\) −10.4909 −0.384873 −0.192436 0.981309i \(-0.561639\pi\)
−0.192436 + 0.981309i \(0.561639\pi\)
\(744\) 0 0
\(745\) −15.9256 −0.583469
\(746\) 6.17311 0.226014
\(747\) 0 0
\(748\) −28.9616 −1.05894
\(749\) 11.8484 0.432933
\(750\) 0 0
\(751\) −32.4188 −1.18298 −0.591489 0.806313i \(-0.701460\pi\)
−0.591489 + 0.806313i \(0.701460\pi\)
\(752\) 29.1260 1.06211
\(753\) 0 0
\(754\) 1.10186 0.0401275
\(755\) 7.40324 0.269432
\(756\) 0 0
\(757\) −46.3828 −1.68581 −0.842906 0.538061i \(-0.819157\pi\)
−0.842906 + 0.538061i \(0.819157\pi\)
\(758\) −3.69975 −0.134381
\(759\) 0 0
\(760\) 13.9573 0.506285
\(761\) −12.2143 −0.442768 −0.221384 0.975187i \(-0.571057\pi\)
−0.221384 + 0.975187i \(0.571057\pi\)
\(762\) 0 0
\(763\) −40.6294 −1.47088
\(764\) −34.7591 −1.25754
\(765\) 0 0
\(766\) −1.76730 −0.0638551
\(767\) 23.2078 0.837985
\(768\) 0 0
\(769\) −2.07966 −0.0749946 −0.0374973 0.999297i \(-0.511939\pi\)
−0.0374973 + 0.999297i \(0.511939\pi\)
\(770\) 4.85896 0.175105
\(771\) 0 0
\(772\) −9.05155 −0.325772
\(773\) 6.91282 0.248637 0.124318 0.992242i \(-0.460326\pi\)
0.124318 + 0.992242i \(0.460326\pi\)
\(774\) 0 0
\(775\) −35.1714 −1.26339
\(776\) −4.94822 −0.177631
\(777\) 0 0
\(778\) −10.7697 −0.386114
\(779\) −0.885858 −0.0317392
\(780\) 0 0
\(781\) 26.9864 0.965649
\(782\) 3.00976 0.107629
\(783\) 0 0
\(784\) 20.8132 0.743329
\(785\) −1.39635 −0.0498377
\(786\) 0 0
\(787\) −12.8364 −0.457570 −0.228785 0.973477i \(-0.573475\pi\)
−0.228785 + 0.973477i \(0.573475\pi\)
\(788\) 41.2906 1.47092
\(789\) 0 0
\(790\) 0.141816 0.00504559
\(791\) −9.05543 −0.321974
\(792\) 0 0
\(793\) 8.18805 0.290766
\(794\) −6.92183 −0.245647
\(795\) 0 0
\(796\) −43.1019 −1.52771
\(797\) 20.2940 0.718849 0.359425 0.933174i \(-0.382973\pi\)
0.359425 + 0.933174i \(0.382973\pi\)
\(798\) 0 0
\(799\) −68.4441 −2.42138
\(800\) 18.9738 0.670824
\(801\) 0 0
\(802\) 13.3865 0.472694
\(803\) 7.84534 0.276856
\(804\) 0 0
\(805\) 4.00459 0.141143
\(806\) −9.91573 −0.349267
\(807\) 0 0
\(808\) 13.6330 0.479606
\(809\) 34.7953 1.22334 0.611669 0.791114i \(-0.290498\pi\)
0.611669 + 0.791114i \(0.290498\pi\)
\(810\) 0 0
\(811\) 23.1603 0.813268 0.406634 0.913591i \(-0.366702\pi\)
0.406634 + 0.913591i \(0.366702\pi\)
\(812\) 6.80721 0.238886
\(813\) 0 0
\(814\) −6.28480 −0.220282
\(815\) 9.98838 0.349878
\(816\) 0 0
\(817\) −22.2531 −0.778537
\(818\) 7.56629 0.264549
\(819\) 0 0
\(820\) 0.219896 0.00767910
\(821\) −49.5815 −1.73041 −0.865203 0.501422i \(-0.832810\pi\)
−0.865203 + 0.501422i \(0.832810\pi\)
\(822\) 0 0
\(823\) −25.0778 −0.874159 −0.437079 0.899423i \(-0.643987\pi\)
−0.437079 + 0.899423i \(0.643987\pi\)
\(824\) −2.34993 −0.0818636
\(825\) 0 0
\(826\) −18.0788 −0.629044
\(827\) −35.7445 −1.24296 −0.621480 0.783430i \(-0.713468\pi\)
−0.621480 + 0.783430i \(0.713468\pi\)
\(828\) 0 0
\(829\) 43.5954 1.51413 0.757066 0.653339i \(-0.226632\pi\)
0.757066 + 0.653339i \(0.226632\pi\)
\(830\) 2.38464 0.0827719
\(831\) 0 0
\(832\) −7.25408 −0.251490
\(833\) −48.9097 −1.69462
\(834\) 0 0
\(835\) 12.6737 0.438593
\(836\) 34.0418 1.17736
\(837\) 0 0
\(838\) −9.51990 −0.328859
\(839\) 18.4127 0.635679 0.317839 0.948145i \(-0.397043\pi\)
0.317839 + 0.948145i \(0.397043\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −2.97123 −0.102395
\(843\) 0 0
\(844\) 23.1890 0.798199
\(845\) −7.91840 −0.272401
\(846\) 0 0
\(847\) −16.9643 −0.582901
\(848\) 14.3224 0.491832
\(849\) 0 0
\(850\) −11.7632 −0.403473
\(851\) −5.17971 −0.177558
\(852\) 0 0
\(853\) −18.4481 −0.631651 −0.315826 0.948817i \(-0.602281\pi\)
−0.315826 + 0.948817i \(0.602281\pi\)
\(854\) −6.37848 −0.218267
\(855\) 0 0
\(856\) 5.52408 0.188809
\(857\) −2.81664 −0.0962147 −0.0481074 0.998842i \(-0.515319\pi\)
−0.0481074 + 0.998842i \(0.515319\pi\)
\(858\) 0 0
\(859\) −45.9546 −1.56795 −0.783975 0.620793i \(-0.786811\pi\)
−0.783975 + 0.620793i \(0.786811\pi\)
\(860\) 5.52387 0.188363
\(861\) 0 0
\(862\) 5.76859 0.196479
\(863\) 18.0449 0.614254 0.307127 0.951669i \(-0.400632\pi\)
0.307127 + 0.951669i \(0.400632\pi\)
\(864\) 0 0
\(865\) 24.9197 0.847296
\(866\) −11.9907 −0.407461
\(867\) 0 0
\(868\) −61.2585 −2.07925
\(869\) 0.735391 0.0249464
\(870\) 0 0
\(871\) 32.8783 1.11404
\(872\) −18.9426 −0.641477
\(873\) 0 0
\(874\) −3.53770 −0.119664
\(875\) −35.6742 −1.20601
\(876\) 0 0
\(877\) 32.0156 1.08109 0.540545 0.841315i \(-0.318218\pi\)
0.540545 + 0.841315i \(0.318218\pi\)
\(878\) −1.52351 −0.0514160
\(879\) 0 0
\(880\) −7.25030 −0.244407
\(881\) −48.5137 −1.63447 −0.817234 0.576306i \(-0.804494\pi\)
−0.817234 + 0.576306i \(0.804494\pi\)
\(882\) 0 0
\(883\) 13.9176 0.468364 0.234182 0.972193i \(-0.424759\pi\)
0.234182 + 0.972193i \(0.424759\pi\)
\(884\) 26.3005 0.884583
\(885\) 0 0
\(886\) 9.75662 0.327780
\(887\) −56.2498 −1.88868 −0.944341 0.328968i \(-0.893299\pi\)
−0.944341 + 0.328968i \(0.893299\pi\)
\(888\) 0 0
\(889\) 54.2099 1.81814
\(890\) −0.559714 −0.0187617
\(891\) 0 0
\(892\) −37.4717 −1.25464
\(893\) 80.4498 2.69215
\(894\) 0 0
\(895\) 21.2265 0.709525
\(896\) 42.8648 1.43201
\(897\) 0 0
\(898\) −15.3181 −0.511170
\(899\) −8.99906 −0.300135
\(900\) 0 0
\(901\) −33.6566 −1.12126
\(902\) −0.143782 −0.00478742
\(903\) 0 0
\(904\) −4.22190 −0.140418
\(905\) 16.3057 0.542019
\(906\) 0 0
\(907\) 24.0052 0.797080 0.398540 0.917151i \(-0.369517\pi\)
0.398540 + 0.917151i \(0.369517\pi\)
\(908\) 50.8580 1.68778
\(909\) 0 0
\(910\) −4.41250 −0.146273
\(911\) −27.2733 −0.903605 −0.451803 0.892118i \(-0.649219\pi\)
−0.451803 + 0.892118i \(0.649219\pi\)
\(912\) 0 0
\(913\) 12.3656 0.409241
\(914\) −0.336793 −0.0111401
\(915\) 0 0
\(916\) 47.1109 1.55659
\(917\) −67.9939 −2.24536
\(918\) 0 0
\(919\) −54.4520 −1.79621 −0.898103 0.439784i \(-0.855055\pi\)
−0.898103 + 0.439784i \(0.855055\pi\)
\(920\) 1.86705 0.0615549
\(921\) 0 0
\(922\) 6.43252 0.211844
\(923\) −24.5068 −0.806650
\(924\) 0 0
\(925\) 20.2441 0.665621
\(926\) −18.7732 −0.616924
\(927\) 0 0
\(928\) 4.85469 0.159363
\(929\) 28.7869 0.944468 0.472234 0.881473i \(-0.343448\pi\)
0.472234 + 0.881473i \(0.343448\pi\)
\(930\) 0 0
\(931\) 57.4889 1.88412
\(932\) −4.15812 −0.136204
\(933\) 0 0
\(934\) 1.80362 0.0590163
\(935\) 17.0377 0.557193
\(936\) 0 0
\(937\) −10.2809 −0.335863 −0.167932 0.985799i \(-0.553709\pi\)
−0.167932 + 0.985799i \(0.553709\pi\)
\(938\) −25.6122 −0.836267
\(939\) 0 0
\(940\) −19.9700 −0.651350
\(941\) 5.08436 0.165745 0.0828726 0.996560i \(-0.473591\pi\)
0.0828726 + 0.996560i \(0.473591\pi\)
\(942\) 0 0
\(943\) −0.118500 −0.00385889
\(944\) 26.9763 0.878005
\(945\) 0 0
\(946\) −3.61186 −0.117432
\(947\) −33.5032 −1.08871 −0.544354 0.838856i \(-0.683225\pi\)
−0.544354 + 0.838856i \(0.683225\pi\)
\(948\) 0 0
\(949\) −7.12448 −0.231270
\(950\) 13.8265 0.448592
\(951\) 0 0
\(952\) −43.5596 −1.41177
\(953\) 51.6427 1.67287 0.836435 0.548065i \(-0.184636\pi\)
0.836435 + 0.548065i \(0.184636\pi\)
\(954\) 0 0
\(955\) 20.4483 0.661692
\(956\) 5.46402 0.176719
\(957\) 0 0
\(958\) 11.6715 0.377090
\(959\) −31.1397 −1.00555
\(960\) 0 0
\(961\) 49.9831 1.61236
\(962\) 5.70733 0.184012
\(963\) 0 0
\(964\) 35.2803 1.13630
\(965\) 5.32490 0.171414
\(966\) 0 0
\(967\) −30.0050 −0.964897 −0.482448 0.875924i \(-0.660252\pi\)
−0.482448 + 0.875924i \(0.660252\pi\)
\(968\) −7.90925 −0.254213
\(969\) 0 0
\(970\) 1.36916 0.0439612
\(971\) 22.9615 0.736869 0.368434 0.929654i \(-0.379894\pi\)
0.368434 + 0.929654i \(0.379894\pi\)
\(972\) 0 0
\(973\) 11.5012 0.368710
\(974\) −13.8239 −0.442945
\(975\) 0 0
\(976\) 9.51764 0.304652
\(977\) 39.9218 1.27721 0.638606 0.769534i \(-0.279512\pi\)
0.638606 + 0.769534i \(0.279512\pi\)
\(978\) 0 0
\(979\) −2.90241 −0.0927615
\(980\) −14.2704 −0.455852
\(981\) 0 0
\(982\) 3.75306 0.119765
\(983\) 16.8911 0.538742 0.269371 0.963036i \(-0.413184\pi\)
0.269371 + 0.963036i \(0.413184\pi\)
\(984\) 0 0
\(985\) −24.2907 −0.773965
\(986\) −3.00976 −0.0958503
\(987\) 0 0
\(988\) −30.9139 −0.983501
\(989\) −2.97677 −0.0946557
\(990\) 0 0
\(991\) −37.1299 −1.17947 −0.589734 0.807597i \(-0.700768\pi\)
−0.589734 + 0.807597i \(0.700768\pi\)
\(992\) −43.6876 −1.38708
\(993\) 0 0
\(994\) 19.0907 0.605521
\(995\) 25.3562 0.803846
\(996\) 0 0
\(997\) 14.6652 0.464452 0.232226 0.972662i \(-0.425399\pi\)
0.232226 + 0.972662i \(0.425399\pi\)
\(998\) −17.2654 −0.546527
\(999\) 0 0
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 6003.2.a.v.1.14 30
3.2 odd 2 6003.2.a.w.1.17 yes 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6003.2.a.v.1.14 30 1.1 even 1 trivial
6003.2.a.w.1.17 yes 30 3.2 odd 2