Properties

Label 2175.2.a.bc.1.8
Level $2175$
Weight $2$
Character 2175.1
Self dual yes
Analytic conductor $17.367$
Analytic rank $0$
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] = [2175,2,Mod(1,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, 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("2175.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(17.3674624396\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 12x^{6} + 23x^{5} + 36x^{4} - 62x^{3} - 15x^{2} + 14x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-2.72810\) of defining polynomial
Character \(\chi\) \(=\) 2175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.72810 q^{2} -1.00000 q^{3} +5.44251 q^{4} -2.72810 q^{6} +4.61695 q^{7} +9.39150 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.72810 q^{2} -1.00000 q^{3} +5.44251 q^{4} -2.72810 q^{6} +4.61695 q^{7} +9.39150 q^{8} +1.00000 q^{9} -1.87758 q^{11} -5.44251 q^{12} +5.48400 q^{13} +12.5955 q^{14} +14.7359 q^{16} -5.74056 q^{17} +2.72810 q^{18} -5.40455 q^{19} -4.61695 q^{21} -5.12221 q^{22} -0.137016 q^{23} -9.39150 q^{24} +14.9609 q^{26} -1.00000 q^{27} +25.1278 q^{28} +1.00000 q^{29} -8.18167 q^{31} +21.4180 q^{32} +1.87758 q^{33} -15.6608 q^{34} +5.44251 q^{36} -4.45785 q^{37} -14.7441 q^{38} -5.48400 q^{39} -0.215243 q^{41} -12.5955 q^{42} +7.17115 q^{43} -10.2187 q^{44} -0.373793 q^{46} -9.91293 q^{47} -14.7359 q^{48} +14.3162 q^{49} +5.74056 q^{51} +29.8467 q^{52} -1.14754 q^{53} -2.72810 q^{54} +43.3601 q^{56} +5.40455 q^{57} +2.72810 q^{58} -0.244913 q^{59} +8.75284 q^{61} -22.3204 q^{62} +4.61695 q^{63} +28.9585 q^{64} +5.12221 q^{66} -6.53295 q^{67} -31.2431 q^{68} +0.137016 q^{69} -0.277412 q^{71} +9.39150 q^{72} -11.4933 q^{73} -12.1614 q^{74} -29.4143 q^{76} -8.66866 q^{77} -14.9609 q^{78} +1.66223 q^{79} +1.00000 q^{81} -0.587203 q^{82} +9.58514 q^{83} -25.1278 q^{84} +19.5636 q^{86} -1.00000 q^{87} -17.6333 q^{88} +16.3332 q^{89} +25.3193 q^{91} -0.745712 q^{92} +8.18167 q^{93} -27.0434 q^{94} -21.4180 q^{96} +8.04128 q^{97} +39.0559 q^{98} -1.87758 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 8 q^{3} + 12 q^{4} + 2 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 8 q^{3} + 12 q^{4} + 2 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9} + 6 q^{11} - 12 q^{12} + 6 q^{13} + 9 q^{14} + 32 q^{16} - 12 q^{17} - 2 q^{18} - 2 q^{21} + 3 q^{22} - 14 q^{23} + 3 q^{24} + 18 q^{26} - 8 q^{27} + 14 q^{28} + 8 q^{29} + 8 q^{31} + 2 q^{32} - 6 q^{33} - 13 q^{34} + 12 q^{36} + 4 q^{37} - 26 q^{38} - 6 q^{39} + 2 q^{41} - 9 q^{42} + 2 q^{43} - 15 q^{44} + 24 q^{46} - 12 q^{47} - 32 q^{48} + 38 q^{49} + 12 q^{51} + 49 q^{52} - 4 q^{53} + 2 q^{54} + 58 q^{56} - 2 q^{58} + 18 q^{59} + 12 q^{61} + 4 q^{62} + 2 q^{63} + 21 q^{64} - 3 q^{66} + 26 q^{67} - 81 q^{68} + 14 q^{69} + 24 q^{71} - 3 q^{72} - 14 q^{73} - 22 q^{74} + 26 q^{77} - 18 q^{78} + 10 q^{79} + 8 q^{81} + 48 q^{82} - 40 q^{83} - 14 q^{84} + 8 q^{86} - 8 q^{87} - 10 q^{88} + 34 q^{89} + 26 q^{91} + 18 q^{92} - 8 q^{93} - 43 q^{94} - 2 q^{96} + 30 q^{97} - 29 q^{98} + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.72810 1.92906 0.964528 0.263981i \(-0.0850357\pi\)
0.964528 + 0.263981i \(0.0850357\pi\)
\(3\) −1.00000 −0.577350
\(4\) 5.44251 2.72126
\(5\) 0 0
\(6\) −2.72810 −1.11374
\(7\) 4.61695 1.74504 0.872521 0.488577i \(-0.162484\pi\)
0.872521 + 0.488577i \(0.162484\pi\)
\(8\) 9.39150 3.32040
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.87758 −0.566110 −0.283055 0.959104i \(-0.591348\pi\)
−0.283055 + 0.959104i \(0.591348\pi\)
\(12\) −5.44251 −1.57112
\(13\) 5.48400 1.52099 0.760494 0.649345i \(-0.224957\pi\)
0.760494 + 0.649345i \(0.224957\pi\)
\(14\) 12.5955 3.36628
\(15\) 0 0
\(16\) 14.7359 3.68398
\(17\) −5.74056 −1.39229 −0.696145 0.717901i \(-0.745103\pi\)
−0.696145 + 0.717901i \(0.745103\pi\)
\(18\) 2.72810 0.643019
\(19\) −5.40455 −1.23989 −0.619944 0.784646i \(-0.712845\pi\)
−0.619944 + 0.784646i \(0.712845\pi\)
\(20\) 0 0
\(21\) −4.61695 −1.00750
\(22\) −5.12221 −1.09206
\(23\) −0.137016 −0.0285698 −0.0142849 0.999898i \(-0.504547\pi\)
−0.0142849 + 0.999898i \(0.504547\pi\)
\(24\) −9.39150 −1.91703
\(25\) 0 0
\(26\) 14.9609 2.93407
\(27\) −1.00000 −0.192450
\(28\) 25.1278 4.74870
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −8.18167 −1.46947 −0.734736 0.678353i \(-0.762694\pi\)
−0.734736 + 0.678353i \(0.762694\pi\)
\(32\) 21.4180 3.78620
\(33\) 1.87758 0.326844
\(34\) −15.6608 −2.68581
\(35\) 0 0
\(36\) 5.44251 0.907085
\(37\) −4.45785 −0.732866 −0.366433 0.930444i \(-0.619421\pi\)
−0.366433 + 0.930444i \(0.619421\pi\)
\(38\) −14.7441 −2.39181
\(39\) −5.48400 −0.878142
\(40\) 0 0
\(41\) −0.215243 −0.0336153 −0.0168076 0.999859i \(-0.505350\pi\)
−0.0168076 + 0.999859i \(0.505350\pi\)
\(42\) −12.5955 −1.94352
\(43\) 7.17115 1.09359 0.546795 0.837266i \(-0.315848\pi\)
0.546795 + 0.837266i \(0.315848\pi\)
\(44\) −10.2187 −1.54053
\(45\) 0 0
\(46\) −0.373793 −0.0551128
\(47\) −9.91293 −1.44595 −0.722975 0.690874i \(-0.757226\pi\)
−0.722975 + 0.690874i \(0.757226\pi\)
\(48\) −14.7359 −2.12695
\(49\) 14.3162 2.04517
\(50\) 0 0
\(51\) 5.74056 0.803839
\(52\) 29.8467 4.13899
\(53\) −1.14754 −0.157627 −0.0788134 0.996889i \(-0.525113\pi\)
−0.0788134 + 0.996889i \(0.525113\pi\)
\(54\) −2.72810 −0.371247
\(55\) 0 0
\(56\) 43.3601 5.79423
\(57\) 5.40455 0.715850
\(58\) 2.72810 0.358217
\(59\) −0.244913 −0.0318849 −0.0159425 0.999873i \(-0.505075\pi\)
−0.0159425 + 0.999873i \(0.505075\pi\)
\(60\) 0 0
\(61\) 8.75284 1.12069 0.560343 0.828260i \(-0.310669\pi\)
0.560343 + 0.828260i \(0.310669\pi\)
\(62\) −22.3204 −2.83469
\(63\) 4.61695 0.581680
\(64\) 28.9585 3.61981
\(65\) 0 0
\(66\) 5.12221 0.630500
\(67\) −6.53295 −0.798127 −0.399064 0.916923i \(-0.630665\pi\)
−0.399064 + 0.916923i \(0.630665\pi\)
\(68\) −31.2431 −3.78878
\(69\) 0.137016 0.0164948
\(70\) 0 0
\(71\) −0.277412 −0.0329227 −0.0164614 0.999865i \(-0.505240\pi\)
−0.0164614 + 0.999865i \(0.505240\pi\)
\(72\) 9.39150 1.10680
\(73\) −11.4933 −1.34519 −0.672594 0.740011i \(-0.734820\pi\)
−0.672594 + 0.740011i \(0.734820\pi\)
\(74\) −12.1614 −1.41374
\(75\) 0 0
\(76\) −29.4143 −3.37405
\(77\) −8.66866 −0.987886
\(78\) −14.9609 −1.69399
\(79\) 1.66223 0.187015 0.0935076 0.995619i \(-0.470192\pi\)
0.0935076 + 0.995619i \(0.470192\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −0.587203 −0.0648457
\(83\) 9.58514 1.05211 0.526053 0.850452i \(-0.323671\pi\)
0.526053 + 0.850452i \(0.323671\pi\)
\(84\) −25.1278 −2.74167
\(85\) 0 0
\(86\) 19.5636 2.10960
\(87\) −1.00000 −0.107211
\(88\) −17.6333 −1.87971
\(89\) 16.3332 1.73132 0.865659 0.500634i \(-0.166900\pi\)
0.865659 + 0.500634i \(0.166900\pi\)
\(90\) 0 0
\(91\) 25.3193 2.65419
\(92\) −0.745712 −0.0777458
\(93\) 8.18167 0.848400
\(94\) −27.0434 −2.78932
\(95\) 0 0
\(96\) −21.4180 −2.18596
\(97\) 8.04128 0.816468 0.408234 0.912877i \(-0.366145\pi\)
0.408234 + 0.912877i \(0.366145\pi\)
\(98\) 39.0559 3.94525
\(99\) −1.87758 −0.188703
\(100\) 0 0
\(101\) −12.3239 −1.22627 −0.613135 0.789978i \(-0.710092\pi\)
−0.613135 + 0.789978i \(0.710092\pi\)
\(102\) 15.6608 1.55065
\(103\) −0.660912 −0.0651216 −0.0325608 0.999470i \(-0.510366\pi\)
−0.0325608 + 0.999470i \(0.510366\pi\)
\(104\) 51.5030 5.05028
\(105\) 0 0
\(106\) −3.13060 −0.304071
\(107\) −14.5868 −1.41016 −0.705079 0.709128i \(-0.749089\pi\)
−0.705079 + 0.709128i \(0.749089\pi\)
\(108\) −5.44251 −0.523706
\(109\) 15.5050 1.48511 0.742557 0.669783i \(-0.233613\pi\)
0.742557 + 0.669783i \(0.233613\pi\)
\(110\) 0 0
\(111\) 4.45785 0.423120
\(112\) 68.0349 6.42869
\(113\) −7.80660 −0.734383 −0.367191 0.930145i \(-0.619681\pi\)
−0.367191 + 0.930145i \(0.619681\pi\)
\(114\) 14.7441 1.38091
\(115\) 0 0
\(116\) 5.44251 0.505325
\(117\) 5.48400 0.506996
\(118\) −0.668145 −0.0615078
\(119\) −26.5038 −2.42960
\(120\) 0 0
\(121\) −7.47471 −0.679519
\(122\) 23.8786 2.16187
\(123\) 0.215243 0.0194078
\(124\) −44.5289 −3.99881
\(125\) 0 0
\(126\) 12.5955 1.12209
\(127\) 19.7106 1.74903 0.874515 0.484999i \(-0.161180\pi\)
0.874515 + 0.484999i \(0.161180\pi\)
\(128\) 36.1656 3.19662
\(129\) −7.17115 −0.631385
\(130\) 0 0
\(131\) −1.05738 −0.0923834 −0.0461917 0.998933i \(-0.514709\pi\)
−0.0461917 + 0.998933i \(0.514709\pi\)
\(132\) 10.2187 0.889426
\(133\) −24.9525 −2.16366
\(134\) −17.8225 −1.53963
\(135\) 0 0
\(136\) −53.9125 −4.62296
\(137\) 3.21467 0.274648 0.137324 0.990526i \(-0.456150\pi\)
0.137324 + 0.990526i \(0.456150\pi\)
\(138\) 0.373793 0.0318194
\(139\) 8.48410 0.719612 0.359806 0.933027i \(-0.382843\pi\)
0.359806 + 0.933027i \(0.382843\pi\)
\(140\) 0 0
\(141\) 9.91293 0.834820
\(142\) −0.756807 −0.0635098
\(143\) −10.2966 −0.861046
\(144\) 14.7359 1.22799
\(145\) 0 0
\(146\) −31.3548 −2.59494
\(147\) −14.3162 −1.18078
\(148\) −24.2619 −1.99432
\(149\) −5.93205 −0.485972 −0.242986 0.970030i \(-0.578127\pi\)
−0.242986 + 0.970030i \(0.578127\pi\)
\(150\) 0 0
\(151\) 8.06281 0.656142 0.328071 0.944653i \(-0.393601\pi\)
0.328071 + 0.944653i \(0.393601\pi\)
\(152\) −50.7568 −4.11692
\(153\) −5.74056 −0.464097
\(154\) −23.6490 −1.90569
\(155\) 0 0
\(156\) −29.8467 −2.38965
\(157\) −9.87127 −0.787813 −0.393906 0.919151i \(-0.628877\pi\)
−0.393906 + 0.919151i \(0.628877\pi\)
\(158\) 4.53472 0.360763
\(159\) 1.14754 0.0910058
\(160\) 0 0
\(161\) −0.632596 −0.0498555
\(162\) 2.72810 0.214340
\(163\) 1.25711 0.0984643 0.0492321 0.998787i \(-0.484323\pi\)
0.0492321 + 0.998787i \(0.484323\pi\)
\(164\) −1.17146 −0.0914757
\(165\) 0 0
\(166\) 26.1492 2.02957
\(167\) −15.1462 −1.17205 −0.586024 0.810294i \(-0.699308\pi\)
−0.586024 + 0.810294i \(0.699308\pi\)
\(168\) −43.3601 −3.34530
\(169\) 17.0742 1.31340
\(170\) 0 0
\(171\) −5.40455 −0.413296
\(172\) 39.0291 2.97594
\(173\) 2.34929 0.178613 0.0893065 0.996004i \(-0.471535\pi\)
0.0893065 + 0.996004i \(0.471535\pi\)
\(174\) −2.72810 −0.206816
\(175\) 0 0
\(176\) −27.6678 −2.08554
\(177\) 0.244913 0.0184088
\(178\) 44.5586 3.33981
\(179\) −17.7814 −1.32904 −0.664520 0.747270i \(-0.731364\pi\)
−0.664520 + 0.747270i \(0.731364\pi\)
\(180\) 0 0
\(181\) −2.69871 −0.200594 −0.100297 0.994958i \(-0.531979\pi\)
−0.100297 + 0.994958i \(0.531979\pi\)
\(182\) 69.0735 5.12007
\(183\) −8.75284 −0.647029
\(184\) −1.28679 −0.0948632
\(185\) 0 0
\(186\) 22.3204 1.63661
\(187\) 10.7783 0.788190
\(188\) −53.9512 −3.93480
\(189\) −4.61695 −0.335833
\(190\) 0 0
\(191\) 10.7997 0.781442 0.390721 0.920509i \(-0.372226\pi\)
0.390721 + 0.920509i \(0.372226\pi\)
\(192\) −28.9585 −2.08990
\(193\) −4.18447 −0.301205 −0.150602 0.988594i \(-0.548121\pi\)
−0.150602 + 0.988594i \(0.548121\pi\)
\(194\) 21.9374 1.57501
\(195\) 0 0
\(196\) 77.9160 5.56543
\(197\) 5.76856 0.410993 0.205496 0.978658i \(-0.434119\pi\)
0.205496 + 0.978658i \(0.434119\pi\)
\(198\) −5.12221 −0.364019
\(199\) −5.45094 −0.386407 −0.193203 0.981159i \(-0.561888\pi\)
−0.193203 + 0.981159i \(0.561888\pi\)
\(200\) 0 0
\(201\) 6.53295 0.460799
\(202\) −33.6207 −2.36554
\(203\) 4.61695 0.324046
\(204\) 31.2431 2.18745
\(205\) 0 0
\(206\) −1.80303 −0.125623
\(207\) −0.137016 −0.00952328
\(208\) 80.8117 5.60328
\(209\) 10.1474 0.701914
\(210\) 0 0
\(211\) 0.863332 0.0594342 0.0297171 0.999558i \(-0.490539\pi\)
0.0297171 + 0.999558i \(0.490539\pi\)
\(212\) −6.24550 −0.428943
\(213\) 0.277412 0.0190080
\(214\) −39.7942 −2.72027
\(215\) 0 0
\(216\) −9.39150 −0.639011
\(217\) −37.7743 −2.56429
\(218\) 42.2993 2.86487
\(219\) 11.4933 0.776645
\(220\) 0 0
\(221\) −31.4812 −2.11766
\(222\) 12.1614 0.816223
\(223\) −0.615197 −0.0411967 −0.0205983 0.999788i \(-0.506557\pi\)
−0.0205983 + 0.999788i \(0.506557\pi\)
\(224\) 98.8856 6.60707
\(225\) 0 0
\(226\) −21.2972 −1.41667
\(227\) −23.7230 −1.57455 −0.787275 0.616602i \(-0.788509\pi\)
−0.787275 + 0.616602i \(0.788509\pi\)
\(228\) 29.4143 1.94801
\(229\) −20.9808 −1.38645 −0.693226 0.720721i \(-0.743811\pi\)
−0.693226 + 0.720721i \(0.743811\pi\)
\(230\) 0 0
\(231\) 8.66866 0.570356
\(232\) 9.39150 0.616582
\(233\) −7.95482 −0.521138 −0.260569 0.965455i \(-0.583910\pi\)
−0.260569 + 0.965455i \(0.583910\pi\)
\(234\) 14.9609 0.978023
\(235\) 0 0
\(236\) −1.33294 −0.0867670
\(237\) −1.66223 −0.107973
\(238\) −72.3051 −4.68684
\(239\) 17.4888 1.13125 0.565627 0.824661i \(-0.308634\pi\)
0.565627 + 0.824661i \(0.308634\pi\)
\(240\) 0 0
\(241\) −14.7743 −0.951699 −0.475849 0.879527i \(-0.657859\pi\)
−0.475849 + 0.879527i \(0.657859\pi\)
\(242\) −20.3917 −1.31083
\(243\) −1.00000 −0.0641500
\(244\) 47.6375 3.04968
\(245\) 0 0
\(246\) 0.587203 0.0374387
\(247\) −29.6385 −1.88585
\(248\) −76.8382 −4.87923
\(249\) −9.58514 −0.607433
\(250\) 0 0
\(251\) −3.30996 −0.208923 −0.104461 0.994529i \(-0.533312\pi\)
−0.104461 + 0.994529i \(0.533312\pi\)
\(252\) 25.1278 1.58290
\(253\) 0.257258 0.0161737
\(254\) 53.7723 3.37398
\(255\) 0 0
\(256\) 40.7463 2.54664
\(257\) −30.8668 −1.92542 −0.962709 0.270539i \(-0.912798\pi\)
−0.962709 + 0.270539i \(0.912798\pi\)
\(258\) −19.5636 −1.21798
\(259\) −20.5816 −1.27888
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) −2.88463 −0.178213
\(263\) 19.4833 1.20139 0.600697 0.799477i \(-0.294890\pi\)
0.600697 + 0.799477i \(0.294890\pi\)
\(264\) 17.6333 1.08525
\(265\) 0 0
\(266\) −68.0728 −4.17381
\(267\) −16.3332 −0.999577
\(268\) −35.5557 −2.17191
\(269\) 20.8385 1.27055 0.635273 0.772287i \(-0.280887\pi\)
0.635273 + 0.772287i \(0.280887\pi\)
\(270\) 0 0
\(271\) 24.5849 1.49343 0.746713 0.665146i \(-0.231631\pi\)
0.746713 + 0.665146i \(0.231631\pi\)
\(272\) −84.5924 −5.12916
\(273\) −25.3193 −1.53239
\(274\) 8.76994 0.529811
\(275\) 0 0
\(276\) 0.745712 0.0448866
\(277\) 1.89904 0.114102 0.0570510 0.998371i \(-0.481830\pi\)
0.0570510 + 0.998371i \(0.481830\pi\)
\(278\) 23.1455 1.38817
\(279\) −8.18167 −0.489824
\(280\) 0 0
\(281\) 32.4916 1.93828 0.969142 0.246503i \(-0.0792814\pi\)
0.969142 + 0.246503i \(0.0792814\pi\)
\(282\) 27.0434 1.61041
\(283\) −11.9151 −0.708279 −0.354140 0.935193i \(-0.615226\pi\)
−0.354140 + 0.935193i \(0.615226\pi\)
\(284\) −1.50982 −0.0895912
\(285\) 0 0
\(286\) −28.0902 −1.66101
\(287\) −0.993764 −0.0586600
\(288\) 21.4180 1.26207
\(289\) 15.9540 0.938472
\(290\) 0 0
\(291\) −8.04128 −0.471388
\(292\) −62.5524 −3.66060
\(293\) 20.7359 1.21140 0.605702 0.795692i \(-0.292892\pi\)
0.605702 + 0.795692i \(0.292892\pi\)
\(294\) −39.0559 −2.27779
\(295\) 0 0
\(296\) −41.8659 −2.43341
\(297\) 1.87758 0.108948
\(298\) −16.1832 −0.937467
\(299\) −0.751396 −0.0434544
\(300\) 0 0
\(301\) 33.1088 1.90836
\(302\) 21.9961 1.26574
\(303\) 12.3239 0.707987
\(304\) −79.6409 −4.56772
\(305\) 0 0
\(306\) −15.6608 −0.895268
\(307\) −16.9755 −0.968842 −0.484421 0.874835i \(-0.660970\pi\)
−0.484421 + 0.874835i \(0.660970\pi\)
\(308\) −47.1793 −2.68829
\(309\) 0.660912 0.0375980
\(310\) 0 0
\(311\) −20.9019 −1.18524 −0.592619 0.805483i \(-0.701906\pi\)
−0.592619 + 0.805483i \(0.701906\pi\)
\(312\) −51.5030 −2.91578
\(313\) 12.7754 0.722110 0.361055 0.932545i \(-0.382417\pi\)
0.361055 + 0.932545i \(0.382417\pi\)
\(314\) −26.9298 −1.51973
\(315\) 0 0
\(316\) 9.04669 0.508916
\(317\) 15.0552 0.845582 0.422791 0.906227i \(-0.361050\pi\)
0.422791 + 0.906227i \(0.361050\pi\)
\(318\) 3.13060 0.175555
\(319\) −1.87758 −0.105124
\(320\) 0 0
\(321\) 14.5868 0.814155
\(322\) −1.72578 −0.0961741
\(323\) 31.0251 1.72628
\(324\) 5.44251 0.302362
\(325\) 0 0
\(326\) 3.42951 0.189943
\(327\) −15.5050 −0.857431
\(328\) −2.02145 −0.111616
\(329\) −45.7675 −2.52324
\(330\) 0 0
\(331\) −17.7325 −0.974668 −0.487334 0.873216i \(-0.662031\pi\)
−0.487334 + 0.873216i \(0.662031\pi\)
\(332\) 52.1672 2.86305
\(333\) −4.45785 −0.244289
\(334\) −41.3203 −2.26095
\(335\) 0 0
\(336\) −68.0349 −3.71161
\(337\) −17.5905 −0.958217 −0.479108 0.877756i \(-0.659040\pi\)
−0.479108 + 0.877756i \(0.659040\pi\)
\(338\) 46.5801 2.53363
\(339\) 7.80660 0.423996
\(340\) 0 0
\(341\) 15.3617 0.831883
\(342\) −14.7441 −0.797271
\(343\) 33.7784 1.82386
\(344\) 67.3479 3.63115
\(345\) 0 0
\(346\) 6.40909 0.344555
\(347\) −21.0789 −1.13157 −0.565787 0.824552i \(-0.691427\pi\)
−0.565787 + 0.824552i \(0.691427\pi\)
\(348\) −5.44251 −0.291749
\(349\) −0.723154 −0.0387096 −0.0193548 0.999813i \(-0.506161\pi\)
−0.0193548 + 0.999813i \(0.506161\pi\)
\(350\) 0 0
\(351\) −5.48400 −0.292714
\(352\) −40.2139 −2.14341
\(353\) 5.93891 0.316096 0.158048 0.987431i \(-0.449480\pi\)
0.158048 + 0.987431i \(0.449480\pi\)
\(354\) 0.668145 0.0355115
\(355\) 0 0
\(356\) 88.8938 4.71136
\(357\) 26.5038 1.40273
\(358\) −48.5092 −2.56379
\(359\) −2.30973 −0.121903 −0.0609514 0.998141i \(-0.519413\pi\)
−0.0609514 + 0.998141i \(0.519413\pi\)
\(360\) 0 0
\(361\) 10.2091 0.537323
\(362\) −7.36234 −0.386956
\(363\) 7.47471 0.392321
\(364\) 137.801 7.22272
\(365\) 0 0
\(366\) −23.8786 −1.24815
\(367\) 22.9235 1.19660 0.598298 0.801274i \(-0.295844\pi\)
0.598298 + 0.801274i \(0.295844\pi\)
\(368\) −2.01906 −0.105251
\(369\) −0.215243 −0.0112051
\(370\) 0 0
\(371\) −5.29813 −0.275065
\(372\) 44.5289 2.30871
\(373\) −2.83182 −0.146626 −0.0733129 0.997309i \(-0.523357\pi\)
−0.0733129 + 0.997309i \(0.523357\pi\)
\(374\) 29.4043 1.52046
\(375\) 0 0
\(376\) −93.0973 −4.80113
\(377\) 5.48400 0.282440
\(378\) −12.5955 −0.647841
\(379\) −29.2539 −1.50267 −0.751335 0.659921i \(-0.770590\pi\)
−0.751335 + 0.659921i \(0.770590\pi\)
\(380\) 0 0
\(381\) −19.7106 −1.00980
\(382\) 29.4627 1.50744
\(383\) −2.99063 −0.152814 −0.0764070 0.997077i \(-0.524345\pi\)
−0.0764070 + 0.997077i \(0.524345\pi\)
\(384\) −36.1656 −1.84557
\(385\) 0 0
\(386\) −11.4156 −0.581040
\(387\) 7.17115 0.364530
\(388\) 43.7648 2.22182
\(389\) −8.57339 −0.434688 −0.217344 0.976095i \(-0.569739\pi\)
−0.217344 + 0.976095i \(0.569739\pi\)
\(390\) 0 0
\(391\) 0.786549 0.0397775
\(392\) 134.451 6.79078
\(393\) 1.05738 0.0533376
\(394\) 15.7372 0.792828
\(395\) 0 0
\(396\) −10.2187 −0.513510
\(397\) 9.29605 0.466556 0.233278 0.972410i \(-0.425055\pi\)
0.233278 + 0.972410i \(0.425055\pi\)
\(398\) −14.8707 −0.745400
\(399\) 24.9525 1.24919
\(400\) 0 0
\(401\) 19.5392 0.975739 0.487870 0.872916i \(-0.337774\pi\)
0.487870 + 0.872916i \(0.337774\pi\)
\(402\) 17.8225 0.888907
\(403\) −44.8683 −2.23505
\(404\) −67.0728 −3.33700
\(405\) 0 0
\(406\) 12.5955 0.625103
\(407\) 8.36995 0.414883
\(408\) 53.9125 2.66907
\(409\) 37.9582 1.87691 0.938456 0.345398i \(-0.112256\pi\)
0.938456 + 0.345398i \(0.112256\pi\)
\(410\) 0 0
\(411\) −3.21467 −0.158568
\(412\) −3.59702 −0.177213
\(413\) −1.13075 −0.0556405
\(414\) −0.373793 −0.0183709
\(415\) 0 0
\(416\) 117.456 5.75876
\(417\) −8.48410 −0.415468
\(418\) 27.6832 1.35403
\(419\) −10.0952 −0.493184 −0.246592 0.969119i \(-0.579311\pi\)
−0.246592 + 0.969119i \(0.579311\pi\)
\(420\) 0 0
\(421\) 13.8521 0.675111 0.337556 0.941306i \(-0.390400\pi\)
0.337556 + 0.941306i \(0.390400\pi\)
\(422\) 2.35525 0.114652
\(423\) −9.91293 −0.481983
\(424\) −10.7771 −0.523384
\(425\) 0 0
\(426\) 0.756807 0.0366674
\(427\) 40.4114 1.95565
\(428\) −79.3888 −3.83740
\(429\) 10.2966 0.497125
\(430\) 0 0
\(431\) 18.2149 0.877380 0.438690 0.898638i \(-0.355443\pi\)
0.438690 + 0.898638i \(0.355443\pi\)
\(432\) −14.7359 −0.708982
\(433\) 6.73476 0.323652 0.161826 0.986819i \(-0.448262\pi\)
0.161826 + 0.986819i \(0.448262\pi\)
\(434\) −103.052 −4.94666
\(435\) 0 0
\(436\) 84.3864 4.04137
\(437\) 0.740510 0.0354234
\(438\) 31.3548 1.49819
\(439\) 30.9590 1.47759 0.738795 0.673930i \(-0.235395\pi\)
0.738795 + 0.673930i \(0.235395\pi\)
\(440\) 0 0
\(441\) 14.3162 0.681723
\(442\) −85.8838 −4.08508
\(443\) 21.4432 1.01880 0.509398 0.860531i \(-0.329868\pi\)
0.509398 + 0.860531i \(0.329868\pi\)
\(444\) 24.2619 1.15142
\(445\) 0 0
\(446\) −1.67832 −0.0794707
\(447\) 5.93205 0.280576
\(448\) 133.700 6.31672
\(449\) 25.6895 1.21236 0.606180 0.795327i \(-0.292701\pi\)
0.606180 + 0.795327i \(0.292701\pi\)
\(450\) 0 0
\(451\) 0.404134 0.0190299
\(452\) −42.4875 −1.99844
\(453\) −8.06281 −0.378824
\(454\) −64.7186 −3.03739
\(455\) 0 0
\(456\) 50.7568 2.37691
\(457\) 30.3278 1.41868 0.709338 0.704868i \(-0.248994\pi\)
0.709338 + 0.704868i \(0.248994\pi\)
\(458\) −57.2377 −2.67454
\(459\) 5.74056 0.267946
\(460\) 0 0
\(461\) −31.0026 −1.44393 −0.721967 0.691928i \(-0.756762\pi\)
−0.721967 + 0.691928i \(0.756762\pi\)
\(462\) 23.6490 1.10025
\(463\) −27.6990 −1.28728 −0.643640 0.765328i \(-0.722577\pi\)
−0.643640 + 0.765328i \(0.722577\pi\)
\(464\) 14.7359 0.684097
\(465\) 0 0
\(466\) −21.7015 −1.00530
\(467\) −4.03243 −0.186598 −0.0932992 0.995638i \(-0.529741\pi\)
−0.0932992 + 0.995638i \(0.529741\pi\)
\(468\) 29.8467 1.37966
\(469\) −30.1623 −1.39277
\(470\) 0 0
\(471\) 9.87127 0.454844
\(472\) −2.30010 −0.105871
\(473\) −13.4644 −0.619093
\(474\) −4.53472 −0.208286
\(475\) 0 0
\(476\) −144.248 −6.61157
\(477\) −1.14754 −0.0525422
\(478\) 47.7110 2.18225
\(479\) 1.32186 0.0603973 0.0301986 0.999544i \(-0.490386\pi\)
0.0301986 + 0.999544i \(0.490386\pi\)
\(480\) 0 0
\(481\) −24.4468 −1.11468
\(482\) −40.3058 −1.83588
\(483\) 0.632596 0.0287841
\(484\) −40.6812 −1.84915
\(485\) 0 0
\(486\) −2.72810 −0.123749
\(487\) −9.89390 −0.448335 −0.224168 0.974551i \(-0.571966\pi\)
−0.224168 + 0.974551i \(0.571966\pi\)
\(488\) 82.2024 3.72113
\(489\) −1.25711 −0.0568484
\(490\) 0 0
\(491\) 17.1305 0.773089 0.386545 0.922271i \(-0.373669\pi\)
0.386545 + 0.922271i \(0.373669\pi\)
\(492\) 1.17146 0.0528135
\(493\) −5.74056 −0.258542
\(494\) −80.8568 −3.63792
\(495\) 0 0
\(496\) −120.564 −5.41350
\(497\) −1.28080 −0.0574516
\(498\) −26.1492 −1.17177
\(499\) −7.95477 −0.356104 −0.178052 0.984021i \(-0.556980\pi\)
−0.178052 + 0.984021i \(0.556980\pi\)
\(500\) 0 0
\(501\) 15.1462 0.676683
\(502\) −9.02989 −0.403024
\(503\) 13.3471 0.595119 0.297559 0.954703i \(-0.403827\pi\)
0.297559 + 0.954703i \(0.403827\pi\)
\(504\) 43.3601 1.93141
\(505\) 0 0
\(506\) 0.701825 0.0311999
\(507\) −17.0742 −0.758293
\(508\) 107.275 4.75956
\(509\) −5.55324 −0.246143 −0.123071 0.992398i \(-0.539274\pi\)
−0.123071 + 0.992398i \(0.539274\pi\)
\(510\) 0 0
\(511\) −53.0639 −2.34741
\(512\) 38.8286 1.71600
\(513\) 5.40455 0.238617
\(514\) −84.2076 −3.71424
\(515\) 0 0
\(516\) −39.0291 −1.71816
\(517\) 18.6123 0.818567
\(518\) −56.1487 −2.46703
\(519\) −2.34929 −0.103122
\(520\) 0 0
\(521\) 41.5095 1.81857 0.909283 0.416179i \(-0.136631\pi\)
0.909283 + 0.416179i \(0.136631\pi\)
\(522\) 2.72810 0.119406
\(523\) −11.6261 −0.508375 −0.254188 0.967155i \(-0.581808\pi\)
−0.254188 + 0.967155i \(0.581808\pi\)
\(524\) −5.75479 −0.251399
\(525\) 0 0
\(526\) 53.1525 2.31756
\(527\) 46.9674 2.04593
\(528\) 27.6678 1.20409
\(529\) −22.9812 −0.999184
\(530\) 0 0
\(531\) −0.244913 −0.0106283
\(532\) −135.804 −5.88786
\(533\) −1.18039 −0.0511284
\(534\) −44.5586 −1.92824
\(535\) 0 0
\(536\) −61.3543 −2.65010
\(537\) 17.7814 0.767322
\(538\) 56.8495 2.45096
\(539\) −26.8797 −1.15779
\(540\) 0 0
\(541\) −5.29830 −0.227792 −0.113896 0.993493i \(-0.536333\pi\)
−0.113896 + 0.993493i \(0.536333\pi\)
\(542\) 67.0700 2.88090
\(543\) 2.69871 0.115813
\(544\) −122.951 −5.27149
\(545\) 0 0
\(546\) −69.0735 −2.95607
\(547\) −6.78093 −0.289932 −0.144966 0.989437i \(-0.546307\pi\)
−0.144966 + 0.989437i \(0.546307\pi\)
\(548\) 17.4959 0.747387
\(549\) 8.75284 0.373562
\(550\) 0 0
\(551\) −5.40455 −0.230241
\(552\) 1.28679 0.0547693
\(553\) 7.67441 0.326349
\(554\) 5.18076 0.220109
\(555\) 0 0
\(556\) 46.1748 1.95825
\(557\) 0.186307 0.00789410 0.00394705 0.999992i \(-0.498744\pi\)
0.00394705 + 0.999992i \(0.498744\pi\)
\(558\) −22.3204 −0.944898
\(559\) 39.3266 1.66334
\(560\) 0 0
\(561\) −10.7783 −0.455062
\(562\) 88.6401 3.73906
\(563\) −27.1382 −1.14374 −0.571869 0.820345i \(-0.693781\pi\)
−0.571869 + 0.820345i \(0.693781\pi\)
\(564\) 53.9512 2.27176
\(565\) 0 0
\(566\) −32.5056 −1.36631
\(567\) 4.61695 0.193893
\(568\) −2.60532 −0.109317
\(569\) 22.8795 0.959157 0.479578 0.877499i \(-0.340790\pi\)
0.479578 + 0.877499i \(0.340790\pi\)
\(570\) 0 0
\(571\) 38.3621 1.60541 0.802703 0.596379i \(-0.203394\pi\)
0.802703 + 0.596379i \(0.203394\pi\)
\(572\) −56.0395 −2.34313
\(573\) −10.7997 −0.451165
\(574\) −2.71108 −0.113158
\(575\) 0 0
\(576\) 28.9585 1.20660
\(577\) −12.0497 −0.501637 −0.250818 0.968034i \(-0.580700\pi\)
−0.250818 + 0.968034i \(0.580700\pi\)
\(578\) 43.5241 1.81036
\(579\) 4.18447 0.173901
\(580\) 0 0
\(581\) 44.2541 1.83597
\(582\) −21.9374 −0.909334
\(583\) 2.15459 0.0892341
\(584\) −107.939 −4.46656
\(585\) 0 0
\(586\) 56.5695 2.33687
\(587\) −23.6704 −0.976982 −0.488491 0.872569i \(-0.662453\pi\)
−0.488491 + 0.872569i \(0.662453\pi\)
\(588\) −77.9160 −3.21320
\(589\) 44.2183 1.82198
\(590\) 0 0
\(591\) −5.76856 −0.237287
\(592\) −65.6905 −2.69986
\(593\) 25.9414 1.06529 0.532643 0.846340i \(-0.321199\pi\)
0.532643 + 0.846340i \(0.321199\pi\)
\(594\) 5.12221 0.210167
\(595\) 0 0
\(596\) −32.2852 −1.32245
\(597\) 5.45094 0.223092
\(598\) −2.04988 −0.0838259
\(599\) −17.2886 −0.706393 −0.353196 0.935549i \(-0.614905\pi\)
−0.353196 + 0.935549i \(0.614905\pi\)
\(600\) 0 0
\(601\) −30.8997 −1.26042 −0.630212 0.776423i \(-0.717032\pi\)
−0.630212 + 0.776423i \(0.717032\pi\)
\(602\) 90.3240 3.68133
\(603\) −6.53295 −0.266042
\(604\) 43.8820 1.78553
\(605\) 0 0
\(606\) 33.6207 1.36575
\(607\) 4.07152 0.165258 0.0826290 0.996580i \(-0.473668\pi\)
0.0826290 + 0.996580i \(0.473668\pi\)
\(608\) −115.754 −4.69446
\(609\) −4.61695 −0.187088
\(610\) 0 0
\(611\) −54.3625 −2.19927
\(612\) −31.2431 −1.26293
\(613\) 32.3595 1.30699 0.653494 0.756931i \(-0.273302\pi\)
0.653494 + 0.756931i \(0.273302\pi\)
\(614\) −46.3108 −1.86895
\(615\) 0 0
\(616\) −81.4118 −3.28017
\(617\) −33.4957 −1.34849 −0.674243 0.738509i \(-0.735530\pi\)
−0.674243 + 0.738509i \(0.735530\pi\)
\(618\) 1.80303 0.0725286
\(619\) −16.6857 −0.670656 −0.335328 0.942101i \(-0.608847\pi\)
−0.335328 + 0.942101i \(0.608847\pi\)
\(620\) 0 0
\(621\) 0.137016 0.00549827
\(622\) −57.0224 −2.28639
\(623\) 75.4096 3.02122
\(624\) −80.8117 −3.23506
\(625\) 0 0
\(626\) 34.8526 1.39299
\(627\) −10.1474 −0.405250
\(628\) −53.7245 −2.14384
\(629\) 25.5905 1.02036
\(630\) 0 0
\(631\) 9.91590 0.394746 0.197373 0.980328i \(-0.436759\pi\)
0.197373 + 0.980328i \(0.436759\pi\)
\(632\) 15.6108 0.620965
\(633\) −0.863332 −0.0343143
\(634\) 41.0719 1.63118
\(635\) 0 0
\(636\) 6.24550 0.247650
\(637\) 78.5099 3.11068
\(638\) −5.12221 −0.202790
\(639\) −0.277412 −0.0109742
\(640\) 0 0
\(641\) −22.9862 −0.907902 −0.453951 0.891027i \(-0.649986\pi\)
−0.453951 + 0.891027i \(0.649986\pi\)
\(642\) 39.7942 1.57055
\(643\) −12.9117 −0.509190 −0.254595 0.967048i \(-0.581942\pi\)
−0.254595 + 0.967048i \(0.581942\pi\)
\(644\) −3.44291 −0.135670
\(645\) 0 0
\(646\) 84.6395 3.33010
\(647\) 16.6096 0.652989 0.326494 0.945199i \(-0.394133\pi\)
0.326494 + 0.945199i \(0.394133\pi\)
\(648\) 9.39150 0.368933
\(649\) 0.459842 0.0180504
\(650\) 0 0
\(651\) 37.7743 1.48049
\(652\) 6.84182 0.267947
\(653\) −21.5638 −0.843858 −0.421929 0.906629i \(-0.638647\pi\)
−0.421929 + 0.906629i \(0.638647\pi\)
\(654\) −42.2993 −1.65403
\(655\) 0 0
\(656\) −3.17180 −0.123838
\(657\) −11.4933 −0.448396
\(658\) −124.858 −4.86747
\(659\) 31.7863 1.23822 0.619110 0.785305i \(-0.287494\pi\)
0.619110 + 0.785305i \(0.287494\pi\)
\(660\) 0 0
\(661\) 30.0046 1.16704 0.583521 0.812098i \(-0.301674\pi\)
0.583521 + 0.812098i \(0.301674\pi\)
\(662\) −48.3760 −1.88019
\(663\) 31.4812 1.22263
\(664\) 90.0189 3.49341
\(665\) 0 0
\(666\) −12.1614 −0.471246
\(667\) −0.137016 −0.00530529
\(668\) −82.4334 −3.18944
\(669\) 0.615197 0.0237849
\(670\) 0 0
\(671\) −16.4341 −0.634432
\(672\) −98.8856 −3.81460
\(673\) −35.0842 −1.35240 −0.676199 0.736719i \(-0.736374\pi\)
−0.676199 + 0.736719i \(0.736374\pi\)
\(674\) −47.9886 −1.84845
\(675\) 0 0
\(676\) 92.9267 3.57410
\(677\) 42.1433 1.61970 0.809849 0.586638i \(-0.199549\pi\)
0.809849 + 0.586638i \(0.199549\pi\)
\(678\) 21.2972 0.817912
\(679\) 37.1261 1.42477
\(680\) 0 0
\(681\) 23.7230 0.909067
\(682\) 41.9082 1.60475
\(683\) −31.8895 −1.22022 −0.610109 0.792318i \(-0.708874\pi\)
−0.610109 + 0.792318i \(0.708874\pi\)
\(684\) −29.4143 −1.12468
\(685\) 0 0
\(686\) 92.1508 3.51833
\(687\) 20.9808 0.800468
\(688\) 105.673 4.02876
\(689\) −6.29311 −0.239748
\(690\) 0 0
\(691\) 11.2963 0.429732 0.214866 0.976644i \(-0.431069\pi\)
0.214866 + 0.976644i \(0.431069\pi\)
\(692\) 12.7860 0.486052
\(693\) −8.66866 −0.329295
\(694\) −57.5052 −2.18287
\(695\) 0 0
\(696\) −9.39150 −0.355984
\(697\) 1.23561 0.0468022
\(698\) −1.97283 −0.0746729
\(699\) 7.95482 0.300879
\(700\) 0 0
\(701\) −1.39354 −0.0526332 −0.0263166 0.999654i \(-0.508378\pi\)
−0.0263166 + 0.999654i \(0.508378\pi\)
\(702\) −14.9609 −0.564662
\(703\) 24.0927 0.908672
\(704\) −54.3717 −2.04921
\(705\) 0 0
\(706\) 16.2019 0.609767
\(707\) −56.8986 −2.13989
\(708\) 1.33294 0.0500949
\(709\) 5.24967 0.197156 0.0985778 0.995129i \(-0.468571\pi\)
0.0985778 + 0.995129i \(0.468571\pi\)
\(710\) 0 0
\(711\) 1.66223 0.0623384
\(712\) 153.394 5.74867
\(713\) 1.12102 0.0419826
\(714\) 72.3051 2.70595
\(715\) 0 0
\(716\) −96.7752 −3.61666
\(717\) −17.4888 −0.653130
\(718\) −6.30116 −0.235157
\(719\) 46.4801 1.73341 0.866707 0.498817i \(-0.166232\pi\)
0.866707 + 0.498817i \(0.166232\pi\)
\(720\) 0 0
\(721\) −3.05140 −0.113640
\(722\) 27.8515 1.03653
\(723\) 14.7743 0.549464
\(724\) −14.6878 −0.545866
\(725\) 0 0
\(726\) 20.3917 0.756808
\(727\) 43.3101 1.60628 0.803141 0.595790i \(-0.203161\pi\)
0.803141 + 0.595790i \(0.203161\pi\)
\(728\) 237.786 8.81295
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −41.1664 −1.52259
\(732\) −47.6375 −1.76073
\(733\) 18.6793 0.689936 0.344968 0.938614i \(-0.387890\pi\)
0.344968 + 0.938614i \(0.387890\pi\)
\(734\) 62.5375 2.30830
\(735\) 0 0
\(736\) −2.93461 −0.108171
\(737\) 12.2661 0.451828
\(738\) −0.587203 −0.0216152
\(739\) 41.3256 1.52019 0.760093 0.649815i \(-0.225153\pi\)
0.760093 + 0.649815i \(0.225153\pi\)
\(740\) 0 0
\(741\) 29.6385 1.08880
\(742\) −14.4538 −0.530616
\(743\) 32.3390 1.18640 0.593202 0.805054i \(-0.297864\pi\)
0.593202 + 0.805054i \(0.297864\pi\)
\(744\) 76.8382 2.81703
\(745\) 0 0
\(746\) −7.72547 −0.282849
\(747\) 9.58514 0.350702
\(748\) 58.6612 2.14487
\(749\) −67.3464 −2.46078
\(750\) 0 0
\(751\) −9.34454 −0.340987 −0.170494 0.985359i \(-0.554536\pi\)
−0.170494 + 0.985359i \(0.554536\pi\)
\(752\) −146.076 −5.32685
\(753\) 3.30996 0.120622
\(754\) 14.9609 0.544843
\(755\) 0 0
\(756\) −25.1278 −0.913888
\(757\) 5.01932 0.182430 0.0912151 0.995831i \(-0.470925\pi\)
0.0912151 + 0.995831i \(0.470925\pi\)
\(758\) −79.8074 −2.89873
\(759\) −0.257258 −0.00933788
\(760\) 0 0
\(761\) −34.9103 −1.26550 −0.632748 0.774358i \(-0.718073\pi\)
−0.632748 + 0.774358i \(0.718073\pi\)
\(762\) −53.7723 −1.94797
\(763\) 71.5859 2.59159
\(764\) 58.7777 2.12650
\(765\) 0 0
\(766\) −8.15872 −0.294787
\(767\) −1.34310 −0.0484965
\(768\) −40.7463 −1.47031
\(769\) 37.4889 1.35189 0.675943 0.736954i \(-0.263737\pi\)
0.675943 + 0.736954i \(0.263737\pi\)
\(770\) 0 0
\(771\) 30.8668 1.11164
\(772\) −22.7740 −0.819655
\(773\) 33.7332 1.21330 0.606649 0.794970i \(-0.292513\pi\)
0.606649 + 0.794970i \(0.292513\pi\)
\(774\) 19.5636 0.703199
\(775\) 0 0
\(776\) 75.5197 2.71100
\(777\) 20.5816 0.738362
\(778\) −23.3890 −0.838538
\(779\) 1.16329 0.0416792
\(780\) 0 0
\(781\) 0.520862 0.0186379
\(782\) 2.14578 0.0767330
\(783\) −1.00000 −0.0357371
\(784\) 210.962 7.53436
\(785\) 0 0
\(786\) 2.88463 0.102891
\(787\) −11.5508 −0.411740 −0.205870 0.978579i \(-0.566002\pi\)
−0.205870 + 0.978579i \(0.566002\pi\)
\(788\) 31.3955 1.11842
\(789\) −19.4833 −0.693626
\(790\) 0 0
\(791\) −36.0426 −1.28153
\(792\) −17.6333 −0.626571
\(793\) 48.0006 1.70455
\(794\) 25.3605 0.900012
\(795\) 0 0
\(796\) −29.6668 −1.05151
\(797\) −9.11795 −0.322974 −0.161487 0.986875i \(-0.551629\pi\)
−0.161487 + 0.986875i \(0.551629\pi\)
\(798\) 68.0728 2.40975
\(799\) 56.9058 2.01318
\(800\) 0 0
\(801\) 16.3332 0.577106
\(802\) 53.3047 1.88226
\(803\) 21.5795 0.761525
\(804\) 35.5557 1.25395
\(805\) 0 0
\(806\) −122.405 −4.31153
\(807\) −20.8385 −0.733551
\(808\) −115.740 −4.07171
\(809\) −11.9809 −0.421227 −0.210613 0.977569i \(-0.567546\pi\)
−0.210613 + 0.977569i \(0.567546\pi\)
\(810\) 0 0
\(811\) −19.2172 −0.674807 −0.337403 0.941360i \(-0.609549\pi\)
−0.337403 + 0.941360i \(0.609549\pi\)
\(812\) 25.1278 0.881812
\(813\) −24.5849 −0.862230
\(814\) 22.8340 0.800332
\(815\) 0 0
\(816\) 84.5924 2.96132
\(817\) −38.7568 −1.35593
\(818\) 103.554 3.62067
\(819\) 25.3193 0.884728
\(820\) 0 0
\(821\) −10.5611 −0.368585 −0.184292 0.982871i \(-0.558999\pi\)
−0.184292 + 0.982871i \(0.558999\pi\)
\(822\) −8.76994 −0.305887
\(823\) 29.1655 1.01665 0.508323 0.861166i \(-0.330266\pi\)
0.508323 + 0.861166i \(0.330266\pi\)
\(824\) −6.20696 −0.216230
\(825\) 0 0
\(826\) −3.08479 −0.107334
\(827\) −40.5140 −1.40881 −0.704405 0.709798i \(-0.748786\pi\)
−0.704405 + 0.709798i \(0.748786\pi\)
\(828\) −0.745712 −0.0259153
\(829\) 51.3925 1.78494 0.892468 0.451111i \(-0.148972\pi\)
0.892468 + 0.451111i \(0.148972\pi\)
\(830\) 0 0
\(831\) −1.89904 −0.0658769
\(832\) 158.808 5.50569
\(833\) −82.1829 −2.84747
\(834\) −23.1455 −0.801461
\(835\) 0 0
\(836\) 55.2276 1.91009
\(837\) 8.18167 0.282800
\(838\) −27.5407 −0.951379
\(839\) −30.2352 −1.04383 −0.521917 0.852996i \(-0.674783\pi\)
−0.521917 + 0.852996i \(0.674783\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 37.7899 1.30233
\(843\) −32.4916 −1.11907
\(844\) 4.69869 0.161736
\(845\) 0 0
\(846\) −27.0434 −0.929773
\(847\) −34.5103 −1.18579
\(848\) −16.9100 −0.580693
\(849\) 11.9151 0.408925
\(850\) 0 0
\(851\) 0.610797 0.0209379
\(852\) 1.50982 0.0517255
\(853\) −0.163336 −0.00559252 −0.00279626 0.999996i \(-0.500890\pi\)
−0.00279626 + 0.999996i \(0.500890\pi\)
\(854\) 110.246 3.77255
\(855\) 0 0
\(856\) −136.992 −4.68229
\(857\) 0.869730 0.0297094 0.0148547 0.999890i \(-0.495271\pi\)
0.0148547 + 0.999890i \(0.495271\pi\)
\(858\) 28.0902 0.958983
\(859\) −50.5362 −1.72427 −0.862136 0.506677i \(-0.830874\pi\)
−0.862136 + 0.506677i \(0.830874\pi\)
\(860\) 0 0
\(861\) 0.993764 0.0338674
\(862\) 49.6920 1.69251
\(863\) −43.1139 −1.46761 −0.733807 0.679358i \(-0.762258\pi\)
−0.733807 + 0.679358i \(0.762258\pi\)
\(864\) −21.4180 −0.728654
\(865\) 0 0
\(866\) 18.3731 0.624342
\(867\) −15.9540 −0.541827
\(868\) −205.587 −6.97809
\(869\) −3.12096 −0.105871
\(870\) 0 0
\(871\) −35.8267 −1.21394
\(872\) 145.616 4.93117
\(873\) 8.04128 0.272156
\(874\) 2.02018 0.0683337
\(875\) 0 0
\(876\) 62.5524 2.11345
\(877\) −3.23901 −0.109374 −0.0546868 0.998504i \(-0.517416\pi\)
−0.0546868 + 0.998504i \(0.517416\pi\)
\(878\) 84.4590 2.85035
\(879\) −20.7359 −0.699404
\(880\) 0 0
\(881\) 9.49218 0.319800 0.159900 0.987133i \(-0.448883\pi\)
0.159900 + 0.987133i \(0.448883\pi\)
\(882\) 39.0559 1.31508
\(883\) −40.4268 −1.36047 −0.680235 0.732994i \(-0.738122\pi\)
−0.680235 + 0.732994i \(0.738122\pi\)
\(884\) −171.337 −5.76268
\(885\) 0 0
\(886\) 58.4991 1.96532
\(887\) 17.5249 0.588430 0.294215 0.955739i \(-0.404942\pi\)
0.294215 + 0.955739i \(0.404942\pi\)
\(888\) 41.8659 1.40493
\(889\) 91.0026 3.05213
\(890\) 0 0
\(891\) −1.87758 −0.0629011
\(892\) −3.34822 −0.112107
\(893\) 53.5749 1.79282
\(894\) 16.1832 0.541247
\(895\) 0 0
\(896\) 166.975 5.57823
\(897\) 0.751396 0.0250884
\(898\) 70.0833 2.33871
\(899\) −8.18167 −0.272874
\(900\) 0 0
\(901\) 6.58752 0.219462
\(902\) 1.10252 0.0367098
\(903\) −33.1088 −1.10179
\(904\) −73.3157 −2.43844
\(905\) 0 0
\(906\) −21.9961 −0.730773
\(907\) 34.8878 1.15843 0.579216 0.815174i \(-0.303359\pi\)
0.579216 + 0.815174i \(0.303359\pi\)
\(908\) −129.113 −4.28475
\(909\) −12.3239 −0.408757
\(910\) 0 0
\(911\) 43.5212 1.44192 0.720960 0.692976i \(-0.243701\pi\)
0.720960 + 0.692976i \(0.243701\pi\)
\(912\) 79.6409 2.63717
\(913\) −17.9968 −0.595608
\(914\) 82.7373 2.73671
\(915\) 0 0
\(916\) −114.188 −3.77289
\(917\) −4.88185 −0.161213
\(918\) 15.6608 0.516883
\(919\) −38.2543 −1.26189 −0.630946 0.775827i \(-0.717333\pi\)
−0.630946 + 0.775827i \(0.717333\pi\)
\(920\) 0 0
\(921\) 16.9755 0.559361
\(922\) −84.5780 −2.78543
\(923\) −1.52133 −0.0500751
\(924\) 47.1793 1.55209
\(925\) 0 0
\(926\) −75.5655 −2.48324
\(927\) −0.660912 −0.0217072
\(928\) 21.4180 0.703079
\(929\) 40.3492 1.32381 0.661906 0.749586i \(-0.269748\pi\)
0.661906 + 0.749586i \(0.269748\pi\)
\(930\) 0 0
\(931\) −77.3725 −2.53578
\(932\) −43.2942 −1.41815
\(933\) 20.9019 0.684298
\(934\) −11.0009 −0.359959
\(935\) 0 0
\(936\) 51.5030 1.68343
\(937\) −4.95384 −0.161835 −0.0809174 0.996721i \(-0.525785\pi\)
−0.0809174 + 0.996721i \(0.525785\pi\)
\(938\) −82.2857 −2.68672
\(939\) −12.7754 −0.416910
\(940\) 0 0
\(941\) −47.5220 −1.54917 −0.774587 0.632467i \(-0.782042\pi\)
−0.774587 + 0.632467i \(0.782042\pi\)
\(942\) 26.9298 0.877419
\(943\) 0.0294917 0.000960383 0
\(944\) −3.60901 −0.117463
\(945\) 0 0
\(946\) −36.7321 −1.19426
\(947\) −37.1547 −1.20736 −0.603682 0.797225i \(-0.706300\pi\)
−0.603682 + 0.797225i \(0.706300\pi\)
\(948\) −9.04669 −0.293823
\(949\) −63.0292 −2.04601
\(950\) 0 0
\(951\) −15.0552 −0.488197
\(952\) −248.911 −8.06725
\(953\) 44.5351 1.44263 0.721316 0.692606i \(-0.243537\pi\)
0.721316 + 0.692606i \(0.243537\pi\)
\(954\) −3.13060 −0.101357
\(955\) 0 0
\(956\) 95.1828 3.07843
\(957\) 1.87758 0.0606934
\(958\) 3.60616 0.116510
\(959\) 14.8420 0.479272
\(960\) 0 0
\(961\) 35.9398 1.15935
\(962\) −66.6933 −2.15028
\(963\) −14.5868 −0.470053
\(964\) −80.4095 −2.58982
\(965\) 0 0
\(966\) 1.72578 0.0555262
\(967\) 48.7291 1.56702 0.783511 0.621379i \(-0.213427\pi\)
0.783511 + 0.621379i \(0.213427\pi\)
\(968\) −70.1988 −2.25627
\(969\) −31.0251 −0.996671
\(970\) 0 0
\(971\) −51.7448 −1.66057 −0.830285 0.557339i \(-0.811822\pi\)
−0.830285 + 0.557339i \(0.811822\pi\)
\(972\) −5.44251 −0.174569
\(973\) 39.1706 1.25575
\(974\) −26.9915 −0.864864
\(975\) 0 0
\(976\) 128.981 4.12858
\(977\) 37.7726 1.20845 0.604227 0.796813i \(-0.293482\pi\)
0.604227 + 0.796813i \(0.293482\pi\)
\(978\) −3.42951 −0.109664
\(979\) −30.6669 −0.980117
\(980\) 0 0
\(981\) 15.5050 0.495038
\(982\) 46.7337 1.49133
\(983\) −21.1318 −0.673999 −0.336999 0.941505i \(-0.609412\pi\)
−0.336999 + 0.941505i \(0.609412\pi\)
\(984\) 2.02145 0.0644416
\(985\) 0 0
\(986\) −15.6608 −0.498742
\(987\) 45.7675 1.45679
\(988\) −161.308 −5.13189
\(989\) −0.982563 −0.0312437
\(990\) 0 0
\(991\) 11.0241 0.350191 0.175096 0.984551i \(-0.443977\pi\)
0.175096 + 0.984551i \(0.443977\pi\)
\(992\) −175.235 −5.56371
\(993\) 17.7325 0.562725
\(994\) −3.49414 −0.110827
\(995\) 0 0
\(996\) −52.1672 −1.65298
\(997\) 28.2785 0.895589 0.447795 0.894136i \(-0.352210\pi\)
0.447795 + 0.894136i \(0.352210\pi\)
\(998\) −21.7014 −0.686945
\(999\) 4.45785 0.141040
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 2175.2.a.bc.1.8 8
3.2 odd 2 6525.2.a.bz.1.1 8
5.2 odd 4 2175.2.c.p.349.16 16
5.3 odd 4 2175.2.c.p.349.1 16
5.4 even 2 2175.2.a.bd.1.1 yes 8
15.14 odd 2 6525.2.a.by.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2175.2.a.bc.1.8 8 1.1 even 1 trivial
2175.2.a.bd.1.1 yes 8 5.4 even 2
2175.2.c.p.349.1 16 5.3 odd 4
2175.2.c.p.349.16 16 5.2 odd 4
6525.2.a.by.1.8 8 15.14 odd 2
6525.2.a.bz.1.1 8 3.2 odd 2