Properties

Label 3234.2.a.bm.1.2
Level $3234$
Weight $2$
Character 3234.1
Self dual yes
Analytic conductor $25.824$
Analytic rank $0$
Dimension $4$
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] = [3234,2,Mod(1,3234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3234, 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("3234.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.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: \(25.8236200137\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.334904\) of defining polynomial
Character \(\chi\) \(=\) 3234.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.473626 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.473626 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -0.473626 q^{10} +1.00000 q^{11} +1.00000 q^{12} +7.03127 q^{13} -0.473626 q^{15} +1.00000 q^{16} -4.64167 q^{17} +1.00000 q^{18} +6.55765 q^{19} -0.473626 q^{20} +1.00000 q^{22} +1.49824 q^{23} +1.00000 q^{24} -4.77568 q^{25} +7.03127 q^{26} +1.00000 q^{27} -2.11882 q^{29} -0.473626 q^{30} +6.83509 q^{31} +1.00000 q^{32} +1.00000 q^{33} -4.64167 q^{34} +1.00000 q^{36} -4.32666 q^{37} +6.55765 q^{38} +7.03127 q^{39} -0.473626 q^{40} -11.2458 q^{41} +0.158619 q^{43} +1.00000 q^{44} -0.473626 q^{45} +1.49824 q^{46} +0.270780 q^{47} +1.00000 q^{48} -4.77568 q^{50} -4.64167 q^{51} +7.03127 q^{52} -9.66628 q^{53} +1.00000 q^{54} -0.473626 q^{55} +6.55765 q^{57} -2.11882 q^{58} +9.82490 q^{59} -0.473626 q^{60} +15.1993 q^{61} +6.83509 q^{62} +1.00000 q^{64} -3.33019 q^{65} +1.00000 q^{66} -1.21137 q^{67} -4.64167 q^{68} +1.49824 q^{69} +15.0098 q^{71} +1.00000 q^{72} -6.13048 q^{73} -4.32666 q^{74} -4.77568 q^{75} +6.55765 q^{76} +7.03127 q^{78} +11.4420 q^{79} -0.473626 q^{80} +1.00000 q^{81} -11.2458 q^{82} -7.54469 q^{83} +2.19841 q^{85} +0.158619 q^{86} -2.11882 q^{87} +1.00000 q^{88} +16.6881 q^{89} -0.473626 q^{90} +1.49824 q^{92} +6.83509 q^{93} +0.270780 q^{94} -3.10587 q^{95} +1.00000 q^{96} +12.9656 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} + 4 q^{8} + 4 q^{9} + 4 q^{10} + 4 q^{11} + 4 q^{12} + 8 q^{13} + 4 q^{15} + 4 q^{16} + 4 q^{17} + 4 q^{18} + 12 q^{19} + 4 q^{20} + 4 q^{22} - 8 q^{23} + 4 q^{24} + 4 q^{25} + 8 q^{26} + 4 q^{27} - 8 q^{29} + 4 q^{30} + 4 q^{31} + 4 q^{32} + 4 q^{33} + 4 q^{34} + 4 q^{36} + 8 q^{37} + 12 q^{38} + 8 q^{39} + 4 q^{40} + 12 q^{41} - 8 q^{43} + 4 q^{44} + 4 q^{45} - 8 q^{46} + 4 q^{47} + 4 q^{48} + 4 q^{50} + 4 q^{51} + 8 q^{52} - 8 q^{53} + 4 q^{54} + 4 q^{55} + 12 q^{57} - 8 q^{58} + 4 q^{60} + 24 q^{61} + 4 q^{62} + 4 q^{64} - 16 q^{65} + 4 q^{66} - 8 q^{67} + 4 q^{68} - 8 q^{69} + 8 q^{71} + 4 q^{72} + 4 q^{73} + 8 q^{74} + 4 q^{75} + 12 q^{76} + 8 q^{78} - 8 q^{79} + 4 q^{80} + 4 q^{81} + 12 q^{82} + 4 q^{83} - 8 q^{85} - 8 q^{86} - 8 q^{87} + 4 q^{88} + 24 q^{89} + 4 q^{90} - 8 q^{92} + 4 q^{93} + 4 q^{94} + 8 q^{95} + 4 q^{96} + 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −0.473626 −0.211812 −0.105906 0.994376i \(-0.533774\pi\)
−0.105906 + 0.994376i \(0.533774\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −0.473626 −0.149774
\(11\) 1.00000 0.301511
\(12\) 1.00000 0.288675
\(13\) 7.03127 1.95012 0.975062 0.221932i \(-0.0712363\pi\)
0.975062 + 0.221932i \(0.0712363\pi\)
\(14\) 0 0
\(15\) −0.473626 −0.122290
\(16\) 1.00000 0.250000
\(17\) −4.64167 −1.12577 −0.562885 0.826535i \(-0.690309\pi\)
−0.562885 + 0.826535i \(0.690309\pi\)
\(18\) 1.00000 0.235702
\(19\) 6.55765 1.50443 0.752214 0.658919i \(-0.228986\pi\)
0.752214 + 0.658919i \(0.228986\pi\)
\(20\) −0.473626 −0.105906
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 1.49824 0.312404 0.156202 0.987725i \(-0.450075\pi\)
0.156202 + 0.987725i \(0.450075\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.77568 −0.955136
\(26\) 7.03127 1.37895
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.11882 −0.393456 −0.196728 0.980458i \(-0.563032\pi\)
−0.196728 + 0.980458i \(0.563032\pi\)
\(30\) −0.473626 −0.0864718
\(31\) 6.83509 1.22762 0.613809 0.789454i \(-0.289636\pi\)
0.613809 + 0.789454i \(0.289636\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) −4.64167 −0.796040
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.32666 −0.711299 −0.355649 0.934619i \(-0.615740\pi\)
−0.355649 + 0.934619i \(0.615740\pi\)
\(38\) 6.55765 1.06379
\(39\) 7.03127 1.12590
\(40\) −0.473626 −0.0748868
\(41\) −11.2458 −1.75629 −0.878147 0.478390i \(-0.841221\pi\)
−0.878147 + 0.478390i \(0.841221\pi\)
\(42\) 0 0
\(43\) 0.158619 0.0241892 0.0120946 0.999927i \(-0.496150\pi\)
0.0120946 + 0.999927i \(0.496150\pi\)
\(44\) 1.00000 0.150756
\(45\) −0.473626 −0.0706040
\(46\) 1.49824 0.220903
\(47\) 0.270780 0.0394973 0.0197486 0.999805i \(-0.493713\pi\)
0.0197486 + 0.999805i \(0.493713\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −4.77568 −0.675383
\(51\) −4.64167 −0.649964
\(52\) 7.03127 0.975062
\(53\) −9.66628 −1.32777 −0.663883 0.747837i \(-0.731093\pi\)
−0.663883 + 0.747837i \(0.731093\pi\)
\(54\) 1.00000 0.136083
\(55\) −0.473626 −0.0638637
\(56\) 0 0
\(57\) 6.55765 0.868582
\(58\) −2.11882 −0.278215
\(59\) 9.82490 1.27909 0.639546 0.768753i \(-0.279122\pi\)
0.639546 + 0.768753i \(0.279122\pi\)
\(60\) −0.473626 −0.0611448
\(61\) 15.1993 1.94607 0.973037 0.230651i \(-0.0740856\pi\)
0.973037 + 0.230651i \(0.0740856\pi\)
\(62\) 6.83509 0.868057
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.33019 −0.413059
\(66\) 1.00000 0.123091
\(67\) −1.21137 −0.147992 −0.0739961 0.997259i \(-0.523575\pi\)
−0.0739961 + 0.997259i \(0.523575\pi\)
\(68\) −4.64167 −0.562885
\(69\) 1.49824 0.180366
\(70\) 0 0
\(71\) 15.0098 1.78134 0.890668 0.454655i \(-0.150237\pi\)
0.890668 + 0.454655i \(0.150237\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.13048 −0.717518 −0.358759 0.933430i \(-0.616800\pi\)
−0.358759 + 0.933430i \(0.616800\pi\)
\(74\) −4.32666 −0.502964
\(75\) −4.77568 −0.551448
\(76\) 6.55765 0.752214
\(77\) 0 0
\(78\) 7.03127 0.796135
\(79\) 11.4420 1.28732 0.643660 0.765311i \(-0.277415\pi\)
0.643660 + 0.765311i \(0.277415\pi\)
\(80\) −0.473626 −0.0529530
\(81\) 1.00000 0.111111
\(82\) −11.2458 −1.24189
\(83\) −7.54469 −0.828138 −0.414069 0.910246i \(-0.635893\pi\)
−0.414069 + 0.910246i \(0.635893\pi\)
\(84\) 0 0
\(85\) 2.19841 0.238451
\(86\) 0.158619 0.0171043
\(87\) −2.11882 −0.227162
\(88\) 1.00000 0.106600
\(89\) 16.6881 1.76894 0.884469 0.466599i \(-0.154521\pi\)
0.884469 + 0.466599i \(0.154521\pi\)
\(90\) −0.473626 −0.0499245
\(91\) 0 0
\(92\) 1.49824 0.156202
\(93\) 6.83509 0.708766
\(94\) 0.270780 0.0279288
\(95\) −3.10587 −0.318656
\(96\) 1.00000 0.102062
\(97\) 12.9656 1.31645 0.658227 0.752819i \(-0.271307\pi\)
0.658227 + 0.752819i \(0.271307\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) −4.77568 −0.477568
\(101\) 0.298919 0.0297436 0.0148718 0.999889i \(-0.495266\pi\)
0.0148718 + 0.999889i \(0.495266\pi\)
\(102\) −4.64167 −0.459594
\(103\) −7.39550 −0.728700 −0.364350 0.931262i \(-0.618709\pi\)
−0.364350 + 0.931262i \(0.618709\pi\)
\(104\) 7.03127 0.689473
\(105\) 0 0
\(106\) −9.66628 −0.938872
\(107\) −15.3231 −1.48134 −0.740672 0.671867i \(-0.765493\pi\)
−0.740672 + 0.671867i \(0.765493\pi\)
\(108\) 1.00000 0.0962250
\(109\) −0.422735 −0.0404907 −0.0202453 0.999795i \(-0.506445\pi\)
−0.0202453 + 0.999795i \(0.506445\pi\)
\(110\) −0.473626 −0.0451584
\(111\) −4.32666 −0.410669
\(112\) 0 0
\(113\) −8.40569 −0.790741 −0.395370 0.918522i \(-0.629384\pi\)
−0.395370 + 0.918522i \(0.629384\pi\)
\(114\) 6.55765 0.614180
\(115\) −0.709603 −0.0661708
\(116\) −2.11882 −0.196728
\(117\) 7.03127 0.650041
\(118\) 9.82490 0.904455
\(119\) 0 0
\(120\) −0.473626 −0.0432359
\(121\) 1.00000 0.0909091
\(122\) 15.1993 1.37608
\(123\) −11.2458 −1.01400
\(124\) 6.83509 0.613809
\(125\) 4.63001 0.414121
\(126\) 0 0
\(127\) 10.1023 0.896438 0.448219 0.893924i \(-0.352059\pi\)
0.448219 + 0.893924i \(0.352059\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0.158619 0.0139656
\(130\) −3.33019 −0.292077
\(131\) 20.8843 1.82467 0.912335 0.409444i \(-0.134277\pi\)
0.912335 + 0.409444i \(0.134277\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) −1.21137 −0.104646
\(135\) −0.473626 −0.0407632
\(136\) −4.64167 −0.398020
\(137\) 7.35294 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(138\) 1.49824 0.127538
\(139\) −4.67647 −0.396653 −0.198327 0.980136i \(-0.563551\pi\)
−0.198327 + 0.980136i \(0.563551\pi\)
\(140\) 0 0
\(141\) 0.270780 0.0228038
\(142\) 15.0098 1.25959
\(143\) 7.03127 0.587985
\(144\) 1.00000 0.0833333
\(145\) 1.00353 0.0833386
\(146\) −6.13048 −0.507362
\(147\) 0 0
\(148\) −4.32666 −0.355649
\(149\) 4.89097 0.400684 0.200342 0.979726i \(-0.435795\pi\)
0.200342 + 0.979726i \(0.435795\pi\)
\(150\) −4.77568 −0.389933
\(151\) −16.6533 −1.35523 −0.677614 0.735418i \(-0.736986\pi\)
−0.677614 + 0.735418i \(0.736986\pi\)
\(152\) 6.55765 0.531895
\(153\) −4.64167 −0.375257
\(154\) 0 0
\(155\) −3.23728 −0.260024
\(156\) 7.03127 0.562952
\(157\) −14.5228 −1.15905 −0.579525 0.814955i \(-0.696762\pi\)
−0.579525 + 0.814955i \(0.696762\pi\)
\(158\) 11.4420 0.910273
\(159\) −9.66628 −0.766586
\(160\) −0.473626 −0.0374434
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −8.33609 −0.652933 −0.326466 0.945209i \(-0.605858\pi\)
−0.326466 + 0.945209i \(0.605858\pi\)
\(164\) −11.2458 −0.878147
\(165\) −0.473626 −0.0368717
\(166\) −7.54469 −0.585582
\(167\) 2.84727 0.220329 0.110164 0.993913i \(-0.464862\pi\)
0.110164 + 0.993913i \(0.464862\pi\)
\(168\) 0 0
\(169\) 36.4388 2.80298
\(170\) 2.19841 0.168611
\(171\) 6.55765 0.501476
\(172\) 0.158619 0.0120946
\(173\) 9.17656 0.697681 0.348841 0.937182i \(-0.386575\pi\)
0.348841 + 0.937182i \(0.386575\pi\)
\(174\) −2.11882 −0.160628
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 9.82490 0.738485
\(178\) 16.6881 1.25083
\(179\) −5.76625 −0.430990 −0.215495 0.976505i \(-0.569137\pi\)
−0.215495 + 0.976505i \(0.569137\pi\)
\(180\) −0.473626 −0.0353020
\(181\) 3.38164 0.251356 0.125678 0.992071i \(-0.459889\pi\)
0.125678 + 0.992071i \(0.459889\pi\)
\(182\) 0 0
\(183\) 15.1993 1.12357
\(184\) 1.49824 0.110451
\(185\) 2.04922 0.150662
\(186\) 6.83509 0.501173
\(187\) −4.64167 −0.339432
\(188\) 0.270780 0.0197486
\(189\) 0 0
\(190\) −3.10587 −0.225324
\(191\) −16.1586 −1.16920 −0.584598 0.811323i \(-0.698748\pi\)
−0.584598 + 0.811323i \(0.698748\pi\)
\(192\) 1.00000 0.0721688
\(193\) −24.8651 −1.78983 −0.894913 0.446240i \(-0.852763\pi\)
−0.894913 + 0.446240i \(0.852763\pi\)
\(194\) 12.9656 0.930874
\(195\) −3.33019 −0.238480
\(196\) 0 0
\(197\) 10.4290 0.743036 0.371518 0.928426i \(-0.378837\pi\)
0.371518 + 0.928426i \(0.378837\pi\)
\(198\) 1.00000 0.0710669
\(199\) −13.2712 −0.940767 −0.470384 0.882462i \(-0.655884\pi\)
−0.470384 + 0.882462i \(0.655884\pi\)
\(200\) −4.77568 −0.337691
\(201\) −1.21137 −0.0854433
\(202\) 0.298919 0.0210319
\(203\) 0 0
\(204\) −4.64167 −0.324982
\(205\) 5.32629 0.372004
\(206\) −7.39550 −0.515269
\(207\) 1.49824 0.104135
\(208\) 7.03127 0.487531
\(209\) 6.55765 0.453602
\(210\) 0 0
\(211\) −24.2837 −1.67176 −0.835880 0.548912i \(-0.815042\pi\)
−0.835880 + 0.548912i \(0.815042\pi\)
\(212\) −9.66628 −0.663883
\(213\) 15.0098 1.02845
\(214\) −15.3231 −1.04747
\(215\) −0.0751261 −0.00512356
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −0.422735 −0.0286312
\(219\) −6.13048 −0.414259
\(220\) −0.473626 −0.0319318
\(221\) −32.6368 −2.19539
\(222\) −4.32666 −0.290387
\(223\) −3.82529 −0.256161 −0.128080 0.991764i \(-0.540882\pi\)
−0.128080 + 0.991764i \(0.540882\pi\)
\(224\) 0 0
\(225\) −4.77568 −0.318379
\(226\) −8.40569 −0.559138
\(227\) −12.3106 −0.817082 −0.408541 0.912740i \(-0.633962\pi\)
−0.408541 + 0.912740i \(0.633962\pi\)
\(228\) 6.55765 0.434291
\(229\) −20.2726 −1.33965 −0.669826 0.742518i \(-0.733631\pi\)
−0.669826 + 0.742518i \(0.733631\pi\)
\(230\) −0.709603 −0.0467898
\(231\) 0 0
\(232\) −2.11882 −0.139108
\(233\) −10.8114 −0.708277 −0.354139 0.935193i \(-0.615226\pi\)
−0.354139 + 0.935193i \(0.615226\pi\)
\(234\) 7.03127 0.459649
\(235\) −0.128248 −0.00836600
\(236\) 9.82490 0.639546
\(237\) 11.4420 0.743235
\(238\) 0 0
\(239\) 4.07959 0.263887 0.131943 0.991257i \(-0.457878\pi\)
0.131943 + 0.991257i \(0.457878\pi\)
\(240\) −0.473626 −0.0305724
\(241\) 21.9946 1.41680 0.708399 0.705812i \(-0.249418\pi\)
0.708399 + 0.705812i \(0.249418\pi\)
\(242\) 1.00000 0.0642824
\(243\) 1.00000 0.0641500
\(244\) 15.1993 0.973037
\(245\) 0 0
\(246\) −11.2458 −0.717004
\(247\) 46.1086 2.93382
\(248\) 6.83509 0.434029
\(249\) −7.54469 −0.478126
\(250\) 4.63001 0.292828
\(251\) −23.1082 −1.45858 −0.729289 0.684205i \(-0.760149\pi\)
−0.729289 + 0.684205i \(0.760149\pi\)
\(252\) 0 0
\(253\) 1.49824 0.0941932
\(254\) 10.1023 0.633877
\(255\) 2.19841 0.137670
\(256\) 1.00000 0.0625000
\(257\) −7.75420 −0.483694 −0.241847 0.970314i \(-0.577753\pi\)
−0.241847 + 0.970314i \(0.577753\pi\)
\(258\) 0.158619 0.00987520
\(259\) 0 0
\(260\) −3.33019 −0.206530
\(261\) −2.11882 −0.131152
\(262\) 20.8843 1.29024
\(263\) −13.7851 −0.850026 −0.425013 0.905187i \(-0.639730\pi\)
−0.425013 + 0.905187i \(0.639730\pi\)
\(264\) 1.00000 0.0615457
\(265\) 4.57820 0.281236
\(266\) 0 0
\(267\) 16.6881 1.02130
\(268\) −1.21137 −0.0739961
\(269\) −6.13048 −0.373782 −0.186891 0.982381i \(-0.559841\pi\)
−0.186891 + 0.982381i \(0.559841\pi\)
\(270\) −0.473626 −0.0288239
\(271\) −11.0661 −0.672216 −0.336108 0.941823i \(-0.609111\pi\)
−0.336108 + 0.941823i \(0.609111\pi\)
\(272\) −4.64167 −0.281443
\(273\) 0 0
\(274\) 7.35294 0.444208
\(275\) −4.77568 −0.287984
\(276\) 1.49824 0.0901832
\(277\) −16.4986 −0.991305 −0.495653 0.868521i \(-0.665071\pi\)
−0.495653 + 0.868521i \(0.665071\pi\)
\(278\) −4.67647 −0.280476
\(279\) 6.83509 0.409206
\(280\) 0 0
\(281\) −17.2341 −1.02810 −0.514051 0.857760i \(-0.671856\pi\)
−0.514051 + 0.857760i \(0.671856\pi\)
\(282\) 0.270780 0.0161247
\(283\) −18.3333 −1.08980 −0.544902 0.838500i \(-0.683433\pi\)
−0.544902 + 0.838500i \(0.683433\pi\)
\(284\) 15.0098 0.890668
\(285\) −3.10587 −0.183976
\(286\) 7.03127 0.415768
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 4.54509 0.267358
\(290\) 1.00353 0.0589293
\(291\) 12.9656 0.760055
\(292\) −6.13048 −0.358759
\(293\) −26.1560 −1.52805 −0.764025 0.645187i \(-0.776779\pi\)
−0.764025 + 0.645187i \(0.776779\pi\)
\(294\) 0 0
\(295\) −4.65332 −0.270927
\(296\) −4.32666 −0.251482
\(297\) 1.00000 0.0580259
\(298\) 4.89097 0.283326
\(299\) 10.5345 0.609226
\(300\) −4.77568 −0.275724
\(301\) 0 0
\(302\) −16.6533 −0.958291
\(303\) 0.298919 0.0171724
\(304\) 6.55765 0.376107
\(305\) −7.19879 −0.412201
\(306\) −4.64167 −0.265347
\(307\) 25.3109 1.44457 0.722286 0.691594i \(-0.243091\pi\)
0.722286 + 0.691594i \(0.243091\pi\)
\(308\) 0 0
\(309\) −7.39550 −0.420715
\(310\) −3.23728 −0.183865
\(311\) −24.6202 −1.39608 −0.698042 0.716057i \(-0.745945\pi\)
−0.698042 + 0.716057i \(0.745945\pi\)
\(312\) 7.03127 0.398067
\(313\) 19.2713 1.08928 0.544639 0.838671i \(-0.316667\pi\)
0.544639 + 0.838671i \(0.316667\pi\)
\(314\) −14.5228 −0.819572
\(315\) 0 0
\(316\) 11.4420 0.643660
\(317\) 4.34551 0.244068 0.122034 0.992526i \(-0.461058\pi\)
0.122034 + 0.992526i \(0.461058\pi\)
\(318\) −9.66628 −0.542058
\(319\) −2.11882 −0.118631
\(320\) −0.473626 −0.0264765
\(321\) −15.3231 −0.855254
\(322\) 0 0
\(323\) −30.4384 −1.69364
\(324\) 1.00000 0.0555556
\(325\) −33.5791 −1.86263
\(326\) −8.33609 −0.461693
\(327\) −0.422735 −0.0233773
\(328\) −11.2458 −0.620944
\(329\) 0 0
\(330\) −0.473626 −0.0260722
\(331\) −5.46786 −0.300541 −0.150270 0.988645i \(-0.548014\pi\)
−0.150270 + 0.988645i \(0.548014\pi\)
\(332\) −7.54469 −0.414069
\(333\) −4.32666 −0.237100
\(334\) 2.84727 0.155796
\(335\) 0.573735 0.0313465
\(336\) 0 0
\(337\) 7.43569 0.405048 0.202524 0.979277i \(-0.435086\pi\)
0.202524 + 0.979277i \(0.435086\pi\)
\(338\) 36.4388 1.98201
\(339\) −8.40569 −0.456535
\(340\) 2.19841 0.119226
\(341\) 6.83509 0.370141
\(342\) 6.55765 0.354597
\(343\) 0 0
\(344\) 0.158619 0.00855217
\(345\) −0.709603 −0.0382037
\(346\) 9.17656 0.493335
\(347\) 5.88508 0.315928 0.157964 0.987445i \(-0.449507\pi\)
0.157964 + 0.987445i \(0.449507\pi\)
\(348\) −2.11882 −0.113581
\(349\) 17.5658 0.940274 0.470137 0.882593i \(-0.344204\pi\)
0.470137 + 0.882593i \(0.344204\pi\)
\(350\) 0 0
\(351\) 7.03127 0.375302
\(352\) 1.00000 0.0533002
\(353\) −1.45401 −0.0773891 −0.0386945 0.999251i \(-0.512320\pi\)
−0.0386945 + 0.999251i \(0.512320\pi\)
\(354\) 9.82490 0.522187
\(355\) −7.10903 −0.377308
\(356\) 16.6881 0.884469
\(357\) 0 0
\(358\) −5.76625 −0.304756
\(359\) 19.6796 1.03865 0.519325 0.854577i \(-0.326183\pi\)
0.519325 + 0.854577i \(0.326183\pi\)
\(360\) −0.473626 −0.0249623
\(361\) 24.0027 1.26330
\(362\) 3.38164 0.177735
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 2.90355 0.151979
\(366\) 15.1993 0.794481
\(367\) −18.0692 −0.943205 −0.471603 0.881811i \(-0.656324\pi\)
−0.471603 + 0.881811i \(0.656324\pi\)
\(368\) 1.49824 0.0781009
\(369\) −11.2458 −0.585432
\(370\) 2.04922 0.106534
\(371\) 0 0
\(372\) 6.83509 0.354383
\(373\) −21.3459 −1.10525 −0.552624 0.833431i \(-0.686373\pi\)
−0.552624 + 0.833431i \(0.686373\pi\)
\(374\) −4.64167 −0.240015
\(375\) 4.63001 0.239093
\(376\) 0.270780 0.0139644
\(377\) −14.8980 −0.767288
\(378\) 0 0
\(379\) −4.63448 −0.238057 −0.119029 0.992891i \(-0.537978\pi\)
−0.119029 + 0.992891i \(0.537978\pi\)
\(380\) −3.10587 −0.159328
\(381\) 10.1023 0.517559
\(382\) −16.1586 −0.826747
\(383\) −7.37981 −0.377091 −0.188545 0.982065i \(-0.560377\pi\)
−0.188545 + 0.982065i \(0.560377\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −24.8651 −1.26560
\(387\) 0.158619 0.00806306
\(388\) 12.9656 0.658227
\(389\) −1.34668 −0.0682792 −0.0341396 0.999417i \(-0.510869\pi\)
−0.0341396 + 0.999417i \(0.510869\pi\)
\(390\) −3.33019 −0.168631
\(391\) −6.95431 −0.351695
\(392\) 0 0
\(393\) 20.8843 1.05347
\(394\) 10.4290 0.525406
\(395\) −5.41921 −0.272670
\(396\) 1.00000 0.0502519
\(397\) −20.7346 −1.04064 −0.520320 0.853972i \(-0.674187\pi\)
−0.520320 + 0.853972i \(0.674187\pi\)
\(398\) −13.2712 −0.665223
\(399\) 0 0
\(400\) −4.77568 −0.238784
\(401\) 7.96077 0.397542 0.198771 0.980046i \(-0.436305\pi\)
0.198771 + 0.980046i \(0.436305\pi\)
\(402\) −1.21137 −0.0604175
\(403\) 48.0594 2.39401
\(404\) 0.298919 0.0148718
\(405\) −0.473626 −0.0235347
\(406\) 0 0
\(407\) −4.32666 −0.214465
\(408\) −4.64167 −0.229797
\(409\) 30.1438 1.49052 0.745258 0.666777i \(-0.232326\pi\)
0.745258 + 0.666777i \(0.232326\pi\)
\(410\) 5.32629 0.263047
\(411\) 7.35294 0.362694
\(412\) −7.39550 −0.364350
\(413\) 0 0
\(414\) 1.49824 0.0736342
\(415\) 3.57336 0.175409
\(416\) 7.03127 0.344737
\(417\) −4.67647 −0.229008
\(418\) 6.55765 0.320745
\(419\) −26.7025 −1.30450 −0.652252 0.758002i \(-0.726176\pi\)
−0.652252 + 0.758002i \(0.726176\pi\)
\(420\) 0 0
\(421\) 8.44158 0.411418 0.205709 0.978613i \(-0.434050\pi\)
0.205709 + 0.978613i \(0.434050\pi\)
\(422\) −24.2837 −1.18211
\(423\) 0.270780 0.0131658
\(424\) −9.66628 −0.469436
\(425\) 22.1671 1.07526
\(426\) 15.0098 0.727227
\(427\) 0 0
\(428\) −15.3231 −0.740672
\(429\) 7.03127 0.339473
\(430\) −0.0751261 −0.00362290
\(431\) 18.5510 0.893569 0.446785 0.894642i \(-0.352569\pi\)
0.446785 + 0.894642i \(0.352569\pi\)
\(432\) 1.00000 0.0481125
\(433\) −4.19969 −0.201824 −0.100912 0.994895i \(-0.532176\pi\)
−0.100912 + 0.994895i \(0.532176\pi\)
\(434\) 0 0
\(435\) 1.00353 0.0481156
\(436\) −0.422735 −0.0202453
\(437\) 9.82490 0.469989
\(438\) −6.13048 −0.292926
\(439\) −3.48175 −0.166175 −0.0830875 0.996542i \(-0.526478\pi\)
−0.0830875 + 0.996542i \(0.526478\pi\)
\(440\) −0.473626 −0.0225792
\(441\) 0 0
\(442\) −32.6368 −1.55238
\(443\) 24.6690 1.17206 0.586030 0.810289i \(-0.300690\pi\)
0.586030 + 0.810289i \(0.300690\pi\)
\(444\) −4.32666 −0.205334
\(445\) −7.90393 −0.374682
\(446\) −3.82529 −0.181133
\(447\) 4.89097 0.231335
\(448\) 0 0
\(449\) 6.39863 0.301970 0.150985 0.988536i \(-0.451755\pi\)
0.150985 + 0.988536i \(0.451755\pi\)
\(450\) −4.77568 −0.225128
\(451\) −11.2458 −0.529543
\(452\) −8.40569 −0.395370
\(453\) −16.6533 −0.782441
\(454\) −12.3106 −0.577764
\(455\) 0 0
\(456\) 6.55765 0.307090
\(457\) 14.5549 0.680849 0.340424 0.940272i \(-0.389429\pi\)
0.340424 + 0.940272i \(0.389429\pi\)
\(458\) −20.2726 −0.947277
\(459\) −4.64167 −0.216655
\(460\) −0.709603 −0.0330854
\(461\) −16.3115 −0.759700 −0.379850 0.925048i \(-0.624024\pi\)
−0.379850 + 0.925048i \(0.624024\pi\)
\(462\) 0 0
\(463\) 8.81138 0.409500 0.204750 0.978814i \(-0.434362\pi\)
0.204750 + 0.978814i \(0.434362\pi\)
\(464\) −2.11882 −0.0983640
\(465\) −3.23728 −0.150125
\(466\) −10.8114 −0.500828
\(467\) 8.20468 0.379667 0.189834 0.981816i \(-0.439205\pi\)
0.189834 + 0.981816i \(0.439205\pi\)
\(468\) 7.03127 0.325021
\(469\) 0 0
\(470\) −0.128248 −0.00591565
\(471\) −14.5228 −0.669177
\(472\) 9.82490 0.452228
\(473\) 0.158619 0.00729332
\(474\) 11.4420 0.525546
\(475\) −31.3172 −1.43693
\(476\) 0 0
\(477\) −9.66628 −0.442588
\(478\) 4.07959 0.186596
\(479\) 6.72920 0.307465 0.153732 0.988113i \(-0.450871\pi\)
0.153732 + 0.988113i \(0.450871\pi\)
\(480\) −0.473626 −0.0216180
\(481\) −30.4219 −1.38712
\(482\) 21.9946 1.00183
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −6.14083 −0.278841
\(486\) 1.00000 0.0453609
\(487\) 18.9870 0.860385 0.430193 0.902737i \(-0.358446\pi\)
0.430193 + 0.902737i \(0.358446\pi\)
\(488\) 15.1993 0.688041
\(489\) −8.33609 −0.376971
\(490\) 0 0
\(491\) 19.1027 0.862093 0.431047 0.902330i \(-0.358144\pi\)
0.431047 + 0.902330i \(0.358144\pi\)
\(492\) −11.2458 −0.506999
\(493\) 9.83488 0.442941
\(494\) 46.1086 2.07452
\(495\) −0.473626 −0.0212879
\(496\) 6.83509 0.306905
\(497\) 0 0
\(498\) −7.54469 −0.338086
\(499\) −38.6243 −1.72906 −0.864530 0.502582i \(-0.832384\pi\)
−0.864530 + 0.502582i \(0.832384\pi\)
\(500\) 4.63001 0.207060
\(501\) 2.84727 0.127207
\(502\) −23.1082 −1.03137
\(503\) 2.53784 0.113157 0.0565784 0.998398i \(-0.481981\pi\)
0.0565784 + 0.998398i \(0.481981\pi\)
\(504\) 0 0
\(505\) −0.141576 −0.00630004
\(506\) 1.49824 0.0666047
\(507\) 36.4388 1.61830
\(508\) 10.1023 0.448219
\(509\) −28.5987 −1.26762 −0.633808 0.773490i \(-0.718509\pi\)
−0.633808 + 0.773490i \(0.718509\pi\)
\(510\) 2.19841 0.0973474
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 6.55765 0.289527
\(514\) −7.75420 −0.342023
\(515\) 3.50270 0.154347
\(516\) 0.158619 0.00698282
\(517\) 0.270780 0.0119089
\(518\) 0 0
\(519\) 9.17656 0.402806
\(520\) −3.33019 −0.146039
\(521\) 15.2789 0.669381 0.334691 0.942328i \(-0.391368\pi\)
0.334691 + 0.942328i \(0.391368\pi\)
\(522\) −2.11882 −0.0927384
\(523\) −34.6031 −1.51309 −0.756545 0.653942i \(-0.773114\pi\)
−0.756545 + 0.653942i \(0.773114\pi\)
\(524\) 20.8843 0.912335
\(525\) 0 0
\(526\) −13.7851 −0.601059
\(527\) −31.7262 −1.38202
\(528\) 1.00000 0.0435194
\(529\) −20.7553 −0.902404
\(530\) 4.57820 0.198864
\(531\) 9.82490 0.426364
\(532\) 0 0
\(533\) −79.0721 −3.42499
\(534\) 16.6881 0.722166
\(535\) 7.25743 0.313766
\(536\) −1.21137 −0.0523231
\(537\) −5.76625 −0.248832
\(538\) −6.13048 −0.264304
\(539\) 0 0
\(540\) −0.473626 −0.0203816
\(541\) 11.1153 0.477884 0.238942 0.971034i \(-0.423199\pi\)
0.238942 + 0.971034i \(0.423199\pi\)
\(542\) −11.0661 −0.475329
\(543\) 3.38164 0.145120
\(544\) −4.64167 −0.199010
\(545\) 0.200218 0.00857641
\(546\) 0 0
\(547\) 41.5576 1.77688 0.888438 0.458997i \(-0.151791\pi\)
0.888438 + 0.458997i \(0.151791\pi\)
\(548\) 7.35294 0.314102
\(549\) 15.1993 0.648691
\(550\) −4.77568 −0.203636
\(551\) −13.8945 −0.591926
\(552\) 1.49824 0.0637691
\(553\) 0 0
\(554\) −16.4986 −0.700959
\(555\) 2.04922 0.0869845
\(556\) −4.67647 −0.198327
\(557\) 4.19841 0.177893 0.0889463 0.996036i \(-0.471650\pi\)
0.0889463 + 0.996036i \(0.471650\pi\)
\(558\) 6.83509 0.289352
\(559\) 1.11529 0.0471719
\(560\) 0 0
\(561\) −4.64167 −0.195971
\(562\) −17.2341 −0.726977
\(563\) 5.10311 0.215070 0.107535 0.994201i \(-0.465704\pi\)
0.107535 + 0.994201i \(0.465704\pi\)
\(564\) 0.270780 0.0114019
\(565\) 3.98115 0.167488
\(566\) −18.3333 −0.770607
\(567\) 0 0
\(568\) 15.0098 0.629797
\(569\) −36.9635 −1.54959 −0.774795 0.632212i \(-0.782147\pi\)
−0.774795 + 0.632212i \(0.782147\pi\)
\(570\) −3.10587 −0.130091
\(571\) 16.7462 0.700808 0.350404 0.936599i \(-0.386044\pi\)
0.350404 + 0.936599i \(0.386044\pi\)
\(572\) 7.03127 0.293992
\(573\) −16.1586 −0.675036
\(574\) 0 0
\(575\) −7.15509 −0.298388
\(576\) 1.00000 0.0416667
\(577\) −20.9889 −0.873779 −0.436889 0.899515i \(-0.643920\pi\)
−0.436889 + 0.899515i \(0.643920\pi\)
\(578\) 4.54509 0.189051
\(579\) −24.8651 −1.03336
\(580\) 1.00353 0.0416693
\(581\) 0 0
\(582\) 12.9656 0.537440
\(583\) −9.66628 −0.400336
\(584\) −6.13048 −0.253681
\(585\) −3.33019 −0.137686
\(586\) −26.1560 −1.08049
\(587\) 2.56747 0.105971 0.0529854 0.998595i \(-0.483126\pi\)
0.0529854 + 0.998595i \(0.483126\pi\)
\(588\) 0 0
\(589\) 44.8221 1.84686
\(590\) −4.65332 −0.191574
\(591\) 10.4290 0.428992
\(592\) −4.32666 −0.177825
\(593\) 27.7177 1.13823 0.569115 0.822258i \(-0.307286\pi\)
0.569115 + 0.822258i \(0.307286\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) 4.89097 0.200342
\(597\) −13.2712 −0.543152
\(598\) 10.5345 0.430788
\(599\) 26.2173 1.07121 0.535604 0.844469i \(-0.320084\pi\)
0.535604 + 0.844469i \(0.320084\pi\)
\(600\) −4.77568 −0.194966
\(601\) 20.5621 0.838745 0.419372 0.907814i \(-0.362250\pi\)
0.419372 + 0.907814i \(0.362250\pi\)
\(602\) 0 0
\(603\) −1.21137 −0.0493307
\(604\) −16.6533 −0.677614
\(605\) −0.473626 −0.0192556
\(606\) 0.298919 0.0121428
\(607\) 22.8839 0.928829 0.464415 0.885618i \(-0.346265\pi\)
0.464415 + 0.885618i \(0.346265\pi\)
\(608\) 6.55765 0.265948
\(609\) 0 0
\(610\) −7.19879 −0.291470
\(611\) 1.90393 0.0770246
\(612\) −4.64167 −0.187628
\(613\) 6.82843 0.275798 0.137899 0.990446i \(-0.455965\pi\)
0.137899 + 0.990446i \(0.455965\pi\)
\(614\) 25.3109 1.02147
\(615\) 5.32629 0.214777
\(616\) 0 0
\(617\) 22.4120 0.902272 0.451136 0.892455i \(-0.351019\pi\)
0.451136 + 0.892455i \(0.351019\pi\)
\(618\) −7.39550 −0.297491
\(619\) −15.6076 −0.627324 −0.313662 0.949535i \(-0.601556\pi\)
−0.313662 + 0.949535i \(0.601556\pi\)
\(620\) −3.23728 −0.130012
\(621\) 1.49824 0.0601221
\(622\) −24.6202 −0.987180
\(623\) 0 0
\(624\) 7.03127 0.281476
\(625\) 21.6855 0.867420
\(626\) 19.2713 0.770236
\(627\) 6.55765 0.261887
\(628\) −14.5228 −0.579525
\(629\) 20.0829 0.800759
\(630\) 0 0
\(631\) −18.4588 −0.734834 −0.367417 0.930056i \(-0.619758\pi\)
−0.367417 + 0.930056i \(0.619758\pi\)
\(632\) 11.4420 0.455137
\(633\) −24.2837 −0.965191
\(634\) 4.34551 0.172582
\(635\) −4.78473 −0.189876
\(636\) −9.66628 −0.383293
\(637\) 0 0
\(638\) −2.11882 −0.0838851
\(639\) 15.0098 0.593778
\(640\) −0.473626 −0.0187217
\(641\) −29.6006 −1.16915 −0.584576 0.811339i \(-0.698739\pi\)
−0.584576 + 0.811339i \(0.698739\pi\)
\(642\) −15.3231 −0.604756
\(643\) −7.93039 −0.312744 −0.156372 0.987698i \(-0.549980\pi\)
−0.156372 + 0.987698i \(0.549980\pi\)
\(644\) 0 0
\(645\) −0.0751261 −0.00295809
\(646\) −30.4384 −1.19758
\(647\) −40.4810 −1.59147 −0.795736 0.605644i \(-0.792916\pi\)
−0.795736 + 0.605644i \(0.792916\pi\)
\(648\) 1.00000 0.0392837
\(649\) 9.82490 0.385661
\(650\) −33.5791 −1.31708
\(651\) 0 0
\(652\) −8.33609 −0.326466
\(653\) −22.8643 −0.894750 −0.447375 0.894346i \(-0.647641\pi\)
−0.447375 + 0.894346i \(0.647641\pi\)
\(654\) −0.422735 −0.0165303
\(655\) −9.89135 −0.386487
\(656\) −11.2458 −0.439074
\(657\) −6.13048 −0.239173
\(658\) 0 0
\(659\) 8.96114 0.349076 0.174538 0.984650i \(-0.444157\pi\)
0.174538 + 0.984650i \(0.444157\pi\)
\(660\) −0.473626 −0.0184359
\(661\) 8.95891 0.348461 0.174231 0.984705i \(-0.444256\pi\)
0.174231 + 0.984705i \(0.444256\pi\)
\(662\) −5.46786 −0.212515
\(663\) −32.6368 −1.26751
\(664\) −7.54469 −0.292791
\(665\) 0 0
\(666\) −4.32666 −0.167655
\(667\) −3.17450 −0.122917
\(668\) 2.84727 0.110164
\(669\) −3.82529 −0.147894
\(670\) 0.573735 0.0221653
\(671\) 15.1993 0.586763
\(672\) 0 0
\(673\) −32.2141 −1.24176 −0.620881 0.783905i \(-0.713225\pi\)
−0.620881 + 0.783905i \(0.713225\pi\)
\(674\) 7.43569 0.286412
\(675\) −4.77568 −0.183816
\(676\) 36.4388 1.40149
\(677\) −2.96482 −0.113947 −0.0569737 0.998376i \(-0.518145\pi\)
−0.0569737 + 0.998376i \(0.518145\pi\)
\(678\) −8.40569 −0.322819
\(679\) 0 0
\(680\) 2.19841 0.0843053
\(681\) −12.3106 −0.471742
\(682\) 6.83509 0.261729
\(683\) 25.4616 0.974259 0.487130 0.873330i \(-0.338044\pi\)
0.487130 + 0.873330i \(0.338044\pi\)
\(684\) 6.55765 0.250738
\(685\) −3.48254 −0.133061
\(686\) 0 0
\(687\) −20.2726 −0.773449
\(688\) 0.158619 0.00604730
\(689\) −67.9662 −2.58931
\(690\) −0.709603 −0.0270141
\(691\) 8.44785 0.321371 0.160686 0.987006i \(-0.448629\pi\)
0.160686 + 0.987006i \(0.448629\pi\)
\(692\) 9.17656 0.348841
\(693\) 0 0
\(694\) 5.88508 0.223395
\(695\) 2.21490 0.0840158
\(696\) −2.11882 −0.0803138
\(697\) 52.1992 1.97718
\(698\) 17.5658 0.664874
\(699\) −10.8114 −0.408924
\(700\) 0 0
\(701\) 49.0886 1.85405 0.927025 0.374999i \(-0.122357\pi\)
0.927025 + 0.374999i \(0.122357\pi\)
\(702\) 7.03127 0.265378
\(703\) −28.3727 −1.07010
\(704\) 1.00000 0.0376889
\(705\) −0.128248 −0.00483011
\(706\) −1.45401 −0.0547223
\(707\) 0 0
\(708\) 9.82490 0.369242
\(709\) −0.167483 −0.00628996 −0.00314498 0.999995i \(-0.501001\pi\)
−0.00314498 + 0.999995i \(0.501001\pi\)
\(710\) −7.10903 −0.266797
\(711\) 11.4420 0.429107
\(712\) 16.6881 0.625414
\(713\) 10.2406 0.383512
\(714\) 0 0
\(715\) −3.33019 −0.124542
\(716\) −5.76625 −0.215495
\(717\) 4.07959 0.152355
\(718\) 19.6796 0.734436
\(719\) −1.83472 −0.0684234 −0.0342117 0.999415i \(-0.510892\pi\)
−0.0342117 + 0.999415i \(0.510892\pi\)
\(720\) −0.473626 −0.0176510
\(721\) 0 0
\(722\) 24.0027 0.893289
\(723\) 21.9946 0.817988
\(724\) 3.38164 0.125678
\(725\) 10.1188 0.375804
\(726\) 1.00000 0.0371135
\(727\) −27.7394 −1.02880 −0.514399 0.857551i \(-0.671985\pi\)
−0.514399 + 0.857551i \(0.671985\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 2.90355 0.107465
\(731\) −0.736258 −0.0272315
\(732\) 15.1993 0.561783
\(733\) 41.5283 1.53388 0.766942 0.641716i \(-0.221777\pi\)
0.766942 + 0.641716i \(0.221777\pi\)
\(734\) −18.0692 −0.666947
\(735\) 0 0
\(736\) 1.49824 0.0552257
\(737\) −1.21137 −0.0446213
\(738\) −11.2458 −0.413963
\(739\) 1.19488 0.0439545 0.0219773 0.999758i \(-0.493004\pi\)
0.0219773 + 0.999758i \(0.493004\pi\)
\(740\) 2.04922 0.0753308
\(741\) 46.1086 1.69384
\(742\) 0 0
\(743\) 4.25650 0.156156 0.0780779 0.996947i \(-0.475122\pi\)
0.0780779 + 0.996947i \(0.475122\pi\)
\(744\) 6.83509 0.250587
\(745\) −2.31649 −0.0848697
\(746\) −21.3459 −0.781528
\(747\) −7.54469 −0.276046
\(748\) −4.64167 −0.169716
\(749\) 0 0
\(750\) 4.63001 0.169064
\(751\) −13.0925 −0.477754 −0.238877 0.971050i \(-0.576779\pi\)
−0.238877 + 0.971050i \(0.576779\pi\)
\(752\) 0.270780 0.00987432
\(753\) −23.1082 −0.842111
\(754\) −14.8980 −0.542554
\(755\) 7.88744 0.287053
\(756\) 0 0
\(757\) −4.21411 −0.153164 −0.0765821 0.997063i \(-0.524401\pi\)
−0.0765821 + 0.997063i \(0.524401\pi\)
\(758\) −4.63448 −0.168332
\(759\) 1.49824 0.0543825
\(760\) −3.10587 −0.112662
\(761\) 45.8587 1.66238 0.831189 0.555990i \(-0.187661\pi\)
0.831189 + 0.555990i \(0.187661\pi\)
\(762\) 10.1023 0.365969
\(763\) 0 0
\(764\) −16.1586 −0.584598
\(765\) 2.19841 0.0794838
\(766\) −7.37981 −0.266643
\(767\) 69.0815 2.49439
\(768\) 1.00000 0.0360844
\(769\) 53.7185 1.93714 0.968569 0.248745i \(-0.0800183\pi\)
0.968569 + 0.248745i \(0.0800183\pi\)
\(770\) 0 0
\(771\) −7.75420 −0.279261
\(772\) −24.8651 −0.894913
\(773\) 6.90523 0.248364 0.124182 0.992259i \(-0.460369\pi\)
0.124182 + 0.992259i \(0.460369\pi\)
\(774\) 0.158619 0.00570145
\(775\) −32.6422 −1.17254
\(776\) 12.9656 0.465437
\(777\) 0 0
\(778\) −1.34668 −0.0482807
\(779\) −73.7458 −2.64222
\(780\) −3.33019 −0.119240
\(781\) 15.0098 0.537093
\(782\) −6.95431 −0.248686
\(783\) −2.11882 −0.0757206
\(784\) 0 0
\(785\) 6.87839 0.245500
\(786\) 20.8843 0.744919
\(787\) −3.06703 −0.109328 −0.0546639 0.998505i \(-0.517409\pi\)
−0.0546639 + 0.998505i \(0.517409\pi\)
\(788\) 10.4290 0.371518
\(789\) −13.7851 −0.490763
\(790\) −5.41921 −0.192807
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) 106.871 3.79508
\(794\) −20.7346 −0.735843
\(795\) 4.57820 0.162372
\(796\) −13.2712 −0.470384
\(797\) −41.1028 −1.45594 −0.727969 0.685610i \(-0.759535\pi\)
−0.727969 + 0.685610i \(0.759535\pi\)
\(798\) 0 0
\(799\) −1.25687 −0.0444649
\(800\) −4.77568 −0.168846
\(801\) 16.6881 0.589646
\(802\) 7.96077 0.281104
\(803\) −6.13048 −0.216340
\(804\) −1.21137 −0.0427216
\(805\) 0 0
\(806\) 48.0594 1.69282
\(807\) −6.13048 −0.215803
\(808\) 0.298919 0.0105159
\(809\) −36.2141 −1.27322 −0.636610 0.771186i \(-0.719664\pi\)
−0.636610 + 0.771186i \(0.719664\pi\)
\(810\) −0.473626 −0.0166415
\(811\) 12.2752 0.431042 0.215521 0.976499i \(-0.430855\pi\)
0.215521 + 0.976499i \(0.430855\pi\)
\(812\) 0 0
\(813\) −11.0661 −0.388104
\(814\) −4.32666 −0.151649
\(815\) 3.94819 0.138299
\(816\) −4.64167 −0.162491
\(817\) 1.04017 0.0363909
\(818\) 30.1438 1.05395
\(819\) 0 0
\(820\) 5.32629 0.186002
\(821\) −43.0697 −1.50314 −0.751572 0.659651i \(-0.770704\pi\)
−0.751572 + 0.659651i \(0.770704\pi\)
\(822\) 7.35294 0.256463
\(823\) 26.9541 0.939560 0.469780 0.882783i \(-0.344333\pi\)
0.469780 + 0.882783i \(0.344333\pi\)
\(824\) −7.39550 −0.257634
\(825\) −4.77568 −0.166268
\(826\) 0 0
\(827\) −34.9902 −1.21673 −0.608364 0.793659i \(-0.708174\pi\)
−0.608364 + 0.793659i \(0.708174\pi\)
\(828\) 1.49824 0.0520673
\(829\) 20.7246 0.719795 0.359898 0.932992i \(-0.382812\pi\)
0.359898 + 0.932992i \(0.382812\pi\)
\(830\) 3.57336 0.124033
\(831\) −16.4986 −0.572330
\(832\) 7.03127 0.243766
\(833\) 0 0
\(834\) −4.67647 −0.161933
\(835\) −1.34854 −0.0466682
\(836\) 6.55765 0.226801
\(837\) 6.83509 0.236255
\(838\) −26.7025 −0.922424
\(839\) −31.3128 −1.08104 −0.540518 0.841332i \(-0.681772\pi\)
−0.540518 + 0.841332i \(0.681772\pi\)
\(840\) 0 0
\(841\) −24.5106 −0.845193
\(842\) 8.44158 0.290916
\(843\) −17.2341 −0.593575
\(844\) −24.2837 −0.835880
\(845\) −17.2584 −0.593705
\(846\) 0.270780 0.00930960
\(847\) 0 0
\(848\) −9.66628 −0.331941
\(849\) −18.3333 −0.629198
\(850\) 22.1671 0.760326
\(851\) −6.48236 −0.222212
\(852\) 15.0098 0.514227
\(853\) −29.5524 −1.01186 −0.505928 0.862576i \(-0.668850\pi\)
−0.505928 + 0.862576i \(0.668850\pi\)
\(854\) 0 0
\(855\) −3.10587 −0.106219
\(856\) −15.3231 −0.523734
\(857\) −36.7050 −1.25382 −0.626909 0.779093i \(-0.715680\pi\)
−0.626909 + 0.779093i \(0.715680\pi\)
\(858\) 7.03127 0.240044
\(859\) 0.326102 0.0111265 0.00556323 0.999985i \(-0.498229\pi\)
0.00556323 + 0.999985i \(0.498229\pi\)
\(860\) −0.0751261 −0.00256178
\(861\) 0 0
\(862\) 18.5510 0.631849
\(863\) 16.8727 0.574353 0.287176 0.957878i \(-0.407283\pi\)
0.287176 + 0.957878i \(0.407283\pi\)
\(864\) 1.00000 0.0340207
\(865\) −4.34626 −0.147777
\(866\) −4.19969 −0.142711
\(867\) 4.54509 0.154359
\(868\) 0 0
\(869\) 11.4420 0.388142
\(870\) 1.00353 0.0340228
\(871\) −8.51746 −0.288603
\(872\) −0.422735 −0.0143156
\(873\) 12.9656 0.438818
\(874\) 9.82490 0.332332
\(875\) 0 0
\(876\) −6.13048 −0.207130
\(877\) −9.71054 −0.327902 −0.163951 0.986469i \(-0.552424\pi\)
−0.163951 + 0.986469i \(0.552424\pi\)
\(878\) −3.48175 −0.117503
\(879\) −26.1560 −0.882220
\(880\) −0.473626 −0.0159659
\(881\) 22.2976 0.751224 0.375612 0.926777i \(-0.377433\pi\)
0.375612 + 0.926777i \(0.377433\pi\)
\(882\) 0 0
\(883\) 6.70589 0.225671 0.112836 0.993614i \(-0.464007\pi\)
0.112836 + 0.993614i \(0.464007\pi\)
\(884\) −32.6368 −1.09770
\(885\) −4.65332 −0.156420
\(886\) 24.6690 0.828772
\(887\) −58.3368 −1.95876 −0.979380 0.202029i \(-0.935247\pi\)
−0.979380 + 0.202029i \(0.935247\pi\)
\(888\) −4.32666 −0.145193
\(889\) 0 0
\(890\) −7.90393 −0.264940
\(891\) 1.00000 0.0335013
\(892\) −3.82529 −0.128080
\(893\) 1.77568 0.0594208
\(894\) 4.89097 0.163579
\(895\) 2.73105 0.0912888
\(896\) 0 0
\(897\) 10.5345 0.351737
\(898\) 6.39863 0.213525
\(899\) −14.4824 −0.483014
\(900\) −4.77568 −0.159189
\(901\) 44.8677 1.49476
\(902\) −11.2458 −0.374443
\(903\) 0 0
\(904\) −8.40569 −0.279569
\(905\) −1.60163 −0.0532401
\(906\) −16.6533 −0.553270
\(907\) 46.2804 1.53671 0.768357 0.640021i \(-0.221074\pi\)
0.768357 + 0.640021i \(0.221074\pi\)
\(908\) −12.3106 −0.408541
\(909\) 0.298919 0.00991452
\(910\) 0 0
\(911\) −55.3864 −1.83503 −0.917517 0.397696i \(-0.869810\pi\)
−0.917517 + 0.397696i \(0.869810\pi\)
\(912\) 6.55765 0.217145
\(913\) −7.54469 −0.249693
\(914\) 14.5549 0.481433
\(915\) −7.19879 −0.237985
\(916\) −20.2726 −0.669826
\(917\) 0 0
\(918\) −4.64167 −0.153198
\(919\) 14.1851 0.467923 0.233961 0.972246i \(-0.424831\pi\)
0.233961 + 0.972246i \(0.424831\pi\)
\(920\) −0.709603 −0.0233949
\(921\) 25.3109 0.834024
\(922\) −16.3115 −0.537189
\(923\) 105.538 3.47383
\(924\) 0 0
\(925\) 20.6627 0.679387
\(926\) 8.81138 0.289560
\(927\) −7.39550 −0.242900
\(928\) −2.11882 −0.0695538
\(929\) −31.0150 −1.01757 −0.508784 0.860894i \(-0.669905\pi\)
−0.508784 + 0.860894i \(0.669905\pi\)
\(930\) −3.23728 −0.106154
\(931\) 0 0
\(932\) −10.8114 −0.354139
\(933\) −24.6202 −0.806029
\(934\) 8.20468 0.268465
\(935\) 2.19841 0.0718958
\(936\) 7.03127 0.229824
\(937\) 33.5238 1.09517 0.547587 0.836749i \(-0.315546\pi\)
0.547587 + 0.836749i \(0.315546\pi\)
\(938\) 0 0
\(939\) 19.2713 0.628895
\(940\) −0.128248 −0.00418300
\(941\) 32.2097 1.05001 0.525003 0.851101i \(-0.324064\pi\)
0.525003 + 0.851101i \(0.324064\pi\)
\(942\) −14.5228 −0.473180
\(943\) −16.8488 −0.548673
\(944\) 9.82490 0.319773
\(945\) 0 0
\(946\) 0.158619 0.00515715
\(947\) −45.4459 −1.47679 −0.738396 0.674367i \(-0.764416\pi\)
−0.738396 + 0.674367i \(0.764416\pi\)
\(948\) 11.4420 0.371617
\(949\) −43.1051 −1.39925
\(950\) −31.3172 −1.01606
\(951\) 4.34551 0.140913
\(952\) 0 0
\(953\) −38.5651 −1.24924 −0.624622 0.780927i \(-0.714747\pi\)
−0.624622 + 0.780927i \(0.714747\pi\)
\(954\) −9.66628 −0.312957
\(955\) 7.65314 0.247650
\(956\) 4.07959 0.131943
\(957\) −2.11882 −0.0684919
\(958\) 6.72920 0.217411
\(959\) 0 0
\(960\) −0.473626 −0.0152862
\(961\) 15.7185 0.507047
\(962\) −30.4219 −0.980843
\(963\) −15.3231 −0.493781
\(964\) 21.9946 0.708399
\(965\) 11.7767 0.379107
\(966\) 0 0
\(967\) 30.4086 0.977875 0.488938 0.872319i \(-0.337385\pi\)
0.488938 + 0.872319i \(0.337385\pi\)
\(968\) 1.00000 0.0321412
\(969\) −30.4384 −0.977823
\(970\) −6.14083 −0.197170
\(971\) −18.1366 −0.582032 −0.291016 0.956718i \(-0.593993\pi\)
−0.291016 + 0.956718i \(0.593993\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 18.9870 0.608384
\(975\) −33.5791 −1.07539
\(976\) 15.1993 0.486518
\(977\) 2.68370 0.0858590 0.0429295 0.999078i \(-0.486331\pi\)
0.0429295 + 0.999078i \(0.486331\pi\)
\(978\) −8.33609 −0.266559
\(979\) 16.6881 0.533355
\(980\) 0 0
\(981\) −0.422735 −0.0134969
\(982\) 19.1027 0.609592
\(983\) 39.2806 1.25286 0.626428 0.779479i \(-0.284516\pi\)
0.626428 + 0.779479i \(0.284516\pi\)
\(984\) −11.2458 −0.358502
\(985\) −4.93944 −0.157384
\(986\) 9.83488 0.313206
\(987\) 0 0
\(988\) 46.1086 1.46691
\(989\) 0.237649 0.00755679
\(990\) −0.473626 −0.0150528
\(991\) −48.8051 −1.55034 −0.775172 0.631750i \(-0.782337\pi\)
−0.775172 + 0.631750i \(0.782337\pi\)
\(992\) 6.83509 0.217014
\(993\) −5.46786 −0.173517
\(994\) 0 0
\(995\) 6.28556 0.199266
\(996\) −7.54469 −0.239063
\(997\) −0.552259 −0.0174902 −0.00874512 0.999962i \(-0.502784\pi\)
−0.00874512 + 0.999962i \(0.502784\pi\)
\(998\) −38.6243 −1.22263
\(999\) −4.32666 −0.136890
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 3234.2.a.bm.1.2 yes 4
3.2 odd 2 9702.2.a.dz.1.3 4
7.6 odd 2 3234.2.a.bl.1.3 4
21.20 even 2 9702.2.a.ea.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.bl.1.3 4 7.6 odd 2
3234.2.a.bm.1.2 yes 4 1.1 even 1 trivial
9702.2.a.dz.1.3 4 3.2 odd 2
9702.2.a.ea.1.2 4 21.20 even 2