Properties

Label 7942.2.a.bo.1.3
Level $7942$
Weight $2$
Character 7942.1
Self dual yes
Analytic conductor $63.417$
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] = [7942,2,Mod(1,7942)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7942, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7942.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.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: \(63.4171892853\)
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: \( x^{8} - 3x^{7} - 14x^{6} + 29x^{5} + 64x^{4} - 50x^{3} - 36x^{2} + 30x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.725465\) of defining polynomial
Character \(\chi\) \(=\) 7942.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.274535 q^{3} +1.00000 q^{4} +0.892569 q^{5} -0.274535 q^{6} +2.55547 q^{7} -1.00000 q^{8} -2.92463 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.274535 q^{3} +1.00000 q^{4} +0.892569 q^{5} -0.274535 q^{6} +2.55547 q^{7} -1.00000 q^{8} -2.92463 q^{9} -0.892569 q^{10} -1.00000 q^{11} +0.274535 q^{12} +2.80434 q^{13} -2.55547 q^{14} +0.245042 q^{15} +1.00000 q^{16} +2.51680 q^{17} +2.92463 q^{18} +0.892569 q^{20} +0.701567 q^{21} +1.00000 q^{22} -4.65781 q^{23} -0.274535 q^{24} -4.20332 q^{25} -2.80434 q^{26} -1.62652 q^{27} +2.55547 q^{28} +4.95536 q^{29} -0.245042 q^{30} -6.85998 q^{31} -1.00000 q^{32} -0.274535 q^{33} -2.51680 q^{34} +2.28093 q^{35} -2.92463 q^{36} +6.39092 q^{37} +0.769891 q^{39} -0.892569 q^{40} +11.0881 q^{41} -0.701567 q^{42} -0.288970 q^{43} -1.00000 q^{44} -2.61044 q^{45} +4.65781 q^{46} -4.66995 q^{47} +0.274535 q^{48} -0.469582 q^{49} +4.20332 q^{50} +0.690951 q^{51} +2.80434 q^{52} +12.4197 q^{53} +1.62652 q^{54} -0.892569 q^{55} -2.55547 q^{56} -4.95536 q^{58} -6.23658 q^{59} +0.245042 q^{60} +3.39707 q^{61} +6.85998 q^{62} -7.47380 q^{63} +1.00000 q^{64} +2.50307 q^{65} +0.274535 q^{66} -7.74670 q^{67} +2.51680 q^{68} -1.27873 q^{69} -2.28093 q^{70} -2.71375 q^{71} +2.92463 q^{72} +6.41177 q^{73} -6.39092 q^{74} -1.15396 q^{75} -2.55547 q^{77} -0.769891 q^{78} +14.5145 q^{79} +0.892569 q^{80} +8.32735 q^{81} -11.0881 q^{82} -3.05634 q^{83} +0.701567 q^{84} +2.24642 q^{85} +0.288970 q^{86} +1.36042 q^{87} +1.00000 q^{88} -0.997910 q^{89} +2.61044 q^{90} +7.16640 q^{91} -4.65781 q^{92} -1.88331 q^{93} +4.66995 q^{94} -0.274535 q^{96} +13.2525 q^{97} +0.469582 q^{98} +2.92463 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 5 q^{3} + 8 q^{4} + q^{5} - 5 q^{6} - 4 q^{7} - 8 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 5 q^{3} + 8 q^{4} + q^{5} - 5 q^{6} - 4 q^{7} - 8 q^{8} + 15 q^{9} - q^{10} - 8 q^{11} + 5 q^{12} + 3 q^{13} + 4 q^{14} + 34 q^{15} + 8 q^{16} - 6 q^{17} - 15 q^{18} + q^{20} - 11 q^{21} + 8 q^{22} + 9 q^{23} - 5 q^{24} + q^{25} - 3 q^{26} + 14 q^{27} - 4 q^{28} + 4 q^{29} - 34 q^{30} + 11 q^{31} - 8 q^{32} - 5 q^{33} + 6 q^{34} - 9 q^{35} + 15 q^{36} - 10 q^{37} + 2 q^{39} - q^{40} + 29 q^{41} + 11 q^{42} + 2 q^{43} - 8 q^{44} + 19 q^{45} - 9 q^{46} + 5 q^{48} - 8 q^{49} - q^{50} + 2 q^{51} + 3 q^{52} + 59 q^{53} - 14 q^{54} - q^{55} + 4 q^{56} - 4 q^{58} + 23 q^{59} + 34 q^{60} - 2 q^{61} - 11 q^{62} - 51 q^{63} + 8 q^{64} + 8 q^{65} + 5 q^{66} - 3 q^{67} - 6 q^{68} + 20 q^{69} + 9 q^{70} + q^{71} - 15 q^{72} - 6 q^{73} + 10 q^{74} - 15 q^{75} + 4 q^{77} - 2 q^{78} + q^{80} + 48 q^{81} - 29 q^{82} + 15 q^{83} - 11 q^{84} - 10 q^{85} - 2 q^{86} - 15 q^{87} + 8 q^{88} + 4 q^{89} - 19 q^{90} + 47 q^{91} + 9 q^{92} + 31 q^{93} - 5 q^{96} + 62 q^{97} + 8 q^{98} - 15 q^{99}+O(q^{100}) \) 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
\(3\) 0.274535 0.158503 0.0792516 0.996855i \(-0.474747\pi\)
0.0792516 + 0.996855i \(0.474747\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.892569 0.399169 0.199585 0.979881i \(-0.436041\pi\)
0.199585 + 0.979881i \(0.436041\pi\)
\(6\) −0.274535 −0.112079
\(7\) 2.55547 0.965876 0.482938 0.875654i \(-0.339570\pi\)
0.482938 + 0.875654i \(0.339570\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.92463 −0.974877
\(10\) −0.892569 −0.282255
\(11\) −1.00000 −0.301511
\(12\) 0.274535 0.0792516
\(13\) 2.80434 0.777784 0.388892 0.921283i \(-0.372858\pi\)
0.388892 + 0.921283i \(0.372858\pi\)
\(14\) −2.55547 −0.682978
\(15\) 0.245042 0.0632696
\(16\) 1.00000 0.250000
\(17\) 2.51680 0.610414 0.305207 0.952286i \(-0.401274\pi\)
0.305207 + 0.952286i \(0.401274\pi\)
\(18\) 2.92463 0.689342
\(19\) 0 0
\(20\) 0.892569 0.199585
\(21\) 0.701567 0.153094
\(22\) 1.00000 0.213201
\(23\) −4.65781 −0.971220 −0.485610 0.874176i \(-0.661402\pi\)
−0.485610 + 0.874176i \(0.661402\pi\)
\(24\) −0.274535 −0.0560393
\(25\) −4.20332 −0.840664
\(26\) −2.80434 −0.549976
\(27\) −1.62652 −0.313024
\(28\) 2.55547 0.482938
\(29\) 4.95536 0.920187 0.460094 0.887870i \(-0.347816\pi\)
0.460094 + 0.887870i \(0.347816\pi\)
\(30\) −0.245042 −0.0447383
\(31\) −6.85998 −1.23209 −0.616044 0.787712i \(-0.711266\pi\)
−0.616044 + 0.787712i \(0.711266\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.274535 −0.0477905
\(34\) −2.51680 −0.431628
\(35\) 2.28093 0.385548
\(36\) −2.92463 −0.487438
\(37\) 6.39092 1.05066 0.525330 0.850898i \(-0.323942\pi\)
0.525330 + 0.850898i \(0.323942\pi\)
\(38\) 0 0
\(39\) 0.769891 0.123281
\(40\) −0.892569 −0.141128
\(41\) 11.0881 1.73167 0.865837 0.500325i \(-0.166786\pi\)
0.865837 + 0.500325i \(0.166786\pi\)
\(42\) −0.701567 −0.108254
\(43\) −0.288970 −0.0440675 −0.0220338 0.999757i \(-0.507014\pi\)
−0.0220338 + 0.999757i \(0.507014\pi\)
\(44\) −1.00000 −0.150756
\(45\) −2.61044 −0.389141
\(46\) 4.65781 0.686756
\(47\) −4.66995 −0.681182 −0.340591 0.940212i \(-0.610627\pi\)
−0.340591 + 0.940212i \(0.610627\pi\)
\(48\) 0.274535 0.0396258
\(49\) −0.469582 −0.0670831
\(50\) 4.20332 0.594439
\(51\) 0.690951 0.0967525
\(52\) 2.80434 0.388892
\(53\) 12.4197 1.70597 0.852985 0.521935i \(-0.174790\pi\)
0.852985 + 0.521935i \(0.174790\pi\)
\(54\) 1.62652 0.221341
\(55\) −0.892569 −0.120354
\(56\) −2.55547 −0.341489
\(57\) 0 0
\(58\) −4.95536 −0.650671
\(59\) −6.23658 −0.811934 −0.405967 0.913888i \(-0.633065\pi\)
−0.405967 + 0.913888i \(0.633065\pi\)
\(60\) 0.245042 0.0316348
\(61\) 3.39707 0.434950 0.217475 0.976066i \(-0.430218\pi\)
0.217475 + 0.976066i \(0.430218\pi\)
\(62\) 6.85998 0.871218
\(63\) −7.47380 −0.941610
\(64\) 1.00000 0.125000
\(65\) 2.50307 0.310467
\(66\) 0.274535 0.0337930
\(67\) −7.74670 −0.946410 −0.473205 0.880952i \(-0.656903\pi\)
−0.473205 + 0.880952i \(0.656903\pi\)
\(68\) 2.51680 0.305207
\(69\) −1.27873 −0.153941
\(70\) −2.28093 −0.272624
\(71\) −2.71375 −0.322063 −0.161032 0.986949i \(-0.551482\pi\)
−0.161032 + 0.986949i \(0.551482\pi\)
\(72\) 2.92463 0.344671
\(73\) 6.41177 0.750441 0.375221 0.926936i \(-0.377567\pi\)
0.375221 + 0.926936i \(0.377567\pi\)
\(74\) −6.39092 −0.742929
\(75\) −1.15396 −0.133248
\(76\) 0 0
\(77\) −2.55547 −0.291223
\(78\) −0.769891 −0.0871730
\(79\) 14.5145 1.63301 0.816505 0.577338i \(-0.195909\pi\)
0.816505 + 0.577338i \(0.195909\pi\)
\(80\) 0.892569 0.0997923
\(81\) 8.32735 0.925261
\(82\) −11.0881 −1.22448
\(83\) −3.05634 −0.335477 −0.167738 0.985832i \(-0.553646\pi\)
−0.167738 + 0.985832i \(0.553646\pi\)
\(84\) 0.701567 0.0765472
\(85\) 2.24642 0.243658
\(86\) 0.288970 0.0311604
\(87\) 1.36042 0.145853
\(88\) 1.00000 0.106600
\(89\) −0.997910 −0.105778 −0.0528891 0.998600i \(-0.516843\pi\)
−0.0528891 + 0.998600i \(0.516843\pi\)
\(90\) 2.61044 0.275164
\(91\) 7.16640 0.751243
\(92\) −4.65781 −0.485610
\(93\) −1.88331 −0.195290
\(94\) 4.66995 0.481668
\(95\) 0 0
\(96\) −0.274535 −0.0280197
\(97\) 13.2525 1.34559 0.672793 0.739831i \(-0.265095\pi\)
0.672793 + 0.739831i \(0.265095\pi\)
\(98\) 0.469582 0.0474349
\(99\) 2.92463 0.293936
\(100\) −4.20332 −0.420332
\(101\) 10.9736 1.09192 0.545958 0.837813i \(-0.316166\pi\)
0.545958 + 0.837813i \(0.316166\pi\)
\(102\) −0.690951 −0.0684143
\(103\) −11.9597 −1.17843 −0.589214 0.807977i \(-0.700562\pi\)
−0.589214 + 0.807977i \(0.700562\pi\)
\(104\) −2.80434 −0.274988
\(105\) 0.626197 0.0611106
\(106\) −12.4197 −1.20630
\(107\) 7.69689 0.744086 0.372043 0.928215i \(-0.378657\pi\)
0.372043 + 0.928215i \(0.378657\pi\)
\(108\) −1.62652 −0.156512
\(109\) −4.32954 −0.414695 −0.207347 0.978267i \(-0.566483\pi\)
−0.207347 + 0.978267i \(0.566483\pi\)
\(110\) 0.892569 0.0851032
\(111\) 1.75453 0.166533
\(112\) 2.55547 0.241469
\(113\) 17.2931 1.62680 0.813401 0.581703i \(-0.197614\pi\)
0.813401 + 0.581703i \(0.197614\pi\)
\(114\) 0 0
\(115\) −4.15742 −0.387681
\(116\) 4.95536 0.460094
\(117\) −8.20166 −0.758244
\(118\) 6.23658 0.574124
\(119\) 6.43160 0.589584
\(120\) −0.245042 −0.0223692
\(121\) 1.00000 0.0909091
\(122\) −3.39707 −0.307556
\(123\) 3.04408 0.274476
\(124\) −6.85998 −0.616044
\(125\) −8.21460 −0.734736
\(126\) 7.47380 0.665819
\(127\) 4.44325 0.394275 0.197137 0.980376i \(-0.436836\pi\)
0.197137 + 0.980376i \(0.436836\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.0793325 −0.00698484
\(130\) −2.50307 −0.219534
\(131\) 12.8733 1.12474 0.562372 0.826884i \(-0.309889\pi\)
0.562372 + 0.826884i \(0.309889\pi\)
\(132\) −0.274535 −0.0238952
\(133\) 0 0
\(134\) 7.74670 0.669213
\(135\) −1.45178 −0.124950
\(136\) −2.51680 −0.215814
\(137\) −4.65730 −0.397900 −0.198950 0.980010i \(-0.563753\pi\)
−0.198950 + 0.980010i \(0.563753\pi\)
\(138\) 1.27873 0.108853
\(139\) 1.05322 0.0893331 0.0446665 0.999002i \(-0.485777\pi\)
0.0446665 + 0.999002i \(0.485777\pi\)
\(140\) 2.28093 0.192774
\(141\) −1.28207 −0.107969
\(142\) 2.71375 0.227733
\(143\) −2.80434 −0.234511
\(144\) −2.92463 −0.243719
\(145\) 4.42300 0.367310
\(146\) −6.41177 −0.530642
\(147\) −0.128917 −0.0106329
\(148\) 6.39092 0.525330
\(149\) −19.6886 −1.61296 −0.806478 0.591264i \(-0.798629\pi\)
−0.806478 + 0.591264i \(0.798629\pi\)
\(150\) 1.15396 0.0942205
\(151\) −6.41598 −0.522125 −0.261063 0.965322i \(-0.584073\pi\)
−0.261063 + 0.965322i \(0.584073\pi\)
\(152\) 0 0
\(153\) −7.36071 −0.595078
\(154\) 2.55547 0.205926
\(155\) −6.12300 −0.491812
\(156\) 0.769891 0.0616406
\(157\) 9.84280 0.785541 0.392770 0.919637i \(-0.371517\pi\)
0.392770 + 0.919637i \(0.371517\pi\)
\(158\) −14.5145 −1.15471
\(159\) 3.40963 0.270402
\(160\) −0.892569 −0.0705638
\(161\) −11.9029 −0.938078
\(162\) −8.32735 −0.654259
\(163\) −19.1942 −1.50340 −0.751701 0.659504i \(-0.770766\pi\)
−0.751701 + 0.659504i \(0.770766\pi\)
\(164\) 11.0881 0.865837
\(165\) −0.245042 −0.0190765
\(166\) 3.05634 0.237218
\(167\) 16.4737 1.27477 0.637387 0.770544i \(-0.280015\pi\)
0.637387 + 0.770544i \(0.280015\pi\)
\(168\) −0.701567 −0.0541270
\(169\) −5.13568 −0.395052
\(170\) −2.24642 −0.172292
\(171\) 0 0
\(172\) −0.288970 −0.0220338
\(173\) 20.3003 1.54340 0.771702 0.635984i \(-0.219405\pi\)
0.771702 + 0.635984i \(0.219405\pi\)
\(174\) −1.36042 −0.103133
\(175\) −10.7415 −0.811977
\(176\) −1.00000 −0.0753778
\(177\) −1.71216 −0.128694
\(178\) 0.997910 0.0747966
\(179\) 5.57802 0.416921 0.208460 0.978031i \(-0.433155\pi\)
0.208460 + 0.978031i \(0.433155\pi\)
\(180\) −2.61044 −0.194570
\(181\) 3.25681 0.242077 0.121038 0.992648i \(-0.461378\pi\)
0.121038 + 0.992648i \(0.461378\pi\)
\(182\) −7.16640 −0.531209
\(183\) 0.932616 0.0689409
\(184\) 4.65781 0.343378
\(185\) 5.70434 0.419391
\(186\) 1.88331 0.138091
\(187\) −2.51680 −0.184047
\(188\) −4.66995 −0.340591
\(189\) −4.15652 −0.302343
\(190\) 0 0
\(191\) 16.6232 1.20281 0.601407 0.798943i \(-0.294607\pi\)
0.601407 + 0.798943i \(0.294607\pi\)
\(192\) 0.274535 0.0198129
\(193\) −1.55689 −0.112068 −0.0560339 0.998429i \(-0.517845\pi\)
−0.0560339 + 0.998429i \(0.517845\pi\)
\(194\) −13.2525 −0.951472
\(195\) 0.687181 0.0492101
\(196\) −0.469582 −0.0335416
\(197\) 0.246728 0.0175786 0.00878931 0.999961i \(-0.497202\pi\)
0.00878931 + 0.999961i \(0.497202\pi\)
\(198\) −2.92463 −0.207844
\(199\) 20.0230 1.41939 0.709696 0.704508i \(-0.248832\pi\)
0.709696 + 0.704508i \(0.248832\pi\)
\(200\) 4.20332 0.297220
\(201\) −2.12674 −0.150009
\(202\) −10.9736 −0.772101
\(203\) 12.6633 0.888787
\(204\) 0.690951 0.0483762
\(205\) 9.89693 0.691231
\(206\) 11.9597 0.833274
\(207\) 13.6224 0.946820
\(208\) 2.80434 0.194446
\(209\) 0 0
\(210\) −0.626197 −0.0432117
\(211\) 13.1424 0.904760 0.452380 0.891825i \(-0.350575\pi\)
0.452380 + 0.891825i \(0.350575\pi\)
\(212\) 12.4197 0.852985
\(213\) −0.745021 −0.0510480
\(214\) −7.69689 −0.526148
\(215\) −0.257926 −0.0175904
\(216\) 1.62652 0.110671
\(217\) −17.5304 −1.19004
\(218\) 4.32954 0.293234
\(219\) 1.76026 0.118947
\(220\) −0.892569 −0.0601770
\(221\) 7.05797 0.474770
\(222\) −1.75453 −0.117757
\(223\) −3.35906 −0.224939 −0.112470 0.993655i \(-0.535876\pi\)
−0.112470 + 0.993655i \(0.535876\pi\)
\(224\) −2.55547 −0.170744
\(225\) 12.2932 0.819544
\(226\) −17.2931 −1.15032
\(227\) −16.3860 −1.08758 −0.543789 0.839222i \(-0.683011\pi\)
−0.543789 + 0.839222i \(0.683011\pi\)
\(228\) 0 0
\(229\) 14.2509 0.941728 0.470864 0.882206i \(-0.343942\pi\)
0.470864 + 0.882206i \(0.343942\pi\)
\(230\) 4.15742 0.274132
\(231\) −0.701567 −0.0461597
\(232\) −4.95536 −0.325335
\(233\) 24.8755 1.62965 0.814823 0.579710i \(-0.196834\pi\)
0.814823 + 0.579710i \(0.196834\pi\)
\(234\) 8.20166 0.536159
\(235\) −4.16825 −0.271907
\(236\) −6.23658 −0.405967
\(237\) 3.98475 0.258837
\(238\) −6.43160 −0.416899
\(239\) 4.31856 0.279345 0.139672 0.990198i \(-0.455395\pi\)
0.139672 + 0.990198i \(0.455395\pi\)
\(240\) 0.245042 0.0158174
\(241\) −20.7016 −1.33351 −0.666753 0.745279i \(-0.732316\pi\)
−0.666753 + 0.745279i \(0.732316\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 7.16572 0.459681
\(244\) 3.39707 0.217475
\(245\) −0.419134 −0.0267775
\(246\) −3.04408 −0.194084
\(247\) 0 0
\(248\) 6.85998 0.435609
\(249\) −0.839073 −0.0531741
\(250\) 8.21460 0.519537
\(251\) −9.10965 −0.574996 −0.287498 0.957781i \(-0.592823\pi\)
−0.287498 + 0.957781i \(0.592823\pi\)
\(252\) −7.47380 −0.470805
\(253\) 4.65781 0.292834
\(254\) −4.44325 −0.278794
\(255\) 0.616722 0.0386206
\(256\) 1.00000 0.0625000
\(257\) −10.4988 −0.654899 −0.327450 0.944869i \(-0.606189\pi\)
−0.327450 + 0.944869i \(0.606189\pi\)
\(258\) 0.0793325 0.00493902
\(259\) 16.3318 1.01481
\(260\) 2.50307 0.155234
\(261\) −14.4926 −0.897069
\(262\) −12.8733 −0.795314
\(263\) 5.40330 0.333182 0.166591 0.986026i \(-0.446724\pi\)
0.166591 + 0.986026i \(0.446724\pi\)
\(264\) 0.274535 0.0168965
\(265\) 11.0854 0.680971
\(266\) 0 0
\(267\) −0.273962 −0.0167662
\(268\) −7.74670 −0.473205
\(269\) 16.8668 1.02839 0.514194 0.857674i \(-0.328091\pi\)
0.514194 + 0.857674i \(0.328091\pi\)
\(270\) 1.45178 0.0883527
\(271\) 18.4177 1.11880 0.559399 0.828898i \(-0.311032\pi\)
0.559399 + 0.828898i \(0.311032\pi\)
\(272\) 2.51680 0.152603
\(273\) 1.96743 0.119074
\(274\) 4.65730 0.281358
\(275\) 4.20332 0.253470
\(276\) −1.27873 −0.0769707
\(277\) −20.3493 −1.22267 −0.611335 0.791372i \(-0.709367\pi\)
−0.611335 + 0.791372i \(0.709367\pi\)
\(278\) −1.05322 −0.0631680
\(279\) 20.0629 1.20113
\(280\) −2.28093 −0.136312
\(281\) 18.9213 1.12875 0.564375 0.825518i \(-0.309117\pi\)
0.564375 + 0.825518i \(0.309117\pi\)
\(282\) 1.28207 0.0763459
\(283\) 14.3130 0.850822 0.425411 0.905000i \(-0.360129\pi\)
0.425411 + 0.905000i \(0.360129\pi\)
\(284\) −2.71375 −0.161032
\(285\) 0 0
\(286\) 2.80434 0.165824
\(287\) 28.3354 1.67258
\(288\) 2.92463 0.172335
\(289\) −10.6657 −0.627395
\(290\) −4.42300 −0.259728
\(291\) 3.63827 0.213279
\(292\) 6.41177 0.375221
\(293\) −7.78857 −0.455013 −0.227506 0.973777i \(-0.573057\pi\)
−0.227506 + 0.973777i \(0.573057\pi\)
\(294\) 0.128917 0.00751858
\(295\) −5.56658 −0.324099
\(296\) −6.39092 −0.371465
\(297\) 1.62652 0.0943803
\(298\) 19.6886 1.14053
\(299\) −13.0621 −0.755399
\(300\) −1.15396 −0.0666239
\(301\) −0.738453 −0.0425638
\(302\) 6.41598 0.369198
\(303\) 3.01265 0.173072
\(304\) 0 0
\(305\) 3.03212 0.173619
\(306\) 7.36071 0.420784
\(307\) 8.08339 0.461343 0.230672 0.973032i \(-0.425908\pi\)
0.230672 + 0.973032i \(0.425908\pi\)
\(308\) −2.55547 −0.145611
\(309\) −3.28337 −0.186784
\(310\) 6.12300 0.347763
\(311\) 24.4319 1.38540 0.692702 0.721224i \(-0.256420\pi\)
0.692702 + 0.721224i \(0.256420\pi\)
\(312\) −0.769891 −0.0435865
\(313\) −10.4738 −0.592013 −0.296006 0.955186i \(-0.595655\pi\)
−0.296006 + 0.955186i \(0.595655\pi\)
\(314\) −9.84280 −0.555461
\(315\) −6.67089 −0.375862
\(316\) 14.5145 0.816505
\(317\) −18.5843 −1.04380 −0.521899 0.853007i \(-0.674776\pi\)
−0.521899 + 0.853007i \(0.674776\pi\)
\(318\) −3.40963 −0.191203
\(319\) −4.95536 −0.277447
\(320\) 0.892569 0.0498961
\(321\) 2.11307 0.117940
\(322\) 11.9029 0.663322
\(323\) 0 0
\(324\) 8.32735 0.462631
\(325\) −11.7875 −0.653855
\(326\) 19.1942 1.06307
\(327\) −1.18861 −0.0657304
\(328\) −11.0881 −0.612240
\(329\) −11.9339 −0.657938
\(330\) 0.245042 0.0134891
\(331\) −21.0755 −1.15841 −0.579206 0.815181i \(-0.696638\pi\)
−0.579206 + 0.815181i \(0.696638\pi\)
\(332\) −3.05634 −0.167738
\(333\) −18.6911 −1.02426
\(334\) −16.4737 −0.901401
\(335\) −6.91446 −0.377778
\(336\) 0.701567 0.0382736
\(337\) 13.4832 0.734476 0.367238 0.930127i \(-0.380303\pi\)
0.367238 + 0.930127i \(0.380303\pi\)
\(338\) 5.13568 0.279344
\(339\) 4.74758 0.257853
\(340\) 2.24642 0.121829
\(341\) 6.85998 0.371488
\(342\) 0 0
\(343\) −19.0883 −1.03067
\(344\) 0.288970 0.0155802
\(345\) −1.14136 −0.0614487
\(346\) −20.3003 −1.09135
\(347\) −15.9885 −0.858306 −0.429153 0.903232i \(-0.641188\pi\)
−0.429153 + 0.903232i \(0.641188\pi\)
\(348\) 1.36042 0.0729263
\(349\) −32.0957 −1.71804 −0.859022 0.511939i \(-0.828927\pi\)
−0.859022 + 0.511939i \(0.828927\pi\)
\(350\) 10.7415 0.574155
\(351\) −4.56132 −0.243465
\(352\) 1.00000 0.0533002
\(353\) 10.1485 0.540148 0.270074 0.962840i \(-0.412952\pi\)
0.270074 + 0.962840i \(0.412952\pi\)
\(354\) 1.71216 0.0910004
\(355\) −2.42221 −0.128558
\(356\) −0.997910 −0.0528891
\(357\) 1.76570 0.0934509
\(358\) −5.57802 −0.294808
\(359\) −5.98929 −0.316103 −0.158051 0.987431i \(-0.550521\pi\)
−0.158051 + 0.987431i \(0.550521\pi\)
\(360\) 2.61044 0.137582
\(361\) 0 0
\(362\) −3.25681 −0.171174
\(363\) 0.274535 0.0144094
\(364\) 7.16640 0.375622
\(365\) 5.72295 0.299553
\(366\) −0.932616 −0.0487486
\(367\) −18.8213 −0.982464 −0.491232 0.871029i \(-0.663453\pi\)
−0.491232 + 0.871029i \(0.663453\pi\)
\(368\) −4.65781 −0.242805
\(369\) −32.4287 −1.68817
\(370\) −5.70434 −0.296554
\(371\) 31.7380 1.64776
\(372\) −1.88331 −0.0976449
\(373\) −6.91364 −0.357975 −0.178987 0.983851i \(-0.557282\pi\)
−0.178987 + 0.983851i \(0.557282\pi\)
\(374\) 2.51680 0.130141
\(375\) −2.25520 −0.116458
\(376\) 4.66995 0.240834
\(377\) 13.8965 0.715707
\(378\) 4.15652 0.213788
\(379\) 22.2824 1.14457 0.572284 0.820056i \(-0.306058\pi\)
0.572284 + 0.820056i \(0.306058\pi\)
\(380\) 0 0
\(381\) 1.21983 0.0624938
\(382\) −16.6232 −0.850519
\(383\) −22.3408 −1.14156 −0.570780 0.821103i \(-0.693359\pi\)
−0.570780 + 0.821103i \(0.693359\pi\)
\(384\) −0.274535 −0.0140098
\(385\) −2.28093 −0.116247
\(386\) 1.55689 0.0792439
\(387\) 0.845130 0.0429604
\(388\) 13.2525 0.672793
\(389\) 25.1591 1.27562 0.637808 0.770195i \(-0.279841\pi\)
0.637808 + 0.770195i \(0.279841\pi\)
\(390\) −0.687181 −0.0347968
\(391\) −11.7228 −0.592846
\(392\) 0.469582 0.0237175
\(393\) 3.53417 0.178275
\(394\) −0.246728 −0.0124300
\(395\) 12.9552 0.651847
\(396\) 2.92463 0.146968
\(397\) 5.11918 0.256924 0.128462 0.991714i \(-0.458996\pi\)
0.128462 + 0.991714i \(0.458996\pi\)
\(398\) −20.0230 −1.00366
\(399\) 0 0
\(400\) −4.20332 −0.210166
\(401\) −3.83325 −0.191424 −0.0957118 0.995409i \(-0.530513\pi\)
−0.0957118 + 0.995409i \(0.530513\pi\)
\(402\) 2.12674 0.106072
\(403\) −19.2377 −0.958298
\(404\) 10.9736 0.545958
\(405\) 7.43274 0.369336
\(406\) −12.6633 −0.628467
\(407\) −6.39092 −0.316786
\(408\) −0.690951 −0.0342072
\(409\) −4.42842 −0.218971 −0.109486 0.993988i \(-0.534920\pi\)
−0.109486 + 0.993988i \(0.534920\pi\)
\(410\) −9.89693 −0.488774
\(411\) −1.27859 −0.0630684
\(412\) −11.9597 −0.589214
\(413\) −15.9374 −0.784228
\(414\) −13.6224 −0.669503
\(415\) −2.72799 −0.133912
\(416\) −2.80434 −0.137494
\(417\) 0.289147 0.0141596
\(418\) 0 0
\(419\) −12.2640 −0.599137 −0.299568 0.954075i \(-0.596843\pi\)
−0.299568 + 0.954075i \(0.596843\pi\)
\(420\) 0.626197 0.0305553
\(421\) −29.8939 −1.45694 −0.728470 0.685078i \(-0.759768\pi\)
−0.728470 + 0.685078i \(0.759768\pi\)
\(422\) −13.1424 −0.639762
\(423\) 13.6579 0.664069
\(424\) −12.4197 −0.603152
\(425\) −10.5789 −0.513153
\(426\) 0.745021 0.0360964
\(427\) 8.68110 0.420108
\(428\) 7.69689 0.372043
\(429\) −0.769891 −0.0371707
\(430\) 0.257926 0.0124383
\(431\) 17.0662 0.822051 0.411026 0.911624i \(-0.365171\pi\)
0.411026 + 0.911624i \(0.365171\pi\)
\(432\) −1.62652 −0.0782560
\(433\) 12.1572 0.584236 0.292118 0.956382i \(-0.405640\pi\)
0.292118 + 0.956382i \(0.405640\pi\)
\(434\) 17.5304 0.841488
\(435\) 1.21427 0.0582198
\(436\) −4.32954 −0.207347
\(437\) 0 0
\(438\) −1.76026 −0.0841084
\(439\) 35.3669 1.68797 0.843986 0.536365i \(-0.180203\pi\)
0.843986 + 0.536365i \(0.180203\pi\)
\(440\) 0.892569 0.0425516
\(441\) 1.37335 0.0653978
\(442\) −7.05797 −0.335713
\(443\) −34.1995 −1.62487 −0.812434 0.583053i \(-0.801858\pi\)
−0.812434 + 0.583053i \(0.801858\pi\)
\(444\) 1.75453 0.0832665
\(445\) −0.890704 −0.0422234
\(446\) 3.35906 0.159056
\(447\) −5.40523 −0.255658
\(448\) 2.55547 0.120735
\(449\) 33.9412 1.60178 0.800892 0.598808i \(-0.204359\pi\)
0.800892 + 0.598808i \(0.204359\pi\)
\(450\) −12.2932 −0.579505
\(451\) −11.0881 −0.522120
\(452\) 17.2931 0.813401
\(453\) −1.76141 −0.0827584
\(454\) 16.3860 0.769033
\(455\) 6.39651 0.299873
\(456\) 0 0
\(457\) 23.5228 1.10035 0.550176 0.835049i \(-0.314561\pi\)
0.550176 + 0.835049i \(0.314561\pi\)
\(458\) −14.2509 −0.665902
\(459\) −4.09363 −0.191074
\(460\) −4.15742 −0.193841
\(461\) 27.9047 1.29965 0.649827 0.760082i \(-0.274841\pi\)
0.649827 + 0.760082i \(0.274841\pi\)
\(462\) 0.701567 0.0326398
\(463\) −30.3458 −1.41029 −0.705145 0.709063i \(-0.749118\pi\)
−0.705145 + 0.709063i \(0.749118\pi\)
\(464\) 4.95536 0.230047
\(465\) −1.68098 −0.0779537
\(466\) −24.8755 −1.15233
\(467\) 24.6301 1.13975 0.569874 0.821732i \(-0.306992\pi\)
0.569874 + 0.821732i \(0.306992\pi\)
\(468\) −8.20166 −0.379122
\(469\) −19.7964 −0.914114
\(470\) 4.16825 0.192267
\(471\) 2.70220 0.124511
\(472\) 6.23658 0.287062
\(473\) 0.288970 0.0132869
\(474\) −3.98475 −0.183026
\(475\) 0 0
\(476\) 6.43160 0.294792
\(477\) −36.3229 −1.66311
\(478\) −4.31856 −0.197526
\(479\) −29.0486 −1.32726 −0.663632 0.748059i \(-0.730986\pi\)
−0.663632 + 0.748059i \(0.730986\pi\)
\(480\) −0.245042 −0.0111846
\(481\) 17.9223 0.817187
\(482\) 20.7016 0.942930
\(483\) −3.26776 −0.148688
\(484\) 1.00000 0.0454545
\(485\) 11.8288 0.537116
\(486\) −7.16572 −0.325043
\(487\) 9.40209 0.426049 0.213025 0.977047i \(-0.431669\pi\)
0.213025 + 0.977047i \(0.431669\pi\)
\(488\) −3.39707 −0.153778
\(489\) −5.26948 −0.238294
\(490\) 0.419134 0.0189346
\(491\) 17.5251 0.790895 0.395448 0.918489i \(-0.370589\pi\)
0.395448 + 0.918489i \(0.370589\pi\)
\(492\) 3.04408 0.137238
\(493\) 12.4717 0.561695
\(494\) 0 0
\(495\) 2.61044 0.117330
\(496\) −6.85998 −0.308022
\(497\) −6.93491 −0.311073
\(498\) 0.839073 0.0375998
\(499\) 16.0414 0.718111 0.359056 0.933316i \(-0.383099\pi\)
0.359056 + 0.933316i \(0.383099\pi\)
\(500\) −8.21460 −0.367368
\(501\) 4.52261 0.202056
\(502\) 9.10965 0.406583
\(503\) −5.65371 −0.252086 −0.126043 0.992025i \(-0.540228\pi\)
−0.126043 + 0.992025i \(0.540228\pi\)
\(504\) 7.47380 0.332910
\(505\) 9.79472 0.435859
\(506\) −4.65781 −0.207065
\(507\) −1.40992 −0.0626170
\(508\) 4.44325 0.197137
\(509\) −12.4761 −0.552993 −0.276497 0.961015i \(-0.589173\pi\)
−0.276497 + 0.961015i \(0.589173\pi\)
\(510\) −0.616722 −0.0273089
\(511\) 16.3851 0.724833
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 10.4988 0.463084
\(515\) −10.6749 −0.470392
\(516\) −0.0793325 −0.00349242
\(517\) 4.66995 0.205384
\(518\) −16.3318 −0.717578
\(519\) 5.57316 0.244634
\(520\) −2.50307 −0.109767
\(521\) 1.96422 0.0860542 0.0430271 0.999074i \(-0.486300\pi\)
0.0430271 + 0.999074i \(0.486300\pi\)
\(522\) 14.4926 0.634324
\(523\) −25.6587 −1.12198 −0.560988 0.827824i \(-0.689579\pi\)
−0.560988 + 0.827824i \(0.689579\pi\)
\(524\) 12.8733 0.562372
\(525\) −2.94891 −0.128701
\(526\) −5.40330 −0.235595
\(527\) −17.2652 −0.752083
\(528\) −0.274535 −0.0119476
\(529\) −1.30483 −0.0567316
\(530\) −11.0854 −0.481519
\(531\) 18.2397 0.791535
\(532\) 0 0
\(533\) 31.0949 1.34687
\(534\) 0.273962 0.0118555
\(535\) 6.87001 0.297016
\(536\) 7.74670 0.334606
\(537\) 1.53136 0.0660833
\(538\) −16.8668 −0.727180
\(539\) 0.469582 0.0202263
\(540\) −1.45178 −0.0624748
\(541\) −40.2006 −1.72836 −0.864180 0.503183i \(-0.832162\pi\)
−0.864180 + 0.503183i \(0.832162\pi\)
\(542\) −18.4177 −0.791110
\(543\) 0.894110 0.0383699
\(544\) −2.51680 −0.107907
\(545\) −3.86442 −0.165533
\(546\) −1.96743 −0.0841983
\(547\) −23.1690 −0.990635 −0.495318 0.868712i \(-0.664948\pi\)
−0.495318 + 0.868712i \(0.664948\pi\)
\(548\) −4.65730 −0.198950
\(549\) −9.93517 −0.424023
\(550\) −4.20332 −0.179230
\(551\) 0 0
\(552\) 1.27873 0.0544265
\(553\) 37.0914 1.57729
\(554\) 20.3493 0.864558
\(555\) 1.56604 0.0664748
\(556\) 1.05322 0.0446665
\(557\) 0.295914 0.0125383 0.00626914 0.999980i \(-0.498004\pi\)
0.00626914 + 0.999980i \(0.498004\pi\)
\(558\) −20.0629 −0.849330
\(559\) −0.810370 −0.0342750
\(560\) 2.28093 0.0963870
\(561\) −0.690951 −0.0291720
\(562\) −18.9213 −0.798147
\(563\) 37.7508 1.59101 0.795504 0.605948i \(-0.207206\pi\)
0.795504 + 0.605948i \(0.207206\pi\)
\(564\) −1.28207 −0.0539847
\(565\) 15.4353 0.649369
\(566\) −14.3130 −0.601622
\(567\) 21.2803 0.893688
\(568\) 2.71375 0.113866
\(569\) 42.4122 1.77801 0.889006 0.457896i \(-0.151397\pi\)
0.889006 + 0.457896i \(0.151397\pi\)
\(570\) 0 0
\(571\) −20.0643 −0.839664 −0.419832 0.907602i \(-0.637911\pi\)
−0.419832 + 0.907602i \(0.637911\pi\)
\(572\) −2.80434 −0.117255
\(573\) 4.56367 0.190650
\(574\) −28.3354 −1.18270
\(575\) 19.5783 0.816470
\(576\) −2.92463 −0.121860
\(577\) −3.91059 −0.162800 −0.0814000 0.996682i \(-0.525939\pi\)
−0.0814000 + 0.996682i \(0.525939\pi\)
\(578\) 10.6657 0.443635
\(579\) −0.427423 −0.0177631
\(580\) 4.42300 0.183655
\(581\) −7.81037 −0.324029
\(582\) −3.63827 −0.150811
\(583\) −12.4197 −0.514369
\(584\) −6.41177 −0.265321
\(585\) −7.32055 −0.302667
\(586\) 7.78857 0.321743
\(587\) 38.5196 1.58988 0.794938 0.606691i \(-0.207504\pi\)
0.794938 + 0.606691i \(0.207504\pi\)
\(588\) −0.128917 −0.00531644
\(589\) 0 0
\(590\) 5.56658 0.229173
\(591\) 0.0677355 0.00278627
\(592\) 6.39092 0.262665
\(593\) −0.961362 −0.0394784 −0.0197392 0.999805i \(-0.506284\pi\)
−0.0197392 + 0.999805i \(0.506284\pi\)
\(594\) −1.62652 −0.0667370
\(595\) 5.74065 0.235344
\(596\) −19.6886 −0.806478
\(597\) 5.49702 0.224978
\(598\) 13.0621 0.534148
\(599\) 13.2725 0.542299 0.271150 0.962537i \(-0.412596\pi\)
0.271150 + 0.962537i \(0.412596\pi\)
\(600\) 1.15396 0.0471102
\(601\) 27.4695 1.12051 0.560253 0.828322i \(-0.310704\pi\)
0.560253 + 0.828322i \(0.310704\pi\)
\(602\) 0.738453 0.0300971
\(603\) 22.6562 0.922633
\(604\) −6.41598 −0.261063
\(605\) 0.892569 0.0362881
\(606\) −3.01265 −0.122380
\(607\) 14.8642 0.603320 0.301660 0.953416i \(-0.402459\pi\)
0.301660 + 0.953416i \(0.402459\pi\)
\(608\) 0 0
\(609\) 3.47651 0.140875
\(610\) −3.03212 −0.122767
\(611\) −13.0961 −0.529813
\(612\) −7.36071 −0.297539
\(613\) −14.0513 −0.567525 −0.283762 0.958895i \(-0.591583\pi\)
−0.283762 + 0.958895i \(0.591583\pi\)
\(614\) −8.08339 −0.326219
\(615\) 2.71706 0.109562
\(616\) 2.55547 0.102963
\(617\) 1.66750 0.0671311 0.0335656 0.999437i \(-0.489314\pi\)
0.0335656 + 0.999437i \(0.489314\pi\)
\(618\) 3.28337 0.132076
\(619\) 21.6054 0.868394 0.434197 0.900818i \(-0.357032\pi\)
0.434197 + 0.900818i \(0.357032\pi\)
\(620\) −6.12300 −0.245906
\(621\) 7.57602 0.304015
\(622\) −24.4319 −0.979629
\(623\) −2.55013 −0.102169
\(624\) 0.769891 0.0308203
\(625\) 13.6845 0.547380
\(626\) 10.4738 0.418616
\(627\) 0 0
\(628\) 9.84280 0.392770
\(629\) 16.0847 0.641338
\(630\) 6.67089 0.265774
\(631\) 11.2648 0.448445 0.224223 0.974538i \(-0.428016\pi\)
0.224223 + 0.974538i \(0.428016\pi\)
\(632\) −14.5145 −0.577356
\(633\) 3.60805 0.143407
\(634\) 18.5843 0.738077
\(635\) 3.96591 0.157382
\(636\) 3.40963 0.135201
\(637\) −1.31687 −0.0521762
\(638\) 4.95536 0.196185
\(639\) 7.93672 0.313972
\(640\) −0.892569 −0.0352819
\(641\) 29.8776 1.18009 0.590046 0.807370i \(-0.299110\pi\)
0.590046 + 0.807370i \(0.299110\pi\)
\(642\) −2.11307 −0.0833962
\(643\) −12.8767 −0.507806 −0.253903 0.967230i \(-0.581714\pi\)
−0.253903 + 0.967230i \(0.581714\pi\)
\(644\) −11.9029 −0.469039
\(645\) −0.0708097 −0.00278813
\(646\) 0 0
\(647\) −12.3391 −0.485100 −0.242550 0.970139i \(-0.577984\pi\)
−0.242550 + 0.970139i \(0.577984\pi\)
\(648\) −8.32735 −0.327129
\(649\) 6.23658 0.244807
\(650\) 11.7875 0.462345
\(651\) −4.81273 −0.188626
\(652\) −19.1942 −0.751701
\(653\) −9.35777 −0.366198 −0.183099 0.983095i \(-0.558613\pi\)
−0.183099 + 0.983095i \(0.558613\pi\)
\(654\) 1.18861 0.0464784
\(655\) 11.4903 0.448963
\(656\) 11.0881 0.432919
\(657\) −18.7521 −0.731588
\(658\) 11.9339 0.465232
\(659\) 3.19319 0.124389 0.0621946 0.998064i \(-0.480190\pi\)
0.0621946 + 0.998064i \(0.480190\pi\)
\(660\) −0.245042 −0.00953824
\(661\) 2.98717 0.116187 0.0580937 0.998311i \(-0.481498\pi\)
0.0580937 + 0.998311i \(0.481498\pi\)
\(662\) 21.0755 0.819122
\(663\) 1.93766 0.0752525
\(664\) 3.05634 0.118609
\(665\) 0 0
\(666\) 18.6911 0.724264
\(667\) −23.0811 −0.893704
\(668\) 16.4737 0.637387
\(669\) −0.922181 −0.0356536
\(670\) 6.91446 0.267129
\(671\) −3.39707 −0.131142
\(672\) −0.701567 −0.0270635
\(673\) 16.1728 0.623414 0.311707 0.950178i \(-0.399099\pi\)
0.311707 + 0.950178i \(0.399099\pi\)
\(674\) −13.4832 −0.519353
\(675\) 6.83679 0.263148
\(676\) −5.13568 −0.197526
\(677\) −37.6666 −1.44765 −0.723823 0.689986i \(-0.757617\pi\)
−0.723823 + 0.689986i \(0.757617\pi\)
\(678\) −4.74758 −0.182330
\(679\) 33.8663 1.29967
\(680\) −2.24642 −0.0861462
\(681\) −4.49854 −0.172384
\(682\) −6.85998 −0.262682
\(683\) 33.2423 1.27198 0.635991 0.771697i \(-0.280592\pi\)
0.635991 + 0.771697i \(0.280592\pi\)
\(684\) 0 0
\(685\) −4.15697 −0.158830
\(686\) 19.0883 0.728794
\(687\) 3.91239 0.149267
\(688\) −0.288970 −0.0110169
\(689\) 34.8289 1.32688
\(690\) 1.14136 0.0434508
\(691\) 44.4889 1.69244 0.846219 0.532835i \(-0.178873\pi\)
0.846219 + 0.532835i \(0.178873\pi\)
\(692\) 20.3003 0.771702
\(693\) 7.47380 0.283906
\(694\) 15.9885 0.606914
\(695\) 0.940073 0.0356590
\(696\) −1.36042 −0.0515667
\(697\) 27.9066 1.05704
\(698\) 32.0957 1.21484
\(699\) 6.82920 0.258304
\(700\) −10.7415 −0.405989
\(701\) 11.6656 0.440605 0.220303 0.975432i \(-0.429296\pi\)
0.220303 + 0.975432i \(0.429296\pi\)
\(702\) 4.56132 0.172156
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) −1.14433 −0.0430981
\(706\) −10.1485 −0.381942
\(707\) 28.0427 1.05466
\(708\) −1.71216 −0.0643470
\(709\) −1.81477 −0.0681552 −0.0340776 0.999419i \(-0.510849\pi\)
−0.0340776 + 0.999419i \(0.510849\pi\)
\(710\) 2.42221 0.0909040
\(711\) −42.4496 −1.59198
\(712\) 0.997910 0.0373983
\(713\) 31.9524 1.19663
\(714\) −1.76570 −0.0660798
\(715\) −2.50307 −0.0936094
\(716\) 5.57802 0.208460
\(717\) 1.18560 0.0442770
\(718\) 5.98929 0.223518
\(719\) −43.9083 −1.63750 −0.818752 0.574147i \(-0.805334\pi\)
−0.818752 + 0.574147i \(0.805334\pi\)
\(720\) −2.61044 −0.0972852
\(721\) −30.5627 −1.13821
\(722\) 0 0
\(723\) −5.68331 −0.211365
\(724\) 3.25681 0.121038
\(725\) −20.8290 −0.773568
\(726\) −0.274535 −0.0101890
\(727\) −29.9225 −1.10976 −0.554882 0.831929i \(-0.687237\pi\)
−0.554882 + 0.831929i \(0.687237\pi\)
\(728\) −7.16640 −0.265605
\(729\) −23.0148 −0.852401
\(730\) −5.72295 −0.211816
\(731\) −0.727280 −0.0268994
\(732\) 0.932616 0.0344705
\(733\) 21.2241 0.783928 0.391964 0.919980i \(-0.371796\pi\)
0.391964 + 0.919980i \(0.371796\pi\)
\(734\) 18.8213 0.694707
\(735\) −0.115067 −0.00424432
\(736\) 4.65781 0.171689
\(737\) 7.74670 0.285353
\(738\) 32.4287 1.19372
\(739\) 34.7099 1.27683 0.638413 0.769694i \(-0.279591\pi\)
0.638413 + 0.769694i \(0.279591\pi\)
\(740\) 5.70434 0.209696
\(741\) 0 0
\(742\) −31.7380 −1.16514
\(743\) −13.2728 −0.486932 −0.243466 0.969909i \(-0.578284\pi\)
−0.243466 + 0.969909i \(0.578284\pi\)
\(744\) 1.88331 0.0690454
\(745\) −17.5735 −0.643842
\(746\) 6.91364 0.253126
\(747\) 8.93866 0.327048
\(748\) −2.51680 −0.0920233
\(749\) 19.6692 0.718695
\(750\) 2.25520 0.0823482
\(751\) 17.9408 0.654669 0.327335 0.944909i \(-0.393850\pi\)
0.327335 + 0.944909i \(0.393850\pi\)
\(752\) −4.66995 −0.170296
\(753\) −2.50092 −0.0911386
\(754\) −13.8965 −0.506081
\(755\) −5.72671 −0.208416
\(756\) −4.15652 −0.151171
\(757\) −20.3788 −0.740682 −0.370341 0.928896i \(-0.620759\pi\)
−0.370341 + 0.928896i \(0.620759\pi\)
\(758\) −22.2824 −0.809332
\(759\) 1.27873 0.0464151
\(760\) 0 0
\(761\) 15.7074 0.569394 0.284697 0.958618i \(-0.408107\pi\)
0.284697 + 0.958618i \(0.408107\pi\)
\(762\) −1.21983 −0.0441898
\(763\) −11.0640 −0.400544
\(764\) 16.6232 0.601407
\(765\) −6.56995 −0.237537
\(766\) 22.3408 0.807205
\(767\) −17.4895 −0.631509
\(768\) 0.274535 0.00990644
\(769\) 30.4053 1.09644 0.548221 0.836334i \(-0.315305\pi\)
0.548221 + 0.836334i \(0.315305\pi\)
\(770\) 2.28093 0.0821991
\(771\) −2.88230 −0.103804
\(772\) −1.55689 −0.0560339
\(773\) 28.6877 1.03182 0.515912 0.856642i \(-0.327453\pi\)
0.515912 + 0.856642i \(0.327453\pi\)
\(774\) −0.845130 −0.0303776
\(775\) 28.8347 1.03577
\(776\) −13.2525 −0.475736
\(777\) 4.48365 0.160850
\(778\) −25.1591 −0.901997
\(779\) 0 0
\(780\) 0.687181 0.0246050
\(781\) 2.71375 0.0971057
\(782\) 11.7228 0.419206
\(783\) −8.06000 −0.288041
\(784\) −0.469582 −0.0167708
\(785\) 8.78538 0.313564
\(786\) −3.53417 −0.126060
\(787\) −42.5311 −1.51607 −0.758036 0.652213i \(-0.773841\pi\)
−0.758036 + 0.652213i \(0.773841\pi\)
\(788\) 0.246728 0.00878931
\(789\) 1.48340 0.0528104
\(790\) −12.9552 −0.460926
\(791\) 44.1921 1.57129
\(792\) −2.92463 −0.103922
\(793\) 9.52653 0.338297
\(794\) −5.11918 −0.181673
\(795\) 3.04334 0.107936
\(796\) 20.0230 0.709696
\(797\) 32.8831 1.16478 0.582390 0.812909i \(-0.302118\pi\)
0.582390 + 0.812909i \(0.302118\pi\)
\(798\) 0 0
\(799\) −11.7533 −0.415803
\(800\) 4.20332 0.148610
\(801\) 2.91852 0.103121
\(802\) 3.83325 0.135357
\(803\) −6.41177 −0.226267
\(804\) −2.12674 −0.0750044
\(805\) −10.6241 −0.374452
\(806\) 19.2377 0.677619
\(807\) 4.63054 0.163003
\(808\) −10.9736 −0.386051
\(809\) 22.0675 0.775851 0.387925 0.921691i \(-0.373192\pi\)
0.387925 + 0.921691i \(0.373192\pi\)
\(810\) −7.43274 −0.261160
\(811\) 26.4855 0.930032 0.465016 0.885302i \(-0.346049\pi\)
0.465016 + 0.885302i \(0.346049\pi\)
\(812\) 12.6633 0.444393
\(813\) 5.05632 0.177333
\(814\) 6.39092 0.224002
\(815\) −17.1321 −0.600112
\(816\) 0.690951 0.0241881
\(817\) 0 0
\(818\) 4.42842 0.154836
\(819\) −20.9591 −0.732369
\(820\) 9.89693 0.345616
\(821\) −33.3961 −1.16553 −0.582766 0.812640i \(-0.698029\pi\)
−0.582766 + 0.812640i \(0.698029\pi\)
\(822\) 1.27859 0.0445961
\(823\) 52.6240 1.83436 0.917179 0.398477i \(-0.130461\pi\)
0.917179 + 0.398477i \(0.130461\pi\)
\(824\) 11.9597 0.416637
\(825\) 1.15396 0.0401757
\(826\) 15.9374 0.554533
\(827\) 17.2375 0.599407 0.299704 0.954032i \(-0.403112\pi\)
0.299704 + 0.954032i \(0.403112\pi\)
\(828\) 13.6224 0.473410
\(829\) −54.9443 −1.90829 −0.954147 0.299339i \(-0.903234\pi\)
−0.954147 + 0.299339i \(0.903234\pi\)
\(830\) 2.72799 0.0946900
\(831\) −5.58660 −0.193797
\(832\) 2.80434 0.0972230
\(833\) −1.18184 −0.0409485
\(834\) −0.289147 −0.0100123
\(835\) 14.7039 0.508850
\(836\) 0 0
\(837\) 11.1579 0.385673
\(838\) 12.2640 0.423654
\(839\) −27.2997 −0.942492 −0.471246 0.882002i \(-0.656196\pi\)
−0.471246 + 0.882002i \(0.656196\pi\)
\(840\) −0.626197 −0.0216058
\(841\) −4.44441 −0.153256
\(842\) 29.8939 1.03021
\(843\) 5.19457 0.178910
\(844\) 13.1424 0.452380
\(845\) −4.58395 −0.157693
\(846\) −13.6579 −0.469567
\(847\) 2.55547 0.0878069
\(848\) 12.4197 0.426493
\(849\) 3.92944 0.134858
\(850\) 10.5789 0.362854
\(851\) −29.7677 −1.02042
\(852\) −0.745021 −0.0255240
\(853\) 23.3089 0.798082 0.399041 0.916933i \(-0.369343\pi\)
0.399041 + 0.916933i \(0.369343\pi\)
\(854\) −8.68110 −0.297061
\(855\) 0 0
\(856\) −7.69689 −0.263074
\(857\) −35.3421 −1.20726 −0.603632 0.797263i \(-0.706281\pi\)
−0.603632 + 0.797263i \(0.706281\pi\)
\(858\) 0.769891 0.0262836
\(859\) 4.57419 0.156069 0.0780347 0.996951i \(-0.475136\pi\)
0.0780347 + 0.996951i \(0.475136\pi\)
\(860\) −0.257926 −0.00879519
\(861\) 7.77906 0.265110
\(862\) −17.0662 −0.581278
\(863\) 3.06254 0.104250 0.0521250 0.998641i \(-0.483401\pi\)
0.0521250 + 0.998641i \(0.483401\pi\)
\(864\) 1.62652 0.0553354
\(865\) 18.1194 0.616080
\(866\) −12.1572 −0.413118
\(867\) −2.92812 −0.0994441
\(868\) −17.5304 −0.595022
\(869\) −14.5145 −0.492371
\(870\) −1.21427 −0.0411676
\(871\) −21.7244 −0.736102
\(872\) 4.32954 0.146617
\(873\) −38.7586 −1.31178
\(874\) 0 0
\(875\) −20.9922 −0.709664
\(876\) 1.76026 0.0594736
\(877\) 18.9114 0.638591 0.319296 0.947655i \(-0.396554\pi\)
0.319296 + 0.947655i \(0.396554\pi\)
\(878\) −35.3669 −1.19358
\(879\) −2.13824 −0.0721210
\(880\) −0.892569 −0.0300885
\(881\) 21.1597 0.712889 0.356445 0.934316i \(-0.383989\pi\)
0.356445 + 0.934316i \(0.383989\pi\)
\(882\) −1.37335 −0.0462432
\(883\) 45.7789 1.54058 0.770292 0.637691i \(-0.220110\pi\)
0.770292 + 0.637691i \(0.220110\pi\)
\(884\) 7.05797 0.237385
\(885\) −1.52822 −0.0513707
\(886\) 34.1995 1.14896
\(887\) 9.25059 0.310604 0.155302 0.987867i \(-0.450365\pi\)
0.155302 + 0.987867i \(0.450365\pi\)
\(888\) −1.75453 −0.0588783
\(889\) 11.3546 0.380820
\(890\) 0.890704 0.0298565
\(891\) −8.32735 −0.278977
\(892\) −3.35906 −0.112470
\(893\) 0 0
\(894\) 5.40523 0.180778
\(895\) 4.97877 0.166422
\(896\) −2.55547 −0.0853722
\(897\) −3.58600 −0.119733
\(898\) −33.9412 −1.13263
\(899\) −33.9936 −1.13375
\(900\) 12.2932 0.409772
\(901\) 31.2578 1.04135
\(902\) 11.0881 0.369194
\(903\) −0.202732 −0.00674649
\(904\) −17.2931 −0.575161
\(905\) 2.90693 0.0966296
\(906\) 1.76141 0.0585191
\(907\) −1.43404 −0.0476167 −0.0238083 0.999717i \(-0.507579\pi\)
−0.0238083 + 0.999717i \(0.507579\pi\)
\(908\) −16.3860 −0.543789
\(909\) −32.0938 −1.06448
\(910\) −6.39651 −0.212042
\(911\) −55.0525 −1.82397 −0.911985 0.410223i \(-0.865451\pi\)
−0.911985 + 0.410223i \(0.865451\pi\)
\(912\) 0 0
\(913\) 3.05634 0.101150
\(914\) −23.5228 −0.778066
\(915\) 0.832424 0.0275191
\(916\) 14.2509 0.470864
\(917\) 32.8973 1.08636
\(918\) 4.09363 0.135110
\(919\) 29.9768 0.988842 0.494421 0.869223i \(-0.335380\pi\)
0.494421 + 0.869223i \(0.335380\pi\)
\(920\) 4.15742 0.137066
\(921\) 2.21918 0.0731243
\(922\) −27.9047 −0.918994
\(923\) −7.61028 −0.250496
\(924\) −0.701567 −0.0230798
\(925\) −26.8631 −0.883252
\(926\) 30.3458 0.997225
\(927\) 34.9778 1.14882
\(928\) −4.95536 −0.162668
\(929\) 13.6036 0.446318 0.223159 0.974782i \(-0.428363\pi\)
0.223159 + 0.974782i \(0.428363\pi\)
\(930\) 1.68098 0.0551216
\(931\) 0 0
\(932\) 24.8755 0.814823
\(933\) 6.70742 0.219591
\(934\) −24.6301 −0.805923
\(935\) −2.24642 −0.0734658
\(936\) 8.20166 0.268080
\(937\) 12.7388 0.416159 0.208079 0.978112i \(-0.433279\pi\)
0.208079 + 0.978112i \(0.433279\pi\)
\(938\) 19.7964 0.646377
\(939\) −2.87542 −0.0938358
\(940\) −4.16825 −0.135953
\(941\) 12.5791 0.410068 0.205034 0.978755i \(-0.434270\pi\)
0.205034 + 0.978755i \(0.434270\pi\)
\(942\) −2.70220 −0.0880423
\(943\) −51.6464 −1.68184
\(944\) −6.23658 −0.202983
\(945\) −3.70999 −0.120686
\(946\) −0.288970 −0.00939522
\(947\) −58.6889 −1.90713 −0.953567 0.301181i \(-0.902619\pi\)
−0.953567 + 0.301181i \(0.902619\pi\)
\(948\) 3.98475 0.129419
\(949\) 17.9808 0.583681
\(950\) 0 0
\(951\) −5.10205 −0.165445
\(952\) −6.43160 −0.208449
\(953\) 22.6435 0.733494 0.366747 0.930321i \(-0.380471\pi\)
0.366747 + 0.930321i \(0.380471\pi\)
\(954\) 36.3229 1.17600
\(955\) 14.8374 0.480127
\(956\) 4.31856 0.139672
\(957\) −1.36042 −0.0439762
\(958\) 29.0486 0.938517
\(959\) −11.9016 −0.384322
\(960\) 0.245042 0.00790869
\(961\) 16.0593 0.518041
\(962\) −17.9223 −0.577838
\(963\) −22.5106 −0.725392
\(964\) −20.7016 −0.666753
\(965\) −1.38964 −0.0447340
\(966\) 3.26776 0.105139
\(967\) 38.2819 1.23106 0.615531 0.788112i \(-0.288941\pi\)
0.615531 + 0.788112i \(0.288941\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −11.8288 −0.379798
\(971\) −21.9845 −0.705516 −0.352758 0.935715i \(-0.614756\pi\)
−0.352758 + 0.935715i \(0.614756\pi\)
\(972\) 7.16572 0.229840
\(973\) 2.69147 0.0862847
\(974\) −9.40209 −0.301262
\(975\) −3.23610 −0.103638
\(976\) 3.39707 0.108737
\(977\) −22.0839 −0.706527 −0.353264 0.935524i \(-0.614928\pi\)
−0.353264 + 0.935524i \(0.614928\pi\)
\(978\) 5.26948 0.168499
\(979\) 0.997910 0.0318934
\(980\) −0.419134 −0.0133888
\(981\) 12.6623 0.404276
\(982\) −17.5251 −0.559247
\(983\) 1.58991 0.0507102 0.0253551 0.999679i \(-0.491928\pi\)
0.0253551 + 0.999679i \(0.491928\pi\)
\(984\) −3.04408 −0.0970419
\(985\) 0.220222 0.00701685
\(986\) −12.4717 −0.397178
\(987\) −3.27628 −0.104285
\(988\) 0 0
\(989\) 1.34597 0.0427992
\(990\) −2.61044 −0.0829651
\(991\) 18.1634 0.576979 0.288490 0.957483i \(-0.406847\pi\)
0.288490 + 0.957483i \(0.406847\pi\)
\(992\) 6.85998 0.217804
\(993\) −5.78597 −0.183612
\(994\) 6.93491 0.219962
\(995\) 17.8719 0.566577
\(996\) −0.839073 −0.0265870
\(997\) −54.3199 −1.72033 −0.860164 0.510018i \(-0.829639\pi\)
−0.860164 + 0.510018i \(0.829639\pi\)
\(998\) −16.0414 −0.507781
\(999\) −10.3950 −0.328882
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 7942.2.a.bo.1.3 8
19.18 odd 2 7942.2.a.bp.1.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7942.2.a.bo.1.3 8 1.1 even 1 trivial
7942.2.a.bp.1.6 yes 8 19.18 odd 2