Properties

Label 6003.2.a.q.1.8
Level $6003$
Weight $2$
Character 6003.1
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6003,2,Mod(1,6003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} - 22 x^{14} + 68 x^{13} + 187 x^{12} - 597 x^{11} - 795 x^{10} + 2592 x^{9} + 1860 x^{8} - 5877 x^{7} - 2496 x^{6} + 6612 x^{5} + 1842 x^{4} - 3011 x^{3} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 667)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(0.445942\) of defining polynomial
Character \(\chi\) \(=\) 6003.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.445942 q^{2} -1.80114 q^{4} -2.46971 q^{5} -2.81741 q^{7} +1.69509 q^{8} +O(q^{10})\) \(q-0.445942 q^{2} -1.80114 q^{4} -2.46971 q^{5} -2.81741 q^{7} +1.69509 q^{8} +1.10135 q^{10} -4.28934 q^{11} +4.41981 q^{13} +1.25640 q^{14} +2.84636 q^{16} -6.60695 q^{17} +4.20884 q^{19} +4.44829 q^{20} +1.91280 q^{22} -1.00000 q^{23} +1.09948 q^{25} -1.97098 q^{26} +5.07453 q^{28} +1.00000 q^{29} +3.00347 q^{31} -4.65949 q^{32} +2.94632 q^{34} +6.95818 q^{35} +0.401833 q^{37} -1.87690 q^{38} -4.18638 q^{40} +2.84424 q^{41} +2.53651 q^{43} +7.72567 q^{44} +0.445942 q^{46} +11.7062 q^{47} +0.937778 q^{49} -0.490303 q^{50} -7.96068 q^{52} -7.37978 q^{53} +10.5934 q^{55} -4.77575 q^{56} -0.445942 q^{58} +1.99814 q^{59} +14.6586 q^{61} -1.33937 q^{62} -3.61486 q^{64} -10.9157 q^{65} -12.4268 q^{67} +11.9000 q^{68} -3.10295 q^{70} +6.86171 q^{71} +3.98002 q^{73} -0.179194 q^{74} -7.58069 q^{76} +12.0848 q^{77} +17.1580 q^{79} -7.02969 q^{80} -1.26837 q^{82} -3.57952 q^{83} +16.3173 q^{85} -1.13114 q^{86} -7.27080 q^{88} +7.34060 q^{89} -12.4524 q^{91} +1.80114 q^{92} -5.22029 q^{94} -10.3946 q^{95} +7.56673 q^{97} -0.418195 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8} - 14 q^{10} - 4 q^{11} + 15 q^{13} - 8 q^{14} + 23 q^{16} - 20 q^{17} - 4 q^{19} - 25 q^{20} + 13 q^{22} - 16 q^{23} + 30 q^{25} - 25 q^{26} - 13 q^{28} + 16 q^{29} + 19 q^{32} - 23 q^{34} - 5 q^{35} + 5 q^{37} - 38 q^{38} - 20 q^{40} - 7 q^{41} - 17 q^{43} + 21 q^{44} + 3 q^{46} - 29 q^{47} + 31 q^{49} + 44 q^{50} + 20 q^{52} - 63 q^{53} + q^{55} + 19 q^{56} - 3 q^{58} - 11 q^{59} - 33 q^{62} + 29 q^{64} - 53 q^{65} - 13 q^{67} - 63 q^{68} - 46 q^{70} + 23 q^{71} - 38 q^{73} + 47 q^{74} - 56 q^{76} - 97 q^{77} - 27 q^{79} - 8 q^{80} + 9 q^{82} - 36 q^{83} + 6 q^{85} + 11 q^{86} - 24 q^{88} + 16 q^{89} - 47 q^{91} - 21 q^{92} + 37 q^{94} + 12 q^{95} - 30 q^{97} + 27 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.445942 −0.315329 −0.157664 0.987493i \(-0.550396\pi\)
−0.157664 + 0.987493i \(0.550396\pi\)
\(3\) 0 0
\(4\) −1.80114 −0.900568
\(5\) −2.46971 −1.10449 −0.552244 0.833682i \(-0.686228\pi\)
−0.552244 + 0.833682i \(0.686228\pi\)
\(6\) 0 0
\(7\) −2.81741 −1.06488 −0.532440 0.846468i \(-0.678725\pi\)
−0.532440 + 0.846468i \(0.678725\pi\)
\(8\) 1.69509 0.599304
\(9\) 0 0
\(10\) 1.10135 0.348277
\(11\) −4.28934 −1.29328 −0.646642 0.762794i \(-0.723827\pi\)
−0.646642 + 0.762794i \(0.723827\pi\)
\(12\) 0 0
\(13\) 4.41981 1.22584 0.612918 0.790147i \(-0.289996\pi\)
0.612918 + 0.790147i \(0.289996\pi\)
\(14\) 1.25640 0.335787
\(15\) 0 0
\(16\) 2.84636 0.711590
\(17\) −6.60695 −1.60242 −0.801210 0.598383i \(-0.795810\pi\)
−0.801210 + 0.598383i \(0.795810\pi\)
\(18\) 0 0
\(19\) 4.20884 0.965574 0.482787 0.875738i \(-0.339625\pi\)
0.482787 + 0.875738i \(0.339625\pi\)
\(20\) 4.44829 0.994667
\(21\) 0 0
\(22\) 1.91280 0.407810
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.09948 0.219895
\(26\) −1.97098 −0.386541
\(27\) 0 0
\(28\) 5.07453 0.958996
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 3.00347 0.539439 0.269719 0.962939i \(-0.413069\pi\)
0.269719 + 0.962939i \(0.413069\pi\)
\(32\) −4.65949 −0.823689
\(33\) 0 0
\(34\) 2.94632 0.505289
\(35\) 6.95818 1.17615
\(36\) 0 0
\(37\) 0.401833 0.0660609 0.0330305 0.999454i \(-0.489484\pi\)
0.0330305 + 0.999454i \(0.489484\pi\)
\(38\) −1.87690 −0.304473
\(39\) 0 0
\(40\) −4.18638 −0.661924
\(41\) 2.84424 0.444196 0.222098 0.975024i \(-0.428709\pi\)
0.222098 + 0.975024i \(0.428709\pi\)
\(42\) 0 0
\(43\) 2.53651 0.386814 0.193407 0.981119i \(-0.438046\pi\)
0.193407 + 0.981119i \(0.438046\pi\)
\(44\) 7.72567 1.16469
\(45\) 0 0
\(46\) 0.445942 0.0657506
\(47\) 11.7062 1.70753 0.853763 0.520662i \(-0.174315\pi\)
0.853763 + 0.520662i \(0.174315\pi\)
\(48\) 0 0
\(49\) 0.937778 0.133968
\(50\) −0.490303 −0.0693393
\(51\) 0 0
\(52\) −7.96068 −1.10395
\(53\) −7.37978 −1.01369 −0.506846 0.862037i \(-0.669189\pi\)
−0.506846 + 0.862037i \(0.669189\pi\)
\(54\) 0 0
\(55\) 10.5934 1.42842
\(56\) −4.77575 −0.638186
\(57\) 0 0
\(58\) −0.445942 −0.0585551
\(59\) 1.99814 0.260136 0.130068 0.991505i \(-0.458480\pi\)
0.130068 + 0.991505i \(0.458480\pi\)
\(60\) 0 0
\(61\) 14.6586 1.87685 0.938423 0.345488i \(-0.112287\pi\)
0.938423 + 0.345488i \(0.112287\pi\)
\(62\) −1.33937 −0.170101
\(63\) 0 0
\(64\) −3.61486 −0.451857
\(65\) −10.9157 −1.35392
\(66\) 0 0
\(67\) −12.4268 −1.51817 −0.759085 0.650992i \(-0.774353\pi\)
−0.759085 + 0.650992i \(0.774353\pi\)
\(68\) 11.9000 1.44309
\(69\) 0 0
\(70\) −3.10295 −0.370873
\(71\) 6.86171 0.814335 0.407168 0.913353i \(-0.366516\pi\)
0.407168 + 0.913353i \(0.366516\pi\)
\(72\) 0 0
\(73\) 3.98002 0.465827 0.232913 0.972498i \(-0.425174\pi\)
0.232913 + 0.972498i \(0.425174\pi\)
\(74\) −0.179194 −0.0208309
\(75\) 0 0
\(76\) −7.58069 −0.869564
\(77\) 12.0848 1.37719
\(78\) 0 0
\(79\) 17.1580 1.93042 0.965212 0.261469i \(-0.0842069\pi\)
0.965212 + 0.261469i \(0.0842069\pi\)
\(80\) −7.02969 −0.785943
\(81\) 0 0
\(82\) −1.26837 −0.140068
\(83\) −3.57952 −0.392903 −0.196452 0.980514i \(-0.562942\pi\)
−0.196452 + 0.980514i \(0.562942\pi\)
\(84\) 0 0
\(85\) 16.3173 1.76985
\(86\) −1.13114 −0.121974
\(87\) 0 0
\(88\) −7.27080 −0.775070
\(89\) 7.34060 0.778102 0.389051 0.921216i \(-0.372803\pi\)
0.389051 + 0.921216i \(0.372803\pi\)
\(90\) 0 0
\(91\) −12.4524 −1.30537
\(92\) 1.80114 0.187781
\(93\) 0 0
\(94\) −5.22029 −0.538432
\(95\) −10.3946 −1.06646
\(96\) 0 0
\(97\) 7.56673 0.768285 0.384143 0.923274i \(-0.374497\pi\)
0.384143 + 0.923274i \(0.374497\pi\)
\(98\) −0.418195 −0.0422441
\(99\) 0 0
\(100\) −1.98030 −0.198030
\(101\) 8.93289 0.888856 0.444428 0.895815i \(-0.353407\pi\)
0.444428 + 0.895815i \(0.353407\pi\)
\(102\) 0 0
\(103\) −17.0586 −1.68083 −0.840415 0.541944i \(-0.817689\pi\)
−0.840415 + 0.541944i \(0.817689\pi\)
\(104\) 7.49197 0.734648
\(105\) 0 0
\(106\) 3.29096 0.319646
\(107\) −1.72224 −0.166496 −0.0832478 0.996529i \(-0.526529\pi\)
−0.0832478 + 0.996529i \(0.526529\pi\)
\(108\) 0 0
\(109\) −10.6626 −1.02129 −0.510644 0.859792i \(-0.670593\pi\)
−0.510644 + 0.859792i \(0.670593\pi\)
\(110\) −4.72406 −0.450421
\(111\) 0 0
\(112\) −8.01935 −0.757757
\(113\) −16.2113 −1.52503 −0.762515 0.646971i \(-0.776036\pi\)
−0.762515 + 0.646971i \(0.776036\pi\)
\(114\) 0 0
\(115\) 2.46971 0.230302
\(116\) −1.80114 −0.167231
\(117\) 0 0
\(118\) −0.891056 −0.0820283
\(119\) 18.6145 1.70638
\(120\) 0 0
\(121\) 7.39840 0.672582
\(122\) −6.53691 −0.591824
\(123\) 0 0
\(124\) −5.40965 −0.485801
\(125\) 9.63317 0.861617
\(126\) 0 0
\(127\) 6.42264 0.569917 0.284958 0.958540i \(-0.408020\pi\)
0.284958 + 0.958540i \(0.408020\pi\)
\(128\) 10.9310 0.966172
\(129\) 0 0
\(130\) 4.86776 0.426931
\(131\) 14.9726 1.30817 0.654083 0.756423i \(-0.273055\pi\)
0.654083 + 0.756423i \(0.273055\pi\)
\(132\) 0 0
\(133\) −11.8580 −1.02822
\(134\) 5.54162 0.478723
\(135\) 0 0
\(136\) −11.1994 −0.960337
\(137\) 0.156153 0.0133410 0.00667052 0.999978i \(-0.497877\pi\)
0.00667052 + 0.999978i \(0.497877\pi\)
\(138\) 0 0
\(139\) −13.5888 −1.15259 −0.576295 0.817241i \(-0.695502\pi\)
−0.576295 + 0.817241i \(0.695502\pi\)
\(140\) −12.5326 −1.05920
\(141\) 0 0
\(142\) −3.05993 −0.256783
\(143\) −18.9581 −1.58535
\(144\) 0 0
\(145\) −2.46971 −0.205098
\(146\) −1.77486 −0.146889
\(147\) 0 0
\(148\) −0.723756 −0.0594924
\(149\) −2.15399 −0.176461 −0.0882307 0.996100i \(-0.528121\pi\)
−0.0882307 + 0.996100i \(0.528121\pi\)
\(150\) 0 0
\(151\) 3.13094 0.254792 0.127396 0.991852i \(-0.459338\pi\)
0.127396 + 0.991852i \(0.459338\pi\)
\(152\) 7.13435 0.578672
\(153\) 0 0
\(154\) −5.38913 −0.434268
\(155\) −7.41770 −0.595804
\(156\) 0 0
\(157\) 16.8952 1.34838 0.674191 0.738557i \(-0.264492\pi\)
0.674191 + 0.738557i \(0.264492\pi\)
\(158\) −7.65147 −0.608718
\(159\) 0 0
\(160\) 11.5076 0.909755
\(161\) 2.81741 0.222043
\(162\) 0 0
\(163\) −22.1137 −1.73208 −0.866038 0.499978i \(-0.833342\pi\)
−0.866038 + 0.499978i \(0.833342\pi\)
\(164\) −5.12287 −0.400029
\(165\) 0 0
\(166\) 1.59626 0.123894
\(167\) −2.95049 −0.228316 −0.114158 0.993463i \(-0.536417\pi\)
−0.114158 + 0.993463i \(0.536417\pi\)
\(168\) 0 0
\(169\) 6.53474 0.502673
\(170\) −7.27656 −0.558086
\(171\) 0 0
\(172\) −4.56860 −0.348352
\(173\) −18.1533 −1.38017 −0.690085 0.723728i \(-0.742427\pi\)
−0.690085 + 0.723728i \(0.742427\pi\)
\(174\) 0 0
\(175\) −3.09767 −0.234162
\(176\) −12.2090 −0.920287
\(177\) 0 0
\(178\) −3.27348 −0.245358
\(179\) 10.1727 0.760346 0.380173 0.924915i \(-0.375864\pi\)
0.380173 + 0.924915i \(0.375864\pi\)
\(180\) 0 0
\(181\) 21.4731 1.59608 0.798040 0.602604i \(-0.205870\pi\)
0.798040 + 0.602604i \(0.205870\pi\)
\(182\) 5.55306 0.411620
\(183\) 0 0
\(184\) −1.69509 −0.124964
\(185\) −0.992412 −0.0729636
\(186\) 0 0
\(187\) 28.3394 2.07238
\(188\) −21.0845 −1.53774
\(189\) 0 0
\(190\) 4.63540 0.336287
\(191\) −21.7372 −1.57285 −0.786425 0.617686i \(-0.788070\pi\)
−0.786425 + 0.617686i \(0.788070\pi\)
\(192\) 0 0
\(193\) −5.77711 −0.415846 −0.207923 0.978145i \(-0.566670\pi\)
−0.207923 + 0.978145i \(0.566670\pi\)
\(194\) −3.37433 −0.242263
\(195\) 0 0
\(196\) −1.68906 −0.120647
\(197\) 16.9138 1.20506 0.602529 0.798097i \(-0.294160\pi\)
0.602529 + 0.798097i \(0.294160\pi\)
\(198\) 0 0
\(199\) −15.4767 −1.09712 −0.548559 0.836112i \(-0.684823\pi\)
−0.548559 + 0.836112i \(0.684823\pi\)
\(200\) 1.86371 0.131784
\(201\) 0 0
\(202\) −3.98355 −0.280282
\(203\) −2.81741 −0.197743
\(204\) 0 0
\(205\) −7.02446 −0.490610
\(206\) 7.60713 0.530014
\(207\) 0 0
\(208\) 12.5804 0.872292
\(209\) −18.0531 −1.24876
\(210\) 0 0
\(211\) 5.84718 0.402536 0.201268 0.979536i \(-0.435494\pi\)
0.201268 + 0.979536i \(0.435494\pi\)
\(212\) 13.2920 0.912897
\(213\) 0 0
\(214\) 0.768022 0.0525009
\(215\) −6.26445 −0.427232
\(216\) 0 0
\(217\) −8.46199 −0.574437
\(218\) 4.75489 0.322042
\(219\) 0 0
\(220\) −19.0802 −1.28639
\(221\) −29.2015 −1.96430
\(222\) 0 0
\(223\) −22.7769 −1.52525 −0.762625 0.646840i \(-0.776090\pi\)
−0.762625 + 0.646840i \(0.776090\pi\)
\(224\) 13.1277 0.877129
\(225\) 0 0
\(226\) 7.22930 0.480886
\(227\) −4.87598 −0.323630 −0.161815 0.986821i \(-0.551735\pi\)
−0.161815 + 0.986821i \(0.551735\pi\)
\(228\) 0 0
\(229\) −10.9661 −0.724661 −0.362330 0.932050i \(-0.618019\pi\)
−0.362330 + 0.932050i \(0.618019\pi\)
\(230\) −1.10135 −0.0726208
\(231\) 0 0
\(232\) 1.69509 0.111288
\(233\) −29.0404 −1.90250 −0.951251 0.308418i \(-0.900200\pi\)
−0.951251 + 0.308418i \(0.900200\pi\)
\(234\) 0 0
\(235\) −28.9109 −1.88594
\(236\) −3.59892 −0.234270
\(237\) 0 0
\(238\) −8.30098 −0.538072
\(239\) −22.0157 −1.42408 −0.712039 0.702140i \(-0.752228\pi\)
−0.712039 + 0.702140i \(0.752228\pi\)
\(240\) 0 0
\(241\) −25.2176 −1.62441 −0.812206 0.583371i \(-0.801733\pi\)
−0.812206 + 0.583371i \(0.801733\pi\)
\(242\) −3.29926 −0.212085
\(243\) 0 0
\(244\) −26.4022 −1.69023
\(245\) −2.31604 −0.147966
\(246\) 0 0
\(247\) 18.6023 1.18363
\(248\) 5.09114 0.323288
\(249\) 0 0
\(250\) −4.29584 −0.271693
\(251\) 21.9563 1.38587 0.692933 0.721002i \(-0.256318\pi\)
0.692933 + 0.721002i \(0.256318\pi\)
\(252\) 0 0
\(253\) 4.28934 0.269668
\(254\) −2.86413 −0.179711
\(255\) 0 0
\(256\) 2.35512 0.147195
\(257\) 23.2366 1.44946 0.724731 0.689032i \(-0.241964\pi\)
0.724731 + 0.689032i \(0.241964\pi\)
\(258\) 0 0
\(259\) −1.13213 −0.0703469
\(260\) 19.6606 1.21930
\(261\) 0 0
\(262\) −6.67694 −0.412503
\(263\) 7.44609 0.459145 0.229573 0.973292i \(-0.426267\pi\)
0.229573 + 0.973292i \(0.426267\pi\)
\(264\) 0 0
\(265\) 18.2259 1.11961
\(266\) 5.28799 0.324227
\(267\) 0 0
\(268\) 22.3823 1.36721
\(269\) −17.2475 −1.05160 −0.525800 0.850608i \(-0.676234\pi\)
−0.525800 + 0.850608i \(0.676234\pi\)
\(270\) 0 0
\(271\) 3.53709 0.214863 0.107431 0.994212i \(-0.465737\pi\)
0.107431 + 0.994212i \(0.465737\pi\)
\(272\) −18.8057 −1.14027
\(273\) 0 0
\(274\) −0.0696352 −0.00420682
\(275\) −4.71602 −0.284387
\(276\) 0 0
\(277\) −5.35010 −0.321456 −0.160728 0.986999i \(-0.551384\pi\)
−0.160728 + 0.986999i \(0.551384\pi\)
\(278\) 6.05984 0.363445
\(279\) 0 0
\(280\) 11.7947 0.704870
\(281\) −2.43662 −0.145356 −0.0726782 0.997355i \(-0.523155\pi\)
−0.0726782 + 0.997355i \(0.523155\pi\)
\(282\) 0 0
\(283\) −14.9942 −0.891313 −0.445656 0.895204i \(-0.647030\pi\)
−0.445656 + 0.895204i \(0.647030\pi\)
\(284\) −12.3589 −0.733364
\(285\) 0 0
\(286\) 8.45420 0.499908
\(287\) −8.01339 −0.473016
\(288\) 0 0
\(289\) 26.6518 1.56775
\(290\) 1.10135 0.0646735
\(291\) 0 0
\(292\) −7.16856 −0.419508
\(293\) 9.90190 0.578475 0.289238 0.957257i \(-0.406598\pi\)
0.289238 + 0.957257i \(0.406598\pi\)
\(294\) 0 0
\(295\) −4.93483 −0.287317
\(296\) 0.681142 0.0395906
\(297\) 0 0
\(298\) 0.960554 0.0556434
\(299\) −4.41981 −0.255604
\(300\) 0 0
\(301\) −7.14638 −0.411910
\(302\) −1.39622 −0.0803433
\(303\) 0 0
\(304\) 11.9799 0.687092
\(305\) −36.2026 −2.07296
\(306\) 0 0
\(307\) 8.06498 0.460293 0.230146 0.973156i \(-0.426079\pi\)
0.230146 + 0.973156i \(0.426079\pi\)
\(308\) −21.7664 −1.24025
\(309\) 0 0
\(310\) 3.30787 0.187874
\(311\) 16.9600 0.961714 0.480857 0.876799i \(-0.340326\pi\)
0.480857 + 0.876799i \(0.340326\pi\)
\(312\) 0 0
\(313\) 30.2949 1.71237 0.856185 0.516669i \(-0.172828\pi\)
0.856185 + 0.516669i \(0.172828\pi\)
\(314\) −7.53428 −0.425184
\(315\) 0 0
\(316\) −30.9038 −1.73848
\(317\) 21.3506 1.19917 0.599584 0.800312i \(-0.295333\pi\)
0.599584 + 0.800312i \(0.295333\pi\)
\(318\) 0 0
\(319\) −4.28934 −0.240157
\(320\) 8.92765 0.499071
\(321\) 0 0
\(322\) −1.25640 −0.0700165
\(323\) −27.8076 −1.54725
\(324\) 0 0
\(325\) 4.85948 0.269555
\(326\) 9.86143 0.546174
\(327\) 0 0
\(328\) 4.82124 0.266209
\(329\) −32.9811 −1.81831
\(330\) 0 0
\(331\) 0.784375 0.0431131 0.0215566 0.999768i \(-0.493138\pi\)
0.0215566 + 0.999768i \(0.493138\pi\)
\(332\) 6.44720 0.353836
\(333\) 0 0
\(334\) 1.31575 0.0719946
\(335\) 30.6905 1.67680
\(336\) 0 0
\(337\) −22.2511 −1.21209 −0.606047 0.795429i \(-0.707246\pi\)
−0.606047 + 0.795429i \(0.707246\pi\)
\(338\) −2.91412 −0.158507
\(339\) 0 0
\(340\) −29.3896 −1.59387
\(341\) −12.8829 −0.697647
\(342\) 0 0
\(343\) 17.0797 0.922219
\(344\) 4.29961 0.231819
\(345\) 0 0
\(346\) 8.09533 0.435207
\(347\) −19.0506 −1.02269 −0.511344 0.859376i \(-0.670852\pi\)
−0.511344 + 0.859376i \(0.670852\pi\)
\(348\) 0 0
\(349\) −13.1162 −0.702096 −0.351048 0.936357i \(-0.614175\pi\)
−0.351048 + 0.936357i \(0.614175\pi\)
\(350\) 1.38138 0.0738380
\(351\) 0 0
\(352\) 19.9861 1.06526
\(353\) −17.5345 −0.933265 −0.466633 0.884451i \(-0.654533\pi\)
−0.466633 + 0.884451i \(0.654533\pi\)
\(354\) 0 0
\(355\) −16.9464 −0.899424
\(356\) −13.2214 −0.700733
\(357\) 0 0
\(358\) −4.53646 −0.239759
\(359\) −21.0184 −1.10931 −0.554653 0.832082i \(-0.687149\pi\)
−0.554653 + 0.832082i \(0.687149\pi\)
\(360\) 0 0
\(361\) −1.28569 −0.0676678
\(362\) −9.57575 −0.503290
\(363\) 0 0
\(364\) 22.4285 1.17557
\(365\) −9.82951 −0.514500
\(366\) 0 0
\(367\) 22.5736 1.17833 0.589165 0.808013i \(-0.299457\pi\)
0.589165 + 0.808013i \(0.299457\pi\)
\(368\) −2.84636 −0.148377
\(369\) 0 0
\(370\) 0.442559 0.0230075
\(371\) 20.7918 1.07946
\(372\) 0 0
\(373\) −28.5809 −1.47986 −0.739931 0.672683i \(-0.765142\pi\)
−0.739931 + 0.672683i \(0.765142\pi\)
\(374\) −12.6377 −0.653482
\(375\) 0 0
\(376\) 19.8430 1.02333
\(377\) 4.41981 0.227632
\(378\) 0 0
\(379\) −8.35998 −0.429423 −0.214712 0.976677i \(-0.568881\pi\)
−0.214712 + 0.976677i \(0.568881\pi\)
\(380\) 18.7221 0.960424
\(381\) 0 0
\(382\) 9.69355 0.495965
\(383\) 1.67118 0.0853932 0.0426966 0.999088i \(-0.486405\pi\)
0.0426966 + 0.999088i \(0.486405\pi\)
\(384\) 0 0
\(385\) −29.8460 −1.52109
\(386\) 2.57626 0.131128
\(387\) 0 0
\(388\) −13.6287 −0.691893
\(389\) 18.1730 0.921405 0.460703 0.887554i \(-0.347597\pi\)
0.460703 + 0.887554i \(0.347597\pi\)
\(390\) 0 0
\(391\) 6.60695 0.334128
\(392\) 1.58962 0.0802877
\(393\) 0 0
\(394\) −7.54258 −0.379990
\(395\) −42.3753 −2.13213
\(396\) 0 0
\(397\) 11.1359 0.558897 0.279449 0.960161i \(-0.409848\pi\)
0.279449 + 0.960161i \(0.409848\pi\)
\(398\) 6.90174 0.345953
\(399\) 0 0
\(400\) 3.12950 0.156475
\(401\) −36.0633 −1.80091 −0.900457 0.434945i \(-0.856768\pi\)
−0.900457 + 0.434945i \(0.856768\pi\)
\(402\) 0 0
\(403\) 13.2748 0.661263
\(404\) −16.0893 −0.800475
\(405\) 0 0
\(406\) 1.25640 0.0623541
\(407\) −1.72360 −0.0854355
\(408\) 0 0
\(409\) −5.09351 −0.251858 −0.125929 0.992039i \(-0.540191\pi\)
−0.125929 + 0.992039i \(0.540191\pi\)
\(410\) 3.13251 0.154703
\(411\) 0 0
\(412\) 30.7248 1.51370
\(413\) −5.62958 −0.277013
\(414\) 0 0
\(415\) 8.84038 0.433957
\(416\) −20.5941 −1.00971
\(417\) 0 0
\(418\) 8.05065 0.393770
\(419\) 18.9896 0.927701 0.463850 0.885914i \(-0.346468\pi\)
0.463850 + 0.885914i \(0.346468\pi\)
\(420\) 0 0
\(421\) −25.9673 −1.26557 −0.632784 0.774328i \(-0.718088\pi\)
−0.632784 + 0.774328i \(0.718088\pi\)
\(422\) −2.60750 −0.126931
\(423\) 0 0
\(424\) −12.5094 −0.607509
\(425\) −7.26418 −0.352364
\(426\) 0 0
\(427\) −41.2993 −1.99862
\(428\) 3.10200 0.149941
\(429\) 0 0
\(430\) 2.79358 0.134719
\(431\) 20.4418 0.984648 0.492324 0.870412i \(-0.336148\pi\)
0.492324 + 0.870412i \(0.336148\pi\)
\(432\) 0 0
\(433\) 2.43016 0.116786 0.0583930 0.998294i \(-0.481402\pi\)
0.0583930 + 0.998294i \(0.481402\pi\)
\(434\) 3.77356 0.181137
\(435\) 0 0
\(436\) 19.2047 0.919739
\(437\) −4.20884 −0.201336
\(438\) 0 0
\(439\) 9.08175 0.433448 0.216724 0.976233i \(-0.430463\pi\)
0.216724 + 0.976233i \(0.430463\pi\)
\(440\) 17.9568 0.856056
\(441\) 0 0
\(442\) 13.0222 0.619402
\(443\) −26.2999 −1.24954 −0.624772 0.780807i \(-0.714808\pi\)
−0.624772 + 0.780807i \(0.714808\pi\)
\(444\) 0 0
\(445\) −18.1292 −0.859405
\(446\) 10.1572 0.480956
\(447\) 0 0
\(448\) 10.1845 0.481173
\(449\) 21.0659 0.994161 0.497081 0.867704i \(-0.334405\pi\)
0.497081 + 0.867704i \(0.334405\pi\)
\(450\) 0 0
\(451\) −12.1999 −0.574472
\(452\) 29.1987 1.37339
\(453\) 0 0
\(454\) 2.17441 0.102050
\(455\) 30.7539 1.44176
\(456\) 0 0
\(457\) −11.7132 −0.547922 −0.273961 0.961741i \(-0.588334\pi\)
−0.273961 + 0.961741i \(0.588334\pi\)
\(458\) 4.89025 0.228506
\(459\) 0 0
\(460\) −4.44829 −0.207402
\(461\) 11.7310 0.546369 0.273184 0.961962i \(-0.411923\pi\)
0.273184 + 0.961962i \(0.411923\pi\)
\(462\) 0 0
\(463\) −10.9534 −0.509049 −0.254525 0.967066i \(-0.581919\pi\)
−0.254525 + 0.967066i \(0.581919\pi\)
\(464\) 2.84636 0.132139
\(465\) 0 0
\(466\) 12.9504 0.599914
\(467\) 3.23636 0.149761 0.0748804 0.997193i \(-0.476143\pi\)
0.0748804 + 0.997193i \(0.476143\pi\)
\(468\) 0 0
\(469\) 35.0112 1.61667
\(470\) 12.8926 0.594692
\(471\) 0 0
\(472\) 3.38702 0.155900
\(473\) −10.8799 −0.500260
\(474\) 0 0
\(475\) 4.62751 0.212325
\(476\) −33.5272 −1.53671
\(477\) 0 0
\(478\) 9.81773 0.449053
\(479\) 19.1033 0.872854 0.436427 0.899740i \(-0.356244\pi\)
0.436427 + 0.899740i \(0.356244\pi\)
\(480\) 0 0
\(481\) 1.77603 0.0809799
\(482\) 11.2456 0.512224
\(483\) 0 0
\(484\) −13.3255 −0.605706
\(485\) −18.6877 −0.848563
\(486\) 0 0
\(487\) −10.3407 −0.468580 −0.234290 0.972167i \(-0.575277\pi\)
−0.234290 + 0.972167i \(0.575277\pi\)
\(488\) 24.8477 1.12480
\(489\) 0 0
\(490\) 1.03282 0.0466581
\(491\) 5.57488 0.251591 0.125795 0.992056i \(-0.459852\pi\)
0.125795 + 0.992056i \(0.459852\pi\)
\(492\) 0 0
\(493\) −6.60695 −0.297562
\(494\) −8.29554 −0.373234
\(495\) 0 0
\(496\) 8.54895 0.383859
\(497\) −19.3322 −0.867169
\(498\) 0 0
\(499\) −15.7898 −0.706847 −0.353423 0.935463i \(-0.614982\pi\)
−0.353423 + 0.935463i \(0.614982\pi\)
\(500\) −17.3506 −0.775944
\(501\) 0 0
\(502\) −9.79123 −0.437004
\(503\) 33.2833 1.48403 0.742015 0.670383i \(-0.233870\pi\)
0.742015 + 0.670383i \(0.233870\pi\)
\(504\) 0 0
\(505\) −22.0617 −0.981731
\(506\) −1.91280 −0.0850342
\(507\) 0 0
\(508\) −11.5680 −0.513249
\(509\) −21.2303 −0.941015 −0.470507 0.882396i \(-0.655929\pi\)
−0.470507 + 0.882396i \(0.655929\pi\)
\(510\) 0 0
\(511\) −11.2133 −0.496049
\(512\) −22.9122 −1.01259
\(513\) 0 0
\(514\) −10.3622 −0.457057
\(515\) 42.1297 1.85646
\(516\) 0 0
\(517\) −50.2118 −2.20831
\(518\) 0.504863 0.0221824
\(519\) 0 0
\(520\) −18.5030 −0.811410
\(521\) 31.4014 1.37572 0.687860 0.725843i \(-0.258550\pi\)
0.687860 + 0.725843i \(0.258550\pi\)
\(522\) 0 0
\(523\) 17.7232 0.774981 0.387491 0.921874i \(-0.373342\pi\)
0.387491 + 0.921874i \(0.373342\pi\)
\(524\) −26.9678 −1.17809
\(525\) 0 0
\(526\) −3.32053 −0.144782
\(527\) −19.8437 −0.864407
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −8.12772 −0.353046
\(531\) 0 0
\(532\) 21.3579 0.925981
\(533\) 12.5710 0.544512
\(534\) 0 0
\(535\) 4.25345 0.183893
\(536\) −21.0644 −0.909845
\(537\) 0 0
\(538\) 7.69140 0.331600
\(539\) −4.02244 −0.173259
\(540\) 0 0
\(541\) 12.4106 0.533575 0.266788 0.963755i \(-0.414038\pi\)
0.266788 + 0.963755i \(0.414038\pi\)
\(542\) −1.57734 −0.0677525
\(543\) 0 0
\(544\) 30.7850 1.31990
\(545\) 26.3334 1.12800
\(546\) 0 0
\(547\) 25.4806 1.08947 0.544737 0.838607i \(-0.316630\pi\)
0.544737 + 0.838607i \(0.316630\pi\)
\(548\) −0.281253 −0.0120145
\(549\) 0 0
\(550\) 2.10307 0.0896754
\(551\) 4.20884 0.179302
\(552\) 0 0
\(553\) −48.3410 −2.05567
\(554\) 2.38584 0.101364
\(555\) 0 0
\(556\) 24.4754 1.03799
\(557\) 3.73579 0.158291 0.0791453 0.996863i \(-0.474781\pi\)
0.0791453 + 0.996863i \(0.474781\pi\)
\(558\) 0 0
\(559\) 11.2109 0.474170
\(560\) 19.8055 0.836934
\(561\) 0 0
\(562\) 1.08659 0.0458351
\(563\) −25.6826 −1.08239 −0.541197 0.840896i \(-0.682029\pi\)
−0.541197 + 0.840896i \(0.682029\pi\)
\(564\) 0 0
\(565\) 40.0372 1.68438
\(566\) 6.68655 0.281057
\(567\) 0 0
\(568\) 11.6312 0.488034
\(569\) −15.4592 −0.648082 −0.324041 0.946043i \(-0.605042\pi\)
−0.324041 + 0.946043i \(0.605042\pi\)
\(570\) 0 0
\(571\) −22.9487 −0.960373 −0.480186 0.877166i \(-0.659431\pi\)
−0.480186 + 0.877166i \(0.659431\pi\)
\(572\) 34.1460 1.42772
\(573\) 0 0
\(574\) 3.57351 0.149156
\(575\) −1.09948 −0.0458513
\(576\) 0 0
\(577\) −8.10729 −0.337511 −0.168755 0.985658i \(-0.553975\pi\)
−0.168755 + 0.985658i \(0.553975\pi\)
\(578\) −11.8852 −0.494357
\(579\) 0 0
\(580\) 4.44829 0.184705
\(581\) 10.0850 0.418394
\(582\) 0 0
\(583\) 31.6544 1.31099
\(584\) 6.74649 0.279172
\(585\) 0 0
\(586\) −4.41568 −0.182410
\(587\) 21.2223 0.875938 0.437969 0.898990i \(-0.355698\pi\)
0.437969 + 0.898990i \(0.355698\pi\)
\(588\) 0 0
\(589\) 12.6411 0.520868
\(590\) 2.20065 0.0905994
\(591\) 0 0
\(592\) 1.14376 0.0470083
\(593\) 4.05015 0.166320 0.0831599 0.996536i \(-0.473499\pi\)
0.0831599 + 0.996536i \(0.473499\pi\)
\(594\) 0 0
\(595\) −45.9723 −1.88468
\(596\) 3.87962 0.158915
\(597\) 0 0
\(598\) 1.97098 0.0805994
\(599\) 35.6449 1.45641 0.728205 0.685359i \(-0.240355\pi\)
0.728205 + 0.685359i \(0.240355\pi\)
\(600\) 0 0
\(601\) −34.0859 −1.39039 −0.695196 0.718820i \(-0.744683\pi\)
−0.695196 + 0.718820i \(0.744683\pi\)
\(602\) 3.18687 0.129887
\(603\) 0 0
\(604\) −5.63924 −0.229457
\(605\) −18.2719 −0.742859
\(606\) 0 0
\(607\) 24.0259 0.975183 0.487592 0.873072i \(-0.337876\pi\)
0.487592 + 0.873072i \(0.337876\pi\)
\(608\) −19.6110 −0.795332
\(609\) 0 0
\(610\) 16.1443 0.653663
\(611\) 51.7392 2.09315
\(612\) 0 0
\(613\) −10.2325 −0.413285 −0.206643 0.978417i \(-0.566254\pi\)
−0.206643 + 0.978417i \(0.566254\pi\)
\(614\) −3.59652 −0.145144
\(615\) 0 0
\(616\) 20.4848 0.825356
\(617\) −21.8083 −0.877969 −0.438985 0.898495i \(-0.644662\pi\)
−0.438985 + 0.898495i \(0.644662\pi\)
\(618\) 0 0
\(619\) 3.64506 0.146507 0.0732537 0.997313i \(-0.476662\pi\)
0.0732537 + 0.997313i \(0.476662\pi\)
\(620\) 13.3603 0.536562
\(621\) 0 0
\(622\) −7.56319 −0.303256
\(623\) −20.6815 −0.828585
\(624\) 0 0
\(625\) −29.2885 −1.17154
\(626\) −13.5098 −0.539960
\(627\) 0 0
\(628\) −30.4305 −1.21431
\(629\) −2.65489 −0.105857
\(630\) 0 0
\(631\) 26.2926 1.04669 0.523346 0.852120i \(-0.324684\pi\)
0.523346 + 0.852120i \(0.324684\pi\)
\(632\) 29.0843 1.15691
\(633\) 0 0
\(634\) −9.52112 −0.378132
\(635\) −15.8621 −0.629467
\(636\) 0 0
\(637\) 4.14480 0.164223
\(638\) 1.91280 0.0757284
\(639\) 0 0
\(640\) −26.9964 −1.06713
\(641\) 0.983922 0.0388626 0.0194313 0.999811i \(-0.493814\pi\)
0.0194313 + 0.999811i \(0.493814\pi\)
\(642\) 0 0
\(643\) 4.65539 0.183591 0.0917953 0.995778i \(-0.470739\pi\)
0.0917953 + 0.995778i \(0.470739\pi\)
\(644\) −5.07453 −0.199964
\(645\) 0 0
\(646\) 12.4006 0.487894
\(647\) −28.6978 −1.12823 −0.564114 0.825697i \(-0.690782\pi\)
−0.564114 + 0.825697i \(0.690782\pi\)
\(648\) 0 0
\(649\) −8.57070 −0.336429
\(650\) −2.16705 −0.0849986
\(651\) 0 0
\(652\) 39.8297 1.55985
\(653\) −9.32695 −0.364992 −0.182496 0.983207i \(-0.558418\pi\)
−0.182496 + 0.983207i \(0.558418\pi\)
\(654\) 0 0
\(655\) −36.9781 −1.44486
\(656\) 8.09574 0.316086
\(657\) 0 0
\(658\) 14.7077 0.573365
\(659\) 27.8179 1.08363 0.541816 0.840497i \(-0.317737\pi\)
0.541816 + 0.840497i \(0.317737\pi\)
\(660\) 0 0
\(661\) −4.22069 −0.164166 −0.0820828 0.996626i \(-0.526157\pi\)
−0.0820828 + 0.996626i \(0.526157\pi\)
\(662\) −0.349786 −0.0135948
\(663\) 0 0
\(664\) −6.06760 −0.235468
\(665\) 29.2859 1.13566
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 5.31424 0.205614
\(669\) 0 0
\(670\) −13.6862 −0.528744
\(671\) −62.8758 −2.42729
\(672\) 0 0
\(673\) 20.7109 0.798348 0.399174 0.916875i \(-0.369297\pi\)
0.399174 + 0.916875i \(0.369297\pi\)
\(674\) 9.92271 0.382208
\(675\) 0 0
\(676\) −11.7700 −0.452691
\(677\) −32.4382 −1.24670 −0.623350 0.781943i \(-0.714229\pi\)
−0.623350 + 0.781943i \(0.714229\pi\)
\(678\) 0 0
\(679\) −21.3186 −0.818131
\(680\) 27.6592 1.06068
\(681\) 0 0
\(682\) 5.74502 0.219988
\(683\) 30.2595 1.15785 0.578923 0.815382i \(-0.303473\pi\)
0.578923 + 0.815382i \(0.303473\pi\)
\(684\) 0 0
\(685\) −0.385653 −0.0147350
\(686\) −7.61658 −0.290802
\(687\) 0 0
\(688\) 7.21982 0.275253
\(689\) −32.6173 −1.24262
\(690\) 0 0
\(691\) 32.4824 1.23569 0.617845 0.786300i \(-0.288006\pi\)
0.617845 + 0.786300i \(0.288006\pi\)
\(692\) 32.6965 1.24294
\(693\) 0 0
\(694\) 8.49545 0.322483
\(695\) 33.5605 1.27302
\(696\) 0 0
\(697\) −18.7918 −0.711789
\(698\) 5.84909 0.221391
\(699\) 0 0
\(700\) 5.57932 0.210879
\(701\) −7.93803 −0.299815 −0.149908 0.988700i \(-0.547898\pi\)
−0.149908 + 0.988700i \(0.547898\pi\)
\(702\) 0 0
\(703\) 1.69125 0.0637867
\(704\) 15.5053 0.584379
\(705\) 0 0
\(706\) 7.81936 0.294286
\(707\) −25.1676 −0.946524
\(708\) 0 0
\(709\) 26.4257 0.992438 0.496219 0.868197i \(-0.334721\pi\)
0.496219 + 0.868197i \(0.334721\pi\)
\(710\) 7.55714 0.283614
\(711\) 0 0
\(712\) 12.4430 0.466320
\(713\) −3.00347 −0.112481
\(714\) 0 0
\(715\) 46.8209 1.75100
\(716\) −18.3225 −0.684743
\(717\) 0 0
\(718\) 9.37298 0.349796
\(719\) 10.2899 0.383747 0.191873 0.981420i \(-0.438544\pi\)
0.191873 + 0.981420i \(0.438544\pi\)
\(720\) 0 0
\(721\) 48.0609 1.78988
\(722\) 0.573343 0.0213376
\(723\) 0 0
\(724\) −38.6759 −1.43738
\(725\) 1.09948 0.0408335
\(726\) 0 0
\(727\) −8.52204 −0.316065 −0.158033 0.987434i \(-0.550515\pi\)
−0.158033 + 0.987434i \(0.550515\pi\)
\(728\) −21.1079 −0.782312
\(729\) 0 0
\(730\) 4.38340 0.162237
\(731\) −16.7586 −0.619839
\(732\) 0 0
\(733\) −44.2153 −1.63313 −0.816566 0.577253i \(-0.804125\pi\)
−0.816566 + 0.577253i \(0.804125\pi\)
\(734\) −10.0665 −0.371562
\(735\) 0 0
\(736\) 4.65949 0.171751
\(737\) 53.3025 1.96342
\(738\) 0 0
\(739\) −12.5582 −0.461959 −0.230980 0.972959i \(-0.574193\pi\)
−0.230980 + 0.972959i \(0.574193\pi\)
\(740\) 1.78747 0.0657086
\(741\) 0 0
\(742\) −9.27197 −0.340385
\(743\) −22.9126 −0.840580 −0.420290 0.907390i \(-0.638072\pi\)
−0.420290 + 0.907390i \(0.638072\pi\)
\(744\) 0 0
\(745\) 5.31973 0.194900
\(746\) 12.7454 0.466643
\(747\) 0 0
\(748\) −51.0431 −1.86632
\(749\) 4.85226 0.177298
\(750\) 0 0
\(751\) −1.20310 −0.0439018 −0.0219509 0.999759i \(-0.506988\pi\)
−0.0219509 + 0.999759i \(0.506988\pi\)
\(752\) 33.3201 1.21506
\(753\) 0 0
\(754\) −1.97098 −0.0717789
\(755\) −7.73251 −0.281415
\(756\) 0 0
\(757\) 3.23336 0.117518 0.0587592 0.998272i \(-0.481286\pi\)
0.0587592 + 0.998272i \(0.481286\pi\)
\(758\) 3.72807 0.135410
\(759\) 0 0
\(760\) −17.6198 −0.639137
\(761\) 42.3343 1.53462 0.767308 0.641279i \(-0.221596\pi\)
0.767308 + 0.641279i \(0.221596\pi\)
\(762\) 0 0
\(763\) 30.0408 1.08755
\(764\) 39.1517 1.41646
\(765\) 0 0
\(766\) −0.745249 −0.0269269
\(767\) 8.83141 0.318884
\(768\) 0 0
\(769\) −28.5767 −1.03050 −0.515251 0.857040i \(-0.672301\pi\)
−0.515251 + 0.857040i \(0.672301\pi\)
\(770\) 13.3096 0.479644
\(771\) 0 0
\(772\) 10.4054 0.374497
\(773\) 14.8823 0.535281 0.267640 0.963519i \(-0.413756\pi\)
0.267640 + 0.963519i \(0.413756\pi\)
\(774\) 0 0
\(775\) 3.30224 0.118620
\(776\) 12.8263 0.460437
\(777\) 0 0
\(778\) −8.10409 −0.290546
\(779\) 11.9710 0.428904
\(780\) 0 0
\(781\) −29.4322 −1.05317
\(782\) −2.94632 −0.105360
\(783\) 0 0
\(784\) 2.66925 0.0953305
\(785\) −41.7262 −1.48927
\(786\) 0 0
\(787\) −7.61476 −0.271437 −0.135719 0.990747i \(-0.543334\pi\)
−0.135719 + 0.990747i \(0.543334\pi\)
\(788\) −30.4640 −1.08524
\(789\) 0 0
\(790\) 18.8969 0.672323
\(791\) 45.6738 1.62397
\(792\) 0 0
\(793\) 64.7884 2.30070
\(794\) −4.96599 −0.176236
\(795\) 0 0
\(796\) 27.8757 0.988029
\(797\) −31.1708 −1.10413 −0.552063 0.833803i \(-0.686159\pi\)
−0.552063 + 0.833803i \(0.686159\pi\)
\(798\) 0 0
\(799\) −77.3423 −2.73617
\(800\) −5.12299 −0.181125
\(801\) 0 0
\(802\) 16.0821 0.567880
\(803\) −17.0717 −0.602446
\(804\) 0 0
\(805\) −6.95818 −0.245244
\(806\) −5.91978 −0.208515
\(807\) 0 0
\(808\) 15.1420 0.532695
\(809\) 17.5131 0.615729 0.307864 0.951430i \(-0.400386\pi\)
0.307864 + 0.951430i \(0.400386\pi\)
\(810\) 0 0
\(811\) 25.4490 0.893634 0.446817 0.894625i \(-0.352558\pi\)
0.446817 + 0.894625i \(0.352558\pi\)
\(812\) 5.07453 0.178081
\(813\) 0 0
\(814\) 0.768625 0.0269403
\(815\) 54.6144 1.91306
\(816\) 0 0
\(817\) 10.6758 0.373497
\(818\) 2.27141 0.0794181
\(819\) 0 0
\(820\) 12.6520 0.441827
\(821\) −46.7718 −1.63235 −0.816174 0.577807i \(-0.803909\pi\)
−0.816174 + 0.577807i \(0.803909\pi\)
\(822\) 0 0
\(823\) −32.5789 −1.13563 −0.567814 0.823157i \(-0.692211\pi\)
−0.567814 + 0.823157i \(0.692211\pi\)
\(824\) −28.9157 −1.00733
\(825\) 0 0
\(826\) 2.51047 0.0873503
\(827\) 39.4266 1.37100 0.685499 0.728074i \(-0.259584\pi\)
0.685499 + 0.728074i \(0.259584\pi\)
\(828\) 0 0
\(829\) −17.3107 −0.601224 −0.300612 0.953746i \(-0.597191\pi\)
−0.300612 + 0.953746i \(0.597191\pi\)
\(830\) −3.94230 −0.136839
\(831\) 0 0
\(832\) −15.9770 −0.553902
\(833\) −6.19585 −0.214673
\(834\) 0 0
\(835\) 7.28686 0.252172
\(836\) 32.5161 1.12459
\(837\) 0 0
\(838\) −8.46825 −0.292531
\(839\) 44.8932 1.54988 0.774942 0.632032i \(-0.217779\pi\)
0.774942 + 0.632032i \(0.217779\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 11.5799 0.399070
\(843\) 0 0
\(844\) −10.5316 −0.362511
\(845\) −16.1389 −0.555196
\(846\) 0 0
\(847\) −20.8443 −0.716219
\(848\) −21.0055 −0.721332
\(849\) 0 0
\(850\) 3.23941 0.111111
\(851\) −0.401833 −0.0137747
\(852\) 0 0
\(853\) 10.6947 0.366180 0.183090 0.983096i \(-0.441390\pi\)
0.183090 + 0.983096i \(0.441390\pi\)
\(854\) 18.4171 0.630221
\(855\) 0 0
\(856\) −2.91936 −0.0997815
\(857\) 9.25319 0.316083 0.158042 0.987432i \(-0.449482\pi\)
0.158042 + 0.987432i \(0.449482\pi\)
\(858\) 0 0
\(859\) 0.609581 0.0207986 0.0103993 0.999946i \(-0.496690\pi\)
0.0103993 + 0.999946i \(0.496690\pi\)
\(860\) 11.2831 0.384751
\(861\) 0 0
\(862\) −9.11587 −0.310488
\(863\) −39.0869 −1.33053 −0.665267 0.746605i \(-0.731682\pi\)
−0.665267 + 0.746605i \(0.731682\pi\)
\(864\) 0 0
\(865\) 44.8334 1.52438
\(866\) −1.08371 −0.0368260
\(867\) 0 0
\(868\) 15.2412 0.517319
\(869\) −73.5963 −2.49659
\(870\) 0 0
\(871\) −54.9239 −1.86103
\(872\) −18.0740 −0.612062
\(873\) 0 0
\(874\) 1.87690 0.0634871
\(875\) −27.1406 −0.917518
\(876\) 0 0
\(877\) 26.4590 0.893456 0.446728 0.894670i \(-0.352589\pi\)
0.446728 + 0.894670i \(0.352589\pi\)
\(878\) −4.04994 −0.136679
\(879\) 0 0
\(880\) 30.1527 1.01645
\(881\) −14.3959 −0.485009 −0.242504 0.970150i \(-0.577969\pi\)
−0.242504 + 0.970150i \(0.577969\pi\)
\(882\) 0 0
\(883\) −31.2893 −1.05297 −0.526485 0.850184i \(-0.676490\pi\)
−0.526485 + 0.850184i \(0.676490\pi\)
\(884\) 52.5958 1.76899
\(885\) 0 0
\(886\) 11.7282 0.394017
\(887\) 2.94486 0.0988786 0.0494393 0.998777i \(-0.484257\pi\)
0.0494393 + 0.998777i \(0.484257\pi\)
\(888\) 0 0
\(889\) −18.0952 −0.606893
\(890\) 8.08456 0.270995
\(891\) 0 0
\(892\) 41.0242 1.37359
\(893\) 49.2695 1.64874
\(894\) 0 0
\(895\) −25.1237 −0.839794
\(896\) −30.7970 −1.02886
\(897\) 0 0
\(898\) −9.39418 −0.313488
\(899\) 3.00347 0.100171
\(900\) 0 0
\(901\) 48.7578 1.62436
\(902\) 5.44046 0.181148
\(903\) 0 0
\(904\) −27.4796 −0.913956
\(905\) −53.0323 −1.76285
\(906\) 0 0
\(907\) 43.2269 1.43533 0.717663 0.696390i \(-0.245212\pi\)
0.717663 + 0.696390i \(0.245212\pi\)
\(908\) 8.78230 0.291451
\(909\) 0 0
\(910\) −13.7144 −0.454630
\(911\) −7.76302 −0.257200 −0.128600 0.991697i \(-0.541048\pi\)
−0.128600 + 0.991697i \(0.541048\pi\)
\(912\) 0 0
\(913\) 15.3538 0.508135
\(914\) 5.22343 0.172776
\(915\) 0 0
\(916\) 19.7514 0.652606
\(917\) −42.1840 −1.39304
\(918\) 0 0
\(919\) −16.8327 −0.555258 −0.277629 0.960688i \(-0.589549\pi\)
−0.277629 + 0.960688i \(0.589549\pi\)
\(920\) 4.18638 0.138021
\(921\) 0 0
\(922\) −5.23137 −0.172286
\(923\) 30.3275 0.998241
\(924\) 0 0
\(925\) 0.441806 0.0145265
\(926\) 4.88460 0.160518
\(927\) 0 0
\(928\) −4.65949 −0.152955
\(929\) −3.48943 −0.114485 −0.0572423 0.998360i \(-0.518231\pi\)
−0.0572423 + 0.998360i \(0.518231\pi\)
\(930\) 0 0
\(931\) 3.94695 0.129356
\(932\) 52.3057 1.71333
\(933\) 0 0
\(934\) −1.44323 −0.0472239
\(935\) −69.9902 −2.28892
\(936\) 0 0
\(937\) −12.1417 −0.396651 −0.198325 0.980136i \(-0.563550\pi\)
−0.198325 + 0.980136i \(0.563550\pi\)
\(938\) −15.6130 −0.509782
\(939\) 0 0
\(940\) 52.0725 1.69842
\(941\) 0.794187 0.0258897 0.0129449 0.999916i \(-0.495879\pi\)
0.0129449 + 0.999916i \(0.495879\pi\)
\(942\) 0 0
\(943\) −2.84424 −0.0926214
\(944\) 5.68743 0.185110
\(945\) 0 0
\(946\) 4.85183 0.157747
\(947\) 4.85051 0.157620 0.0788102 0.996890i \(-0.474888\pi\)
0.0788102 + 0.996890i \(0.474888\pi\)
\(948\) 0 0
\(949\) 17.5910 0.571027
\(950\) −2.06361 −0.0669522
\(951\) 0 0
\(952\) 31.5531 1.02264
\(953\) −45.0348 −1.45882 −0.729410 0.684077i \(-0.760205\pi\)
−0.729410 + 0.684077i \(0.760205\pi\)
\(954\) 0 0
\(955\) 53.6847 1.73720
\(956\) 39.6532 1.28248
\(957\) 0 0
\(958\) −8.51899 −0.275236
\(959\) −0.439946 −0.0142066
\(960\) 0 0
\(961\) −21.9792 −0.709006
\(962\) −0.792006 −0.0255353
\(963\) 0 0
\(964\) 45.4204 1.46289
\(965\) 14.2678 0.459297
\(966\) 0 0
\(967\) 12.2318 0.393348 0.196674 0.980469i \(-0.436986\pi\)
0.196674 + 0.980469i \(0.436986\pi\)
\(968\) 12.5409 0.403081
\(969\) 0 0
\(970\) 8.33362 0.267576
\(971\) −13.2141 −0.424062 −0.212031 0.977263i \(-0.568008\pi\)
−0.212031 + 0.977263i \(0.568008\pi\)
\(972\) 0 0
\(973\) 38.2853 1.22737
\(974\) 4.61134 0.147757
\(975\) 0 0
\(976\) 41.7238 1.33554
\(977\) 34.4548 1.10231 0.551153 0.834404i \(-0.314188\pi\)
0.551153 + 0.834404i \(0.314188\pi\)
\(978\) 0 0
\(979\) −31.4863 −1.00631
\(980\) 4.17150 0.133254
\(981\) 0 0
\(982\) −2.48607 −0.0793338
\(983\) −6.12679 −0.195414 −0.0977071 0.995215i \(-0.531151\pi\)
−0.0977071 + 0.995215i \(0.531151\pi\)
\(984\) 0 0
\(985\) −41.7722 −1.33097
\(986\) 2.94632 0.0938299
\(987\) 0 0
\(988\) −33.5052 −1.06594
\(989\) −2.53651 −0.0806563
\(990\) 0 0
\(991\) −51.7917 −1.64522 −0.822609 0.568608i \(-0.807482\pi\)
−0.822609 + 0.568608i \(0.807482\pi\)
\(992\) −13.9946 −0.444329
\(993\) 0 0
\(994\) 8.62106 0.273443
\(995\) 38.2231 1.21175
\(996\) 0 0
\(997\) 40.0083 1.26708 0.633538 0.773711i \(-0.281602\pi\)
0.633538 + 0.773711i \(0.281602\pi\)
\(998\) 7.04132 0.222889
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6003.2.a.q.1.8 16
3.2 odd 2 667.2.a.d.1.9 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
667.2.a.d.1.9 16 3.2 odd 2
6003.2.a.q.1.8 16 1.1 even 1 trivial