Properties

Label 6003.2.a.i.1.4
Level $6003$
Weight $2$
Character 6003.1
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 9x^{5} + 10x^{4} + 19x^{3} - 20x^{2} - 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2001)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.64802\) 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)
 
\(f(q)\) \(=\) \(q+0.865893 q^{2} -1.25023 q^{4} -1.66575 q^{5} -0.715958 q^{7} -2.81435 q^{8} +O(q^{10})\) \(q+0.865893 q^{2} -1.25023 q^{4} -1.66575 q^{5} -0.715958 q^{7} -2.81435 q^{8} -1.44236 q^{10} +0.656890 q^{11} -4.37251 q^{13} -0.619943 q^{14} +0.0635329 q^{16} +7.15038 q^{17} +6.56857 q^{19} +2.08256 q^{20} +0.568796 q^{22} +1.00000 q^{23} -2.22529 q^{25} -3.78613 q^{26} +0.895112 q^{28} -1.00000 q^{29} +0.705852 q^{31} +5.68371 q^{32} +6.19146 q^{34} +1.19260 q^{35} -7.06983 q^{37} +5.68768 q^{38} +4.68799 q^{40} +12.6781 q^{41} -2.05769 q^{43} -0.821263 q^{44} +0.865893 q^{46} -8.50359 q^{47} -6.48740 q^{49} -1.92686 q^{50} +5.46665 q^{52} -3.04227 q^{53} -1.09421 q^{55} +2.01496 q^{56} -0.865893 q^{58} +4.81329 q^{59} +2.51798 q^{61} +0.611192 q^{62} +4.79442 q^{64} +7.28350 q^{65} -2.81077 q^{67} -8.93961 q^{68} +1.03267 q^{70} -0.253574 q^{71} +13.1983 q^{73} -6.12171 q^{74} -8.21222 q^{76} -0.470305 q^{77} +0.611697 q^{79} -0.105830 q^{80} +10.9778 q^{82} -2.06729 q^{83} -11.9107 q^{85} -1.78174 q^{86} -1.84872 q^{88} +12.6267 q^{89} +3.13054 q^{91} -1.25023 q^{92} -7.36320 q^{94} -10.9416 q^{95} -16.3343 q^{97} -5.61740 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8} - 11 q^{10} + 12 q^{11} - 13 q^{13} + 3 q^{14} - 13 q^{16} + 12 q^{17} - 5 q^{19} + 8 q^{20} - q^{22} + 7 q^{23} - 4 q^{25} - 2 q^{26} - 21 q^{28} - 7 q^{29} - 8 q^{31} + 5 q^{32} - 28 q^{34} - 5 q^{35} - 24 q^{37} + 6 q^{38} - 20 q^{40} - 9 q^{41} - q^{43} + 23 q^{44} + q^{46} - 27 q^{47} - 14 q^{49} - 7 q^{50} - 9 q^{52} + q^{53} - 11 q^{55} + 20 q^{56} - q^{58} - 8 q^{59} + q^{61} + 3 q^{64} - 12 q^{65} - 16 q^{67} - 15 q^{68} + 40 q^{70} + 13 q^{71} - 23 q^{73} + 8 q^{74} - 2 q^{76} - 13 q^{77} - 44 q^{79} - 30 q^{80} - 10 q^{82} - 21 q^{83} - 6 q^{86} + 21 q^{88} + 5 q^{89} - 18 q^{91} + 7 q^{92} + 28 q^{94} - 9 q^{95} - 55 q^{97} - 36 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.865893 0.612279 0.306139 0.951987i \(-0.400963\pi\)
0.306139 + 0.951987i \(0.400963\pi\)
\(3\) 0 0
\(4\) −1.25023 −0.625115
\(5\) −1.66575 −0.744944 −0.372472 0.928043i \(-0.621490\pi\)
−0.372472 + 0.928043i \(0.621490\pi\)
\(6\) 0 0
\(7\) −0.715958 −0.270607 −0.135303 0.990804i \(-0.543201\pi\)
−0.135303 + 0.990804i \(0.543201\pi\)
\(8\) −2.81435 −0.995023
\(9\) 0 0
\(10\) −1.44236 −0.456114
\(11\) 0.656890 0.198060 0.0990298 0.995084i \(-0.468426\pi\)
0.0990298 + 0.995084i \(0.468426\pi\)
\(12\) 0 0
\(13\) −4.37251 −1.21272 −0.606359 0.795191i \(-0.707370\pi\)
−0.606359 + 0.795191i \(0.707370\pi\)
\(14\) −0.619943 −0.165687
\(15\) 0 0
\(16\) 0.0635329 0.0158832
\(17\) 7.15038 1.73422 0.867111 0.498116i \(-0.165975\pi\)
0.867111 + 0.498116i \(0.165975\pi\)
\(18\) 0 0
\(19\) 6.56857 1.50693 0.753467 0.657486i \(-0.228380\pi\)
0.753467 + 0.657486i \(0.228380\pi\)
\(20\) 2.08256 0.465676
\(21\) 0 0
\(22\) 0.568796 0.121268
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −2.22529 −0.445058
\(26\) −3.78613 −0.742521
\(27\) 0 0
\(28\) 0.895112 0.169160
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 0.705852 0.126775 0.0633874 0.997989i \(-0.479810\pi\)
0.0633874 + 0.997989i \(0.479810\pi\)
\(32\) 5.68371 1.00475
\(33\) 0 0
\(34\) 6.19146 1.06183
\(35\) 1.19260 0.201587
\(36\) 0 0
\(37\) −7.06983 −1.16227 −0.581136 0.813806i \(-0.697392\pi\)
−0.581136 + 0.813806i \(0.697392\pi\)
\(38\) 5.68768 0.922663
\(39\) 0 0
\(40\) 4.68799 0.741237
\(41\) 12.6781 1.97998 0.989990 0.141139i \(-0.0450763\pi\)
0.989990 + 0.141139i \(0.0450763\pi\)
\(42\) 0 0
\(43\) −2.05769 −0.313794 −0.156897 0.987615i \(-0.550149\pi\)
−0.156897 + 0.987615i \(0.550149\pi\)
\(44\) −0.821263 −0.123810
\(45\) 0 0
\(46\) 0.865893 0.127669
\(47\) −8.50359 −1.24038 −0.620188 0.784453i \(-0.712944\pi\)
−0.620188 + 0.784453i \(0.712944\pi\)
\(48\) 0 0
\(49\) −6.48740 −0.926772
\(50\) −1.92686 −0.272500
\(51\) 0 0
\(52\) 5.46665 0.758087
\(53\) −3.04227 −0.417888 −0.208944 0.977928i \(-0.567003\pi\)
−0.208944 + 0.977928i \(0.567003\pi\)
\(54\) 0 0
\(55\) −1.09421 −0.147543
\(56\) 2.01496 0.269260
\(57\) 0 0
\(58\) −0.865893 −0.113697
\(59\) 4.81329 0.626637 0.313318 0.949648i \(-0.398559\pi\)
0.313318 + 0.949648i \(0.398559\pi\)
\(60\) 0 0
\(61\) 2.51798 0.322395 0.161197 0.986922i \(-0.448464\pi\)
0.161197 + 0.986922i \(0.448464\pi\)
\(62\) 0.611192 0.0776215
\(63\) 0 0
\(64\) 4.79442 0.599303
\(65\) 7.28350 0.903407
\(66\) 0 0
\(67\) −2.81077 −0.343390 −0.171695 0.985150i \(-0.554924\pi\)
−0.171695 + 0.985150i \(0.554924\pi\)
\(68\) −8.93961 −1.08409
\(69\) 0 0
\(70\) 1.03267 0.123427
\(71\) −0.253574 −0.0300937 −0.0150469 0.999887i \(-0.504790\pi\)
−0.0150469 + 0.999887i \(0.504790\pi\)
\(72\) 0 0
\(73\) 13.1983 1.54475 0.772374 0.635168i \(-0.219069\pi\)
0.772374 + 0.635168i \(0.219069\pi\)
\(74\) −6.12171 −0.711635
\(75\) 0 0
\(76\) −8.21222 −0.942006
\(77\) −0.470305 −0.0535963
\(78\) 0 0
\(79\) 0.611697 0.0688213 0.0344106 0.999408i \(-0.489045\pi\)
0.0344106 + 0.999408i \(0.489045\pi\)
\(80\) −0.105830 −0.0118321
\(81\) 0 0
\(82\) 10.9778 1.21230
\(83\) −2.06729 −0.226915 −0.113457 0.993543i \(-0.536193\pi\)
−0.113457 + 0.993543i \(0.536193\pi\)
\(84\) 0 0
\(85\) −11.9107 −1.29190
\(86\) −1.78174 −0.192130
\(87\) 0 0
\(88\) −1.84872 −0.197074
\(89\) 12.6267 1.33842 0.669212 0.743071i \(-0.266632\pi\)
0.669212 + 0.743071i \(0.266632\pi\)
\(90\) 0 0
\(91\) 3.13054 0.328169
\(92\) −1.25023 −0.130345
\(93\) 0 0
\(94\) −7.36320 −0.759456
\(95\) −10.9416 −1.12258
\(96\) 0 0
\(97\) −16.3343 −1.65850 −0.829248 0.558881i \(-0.811231\pi\)
−0.829248 + 0.558881i \(0.811231\pi\)
\(98\) −5.61740 −0.567443
\(99\) 0 0
\(100\) 2.78212 0.278212
\(101\) 0.174650 0.0173784 0.00868919 0.999962i \(-0.497234\pi\)
0.00868919 + 0.999962i \(0.497234\pi\)
\(102\) 0 0
\(103\) 1.40140 0.138084 0.0690419 0.997614i \(-0.478006\pi\)
0.0690419 + 0.997614i \(0.478006\pi\)
\(104\) 12.3058 1.20668
\(105\) 0 0
\(106\) −2.63428 −0.255864
\(107\) −8.77557 −0.848366 −0.424183 0.905576i \(-0.639439\pi\)
−0.424183 + 0.905576i \(0.639439\pi\)
\(108\) 0 0
\(109\) −9.29618 −0.890413 −0.445206 0.895428i \(-0.646870\pi\)
−0.445206 + 0.895428i \(0.646870\pi\)
\(110\) −0.947470 −0.0903377
\(111\) 0 0
\(112\) −0.0454869 −0.00429810
\(113\) −11.1470 −1.04862 −0.524312 0.851527i \(-0.675677\pi\)
−0.524312 + 0.851527i \(0.675677\pi\)
\(114\) 0 0
\(115\) −1.66575 −0.155332
\(116\) 1.25023 0.116081
\(117\) 0 0
\(118\) 4.16779 0.383676
\(119\) −5.11937 −0.469292
\(120\) 0 0
\(121\) −10.5685 −0.960772
\(122\) 2.18031 0.197396
\(123\) 0 0
\(124\) −0.882477 −0.0792488
\(125\) 12.0355 1.07649
\(126\) 0 0
\(127\) −21.2129 −1.88234 −0.941171 0.337932i \(-0.890273\pi\)
−0.941171 + 0.337932i \(0.890273\pi\)
\(128\) −7.21597 −0.637808
\(129\) 0 0
\(130\) 6.30673 0.553137
\(131\) 19.0666 1.66586 0.832928 0.553381i \(-0.186663\pi\)
0.832928 + 0.553381i \(0.186663\pi\)
\(132\) 0 0
\(133\) −4.70282 −0.407786
\(134\) −2.43382 −0.210250
\(135\) 0 0
\(136\) −20.1237 −1.72559
\(137\) −11.5218 −0.984370 −0.492185 0.870491i \(-0.663802\pi\)
−0.492185 + 0.870491i \(0.663802\pi\)
\(138\) 0 0
\(139\) 0.933345 0.0791653 0.0395826 0.999216i \(-0.487397\pi\)
0.0395826 + 0.999216i \(0.487397\pi\)
\(140\) −1.49103 −0.126015
\(141\) 0 0
\(142\) −0.219568 −0.0184257
\(143\) −2.87226 −0.240190
\(144\) 0 0
\(145\) 1.66575 0.138333
\(146\) 11.4283 0.945816
\(147\) 0 0
\(148\) 8.83891 0.726554
\(149\) −9.85776 −0.807579 −0.403790 0.914852i \(-0.632307\pi\)
−0.403790 + 0.914852i \(0.632307\pi\)
\(150\) 0 0
\(151\) −14.4385 −1.17499 −0.587495 0.809228i \(-0.699886\pi\)
−0.587495 + 0.809228i \(0.699886\pi\)
\(152\) −18.4863 −1.49943
\(153\) 0 0
\(154\) −0.407234 −0.0328159
\(155\) −1.17577 −0.0944402
\(156\) 0 0
\(157\) −5.07449 −0.404988 −0.202494 0.979283i \(-0.564905\pi\)
−0.202494 + 0.979283i \(0.564905\pi\)
\(158\) 0.529664 0.0421378
\(159\) 0 0
\(160\) −9.46762 −0.748481
\(161\) −0.715958 −0.0564254
\(162\) 0 0
\(163\) −9.44012 −0.739407 −0.369704 0.929150i \(-0.620541\pi\)
−0.369704 + 0.929150i \(0.620541\pi\)
\(164\) −15.8505 −1.23771
\(165\) 0 0
\(166\) −1.79005 −0.138935
\(167\) −20.7134 −1.60285 −0.801426 0.598094i \(-0.795925\pi\)
−0.801426 + 0.598094i \(0.795925\pi\)
\(168\) 0 0
\(169\) 6.11888 0.470683
\(170\) −10.3134 −0.791002
\(171\) 0 0
\(172\) 2.57258 0.196158
\(173\) −4.97238 −0.378043 −0.189022 0.981973i \(-0.560532\pi\)
−0.189022 + 0.981973i \(0.560532\pi\)
\(174\) 0 0
\(175\) 1.59321 0.120436
\(176\) 0.0417341 0.00314582
\(177\) 0 0
\(178\) 10.9333 0.819489
\(179\) 21.5558 1.61116 0.805578 0.592490i \(-0.201855\pi\)
0.805578 + 0.592490i \(0.201855\pi\)
\(180\) 0 0
\(181\) −20.0768 −1.49230 −0.746150 0.665778i \(-0.768100\pi\)
−0.746150 + 0.665778i \(0.768100\pi\)
\(182\) 2.71071 0.200931
\(183\) 0 0
\(184\) −2.81435 −0.207477
\(185\) 11.7765 0.865828
\(186\) 0 0
\(187\) 4.69701 0.343479
\(188\) 10.6314 0.775378
\(189\) 0 0
\(190\) −9.47423 −0.687333
\(191\) −0.833088 −0.0602801 −0.0301401 0.999546i \(-0.509595\pi\)
−0.0301401 + 0.999546i \(0.509595\pi\)
\(192\) 0 0
\(193\) 10.9407 0.787532 0.393766 0.919211i \(-0.371172\pi\)
0.393766 + 0.919211i \(0.371172\pi\)
\(194\) −14.1437 −1.01546
\(195\) 0 0
\(196\) 8.11074 0.579339
\(197\) −0.438846 −0.0312665 −0.0156332 0.999878i \(-0.504976\pi\)
−0.0156332 + 0.999878i \(0.504976\pi\)
\(198\) 0 0
\(199\) −5.39837 −0.382680 −0.191340 0.981524i \(-0.561283\pi\)
−0.191340 + 0.981524i \(0.561283\pi\)
\(200\) 6.26275 0.442843
\(201\) 0 0
\(202\) 0.151229 0.0106404
\(203\) 0.715958 0.0502504
\(204\) 0 0
\(205\) −21.1184 −1.47497
\(206\) 1.21346 0.0845457
\(207\) 0 0
\(208\) −0.277798 −0.0192618
\(209\) 4.31482 0.298463
\(210\) 0 0
\(211\) −11.8593 −0.816425 −0.408212 0.912887i \(-0.633848\pi\)
−0.408212 + 0.912887i \(0.633848\pi\)
\(212\) 3.80354 0.261228
\(213\) 0 0
\(214\) −7.59870 −0.519436
\(215\) 3.42759 0.233759
\(216\) 0 0
\(217\) −0.505360 −0.0343061
\(218\) −8.04950 −0.545181
\(219\) 0 0
\(220\) 1.36802 0.0922316
\(221\) −31.2651 −2.10312
\(222\) 0 0
\(223\) −17.6698 −1.18325 −0.591627 0.806212i \(-0.701514\pi\)
−0.591627 + 0.806212i \(0.701514\pi\)
\(224\) −4.06930 −0.271892
\(225\) 0 0
\(226\) −9.65212 −0.642050
\(227\) −19.2183 −1.27556 −0.637782 0.770217i \(-0.720148\pi\)
−0.637782 + 0.770217i \(0.720148\pi\)
\(228\) 0 0
\(229\) 0.0207973 0.00137433 0.000687163 1.00000i \(-0.499781\pi\)
0.000687163 1.00000i \(0.499781\pi\)
\(230\) −1.44236 −0.0951062
\(231\) 0 0
\(232\) 2.81435 0.184771
\(233\) −10.5644 −0.692095 −0.346048 0.938217i \(-0.612476\pi\)
−0.346048 + 0.938217i \(0.612476\pi\)
\(234\) 0 0
\(235\) 14.1648 0.924011
\(236\) −6.01771 −0.391720
\(237\) 0 0
\(238\) −4.43283 −0.287337
\(239\) −1.89477 −0.122562 −0.0612812 0.998121i \(-0.519519\pi\)
−0.0612812 + 0.998121i \(0.519519\pi\)
\(240\) 0 0
\(241\) 2.61371 0.168364 0.0841820 0.996450i \(-0.473172\pi\)
0.0841820 + 0.996450i \(0.473172\pi\)
\(242\) −9.15119 −0.588260
\(243\) 0 0
\(244\) −3.14806 −0.201534
\(245\) 10.8064 0.690394
\(246\) 0 0
\(247\) −28.7212 −1.82748
\(248\) −1.98652 −0.126144
\(249\) 0 0
\(250\) 10.4215 0.659111
\(251\) 14.8239 0.935674 0.467837 0.883815i \(-0.345033\pi\)
0.467837 + 0.883815i \(0.345033\pi\)
\(252\) 0 0
\(253\) 0.656890 0.0412983
\(254\) −18.3681 −1.15252
\(255\) 0 0
\(256\) −15.8371 −0.989819
\(257\) −6.92245 −0.431810 −0.215905 0.976414i \(-0.569270\pi\)
−0.215905 + 0.976414i \(0.569270\pi\)
\(258\) 0 0
\(259\) 5.06170 0.314519
\(260\) −9.10604 −0.564733
\(261\) 0 0
\(262\) 16.5096 1.01997
\(263\) 0.368207 0.0227046 0.0113523 0.999936i \(-0.496386\pi\)
0.0113523 + 0.999936i \(0.496386\pi\)
\(264\) 0 0
\(265\) 5.06766 0.311304
\(266\) −4.07214 −0.249679
\(267\) 0 0
\(268\) 3.51410 0.214658
\(269\) 7.66062 0.467076 0.233538 0.972348i \(-0.424970\pi\)
0.233538 + 0.972348i \(0.424970\pi\)
\(270\) 0 0
\(271\) 20.2402 1.22951 0.614753 0.788719i \(-0.289256\pi\)
0.614753 + 0.788719i \(0.289256\pi\)
\(272\) 0.454284 0.0275450
\(273\) 0 0
\(274\) −9.97660 −0.602709
\(275\) −1.46177 −0.0881480
\(276\) 0 0
\(277\) 31.4385 1.88896 0.944479 0.328572i \(-0.106567\pi\)
0.944479 + 0.328572i \(0.106567\pi\)
\(278\) 0.808176 0.0484712
\(279\) 0 0
\(280\) −3.35641 −0.200584
\(281\) 2.99818 0.178856 0.0894281 0.995993i \(-0.471496\pi\)
0.0894281 + 0.995993i \(0.471496\pi\)
\(282\) 0 0
\(283\) −25.4423 −1.51239 −0.756193 0.654348i \(-0.772943\pi\)
−0.756193 + 0.654348i \(0.772943\pi\)
\(284\) 0.317026 0.0188120
\(285\) 0 0
\(286\) −2.48707 −0.147063
\(287\) −9.07695 −0.535796
\(288\) 0 0
\(289\) 34.1279 2.00752
\(290\) 1.44236 0.0846982
\(291\) 0 0
\(292\) −16.5009 −0.965645
\(293\) 24.9329 1.45659 0.728297 0.685261i \(-0.240312\pi\)
0.728297 + 0.685261i \(0.240312\pi\)
\(294\) 0 0
\(295\) −8.01771 −0.466809
\(296\) 19.8970 1.15649
\(297\) 0 0
\(298\) −8.53577 −0.494464
\(299\) −4.37251 −0.252869
\(300\) 0 0
\(301\) 1.47322 0.0849149
\(302\) −12.5022 −0.719421
\(303\) 0 0
\(304\) 0.417320 0.0239349
\(305\) −4.19432 −0.240166
\(306\) 0 0
\(307\) 7.48358 0.427111 0.213555 0.976931i \(-0.431496\pi\)
0.213555 + 0.976931i \(0.431496\pi\)
\(308\) 0.587990 0.0335038
\(309\) 0 0
\(310\) −1.01809 −0.0578237
\(311\) −7.32248 −0.415220 −0.207610 0.978212i \(-0.566568\pi\)
−0.207610 + 0.978212i \(0.566568\pi\)
\(312\) 0 0
\(313\) 32.0436 1.81121 0.905605 0.424122i \(-0.139417\pi\)
0.905605 + 0.424122i \(0.139417\pi\)
\(314\) −4.39396 −0.247966
\(315\) 0 0
\(316\) −0.764761 −0.0430212
\(317\) −17.6048 −0.988784 −0.494392 0.869239i \(-0.664609\pi\)
−0.494392 + 0.869239i \(0.664609\pi\)
\(318\) 0 0
\(319\) −0.656890 −0.0367788
\(320\) −7.98629 −0.446447
\(321\) 0 0
\(322\) −0.619943 −0.0345481
\(323\) 46.9677 2.61336
\(324\) 0 0
\(325\) 9.73011 0.539730
\(326\) −8.17413 −0.452723
\(327\) 0 0
\(328\) −35.6805 −1.97013
\(329\) 6.08821 0.335654
\(330\) 0 0
\(331\) −21.2917 −1.17030 −0.585148 0.810927i \(-0.698963\pi\)
−0.585148 + 0.810927i \(0.698963\pi\)
\(332\) 2.58459 0.141848
\(333\) 0 0
\(334\) −17.9356 −0.981392
\(335\) 4.68203 0.255806
\(336\) 0 0
\(337\) 15.5478 0.846941 0.423470 0.905910i \(-0.360812\pi\)
0.423470 + 0.905910i \(0.360812\pi\)
\(338\) 5.29830 0.288189
\(339\) 0 0
\(340\) 14.8911 0.807585
\(341\) 0.463667 0.0251090
\(342\) 0 0
\(343\) 9.65641 0.521397
\(344\) 5.79105 0.312233
\(345\) 0 0
\(346\) −4.30555 −0.231468
\(347\) −12.8576 −0.690230 −0.345115 0.938560i \(-0.612160\pi\)
−0.345115 + 0.938560i \(0.612160\pi\)
\(348\) 0 0
\(349\) 24.0074 1.28509 0.642544 0.766249i \(-0.277879\pi\)
0.642544 + 0.766249i \(0.277879\pi\)
\(350\) 1.37955 0.0737402
\(351\) 0 0
\(352\) 3.73357 0.199000
\(353\) 9.74926 0.518901 0.259450 0.965756i \(-0.416459\pi\)
0.259450 + 0.965756i \(0.416459\pi\)
\(354\) 0 0
\(355\) 0.422390 0.0224181
\(356\) −15.7862 −0.836669
\(357\) 0 0
\(358\) 18.6650 0.986477
\(359\) 3.66746 0.193561 0.0967806 0.995306i \(-0.469145\pi\)
0.0967806 + 0.995306i \(0.469145\pi\)
\(360\) 0 0
\(361\) 24.1461 1.27085
\(362\) −17.3844 −0.913703
\(363\) 0 0
\(364\) −3.91389 −0.205144
\(365\) −21.9851 −1.15075
\(366\) 0 0
\(367\) 2.11364 0.110331 0.0551657 0.998477i \(-0.482431\pi\)
0.0551657 + 0.998477i \(0.482431\pi\)
\(368\) 0.0635329 0.00331188
\(369\) 0 0
\(370\) 10.1972 0.530128
\(371\) 2.17814 0.113083
\(372\) 0 0
\(373\) −11.8307 −0.612568 −0.306284 0.951940i \(-0.599086\pi\)
−0.306284 + 0.951940i \(0.599086\pi\)
\(374\) 4.06711 0.210305
\(375\) 0 0
\(376\) 23.9321 1.23420
\(377\) 4.37251 0.225196
\(378\) 0 0
\(379\) −12.2695 −0.630239 −0.315120 0.949052i \(-0.602045\pi\)
−0.315120 + 0.949052i \(0.602045\pi\)
\(380\) 13.6795 0.701742
\(381\) 0 0
\(382\) −0.721365 −0.0369082
\(383\) 18.8237 0.961846 0.480923 0.876763i \(-0.340302\pi\)
0.480923 + 0.876763i \(0.340302\pi\)
\(384\) 0 0
\(385\) 0.783409 0.0399262
\(386\) 9.47351 0.482189
\(387\) 0 0
\(388\) 20.4216 1.03675
\(389\) 0.609090 0.0308821 0.0154410 0.999881i \(-0.495085\pi\)
0.0154410 + 0.999881i \(0.495085\pi\)
\(390\) 0 0
\(391\) 7.15038 0.361610
\(392\) 18.2578 0.922160
\(393\) 0 0
\(394\) −0.379994 −0.0191438
\(395\) −1.01893 −0.0512680
\(396\) 0 0
\(397\) −8.50375 −0.426791 −0.213396 0.976966i \(-0.568452\pi\)
−0.213396 + 0.976966i \(0.568452\pi\)
\(398\) −4.67441 −0.234307
\(399\) 0 0
\(400\) −0.141379 −0.00706895
\(401\) −38.0500 −1.90012 −0.950062 0.312060i \(-0.898981\pi\)
−0.950062 + 0.312060i \(0.898981\pi\)
\(402\) 0 0
\(403\) −3.08635 −0.153742
\(404\) −0.218353 −0.0108635
\(405\) 0 0
\(406\) 0.619943 0.0307672
\(407\) −4.64410 −0.230199
\(408\) 0 0
\(409\) 15.1451 0.748878 0.374439 0.927251i \(-0.377835\pi\)
0.374439 + 0.927251i \(0.377835\pi\)
\(410\) −18.2863 −0.903096
\(411\) 0 0
\(412\) −1.75207 −0.0863182
\(413\) −3.44611 −0.169572
\(414\) 0 0
\(415\) 3.44358 0.169039
\(416\) −24.8521 −1.21848
\(417\) 0 0
\(418\) 3.73618 0.182742
\(419\) −19.6534 −0.960133 −0.480067 0.877232i \(-0.659388\pi\)
−0.480067 + 0.877232i \(0.659388\pi\)
\(420\) 0 0
\(421\) −26.7076 −1.30165 −0.650824 0.759228i \(-0.725577\pi\)
−0.650824 + 0.759228i \(0.725577\pi\)
\(422\) −10.2688 −0.499879
\(423\) 0 0
\(424\) 8.56202 0.415809
\(425\) −15.9117 −0.771829
\(426\) 0 0
\(427\) −1.80277 −0.0872422
\(428\) 10.9715 0.530326
\(429\) 0 0
\(430\) 2.96792 0.143126
\(431\) 33.5313 1.61514 0.807572 0.589768i \(-0.200781\pi\)
0.807572 + 0.589768i \(0.200781\pi\)
\(432\) 0 0
\(433\) 24.5137 1.17805 0.589026 0.808114i \(-0.299512\pi\)
0.589026 + 0.808114i \(0.299512\pi\)
\(434\) −0.437588 −0.0210049
\(435\) 0 0
\(436\) 11.6224 0.556610
\(437\) 6.56857 0.314217
\(438\) 0 0
\(439\) 13.7324 0.655410 0.327705 0.944780i \(-0.393725\pi\)
0.327705 + 0.944780i \(0.393725\pi\)
\(440\) 3.07949 0.146809
\(441\) 0 0
\(442\) −27.0723 −1.28770
\(443\) −31.3119 −1.48767 −0.743837 0.668361i \(-0.766996\pi\)
−0.743837 + 0.668361i \(0.766996\pi\)
\(444\) 0 0
\(445\) −21.0328 −0.997052
\(446\) −15.3001 −0.724481
\(447\) 0 0
\(448\) −3.43260 −0.162175
\(449\) −8.62460 −0.407020 −0.203510 0.979073i \(-0.565235\pi\)
−0.203510 + 0.979073i \(0.565235\pi\)
\(450\) 0 0
\(451\) 8.32808 0.392154
\(452\) 13.9363 0.655510
\(453\) 0 0
\(454\) −16.6410 −0.781001
\(455\) −5.21468 −0.244468
\(456\) 0 0
\(457\) −29.4709 −1.37859 −0.689295 0.724481i \(-0.742080\pi\)
−0.689295 + 0.724481i \(0.742080\pi\)
\(458\) 0.0180083 0.000841470 0
\(459\) 0 0
\(460\) 2.08256 0.0971001
\(461\) −40.0274 −1.86426 −0.932132 0.362120i \(-0.882053\pi\)
−0.932132 + 0.362120i \(0.882053\pi\)
\(462\) 0 0
\(463\) −30.9773 −1.43964 −0.719819 0.694162i \(-0.755775\pi\)
−0.719819 + 0.694162i \(0.755775\pi\)
\(464\) −0.0635329 −0.00294944
\(465\) 0 0
\(466\) −9.14762 −0.423755
\(467\) −32.6472 −1.51073 −0.755367 0.655302i \(-0.772541\pi\)
−0.755367 + 0.655302i \(0.772541\pi\)
\(468\) 0 0
\(469\) 2.01239 0.0929236
\(470\) 12.2652 0.565753
\(471\) 0 0
\(472\) −13.5463 −0.623518
\(473\) −1.35167 −0.0621500
\(474\) 0 0
\(475\) −14.6170 −0.670673
\(476\) 6.40039 0.293361
\(477\) 0 0
\(478\) −1.64067 −0.0750424
\(479\) −10.2273 −0.467298 −0.233649 0.972321i \(-0.575067\pi\)
−0.233649 + 0.972321i \(0.575067\pi\)
\(480\) 0 0
\(481\) 30.9129 1.40951
\(482\) 2.26319 0.103086
\(483\) 0 0
\(484\) 13.2130 0.600593
\(485\) 27.2088 1.23549
\(486\) 0 0
\(487\) 15.4717 0.701089 0.350544 0.936546i \(-0.385997\pi\)
0.350544 + 0.936546i \(0.385997\pi\)
\(488\) −7.08649 −0.320790
\(489\) 0 0
\(490\) 9.35716 0.422713
\(491\) −17.4990 −0.789718 −0.394859 0.918742i \(-0.629207\pi\)
−0.394859 + 0.918742i \(0.629207\pi\)
\(492\) 0 0
\(493\) −7.15038 −0.322037
\(494\) −24.8694 −1.11893
\(495\) 0 0
\(496\) 0.0448448 0.00201359
\(497\) 0.181548 0.00814356
\(498\) 0 0
\(499\) 17.6612 0.790625 0.395312 0.918547i \(-0.370636\pi\)
0.395312 + 0.918547i \(0.370636\pi\)
\(500\) −15.0471 −0.672928
\(501\) 0 0
\(502\) 12.8359 0.572893
\(503\) −20.1271 −0.897424 −0.448712 0.893677i \(-0.648117\pi\)
−0.448712 + 0.893677i \(0.648117\pi\)
\(504\) 0 0
\(505\) −0.290923 −0.0129459
\(506\) 0.568796 0.0252861
\(507\) 0 0
\(508\) 26.5210 1.17668
\(509\) 17.3554 0.769266 0.384633 0.923070i \(-0.374328\pi\)
0.384633 + 0.923070i \(0.374328\pi\)
\(510\) 0 0
\(511\) −9.44945 −0.418019
\(512\) 0.718710 0.0317628
\(513\) 0 0
\(514\) −5.99410 −0.264388
\(515\) −2.33437 −0.102865
\(516\) 0 0
\(517\) −5.58592 −0.245669
\(518\) 4.38289 0.192573
\(519\) 0 0
\(520\) −20.4983 −0.898911
\(521\) −19.9195 −0.872690 −0.436345 0.899779i \(-0.643727\pi\)
−0.436345 + 0.899779i \(0.643727\pi\)
\(522\) 0 0
\(523\) −3.54078 −0.154827 −0.0774136 0.996999i \(-0.524666\pi\)
−0.0774136 + 0.996999i \(0.524666\pi\)
\(524\) −23.8376 −1.04135
\(525\) 0 0
\(526\) 0.318828 0.0139016
\(527\) 5.04711 0.219856
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 4.38805 0.190605
\(531\) 0 0
\(532\) 5.87960 0.254913
\(533\) −55.4350 −2.40116
\(534\) 0 0
\(535\) 14.6179 0.631985
\(536\) 7.91049 0.341681
\(537\) 0 0
\(538\) 6.63327 0.285981
\(539\) −4.26151 −0.183556
\(540\) 0 0
\(541\) −34.2816 −1.47388 −0.736941 0.675957i \(-0.763731\pi\)
−0.736941 + 0.675957i \(0.763731\pi\)
\(542\) 17.5259 0.752801
\(543\) 0 0
\(544\) 40.6407 1.74246
\(545\) 15.4851 0.663308
\(546\) 0 0
\(547\) −2.28458 −0.0976815 −0.0488407 0.998807i \(-0.515553\pi\)
−0.0488407 + 0.998807i \(0.515553\pi\)
\(548\) 14.4048 0.615344
\(549\) 0 0
\(550\) −1.26574 −0.0539712
\(551\) −6.56857 −0.279830
\(552\) 0 0
\(553\) −0.437949 −0.0186235
\(554\) 27.2224 1.15657
\(555\) 0 0
\(556\) −1.16690 −0.0494874
\(557\) −27.4491 −1.16306 −0.581528 0.813526i \(-0.697545\pi\)
−0.581528 + 0.813526i \(0.697545\pi\)
\(558\) 0 0
\(559\) 8.99727 0.380544
\(560\) 0.0757695 0.00320185
\(561\) 0 0
\(562\) 2.59610 0.109510
\(563\) −35.1985 −1.48344 −0.741719 0.670710i \(-0.765989\pi\)
−0.741719 + 0.670710i \(0.765989\pi\)
\(564\) 0 0
\(565\) 18.5681 0.781166
\(566\) −22.0303 −0.926002
\(567\) 0 0
\(568\) 0.713646 0.0299439
\(569\) 12.8457 0.538519 0.269260 0.963068i \(-0.413221\pi\)
0.269260 + 0.963068i \(0.413221\pi\)
\(570\) 0 0
\(571\) 31.0342 1.29874 0.649371 0.760472i \(-0.275032\pi\)
0.649371 + 0.760472i \(0.275032\pi\)
\(572\) 3.59098 0.150147
\(573\) 0 0
\(574\) −7.85967 −0.328056
\(575\) −2.22529 −0.0928010
\(576\) 0 0
\(577\) 1.32689 0.0552393 0.0276197 0.999619i \(-0.491207\pi\)
0.0276197 + 0.999619i \(0.491207\pi\)
\(578\) 29.5511 1.22916
\(579\) 0 0
\(580\) −2.08256 −0.0864738
\(581\) 1.48009 0.0614046
\(582\) 0 0
\(583\) −1.99844 −0.0827668
\(584\) −37.1447 −1.53706
\(585\) 0 0
\(586\) 21.5892 0.891842
\(587\) −7.82602 −0.323015 −0.161507 0.986872i \(-0.551636\pi\)
−0.161507 + 0.986872i \(0.551636\pi\)
\(588\) 0 0
\(589\) 4.63644 0.191041
\(590\) −6.94248 −0.285817
\(591\) 0 0
\(592\) −0.449166 −0.0184606
\(593\) 17.9167 0.735750 0.367875 0.929875i \(-0.380085\pi\)
0.367875 + 0.929875i \(0.380085\pi\)
\(594\) 0 0
\(595\) 8.52757 0.349596
\(596\) 12.3245 0.504830
\(597\) 0 0
\(598\) −3.78613 −0.154826
\(599\) 28.4367 1.16189 0.580946 0.813942i \(-0.302683\pi\)
0.580946 + 0.813942i \(0.302683\pi\)
\(600\) 0 0
\(601\) −2.12781 −0.0867952 −0.0433976 0.999058i \(-0.513818\pi\)
−0.0433976 + 0.999058i \(0.513818\pi\)
\(602\) 1.27565 0.0519916
\(603\) 0 0
\(604\) 18.0515 0.734504
\(605\) 17.6044 0.715722
\(606\) 0 0
\(607\) −3.24329 −0.131641 −0.0658205 0.997831i \(-0.520966\pi\)
−0.0658205 + 0.997831i \(0.520966\pi\)
\(608\) 37.3339 1.51409
\(609\) 0 0
\(610\) −3.63183 −0.147049
\(611\) 37.1821 1.50423
\(612\) 0 0
\(613\) 4.85928 0.196264 0.0981322 0.995173i \(-0.468713\pi\)
0.0981322 + 0.995173i \(0.468713\pi\)
\(614\) 6.47998 0.261511
\(615\) 0 0
\(616\) 1.32360 0.0533295
\(617\) −18.6045 −0.748989 −0.374495 0.927229i \(-0.622184\pi\)
−0.374495 + 0.927229i \(0.622184\pi\)
\(618\) 0 0
\(619\) −1.75731 −0.0706323 −0.0353161 0.999376i \(-0.511244\pi\)
−0.0353161 + 0.999376i \(0.511244\pi\)
\(620\) 1.46998 0.0590359
\(621\) 0 0
\(622\) −6.34048 −0.254230
\(623\) −9.04016 −0.362187
\(624\) 0 0
\(625\) −8.92163 −0.356865
\(626\) 27.7463 1.10897
\(627\) 0 0
\(628\) 6.34427 0.253164
\(629\) −50.5519 −2.01564
\(630\) 0 0
\(631\) 39.6157 1.57708 0.788539 0.614985i \(-0.210838\pi\)
0.788539 + 0.614985i \(0.210838\pi\)
\(632\) −1.72153 −0.0684788
\(633\) 0 0
\(634\) −15.2439 −0.605412
\(635\) 35.3353 1.40224
\(636\) 0 0
\(637\) 28.3663 1.12391
\(638\) −0.568796 −0.0225189
\(639\) 0 0
\(640\) 12.0200 0.475131
\(641\) −1.34325 −0.0530553 −0.0265276 0.999648i \(-0.508445\pi\)
−0.0265276 + 0.999648i \(0.508445\pi\)
\(642\) 0 0
\(643\) 0.605211 0.0238672 0.0119336 0.999929i \(-0.496201\pi\)
0.0119336 + 0.999929i \(0.496201\pi\)
\(644\) 0.895112 0.0352723
\(645\) 0 0
\(646\) 40.6690 1.60010
\(647\) 41.4111 1.62804 0.814019 0.580838i \(-0.197275\pi\)
0.814019 + 0.580838i \(0.197275\pi\)
\(648\) 0 0
\(649\) 3.16180 0.124111
\(650\) 8.42524 0.330465
\(651\) 0 0
\(652\) 11.8023 0.462214
\(653\) 38.2945 1.49858 0.749290 0.662242i \(-0.230395\pi\)
0.749290 + 0.662242i \(0.230395\pi\)
\(654\) 0 0
\(655\) −31.7601 −1.24097
\(656\) 0.805473 0.0314484
\(657\) 0 0
\(658\) 5.27174 0.205514
\(659\) −7.30554 −0.284584 −0.142292 0.989825i \(-0.545447\pi\)
−0.142292 + 0.989825i \(0.545447\pi\)
\(660\) 0 0
\(661\) −12.7358 −0.495364 −0.247682 0.968841i \(-0.579669\pi\)
−0.247682 + 0.968841i \(0.579669\pi\)
\(662\) −18.4363 −0.716547
\(663\) 0 0
\(664\) 5.81808 0.225785
\(665\) 7.83370 0.303778
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 25.8965 1.00197
\(669\) 0 0
\(670\) 4.05413 0.156625
\(671\) 1.65404 0.0638534
\(672\) 0 0
\(673\) 24.6573 0.950470 0.475235 0.879859i \(-0.342363\pi\)
0.475235 + 0.879859i \(0.342363\pi\)
\(674\) 13.4627 0.518564
\(675\) 0 0
\(676\) −7.65001 −0.294231
\(677\) −38.8256 −1.49219 −0.746095 0.665840i \(-0.768073\pi\)
−0.746095 + 0.665840i \(0.768073\pi\)
\(678\) 0 0
\(679\) 11.6947 0.448800
\(680\) 33.5209 1.28547
\(681\) 0 0
\(682\) 0.401486 0.0153737
\(683\) 27.9756 1.07046 0.535229 0.844707i \(-0.320225\pi\)
0.535229 + 0.844707i \(0.320225\pi\)
\(684\) 0 0
\(685\) 19.1923 0.733301
\(686\) 8.36142 0.319240
\(687\) 0 0
\(688\) −0.130731 −0.00498406
\(689\) 13.3024 0.506780
\(690\) 0 0
\(691\) −39.5369 −1.50405 −0.752027 0.659133i \(-0.770924\pi\)
−0.752027 + 0.659133i \(0.770924\pi\)
\(692\) 6.21662 0.236320
\(693\) 0 0
\(694\) −11.1333 −0.422613
\(695\) −1.55472 −0.0589737
\(696\) 0 0
\(697\) 90.6529 3.43372
\(698\) 20.7879 0.786832
\(699\) 0 0
\(700\) −1.99188 −0.0752861
\(701\) −12.3176 −0.465229 −0.232615 0.972569i \(-0.574728\pi\)
−0.232615 + 0.972569i \(0.574728\pi\)
\(702\) 0 0
\(703\) −46.4386 −1.75147
\(704\) 3.14941 0.118698
\(705\) 0 0
\(706\) 8.44182 0.317712
\(707\) −0.125042 −0.00470270
\(708\) 0 0
\(709\) −12.0155 −0.451253 −0.225626 0.974214i \(-0.572443\pi\)
−0.225626 + 0.974214i \(0.572443\pi\)
\(710\) 0.365744 0.0137261
\(711\) 0 0
\(712\) −35.5359 −1.33176
\(713\) 0.705852 0.0264344
\(714\) 0 0
\(715\) 4.78445 0.178928
\(716\) −26.9497 −1.00716
\(717\) 0 0
\(718\) 3.17563 0.118513
\(719\) −16.8456 −0.628234 −0.314117 0.949384i \(-0.601708\pi\)
−0.314117 + 0.949384i \(0.601708\pi\)
\(720\) 0 0
\(721\) −1.00334 −0.0373664
\(722\) 20.9079 0.778113
\(723\) 0 0
\(724\) 25.1007 0.932858
\(725\) 2.22529 0.0826452
\(726\) 0 0
\(727\) 9.37404 0.347664 0.173832 0.984775i \(-0.444385\pi\)
0.173832 + 0.984775i \(0.444385\pi\)
\(728\) −8.81043 −0.326536
\(729\) 0 0
\(730\) −19.0367 −0.704580
\(731\) −14.7132 −0.544189
\(732\) 0 0
\(733\) −26.7121 −0.986634 −0.493317 0.869850i \(-0.664216\pi\)
−0.493317 + 0.869850i \(0.664216\pi\)
\(734\) 1.83019 0.0675535
\(735\) 0 0
\(736\) 5.68371 0.209504
\(737\) −1.84636 −0.0680117
\(738\) 0 0
\(739\) 5.88668 0.216545 0.108273 0.994121i \(-0.465468\pi\)
0.108273 + 0.994121i \(0.465468\pi\)
\(740\) −14.7234 −0.541242
\(741\) 0 0
\(742\) 1.88604 0.0692385
\(743\) 16.1024 0.590741 0.295370 0.955383i \(-0.404557\pi\)
0.295370 + 0.955383i \(0.404557\pi\)
\(744\) 0 0
\(745\) 16.4205 0.601602
\(746\) −10.2441 −0.375062
\(747\) 0 0
\(748\) −5.87234 −0.214714
\(749\) 6.28294 0.229573
\(750\) 0 0
\(751\) −34.6373 −1.26393 −0.631967 0.774995i \(-0.717752\pi\)
−0.631967 + 0.774995i \(0.717752\pi\)
\(752\) −0.540258 −0.0197012
\(753\) 0 0
\(754\) 3.78613 0.137883
\(755\) 24.0509 0.875302
\(756\) 0 0
\(757\) −28.2931 −1.02833 −0.514165 0.857691i \(-0.671898\pi\)
−0.514165 + 0.857691i \(0.671898\pi\)
\(758\) −10.6240 −0.385882
\(759\) 0 0
\(760\) 30.7934 1.11699
\(761\) −3.96877 −0.143868 −0.0719339 0.997409i \(-0.522917\pi\)
−0.0719339 + 0.997409i \(0.522917\pi\)
\(762\) 0 0
\(763\) 6.65568 0.240952
\(764\) 1.04155 0.0376820
\(765\) 0 0
\(766\) 16.2993 0.588918
\(767\) −21.0462 −0.759933
\(768\) 0 0
\(769\) −7.71123 −0.278074 −0.139037 0.990287i \(-0.544401\pi\)
−0.139037 + 0.990287i \(0.544401\pi\)
\(770\) 0.678348 0.0244460
\(771\) 0 0
\(772\) −13.6784 −0.492298
\(773\) 47.4137 1.70535 0.852677 0.522439i \(-0.174978\pi\)
0.852677 + 0.522439i \(0.174978\pi\)
\(774\) 0 0
\(775\) −1.57073 −0.0564221
\(776\) 45.9704 1.65024
\(777\) 0 0
\(778\) 0.527406 0.0189084
\(779\) 83.2767 2.98370
\(780\) 0 0
\(781\) −0.166570 −0.00596035
\(782\) 6.19146 0.221406
\(783\) 0 0
\(784\) −0.412163 −0.0147201
\(785\) 8.45281 0.301694
\(786\) 0 0
\(787\) 14.3049 0.509914 0.254957 0.966952i \(-0.417939\pi\)
0.254957 + 0.966952i \(0.417939\pi\)
\(788\) 0.548658 0.0195451
\(789\) 0 0
\(790\) −0.882286 −0.0313903
\(791\) 7.98079 0.283764
\(792\) 0 0
\(793\) −11.0099 −0.390974
\(794\) −7.36334 −0.261315
\(795\) 0 0
\(796\) 6.74920 0.239219
\(797\) −11.6866 −0.413961 −0.206981 0.978345i \(-0.566364\pi\)
−0.206981 + 0.978345i \(0.566364\pi\)
\(798\) 0 0
\(799\) −60.8039 −2.15109
\(800\) −12.6479 −0.447171
\(801\) 0 0
\(802\) −32.9472 −1.16341
\(803\) 8.66985 0.305952
\(804\) 0 0
\(805\) 1.19260 0.0420338
\(806\) −2.67245 −0.0941330
\(807\) 0 0
\(808\) −0.491528 −0.0172919
\(809\) −10.9340 −0.384420 −0.192210 0.981354i \(-0.561566\pi\)
−0.192210 + 0.981354i \(0.561566\pi\)
\(810\) 0 0
\(811\) −15.2080 −0.534023 −0.267012 0.963693i \(-0.586036\pi\)
−0.267012 + 0.963693i \(0.586036\pi\)
\(812\) −0.895112 −0.0314123
\(813\) 0 0
\(814\) −4.02129 −0.140946
\(815\) 15.7248 0.550817
\(816\) 0 0
\(817\) −13.5161 −0.472867
\(818\) 13.1141 0.458522
\(819\) 0 0
\(820\) 26.4029 0.922028
\(821\) 40.4608 1.41209 0.706046 0.708166i \(-0.250477\pi\)
0.706046 + 0.708166i \(0.250477\pi\)
\(822\) 0 0
\(823\) 19.1656 0.668070 0.334035 0.942561i \(-0.391590\pi\)
0.334035 + 0.942561i \(0.391590\pi\)
\(824\) −3.94402 −0.137397
\(825\) 0 0
\(826\) −2.98396 −0.103825
\(827\) −51.5955 −1.79415 −0.897076 0.441877i \(-0.854313\pi\)
−0.897076 + 0.441877i \(0.854313\pi\)
\(828\) 0 0
\(829\) 22.8118 0.792288 0.396144 0.918188i \(-0.370348\pi\)
0.396144 + 0.918188i \(0.370348\pi\)
\(830\) 2.98177 0.103499
\(831\) 0 0
\(832\) −20.9637 −0.726785
\(833\) −46.3874 −1.60723
\(834\) 0 0
\(835\) 34.5033 1.19403
\(836\) −5.39452 −0.186573
\(837\) 0 0
\(838\) −17.0178 −0.587869
\(839\) 22.8445 0.788680 0.394340 0.918965i \(-0.370973\pi\)
0.394340 + 0.918965i \(0.370973\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −23.1259 −0.796972
\(843\) 0 0
\(844\) 14.8268 0.510359
\(845\) −10.1925 −0.350633
\(846\) 0 0
\(847\) 7.56660 0.259991
\(848\) −0.193284 −0.00663741
\(849\) 0 0
\(850\) −13.7778 −0.472575
\(851\) −7.06983 −0.242351
\(852\) 0 0
\(853\) 10.6600 0.364993 0.182497 0.983206i \(-0.441582\pi\)
0.182497 + 0.983206i \(0.441582\pi\)
\(854\) −1.56101 −0.0534165
\(855\) 0 0
\(856\) 24.6975 0.844144
\(857\) −29.0376 −0.991904 −0.495952 0.868350i \(-0.665181\pi\)
−0.495952 + 0.868350i \(0.665181\pi\)
\(858\) 0 0
\(859\) 38.0437 1.29804 0.649018 0.760773i \(-0.275180\pi\)
0.649018 + 0.760773i \(0.275180\pi\)
\(860\) −4.28527 −0.146126
\(861\) 0 0
\(862\) 29.0345 0.988919
\(863\) 34.2406 1.16556 0.582781 0.812629i \(-0.301964\pi\)
0.582781 + 0.812629i \(0.301964\pi\)
\(864\) 0 0
\(865\) 8.28273 0.281621
\(866\) 21.2262 0.721296
\(867\) 0 0
\(868\) 0.631817 0.0214453
\(869\) 0.401817 0.0136307
\(870\) 0 0
\(871\) 12.2901 0.416435
\(872\) 26.1627 0.885982
\(873\) 0 0
\(874\) 5.68768 0.192389
\(875\) −8.61691 −0.291305
\(876\) 0 0
\(877\) −53.2117 −1.79683 −0.898415 0.439148i \(-0.855281\pi\)
−0.898415 + 0.439148i \(0.855281\pi\)
\(878\) 11.8908 0.401294
\(879\) 0 0
\(880\) −0.0695184 −0.00234346
\(881\) 29.7942 1.00379 0.501896 0.864928i \(-0.332636\pi\)
0.501896 + 0.864928i \(0.332636\pi\)
\(882\) 0 0
\(883\) 18.4786 0.621854 0.310927 0.950434i \(-0.399361\pi\)
0.310927 + 0.950434i \(0.399361\pi\)
\(884\) 39.0886 1.31469
\(885\) 0 0
\(886\) −27.1128 −0.910871
\(887\) −46.6348 −1.56584 −0.782922 0.622119i \(-0.786272\pi\)
−0.782922 + 0.622119i \(0.786272\pi\)
\(888\) 0 0
\(889\) 15.1875 0.509374
\(890\) −18.2122 −0.610473
\(891\) 0 0
\(892\) 22.0912 0.739670
\(893\) −55.8564 −1.86916
\(894\) 0 0
\(895\) −35.9065 −1.20022
\(896\) 5.16633 0.172595
\(897\) 0 0
\(898\) −7.46798 −0.249210
\(899\) −0.705852 −0.0235415
\(900\) 0 0
\(901\) −21.7534 −0.724711
\(902\) 7.21123 0.240108
\(903\) 0 0
\(904\) 31.3716 1.04340
\(905\) 33.4429 1.11168
\(906\) 0 0
\(907\) 51.1950 1.69990 0.849951 0.526861i \(-0.176631\pi\)
0.849951 + 0.526861i \(0.176631\pi\)
\(908\) 24.0273 0.797374
\(909\) 0 0
\(910\) −4.51535 −0.149682
\(911\) −9.25114 −0.306504 −0.153252 0.988187i \(-0.548975\pi\)
−0.153252 + 0.988187i \(0.548975\pi\)
\(912\) 0 0
\(913\) −1.35798 −0.0449427
\(914\) −25.5186 −0.844081
\(915\) 0 0
\(916\) −0.0260014 −0.000859111 0
\(917\) −13.6509 −0.450792
\(918\) 0 0
\(919\) 42.8124 1.41225 0.706125 0.708087i \(-0.250441\pi\)
0.706125 + 0.708087i \(0.250441\pi\)
\(920\) 4.68799 0.154559
\(921\) 0 0
\(922\) −34.6595 −1.14145
\(923\) 1.10876 0.0364952
\(924\) 0 0
\(925\) 15.7324 0.517279
\(926\) −26.8230 −0.881459
\(927\) 0 0
\(928\) −5.68371 −0.186577
\(929\) −40.4332 −1.32657 −0.663286 0.748366i \(-0.730838\pi\)
−0.663286 + 0.748366i \(0.730838\pi\)
\(930\) 0 0
\(931\) −42.6130 −1.39658
\(932\) 13.2079 0.432639
\(933\) 0 0
\(934\) −28.2690 −0.924990
\(935\) −7.82402 −0.255873
\(936\) 0 0
\(937\) −16.4011 −0.535802 −0.267901 0.963447i \(-0.586330\pi\)
−0.267901 + 0.963447i \(0.586330\pi\)
\(938\) 1.74252 0.0568951
\(939\) 0 0
\(940\) −17.7093 −0.577613
\(941\) 26.4355 0.861772 0.430886 0.902406i \(-0.358201\pi\)
0.430886 + 0.902406i \(0.358201\pi\)
\(942\) 0 0
\(943\) 12.6781 0.412854
\(944\) 0.305802 0.00995300
\(945\) 0 0
\(946\) −1.17040 −0.0380531
\(947\) −31.4428 −1.02175 −0.510876 0.859654i \(-0.670679\pi\)
−0.510876 + 0.859654i \(0.670679\pi\)
\(948\) 0 0
\(949\) −57.7099 −1.87334
\(950\) −12.6567 −0.410639
\(951\) 0 0
\(952\) 14.4077 0.466956
\(953\) 23.4157 0.758509 0.379254 0.925292i \(-0.376181\pi\)
0.379254 + 0.925292i \(0.376181\pi\)
\(954\) 0 0
\(955\) 1.38771 0.0449053
\(956\) 2.36890 0.0766156
\(957\) 0 0
\(958\) −8.85575 −0.286116
\(959\) 8.24909 0.266377
\(960\) 0 0
\(961\) −30.5018 −0.983928
\(962\) 26.7673 0.863012
\(963\) 0 0
\(964\) −3.26774 −0.105247
\(965\) −18.2245 −0.586667
\(966\) 0 0
\(967\) 1.40682 0.0452402 0.0226201 0.999744i \(-0.492799\pi\)
0.0226201 + 0.999744i \(0.492799\pi\)
\(968\) 29.7435 0.955991
\(969\) 0 0
\(970\) 23.5599 0.756463
\(971\) 41.3493 1.32696 0.663481 0.748193i \(-0.269078\pi\)
0.663481 + 0.748193i \(0.269078\pi\)
\(972\) 0 0
\(973\) −0.668235 −0.0214226
\(974\) 13.3968 0.429262
\(975\) 0 0
\(976\) 0.159975 0.00512067
\(977\) 16.1169 0.515626 0.257813 0.966195i \(-0.416998\pi\)
0.257813 + 0.966195i \(0.416998\pi\)
\(978\) 0 0
\(979\) 8.29433 0.265088
\(980\) −13.5104 −0.431575
\(981\) 0 0
\(982\) −15.1522 −0.483527
\(983\) 7.74532 0.247037 0.123519 0.992342i \(-0.460582\pi\)
0.123519 + 0.992342i \(0.460582\pi\)
\(984\) 0 0
\(985\) 0.731006 0.0232918
\(986\) −6.19146 −0.197176
\(987\) 0 0
\(988\) 35.9080 1.14239
\(989\) −2.05769 −0.0654307
\(990\) 0 0
\(991\) −17.9134 −0.569038 −0.284519 0.958670i \(-0.591834\pi\)
−0.284519 + 0.958670i \(0.591834\pi\)
\(992\) 4.01186 0.127377
\(993\) 0 0
\(994\) 0.157201 0.00498613
\(995\) 8.99231 0.285075
\(996\) 0 0
\(997\) −46.5539 −1.47438 −0.737188 0.675687i \(-0.763847\pi\)
−0.737188 + 0.675687i \(0.763847\pi\)
\(998\) 15.2927 0.484083
\(999\) 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.i.1.4 7
3.2 odd 2 2001.2.a.j.1.4 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.j.1.4 7 3.2 odd 2
6003.2.a.i.1.4 7 1.1 even 1 trivial