Properties

Label 8034.2.a.w.1.4
Level 8034
Weight 2
Character 8034.1
Self dual yes
Analytic conductor 64.152
Analytic rank 1
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 6 x^{11} - 8 x^{10} + 94 x^{9} - 7 x^{8} - 580 x^{7} + 180 x^{6} + 1787 x^{5} - 308 x^{4} - 2790 x^{3} - 352 x^{2} + 1768 x + 768\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.08335\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.71945 q^{5} -1.00000 q^{6} +2.07930 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.71945 q^{5} -1.00000 q^{6} +2.07930 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.71945 q^{10} -4.17067 q^{11} +1.00000 q^{12} +1.00000 q^{13} -2.07930 q^{14} -1.71945 q^{15} +1.00000 q^{16} -3.34293 q^{17} -1.00000 q^{18} +3.21672 q^{19} -1.71945 q^{20} +2.07930 q^{21} +4.17067 q^{22} -3.16834 q^{23} -1.00000 q^{24} -2.04350 q^{25} -1.00000 q^{26} +1.00000 q^{27} +2.07930 q^{28} +7.68812 q^{29} +1.71945 q^{30} -4.89876 q^{31} -1.00000 q^{32} -4.17067 q^{33} +3.34293 q^{34} -3.57525 q^{35} +1.00000 q^{36} +6.91253 q^{37} -3.21672 q^{38} +1.00000 q^{39} +1.71945 q^{40} +2.12821 q^{41} -2.07930 q^{42} +0.0584673 q^{43} -4.17067 q^{44} -1.71945 q^{45} +3.16834 q^{46} +8.24152 q^{47} +1.00000 q^{48} -2.67649 q^{49} +2.04350 q^{50} -3.34293 q^{51} +1.00000 q^{52} -4.72692 q^{53} -1.00000 q^{54} +7.17124 q^{55} -2.07930 q^{56} +3.21672 q^{57} -7.68812 q^{58} +3.20023 q^{59} -1.71945 q^{60} +1.94455 q^{61} +4.89876 q^{62} +2.07930 q^{63} +1.00000 q^{64} -1.71945 q^{65} +4.17067 q^{66} -8.47636 q^{67} -3.34293 q^{68} -3.16834 q^{69} +3.57525 q^{70} -1.76140 q^{71} -1.00000 q^{72} +10.0154 q^{73} -6.91253 q^{74} -2.04350 q^{75} +3.21672 q^{76} -8.67209 q^{77} -1.00000 q^{78} +5.26682 q^{79} -1.71945 q^{80} +1.00000 q^{81} -2.12821 q^{82} -16.0174 q^{83} +2.07930 q^{84} +5.74799 q^{85} -0.0584673 q^{86} +7.68812 q^{87} +4.17067 q^{88} -7.65296 q^{89} +1.71945 q^{90} +2.07930 q^{91} -3.16834 q^{92} -4.89876 q^{93} -8.24152 q^{94} -5.53097 q^{95} -1.00000 q^{96} -2.20578 q^{97} +2.67649 q^{98} -4.17067 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + 4q^{10} - 9q^{11} + 12q^{12} + 12q^{13} - 4q^{15} + 12q^{16} - 20q^{17} - 12q^{18} + 4q^{19} - 4q^{20} + 9q^{22} - 30q^{23} - 12q^{24} + 14q^{25} - 12q^{26} + 12q^{27} - 29q^{29} + 4q^{30} + 6q^{31} - 12q^{32} - 9q^{33} + 20q^{34} - 22q^{35} + 12q^{36} + 7q^{37} - 4q^{38} + 12q^{39} + 4q^{40} - 8q^{41} - 8q^{43} - 9q^{44} - 4q^{45} + 30q^{46} - 16q^{47} + 12q^{48} + 10q^{49} - 14q^{50} - 20q^{51} + 12q^{52} - 9q^{53} - 12q^{54} - 20q^{55} + 4q^{57} + 29q^{58} - 29q^{59} - 4q^{60} - 26q^{61} - 6q^{62} + 12q^{64} - 4q^{65} + 9q^{66} + 12q^{67} - 20q^{68} - 30q^{69} + 22q^{70} - 35q^{71} - 12q^{72} + 18q^{73} - 7q^{74} + 14q^{75} + 4q^{76} - 25q^{77} - 12q^{78} - 37q^{79} - 4q^{80} + 12q^{81} + 8q^{82} - 24q^{83} - 17q^{85} + 8q^{86} - 29q^{87} + 9q^{88} + 15q^{89} + 4q^{90} - 30q^{92} + 6q^{93} + 16q^{94} - 54q^{95} - 12q^{96} - 11q^{97} - 10q^{98} - 9q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.71945 −0.768960 −0.384480 0.923133i \(-0.625619\pi\)
−0.384480 + 0.923133i \(0.625619\pi\)
\(6\) −1.00000 −0.408248
\(7\) 2.07930 0.785903 0.392952 0.919559i \(-0.371454\pi\)
0.392952 + 0.919559i \(0.371454\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.71945 0.543737
\(11\) −4.17067 −1.25750 −0.628752 0.777606i \(-0.716434\pi\)
−0.628752 + 0.777606i \(0.716434\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) −2.07930 −0.555718
\(15\) −1.71945 −0.443959
\(16\) 1.00000 0.250000
\(17\) −3.34293 −0.810779 −0.405390 0.914144i \(-0.632864\pi\)
−0.405390 + 0.914144i \(0.632864\pi\)
\(18\) −1.00000 −0.235702
\(19\) 3.21672 0.737966 0.368983 0.929436i \(-0.379706\pi\)
0.368983 + 0.929436i \(0.379706\pi\)
\(20\) −1.71945 −0.384480
\(21\) 2.07930 0.453741
\(22\) 4.17067 0.889189
\(23\) −3.16834 −0.660645 −0.330322 0.943868i \(-0.607157\pi\)
−0.330322 + 0.943868i \(0.607157\pi\)
\(24\) −1.00000 −0.204124
\(25\) −2.04350 −0.408701
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 2.07930 0.392952
\(29\) 7.68812 1.42765 0.713824 0.700325i \(-0.246962\pi\)
0.713824 + 0.700325i \(0.246962\pi\)
\(30\) 1.71945 0.313927
\(31\) −4.89876 −0.879843 −0.439922 0.898036i \(-0.644994\pi\)
−0.439922 + 0.898036i \(0.644994\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.17067 −0.726020
\(34\) 3.34293 0.573308
\(35\) −3.57525 −0.604328
\(36\) 1.00000 0.166667
\(37\) 6.91253 1.13641 0.568207 0.822886i \(-0.307637\pi\)
0.568207 + 0.822886i \(0.307637\pi\)
\(38\) −3.21672 −0.521821
\(39\) 1.00000 0.160128
\(40\) 1.71945 0.271868
\(41\) 2.12821 0.332371 0.166186 0.986095i \(-0.446855\pi\)
0.166186 + 0.986095i \(0.446855\pi\)
\(42\) −2.07930 −0.320844
\(43\) 0.0584673 0.00891618 0.00445809 0.999990i \(-0.498581\pi\)
0.00445809 + 0.999990i \(0.498581\pi\)
\(44\) −4.17067 −0.628752
\(45\) −1.71945 −0.256320
\(46\) 3.16834 0.467146
\(47\) 8.24152 1.20215 0.601075 0.799193i \(-0.294739\pi\)
0.601075 + 0.799193i \(0.294739\pi\)
\(48\) 1.00000 0.144338
\(49\) −2.67649 −0.382356
\(50\) 2.04350 0.288995
\(51\) −3.34293 −0.468104
\(52\) 1.00000 0.138675
\(53\) −4.72692 −0.649293 −0.324646 0.945835i \(-0.605245\pi\)
−0.324646 + 0.945835i \(0.605245\pi\)
\(54\) −1.00000 −0.136083
\(55\) 7.17124 0.966970
\(56\) −2.07930 −0.277859
\(57\) 3.21672 0.426065
\(58\) −7.68812 −1.00950
\(59\) 3.20023 0.416635 0.208317 0.978061i \(-0.433201\pi\)
0.208317 + 0.978061i \(0.433201\pi\)
\(60\) −1.71945 −0.221980
\(61\) 1.94455 0.248974 0.124487 0.992221i \(-0.460271\pi\)
0.124487 + 0.992221i \(0.460271\pi\)
\(62\) 4.89876 0.622143
\(63\) 2.07930 0.261968
\(64\) 1.00000 0.125000
\(65\) −1.71945 −0.213271
\(66\) 4.17067 0.513374
\(67\) −8.47636 −1.03555 −0.517776 0.855516i \(-0.673240\pi\)
−0.517776 + 0.855516i \(0.673240\pi\)
\(68\) −3.34293 −0.405390
\(69\) −3.16834 −0.381423
\(70\) 3.57525 0.427324
\(71\) −1.76140 −0.209040 −0.104520 0.994523i \(-0.533331\pi\)
−0.104520 + 0.994523i \(0.533331\pi\)
\(72\) −1.00000 −0.117851
\(73\) 10.0154 1.17221 0.586105 0.810235i \(-0.300660\pi\)
0.586105 + 0.810235i \(0.300660\pi\)
\(74\) −6.91253 −0.803566
\(75\) −2.04350 −0.235964
\(76\) 3.21672 0.368983
\(77\) −8.67209 −0.988276
\(78\) −1.00000 −0.113228
\(79\) 5.26682 0.592564 0.296282 0.955101i \(-0.404253\pi\)
0.296282 + 0.955101i \(0.404253\pi\)
\(80\) −1.71945 −0.192240
\(81\) 1.00000 0.111111
\(82\) −2.12821 −0.235022
\(83\) −16.0174 −1.75814 −0.879070 0.476694i \(-0.841835\pi\)
−0.879070 + 0.476694i \(0.841835\pi\)
\(84\) 2.07930 0.226871
\(85\) 5.74799 0.623457
\(86\) −0.0584673 −0.00630469
\(87\) 7.68812 0.824253
\(88\) 4.17067 0.444595
\(89\) −7.65296 −0.811212 −0.405606 0.914048i \(-0.632939\pi\)
−0.405606 + 0.914048i \(0.632939\pi\)
\(90\) 1.71945 0.181246
\(91\) 2.07930 0.217970
\(92\) −3.16834 −0.330322
\(93\) −4.89876 −0.507978
\(94\) −8.24152 −0.850048
\(95\) −5.53097 −0.567466
\(96\) −1.00000 −0.102062
\(97\) −2.20578 −0.223964 −0.111982 0.993710i \(-0.535720\pi\)
−0.111982 + 0.993710i \(0.535720\pi\)
\(98\) 2.67649 0.270367
\(99\) −4.17067 −0.419168
\(100\) −2.04350 −0.204350
\(101\) −18.0449 −1.79553 −0.897765 0.440474i \(-0.854810\pi\)
−0.897765 + 0.440474i \(0.854810\pi\)
\(102\) 3.34293 0.330999
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −3.57525 −0.348909
\(106\) 4.72692 0.459119
\(107\) −12.7003 −1.22778 −0.613890 0.789391i \(-0.710396\pi\)
−0.613890 + 0.789391i \(0.710396\pi\)
\(108\) 1.00000 0.0962250
\(109\) 6.46568 0.619300 0.309650 0.950851i \(-0.399788\pi\)
0.309650 + 0.950851i \(0.399788\pi\)
\(110\) −7.17124 −0.683751
\(111\) 6.91253 0.656109
\(112\) 2.07930 0.196476
\(113\) 12.2618 1.15350 0.576748 0.816922i \(-0.304321\pi\)
0.576748 + 0.816922i \(0.304321\pi\)
\(114\) −3.21672 −0.301273
\(115\) 5.44779 0.508009
\(116\) 7.68812 0.713824
\(117\) 1.00000 0.0924500
\(118\) −3.20023 −0.294605
\(119\) −6.95097 −0.637194
\(120\) 1.71945 0.156963
\(121\) 6.39447 0.581315
\(122\) −1.94455 −0.176051
\(123\) 2.12821 0.191895
\(124\) −4.89876 −0.439922
\(125\) 12.1109 1.08323
\(126\) −2.07930 −0.185239
\(127\) 2.84286 0.252263 0.126132 0.992014i \(-0.459744\pi\)
0.126132 + 0.992014i \(0.459744\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0.0584673 0.00514776
\(130\) 1.71945 0.150805
\(131\) −19.9421 −1.74235 −0.871176 0.490971i \(-0.836642\pi\)
−0.871176 + 0.490971i \(0.836642\pi\)
\(132\) −4.17067 −0.363010
\(133\) 6.68854 0.579970
\(134\) 8.47636 0.732246
\(135\) −1.71945 −0.147986
\(136\) 3.34293 0.286654
\(137\) −13.6536 −1.16650 −0.583252 0.812291i \(-0.698220\pi\)
−0.583252 + 0.812291i \(0.698220\pi\)
\(138\) 3.16834 0.269707
\(139\) 15.7883 1.33915 0.669574 0.742745i \(-0.266477\pi\)
0.669574 + 0.742745i \(0.266477\pi\)
\(140\) −3.57525 −0.302164
\(141\) 8.24152 0.694061
\(142\) 1.76140 0.147814
\(143\) −4.17067 −0.348769
\(144\) 1.00000 0.0833333
\(145\) −13.2193 −1.09780
\(146\) −10.0154 −0.828878
\(147\) −2.67649 −0.220753
\(148\) 6.91253 0.568207
\(149\) 5.60769 0.459400 0.229700 0.973262i \(-0.426226\pi\)
0.229700 + 0.973262i \(0.426226\pi\)
\(150\) 2.04350 0.166851
\(151\) −21.8696 −1.77972 −0.889862 0.456229i \(-0.849200\pi\)
−0.889862 + 0.456229i \(0.849200\pi\)
\(152\) −3.21672 −0.260910
\(153\) −3.34293 −0.270260
\(154\) 8.67209 0.698817
\(155\) 8.42315 0.676564
\(156\) 1.00000 0.0800641
\(157\) −6.73068 −0.537167 −0.268583 0.963256i \(-0.586555\pi\)
−0.268583 + 0.963256i \(0.586555\pi\)
\(158\) −5.26682 −0.419006
\(159\) −4.72692 −0.374869
\(160\) 1.71945 0.135934
\(161\) −6.58795 −0.519203
\(162\) −1.00000 −0.0785674
\(163\) −5.72060 −0.448072 −0.224036 0.974581i \(-0.571923\pi\)
−0.224036 + 0.974581i \(0.571923\pi\)
\(164\) 2.12821 0.166186
\(165\) 7.17124 0.558280
\(166\) 16.0174 1.24319
\(167\) 14.9623 1.15782 0.578910 0.815391i \(-0.303478\pi\)
0.578910 + 0.815391i \(0.303478\pi\)
\(168\) −2.07930 −0.160422
\(169\) 1.00000 0.0769231
\(170\) −5.74799 −0.440850
\(171\) 3.21672 0.245989
\(172\) 0.0584673 0.00445809
\(173\) −20.0856 −1.52708 −0.763539 0.645762i \(-0.776540\pi\)
−0.763539 + 0.645762i \(0.776540\pi\)
\(174\) −7.68812 −0.582835
\(175\) −4.24907 −0.321199
\(176\) −4.17067 −0.314376
\(177\) 3.20023 0.240544
\(178\) 7.65296 0.573613
\(179\) −19.1234 −1.42935 −0.714675 0.699457i \(-0.753425\pi\)
−0.714675 + 0.699457i \(0.753425\pi\)
\(180\) −1.71945 −0.128160
\(181\) −6.63039 −0.492833 −0.246417 0.969164i \(-0.579253\pi\)
−0.246417 + 0.969164i \(0.579253\pi\)
\(182\) −2.07930 −0.154128
\(183\) 1.94455 0.143745
\(184\) 3.16834 0.233573
\(185\) −11.8857 −0.873856
\(186\) 4.89876 0.359194
\(187\) 13.9422 1.01956
\(188\) 8.24152 0.601075
\(189\) 2.07930 0.151247
\(190\) 5.53097 0.401259
\(191\) −9.22135 −0.667233 −0.333617 0.942709i \(-0.608269\pi\)
−0.333617 + 0.942709i \(0.608269\pi\)
\(192\) 1.00000 0.0721688
\(193\) −4.66229 −0.335599 −0.167800 0.985821i \(-0.553666\pi\)
−0.167800 + 0.985821i \(0.553666\pi\)
\(194\) 2.20578 0.158366
\(195\) −1.71945 −0.123132
\(196\) −2.67649 −0.191178
\(197\) 8.89204 0.633532 0.316766 0.948504i \(-0.397403\pi\)
0.316766 + 0.948504i \(0.397403\pi\)
\(198\) 4.17067 0.296396
\(199\) −13.4916 −0.956391 −0.478195 0.878254i \(-0.658709\pi\)
−0.478195 + 0.878254i \(0.658709\pi\)
\(200\) 2.04350 0.144498
\(201\) −8.47636 −0.597876
\(202\) 18.0449 1.26963
\(203\) 15.9859 1.12199
\(204\) −3.34293 −0.234052
\(205\) −3.65935 −0.255580
\(206\) 1.00000 0.0696733
\(207\) −3.16834 −0.220215
\(208\) 1.00000 0.0693375
\(209\) −13.4159 −0.927995
\(210\) 3.57525 0.246716
\(211\) −7.92962 −0.545898 −0.272949 0.962029i \(-0.587999\pi\)
−0.272949 + 0.962029i \(0.587999\pi\)
\(212\) −4.72692 −0.324646
\(213\) −1.76140 −0.120689
\(214\) 12.7003 0.868172
\(215\) −0.100531 −0.00685619
\(216\) −1.00000 −0.0680414
\(217\) −10.1860 −0.691472
\(218\) −6.46568 −0.437911
\(219\) 10.0154 0.676776
\(220\) 7.17124 0.483485
\(221\) −3.34293 −0.224870
\(222\) −6.91253 −0.463939
\(223\) −3.35526 −0.224685 −0.112342 0.993670i \(-0.535835\pi\)
−0.112342 + 0.993670i \(0.535835\pi\)
\(224\) −2.07930 −0.138929
\(225\) −2.04350 −0.136234
\(226\) −12.2618 −0.815645
\(227\) −1.28365 −0.0851986 −0.0425993 0.999092i \(-0.513564\pi\)
−0.0425993 + 0.999092i \(0.513564\pi\)
\(228\) 3.21672 0.213032
\(229\) −6.54398 −0.432438 −0.216219 0.976345i \(-0.569373\pi\)
−0.216219 + 0.976345i \(0.569373\pi\)
\(230\) −5.44779 −0.359217
\(231\) −8.67209 −0.570582
\(232\) −7.68812 −0.504750
\(233\) −1.47551 −0.0966637 −0.0483318 0.998831i \(-0.515390\pi\)
−0.0483318 + 0.998831i \(0.515390\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −14.1708 −0.924404
\(236\) 3.20023 0.208317
\(237\) 5.26682 0.342117
\(238\) 6.95097 0.450564
\(239\) 18.4065 1.19062 0.595309 0.803497i \(-0.297030\pi\)
0.595309 + 0.803497i \(0.297030\pi\)
\(240\) −1.71945 −0.110990
\(241\) −7.89644 −0.508654 −0.254327 0.967118i \(-0.581854\pi\)
−0.254327 + 0.967118i \(0.581854\pi\)
\(242\) −6.39447 −0.411052
\(243\) 1.00000 0.0641500
\(244\) 1.94455 0.124487
\(245\) 4.60208 0.294016
\(246\) −2.12821 −0.135690
\(247\) 3.21672 0.204675
\(248\) 4.89876 0.311072
\(249\) −16.0174 −1.01506
\(250\) −12.1109 −0.765962
\(251\) 13.4455 0.848674 0.424337 0.905504i \(-0.360507\pi\)
0.424337 + 0.905504i \(0.360507\pi\)
\(252\) 2.07930 0.130984
\(253\) 13.2141 0.830763
\(254\) −2.84286 −0.178377
\(255\) 5.74799 0.359953
\(256\) 1.00000 0.0625000
\(257\) 25.5686 1.59492 0.797462 0.603369i \(-0.206175\pi\)
0.797462 + 0.603369i \(0.206175\pi\)
\(258\) −0.0584673 −0.00364002
\(259\) 14.3733 0.893111
\(260\) −1.71945 −0.106636
\(261\) 7.68812 0.475882
\(262\) 19.9421 1.23203
\(263\) −9.78932 −0.603635 −0.301818 0.953366i \(-0.597593\pi\)
−0.301818 + 0.953366i \(0.597593\pi\)
\(264\) 4.17067 0.256687
\(265\) 8.12769 0.499280
\(266\) −6.68854 −0.410101
\(267\) −7.65296 −0.468353
\(268\) −8.47636 −0.517776
\(269\) 28.9797 1.76692 0.883462 0.468503i \(-0.155207\pi\)
0.883462 + 0.468503i \(0.155207\pi\)
\(270\) 1.71945 0.104642
\(271\) −1.76169 −0.107015 −0.0535074 0.998567i \(-0.517040\pi\)
−0.0535074 + 0.998567i \(0.517040\pi\)
\(272\) −3.34293 −0.202695
\(273\) 2.07930 0.125845
\(274\) 13.6536 0.824843
\(275\) 8.52278 0.513943
\(276\) −3.16834 −0.190712
\(277\) 20.3481 1.22260 0.611300 0.791399i \(-0.290647\pi\)
0.611300 + 0.791399i \(0.290647\pi\)
\(278\) −15.7883 −0.946921
\(279\) −4.89876 −0.293281
\(280\) 3.57525 0.213662
\(281\) −2.83942 −0.169385 −0.0846927 0.996407i \(-0.526991\pi\)
−0.0846927 + 0.996407i \(0.526991\pi\)
\(282\) −8.24152 −0.490775
\(283\) −12.5214 −0.744322 −0.372161 0.928168i \(-0.621383\pi\)
−0.372161 + 0.928168i \(0.621383\pi\)
\(284\) −1.76140 −0.104520
\(285\) −5.53097 −0.327627
\(286\) 4.17067 0.246617
\(287\) 4.42520 0.261211
\(288\) −1.00000 −0.0589256
\(289\) −5.82483 −0.342637
\(290\) 13.2193 0.776264
\(291\) −2.20578 −0.129305
\(292\) 10.0154 0.586105
\(293\) −0.436085 −0.0254764 −0.0127382 0.999919i \(-0.504055\pi\)
−0.0127382 + 0.999919i \(0.504055\pi\)
\(294\) 2.67649 0.156096
\(295\) −5.50263 −0.320375
\(296\) −6.91253 −0.401783
\(297\) −4.17067 −0.242007
\(298\) −5.60769 −0.324845
\(299\) −3.16834 −0.183230
\(300\) −2.04350 −0.117982
\(301\) 0.121571 0.00700726
\(302\) 21.8696 1.25846
\(303\) −18.0449 −1.03665
\(304\) 3.21672 0.184491
\(305\) −3.34355 −0.191451
\(306\) 3.34293 0.191103
\(307\) −14.4308 −0.823607 −0.411804 0.911273i \(-0.635101\pi\)
−0.411804 + 0.911273i \(0.635101\pi\)
\(308\) −8.67209 −0.494138
\(309\) −1.00000 −0.0568880
\(310\) −8.42315 −0.478403
\(311\) −29.3998 −1.66711 −0.833556 0.552435i \(-0.813699\pi\)
−0.833556 + 0.552435i \(0.813699\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −19.8482 −1.12188 −0.560942 0.827855i \(-0.689561\pi\)
−0.560942 + 0.827855i \(0.689561\pi\)
\(314\) 6.73068 0.379834
\(315\) −3.57525 −0.201443
\(316\) 5.26682 0.296282
\(317\) −15.1910 −0.853212 −0.426606 0.904438i \(-0.640291\pi\)
−0.426606 + 0.904438i \(0.640291\pi\)
\(318\) 4.72692 0.265073
\(319\) −32.0646 −1.79527
\(320\) −1.71945 −0.0961200
\(321\) −12.7003 −0.708859
\(322\) 6.58795 0.367132
\(323\) −10.7533 −0.598327
\(324\) 1.00000 0.0555556
\(325\) −2.04350 −0.113353
\(326\) 5.72060 0.316835
\(327\) 6.46568 0.357553
\(328\) −2.12821 −0.117511
\(329\) 17.1366 0.944773
\(330\) −7.17124 −0.394764
\(331\) 6.93505 0.381185 0.190592 0.981669i \(-0.438959\pi\)
0.190592 + 0.981669i \(0.438959\pi\)
\(332\) −16.0174 −0.879070
\(333\) 6.91253 0.378804
\(334\) −14.9623 −0.818702
\(335\) 14.5746 0.796298
\(336\) 2.07930 0.113435
\(337\) −8.13952 −0.443388 −0.221694 0.975116i \(-0.571159\pi\)
−0.221694 + 0.975116i \(0.571159\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 12.2618 0.665972
\(340\) 5.74799 0.311728
\(341\) 20.4311 1.10641
\(342\) −3.21672 −0.173940
\(343\) −20.1204 −1.08640
\(344\) −0.0584673 −0.00315235
\(345\) 5.44779 0.293299
\(346\) 20.0856 1.07981
\(347\) −1.37702 −0.0739221 −0.0369611 0.999317i \(-0.511768\pi\)
−0.0369611 + 0.999317i \(0.511768\pi\)
\(348\) 7.68812 0.412126
\(349\) 26.4328 1.41491 0.707457 0.706756i \(-0.249842\pi\)
0.707457 + 0.706756i \(0.249842\pi\)
\(350\) 4.24907 0.227122
\(351\) 1.00000 0.0533761
\(352\) 4.17067 0.222297
\(353\) −16.4525 −0.875680 −0.437840 0.899053i \(-0.644256\pi\)
−0.437840 + 0.899053i \(0.644256\pi\)
\(354\) −3.20023 −0.170090
\(355\) 3.02864 0.160744
\(356\) −7.65296 −0.405606
\(357\) −6.95097 −0.367884
\(358\) 19.1234 1.01070
\(359\) 10.5179 0.555112 0.277556 0.960709i \(-0.410476\pi\)
0.277556 + 0.960709i \(0.410476\pi\)
\(360\) 1.71945 0.0906228
\(361\) −8.65272 −0.455407
\(362\) 6.63039 0.348486
\(363\) 6.39447 0.335623
\(364\) 2.07930 0.108985
\(365\) −17.2209 −0.901383
\(366\) −1.94455 −0.101643
\(367\) 5.07178 0.264745 0.132372 0.991200i \(-0.457741\pi\)
0.132372 + 0.991200i \(0.457741\pi\)
\(368\) −3.16834 −0.165161
\(369\) 2.12821 0.110790
\(370\) 11.8857 0.617910
\(371\) −9.82871 −0.510281
\(372\) −4.89876 −0.253989
\(373\) −17.1209 −0.886485 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(374\) −13.9422 −0.720936
\(375\) 12.1109 0.625406
\(376\) −8.24152 −0.425024
\(377\) 7.68812 0.395958
\(378\) −2.07930 −0.106948
\(379\) 15.0132 0.771175 0.385587 0.922671i \(-0.373999\pi\)
0.385587 + 0.922671i \(0.373999\pi\)
\(380\) −5.53097 −0.283733
\(381\) 2.84286 0.145644
\(382\) 9.22135 0.471805
\(383\) 23.8022 1.21623 0.608117 0.793848i \(-0.291925\pi\)
0.608117 + 0.793848i \(0.291925\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 14.9112 0.759945
\(386\) 4.66229 0.237304
\(387\) 0.0584673 0.00297206
\(388\) −2.20578 −0.111982
\(389\) −28.7020 −1.45525 −0.727623 0.685977i \(-0.759375\pi\)
−0.727623 + 0.685977i \(0.759375\pi\)
\(390\) 1.71945 0.0870675
\(391\) 10.5915 0.535637
\(392\) 2.67649 0.135183
\(393\) −19.9421 −1.00595
\(394\) −8.89204 −0.447975
\(395\) −9.05601 −0.455658
\(396\) −4.17067 −0.209584
\(397\) 1.63976 0.0822974 0.0411487 0.999153i \(-0.486898\pi\)
0.0411487 + 0.999153i \(0.486898\pi\)
\(398\) 13.4916 0.676270
\(399\) 6.68854 0.334846
\(400\) −2.04350 −0.102175
\(401\) −1.73416 −0.0865996 −0.0432998 0.999062i \(-0.513787\pi\)
−0.0432998 + 0.999062i \(0.513787\pi\)
\(402\) 8.47636 0.422762
\(403\) −4.89876 −0.244025
\(404\) −18.0449 −0.897765
\(405\) −1.71945 −0.0854400
\(406\) −15.9859 −0.793369
\(407\) −28.8299 −1.42904
\(408\) 3.34293 0.165500
\(409\) −16.4451 −0.813160 −0.406580 0.913615i \(-0.633279\pi\)
−0.406580 + 0.913615i \(0.633279\pi\)
\(410\) 3.65935 0.180722
\(411\) −13.6536 −0.673481
\(412\) −1.00000 −0.0492665
\(413\) 6.65426 0.327435
\(414\) 3.16834 0.155715
\(415\) 27.5411 1.35194
\(416\) −1.00000 −0.0490290
\(417\) 15.7883 0.773158
\(418\) 13.4159 0.656191
\(419\) −4.04844 −0.197779 −0.0988897 0.995098i \(-0.531529\pi\)
−0.0988897 + 0.995098i \(0.531529\pi\)
\(420\) −3.57525 −0.174454
\(421\) −25.1134 −1.22395 −0.611975 0.790877i \(-0.709625\pi\)
−0.611975 + 0.790877i \(0.709625\pi\)
\(422\) 7.92962 0.386008
\(423\) 8.24152 0.400716
\(424\) 4.72692 0.229560
\(425\) 6.83129 0.331366
\(426\) 1.76140 0.0853403
\(427\) 4.04331 0.195670
\(428\) −12.7003 −0.613890
\(429\) −4.17067 −0.201362
\(430\) 0.100531 0.00484806
\(431\) 13.7857 0.664035 0.332018 0.943273i \(-0.392271\pi\)
0.332018 + 0.943273i \(0.392271\pi\)
\(432\) 1.00000 0.0481125
\(433\) −27.8209 −1.33699 −0.668493 0.743718i \(-0.733060\pi\)
−0.668493 + 0.743718i \(0.733060\pi\)
\(434\) 10.1860 0.488944
\(435\) −13.2193 −0.633817
\(436\) 6.46568 0.309650
\(437\) −10.1917 −0.487533
\(438\) −10.0154 −0.478553
\(439\) −0.721623 −0.0344412 −0.0172206 0.999852i \(-0.505482\pi\)
−0.0172206 + 0.999852i \(0.505482\pi\)
\(440\) −7.17124 −0.341875
\(441\) −2.67649 −0.127452
\(442\) 3.34293 0.159007
\(443\) −28.5213 −1.35509 −0.677543 0.735483i \(-0.736955\pi\)
−0.677543 + 0.735483i \(0.736955\pi\)
\(444\) 6.91253 0.328054
\(445\) 13.1588 0.623789
\(446\) 3.35526 0.158876
\(447\) 5.60769 0.265235
\(448\) 2.07930 0.0982379
\(449\) −21.0796 −0.994808 −0.497404 0.867519i \(-0.665713\pi\)
−0.497404 + 0.867519i \(0.665713\pi\)
\(450\) 2.04350 0.0963317
\(451\) −8.87607 −0.417958
\(452\) 12.2618 0.576748
\(453\) −21.8696 −1.02752
\(454\) 1.28365 0.0602445
\(455\) −3.57525 −0.167610
\(456\) −3.21672 −0.150637
\(457\) 37.2739 1.74360 0.871799 0.489863i \(-0.162953\pi\)
0.871799 + 0.489863i \(0.162953\pi\)
\(458\) 6.54398 0.305780
\(459\) −3.34293 −0.156035
\(460\) 5.44779 0.254005
\(461\) −27.0955 −1.26197 −0.630983 0.775797i \(-0.717348\pi\)
−0.630983 + 0.775797i \(0.717348\pi\)
\(462\) 8.67209 0.403462
\(463\) 4.26509 0.198215 0.0991077 0.995077i \(-0.468401\pi\)
0.0991077 + 0.995077i \(0.468401\pi\)
\(464\) 7.68812 0.356912
\(465\) 8.42315 0.390614
\(466\) 1.47551 0.0683515
\(467\) −10.4283 −0.482563 −0.241282 0.970455i \(-0.577568\pi\)
−0.241282 + 0.970455i \(0.577568\pi\)
\(468\) 1.00000 0.0462250
\(469\) −17.6249 −0.813844
\(470\) 14.1708 0.653653
\(471\) −6.73068 −0.310133
\(472\) −3.20023 −0.147303
\(473\) −0.243848 −0.0112121
\(474\) −5.26682 −0.241913
\(475\) −6.57338 −0.301607
\(476\) −6.95097 −0.318597
\(477\) −4.72692 −0.216431
\(478\) −18.4065 −0.841894
\(479\) 28.8986 1.32041 0.660206 0.751084i \(-0.270469\pi\)
0.660206 + 0.751084i \(0.270469\pi\)
\(480\) 1.71945 0.0784816
\(481\) 6.91253 0.315184
\(482\) 7.89644 0.359673
\(483\) −6.58795 −0.299762
\(484\) 6.39447 0.290658
\(485\) 3.79273 0.172219
\(486\) −1.00000 −0.0453609
\(487\) 12.2855 0.556709 0.278354 0.960478i \(-0.410211\pi\)
0.278354 + 0.960478i \(0.410211\pi\)
\(488\) −1.94455 −0.0880257
\(489\) −5.72060 −0.258695
\(490\) −4.60208 −0.207901
\(491\) −12.1091 −0.546478 −0.273239 0.961946i \(-0.588095\pi\)
−0.273239 + 0.961946i \(0.588095\pi\)
\(492\) 2.12821 0.0959473
\(493\) −25.7008 −1.15751
\(494\) −3.21672 −0.144727
\(495\) 7.17124 0.322323
\(496\) −4.89876 −0.219961
\(497\) −3.66250 −0.164285
\(498\) 16.0174 0.717757
\(499\) 22.1382 0.991043 0.495522 0.868596i \(-0.334977\pi\)
0.495522 + 0.868596i \(0.334977\pi\)
\(500\) 12.1109 0.541617
\(501\) 14.9623 0.668468
\(502\) −13.4455 −0.600103
\(503\) 8.26073 0.368328 0.184164 0.982896i \(-0.441042\pi\)
0.184164 + 0.982896i \(0.441042\pi\)
\(504\) −2.07930 −0.0926196
\(505\) 31.0272 1.38069
\(506\) −13.2141 −0.587438
\(507\) 1.00000 0.0444116
\(508\) 2.84286 0.126132
\(509\) −25.7853 −1.14291 −0.571456 0.820632i \(-0.693621\pi\)
−0.571456 + 0.820632i \(0.693621\pi\)
\(510\) −5.74799 −0.254525
\(511\) 20.8250 0.921244
\(512\) −1.00000 −0.0441942
\(513\) 3.21672 0.142022
\(514\) −25.5686 −1.12778
\(515\) 1.71945 0.0757679
\(516\) 0.0584673 0.00257388
\(517\) −34.3726 −1.51171
\(518\) −14.3733 −0.631525
\(519\) −20.0856 −0.881659
\(520\) 1.71945 0.0754027
\(521\) −14.2152 −0.622778 −0.311389 0.950283i \(-0.600794\pi\)
−0.311389 + 0.950283i \(0.600794\pi\)
\(522\) −7.68812 −0.336500
\(523\) −33.4547 −1.46287 −0.731435 0.681911i \(-0.761149\pi\)
−0.731435 + 0.681911i \(0.761149\pi\)
\(524\) −19.9421 −0.871176
\(525\) −4.24907 −0.185445
\(526\) 9.78932 0.426835
\(527\) 16.3762 0.713359
\(528\) −4.17067 −0.181505
\(529\) −12.9616 −0.563548
\(530\) −8.12769 −0.353044
\(531\) 3.20023 0.138878
\(532\) 6.68854 0.289985
\(533\) 2.12821 0.0921831
\(534\) 7.65296 0.331176
\(535\) 21.8374 0.944114
\(536\) 8.47636 0.366123
\(537\) −19.1234 −0.825235
\(538\) −28.9797 −1.24940
\(539\) 11.1628 0.480814
\(540\) −1.71945 −0.0739932
\(541\) −10.5204 −0.452307 −0.226154 0.974092i \(-0.572615\pi\)
−0.226154 + 0.974092i \(0.572615\pi\)
\(542\) 1.76169 0.0756709
\(543\) −6.63039 −0.284537
\(544\) 3.34293 0.143327
\(545\) −11.1174 −0.476217
\(546\) −2.07930 −0.0889860
\(547\) −30.4223 −1.30076 −0.650381 0.759608i \(-0.725391\pi\)
−0.650381 + 0.759608i \(0.725391\pi\)
\(548\) −13.6536 −0.583252
\(549\) 1.94455 0.0829914
\(550\) −8.52278 −0.363412
\(551\) 24.7305 1.05355
\(552\) 3.16834 0.134854
\(553\) 10.9513 0.465698
\(554\) −20.3481 −0.864509
\(555\) −11.8857 −0.504521
\(556\) 15.7883 0.669574
\(557\) 10.0678 0.426586 0.213293 0.976988i \(-0.431581\pi\)
0.213293 + 0.976988i \(0.431581\pi\)
\(558\) 4.89876 0.207381
\(559\) 0.0584673 0.00247290
\(560\) −3.57525 −0.151082
\(561\) 13.9422 0.588642
\(562\) 2.83942 0.119774
\(563\) 12.6046 0.531222 0.265611 0.964080i \(-0.414426\pi\)
0.265611 + 0.964080i \(0.414426\pi\)
\(564\) 8.24152 0.347031
\(565\) −21.0836 −0.886993
\(566\) 12.5214 0.526315
\(567\) 2.07930 0.0873226
\(568\) 1.76140 0.0739069
\(569\) 34.6331 1.45190 0.725948 0.687750i \(-0.241401\pi\)
0.725948 + 0.687750i \(0.241401\pi\)
\(570\) 5.53097 0.231667
\(571\) 16.8351 0.704527 0.352263 0.935901i \(-0.385412\pi\)
0.352263 + 0.935901i \(0.385412\pi\)
\(572\) −4.17067 −0.174384
\(573\) −9.22135 −0.385227
\(574\) −4.42520 −0.184704
\(575\) 6.47452 0.270006
\(576\) 1.00000 0.0416667
\(577\) 11.1484 0.464112 0.232056 0.972702i \(-0.425455\pi\)
0.232056 + 0.972702i \(0.425455\pi\)
\(578\) 5.82483 0.242281
\(579\) −4.66229 −0.193758
\(580\) −13.2193 −0.548902
\(581\) −33.3051 −1.38173
\(582\) 2.20578 0.0914327
\(583\) 19.7144 0.816488
\(584\) −10.0154 −0.414439
\(585\) −1.71945 −0.0710904
\(586\) 0.436085 0.0180145
\(587\) 18.7838 0.775289 0.387644 0.921809i \(-0.373289\pi\)
0.387644 + 0.921809i \(0.373289\pi\)
\(588\) −2.67649 −0.110377
\(589\) −15.7579 −0.649294
\(590\) 5.50263 0.226540
\(591\) 8.89204 0.365770
\(592\) 6.91253 0.284103
\(593\) 3.38313 0.138928 0.0694642 0.997584i \(-0.477871\pi\)
0.0694642 + 0.997584i \(0.477871\pi\)
\(594\) 4.17067 0.171125
\(595\) 11.9518 0.489977
\(596\) 5.60769 0.229700
\(597\) −13.4916 −0.552172
\(598\) 3.16834 0.129563
\(599\) 4.98938 0.203861 0.101930 0.994792i \(-0.467498\pi\)
0.101930 + 0.994792i \(0.467498\pi\)
\(600\) 2.04350 0.0834257
\(601\) 15.8393 0.646100 0.323050 0.946382i \(-0.395292\pi\)
0.323050 + 0.946382i \(0.395292\pi\)
\(602\) −0.121571 −0.00495488
\(603\) −8.47636 −0.345184
\(604\) −21.8696 −0.889862
\(605\) −10.9949 −0.447008
\(606\) 18.0449 0.733022
\(607\) −16.6413 −0.675449 −0.337725 0.941245i \(-0.609657\pi\)
−0.337725 + 0.941245i \(0.609657\pi\)
\(608\) −3.21672 −0.130455
\(609\) 15.9859 0.647783
\(610\) 3.34355 0.135376
\(611\) 8.24152 0.333416
\(612\) −3.34293 −0.135130
\(613\) 34.6382 1.39902 0.699512 0.714621i \(-0.253401\pi\)
0.699512 + 0.714621i \(0.253401\pi\)
\(614\) 14.4308 0.582378
\(615\) −3.65935 −0.147559
\(616\) 8.67209 0.349408
\(617\) −10.2630 −0.413172 −0.206586 0.978428i \(-0.566235\pi\)
−0.206586 + 0.978428i \(0.566235\pi\)
\(618\) 1.00000 0.0402259
\(619\) 2.78018 0.111745 0.0558725 0.998438i \(-0.482206\pi\)
0.0558725 + 0.998438i \(0.482206\pi\)
\(620\) 8.42315 0.338282
\(621\) −3.16834 −0.127141
\(622\) 29.3998 1.17883
\(623\) −15.9128 −0.637534
\(624\) 1.00000 0.0400320
\(625\) −10.6066 −0.424263
\(626\) 19.8482 0.793292
\(627\) −13.4159 −0.535778
\(628\) −6.73068 −0.268583
\(629\) −23.1081 −0.921380
\(630\) 3.57525 0.142441
\(631\) −30.4086 −1.21055 −0.605273 0.796018i \(-0.706936\pi\)
−0.605273 + 0.796018i \(0.706936\pi\)
\(632\) −5.26682 −0.209503
\(633\) −7.92962 −0.315174
\(634\) 15.1910 0.603312
\(635\) −4.88815 −0.193980
\(636\) −4.72692 −0.187435
\(637\) −2.67649 −0.106046
\(638\) 32.0646 1.26945
\(639\) −1.76140 −0.0696801
\(640\) 1.71945 0.0679671
\(641\) −3.64584 −0.144002 −0.0720009 0.997405i \(-0.522938\pi\)
−0.0720009 + 0.997405i \(0.522938\pi\)
\(642\) 12.7003 0.501239
\(643\) −14.2838 −0.563299 −0.281649 0.959517i \(-0.590882\pi\)
−0.281649 + 0.959517i \(0.590882\pi\)
\(644\) −6.58795 −0.259601
\(645\) −0.100531 −0.00395842
\(646\) 10.7533 0.423081
\(647\) −29.0889 −1.14360 −0.571801 0.820392i \(-0.693755\pi\)
−0.571801 + 0.820392i \(0.693755\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −13.3471 −0.523920
\(650\) 2.04350 0.0801528
\(651\) −10.1860 −0.399221
\(652\) −5.72060 −0.224036
\(653\) −43.4883 −1.70183 −0.850915 0.525304i \(-0.823952\pi\)
−0.850915 + 0.525304i \(0.823952\pi\)
\(654\) −6.46568 −0.252828
\(655\) 34.2894 1.33980
\(656\) 2.12821 0.0830928
\(657\) 10.0154 0.390737
\(658\) −17.1366 −0.668055
\(659\) 35.2543 1.37331 0.686657 0.726982i \(-0.259077\pi\)
0.686657 + 0.726982i \(0.259077\pi\)
\(660\) 7.17124 0.279140
\(661\) 18.7162 0.727975 0.363987 0.931404i \(-0.381415\pi\)
0.363987 + 0.931404i \(0.381415\pi\)
\(662\) −6.93505 −0.269538
\(663\) −3.34293 −0.129829
\(664\) 16.0174 0.621596
\(665\) −11.5006 −0.445973
\(666\) −6.91253 −0.267855
\(667\) −24.3586 −0.943168
\(668\) 14.9623 0.578910
\(669\) −3.35526 −0.129722
\(670\) −14.5746 −0.563068
\(671\) −8.11008 −0.313086
\(672\) −2.07930 −0.0802109
\(673\) 38.1951 1.47231 0.736156 0.676812i \(-0.236639\pi\)
0.736156 + 0.676812i \(0.236639\pi\)
\(674\) 8.13952 0.313523
\(675\) −2.04350 −0.0786545
\(676\) 1.00000 0.0384615
\(677\) 48.1041 1.84879 0.924395 0.381437i \(-0.124571\pi\)
0.924395 + 0.381437i \(0.124571\pi\)
\(678\) −12.2618 −0.470913
\(679\) −4.58650 −0.176014
\(680\) −5.74799 −0.220425
\(681\) −1.28365 −0.0491894
\(682\) −20.4311 −0.782347
\(683\) −17.2269 −0.659170 −0.329585 0.944126i \(-0.606909\pi\)
−0.329585 + 0.944126i \(0.606909\pi\)
\(684\) 3.21672 0.122994
\(685\) 23.4766 0.896994
\(686\) 20.1204 0.768199
\(687\) −6.54398 −0.249668
\(688\) 0.0584673 0.00222905
\(689\) −4.72692 −0.180081
\(690\) −5.44779 −0.207394
\(691\) −9.00548 −0.342585 −0.171292 0.985220i \(-0.554794\pi\)
−0.171292 + 0.985220i \(0.554794\pi\)
\(692\) −20.0856 −0.763539
\(693\) −8.67209 −0.329425
\(694\) 1.37702 0.0522708
\(695\) −27.1472 −1.02975
\(696\) −7.68812 −0.291417
\(697\) −7.11446 −0.269480
\(698\) −26.4328 −1.00050
\(699\) −1.47551 −0.0558088
\(700\) −4.24907 −0.160600
\(701\) 32.0734 1.21140 0.605698 0.795695i \(-0.292894\pi\)
0.605698 + 0.795695i \(0.292894\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 22.2357 0.838634
\(704\) −4.17067 −0.157188
\(705\) −14.1708 −0.533705
\(706\) 16.4525 0.619199
\(707\) −37.5208 −1.41111
\(708\) 3.20023 0.120272
\(709\) 27.1827 1.02087 0.510433 0.859917i \(-0.329485\pi\)
0.510433 + 0.859917i \(0.329485\pi\)
\(710\) −3.02864 −0.113663
\(711\) 5.26682 0.197521
\(712\) 7.65296 0.286807
\(713\) 15.5209 0.581264
\(714\) 6.95097 0.260133
\(715\) 7.17124 0.268189
\(716\) −19.1234 −0.714675
\(717\) 18.4065 0.687404
\(718\) −10.5179 −0.392523
\(719\) 2.28008 0.0850325 0.0425163 0.999096i \(-0.486463\pi\)
0.0425163 + 0.999096i \(0.486463\pi\)
\(720\) −1.71945 −0.0640800
\(721\) −2.07930 −0.0774374
\(722\) 8.65272 0.322021
\(723\) −7.89644 −0.293672
\(724\) −6.63039 −0.246417
\(725\) −15.7107 −0.583481
\(726\) −6.39447 −0.237321
\(727\) −40.2051 −1.49113 −0.745563 0.666435i \(-0.767819\pi\)
−0.745563 + 0.666435i \(0.767819\pi\)
\(728\) −2.07930 −0.0770642
\(729\) 1.00000 0.0370370
\(730\) 17.2209 0.637374
\(731\) −0.195452 −0.00722906
\(732\) 1.94455 0.0718727
\(733\) −33.7731 −1.24744 −0.623718 0.781649i \(-0.714379\pi\)
−0.623718 + 0.781649i \(0.714379\pi\)
\(734\) −5.07178 −0.187203
\(735\) 4.60208 0.169750
\(736\) 3.16834 0.116787
\(737\) 35.3521 1.30221
\(738\) −2.12821 −0.0783406
\(739\) −8.14184 −0.299503 −0.149751 0.988724i \(-0.547847\pi\)
−0.149751 + 0.988724i \(0.547847\pi\)
\(740\) −11.8857 −0.436928
\(741\) 3.21672 0.118169
\(742\) 9.82871 0.360823
\(743\) 13.1760 0.483380 0.241690 0.970354i \(-0.422298\pi\)
0.241690 + 0.970354i \(0.422298\pi\)
\(744\) 4.89876 0.179597
\(745\) −9.64212 −0.353260
\(746\) 17.1209 0.626839
\(747\) −16.0174 −0.586046
\(748\) 13.9422 0.509779
\(749\) −26.4077 −0.964917
\(750\) −12.1109 −0.442229
\(751\) −32.4857 −1.18542 −0.592709 0.805416i \(-0.701942\pi\)
−0.592709 + 0.805416i \(0.701942\pi\)
\(752\) 8.24152 0.300537
\(753\) 13.4455 0.489982
\(754\) −7.68812 −0.279985
\(755\) 37.6036 1.36854
\(756\) 2.07930 0.0756236
\(757\) 28.5693 1.03837 0.519184 0.854662i \(-0.326236\pi\)
0.519184 + 0.854662i \(0.326236\pi\)
\(758\) −15.0132 −0.545303
\(759\) 13.2141 0.479641
\(760\) 5.53097 0.200630
\(761\) −16.9883 −0.615826 −0.307913 0.951414i \(-0.599631\pi\)
−0.307913 + 0.951414i \(0.599631\pi\)
\(762\) −2.84286 −0.102986
\(763\) 13.4441 0.486710
\(764\) −9.22135 −0.333617
\(765\) 5.74799 0.207819
\(766\) −23.8022 −0.860007
\(767\) 3.20023 0.115554
\(768\) 1.00000 0.0360844
\(769\) −27.7040 −0.999031 −0.499515 0.866305i \(-0.666489\pi\)
−0.499515 + 0.866305i \(0.666489\pi\)
\(770\) −14.9112 −0.537362
\(771\) 25.5686 0.920830
\(772\) −4.66229 −0.167800
\(773\) −19.0774 −0.686167 −0.343084 0.939305i \(-0.611471\pi\)
−0.343084 + 0.939305i \(0.611471\pi\)
\(774\) −0.0584673 −0.00210156
\(775\) 10.0106 0.359593
\(776\) 2.20578 0.0791831
\(777\) 14.3733 0.515638
\(778\) 28.7020 1.02901
\(779\) 6.84586 0.245278
\(780\) −1.71945 −0.0615661
\(781\) 7.34623 0.262869
\(782\) −10.5915 −0.378753
\(783\) 7.68812 0.274751
\(784\) −2.67649 −0.0955890
\(785\) 11.5730 0.413059
\(786\) 19.9421 0.711312
\(787\) 23.7380 0.846168 0.423084 0.906090i \(-0.360947\pi\)
0.423084 + 0.906090i \(0.360947\pi\)
\(788\) 8.89204 0.316766
\(789\) −9.78932 −0.348509
\(790\) 9.05601 0.322199
\(791\) 25.4961 0.906537
\(792\) 4.17067 0.148198
\(793\) 1.94455 0.0690530
\(794\) −1.63976 −0.0581930
\(795\) 8.12769 0.288259
\(796\) −13.4916 −0.478195
\(797\) 11.4045 0.403969 0.201984 0.979389i \(-0.435261\pi\)
0.201984 + 0.979389i \(0.435261\pi\)
\(798\) −6.68854 −0.236772
\(799\) −27.5508 −0.974678
\(800\) 2.04350 0.0722488
\(801\) −7.65296 −0.270404
\(802\) 1.73416 0.0612352
\(803\) −41.7708 −1.47406
\(804\) −8.47636 −0.298938
\(805\) 11.3276 0.399246
\(806\) 4.89876 0.172551
\(807\) 28.9797 1.02013
\(808\) 18.0449 0.634816
\(809\) −30.3731 −1.06786 −0.533931 0.845528i \(-0.679286\pi\)
−0.533931 + 0.845528i \(0.679286\pi\)
\(810\) 1.71945 0.0604152
\(811\) 39.4757 1.38618 0.693090 0.720851i \(-0.256249\pi\)
0.693090 + 0.720851i \(0.256249\pi\)
\(812\) 15.9859 0.560996
\(813\) −1.76169 −0.0617851
\(814\) 28.8299 1.01049
\(815\) 9.83627 0.344549
\(816\) −3.34293 −0.117026
\(817\) 0.188073 0.00657984
\(818\) 16.4451 0.574991
\(819\) 2.07930 0.0726568
\(820\) −3.65935 −0.127790
\(821\) −17.0232 −0.594114 −0.297057 0.954860i \(-0.596005\pi\)
−0.297057 + 0.954860i \(0.596005\pi\)
\(822\) 13.6536 0.476223
\(823\) −8.37425 −0.291908 −0.145954 0.989291i \(-0.546625\pi\)
−0.145954 + 0.989291i \(0.546625\pi\)
\(824\) 1.00000 0.0348367
\(825\) 8.52278 0.296725
\(826\) −6.65426 −0.231531
\(827\) 40.5432 1.40982 0.704912 0.709295i \(-0.250987\pi\)
0.704912 + 0.709295i \(0.250987\pi\)
\(828\) −3.16834 −0.110107
\(829\) 17.5392 0.609161 0.304580 0.952487i \(-0.401484\pi\)
0.304580 + 0.952487i \(0.401484\pi\)
\(830\) −27.5411 −0.955965
\(831\) 20.3481 0.705869
\(832\) 1.00000 0.0346688
\(833\) 8.94732 0.310006
\(834\) −15.7883 −0.546705
\(835\) −25.7269 −0.890317
\(836\) −13.4159 −0.463997
\(837\) −4.89876 −0.169326
\(838\) 4.04844 0.139851
\(839\) 34.2631 1.18289 0.591446 0.806344i \(-0.298557\pi\)
0.591446 + 0.806344i \(0.298557\pi\)
\(840\) 3.57525 0.123358
\(841\) 30.1071 1.03818
\(842\) 25.1134 0.865464
\(843\) −2.83942 −0.0977948
\(844\) −7.92962 −0.272949
\(845\) −1.71945 −0.0591508
\(846\) −8.24152 −0.283349
\(847\) 13.2960 0.456858
\(848\) −4.72692 −0.162323
\(849\) −12.5214 −0.429735
\(850\) −6.83129 −0.234311
\(851\) −21.9013 −0.750766
\(852\) −1.76140 −0.0603447
\(853\) −30.3174 −1.03805 −0.519025 0.854759i \(-0.673705\pi\)
−0.519025 + 0.854759i \(0.673705\pi\)
\(854\) −4.04331 −0.138359
\(855\) −5.53097 −0.189155
\(856\) 12.7003 0.434086
\(857\) −52.1434 −1.78118 −0.890592 0.454804i \(-0.849709\pi\)
−0.890592 + 0.454804i \(0.849709\pi\)
\(858\) 4.17067 0.142384
\(859\) 3.83046 0.130694 0.0653468 0.997863i \(-0.479185\pi\)
0.0653468 + 0.997863i \(0.479185\pi\)
\(860\) −0.100531 −0.00342809
\(861\) 4.42520 0.150811
\(862\) −13.7857 −0.469544
\(863\) −24.5175 −0.834585 −0.417293 0.908772i \(-0.637021\pi\)
−0.417293 + 0.908772i \(0.637021\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 34.5361 1.17426
\(866\) 27.8209 0.945392
\(867\) −5.82483 −0.197821
\(868\) −10.1860 −0.345736
\(869\) −21.9662 −0.745151
\(870\) 13.2193 0.448176
\(871\) −8.47636 −0.287210
\(872\) −6.46568 −0.218956
\(873\) −2.20578 −0.0746545
\(874\) 10.1917 0.344738
\(875\) 25.1823 0.851317
\(876\) 10.0154 0.338388
\(877\) −34.2882 −1.15783 −0.578916 0.815387i \(-0.696524\pi\)
−0.578916 + 0.815387i \(0.696524\pi\)
\(878\) 0.721623 0.0243536
\(879\) −0.436085 −0.0147088
\(880\) 7.17124 0.241742
\(881\) −42.2080 −1.42202 −0.711011 0.703181i \(-0.751763\pi\)
−0.711011 + 0.703181i \(0.751763\pi\)
\(882\) 2.67649 0.0901222
\(883\) 35.5092 1.19498 0.597490 0.801876i \(-0.296165\pi\)
0.597490 + 0.801876i \(0.296165\pi\)
\(884\) −3.34293 −0.112435
\(885\) −5.50263 −0.184969
\(886\) 28.5213 0.958190
\(887\) −13.6487 −0.458280 −0.229140 0.973394i \(-0.573591\pi\)
−0.229140 + 0.973394i \(0.573591\pi\)
\(888\) −6.91253 −0.231969
\(889\) 5.91118 0.198254
\(890\) −13.1588 −0.441086
\(891\) −4.17067 −0.139723
\(892\) −3.35526 −0.112342
\(893\) 26.5106 0.887145
\(894\) −5.60769 −0.187549
\(895\) 32.8816 1.09911
\(896\) −2.07930 −0.0694647
\(897\) −3.16834 −0.105788
\(898\) 21.0796 0.703435
\(899\) −37.6622 −1.25611
\(900\) −2.04350 −0.0681168
\(901\) 15.8018 0.526433
\(902\) 8.87607 0.295541
\(903\) 0.121571 0.00404564
\(904\) −12.2618 −0.407823
\(905\) 11.4006 0.378969
\(906\) 21.8696 0.726570
\(907\) −18.7493 −0.622561 −0.311280 0.950318i \(-0.600758\pi\)
−0.311280 + 0.950318i \(0.600758\pi\)
\(908\) −1.28365 −0.0425993
\(909\) −18.0449 −0.598510
\(910\) 3.57525 0.118518
\(911\) −27.0544 −0.896351 −0.448176 0.893946i \(-0.647926\pi\)
−0.448176 + 0.893946i \(0.647926\pi\)
\(912\) 3.21672 0.106516
\(913\) 66.8033 2.21087
\(914\) −37.2739 −1.23291
\(915\) −3.34355 −0.110534
\(916\) −6.54398 −0.216219
\(917\) −41.4658 −1.36932
\(918\) 3.34293 0.110333
\(919\) 29.4764 0.972337 0.486169 0.873865i \(-0.338394\pi\)
0.486169 + 0.873865i \(0.338394\pi\)
\(920\) −5.44779 −0.179608
\(921\) −14.4308 −0.475510
\(922\) 27.0955 0.892344
\(923\) −1.76140 −0.0579773
\(924\) −8.67209 −0.285291
\(925\) −14.1258 −0.464453
\(926\) −4.26509 −0.140160
\(927\) −1.00000 −0.0328443
\(928\) −7.68812 −0.252375
\(929\) −11.3610 −0.372742 −0.186371 0.982479i \(-0.559673\pi\)
−0.186371 + 0.982479i \(0.559673\pi\)
\(930\) −8.42315 −0.276206
\(931\) −8.60952 −0.282166
\(932\) −1.47551 −0.0483318
\(933\) −29.3998 −0.962507
\(934\) 10.4283 0.341224
\(935\) −23.9729 −0.783999
\(936\) −1.00000 −0.0326860
\(937\) 14.4433 0.471842 0.235921 0.971772i \(-0.424189\pi\)
0.235921 + 0.971772i \(0.424189\pi\)
\(938\) 17.6249 0.575474
\(939\) −19.8482 −0.647721
\(940\) −14.1708 −0.462202
\(941\) 53.6009 1.74734 0.873671 0.486518i \(-0.161733\pi\)
0.873671 + 0.486518i \(0.161733\pi\)
\(942\) 6.73068 0.219297
\(943\) −6.74291 −0.219579
\(944\) 3.20023 0.104159
\(945\) −3.57525 −0.116303
\(946\) 0.243848 0.00792817
\(947\) −12.5622 −0.408218 −0.204109 0.978948i \(-0.565430\pi\)
−0.204109 + 0.978948i \(0.565430\pi\)
\(948\) 5.26682 0.171058
\(949\) 10.0154 0.325113
\(950\) 6.57338 0.213269
\(951\) −15.1910 −0.492602
\(952\) 6.95097 0.225282
\(953\) 38.6404 1.25169 0.625843 0.779949i \(-0.284755\pi\)
0.625843 + 0.779949i \(0.284755\pi\)
\(954\) 4.72692 0.153040
\(955\) 15.8556 0.513076
\(956\) 18.4065 0.595309
\(957\) −32.0646 −1.03650
\(958\) −28.8986 −0.933673
\(959\) −28.3899 −0.916759
\(960\) −1.71945 −0.0554949
\(961\) −7.00216 −0.225876
\(962\) −6.91253 −0.222869
\(963\) −12.7003 −0.409260
\(964\) −7.89644 −0.254327
\(965\) 8.01656 0.258062
\(966\) 6.58795 0.211964
\(967\) −11.8396 −0.380735 −0.190367 0.981713i \(-0.560968\pi\)
−0.190367 + 0.981713i \(0.560968\pi\)
\(968\) −6.39447 −0.205526
\(969\) −10.7533 −0.345444
\(970\) −3.79273 −0.121777
\(971\) −7.00827 −0.224906 −0.112453 0.993657i \(-0.535871\pi\)
−0.112453 + 0.993657i \(0.535871\pi\)
\(972\) 1.00000 0.0320750
\(973\) 32.8288 1.05244
\(974\) −12.2855 −0.393653
\(975\) −2.04350 −0.0654445
\(976\) 1.94455 0.0622436
\(977\) 21.2603 0.680178 0.340089 0.940393i \(-0.389543\pi\)
0.340089 + 0.940393i \(0.389543\pi\)
\(978\) 5.72060 0.182925
\(979\) 31.9179 1.02010
\(980\) 4.60208 0.147008
\(981\) 6.46568 0.206433
\(982\) 12.1091 0.386418
\(983\) 3.68168 0.117427 0.0587136 0.998275i \(-0.481300\pi\)
0.0587136 + 0.998275i \(0.481300\pi\)
\(984\) −2.12821 −0.0678450
\(985\) −15.2894 −0.487161
\(986\) 25.7008 0.818481
\(987\) 17.1366 0.545465
\(988\) 3.21672 0.102337
\(989\) −0.185244 −0.00589043
\(990\) −7.17124 −0.227917
\(991\) −25.9361 −0.823886 −0.411943 0.911210i \(-0.635150\pi\)
−0.411943 + 0.911210i \(0.635150\pi\)
\(992\) 4.89876 0.155536
\(993\) 6.93505 0.220077
\(994\) 3.66250 0.116167
\(995\) 23.1980 0.735426
\(996\) −16.0174 −0.507531
\(997\) 10.1865 0.322609 0.161305 0.986905i \(-0.448430\pi\)
0.161305 + 0.986905i \(0.448430\pi\)
\(998\) −22.1382 −0.700774
\(999\) 6.91253 0.218703
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 8034.2.a.w.1.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.w.1.4 12 1.1 even 1 trivial