Properties

Label 8004.2.a.g.1.5
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} - 1156 x^{3} - 616 x^{2} + 136 x + 16\)
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 1.5
Root \(0.425546\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.425546 q^{5} -4.01837 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.425546 q^{5} -4.01837 q^{7} +1.00000 q^{9} +2.74669 q^{11} +2.75061 q^{13} +0.425546 q^{15} -0.750311 q^{17} -4.08585 q^{19} +4.01837 q^{21} +1.00000 q^{23} -4.81891 q^{25} -1.00000 q^{27} -1.00000 q^{29} +3.50717 q^{31} -2.74669 q^{33} +1.71000 q^{35} -5.25315 q^{37} -2.75061 q^{39} +11.5128 q^{41} +7.71894 q^{43} -0.425546 q^{45} +3.88894 q^{47} +9.14727 q^{49} +0.750311 q^{51} -0.561794 q^{53} -1.16884 q^{55} +4.08585 q^{57} -11.0832 q^{59} -14.0362 q^{61} -4.01837 q^{63} -1.17051 q^{65} +11.7333 q^{67} -1.00000 q^{69} +11.6376 q^{71} -8.17001 q^{73} +4.81891 q^{75} -11.0372 q^{77} -1.31646 q^{79} +1.00000 q^{81} +8.92356 q^{83} +0.319292 q^{85} +1.00000 q^{87} +2.90607 q^{89} -11.0530 q^{91} -3.50717 q^{93} +1.73872 q^{95} -17.7139 q^{97} +2.74669 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} - 5q^{11} - 6q^{13} + 3q^{15} - 7q^{17} - 3q^{19} - 4q^{21} + 12q^{23} + 11q^{25} - 12q^{27} - 12q^{29} + 2q^{31} + 5q^{33} - 9q^{35} - 20q^{37} + 6q^{39} - 3q^{41} + 5q^{43} - 3q^{45} - 2q^{49} + 7q^{51} - 3q^{53} + 19q^{55} + 3q^{57} - 20q^{59} - 17q^{61} + 4q^{63} - 4q^{65} - 9q^{67} - 12q^{69} + 7q^{71} - 9q^{73} - 11q^{75} - 34q^{77} + 14q^{79} + 12q^{81} + 5q^{83} - 12q^{85} + 12q^{87} - 22q^{89} - 3q^{91} - 2q^{93} - 27q^{95} + 17q^{97} - 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −0.425546 −0.190310 −0.0951549 0.995462i \(-0.530335\pi\)
−0.0951549 + 0.995462i \(0.530335\pi\)
\(6\) 0 0
\(7\) −4.01837 −1.51880 −0.759400 0.650624i \(-0.774507\pi\)
−0.759400 + 0.650624i \(0.774507\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.74669 0.828158 0.414079 0.910241i \(-0.364104\pi\)
0.414079 + 0.910241i \(0.364104\pi\)
\(12\) 0 0
\(13\) 2.75061 0.762883 0.381441 0.924393i \(-0.375428\pi\)
0.381441 + 0.924393i \(0.375428\pi\)
\(14\) 0 0
\(15\) 0.425546 0.109875
\(16\) 0 0
\(17\) −0.750311 −0.181977 −0.0909886 0.995852i \(-0.529003\pi\)
−0.0909886 + 0.995852i \(0.529003\pi\)
\(18\) 0 0
\(19\) −4.08585 −0.937359 −0.468679 0.883368i \(-0.655270\pi\)
−0.468679 + 0.883368i \(0.655270\pi\)
\(20\) 0 0
\(21\) 4.01837 0.876880
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.81891 −0.963782
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 3.50717 0.629906 0.314953 0.949107i \(-0.398011\pi\)
0.314953 + 0.949107i \(0.398011\pi\)
\(32\) 0 0
\(33\) −2.74669 −0.478137
\(34\) 0 0
\(35\) 1.71000 0.289043
\(36\) 0 0
\(37\) −5.25315 −0.863612 −0.431806 0.901966i \(-0.642124\pi\)
−0.431806 + 0.901966i \(0.642124\pi\)
\(38\) 0 0
\(39\) −2.75061 −0.440451
\(40\) 0 0
\(41\) 11.5128 1.79799 0.898997 0.437955i \(-0.144297\pi\)
0.898997 + 0.437955i \(0.144297\pi\)
\(42\) 0 0
\(43\) 7.71894 1.17713 0.588564 0.808451i \(-0.299694\pi\)
0.588564 + 0.808451i \(0.299694\pi\)
\(44\) 0 0
\(45\) −0.425546 −0.0634366
\(46\) 0 0
\(47\) 3.88894 0.567261 0.283630 0.958934i \(-0.408461\pi\)
0.283630 + 0.958934i \(0.408461\pi\)
\(48\) 0 0
\(49\) 9.14727 1.30675
\(50\) 0 0
\(51\) 0.750311 0.105065
\(52\) 0 0
\(53\) −0.561794 −0.0771683 −0.0385841 0.999255i \(-0.512285\pi\)
−0.0385841 + 0.999255i \(0.512285\pi\)
\(54\) 0 0
\(55\) −1.16884 −0.157607
\(56\) 0 0
\(57\) 4.08585 0.541184
\(58\) 0 0
\(59\) −11.0832 −1.44291 −0.721456 0.692460i \(-0.756527\pi\)
−0.721456 + 0.692460i \(0.756527\pi\)
\(60\) 0 0
\(61\) −14.0362 −1.79715 −0.898574 0.438821i \(-0.855396\pi\)
−0.898574 + 0.438821i \(0.855396\pi\)
\(62\) 0 0
\(63\) −4.01837 −0.506267
\(64\) 0 0
\(65\) −1.17051 −0.145184
\(66\) 0 0
\(67\) 11.7333 1.43345 0.716724 0.697357i \(-0.245641\pi\)
0.716724 + 0.697357i \(0.245641\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 11.6376 1.38113 0.690563 0.723272i \(-0.257363\pi\)
0.690563 + 0.723272i \(0.257363\pi\)
\(72\) 0 0
\(73\) −8.17001 −0.956228 −0.478114 0.878298i \(-0.658679\pi\)
−0.478114 + 0.878298i \(0.658679\pi\)
\(74\) 0 0
\(75\) 4.81891 0.556440
\(76\) 0 0
\(77\) −11.0372 −1.25781
\(78\) 0 0
\(79\) −1.31646 −0.148113 −0.0740565 0.997254i \(-0.523595\pi\)
−0.0740565 + 0.997254i \(0.523595\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.92356 0.979488 0.489744 0.871866i \(-0.337090\pi\)
0.489744 + 0.871866i \(0.337090\pi\)
\(84\) 0 0
\(85\) 0.319292 0.0346321
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) 2.90607 0.308043 0.154021 0.988068i \(-0.450778\pi\)
0.154021 + 0.988068i \(0.450778\pi\)
\(90\) 0 0
\(91\) −11.0530 −1.15867
\(92\) 0 0
\(93\) −3.50717 −0.363677
\(94\) 0 0
\(95\) 1.73872 0.178389
\(96\) 0 0
\(97\) −17.7139 −1.79857 −0.899286 0.437360i \(-0.855913\pi\)
−0.899286 + 0.437360i \(0.855913\pi\)
\(98\) 0 0
\(99\) 2.74669 0.276053
\(100\) 0 0
\(101\) −10.1187 −1.00685 −0.503426 0.864038i \(-0.667927\pi\)
−0.503426 + 0.864038i \(0.667927\pi\)
\(102\) 0 0
\(103\) 14.6364 1.44217 0.721083 0.692849i \(-0.243645\pi\)
0.721083 + 0.692849i \(0.243645\pi\)
\(104\) 0 0
\(105\) −1.71000 −0.166879
\(106\) 0 0
\(107\) 4.56932 0.441733 0.220866 0.975304i \(-0.429111\pi\)
0.220866 + 0.975304i \(0.429111\pi\)
\(108\) 0 0
\(109\) 6.65094 0.637045 0.318522 0.947915i \(-0.396813\pi\)
0.318522 + 0.947915i \(0.396813\pi\)
\(110\) 0 0
\(111\) 5.25315 0.498607
\(112\) 0 0
\(113\) −7.23728 −0.680826 −0.340413 0.940276i \(-0.610567\pi\)
−0.340413 + 0.940276i \(0.610567\pi\)
\(114\) 0 0
\(115\) −0.425546 −0.0396823
\(116\) 0 0
\(117\) 2.75061 0.254294
\(118\) 0 0
\(119\) 3.01503 0.276387
\(120\) 0 0
\(121\) −3.45569 −0.314154
\(122\) 0 0
\(123\) −11.5128 −1.03807
\(124\) 0 0
\(125\) 4.17840 0.373727
\(126\) 0 0
\(127\) 6.27278 0.556619 0.278310 0.960491i \(-0.410226\pi\)
0.278310 + 0.960491i \(0.410226\pi\)
\(128\) 0 0
\(129\) −7.71894 −0.679615
\(130\) 0 0
\(131\) 4.38075 0.382748 0.191374 0.981517i \(-0.438706\pi\)
0.191374 + 0.981517i \(0.438706\pi\)
\(132\) 0 0
\(133\) 16.4185 1.42366
\(134\) 0 0
\(135\) 0.425546 0.0366251
\(136\) 0 0
\(137\) 1.73154 0.147935 0.0739676 0.997261i \(-0.476434\pi\)
0.0739676 + 0.997261i \(0.476434\pi\)
\(138\) 0 0
\(139\) 7.32975 0.621701 0.310851 0.950459i \(-0.399386\pi\)
0.310851 + 0.950459i \(0.399386\pi\)
\(140\) 0 0
\(141\) −3.88894 −0.327508
\(142\) 0 0
\(143\) 7.55508 0.631788
\(144\) 0 0
\(145\) 0.425546 0.0353396
\(146\) 0 0
\(147\) −9.14727 −0.754454
\(148\) 0 0
\(149\) −2.50672 −0.205358 −0.102679 0.994715i \(-0.532742\pi\)
−0.102679 + 0.994715i \(0.532742\pi\)
\(150\) 0 0
\(151\) −21.6115 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(152\) 0 0
\(153\) −0.750311 −0.0606591
\(154\) 0 0
\(155\) −1.49246 −0.119877
\(156\) 0 0
\(157\) 14.6172 1.16658 0.583289 0.812265i \(-0.301766\pi\)
0.583289 + 0.812265i \(0.301766\pi\)
\(158\) 0 0
\(159\) 0.561794 0.0445531
\(160\) 0 0
\(161\) −4.01837 −0.316692
\(162\) 0 0
\(163\) −0.0491942 −0.00385319 −0.00192659 0.999998i \(-0.500613\pi\)
−0.00192659 + 0.999998i \(0.500613\pi\)
\(164\) 0 0
\(165\) 1.16884 0.0909942
\(166\) 0 0
\(167\) 0.101312 0.00783979 0.00391989 0.999992i \(-0.498752\pi\)
0.00391989 + 0.999992i \(0.498752\pi\)
\(168\) 0 0
\(169\) −5.43413 −0.418010
\(170\) 0 0
\(171\) −4.08585 −0.312453
\(172\) 0 0
\(173\) 25.4548 1.93529 0.967646 0.252312i \(-0.0811909\pi\)
0.967646 + 0.252312i \(0.0811909\pi\)
\(174\) 0 0
\(175\) 19.3642 1.46379
\(176\) 0 0
\(177\) 11.0832 0.833066
\(178\) 0 0
\(179\) −21.2235 −1.58632 −0.793161 0.609012i \(-0.791566\pi\)
−0.793161 + 0.609012i \(0.791566\pi\)
\(180\) 0 0
\(181\) −5.95457 −0.442599 −0.221300 0.975206i \(-0.571030\pi\)
−0.221300 + 0.975206i \(0.571030\pi\)
\(182\) 0 0
\(183\) 14.0362 1.03758
\(184\) 0 0
\(185\) 2.23546 0.164354
\(186\) 0 0
\(187\) −2.06087 −0.150706
\(188\) 0 0
\(189\) 4.01837 0.292293
\(190\) 0 0
\(191\) 22.3732 1.61887 0.809434 0.587210i \(-0.199774\pi\)
0.809434 + 0.587210i \(0.199774\pi\)
\(192\) 0 0
\(193\) −14.1935 −1.02167 −0.510834 0.859679i \(-0.670663\pi\)
−0.510834 + 0.859679i \(0.670663\pi\)
\(194\) 0 0
\(195\) 1.17051 0.0838221
\(196\) 0 0
\(197\) −16.4768 −1.17392 −0.586962 0.809614i \(-0.699676\pi\)
−0.586962 + 0.809614i \(0.699676\pi\)
\(198\) 0 0
\(199\) 11.4091 0.808769 0.404384 0.914589i \(-0.367486\pi\)
0.404384 + 0.914589i \(0.367486\pi\)
\(200\) 0 0
\(201\) −11.7333 −0.827602
\(202\) 0 0
\(203\) 4.01837 0.282034
\(204\) 0 0
\(205\) −4.89921 −0.342176
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −11.2226 −0.776282
\(210\) 0 0
\(211\) −13.0094 −0.895606 −0.447803 0.894132i \(-0.647793\pi\)
−0.447803 + 0.894132i \(0.647793\pi\)
\(212\) 0 0
\(213\) −11.6376 −0.797393
\(214\) 0 0
\(215\) −3.28476 −0.224019
\(216\) 0 0
\(217\) −14.0931 −0.956701
\(218\) 0 0
\(219\) 8.17001 0.552078
\(220\) 0 0
\(221\) −2.06382 −0.138827
\(222\) 0 0
\(223\) −25.1539 −1.68443 −0.842214 0.539143i \(-0.818748\pi\)
−0.842214 + 0.539143i \(0.818748\pi\)
\(224\) 0 0
\(225\) −4.81891 −0.321261
\(226\) 0 0
\(227\) −21.9530 −1.45707 −0.728535 0.685009i \(-0.759798\pi\)
−0.728535 + 0.685009i \(0.759798\pi\)
\(228\) 0 0
\(229\) 25.3691 1.67644 0.838220 0.545333i \(-0.183597\pi\)
0.838220 + 0.545333i \(0.183597\pi\)
\(230\) 0 0
\(231\) 11.0372 0.726195
\(232\) 0 0
\(233\) 12.1740 0.797546 0.398773 0.917050i \(-0.369436\pi\)
0.398773 + 0.917050i \(0.369436\pi\)
\(234\) 0 0
\(235\) −1.65492 −0.107955
\(236\) 0 0
\(237\) 1.31646 0.0855131
\(238\) 0 0
\(239\) −22.5942 −1.46150 −0.730749 0.682647i \(-0.760829\pi\)
−0.730749 + 0.682647i \(0.760829\pi\)
\(240\) 0 0
\(241\) −9.98289 −0.643054 −0.321527 0.946900i \(-0.604196\pi\)
−0.321527 + 0.946900i \(0.604196\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −3.89258 −0.248688
\(246\) 0 0
\(247\) −11.2386 −0.715095
\(248\) 0 0
\(249\) −8.92356 −0.565508
\(250\) 0 0
\(251\) −9.85136 −0.621812 −0.310906 0.950441i \(-0.600632\pi\)
−0.310906 + 0.950441i \(0.600632\pi\)
\(252\) 0 0
\(253\) 2.74669 0.172683
\(254\) 0 0
\(255\) −0.319292 −0.0199948
\(256\) 0 0
\(257\) 1.69890 0.105975 0.0529873 0.998595i \(-0.483126\pi\)
0.0529873 + 0.998595i \(0.483126\pi\)
\(258\) 0 0
\(259\) 21.1091 1.31165
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) −7.43928 −0.458726 −0.229363 0.973341i \(-0.573664\pi\)
−0.229363 + 0.973341i \(0.573664\pi\)
\(264\) 0 0
\(265\) 0.239069 0.0146859
\(266\) 0 0
\(267\) −2.90607 −0.177849
\(268\) 0 0
\(269\) −29.6573 −1.80824 −0.904120 0.427279i \(-0.859472\pi\)
−0.904120 + 0.427279i \(0.859472\pi\)
\(270\) 0 0
\(271\) 17.2669 1.04889 0.524443 0.851445i \(-0.324273\pi\)
0.524443 + 0.851445i \(0.324273\pi\)
\(272\) 0 0
\(273\) 11.0530 0.668956
\(274\) 0 0
\(275\) −13.2361 −0.798164
\(276\) 0 0
\(277\) −22.6426 −1.36046 −0.680231 0.732998i \(-0.738121\pi\)
−0.680231 + 0.732998i \(0.738121\pi\)
\(278\) 0 0
\(279\) 3.50717 0.209969
\(280\) 0 0
\(281\) −26.8833 −1.60372 −0.801861 0.597510i \(-0.796157\pi\)
−0.801861 + 0.597510i \(0.796157\pi\)
\(282\) 0 0
\(283\) 11.4876 0.682864 0.341432 0.939906i \(-0.389088\pi\)
0.341432 + 0.939906i \(0.389088\pi\)
\(284\) 0 0
\(285\) −1.73872 −0.102993
\(286\) 0 0
\(287\) −46.2626 −2.73079
\(288\) 0 0
\(289\) −16.4370 −0.966884
\(290\) 0 0
\(291\) 17.7139 1.03841
\(292\) 0 0
\(293\) 19.4370 1.13552 0.567761 0.823193i \(-0.307810\pi\)
0.567761 + 0.823193i \(0.307810\pi\)
\(294\) 0 0
\(295\) 4.71642 0.274600
\(296\) 0 0
\(297\) −2.74669 −0.159379
\(298\) 0 0
\(299\) 2.75061 0.159072
\(300\) 0 0
\(301\) −31.0175 −1.78782
\(302\) 0 0
\(303\) 10.1187 0.581307
\(304\) 0 0
\(305\) 5.97304 0.342015
\(306\) 0 0
\(307\) 5.58977 0.319025 0.159513 0.987196i \(-0.449008\pi\)
0.159513 + 0.987196i \(0.449008\pi\)
\(308\) 0 0
\(309\) −14.6364 −0.832635
\(310\) 0 0
\(311\) −28.5683 −1.61996 −0.809981 0.586456i \(-0.800523\pi\)
−0.809981 + 0.586456i \(0.800523\pi\)
\(312\) 0 0
\(313\) −12.0076 −0.678708 −0.339354 0.940659i \(-0.610208\pi\)
−0.339354 + 0.940659i \(0.610208\pi\)
\(314\) 0 0
\(315\) 1.71000 0.0963475
\(316\) 0 0
\(317\) −21.6627 −1.21670 −0.608348 0.793670i \(-0.708168\pi\)
−0.608348 + 0.793670i \(0.708168\pi\)
\(318\) 0 0
\(319\) −2.74669 −0.153785
\(320\) 0 0
\(321\) −4.56932 −0.255035
\(322\) 0 0
\(323\) 3.06566 0.170578
\(324\) 0 0
\(325\) −13.2550 −0.735253
\(326\) 0 0
\(327\) −6.65094 −0.367798
\(328\) 0 0
\(329\) −15.6272 −0.861556
\(330\) 0 0
\(331\) −8.50054 −0.467232 −0.233616 0.972329i \(-0.575056\pi\)
−0.233616 + 0.972329i \(0.575056\pi\)
\(332\) 0 0
\(333\) −5.25315 −0.287871
\(334\) 0 0
\(335\) −4.99305 −0.272799
\(336\) 0 0
\(337\) −20.8670 −1.13670 −0.568350 0.822787i \(-0.692418\pi\)
−0.568350 + 0.822787i \(0.692418\pi\)
\(338\) 0 0
\(339\) 7.23728 0.393075
\(340\) 0 0
\(341\) 9.63311 0.521662
\(342\) 0 0
\(343\) −8.62852 −0.465896
\(344\) 0 0
\(345\) 0.425546 0.0229106
\(346\) 0 0
\(347\) 15.6962 0.842616 0.421308 0.906918i \(-0.361571\pi\)
0.421308 + 0.906918i \(0.361571\pi\)
\(348\) 0 0
\(349\) −36.1278 −1.93388 −0.966938 0.255011i \(-0.917921\pi\)
−0.966938 + 0.255011i \(0.917921\pi\)
\(350\) 0 0
\(351\) −2.75061 −0.146817
\(352\) 0 0
\(353\) 16.4759 0.876924 0.438462 0.898750i \(-0.355523\pi\)
0.438462 + 0.898750i \(0.355523\pi\)
\(354\) 0 0
\(355\) −4.95232 −0.262842
\(356\) 0 0
\(357\) −3.01503 −0.159572
\(358\) 0 0
\(359\) 9.04694 0.477479 0.238740 0.971084i \(-0.423266\pi\)
0.238740 + 0.971084i \(0.423266\pi\)
\(360\) 0 0
\(361\) −2.30581 −0.121358
\(362\) 0 0
\(363\) 3.45569 0.181377
\(364\) 0 0
\(365\) 3.47671 0.181980
\(366\) 0 0
\(367\) 28.9960 1.51358 0.756791 0.653657i \(-0.226766\pi\)
0.756791 + 0.653657i \(0.226766\pi\)
\(368\) 0 0
\(369\) 11.5128 0.599331
\(370\) 0 0
\(371\) 2.25749 0.117203
\(372\) 0 0
\(373\) 24.6416 1.27589 0.637946 0.770081i \(-0.279784\pi\)
0.637946 + 0.770081i \(0.279784\pi\)
\(374\) 0 0
\(375\) −4.17840 −0.215771
\(376\) 0 0
\(377\) −2.75061 −0.141664
\(378\) 0 0
\(379\) 8.85063 0.454627 0.227313 0.973822i \(-0.427006\pi\)
0.227313 + 0.973822i \(0.427006\pi\)
\(380\) 0 0
\(381\) −6.27278 −0.321364
\(382\) 0 0
\(383\) 21.9176 1.11994 0.559969 0.828513i \(-0.310813\pi\)
0.559969 + 0.828513i \(0.310813\pi\)
\(384\) 0 0
\(385\) 4.69684 0.239373
\(386\) 0 0
\(387\) 7.71894 0.392376
\(388\) 0 0
\(389\) −23.2463 −1.17863 −0.589316 0.807902i \(-0.700603\pi\)
−0.589316 + 0.807902i \(0.700603\pi\)
\(390\) 0 0
\(391\) −0.750311 −0.0379449
\(392\) 0 0
\(393\) −4.38075 −0.220980
\(394\) 0 0
\(395\) 0.560213 0.0281874
\(396\) 0 0
\(397\) −19.2387 −0.965563 −0.482781 0.875741i \(-0.660373\pi\)
−0.482781 + 0.875741i \(0.660373\pi\)
\(398\) 0 0
\(399\) −16.4185 −0.821951
\(400\) 0 0
\(401\) −8.72345 −0.435628 −0.217814 0.975990i \(-0.569893\pi\)
−0.217814 + 0.975990i \(0.569893\pi\)
\(402\) 0 0
\(403\) 9.64687 0.480545
\(404\) 0 0
\(405\) −0.425546 −0.0211455
\(406\) 0 0
\(407\) −14.4288 −0.715208
\(408\) 0 0
\(409\) 16.3605 0.808977 0.404489 0.914543i \(-0.367450\pi\)
0.404489 + 0.914543i \(0.367450\pi\)
\(410\) 0 0
\(411\) −1.73154 −0.0854104
\(412\) 0 0
\(413\) 44.5364 2.19149
\(414\) 0 0
\(415\) −3.79738 −0.186406
\(416\) 0 0
\(417\) −7.32975 −0.358940
\(418\) 0 0
\(419\) −19.4575 −0.950563 −0.475282 0.879834i \(-0.657654\pi\)
−0.475282 + 0.879834i \(0.657654\pi\)
\(420\) 0 0
\(421\) −6.28820 −0.306468 −0.153234 0.988190i \(-0.548969\pi\)
−0.153234 + 0.988190i \(0.548969\pi\)
\(422\) 0 0
\(423\) 3.88894 0.189087
\(424\) 0 0
\(425\) 3.61568 0.175386
\(426\) 0 0
\(427\) 56.4025 2.72951
\(428\) 0 0
\(429\) −7.55508 −0.364763
\(430\) 0 0
\(431\) −7.41978 −0.357398 −0.178699 0.983904i \(-0.557189\pi\)
−0.178699 + 0.983904i \(0.557189\pi\)
\(432\) 0 0
\(433\) −18.8000 −0.903471 −0.451736 0.892152i \(-0.649195\pi\)
−0.451736 + 0.892152i \(0.649195\pi\)
\(434\) 0 0
\(435\) −0.425546 −0.0204034
\(436\) 0 0
\(437\) −4.08585 −0.195453
\(438\) 0 0
\(439\) −19.3463 −0.923350 −0.461675 0.887049i \(-0.652751\pi\)
−0.461675 + 0.887049i \(0.652751\pi\)
\(440\) 0 0
\(441\) 9.14727 0.435584
\(442\) 0 0
\(443\) −21.8430 −1.03779 −0.518896 0.854837i \(-0.673657\pi\)
−0.518896 + 0.854837i \(0.673657\pi\)
\(444\) 0 0
\(445\) −1.23667 −0.0586236
\(446\) 0 0
\(447\) 2.50672 0.118564
\(448\) 0 0
\(449\) −1.37864 −0.0650618 −0.0325309 0.999471i \(-0.510357\pi\)
−0.0325309 + 0.999471i \(0.510357\pi\)
\(450\) 0 0
\(451\) 31.6220 1.48902
\(452\) 0 0
\(453\) 21.6115 1.01539
\(454\) 0 0
\(455\) 4.70355 0.220506
\(456\) 0 0
\(457\) 8.11178 0.379453 0.189727 0.981837i \(-0.439240\pi\)
0.189727 + 0.981837i \(0.439240\pi\)
\(458\) 0 0
\(459\) 0.750311 0.0350215
\(460\) 0 0
\(461\) −12.4751 −0.581022 −0.290511 0.956872i \(-0.593825\pi\)
−0.290511 + 0.956872i \(0.593825\pi\)
\(462\) 0 0
\(463\) −33.1911 −1.54252 −0.771261 0.636519i \(-0.780374\pi\)
−0.771261 + 0.636519i \(0.780374\pi\)
\(464\) 0 0
\(465\) 1.49246 0.0692112
\(466\) 0 0
\(467\) −5.38720 −0.249290 −0.124645 0.992201i \(-0.539779\pi\)
−0.124645 + 0.992201i \(0.539779\pi\)
\(468\) 0 0
\(469\) −47.1486 −2.17712
\(470\) 0 0
\(471\) −14.6172 −0.673524
\(472\) 0 0
\(473\) 21.2015 0.974848
\(474\) 0 0
\(475\) 19.6894 0.903410
\(476\) 0 0
\(477\) −0.561794 −0.0257228
\(478\) 0 0
\(479\) −26.5235 −1.21189 −0.605945 0.795506i \(-0.707205\pi\)
−0.605945 + 0.795506i \(0.707205\pi\)
\(480\) 0 0
\(481\) −14.4494 −0.658835
\(482\) 0 0
\(483\) 4.01837 0.182842
\(484\) 0 0
\(485\) 7.53807 0.342286
\(486\) 0 0
\(487\) −34.1450 −1.54726 −0.773630 0.633638i \(-0.781561\pi\)
−0.773630 + 0.633638i \(0.781561\pi\)
\(488\) 0 0
\(489\) 0.0491942 0.00222464
\(490\) 0 0
\(491\) −29.3107 −1.32277 −0.661387 0.750044i \(-0.730032\pi\)
−0.661387 + 0.750044i \(0.730032\pi\)
\(492\) 0 0
\(493\) 0.750311 0.0337923
\(494\) 0 0
\(495\) −1.16884 −0.0525356
\(496\) 0 0
\(497\) −46.7640 −2.09765
\(498\) 0 0
\(499\) −5.79016 −0.259203 −0.129602 0.991566i \(-0.541370\pi\)
−0.129602 + 0.991566i \(0.541370\pi\)
\(500\) 0 0
\(501\) −0.101312 −0.00452630
\(502\) 0 0
\(503\) −35.0276 −1.56180 −0.780902 0.624653i \(-0.785240\pi\)
−0.780902 + 0.624653i \(0.785240\pi\)
\(504\) 0 0
\(505\) 4.30599 0.191614
\(506\) 0 0
\(507\) 5.43413 0.241338
\(508\) 0 0
\(509\) 28.0283 1.24233 0.621166 0.783679i \(-0.286659\pi\)
0.621166 + 0.783679i \(0.286659\pi\)
\(510\) 0 0
\(511\) 32.8301 1.45232
\(512\) 0 0
\(513\) 4.08585 0.180395
\(514\) 0 0
\(515\) −6.22845 −0.274458
\(516\) 0 0
\(517\) 10.6817 0.469782
\(518\) 0 0
\(519\) −25.4548 −1.11734
\(520\) 0 0
\(521\) 5.66552 0.248211 0.124106 0.992269i \(-0.460394\pi\)
0.124106 + 0.992269i \(0.460394\pi\)
\(522\) 0 0
\(523\) 0.708170 0.0309661 0.0154831 0.999880i \(-0.495071\pi\)
0.0154831 + 0.999880i \(0.495071\pi\)
\(524\) 0 0
\(525\) −19.3642 −0.845121
\(526\) 0 0
\(527\) −2.63147 −0.114629
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −11.0832 −0.480971
\(532\) 0 0
\(533\) 31.6672 1.37166
\(534\) 0 0
\(535\) −1.94445 −0.0840661
\(536\) 0 0
\(537\) 21.2235 0.915863
\(538\) 0 0
\(539\) 25.1247 1.08220
\(540\) 0 0
\(541\) 38.9948 1.67652 0.838260 0.545271i \(-0.183573\pi\)
0.838260 + 0.545271i \(0.183573\pi\)
\(542\) 0 0
\(543\) 5.95457 0.255535
\(544\) 0 0
\(545\) −2.83028 −0.121236
\(546\) 0 0
\(547\) −10.6089 −0.453605 −0.226802 0.973941i \(-0.572827\pi\)
−0.226802 + 0.973941i \(0.572827\pi\)
\(548\) 0 0
\(549\) −14.0362 −0.599050
\(550\) 0 0
\(551\) 4.08585 0.174063
\(552\) 0 0
\(553\) 5.29001 0.224954
\(554\) 0 0
\(555\) −2.23546 −0.0948898
\(556\) 0 0
\(557\) 14.3143 0.606517 0.303259 0.952908i \(-0.401925\pi\)
0.303259 + 0.952908i \(0.401925\pi\)
\(558\) 0 0
\(559\) 21.2318 0.898010
\(560\) 0 0
\(561\) 2.06087 0.0870101
\(562\) 0 0
\(563\) −9.43658 −0.397704 −0.198852 0.980029i \(-0.563721\pi\)
−0.198852 + 0.980029i \(0.563721\pi\)
\(564\) 0 0
\(565\) 3.07979 0.129568
\(566\) 0 0
\(567\) −4.01837 −0.168756
\(568\) 0 0
\(569\) −8.71454 −0.365332 −0.182666 0.983175i \(-0.558473\pi\)
−0.182666 + 0.983175i \(0.558473\pi\)
\(570\) 0 0
\(571\) −34.4748 −1.44272 −0.721362 0.692558i \(-0.756483\pi\)
−0.721362 + 0.692558i \(0.756483\pi\)
\(572\) 0 0
\(573\) −22.3732 −0.934654
\(574\) 0 0
\(575\) −4.81891 −0.200962
\(576\) 0 0
\(577\) −6.60547 −0.274989 −0.137495 0.990502i \(-0.543905\pi\)
−0.137495 + 0.990502i \(0.543905\pi\)
\(578\) 0 0
\(579\) 14.1935 0.589860
\(580\) 0 0
\(581\) −35.8581 −1.48765
\(582\) 0 0
\(583\) −1.54307 −0.0639076
\(584\) 0 0
\(585\) −1.17051 −0.0483947
\(586\) 0 0
\(587\) −36.2328 −1.49549 −0.747744 0.663987i \(-0.768863\pi\)
−0.747744 + 0.663987i \(0.768863\pi\)
\(588\) 0 0
\(589\) −14.3298 −0.590448
\(590\) 0 0
\(591\) 16.4768 0.677766
\(592\) 0 0
\(593\) 2.12562 0.0872888 0.0436444 0.999047i \(-0.486103\pi\)
0.0436444 + 0.999047i \(0.486103\pi\)
\(594\) 0 0
\(595\) −1.28303 −0.0525992
\(596\) 0 0
\(597\) −11.4091 −0.466943
\(598\) 0 0
\(599\) −1.29513 −0.0529177 −0.0264588 0.999650i \(-0.508423\pi\)
−0.0264588 + 0.999650i \(0.508423\pi\)
\(600\) 0 0
\(601\) 0.407705 0.0166306 0.00831532 0.999965i \(-0.497353\pi\)
0.00831532 + 0.999965i \(0.497353\pi\)
\(602\) 0 0
\(603\) 11.7333 0.477816
\(604\) 0 0
\(605\) 1.47055 0.0597866
\(606\) 0 0
\(607\) 23.5993 0.957865 0.478932 0.877852i \(-0.341024\pi\)
0.478932 + 0.877852i \(0.341024\pi\)
\(608\) 0 0
\(609\) −4.01837 −0.162832
\(610\) 0 0
\(611\) 10.6970 0.432754
\(612\) 0 0
\(613\) 26.3821 1.06556 0.532782 0.846253i \(-0.321147\pi\)
0.532782 + 0.846253i \(0.321147\pi\)
\(614\) 0 0
\(615\) 4.89921 0.197555
\(616\) 0 0
\(617\) −11.7704 −0.473857 −0.236928 0.971527i \(-0.576141\pi\)
−0.236928 + 0.971527i \(0.576141\pi\)
\(618\) 0 0
\(619\) 21.5097 0.864550 0.432275 0.901742i \(-0.357711\pi\)
0.432275 + 0.901742i \(0.357711\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −11.6777 −0.467855
\(624\) 0 0
\(625\) 22.3165 0.892658
\(626\) 0 0
\(627\) 11.2226 0.448186
\(628\) 0 0
\(629\) 3.94150 0.157158
\(630\) 0 0
\(631\) 35.0626 1.39582 0.697910 0.716186i \(-0.254114\pi\)
0.697910 + 0.716186i \(0.254114\pi\)
\(632\) 0 0
\(633\) 13.0094 0.517078
\(634\) 0 0
\(635\) −2.66935 −0.105930
\(636\) 0 0
\(637\) 25.1606 0.996899
\(638\) 0 0
\(639\) 11.6376 0.460375
\(640\) 0 0
\(641\) −34.8561 −1.37673 −0.688367 0.725362i \(-0.741672\pi\)
−0.688367 + 0.725362i \(0.741672\pi\)
\(642\) 0 0
\(643\) 33.0448 1.30316 0.651579 0.758581i \(-0.274107\pi\)
0.651579 + 0.758581i \(0.274107\pi\)
\(644\) 0 0
\(645\) 3.28476 0.129337
\(646\) 0 0
\(647\) 3.65962 0.143874 0.0719372 0.997409i \(-0.477082\pi\)
0.0719372 + 0.997409i \(0.477082\pi\)
\(648\) 0 0
\(649\) −30.4422 −1.19496
\(650\) 0 0
\(651\) 14.0931 0.552352
\(652\) 0 0
\(653\) 15.3181 0.599444 0.299722 0.954027i \(-0.403106\pi\)
0.299722 + 0.954027i \(0.403106\pi\)
\(654\) 0 0
\(655\) −1.86421 −0.0728407
\(656\) 0 0
\(657\) −8.17001 −0.318743
\(658\) 0 0
\(659\) 19.1016 0.744093 0.372046 0.928214i \(-0.378656\pi\)
0.372046 + 0.928214i \(0.378656\pi\)
\(660\) 0 0
\(661\) −15.0959 −0.587164 −0.293582 0.955934i \(-0.594847\pi\)
−0.293582 + 0.955934i \(0.594847\pi\)
\(662\) 0 0
\(663\) 2.06382 0.0801520
\(664\) 0 0
\(665\) −6.98680 −0.270937
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) 25.1539 0.972505
\(670\) 0 0
\(671\) −38.5530 −1.48832
\(672\) 0 0
\(673\) 44.6238 1.72012 0.860061 0.510191i \(-0.170425\pi\)
0.860061 + 0.510191i \(0.170425\pi\)
\(674\) 0 0
\(675\) 4.81891 0.185480
\(676\) 0 0
\(677\) 35.6671 1.37080 0.685400 0.728167i \(-0.259628\pi\)
0.685400 + 0.728167i \(0.259628\pi\)
\(678\) 0 0
\(679\) 71.1809 2.73167
\(680\) 0 0
\(681\) 21.9530 0.841240
\(682\) 0 0
\(683\) −18.3501 −0.702146 −0.351073 0.936348i \(-0.614183\pi\)
−0.351073 + 0.936348i \(0.614183\pi\)
\(684\) 0 0
\(685\) −0.736848 −0.0281535
\(686\) 0 0
\(687\) −25.3691 −0.967893
\(688\) 0 0
\(689\) −1.54528 −0.0588704
\(690\) 0 0
\(691\) 45.4845 1.73031 0.865155 0.501504i \(-0.167220\pi\)
0.865155 + 0.501504i \(0.167220\pi\)
\(692\) 0 0
\(693\) −11.0372 −0.419269
\(694\) 0 0
\(695\) −3.11914 −0.118316
\(696\) 0 0
\(697\) −8.63816 −0.327194
\(698\) 0 0
\(699\) −12.1740 −0.460463
\(700\) 0 0
\(701\) −19.9987 −0.755339 −0.377670 0.925940i \(-0.623275\pi\)
−0.377670 + 0.925940i \(0.623275\pi\)
\(702\) 0 0
\(703\) 21.4636 0.809515
\(704\) 0 0
\(705\) 1.65492 0.0623280
\(706\) 0 0
\(707\) 40.6608 1.52921
\(708\) 0 0
\(709\) 1.71473 0.0643979 0.0321989 0.999481i \(-0.489749\pi\)
0.0321989 + 0.999481i \(0.489749\pi\)
\(710\) 0 0
\(711\) −1.31646 −0.0493710
\(712\) 0 0
\(713\) 3.50717 0.131345
\(714\) 0 0
\(715\) −3.21503 −0.120235
\(716\) 0 0
\(717\) 22.5942 0.843796
\(718\) 0 0
\(719\) −12.0054 −0.447725 −0.223862 0.974621i \(-0.571867\pi\)
−0.223862 + 0.974621i \(0.571867\pi\)
\(720\) 0 0
\(721\) −58.8143 −2.19036
\(722\) 0 0
\(723\) 9.98289 0.371268
\(724\) 0 0
\(725\) 4.81891 0.178970
\(726\) 0 0
\(727\) −48.7484 −1.80798 −0.903989 0.427557i \(-0.859374\pi\)
−0.903989 + 0.427557i \(0.859374\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −5.79161 −0.214210
\(732\) 0 0
\(733\) 17.4743 0.645428 0.322714 0.946497i \(-0.395405\pi\)
0.322714 + 0.946497i \(0.395405\pi\)
\(734\) 0 0
\(735\) 3.89258 0.143580
\(736\) 0 0
\(737\) 32.2277 1.18712
\(738\) 0 0
\(739\) −7.07816 −0.260374 −0.130187 0.991489i \(-0.541558\pi\)
−0.130187 + 0.991489i \(0.541558\pi\)
\(740\) 0 0
\(741\) 11.2386 0.412860
\(742\) 0 0
\(743\) 31.5073 1.15589 0.577945 0.816076i \(-0.303855\pi\)
0.577945 + 0.816076i \(0.303855\pi\)
\(744\) 0 0
\(745\) 1.06672 0.0390817
\(746\) 0 0
\(747\) 8.92356 0.326496
\(748\) 0 0
\(749\) −18.3612 −0.670904
\(750\) 0 0
\(751\) 17.9447 0.654812 0.327406 0.944884i \(-0.393825\pi\)
0.327406 + 0.944884i \(0.393825\pi\)
\(752\) 0 0
\(753\) 9.85136 0.359003
\(754\) 0 0
\(755\) 9.19666 0.334701
\(756\) 0 0
\(757\) 4.40530 0.160113 0.0800566 0.996790i \(-0.474490\pi\)
0.0800566 + 0.996790i \(0.474490\pi\)
\(758\) 0 0
\(759\) −2.74669 −0.0996985
\(760\) 0 0
\(761\) −4.46186 −0.161742 −0.0808712 0.996725i \(-0.525770\pi\)
−0.0808712 + 0.996725i \(0.525770\pi\)
\(762\) 0 0
\(763\) −26.7259 −0.967544
\(764\) 0 0
\(765\) 0.319292 0.0115440
\(766\) 0 0
\(767\) −30.4856 −1.10077
\(768\) 0 0
\(769\) 24.2390 0.874080 0.437040 0.899442i \(-0.356027\pi\)
0.437040 + 0.899442i \(0.356027\pi\)
\(770\) 0 0
\(771\) −1.69890 −0.0611844
\(772\) 0 0
\(773\) −5.22313 −0.187863 −0.0939315 0.995579i \(-0.529943\pi\)
−0.0939315 + 0.995579i \(0.529943\pi\)
\(774\) 0 0
\(775\) −16.9007 −0.607092
\(776\) 0 0
\(777\) −21.1091 −0.757284
\(778\) 0 0
\(779\) −47.0395 −1.68536
\(780\) 0 0
\(781\) 31.9648 1.14379
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) −6.22028 −0.222011
\(786\) 0 0
\(787\) −26.6085 −0.948492 −0.474246 0.880392i \(-0.657279\pi\)
−0.474246 + 0.880392i \(0.657279\pi\)
\(788\) 0 0
\(789\) 7.43928 0.264845
\(790\) 0 0
\(791\) 29.0821 1.03404
\(792\) 0 0
\(793\) −38.6081 −1.37101
\(794\) 0 0
\(795\) −0.239069 −0.00847890
\(796\) 0 0
\(797\) −24.7448 −0.876507 −0.438253 0.898852i \(-0.644403\pi\)
−0.438253 + 0.898852i \(0.644403\pi\)
\(798\) 0 0
\(799\) −2.91792 −0.103229
\(800\) 0 0
\(801\) 2.90607 0.102681
\(802\) 0 0
\(803\) −22.4405 −0.791908
\(804\) 0 0
\(805\) 1.71000 0.0602695
\(806\) 0 0
\(807\) 29.6573 1.04399
\(808\) 0 0
\(809\) −6.39850 −0.224959 −0.112480 0.993654i \(-0.535879\pi\)
−0.112480 + 0.993654i \(0.535879\pi\)
\(810\) 0 0
\(811\) 12.3958 0.435276 0.217638 0.976030i \(-0.430165\pi\)
0.217638 + 0.976030i \(0.430165\pi\)
\(812\) 0 0
\(813\) −17.2669 −0.605575
\(814\) 0 0
\(815\) 0.0209344 0.000733300 0
\(816\) 0 0
\(817\) −31.5385 −1.10339
\(818\) 0 0
\(819\) −11.0530 −0.386222
\(820\) 0 0
\(821\) −14.8206 −0.517242 −0.258621 0.965979i \(-0.583268\pi\)
−0.258621 + 0.965979i \(0.583268\pi\)
\(822\) 0 0
\(823\) −10.5541 −0.367894 −0.183947 0.982936i \(-0.558888\pi\)
−0.183947 + 0.982936i \(0.558888\pi\)
\(824\) 0 0
\(825\) 13.2361 0.460820
\(826\) 0 0
\(827\) −45.1063 −1.56850 −0.784250 0.620445i \(-0.786952\pi\)
−0.784250 + 0.620445i \(0.786952\pi\)
\(828\) 0 0
\(829\) 20.4287 0.709518 0.354759 0.934958i \(-0.384563\pi\)
0.354759 + 0.934958i \(0.384563\pi\)
\(830\) 0 0
\(831\) 22.6426 0.785463
\(832\) 0 0
\(833\) −6.86330 −0.237799
\(834\) 0 0
\(835\) −0.0431130 −0.00149199
\(836\) 0 0
\(837\) −3.50717 −0.121226
\(838\) 0 0
\(839\) 4.66753 0.161141 0.0805705 0.996749i \(-0.474326\pi\)
0.0805705 + 0.996749i \(0.474326\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 26.8833 0.925910
\(844\) 0 0
\(845\) 2.31247 0.0795514
\(846\) 0 0
\(847\) 13.8862 0.477137
\(848\) 0 0
\(849\) −11.4876 −0.394252
\(850\) 0 0
\(851\) −5.25315 −0.180076
\(852\) 0 0
\(853\) −44.5932 −1.52684 −0.763421 0.645901i \(-0.776482\pi\)
−0.763421 + 0.645901i \(0.776482\pi\)
\(854\) 0 0
\(855\) 1.73872 0.0594629
\(856\) 0 0
\(857\) 3.26798 0.111632 0.0558160 0.998441i \(-0.482224\pi\)
0.0558160 + 0.998441i \(0.482224\pi\)
\(858\) 0 0
\(859\) −21.7938 −0.743596 −0.371798 0.928314i \(-0.621259\pi\)
−0.371798 + 0.928314i \(0.621259\pi\)
\(860\) 0 0
\(861\) 46.2626 1.57662
\(862\) 0 0
\(863\) −31.5587 −1.07427 −0.537135 0.843496i \(-0.680494\pi\)
−0.537135 + 0.843496i \(0.680494\pi\)
\(864\) 0 0
\(865\) −10.8322 −0.368305
\(866\) 0 0
\(867\) 16.4370 0.558231
\(868\) 0 0
\(869\) −3.61590 −0.122661
\(870\) 0 0
\(871\) 32.2737 1.09355
\(872\) 0 0
\(873\) −17.7139 −0.599524
\(874\) 0 0
\(875\) −16.7903 −0.567617
\(876\) 0 0
\(877\) −3.24755 −0.109662 −0.0548310 0.998496i \(-0.517462\pi\)
−0.0548310 + 0.998496i \(0.517462\pi\)
\(878\) 0 0
\(879\) −19.4370 −0.655594
\(880\) 0 0
\(881\) 7.16127 0.241269 0.120635 0.992697i \(-0.461507\pi\)
0.120635 + 0.992697i \(0.461507\pi\)
\(882\) 0 0
\(883\) −24.0619 −0.809747 −0.404874 0.914373i \(-0.632684\pi\)
−0.404874 + 0.914373i \(0.632684\pi\)
\(884\) 0 0
\(885\) −4.71642 −0.158541
\(886\) 0 0
\(887\) −41.5681 −1.39572 −0.697860 0.716234i \(-0.745864\pi\)
−0.697860 + 0.716234i \(0.745864\pi\)
\(888\) 0 0
\(889\) −25.2063 −0.845393
\(890\) 0 0
\(891\) 2.74669 0.0920176
\(892\) 0 0
\(893\) −15.8897 −0.531727
\(894\) 0 0
\(895\) 9.03159 0.301893
\(896\) 0 0
\(897\) −2.75061 −0.0918403
\(898\) 0 0
\(899\) −3.50717 −0.116971
\(900\) 0 0
\(901\) 0.421520 0.0140429
\(902\) 0 0
\(903\) 31.0175 1.03220
\(904\) 0 0
\(905\) 2.53394 0.0842310
\(906\) 0 0
\(907\) 40.6957 1.35128 0.675640 0.737232i \(-0.263868\pi\)
0.675640 + 0.737232i \(0.263868\pi\)
\(908\) 0 0
\(909\) −10.1187 −0.335617
\(910\) 0 0
\(911\) 56.4805 1.87128 0.935642 0.352951i \(-0.114822\pi\)
0.935642 + 0.352951i \(0.114822\pi\)
\(912\) 0 0
\(913\) 24.5103 0.811171
\(914\) 0 0
\(915\) −5.97304 −0.197462
\(916\) 0 0
\(917\) −17.6035 −0.581318
\(918\) 0 0
\(919\) −27.7336 −0.914847 −0.457424 0.889249i \(-0.651228\pi\)
−0.457424 + 0.889249i \(0.651228\pi\)
\(920\) 0 0
\(921\) −5.58977 −0.184189
\(922\) 0 0
\(923\) 32.0105 1.05364
\(924\) 0 0
\(925\) 25.3145 0.832334
\(926\) 0 0
\(927\) 14.6364 0.480722
\(928\) 0 0
\(929\) −25.2779 −0.829342 −0.414671 0.909971i \(-0.636103\pi\)
−0.414671 + 0.909971i \(0.636103\pi\)
\(930\) 0 0
\(931\) −37.3744 −1.22490
\(932\) 0 0
\(933\) 28.5683 0.935286
\(934\) 0 0
\(935\) 0.876996 0.0286808
\(936\) 0 0
\(937\) 35.3248 1.15401 0.577006 0.816740i \(-0.304221\pi\)
0.577006 + 0.816740i \(0.304221\pi\)
\(938\) 0 0
\(939\) 12.0076 0.391852
\(940\) 0 0
\(941\) −23.6560 −0.771164 −0.385582 0.922674i \(-0.625999\pi\)
−0.385582 + 0.922674i \(0.625999\pi\)
\(942\) 0 0
\(943\) 11.5128 0.374908
\(944\) 0 0
\(945\) −1.71000 −0.0556263
\(946\) 0 0
\(947\) 43.9569 1.42841 0.714203 0.699938i \(-0.246789\pi\)
0.714203 + 0.699938i \(0.246789\pi\)
\(948\) 0 0
\(949\) −22.4726 −0.729490
\(950\) 0 0
\(951\) 21.6627 0.702460
\(952\) 0 0
\(953\) −11.9240 −0.386256 −0.193128 0.981174i \(-0.561863\pi\)
−0.193128 + 0.981174i \(0.561863\pi\)
\(954\) 0 0
\(955\) −9.52082 −0.308087
\(956\) 0 0
\(957\) 2.74669 0.0887879
\(958\) 0 0
\(959\) −6.95795 −0.224684
\(960\) 0 0
\(961\) −18.6998 −0.603218
\(962\) 0 0
\(963\) 4.56932 0.147244
\(964\) 0 0
\(965\) 6.03997 0.194433
\(966\) 0 0
\(967\) 35.6155 1.14532 0.572658 0.819794i \(-0.305912\pi\)
0.572658 + 0.819794i \(0.305912\pi\)
\(968\) 0 0
\(969\) −3.06566 −0.0984832
\(970\) 0 0
\(971\) −53.8649 −1.72861 −0.864304 0.502970i \(-0.832241\pi\)
−0.864304 + 0.502970i \(0.832241\pi\)
\(972\) 0 0
\(973\) −29.4536 −0.944240
\(974\) 0 0
\(975\) 13.2550 0.424498
\(976\) 0 0
\(977\) −33.3196 −1.06599 −0.532994 0.846119i \(-0.678933\pi\)
−0.532994 + 0.846119i \(0.678933\pi\)
\(978\) 0 0
\(979\) 7.98207 0.255108
\(980\) 0 0
\(981\) 6.65094 0.212348
\(982\) 0 0
\(983\) 9.83804 0.313785 0.156892 0.987616i \(-0.449852\pi\)
0.156892 + 0.987616i \(0.449852\pi\)
\(984\) 0 0
\(985\) 7.01164 0.223409
\(986\) 0 0
\(987\) 15.6272 0.497419
\(988\) 0 0
\(989\) 7.71894 0.245448
\(990\) 0 0
\(991\) −2.64379 −0.0839827 −0.0419914 0.999118i \(-0.513370\pi\)
−0.0419914 + 0.999118i \(0.513370\pi\)
\(992\) 0 0
\(993\) 8.50054 0.269756
\(994\) 0 0
\(995\) −4.85509 −0.153917
\(996\) 0 0
\(997\) −44.8077 −1.41908 −0.709538 0.704668i \(-0.751096\pi\)
−0.709538 + 0.704668i \(0.751096\pi\)
\(998\) 0 0
\(999\) 5.25315 0.166202
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 8004.2.a.g.1.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.g.1.5 12 1.1 even 1 trivial