Properties

Label 4560.2.a.br.1.2
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Defining polynomial: \(x^{3} - x^{2} - 5 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1140)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.08613\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -0.648061 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -0.648061 q^{7} +1.00000 q^{9} +3.52420 q^{11} +1.35194 q^{13} -1.00000 q^{15} +6.87614 q^{17} +1.00000 q^{19} +0.648061 q^{21} -5.46838 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.52420 q^{29} +3.46838 q^{31} -3.52420 q^{33} -0.648061 q^{35} +0.0558176 q^{37} -1.35194 q^{39} +9.52420 q^{41} -7.69646 q^{43} +1.00000 q^{45} +1.46838 q^{47} -6.58002 q^{49} -6.87614 q^{51} -13.2207 q^{53} +3.52420 q^{55} -1.00000 q^{57} +9.64064 q^{59} -0.172260 q^{61} -0.648061 q^{63} +1.35194 q^{65} +5.40776 q^{67} +5.46838 q^{69} -13.6406 q^{71} +6.34452 q^{73} -1.00000 q^{75} -2.28390 q^{77} -4.34452 q^{79} +1.00000 q^{81} +4.17226 q^{83} +6.87614 q^{85} -5.52420 q^{87} +5.52420 q^{89} -0.876139 q^{91} -3.46838 q^{93} +1.00000 q^{95} -2.64806 q^{97} +3.52420 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 6 q^{11} + 2 q^{13} - 3 q^{15} + 2 q^{17} + 3 q^{19} + 4 q^{21} - 6 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{33} - 4 q^{35} - 6 q^{37} - 2 q^{39} + 12 q^{41} + 8 q^{43} + 3 q^{45} - 6 q^{47} + 3 q^{49} - 2 q^{51} + 8 q^{53} - 6 q^{55} - 3 q^{57} + 4 q^{59} + 14 q^{61} - 4 q^{63} + 2 q^{65} + 8 q^{67} + 6 q^{69} - 16 q^{71} - 10 q^{73} - 3 q^{75} + 20 q^{77} + 16 q^{79} + 3 q^{81} - 2 q^{83} + 2 q^{85} + 16 q^{91} + 3 q^{95} - 10 q^{97} - 6 q^{99} + O(q^{100}) \)

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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.648061 −0.244944 −0.122472 0.992472i \(-0.539082\pi\)
−0.122472 + 0.992472i \(0.539082\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.52420 1.06259 0.531293 0.847188i \(-0.321706\pi\)
0.531293 + 0.847188i \(0.321706\pi\)
\(12\) 0 0
\(13\) 1.35194 0.374960 0.187480 0.982268i \(-0.439968\pi\)
0.187480 + 0.982268i \(0.439968\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 6.87614 1.66771 0.833854 0.551985i \(-0.186129\pi\)
0.833854 + 0.551985i \(0.186129\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.648061 0.141418
\(22\) 0 0
\(23\) −5.46838 −1.14024 −0.570118 0.821563i \(-0.693103\pi\)
−0.570118 + 0.821563i \(0.693103\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.52420 1.02582 0.512909 0.858443i \(-0.328568\pi\)
0.512909 + 0.858443i \(0.328568\pi\)
\(30\) 0 0
\(31\) 3.46838 0.622940 0.311470 0.950256i \(-0.399179\pi\)
0.311470 + 0.950256i \(0.399179\pi\)
\(32\) 0 0
\(33\) −3.52420 −0.613484
\(34\) 0 0
\(35\) −0.648061 −0.109542
\(36\) 0 0
\(37\) 0.0558176 0.00917636 0.00458818 0.999989i \(-0.498540\pi\)
0.00458818 + 0.999989i \(0.498540\pi\)
\(38\) 0 0
\(39\) −1.35194 −0.216484
\(40\) 0 0
\(41\) 9.52420 1.48743 0.743715 0.668497i \(-0.233062\pi\)
0.743715 + 0.668497i \(0.233062\pi\)
\(42\) 0 0
\(43\) −7.69646 −1.17370 −0.586850 0.809696i \(-0.699632\pi\)
−0.586850 + 0.809696i \(0.699632\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 1.46838 0.214186 0.107093 0.994249i \(-0.465846\pi\)
0.107093 + 0.994249i \(0.465846\pi\)
\(48\) 0 0
\(49\) −6.58002 −0.940002
\(50\) 0 0
\(51\) −6.87614 −0.962852
\(52\) 0 0
\(53\) −13.2207 −1.81600 −0.907999 0.418973i \(-0.862390\pi\)
−0.907999 + 0.418973i \(0.862390\pi\)
\(54\) 0 0
\(55\) 3.52420 0.475203
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 9.64064 1.25510 0.627552 0.778574i \(-0.284057\pi\)
0.627552 + 0.778574i \(0.284057\pi\)
\(60\) 0 0
\(61\) −0.172260 −0.0220557 −0.0110278 0.999939i \(-0.503510\pi\)
−0.0110278 + 0.999939i \(0.503510\pi\)
\(62\) 0 0
\(63\) −0.648061 −0.0816480
\(64\) 0 0
\(65\) 1.35194 0.167687
\(66\) 0 0
\(67\) 5.40776 0.660663 0.330331 0.943865i \(-0.392840\pi\)
0.330331 + 0.943865i \(0.392840\pi\)
\(68\) 0 0
\(69\) 5.46838 0.658316
\(70\) 0 0
\(71\) −13.6406 −1.61885 −0.809423 0.587226i \(-0.800220\pi\)
−0.809423 + 0.587226i \(0.800220\pi\)
\(72\) 0 0
\(73\) 6.34452 0.742570 0.371285 0.928519i \(-0.378917\pi\)
0.371285 + 0.928519i \(0.378917\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −2.28390 −0.260274
\(78\) 0 0
\(79\) −4.34452 −0.488797 −0.244398 0.969675i \(-0.578590\pi\)
−0.244398 + 0.969675i \(0.578590\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.17226 0.457965 0.228983 0.973431i \(-0.426460\pi\)
0.228983 + 0.973431i \(0.426460\pi\)
\(84\) 0 0
\(85\) 6.87614 0.745822
\(86\) 0 0
\(87\) −5.52420 −0.592256
\(88\) 0 0
\(89\) 5.52420 0.585564 0.292782 0.956179i \(-0.405419\pi\)
0.292782 + 0.956179i \(0.405419\pi\)
\(90\) 0 0
\(91\) −0.876139 −0.0918443
\(92\) 0 0
\(93\) −3.46838 −0.359654
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −2.64806 −0.268870 −0.134435 0.990922i \(-0.542922\pi\)
−0.134435 + 0.990922i \(0.542922\pi\)
\(98\) 0 0
\(99\) 3.52420 0.354195
\(100\) 0 0
\(101\) 8.70388 0.866068 0.433034 0.901378i \(-0.357443\pi\)
0.433034 + 0.901378i \(0.357443\pi\)
\(102\) 0 0
\(103\) 12.3445 1.21634 0.608171 0.793806i \(-0.291904\pi\)
0.608171 + 0.793806i \(0.291904\pi\)
\(104\) 0 0
\(105\) 0.648061 0.0632443
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −11.6406 −1.11497 −0.557486 0.830187i \(-0.688234\pi\)
−0.557486 + 0.830187i \(0.688234\pi\)
\(110\) 0 0
\(111\) −0.0558176 −0.00529797
\(112\) 0 0
\(113\) 0.764504 0.0719184 0.0359592 0.999353i \(-0.488551\pi\)
0.0359592 + 0.999353i \(0.488551\pi\)
\(114\) 0 0
\(115\) −5.46838 −0.509929
\(116\) 0 0
\(117\) 1.35194 0.124987
\(118\) 0 0
\(119\) −4.45616 −0.408495
\(120\) 0 0
\(121\) 1.41998 0.129089
\(122\) 0 0
\(123\) −9.52420 −0.858768
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) 7.69646 0.677636
\(130\) 0 0
\(131\) −11.5242 −1.00687 −0.503437 0.864032i \(-0.667931\pi\)
−0.503437 + 0.864032i \(0.667931\pi\)
\(132\) 0 0
\(133\) −0.648061 −0.0561940
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 23.0484 1.95494 0.977470 0.211075i \(-0.0676965\pi\)
0.977470 + 0.211075i \(0.0676965\pi\)
\(140\) 0 0
\(141\) −1.46838 −0.123660
\(142\) 0 0
\(143\) 4.76450 0.398428
\(144\) 0 0
\(145\) 5.52420 0.458760
\(146\) 0 0
\(147\) 6.58002 0.542711
\(148\) 0 0
\(149\) −6.34452 −0.519763 −0.259882 0.965640i \(-0.583684\pi\)
−0.259882 + 0.965640i \(0.583684\pi\)
\(150\) 0 0
\(151\) 1.93937 0.157824 0.0789120 0.996882i \(-0.474855\pi\)
0.0789120 + 0.996882i \(0.474855\pi\)
\(152\) 0 0
\(153\) 6.87614 0.555903
\(154\) 0 0
\(155\) 3.46838 0.278587
\(156\) 0 0
\(157\) −18.4562 −1.47296 −0.736481 0.676458i \(-0.763514\pi\)
−0.736481 + 0.676458i \(0.763514\pi\)
\(158\) 0 0
\(159\) 13.2207 1.04847
\(160\) 0 0
\(161\) 3.54384 0.279294
\(162\) 0 0
\(163\) 5.94418 0.465584 0.232792 0.972527i \(-0.425214\pi\)
0.232792 + 0.972527i \(0.425214\pi\)
\(164\) 0 0
\(165\) −3.52420 −0.274359
\(166\) 0 0
\(167\) 1.82774 0.141435 0.0707174 0.997496i \(-0.477471\pi\)
0.0707174 + 0.997496i \(0.477471\pi\)
\(168\) 0 0
\(169\) −11.1723 −0.859405
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) 20.2691 1.54103 0.770514 0.637423i \(-0.220000\pi\)
0.770514 + 0.637423i \(0.220000\pi\)
\(174\) 0 0
\(175\) −0.648061 −0.0489888
\(176\) 0 0
\(177\) −9.64064 −0.724635
\(178\) 0 0
\(179\) 8.11164 0.606292 0.303146 0.952944i \(-0.401963\pi\)
0.303146 + 0.952944i \(0.401963\pi\)
\(180\) 0 0
\(181\) 17.3929 1.29281 0.646403 0.762996i \(-0.276273\pi\)
0.646403 + 0.762996i \(0.276273\pi\)
\(182\) 0 0
\(183\) 0.172260 0.0127339
\(184\) 0 0
\(185\) 0.0558176 0.00410379
\(186\) 0 0
\(187\) 24.2329 1.77208
\(188\) 0 0
\(189\) 0.648061 0.0471395
\(190\) 0 0
\(191\) −7.17968 −0.519503 −0.259752 0.965675i \(-0.583641\pi\)
−0.259752 + 0.965675i \(0.583641\pi\)
\(192\) 0 0
\(193\) −12.2887 −0.884560 −0.442280 0.896877i \(-0.645830\pi\)
−0.442280 + 0.896877i \(0.645830\pi\)
\(194\) 0 0
\(195\) −1.35194 −0.0968144
\(196\) 0 0
\(197\) 20.6284 1.46971 0.734857 0.678222i \(-0.237249\pi\)
0.734857 + 0.678222i \(0.237249\pi\)
\(198\) 0 0
\(199\) −1.64064 −0.116302 −0.0581510 0.998308i \(-0.518520\pi\)
−0.0581510 + 0.998308i \(0.518520\pi\)
\(200\) 0 0
\(201\) −5.40776 −0.381434
\(202\) 0 0
\(203\) −3.58002 −0.251268
\(204\) 0 0
\(205\) 9.52420 0.665199
\(206\) 0 0
\(207\) −5.46838 −0.380079
\(208\) 0 0
\(209\) 3.52420 0.243774
\(210\) 0 0
\(211\) 16.3445 1.12520 0.562602 0.826728i \(-0.309801\pi\)
0.562602 + 0.826728i \(0.309801\pi\)
\(212\) 0 0
\(213\) 13.6406 0.934641
\(214\) 0 0
\(215\) −7.69646 −0.524894
\(216\) 0 0
\(217\) −2.24772 −0.152585
\(218\) 0 0
\(219\) −6.34452 −0.428723
\(220\) 0 0
\(221\) 9.29612 0.625325
\(222\) 0 0
\(223\) 12.3445 0.826650 0.413325 0.910584i \(-0.364367\pi\)
0.413325 + 0.910584i \(0.364367\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 11.2355 0.745726 0.372863 0.927886i \(-0.378376\pi\)
0.372863 + 0.927886i \(0.378376\pi\)
\(228\) 0 0
\(229\) 17.9245 1.18449 0.592243 0.805759i \(-0.298242\pi\)
0.592243 + 0.805759i \(0.298242\pi\)
\(230\) 0 0
\(231\) 2.28390 0.150269
\(232\) 0 0
\(233\) −22.3445 −1.46384 −0.731919 0.681392i \(-0.761375\pi\)
−0.731919 + 0.681392i \(0.761375\pi\)
\(234\) 0 0
\(235\) 1.46838 0.0957867
\(236\) 0 0
\(237\) 4.34452 0.282207
\(238\) 0 0
\(239\) 26.6842 1.72606 0.863030 0.505153i \(-0.168564\pi\)
0.863030 + 0.505153i \(0.168564\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −6.58002 −0.420382
\(246\) 0 0
\(247\) 1.35194 0.0860218
\(248\) 0 0
\(249\) −4.17226 −0.264406
\(250\) 0 0
\(251\) −9.88356 −0.623845 −0.311922 0.950108i \(-0.600973\pi\)
−0.311922 + 0.950108i \(0.600973\pi\)
\(252\) 0 0
\(253\) −19.2717 −1.21160
\(254\) 0 0
\(255\) −6.87614 −0.430601
\(256\) 0 0
\(257\) −30.5168 −1.90358 −0.951792 0.306743i \(-0.900761\pi\)
−0.951792 + 0.306743i \(0.900761\pi\)
\(258\) 0 0
\(259\) −0.0361732 −0.00224769
\(260\) 0 0
\(261\) 5.52420 0.341939
\(262\) 0 0
\(263\) 5.23550 0.322835 0.161417 0.986886i \(-0.448394\pi\)
0.161417 + 0.986886i \(0.448394\pi\)
\(264\) 0 0
\(265\) −13.2207 −0.812139
\(266\) 0 0
\(267\) −5.52420 −0.338076
\(268\) 0 0
\(269\) 11.7719 0.717747 0.358873 0.933386i \(-0.383161\pi\)
0.358873 + 0.933386i \(0.383161\pi\)
\(270\) 0 0
\(271\) 27.3929 1.66400 0.832001 0.554775i \(-0.187196\pi\)
0.832001 + 0.554775i \(0.187196\pi\)
\(272\) 0 0
\(273\) 0.876139 0.0530263
\(274\) 0 0
\(275\) 3.52420 0.212517
\(276\) 0 0
\(277\) −0.247722 −0.0148842 −0.00744210 0.999972i \(-0.502369\pi\)
−0.00744210 + 0.999972i \(0.502369\pi\)
\(278\) 0 0
\(279\) 3.46838 0.207647
\(280\) 0 0
\(281\) −26.2132 −1.56375 −0.781875 0.623435i \(-0.785737\pi\)
−0.781875 + 0.623435i \(0.785737\pi\)
\(282\) 0 0
\(283\) −19.0894 −1.13475 −0.567373 0.823461i \(-0.692040\pi\)
−0.567373 + 0.823461i \(0.692040\pi\)
\(284\) 0 0
\(285\) −1.00000 −0.0592349
\(286\) 0 0
\(287\) −6.17226 −0.364337
\(288\) 0 0
\(289\) 30.2813 1.78125
\(290\) 0 0
\(291\) 2.64806 0.155232
\(292\) 0 0
\(293\) −19.5800 −1.14388 −0.571938 0.820297i \(-0.693808\pi\)
−0.571938 + 0.820297i \(0.693808\pi\)
\(294\) 0 0
\(295\) 9.64064 0.561300
\(296\) 0 0
\(297\) −3.52420 −0.204495
\(298\) 0 0
\(299\) −7.39292 −0.427544
\(300\) 0 0
\(301\) 4.98777 0.287491
\(302\) 0 0
\(303\) −8.70388 −0.500025
\(304\) 0 0
\(305\) −0.172260 −0.00986360
\(306\) 0 0
\(307\) 31.0484 1.77203 0.886013 0.463661i \(-0.153464\pi\)
0.886013 + 0.463661i \(0.153464\pi\)
\(308\) 0 0
\(309\) −12.3445 −0.702255
\(310\) 0 0
\(311\) −29.2765 −1.66012 −0.830058 0.557677i \(-0.811693\pi\)
−0.830058 + 0.557677i \(0.811693\pi\)
\(312\) 0 0
\(313\) −7.98516 −0.451348 −0.225674 0.974203i \(-0.572458\pi\)
−0.225674 + 0.974203i \(0.572458\pi\)
\(314\) 0 0
\(315\) −0.648061 −0.0365141
\(316\) 0 0
\(317\) −18.9729 −1.06563 −0.532813 0.846233i \(-0.678865\pi\)
−0.532813 + 0.846233i \(0.678865\pi\)
\(318\) 0 0
\(319\) 19.4684 1.09002
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.87614 0.382599
\(324\) 0 0
\(325\) 1.35194 0.0749921
\(326\) 0 0
\(327\) 11.6406 0.643729
\(328\) 0 0
\(329\) −0.951601 −0.0524635
\(330\) 0 0
\(331\) 5.93937 0.326458 0.163229 0.986588i \(-0.447809\pi\)
0.163229 + 0.986588i \(0.447809\pi\)
\(332\) 0 0
\(333\) 0.0558176 0.00305879
\(334\) 0 0
\(335\) 5.40776 0.295457
\(336\) 0 0
\(337\) 27.7933 1.51400 0.756998 0.653418i \(-0.226665\pi\)
0.756998 + 0.653418i \(0.226665\pi\)
\(338\) 0 0
\(339\) −0.764504 −0.0415221
\(340\) 0 0
\(341\) 12.2233 0.661927
\(342\) 0 0
\(343\) 8.80068 0.475192
\(344\) 0 0
\(345\) 5.46838 0.294408
\(346\) 0 0
\(347\) 17.2355 0.925250 0.462625 0.886554i \(-0.346908\pi\)
0.462625 + 0.886554i \(0.346908\pi\)
\(348\) 0 0
\(349\) 27.4078 1.46710 0.733552 0.679634i \(-0.237861\pi\)
0.733552 + 0.679634i \(0.237861\pi\)
\(350\) 0 0
\(351\) −1.35194 −0.0721612
\(352\) 0 0
\(353\) −16.5922 −0.883116 −0.441558 0.897233i \(-0.645574\pi\)
−0.441558 + 0.897233i \(0.645574\pi\)
\(354\) 0 0
\(355\) −13.6406 −0.723970
\(356\) 0 0
\(357\) 4.45616 0.235845
\(358\) 0 0
\(359\) 17.5094 0.924109 0.462054 0.886852i \(-0.347112\pi\)
0.462054 + 0.886852i \(0.347112\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −1.41998 −0.0745298
\(364\) 0 0
\(365\) 6.34452 0.332087
\(366\) 0 0
\(367\) −17.7933 −0.928801 −0.464400 0.885625i \(-0.653730\pi\)
−0.464400 + 0.885625i \(0.653730\pi\)
\(368\) 0 0
\(369\) 9.52420 0.495810
\(370\) 0 0
\(371\) 8.56779 0.444818
\(372\) 0 0
\(373\) 3.59966 0.186383 0.0931917 0.995648i \(-0.470293\pi\)
0.0931917 + 0.995648i \(0.470293\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 7.46838 0.384641
\(378\) 0 0
\(379\) 30.4051 1.56181 0.780904 0.624651i \(-0.214759\pi\)
0.780904 + 0.624651i \(0.214759\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) −2.93676 −0.150062 −0.0750308 0.997181i \(-0.523906\pi\)
−0.0750308 + 0.997181i \(0.523906\pi\)
\(384\) 0 0
\(385\) −2.28390 −0.116398
\(386\) 0 0
\(387\) −7.69646 −0.391233
\(388\) 0 0
\(389\) 1.65548 0.0839361 0.0419681 0.999119i \(-0.486637\pi\)
0.0419681 + 0.999119i \(0.486637\pi\)
\(390\) 0 0
\(391\) −37.6014 −1.90158
\(392\) 0 0
\(393\) 11.5242 0.581319
\(394\) 0 0
\(395\) −4.34452 −0.218597
\(396\) 0 0
\(397\) −34.9123 −1.75220 −0.876099 0.482131i \(-0.839863\pi\)
−0.876099 + 0.482131i \(0.839863\pi\)
\(398\) 0 0
\(399\) 0.648061 0.0324436
\(400\) 0 0
\(401\) 32.9171 1.64380 0.821901 0.569630i \(-0.192913\pi\)
0.821901 + 0.569630i \(0.192913\pi\)
\(402\) 0 0
\(403\) 4.68904 0.233578
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 0.196712 0.00975067
\(408\) 0 0
\(409\) −27.1452 −1.34224 −0.671122 0.741347i \(-0.734187\pi\)
−0.671122 + 0.741347i \(0.734187\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) −6.24772 −0.307430
\(414\) 0 0
\(415\) 4.17226 0.204808
\(416\) 0 0
\(417\) −23.0484 −1.12868
\(418\) 0 0
\(419\) −1.88356 −0.0920178 −0.0460089 0.998941i \(-0.514650\pi\)
−0.0460089 + 0.998941i \(0.514650\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 1.46838 0.0713952
\(424\) 0 0
\(425\) 6.87614 0.333542
\(426\) 0 0
\(427\) 0.111635 0.00540241
\(428\) 0 0
\(429\) −4.76450 −0.230032
\(430\) 0 0
\(431\) −18.7039 −0.900934 −0.450467 0.892793i \(-0.648743\pi\)
−0.450467 + 0.892793i \(0.648743\pi\)
\(432\) 0 0
\(433\) 37.0894 1.78240 0.891201 0.453609i \(-0.149864\pi\)
0.891201 + 0.453609i \(0.149864\pi\)
\(434\) 0 0
\(435\) −5.52420 −0.264865
\(436\) 0 0
\(437\) −5.46838 −0.261588
\(438\) 0 0
\(439\) 8.22327 0.392475 0.196238 0.980556i \(-0.437128\pi\)
0.196238 + 0.980556i \(0.437128\pi\)
\(440\) 0 0
\(441\) −6.58002 −0.313334
\(442\) 0 0
\(443\) 12.0606 0.573018 0.286509 0.958078i \(-0.407505\pi\)
0.286509 + 0.958078i \(0.407505\pi\)
\(444\) 0 0
\(445\) 5.52420 0.261872
\(446\) 0 0
\(447\) 6.34452 0.300086
\(448\) 0 0
\(449\) 23.2765 1.09848 0.549242 0.835663i \(-0.314916\pi\)
0.549242 + 0.835663i \(0.314916\pi\)
\(450\) 0 0
\(451\) 33.5652 1.58052
\(452\) 0 0
\(453\) −1.93937 −0.0911198
\(454\) 0 0
\(455\) −0.876139 −0.0410740
\(456\) 0 0
\(457\) −6.45616 −0.302006 −0.151003 0.988533i \(-0.548250\pi\)
−0.151003 + 0.988533i \(0.548250\pi\)
\(458\) 0 0
\(459\) −6.87614 −0.320951
\(460\) 0 0
\(461\) 22.6890 1.05673 0.528367 0.849016i \(-0.322805\pi\)
0.528367 + 0.849016i \(0.322805\pi\)
\(462\) 0 0
\(463\) −30.2887 −1.40764 −0.703818 0.710381i \(-0.748523\pi\)
−0.703818 + 0.710381i \(0.748523\pi\)
\(464\) 0 0
\(465\) −3.46838 −0.160842
\(466\) 0 0
\(467\) 1.12386 0.0520061 0.0260030 0.999662i \(-0.491722\pi\)
0.0260030 + 0.999662i \(0.491722\pi\)
\(468\) 0 0
\(469\) −3.50456 −0.161825
\(470\) 0 0
\(471\) 18.4562 0.850415
\(472\) 0 0
\(473\) −27.1239 −1.24716
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) −13.2207 −0.605332
\(478\) 0 0
\(479\) −10.3397 −0.472434 −0.236217 0.971700i \(-0.575908\pi\)
−0.236217 + 0.971700i \(0.575908\pi\)
\(480\) 0 0
\(481\) 0.0754620 0.00344077
\(482\) 0 0
\(483\) −3.54384 −0.161250
\(484\) 0 0
\(485\) −2.64806 −0.120242
\(486\) 0 0
\(487\) −3.54384 −0.160587 −0.0802935 0.996771i \(-0.525586\pi\)
−0.0802935 + 0.996771i \(0.525586\pi\)
\(488\) 0 0
\(489\) −5.94418 −0.268805
\(490\) 0 0
\(491\) 2.22808 0.100552 0.0502759 0.998735i \(-0.483990\pi\)
0.0502759 + 0.998735i \(0.483990\pi\)
\(492\) 0 0
\(493\) 37.9852 1.71077
\(494\) 0 0
\(495\) 3.52420 0.158401
\(496\) 0 0
\(497\) 8.83997 0.396527
\(498\) 0 0
\(499\) −29.1452 −1.30472 −0.652359 0.757910i \(-0.726221\pi\)
−0.652359 + 0.757910i \(0.726221\pi\)
\(500\) 0 0
\(501\) −1.82774 −0.0816574
\(502\) 0 0
\(503\) −40.1723 −1.79119 −0.895596 0.444868i \(-0.853251\pi\)
−0.895596 + 0.444868i \(0.853251\pi\)
\(504\) 0 0
\(505\) 8.70388 0.387318
\(506\) 0 0
\(507\) 11.1723 0.496178
\(508\) 0 0
\(509\) −0.835158 −0.0370177 −0.0185089 0.999829i \(-0.505892\pi\)
−0.0185089 + 0.999829i \(0.505892\pi\)
\(510\) 0 0
\(511\) −4.11164 −0.181888
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 12.3445 0.543965
\(516\) 0 0
\(517\) 5.17487 0.227591
\(518\) 0 0
\(519\) −20.2691 −0.889713
\(520\) 0 0
\(521\) 6.82032 0.298804 0.149402 0.988777i \(-0.452265\pi\)
0.149402 + 0.988777i \(0.452265\pi\)
\(522\) 0 0
\(523\) −13.6406 −0.596464 −0.298232 0.954493i \(-0.596397\pi\)
−0.298232 + 0.954493i \(0.596397\pi\)
\(524\) 0 0
\(525\) 0.648061 0.0282837
\(526\) 0 0
\(527\) 23.8491 1.03888
\(528\) 0 0
\(529\) 6.90320 0.300139
\(530\) 0 0
\(531\) 9.64064 0.418368
\(532\) 0 0
\(533\) 12.8761 0.557727
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −8.11164 −0.350043
\(538\) 0 0
\(539\) −23.1893 −0.998834
\(540\) 0 0
\(541\) 29.5046 1.26850 0.634250 0.773128i \(-0.281309\pi\)
0.634250 + 0.773128i \(0.281309\pi\)
\(542\) 0 0
\(543\) −17.3929 −0.746402
\(544\) 0 0
\(545\) −11.6406 −0.498630
\(546\) 0 0
\(547\) 39.7374 1.69905 0.849525 0.527548i \(-0.176889\pi\)
0.849525 + 0.527548i \(0.176889\pi\)
\(548\) 0 0
\(549\) −0.172260 −0.00735189
\(550\) 0 0
\(551\) 5.52420 0.235339
\(552\) 0 0
\(553\) 2.81551 0.119728
\(554\) 0 0
\(555\) −0.0558176 −0.00236933
\(556\) 0 0
\(557\) 11.7523 0.497960 0.248980 0.968509i \(-0.419905\pi\)
0.248980 + 0.968509i \(0.419905\pi\)
\(558\) 0 0
\(559\) −10.4051 −0.440091
\(560\) 0 0
\(561\) −24.2329 −1.02311
\(562\) 0 0
\(563\) 36.6136 1.54308 0.771539 0.636182i \(-0.219487\pi\)
0.771539 + 0.636182i \(0.219487\pi\)
\(564\) 0 0
\(565\) 0.764504 0.0321629
\(566\) 0 0
\(567\) −0.648061 −0.0272160
\(568\) 0 0
\(569\) −15.0436 −0.630660 −0.315330 0.948982i \(-0.602115\pi\)
−0.315330 + 0.948982i \(0.602115\pi\)
\(570\) 0 0
\(571\) −23.3929 −0.978963 −0.489482 0.872014i \(-0.662814\pi\)
−0.489482 + 0.872014i \(0.662814\pi\)
\(572\) 0 0
\(573\) 7.17968 0.299935
\(574\) 0 0
\(575\) −5.46838 −0.228047
\(576\) 0 0
\(577\) 32.7039 1.36148 0.680740 0.732525i \(-0.261658\pi\)
0.680740 + 0.732525i \(0.261658\pi\)
\(578\) 0 0
\(579\) 12.2887 0.510701
\(580\) 0 0
\(581\) −2.70388 −0.112176
\(582\) 0 0
\(583\) −46.5922 −1.92965
\(584\) 0 0
\(585\) 1.35194 0.0558958
\(586\) 0 0
\(587\) −9.46838 −0.390802 −0.195401 0.980723i \(-0.562601\pi\)
−0.195401 + 0.980723i \(0.562601\pi\)
\(588\) 0 0
\(589\) 3.46838 0.142912
\(590\) 0 0
\(591\) −20.6284 −0.848540
\(592\) 0 0
\(593\) −27.7523 −1.13965 −0.569825 0.821766i \(-0.692989\pi\)
−0.569825 + 0.821766i \(0.692989\pi\)
\(594\) 0 0
\(595\) −4.45616 −0.182685
\(596\) 0 0
\(597\) 1.64064 0.0671470
\(598\) 0 0
\(599\) −28.6890 −1.17220 −0.586101 0.810238i \(-0.699338\pi\)
−0.586101 + 0.810238i \(0.699338\pi\)
\(600\) 0 0
\(601\) 11.2961 0.460778 0.230389 0.973099i \(-0.426000\pi\)
0.230389 + 0.973099i \(0.426000\pi\)
\(602\) 0 0
\(603\) 5.40776 0.220221
\(604\) 0 0
\(605\) 1.41998 0.0577305
\(606\) 0 0
\(607\) −8.11164 −0.329241 −0.164621 0.986357i \(-0.552640\pi\)
−0.164621 + 0.986357i \(0.552640\pi\)
\(608\) 0 0
\(609\) 3.58002 0.145070
\(610\) 0 0
\(611\) 1.98516 0.0803111
\(612\) 0 0
\(613\) −17.1696 −0.693476 −0.346738 0.937962i \(-0.612711\pi\)
−0.346738 + 0.937962i \(0.612711\pi\)
\(614\) 0 0
\(615\) −9.52420 −0.384053
\(616\) 0 0
\(617\) 26.3807 1.06205 0.531023 0.847357i \(-0.321808\pi\)
0.531023 + 0.847357i \(0.321808\pi\)
\(618\) 0 0
\(619\) 38.6742 1.55445 0.777224 0.629224i \(-0.216627\pi\)
0.777224 + 0.629224i \(0.216627\pi\)
\(620\) 0 0
\(621\) 5.46838 0.219439
\(622\) 0 0
\(623\) −3.58002 −0.143430
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −3.52420 −0.140743
\(628\) 0 0
\(629\) 0.383810 0.0153035
\(630\) 0 0
\(631\) −21.4078 −0.852229 −0.426115 0.904669i \(-0.640118\pi\)
−0.426115 + 0.904669i \(0.640118\pi\)
\(632\) 0 0
\(633\) −16.3445 −0.649636
\(634\) 0 0
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) −8.89578 −0.352464
\(638\) 0 0
\(639\) −13.6406 −0.539615
\(640\) 0 0
\(641\) −19.1648 −0.756966 −0.378483 0.925608i \(-0.623554\pi\)
−0.378483 + 0.925608i \(0.623554\pi\)
\(642\) 0 0
\(643\) −8.99258 −0.354633 −0.177316 0.984154i \(-0.556742\pi\)
−0.177316 + 0.984154i \(0.556742\pi\)
\(644\) 0 0
\(645\) 7.69646 0.303048
\(646\) 0 0
\(647\) −38.9878 −1.53277 −0.766384 0.642383i \(-0.777946\pi\)
−0.766384 + 0.642383i \(0.777946\pi\)
\(648\) 0 0
\(649\) 33.9755 1.33366
\(650\) 0 0
\(651\) 2.24772 0.0880952
\(652\) 0 0
\(653\) −33.8129 −1.32320 −0.661601 0.749856i \(-0.730123\pi\)
−0.661601 + 0.749856i \(0.730123\pi\)
\(654\) 0 0
\(655\) −11.5242 −0.450288
\(656\) 0 0
\(657\) 6.34452 0.247523
\(658\) 0 0
\(659\) −1.75228 −0.0682590 −0.0341295 0.999417i \(-0.510866\pi\)
−0.0341295 + 0.999417i \(0.510866\pi\)
\(660\) 0 0
\(661\) −16.9271 −0.658390 −0.329195 0.944262i \(-0.606777\pi\)
−0.329195 + 0.944262i \(0.606777\pi\)
\(662\) 0 0
\(663\) −9.29612 −0.361031
\(664\) 0 0
\(665\) −0.648061 −0.0251307
\(666\) 0 0
\(667\) −30.2084 −1.16968
\(668\) 0 0
\(669\) −12.3445 −0.477267
\(670\) 0 0
\(671\) −0.607080 −0.0234361
\(672\) 0 0
\(673\) −0.0558176 −0.00215161 −0.00107581 0.999999i \(-0.500342\pi\)
−0.00107581 + 0.999999i \(0.500342\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 19.4684 0.748231 0.374115 0.927382i \(-0.377946\pi\)
0.374115 + 0.927382i \(0.377946\pi\)
\(678\) 0 0
\(679\) 1.71610 0.0658580
\(680\) 0 0
\(681\) −11.2355 −0.430545
\(682\) 0 0
\(683\) −16.2233 −0.620766 −0.310383 0.950612i \(-0.600457\pi\)
−0.310383 + 0.950612i \(0.600457\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) −17.9245 −0.683864
\(688\) 0 0
\(689\) −17.8735 −0.680927
\(690\) 0 0
\(691\) −19.8884 −0.756589 −0.378295 0.925685i \(-0.623489\pi\)
−0.378295 + 0.925685i \(0.623489\pi\)
\(692\) 0 0
\(693\) −2.28390 −0.0867580
\(694\) 0 0
\(695\) 23.0484 0.874276
\(696\) 0 0
\(697\) 65.4897 2.48060
\(698\) 0 0
\(699\) 22.3445 0.845147
\(700\) 0 0
\(701\) 7.75228 0.292799 0.146400 0.989226i \(-0.453231\pi\)
0.146400 + 0.989226i \(0.453231\pi\)
\(702\) 0 0
\(703\) 0.0558176 0.00210520
\(704\) 0 0
\(705\) −1.46838 −0.0553025
\(706\) 0 0
\(707\) −5.64064 −0.212138
\(708\) 0 0
\(709\) 29.9245 1.12384 0.561920 0.827192i \(-0.310063\pi\)
0.561920 + 0.827192i \(0.310063\pi\)
\(710\) 0 0
\(711\) −4.34452 −0.162932
\(712\) 0 0
\(713\) −18.9664 −0.710299
\(714\) 0 0
\(715\) 4.76450 0.178182
\(716\) 0 0
\(717\) −26.6842 −0.996541
\(718\) 0 0
\(719\) −11.2913 −0.421095 −0.210547 0.977584i \(-0.567525\pi\)
−0.210547 + 0.977584i \(0.567525\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) −14.0000 −0.520666
\(724\) 0 0
\(725\) 5.52420 0.205164
\(726\) 0 0
\(727\) −9.59966 −0.356032 −0.178016 0.984028i \(-0.556968\pi\)
−0.178016 + 0.984028i \(0.556968\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −52.9219 −1.95739
\(732\) 0 0
\(733\) 12.9368 0.477830 0.238915 0.971040i \(-0.423208\pi\)
0.238915 + 0.971040i \(0.423208\pi\)
\(734\) 0 0
\(735\) 6.58002 0.242708
\(736\) 0 0
\(737\) 19.0580 0.702011
\(738\) 0 0
\(739\) 28.0000 1.03000 0.514998 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 0 0
\(741\) −1.35194 −0.0496647
\(742\) 0 0
\(743\) 14.4413 0.529801 0.264900 0.964276i \(-0.414661\pi\)
0.264900 + 0.964276i \(0.414661\pi\)
\(744\) 0 0
\(745\) −6.34452 −0.232445
\(746\) 0 0
\(747\) 4.17226 0.152655
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 25.5652 0.932887 0.466443 0.884551i \(-0.345535\pi\)
0.466443 + 0.884551i \(0.345535\pi\)
\(752\) 0 0
\(753\) 9.88356 0.360177
\(754\) 0 0
\(755\) 1.93937 0.0705811
\(756\) 0 0
\(757\) 3.41737 0.124206 0.0621032 0.998070i \(-0.480219\pi\)
0.0621032 + 0.998070i \(0.480219\pi\)
\(758\) 0 0
\(759\) 19.2717 0.699517
\(760\) 0 0
\(761\) −20.9368 −0.758957 −0.379479 0.925201i \(-0.623897\pi\)
−0.379479 + 0.925201i \(0.623897\pi\)
\(762\) 0 0
\(763\) 7.54384 0.273105
\(764\) 0 0
\(765\) 6.87614 0.248607
\(766\) 0 0
\(767\) 13.0336 0.470615
\(768\) 0 0
\(769\) 0.592243 0.0213568 0.0106784 0.999943i \(-0.496601\pi\)
0.0106784 + 0.999943i \(0.496601\pi\)
\(770\) 0 0
\(771\) 30.5168 1.09904
\(772\) 0 0
\(773\) −7.69165 −0.276650 −0.138325 0.990387i \(-0.544172\pi\)
−0.138325 + 0.990387i \(0.544172\pi\)
\(774\) 0 0
\(775\) 3.46838 0.124588
\(776\) 0 0
\(777\) 0.0361732 0.00129771
\(778\) 0 0
\(779\) 9.52420 0.341240
\(780\) 0 0
\(781\) −48.0723 −1.72016
\(782\) 0 0
\(783\) −5.52420 −0.197419
\(784\) 0 0
\(785\) −18.4562 −0.658728
\(786\) 0 0
\(787\) 38.3297 1.36631 0.683153 0.730275i \(-0.260608\pi\)
0.683153 + 0.730275i \(0.260608\pi\)
\(788\) 0 0
\(789\) −5.23550 −0.186389
\(790\) 0 0
\(791\) −0.495445 −0.0176160
\(792\) 0 0
\(793\) −0.232886 −0.00827001
\(794\) 0 0
\(795\) 13.2207 0.468888
\(796\) 0 0
\(797\) −10.2935 −0.364615 −0.182307 0.983242i \(-0.558357\pi\)
−0.182307 + 0.983242i \(0.558357\pi\)
\(798\) 0 0
\(799\) 10.0968 0.357199
\(800\) 0 0
\(801\) 5.52420 0.195188
\(802\) 0 0
\(803\) 22.3594 0.789045
\(804\) 0 0
\(805\) 3.54384 0.124904
\(806\) 0 0
\(807\) −11.7719 −0.414391
\(808\) 0 0
\(809\) 1.16003 0.0407846 0.0203923 0.999792i \(-0.493508\pi\)
0.0203923 + 0.999792i \(0.493508\pi\)
\(810\) 0 0
\(811\) −0.495445 −0.0173974 −0.00869871 0.999962i \(-0.502769\pi\)
−0.00869871 + 0.999962i \(0.502769\pi\)
\(812\) 0 0
\(813\) −27.3929 −0.960712
\(814\) 0 0
\(815\) 5.94418 0.208216
\(816\) 0 0
\(817\) −7.69646 −0.269265
\(818\) 0 0
\(819\) −0.876139 −0.0306148
\(820\) 0 0
\(821\) −5.76711 −0.201274 −0.100637 0.994923i \(-0.532088\pi\)
−0.100637 + 0.994923i \(0.532088\pi\)
\(822\) 0 0
\(823\) 1.83255 0.0638786 0.0319393 0.999490i \(-0.489832\pi\)
0.0319393 + 0.999490i \(0.489832\pi\)
\(824\) 0 0
\(825\) −3.52420 −0.122697
\(826\) 0 0
\(827\) 5.94899 0.206867 0.103433 0.994636i \(-0.467017\pi\)
0.103433 + 0.994636i \(0.467017\pi\)
\(828\) 0 0
\(829\) 6.11164 0.212266 0.106133 0.994352i \(-0.466153\pi\)
0.106133 + 0.994352i \(0.466153\pi\)
\(830\) 0 0
\(831\) 0.247722 0.00859339
\(832\) 0 0
\(833\) −45.2451 −1.56765
\(834\) 0 0
\(835\) 1.82774 0.0632515
\(836\) 0 0
\(837\) −3.46838 −0.119885
\(838\) 0 0
\(839\) −29.6406 −1.02331 −0.511654 0.859191i \(-0.670967\pi\)
−0.511654 + 0.859191i \(0.670967\pi\)
\(840\) 0 0
\(841\) 1.51678 0.0523028
\(842\) 0 0
\(843\) 26.2132 0.902832
\(844\) 0 0
\(845\) −11.1723 −0.384337
\(846\) 0 0
\(847\) −0.920235 −0.0316197
\(848\) 0 0
\(849\) 19.0894 0.655146
\(850\) 0 0
\(851\) −0.305232 −0.0104632
\(852\) 0 0
\(853\) −22.9516 −0.785848 −0.392924 0.919571i \(-0.628536\pi\)
−0.392924 + 0.919571i \(0.628536\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) 0.308348 0.0105330 0.00526648 0.999986i \(-0.498324\pi\)
0.00526648 + 0.999986i \(0.498324\pi\)
\(858\) 0 0
\(859\) −30.2180 −1.03103 −0.515513 0.856882i \(-0.672399\pi\)
−0.515513 + 0.856882i \(0.672399\pi\)
\(860\) 0 0
\(861\) 6.17226 0.210350
\(862\) 0 0
\(863\) −28.6890 −0.976586 −0.488293 0.872680i \(-0.662380\pi\)
−0.488293 + 0.872680i \(0.662380\pi\)
\(864\) 0 0
\(865\) 20.2691 0.689169
\(866\) 0 0
\(867\) −30.2813 −1.02841
\(868\) 0 0
\(869\) −15.3110 −0.519389
\(870\) 0 0
\(871\) 7.31096 0.247722
\(872\) 0 0
\(873\) −2.64806 −0.0896233
\(874\) 0 0
\(875\) −0.648061 −0.0219085
\(876\) 0 0
\(877\) 12.1675 0.410866 0.205433 0.978671i \(-0.434140\pi\)
0.205433 + 0.978671i \(0.434140\pi\)
\(878\) 0 0
\(879\) 19.5800 0.660418
\(880\) 0 0
\(881\) −24.3297 −0.819688 −0.409844 0.912156i \(-0.634417\pi\)
−0.409844 + 0.912156i \(0.634417\pi\)
\(882\) 0 0
\(883\) −58.7449 −1.97692 −0.988461 0.151476i \(-0.951597\pi\)
−0.988461 + 0.151476i \(0.951597\pi\)
\(884\) 0 0
\(885\) −9.64064 −0.324067
\(886\) 0 0
\(887\) −38.5626 −1.29480 −0.647402 0.762149i \(-0.724145\pi\)
−0.647402 + 0.762149i \(0.724145\pi\)
\(888\) 0 0
\(889\) −2.59224 −0.0869410
\(890\) 0 0
\(891\) 3.52420 0.118065
\(892\) 0 0
\(893\) 1.46838 0.0491375
\(894\) 0 0
\(895\) 8.11164 0.271142
\(896\) 0 0
\(897\) 7.39292 0.246842
\(898\) 0 0
\(899\) 19.1600 0.639023
\(900\) 0 0
\(901\) −90.9071 −3.02855
\(902\) 0 0
\(903\) −4.98777 −0.165983
\(904\) 0 0
\(905\) 17.3929 0.578160
\(906\) 0 0
\(907\) −29.4897 −0.979190 −0.489595 0.871950i \(-0.662855\pi\)
−0.489595 + 0.871950i \(0.662855\pi\)
\(908\) 0 0
\(909\) 8.70388 0.288689
\(910\) 0 0
\(911\) −18.2180 −0.603591 −0.301795 0.953373i \(-0.597586\pi\)
−0.301795 + 0.953373i \(0.597586\pi\)
\(912\) 0 0
\(913\) 14.7039 0.486627
\(914\) 0 0
\(915\) 0.172260 0.00569475
\(916\) 0 0
\(917\) 7.46838 0.246628
\(918\) 0 0
\(919\) −55.1600 −1.81956 −0.909781 0.415089i \(-0.863750\pi\)
−0.909781 + 0.415089i \(0.863750\pi\)
\(920\) 0 0
\(921\) −31.0484 −1.02308
\(922\) 0 0
\(923\) −18.4413 −0.607003
\(924\) 0 0
\(925\) 0.0558176 0.00183527
\(926\) 0 0
\(927\) 12.3445 0.405447
\(928\) 0 0
\(929\) −42.4562 −1.39294 −0.696471 0.717585i \(-0.745247\pi\)
−0.696471 + 0.717585i \(0.745247\pi\)
\(930\) 0 0
\(931\) −6.58002 −0.215651
\(932\) 0 0
\(933\) 29.2765 0.958469
\(934\) 0 0
\(935\) 24.2329 0.792500
\(936\) 0 0
\(937\) −11.6406 −0.380283 −0.190142 0.981757i \(-0.560895\pi\)
−0.190142 + 0.981757i \(0.560895\pi\)
\(938\) 0 0
\(939\) 7.98516 0.260586
\(940\) 0 0
\(941\) −41.4126 −1.35001 −0.675006 0.737813i \(-0.735859\pi\)
−0.675006 + 0.737813i \(0.735859\pi\)
\(942\) 0 0
\(943\) −52.0820 −1.69602
\(944\) 0 0
\(945\) 0.648061 0.0210814
\(946\) 0 0
\(947\) −20.3955 −0.662766 −0.331383 0.943496i \(-0.607515\pi\)
−0.331383 + 0.943496i \(0.607515\pi\)
\(948\) 0 0
\(949\) 8.57741 0.278434
\(950\) 0 0
\(951\) 18.9729 0.615240
\(952\) 0 0
\(953\) −41.3323 −1.33888 −0.669442 0.742864i \(-0.733467\pi\)
−0.669442 + 0.742864i \(0.733467\pi\)
\(954\) 0 0
\(955\) −7.17968 −0.232329
\(956\) 0 0
\(957\) −19.4684 −0.629323
\(958\) 0 0
\(959\) −1.29612 −0.0418539
\(960\) 0 0
\(961\) −18.9703 −0.611946
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −12.2887 −0.395587
\(966\) 0 0
\(967\) 4.15262 0.133539 0.0667696 0.997768i \(-0.478731\pi\)
0.0667696 + 0.997768i \(0.478731\pi\)
\(968\) 0 0
\(969\) −6.87614 −0.220893
\(970\) 0 0
\(971\) −41.7523 −1.33989 −0.669947 0.742409i \(-0.733683\pi\)
−0.669947 + 0.742409i \(0.733683\pi\)
\(972\) 0 0
\(973\) −14.9368 −0.478851
\(974\) 0 0
\(975\) −1.35194 −0.0432967
\(976\) 0 0
\(977\) −17.2207 −0.550938 −0.275469 0.961310i \(-0.588833\pi\)
−0.275469 + 0.961310i \(0.588833\pi\)
\(978\) 0 0
\(979\) 19.4684 0.622212
\(980\) 0 0
\(981\) −11.6406 −0.371657
\(982\) 0 0
\(983\) 25.6768 0.818963 0.409482 0.912318i \(-0.365710\pi\)
0.409482 + 0.912318i \(0.365710\pi\)
\(984\) 0 0
\(985\) 20.6284 0.657276
\(986\) 0 0
\(987\) 0.951601 0.0302898
\(988\) 0 0
\(989\) 42.0872 1.33829
\(990\) 0 0
\(991\) 9.90320 0.314586 0.157293 0.987552i \(-0.449723\pi\)
0.157293 + 0.987552i \(0.449723\pi\)
\(992\) 0 0
\(993\) −5.93937 −0.188480
\(994\) 0 0
\(995\) −1.64064 −0.0520119
\(996\) 0 0
\(997\) −41.0187 −1.29908 −0.649538 0.760329i \(-0.725038\pi\)
−0.649538 + 0.760329i \(0.725038\pi\)
\(998\) 0 0
\(999\) −0.0558176 −0.00176599
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 4560.2.a.br.1.2 3
4.3 odd 2 1140.2.a.g.1.2 3
12.11 even 2 3420.2.a.m.1.2 3
20.3 even 4 5700.2.f.q.3649.5 6
20.7 even 4 5700.2.f.q.3649.2 6
20.19 odd 2 5700.2.a.w.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.a.g.1.2 3 4.3 odd 2
3420.2.a.m.1.2 3 12.11 even 2
4560.2.a.br.1.2 3 1.1 even 1 trivial
5700.2.a.w.1.2 3 20.19 odd 2
5700.2.f.q.3649.2 6 20.7 even 4
5700.2.f.q.3649.5 6 20.3 even 4