Properties

Label 8470.2.a.cc.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.61803 q^{6} +1.00000 q^{7} +1.00000 q^{8} -0.381966 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.61803 q^{6} +1.00000 q^{7} +1.00000 q^{8} -0.381966 q^{9} +1.00000 q^{10} +1.61803 q^{12} +5.61803 q^{13} +1.00000 q^{14} +1.61803 q^{15} +1.00000 q^{16} +4.61803 q^{17} -0.381966 q^{18} -5.23607 q^{19} +1.00000 q^{20} +1.61803 q^{21} +8.47214 q^{23} +1.61803 q^{24} +1.00000 q^{25} +5.61803 q^{26} -5.47214 q^{27} +1.00000 q^{28} -5.85410 q^{29} +1.61803 q^{30} +5.23607 q^{31} +1.00000 q^{32} +4.61803 q^{34} +1.00000 q^{35} -0.381966 q^{36} -3.70820 q^{37} -5.23607 q^{38} +9.09017 q^{39} +1.00000 q^{40} +8.94427 q^{41} +1.61803 q^{42} +3.23607 q^{43} -0.381966 q^{45} +8.47214 q^{46} -13.0902 q^{47} +1.61803 q^{48} +1.00000 q^{49} +1.00000 q^{50} +7.47214 q^{51} +5.61803 q^{52} +6.47214 q^{53} -5.47214 q^{54} +1.00000 q^{56} -8.47214 q^{57} -5.85410 q^{58} +4.76393 q^{59} +1.61803 q^{60} -11.7082 q^{61} +5.23607 q^{62} -0.381966 q^{63} +1.00000 q^{64} +5.61803 q^{65} -1.23607 q^{67} +4.61803 q^{68} +13.7082 q^{69} +1.00000 q^{70} -7.32624 q^{71} -0.381966 q^{72} +4.32624 q^{73} -3.70820 q^{74} +1.61803 q^{75} -5.23607 q^{76} +9.09017 q^{78} -0.854102 q^{79} +1.00000 q^{80} -7.70820 q^{81} +8.94427 q^{82} +13.6180 q^{83} +1.61803 q^{84} +4.61803 q^{85} +3.23607 q^{86} -9.47214 q^{87} +0.291796 q^{89} -0.381966 q^{90} +5.61803 q^{91} +8.47214 q^{92} +8.47214 q^{93} -13.0902 q^{94} -5.23607 q^{95} +1.61803 q^{96} -3.38197 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + q^{3} + 2q^{4} + 2q^{5} + q^{6} + 2q^{7} + 2q^{8} - 3q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + q^{3} + 2q^{4} + 2q^{5} + q^{6} + 2q^{7} + 2q^{8} - 3q^{9} + 2q^{10} + q^{12} + 9q^{13} + 2q^{14} + q^{15} + 2q^{16} + 7q^{17} - 3q^{18} - 6q^{19} + 2q^{20} + q^{21} + 8q^{23} + q^{24} + 2q^{25} + 9q^{26} - 2q^{27} + 2q^{28} - 5q^{29} + q^{30} + 6q^{31} + 2q^{32} + 7q^{34} + 2q^{35} - 3q^{36} + 6q^{37} - 6q^{38} + 7q^{39} + 2q^{40} + q^{42} + 2q^{43} - 3q^{45} + 8q^{46} - 15q^{47} + q^{48} + 2q^{49} + 2q^{50} + 6q^{51} + 9q^{52} + 4q^{53} - 2q^{54} + 2q^{56} - 8q^{57} - 5q^{58} + 14q^{59} + q^{60} - 10q^{61} + 6q^{62} - 3q^{63} + 2q^{64} + 9q^{65} + 2q^{67} + 7q^{68} + 14q^{69} + 2q^{70} + q^{71} - 3q^{72} - 7q^{73} + 6q^{74} + q^{75} - 6q^{76} + 7q^{78} + 5q^{79} + 2q^{80} - 2q^{81} + 25q^{83} + q^{84} + 7q^{85} + 2q^{86} - 10q^{87} + 14q^{89} - 3q^{90} + 9q^{91} + 8q^{92} + 8q^{93} - 15q^{94} - 6q^{95} + q^{96} - 9q^{97} + 2q^{98} + O(q^{100}) \)

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
\(3\) 1.61803 0.934172 0.467086 0.884212i \(-0.345304\pi\)
0.467086 + 0.884212i \(0.345304\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.61803 0.660560
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −0.381966 −0.127322
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 1.61803 0.467086
\(13\) 5.61803 1.55816 0.779081 0.626923i \(-0.215686\pi\)
0.779081 + 0.626923i \(0.215686\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.61803 0.417775
\(16\) 1.00000 0.250000
\(17\) 4.61803 1.12004 0.560019 0.828480i \(-0.310794\pi\)
0.560019 + 0.828480i \(0.310794\pi\)
\(18\) −0.381966 −0.0900303
\(19\) −5.23607 −1.20124 −0.600618 0.799536i \(-0.705079\pi\)
−0.600618 + 0.799536i \(0.705079\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.61803 0.353084
\(22\) 0 0
\(23\) 8.47214 1.76656 0.883281 0.468844i \(-0.155329\pi\)
0.883281 + 0.468844i \(0.155329\pi\)
\(24\) 1.61803 0.330280
\(25\) 1.00000 0.200000
\(26\) 5.61803 1.10179
\(27\) −5.47214 −1.05311
\(28\) 1.00000 0.188982
\(29\) −5.85410 −1.08708 −0.543540 0.839383i \(-0.682916\pi\)
−0.543540 + 0.839383i \(0.682916\pi\)
\(30\) 1.61803 0.295411
\(31\) 5.23607 0.940426 0.470213 0.882553i \(-0.344177\pi\)
0.470213 + 0.882553i \(0.344177\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.61803 0.791986
\(35\) 1.00000 0.169031
\(36\) −0.381966 −0.0636610
\(37\) −3.70820 −0.609625 −0.304812 0.952412i \(-0.598594\pi\)
−0.304812 + 0.952412i \(0.598594\pi\)
\(38\) −5.23607 −0.849402
\(39\) 9.09017 1.45559
\(40\) 1.00000 0.158114
\(41\) 8.94427 1.39686 0.698430 0.715678i \(-0.253882\pi\)
0.698430 + 0.715678i \(0.253882\pi\)
\(42\) 1.61803 0.249668
\(43\) 3.23607 0.493496 0.246748 0.969080i \(-0.420638\pi\)
0.246748 + 0.969080i \(0.420638\pi\)
\(44\) 0 0
\(45\) −0.381966 −0.0569401
\(46\) 8.47214 1.24915
\(47\) −13.0902 −1.90940 −0.954699 0.297574i \(-0.903823\pi\)
−0.954699 + 0.297574i \(0.903823\pi\)
\(48\) 1.61803 0.233543
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 7.47214 1.04631
\(52\) 5.61803 0.779081
\(53\) 6.47214 0.889016 0.444508 0.895775i \(-0.353378\pi\)
0.444508 + 0.895775i \(0.353378\pi\)
\(54\) −5.47214 −0.744663
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −8.47214 −1.12216
\(58\) −5.85410 −0.768681
\(59\) 4.76393 0.620211 0.310106 0.950702i \(-0.399636\pi\)
0.310106 + 0.950702i \(0.399636\pi\)
\(60\) 1.61803 0.208887
\(61\) −11.7082 −1.49908 −0.749541 0.661958i \(-0.769726\pi\)
−0.749541 + 0.661958i \(0.769726\pi\)
\(62\) 5.23607 0.664981
\(63\) −0.381966 −0.0481232
\(64\) 1.00000 0.125000
\(65\) 5.61803 0.696831
\(66\) 0 0
\(67\) −1.23607 −0.151010 −0.0755049 0.997145i \(-0.524057\pi\)
−0.0755049 + 0.997145i \(0.524057\pi\)
\(68\) 4.61803 0.560019
\(69\) 13.7082 1.65027
\(70\) 1.00000 0.119523
\(71\) −7.32624 −0.869464 −0.434732 0.900560i \(-0.643157\pi\)
−0.434732 + 0.900560i \(0.643157\pi\)
\(72\) −0.381966 −0.0450151
\(73\) 4.32624 0.506348 0.253174 0.967421i \(-0.418526\pi\)
0.253174 + 0.967421i \(0.418526\pi\)
\(74\) −3.70820 −0.431070
\(75\) 1.61803 0.186834
\(76\) −5.23607 −0.600618
\(77\) 0 0
\(78\) 9.09017 1.02926
\(79\) −0.854102 −0.0960940 −0.0480470 0.998845i \(-0.515300\pi\)
−0.0480470 + 0.998845i \(0.515300\pi\)
\(80\) 1.00000 0.111803
\(81\) −7.70820 −0.856467
\(82\) 8.94427 0.987730
\(83\) 13.6180 1.49477 0.747387 0.664389i \(-0.231308\pi\)
0.747387 + 0.664389i \(0.231308\pi\)
\(84\) 1.61803 0.176542
\(85\) 4.61803 0.500896
\(86\) 3.23607 0.348954
\(87\) −9.47214 −1.01552
\(88\) 0 0
\(89\) 0.291796 0.0309303 0.0154652 0.999880i \(-0.495077\pi\)
0.0154652 + 0.999880i \(0.495077\pi\)
\(90\) −0.381966 −0.0402628
\(91\) 5.61803 0.588930
\(92\) 8.47214 0.883281
\(93\) 8.47214 0.878520
\(94\) −13.0902 −1.35015
\(95\) −5.23607 −0.537209
\(96\) 1.61803 0.165140
\(97\) −3.38197 −0.343387 −0.171693 0.985150i \(-0.554924\pi\)
−0.171693 + 0.985150i \(0.554924\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −5.23607 −0.521008 −0.260504 0.965473i \(-0.583889\pi\)
−0.260504 + 0.965473i \(0.583889\pi\)
\(102\) 7.47214 0.739852
\(103\) 3.56231 0.351004 0.175502 0.984479i \(-0.443845\pi\)
0.175502 + 0.984479i \(0.443845\pi\)
\(104\) 5.61803 0.550894
\(105\) 1.61803 0.157904
\(106\) 6.47214 0.628629
\(107\) −10.7639 −1.04059 −0.520294 0.853987i \(-0.674178\pi\)
−0.520294 + 0.853987i \(0.674178\pi\)
\(108\) −5.47214 −0.526557
\(109\) 3.52786 0.337908 0.168954 0.985624i \(-0.445961\pi\)
0.168954 + 0.985624i \(0.445961\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 1.00000 0.0944911
\(113\) −1.52786 −0.143729 −0.0718647 0.997414i \(-0.522895\pi\)
−0.0718647 + 0.997414i \(0.522895\pi\)
\(114\) −8.47214 −0.793488
\(115\) 8.47214 0.790031
\(116\) −5.85410 −0.543540
\(117\) −2.14590 −0.198388
\(118\) 4.76393 0.438555
\(119\) 4.61803 0.423334
\(120\) 1.61803 0.147706
\(121\) 0 0
\(122\) −11.7082 −1.06001
\(123\) 14.4721 1.30491
\(124\) 5.23607 0.470213
\(125\) 1.00000 0.0894427
\(126\) −0.381966 −0.0340282
\(127\) 20.1803 1.79072 0.895358 0.445348i \(-0.146920\pi\)
0.895358 + 0.445348i \(0.146920\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.23607 0.461010
\(130\) 5.61803 0.492734
\(131\) −11.4164 −0.997456 −0.498728 0.866758i \(-0.666199\pi\)
−0.498728 + 0.866758i \(0.666199\pi\)
\(132\) 0 0
\(133\) −5.23607 −0.454025
\(134\) −1.23607 −0.106780
\(135\) −5.47214 −0.470966
\(136\) 4.61803 0.395993
\(137\) −16.0000 −1.36697 −0.683486 0.729964i \(-0.739537\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(138\) 13.7082 1.16692
\(139\) 8.94427 0.758643 0.379322 0.925265i \(-0.376157\pi\)
0.379322 + 0.925265i \(0.376157\pi\)
\(140\) 1.00000 0.0845154
\(141\) −21.1803 −1.78371
\(142\) −7.32624 −0.614804
\(143\) 0 0
\(144\) −0.381966 −0.0318305
\(145\) −5.85410 −0.486157
\(146\) 4.32624 0.358042
\(147\) 1.61803 0.133453
\(148\) −3.70820 −0.304812
\(149\) 20.8541 1.70843 0.854217 0.519916i \(-0.174037\pi\)
0.854217 + 0.519916i \(0.174037\pi\)
\(150\) 1.61803 0.132112
\(151\) 21.3820 1.74004 0.870020 0.493017i \(-0.164106\pi\)
0.870020 + 0.493017i \(0.164106\pi\)
\(152\) −5.23607 −0.424701
\(153\) −1.76393 −0.142605
\(154\) 0 0
\(155\) 5.23607 0.420571
\(156\) 9.09017 0.727796
\(157\) −11.0902 −0.885092 −0.442546 0.896746i \(-0.645925\pi\)
−0.442546 + 0.896746i \(0.645925\pi\)
\(158\) −0.854102 −0.0679487
\(159\) 10.4721 0.830494
\(160\) 1.00000 0.0790569
\(161\) 8.47214 0.667698
\(162\) −7.70820 −0.605614
\(163\) −14.6525 −1.14767 −0.573835 0.818971i \(-0.694545\pi\)
−0.573835 + 0.818971i \(0.694545\pi\)
\(164\) 8.94427 0.698430
\(165\) 0 0
\(166\) 13.6180 1.05696
\(167\) 20.0000 1.54765 0.773823 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) 1.61803 0.124834
\(169\) 18.5623 1.42787
\(170\) 4.61803 0.354187
\(171\) 2.00000 0.152944
\(172\) 3.23607 0.246748
\(173\) −15.5066 −1.17894 −0.589472 0.807789i \(-0.700664\pi\)
−0.589472 + 0.807789i \(0.700664\pi\)
\(174\) −9.47214 −0.718081
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 7.70820 0.579384
\(178\) 0.291796 0.0218710
\(179\) −19.5623 −1.46216 −0.731078 0.682294i \(-0.760982\pi\)
−0.731078 + 0.682294i \(0.760982\pi\)
\(180\) −0.381966 −0.0284701
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 5.61803 0.416436
\(183\) −18.9443 −1.40040
\(184\) 8.47214 0.624574
\(185\) −3.70820 −0.272633
\(186\) 8.47214 0.621207
\(187\) 0 0
\(188\) −13.0902 −0.954699
\(189\) −5.47214 −0.398039
\(190\) −5.23607 −0.379864
\(191\) 0.854102 0.0618006 0.0309003 0.999522i \(-0.490163\pi\)
0.0309003 + 0.999522i \(0.490163\pi\)
\(192\) 1.61803 0.116772
\(193\) 12.9443 0.931749 0.465875 0.884851i \(-0.345740\pi\)
0.465875 + 0.884851i \(0.345740\pi\)
\(194\) −3.38197 −0.242811
\(195\) 9.09017 0.650961
\(196\) 1.00000 0.0714286
\(197\) −15.4164 −1.09837 −0.549187 0.835700i \(-0.685062\pi\)
−0.549187 + 0.835700i \(0.685062\pi\)
\(198\) 0 0
\(199\) −10.9443 −0.775819 −0.387909 0.921697i \(-0.626803\pi\)
−0.387909 + 0.921697i \(0.626803\pi\)
\(200\) 1.00000 0.0707107
\(201\) −2.00000 −0.141069
\(202\) −5.23607 −0.368408
\(203\) −5.85410 −0.410877
\(204\) 7.47214 0.523154
\(205\) 8.94427 0.624695
\(206\) 3.56231 0.248198
\(207\) −3.23607 −0.224922
\(208\) 5.61803 0.389541
\(209\) 0 0
\(210\) 1.61803 0.111655
\(211\) 15.3262 1.05510 0.527551 0.849523i \(-0.323110\pi\)
0.527551 + 0.849523i \(0.323110\pi\)
\(212\) 6.47214 0.444508
\(213\) −11.8541 −0.812230
\(214\) −10.7639 −0.735807
\(215\) 3.23607 0.220698
\(216\) −5.47214 −0.372332
\(217\) 5.23607 0.355447
\(218\) 3.52786 0.238937
\(219\) 7.00000 0.473016
\(220\) 0 0
\(221\) 25.9443 1.74520
\(222\) −6.00000 −0.402694
\(223\) −18.4721 −1.23699 −0.618493 0.785790i \(-0.712256\pi\)
−0.618493 + 0.785790i \(0.712256\pi\)
\(224\) 1.00000 0.0668153
\(225\) −0.381966 −0.0254644
\(226\) −1.52786 −0.101632
\(227\) 7.14590 0.474290 0.237145 0.971474i \(-0.423788\pi\)
0.237145 + 0.971474i \(0.423788\pi\)
\(228\) −8.47214 −0.561081
\(229\) 7.70820 0.509372 0.254686 0.967024i \(-0.418028\pi\)
0.254686 + 0.967024i \(0.418028\pi\)
\(230\) 8.47214 0.558636
\(231\) 0 0
\(232\) −5.85410 −0.384341
\(233\) −16.6525 −1.09094 −0.545470 0.838130i \(-0.683649\pi\)
−0.545470 + 0.838130i \(0.683649\pi\)
\(234\) −2.14590 −0.140282
\(235\) −13.0902 −0.853909
\(236\) 4.76393 0.310106
\(237\) −1.38197 −0.0897683
\(238\) 4.61803 0.299343
\(239\) 8.85410 0.572724 0.286362 0.958121i \(-0.407554\pi\)
0.286362 + 0.958121i \(0.407554\pi\)
\(240\) 1.61803 0.104444
\(241\) −3.70820 −0.238866 −0.119433 0.992842i \(-0.538108\pi\)
−0.119433 + 0.992842i \(0.538108\pi\)
\(242\) 0 0
\(243\) 3.94427 0.253025
\(244\) −11.7082 −0.749541
\(245\) 1.00000 0.0638877
\(246\) 14.4721 0.922710
\(247\) −29.4164 −1.87172
\(248\) 5.23607 0.332491
\(249\) 22.0344 1.39638
\(250\) 1.00000 0.0632456
\(251\) 0.763932 0.0482190 0.0241095 0.999709i \(-0.492325\pi\)
0.0241095 + 0.999709i \(0.492325\pi\)
\(252\) −0.381966 −0.0240616
\(253\) 0 0
\(254\) 20.1803 1.26623
\(255\) 7.47214 0.467923
\(256\) 1.00000 0.0625000
\(257\) −13.2705 −0.827792 −0.413896 0.910324i \(-0.635832\pi\)
−0.413896 + 0.910324i \(0.635832\pi\)
\(258\) 5.23607 0.325983
\(259\) −3.70820 −0.230417
\(260\) 5.61803 0.348416
\(261\) 2.23607 0.138409
\(262\) −11.4164 −0.705308
\(263\) 14.2918 0.881270 0.440635 0.897686i \(-0.354753\pi\)
0.440635 + 0.897686i \(0.354753\pi\)
\(264\) 0 0
\(265\) 6.47214 0.397580
\(266\) −5.23607 −0.321044
\(267\) 0.472136 0.0288943
\(268\) −1.23607 −0.0755049
\(269\) 19.7082 1.20163 0.600815 0.799388i \(-0.294843\pi\)
0.600815 + 0.799388i \(0.294843\pi\)
\(270\) −5.47214 −0.333024
\(271\) 15.1246 0.918755 0.459377 0.888241i \(-0.348073\pi\)
0.459377 + 0.888241i \(0.348073\pi\)
\(272\) 4.61803 0.280009
\(273\) 9.09017 0.550162
\(274\) −16.0000 −0.966595
\(275\) 0 0
\(276\) 13.7082 0.825137
\(277\) 13.4164 0.806114 0.403057 0.915175i \(-0.367948\pi\)
0.403057 + 0.915175i \(0.367948\pi\)
\(278\) 8.94427 0.536442
\(279\) −2.00000 −0.119737
\(280\) 1.00000 0.0597614
\(281\) 18.3607 1.09531 0.547653 0.836705i \(-0.315521\pi\)
0.547653 + 0.836705i \(0.315521\pi\)
\(282\) −21.1803 −1.26127
\(283\) −14.3262 −0.851606 −0.425803 0.904816i \(-0.640008\pi\)
−0.425803 + 0.904816i \(0.640008\pi\)
\(284\) −7.32624 −0.434732
\(285\) −8.47214 −0.501846
\(286\) 0 0
\(287\) 8.94427 0.527964
\(288\) −0.381966 −0.0225076
\(289\) 4.32624 0.254485
\(290\) −5.85410 −0.343765
\(291\) −5.47214 −0.320782
\(292\) 4.32624 0.253174
\(293\) −25.4164 −1.48484 −0.742421 0.669933i \(-0.766323\pi\)
−0.742421 + 0.669933i \(0.766323\pi\)
\(294\) 1.61803 0.0943657
\(295\) 4.76393 0.277367
\(296\) −3.70820 −0.215535
\(297\) 0 0
\(298\) 20.8541 1.20805
\(299\) 47.5967 2.75259
\(300\) 1.61803 0.0934172
\(301\) 3.23607 0.186524
\(302\) 21.3820 1.23039
\(303\) −8.47214 −0.486711
\(304\) −5.23607 −0.300309
\(305\) −11.7082 −0.670410
\(306\) −1.76393 −0.100837
\(307\) −33.4508 −1.90914 −0.954570 0.297985i \(-0.903685\pi\)
−0.954570 + 0.297985i \(0.903685\pi\)
\(308\) 0 0
\(309\) 5.76393 0.327899
\(310\) 5.23607 0.297389
\(311\) 10.1803 0.577274 0.288637 0.957439i \(-0.406798\pi\)
0.288637 + 0.957439i \(0.406798\pi\)
\(312\) 9.09017 0.514630
\(313\) 11.8885 0.671980 0.335990 0.941866i \(-0.390929\pi\)
0.335990 + 0.941866i \(0.390929\pi\)
\(314\) −11.0902 −0.625854
\(315\) −0.381966 −0.0215213
\(316\) −0.854102 −0.0480470
\(317\) 17.8885 1.00472 0.502360 0.864658i \(-0.332465\pi\)
0.502360 + 0.864658i \(0.332465\pi\)
\(318\) 10.4721 0.587248
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −17.4164 −0.972089
\(322\) 8.47214 0.472134
\(323\) −24.1803 −1.34543
\(324\) −7.70820 −0.428234
\(325\) 5.61803 0.311632
\(326\) −14.6525 −0.811526
\(327\) 5.70820 0.315664
\(328\) 8.94427 0.493865
\(329\) −13.0902 −0.721684
\(330\) 0 0
\(331\) −6.38197 −0.350785 −0.175392 0.984499i \(-0.556119\pi\)
−0.175392 + 0.984499i \(0.556119\pi\)
\(332\) 13.6180 0.747387
\(333\) 1.41641 0.0776187
\(334\) 20.0000 1.09435
\(335\) −1.23607 −0.0675336
\(336\) 1.61803 0.0882710
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 18.5623 1.00966
\(339\) −2.47214 −0.134268
\(340\) 4.61803 0.250448
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) 1.00000 0.0539949
\(344\) 3.23607 0.174477
\(345\) 13.7082 0.738025
\(346\) −15.5066 −0.833639
\(347\) 25.8885 1.38977 0.694885 0.719121i \(-0.255455\pi\)
0.694885 + 0.719121i \(0.255455\pi\)
\(348\) −9.47214 −0.507760
\(349\) −26.9443 −1.44229 −0.721147 0.692782i \(-0.756385\pi\)
−0.721147 + 0.692782i \(0.756385\pi\)
\(350\) 1.00000 0.0534522
\(351\) −30.7426 −1.64092
\(352\) 0 0
\(353\) −24.4508 −1.30139 −0.650694 0.759340i \(-0.725522\pi\)
−0.650694 + 0.759340i \(0.725522\pi\)
\(354\) 7.70820 0.409686
\(355\) −7.32624 −0.388836
\(356\) 0.291796 0.0154652
\(357\) 7.47214 0.395467
\(358\) −19.5623 −1.03390
\(359\) 17.9787 0.948880 0.474440 0.880288i \(-0.342651\pi\)
0.474440 + 0.880288i \(0.342651\pi\)
\(360\) −0.381966 −0.0201314
\(361\) 8.41641 0.442969
\(362\) 0 0
\(363\) 0 0
\(364\) 5.61803 0.294465
\(365\) 4.32624 0.226446
\(366\) −18.9443 −0.990233
\(367\) 35.3262 1.84401 0.922007 0.387172i \(-0.126548\pi\)
0.922007 + 0.387172i \(0.126548\pi\)
\(368\) 8.47214 0.441641
\(369\) −3.41641 −0.177851
\(370\) −3.70820 −0.192780
\(371\) 6.47214 0.336017
\(372\) 8.47214 0.439260
\(373\) −2.65248 −0.137340 −0.0686700 0.997639i \(-0.521876\pi\)
−0.0686700 + 0.997639i \(0.521876\pi\)
\(374\) 0 0
\(375\) 1.61803 0.0835549
\(376\) −13.0902 −0.675074
\(377\) −32.8885 −1.69385
\(378\) −5.47214 −0.281456
\(379\) 16.1459 0.829359 0.414680 0.909968i \(-0.363894\pi\)
0.414680 + 0.909968i \(0.363894\pi\)
\(380\) −5.23607 −0.268605
\(381\) 32.6525 1.67284
\(382\) 0.854102 0.0436997
\(383\) 8.85410 0.452423 0.226212 0.974078i \(-0.427366\pi\)
0.226212 + 0.974078i \(0.427366\pi\)
\(384\) 1.61803 0.0825700
\(385\) 0 0
\(386\) 12.9443 0.658846
\(387\) −1.23607 −0.0628329
\(388\) −3.38197 −0.171693
\(389\) −20.2705 −1.02776 −0.513878 0.857863i \(-0.671792\pi\)
−0.513878 + 0.857863i \(0.671792\pi\)
\(390\) 9.09017 0.460299
\(391\) 39.1246 1.97862
\(392\) 1.00000 0.0505076
\(393\) −18.4721 −0.931796
\(394\) −15.4164 −0.776667
\(395\) −0.854102 −0.0429745
\(396\) 0 0
\(397\) −30.5623 −1.53388 −0.766939 0.641720i \(-0.778221\pi\)
−0.766939 + 0.641720i \(0.778221\pi\)
\(398\) −10.9443 −0.548587
\(399\) −8.47214 −0.424137
\(400\) 1.00000 0.0500000
\(401\) −36.6180 −1.82862 −0.914309 0.405018i \(-0.867265\pi\)
−0.914309 + 0.405018i \(0.867265\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 29.4164 1.46534
\(404\) −5.23607 −0.260504
\(405\) −7.70820 −0.383024
\(406\) −5.85410 −0.290534
\(407\) 0 0
\(408\) 7.47214 0.369926
\(409\) 5.23607 0.258907 0.129453 0.991586i \(-0.458678\pi\)
0.129453 + 0.991586i \(0.458678\pi\)
\(410\) 8.94427 0.441726
\(411\) −25.8885 −1.27699
\(412\) 3.56231 0.175502
\(413\) 4.76393 0.234418
\(414\) −3.23607 −0.159044
\(415\) 13.6180 0.668483
\(416\) 5.61803 0.275447
\(417\) 14.4721 0.708704
\(418\) 0 0
\(419\) 6.94427 0.339250 0.169625 0.985509i \(-0.445744\pi\)
0.169625 + 0.985509i \(0.445744\pi\)
\(420\) 1.61803 0.0789520
\(421\) −11.3262 −0.552007 −0.276004 0.961157i \(-0.589010\pi\)
−0.276004 + 0.961157i \(0.589010\pi\)
\(422\) 15.3262 0.746070
\(423\) 5.00000 0.243108
\(424\) 6.47214 0.314315
\(425\) 4.61803 0.224008
\(426\) −11.8541 −0.574333
\(427\) −11.7082 −0.566600
\(428\) −10.7639 −0.520294
\(429\) 0 0
\(430\) 3.23607 0.156057
\(431\) −15.5623 −0.749610 −0.374805 0.927104i \(-0.622290\pi\)
−0.374805 + 0.927104i \(0.622290\pi\)
\(432\) −5.47214 −0.263278
\(433\) 4.09017 0.196561 0.0982805 0.995159i \(-0.468666\pi\)
0.0982805 + 0.995159i \(0.468666\pi\)
\(434\) 5.23607 0.251339
\(435\) −9.47214 −0.454154
\(436\) 3.52786 0.168954
\(437\) −44.3607 −2.12206
\(438\) 7.00000 0.334473
\(439\) −10.6525 −0.508415 −0.254207 0.967150i \(-0.581815\pi\)
−0.254207 + 0.967150i \(0.581815\pi\)
\(440\) 0 0
\(441\) −0.381966 −0.0181889
\(442\) 25.9443 1.23404
\(443\) −15.2361 −0.723887 −0.361944 0.932200i \(-0.617887\pi\)
−0.361944 + 0.932200i \(0.617887\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0.291796 0.0138325
\(446\) −18.4721 −0.874681
\(447\) 33.7426 1.59597
\(448\) 1.00000 0.0472456
\(449\) −19.5066 −0.920572 −0.460286 0.887771i \(-0.652253\pi\)
−0.460286 + 0.887771i \(0.652253\pi\)
\(450\) −0.381966 −0.0180061
\(451\) 0 0
\(452\) −1.52786 −0.0718647
\(453\) 34.5967 1.62550
\(454\) 7.14590 0.335374
\(455\) 5.61803 0.263377
\(456\) −8.47214 −0.396744
\(457\) −14.0000 −0.654892 −0.327446 0.944870i \(-0.606188\pi\)
−0.327446 + 0.944870i \(0.606188\pi\)
\(458\) 7.70820 0.360181
\(459\) −25.2705 −1.17953
\(460\) 8.47214 0.395015
\(461\) 25.1246 1.17017 0.585085 0.810972i \(-0.301061\pi\)
0.585085 + 0.810972i \(0.301061\pi\)
\(462\) 0 0
\(463\) −14.1803 −0.659016 −0.329508 0.944153i \(-0.606883\pi\)
−0.329508 + 0.944153i \(0.606883\pi\)
\(464\) −5.85410 −0.271770
\(465\) 8.47214 0.392886
\(466\) −16.6525 −0.771411
\(467\) 21.6180 1.00036 0.500182 0.865920i \(-0.333267\pi\)
0.500182 + 0.865920i \(0.333267\pi\)
\(468\) −2.14590 −0.0991942
\(469\) −1.23607 −0.0570763
\(470\) −13.0902 −0.603805
\(471\) −17.9443 −0.826828
\(472\) 4.76393 0.219278
\(473\) 0 0
\(474\) −1.38197 −0.0634758
\(475\) −5.23607 −0.240247
\(476\) 4.61803 0.211667
\(477\) −2.47214 −0.113191
\(478\) 8.85410 0.404977
\(479\) −35.1246 −1.60488 −0.802442 0.596730i \(-0.796466\pi\)
−0.802442 + 0.596730i \(0.796466\pi\)
\(480\) 1.61803 0.0738528
\(481\) −20.8328 −0.949895
\(482\) −3.70820 −0.168904
\(483\) 13.7082 0.623745
\(484\) 0 0
\(485\) −3.38197 −0.153567
\(486\) 3.94427 0.178916
\(487\) 11.8197 0.535600 0.267800 0.963475i \(-0.413703\pi\)
0.267800 + 0.963475i \(0.413703\pi\)
\(488\) −11.7082 −0.530005
\(489\) −23.7082 −1.07212
\(490\) 1.00000 0.0451754
\(491\) 0.944272 0.0426144 0.0213072 0.999773i \(-0.493217\pi\)
0.0213072 + 0.999773i \(0.493217\pi\)
\(492\) 14.4721 0.652454
\(493\) −27.0344 −1.21757
\(494\) −29.4164 −1.32351
\(495\) 0 0
\(496\) 5.23607 0.235106
\(497\) −7.32624 −0.328627
\(498\) 22.0344 0.987387
\(499\) −24.7984 −1.11013 −0.555064 0.831808i \(-0.687306\pi\)
−0.555064 + 0.831808i \(0.687306\pi\)
\(500\) 1.00000 0.0447214
\(501\) 32.3607 1.44577
\(502\) 0.763932 0.0340960
\(503\) 25.9787 1.15833 0.579167 0.815209i \(-0.303378\pi\)
0.579167 + 0.815209i \(0.303378\pi\)
\(504\) −0.381966 −0.0170141
\(505\) −5.23607 −0.233002
\(506\) 0 0
\(507\) 30.0344 1.33388
\(508\) 20.1803 0.895358
\(509\) −19.8885 −0.881544 −0.440772 0.897619i \(-0.645295\pi\)
−0.440772 + 0.897619i \(0.645295\pi\)
\(510\) 7.47214 0.330872
\(511\) 4.32624 0.191381
\(512\) 1.00000 0.0441942
\(513\) 28.6525 1.26504
\(514\) −13.2705 −0.585337
\(515\) 3.56231 0.156974
\(516\) 5.23607 0.230505
\(517\) 0 0
\(518\) −3.70820 −0.162929
\(519\) −25.0902 −1.10134
\(520\) 5.61803 0.246367
\(521\) −32.5410 −1.42565 −0.712824 0.701343i \(-0.752584\pi\)
−0.712824 + 0.701343i \(0.752584\pi\)
\(522\) 2.23607 0.0978700
\(523\) −9.74265 −0.426016 −0.213008 0.977050i \(-0.568326\pi\)
−0.213008 + 0.977050i \(0.568326\pi\)
\(524\) −11.4164 −0.498728
\(525\) 1.61803 0.0706168
\(526\) 14.2918 0.623152
\(527\) 24.1803 1.05331
\(528\) 0 0
\(529\) 48.7771 2.12074
\(530\) 6.47214 0.281132
\(531\) −1.81966 −0.0789665
\(532\) −5.23607 −0.227012
\(533\) 50.2492 2.17654
\(534\) 0.472136 0.0204313
\(535\) −10.7639 −0.465365
\(536\) −1.23607 −0.0533900
\(537\) −31.6525 −1.36591
\(538\) 19.7082 0.849681
\(539\) 0 0
\(540\) −5.47214 −0.235483
\(541\) −8.09017 −0.347824 −0.173912 0.984761i \(-0.555641\pi\)
−0.173912 + 0.984761i \(0.555641\pi\)
\(542\) 15.1246 0.649658
\(543\) 0 0
\(544\) 4.61803 0.197997
\(545\) 3.52786 0.151117
\(546\) 9.09017 0.389023
\(547\) −22.1803 −0.948363 −0.474181 0.880427i \(-0.657256\pi\)
−0.474181 + 0.880427i \(0.657256\pi\)
\(548\) −16.0000 −0.683486
\(549\) 4.47214 0.190866
\(550\) 0 0
\(551\) 30.6525 1.30584
\(552\) 13.7082 0.583460
\(553\) −0.854102 −0.0363201
\(554\) 13.4164 0.570009
\(555\) −6.00000 −0.254686
\(556\) 8.94427 0.379322
\(557\) −45.5967 −1.93200 −0.965998 0.258549i \(-0.916756\pi\)
−0.965998 + 0.258549i \(0.916756\pi\)
\(558\) −2.00000 −0.0846668
\(559\) 18.1803 0.768946
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 18.3607 0.774499
\(563\) −23.5623 −0.993033 −0.496516 0.868027i \(-0.665388\pi\)
−0.496516 + 0.868027i \(0.665388\pi\)
\(564\) −21.1803 −0.891853
\(565\) −1.52786 −0.0642777
\(566\) −14.3262 −0.602177
\(567\) −7.70820 −0.323714
\(568\) −7.32624 −0.307402
\(569\) 18.7984 0.788069 0.394034 0.919096i \(-0.371079\pi\)
0.394034 + 0.919096i \(0.371079\pi\)
\(570\) −8.47214 −0.354859
\(571\) −43.8541 −1.83524 −0.917619 0.397462i \(-0.869891\pi\)
−0.917619 + 0.397462i \(0.869891\pi\)
\(572\) 0 0
\(573\) 1.38197 0.0577325
\(574\) 8.94427 0.373327
\(575\) 8.47214 0.353312
\(576\) −0.381966 −0.0159153
\(577\) 42.2148 1.75742 0.878712 0.477352i \(-0.158403\pi\)
0.878712 + 0.477352i \(0.158403\pi\)
\(578\) 4.32624 0.179948
\(579\) 20.9443 0.870414
\(580\) −5.85410 −0.243078
\(581\) 13.6180 0.564971
\(582\) −5.47214 −0.226827
\(583\) 0 0
\(584\) 4.32624 0.179021
\(585\) −2.14590 −0.0887220
\(586\) −25.4164 −1.04994
\(587\) 19.5066 0.805123 0.402561 0.915393i \(-0.368120\pi\)
0.402561 + 0.915393i \(0.368120\pi\)
\(588\) 1.61803 0.0667266
\(589\) −27.4164 −1.12967
\(590\) 4.76393 0.196128
\(591\) −24.9443 −1.02607
\(592\) −3.70820 −0.152406
\(593\) 14.7984 0.607696 0.303848 0.952720i \(-0.401728\pi\)
0.303848 + 0.952720i \(0.401728\pi\)
\(594\) 0 0
\(595\) 4.61803 0.189321
\(596\) 20.8541 0.854217
\(597\) −17.7082 −0.724749
\(598\) 47.5967 1.94638
\(599\) 38.3951 1.56878 0.784391 0.620267i \(-0.212976\pi\)
0.784391 + 0.620267i \(0.212976\pi\)
\(600\) 1.61803 0.0660560
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 3.23607 0.131892
\(603\) 0.472136 0.0192269
\(604\) 21.3820 0.870020
\(605\) 0 0
\(606\) −8.47214 −0.344157
\(607\) −38.6869 −1.57025 −0.785127 0.619335i \(-0.787402\pi\)
−0.785127 + 0.619335i \(0.787402\pi\)
\(608\) −5.23607 −0.212351
\(609\) −9.47214 −0.383830
\(610\) −11.7082 −0.474051
\(611\) −73.5410 −2.97515
\(612\) −1.76393 −0.0713027
\(613\) −22.9443 −0.926710 −0.463355 0.886173i \(-0.653355\pi\)
−0.463355 + 0.886173i \(0.653355\pi\)
\(614\) −33.4508 −1.34997
\(615\) 14.4721 0.583573
\(616\) 0 0
\(617\) −3.81966 −0.153774 −0.0768869 0.997040i \(-0.524498\pi\)
−0.0768869 + 0.997040i \(0.524498\pi\)
\(618\) 5.76393 0.231859
\(619\) 0.583592 0.0234565 0.0117283 0.999931i \(-0.496267\pi\)
0.0117283 + 0.999931i \(0.496267\pi\)
\(620\) 5.23607 0.210286
\(621\) −46.3607 −1.86039
\(622\) 10.1803 0.408194
\(623\) 0.291796 0.0116906
\(624\) 9.09017 0.363898
\(625\) 1.00000 0.0400000
\(626\) 11.8885 0.475162
\(627\) 0 0
\(628\) −11.0902 −0.442546
\(629\) −17.1246 −0.682803
\(630\) −0.381966 −0.0152179
\(631\) 9.45085 0.376232 0.188116 0.982147i \(-0.439762\pi\)
0.188116 + 0.982147i \(0.439762\pi\)
\(632\) −0.854102 −0.0339744
\(633\) 24.7984 0.985647
\(634\) 17.8885 0.710445
\(635\) 20.1803 0.800832
\(636\) 10.4721 0.415247
\(637\) 5.61803 0.222595
\(638\) 0 0
\(639\) 2.79837 0.110702
\(640\) 1.00000 0.0395285
\(641\) 9.79837 0.387012 0.193506 0.981099i \(-0.438014\pi\)
0.193506 + 0.981099i \(0.438014\pi\)
\(642\) −17.4164 −0.687371
\(643\) −15.1459 −0.597296 −0.298648 0.954363i \(-0.596536\pi\)
−0.298648 + 0.954363i \(0.596536\pi\)
\(644\) 8.47214 0.333849
\(645\) 5.23607 0.206170
\(646\) −24.1803 −0.951363
\(647\) −41.7426 −1.64107 −0.820536 0.571594i \(-0.806325\pi\)
−0.820536 + 0.571594i \(0.806325\pi\)
\(648\) −7.70820 −0.302807
\(649\) 0 0
\(650\) 5.61803 0.220357
\(651\) 8.47214 0.332049
\(652\) −14.6525 −0.573835
\(653\) −14.2918 −0.559281 −0.279641 0.960105i \(-0.590215\pi\)
−0.279641 + 0.960105i \(0.590215\pi\)
\(654\) 5.70820 0.223208
\(655\) −11.4164 −0.446076
\(656\) 8.94427 0.349215
\(657\) −1.65248 −0.0644692
\(658\) −13.0902 −0.510308
\(659\) 5.25735 0.204797 0.102399 0.994743i \(-0.467348\pi\)
0.102399 + 0.994743i \(0.467348\pi\)
\(660\) 0 0
\(661\) −18.9443 −0.736847 −0.368423 0.929658i \(-0.620102\pi\)
−0.368423 + 0.929658i \(0.620102\pi\)
\(662\) −6.38197 −0.248042
\(663\) 41.9787 1.63032
\(664\) 13.6180 0.528482
\(665\) −5.23607 −0.203046
\(666\) 1.41641 0.0548847
\(667\) −49.5967 −1.92039
\(668\) 20.0000 0.773823
\(669\) −29.8885 −1.15556
\(670\) −1.23607 −0.0477535
\(671\) 0 0
\(672\) 1.61803 0.0624170
\(673\) −4.58359 −0.176684 −0.0883422 0.996090i \(-0.528157\pi\)
−0.0883422 + 0.996090i \(0.528157\pi\)
\(674\) 8.00000 0.308148
\(675\) −5.47214 −0.210623
\(676\) 18.5623 0.713935
\(677\) −13.9230 −0.535104 −0.267552 0.963543i \(-0.586215\pi\)
−0.267552 + 0.963543i \(0.586215\pi\)
\(678\) −2.47214 −0.0949418
\(679\) −3.38197 −0.129788
\(680\) 4.61803 0.177094
\(681\) 11.5623 0.443069
\(682\) 0 0
\(683\) −43.5967 −1.66818 −0.834092 0.551626i \(-0.814008\pi\)
−0.834092 + 0.551626i \(0.814008\pi\)
\(684\) 2.00000 0.0764719
\(685\) −16.0000 −0.611329
\(686\) 1.00000 0.0381802
\(687\) 12.4721 0.475842
\(688\) 3.23607 0.123374
\(689\) 36.3607 1.38523
\(690\) 13.7082 0.521862
\(691\) 2.47214 0.0940445 0.0470222 0.998894i \(-0.485027\pi\)
0.0470222 + 0.998894i \(0.485027\pi\)
\(692\) −15.5066 −0.589472
\(693\) 0 0
\(694\) 25.8885 0.982716
\(695\) 8.94427 0.339276
\(696\) −9.47214 −0.359040
\(697\) 41.3050 1.56454
\(698\) −26.9443 −1.01986
\(699\) −26.9443 −1.01913
\(700\) 1.00000 0.0377964
\(701\) 39.5279 1.49295 0.746473 0.665415i \(-0.231745\pi\)
0.746473 + 0.665415i \(0.231745\pi\)
\(702\) −30.7426 −1.16031
\(703\) 19.4164 0.732304
\(704\) 0 0
\(705\) −21.1803 −0.797698
\(706\) −24.4508 −0.920220
\(707\) −5.23607 −0.196923
\(708\) 7.70820 0.289692
\(709\) 6.96556 0.261597 0.130799 0.991409i \(-0.458246\pi\)
0.130799 + 0.991409i \(0.458246\pi\)
\(710\) −7.32624 −0.274949
\(711\) 0.326238 0.0122349
\(712\) 0.291796 0.0109355
\(713\) 44.3607 1.66132
\(714\) 7.47214 0.279638
\(715\) 0 0
\(716\) −19.5623 −0.731078
\(717\) 14.3262 0.535023
\(718\) 17.9787 0.670960
\(719\) −32.3607 −1.20685 −0.603425 0.797420i \(-0.706198\pi\)
−0.603425 + 0.797420i \(0.706198\pi\)
\(720\) −0.381966 −0.0142350
\(721\) 3.56231 0.132667
\(722\) 8.41641 0.313226
\(723\) −6.00000 −0.223142
\(724\) 0 0
\(725\) −5.85410 −0.217416
\(726\) 0 0
\(727\) 39.3262 1.45853 0.729265 0.684232i \(-0.239862\pi\)
0.729265 + 0.684232i \(0.239862\pi\)
\(728\) 5.61803 0.208218
\(729\) 29.5066 1.09284
\(730\) 4.32624 0.160121
\(731\) 14.9443 0.552734
\(732\) −18.9443 −0.700200
\(733\) 5.25735 0.194185 0.0970924 0.995275i \(-0.469046\pi\)
0.0970924 + 0.995275i \(0.469046\pi\)
\(734\) 35.3262 1.30392
\(735\) 1.61803 0.0596821
\(736\) 8.47214 0.312287
\(737\) 0 0
\(738\) −3.41641 −0.125760
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) −3.70820 −0.136316
\(741\) −47.5967 −1.74851
\(742\) 6.47214 0.237600
\(743\) −19.1246 −0.701614 −0.350807 0.936448i \(-0.614093\pi\)
−0.350807 + 0.936448i \(0.614093\pi\)
\(744\) 8.47214 0.310604
\(745\) 20.8541 0.764035
\(746\) −2.65248 −0.0971140
\(747\) −5.20163 −0.190318
\(748\) 0 0
\(749\) −10.7639 −0.393306
\(750\) 1.61803 0.0590822
\(751\) −15.2705 −0.557229 −0.278614 0.960403i \(-0.589875\pi\)
−0.278614 + 0.960403i \(0.589875\pi\)
\(752\) −13.0902 −0.477349
\(753\) 1.23607 0.0450448
\(754\) −32.8885 −1.19773
\(755\) 21.3820 0.778169
\(756\) −5.47214 −0.199020
\(757\) −12.6525 −0.459862 −0.229931 0.973207i \(-0.573850\pi\)
−0.229931 + 0.973207i \(0.573850\pi\)
\(758\) 16.1459 0.586445
\(759\) 0 0
\(760\) −5.23607 −0.189932
\(761\) 18.5836 0.673655 0.336827 0.941566i \(-0.390646\pi\)
0.336827 + 0.941566i \(0.390646\pi\)
\(762\) 32.6525 1.18287
\(763\) 3.52786 0.127717
\(764\) 0.854102 0.0309003
\(765\) −1.76393 −0.0637751
\(766\) 8.85410 0.319912
\(767\) 26.7639 0.966390
\(768\) 1.61803 0.0583858
\(769\) −21.7082 −0.782818 −0.391409 0.920217i \(-0.628012\pi\)
−0.391409 + 0.920217i \(0.628012\pi\)
\(770\) 0 0
\(771\) −21.4721 −0.773300
\(772\) 12.9443 0.465875
\(773\) 39.5066 1.42095 0.710476 0.703721i \(-0.248479\pi\)
0.710476 + 0.703721i \(0.248479\pi\)
\(774\) −1.23607 −0.0444295
\(775\) 5.23607 0.188085
\(776\) −3.38197 −0.121406
\(777\) −6.00000 −0.215249
\(778\) −20.2705 −0.726733
\(779\) −46.8328 −1.67796
\(780\) 9.09017 0.325480
\(781\) 0 0
\(782\) 39.1246 1.39909
\(783\) 32.0344 1.14482
\(784\) 1.00000 0.0357143
\(785\) −11.0902 −0.395825
\(786\) −18.4721 −0.658879
\(787\) −15.9098 −0.567124 −0.283562 0.958954i \(-0.591516\pi\)
−0.283562 + 0.958954i \(0.591516\pi\)
\(788\) −15.4164 −0.549187
\(789\) 23.1246 0.823258
\(790\) −0.854102 −0.0303876
\(791\) −1.52786 −0.0543246
\(792\) 0 0
\(793\) −65.7771 −2.33581
\(794\) −30.5623 −1.08462
\(795\) 10.4721 0.371408
\(796\) −10.9443 −0.387909
\(797\) −32.6738 −1.15736 −0.578682 0.815553i \(-0.696433\pi\)
−0.578682 + 0.815553i \(0.696433\pi\)
\(798\) −8.47214 −0.299910
\(799\) −60.4508 −2.13860
\(800\) 1.00000 0.0353553
\(801\) −0.111456 −0.00393811
\(802\) −36.6180 −1.29303
\(803\) 0 0
\(804\) −2.00000 −0.0705346
\(805\) 8.47214 0.298604
\(806\) 29.4164 1.03615
\(807\) 31.8885 1.12253
\(808\) −5.23607 −0.184204
\(809\) −27.6869 −0.973420 −0.486710 0.873564i \(-0.661803\pi\)
−0.486710 + 0.873564i \(0.661803\pi\)
\(810\) −7.70820 −0.270839
\(811\) −18.1115 −0.635979 −0.317990 0.948094i \(-0.603008\pi\)
−0.317990 + 0.948094i \(0.603008\pi\)
\(812\) −5.85410 −0.205439
\(813\) 24.4721 0.858275
\(814\) 0 0
\(815\) −14.6525 −0.513254
\(816\) 7.47214 0.261577
\(817\) −16.9443 −0.592805
\(818\) 5.23607 0.183075
\(819\) −2.14590 −0.0749837
\(820\) 8.94427 0.312348
\(821\) 14.2148 0.496099 0.248050 0.968747i \(-0.420210\pi\)
0.248050 + 0.968747i \(0.420210\pi\)
\(822\) −25.8885 −0.902967
\(823\) −40.3607 −1.40688 −0.703442 0.710752i \(-0.748355\pi\)
−0.703442 + 0.710752i \(0.748355\pi\)
\(824\) 3.56231 0.124099
\(825\) 0 0
\(826\) 4.76393 0.165758
\(827\) −32.9443 −1.14558 −0.572792 0.819701i \(-0.694140\pi\)
−0.572792 + 0.819701i \(0.694140\pi\)
\(828\) −3.23607 −0.112461
\(829\) 29.8885 1.03807 0.519036 0.854752i \(-0.326291\pi\)
0.519036 + 0.854752i \(0.326291\pi\)
\(830\) 13.6180 0.472689
\(831\) 21.7082 0.753049
\(832\) 5.61803 0.194770
\(833\) 4.61803 0.160005
\(834\) 14.4721 0.501129
\(835\) 20.0000 0.692129
\(836\) 0 0
\(837\) −28.6525 −0.990374
\(838\) 6.94427 0.239886
\(839\) −0.360680 −0.0124520 −0.00622602 0.999981i \(-0.501982\pi\)
−0.00622602 + 0.999981i \(0.501982\pi\)
\(840\) 1.61803 0.0558275
\(841\) 5.27051 0.181742
\(842\) −11.3262 −0.390328
\(843\) 29.7082 1.02320
\(844\) 15.3262 0.527551
\(845\) 18.5623 0.638563
\(846\) 5.00000 0.171904
\(847\) 0 0
\(848\) 6.47214 0.222254
\(849\) −23.1803 −0.795547
\(850\) 4.61803 0.158397
\(851\) −31.4164 −1.07694
\(852\) −11.8541 −0.406115
\(853\) 8.27051 0.283177 0.141588 0.989926i \(-0.454779\pi\)
0.141588 + 0.989926i \(0.454779\pi\)
\(854\) −11.7082 −0.400646
\(855\) 2.00000 0.0683986
\(856\) −10.7639 −0.367904
\(857\) 33.4164 1.14148 0.570741 0.821130i \(-0.306656\pi\)
0.570741 + 0.821130i \(0.306656\pi\)
\(858\) 0 0
\(859\) 55.4164 1.89078 0.945392 0.325936i \(-0.105680\pi\)
0.945392 + 0.325936i \(0.105680\pi\)
\(860\) 3.23607 0.110349
\(861\) 14.4721 0.493209
\(862\) −15.5623 −0.530054
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −5.47214 −0.186166
\(865\) −15.5066 −0.527239
\(866\) 4.09017 0.138990
\(867\) 7.00000 0.237732
\(868\) 5.23607 0.177724
\(869\) 0 0
\(870\) −9.47214 −0.321135
\(871\) −6.94427 −0.235298
\(872\) 3.52786 0.119469
\(873\) 1.29180 0.0437207
\(874\) −44.3607 −1.50052
\(875\) 1.00000 0.0338062
\(876\) 7.00000 0.236508
\(877\) 26.5410 0.896227 0.448113 0.893977i \(-0.352096\pi\)
0.448113 + 0.893977i \(0.352096\pi\)
\(878\) −10.6525 −0.359504
\(879\) −41.1246 −1.38710
\(880\) 0 0
\(881\) 37.4853 1.26291 0.631456 0.775412i \(-0.282458\pi\)
0.631456 + 0.775412i \(0.282458\pi\)
\(882\) −0.381966 −0.0128615
\(883\) −43.2361 −1.45501 −0.727505 0.686103i \(-0.759320\pi\)
−0.727505 + 0.686103i \(0.759320\pi\)
\(884\) 25.9443 0.872600
\(885\) 7.70820 0.259108
\(886\) −15.2361 −0.511866
\(887\) 35.8541 1.20386 0.601931 0.798548i \(-0.294398\pi\)
0.601931 + 0.798548i \(0.294398\pi\)
\(888\) −6.00000 −0.201347
\(889\) 20.1803 0.676827
\(890\) 0.291796 0.00978103
\(891\) 0 0
\(892\) −18.4721 −0.618493
\(893\) 68.5410 2.29364
\(894\) 33.7426 1.12852
\(895\) −19.5623 −0.653896
\(896\) 1.00000 0.0334077
\(897\) 77.0132 2.57139
\(898\) −19.5066 −0.650943
\(899\) −30.6525 −1.02232
\(900\) −0.381966 −0.0127322
\(901\) 29.8885 0.995732
\(902\) 0 0
\(903\) 5.23607 0.174245
\(904\) −1.52786 −0.0508160
\(905\) 0 0
\(906\) 34.5967 1.14940
\(907\) −30.3607 −1.00811 −0.504055 0.863672i \(-0.668159\pi\)
−0.504055 + 0.863672i \(0.668159\pi\)
\(908\) 7.14590 0.237145
\(909\) 2.00000 0.0663358
\(910\) 5.61803 0.186236
\(911\) 17.4508 0.578172 0.289086 0.957303i \(-0.406649\pi\)
0.289086 + 0.957303i \(0.406649\pi\)
\(912\) −8.47214 −0.280540
\(913\) 0 0
\(914\) −14.0000 −0.463079
\(915\) −18.9443 −0.626278
\(916\) 7.70820 0.254686
\(917\) −11.4164 −0.377003
\(918\) −25.2705 −0.834051
\(919\) 52.5066 1.73203 0.866016 0.500016i \(-0.166673\pi\)
0.866016 + 0.500016i \(0.166673\pi\)
\(920\) 8.47214 0.279318
\(921\) −54.1246 −1.78347
\(922\) 25.1246 0.827435
\(923\) −41.1591 −1.35477
\(924\) 0 0
\(925\) −3.70820 −0.121925
\(926\) −14.1803 −0.465995
\(927\) −1.36068 −0.0446906
\(928\) −5.85410 −0.192170
\(929\) 27.7771 0.911337 0.455668 0.890150i \(-0.349400\pi\)
0.455668 + 0.890150i \(0.349400\pi\)
\(930\) 8.47214 0.277812
\(931\) −5.23607 −0.171605
\(932\) −16.6525 −0.545470
\(933\) 16.4721 0.539274
\(934\) 21.6180 0.707364
\(935\) 0 0
\(936\) −2.14590 −0.0701409
\(937\) −37.3951 −1.22165 −0.610823 0.791767i \(-0.709161\pi\)
−0.610823 + 0.791767i \(0.709161\pi\)
\(938\) −1.23607 −0.0403591
\(939\) 19.2361 0.627745
\(940\) −13.0902 −0.426954
\(941\) 42.6525 1.39043 0.695215 0.718801i \(-0.255309\pi\)
0.695215 + 0.718801i \(0.255309\pi\)
\(942\) −17.9443 −0.584656
\(943\) 75.7771 2.46764
\(944\) 4.76393 0.155053
\(945\) −5.47214 −0.178009
\(946\) 0 0
\(947\) 32.6525 1.06106 0.530531 0.847665i \(-0.321992\pi\)
0.530531 + 0.847665i \(0.321992\pi\)
\(948\) −1.38197 −0.0448842
\(949\) 24.3050 0.788972
\(950\) −5.23607 −0.169880
\(951\) 28.9443 0.938582
\(952\) 4.61803 0.149671
\(953\) 2.58359 0.0836908 0.0418454 0.999124i \(-0.486676\pi\)
0.0418454 + 0.999124i \(0.486676\pi\)
\(954\) −2.47214 −0.0800384
\(955\) 0.854102 0.0276381
\(956\) 8.85410 0.286362
\(957\) 0 0
\(958\) −35.1246 −1.13482
\(959\) −16.0000 −0.516667
\(960\) 1.61803 0.0522218
\(961\) −3.58359 −0.115600
\(962\) −20.8328 −0.671677
\(963\) 4.11146 0.132490
\(964\) −3.70820 −0.119433
\(965\) 12.9443 0.416691
\(966\) 13.7082 0.441054
\(967\) −4.29180 −0.138015 −0.0690074 0.997616i \(-0.521983\pi\)
−0.0690074 + 0.997616i \(0.521983\pi\)
\(968\) 0 0
\(969\) −39.1246 −1.25686
\(970\) −3.38197 −0.108588
\(971\) 42.3607 1.35942 0.679709 0.733481i \(-0.262106\pi\)
0.679709 + 0.733481i \(0.262106\pi\)
\(972\) 3.94427 0.126513
\(973\) 8.94427 0.286740
\(974\) 11.8197 0.378726
\(975\) 9.09017 0.291118
\(976\) −11.7082 −0.374770
\(977\) 52.9017 1.69248 0.846238 0.532806i \(-0.178862\pi\)
0.846238 + 0.532806i \(0.178862\pi\)
\(978\) −23.7082 −0.758105
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −1.34752 −0.0430231
\(982\) 0.944272 0.0301329
\(983\) −30.0344 −0.957950 −0.478975 0.877829i \(-0.658992\pi\)
−0.478975 + 0.877829i \(0.658992\pi\)
\(984\) 14.4721 0.461355
\(985\) −15.4164 −0.491208
\(986\) −27.0344 −0.860952
\(987\) −21.1803 −0.674178
\(988\) −29.4164 −0.935861
\(989\) 27.4164 0.871791
\(990\) 0 0
\(991\) 10.9230 0.346980 0.173490 0.984836i \(-0.444496\pi\)
0.173490 + 0.984836i \(0.444496\pi\)
\(992\) 5.23607 0.166245
\(993\) −10.3262 −0.327693
\(994\) −7.32624 −0.232374
\(995\) −10.9443 −0.346957
\(996\) 22.0344 0.698188
\(997\) 17.5623 0.556204 0.278102 0.960552i \(-0.410295\pi\)
0.278102 + 0.960552i \(0.410295\pi\)
\(998\) −24.7984 −0.784979
\(999\) 20.2918 0.642004
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 8470.2.a.cc.1.2 2
11.3 even 5 770.2.n.a.141.1 yes 4
11.4 even 5 770.2.n.a.71.1 4
11.10 odd 2 8470.2.a.bq.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.a.71.1 4 11.4 even 5
770.2.n.a.141.1 yes 4 11.3 even 5
8470.2.a.bq.1.2 2 11.10 odd 2
8470.2.a.cc.1.2 2 1.1 even 1 trivial