Properties

Label 4235.2.a.k.1.2
Level $4235$
Weight $2$
Character 4235.1
Self dual yes
Analytic conductor $33.817$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.8166452560\)
Analytic rank: \(1\)
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 385)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 4235.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.23607 q^{2} +0.618034 q^{3} +3.00000 q^{4} -1.00000 q^{5} +1.38197 q^{6} +1.00000 q^{7} +2.23607 q^{8} -2.61803 q^{9} +O(q^{10})\) \(q+2.23607 q^{2} +0.618034 q^{3} +3.00000 q^{4} -1.00000 q^{5} +1.38197 q^{6} +1.00000 q^{7} +2.23607 q^{8} -2.61803 q^{9} -2.23607 q^{10} +1.85410 q^{12} -4.61803 q^{13} +2.23607 q^{14} -0.618034 q^{15} -1.00000 q^{16} -5.61803 q^{17} -5.85410 q^{18} +3.23607 q^{19} -3.00000 q^{20} +0.618034 q^{21} -4.47214 q^{23} +1.38197 q^{24} +1.00000 q^{25} -10.3262 q^{26} -3.47214 q^{27} +3.00000 q^{28} -0.854102 q^{29} -1.38197 q^{30} +3.23607 q^{31} -6.70820 q^{32} -12.5623 q^{34} -1.00000 q^{35} -7.85410 q^{36} -7.23607 q^{37} +7.23607 q^{38} -2.85410 q^{39} -2.23607 q^{40} -2.47214 q^{41} +1.38197 q^{42} +5.23607 q^{43} +2.61803 q^{45} -10.0000 q^{46} -1.14590 q^{47} -0.618034 q^{48} +1.00000 q^{49} +2.23607 q^{50} -3.47214 q^{51} -13.8541 q^{52} -1.52786 q^{53} -7.76393 q^{54} +2.23607 q^{56} +2.00000 q^{57} -1.90983 q^{58} -9.23607 q^{59} -1.85410 q^{60} +4.76393 q^{61} +7.23607 q^{62} -2.61803 q^{63} -13.0000 q^{64} +4.61803 q^{65} +8.18034 q^{67} -16.8541 q^{68} -2.76393 q^{69} -2.23607 q^{70} +16.3262 q^{71} -5.85410 q^{72} -6.38197 q^{73} -16.1803 q^{74} +0.618034 q^{75} +9.70820 q^{76} -6.38197 q^{78} -10.7984 q^{79} +1.00000 q^{80} +5.70820 q^{81} -5.52786 q^{82} -9.56231 q^{83} +1.85410 q^{84} +5.61803 q^{85} +11.7082 q^{86} -0.527864 q^{87} +12.7639 q^{89} +5.85410 q^{90} -4.61803 q^{91} -13.4164 q^{92} +2.00000 q^{93} -2.56231 q^{94} -3.23607 q^{95} -4.14590 q^{96} -15.3262 q^{97} +2.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 6 q^{4} - 2 q^{5} + 5 q^{6} + 2 q^{7} - 3 q^{9} + O(q^{10}) \) \( 2 q - q^{3} + 6 q^{4} - 2 q^{5} + 5 q^{6} + 2 q^{7} - 3 q^{9} - 3 q^{12} - 7 q^{13} + q^{15} - 2 q^{16} - 9 q^{17} - 5 q^{18} + 2 q^{19} - 6 q^{20} - q^{21} + 5 q^{24} + 2 q^{25} - 5 q^{26} + 2 q^{27} + 6 q^{28} + 5 q^{29} - 5 q^{30} + 2 q^{31} - 5 q^{34} - 2 q^{35} - 9 q^{36} - 10 q^{37} + 10 q^{38} + q^{39} + 4 q^{41} + 5 q^{42} + 6 q^{43} + 3 q^{45} - 20 q^{46} - 9 q^{47} + q^{48} + 2 q^{49} + 2 q^{51} - 21 q^{52} - 12 q^{53} - 20 q^{54} + 4 q^{57} - 15 q^{58} - 14 q^{59} + 3 q^{60} + 14 q^{61} + 10 q^{62} - 3 q^{63} - 26 q^{64} + 7 q^{65} - 6 q^{67} - 27 q^{68} - 10 q^{69} + 17 q^{71} - 5 q^{72} - 15 q^{73} - 10 q^{74} - q^{75} + 6 q^{76} - 15 q^{78} + 3 q^{79} + 2 q^{80} - 2 q^{81} - 20 q^{82} + q^{83} - 3 q^{84} + 9 q^{85} + 10 q^{86} - 10 q^{87} + 30 q^{89} + 5 q^{90} - 7 q^{91} + 4 q^{93} + 15 q^{94} - 2 q^{95} - 15 q^{96} - 15 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23607 1.58114 0.790569 0.612372i \(-0.209785\pi\)
0.790569 + 0.612372i \(0.209785\pi\)
\(3\) 0.618034 0.356822 0.178411 0.983956i \(-0.442904\pi\)
0.178411 + 0.983956i \(0.442904\pi\)
\(4\) 3.00000 1.50000
\(5\) −1.00000 −0.447214
\(6\) 1.38197 0.564185
\(7\) 1.00000 0.377964
\(8\) 2.23607 0.790569
\(9\) −2.61803 −0.872678
\(10\) −2.23607 −0.707107
\(11\) 0 0
\(12\) 1.85410 0.535233
\(13\) −4.61803 −1.28081 −0.640406 0.768036i \(-0.721234\pi\)
−0.640406 + 0.768036i \(0.721234\pi\)
\(14\) 2.23607 0.597614
\(15\) −0.618034 −0.159576
\(16\) −1.00000 −0.250000
\(17\) −5.61803 −1.36257 −0.681287 0.732017i \(-0.738579\pi\)
−0.681287 + 0.732017i \(0.738579\pi\)
\(18\) −5.85410 −1.37983
\(19\) 3.23607 0.742405 0.371202 0.928552i \(-0.378946\pi\)
0.371202 + 0.928552i \(0.378946\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0.618034 0.134866
\(22\) 0 0
\(23\) −4.47214 −0.932505 −0.466252 0.884652i \(-0.654396\pi\)
−0.466252 + 0.884652i \(0.654396\pi\)
\(24\) 1.38197 0.282093
\(25\) 1.00000 0.200000
\(26\) −10.3262 −2.02514
\(27\) −3.47214 −0.668213
\(28\) 3.00000 0.566947
\(29\) −0.854102 −0.158603 −0.0793014 0.996851i \(-0.525269\pi\)
−0.0793014 + 0.996851i \(0.525269\pi\)
\(30\) −1.38197 −0.252311
\(31\) 3.23607 0.581215 0.290607 0.956842i \(-0.406143\pi\)
0.290607 + 0.956842i \(0.406143\pi\)
\(32\) −6.70820 −1.18585
\(33\) 0 0
\(34\) −12.5623 −2.15442
\(35\) −1.00000 −0.169031
\(36\) −7.85410 −1.30902
\(37\) −7.23607 −1.18960 −0.594801 0.803873i \(-0.702769\pi\)
−0.594801 + 0.803873i \(0.702769\pi\)
\(38\) 7.23607 1.17385
\(39\) −2.85410 −0.457022
\(40\) −2.23607 −0.353553
\(41\) −2.47214 −0.386083 −0.193041 0.981191i \(-0.561835\pi\)
−0.193041 + 0.981191i \(0.561835\pi\)
\(42\) 1.38197 0.213242
\(43\) 5.23607 0.798493 0.399246 0.916844i \(-0.369272\pi\)
0.399246 + 0.916844i \(0.369272\pi\)
\(44\) 0 0
\(45\) 2.61803 0.390273
\(46\) −10.0000 −1.47442
\(47\) −1.14590 −0.167146 −0.0835732 0.996502i \(-0.526633\pi\)
−0.0835732 + 0.996502i \(0.526633\pi\)
\(48\) −0.618034 −0.0892055
\(49\) 1.00000 0.142857
\(50\) 2.23607 0.316228
\(51\) −3.47214 −0.486196
\(52\) −13.8541 −1.92122
\(53\) −1.52786 −0.209868 −0.104934 0.994479i \(-0.533463\pi\)
−0.104934 + 0.994479i \(0.533463\pi\)
\(54\) −7.76393 −1.05654
\(55\) 0 0
\(56\) 2.23607 0.298807
\(57\) 2.00000 0.264906
\(58\) −1.90983 −0.250773
\(59\) −9.23607 −1.20243 −0.601217 0.799086i \(-0.705317\pi\)
−0.601217 + 0.799086i \(0.705317\pi\)
\(60\) −1.85410 −0.239364
\(61\) 4.76393 0.609959 0.304979 0.952359i \(-0.401350\pi\)
0.304979 + 0.952359i \(0.401350\pi\)
\(62\) 7.23607 0.918982
\(63\) −2.61803 −0.329841
\(64\) −13.0000 −1.62500
\(65\) 4.61803 0.572797
\(66\) 0 0
\(67\) 8.18034 0.999388 0.499694 0.866202i \(-0.333446\pi\)
0.499694 + 0.866202i \(0.333446\pi\)
\(68\) −16.8541 −2.04386
\(69\) −2.76393 −0.332738
\(70\) −2.23607 −0.267261
\(71\) 16.3262 1.93757 0.968784 0.247906i \(-0.0797425\pi\)
0.968784 + 0.247906i \(0.0797425\pi\)
\(72\) −5.85410 −0.689913
\(73\) −6.38197 −0.746953 −0.373476 0.927640i \(-0.621834\pi\)
−0.373476 + 0.927640i \(0.621834\pi\)
\(74\) −16.1803 −1.88093
\(75\) 0.618034 0.0713644
\(76\) 9.70820 1.11361
\(77\) 0 0
\(78\) −6.38197 −0.722615
\(79\) −10.7984 −1.21491 −0.607456 0.794353i \(-0.707810\pi\)
−0.607456 + 0.794353i \(0.707810\pi\)
\(80\) 1.00000 0.111803
\(81\) 5.70820 0.634245
\(82\) −5.52786 −0.610450
\(83\) −9.56231 −1.04960 −0.524800 0.851226i \(-0.675860\pi\)
−0.524800 + 0.851226i \(0.675860\pi\)
\(84\) 1.85410 0.202299
\(85\) 5.61803 0.609361
\(86\) 11.7082 1.26253
\(87\) −0.527864 −0.0565930
\(88\) 0 0
\(89\) 12.7639 1.35297 0.676487 0.736455i \(-0.263501\pi\)
0.676487 + 0.736455i \(0.263501\pi\)
\(90\) 5.85410 0.617077
\(91\) −4.61803 −0.484102
\(92\) −13.4164 −1.39876
\(93\) 2.00000 0.207390
\(94\) −2.56231 −0.264282
\(95\) −3.23607 −0.332014
\(96\) −4.14590 −0.423139
\(97\) −15.3262 −1.55614 −0.778072 0.628175i \(-0.783802\pi\)
−0.778072 + 0.628175i \(0.783802\pi\)
\(98\) 2.23607 0.225877
\(99\) 0 0
\(100\) 3.00000 0.300000
\(101\) 13.7082 1.36402 0.682009 0.731344i \(-0.261107\pi\)
0.682009 + 0.731344i \(0.261107\pi\)
\(102\) −7.76393 −0.768744
\(103\) −17.3262 −1.70720 −0.853602 0.520925i \(-0.825587\pi\)
−0.853602 + 0.520925i \(0.825587\pi\)
\(104\) −10.3262 −1.01257
\(105\) −0.618034 −0.0603139
\(106\) −3.41641 −0.331831
\(107\) 7.23607 0.699537 0.349769 0.936836i \(-0.386260\pi\)
0.349769 + 0.936836i \(0.386260\pi\)
\(108\) −10.4164 −1.00232
\(109\) 11.5279 1.10417 0.552085 0.833788i \(-0.313833\pi\)
0.552085 + 0.833788i \(0.313833\pi\)
\(110\) 0 0
\(111\) −4.47214 −0.424476
\(112\) −1.00000 −0.0944911
\(113\) 10.4721 0.985136 0.492568 0.870274i \(-0.336058\pi\)
0.492568 + 0.870274i \(0.336058\pi\)
\(114\) 4.47214 0.418854
\(115\) 4.47214 0.417029
\(116\) −2.56231 −0.237904
\(117\) 12.0902 1.11774
\(118\) −20.6525 −1.90121
\(119\) −5.61803 −0.515004
\(120\) −1.38197 −0.126156
\(121\) 0 0
\(122\) 10.6525 0.964430
\(123\) −1.52786 −0.137763
\(124\) 9.70820 0.871822
\(125\) −1.00000 −0.0894427
\(126\) −5.85410 −0.521525
\(127\) −9.23607 −0.819569 −0.409784 0.912182i \(-0.634396\pi\)
−0.409784 + 0.912182i \(0.634396\pi\)
\(128\) −15.6525 −1.38350
\(129\) 3.23607 0.284920
\(130\) 10.3262 0.905671
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 3.23607 0.280603
\(134\) 18.2918 1.58017
\(135\) 3.47214 0.298834
\(136\) −12.5623 −1.07721
\(137\) −11.4164 −0.975370 −0.487685 0.873020i \(-0.662158\pi\)
−0.487685 + 0.873020i \(0.662158\pi\)
\(138\) −6.18034 −0.526105
\(139\) −5.52786 −0.468867 −0.234434 0.972132i \(-0.575324\pi\)
−0.234434 + 0.972132i \(0.575324\pi\)
\(140\) −3.00000 −0.253546
\(141\) −0.708204 −0.0596415
\(142\) 36.5066 3.06356
\(143\) 0 0
\(144\) 2.61803 0.218169
\(145\) 0.854102 0.0709293
\(146\) −14.2705 −1.18104
\(147\) 0.618034 0.0509746
\(148\) −21.7082 −1.78440
\(149\) 17.8541 1.46267 0.731333 0.682021i \(-0.238899\pi\)
0.731333 + 0.682021i \(0.238899\pi\)
\(150\) 1.38197 0.112837
\(151\) 0.381966 0.0310840 0.0155420 0.999879i \(-0.495053\pi\)
0.0155420 + 0.999879i \(0.495053\pi\)
\(152\) 7.23607 0.586923
\(153\) 14.7082 1.18909
\(154\) 0 0
\(155\) −3.23607 −0.259927
\(156\) −8.56231 −0.685533
\(157\) −16.0902 −1.28414 −0.642068 0.766648i \(-0.721923\pi\)
−0.642068 + 0.766648i \(0.721923\pi\)
\(158\) −24.1459 −1.92094
\(159\) −0.944272 −0.0748856
\(160\) 6.70820 0.530330
\(161\) −4.47214 −0.352454
\(162\) 12.7639 1.00283
\(163\) −22.1803 −1.73730 −0.868649 0.495428i \(-0.835011\pi\)
−0.868649 + 0.495428i \(0.835011\pi\)
\(164\) −7.41641 −0.579124
\(165\) 0 0
\(166\) −21.3820 −1.65956
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 1.38197 0.106621
\(169\) 8.32624 0.640480
\(170\) 12.5623 0.963485
\(171\) −8.47214 −0.647880
\(172\) 15.7082 1.19774
\(173\) 1.56231 0.118780 0.0593900 0.998235i \(-0.481084\pi\)
0.0593900 + 0.998235i \(0.481084\pi\)
\(174\) −1.18034 −0.0894813
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −5.70820 −0.429055
\(178\) 28.5410 2.13924
\(179\) 19.6180 1.46632 0.733160 0.680056i \(-0.238044\pi\)
0.733160 + 0.680056i \(0.238044\pi\)
\(180\) 7.85410 0.585410
\(181\) 5.52786 0.410883 0.205441 0.978669i \(-0.434137\pi\)
0.205441 + 0.978669i \(0.434137\pi\)
\(182\) −10.3262 −0.765432
\(183\) 2.94427 0.217647
\(184\) −10.0000 −0.737210
\(185\) 7.23607 0.532006
\(186\) 4.47214 0.327913
\(187\) 0 0
\(188\) −3.43769 −0.250720
\(189\) −3.47214 −0.252561
\(190\) −7.23607 −0.524960
\(191\) 23.0902 1.67075 0.835373 0.549683i \(-0.185252\pi\)
0.835373 + 0.549683i \(0.185252\pi\)
\(192\) −8.03444 −0.579836
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) −34.2705 −2.46048
\(195\) 2.85410 0.204386
\(196\) 3.00000 0.214286
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) 2.23607 0.158114
\(201\) 5.05573 0.356604
\(202\) 30.6525 2.15670
\(203\) −0.854102 −0.0599462
\(204\) −10.4164 −0.729294
\(205\) 2.47214 0.172661
\(206\) −38.7426 −2.69933
\(207\) 11.7082 0.813776
\(208\) 4.61803 0.320203
\(209\) 0 0
\(210\) −1.38197 −0.0953647
\(211\) −7.67376 −0.528284 −0.264142 0.964484i \(-0.585089\pi\)
−0.264142 + 0.964484i \(0.585089\pi\)
\(212\) −4.58359 −0.314802
\(213\) 10.0902 0.691367
\(214\) 16.1803 1.10607
\(215\) −5.23607 −0.357097
\(216\) −7.76393 −0.528269
\(217\) 3.23607 0.219679
\(218\) 25.7771 1.74584
\(219\) −3.94427 −0.266529
\(220\) 0 0
\(221\) 25.9443 1.74520
\(222\) −10.0000 −0.671156
\(223\) 1.52786 0.102313 0.0511567 0.998691i \(-0.483709\pi\)
0.0511567 + 0.998691i \(0.483709\pi\)
\(224\) −6.70820 −0.448211
\(225\) −2.61803 −0.174536
\(226\) 23.4164 1.55764
\(227\) 10.7984 0.716713 0.358357 0.933585i \(-0.383337\pi\)
0.358357 + 0.933585i \(0.383337\pi\)
\(228\) 6.00000 0.397360
\(229\) −24.7639 −1.63645 −0.818223 0.574900i \(-0.805041\pi\)
−0.818223 + 0.574900i \(0.805041\pi\)
\(230\) 10.0000 0.659380
\(231\) 0 0
\(232\) −1.90983 −0.125386
\(233\) −6.65248 −0.435818 −0.217909 0.975969i \(-0.569924\pi\)
−0.217909 + 0.975969i \(0.569924\pi\)
\(234\) 27.0344 1.76730
\(235\) 1.14590 0.0747501
\(236\) −27.7082 −1.80365
\(237\) −6.67376 −0.433507
\(238\) −12.5623 −0.814293
\(239\) −24.9787 −1.61574 −0.807869 0.589362i \(-0.799379\pi\)
−0.807869 + 0.589362i \(0.799379\pi\)
\(240\) 0.618034 0.0398939
\(241\) 17.1246 1.10309 0.551547 0.834144i \(-0.314038\pi\)
0.551547 + 0.834144i \(0.314038\pi\)
\(242\) 0 0
\(243\) 13.9443 0.894525
\(244\) 14.2918 0.914938
\(245\) −1.00000 −0.0638877
\(246\) −3.41641 −0.217822
\(247\) −14.9443 −0.950881
\(248\) 7.23607 0.459491
\(249\) −5.90983 −0.374520
\(250\) −2.23607 −0.141421
\(251\) −7.70820 −0.486538 −0.243269 0.969959i \(-0.578220\pi\)
−0.243269 + 0.969959i \(0.578220\pi\)
\(252\) −7.85410 −0.494762
\(253\) 0 0
\(254\) −20.6525 −1.29585
\(255\) 3.47214 0.217434
\(256\) −9.00000 −0.562500
\(257\) −2.38197 −0.148583 −0.0742915 0.997237i \(-0.523670\pi\)
−0.0742915 + 0.997237i \(0.523670\pi\)
\(258\) 7.23607 0.450498
\(259\) −7.23607 −0.449627
\(260\) 13.8541 0.859195
\(261\) 2.23607 0.138409
\(262\) −17.8885 −1.10516
\(263\) −14.1803 −0.874397 −0.437199 0.899365i \(-0.644029\pi\)
−0.437199 + 0.899365i \(0.644029\pi\)
\(264\) 0 0
\(265\) 1.52786 0.0938559
\(266\) 7.23607 0.443672
\(267\) 7.88854 0.482771
\(268\) 24.5410 1.49908
\(269\) −18.6525 −1.13726 −0.568631 0.822593i \(-0.692527\pi\)
−0.568631 + 0.822593i \(0.692527\pi\)
\(270\) 7.76393 0.472498
\(271\) −23.2361 −1.41149 −0.705745 0.708466i \(-0.749388\pi\)
−0.705745 + 0.708466i \(0.749388\pi\)
\(272\) 5.61803 0.340643
\(273\) −2.85410 −0.172738
\(274\) −25.5279 −1.54219
\(275\) 0 0
\(276\) −8.29180 −0.499107
\(277\) 9.41641 0.565777 0.282889 0.959153i \(-0.408707\pi\)
0.282889 + 0.959153i \(0.408707\pi\)
\(278\) −12.3607 −0.741344
\(279\) −8.47214 −0.507214
\(280\) −2.23607 −0.133631
\(281\) 8.47214 0.505405 0.252703 0.967544i \(-0.418681\pi\)
0.252703 + 0.967544i \(0.418681\pi\)
\(282\) −1.58359 −0.0943015
\(283\) 7.43769 0.442125 0.221063 0.975260i \(-0.429048\pi\)
0.221063 + 0.975260i \(0.429048\pi\)
\(284\) 48.9787 2.90635
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) −2.47214 −0.145926
\(288\) 17.5623 1.03487
\(289\) 14.5623 0.856606
\(290\) 1.90983 0.112149
\(291\) −9.47214 −0.555266
\(292\) −19.1459 −1.12043
\(293\) 9.41641 0.550112 0.275056 0.961428i \(-0.411304\pi\)
0.275056 + 0.961428i \(0.411304\pi\)
\(294\) 1.38197 0.0805979
\(295\) 9.23607 0.537745
\(296\) −16.1803 −0.940463
\(297\) 0 0
\(298\) 39.9230 2.31268
\(299\) 20.6525 1.19436
\(300\) 1.85410 0.107047
\(301\) 5.23607 0.301802
\(302\) 0.854102 0.0491480
\(303\) 8.47214 0.486711
\(304\) −3.23607 −0.185601
\(305\) −4.76393 −0.272782
\(306\) 32.8885 1.88011
\(307\) 20.5623 1.17355 0.586776 0.809749i \(-0.300397\pi\)
0.586776 + 0.809749i \(0.300397\pi\)
\(308\) 0 0
\(309\) −10.7082 −0.609168
\(310\) −7.23607 −0.410981
\(311\) 13.1246 0.744228 0.372114 0.928187i \(-0.378633\pi\)
0.372114 + 0.928187i \(0.378633\pi\)
\(312\) −6.38197 −0.361308
\(313\) −18.0000 −1.01742 −0.508710 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) −35.9787 −2.03040
\(315\) 2.61803 0.147510
\(316\) −32.3951 −1.82237
\(317\) −21.5279 −1.20913 −0.604563 0.796558i \(-0.706652\pi\)
−0.604563 + 0.796558i \(0.706652\pi\)
\(318\) −2.11146 −0.118405
\(319\) 0 0
\(320\) 13.0000 0.726722
\(321\) 4.47214 0.249610
\(322\) −10.0000 −0.557278
\(323\) −18.1803 −1.01158
\(324\) 17.1246 0.951367
\(325\) −4.61803 −0.256162
\(326\) −49.5967 −2.74691
\(327\) 7.12461 0.393992
\(328\) −5.52786 −0.305225
\(329\) −1.14590 −0.0631754
\(330\) 0 0
\(331\) −18.5066 −1.01721 −0.508607 0.860999i \(-0.669839\pi\)
−0.508607 + 0.860999i \(0.669839\pi\)
\(332\) −28.6869 −1.57440
\(333\) 18.9443 1.03814
\(334\) 26.8328 1.46823
\(335\) −8.18034 −0.446940
\(336\) −0.618034 −0.0337165
\(337\) 31.4164 1.71136 0.855680 0.517505i \(-0.173139\pi\)
0.855680 + 0.517505i \(0.173139\pi\)
\(338\) 18.6180 1.01269
\(339\) 6.47214 0.351518
\(340\) 16.8541 0.914042
\(341\) 0 0
\(342\) −18.9443 −1.02439
\(343\) 1.00000 0.0539949
\(344\) 11.7082 0.631264
\(345\) 2.76393 0.148805
\(346\) 3.49342 0.187808
\(347\) 1.52786 0.0820200 0.0410100 0.999159i \(-0.486942\pi\)
0.0410100 + 0.999159i \(0.486942\pi\)
\(348\) −1.58359 −0.0848894
\(349\) −3.88854 −0.208149 −0.104074 0.994570i \(-0.533188\pi\)
−0.104074 + 0.994570i \(0.533188\pi\)
\(350\) 2.23607 0.119523
\(351\) 16.0344 0.855855
\(352\) 0 0
\(353\) −13.5623 −0.721849 −0.360924 0.932595i \(-0.617539\pi\)
−0.360924 + 0.932595i \(0.617539\pi\)
\(354\) −12.7639 −0.678395
\(355\) −16.3262 −0.866507
\(356\) 38.2918 2.02946
\(357\) −3.47214 −0.183765
\(358\) 43.8673 2.31846
\(359\) 12.9787 0.684990 0.342495 0.939520i \(-0.388728\pi\)
0.342495 + 0.939520i \(0.388728\pi\)
\(360\) 5.85410 0.308538
\(361\) −8.52786 −0.448835
\(362\) 12.3607 0.649663
\(363\) 0 0
\(364\) −13.8541 −0.726152
\(365\) 6.38197 0.334047
\(366\) 6.58359 0.344130
\(367\) −31.4508 −1.64172 −0.820860 0.571129i \(-0.806506\pi\)
−0.820860 + 0.571129i \(0.806506\pi\)
\(368\) 4.47214 0.233126
\(369\) 6.47214 0.336926
\(370\) 16.1803 0.841176
\(371\) −1.52786 −0.0793227
\(372\) 6.00000 0.311086
\(373\) −11.1246 −0.576011 −0.288005 0.957629i \(-0.592992\pi\)
−0.288005 + 0.957629i \(0.592992\pi\)
\(374\) 0 0
\(375\) −0.618034 −0.0319151
\(376\) −2.56231 −0.132141
\(377\) 3.94427 0.203140
\(378\) −7.76393 −0.399334
\(379\) 24.7426 1.27094 0.635472 0.772124i \(-0.280805\pi\)
0.635472 + 0.772124i \(0.280805\pi\)
\(380\) −9.70820 −0.498020
\(381\) −5.70820 −0.292440
\(382\) 51.6312 2.64168
\(383\) 26.7984 1.36933 0.684666 0.728857i \(-0.259948\pi\)
0.684666 + 0.728857i \(0.259948\pi\)
\(384\) −9.67376 −0.493662
\(385\) 0 0
\(386\) −8.94427 −0.455251
\(387\) −13.7082 −0.696827
\(388\) −45.9787 −2.33422
\(389\) −33.5623 −1.70168 −0.850838 0.525428i \(-0.823905\pi\)
−0.850838 + 0.525428i \(0.823905\pi\)
\(390\) 6.38197 0.323163
\(391\) 25.1246 1.27061
\(392\) 2.23607 0.112938
\(393\) −4.94427 −0.249406
\(394\) 26.8328 1.35182
\(395\) 10.7984 0.543325
\(396\) 0 0
\(397\) 35.1591 1.76458 0.882291 0.470704i \(-0.156000\pi\)
0.882291 + 0.470704i \(0.156000\pi\)
\(398\) −40.2492 −2.01751
\(399\) 2.00000 0.100125
\(400\) −1.00000 −0.0500000
\(401\) 15.5066 0.774362 0.387181 0.922004i \(-0.373449\pi\)
0.387181 + 0.922004i \(0.373449\pi\)
\(402\) 11.3050 0.563840
\(403\) −14.9443 −0.744427
\(404\) 41.1246 2.04603
\(405\) −5.70820 −0.283643
\(406\) −1.90983 −0.0947833
\(407\) 0 0
\(408\) −7.76393 −0.384372
\(409\) 3.23607 0.160013 0.0800066 0.996794i \(-0.474506\pi\)
0.0800066 + 0.996794i \(0.474506\pi\)
\(410\) 5.52786 0.273002
\(411\) −7.05573 −0.348033
\(412\) −51.9787 −2.56081
\(413\) −9.23607 −0.454477
\(414\) 26.1803 1.28669
\(415\) 9.56231 0.469395
\(416\) 30.9787 1.51886
\(417\) −3.41641 −0.167302
\(418\) 0 0
\(419\) −35.3050 −1.72476 −0.862380 0.506262i \(-0.831027\pi\)
−0.862380 + 0.506262i \(0.831027\pi\)
\(420\) −1.85410 −0.0904709
\(421\) −34.5066 −1.68175 −0.840874 0.541231i \(-0.817958\pi\)
−0.840874 + 0.541231i \(0.817958\pi\)
\(422\) −17.1591 −0.835290
\(423\) 3.00000 0.145865
\(424\) −3.41641 −0.165915
\(425\) −5.61803 −0.272515
\(426\) 22.5623 1.09315
\(427\) 4.76393 0.230543
\(428\) 21.7082 1.04931
\(429\) 0 0
\(430\) −11.7082 −0.564620
\(431\) 5.32624 0.256556 0.128278 0.991738i \(-0.459055\pi\)
0.128278 + 0.991738i \(0.459055\pi\)
\(432\) 3.47214 0.167053
\(433\) 12.0344 0.578338 0.289169 0.957278i \(-0.406621\pi\)
0.289169 + 0.957278i \(0.406621\pi\)
\(434\) 7.23607 0.347342
\(435\) 0.527864 0.0253091
\(436\) 34.5836 1.65625
\(437\) −14.4721 −0.692296
\(438\) −8.81966 −0.421420
\(439\) −4.29180 −0.204836 −0.102418 0.994741i \(-0.532658\pi\)
−0.102418 + 0.994741i \(0.532658\pi\)
\(440\) 0 0
\(441\) −2.61803 −0.124668
\(442\) 58.0132 2.75940
\(443\) −3.34752 −0.159046 −0.0795228 0.996833i \(-0.525340\pi\)
−0.0795228 + 0.996833i \(0.525340\pi\)
\(444\) −13.4164 −0.636715
\(445\) −12.7639 −0.605068
\(446\) 3.41641 0.161772
\(447\) 11.0344 0.521911
\(448\) −13.0000 −0.614192
\(449\) −9.27051 −0.437502 −0.218751 0.975781i \(-0.570198\pi\)
−0.218751 + 0.975781i \(0.570198\pi\)
\(450\) −5.85410 −0.275965
\(451\) 0 0
\(452\) 31.4164 1.47770
\(453\) 0.236068 0.0110914
\(454\) 24.1459 1.13322
\(455\) 4.61803 0.216497
\(456\) 4.47214 0.209427
\(457\) 37.7771 1.76714 0.883569 0.468301i \(-0.155134\pi\)
0.883569 + 0.468301i \(0.155134\pi\)
\(458\) −55.3738 −2.58745
\(459\) 19.5066 0.910489
\(460\) 13.4164 0.625543
\(461\) −16.2918 −0.758785 −0.379392 0.925236i \(-0.623867\pi\)
−0.379392 + 0.925236i \(0.623867\pi\)
\(462\) 0 0
\(463\) −17.1246 −0.795848 −0.397924 0.917418i \(-0.630269\pi\)
−0.397924 + 0.917418i \(0.630269\pi\)
\(464\) 0.854102 0.0396507
\(465\) −2.00000 −0.0927478
\(466\) −14.8754 −0.689089
\(467\) −32.3262 −1.49588 −0.747940 0.663766i \(-0.768957\pi\)
−0.747940 + 0.663766i \(0.768957\pi\)
\(468\) 36.2705 1.67660
\(469\) 8.18034 0.377733
\(470\) 2.56231 0.118190
\(471\) −9.94427 −0.458208
\(472\) −20.6525 −0.950607
\(473\) 0 0
\(474\) −14.9230 −0.685435
\(475\) 3.23607 0.148481
\(476\) −16.8541 −0.772506
\(477\) 4.00000 0.183147
\(478\) −55.8541 −2.55471
\(479\) −22.6525 −1.03502 −0.517509 0.855678i \(-0.673141\pi\)
−0.517509 + 0.855678i \(0.673141\pi\)
\(480\) 4.14590 0.189233
\(481\) 33.4164 1.52366
\(482\) 38.2918 1.74414
\(483\) −2.76393 −0.125763
\(484\) 0 0
\(485\) 15.3262 0.695929
\(486\) 31.1803 1.41437
\(487\) −1.81966 −0.0824567 −0.0412283 0.999150i \(-0.513127\pi\)
−0.0412283 + 0.999150i \(0.513127\pi\)
\(488\) 10.6525 0.482215
\(489\) −13.7082 −0.619906
\(490\) −2.23607 −0.101015
\(491\) −2.11146 −0.0952887 −0.0476443 0.998864i \(-0.515171\pi\)
−0.0476443 + 0.998864i \(0.515171\pi\)
\(492\) −4.58359 −0.206644
\(493\) 4.79837 0.216108
\(494\) −33.4164 −1.50348
\(495\) 0 0
\(496\) −3.23607 −0.145304
\(497\) 16.3262 0.732332
\(498\) −13.2148 −0.592169
\(499\) −23.0344 −1.03116 −0.515582 0.856840i \(-0.672424\pi\)
−0.515582 + 0.856840i \(0.672424\pi\)
\(500\) −3.00000 −0.134164
\(501\) 7.41641 0.331341
\(502\) −17.2361 −0.769283
\(503\) −3.09017 −0.137784 −0.0688919 0.997624i \(-0.521946\pi\)
−0.0688919 + 0.997624i \(0.521946\pi\)
\(504\) −5.85410 −0.260762
\(505\) −13.7082 −0.610007
\(506\) 0 0
\(507\) 5.14590 0.228537
\(508\) −27.7082 −1.22935
\(509\) 26.3607 1.16842 0.584208 0.811604i \(-0.301405\pi\)
0.584208 + 0.811604i \(0.301405\pi\)
\(510\) 7.76393 0.343793
\(511\) −6.38197 −0.282322
\(512\) 11.1803 0.494106
\(513\) −11.2361 −0.496085
\(514\) −5.32624 −0.234930
\(515\) 17.3262 0.763485
\(516\) 9.70820 0.427380
\(517\) 0 0
\(518\) −16.1803 −0.710923
\(519\) 0.965558 0.0423833
\(520\) 10.3262 0.452835
\(521\) −14.1803 −0.621252 −0.310626 0.950532i \(-0.600539\pi\)
−0.310626 + 0.950532i \(0.600539\pi\)
\(522\) 5.00000 0.218844
\(523\) −1.14590 −0.0501066 −0.0250533 0.999686i \(-0.507976\pi\)
−0.0250533 + 0.999686i \(0.507976\pi\)
\(524\) −24.0000 −1.04844
\(525\) 0.618034 0.0269732
\(526\) −31.7082 −1.38254
\(527\) −18.1803 −0.791948
\(528\) 0 0
\(529\) −3.00000 −0.130435
\(530\) 3.41641 0.148399
\(531\) 24.1803 1.04934
\(532\) 9.70820 0.420904
\(533\) 11.4164 0.494500
\(534\) 17.6393 0.763328
\(535\) −7.23607 −0.312842
\(536\) 18.2918 0.790085
\(537\) 12.1246 0.523216
\(538\) −41.7082 −1.79817
\(539\) 0 0
\(540\) 10.4164 0.448251
\(541\) −6.14590 −0.264233 −0.132116 0.991234i \(-0.542177\pi\)
−0.132116 + 0.991234i \(0.542177\pi\)
\(542\) −51.9574 −2.23176
\(543\) 3.41641 0.146612
\(544\) 37.6869 1.61581
\(545\) −11.5279 −0.493799
\(546\) −6.38197 −0.273123
\(547\) 18.0689 0.772570 0.386285 0.922380i \(-0.373758\pi\)
0.386285 + 0.922380i \(0.373758\pi\)
\(548\) −34.2492 −1.46305
\(549\) −12.4721 −0.532298
\(550\) 0 0
\(551\) −2.76393 −0.117747
\(552\) −6.18034 −0.263053
\(553\) −10.7984 −0.459194
\(554\) 21.0557 0.894572
\(555\) 4.47214 0.189832
\(556\) −16.5836 −0.703301
\(557\) 17.7082 0.750321 0.375160 0.926960i \(-0.377588\pi\)
0.375160 + 0.926960i \(0.377588\pi\)
\(558\) −18.9443 −0.801975
\(559\) −24.1803 −1.02272
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 18.9443 0.799116
\(563\) 23.6180 0.995381 0.497691 0.867355i \(-0.334181\pi\)
0.497691 + 0.867355i \(0.334181\pi\)
\(564\) −2.12461 −0.0894623
\(565\) −10.4721 −0.440566
\(566\) 16.6312 0.699061
\(567\) 5.70820 0.239722
\(568\) 36.5066 1.53178
\(569\) 37.7984 1.58459 0.792295 0.610138i \(-0.208886\pi\)
0.792295 + 0.610138i \(0.208886\pi\)
\(570\) −4.47214 −0.187317
\(571\) 35.9787 1.50566 0.752831 0.658214i \(-0.228688\pi\)
0.752831 + 0.658214i \(0.228688\pi\)
\(572\) 0 0
\(573\) 14.2705 0.596159
\(574\) −5.52786 −0.230729
\(575\) −4.47214 −0.186501
\(576\) 34.0344 1.41810
\(577\) 33.2148 1.38275 0.691375 0.722496i \(-0.257005\pi\)
0.691375 + 0.722496i \(0.257005\pi\)
\(578\) 32.5623 1.35441
\(579\) −2.47214 −0.102738
\(580\) 2.56231 0.106394
\(581\) −9.56231 −0.396711
\(582\) −21.1803 −0.877953
\(583\) 0 0
\(584\) −14.2705 −0.590518
\(585\) −12.0902 −0.499867
\(586\) 21.0557 0.869804
\(587\) 6.72949 0.277756 0.138878 0.990310i \(-0.455650\pi\)
0.138878 + 0.990310i \(0.455650\pi\)
\(588\) 1.85410 0.0764619
\(589\) 10.4721 0.431497
\(590\) 20.6525 0.850249
\(591\) 7.41641 0.305070
\(592\) 7.23607 0.297401
\(593\) −9.79837 −0.402371 −0.201185 0.979553i \(-0.564479\pi\)
−0.201185 + 0.979553i \(0.564479\pi\)
\(594\) 0 0
\(595\) 5.61803 0.230317
\(596\) 53.5623 2.19400
\(597\) −11.1246 −0.455300
\(598\) 46.1803 1.88845
\(599\) 26.2705 1.07338 0.536692 0.843778i \(-0.319674\pi\)
0.536692 + 0.843778i \(0.319674\pi\)
\(600\) 1.38197 0.0564185
\(601\) 15.4164 0.628848 0.314424 0.949283i \(-0.398189\pi\)
0.314424 + 0.949283i \(0.398189\pi\)
\(602\) 11.7082 0.477191
\(603\) −21.4164 −0.872144
\(604\) 1.14590 0.0466259
\(605\) 0 0
\(606\) 18.9443 0.769558
\(607\) 1.90983 0.0775176 0.0387588 0.999249i \(-0.487660\pi\)
0.0387588 + 0.999249i \(0.487660\pi\)
\(608\) −21.7082 −0.880384
\(609\) −0.527864 −0.0213901
\(610\) −10.6525 −0.431306
\(611\) 5.29180 0.214083
\(612\) 44.1246 1.78363
\(613\) 40.8328 1.64922 0.824611 0.565700i \(-0.191394\pi\)
0.824611 + 0.565700i \(0.191394\pi\)
\(614\) 45.9787 1.85555
\(615\) 1.52786 0.0616094
\(616\) 0 0
\(617\) −2.76393 −0.111272 −0.0556359 0.998451i \(-0.517719\pi\)
−0.0556359 + 0.998451i \(0.517719\pi\)
\(618\) −23.9443 −0.963180
\(619\) −22.4721 −0.903231 −0.451616 0.892213i \(-0.649152\pi\)
−0.451616 + 0.892213i \(0.649152\pi\)
\(620\) −9.70820 −0.389891
\(621\) 15.5279 0.623112
\(622\) 29.3475 1.17673
\(623\) 12.7639 0.511376
\(624\) 2.85410 0.114256
\(625\) 1.00000 0.0400000
\(626\) −40.2492 −1.60868
\(627\) 0 0
\(628\) −48.2705 −1.92620
\(629\) 40.6525 1.62092
\(630\) 5.85410 0.233233
\(631\) −9.50658 −0.378451 −0.189225 0.981934i \(-0.560598\pi\)
−0.189225 + 0.981934i \(0.560598\pi\)
\(632\) −24.1459 −0.960472
\(633\) −4.74265 −0.188503
\(634\) −48.1378 −1.91179
\(635\) 9.23607 0.366522
\(636\) −2.83282 −0.112328
\(637\) −4.61803 −0.182973
\(638\) 0 0
\(639\) −42.7426 −1.69087
\(640\) 15.6525 0.618718
\(641\) 20.9787 0.828609 0.414305 0.910138i \(-0.364025\pi\)
0.414305 + 0.910138i \(0.364025\pi\)
\(642\) 10.0000 0.394669
\(643\) 10.7984 0.425846 0.212923 0.977069i \(-0.431702\pi\)
0.212923 + 0.977069i \(0.431702\pi\)
\(644\) −13.4164 −0.528681
\(645\) −3.23607 −0.127420
\(646\) −40.6525 −1.59945
\(647\) −46.8541 −1.84202 −0.921012 0.389533i \(-0.872636\pi\)
−0.921012 + 0.389533i \(0.872636\pi\)
\(648\) 12.7639 0.501415
\(649\) 0 0
\(650\) −10.3262 −0.405028
\(651\) 2.00000 0.0783862
\(652\) −66.5410 −2.60595
\(653\) −38.1803 −1.49411 −0.747056 0.664761i \(-0.768533\pi\)
−0.747056 + 0.664761i \(0.768533\pi\)
\(654\) 15.9311 0.622956
\(655\) 8.00000 0.312586
\(656\) 2.47214 0.0965207
\(657\) 16.7082 0.651849
\(658\) −2.56231 −0.0998891
\(659\) 4.03444 0.157160 0.0785798 0.996908i \(-0.474961\pi\)
0.0785798 + 0.996908i \(0.474961\pi\)
\(660\) 0 0
\(661\) −6.58359 −0.256072 −0.128036 0.991770i \(-0.540867\pi\)
−0.128036 + 0.991770i \(0.540867\pi\)
\(662\) −41.3820 −1.60836
\(663\) 16.0344 0.622726
\(664\) −21.3820 −0.829781
\(665\) −3.23607 −0.125489
\(666\) 42.3607 1.64144
\(667\) 3.81966 0.147898
\(668\) 36.0000 1.39288
\(669\) 0.944272 0.0365077
\(670\) −18.2918 −0.706674
\(671\) 0 0
\(672\) −4.14590 −0.159931
\(673\) −21.8885 −0.843741 −0.421871 0.906656i \(-0.638626\pi\)
−0.421871 + 0.906656i \(0.638626\pi\)
\(674\) 70.2492 2.70590
\(675\) −3.47214 −0.133643
\(676\) 24.9787 0.960720
\(677\) −48.8541 −1.87762 −0.938808 0.344441i \(-0.888068\pi\)
−0.938808 + 0.344441i \(0.888068\pi\)
\(678\) 14.4721 0.555799
\(679\) −15.3262 −0.588167
\(680\) 12.5623 0.481742
\(681\) 6.67376 0.255739
\(682\) 0 0
\(683\) 34.7639 1.33020 0.665102 0.746752i \(-0.268388\pi\)
0.665102 + 0.746752i \(0.268388\pi\)
\(684\) −25.4164 −0.971821
\(685\) 11.4164 0.436199
\(686\) 2.23607 0.0853735
\(687\) −15.3050 −0.583920
\(688\) −5.23607 −0.199623
\(689\) 7.05573 0.268802
\(690\) 6.18034 0.235282
\(691\) −26.8328 −1.02077 −0.510384 0.859946i \(-0.670497\pi\)
−0.510384 + 0.859946i \(0.670497\pi\)
\(692\) 4.68692 0.178170
\(693\) 0 0
\(694\) 3.41641 0.129685
\(695\) 5.52786 0.209684
\(696\) −1.18034 −0.0447407
\(697\) 13.8885 0.526066
\(698\) −8.69505 −0.329112
\(699\) −4.11146 −0.155510
\(700\) 3.00000 0.113389
\(701\) 51.3050 1.93776 0.968881 0.247528i \(-0.0796184\pi\)
0.968881 + 0.247528i \(0.0796184\pi\)
\(702\) 35.8541 1.35323
\(703\) −23.4164 −0.883167
\(704\) 0 0
\(705\) 0.708204 0.0266725
\(706\) −30.3262 −1.14134
\(707\) 13.7082 0.515550
\(708\) −17.1246 −0.643582
\(709\) 0.257354 0.00966514 0.00483257 0.999988i \(-0.498462\pi\)
0.00483257 + 0.999988i \(0.498462\pi\)
\(710\) −36.5066 −1.37007
\(711\) 28.2705 1.06023
\(712\) 28.5410 1.06962
\(713\) −14.4721 −0.541986
\(714\) −7.76393 −0.290558
\(715\) 0 0
\(716\) 58.8541 2.19948
\(717\) −15.4377 −0.576531
\(718\) 29.0213 1.08306
\(719\) 21.8885 0.816305 0.408152 0.912914i \(-0.366173\pi\)
0.408152 + 0.912914i \(0.366173\pi\)
\(720\) −2.61803 −0.0975684
\(721\) −17.3262 −0.645263
\(722\) −19.0689 −0.709670
\(723\) 10.5836 0.393608
\(724\) 16.5836 0.616324
\(725\) −0.854102 −0.0317206
\(726\) 0 0
\(727\) −41.5623 −1.54146 −0.770730 0.637162i \(-0.780108\pi\)
−0.770730 + 0.637162i \(0.780108\pi\)
\(728\) −10.3262 −0.382716
\(729\) −8.50658 −0.315058
\(730\) 14.2705 0.528175
\(731\) −29.4164 −1.08801
\(732\) 8.83282 0.326470
\(733\) −34.8673 −1.28785 −0.643926 0.765088i \(-0.722695\pi\)
−0.643926 + 0.765088i \(0.722695\pi\)
\(734\) −70.3262 −2.59579
\(735\) −0.618034 −0.0227965
\(736\) 30.0000 1.10581
\(737\) 0 0
\(738\) 14.4721 0.532727
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 21.7082 0.798009
\(741\) −9.23607 −0.339295
\(742\) −3.41641 −0.125420
\(743\) −5.34752 −0.196182 −0.0980908 0.995177i \(-0.531274\pi\)
−0.0980908 + 0.995177i \(0.531274\pi\)
\(744\) 4.47214 0.163956
\(745\) −17.8541 −0.654124
\(746\) −24.8754 −0.910753
\(747\) 25.0344 0.915962
\(748\) 0 0
\(749\) 7.23607 0.264400
\(750\) −1.38197 −0.0504623
\(751\) 37.3262 1.36205 0.681027 0.732258i \(-0.261534\pi\)
0.681027 + 0.732258i \(0.261534\pi\)
\(752\) 1.14590 0.0417866
\(753\) −4.76393 −0.173607
\(754\) 8.81966 0.321193
\(755\) −0.381966 −0.0139012
\(756\) −10.4164 −0.378841
\(757\) 4.18034 0.151937 0.0759685 0.997110i \(-0.475795\pi\)
0.0759685 + 0.997110i \(0.475795\pi\)
\(758\) 55.3262 2.00954
\(759\) 0 0
\(760\) −7.23607 −0.262480
\(761\) −48.8328 −1.77019 −0.885094 0.465412i \(-0.845906\pi\)
−0.885094 + 0.465412i \(0.845906\pi\)
\(762\) −12.7639 −0.462388
\(763\) 11.5279 0.417337
\(764\) 69.2705 2.50612
\(765\) −14.7082 −0.531776
\(766\) 59.9230 2.16511
\(767\) 42.6525 1.54009
\(768\) −5.56231 −0.200712
\(769\) −11.7082 −0.422209 −0.211104 0.977464i \(-0.567706\pi\)
−0.211104 + 0.977464i \(0.567706\pi\)
\(770\) 0 0
\(771\) −1.47214 −0.0530177
\(772\) −12.0000 −0.431889
\(773\) −27.3820 −0.984861 −0.492430 0.870352i \(-0.663891\pi\)
−0.492430 + 0.870352i \(0.663891\pi\)
\(774\) −30.6525 −1.10178
\(775\) 3.23607 0.116243
\(776\) −34.2705 −1.23024
\(777\) −4.47214 −0.160437
\(778\) −75.0476 −2.69059
\(779\) −8.00000 −0.286630
\(780\) 8.56231 0.306580
\(781\) 0 0
\(782\) 56.1803 2.00900
\(783\) 2.96556 0.105980
\(784\) −1.00000 −0.0357143
\(785\) 16.0902 0.574283
\(786\) −11.0557 −0.394345
\(787\) −40.0344 −1.42707 −0.713537 0.700618i \(-0.752908\pi\)
−0.713537 + 0.700618i \(0.752908\pi\)
\(788\) 36.0000 1.28245
\(789\) −8.76393 −0.312004
\(790\) 24.1459 0.859072
\(791\) 10.4721 0.372346
\(792\) 0 0
\(793\) −22.0000 −0.781243
\(794\) 78.6180 2.79005
\(795\) 0.944272 0.0334899
\(796\) −54.0000 −1.91398
\(797\) 44.1033 1.56222 0.781110 0.624393i \(-0.214654\pi\)
0.781110 + 0.624393i \(0.214654\pi\)
\(798\) 4.47214 0.158312
\(799\) 6.43769 0.227749
\(800\) −6.70820 −0.237171
\(801\) −33.4164 −1.18071
\(802\) 34.6738 1.22437
\(803\) 0 0
\(804\) 15.1672 0.534905
\(805\) 4.47214 0.157622
\(806\) −33.4164 −1.17704
\(807\) −11.5279 −0.405800
\(808\) 30.6525 1.07835
\(809\) 3.09017 0.108645 0.0543223 0.998523i \(-0.482700\pi\)
0.0543223 + 0.998523i \(0.482700\pi\)
\(810\) −12.7639 −0.448479
\(811\) −31.4164 −1.10318 −0.551590 0.834116i \(-0.685979\pi\)
−0.551590 + 0.834116i \(0.685979\pi\)
\(812\) −2.56231 −0.0899193
\(813\) −14.3607 −0.503651
\(814\) 0 0
\(815\) 22.1803 0.776943
\(816\) 3.47214 0.121549
\(817\) 16.9443 0.592805
\(818\) 7.23607 0.253003
\(819\) 12.0902 0.422465
\(820\) 7.41641 0.258992
\(821\) −12.6738 −0.442317 −0.221159 0.975238i \(-0.570984\pi\)
−0.221159 + 0.975238i \(0.570984\pi\)
\(822\) −15.7771 −0.550289
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) −38.7426 −1.34966
\(825\) 0 0
\(826\) −20.6525 −0.718592
\(827\) 37.8885 1.31751 0.658757 0.752356i \(-0.271083\pi\)
0.658757 + 0.752356i \(0.271083\pi\)
\(828\) 35.1246 1.22066
\(829\) 33.8885 1.17700 0.588499 0.808498i \(-0.299719\pi\)
0.588499 + 0.808498i \(0.299719\pi\)
\(830\) 21.3820 0.742179
\(831\) 5.81966 0.201882
\(832\) 60.0344 2.08132
\(833\) −5.61803 −0.194653
\(834\) −7.63932 −0.264528
\(835\) −12.0000 −0.415277
\(836\) 0 0
\(837\) −11.2361 −0.388375
\(838\) −78.9443 −2.72708
\(839\) 7.63932 0.263739 0.131869 0.991267i \(-0.457902\pi\)
0.131869 + 0.991267i \(0.457902\pi\)
\(840\) −1.38197 −0.0476824
\(841\) −28.2705 −0.974845
\(842\) −77.1591 −2.65908
\(843\) 5.23607 0.180340
\(844\) −23.0213 −0.792425
\(845\) −8.32624 −0.286431
\(846\) 6.70820 0.230633
\(847\) 0 0
\(848\) 1.52786 0.0524671
\(849\) 4.59675 0.157760
\(850\) −12.5623 −0.430884
\(851\) 32.3607 1.10931
\(852\) 30.2705 1.03705
\(853\) −18.4377 −0.631295 −0.315647 0.948877i \(-0.602222\pi\)
−0.315647 + 0.948877i \(0.602222\pi\)
\(854\) 10.6525 0.364520
\(855\) 8.47214 0.289741
\(856\) 16.1803 0.553033
\(857\) 57.1935 1.95369 0.976846 0.213942i \(-0.0686305\pi\)
0.976846 + 0.213942i \(0.0686305\pi\)
\(858\) 0 0
\(859\) −11.0557 −0.377217 −0.188608 0.982052i \(-0.560398\pi\)
−0.188608 + 0.982052i \(0.560398\pi\)
\(860\) −15.7082 −0.535645
\(861\) −1.52786 −0.0520695
\(862\) 11.9098 0.405650
\(863\) 3.05573 0.104018 0.0520091 0.998647i \(-0.483438\pi\)
0.0520091 + 0.998647i \(0.483438\pi\)
\(864\) 23.2918 0.792403
\(865\) −1.56231 −0.0531200
\(866\) 26.9098 0.914433
\(867\) 9.00000 0.305656
\(868\) 9.70820 0.329518
\(869\) 0 0
\(870\) 1.18034 0.0400173
\(871\) −37.7771 −1.28003
\(872\) 25.7771 0.872922
\(873\) 40.1246 1.35801
\(874\) −32.3607 −1.09462
\(875\) −1.00000 −0.0338062
\(876\) −11.8328 −0.399794
\(877\) −55.0132 −1.85766 −0.928831 0.370503i \(-0.879185\pi\)
−0.928831 + 0.370503i \(0.879185\pi\)
\(878\) −9.59675 −0.323875
\(879\) 5.81966 0.196292
\(880\) 0 0
\(881\) −5.23607 −0.176408 −0.0882038 0.996102i \(-0.528113\pi\)
−0.0882038 + 0.996102i \(0.528113\pi\)
\(882\) −5.85410 −0.197118
\(883\) −10.1803 −0.342596 −0.171298 0.985219i \(-0.554796\pi\)
−0.171298 + 0.985219i \(0.554796\pi\)
\(884\) 77.8328 2.61780
\(885\) 5.70820 0.191879
\(886\) −7.48529 −0.251473
\(887\) 21.1459 0.710010 0.355005 0.934864i \(-0.384479\pi\)
0.355005 + 0.934864i \(0.384479\pi\)
\(888\) −10.0000 −0.335578
\(889\) −9.23607 −0.309768
\(890\) −28.5410 −0.956697
\(891\) 0 0
\(892\) 4.58359 0.153470
\(893\) −3.70820 −0.124090
\(894\) 24.6738 0.825214
\(895\) −19.6180 −0.655759
\(896\) −15.6525 −0.522913
\(897\) 12.7639 0.426175
\(898\) −20.7295 −0.691752
\(899\) −2.76393 −0.0921823
\(900\) −7.85410 −0.261803
\(901\) 8.58359 0.285961
\(902\) 0 0
\(903\) 3.23607 0.107690
\(904\) 23.4164 0.778818
\(905\) −5.52786 −0.183752
\(906\) 0.527864 0.0175371
\(907\) 27.8885 0.926024 0.463012 0.886352i \(-0.346769\pi\)
0.463012 + 0.886352i \(0.346769\pi\)
\(908\) 32.3951 1.07507
\(909\) −35.8885 −1.19035
\(910\) 10.3262 0.342311
\(911\) −1.50658 −0.0499151 −0.0249576 0.999689i \(-0.507945\pi\)
−0.0249576 + 0.999689i \(0.507945\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 0 0
\(914\) 84.4721 2.79409
\(915\) −2.94427 −0.0973346
\(916\) −74.2918 −2.45467
\(917\) −8.00000 −0.264183
\(918\) 43.6180 1.43961
\(919\) −2.27051 −0.0748972 −0.0374486 0.999299i \(-0.511923\pi\)
−0.0374486 + 0.999299i \(0.511923\pi\)
\(920\) 10.0000 0.329690
\(921\) 12.7082 0.418750
\(922\) −36.4296 −1.19974
\(923\) −75.3951 −2.48166
\(924\) 0 0
\(925\) −7.23607 −0.237920
\(926\) −38.2918 −1.25835
\(927\) 45.3607 1.48984
\(928\) 5.72949 0.188080
\(929\) −17.8885 −0.586904 −0.293452 0.955974i \(-0.594804\pi\)
−0.293452 + 0.955974i \(0.594804\pi\)
\(930\) −4.47214 −0.146647
\(931\) 3.23607 0.106058
\(932\) −19.9574 −0.653727
\(933\) 8.11146 0.265557
\(934\) −72.2837 −2.36519
\(935\) 0 0
\(936\) 27.0344 0.883648
\(937\) −0.549150 −0.0179400 −0.00896998 0.999960i \(-0.502855\pi\)
−0.00896998 + 0.999960i \(0.502855\pi\)
\(938\) 18.2918 0.597248
\(939\) −11.1246 −0.363038
\(940\) 3.43769 0.112125
\(941\) 30.5410 0.995609 0.497804 0.867289i \(-0.334140\pi\)
0.497804 + 0.867289i \(0.334140\pi\)
\(942\) −22.2361 −0.724490
\(943\) 11.0557 0.360024
\(944\) 9.23607 0.300608
\(945\) 3.47214 0.112949
\(946\) 0 0
\(947\) 34.0689 1.10709 0.553545 0.832819i \(-0.313275\pi\)
0.553545 + 0.832819i \(0.313275\pi\)
\(948\) −20.0213 −0.650261
\(949\) 29.4721 0.956706
\(950\) 7.23607 0.234769
\(951\) −13.3050 −0.431443
\(952\) −12.5623 −0.407147
\(953\) −42.9443 −1.39110 −0.695551 0.718477i \(-0.744840\pi\)
−0.695551 + 0.718477i \(0.744840\pi\)
\(954\) 8.94427 0.289581
\(955\) −23.0902 −0.747180
\(956\) −74.9361 −2.42361
\(957\) 0 0
\(958\) −50.6525 −1.63651
\(959\) −11.4164 −0.368655
\(960\) 8.03444 0.259310
\(961\) −20.5279 −0.662189
\(962\) 74.7214 2.40911
\(963\) −18.9443 −0.610471
\(964\) 51.3738 1.65464
\(965\) 4.00000 0.128765
\(966\) −6.18034 −0.198849
\(967\) 16.7639 0.539092 0.269546 0.962988i \(-0.413126\pi\)
0.269546 + 0.962988i \(0.413126\pi\)
\(968\) 0 0
\(969\) −11.2361 −0.360955
\(970\) 34.2705 1.10036
\(971\) −2.58359 −0.0829114 −0.0414557 0.999140i \(-0.513200\pi\)
−0.0414557 + 0.999140i \(0.513200\pi\)
\(972\) 41.8328 1.34179
\(973\) −5.52786 −0.177215
\(974\) −4.06888 −0.130375
\(975\) −2.85410 −0.0914044
\(976\) −4.76393 −0.152490
\(977\) 41.2361 1.31926 0.659629 0.751591i \(-0.270713\pi\)
0.659629 + 0.751591i \(0.270713\pi\)
\(978\) −30.6525 −0.980158
\(979\) 0 0
\(980\) −3.00000 −0.0958315
\(981\) −30.1803 −0.963584
\(982\) −4.72136 −0.150665
\(983\) −27.9787 −0.892382 −0.446191 0.894938i \(-0.647220\pi\)
−0.446191 + 0.894938i \(0.647220\pi\)
\(984\) −3.41641 −0.108911
\(985\) −12.0000 −0.382352
\(986\) 10.7295 0.341697
\(987\) −0.708204 −0.0225424
\(988\) −44.8328 −1.42632
\(989\) −23.4164 −0.744598
\(990\) 0 0
\(991\) 13.8541 0.440090 0.220045 0.975490i \(-0.429380\pi\)
0.220045 + 0.975490i \(0.429380\pi\)
\(992\) −21.7082 −0.689236
\(993\) −11.4377 −0.362964
\(994\) 36.5066 1.15792
\(995\) 18.0000 0.570638
\(996\) −17.7295 −0.561780
\(997\) 25.2148 0.798560 0.399280 0.916829i \(-0.369260\pi\)
0.399280 + 0.916829i \(0.369260\pi\)
\(998\) −51.5066 −1.63041
\(999\) 25.1246 0.794908
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 4235.2.a.k.1.2 2
11.2 odd 10 385.2.n.b.246.1 yes 4
11.6 odd 10 385.2.n.b.36.1 4
11.10 odd 2 4235.2.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.n.b.36.1 4 11.6 odd 10
385.2.n.b.246.1 yes 4 11.2 odd 10
4235.2.a.j.1.1 2 11.10 odd 2
4235.2.a.k.1.2 2 1.1 even 1 trivial