Properties

Label 9075.2.a.bh.1.1
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4642398343\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 9075.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} +2.00000 q^{7} +1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} +2.00000 q^{7} +1.73205 q^{8} +1.00000 q^{9} -1.00000 q^{12} +5.46410 q^{13} -3.46410 q^{14} -5.00000 q^{16} -1.73205 q^{18} -5.46410 q^{19} -2.00000 q^{21} -6.92820 q^{23} -1.73205 q^{24} -9.46410 q^{26} -1.00000 q^{27} +2.00000 q^{28} +3.46410 q^{29} -10.9282 q^{31} +5.19615 q^{32} +1.00000 q^{36} +4.92820 q^{37} +9.46410 q^{38} -5.46410 q^{39} -3.46410 q^{41} +3.46410 q^{42} -4.92820 q^{43} +12.0000 q^{46} +6.92820 q^{47} +5.00000 q^{48} -3.00000 q^{49} +5.46410 q^{52} -0.928203 q^{53} +1.73205 q^{54} +3.46410 q^{56} +5.46410 q^{57} -6.00000 q^{58} -6.92820 q^{59} -2.00000 q^{61} +18.9282 q^{62} +2.00000 q^{63} +1.00000 q^{64} -8.00000 q^{67} +6.92820 q^{69} +13.8564 q^{71} +1.73205 q^{72} -8.39230 q^{73} -8.53590 q^{74} -5.46410 q^{76} +9.46410 q^{78} +6.53590 q^{79} +1.00000 q^{81} +6.00000 q^{82} +8.53590 q^{83} -2.00000 q^{84} +8.53590 q^{86} -3.46410 q^{87} +0.928203 q^{89} +10.9282 q^{91} -6.92820 q^{92} +10.9282 q^{93} -12.0000 q^{94} -5.19615 q^{96} +10.0000 q^{97} +5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{4} + 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{4} + 4q^{7} + 2q^{9} - 2q^{12} + 4q^{13} - 10q^{16} - 4q^{19} - 4q^{21} - 12q^{26} - 2q^{27} + 4q^{28} - 8q^{31} + 2q^{36} - 4q^{37} + 12q^{38} - 4q^{39} + 4q^{43} + 24q^{46} + 10q^{48} - 6q^{49} + 4q^{52} + 12q^{53} + 4q^{57} - 12q^{58} - 4q^{61} + 24q^{62} + 4q^{63} + 2q^{64} - 16q^{67} + 4q^{73} - 24q^{74} - 4q^{76} + 12q^{78} + 20q^{79} + 2q^{81} + 12q^{82} + 24q^{83} - 4q^{84} + 24q^{86} - 12q^{89} + 8q^{91} + 8q^{93} - 24q^{94} + 20q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 −1.22474 −0.612372 0.790569i \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.73205 0.707107
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.73205 0.612372
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) −1.00000 −0.288675
\(13\) 5.46410 1.51547 0.757735 0.652563i \(-0.226306\pi\)
0.757735 + 0.652563i \(0.226306\pi\)
\(14\) −3.46410 −0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.73205 −0.408248
\(19\) −5.46410 −1.25355 −0.626775 0.779200i \(-0.715626\pi\)
−0.626775 + 0.779200i \(0.715626\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) −1.73205 −0.353553
\(25\) 0 0
\(26\) −9.46410 −1.85606
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) 3.46410 0.643268 0.321634 0.946864i \(-0.395768\pi\)
0.321634 + 0.946864i \(0.395768\pi\)
\(30\) 0 0
\(31\) −10.9282 −1.96276 −0.981382 0.192068i \(-0.938481\pi\)
−0.981382 + 0.192068i \(0.938481\pi\)
\(32\) 5.19615 0.918559
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.92820 0.810192 0.405096 0.914274i \(-0.367238\pi\)
0.405096 + 0.914274i \(0.367238\pi\)
\(38\) 9.46410 1.53528
\(39\) −5.46410 −0.874957
\(40\) 0 0
\(41\) −3.46410 −0.541002 −0.270501 0.962720i \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) 3.46410 0.534522
\(43\) −4.92820 −0.751544 −0.375772 0.926712i \(-0.622622\pi\)
−0.375772 + 0.926712i \(0.622622\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 12.0000 1.76930
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 5.00000 0.721688
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 5.46410 0.757735
\(53\) −0.928203 −0.127499 −0.0637493 0.997966i \(-0.520306\pi\)
−0.0637493 + 0.997966i \(0.520306\pi\)
\(54\) 1.73205 0.235702
\(55\) 0 0
\(56\) 3.46410 0.462910
\(57\) 5.46410 0.723738
\(58\) −6.00000 −0.787839
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 18.9282 2.40388
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 6.92820 0.834058
\(70\) 0 0
\(71\) 13.8564 1.64445 0.822226 0.569160i \(-0.192732\pi\)
0.822226 + 0.569160i \(0.192732\pi\)
\(72\) 1.73205 0.204124
\(73\) −8.39230 −0.982245 −0.491122 0.871091i \(-0.663413\pi\)
−0.491122 + 0.871091i \(0.663413\pi\)
\(74\) −8.53590 −0.992278
\(75\) 0 0
\(76\) −5.46410 −0.626775
\(77\) 0 0
\(78\) 9.46410 1.07160
\(79\) 6.53590 0.735346 0.367673 0.929955i \(-0.380155\pi\)
0.367673 + 0.929955i \(0.380155\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 8.53590 0.936937 0.468468 0.883480i \(-0.344806\pi\)
0.468468 + 0.883480i \(0.344806\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 8.53590 0.920450
\(87\) −3.46410 −0.371391
\(88\) 0 0
\(89\) 0.928203 0.0983893 0.0491947 0.998789i \(-0.484335\pi\)
0.0491947 + 0.998789i \(0.484335\pi\)
\(90\) 0 0
\(91\) 10.9282 1.14559
\(92\) −6.92820 −0.722315
\(93\) 10.9282 1.13320
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) −5.19615 −0.530330
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 5.19615 0.524891
\(99\) 0 0
\(100\) 0 0
\(101\) 10.3923 1.03407 0.517036 0.855963i \(-0.327035\pi\)
0.517036 + 0.855963i \(0.327035\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 9.46410 0.928032
\(105\) 0 0
\(106\) 1.60770 0.156153
\(107\) 8.53590 0.825196 0.412598 0.910913i \(-0.364621\pi\)
0.412598 + 0.910913i \(0.364621\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −4.92820 −0.467764
\(112\) −10.0000 −0.944911
\(113\) 12.9282 1.21618 0.608092 0.793867i \(-0.291935\pi\)
0.608092 + 0.793867i \(0.291935\pi\)
\(114\) −9.46410 −0.886394
\(115\) 0 0
\(116\) 3.46410 0.321634
\(117\) 5.46410 0.505156
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 3.46410 0.313625
\(123\) 3.46410 0.312348
\(124\) −10.9282 −0.981382
\(125\) 0 0
\(126\) −3.46410 −0.308607
\(127\) 8.92820 0.792250 0.396125 0.918197i \(-0.370355\pi\)
0.396125 + 0.918197i \(0.370355\pi\)
\(128\) −12.1244 −1.07165
\(129\) 4.92820 0.433904
\(130\) 0 0
\(131\) −18.9282 −1.65376 −0.826882 0.562375i \(-0.809888\pi\)
−0.826882 + 0.562375i \(0.809888\pi\)
\(132\) 0 0
\(133\) −10.9282 −0.947595
\(134\) 13.8564 1.19701
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) −12.0000 −1.02151
\(139\) −12.3923 −1.05110 −0.525551 0.850762i \(-0.676141\pi\)
−0.525551 + 0.850762i \(0.676141\pi\)
\(140\) 0 0
\(141\) −6.92820 −0.583460
\(142\) −24.0000 −2.01404
\(143\) 0 0
\(144\) −5.00000 −0.416667
\(145\) 0 0
\(146\) 14.5359 1.20300
\(147\) 3.00000 0.247436
\(148\) 4.92820 0.405096
\(149\) 15.4641 1.26687 0.633434 0.773796i \(-0.281645\pi\)
0.633434 + 0.773796i \(0.281645\pi\)
\(150\) 0 0
\(151\) 20.3923 1.65950 0.829751 0.558134i \(-0.188482\pi\)
0.829751 + 0.558134i \(0.188482\pi\)
\(152\) −9.46410 −0.767640
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −5.46410 −0.437478
\(157\) 3.07180 0.245156 0.122578 0.992459i \(-0.460884\pi\)
0.122578 + 0.992459i \(0.460884\pi\)
\(158\) −11.3205 −0.900611
\(159\) 0.928203 0.0736113
\(160\) 0 0
\(161\) −13.8564 −1.09204
\(162\) −1.73205 −0.136083
\(163\) −9.85641 −0.772013 −0.386007 0.922496i \(-0.626146\pi\)
−0.386007 + 0.922496i \(0.626146\pi\)
\(164\) −3.46410 −0.270501
\(165\) 0 0
\(166\) −14.7846 −1.14751
\(167\) −10.3923 −0.804181 −0.402090 0.915600i \(-0.631716\pi\)
−0.402090 + 0.915600i \(0.631716\pi\)
\(168\) −3.46410 −0.267261
\(169\) 16.8564 1.29665
\(170\) 0 0
\(171\) −5.46410 −0.417850
\(172\) −4.92820 −0.375772
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 0 0
\(177\) 6.92820 0.520756
\(178\) −1.60770 −0.120502
\(179\) 6.92820 0.517838 0.258919 0.965899i \(-0.416634\pi\)
0.258919 + 0.965899i \(0.416634\pi\)
\(180\) 0 0
\(181\) 15.8564 1.17860 0.589299 0.807915i \(-0.299404\pi\)
0.589299 + 0.807915i \(0.299404\pi\)
\(182\) −18.9282 −1.40305
\(183\) 2.00000 0.147844
\(184\) −12.0000 −0.884652
\(185\) 0 0
\(186\) −18.9282 −1.38788
\(187\) 0 0
\(188\) 6.92820 0.505291
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) −18.9282 −1.36960 −0.684798 0.728733i \(-0.740110\pi\)
−0.684798 + 0.728733i \(0.740110\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 24.3923 1.75580 0.877898 0.478847i \(-0.158945\pi\)
0.877898 + 0.478847i \(0.158945\pi\)
\(194\) −17.3205 −1.24354
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) −24.7846 −1.75693 −0.878467 0.477803i \(-0.841433\pi\)
−0.878467 + 0.477803i \(0.841433\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) −18.0000 −1.26648
\(203\) 6.92820 0.486265
\(204\) 0 0
\(205\) 0 0
\(206\) 13.8564 0.965422
\(207\) −6.92820 −0.481543
\(208\) −27.3205 −1.89434
\(209\) 0 0
\(210\) 0 0
\(211\) 8.39230 0.577750 0.288875 0.957367i \(-0.406719\pi\)
0.288875 + 0.957367i \(0.406719\pi\)
\(212\) −0.928203 −0.0637493
\(213\) −13.8564 −0.949425
\(214\) −14.7846 −1.01066
\(215\) 0 0
\(216\) −1.73205 −0.117851
\(217\) −21.8564 −1.48371
\(218\) −17.3205 −1.17309
\(219\) 8.39230 0.567099
\(220\) 0 0
\(221\) 0 0
\(222\) 8.53590 0.572892
\(223\) −9.85641 −0.660034 −0.330017 0.943975i \(-0.607054\pi\)
−0.330017 + 0.943975i \(0.607054\pi\)
\(224\) 10.3923 0.694365
\(225\) 0 0
\(226\) −22.3923 −1.48951
\(227\) 15.4641 1.02639 0.513194 0.858272i \(-0.328462\pi\)
0.513194 + 0.858272i \(0.328462\pi\)
\(228\) 5.46410 0.361869
\(229\) −23.8564 −1.57648 −0.788238 0.615371i \(-0.789006\pi\)
−0.788238 + 0.615371i \(0.789006\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) −9.46410 −0.618688
\(235\) 0 0
\(236\) −6.92820 −0.450988
\(237\) −6.53590 −0.424552
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −0.143594 −0.00924967 −0.00462484 0.999989i \(-0.501472\pi\)
−0.00462484 + 0.999989i \(0.501472\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) −29.8564 −1.89972
\(248\) −18.9282 −1.20194
\(249\) −8.53590 −0.540941
\(250\) 0 0
\(251\) −1.85641 −0.117175 −0.0585877 0.998282i \(-0.518660\pi\)
−0.0585877 + 0.998282i \(0.518660\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) −15.4641 −0.970304
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) 19.8564 1.23861 0.619304 0.785151i \(-0.287415\pi\)
0.619304 + 0.785151i \(0.287415\pi\)
\(258\) −8.53590 −0.531422
\(259\) 9.85641 0.612447
\(260\) 0 0
\(261\) 3.46410 0.214423
\(262\) 32.7846 2.02544
\(263\) 20.5359 1.26630 0.633149 0.774030i \(-0.281762\pi\)
0.633149 + 0.774030i \(0.281762\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 18.9282 1.16056
\(267\) −0.928203 −0.0568051
\(268\) −8.00000 −0.488678
\(269\) 19.8564 1.21067 0.605333 0.795972i \(-0.293040\pi\)
0.605333 + 0.795972i \(0.293040\pi\)
\(270\) 0 0
\(271\) 11.6077 0.705117 0.352559 0.935790i \(-0.385312\pi\)
0.352559 + 0.935790i \(0.385312\pi\)
\(272\) 0 0
\(273\) −10.9282 −0.661405
\(274\) −31.1769 −1.88347
\(275\) 0 0
\(276\) 6.92820 0.417029
\(277\) 29.4641 1.77033 0.885163 0.465281i \(-0.154047\pi\)
0.885163 + 0.465281i \(0.154047\pi\)
\(278\) 21.4641 1.28733
\(279\) −10.9282 −0.654254
\(280\) 0 0
\(281\) −3.46410 −0.206651 −0.103325 0.994648i \(-0.532948\pi\)
−0.103325 + 0.994648i \(0.532948\pi\)
\(282\) 12.0000 0.714590
\(283\) −4.92820 −0.292951 −0.146476 0.989214i \(-0.546793\pi\)
−0.146476 + 0.989214i \(0.546793\pi\)
\(284\) 13.8564 0.822226
\(285\) 0 0
\(286\) 0 0
\(287\) −6.92820 −0.408959
\(288\) 5.19615 0.306186
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) −8.39230 −0.491122
\(293\) −13.8564 −0.809500 −0.404750 0.914427i \(-0.632641\pi\)
−0.404750 + 0.914427i \(0.632641\pi\)
\(294\) −5.19615 −0.303046
\(295\) 0 0
\(296\) 8.53590 0.496139
\(297\) 0 0
\(298\) −26.7846 −1.55159
\(299\) −37.8564 −2.18929
\(300\) 0 0
\(301\) −9.85641 −0.568114
\(302\) −35.3205 −2.03247
\(303\) −10.3923 −0.597022
\(304\) 27.3205 1.56694
\(305\) 0 0
\(306\) 0 0
\(307\) 14.0000 0.799022 0.399511 0.916728i \(-0.369180\pi\)
0.399511 + 0.916728i \(0.369180\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 5.07180 0.287595 0.143798 0.989607i \(-0.454069\pi\)
0.143798 + 0.989607i \(0.454069\pi\)
\(312\) −9.46410 −0.535799
\(313\) −20.9282 −1.18293 −0.591466 0.806330i \(-0.701451\pi\)
−0.591466 + 0.806330i \(0.701451\pi\)
\(314\) −5.32051 −0.300254
\(315\) 0 0
\(316\) 6.53590 0.367673
\(317\) 24.9282 1.40011 0.700054 0.714090i \(-0.253159\pi\)
0.700054 + 0.714090i \(0.253159\pi\)
\(318\) −1.60770 −0.0901551
\(319\) 0 0
\(320\) 0 0
\(321\) −8.53590 −0.476427
\(322\) 24.0000 1.33747
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 17.0718 0.945519
\(327\) −10.0000 −0.553001
\(328\) −6.00000 −0.331295
\(329\) 13.8564 0.763928
\(330\) 0 0
\(331\) 9.85641 0.541757 0.270879 0.962614i \(-0.412686\pi\)
0.270879 + 0.962614i \(0.412686\pi\)
\(332\) 8.53590 0.468468
\(333\) 4.92820 0.270064
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) 10.0000 0.545545
\(337\) 33.1769 1.80726 0.903631 0.428312i \(-0.140892\pi\)
0.903631 + 0.428312i \(0.140892\pi\)
\(338\) −29.1962 −1.58806
\(339\) −12.9282 −0.702164
\(340\) 0 0
\(341\) 0 0
\(342\) 9.46410 0.511760
\(343\) −20.0000 −1.07990
\(344\) −8.53590 −0.460225
\(345\) 0 0
\(346\) 20.7846 1.11739
\(347\) 22.3923 1.20208 0.601041 0.799218i \(-0.294753\pi\)
0.601041 + 0.799218i \(0.294753\pi\)
\(348\) −3.46410 −0.185695
\(349\) 8.14359 0.435917 0.217958 0.975958i \(-0.430060\pi\)
0.217958 + 0.975958i \(0.430060\pi\)
\(350\) 0 0
\(351\) −5.46410 −0.291652
\(352\) 0 0
\(353\) −12.9282 −0.688099 −0.344049 0.938952i \(-0.611799\pi\)
−0.344049 + 0.938952i \(0.611799\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 0.928203 0.0491947
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 20.7846 1.09697 0.548485 0.836160i \(-0.315205\pi\)
0.548485 + 0.836160i \(0.315205\pi\)
\(360\) 0 0
\(361\) 10.8564 0.571390
\(362\) −27.4641 −1.44348
\(363\) 0 0
\(364\) 10.9282 0.572793
\(365\) 0 0
\(366\) −3.46410 −0.181071
\(367\) −20.0000 −1.04399 −0.521996 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) 34.6410 1.80579
\(369\) −3.46410 −0.180334
\(370\) 0 0
\(371\) −1.85641 −0.0963798
\(372\) 10.9282 0.566601
\(373\) −20.3923 −1.05587 −0.527937 0.849284i \(-0.677034\pi\)
−0.527937 + 0.849284i \(0.677034\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) 18.9282 0.974852
\(378\) 3.46410 0.178174
\(379\) −17.8564 −0.917222 −0.458611 0.888637i \(-0.651653\pi\)
−0.458611 + 0.888637i \(0.651653\pi\)
\(380\) 0 0
\(381\) −8.92820 −0.457406
\(382\) 32.7846 1.67741
\(383\) 13.8564 0.708029 0.354015 0.935240i \(-0.384816\pi\)
0.354015 + 0.935240i \(0.384816\pi\)
\(384\) 12.1244 0.618718
\(385\) 0 0
\(386\) −42.2487 −2.15040
\(387\) −4.92820 −0.250515
\(388\) 10.0000 0.507673
\(389\) 11.0718 0.561362 0.280681 0.959801i \(-0.409440\pi\)
0.280681 + 0.959801i \(0.409440\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −5.19615 −0.262445
\(393\) 18.9282 0.954802
\(394\) 20.7846 1.04711
\(395\) 0 0
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 42.9282 2.15180
\(399\) 10.9282 0.547094
\(400\) 0 0
\(401\) −7.85641 −0.392330 −0.196165 0.980571i \(-0.562849\pi\)
−0.196165 + 0.980571i \(0.562849\pi\)
\(402\) −13.8564 −0.691095
\(403\) −59.7128 −2.97451
\(404\) 10.3923 0.517036
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) 0 0
\(408\) 0 0
\(409\) 6.78461 0.335477 0.167739 0.985831i \(-0.446354\pi\)
0.167739 + 0.985831i \(0.446354\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) −8.00000 −0.394132
\(413\) −13.8564 −0.681829
\(414\) 12.0000 0.589768
\(415\) 0 0
\(416\) 28.3923 1.39205
\(417\) 12.3923 0.606854
\(418\) 0 0
\(419\) 30.9282 1.51094 0.755471 0.655182i \(-0.227408\pi\)
0.755471 + 0.655182i \(0.227408\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −14.5359 −0.707596
\(423\) 6.92820 0.336861
\(424\) −1.60770 −0.0780766
\(425\) 0 0
\(426\) 24.0000 1.16280
\(427\) −4.00000 −0.193574
\(428\) 8.53590 0.412598
\(429\) 0 0
\(430\) 0 0
\(431\) −8.78461 −0.423140 −0.211570 0.977363i \(-0.567858\pi\)
−0.211570 + 0.977363i \(0.567858\pi\)
\(432\) 5.00000 0.240563
\(433\) −0.143594 −0.00690067 −0.00345033 0.999994i \(-0.501098\pi\)
−0.00345033 + 0.999994i \(0.501098\pi\)
\(434\) 37.8564 1.81717
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 37.8564 1.81092
\(438\) −14.5359 −0.694552
\(439\) −33.1769 −1.58345 −0.791724 0.610879i \(-0.790816\pi\)
−0.791724 + 0.610879i \(0.790816\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −4.92820 −0.233882
\(445\) 0 0
\(446\) 17.0718 0.808373
\(447\) −15.4641 −0.731427
\(448\) 2.00000 0.0944911
\(449\) −26.7846 −1.26404 −0.632022 0.774950i \(-0.717775\pi\)
−0.632022 + 0.774950i \(0.717775\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 12.9282 0.608092
\(453\) −20.3923 −0.958114
\(454\) −26.7846 −1.25706
\(455\) 0 0
\(456\) 9.46410 0.443197
\(457\) 12.3923 0.579688 0.289844 0.957074i \(-0.406397\pi\)
0.289844 + 0.957074i \(0.406397\pi\)
\(458\) 41.3205 1.93078
\(459\) 0 0
\(460\) 0 0
\(461\) −36.2487 −1.68827 −0.844135 0.536130i \(-0.819886\pi\)
−0.844135 + 0.536130i \(0.819886\pi\)
\(462\) 0 0
\(463\) 28.0000 1.30127 0.650635 0.759390i \(-0.274503\pi\)
0.650635 + 0.759390i \(0.274503\pi\)
\(464\) −17.3205 −0.804084
\(465\) 0 0
\(466\) −20.7846 −0.962828
\(467\) −5.07180 −0.234695 −0.117347 0.993091i \(-0.537439\pi\)
−0.117347 + 0.993091i \(0.537439\pi\)
\(468\) 5.46410 0.252578
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) −3.07180 −0.141541
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 11.3205 0.519968
\(475\) 0 0
\(476\) 0 0
\(477\) −0.928203 −0.0424995
\(478\) −20.7846 −0.950666
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) 26.9282 1.22782
\(482\) 0.248711 0.0113285
\(483\) 13.8564 0.630488
\(484\) 0 0
\(485\) 0 0
\(486\) 1.73205 0.0785674
\(487\) 31.7128 1.43704 0.718522 0.695504i \(-0.244819\pi\)
0.718522 + 0.695504i \(0.244819\pi\)
\(488\) −3.46410 −0.156813
\(489\) 9.85641 0.445722
\(490\) 0 0
\(491\) 30.9282 1.39577 0.697885 0.716210i \(-0.254125\pi\)
0.697885 + 0.716210i \(0.254125\pi\)
\(492\) 3.46410 0.156174
\(493\) 0 0
\(494\) 51.7128 2.32667
\(495\) 0 0
\(496\) 54.6410 2.45345
\(497\) 27.7128 1.24309
\(498\) 14.7846 0.662514
\(499\) 28.7846 1.28858 0.644288 0.764783i \(-0.277154\pi\)
0.644288 + 0.764783i \(0.277154\pi\)
\(500\) 0 0
\(501\) 10.3923 0.464294
\(502\) 3.21539 0.143510
\(503\) 31.1769 1.39011 0.695055 0.718957i \(-0.255380\pi\)
0.695055 + 0.718957i \(0.255380\pi\)
\(504\) 3.46410 0.154303
\(505\) 0 0
\(506\) 0 0
\(507\) −16.8564 −0.748619
\(508\) 8.92820 0.396125
\(509\) 19.8564 0.880120 0.440060 0.897968i \(-0.354957\pi\)
0.440060 + 0.897968i \(0.354957\pi\)
\(510\) 0 0
\(511\) −16.7846 −0.742507
\(512\) −8.66025 −0.382733
\(513\) 5.46410 0.241246
\(514\) −34.3923 −1.51698
\(515\) 0 0
\(516\) 4.92820 0.216952
\(517\) 0 0
\(518\) −17.0718 −0.750092
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) −6.00000 −0.262613
\(523\) −22.0000 −0.961993 −0.480996 0.876723i \(-0.659725\pi\)
−0.480996 + 0.876723i \(0.659725\pi\)
\(524\) −18.9282 −0.826882
\(525\) 0 0
\(526\) −35.5692 −1.55089
\(527\) 0 0
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) −6.92820 −0.300658
\(532\) −10.9282 −0.473798
\(533\) −18.9282 −0.819871
\(534\) 1.60770 0.0695718
\(535\) 0 0
\(536\) −13.8564 −0.598506
\(537\) −6.92820 −0.298974
\(538\) −34.3923 −1.48276
\(539\) 0 0
\(540\) 0 0
\(541\) −27.8564 −1.19764 −0.598820 0.800883i \(-0.704364\pi\)
−0.598820 + 0.800883i \(0.704364\pi\)
\(542\) −20.1051 −0.863589
\(543\) −15.8564 −0.680464
\(544\) 0 0
\(545\) 0 0
\(546\) 18.9282 0.810052
\(547\) 2.00000 0.0855138 0.0427569 0.999086i \(-0.486386\pi\)
0.0427569 + 0.999086i \(0.486386\pi\)
\(548\) 18.0000 0.768922
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −18.9282 −0.806369
\(552\) 12.0000 0.510754
\(553\) 13.0718 0.555869
\(554\) −51.0333 −2.16820
\(555\) 0 0
\(556\) −12.3923 −0.525551
\(557\) 3.21539 0.136240 0.0681202 0.997677i \(-0.478300\pi\)
0.0681202 + 0.997677i \(0.478300\pi\)
\(558\) 18.9282 0.801295
\(559\) −26.9282 −1.13894
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) 10.3923 0.437983 0.218992 0.975727i \(-0.429723\pi\)
0.218992 + 0.975727i \(0.429723\pi\)
\(564\) −6.92820 −0.291730
\(565\) 0 0
\(566\) 8.53590 0.358791
\(567\) 2.00000 0.0839921
\(568\) 24.0000 1.00702
\(569\) −5.32051 −0.223047 −0.111524 0.993762i \(-0.535573\pi\)
−0.111524 + 0.993762i \(0.535573\pi\)
\(570\) 0 0
\(571\) −3.60770 −0.150977 −0.0754887 0.997147i \(-0.524052\pi\)
−0.0754887 + 0.997147i \(0.524052\pi\)
\(572\) 0 0
\(573\) 18.9282 0.790737
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 18.7846 0.782014 0.391007 0.920388i \(-0.372127\pi\)
0.391007 + 0.920388i \(0.372127\pi\)
\(578\) 29.4449 1.22474
\(579\) −24.3923 −1.01371
\(580\) 0 0
\(581\) 17.0718 0.708257
\(582\) 17.3205 0.717958
\(583\) 0 0
\(584\) −14.5359 −0.601500
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 18.9282 0.781251 0.390625 0.920550i \(-0.372259\pi\)
0.390625 + 0.920550i \(0.372259\pi\)
\(588\) 3.00000 0.123718
\(589\) 59.7128 2.46042
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) −24.6410 −1.01274
\(593\) 8.78461 0.360741 0.180370 0.983599i \(-0.442270\pi\)
0.180370 + 0.983599i \(0.442270\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.4641 0.633434
\(597\) 24.7846 1.01437
\(598\) 65.5692 2.68132
\(599\) −37.8564 −1.54677 −0.773385 0.633936i \(-0.781438\pi\)
−0.773385 + 0.633936i \(0.781438\pi\)
\(600\) 0 0
\(601\) 32.6410 1.33145 0.665727 0.746195i \(-0.268121\pi\)
0.665727 + 0.746195i \(0.268121\pi\)
\(602\) 17.0718 0.695794
\(603\) −8.00000 −0.325785
\(604\) 20.3923 0.829751
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) −18.7846 −0.762444 −0.381222 0.924484i \(-0.624497\pi\)
−0.381222 + 0.924484i \(0.624497\pi\)
\(608\) −28.3923 −1.15146
\(609\) −6.92820 −0.280745
\(610\) 0 0
\(611\) 37.8564 1.53151
\(612\) 0 0
\(613\) −20.3923 −0.823637 −0.411819 0.911266i \(-0.635106\pi\)
−0.411819 + 0.911266i \(0.635106\pi\)
\(614\) −24.2487 −0.978598
\(615\) 0 0
\(616\) 0 0
\(617\) 36.9282 1.48667 0.743337 0.668917i \(-0.233242\pi\)
0.743337 + 0.668917i \(0.233242\pi\)
\(618\) −13.8564 −0.557386
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 6.92820 0.278019
\(622\) −8.78461 −0.352231
\(623\) 1.85641 0.0743754
\(624\) 27.3205 1.09370
\(625\) 0 0
\(626\) 36.2487 1.44879
\(627\) 0 0
\(628\) 3.07180 0.122578
\(629\) 0 0
\(630\) 0 0
\(631\) −21.0718 −0.838855 −0.419427 0.907789i \(-0.637769\pi\)
−0.419427 + 0.907789i \(0.637769\pi\)
\(632\) 11.3205 0.450306
\(633\) −8.39230 −0.333564
\(634\) −43.1769 −1.71477
\(635\) 0 0
\(636\) 0.928203 0.0368057
\(637\) −16.3923 −0.649487
\(638\) 0 0
\(639\) 13.8564 0.548151
\(640\) 0 0
\(641\) −12.9282 −0.510633 −0.255317 0.966857i \(-0.582180\pi\)
−0.255317 + 0.966857i \(0.582180\pi\)
\(642\) 14.7846 0.583502
\(643\) −37.5692 −1.48159 −0.740793 0.671734i \(-0.765550\pi\)
−0.740793 + 0.671734i \(0.765550\pi\)
\(644\) −13.8564 −0.546019
\(645\) 0 0
\(646\) 0 0
\(647\) −27.7128 −1.08950 −0.544752 0.838597i \(-0.683376\pi\)
−0.544752 + 0.838597i \(0.683376\pi\)
\(648\) 1.73205 0.0680414
\(649\) 0 0
\(650\) 0 0
\(651\) 21.8564 0.856620
\(652\) −9.85641 −0.386007
\(653\) −19.8564 −0.777041 −0.388521 0.921440i \(-0.627014\pi\)
−0.388521 + 0.921440i \(0.627014\pi\)
\(654\) 17.3205 0.677285
\(655\) 0 0
\(656\) 17.3205 0.676252
\(657\) −8.39230 −0.327415
\(658\) −24.0000 −0.935617
\(659\) 15.7128 0.612084 0.306042 0.952018i \(-0.400995\pi\)
0.306042 + 0.952018i \(0.400995\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) −17.0718 −0.663514
\(663\) 0 0
\(664\) 14.7846 0.573754
\(665\) 0 0
\(666\) −8.53590 −0.330759
\(667\) −24.0000 −0.929284
\(668\) −10.3923 −0.402090
\(669\) 9.85641 0.381071
\(670\) 0 0
\(671\) 0 0
\(672\) −10.3923 −0.400892
\(673\) −3.32051 −0.127996 −0.0639981 0.997950i \(-0.520385\pi\)
−0.0639981 + 0.997950i \(0.520385\pi\)
\(674\) −57.4641 −2.21343
\(675\) 0 0
\(676\) 16.8564 0.648323
\(677\) 8.78461 0.337620 0.168810 0.985649i \(-0.446008\pi\)
0.168810 + 0.985649i \(0.446008\pi\)
\(678\) 22.3923 0.859971
\(679\) 20.0000 0.767530
\(680\) 0 0
\(681\) −15.4641 −0.592586
\(682\) 0 0
\(683\) 32.7846 1.25447 0.627234 0.778831i \(-0.284187\pi\)
0.627234 + 0.778831i \(0.284187\pi\)
\(684\) −5.46410 −0.208925
\(685\) 0 0
\(686\) 34.6410 1.32260
\(687\) 23.8564 0.910179
\(688\) 24.6410 0.939430
\(689\) −5.07180 −0.193220
\(690\) 0 0
\(691\) 47.7128 1.81508 0.907540 0.419965i \(-0.137958\pi\)
0.907540 + 0.419965i \(0.137958\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) −38.7846 −1.47224
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) 0 0
\(698\) −14.1051 −0.533887
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) −39.4641 −1.49054 −0.745269 0.666764i \(-0.767679\pi\)
−0.745269 + 0.666764i \(0.767679\pi\)
\(702\) 9.46410 0.357199
\(703\) −26.9282 −1.01562
\(704\) 0 0
\(705\) 0 0
\(706\) 22.3923 0.842746
\(707\) 20.7846 0.781686
\(708\) 6.92820 0.260378
\(709\) −11.8564 −0.445277 −0.222638 0.974901i \(-0.571467\pi\)
−0.222638 + 0.974901i \(0.571467\pi\)
\(710\) 0 0
\(711\) 6.53590 0.245115
\(712\) 1.60770 0.0602509
\(713\) 75.7128 2.83547
\(714\) 0 0
\(715\) 0 0
\(716\) 6.92820 0.258919
\(717\) −12.0000 −0.448148
\(718\) −36.0000 −1.34351
\(719\) −5.07180 −0.189146 −0.0945731 0.995518i \(-0.530149\pi\)
−0.0945731 + 0.995518i \(0.530149\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) −18.8038 −0.699807
\(723\) 0.143594 0.00534030
\(724\) 15.8564 0.589299
\(725\) 0 0
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 18.9282 0.701526
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 2.00000 0.0739221
\(733\) 53.9615 1.99311 0.996557 0.0829082i \(-0.0264208\pi\)
0.996557 + 0.0829082i \(0.0264208\pi\)
\(734\) 34.6410 1.27862
\(735\) 0 0
\(736\) −36.0000 −1.32698
\(737\) 0 0
\(738\) 6.00000 0.220863
\(739\) −17.4641 −0.642427 −0.321214 0.947007i \(-0.604091\pi\)
−0.321214 + 0.947007i \(0.604091\pi\)
\(740\) 0 0
\(741\) 29.8564 1.09680
\(742\) 3.21539 0.118041
\(743\) 25.6077 0.939455 0.469728 0.882811i \(-0.344352\pi\)
0.469728 + 0.882811i \(0.344352\pi\)
\(744\) 18.9282 0.693942
\(745\) 0 0
\(746\) 35.3205 1.29318
\(747\) 8.53590 0.312312
\(748\) 0 0
\(749\) 17.0718 0.623790
\(750\) 0 0
\(751\) 26.9282 0.982624 0.491312 0.870984i \(-0.336517\pi\)
0.491312 + 0.870984i \(0.336517\pi\)
\(752\) −34.6410 −1.26323
\(753\) 1.85641 0.0676512
\(754\) −32.7846 −1.19395
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) −34.7846 −1.26427 −0.632134 0.774859i \(-0.717821\pi\)
−0.632134 + 0.774859i \(0.717821\pi\)
\(758\) 30.9282 1.12336
\(759\) 0 0
\(760\) 0 0
\(761\) 32.5359 1.17943 0.589713 0.807613i \(-0.299241\pi\)
0.589713 + 0.807613i \(0.299241\pi\)
\(762\) 15.4641 0.560205
\(763\) 20.0000 0.724049
\(764\) −18.9282 −0.684798
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) −37.8564 −1.36692
\(768\) −19.0000 −0.685603
\(769\) −50.4974 −1.82098 −0.910492 0.413527i \(-0.864297\pi\)
−0.910492 + 0.413527i \(0.864297\pi\)
\(770\) 0 0
\(771\) −19.8564 −0.715111
\(772\) 24.3923 0.877898
\(773\) 4.14359 0.149035 0.0745174 0.997220i \(-0.476258\pi\)
0.0745174 + 0.997220i \(0.476258\pi\)
\(774\) 8.53590 0.306817
\(775\) 0 0
\(776\) 17.3205 0.621770
\(777\) −9.85641 −0.353597
\(778\) −19.1769 −0.687526
\(779\) 18.9282 0.678173
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −3.46410 −0.123797
\(784\) 15.0000 0.535714
\(785\) 0 0
\(786\) −32.7846 −1.16939
\(787\) 22.7846 0.812184 0.406092 0.913832i \(-0.366891\pi\)
0.406092 + 0.913832i \(0.366891\pi\)
\(788\) −12.0000 −0.427482
\(789\) −20.5359 −0.731097
\(790\) 0 0
\(791\) 25.8564 0.919348
\(792\) 0 0
\(793\) −10.9282 −0.388072
\(794\) 3.46410 0.122936
\(795\) 0 0
\(796\) −24.7846 −0.878467
\(797\) 52.6410 1.86464 0.932320 0.361634i \(-0.117781\pi\)
0.932320 + 0.361634i \(0.117781\pi\)
\(798\) −18.9282 −0.670051
\(799\) 0 0
\(800\) 0 0
\(801\) 0.928203 0.0327964
\(802\) 13.6077 0.480504
\(803\) 0 0
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 103.426 3.64301
\(807\) −19.8564 −0.698979
\(808\) 18.0000 0.633238
\(809\) 15.4641 0.543689 0.271844 0.962341i \(-0.412366\pi\)
0.271844 + 0.962341i \(0.412366\pi\)
\(810\) 0 0
\(811\) −12.3923 −0.435153 −0.217576 0.976043i \(-0.569815\pi\)
−0.217576 + 0.976043i \(0.569815\pi\)
\(812\) 6.92820 0.243132
\(813\) −11.6077 −0.407100
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 26.9282 0.942099
\(818\) −11.7513 −0.410874
\(819\) 10.9282 0.381862
\(820\) 0 0
\(821\) −20.5359 −0.716708 −0.358354 0.933586i \(-0.616662\pi\)
−0.358354 + 0.933586i \(0.616662\pi\)
\(822\) 31.1769 1.08742
\(823\) 33.5692 1.17015 0.585075 0.810979i \(-0.301065\pi\)
0.585075 + 0.810979i \(0.301065\pi\)
\(824\) −13.8564 −0.482711
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) 22.3923 0.778657 0.389328 0.921099i \(-0.372707\pi\)
0.389328 + 0.921099i \(0.372707\pi\)
\(828\) −6.92820 −0.240772
\(829\) 29.7128 1.03197 0.515984 0.856598i \(-0.327426\pi\)
0.515984 + 0.856598i \(0.327426\pi\)
\(830\) 0 0
\(831\) −29.4641 −1.02210
\(832\) 5.46410 0.189434
\(833\) 0 0
\(834\) −21.4641 −0.743241
\(835\) 0 0
\(836\) 0 0
\(837\) 10.9282 0.377734
\(838\) −53.5692 −1.85052
\(839\) 56.7846 1.96042 0.980211 0.197954i \(-0.0634298\pi\)
0.980211 + 0.197954i \(0.0634298\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) −3.46410 −0.119381
\(843\) 3.46410 0.119310
\(844\) 8.39230 0.288875
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) 4.64102 0.159373
\(849\) 4.92820 0.169135
\(850\) 0 0
\(851\) −34.1436 −1.17043
\(852\) −13.8564 −0.474713
\(853\) 3.60770 0.123525 0.0617626 0.998091i \(-0.480328\pi\)
0.0617626 + 0.998091i \(0.480328\pi\)
\(854\) 6.92820 0.237078
\(855\) 0 0
\(856\) 14.7846 0.505328
\(857\) −37.8564 −1.29315 −0.646575 0.762850i \(-0.723799\pi\)
−0.646575 + 0.762850i \(0.723799\pi\)
\(858\) 0 0
\(859\) −7.71281 −0.263158 −0.131579 0.991306i \(-0.542005\pi\)
−0.131579 + 0.991306i \(0.542005\pi\)
\(860\) 0 0
\(861\) 6.92820 0.236113
\(862\) 15.2154 0.518238
\(863\) 37.8564 1.28865 0.644324 0.764753i \(-0.277139\pi\)
0.644324 + 0.764753i \(0.277139\pi\)
\(864\) −5.19615 −0.176777
\(865\) 0 0
\(866\) 0.248711 0.00845155
\(867\) 17.0000 0.577350
\(868\) −21.8564 −0.741855
\(869\) 0 0
\(870\) 0 0
\(871\) −43.7128 −1.48115
\(872\) 17.3205 0.586546
\(873\) 10.0000 0.338449
\(874\) −65.5692 −2.21791
\(875\) 0 0
\(876\) 8.39230 0.283550
\(877\) −34.2487 −1.15650 −0.578248 0.815861i \(-0.696264\pi\)
−0.578248 + 0.815861i \(0.696264\pi\)
\(878\) 57.4641 1.93932
\(879\) 13.8564 0.467365
\(880\) 0 0
\(881\) −0.928203 −0.0312720 −0.0156360 0.999878i \(-0.504977\pi\)
−0.0156360 + 0.999878i \(0.504977\pi\)
\(882\) 5.19615 0.174964
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 20.7846 0.698273
\(887\) 12.2487 0.411271 0.205636 0.978629i \(-0.434074\pi\)
0.205636 + 0.978629i \(0.434074\pi\)
\(888\) −8.53590 −0.286446
\(889\) 17.8564 0.598885
\(890\) 0 0
\(891\) 0 0
\(892\) −9.85641 −0.330017
\(893\) −37.8564 −1.26682
\(894\) 26.7846 0.895811
\(895\) 0 0
\(896\) −24.2487 −0.810093
\(897\) 37.8564 1.26399
\(898\) 46.3923 1.54813
\(899\) −37.8564 −1.26258
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 9.85641 0.328001
\(904\) 22.3923 0.744757
\(905\) 0 0
\(906\) 35.3205 1.17345
\(907\) −18.1436 −0.602448 −0.301224 0.953553i \(-0.597395\pi\)
−0.301224 + 0.953553i \(0.597395\pi\)
\(908\) 15.4641 0.513194
\(909\) 10.3923 0.344691
\(910\) 0 0
\(911\) 18.9282 0.627119 0.313560 0.949568i \(-0.398478\pi\)
0.313560 + 0.949568i \(0.398478\pi\)
\(912\) −27.3205 −0.904672
\(913\) 0 0
\(914\) −21.4641 −0.709969
\(915\) 0 0
\(916\) −23.8564 −0.788238
\(917\) −37.8564 −1.25013
\(918\) 0 0
\(919\) 32.3923 1.06852 0.534262 0.845319i \(-0.320590\pi\)
0.534262 + 0.845319i \(0.320590\pi\)
\(920\) 0 0
\(921\) −14.0000 −0.461316
\(922\) 62.7846 2.06770
\(923\) 75.7128 2.49212
\(924\) 0 0
\(925\) 0 0
\(926\) −48.4974 −1.59372
\(927\) −8.00000 −0.262754
\(928\) 18.0000 0.590879
\(929\) 2.78461 0.0913601 0.0456800 0.998956i \(-0.485455\pi\)
0.0456800 + 0.998956i \(0.485455\pi\)
\(930\) 0 0
\(931\) 16.3923 0.537236
\(932\) 12.0000 0.393073
\(933\) −5.07180 −0.166043
\(934\) 8.78461 0.287441
\(935\) 0 0
\(936\) 9.46410 0.309344
\(937\) −20.3923 −0.666188 −0.333094 0.942894i \(-0.608093\pi\)
−0.333094 + 0.942894i \(0.608093\pi\)
\(938\) 27.7128 0.904855
\(939\) 20.9282 0.682966
\(940\) 0 0
\(941\) −27.4641 −0.895304 −0.447652 0.894208i \(-0.647740\pi\)
−0.447652 + 0.894208i \(0.647740\pi\)
\(942\) 5.32051 0.173352
\(943\) 24.0000 0.781548
\(944\) 34.6410 1.12747
\(945\) 0 0
\(946\) 0 0
\(947\) 18.9282 0.615084 0.307542 0.951535i \(-0.400494\pi\)
0.307542 + 0.951535i \(0.400494\pi\)
\(948\) −6.53590 −0.212276
\(949\) −45.8564 −1.48856
\(950\) 0 0
\(951\) −24.9282 −0.808352
\(952\) 0 0
\(953\) −3.21539 −0.104157 −0.0520784 0.998643i \(-0.516585\pi\)
−0.0520784 + 0.998643i \(0.516585\pi\)
\(954\) 1.60770 0.0520511
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 20.7846 0.671520
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) 88.4256 2.85244
\(962\) −46.6410 −1.50377
\(963\) 8.53590 0.275065
\(964\) −0.143594 −0.00462484
\(965\) 0 0
\(966\) −24.0000 −0.772187
\(967\) 22.7846 0.732704 0.366352 0.930476i \(-0.380607\pi\)
0.366352 + 0.930476i \(0.380607\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 25.8564 0.829772 0.414886 0.909873i \(-0.363822\pi\)
0.414886 + 0.909873i \(0.363822\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −24.7846 −0.794558
\(974\) −54.9282 −1.76001
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −47.5692 −1.52187 −0.760937 0.648826i \(-0.775260\pi\)
−0.760937 + 0.648826i \(0.775260\pi\)
\(978\) −17.0718 −0.545896
\(979\) 0 0
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) −53.5692 −1.70946
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 0 0
\(987\) −13.8564 −0.441054
\(988\) −29.8564 −0.949859
\(989\) 34.1436 1.08570
\(990\) 0 0
\(991\) −7.21539 −0.229204 −0.114602 0.993411i \(-0.536559\pi\)
−0.114602 + 0.993411i \(0.536559\pi\)
\(992\) −56.7846 −1.80291
\(993\) −9.85641 −0.312784
\(994\) −48.0000 −1.52247
\(995\) 0 0
\(996\) −8.53590 −0.270470
\(997\) 27.6077 0.874344 0.437172 0.899378i \(-0.355980\pi\)
0.437172 + 0.899378i \(0.355980\pi\)
\(998\) −49.8564 −1.57818
\(999\) −4.92820 −0.155921
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 9075.2.a.bh.1.1 2
5.4 even 2 1815.2.a.i.1.2 2
11.10 odd 2 825.2.a.e.1.2 2
15.14 odd 2 5445.2.a.s.1.1 2
33.32 even 2 2475.2.a.r.1.1 2
55.32 even 4 825.2.c.c.199.4 4
55.43 even 4 825.2.c.c.199.1 4
55.54 odd 2 165.2.a.b.1.1 2
165.32 odd 4 2475.2.c.n.199.1 4
165.98 odd 4 2475.2.c.n.199.4 4
165.164 even 2 495.2.a.c.1.2 2
220.219 even 2 2640.2.a.x.1.2 2
385.384 even 2 8085.2.a.bd.1.1 2
660.659 odd 2 7920.2.a.bz.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.b.1.1 2 55.54 odd 2
495.2.a.c.1.2 2 165.164 even 2
825.2.a.e.1.2 2 11.10 odd 2
825.2.c.c.199.1 4 55.43 even 4
825.2.c.c.199.4 4 55.32 even 4
1815.2.a.i.1.2 2 5.4 even 2
2475.2.a.r.1.1 2 33.32 even 2
2475.2.c.n.199.1 4 165.32 odd 4
2475.2.c.n.199.4 4 165.98 odd 4
2640.2.a.x.1.2 2 220.219 even 2
5445.2.a.s.1.1 2 15.14 odd 2
7920.2.a.bz.1.2 2 660.659 odd 2
8085.2.a.bd.1.1 2 385.384 even 2
9075.2.a.bh.1.1 2 1.1 even 1 trivial