Properties

Label 6930.2.a.bt.1.2
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.3363286007\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2310)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6930.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.46410 q^{17} +1.00000 q^{20} +1.00000 q^{22} +1.46410 q^{23} +1.00000 q^{25} -3.46410 q^{26} +1.00000 q^{28} -2.00000 q^{29} +6.92820 q^{31} -1.00000 q^{32} -3.46410 q^{34} +1.00000 q^{35} +7.46410 q^{37} -1.00000 q^{40} +8.92820 q^{41} -2.92820 q^{43} -1.00000 q^{44} -1.46410 q^{46} -6.92820 q^{47} +1.00000 q^{49} -1.00000 q^{50} +3.46410 q^{52} -12.9282 q^{53} -1.00000 q^{55} -1.00000 q^{56} +2.00000 q^{58} +2.92820 q^{59} -4.92820 q^{61} -6.92820 q^{62} +1.00000 q^{64} +3.46410 q^{65} +9.46410 q^{67} +3.46410 q^{68} -1.00000 q^{70} +4.00000 q^{71} -0.928203 q^{73} -7.46410 q^{74} -1.00000 q^{77} +8.00000 q^{79} +1.00000 q^{80} -8.92820 q^{82} -6.92820 q^{83} +3.46410 q^{85} +2.92820 q^{86} +1.00000 q^{88} -3.46410 q^{89} +3.46410 q^{91} +1.46410 q^{92} +6.92820 q^{94} -11.8564 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8} - 2 q^{10} - 2 q^{11} - 2 q^{14} + 2 q^{16} + 2 q^{20} + 2 q^{22} - 4 q^{23} + 2 q^{25} + 2 q^{28} - 4 q^{29} - 2 q^{32} + 2 q^{35} + 8 q^{37} - 2 q^{40} + 4 q^{41} + 8 q^{43} - 2 q^{44} + 4 q^{46} + 2 q^{49} - 2 q^{50} - 12 q^{53} - 2 q^{55} - 2 q^{56} + 4 q^{58} - 8 q^{59} + 4 q^{61} + 2 q^{64} + 12 q^{67} - 2 q^{70} + 8 q^{71} + 12 q^{73} - 8 q^{74} - 2 q^{77} + 16 q^{79} + 2 q^{80} - 4 q^{82} - 8 q^{86} + 2 q^{88} - 4 q^{92} + 4 q^{97} - 2 q^{98} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 3.46410 0.960769 0.480384 0.877058i \(-0.340497\pi\)
0.480384 + 0.877058i \(0.340497\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 1.46410 0.305286 0.152643 0.988281i \(-0.451221\pi\)
0.152643 + 0.988281i \(0.451221\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −3.46410 −0.679366
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 6.92820 1.24434 0.622171 0.782881i \(-0.286251\pi\)
0.622171 + 0.782881i \(0.286251\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −3.46410 −0.594089
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 7.46410 1.22709 0.613545 0.789659i \(-0.289743\pi\)
0.613545 + 0.789659i \(0.289743\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 8.92820 1.39435 0.697176 0.716900i \(-0.254440\pi\)
0.697176 + 0.716900i \(0.254440\pi\)
\(42\) 0 0
\(43\) −2.92820 −0.446547 −0.223273 0.974756i \(-0.571674\pi\)
−0.223273 + 0.974756i \(0.571674\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −1.46410 −0.215870
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 3.46410 0.480384
\(53\) −12.9282 −1.77583 −0.887913 0.460012i \(-0.847845\pi\)
−0.887913 + 0.460012i \(0.847845\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 2.92820 0.381220 0.190610 0.981666i \(-0.438953\pi\)
0.190610 + 0.981666i \(0.438953\pi\)
\(60\) 0 0
\(61\) −4.92820 −0.630992 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(62\) −6.92820 −0.879883
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.46410 0.429669
\(66\) 0 0
\(67\) 9.46410 1.15622 0.578112 0.815957i \(-0.303790\pi\)
0.578112 + 0.815957i \(0.303790\pi\)
\(68\) 3.46410 0.420084
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) −0.928203 −0.108638 −0.0543190 0.998524i \(-0.517299\pi\)
−0.0543190 + 0.998524i \(0.517299\pi\)
\(74\) −7.46410 −0.867684
\(75\) 0 0
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −8.92820 −0.985955
\(83\) −6.92820 −0.760469 −0.380235 0.924890i \(-0.624157\pi\)
−0.380235 + 0.924890i \(0.624157\pi\)
\(84\) 0 0
\(85\) 3.46410 0.375735
\(86\) 2.92820 0.315756
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −3.46410 −0.367194 −0.183597 0.983002i \(-0.558774\pi\)
−0.183597 + 0.983002i \(0.558774\pi\)
\(90\) 0 0
\(91\) 3.46410 0.363137
\(92\) 1.46410 0.152643
\(93\) 0 0
\(94\) 6.92820 0.714590
\(95\) 0 0
\(96\) 0 0
\(97\) −11.8564 −1.20384 −0.601918 0.798558i \(-0.705597\pi\)
−0.601918 + 0.798558i \(0.705597\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −11.8564 −1.17976 −0.589878 0.807492i \(-0.700824\pi\)
−0.589878 + 0.807492i \(0.700824\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) −3.46410 −0.339683
\(105\) 0 0
\(106\) 12.9282 1.25570
\(107\) 6.92820 0.669775 0.334887 0.942258i \(-0.391302\pi\)
0.334887 + 0.942258i \(0.391302\pi\)
\(108\) 0 0
\(109\) 14.3923 1.37853 0.689266 0.724508i \(-0.257933\pi\)
0.689266 + 0.724508i \(0.257933\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −11.4641 −1.07845 −0.539226 0.842161i \(-0.681283\pi\)
−0.539226 + 0.842161i \(0.681283\pi\)
\(114\) 0 0
\(115\) 1.46410 0.136528
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) −2.92820 −0.269563
\(119\) 3.46410 0.317554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 4.92820 0.446179
\(123\) 0 0
\(124\) 6.92820 0.622171
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 10.9282 0.969721 0.484861 0.874591i \(-0.338870\pi\)
0.484861 + 0.874591i \(0.338870\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −3.46410 −0.303822
\(131\) 1.07180 0.0936433 0.0468217 0.998903i \(-0.485091\pi\)
0.0468217 + 0.998903i \(0.485091\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −9.46410 −0.817574
\(135\) 0 0
\(136\) −3.46410 −0.297044
\(137\) 4.53590 0.387528 0.193764 0.981048i \(-0.437930\pi\)
0.193764 + 0.981048i \(0.437930\pi\)
\(138\) 0 0
\(139\) −2.92820 −0.248367 −0.124183 0.992259i \(-0.539631\pi\)
−0.124183 + 0.992259i \(0.539631\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −4.00000 −0.335673
\(143\) −3.46410 −0.289683
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0.928203 0.0768186
\(147\) 0 0
\(148\) 7.46410 0.613545
\(149\) −12.9282 −1.05912 −0.529560 0.848273i \(-0.677643\pi\)
−0.529560 + 0.848273i \(0.677643\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) 6.92820 0.556487
\(156\) 0 0
\(157\) −18.7846 −1.49918 −0.749588 0.661905i \(-0.769748\pi\)
−0.749588 + 0.661905i \(0.769748\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 1.46410 0.115387
\(162\) 0 0
\(163\) 9.46410 0.741286 0.370643 0.928775i \(-0.379137\pi\)
0.370643 + 0.928775i \(0.379137\pi\)
\(164\) 8.92820 0.697176
\(165\) 0 0
\(166\) 6.92820 0.537733
\(167\) 9.46410 0.732354 0.366177 0.930545i \(-0.380666\pi\)
0.366177 + 0.930545i \(0.380666\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) −3.46410 −0.265684
\(171\) 0 0
\(172\) −2.92820 −0.223273
\(173\) 8.92820 0.678799 0.339399 0.940642i \(-0.389776\pi\)
0.339399 + 0.940642i \(0.389776\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 3.46410 0.259645
\(179\) 20.7846 1.55351 0.776757 0.629800i \(-0.216863\pi\)
0.776757 + 0.629800i \(0.216863\pi\)
\(180\) 0 0
\(181\) −16.5359 −1.22910 −0.614552 0.788876i \(-0.710663\pi\)
−0.614552 + 0.788876i \(0.710663\pi\)
\(182\) −3.46410 −0.256776
\(183\) 0 0
\(184\) −1.46410 −0.107935
\(185\) 7.46410 0.548772
\(186\) 0 0
\(187\) −3.46410 −0.253320
\(188\) −6.92820 −0.505291
\(189\) 0 0
\(190\) 0 0
\(191\) −14.9282 −1.08017 −0.540083 0.841611i \(-0.681607\pi\)
−0.540083 + 0.841611i \(0.681607\pi\)
\(192\) 0 0
\(193\) 11.0718 0.796965 0.398483 0.917176i \(-0.369537\pi\)
0.398483 + 0.917176i \(0.369537\pi\)
\(194\) 11.8564 0.851240
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −6.92820 −0.491127 −0.245564 0.969380i \(-0.578973\pi\)
−0.245564 + 0.969380i \(0.578973\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 11.8564 0.834214
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) 8.92820 0.623573
\(206\) −16.0000 −1.11477
\(207\) 0 0
\(208\) 3.46410 0.240192
\(209\) 0 0
\(210\) 0 0
\(211\) −23.3205 −1.60545 −0.802725 0.596349i \(-0.796617\pi\)
−0.802725 + 0.596349i \(0.796617\pi\)
\(212\) −12.9282 −0.887913
\(213\) 0 0
\(214\) −6.92820 −0.473602
\(215\) −2.92820 −0.199702
\(216\) 0 0
\(217\) 6.92820 0.470317
\(218\) −14.3923 −0.974770
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 24.7846 1.65970 0.829850 0.557986i \(-0.188426\pi\)
0.829850 + 0.557986i \(0.188426\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 11.4641 0.762581
\(227\) −6.92820 −0.459841 −0.229920 0.973209i \(-0.573847\pi\)
−0.229920 + 0.973209i \(0.573847\pi\)
\(228\) 0 0
\(229\) 2.39230 0.158088 0.0790440 0.996871i \(-0.474813\pi\)
0.0790440 + 0.996871i \(0.474813\pi\)
\(230\) −1.46410 −0.0965400
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) −2.00000 −0.131024 −0.0655122 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) 0 0
\(235\) −6.92820 −0.451946
\(236\) 2.92820 0.190610
\(237\) 0 0
\(238\) −3.46410 −0.224544
\(239\) −4.39230 −0.284115 −0.142057 0.989858i \(-0.545372\pi\)
−0.142057 + 0.989858i \(0.545372\pi\)
\(240\) 0 0
\(241\) −7.07180 −0.455534 −0.227767 0.973716i \(-0.573143\pi\)
−0.227767 + 0.973716i \(0.573143\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) −4.92820 −0.315496
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) −6.92820 −0.439941
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 5.07180 0.320129 0.160064 0.987107i \(-0.448830\pi\)
0.160064 + 0.987107i \(0.448830\pi\)
\(252\) 0 0
\(253\) −1.46410 −0.0920473
\(254\) −10.9282 −0.685696
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 16.9282 1.05595 0.527976 0.849259i \(-0.322951\pi\)
0.527976 + 0.849259i \(0.322951\pi\)
\(258\) 0 0
\(259\) 7.46410 0.463797
\(260\) 3.46410 0.214834
\(261\) 0 0
\(262\) −1.07180 −0.0662158
\(263\) 16.7846 1.03498 0.517492 0.855688i \(-0.326866\pi\)
0.517492 + 0.855688i \(0.326866\pi\)
\(264\) 0 0
\(265\) −12.9282 −0.794173
\(266\) 0 0
\(267\) 0 0
\(268\) 9.46410 0.578112
\(269\) 16.9282 1.03213 0.516065 0.856549i \(-0.327396\pi\)
0.516065 + 0.856549i \(0.327396\pi\)
\(270\) 0 0
\(271\) −5.85641 −0.355751 −0.177876 0.984053i \(-0.556922\pi\)
−0.177876 + 0.984053i \(0.556922\pi\)
\(272\) 3.46410 0.210042
\(273\) 0 0
\(274\) −4.53590 −0.274024
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 2.92820 0.175622
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) 14.3923 0.858573 0.429286 0.903168i \(-0.358765\pi\)
0.429286 + 0.903168i \(0.358765\pi\)
\(282\) 0 0
\(283\) 1.46410 0.0870318 0.0435159 0.999053i \(-0.486144\pi\)
0.0435159 + 0.999053i \(0.486144\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) 3.46410 0.204837
\(287\) 8.92820 0.527015
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 2.00000 0.117444
\(291\) 0 0
\(292\) −0.928203 −0.0543190
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 2.92820 0.170487
\(296\) −7.46410 −0.433842
\(297\) 0 0
\(298\) 12.9282 0.748911
\(299\) 5.07180 0.293310
\(300\) 0 0
\(301\) −2.92820 −0.168779
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) 0 0
\(305\) −4.92820 −0.282188
\(306\) 0 0
\(307\) 7.32051 0.417803 0.208902 0.977937i \(-0.433011\pi\)
0.208902 + 0.977937i \(0.433011\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) −6.92820 −0.393496
\(311\) −6.53590 −0.370617 −0.185308 0.982680i \(-0.559328\pi\)
−0.185308 + 0.982680i \(0.559328\pi\)
\(312\) 0 0
\(313\) 21.7128 1.22728 0.613640 0.789586i \(-0.289704\pi\)
0.613640 + 0.789586i \(0.289704\pi\)
\(314\) 18.7846 1.06008
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 3.07180 0.172529 0.0862646 0.996272i \(-0.472507\pi\)
0.0862646 + 0.996272i \(0.472507\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −1.46410 −0.0815912
\(323\) 0 0
\(324\) 0 0
\(325\) 3.46410 0.192154
\(326\) −9.46410 −0.524168
\(327\) 0 0
\(328\) −8.92820 −0.492978
\(329\) −6.92820 −0.381964
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −6.92820 −0.380235
\(333\) 0 0
\(334\) −9.46410 −0.517853
\(335\) 9.46410 0.517079
\(336\) 0 0
\(337\) 22.7846 1.24116 0.620578 0.784144i \(-0.286898\pi\)
0.620578 + 0.784144i \(0.286898\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) 3.46410 0.187867
\(341\) −6.92820 −0.375183
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 2.92820 0.157878
\(345\) 0 0
\(346\) −8.92820 −0.479983
\(347\) 1.07180 0.0575371 0.0287685 0.999586i \(-0.490841\pi\)
0.0287685 + 0.999586i \(0.490841\pi\)
\(348\) 0 0
\(349\) −28.9282 −1.54849 −0.774246 0.632885i \(-0.781870\pi\)
−0.774246 + 0.632885i \(0.781870\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −20.9282 −1.11390 −0.556948 0.830547i \(-0.688028\pi\)
−0.556948 + 0.830547i \(0.688028\pi\)
\(354\) 0 0
\(355\) 4.00000 0.212298
\(356\) −3.46410 −0.183597
\(357\) 0 0
\(358\) −20.7846 −1.09850
\(359\) −9.46410 −0.499496 −0.249748 0.968311i \(-0.580348\pi\)
−0.249748 + 0.968311i \(0.580348\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 16.5359 0.869108
\(363\) 0 0
\(364\) 3.46410 0.181568
\(365\) −0.928203 −0.0485844
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 1.46410 0.0763216
\(369\) 0 0
\(370\) −7.46410 −0.388040
\(371\) −12.9282 −0.671199
\(372\) 0 0
\(373\) −4.92820 −0.255173 −0.127586 0.991827i \(-0.540723\pi\)
−0.127586 + 0.991827i \(0.540723\pi\)
\(374\) 3.46410 0.179124
\(375\) 0 0
\(376\) 6.92820 0.357295
\(377\) −6.92820 −0.356821
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 14.9282 0.763793
\(383\) 12.7846 0.653263 0.326632 0.945152i \(-0.394086\pi\)
0.326632 + 0.945152i \(0.394086\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) −11.0718 −0.563540
\(387\) 0 0
\(388\) −11.8564 −0.601918
\(389\) −21.7128 −1.10088 −0.550442 0.834874i \(-0.685541\pi\)
−0.550442 + 0.834874i \(0.685541\pi\)
\(390\) 0 0
\(391\) 5.07180 0.256492
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −10.0000 −0.501886 −0.250943 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) 6.92820 0.347279
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 25.7128 1.28404 0.642018 0.766689i \(-0.278097\pi\)
0.642018 + 0.766689i \(0.278097\pi\)
\(402\) 0 0
\(403\) 24.0000 1.19553
\(404\) −11.8564 −0.589878
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) −7.46410 −0.369982
\(408\) 0 0
\(409\) −12.9282 −0.639259 −0.319629 0.947543i \(-0.603558\pi\)
−0.319629 + 0.947543i \(0.603558\pi\)
\(410\) −8.92820 −0.440933
\(411\) 0 0
\(412\) 16.0000 0.788263
\(413\) 2.92820 0.144087
\(414\) 0 0
\(415\) −6.92820 −0.340092
\(416\) −3.46410 −0.169842
\(417\) 0 0
\(418\) 0 0
\(419\) −13.8564 −0.676930 −0.338465 0.940979i \(-0.609908\pi\)
−0.338465 + 0.940979i \(0.609908\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 23.3205 1.13522
\(423\) 0 0
\(424\) 12.9282 0.627849
\(425\) 3.46410 0.168034
\(426\) 0 0
\(427\) −4.92820 −0.238492
\(428\) 6.92820 0.334887
\(429\) 0 0
\(430\) 2.92820 0.141210
\(431\) 23.3205 1.12331 0.561655 0.827372i \(-0.310165\pi\)
0.561655 + 0.827372i \(0.310165\pi\)
\(432\) 0 0
\(433\) −17.7128 −0.851223 −0.425612 0.904906i \(-0.639941\pi\)
−0.425612 + 0.904906i \(0.639941\pi\)
\(434\) −6.92820 −0.332564
\(435\) 0 0
\(436\) 14.3923 0.689266
\(437\) 0 0
\(438\) 0 0
\(439\) −16.7846 −0.801086 −0.400543 0.916278i \(-0.631178\pi\)
−0.400543 + 0.916278i \(0.631178\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) −26.9282 −1.27940 −0.639699 0.768626i \(-0.720941\pi\)
−0.639699 + 0.768626i \(0.720941\pi\)
\(444\) 0 0
\(445\) −3.46410 −0.164214
\(446\) −24.7846 −1.17359
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) −8.92820 −0.420413
\(452\) −11.4641 −0.539226
\(453\) 0 0
\(454\) 6.92820 0.325157
\(455\) 3.46410 0.162400
\(456\) 0 0
\(457\) 19.0718 0.892141 0.446071 0.894998i \(-0.352823\pi\)
0.446071 + 0.894998i \(0.352823\pi\)
\(458\) −2.39230 −0.111785
\(459\) 0 0
\(460\) 1.46410 0.0682641
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 20.7846 0.965943 0.482971 0.875636i \(-0.339558\pi\)
0.482971 + 0.875636i \(0.339558\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 2.00000 0.0926482
\(467\) 30.9282 1.43119 0.715593 0.698517i \(-0.246156\pi\)
0.715593 + 0.698517i \(0.246156\pi\)
\(468\) 0 0
\(469\) 9.46410 0.437012
\(470\) 6.92820 0.319574
\(471\) 0 0
\(472\) −2.92820 −0.134781
\(473\) 2.92820 0.134639
\(474\) 0 0
\(475\) 0 0
\(476\) 3.46410 0.158777
\(477\) 0 0
\(478\) 4.39230 0.200899
\(479\) −13.8564 −0.633115 −0.316558 0.948573i \(-0.602527\pi\)
−0.316558 + 0.948573i \(0.602527\pi\)
\(480\) 0 0
\(481\) 25.8564 1.17895
\(482\) 7.07180 0.322112
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −11.8564 −0.538372
\(486\) 0 0
\(487\) 1.07180 0.0485677 0.0242839 0.999705i \(-0.492269\pi\)
0.0242839 + 0.999705i \(0.492269\pi\)
\(488\) 4.92820 0.223089
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) −1.85641 −0.0837785 −0.0418892 0.999122i \(-0.513338\pi\)
−0.0418892 + 0.999122i \(0.513338\pi\)
\(492\) 0 0
\(493\) −6.92820 −0.312031
\(494\) 0 0
\(495\) 0 0
\(496\) 6.92820 0.311086
\(497\) 4.00000 0.179425
\(498\) 0 0
\(499\) 17.0718 0.764239 0.382119 0.924113i \(-0.375194\pi\)
0.382119 + 0.924113i \(0.375194\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −5.07180 −0.226365
\(503\) −12.3923 −0.552546 −0.276273 0.961079i \(-0.589099\pi\)
−0.276273 + 0.961079i \(0.589099\pi\)
\(504\) 0 0
\(505\) −11.8564 −0.527603
\(506\) 1.46410 0.0650873
\(507\) 0 0
\(508\) 10.9282 0.484861
\(509\) 32.9282 1.45952 0.729758 0.683705i \(-0.239633\pi\)
0.729758 + 0.683705i \(0.239633\pi\)
\(510\) 0 0
\(511\) −0.928203 −0.0410613
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −16.9282 −0.746671
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) 6.92820 0.304702
\(518\) −7.46410 −0.327954
\(519\) 0 0
\(520\) −3.46410 −0.151911
\(521\) −27.4641 −1.20322 −0.601612 0.798788i \(-0.705475\pi\)
−0.601612 + 0.798788i \(0.705475\pi\)
\(522\) 0 0
\(523\) 44.3923 1.94114 0.970570 0.240819i \(-0.0774161\pi\)
0.970570 + 0.240819i \(0.0774161\pi\)
\(524\) 1.07180 0.0468217
\(525\) 0 0
\(526\) −16.7846 −0.731844
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) −20.8564 −0.906800
\(530\) 12.9282 0.561565
\(531\) 0 0
\(532\) 0 0
\(533\) 30.9282 1.33965
\(534\) 0 0
\(535\) 6.92820 0.299532
\(536\) −9.46410 −0.408787
\(537\) 0 0
\(538\) −16.9282 −0.729827
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 9.32051 0.400720 0.200360 0.979722i \(-0.435789\pi\)
0.200360 + 0.979722i \(0.435789\pi\)
\(542\) 5.85641 0.251554
\(543\) 0 0
\(544\) −3.46410 −0.148522
\(545\) 14.3923 0.616499
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 4.53590 0.193764
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 0 0
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −2.92820 −0.124183
\(557\) 31.8564 1.34980 0.674900 0.737910i \(-0.264187\pi\)
0.674900 + 0.737910i \(0.264187\pi\)
\(558\) 0 0
\(559\) −10.1436 −0.429028
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −14.3923 −0.607103
\(563\) 31.7128 1.33654 0.668268 0.743921i \(-0.267036\pi\)
0.668268 + 0.743921i \(0.267036\pi\)
\(564\) 0 0
\(565\) −11.4641 −0.482298
\(566\) −1.46410 −0.0615408
\(567\) 0 0
\(568\) −4.00000 −0.167836
\(569\) 39.1769 1.64238 0.821191 0.570654i \(-0.193310\pi\)
0.821191 + 0.570654i \(0.193310\pi\)
\(570\) 0 0
\(571\) 23.3205 0.975933 0.487966 0.872862i \(-0.337739\pi\)
0.487966 + 0.872862i \(0.337739\pi\)
\(572\) −3.46410 −0.144841
\(573\) 0 0
\(574\) −8.92820 −0.372656
\(575\) 1.46410 0.0610573
\(576\) 0 0
\(577\) 45.7128 1.90305 0.951525 0.307572i \(-0.0995167\pi\)
0.951525 + 0.307572i \(0.0995167\pi\)
\(578\) 5.00000 0.207973
\(579\) 0 0
\(580\) −2.00000 −0.0830455
\(581\) −6.92820 −0.287430
\(582\) 0 0
\(583\) 12.9282 0.535431
\(584\) 0.928203 0.0384093
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 20.7846 0.857873 0.428936 0.903335i \(-0.358888\pi\)
0.428936 + 0.903335i \(0.358888\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −2.92820 −0.120552
\(591\) 0 0
\(592\) 7.46410 0.306773
\(593\) −27.1769 −1.11602 −0.558011 0.829834i \(-0.688435\pi\)
−0.558011 + 0.829834i \(0.688435\pi\)
\(594\) 0 0
\(595\) 3.46410 0.142014
\(596\) −12.9282 −0.529560
\(597\) 0 0
\(598\) −5.07180 −0.207401
\(599\) 3.21539 0.131377 0.0656886 0.997840i \(-0.479076\pi\)
0.0656886 + 0.997840i \(0.479076\pi\)
\(600\) 0 0
\(601\) −23.0718 −0.941118 −0.470559 0.882368i \(-0.655948\pi\)
−0.470559 + 0.882368i \(0.655948\pi\)
\(602\) 2.92820 0.119345
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 4.92820 0.199537
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −7.32051 −0.295432
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 2.39230 0.0963106 0.0481553 0.998840i \(-0.484666\pi\)
0.0481553 + 0.998840i \(0.484666\pi\)
\(618\) 0 0
\(619\) 15.3205 0.615783 0.307892 0.951421i \(-0.400377\pi\)
0.307892 + 0.951421i \(0.400377\pi\)
\(620\) 6.92820 0.278243
\(621\) 0 0
\(622\) 6.53590 0.262066
\(623\) −3.46410 −0.138786
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −21.7128 −0.867819
\(627\) 0 0
\(628\) −18.7846 −0.749588
\(629\) 25.8564 1.03096
\(630\) 0 0
\(631\) 16.7846 0.668185 0.334092 0.942540i \(-0.391570\pi\)
0.334092 + 0.942540i \(0.391570\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) −3.07180 −0.121997
\(635\) 10.9282 0.433673
\(636\) 0 0
\(637\) 3.46410 0.137253
\(638\) −2.00000 −0.0791808
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 0 0
\(643\) −23.7128 −0.935142 −0.467571 0.883956i \(-0.654871\pi\)
−0.467571 + 0.883956i \(0.654871\pi\)
\(644\) 1.46410 0.0576937
\(645\) 0 0
\(646\) 0 0
\(647\) 22.9282 0.901401 0.450700 0.892675i \(-0.351174\pi\)
0.450700 + 0.892675i \(0.351174\pi\)
\(648\) 0 0
\(649\) −2.92820 −0.114942
\(650\) −3.46410 −0.135873
\(651\) 0 0
\(652\) 9.46410 0.370643
\(653\) −31.8564 −1.24664 −0.623319 0.781968i \(-0.714216\pi\)
−0.623319 + 0.781968i \(0.714216\pi\)
\(654\) 0 0
\(655\) 1.07180 0.0418786
\(656\) 8.92820 0.348588
\(657\) 0 0
\(658\) 6.92820 0.270089
\(659\) 39.7128 1.54699 0.773496 0.633801i \(-0.218506\pi\)
0.773496 + 0.633801i \(0.218506\pi\)
\(660\) 0 0
\(661\) 1.60770 0.0625321 0.0312660 0.999511i \(-0.490046\pi\)
0.0312660 + 0.999511i \(0.490046\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) 6.92820 0.268866
\(665\) 0 0
\(666\) 0 0
\(667\) −2.92820 −0.113380
\(668\) 9.46410 0.366177
\(669\) 0 0
\(670\) −9.46410 −0.365630
\(671\) 4.92820 0.190251
\(672\) 0 0
\(673\) −15.0718 −0.580975 −0.290488 0.956879i \(-0.593817\pi\)
−0.290488 + 0.956879i \(0.593817\pi\)
\(674\) −22.7846 −0.877630
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) 40.9282 1.57300 0.786499 0.617591i \(-0.211891\pi\)
0.786499 + 0.617591i \(0.211891\pi\)
\(678\) 0 0
\(679\) −11.8564 −0.455007
\(680\) −3.46410 −0.132842
\(681\) 0 0
\(682\) 6.92820 0.265295
\(683\) −26.9282 −1.03038 −0.515190 0.857076i \(-0.672278\pi\)
−0.515190 + 0.857076i \(0.672278\pi\)
\(684\) 0 0
\(685\) 4.53590 0.173308
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −2.92820 −0.111637
\(689\) −44.7846 −1.70616
\(690\) 0 0
\(691\) 3.60770 0.137243 0.0686216 0.997643i \(-0.478140\pi\)
0.0686216 + 0.997643i \(0.478140\pi\)
\(692\) 8.92820 0.339399
\(693\) 0 0
\(694\) −1.07180 −0.0406848
\(695\) −2.92820 −0.111073
\(696\) 0 0
\(697\) 30.9282 1.17149
\(698\) 28.9282 1.09495
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 20.9282 0.787643
\(707\) −11.8564 −0.445906
\(708\) 0 0
\(709\) −7.85641 −0.295054 −0.147527 0.989058i \(-0.547131\pi\)
−0.147527 + 0.989058i \(0.547131\pi\)
\(710\) −4.00000 −0.150117
\(711\) 0 0
\(712\) 3.46410 0.129823
\(713\) 10.1436 0.379881
\(714\) 0 0
\(715\) −3.46410 −0.129550
\(716\) 20.7846 0.776757
\(717\) 0 0
\(718\) 9.46410 0.353197
\(719\) 4.39230 0.163805 0.0819027 0.996640i \(-0.473900\pi\)
0.0819027 + 0.996640i \(0.473900\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 19.0000 0.707107
\(723\) 0 0
\(724\) −16.5359 −0.614552
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 13.0718 0.484806 0.242403 0.970176i \(-0.422064\pi\)
0.242403 + 0.970176i \(0.422064\pi\)
\(728\) −3.46410 −0.128388
\(729\) 0 0
\(730\) 0.928203 0.0343543
\(731\) −10.1436 −0.375174
\(732\) 0 0
\(733\) −24.2487 −0.895647 −0.447823 0.894122i \(-0.647801\pi\)
−0.447823 + 0.894122i \(0.647801\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) −1.46410 −0.0539675
\(737\) −9.46410 −0.348615
\(738\) 0 0
\(739\) 15.3205 0.563574 0.281787 0.959477i \(-0.409073\pi\)
0.281787 + 0.959477i \(0.409073\pi\)
\(740\) 7.46410 0.274386
\(741\) 0 0
\(742\) 12.9282 0.474609
\(743\) −13.8564 −0.508342 −0.254171 0.967159i \(-0.581803\pi\)
−0.254171 + 0.967159i \(0.581803\pi\)
\(744\) 0 0
\(745\) −12.9282 −0.473653
\(746\) 4.92820 0.180434
\(747\) 0 0
\(748\) −3.46410 −0.126660
\(749\) 6.92820 0.253151
\(750\) 0 0
\(751\) −0.784610 −0.0286308 −0.0143154 0.999898i \(-0.504557\pi\)
−0.0143154 + 0.999898i \(0.504557\pi\)
\(752\) −6.92820 −0.252646
\(753\) 0 0
\(754\) 6.92820 0.252310
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −50.1051 −1.82110 −0.910551 0.413397i \(-0.864342\pi\)
−0.910551 + 0.413397i \(0.864342\pi\)
\(758\) 28.0000 1.01701
\(759\) 0 0
\(760\) 0 0
\(761\) 38.7846 1.40594 0.702971 0.711219i \(-0.251857\pi\)
0.702971 + 0.711219i \(0.251857\pi\)
\(762\) 0 0
\(763\) 14.3923 0.521036
\(764\) −14.9282 −0.540083
\(765\) 0 0
\(766\) −12.7846 −0.461927
\(767\) 10.1436 0.366264
\(768\) 0 0
\(769\) 38.7846 1.39861 0.699304 0.714824i \(-0.253493\pi\)
0.699304 + 0.714824i \(0.253493\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) 11.0718 0.398483
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 0 0
\(775\) 6.92820 0.248868
\(776\) 11.8564 0.425620
\(777\) 0 0
\(778\) 21.7128 0.778442
\(779\) 0 0
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) −5.07180 −0.181367
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −18.7846 −0.670451
\(786\) 0 0
\(787\) 41.4641 1.47804 0.739018 0.673686i \(-0.235290\pi\)
0.739018 + 0.673686i \(0.235290\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) −11.4641 −0.407617
\(792\) 0 0
\(793\) −17.0718 −0.606237
\(794\) 10.0000 0.354887
\(795\) 0 0
\(796\) −6.92820 −0.245564
\(797\) −31.8564 −1.12841 −0.564206 0.825634i \(-0.690818\pi\)
−0.564206 + 0.825634i \(0.690818\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −25.7128 −0.907951
\(803\) 0.928203 0.0327556
\(804\) 0 0
\(805\) 1.46410 0.0516028
\(806\) −24.0000 −0.845364
\(807\) 0 0
\(808\) 11.8564 0.417107
\(809\) 11.4641 0.403056 0.201528 0.979483i \(-0.435409\pi\)
0.201528 + 0.979483i \(0.435409\pi\)
\(810\) 0 0
\(811\) 13.8564 0.486564 0.243282 0.969956i \(-0.421776\pi\)
0.243282 + 0.969956i \(0.421776\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 0 0
\(814\) 7.46410 0.261617
\(815\) 9.46410 0.331513
\(816\) 0 0
\(817\) 0 0
\(818\) 12.9282 0.452024
\(819\) 0 0
\(820\) 8.92820 0.311786
\(821\) −12.1436 −0.423814 −0.211907 0.977290i \(-0.567967\pi\)
−0.211907 + 0.977290i \(0.567967\pi\)
\(822\) 0 0
\(823\) 49.8564 1.73789 0.868943 0.494913i \(-0.164800\pi\)
0.868943 + 0.494913i \(0.164800\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) −2.92820 −0.101885
\(827\) −20.7846 −0.722752 −0.361376 0.932420i \(-0.617693\pi\)
−0.361376 + 0.932420i \(0.617693\pi\)
\(828\) 0 0
\(829\) 51.1769 1.77745 0.888724 0.458443i \(-0.151593\pi\)
0.888724 + 0.458443i \(0.151593\pi\)
\(830\) 6.92820 0.240481
\(831\) 0 0
\(832\) 3.46410 0.120096
\(833\) 3.46410 0.120024
\(834\) 0 0
\(835\) 9.46410 0.327519
\(836\) 0 0
\(837\) 0 0
\(838\) 13.8564 0.478662
\(839\) −30.5359 −1.05422 −0.527108 0.849798i \(-0.676724\pi\)
−0.527108 + 0.849798i \(0.676724\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) 0 0
\(844\) −23.3205 −0.802725
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) −12.9282 −0.443956
\(849\) 0 0
\(850\) −3.46410 −0.118818
\(851\) 10.9282 0.374614
\(852\) 0 0
\(853\) 15.1769 0.519648 0.259824 0.965656i \(-0.416336\pi\)
0.259824 + 0.965656i \(0.416336\pi\)
\(854\) 4.92820 0.168640
\(855\) 0 0
\(856\) −6.92820 −0.236801
\(857\) 0.535898 0.0183059 0.00915297 0.999958i \(-0.497086\pi\)
0.00915297 + 0.999958i \(0.497086\pi\)
\(858\) 0 0
\(859\) 20.3923 0.695776 0.347888 0.937536i \(-0.386899\pi\)
0.347888 + 0.937536i \(0.386899\pi\)
\(860\) −2.92820 −0.0998509
\(861\) 0 0
\(862\) −23.3205 −0.794300
\(863\) −42.2487 −1.43816 −0.719081 0.694926i \(-0.755437\pi\)
−0.719081 + 0.694926i \(0.755437\pi\)
\(864\) 0 0
\(865\) 8.92820 0.303568
\(866\) 17.7128 0.601906
\(867\) 0 0
\(868\) 6.92820 0.235159
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) 32.7846 1.11086
\(872\) −14.3923 −0.487385
\(873\) 0 0
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −7.85641 −0.265292 −0.132646 0.991163i \(-0.542347\pi\)
−0.132646 + 0.991163i \(0.542347\pi\)
\(878\) 16.7846 0.566453
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) −17.3205 −0.583543 −0.291771 0.956488i \(-0.594245\pi\)
−0.291771 + 0.956488i \(0.594245\pi\)
\(882\) 0 0
\(883\) −48.1051 −1.61887 −0.809433 0.587212i \(-0.800225\pi\)
−0.809433 + 0.587212i \(0.800225\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) 26.9282 0.904671
\(887\) 16.6795 0.560043 0.280021 0.959994i \(-0.409658\pi\)
0.280021 + 0.959994i \(0.409658\pi\)
\(888\) 0 0
\(889\) 10.9282 0.366520
\(890\) 3.46410 0.116117
\(891\) 0 0
\(892\) 24.7846 0.829850
\(893\) 0 0
\(894\) 0 0
\(895\) 20.7846 0.694753
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −22.0000 −0.734150
\(899\) −13.8564 −0.462137
\(900\) 0 0
\(901\) −44.7846 −1.49199
\(902\) 8.92820 0.297277
\(903\) 0 0
\(904\) 11.4641 0.381290
\(905\) −16.5359 −0.549672
\(906\) 0 0
\(907\) −21.1769 −0.703168 −0.351584 0.936156i \(-0.614357\pi\)
−0.351584 + 0.936156i \(0.614357\pi\)
\(908\) −6.92820 −0.229920
\(909\) 0 0
\(910\) −3.46410 −0.114834
\(911\) 4.00000 0.132526 0.0662630 0.997802i \(-0.478892\pi\)
0.0662630 + 0.997802i \(0.478892\pi\)
\(912\) 0 0
\(913\) 6.92820 0.229290
\(914\) −19.0718 −0.630839
\(915\) 0 0
\(916\) 2.39230 0.0790440
\(917\) 1.07180 0.0353938
\(918\) 0 0
\(919\) 45.0718 1.48678 0.743391 0.668857i \(-0.233216\pi\)
0.743391 + 0.668857i \(0.233216\pi\)
\(920\) −1.46410 −0.0482700
\(921\) 0 0
\(922\) −18.0000 −0.592798
\(923\) 13.8564 0.456089
\(924\) 0 0
\(925\) 7.46410 0.245418
\(926\) −20.7846 −0.683025
\(927\) 0 0
\(928\) 2.00000 0.0656532
\(929\) 25.6077 0.840161 0.420081 0.907487i \(-0.362002\pi\)
0.420081 + 0.907487i \(0.362002\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −2.00000 −0.0655122
\(933\) 0 0
\(934\) −30.9282 −1.01200
\(935\) −3.46410 −0.113288
\(936\) 0 0
\(937\) −3.85641 −0.125983 −0.0629917 0.998014i \(-0.520064\pi\)
−0.0629917 + 0.998014i \(0.520064\pi\)
\(938\) −9.46410 −0.309014
\(939\) 0 0
\(940\) −6.92820 −0.225973
\(941\) −22.0000 −0.717180 −0.358590 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) 0 0
\(943\) 13.0718 0.425676
\(944\) 2.92820 0.0953049
\(945\) 0 0
\(946\) −2.92820 −0.0952041
\(947\) 10.1436 0.329622 0.164811 0.986325i \(-0.447299\pi\)
0.164811 + 0.986325i \(0.447299\pi\)
\(948\) 0 0
\(949\) −3.21539 −0.104376
\(950\) 0 0
\(951\) 0 0
\(952\) −3.46410 −0.112272
\(953\) 28.6410 0.927774 0.463887 0.885895i \(-0.346454\pi\)
0.463887 + 0.885895i \(0.346454\pi\)
\(954\) 0 0
\(955\) −14.9282 −0.483065
\(956\) −4.39230 −0.142057
\(957\) 0 0
\(958\) 13.8564 0.447680
\(959\) 4.53590 0.146472
\(960\) 0 0
\(961\) 17.0000 0.548387
\(962\) −25.8564 −0.833644
\(963\) 0 0
\(964\) −7.07180 −0.227767
\(965\) 11.0718 0.356414
\(966\) 0 0
\(967\) −26.9282 −0.865953 −0.432976 0.901405i \(-0.642537\pi\)
−0.432976 + 0.901405i \(0.642537\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 11.8564 0.380686
\(971\) −35.7128 −1.14608 −0.573039 0.819528i \(-0.694236\pi\)
−0.573039 + 0.819528i \(0.694236\pi\)
\(972\) 0 0
\(973\) −2.92820 −0.0938739
\(974\) −1.07180 −0.0343426
\(975\) 0 0
\(976\) −4.92820 −0.157748
\(977\) 37.3205 1.19399 0.596994 0.802245i \(-0.296361\pi\)
0.596994 + 0.802245i \(0.296361\pi\)
\(978\) 0 0
\(979\) 3.46410 0.110713
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 1.85641 0.0592403
\(983\) −3.21539 −0.102555 −0.0512775 0.998684i \(-0.516329\pi\)
−0.0512775 + 0.998684i \(0.516329\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 6.92820 0.220639
\(987\) 0 0
\(988\) 0 0
\(989\) −4.28719 −0.136325
\(990\) 0 0
\(991\) −40.7846 −1.29557 −0.647783 0.761825i \(-0.724304\pi\)
−0.647783 + 0.761825i \(0.724304\pi\)
\(992\) −6.92820 −0.219971
\(993\) 0 0
\(994\) −4.00000 −0.126872
\(995\) −6.92820 −0.219639
\(996\) 0 0
\(997\) 18.6795 0.591585 0.295793 0.955252i \(-0.404416\pi\)
0.295793 + 0.955252i \(0.404416\pi\)
\(998\) −17.0718 −0.540398
\(999\) 0 0
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 6930.2.a.bt.1.2 2
3.2 odd 2 2310.2.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.bb.1.2 2 3.2 odd 2
6930.2.a.bt.1.2 2 1.1 even 1 trivial