Properties

Label 2070.2.a.x.1.2
Level $2070$
Weight $2$
Character 2070.1
Self dual yes
Analytic conductor $16.529$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \(x^{2} - x - 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.79129\) of defining polynomial
Character \(\chi\) \(=\) 2070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.79129 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.79129 q^{7} +1.00000 q^{8} +1.00000 q^{10} -3.79129 q^{11} +1.20871 q^{13} +2.79129 q^{14} +1.00000 q^{16} +3.79129 q^{17} +1.20871 q^{19} +1.00000 q^{20} -3.79129 q^{22} -1.00000 q^{23} +1.00000 q^{25} +1.20871 q^{26} +2.79129 q^{28} +1.58258 q^{29} +10.3739 q^{31} +1.00000 q^{32} +3.79129 q^{34} +2.79129 q^{35} -4.00000 q^{37} +1.20871 q^{38} +1.00000 q^{40} +2.20871 q^{41} -7.16515 q^{43} -3.79129 q^{44} -1.00000 q^{46} +13.5826 q^{47} +0.791288 q^{49} +1.00000 q^{50} +1.20871 q^{52} -6.00000 q^{53} -3.79129 q^{55} +2.79129 q^{56} +1.58258 q^{58} +4.41742 q^{59} -3.37386 q^{61} +10.3739 q^{62} +1.00000 q^{64} +1.20871 q^{65} -7.16515 q^{67} +3.79129 q^{68} +2.79129 q^{70} +5.37386 q^{71} -14.7477 q^{73} -4.00000 q^{74} +1.20871 q^{76} -10.5826 q^{77} +8.00000 q^{79} +1.00000 q^{80} +2.20871 q^{82} +6.00000 q^{83} +3.79129 q^{85} -7.16515 q^{86} -3.79129 q^{88} +3.16515 q^{89} +3.37386 q^{91} -1.00000 q^{92} +13.5826 q^{94} +1.20871 q^{95} +14.9564 q^{97} +0.791288 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + q^{7} + 2 q^{8} + 2 q^{10} - 3 q^{11} + 7 q^{13} + q^{14} + 2 q^{16} + 3 q^{17} + 7 q^{19} + 2 q^{20} - 3 q^{22} - 2 q^{23} + 2 q^{25} + 7 q^{26} + q^{28} - 6 q^{29} + 7 q^{31} + 2 q^{32} + 3 q^{34} + q^{35} - 8 q^{37} + 7 q^{38} + 2 q^{40} + 9 q^{41} + 4 q^{43} - 3 q^{44} - 2 q^{46} + 18 q^{47} - 3 q^{49} + 2 q^{50} + 7 q^{52} - 12 q^{53} - 3 q^{55} + q^{56} - 6 q^{58} + 18 q^{59} + 7 q^{61} + 7 q^{62} + 2 q^{64} + 7 q^{65} + 4 q^{67} + 3 q^{68} + q^{70} - 3 q^{71} - 2 q^{73} - 8 q^{74} + 7 q^{76} - 12 q^{77} + 16 q^{79} + 2 q^{80} + 9 q^{82} + 12 q^{83} + 3 q^{85} + 4 q^{86} - 3 q^{88} - 12 q^{89} - 7 q^{91} - 2 q^{92} + 18 q^{94} + 7 q^{95} + 7 q^{97} - 3 q^{98} + O(q^{100}) \)

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
\(7\) 2.79129 1.05501 0.527504 0.849553i \(-0.323128\pi\)
0.527504 + 0.849553i \(0.323128\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −3.79129 −1.14312 −0.571558 0.820562i \(-0.693661\pi\)
−0.571558 + 0.820562i \(0.693661\pi\)
\(12\) 0 0
\(13\) 1.20871 0.335236 0.167618 0.985852i \(-0.446392\pi\)
0.167618 + 0.985852i \(0.446392\pi\)
\(14\) 2.79129 0.746003
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.79129 0.919522 0.459761 0.888043i \(-0.347935\pi\)
0.459761 + 0.888043i \(0.347935\pi\)
\(18\) 0 0
\(19\) 1.20871 0.277298 0.138649 0.990342i \(-0.455724\pi\)
0.138649 + 0.990342i \(0.455724\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −3.79129 −0.808305
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.20871 0.237048
\(27\) 0 0
\(28\) 2.79129 0.527504
\(29\) 1.58258 0.293877 0.146938 0.989146i \(-0.453058\pi\)
0.146938 + 0.989146i \(0.453058\pi\)
\(30\) 0 0
\(31\) 10.3739 1.86320 0.931600 0.363484i \(-0.118413\pi\)
0.931600 + 0.363484i \(0.118413\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.79129 0.650201
\(35\) 2.79129 0.471814
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 1.20871 0.196079
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 2.20871 0.344943 0.172471 0.985015i \(-0.444825\pi\)
0.172471 + 0.985015i \(0.444825\pi\)
\(42\) 0 0
\(43\) −7.16515 −1.09268 −0.546338 0.837565i \(-0.683978\pi\)
−0.546338 + 0.837565i \(0.683978\pi\)
\(44\) −3.79129 −0.571558
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 13.5826 1.98122 0.990611 0.136710i \(-0.0436528\pi\)
0.990611 + 0.136710i \(0.0436528\pi\)
\(48\) 0 0
\(49\) 0.791288 0.113041
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 1.20871 0.167618
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −3.79129 −0.511217
\(56\) 2.79129 0.373002
\(57\) 0 0
\(58\) 1.58258 0.207802
\(59\) 4.41742 0.575100 0.287550 0.957766i \(-0.407159\pi\)
0.287550 + 0.957766i \(0.407159\pi\)
\(60\) 0 0
\(61\) −3.37386 −0.431979 −0.215989 0.976396i \(-0.569298\pi\)
−0.215989 + 0.976396i \(0.569298\pi\)
\(62\) 10.3739 1.31748
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.20871 0.149922
\(66\) 0 0
\(67\) −7.16515 −0.875363 −0.437681 0.899130i \(-0.644200\pi\)
−0.437681 + 0.899130i \(0.644200\pi\)
\(68\) 3.79129 0.459761
\(69\) 0 0
\(70\) 2.79129 0.333623
\(71\) 5.37386 0.637760 0.318880 0.947795i \(-0.396693\pi\)
0.318880 + 0.947795i \(0.396693\pi\)
\(72\) 0 0
\(73\) −14.7477 −1.72609 −0.863045 0.505126i \(-0.831446\pi\)
−0.863045 + 0.505126i \(0.831446\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 1.20871 0.138649
\(77\) −10.5826 −1.20600
\(78\) 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
\(82\) 2.20871 0.243911
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 3.79129 0.411223
\(86\) −7.16515 −0.772638
\(87\) 0 0
\(88\) −3.79129 −0.404153
\(89\) 3.16515 0.335505 0.167753 0.985829i \(-0.446349\pi\)
0.167753 + 0.985829i \(0.446349\pi\)
\(90\) 0 0
\(91\) 3.37386 0.353677
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 13.5826 1.40094
\(95\) 1.20871 0.124011
\(96\) 0 0
\(97\) 14.9564 1.51860 0.759298 0.650743i \(-0.225542\pi\)
0.759298 + 0.650743i \(0.225542\pi\)
\(98\) 0.791288 0.0799321
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −13.5826 −1.35152 −0.675758 0.737123i \(-0.736184\pi\)
−0.675758 + 0.737123i \(0.736184\pi\)
\(102\) 0 0
\(103\) 7.37386 0.726568 0.363284 0.931678i \(-0.381655\pi\)
0.363284 + 0.931678i \(0.381655\pi\)
\(104\) 1.20871 0.118524
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −13.5826 −1.31308 −0.656539 0.754292i \(-0.727980\pi\)
−0.656539 + 0.754292i \(0.727980\pi\)
\(108\) 0 0
\(109\) 10.3739 0.993636 0.496818 0.867855i \(-0.334502\pi\)
0.496818 + 0.867855i \(0.334502\pi\)
\(110\) −3.79129 −0.361485
\(111\) 0 0
\(112\) 2.79129 0.263752
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 1.58258 0.146938
\(117\) 0 0
\(118\) 4.41742 0.406657
\(119\) 10.5826 0.970103
\(120\) 0 0
\(121\) 3.37386 0.306715
\(122\) −3.37386 −0.305455
\(123\) 0 0
\(124\) 10.3739 0.931600
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −14.7477 −1.30865 −0.654325 0.756214i \(-0.727047\pi\)
−0.654325 + 0.756214i \(0.727047\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 1.20871 0.106011
\(131\) −9.16515 −0.800763 −0.400381 0.916349i \(-0.631122\pi\)
−0.400381 + 0.916349i \(0.631122\pi\)
\(132\) 0 0
\(133\) 3.37386 0.292551
\(134\) −7.16515 −0.618975
\(135\) 0 0
\(136\) 3.79129 0.325100
\(137\) 0.791288 0.0676043 0.0338021 0.999429i \(-0.489238\pi\)
0.0338021 + 0.999429i \(0.489238\pi\)
\(138\) 0 0
\(139\) −14.7477 −1.25089 −0.625443 0.780270i \(-0.715082\pi\)
−0.625443 + 0.780270i \(0.715082\pi\)
\(140\) 2.79129 0.235907
\(141\) 0 0
\(142\) 5.37386 0.450965
\(143\) −4.58258 −0.383214
\(144\) 0 0
\(145\) 1.58258 0.131426
\(146\) −14.7477 −1.22053
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) −12.7913 −1.04790 −0.523952 0.851748i \(-0.675543\pi\)
−0.523952 + 0.851748i \(0.675543\pi\)
\(150\) 0 0
\(151\) −6.20871 −0.505258 −0.252629 0.967563i \(-0.581295\pi\)
−0.252629 + 0.967563i \(0.581295\pi\)
\(152\) 1.20871 0.0980395
\(153\) 0 0
\(154\) −10.5826 −0.852768
\(155\) 10.3739 0.833249
\(156\) 0 0
\(157\) 12.7477 1.01738 0.508690 0.860950i \(-0.330130\pi\)
0.508690 + 0.860950i \(0.330130\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −2.79129 −0.219984
\(162\) 0 0
\(163\) 22.3739 1.75246 0.876228 0.481897i \(-0.160052\pi\)
0.876228 + 0.481897i \(0.160052\pi\)
\(164\) 2.20871 0.172471
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 18.3303 1.41844 0.709221 0.704987i \(-0.249047\pi\)
0.709221 + 0.704987i \(0.249047\pi\)
\(168\) 0 0
\(169\) −11.5390 −0.887617
\(170\) 3.79129 0.290779
\(171\) 0 0
\(172\) −7.16515 −0.546338
\(173\) 14.2087 1.08027 0.540134 0.841579i \(-0.318373\pi\)
0.540134 + 0.841579i \(0.318373\pi\)
\(174\) 0 0
\(175\) 2.79129 0.211002
\(176\) −3.79129 −0.285779
\(177\) 0 0
\(178\) 3.16515 0.237238
\(179\) 16.7477 1.25178 0.625892 0.779910i \(-0.284735\pi\)
0.625892 + 0.779910i \(0.284735\pi\)
\(180\) 0 0
\(181\) 13.5390 1.00635 0.503174 0.864185i \(-0.332166\pi\)
0.503174 + 0.864185i \(0.332166\pi\)
\(182\) 3.37386 0.250087
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) −14.3739 −1.05112
\(188\) 13.5826 0.990611
\(189\) 0 0
\(190\) 1.20871 0.0876892
\(191\) −16.4174 −1.18792 −0.593962 0.804493i \(-0.702437\pi\)
−0.593962 + 0.804493i \(0.702437\pi\)
\(192\) 0 0
\(193\) 6.74773 0.485712 0.242856 0.970062i \(-0.421916\pi\)
0.242856 + 0.970062i \(0.421916\pi\)
\(194\) 14.9564 1.07381
\(195\) 0 0
\(196\) 0.791288 0.0565206
\(197\) −20.5390 −1.46334 −0.731672 0.681657i \(-0.761260\pi\)
−0.731672 + 0.681657i \(0.761260\pi\)
\(198\) 0 0
\(199\) 20.3303 1.44118 0.720588 0.693363i \(-0.243872\pi\)
0.720588 + 0.693363i \(0.243872\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −13.5826 −0.955667
\(203\) 4.41742 0.310042
\(204\) 0 0
\(205\) 2.20871 0.154263
\(206\) 7.37386 0.513761
\(207\) 0 0
\(208\) 1.20871 0.0838091
\(209\) −4.58258 −0.316983
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −13.5826 −0.928486
\(215\) −7.16515 −0.488659
\(216\) 0 0
\(217\) 28.9564 1.96569
\(218\) 10.3739 0.702607
\(219\) 0 0
\(220\) −3.79129 −0.255609
\(221\) 4.58258 0.308257
\(222\) 0 0
\(223\) 11.1652 0.747674 0.373837 0.927494i \(-0.378042\pi\)
0.373837 + 0.927494i \(0.378042\pi\)
\(224\) 2.79129 0.186501
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −4.74773 −0.315118 −0.157559 0.987510i \(-0.550362\pi\)
−0.157559 + 0.987510i \(0.550362\pi\)
\(228\) 0 0
\(229\) −16.3303 −1.07914 −0.539568 0.841942i \(-0.681413\pi\)
−0.539568 + 0.841942i \(0.681413\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) 1.58258 0.103901
\(233\) −7.58258 −0.496751 −0.248376 0.968664i \(-0.579897\pi\)
−0.248376 + 0.968664i \(0.579897\pi\)
\(234\) 0 0
\(235\) 13.5826 0.886030
\(236\) 4.41742 0.287550
\(237\) 0 0
\(238\) 10.5826 0.685966
\(239\) −3.16515 −0.204737 −0.102368 0.994747i \(-0.532642\pi\)
−0.102368 + 0.994747i \(0.532642\pi\)
\(240\) 0 0
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) 3.37386 0.216880
\(243\) 0 0
\(244\) −3.37386 −0.215989
\(245\) 0.791288 0.0505535
\(246\) 0 0
\(247\) 1.46099 0.0929603
\(248\) 10.3739 0.658741
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −30.7913 −1.94353 −0.971764 0.235953i \(-0.924179\pi\)
−0.971764 + 0.235953i \(0.924179\pi\)
\(252\) 0 0
\(253\) 3.79129 0.238356
\(254\) −14.7477 −0.925355
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −22.7477 −1.41896 −0.709482 0.704723i \(-0.751071\pi\)
−0.709482 + 0.704723i \(0.751071\pi\)
\(258\) 0 0
\(259\) −11.1652 −0.693769
\(260\) 1.20871 0.0749611
\(261\) 0 0
\(262\) −9.16515 −0.566225
\(263\) −15.7913 −0.973733 −0.486866 0.873477i \(-0.661860\pi\)
−0.486866 + 0.873477i \(0.661860\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 3.37386 0.206865
\(267\) 0 0
\(268\) −7.16515 −0.437681
\(269\) −16.7477 −1.02113 −0.510563 0.859840i \(-0.670563\pi\)
−0.510563 + 0.859840i \(0.670563\pi\)
\(270\) 0 0
\(271\) −23.1216 −1.40454 −0.702268 0.711912i \(-0.747829\pi\)
−0.702268 + 0.711912i \(0.747829\pi\)
\(272\) 3.79129 0.229881
\(273\) 0 0
\(274\) 0.791288 0.0478034
\(275\) −3.79129 −0.228623
\(276\) 0 0
\(277\) −1.16515 −0.0700072 −0.0350036 0.999387i \(-0.511144\pi\)
−0.0350036 + 0.999387i \(0.511144\pi\)
\(278\) −14.7477 −0.884510
\(279\) 0 0
\(280\) 2.79129 0.166811
\(281\) 16.7477 0.999086 0.499543 0.866289i \(-0.333501\pi\)
0.499543 + 0.866289i \(0.333501\pi\)
\(282\) 0 0
\(283\) −28.3303 −1.68406 −0.842031 0.539429i \(-0.818640\pi\)
−0.842031 + 0.539429i \(0.818640\pi\)
\(284\) 5.37386 0.318880
\(285\) 0 0
\(286\) −4.58258 −0.270973
\(287\) 6.16515 0.363917
\(288\) 0 0
\(289\) −2.62614 −0.154479
\(290\) 1.58258 0.0929320
\(291\) 0 0
\(292\) −14.7477 −0.863045
\(293\) 27.4955 1.60630 0.803151 0.595776i \(-0.203155\pi\)
0.803151 + 0.595776i \(0.203155\pi\)
\(294\) 0 0
\(295\) 4.41742 0.257192
\(296\) −4.00000 −0.232495
\(297\) 0 0
\(298\) −12.7913 −0.740979
\(299\) −1.20871 −0.0699016
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) −6.20871 −0.357271
\(303\) 0 0
\(304\) 1.20871 0.0693244
\(305\) −3.37386 −0.193187
\(306\) 0 0
\(307\) 16.5390 0.943931 0.471966 0.881617i \(-0.343545\pi\)
0.471966 + 0.881617i \(0.343545\pi\)
\(308\) −10.5826 −0.602998
\(309\) 0 0
\(310\) 10.3739 0.589196
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −18.3739 −1.03855 −0.519276 0.854607i \(-0.673798\pi\)
−0.519276 + 0.854607i \(0.673798\pi\)
\(314\) 12.7477 0.719396
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −5.20871 −0.292550 −0.146275 0.989244i \(-0.546729\pi\)
−0.146275 + 0.989244i \(0.546729\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −2.79129 −0.155552
\(323\) 4.58258 0.254981
\(324\) 0 0
\(325\) 1.20871 0.0670473
\(326\) 22.3739 1.23917
\(327\) 0 0
\(328\) 2.20871 0.121956
\(329\) 37.9129 2.09020
\(330\) 0 0
\(331\) 6.74773 0.370889 0.185444 0.982655i \(-0.440628\pi\)
0.185444 + 0.982655i \(0.440628\pi\)
\(332\) 6.00000 0.329293
\(333\) 0 0
\(334\) 18.3303 1.00299
\(335\) −7.16515 −0.391474
\(336\) 0 0
\(337\) −16.7913 −0.914680 −0.457340 0.889292i \(-0.651198\pi\)
−0.457340 + 0.889292i \(0.651198\pi\)
\(338\) −11.5390 −0.627640
\(339\) 0 0
\(340\) 3.79129 0.205611
\(341\) −39.3303 −2.12986
\(342\) 0 0
\(343\) −17.3303 −0.935748
\(344\) −7.16515 −0.386319
\(345\) 0 0
\(346\) 14.2087 0.763865
\(347\) −9.79129 −0.525624 −0.262812 0.964847i \(-0.584650\pi\)
−0.262812 + 0.964847i \(0.584650\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 2.79129 0.149201
\(351\) 0 0
\(352\) −3.79129 −0.202076
\(353\) 15.1652 0.807160 0.403580 0.914944i \(-0.367766\pi\)
0.403580 + 0.914944i \(0.367766\pi\)
\(354\) 0 0
\(355\) 5.37386 0.285215
\(356\) 3.16515 0.167753
\(357\) 0 0
\(358\) 16.7477 0.885145
\(359\) 9.16515 0.483718 0.241859 0.970311i \(-0.422243\pi\)
0.241859 + 0.970311i \(0.422243\pi\)
\(360\) 0 0
\(361\) −17.5390 −0.923106
\(362\) 13.5390 0.711595
\(363\) 0 0
\(364\) 3.37386 0.176838
\(365\) −14.7477 −0.771931
\(366\) 0 0
\(367\) −0.834849 −0.0435787 −0.0217894 0.999763i \(-0.506936\pi\)
−0.0217894 + 0.999763i \(0.506936\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) −4.00000 −0.207950
\(371\) −16.7477 −0.869499
\(372\) 0 0
\(373\) −14.7477 −0.763608 −0.381804 0.924243i \(-0.624697\pi\)
−0.381804 + 0.924243i \(0.624697\pi\)
\(374\) −14.3739 −0.743255
\(375\) 0 0
\(376\) 13.5826 0.700468
\(377\) 1.91288 0.0985183
\(378\) 0 0
\(379\) 7.37386 0.378770 0.189385 0.981903i \(-0.439351\pi\)
0.189385 + 0.981903i \(0.439351\pi\)
\(380\) 1.20871 0.0620056
\(381\) 0 0
\(382\) −16.4174 −0.839989
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) −10.5826 −0.539338
\(386\) 6.74773 0.343450
\(387\) 0 0
\(388\) 14.9564 0.759298
\(389\) −29.7042 −1.50606 −0.753031 0.657986i \(-0.771409\pi\)
−0.753031 + 0.657986i \(0.771409\pi\)
\(390\) 0 0
\(391\) −3.79129 −0.191734
\(392\) 0.791288 0.0399661
\(393\) 0 0
\(394\) −20.5390 −1.03474
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 16.5390 0.830069 0.415035 0.909806i \(-0.363769\pi\)
0.415035 + 0.909806i \(0.363769\pi\)
\(398\) 20.3303 1.01907
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −22.7477 −1.13597 −0.567984 0.823040i \(-0.692276\pi\)
−0.567984 + 0.823040i \(0.692276\pi\)
\(402\) 0 0
\(403\) 12.5390 0.624613
\(404\) −13.5826 −0.675758
\(405\) 0 0
\(406\) 4.41742 0.219233
\(407\) 15.1652 0.751709
\(408\) 0 0
\(409\) −22.7913 −1.12696 −0.563478 0.826131i \(-0.690537\pi\)
−0.563478 + 0.826131i \(0.690537\pi\)
\(410\) 2.20871 0.109081
\(411\) 0 0
\(412\) 7.37386 0.363284
\(413\) 12.3303 0.606735
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 1.20871 0.0592620
\(417\) 0 0
\(418\) −4.58258 −0.224141
\(419\) −39.1652 −1.91334 −0.956671 0.291170i \(-0.905956\pi\)
−0.956671 + 0.291170i \(0.905956\pi\)
\(420\) 0 0
\(421\) −23.1216 −1.12688 −0.563439 0.826158i \(-0.690522\pi\)
−0.563439 + 0.826158i \(0.690522\pi\)
\(422\) −10.0000 −0.486792
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 3.79129 0.183904
\(426\) 0 0
\(427\) −9.41742 −0.455741
\(428\) −13.5826 −0.656539
\(429\) 0 0
\(430\) −7.16515 −0.345534
\(431\) 19.9129 0.959170 0.479585 0.877496i \(-0.340787\pi\)
0.479585 + 0.877496i \(0.340787\pi\)
\(432\) 0 0
\(433\) 1.53901 0.0739603 0.0369802 0.999316i \(-0.488226\pi\)
0.0369802 + 0.999316i \(0.488226\pi\)
\(434\) 28.9564 1.38995
\(435\) 0 0
\(436\) 10.3739 0.496818
\(437\) −1.20871 −0.0578205
\(438\) 0 0
\(439\) 25.5390 1.21891 0.609455 0.792820i \(-0.291388\pi\)
0.609455 + 0.792820i \(0.291388\pi\)
\(440\) −3.79129 −0.180743
\(441\) 0 0
\(442\) 4.58258 0.217971
\(443\) 35.2087 1.67282 0.836408 0.548107i \(-0.184651\pi\)
0.836408 + 0.548107i \(0.184651\pi\)
\(444\) 0 0
\(445\) 3.16515 0.150043
\(446\) 11.1652 0.528685
\(447\) 0 0
\(448\) 2.79129 0.131876
\(449\) −25.1216 −1.18556 −0.592781 0.805364i \(-0.701970\pi\)
−0.592781 + 0.805364i \(0.701970\pi\)
\(450\) 0 0
\(451\) −8.37386 −0.394310
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) −4.74773 −0.222822
\(455\) 3.37386 0.158169
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −16.3303 −0.763065
\(459\) 0 0
\(460\) −1.00000 −0.0466252
\(461\) 1.25227 0.0583242 0.0291621 0.999575i \(-0.490716\pi\)
0.0291621 + 0.999575i \(0.490716\pi\)
\(462\) 0 0
\(463\) −10.0000 −0.464739 −0.232370 0.972628i \(-0.574648\pi\)
−0.232370 + 0.972628i \(0.574648\pi\)
\(464\) 1.58258 0.0734692
\(465\) 0 0
\(466\) −7.58258 −0.351256
\(467\) −25.9129 −1.19911 −0.599553 0.800335i \(-0.704655\pi\)
−0.599553 + 0.800335i \(0.704655\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) 13.5826 0.626517
\(471\) 0 0
\(472\) 4.41742 0.203328
\(473\) 27.1652 1.24905
\(474\) 0 0
\(475\) 1.20871 0.0554595
\(476\) 10.5826 0.485052
\(477\) 0 0
\(478\) −3.16515 −0.144771
\(479\) 39.4955 1.80459 0.902297 0.431116i \(-0.141880\pi\)
0.902297 + 0.431116i \(0.141880\pi\)
\(480\) 0 0
\(481\) −4.83485 −0.220450
\(482\) −28.0000 −1.27537
\(483\) 0 0
\(484\) 3.37386 0.153357
\(485\) 14.9564 0.679137
\(486\) 0 0
\(487\) 15.5826 0.706114 0.353057 0.935602i \(-0.385142\pi\)
0.353057 + 0.935602i \(0.385142\pi\)
\(488\) −3.37386 −0.152728
\(489\) 0 0
\(490\) 0.791288 0.0357467
\(491\) 16.7477 0.755814 0.377907 0.925843i \(-0.376644\pi\)
0.377907 + 0.925843i \(0.376644\pi\)
\(492\) 0 0
\(493\) 6.00000 0.270226
\(494\) 1.46099 0.0657328
\(495\) 0 0
\(496\) 10.3739 0.465800
\(497\) 15.0000 0.672842
\(498\) 0 0
\(499\) 23.1652 1.03701 0.518507 0.855073i \(-0.326488\pi\)
0.518507 + 0.855073i \(0.326488\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −30.7913 −1.37428
\(503\) −18.7913 −0.837862 −0.418931 0.908018i \(-0.637595\pi\)
−0.418931 + 0.908018i \(0.637595\pi\)
\(504\) 0 0
\(505\) −13.5826 −0.604417
\(506\) 3.79129 0.168543
\(507\) 0 0
\(508\) −14.7477 −0.654325
\(509\) −7.25227 −0.321451 −0.160726 0.986999i \(-0.551383\pi\)
−0.160726 + 0.986999i \(0.551383\pi\)
\(510\) 0 0
\(511\) −41.1652 −1.82104
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −22.7477 −1.00336
\(515\) 7.37386 0.324931
\(516\) 0 0
\(517\) −51.4955 −2.26477
\(518\) −11.1652 −0.490569
\(519\) 0 0
\(520\) 1.20871 0.0530055
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −1.16515 −0.0509485 −0.0254743 0.999675i \(-0.508110\pi\)
−0.0254743 + 0.999675i \(0.508110\pi\)
\(524\) −9.16515 −0.400381
\(525\) 0 0
\(526\) −15.7913 −0.688533
\(527\) 39.3303 1.71325
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) 3.37386 0.146276
\(533\) 2.66970 0.115637
\(534\) 0 0
\(535\) −13.5826 −0.587226
\(536\) −7.16515 −0.309487
\(537\) 0 0
\(538\) −16.7477 −0.722046
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 38.3303 1.64795 0.823974 0.566627i \(-0.191752\pi\)
0.823974 + 0.566627i \(0.191752\pi\)
\(542\) −23.1216 −0.993157
\(543\) 0 0
\(544\) 3.79129 0.162550
\(545\) 10.3739 0.444367
\(546\) 0 0
\(547\) 15.1216 0.646553 0.323276 0.946305i \(-0.395216\pi\)
0.323276 + 0.946305i \(0.395216\pi\)
\(548\) 0.791288 0.0338021
\(549\) 0 0
\(550\) −3.79129 −0.161661
\(551\) 1.91288 0.0814914
\(552\) 0 0
\(553\) 22.3303 0.949581
\(554\) −1.16515 −0.0495025
\(555\) 0 0
\(556\) −14.7477 −0.625443
\(557\) −30.3303 −1.28514 −0.642568 0.766229i \(-0.722131\pi\)
−0.642568 + 0.766229i \(0.722131\pi\)
\(558\) 0 0
\(559\) −8.66061 −0.366305
\(560\) 2.79129 0.117953
\(561\) 0 0
\(562\) 16.7477 0.706460
\(563\) −3.16515 −0.133395 −0.0666976 0.997773i \(-0.521246\pi\)
−0.0666976 + 0.997773i \(0.521246\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) −28.3303 −1.19081
\(567\) 0 0
\(568\) 5.37386 0.225482
\(569\) −15.4955 −0.649603 −0.324802 0.945782i \(-0.605298\pi\)
−0.324802 + 0.945782i \(0.605298\pi\)
\(570\) 0 0
\(571\) 30.1216 1.26055 0.630275 0.776372i \(-0.282942\pi\)
0.630275 + 0.776372i \(0.282942\pi\)
\(572\) −4.58258 −0.191607
\(573\) 0 0
\(574\) 6.16515 0.257328
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 22.8348 0.950627 0.475314 0.879816i \(-0.342335\pi\)
0.475314 + 0.879816i \(0.342335\pi\)
\(578\) −2.62614 −0.109233
\(579\) 0 0
\(580\) 1.58258 0.0657129
\(581\) 16.7477 0.694813
\(582\) 0 0
\(583\) 22.7477 0.942115
\(584\) −14.7477 −0.610265
\(585\) 0 0
\(586\) 27.4955 1.13583
\(587\) 26.2087 1.08175 0.540875 0.841103i \(-0.318093\pi\)
0.540875 + 0.841103i \(0.318093\pi\)
\(588\) 0 0
\(589\) 12.5390 0.516661
\(590\) 4.41742 0.181862
\(591\) 0 0
\(592\) −4.00000 −0.164399
\(593\) −13.9129 −0.571333 −0.285667 0.958329i \(-0.592215\pi\)
−0.285667 + 0.958329i \(0.592215\pi\)
\(594\) 0 0
\(595\) 10.5826 0.433843
\(596\) −12.7913 −0.523952
\(597\) 0 0
\(598\) −1.20871 −0.0494279
\(599\) 40.1216 1.63932 0.819662 0.572848i \(-0.194161\pi\)
0.819662 + 0.572848i \(0.194161\pi\)
\(600\) 0 0
\(601\) −22.7913 −0.929676 −0.464838 0.885396i \(-0.653887\pi\)
−0.464838 + 0.885396i \(0.653887\pi\)
\(602\) −20.0000 −0.815139
\(603\) 0 0
\(604\) −6.20871 −0.252629
\(605\) 3.37386 0.137167
\(606\) 0 0
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 1.20871 0.0490198
\(609\) 0 0
\(610\) −3.37386 −0.136604
\(611\) 16.4174 0.664178
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 16.5390 0.667460
\(615\) 0 0
\(616\) −10.5826 −0.426384
\(617\) −44.8693 −1.80637 −0.903185 0.429251i \(-0.858778\pi\)
−0.903185 + 0.429251i \(0.858778\pi\)
\(618\) 0 0
\(619\) 2.79129 0.112191 0.0560957 0.998425i \(-0.482135\pi\)
0.0560957 + 0.998425i \(0.482135\pi\)
\(620\) 10.3739 0.416624
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) 8.83485 0.353961
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −18.3739 −0.734367
\(627\) 0 0
\(628\) 12.7477 0.508690
\(629\) −15.1652 −0.604674
\(630\) 0 0
\(631\) −17.9129 −0.713100 −0.356550 0.934276i \(-0.616047\pi\)
−0.356550 + 0.934276i \(0.616047\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) −5.20871 −0.206864
\(635\) −14.7477 −0.585246
\(636\) 0 0
\(637\) 0.956439 0.0378955
\(638\) −6.00000 −0.237542
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 3.16515 0.125016 0.0625080 0.998044i \(-0.480090\pi\)
0.0625080 + 0.998044i \(0.480090\pi\)
\(642\) 0 0
\(643\) −20.7477 −0.818210 −0.409105 0.912487i \(-0.634159\pi\)
−0.409105 + 0.912487i \(0.634159\pi\)
\(644\) −2.79129 −0.109992
\(645\) 0 0
\(646\) 4.58258 0.180299
\(647\) −2.83485 −0.111449 −0.0557247 0.998446i \(-0.517747\pi\)
−0.0557247 + 0.998446i \(0.517747\pi\)
\(648\) 0 0
\(649\) −16.7477 −0.657406
\(650\) 1.20871 0.0474096
\(651\) 0 0
\(652\) 22.3739 0.876228
\(653\) −35.5390 −1.39075 −0.695375 0.718647i \(-0.744762\pi\)
−0.695375 + 0.718647i \(0.744762\pi\)
\(654\) 0 0
\(655\) −9.16515 −0.358112
\(656\) 2.20871 0.0862357
\(657\) 0 0
\(658\) 37.9129 1.47800
\(659\) 27.1652 1.05820 0.529102 0.848558i \(-0.322529\pi\)
0.529102 + 0.848558i \(0.322529\pi\)
\(660\) 0 0
\(661\) −39.3739 −1.53147 −0.765733 0.643159i \(-0.777624\pi\)
−0.765733 + 0.643159i \(0.777624\pi\)
\(662\) 6.74773 0.262258
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) 3.37386 0.130833
\(666\) 0 0
\(667\) −1.58258 −0.0612776
\(668\) 18.3303 0.709221
\(669\) 0 0
\(670\) −7.16515 −0.276814
\(671\) 12.7913 0.493802
\(672\) 0 0
\(673\) 38.0000 1.46479 0.732396 0.680879i \(-0.238402\pi\)
0.732396 + 0.680879i \(0.238402\pi\)
\(674\) −16.7913 −0.646776
\(675\) 0 0
\(676\) −11.5390 −0.443808
\(677\) 30.6606 1.17838 0.589191 0.807993i \(-0.299446\pi\)
0.589191 + 0.807993i \(0.299446\pi\)
\(678\) 0 0
\(679\) 41.7477 1.60213
\(680\) 3.79129 0.145389
\(681\) 0 0
\(682\) −39.3303 −1.50604
\(683\) −2.37386 −0.0908334 −0.0454167 0.998968i \(-0.514462\pi\)
−0.0454167 + 0.998968i \(0.514462\pi\)
\(684\) 0 0
\(685\) 0.791288 0.0302336
\(686\) −17.3303 −0.661674
\(687\) 0 0
\(688\) −7.16515 −0.273169
\(689\) −7.25227 −0.276290
\(690\) 0 0
\(691\) 15.2523 0.580224 0.290112 0.956993i \(-0.406307\pi\)
0.290112 + 0.956993i \(0.406307\pi\)
\(692\) 14.2087 0.540134
\(693\) 0 0
\(694\) −9.79129 −0.371672
\(695\) −14.7477 −0.559413
\(696\) 0 0
\(697\) 8.37386 0.317183
\(698\) 26.0000 0.984115
\(699\) 0 0
\(700\) 2.79129 0.105501
\(701\) −9.62614 −0.363574 −0.181787 0.983338i \(-0.558188\pi\)
−0.181787 + 0.983338i \(0.558188\pi\)
\(702\) 0 0
\(703\) −4.83485 −0.182350
\(704\) −3.79129 −0.142890
\(705\) 0 0
\(706\) 15.1652 0.570748
\(707\) −37.9129 −1.42586
\(708\) 0 0
\(709\) 34.5390 1.29714 0.648570 0.761155i \(-0.275367\pi\)
0.648570 + 0.761155i \(0.275367\pi\)
\(710\) 5.37386 0.201678
\(711\) 0 0
\(712\) 3.16515 0.118619
\(713\) −10.3739 −0.388504
\(714\) 0 0
\(715\) −4.58258 −0.171379
\(716\) 16.7477 0.625892
\(717\) 0 0
\(718\) 9.16515 0.342040
\(719\) 29.5390 1.10162 0.550810 0.834631i \(-0.314319\pi\)
0.550810 + 0.834631i \(0.314319\pi\)
\(720\) 0 0
\(721\) 20.5826 0.766535
\(722\) −17.5390 −0.652735
\(723\) 0 0
\(724\) 13.5390 0.503174
\(725\) 1.58258 0.0587754
\(726\) 0 0
\(727\) −2.12159 −0.0786854 −0.0393427 0.999226i \(-0.512526\pi\)
−0.0393427 + 0.999226i \(0.512526\pi\)
\(728\) 3.37386 0.125044
\(729\) 0 0
\(730\) −14.7477 −0.545838
\(731\) −27.1652 −1.00474
\(732\) 0 0
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) −0.834849 −0.0308148
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 27.1652 1.00064
\(738\) 0 0
\(739\) 8.00000 0.294285 0.147142 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) −16.7477 −0.614828
\(743\) −9.95644 −0.365266 −0.182633 0.983181i \(-0.558462\pi\)
−0.182633 + 0.983181i \(0.558462\pi\)
\(744\) 0 0
\(745\) −12.7913 −0.468637
\(746\) −14.7477 −0.539953
\(747\) 0 0
\(748\) −14.3739 −0.525561
\(749\) −37.9129 −1.38531
\(750\) 0 0
\(751\) 18.7477 0.684114 0.342057 0.939679i \(-0.388876\pi\)
0.342057 + 0.939679i \(0.388876\pi\)
\(752\) 13.5826 0.495306
\(753\) 0 0
\(754\) 1.91288 0.0696629
\(755\) −6.20871 −0.225958
\(756\) 0 0
\(757\) 26.3303 0.956991 0.478496 0.878090i \(-0.341182\pi\)
0.478496 + 0.878090i \(0.341182\pi\)
\(758\) 7.37386 0.267831
\(759\) 0 0
\(760\) 1.20871 0.0438446
\(761\) 33.9564 1.23092 0.615460 0.788168i \(-0.288970\pi\)
0.615460 + 0.788168i \(0.288970\pi\)
\(762\) 0 0
\(763\) 28.9564 1.04829
\(764\) −16.4174 −0.593962
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 5.33939 0.192794
\(768\) 0 0
\(769\) −3.66970 −0.132333 −0.0661663 0.997809i \(-0.521077\pi\)
−0.0661663 + 0.997809i \(0.521077\pi\)
\(770\) −10.5826 −0.381370
\(771\) 0 0
\(772\) 6.74773 0.242856
\(773\) 21.4955 0.773138 0.386569 0.922261i \(-0.373660\pi\)
0.386569 + 0.922261i \(0.373660\pi\)
\(774\) 0 0
\(775\) 10.3739 0.372640
\(776\) 14.9564 0.536905
\(777\) 0 0
\(778\) −29.7042 −1.06495
\(779\) 2.66970 0.0956518
\(780\) 0 0
\(781\) −20.3739 −0.729034
\(782\) −3.79129 −0.135576
\(783\) 0 0
\(784\) 0.791288 0.0282603
\(785\) 12.7477 0.454986
\(786\) 0 0
\(787\) −8.41742 −0.300049 −0.150024 0.988682i \(-0.547935\pi\)
−0.150024 + 0.988682i \(0.547935\pi\)
\(788\) −20.5390 −0.731672
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) −16.7477 −0.595481
\(792\) 0 0
\(793\) −4.07803 −0.144815
\(794\) 16.5390 0.586948
\(795\) 0 0
\(796\) 20.3303 0.720588
\(797\) 49.9129 1.76800 0.884002 0.467482i \(-0.154839\pi\)
0.884002 + 0.467482i \(0.154839\pi\)
\(798\) 0 0
\(799\) 51.4955 1.82178
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −22.7477 −0.803250
\(803\) 55.9129 1.97312
\(804\) 0 0
\(805\) −2.79129 −0.0983800
\(806\) 12.5390 0.441668
\(807\) 0 0
\(808\) −13.5826 −0.477833
\(809\) −11.0436 −0.388271 −0.194135 0.980975i \(-0.562190\pi\)
−0.194135 + 0.980975i \(0.562190\pi\)
\(810\) 0 0
\(811\) −47.9129 −1.68245 −0.841224 0.540686i \(-0.818165\pi\)
−0.841224 + 0.540686i \(0.818165\pi\)
\(812\) 4.41742 0.155021
\(813\) 0 0
\(814\) 15.1652 0.531538
\(815\) 22.3739 0.783722
\(816\) 0 0
\(817\) −8.66061 −0.302996
\(818\) −22.7913 −0.796879
\(819\) 0 0
\(820\) 2.20871 0.0771316
\(821\) −2.83485 −0.0989369 −0.0494684 0.998776i \(-0.515753\pi\)
−0.0494684 + 0.998776i \(0.515753\pi\)
\(822\) 0 0
\(823\) 41.1652 1.43493 0.717463 0.696596i \(-0.245303\pi\)
0.717463 + 0.696596i \(0.245303\pi\)
\(824\) 7.37386 0.256881
\(825\) 0 0
\(826\) 12.3303 0.429026
\(827\) −41.0780 −1.42842 −0.714212 0.699930i \(-0.753215\pi\)
−0.714212 + 0.699930i \(0.753215\pi\)
\(828\) 0 0
\(829\) −31.4955 −1.09388 −0.546941 0.837171i \(-0.684208\pi\)
−0.546941 + 0.837171i \(0.684208\pi\)
\(830\) 6.00000 0.208263
\(831\) 0 0
\(832\) 1.20871 0.0419046
\(833\) 3.00000 0.103944
\(834\) 0 0
\(835\) 18.3303 0.634346
\(836\) −4.58258 −0.158492
\(837\) 0 0
\(838\) −39.1652 −1.35294
\(839\) 22.4174 0.773935 0.386968 0.922093i \(-0.373522\pi\)
0.386968 + 0.922093i \(0.373522\pi\)
\(840\) 0 0
\(841\) −26.4955 −0.913636
\(842\) −23.1216 −0.796823
\(843\) 0 0
\(844\) −10.0000 −0.344214
\(845\) −11.5390 −0.396954
\(846\) 0 0
\(847\) 9.41742 0.323587
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 3.79129 0.130040
\(851\) 4.00000 0.137118
\(852\) 0 0
\(853\) 8.46099 0.289699 0.144849 0.989454i \(-0.453730\pi\)
0.144849 + 0.989454i \(0.453730\pi\)
\(854\) −9.41742 −0.322258
\(855\) 0 0
\(856\) −13.5826 −0.464243
\(857\) −9.16515 −0.313076 −0.156538 0.987672i \(-0.550033\pi\)
−0.156538 + 0.987672i \(0.550033\pi\)
\(858\) 0 0
\(859\) 0.747727 0.0255121 0.0127561 0.999919i \(-0.495940\pi\)
0.0127561 + 0.999919i \(0.495940\pi\)
\(860\) −7.16515 −0.244330
\(861\) 0 0
\(862\) 19.9129 0.678235
\(863\) 31.5826 1.07508 0.537542 0.843237i \(-0.319353\pi\)
0.537542 + 0.843237i \(0.319353\pi\)
\(864\) 0 0
\(865\) 14.2087 0.483111
\(866\) 1.53901 0.0522979
\(867\) 0 0
\(868\) 28.9564 0.982846
\(869\) −30.3303 −1.02889
\(870\) 0 0
\(871\) −8.66061 −0.293453
\(872\) 10.3739 0.351303
\(873\) 0 0
\(874\) −1.20871 −0.0408853
\(875\) 2.79129 0.0943628
\(876\) 0 0
\(877\) 7.70417 0.260151 0.130076 0.991504i \(-0.458478\pi\)
0.130076 + 0.991504i \(0.458478\pi\)
\(878\) 25.5390 0.861900
\(879\) 0 0
\(880\) −3.79129 −0.127804
\(881\) 6.33030 0.213273 0.106637 0.994298i \(-0.465992\pi\)
0.106637 + 0.994298i \(0.465992\pi\)
\(882\) 0 0
\(883\) −12.0436 −0.405298 −0.202649 0.979251i \(-0.564955\pi\)
−0.202649 + 0.979251i \(0.564955\pi\)
\(884\) 4.58258 0.154129
\(885\) 0 0
\(886\) 35.2087 1.18286
\(887\) 3.16515 0.106275 0.0531377 0.998587i \(-0.483078\pi\)
0.0531377 + 0.998587i \(0.483078\pi\)
\(888\) 0 0
\(889\) −41.1652 −1.38063
\(890\) 3.16515 0.106096
\(891\) 0 0
\(892\) 11.1652 0.373837
\(893\) 16.4174 0.549388
\(894\) 0 0
\(895\) 16.7477 0.559815
\(896\) 2.79129 0.0932504
\(897\) 0 0
\(898\) −25.1216 −0.838318
\(899\) 16.4174 0.547552
\(900\) 0 0
\(901\) −22.7477 −0.757837
\(902\) −8.37386 −0.278819
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 13.5390 0.450052
\(906\) 0 0
\(907\) −20.7477 −0.688917 −0.344458 0.938802i \(-0.611937\pi\)
−0.344458 + 0.938802i \(0.611937\pi\)
\(908\) −4.74773 −0.157559
\(909\) 0 0
\(910\) 3.37386 0.111842
\(911\) −13.5826 −0.450011 −0.225005 0.974358i \(-0.572240\pi\)
−0.225005 + 0.974358i \(0.572240\pi\)
\(912\) 0 0
\(913\) −22.7477 −0.752840
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −16.3303 −0.539568
\(917\) −25.5826 −0.844811
\(918\) 0 0
\(919\) −36.8348 −1.21507 −0.607535 0.794293i \(-0.707841\pi\)
−0.607535 + 0.794293i \(0.707841\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 0 0
\(922\) 1.25227 0.0412414
\(923\) 6.49545 0.213800
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) −10.0000 −0.328620
\(927\) 0 0
\(928\) 1.58258 0.0519506
\(929\) 39.4955 1.29580 0.647902 0.761724i \(-0.275647\pi\)
0.647902 + 0.761724i \(0.275647\pi\)
\(930\) 0 0
\(931\) 0.956439 0.0313460
\(932\) −7.58258 −0.248376
\(933\) 0 0
\(934\) −25.9129 −0.847895
\(935\) −14.3739 −0.470076
\(936\) 0 0
\(937\) 58.3739 1.90699 0.953495 0.301407i \(-0.0974564\pi\)
0.953495 + 0.301407i \(0.0974564\pi\)
\(938\) −20.0000 −0.653023
\(939\) 0 0
\(940\) 13.5826 0.443015
\(941\) −54.9564 −1.79153 −0.895764 0.444529i \(-0.853371\pi\)
−0.895764 + 0.444529i \(0.853371\pi\)
\(942\) 0 0
\(943\) −2.20871 −0.0719256
\(944\) 4.41742 0.143775
\(945\) 0 0
\(946\) 27.1652 0.883215
\(947\) 29.5390 0.959889 0.479945 0.877299i \(-0.340657\pi\)
0.479945 + 0.877299i \(0.340657\pi\)
\(948\) 0 0
\(949\) −17.8258 −0.578649
\(950\) 1.20871 0.0392158
\(951\) 0 0
\(952\) 10.5826 0.342983
\(953\) 26.5390 0.859683 0.429842 0.902904i \(-0.358569\pi\)
0.429842 + 0.902904i \(0.358569\pi\)
\(954\) 0 0
\(955\) −16.4174 −0.531255
\(956\) −3.16515 −0.102368
\(957\) 0 0
\(958\) 39.4955 1.27604
\(959\) 2.20871 0.0713230
\(960\) 0 0
\(961\) 76.6170 2.47152
\(962\) −4.83485 −0.155882
\(963\) 0 0
\(964\) −28.0000 −0.901819
\(965\) 6.74773 0.217217
\(966\) 0 0
\(967\) −5.25227 −0.168902 −0.0844509 0.996428i \(-0.526914\pi\)
−0.0844509 + 0.996428i \(0.526914\pi\)
\(968\) 3.37386 0.108440
\(969\) 0 0
\(970\) 14.9564 0.480222
\(971\) 6.95644 0.223243 0.111621 0.993751i \(-0.464396\pi\)
0.111621 + 0.993751i \(0.464396\pi\)
\(972\) 0 0
\(973\) −41.1652 −1.31969
\(974\) 15.5826 0.499298
\(975\) 0 0
\(976\) −3.37386 −0.107995
\(977\) −7.12159 −0.227840 −0.113920 0.993490i \(-0.536341\pi\)
−0.113920 + 0.993490i \(0.536341\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) 0.791288 0.0252768
\(981\) 0 0
\(982\) 16.7477 0.534441
\(983\) 0.626136 0.0199707 0.00998533 0.999950i \(-0.496822\pi\)
0.00998533 + 0.999950i \(0.496822\pi\)
\(984\) 0 0
\(985\) −20.5390 −0.654427
\(986\) 6.00000 0.191079
\(987\) 0 0
\(988\) 1.46099 0.0464801
\(989\) 7.16515 0.227839
\(990\) 0 0
\(991\) −37.7913 −1.20048 −0.600240 0.799820i \(-0.704928\pi\)
−0.600240 + 0.799820i \(0.704928\pi\)
\(992\) 10.3739 0.329370
\(993\) 0 0
\(994\) 15.0000 0.475771
\(995\) 20.3303 0.644514
\(996\) 0 0
\(997\) 11.4955 0.364065 0.182032 0.983293i \(-0.441732\pi\)
0.182032 + 0.983293i \(0.441732\pi\)
\(998\) 23.1652 0.733280
\(999\) 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 2070.2.a.x.1.2 2
3.2 odd 2 230.2.a.a.1.2 2
12.11 even 2 1840.2.a.n.1.1 2
15.2 even 4 1150.2.b.g.599.1 4
15.8 even 4 1150.2.b.g.599.4 4
15.14 odd 2 1150.2.a.o.1.1 2
24.5 odd 2 7360.2.a.bq.1.1 2
24.11 even 2 7360.2.a.bk.1.2 2
60.59 even 2 9200.2.a.bs.1.2 2
69.68 even 2 5290.2.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.a.1.2 2 3.2 odd 2
1150.2.a.o.1.1 2 15.14 odd 2
1150.2.b.g.599.1 4 15.2 even 4
1150.2.b.g.599.4 4 15.8 even 4
1840.2.a.n.1.1 2 12.11 even 2
2070.2.a.x.1.2 2 1.1 even 1 trivial
5290.2.a.e.1.2 2 69.68 even 2
7360.2.a.bk.1.2 2 24.11 even 2
7360.2.a.bq.1.1 2 24.5 odd 2
9200.2.a.bs.1.2 2 60.59 even 2