Properties

Label 2790.2.a.bk.1.4
Level $2790$
Weight $2$
Character 2790.1
Self dual yes
Analytic conductor $22.278$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2790.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.2782621639\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.17428.1
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.52616\) of defining polynomial
Character \(\chi\) \(=\) 2790.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.54997 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.54997 q^{7} +1.00000 q^{8} +1.00000 q^{10} -2.54997 q^{11} -1.05232 q^{13} +4.54997 q^{14} +1.00000 q^{16} -4.39400 q^{17} +7.60228 q^{19} +1.00000 q^{20} -2.54997 q^{22} -1.20828 q^{23} +1.00000 q^{25} -1.05232 q^{26} +4.54997 q^{28} +6.10463 q^{29} +1.00000 q^{31} +1.00000 q^{32} -4.39400 q^{34} +4.54997 q^{35} +7.73569 q^{37} +7.60228 q^{38} +1.00000 q^{40} +6.10463 q^{41} -6.28566 q^{43} -2.54997 q^{44} -1.20828 q^{46} +2.68337 q^{47} +13.7022 q^{49} +1.00000 q^{50} -1.05232 q^{52} -0.502345 q^{53} -2.54997 q^{55} +4.54997 q^{56} +6.10463 q^{58} -10.8356 q^{59} -9.80588 q^{61} +1.00000 q^{62} +1.00000 q^{64} -1.05232 q^{65} +7.46888 q^{67} -4.39400 q^{68} +4.54997 q^{70} -8.65460 q^{71} -8.70222 q^{73} +7.73569 q^{74} +7.60228 q^{76} -11.6023 q^{77} -0.918913 q^{79} +1.00000 q^{80} +6.10463 q^{82} +2.41657 q^{83} -4.39400 q^{85} -6.28566 q^{86} -2.54997 q^{88} +16.4379 q^{89} -4.78800 q^{91} -1.20828 q^{92} +2.68337 q^{94} +7.60228 q^{95} -9.20457 q^{97} +13.7022 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 6 q^{13} + 5 q^{14} + 4 q^{16} + 7 q^{19} + 4 q^{20} + 3 q^{22} + q^{23} + 4 q^{25} + 6 q^{26} + 5 q^{28} + 4 q^{29} + 4 q^{31} + 4 q^{32} + 5 q^{35} + 6 q^{37} + 7 q^{38} + 4 q^{40} + 4 q^{41} + 13 q^{43} + 3 q^{44} + q^{46} - 4 q^{47} + 5 q^{49} + 4 q^{50} + 6 q^{52} - 5 q^{53} + 3 q^{55} + 5 q^{56} + 4 q^{58} + 8 q^{59} - 4 q^{61} + 4 q^{62} + 4 q^{64} + 6 q^{65} + 8 q^{67} + 5 q^{70} - q^{71} + 15 q^{73} + 6 q^{74} + 7 q^{76} - 23 q^{77} + 5 q^{79} + 4 q^{80} + 4 q^{82} - 2 q^{83} + 13 q^{86} + 3 q^{88} - 9 q^{89} + 16 q^{91} + q^{92} - 4 q^{94} + 7 q^{95} + 10 q^{97} + 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) 4.54997 1.71973 0.859863 0.510524i \(-0.170549\pi\)
0.859863 + 0.510524i \(0.170549\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −2.54997 −0.768845 −0.384422 0.923157i \(-0.625599\pi\)
−0.384422 + 0.923157i \(0.625599\pi\)
\(12\) 0 0
\(13\) −1.05232 −0.291860 −0.145930 0.989295i \(-0.546617\pi\)
−0.145930 + 0.989295i \(0.546617\pi\)
\(14\) 4.54997 1.21603
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.39400 −1.06570 −0.532851 0.846209i \(-0.678879\pi\)
−0.532851 + 0.846209i \(0.678879\pi\)
\(18\) 0 0
\(19\) 7.60228 1.74408 0.872042 0.489431i \(-0.162796\pi\)
0.872042 + 0.489431i \(0.162796\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −2.54997 −0.543655
\(23\) −1.20828 −0.251945 −0.125972 0.992034i \(-0.540205\pi\)
−0.125972 + 0.992034i \(0.540205\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.05232 −0.206376
\(27\) 0 0
\(28\) 4.54997 0.859863
\(29\) 6.10463 1.13360 0.566801 0.823855i \(-0.308181\pi\)
0.566801 + 0.823855i \(0.308181\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.39400 −0.753565
\(35\) 4.54997 0.769085
\(36\) 0 0
\(37\) 7.73569 1.27174 0.635870 0.771797i \(-0.280642\pi\)
0.635870 + 0.771797i \(0.280642\pi\)
\(38\) 7.60228 1.23325
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 6.10463 0.953383 0.476692 0.879071i \(-0.341836\pi\)
0.476692 + 0.879071i \(0.341836\pi\)
\(42\) 0 0
\(43\) −6.28566 −0.958554 −0.479277 0.877664i \(-0.659101\pi\)
−0.479277 + 0.877664i \(0.659101\pi\)
\(44\) −2.54997 −0.384422
\(45\) 0 0
\(46\) −1.20828 −0.178152
\(47\) 2.68337 0.391410 0.195705 0.980663i \(-0.437300\pi\)
0.195705 + 0.980663i \(0.437300\pi\)
\(48\) 0 0
\(49\) 13.7022 1.95746
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −1.05232 −0.145930
\(53\) −0.502345 −0.0690025 −0.0345012 0.999405i \(-0.510984\pi\)
−0.0345012 + 0.999405i \(0.510984\pi\)
\(54\) 0 0
\(55\) −2.54997 −0.343838
\(56\) 4.54997 0.608015
\(57\) 0 0
\(58\) 6.10463 0.801577
\(59\) −10.8356 −1.41068 −0.705339 0.708870i \(-0.749205\pi\)
−0.705339 + 0.708870i \(0.749205\pi\)
\(60\) 0 0
\(61\) −9.80588 −1.25551 −0.627757 0.778409i \(-0.716027\pi\)
−0.627757 + 0.778409i \(0.716027\pi\)
\(62\) 1.00000 0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.05232 −0.130524
\(66\) 0 0
\(67\) 7.46888 0.912469 0.456235 0.889860i \(-0.349198\pi\)
0.456235 + 0.889860i \(0.349198\pi\)
\(68\) −4.39400 −0.532851
\(69\) 0 0
\(70\) 4.54997 0.543825
\(71\) −8.65460 −1.02711 −0.513556 0.858056i \(-0.671672\pi\)
−0.513556 + 0.858056i \(0.671672\pi\)
\(72\) 0 0
\(73\) −8.70222 −1.01852 −0.509259 0.860613i \(-0.670081\pi\)
−0.509259 + 0.860613i \(0.670081\pi\)
\(74\) 7.73569 0.899255
\(75\) 0 0
\(76\) 7.60228 0.872042
\(77\) −11.6023 −1.32220
\(78\) 0 0
\(79\) −0.918913 −0.103386 −0.0516929 0.998663i \(-0.516462\pi\)
−0.0516929 + 0.998663i \(0.516462\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 6.10463 0.674144
\(83\) 2.41657 0.265253 0.132626 0.991166i \(-0.457659\pi\)
0.132626 + 0.991166i \(0.457659\pi\)
\(84\) 0 0
\(85\) −4.39400 −0.476596
\(86\) −6.28566 −0.677800
\(87\) 0 0
\(88\) −2.54997 −0.271828
\(89\) 16.4379 1.74242 0.871208 0.490915i \(-0.163337\pi\)
0.871208 + 0.490915i \(0.163337\pi\)
\(90\) 0 0
\(91\) −4.78800 −0.501919
\(92\) −1.20828 −0.125972
\(93\) 0 0
\(94\) 2.68337 0.276769
\(95\) 7.60228 0.779978
\(96\) 0 0
\(97\) −9.20457 −0.934582 −0.467291 0.884103i \(-0.654770\pi\)
−0.467291 + 0.884103i \(0.654770\pi\)
\(98\) 13.7022 1.38413
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 3.18572 0.316991 0.158495 0.987360i \(-0.449336\pi\)
0.158495 + 0.987360i \(0.449336\pi\)
\(102\) 0 0
\(103\) 18.8356 1.85593 0.927965 0.372668i \(-0.121557\pi\)
0.927965 + 0.372668i \(0.121557\pi\)
\(104\) −1.05232 −0.103188
\(105\) 0 0
\(106\) −0.502345 −0.0487921
\(107\) −14.2857 −1.38105 −0.690523 0.723310i \(-0.742620\pi\)
−0.690523 + 0.723310i \(0.742620\pi\)
\(108\) 0 0
\(109\) −5.78331 −0.553941 −0.276970 0.960878i \(-0.589330\pi\)
−0.276970 + 0.960878i \(0.589330\pi\)
\(110\) −2.54997 −0.243130
\(111\) 0 0
\(112\) 4.54997 0.429932
\(113\) 8.01885 0.754350 0.377175 0.926142i \(-0.376895\pi\)
0.377175 + 0.926142i \(0.376895\pi\)
\(114\) 0 0
\(115\) −1.20828 −0.112673
\(116\) 6.10463 0.566801
\(117\) 0 0
\(118\) −10.8356 −0.997500
\(119\) −19.9926 −1.83272
\(120\) 0 0
\(121\) −4.49765 −0.408878
\(122\) −9.80588 −0.887782
\(123\) 0 0
\(124\) 1.00000 0.0898027
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −7.15695 −0.635076 −0.317538 0.948246i \(-0.602856\pi\)
−0.317538 + 0.948246i \(0.602856\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −1.05232 −0.0922941
\(131\) −4.46419 −0.390038 −0.195019 0.980799i \(-0.562477\pi\)
−0.195019 + 0.980799i \(0.562477\pi\)
\(132\) 0 0
\(133\) 34.5902 2.99935
\(134\) 7.46888 0.645213
\(135\) 0 0
\(136\) −4.39400 −0.376782
\(137\) 17.0104 1.45330 0.726650 0.687008i \(-0.241076\pi\)
0.726650 + 0.687008i \(0.241076\pi\)
\(138\) 0 0
\(139\) −7.49394 −0.635628 −0.317814 0.948153i \(-0.602949\pi\)
−0.317814 + 0.948153i \(0.602949\pi\)
\(140\) 4.54997 0.384543
\(141\) 0 0
\(142\) −8.65460 −0.726278
\(143\) 2.68337 0.224395
\(144\) 0 0
\(145\) 6.10463 0.506962
\(146\) −8.70222 −0.720201
\(147\) 0 0
\(148\) 7.73569 0.635870
\(149\) 2.18103 0.178677 0.0893383 0.996001i \(-0.471525\pi\)
0.0893383 + 0.996001i \(0.471525\pi\)
\(150\) 0 0
\(151\) 4.78800 0.389642 0.194821 0.980839i \(-0.437587\pi\)
0.194821 + 0.980839i \(0.437587\pi\)
\(152\) 7.60228 0.616627
\(153\) 0 0
\(154\) −11.6023 −0.934939
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) 10.0189 0.799591 0.399796 0.916604i \(-0.369081\pi\)
0.399796 + 0.916604i \(0.369081\pi\)
\(158\) −0.918913 −0.0731048
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −5.49765 −0.433276
\(162\) 0 0
\(163\) 13.6311 1.06767 0.533833 0.845590i \(-0.320751\pi\)
0.533833 + 0.845590i \(0.320751\pi\)
\(164\) 6.10463 0.476692
\(165\) 0 0
\(166\) 2.41657 0.187562
\(167\) −19.9869 −1.54663 −0.773317 0.634020i \(-0.781404\pi\)
−0.773317 + 0.634020i \(0.781404\pi\)
\(168\) 0 0
\(169\) −11.8926 −0.914818
\(170\) −4.39400 −0.337005
\(171\) 0 0
\(172\) −6.28566 −0.479277
\(173\) −14.5212 −1.10403 −0.552013 0.833835i \(-0.686140\pi\)
−0.552013 + 0.833835i \(0.686140\pi\)
\(174\) 0 0
\(175\) 4.54997 0.343945
\(176\) −2.54997 −0.192211
\(177\) 0 0
\(178\) 16.4379 1.23207
\(179\) −16.5688 −1.23841 −0.619206 0.785229i \(-0.712545\pi\)
−0.619206 + 0.785229i \(0.712545\pi\)
\(180\) 0 0
\(181\) 9.31291 0.692223 0.346112 0.938193i \(-0.387502\pi\)
0.346112 + 0.938193i \(0.387502\pi\)
\(182\) −4.78800 −0.354910
\(183\) 0 0
\(184\) −1.20828 −0.0890759
\(185\) 7.73569 0.568739
\(186\) 0 0
\(187\) 11.2046 0.819359
\(188\) 2.68337 0.195705
\(189\) 0 0
\(190\) 7.60228 0.551528
\(191\) 10.4642 0.757162 0.378581 0.925568i \(-0.376412\pi\)
0.378581 + 0.925568i \(0.376412\pi\)
\(192\) 0 0
\(193\) −8.19988 −0.590240 −0.295120 0.955460i \(-0.595360\pi\)
−0.295120 + 0.955460i \(0.595360\pi\)
\(194\) −9.20457 −0.660850
\(195\) 0 0
\(196\) 13.7022 0.978730
\(197\) −11.8954 −0.847510 −0.423755 0.905777i \(-0.639288\pi\)
−0.423755 + 0.905777i \(0.639288\pi\)
\(198\) 0 0
\(199\) 8.60698 0.610132 0.305066 0.952331i \(-0.401321\pi\)
0.305066 + 0.952331i \(0.401321\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 3.18572 0.224146
\(203\) 27.7759 1.94948
\(204\) 0 0
\(205\) 6.10463 0.426366
\(206\) 18.8356 1.31234
\(207\) 0 0
\(208\) −1.05232 −0.0729649
\(209\) −19.3856 −1.34093
\(210\) 0 0
\(211\) −1.92360 −0.132426 −0.0662132 0.997805i \(-0.521092\pi\)
−0.0662132 + 0.997805i \(0.521092\pi\)
\(212\) −0.502345 −0.0345012
\(213\) 0 0
\(214\) −14.2857 −0.976547
\(215\) −6.28566 −0.428678
\(216\) 0 0
\(217\) 4.54997 0.308872
\(218\) −5.78331 −0.391695
\(219\) 0 0
\(220\) −2.54997 −0.171919
\(221\) 4.62387 0.311035
\(222\) 0 0
\(223\) 21.8855 1.46556 0.732779 0.680467i \(-0.238223\pi\)
0.732779 + 0.680467i \(0.238223\pi\)
\(224\) 4.54997 0.304008
\(225\) 0 0
\(226\) 8.01885 0.533406
\(227\) 13.4902 0.895378 0.447689 0.894189i \(-0.352247\pi\)
0.447689 + 0.894189i \(0.352247\pi\)
\(228\) 0 0
\(229\) 23.4677 1.55079 0.775393 0.631479i \(-0.217552\pi\)
0.775393 + 0.631479i \(0.217552\pi\)
\(230\) −1.20828 −0.0796719
\(231\) 0 0
\(232\) 6.10463 0.400789
\(233\) −21.9643 −1.43893 −0.719466 0.694528i \(-0.755613\pi\)
−0.719466 + 0.694528i \(0.755613\pi\)
\(234\) 0 0
\(235\) 2.68337 0.175044
\(236\) −10.8356 −0.705339
\(237\) 0 0
\(238\) −19.9926 −1.29593
\(239\) 12.6164 0.816090 0.408045 0.912962i \(-0.366211\pi\)
0.408045 + 0.912962i \(0.366211\pi\)
\(240\) 0 0
\(241\) −18.7880 −1.21024 −0.605121 0.796134i \(-0.706875\pi\)
−0.605121 + 0.796134i \(0.706875\pi\)
\(242\) −4.49765 −0.289120
\(243\) 0 0
\(244\) −9.80588 −0.627757
\(245\) 13.7022 0.875403
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 1.00000 0.0635001
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 16.5688 1.04581 0.522907 0.852390i \(-0.324847\pi\)
0.522907 + 0.852390i \(0.324847\pi\)
\(252\) 0 0
\(253\) 3.08109 0.193706
\(254\) −7.15695 −0.449067
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −29.8115 −1.85959 −0.929797 0.368074i \(-0.880017\pi\)
−0.929797 + 0.368074i \(0.880017\pi\)
\(258\) 0 0
\(259\) 35.1971 2.18704
\(260\) −1.05232 −0.0652618
\(261\) 0 0
\(262\) −4.46419 −0.275799
\(263\) 12.0151 0.740885 0.370443 0.928855i \(-0.379206\pi\)
0.370443 + 0.928855i \(0.379206\pi\)
\(264\) 0 0
\(265\) −0.502345 −0.0308588
\(266\) 34.5902 2.12086
\(267\) 0 0
\(268\) 7.46888 0.456235
\(269\) −23.2046 −1.41481 −0.707404 0.706810i \(-0.750134\pi\)
−0.707404 + 0.706810i \(0.750134\pi\)
\(270\) 0 0
\(271\) −10.7116 −0.650684 −0.325342 0.945596i \(-0.605479\pi\)
−0.325342 + 0.945596i \(0.605479\pi\)
\(272\) −4.39400 −0.266425
\(273\) 0 0
\(274\) 17.0104 1.02764
\(275\) −2.54997 −0.153769
\(276\) 0 0
\(277\) 9.14756 0.549624 0.274812 0.961498i \(-0.411384\pi\)
0.274812 + 0.961498i \(0.411384\pi\)
\(278\) −7.49394 −0.449457
\(279\) 0 0
\(280\) 4.54997 0.271913
\(281\) −12.2093 −0.728343 −0.364172 0.931332i \(-0.618648\pi\)
−0.364172 + 0.931332i \(0.618648\pi\)
\(282\) 0 0
\(283\) 8.26431 0.491262 0.245631 0.969363i \(-0.421005\pi\)
0.245631 + 0.969363i \(0.421005\pi\)
\(284\) −8.65460 −0.513556
\(285\) 0 0
\(286\) 2.68337 0.158671
\(287\) 27.7759 1.63956
\(288\) 0 0
\(289\) 2.30725 0.135720
\(290\) 6.10463 0.358476
\(291\) 0 0
\(292\) −8.70222 −0.509259
\(293\) −9.68806 −0.565983 −0.282991 0.959122i \(-0.591327\pi\)
−0.282991 + 0.959122i \(0.591327\pi\)
\(294\) 0 0
\(295\) −10.8356 −0.630875
\(296\) 7.73569 0.449628
\(297\) 0 0
\(298\) 2.18103 0.126343
\(299\) 1.27150 0.0735325
\(300\) 0 0
\(301\) −28.5995 −1.64845
\(302\) 4.78800 0.275519
\(303\) 0 0
\(304\) 7.60228 0.436021
\(305\) −9.80588 −0.561483
\(306\) 0 0
\(307\) −24.6641 −1.40765 −0.703826 0.710372i \(-0.748527\pi\)
−0.703826 + 0.710372i \(0.748527\pi\)
\(308\) −11.6023 −0.661102
\(309\) 0 0
\(310\) 1.00000 0.0567962
\(311\) 7.64044 0.433250 0.216625 0.976255i \(-0.430495\pi\)
0.216625 + 0.976255i \(0.430495\pi\)
\(312\) 0 0
\(313\) −13.7927 −0.779609 −0.389805 0.920898i \(-0.627457\pi\)
−0.389805 + 0.920898i \(0.627457\pi\)
\(314\) 10.0189 0.565397
\(315\) 0 0
\(316\) −0.918913 −0.0516929
\(317\) 4.19988 0.235889 0.117944 0.993020i \(-0.462370\pi\)
0.117944 + 0.993020i \(0.462370\pi\)
\(318\) 0 0
\(319\) −15.5666 −0.871564
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −5.49765 −0.306372
\(323\) −33.4044 −1.85867
\(324\) 0 0
\(325\) −1.05232 −0.0583719
\(326\) 13.6311 0.754954
\(327\) 0 0
\(328\) 6.10463 0.337072
\(329\) 12.2093 0.673118
\(330\) 0 0
\(331\) 1.50332 0.0826301 0.0413150 0.999146i \(-0.486845\pi\)
0.0413150 + 0.999146i \(0.486845\pi\)
\(332\) 2.41657 0.132626
\(333\) 0 0
\(334\) −19.9869 −1.09363
\(335\) 7.46888 0.408069
\(336\) 0 0
\(337\) −23.7833 −1.29556 −0.647780 0.761828i \(-0.724302\pi\)
−0.647780 + 0.761828i \(0.724302\pi\)
\(338\) −11.8926 −0.646874
\(339\) 0 0
\(340\) −4.39400 −0.238298
\(341\) −2.54997 −0.138089
\(342\) 0 0
\(343\) 30.4949 1.64657
\(344\) −6.28566 −0.338900
\(345\) 0 0
\(346\) −14.5212 −0.780664
\(347\) −30.9879 −1.66352 −0.831758 0.555138i \(-0.812665\pi\)
−0.831758 + 0.555138i \(0.812665\pi\)
\(348\) 0 0
\(349\) −30.6090 −1.63846 −0.819232 0.573463i \(-0.805600\pi\)
−0.819232 + 0.573463i \(0.805600\pi\)
\(350\) 4.54997 0.243206
\(351\) 0 0
\(352\) −2.54997 −0.135914
\(353\) −25.3224 −1.34777 −0.673887 0.738834i \(-0.735377\pi\)
−0.673887 + 0.738834i \(0.735377\pi\)
\(354\) 0 0
\(355\) −8.65460 −0.459338
\(356\) 16.4379 0.871208
\(357\) 0 0
\(358\) −16.5688 −0.875689
\(359\) 17.8498 0.942076 0.471038 0.882113i \(-0.343880\pi\)
0.471038 + 0.882113i \(0.343880\pi\)
\(360\) 0 0
\(361\) 38.7947 2.04183
\(362\) 9.31291 0.489476
\(363\) 0 0
\(364\) −4.78800 −0.250960
\(365\) −8.70222 −0.455495
\(366\) 0 0
\(367\) −23.9901 −1.25227 −0.626136 0.779714i \(-0.715365\pi\)
−0.626136 + 0.779714i \(0.715365\pi\)
\(368\) −1.20828 −0.0629861
\(369\) 0 0
\(370\) 7.73569 0.402159
\(371\) −2.28566 −0.118665
\(372\) 0 0
\(373\) 5.92360 0.306713 0.153356 0.988171i \(-0.450992\pi\)
0.153356 + 0.988171i \(0.450992\pi\)
\(374\) 11.2046 0.579375
\(375\) 0 0
\(376\) 2.68337 0.138384
\(377\) −6.42400 −0.330853
\(378\) 0 0
\(379\) −19.9068 −1.02254 −0.511272 0.859419i \(-0.670825\pi\)
−0.511272 + 0.859419i \(0.670825\pi\)
\(380\) 7.60228 0.389989
\(381\) 0 0
\(382\) 10.4642 0.535395
\(383\) −32.8031 −1.67616 −0.838081 0.545546i \(-0.816322\pi\)
−0.838081 + 0.545546i \(0.816322\pi\)
\(384\) 0 0
\(385\) −11.6023 −0.591307
\(386\) −8.19988 −0.417363
\(387\) 0 0
\(388\) −9.20457 −0.467291
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) 5.30920 0.268498
\(392\) 13.7022 0.692067
\(393\) 0 0
\(394\) −11.8954 −0.599280
\(395\) −0.918913 −0.0462355
\(396\) 0 0
\(397\) 11.2903 0.566646 0.283323 0.959024i \(-0.408563\pi\)
0.283323 + 0.959024i \(0.408563\pi\)
\(398\) 8.60698 0.431429
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 27.7000 1.38327 0.691637 0.722246i \(-0.256890\pi\)
0.691637 + 0.722246i \(0.256890\pi\)
\(402\) 0 0
\(403\) −1.05232 −0.0524196
\(404\) 3.18572 0.158495
\(405\) 0 0
\(406\) 27.7759 1.37849
\(407\) −19.7258 −0.977770
\(408\) 0 0
\(409\) 22.1905 1.09725 0.548625 0.836069i \(-0.315152\pi\)
0.548625 + 0.836069i \(0.315152\pi\)
\(410\) 6.10463 0.301486
\(411\) 0 0
\(412\) 18.8356 0.927965
\(413\) −49.3018 −2.42598
\(414\) 0 0
\(415\) 2.41657 0.118625
\(416\) −1.05232 −0.0515940
\(417\) 0 0
\(418\) −19.3856 −0.948181
\(419\) 23.0449 1.12582 0.562908 0.826519i \(-0.309682\pi\)
0.562908 + 0.826519i \(0.309682\pi\)
\(420\) 0 0
\(421\) −26.3621 −1.28481 −0.642404 0.766366i \(-0.722063\pi\)
−0.642404 + 0.766366i \(0.722063\pi\)
\(422\) −1.92360 −0.0936396
\(423\) 0 0
\(424\) −0.502345 −0.0243961
\(425\) −4.39400 −0.213140
\(426\) 0 0
\(427\) −44.6164 −2.15914
\(428\) −14.2857 −0.690523
\(429\) 0 0
\(430\) −6.28566 −0.303121
\(431\) 12.2643 0.590751 0.295376 0.955381i \(-0.404555\pi\)
0.295376 + 0.955381i \(0.404555\pi\)
\(432\) 0 0
\(433\) −25.0643 −1.20451 −0.602256 0.798303i \(-0.705731\pi\)
−0.602256 + 0.798303i \(0.705731\pi\)
\(434\) 4.54997 0.218406
\(435\) 0 0
\(436\) −5.78331 −0.276970
\(437\) −9.18572 −0.439412
\(438\) 0 0
\(439\) 13.3092 0.635213 0.317607 0.948223i \(-0.397121\pi\)
0.317607 + 0.948223i \(0.397121\pi\)
\(440\) −2.54997 −0.121565
\(441\) 0 0
\(442\) 4.62387 0.219935
\(443\) −35.4808 −1.68575 −0.842873 0.538113i \(-0.819137\pi\)
−0.842873 + 0.538113i \(0.819137\pi\)
\(444\) 0 0
\(445\) 16.4379 0.779232
\(446\) 21.8855 1.03631
\(447\) 0 0
\(448\) 4.54997 0.214966
\(449\) −32.0476 −1.51242 −0.756210 0.654328i \(-0.772951\pi\)
−0.756210 + 0.654328i \(0.772951\pi\)
\(450\) 0 0
\(451\) −15.5666 −0.733004
\(452\) 8.01885 0.377175
\(453\) 0 0
\(454\) 13.4902 0.633128
\(455\) −4.78800 −0.224465
\(456\) 0 0
\(457\) −3.38355 −0.158276 −0.0791380 0.996864i \(-0.525217\pi\)
−0.0791380 + 0.996864i \(0.525217\pi\)
\(458\) 23.4677 1.09657
\(459\) 0 0
\(460\) −1.20828 −0.0563365
\(461\) 4.83314 0.225102 0.112551 0.993646i \(-0.464098\pi\)
0.112551 + 0.993646i \(0.464098\pi\)
\(462\) 0 0
\(463\) 26.5143 1.23222 0.616112 0.787658i \(-0.288707\pi\)
0.616112 + 0.787658i \(0.288707\pi\)
\(464\) 6.10463 0.283400
\(465\) 0 0
\(466\) −21.9643 −1.01748
\(467\) −21.4620 −0.993143 −0.496571 0.867996i \(-0.665408\pi\)
−0.496571 + 0.867996i \(0.665408\pi\)
\(468\) 0 0
\(469\) 33.9832 1.56920
\(470\) 2.68337 0.123775
\(471\) 0 0
\(472\) −10.8356 −0.498750
\(473\) 16.0282 0.736979
\(474\) 0 0
\(475\) 7.60228 0.348817
\(476\) −19.9926 −0.916358
\(477\) 0 0
\(478\) 12.6164 0.577063
\(479\) 13.7545 0.628461 0.314230 0.949347i \(-0.398254\pi\)
0.314230 + 0.949347i \(0.398254\pi\)
\(480\) 0 0
\(481\) −8.14038 −0.371169
\(482\) −18.7880 −0.855770
\(483\) 0 0
\(484\) −4.49765 −0.204439
\(485\) −9.20457 −0.417958
\(486\) 0 0
\(487\) 11.3662 0.515052 0.257526 0.966271i \(-0.417093\pi\)
0.257526 + 0.966271i \(0.417093\pi\)
\(488\) −9.80588 −0.443891
\(489\) 0 0
\(490\) 13.7022 0.619003
\(491\) 37.7545 1.70384 0.851919 0.523673i \(-0.175439\pi\)
0.851919 + 0.523673i \(0.175439\pi\)
\(492\) 0 0
\(493\) −26.8238 −1.20808
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) −39.3782 −1.76635
\(498\) 0 0
\(499\) −2.92263 −0.130835 −0.0654174 0.997858i \(-0.520838\pi\)
−0.0654174 + 0.997858i \(0.520838\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 16.5688 0.739503
\(503\) −6.31389 −0.281522 −0.140761 0.990044i \(-0.544955\pi\)
−0.140761 + 0.990044i \(0.544955\pi\)
\(504\) 0 0
\(505\) 3.18572 0.141763
\(506\) 3.08109 0.136971
\(507\) 0 0
\(508\) −7.15695 −0.317538
\(509\) 21.4044 0.948736 0.474368 0.880327i \(-0.342677\pi\)
0.474368 + 0.880327i \(0.342677\pi\)
\(510\) 0 0
\(511\) −39.5949 −1.75157
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −29.8115 −1.31493
\(515\) 18.8356 0.829997
\(516\) 0 0
\(517\) −6.84252 −0.300934
\(518\) 35.1971 1.54647
\(519\) 0 0
\(520\) −1.05232 −0.0461471
\(521\) 16.0952 0.705146 0.352573 0.935784i \(-0.385307\pi\)
0.352573 + 0.935784i \(0.385307\pi\)
\(522\) 0 0
\(523\) 1.23085 0.0538213 0.0269107 0.999638i \(-0.491433\pi\)
0.0269107 + 0.999638i \(0.491433\pi\)
\(524\) −4.46419 −0.195019
\(525\) 0 0
\(526\) 12.0151 0.523885
\(527\) −4.39400 −0.191406
\(528\) 0 0
\(529\) −21.5401 −0.936524
\(530\) −0.502345 −0.0218205
\(531\) 0 0
\(532\) 34.5902 1.49967
\(533\) −6.42400 −0.278254
\(534\) 0 0
\(535\) −14.2857 −0.617623
\(536\) 7.46888 0.322607
\(537\) 0 0
\(538\) −23.2046 −1.00042
\(539\) −34.9403 −1.50498
\(540\) 0 0
\(541\) 11.7927 0.507007 0.253504 0.967334i \(-0.418417\pi\)
0.253504 + 0.967334i \(0.418417\pi\)
\(542\) −10.7116 −0.460103
\(543\) 0 0
\(544\) −4.39400 −0.188391
\(545\) −5.78331 −0.247730
\(546\) 0 0
\(547\) −4.53112 −0.193737 −0.0968683 0.995297i \(-0.530883\pi\)
−0.0968683 + 0.995297i \(0.530883\pi\)
\(548\) 17.0104 0.726650
\(549\) 0 0
\(550\) −2.54997 −0.108731
\(551\) 46.4091 1.97710
\(552\) 0 0
\(553\) −4.18103 −0.177795
\(554\) 9.14756 0.388643
\(555\) 0 0
\(556\) −7.49394 −0.317814
\(557\) −3.71434 −0.157382 −0.0786909 0.996899i \(-0.525074\pi\)
−0.0786909 + 0.996899i \(0.525074\pi\)
\(558\) 0 0
\(559\) 6.61449 0.279763
\(560\) 4.54997 0.192271
\(561\) 0 0
\(562\) −12.2093 −0.515017
\(563\) 32.3998 1.36549 0.682743 0.730658i \(-0.260787\pi\)
0.682743 + 0.730658i \(0.260787\pi\)
\(564\) 0 0
\(565\) 8.01885 0.337356
\(566\) 8.26431 0.347375
\(567\) 0 0
\(568\) −8.65460 −0.363139
\(569\) 17.8498 0.748302 0.374151 0.927368i \(-0.377934\pi\)
0.374151 + 0.927368i \(0.377934\pi\)
\(570\) 0 0
\(571\) 40.6817 1.70248 0.851238 0.524780i \(-0.175852\pi\)
0.851238 + 0.524780i \(0.175852\pi\)
\(572\) 2.68337 0.112197
\(573\) 0 0
\(574\) 27.7759 1.15934
\(575\) −1.20828 −0.0503889
\(576\) 0 0
\(577\) −6.17156 −0.256925 −0.128463 0.991714i \(-0.541004\pi\)
−0.128463 + 0.991714i \(0.541004\pi\)
\(578\) 2.30725 0.0959688
\(579\) 0 0
\(580\) 6.10463 0.253481
\(581\) 10.9953 0.456162
\(582\) 0 0
\(583\) 1.28097 0.0530522
\(584\) −8.70222 −0.360101
\(585\) 0 0
\(586\) −9.68806 −0.400210
\(587\) −6.31389 −0.260602 −0.130301 0.991474i \(-0.541594\pi\)
−0.130301 + 0.991474i \(0.541594\pi\)
\(588\) 0 0
\(589\) 7.60228 0.313247
\(590\) −10.8356 −0.446096
\(591\) 0 0
\(592\) 7.73569 0.317935
\(593\) 32.3640 1.32903 0.664515 0.747275i \(-0.268638\pi\)
0.664515 + 0.747275i \(0.268638\pi\)
\(594\) 0 0
\(595\) −19.9926 −0.819616
\(596\) 2.18103 0.0893383
\(597\) 0 0
\(598\) 1.27150 0.0519953
\(599\) −13.9638 −0.570545 −0.285273 0.958446i \(-0.592084\pi\)
−0.285273 + 0.958446i \(0.592084\pi\)
\(600\) 0 0
\(601\) −7.84525 −0.320015 −0.160007 0.987116i \(-0.551152\pi\)
−0.160007 + 0.987116i \(0.551152\pi\)
\(602\) −28.5995 −1.16563
\(603\) 0 0
\(604\) 4.78800 0.194821
\(605\) −4.49765 −0.182856
\(606\) 0 0
\(607\) −26.8262 −1.08884 −0.544420 0.838813i \(-0.683250\pi\)
−0.544420 + 0.838813i \(0.683250\pi\)
\(608\) 7.60228 0.308313
\(609\) 0 0
\(610\) −9.80588 −0.397028
\(611\) −2.82375 −0.114237
\(612\) 0 0
\(613\) −31.6330 −1.27765 −0.638823 0.769354i \(-0.720578\pi\)
−0.638823 + 0.769354i \(0.720578\pi\)
\(614\) −24.6641 −0.995361
\(615\) 0 0
\(616\) −11.6023 −0.467469
\(617\) −0.235541 −0.00948253 −0.00474126 0.999989i \(-0.501509\pi\)
−0.00474126 + 0.999989i \(0.501509\pi\)
\(618\) 0 0
\(619\) −36.3271 −1.46011 −0.730054 0.683389i \(-0.760505\pi\)
−0.730054 + 0.683389i \(0.760505\pi\)
\(620\) 1.00000 0.0401610
\(621\) 0 0
\(622\) 7.64044 0.306354
\(623\) 74.7920 2.99648
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −13.7927 −0.551267
\(627\) 0 0
\(628\) 10.0189 0.399796
\(629\) −33.9906 −1.35529
\(630\) 0 0
\(631\) −28.4280 −1.13170 −0.565850 0.824508i \(-0.691452\pi\)
−0.565850 + 0.824508i \(0.691452\pi\)
\(632\) −0.918913 −0.0365524
\(633\) 0 0
\(634\) 4.19988 0.166798
\(635\) −7.15695 −0.284015
\(636\) 0 0
\(637\) −14.4191 −0.571304
\(638\) −15.5666 −0.616288
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 30.1071 1.18916 0.594580 0.804037i \(-0.297318\pi\)
0.594580 + 0.804037i \(0.297318\pi\)
\(642\) 0 0
\(643\) −7.22147 −0.284787 −0.142393 0.989810i \(-0.545480\pi\)
−0.142393 + 0.989810i \(0.545480\pi\)
\(644\) −5.49765 −0.216638
\(645\) 0 0
\(646\) −33.4044 −1.31428
\(647\) 42.3960 1.66676 0.833380 0.552700i \(-0.186402\pi\)
0.833380 + 0.552700i \(0.186402\pi\)
\(648\) 0 0
\(649\) 27.6305 1.08459
\(650\) −1.05232 −0.0412752
\(651\) 0 0
\(652\) 13.6311 0.533833
\(653\) 8.68337 0.339807 0.169903 0.985461i \(-0.445654\pi\)
0.169903 + 0.985461i \(0.445654\pi\)
\(654\) 0 0
\(655\) −4.46419 −0.174430
\(656\) 6.10463 0.238346
\(657\) 0 0
\(658\) 12.2093 0.475967
\(659\) −34.5163 −1.34456 −0.672281 0.740296i \(-0.734685\pi\)
−0.672281 + 0.740296i \(0.734685\pi\)
\(660\) 0 0
\(661\) −38.7806 −1.50839 −0.754195 0.656651i \(-0.771973\pi\)
−0.754195 + 0.656651i \(0.771973\pi\)
\(662\) 1.50332 0.0584283
\(663\) 0 0
\(664\) 2.41657 0.0937810
\(665\) 34.5902 1.34135
\(666\) 0 0
\(667\) −7.37613 −0.285605
\(668\) −19.9869 −0.773317
\(669\) 0 0
\(670\) 7.46888 0.288548
\(671\) 25.0047 0.965295
\(672\) 0 0
\(673\) −24.5212 −0.945223 −0.472611 0.881271i \(-0.656689\pi\)
−0.472611 + 0.881271i \(0.656689\pi\)
\(674\) −23.7833 −0.916099
\(675\) 0 0
\(676\) −11.8926 −0.457409
\(677\) 36.1254 1.38841 0.694207 0.719776i \(-0.255755\pi\)
0.694207 + 0.719776i \(0.255755\pi\)
\(678\) 0 0
\(679\) −41.8805 −1.60723
\(680\) −4.39400 −0.168502
\(681\) 0 0
\(682\) −2.54997 −0.0976434
\(683\) 40.3140 1.54257 0.771286 0.636489i \(-0.219614\pi\)
0.771286 + 0.636489i \(0.219614\pi\)
\(684\) 0 0
\(685\) 17.0104 0.649936
\(686\) 30.4949 1.16430
\(687\) 0 0
\(688\) −6.28566 −0.239638
\(689\) 0.528626 0.0201390
\(690\) 0 0
\(691\) −9.29973 −0.353778 −0.176889 0.984231i \(-0.556603\pi\)
−0.176889 + 0.984231i \(0.556603\pi\)
\(692\) −14.5212 −0.552013
\(693\) 0 0
\(694\) −30.9879 −1.17628
\(695\) −7.49394 −0.284261
\(696\) 0 0
\(697\) −26.8238 −1.01602
\(698\) −30.6090 −1.15857
\(699\) 0 0
\(700\) 4.54997 0.171973
\(701\) −34.5808 −1.30610 −0.653049 0.757316i \(-0.726510\pi\)
−0.653049 + 0.757316i \(0.726510\pi\)
\(702\) 0 0
\(703\) 58.8089 2.21802
\(704\) −2.54997 −0.0961056
\(705\) 0 0
\(706\) −25.3224 −0.953021
\(707\) 14.4949 0.545137
\(708\) 0 0
\(709\) −6.16589 −0.231565 −0.115782 0.993275i \(-0.536938\pi\)
−0.115782 + 0.993275i \(0.536938\pi\)
\(710\) −8.65460 −0.324801
\(711\) 0 0
\(712\) 16.4379 0.616037
\(713\) −1.20828 −0.0452506
\(714\) 0 0
\(715\) 2.68337 0.100352
\(716\) −16.5688 −0.619206
\(717\) 0 0
\(718\) 17.8498 0.666148
\(719\) 3.63052 0.135396 0.0676978 0.997706i \(-0.478435\pi\)
0.0676978 + 0.997706i \(0.478435\pi\)
\(720\) 0 0
\(721\) 85.7015 3.19169
\(722\) 38.7947 1.44379
\(723\) 0 0
\(724\) 9.31291 0.346112
\(725\) 6.10463 0.226720
\(726\) 0 0
\(727\) 31.2636 1.15950 0.579752 0.814793i \(-0.303150\pi\)
0.579752 + 0.814793i \(0.303150\pi\)
\(728\) −4.78800 −0.177455
\(729\) 0 0
\(730\) −8.70222 −0.322084
\(731\) 27.6192 1.02153
\(732\) 0 0
\(733\) −9.30920 −0.343843 −0.171922 0.985111i \(-0.554998\pi\)
−0.171922 + 0.985111i \(0.554998\pi\)
\(734\) −23.9901 −0.885490
\(735\) 0 0
\(736\) −1.20828 −0.0445379
\(737\) −19.0454 −0.701547
\(738\) 0 0
\(739\) 7.13189 0.262351 0.131175 0.991359i \(-0.458125\pi\)
0.131175 + 0.991359i \(0.458125\pi\)
\(740\) 7.73569 0.284370
\(741\) 0 0
\(742\) −2.28566 −0.0839091
\(743\) 23.1989 0.851085 0.425543 0.904938i \(-0.360083\pi\)
0.425543 + 0.904938i \(0.360083\pi\)
\(744\) 0 0
\(745\) 2.18103 0.0799066
\(746\) 5.92360 0.216879
\(747\) 0 0
\(748\) 11.2046 0.409680
\(749\) −64.9993 −2.37502
\(750\) 0 0
\(751\) −37.6713 −1.37464 −0.687322 0.726353i \(-0.741214\pi\)
−0.687322 + 0.726353i \(0.741214\pi\)
\(752\) 2.68337 0.0978525
\(753\) 0 0
\(754\) −6.42400 −0.233948
\(755\) 4.78800 0.174253
\(756\) 0 0
\(757\) 9.41437 0.342171 0.171086 0.985256i \(-0.445273\pi\)
0.171086 + 0.985256i \(0.445273\pi\)
\(758\) −19.9068 −0.723047
\(759\) 0 0
\(760\) 7.60228 0.275764
\(761\) 1.39991 0.0507469 0.0253734 0.999678i \(-0.491923\pi\)
0.0253734 + 0.999678i \(0.491923\pi\)
\(762\) 0 0
\(763\) −26.3139 −0.952627
\(764\) 10.4642 0.378581
\(765\) 0 0
\(766\) −32.8031 −1.18523
\(767\) 11.4025 0.411720
\(768\) 0 0
\(769\) 51.4327 1.85471 0.927355 0.374183i \(-0.122077\pi\)
0.927355 + 0.374183i \(0.122077\pi\)
\(770\) −11.6023 −0.418117
\(771\) 0 0
\(772\) −8.19988 −0.295120
\(773\) −41.4283 −1.49007 −0.745036 0.667024i \(-0.767568\pi\)
−0.745036 + 0.667024i \(0.767568\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) −9.20457 −0.330425
\(777\) 0 0
\(778\) −12.0000 −0.430221
\(779\) 46.4091 1.66278
\(780\) 0 0
\(781\) 22.0690 0.789690
\(782\) 5.30920 0.189857
\(783\) 0 0
\(784\) 13.7022 0.489365
\(785\) 10.0189 0.357588
\(786\) 0 0
\(787\) −20.0858 −0.715981 −0.357990 0.933725i \(-0.616538\pi\)
−0.357990 + 0.933725i \(0.616538\pi\)
\(788\) −11.8954 −0.423755
\(789\) 0 0
\(790\) −0.918913 −0.0326935
\(791\) 36.4855 1.29728
\(792\) 0 0
\(793\) 10.3189 0.366434
\(794\) 11.2903 0.400679
\(795\) 0 0
\(796\) 8.60698 0.305066
\(797\) −39.1376 −1.38633 −0.693163 0.720781i \(-0.743784\pi\)
−0.693163 + 0.720781i \(0.743784\pi\)
\(798\) 0 0
\(799\) −11.7907 −0.417126
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 27.7000 0.978122
\(803\) 22.1904 0.783083
\(804\) 0 0
\(805\) −5.49765 −0.193767
\(806\) −1.05232 −0.0370662
\(807\) 0 0
\(808\) 3.18572 0.112073
\(809\) −18.1805 −0.639192 −0.319596 0.947554i \(-0.603547\pi\)
−0.319596 + 0.947554i \(0.603547\pi\)
\(810\) 0 0
\(811\) −2.06897 −0.0726513 −0.0363256 0.999340i \(-0.511565\pi\)
−0.0363256 + 0.999340i \(0.511565\pi\)
\(812\) 27.7759 0.974742
\(813\) 0 0
\(814\) −19.7258 −0.691388
\(815\) 13.6311 0.477475
\(816\) 0 0
\(817\) −47.7854 −1.67180
\(818\) 22.1905 0.775873
\(819\) 0 0
\(820\) 6.10463 0.213183
\(821\) −10.9953 −0.383739 −0.191869 0.981420i \(-0.561455\pi\)
−0.191869 + 0.981420i \(0.561455\pi\)
\(822\) 0 0
\(823\) 29.7376 1.03659 0.518295 0.855202i \(-0.326567\pi\)
0.518295 + 0.855202i \(0.326567\pi\)
\(824\) 18.8356 0.656170
\(825\) 0 0
\(826\) −49.3018 −1.71543
\(827\) −17.3092 −0.601900 −0.300950 0.953640i \(-0.597304\pi\)
−0.300950 + 0.953640i \(0.597304\pi\)
\(828\) 0 0
\(829\) 39.7722 1.38134 0.690672 0.723168i \(-0.257315\pi\)
0.690672 + 0.723168i \(0.257315\pi\)
\(830\) 2.41657 0.0838803
\(831\) 0 0
\(832\) −1.05232 −0.0364825
\(833\) −60.2076 −2.08607
\(834\) 0 0
\(835\) −19.9869 −0.691675
\(836\) −19.3856 −0.670465
\(837\) 0 0
\(838\) 23.0449 0.796072
\(839\) 18.3834 0.634665 0.317333 0.948314i \(-0.397213\pi\)
0.317333 + 0.948314i \(0.397213\pi\)
\(840\) 0 0
\(841\) 8.26651 0.285052
\(842\) −26.3621 −0.908496
\(843\) 0 0
\(844\) −1.92360 −0.0662132
\(845\) −11.8926 −0.409119
\(846\) 0 0
\(847\) −20.4642 −0.703158
\(848\) −0.502345 −0.0172506
\(849\) 0 0
\(850\) −4.39400 −0.150713
\(851\) −9.34691 −0.320408
\(852\) 0 0
\(853\) −29.9663 −1.02603 −0.513013 0.858381i \(-0.671471\pi\)
−0.513013 + 0.858381i \(0.671471\pi\)
\(854\) −44.6164 −1.52674
\(855\) 0 0
\(856\) −14.2857 −0.488274
\(857\) −38.8183 −1.32601 −0.663004 0.748616i \(-0.730719\pi\)
−0.663004 + 0.748616i \(0.730719\pi\)
\(858\) 0 0
\(859\) −52.3009 −1.78448 −0.892242 0.451558i \(-0.850868\pi\)
−0.892242 + 0.451558i \(0.850868\pi\)
\(860\) −6.28566 −0.214339
\(861\) 0 0
\(862\) 12.2643 0.417724
\(863\) 15.0084 0.510892 0.255446 0.966823i \(-0.417778\pi\)
0.255446 + 0.966823i \(0.417778\pi\)
\(864\) 0 0
\(865\) −14.5212 −0.493736
\(866\) −25.0643 −0.851719
\(867\) 0 0
\(868\) 4.54997 0.154436
\(869\) 2.34320 0.0794876
\(870\) 0 0
\(871\) −7.85962 −0.266313
\(872\) −5.78331 −0.195848
\(873\) 0 0
\(874\) −9.18572 −0.310712
\(875\) 4.54997 0.153817
\(876\) 0 0
\(877\) −47.0757 −1.58963 −0.794817 0.606850i \(-0.792433\pi\)
−0.794817 + 0.606850i \(0.792433\pi\)
\(878\) 13.3092 0.449164
\(879\) 0 0
\(880\) −2.54997 −0.0859595
\(881\) 55.3400 1.86445 0.932226 0.361876i \(-0.117864\pi\)
0.932226 + 0.361876i \(0.117864\pi\)
\(882\) 0 0
\(883\) −41.3112 −1.39023 −0.695117 0.718897i \(-0.744647\pi\)
−0.695117 + 0.718897i \(0.744647\pi\)
\(884\) 4.62387 0.155518
\(885\) 0 0
\(886\) −35.4808 −1.19200
\(887\) −17.4546 −0.586067 −0.293033 0.956102i \(-0.594665\pi\)
−0.293033 + 0.956102i \(0.594665\pi\)
\(888\) 0 0
\(889\) −32.5639 −1.09216
\(890\) 16.4379 0.551000
\(891\) 0 0
\(892\) 21.8855 0.732779
\(893\) 20.3998 0.682652
\(894\) 0 0
\(895\) −16.5688 −0.553835
\(896\) 4.54997 0.152004
\(897\) 0 0
\(898\) −32.0476 −1.06944
\(899\) 6.10463 0.203601
\(900\) 0 0
\(901\) 2.20731 0.0735360
\(902\) −15.5666 −0.518312
\(903\) 0 0
\(904\) 8.01885 0.266703
\(905\) 9.31291 0.309572
\(906\) 0 0
\(907\) −1.45950 −0.0484619 −0.0242310 0.999706i \(-0.507714\pi\)
−0.0242310 + 0.999706i \(0.507714\pi\)
\(908\) 13.4902 0.447689
\(909\) 0 0
\(910\) −4.78800 −0.158721
\(911\) −22.6710 −0.751122 −0.375561 0.926798i \(-0.622550\pi\)
−0.375561 + 0.926798i \(0.622550\pi\)
\(912\) 0 0
\(913\) −6.16217 −0.203938
\(914\) −3.38355 −0.111918
\(915\) 0 0
\(916\) 23.4677 0.775393
\(917\) −20.3119 −0.670759
\(918\) 0 0
\(919\) −19.2423 −0.634744 −0.317372 0.948301i \(-0.602800\pi\)
−0.317372 + 0.948301i \(0.602800\pi\)
\(920\) −1.20828 −0.0398359
\(921\) 0 0
\(922\) 4.83314 0.159171
\(923\) 9.10737 0.299773
\(924\) 0 0
\(925\) 7.73569 0.254348
\(926\) 26.5143 0.871314
\(927\) 0 0
\(928\) 6.10463 0.200394
\(929\) 42.0015 1.37802 0.689012 0.724750i \(-0.258045\pi\)
0.689012 + 0.724750i \(0.258045\pi\)
\(930\) 0 0
\(931\) 104.168 3.41398
\(932\) −21.9643 −0.719466
\(933\) 0 0
\(934\) −21.4620 −0.702258
\(935\) 11.2046 0.366429
\(936\) 0 0
\(937\) −8.40914 −0.274715 −0.137357 0.990522i \(-0.543861\pi\)
−0.137357 + 0.990522i \(0.543861\pi\)
\(938\) 33.9832 1.10959
\(939\) 0 0
\(940\) 2.68337 0.0875219
\(941\) 21.4044 0.697765 0.348883 0.937166i \(-0.386561\pi\)
0.348883 + 0.937166i \(0.386561\pi\)
\(942\) 0 0
\(943\) −7.37613 −0.240200
\(944\) −10.8356 −0.352670
\(945\) 0 0
\(946\) 16.0282 0.521123
\(947\) 18.5118 0.601553 0.300777 0.953695i \(-0.402754\pi\)
0.300777 + 0.953695i \(0.402754\pi\)
\(948\) 0 0
\(949\) 9.15748 0.297264
\(950\) 7.60228 0.246651
\(951\) 0 0
\(952\) −19.9926 −0.647963
\(953\) 44.1123 1.42894 0.714469 0.699667i \(-0.246668\pi\)
0.714469 + 0.699667i \(0.246668\pi\)
\(954\) 0 0
\(955\) 10.4642 0.338613
\(956\) 12.6164 0.408045
\(957\) 0 0
\(958\) 13.7545 0.444389
\(959\) 77.3970 2.49928
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −8.14038 −0.262456
\(963\) 0 0
\(964\) −18.7880 −0.605121
\(965\) −8.19988 −0.263963
\(966\) 0 0
\(967\) 49.3995 1.58858 0.794291 0.607538i \(-0.207843\pi\)
0.794291 + 0.607538i \(0.207843\pi\)
\(968\) −4.49765 −0.144560
\(969\) 0 0
\(970\) −9.20457 −0.295541
\(971\) −14.8733 −0.477308 −0.238654 0.971105i \(-0.576706\pi\)
−0.238654 + 0.971105i \(0.576706\pi\)
\(972\) 0 0
\(973\) −34.0972 −1.09311
\(974\) 11.3662 0.364197
\(975\) 0 0
\(976\) −9.80588 −0.313878
\(977\) −4.58812 −0.146787 −0.0733935 0.997303i \(-0.523383\pi\)
−0.0733935 + 0.997303i \(0.523383\pi\)
\(978\) 0 0
\(979\) −41.9162 −1.33965
\(980\) 13.7022 0.437702
\(981\) 0 0
\(982\) 37.7545 1.20480
\(983\) 13.2178 0.421581 0.210790 0.977531i \(-0.432396\pi\)
0.210790 + 0.977531i \(0.432396\pi\)
\(984\) 0 0
\(985\) −11.8954 −0.379018
\(986\) −26.8238 −0.854242
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 7.59486 0.241502
\(990\) 0 0
\(991\) 6.45721 0.205120 0.102560 0.994727i \(-0.467297\pi\)
0.102560 + 0.994727i \(0.467297\pi\)
\(992\) 1.00000 0.0317500
\(993\) 0 0
\(994\) −39.3782 −1.24900
\(995\) 8.60698 0.272859
\(996\) 0 0
\(997\) 8.26680 0.261812 0.130906 0.991395i \(-0.458211\pi\)
0.130906 + 0.991395i \(0.458211\pi\)
\(998\) −2.92263 −0.0925141
\(999\) 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 2790.2.a.bk.1.4 yes 4
3.2 odd 2 2790.2.a.bj.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2790.2.a.bj.1.4 4 3.2 odd 2
2790.2.a.bk.1.4 yes 4 1.1 even 1 trivial