Properties

Label 4598.2.a.bx.1.5
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 2x^{7} - 10x^{6} + 16x^{5} + 26x^{4} - 32x^{3} - 16x^{2} + 20x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.03422\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.912987 q^{3} +1.00000 q^{4} -1.77724 q^{5} -0.912987 q^{6} +5.04753 q^{7} -1.00000 q^{8} -2.16645 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.912987 q^{3} +1.00000 q^{4} -1.77724 q^{5} -0.912987 q^{6} +5.04753 q^{7} -1.00000 q^{8} -2.16645 q^{9} +1.77724 q^{10} +0.912987 q^{12} -5.42371 q^{13} -5.04753 q^{14} -1.62260 q^{15} +1.00000 q^{16} +1.82249 q^{17} +2.16645 q^{18} +1.00000 q^{19} -1.77724 q^{20} +4.60833 q^{21} +5.61764 q^{23} -0.912987 q^{24} -1.84141 q^{25} +5.42371 q^{26} -4.71691 q^{27} +5.04753 q^{28} -2.23982 q^{29} +1.62260 q^{30} +0.386531 q^{31} -1.00000 q^{32} -1.82249 q^{34} -8.97068 q^{35} -2.16645 q^{36} -10.0927 q^{37} -1.00000 q^{38} -4.95177 q^{39} +1.77724 q^{40} -9.28332 q^{41} -4.60833 q^{42} -5.45957 q^{43} +3.85031 q^{45} -5.61764 q^{46} +10.8132 q^{47} +0.912987 q^{48} +18.4776 q^{49} +1.84141 q^{50} +1.66391 q^{51} -5.42371 q^{52} +6.85692 q^{53} +4.71691 q^{54} -5.04753 q^{56} +0.912987 q^{57} +2.23982 q^{58} +3.70481 q^{59} -1.62260 q^{60} -2.89301 q^{61} -0.386531 q^{62} -10.9352 q^{63} +1.00000 q^{64} +9.63923 q^{65} -11.5655 q^{67} +1.82249 q^{68} +5.12883 q^{69} +8.97068 q^{70} -0.872761 q^{71} +2.16645 q^{72} -9.78216 q^{73} +10.0927 q^{74} -1.68119 q^{75} +1.00000 q^{76} +4.95177 q^{78} -9.31807 q^{79} -1.77724 q^{80} +2.19288 q^{81} +9.28332 q^{82} +4.28702 q^{83} +4.60833 q^{84} -3.23901 q^{85} +5.45957 q^{86} -2.04493 q^{87} -5.60537 q^{89} -3.85031 q^{90} -27.3763 q^{91} +5.61764 q^{92} +0.352898 q^{93} -10.8132 q^{94} -1.77724 q^{95} -0.912987 q^{96} -2.90528 q^{97} -18.4776 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} + 2 q^{5} - 8 q^{7} - 8 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{4} + 2 q^{5} - 8 q^{7} - 8 q^{8} + 20 q^{9} - 2 q^{10} - 18 q^{13} + 8 q^{14} + 10 q^{15} + 8 q^{16} - 4 q^{17} - 20 q^{18} + 8 q^{19} + 2 q^{20} - 14 q^{21} + 12 q^{23} + 18 q^{26} - 24 q^{27} - 8 q^{28} - 14 q^{29} - 10 q^{30} - 2 q^{31} - 8 q^{32} + 4 q^{34} - 40 q^{35} + 20 q^{36} - 22 q^{37} - 8 q^{38} + 4 q^{39} - 2 q^{40} - 8 q^{41} + 14 q^{42} - 28 q^{43} - 28 q^{45} - 12 q^{46} + 6 q^{47} + 32 q^{49} + 12 q^{51} - 18 q^{52} - 24 q^{53} + 24 q^{54} + 8 q^{56} + 14 q^{58} + 46 q^{59} + 10 q^{60} + 24 q^{61} + 2 q^{62} - 30 q^{63} + 8 q^{64} + 16 q^{65} - 22 q^{67} - 4 q^{68} - 38 q^{69} + 40 q^{70} + 8 q^{71} - 20 q^{72} - 16 q^{73} + 22 q^{74} + 6 q^{75} + 8 q^{76} - 4 q^{78} - 4 q^{79} + 2 q^{80} + 28 q^{81} + 8 q^{82} - 12 q^{83} - 14 q^{84} - 48 q^{85} + 28 q^{86} - 42 q^{87} - 28 q^{89} + 28 q^{90} - 12 q^{91} + 12 q^{92} + 22 q^{93} - 6 q^{94} + 2 q^{95} - 22 q^{97} - 32 q^{98}+O(q^{100}) \) 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
\(3\) 0.912987 0.527114 0.263557 0.964644i \(-0.415104\pi\)
0.263557 + 0.964644i \(0.415104\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.77724 −0.794806 −0.397403 0.917644i \(-0.630089\pi\)
−0.397403 + 0.917644i \(0.630089\pi\)
\(6\) −0.912987 −0.372726
\(7\) 5.04753 1.90779 0.953894 0.300145i \(-0.0970351\pi\)
0.953894 + 0.300145i \(0.0970351\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.16645 −0.722151
\(10\) 1.77724 0.562013
\(11\) 0 0
\(12\) 0.912987 0.263557
\(13\) −5.42371 −1.50427 −0.752133 0.659012i \(-0.770975\pi\)
−0.752133 + 0.659012i \(0.770975\pi\)
\(14\) −5.04753 −1.34901
\(15\) −1.62260 −0.418953
\(16\) 1.00000 0.250000
\(17\) 1.82249 0.442020 0.221010 0.975272i \(-0.429065\pi\)
0.221010 + 0.975272i \(0.429065\pi\)
\(18\) 2.16645 0.510638
\(19\) 1.00000 0.229416
\(20\) −1.77724 −0.397403
\(21\) 4.60833 1.00562
\(22\) 0 0
\(23\) 5.61764 1.17136 0.585679 0.810543i \(-0.300828\pi\)
0.585679 + 0.810543i \(0.300828\pi\)
\(24\) −0.912987 −0.186363
\(25\) −1.84141 −0.368283
\(26\) 5.42371 1.06368
\(27\) −4.71691 −0.907769
\(28\) 5.04753 0.953894
\(29\) −2.23982 −0.415924 −0.207962 0.978137i \(-0.566683\pi\)
−0.207962 + 0.978137i \(0.566683\pi\)
\(30\) 1.62260 0.296245
\(31\) 0.386531 0.0694229 0.0347115 0.999397i \(-0.488949\pi\)
0.0347115 + 0.999397i \(0.488949\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −1.82249 −0.312555
\(35\) −8.97068 −1.51632
\(36\) −2.16645 −0.361076
\(37\) −10.0927 −1.65923 −0.829615 0.558335i \(-0.811440\pi\)
−0.829615 + 0.558335i \(0.811440\pi\)
\(38\) −1.00000 −0.162221
\(39\) −4.95177 −0.792919
\(40\) 1.77724 0.281006
\(41\) −9.28332 −1.44981 −0.724906 0.688848i \(-0.758117\pi\)
−0.724906 + 0.688848i \(0.758117\pi\)
\(42\) −4.60833 −0.711081
\(43\) −5.45957 −0.832576 −0.416288 0.909233i \(-0.636669\pi\)
−0.416288 + 0.909233i \(0.636669\pi\)
\(44\) 0 0
\(45\) 3.85031 0.573970
\(46\) −5.61764 −0.828275
\(47\) 10.8132 1.57727 0.788636 0.614860i \(-0.210787\pi\)
0.788636 + 0.614860i \(0.210787\pi\)
\(48\) 0.912987 0.131778
\(49\) 18.4776 2.63965
\(50\) 1.84141 0.260415
\(51\) 1.66391 0.232995
\(52\) −5.42371 −0.752133
\(53\) 6.85692 0.941870 0.470935 0.882168i \(-0.343917\pi\)
0.470935 + 0.882168i \(0.343917\pi\)
\(54\) 4.71691 0.641890
\(55\) 0 0
\(56\) −5.04753 −0.674505
\(57\) 0.912987 0.120928
\(58\) 2.23982 0.294103
\(59\) 3.70481 0.482325 0.241163 0.970485i \(-0.422471\pi\)
0.241163 + 0.970485i \(0.422471\pi\)
\(60\) −1.62260 −0.209477
\(61\) −2.89301 −0.370412 −0.185206 0.982700i \(-0.559295\pi\)
−0.185206 + 0.982700i \(0.559295\pi\)
\(62\) −0.386531 −0.0490894
\(63\) −10.9352 −1.37771
\(64\) 1.00000 0.125000
\(65\) 9.63923 1.19560
\(66\) 0 0
\(67\) −11.5655 −1.41295 −0.706474 0.707739i \(-0.749715\pi\)
−0.706474 + 0.707739i \(0.749715\pi\)
\(68\) 1.82249 0.221010
\(69\) 5.12883 0.617439
\(70\) 8.97068 1.07220
\(71\) −0.872761 −0.103578 −0.0517888 0.998658i \(-0.516492\pi\)
−0.0517888 + 0.998658i \(0.516492\pi\)
\(72\) 2.16645 0.255319
\(73\) −9.78216 −1.14491 −0.572457 0.819934i \(-0.694010\pi\)
−0.572457 + 0.819934i \(0.694010\pi\)
\(74\) 10.0927 1.17325
\(75\) −1.68119 −0.194127
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 4.95177 0.560678
\(79\) −9.31807 −1.04836 −0.524182 0.851606i \(-0.675629\pi\)
−0.524182 + 0.851606i \(0.675629\pi\)
\(80\) −1.77724 −0.198702
\(81\) 2.19288 0.243654
\(82\) 9.28332 1.02517
\(83\) 4.28702 0.470561 0.235281 0.971927i \(-0.424399\pi\)
0.235281 + 0.971927i \(0.424399\pi\)
\(84\) 4.60833 0.502810
\(85\) −3.23901 −0.351320
\(86\) 5.45957 0.588720
\(87\) −2.04493 −0.219239
\(88\) 0 0
\(89\) −5.60537 −0.594169 −0.297084 0.954851i \(-0.596014\pi\)
−0.297084 + 0.954851i \(0.596014\pi\)
\(90\) −3.85031 −0.405858
\(91\) −27.3763 −2.86982
\(92\) 5.61764 0.585679
\(93\) 0.352898 0.0365938
\(94\) −10.8132 −1.11530
\(95\) −1.77724 −0.182341
\(96\) −0.912987 −0.0931814
\(97\) −2.90528 −0.294987 −0.147493 0.989063i \(-0.547121\pi\)
−0.147493 + 0.989063i \(0.547121\pi\)
\(98\) −18.4776 −1.86652
\(99\) 0 0
\(100\) −1.84141 −0.184141
\(101\) 0.530731 0.0528097 0.0264049 0.999651i \(-0.491594\pi\)
0.0264049 + 0.999651i \(0.491594\pi\)
\(102\) −1.66391 −0.164752
\(103\) −3.63890 −0.358552 −0.179276 0.983799i \(-0.557376\pi\)
−0.179276 + 0.983799i \(0.557376\pi\)
\(104\) 5.42371 0.531838
\(105\) −8.19012 −0.799273
\(106\) −6.85692 −0.666003
\(107\) −8.86825 −0.857326 −0.428663 0.903464i \(-0.641015\pi\)
−0.428663 + 0.903464i \(0.641015\pi\)
\(108\) −4.71691 −0.453885
\(109\) 2.02691 0.194143 0.0970716 0.995277i \(-0.469052\pi\)
0.0970716 + 0.995277i \(0.469052\pi\)
\(110\) 0 0
\(111\) −9.21451 −0.874603
\(112\) 5.04753 0.476947
\(113\) 1.61136 0.151584 0.0757919 0.997124i \(-0.475852\pi\)
0.0757919 + 0.997124i \(0.475852\pi\)
\(114\) −0.912987 −0.0855091
\(115\) −9.98389 −0.931003
\(116\) −2.23982 −0.207962
\(117\) 11.7502 1.08631
\(118\) −3.70481 −0.341055
\(119\) 9.19910 0.843280
\(120\) 1.62260 0.148122
\(121\) 0 0
\(122\) 2.89301 0.261920
\(123\) −8.47556 −0.764215
\(124\) 0.386531 0.0347115
\(125\) 12.1588 1.08752
\(126\) 10.9352 0.974189
\(127\) −15.7073 −1.39380 −0.696901 0.717168i \(-0.745438\pi\)
−0.696901 + 0.717168i \(0.745438\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.98452 −0.438862
\(130\) −9.63923 −0.845416
\(131\) −13.6488 −1.19250 −0.596250 0.802798i \(-0.703343\pi\)
−0.596250 + 0.802798i \(0.703343\pi\)
\(132\) 0 0
\(133\) 5.04753 0.437676
\(134\) 11.5655 0.999105
\(135\) 8.38308 0.721501
\(136\) −1.82249 −0.156278
\(137\) −9.02937 −0.771432 −0.385716 0.922618i \(-0.626045\pi\)
−0.385716 + 0.922618i \(0.626045\pi\)
\(138\) −5.12883 −0.436595
\(139\) −10.3574 −0.878500 −0.439250 0.898365i \(-0.644756\pi\)
−0.439250 + 0.898365i \(0.644756\pi\)
\(140\) −8.97068 −0.758161
\(141\) 9.87235 0.831402
\(142\) 0.872761 0.0732405
\(143\) 0 0
\(144\) −2.16645 −0.180538
\(145\) 3.98070 0.330579
\(146\) 9.78216 0.809577
\(147\) 16.8698 1.39140
\(148\) −10.0927 −0.829615
\(149\) −14.2335 −1.16605 −0.583026 0.812454i \(-0.698131\pi\)
−0.583026 + 0.812454i \(0.698131\pi\)
\(150\) 1.68119 0.137268
\(151\) 14.2150 1.15680 0.578398 0.815754i \(-0.303678\pi\)
0.578398 + 0.815754i \(0.303678\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −3.94835 −0.319205
\(154\) 0 0
\(155\) −0.686958 −0.0551778
\(156\) −4.95177 −0.396459
\(157\) 13.8523 1.10553 0.552767 0.833336i \(-0.313572\pi\)
0.552767 + 0.833336i \(0.313572\pi\)
\(158\) 9.31807 0.741306
\(159\) 6.26028 0.496473
\(160\) 1.77724 0.140503
\(161\) 28.3552 2.23470
\(162\) −2.19288 −0.172289
\(163\) 10.9348 0.856480 0.428240 0.903665i \(-0.359134\pi\)
0.428240 + 0.903665i \(0.359134\pi\)
\(164\) −9.28332 −0.724906
\(165\) 0 0
\(166\) −4.28702 −0.332737
\(167\) −3.90441 −0.302132 −0.151066 0.988524i \(-0.548271\pi\)
−0.151066 + 0.988524i \(0.548271\pi\)
\(168\) −4.60833 −0.355541
\(169\) 16.4166 1.26281
\(170\) 3.23901 0.248421
\(171\) −2.16645 −0.165673
\(172\) −5.45957 −0.416288
\(173\) 8.63863 0.656783 0.328392 0.944542i \(-0.393493\pi\)
0.328392 + 0.944542i \(0.393493\pi\)
\(174\) 2.04493 0.155026
\(175\) −9.29460 −0.702605
\(176\) 0 0
\(177\) 3.38244 0.254240
\(178\) 5.60537 0.420141
\(179\) −6.23923 −0.466342 −0.233171 0.972436i \(-0.574910\pi\)
−0.233171 + 0.972436i \(0.574910\pi\)
\(180\) 3.85031 0.286985
\(181\) −26.1545 −1.94405 −0.972023 0.234884i \(-0.924529\pi\)
−0.972023 + 0.234884i \(0.924529\pi\)
\(182\) 27.3763 2.02927
\(183\) −2.64128 −0.195249
\(184\) −5.61764 −0.414138
\(185\) 17.9372 1.31877
\(186\) −0.352898 −0.0258757
\(187\) 0 0
\(188\) 10.8132 0.788636
\(189\) −23.8087 −1.73183
\(190\) 1.77724 0.128935
\(191\) 25.9985 1.88119 0.940594 0.339534i \(-0.110269\pi\)
0.940594 + 0.339534i \(0.110269\pi\)
\(192\) 0.912987 0.0658892
\(193\) −9.15782 −0.659195 −0.329597 0.944122i \(-0.606913\pi\)
−0.329597 + 0.944122i \(0.606913\pi\)
\(194\) 2.90528 0.208587
\(195\) 8.80050 0.630217
\(196\) 18.4776 1.31983
\(197\) 19.5680 1.39416 0.697080 0.716994i \(-0.254482\pi\)
0.697080 + 0.716994i \(0.254482\pi\)
\(198\) 0 0
\(199\) −14.8418 −1.05211 −0.526055 0.850451i \(-0.676329\pi\)
−0.526055 + 0.850451i \(0.676329\pi\)
\(200\) 1.84141 0.130208
\(201\) −10.5591 −0.744784
\(202\) −0.530731 −0.0373421
\(203\) −11.3056 −0.793495
\(204\) 1.66391 0.116497
\(205\) 16.4987 1.15232
\(206\) 3.63890 0.253535
\(207\) −12.1704 −0.845898
\(208\) −5.42371 −0.376066
\(209\) 0 0
\(210\) 8.19012 0.565172
\(211\) 17.6257 1.21340 0.606702 0.794930i \(-0.292492\pi\)
0.606702 + 0.794930i \(0.292492\pi\)
\(212\) 6.85692 0.470935
\(213\) −0.796820 −0.0545972
\(214\) 8.86825 0.606221
\(215\) 9.70297 0.661737
\(216\) 4.71691 0.320945
\(217\) 1.95102 0.132444
\(218\) −2.02691 −0.137280
\(219\) −8.93099 −0.603500
\(220\) 0 0
\(221\) −9.88467 −0.664915
\(222\) 9.21451 0.618438
\(223\) 10.1395 0.678989 0.339495 0.940608i \(-0.389744\pi\)
0.339495 + 0.940608i \(0.389744\pi\)
\(224\) −5.04753 −0.337252
\(225\) 3.98934 0.265956
\(226\) −1.61136 −0.107186
\(227\) −9.16956 −0.608605 −0.304302 0.952575i \(-0.598423\pi\)
−0.304302 + 0.952575i \(0.598423\pi\)
\(228\) 0.912987 0.0604641
\(229\) −0.458204 −0.0302789 −0.0151395 0.999885i \(-0.504819\pi\)
−0.0151395 + 0.999885i \(0.504819\pi\)
\(230\) 9.98389 0.658318
\(231\) 0 0
\(232\) 2.23982 0.147051
\(233\) −5.83823 −0.382475 −0.191237 0.981544i \(-0.561250\pi\)
−0.191237 + 0.981544i \(0.561250\pi\)
\(234\) −11.7502 −0.768135
\(235\) −19.2177 −1.25363
\(236\) 3.70481 0.241163
\(237\) −8.50728 −0.552607
\(238\) −9.19910 −0.596289
\(239\) −7.69554 −0.497783 −0.248892 0.968531i \(-0.580066\pi\)
−0.248892 + 0.968531i \(0.580066\pi\)
\(240\) −1.62260 −0.104738
\(241\) 11.2195 0.722714 0.361357 0.932428i \(-0.382314\pi\)
0.361357 + 0.932428i \(0.382314\pi\)
\(242\) 0 0
\(243\) 16.1528 1.03620
\(244\) −2.89301 −0.185206
\(245\) −32.8391 −2.09801
\(246\) 8.47556 0.540382
\(247\) −5.42371 −0.345102
\(248\) −0.386531 −0.0245447
\(249\) 3.91399 0.248039
\(250\) −12.1588 −0.768993
\(251\) −2.94541 −0.185913 −0.0929563 0.995670i \(-0.529632\pi\)
−0.0929563 + 0.995670i \(0.529632\pi\)
\(252\) −10.9352 −0.688856
\(253\) 0 0
\(254\) 15.7073 0.985566
\(255\) −2.95718 −0.185186
\(256\) 1.00000 0.0625000
\(257\) 21.1881 1.32168 0.660839 0.750528i \(-0.270201\pi\)
0.660839 + 0.750528i \(0.270201\pi\)
\(258\) 4.98452 0.310322
\(259\) −50.9432 −3.16546
\(260\) 9.63923 0.597800
\(261\) 4.85247 0.300360
\(262\) 13.6488 0.843226
\(263\) −3.78559 −0.233430 −0.116715 0.993165i \(-0.537236\pi\)
−0.116715 + 0.993165i \(0.537236\pi\)
\(264\) 0 0
\(265\) −12.1864 −0.748604
\(266\) −5.04753 −0.309484
\(267\) −5.11764 −0.313194
\(268\) −11.5655 −0.706474
\(269\) −29.0402 −1.77061 −0.885305 0.465010i \(-0.846051\pi\)
−0.885305 + 0.465010i \(0.846051\pi\)
\(270\) −8.38308 −0.510178
\(271\) −16.5540 −1.00559 −0.502793 0.864407i \(-0.667694\pi\)
−0.502793 + 0.864407i \(0.667694\pi\)
\(272\) 1.82249 0.110505
\(273\) −24.9942 −1.51272
\(274\) 9.02937 0.545484
\(275\) 0 0
\(276\) 5.12883 0.308719
\(277\) −20.9258 −1.25731 −0.628656 0.777684i \(-0.716395\pi\)
−0.628656 + 0.777684i \(0.716395\pi\)
\(278\) 10.3574 0.621194
\(279\) −0.837401 −0.0501339
\(280\) 8.97068 0.536101
\(281\) 8.63480 0.515109 0.257554 0.966264i \(-0.417083\pi\)
0.257554 + 0.966264i \(0.417083\pi\)
\(282\) −9.87235 −0.587890
\(283\) −6.31226 −0.375225 −0.187613 0.982243i \(-0.560075\pi\)
−0.187613 + 0.982243i \(0.560075\pi\)
\(284\) −0.872761 −0.0517888
\(285\) −1.62260 −0.0961144
\(286\) 0 0
\(287\) −46.8578 −2.76593
\(288\) 2.16645 0.127660
\(289\) −13.6785 −0.804619
\(290\) −3.98070 −0.233755
\(291\) −2.65249 −0.155492
\(292\) −9.78216 −0.572457
\(293\) 12.6769 0.740590 0.370295 0.928914i \(-0.379257\pi\)
0.370295 + 0.928914i \(0.379257\pi\)
\(294\) −16.8698 −0.983866
\(295\) −6.58434 −0.383355
\(296\) 10.0927 0.586627
\(297\) 0 0
\(298\) 14.2335 0.824523
\(299\) −30.4684 −1.76203
\(300\) −1.68119 −0.0970635
\(301\) −27.5573 −1.58838
\(302\) −14.2150 −0.817979
\(303\) 0.484551 0.0278367
\(304\) 1.00000 0.0573539
\(305\) 5.14157 0.294405
\(306\) 3.94835 0.225712
\(307\) −26.6423 −1.52056 −0.760279 0.649597i \(-0.774938\pi\)
−0.760279 + 0.649597i \(0.774938\pi\)
\(308\) 0 0
\(309\) −3.32227 −0.188998
\(310\) 0.686958 0.0390166
\(311\) −25.0841 −1.42239 −0.711194 0.702995i \(-0.751845\pi\)
−0.711194 + 0.702995i \(0.751845\pi\)
\(312\) 4.95177 0.280339
\(313\) 34.1330 1.92931 0.964655 0.263518i \(-0.0848827\pi\)
0.964655 + 0.263518i \(0.0848827\pi\)
\(314\) −13.8523 −0.781730
\(315\) 19.4346 1.09501
\(316\) −9.31807 −0.524182
\(317\) −30.1730 −1.69468 −0.847342 0.531048i \(-0.821798\pi\)
−0.847342 + 0.531048i \(0.821798\pi\)
\(318\) −6.26028 −0.351059
\(319\) 0 0
\(320\) −1.77724 −0.0993508
\(321\) −8.09660 −0.451908
\(322\) −28.3552 −1.58017
\(323\) 1.82249 0.101406
\(324\) 2.19288 0.121827
\(325\) 9.98729 0.553995
\(326\) −10.9348 −0.605623
\(327\) 1.85055 0.102335
\(328\) 9.28332 0.512586
\(329\) 54.5801 3.00910
\(330\) 0 0
\(331\) 10.7721 0.592091 0.296045 0.955174i \(-0.404332\pi\)
0.296045 + 0.955174i \(0.404332\pi\)
\(332\) 4.28702 0.235281
\(333\) 21.8654 1.19822
\(334\) 3.90441 0.213640
\(335\) 20.5546 1.12302
\(336\) 4.60833 0.251405
\(337\) 4.86374 0.264945 0.132472 0.991187i \(-0.457708\pi\)
0.132472 + 0.991187i \(0.457708\pi\)
\(338\) −16.4166 −0.892944
\(339\) 1.47115 0.0799019
\(340\) −3.23901 −0.175660
\(341\) 0 0
\(342\) 2.16645 0.117148
\(343\) 57.9333 3.12811
\(344\) 5.45957 0.294360
\(345\) −9.11517 −0.490744
\(346\) −8.63863 −0.464416
\(347\) −8.80331 −0.472586 −0.236293 0.971682i \(-0.575933\pi\)
−0.236293 + 0.971682i \(0.575933\pi\)
\(348\) −2.04493 −0.109620
\(349\) −22.3320 −1.19540 −0.597702 0.801718i \(-0.703920\pi\)
−0.597702 + 0.801718i \(0.703920\pi\)
\(350\) 9.29460 0.496817
\(351\) 25.5831 1.36553
\(352\) 0 0
\(353\) −7.98769 −0.425142 −0.212571 0.977146i \(-0.568184\pi\)
−0.212571 + 0.977146i \(0.568184\pi\)
\(354\) −3.38244 −0.179775
\(355\) 1.55111 0.0823242
\(356\) −5.60537 −0.297084
\(357\) 8.39866 0.444504
\(358\) 6.23923 0.329754
\(359\) 26.2639 1.38615 0.693077 0.720863i \(-0.256254\pi\)
0.693077 + 0.720863i \(0.256254\pi\)
\(360\) −3.85031 −0.202929
\(361\) 1.00000 0.0526316
\(362\) 26.1545 1.37465
\(363\) 0 0
\(364\) −27.3763 −1.43491
\(365\) 17.3852 0.909985
\(366\) 2.64128 0.138062
\(367\) −15.5072 −0.809468 −0.404734 0.914434i \(-0.632636\pi\)
−0.404734 + 0.914434i \(0.632636\pi\)
\(368\) 5.61764 0.292840
\(369\) 20.1119 1.04698
\(370\) −17.9372 −0.932509
\(371\) 34.6105 1.79689
\(372\) 0.352898 0.0182969
\(373\) 17.4626 0.904179 0.452090 0.891972i \(-0.350679\pi\)
0.452090 + 0.891972i \(0.350679\pi\)
\(374\) 0 0
\(375\) 11.1009 0.573246
\(376\) −10.8132 −0.557650
\(377\) 12.1481 0.625660
\(378\) 23.8087 1.22459
\(379\) −31.6595 −1.62624 −0.813119 0.582097i \(-0.802232\pi\)
−0.813119 + 0.582097i \(0.802232\pi\)
\(380\) −1.77724 −0.0911705
\(381\) −14.3406 −0.734692
\(382\) −25.9985 −1.33020
\(383\) −6.09885 −0.311637 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(384\) −0.912987 −0.0465907
\(385\) 0 0
\(386\) 9.15782 0.466121
\(387\) 11.8279 0.601246
\(388\) −2.90528 −0.147493
\(389\) −32.0224 −1.62360 −0.811801 0.583934i \(-0.801513\pi\)
−0.811801 + 0.583934i \(0.801513\pi\)
\(390\) −8.80050 −0.445630
\(391\) 10.2381 0.517764
\(392\) −18.4776 −0.933258
\(393\) −12.4612 −0.628583
\(394\) −19.5680 −0.985819
\(395\) 16.5605 0.833247
\(396\) 0 0
\(397\) 10.0101 0.502395 0.251197 0.967936i \(-0.419176\pi\)
0.251197 + 0.967936i \(0.419176\pi\)
\(398\) 14.8418 0.743954
\(399\) 4.60833 0.230705
\(400\) −1.84141 −0.0920707
\(401\) −5.31105 −0.265221 −0.132611 0.991168i \(-0.542336\pi\)
−0.132611 + 0.991168i \(0.542336\pi\)
\(402\) 10.5591 0.526642
\(403\) −2.09643 −0.104431
\(404\) 0.530731 0.0264049
\(405\) −3.89728 −0.193658
\(406\) 11.3056 0.561085
\(407\) 0 0
\(408\) −1.66391 −0.0823760
\(409\) −18.5420 −0.916843 −0.458421 0.888735i \(-0.651585\pi\)
−0.458421 + 0.888735i \(0.651585\pi\)
\(410\) −16.4987 −0.814813
\(411\) −8.24371 −0.406632
\(412\) −3.63890 −0.179276
\(413\) 18.7001 0.920174
\(414\) 12.1704 0.598140
\(415\) −7.61906 −0.374005
\(416\) 5.42371 0.265919
\(417\) −9.45615 −0.463069
\(418\) 0 0
\(419\) 11.5438 0.563953 0.281977 0.959421i \(-0.409010\pi\)
0.281977 + 0.959421i \(0.409010\pi\)
\(420\) −8.19012 −0.399637
\(421\) 33.7279 1.64380 0.821899 0.569633i \(-0.192914\pi\)
0.821899 + 0.569633i \(0.192914\pi\)
\(422\) −17.6257 −0.858006
\(423\) −23.4264 −1.13903
\(424\) −6.85692 −0.333001
\(425\) −3.35597 −0.162788
\(426\) 0.796820 0.0386060
\(427\) −14.6025 −0.706666
\(428\) −8.86825 −0.428663
\(429\) 0 0
\(430\) −9.70297 −0.467919
\(431\) 10.0313 0.483189 0.241595 0.970377i \(-0.422330\pi\)
0.241595 + 0.970377i \(0.422330\pi\)
\(432\) −4.71691 −0.226942
\(433\) −7.90731 −0.380001 −0.190001 0.981784i \(-0.560849\pi\)
−0.190001 + 0.981784i \(0.560849\pi\)
\(434\) −1.95102 −0.0936522
\(435\) 3.63433 0.174253
\(436\) 2.02691 0.0970716
\(437\) 5.61764 0.268728
\(438\) 8.93099 0.426739
\(439\) −25.7405 −1.22853 −0.614263 0.789102i \(-0.710547\pi\)
−0.614263 + 0.789102i \(0.710547\pi\)
\(440\) 0 0
\(441\) −40.0308 −1.90623
\(442\) 9.88467 0.470166
\(443\) −24.2023 −1.14988 −0.574942 0.818194i \(-0.694975\pi\)
−0.574942 + 0.818194i \(0.694975\pi\)
\(444\) −9.21451 −0.437301
\(445\) 9.96210 0.472249
\(446\) −10.1395 −0.480118
\(447\) −12.9950 −0.614641
\(448\) 5.04753 0.238473
\(449\) 36.5068 1.72286 0.861430 0.507876i \(-0.169569\pi\)
0.861430 + 0.507876i \(0.169569\pi\)
\(450\) −3.98934 −0.188059
\(451\) 0 0
\(452\) 1.61136 0.0757919
\(453\) 12.9781 0.609763
\(454\) 9.16956 0.430349
\(455\) 48.6543 2.28095
\(456\) −0.912987 −0.0427546
\(457\) −15.7612 −0.737277 −0.368639 0.929573i \(-0.620176\pi\)
−0.368639 + 0.929573i \(0.620176\pi\)
\(458\) 0.458204 0.0214105
\(459\) −8.59654 −0.401252
\(460\) −9.98389 −0.465501
\(461\) −6.61141 −0.307924 −0.153962 0.988077i \(-0.549203\pi\)
−0.153962 + 0.988077i \(0.549203\pi\)
\(462\) 0 0
\(463\) −32.4747 −1.50923 −0.754614 0.656169i \(-0.772176\pi\)
−0.754614 + 0.656169i \(0.772176\pi\)
\(464\) −2.23982 −0.103981
\(465\) −0.627184 −0.0290850
\(466\) 5.83823 0.270451
\(467\) 38.0595 1.76118 0.880592 0.473876i \(-0.157146\pi\)
0.880592 + 0.473876i \(0.157146\pi\)
\(468\) 11.7502 0.543154
\(469\) −58.3771 −2.69560
\(470\) 19.2177 0.886448
\(471\) 12.6470 0.582742
\(472\) −3.70481 −0.170528
\(473\) 0 0
\(474\) 8.50728 0.390752
\(475\) −1.84141 −0.0844899
\(476\) 9.19910 0.421640
\(477\) −14.8552 −0.680173
\(478\) 7.69554 0.351986
\(479\) −30.0892 −1.37481 −0.687405 0.726274i \(-0.741250\pi\)
−0.687405 + 0.726274i \(0.741250\pi\)
\(480\) 1.62260 0.0740612
\(481\) 54.7399 2.49592
\(482\) −11.2195 −0.511036
\(483\) 25.8879 1.17794
\(484\) 0 0
\(485\) 5.16339 0.234457
\(486\) −16.1528 −0.732706
\(487\) −2.69394 −0.122074 −0.0610370 0.998136i \(-0.519441\pi\)
−0.0610370 + 0.998136i \(0.519441\pi\)
\(488\) 2.89301 0.130960
\(489\) 9.98334 0.451462
\(490\) 32.8391 1.48352
\(491\) −27.8830 −1.25834 −0.629171 0.777267i \(-0.716606\pi\)
−0.629171 + 0.777267i \(0.716606\pi\)
\(492\) −8.47556 −0.382108
\(493\) −4.08206 −0.183847
\(494\) 5.42371 0.244024
\(495\) 0 0
\(496\) 0.386531 0.0173557
\(497\) −4.40529 −0.197604
\(498\) −3.91399 −0.175390
\(499\) 21.0031 0.940229 0.470115 0.882605i \(-0.344213\pi\)
0.470115 + 0.882605i \(0.344213\pi\)
\(500\) 12.1588 0.543760
\(501\) −3.56467 −0.159258
\(502\) 2.94541 0.131460
\(503\) 10.6716 0.475822 0.237911 0.971287i \(-0.423537\pi\)
0.237911 + 0.971287i \(0.423537\pi\)
\(504\) 10.9352 0.487094
\(505\) −0.943237 −0.0419735
\(506\) 0 0
\(507\) 14.9881 0.665646
\(508\) −15.7073 −0.696901
\(509\) −13.7570 −0.609767 −0.304884 0.952390i \(-0.598618\pi\)
−0.304884 + 0.952390i \(0.598618\pi\)
\(510\) 2.95718 0.130946
\(511\) −49.3757 −2.18425
\(512\) −1.00000 −0.0441942
\(513\) −4.71691 −0.208257
\(514\) −21.1881 −0.934567
\(515\) 6.46721 0.284979
\(516\) −4.98452 −0.219431
\(517\) 0 0
\(518\) 50.9432 2.23832
\(519\) 7.88696 0.346199
\(520\) −9.63923 −0.422708
\(521\) −5.73245 −0.251143 −0.125572 0.992085i \(-0.540076\pi\)
−0.125572 + 0.992085i \(0.540076\pi\)
\(522\) −4.85247 −0.212387
\(523\) −13.7126 −0.599612 −0.299806 0.954000i \(-0.596922\pi\)
−0.299806 + 0.954000i \(0.596922\pi\)
\(524\) −13.6488 −0.596250
\(525\) −8.48585 −0.370353
\(526\) 3.78559 0.165060
\(527\) 0.704450 0.0306863
\(528\) 0 0
\(529\) 8.55784 0.372080
\(530\) 12.1864 0.529343
\(531\) −8.02630 −0.348312
\(532\) 5.04753 0.218838
\(533\) 50.3500 2.18090
\(534\) 5.11764 0.221462
\(535\) 15.7610 0.681408
\(536\) 11.5655 0.499552
\(537\) −5.69634 −0.245815
\(538\) 29.0402 1.25201
\(539\) 0 0
\(540\) 8.38308 0.360750
\(541\) 9.98640 0.429349 0.214674 0.976686i \(-0.431131\pi\)
0.214674 + 0.976686i \(0.431131\pi\)
\(542\) 16.5540 0.711057
\(543\) −23.8787 −1.02473
\(544\) −1.82249 −0.0781388
\(545\) −3.60231 −0.154306
\(546\) 24.9942 1.06965
\(547\) 1.14622 0.0490088 0.0245044 0.999700i \(-0.492199\pi\)
0.0245044 + 0.999700i \(0.492199\pi\)
\(548\) −9.02937 −0.385716
\(549\) 6.26757 0.267493
\(550\) 0 0
\(551\) −2.23982 −0.0954195
\(552\) −5.12883 −0.218298
\(553\) −47.0332 −2.00006
\(554\) 20.9258 0.889054
\(555\) 16.3764 0.695140
\(556\) −10.3574 −0.439250
\(557\) 29.4129 1.24626 0.623132 0.782117i \(-0.285860\pi\)
0.623132 + 0.782117i \(0.285860\pi\)
\(558\) 0.837401 0.0354500
\(559\) 29.6111 1.25242
\(560\) −8.97068 −0.379080
\(561\) 0 0
\(562\) −8.63480 −0.364237
\(563\) −3.52757 −0.148669 −0.0743347 0.997233i \(-0.523683\pi\)
−0.0743347 + 0.997233i \(0.523683\pi\)
\(564\) 9.87235 0.415701
\(565\) −2.86377 −0.120480
\(566\) 6.31226 0.265324
\(567\) 11.0687 0.464840
\(568\) 0.872761 0.0366202
\(569\) −23.4396 −0.982638 −0.491319 0.870980i \(-0.663485\pi\)
−0.491319 + 0.870980i \(0.663485\pi\)
\(570\) 1.62260 0.0679632
\(571\) −23.7470 −0.993782 −0.496891 0.867813i \(-0.665525\pi\)
−0.496891 + 0.867813i \(0.665525\pi\)
\(572\) 0 0
\(573\) 23.7363 0.991599
\(574\) 46.8578 1.95581
\(575\) −10.3444 −0.431391
\(576\) −2.16645 −0.0902689
\(577\) −12.3050 −0.512263 −0.256131 0.966642i \(-0.582448\pi\)
−0.256131 + 0.966642i \(0.582448\pi\)
\(578\) 13.6785 0.568951
\(579\) −8.36098 −0.347470
\(580\) 3.98070 0.165290
\(581\) 21.6389 0.897731
\(582\) 2.65249 0.109949
\(583\) 0 0
\(584\) 9.78216 0.404789
\(585\) −20.8830 −0.863404
\(586\) −12.6769 −0.523676
\(587\) −25.7463 −1.06266 −0.531332 0.847164i \(-0.678308\pi\)
−0.531332 + 0.847164i \(0.678308\pi\)
\(588\) 16.8698 0.695698
\(589\) 0.386531 0.0159267
\(590\) 6.58434 0.271073
\(591\) 17.8653 0.734880
\(592\) −10.0927 −0.414808
\(593\) −13.4292 −0.551469 −0.275735 0.961234i \(-0.588921\pi\)
−0.275735 + 0.961234i \(0.588921\pi\)
\(594\) 0 0
\(595\) −16.3490 −0.670244
\(596\) −14.2335 −0.583026
\(597\) −13.5504 −0.554581
\(598\) 30.4684 1.24595
\(599\) 33.1006 1.35245 0.676227 0.736693i \(-0.263614\pi\)
0.676227 + 0.736693i \(0.263614\pi\)
\(600\) 1.68119 0.0686342
\(601\) 34.6197 1.41217 0.706084 0.708128i \(-0.250460\pi\)
0.706084 + 0.708128i \(0.250460\pi\)
\(602\) 27.5573 1.12315
\(603\) 25.0561 1.02036
\(604\) 14.2150 0.578398
\(605\) 0 0
\(606\) −0.484551 −0.0196835
\(607\) 39.6229 1.60824 0.804122 0.594465i \(-0.202636\pi\)
0.804122 + 0.594465i \(0.202636\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −10.3218 −0.418262
\(610\) −5.14157 −0.208176
\(611\) −58.6478 −2.37264
\(612\) −3.94835 −0.159603
\(613\) −21.7299 −0.877661 −0.438831 0.898570i \(-0.644607\pi\)
−0.438831 + 0.898570i \(0.644607\pi\)
\(614\) 26.6423 1.07520
\(615\) 15.0631 0.607403
\(616\) 0 0
\(617\) 7.82581 0.315055 0.157528 0.987515i \(-0.449648\pi\)
0.157528 + 0.987515i \(0.449648\pi\)
\(618\) 3.32227 0.133641
\(619\) −13.4660 −0.541242 −0.270621 0.962686i \(-0.587229\pi\)
−0.270621 + 0.962686i \(0.587229\pi\)
\(620\) −0.686958 −0.0275889
\(621\) −26.4979 −1.06332
\(622\) 25.0841 1.00578
\(623\) −28.2933 −1.13355
\(624\) −4.95177 −0.198230
\(625\) −12.4021 −0.496085
\(626\) −34.1330 −1.36423
\(627\) 0 0
\(628\) 13.8523 0.552767
\(629\) −18.3939 −0.733413
\(630\) −19.4346 −0.774291
\(631\) 17.7750 0.707611 0.353806 0.935319i \(-0.384887\pi\)
0.353806 + 0.935319i \(0.384887\pi\)
\(632\) 9.31807 0.370653
\(633\) 16.0920 0.639601
\(634\) 30.1730 1.19832
\(635\) 27.9157 1.10780
\(636\) 6.26028 0.248236
\(637\) −100.217 −3.97074
\(638\) 0 0
\(639\) 1.89080 0.0747988
\(640\) 1.77724 0.0702516
\(641\) 29.8405 1.17863 0.589315 0.807903i \(-0.299398\pi\)
0.589315 + 0.807903i \(0.299398\pi\)
\(642\) 8.09660 0.319547
\(643\) −19.4161 −0.765698 −0.382849 0.923811i \(-0.625057\pi\)
−0.382849 + 0.923811i \(0.625057\pi\)
\(644\) 28.3552 1.11735
\(645\) 8.85869 0.348810
\(646\) −1.82249 −0.0717051
\(647\) −22.9533 −0.902388 −0.451194 0.892426i \(-0.649002\pi\)
−0.451194 + 0.892426i \(0.649002\pi\)
\(648\) −2.19288 −0.0861446
\(649\) 0 0
\(650\) −9.98729 −0.391734
\(651\) 1.78126 0.0698131
\(652\) 10.9348 0.428240
\(653\) 35.1675 1.37621 0.688105 0.725611i \(-0.258443\pi\)
0.688105 + 0.725611i \(0.258443\pi\)
\(654\) −1.85055 −0.0723621
\(655\) 24.2572 0.947807
\(656\) −9.28332 −0.362453
\(657\) 21.1926 0.826802
\(658\) −54.5801 −2.12776
\(659\) −14.4098 −0.561325 −0.280662 0.959807i \(-0.590554\pi\)
−0.280662 + 0.959807i \(0.590554\pi\)
\(660\) 0 0
\(661\) −19.8395 −0.771668 −0.385834 0.922568i \(-0.626086\pi\)
−0.385834 + 0.922568i \(0.626086\pi\)
\(662\) −10.7721 −0.418671
\(663\) −9.02458 −0.350486
\(664\) −4.28702 −0.166369
\(665\) −8.97068 −0.347868
\(666\) −21.8654 −0.847266
\(667\) −12.5825 −0.487196
\(668\) −3.90441 −0.151066
\(669\) 9.25721 0.357905
\(670\) −20.5546 −0.794095
\(671\) 0 0
\(672\) −4.60833 −0.177770
\(673\) −17.3731 −0.669685 −0.334843 0.942274i \(-0.608683\pi\)
−0.334843 + 0.942274i \(0.608683\pi\)
\(674\) −4.86374 −0.187344
\(675\) 8.68578 0.334316
\(676\) 16.4166 0.631407
\(677\) 12.7384 0.489577 0.244789 0.969576i \(-0.421281\pi\)
0.244789 + 0.969576i \(0.421281\pi\)
\(678\) −1.47115 −0.0564992
\(679\) −14.6645 −0.562772
\(680\) 3.23901 0.124210
\(681\) −8.37169 −0.320804
\(682\) 0 0
\(683\) 3.11068 0.119027 0.0595134 0.998228i \(-0.481045\pi\)
0.0595134 + 0.998228i \(0.481045\pi\)
\(684\) −2.16645 −0.0828364
\(685\) 16.0474 0.613139
\(686\) −57.9333 −2.21191
\(687\) −0.418334 −0.0159604
\(688\) −5.45957 −0.208144
\(689\) −37.1899 −1.41682
\(690\) 9.11517 0.347009
\(691\) 34.6675 1.31881 0.659407 0.751786i \(-0.270807\pi\)
0.659407 + 0.751786i \(0.270807\pi\)
\(692\) 8.63863 0.328392
\(693\) 0 0
\(694\) 8.80331 0.334169
\(695\) 18.4075 0.698238
\(696\) 2.04493 0.0775128
\(697\) −16.9188 −0.640845
\(698\) 22.3320 0.845279
\(699\) −5.33023 −0.201608
\(700\) −9.29460 −0.351303
\(701\) 1.33992 0.0506079 0.0253040 0.999680i \(-0.491945\pi\)
0.0253040 + 0.999680i \(0.491945\pi\)
\(702\) −25.5831 −0.965573
\(703\) −10.0927 −0.380654
\(704\) 0 0
\(705\) −17.5455 −0.660803
\(706\) 7.98769 0.300621
\(707\) 2.67888 0.100750
\(708\) 3.38244 0.127120
\(709\) −14.2636 −0.535681 −0.267841 0.963463i \(-0.586310\pi\)
−0.267841 + 0.963463i \(0.586310\pi\)
\(710\) −1.55111 −0.0582120
\(711\) 20.1872 0.757078
\(712\) 5.60537 0.210070
\(713\) 2.17139 0.0813191
\(714\) −8.39866 −0.314312
\(715\) 0 0
\(716\) −6.23923 −0.233171
\(717\) −7.02593 −0.262388
\(718\) −26.2639 −0.980160
\(719\) −32.6745 −1.21855 −0.609276 0.792958i \(-0.708540\pi\)
−0.609276 + 0.792958i \(0.708540\pi\)
\(720\) 3.85031 0.143493
\(721\) −18.3675 −0.684041
\(722\) −1.00000 −0.0372161
\(723\) 10.2433 0.380952
\(724\) −26.1545 −0.972023
\(725\) 4.12444 0.153178
\(726\) 0 0
\(727\) −7.95801 −0.295146 −0.147573 0.989051i \(-0.547146\pi\)
−0.147573 + 0.989051i \(0.547146\pi\)
\(728\) 27.3763 1.01463
\(729\) 8.16865 0.302543
\(730\) −17.3852 −0.643457
\(731\) −9.95003 −0.368015
\(732\) −2.64128 −0.0976245
\(733\) 23.8742 0.881812 0.440906 0.897553i \(-0.354657\pi\)
0.440906 + 0.897553i \(0.354657\pi\)
\(734\) 15.5072 0.572380
\(735\) −29.9817 −1.10589
\(736\) −5.61764 −0.207069
\(737\) 0 0
\(738\) −20.1119 −0.740329
\(739\) 8.25266 0.303579 0.151790 0.988413i \(-0.451496\pi\)
0.151790 + 0.988413i \(0.451496\pi\)
\(740\) 17.9372 0.659383
\(741\) −4.95177 −0.181908
\(742\) −34.6105 −1.27059
\(743\) 4.22346 0.154944 0.0774718 0.996995i \(-0.475315\pi\)
0.0774718 + 0.996995i \(0.475315\pi\)
\(744\) −0.352898 −0.0129379
\(745\) 25.2963 0.926785
\(746\) −17.4626 −0.639351
\(747\) −9.28763 −0.339817
\(748\) 0 0
\(749\) −44.7628 −1.63560
\(750\) −11.1009 −0.405346
\(751\) 35.6180 1.29972 0.649859 0.760055i \(-0.274828\pi\)
0.649859 + 0.760055i \(0.274828\pi\)
\(752\) 10.8132 0.394318
\(753\) −2.68912 −0.0979970
\(754\) −12.1481 −0.442408
\(755\) −25.2634 −0.919429
\(756\) −23.8087 −0.865915
\(757\) 38.4502 1.39749 0.698747 0.715369i \(-0.253741\pi\)
0.698747 + 0.715369i \(0.253741\pi\)
\(758\) 31.6595 1.14992
\(759\) 0 0
\(760\) 1.77724 0.0644673
\(761\) −2.50805 −0.0909168 −0.0454584 0.998966i \(-0.514475\pi\)
−0.0454584 + 0.998966i \(0.514475\pi\)
\(762\) 14.3406 0.519505
\(763\) 10.2309 0.370384
\(764\) 25.9985 0.940594
\(765\) 7.01717 0.253706
\(766\) 6.09885 0.220360
\(767\) −20.0938 −0.725545
\(768\) 0.912987 0.0329446
\(769\) −1.24222 −0.0447955 −0.0223978 0.999749i \(-0.507130\pi\)
−0.0223978 + 0.999749i \(0.507130\pi\)
\(770\) 0 0
\(771\) 19.3445 0.696674
\(772\) −9.15782 −0.329597
\(773\) 26.8350 0.965189 0.482595 0.875844i \(-0.339694\pi\)
0.482595 + 0.875844i \(0.339694\pi\)
\(774\) −11.8279 −0.425145
\(775\) −0.711763 −0.0255673
\(776\) 2.90528 0.104294
\(777\) −46.5105 −1.66856
\(778\) 32.0224 1.14806
\(779\) −9.28332 −0.332609
\(780\) 8.80050 0.315108
\(781\) 0 0
\(782\) −10.2381 −0.366114
\(783\) 10.5650 0.377563
\(784\) 18.4776 0.659913
\(785\) −24.6189 −0.878685
\(786\) 12.4612 0.444476
\(787\) −4.79827 −0.171040 −0.0855200 0.996336i \(-0.527255\pi\)
−0.0855200 + 0.996336i \(0.527255\pi\)
\(788\) 19.5680 0.697080
\(789\) −3.45620 −0.123044
\(790\) −16.5605 −0.589195
\(791\) 8.13338 0.289190
\(792\) 0 0
\(793\) 15.6908 0.557197
\(794\) −10.0101 −0.355247
\(795\) −11.1260 −0.394600
\(796\) −14.8418 −0.526055
\(797\) −4.15409 −0.147146 −0.0735728 0.997290i \(-0.523440\pi\)
−0.0735728 + 0.997290i \(0.523440\pi\)
\(798\) −4.60833 −0.163133
\(799\) 19.7071 0.697186
\(800\) 1.84141 0.0651038
\(801\) 12.1438 0.429080
\(802\) 5.31105 0.187540
\(803\) 0 0
\(804\) −10.5591 −0.372392
\(805\) −50.3940 −1.77616
\(806\) 2.09643 0.0738435
\(807\) −26.5133 −0.933313
\(808\) −0.530731 −0.0186711
\(809\) 17.9486 0.631039 0.315520 0.948919i \(-0.397821\pi\)
0.315520 + 0.948919i \(0.397821\pi\)
\(810\) 3.89728 0.136937
\(811\) 8.31396 0.291943 0.145971 0.989289i \(-0.453369\pi\)
0.145971 + 0.989289i \(0.453369\pi\)
\(812\) −11.3056 −0.396747
\(813\) −15.1136 −0.530058
\(814\) 0 0
\(815\) −19.4338 −0.680736
\(816\) 1.66391 0.0582487
\(817\) −5.45957 −0.191006
\(818\) 18.5420 0.648306
\(819\) 59.3095 2.07244
\(820\) 16.4987 0.576160
\(821\) −50.0923 −1.74823 −0.874116 0.485716i \(-0.838559\pi\)
−0.874116 + 0.485716i \(0.838559\pi\)
\(822\) 8.24371 0.287532
\(823\) 26.9086 0.937974 0.468987 0.883205i \(-0.344619\pi\)
0.468987 + 0.883205i \(0.344619\pi\)
\(824\) 3.63890 0.126767
\(825\) 0 0
\(826\) −18.7001 −0.650661
\(827\) −6.87811 −0.239175 −0.119588 0.992824i \(-0.538157\pi\)
−0.119588 + 0.992824i \(0.538157\pi\)
\(828\) −12.1704 −0.422949
\(829\) 26.0817 0.905856 0.452928 0.891547i \(-0.350379\pi\)
0.452928 + 0.891547i \(0.350379\pi\)
\(830\) 7.61906 0.264462
\(831\) −19.1050 −0.662746
\(832\) −5.42371 −0.188033
\(833\) 33.6753 1.16678
\(834\) 9.45615 0.327440
\(835\) 6.93907 0.240136
\(836\) 0 0
\(837\) −1.82323 −0.0630200
\(838\) −11.5438 −0.398775
\(839\) 29.8292 1.02982 0.514910 0.857244i \(-0.327825\pi\)
0.514910 + 0.857244i \(0.327825\pi\)
\(840\) 8.19012 0.282586
\(841\) −23.9832 −0.827007
\(842\) −33.7279 −1.16234
\(843\) 7.88346 0.271521
\(844\) 17.6257 0.606702
\(845\) −29.1762 −1.00369
\(846\) 23.4264 0.805416
\(847\) 0 0
\(848\) 6.85692 0.235468
\(849\) −5.76302 −0.197786
\(850\) 3.35597 0.115109
\(851\) −56.6972 −1.94355
\(852\) −0.796820 −0.0272986
\(853\) 35.7549 1.22423 0.612113 0.790770i \(-0.290320\pi\)
0.612113 + 0.790770i \(0.290320\pi\)
\(854\) 14.6025 0.499689
\(855\) 3.85031 0.131678
\(856\) 8.86825 0.303111
\(857\) 53.7988 1.83773 0.918867 0.394568i \(-0.129106\pi\)
0.918867 + 0.394568i \(0.129106\pi\)
\(858\) 0 0
\(859\) 40.7851 1.39157 0.695785 0.718250i \(-0.255057\pi\)
0.695785 + 0.718250i \(0.255057\pi\)
\(860\) 9.70297 0.330868
\(861\) −42.7806 −1.45796
\(862\) −10.0313 −0.341666
\(863\) −34.9318 −1.18909 −0.594546 0.804062i \(-0.702668\pi\)
−0.594546 + 0.804062i \(0.702668\pi\)
\(864\) 4.71691 0.160472
\(865\) −15.3529 −0.522015
\(866\) 7.90731 0.268701
\(867\) −12.4883 −0.424125
\(868\) 1.95102 0.0662221
\(869\) 0 0
\(870\) −3.63433 −0.123215
\(871\) 62.7277 2.12545
\(872\) −2.02691 −0.0686400
\(873\) 6.29416 0.213025
\(874\) −5.61764 −0.190019
\(875\) 61.3721 2.07476
\(876\) −8.93099 −0.301750
\(877\) 2.35793 0.0796218 0.0398109 0.999207i \(-0.487324\pi\)
0.0398109 + 0.999207i \(0.487324\pi\)
\(878\) 25.7405 0.868699
\(879\) 11.5738 0.390375
\(880\) 0 0
\(881\) 24.8750 0.838060 0.419030 0.907972i \(-0.362370\pi\)
0.419030 + 0.907972i \(0.362370\pi\)
\(882\) 40.0308 1.34791
\(883\) 35.8838 1.20759 0.603793 0.797141i \(-0.293655\pi\)
0.603793 + 0.797141i \(0.293655\pi\)
\(884\) −9.88467 −0.332457
\(885\) −6.01142 −0.202072
\(886\) 24.2023 0.813091
\(887\) 49.9066 1.67570 0.837850 0.545901i \(-0.183812\pi\)
0.837850 + 0.545901i \(0.183812\pi\)
\(888\) 9.21451 0.309219
\(889\) −79.2833 −2.65908
\(890\) −9.96210 −0.333930
\(891\) 0 0
\(892\) 10.1395 0.339495
\(893\) 10.8132 0.361851
\(894\) 12.9950 0.434617
\(895\) 11.0886 0.370652
\(896\) −5.04753 −0.168626
\(897\) −27.8173 −0.928792
\(898\) −36.5068 −1.21825
\(899\) −0.865759 −0.0288747
\(900\) 3.98934 0.132978
\(901\) 12.4967 0.416325
\(902\) 0 0
\(903\) −25.1595 −0.837256
\(904\) −1.61136 −0.0535930
\(905\) 46.4828 1.54514
\(906\) −12.9781 −0.431168
\(907\) −4.36413 −0.144909 −0.0724543 0.997372i \(-0.523083\pi\)
−0.0724543 + 0.997372i \(0.523083\pi\)
\(908\) −9.16956 −0.304302
\(909\) −1.14980 −0.0381366
\(910\) −48.6543 −1.61287
\(911\) 45.2710 1.49989 0.749947 0.661498i \(-0.230079\pi\)
0.749947 + 0.661498i \(0.230079\pi\)
\(912\) 0.912987 0.0302320
\(913\) 0 0
\(914\) 15.7612 0.521334
\(915\) 4.69419 0.155185
\(916\) −0.458204 −0.0151395
\(917\) −68.8927 −2.27504
\(918\) 8.59654 0.283728
\(919\) 50.6574 1.67103 0.835517 0.549464i \(-0.185168\pi\)
0.835517 + 0.549464i \(0.185168\pi\)
\(920\) 9.98389 0.329159
\(921\) −24.3241 −0.801507
\(922\) 6.61141 0.217735
\(923\) 4.73360 0.155808
\(924\) 0 0
\(925\) 18.5849 0.611066
\(926\) 32.4747 1.06719
\(927\) 7.88352 0.258929
\(928\) 2.23982 0.0735257
\(929\) −1.41717 −0.0464958 −0.0232479 0.999730i \(-0.507401\pi\)
−0.0232479 + 0.999730i \(0.507401\pi\)
\(930\) 0.627184 0.0205662
\(931\) 18.4776 0.605578
\(932\) −5.83823 −0.191237
\(933\) −22.9015 −0.749760
\(934\) −38.0595 −1.24534
\(935\) 0 0
\(936\) −11.7502 −0.384068
\(937\) −0.392294 −0.0128157 −0.00640784 0.999979i \(-0.502040\pi\)
−0.00640784 + 0.999979i \(0.502040\pi\)
\(938\) 58.3771 1.90608
\(939\) 31.1630 1.01696
\(940\) −19.2177 −0.626813
\(941\) −24.7594 −0.807132 −0.403566 0.914950i \(-0.632229\pi\)
−0.403566 + 0.914950i \(0.632229\pi\)
\(942\) −12.6470 −0.412060
\(943\) −52.1503 −1.69825
\(944\) 3.70481 0.120581
\(945\) 42.3139 1.37647
\(946\) 0 0
\(947\) 30.8771 1.00337 0.501685 0.865050i \(-0.332714\pi\)
0.501685 + 0.865050i \(0.332714\pi\)
\(948\) −8.50728 −0.276304
\(949\) 53.0555 1.72226
\(950\) 1.84141 0.0597434
\(951\) −27.5475 −0.893290
\(952\) −9.19910 −0.298144
\(953\) −7.78750 −0.252262 −0.126131 0.992014i \(-0.540256\pi\)
−0.126131 + 0.992014i \(0.540256\pi\)
\(954\) 14.8552 0.480955
\(955\) −46.2057 −1.49518
\(956\) −7.69554 −0.248892
\(957\) 0 0
\(958\) 30.0892 0.972138
\(959\) −45.5760 −1.47173
\(960\) −1.62260 −0.0523691
\(961\) −30.8506 −0.995180
\(962\) −54.7399 −1.76488
\(963\) 19.2127 0.619119
\(964\) 11.2195 0.361357
\(965\) 16.2757 0.523932
\(966\) −25.8879 −0.832931
\(967\) −13.5476 −0.435662 −0.217831 0.975987i \(-0.569898\pi\)
−0.217831 + 0.975987i \(0.569898\pi\)
\(968\) 0 0
\(969\) 1.66391 0.0534526
\(970\) −5.16339 −0.165786
\(971\) 45.5661 1.46229 0.731143 0.682224i \(-0.238987\pi\)
0.731143 + 0.682224i \(0.238987\pi\)
\(972\) 16.1528 0.518101
\(973\) −52.2791 −1.67599
\(974\) 2.69394 0.0863193
\(975\) 9.11827 0.292018
\(976\) −2.89301 −0.0926029
\(977\) −21.4985 −0.687796 −0.343898 0.939007i \(-0.611748\pi\)
−0.343898 + 0.939007i \(0.611748\pi\)
\(978\) −9.98334 −0.319232
\(979\) 0 0
\(980\) −32.8391 −1.04901
\(981\) −4.39122 −0.140201
\(982\) 27.8830 0.889783
\(983\) 13.6341 0.434862 0.217431 0.976076i \(-0.430232\pi\)
0.217431 + 0.976076i \(0.430232\pi\)
\(984\) 8.47556 0.270191
\(985\) −34.7770 −1.10809
\(986\) 4.08206 0.129999
\(987\) 49.8310 1.58614
\(988\) −5.42371 −0.172551
\(989\) −30.6699 −0.975245
\(990\) 0 0
\(991\) 16.9742 0.539202 0.269601 0.962972i \(-0.413108\pi\)
0.269601 + 0.962972i \(0.413108\pi\)
\(992\) −0.386531 −0.0122724
\(993\) 9.83483 0.312099
\(994\) 4.40529 0.139727
\(995\) 26.3775 0.836223
\(996\) 3.91399 0.124020
\(997\) 40.8048 1.29230 0.646150 0.763211i \(-0.276378\pi\)
0.646150 + 0.763211i \(0.276378\pi\)
\(998\) −21.0031 −0.664843
\(999\) 47.6064 1.50620
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 4598.2.a.bx.1.5 8
11.7 odd 10 418.2.f.f.115.2 16
11.8 odd 10 418.2.f.f.229.2 yes 16
11.10 odd 2 4598.2.a.ca.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.f.f.115.2 16 11.7 odd 10
418.2.f.f.229.2 yes 16 11.8 odd 10
4598.2.a.bx.1.5 8 1.1 even 1 trivial
4598.2.a.ca.1.5 8 11.10 odd 2