Properties

Label 6030.2.a.bu.1.1
Level 6030
Weight 2
Character 6030.1
Self dual yes
Analytic conductor 48.150
Analytic rank 0
Dimension 4
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6030.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1497924188\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.70292.1
Defining polynomial: \(x^{4} - x^{3} - 8 x^{2} + 10 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2010)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.175890\) of defining polynomial
Character \(\chi\) \(=\) 6030.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.96906 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.96906 q^{7} +1.00000 q^{8} +1.00000 q^{10} -3.04990 q^{11} +0.919164 q^{13} -3.96906 q^{14} +1.00000 q^{16} -0.351780 q^{17} -0.351780 q^{19} +1.00000 q^{20} -3.04990 q^{22} +3.43262 q^{23} +1.00000 q^{25} +0.919164 q^{26} -3.96906 q^{28} -9.37074 q^{29} +2.91916 q^{31} +1.00000 q^{32} -0.351780 q^{34} -3.96906 q^{35} +5.40168 q^{37} -0.351780 q^{38} +1.00000 q^{40} +8.45158 q^{41} +6.09980 q^{43} -3.04990 q^{44} +3.43262 q^{46} +13.2091 q^{47} +8.75346 q^{49} +1.00000 q^{50} +0.919164 q^{52} -5.93813 q^{53} -3.04990 q^{55} -3.96906 q^{56} -9.37074 q^{58} +8.45158 q^{59} -1.96906 q^{61} +2.91916 q^{62} +1.00000 q^{64} +0.919164 q^{65} -1.00000 q^{67} -0.351780 q^{68} -3.96906 q^{70} +5.61728 q^{71} -12.8034 q^{73} +5.40168 q^{74} -0.351780 q^{76} +12.1052 q^{77} +11.8842 q^{79} +1.00000 q^{80} +8.45158 q^{82} -0.105239 q^{83} -0.351780 q^{85} +6.09980 q^{86} -3.04990 q^{88} +16.2050 q^{89} -3.64822 q^{91} +3.43262 q^{92} +13.2091 q^{94} -0.351780 q^{95} +3.04990 q^{97} +8.75346 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{4} + 4q^{5} + q^{7} + 4q^{8} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{4} + 4q^{5} + q^{7} + 4q^{8} + 4q^{10} + 3q^{11} + 2q^{13} + q^{14} + 4q^{16} + 2q^{17} + 2q^{19} + 4q^{20} + 3q^{22} + 12q^{23} + 4q^{25} + 2q^{26} + q^{28} - 2q^{29} + 10q^{31} + 4q^{32} + 2q^{34} + q^{35} + 3q^{37} + 2q^{38} + 4q^{40} - 6q^{43} + 3q^{44} + 12q^{46} + 14q^{47} + 13q^{49} + 4q^{50} + 2q^{52} + 10q^{53} + 3q^{55} + q^{56} - 2q^{58} + 9q^{61} + 10q^{62} + 4q^{64} + 2q^{65} - 4q^{67} + 2q^{68} + q^{70} + 9q^{71} - 14q^{73} + 3q^{74} + 2q^{76} + 23q^{77} + 12q^{79} + 4q^{80} + 25q^{83} + 2q^{85} - 6q^{86} + 3q^{88} + 9q^{89} - 18q^{91} + 12q^{92} + 14q^{94} + 2q^{95} - 3q^{97} + 13q^{98} + O(q^{100}) \)

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
\(7\) −3.96906 −1.50016 −0.750082 0.661344i \(-0.769986\pi\)
−0.750082 + 0.661344i \(0.769986\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −3.04990 −0.919579 −0.459790 0.888028i \(-0.652075\pi\)
−0.459790 + 0.888028i \(0.652075\pi\)
\(12\) 0 0
\(13\) 0.919164 0.254930 0.127465 0.991843i \(-0.459316\pi\)
0.127465 + 0.991843i \(0.459316\pi\)
\(14\) −3.96906 −1.06078
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.351780 −0.0853192 −0.0426596 0.999090i \(-0.513583\pi\)
−0.0426596 + 0.999090i \(0.513583\pi\)
\(18\) 0 0
\(19\) −0.351780 −0.0807039 −0.0403519 0.999186i \(-0.512848\pi\)
−0.0403519 + 0.999186i \(0.512848\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −3.04990 −0.650241
\(23\) 3.43262 0.715750 0.357875 0.933770i \(-0.383501\pi\)
0.357875 + 0.933770i \(0.383501\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0.919164 0.180263
\(27\) 0 0
\(28\) −3.96906 −0.750082
\(29\) −9.37074 −1.74010 −0.870051 0.492961i \(-0.835915\pi\)
−0.870051 + 0.492961i \(0.835915\pi\)
\(30\) 0 0
\(31\) 2.91916 0.524297 0.262149 0.965027i \(-0.415569\pi\)
0.262149 + 0.965027i \(0.415569\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −0.351780 −0.0603298
\(35\) −3.96906 −0.670894
\(36\) 0 0
\(37\) 5.40168 0.888030 0.444015 0.896019i \(-0.353554\pi\)
0.444015 + 0.896019i \(0.353554\pi\)
\(38\) −0.351780 −0.0570663
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 8.45158 1.31991 0.659957 0.751303i \(-0.270574\pi\)
0.659957 + 0.751303i \(0.270574\pi\)
\(42\) 0 0
\(43\) 6.09980 0.930210 0.465105 0.885255i \(-0.346017\pi\)
0.465105 + 0.885255i \(0.346017\pi\)
\(44\) −3.04990 −0.459790
\(45\) 0 0
\(46\) 3.43262 0.506112
\(47\) 13.2091 1.92674 0.963370 0.268174i \(-0.0864203\pi\)
0.963370 + 0.268174i \(0.0864203\pi\)
\(48\) 0 0
\(49\) 8.75346 1.25049
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 0.919164 0.127465
\(53\) −5.93813 −0.815664 −0.407832 0.913057i \(-0.633715\pi\)
−0.407832 + 0.913057i \(0.633715\pi\)
\(54\) 0 0
\(55\) −3.04990 −0.411248
\(56\) −3.96906 −0.530388
\(57\) 0 0
\(58\) −9.37074 −1.23044
\(59\) 8.45158 1.10030 0.550151 0.835065i \(-0.314570\pi\)
0.550151 + 0.835065i \(0.314570\pi\)
\(60\) 0 0
\(61\) −1.96906 −0.252113 −0.126056 0.992023i \(-0.540232\pi\)
−0.126056 + 0.992023i \(0.540232\pi\)
\(62\) 2.91916 0.370734
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0.919164 0.114008
\(66\) 0 0
\(67\) −1.00000 −0.122169
\(68\) −0.351780 −0.0426596
\(69\) 0 0
\(70\) −3.96906 −0.474394
\(71\) 5.61728 0.666649 0.333324 0.942812i \(-0.391830\pi\)
0.333324 + 0.942812i \(0.391830\pi\)
\(72\) 0 0
\(73\) −12.8034 −1.49852 −0.749260 0.662276i \(-0.769591\pi\)
−0.749260 + 0.662276i \(0.769591\pi\)
\(74\) 5.40168 0.627932
\(75\) 0 0
\(76\) −0.351780 −0.0403519
\(77\) 12.1052 1.37952
\(78\) 0 0
\(79\) 11.8842 1.33708 0.668538 0.743678i \(-0.266920\pi\)
0.668538 + 0.743678i \(0.266920\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 8.45158 0.933321
\(83\) −0.105239 −0.0115515 −0.00577573 0.999983i \(-0.501838\pi\)
−0.00577573 + 0.999983i \(0.501838\pi\)
\(84\) 0 0
\(85\) −0.351780 −0.0381559
\(86\) 6.09980 0.657758
\(87\) 0 0
\(88\) −3.04990 −0.325120
\(89\) 16.2050 1.71773 0.858865 0.512202i \(-0.171170\pi\)
0.858865 + 0.512202i \(0.171170\pi\)
\(90\) 0 0
\(91\) −3.64822 −0.382437
\(92\) 3.43262 0.357875
\(93\) 0 0
\(94\) 13.2091 1.36241
\(95\) −0.351780 −0.0360919
\(96\) 0 0
\(97\) 3.04990 0.309670 0.154835 0.987940i \(-0.450515\pi\)
0.154835 + 0.987940i \(0.450515\pi\)
\(98\) 8.75346 0.884233
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 4.69812 0.467480 0.233740 0.972299i \(-0.424904\pi\)
0.233740 + 0.972299i \(0.424904\pi\)
\(102\) 0 0
\(103\) 2.09980 0.206899 0.103450 0.994635i \(-0.467012\pi\)
0.103450 + 0.994635i \(0.467012\pi\)
\(104\) 0.919164 0.0901315
\(105\) 0 0
\(106\) −5.93813 −0.576762
\(107\) 9.83833 0.951107 0.475554 0.879687i \(-0.342248\pi\)
0.475554 + 0.879687i \(0.342248\pi\)
\(108\) 0 0
\(109\) 1.26550 0.121213 0.0606066 0.998162i \(-0.480696\pi\)
0.0606066 + 0.998162i \(0.480696\pi\)
\(110\) −3.04990 −0.290796
\(111\) 0 0
\(112\) −3.96906 −0.375041
\(113\) −9.69158 −0.911708 −0.455854 0.890055i \(-0.650666\pi\)
−0.455854 + 0.890055i \(0.650666\pi\)
\(114\) 0 0
\(115\) 3.43262 0.320093
\(116\) −9.37074 −0.870051
\(117\) 0 0
\(118\) 8.45158 0.778031
\(119\) 1.39624 0.127993
\(120\) 0 0
\(121\) −1.69812 −0.154374
\(122\) −1.96906 −0.178271
\(123\) 0 0
\(124\) 2.91916 0.262149
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −12.9650 −1.15046 −0.575230 0.817992i \(-0.695088\pi\)
−0.575230 + 0.817992i \(0.695088\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0.919164 0.0806160
\(131\) 12.5135 1.09331 0.546653 0.837359i \(-0.315902\pi\)
0.546653 + 0.837359i \(0.315902\pi\)
\(132\) 0 0
\(133\) 1.39624 0.121069
\(134\) −1.00000 −0.0863868
\(135\) 0 0
\(136\) −0.351780 −0.0301649
\(137\) −1.40168 −0.119753 −0.0598767 0.998206i \(-0.519071\pi\)
−0.0598767 + 0.998206i \(0.519071\pi\)
\(138\) 0 0
\(139\) 6.02844 0.511325 0.255663 0.966766i \(-0.417706\pi\)
0.255663 + 0.966766i \(0.417706\pi\)
\(140\) −3.96906 −0.335447
\(141\) 0 0
\(142\) 5.61728 0.471392
\(143\) −2.80336 −0.234429
\(144\) 0 0
\(145\) −9.37074 −0.778198
\(146\) −12.8034 −1.05961
\(147\) 0 0
\(148\) 5.40168 0.444015
\(149\) −9.37074 −0.767681 −0.383841 0.923399i \(-0.625399\pi\)
−0.383841 + 0.923399i \(0.625399\pi\)
\(150\) 0 0
\(151\) −3.14021 −0.255547 −0.127773 0.991803i \(-0.540783\pi\)
−0.127773 + 0.991803i \(0.540783\pi\)
\(152\) −0.351780 −0.0285331
\(153\) 0 0
\(154\) 12.1052 0.975468
\(155\) 2.91916 0.234473
\(156\) 0 0
\(157\) 14.5798 1.16360 0.581798 0.813333i \(-0.302350\pi\)
0.581798 + 0.813333i \(0.302350\pi\)
\(158\) 11.8842 0.945456
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −13.6243 −1.07374
\(162\) 0 0
\(163\) −4.59832 −0.360168 −0.180084 0.983651i \(-0.557637\pi\)
−0.180084 + 0.983651i \(0.557637\pi\)
\(164\) 8.45158 0.659957
\(165\) 0 0
\(166\) −0.105239 −0.00816811
\(167\) 6.40571 0.495689 0.247844 0.968800i \(-0.420278\pi\)
0.247844 + 0.968800i \(0.420278\pi\)
\(168\) 0 0
\(169\) −12.1551 −0.935011
\(170\) −0.351780 −0.0269803
\(171\) 0 0
\(172\) 6.09980 0.465105
\(173\) 24.9625 1.89787 0.948933 0.315478i \(-0.102165\pi\)
0.948933 + 0.315478i \(0.102165\pi\)
\(174\) 0 0
\(175\) −3.96906 −0.300033
\(176\) −3.04990 −0.229895
\(177\) 0 0
\(178\) 16.2050 1.21462
\(179\) 14.1282 1.05599 0.527997 0.849246i \(-0.322943\pi\)
0.527997 + 0.849246i \(0.322943\pi\)
\(180\) 0 0
\(181\) −11.0729 −0.823042 −0.411521 0.911400i \(-0.635002\pi\)
−0.411521 + 0.911400i \(0.635002\pi\)
\(182\) −3.64822 −0.270424
\(183\) 0 0
\(184\) 3.43262 0.253056
\(185\) 5.40168 0.397139
\(186\) 0 0
\(187\) 1.07289 0.0784577
\(188\) 13.2091 0.963370
\(189\) 0 0
\(190\) −0.351780 −0.0255208
\(191\) 18.0998 1.30966 0.654828 0.755778i \(-0.272741\pi\)
0.654828 + 0.755778i \(0.272741\pi\)
\(192\) 0 0
\(193\) −11.6861 −0.841187 −0.420593 0.907249i \(-0.638178\pi\)
−0.420593 + 0.907249i \(0.638178\pi\)
\(194\) 3.04990 0.218970
\(195\) 0 0
\(196\) 8.75346 0.625247
\(197\) 0.161672 0.0115186 0.00575932 0.999983i \(-0.498167\pi\)
0.00575932 + 0.999983i \(0.498167\pi\)
\(198\) 0 0
\(199\) −8.10524 −0.574565 −0.287283 0.957846i \(-0.592752\pi\)
−0.287283 + 0.957846i \(0.592752\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 4.69812 0.330558
\(203\) 37.1931 2.61044
\(204\) 0 0
\(205\) 8.45158 0.590284
\(206\) 2.09980 0.146300
\(207\) 0 0
\(208\) 0.919164 0.0637326
\(209\) 1.07289 0.0742136
\(210\) 0 0
\(211\) 18.3278 1.26174 0.630870 0.775889i \(-0.282698\pi\)
0.630870 + 0.775889i \(0.282698\pi\)
\(212\) −5.93813 −0.407832
\(213\) 0 0
\(214\) 9.83833 0.672534
\(215\) 6.09980 0.416003
\(216\) 0 0
\(217\) −11.5863 −0.786532
\(218\) 1.26550 0.0857107
\(219\) 0 0
\(220\) −3.04990 −0.205624
\(221\) −0.323344 −0.0217504
\(222\) 0 0
\(223\) −0.442091 −0.0296046 −0.0148023 0.999890i \(-0.504712\pi\)
−0.0148023 + 0.999890i \(0.504712\pi\)
\(224\) −3.96906 −0.265194
\(225\) 0 0
\(226\) −9.69158 −0.644675
\(227\) 25.3344 1.68150 0.840750 0.541423i \(-0.182114\pi\)
0.840750 + 0.541423i \(0.182114\pi\)
\(228\) 0 0
\(229\) 8.06886 0.533205 0.266603 0.963807i \(-0.414099\pi\)
0.266603 + 0.963807i \(0.414099\pi\)
\(230\) 3.43262 0.226340
\(231\) 0 0
\(232\) −9.37074 −0.615219
\(233\) −3.67121 −0.240509 −0.120255 0.992743i \(-0.538371\pi\)
−0.120255 + 0.992743i \(0.538371\pi\)
\(234\) 0 0
\(235\) 13.2091 0.861665
\(236\) 8.45158 0.550151
\(237\) 0 0
\(238\) 1.39624 0.0905046
\(239\) −12.9650 −0.838638 −0.419319 0.907839i \(-0.637731\pi\)
−0.419319 + 0.907839i \(0.637731\pi\)
\(240\) 0 0
\(241\) −21.9531 −1.41412 −0.707060 0.707153i \(-0.749979\pi\)
−0.707060 + 0.707153i \(0.749979\pi\)
\(242\) −1.69812 −0.109159
\(243\) 0 0
\(244\) −1.96906 −0.126056
\(245\) 8.75346 0.559238
\(246\) 0 0
\(247\) −0.323344 −0.0205739
\(248\) 2.91916 0.185367
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 4.59832 0.290243 0.145122 0.989414i \(-0.453643\pi\)
0.145122 + 0.989414i \(0.453643\pi\)
\(252\) 0 0
\(253\) −10.4691 −0.658189
\(254\) −12.9650 −0.813498
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −19.7968 −1.23489 −0.617446 0.786613i \(-0.711833\pi\)
−0.617446 + 0.786613i \(0.711833\pi\)
\(258\) 0 0
\(259\) −21.4396 −1.33219
\(260\) 0.919164 0.0570041
\(261\) 0 0
\(262\) 12.5135 0.773084
\(263\) 17.4705 1.07728 0.538640 0.842536i \(-0.318938\pi\)
0.538640 + 0.842536i \(0.318938\pi\)
\(264\) 0 0
\(265\) −5.93813 −0.364776
\(266\) 1.39624 0.0856088
\(267\) 0 0
\(268\) −1.00000 −0.0610847
\(269\) −7.33282 −0.447090 −0.223545 0.974694i \(-0.571763\pi\)
−0.223545 + 0.974694i \(0.571763\pi\)
\(270\) 0 0
\(271\) −14.1537 −0.859778 −0.429889 0.902882i \(-0.641447\pi\)
−0.429889 + 0.902882i \(0.641447\pi\)
\(272\) −0.351780 −0.0213298
\(273\) 0 0
\(274\) −1.40168 −0.0846785
\(275\) −3.04990 −0.183916
\(276\) 0 0
\(277\) 13.6631 0.820939 0.410469 0.911874i \(-0.365365\pi\)
0.410469 + 0.911874i \(0.365365\pi\)
\(278\) 6.02844 0.361562
\(279\) 0 0
\(280\) −3.96906 −0.237197
\(281\) 1.67666 0.100021 0.0500105 0.998749i \(-0.484075\pi\)
0.0500105 + 0.998749i \(0.484075\pi\)
\(282\) 0 0
\(283\) 0.978538 0.0581680 0.0290840 0.999577i \(-0.490741\pi\)
0.0290840 + 0.999577i \(0.490741\pi\)
\(284\) 5.61728 0.333324
\(285\) 0 0
\(286\) −2.80336 −0.165766
\(287\) −33.5448 −1.98009
\(288\) 0 0
\(289\) −16.8763 −0.992721
\(290\) −9.37074 −0.550269
\(291\) 0 0
\(292\) −12.8034 −0.749260
\(293\) −16.5823 −0.968749 −0.484374 0.874861i \(-0.660953\pi\)
−0.484374 + 0.874861i \(0.660953\pi\)
\(294\) 0 0
\(295\) 8.45158 0.492070
\(296\) 5.40168 0.313966
\(297\) 0 0
\(298\) −9.37074 −0.542832
\(299\) 3.15514 0.182466
\(300\) 0 0
\(301\) −24.2105 −1.39547
\(302\) −3.14021 −0.180699
\(303\) 0 0
\(304\) −0.351780 −0.0201760
\(305\) −1.96906 −0.112748
\(306\) 0 0
\(307\) −24.3843 −1.39168 −0.695842 0.718195i \(-0.744968\pi\)
−0.695842 + 0.718195i \(0.744968\pi\)
\(308\) 12.1052 0.689760
\(309\) 0 0
\(310\) 2.91916 0.165797
\(311\) −6.03792 −0.342379 −0.171190 0.985238i \(-0.554761\pi\)
−0.171190 + 0.985238i \(0.554761\pi\)
\(312\) 0 0
\(313\) 11.4301 0.646068 0.323034 0.946387i \(-0.395297\pi\)
0.323034 + 0.946387i \(0.395297\pi\)
\(314\) 14.5798 0.822786
\(315\) 0 0
\(316\) 11.8842 0.668538
\(317\) 29.3911 1.65077 0.825385 0.564571i \(-0.190958\pi\)
0.825385 + 0.564571i \(0.190958\pi\)
\(318\) 0 0
\(319\) 28.5798 1.60016
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −13.6243 −0.759251
\(323\) 0.123749 0.00688559
\(324\) 0 0
\(325\) 0.919164 0.0509861
\(326\) −4.59832 −0.254677
\(327\) 0 0
\(328\) 8.45158 0.466660
\(329\) −52.4276 −2.89043
\(330\) 0 0
\(331\) −4.60781 −0.253268 −0.126634 0.991950i \(-0.540417\pi\)
−0.126634 + 0.991950i \(0.540417\pi\)
\(332\) −0.105239 −0.00577573
\(333\) 0 0
\(334\) 6.40571 0.350505
\(335\) −1.00000 −0.0546358
\(336\) 0 0
\(337\) 18.0095 0.981039 0.490520 0.871430i \(-0.336807\pi\)
0.490520 + 0.871430i \(0.336807\pi\)
\(338\) −12.1551 −0.661152
\(339\) 0 0
\(340\) −0.351780 −0.0190780
\(341\) −8.90315 −0.482133
\(342\) 0 0
\(343\) −6.95959 −0.375782
\(344\) 6.09980 0.328879
\(345\) 0 0
\(346\) 24.9625 1.34199
\(347\) 10.7239 0.575691 0.287845 0.957677i \(-0.407061\pi\)
0.287845 + 0.957677i \(0.407061\pi\)
\(348\) 0 0
\(349\) 18.1996 0.974202 0.487101 0.873346i \(-0.338054\pi\)
0.487101 + 0.873346i \(0.338054\pi\)
\(350\) −3.96906 −0.212155
\(351\) 0 0
\(352\) −3.04990 −0.162560
\(353\) −25.5863 −1.36182 −0.680912 0.732365i \(-0.738416\pi\)
−0.680912 + 0.732365i \(0.738416\pi\)
\(354\) 0 0
\(355\) 5.61728 0.298134
\(356\) 16.2050 0.858865
\(357\) 0 0
\(358\) 14.1282 0.746700
\(359\) −4.59038 −0.242271 −0.121135 0.992636i \(-0.538654\pi\)
−0.121135 + 0.992636i \(0.538654\pi\)
\(360\) 0 0
\(361\) −18.8763 −0.993487
\(362\) −11.0729 −0.581978
\(363\) 0 0
\(364\) −3.64822 −0.191219
\(365\) −12.8034 −0.670158
\(366\) 0 0
\(367\) 26.9855 1.40863 0.704316 0.709886i \(-0.251254\pi\)
0.704316 + 0.709886i \(0.251254\pi\)
\(368\) 3.43262 0.178937
\(369\) 0 0
\(370\) 5.40168 0.280820
\(371\) 23.5688 1.22363
\(372\) 0 0
\(373\) −16.7954 −0.869634 −0.434817 0.900519i \(-0.643187\pi\)
−0.434817 + 0.900519i \(0.643187\pi\)
\(374\) 1.07289 0.0554780
\(375\) 0 0
\(376\) 13.2091 0.681206
\(377\) −8.61325 −0.443605
\(378\) 0 0
\(379\) −21.9047 −1.12517 −0.562584 0.826740i \(-0.690193\pi\)
−0.562584 + 0.826740i \(0.690193\pi\)
\(380\) −0.351780 −0.0180459
\(381\) 0 0
\(382\) 18.0998 0.926066
\(383\) 4.47457 0.228640 0.114320 0.993444i \(-0.463531\pi\)
0.114320 + 0.993444i \(0.463531\pi\)
\(384\) 0 0
\(385\) 12.1052 0.616940
\(386\) −11.6861 −0.594809
\(387\) 0 0
\(388\) 3.04990 0.154835
\(389\) −30.4545 −1.54411 −0.772053 0.635559i \(-0.780770\pi\)
−0.772053 + 0.635559i \(0.780770\pi\)
\(390\) 0 0
\(391\) −1.20753 −0.0610672
\(392\) 8.75346 0.442116
\(393\) 0 0
\(394\) 0.161672 0.00814491
\(395\) 11.8842 0.597959
\(396\) 0 0
\(397\) 27.3777 1.37405 0.687024 0.726634i \(-0.258917\pi\)
0.687024 + 0.726634i \(0.258917\pi\)
\(398\) −8.10524 −0.406279
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 34.0867 1.70221 0.851105 0.524996i \(-0.175933\pi\)
0.851105 + 0.524996i \(0.175933\pi\)
\(402\) 0 0
\(403\) 2.68319 0.133659
\(404\) 4.69812 0.233740
\(405\) 0 0
\(406\) 37.1931 1.84586
\(407\) −16.4746 −0.816614
\(408\) 0 0
\(409\) 8.16167 0.403569 0.201784 0.979430i \(-0.435326\pi\)
0.201784 + 0.979430i \(0.435326\pi\)
\(410\) 8.45158 0.417394
\(411\) 0 0
\(412\) 2.09980 0.103450
\(413\) −33.5448 −1.65063
\(414\) 0 0
\(415\) −0.105239 −0.00516597
\(416\) 0.919164 0.0450657
\(417\) 0 0
\(418\) 1.07289 0.0524769
\(419\) −27.3628 −1.33676 −0.668380 0.743820i \(-0.733012\pi\)
−0.668380 + 0.743820i \(0.733012\pi\)
\(420\) 0 0
\(421\) −15.9760 −0.778625 −0.389312 0.921106i \(-0.627287\pi\)
−0.389312 + 0.921106i \(0.627287\pi\)
\(422\) 18.3278 0.892185
\(423\) 0 0
\(424\) −5.93813 −0.288381
\(425\) −0.351780 −0.0170638
\(426\) 0 0
\(427\) 7.81533 0.378210
\(428\) 9.83833 0.475554
\(429\) 0 0
\(430\) 6.09980 0.294158
\(431\) −16.7900 −0.808745 −0.404372 0.914594i \(-0.632510\pi\)
−0.404372 + 0.914594i \(0.632510\pi\)
\(432\) 0 0
\(433\) 29.6352 1.42417 0.712087 0.702091i \(-0.247750\pi\)
0.712087 + 0.702091i \(0.247750\pi\)
\(434\) −11.5863 −0.556162
\(435\) 0 0
\(436\) 1.26550 0.0606066
\(437\) −1.20753 −0.0577638
\(438\) 0 0
\(439\) 1.99456 0.0951951 0.0475975 0.998867i \(-0.484844\pi\)
0.0475975 + 0.998867i \(0.484844\pi\)
\(440\) −3.04990 −0.145398
\(441\) 0 0
\(442\) −0.323344 −0.0153799
\(443\) 3.29644 0.156619 0.0783093 0.996929i \(-0.475048\pi\)
0.0783093 + 0.996929i \(0.475048\pi\)
\(444\) 0 0
\(445\) 16.2050 0.768192
\(446\) −0.442091 −0.0209336
\(447\) 0 0
\(448\) −3.96906 −0.187521
\(449\) 13.9196 0.656907 0.328454 0.944520i \(-0.393473\pi\)
0.328454 + 0.944520i \(0.393473\pi\)
\(450\) 0 0
\(451\) −25.7765 −1.21377
\(452\) −9.69158 −0.455854
\(453\) 0 0
\(454\) 25.3344 1.18900
\(455\) −3.64822 −0.171031
\(456\) 0 0
\(457\) 12.3897 0.579566 0.289783 0.957092i \(-0.406417\pi\)
0.289783 + 0.957092i \(0.406417\pi\)
\(458\) 8.06886 0.377033
\(459\) 0 0
\(460\) 3.43262 0.160047
\(461\) −38.8537 −1.80960 −0.904799 0.425839i \(-0.859979\pi\)
−0.904799 + 0.425839i \(0.859979\pi\)
\(462\) 0 0
\(463\) −5.85076 −0.271908 −0.135954 0.990715i \(-0.543410\pi\)
−0.135954 + 0.990715i \(0.543410\pi\)
\(464\) −9.37074 −0.435026
\(465\) 0 0
\(466\) −3.67121 −0.170066
\(467\) 6.37477 0.294989 0.147495 0.989063i \(-0.452879\pi\)
0.147495 + 0.989063i \(0.452879\pi\)
\(468\) 0 0
\(469\) 3.96906 0.183274
\(470\) 13.2091 0.609289
\(471\) 0 0
\(472\) 8.45158 0.389015
\(473\) −18.6038 −0.855402
\(474\) 0 0
\(475\) −0.351780 −0.0161408
\(476\) 1.39624 0.0639964
\(477\) 0 0
\(478\) −12.9650 −0.593007
\(479\) −13.5094 −0.617261 −0.308631 0.951182i \(-0.599871\pi\)
−0.308631 + 0.951182i \(0.599871\pi\)
\(480\) 0 0
\(481\) 4.96503 0.226386
\(482\) −21.9531 −0.999934
\(483\) 0 0
\(484\) −1.69812 −0.0771872
\(485\) 3.04990 0.138489
\(486\) 0 0
\(487\) 24.4920 1.10984 0.554919 0.831904i \(-0.312749\pi\)
0.554919 + 0.831904i \(0.312749\pi\)
\(488\) −1.96906 −0.0891353
\(489\) 0 0
\(490\) 8.75346 0.395441
\(491\) −1.59723 −0.0720819 −0.0360410 0.999350i \(-0.511475\pi\)
−0.0360410 + 0.999350i \(0.511475\pi\)
\(492\) 0 0
\(493\) 3.29644 0.148464
\(494\) −0.323344 −0.0145479
\(495\) 0 0
\(496\) 2.91916 0.131074
\(497\) −22.2953 −1.00008
\(498\) 0 0
\(499\) 21.5837 0.966220 0.483110 0.875560i \(-0.339507\pi\)
0.483110 + 0.875560i \(0.339507\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 4.59832 0.205233
\(503\) 34.3428 1.53127 0.765634 0.643277i \(-0.222425\pi\)
0.765634 + 0.643277i \(0.222425\pi\)
\(504\) 0 0
\(505\) 4.69812 0.209064
\(506\) −10.4691 −0.465410
\(507\) 0 0
\(508\) −12.9650 −0.575230
\(509\) −25.3089 −1.12180 −0.560898 0.827885i \(-0.689544\pi\)
−0.560898 + 0.827885i \(0.689544\pi\)
\(510\) 0 0
\(511\) 50.8173 2.24803
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −19.7968 −0.873200
\(515\) 2.09980 0.0925281
\(516\) 0 0
\(517\) −40.2863 −1.77179
\(518\) −21.4396 −0.942002
\(519\) 0 0
\(520\) 0.919164 0.0403080
\(521\) 23.4545 1.02756 0.513781 0.857922i \(-0.328244\pi\)
0.513781 + 0.857922i \(0.328244\pi\)
\(522\) 0 0
\(523\) −19.4586 −0.850863 −0.425432 0.904991i \(-0.639878\pi\)
−0.425432 + 0.904991i \(0.639878\pi\)
\(524\) 12.5135 0.546653
\(525\) 0 0
\(526\) 17.4705 0.761752
\(527\) −1.02690 −0.0447326
\(528\) 0 0
\(529\) −11.2171 −0.487702
\(530\) −5.93813 −0.257936
\(531\) 0 0
\(532\) 1.39624 0.0605346
\(533\) 7.76839 0.336486
\(534\) 0 0
\(535\) 9.83833 0.425348
\(536\) −1.00000 −0.0431934
\(537\) 0 0
\(538\) −7.33282 −0.316140
\(539\) −26.6972 −1.14993
\(540\) 0 0
\(541\) −39.2389 −1.68701 −0.843507 0.537119i \(-0.819513\pi\)
−0.843507 + 0.537119i \(0.819513\pi\)
\(542\) −14.1537 −0.607955
\(543\) 0 0
\(544\) −0.351780 −0.0150824
\(545\) 1.26550 0.0542082
\(546\) 0 0
\(547\) −26.1865 −1.11965 −0.559827 0.828609i \(-0.689133\pi\)
−0.559827 + 0.828609i \(0.689133\pi\)
\(548\) −1.40168 −0.0598767
\(549\) 0 0
\(550\) −3.04990 −0.130048
\(551\) 3.29644 0.140433
\(552\) 0 0
\(553\) −47.1691 −2.00583
\(554\) 13.6631 0.580492
\(555\) 0 0
\(556\) 6.02844 0.255663
\(557\) 40.1326 1.70047 0.850236 0.526401i \(-0.176459\pi\)
0.850236 + 0.526401i \(0.176459\pi\)
\(558\) 0 0
\(559\) 5.60671 0.237139
\(560\) −3.96906 −0.167724
\(561\) 0 0
\(562\) 1.67666 0.0707255
\(563\) 18.0998 0.762816 0.381408 0.924407i \(-0.375439\pi\)
0.381408 + 0.924407i \(0.375439\pi\)
\(564\) 0 0
\(565\) −9.69158 −0.407728
\(566\) 0.978538 0.0411310
\(567\) 0 0
\(568\) 5.61728 0.235696
\(569\) −14.4905 −0.607472 −0.303736 0.952756i \(-0.598234\pi\)
−0.303736 + 0.952756i \(0.598234\pi\)
\(570\) 0 0
\(571\) 0.703560 0.0294431 0.0147215 0.999892i \(-0.495314\pi\)
0.0147215 + 0.999892i \(0.495314\pi\)
\(572\) −2.80336 −0.117214
\(573\) 0 0
\(574\) −33.5448 −1.40013
\(575\) 3.43262 0.143150
\(576\) 0 0
\(577\) −32.8237 −1.36647 −0.683235 0.730199i \(-0.739427\pi\)
−0.683235 + 0.730199i \(0.739427\pi\)
\(578\) −16.8763 −0.701959
\(579\) 0 0
\(580\) −9.37074 −0.389099
\(581\) 0.417699 0.0173291
\(582\) 0 0
\(583\) 18.1107 0.750068
\(584\) −12.8034 −0.529807
\(585\) 0 0
\(586\) −16.5823 −0.685009
\(587\) 39.2929 1.62179 0.810895 0.585192i \(-0.198981\pi\)
0.810895 + 0.585192i \(0.198981\pi\)
\(588\) 0 0
\(589\) −1.02690 −0.0423128
\(590\) 8.45158 0.347946
\(591\) 0 0
\(592\) 5.40168 0.222008
\(593\) −10.3048 −0.423169 −0.211584 0.977360i \(-0.567862\pi\)
−0.211584 + 0.977360i \(0.567862\pi\)
\(594\) 0 0
\(595\) 1.39624 0.0572401
\(596\) −9.37074 −0.383841
\(597\) 0 0
\(598\) 3.15514 0.129023
\(599\) −28.3183 −1.15706 −0.578528 0.815662i \(-0.696373\pi\)
−0.578528 + 0.815662i \(0.696373\pi\)
\(600\) 0 0
\(601\) 24.2335 0.988504 0.494252 0.869319i \(-0.335442\pi\)
0.494252 + 0.869319i \(0.335442\pi\)
\(602\) −24.2105 −0.986745
\(603\) 0 0
\(604\) −3.14021 −0.127773
\(605\) −1.69812 −0.0690383
\(606\) 0 0
\(607\) 27.6148 1.12085 0.560425 0.828205i \(-0.310638\pi\)
0.560425 + 0.828205i \(0.310638\pi\)
\(608\) −0.351780 −0.0142666
\(609\) 0 0
\(610\) −1.96906 −0.0797250
\(611\) 12.1413 0.491185
\(612\) 0 0
\(613\) 37.7525 1.52481 0.762405 0.647101i \(-0.224019\pi\)
0.762405 + 0.647101i \(0.224019\pi\)
\(614\) −24.3843 −0.984069
\(615\) 0 0
\(616\) 12.1052 0.487734
\(617\) 25.5733 1.02954 0.514771 0.857328i \(-0.327877\pi\)
0.514771 + 0.857328i \(0.327877\pi\)
\(618\) 0 0
\(619\) −20.3518 −0.818007 −0.409004 0.912533i \(-0.634124\pi\)
−0.409004 + 0.912533i \(0.634124\pi\)
\(620\) 2.91916 0.117236
\(621\) 0 0
\(622\) −6.03792 −0.242099
\(623\) −64.3188 −2.57688
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 11.4301 0.456839
\(627\) 0 0
\(628\) 14.5798 0.581798
\(629\) −1.90020 −0.0757660
\(630\) 0 0
\(631\) −17.7844 −0.707986 −0.353993 0.935248i \(-0.615176\pi\)
−0.353993 + 0.935248i \(0.615176\pi\)
\(632\) 11.8842 0.472728
\(633\) 0 0
\(634\) 29.3911 1.16727
\(635\) −12.9650 −0.514501
\(636\) 0 0
\(637\) 8.04586 0.318789
\(638\) 28.5798 1.13149
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 37.4292 1.47836 0.739181 0.673506i \(-0.235213\pi\)
0.739181 + 0.673506i \(0.235213\pi\)
\(642\) 0 0
\(643\) 25.7729 1.01638 0.508191 0.861244i \(-0.330314\pi\)
0.508191 + 0.861244i \(0.330314\pi\)
\(644\) −13.6243 −0.536871
\(645\) 0 0
\(646\) 0.123749 0.00486885
\(647\) 29.8658 1.17415 0.587073 0.809534i \(-0.300280\pi\)
0.587073 + 0.809534i \(0.300280\pi\)
\(648\) 0 0
\(649\) −25.7765 −1.01181
\(650\) 0.919164 0.0360526
\(651\) 0 0
\(652\) −4.59832 −0.180084
\(653\) 48.8282 1.91080 0.955398 0.295322i \(-0.0954269\pi\)
0.955398 + 0.295322i \(0.0954269\pi\)
\(654\) 0 0
\(655\) 12.5135 0.488941
\(656\) 8.45158 0.329979
\(657\) 0 0
\(658\) −52.4276 −2.04384
\(659\) −8.24110 −0.321028 −0.160514 0.987034i \(-0.551315\pi\)
−0.160514 + 0.987034i \(0.551315\pi\)
\(660\) 0 0
\(661\) −24.6053 −0.957036 −0.478518 0.878078i \(-0.658826\pi\)
−0.478518 + 0.878078i \(0.658826\pi\)
\(662\) −4.60781 −0.179088
\(663\) 0 0
\(664\) −0.105239 −0.00408406
\(665\) 1.39624 0.0541438
\(666\) 0 0
\(667\) −32.1662 −1.24548
\(668\) 6.40571 0.247844
\(669\) 0 0
\(670\) −1.00000 −0.0386334
\(671\) 6.00544 0.231838
\(672\) 0 0
\(673\) −14.0822 −0.542831 −0.271415 0.962462i \(-0.587492\pi\)
−0.271415 + 0.962462i \(0.587492\pi\)
\(674\) 18.0095 0.693699
\(675\) 0 0
\(676\) −12.1551 −0.467505
\(677\) −5.67666 −0.218172 −0.109086 0.994032i \(-0.534792\pi\)
−0.109086 + 0.994032i \(0.534792\pi\)
\(678\) 0 0
\(679\) −12.1052 −0.464556
\(680\) −0.351780 −0.0134901
\(681\) 0 0
\(682\) −8.90315 −0.340919
\(683\) −3.63067 −0.138924 −0.0694618 0.997585i \(-0.522128\pi\)
−0.0694618 + 0.997585i \(0.522128\pi\)
\(684\) 0 0
\(685\) −1.40168 −0.0535554
\(686\) −6.95959 −0.265718
\(687\) 0 0
\(688\) 6.09980 0.232553
\(689\) −5.45811 −0.207937
\(690\) 0 0
\(691\) 13.2374 0.503574 0.251787 0.967783i \(-0.418982\pi\)
0.251787 + 0.967783i \(0.418982\pi\)
\(692\) 24.9625 0.948933
\(693\) 0 0
\(694\) 10.7239 0.407075
\(695\) 6.02844 0.228672
\(696\) 0 0
\(697\) −2.97310 −0.112614
\(698\) 18.1996 0.688865
\(699\) 0 0
\(700\) −3.96906 −0.150016
\(701\) 20.5125 0.774746 0.387373 0.921923i \(-0.373383\pi\)
0.387373 + 0.921923i \(0.373383\pi\)
\(702\) 0 0
\(703\) −1.90020 −0.0716675
\(704\) −3.04990 −0.114947
\(705\) 0 0
\(706\) −25.5863 −0.962955
\(707\) −18.6471 −0.701297
\(708\) 0 0
\(709\) 18.1946 0.683312 0.341656 0.939825i \(-0.389012\pi\)
0.341656 + 0.939825i \(0.389012\pi\)
\(710\) 5.61728 0.210813
\(711\) 0 0
\(712\) 16.2050 0.607309
\(713\) 10.0204 0.375266
\(714\) 0 0
\(715\) −2.80336 −0.104840
\(716\) 14.1282 0.527997
\(717\) 0 0
\(718\) −4.59038 −0.171311
\(719\) −11.4990 −0.428839 −0.214420 0.976742i \(-0.568786\pi\)
−0.214420 + 0.976742i \(0.568786\pi\)
\(720\) 0 0
\(721\) −8.33423 −0.310383
\(722\) −18.8763 −0.702501
\(723\) 0 0
\(724\) −11.0729 −0.411521
\(725\) −9.37074 −0.348021
\(726\) 0 0
\(727\) 25.8508 0.958751 0.479376 0.877610i \(-0.340863\pi\)
0.479376 + 0.877610i \(0.340863\pi\)
\(728\) −3.64822 −0.135212
\(729\) 0 0
\(730\) −12.8034 −0.473874
\(731\) −2.14579 −0.0793648
\(732\) 0 0
\(733\) 32.1836 1.18873 0.594364 0.804196i \(-0.297404\pi\)
0.594364 + 0.804196i \(0.297404\pi\)
\(734\) 26.9855 0.996054
\(735\) 0 0
\(736\) 3.43262 0.126528
\(737\) 3.04990 0.112344
\(738\) 0 0
\(739\) −9.32488 −0.343021 −0.171511 0.985182i \(-0.554865\pi\)
−0.171511 + 0.985182i \(0.554865\pi\)
\(740\) 5.40168 0.198570
\(741\) 0 0
\(742\) 23.5688 0.865238
\(743\) 12.5624 0.460869 0.230435 0.973088i \(-0.425985\pi\)
0.230435 + 0.973088i \(0.425985\pi\)
\(744\) 0 0
\(745\) −9.37074 −0.343317
\(746\) −16.7954 −0.614924
\(747\) 0 0
\(748\) 1.07289 0.0392289
\(749\) −39.0489 −1.42682
\(750\) 0 0
\(751\) −16.1996 −0.591132 −0.295566 0.955322i \(-0.595508\pi\)
−0.295566 + 0.955322i \(0.595508\pi\)
\(752\) 13.2091 0.481685
\(753\) 0 0
\(754\) −8.61325 −0.313676
\(755\) −3.14021 −0.114284
\(756\) 0 0
\(757\) 16.8682 0.613084 0.306542 0.951857i \(-0.400828\pi\)
0.306542 + 0.951857i \(0.400828\pi\)
\(758\) −21.9047 −0.795614
\(759\) 0 0
\(760\) −0.351780 −0.0127604
\(761\) −21.1701 −0.767414 −0.383707 0.923455i \(-0.625353\pi\)
−0.383707 + 0.923455i \(0.625353\pi\)
\(762\) 0 0
\(763\) −5.02286 −0.181840
\(764\) 18.0998 0.654828
\(765\) 0 0
\(766\) 4.47457 0.161673
\(767\) 7.76839 0.280500
\(768\) 0 0
\(769\) −11.2884 −0.407069 −0.203535 0.979068i \(-0.565243\pi\)
−0.203535 + 0.979068i \(0.565243\pi\)
\(770\) 12.1052 0.436243
\(771\) 0 0
\(772\) −11.6861 −0.420593
\(773\) −48.8068 −1.75546 −0.877729 0.479158i \(-0.840942\pi\)
−0.877729 + 0.479158i \(0.840942\pi\)
\(774\) 0 0
\(775\) 2.91916 0.104859
\(776\) 3.04990 0.109485
\(777\) 0 0
\(778\) −30.4545 −1.09185
\(779\) −2.97310 −0.106522
\(780\) 0 0
\(781\) −17.1321 −0.613036
\(782\) −1.20753 −0.0431810
\(783\) 0 0
\(784\) 8.75346 0.312624
\(785\) 14.5798 0.520376
\(786\) 0 0
\(787\) 17.4880 0.623379 0.311689 0.950184i \(-0.399105\pi\)
0.311689 + 0.950184i \(0.399105\pi\)
\(788\) 0.161672 0.00575932
\(789\) 0 0
\(790\) 11.8842 0.422821
\(791\) 38.4665 1.36771
\(792\) 0 0
\(793\) −1.80989 −0.0642711
\(794\) 27.3777 0.971599
\(795\) 0 0
\(796\) −8.10524 −0.287283
\(797\) −40.7200 −1.44238 −0.721189 0.692739i \(-0.756404\pi\)
−0.721189 + 0.692739i \(0.756404\pi\)
\(798\) 0 0
\(799\) −4.64669 −0.164388
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 34.0867 1.20364
\(803\) 39.0489 1.37801
\(804\) 0 0
\(805\) −13.6243 −0.480192
\(806\) 2.68319 0.0945114
\(807\) 0 0
\(808\) 4.69812 0.165279
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) −1.32725 −0.0466061 −0.0233031 0.999728i \(-0.507418\pi\)
−0.0233031 + 0.999728i \(0.507418\pi\)
\(812\) 37.1931 1.30522
\(813\) 0 0
\(814\) −16.4746 −0.577433
\(815\) −4.59832 −0.161072
\(816\) 0 0
\(817\) −2.14579 −0.0750716
\(818\) 8.16167 0.285366
\(819\) 0 0
\(820\) 8.45158 0.295142
\(821\) 0.775046 0.0270493 0.0135246 0.999909i \(-0.495695\pi\)
0.0135246 + 0.999909i \(0.495695\pi\)
\(822\) 0 0
\(823\) −7.00295 −0.244108 −0.122054 0.992523i \(-0.538948\pi\)
−0.122054 + 0.992523i \(0.538948\pi\)
\(824\) 2.09980 0.0731499
\(825\) 0 0
\(826\) −33.5448 −1.16717
\(827\) −8.24296 −0.286636 −0.143318 0.989677i \(-0.545777\pi\)
−0.143318 + 0.989677i \(0.545777\pi\)
\(828\) 0 0
\(829\) −17.2346 −0.598581 −0.299291 0.954162i \(-0.596750\pi\)
−0.299291 + 0.954162i \(0.596750\pi\)
\(830\) −0.105239 −0.00365289
\(831\) 0 0
\(832\) 0.919164 0.0318663
\(833\) −3.07929 −0.106691
\(834\) 0 0
\(835\) 6.40571 0.221679
\(836\) 1.07289 0.0371068
\(837\) 0 0
\(838\) −27.3628 −0.945232
\(839\) 11.8951 0.410664 0.205332 0.978692i \(-0.434173\pi\)
0.205332 + 0.978692i \(0.434173\pi\)
\(840\) 0 0
\(841\) 58.8108 2.02796
\(842\) −15.9760 −0.550571
\(843\) 0 0
\(844\) 18.3278 0.630870
\(845\) −12.1551 −0.418149
\(846\) 0 0
\(847\) 6.73994 0.231587
\(848\) −5.93813 −0.203916
\(849\) 0 0
\(850\) −0.351780 −0.0120660
\(851\) 18.5419 0.635608
\(852\) 0 0
\(853\) −25.9841 −0.889679 −0.444840 0.895610i \(-0.646739\pi\)
−0.444840 + 0.895610i \(0.646739\pi\)
\(854\) 7.81533 0.267435
\(855\) 0 0
\(856\) 9.83833 0.336267
\(857\) −43.6785 −1.49203 −0.746015 0.665929i \(-0.768035\pi\)
−0.746015 + 0.665929i \(0.768035\pi\)
\(858\) 0 0
\(859\) −13.1682 −0.449293 −0.224647 0.974440i \(-0.572123\pi\)
−0.224647 + 0.974440i \(0.572123\pi\)
\(860\) 6.09980 0.208001
\(861\) 0 0
\(862\) −16.7900 −0.571869
\(863\) 12.3059 0.418898 0.209449 0.977820i \(-0.432833\pi\)
0.209449 + 0.977820i \(0.432833\pi\)
\(864\) 0 0
\(865\) 24.9625 0.848751
\(866\) 29.6352 1.00704
\(867\) 0 0
\(868\) −11.5863 −0.393266
\(869\) −36.2456 −1.22955
\(870\) 0 0
\(871\) −0.919164 −0.0311447
\(872\) 1.26550 0.0428553
\(873\) 0 0
\(874\) −1.20753 −0.0408452
\(875\) −3.96906 −0.134179
\(876\) 0 0
\(877\) 41.1407 1.38922 0.694611 0.719386i \(-0.255577\pi\)
0.694611 + 0.719386i \(0.255577\pi\)
\(878\) 1.99456 0.0673131
\(879\) 0 0
\(880\) −3.04990 −0.102812
\(881\) 25.6122 0.862895 0.431448 0.902138i \(-0.358003\pi\)
0.431448 + 0.902138i \(0.358003\pi\)
\(882\) 0 0
\(883\) −5.96208 −0.200640 −0.100320 0.994955i \(-0.531987\pi\)
−0.100320 + 0.994955i \(0.531987\pi\)
\(884\) −0.323344 −0.0108752
\(885\) 0 0
\(886\) 3.29644 0.110746
\(887\) 1.79388 0.0602327 0.0301163 0.999546i \(-0.490412\pi\)
0.0301163 + 0.999546i \(0.490412\pi\)
\(888\) 0 0
\(889\) 51.4590 1.72588
\(890\) 16.2050 0.543194
\(891\) 0 0
\(892\) −0.442091 −0.0148023
\(893\) −4.64669 −0.155495
\(894\) 0 0
\(895\) 14.1282 0.472255
\(896\) −3.96906 −0.132597
\(897\) 0 0
\(898\) 13.9196 0.464504
\(899\) −27.3547 −0.912331
\(900\) 0 0
\(901\) 2.08891 0.0695918
\(902\) −25.7765 −0.858262
\(903\) 0 0
\(904\) −9.69158 −0.322337
\(905\) −11.0729 −0.368075
\(906\) 0 0
\(907\) −54.3603 −1.80500 −0.902502 0.430685i \(-0.858272\pi\)
−0.902502 + 0.430685i \(0.858272\pi\)
\(908\) 25.3344 0.840750
\(909\) 0 0
\(910\) −3.64822 −0.120937
\(911\) −20.1890 −0.668892 −0.334446 0.942415i \(-0.608549\pi\)
−0.334446 + 0.942415i \(0.608549\pi\)
\(912\) 0 0
\(913\) 0.320968 0.0106225
\(914\) 12.3897 0.409815
\(915\) 0 0
\(916\) 8.06886 0.266603
\(917\) −49.6667 −1.64014
\(918\) 0 0
\(919\) −37.9221 −1.25094 −0.625468 0.780250i \(-0.715092\pi\)
−0.625468 + 0.780250i \(0.715092\pi\)
\(920\) 3.43262 0.113170
\(921\) 0 0
\(922\) −38.8537 −1.27958
\(923\) 5.16320 0.169949
\(924\) 0 0
\(925\) 5.40168 0.177606
\(926\) −5.85076 −0.192268
\(927\) 0 0
\(928\) −9.37074 −0.307610
\(929\) 51.0074 1.67350 0.836750 0.547585i \(-0.184453\pi\)
0.836750 + 0.547585i \(0.184453\pi\)
\(930\) 0 0
\(931\) −3.07929 −0.100920
\(932\) −3.67121 −0.120255
\(933\) 0 0
\(934\) 6.37477 0.208589
\(935\) 1.07289 0.0350874
\(936\) 0 0
\(937\) −36.9290 −1.20642 −0.603208 0.797584i \(-0.706111\pi\)
−0.603208 + 0.797584i \(0.706111\pi\)
\(938\) 3.96906 0.129594
\(939\) 0 0
\(940\) 13.2091 0.430832
\(941\) 45.0489 1.46855 0.734277 0.678850i \(-0.237521\pi\)
0.734277 + 0.678850i \(0.237521\pi\)
\(942\) 0 0
\(943\) 29.0110 0.944729
\(944\) 8.45158 0.275075
\(945\) 0 0
\(946\) −18.6038 −0.604861
\(947\) 5.68210 0.184643 0.0923217 0.995729i \(-0.470571\pi\)
0.0923217 + 0.995729i \(0.470571\pi\)
\(948\) 0 0
\(949\) −11.7684 −0.382018
\(950\) −0.351780 −0.0114133
\(951\) 0 0
\(952\) 1.39624 0.0452523
\(953\) −34.3738 −1.11348 −0.556739 0.830688i \(-0.687947\pi\)
−0.556739 + 0.830688i \(0.687947\pi\)
\(954\) 0 0
\(955\) 18.0998 0.585696
\(956\) −12.9650 −0.419319
\(957\) 0 0
\(958\) −13.5094 −0.436469
\(959\) 5.56335 0.179650
\(960\) 0 0
\(961\) −22.4785 −0.725112
\(962\) 4.96503 0.160079
\(963\) 0 0
\(964\) −21.9531 −0.707060
\(965\) −11.6861 −0.376190
\(966\) 0 0
\(967\) −55.7314 −1.79220 −0.896100 0.443852i \(-0.853611\pi\)
−0.896100 + 0.443852i \(0.853611\pi\)
\(968\) −1.69812 −0.0545796
\(969\) 0 0
\(970\) 3.04990 0.0979263
\(971\) 41.3438 1.32679 0.663394 0.748271i \(-0.269116\pi\)
0.663394 + 0.748271i \(0.269116\pi\)
\(972\) 0 0
\(973\) −23.9272 −0.767072
\(974\) 24.4920 0.784774
\(975\) 0 0
\(976\) −1.96906 −0.0630282
\(977\) 61.0643 1.95362 0.976810 0.214107i \(-0.0686842\pi\)
0.976810 + 0.214107i \(0.0686842\pi\)
\(978\) 0 0
\(979\) −49.4237 −1.57959
\(980\) 8.75346 0.279619
\(981\) 0 0
\(982\) −1.59723 −0.0509696
\(983\) −5.76557 −0.183893 −0.0919466 0.995764i \(-0.529309\pi\)
−0.0919466 + 0.995764i \(0.529309\pi\)
\(984\) 0 0
\(985\) 0.161672 0.00515129
\(986\) 3.29644 0.104980
\(987\) 0 0
\(988\) −0.323344 −0.0102869
\(989\) 20.9383 0.665798
\(990\) 0 0
\(991\) 22.5339 0.715814 0.357907 0.933757i \(-0.383490\pi\)
0.357907 + 0.933757i \(0.383490\pi\)
\(992\) 2.91916 0.0926836
\(993\) 0 0
\(994\) −22.2953 −0.707165
\(995\) −8.10524 −0.256953
\(996\) 0 0
\(997\) 6.21310 0.196771 0.0983855 0.995148i \(-0.468632\pi\)
0.0983855 + 0.995148i \(0.468632\pi\)
\(998\) 21.5837 0.683221
\(999\) 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 6030.2.a.bu.1.1 4
3.2 odd 2 2010.2.a.r.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2010.2.a.r.1.1 4 3.2 odd 2
6030.2.a.bu.1.1 4 1.1 even 1 trivial