Properties

Label 6724.2.a.c.1.2
Level $6724$
Weight $2$
Character 6724.1
Self dual yes
Analytic conductor $53.691$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6724 = 2^{2} \cdot 41^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6724.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.6914103191\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.25808.1
Defining polynomial: \(x^{4} - 10 x^{2} - 6 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 164)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.46810\) of defining polynomial
Character \(\chi\) \(=\) 6724.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.21551 q^{3} +3.59669 q^{5} +5.06479 q^{7} +1.90849 q^{9} +O(q^{10})\) \(q-2.21551 q^{3} +3.59669 q^{5} +5.06479 q^{7} +1.90849 q^{9} +2.55961 q^{11} +4.93620 q^{13} -7.96851 q^{15} -2.68821 q^{17} -4.72069 q^{19} -11.2211 q^{21} -1.49482 q^{23} +7.93620 q^{25} +2.41826 q^{27} -2.43102 q^{29} +3.19339 q^{31} -5.67085 q^{33} +18.2165 q^{35} +2.90849 q^{37} -10.9362 q^{39} -7.62441 q^{43} +6.86424 q^{45} -5.15171 q^{47} +18.6521 q^{49} +5.95575 q^{51} +11.1192 q^{53} +9.20614 q^{55} +10.4587 q^{57} +1.49482 q^{59} +1.06380 q^{61} +9.66608 q^{63} +17.7540 q^{65} -10.5596 q^{67} +3.31179 q^{69} +12.6023 q^{71} +8.28490 q^{73} -17.5827 q^{75} +12.9639 q^{77} +4.28967 q^{79} -11.0831 q^{81} +10.5606 q^{83} -9.66866 q^{85} +5.38595 q^{87} -9.36722 q^{89} +25.0008 q^{91} -7.07498 q^{93} -16.9789 q^{95} +3.37641 q^{97} +4.88499 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + 4q^{5} + 12q^{9} + O(q^{10}) \) \( 4q - 2q^{3} + 4q^{5} + 12q^{9} - 4q^{11} + 10q^{15} + 4q^{17} - 6q^{19} - 12q^{23} + 12q^{25} + 10q^{27} + 4q^{29} - 8q^{31} - 20q^{33} + 26q^{35} + 16q^{37} - 24q^{39} + 4q^{43} + 4q^{45} + 6q^{47} + 16q^{49} - 4q^{51} + 16q^{53} + 2q^{55} + 4q^{57} + 12q^{59} + 24q^{61} + 10q^{63} - 4q^{65} - 28q^{67} + 28q^{69} + 2q^{71} + 8q^{73} - 30q^{75} + 8q^{77} + 18q^{79} + 28q^{81} - 12q^{83} - 32q^{85} + 44q^{87} - 4q^{89} + 36q^{91} + 28q^{93} - 14q^{95} - 16q^{97} - 58q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.21551 −1.27913 −0.639563 0.768739i \(-0.720885\pi\)
−0.639563 + 0.768739i \(0.720885\pi\)
\(4\) 0 0
\(5\) 3.59669 1.60849 0.804245 0.594298i \(-0.202570\pi\)
0.804245 + 0.594298i \(0.202570\pi\)
\(6\) 0 0
\(7\) 5.06479 1.91431 0.957156 0.289574i \(-0.0935135\pi\)
0.957156 + 0.289574i \(0.0935135\pi\)
\(8\) 0 0
\(9\) 1.90849 0.636162
\(10\) 0 0
\(11\) 2.55961 0.771753 0.385876 0.922551i \(-0.373899\pi\)
0.385876 + 0.922551i \(0.373899\pi\)
\(12\) 0 0
\(13\) 4.93620 1.36906 0.684528 0.728987i \(-0.260009\pi\)
0.684528 + 0.728987i \(0.260009\pi\)
\(14\) 0 0
\(15\) −7.96851 −2.05746
\(16\) 0 0
\(17\) −2.68821 −0.651986 −0.325993 0.945372i \(-0.605699\pi\)
−0.325993 + 0.945372i \(0.605699\pi\)
\(18\) 0 0
\(19\) −4.72069 −1.08300 −0.541500 0.840701i \(-0.682143\pi\)
−0.541500 + 0.840701i \(0.682143\pi\)
\(20\) 0 0
\(21\) −11.2211 −2.44864
\(22\) 0 0
\(23\) −1.49482 −0.311692 −0.155846 0.987781i \(-0.549810\pi\)
−0.155846 + 0.987781i \(0.549810\pi\)
\(24\) 0 0
\(25\) 7.93620 1.58724
\(26\) 0 0
\(27\) 2.41826 0.465395
\(28\) 0 0
\(29\) −2.43102 −0.451429 −0.225715 0.974193i \(-0.572472\pi\)
−0.225715 + 0.974193i \(0.572472\pi\)
\(30\) 0 0
\(31\) 3.19339 0.573549 0.286774 0.957998i \(-0.407417\pi\)
0.286774 + 0.957998i \(0.407417\pi\)
\(32\) 0 0
\(33\) −5.67085 −0.987168
\(34\) 0 0
\(35\) 18.2165 3.07915
\(36\) 0 0
\(37\) 2.90849 0.478152 0.239076 0.971001i \(-0.423155\pi\)
0.239076 + 0.971001i \(0.423155\pi\)
\(38\) 0 0
\(39\) −10.9362 −1.75119
\(40\) 0 0
\(41\) 0 0
\(42\) 0 0
\(43\) −7.62441 −1.16271 −0.581356 0.813650i \(-0.697477\pi\)
−0.581356 + 0.813650i \(0.697477\pi\)
\(44\) 0 0
\(45\) 6.86424 1.02326
\(46\) 0 0
\(47\) −5.15171 −0.751454 −0.375727 0.926730i \(-0.622607\pi\)
−0.375727 + 0.926730i \(0.622607\pi\)
\(48\) 0 0
\(49\) 18.6521 2.66459
\(50\) 0 0
\(51\) 5.95575 0.833972
\(52\) 0 0
\(53\) 11.1192 1.52734 0.763672 0.645605i \(-0.223395\pi\)
0.763672 + 0.645605i \(0.223395\pi\)
\(54\) 0 0
\(55\) 9.20614 1.24136
\(56\) 0 0
\(57\) 10.4587 1.38529
\(58\) 0 0
\(59\) 1.49482 0.194609 0.0973046 0.995255i \(-0.468978\pi\)
0.0973046 + 0.995255i \(0.468978\pi\)
\(60\) 0 0
\(61\) 1.06380 0.136206 0.0681029 0.997678i \(-0.478305\pi\)
0.0681029 + 0.997678i \(0.478305\pi\)
\(62\) 0 0
\(63\) 9.66608 1.21781
\(64\) 0 0
\(65\) 17.7540 2.20211
\(66\) 0 0
\(67\) −10.5596 −1.29006 −0.645031 0.764156i \(-0.723156\pi\)
−0.645031 + 0.764156i \(0.723156\pi\)
\(68\) 0 0
\(69\) 3.31179 0.398693
\(70\) 0 0
\(71\) 12.6023 1.49562 0.747808 0.663915i \(-0.231106\pi\)
0.747808 + 0.663915i \(0.231106\pi\)
\(72\) 0 0
\(73\) 8.28490 0.969674 0.484837 0.874604i \(-0.338879\pi\)
0.484837 + 0.874604i \(0.338879\pi\)
\(74\) 0 0
\(75\) −17.5827 −2.03028
\(76\) 0 0
\(77\) 12.9639 1.47737
\(78\) 0 0
\(79\) 4.28967 0.482625 0.241313 0.970447i \(-0.422422\pi\)
0.241313 + 0.970447i \(0.422422\pi\)
\(80\) 0 0
\(81\) −11.0831 −1.23146
\(82\) 0 0
\(83\) 10.5606 1.15918 0.579588 0.814909i \(-0.303213\pi\)
0.579588 + 0.814909i \(0.303213\pi\)
\(84\) 0 0
\(85\) −9.66866 −1.04871
\(86\) 0 0
\(87\) 5.38595 0.577435
\(88\) 0 0
\(89\) −9.36722 −0.992923 −0.496462 0.868059i \(-0.665368\pi\)
−0.496462 + 0.868059i \(0.665368\pi\)
\(90\) 0 0
\(91\) 25.0008 2.62080
\(92\) 0 0
\(93\) −7.07498 −0.733641
\(94\) 0 0
\(95\) −16.9789 −1.74199
\(96\) 0 0
\(97\) 3.37641 0.342823 0.171411 0.985200i \(-0.445167\pi\)
0.171411 + 0.985200i \(0.445167\pi\)
\(98\) 0 0
\(99\) 4.88499 0.490960
\(100\) 0 0
\(101\) 4.24799 0.422691 0.211345 0.977411i \(-0.432215\pi\)
0.211345 + 0.977411i \(0.432215\pi\)
\(102\) 0 0
\(103\) 1.11923 0.110281 0.0551404 0.998479i \(-0.482439\pi\)
0.0551404 + 0.998479i \(0.482439\pi\)
\(104\) 0 0
\(105\) −40.3588 −3.93862
\(106\) 0 0
\(107\) −0.322149 −0.0311434 −0.0155717 0.999879i \(-0.504957\pi\)
−0.0155717 + 0.999879i \(0.504957\pi\)
\(108\) 0 0
\(109\) −3.23763 −0.310109 −0.155055 0.987906i \(-0.549555\pi\)
−0.155055 + 0.987906i \(0.549555\pi\)
\(110\) 0 0
\(111\) −6.44378 −0.611616
\(112\) 0 0
\(113\) 2.10187 0.197727 0.0988637 0.995101i \(-0.468479\pi\)
0.0988637 + 0.995101i \(0.468479\pi\)
\(114\) 0 0
\(115\) −5.37641 −0.501353
\(116\) 0 0
\(117\) 9.42066 0.870941
\(118\) 0 0
\(119\) −13.6152 −1.24810
\(120\) 0 0
\(121\) −4.44838 −0.404398
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10.5606 0.944569
\(126\) 0 0
\(127\) −13.0658 −1.15940 −0.579700 0.814830i \(-0.696830\pi\)
−0.579700 + 0.814830i \(0.696830\pi\)
\(128\) 0 0
\(129\) 16.8919 1.48725
\(130\) 0 0
\(131\) 8.43102 0.736622 0.368311 0.929703i \(-0.379936\pi\)
0.368311 + 0.929703i \(0.379936\pi\)
\(132\) 0 0
\(133\) −23.9093 −2.07320
\(134\) 0 0
\(135\) 8.69774 0.748583
\(136\) 0 0
\(137\) 16.5606 1.41487 0.707434 0.706779i \(-0.249853\pi\)
0.707434 + 0.706779i \(0.249853\pi\)
\(138\) 0 0
\(139\) −2.18303 −0.185162 −0.0925810 0.995705i \(-0.529512\pi\)
−0.0925810 + 0.995705i \(0.529512\pi\)
\(140\) 0 0
\(141\) 11.4137 0.961204
\(142\) 0 0
\(143\) 12.6348 1.05657
\(144\) 0 0
\(145\) −8.74363 −0.726119
\(146\) 0 0
\(147\) −41.3240 −3.40834
\(148\) 0 0
\(149\) −2.43102 −0.199157 −0.0995785 0.995030i \(-0.531749\pi\)
−0.0995785 + 0.995030i \(0.531749\pi\)
\(150\) 0 0
\(151\) 0.343113 0.0279221 0.0139611 0.999903i \(-0.495556\pi\)
0.0139611 + 0.999903i \(0.495556\pi\)
\(152\) 0 0
\(153\) −5.13040 −0.414769
\(154\) 0 0
\(155\) 11.4856 0.922548
\(156\) 0 0
\(157\) 10.6348 0.848746 0.424373 0.905487i \(-0.360494\pi\)
0.424373 + 0.905487i \(0.360494\pi\)
\(158\) 0 0
\(159\) −24.6348 −1.95366
\(160\) 0 0
\(161\) −7.57096 −0.596675
\(162\) 0 0
\(163\) −0.173834 −0.0136157 −0.00680785 0.999977i \(-0.502167\pi\)
−0.00680785 + 0.999977i \(0.502167\pi\)
\(164\) 0 0
\(165\) −20.3963 −1.58785
\(166\) 0 0
\(167\) −14.5596 −1.12666 −0.563328 0.826233i \(-0.690479\pi\)
−0.563328 + 0.826233i \(0.690479\pi\)
\(168\) 0 0
\(169\) 11.3661 0.874312
\(170\) 0 0
\(171\) −9.00937 −0.688963
\(172\) 0 0
\(173\) −22.2592 −1.69233 −0.846167 0.532918i \(-0.821095\pi\)
−0.846167 + 0.532918i \(0.821095\pi\)
\(174\) 0 0
\(175\) 40.1952 3.03847
\(176\) 0 0
\(177\) −3.31179 −0.248930
\(178\) 0 0
\(179\) 17.9268 1.33991 0.669957 0.742400i \(-0.266312\pi\)
0.669957 + 0.742400i \(0.266312\pi\)
\(180\) 0 0
\(181\) −20.9916 −1.56030 −0.780148 0.625595i \(-0.784856\pi\)
−0.780148 + 0.625595i \(0.784856\pi\)
\(182\) 0 0
\(183\) −2.35686 −0.174224
\(184\) 0 0
\(185\) 10.4609 0.769103
\(186\) 0 0
\(187\) −6.88077 −0.503172
\(188\) 0 0
\(189\) 12.2480 0.890910
\(190\) 0 0
\(191\) −6.45074 −0.466759 −0.233380 0.972386i \(-0.574978\pi\)
−0.233380 + 0.972386i \(0.574978\pi\)
\(192\) 0 0
\(193\) −6.20374 −0.446555 −0.223278 0.974755i \(-0.571676\pi\)
−0.223278 + 0.974755i \(0.571676\pi\)
\(194\) 0 0
\(195\) −39.3341 −2.81678
\(196\) 0 0
\(197\) 14.1850 1.01064 0.505320 0.862932i \(-0.331374\pi\)
0.505320 + 0.862932i \(0.331374\pi\)
\(198\) 0 0
\(199\) −18.7099 −1.32631 −0.663155 0.748482i \(-0.730783\pi\)
−0.663155 + 0.748482i \(0.730783\pi\)
\(200\) 0 0
\(201\) 23.3949 1.65015
\(202\) 0 0
\(203\) −12.3126 −0.864176
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2.85285 −0.198286
\(208\) 0 0
\(209\) −12.0831 −0.835808
\(210\) 0 0
\(211\) −14.6984 −1.01188 −0.505940 0.862569i \(-0.668854\pi\)
−0.505940 + 0.862569i \(0.668854\pi\)
\(212\) 0 0
\(213\) −27.9205 −1.91308
\(214\) 0 0
\(215\) −27.4226 −1.87021
\(216\) 0 0
\(217\) 16.1738 1.09795
\(218\) 0 0
\(219\) −18.3553 −1.24033
\(220\) 0 0
\(221\) −13.2695 −0.892605
\(222\) 0 0
\(223\) 20.1109 1.34672 0.673361 0.739314i \(-0.264850\pi\)
0.673361 + 0.739314i \(0.264850\pi\)
\(224\) 0 0
\(225\) 15.1461 1.00974
\(226\) 0 0
\(227\) 6.71149 0.445457 0.222729 0.974880i \(-0.428504\pi\)
0.222729 + 0.974880i \(0.428504\pi\)
\(228\) 0 0
\(229\) −12.7624 −0.843361 −0.421680 0.906745i \(-0.638560\pi\)
−0.421680 + 0.906745i \(0.638560\pi\)
\(230\) 0 0
\(231\) −28.7217 −1.88975
\(232\) 0 0
\(233\) 29.0750 1.90477 0.952383 0.304906i \(-0.0986249\pi\)
0.952383 + 0.304906i \(0.0986249\pi\)
\(234\) 0 0
\(235\) −18.5291 −1.20871
\(236\) 0 0
\(237\) −9.50380 −0.617338
\(238\) 0 0
\(239\) −26.0971 −1.68808 −0.844041 0.536279i \(-0.819829\pi\)
−0.844041 + 0.536279i \(0.819829\pi\)
\(240\) 0 0
\(241\) 1.06380 0.0685255 0.0342627 0.999413i \(-0.489092\pi\)
0.0342627 + 0.999413i \(0.489092\pi\)
\(242\) 0 0
\(243\) 17.3000 1.10980
\(244\) 0 0
\(245\) 67.0859 4.28596
\(246\) 0 0
\(247\) −23.3023 −1.48269
\(248\) 0 0
\(249\) −23.3971 −1.48273
\(250\) 0 0
\(251\) −7.82815 −0.494108 −0.247054 0.969002i \(-0.579463\pi\)
−0.247054 + 0.969002i \(0.579463\pi\)
\(252\) 0 0
\(253\) −3.82617 −0.240549
\(254\) 0 0
\(255\) 21.4210 1.34144
\(256\) 0 0
\(257\) −7.49482 −0.467514 −0.233757 0.972295i \(-0.575102\pi\)
−0.233757 + 0.972295i \(0.575102\pi\)
\(258\) 0 0
\(259\) 14.7309 0.915332
\(260\) 0 0
\(261\) −4.63957 −0.287182
\(262\) 0 0
\(263\) −15.5919 −0.961439 −0.480720 0.876874i \(-0.659625\pi\)
−0.480720 + 0.876874i \(0.659625\pi\)
\(264\) 0 0
\(265\) 39.9924 2.45672
\(266\) 0 0
\(267\) 20.7532 1.27007
\(268\) 0 0
\(269\) 8.80860 0.537070 0.268535 0.963270i \(-0.413461\pi\)
0.268535 + 0.963270i \(0.413461\pi\)
\(270\) 0 0
\(271\) −19.8816 −1.20772 −0.603860 0.797090i \(-0.706372\pi\)
−0.603860 + 0.797090i \(0.706372\pi\)
\(272\) 0 0
\(273\) −55.3896 −3.35233
\(274\) 0 0
\(275\) 20.3136 1.22496
\(276\) 0 0
\(277\) −7.08232 −0.425535 −0.212768 0.977103i \(-0.568248\pi\)
−0.212768 + 0.977103i \(0.568248\pi\)
\(278\) 0 0
\(279\) 6.09453 0.364870
\(280\) 0 0
\(281\) 31.2695 1.86538 0.932692 0.360675i \(-0.117453\pi\)
0.932692 + 0.360675i \(0.117453\pi\)
\(282\) 0 0
\(283\) −2.18303 −0.129768 −0.0648838 0.997893i \(-0.520668\pi\)
−0.0648838 + 0.997893i \(0.520668\pi\)
\(284\) 0 0
\(285\) 37.6169 2.22823
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −9.77354 −0.574914
\(290\) 0 0
\(291\) −7.48048 −0.438514
\(292\) 0 0
\(293\) 33.6798 1.96760 0.983798 0.179278i \(-0.0573762\pi\)
0.983798 + 0.179278i \(0.0573762\pi\)
\(294\) 0 0
\(295\) 5.37641 0.313027
\(296\) 0 0
\(297\) 6.18982 0.359170
\(298\) 0 0
\(299\) −7.37874 −0.426723
\(300\) 0 0
\(301\) −38.6160 −2.22579
\(302\) 0 0
\(303\) −9.41147 −0.540675
\(304\) 0 0
\(305\) 3.82617 0.219086
\(306\) 0 0
\(307\) −12.7532 −0.727862 −0.363931 0.931426i \(-0.618566\pi\)
−0.363931 + 0.931426i \(0.618566\pi\)
\(308\) 0 0
\(309\) −2.47966 −0.141063
\(310\) 0 0
\(311\) −2.18204 −0.123732 −0.0618660 0.998084i \(-0.519705\pi\)
−0.0618660 + 0.998084i \(0.519705\pi\)
\(312\) 0 0
\(313\) 29.3042 1.65637 0.828187 0.560452i \(-0.189373\pi\)
0.828187 + 0.560452i \(0.189373\pi\)
\(314\) 0 0
\(315\) 34.7659 1.95884
\(316\) 0 0
\(317\) 28.5511 1.60359 0.801794 0.597601i \(-0.203879\pi\)
0.801794 + 0.597601i \(0.203879\pi\)
\(318\) 0 0
\(319\) −6.22247 −0.348392
\(320\) 0 0
\(321\) 0.713725 0.0398363
\(322\) 0 0
\(323\) 12.6902 0.706101
\(324\) 0 0
\(325\) 39.1747 2.17302
\(326\) 0 0
\(327\) 7.17301 0.396669
\(328\) 0 0
\(329\) −26.0923 −1.43852
\(330\) 0 0
\(331\) 4.80761 0.264250 0.132125 0.991233i \(-0.457820\pi\)
0.132125 + 0.991233i \(0.457820\pi\)
\(332\) 0 0
\(333\) 5.55080 0.304182
\(334\) 0 0
\(335\) −37.9797 −2.07505
\(336\) 0 0
\(337\) 16.3148 0.888724 0.444362 0.895847i \(-0.353430\pi\)
0.444362 + 0.895847i \(0.353430\pi\)
\(338\) 0 0
\(339\) −4.65672 −0.252918
\(340\) 0 0
\(341\) 8.17383 0.442638
\(342\) 0 0
\(343\) 59.0156 3.18654
\(344\) 0 0
\(345\) 11.9115 0.641294
\(346\) 0 0
\(347\) −31.6034 −1.69656 −0.848281 0.529547i \(-0.822362\pi\)
−0.848281 + 0.529547i \(0.822362\pi\)
\(348\) 0 0
\(349\) 4.97311 0.266204 0.133102 0.991102i \(-0.457506\pi\)
0.133102 + 0.991102i \(0.457506\pi\)
\(350\) 0 0
\(351\) 11.9370 0.637151
\(352\) 0 0
\(353\) 1.72430 0.0917750 0.0458875 0.998947i \(-0.485388\pi\)
0.0458875 + 0.998947i \(0.485388\pi\)
\(354\) 0 0
\(355\) 45.3265 2.40568
\(356\) 0 0
\(357\) 30.1646 1.59648
\(358\) 0 0
\(359\) −7.13796 −0.376727 −0.188364 0.982099i \(-0.560318\pi\)
−0.188364 + 0.982099i \(0.560318\pi\)
\(360\) 0 0
\(361\) 3.28490 0.172889
\(362\) 0 0
\(363\) 9.85542 0.517276
\(364\) 0 0
\(365\) 29.7982 1.55971
\(366\) 0 0
\(367\) −9.38595 −0.489943 −0.244971 0.969530i \(-0.578779\pi\)
−0.244971 + 0.969530i \(0.578779\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 56.3166 2.92381
\(372\) 0 0
\(373\) 23.1212 1.19717 0.598585 0.801059i \(-0.295730\pi\)
0.598585 + 0.801059i \(0.295730\pi\)
\(374\) 0 0
\(375\) −23.3971 −1.20822
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 5.99080 0.307727 0.153863 0.988092i \(-0.450828\pi\)
0.153863 + 0.988092i \(0.450828\pi\)
\(380\) 0 0
\(381\) 28.9474 1.48302
\(382\) 0 0
\(383\) 29.5811 1.51153 0.755763 0.654845i \(-0.227266\pi\)
0.755763 + 0.654845i \(0.227266\pi\)
\(384\) 0 0
\(385\) 46.6272 2.37634
\(386\) 0 0
\(387\) −14.5511 −0.739672
\(388\) 0 0
\(389\) 22.8620 1.15915 0.579576 0.814918i \(-0.303218\pi\)
0.579576 + 0.814918i \(0.303218\pi\)
\(390\) 0 0
\(391\) 4.01839 0.203219
\(392\) 0 0
\(393\) −18.6790 −0.942232
\(394\) 0 0
\(395\) 15.4286 0.776298
\(396\) 0 0
\(397\) 0.588531 0.0295375 0.0147688 0.999891i \(-0.495299\pi\)
0.0147688 + 0.999891i \(0.495299\pi\)
\(398\) 0 0
\(399\) 52.9713 2.65188
\(400\) 0 0
\(401\) −0.0926758 −0.00462801 −0.00231400 0.999997i \(-0.500737\pi\)
−0.00231400 + 0.999997i \(0.500737\pi\)
\(402\) 0 0
\(403\) 15.7632 0.785220
\(404\) 0 0
\(405\) −39.8626 −1.98079
\(406\) 0 0
\(407\) 7.44460 0.369015
\(408\) 0 0
\(409\) −1.29526 −0.0640463 −0.0320232 0.999487i \(-0.510195\pi\)
−0.0320232 + 0.999487i \(0.510195\pi\)
\(410\) 0 0
\(411\) −36.6902 −1.80979
\(412\) 0 0
\(413\) 7.57096 0.372543
\(414\) 0 0
\(415\) 37.9833 1.86452
\(416\) 0 0
\(417\) 4.83652 0.236846
\(418\) 0 0
\(419\) 1.75201 0.0855912 0.0427956 0.999084i \(-0.486374\pi\)
0.0427956 + 0.999084i \(0.486374\pi\)
\(420\) 0 0
\(421\) −36.7364 −1.79042 −0.895212 0.445641i \(-0.852976\pi\)
−0.895212 + 0.445641i \(0.852976\pi\)
\(422\) 0 0
\(423\) −9.83196 −0.478046
\(424\) 0 0
\(425\) −21.3341 −1.03486
\(426\) 0 0
\(427\) 5.38793 0.260740
\(428\) 0 0
\(429\) −27.9924 −1.35149
\(430\) 0 0
\(431\) 12.3126 0.593078 0.296539 0.955021i \(-0.404168\pi\)
0.296539 + 0.955021i \(0.404168\pi\)
\(432\) 0 0
\(433\) −17.7632 −0.853644 −0.426822 0.904336i \(-0.640367\pi\)
−0.426822 + 0.904336i \(0.640367\pi\)
\(434\) 0 0
\(435\) 19.3716 0.928798
\(436\) 0 0
\(437\) 7.05659 0.337562
\(438\) 0 0
\(439\) −3.22230 −0.153792 −0.0768961 0.997039i \(-0.524501\pi\)
−0.0768961 + 0.997039i \(0.524501\pi\)
\(440\) 0 0
\(441\) 35.5973 1.69511
\(442\) 0 0
\(443\) −35.8537 −1.70346 −0.851730 0.523982i \(-0.824446\pi\)
−0.851730 + 0.523982i \(0.824446\pi\)
\(444\) 0 0
\(445\) −33.6910 −1.59711
\(446\) 0 0
\(447\) 5.38595 0.254747
\(448\) 0 0
\(449\) −15.3437 −0.724113 −0.362057 0.932156i \(-0.617925\pi\)
−0.362057 + 0.932156i \(0.617925\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −0.760170 −0.0357159
\(454\) 0 0
\(455\) 89.9203 4.21553
\(456\) 0 0
\(457\) −27.5917 −1.29068 −0.645342 0.763894i \(-0.723285\pi\)
−0.645342 + 0.763894i \(0.723285\pi\)
\(458\) 0 0
\(459\) −6.50079 −0.303431
\(460\) 0 0
\(461\) −8.08116 −0.376377 −0.188189 0.982133i \(-0.560262\pi\)
−0.188189 + 0.982133i \(0.560262\pi\)
\(462\) 0 0
\(463\) 8.23622 0.382770 0.191385 0.981515i \(-0.438702\pi\)
0.191385 + 0.981515i \(0.438702\pi\)
\(464\) 0 0
\(465\) −25.4465 −1.18005
\(466\) 0 0
\(467\) −23.7333 −1.09825 −0.549123 0.835742i \(-0.685038\pi\)
−0.549123 + 0.835742i \(0.685038\pi\)
\(468\) 0 0
\(469\) −53.4822 −2.46958
\(470\) 0 0
\(471\) −23.5614 −1.08565
\(472\) 0 0
\(473\) −19.5155 −0.897325
\(474\) 0 0
\(475\) −37.4643 −1.71898
\(476\) 0 0
\(477\) 21.2209 0.971638
\(478\) 0 0
\(479\) 16.6987 0.762985 0.381492 0.924372i \(-0.375410\pi\)
0.381492 + 0.924372i \(0.375410\pi\)
\(480\) 0 0
\(481\) 14.3569 0.654617
\(482\) 0 0
\(483\) 16.7735 0.763223
\(484\) 0 0
\(485\) 12.1439 0.551427
\(486\) 0 0
\(487\) 0.927003 0.0420065 0.0210033 0.999779i \(-0.493314\pi\)
0.0210033 + 0.999779i \(0.493314\pi\)
\(488\) 0 0
\(489\) 0.385130 0.0174162
\(490\) 0 0
\(491\) −26.8752 −1.21286 −0.606430 0.795137i \(-0.707399\pi\)
−0.606430 + 0.795137i \(0.707399\pi\)
\(492\) 0 0
\(493\) 6.53509 0.294326
\(494\) 0 0
\(495\) 17.5698 0.789703
\(496\) 0 0
\(497\) 63.8279 2.86307
\(498\) 0 0
\(499\) −18.2155 −0.815438 −0.407719 0.913107i \(-0.633676\pi\)
−0.407719 + 0.913107i \(0.633676\pi\)
\(500\) 0 0
\(501\) 32.2570 1.44114
\(502\) 0 0
\(503\) −18.9591 −0.845346 −0.422673 0.906282i \(-0.638908\pi\)
−0.422673 + 0.906282i \(0.638908\pi\)
\(504\) 0 0
\(505\) 15.2787 0.679894
\(506\) 0 0
\(507\) −25.1816 −1.11835
\(508\) 0 0
\(509\) −12.4218 −0.550588 −0.275294 0.961360i \(-0.588775\pi\)
−0.275294 + 0.961360i \(0.588775\pi\)
\(510\) 0 0
\(511\) 41.9613 1.85626
\(512\) 0 0
\(513\) −11.4159 −0.504022
\(514\) 0 0
\(515\) 4.02552 0.177386
\(516\) 0 0
\(517\) −13.1864 −0.579937
\(518\) 0 0
\(519\) 49.3154 2.16471
\(520\) 0 0
\(521\) 7.17267 0.314240 0.157120 0.987579i \(-0.449779\pi\)
0.157120 + 0.987579i \(0.449779\pi\)
\(522\) 0 0
\(523\) 35.6563 1.55914 0.779570 0.626315i \(-0.215437\pi\)
0.779570 + 0.626315i \(0.215437\pi\)
\(524\) 0 0
\(525\) −89.0529 −3.88659
\(526\) 0 0
\(527\) −8.58448 −0.373946
\(528\) 0 0
\(529\) −20.7655 −0.902848
\(530\) 0 0
\(531\) 2.85285 0.123803
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −1.15867 −0.0500938
\(536\) 0 0
\(537\) −39.7171 −1.71392
\(538\) 0 0
\(539\) 47.7422 2.05640
\(540\) 0 0
\(541\) 28.9731 1.24565 0.622826 0.782361i \(-0.285985\pi\)
0.622826 + 0.782361i \(0.285985\pi\)
\(542\) 0 0
\(543\) 46.5072 1.99581
\(544\) 0 0
\(545\) −11.6448 −0.498807
\(546\) 0 0
\(547\) −16.1178 −0.689148 −0.344574 0.938759i \(-0.611977\pi\)
−0.344574 + 0.938759i \(0.611977\pi\)
\(548\) 0 0
\(549\) 2.03025 0.0866489
\(550\) 0 0
\(551\) 11.4761 0.488898
\(552\) 0 0
\(553\) 21.7263 0.923895
\(554\) 0 0
\(555\) −23.1763 −0.983779
\(556\) 0 0
\(557\) 27.2675 1.15536 0.577681 0.816262i \(-0.303958\pi\)
0.577681 + 0.816262i \(0.303958\pi\)
\(558\) 0 0
\(559\) −37.6356 −1.59182
\(560\) 0 0
\(561\) 15.2444 0.643620
\(562\) 0 0
\(563\) −41.9173 −1.76660 −0.883302 0.468805i \(-0.844685\pi\)
−0.883302 + 0.468805i \(0.844685\pi\)
\(564\) 0 0
\(565\) 7.55978 0.318043
\(566\) 0 0
\(567\) −56.1338 −2.35740
\(568\) 0 0
\(569\) 37.3762 1.56689 0.783446 0.621460i \(-0.213460\pi\)
0.783446 + 0.621460i \(0.213460\pi\)
\(570\) 0 0
\(571\) 26.3228 1.10157 0.550787 0.834646i \(-0.314327\pi\)
0.550787 + 0.834646i \(0.314327\pi\)
\(572\) 0 0
\(573\) 14.2917 0.597044
\(574\) 0 0
\(575\) −11.8632 −0.494730
\(576\) 0 0
\(577\) 13.8262 0.575591 0.287795 0.957692i \(-0.407078\pi\)
0.287795 + 0.957692i \(0.407078\pi\)
\(578\) 0 0
\(579\) 13.7445 0.571200
\(580\) 0 0
\(581\) 53.4873 2.21903
\(582\) 0 0
\(583\) 28.4609 1.17873
\(584\) 0 0
\(585\) 33.8832 1.40090
\(586\) 0 0
\(587\) −22.7004 −0.936945 −0.468472 0.883478i \(-0.655195\pi\)
−0.468472 + 0.883478i \(0.655195\pi\)
\(588\) 0 0
\(589\) −15.0750 −0.621154
\(590\) 0 0
\(591\) −31.4270 −1.29274
\(592\) 0 0
\(593\) −16.4167 −0.674152 −0.337076 0.941477i \(-0.609438\pi\)
−0.337076 + 0.941477i \(0.609438\pi\)
\(594\) 0 0
\(595\) −48.9697 −2.00756
\(596\) 0 0
\(597\) 41.4520 1.69652
\(598\) 0 0
\(599\) −6.54909 −0.267588 −0.133794 0.991009i \(-0.542716\pi\)
−0.133794 + 0.991009i \(0.542716\pi\)
\(600\) 0 0
\(601\) 24.7783 1.01073 0.505365 0.862906i \(-0.331358\pi\)
0.505365 + 0.862906i \(0.331358\pi\)
\(602\) 0 0
\(603\) −20.1529 −0.820688
\(604\) 0 0
\(605\) −15.9994 −0.650470
\(606\) 0 0
\(607\) −15.7982 −0.641231 −0.320615 0.947209i \(-0.603890\pi\)
−0.320615 + 0.947209i \(0.603890\pi\)
\(608\) 0 0
\(609\) 27.2787 1.10539
\(610\) 0 0
\(611\) −25.4299 −1.02878
\(612\) 0 0
\(613\) 39.3415 1.58899 0.794494 0.607272i \(-0.207736\pi\)
0.794494 + 0.607272i \(0.207736\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −26.4075 −1.06313 −0.531563 0.847019i \(-0.678395\pi\)
−0.531563 + 0.847019i \(0.678395\pi\)
\(618\) 0 0
\(619\) 33.9278 1.36367 0.681837 0.731504i \(-0.261181\pi\)
0.681837 + 0.731504i \(0.261181\pi\)
\(620\) 0 0
\(621\) −3.61487 −0.145060
\(622\) 0 0
\(623\) −47.4430 −1.90076
\(624\) 0 0
\(625\) −1.69774 −0.0679097
\(626\) 0 0
\(627\) 26.7703 1.06910
\(628\) 0 0
\(629\) −7.81861 −0.311748
\(630\) 0 0
\(631\) −5.68017 −0.226124 −0.113062 0.993588i \(-0.536066\pi\)
−0.113062 + 0.993588i \(0.536066\pi\)
\(632\) 0 0
\(633\) 32.5644 1.29432
\(634\) 0 0
\(635\) −46.9936 −1.86488
\(636\) 0 0
\(637\) 92.0706 3.64797
\(638\) 0 0
\(639\) 24.0513 0.951454
\(640\) 0 0
\(641\) −15.5802 −0.615379 −0.307690 0.951487i \(-0.599556\pi\)
−0.307690 + 0.951487i \(0.599556\pi\)
\(642\) 0 0
\(643\) −22.9372 −0.904554 −0.452277 0.891877i \(-0.649388\pi\)
−0.452277 + 0.891877i \(0.649388\pi\)
\(644\) 0 0
\(645\) 60.7552 2.39223
\(646\) 0 0
\(647\) 15.0546 0.591858 0.295929 0.955210i \(-0.404371\pi\)
0.295929 + 0.955210i \(0.404371\pi\)
\(648\) 0 0
\(649\) 3.82617 0.150190
\(650\) 0 0
\(651\) −35.8333 −1.40442
\(652\) 0 0
\(653\) 45.8652 1.79484 0.897422 0.441174i \(-0.145438\pi\)
0.897422 + 0.441174i \(0.145438\pi\)
\(654\) 0 0
\(655\) 30.3238 1.18485
\(656\) 0 0
\(657\) 15.8116 0.616870
\(658\) 0 0
\(659\) 4.37302 0.170349 0.0851744 0.996366i \(-0.472855\pi\)
0.0851744 + 0.996366i \(0.472855\pi\)
\(660\) 0 0
\(661\) −25.4946 −0.991625 −0.495813 0.868430i \(-0.665130\pi\)
−0.495813 + 0.868430i \(0.665130\pi\)
\(662\) 0 0
\(663\) 29.3988 1.14175
\(664\) 0 0
\(665\) −85.9944 −3.33472
\(666\) 0 0
\(667\) 3.63394 0.140707
\(668\) 0 0
\(669\) −44.5558 −1.72263
\(670\) 0 0
\(671\) 2.72292 0.105117
\(672\) 0 0
\(673\) −31.7540 −1.22403 −0.612013 0.790848i \(-0.709640\pi\)
−0.612013 + 0.790848i \(0.709640\pi\)
\(674\) 0 0
\(675\) 19.1918 0.738693
\(676\) 0 0
\(677\) −27.8998 −1.07228 −0.536138 0.844131i \(-0.680117\pi\)
−0.536138 + 0.844131i \(0.680117\pi\)
\(678\) 0 0
\(679\) 17.1008 0.656270
\(680\) 0 0
\(681\) −14.8694 −0.569796
\(682\) 0 0
\(683\) −8.11782 −0.310620 −0.155310 0.987866i \(-0.549638\pi\)
−0.155310 + 0.987866i \(0.549638\pi\)
\(684\) 0 0
\(685\) 59.5634 2.27580
\(686\) 0 0
\(687\) 28.2752 1.07876
\(688\) 0 0
\(689\) 54.8867 2.09102
\(690\) 0 0
\(691\) 26.6390 1.01339 0.506697 0.862124i \(-0.330866\pi\)
0.506697 + 0.862124i \(0.330866\pi\)
\(692\) 0 0
\(693\) 24.7414 0.939850
\(694\) 0 0
\(695\) −7.85168 −0.297831
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −64.4159 −2.43643
\(700\) 0 0
\(701\) 43.1302 1.62900 0.814502 0.580160i \(-0.197010\pi\)
0.814502 + 0.580160i \(0.197010\pi\)
\(702\) 0 0
\(703\) −13.7301 −0.517839
\(704\) 0 0
\(705\) 41.0514 1.54609
\(706\) 0 0
\(707\) 21.5152 0.809162
\(708\) 0 0
\(709\) −49.5175 −1.85967 −0.929835 0.367978i \(-0.880050\pi\)
−0.929835 + 0.367978i \(0.880050\pi\)
\(710\) 0 0
\(711\) 8.18677 0.307028
\(712\) 0 0
\(713\) −4.77354 −0.178771
\(714\) 0 0
\(715\) 45.4434 1.69949
\(716\) 0 0
\(717\) 57.8184 2.15927
\(718\) 0 0
\(719\) 16.4839 0.614745 0.307372 0.951589i \(-0.400550\pi\)
0.307372 + 0.951589i \(0.400550\pi\)
\(720\) 0 0
\(721\) 5.66866 0.211112
\(722\) 0 0
\(723\) −2.35686 −0.0876527
\(724\) 0 0
\(725\) −19.2931 −0.716526
\(726\) 0 0
\(727\) −48.5947 −1.80228 −0.901139 0.433530i \(-0.857268\pi\)
−0.901139 + 0.433530i \(0.857268\pi\)
\(728\) 0 0
\(729\) −5.07896 −0.188110
\(730\) 0 0
\(731\) 20.4960 0.758071
\(732\) 0 0
\(733\) 11.8166 0.436457 0.218229 0.975898i \(-0.429972\pi\)
0.218229 + 0.975898i \(0.429972\pi\)
\(734\) 0 0
\(735\) −148.630 −5.48228
\(736\) 0 0
\(737\) −27.0285 −0.995609
\(738\) 0 0
\(739\) −5.68017 −0.208949 −0.104474 0.994528i \(-0.533316\pi\)
−0.104474 + 0.994528i \(0.533316\pi\)
\(740\) 0 0
\(741\) 51.6264 1.89654
\(742\) 0 0
\(743\) −24.4980 −0.898743 −0.449372 0.893345i \(-0.648352\pi\)
−0.449372 + 0.893345i \(0.648352\pi\)
\(744\) 0 0
\(745\) −8.74363 −0.320342
\(746\) 0 0
\(747\) 20.1548 0.737424
\(748\) 0 0
\(749\) −1.63162 −0.0596181
\(750\) 0 0
\(751\) 35.7897 1.30598 0.652992 0.757365i \(-0.273513\pi\)
0.652992 + 0.757365i \(0.273513\pi\)
\(752\) 0 0
\(753\) 17.3433 0.632027
\(754\) 0 0
\(755\) 1.23407 0.0449124
\(756\) 0 0
\(757\) −37.7169 −1.37084 −0.685421 0.728147i \(-0.740382\pi\)
−0.685421 + 0.728147i \(0.740382\pi\)
\(758\) 0 0
\(759\) 8.47691 0.307692
\(760\) 0 0
\(761\) −22.6533 −0.821181 −0.410590 0.911820i \(-0.634677\pi\)
−0.410590 + 0.911820i \(0.634677\pi\)
\(762\) 0 0
\(763\) −16.3979 −0.593646
\(764\) 0 0
\(765\) −18.4525 −0.667151
\(766\) 0 0
\(767\) 7.37874 0.266431
\(768\) 0 0
\(769\) −42.6292 −1.53725 −0.768624 0.639701i \(-0.779058\pi\)
−0.768624 + 0.639701i \(0.779058\pi\)
\(770\) 0 0
\(771\) 16.6049 0.598009
\(772\) 0 0
\(773\) −11.7635 −0.423105 −0.211552 0.977367i \(-0.567852\pi\)
−0.211552 + 0.977367i \(0.567852\pi\)
\(774\) 0 0
\(775\) 25.3433 0.910360
\(776\) 0 0
\(777\) −32.6364 −1.17082
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 32.2570 1.15425
\(782\) 0 0
\(783\) −5.87884 −0.210093
\(784\) 0 0
\(785\) 38.2500 1.36520
\(786\) 0 0
\(787\) 34.4517 1.22807 0.614036 0.789278i \(-0.289545\pi\)
0.614036 + 0.789278i \(0.289545\pi\)
\(788\) 0 0
\(789\) 34.5441 1.22980
\(790\) 0 0
\(791\) 10.6455 0.378512
\(792\) 0 0
\(793\) 5.25113 0.186473
\(794\) 0 0
\(795\) −88.6037 −3.14245
\(796\) 0 0
\(797\) 41.2161 1.45995 0.729974 0.683475i \(-0.239532\pi\)
0.729974 + 0.683475i \(0.239532\pi\)
\(798\) 0 0
\(799\) 13.8489 0.489937
\(800\) 0 0
\(801\) −17.8772 −0.631660
\(802\) 0 0
\(803\) 21.2061 0.748349
\(804\) 0 0
\(805\) −27.2304 −0.959746
\(806\) 0 0
\(807\) −19.5155 −0.686979
\(808\) 0 0
\(809\) −12.4123 −0.436393 −0.218196 0.975905i \(-0.570017\pi\)
−0.218196 + 0.975905i \(0.570017\pi\)
\(810\) 0 0
\(811\) 3.91665 0.137532 0.0687660 0.997633i \(-0.478094\pi\)
0.0687660 + 0.997633i \(0.478094\pi\)
\(812\) 0 0
\(813\) 44.0479 1.54483
\(814\) 0 0
\(815\) −0.625226 −0.0219007
\(816\) 0 0
\(817\) 35.9924 1.25922
\(818\) 0 0
\(819\) 47.7137 1.66725
\(820\) 0 0
\(821\) 28.6326 0.999283 0.499642 0.866232i \(-0.333465\pi\)
0.499642 + 0.866232i \(0.333465\pi\)
\(822\) 0 0
\(823\) −43.8212 −1.52751 −0.763755 0.645506i \(-0.776647\pi\)
−0.763755 + 0.645506i \(0.776647\pi\)
\(824\) 0 0
\(825\) −45.0050 −1.56687
\(826\) 0 0
\(827\) −11.1924 −0.389198 −0.194599 0.980883i \(-0.562341\pi\)
−0.194599 + 0.980883i \(0.562341\pi\)
\(828\) 0 0
\(829\) 26.6278 0.924820 0.462410 0.886666i \(-0.346985\pi\)
0.462410 + 0.886666i \(0.346985\pi\)
\(830\) 0 0
\(831\) 15.6910 0.544313
\(832\) 0 0
\(833\) −50.1408 −1.73727
\(834\) 0 0
\(835\) −52.3665 −1.81222
\(836\) 0 0
\(837\) 7.72244 0.266927
\(838\) 0 0
\(839\) −9.43085 −0.325589 −0.162795 0.986660i \(-0.552051\pi\)
−0.162795 + 0.986660i \(0.552051\pi\)
\(840\) 0 0
\(841\) −23.0901 −0.796212
\(842\) 0 0
\(843\) −69.2780 −2.38606
\(844\) 0 0
\(845\) 40.8802 1.40632
\(846\) 0 0
\(847\) −22.5301 −0.774144
\(848\) 0 0
\(849\) 4.83652 0.165989
\(850\) 0 0
\(851\) −4.34767 −0.149036
\(852\) 0 0
\(853\) 32.6045 1.11636 0.558179 0.829721i \(-0.311500\pi\)
0.558179 + 0.829721i \(0.311500\pi\)
\(854\) 0 0
\(855\) −32.4039 −1.10819
\(856\) 0 0
\(857\) −4.25499 −0.145348 −0.0726739 0.997356i \(-0.523153\pi\)
−0.0726739 + 0.997356i \(0.523153\pi\)
\(858\) 0 0
\(859\) 24.5164 0.836487 0.418244 0.908335i \(-0.362646\pi\)
0.418244 + 0.908335i \(0.362646\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −4.11643 −0.140125 −0.0700624 0.997543i \(-0.522320\pi\)
−0.0700624 + 0.997543i \(0.522320\pi\)
\(864\) 0 0
\(865\) −80.0594 −2.72210
\(866\) 0 0
\(867\) 21.6534 0.735387
\(868\) 0 0
\(869\) 10.9799 0.372467
\(870\) 0 0
\(871\) −52.1244 −1.76617
\(872\) 0 0
\(873\) 6.44384 0.218091
\(874\) 0 0
\(875\) 53.4873 1.80820
\(876\) 0 0
\(877\) −3.17301 −0.107145 −0.0535725 0.998564i \(-0.517061\pi\)
−0.0535725 + 0.998564i \(0.517061\pi\)
\(878\) 0 0
\(879\) −74.6180 −2.51680
\(880\) 0 0
\(881\) 16.0391 0.540371 0.270186 0.962808i \(-0.412915\pi\)
0.270186 + 0.962808i \(0.412915\pi\)
\(882\) 0 0
\(883\) 36.5648 1.23050 0.615252 0.788330i \(-0.289054\pi\)
0.615252 + 0.788330i \(0.289054\pi\)
\(884\) 0 0
\(885\) −11.9115 −0.400401
\(886\) 0 0
\(887\) 44.6498 1.49919 0.749596 0.661896i \(-0.230248\pi\)
0.749596 + 0.661896i \(0.230248\pi\)
\(888\) 0 0
\(889\) −66.1755 −2.21945
\(890\) 0 0
\(891\) −28.3686 −0.950382
\(892\) 0 0
\(893\) 24.3196 0.813825
\(894\) 0 0
\(895\) 64.4773 2.15524
\(896\) 0 0
\(897\) 16.3477 0.545833
\(898\) 0 0
\(899\) −7.76319 −0.258917
\(900\) 0 0
\(901\) −29.8908 −0.995807
\(902\) 0 0
\(903\) 85.5542 2.84707
\(904\) 0 0
\(905\) −75.5004 −2.50972
\(906\) 0 0
\(907\) −43.2692 −1.43673 −0.718365 0.695667i \(-0.755109\pi\)
−0.718365 + 0.695667i \(0.755109\pi\)
\(908\) 0 0
\(909\) 8.10723 0.268900
\(910\) 0 0
\(911\) −1.64116 −0.0543739 −0.0271870 0.999630i \(-0.508655\pi\)
−0.0271870 + 0.999630i \(0.508655\pi\)
\(912\) 0 0
\(913\) 27.0311 0.894598
\(914\) 0 0
\(915\) −8.47691 −0.280238
\(916\) 0 0
\(917\) 42.7014 1.41012
\(918\) 0 0
\(919\) −6.67802 −0.220288 −0.110144 0.993916i \(-0.535131\pi\)
−0.110144 + 0.993916i \(0.535131\pi\)
\(920\) 0 0
\(921\) 28.2548 0.931027
\(922\) 0 0
\(923\) 62.2074 2.04758
\(924\) 0 0
\(925\) 23.0823 0.758942
\(926\) 0 0
\(927\) 2.13603 0.0701564
\(928\) 0 0
\(929\) 23.3505 0.766104 0.383052 0.923727i \(-0.374873\pi\)
0.383052 + 0.923727i \(0.374873\pi\)
\(930\) 0 0
\(931\) −88.0508 −2.88575
\(932\) 0 0
\(933\) 4.83433 0.158269
\(934\) 0 0
\(935\) −24.7480 −0.809347
\(936\) 0 0
\(937\) −45.9295 −1.50045 −0.750225 0.661182i \(-0.770055\pi\)
−0.750225 + 0.661182i \(0.770055\pi\)
\(938\) 0 0
\(939\) −64.9238 −2.11871
\(940\) 0 0
\(941\) −13.2488 −0.431899 −0.215949 0.976405i \(-0.569285\pi\)
−0.215949 + 0.976405i \(0.569285\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 44.0523 1.43302
\(946\) 0 0
\(947\) −22.1017 −0.718207 −0.359104 0.933298i \(-0.616918\pi\)
−0.359104 + 0.933298i \(0.616918\pi\)
\(948\) 0 0
\(949\) 40.8959 1.32754
\(950\) 0 0
\(951\) −63.2552 −2.05119
\(952\) 0 0
\(953\) −18.2642 −0.591635 −0.295818 0.955244i \(-0.595592\pi\)
−0.295818 + 0.955244i \(0.595592\pi\)
\(954\) 0 0
\(955\) −23.2013 −0.750778
\(956\) 0 0
\(957\) 13.7860 0.445637
\(958\) 0 0
\(959\) 83.8760 2.70850
\(960\) 0 0
\(961\) −20.8023 −0.671042
\(962\) 0 0
\(963\) −0.614817 −0.0198122
\(964\) 0 0
\(965\) −22.3130 −0.718279
\(966\) 0 0
\(967\) 28.0843 0.903132 0.451566 0.892238i \(-0.350866\pi\)
0.451566 + 0.892238i \(0.350866\pi\)
\(968\) 0 0
\(969\) −28.1152 −0.903192
\(970\) 0 0
\(971\) 34.1505 1.09594 0.547972 0.836497i \(-0.315400\pi\)
0.547972 + 0.836497i \(0.315400\pi\)
\(972\) 0 0
\(973\) −11.0566 −0.354458
\(974\) 0 0
\(975\) −86.7918 −2.77956
\(976\) 0 0
\(977\) −41.4597 −1.32641 −0.663206 0.748437i \(-0.730805\pi\)
−0.663206 + 0.748437i \(0.730805\pi\)
\(978\) 0 0
\(979\) −23.9765 −0.766291
\(980\) 0 0
\(981\) −6.17898 −0.197280
\(982\) 0 0
\(983\) −34.1240 −1.08839 −0.544193 0.838960i \(-0.683164\pi\)
−0.544193 + 0.838960i \(0.683164\pi\)
\(984\) 0 0
\(985\) 51.0191 1.62560
\(986\) 0 0
\(987\) 57.8078 1.84004
\(988\) 0 0
\(989\) 11.3971 0.362408
\(990\) 0 0
\(991\) 21.3535 0.678315 0.339158 0.940730i \(-0.389858\pi\)
0.339158 + 0.940730i \(0.389858\pi\)
\(992\) 0 0
\(993\) −10.6513 −0.338009
\(994\) 0 0
\(995\) −67.2938 −2.13336
\(996\) 0 0
\(997\) 19.3560 0.613012 0.306506 0.951869i \(-0.400840\pi\)
0.306506 + 0.951869i \(0.400840\pi\)
\(998\) 0 0
\(999\) 7.03348 0.222529
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 6724.2.a.c.1.2 4
41.40 even 2 164.2.a.a.1.3 4
123.122 odd 2 1476.2.a.g.1.1 4
164.163 odd 2 656.2.a.i.1.2 4
205.122 odd 4 4100.2.d.c.1149.3 8
205.163 odd 4 4100.2.d.c.1149.6 8
205.204 even 2 4100.2.a.c.1.2 4
287.286 odd 2 8036.2.a.i.1.2 4
328.163 odd 2 2624.2.a.y.1.3 4
328.245 even 2 2624.2.a.v.1.2 4
492.491 even 2 5904.2.a.bp.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.2.a.a.1.3 4 41.40 even 2
656.2.a.i.1.2 4 164.163 odd 2
1476.2.a.g.1.1 4 123.122 odd 2
2624.2.a.v.1.2 4 328.245 even 2
2624.2.a.y.1.3 4 328.163 odd 2
4100.2.a.c.1.2 4 205.204 even 2
4100.2.d.c.1149.3 8 205.122 odd 4
4100.2.d.c.1149.6 8 205.163 odd 4
5904.2.a.bp.1.1 4 492.491 even 2
6724.2.a.c.1.2 4 1.1 even 1 trivial
8036.2.a.i.1.2 4 287.286 odd 2