Properties

Label 1815.2.a.y.1.5
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.437199552.1
Defining polynomial: \(x^{6} - 13 x^{4} + 49 x^{2} - 48\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.13353\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.13353 q^{2} -1.00000 q^{3} +2.55193 q^{4} +1.00000 q^{5} -2.13353 q^{6} +4.82155 q^{7} +1.17756 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.13353 q^{2} -1.00000 q^{3} +2.55193 q^{4} +1.00000 q^{5} -2.13353 q^{6} +4.82155 q^{7} +1.17756 q^{8} +1.00000 q^{9} +2.13353 q^{10} -2.55193 q^{12} +4.26705 q^{13} +10.2869 q^{14} -1.00000 q^{15} -2.59152 q^{16} +4.64166 q^{17} +2.13353 q^{18} -6.37371 q^{19} +2.55193 q^{20} -4.82155 q^{21} -5.14344 q^{23} -1.17756 q^{24} +1.00000 q^{25} +9.10386 q^{26} -1.00000 q^{27} +12.3042 q^{28} +4.26705 q^{29} -2.13353 q^{30} -6.39075 q^{31} -7.88417 q^{32} +9.90309 q^{34} +4.82155 q^{35} +2.55193 q^{36} +6.14344 q^{37} -13.5985 q^{38} -4.26705 q^{39} +1.17756 q^{40} +3.46410 q^{41} -10.2869 q^{42} -1.55216 q^{43} +1.00000 q^{45} -10.9737 q^{46} +5.35116 q^{47} +2.59152 q^{48} +16.2473 q^{49} +2.13353 q^{50} -4.64166 q^{51} +10.8892 q^{52} +2.24730 q^{53} -2.13353 q^{54} +5.67764 q^{56} +6.37371 q^{57} +9.10386 q^{58} -13.5985 q^{59} -2.55193 q^{60} -6.80205 q^{61} -13.6348 q^{62} +4.82155 q^{63} -11.6381 q^{64} +4.26705 q^{65} +6.14344 q^{67} +11.8452 q^{68} +5.14344 q^{69} +10.2869 q^{70} -10.4946 q^{71} +1.17756 q^{72} +5.62449 q^{73} +13.1072 q^{74} -1.00000 q^{75} -16.2653 q^{76} -9.10386 q^{78} +16.3914 q^{79} -2.59152 q^{80} +1.00000 q^{81} +7.39075 q^{82} +6.17899 q^{83} -12.3042 q^{84} +4.64166 q^{85} -3.31158 q^{86} -4.26705 q^{87} -1.39075 q^{89} +2.13353 q^{90} +20.5738 q^{91} -13.1257 q^{92} +6.39075 q^{93} +11.4168 q^{94} -6.37371 q^{95} +7.88417 q^{96} +3.93573 q^{97} +34.6640 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} + 14q^{4} + 6q^{5} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} + 14q^{4} + 6q^{5} + 6q^{9} - 14q^{12} + 8q^{14} - 6q^{15} + 10q^{16} + 14q^{20} - 4q^{23} + 6q^{25} + 52q^{26} - 6q^{27} + 18q^{31} + 26q^{34} + 14q^{36} + 10q^{37} - 20q^{38} - 8q^{42} + 6q^{45} - 10q^{48} + 68q^{49} - 16q^{53} - 76q^{56} + 52q^{58} - 20q^{59} - 14q^{60} + 16q^{64} + 10q^{67} + 4q^{69} + 8q^{70} - 4q^{71} - 6q^{75} - 52q^{78} + 10q^{80} + 6q^{81} - 12q^{82} - 12q^{86} + 48q^{89} + 16q^{91} + 30q^{92} - 18q^{93} + 2q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.13353 1.50863 0.754315 0.656513i \(-0.227969\pi\)
0.754315 + 0.656513i \(0.227969\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.55193 1.27596
\(5\) 1.00000 0.447214
\(6\) −2.13353 −0.871008
\(7\) 4.82155 1.82237 0.911186 0.411994i \(-0.135168\pi\)
0.911186 + 0.411994i \(0.135168\pi\)
\(8\) 1.17756 0.416329
\(9\) 1.00000 0.333333
\(10\) 2.13353 0.674680
\(11\) 0 0
\(12\) −2.55193 −0.736679
\(13\) 4.26705 1.18347 0.591733 0.806134i \(-0.298444\pi\)
0.591733 + 0.806134i \(0.298444\pi\)
\(14\) 10.2869 2.74929
\(15\) −1.00000 −0.258199
\(16\) −2.59152 −0.647879
\(17\) 4.64166 1.12577 0.562884 0.826536i \(-0.309692\pi\)
0.562884 + 0.826536i \(0.309692\pi\)
\(18\) 2.13353 0.502877
\(19\) −6.37371 −1.46223 −0.731114 0.682255i \(-0.760999\pi\)
−0.731114 + 0.682255i \(0.760999\pi\)
\(20\) 2.55193 0.570629
\(21\) −4.82155 −1.05215
\(22\) 0 0
\(23\) −5.14344 −1.07248 −0.536241 0.844065i \(-0.680156\pi\)
−0.536241 + 0.844065i \(0.680156\pi\)
\(24\) −1.17756 −0.240367
\(25\) 1.00000 0.200000
\(26\) 9.10386 1.78541
\(27\) −1.00000 −0.192450
\(28\) 12.3042 2.32528
\(29\) 4.26705 0.792371 0.396186 0.918170i \(-0.370334\pi\)
0.396186 + 0.918170i \(0.370334\pi\)
\(30\) −2.13353 −0.389527
\(31\) −6.39075 −1.14781 −0.573906 0.818921i \(-0.694573\pi\)
−0.573906 + 0.818921i \(0.694573\pi\)
\(32\) −7.88417 −1.39374
\(33\) 0 0
\(34\) 9.90309 1.69837
\(35\) 4.82155 0.814990
\(36\) 2.55193 0.425322
\(37\) 6.14344 1.00998 0.504988 0.863126i \(-0.331497\pi\)
0.504988 + 0.863126i \(0.331497\pi\)
\(38\) −13.5985 −2.20596
\(39\) −4.26705 −0.683275
\(40\) 1.17756 0.186188
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) −10.2869 −1.58730
\(43\) −1.55216 −0.236702 −0.118351 0.992972i \(-0.537761\pi\)
−0.118351 + 0.992972i \(0.537761\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) −10.9737 −1.61798
\(47\) 5.35116 0.780547 0.390274 0.920699i \(-0.372380\pi\)
0.390274 + 0.920699i \(0.372380\pi\)
\(48\) 2.59152 0.374053
\(49\) 16.2473 2.32104
\(50\) 2.13353 0.301726
\(51\) −4.64166 −0.649962
\(52\) 10.8892 1.51006
\(53\) 2.24730 0.308691 0.154345 0.988017i \(-0.450673\pi\)
0.154345 + 0.988017i \(0.450673\pi\)
\(54\) −2.13353 −0.290336
\(55\) 0 0
\(56\) 5.67764 0.758706
\(57\) 6.37371 0.844218
\(58\) 9.10386 1.19540
\(59\) −13.5985 −1.77037 −0.885185 0.465240i \(-0.845968\pi\)
−0.885185 + 0.465240i \(0.845968\pi\)
\(60\) −2.55193 −0.329453
\(61\) −6.80205 −0.870913 −0.435457 0.900210i \(-0.643413\pi\)
−0.435457 + 0.900210i \(0.643413\pi\)
\(62\) −13.6348 −1.73162
\(63\) 4.82155 0.607458
\(64\) −11.6381 −1.45476
\(65\) 4.26705 0.529262
\(66\) 0 0
\(67\) 6.14344 0.750541 0.375271 0.926915i \(-0.377550\pi\)
0.375271 + 0.926915i \(0.377550\pi\)
\(68\) 11.8452 1.43644
\(69\) 5.14344 0.619198
\(70\) 10.2869 1.22952
\(71\) −10.4946 −1.24548 −0.622740 0.782429i \(-0.713981\pi\)
−0.622740 + 0.782429i \(0.713981\pi\)
\(72\) 1.17756 0.138776
\(73\) 5.62449 0.658297 0.329149 0.944278i \(-0.393238\pi\)
0.329149 + 0.944278i \(0.393238\pi\)
\(74\) 13.1072 1.52368
\(75\) −1.00000 −0.115470
\(76\) −16.2653 −1.86575
\(77\) 0 0
\(78\) −9.10386 −1.03081
\(79\) 16.3914 1.84418 0.922089 0.386979i \(-0.126481\pi\)
0.922089 + 0.386979i \(0.126481\pi\)
\(80\) −2.59152 −0.289740
\(81\) 1.00000 0.111111
\(82\) 7.39075 0.816172
\(83\) 6.17899 0.678232 0.339116 0.940745i \(-0.389872\pi\)
0.339116 + 0.940745i \(0.389872\pi\)
\(84\) −12.3042 −1.34250
\(85\) 4.64166 0.503458
\(86\) −3.31158 −0.357096
\(87\) −4.26705 −0.457476
\(88\) 0 0
\(89\) −1.39075 −0.147419 −0.0737095 0.997280i \(-0.523484\pi\)
−0.0737095 + 0.997280i \(0.523484\pi\)
\(90\) 2.13353 0.224893
\(91\) 20.5738 2.15672
\(92\) −13.1257 −1.36845
\(93\) 6.39075 0.662690
\(94\) 11.4168 1.17756
\(95\) −6.37371 −0.653929
\(96\) 7.88417 0.804675
\(97\) 3.93573 0.399613 0.199806 0.979835i \(-0.435969\pi\)
0.199806 + 0.979835i \(0.435969\pi\)
\(98\) 34.6640 3.50160
\(99\) 0 0
\(100\) 2.55193 0.255193
\(101\) −15.4623 −1.53856 −0.769278 0.638914i \(-0.779384\pi\)
−0.769278 + 0.638914i \(0.779384\pi\)
\(102\) −9.90309 −0.980552
\(103\) 7.24730 0.714098 0.357049 0.934086i \(-0.383783\pi\)
0.357049 + 0.934086i \(0.383783\pi\)
\(104\) 5.02469 0.492711
\(105\) −4.82155 −0.470535
\(106\) 4.79468 0.465700
\(107\) −5.44461 −0.526350 −0.263175 0.964748i \(-0.584770\pi\)
−0.263175 + 0.964748i \(0.584770\pi\)
\(108\) −2.55193 −0.245560
\(109\) −2.90961 −0.278690 −0.139345 0.990244i \(-0.544500\pi\)
−0.139345 + 0.990244i \(0.544500\pi\)
\(110\) 0 0
\(111\) −6.14344 −0.583110
\(112\) −12.4951 −1.18068
\(113\) −2.45502 −0.230949 −0.115474 0.993310i \(-0.536839\pi\)
−0.115474 + 0.993310i \(0.536839\pi\)
\(114\) 13.5985 1.27361
\(115\) −5.14344 −0.479629
\(116\) 10.8892 1.01104
\(117\) 4.26705 0.394489
\(118\) −29.0127 −2.67083
\(119\) 22.3800 2.05157
\(120\) −1.17756 −0.107496
\(121\) 0 0
\(122\) −14.5123 −1.31389
\(123\) −3.46410 −0.312348
\(124\) −16.3087 −1.46457
\(125\) 1.00000 0.0894427
\(126\) 10.2869 0.916429
\(127\) −9.89154 −0.877733 −0.438866 0.898552i \(-0.644620\pi\)
−0.438866 + 0.898552i \(0.644620\pi\)
\(128\) −9.06173 −0.800951
\(129\) 1.55216 0.136660
\(130\) 9.10386 0.798461
\(131\) 3.46410 0.302660 0.151330 0.988483i \(-0.451644\pi\)
0.151330 + 0.988483i \(0.451644\pi\)
\(132\) 0 0
\(133\) −30.7311 −2.66473
\(134\) 13.1072 1.13229
\(135\) −1.00000 −0.0860663
\(136\) 5.46581 0.468689
\(137\) −6.53419 −0.558254 −0.279127 0.960254i \(-0.590045\pi\)
−0.279127 + 0.960254i \(0.590045\pi\)
\(138\) 10.9737 0.934141
\(139\) −19.6608 −1.66761 −0.833803 0.552062i \(-0.813841\pi\)
−0.833803 + 0.552062i \(0.813841\pi\)
\(140\) 12.3042 1.03990
\(141\) −5.35116 −0.450649
\(142\) −22.3905 −1.87897
\(143\) 0 0
\(144\) −2.59152 −0.215960
\(145\) 4.26705 0.354359
\(146\) 12.0000 0.993127
\(147\) −16.2473 −1.34005
\(148\) 15.6776 1.28869
\(149\) −12.3042 −1.00800 −0.504001 0.863703i \(-0.668139\pi\)
−0.504001 + 0.863703i \(0.668139\pi\)
\(150\) −2.13353 −0.174202
\(151\) −5.80438 −0.472354 −0.236177 0.971710i \(-0.575895\pi\)
−0.236177 + 0.971710i \(0.575895\pi\)
\(152\) −7.50539 −0.608768
\(153\) 4.64166 0.375256
\(154\) 0 0
\(155\) −6.39075 −0.513317
\(156\) −10.8892 −0.871835
\(157\) 0.351162 0.0280258 0.0140129 0.999902i \(-0.495539\pi\)
0.0140129 + 0.999902i \(0.495539\pi\)
\(158\) 34.9715 2.78218
\(159\) −2.24730 −0.178223
\(160\) −7.88417 −0.623299
\(161\) −24.7994 −1.95446
\(162\) 2.13353 0.167626
\(163\) 14.6381 1.14654 0.573270 0.819366i \(-0.305674\pi\)
0.573270 + 0.819366i \(0.305674\pi\)
\(164\) 8.84014 0.690299
\(165\) 0 0
\(166\) 13.1830 1.02320
\(167\) 14.2310 1.10123 0.550614 0.834760i \(-0.314393\pi\)
0.550614 + 0.834760i \(0.314393\pi\)
\(168\) −5.67764 −0.438039
\(169\) 5.20772 0.400594
\(170\) 9.90309 0.759532
\(171\) −6.37371 −0.487410
\(172\) −3.96101 −0.302024
\(173\) −18.1772 −1.38199 −0.690993 0.722861i \(-0.742827\pi\)
−0.690993 + 0.722861i \(0.742827\pi\)
\(174\) −9.10386 −0.690162
\(175\) 4.82155 0.364475
\(176\) 0 0
\(177\) 13.5985 1.02212
\(178\) −2.96720 −0.222401
\(179\) −9.10386 −0.680454 −0.340227 0.940343i \(-0.610504\pi\)
−0.340227 + 0.940343i \(0.610504\pi\)
\(180\) 2.55193 0.190210
\(181\) 13.2473 0.984664 0.492332 0.870407i \(-0.336145\pi\)
0.492332 + 0.870407i \(0.336145\pi\)
\(182\) 43.8947 3.25369
\(183\) 6.80205 0.502822
\(184\) −6.05669 −0.446505
\(185\) 6.14344 0.451675
\(186\) 13.6348 0.999754
\(187\) 0 0
\(188\) 13.6558 0.995951
\(189\) −4.82155 −0.350716
\(190\) −13.5985 −0.986536
\(191\) −18.0931 −1.30917 −0.654584 0.755989i \(-0.727156\pi\)
−0.654584 + 0.755989i \(0.727156\pi\)
\(192\) 11.6381 0.839904
\(193\) −16.0705 −1.15678 −0.578391 0.815760i \(-0.696319\pi\)
−0.578391 + 0.815760i \(0.696319\pi\)
\(194\) 8.39697 0.602867
\(195\) −4.26705 −0.305570
\(196\) 41.4620 2.96157
\(197\) −15.4623 −1.10164 −0.550822 0.834623i \(-0.685686\pi\)
−0.550822 + 0.834623i \(0.685686\pi\)
\(198\) 0 0
\(199\) 20.3907 1.44546 0.722731 0.691130i \(-0.242887\pi\)
0.722731 + 0.691130i \(0.242887\pi\)
\(200\) 1.17756 0.0832657
\(201\) −6.14344 −0.433325
\(202\) −32.9892 −2.32111
\(203\) 20.5738 1.44400
\(204\) −11.8452 −0.829329
\(205\) 3.46410 0.241943
\(206\) 15.4623 1.07731
\(207\) −5.14344 −0.357494
\(208\) −11.0581 −0.766743
\(209\) 0 0
\(210\) −10.2869 −0.709863
\(211\) −4.14090 −0.285071 −0.142536 0.989790i \(-0.545526\pi\)
−0.142536 + 0.989790i \(0.545526\pi\)
\(212\) 5.73496 0.393879
\(213\) 10.4946 0.719079
\(214\) −11.6162 −0.794067
\(215\) −1.55216 −0.105857
\(216\) −1.17756 −0.0801225
\(217\) −30.8133 −2.09174
\(218\) −6.20772 −0.420440
\(219\) −5.62449 −0.380068
\(220\) 0 0
\(221\) 19.8062 1.33231
\(222\) −13.1072 −0.879697
\(223\) −25.5342 −1.70990 −0.854948 0.518714i \(-0.826411\pi\)
−0.854948 + 0.518714i \(0.826411\pi\)
\(224\) −38.0139 −2.53991
\(225\) 1.00000 0.0666667
\(226\) −5.23785 −0.348417
\(227\) −20.1577 −1.33791 −0.668957 0.743301i \(-0.733259\pi\)
−0.668957 + 0.743301i \(0.733259\pi\)
\(228\) 16.2653 1.07719
\(229\) 7.96041 0.526039 0.263019 0.964791i \(-0.415282\pi\)
0.263019 + 0.964791i \(0.415282\pi\)
\(230\) −10.9737 −0.723582
\(231\) 0 0
\(232\) 5.02469 0.329887
\(233\) −0.428342 −0.0280616 −0.0140308 0.999902i \(-0.504466\pi\)
−0.0140308 + 0.999902i \(0.504466\pi\)
\(234\) 9.10386 0.595138
\(235\) 5.35116 0.349071
\(236\) −34.7023 −2.25893
\(237\) −16.3914 −1.06474
\(238\) 47.7482 3.09506
\(239\) −18.1235 −1.17231 −0.586154 0.810199i \(-0.699359\pi\)
−0.586154 + 0.810199i \(0.699359\pi\)
\(240\) 2.59152 0.167282
\(241\) −20.4637 −1.31819 −0.659093 0.752062i \(-0.729059\pi\)
−0.659093 + 0.752062i \(0.729059\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −17.3584 −1.11125
\(245\) 16.2473 1.03800
\(246\) −7.39075 −0.471217
\(247\) −27.1969 −1.73050
\(248\) −7.52546 −0.477867
\(249\) −6.17899 −0.391578
\(250\) 2.13353 0.134936
\(251\) 24.0931 1.52074 0.760371 0.649489i \(-0.225017\pi\)
0.760371 + 0.649489i \(0.225017\pi\)
\(252\) 12.3042 0.775095
\(253\) 0 0
\(254\) −21.1039 −1.32417
\(255\) −4.64166 −0.290672
\(256\) 3.94268 0.246417
\(257\) 14.6627 0.914637 0.457318 0.889303i \(-0.348810\pi\)
0.457318 + 0.889303i \(0.348810\pi\)
\(258\) 3.31158 0.206170
\(259\) 29.6209 1.84055
\(260\) 10.8892 0.675320
\(261\) 4.26705 0.264124
\(262\) 7.39075 0.456602
\(263\) 6.55360 0.404112 0.202056 0.979374i \(-0.435238\pi\)
0.202056 + 0.979374i \(0.435238\pi\)
\(264\) 0 0
\(265\) 2.24730 0.138051
\(266\) −65.5656 −4.02009
\(267\) 1.39075 0.0851124
\(268\) 15.6776 0.957664
\(269\) 26.7815 1.63290 0.816448 0.577419i \(-0.195940\pi\)
0.816448 + 0.577419i \(0.195940\pi\)
\(270\) −2.13353 −0.129842
\(271\) 5.08483 0.308881 0.154441 0.988002i \(-0.450642\pi\)
0.154441 + 0.988002i \(0.450642\pi\)
\(272\) −12.0289 −0.729361
\(273\) −20.5738 −1.24518
\(274\) −13.9409 −0.842198
\(275\) 0 0
\(276\) 13.1257 0.790075
\(277\) 19.4809 1.17049 0.585247 0.810855i \(-0.300998\pi\)
0.585247 + 0.810855i \(0.300998\pi\)
\(278\) −41.9468 −2.51580
\(279\) −6.39075 −0.382604
\(280\) 5.67764 0.339304
\(281\) 4.62683 0.276013 0.138007 0.990431i \(-0.455930\pi\)
0.138007 + 0.990431i \(0.455930\pi\)
\(282\) −11.4168 −0.679863
\(283\) 19.1211 1.13663 0.568316 0.822810i \(-0.307595\pi\)
0.568316 + 0.822810i \(0.307595\pi\)
\(284\) −26.7815 −1.58919
\(285\) 6.37371 0.377546
\(286\) 0 0
\(287\) 16.7023 0.985907
\(288\) −7.88417 −0.464579
\(289\) 4.54498 0.267352
\(290\) 9.10386 0.534597
\(291\) −3.93573 −0.230716
\(292\) 14.3533 0.839964
\(293\) −9.21475 −0.538331 −0.269166 0.963094i \(-0.586748\pi\)
−0.269166 + 0.963094i \(0.586748\pi\)
\(294\) −34.6640 −2.02165
\(295\) −13.5985 −0.791733
\(296\) 7.23425 0.420482
\(297\) 0 0
\(298\) −26.2514 −1.52070
\(299\) −21.9473 −1.26925
\(300\) −2.55193 −0.147336
\(301\) −7.48382 −0.431360
\(302\) −12.3838 −0.712607
\(303\) 15.4623 0.888286
\(304\) 16.5176 0.947347
\(305\) −6.80205 −0.389484
\(306\) 9.90309 0.566122
\(307\) −3.65882 −0.208820 −0.104410 0.994534i \(-0.533295\pi\)
−0.104410 + 0.994534i \(0.533295\pi\)
\(308\) 0 0
\(309\) −7.24730 −0.412285
\(310\) −13.6348 −0.774406
\(311\) −12.9753 −0.735762 −0.367881 0.929873i \(-0.619917\pi\)
−0.367881 + 0.929873i \(0.619917\pi\)
\(312\) −5.02469 −0.284467
\(313\) −4.48071 −0.253264 −0.126632 0.991950i \(-0.540417\pi\)
−0.126632 + 0.991950i \(0.540417\pi\)
\(314\) 0.749213 0.0422806
\(315\) 4.82155 0.271663
\(316\) 41.8297 2.35311
\(317\) −14.2473 −0.800208 −0.400104 0.916470i \(-0.631026\pi\)
−0.400104 + 0.916470i \(0.631026\pi\)
\(318\) −4.79468 −0.268872
\(319\) 0 0
\(320\) −11.6381 −0.650587
\(321\) 5.44461 0.303888
\(322\) −52.9100 −2.94856
\(323\) −29.5846 −1.64613
\(324\) 2.55193 0.141774
\(325\) 4.26705 0.236693
\(326\) 31.2306 1.72971
\(327\) 2.90961 0.160902
\(328\) 4.07917 0.225235
\(329\) 25.8009 1.42245
\(330\) 0 0
\(331\) 1.68842 0.0928041 0.0464021 0.998923i \(-0.485224\pi\)
0.0464021 + 0.998923i \(0.485224\pi\)
\(332\) 15.7683 0.865400
\(333\) 6.14344 0.336659
\(334\) 30.3622 1.66135
\(335\) 6.14344 0.335652
\(336\) 12.4951 0.681664
\(337\) −5.26472 −0.286787 −0.143394 0.989666i \(-0.545802\pi\)
−0.143394 + 0.989666i \(0.545802\pi\)
\(338\) 11.1108 0.604348
\(339\) 2.45502 0.133338
\(340\) 11.8452 0.642395
\(341\) 0 0
\(342\) −13.5985 −0.735321
\(343\) 44.5863 2.40743
\(344\) −1.82776 −0.0985460
\(345\) 5.14344 0.276914
\(346\) −38.7815 −2.08491
\(347\) −30.5500 −1.64001 −0.820005 0.572356i \(-0.806029\pi\)
−0.820005 + 0.572356i \(0.806029\pi\)
\(348\) −10.8892 −0.583723
\(349\) 34.6223 1.85329 0.926646 0.375936i \(-0.122679\pi\)
0.926646 + 0.375936i \(0.122679\pi\)
\(350\) 10.2869 0.549857
\(351\) −4.26705 −0.227758
\(352\) 0 0
\(353\) 1.83187 0.0975005 0.0487502 0.998811i \(-0.484476\pi\)
0.0487502 + 0.998811i \(0.484476\pi\)
\(354\) 29.0127 1.54201
\(355\) −10.4946 −0.556996
\(356\) −3.54909 −0.188101
\(357\) −22.3800 −1.18447
\(358\) −19.4233 −1.02655
\(359\) 16.5713 0.874599 0.437300 0.899316i \(-0.355935\pi\)
0.437300 + 0.899316i \(0.355935\pi\)
\(360\) 1.17756 0.0620626
\(361\) 21.6242 1.13811
\(362\) 28.2635 1.48549
\(363\) 0 0
\(364\) 52.5028 2.75190
\(365\) 5.62449 0.294399
\(366\) 14.5123 0.758572
\(367\) −4.11465 −0.214783 −0.107391 0.994217i \(-0.534250\pi\)
−0.107391 + 0.994217i \(0.534250\pi\)
\(368\) 13.3293 0.694839
\(369\) 3.46410 0.180334
\(370\) 13.1072 0.681411
\(371\) 10.8355 0.562550
\(372\) 16.3087 0.845569
\(373\) −21.0868 −1.09183 −0.545917 0.837840i \(-0.683818\pi\)
−0.545917 + 0.837840i \(0.683818\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 6.30129 0.324964
\(377\) 18.2077 0.937745
\(378\) −10.2869 −0.529100
\(379\) −3.14344 −0.161468 −0.0807339 0.996736i \(-0.525726\pi\)
−0.0807339 + 0.996736i \(0.525726\pi\)
\(380\) −16.2653 −0.834390
\(381\) 9.89154 0.506759
\(382\) −38.6020 −1.97505
\(383\) −10.9100 −0.557477 −0.278739 0.960367i \(-0.589916\pi\)
−0.278739 + 0.960367i \(0.589916\pi\)
\(384\) 9.06173 0.462429
\(385\) 0 0
\(386\) −34.2869 −1.74516
\(387\) −1.55216 −0.0789008
\(388\) 10.0437 0.509891
\(389\) −29.4699 −1.49418 −0.747092 0.664721i \(-0.768551\pi\)
−0.747092 + 0.664721i \(0.768551\pi\)
\(390\) −9.10386 −0.460992
\(391\) −23.8741 −1.20737
\(392\) 19.1321 0.966317
\(393\) −3.46410 −0.174741
\(394\) −32.9892 −1.66197
\(395\) 16.3914 0.824741
\(396\) 0 0
\(397\) 7.03959 0.353307 0.176653 0.984273i \(-0.443473\pi\)
0.176653 + 0.984273i \(0.443473\pi\)
\(398\) 43.5042 2.18067
\(399\) 30.7311 1.53848
\(400\) −2.59152 −0.129576
\(401\) −1.50539 −0.0751758 −0.0375879 0.999293i \(-0.511967\pi\)
−0.0375879 + 0.999293i \(0.511967\pi\)
\(402\) −13.1072 −0.653727
\(403\) −27.2696 −1.35840
\(404\) −39.4587 −1.96314
\(405\) 1.00000 0.0496904
\(406\) 43.8947 2.17846
\(407\) 0 0
\(408\) −5.46581 −0.270598
\(409\) −37.2298 −1.84089 −0.920446 0.390869i \(-0.872175\pi\)
−0.920446 + 0.390869i \(0.872175\pi\)
\(410\) 7.39075 0.365003
\(411\) 6.53419 0.322308
\(412\) 18.4946 0.911164
\(413\) −65.5656 −3.22627
\(414\) −10.9737 −0.539326
\(415\) 6.17899 0.303315
\(416\) −33.6422 −1.64944
\(417\) 19.6608 0.962793
\(418\) 0 0
\(419\) −25.4838 −1.24497 −0.622483 0.782633i \(-0.713876\pi\)
−0.622483 + 0.782633i \(0.713876\pi\)
\(420\) −12.3042 −0.600386
\(421\) 21.0288 1.02488 0.512440 0.858723i \(-0.328742\pi\)
0.512440 + 0.858723i \(0.328742\pi\)
\(422\) −8.83471 −0.430067
\(423\) 5.35116 0.260182
\(424\) 2.64632 0.128517
\(425\) 4.64166 0.225153
\(426\) 22.3905 1.08482
\(427\) −32.7964 −1.58713
\(428\) −13.8942 −0.671604
\(429\) 0 0
\(430\) −3.31158 −0.159698
\(431\) 31.5904 1.52166 0.760829 0.648953i \(-0.224793\pi\)
0.760829 + 0.648953i \(0.224793\pi\)
\(432\) 2.59152 0.124684
\(433\) 9.93573 0.477481 0.238740 0.971083i \(-0.423266\pi\)
0.238740 + 0.971083i \(0.423266\pi\)
\(434\) −65.7409 −3.15566
\(435\) −4.26705 −0.204589
\(436\) −7.42511 −0.355598
\(437\) 32.7828 1.56821
\(438\) −12.0000 −0.573382
\(439\) 29.2463 1.39585 0.697925 0.716171i \(-0.254107\pi\)
0.697925 + 0.716171i \(0.254107\pi\)
\(440\) 0 0
\(441\) 16.2473 0.773681
\(442\) 42.2570 2.00996
\(443\) −28.7023 −1.36369 −0.681844 0.731497i \(-0.738822\pi\)
−0.681844 + 0.731497i \(0.738822\pi\)
\(444\) −15.6776 −0.744028
\(445\) −1.39075 −0.0659278
\(446\) −54.4778 −2.57960
\(447\) 12.3042 0.581971
\(448\) −56.1134 −2.65111
\(449\) −12.2077 −0.576118 −0.288059 0.957613i \(-0.593010\pi\)
−0.288059 + 0.957613i \(0.593010\pi\)
\(450\) 2.13353 0.100575
\(451\) 0 0
\(452\) −6.26504 −0.294683
\(453\) 5.80438 0.272714
\(454\) −43.0070 −2.01842
\(455\) 20.5738 0.964514
\(456\) 7.50539 0.351472
\(457\) 13.1072 0.613129 0.306564 0.951850i \(-0.400821\pi\)
0.306564 + 0.951850i \(0.400821\pi\)
\(458\) 16.9837 0.793598
\(459\) −4.64166 −0.216654
\(460\) −13.1257 −0.611989
\(461\) 3.77014 0.175593 0.0877966 0.996138i \(-0.472017\pi\)
0.0877966 + 0.996138i \(0.472017\pi\)
\(462\) 0 0
\(463\) −1.87145 −0.0869738 −0.0434869 0.999054i \(-0.513847\pi\)
−0.0434869 + 0.999054i \(0.513847\pi\)
\(464\) −11.0581 −0.513361
\(465\) 6.39075 0.296364
\(466\) −0.913878 −0.0423346
\(467\) 27.8458 1.28855 0.644274 0.764795i \(-0.277160\pi\)
0.644274 + 0.764795i \(0.277160\pi\)
\(468\) 10.8892 0.503354
\(469\) 29.6209 1.36777
\(470\) 11.4168 0.526620
\(471\) −0.351162 −0.0161807
\(472\) −16.0129 −0.737056
\(473\) 0 0
\(474\) −34.9715 −1.60629
\(475\) −6.37371 −0.292446
\(476\) 57.1121 2.61773
\(477\) 2.24730 0.102897
\(478\) −38.6669 −1.76858
\(479\) 2.66115 0.121591 0.0607956 0.998150i \(-0.480636\pi\)
0.0607956 + 0.998150i \(0.480636\pi\)
\(480\) 7.88417 0.359862
\(481\) 26.2144 1.19527
\(482\) −43.6599 −1.98865
\(483\) 24.7994 1.12841
\(484\) 0 0
\(485\) 3.93573 0.178712
\(486\) −2.13353 −0.0967787
\(487\) 11.8062 0.534989 0.267495 0.963559i \(-0.413804\pi\)
0.267495 + 0.963559i \(0.413804\pi\)
\(488\) −8.00979 −0.362586
\(489\) −14.6381 −0.661956
\(490\) 34.6640 1.56596
\(491\) 29.8156 1.34556 0.672780 0.739843i \(-0.265100\pi\)
0.672780 + 0.739843i \(0.265100\pi\)
\(492\) −8.84014 −0.398544
\(493\) 19.8062 0.892026
\(494\) −58.0253 −2.61068
\(495\) 0 0
\(496\) 16.5617 0.743643
\(497\) −50.6002 −2.26973
\(498\) −13.1830 −0.590746
\(499\) −10.0288 −0.448951 −0.224475 0.974480i \(-0.572067\pi\)
−0.224475 + 0.974480i \(0.572067\pi\)
\(500\) 2.55193 0.114126
\(501\) −14.2310 −0.635795
\(502\) 51.4032 2.29424
\(503\) 26.1996 1.16818 0.584090 0.811689i \(-0.301451\pi\)
0.584090 + 0.811689i \(0.301451\pi\)
\(504\) 5.67764 0.252902
\(505\) −15.4623 −0.688063
\(506\) 0 0
\(507\) −5.20772 −0.231283
\(508\) −25.2425 −1.11996
\(509\) −17.6776 −0.783547 −0.391774 0.920062i \(-0.628138\pi\)
−0.391774 + 0.920062i \(0.628138\pi\)
\(510\) −9.90309 −0.438516
\(511\) 27.1188 1.19966
\(512\) 26.5353 1.17270
\(513\) 6.37371 0.281406
\(514\) 31.2833 1.37985
\(515\) 7.24730 0.319354
\(516\) 3.96101 0.174374
\(517\) 0 0
\(518\) 63.1969 2.77671
\(519\) 18.1772 0.797890
\(520\) 5.02469 0.220347
\(521\) 33.7568 1.47891 0.739456 0.673205i \(-0.235083\pi\)
0.739456 + 0.673205i \(0.235083\pi\)
\(522\) 9.10386 0.398465
\(523\) −3.71255 −0.162339 −0.0811693 0.996700i \(-0.525865\pi\)
−0.0811693 + 0.996700i \(0.525865\pi\)
\(524\) 8.84014 0.386183
\(525\) −4.82155 −0.210430
\(526\) 13.9823 0.609656
\(527\) −29.6637 −1.29217
\(528\) 0 0
\(529\) 3.45502 0.150218
\(530\) 4.79468 0.208268
\(531\) −13.5985 −0.590123
\(532\) −78.4237 −3.40010
\(533\) 14.7815 0.640258
\(534\) 2.96720 0.128403
\(535\) −5.44461 −0.235391
\(536\) 7.23425 0.312472
\(537\) 9.10386 0.392861
\(538\) 57.1390 2.46344
\(539\) 0 0
\(540\) −2.55193 −0.109818
\(541\) −22.3905 −0.962643 −0.481322 0.876544i \(-0.659843\pi\)
−0.481322 + 0.876544i \(0.659843\pi\)
\(542\) 10.8486 0.465988
\(543\) −13.2473 −0.568496
\(544\) −36.5956 −1.56902
\(545\) −2.90961 −0.124634
\(546\) −43.8947 −1.87852
\(547\) −19.2324 −0.822320 −0.411160 0.911563i \(-0.634876\pi\)
−0.411160 + 0.911563i \(0.634876\pi\)
\(548\) −16.6748 −0.712312
\(549\) −6.80205 −0.290304
\(550\) 0 0
\(551\) −27.1969 −1.15863
\(552\) 6.05669 0.257790
\(553\) 79.0319 3.36078
\(554\) 41.5630 1.76584
\(555\) −6.14344 −0.260775
\(556\) −50.1729 −2.12781
\(557\) −37.9214 −1.60678 −0.803390 0.595453i \(-0.796973\pi\)
−0.803390 + 0.595453i \(0.796973\pi\)
\(558\) −13.6348 −0.577208
\(559\) −6.62315 −0.280130
\(560\) −12.4951 −0.528015
\(561\) 0 0
\(562\) 9.87145 0.416402
\(563\) −2.46258 −0.103785 −0.0518927 0.998653i \(-0.516525\pi\)
−0.0518927 + 0.998653i \(0.516525\pi\)
\(564\) −13.6558 −0.575012
\(565\) −2.45502 −0.103284
\(566\) 40.7954 1.71476
\(567\) 4.82155 0.202486
\(568\) −12.3580 −0.518529
\(569\) −6.48503 −0.271867 −0.135933 0.990718i \(-0.543403\pi\)
−0.135933 + 0.990718i \(0.543403\pi\)
\(570\) 13.5985 0.569577
\(571\) −31.4643 −1.31674 −0.658369 0.752695i \(-0.728754\pi\)
−0.658369 + 0.752695i \(0.728754\pi\)
\(572\) 0 0
\(573\) 18.0931 0.755849
\(574\) 35.6348 1.48737
\(575\) −5.14344 −0.214496
\(576\) −11.6381 −0.484919
\(577\) 12.4658 0.518958 0.259479 0.965749i \(-0.416449\pi\)
0.259479 + 0.965749i \(0.416449\pi\)
\(578\) 9.69683 0.403335
\(579\) 16.0705 0.667869
\(580\) 10.8892 0.452150
\(581\) 29.7923 1.23599
\(582\) −8.39697 −0.348066
\(583\) 0 0
\(584\) 6.62315 0.274068
\(585\) 4.26705 0.176421
\(586\) −19.6599 −0.812143
\(587\) 30.4195 1.25555 0.627775 0.778395i \(-0.283966\pi\)
0.627775 + 0.778395i \(0.283966\pi\)
\(588\) −41.4620 −1.70986
\(589\) 40.7328 1.67836
\(590\) −29.0127 −1.19443
\(591\) 15.4623 0.636034
\(592\) −15.9208 −0.654342
\(593\) 19.6756 0.807981 0.403990 0.914763i \(-0.367623\pi\)
0.403990 + 0.914763i \(0.367623\pi\)
\(594\) 0 0
\(595\) 22.3800 0.917489
\(596\) −31.3996 −1.28618
\(597\) −20.3907 −0.834538
\(598\) −46.8252 −1.91482
\(599\) 8.01390 0.327439 0.163720 0.986507i \(-0.447651\pi\)
0.163720 + 0.986507i \(0.447651\pi\)
\(600\) −1.17756 −0.0480735
\(601\) 30.7299 1.25350 0.626749 0.779221i \(-0.284385\pi\)
0.626749 + 0.779221i \(0.284385\pi\)
\(602\) −15.9669 −0.650763
\(603\) 6.14344 0.250180
\(604\) −14.8124 −0.602707
\(605\) 0 0
\(606\) 32.9892 1.34010
\(607\) 33.2260 1.34860 0.674301 0.738457i \(-0.264445\pi\)
0.674301 + 0.738457i \(0.264445\pi\)
\(608\) 50.2514 2.03796
\(609\) −20.5738 −0.833692
\(610\) −14.5123 −0.587588
\(611\) 22.8337 0.923752
\(612\) 11.8452 0.478813
\(613\) 20.9793 0.847347 0.423674 0.905815i \(-0.360740\pi\)
0.423674 + 0.905815i \(0.360740\pi\)
\(614\) −7.80618 −0.315032
\(615\) −3.46410 −0.139686
\(616\) 0 0
\(617\) 14.5738 0.586718 0.293359 0.956002i \(-0.405227\pi\)
0.293359 + 0.956002i \(0.405227\pi\)
\(618\) −15.4623 −0.621985
\(619\) 5.21850 0.209749 0.104875 0.994485i \(-0.466556\pi\)
0.104875 + 0.994485i \(0.466556\pi\)
\(620\) −16.3087 −0.654975
\(621\) 5.14344 0.206399
\(622\) −27.6832 −1.10999
\(623\) −6.70555 −0.268652
\(624\) 11.0581 0.442679
\(625\) 1.00000 0.0400000
\(626\) −9.55970 −0.382082
\(627\) 0 0
\(628\) 0.896141 0.0357599
\(629\) 28.5158 1.13700
\(630\) 10.2869 0.409839
\(631\) 13.8458 0.551191 0.275596 0.961274i \(-0.411125\pi\)
0.275596 + 0.961274i \(0.411125\pi\)
\(632\) 19.3018 0.767784
\(633\) 4.14090 0.164586
\(634\) −30.3970 −1.20722
\(635\) −9.89154 −0.392534
\(636\) −5.73496 −0.227406
\(637\) 69.3281 2.74688
\(638\) 0 0
\(639\) −10.4946 −0.415160
\(640\) −9.06173 −0.358196
\(641\) −49.5769 −1.95817 −0.979085 0.203453i \(-0.934784\pi\)
−0.979085 + 0.203453i \(0.934784\pi\)
\(642\) 11.6162 0.458455
\(643\) 4.25809 0.167923 0.0839613 0.996469i \(-0.473243\pi\)
0.0839613 + 0.996469i \(0.473243\pi\)
\(644\) −63.2862 −2.49383
\(645\) 1.55216 0.0611163
\(646\) −63.1194 −2.48340
\(647\) −24.6273 −0.968198 −0.484099 0.875013i \(-0.660852\pi\)
−0.484099 + 0.875013i \(0.660852\pi\)
\(648\) 1.17756 0.0462587
\(649\) 0 0
\(650\) 9.10386 0.357083
\(651\) 30.8133 1.20767
\(652\) 37.3553 1.46295
\(653\) −2.98921 −0.116977 −0.0584885 0.998288i \(-0.518628\pi\)
−0.0584885 + 0.998288i \(0.518628\pi\)
\(654\) 6.20772 0.242741
\(655\) 3.46410 0.135354
\(656\) −8.97727 −0.350504
\(657\) 5.62449 0.219432
\(658\) 55.0468 2.14595
\(659\) −28.5158 −1.11082 −0.555408 0.831578i \(-0.687438\pi\)
−0.555408 + 0.831578i \(0.687438\pi\)
\(660\) 0 0
\(661\) 36.0288 1.40136 0.700679 0.713477i \(-0.252881\pi\)
0.700679 + 0.713477i \(0.252881\pi\)
\(662\) 3.60229 0.140007
\(663\) −19.8062 −0.769208
\(664\) 7.27610 0.282368
\(665\) −30.7311 −1.19170
\(666\) 13.1072 0.507893
\(667\) −21.9473 −0.849804
\(668\) 36.3165 1.40513
\(669\) 25.5342 0.987209
\(670\) 13.1072 0.506375
\(671\) 0 0
\(672\) 38.0139 1.46642
\(673\) 6.92435 0.266914 0.133457 0.991055i \(-0.457392\pi\)
0.133457 + 0.991055i \(0.457392\pi\)
\(674\) −11.2324 −0.432656
\(675\) −1.00000 −0.0384900
\(676\) 13.2897 0.511143
\(677\) 20.1725 0.775293 0.387647 0.921808i \(-0.373288\pi\)
0.387647 + 0.921808i \(0.373288\pi\)
\(678\) 5.23785 0.201158
\(679\) 18.9763 0.728243
\(680\) 5.46581 0.209604
\(681\) 20.1577 0.772445
\(682\) 0 0
\(683\) 43.4838 1.66386 0.831931 0.554879i \(-0.187235\pi\)
0.831931 + 0.554879i \(0.187235\pi\)
\(684\) −16.2653 −0.621917
\(685\) −6.53419 −0.249659
\(686\) 95.1260 3.63193
\(687\) −7.96041 −0.303709
\(688\) 4.02245 0.153355
\(689\) 9.58936 0.365325
\(690\) 10.9737 0.417760
\(691\) 16.6776 0.634447 0.317224 0.948351i \(-0.397249\pi\)
0.317224 + 0.948351i \(0.397249\pi\)
\(692\) −46.3869 −1.76337
\(693\) 0 0
\(694\) −65.1792 −2.47417
\(695\) −19.6608 −0.745776
\(696\) −5.02469 −0.190460
\(697\) 16.0792 0.609042
\(698\) 73.8676 2.79593
\(699\) 0.428342 0.0162014
\(700\) 12.3042 0.465057
\(701\) −3.90727 −0.147576 −0.0737878 0.997274i \(-0.523509\pi\)
−0.0737878 + 0.997274i \(0.523509\pi\)
\(702\) −9.10386 −0.343603
\(703\) −39.1565 −1.47682
\(704\) 0 0
\(705\) −5.35116 −0.201536
\(706\) 3.90834 0.147092
\(707\) −74.5522 −2.80382
\(708\) 34.7023 1.30419
\(709\) 41.6026 1.56242 0.781209 0.624270i \(-0.214603\pi\)
0.781209 + 0.624270i \(0.214603\pi\)
\(710\) −22.3905 −0.840301
\(711\) 16.3914 0.614726
\(712\) −1.63768 −0.0613747
\(713\) 32.8705 1.23101
\(714\) −47.7482 −1.78693
\(715\) 0 0
\(716\) −23.2324 −0.868236
\(717\) 18.1235 0.676833
\(718\) 35.3553 1.31945
\(719\) 4.91004 0.183114 0.0915568 0.995800i \(-0.470816\pi\)
0.0915568 + 0.995800i \(0.470816\pi\)
\(720\) −2.59152 −0.0965801
\(721\) 34.9432 1.30135
\(722\) 46.1357 1.71699
\(723\) 20.4637 0.761055
\(724\) 33.8062 1.25640
\(725\) 4.26705 0.158474
\(726\) 0 0
\(727\) 17.7131 0.656943 0.328471 0.944514i \(-0.393467\pi\)
0.328471 + 0.944514i \(0.393467\pi\)
\(728\) 24.2268 0.897904
\(729\) 1.00000 0.0370370
\(730\) 12.0000 0.444140
\(731\) −7.20460 −0.266472
\(732\) 17.3584 0.641583
\(733\) −14.7131 −0.543440 −0.271720 0.962376i \(-0.587593\pi\)
−0.271720 + 0.962376i \(0.587593\pi\)
\(734\) −8.77870 −0.324028
\(735\) −16.2473 −0.599291
\(736\) 40.5518 1.49476
\(737\) 0 0
\(738\) 7.39075 0.272057
\(739\) 14.9226 0.548938 0.274469 0.961596i \(-0.411498\pi\)
0.274469 + 0.961596i \(0.411498\pi\)
\(740\) 15.6776 0.576321
\(741\) 27.1969 0.999104
\(742\) 23.1178 0.848680
\(743\) 52.6882 1.93294 0.966471 0.256775i \(-0.0826598\pi\)
0.966471 + 0.256775i \(0.0826598\pi\)
\(744\) 7.52546 0.275897
\(745\) −12.3042 −0.450793
\(746\) −44.9892 −1.64717
\(747\) 6.17899 0.226077
\(748\) 0 0
\(749\) −26.2514 −0.959206
\(750\) −2.13353 −0.0779053
\(751\) 14.5985 0.532706 0.266353 0.963876i \(-0.414181\pi\)
0.266353 + 0.963876i \(0.414181\pi\)
\(752\) −13.8676 −0.505700
\(753\) −24.0931 −0.878000
\(754\) 38.8466 1.41471
\(755\) −5.80438 −0.211243
\(756\) −12.3042 −0.447501
\(757\) 10.0782 0.366297 0.183149 0.983085i \(-0.441371\pi\)
0.183149 + 0.983085i \(0.441371\pi\)
\(758\) −6.70662 −0.243595
\(759\) 0 0
\(760\) −7.50539 −0.272249
\(761\) −18.5129 −0.671092 −0.335546 0.942024i \(-0.608921\pi\)
−0.335546 + 0.942024i \(0.608921\pi\)
\(762\) 21.1039 0.764512
\(763\) −14.0288 −0.507877
\(764\) −46.1722 −1.67045
\(765\) 4.64166 0.167819
\(766\) −23.2768 −0.841027
\(767\) −58.0253 −2.09517
\(768\) −3.94268 −0.142269
\(769\) 21.7674 0.784954 0.392477 0.919762i \(-0.371618\pi\)
0.392477 + 0.919762i \(0.371618\pi\)
\(770\) 0 0
\(771\) −14.6627 −0.528066
\(772\) −41.0109 −1.47601
\(773\) −44.9003 −1.61495 −0.807475 0.589902i \(-0.799166\pi\)
−0.807475 + 0.589902i \(0.799166\pi\)
\(774\) −3.31158 −0.119032
\(775\) −6.39075 −0.229562
\(776\) 4.63454 0.166370
\(777\) −29.6209 −1.06264
\(778\) −62.8748 −2.25417
\(779\) −22.0792 −0.791068
\(780\) −10.8892 −0.389896
\(781\) 0 0
\(782\) −50.9360 −1.82147
\(783\) −4.26705 −0.152492
\(784\) −42.1051 −1.50375
\(785\) 0.351162 0.0125335
\(786\) −7.39075 −0.263619
\(787\) −25.1054 −0.894911 −0.447455 0.894306i \(-0.647670\pi\)
−0.447455 + 0.894306i \(0.647670\pi\)
\(788\) −39.4587 −1.40566
\(789\) −6.55360 −0.233314
\(790\) 34.9715 1.24423
\(791\) −11.8370 −0.420875
\(792\) 0 0
\(793\) −29.0247 −1.03070
\(794\) 15.0191 0.533009
\(795\) −2.24730 −0.0797036
\(796\) 52.0357 1.84436
\(797\) 2.98921 0.105883 0.0529417 0.998598i \(-0.483140\pi\)
0.0529417 + 0.998598i \(0.483140\pi\)
\(798\) 65.5656 2.32100
\(799\) 24.8383 0.878714
\(800\) −7.88417 −0.278748
\(801\) −1.39075 −0.0491397
\(802\) −3.21179 −0.113412
\(803\) 0 0
\(804\) −15.6776 −0.552908
\(805\) −24.7994 −0.874062
\(806\) −58.1805 −2.04932
\(807\) −26.7815 −0.942753
\(808\) −18.2077 −0.640545
\(809\) −17.2909 −0.607914 −0.303957 0.952686i \(-0.598308\pi\)
−0.303957 + 0.952686i \(0.598308\pi\)
\(810\) 2.13353 0.0749644
\(811\) −43.4625 −1.52617 −0.763087 0.646296i \(-0.776317\pi\)
−0.763087 + 0.646296i \(0.776317\pi\)
\(812\) 52.5028 1.84249
\(813\) −5.08483 −0.178333
\(814\) 0 0
\(815\) 14.6381 0.512749
\(816\) 12.0289 0.421097
\(817\) 9.89303 0.346113
\(818\) −79.4306 −2.77723
\(819\) 20.5738 0.718906
\(820\) 8.84014 0.308711
\(821\) 25.1351 0.877219 0.438610 0.898678i \(-0.355471\pi\)
0.438610 + 0.898678i \(0.355471\pi\)
\(822\) 13.9409 0.486243
\(823\) −37.7419 −1.31560 −0.657800 0.753193i \(-0.728513\pi\)
−0.657800 + 0.753193i \(0.728513\pi\)
\(824\) 8.53410 0.297299
\(825\) 0 0
\(826\) −139.886 −4.86725
\(827\) −16.3190 −0.567467 −0.283733 0.958903i \(-0.591573\pi\)
−0.283733 + 0.958903i \(0.591573\pi\)
\(828\) −13.1257 −0.456150
\(829\) 8.66374 0.300904 0.150452 0.988617i \(-0.451927\pi\)
0.150452 + 0.988617i \(0.451927\pi\)
\(830\) 13.1830 0.457590
\(831\) −19.4809 −0.675785
\(832\) −49.6601 −1.72166
\(833\) 75.4144 2.61295
\(834\) 41.9468 1.45250
\(835\) 14.2310 0.492485
\(836\) 0 0
\(837\) 6.39075 0.220897
\(838\) −54.3704 −1.87819
\(839\) 11.5846 0.399944 0.199972 0.979802i \(-0.435915\pi\)
0.199972 + 0.979802i \(0.435915\pi\)
\(840\) −5.67764 −0.195897
\(841\) −10.7923 −0.372148
\(842\) 44.8655 1.54617
\(843\) −4.62683 −0.159356
\(844\) −10.5673 −0.363741
\(845\) 5.20772 0.179151
\(846\) 11.4168 0.392519
\(847\) 0 0
\(848\) −5.82392 −0.199994
\(849\) −19.1211 −0.656235
\(850\) 9.90309 0.339673
\(851\) −31.5985 −1.08318
\(852\) 26.7815 0.917519
\(853\) 18.0121 0.616724 0.308362 0.951269i \(-0.400219\pi\)
0.308362 + 0.951269i \(0.400219\pi\)
\(854\) −69.9719 −2.39439
\(855\) −6.37371 −0.217976
\(856\) −6.41132 −0.219135
\(857\) 13.0386 0.445391 0.222696 0.974888i \(-0.428514\pi\)
0.222696 + 0.974888i \(0.428514\pi\)
\(858\) 0 0
\(859\) 22.4442 0.765787 0.382894 0.923792i \(-0.374928\pi\)
0.382894 + 0.923792i \(0.374928\pi\)
\(860\) −3.96101 −0.135069
\(861\) −16.7023 −0.569214
\(862\) 67.3990 2.29562
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 7.88417 0.268225
\(865\) −18.1772 −0.618043
\(866\) 21.1981 0.720342
\(867\) −4.54498 −0.154356
\(868\) −78.6333 −2.66899
\(869\) 0 0
\(870\) −9.10386 −0.308650
\(871\) 26.2144 0.888241
\(872\) −3.42622 −0.116027
\(873\) 3.93573 0.133204
\(874\) 69.9430 2.36586
\(875\) 4.82155 0.162998
\(876\) −14.3533 −0.484953
\(877\) 41.0147 1.38497 0.692484 0.721433i \(-0.256516\pi\)
0.692484 + 0.721433i \(0.256516\pi\)
\(878\) 62.3977 2.10582
\(879\) 9.21475 0.310806
\(880\) 0 0
\(881\) 18.6232 0.627430 0.313715 0.949517i \(-0.398426\pi\)
0.313715 + 0.949517i \(0.398426\pi\)
\(882\) 34.6640 1.16720
\(883\) −32.8458 −1.10535 −0.552674 0.833397i \(-0.686393\pi\)
−0.552674 + 0.833397i \(0.686393\pi\)
\(884\) 50.5440 1.69998
\(885\) 13.5985 0.457107
\(886\) −61.2371 −2.05730
\(887\) 29.5710 0.992898 0.496449 0.868066i \(-0.334637\pi\)
0.496449 + 0.868066i \(0.334637\pi\)
\(888\) −7.23425 −0.242765
\(889\) −47.6925 −1.59956
\(890\) −2.96720 −0.0994606
\(891\) 0 0
\(892\) −65.1615 −2.18177
\(893\) −34.1067 −1.14134
\(894\) 26.2514 0.877979
\(895\) −9.10386 −0.304308
\(896\) −43.6915 −1.45963
\(897\) 21.9473 0.732800
\(898\) −26.0455 −0.869149
\(899\) −27.2696 −0.909493
\(900\) 2.55193 0.0850643
\(901\) 10.4312 0.347514
\(902\) 0 0
\(903\) 7.48382 0.249046
\(904\) −2.89092 −0.0961507
\(905\) 13.2473 0.440355
\(906\) 12.3838 0.411424
\(907\) 15.1681 0.503650 0.251825 0.967773i \(-0.418969\pi\)
0.251825 + 0.967773i \(0.418969\pi\)
\(908\) −51.4410 −1.70713
\(909\) −15.4623 −0.512852
\(910\) 43.8947 1.45509
\(911\) −26.8961 −0.891109 −0.445554 0.895255i \(-0.646993\pi\)
−0.445554 + 0.895255i \(0.646993\pi\)
\(912\) −16.5176 −0.546951
\(913\) 0 0
\(914\) 27.9645 0.924984
\(915\) 6.80205 0.224869
\(916\) 20.3144 0.671207
\(917\) 16.7023 0.551559
\(918\) −9.90309 −0.326851
\(919\) 37.6878 1.24320 0.621602 0.783333i \(-0.286482\pi\)
0.621602 + 0.783333i \(0.286482\pi\)
\(920\) −6.05669 −0.199683
\(921\) 3.65882 0.120562
\(922\) 8.04370 0.264905
\(923\) −44.7810 −1.47399
\(924\) 0 0
\(925\) 6.14344 0.201995
\(926\) −3.99279 −0.131211
\(927\) 7.24730 0.238033
\(928\) −33.6422 −1.10436
\(929\) 46.7321 1.53323 0.766616 0.642106i \(-0.221939\pi\)
0.766616 + 0.642106i \(0.221939\pi\)
\(930\) 13.6348 0.447103
\(931\) −103.556 −3.39390
\(932\) −1.09310 −0.0358056
\(933\) 12.9753 0.424793
\(934\) 59.4096 1.94394
\(935\) 0 0
\(936\) 5.02469 0.164237
\(937\) 4.20946 0.137517 0.0687585 0.997633i \(-0.478096\pi\)
0.0687585 + 0.997633i \(0.478096\pi\)
\(938\) 63.1969 2.06345
\(939\) 4.48071 0.146222
\(940\) 13.6558 0.445403
\(941\) 36.4081 1.18687 0.593435 0.804882i \(-0.297771\pi\)
0.593435 + 0.804882i \(0.297771\pi\)
\(942\) −0.749213 −0.0244107
\(943\) −17.8174 −0.580215
\(944\) 35.2406 1.14698
\(945\) −4.82155 −0.156845
\(946\) 0 0
\(947\) −47.5589 −1.54546 −0.772728 0.634737i \(-0.781108\pi\)
−0.772728 + 0.634737i \(0.781108\pi\)
\(948\) −41.8297 −1.35857
\(949\) 24.0000 0.779073
\(950\) −13.5985 −0.441192
\(951\) 14.2473 0.462000
\(952\) 26.3536 0.854126
\(953\) −14.9654 −0.484777 −0.242388 0.970179i \(-0.577931\pi\)
−0.242388 + 0.970179i \(0.577931\pi\)
\(954\) 4.79468 0.155233
\(955\) −18.0931 −0.585478
\(956\) −46.2498 −1.49582
\(957\) 0 0
\(958\) 5.67764 0.183436
\(959\) −31.5049 −1.01735
\(960\) 11.6381 0.375616
\(961\) 9.84166 0.317473
\(962\) 55.9291 1.80322
\(963\) −5.44461 −0.175450
\(964\) −52.2220 −1.68196
\(965\) −16.0705 −0.517329
\(966\) 52.9100 1.70235
\(967\) −28.6271 −0.920585 −0.460293 0.887767i \(-0.652255\pi\)
−0.460293 + 0.887767i \(0.652255\pi\)
\(968\) 0 0
\(969\) 29.5846 0.950393
\(970\) 8.39697 0.269611
\(971\) 8.89614 0.285491 0.142745 0.989759i \(-0.454407\pi\)
0.142745 + 0.989759i \(0.454407\pi\)
\(972\) −2.55193 −0.0818532
\(973\) −94.7954 −3.03900
\(974\) 25.1888 0.807101
\(975\) −4.26705 −0.136655
\(976\) 17.6276 0.564246
\(977\) 34.8211 1.11403 0.557013 0.830504i \(-0.311948\pi\)
0.557013 + 0.830504i \(0.311948\pi\)
\(978\) −31.2306 −0.998646
\(979\) 0 0
\(980\) 41.4620 1.32445
\(981\) −2.90961 −0.0928966
\(982\) 63.6124 2.02995
\(983\) 14.3619 0.458075 0.229038 0.973418i \(-0.426442\pi\)
0.229038 + 0.973418i \(0.426442\pi\)
\(984\) −4.07917 −0.130039
\(985\) −15.4623 −0.492670
\(986\) 42.2570 1.34574
\(987\) −25.8009 −0.821251
\(988\) −69.4046 −2.20806
\(989\) 7.98346 0.253859
\(990\) 0 0
\(991\) 16.2118 0.514986 0.257493 0.966280i \(-0.417104\pi\)
0.257493 + 0.966280i \(0.417104\pi\)
\(992\) 50.3858 1.59975
\(993\) −1.68842 −0.0535805
\(994\) −107.957 −3.42418
\(995\) 20.3907 0.646430
\(996\) −15.7683 −0.499639
\(997\) −25.9659 −0.822349 −0.411175 0.911557i \(-0.634881\pi\)
−0.411175 + 0.911557i \(0.634881\pi\)
\(998\) −21.3967 −0.677301
\(999\) −6.14344 −0.194370
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 1815.2.a.y.1.5 yes 6
3.2 odd 2 5445.2.a.bz.1.2 6
5.4 even 2 9075.2.a.dq.1.2 6
11.10 odd 2 inner 1815.2.a.y.1.2 6
33.32 even 2 5445.2.a.bz.1.5 6
55.54 odd 2 9075.2.a.dq.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.y.1.2 6 11.10 odd 2 inner
1815.2.a.y.1.5 yes 6 1.1 even 1 trivial
5445.2.a.bz.1.2 6 3.2 odd 2
5445.2.a.bz.1.5 6 33.32 even 2
9075.2.a.dq.1.2 6 5.4 even 2
9075.2.a.dq.1.5 6 55.54 odd 2