Properties

Label 1617.2.a.r.1.1
Level $1617$
Weight $2$
Character 1617.1
Self dual yes
Analytic conductor $12.912$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1617.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.9118100068\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.86081\) of defining polynomial
Character \(\chi\) \(=\) 1617.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.86081 q^{2} +1.00000 q^{3} +1.46260 q^{4} -1.32340 q^{5} -1.86081 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.86081 q^{2} +1.00000 q^{3} +1.46260 q^{4} -1.32340 q^{5} -1.86081 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.46260 q^{10} +1.00000 q^{11} +1.46260 q^{12} -0.398207 q^{13} -1.32340 q^{15} -4.78600 q^{16} +6.64681 q^{17} -1.86081 q^{18} -1.32340 q^{19} -1.93561 q^{20} -1.86081 q^{22} +1.00000 q^{24} -3.24860 q^{25} +0.740987 q^{26} +1.00000 q^{27} +1.47301 q^{29} +2.46260 q^{30} +4.79641 q^{31} +6.90582 q^{32} +1.00000 q^{33} -12.3684 q^{34} +1.46260 q^{36} -1.60179 q^{37} +2.46260 q^{38} -0.398207 q^{39} -1.32340 q^{40} -4.79641 q^{41} -4.79641 q^{43} +1.46260 q^{44} -1.32340 q^{45} -3.04502 q^{47} -4.78600 q^{48} +6.04502 q^{50} +6.64681 q^{51} -0.582418 q^{52} +8.64681 q^{53} -1.86081 q^{54} -1.32340 q^{55} -1.32340 q^{57} -2.74099 q^{58} +13.6918 q^{59} -1.93561 q^{60} +3.20359 q^{61} -8.92520 q^{62} -3.27839 q^{64} +0.526989 q^{65} -1.86081 q^{66} -10.6170 q^{67} +9.72161 q^{68} +15.9404 q^{71} +1.00000 q^{72} -1.45219 q^{73} +2.98062 q^{74} -3.24860 q^{75} -1.93561 q^{76} +0.740987 q^{78} +3.44322 q^{79} +6.33382 q^{80} +1.00000 q^{81} +8.92520 q^{82} +0.796415 q^{83} -8.79641 q^{85} +8.92520 q^{86} +1.47301 q^{87} +1.00000 q^{88} +10.6468 q^{89} +2.46260 q^{90} +4.79641 q^{93} +5.66618 q^{94} +1.75140 q^{95} +6.90582 q^{96} +12.2396 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 2 q^{4} + 4 q^{5} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 2 q^{4} + 4 q^{5} + 3 q^{8} + 3 q^{9} + 5 q^{10} + 3 q^{11} + 2 q^{12} + 2 q^{13} + 4 q^{15} - 4 q^{16} + 4 q^{17} + 4 q^{19} - 5 q^{20} + 3 q^{24} + 3 q^{25} + 11 q^{26} + 3 q^{27} + 6 q^{29} + 5 q^{30} + 8 q^{31} - 4 q^{32} + 3 q^{33} - 10 q^{34} + 2 q^{36} - 8 q^{37} + 5 q^{38} + 2 q^{39} + 4 q^{40} - 8 q^{41} - 8 q^{43} + 2 q^{44} + 4 q^{45} + 10 q^{47} - 4 q^{48} - q^{50} + 4 q^{51} + 15 q^{52} + 10 q^{53} + 4 q^{55} + 4 q^{57} - 17 q^{58} + 6 q^{59} - 5 q^{60} + 16 q^{61} - 22 q^{62} - 21 q^{64} + 8 q^{67} + 18 q^{68} + 3 q^{72} + 2 q^{73} - 11 q^{74} + 3 q^{75} - 5 q^{76} + 11 q^{78} - 12 q^{79} + 15 q^{80} + 3 q^{81} + 22 q^{82} - 4 q^{83} - 20 q^{85} + 22 q^{86} + 6 q^{87} + 3 q^{88} + 16 q^{89} + 5 q^{90} + 8 q^{93} + 21 q^{94} + 18 q^{95} - 4 q^{96} + 8 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.86081 −1.31579 −0.657894 0.753110i \(-0.728553\pi\)
−0.657894 + 0.753110i \(0.728553\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.46260 0.731299
\(5\) −1.32340 −0.591844 −0.295922 0.955212i \(-0.595627\pi\)
−0.295922 + 0.955212i \(0.595627\pi\)
\(6\) −1.86081 −0.759671
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.46260 0.778742
\(11\) 1.00000 0.301511
\(12\) 1.46260 0.422216
\(13\) −0.398207 −0.110443 −0.0552214 0.998474i \(-0.517586\pi\)
−0.0552214 + 0.998474i \(0.517586\pi\)
\(14\) 0 0
\(15\) −1.32340 −0.341702
\(16\) −4.78600 −1.19650
\(17\) 6.64681 1.61209 0.806044 0.591856i \(-0.201604\pi\)
0.806044 + 0.591856i \(0.201604\pi\)
\(18\) −1.86081 −0.438596
\(19\) −1.32340 −0.303610 −0.151805 0.988410i \(-0.548509\pi\)
−0.151805 + 0.988410i \(0.548509\pi\)
\(20\) −1.93561 −0.432815
\(21\) 0 0
\(22\) −1.86081 −0.396725
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) −3.24860 −0.649720
\(26\) 0.740987 0.145319
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.47301 0.273531 0.136766 0.990603i \(-0.456329\pi\)
0.136766 + 0.990603i \(0.456329\pi\)
\(30\) 2.46260 0.449607
\(31\) 4.79641 0.861462 0.430731 0.902480i \(-0.358256\pi\)
0.430731 + 0.902480i \(0.358256\pi\)
\(32\) 6.90582 1.22079
\(33\) 1.00000 0.174078
\(34\) −12.3684 −2.12117
\(35\) 0 0
\(36\) 1.46260 0.243766
\(37\) −1.60179 −0.263333 −0.131667 0.991294i \(-0.542033\pi\)
−0.131667 + 0.991294i \(0.542033\pi\)
\(38\) 2.46260 0.399486
\(39\) −0.398207 −0.0637642
\(40\) −1.32340 −0.209249
\(41\) −4.79641 −0.749074 −0.374537 0.927212i \(-0.622198\pi\)
−0.374537 + 0.927212i \(0.622198\pi\)
\(42\) 0 0
\(43\) −4.79641 −0.731446 −0.365723 0.930724i \(-0.619178\pi\)
−0.365723 + 0.930724i \(0.619178\pi\)
\(44\) 1.46260 0.220495
\(45\) −1.32340 −0.197281
\(46\) 0 0
\(47\) −3.04502 −0.444161 −0.222081 0.975028i \(-0.571285\pi\)
−0.222081 + 0.975028i \(0.571285\pi\)
\(48\) −4.78600 −0.690800
\(49\) 0 0
\(50\) 6.04502 0.854894
\(51\) 6.64681 0.930739
\(52\) −0.582418 −0.0807668
\(53\) 8.64681 1.18773 0.593865 0.804565i \(-0.297601\pi\)
0.593865 + 0.804565i \(0.297601\pi\)
\(54\) −1.86081 −0.253224
\(55\) −1.32340 −0.178448
\(56\) 0 0
\(57\) −1.32340 −0.175289
\(58\) −2.74099 −0.359909
\(59\) 13.6918 1.78252 0.891262 0.453489i \(-0.149821\pi\)
0.891262 + 0.453489i \(0.149821\pi\)
\(60\) −1.93561 −0.249886
\(61\) 3.20359 0.410177 0.205089 0.978743i \(-0.434252\pi\)
0.205089 + 0.978743i \(0.434252\pi\)
\(62\) −8.92520 −1.13350
\(63\) 0 0
\(64\) −3.27839 −0.409799
\(65\) 0.526989 0.0653650
\(66\) −1.86081 −0.229049
\(67\) −10.6170 −1.29708 −0.648538 0.761182i \(-0.724619\pi\)
−0.648538 + 0.761182i \(0.724619\pi\)
\(68\) 9.72161 1.17892
\(69\) 0 0
\(70\) 0 0
\(71\) 15.9404 1.89178 0.945890 0.324487i \(-0.105192\pi\)
0.945890 + 0.324487i \(0.105192\pi\)
\(72\) 1.00000 0.117851
\(73\) −1.45219 −0.169966 −0.0849828 0.996382i \(-0.527084\pi\)
−0.0849828 + 0.996382i \(0.527084\pi\)
\(74\) 2.98062 0.346491
\(75\) −3.24860 −0.375116
\(76\) −1.93561 −0.222030
\(77\) 0 0
\(78\) 0.740987 0.0839002
\(79\) 3.44322 0.387393 0.193696 0.981062i \(-0.437952\pi\)
0.193696 + 0.981062i \(0.437952\pi\)
\(80\) 6.33382 0.708142
\(81\) 1.00000 0.111111
\(82\) 8.92520 0.985623
\(83\) 0.796415 0.0874179 0.0437089 0.999044i \(-0.486083\pi\)
0.0437089 + 0.999044i \(0.486083\pi\)
\(84\) 0 0
\(85\) −8.79641 −0.954105
\(86\) 8.92520 0.962429
\(87\) 1.47301 0.157923
\(88\) 1.00000 0.106600
\(89\) 10.6468 1.12856 0.564280 0.825584i \(-0.309154\pi\)
0.564280 + 0.825584i \(0.309154\pi\)
\(90\) 2.46260 0.259581
\(91\) 0 0
\(92\) 0 0
\(93\) 4.79641 0.497365
\(94\) 5.66618 0.584422
\(95\) 1.75140 0.179690
\(96\) 6.90582 0.704822
\(97\) 12.2396 1.24275 0.621373 0.783515i \(-0.286575\pi\)
0.621373 + 0.783515i \(0.286575\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) −4.75140 −0.475140
\(101\) −16.4972 −1.64153 −0.820766 0.571264i \(-0.806453\pi\)
−0.820766 + 0.571264i \(0.806453\pi\)
\(102\) −12.3684 −1.22466
\(103\) 13.8504 1.36472 0.682360 0.731016i \(-0.260954\pi\)
0.682360 + 0.731016i \(0.260954\pi\)
\(104\) −0.398207 −0.0390475
\(105\) 0 0
\(106\) −16.0900 −1.56280
\(107\) −5.69182 −0.550249 −0.275125 0.961409i \(-0.588719\pi\)
−0.275125 + 0.961409i \(0.588719\pi\)
\(108\) 1.46260 0.140739
\(109\) −4.14961 −0.397460 −0.198730 0.980054i \(-0.563682\pi\)
−0.198730 + 0.980054i \(0.563682\pi\)
\(110\) 2.46260 0.234800
\(111\) −1.60179 −0.152035
\(112\) 0 0
\(113\) 0.946021 0.0889942 0.0444971 0.999010i \(-0.485831\pi\)
0.0444971 + 0.999010i \(0.485831\pi\)
\(114\) 2.46260 0.230643
\(115\) 0 0
\(116\) 2.15442 0.200033
\(117\) −0.398207 −0.0368143
\(118\) −25.4778 −2.34542
\(119\) 0 0
\(120\) −1.32340 −0.120810
\(121\) 1.00000 0.0909091
\(122\) −5.96125 −0.539706
\(123\) −4.79641 −0.432478
\(124\) 7.01523 0.629986
\(125\) 10.9162 0.976378
\(126\) 0 0
\(127\) −4.55678 −0.404349 −0.202174 0.979350i \(-0.564801\pi\)
−0.202174 + 0.979350i \(0.564801\pi\)
\(128\) −7.71120 −0.681580
\(129\) −4.79641 −0.422301
\(130\) −0.980625 −0.0860065
\(131\) −1.05398 −0.0920866 −0.0460433 0.998939i \(-0.514661\pi\)
−0.0460433 + 0.998939i \(0.514661\pi\)
\(132\) 1.46260 0.127303
\(133\) 0 0
\(134\) 19.7562 1.70668
\(135\) −1.32340 −0.113901
\(136\) 6.64681 0.569959
\(137\) 1.44322 0.123303 0.0616514 0.998098i \(-0.480363\pi\)
0.0616514 + 0.998098i \(0.480363\pi\)
\(138\) 0 0
\(139\) 18.5872 1.57655 0.788274 0.615324i \(-0.210975\pi\)
0.788274 + 0.615324i \(0.210975\pi\)
\(140\) 0 0
\(141\) −3.04502 −0.256437
\(142\) −29.6620 −2.48918
\(143\) −0.398207 −0.0332998
\(144\) −4.78600 −0.398834
\(145\) −1.94939 −0.161888
\(146\) 2.70224 0.223639
\(147\) 0 0
\(148\) −2.34278 −0.192575
\(149\) 3.88018 0.317877 0.158938 0.987289i \(-0.449193\pi\)
0.158938 + 0.987289i \(0.449193\pi\)
\(150\) 6.04502 0.493573
\(151\) 10.6468 0.866425 0.433212 0.901292i \(-0.357380\pi\)
0.433212 + 0.901292i \(0.357380\pi\)
\(152\) −1.32340 −0.107342
\(153\) 6.64681 0.537363
\(154\) 0 0
\(155\) −6.34760 −0.509851
\(156\) −0.582418 −0.0466307
\(157\) 16.7368 1.33575 0.667873 0.744276i \(-0.267205\pi\)
0.667873 + 0.744276i \(0.267205\pi\)
\(158\) −6.40717 −0.509727
\(159\) 8.64681 0.685737
\(160\) −9.13919 −0.722517
\(161\) 0 0
\(162\) −1.86081 −0.146199
\(163\) 2.67660 0.209647 0.104824 0.994491i \(-0.466572\pi\)
0.104824 + 0.994491i \(0.466572\pi\)
\(164\) −7.01523 −0.547797
\(165\) −1.32340 −0.103027
\(166\) −1.48197 −0.115023
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) −12.8414 −0.987802
\(170\) 16.3684 1.25540
\(171\) −1.32340 −0.101203
\(172\) −7.01523 −0.534906
\(173\) 12.7368 0.968364 0.484182 0.874967i \(-0.339117\pi\)
0.484182 + 0.874967i \(0.339117\pi\)
\(174\) −2.74099 −0.207794
\(175\) 0 0
\(176\) −4.78600 −0.360759
\(177\) 13.6918 1.02914
\(178\) −19.8116 −1.48495
\(179\) 5.85039 0.437279 0.218639 0.975806i \(-0.429838\pi\)
0.218639 + 0.975806i \(0.429838\pi\)
\(180\) −1.93561 −0.144272
\(181\) 16.7964 1.24847 0.624234 0.781238i \(-0.285411\pi\)
0.624234 + 0.781238i \(0.285411\pi\)
\(182\) 0 0
\(183\) 3.20359 0.236816
\(184\) 0 0
\(185\) 2.11982 0.155852
\(186\) −8.92520 −0.654427
\(187\) 6.64681 0.486063
\(188\) −4.45364 −0.324815
\(189\) 0 0
\(190\) −3.25901 −0.236434
\(191\) −16.7368 −1.21104 −0.605518 0.795832i \(-0.707034\pi\)
−0.605518 + 0.795832i \(0.707034\pi\)
\(192\) −3.27839 −0.236597
\(193\) 9.94043 0.715527 0.357764 0.933812i \(-0.383539\pi\)
0.357764 + 0.933812i \(0.383539\pi\)
\(194\) −22.7756 −1.63519
\(195\) 0.526989 0.0377385
\(196\) 0 0
\(197\) 15.8504 1.12929 0.564647 0.825333i \(-0.309012\pi\)
0.564647 + 0.825333i \(0.309012\pi\)
\(198\) −1.86081 −0.132242
\(199\) −6.38924 −0.452922 −0.226461 0.974020i \(-0.572716\pi\)
−0.226461 + 0.974020i \(0.572716\pi\)
\(200\) −3.24860 −0.229711
\(201\) −10.6170 −0.748867
\(202\) 30.6981 2.15991
\(203\) 0 0
\(204\) 9.72161 0.680649
\(205\) 6.34760 0.443335
\(206\) −25.7729 −1.79568
\(207\) 0 0
\(208\) 1.90582 0.132145
\(209\) −1.32340 −0.0915418
\(210\) 0 0
\(211\) −26.0900 −1.79611 −0.898056 0.439881i \(-0.855021\pi\)
−0.898056 + 0.439881i \(0.855021\pi\)
\(212\) 12.6468 0.868586
\(213\) 15.9404 1.09222
\(214\) 10.5914 0.724012
\(215\) 6.34760 0.432902
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 7.72161 0.522974
\(219\) −1.45219 −0.0981297
\(220\) −1.93561 −0.130499
\(221\) −2.64681 −0.178044
\(222\) 2.98062 0.200046
\(223\) 18.0900 1.21140 0.605699 0.795694i \(-0.292894\pi\)
0.605699 + 0.795694i \(0.292894\pi\)
\(224\) 0 0
\(225\) −3.24860 −0.216573
\(226\) −1.76036 −0.117098
\(227\) 28.1801 1.87038 0.935188 0.354151i \(-0.115230\pi\)
0.935188 + 0.354151i \(0.115230\pi\)
\(228\) −1.93561 −0.128189
\(229\) −8.00000 −0.528655 −0.264327 0.964433i \(-0.585150\pi\)
−0.264327 + 0.964433i \(0.585150\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.47301 0.0967079
\(233\) −21.4432 −1.40479 −0.702396 0.711786i \(-0.747886\pi\)
−0.702396 + 0.711786i \(0.747886\pi\)
\(234\) 0.740987 0.0484398
\(235\) 4.02979 0.262874
\(236\) 20.0256 1.30356
\(237\) 3.44322 0.223661
\(238\) 0 0
\(239\) 11.5422 0.746604 0.373302 0.927710i \(-0.378226\pi\)
0.373302 + 0.927710i \(0.378226\pi\)
\(240\) 6.33382 0.408846
\(241\) −12.9550 −0.834504 −0.417252 0.908791i \(-0.637007\pi\)
−0.417252 + 0.908791i \(0.637007\pi\)
\(242\) −1.86081 −0.119617
\(243\) 1.00000 0.0641500
\(244\) 4.68556 0.299962
\(245\) 0 0
\(246\) 8.92520 0.569050
\(247\) 0.526989 0.0335315
\(248\) 4.79641 0.304573
\(249\) 0.796415 0.0504707
\(250\) −20.3130 −1.28471
\(251\) 28.8954 1.82386 0.911931 0.410343i \(-0.134591\pi\)
0.911931 + 0.410343i \(0.134591\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 8.47928 0.532037
\(255\) −8.79641 −0.550853
\(256\) 20.9058 1.30661
\(257\) 12.2278 0.762748 0.381374 0.924421i \(-0.375451\pi\)
0.381374 + 0.924421i \(0.375451\pi\)
\(258\) 8.92520 0.555658
\(259\) 0 0
\(260\) 0.770774 0.0478014
\(261\) 1.47301 0.0911771
\(262\) 1.96125 0.121166
\(263\) −0.954984 −0.0588868 −0.0294434 0.999566i \(-0.509373\pi\)
−0.0294434 + 0.999566i \(0.509373\pi\)
\(264\) 1.00000 0.0615457
\(265\) −11.4432 −0.702952
\(266\) 0 0
\(267\) 10.6468 0.651574
\(268\) −15.5284 −0.948550
\(269\) −20.2396 −1.23403 −0.617016 0.786950i \(-0.711659\pi\)
−0.617016 + 0.786950i \(0.711659\pi\)
\(270\) 2.46260 0.149869
\(271\) −16.4674 −1.00032 −0.500162 0.865932i \(-0.666726\pi\)
−0.500162 + 0.865932i \(0.666726\pi\)
\(272\) −31.8116 −1.92886
\(273\) 0 0
\(274\) −2.68556 −0.162241
\(275\) −3.24860 −0.195898
\(276\) 0 0
\(277\) 18.1801 1.09233 0.546167 0.837676i \(-0.316086\pi\)
0.546167 + 0.837676i \(0.316086\pi\)
\(278\) −34.5872 −2.07440
\(279\) 4.79641 0.287154
\(280\) 0 0
\(281\) 24.3178 1.45068 0.725339 0.688391i \(-0.241683\pi\)
0.725339 + 0.688391i \(0.241683\pi\)
\(282\) 5.66618 0.337416
\(283\) −21.5035 −1.27825 −0.639124 0.769103i \(-0.720703\pi\)
−0.639124 + 0.769103i \(0.720703\pi\)
\(284\) 23.3144 1.38346
\(285\) 1.75140 0.103744
\(286\) 0.740987 0.0438155
\(287\) 0 0
\(288\) 6.90582 0.406929
\(289\) 27.1801 1.59883
\(290\) 3.62743 0.213010
\(291\) 12.2396 0.717500
\(292\) −2.12397 −0.124296
\(293\) 5.59283 0.326737 0.163368 0.986565i \(-0.447764\pi\)
0.163368 + 0.986565i \(0.447764\pi\)
\(294\) 0 0
\(295\) −18.1198 −1.05498
\(296\) −1.60179 −0.0931023
\(297\) 1.00000 0.0580259
\(298\) −7.22026 −0.418259
\(299\) 0 0
\(300\) −4.75140 −0.274322
\(301\) 0 0
\(302\) −19.8116 −1.14003
\(303\) −16.4972 −0.947740
\(304\) 6.33382 0.363269
\(305\) −4.23964 −0.242761
\(306\) −12.3684 −0.707056
\(307\) −10.5872 −0.604245 −0.302123 0.953269i \(-0.597695\pi\)
−0.302123 + 0.953269i \(0.597695\pi\)
\(308\) 0 0
\(309\) 13.8504 0.787921
\(310\) 11.8116 0.670856
\(311\) 5.59283 0.317140 0.158570 0.987348i \(-0.449312\pi\)
0.158570 + 0.987348i \(0.449312\pi\)
\(312\) −0.398207 −0.0225441
\(313\) 1.90997 0.107958 0.0539789 0.998542i \(-0.482810\pi\)
0.0539789 + 0.998542i \(0.482810\pi\)
\(314\) −31.1440 −1.75756
\(315\) 0 0
\(316\) 5.03605 0.283300
\(317\) 9.44322 0.530384 0.265192 0.964196i \(-0.414565\pi\)
0.265192 + 0.964196i \(0.414565\pi\)
\(318\) −16.0900 −0.902284
\(319\) 1.47301 0.0824728
\(320\) 4.33863 0.242537
\(321\) −5.69182 −0.317687
\(322\) 0 0
\(323\) −8.79641 −0.489446
\(324\) 1.46260 0.0812555
\(325\) 1.29362 0.0717570
\(326\) −4.98062 −0.275851
\(327\) −4.14961 −0.229474
\(328\) −4.79641 −0.264838
\(329\) 0 0
\(330\) 2.46260 0.135562
\(331\) −20.4793 −1.12564 −0.562821 0.826579i \(-0.690284\pi\)
−0.562821 + 0.826579i \(0.690284\pi\)
\(332\) 1.16484 0.0639286
\(333\) −1.60179 −0.0877777
\(334\) −29.7729 −1.62910
\(335\) 14.0506 0.767667
\(336\) 0 0
\(337\) 17.3836 0.946948 0.473474 0.880808i \(-0.343000\pi\)
0.473474 + 0.880808i \(0.343000\pi\)
\(338\) 23.8954 1.29974
\(339\) 0.946021 0.0513808
\(340\) −12.8656 −0.697736
\(341\) 4.79641 0.259740
\(342\) 2.46260 0.133162
\(343\) 0 0
\(344\) −4.79641 −0.258605
\(345\) 0 0
\(346\) −23.7008 −1.27416
\(347\) −30.5872 −1.64201 −0.821004 0.570922i \(-0.806586\pi\)
−0.821004 + 0.570922i \(0.806586\pi\)
\(348\) 2.15442 0.115489
\(349\) −32.8775 −1.75989 −0.879946 0.475074i \(-0.842421\pi\)
−0.879946 + 0.475074i \(0.842421\pi\)
\(350\) 0 0
\(351\) −0.398207 −0.0212547
\(352\) 6.90582 0.368082
\(353\) −33.0063 −1.75675 −0.878373 0.477976i \(-0.841371\pi\)
−0.878373 + 0.477976i \(0.841371\pi\)
\(354\) −25.4778 −1.35413
\(355\) −21.0956 −1.11964
\(356\) 15.5720 0.825315
\(357\) 0 0
\(358\) −10.8864 −0.575367
\(359\) −19.7008 −1.03977 −0.519884 0.854237i \(-0.674025\pi\)
−0.519884 + 0.854237i \(0.674025\pi\)
\(360\) −1.32340 −0.0697495
\(361\) −17.2486 −0.907821
\(362\) −31.2549 −1.64272
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 1.92183 0.100593
\(366\) −5.96125 −0.311600
\(367\) 31.1261 1.62477 0.812384 0.583123i \(-0.198169\pi\)
0.812384 + 0.583123i \(0.198169\pi\)
\(368\) 0 0
\(369\) −4.79641 −0.249691
\(370\) −3.94457 −0.205069
\(371\) 0 0
\(372\) 7.01523 0.363723
\(373\) −8.94602 −0.463207 −0.231604 0.972810i \(-0.574397\pi\)
−0.231604 + 0.972810i \(0.574397\pi\)
\(374\) −12.3684 −0.639556
\(375\) 10.9162 0.563712
\(376\) −3.04502 −0.157035
\(377\) −0.586564 −0.0302096
\(378\) 0 0
\(379\) 4.20985 0.216246 0.108123 0.994138i \(-0.465516\pi\)
0.108123 + 0.994138i \(0.465516\pi\)
\(380\) 2.56159 0.131407
\(381\) −4.55678 −0.233451
\(382\) 31.1440 1.59347
\(383\) −14.5872 −0.745373 −0.372686 0.927957i \(-0.621563\pi\)
−0.372686 + 0.927957i \(0.621563\pi\)
\(384\) −7.71120 −0.393511
\(385\) 0 0
\(386\) −18.4972 −0.941483
\(387\) −4.79641 −0.243815
\(388\) 17.9017 0.908820
\(389\) −30.4793 −1.54536 −0.772680 0.634795i \(-0.781084\pi\)
−0.772680 + 0.634795i \(0.781084\pi\)
\(390\) −0.980625 −0.0496559
\(391\) 0 0
\(392\) 0 0
\(393\) −1.05398 −0.0531662
\(394\) −29.4945 −1.48591
\(395\) −4.55678 −0.229276
\(396\) 1.46260 0.0734983
\(397\) 10.9044 0.547275 0.273637 0.961833i \(-0.411773\pi\)
0.273637 + 0.961833i \(0.411773\pi\)
\(398\) 11.8891 0.595949
\(399\) 0 0
\(400\) 15.5478 0.777391
\(401\) 12.4072 0.619585 0.309792 0.950804i \(-0.399740\pi\)
0.309792 + 0.950804i \(0.399740\pi\)
\(402\) 19.7562 0.985350
\(403\) −1.90997 −0.0951423
\(404\) −24.1288 −1.20045
\(405\) −1.32340 −0.0657605
\(406\) 0 0
\(407\) −1.60179 −0.0793979
\(408\) 6.64681 0.329066
\(409\) −31.3836 −1.55182 −0.775911 0.630843i \(-0.782709\pi\)
−0.775911 + 0.630843i \(0.782709\pi\)
\(410\) −11.8116 −0.583336
\(411\) 1.44322 0.0711890
\(412\) 20.2576 0.998019
\(413\) 0 0
\(414\) 0 0
\(415\) −1.05398 −0.0517378
\(416\) −2.74995 −0.134827
\(417\) 18.5872 0.910221
\(418\) 2.46260 0.120450
\(419\) −30.1711 −1.47395 −0.736977 0.675917i \(-0.763748\pi\)
−0.736977 + 0.675917i \(0.763748\pi\)
\(420\) 0 0
\(421\) −21.2847 −1.03735 −0.518675 0.854971i \(-0.673575\pi\)
−0.518675 + 0.854971i \(0.673575\pi\)
\(422\) 48.5485 2.36330
\(423\) −3.04502 −0.148054
\(424\) 8.64681 0.419926
\(425\) −21.5928 −1.04741
\(426\) −29.6620 −1.43713
\(427\) 0 0
\(428\) −8.32485 −0.402397
\(429\) −0.398207 −0.0192256
\(430\) −11.8116 −0.569608
\(431\) −14.7279 −0.709417 −0.354708 0.934977i \(-0.615420\pi\)
−0.354708 + 0.934977i \(0.615420\pi\)
\(432\) −4.78600 −0.230267
\(433\) 14.1496 0.679987 0.339993 0.940428i \(-0.389575\pi\)
0.339993 + 0.940428i \(0.389575\pi\)
\(434\) 0 0
\(435\) −1.94939 −0.0934660
\(436\) −6.06921 −0.290662
\(437\) 0 0
\(438\) 2.70224 0.129118
\(439\) 26.0007 1.24094 0.620472 0.784228i \(-0.286941\pi\)
0.620472 + 0.784228i \(0.286941\pi\)
\(440\) −1.32340 −0.0630908
\(441\) 0 0
\(442\) 4.92520 0.234268
\(443\) −32.9765 −1.56676 −0.783380 0.621543i \(-0.786506\pi\)
−0.783380 + 0.621543i \(0.786506\pi\)
\(444\) −2.34278 −0.111183
\(445\) −14.0900 −0.667932
\(446\) −33.6620 −1.59394
\(447\) 3.88018 0.183526
\(448\) 0 0
\(449\) 4.96395 0.234263 0.117132 0.993116i \(-0.462630\pi\)
0.117132 + 0.993116i \(0.462630\pi\)
\(450\) 6.04502 0.284965
\(451\) −4.79641 −0.225854
\(452\) 1.38365 0.0650814
\(453\) 10.6468 0.500231
\(454\) −52.4376 −2.46102
\(455\) 0 0
\(456\) −1.32340 −0.0619741
\(457\) −22.1980 −1.03838 −0.519189 0.854659i \(-0.673766\pi\)
−0.519189 + 0.854659i \(0.673766\pi\)
\(458\) 14.8864 0.695598
\(459\) 6.64681 0.310246
\(460\) 0 0
\(461\) −12.5389 −0.583993 −0.291996 0.956419i \(-0.594319\pi\)
−0.291996 + 0.956419i \(0.594319\pi\)
\(462\) 0 0
\(463\) 17.5214 0.814288 0.407144 0.913364i \(-0.366525\pi\)
0.407144 + 0.913364i \(0.366525\pi\)
\(464\) −7.04983 −0.327280
\(465\) −6.34760 −0.294363
\(466\) 39.9017 1.84841
\(467\) −11.7819 −0.545199 −0.272600 0.962128i \(-0.587883\pi\)
−0.272600 + 0.962128i \(0.587883\pi\)
\(468\) −0.582418 −0.0269223
\(469\) 0 0
\(470\) −7.49865 −0.345887
\(471\) 16.7368 0.771193
\(472\) 13.6918 0.630217
\(473\) −4.79641 −0.220539
\(474\) −6.40717 −0.294291
\(475\) 4.29921 0.197261
\(476\) 0 0
\(477\) 8.64681 0.395910
\(478\) −21.4778 −0.982373
\(479\) −5.03605 −0.230103 −0.115052 0.993360i \(-0.536703\pi\)
−0.115052 + 0.993360i \(0.536703\pi\)
\(480\) −9.13919 −0.417145
\(481\) 0.637846 0.0290833
\(482\) 24.1067 1.09803
\(483\) 0 0
\(484\) 1.46260 0.0664817
\(485\) −16.1980 −0.735513
\(486\) −1.86081 −0.0844079
\(487\) 20.1801 0.914446 0.457223 0.889352i \(-0.348844\pi\)
0.457223 + 0.889352i \(0.348844\pi\)
\(488\) 3.20359 0.145019
\(489\) 2.67660 0.121040
\(490\) 0 0
\(491\) −25.3926 −1.14595 −0.572976 0.819572i \(-0.694211\pi\)
−0.572976 + 0.819572i \(0.694211\pi\)
\(492\) −7.01523 −0.316271
\(493\) 9.79082 0.440956
\(494\) −0.980625 −0.0441204
\(495\) −1.32340 −0.0594826
\(496\) −22.9557 −1.03074
\(497\) 0 0
\(498\) −1.48197 −0.0664088
\(499\) 14.9162 0.667742 0.333871 0.942619i \(-0.391645\pi\)
0.333871 + 0.942619i \(0.391645\pi\)
\(500\) 15.9661 0.714024
\(501\) 16.0000 0.714827
\(502\) −53.7687 −2.39982
\(503\) −28.9765 −1.29200 −0.645999 0.763339i \(-0.723559\pi\)
−0.645999 + 0.763339i \(0.723559\pi\)
\(504\) 0 0
\(505\) 21.8325 0.971532
\(506\) 0 0
\(507\) −12.8414 −0.570308
\(508\) −6.66473 −0.295700
\(509\) 1.05398 0.0467168 0.0233584 0.999727i \(-0.492564\pi\)
0.0233584 + 0.999727i \(0.492564\pi\)
\(510\) 16.3684 0.724806
\(511\) 0 0
\(512\) −23.4793 −1.03765
\(513\) −1.32340 −0.0584297
\(514\) −22.7535 −1.00361
\(515\) −18.3297 −0.807702
\(516\) −7.01523 −0.308828
\(517\) −3.04502 −0.133920
\(518\) 0 0
\(519\) 12.7368 0.559085
\(520\) 0.526989 0.0231100
\(521\) 14.2999 0.626489 0.313245 0.949672i \(-0.398584\pi\)
0.313245 + 0.949672i \(0.398584\pi\)
\(522\) −2.74099 −0.119970
\(523\) −34.8567 −1.52418 −0.762088 0.647474i \(-0.775825\pi\)
−0.762088 + 0.647474i \(0.775825\pi\)
\(524\) −1.54155 −0.0673428
\(525\) 0 0
\(526\) 1.77704 0.0774826
\(527\) 31.8809 1.38875
\(528\) −4.78600 −0.208284
\(529\) −23.0000 −1.00000
\(530\) 21.2936 0.924936
\(531\) 13.6918 0.594175
\(532\) 0 0
\(533\) 1.90997 0.0827299
\(534\) −19.8116 −0.857334
\(535\) 7.53258 0.325662
\(536\) −10.6170 −0.458585
\(537\) 5.85039 0.252463
\(538\) 37.6620 1.62373
\(539\) 0 0
\(540\) −1.93561 −0.0832953
\(541\) 39.2161 1.68603 0.843016 0.537888i \(-0.180778\pi\)
0.843016 + 0.537888i \(0.180778\pi\)
\(542\) 30.6427 1.31622
\(543\) 16.7964 0.720803
\(544\) 45.9017 1.96802
\(545\) 5.49161 0.235235
\(546\) 0 0
\(547\) −38.5277 −1.64732 −0.823662 0.567081i \(-0.808073\pi\)
−0.823662 + 0.567081i \(0.808073\pi\)
\(548\) 2.11086 0.0901713
\(549\) 3.20359 0.136726
\(550\) 6.04502 0.257760
\(551\) −1.94939 −0.0830467
\(552\) 0 0
\(553\) 0 0
\(554\) −33.8296 −1.43728
\(555\) 2.11982 0.0899813
\(556\) 27.1857 1.15293
\(557\) 43.7610 1.85421 0.927107 0.374796i \(-0.122287\pi\)
0.927107 + 0.374796i \(0.122287\pi\)
\(558\) −8.92520 −0.377834
\(559\) 1.90997 0.0807830
\(560\) 0 0
\(561\) 6.64681 0.280628
\(562\) −45.2507 −1.90879
\(563\) −19.9821 −0.842144 −0.421072 0.907027i \(-0.638346\pi\)
−0.421072 + 0.907027i \(0.638346\pi\)
\(564\) −4.45364 −0.187532
\(565\) −1.25197 −0.0526707
\(566\) 40.0138 1.68190
\(567\) 0 0
\(568\) 15.9404 0.668845
\(569\) 31.7312 1.33024 0.665121 0.746735i \(-0.268380\pi\)
0.665121 + 0.746735i \(0.268380\pi\)
\(570\) −3.25901 −0.136505
\(571\) −10.9460 −0.458077 −0.229038 0.973417i \(-0.573558\pi\)
−0.229038 + 0.973417i \(0.573558\pi\)
\(572\) −0.582418 −0.0243521
\(573\) −16.7368 −0.699192
\(574\) 0 0
\(575\) 0 0
\(576\) −3.27839 −0.136600
\(577\) −3.24523 −0.135101 −0.0675504 0.997716i \(-0.521518\pi\)
−0.0675504 + 0.997716i \(0.521518\pi\)
\(578\) −50.5768 −2.10372
\(579\) 9.94043 0.413110
\(580\) −2.85117 −0.118388
\(581\) 0 0
\(582\) −22.7756 −0.944079
\(583\) 8.64681 0.358114
\(584\) −1.45219 −0.0600919
\(585\) 0.526989 0.0217883
\(586\) −10.4072 −0.429916
\(587\) 26.7875 1.10564 0.552818 0.833302i \(-0.313552\pi\)
0.552818 + 0.833302i \(0.313552\pi\)
\(588\) 0 0
\(589\) −6.34760 −0.261548
\(590\) 33.7175 1.38813
\(591\) 15.8504 0.651998
\(592\) 7.66618 0.315078
\(593\) 32.2396 1.32392 0.661962 0.749538i \(-0.269724\pi\)
0.661962 + 0.749538i \(0.269724\pi\)
\(594\) −1.86081 −0.0763498
\(595\) 0 0
\(596\) 5.67515 0.232463
\(597\) −6.38924 −0.261494
\(598\) 0 0
\(599\) −7.14401 −0.291896 −0.145948 0.989292i \(-0.546623\pi\)
−0.145948 + 0.989292i \(0.546623\pi\)
\(600\) −3.24860 −0.132624
\(601\) −21.7514 −0.887258 −0.443629 0.896211i \(-0.646309\pi\)
−0.443629 + 0.896211i \(0.646309\pi\)
\(602\) 0 0
\(603\) −10.6170 −0.432359
\(604\) 15.5720 0.633616
\(605\) −1.32340 −0.0538040
\(606\) 30.6981 1.24702
\(607\) 39.4134 1.59974 0.799871 0.600172i \(-0.204901\pi\)
0.799871 + 0.600172i \(0.204901\pi\)
\(608\) −9.13919 −0.370643
\(609\) 0 0
\(610\) 7.88914 0.319422
\(611\) 1.21255 0.0490544
\(612\) 9.72161 0.392973
\(613\) −22.1980 −0.896568 −0.448284 0.893891i \(-0.647965\pi\)
−0.448284 + 0.893891i \(0.647965\pi\)
\(614\) 19.7008 0.795059
\(615\) 6.34760 0.255960
\(616\) 0 0
\(617\) −37.4432 −1.50741 −0.753704 0.657214i \(-0.771735\pi\)
−0.753704 + 0.657214i \(0.771735\pi\)
\(618\) −25.7729 −1.03674
\(619\) 15.7008 0.631068 0.315534 0.948914i \(-0.397816\pi\)
0.315534 + 0.948914i \(0.397816\pi\)
\(620\) −9.28398 −0.372854
\(621\) 0 0
\(622\) −10.4072 −0.417290
\(623\) 0 0
\(624\) 1.90582 0.0762939
\(625\) 1.79641 0.0718566
\(626\) −3.55408 −0.142050
\(627\) −1.32340 −0.0528517
\(628\) 24.4793 0.976829
\(629\) −10.6468 −0.424516
\(630\) 0 0
\(631\) 35.0665 1.39598 0.697988 0.716110i \(-0.254079\pi\)
0.697988 + 0.716110i \(0.254079\pi\)
\(632\) 3.44322 0.136964
\(633\) −26.0900 −1.03699
\(634\) −17.5720 −0.697873
\(635\) 6.03046 0.239311
\(636\) 12.6468 0.501479
\(637\) 0 0
\(638\) −2.74099 −0.108517
\(639\) 15.9404 0.630593
\(640\) 10.2050 0.403389
\(641\) −30.1801 −1.19204 −0.596020 0.802969i \(-0.703252\pi\)
−0.596020 + 0.802969i \(0.703252\pi\)
\(642\) 10.5914 0.418008
\(643\) 36.7964 1.45111 0.725554 0.688165i \(-0.241583\pi\)
0.725554 + 0.688165i \(0.241583\pi\)
\(644\) 0 0
\(645\) 6.34760 0.249936
\(646\) 16.3684 0.644007
\(647\) 33.4522 1.31514 0.657571 0.753393i \(-0.271584\pi\)
0.657571 + 0.753393i \(0.271584\pi\)
\(648\) 1.00000 0.0392837
\(649\) 13.6918 0.537451
\(650\) −2.40717 −0.0944170
\(651\) 0 0
\(652\) 3.91478 0.153315
\(653\) 5.50280 0.215341 0.107671 0.994187i \(-0.465661\pi\)
0.107671 + 0.994187i \(0.465661\pi\)
\(654\) 7.72161 0.301939
\(655\) 1.39484 0.0545009
\(656\) 22.9557 0.896268
\(657\) −1.45219 −0.0566552
\(658\) 0 0
\(659\) 33.1946 1.29308 0.646539 0.762881i \(-0.276216\pi\)
0.646539 + 0.762881i \(0.276216\pi\)
\(660\) −1.93561 −0.0753435
\(661\) −43.3241 −1.68511 −0.842556 0.538609i \(-0.818950\pi\)
−0.842556 + 0.538609i \(0.818950\pi\)
\(662\) 38.1080 1.48111
\(663\) −2.64681 −0.102794
\(664\) 0.796415 0.0309069
\(665\) 0 0
\(666\) 2.98062 0.115497
\(667\) 0 0
\(668\) 23.4016 0.905434
\(669\) 18.0900 0.699401
\(670\) −26.1455 −1.01009
\(671\) 3.20359 0.123673
\(672\) 0 0
\(673\) −28.0721 −1.08210 −0.541050 0.840990i \(-0.681973\pi\)
−0.541050 + 0.840990i \(0.681973\pi\)
\(674\) −32.3476 −1.24598
\(675\) −3.24860 −0.125039
\(676\) −18.7819 −0.722379
\(677\) −22.8448 −0.877997 −0.438998 0.898488i \(-0.644667\pi\)
−0.438998 + 0.898488i \(0.644667\pi\)
\(678\) −1.76036 −0.0676063
\(679\) 0 0
\(680\) −8.79641 −0.337327
\(681\) 28.1801 1.07986
\(682\) −8.92520 −0.341763
\(683\) 11.4016 0.436269 0.218135 0.975919i \(-0.430003\pi\)
0.218135 + 0.975919i \(0.430003\pi\)
\(684\) −1.93561 −0.0740099
\(685\) −1.90997 −0.0729761
\(686\) 0 0
\(687\) −8.00000 −0.305219
\(688\) 22.9557 0.875176
\(689\) −3.44322 −0.131176
\(690\) 0 0
\(691\) 11.9404 0.454235 0.227118 0.973867i \(-0.427070\pi\)
0.227118 + 0.973867i \(0.427070\pi\)
\(692\) 18.6289 0.708164
\(693\) 0 0
\(694\) 56.9169 2.16054
\(695\) −24.5984 −0.933071
\(696\) 1.47301 0.0558343
\(697\) −31.8809 −1.20757
\(698\) 61.1786 2.31564
\(699\) −21.4432 −0.811057
\(700\) 0 0
\(701\) 17.1440 0.647520 0.323760 0.946139i \(-0.395053\pi\)
0.323760 + 0.946139i \(0.395053\pi\)
\(702\) 0.740987 0.0279667
\(703\) 2.11982 0.0799505
\(704\) −3.27839 −0.123559
\(705\) 4.02979 0.151771
\(706\) 61.4183 2.31151
\(707\) 0 0
\(708\) 20.0256 0.752610
\(709\) −50.7583 −1.90627 −0.953135 0.302546i \(-0.902163\pi\)
−0.953135 + 0.302546i \(0.902163\pi\)
\(710\) 39.2549 1.47321
\(711\) 3.44322 0.129131
\(712\) 10.6468 0.399006
\(713\) 0 0
\(714\) 0 0
\(715\) 0.526989 0.0197083
\(716\) 8.55678 0.319782
\(717\) 11.5422 0.431052
\(718\) 36.6593 1.36811
\(719\) −4.35656 −0.162472 −0.0812361 0.996695i \(-0.525887\pi\)
−0.0812361 + 0.996695i \(0.525887\pi\)
\(720\) 6.33382 0.236047
\(721\) 0 0
\(722\) 32.0963 1.19450
\(723\) −12.9550 −0.481801
\(724\) 24.5664 0.913003
\(725\) −4.78522 −0.177719
\(726\) −1.86081 −0.0690610
\(727\) 33.1745 1.23037 0.615186 0.788382i \(-0.289081\pi\)
0.615186 + 0.788382i \(0.289081\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −3.57615 −0.132359
\(731\) −31.8809 −1.17916
\(732\) 4.68556 0.173183
\(733\) 31.3836 1.15918 0.579591 0.814908i \(-0.303212\pi\)
0.579591 + 0.814908i \(0.303212\pi\)
\(734\) −57.9196 −2.13785
\(735\) 0 0
\(736\) 0 0
\(737\) −10.6170 −0.391083
\(738\) 8.92520 0.328541
\(739\) 29.4737 1.08421 0.542103 0.840312i \(-0.317628\pi\)
0.542103 + 0.840312i \(0.317628\pi\)
\(740\) 3.10044 0.113975
\(741\) 0.526989 0.0193594
\(742\) 0 0
\(743\) −15.8414 −0.581166 −0.290583 0.956850i \(-0.593849\pi\)
−0.290583 + 0.956850i \(0.593849\pi\)
\(744\) 4.79641 0.175845
\(745\) −5.13505 −0.188134
\(746\) 16.6468 0.609483
\(747\) 0.796415 0.0291393
\(748\) 9.72161 0.355457
\(749\) 0 0
\(750\) −20.3130 −0.741726
\(751\) 44.1919 1.61259 0.806293 0.591516i \(-0.201470\pi\)
0.806293 + 0.591516i \(0.201470\pi\)
\(752\) 14.5735 0.531439
\(753\) 28.8954 1.05301
\(754\) 1.09148 0.0397494
\(755\) −14.0900 −0.512789
\(756\) 0 0
\(757\) 44.1711 1.60543 0.802713 0.596366i \(-0.203389\pi\)
0.802713 + 0.596366i \(0.203389\pi\)
\(758\) −7.83372 −0.284533
\(759\) 0 0
\(760\) 1.75140 0.0635299
\(761\) 41.0361 1.48756 0.743778 0.668427i \(-0.233032\pi\)
0.743778 + 0.668427i \(0.233032\pi\)
\(762\) 8.47928 0.307172
\(763\) 0 0
\(764\) −24.4793 −0.885629
\(765\) −8.79641 −0.318035
\(766\) 27.1440 0.980753
\(767\) −5.45219 −0.196867
\(768\) 20.9058 0.754374
\(769\) −52.8358 −1.90531 −0.952654 0.304055i \(-0.901659\pi\)
−0.952654 + 0.304055i \(0.901659\pi\)
\(770\) 0 0
\(771\) 12.2278 0.440373
\(772\) 14.5389 0.523265
\(773\) −32.0907 −1.15422 −0.577111 0.816666i \(-0.695820\pi\)
−0.577111 + 0.816666i \(0.695820\pi\)
\(774\) 8.92520 0.320810
\(775\) −15.5816 −0.559709
\(776\) 12.2396 0.439377
\(777\) 0 0
\(778\) 56.7160 2.03337
\(779\) 6.34760 0.227426
\(780\) 0.770774 0.0275981
\(781\) 15.9404 0.570393
\(782\) 0 0
\(783\) 1.47301 0.0526411
\(784\) 0 0
\(785\) −22.1496 −0.790553
\(786\) 1.96125 0.0699555
\(787\) −2.37738 −0.0847446 −0.0423723 0.999102i \(-0.513492\pi\)
−0.0423723 + 0.999102i \(0.513492\pi\)
\(788\) 23.1828 0.825852
\(789\) −0.954984 −0.0339983
\(790\) 8.47928 0.301679
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) −1.27569 −0.0453011
\(794\) −20.2909 −0.720098
\(795\) −11.4432 −0.405849
\(796\) −9.34490 −0.331221
\(797\) 20.5091 0.726468 0.363234 0.931698i \(-0.381673\pi\)
0.363234 + 0.931698i \(0.381673\pi\)
\(798\) 0 0
\(799\) −20.2396 −0.716027
\(800\) −22.4343 −0.793171
\(801\) 10.6468 0.376186
\(802\) −23.0873 −0.815242
\(803\) −1.45219 −0.0512465
\(804\) −15.5284 −0.547646
\(805\) 0 0
\(806\) 3.55408 0.125187
\(807\) −20.2396 −0.712469
\(808\) −16.4972 −0.580370
\(809\) −8.85666 −0.311384 −0.155692 0.987806i \(-0.549761\pi\)
−0.155692 + 0.987806i \(0.549761\pi\)
\(810\) 2.46260 0.0865269
\(811\) −8.26943 −0.290379 −0.145189 0.989404i \(-0.546379\pi\)
−0.145189 + 0.989404i \(0.546379\pi\)
\(812\) 0 0
\(813\) −16.4674 −0.577537
\(814\) 2.98062 0.104471
\(815\) −3.54222 −0.124078
\(816\) −31.8116 −1.11363
\(817\) 6.34760 0.222074
\(818\) 58.3989 2.04187
\(819\) 0 0
\(820\) 9.28398 0.324211
\(821\) −32.3595 −1.12935 −0.564676 0.825312i \(-0.690999\pi\)
−0.564676 + 0.825312i \(0.690999\pi\)
\(822\) −2.68556 −0.0936696
\(823\) −45.2459 −1.57717 −0.788587 0.614924i \(-0.789187\pi\)
−0.788587 + 0.614924i \(0.789187\pi\)
\(824\) 13.8504 0.482501
\(825\) −3.24860 −0.113102
\(826\) 0 0
\(827\) 20.8954 0.726605 0.363302 0.931671i \(-0.381649\pi\)
0.363302 + 0.931671i \(0.381649\pi\)
\(828\) 0 0
\(829\) −44.6952 −1.55233 −0.776164 0.630531i \(-0.782837\pi\)
−0.776164 + 0.630531i \(0.782837\pi\)
\(830\) 1.96125 0.0680760
\(831\) 18.1801 0.630659
\(832\) 1.30548 0.0452593
\(833\) 0 0
\(834\) −34.5872 −1.19766
\(835\) −21.1745 −0.732773
\(836\) −1.93561 −0.0669444
\(837\) 4.79641 0.165788
\(838\) 56.1426 1.93941
\(839\) −0.457782 −0.0158044 −0.00790219 0.999969i \(-0.502515\pi\)
−0.00790219 + 0.999969i \(0.502515\pi\)
\(840\) 0 0
\(841\) −26.8302 −0.925181
\(842\) 39.6066 1.36493
\(843\) 24.3178 0.837550
\(844\) −38.1592 −1.31350
\(845\) 16.9944 0.584625
\(846\) 5.66618 0.194807
\(847\) 0 0
\(848\) −41.3836 −1.42112
\(849\) −21.5035 −0.737997
\(850\) 40.1801 1.37816
\(851\) 0 0
\(852\) 23.3144 0.798740
\(853\) −26.0900 −0.893306 −0.446653 0.894707i \(-0.647384\pi\)
−0.446653 + 0.894707i \(0.647384\pi\)
\(854\) 0 0
\(855\) 1.75140 0.0598966
\(856\) −5.69182 −0.194543
\(857\) 6.38924 0.218252 0.109126 0.994028i \(-0.465195\pi\)
0.109126 + 0.994028i \(0.465195\pi\)
\(858\) 0.740987 0.0252969
\(859\) −3.26316 −0.111338 −0.0556688 0.998449i \(-0.517729\pi\)
−0.0556688 + 0.998449i \(0.517729\pi\)
\(860\) 9.28398 0.316581
\(861\) 0 0
\(862\) 27.4057 0.933443
\(863\) −12.6164 −0.429466 −0.214733 0.976673i \(-0.568888\pi\)
−0.214733 + 0.976673i \(0.568888\pi\)
\(864\) 6.90582 0.234941
\(865\) −16.8560 −0.573121
\(866\) −26.3297 −0.894719
\(867\) 27.1801 0.923083
\(868\) 0 0
\(869\) 3.44322 0.116803
\(870\) 3.62743 0.122982
\(871\) 4.22778 0.143253
\(872\) −4.14961 −0.140523
\(873\) 12.2396 0.414249
\(874\) 0 0
\(875\) 0 0
\(876\) −2.12397 −0.0717621
\(877\) −22.7964 −0.769780 −0.384890 0.922962i \(-0.625761\pi\)
−0.384890 + 0.922962i \(0.625761\pi\)
\(878\) −48.3822 −1.63282
\(879\) 5.59283 0.188641
\(880\) 6.33382 0.213513
\(881\) −15.9881 −0.538654 −0.269327 0.963049i \(-0.586801\pi\)
−0.269327 + 0.963049i \(0.586801\pi\)
\(882\) 0 0
\(883\) 30.9162 1.04041 0.520207 0.854040i \(-0.325855\pi\)
0.520207 + 0.854040i \(0.325855\pi\)
\(884\) −3.87122 −0.130203
\(885\) −18.1198 −0.609091
\(886\) 61.3628 2.06152
\(887\) 27.7008 0.930101 0.465051 0.885284i \(-0.346036\pi\)
0.465051 + 0.885284i \(0.346036\pi\)
\(888\) −1.60179 −0.0537526
\(889\) 0 0
\(890\) 26.2188 0.878857
\(891\) 1.00000 0.0335013
\(892\) 26.4585 0.885895
\(893\) 4.02979 0.134852
\(894\) −7.22026 −0.241482
\(895\) −7.74244 −0.258801
\(896\) 0 0
\(897\) 0 0
\(898\) −9.23694 −0.308241
\(899\) 7.06517 0.235637
\(900\) −4.75140 −0.158380
\(901\) 57.4737 1.91473
\(902\) 8.92520 0.297177
\(903\) 0 0
\(904\) 0.946021 0.0314642
\(905\) −22.2284 −0.738899
\(906\) −19.8116 −0.658198
\(907\) 17.2936 0.574225 0.287113 0.957897i \(-0.407305\pi\)
0.287113 + 0.957897i \(0.407305\pi\)
\(908\) 41.2161 1.36780
\(909\) −16.4972 −0.547178
\(910\) 0 0
\(911\) 8.85599 0.293412 0.146706 0.989180i \(-0.453133\pi\)
0.146706 + 0.989180i \(0.453133\pi\)
\(912\) 6.33382 0.209734
\(913\) 0.796415 0.0263575
\(914\) 41.3061 1.36629
\(915\) −4.23964 −0.140158
\(916\) −11.7008 −0.386605
\(917\) 0 0
\(918\) −12.3684 −0.408219
\(919\) 54.0305 1.78230 0.891150 0.453708i \(-0.149899\pi\)
0.891150 + 0.453708i \(0.149899\pi\)
\(920\) 0 0
\(921\) −10.5872 −0.348861
\(922\) 23.3324 0.768411
\(923\) −6.34760 −0.208934
\(924\) 0 0
\(925\) 5.20359 0.171093
\(926\) −32.6039 −1.07143
\(927\) 13.8504 0.454907
\(928\) 10.1723 0.333924
\(929\) 20.7666 0.681331 0.340665 0.940185i \(-0.389348\pi\)
0.340665 + 0.940185i \(0.389348\pi\)
\(930\) 11.8116 0.387319
\(931\) 0 0
\(932\) −31.3628 −1.02732
\(933\) 5.59283 0.183101
\(934\) 21.9237 0.717367
\(935\) −8.79641 −0.287674
\(936\) −0.398207 −0.0130158
\(937\) −3.20359 −0.104657 −0.0523283 0.998630i \(-0.516664\pi\)
−0.0523283 + 0.998630i \(0.516664\pi\)
\(938\) 0 0
\(939\) 1.90997 0.0623295
\(940\) 5.89396 0.192240
\(941\) −43.0249 −1.40257 −0.701285 0.712881i \(-0.747390\pi\)
−0.701285 + 0.712881i \(0.747390\pi\)
\(942\) −31.1440 −1.01473
\(943\) 0 0
\(944\) −65.5291 −2.13279
\(945\) 0 0
\(946\) 8.92520 0.290183
\(947\) 17.5749 0.571108 0.285554 0.958363i \(-0.407822\pi\)
0.285554 + 0.958363i \(0.407822\pi\)
\(948\) 5.03605 0.163563
\(949\) 0.578271 0.0187715
\(950\) −8.00000 −0.259554
\(951\) 9.44322 0.306217
\(952\) 0 0
\(953\) −14.7846 −0.478919 −0.239459 0.970906i \(-0.576970\pi\)
−0.239459 + 0.970906i \(0.576970\pi\)
\(954\) −16.0900 −0.520934
\(955\) 22.1496 0.716744
\(956\) 16.8816 0.545991
\(957\) 1.47301 0.0476157
\(958\) 9.37112 0.302767
\(959\) 0 0
\(960\) 4.33863 0.140029
\(961\) −7.99440 −0.257884
\(962\) −1.18691 −0.0382674
\(963\) −5.69182 −0.183416
\(964\) −18.9479 −0.610272
\(965\) −13.1552 −0.423481
\(966\) 0 0
\(967\) −23.1440 −0.744261 −0.372131 0.928180i \(-0.621373\pi\)
−0.372131 + 0.928180i \(0.621373\pi\)
\(968\) 1.00000 0.0321412
\(969\) −8.79641 −0.282582
\(970\) 30.1413 0.967779
\(971\) −36.5783 −1.17385 −0.586926 0.809640i \(-0.699662\pi\)
−0.586926 + 0.809640i \(0.699662\pi\)
\(972\) 1.46260 0.0469129
\(973\) 0 0
\(974\) −37.5512 −1.20322
\(975\) 1.29362 0.0414289
\(976\) −15.3324 −0.490777
\(977\) 54.1621 1.73280 0.866400 0.499350i \(-0.166428\pi\)
0.866400 + 0.499350i \(0.166428\pi\)
\(978\) −4.98062 −0.159263
\(979\) 10.6468 0.340273
\(980\) 0 0
\(981\) −4.14961 −0.132487
\(982\) 47.2507 1.50783
\(983\) −1.29362 −0.0412600 −0.0206300 0.999787i \(-0.506567\pi\)
−0.0206300 + 0.999787i \(0.506567\pi\)
\(984\) −4.79641 −0.152904
\(985\) −20.9765 −0.668366
\(986\) −18.2188 −0.580205
\(987\) 0 0
\(988\) 0.770774 0.0245216
\(989\) 0 0
\(990\) 2.46260 0.0782665
\(991\) −15.9523 −0.506741 −0.253371 0.967369i \(-0.581539\pi\)
−0.253371 + 0.967369i \(0.581539\pi\)
\(992\) 33.1232 1.05166
\(993\) −20.4793 −0.649890
\(994\) 0 0
\(995\) 8.45555 0.268059
\(996\) 1.16484 0.0369092
\(997\) 23.3836 0.740568 0.370284 0.928919i \(-0.379260\pi\)
0.370284 + 0.928919i \(0.379260\pi\)
\(998\) −27.7562 −0.878608
\(999\) −1.60179 −0.0506785
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 1617.2.a.r.1.1 yes 3
3.2 odd 2 4851.2.a.bl.1.3 3
7.6 odd 2 1617.2.a.q.1.1 3
21.20 even 2 4851.2.a.bm.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.2.a.q.1.1 3 7.6 odd 2
1617.2.a.r.1.1 yes 3 1.1 even 1 trivial
4851.2.a.bl.1.3 3 3.2 odd 2
4851.2.a.bm.1.3 3 21.20 even 2