Properties

Label 3969.2.a.bc.1.1
Level $3969$
Weight $2$
Character 3969.1
Self dual yes
Analytic conductor $31.693$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3969,2,Mod(1,3969)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3969, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3969.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3969.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.6926245622\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.574857.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 9x^{2} + 3x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.05365\) of defining polynomial
Character \(\chi\) \(=\) 3969.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.05365 q^{2} +2.21746 q^{4} +0.146246 q^{5} -0.446582 q^{8} +O(q^{10})\) \(q-2.05365 q^{2} +2.21746 q^{4} +0.146246 q^{5} -0.446582 q^{8} -0.300337 q^{10} -1.66404 q^{11} -0.199891 q^{13} -3.51780 q^{16} +6.27110 q^{17} -6.91758 q^{19} +0.324294 q^{20} +3.41735 q^{22} +6.18184 q^{23} -4.97861 q^{25} +0.410505 q^{26} -4.93514 q^{29} -2.51780 q^{31} +8.11747 q^{32} -12.8786 q^{34} +7.00046 q^{37} +14.2062 q^{38} -0.0653107 q^{40} +2.31790 q^{41} +1.88199 q^{43} -3.68994 q^{44} -12.6953 q^{46} +1.81177 q^{47} +10.2243 q^{50} -0.443250 q^{52} -5.34614 q^{53} -0.243359 q^{55} +10.1350 q^{58} +4.57099 q^{59} -0.678276 q^{61} +5.17066 q^{62} -9.63481 q^{64} -0.0292332 q^{65} -6.18684 q^{67} +13.9059 q^{68} -1.27749 q^{71} +1.55721 q^{73} -14.3765 q^{74} -15.3394 q^{76} +12.7957 q^{79} -0.514462 q^{80} -4.76015 q^{82} +7.51374 q^{83} +0.917122 q^{85} -3.86493 q^{86} +0.743131 q^{88} +9.06788 q^{89} +13.7080 q^{92} -3.72074 q^{94} -1.01167 q^{95} +7.97028 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + 4 q^{4} + 4 q^{5} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{2} + 4 q^{4} + 4 q^{5} + 3 q^{8} + 7 q^{10} + 4 q^{11} + 8 q^{13} - 2 q^{16} + 12 q^{17} - q^{19} + 5 q^{20} + q^{22} + 3 q^{23} + q^{25} + 11 q^{26} + 7 q^{29} + 3 q^{31} - 2 q^{32} - 3 q^{34} + 20 q^{38} + 3 q^{40} + 5 q^{41} + 7 q^{43} - 10 q^{44} - 3 q^{46} + 27 q^{47} + 19 q^{50} + 10 q^{52} - 21 q^{53} + 2 q^{55} + 10 q^{58} + 30 q^{59} + 14 q^{61} + 6 q^{62} - 25 q^{64} - 11 q^{65} + 2 q^{67} + 27 q^{68} + 3 q^{71} - 15 q^{73} - 36 q^{74} - 5 q^{76} + 4 q^{79} + 20 q^{80} + 5 q^{82} + 9 q^{83} + 6 q^{85} - 8 q^{86} + 18 q^{88} + 28 q^{89} + 27 q^{92} + 3 q^{94} - 14 q^{95} + 12 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.05365 −1.45215 −0.726073 0.687617i \(-0.758657\pi\)
−0.726073 + 0.687617i \(0.758657\pi\)
\(3\) 0 0
\(4\) 2.21746 1.10873
\(5\) 0.146246 0.0654030 0.0327015 0.999465i \(-0.489589\pi\)
0.0327015 + 0.999465i \(0.489589\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −0.446582 −0.157891
\(9\) 0 0
\(10\) −0.300337 −0.0949748
\(11\) −1.66404 −0.501727 −0.250864 0.968022i \(-0.580715\pi\)
−0.250864 + 0.968022i \(0.580715\pi\)
\(12\) 0 0
\(13\) −0.199891 −0.0554397 −0.0277199 0.999616i \(-0.508825\pi\)
−0.0277199 + 0.999616i \(0.508825\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −3.51780 −0.879449
\(17\) 6.27110 1.52097 0.760483 0.649358i \(-0.224962\pi\)
0.760483 + 0.649358i \(0.224962\pi\)
\(18\) 0 0
\(19\) −6.91758 −1.58700 −0.793500 0.608570i \(-0.791744\pi\)
−0.793500 + 0.608570i \(0.791744\pi\)
\(20\) 0.324294 0.0725143
\(21\) 0 0
\(22\) 3.41735 0.728581
\(23\) 6.18184 1.28900 0.644501 0.764604i \(-0.277065\pi\)
0.644501 + 0.764604i \(0.277065\pi\)
\(24\) 0 0
\(25\) −4.97861 −0.995722
\(26\) 0.410505 0.0805066
\(27\) 0 0
\(28\) 0 0
\(29\) −4.93514 −0.916433 −0.458217 0.888841i \(-0.651512\pi\)
−0.458217 + 0.888841i \(0.651512\pi\)
\(30\) 0 0
\(31\) −2.51780 −0.452209 −0.226105 0.974103i \(-0.572599\pi\)
−0.226105 + 0.974103i \(0.572599\pi\)
\(32\) 8.11747 1.43498
\(33\) 0 0
\(34\) −12.8786 −2.20867
\(35\) 0 0
\(36\) 0 0
\(37\) 7.00046 1.15087 0.575434 0.817848i \(-0.304833\pi\)
0.575434 + 0.817848i \(0.304833\pi\)
\(38\) 14.2062 2.30456
\(39\) 0 0
\(40\) −0.0653107 −0.0103265
\(41\) 2.31790 0.361996 0.180998 0.983483i \(-0.442067\pi\)
0.180998 + 0.983483i \(0.442067\pi\)
\(42\) 0 0
\(43\) 1.88199 0.287000 0.143500 0.989650i \(-0.454164\pi\)
0.143500 + 0.989650i \(0.454164\pi\)
\(44\) −3.68994 −0.556280
\(45\) 0 0
\(46\) −12.6953 −1.87182
\(47\) 1.81177 0.264275 0.132137 0.991231i \(-0.457816\pi\)
0.132137 + 0.991231i \(0.457816\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 10.2243 1.44593
\(51\) 0 0
\(52\) −0.443250 −0.0614677
\(53\) −5.34614 −0.734348 −0.367174 0.930152i \(-0.619675\pi\)
−0.367174 + 0.930152i \(0.619675\pi\)
\(54\) 0 0
\(55\) −0.243359 −0.0328145
\(56\) 0 0
\(57\) 0 0
\(58\) 10.1350 1.33080
\(59\) 4.57099 0.595092 0.297546 0.954708i \(-0.403832\pi\)
0.297546 + 0.954708i \(0.403832\pi\)
\(60\) 0 0
\(61\) −0.678276 −0.0868443 −0.0434221 0.999057i \(-0.513826\pi\)
−0.0434221 + 0.999057i \(0.513826\pi\)
\(62\) 5.17066 0.656674
\(63\) 0 0
\(64\) −9.63481 −1.20435
\(65\) −0.0292332 −0.00362593
\(66\) 0 0
\(67\) −6.18684 −0.755842 −0.377921 0.925838i \(-0.623361\pi\)
−0.377921 + 0.925838i \(0.623361\pi\)
\(68\) 13.9059 1.68634
\(69\) 0 0
\(70\) 0 0
\(71\) −1.27749 −0.151611 −0.0758053 0.997123i \(-0.524153\pi\)
−0.0758053 + 0.997123i \(0.524153\pi\)
\(72\) 0 0
\(73\) 1.55721 0.182257 0.0911286 0.995839i \(-0.470953\pi\)
0.0911286 + 0.995839i \(0.470953\pi\)
\(74\) −14.3765 −1.67123
\(75\) 0 0
\(76\) −15.3394 −1.75955
\(77\) 0 0
\(78\) 0 0
\(79\) 12.7957 1.43963 0.719817 0.694164i \(-0.244226\pi\)
0.719817 + 0.694164i \(0.244226\pi\)
\(80\) −0.514462 −0.0575186
\(81\) 0 0
\(82\) −4.76015 −0.525671
\(83\) 7.51374 0.824740 0.412370 0.911016i \(-0.364701\pi\)
0.412370 + 0.911016i \(0.364701\pi\)
\(84\) 0 0
\(85\) 0.917122 0.0994758
\(86\) −3.86493 −0.416766
\(87\) 0 0
\(88\) 0.743131 0.0792181
\(89\) 9.06788 0.961193 0.480597 0.876942i \(-0.340420\pi\)
0.480597 + 0.876942i \(0.340420\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 13.7080 1.42915
\(93\) 0 0
\(94\) −3.72074 −0.383765
\(95\) −1.01167 −0.103795
\(96\) 0 0
\(97\) 7.97028 0.809259 0.404630 0.914481i \(-0.367400\pi\)
0.404630 + 0.914481i \(0.367400\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −11.0399 −1.10399
\(101\) −14.8430 −1.47693 −0.738467 0.674290i \(-0.764450\pi\)
−0.738467 + 0.674290i \(0.764450\pi\)
\(102\) 0 0
\(103\) −0.203948 −0.0200956 −0.0100478 0.999950i \(-0.503198\pi\)
−0.0100478 + 0.999950i \(0.503198\pi\)
\(104\) 0.0892677 0.00875342
\(105\) 0 0
\(106\) 10.9791 1.06638
\(107\) 6.96889 0.673708 0.336854 0.941557i \(-0.390637\pi\)
0.336854 + 0.941557i \(0.390637\pi\)
\(108\) 0 0
\(109\) −6.66116 −0.638024 −0.319012 0.947751i \(-0.603351\pi\)
−0.319012 + 0.947751i \(0.603351\pi\)
\(110\) 0.499772 0.0476514
\(111\) 0 0
\(112\) 0 0
\(113\) −0.0386468 −0.00363558 −0.00181779 0.999998i \(-0.500579\pi\)
−0.00181779 + 0.999998i \(0.500579\pi\)
\(114\) 0 0
\(115\) 0.904067 0.0843047
\(116\) −10.9435 −1.01608
\(117\) 0 0
\(118\) −9.38718 −0.864160
\(119\) 0 0
\(120\) 0 0
\(121\) −8.23097 −0.748270
\(122\) 1.39294 0.126111
\(123\) 0 0
\(124\) −5.58311 −0.501378
\(125\) −1.45933 −0.130526
\(126\) 0 0
\(127\) 13.4788 1.19605 0.598027 0.801476i \(-0.295952\pi\)
0.598027 + 0.801476i \(0.295952\pi\)
\(128\) 3.55154 0.313915
\(129\) 0 0
\(130\) 0.0600345 0.00526538
\(131\) 19.8333 1.73284 0.866422 0.499312i \(-0.166414\pi\)
0.866422 + 0.499312i \(0.166414\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 12.7056 1.09759
\(135\) 0 0
\(136\) −2.80056 −0.240146
\(137\) 6.44509 0.550642 0.275321 0.961352i \(-0.411216\pi\)
0.275321 + 0.961352i \(0.411216\pi\)
\(138\) 0 0
\(139\) −12.5305 −1.06283 −0.531413 0.847113i \(-0.678339\pi\)
−0.531413 + 0.847113i \(0.678339\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.62352 0.220161
\(143\) 0.332626 0.0278156
\(144\) 0 0
\(145\) −0.721743 −0.0599375
\(146\) −3.19795 −0.264664
\(147\) 0 0
\(148\) 15.5232 1.27600
\(149\) −17.7673 −1.45555 −0.727776 0.685815i \(-0.759446\pi\)
−0.727776 + 0.685815i \(0.759446\pi\)
\(150\) 0 0
\(151\) 8.46599 0.688953 0.344476 0.938795i \(-0.388056\pi\)
0.344476 + 0.938795i \(0.388056\pi\)
\(152\) 3.08927 0.250573
\(153\) 0 0
\(154\) 0 0
\(155\) −0.368217 −0.0295759
\(156\) 0 0
\(157\) 5.69935 0.454858 0.227429 0.973795i \(-0.426968\pi\)
0.227429 + 0.973795i \(0.426968\pi\)
\(158\) −26.2779 −2.09056
\(159\) 0 0
\(160\) 1.18714 0.0938520
\(161\) 0 0
\(162\) 0 0
\(163\) 2.12535 0.166470 0.0832349 0.996530i \(-0.473475\pi\)
0.0832349 + 0.996530i \(0.473475\pi\)
\(164\) 5.13986 0.401355
\(165\) 0 0
\(166\) −15.4306 −1.19764
\(167\) −11.5745 −0.895659 −0.447829 0.894119i \(-0.647803\pi\)
−0.447829 + 0.894119i \(0.647803\pi\)
\(168\) 0 0
\(169\) −12.9600 −0.996926
\(170\) −1.88344 −0.144453
\(171\) 0 0
\(172\) 4.17323 0.318206
\(173\) 15.9109 1.20968 0.604842 0.796345i \(-0.293236\pi\)
0.604842 + 0.796345i \(0.293236\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.85375 0.441243
\(177\) 0 0
\(178\) −18.6222 −1.39579
\(179\) 7.75331 0.579509 0.289755 0.957101i \(-0.406426\pi\)
0.289755 + 0.957101i \(0.406426\pi\)
\(180\) 0 0
\(181\) −12.1618 −0.903982 −0.451991 0.892022i \(-0.649286\pi\)
−0.451991 + 0.892022i \(0.649286\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2.76070 −0.203521
\(185\) 1.02379 0.0752703
\(186\) 0 0
\(187\) −10.4354 −0.763110
\(188\) 4.01754 0.293009
\(189\) 0 0
\(190\) 2.07760 0.150725
\(191\) 4.96765 0.359447 0.179723 0.983717i \(-0.442480\pi\)
0.179723 + 0.983717i \(0.442480\pi\)
\(192\) 0 0
\(193\) −14.9044 −1.07284 −0.536422 0.843950i \(-0.680224\pi\)
−0.536422 + 0.843950i \(0.680224\pi\)
\(194\) −16.3681 −1.17516
\(195\) 0 0
\(196\) 0 0
\(197\) 21.2608 1.51477 0.757386 0.652968i \(-0.226476\pi\)
0.757386 + 0.652968i \(0.226476\pi\)
\(198\) 0 0
\(199\) 19.9442 1.41380 0.706902 0.707311i \(-0.250092\pi\)
0.706902 + 0.707311i \(0.250092\pi\)
\(200\) 2.22336 0.157215
\(201\) 0 0
\(202\) 30.4823 2.14472
\(203\) 0 0
\(204\) 0 0
\(205\) 0.338983 0.0236756
\(206\) 0.418838 0.0291818
\(207\) 0 0
\(208\) 0.703175 0.0487564
\(209\) 11.5111 0.796241
\(210\) 0 0
\(211\) −23.5139 −1.61876 −0.809381 0.587284i \(-0.800197\pi\)
−0.809381 + 0.587284i \(0.800197\pi\)
\(212\) −11.8548 −0.814193
\(213\) 0 0
\(214\) −14.3116 −0.978323
\(215\) 0.275232 0.0187707
\(216\) 0 0
\(217\) 0 0
\(218\) 13.6797 0.926504
\(219\) 0 0
\(220\) −0.539638 −0.0363824
\(221\) −1.25354 −0.0843220
\(222\) 0 0
\(223\) −4.06104 −0.271947 −0.135974 0.990712i \(-0.543416\pi\)
−0.135974 + 0.990712i \(0.543416\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0.0793667 0.00527940
\(227\) 3.85285 0.255723 0.127861 0.991792i \(-0.459189\pi\)
0.127861 + 0.991792i \(0.459189\pi\)
\(228\) 0 0
\(229\) 13.1162 0.866746 0.433373 0.901215i \(-0.357323\pi\)
0.433373 + 0.901215i \(0.357323\pi\)
\(230\) −1.85663 −0.122423
\(231\) 0 0
\(232\) 2.20395 0.144696
\(233\) −17.5023 −1.14661 −0.573307 0.819340i \(-0.694340\pi\)
−0.573307 + 0.819340i \(0.694340\pi\)
\(234\) 0 0
\(235\) 0.264964 0.0172844
\(236\) 10.1360 0.659795
\(237\) 0 0
\(238\) 0 0
\(239\) 7.31714 0.473306 0.236653 0.971594i \(-0.423949\pi\)
0.236653 + 0.971594i \(0.423949\pi\)
\(240\) 0 0
\(241\) 6.23107 0.401378 0.200689 0.979655i \(-0.435682\pi\)
0.200689 + 0.979655i \(0.435682\pi\)
\(242\) 16.9035 1.08660
\(243\) 0 0
\(244\) −1.50405 −0.0962868
\(245\) 0 0
\(246\) 0 0
\(247\) 1.38276 0.0879829
\(248\) 1.12440 0.0713997
\(249\) 0 0
\(250\) 2.99694 0.189543
\(251\) −5.65283 −0.356803 −0.178402 0.983958i \(-0.557093\pi\)
−0.178402 + 0.983958i \(0.557093\pi\)
\(252\) 0 0
\(253\) −10.2868 −0.646727
\(254\) −27.6808 −1.73684
\(255\) 0 0
\(256\) 11.9760 0.748501
\(257\) −11.8016 −0.736166 −0.368083 0.929793i \(-0.619986\pi\)
−0.368083 + 0.929793i \(0.619986\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −0.0648233 −0.00402017
\(261\) 0 0
\(262\) −40.7306 −2.51634
\(263\) 22.2401 1.37138 0.685691 0.727893i \(-0.259500\pi\)
0.685691 + 0.727893i \(0.259500\pi\)
\(264\) 0 0
\(265\) −0.781849 −0.0480286
\(266\) 0 0
\(267\) 0 0
\(268\) −13.7191 −0.838024
\(269\) −2.38884 −0.145650 −0.0728251 0.997345i \(-0.523201\pi\)
−0.0728251 + 0.997345i \(0.523201\pi\)
\(270\) 0 0
\(271\) 23.2258 1.41087 0.705435 0.708775i \(-0.250752\pi\)
0.705435 + 0.708775i \(0.250752\pi\)
\(272\) −22.0605 −1.33761
\(273\) 0 0
\(274\) −13.2359 −0.799612
\(275\) 8.28461 0.499581
\(276\) 0 0
\(277\) −4.61800 −0.277469 −0.138734 0.990330i \(-0.544303\pi\)
−0.138734 + 0.990330i \(0.544303\pi\)
\(278\) 25.7333 1.54338
\(279\) 0 0
\(280\) 0 0
\(281\) 11.8168 0.704933 0.352466 0.935825i \(-0.385343\pi\)
0.352466 + 0.935825i \(0.385343\pi\)
\(282\) 0 0
\(283\) 15.8497 0.942165 0.471082 0.882089i \(-0.343863\pi\)
0.471082 + 0.882089i \(0.343863\pi\)
\(284\) −2.83279 −0.168095
\(285\) 0 0
\(286\) −0.683097 −0.0403924
\(287\) 0 0
\(288\) 0 0
\(289\) 22.3267 1.31334
\(290\) 1.48220 0.0870381
\(291\) 0 0
\(292\) 3.45304 0.202074
\(293\) 14.0961 0.823502 0.411751 0.911296i \(-0.364917\pi\)
0.411751 + 0.911296i \(0.364917\pi\)
\(294\) 0 0
\(295\) 0.668487 0.0389208
\(296\) −3.12628 −0.181711
\(297\) 0 0
\(298\) 36.4877 2.11367
\(299\) −1.23569 −0.0714619
\(300\) 0 0
\(301\) 0 0
\(302\) −17.3861 −1.00046
\(303\) 0 0
\(304\) 24.3346 1.39569
\(305\) −0.0991949 −0.00567988
\(306\) 0 0
\(307\) 27.3916 1.56332 0.781660 0.623704i \(-0.214373\pi\)
0.781660 + 0.623704i \(0.214373\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0.756186 0.0429485
\(311\) 14.0557 0.797026 0.398513 0.917163i \(-0.369526\pi\)
0.398513 + 0.917163i \(0.369526\pi\)
\(312\) 0 0
\(313\) 21.7446 1.22908 0.614540 0.788886i \(-0.289342\pi\)
0.614540 + 0.788886i \(0.289342\pi\)
\(314\) −11.7045 −0.660520
\(315\) 0 0
\(316\) 28.3740 1.59616
\(317\) −8.56297 −0.480944 −0.240472 0.970656i \(-0.577302\pi\)
−0.240472 + 0.970656i \(0.577302\pi\)
\(318\) 0 0
\(319\) 8.21228 0.459799
\(320\) −1.40905 −0.0787682
\(321\) 0 0
\(322\) 0 0
\(323\) −43.3808 −2.41377
\(324\) 0 0
\(325\) 0.995179 0.0552026
\(326\) −4.36471 −0.241739
\(327\) 0 0
\(328\) −1.03514 −0.0571558
\(329\) 0 0
\(330\) 0 0
\(331\) 10.8472 0.596216 0.298108 0.954532i \(-0.403644\pi\)
0.298108 + 0.954532i \(0.403644\pi\)
\(332\) 16.6614 0.914413
\(333\) 0 0
\(334\) 23.7698 1.30063
\(335\) −0.904798 −0.0494344
\(336\) 0 0
\(337\) −3.34822 −0.182389 −0.0911945 0.995833i \(-0.529069\pi\)
−0.0911945 + 0.995833i \(0.529069\pi\)
\(338\) 26.6153 1.44768
\(339\) 0 0
\(340\) 2.03368 0.110292
\(341\) 4.18971 0.226886
\(342\) 0 0
\(343\) 0 0
\(344\) −0.840462 −0.0453147
\(345\) 0 0
\(346\) −32.6754 −1.75664
\(347\) 11.5330 0.619126 0.309563 0.950879i \(-0.399817\pi\)
0.309563 + 0.950879i \(0.399817\pi\)
\(348\) 0 0
\(349\) 8.89834 0.476317 0.238159 0.971226i \(-0.423456\pi\)
0.238159 + 0.971226i \(0.423456\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −13.5078 −0.719968
\(353\) 2.64699 0.140885 0.0704424 0.997516i \(-0.477559\pi\)
0.0704424 + 0.997516i \(0.477559\pi\)
\(354\) 0 0
\(355\) −0.186828 −0.00991579
\(356\) 20.1076 1.06570
\(357\) 0 0
\(358\) −15.9225 −0.841533
\(359\) −25.9671 −1.37049 −0.685245 0.728312i \(-0.740305\pi\)
−0.685245 + 0.728312i \(0.740305\pi\)
\(360\) 0 0
\(361\) 28.8529 1.51857
\(362\) 24.9761 1.31271
\(363\) 0 0
\(364\) 0 0
\(365\) 0.227735 0.0119202
\(366\) 0 0
\(367\) 17.5874 0.918056 0.459028 0.888422i \(-0.348198\pi\)
0.459028 + 0.888422i \(0.348198\pi\)
\(368\) −21.7464 −1.13361
\(369\) 0 0
\(370\) −2.10249 −0.109303
\(371\) 0 0
\(372\) 0 0
\(373\) 0.815075 0.0422030 0.0211015 0.999777i \(-0.493283\pi\)
0.0211015 + 0.999777i \(0.493283\pi\)
\(374\) 21.4306 1.10815
\(375\) 0 0
\(376\) −0.809107 −0.0417265
\(377\) 0.986490 0.0508068
\(378\) 0 0
\(379\) −20.4312 −1.04948 −0.524741 0.851262i \(-0.675838\pi\)
−0.524741 + 0.851262i \(0.675838\pi\)
\(380\) −2.24333 −0.115080
\(381\) 0 0
\(382\) −10.2018 −0.521969
\(383\) −17.8928 −0.914278 −0.457139 0.889395i \(-0.651126\pi\)
−0.457139 + 0.889395i \(0.651126\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 30.6084 1.55793
\(387\) 0 0
\(388\) 17.6738 0.897249
\(389\) −15.6278 −0.792363 −0.396181 0.918172i \(-0.629665\pi\)
−0.396181 + 0.918172i \(0.629665\pi\)
\(390\) 0 0
\(391\) 38.7669 1.96053
\(392\) 0 0
\(393\) 0 0
\(394\) −43.6622 −2.19967
\(395\) 1.87132 0.0941564
\(396\) 0 0
\(397\) −19.2613 −0.966696 −0.483348 0.875428i \(-0.660579\pi\)
−0.483348 + 0.875428i \(0.660579\pi\)
\(398\) −40.9582 −2.05305
\(399\) 0 0
\(400\) 17.5137 0.875687
\(401\) −14.3013 −0.714172 −0.357086 0.934072i \(-0.616230\pi\)
−0.357086 + 0.934072i \(0.616230\pi\)
\(402\) 0 0
\(403\) 0.503284 0.0250704
\(404\) −32.9137 −1.63752
\(405\) 0 0
\(406\) 0 0
\(407\) −11.6490 −0.577422
\(408\) 0 0
\(409\) 31.8610 1.57542 0.787712 0.616044i \(-0.211266\pi\)
0.787712 + 0.616044i \(0.211266\pi\)
\(410\) −0.696152 −0.0343805
\(411\) 0 0
\(412\) −0.452247 −0.0222806
\(413\) 0 0
\(414\) 0 0
\(415\) 1.09885 0.0539405
\(416\) −1.62261 −0.0795549
\(417\) 0 0
\(418\) −23.6398 −1.15626
\(419\) 23.8960 1.16739 0.583697 0.811971i \(-0.301605\pi\)
0.583697 + 0.811971i \(0.301605\pi\)
\(420\) 0 0
\(421\) 2.44501 0.119163 0.0595813 0.998223i \(-0.481023\pi\)
0.0595813 + 0.998223i \(0.481023\pi\)
\(422\) 48.2892 2.35068
\(423\) 0 0
\(424\) 2.38749 0.115947
\(425\) −31.2214 −1.51446
\(426\) 0 0
\(427\) 0 0
\(428\) 15.4532 0.746960
\(429\) 0 0
\(430\) −0.565230 −0.0272578
\(431\) 4.92764 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(432\) 0 0
\(433\) 30.8539 1.48274 0.741371 0.671095i \(-0.234176\pi\)
0.741371 + 0.671095i \(0.234176\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −14.7709 −0.707395
\(437\) −42.7633 −2.04565
\(438\) 0 0
\(439\) 2.44822 0.116847 0.0584235 0.998292i \(-0.481393\pi\)
0.0584235 + 0.998292i \(0.481393\pi\)
\(440\) 0.108680 0.00518110
\(441\) 0 0
\(442\) 2.57432 0.122448
\(443\) 26.2950 1.24931 0.624657 0.780899i \(-0.285239\pi\)
0.624657 + 0.780899i \(0.285239\pi\)
\(444\) 0 0
\(445\) 1.32614 0.0628650
\(446\) 8.33993 0.394907
\(447\) 0 0
\(448\) 0 0
\(449\) 38.7077 1.82673 0.913365 0.407141i \(-0.133474\pi\)
0.913365 + 0.407141i \(0.133474\pi\)
\(450\) 0 0
\(451\) −3.85709 −0.181623
\(452\) −0.0856976 −0.00403087
\(453\) 0 0
\(454\) −7.91239 −0.371347
\(455\) 0 0
\(456\) 0 0
\(457\) −9.15511 −0.428258 −0.214129 0.976805i \(-0.568691\pi\)
−0.214129 + 0.976805i \(0.568691\pi\)
\(458\) −26.9361 −1.25864
\(459\) 0 0
\(460\) 2.00473 0.0934710
\(461\) 29.2304 1.36140 0.680698 0.732564i \(-0.261676\pi\)
0.680698 + 0.732564i \(0.261676\pi\)
\(462\) 0 0
\(463\) 16.4206 0.763131 0.381565 0.924342i \(-0.375385\pi\)
0.381565 + 0.924342i \(0.375385\pi\)
\(464\) 17.3608 0.805956
\(465\) 0 0
\(466\) 35.9435 1.66505
\(467\) 15.3726 0.711361 0.355680 0.934608i \(-0.384249\pi\)
0.355680 + 0.934608i \(0.384249\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −0.544142 −0.0250994
\(471\) 0 0
\(472\) −2.04132 −0.0939594
\(473\) −3.13170 −0.143996
\(474\) 0 0
\(475\) 34.4399 1.58021
\(476\) 0 0
\(477\) 0 0
\(478\) −15.0268 −0.687310
\(479\) 37.9291 1.73303 0.866513 0.499155i \(-0.166356\pi\)
0.866513 + 0.499155i \(0.166356\pi\)
\(480\) 0 0
\(481\) −1.39933 −0.0638038
\(482\) −12.7964 −0.582860
\(483\) 0 0
\(484\) −18.2518 −0.829629
\(485\) 1.16562 0.0529280
\(486\) 0 0
\(487\) −4.60495 −0.208670 −0.104335 0.994542i \(-0.533271\pi\)
−0.104335 + 0.994542i \(0.533271\pi\)
\(488\) 0.302906 0.0137119
\(489\) 0 0
\(490\) 0 0
\(491\) −30.3751 −1.37081 −0.685405 0.728162i \(-0.740375\pi\)
−0.685405 + 0.728162i \(0.740375\pi\)
\(492\) 0 0
\(493\) −30.9488 −1.39386
\(494\) −2.83970 −0.127764
\(495\) 0 0
\(496\) 8.85709 0.397695
\(497\) 0 0
\(498\) 0 0
\(499\) 9.26871 0.414925 0.207462 0.978243i \(-0.433480\pi\)
0.207462 + 0.978243i \(0.433480\pi\)
\(500\) −3.23600 −0.144718
\(501\) 0 0
\(502\) 11.6089 0.518131
\(503\) 22.4230 0.999791 0.499896 0.866086i \(-0.333372\pi\)
0.499896 + 0.866086i \(0.333372\pi\)
\(504\) 0 0
\(505\) −2.17072 −0.0965960
\(506\) 21.1255 0.939143
\(507\) 0 0
\(508\) 29.8888 1.32610
\(509\) 37.6414 1.66843 0.834213 0.551443i \(-0.185923\pi\)
0.834213 + 0.551443i \(0.185923\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −31.6976 −1.40085
\(513\) 0 0
\(514\) 24.2364 1.06902
\(515\) −0.0298266 −0.00131432
\(516\) 0 0
\(517\) −3.01487 −0.132594
\(518\) 0 0
\(519\) 0 0
\(520\) 0.0130550 0.000572500 0
\(521\) 34.9283 1.53023 0.765117 0.643891i \(-0.222681\pi\)
0.765117 + 0.643891i \(0.222681\pi\)
\(522\) 0 0
\(523\) 23.7471 1.03839 0.519194 0.854656i \(-0.326232\pi\)
0.519194 + 0.854656i \(0.326232\pi\)
\(524\) 43.9795 1.92126
\(525\) 0 0
\(526\) −45.6732 −1.99145
\(527\) −15.7894 −0.687795
\(528\) 0 0
\(529\) 15.2151 0.661526
\(530\) 1.60564 0.0697446
\(531\) 0 0
\(532\) 0 0
\(533\) −0.463328 −0.0200690
\(534\) 0 0
\(535\) 1.01917 0.0440626
\(536\) 2.76293 0.119340
\(537\) 0 0
\(538\) 4.90583 0.211505
\(539\) 0 0
\(540\) 0 0
\(541\) −17.1708 −0.738232 −0.369116 0.929383i \(-0.620340\pi\)
−0.369116 + 0.929383i \(0.620340\pi\)
\(542\) −47.6976 −2.04879
\(543\) 0 0
\(544\) 50.9055 2.18255
\(545\) −0.974166 −0.0417287
\(546\) 0 0
\(547\) 20.0091 0.855529 0.427765 0.903890i \(-0.359301\pi\)
0.427765 + 0.903890i \(0.359301\pi\)
\(548\) 14.2917 0.610512
\(549\) 0 0
\(550\) −17.0137 −0.725465
\(551\) 34.1392 1.45438
\(552\) 0 0
\(553\) 0 0
\(554\) 9.48374 0.402926
\(555\) 0 0
\(556\) −27.7860 −1.17839
\(557\) −0.245481 −0.0104014 −0.00520068 0.999986i \(-0.501655\pi\)
−0.00520068 + 0.999986i \(0.501655\pi\)
\(558\) 0 0
\(559\) −0.376192 −0.0159112
\(560\) 0 0
\(561\) 0 0
\(562\) −24.2676 −1.02367
\(563\) 44.2509 1.86495 0.932477 0.361230i \(-0.117643\pi\)
0.932477 + 0.361230i \(0.117643\pi\)
\(564\) 0 0
\(565\) −0.00565192 −0.000237778 0
\(566\) −32.5496 −1.36816
\(567\) 0 0
\(568\) 0.570506 0.0239379
\(569\) 5.53533 0.232053 0.116027 0.993246i \(-0.462984\pi\)
0.116027 + 0.993246i \(0.462984\pi\)
\(570\) 0 0
\(571\) −4.10381 −0.171739 −0.0858696 0.996306i \(-0.527367\pi\)
−0.0858696 + 0.996306i \(0.527367\pi\)
\(572\) 0.737585 0.0308400
\(573\) 0 0
\(574\) 0 0
\(575\) −30.7770 −1.28349
\(576\) 0 0
\(577\) 5.64550 0.235025 0.117513 0.993071i \(-0.462508\pi\)
0.117513 + 0.993071i \(0.462508\pi\)
\(578\) −45.8512 −1.90716
\(579\) 0 0
\(580\) −1.60044 −0.0664545
\(581\) 0 0
\(582\) 0 0
\(583\) 8.89619 0.368442
\(584\) −0.695420 −0.0287767
\(585\) 0 0
\(586\) −28.9483 −1.19585
\(587\) 18.7329 0.773189 0.386595 0.922250i \(-0.373651\pi\)
0.386595 + 0.922250i \(0.373651\pi\)
\(588\) 0 0
\(589\) 17.4170 0.717657
\(590\) −1.37283 −0.0565187
\(591\) 0 0
\(592\) −24.6262 −1.01213
\(593\) −18.8703 −0.774912 −0.387456 0.921888i \(-0.626646\pi\)
−0.387456 + 0.921888i \(0.626646\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −39.3982 −1.61381
\(597\) 0 0
\(598\) 2.53767 0.103773
\(599\) −2.67451 −0.109278 −0.0546388 0.998506i \(-0.517401\pi\)
−0.0546388 + 0.998506i \(0.517401\pi\)
\(600\) 0 0
\(601\) 13.2143 0.539023 0.269511 0.962997i \(-0.413138\pi\)
0.269511 + 0.962997i \(0.413138\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 18.7730 0.763862
\(605\) −1.20374 −0.0489391
\(606\) 0 0
\(607\) 25.8052 1.04740 0.523701 0.851902i \(-0.324551\pi\)
0.523701 + 0.851902i \(0.324551\pi\)
\(608\) −56.1532 −2.27731
\(609\) 0 0
\(610\) 0.203711 0.00824802
\(611\) −0.362157 −0.0146513
\(612\) 0 0
\(613\) −26.9533 −1.08863 −0.544316 0.838880i \(-0.683211\pi\)
−0.544316 + 0.838880i \(0.683211\pi\)
\(614\) −56.2526 −2.27017
\(615\) 0 0
\(616\) 0 0
\(617\) −9.53175 −0.383734 −0.191867 0.981421i \(-0.561454\pi\)
−0.191867 + 0.981421i \(0.561454\pi\)
\(618\) 0 0
\(619\) 34.7071 1.39500 0.697499 0.716586i \(-0.254296\pi\)
0.697499 + 0.716586i \(0.254296\pi\)
\(620\) −0.816505 −0.0327916
\(621\) 0 0
\(622\) −28.8654 −1.15740
\(623\) 0 0
\(624\) 0 0
\(625\) 24.6796 0.987186
\(626\) −44.6558 −1.78480
\(627\) 0 0
\(628\) 12.6381 0.504314
\(629\) 43.9006 1.75043
\(630\) 0 0
\(631\) −36.7963 −1.46484 −0.732419 0.680854i \(-0.761609\pi\)
−0.732419 + 0.680854i \(0.761609\pi\)
\(632\) −5.71435 −0.227305
\(633\) 0 0
\(634\) 17.5853 0.698401
\(635\) 1.97122 0.0782256
\(636\) 0 0
\(637\) 0 0
\(638\) −16.8651 −0.667696
\(639\) 0 0
\(640\) 0.519397 0.0205310
\(641\) 44.1844 1.74518 0.872590 0.488454i \(-0.162439\pi\)
0.872590 + 0.488454i \(0.162439\pi\)
\(642\) 0 0
\(643\) −14.4813 −0.571087 −0.285543 0.958366i \(-0.592174\pi\)
−0.285543 + 0.958366i \(0.592174\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 89.0889 3.50515
\(647\) −33.3071 −1.30944 −0.654719 0.755872i \(-0.727213\pi\)
−0.654719 + 0.755872i \(0.727213\pi\)
\(648\) 0 0
\(649\) −7.60631 −0.298574
\(650\) −2.04374 −0.0801622
\(651\) 0 0
\(652\) 4.71286 0.184570
\(653\) 9.06643 0.354797 0.177398 0.984139i \(-0.443232\pi\)
0.177398 + 0.984139i \(0.443232\pi\)
\(654\) 0 0
\(655\) 2.90054 0.113333
\(656\) −8.15391 −0.318357
\(657\) 0 0
\(658\) 0 0
\(659\) 32.3611 1.26061 0.630305 0.776348i \(-0.282930\pi\)
0.630305 + 0.776348i \(0.282930\pi\)
\(660\) 0 0
\(661\) −8.65915 −0.336802 −0.168401 0.985719i \(-0.553860\pi\)
−0.168401 + 0.985719i \(0.553860\pi\)
\(662\) −22.2763 −0.865794
\(663\) 0 0
\(664\) −3.35550 −0.130219
\(665\) 0 0
\(666\) 0 0
\(667\) −30.5083 −1.18128
\(668\) −25.6659 −0.993043
\(669\) 0 0
\(670\) 1.85813 0.0717860
\(671\) 1.12868 0.0435721
\(672\) 0 0
\(673\) −14.4968 −0.558812 −0.279406 0.960173i \(-0.590138\pi\)
−0.279406 + 0.960173i \(0.590138\pi\)
\(674\) 6.87605 0.264856
\(675\) 0 0
\(676\) −28.7384 −1.10532
\(677\) −38.3315 −1.47320 −0.736600 0.676329i \(-0.763570\pi\)
−0.736600 + 0.676329i \(0.763570\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −0.409570 −0.0157063
\(681\) 0 0
\(682\) −8.60418 −0.329471
\(683\) −6.63318 −0.253812 −0.126906 0.991915i \(-0.540505\pi\)
−0.126906 + 0.991915i \(0.540505\pi\)
\(684\) 0 0
\(685\) 0.942567 0.0360136
\(686\) 0 0
\(687\) 0 0
\(688\) −6.62044 −0.252402
\(689\) 1.06864 0.0407121
\(690\) 0 0
\(691\) −23.3875 −0.889704 −0.444852 0.895604i \(-0.646744\pi\)
−0.444852 + 0.895604i \(0.646744\pi\)
\(692\) 35.2818 1.34121
\(693\) 0 0
\(694\) −23.6848 −0.899061
\(695\) −1.83254 −0.0695121
\(696\) 0 0
\(697\) 14.5358 0.550583
\(698\) −18.2740 −0.691683
\(699\) 0 0
\(700\) 0 0
\(701\) −9.26736 −0.350023 −0.175012 0.984566i \(-0.555996\pi\)
−0.175012 + 0.984566i \(0.555996\pi\)
\(702\) 0 0
\(703\) −48.4262 −1.82643
\(704\) 16.0327 0.604256
\(705\) 0 0
\(706\) −5.43597 −0.204585
\(707\) 0 0
\(708\) 0 0
\(709\) 14.2355 0.534626 0.267313 0.963610i \(-0.413864\pi\)
0.267313 + 0.963610i \(0.413864\pi\)
\(710\) 0.383678 0.0143992
\(711\) 0 0
\(712\) −4.04956 −0.151763
\(713\) −15.5646 −0.582899
\(714\) 0 0
\(715\) 0.0486452 0.00181923
\(716\) 17.1926 0.642519
\(717\) 0 0
\(718\) 53.3272 1.99015
\(719\) 13.8570 0.516777 0.258389 0.966041i \(-0.416808\pi\)
0.258389 + 0.966041i \(0.416808\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −59.2536 −2.20519
\(723\) 0 0
\(724\) −26.9684 −1.00227
\(725\) 24.5702 0.912513
\(726\) 0 0
\(727\) −31.4000 −1.16456 −0.582280 0.812988i \(-0.697839\pi\)
−0.582280 + 0.812988i \(0.697839\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −0.467686 −0.0173098
\(731\) 11.8021 0.436518
\(732\) 0 0
\(733\) −26.6006 −0.982515 −0.491257 0.871014i \(-0.663463\pi\)
−0.491257 + 0.871014i \(0.663463\pi\)
\(734\) −36.1183 −1.33315
\(735\) 0 0
\(736\) 50.1809 1.84969
\(737\) 10.2951 0.379227
\(738\) 0 0
\(739\) −33.0039 −1.21407 −0.607034 0.794676i \(-0.707641\pi\)
−0.607034 + 0.794676i \(0.707641\pi\)
\(740\) 2.27020 0.0834543
\(741\) 0 0
\(742\) 0 0
\(743\) 38.6015 1.41615 0.708076 0.706136i \(-0.249563\pi\)
0.708076 + 0.706136i \(0.249563\pi\)
\(744\) 0 0
\(745\) −2.59839 −0.0951975
\(746\) −1.67388 −0.0612849
\(747\) 0 0
\(748\) −23.1400 −0.846082
\(749\) 0 0
\(750\) 0 0
\(751\) −37.8996 −1.38297 −0.691487 0.722389i \(-0.743044\pi\)
−0.691487 + 0.722389i \(0.743044\pi\)
\(752\) −6.37345 −0.232416
\(753\) 0 0
\(754\) −2.02590 −0.0737789
\(755\) 1.23811 0.0450596
\(756\) 0 0
\(757\) 22.5927 0.821147 0.410573 0.911828i \(-0.365329\pi\)
0.410573 + 0.911828i \(0.365329\pi\)
\(758\) 41.9585 1.52400
\(759\) 0 0
\(760\) 0.451792 0.0163882
\(761\) −27.7470 −1.00583 −0.502913 0.864337i \(-0.667739\pi\)
−0.502913 + 0.864337i \(0.667739\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 11.0156 0.398529
\(765\) 0 0
\(766\) 36.7454 1.32766
\(767\) −0.913698 −0.0329917
\(768\) 0 0
\(769\) 12.1534 0.438262 0.219131 0.975695i \(-0.429678\pi\)
0.219131 + 0.975695i \(0.429678\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −33.0499 −1.18949
\(773\) −41.5591 −1.49478 −0.747388 0.664388i \(-0.768692\pi\)
−0.747388 + 0.664388i \(0.768692\pi\)
\(774\) 0 0
\(775\) 12.5351 0.450275
\(776\) −3.55939 −0.127775
\(777\) 0 0
\(778\) 32.0940 1.15063
\(779\) −16.0343 −0.574488
\(780\) 0 0
\(781\) 2.12580 0.0760671
\(782\) −79.6135 −2.84697
\(783\) 0 0
\(784\) 0 0
\(785\) 0.833506 0.0297491
\(786\) 0 0
\(787\) −20.8969 −0.744893 −0.372446 0.928054i \(-0.621481\pi\)
−0.372446 + 0.928054i \(0.621481\pi\)
\(788\) 47.1450 1.67947
\(789\) 0 0
\(790\) −3.84303 −0.136729
\(791\) 0 0
\(792\) 0 0
\(793\) 0.135581 0.00481462
\(794\) 39.5558 1.40378
\(795\) 0 0
\(796\) 44.2254 1.56753
\(797\) −0.638766 −0.0226263 −0.0113131 0.999936i \(-0.503601\pi\)
−0.0113131 + 0.999936i \(0.503601\pi\)
\(798\) 0 0
\(799\) 11.3618 0.401953
\(800\) −40.4137 −1.42884
\(801\) 0 0
\(802\) 29.3698 1.03708
\(803\) −2.59125 −0.0914433
\(804\) 0 0
\(805\) 0 0
\(806\) −1.03357 −0.0364058
\(807\) 0 0
\(808\) 6.62862 0.233194
\(809\) 50.5592 1.77757 0.888783 0.458327i \(-0.151551\pi\)
0.888783 + 0.458327i \(0.151551\pi\)
\(810\) 0 0
\(811\) −0.784071 −0.0275325 −0.0137662 0.999905i \(-0.504382\pi\)
−0.0137662 + 0.999905i \(0.504382\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 23.9230 0.838501
\(815\) 0.310823 0.0108876
\(816\) 0 0
\(817\) −13.0188 −0.455470
\(818\) −65.4311 −2.28775
\(819\) 0 0
\(820\) 0.751682 0.0262499
\(821\) −43.4413 −1.51611 −0.758056 0.652189i \(-0.773851\pi\)
−0.758056 + 0.652189i \(0.773851\pi\)
\(822\) 0 0
\(823\) 3.96546 0.138227 0.0691136 0.997609i \(-0.477983\pi\)
0.0691136 + 0.997609i \(0.477983\pi\)
\(824\) 0.0910797 0.00317291
\(825\) 0 0
\(826\) 0 0
\(827\) −29.3159 −1.01941 −0.509707 0.860348i \(-0.670246\pi\)
−0.509707 + 0.860348i \(0.670246\pi\)
\(828\) 0 0
\(829\) 35.0427 1.21708 0.608541 0.793522i \(-0.291755\pi\)
0.608541 + 0.793522i \(0.291755\pi\)
\(830\) −2.25665 −0.0783295
\(831\) 0 0
\(832\) 1.92591 0.0667689
\(833\) 0 0
\(834\) 0 0
\(835\) −1.69272 −0.0585788
\(836\) 25.5255 0.882816
\(837\) 0 0
\(838\) −49.0738 −1.69523
\(839\) −37.5843 −1.29755 −0.648777 0.760979i \(-0.724719\pi\)
−0.648777 + 0.760979i \(0.724719\pi\)
\(840\) 0 0
\(841\) −4.64435 −0.160150
\(842\) −5.02119 −0.173042
\(843\) 0 0
\(844\) −52.1411 −1.79477
\(845\) −1.89535 −0.0652020
\(846\) 0 0
\(847\) 0 0
\(848\) 18.8066 0.645822
\(849\) 0 0
\(850\) 64.1177 2.19922
\(851\) 43.2757 1.48347
\(852\) 0 0
\(853\) −32.7699 −1.12202 −0.561009 0.827810i \(-0.689587\pi\)
−0.561009 + 0.827810i \(0.689587\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −3.11218 −0.106372
\(857\) −27.5347 −0.940566 −0.470283 0.882516i \(-0.655848\pi\)
−0.470283 + 0.882516i \(0.655848\pi\)
\(858\) 0 0
\(859\) −46.5101 −1.58690 −0.793451 0.608634i \(-0.791718\pi\)
−0.793451 + 0.608634i \(0.791718\pi\)
\(860\) 0.610316 0.0208116
\(861\) 0 0
\(862\) −10.1196 −0.344675
\(863\) 4.88014 0.166122 0.0830610 0.996544i \(-0.473530\pi\)
0.0830610 + 0.996544i \(0.473530\pi\)
\(864\) 0 0
\(865\) 2.32690 0.0791170
\(866\) −63.3629 −2.15316
\(867\) 0 0
\(868\) 0 0
\(869\) −21.2926 −0.722303
\(870\) 0 0
\(871\) 1.23669 0.0419037
\(872\) 2.97476 0.100738
\(873\) 0 0
\(874\) 87.8207 2.97058
\(875\) 0 0
\(876\) 0 0
\(877\) 39.2892 1.32670 0.663352 0.748308i \(-0.269133\pi\)
0.663352 + 0.748308i \(0.269133\pi\)
\(878\) −5.02777 −0.169679
\(879\) 0 0
\(880\) 0.856086 0.0288587
\(881\) −47.3713 −1.59598 −0.797990 0.602670i \(-0.794103\pi\)
−0.797990 + 0.602670i \(0.794103\pi\)
\(882\) 0 0
\(883\) −2.67206 −0.0899221 −0.0449610 0.998989i \(-0.514316\pi\)
−0.0449610 + 0.998989i \(0.514316\pi\)
\(884\) −2.77966 −0.0934902
\(885\) 0 0
\(886\) −54.0007 −1.81419
\(887\) 22.9600 0.770922 0.385461 0.922724i \(-0.374042\pi\)
0.385461 + 0.922724i \(0.374042\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −2.72342 −0.0912891
\(891\) 0 0
\(892\) −9.00518 −0.301516
\(893\) −12.5331 −0.419404
\(894\) 0 0
\(895\) 1.13389 0.0379017
\(896\) 0 0
\(897\) 0 0
\(898\) −79.4920 −2.65268
\(899\) 12.4257 0.414420
\(900\) 0 0
\(901\) −33.5262 −1.11692
\(902\) 7.92109 0.263743
\(903\) 0 0
\(904\) 0.0172590 0.000574024 0
\(905\) −1.77862 −0.0591232
\(906\) 0 0
\(907\) −27.8982 −0.926345 −0.463173 0.886268i \(-0.653289\pi\)
−0.463173 + 0.886268i \(0.653289\pi\)
\(908\) 8.54354 0.283527
\(909\) 0 0
\(910\) 0 0
\(911\) −37.4762 −1.24164 −0.620820 0.783953i \(-0.713200\pi\)
−0.620820 + 0.783953i \(0.713200\pi\)
\(912\) 0 0
\(913\) −12.5032 −0.413794
\(914\) 18.8014 0.621894
\(915\) 0 0
\(916\) 29.0847 0.960987
\(917\) 0 0
\(918\) 0 0
\(919\) 30.2147 0.996691 0.498345 0.866979i \(-0.333941\pi\)
0.498345 + 0.866979i \(0.333941\pi\)
\(920\) −0.403740 −0.0133109
\(921\) 0 0
\(922\) −60.0289 −1.97695
\(923\) 0.255359 0.00840525
\(924\) 0 0
\(925\) −34.8526 −1.14594
\(926\) −33.7221 −1.10818
\(927\) 0 0
\(928\) −40.0609 −1.31506
\(929\) 45.9351 1.50708 0.753540 0.657402i \(-0.228344\pi\)
0.753540 + 0.657402i \(0.228344\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −38.8106 −1.27128
\(933\) 0 0
\(934\) −31.5699 −1.03300
\(935\) −1.52613 −0.0499097
\(936\) 0 0
\(937\) −45.3797 −1.48249 −0.741245 0.671235i \(-0.765764\pi\)
−0.741245 + 0.671235i \(0.765764\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0.587547 0.0191637
\(941\) −49.4003 −1.61040 −0.805202 0.593000i \(-0.797943\pi\)
−0.805202 + 0.593000i \(0.797943\pi\)
\(942\) 0 0
\(943\) 14.3289 0.466613
\(944\) −16.0798 −0.523353
\(945\) 0 0
\(946\) 6.43141 0.209103
\(947\) −31.6505 −1.02850 −0.514252 0.857639i \(-0.671930\pi\)
−0.514252 + 0.857639i \(0.671930\pi\)
\(948\) 0 0
\(949\) −0.311271 −0.0101043
\(950\) −70.7274 −2.29470
\(951\) 0 0
\(952\) 0 0
\(953\) 19.1237 0.619477 0.309739 0.950822i \(-0.399758\pi\)
0.309739 + 0.950822i \(0.399758\pi\)
\(954\) 0 0
\(955\) 0.726498 0.0235089
\(956\) 16.2255 0.524769
\(957\) 0 0
\(958\) −77.8929 −2.51661
\(959\) 0 0
\(960\) 0 0
\(961\) −24.6607 −0.795507
\(962\) 2.87372 0.0926525
\(963\) 0 0
\(964\) 13.8171 0.445020
\(965\) −2.17971 −0.0701673
\(966\) 0 0
\(967\) −9.97050 −0.320630 −0.160315 0.987066i \(-0.551251\pi\)
−0.160315 + 0.987066i \(0.551251\pi\)
\(968\) 3.67581 0.118145
\(969\) 0 0
\(970\) −2.39377 −0.0768592
\(971\) 1.04511 0.0335391 0.0167695 0.999859i \(-0.494662\pi\)
0.0167695 + 0.999859i \(0.494662\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 9.45693 0.303020
\(975\) 0 0
\(976\) 2.38603 0.0763751
\(977\) 18.8862 0.604222 0.302111 0.953273i \(-0.402309\pi\)
0.302111 + 0.953273i \(0.402309\pi\)
\(978\) 0 0
\(979\) −15.0893 −0.482257
\(980\) 0 0
\(981\) 0 0
\(982\) 62.3797 1.99062
\(983\) −2.28891 −0.0730050 −0.0365025 0.999334i \(-0.511622\pi\)
−0.0365025 + 0.999334i \(0.511622\pi\)
\(984\) 0 0
\(985\) 3.10930 0.0990707
\(986\) 63.5579 2.02409
\(987\) 0 0
\(988\) 3.06621 0.0975492
\(989\) 11.6341 0.369944
\(990\) 0 0
\(991\) 19.0698 0.605773 0.302886 0.953027i \(-0.402050\pi\)
0.302886 + 0.953027i \(0.402050\pi\)
\(992\) −20.4381 −0.648911
\(993\) 0 0
\(994\) 0 0
\(995\) 2.91675 0.0924671
\(996\) 0 0
\(997\) 37.0151 1.17228 0.586139 0.810210i \(-0.300647\pi\)
0.586139 + 0.810210i \(0.300647\pi\)
\(998\) −19.0346 −0.602531
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3969.2.a.bc.1.1 5
3.2 odd 2 3969.2.a.z.1.5 5
7.2 even 3 567.2.e.e.487.5 10
7.4 even 3 567.2.e.e.163.5 10
7.6 odd 2 3969.2.a.bb.1.1 5
9.2 odd 6 441.2.f.e.148.1 10
9.4 even 3 1323.2.f.e.883.5 10
9.5 odd 6 441.2.f.e.295.1 10
9.7 even 3 1323.2.f.e.442.5 10
21.2 odd 6 567.2.e.f.487.1 10
21.11 odd 6 567.2.e.f.163.1 10
21.20 even 2 3969.2.a.ba.1.5 5
63.2 odd 6 63.2.g.b.4.1 10
63.4 even 3 189.2.g.b.100.5 10
63.5 even 6 441.2.h.f.214.5 10
63.11 odd 6 63.2.h.b.58.5 yes 10
63.13 odd 6 1323.2.f.f.883.5 10
63.16 even 3 189.2.g.b.172.5 10
63.20 even 6 441.2.f.f.148.1 10
63.23 odd 6 63.2.h.b.25.5 yes 10
63.25 even 3 189.2.h.b.37.1 10
63.31 odd 6 1323.2.g.f.667.5 10
63.32 odd 6 63.2.g.b.16.1 yes 10
63.34 odd 6 1323.2.f.f.442.5 10
63.38 even 6 441.2.h.f.373.5 10
63.40 odd 6 1323.2.h.f.802.1 10
63.41 even 6 441.2.f.f.295.1 10
63.47 even 6 441.2.g.f.67.1 10
63.52 odd 6 1323.2.h.f.226.1 10
63.58 even 3 189.2.h.b.46.1 10
63.59 even 6 441.2.g.f.79.1 10
63.61 odd 6 1323.2.g.f.361.5 10
252.11 even 6 1008.2.q.i.625.5 10
252.23 even 6 1008.2.q.i.529.5 10
252.67 odd 6 3024.2.t.i.289.3 10
252.79 odd 6 3024.2.t.i.1873.3 10
252.95 even 6 1008.2.t.i.961.2 10
252.151 odd 6 3024.2.q.i.2305.3 10
252.191 even 6 1008.2.t.i.193.2 10
252.247 odd 6 3024.2.q.i.2881.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.g.b.4.1 10 63.2 odd 6
63.2.g.b.16.1 yes 10 63.32 odd 6
63.2.h.b.25.5 yes 10 63.23 odd 6
63.2.h.b.58.5 yes 10 63.11 odd 6
189.2.g.b.100.5 10 63.4 even 3
189.2.g.b.172.5 10 63.16 even 3
189.2.h.b.37.1 10 63.25 even 3
189.2.h.b.46.1 10 63.58 even 3
441.2.f.e.148.1 10 9.2 odd 6
441.2.f.e.295.1 10 9.5 odd 6
441.2.f.f.148.1 10 63.20 even 6
441.2.f.f.295.1 10 63.41 even 6
441.2.g.f.67.1 10 63.47 even 6
441.2.g.f.79.1 10 63.59 even 6
441.2.h.f.214.5 10 63.5 even 6
441.2.h.f.373.5 10 63.38 even 6
567.2.e.e.163.5 10 7.4 even 3
567.2.e.e.487.5 10 7.2 even 3
567.2.e.f.163.1 10 21.11 odd 6
567.2.e.f.487.1 10 21.2 odd 6
1008.2.q.i.529.5 10 252.23 even 6
1008.2.q.i.625.5 10 252.11 even 6
1008.2.t.i.193.2 10 252.191 even 6
1008.2.t.i.961.2 10 252.95 even 6
1323.2.f.e.442.5 10 9.7 even 3
1323.2.f.e.883.5 10 9.4 even 3
1323.2.f.f.442.5 10 63.34 odd 6
1323.2.f.f.883.5 10 63.13 odd 6
1323.2.g.f.361.5 10 63.61 odd 6
1323.2.g.f.667.5 10 63.31 odd 6
1323.2.h.f.226.1 10 63.52 odd 6
1323.2.h.f.802.1 10 63.40 odd 6
3024.2.q.i.2305.3 10 252.151 odd 6
3024.2.q.i.2881.3 10 252.247 odd 6
3024.2.t.i.289.3 10 252.67 odd 6
3024.2.t.i.1873.3 10 252.79 odd 6
3969.2.a.z.1.5 5 3.2 odd 2
3969.2.a.ba.1.5 5 21.20 even 2
3969.2.a.bb.1.1 5 7.6 odd 2
3969.2.a.bc.1.1 5 1.1 even 1 trivial