Properties

Label 1859.2.a.f.1.3
Level $1859$
Weight $2$
Character 1859.1
Self dual yes
Analytic conductor $14.844$
Analytic rank $0$
Dimension $3$
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] = [1859,2,Mod(1,1859)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1859, 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("1859.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1859.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: \(14.8441897358\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.361.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 143)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.50702\) of defining polynomial
Character \(\chi\) \(=\) 1859.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.50702 q^{2} +2.28514 q^{3} +4.28514 q^{4} -1.22188 q^{5} +5.72889 q^{6} +0.778124 q^{7} +5.72889 q^{8} +2.22188 q^{9} +O(q^{10})\) \(q+2.50702 q^{2} +2.28514 q^{3} +4.28514 q^{4} -1.22188 q^{5} +5.72889 q^{6} +0.778124 q^{7} +5.72889 q^{8} +2.22188 q^{9} -3.06327 q^{10} +1.00000 q^{11} +9.79216 q^{12} +1.95077 q^{14} -2.79216 q^{15} +5.79216 q^{16} -4.28514 q^{17} +5.57028 q^{18} +8.29918 q^{19} -5.23591 q^{20} +1.77812 q^{21} +2.50702 q^{22} -2.72889 q^{23} +13.0913 q^{24} -3.50702 q^{25} -1.77812 q^{27} +3.33437 q^{28} +7.36245 q^{29} -7.00000 q^{30} -1.44375 q^{31} +3.06327 q^{32} +2.28514 q^{33} -10.7429 q^{34} -0.950771 q^{35} +9.52106 q^{36} -0.792161 q^{37} +20.8062 q^{38} -7.00000 q^{40} +9.74293 q^{41} +4.45779 q^{42} -9.67967 q^{43} +4.28514 q^{44} -2.71486 q^{45} -6.84139 q^{46} -3.38049 q^{47} +13.2359 q^{48} -6.39452 q^{49} -8.79216 q^{50} -9.79216 q^{51} +1.79216 q^{53} -4.45779 q^{54} -1.22188 q^{55} +4.45779 q^{56} +18.9648 q^{57} +18.4578 q^{58} +5.20784 q^{59} -11.9648 q^{60} -14.5211 q^{61} -3.61951 q^{62} +1.72889 q^{63} -3.90466 q^{64} +5.72889 q^{66} +6.06327 q^{67} -18.3624 q^{68} -6.23591 q^{69} -2.38360 q^{70} -6.50702 q^{71} +12.7289 q^{72} -2.00000 q^{73} -1.98596 q^{74} -8.01404 q^{75} +35.5632 q^{76} +0.778124 q^{77} +13.5843 q^{79} -7.07730 q^{80} -10.7289 q^{81} +24.4257 q^{82} -13.3624 q^{83} +7.61951 q^{84} +5.23591 q^{85} -24.2671 q^{86} +16.8242 q^{87} +5.72889 q^{88} -4.77812 q^{89} -6.80620 q^{90} -11.6937 q^{92} -3.29918 q^{93} -8.47494 q^{94} -10.1406 q^{95} +7.00000 q^{96} -4.14057 q^{97} -16.0312 q^{98} +2.22188 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + q^{3} + 7 q^{4} - q^{5} + 6 q^{6} + 5 q^{7} + 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + q^{3} + 7 q^{4} - q^{5} + 6 q^{6} + 5 q^{7} + 6 q^{8} + 4 q^{9} - 6 q^{10} + 3 q^{11} + 15 q^{12} - 8 q^{14} + 6 q^{15} + 3 q^{16} - 7 q^{17} + 5 q^{18} + 2 q^{19} + 4 q^{20} + 8 q^{21} - q^{22} + 3 q^{23} + 2 q^{24} - 2 q^{25} - 8 q^{27} + 18 q^{28} - 4 q^{29} - 21 q^{30} + q^{31} + 6 q^{32} + q^{33} - 4 q^{34} + 11 q^{35} + 3 q^{36} + 12 q^{37} + 31 q^{38} - 21 q^{40} + q^{41} - 9 q^{42} - 4 q^{43} + 7 q^{44} - 14 q^{45} - 20 q^{46} - 8 q^{47} + 20 q^{48} - 12 q^{50} - 15 q^{51} - 9 q^{53} + 9 q^{54} - q^{55} - 9 q^{56} + 26 q^{57} + 33 q^{58} + 30 q^{59} - 5 q^{60} - 18 q^{61} - 13 q^{62} - 6 q^{63} - 8 q^{64} + 6 q^{66} + 15 q^{67} - 29 q^{68} + q^{69} - 29 q^{70} - 11 q^{71} + 27 q^{72} - 6 q^{73} - 23 q^{74} - 7 q^{75} + 30 q^{76} + 5 q^{77} + 12 q^{79} - q^{80} - 21 q^{81} + 44 q^{82} - 14 q^{83} + 25 q^{84} - 4 q^{85} - 43 q^{86} + 43 q^{87} + 6 q^{88} - 17 q^{89} + 11 q^{90} + 7 q^{92} + 13 q^{93} - 10 q^{94} - 7 q^{95} + 21 q^{96} + 11 q^{97} - 38 q^{98} + 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.50702 1.77273 0.886365 0.462987i \(-0.153222\pi\)
0.886365 + 0.462987i \(0.153222\pi\)
\(3\) 2.28514 1.31933 0.659664 0.751561i \(-0.270699\pi\)
0.659664 + 0.751561i \(0.270699\pi\)
\(4\) 4.28514 2.14257
\(5\) −1.22188 −0.546440 −0.273220 0.961952i \(-0.588089\pi\)
−0.273220 + 0.961952i \(0.588089\pi\)
\(6\) 5.72889 2.33881
\(7\) 0.778124 0.294103 0.147052 0.989129i \(-0.453022\pi\)
0.147052 + 0.989129i \(0.453022\pi\)
\(8\) 5.72889 2.02547
\(9\) 2.22188 0.740625
\(10\) −3.06327 −0.968690
\(11\) 1.00000 0.301511
\(12\) 9.79216 2.82675
\(13\) 0 0
\(14\) 1.95077 0.521365
\(15\) −2.79216 −0.720933
\(16\) 5.79216 1.44804
\(17\) −4.28514 −1.03930 −0.519650 0.854379i \(-0.673938\pi\)
−0.519650 + 0.854379i \(0.673938\pi\)
\(18\) 5.57028 1.31293
\(19\) 8.29918 1.90396 0.951981 0.306156i \(-0.0990431\pi\)
0.951981 + 0.306156i \(0.0990431\pi\)
\(20\) −5.23591 −1.17079
\(21\) 1.77812 0.388018
\(22\) 2.50702 0.534498
\(23\) −2.72889 −0.569014 −0.284507 0.958674i \(-0.591830\pi\)
−0.284507 + 0.958674i \(0.591830\pi\)
\(24\) 13.0913 2.67226
\(25\) −3.50702 −0.701404
\(26\) 0 0
\(27\) −1.77812 −0.342200
\(28\) 3.33437 0.630137
\(29\) 7.36245 1.36717 0.683586 0.729870i \(-0.260419\pi\)
0.683586 + 0.729870i \(0.260419\pi\)
\(30\) −7.00000 −1.27802
\(31\) −1.44375 −0.259306 −0.129653 0.991559i \(-0.541386\pi\)
−0.129653 + 0.991559i \(0.541386\pi\)
\(32\) 3.06327 0.541514
\(33\) 2.28514 0.397792
\(34\) −10.7429 −1.84240
\(35\) −0.950771 −0.160710
\(36\) 9.52106 1.58684
\(37\) −0.792161 −0.130230 −0.0651152 0.997878i \(-0.520742\pi\)
−0.0651152 + 0.997878i \(0.520742\pi\)
\(38\) 20.8062 3.37521
\(39\) 0 0
\(40\) −7.00000 −1.10680
\(41\) 9.74293 1.52159 0.760795 0.648992i \(-0.224809\pi\)
0.760795 + 0.648992i \(0.224809\pi\)
\(42\) 4.45779 0.687852
\(43\) −9.67967 −1.47614 −0.738068 0.674727i \(-0.764261\pi\)
−0.738068 + 0.674727i \(0.764261\pi\)
\(44\) 4.28514 0.646010
\(45\) −2.71486 −0.404707
\(46\) −6.84139 −1.00871
\(47\) −3.38049 −0.493095 −0.246547 0.969131i \(-0.579296\pi\)
−0.246547 + 0.969131i \(0.579296\pi\)
\(48\) 13.2359 1.91044
\(49\) −6.39452 −0.913503
\(50\) −8.79216 −1.24340
\(51\) −9.79216 −1.37118
\(52\) 0 0
\(53\) 1.79216 0.246172 0.123086 0.992396i \(-0.460721\pi\)
0.123086 + 0.992396i \(0.460721\pi\)
\(54\) −4.45779 −0.606628
\(55\) −1.22188 −0.164758
\(56\) 4.45779 0.595697
\(57\) 18.9648 2.51195
\(58\) 18.4578 2.42363
\(59\) 5.20784 0.678003 0.339001 0.940786i \(-0.389911\pi\)
0.339001 + 0.940786i \(0.389911\pi\)
\(60\) −11.9648 −1.54465
\(61\) −14.5211 −1.85923 −0.929615 0.368531i \(-0.879861\pi\)
−0.929615 + 0.368531i \(0.879861\pi\)
\(62\) −3.61951 −0.459679
\(63\) 1.72889 0.217820
\(64\) −3.90466 −0.488082
\(65\) 0 0
\(66\) 5.72889 0.705178
\(67\) 6.06327 0.740746 0.370373 0.928883i \(-0.379230\pi\)
0.370373 + 0.928883i \(0.379230\pi\)
\(68\) −18.3624 −2.22677
\(69\) −6.23591 −0.750716
\(70\) −2.38360 −0.284895
\(71\) −6.50702 −0.772241 −0.386121 0.922448i \(-0.626185\pi\)
−0.386121 + 0.922448i \(0.626185\pi\)
\(72\) 12.7289 1.50011
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −1.98596 −0.230863
\(75\) −8.01404 −0.925381
\(76\) 35.5632 4.07938
\(77\) 0.778124 0.0886754
\(78\) 0 0
\(79\) 13.5843 1.52836 0.764178 0.645006i \(-0.223145\pi\)
0.764178 + 0.645006i \(0.223145\pi\)
\(80\) −7.07730 −0.791267
\(81\) −10.7289 −1.19210
\(82\) 24.4257 2.69737
\(83\) −13.3624 −1.46672 −0.733360 0.679841i \(-0.762049\pi\)
−0.733360 + 0.679841i \(0.762049\pi\)
\(84\) 7.61951 0.831357
\(85\) 5.23591 0.567915
\(86\) −24.2671 −2.61679
\(87\) 16.8242 1.80375
\(88\) 5.72889 0.610702
\(89\) −4.77812 −0.506480 −0.253240 0.967403i \(-0.581496\pi\)
−0.253240 + 0.967403i \(0.581496\pi\)
\(90\) −6.80620 −0.717436
\(91\) 0 0
\(92\) −11.6937 −1.21915
\(93\) −3.29918 −0.342109
\(94\) −8.47494 −0.874123
\(95\) −10.1406 −1.04040
\(96\) 7.00000 0.714435
\(97\) −4.14057 −0.420411 −0.210206 0.977657i \(-0.567413\pi\)
−0.210206 + 0.977657i \(0.567413\pi\)
\(98\) −16.0312 −1.61939
\(99\) 2.22188 0.223307
\(100\) −15.0281 −1.50281
\(101\) 4.82424 0.480030 0.240015 0.970769i \(-0.422848\pi\)
0.240015 + 0.970769i \(0.422848\pi\)
\(102\) −24.5491 −2.43073
\(103\) 9.76008 0.961690 0.480845 0.876806i \(-0.340330\pi\)
0.480845 + 0.876806i \(0.340330\pi\)
\(104\) 0 0
\(105\) −2.17265 −0.212029
\(106\) 4.49298 0.436397
\(107\) 2.36245 0.228386 0.114193 0.993459i \(-0.463572\pi\)
0.114193 + 0.993459i \(0.463572\pi\)
\(108\) −7.61951 −0.733188
\(109\) −2.86946 −0.274845 −0.137422 0.990513i \(-0.543882\pi\)
−0.137422 + 0.990513i \(0.543882\pi\)
\(110\) −3.06327 −0.292071
\(111\) −1.81020 −0.171817
\(112\) 4.50702 0.425873
\(113\) −5.60236 −0.527026 −0.263513 0.964656i \(-0.584881\pi\)
−0.263513 + 0.964656i \(0.584881\pi\)
\(114\) 47.5451 4.45301
\(115\) 3.33437 0.310932
\(116\) 31.5491 2.92926
\(117\) 0 0
\(118\) 13.0561 1.20192
\(119\) −3.33437 −0.305661
\(120\) −15.9960 −1.46023
\(121\) 1.00000 0.0909091
\(122\) −36.4046 −3.29591
\(123\) 22.2640 2.00748
\(124\) −6.18668 −0.555581
\(125\) 10.3945 0.929714
\(126\) 4.33437 0.386137
\(127\) 7.23591 0.642083 0.321042 0.947065i \(-0.395967\pi\)
0.321042 + 0.947065i \(0.395967\pi\)
\(128\) −15.9156 −1.40675
\(129\) −22.1194 −1.94751
\(130\) 0 0
\(131\) −10.3484 −0.904145 −0.452072 0.891981i \(-0.649315\pi\)
−0.452072 + 0.891981i \(0.649315\pi\)
\(132\) 9.79216 0.852298
\(133\) 6.45779 0.559961
\(134\) 15.2007 1.31314
\(135\) 2.17265 0.186992
\(136\) −24.5491 −2.10507
\(137\) −4.93673 −0.421774 −0.210887 0.977510i \(-0.567635\pi\)
−0.210887 + 0.977510i \(0.567635\pi\)
\(138\) −15.6336 −1.33082
\(139\) 16.4718 1.39712 0.698561 0.715550i \(-0.253824\pi\)
0.698561 + 0.715550i \(0.253824\pi\)
\(140\) −4.07419 −0.344332
\(141\) −7.72489 −0.650553
\(142\) −16.3132 −1.36897
\(143\) 0 0
\(144\) 12.8695 1.07246
\(145\) −8.99600 −0.747077
\(146\) −5.01404 −0.414965
\(147\) −14.6124 −1.20521
\(148\) −3.39452 −0.279028
\(149\) −1.90466 −0.156036 −0.0780178 0.996952i \(-0.524859\pi\)
−0.0780178 + 0.996952i \(0.524859\pi\)
\(150\) −20.0913 −1.64045
\(151\) 18.4086 1.49807 0.749034 0.662532i \(-0.230518\pi\)
0.749034 + 0.662532i \(0.230518\pi\)
\(152\) 47.5451 3.85642
\(153\) −9.52106 −0.769732
\(154\) 1.95077 0.157198
\(155\) 1.76409 0.141695
\(156\) 0 0
\(157\) −12.1054 −0.966114 −0.483057 0.875589i \(-0.660474\pi\)
−0.483057 + 0.875589i \(0.660474\pi\)
\(158\) 34.0561 2.70936
\(159\) 4.09534 0.324782
\(160\) −3.74293 −0.295905
\(161\) −2.12342 −0.167349
\(162\) −26.8975 −2.11327
\(163\) 1.91869 0.150284 0.0751418 0.997173i \(-0.476059\pi\)
0.0751418 + 0.997173i \(0.476059\pi\)
\(164\) 41.7499 3.26012
\(165\) −2.79216 −0.217369
\(166\) −33.4999 −2.60010
\(167\) −7.06015 −0.546331 −0.273165 0.961967i \(-0.588071\pi\)
−0.273165 + 0.961967i \(0.588071\pi\)
\(168\) 10.1867 0.785920
\(169\) 0 0
\(170\) 13.1265 1.00676
\(171\) 18.4397 1.41012
\(172\) −41.4787 −3.16272
\(173\) 1.64759 0.125264 0.0626319 0.998037i \(-0.480051\pi\)
0.0626319 + 0.998037i \(0.480051\pi\)
\(174\) 42.1787 3.19756
\(175\) −2.72889 −0.206285
\(176\) 5.79216 0.436601
\(177\) 11.9007 0.894508
\(178\) −11.9788 −0.897852
\(179\) 7.33437 0.548197 0.274098 0.961702i \(-0.411621\pi\)
0.274098 + 0.961702i \(0.411621\pi\)
\(180\) −11.6336 −0.867114
\(181\) 11.7750 0.875230 0.437615 0.899163i \(-0.355823\pi\)
0.437615 + 0.899163i \(0.355823\pi\)
\(182\) 0 0
\(183\) −33.1827 −2.45293
\(184\) −15.6336 −1.15252
\(185\) 0.967923 0.0711631
\(186\) −8.27111 −0.606467
\(187\) −4.28514 −0.313361
\(188\) −14.4859 −1.05649
\(189\) −1.38360 −0.100642
\(190\) −25.4226 −1.84435
\(191\) 14.7429 1.06676 0.533381 0.845875i \(-0.320921\pi\)
0.533381 + 0.845875i \(0.320921\pi\)
\(192\) −8.92270 −0.643940
\(193\) −8.90154 −0.640747 −0.320374 0.947291i \(-0.603808\pi\)
−0.320374 + 0.947291i \(0.603808\pi\)
\(194\) −10.3805 −0.745275
\(195\) 0 0
\(196\) −27.4014 −1.95725
\(197\) −6.11250 −0.435497 −0.217749 0.976005i \(-0.569871\pi\)
−0.217749 + 0.976005i \(0.569871\pi\)
\(198\) 5.57028 0.395863
\(199\) 9.34529 0.662470 0.331235 0.943548i \(-0.392535\pi\)
0.331235 + 0.943548i \(0.392535\pi\)
\(200\) −20.0913 −1.42067
\(201\) 13.8554 0.977287
\(202\) 12.0945 0.850963
\(203\) 5.72889 0.402090
\(204\) −41.9608 −2.93784
\(205\) −11.9047 −0.831457
\(206\) 24.4687 1.70482
\(207\) −6.06327 −0.421426
\(208\) 0 0
\(209\) 8.29918 0.574066
\(210\) −5.44687 −0.375870
\(211\) −5.18668 −0.357066 −0.178533 0.983934i \(-0.557135\pi\)
−0.178533 + 0.983934i \(0.557135\pi\)
\(212\) 7.67967 0.527442
\(213\) −14.8695 −1.01884
\(214\) 5.92270 0.404867
\(215\) 11.8274 0.806619
\(216\) −10.1867 −0.693116
\(217\) −1.12342 −0.0762626
\(218\) −7.19380 −0.487226
\(219\) −4.57028 −0.308831
\(220\) −5.23591 −0.353005
\(221\) 0 0
\(222\) −4.53821 −0.304585
\(223\) 16.8343 1.12731 0.563653 0.826012i \(-0.309395\pi\)
0.563653 + 0.826012i \(0.309395\pi\)
\(224\) 2.38360 0.159261
\(225\) −7.79216 −0.519477
\(226\) −14.0452 −0.934275
\(227\) −17.6164 −1.16924 −0.584621 0.811307i \(-0.698757\pi\)
−0.584621 + 0.811307i \(0.698757\pi\)
\(228\) 81.2669 5.38203
\(229\) 26.8554 1.77466 0.887328 0.461138i \(-0.152559\pi\)
0.887328 + 0.461138i \(0.152559\pi\)
\(230\) 8.35933 0.551198
\(231\) 1.77812 0.116992
\(232\) 42.1787 2.76917
\(233\) −27.1406 −1.77804 −0.889019 0.457870i \(-0.848612\pi\)
−0.889019 + 0.457870i \(0.848612\pi\)
\(234\) 0 0
\(235\) 4.13054 0.269446
\(236\) 22.3163 1.45267
\(237\) 31.0421 2.01640
\(238\) −8.35933 −0.541855
\(239\) 0.746047 0.0482577 0.0241289 0.999709i \(-0.492319\pi\)
0.0241289 + 0.999709i \(0.492319\pi\)
\(240\) −16.1726 −1.04394
\(241\) 16.1726 1.04177 0.520886 0.853626i \(-0.325602\pi\)
0.520886 + 0.853626i \(0.325602\pi\)
\(242\) 2.50702 0.161157
\(243\) −19.1827 −1.23057
\(244\) −62.2248 −3.98353
\(245\) 7.81332 0.499174
\(246\) 55.8162 3.55871
\(247\) 0 0
\(248\) −8.27111 −0.525216
\(249\) −30.5351 −1.93508
\(250\) 26.0593 1.64813
\(251\) −11.0281 −0.696086 −0.348043 0.937479i \(-0.613154\pi\)
−0.348043 + 0.937479i \(0.613154\pi\)
\(252\) 7.40856 0.466695
\(253\) −2.72889 −0.171564
\(254\) 18.1406 1.13824
\(255\) 11.9648 0.749265
\(256\) −32.0913 −2.00571
\(257\) 18.8523 1.17597 0.587987 0.808870i \(-0.299920\pi\)
0.587987 + 0.808870i \(0.299920\pi\)
\(258\) −55.4538 −3.45240
\(259\) −0.616399 −0.0383012
\(260\) 0 0
\(261\) 16.3584 1.01256
\(262\) −25.9437 −1.60280
\(263\) −15.5522 −0.958993 −0.479496 0.877544i \(-0.659181\pi\)
−0.479496 + 0.877544i \(0.659181\pi\)
\(264\) 13.0913 0.805716
\(265\) −2.18980 −0.134518
\(266\) 16.1898 0.992660
\(267\) −10.9187 −0.668213
\(268\) 25.9820 1.58710
\(269\) −5.22188 −0.318383 −0.159192 0.987248i \(-0.550889\pi\)
−0.159192 + 0.987248i \(0.550889\pi\)
\(270\) 5.44687 0.331486
\(271\) 25.9367 1.57554 0.787772 0.615967i \(-0.211234\pi\)
0.787772 + 0.615967i \(0.211234\pi\)
\(272\) −24.8202 −1.50495
\(273\) 0 0
\(274\) −12.3765 −0.747691
\(275\) −3.50702 −0.211481
\(276\) −26.7218 −1.60846
\(277\) −17.7741 −1.06794 −0.533972 0.845502i \(-0.679301\pi\)
−0.533972 + 0.845502i \(0.679301\pi\)
\(278\) 41.2952 2.47672
\(279\) −3.20784 −0.192048
\(280\) −5.44687 −0.325513
\(281\) 7.41168 0.442143 0.221072 0.975258i \(-0.429045\pi\)
0.221072 + 0.975258i \(0.429045\pi\)
\(282\) −19.3664 −1.15326
\(283\) 9.31722 0.553851 0.276926 0.960891i \(-0.410684\pi\)
0.276926 + 0.960891i \(0.410684\pi\)
\(284\) −27.8835 −1.65458
\(285\) −23.1726 −1.37263
\(286\) 0 0
\(287\) 7.58121 0.447505
\(288\) 6.80620 0.401059
\(289\) 1.36245 0.0801439
\(290\) −22.5531 −1.32437
\(291\) −9.46179 −0.554660
\(292\) −8.57028 −0.501538
\(293\) −28.7530 −1.67977 −0.839883 0.542767i \(-0.817377\pi\)
−0.839883 + 0.542767i \(0.817377\pi\)
\(294\) −36.6336 −2.13651
\(295\) −6.36333 −0.370488
\(296\) −4.53821 −0.263778
\(297\) −1.77812 −0.103177
\(298\) −4.77501 −0.276609
\(299\) 0 0
\(300\) −34.3413 −1.98270
\(301\) −7.53198 −0.434136
\(302\) 46.1506 2.65567
\(303\) 11.0241 0.633316
\(304\) 48.0702 2.75701
\(305\) 17.7429 1.01596
\(306\) −23.8695 −1.36453
\(307\) −0.792161 −0.0452110 −0.0226055 0.999744i \(-0.507196\pi\)
−0.0226055 + 0.999744i \(0.507196\pi\)
\(308\) 3.33437 0.189993
\(309\) 22.3032 1.26878
\(310\) 4.42260 0.251187
\(311\) 25.8022 1.46311 0.731554 0.681783i \(-0.238795\pi\)
0.731554 + 0.681783i \(0.238795\pi\)
\(312\) 0 0
\(313\) −26.2780 −1.48532 −0.742661 0.669668i \(-0.766437\pi\)
−0.742661 + 0.669668i \(0.766437\pi\)
\(314\) −30.3484 −1.71266
\(315\) −2.11250 −0.119026
\(316\) 58.2108 3.27461
\(317\) 10.9788 0.616633 0.308317 0.951284i \(-0.400234\pi\)
0.308317 + 0.951284i \(0.400234\pi\)
\(318\) 10.2671 0.575751
\(319\) 7.36245 0.412218
\(320\) 4.77101 0.266707
\(321\) 5.39853 0.301316
\(322\) −5.32345 −0.296664
\(323\) −35.5632 −1.97879
\(324\) −45.9748 −2.55416
\(325\) 0 0
\(326\) 4.81020 0.266412
\(327\) −6.55714 −0.362610
\(328\) 55.8162 3.08194
\(329\) −2.63044 −0.145021
\(330\) −7.00000 −0.385337
\(331\) 10.0561 0.552736 0.276368 0.961052i \(-0.410869\pi\)
0.276368 + 0.961052i \(0.410869\pi\)
\(332\) −57.2600 −3.14255
\(333\) −1.76008 −0.0964520
\(334\) −17.6999 −0.968497
\(335\) −7.40856 −0.404773
\(336\) 10.2992 0.561866
\(337\) 19.7398 1.07530 0.537648 0.843169i \(-0.319313\pi\)
0.537648 + 0.843169i \(0.319313\pi\)
\(338\) 0 0
\(339\) −12.8022 −0.695320
\(340\) 22.4366 1.21680
\(341\) −1.44375 −0.0781836
\(342\) 46.2288 2.49977
\(343\) −10.4226 −0.562767
\(344\) −55.4538 −2.98987
\(345\) 7.61951 0.410221
\(346\) 4.13054 0.222059
\(347\) 18.3132 0.983105 0.491553 0.870848i \(-0.336430\pi\)
0.491553 + 0.870848i \(0.336430\pi\)
\(348\) 72.0943 3.86466
\(349\) 17.3765 0.930142 0.465071 0.885273i \(-0.346029\pi\)
0.465071 + 0.885273i \(0.346029\pi\)
\(350\) −6.84139 −0.365688
\(351\) 0 0
\(352\) 3.06327 0.163273
\(353\) 33.7670 1.79724 0.898618 0.438732i \(-0.144572\pi\)
0.898618 + 0.438732i \(0.144572\pi\)
\(354\) 29.8352 1.58572
\(355\) 7.95077 0.421983
\(356\) −20.4749 −1.08517
\(357\) −7.61951 −0.403267
\(358\) 18.3874 0.971805
\(359\) −4.72889 −0.249582 −0.124791 0.992183i \(-0.539826\pi\)
−0.124791 + 0.992183i \(0.539826\pi\)
\(360\) −15.5531 −0.819722
\(361\) 49.8764 2.62507
\(362\) 29.5202 1.55155
\(363\) 2.28514 0.119939
\(364\) 0 0
\(365\) 2.44375 0.127912
\(366\) −83.1896 −4.34839
\(367\) 6.42660 0.335466 0.167733 0.985832i \(-0.446355\pi\)
0.167733 + 0.985832i \(0.446355\pi\)
\(368\) −15.8062 −0.823955
\(369\) 21.6476 1.12693
\(370\) 2.42660 0.126153
\(371\) 1.39452 0.0724000
\(372\) −14.1375 −0.732993
\(373\) −4.41168 −0.228428 −0.114214 0.993456i \(-0.536435\pi\)
−0.114214 + 0.993456i \(0.536435\pi\)
\(374\) −10.7429 −0.555504
\(375\) 23.7530 1.22660
\(376\) −19.3664 −0.998748
\(377\) 0 0
\(378\) −3.46871 −0.178411
\(379\) 19.5070 1.00201 0.501004 0.865445i \(-0.332964\pi\)
0.501004 + 0.865445i \(0.332964\pi\)
\(380\) −43.4538 −2.22913
\(381\) 16.5351 0.847118
\(382\) 36.9608 1.89108
\(383\) 27.6304 1.41185 0.705925 0.708287i \(-0.250532\pi\)
0.705925 + 0.708287i \(0.250532\pi\)
\(384\) −36.3694 −1.85597
\(385\) −0.950771 −0.0484558
\(386\) −22.3163 −1.13587
\(387\) −21.5070 −1.09326
\(388\) −17.7429 −0.900761
\(389\) −28.9960 −1.47016 −0.735078 0.677983i \(-0.762854\pi\)
−0.735078 + 0.677983i \(0.762854\pi\)
\(390\) 0 0
\(391\) 11.6937 0.591376
\(392\) −36.6336 −1.85027
\(393\) −23.6476 −1.19286
\(394\) −15.3241 −0.772019
\(395\) −16.5984 −0.835154
\(396\) 9.52106 0.478451
\(397\) 34.6585 1.73946 0.869730 0.493527i \(-0.164293\pi\)
0.869730 + 0.493527i \(0.164293\pi\)
\(398\) 23.4288 1.17438
\(399\) 14.7570 0.738773
\(400\) −20.3132 −1.01566
\(401\) −14.5874 −0.728462 −0.364231 0.931309i \(-0.618668\pi\)
−0.364231 + 0.931309i \(0.618668\pi\)
\(402\) 34.7358 1.73246
\(403\) 0 0
\(404\) 20.6725 1.02850
\(405\) 13.1094 0.651410
\(406\) 14.3624 0.712796
\(407\) −0.792161 −0.0392660
\(408\) −56.0983 −2.77728
\(409\) 16.6015 0.820890 0.410445 0.911885i \(-0.365373\pi\)
0.410445 + 0.911885i \(0.365373\pi\)
\(410\) −29.8452 −1.47395
\(411\) −11.2811 −0.556458
\(412\) 41.8234 2.06049
\(413\) 4.05234 0.199403
\(414\) −15.2007 −0.747075
\(415\) 16.3273 0.801473
\(416\) 0 0
\(417\) 37.6405 1.84326
\(418\) 20.8062 1.01766
\(419\) −22.2952 −1.08919 −0.544595 0.838699i \(-0.683317\pi\)
−0.544595 + 0.838699i \(0.683317\pi\)
\(420\) −9.31010 −0.454286
\(421\) −3.45779 −0.168522 −0.0842612 0.996444i \(-0.526853\pi\)
−0.0842612 + 0.996444i \(0.526853\pi\)
\(422\) −13.0031 −0.632982
\(423\) −7.51102 −0.365198
\(424\) 10.2671 0.498615
\(425\) 15.0281 0.728969
\(426\) −37.2780 −1.80613
\(427\) −11.2992 −0.546806
\(428\) 10.1234 0.489334
\(429\) 0 0
\(430\) 29.6514 1.42992
\(431\) 25.8804 1.24661 0.623307 0.781977i \(-0.285789\pi\)
0.623307 + 0.781977i \(0.285789\pi\)
\(432\) −10.2992 −0.495520
\(433\) 14.3805 0.691082 0.345541 0.938404i \(-0.387695\pi\)
0.345541 + 0.938404i \(0.387695\pi\)
\(434\) −2.81643 −0.135193
\(435\) −20.5571 −0.985639
\(436\) −12.2961 −0.588875
\(437\) −22.6476 −1.08338
\(438\) −11.4578 −0.547474
\(439\) 4.56628 0.217937 0.108968 0.994045i \(-0.465245\pi\)
0.108968 + 0.994045i \(0.465245\pi\)
\(440\) −7.00000 −0.333712
\(441\) −14.2078 −0.676564
\(442\) 0 0
\(443\) 18.1718 0.863366 0.431683 0.902025i \(-0.357920\pi\)
0.431683 + 0.902025i \(0.357920\pi\)
\(444\) −7.75697 −0.368129
\(445\) 5.83828 0.276761
\(446\) 42.2038 1.99841
\(447\) −4.35241 −0.205862
\(448\) −3.03831 −0.143546
\(449\) −12.6295 −0.596025 −0.298013 0.954562i \(-0.596324\pi\)
−0.298013 + 0.954562i \(0.596324\pi\)
\(450\) −19.5351 −0.920893
\(451\) 9.74293 0.458777
\(452\) −24.0069 −1.12919
\(453\) 42.0662 1.97644
\(454\) −44.1646 −2.07275
\(455\) 0 0
\(456\) 108.647 5.08788
\(457\) −25.7882 −1.20632 −0.603160 0.797621i \(-0.706092\pi\)
−0.603160 + 0.797621i \(0.706092\pi\)
\(458\) 67.3271 3.14599
\(459\) 7.61951 0.355648
\(460\) 14.2883 0.666193
\(461\) −12.6336 −0.588403 −0.294202 0.955743i \(-0.595054\pi\)
−0.294202 + 0.955743i \(0.595054\pi\)
\(462\) 4.45779 0.207395
\(463\) −3.47494 −0.161494 −0.0807471 0.996735i \(-0.525731\pi\)
−0.0807471 + 0.996735i \(0.525731\pi\)
\(464\) 42.6445 1.97972
\(465\) 4.03119 0.186942
\(466\) −68.0419 −3.15198
\(467\) −18.7258 −0.866526 −0.433263 0.901268i \(-0.642638\pi\)
−0.433263 + 0.901268i \(0.642638\pi\)
\(468\) 0 0
\(469\) 4.71797 0.217856
\(470\) 10.3553 0.477656
\(471\) −27.6625 −1.27462
\(472\) 29.8352 1.37327
\(473\) −9.67967 −0.445072
\(474\) 77.8232 3.57454
\(475\) −29.1054 −1.33545
\(476\) −14.2883 −0.654901
\(477\) 3.98196 0.182321
\(478\) 1.87035 0.0855479
\(479\) 9.28514 0.424249 0.212124 0.977243i \(-0.431962\pi\)
0.212124 + 0.977243i \(0.431962\pi\)
\(480\) −8.55313 −0.390395
\(481\) 0 0
\(482\) 40.5451 1.84678
\(483\) −4.85231 −0.220788
\(484\) 4.28514 0.194779
\(485\) 5.05926 0.229729
\(486\) −48.0913 −2.18147
\(487\) −39.9296 −1.80938 −0.904692 0.426067i \(-0.859899\pi\)
−0.904692 + 0.426067i \(0.859899\pi\)
\(488\) −83.1896 −3.76582
\(489\) 4.38449 0.198273
\(490\) 19.5881 0.884901
\(491\) −41.1827 −1.85855 −0.929274 0.369391i \(-0.879566\pi\)
−0.929274 + 0.369391i \(0.879566\pi\)
\(492\) 95.4044 4.30116
\(493\) −31.5491 −1.42090
\(494\) 0 0
\(495\) −2.71486 −0.122024
\(496\) −8.36245 −0.375485
\(497\) −5.06327 −0.227119
\(498\) −76.5520 −3.43038
\(499\) 19.3132 0.864578 0.432289 0.901735i \(-0.357706\pi\)
0.432289 + 0.901735i \(0.357706\pi\)
\(500\) 44.5420 1.99198
\(501\) −16.1335 −0.720790
\(502\) −27.6476 −1.23397
\(503\) 19.6897 0.877920 0.438960 0.898507i \(-0.355347\pi\)
0.438960 + 0.898507i \(0.355347\pi\)
\(504\) 9.90466 0.441188
\(505\) −5.89462 −0.262307
\(506\) −6.84139 −0.304137
\(507\) 0 0
\(508\) 31.0069 1.37571
\(509\) 13.4749 0.597266 0.298633 0.954368i \(-0.403469\pi\)
0.298633 + 0.954368i \(0.403469\pi\)
\(510\) 29.9960 1.32825
\(511\) −1.55625 −0.0688443
\(512\) −48.6224 −2.14883
\(513\) −14.7570 −0.651536
\(514\) 47.2631 2.08469
\(515\) −11.9256 −0.525505
\(516\) −94.7848 −4.17267
\(517\) −3.38049 −0.148674
\(518\) −1.54532 −0.0678977
\(519\) 3.76497 0.165264
\(520\) 0 0
\(521\) 19.4266 0.851095 0.425547 0.904936i \(-0.360082\pi\)
0.425547 + 0.904936i \(0.360082\pi\)
\(522\) 41.0109 1.79500
\(523\) 10.9227 0.477616 0.238808 0.971067i \(-0.423243\pi\)
0.238808 + 0.971067i \(0.423243\pi\)
\(524\) −44.3444 −1.93719
\(525\) −6.23591 −0.272158
\(526\) −38.9898 −1.70003
\(527\) 6.18668 0.269496
\(528\) 13.2359 0.576019
\(529\) −15.5531 −0.676223
\(530\) −5.48987 −0.238465
\(531\) 11.5712 0.502146
\(532\) 27.6725 1.19976
\(533\) 0 0
\(534\) −27.3734 −1.18456
\(535\) −2.88662 −0.124799
\(536\) 34.7358 1.50036
\(537\) 16.7601 0.723251
\(538\) −13.0913 −0.564408
\(539\) −6.39452 −0.275432
\(540\) 9.31010 0.400643
\(541\) −16.9820 −0.730111 −0.365056 0.930986i \(-0.618950\pi\)
−0.365056 + 0.930986i \(0.618950\pi\)
\(542\) 65.0239 2.79301
\(543\) 26.9076 1.15471
\(544\) −13.1265 −0.562795
\(545\) 3.50613 0.150186
\(546\) 0 0
\(547\) 1.13345 0.0484629 0.0242315 0.999706i \(-0.492286\pi\)
0.0242315 + 0.999706i \(0.492286\pi\)
\(548\) −21.1546 −0.903680
\(549\) −32.2640 −1.37699
\(550\) −8.79216 −0.374899
\(551\) 61.1023 2.60304
\(552\) −35.7249 −1.52055
\(553\) 10.5703 0.449494
\(554\) −44.5601 −1.89318
\(555\) 2.21184 0.0938874
\(556\) 70.5841 2.99343
\(557\) −20.2671 −0.858745 −0.429372 0.903128i \(-0.641265\pi\)
−0.429372 + 0.903128i \(0.641265\pi\)
\(558\) −8.04211 −0.340450
\(559\) 0 0
\(560\) −5.50702 −0.232714
\(561\) −9.79216 −0.413425
\(562\) 18.5812 0.783801
\(563\) 0.313217 0.0132005 0.00660026 0.999978i \(-0.497899\pi\)
0.00660026 + 0.999978i \(0.497899\pi\)
\(564\) −33.1023 −1.39386
\(565\) 6.84539 0.287988
\(566\) 23.3584 0.981829
\(567\) −8.34841 −0.350600
\(568\) −37.2780 −1.56415
\(569\) −11.0842 −0.464675 −0.232337 0.972635i \(-0.574637\pi\)
−0.232337 + 0.972635i \(0.574637\pi\)
\(570\) −58.0943 −2.43330
\(571\) 41.7037 1.74525 0.872624 0.488393i \(-0.162417\pi\)
0.872624 + 0.488393i \(0.162417\pi\)
\(572\) 0 0
\(573\) 33.6897 1.40741
\(574\) 19.0062 0.793305
\(575\) 9.57028 0.399108
\(576\) −8.67566 −0.361486
\(577\) 19.1867 0.798752 0.399376 0.916787i \(-0.369227\pi\)
0.399376 + 0.916787i \(0.369227\pi\)
\(578\) 3.41568 0.142073
\(579\) −20.3413 −0.845355
\(580\) −38.5491 −1.60067
\(581\) −10.3976 −0.431367
\(582\) −23.7209 −0.983263
\(583\) 1.79216 0.0742237
\(584\) −11.4578 −0.474127
\(585\) 0 0
\(586\) −72.0842 −2.97777
\(587\) −12.5843 −0.519411 −0.259705 0.965688i \(-0.583625\pi\)
−0.259705 + 0.965688i \(0.583625\pi\)
\(588\) −62.6162 −2.58225
\(589\) −11.9820 −0.493708
\(590\) −15.9530 −0.656775
\(591\) −13.9679 −0.574564
\(592\) −4.58832 −0.188579
\(593\) −13.7117 −0.563074 −0.281537 0.959550i \(-0.590844\pi\)
−0.281537 + 0.959550i \(0.590844\pi\)
\(594\) −4.45779 −0.182905
\(595\) 4.07419 0.167025
\(596\) −8.16172 −0.334317
\(597\) 21.3553 0.874015
\(598\) 0 0
\(599\) −15.3905 −0.628840 −0.314420 0.949284i \(-0.601810\pi\)
−0.314420 + 0.949284i \(0.601810\pi\)
\(600\) −45.9116 −1.87433
\(601\) −16.2148 −0.661414 −0.330707 0.943734i \(-0.607287\pi\)
−0.330707 + 0.943734i \(0.607287\pi\)
\(602\) −18.8828 −0.769606
\(603\) 13.4718 0.548615
\(604\) 78.8833 3.20972
\(605\) −1.22188 −0.0496763
\(606\) 27.6376 1.12270
\(607\) 2.68278 0.108891 0.0544453 0.998517i \(-0.482661\pi\)
0.0544453 + 0.998517i \(0.482661\pi\)
\(608\) 25.4226 1.03102
\(609\) 13.0913 0.530488
\(610\) 44.4819 1.80102
\(611\) 0 0
\(612\) −40.7991 −1.64921
\(613\) −36.0913 −1.45772 −0.728858 0.684665i \(-0.759948\pi\)
−0.728858 + 0.684665i \(0.759948\pi\)
\(614\) −1.98596 −0.0801469
\(615\) −27.2038 −1.09696
\(616\) 4.45779 0.179609
\(617\) −19.1718 −0.771826 −0.385913 0.922535i \(-0.626114\pi\)
−0.385913 + 0.922535i \(0.626114\pi\)
\(618\) 55.9145 2.24921
\(619\) 6.38049 0.256453 0.128227 0.991745i \(-0.459071\pi\)
0.128227 + 0.991745i \(0.459071\pi\)
\(620\) 7.55936 0.303591
\(621\) 4.85231 0.194717
\(622\) 64.6866 2.59370
\(623\) −3.71797 −0.148957
\(624\) 0 0
\(625\) 4.83427 0.193371
\(626\) −65.8795 −2.63307
\(627\) 18.9648 0.757381
\(628\) −51.8733 −2.06997
\(629\) 3.39452 0.135349
\(630\) −5.29607 −0.211000
\(631\) −20.9007 −0.832042 −0.416021 0.909355i \(-0.636576\pi\)
−0.416021 + 0.909355i \(0.636576\pi\)
\(632\) 77.8232 3.09564
\(633\) −11.8523 −0.471087
\(634\) 27.5242 1.09312
\(635\) −8.84139 −0.350860
\(636\) 17.5491 0.695868
\(637\) 0 0
\(638\) 18.4578 0.730751
\(639\) −14.4578 −0.571941
\(640\) 19.4469 0.768705
\(641\) 19.1085 0.754740 0.377370 0.926063i \(-0.376828\pi\)
0.377370 + 0.926063i \(0.376828\pi\)
\(642\) 13.5342 0.534152
\(643\) −3.50702 −0.138303 −0.0691517 0.997606i \(-0.522029\pi\)
−0.0691517 + 0.997606i \(0.522029\pi\)
\(644\) −9.09915 −0.358557
\(645\) 27.0272 1.06419
\(646\) −89.1575 −3.50786
\(647\) 20.8343 0.819080 0.409540 0.912292i \(-0.365689\pi\)
0.409540 + 0.912292i \(0.365689\pi\)
\(648\) −61.4647 −2.41456
\(649\) 5.20784 0.204426
\(650\) 0 0
\(651\) −2.56717 −0.100615
\(652\) 8.22188 0.321994
\(653\) −12.6757 −0.496037 −0.248019 0.968755i \(-0.579779\pi\)
−0.248019 + 0.968755i \(0.579779\pi\)
\(654\) −16.4389 −0.642810
\(655\) 12.6445 0.494060
\(656\) 56.4326 2.20332
\(657\) −4.44375 −0.173367
\(658\) −6.59455 −0.257082
\(659\) 4.37960 0.170605 0.0853025 0.996355i \(-0.472814\pi\)
0.0853025 + 0.996355i \(0.472814\pi\)
\(660\) −11.9648 −0.465730
\(661\) −1.12253 −0.0436614 −0.0218307 0.999762i \(-0.506949\pi\)
−0.0218307 + 0.999762i \(0.506949\pi\)
\(662\) 25.2110 0.979852
\(663\) 0 0
\(664\) −76.5520 −2.97080
\(665\) −7.89062 −0.305985
\(666\) −4.41256 −0.170983
\(667\) −20.0913 −0.777940
\(668\) −30.2538 −1.17055
\(669\) 38.4687 1.48729
\(670\) −18.5734 −0.717553
\(671\) −14.5211 −0.560579
\(672\) 5.44687 0.210117
\(673\) 18.0492 0.695747 0.347873 0.937542i \(-0.386904\pi\)
0.347873 + 0.937542i \(0.386904\pi\)
\(674\) 49.4881 1.90621
\(675\) 6.23591 0.240020
\(676\) 0 0
\(677\) −26.7037 −1.02631 −0.513154 0.858297i \(-0.671523\pi\)
−0.513154 + 0.858297i \(0.671523\pi\)
\(678\) −32.0953 −1.23261
\(679\) −3.22188 −0.123644
\(680\) 29.9960 1.15029
\(681\) −40.2560 −1.54261
\(682\) −3.61951 −0.138598
\(683\) 30.9788 1.18537 0.592686 0.805433i \(-0.298067\pi\)
0.592686 + 0.805433i \(0.298067\pi\)
\(684\) 79.0170 3.02129
\(685\) 6.03208 0.230474
\(686\) −26.1296 −0.997635
\(687\) 61.3685 2.34135
\(688\) −56.0662 −2.13750
\(689\) 0 0
\(690\) 19.1023 0.727211
\(691\) −2.33126 −0.0886852 −0.0443426 0.999016i \(-0.514119\pi\)
−0.0443426 + 0.999016i \(0.514119\pi\)
\(692\) 7.06015 0.268387
\(693\) 1.72889 0.0656753
\(694\) 45.9116 1.74278
\(695\) −20.1265 −0.763443
\(696\) 96.3843 3.65344
\(697\) −41.7499 −1.58139
\(698\) 43.5632 1.64889
\(699\) −62.0201 −2.34581
\(700\) −11.6937 −0.441980
\(701\) −22.5834 −0.852965 −0.426482 0.904496i \(-0.640247\pi\)
−0.426482 + 0.904496i \(0.640247\pi\)
\(702\) 0 0
\(703\) −6.57429 −0.247954
\(704\) −3.90466 −0.147162
\(705\) 9.43886 0.355488
\(706\) 84.6545 3.18601
\(707\) 3.75385 0.141178
\(708\) 50.9960 1.91655
\(709\) −15.7218 −0.590444 −0.295222 0.955429i \(-0.595394\pi\)
−0.295222 + 0.955429i \(0.595394\pi\)
\(710\) 19.9327 0.748062
\(711\) 30.1827 1.13194
\(712\) −27.3734 −1.02586
\(713\) 3.93985 0.147548
\(714\) −19.1023 −0.714884
\(715\) 0 0
\(716\) 31.4288 1.17455
\(717\) 1.70482 0.0636678
\(718\) −11.8554 −0.442441
\(719\) −29.8303 −1.11248 −0.556241 0.831021i \(-0.687757\pi\)
−0.556241 + 0.831021i \(0.687757\pi\)
\(720\) −15.7249 −0.586032
\(721\) 7.59455 0.282836
\(722\) 125.041 4.65355
\(723\) 36.9568 1.37444
\(724\) 50.4576 1.87524
\(725\) −25.8202 −0.958939
\(726\) 5.72889 0.212619
\(727\) −36.6053 −1.35761 −0.678807 0.734316i \(-0.737503\pi\)
−0.678807 + 0.734316i \(0.737503\pi\)
\(728\) 0 0
\(729\) −11.6485 −0.431425
\(730\) 6.12653 0.226753
\(731\) 41.4787 1.53415
\(732\) −142.193 −5.25559
\(733\) 29.8695 1.10325 0.551627 0.834091i \(-0.314007\pi\)
0.551627 + 0.834091i \(0.314007\pi\)
\(734\) 16.1116 0.594690
\(735\) 17.8545 0.658575
\(736\) −8.35933 −0.308129
\(737\) 6.06327 0.223343
\(738\) 54.2709 1.99774
\(739\) 30.2569 1.11302 0.556508 0.830842i \(-0.312141\pi\)
0.556508 + 0.830842i \(0.312141\pi\)
\(740\) 4.14769 0.152472
\(741\) 0 0
\(742\) 3.49610 0.128346
\(743\) 39.2419 1.43965 0.719824 0.694157i \(-0.244223\pi\)
0.719824 + 0.694157i \(0.244223\pi\)
\(744\) −18.9007 −0.692932
\(745\) 2.32725 0.0852640
\(746\) −11.0602 −0.404941
\(747\) −29.6897 −1.08629
\(748\) −18.3624 −0.671398
\(749\) 1.83828 0.0671691
\(750\) 59.5491 2.17443
\(751\) 6.82335 0.248988 0.124494 0.992220i \(-0.460269\pi\)
0.124494 + 0.992220i \(0.460269\pi\)
\(752\) −19.5803 −0.714021
\(753\) −25.2007 −0.918365
\(754\) 0 0
\(755\) −22.4930 −0.818603
\(756\) −5.92893 −0.215633
\(757\) −0.752967 −0.0273670 −0.0136835 0.999906i \(-0.504356\pi\)
−0.0136835 + 0.999906i \(0.504356\pi\)
\(758\) 48.9045 1.77629
\(759\) −6.23591 −0.226349
\(760\) −58.0943 −2.10730
\(761\) 34.9748 1.26784 0.633919 0.773400i \(-0.281445\pi\)
0.633919 + 0.773400i \(0.281445\pi\)
\(762\) 41.4538 1.50171
\(763\) −2.23280 −0.0808327
\(764\) 63.1756 2.28561
\(765\) 11.6336 0.420612
\(766\) 69.2700 2.50283
\(767\) 0 0
\(768\) −73.3333 −2.64619
\(769\) 15.4649 0.557679 0.278839 0.960338i \(-0.410050\pi\)
0.278839 + 0.960338i \(0.410050\pi\)
\(770\) −2.38360 −0.0858990
\(771\) 43.0802 1.55150
\(772\) −38.1444 −1.37285
\(773\) −36.1366 −1.29974 −0.649871 0.760045i \(-0.725177\pi\)
−0.649871 + 0.760045i \(0.725177\pi\)
\(774\) −53.9185 −1.93806
\(775\) 5.06327 0.181878
\(776\) −23.7209 −0.851530
\(777\) −1.40856 −0.0505318
\(778\) −72.6935 −2.60619
\(779\) 80.8583 2.89705
\(780\) 0 0
\(781\) −6.50702 −0.232839
\(782\) 29.3163 1.04835
\(783\) −13.0913 −0.467846
\(784\) −37.0381 −1.32279
\(785\) 14.7913 0.527923
\(786\) −59.2849 −2.11462
\(787\) −22.9499 −0.818075 −0.409037 0.912518i \(-0.634135\pi\)
−0.409037 + 0.912518i \(0.634135\pi\)
\(788\) −26.1929 −0.933084
\(789\) −35.5391 −1.26523
\(790\) −41.6124 −1.48050
\(791\) −4.35933 −0.155000
\(792\) 12.7289 0.452302
\(793\) 0 0
\(794\) 86.8895 3.08359
\(795\) −5.00400 −0.177474
\(796\) 40.0459 1.41939
\(797\) 43.4498 1.53907 0.769535 0.638604i \(-0.220488\pi\)
0.769535 + 0.638604i \(0.220488\pi\)
\(798\) 36.9960 1.30964
\(799\) 14.4859 0.512473
\(800\) −10.7429 −0.379820
\(801\) −10.6164 −0.375112
\(802\) −36.5710 −1.29137
\(803\) −2.00000 −0.0705785
\(804\) 59.3725 2.09391
\(805\) 2.59455 0.0914460
\(806\) 0 0
\(807\) −11.9327 −0.420052
\(808\) 27.6376 0.972286
\(809\) −43.1094 −1.51565 −0.757823 0.652461i \(-0.773737\pi\)
−0.757823 + 0.652461i \(0.773737\pi\)
\(810\) 32.8655 1.15477
\(811\) −17.9428 −0.630056 −0.315028 0.949082i \(-0.602014\pi\)
−0.315028 + 0.949082i \(0.602014\pi\)
\(812\) 24.5491 0.861506
\(813\) 59.2691 2.07866
\(814\) −1.98596 −0.0696080
\(815\) −2.34441 −0.0821210
\(816\) −56.7178 −1.98552
\(817\) −80.3333 −2.81051
\(818\) 41.6202 1.45522
\(819\) 0 0
\(820\) −51.0131 −1.78146
\(821\) 15.6647 0.546703 0.273352 0.961914i \(-0.411868\pi\)
0.273352 + 0.961914i \(0.411868\pi\)
\(822\) −28.2820 −0.986449
\(823\) 3.69771 0.128894 0.0644470 0.997921i \(-0.479472\pi\)
0.0644470 + 0.997921i \(0.479472\pi\)
\(824\) 55.9145 1.94787
\(825\) −8.01404 −0.279013
\(826\) 10.1593 0.353487
\(827\) −42.3201 −1.47162 −0.735808 0.677191i \(-0.763197\pi\)
−0.735808 + 0.677191i \(0.763197\pi\)
\(828\) −25.9820 −0.902936
\(829\) 7.15461 0.248490 0.124245 0.992252i \(-0.460349\pi\)
0.124245 + 0.992252i \(0.460349\pi\)
\(830\) 40.9327 1.42080
\(831\) −40.6164 −1.40897
\(832\) 0 0
\(833\) 27.4014 0.949404
\(834\) 94.3654 3.26761
\(835\) 8.62663 0.298537
\(836\) 35.5632 1.22998
\(837\) 2.56717 0.0887344
\(838\) −55.8944 −1.93084
\(839\) 2.76008 0.0952887 0.0476443 0.998864i \(-0.484829\pi\)
0.0476443 + 0.998864i \(0.484829\pi\)
\(840\) −12.4469 −0.429458
\(841\) 25.2056 0.869159
\(842\) −8.66874 −0.298745
\(843\) 16.9367 0.583332
\(844\) −22.2257 −0.765040
\(845\) 0 0
\(846\) −18.8303 −0.647398
\(847\) 0.778124 0.0267367
\(848\) 10.3805 0.356467
\(849\) 21.2912 0.730711
\(850\) 37.6757 1.29226
\(851\) 2.16172 0.0741030
\(852\) −63.7178 −2.18294
\(853\) 46.1615 1.58054 0.790270 0.612758i \(-0.209940\pi\)
0.790270 + 0.612758i \(0.209940\pi\)
\(854\) −28.3273 −0.969339
\(855\) −22.5311 −0.770547
\(856\) 13.5342 0.462590
\(857\) 26.2680 0.897297 0.448649 0.893708i \(-0.351905\pi\)
0.448649 + 0.893708i \(0.351905\pi\)
\(858\) 0 0
\(859\) −32.0381 −1.09313 −0.546563 0.837418i \(-0.684064\pi\)
−0.546563 + 0.837418i \(0.684064\pi\)
\(860\) 50.6819 1.72824
\(861\) 17.3241 0.590405
\(862\) 64.8826 2.20991
\(863\) −24.2078 −0.824044 −0.412022 0.911174i \(-0.635177\pi\)
−0.412022 + 0.911174i \(0.635177\pi\)
\(864\) −5.44687 −0.185306
\(865\) −2.01315 −0.0684491
\(866\) 36.0521 1.22510
\(867\) 3.11338 0.105736
\(868\) −4.81401 −0.163398
\(869\) 13.5843 0.460817
\(870\) −51.5371 −1.74727
\(871\) 0 0
\(872\) −16.4389 −0.556690
\(873\) −9.19983 −0.311367
\(874\) −56.7779 −1.92054
\(875\) 8.08823 0.273432
\(876\) −19.5843 −0.661693
\(877\) 1.05635 0.0356703 0.0178351 0.999841i \(-0.494323\pi\)
0.0178351 + 0.999841i \(0.494323\pi\)
\(878\) 11.4478 0.386343
\(879\) −65.7046 −2.21616
\(880\) −7.07730 −0.238576
\(881\) −24.1655 −0.814157 −0.407079 0.913393i \(-0.633453\pi\)
−0.407079 + 0.913393i \(0.633453\pi\)
\(882\) −35.6193 −1.19936
\(883\) 5.84831 0.196811 0.0984057 0.995146i \(-0.468626\pi\)
0.0984057 + 0.995146i \(0.468626\pi\)
\(884\) 0 0
\(885\) −14.5411 −0.488795
\(886\) 45.5569 1.53052
\(887\) −8.67566 −0.291300 −0.145650 0.989336i \(-0.546527\pi\)
−0.145650 + 0.989336i \(0.546527\pi\)
\(888\) −10.3705 −0.348010
\(889\) 5.63044 0.188839
\(890\) 14.6367 0.490622
\(891\) −10.7289 −0.359431
\(892\) 72.1373 2.41533
\(893\) −28.0553 −0.938834
\(894\) −10.9116 −0.364938
\(895\) −8.96169 −0.299556
\(896\) −12.3843 −0.413730
\(897\) 0 0
\(898\) −31.6625 −1.05659
\(899\) −10.6295 −0.354515
\(900\) −33.3905 −1.11302
\(901\) −7.67967 −0.255847
\(902\) 24.4257 0.813287
\(903\) −17.2116 −0.572768
\(904\) −32.0953 −1.06748
\(905\) −14.3876 −0.478260
\(906\) 105.461 3.50370
\(907\) −15.5952 −0.517832 −0.258916 0.965900i \(-0.583365\pi\)
−0.258916 + 0.965900i \(0.583365\pi\)
\(908\) −75.4888 −2.50518
\(909\) 10.7189 0.355522
\(910\) 0 0
\(911\) 24.5672 0.813947 0.406973 0.913440i \(-0.366584\pi\)
0.406973 + 0.913440i \(0.366584\pi\)
\(912\) 109.847 3.63741
\(913\) −13.3624 −0.442232
\(914\) −64.6514 −2.13848
\(915\) 40.5451 1.34038
\(916\) 115.079 3.80233
\(917\) −8.05234 −0.265912
\(918\) 19.1023 0.630469
\(919\) 17.2811 0.570052 0.285026 0.958520i \(-0.407998\pi\)
0.285026 + 0.958520i \(0.407998\pi\)
\(920\) 19.1023 0.629783
\(921\) −1.81020 −0.0596482
\(922\) −31.6725 −1.04308
\(923\) 0 0
\(924\) 7.61951 0.250664
\(925\) 2.77812 0.0913441
\(926\) −8.71174 −0.286286
\(927\) 21.6857 0.712252
\(928\) 22.5531 0.740343
\(929\) −48.2569 −1.58326 −0.791628 0.611003i \(-0.790766\pi\)
−0.791628 + 0.611003i \(0.790766\pi\)
\(930\) 10.1063 0.331398
\(931\) −53.0693 −1.73928
\(932\) −116.301 −3.80957
\(933\) 58.9617 1.93032
\(934\) −46.9459 −1.53612
\(935\) 5.23591 0.171233
\(936\) 0 0
\(937\) −17.7209 −0.578916 −0.289458 0.957191i \(-0.593475\pi\)
−0.289458 + 0.957191i \(0.593475\pi\)
\(938\) 11.8280 0.386199
\(939\) −60.0490 −1.95963
\(940\) 17.6999 0.577308
\(941\) 7.96392 0.259616 0.129808 0.991539i \(-0.458564\pi\)
0.129808 + 0.991539i \(0.458564\pi\)
\(942\) −69.3504 −2.25956
\(943\) −26.5874 −0.865806
\(944\) 30.1646 0.981775
\(945\) 1.69059 0.0549948
\(946\) −24.2671 −0.788992
\(947\) 15.5632 0.505735 0.252867 0.967501i \(-0.418626\pi\)
0.252867 + 0.967501i \(0.418626\pi\)
\(948\) 133.020 4.32028
\(949\) 0 0
\(950\) −72.9677 −2.36739
\(951\) 25.0882 0.813541
\(952\) −19.1023 −0.619108
\(953\) −31.8483 −1.03167 −0.515834 0.856689i \(-0.672518\pi\)
−0.515834 + 0.856689i \(0.672518\pi\)
\(954\) 9.98285 0.323207
\(955\) −18.0140 −0.582921
\(956\) 3.19692 0.103396
\(957\) 16.8242 0.543850
\(958\) 23.2780 0.752079
\(959\) −3.84139 −0.124045
\(960\) 10.9024 0.351874
\(961\) −28.9156 −0.932761
\(962\) 0 0
\(963\) 5.24906 0.169149
\(964\) 69.3021 2.23207
\(965\) 10.8766 0.350130
\(966\) −12.1648 −0.391397
\(967\) −46.1023 −1.48255 −0.741274 0.671202i \(-0.765778\pi\)
−0.741274 + 0.671202i \(0.765778\pi\)
\(968\) 5.72889 0.184134
\(969\) −81.2669 −2.61067
\(970\) 12.6837 0.407248
\(971\) −14.7882 −0.474575 −0.237287 0.971440i \(-0.576258\pi\)
−0.237287 + 0.971440i \(0.576258\pi\)
\(972\) −82.2005 −2.63658
\(973\) 12.8171 0.410898
\(974\) −100.104 −3.20755
\(975\) 0 0
\(976\) −84.1083 −2.69224
\(977\) 1.35933 0.0434889 0.0217444 0.999764i \(-0.493078\pi\)
0.0217444 + 0.999764i \(0.493078\pi\)
\(978\) 10.9920 0.351485
\(979\) −4.77812 −0.152710
\(980\) 33.4812 1.06952
\(981\) −6.37560 −0.203557
\(982\) −103.246 −3.29470
\(983\) 27.4709 0.876187 0.438093 0.898929i \(-0.355654\pi\)
0.438093 + 0.898929i \(0.355654\pi\)
\(984\) 127.548 4.06608
\(985\) 7.46871 0.237973
\(986\) −79.0943 −2.51887
\(987\) −6.01092 −0.191330
\(988\) 0 0
\(989\) 26.4148 0.839941
\(990\) −6.80620 −0.216315
\(991\) 0.173535 0.00551253 0.00275626 0.999996i \(-0.499123\pi\)
0.00275626 + 0.999996i \(0.499123\pi\)
\(992\) −4.42260 −0.140418
\(993\) 22.9797 0.729240
\(994\) −12.6937 −0.402620
\(995\) −11.4188 −0.362000
\(996\) −130.847 −4.14605
\(997\) 44.1896 1.39950 0.699749 0.714388i \(-0.253295\pi\)
0.699749 + 0.714388i \(0.253295\pi\)
\(998\) 48.4186 1.53266
\(999\) 1.40856 0.0445649
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 1859.2.a.f.1.3 3
13.3 even 3 143.2.e.b.100.1 6
13.9 even 3 143.2.e.b.133.1 yes 6
13.12 even 2 1859.2.a.g.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
143.2.e.b.100.1 6 13.3 even 3
143.2.e.b.133.1 yes 6 13.9 even 3
1859.2.a.f.1.3 3 1.1 even 1 trivial
1859.2.a.g.1.1 3 13.12 even 2