Properties

Label 2175.2.a.v.1.1
Level $2175$
Weight $2$
Character 2175.1
Self dual yes
Analytic conductor $17.367$
Analytic rank $0$
Dimension $4$
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] = [2175,2,Mod(1,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2175.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(17.3674624396\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.43828\) of defining polynomial
Character \(\chi\) \(=\) 2175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.43828 q^{2} +1.00000 q^{3} +0.0686587 q^{4} -1.43828 q^{6} +2.74301 q^{7} +2.77782 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.43828 q^{2} +1.00000 q^{3} +0.0686587 q^{4} -1.43828 q^{6} +2.74301 q^{7} +2.77782 q^{8} +1.00000 q^{9} +2.74301 q^{11} +0.0686587 q^{12} +5.14744 q^{13} -3.94523 q^{14} -4.13260 q^{16} +3.72913 q^{17} -1.43828 q^{18} -0.404431 q^{19} +2.74301 q^{21} -3.94523 q^{22} +5.45825 q^{23} +2.77782 q^{24} -7.40348 q^{26} +1.00000 q^{27} +0.188331 q^{28} -1.00000 q^{29} +1.45825 q^{31} +0.388222 q^{32} +2.74301 q^{33} -5.36354 q^{34} +0.0686587 q^{36} -6.76702 q^{37} +0.581686 q^{38} +5.14744 q^{39} +9.78090 q^{41} -3.94523 q^{42} -4.43220 q^{43} +0.188331 q^{44} -7.85051 q^{46} -2.60569 q^{47} -4.13260 q^{48} +0.524103 q^{49} +3.72913 q^{51} +0.353416 q^{52} +6.43220 q^{53} -1.43828 q^{54} +7.61958 q^{56} -0.404431 q^{57} +1.43828 q^{58} -9.91822 q^{59} -13.0816 q^{61} -2.09738 q^{62} +2.74301 q^{63} +7.70683 q^{64} -3.94523 q^{66} -12.4961 q^{67} +0.256037 q^{68} +5.45825 q^{69} -11.3487 q^{71} +2.77782 q^{72} +10.7670 q^{73} +9.73289 q^{74} -0.0277677 q^{76} +7.52410 q^{77} -7.40348 q^{78} +14.1576 q^{79} +1.00000 q^{81} -14.0677 q^{82} -1.62334 q^{83} +0.188331 q^{84} +6.37476 q^{86} -1.00000 q^{87} +7.61958 q^{88} -8.87281 q^{89} +14.1195 q^{91} +0.374756 q^{92} +1.45825 q^{93} +3.74772 q^{94} +0.388222 q^{96} -7.82084 q^{97} -0.753809 q^{98} +2.74301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 4 q^{3} + 5 q^{4} + 3 q^{6} - 2 q^{7} + 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} + 4 q^{3} + 5 q^{4} + 3 q^{6} - 2 q^{7} + 12 q^{8} + 4 q^{9} - 2 q^{11} + 5 q^{12} + 8 q^{13} - 3 q^{14} + 11 q^{16} + 10 q^{17} + 3 q^{18} - 2 q^{19} - 2 q^{21} - 3 q^{22} + 12 q^{23} + 12 q^{24} - 7 q^{26} + 4 q^{27} + 9 q^{28} - 4 q^{29} - 4 q^{31} + 17 q^{32} - 2 q^{33} - q^{34} + 5 q^{36} + 16 q^{37} + 10 q^{38} + 8 q^{39} - 12 q^{41} - 3 q^{42} - 2 q^{43} + 9 q^{44} - 8 q^{46} + 12 q^{47} + 11 q^{48} + 6 q^{49} + 10 q^{51} + 3 q^{52} + 10 q^{53} + 3 q^{54} - 2 q^{57} - 3 q^{58} + 2 q^{59} - 26 q^{61} - 20 q^{62} - 2 q^{63} + 34 q^{64} - 3 q^{66} - 2 q^{67} - 9 q^{68} + 12 q^{69} - 10 q^{71} + 12 q^{72} + 48 q^{74} + 16 q^{76} + 34 q^{77} - 7 q^{78} + 22 q^{79} + 4 q^{81} - 38 q^{82} + 10 q^{83} + 9 q^{84} - 4 q^{86} - 4 q^{87} - 4 q^{89} - 8 q^{91} - 28 q^{92} - 4 q^{93} + 39 q^{94} + 17 q^{96} + 22 q^{97} - 34 q^{98} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.43828 −1.01702 −0.508510 0.861056i \(-0.669803\pi\)
−0.508510 + 0.861056i \(0.669803\pi\)
\(3\) 1.00000 0.577350
\(4\) 0.0686587 0.0343293
\(5\) 0 0
\(6\) −1.43828 −0.587177
\(7\) 2.74301 1.03676 0.518380 0.855150i \(-0.326535\pi\)
0.518380 + 0.855150i \(0.326535\pi\)
\(8\) 2.77782 0.982106
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.74301 0.827049 0.413524 0.910493i \(-0.364298\pi\)
0.413524 + 0.910493i \(0.364298\pi\)
\(12\) 0.0686587 0.0198201
\(13\) 5.14744 1.42764 0.713822 0.700328i \(-0.246963\pi\)
0.713822 + 0.700328i \(0.246963\pi\)
\(14\) −3.94523 −1.05441
\(15\) 0 0
\(16\) −4.13260 −1.03315
\(17\) 3.72913 0.904446 0.452223 0.891905i \(-0.350631\pi\)
0.452223 + 0.891905i \(0.350631\pi\)
\(18\) −1.43828 −0.339007
\(19\) −0.404431 −0.0927827 −0.0463914 0.998923i \(-0.514772\pi\)
−0.0463914 + 0.998923i \(0.514772\pi\)
\(20\) 0 0
\(21\) 2.74301 0.598574
\(22\) −3.94523 −0.841125
\(23\) 5.45825 1.13812 0.569062 0.822295i \(-0.307306\pi\)
0.569062 + 0.822295i \(0.307306\pi\)
\(24\) 2.77782 0.567019
\(25\) 0 0
\(26\) −7.40348 −1.45194
\(27\) 1.00000 0.192450
\(28\) 0.188331 0.0355913
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 1.45825 0.261910 0.130955 0.991388i \(-0.458196\pi\)
0.130955 + 0.991388i \(0.458196\pi\)
\(32\) 0.388222 0.0686287
\(33\) 2.74301 0.477497
\(34\) −5.36354 −0.919839
\(35\) 0 0
\(36\) 0.0686587 0.0114431
\(37\) −6.76702 −1.11249 −0.556245 0.831018i \(-0.687759\pi\)
−0.556245 + 0.831018i \(0.687759\pi\)
\(38\) 0.581686 0.0943619
\(39\) 5.14744 0.824250
\(40\) 0 0
\(41\) 9.78090 1.52752 0.763760 0.645500i \(-0.223351\pi\)
0.763760 + 0.645500i \(0.223351\pi\)
\(42\) −3.94523 −0.608761
\(43\) −4.43220 −0.675904 −0.337952 0.941163i \(-0.609734\pi\)
−0.337952 + 0.941163i \(0.609734\pi\)
\(44\) 0.188331 0.0283920
\(45\) 0 0
\(46\) −7.85051 −1.15749
\(47\) −2.60569 −0.380079 −0.190040 0.981776i \(-0.560862\pi\)
−0.190040 + 0.981776i \(0.560862\pi\)
\(48\) −4.13260 −0.596490
\(49\) 0.524103 0.0748719
\(50\) 0 0
\(51\) 3.72913 0.522182
\(52\) 0.353416 0.0490100
\(53\) 6.43220 0.883530 0.441765 0.897131i \(-0.354352\pi\)
0.441765 + 0.897131i \(0.354352\pi\)
\(54\) −1.43828 −0.195726
\(55\) 0 0
\(56\) 7.61958 1.01821
\(57\) −0.404431 −0.0535681
\(58\) 1.43828 0.188856
\(59\) −9.91822 −1.29124 −0.645621 0.763658i \(-0.723401\pi\)
−0.645621 + 0.763658i \(0.723401\pi\)
\(60\) 0 0
\(61\) −13.0816 −1.67493 −0.837463 0.546494i \(-0.815962\pi\)
−0.837463 + 0.546494i \(0.815962\pi\)
\(62\) −2.09738 −0.266367
\(63\) 2.74301 0.345587
\(64\) 7.70683 0.963354
\(65\) 0 0
\(66\) −3.94523 −0.485624
\(67\) −12.4961 −1.52665 −0.763323 0.646017i \(-0.776434\pi\)
−0.763323 + 0.646017i \(0.776434\pi\)
\(68\) 0.256037 0.0310490
\(69\) 5.45825 0.657096
\(70\) 0 0
\(71\) −11.3487 −1.34684 −0.673422 0.739259i \(-0.735176\pi\)
−0.673422 + 0.739259i \(0.735176\pi\)
\(72\) 2.77782 0.327369
\(73\) 10.7670 1.26018 0.630092 0.776520i \(-0.283017\pi\)
0.630092 + 0.776520i \(0.283017\pi\)
\(74\) 9.73289 1.13143
\(75\) 0 0
\(76\) −0.0277677 −0.00318517
\(77\) 7.52410 0.857451
\(78\) −7.40348 −0.838279
\(79\) 14.1576 1.59285 0.796425 0.604737i \(-0.206722\pi\)
0.796425 + 0.604737i \(0.206722\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −14.0677 −1.55352
\(83\) −1.62334 −0.178184 −0.0890922 0.996023i \(-0.528397\pi\)
−0.0890922 + 0.996023i \(0.528397\pi\)
\(84\) 0.188331 0.0205486
\(85\) 0 0
\(86\) 6.37476 0.687408
\(87\) −1.00000 −0.107211
\(88\) 7.61958 0.812250
\(89\) −8.87281 −0.940516 −0.470258 0.882529i \(-0.655839\pi\)
−0.470258 + 0.882529i \(0.655839\pi\)
\(90\) 0 0
\(91\) 14.1195 1.48012
\(92\) 0.374756 0.0390711
\(93\) 1.45825 0.151214
\(94\) 3.74772 0.386548
\(95\) 0 0
\(96\) 0.388222 0.0396228
\(97\) −7.82084 −0.794086 −0.397043 0.917800i \(-0.629964\pi\)
−0.397043 + 0.917800i \(0.629964\pi\)
\(98\) −0.753809 −0.0761462
\(99\) 2.74301 0.275683
\(100\) 0 0
\(101\) −4.88033 −0.485611 −0.242805 0.970075i \(-0.578068\pi\)
−0.242805 + 0.970075i \(0.578068\pi\)
\(102\) −5.36354 −0.531070
\(103\) 0.294881 0.0290555 0.0145277 0.999894i \(-0.495376\pi\)
0.0145277 + 0.999894i \(0.495376\pi\)
\(104\) 14.2986 1.40210
\(105\) 0 0
\(106\) −9.25132 −0.898568
\(107\) 13.7809 1.33225 0.666125 0.745840i \(-0.267952\pi\)
0.666125 + 0.745840i \(0.267952\pi\)
\(108\) 0.0686587 0.00660668
\(109\) −6.20126 −0.593973 −0.296987 0.954882i \(-0.595982\pi\)
−0.296987 + 0.954882i \(0.595982\pi\)
\(110\) 0 0
\(111\) −6.76702 −0.642297
\(112\) −11.3358 −1.07113
\(113\) 10.5658 0.993943 0.496971 0.867767i \(-0.334445\pi\)
0.496971 + 0.867767i \(0.334445\pi\)
\(114\) 0.581686 0.0544799
\(115\) 0 0
\(116\) −0.0686587 −0.00637480
\(117\) 5.14744 0.475881
\(118\) 14.2652 1.31322
\(119\) 10.2290 0.937694
\(120\) 0 0
\(121\) −3.47590 −0.315991
\(122\) 18.8150 1.70343
\(123\) 9.78090 0.881914
\(124\) 0.100122 0.00899119
\(125\) 0 0
\(126\) −3.94523 −0.351469
\(127\) −8.54175 −0.757958 −0.378979 0.925405i \(-0.623725\pi\)
−0.378979 + 0.925405i \(0.623725\pi\)
\(128\) −11.8611 −1.04838
\(129\) −4.43220 −0.390233
\(130\) 0 0
\(131\) 13.3050 1.16246 0.581232 0.813738i \(-0.302571\pi\)
0.581232 + 0.813738i \(0.302571\pi\)
\(132\) 0.188331 0.0163921
\(133\) −1.10936 −0.0961934
\(134\) 17.9730 1.55263
\(135\) 0 0
\(136\) 10.3588 0.888262
\(137\) 18.2253 1.55709 0.778545 0.627589i \(-0.215958\pi\)
0.778545 + 0.627589i \(0.215958\pi\)
\(138\) −7.85051 −0.668280
\(139\) −5.66142 −0.480195 −0.240098 0.970749i \(-0.577179\pi\)
−0.240098 + 0.970749i \(0.577179\pi\)
\(140\) 0 0
\(141\) −2.60569 −0.219439
\(142\) 16.3226 1.36977
\(143\) 14.1195 1.18073
\(144\) −4.13260 −0.344384
\(145\) 0 0
\(146\) −15.4860 −1.28163
\(147\) 0.524103 0.0432273
\(148\) −0.464614 −0.0381911
\(149\) −11.9182 −0.976378 −0.488189 0.872738i \(-0.662342\pi\)
−0.488189 + 0.872738i \(0.662342\pi\)
\(150\) 0 0
\(151\) 7.19114 0.585207 0.292603 0.956234i \(-0.405478\pi\)
0.292603 + 0.956234i \(0.405478\pi\)
\(152\) −1.12343 −0.0911225
\(153\) 3.72913 0.301482
\(154\) −10.8218 −0.872045
\(155\) 0 0
\(156\) 0.353416 0.0282960
\(157\) 4.31457 0.344340 0.172170 0.985067i \(-0.444922\pi\)
0.172170 + 0.985067i \(0.444922\pi\)
\(158\) −20.3626 −1.61996
\(159\) 6.43220 0.510106
\(160\) 0 0
\(161\) 14.9720 1.17996
\(162\) −1.43828 −0.113002
\(163\) −21.6436 −1.69526 −0.847628 0.530591i \(-0.821970\pi\)
−0.847628 + 0.530591i \(0.821970\pi\)
\(164\) 0.671544 0.0524388
\(165\) 0 0
\(166\) 2.33482 0.181217
\(167\) 16.4303 1.27141 0.635707 0.771930i \(-0.280709\pi\)
0.635707 + 0.771930i \(0.280709\pi\)
\(168\) 7.61958 0.587863
\(169\) 13.4961 1.03816
\(170\) 0 0
\(171\) −0.404431 −0.0309276
\(172\) −0.304309 −0.0232033
\(173\) 4.64939 0.353487 0.176743 0.984257i \(-0.443444\pi\)
0.176743 + 0.984257i \(0.443444\pi\)
\(174\) 1.43828 0.109036
\(175\) 0 0
\(176\) −11.3358 −0.854466
\(177\) −9.91822 −0.745499
\(178\) 12.7616 0.956523
\(179\) 7.62334 0.569795 0.284897 0.958558i \(-0.408040\pi\)
0.284897 + 0.958558i \(0.408040\pi\)
\(180\) 0 0
\(181\) −21.3050 −1.58359 −0.791794 0.610788i \(-0.790853\pi\)
−0.791794 + 0.610788i \(0.790853\pi\)
\(182\) −20.3078 −1.50532
\(183\) −13.0816 −0.967019
\(184\) 15.1620 1.11776
\(185\) 0 0
\(186\) −2.09738 −0.153787
\(187\) 10.2290 0.748021
\(188\) −0.178903 −0.0130479
\(189\) 2.74301 0.199525
\(190\) 0 0
\(191\) 3.07597 0.222570 0.111285 0.993789i \(-0.464503\pi\)
0.111285 + 0.993789i \(0.464503\pi\)
\(192\) 7.70683 0.556193
\(193\) −6.77454 −0.487642 −0.243821 0.969820i \(-0.578401\pi\)
−0.243821 + 0.969820i \(0.578401\pi\)
\(194\) 11.2486 0.807601
\(195\) 0 0
\(196\) 0.0359842 0.00257030
\(197\) 13.6455 0.972201 0.486100 0.873903i \(-0.338419\pi\)
0.486100 + 0.873903i \(0.338419\pi\)
\(198\) −3.94523 −0.280375
\(199\) −6.33858 −0.449330 −0.224665 0.974436i \(-0.572129\pi\)
−0.224665 + 0.974436i \(0.572129\pi\)
\(200\) 0 0
\(201\) −12.4961 −0.881410
\(202\) 7.01929 0.493876
\(203\) −2.74301 −0.192522
\(204\) 0.256037 0.0179262
\(205\) 0 0
\(206\) −0.424122 −0.0295500
\(207\) 5.45825 0.379375
\(208\) −21.2723 −1.47497
\(209\) −1.10936 −0.0767358
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 0.441626 0.0303310
\(213\) −11.3487 −0.777600
\(214\) −19.8208 −1.35492
\(215\) 0 0
\(216\) 2.77782 0.189006
\(217\) 4.00000 0.271538
\(218\) 8.91917 0.604082
\(219\) 10.7670 0.727568
\(220\) 0 0
\(221\) 19.1955 1.29123
\(222\) 9.73289 0.653229
\(223\) 2.20126 0.147407 0.0737037 0.997280i \(-0.476518\pi\)
0.0737037 + 0.997280i \(0.476518\pi\)
\(224\) 1.06490 0.0711515
\(225\) 0 0
\(226\) −15.1965 −1.01086
\(227\) −12.1853 −0.808769 −0.404384 0.914589i \(-0.632514\pi\)
−0.404384 + 0.914589i \(0.632514\pi\)
\(228\) −0.0277677 −0.00183896
\(229\) −3.16337 −0.209041 −0.104521 0.994523i \(-0.533331\pi\)
−0.104521 + 0.994523i \(0.533331\pi\)
\(230\) 0 0
\(231\) 7.52410 0.495050
\(232\) −2.77782 −0.182373
\(233\) −23.8087 −1.55976 −0.779879 0.625930i \(-0.784719\pi\)
−0.779879 + 0.625930i \(0.784719\pi\)
\(234\) −7.40348 −0.483980
\(235\) 0 0
\(236\) −0.680972 −0.0443275
\(237\) 14.1576 0.919633
\(238\) −14.7122 −0.953653
\(239\) −6.02025 −0.389417 −0.194709 0.980861i \(-0.562376\pi\)
−0.194709 + 0.980861i \(0.562376\pi\)
\(240\) 0 0
\(241\) 24.5517 1.58151 0.790756 0.612131i \(-0.209688\pi\)
0.790756 + 0.612131i \(0.209688\pi\)
\(242\) 4.99932 0.321369
\(243\) 1.00000 0.0641500
\(244\) −0.898165 −0.0574991
\(245\) 0 0
\(246\) −14.0677 −0.896924
\(247\) −2.08178 −0.132461
\(248\) 4.05076 0.257223
\(249\) −1.62334 −0.102875
\(250\) 0 0
\(251\) −27.5162 −1.73681 −0.868403 0.495858i \(-0.834854\pi\)
−0.868403 + 0.495858i \(0.834854\pi\)
\(252\) 0.188331 0.0118638
\(253\) 14.9720 0.941284
\(254\) 12.2855 0.770858
\(255\) 0 0
\(256\) 1.64589 0.102868
\(257\) −15.6436 −0.975820 −0.487910 0.872894i \(-0.662241\pi\)
−0.487910 + 0.872894i \(0.662241\pi\)
\(258\) 6.37476 0.396875
\(259\) −18.5620 −1.15339
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) −19.1364 −1.18225
\(263\) 18.6175 1.14801 0.574003 0.818853i \(-0.305390\pi\)
0.574003 + 0.818853i \(0.305390\pi\)
\(264\) 7.61958 0.468953
\(265\) 0 0
\(266\) 1.59557 0.0978306
\(267\) −8.87281 −0.543007
\(268\) −0.857969 −0.0524088
\(269\) 10.9006 0.664620 0.332310 0.943170i \(-0.392172\pi\)
0.332310 + 0.943170i \(0.392172\pi\)
\(270\) 0 0
\(271\) 17.7809 1.08011 0.540056 0.841629i \(-0.318403\pi\)
0.540056 + 0.841629i \(0.318403\pi\)
\(272\) −15.4110 −0.934429
\(273\) 14.1195 0.854550
\(274\) −26.2131 −1.58359
\(275\) 0 0
\(276\) 0.374756 0.0225577
\(277\) 20.6612 1.24141 0.620706 0.784043i \(-0.286846\pi\)
0.620706 + 0.784043i \(0.286846\pi\)
\(278\) 8.14273 0.488368
\(279\) 1.45825 0.0873033
\(280\) 0 0
\(281\) 28.2652 1.68616 0.843080 0.537787i \(-0.180740\pi\)
0.843080 + 0.537787i \(0.180740\pi\)
\(282\) 3.74772 0.223174
\(283\) 3.21139 0.190897 0.0954485 0.995434i \(-0.469571\pi\)
0.0954485 + 0.995434i \(0.469571\pi\)
\(284\) −0.779187 −0.0462362
\(285\) 0 0
\(286\) −20.3078 −1.20083
\(287\) 26.8291 1.58367
\(288\) 0.388222 0.0228762
\(289\) −3.09362 −0.181978
\(290\) 0 0
\(291\) −7.82084 −0.458466
\(292\) 0.739249 0.0432613
\(293\) 3.99624 0.233463 0.116731 0.993164i \(-0.462758\pi\)
0.116731 + 0.993164i \(0.462758\pi\)
\(294\) −0.753809 −0.0439630
\(295\) 0 0
\(296\) −18.7975 −1.09258
\(297\) 2.74301 0.159166
\(298\) 17.1418 0.992996
\(299\) 28.0960 1.62484
\(300\) 0 0
\(301\) −12.1576 −0.700750
\(302\) −10.3429 −0.595167
\(303\) −4.88033 −0.280367
\(304\) 1.67135 0.0958586
\(305\) 0 0
\(306\) −5.36354 −0.306613
\(307\) 12.0555 0.688046 0.344023 0.938961i \(-0.388210\pi\)
0.344023 + 0.938961i \(0.388210\pi\)
\(308\) 0.516595 0.0294357
\(309\) 0.294881 0.0167752
\(310\) 0 0
\(311\) 15.6873 0.889544 0.444772 0.895644i \(-0.353285\pi\)
0.444772 + 0.895644i \(0.353285\pi\)
\(312\) 14.2986 0.809501
\(313\) 33.8726 1.91459 0.957297 0.289107i \(-0.0933585\pi\)
0.957297 + 0.289107i \(0.0933585\pi\)
\(314\) −6.20558 −0.350201
\(315\) 0 0
\(316\) 0.972040 0.0546815
\(317\) −1.75689 −0.0986770 −0.0493385 0.998782i \(-0.515711\pi\)
−0.0493385 + 0.998782i \(0.515711\pi\)
\(318\) −9.25132 −0.518788
\(319\) −2.74301 −0.153579
\(320\) 0 0
\(321\) 13.7809 0.769175
\(322\) −21.5340 −1.20004
\(323\) −1.50817 −0.0839170
\(324\) 0.0686587 0.00381437
\(325\) 0 0
\(326\) 31.1296 1.72411
\(327\) −6.20126 −0.342931
\(328\) 27.1695 1.50019
\(329\) −7.14744 −0.394051
\(330\) 0 0
\(331\) −22.4505 −1.23399 −0.616997 0.786966i \(-0.711651\pi\)
−0.616997 + 0.786966i \(0.711651\pi\)
\(332\) −0.111456 −0.00611695
\(333\) −6.76702 −0.370830
\(334\) −23.6314 −1.29305
\(335\) 0 0
\(336\) −11.3358 −0.618417
\(337\) −15.3886 −0.838273 −0.419136 0.907923i \(-0.637667\pi\)
−0.419136 + 0.907923i \(0.637667\pi\)
\(338\) −19.4113 −1.05583
\(339\) 10.5658 0.573853
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 0.581686 0.0314540
\(343\) −17.7634 −0.959136
\(344\) −12.3118 −0.663809
\(345\) 0 0
\(346\) −6.68714 −0.359503
\(347\) 12.3504 0.663005 0.331503 0.943454i \(-0.392444\pi\)
0.331503 + 0.943454i \(0.392444\pi\)
\(348\) −0.0686587 −0.00368049
\(349\) −1.94446 −0.104085 −0.0520424 0.998645i \(-0.516573\pi\)
−0.0520424 + 0.998645i \(0.516573\pi\)
\(350\) 0 0
\(351\) 5.14744 0.274750
\(352\) 1.06490 0.0567592
\(353\) −6.72517 −0.357945 −0.178972 0.983854i \(-0.557277\pi\)
−0.178972 + 0.983854i \(0.557277\pi\)
\(354\) 14.2652 0.758187
\(355\) 0 0
\(356\) −0.609195 −0.0322873
\(357\) 10.2290 0.541378
\(358\) −10.9645 −0.579493
\(359\) 15.2114 0.802826 0.401413 0.915897i \(-0.368519\pi\)
0.401413 + 0.915897i \(0.368519\pi\)
\(360\) 0 0
\(361\) −18.8364 −0.991391
\(362\) 30.6426 1.61054
\(363\) −3.47590 −0.182437
\(364\) 0.969425 0.0508117
\(365\) 0 0
\(366\) 18.8150 0.983477
\(367\) −13.2189 −0.690021 −0.345011 0.938599i \(-0.612125\pi\)
−0.345011 + 0.938599i \(0.612125\pi\)
\(368\) −22.5568 −1.17585
\(369\) 9.78090 0.509173
\(370\) 0 0
\(371\) 17.6436 0.916009
\(372\) 0.100122 0.00519107
\(373\) −26.8050 −1.38791 −0.693956 0.720017i \(-0.744134\pi\)
−0.693956 + 0.720017i \(0.744134\pi\)
\(374\) −14.7122 −0.760752
\(375\) 0 0
\(376\) −7.23813 −0.373278
\(377\) −5.14744 −0.265107
\(378\) −3.94523 −0.202920
\(379\) −24.4228 −1.25451 −0.627257 0.778813i \(-0.715822\pi\)
−0.627257 + 0.778813i \(0.715822\pi\)
\(380\) 0 0
\(381\) −8.54175 −0.437607
\(382\) −4.42412 −0.226358
\(383\) −27.2372 −1.39176 −0.695879 0.718159i \(-0.744985\pi\)
−0.695879 + 0.718159i \(0.744985\pi\)
\(384\) −11.8611 −0.605282
\(385\) 0 0
\(386\) 9.74370 0.495942
\(387\) −4.43220 −0.225301
\(388\) −0.536968 −0.0272604
\(389\) −12.1994 −0.618532 −0.309266 0.950976i \(-0.600083\pi\)
−0.309266 + 0.950976i \(0.600083\pi\)
\(390\) 0 0
\(391\) 20.3545 1.02937
\(392\) 1.45586 0.0735322
\(393\) 13.3050 0.671149
\(394\) −19.6261 −0.988748
\(395\) 0 0
\(396\) 0.188331 0.00946401
\(397\) 17.7254 0.889611 0.444805 0.895627i \(-0.353273\pi\)
0.444805 + 0.895627i \(0.353273\pi\)
\(398\) 9.11667 0.456977
\(399\) −1.10936 −0.0555373
\(400\) 0 0
\(401\) −2.48773 −0.124231 −0.0621157 0.998069i \(-0.519785\pi\)
−0.0621157 + 0.998069i \(0.519785\pi\)
\(402\) 17.9730 0.896411
\(403\) 7.50627 0.373914
\(404\) −0.335077 −0.0166707
\(405\) 0 0
\(406\) 3.94523 0.195798
\(407\) −18.5620 −0.920084
\(408\) 10.3588 0.512838
\(409\) 37.7194 1.86510 0.932551 0.361038i \(-0.117577\pi\)
0.932551 + 0.361038i \(0.117577\pi\)
\(410\) 0 0
\(411\) 18.2253 0.898986
\(412\) 0.0202461 0.000997455 0
\(413\) −27.2058 −1.33871
\(414\) −7.85051 −0.385832
\(415\) 0 0
\(416\) 1.99835 0.0979773
\(417\) −5.66142 −0.277241
\(418\) 1.59557 0.0780419
\(419\) −2.29488 −0.112112 −0.0560561 0.998428i \(-0.517853\pi\)
−0.0560561 + 0.998428i \(0.517853\pi\)
\(420\) 0 0
\(421\) 35.9662 1.75289 0.876443 0.481505i \(-0.159910\pi\)
0.876443 + 0.481505i \(0.159910\pi\)
\(422\) −2.87657 −0.140029
\(423\) −2.60569 −0.126693
\(424\) 17.8675 0.867721
\(425\) 0 0
\(426\) 16.3226 0.790835
\(427\) −35.8829 −1.73650
\(428\) 0.946178 0.0457353
\(429\) 14.1195 0.681695
\(430\) 0 0
\(431\) 22.6974 1.09330 0.546648 0.837363i \(-0.315904\pi\)
0.546648 + 0.837363i \(0.315904\pi\)
\(432\) −4.13260 −0.198830
\(433\) 11.4108 0.548368 0.274184 0.961677i \(-0.411592\pi\)
0.274184 + 0.961677i \(0.411592\pi\)
\(434\) −5.75313 −0.276159
\(435\) 0 0
\(436\) −0.425770 −0.0203907
\(437\) −2.20748 −0.105598
\(438\) −15.4860 −0.739951
\(439\) −7.36654 −0.351586 −0.175793 0.984427i \(-0.556249\pi\)
−0.175793 + 0.984427i \(0.556249\pi\)
\(440\) 0 0
\(441\) 0.524103 0.0249573
\(442\) −27.6085 −1.31320
\(443\) −36.5423 −1.73617 −0.868087 0.496411i \(-0.834651\pi\)
−0.868087 + 0.496411i \(0.834651\pi\)
\(444\) −0.464614 −0.0220496
\(445\) 0 0
\(446\) −3.16604 −0.149916
\(447\) −11.9182 −0.563712
\(448\) 21.1399 0.998767
\(449\) −39.6182 −1.86970 −0.934850 0.355043i \(-0.884466\pi\)
−0.934850 + 0.355043i \(0.884466\pi\)
\(450\) 0 0
\(451\) 26.8291 1.26333
\(452\) 0.725431 0.0341214
\(453\) 7.19114 0.337869
\(454\) 17.5260 0.822534
\(455\) 0 0
\(456\) −1.12343 −0.0526096
\(457\) −28.5220 −1.33420 −0.667102 0.744967i \(-0.732465\pi\)
−0.667102 + 0.744967i \(0.732465\pi\)
\(458\) 4.54982 0.212599
\(459\) 3.72913 0.174061
\(460\) 0 0
\(461\) −22.6696 −1.05583 −0.527915 0.849297i \(-0.677026\pi\)
−0.527915 + 0.849297i \(0.677026\pi\)
\(462\) −10.8218 −0.503475
\(463\) −28.5517 −1.32691 −0.663455 0.748217i \(-0.730910\pi\)
−0.663455 + 0.748217i \(0.730910\pi\)
\(464\) 4.13260 0.191851
\(465\) 0 0
\(466\) 34.2436 1.58630
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0.353416 0.0163367
\(469\) −34.2770 −1.58277
\(470\) 0 0
\(471\) 4.31457 0.198805
\(472\) −27.5510 −1.26814
\(473\) −12.1576 −0.559005
\(474\) −20.3626 −0.935285
\(475\) 0 0
\(476\) 0.702312 0.0321904
\(477\) 6.43220 0.294510
\(478\) 8.65882 0.396045
\(479\) −4.43410 −0.202599 −0.101300 0.994856i \(-0.532300\pi\)
−0.101300 + 0.994856i \(0.532300\pi\)
\(480\) 0 0
\(481\) −34.8328 −1.58824
\(482\) −35.3123 −1.60843
\(483\) 14.9720 0.681251
\(484\) −0.238650 −0.0108477
\(485\) 0 0
\(486\) −1.43828 −0.0652419
\(487\) −14.1035 −0.639093 −0.319546 0.947571i \(-0.603531\pi\)
−0.319546 + 0.947571i \(0.603531\pi\)
\(488\) −36.3382 −1.64496
\(489\) −21.6436 −0.978757
\(490\) 0 0
\(491\) 39.6376 1.78882 0.894410 0.447249i \(-0.147596\pi\)
0.894410 + 0.447249i \(0.147596\pi\)
\(492\) 0.671544 0.0302755
\(493\) −3.72913 −0.167951
\(494\) 2.99419 0.134715
\(495\) 0 0
\(496\) −6.02638 −0.270592
\(497\) −31.1296 −1.39635
\(498\) 2.33482 0.104626
\(499\) 6.33858 0.283754 0.141877 0.989884i \(-0.454686\pi\)
0.141877 + 0.989884i \(0.454686\pi\)
\(500\) 0 0
\(501\) 16.4303 0.734051
\(502\) 39.5761 1.76637
\(503\) 19.0638 0.850011 0.425005 0.905191i \(-0.360272\pi\)
0.425005 + 0.905191i \(0.360272\pi\)
\(504\) 7.61958 0.339403
\(505\) 0 0
\(506\) −21.5340 −0.957305
\(507\) 13.4961 0.599385
\(508\) −0.586465 −0.0260202
\(509\) −12.8145 −0.567992 −0.283996 0.958826i \(-0.591660\pi\)
−0.283996 + 0.958826i \(0.591660\pi\)
\(510\) 0 0
\(511\) 29.5340 1.30651
\(512\) 21.3549 0.943760
\(513\) −0.404431 −0.0178560
\(514\) 22.4999 0.992428
\(515\) 0 0
\(516\) −0.304309 −0.0133965
\(517\) −7.14744 −0.314344
\(518\) 26.6974 1.17302
\(519\) 4.64939 0.204086
\(520\) 0 0
\(521\) −35.4800 −1.55441 −0.777204 0.629249i \(-0.783363\pi\)
−0.777204 + 0.629249i \(0.783363\pi\)
\(522\) 1.43828 0.0629519
\(523\) 14.9227 0.652525 0.326263 0.945279i \(-0.394211\pi\)
0.326263 + 0.945279i \(0.394211\pi\)
\(524\) 0.913504 0.0399066
\(525\) 0 0
\(526\) −26.7773 −1.16754
\(527\) 5.43801 0.236883
\(528\) −11.3358 −0.493326
\(529\) 6.79252 0.295327
\(530\) 0 0
\(531\) −9.91822 −0.430414
\(532\) −0.0761670 −0.00330226
\(533\) 50.3466 2.18075
\(534\) 12.7616 0.552249
\(535\) 0 0
\(536\) −34.7120 −1.49933
\(537\) 7.62334 0.328971
\(538\) −15.6781 −0.675931
\(539\) 1.43762 0.0619227
\(540\) 0 0
\(541\) −22.8644 −0.983017 −0.491509 0.870873i \(-0.663554\pi\)
−0.491509 + 0.870873i \(0.663554\pi\)
\(542\) −25.5740 −1.09850
\(543\) −21.3050 −0.914285
\(544\) 1.44773 0.0620709
\(545\) 0 0
\(546\) −20.3078 −0.869094
\(547\) 35.3290 1.51056 0.755279 0.655404i \(-0.227502\pi\)
0.755279 + 0.655404i \(0.227502\pi\)
\(548\) 1.25132 0.0534539
\(549\) −13.0816 −0.558309
\(550\) 0 0
\(551\) 0.404431 0.0172293
\(552\) 15.1620 0.645338
\(553\) 38.8343 1.65140
\(554\) −29.7167 −1.26254
\(555\) 0 0
\(556\) −0.388706 −0.0164848
\(557\) −32.7994 −1.38976 −0.694878 0.719127i \(-0.744542\pi\)
−0.694878 + 0.719127i \(0.744542\pi\)
\(558\) −2.09738 −0.0887892
\(559\) −22.8145 −0.964950
\(560\) 0 0
\(561\) 10.2290 0.431870
\(562\) −40.6534 −1.71486
\(563\) −16.8169 −0.708747 −0.354374 0.935104i \(-0.615306\pi\)
−0.354374 + 0.935104i \(0.615306\pi\)
\(564\) −0.178903 −0.00753319
\(565\) 0 0
\(566\) −4.61888 −0.194146
\(567\) 2.74301 0.115196
\(568\) −31.5246 −1.32274
\(569\) 8.88033 0.372283 0.186141 0.982523i \(-0.440402\pi\)
0.186141 + 0.982523i \(0.440402\pi\)
\(570\) 0 0
\(571\) 25.1240 1.05141 0.525703 0.850668i \(-0.323802\pi\)
0.525703 + 0.850668i \(0.323802\pi\)
\(572\) 0.969425 0.0405337
\(573\) 3.07597 0.128501
\(574\) −38.5879 −1.61063
\(575\) 0 0
\(576\) 7.70683 0.321118
\(577\) 13.9026 0.578774 0.289387 0.957212i \(-0.406549\pi\)
0.289387 + 0.957212i \(0.406549\pi\)
\(578\) 4.44950 0.185075
\(579\) −6.77454 −0.281540
\(580\) 0 0
\(581\) −4.45283 −0.184735
\(582\) 11.2486 0.466269
\(583\) 17.6436 0.730723
\(584\) 29.9088 1.23763
\(585\) 0 0
\(586\) −5.74772 −0.237436
\(587\) 20.9942 0.866523 0.433262 0.901268i \(-0.357363\pi\)
0.433262 + 0.901268i \(0.357363\pi\)
\(588\) 0.0359842 0.00148396
\(589\) −0.589762 −0.0243007
\(590\) 0 0
\(591\) 13.6455 0.561300
\(592\) 27.9654 1.14937
\(593\) −3.54003 −0.145372 −0.0726859 0.997355i \(-0.523157\pi\)
−0.0726859 + 0.997355i \(0.523157\pi\)
\(594\) −3.94523 −0.161875
\(595\) 0 0
\(596\) −0.818289 −0.0335184
\(597\) −6.33858 −0.259421
\(598\) −40.4100 −1.65249
\(599\) −32.4886 −1.32745 −0.663725 0.747977i \(-0.731025\pi\)
−0.663725 + 0.747977i \(0.731025\pi\)
\(600\) 0 0
\(601\) −12.5620 −0.512414 −0.256207 0.966622i \(-0.582473\pi\)
−0.256207 + 0.966622i \(0.582473\pi\)
\(602\) 17.4860 0.712677
\(603\) −12.4961 −0.508882
\(604\) 0.493734 0.0200898
\(605\) 0 0
\(606\) 7.01929 0.285139
\(607\) 0.369141 0.0149830 0.00749149 0.999972i \(-0.497615\pi\)
0.00749149 + 0.999972i \(0.497615\pi\)
\(608\) −0.157009 −0.00636756
\(609\) −2.74301 −0.111152
\(610\) 0 0
\(611\) −13.4126 −0.542618
\(612\) 0.256037 0.0103497
\(613\) −44.7017 −1.80549 −0.902743 0.430181i \(-0.858450\pi\)
−0.902743 + 0.430181i \(0.858450\pi\)
\(614\) −17.3393 −0.699756
\(615\) 0 0
\(616\) 20.9006 0.842108
\(617\) 5.91014 0.237933 0.118967 0.992898i \(-0.462042\pi\)
0.118967 + 0.992898i \(0.462042\pi\)
\(618\) −0.424122 −0.0170607
\(619\) −44.2187 −1.77730 −0.888650 0.458586i \(-0.848356\pi\)
−0.888650 + 0.458586i \(0.848356\pi\)
\(620\) 0 0
\(621\) 5.45825 0.219032
\(622\) −22.5628 −0.904684
\(623\) −24.3382 −0.975089
\(624\) −21.2723 −0.851575
\(625\) 0 0
\(626\) −48.7184 −1.94718
\(627\) −1.10936 −0.0443034
\(628\) 0.296233 0.0118210
\(629\) −25.2351 −1.00619
\(630\) 0 0
\(631\) −29.5702 −1.17717 −0.588586 0.808435i \(-0.700315\pi\)
−0.588586 + 0.808435i \(0.700315\pi\)
\(632\) 39.3271 1.56435
\(633\) 2.00000 0.0794929
\(634\) 2.52691 0.100356
\(635\) 0 0
\(636\) 0.441626 0.0175116
\(637\) 2.69779 0.106890
\(638\) 3.94523 0.156193
\(639\) −11.3487 −0.448948
\(640\) 0 0
\(641\) 26.6335 1.05196 0.525979 0.850497i \(-0.323699\pi\)
0.525979 + 0.850497i \(0.323699\pi\)
\(642\) −19.8208 −0.782266
\(643\) −8.16597 −0.322035 −0.161017 0.986952i \(-0.551477\pi\)
−0.161017 + 0.986952i \(0.551477\pi\)
\(644\) 1.02796 0.0405073
\(645\) 0 0
\(646\) 2.16918 0.0853452
\(647\) −20.8381 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(648\) 2.77782 0.109123
\(649\) −27.2058 −1.06792
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) −1.48602 −0.0581970
\(653\) −13.4267 −0.525428 −0.262714 0.964874i \(-0.584618\pi\)
−0.262714 + 0.964874i \(0.584618\pi\)
\(654\) 8.91917 0.348767
\(655\) 0 0
\(656\) −40.4206 −1.57816
\(657\) 10.7670 0.420061
\(658\) 10.2800 0.400758
\(659\) −42.9744 −1.67405 −0.837023 0.547167i \(-0.815706\pi\)
−0.837023 + 0.547167i \(0.815706\pi\)
\(660\) 0 0
\(661\) −10.0178 −0.389649 −0.194824 0.980838i \(-0.562414\pi\)
−0.194824 + 0.980838i \(0.562414\pi\)
\(662\) 32.2902 1.25500
\(663\) 19.1955 0.745490
\(664\) −4.50933 −0.174996
\(665\) 0 0
\(666\) 9.73289 0.377142
\(667\) −5.45825 −0.211344
\(668\) 1.12808 0.0436468
\(669\) 2.20126 0.0851057
\(670\) 0 0
\(671\) −35.8829 −1.38525
\(672\) 1.06490 0.0410793
\(673\) −23.0878 −0.889970 −0.444985 0.895538i \(-0.646791\pi\)
−0.444985 + 0.895538i \(0.646791\pi\)
\(674\) 22.1332 0.852540
\(675\) 0 0
\(676\) 0.926627 0.0356395
\(677\) −4.95555 −0.190457 −0.0952287 0.995455i \(-0.530358\pi\)
−0.0952287 + 0.995455i \(0.530358\pi\)
\(678\) −15.1965 −0.583620
\(679\) −21.4526 −0.823277
\(680\) 0 0
\(681\) −12.1853 −0.466943
\(682\) −5.75313 −0.220299
\(683\) 36.8010 1.40815 0.704075 0.710126i \(-0.251362\pi\)
0.704075 + 0.710126i \(0.251362\pi\)
\(684\) −0.0277677 −0.00106172
\(685\) 0 0
\(686\) 25.5489 0.975460
\(687\) −3.16337 −0.120690
\(688\) 18.3165 0.698311
\(689\) 33.1094 1.26137
\(690\) 0 0
\(691\) 15.8246 0.601996 0.300998 0.953625i \(-0.402680\pi\)
0.300998 + 0.953625i \(0.402680\pi\)
\(692\) 0.319221 0.0121350
\(693\) 7.52410 0.285817
\(694\) −17.7634 −0.674289
\(695\) 0 0
\(696\) −2.77782 −0.105293
\(697\) 36.4742 1.38156
\(698\) 2.79669 0.105856
\(699\) −23.8087 −0.900527
\(700\) 0 0
\(701\) −42.6156 −1.60957 −0.804785 0.593567i \(-0.797719\pi\)
−0.804785 + 0.593567i \(0.797719\pi\)
\(702\) −7.40348 −0.279426
\(703\) 2.73679 0.103220
\(704\) 21.1399 0.796741
\(705\) 0 0
\(706\) 9.67270 0.364037
\(707\) −13.3868 −0.503462
\(708\) −0.680972 −0.0255925
\(709\) 35.1795 1.32119 0.660597 0.750740i \(-0.270303\pi\)
0.660597 + 0.750740i \(0.270303\pi\)
\(710\) 0 0
\(711\) 14.1576 0.530950
\(712\) −24.6470 −0.923686
\(713\) 7.95951 0.298086
\(714\) −14.7122 −0.550592
\(715\) 0 0
\(716\) 0.523408 0.0195607
\(717\) −6.02025 −0.224830
\(718\) −21.8783 −0.816490
\(719\) −29.4563 −1.09854 −0.549268 0.835646i \(-0.685093\pi\)
−0.549268 + 0.835646i \(0.685093\pi\)
\(720\) 0 0
\(721\) 0.808861 0.0301236
\(722\) 27.0921 1.00826
\(723\) 24.5517 0.913087
\(724\) −1.46277 −0.0543635
\(725\) 0 0
\(726\) 4.99932 0.185542
\(727\) −46.3391 −1.71862 −0.859311 0.511454i \(-0.829107\pi\)
−0.859311 + 0.511454i \(0.829107\pi\)
\(728\) 39.2213 1.45364
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −16.5282 −0.611319
\(732\) −0.898165 −0.0331971
\(733\) 3.73344 0.137898 0.0689489 0.997620i \(-0.478035\pi\)
0.0689489 + 0.997620i \(0.478035\pi\)
\(734\) 19.0125 0.701765
\(735\) 0 0
\(736\) 2.11902 0.0781080
\(737\) −34.2770 −1.26261
\(738\) −14.0677 −0.517839
\(739\) −6.48021 −0.238378 −0.119189 0.992872i \(-0.538030\pi\)
−0.119189 + 0.992872i \(0.538030\pi\)
\(740\) 0 0
\(741\) −2.08178 −0.0764762
\(742\) −25.3765 −0.931600
\(743\) 51.0638 1.87335 0.936674 0.350203i \(-0.113888\pi\)
0.936674 + 0.350203i \(0.113888\pi\)
\(744\) 4.05076 0.148508
\(745\) 0 0
\(746\) 38.5533 1.41153
\(747\) −1.62334 −0.0593948
\(748\) 0.702312 0.0256791
\(749\) 37.8011 1.38122
\(750\) 0 0
\(751\) 24.6494 0.899469 0.449735 0.893162i \(-0.351519\pi\)
0.449735 + 0.893162i \(0.351519\pi\)
\(752\) 10.7683 0.392679
\(753\) −27.5162 −1.00275
\(754\) 7.40348 0.269619
\(755\) 0 0
\(756\) 0.188331 0.00684955
\(757\) 38.6833 1.40597 0.702985 0.711205i \(-0.251850\pi\)
0.702985 + 0.711205i \(0.251850\pi\)
\(758\) 35.1269 1.27587
\(759\) 14.9720 0.543451
\(760\) 0 0
\(761\) −3.23725 −0.117350 −0.0586750 0.998277i \(-0.518688\pi\)
−0.0586750 + 0.998277i \(0.518688\pi\)
\(762\) 12.2855 0.445055
\(763\) −17.0101 −0.615808
\(764\) 0.211192 0.00764067
\(765\) 0 0
\(766\) 39.1749 1.41545
\(767\) −51.0534 −1.84343
\(768\) 1.64589 0.0593909
\(769\) 46.3166 1.67022 0.835111 0.550082i \(-0.185404\pi\)
0.835111 + 0.550082i \(0.185404\pi\)
\(770\) 0 0
\(771\) −15.6436 −0.563390
\(772\) −0.465131 −0.0167404
\(773\) 27.0341 0.972350 0.486175 0.873861i \(-0.338392\pi\)
0.486175 + 0.873861i \(0.338392\pi\)
\(774\) 6.37476 0.229136
\(775\) 0 0
\(776\) −21.7248 −0.779877
\(777\) −18.5620 −0.665908
\(778\) 17.5461 0.629059
\(779\) −3.95569 −0.141727
\(780\) 0 0
\(781\) −31.1296 −1.11390
\(782\) −29.2756 −1.04689
\(783\) −1.00000 −0.0357371
\(784\) −2.16591 −0.0773540
\(785\) 0 0
\(786\) −19.1364 −0.682572
\(787\) 39.1870 1.39687 0.698434 0.715675i \(-0.253881\pi\)
0.698434 + 0.715675i \(0.253881\pi\)
\(788\) 0.936881 0.0333750
\(789\) 18.6175 0.662802
\(790\) 0 0
\(791\) 28.9820 1.03048
\(792\) 7.61958 0.270750
\(793\) −67.3367 −2.39120
\(794\) −25.4941 −0.904752
\(795\) 0 0
\(796\) −0.435198 −0.0154252
\(797\) 53.2933 1.88775 0.943873 0.330307i \(-0.107152\pi\)
0.943873 + 0.330307i \(0.107152\pi\)
\(798\) 1.59557 0.0564825
\(799\) −9.71696 −0.343761
\(800\) 0 0
\(801\) −8.87281 −0.313505
\(802\) 3.57807 0.126346
\(803\) 29.5340 1.04223
\(804\) −0.857969 −0.0302582
\(805\) 0 0
\(806\) −10.7961 −0.380278
\(807\) 10.9006 0.383718
\(808\) −13.5567 −0.476921
\(809\) 0.613214 0.0215595 0.0107797 0.999942i \(-0.496569\pi\)
0.0107797 + 0.999942i \(0.496569\pi\)
\(810\) 0 0
\(811\) −0.230936 −0.00810927 −0.00405463 0.999992i \(-0.501291\pi\)
−0.00405463 + 0.999992i \(0.501291\pi\)
\(812\) −0.188331 −0.00660914
\(813\) 17.7809 0.623603
\(814\) 26.6974 0.935744
\(815\) 0 0
\(816\) −15.4110 −0.539493
\(817\) 1.79252 0.0627122
\(818\) −54.2511 −1.89685
\(819\) 14.1195 0.493375
\(820\) 0 0
\(821\) 25.7254 0.897821 0.448911 0.893577i \(-0.351812\pi\)
0.448911 + 0.893577i \(0.351812\pi\)
\(822\) −26.2131 −0.914287
\(823\) 38.2258 1.33247 0.666234 0.745743i \(-0.267905\pi\)
0.666234 + 0.745743i \(0.267905\pi\)
\(824\) 0.819125 0.0285356
\(825\) 0 0
\(826\) 39.1296 1.36149
\(827\) −24.9199 −0.866551 −0.433275 0.901262i \(-0.642642\pi\)
−0.433275 + 0.901262i \(0.642642\pi\)
\(828\) 0.374756 0.0130237
\(829\) −1.65301 −0.0574115 −0.0287057 0.999588i \(-0.509139\pi\)
−0.0287057 + 0.999588i \(0.509139\pi\)
\(830\) 0 0
\(831\) 20.6612 0.716730
\(832\) 39.6705 1.37533
\(833\) 1.95445 0.0677176
\(834\) 8.14273 0.281960
\(835\) 0 0
\(836\) −0.0761670 −0.00263429
\(837\) 1.45825 0.0504046
\(838\) 3.30069 0.114020
\(839\) −31.4757 −1.08666 −0.543331 0.839519i \(-0.682837\pi\)
−0.543331 + 0.839519i \(0.682837\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −51.7296 −1.78272
\(843\) 28.2652 0.973505
\(844\) 0.137317 0.00472666
\(845\) 0 0
\(846\) 3.74772 0.128849
\(847\) −9.53442 −0.327607
\(848\) −26.5817 −0.912820
\(849\) 3.21139 0.110214
\(850\) 0 0
\(851\) −36.9361 −1.26615
\(852\) −0.779187 −0.0266945
\(853\) 30.3753 1.04003 0.520015 0.854157i \(-0.325926\pi\)
0.520015 + 0.854157i \(0.325926\pi\)
\(854\) 51.6098 1.76605
\(855\) 0 0
\(856\) 38.2808 1.30841
\(857\) 6.53423 0.223205 0.111602 0.993753i \(-0.464402\pi\)
0.111602 + 0.993753i \(0.464402\pi\)
\(858\) −20.3078 −0.693297
\(859\) 27.0220 0.921977 0.460989 0.887406i \(-0.347495\pi\)
0.460989 + 0.887406i \(0.347495\pi\)
\(860\) 0 0
\(861\) 26.8291 0.914334
\(862\) −32.6453 −1.11190
\(863\) −31.9587 −1.08789 −0.543944 0.839122i \(-0.683069\pi\)
−0.543944 + 0.839122i \(0.683069\pi\)
\(864\) 0.388222 0.0132076
\(865\) 0 0
\(866\) −16.4120 −0.557701
\(867\) −3.09362 −0.105065
\(868\) 0.274635 0.00932171
\(869\) 38.8343 1.31736
\(870\) 0 0
\(871\) −64.3232 −2.17951
\(872\) −17.2260 −0.583345
\(873\) −7.82084 −0.264695
\(874\) 3.17499 0.107396
\(875\) 0 0
\(876\) 0.739249 0.0249769
\(877\) 50.0679 1.69067 0.845336 0.534235i \(-0.179400\pi\)
0.845336 + 0.534235i \(0.179400\pi\)
\(878\) 10.5952 0.357570
\(879\) 3.99624 0.134790
\(880\) 0 0
\(881\) −13.6817 −0.460947 −0.230474 0.973079i \(-0.574028\pi\)
−0.230474 + 0.973079i \(0.574028\pi\)
\(882\) −0.753809 −0.0253821
\(883\) −29.5693 −0.995087 −0.497543 0.867439i \(-0.665764\pi\)
−0.497543 + 0.867439i \(0.665764\pi\)
\(884\) 1.31793 0.0443269
\(885\) 0 0
\(886\) 52.5581 1.76572
\(887\) −4.35130 −0.146102 −0.0730512 0.997328i \(-0.523274\pi\)
−0.0730512 + 0.997328i \(0.523274\pi\)
\(888\) −18.7975 −0.630804
\(889\) −23.4301 −0.785820
\(890\) 0 0
\(891\) 2.74301 0.0918943
\(892\) 0.151136 0.00506040
\(893\) 1.05382 0.0352648
\(894\) 17.1418 0.573307
\(895\) 0 0
\(896\) −32.5350 −1.08692
\(897\) 28.0960 0.938099
\(898\) 56.9822 1.90152
\(899\) −1.45825 −0.0486354
\(900\) 0 0
\(901\) 23.9865 0.799105
\(902\) −38.5879 −1.28484
\(903\) −12.1576 −0.404578
\(904\) 29.3497 0.976157
\(905\) 0 0
\(906\) −10.3429 −0.343620
\(907\) 52.3965 1.73980 0.869899 0.493230i \(-0.164184\pi\)
0.869899 + 0.493230i \(0.164184\pi\)
\(908\) −0.836629 −0.0277645
\(909\) −4.88033 −0.161870
\(910\) 0 0
\(911\) 7.85085 0.260110 0.130055 0.991507i \(-0.458485\pi\)
0.130055 + 0.991507i \(0.458485\pi\)
\(912\) 1.67135 0.0553440
\(913\) −4.45283 −0.147367
\(914\) 41.0227 1.35691
\(915\) 0 0
\(916\) −0.217193 −0.00717625
\(917\) 36.4958 1.20520
\(918\) −5.36354 −0.177023
\(919\) 13.4979 0.445253 0.222627 0.974904i \(-0.428537\pi\)
0.222627 + 0.974904i \(0.428537\pi\)
\(920\) 0 0
\(921\) 12.0555 0.397243
\(922\) 32.6054 1.07380
\(923\) −58.4168 −1.92281
\(924\) 0.516595 0.0169947
\(925\) 0 0
\(926\) 41.0654 1.34949
\(927\) 0.294881 0.00968516
\(928\) −0.388222 −0.0127440
\(929\) 11.6177 0.381165 0.190583 0.981671i \(-0.438962\pi\)
0.190583 + 0.981671i \(0.438962\pi\)
\(930\) 0 0
\(931\) −0.211963 −0.00694682
\(932\) −1.63467 −0.0535455
\(933\) 15.6873 0.513579
\(934\) −11.5063 −0.376497
\(935\) 0 0
\(936\) 14.2986 0.467366
\(937\) 42.6740 1.39410 0.697049 0.717024i \(-0.254496\pi\)
0.697049 + 0.717024i \(0.254496\pi\)
\(938\) 49.3001 1.60971
\(939\) 33.8726 1.10539
\(940\) 0 0
\(941\) −5.91479 −0.192817 −0.0964083 0.995342i \(-0.530735\pi\)
−0.0964083 + 0.995342i \(0.530735\pi\)
\(942\) −6.20558 −0.202189
\(943\) 53.3866 1.73851
\(944\) 40.9881 1.33405
\(945\) 0 0
\(946\) 17.4860 0.568520
\(947\) 45.0915 1.46528 0.732639 0.680618i \(-0.238289\pi\)
0.732639 + 0.680618i \(0.238289\pi\)
\(948\) 0.972040 0.0315704
\(949\) 55.4226 1.79909
\(950\) 0 0
\(951\) −1.75689 −0.0569712
\(952\) 28.4144 0.920915
\(953\) −30.3166 −0.982053 −0.491026 0.871145i \(-0.663378\pi\)
−0.491026 + 0.871145i \(0.663378\pi\)
\(954\) −9.25132 −0.299523
\(955\) 0 0
\(956\) −0.413342 −0.0133684
\(957\) −2.74301 −0.0886689
\(958\) 6.37750 0.206048
\(959\) 49.9921 1.61433
\(960\) 0 0
\(961\) −28.8735 −0.931403
\(962\) 50.0995 1.61527
\(963\) 13.7809 0.444083
\(964\) 1.68569 0.0542923
\(965\) 0 0
\(966\) −21.5340 −0.692846
\(967\) −5.99809 −0.192886 −0.0964428 0.995339i \(-0.530746\pi\)
−0.0964428 + 0.995339i \(0.530746\pi\)
\(968\) −9.65540 −0.310336
\(969\) −1.50817 −0.0484495
\(970\) 0 0
\(971\) 28.5140 0.915057 0.457529 0.889195i \(-0.348735\pi\)
0.457529 + 0.889195i \(0.348735\pi\)
\(972\) 0.0686587 0.00220223
\(973\) −15.5293 −0.497848
\(974\) 20.2849 0.649970
\(975\) 0 0
\(976\) 54.0610 1.73045
\(977\) 5.53813 0.177180 0.0885902 0.996068i \(-0.471764\pi\)
0.0885902 + 0.996068i \(0.471764\pi\)
\(978\) 31.1296 0.995415
\(979\) −24.3382 −0.777852
\(980\) 0 0
\(981\) −6.20126 −0.197991
\(982\) −57.0101 −1.81926
\(983\) −7.37086 −0.235094 −0.117547 0.993067i \(-0.537503\pi\)
−0.117547 + 0.993067i \(0.537503\pi\)
\(984\) 27.1695 0.866133
\(985\) 0 0
\(986\) 5.36354 0.170810
\(987\) −7.14744 −0.227506
\(988\) −0.142932 −0.00454729
\(989\) −24.1921 −0.769263
\(990\) 0 0
\(991\) 3.68167 0.116952 0.0584760 0.998289i \(-0.481376\pi\)
0.0584760 + 0.998289i \(0.481376\pi\)
\(992\) 0.566126 0.0179745
\(993\) −22.4505 −0.712446
\(994\) 44.7732 1.42012
\(995\) 0 0
\(996\) −0.111456 −0.00353162
\(997\) −36.8747 −1.16783 −0.583916 0.811814i \(-0.698480\pi\)
−0.583916 + 0.811814i \(0.698480\pi\)
\(998\) −9.11667 −0.288583
\(999\) −6.76702 −0.214099
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 2175.2.a.v.1.1 4
3.2 odd 2 6525.2.a.bi.1.4 4
5.2 odd 4 2175.2.c.n.349.3 8
5.3 odd 4 2175.2.c.n.349.6 8
5.4 even 2 435.2.a.j.1.4 4
15.14 odd 2 1305.2.a.r.1.1 4
20.19 odd 2 6960.2.a.co.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.4 4 5.4 even 2
1305.2.a.r.1.1 4 15.14 odd 2
2175.2.a.v.1.1 4 1.1 even 1 trivial
2175.2.c.n.349.3 8 5.2 odd 4
2175.2.c.n.349.6 8 5.3 odd 4
6525.2.a.bi.1.4 4 3.2 odd 2
6960.2.a.co.1.4 4 20.19 odd 2