Properties

Label 1875.2.a.o.1.6
Level $1875$
Weight $2$
Character 1875.1
Self dual yes
Analytic conductor $14.972$
Analytic rank $0$
Dimension $8$
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] = [1875,2,Mod(1,1875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, 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("1875.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.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: \(14.9719503790\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.13366265625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 12x^{6} + 10x^{5} + 41x^{4} - 20x^{3} - 48x^{2} + 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(1.31354\) of defining polynomial
Character \(\chi\) \(=\) 1875.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.31354 q^{2} +1.00000 q^{3} -0.274605 q^{4} +1.31354 q^{6} +4.19091 q^{7} -2.98779 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.31354 q^{2} +1.00000 q^{3} -0.274605 q^{4} +1.31354 q^{6} +4.19091 q^{7} -2.98779 q^{8} +1.00000 q^{9} -0.167760 q^{11} -0.274605 q^{12} -3.39519 q^{13} +5.50493 q^{14} -3.37538 q^{16} +4.57476 q^{17} +1.31354 q^{18} +3.78258 q^{19} +4.19091 q^{21} -0.220360 q^{22} +8.31374 q^{23} -2.98779 q^{24} -4.45973 q^{26} +1.00000 q^{27} -1.15084 q^{28} +2.74553 q^{29} -7.71771 q^{31} +1.54187 q^{32} -0.167760 q^{33} +6.00914 q^{34} -0.274605 q^{36} +2.07700 q^{37} +4.96858 q^{38} -3.39519 q^{39} -1.28043 q^{41} +5.50493 q^{42} +11.6226 q^{43} +0.0460679 q^{44} +10.9204 q^{46} -9.99490 q^{47} -3.37538 q^{48} +10.5637 q^{49} +4.57476 q^{51} +0.932338 q^{52} -1.07165 q^{53} +1.31354 q^{54} -12.5216 q^{56} +3.78258 q^{57} +3.60638 q^{58} -4.95536 q^{59} +2.36706 q^{61} -10.1375 q^{62} +4.19091 q^{63} +8.77608 q^{64} -0.220360 q^{66} +12.2369 q^{67} -1.25625 q^{68} +8.31374 q^{69} +14.1746 q^{71} -2.98779 q^{72} -8.67832 q^{73} +2.72823 q^{74} -1.03872 q^{76} -0.703068 q^{77} -4.45973 q^{78} -2.64466 q^{79} +1.00000 q^{81} -1.68189 q^{82} -14.3131 q^{83} -1.15084 q^{84} +15.2668 q^{86} +2.74553 q^{87} +0.501233 q^{88} -4.06867 q^{89} -14.2289 q^{91} -2.28300 q^{92} -7.71771 q^{93} -13.1287 q^{94} +1.54187 q^{96} -2.78196 q^{97} +13.8759 q^{98} -0.167760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 8 q^{3} + 9 q^{4} + q^{6} + 12 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + 8 q^{3} + 9 q^{4} + q^{6} + 12 q^{7} + 3 q^{8} + 8 q^{9} + 12 q^{11} + 9 q^{12} + 14 q^{13} + 16 q^{14} + 15 q^{16} - q^{17} + q^{18} + 16 q^{19} + 12 q^{21} + 18 q^{22} - 4 q^{23} + 3 q^{24} - 34 q^{26} + 8 q^{27} - 21 q^{28} + 2 q^{29} + 13 q^{31} - 18 q^{32} + 12 q^{33} - 37 q^{34} + 9 q^{36} - 8 q^{37} - 24 q^{38} + 14 q^{39} - 12 q^{41} + 16 q^{42} + 20 q^{43} + 47 q^{44} + 33 q^{46} - 15 q^{47} + 15 q^{48} + 30 q^{49} - q^{51} - q^{52} - 4 q^{53} + q^{54} + 60 q^{56} + 16 q^{57} + 2 q^{58} + 14 q^{59} + 10 q^{61} + 4 q^{62} + 12 q^{63} + 41 q^{64} + 18 q^{66} + 19 q^{67} - 33 q^{68} - 4 q^{69} + 21 q^{71} + 3 q^{72} - 19 q^{73} - 9 q^{74} - q^{76} - 11 q^{77} - 34 q^{78} + 10 q^{79} + 8 q^{81} + 24 q^{82} - 27 q^{83} - 21 q^{84} + 42 q^{86} + 2 q^{87} + 53 q^{88} - 9 q^{89} - 12 q^{91} - 63 q^{92} + 13 q^{93} + 14 q^{94} - 18 q^{96} + 24 q^{97} - 24 q^{98} + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.31354 0.928815 0.464408 0.885622i \(-0.346267\pi\)
0.464408 + 0.885622i \(0.346267\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.274605 −0.137303
\(5\) 0 0
\(6\) 1.31354 0.536252
\(7\) 4.19091 1.58401 0.792007 0.610512i \(-0.209036\pi\)
0.792007 + 0.610512i \(0.209036\pi\)
\(8\) −2.98779 −1.05634
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.167760 −0.0505816 −0.0252908 0.999680i \(-0.508051\pi\)
−0.0252908 + 0.999680i \(0.508051\pi\)
\(12\) −0.274605 −0.0792717
\(13\) −3.39519 −0.941658 −0.470829 0.882225i \(-0.656045\pi\)
−0.470829 + 0.882225i \(0.656045\pi\)
\(14\) 5.50493 1.47126
\(15\) 0 0
\(16\) −3.37538 −0.843845
\(17\) 4.57476 1.10954 0.554771 0.832003i \(-0.312806\pi\)
0.554771 + 0.832003i \(0.312806\pi\)
\(18\) 1.31354 0.309605
\(19\) 3.78258 0.867784 0.433892 0.900965i \(-0.357140\pi\)
0.433892 + 0.900965i \(0.357140\pi\)
\(20\) 0 0
\(21\) 4.19091 0.914531
\(22\) −0.220360 −0.0469810
\(23\) 8.31374 1.73353 0.866767 0.498713i \(-0.166194\pi\)
0.866767 + 0.498713i \(0.166194\pi\)
\(24\) −2.98779 −0.609880
\(25\) 0 0
\(26\) −4.45973 −0.874626
\(27\) 1.00000 0.192450
\(28\) −1.15084 −0.217489
\(29\) 2.74553 0.509833 0.254916 0.966963i \(-0.417952\pi\)
0.254916 + 0.966963i \(0.417952\pi\)
\(30\) 0 0
\(31\) −7.71771 −1.38614 −0.693070 0.720870i \(-0.743742\pi\)
−0.693070 + 0.720870i \(0.743742\pi\)
\(32\) 1.54187 0.272567
\(33\) −0.167760 −0.0292033
\(34\) 6.00914 1.03056
\(35\) 0 0
\(36\) −0.274605 −0.0457675
\(37\) 2.07700 0.341457 0.170728 0.985318i \(-0.445388\pi\)
0.170728 + 0.985318i \(0.445388\pi\)
\(38\) 4.96858 0.806011
\(39\) −3.39519 −0.543666
\(40\) 0 0
\(41\) −1.28043 −0.199969 −0.0999845 0.994989i \(-0.531879\pi\)
−0.0999845 + 0.994989i \(0.531879\pi\)
\(42\) 5.50493 0.849430
\(43\) 11.6226 1.77243 0.886217 0.463270i \(-0.153324\pi\)
0.886217 + 0.463270i \(0.153324\pi\)
\(44\) 0.0460679 0.00694499
\(45\) 0 0
\(46\) 10.9204 1.61013
\(47\) −9.99490 −1.45791 −0.728953 0.684564i \(-0.759993\pi\)
−0.728953 + 0.684564i \(0.759993\pi\)
\(48\) −3.37538 −0.487194
\(49\) 10.5637 1.50910
\(50\) 0 0
\(51\) 4.57476 0.640594
\(52\) 0.932338 0.129292
\(53\) −1.07165 −0.147203 −0.0736015 0.997288i \(-0.523449\pi\)
−0.0736015 + 0.997288i \(0.523449\pi\)
\(54\) 1.31354 0.178751
\(55\) 0 0
\(56\) −12.5216 −1.67326
\(57\) 3.78258 0.501015
\(58\) 3.60638 0.473540
\(59\) −4.95536 −0.645133 −0.322566 0.946547i \(-0.604546\pi\)
−0.322566 + 0.946547i \(0.604546\pi\)
\(60\) 0 0
\(61\) 2.36706 0.303071 0.151536 0.988452i \(-0.451578\pi\)
0.151536 + 0.988452i \(0.451578\pi\)
\(62\) −10.1375 −1.28747
\(63\) 4.19091 0.528004
\(64\) 8.77608 1.09701
\(65\) 0 0
\(66\) −0.220360 −0.0271245
\(67\) 12.2369 1.49497 0.747486 0.664277i \(-0.231261\pi\)
0.747486 + 0.664277i \(0.231261\pi\)
\(68\) −1.25625 −0.152343
\(69\) 8.31374 1.00086
\(70\) 0 0
\(71\) 14.1746 1.68222 0.841109 0.540866i \(-0.181903\pi\)
0.841109 + 0.540866i \(0.181903\pi\)
\(72\) −2.98779 −0.352115
\(73\) −8.67832 −1.01572 −0.507860 0.861439i \(-0.669563\pi\)
−0.507860 + 0.861439i \(0.669563\pi\)
\(74\) 2.72823 0.317150
\(75\) 0 0
\(76\) −1.03872 −0.119149
\(77\) −0.703068 −0.0801220
\(78\) −4.45973 −0.504965
\(79\) −2.64466 −0.297547 −0.148774 0.988871i \(-0.547533\pi\)
−0.148774 + 0.988871i \(0.547533\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −1.68189 −0.185734
\(83\) −14.3131 −1.57106 −0.785532 0.618821i \(-0.787611\pi\)
−0.785532 + 0.618821i \(0.787611\pi\)
\(84\) −1.15084 −0.125567
\(85\) 0 0
\(86\) 15.2668 1.64626
\(87\) 2.74553 0.294352
\(88\) 0.501233 0.0534316
\(89\) −4.06867 −0.431278 −0.215639 0.976473i \(-0.569183\pi\)
−0.215639 + 0.976473i \(0.569183\pi\)
\(90\) 0 0
\(91\) −14.2289 −1.49160
\(92\) −2.28300 −0.238019
\(93\) −7.71771 −0.800289
\(94\) −13.1287 −1.35412
\(95\) 0 0
\(96\) 1.54187 0.157367
\(97\) −2.78196 −0.282466 −0.141233 0.989976i \(-0.545107\pi\)
−0.141233 + 0.989976i \(0.545107\pi\)
\(98\) 13.8759 1.40167
\(99\) −0.167760 −0.0168605
\(100\) 0 0
\(101\) −10.7004 −1.06473 −0.532366 0.846514i \(-0.678697\pi\)
−0.532366 + 0.846514i \(0.678697\pi\)
\(102\) 6.00914 0.594993
\(103\) −3.61627 −0.356322 −0.178161 0.984001i \(-0.557015\pi\)
−0.178161 + 0.984001i \(0.557015\pi\)
\(104\) 10.1441 0.994714
\(105\) 0 0
\(106\) −1.40766 −0.136724
\(107\) 0.522034 0.0504669 0.0252335 0.999682i \(-0.491967\pi\)
0.0252335 + 0.999682i \(0.491967\pi\)
\(108\) −0.274605 −0.0264239
\(109\) 11.5033 1.10181 0.550907 0.834566i \(-0.314282\pi\)
0.550907 + 0.834566i \(0.314282\pi\)
\(110\) 0 0
\(111\) 2.07700 0.197140
\(112\) −14.1459 −1.33666
\(113\) −20.1421 −1.89481 −0.947404 0.320041i \(-0.896303\pi\)
−0.947404 + 0.320041i \(0.896303\pi\)
\(114\) 4.96858 0.465350
\(115\) 0 0
\(116\) −0.753938 −0.0700014
\(117\) −3.39519 −0.313886
\(118\) −6.50908 −0.599209
\(119\) 19.1724 1.75753
\(120\) 0 0
\(121\) −10.9719 −0.997441
\(122\) 3.10924 0.281497
\(123\) −1.28043 −0.115452
\(124\) 2.11932 0.190321
\(125\) 0 0
\(126\) 5.50493 0.490418
\(127\) 16.3345 1.44945 0.724726 0.689037i \(-0.241966\pi\)
0.724726 + 0.689037i \(0.241966\pi\)
\(128\) 8.44401 0.746352
\(129\) 11.6226 1.02332
\(130\) 0 0
\(131\) 18.9147 1.65259 0.826294 0.563239i \(-0.190445\pi\)
0.826294 + 0.563239i \(0.190445\pi\)
\(132\) 0.0460679 0.00400969
\(133\) 15.8524 1.37458
\(134\) 16.0737 1.38855
\(135\) 0 0
\(136\) −13.6684 −1.17206
\(137\) −13.5975 −1.16171 −0.580856 0.814006i \(-0.697282\pi\)
−0.580856 + 0.814006i \(0.697282\pi\)
\(138\) 10.9204 0.929610
\(139\) −3.36103 −0.285079 −0.142540 0.989789i \(-0.545527\pi\)
−0.142540 + 0.989789i \(0.545527\pi\)
\(140\) 0 0
\(141\) −9.99490 −0.841722
\(142\) 18.6190 1.56247
\(143\) 0.569579 0.0476306
\(144\) −3.37538 −0.281282
\(145\) 0 0
\(146\) −11.3993 −0.943417
\(147\) 10.5637 0.871278
\(148\) −0.570355 −0.0468829
\(149\) −6.33430 −0.518926 −0.259463 0.965753i \(-0.583546\pi\)
−0.259463 + 0.965753i \(0.583546\pi\)
\(150\) 0 0
\(151\) 0.375265 0.0305386 0.0152693 0.999883i \(-0.495139\pi\)
0.0152693 + 0.999883i \(0.495139\pi\)
\(152\) −11.3016 −0.916678
\(153\) 4.57476 0.369847
\(154\) −0.923510 −0.0744185
\(155\) 0 0
\(156\) 0.932338 0.0746468
\(157\) 10.5002 0.838004 0.419002 0.907985i \(-0.362380\pi\)
0.419002 + 0.907985i \(0.362380\pi\)
\(158\) −3.47387 −0.276366
\(159\) −1.07165 −0.0849877
\(160\) 0 0
\(161\) 34.8421 2.74594
\(162\) 1.31354 0.103202
\(163\) −13.8457 −1.08448 −0.542241 0.840223i \(-0.682424\pi\)
−0.542241 + 0.840223i \(0.682424\pi\)
\(164\) 0.351612 0.0274563
\(165\) 0 0
\(166\) −18.8008 −1.45923
\(167\) −1.08114 −0.0836611 −0.0418306 0.999125i \(-0.513319\pi\)
−0.0418306 + 0.999125i \(0.513319\pi\)
\(168\) −12.5216 −0.966059
\(169\) −1.47265 −0.113281
\(170\) 0 0
\(171\) 3.78258 0.289261
\(172\) −3.19163 −0.243360
\(173\) −18.2635 −1.38854 −0.694272 0.719712i \(-0.744274\pi\)
−0.694272 + 0.719712i \(0.744274\pi\)
\(174\) 3.60638 0.273399
\(175\) 0 0
\(176\) 0.566255 0.0426831
\(177\) −4.95536 −0.372468
\(178\) −5.34437 −0.400578
\(179\) −6.93140 −0.518077 −0.259039 0.965867i \(-0.583406\pi\)
−0.259039 + 0.965867i \(0.583406\pi\)
\(180\) 0 0
\(181\) 13.1858 0.980091 0.490045 0.871697i \(-0.336980\pi\)
0.490045 + 0.871697i \(0.336980\pi\)
\(182\) −18.6903 −1.38542
\(183\) 2.36706 0.174978
\(184\) −24.8397 −1.83121
\(185\) 0 0
\(186\) −10.1375 −0.743320
\(187\) −0.767463 −0.0561224
\(188\) 2.74465 0.200174
\(189\) 4.19091 0.304844
\(190\) 0 0
\(191\) 23.3938 1.69272 0.846358 0.532614i \(-0.178790\pi\)
0.846358 + 0.532614i \(0.178790\pi\)
\(192\) 8.77608 0.633359
\(193\) −9.06435 −0.652466 −0.326233 0.945289i \(-0.605779\pi\)
−0.326233 + 0.945289i \(0.605779\pi\)
\(194\) −3.65423 −0.262358
\(195\) 0 0
\(196\) −2.90084 −0.207203
\(197\) −13.2823 −0.946327 −0.473163 0.880975i \(-0.656888\pi\)
−0.473163 + 0.880975i \(0.656888\pi\)
\(198\) −0.220360 −0.0156603
\(199\) 23.9985 1.70121 0.850606 0.525804i \(-0.176235\pi\)
0.850606 + 0.525804i \(0.176235\pi\)
\(200\) 0 0
\(201\) 12.2369 0.863123
\(202\) −14.0555 −0.988938
\(203\) 11.5063 0.807582
\(204\) −1.25625 −0.0879552
\(205\) 0 0
\(206\) −4.75013 −0.330957
\(207\) 8.31374 0.577845
\(208\) 11.4601 0.794613
\(209\) −0.634567 −0.0438939
\(210\) 0 0
\(211\) −7.75260 −0.533711 −0.266855 0.963737i \(-0.585985\pi\)
−0.266855 + 0.963737i \(0.585985\pi\)
\(212\) 0.294282 0.0202114
\(213\) 14.1746 0.971229
\(214\) 0.685714 0.0468744
\(215\) 0 0
\(216\) −2.98779 −0.203293
\(217\) −32.3442 −2.19567
\(218\) 15.1101 1.02338
\(219\) −8.67832 −0.586427
\(220\) 0 0
\(221\) −15.5322 −1.04481
\(222\) 2.72823 0.183107
\(223\) −22.1577 −1.48379 −0.741895 0.670517i \(-0.766073\pi\)
−0.741895 + 0.670517i \(0.766073\pi\)
\(224\) 6.46185 0.431751
\(225\) 0 0
\(226\) −26.4575 −1.75993
\(227\) −11.9286 −0.791729 −0.395864 0.918309i \(-0.629555\pi\)
−0.395864 + 0.918309i \(0.629555\pi\)
\(228\) −1.03872 −0.0687907
\(229\) −7.73897 −0.511406 −0.255703 0.966755i \(-0.582307\pi\)
−0.255703 + 0.966755i \(0.582307\pi\)
\(230\) 0 0
\(231\) −0.703068 −0.0462585
\(232\) −8.20308 −0.538559
\(233\) −2.85168 −0.186820 −0.0934099 0.995628i \(-0.529777\pi\)
−0.0934099 + 0.995628i \(0.529777\pi\)
\(234\) −4.45973 −0.291542
\(235\) 0 0
\(236\) 1.36077 0.0885784
\(237\) −2.64466 −0.171789
\(238\) 25.1837 1.63242
\(239\) −19.9126 −1.28804 −0.644020 0.765009i \(-0.722735\pi\)
−0.644020 + 0.765009i \(0.722735\pi\)
\(240\) 0 0
\(241\) 6.61379 0.426032 0.213016 0.977049i \(-0.431671\pi\)
0.213016 + 0.977049i \(0.431671\pi\)
\(242\) −14.4120 −0.926439
\(243\) 1.00000 0.0641500
\(244\) −0.650008 −0.0416125
\(245\) 0 0
\(246\) −1.68189 −0.107234
\(247\) −12.8426 −0.817155
\(248\) 23.0589 1.46424
\(249\) −14.3131 −0.907054
\(250\) 0 0
\(251\) −12.0644 −0.761501 −0.380750 0.924678i \(-0.624334\pi\)
−0.380750 + 0.924678i \(0.624334\pi\)
\(252\) −1.15084 −0.0724964
\(253\) −1.39472 −0.0876850
\(254\) 21.4561 1.34627
\(255\) 0 0
\(256\) −6.46059 −0.403787
\(257\) −12.7951 −0.798136 −0.399068 0.916921i \(-0.630666\pi\)
−0.399068 + 0.916921i \(0.630666\pi\)
\(258\) 15.2668 0.950471
\(259\) 8.70451 0.540872
\(260\) 0 0
\(261\) 2.74553 0.169944
\(262\) 24.8453 1.53495
\(263\) −10.5225 −0.648843 −0.324422 0.945913i \(-0.605170\pi\)
−0.324422 + 0.945913i \(0.605170\pi\)
\(264\) 0.501233 0.0308488
\(265\) 0 0
\(266\) 20.8229 1.27673
\(267\) −4.06867 −0.248999
\(268\) −3.36031 −0.205264
\(269\) 6.80128 0.414681 0.207341 0.978269i \(-0.433519\pi\)
0.207341 + 0.978269i \(0.433519\pi\)
\(270\) 0 0
\(271\) 5.66981 0.344416 0.172208 0.985061i \(-0.444910\pi\)
0.172208 + 0.985061i \(0.444910\pi\)
\(272\) −15.4415 −0.936281
\(273\) −14.2289 −0.861175
\(274\) −17.8609 −1.07902
\(275\) 0 0
\(276\) −2.28300 −0.137420
\(277\) 0.874982 0.0525726 0.0262863 0.999654i \(-0.491632\pi\)
0.0262863 + 0.999654i \(0.491632\pi\)
\(278\) −4.41486 −0.264786
\(279\) −7.71771 −0.462047
\(280\) 0 0
\(281\) −4.85806 −0.289808 −0.144904 0.989446i \(-0.546287\pi\)
−0.144904 + 0.989446i \(0.546287\pi\)
\(282\) −13.1287 −0.781804
\(283\) 11.4230 0.679024 0.339512 0.940602i \(-0.389738\pi\)
0.339512 + 0.940602i \(0.389738\pi\)
\(284\) −3.89242 −0.230973
\(285\) 0 0
\(286\) 0.748167 0.0442400
\(287\) −5.36615 −0.316754
\(288\) 1.54187 0.0908558
\(289\) 3.92839 0.231082
\(290\) 0 0
\(291\) −2.78196 −0.163082
\(292\) 2.38311 0.139461
\(293\) 0.0808664 0.00472427 0.00236213 0.999997i \(-0.499248\pi\)
0.00236213 + 0.999997i \(0.499248\pi\)
\(294\) 13.8759 0.809256
\(295\) 0 0
\(296\) −6.20564 −0.360696
\(297\) −0.167760 −0.00973444
\(298\) −8.32038 −0.481987
\(299\) −28.2268 −1.63240
\(300\) 0 0
\(301\) 48.7093 2.80756
\(302\) 0.492927 0.0283647
\(303\) −10.7004 −0.614723
\(304\) −12.7677 −0.732275
\(305\) 0 0
\(306\) 6.00914 0.343520
\(307\) −7.77807 −0.443918 −0.221959 0.975056i \(-0.571245\pi\)
−0.221959 + 0.975056i \(0.571245\pi\)
\(308\) 0.193066 0.0110010
\(309\) −3.61627 −0.205723
\(310\) 0 0
\(311\) −13.6181 −0.772213 −0.386106 0.922454i \(-0.626180\pi\)
−0.386106 + 0.922454i \(0.626180\pi\)
\(312\) 10.1441 0.574299
\(313\) 4.85037 0.274159 0.137079 0.990560i \(-0.456228\pi\)
0.137079 + 0.990560i \(0.456228\pi\)
\(314\) 13.7924 0.778351
\(315\) 0 0
\(316\) 0.726237 0.0408540
\(317\) −22.3020 −1.25260 −0.626302 0.779580i \(-0.715432\pi\)
−0.626302 + 0.779580i \(0.715432\pi\)
\(318\) −1.40766 −0.0789378
\(319\) −0.460592 −0.0257882
\(320\) 0 0
\(321\) 0.522034 0.0291371
\(322\) 45.7666 2.55047
\(323\) 17.3044 0.962842
\(324\) −0.274605 −0.0152558
\(325\) 0 0
\(326\) −18.1870 −1.00728
\(327\) 11.5033 0.636133
\(328\) 3.82565 0.211236
\(329\) −41.8877 −2.30934
\(330\) 0 0
\(331\) 0.144235 0.00792788 0.00396394 0.999992i \(-0.498738\pi\)
0.00396394 + 0.999992i \(0.498738\pi\)
\(332\) 3.93045 0.215711
\(333\) 2.07700 0.113819
\(334\) −1.42012 −0.0777057
\(335\) 0 0
\(336\) −14.1459 −0.771722
\(337\) 10.1832 0.554717 0.277358 0.960767i \(-0.410541\pi\)
0.277358 + 0.960767i \(0.410541\pi\)
\(338\) −1.93439 −0.105217
\(339\) −20.1421 −1.09397
\(340\) 0 0
\(341\) 1.29472 0.0701133
\(342\) 4.96858 0.268670
\(343\) 14.9351 0.806418
\(344\) −34.7260 −1.87230
\(345\) 0 0
\(346\) −23.9898 −1.28970
\(347\) 10.5782 0.567870 0.283935 0.958844i \(-0.408360\pi\)
0.283935 + 0.958844i \(0.408360\pi\)
\(348\) −0.753938 −0.0404153
\(349\) −29.1511 −1.56042 −0.780210 0.625517i \(-0.784888\pi\)
−0.780210 + 0.625517i \(0.784888\pi\)
\(350\) 0 0
\(351\) −3.39519 −0.181222
\(352\) −0.258665 −0.0137869
\(353\) 3.81907 0.203268 0.101634 0.994822i \(-0.467593\pi\)
0.101634 + 0.994822i \(0.467593\pi\)
\(354\) −6.50908 −0.345953
\(355\) 0 0
\(356\) 1.11728 0.0592156
\(357\) 19.1724 1.01471
\(358\) −9.10469 −0.481198
\(359\) −10.9849 −0.579759 −0.289879 0.957063i \(-0.593615\pi\)
−0.289879 + 0.957063i \(0.593615\pi\)
\(360\) 0 0
\(361\) −4.69207 −0.246951
\(362\) 17.3201 0.910323
\(363\) −10.9719 −0.575873
\(364\) 3.90734 0.204800
\(365\) 0 0
\(366\) 3.10924 0.162522
\(367\) 4.07169 0.212541 0.106270 0.994337i \(-0.466109\pi\)
0.106270 + 0.994337i \(0.466109\pi\)
\(368\) −28.0620 −1.46283
\(369\) −1.28043 −0.0666563
\(370\) 0 0
\(371\) −4.49120 −0.233171
\(372\) 2.11932 0.109882
\(373\) 1.92973 0.0999175 0.0499588 0.998751i \(-0.484091\pi\)
0.0499588 + 0.998751i \(0.484091\pi\)
\(374\) −1.00810 −0.0521274
\(375\) 0 0
\(376\) 29.8627 1.54005
\(377\) −9.32162 −0.480088
\(378\) 5.50493 0.283143
\(379\) 14.6050 0.750210 0.375105 0.926982i \(-0.377607\pi\)
0.375105 + 0.926982i \(0.377607\pi\)
\(380\) 0 0
\(381\) 16.3345 0.836842
\(382\) 30.7288 1.57222
\(383\) −31.5290 −1.61106 −0.805530 0.592556i \(-0.798119\pi\)
−0.805530 + 0.592556i \(0.798119\pi\)
\(384\) 8.44401 0.430907
\(385\) 0 0
\(386\) −11.9064 −0.606021
\(387\) 11.6226 0.590811
\(388\) 0.763942 0.0387833
\(389\) 14.6046 0.740485 0.370242 0.928935i \(-0.379275\pi\)
0.370242 + 0.928935i \(0.379275\pi\)
\(390\) 0 0
\(391\) 38.0333 1.92343
\(392\) −31.5621 −1.59413
\(393\) 18.9147 0.954122
\(394\) −17.4469 −0.878962
\(395\) 0 0
\(396\) 0.0460679 0.00231500
\(397\) 15.2793 0.766845 0.383422 0.923573i \(-0.374745\pi\)
0.383422 + 0.923573i \(0.374745\pi\)
\(398\) 31.5231 1.58011
\(399\) 15.8524 0.793615
\(400\) 0 0
\(401\) −15.7148 −0.784760 −0.392380 0.919803i \(-0.628348\pi\)
−0.392380 + 0.919803i \(0.628348\pi\)
\(402\) 16.0737 0.801682
\(403\) 26.2031 1.30527
\(404\) 2.93839 0.146190
\(405\) 0 0
\(406\) 15.1140 0.750094
\(407\) −0.348438 −0.0172714
\(408\) −13.6684 −0.676687
\(409\) 12.4187 0.614064 0.307032 0.951699i \(-0.400664\pi\)
0.307032 + 0.951699i \(0.400664\pi\)
\(410\) 0 0
\(411\) −13.5975 −0.670715
\(412\) 0.993047 0.0489239
\(413\) −20.7674 −1.02190
\(414\) 10.9204 0.536711
\(415\) 0 0
\(416\) −5.23496 −0.256665
\(417\) −3.36103 −0.164590
\(418\) −0.833531 −0.0407693
\(419\) −9.08687 −0.443923 −0.221961 0.975055i \(-0.571246\pi\)
−0.221961 + 0.975055i \(0.571246\pi\)
\(420\) 0 0
\(421\) 5.15620 0.251298 0.125649 0.992075i \(-0.459899\pi\)
0.125649 + 0.992075i \(0.459899\pi\)
\(422\) −10.1834 −0.495719
\(423\) −9.99490 −0.485968
\(424\) 3.20188 0.155497
\(425\) 0 0
\(426\) 18.6190 0.902092
\(427\) 9.92014 0.480069
\(428\) −0.143353 −0.00692924
\(429\) 0.569579 0.0274995
\(430\) 0 0
\(431\) 3.66953 0.176755 0.0883776 0.996087i \(-0.471832\pi\)
0.0883776 + 0.996087i \(0.471832\pi\)
\(432\) −3.37538 −0.162398
\(433\) −17.7340 −0.852242 −0.426121 0.904666i \(-0.640120\pi\)
−0.426121 + 0.904666i \(0.640120\pi\)
\(434\) −42.4855 −2.03937
\(435\) 0 0
\(436\) −3.15886 −0.151282
\(437\) 31.4474 1.50433
\(438\) −11.3993 −0.544682
\(439\) −26.1823 −1.24961 −0.624806 0.780780i \(-0.714822\pi\)
−0.624806 + 0.780780i \(0.714822\pi\)
\(440\) 0 0
\(441\) 10.5637 0.503033
\(442\) −20.4022 −0.970433
\(443\) 19.2199 0.913166 0.456583 0.889681i \(-0.349073\pi\)
0.456583 + 0.889681i \(0.349073\pi\)
\(444\) −0.570355 −0.0270679
\(445\) 0 0
\(446\) −29.1051 −1.37817
\(447\) −6.33430 −0.299602
\(448\) 36.7797 1.73768
\(449\) 33.6407 1.58760 0.793802 0.608177i \(-0.208099\pi\)
0.793802 + 0.608177i \(0.208099\pi\)
\(450\) 0 0
\(451\) 0.214805 0.0101148
\(452\) 5.53112 0.260162
\(453\) 0.375265 0.0176315
\(454\) −15.6687 −0.735369
\(455\) 0 0
\(456\) −11.3016 −0.529244
\(457\) −24.0255 −1.12387 −0.561933 0.827183i \(-0.689942\pi\)
−0.561933 + 0.827183i \(0.689942\pi\)
\(458\) −10.1655 −0.475001
\(459\) 4.57476 0.213531
\(460\) 0 0
\(461\) −14.4282 −0.671990 −0.335995 0.941864i \(-0.609073\pi\)
−0.335995 + 0.941864i \(0.609073\pi\)
\(462\) −0.923510 −0.0429656
\(463\) −2.95599 −0.137377 −0.0686883 0.997638i \(-0.521881\pi\)
−0.0686883 + 0.997638i \(0.521881\pi\)
\(464\) −9.26722 −0.430220
\(465\) 0 0
\(466\) −3.74581 −0.173521
\(467\) 37.1933 1.72110 0.860551 0.509364i \(-0.170119\pi\)
0.860551 + 0.509364i \(0.170119\pi\)
\(468\) 0.932338 0.0430974
\(469\) 51.2836 2.36806
\(470\) 0 0
\(471\) 10.5002 0.483822
\(472\) 14.8056 0.681482
\(473\) −1.94982 −0.0896526
\(474\) −3.47387 −0.159560
\(475\) 0 0
\(476\) −5.26483 −0.241313
\(477\) −1.07165 −0.0490677
\(478\) −26.1561 −1.19635
\(479\) −8.65394 −0.395409 −0.197704 0.980262i \(-0.563349\pi\)
−0.197704 + 0.980262i \(0.563349\pi\)
\(480\) 0 0
\(481\) −7.05182 −0.321535
\(482\) 8.68750 0.395705
\(483\) 34.8421 1.58537
\(484\) 3.01293 0.136951
\(485\) 0 0
\(486\) 1.31354 0.0595835
\(487\) 38.4645 1.74299 0.871496 0.490402i \(-0.163150\pi\)
0.871496 + 0.490402i \(0.163150\pi\)
\(488\) −7.07229 −0.320148
\(489\) −13.8457 −0.626126
\(490\) 0 0
\(491\) 30.0853 1.35773 0.678865 0.734263i \(-0.262472\pi\)
0.678865 + 0.734263i \(0.262472\pi\)
\(492\) 0.351612 0.0158519
\(493\) 12.5601 0.565680
\(494\) −16.8693 −0.758986
\(495\) 0 0
\(496\) 26.0502 1.16969
\(497\) 59.4045 2.66466
\(498\) −18.8008 −0.842486
\(499\) −19.6343 −0.878951 −0.439476 0.898254i \(-0.644836\pi\)
−0.439476 + 0.898254i \(0.644836\pi\)
\(500\) 0 0
\(501\) −1.08114 −0.0483018
\(502\) −15.8472 −0.707293
\(503\) −30.4802 −1.35904 −0.679522 0.733655i \(-0.737813\pi\)
−0.679522 + 0.733655i \(0.737813\pi\)
\(504\) −12.5216 −0.557754
\(505\) 0 0
\(506\) −1.83202 −0.0814431
\(507\) −1.47265 −0.0654027
\(508\) −4.48554 −0.199014
\(509\) 11.4371 0.506942 0.253471 0.967343i \(-0.418428\pi\)
0.253471 + 0.967343i \(0.418428\pi\)
\(510\) 0 0
\(511\) −36.3700 −1.60892
\(512\) −25.3743 −1.12140
\(513\) 3.78258 0.167005
\(514\) −16.8069 −0.741321
\(515\) 0 0
\(516\) −3.19163 −0.140504
\(517\) 1.67675 0.0737433
\(518\) 11.4337 0.502370
\(519\) −18.2635 −0.801677
\(520\) 0 0
\(521\) 16.6733 0.730471 0.365236 0.930915i \(-0.380988\pi\)
0.365236 + 0.930915i \(0.380988\pi\)
\(522\) 3.60638 0.157847
\(523\) 2.54272 0.111185 0.0555926 0.998454i \(-0.482295\pi\)
0.0555926 + 0.998454i \(0.482295\pi\)
\(524\) −5.19409 −0.226905
\(525\) 0 0
\(526\) −13.8217 −0.602656
\(527\) −35.3066 −1.53798
\(528\) 0.566255 0.0246431
\(529\) 46.1182 2.00514
\(530\) 0 0
\(531\) −4.95536 −0.215044
\(532\) −4.35316 −0.188734
\(533\) 4.34730 0.188302
\(534\) −5.34437 −0.231274
\(535\) 0 0
\(536\) −36.5612 −1.57921
\(537\) −6.93140 −0.299112
\(538\) 8.93377 0.385162
\(539\) −1.77217 −0.0763327
\(540\) 0 0
\(541\) 26.3649 1.13352 0.566758 0.823884i \(-0.308197\pi\)
0.566758 + 0.823884i \(0.308197\pi\)
\(542\) 7.44753 0.319899
\(543\) 13.1858 0.565856
\(544\) 7.05370 0.302425
\(545\) 0 0
\(546\) −18.6903 −0.799872
\(547\) −14.8319 −0.634165 −0.317082 0.948398i \(-0.602703\pi\)
−0.317082 + 0.948398i \(0.602703\pi\)
\(548\) 3.73394 0.159506
\(549\) 2.36706 0.101024
\(550\) 0 0
\(551\) 10.3852 0.442425
\(552\) −24.8397 −1.05725
\(553\) −11.0835 −0.471319
\(554\) 1.14933 0.0488302
\(555\) 0 0
\(556\) 0.922957 0.0391421
\(557\) −18.7407 −0.794069 −0.397034 0.917804i \(-0.629961\pi\)
−0.397034 + 0.917804i \(0.629961\pi\)
\(558\) −10.1375 −0.429156
\(559\) −39.4611 −1.66903
\(560\) 0 0
\(561\) −0.767463 −0.0324023
\(562\) −6.38127 −0.269178
\(563\) 32.1447 1.35474 0.677368 0.735644i \(-0.263121\pi\)
0.677368 + 0.735644i \(0.263121\pi\)
\(564\) 2.74465 0.115571
\(565\) 0 0
\(566\) 15.0045 0.630688
\(567\) 4.19091 0.176001
\(568\) −42.3508 −1.77700
\(569\) −2.14141 −0.0897725 −0.0448862 0.998992i \(-0.514293\pi\)
−0.0448862 + 0.998992i \(0.514293\pi\)
\(570\) 0 0
\(571\) 11.8476 0.495805 0.247902 0.968785i \(-0.420259\pi\)
0.247902 + 0.968785i \(0.420259\pi\)
\(572\) −0.156409 −0.00653981
\(573\) 23.3938 0.977290
\(574\) −7.04866 −0.294205
\(575\) 0 0
\(576\) 8.77608 0.365670
\(577\) 15.6985 0.653539 0.326769 0.945104i \(-0.394040\pi\)
0.326769 + 0.945104i \(0.394040\pi\)
\(578\) 5.16011 0.214632
\(579\) −9.06435 −0.376702
\(580\) 0 0
\(581\) −59.9847 −2.48859
\(582\) −3.65423 −0.151473
\(583\) 0.179781 0.00744577
\(584\) 25.9290 1.07295
\(585\) 0 0
\(586\) 0.106221 0.00438797
\(587\) 2.32845 0.0961055 0.0480527 0.998845i \(-0.484698\pi\)
0.0480527 + 0.998845i \(0.484698\pi\)
\(588\) −2.90084 −0.119629
\(589\) −29.1929 −1.20287
\(590\) 0 0
\(591\) −13.2823 −0.546362
\(592\) −7.01067 −0.288137
\(593\) 34.0275 1.39734 0.698672 0.715442i \(-0.253775\pi\)
0.698672 + 0.715442i \(0.253775\pi\)
\(594\) −0.220360 −0.00904150
\(595\) 0 0
\(596\) 1.73943 0.0712499
\(597\) 23.9985 0.982195
\(598\) −37.0771 −1.51619
\(599\) −2.93932 −0.120097 −0.0600486 0.998195i \(-0.519126\pi\)
−0.0600486 + 0.998195i \(0.519126\pi\)
\(600\) 0 0
\(601\) 5.17826 0.211225 0.105613 0.994407i \(-0.466320\pi\)
0.105613 + 0.994407i \(0.466320\pi\)
\(602\) 63.9818 2.60770
\(603\) 12.2369 0.498324
\(604\) −0.103050 −0.00419303
\(605\) 0 0
\(606\) −14.0555 −0.570964
\(607\) 1.45215 0.0589409 0.0294705 0.999566i \(-0.490618\pi\)
0.0294705 + 0.999566i \(0.490618\pi\)
\(608\) 5.83227 0.236530
\(609\) 11.5063 0.466258
\(610\) 0 0
\(611\) 33.9346 1.37285
\(612\) −1.25625 −0.0507810
\(613\) 41.1889 1.66361 0.831803 0.555072i \(-0.187309\pi\)
0.831803 + 0.555072i \(0.187309\pi\)
\(614\) −10.2168 −0.412318
\(615\) 0 0
\(616\) 2.10062 0.0846364
\(617\) −33.9582 −1.36711 −0.683553 0.729901i \(-0.739566\pi\)
−0.683553 + 0.729901i \(0.739566\pi\)
\(618\) −4.75013 −0.191078
\(619\) −31.1879 −1.25355 −0.626774 0.779201i \(-0.715625\pi\)
−0.626774 + 0.779201i \(0.715625\pi\)
\(620\) 0 0
\(621\) 8.31374 0.333619
\(622\) −17.8880 −0.717243
\(623\) −17.0514 −0.683151
\(624\) 11.4601 0.458770
\(625\) 0 0
\(626\) 6.37117 0.254643
\(627\) −0.634567 −0.0253422
\(628\) −2.88340 −0.115060
\(629\) 9.50177 0.378860
\(630\) 0 0
\(631\) 21.9274 0.872917 0.436459 0.899724i \(-0.356233\pi\)
0.436459 + 0.899724i \(0.356233\pi\)
\(632\) 7.90168 0.314312
\(633\) −7.75260 −0.308138
\(634\) −29.2946 −1.16344
\(635\) 0 0
\(636\) 0.294282 0.0116690
\(637\) −35.8658 −1.42105
\(638\) −0.605007 −0.0239524
\(639\) 14.1746 0.560739
\(640\) 0 0
\(641\) −19.2185 −0.759083 −0.379542 0.925175i \(-0.623918\pi\)
−0.379542 + 0.925175i \(0.623918\pi\)
\(642\) 0.685714 0.0270630
\(643\) −13.6611 −0.538739 −0.269370 0.963037i \(-0.586815\pi\)
−0.269370 + 0.963037i \(0.586815\pi\)
\(644\) −9.56782 −0.377025
\(645\) 0 0
\(646\) 22.7301 0.894302
\(647\) −30.7060 −1.20718 −0.603588 0.797296i \(-0.706263\pi\)
−0.603588 + 0.797296i \(0.706263\pi\)
\(648\) −2.98779 −0.117372
\(649\) 0.831313 0.0326319
\(650\) 0 0
\(651\) −32.3442 −1.26767
\(652\) 3.80211 0.148902
\(653\) −5.75846 −0.225346 −0.112673 0.993632i \(-0.535941\pi\)
−0.112673 + 0.993632i \(0.535941\pi\)
\(654\) 15.1101 0.590850
\(655\) 0 0
\(656\) 4.32193 0.168743
\(657\) −8.67832 −0.338574
\(658\) −55.0212 −2.14495
\(659\) 15.9750 0.622298 0.311149 0.950361i \(-0.399286\pi\)
0.311149 + 0.950361i \(0.399286\pi\)
\(660\) 0 0
\(661\) 31.1613 1.21203 0.606017 0.795451i \(-0.292766\pi\)
0.606017 + 0.795451i \(0.292766\pi\)
\(662\) 0.189459 0.00736354
\(663\) −15.5322 −0.603220
\(664\) 42.7645 1.65958
\(665\) 0 0
\(666\) 2.72823 0.105717
\(667\) 22.8256 0.883812
\(668\) 0.296887 0.0114869
\(669\) −22.1577 −0.856666
\(670\) 0 0
\(671\) −0.397099 −0.0153298
\(672\) 6.46185 0.249271
\(673\) 19.1758 0.739174 0.369587 0.929196i \(-0.379499\pi\)
0.369587 + 0.929196i \(0.379499\pi\)
\(674\) 13.3761 0.515229
\(675\) 0 0
\(676\) 0.404398 0.0155538
\(677\) −25.2153 −0.969105 −0.484552 0.874762i \(-0.661017\pi\)
−0.484552 + 0.874762i \(0.661017\pi\)
\(678\) −26.4575 −1.01609
\(679\) −11.6590 −0.447429
\(680\) 0 0
\(681\) −11.9286 −0.457105
\(682\) 1.70068 0.0651223
\(683\) 28.1160 1.07583 0.537914 0.843000i \(-0.319212\pi\)
0.537914 + 0.843000i \(0.319212\pi\)
\(684\) −1.03872 −0.0397163
\(685\) 0 0
\(686\) 19.6179 0.749013
\(687\) −7.73897 −0.295260
\(688\) −39.2308 −1.49566
\(689\) 3.63847 0.138615
\(690\) 0 0
\(691\) 11.8688 0.451511 0.225756 0.974184i \(-0.427515\pi\)
0.225756 + 0.974184i \(0.427515\pi\)
\(692\) 5.01524 0.190651
\(693\) −0.703068 −0.0267073
\(694\) 13.8950 0.527446
\(695\) 0 0
\(696\) −8.20308 −0.310937
\(697\) −5.85764 −0.221874
\(698\) −38.2912 −1.44934
\(699\) −2.85168 −0.107860
\(700\) 0 0
\(701\) 26.1748 0.988608 0.494304 0.869289i \(-0.335423\pi\)
0.494304 + 0.869289i \(0.335423\pi\)
\(702\) −4.45973 −0.168322
\(703\) 7.85642 0.296311
\(704\) −1.47228 −0.0554886
\(705\) 0 0
\(706\) 5.01651 0.188799
\(707\) −44.8444 −1.68655
\(708\) 1.36077 0.0511408
\(709\) 19.8934 0.747111 0.373556 0.927608i \(-0.378138\pi\)
0.373556 + 0.927608i \(0.378138\pi\)
\(710\) 0 0
\(711\) −2.64466 −0.0991824
\(712\) 12.1563 0.455578
\(713\) −64.1630 −2.40292
\(714\) 25.1837 0.942477
\(715\) 0 0
\(716\) 1.90340 0.0711333
\(717\) −19.9126 −0.743650
\(718\) −14.4291 −0.538489
\(719\) 53.3070 1.98802 0.994008 0.109309i \(-0.0348638\pi\)
0.994008 + 0.109309i \(0.0348638\pi\)
\(720\) 0 0
\(721\) −15.1555 −0.564419
\(722\) −6.16324 −0.229372
\(723\) 6.61379 0.245970
\(724\) −3.62088 −0.134569
\(725\) 0 0
\(726\) −14.4120 −0.534880
\(727\) 21.0186 0.779537 0.389768 0.920913i \(-0.372555\pi\)
0.389768 + 0.920913i \(0.372555\pi\)
\(728\) 42.5131 1.57564
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 53.1707 1.96659
\(732\) −0.650008 −0.0240250
\(733\) −24.4993 −0.904901 −0.452451 0.891789i \(-0.649450\pi\)
−0.452451 + 0.891789i \(0.649450\pi\)
\(734\) 5.34834 0.197411
\(735\) 0 0
\(736\) 12.8187 0.472505
\(737\) −2.05286 −0.0756182
\(738\) −1.68189 −0.0619114
\(739\) −11.3415 −0.417203 −0.208601 0.978001i \(-0.566891\pi\)
−0.208601 + 0.978001i \(0.566891\pi\)
\(740\) 0 0
\(741\) −12.8426 −0.471785
\(742\) −5.89938 −0.216573
\(743\) −20.0546 −0.735732 −0.367866 0.929879i \(-0.619912\pi\)
−0.367866 + 0.929879i \(0.619912\pi\)
\(744\) 23.0589 0.845380
\(745\) 0 0
\(746\) 2.53478 0.0928049
\(747\) −14.3131 −0.523688
\(748\) 0.210749 0.00770576
\(749\) 2.18779 0.0799403
\(750\) 0 0
\(751\) −10.7620 −0.392711 −0.196355 0.980533i \(-0.562911\pi\)
−0.196355 + 0.980533i \(0.562911\pi\)
\(752\) 33.7366 1.23025
\(753\) −12.0644 −0.439653
\(754\) −12.2443 −0.445913
\(755\) 0 0
\(756\) −1.15084 −0.0418558
\(757\) 36.8528 1.33944 0.669718 0.742615i \(-0.266415\pi\)
0.669718 + 0.742615i \(0.266415\pi\)
\(758\) 19.1843 0.696806
\(759\) −1.39472 −0.0506250
\(760\) 0 0
\(761\) −15.5973 −0.565403 −0.282701 0.959208i \(-0.591231\pi\)
−0.282701 + 0.959208i \(0.591231\pi\)
\(762\) 21.4561 0.777271
\(763\) 48.2092 1.74529
\(764\) −6.42406 −0.232414
\(765\) 0 0
\(766\) −41.4148 −1.49638
\(767\) 16.8244 0.607494
\(768\) −6.46059 −0.233127
\(769\) −10.4385 −0.376423 −0.188212 0.982128i \(-0.560269\pi\)
−0.188212 + 0.982128i \(0.560269\pi\)
\(770\) 0 0
\(771\) −12.7951 −0.460804
\(772\) 2.48912 0.0895853
\(773\) −29.1248 −1.04755 −0.523773 0.851858i \(-0.675476\pi\)
−0.523773 + 0.851858i \(0.675476\pi\)
\(774\) 15.2668 0.548754
\(775\) 0 0
\(776\) 8.31193 0.298381
\(777\) 8.70451 0.312273
\(778\) 19.1838 0.687773
\(779\) −4.84332 −0.173530
\(780\) 0 0
\(781\) −2.37794 −0.0850894
\(782\) 49.9584 1.78651
\(783\) 2.74553 0.0981173
\(784\) −35.6565 −1.27345
\(785\) 0 0
\(786\) 24.8453 0.886203
\(787\) −50.9945 −1.81776 −0.908879 0.417060i \(-0.863060\pi\)
−0.908879 + 0.417060i \(0.863060\pi\)
\(788\) 3.64740 0.129933
\(789\) −10.5225 −0.374610
\(790\) 0 0
\(791\) −84.4135 −3.00140
\(792\) 0.501233 0.0178105
\(793\) −8.03664 −0.285389
\(794\) 20.0700 0.712257
\(795\) 0 0
\(796\) −6.59013 −0.233581
\(797\) 36.6176 1.29706 0.648532 0.761188i \(-0.275383\pi\)
0.648532 + 0.761188i \(0.275383\pi\)
\(798\) 20.8229 0.737121
\(799\) −45.7242 −1.61761
\(800\) 0 0
\(801\) −4.06867 −0.143759
\(802\) −20.6421 −0.728897
\(803\) 1.45588 0.0513768
\(804\) −3.36031 −0.118509
\(805\) 0 0
\(806\) 34.4189 1.21235
\(807\) 6.80128 0.239416
\(808\) 31.9706 1.12472
\(809\) 34.4639 1.21168 0.605842 0.795585i \(-0.292836\pi\)
0.605842 + 0.795585i \(0.292836\pi\)
\(810\) 0 0
\(811\) 35.3834 1.24248 0.621240 0.783620i \(-0.286629\pi\)
0.621240 + 0.783620i \(0.286629\pi\)
\(812\) −3.15968 −0.110883
\(813\) 5.66981 0.198849
\(814\) −0.457689 −0.0160420
\(815\) 0 0
\(816\) −15.4415 −0.540562
\(817\) 43.9635 1.53809
\(818\) 16.3125 0.570352
\(819\) −14.2289 −0.497199
\(820\) 0 0
\(821\) −2.15613 −0.0752493 −0.0376246 0.999292i \(-0.511979\pi\)
−0.0376246 + 0.999292i \(0.511979\pi\)
\(822\) −17.8609 −0.622970
\(823\) 34.5492 1.20431 0.602154 0.798380i \(-0.294309\pi\)
0.602154 + 0.798380i \(0.294309\pi\)
\(824\) 10.8047 0.376398
\(825\) 0 0
\(826\) −27.2789 −0.949155
\(827\) −21.2543 −0.739083 −0.369542 0.929214i \(-0.620485\pi\)
−0.369542 + 0.929214i \(0.620485\pi\)
\(828\) −2.28300 −0.0793396
\(829\) 28.8519 1.00207 0.501035 0.865427i \(-0.332953\pi\)
0.501035 + 0.865427i \(0.332953\pi\)
\(830\) 0 0
\(831\) 0.874982 0.0303528
\(832\) −29.7965 −1.03301
\(833\) 48.3263 1.67441
\(834\) −4.41486 −0.152874
\(835\) 0 0
\(836\) 0.174255 0.00602675
\(837\) −7.71771 −0.266763
\(838\) −11.9360 −0.412322
\(839\) 0.443198 0.0153009 0.00765045 0.999971i \(-0.497565\pi\)
0.00765045 + 0.999971i \(0.497565\pi\)
\(840\) 0 0
\(841\) −21.4620 −0.740071
\(842\) 6.77289 0.233409
\(843\) −4.85806 −0.167321
\(844\) 2.12890 0.0732799
\(845\) 0 0
\(846\) −13.1287 −0.451375
\(847\) −45.9820 −1.57996
\(848\) 3.61724 0.124217
\(849\) 11.4230 0.392035
\(850\) 0 0
\(851\) 17.2676 0.591927
\(852\) −3.89242 −0.133352
\(853\) 29.0351 0.994142 0.497071 0.867710i \(-0.334409\pi\)
0.497071 + 0.867710i \(0.334409\pi\)
\(854\) 13.0305 0.445895
\(855\) 0 0
\(856\) −1.55973 −0.0533104
\(857\) 44.1483 1.50808 0.754039 0.656830i \(-0.228103\pi\)
0.754039 + 0.656830i \(0.228103\pi\)
\(858\) 0.748167 0.0255420
\(859\) 51.1393 1.74485 0.872426 0.488747i \(-0.162546\pi\)
0.872426 + 0.488747i \(0.162546\pi\)
\(860\) 0 0
\(861\) −5.36615 −0.182878
\(862\) 4.82009 0.164173
\(863\) −2.01515 −0.0685967 −0.0342983 0.999412i \(-0.510920\pi\)
−0.0342983 + 0.999412i \(0.510920\pi\)
\(864\) 1.54187 0.0524556
\(865\) 0 0
\(866\) −23.2944 −0.791575
\(867\) 3.92839 0.133415
\(868\) 8.88188 0.301471
\(869\) 0.443669 0.0150504
\(870\) 0 0
\(871\) −41.5466 −1.40775
\(872\) −34.3694 −1.16390
\(873\) −2.78196 −0.0941552
\(874\) 41.3075 1.39725
\(875\) 0 0
\(876\) 2.38311 0.0805179
\(877\) −5.02192 −0.169578 −0.0847891 0.996399i \(-0.527022\pi\)
−0.0847891 + 0.996399i \(0.527022\pi\)
\(878\) −34.3916 −1.16066
\(879\) 0.0808664 0.00272756
\(880\) 0 0
\(881\) −42.2541 −1.42358 −0.711789 0.702393i \(-0.752115\pi\)
−0.711789 + 0.702393i \(0.752115\pi\)
\(882\) 13.8759 0.467224
\(883\) 56.8687 1.91379 0.956893 0.290442i \(-0.0938024\pi\)
0.956893 + 0.290442i \(0.0938024\pi\)
\(884\) 4.26522 0.143455
\(885\) 0 0
\(886\) 25.2462 0.848162
\(887\) −17.1287 −0.575127 −0.287563 0.957762i \(-0.592845\pi\)
−0.287563 + 0.957762i \(0.592845\pi\)
\(888\) −6.20564 −0.208248
\(889\) 68.4563 2.29595
\(890\) 0 0
\(891\) −0.167760 −0.00562018
\(892\) 6.08462 0.203728
\(893\) −37.8065 −1.26515
\(894\) −8.32038 −0.278275
\(895\) 0 0
\(896\) 35.3880 1.18223
\(897\) −28.2268 −0.942464
\(898\) 44.1885 1.47459
\(899\) −21.1892 −0.706700
\(900\) 0 0
\(901\) −4.90255 −0.163328
\(902\) 0.282155 0.00939474
\(903\) 48.7093 1.62095
\(904\) 60.1803 2.00157
\(905\) 0 0
\(906\) 0.492927 0.0163764
\(907\) −57.9111 −1.92291 −0.961454 0.274967i \(-0.911333\pi\)
−0.961454 + 0.274967i \(0.911333\pi\)
\(908\) 3.27565 0.108706
\(909\) −10.7004 −0.354910
\(910\) 0 0
\(911\) 7.89167 0.261463 0.130731 0.991418i \(-0.458267\pi\)
0.130731 + 0.991418i \(0.458267\pi\)
\(912\) −12.7677 −0.422779
\(913\) 2.40117 0.0794670
\(914\) −31.5585 −1.04386
\(915\) 0 0
\(916\) 2.12516 0.0702173
\(917\) 79.2699 2.61772
\(918\) 6.00914 0.198331
\(919\) −3.66345 −0.120846 −0.0604230 0.998173i \(-0.519245\pi\)
−0.0604230 + 0.998173i \(0.519245\pi\)
\(920\) 0 0
\(921\) −7.77807 −0.256296
\(922\) −18.9521 −0.624155
\(923\) −48.1256 −1.58407
\(924\) 0.193066 0.00635141
\(925\) 0 0
\(926\) −3.88282 −0.127597
\(927\) −3.61627 −0.118774
\(928\) 4.23327 0.138964
\(929\) −49.7787 −1.63319 −0.816594 0.577213i \(-0.804140\pi\)
−0.816594 + 0.577213i \(0.804140\pi\)
\(930\) 0 0
\(931\) 39.9580 1.30957
\(932\) 0.783087 0.0256509
\(933\) −13.6181 −0.445837
\(934\) 48.8550 1.59859
\(935\) 0 0
\(936\) 10.1441 0.331571
\(937\) −9.69363 −0.316677 −0.158338 0.987385i \(-0.550614\pi\)
−0.158338 + 0.987385i \(0.550614\pi\)
\(938\) 67.3632 2.19949
\(939\) 4.85037 0.158286
\(940\) 0 0
\(941\) 27.4206 0.893886 0.446943 0.894562i \(-0.352513\pi\)
0.446943 + 0.894562i \(0.352513\pi\)
\(942\) 13.7924 0.449381
\(943\) −10.6451 −0.346653
\(944\) 16.7262 0.544392
\(945\) 0 0
\(946\) −2.56117 −0.0832707
\(947\) −12.7622 −0.414716 −0.207358 0.978265i \(-0.566486\pi\)
−0.207358 + 0.978265i \(0.566486\pi\)
\(948\) 0.726237 0.0235871
\(949\) 29.4646 0.956461
\(950\) 0 0
\(951\) −22.3020 −0.723192
\(952\) −57.2830 −1.85655
\(953\) 22.3816 0.725012 0.362506 0.931981i \(-0.381921\pi\)
0.362506 + 0.931981i \(0.381921\pi\)
\(954\) −1.40766 −0.0455748
\(955\) 0 0
\(956\) 5.46811 0.176851
\(957\) −0.460592 −0.0148888
\(958\) −11.3673 −0.367262
\(959\) −56.9858 −1.84017
\(960\) 0 0
\(961\) 28.5630 0.921386
\(962\) −9.26287 −0.298647
\(963\) 0.522034 0.0168223
\(964\) −1.81618 −0.0584953
\(965\) 0 0
\(966\) 45.7666 1.47252
\(967\) −41.4419 −1.33268 −0.666342 0.745647i \(-0.732141\pi\)
−0.666342 + 0.745647i \(0.732141\pi\)
\(968\) 32.7816 1.05364
\(969\) 17.3044 0.555897
\(970\) 0 0
\(971\) 40.7475 1.30765 0.653824 0.756646i \(-0.273164\pi\)
0.653824 + 0.756646i \(0.273164\pi\)
\(972\) −0.274605 −0.00880797
\(973\) −14.0858 −0.451569
\(974\) 50.5248 1.61892
\(975\) 0 0
\(976\) −7.98974 −0.255745
\(977\) −21.8246 −0.698231 −0.349116 0.937080i \(-0.613518\pi\)
−0.349116 + 0.937080i \(0.613518\pi\)
\(978\) −18.1870 −0.581555
\(979\) 0.682562 0.0218148
\(980\) 0 0
\(981\) 11.5033 0.367272
\(982\) 39.5183 1.26108
\(983\) −15.8023 −0.504016 −0.252008 0.967725i \(-0.581091\pi\)
−0.252008 + 0.967725i \(0.581091\pi\)
\(984\) 3.82565 0.121957
\(985\) 0 0
\(986\) 16.4983 0.525412
\(987\) −41.8877 −1.33330
\(988\) 3.52665 0.112198
\(989\) 96.6275 3.07257
\(990\) 0 0
\(991\) 7.78545 0.247313 0.123657 0.992325i \(-0.460538\pi\)
0.123657 + 0.992325i \(0.460538\pi\)
\(992\) −11.8997 −0.377817
\(993\) 0.144235 0.00457716
\(994\) 78.0303 2.47497
\(995\) 0 0
\(996\) 3.93045 0.124541
\(997\) 20.9919 0.664820 0.332410 0.943135i \(-0.392138\pi\)
0.332410 + 0.943135i \(0.392138\pi\)
\(998\) −25.7905 −0.816383
\(999\) 2.07700 0.0657134
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 1875.2.a.o.1.6 yes 8
3.2 odd 2 5625.2.a.u.1.3 8
5.2 odd 4 1875.2.b.g.1249.12 16
5.3 odd 4 1875.2.b.g.1249.5 16
5.4 even 2 1875.2.a.n.1.3 8
15.14 odd 2 5625.2.a.bc.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.n.1.3 8 5.4 even 2
1875.2.a.o.1.6 yes 8 1.1 even 1 trivial
1875.2.b.g.1249.5 16 5.3 odd 4
1875.2.b.g.1249.12 16 5.2 odd 4
5625.2.a.u.1.3 8 3.2 odd 2
5625.2.a.bc.1.6 8 15.14 odd 2