Properties

Label 1875.2.a.p.1.4
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: 8.8.5444000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.0898194\) 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)
 
\(f(q)\) \(=\) \(q-0.0898194 q^{2} -1.00000 q^{3} -1.99193 q^{4} +0.0898194 q^{6} +4.36070 q^{7} +0.358553 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.0898194 q^{2} -1.00000 q^{3} -1.99193 q^{4} +0.0898194 q^{6} +4.36070 q^{7} +0.358553 q^{8} +1.00000 q^{9} +4.39094 q^{11} +1.99193 q^{12} -1.98166 q^{13} -0.391676 q^{14} +3.95166 q^{16} +0.997022 q^{17} -0.0898194 q^{18} +1.35096 q^{19} -4.36070 q^{21} -0.394392 q^{22} +2.35651 q^{23} -0.358553 q^{24} +0.177991 q^{26} -1.00000 q^{27} -8.68622 q^{28} -7.97856 q^{29} -3.67761 q^{31} -1.07204 q^{32} -4.39094 q^{33} -0.0895519 q^{34} -1.99193 q^{36} -1.43706 q^{37} -0.121342 q^{38} +1.98166 q^{39} -5.98248 q^{41} +0.391676 q^{42} +2.68554 q^{43} -8.74646 q^{44} -0.211660 q^{46} +10.9393 q^{47} -3.95166 q^{48} +12.0157 q^{49} -0.997022 q^{51} +3.94732 q^{52} +11.0510 q^{53} +0.0898194 q^{54} +1.56354 q^{56} -1.35096 q^{57} +0.716629 q^{58} +6.68895 q^{59} -9.45570 q^{61} +0.330321 q^{62} +4.36070 q^{63} -7.80703 q^{64} +0.394392 q^{66} +12.9219 q^{67} -1.98600 q^{68} -2.35651 q^{69} +7.32257 q^{71} +0.358553 q^{72} -0.424804 q^{73} +0.129076 q^{74} -2.69101 q^{76} +19.1476 q^{77} -0.177991 q^{78} -6.35531 q^{79} +1.00000 q^{81} +0.537343 q^{82} +0.737011 q^{83} +8.68622 q^{84} -0.241213 q^{86} +7.97856 q^{87} +1.57439 q^{88} -9.78736 q^{89} -8.64141 q^{91} -4.69401 q^{92} +3.67761 q^{93} -0.982560 q^{94} +1.07204 q^{96} +0.0337081 q^{97} -1.07925 q^{98} +4.39094 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 8 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 8 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9} + 2 q^{11} - 4 q^{12} + 16 q^{13} + 6 q^{14} + 16 q^{17} + 4 q^{18} - 14 q^{19} - 8 q^{21} + 12 q^{22} + 14 q^{23} - 12 q^{24} + 6 q^{26} - 8 q^{27} + 16 q^{28} + 2 q^{29} - 22 q^{31} - 2 q^{32} - 2 q^{33} - 12 q^{34} + 4 q^{36} + 28 q^{37} - 16 q^{38} - 16 q^{39} + 8 q^{41} - 6 q^{42} + 20 q^{43} + 22 q^{44} - 2 q^{46} + 10 q^{47} - 16 q^{51} + 16 q^{52} + 44 q^{53} - 4 q^{54} + 30 q^{56} + 14 q^{57} + 8 q^{58} + 14 q^{59} - 20 q^{61} + 16 q^{62} + 8 q^{63} + 6 q^{64} - 12 q^{66} + 16 q^{67} - 2 q^{68} - 14 q^{69} + 16 q^{71} + 12 q^{72} + 24 q^{73} + 26 q^{74} - 16 q^{76} + 46 q^{77} - 6 q^{78} - 30 q^{79} + 8 q^{81} + 16 q^{82} + 12 q^{83} - 16 q^{84} + 32 q^{86} - 2 q^{87} + 32 q^{88} + 16 q^{89} - 12 q^{91} - 2 q^{92} + 22 q^{93} + 14 q^{94} + 2 q^{96} + 16 q^{97} + 4 q^{98} + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0898194 −0.0635119 −0.0317560 0.999496i \(-0.510110\pi\)
−0.0317560 + 0.999496i \(0.510110\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.99193 −0.995966
\(5\) 0 0
\(6\) 0.0898194 0.0366686
\(7\) 4.36070 1.64819 0.824095 0.566451i \(-0.191684\pi\)
0.824095 + 0.566451i \(0.191684\pi\)
\(8\) 0.358553 0.126768
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.39094 1.32392 0.661959 0.749540i \(-0.269725\pi\)
0.661959 + 0.749540i \(0.269725\pi\)
\(12\) 1.99193 0.575021
\(13\) −1.98166 −0.549612 −0.274806 0.961500i \(-0.588614\pi\)
−0.274806 + 0.961500i \(0.588614\pi\)
\(14\) −0.391676 −0.104680
\(15\) 0 0
\(16\) 3.95166 0.987915
\(17\) 0.997022 0.241813 0.120907 0.992664i \(-0.461420\pi\)
0.120907 + 0.992664i \(0.461420\pi\)
\(18\) −0.0898194 −0.0211706
\(19\) 1.35096 0.309931 0.154965 0.987920i \(-0.450473\pi\)
0.154965 + 0.987920i \(0.450473\pi\)
\(20\) 0 0
\(21\) −4.36070 −0.951583
\(22\) −0.394392 −0.0840846
\(23\) 2.35651 0.491366 0.245683 0.969350i \(-0.420988\pi\)
0.245683 + 0.969350i \(0.420988\pi\)
\(24\) −0.358553 −0.0731893
\(25\) 0 0
\(26\) 0.177991 0.0349069
\(27\) −1.00000 −0.192450
\(28\) −8.68622 −1.64154
\(29\) −7.97856 −1.48158 −0.740790 0.671736i \(-0.765549\pi\)
−0.740790 + 0.671736i \(0.765549\pi\)
\(30\) 0 0
\(31\) −3.67761 −0.660519 −0.330259 0.943890i \(-0.607136\pi\)
−0.330259 + 0.943890i \(0.607136\pi\)
\(32\) −1.07204 −0.189512
\(33\) −4.39094 −0.764365
\(34\) −0.0895519 −0.0153580
\(35\) 0 0
\(36\) −1.99193 −0.331989
\(37\) −1.43706 −0.236251 −0.118125 0.992999i \(-0.537688\pi\)
−0.118125 + 0.992999i \(0.537688\pi\)
\(38\) −0.121342 −0.0196843
\(39\) 1.98166 0.317319
\(40\) 0 0
\(41\) −5.98248 −0.934306 −0.467153 0.884177i \(-0.654720\pi\)
−0.467153 + 0.884177i \(0.654720\pi\)
\(42\) 0.391676 0.0604369
\(43\) 2.68554 0.409541 0.204770 0.978810i \(-0.434355\pi\)
0.204770 + 0.978810i \(0.434355\pi\)
\(44\) −8.74646 −1.31858
\(45\) 0 0
\(46\) −0.211660 −0.0312076
\(47\) 10.9393 1.59566 0.797829 0.602883i \(-0.205982\pi\)
0.797829 + 0.602883i \(0.205982\pi\)
\(48\) −3.95166 −0.570373
\(49\) 12.0157 1.71653
\(50\) 0 0
\(51\) −0.997022 −0.139611
\(52\) 3.94732 0.547395
\(53\) 11.0510 1.51798 0.758989 0.651104i \(-0.225694\pi\)
0.758989 + 0.651104i \(0.225694\pi\)
\(54\) 0.0898194 0.0122229
\(55\) 0 0
\(56\) 1.56354 0.208937
\(57\) −1.35096 −0.178938
\(58\) 0.716629 0.0940980
\(59\) 6.68895 0.870827 0.435414 0.900231i \(-0.356602\pi\)
0.435414 + 0.900231i \(0.356602\pi\)
\(60\) 0 0
\(61\) −9.45570 −1.21068 −0.605339 0.795967i \(-0.706963\pi\)
−0.605339 + 0.795967i \(0.706963\pi\)
\(62\) 0.330321 0.0419508
\(63\) 4.36070 0.549397
\(64\) −7.80703 −0.975879
\(65\) 0 0
\(66\) 0.394392 0.0485463
\(67\) 12.9219 1.57866 0.789328 0.613972i \(-0.210429\pi\)
0.789328 + 0.613972i \(0.210429\pi\)
\(68\) −1.98600 −0.240838
\(69\) −2.35651 −0.283690
\(70\) 0 0
\(71\) 7.32257 0.869029 0.434515 0.900665i \(-0.356920\pi\)
0.434515 + 0.900665i \(0.356920\pi\)
\(72\) 0.358553 0.0422559
\(73\) −0.424804 −0.0497195 −0.0248598 0.999691i \(-0.507914\pi\)
−0.0248598 + 0.999691i \(0.507914\pi\)
\(74\) 0.129076 0.0150047
\(75\) 0 0
\(76\) −2.69101 −0.308680
\(77\) 19.1476 2.18207
\(78\) −0.177991 −0.0201535
\(79\) −6.35531 −0.715028 −0.357514 0.933908i \(-0.616376\pi\)
−0.357514 + 0.933908i \(0.616376\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0.537343 0.0593396
\(83\) 0.737011 0.0808975 0.0404487 0.999182i \(-0.487121\pi\)
0.0404487 + 0.999182i \(0.487121\pi\)
\(84\) 8.68622 0.947745
\(85\) 0 0
\(86\) −0.241213 −0.0260107
\(87\) 7.97856 0.855391
\(88\) 1.57439 0.167830
\(89\) −9.78736 −1.03746 −0.518729 0.854939i \(-0.673595\pi\)
−0.518729 + 0.854939i \(0.673595\pi\)
\(90\) 0 0
\(91\) −8.64141 −0.905866
\(92\) −4.69401 −0.489384
\(93\) 3.67761 0.381351
\(94\) −0.982560 −0.101343
\(95\) 0 0
\(96\) 1.07204 0.109415
\(97\) 0.0337081 0.00342254 0.00171127 0.999999i \(-0.499455\pi\)
0.00171127 + 0.999999i \(0.499455\pi\)
\(98\) −1.07925 −0.109020
\(99\) 4.39094 0.441306
\(100\) 0 0
\(101\) 3.19390 0.317805 0.158902 0.987294i \(-0.449204\pi\)
0.158902 + 0.987294i \(0.449204\pi\)
\(102\) 0.0895519 0.00886696
\(103\) 8.55342 0.842794 0.421397 0.906876i \(-0.361540\pi\)
0.421397 + 0.906876i \(0.361540\pi\)
\(104\) −0.710529 −0.0696731
\(105\) 0 0
\(106\) −0.992598 −0.0964097
\(107\) 2.22136 0.214747 0.107373 0.994219i \(-0.465756\pi\)
0.107373 + 0.994219i \(0.465756\pi\)
\(108\) 1.99193 0.191674
\(109\) 11.0023 1.05383 0.526916 0.849917i \(-0.323348\pi\)
0.526916 + 0.849917i \(0.323348\pi\)
\(110\) 0 0
\(111\) 1.43706 0.136399
\(112\) 17.2320 1.62827
\(113\) 1.71021 0.160883 0.0804415 0.996759i \(-0.474367\pi\)
0.0804415 + 0.996759i \(0.474367\pi\)
\(114\) 0.121342 0.0113647
\(115\) 0 0
\(116\) 15.8927 1.47560
\(117\) −1.98166 −0.183204
\(118\) −0.600798 −0.0553079
\(119\) 4.34771 0.398554
\(120\) 0 0
\(121\) 8.28037 0.752761
\(122\) 0.849306 0.0768925
\(123\) 5.98248 0.539422
\(124\) 7.32556 0.657855
\(125\) 0 0
\(126\) −0.391676 −0.0348933
\(127\) −12.5570 −1.11425 −0.557125 0.830429i \(-0.688095\pi\)
−0.557125 + 0.830429i \(0.688095\pi\)
\(128\) 2.84531 0.251492
\(129\) −2.68554 −0.236448
\(130\) 0 0
\(131\) 16.4718 1.43915 0.719574 0.694416i \(-0.244337\pi\)
0.719574 + 0.694416i \(0.244337\pi\)
\(132\) 8.74646 0.761282
\(133\) 5.89112 0.510825
\(134\) −1.16063 −0.100263
\(135\) 0 0
\(136\) 0.357485 0.0306541
\(137\) −9.66732 −0.825935 −0.412967 0.910746i \(-0.635508\pi\)
−0.412967 + 0.910746i \(0.635508\pi\)
\(138\) 0.211660 0.0180177
\(139\) 13.5327 1.14783 0.573913 0.818916i \(-0.305425\pi\)
0.573913 + 0.818916i \(0.305425\pi\)
\(140\) 0 0
\(141\) −10.9393 −0.921254
\(142\) −0.657709 −0.0551937
\(143\) −8.70133 −0.727642
\(144\) 3.95166 0.329305
\(145\) 0 0
\(146\) 0.0381557 0.00315778
\(147\) −12.0157 −0.991040
\(148\) 2.86252 0.235298
\(149\) 13.6843 1.12106 0.560529 0.828134i \(-0.310598\pi\)
0.560529 + 0.828134i \(0.310598\pi\)
\(150\) 0 0
\(151\) −11.3204 −0.921237 −0.460619 0.887598i \(-0.652372\pi\)
−0.460619 + 0.887598i \(0.652372\pi\)
\(152\) 0.484389 0.0392892
\(153\) 0.997022 0.0806044
\(154\) −1.71983 −0.138587
\(155\) 0 0
\(156\) −3.94732 −0.316039
\(157\) 8.56070 0.683219 0.341609 0.939842i \(-0.389028\pi\)
0.341609 + 0.939842i \(0.389028\pi\)
\(158\) 0.570830 0.0454128
\(159\) −11.0510 −0.876405
\(160\) 0 0
\(161\) 10.2760 0.809865
\(162\) −0.0898194 −0.00705688
\(163\) 4.58509 0.359132 0.179566 0.983746i \(-0.442531\pi\)
0.179566 + 0.983746i \(0.442531\pi\)
\(164\) 11.9167 0.930537
\(165\) 0 0
\(166\) −0.0661979 −0.00513795
\(167\) 7.21792 0.558540 0.279270 0.960213i \(-0.409908\pi\)
0.279270 + 0.960213i \(0.409908\pi\)
\(168\) −1.56354 −0.120630
\(169\) −9.07304 −0.697926
\(170\) 0 0
\(171\) 1.35096 0.103310
\(172\) −5.34941 −0.407889
\(173\) −4.65009 −0.353540 −0.176770 0.984252i \(-0.556565\pi\)
−0.176770 + 0.984252i \(0.556565\pi\)
\(174\) −0.716629 −0.0543275
\(175\) 0 0
\(176\) 17.3515 1.30792
\(177\) −6.68895 −0.502772
\(178\) 0.879095 0.0658909
\(179\) −7.20338 −0.538406 −0.269203 0.963083i \(-0.586760\pi\)
−0.269203 + 0.963083i \(0.586760\pi\)
\(180\) 0 0
\(181\) −3.80424 −0.282767 −0.141383 0.989955i \(-0.545155\pi\)
−0.141383 + 0.989955i \(0.545155\pi\)
\(182\) 0.776167 0.0575333
\(183\) 9.45570 0.698986
\(184\) 0.844934 0.0622893
\(185\) 0 0
\(186\) −0.330321 −0.0242203
\(187\) 4.37786 0.320141
\(188\) −21.7903 −1.58922
\(189\) −4.36070 −0.317194
\(190\) 0 0
\(191\) 21.5541 1.55960 0.779801 0.626027i \(-0.215320\pi\)
0.779801 + 0.626027i \(0.215320\pi\)
\(192\) 7.80703 0.563424
\(193\) 3.15029 0.226763 0.113381 0.993552i \(-0.463832\pi\)
0.113381 + 0.993552i \(0.463832\pi\)
\(194\) −0.00302764 −0.000217372 0
\(195\) 0 0
\(196\) −23.9345 −1.70961
\(197\) 26.0837 1.85839 0.929195 0.369590i \(-0.120502\pi\)
0.929195 + 0.369590i \(0.120502\pi\)
\(198\) −0.394392 −0.0280282
\(199\) −24.2662 −1.72018 −0.860092 0.510139i \(-0.829594\pi\)
−0.860092 + 0.510139i \(0.829594\pi\)
\(200\) 0 0
\(201\) −12.9219 −0.911437
\(202\) −0.286874 −0.0201844
\(203\) −34.7921 −2.44193
\(204\) 1.98600 0.139048
\(205\) 0 0
\(206\) −0.768264 −0.0535275
\(207\) 2.35651 0.163789
\(208\) −7.83083 −0.542970
\(209\) 5.93197 0.410323
\(210\) 0 0
\(211\) 16.3783 1.12753 0.563765 0.825935i \(-0.309352\pi\)
0.563765 + 0.825935i \(0.309352\pi\)
\(212\) −22.0129 −1.51185
\(213\) −7.32257 −0.501734
\(214\) −0.199521 −0.0136390
\(215\) 0 0
\(216\) −0.358553 −0.0243964
\(217\) −16.0370 −1.08866
\(218\) −0.988224 −0.0669310
\(219\) 0.424804 0.0287056
\(220\) 0 0
\(221\) −1.97575 −0.132904
\(222\) −0.129076 −0.00866299
\(223\) −5.68295 −0.380558 −0.190279 0.981730i \(-0.560939\pi\)
−0.190279 + 0.981730i \(0.560939\pi\)
\(224\) −4.67486 −0.312352
\(225\) 0 0
\(226\) −0.153610 −0.0102180
\(227\) 3.64374 0.241844 0.120922 0.992662i \(-0.461415\pi\)
0.120922 + 0.992662i \(0.461415\pi\)
\(228\) 2.69101 0.178217
\(229\) 1.66125 0.109779 0.0548893 0.998492i \(-0.482519\pi\)
0.0548893 + 0.998492i \(0.482519\pi\)
\(230\) 0 0
\(231\) −19.1476 −1.25982
\(232\) −2.86074 −0.187817
\(233\) 7.85899 0.514860 0.257430 0.966297i \(-0.417124\pi\)
0.257430 + 0.966297i \(0.417124\pi\)
\(234\) 0.177991 0.0116356
\(235\) 0 0
\(236\) −13.3239 −0.867314
\(237\) 6.35531 0.412821
\(238\) −0.390509 −0.0253130
\(239\) −0.567301 −0.0366956 −0.0183478 0.999832i \(-0.505841\pi\)
−0.0183478 + 0.999832i \(0.505841\pi\)
\(240\) 0 0
\(241\) −19.0081 −1.22442 −0.612211 0.790695i \(-0.709720\pi\)
−0.612211 + 0.790695i \(0.709720\pi\)
\(242\) −0.743738 −0.0478093
\(243\) −1.00000 −0.0641500
\(244\) 18.8351 1.20580
\(245\) 0 0
\(246\) −0.537343 −0.0342597
\(247\) −2.67713 −0.170342
\(248\) −1.31862 −0.0837324
\(249\) −0.737011 −0.0467062
\(250\) 0 0
\(251\) 3.02533 0.190957 0.0954787 0.995431i \(-0.469562\pi\)
0.0954787 + 0.995431i \(0.469562\pi\)
\(252\) −8.68622 −0.547181
\(253\) 10.3473 0.650529
\(254\) 1.12786 0.0707681
\(255\) 0 0
\(256\) 15.3585 0.959906
\(257\) −19.8613 −1.23891 −0.619456 0.785032i \(-0.712647\pi\)
−0.619456 + 0.785032i \(0.712647\pi\)
\(258\) 0.241213 0.0150173
\(259\) −6.26658 −0.389386
\(260\) 0 0
\(261\) −7.97856 −0.493860
\(262\) −1.47949 −0.0914030
\(263\) 22.8299 1.40775 0.703876 0.710323i \(-0.251451\pi\)
0.703876 + 0.710323i \(0.251451\pi\)
\(264\) −1.57439 −0.0968968
\(265\) 0 0
\(266\) −0.529137 −0.0324435
\(267\) 9.78736 0.598977
\(268\) −25.7395 −1.57229
\(269\) 14.8324 0.904347 0.452174 0.891930i \(-0.350649\pi\)
0.452174 + 0.891930i \(0.350649\pi\)
\(270\) 0 0
\(271\) −6.43720 −0.391032 −0.195516 0.980701i \(-0.562638\pi\)
−0.195516 + 0.980701i \(0.562638\pi\)
\(272\) 3.93989 0.238891
\(273\) 8.64141 0.523002
\(274\) 0.868313 0.0524567
\(275\) 0 0
\(276\) 4.69401 0.282546
\(277\) 6.70976 0.403150 0.201575 0.979473i \(-0.435394\pi\)
0.201575 + 0.979473i \(0.435394\pi\)
\(278\) −1.21550 −0.0729006
\(279\) −3.67761 −0.220173
\(280\) 0 0
\(281\) −20.4867 −1.22214 −0.611068 0.791578i \(-0.709260\pi\)
−0.611068 + 0.791578i \(0.709260\pi\)
\(282\) 0.982560 0.0585106
\(283\) −11.4177 −0.678715 −0.339357 0.940658i \(-0.610210\pi\)
−0.339357 + 0.940658i \(0.610210\pi\)
\(284\) −14.5861 −0.865524
\(285\) 0 0
\(286\) 0.781549 0.0462140
\(287\) −26.0878 −1.53991
\(288\) −1.07204 −0.0631707
\(289\) −16.0059 −0.941526
\(290\) 0 0
\(291\) −0.0337081 −0.00197600
\(292\) 0.846181 0.0495190
\(293\) 28.5505 1.66794 0.833968 0.551812i \(-0.186063\pi\)
0.833968 + 0.551812i \(0.186063\pi\)
\(294\) 1.07925 0.0629429
\(295\) 0 0
\(296\) −0.515261 −0.0299489
\(297\) −4.39094 −0.254788
\(298\) −1.22911 −0.0712006
\(299\) −4.66979 −0.270061
\(300\) 0 0
\(301\) 11.7108 0.675001
\(302\) 1.01679 0.0585095
\(303\) −3.19390 −0.183485
\(304\) 5.33852 0.306185
\(305\) 0 0
\(306\) −0.0895519 −0.00511934
\(307\) −20.5417 −1.17238 −0.586188 0.810175i \(-0.699372\pi\)
−0.586188 + 0.810175i \(0.699372\pi\)
\(308\) −38.1407 −2.17327
\(309\) −8.55342 −0.486587
\(310\) 0 0
\(311\) 17.5496 0.995146 0.497573 0.867422i \(-0.334225\pi\)
0.497573 + 0.867422i \(0.334225\pi\)
\(312\) 0.710529 0.0402258
\(313\) −2.98564 −0.168758 −0.0843790 0.996434i \(-0.526891\pi\)
−0.0843790 + 0.996434i \(0.526891\pi\)
\(314\) −0.768918 −0.0433925
\(315\) 0 0
\(316\) 12.6593 0.712143
\(317\) −16.1708 −0.908244 −0.454122 0.890940i \(-0.650047\pi\)
−0.454122 + 0.890940i \(0.650047\pi\)
\(318\) 0.992598 0.0556621
\(319\) −35.0334 −1.96149
\(320\) 0 0
\(321\) −2.22136 −0.123984
\(322\) −0.922988 −0.0514361
\(323\) 1.34693 0.0749453
\(324\) −1.99193 −0.110663
\(325\) 0 0
\(326\) −0.411830 −0.0228092
\(327\) −11.0023 −0.608431
\(328\) −2.14504 −0.118440
\(329\) 47.7030 2.62995
\(330\) 0 0
\(331\) −13.0705 −0.718418 −0.359209 0.933257i \(-0.616953\pi\)
−0.359209 + 0.933257i \(0.616953\pi\)
\(332\) −1.46808 −0.0805711
\(333\) −1.43706 −0.0787502
\(334\) −0.648310 −0.0354739
\(335\) 0 0
\(336\) −17.2320 −0.940083
\(337\) 26.8049 1.46015 0.730077 0.683365i \(-0.239484\pi\)
0.730077 + 0.683365i \(0.239484\pi\)
\(338\) 0.814935 0.0443266
\(339\) −1.71021 −0.0928858
\(340\) 0 0
\(341\) −16.1482 −0.874473
\(342\) −0.121342 −0.00656143
\(343\) 21.8721 1.18098
\(344\) 0.962908 0.0519165
\(345\) 0 0
\(346\) 0.417669 0.0224540
\(347\) −25.4859 −1.36815 −0.684077 0.729410i \(-0.739795\pi\)
−0.684077 + 0.729410i \(0.739795\pi\)
\(348\) −15.8927 −0.851941
\(349\) −28.0435 −1.50113 −0.750566 0.660795i \(-0.770219\pi\)
−0.750566 + 0.660795i \(0.770219\pi\)
\(350\) 0 0
\(351\) 1.98166 0.105773
\(352\) −4.70727 −0.250899
\(353\) 14.6667 0.780630 0.390315 0.920681i \(-0.372366\pi\)
0.390315 + 0.920681i \(0.372366\pi\)
\(354\) 0.600798 0.0319320
\(355\) 0 0
\(356\) 19.4958 1.03327
\(357\) −4.34771 −0.230105
\(358\) 0.647004 0.0341952
\(359\) −14.6205 −0.771642 −0.385821 0.922574i \(-0.626082\pi\)
−0.385821 + 0.922574i \(0.626082\pi\)
\(360\) 0 0
\(361\) −17.1749 −0.903943
\(362\) 0.341694 0.0179591
\(363\) −8.28037 −0.434607
\(364\) 17.2131 0.902212
\(365\) 0 0
\(366\) −0.849306 −0.0443939
\(367\) −17.6940 −0.923617 −0.461808 0.886980i \(-0.652799\pi\)
−0.461808 + 0.886980i \(0.652799\pi\)
\(368\) 9.31212 0.485428
\(369\) −5.98248 −0.311435
\(370\) 0 0
\(371\) 48.1903 2.50192
\(372\) −7.32556 −0.379813
\(373\) 12.2293 0.633210 0.316605 0.948557i \(-0.397457\pi\)
0.316605 + 0.948557i \(0.397457\pi\)
\(374\) −0.393217 −0.0203328
\(375\) 0 0
\(376\) 3.92231 0.202278
\(377\) 15.8107 0.814295
\(378\) 0.391676 0.0201456
\(379\) 28.0951 1.44315 0.721574 0.692338i \(-0.243419\pi\)
0.721574 + 0.692338i \(0.243419\pi\)
\(380\) 0 0
\(381\) 12.5570 0.643312
\(382\) −1.93598 −0.0990533
\(383\) 34.4334 1.75947 0.879733 0.475468i \(-0.157721\pi\)
0.879733 + 0.475468i \(0.157721\pi\)
\(384\) −2.84531 −0.145199
\(385\) 0 0
\(386\) −0.282957 −0.0144021
\(387\) 2.68554 0.136514
\(388\) −0.0671442 −0.00340873
\(389\) 13.5225 0.685619 0.342809 0.939405i \(-0.388622\pi\)
0.342809 + 0.939405i \(0.388622\pi\)
\(390\) 0 0
\(391\) 2.34949 0.118819
\(392\) 4.30827 0.217601
\(393\) −16.4718 −0.830892
\(394\) −2.34283 −0.118030
\(395\) 0 0
\(396\) −8.74646 −0.439526
\(397\) −35.9744 −1.80550 −0.902751 0.430164i \(-0.858456\pi\)
−0.902751 + 0.430164i \(0.858456\pi\)
\(398\) 2.17958 0.109252
\(399\) −5.89112 −0.294925
\(400\) 0 0
\(401\) 4.35977 0.217717 0.108858 0.994057i \(-0.465281\pi\)
0.108858 + 0.994057i \(0.465281\pi\)
\(402\) 1.16063 0.0578871
\(403\) 7.28776 0.363029
\(404\) −6.36203 −0.316523
\(405\) 0 0
\(406\) 3.12501 0.155092
\(407\) −6.31003 −0.312777
\(408\) −0.357485 −0.0176982
\(409\) −18.1138 −0.895668 −0.447834 0.894117i \(-0.647804\pi\)
−0.447834 + 0.894117i \(0.647804\pi\)
\(410\) 0 0
\(411\) 9.66732 0.476854
\(412\) −17.0378 −0.839394
\(413\) 29.1685 1.43529
\(414\) −0.211660 −0.0104025
\(415\) 0 0
\(416\) 2.12442 0.104158
\(417\) −13.5327 −0.662698
\(418\) −0.532806 −0.0260604
\(419\) 0.527867 0.0257880 0.0128940 0.999917i \(-0.495896\pi\)
0.0128940 + 0.999917i \(0.495896\pi\)
\(420\) 0 0
\(421\) 18.6586 0.909365 0.454682 0.890654i \(-0.349753\pi\)
0.454682 + 0.890654i \(0.349753\pi\)
\(422\) −1.47109 −0.0716117
\(423\) 10.9393 0.531886
\(424\) 3.96239 0.192430
\(425\) 0 0
\(426\) 0.657709 0.0318661
\(427\) −41.2335 −1.99543
\(428\) −4.42480 −0.213881
\(429\) 8.70133 0.420104
\(430\) 0 0
\(431\) 20.9913 1.01112 0.505559 0.862792i \(-0.331286\pi\)
0.505559 + 0.862792i \(0.331286\pi\)
\(432\) −3.95166 −0.190124
\(433\) −13.6639 −0.656647 −0.328324 0.944565i \(-0.606484\pi\)
−0.328324 + 0.944565i \(0.606484\pi\)
\(434\) 1.44043 0.0691430
\(435\) 0 0
\(436\) −21.9159 −1.04958
\(437\) 3.18354 0.152289
\(438\) −0.0381557 −0.00182315
\(439\) −4.33339 −0.206821 −0.103411 0.994639i \(-0.532976\pi\)
−0.103411 + 0.994639i \(0.532976\pi\)
\(440\) 0 0
\(441\) 12.0157 0.572177
\(442\) 0.177461 0.00844096
\(443\) −1.60742 −0.0763707 −0.0381854 0.999271i \(-0.512158\pi\)
−0.0381854 + 0.999271i \(0.512158\pi\)
\(444\) −2.86252 −0.135849
\(445\) 0 0
\(446\) 0.510439 0.0241700
\(447\) −13.6843 −0.647244
\(448\) −34.0441 −1.60843
\(449\) −13.8291 −0.652634 −0.326317 0.945260i \(-0.605808\pi\)
−0.326317 + 0.945260i \(0.605808\pi\)
\(450\) 0 0
\(451\) −26.2687 −1.23694
\(452\) −3.40662 −0.160234
\(453\) 11.3204 0.531876
\(454\) −0.327279 −0.0153600
\(455\) 0 0
\(456\) −0.484389 −0.0226836
\(457\) 20.1345 0.941850 0.470925 0.882173i \(-0.343920\pi\)
0.470925 + 0.882173i \(0.343920\pi\)
\(458\) −0.149213 −0.00697225
\(459\) −0.997022 −0.0465370
\(460\) 0 0
\(461\) 31.5264 1.46833 0.734166 0.678970i \(-0.237573\pi\)
0.734166 + 0.678970i \(0.237573\pi\)
\(462\) 1.71983 0.0800135
\(463\) −0.0451697 −0.00209921 −0.00104961 0.999999i \(-0.500334\pi\)
−0.00104961 + 0.999999i \(0.500334\pi\)
\(464\) −31.5285 −1.46368
\(465\) 0 0
\(466\) −0.705890 −0.0326997
\(467\) −32.8349 −1.51942 −0.759708 0.650265i \(-0.774658\pi\)
−0.759708 + 0.650265i \(0.774658\pi\)
\(468\) 3.94732 0.182465
\(469\) 56.3484 2.60193
\(470\) 0 0
\(471\) −8.56070 −0.394456
\(472\) 2.39834 0.110393
\(473\) 11.7920 0.542198
\(474\) −0.570830 −0.0262191
\(475\) 0 0
\(476\) −8.66035 −0.396947
\(477\) 11.0510 0.505992
\(478\) 0.0509546 0.00233061
\(479\) 7.48576 0.342033 0.171017 0.985268i \(-0.445295\pi\)
0.171017 + 0.985268i \(0.445295\pi\)
\(480\) 0 0
\(481\) 2.84775 0.129846
\(482\) 1.70730 0.0777654
\(483\) −10.2760 −0.467576
\(484\) −16.4939 −0.749724
\(485\) 0 0
\(486\) 0.0898194 0.00407429
\(487\) −25.4295 −1.15232 −0.576160 0.817337i \(-0.695449\pi\)
−0.576160 + 0.817337i \(0.695449\pi\)
\(488\) −3.39037 −0.153475
\(489\) −4.58509 −0.207345
\(490\) 0 0
\(491\) −11.0668 −0.499438 −0.249719 0.968318i \(-0.580338\pi\)
−0.249719 + 0.968318i \(0.580338\pi\)
\(492\) −11.9167 −0.537246
\(493\) −7.95479 −0.358266
\(494\) 0.240458 0.0108187
\(495\) 0 0
\(496\) −14.5327 −0.652537
\(497\) 31.9315 1.43233
\(498\) 0.0661979 0.00296640
\(499\) 4.68157 0.209576 0.104788 0.994495i \(-0.466584\pi\)
0.104788 + 0.994495i \(0.466584\pi\)
\(500\) 0 0
\(501\) −7.21792 −0.322473
\(502\) −0.271734 −0.0121281
\(503\) 10.8285 0.482818 0.241409 0.970423i \(-0.422391\pi\)
0.241409 + 0.970423i \(0.422391\pi\)
\(504\) 1.56354 0.0696458
\(505\) 0 0
\(506\) −0.929388 −0.0413163
\(507\) 9.07304 0.402948
\(508\) 25.0126 1.10975
\(509\) 11.5007 0.509757 0.254879 0.966973i \(-0.417964\pi\)
0.254879 + 0.966973i \(0.417964\pi\)
\(510\) 0 0
\(511\) −1.85244 −0.0819473
\(512\) −7.07011 −0.312457
\(513\) −1.35096 −0.0596462
\(514\) 1.78393 0.0786856
\(515\) 0 0
\(516\) 5.34941 0.235495
\(517\) 48.0338 2.11252
\(518\) 0.562860 0.0247307
\(519\) 4.65009 0.204116
\(520\) 0 0
\(521\) 0.797807 0.0349525 0.0174763 0.999847i \(-0.494437\pi\)
0.0174763 + 0.999847i \(0.494437\pi\)
\(522\) 0.716629 0.0313660
\(523\) 40.1429 1.75533 0.877663 0.479279i \(-0.159102\pi\)
0.877663 + 0.479279i \(0.159102\pi\)
\(524\) −32.8107 −1.43334
\(525\) 0 0
\(526\) −2.05057 −0.0894090
\(527\) −3.66666 −0.159722
\(528\) −17.3515 −0.755127
\(529\) −17.4469 −0.758559
\(530\) 0 0
\(531\) 6.68895 0.290276
\(532\) −11.7347 −0.508764
\(533\) 11.8552 0.513506
\(534\) −0.879095 −0.0380422
\(535\) 0 0
\(536\) 4.63317 0.200123
\(537\) 7.20338 0.310849
\(538\) −1.33224 −0.0574369
\(539\) 52.7603 2.27255
\(540\) 0 0
\(541\) 1.41016 0.0606277 0.0303138 0.999540i \(-0.490349\pi\)
0.0303138 + 0.999540i \(0.490349\pi\)
\(542\) 0.578185 0.0248352
\(543\) 3.80424 0.163255
\(544\) −1.06885 −0.0458265
\(545\) 0 0
\(546\) −0.776167 −0.0332169
\(547\) 15.4621 0.661110 0.330555 0.943787i \(-0.392764\pi\)
0.330555 + 0.943787i \(0.392764\pi\)
\(548\) 19.2566 0.822603
\(549\) −9.45570 −0.403560
\(550\) 0 0
\(551\) −10.7787 −0.459187
\(552\) −0.844934 −0.0359628
\(553\) −27.7136 −1.17850
\(554\) −0.602667 −0.0256049
\(555\) 0 0
\(556\) −26.9562 −1.14320
\(557\) −18.0445 −0.764568 −0.382284 0.924045i \(-0.624862\pi\)
−0.382284 + 0.924045i \(0.624862\pi\)
\(558\) 0.330321 0.0139836
\(559\) −5.32181 −0.225089
\(560\) 0 0
\(561\) −4.37786 −0.184834
\(562\) 1.84011 0.0776202
\(563\) −28.5327 −1.20251 −0.601254 0.799058i \(-0.705332\pi\)
−0.601254 + 0.799058i \(0.705332\pi\)
\(564\) 21.7903 0.917538
\(565\) 0 0
\(566\) 1.02554 0.0431065
\(567\) 4.36070 0.183132
\(568\) 2.62553 0.110165
\(569\) 16.4072 0.687826 0.343913 0.939001i \(-0.388247\pi\)
0.343913 + 0.939001i \(0.388247\pi\)
\(570\) 0 0
\(571\) −8.34705 −0.349313 −0.174657 0.984629i \(-0.555882\pi\)
−0.174657 + 0.984629i \(0.555882\pi\)
\(572\) 17.3325 0.724707
\(573\) −21.5541 −0.900437
\(574\) 2.34319 0.0978029
\(575\) 0 0
\(576\) −7.80703 −0.325293
\(577\) 8.43531 0.351167 0.175583 0.984465i \(-0.443819\pi\)
0.175583 + 0.984465i \(0.443819\pi\)
\(578\) 1.43765 0.0597982
\(579\) −3.15029 −0.130921
\(580\) 0 0
\(581\) 3.21389 0.133334
\(582\) 0.00302764 0.000125500 0
\(583\) 48.5245 2.00968
\(584\) −0.152315 −0.00630283
\(585\) 0 0
\(586\) −2.56439 −0.105934
\(587\) −24.2852 −1.00236 −0.501179 0.865344i \(-0.667100\pi\)
−0.501179 + 0.865344i \(0.667100\pi\)
\(588\) 23.9345 0.987042
\(589\) −4.96829 −0.204715
\(590\) 0 0
\(591\) −26.0837 −1.07294
\(592\) −5.67876 −0.233396
\(593\) −28.4653 −1.16893 −0.584466 0.811418i \(-0.698696\pi\)
−0.584466 + 0.811418i \(0.698696\pi\)
\(594\) 0.394392 0.0161821
\(595\) 0 0
\(596\) −27.2581 −1.11654
\(597\) 24.2662 0.993149
\(598\) 0.419438 0.0171521
\(599\) 16.0387 0.655323 0.327662 0.944795i \(-0.393739\pi\)
0.327662 + 0.944795i \(0.393739\pi\)
\(600\) 0 0
\(601\) −8.09005 −0.330000 −0.165000 0.986294i \(-0.552762\pi\)
−0.165000 + 0.986294i \(0.552762\pi\)
\(602\) −1.05186 −0.0428706
\(603\) 12.9219 0.526219
\(604\) 22.5494 0.917521
\(605\) 0 0
\(606\) 0.286874 0.0116535
\(607\) −0.434608 −0.0176402 −0.00882010 0.999961i \(-0.502808\pi\)
−0.00882010 + 0.999961i \(0.502808\pi\)
\(608\) −1.44828 −0.0587356
\(609\) 34.7921 1.40985
\(610\) 0 0
\(611\) −21.6779 −0.876994
\(612\) −1.98600 −0.0802793
\(613\) −39.3962 −1.59120 −0.795599 0.605824i \(-0.792844\pi\)
−0.795599 + 0.605824i \(0.792844\pi\)
\(614\) 1.84504 0.0744599
\(615\) 0 0
\(616\) 6.86543 0.276616
\(617\) 15.4146 0.620568 0.310284 0.950644i \(-0.399576\pi\)
0.310284 + 0.950644i \(0.399576\pi\)
\(618\) 0.768264 0.0309041
\(619\) −10.6718 −0.428935 −0.214468 0.976731i \(-0.568802\pi\)
−0.214468 + 0.976731i \(0.568802\pi\)
\(620\) 0 0
\(621\) −2.35651 −0.0945635
\(622\) −1.57629 −0.0632036
\(623\) −42.6797 −1.70993
\(624\) 7.83083 0.313484
\(625\) 0 0
\(626\) 0.268168 0.0107182
\(627\) −5.93197 −0.236900
\(628\) −17.0523 −0.680463
\(629\) −1.43278 −0.0571285
\(630\) 0 0
\(631\) −18.4347 −0.733874 −0.366937 0.930246i \(-0.619594\pi\)
−0.366937 + 0.930246i \(0.619594\pi\)
\(632\) −2.27871 −0.0906424
\(633\) −16.3783 −0.650980
\(634\) 1.45245 0.0576843
\(635\) 0 0
\(636\) 22.0129 0.872869
\(637\) −23.8110 −0.943427
\(638\) 3.14668 0.124578
\(639\) 7.32257 0.289676
\(640\) 0 0
\(641\) 12.4281 0.490882 0.245441 0.969412i \(-0.421067\pi\)
0.245441 + 0.969412i \(0.421067\pi\)
\(642\) 0.199521 0.00787447
\(643\) −1.84657 −0.0728218 −0.0364109 0.999337i \(-0.511593\pi\)
−0.0364109 + 0.999337i \(0.511593\pi\)
\(644\) −20.4692 −0.806598
\(645\) 0 0
\(646\) −0.120981 −0.00475992
\(647\) −38.9760 −1.53230 −0.766152 0.642660i \(-0.777831\pi\)
−0.766152 + 0.642660i \(0.777831\pi\)
\(648\) 0.358553 0.0140853
\(649\) 29.3708 1.15290
\(650\) 0 0
\(651\) 16.0370 0.628539
\(652\) −9.13319 −0.357683
\(653\) 28.3389 1.10899 0.554493 0.832189i \(-0.312912\pi\)
0.554493 + 0.832189i \(0.312912\pi\)
\(654\) 0.988224 0.0386426
\(655\) 0 0
\(656\) −23.6407 −0.923015
\(657\) −0.424804 −0.0165732
\(658\) −4.28465 −0.167033
\(659\) −22.3561 −0.870868 −0.435434 0.900221i \(-0.643405\pi\)
−0.435434 + 0.900221i \(0.643405\pi\)
\(660\) 0 0
\(661\) 29.2143 1.13631 0.568153 0.822923i \(-0.307658\pi\)
0.568153 + 0.822923i \(0.307658\pi\)
\(662\) 1.17398 0.0456281
\(663\) 1.97575 0.0767319
\(664\) 0.264258 0.0102552
\(665\) 0 0
\(666\) 0.129076 0.00500158
\(667\) −18.8015 −0.727998
\(668\) −14.3776 −0.556287
\(669\) 5.68295 0.219715
\(670\) 0 0
\(671\) −41.5194 −1.60284
\(672\) 4.67486 0.180336
\(673\) 3.21546 0.123947 0.0619735 0.998078i \(-0.480261\pi\)
0.0619735 + 0.998078i \(0.480261\pi\)
\(674\) −2.40760 −0.0927372
\(675\) 0 0
\(676\) 18.0729 0.695111
\(677\) −47.2470 −1.81585 −0.907925 0.419133i \(-0.862334\pi\)
−0.907925 + 0.419133i \(0.862334\pi\)
\(678\) 0.153610 0.00589936
\(679\) 0.146991 0.00564099
\(680\) 0 0
\(681\) −3.64374 −0.139629
\(682\) 1.45042 0.0555395
\(683\) −37.2527 −1.42543 −0.712717 0.701452i \(-0.752536\pi\)
−0.712717 + 0.701452i \(0.752536\pi\)
\(684\) −2.69101 −0.102893
\(685\) 0 0
\(686\) −1.96454 −0.0750064
\(687\) −1.66125 −0.0633807
\(688\) 10.6123 0.404591
\(689\) −21.8994 −0.834299
\(690\) 0 0
\(691\) −10.9827 −0.417803 −0.208901 0.977937i \(-0.566989\pi\)
−0.208901 + 0.977937i \(0.566989\pi\)
\(692\) 9.26267 0.352114
\(693\) 19.1476 0.727357
\(694\) 2.28913 0.0868941
\(695\) 0 0
\(696\) 2.86074 0.108436
\(697\) −5.96466 −0.225928
\(698\) 2.51885 0.0953399
\(699\) −7.85899 −0.297254
\(700\) 0 0
\(701\) 22.4086 0.846361 0.423180 0.906046i \(-0.360914\pi\)
0.423180 + 0.906046i \(0.360914\pi\)
\(702\) −0.177991 −0.00671784
\(703\) −1.94140 −0.0732213
\(704\) −34.2802 −1.29198
\(705\) 0 0
\(706\) −1.31736 −0.0495793
\(707\) 13.9276 0.523803
\(708\) 13.3239 0.500744
\(709\) −14.6297 −0.549431 −0.274716 0.961526i \(-0.588584\pi\)
−0.274716 + 0.961526i \(0.588584\pi\)
\(710\) 0 0
\(711\) −6.35531 −0.238343
\(712\) −3.50929 −0.131516
\(713\) −8.66633 −0.324557
\(714\) 0.390509 0.0146144
\(715\) 0 0
\(716\) 14.3486 0.536234
\(717\) 0.567301 0.0211862
\(718\) 1.31321 0.0490085
\(719\) −28.4977 −1.06278 −0.531392 0.847126i \(-0.678331\pi\)
−0.531392 + 0.847126i \(0.678331\pi\)
\(720\) 0 0
\(721\) 37.2989 1.38908
\(722\) 1.54264 0.0574112
\(723\) 19.0081 0.706920
\(724\) 7.57778 0.281626
\(725\) 0 0
\(726\) 0.743738 0.0276027
\(727\) 44.0064 1.63211 0.816054 0.577976i \(-0.196157\pi\)
0.816054 + 0.577976i \(0.196157\pi\)
\(728\) −3.09840 −0.114835
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 2.67754 0.0990324
\(732\) −18.8351 −0.696166
\(733\) −27.3227 −1.00919 −0.504593 0.863357i \(-0.668357\pi\)
−0.504593 + 0.863357i \(0.668357\pi\)
\(734\) 1.58926 0.0586607
\(735\) 0 0
\(736\) −2.52628 −0.0931198
\(737\) 56.7391 2.09001
\(738\) 0.537343 0.0197799
\(739\) 16.4678 0.605776 0.302888 0.953026i \(-0.402049\pi\)
0.302888 + 0.953026i \(0.402049\pi\)
\(740\) 0 0
\(741\) 2.67713 0.0983468
\(742\) −4.32843 −0.158901
\(743\) −35.6012 −1.30608 −0.653041 0.757322i \(-0.726507\pi\)
−0.653041 + 0.757322i \(0.726507\pi\)
\(744\) 1.31862 0.0483430
\(745\) 0 0
\(746\) −1.09843 −0.0402164
\(747\) 0.737011 0.0269658
\(748\) −8.72041 −0.318850
\(749\) 9.68668 0.353944
\(750\) 0 0
\(751\) 46.0748 1.68129 0.840647 0.541583i \(-0.182175\pi\)
0.840647 + 0.541583i \(0.182175\pi\)
\(752\) 43.2283 1.57638
\(753\) −3.02533 −0.110249
\(754\) −1.42011 −0.0517174
\(755\) 0 0
\(756\) 8.68622 0.315915
\(757\) 36.6482 1.33200 0.666000 0.745951i \(-0.268005\pi\)
0.666000 + 0.745951i \(0.268005\pi\)
\(758\) −2.52348 −0.0916571
\(759\) −10.3473 −0.375583
\(760\) 0 0
\(761\) −27.2078 −0.986282 −0.493141 0.869949i \(-0.664151\pi\)
−0.493141 + 0.869949i \(0.664151\pi\)
\(762\) −1.12786 −0.0408580
\(763\) 47.9779 1.73692
\(764\) −42.9344 −1.55331
\(765\) 0 0
\(766\) −3.09279 −0.111747
\(767\) −13.2552 −0.478617
\(768\) −15.3585 −0.554202
\(769\) 26.5327 0.956794 0.478397 0.878144i \(-0.341218\pi\)
0.478397 + 0.878144i \(0.341218\pi\)
\(770\) 0 0
\(771\) 19.8613 0.715286
\(772\) −6.27516 −0.225848
\(773\) −28.7677 −1.03470 −0.517351 0.855773i \(-0.673082\pi\)
−0.517351 + 0.855773i \(0.673082\pi\)
\(774\) −0.241213 −0.00867024
\(775\) 0 0
\(776\) 0.0120861 0.000433867 0
\(777\) 6.26658 0.224812
\(778\) −1.21458 −0.0435450
\(779\) −8.08206 −0.289570
\(780\) 0 0
\(781\) 32.1530 1.15052
\(782\) −0.211030 −0.00754641
\(783\) 7.97856 0.285130
\(784\) 47.4820 1.69579
\(785\) 0 0
\(786\) 1.47949 0.0527716
\(787\) 24.8663 0.886387 0.443193 0.896426i \(-0.353845\pi\)
0.443193 + 0.896426i \(0.353845\pi\)
\(788\) −51.9571 −1.85089
\(789\) −22.8299 −0.812766
\(790\) 0 0
\(791\) 7.45771 0.265166
\(792\) 1.57439 0.0559434
\(793\) 18.7379 0.665404
\(794\) 3.23120 0.114671
\(795\) 0 0
\(796\) 48.3366 1.71325
\(797\) −31.4206 −1.11298 −0.556488 0.830856i \(-0.687851\pi\)
−0.556488 + 0.830856i \(0.687851\pi\)
\(798\) 0.529137 0.0187312
\(799\) 10.9067 0.385851
\(800\) 0 0
\(801\) −9.78736 −0.345819
\(802\) −0.391592 −0.0138276
\(803\) −1.86529 −0.0658246
\(804\) 25.7395 0.907761
\(805\) 0 0
\(806\) −0.654583 −0.0230567
\(807\) −14.8324 −0.522125
\(808\) 1.14518 0.0402874
\(809\) 50.5027 1.77558 0.887790 0.460249i \(-0.152240\pi\)
0.887790 + 0.460249i \(0.152240\pi\)
\(810\) 0 0
\(811\) 25.1612 0.883528 0.441764 0.897131i \(-0.354353\pi\)
0.441764 + 0.897131i \(0.354353\pi\)
\(812\) 69.3035 2.43208
\(813\) 6.43720 0.225762
\(814\) 0.566763 0.0198650
\(815\) 0 0
\(816\) −3.93989 −0.137924
\(817\) 3.62804 0.126929
\(818\) 1.62697 0.0568856
\(819\) −8.64141 −0.301955
\(820\) 0 0
\(821\) −20.6307 −0.720017 −0.360009 0.932949i \(-0.617226\pi\)
−0.360009 + 0.932949i \(0.617226\pi\)
\(822\) −0.868313 −0.0302859
\(823\) −35.8618 −1.25006 −0.625032 0.780599i \(-0.714914\pi\)
−0.625032 + 0.780599i \(0.714914\pi\)
\(824\) 3.06686 0.106839
\(825\) 0 0
\(826\) −2.61990 −0.0911580
\(827\) 4.73642 0.164701 0.0823507 0.996603i \(-0.473757\pi\)
0.0823507 + 0.996603i \(0.473757\pi\)
\(828\) −4.69401 −0.163128
\(829\) −28.4768 −0.989039 −0.494520 0.869166i \(-0.664656\pi\)
−0.494520 + 0.869166i \(0.664656\pi\)
\(830\) 0 0
\(831\) −6.70976 −0.232759
\(832\) 15.4708 0.536355
\(833\) 11.9799 0.415080
\(834\) 1.21550 0.0420892
\(835\) 0 0
\(836\) −11.8161 −0.408668
\(837\) 3.67761 0.127117
\(838\) −0.0474127 −0.00163784
\(839\) −0.765715 −0.0264354 −0.0132177 0.999913i \(-0.504207\pi\)
−0.0132177 + 0.999913i \(0.504207\pi\)
\(840\) 0 0
\(841\) 34.6574 1.19508
\(842\) −1.67591 −0.0577555
\(843\) 20.4867 0.705600
\(844\) −32.6245 −1.12298
\(845\) 0 0
\(846\) −0.982560 −0.0337811
\(847\) 36.1082 1.24069
\(848\) 43.6700 1.49963
\(849\) 11.4177 0.391856
\(850\) 0 0
\(851\) −3.38644 −0.116086
\(852\) 14.5861 0.499710
\(853\) 34.7289 1.18910 0.594548 0.804060i \(-0.297331\pi\)
0.594548 + 0.804060i \(0.297331\pi\)
\(854\) 3.70357 0.126734
\(855\) 0 0
\(856\) 0.796475 0.0272230
\(857\) 54.2561 1.85335 0.926676 0.375860i \(-0.122653\pi\)
0.926676 + 0.375860i \(0.122653\pi\)
\(858\) −0.781549 −0.0266816
\(859\) 15.9095 0.542824 0.271412 0.962463i \(-0.412509\pi\)
0.271412 + 0.962463i \(0.412509\pi\)
\(860\) 0 0
\(861\) 26.0878 0.889070
\(862\) −1.88543 −0.0642180
\(863\) 22.4714 0.764935 0.382467 0.923969i \(-0.375074\pi\)
0.382467 + 0.923969i \(0.375074\pi\)
\(864\) 1.07204 0.0364716
\(865\) 0 0
\(866\) 1.22729 0.0417049
\(867\) 16.0059 0.543590
\(868\) 31.9446 1.08427
\(869\) −27.9058 −0.946639
\(870\) 0 0
\(871\) −25.6067 −0.867649
\(872\) 3.94492 0.133592
\(873\) 0.0337081 0.00114085
\(874\) −0.285944 −0.00967219
\(875\) 0 0
\(876\) −0.846181 −0.0285898
\(877\) 27.0756 0.914278 0.457139 0.889395i \(-0.348874\pi\)
0.457139 + 0.889395i \(0.348874\pi\)
\(878\) 0.389222 0.0131356
\(879\) −28.5505 −0.962984
\(880\) 0 0
\(881\) −7.81317 −0.263232 −0.131616 0.991301i \(-0.542017\pi\)
−0.131616 + 0.991301i \(0.542017\pi\)
\(882\) −1.07925 −0.0363401
\(883\) −58.3010 −1.96199 −0.980993 0.194042i \(-0.937840\pi\)
−0.980993 + 0.194042i \(0.937840\pi\)
\(884\) 3.93557 0.132367
\(885\) 0 0
\(886\) 0.144377 0.00485045
\(887\) −8.64610 −0.290307 −0.145154 0.989409i \(-0.546368\pi\)
−0.145154 + 0.989409i \(0.546368\pi\)
\(888\) 0.515261 0.0172910
\(889\) −54.7571 −1.83650
\(890\) 0 0
\(891\) 4.39094 0.147102
\(892\) 11.3200 0.379023
\(893\) 14.7785 0.494543
\(894\) 1.22911 0.0411077
\(895\) 0 0
\(896\) 12.4075 0.414507
\(897\) 4.66979 0.155920
\(898\) 1.24212 0.0414500
\(899\) 29.3420 0.978612
\(900\) 0 0
\(901\) 11.0181 0.367067
\(902\) 2.35944 0.0785608
\(903\) −11.7108 −0.389712
\(904\) 0.613201 0.0203948
\(905\) 0 0
\(906\) −1.01679 −0.0337805
\(907\) −40.4367 −1.34268 −0.671339 0.741151i \(-0.734280\pi\)
−0.671339 + 0.741151i \(0.734280\pi\)
\(908\) −7.25809 −0.240868
\(909\) 3.19390 0.105935
\(910\) 0 0
\(911\) −50.5643 −1.67527 −0.837635 0.546230i \(-0.816063\pi\)
−0.837635 + 0.546230i \(0.816063\pi\)
\(912\) −5.33852 −0.176776
\(913\) 3.23617 0.107102
\(914\) −1.80847 −0.0598187
\(915\) 0 0
\(916\) −3.30910 −0.109336
\(917\) 71.8286 2.37199
\(918\) 0.0895519 0.00295565
\(919\) −43.3566 −1.43020 −0.715101 0.699021i \(-0.753619\pi\)
−0.715101 + 0.699021i \(0.753619\pi\)
\(920\) 0 0
\(921\) 20.5417 0.676871
\(922\) −2.83169 −0.0932566
\(923\) −14.5108 −0.477629
\(924\) 38.1407 1.25474
\(925\) 0 0
\(926\) 0.00405712 0.000133325 0
\(927\) 8.55342 0.280931
\(928\) 8.55335 0.280777
\(929\) −34.1746 −1.12123 −0.560617 0.828075i \(-0.689436\pi\)
−0.560617 + 0.828075i \(0.689436\pi\)
\(930\) 0 0
\(931\) 16.2327 0.532006
\(932\) −15.6546 −0.512783
\(933\) −17.5496 −0.574548
\(934\) 2.94921 0.0965010
\(935\) 0 0
\(936\) −0.710529 −0.0232244
\(937\) 27.8996 0.911441 0.455720 0.890123i \(-0.349382\pi\)
0.455720 + 0.890123i \(0.349382\pi\)
\(938\) −5.06118 −0.165253
\(939\) 2.98564 0.0974325
\(940\) 0 0
\(941\) −54.3261 −1.77098 −0.885490 0.464659i \(-0.846177\pi\)
−0.885490 + 0.464659i \(0.846177\pi\)
\(942\) 0.768918 0.0250527
\(943\) −14.0978 −0.459086
\(944\) 26.4325 0.860303
\(945\) 0 0
\(946\) −1.05915 −0.0344361
\(947\) 4.23171 0.137512 0.0687561 0.997633i \(-0.478097\pi\)
0.0687561 + 0.997633i \(0.478097\pi\)
\(948\) −12.6593 −0.411156
\(949\) 0.841815 0.0273265
\(950\) 0 0
\(951\) 16.1708 0.524375
\(952\) 1.55889 0.0505238
\(953\) −47.9307 −1.55263 −0.776314 0.630346i \(-0.782913\pi\)
−0.776314 + 0.630346i \(0.782913\pi\)
\(954\) −0.992598 −0.0321366
\(955\) 0 0
\(956\) 1.13002 0.0365476
\(957\) 35.0334 1.13247
\(958\) −0.672367 −0.0217232
\(959\) −42.1563 −1.36130
\(960\) 0 0
\(961\) −17.4752 −0.563715
\(962\) −0.255783 −0.00824679
\(963\) 2.22136 0.0715823
\(964\) 37.8629 1.21948
\(965\) 0 0
\(966\) 0.922988 0.0296966
\(967\) −20.8827 −0.671544 −0.335772 0.941943i \(-0.608997\pi\)
−0.335772 + 0.941943i \(0.608997\pi\)
\(968\) 2.96895 0.0954257
\(969\) −1.34693 −0.0432697
\(970\) 0 0
\(971\) 16.0740 0.515838 0.257919 0.966167i \(-0.416963\pi\)
0.257919 + 0.966167i \(0.416963\pi\)
\(972\) 1.99193 0.0638913
\(973\) 59.0119 1.89184
\(974\) 2.28406 0.0731860
\(975\) 0 0
\(976\) −37.3657 −1.19605
\(977\) −35.5665 −1.13787 −0.568937 0.822381i \(-0.692645\pi\)
−0.568937 + 0.822381i \(0.692645\pi\)
\(978\) 0.411830 0.0131689
\(979\) −42.9757 −1.37351
\(980\) 0 0
\(981\) 11.0023 0.351278
\(982\) 0.994014 0.0317202
\(983\) 18.4711 0.589138 0.294569 0.955630i \(-0.404824\pi\)
0.294569 + 0.955630i \(0.404824\pi\)
\(984\) 2.14504 0.0683812
\(985\) 0 0
\(986\) 0.714495 0.0227542
\(987\) −47.7030 −1.51840
\(988\) 5.33266 0.169655
\(989\) 6.32849 0.201234
\(990\) 0 0
\(991\) −41.5907 −1.32117 −0.660586 0.750750i \(-0.729692\pi\)
−0.660586 + 0.750750i \(0.729692\pi\)
\(992\) 3.94256 0.125176
\(993\) 13.0705 0.414779
\(994\) −2.86807 −0.0909698
\(995\) 0 0
\(996\) 1.46808 0.0465178
\(997\) 31.8374 1.00830 0.504151 0.863616i \(-0.331806\pi\)
0.504151 + 0.863616i \(0.331806\pi\)
\(998\) −0.420496 −0.0133106
\(999\) 1.43706 0.0454665
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.p.1.4 8
3.2 odd 2 5625.2.a.t.1.5 8
5.2 odd 4 1875.2.b.h.1249.8 16
5.3 odd 4 1875.2.b.h.1249.9 16
5.4 even 2 1875.2.a.m.1.5 8
15.14 odd 2 5625.2.a.bd.1.4 8
25.3 odd 20 75.2.i.a.34.3 16
25.4 even 10 375.2.g.e.76.2 16
25.6 even 5 375.2.g.d.301.3 16
25.8 odd 20 375.2.i.c.199.2 16
25.17 odd 20 75.2.i.a.64.3 yes 16
25.19 even 10 375.2.g.e.301.2 16
25.21 even 5 375.2.g.d.76.3 16
25.22 odd 20 375.2.i.c.49.2 16
75.17 even 20 225.2.m.b.64.2 16
75.53 even 20 225.2.m.b.109.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.i.a.34.3 16 25.3 odd 20
75.2.i.a.64.3 yes 16 25.17 odd 20
225.2.m.b.64.2 16 75.17 even 20
225.2.m.b.109.2 16 75.53 even 20
375.2.g.d.76.3 16 25.21 even 5
375.2.g.d.301.3 16 25.6 even 5
375.2.g.e.76.2 16 25.4 even 10
375.2.g.e.301.2 16 25.19 even 10
375.2.i.c.49.2 16 25.22 odd 20
375.2.i.c.199.2 16 25.8 odd 20
1875.2.a.m.1.5 8 5.4 even 2
1875.2.a.p.1.4 8 1.1 even 1 trivial
1875.2.b.h.1249.8 16 5.2 odd 4
1875.2.b.h.1249.9 16 5.3 odd 4
5625.2.a.t.1.5 8 3.2 odd 2
5625.2.a.bd.1.4 8 15.14 odd 2