Properties

Label 1875.2.a.m.1.5
Level $1875$
Weight $2$
Character 1875.1
Self dual yes
Analytic conductor $14.972$
Analytic rank $1$
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: \(1\)
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 1.5
Root \(1.08982\) 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 \(-0.489890\pi\)
0.0317560 + 0.999496i \(0.489890\pi\)
\(3\) 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 \(-0.808316\pi\)
−0.824095 + 0.566451i \(0.808316\pi\)
\(8\) −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 \(-0.411386\pi\)
0.274806 + 0.961500i \(0.411386\pi\)
\(14\) −0.391676 −0.104680
\(15\) 0 0
\(16\) 3.95166 0.987915
\(17\) −0.997022 −0.241813 −0.120907 0.992664i \(-0.538580\pi\)
−0.120907 + 0.992664i \(0.538580\pi\)
\(18\) 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 \(-0.579012\pi\)
−0.245683 + 0.969350i \(0.579012\pi\)
\(24\) −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 \(-0.462312\pi\)
0.118125 + 0.992999i \(0.462312\pi\)
\(38\) 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 \(-0.565645\pi\)
−0.204770 + 0.978810i \(0.565645\pi\)
\(44\) −8.74646 −1.31858
\(45\) 0 0
\(46\) −0.211660 −0.0312076
\(47\) −10.9393 −1.59566 −0.797829 0.602883i \(-0.794018\pi\)
−0.797829 + 0.602883i \(0.794018\pi\)
\(48\) 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 \(-0.774306\pi\)
−0.758989 + 0.651104i \(0.774306\pi\)
\(54\) 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 \(-0.789571\pi\)
−0.789328 + 0.613972i \(0.789571\pi\)
\(68\) 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 \(-0.492086\pi\)
0.0248598 + 0.999691i \(0.492086\pi\)
\(74\) 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 \(-0.512879\pi\)
−0.0404487 + 0.999182i \(0.512879\pi\)
\(84\) 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 \(-0.500545\pi\)
−0.00171127 + 0.999999i \(0.500545\pi\)
\(98\) 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 \(-0.638460\pi\)
−0.421397 + 0.906876i \(0.638460\pi\)
\(104\) −0.710529 −0.0696731
\(105\) 0 0
\(106\) −0.992598 −0.0964097
\(107\) −2.22136 −0.214747 −0.107373 0.994219i \(-0.534244\pi\)
−0.107373 + 0.994219i \(0.534244\pi\)
\(108\) −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 \(-0.525633\pi\)
−0.0804415 + 0.996759i \(0.525633\pi\)
\(114\) 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 \(-0.311905\pi\)
0.557125 + 0.830429i \(0.311905\pi\)
\(128\) −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 \(-0.364492\pi\)
0.412967 + 0.910746i \(0.364492\pi\)
\(138\) −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 \(-0.610972\pi\)
−0.341609 + 0.939842i \(0.610972\pi\)
\(158\) −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 \(-0.557469\pi\)
−0.179566 + 0.983746i \(0.557469\pi\)
\(164\) 11.9167 0.930537
\(165\) 0 0
\(166\) −0.0661979 −0.00513795
\(167\) −7.21792 −0.558540 −0.279270 0.960213i \(-0.590092\pi\)
−0.279270 + 0.960213i \(0.590092\pi\)
\(168\) 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 \(-0.443435\pi\)
0.176770 + 0.984252i \(0.443435\pi\)
\(174\) −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 \(-0.536168\pi\)
−0.113381 + 0.993552i \(0.536168\pi\)
\(194\) −0.00302764 −0.000217372 0
\(195\) 0 0
\(196\) −23.9345 −1.70961
\(197\) −26.0837 −1.85839 −0.929195 0.369590i \(-0.879498\pi\)
−0.929195 + 0.369590i \(0.879498\pi\)
\(198\) 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 \(-0.439061\pi\)
0.190279 + 0.981730i \(0.439061\pi\)
\(224\) −4.67486 −0.312352
\(225\) 0 0
\(226\) −0.153610 −0.0102180
\(227\) −3.64374 −0.241844 −0.120922 0.992662i \(-0.538585\pi\)
−0.120922 + 0.992662i \(0.538585\pi\)
\(228\) −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 \(-0.582876\pi\)
−0.257430 + 0.966297i \(0.582876\pi\)
\(234\) 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 \(-0.287353\pi\)
0.619456 + 0.785032i \(0.287353\pi\)
\(258\) −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 \(-0.748549\pi\)
−0.703876 + 0.710323i \(0.748549\pi\)
\(264\) −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 \(-0.564606\pi\)
−0.201575 + 0.979473i \(0.564606\pi\)
\(278\) 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 \(-0.389790\pi\)
0.339357 + 0.940658i \(0.389790\pi\)
\(284\) −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 \(-0.813937\pi\)
−0.833968 + 0.551812i \(0.813937\pi\)
\(294\) 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 \(-0.300628\pi\)
0.586188 + 0.810175i \(0.300628\pi\)
\(308\) 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 \(-0.473109\pi\)
0.0843790 + 0.996434i \(0.473109\pi\)
\(314\) −0.768918 −0.0433925
\(315\) 0 0
\(316\) 12.6593 0.712143
\(317\) 16.1708 0.908244 0.454122 0.890940i \(-0.349953\pi\)
0.454122 + 0.890940i \(0.349953\pi\)
\(318\) −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 \(-0.760516\pi\)
−0.730077 + 0.683365i \(0.760516\pi\)
\(338\) −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 \(-0.260205\pi\)
0.684077 + 0.729410i \(0.260205\pi\)
\(348\) 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 \(-0.627634\pi\)
−0.390315 + 0.920681i \(0.627634\pi\)
\(354\) 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 \(-0.347201\pi\)
0.461808 + 0.886980i \(0.347201\pi\)
\(368\) −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 \(-0.602543\pi\)
−0.316605 + 0.948557i \(0.602543\pi\)
\(374\) −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 \(-0.842279\pi\)
−0.879733 + 0.475468i \(0.842279\pi\)
\(384\) −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 \(-0.141544\pi\)
0.902751 + 0.430164i \(0.141544\pi\)
\(398\) −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 \(-0.393516\pi\)
0.328324 + 0.944565i \(0.393516\pi\)
\(434\) 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 \(-0.487842\pi\)
0.0381854 + 0.999271i \(0.487842\pi\)
\(444\) −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 \(-0.656080\pi\)
−0.470925 + 0.882173i \(0.656080\pi\)
\(458\) 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 \(-0.499666\pi\)
0.00104961 + 0.999999i \(0.499666\pi\)
\(464\) −31.5285 −1.46368
\(465\) 0 0
\(466\) −0.705890 −0.0326997
\(467\) 32.8349 1.51942 0.759708 0.650265i \(-0.225342\pi\)
0.759708 + 0.650265i \(0.225342\pi\)
\(468\) −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 \(-0.304551\pi\)
0.576160 + 0.817337i \(0.304551\pi\)
\(488\) 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 \(-0.577609\pi\)
−0.241409 + 0.970423i \(0.577609\pi\)
\(504\) 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 \(-0.840898\pi\)
−0.877663 + 0.479279i \(0.840898\pi\)
\(524\) −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 \(-0.607236\pi\)
−0.330555 + 0.943787i \(0.607236\pi\)
\(548\) −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 \(-0.375138\pi\)
0.382284 + 0.924045i \(0.375138\pi\)
\(558\) −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 \(-0.294668\pi\)
0.601254 + 0.799058i \(0.294668\pi\)
\(564\) 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 \(-0.556181\pi\)
−0.175583 + 0.984465i \(0.556181\pi\)
\(578\) −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 \(-0.332900\pi\)
0.501179 + 0.865344i \(0.332900\pi\)
\(588\) −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 \(-0.301304\pi\)
0.584466 + 0.811418i \(0.301304\pi\)
\(594\) 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 \(-0.497192\pi\)
0.00882010 + 0.999961i \(0.497192\pi\)
\(608\) 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 \(-0.207156\pi\)
0.795599 + 0.605824i \(0.207156\pi\)
\(614\) 1.84504 0.0744599
\(615\) 0 0
\(616\) 6.86543 0.276616
\(617\) −15.4146 −0.620568 −0.310284 0.950644i \(-0.600424\pi\)
−0.310284 + 0.950644i \(0.600424\pi\)
\(618\) −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 \(-0.488407\pi\)
0.0364109 + 0.999337i \(0.488407\pi\)
\(644\) −20.4692 −0.806598
\(645\) 0 0
\(646\) −0.120981 −0.00475992
\(647\) 38.9760 1.53230 0.766152 0.642660i \(-0.222169\pi\)
0.766152 + 0.642660i \(0.222169\pi\)
\(648\) −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 \(-0.687088\pi\)
−0.554493 + 0.832189i \(0.687088\pi\)
\(654\) 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 \(-0.519739\pi\)
−0.0619735 + 0.998078i \(0.519739\pi\)
\(674\) −2.40760 −0.0927372
\(675\) 0 0
\(676\) 18.0729 0.695111
\(677\) 47.2470 1.81585 0.907925 0.419133i \(-0.137666\pi\)
0.907925 + 0.419133i \(0.137666\pi\)
\(678\) −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 \(-0.247464\pi\)
0.712717 + 0.701452i \(0.247464\pi\)
\(684\) −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 \(-0.803843\pi\)
−0.816054 + 0.577976i \(0.803843\pi\)
\(728\) 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 \(-0.331643\pi\)
0.504593 + 0.863357i \(0.331643\pi\)
\(734\) 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 \(-0.273493\pi\)
0.653041 + 0.757322i \(0.273493\pi\)
\(744\) 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 \(-0.731995\pi\)
−0.666000 + 0.745951i \(0.731995\pi\)
\(758\) 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 \(-0.326918\pi\)
0.517351 + 0.855773i \(0.326918\pi\)
\(774\) −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 \(-0.646155\pi\)
−0.443193 + 0.896426i \(0.646155\pi\)
\(788\) 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 \(-0.312149\pi\)
0.556488 + 0.830856i \(0.312149\pi\)
\(798\) −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 \(-0.285086\pi\)
0.625032 + 0.780599i \(0.285086\pi\)
\(824\) 3.06686 0.106839
\(825\) 0 0
\(826\) −2.61990 −0.0911580
\(827\) −4.73642 −0.164701 −0.0823507 0.996603i \(-0.526243\pi\)
−0.0823507 + 0.996603i \(0.526243\pi\)
\(828\) 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 \(-0.702669\pi\)
−0.594548 + 0.804060i \(0.702669\pi\)
\(854\) 3.70357 0.126734
\(855\) 0 0
\(856\) 0.796475 0.0272230
\(857\) −54.2561 −1.85335 −0.926676 0.375860i \(-0.877347\pi\)
−0.926676 + 0.375860i \(0.877347\pi\)
\(858\) 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 \(-0.624926\pi\)
−0.382467 + 0.923969i \(0.624926\pi\)
\(864\) 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 \(-0.651126\pi\)
−0.457139 + 0.889395i \(0.651126\pi\)
\(878\) −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 \(-0.0621598\pi\)
0.980993 + 0.194042i \(0.0621598\pi\)
\(884\) 3.93557 0.132367
\(885\) 0 0
\(886\) 0.144377 0.00485045
\(887\) 8.64610 0.290307 0.145154 0.989409i \(-0.453632\pi\)
0.145154 + 0.989409i \(0.453632\pi\)
\(888\) −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 \(-0.265720\pi\)
0.671339 + 0.741151i \(0.265720\pi\)
\(908\) 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 \(-0.650618\pi\)
−0.455720 + 0.890123i \(0.650618\pi\)
\(938\) 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 \(-0.521903\pi\)
−0.0687561 + 0.997633i \(0.521903\pi\)
\(948\) 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 \(-0.217087\pi\)
0.776314 + 0.630346i \(0.217087\pi\)
\(954\) −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 \(-0.391003\pi\)
0.335772 + 0.941943i \(0.391003\pi\)
\(968\) −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 \(-0.307355\pi\)
0.568937 + 0.822381i \(0.307355\pi\)
\(978\) −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 \(-0.595176\pi\)
−0.294569 + 0.955630i \(0.595176\pi\)
\(984\) 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 \(-0.668194\pi\)
−0.504151 + 0.863616i \(0.668194\pi\)
\(998\) 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.m.1.5 8
3.2 odd 2 5625.2.a.bd.1.4 8
5.2 odd 4 1875.2.b.h.1249.9 16
5.3 odd 4 1875.2.b.h.1249.8 16
5.4 even 2 1875.2.a.p.1.4 8
15.14 odd 2 5625.2.a.t.1.5 8
25.3 odd 20 375.2.i.c.49.2 16
25.4 even 10 375.2.g.d.76.3 16
25.6 even 5 375.2.g.e.301.2 16
25.8 odd 20 75.2.i.a.64.3 yes 16
25.17 odd 20 375.2.i.c.199.2 16
25.19 even 10 375.2.g.d.301.3 16
25.21 even 5 375.2.g.e.76.2 16
25.22 odd 20 75.2.i.a.34.3 16
75.8 even 20 225.2.m.b.64.2 16
75.47 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 25.22 odd 20
75.2.i.a.64.3 yes 16 25.8 odd 20
225.2.m.b.64.2 16 75.8 even 20
225.2.m.b.109.2 16 75.47 even 20
375.2.g.d.76.3 16 25.4 even 10
375.2.g.d.301.3 16 25.19 even 10
375.2.g.e.76.2 16 25.21 even 5
375.2.g.e.301.2 16 25.6 even 5
375.2.i.c.49.2 16 25.3 odd 20
375.2.i.c.199.2 16 25.17 odd 20
1875.2.a.m.1.5 8 1.1 even 1 trivial
1875.2.a.p.1.4 8 5.4 even 2
1875.2.b.h.1249.8 16 5.3 odd 4
1875.2.b.h.1249.9 16 5.2 odd 4
5625.2.a.t.1.5 8 15.14 odd 2
5625.2.a.bd.1.4 8 3.2 odd 2