Properties

Label 625.2.a.f.1.5
Level $625$
Weight $2$
Character 625.1
Self dual yes
Analytic conductor $4.991$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [625,2,Mod(1,625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(625, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("625.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.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: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.14884000000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.183172\) of defining polynomial
Character \(\chi\) \(=\) 625.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.183172 q^{2} +1.47195 q^{3} -1.96645 q^{4} +0.269620 q^{6} +3.26086 q^{7} -0.726543 q^{8} -0.833366 q^{9} +O(q^{10})\) \(q+0.183172 q^{2} +1.47195 q^{3} -1.96645 q^{4} +0.269620 q^{6} +3.26086 q^{7} -0.726543 q^{8} -0.833366 q^{9} +2.00000 q^{11} -2.89451 q^{12} +0.296379 q^{13} +0.597298 q^{14} +3.79981 q^{16} +5.16297 q^{17} -0.152649 q^{18} +1.73038 q^{19} +4.79981 q^{21} +0.366344 q^{22} -0.879192 q^{23} -1.06943 q^{24} +0.0542883 q^{26} -5.64252 q^{27} -6.41230 q^{28} +5.91216 q^{29} +6.09953 q^{31} +2.14910 q^{32} +2.94390 q^{33} +0.945712 q^{34} +1.63877 q^{36} -8.08800 q^{37} +0.316957 q^{38} +0.436254 q^{39} -1.01515 q^{41} +0.879192 q^{42} +3.24199 q^{43} -3.93290 q^{44} -0.161043 q^{46} -4.21996 q^{47} +5.59313 q^{48} +3.63318 q^{49} +7.59963 q^{51} -0.582813 q^{52} +8.10072 q^{53} -1.03355 q^{54} -2.36915 q^{56} +2.54703 q^{57} +1.08294 q^{58} +5.93635 q^{59} +0.915615 q^{61} +1.11726 q^{62} -2.71749 q^{63} -7.20597 q^{64} +0.539240 q^{66} -6.88806 q^{67} -10.1527 q^{68} -1.29413 q^{69} -5.96878 q^{71} +0.605476 q^{72} -8.83341 q^{73} -1.48150 q^{74} -3.40270 q^{76} +6.52171 q^{77} +0.0799096 q^{78} -7.76067 q^{79} -5.80540 q^{81} -0.185946 q^{82} -14.5154 q^{83} -9.43858 q^{84} +0.593842 q^{86} +8.70240 q^{87} -1.45309 q^{88} +7.52642 q^{89} +0.966448 q^{91} +1.72889 q^{92} +8.97820 q^{93} -0.772978 q^{94} +3.16337 q^{96} +6.72649 q^{97} +0.665497 q^{98} -1.66673 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{4} + 6 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{4} + 6 q^{6} + 4 q^{9} + 16 q^{11} + 12 q^{14} - 2 q^{16} + 10 q^{19} + 6 q^{21} + 20 q^{24} + 6 q^{26} + 20 q^{29} + 16 q^{31} + 2 q^{34} - 12 q^{36} + 18 q^{39} + 26 q^{41} + 12 q^{44} + 6 q^{46} - 14 q^{49} - 4 q^{51} - 30 q^{54} + 10 q^{56} + 30 q^{59} + 6 q^{61} - 44 q^{64} + 12 q^{66} + 8 q^{69} + 46 q^{71} + 12 q^{74} - 20 q^{76} + 10 q^{79} - 32 q^{81} - 18 q^{84} - 14 q^{86} + 30 q^{89} - 14 q^{91} - 68 q^{94} - 54 q^{96} + 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.183172 0.129522 0.0647611 0.997901i \(-0.479371\pi\)
0.0647611 + 0.997901i \(0.479371\pi\)
\(3\) 1.47195 0.849830 0.424915 0.905233i \(-0.360304\pi\)
0.424915 + 0.905233i \(0.360304\pi\)
\(4\) −1.96645 −0.983224
\(5\) 0 0
\(6\) 0.269620 0.110072
\(7\) 3.26086 1.23249 0.616244 0.787555i \(-0.288654\pi\)
0.616244 + 0.787555i \(0.288654\pi\)
\(8\) −0.726543 −0.256872
\(9\) −0.833366 −0.277789
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −2.89451 −0.835573
\(13\) 0.296379 0.0822006 0.0411003 0.999155i \(-0.486914\pi\)
0.0411003 + 0.999155i \(0.486914\pi\)
\(14\) 0.597298 0.159635
\(15\) 0 0
\(16\) 3.79981 0.949953
\(17\) 5.16297 1.25220 0.626102 0.779741i \(-0.284649\pi\)
0.626102 + 0.779741i \(0.284649\pi\)
\(18\) −0.152649 −0.0359798
\(19\) 1.73038 0.396976 0.198488 0.980103i \(-0.436397\pi\)
0.198488 + 0.980103i \(0.436397\pi\)
\(20\) 0 0
\(21\) 4.79981 1.04741
\(22\) 0.366344 0.0781048
\(23\) −0.879192 −0.183324 −0.0916621 0.995790i \(-0.529218\pi\)
−0.0916621 + 0.995790i \(0.529218\pi\)
\(24\) −1.06943 −0.218297
\(25\) 0 0
\(26\) 0.0542883 0.0106468
\(27\) −5.64252 −1.08590
\(28\) −6.41230 −1.21181
\(29\) 5.91216 1.09786 0.548930 0.835868i \(-0.315035\pi\)
0.548930 + 0.835868i \(0.315035\pi\)
\(30\) 0 0
\(31\) 6.09953 1.09551 0.547754 0.836639i \(-0.315483\pi\)
0.547754 + 0.836639i \(0.315483\pi\)
\(32\) 2.14910 0.379912
\(33\) 2.94390 0.512467
\(34\) 0.945712 0.162188
\(35\) 0 0
\(36\) 1.63877 0.273128
\(37\) −8.08800 −1.32966 −0.664830 0.746995i \(-0.731496\pi\)
−0.664830 + 0.746995i \(0.731496\pi\)
\(38\) 0.316957 0.0514173
\(39\) 0.436254 0.0698566
\(40\) 0 0
\(41\) −1.01515 −0.158539 −0.0792695 0.996853i \(-0.525259\pi\)
−0.0792695 + 0.996853i \(0.525259\pi\)
\(42\) 0.879192 0.135662
\(43\) 3.24199 0.494399 0.247200 0.968965i \(-0.420490\pi\)
0.247200 + 0.968965i \(0.420490\pi\)
\(44\) −3.93290 −0.592906
\(45\) 0 0
\(46\) −0.161043 −0.0237446
\(47\) −4.21996 −0.615544 −0.307772 0.951460i \(-0.599583\pi\)
−0.307772 + 0.951460i \(0.599583\pi\)
\(48\) 5.59313 0.807299
\(49\) 3.63318 0.519026
\(50\) 0 0
\(51\) 7.59963 1.06416
\(52\) −0.582813 −0.0808216
\(53\) 8.10072 1.11272 0.556360 0.830941i \(-0.312198\pi\)
0.556360 + 0.830941i \(0.312198\pi\)
\(54\) −1.03355 −0.140649
\(55\) 0 0
\(56\) −2.36915 −0.316591
\(57\) 2.54703 0.337363
\(58\) 1.08294 0.142197
\(59\) 5.93635 0.772847 0.386424 0.922321i \(-0.373710\pi\)
0.386424 + 0.922321i \(0.373710\pi\)
\(60\) 0 0
\(61\) 0.915615 0.117232 0.0586162 0.998281i \(-0.481331\pi\)
0.0586162 + 0.998281i \(0.481331\pi\)
\(62\) 1.11726 0.141893
\(63\) −2.71749 −0.342371
\(64\) −7.20597 −0.900746
\(65\) 0 0
\(66\) 0.539240 0.0663759
\(67\) −6.88806 −0.841510 −0.420755 0.907174i \(-0.638235\pi\)
−0.420755 + 0.907174i \(0.638235\pi\)
\(68\) −10.1527 −1.23120
\(69\) −1.29413 −0.155794
\(70\) 0 0
\(71\) −5.96878 −0.708364 −0.354182 0.935177i \(-0.615241\pi\)
−0.354182 + 0.935177i \(0.615241\pi\)
\(72\) 0.605476 0.0713560
\(73\) −8.83341 −1.03387 −0.516936 0.856024i \(-0.672928\pi\)
−0.516936 + 0.856024i \(0.672928\pi\)
\(74\) −1.48150 −0.172220
\(75\) 0 0
\(76\) −3.40270 −0.390317
\(77\) 6.52171 0.743218
\(78\) 0.0799096 0.00904798
\(79\) −7.76067 −0.873144 −0.436572 0.899669i \(-0.643808\pi\)
−0.436572 + 0.899669i \(0.643808\pi\)
\(80\) 0 0
\(81\) −5.80540 −0.645045
\(82\) −0.185946 −0.0205343
\(83\) −14.5154 −1.59327 −0.796635 0.604461i \(-0.793388\pi\)
−0.796635 + 0.604461i \(0.793388\pi\)
\(84\) −9.43858 −1.02983
\(85\) 0 0
\(86\) 0.593842 0.0640357
\(87\) 8.70240 0.932995
\(88\) −1.45309 −0.154899
\(89\) 7.52642 0.797799 0.398900 0.916995i \(-0.369392\pi\)
0.398900 + 0.916995i \(0.369392\pi\)
\(90\) 0 0
\(91\) 0.966448 0.101311
\(92\) 1.72889 0.180249
\(93\) 8.97820 0.930996
\(94\) −0.772978 −0.0797266
\(95\) 0 0
\(96\) 3.16337 0.322860
\(97\) 6.72649 0.682971 0.341486 0.939887i \(-0.389070\pi\)
0.341486 + 0.939887i \(0.389070\pi\)
\(98\) 0.665497 0.0672254
\(99\) −1.66673 −0.167513
\(100\) 0 0
\(101\) −12.1955 −1.21350 −0.606748 0.794894i \(-0.707526\pi\)
−0.606748 + 0.794894i \(0.707526\pi\)
\(102\) 1.39204 0.137832
\(103\) −1.38140 −0.136114 −0.0680569 0.997681i \(-0.521680\pi\)
−0.0680569 + 0.997681i \(0.521680\pi\)
\(104\) −0.215332 −0.0211150
\(105\) 0 0
\(106\) 1.48383 0.144122
\(107\) −15.8285 −1.53020 −0.765101 0.643911i \(-0.777311\pi\)
−0.765101 + 0.643911i \(0.777311\pi\)
\(108\) 11.0957 1.06769
\(109\) −2.00377 −0.191926 −0.0959632 0.995385i \(-0.530593\pi\)
−0.0959632 + 0.995385i \(0.530593\pi\)
\(110\) 0 0
\(111\) −11.9051 −1.12998
\(112\) 12.3906 1.17081
\(113\) 10.4275 0.980935 0.490468 0.871459i \(-0.336826\pi\)
0.490468 + 0.871459i \(0.336826\pi\)
\(114\) 0.466545 0.0436959
\(115\) 0 0
\(116\) −11.6260 −1.07944
\(117\) −0.246992 −0.0228344
\(118\) 1.08737 0.100101
\(119\) 16.8357 1.54333
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0.167715 0.0151842
\(123\) −1.49424 −0.134731
\(124\) −11.9944 −1.07713
\(125\) 0 0
\(126\) −0.497767 −0.0443446
\(127\) 5.85007 0.519110 0.259555 0.965728i \(-0.416424\pi\)
0.259555 + 0.965728i \(0.416424\pi\)
\(128\) −5.61814 −0.496578
\(129\) 4.77205 0.420155
\(130\) 0 0
\(131\) −1.49664 −0.130762 −0.0653811 0.997860i \(-0.520826\pi\)
−0.0653811 + 0.997860i \(0.520826\pi\)
\(132\) −5.78902 −0.503870
\(133\) 5.64252 0.489268
\(134\) −1.26170 −0.108994
\(135\) 0 0
\(136\) −3.75112 −0.321656
\(137\) 7.84887 0.670574 0.335287 0.942116i \(-0.391167\pi\)
0.335287 + 0.942116i \(0.391167\pi\)
\(138\) −0.237048 −0.0201788
\(139\) 5.39152 0.457303 0.228651 0.973508i \(-0.426568\pi\)
0.228651 + 0.973508i \(0.426568\pi\)
\(140\) 0 0
\(141\) −6.21156 −0.523108
\(142\) −1.09331 −0.0917488
\(143\) 0.592757 0.0495689
\(144\) −3.16663 −0.263886
\(145\) 0 0
\(146\) −1.61803 −0.133909
\(147\) 5.34786 0.441084
\(148\) 15.9046 1.30735
\(149\) 18.8229 1.54203 0.771015 0.636817i \(-0.219749\pi\)
0.771015 + 0.636817i \(0.219749\pi\)
\(150\) 0 0
\(151\) −3.88797 −0.316398 −0.158199 0.987407i \(-0.550569\pi\)
−0.158199 + 0.987407i \(0.550569\pi\)
\(152\) −1.25719 −0.101972
\(153\) −4.30264 −0.347848
\(154\) 1.19460 0.0962632
\(155\) 0 0
\(156\) −0.857871 −0.0686847
\(157\) −4.28378 −0.341883 −0.170941 0.985281i \(-0.554681\pi\)
−0.170941 + 0.985281i \(0.554681\pi\)
\(158\) −1.42154 −0.113092
\(159\) 11.9238 0.945623
\(160\) 0 0
\(161\) −2.86692 −0.225945
\(162\) −1.06339 −0.0835477
\(163\) −15.7263 −1.23178 −0.615890 0.787832i \(-0.711204\pi\)
−0.615890 + 0.787832i \(0.711204\pi\)
\(164\) 1.99623 0.155879
\(165\) 0 0
\(166\) −2.65881 −0.206364
\(167\) −21.0330 −1.62758 −0.813792 0.581156i \(-0.802601\pi\)
−0.813792 + 0.581156i \(0.802601\pi\)
\(168\) −3.48727 −0.269049
\(169\) −12.9122 −0.993243
\(170\) 0 0
\(171\) −1.44204 −0.110276
\(172\) −6.37521 −0.486105
\(173\) −7.16663 −0.544869 −0.272434 0.962174i \(-0.587829\pi\)
−0.272434 + 0.962174i \(0.587829\pi\)
\(174\) 1.59404 0.120844
\(175\) 0 0
\(176\) 7.59963 0.572843
\(177\) 8.73801 0.656789
\(178\) 1.37863 0.103333
\(179\) 8.03934 0.600888 0.300444 0.953799i \(-0.402865\pi\)
0.300444 + 0.953799i \(0.402865\pi\)
\(180\) 0 0
\(181\) −20.6171 −1.53246 −0.766229 0.642568i \(-0.777869\pi\)
−0.766229 + 0.642568i \(0.777869\pi\)
\(182\) 0.177026 0.0131221
\(183\) 1.34774 0.0996277
\(184\) 0.638770 0.0470908
\(185\) 0 0
\(186\) 1.64456 0.120585
\(187\) 10.3259 0.755107
\(188\) 8.29833 0.605218
\(189\) −18.3994 −1.33836
\(190\) 0 0
\(191\) 18.0303 1.30463 0.652313 0.757950i \(-0.273799\pi\)
0.652313 + 0.757950i \(0.273799\pi\)
\(192\) −10.6068 −0.765482
\(193\) −6.78859 −0.488653 −0.244327 0.969693i \(-0.578567\pi\)
−0.244327 + 0.969693i \(0.578567\pi\)
\(194\) 1.23210 0.0884599
\(195\) 0 0
\(196\) −7.14446 −0.510318
\(197\) −7.99537 −0.569647 −0.284823 0.958580i \(-0.591935\pi\)
−0.284823 + 0.958580i \(0.591935\pi\)
\(198\) −0.305299 −0.0216966
\(199\) −5.20485 −0.368962 −0.184481 0.982836i \(-0.559060\pi\)
−0.184481 + 0.982836i \(0.559060\pi\)
\(200\) 0 0
\(201\) −10.1389 −0.715141
\(202\) −2.23387 −0.157175
\(203\) 19.2787 1.35310
\(204\) −14.9443 −1.04631
\(205\) 0 0
\(206\) −0.253035 −0.0176298
\(207\) 0.732688 0.0509254
\(208\) 1.12618 0.0780868
\(209\) 3.46076 0.239386
\(210\) 0 0
\(211\) 16.6020 1.14293 0.571463 0.820628i \(-0.306376\pi\)
0.571463 + 0.820628i \(0.306376\pi\)
\(212\) −15.9296 −1.09405
\(213\) −8.78574 −0.601989
\(214\) −2.89934 −0.198195
\(215\) 0 0
\(216\) 4.09953 0.278938
\(217\) 19.8897 1.35020
\(218\) −0.367035 −0.0248587
\(219\) −13.0023 −0.878616
\(220\) 0 0
\(221\) 1.53019 0.102932
\(222\) −2.18069 −0.146358
\(223\) −6.62808 −0.443849 −0.221925 0.975064i \(-0.571234\pi\)
−0.221925 + 0.975064i \(0.571234\pi\)
\(224\) 7.00792 0.468236
\(225\) 0 0
\(226\) 1.91002 0.127053
\(227\) −13.3919 −0.888853 −0.444427 0.895815i \(-0.646593\pi\)
−0.444427 + 0.895815i \(0.646593\pi\)
\(228\) −5.00860 −0.331703
\(229\) 10.0867 0.666549 0.333274 0.942830i \(-0.391846\pi\)
0.333274 + 0.942830i \(0.391846\pi\)
\(230\) 0 0
\(231\) 9.59963 0.631609
\(232\) −4.29544 −0.282009
\(233\) −22.0055 −1.44163 −0.720814 0.693129i \(-0.756232\pi\)
−0.720814 + 0.693129i \(0.756232\pi\)
\(234\) −0.0452420 −0.00295756
\(235\) 0 0
\(236\) −11.6735 −0.759882
\(237\) −11.4233 −0.742024
\(238\) 3.08383 0.199895
\(239\) 7.56029 0.489035 0.244517 0.969645i \(-0.421371\pi\)
0.244517 + 0.969645i \(0.421371\pi\)
\(240\) 0 0
\(241\) −20.3945 −1.31373 −0.656864 0.754009i \(-0.728117\pi\)
−0.656864 + 0.754009i \(0.728117\pi\)
\(242\) −1.28220 −0.0824232
\(243\) 8.38230 0.537725
\(244\) −1.80051 −0.115266
\(245\) 0 0
\(246\) −0.273703 −0.0174507
\(247\) 0.512848 0.0326317
\(248\) −4.43157 −0.281405
\(249\) −21.3659 −1.35401
\(250\) 0 0
\(251\) 10.5717 0.667278 0.333639 0.942701i \(-0.391723\pi\)
0.333639 + 0.942701i \(0.391723\pi\)
\(252\) 5.34379 0.336627
\(253\) −1.75838 −0.110549
\(254\) 1.07157 0.0672362
\(255\) 0 0
\(256\) 13.3829 0.836428
\(257\) 20.2700 1.26441 0.632205 0.774801i \(-0.282150\pi\)
0.632205 + 0.774801i \(0.282150\pi\)
\(258\) 0.874106 0.0544194
\(259\) −26.3738 −1.63879
\(260\) 0 0
\(261\) −4.92699 −0.304973
\(262\) −0.274143 −0.0169366
\(263\) 28.0909 1.73216 0.866079 0.499906i \(-0.166632\pi\)
0.866079 + 0.499906i \(0.166632\pi\)
\(264\) −2.13887 −0.131638
\(265\) 0 0
\(266\) 1.03355 0.0633711
\(267\) 11.0785 0.677994
\(268\) 13.5450 0.827393
\(269\) 20.3230 1.23911 0.619557 0.784952i \(-0.287312\pi\)
0.619557 + 0.784952i \(0.287312\pi\)
\(270\) 0 0
\(271\) 31.4467 1.91025 0.955125 0.296202i \(-0.0957200\pi\)
0.955125 + 0.296202i \(0.0957200\pi\)
\(272\) 19.6183 1.18954
\(273\) 1.42256 0.0860974
\(274\) 1.43769 0.0868543
\(275\) 0 0
\(276\) 2.54483 0.153181
\(277\) −13.9305 −0.837001 −0.418501 0.908217i \(-0.637444\pi\)
−0.418501 + 0.908217i \(0.637444\pi\)
\(278\) 0.987576 0.0592309
\(279\) −5.08314 −0.304320
\(280\) 0 0
\(281\) 25.8777 1.54373 0.771866 0.635785i \(-0.219323\pi\)
0.771866 + 0.635785i \(0.219323\pi\)
\(282\) −1.13778 −0.0677541
\(283\) 23.7316 1.41070 0.705350 0.708859i \(-0.250790\pi\)
0.705350 + 0.708859i \(0.250790\pi\)
\(284\) 11.7373 0.696480
\(285\) 0 0
\(286\) 0.108577 0.00642027
\(287\) −3.31024 −0.195397
\(288\) −1.79099 −0.105535
\(289\) 9.65625 0.568014
\(290\) 0 0
\(291\) 9.90105 0.580410
\(292\) 17.3704 1.01653
\(293\) −12.3029 −0.718742 −0.359371 0.933195i \(-0.617009\pi\)
−0.359371 + 0.933195i \(0.617009\pi\)
\(294\) 0.979578 0.0571301
\(295\) 0 0
\(296\) 5.87628 0.341552
\(297\) −11.2850 −0.654824
\(298\) 3.44783 0.199727
\(299\) −0.260574 −0.0150694
\(300\) 0 0
\(301\) 10.5717 0.609341
\(302\) −0.712167 −0.0409806
\(303\) −17.9511 −1.03127
\(304\) 6.57512 0.377109
\(305\) 0 0
\(306\) −0.788124 −0.0450540
\(307\) 4.28249 0.244415 0.122207 0.992505i \(-0.461003\pi\)
0.122207 + 0.992505i \(0.461003\pi\)
\(308\) −12.8246 −0.730750
\(309\) −2.03336 −0.115674
\(310\) 0 0
\(311\) 25.6467 1.45429 0.727145 0.686484i \(-0.240847\pi\)
0.727145 + 0.686484i \(0.240847\pi\)
\(312\) −0.316957 −0.0179442
\(313\) −22.3513 −1.26337 −0.631684 0.775226i \(-0.717636\pi\)
−0.631684 + 0.775226i \(0.717636\pi\)
\(314\) −0.784668 −0.0442814
\(315\) 0 0
\(316\) 15.2610 0.858496
\(317\) 21.9228 1.23131 0.615655 0.788016i \(-0.288892\pi\)
0.615655 + 0.788016i \(0.288892\pi\)
\(318\) 2.18412 0.122479
\(319\) 11.8243 0.662035
\(320\) 0 0
\(321\) −23.2988 −1.30041
\(322\) −0.525139 −0.0292649
\(323\) 8.93390 0.497095
\(324\) 11.4160 0.634224
\(325\) 0 0
\(326\) −2.88062 −0.159543
\(327\) −2.94945 −0.163105
\(328\) 0.737546 0.0407242
\(329\) −13.7607 −0.758650
\(330\) 0 0
\(331\) −8.96299 −0.492651 −0.246325 0.969187i \(-0.579223\pi\)
−0.246325 + 0.969187i \(0.579223\pi\)
\(332\) 28.5437 1.56654
\(333\) 6.74026 0.369364
\(334\) −3.85266 −0.210808
\(335\) 0 0
\(336\) 18.2384 0.994986
\(337\) −29.0836 −1.58428 −0.792141 0.610338i \(-0.791034\pi\)
−0.792141 + 0.610338i \(0.791034\pi\)
\(338\) −2.36515 −0.128647
\(339\) 15.3487 0.833628
\(340\) 0 0
\(341\) 12.1991 0.660616
\(342\) −0.264141 −0.0142831
\(343\) −10.9787 −0.592795
\(344\) −2.35544 −0.126997
\(345\) 0 0
\(346\) −1.31273 −0.0705726
\(347\) 14.2972 0.767513 0.383757 0.923434i \(-0.374630\pi\)
0.383757 + 0.923434i \(0.374630\pi\)
\(348\) −17.1128 −0.917343
\(349\) 5.62382 0.301036 0.150518 0.988607i \(-0.451906\pi\)
0.150518 + 0.988607i \(0.451906\pi\)
\(350\) 0 0
\(351\) −1.67232 −0.0892620
\(352\) 4.29821 0.229095
\(353\) 1.90997 0.101658 0.0508288 0.998707i \(-0.483814\pi\)
0.0508288 + 0.998707i \(0.483814\pi\)
\(354\) 1.60056 0.0850688
\(355\) 0 0
\(356\) −14.8003 −0.784415
\(357\) 24.7813 1.31156
\(358\) 1.47258 0.0778284
\(359\) 22.2047 1.17192 0.585958 0.810341i \(-0.300718\pi\)
0.585958 + 0.810341i \(0.300718\pi\)
\(360\) 0 0
\(361\) −16.0058 −0.842410
\(362\) −3.77648 −0.198487
\(363\) −10.3036 −0.540801
\(364\) −1.90047 −0.0996117
\(365\) 0 0
\(366\) 0.246868 0.0129040
\(367\) −21.4450 −1.11942 −0.559709 0.828689i \(-0.689087\pi\)
−0.559709 + 0.828689i \(0.689087\pi\)
\(368\) −3.34077 −0.174149
\(369\) 0.845987 0.0440403
\(370\) 0 0
\(371\) 26.4153 1.37141
\(372\) −17.6552 −0.915377
\(373\) 23.1399 1.19814 0.599070 0.800697i \(-0.295537\pi\)
0.599070 + 0.800697i \(0.295537\pi\)
\(374\) 1.89142 0.0978032
\(375\) 0 0
\(376\) 3.06598 0.158116
\(377\) 1.75224 0.0902448
\(378\) −3.37026 −0.173348
\(379\) −21.6501 −1.11209 −0.556047 0.831151i \(-0.687682\pi\)
−0.556047 + 0.831151i \(0.687682\pi\)
\(380\) 0 0
\(381\) 8.61100 0.441155
\(382\) 3.30265 0.168978
\(383\) 5.65481 0.288947 0.144474 0.989509i \(-0.453851\pi\)
0.144474 + 0.989509i \(0.453851\pi\)
\(384\) −8.26962 −0.422007
\(385\) 0 0
\(386\) −1.24348 −0.0632915
\(387\) −2.70176 −0.137338
\(388\) −13.2273 −0.671514
\(389\) −8.12605 −0.412007 −0.206004 0.978551i \(-0.566046\pi\)
−0.206004 + 0.978551i \(0.566046\pi\)
\(390\) 0 0
\(391\) −4.53924 −0.229559
\(392\) −2.63966 −0.133323
\(393\) −2.20298 −0.111126
\(394\) −1.46453 −0.0737819
\(395\) 0 0
\(396\) 3.27754 0.164703
\(397\) −20.7689 −1.04236 −0.521180 0.853447i \(-0.674508\pi\)
−0.521180 + 0.853447i \(0.674508\pi\)
\(398\) −0.953382 −0.0477887
\(399\) 8.30550 0.415795
\(400\) 0 0
\(401\) 30.1195 1.50410 0.752049 0.659107i \(-0.229066\pi\)
0.752049 + 0.659107i \(0.229066\pi\)
\(402\) −1.85716 −0.0926266
\(403\) 1.80777 0.0900515
\(404\) 23.9818 1.19314
\(405\) 0 0
\(406\) 3.53132 0.175256
\(407\) −16.1760 −0.801815
\(408\) −5.52145 −0.273353
\(409\) 11.0452 0.546152 0.273076 0.961992i \(-0.411959\pi\)
0.273076 + 0.961992i \(0.411959\pi\)
\(410\) 0 0
\(411\) 11.5531 0.569874
\(412\) 2.71646 0.133830
\(413\) 19.3576 0.952524
\(414\) 0.134208 0.00659597
\(415\) 0 0
\(416\) 0.636949 0.0312290
\(417\) 7.93604 0.388630
\(418\) 0.633915 0.0310058
\(419\) −32.9212 −1.60830 −0.804152 0.594424i \(-0.797380\pi\)
−0.804152 + 0.594424i \(0.797380\pi\)
\(420\) 0 0
\(421\) −20.2647 −0.987642 −0.493821 0.869564i \(-0.664400\pi\)
−0.493821 + 0.869564i \(0.664400\pi\)
\(422\) 3.04101 0.148034
\(423\) 3.51677 0.170991
\(424\) −5.88552 −0.285826
\(425\) 0 0
\(426\) −1.60930 −0.0779709
\(427\) 2.98569 0.144488
\(428\) 31.1260 1.50453
\(429\) 0.872509 0.0421251
\(430\) 0 0
\(431\) 7.15526 0.344657 0.172328 0.985040i \(-0.444871\pi\)
0.172328 + 0.985040i \(0.444871\pi\)
\(432\) −21.4405 −1.03156
\(433\) −5.48264 −0.263479 −0.131739 0.991284i \(-0.542056\pi\)
−0.131739 + 0.991284i \(0.542056\pi\)
\(434\) 3.64324 0.174881
\(435\) 0 0
\(436\) 3.94031 0.188707
\(437\) −1.52134 −0.0727754
\(438\) −2.38166 −0.113800
\(439\) −30.7561 −1.46791 −0.733954 0.679199i \(-0.762327\pi\)
−0.733954 + 0.679199i \(0.762327\pi\)
\(440\) 0 0
\(441\) −3.02777 −0.144179
\(442\) 0.280289 0.0133320
\(443\) 11.3527 0.539381 0.269691 0.962947i \(-0.413079\pi\)
0.269691 + 0.962947i \(0.413079\pi\)
\(444\) 23.4108 1.11103
\(445\) 0 0
\(446\) −1.21408 −0.0574883
\(447\) 27.7063 1.31046
\(448\) −23.4976 −1.11016
\(449\) −15.7661 −0.744050 −0.372025 0.928223i \(-0.621336\pi\)
−0.372025 + 0.928223i \(0.621336\pi\)
\(450\) 0 0
\(451\) −2.03029 −0.0956027
\(452\) −20.5051 −0.964479
\(453\) −5.72289 −0.268885
\(454\) −2.45303 −0.115126
\(455\) 0 0
\(456\) −1.85053 −0.0866588
\(457\) −4.16714 −0.194931 −0.0974653 0.995239i \(-0.531074\pi\)
−0.0974653 + 0.995239i \(0.531074\pi\)
\(458\) 1.84760 0.0863329
\(459\) −29.1322 −1.35977
\(460\) 0 0
\(461\) −23.9714 −1.11646 −0.558230 0.829686i \(-0.688519\pi\)
−0.558230 + 0.829686i \(0.688519\pi\)
\(462\) 1.75838 0.0818074
\(463\) 41.3247 1.92052 0.960260 0.279107i \(-0.0900385\pi\)
0.960260 + 0.279107i \(0.0900385\pi\)
\(464\) 22.4651 1.04292
\(465\) 0 0
\(466\) −4.03079 −0.186723
\(467\) −10.3966 −0.481097 −0.240548 0.970637i \(-0.577327\pi\)
−0.240548 + 0.970637i \(0.577327\pi\)
\(468\) 0.485697 0.0224513
\(469\) −22.4610 −1.03715
\(470\) 0 0
\(471\) −6.30550 −0.290542
\(472\) −4.31301 −0.198522
\(473\) 6.48398 0.298134
\(474\) −2.09243 −0.0961086
\(475\) 0 0
\(476\) −33.1065 −1.51743
\(477\) −6.75086 −0.309101
\(478\) 1.38483 0.0633408
\(479\) 36.0081 1.64525 0.822626 0.568582i \(-0.192508\pi\)
0.822626 + 0.568582i \(0.192508\pi\)
\(480\) 0 0
\(481\) −2.39711 −0.109299
\(482\) −3.73571 −0.170157
\(483\) −4.21996 −0.192015
\(484\) 13.7651 0.625688
\(485\) 0 0
\(486\) 1.53540 0.0696473
\(487\) 10.6378 0.482045 0.241023 0.970520i \(-0.422517\pi\)
0.241023 + 0.970520i \(0.422517\pi\)
\(488\) −0.665233 −0.0301137
\(489\) −23.1483 −1.04680
\(490\) 0 0
\(491\) −17.6693 −0.797402 −0.398701 0.917081i \(-0.630539\pi\)
−0.398701 + 0.917081i \(0.630539\pi\)
\(492\) 2.93835 0.132471
\(493\) 30.5243 1.37475
\(494\) 0.0939394 0.00422653
\(495\) 0 0
\(496\) 23.1771 1.04068
\(497\) −19.4633 −0.873049
\(498\) −3.91363 −0.175374
\(499\) −9.41734 −0.421578 −0.210789 0.977532i \(-0.567603\pi\)
−0.210789 + 0.977532i \(0.567603\pi\)
\(500\) 0 0
\(501\) −30.9595 −1.38317
\(502\) 1.93643 0.0864273
\(503\) −18.0133 −0.803172 −0.401586 0.915821i \(-0.631541\pi\)
−0.401586 + 0.915821i \(0.631541\pi\)
\(504\) 1.97437 0.0879454
\(505\) 0 0
\(506\) −0.322087 −0.0143185
\(507\) −19.0060 −0.844088
\(508\) −11.5039 −0.510401
\(509\) −16.0485 −0.711337 −0.355669 0.934612i \(-0.615747\pi\)
−0.355669 + 0.934612i \(0.615747\pi\)
\(510\) 0 0
\(511\) −28.8045 −1.27423
\(512\) 13.6877 0.604914
\(513\) −9.76370 −0.431078
\(514\) 3.71291 0.163769
\(515\) 0 0
\(516\) −9.38398 −0.413107
\(517\) −8.43991 −0.371187
\(518\) −4.83095 −0.212260
\(519\) −10.5489 −0.463046
\(520\) 0 0
\(521\) −2.20069 −0.0964142 −0.0482071 0.998837i \(-0.515351\pi\)
−0.0482071 + 0.998837i \(0.515351\pi\)
\(522\) −0.902487 −0.0395008
\(523\) −7.43588 −0.325148 −0.162574 0.986696i \(-0.551980\pi\)
−0.162574 + 0.986696i \(0.551980\pi\)
\(524\) 2.94307 0.128569
\(525\) 0 0
\(526\) 5.14547 0.224353
\(527\) 31.4917 1.37180
\(528\) 11.1863 0.486820
\(529\) −22.2270 −0.966392
\(530\) 0 0
\(531\) −4.94715 −0.214688
\(532\) −11.0957 −0.481061
\(533\) −0.300867 −0.0130320
\(534\) 2.02927 0.0878153
\(535\) 0 0
\(536\) 5.00447 0.216160
\(537\) 11.8335 0.510653
\(538\) 3.72260 0.160493
\(539\) 7.26636 0.312984
\(540\) 0 0
\(541\) 20.6474 0.887701 0.443850 0.896101i \(-0.353612\pi\)
0.443850 + 0.896101i \(0.353612\pi\)
\(542\) 5.76016 0.247420
\(543\) −30.3473 −1.30233
\(544\) 11.0958 0.475727
\(545\) 0 0
\(546\) 0.260574 0.0111515
\(547\) 31.3762 1.34155 0.670774 0.741662i \(-0.265962\pi\)
0.670774 + 0.741662i \(0.265962\pi\)
\(548\) −15.4344 −0.659325
\(549\) −0.763042 −0.0325658
\(550\) 0 0
\(551\) 10.2303 0.435825
\(552\) 0.940237 0.0400192
\(553\) −25.3064 −1.07614
\(554\) −2.55167 −0.108410
\(555\) 0 0
\(556\) −10.6021 −0.449631
\(557\) −22.3515 −0.947064 −0.473532 0.880776i \(-0.657021\pi\)
−0.473532 + 0.880776i \(0.657021\pi\)
\(558\) −0.931089 −0.0394161
\(559\) 0.960857 0.0406399
\(560\) 0 0
\(561\) 15.1993 0.641713
\(562\) 4.74007 0.199948
\(563\) −34.3663 −1.44837 −0.724184 0.689607i \(-0.757783\pi\)
−0.724184 + 0.689607i \(0.757783\pi\)
\(564\) 12.2147 0.514332
\(565\) 0 0
\(566\) 4.34697 0.182717
\(567\) −18.9306 −0.795010
\(568\) 4.33657 0.181958
\(569\) 32.1662 1.34848 0.674239 0.738513i \(-0.264472\pi\)
0.674239 + 0.738513i \(0.264472\pi\)
\(570\) 0 0
\(571\) −27.1668 −1.13690 −0.568448 0.822719i \(-0.692456\pi\)
−0.568448 + 0.822719i \(0.692456\pi\)
\(572\) −1.16563 −0.0487373
\(573\) 26.5397 1.10871
\(574\) −0.606344 −0.0253083
\(575\) 0 0
\(576\) 6.00521 0.250217
\(577\) 13.8503 0.576596 0.288298 0.957541i \(-0.406911\pi\)
0.288298 + 0.957541i \(0.406911\pi\)
\(578\) 1.76875 0.0735705
\(579\) −9.99246 −0.415273
\(580\) 0 0
\(581\) −47.3325 −1.96368
\(582\) 1.81360 0.0751759
\(583\) 16.2014 0.670995
\(584\) 6.41785 0.265572
\(585\) 0 0
\(586\) −2.25354 −0.0930930
\(587\) 44.1499 1.82226 0.911131 0.412118i \(-0.135211\pi\)
0.911131 + 0.412118i \(0.135211\pi\)
\(588\) −10.5163 −0.433684
\(589\) 10.5545 0.434891
\(590\) 0 0
\(591\) −11.7688 −0.484103
\(592\) −30.7329 −1.26311
\(593\) −16.2531 −0.667437 −0.333718 0.942673i \(-0.608303\pi\)
−0.333718 + 0.942673i \(0.608303\pi\)
\(594\) −2.06710 −0.0848143
\(595\) 0 0
\(596\) −37.0142 −1.51616
\(597\) −7.66127 −0.313555
\(598\) −0.0477298 −0.00195182
\(599\) 30.4822 1.24547 0.622734 0.782433i \(-0.286022\pi\)
0.622734 + 0.782433i \(0.286022\pi\)
\(600\) 0 0
\(601\) −28.9162 −1.17952 −0.589758 0.807580i \(-0.700777\pi\)
−0.589758 + 0.807580i \(0.700777\pi\)
\(602\) 1.93643 0.0789232
\(603\) 5.74027 0.233762
\(604\) 7.64549 0.311090
\(605\) 0 0
\(606\) −3.28815 −0.133572
\(607\) 8.23276 0.334157 0.167079 0.985944i \(-0.446567\pi\)
0.167079 + 0.985944i \(0.446567\pi\)
\(608\) 3.71877 0.150816
\(609\) 28.3773 1.14990
\(610\) 0 0
\(611\) −1.25071 −0.0505981
\(612\) 8.46092 0.342012
\(613\) 4.79811 0.193794 0.0968969 0.995294i \(-0.469108\pi\)
0.0968969 + 0.995294i \(0.469108\pi\)
\(614\) 0.784432 0.0316571
\(615\) 0 0
\(616\) −4.73830 −0.190912
\(617\) −2.03184 −0.0817986 −0.0408993 0.999163i \(-0.513022\pi\)
−0.0408993 + 0.999163i \(0.513022\pi\)
\(618\) −0.372454 −0.0149823
\(619\) 8.14100 0.327215 0.163607 0.986526i \(-0.447687\pi\)
0.163607 + 0.986526i \(0.447687\pi\)
\(620\) 0 0
\(621\) 4.96086 0.199072
\(622\) 4.69776 0.188363
\(623\) 24.5426 0.983278
\(624\) 1.65769 0.0663605
\(625\) 0 0
\(626\) −4.09413 −0.163634
\(627\) 5.09406 0.203437
\(628\) 8.42382 0.336147
\(629\) −41.7581 −1.66500
\(630\) 0 0
\(631\) 33.2847 1.32504 0.662522 0.749042i \(-0.269486\pi\)
0.662522 + 0.749042i \(0.269486\pi\)
\(632\) 5.63846 0.224286
\(633\) 24.4372 0.971293
\(634\) 4.01565 0.159482
\(635\) 0 0
\(636\) −23.4476 −0.929759
\(637\) 1.07680 0.0426642
\(638\) 2.16589 0.0857482
\(639\) 4.97417 0.196775
\(640\) 0 0
\(641\) −40.0481 −1.58180 −0.790902 0.611943i \(-0.790388\pi\)
−0.790902 + 0.611943i \(0.790388\pi\)
\(642\) −4.26769 −0.168432
\(643\) −11.6870 −0.460890 −0.230445 0.973085i \(-0.574018\pi\)
−0.230445 + 0.973085i \(0.574018\pi\)
\(644\) 5.63764 0.222154
\(645\) 0 0
\(646\) 1.63644 0.0643849
\(647\) −7.94936 −0.312522 −0.156261 0.987716i \(-0.549944\pi\)
−0.156261 + 0.987716i \(0.549944\pi\)
\(648\) 4.21787 0.165694
\(649\) 11.8727 0.466044
\(650\) 0 0
\(651\) 29.2766 1.14744
\(652\) 30.9250 1.21112
\(653\) 3.33736 0.130601 0.0653005 0.997866i \(-0.479199\pi\)
0.0653005 + 0.997866i \(0.479199\pi\)
\(654\) −0.540256 −0.0211257
\(655\) 0 0
\(656\) −3.85736 −0.150605
\(657\) 7.36146 0.287198
\(658\) −2.52057 −0.0982621
\(659\) −30.0508 −1.17061 −0.585306 0.810812i \(-0.699026\pi\)
−0.585306 + 0.810812i \(0.699026\pi\)
\(660\) 0 0
\(661\) −6.58417 −0.256094 −0.128047 0.991768i \(-0.540871\pi\)
−0.128047 + 0.991768i \(0.540871\pi\)
\(662\) −1.64177 −0.0638092
\(663\) 2.25237 0.0874747
\(664\) 10.5460 0.409266
\(665\) 0 0
\(666\) 1.23463 0.0478409
\(667\) −5.19792 −0.201264
\(668\) 41.3603 1.60028
\(669\) −9.75620 −0.377196
\(670\) 0 0
\(671\) 1.83123 0.0706939
\(672\) 10.3153 0.397921
\(673\) 6.73140 0.259476 0.129738 0.991548i \(-0.458586\pi\)
0.129738 + 0.991548i \(0.458586\pi\)
\(674\) −5.32730 −0.205200
\(675\) 0 0
\(676\) 25.3911 0.976580
\(677\) 13.6478 0.524529 0.262265 0.964996i \(-0.415531\pi\)
0.262265 + 0.964996i \(0.415531\pi\)
\(678\) 2.81146 0.107973
\(679\) 21.9341 0.841753
\(680\) 0 0
\(681\) −19.7122 −0.755374
\(682\) 2.23453 0.0855645
\(683\) 1.17220 0.0448529 0.0224265 0.999748i \(-0.492861\pi\)
0.0224265 + 0.999748i \(0.492861\pi\)
\(684\) 2.83570 0.108426
\(685\) 0 0
\(686\) −2.01099 −0.0767801
\(687\) 14.8471 0.566453
\(688\) 12.3190 0.469656
\(689\) 2.40088 0.0914663
\(690\) 0 0
\(691\) 12.2674 0.466673 0.233336 0.972396i \(-0.425036\pi\)
0.233336 + 0.972396i \(0.425036\pi\)
\(692\) 14.0928 0.535728
\(693\) −5.43497 −0.206457
\(694\) 2.61885 0.0994100
\(695\) 0 0
\(696\) −6.32266 −0.239660
\(697\) −5.24116 −0.198523
\(698\) 1.03013 0.0389909
\(699\) −32.3910 −1.22514
\(700\) 0 0
\(701\) −20.0271 −0.756415 −0.378207 0.925721i \(-0.623459\pi\)
−0.378207 + 0.925721i \(0.623459\pi\)
\(702\) −0.306323 −0.0115614
\(703\) −13.9953 −0.527843
\(704\) −14.4119 −0.543171
\(705\) 0 0
\(706\) 0.349854 0.0131669
\(707\) −39.7677 −1.49562
\(708\) −17.1828 −0.645771
\(709\) −4.39510 −0.165061 −0.0825306 0.996589i \(-0.526300\pi\)
−0.0825306 + 0.996589i \(0.526300\pi\)
\(710\) 0 0
\(711\) 6.46748 0.242549
\(712\) −5.46827 −0.204932
\(713\) −5.36266 −0.200833
\(714\) 4.53924 0.169877
\(715\) 0 0
\(716\) −15.8089 −0.590808
\(717\) 11.1284 0.415596
\(718\) 4.06727 0.151789
\(719\) −10.9283 −0.407557 −0.203779 0.979017i \(-0.565322\pi\)
−0.203779 + 0.979017i \(0.565322\pi\)
\(720\) 0 0
\(721\) −4.50456 −0.167759
\(722\) −2.93181 −0.109111
\(723\) −30.0197 −1.11645
\(724\) 40.5425 1.50675
\(725\) 0 0
\(726\) −1.88734 −0.0700458
\(727\) −32.9032 −1.22031 −0.610156 0.792281i \(-0.708893\pi\)
−0.610156 + 0.792281i \(0.708893\pi\)
\(728\) −0.702166 −0.0260240
\(729\) 29.7545 1.10202
\(730\) 0 0
\(731\) 16.7383 0.619088
\(732\) −2.65026 −0.0979564
\(733\) −13.7498 −0.507859 −0.253929 0.967223i \(-0.581723\pi\)
−0.253929 + 0.967223i \(0.581723\pi\)
\(734\) −3.92812 −0.144990
\(735\) 0 0
\(736\) −1.88948 −0.0696470
\(737\) −13.7761 −0.507450
\(738\) 0.154961 0.00570420
\(739\) −43.1893 −1.58874 −0.794372 0.607432i \(-0.792200\pi\)
−0.794372 + 0.607432i \(0.792200\pi\)
\(740\) 0 0
\(741\) 0.754886 0.0277314
\(742\) 4.83854 0.177628
\(743\) −31.8479 −1.16838 −0.584192 0.811615i \(-0.698589\pi\)
−0.584192 + 0.811615i \(0.698589\pi\)
\(744\) −6.52304 −0.239146
\(745\) 0 0
\(746\) 4.23859 0.155186
\(747\) 12.0966 0.442592
\(748\) −20.3054 −0.742440
\(749\) −51.6145 −1.88595
\(750\) 0 0
\(751\) −29.5952 −1.07995 −0.539973 0.841682i \(-0.681565\pi\)
−0.539973 + 0.841682i \(0.681565\pi\)
\(752\) −16.0351 −0.584738
\(753\) 15.5610 0.567073
\(754\) 0.320961 0.0116887
\(755\) 0 0
\(756\) 36.1815 1.31591
\(757\) 0.0984401 0.00357786 0.00178893 0.999998i \(-0.499431\pi\)
0.00178893 + 0.999998i \(0.499431\pi\)
\(758\) −3.96570 −0.144041
\(759\) −2.58825 −0.0939476
\(760\) 0 0
\(761\) −3.54402 −0.128471 −0.0642353 0.997935i \(-0.520461\pi\)
−0.0642353 + 0.997935i \(0.520461\pi\)
\(762\) 1.57730 0.0571394
\(763\) −6.53400 −0.236547
\(764\) −35.4556 −1.28274
\(765\) 0 0
\(766\) 1.03580 0.0374251
\(767\) 1.75941 0.0635285
\(768\) 19.6989 0.710822
\(769\) 1.42759 0.0514802 0.0257401 0.999669i \(-0.491806\pi\)
0.0257401 + 0.999669i \(0.491806\pi\)
\(770\) 0 0
\(771\) 29.8365 1.07453
\(772\) 13.3494 0.480456
\(773\) −33.7807 −1.21501 −0.607504 0.794317i \(-0.707829\pi\)
−0.607504 + 0.794317i \(0.707829\pi\)
\(774\) −0.494888 −0.0177884
\(775\) 0 0
\(776\) −4.88708 −0.175436
\(777\) −38.8209 −1.39269
\(778\) −1.48847 −0.0533641
\(779\) −1.75659 −0.0629363
\(780\) 0 0
\(781\) −11.9376 −0.427159
\(782\) −0.831462 −0.0297330
\(783\) −33.3595 −1.19217
\(784\) 13.8054 0.493050
\(785\) 0 0
\(786\) −0.403525 −0.0143932
\(787\) −2.18030 −0.0777192 −0.0388596 0.999245i \(-0.512373\pi\)
−0.0388596 + 0.999245i \(0.512373\pi\)
\(788\) 15.7225 0.560090
\(789\) 41.3484 1.47204
\(790\) 0 0
\(791\) 34.0025 1.20899
\(792\) 1.21095 0.0430293
\(793\) 0.271369 0.00963659
\(794\) −3.80428 −0.135009
\(795\) 0 0
\(796\) 10.2351 0.362772
\(797\) −23.4919 −0.832124 −0.416062 0.909336i \(-0.636590\pi\)
−0.416062 + 0.909336i \(0.636590\pi\)
\(798\) 1.52134 0.0538547
\(799\) −21.7875 −0.770787
\(800\) 0 0
\(801\) −6.27226 −0.221620
\(802\) 5.51706 0.194814
\(803\) −17.6668 −0.623449
\(804\) 19.9376 0.703143
\(805\) 0 0
\(806\) 0.331133 0.0116637
\(807\) 29.9144 1.05304
\(808\) 8.86054 0.311713
\(809\) −38.1509 −1.34132 −0.670658 0.741767i \(-0.733988\pi\)
−0.670658 + 0.741767i \(0.733988\pi\)
\(810\) 0 0
\(811\) 47.9069 1.68224 0.841120 0.540849i \(-0.181897\pi\)
0.841120 + 0.540849i \(0.181897\pi\)
\(812\) −37.9106 −1.33040
\(813\) 46.2879 1.62339
\(814\) −2.96299 −0.103853
\(815\) 0 0
\(816\) 28.8772 1.01090
\(817\) 5.60988 0.196265
\(818\) 2.02318 0.0707388
\(819\) −0.805405 −0.0281431
\(820\) 0 0
\(821\) 18.9808 0.662434 0.331217 0.943555i \(-0.392541\pi\)
0.331217 + 0.943555i \(0.392541\pi\)
\(822\) 2.11621 0.0738114
\(823\) 22.1681 0.772732 0.386366 0.922345i \(-0.373730\pi\)
0.386366 + 0.922345i \(0.373730\pi\)
\(824\) 1.00365 0.0349638
\(825\) 0 0
\(826\) 3.54577 0.123373
\(827\) 4.72938 0.164457 0.0822283 0.996614i \(-0.473796\pi\)
0.0822283 + 0.996614i \(0.473796\pi\)
\(828\) −1.44079 −0.0500710
\(829\) 16.4400 0.570986 0.285493 0.958381i \(-0.407843\pi\)
0.285493 + 0.958381i \(0.407843\pi\)
\(830\) 0 0
\(831\) −20.5050 −0.711309
\(832\) −2.13570 −0.0740419
\(833\) 18.7580 0.649926
\(834\) 1.45366 0.0503362
\(835\) 0 0
\(836\) −6.80540 −0.235370
\(837\) −34.4167 −1.18962
\(838\) −6.03024 −0.208311
\(839\) −5.60083 −0.193362 −0.0966811 0.995315i \(-0.530823\pi\)
−0.0966811 + 0.995315i \(0.530823\pi\)
\(840\) 0 0
\(841\) 5.95363 0.205298
\(842\) −3.71193 −0.127922
\(843\) 38.0906 1.31191
\(844\) −32.6469 −1.12375
\(845\) 0 0
\(846\) 0.644174 0.0221471
\(847\) −22.8260 −0.784310
\(848\) 30.7812 1.05703
\(849\) 34.9318 1.19886
\(850\) 0 0
\(851\) 7.11091 0.243759
\(852\) 17.2767 0.591890
\(853\) 17.8905 0.612559 0.306279 0.951942i \(-0.400916\pi\)
0.306279 + 0.951942i \(0.400916\pi\)
\(854\) 0.546895 0.0187144
\(855\) 0 0
\(856\) 11.5001 0.393065
\(857\) −3.19536 −0.109151 −0.0545757 0.998510i \(-0.517381\pi\)
−0.0545757 + 0.998510i \(0.517381\pi\)
\(858\) 0.159819 0.00545614
\(859\) 43.4196 1.48146 0.740728 0.671805i \(-0.234481\pi\)
0.740728 + 0.671805i \(0.234481\pi\)
\(860\) 0 0
\(861\) −4.87251 −0.166055
\(862\) 1.31064 0.0446407
\(863\) −43.2724 −1.47301 −0.736504 0.676433i \(-0.763525\pi\)
−0.736504 + 0.676433i \(0.763525\pi\)
\(864\) −12.1264 −0.412547
\(865\) 0 0
\(866\) −1.00427 −0.0341264
\(867\) 14.2135 0.482716
\(868\) −39.1120 −1.32755
\(869\) −15.5213 −0.526525
\(870\) 0 0
\(871\) −2.04147 −0.0691727
\(872\) 1.45582 0.0493004
\(873\) −5.60562 −0.189722
\(874\) −0.278666 −0.00942603
\(875\) 0 0
\(876\) 25.5684 0.863876
\(877\) −34.2339 −1.15600 −0.577999 0.816038i \(-0.696166\pi\)
−0.577999 + 0.816038i \(0.696166\pi\)
\(878\) −5.63366 −0.190127
\(879\) −18.1092 −0.610808
\(880\) 0 0
\(881\) 27.6270 0.930779 0.465389 0.885106i \(-0.345914\pi\)
0.465389 + 0.885106i \(0.345914\pi\)
\(882\) −0.554602 −0.0186744
\(883\) −25.8990 −0.871571 −0.435786 0.900051i \(-0.643529\pi\)
−0.435786 + 0.900051i \(0.643529\pi\)
\(884\) −3.00905 −0.101205
\(885\) 0 0
\(886\) 2.07949 0.0698618
\(887\) −17.2449 −0.579027 −0.289514 0.957174i \(-0.593494\pi\)
−0.289514 + 0.957174i \(0.593494\pi\)
\(888\) 8.64958 0.290261
\(889\) 19.0762 0.639796
\(890\) 0 0
\(891\) −11.6108 −0.388977
\(892\) 13.0338 0.436403
\(893\) −7.30213 −0.244356
\(894\) 5.07502 0.169734
\(895\) 0 0
\(896\) −18.3200 −0.612027
\(897\) −0.383551 −0.0128064
\(898\) −2.88792 −0.0963710
\(899\) 36.0614 1.20271
\(900\) 0 0
\(901\) 41.8238 1.39335
\(902\) −0.371893 −0.0123827
\(903\) 15.5610 0.517836
\(904\) −7.57601 −0.251974
\(905\) 0 0
\(906\) −1.04827 −0.0348266
\(907\) 57.0465 1.89420 0.947099 0.320940i \(-0.103999\pi\)
0.947099 + 0.320940i \(0.103999\pi\)
\(908\) 26.3345 0.873942
\(909\) 10.1633 0.337095
\(910\) 0 0
\(911\) 19.8683 0.658267 0.329133 0.944283i \(-0.393243\pi\)
0.329133 + 0.944283i \(0.393243\pi\)
\(912\) 9.67824 0.320479
\(913\) −29.0307 −0.960777
\(914\) −0.763304 −0.0252478
\(915\) 0 0
\(916\) −19.8350 −0.655367
\(917\) −4.88033 −0.161163
\(918\) −5.33620 −0.176121
\(919\) 27.9785 0.922924 0.461462 0.887160i \(-0.347325\pi\)
0.461462 + 0.887160i \(0.347325\pi\)
\(920\) 0 0
\(921\) 6.30361 0.207711
\(922\) −4.39090 −0.144606
\(923\) −1.76902 −0.0582279
\(924\) −18.8772 −0.621013
\(925\) 0 0
\(926\) 7.56952 0.248750
\(927\) 1.15122 0.0378109
\(928\) 12.7059 0.417090
\(929\) −14.1249 −0.463424 −0.231712 0.972784i \(-0.574433\pi\)
−0.231712 + 0.972784i \(0.574433\pi\)
\(930\) 0 0
\(931\) 6.28678 0.206041
\(932\) 43.2727 1.41744
\(933\) 37.7506 1.23590
\(934\) −1.90437 −0.0623127
\(935\) 0 0
\(936\) 0.179450 0.00586551
\(937\) −35.9546 −1.17459 −0.587293 0.809374i \(-0.699806\pi\)
−0.587293 + 0.809374i \(0.699806\pi\)
\(938\) −4.11422 −0.134334
\(939\) −32.8999 −1.07365
\(940\) 0 0
\(941\) 26.6979 0.870327 0.435164 0.900351i \(-0.356691\pi\)
0.435164 + 0.900351i \(0.356691\pi\)
\(942\) −1.15499 −0.0376317
\(943\) 0.892508 0.0290640
\(944\) 22.5570 0.734169
\(945\) 0 0
\(946\) 1.18768 0.0386150
\(947\) −17.3102 −0.562506 −0.281253 0.959634i \(-0.590750\pi\)
−0.281253 + 0.959634i \(0.590750\pi\)
\(948\) 22.4634 0.729576
\(949\) −2.61803 −0.0849850
\(950\) 0 0
\(951\) 32.2693 1.04640
\(952\) −12.2318 −0.396436
\(953\) −24.2622 −0.785931 −0.392965 0.919553i \(-0.628551\pi\)
−0.392965 + 0.919553i \(0.628551\pi\)
\(954\) −1.23657 −0.0400354
\(955\) 0 0
\(956\) −14.8669 −0.480830
\(957\) 17.4048 0.562617
\(958\) 6.59568 0.213097
\(959\) 25.5940 0.826475
\(960\) 0 0
\(961\) 6.20427 0.200138
\(962\) −0.439084 −0.0141566
\(963\) 13.1910 0.425072
\(964\) 40.1048 1.29169
\(965\) 0 0
\(966\) −0.772978 −0.0248702
\(967\) 31.1143 1.00057 0.500284 0.865861i \(-0.333229\pi\)
0.500284 + 0.865861i \(0.333229\pi\)
\(968\) 5.08580 0.163464
\(969\) 13.1502 0.422447
\(970\) 0 0
\(971\) −17.3721 −0.557497 −0.278749 0.960364i \(-0.589920\pi\)
−0.278749 + 0.960364i \(0.589920\pi\)
\(972\) −16.4834 −0.528704
\(973\) 17.5810 0.563620
\(974\) 1.94855 0.0624356
\(975\) 0 0
\(976\) 3.47917 0.111365
\(977\) 42.8929 1.37227 0.686133 0.727476i \(-0.259307\pi\)
0.686133 + 0.727476i \(0.259307\pi\)
\(978\) −4.24013 −0.135584
\(979\) 15.0528 0.481091
\(980\) 0 0
\(981\) 1.66987 0.0533149
\(982\) −3.23651 −0.103281
\(983\) 37.4533 1.19457 0.597287 0.802028i \(-0.296245\pi\)
0.597287 + 0.802028i \(0.296245\pi\)
\(984\) 1.08563 0.0346086
\(985\) 0 0
\(986\) 5.59120 0.178060
\(987\) −20.2550 −0.644724
\(988\) −1.00849 −0.0320843
\(989\) −2.85033 −0.0906353
\(990\) 0 0
\(991\) 16.7624 0.532476 0.266238 0.963907i \(-0.414219\pi\)
0.266238 + 0.963907i \(0.414219\pi\)
\(992\) 13.1085 0.416196
\(993\) −13.1931 −0.418669
\(994\) −3.56514 −0.113079
\(995\) 0 0
\(996\) 42.0149 1.33129
\(997\) 59.8595 1.89577 0.947884 0.318615i \(-0.103218\pi\)
0.947884 + 0.318615i \(0.103218\pi\)
\(998\) −1.72499 −0.0546037
\(999\) 45.6367 1.44388
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 625.2.a.f.1.5 8
3.2 odd 2 5625.2.a.x.1.4 8
4.3 odd 2 10000.2.a.bj.1.3 8
5.2 odd 4 625.2.b.c.624.5 8
5.3 odd 4 625.2.b.c.624.4 8
5.4 even 2 inner 625.2.a.f.1.4 8
15.14 odd 2 5625.2.a.x.1.5 8
20.19 odd 2 10000.2.a.bj.1.6 8
25.2 odd 20 25.2.e.a.4.2 8
25.3 odd 20 625.2.e.i.249.1 8
25.4 even 10 625.2.d.o.376.3 16
25.6 even 5 625.2.d.o.251.2 16
25.8 odd 20 625.2.e.a.374.2 8
25.9 even 10 125.2.d.b.26.2 16
25.11 even 5 125.2.d.b.101.3 16
25.12 odd 20 125.2.e.b.99.1 8
25.13 odd 20 25.2.e.a.19.2 yes 8
25.14 even 10 125.2.d.b.101.2 16
25.16 even 5 125.2.d.b.26.3 16
25.17 odd 20 625.2.e.i.374.1 8
25.19 even 10 625.2.d.o.251.3 16
25.21 even 5 625.2.d.o.376.2 16
25.22 odd 20 625.2.e.a.249.2 8
25.23 odd 20 125.2.e.b.24.1 8
75.2 even 20 225.2.m.a.154.1 8
75.38 even 20 225.2.m.a.19.1 8
100.27 even 20 400.2.y.c.129.2 8
100.63 even 20 400.2.y.c.369.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.2.e.a.4.2 8 25.2 odd 20
25.2.e.a.19.2 yes 8 25.13 odd 20
125.2.d.b.26.2 16 25.9 even 10
125.2.d.b.26.3 16 25.16 even 5
125.2.d.b.101.2 16 25.14 even 10
125.2.d.b.101.3 16 25.11 even 5
125.2.e.b.24.1 8 25.23 odd 20
125.2.e.b.99.1 8 25.12 odd 20
225.2.m.a.19.1 8 75.38 even 20
225.2.m.a.154.1 8 75.2 even 20
400.2.y.c.129.2 8 100.27 even 20
400.2.y.c.369.2 8 100.63 even 20
625.2.a.f.1.4 8 5.4 even 2 inner
625.2.a.f.1.5 8 1.1 even 1 trivial
625.2.b.c.624.4 8 5.3 odd 4
625.2.b.c.624.5 8 5.2 odd 4
625.2.d.o.251.2 16 25.6 even 5
625.2.d.o.251.3 16 25.19 even 10
625.2.d.o.376.2 16 25.21 even 5
625.2.d.o.376.3 16 25.4 even 10
625.2.e.a.249.2 8 25.22 odd 20
625.2.e.a.374.2 8 25.8 odd 20
625.2.e.i.249.1 8 25.3 odd 20
625.2.e.i.374.1 8 25.17 odd 20
5625.2.a.x.1.4 8 3.2 odd 2
5625.2.a.x.1.5 8 15.14 odd 2
10000.2.a.bj.1.3 8 4.3 odd 2
10000.2.a.bj.1.6 8 20.19 odd 2