Properties

Label 5225.2.a.z.1.8
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 19x^{14} + 144x^{12} - 552x^{10} + 1119x^{8} - 1146x^{6} + 524x^{4} - 83x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-0.301154\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.301154 q^{2} -1.85377 q^{3} -1.90931 q^{4} +0.558272 q^{6} +1.94668 q^{7} +1.17730 q^{8} +0.436480 q^{9} +O(q^{10})\) \(q-0.301154 q^{2} -1.85377 q^{3} -1.90931 q^{4} +0.558272 q^{6} +1.94668 q^{7} +1.17730 q^{8} +0.436480 q^{9} -1.00000 q^{11} +3.53942 q^{12} -3.75832 q^{13} -0.586251 q^{14} +3.46406 q^{16} -2.16098 q^{17} -0.131448 q^{18} +1.00000 q^{19} -3.60871 q^{21} +0.301154 q^{22} +4.03747 q^{23} -2.18246 q^{24} +1.13183 q^{26} +4.75219 q^{27} -3.71681 q^{28} -2.18930 q^{29} +3.21529 q^{31} -3.39783 q^{32} +1.85377 q^{33} +0.650787 q^{34} -0.833374 q^{36} -5.64900 q^{37} -0.301154 q^{38} +6.96707 q^{39} -3.52926 q^{41} +1.08678 q^{42} +2.74094 q^{43} +1.90931 q^{44} -1.21590 q^{46} -1.87707 q^{47} -6.42159 q^{48} -3.21044 q^{49} +4.00596 q^{51} +7.17578 q^{52} +4.59639 q^{53} -1.43114 q^{54} +2.29184 q^{56} -1.85377 q^{57} +0.659317 q^{58} +3.10100 q^{59} -10.0971 q^{61} -0.968298 q^{62} +0.849687 q^{63} -5.90485 q^{64} -0.558272 q^{66} +12.0622 q^{67} +4.12596 q^{68} -7.48457 q^{69} +6.42317 q^{71} +0.513870 q^{72} +10.7452 q^{73} +1.70122 q^{74} -1.90931 q^{76} -1.94668 q^{77} -2.09816 q^{78} +10.8068 q^{79} -10.1189 q^{81} +1.06285 q^{82} +8.12591 q^{83} +6.89012 q^{84} -0.825447 q^{86} +4.05847 q^{87} -1.17730 q^{88} -11.1151 q^{89} -7.31624 q^{91} -7.70877 q^{92} -5.96042 q^{93} +0.565288 q^{94} +6.29880 q^{96} +11.4492 q^{97} +0.966837 q^{98} -0.436480 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{4} - 8 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{4} - 8 q^{6} + 8 q^{9} - 16 q^{11} - 4 q^{14} - 18 q^{16} + 16 q^{19} - 10 q^{21} - 10 q^{24} - 24 q^{26} - 2 q^{29} - 32 q^{31} + 16 q^{34} + 18 q^{36} - 40 q^{39} + 6 q^{41} - 6 q^{44} - 38 q^{49} - 16 q^{51} - 18 q^{54} + 12 q^{56} - 24 q^{59} - 42 q^{61} - 62 q^{64} + 8 q^{66} - 30 q^{69} - 46 q^{71} + 2 q^{74} + 6 q^{76} - 74 q^{79} - 56 q^{81} - 34 q^{84} + 8 q^{86} - 14 q^{89} - 24 q^{91} - 64 q^{94} + 54 q^{96} - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.301154 −0.212948 −0.106474 0.994315i \(-0.533956\pi\)
−0.106474 + 0.994315i \(0.533956\pi\)
\(3\) −1.85377 −1.07028 −0.535139 0.844764i \(-0.679741\pi\)
−0.535139 + 0.844764i \(0.679741\pi\)
\(4\) −1.90931 −0.954653
\(5\) 0 0
\(6\) 0.558272 0.227914
\(7\) 1.94668 0.735776 0.367888 0.929870i \(-0.380081\pi\)
0.367888 + 0.929870i \(0.380081\pi\)
\(8\) 1.17730 0.416240
\(9\) 0.436480 0.145493
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 3.53942 1.02174
\(13\) −3.75832 −1.04237 −0.521185 0.853444i \(-0.674510\pi\)
−0.521185 + 0.853444i \(0.674510\pi\)
\(14\) −0.586251 −0.156682
\(15\) 0 0
\(16\) 3.46406 0.866015
\(17\) −2.16098 −0.524114 −0.262057 0.965052i \(-0.584401\pi\)
−0.262057 + 0.965052i \(0.584401\pi\)
\(18\) −0.131448 −0.0309825
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −3.60871 −0.787484
\(22\) 0.301154 0.0642063
\(23\) 4.03747 0.841872 0.420936 0.907090i \(-0.361702\pi\)
0.420936 + 0.907090i \(0.361702\pi\)
\(24\) −2.18246 −0.445492
\(25\) 0 0
\(26\) 1.13183 0.221971
\(27\) 4.75219 0.914559
\(28\) −3.71681 −0.702411
\(29\) −2.18930 −0.406543 −0.203271 0.979122i \(-0.565157\pi\)
−0.203271 + 0.979122i \(0.565157\pi\)
\(30\) 0 0
\(31\) 3.21529 0.577483 0.288741 0.957407i \(-0.406763\pi\)
0.288741 + 0.957407i \(0.406763\pi\)
\(32\) −3.39783 −0.600656
\(33\) 1.85377 0.322701
\(34\) 0.650787 0.111609
\(35\) 0 0
\(36\) −0.833374 −0.138896
\(37\) −5.64900 −0.928690 −0.464345 0.885654i \(-0.653710\pi\)
−0.464345 + 0.885654i \(0.653710\pi\)
\(38\) −0.301154 −0.0488537
\(39\) 6.96707 1.11562
\(40\) 0 0
\(41\) −3.52926 −0.551177 −0.275589 0.961276i \(-0.588873\pi\)
−0.275589 + 0.961276i \(0.588873\pi\)
\(42\) 1.08678 0.167693
\(43\) 2.74094 0.417990 0.208995 0.977917i \(-0.432981\pi\)
0.208995 + 0.977917i \(0.432981\pi\)
\(44\) 1.90931 0.287839
\(45\) 0 0
\(46\) −1.21590 −0.179275
\(47\) −1.87707 −0.273799 −0.136899 0.990585i \(-0.543714\pi\)
−0.136899 + 0.990585i \(0.543714\pi\)
\(48\) −6.42159 −0.926877
\(49\) −3.21044 −0.458634
\(50\) 0 0
\(51\) 4.00596 0.560947
\(52\) 7.17578 0.995101
\(53\) 4.59639 0.631362 0.315681 0.948865i \(-0.397767\pi\)
0.315681 + 0.948865i \(0.397767\pi\)
\(54\) −1.43114 −0.194754
\(55\) 0 0
\(56\) 2.29184 0.306259
\(57\) −1.85377 −0.245538
\(58\) 0.659317 0.0865726
\(59\) 3.10100 0.403715 0.201858 0.979415i \(-0.435302\pi\)
0.201858 + 0.979415i \(0.435302\pi\)
\(60\) 0 0
\(61\) −10.0971 −1.29281 −0.646403 0.762996i \(-0.723728\pi\)
−0.646403 + 0.762996i \(0.723728\pi\)
\(62\) −0.968298 −0.122974
\(63\) 0.849687 0.107050
\(64\) −5.90485 −0.738107
\(65\) 0 0
\(66\) −0.558272 −0.0687186
\(67\) 12.0622 1.47363 0.736814 0.676096i \(-0.236329\pi\)
0.736814 + 0.676096i \(0.236329\pi\)
\(68\) 4.12596 0.500347
\(69\) −7.48457 −0.901036
\(70\) 0 0
\(71\) 6.42317 0.762291 0.381145 0.924515i \(-0.375530\pi\)
0.381145 + 0.924515i \(0.375530\pi\)
\(72\) 0.513870 0.0605601
\(73\) 10.7452 1.25763 0.628814 0.777556i \(-0.283541\pi\)
0.628814 + 0.777556i \(0.283541\pi\)
\(74\) 1.70122 0.197763
\(75\) 0 0
\(76\) −1.90931 −0.219012
\(77\) −1.94668 −0.221845
\(78\) −2.09816 −0.237570
\(79\) 10.8068 1.21586 0.607932 0.793989i \(-0.291999\pi\)
0.607932 + 0.793989i \(0.291999\pi\)
\(80\) 0 0
\(81\) −10.1189 −1.12432
\(82\) 1.06285 0.117372
\(83\) 8.12591 0.891934 0.445967 0.895049i \(-0.352860\pi\)
0.445967 + 0.895049i \(0.352860\pi\)
\(84\) 6.89012 0.751774
\(85\) 0 0
\(86\) −0.825447 −0.0890103
\(87\) 4.05847 0.435114
\(88\) −1.17730 −0.125501
\(89\) −11.1151 −1.17820 −0.589101 0.808060i \(-0.700518\pi\)
−0.589101 + 0.808060i \(0.700518\pi\)
\(90\) 0 0
\(91\) −7.31624 −0.766950
\(92\) −7.70877 −0.803695
\(93\) −5.96042 −0.618066
\(94\) 0.565288 0.0583050
\(95\) 0 0
\(96\) 6.29880 0.642869
\(97\) 11.4492 1.16249 0.581246 0.813728i \(-0.302565\pi\)
0.581246 + 0.813728i \(0.302565\pi\)
\(98\) 0.966837 0.0976653
\(99\) −0.436480 −0.0438679
\(100\) 0 0
\(101\) 6.86482 0.683075 0.341537 0.939868i \(-0.389052\pi\)
0.341537 + 0.939868i \(0.389052\pi\)
\(102\) −1.20641 −0.119453
\(103\) −13.7258 −1.35244 −0.676220 0.736699i \(-0.736383\pi\)
−0.676220 + 0.736699i \(0.736383\pi\)
\(104\) −4.42468 −0.433876
\(105\) 0 0
\(106\) −1.38422 −0.134447
\(107\) 14.8545 1.43603 0.718017 0.696025i \(-0.245050\pi\)
0.718017 + 0.696025i \(0.245050\pi\)
\(108\) −9.07338 −0.873087
\(109\) −7.13713 −0.683613 −0.341806 0.939770i \(-0.611039\pi\)
−0.341806 + 0.939770i \(0.611039\pi\)
\(110\) 0 0
\(111\) 10.4720 0.993956
\(112\) 6.74342 0.637193
\(113\) −1.22524 −0.115261 −0.0576303 0.998338i \(-0.518354\pi\)
−0.0576303 + 0.998338i \(0.518354\pi\)
\(114\) 0.558272 0.0522870
\(115\) 0 0
\(116\) 4.18004 0.388107
\(117\) −1.64043 −0.151658
\(118\) −0.933879 −0.0859705
\(119\) −4.20673 −0.385630
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 3.04080 0.275301
\(123\) 6.54245 0.589913
\(124\) −6.13897 −0.551296
\(125\) 0 0
\(126\) −0.255887 −0.0227962
\(127\) −1.78182 −0.158111 −0.0790554 0.996870i \(-0.525190\pi\)
−0.0790554 + 0.996870i \(0.525190\pi\)
\(128\) 8.57392 0.757835
\(129\) −5.08109 −0.447365
\(130\) 0 0
\(131\) 18.4204 1.60939 0.804697 0.593685i \(-0.202328\pi\)
0.804697 + 0.593685i \(0.202328\pi\)
\(132\) −3.53942 −0.308067
\(133\) 1.94668 0.168799
\(134\) −3.63257 −0.313807
\(135\) 0 0
\(136\) −2.54413 −0.218157
\(137\) −13.8926 −1.18693 −0.593463 0.804861i \(-0.702240\pi\)
−0.593463 + 0.804861i \(0.702240\pi\)
\(138\) 2.25401 0.191874
\(139\) −4.35165 −0.369102 −0.184551 0.982823i \(-0.559083\pi\)
−0.184551 + 0.982823i \(0.559083\pi\)
\(140\) 0 0
\(141\) 3.47967 0.293041
\(142\) −1.93437 −0.162328
\(143\) 3.75832 0.314286
\(144\) 1.51199 0.125999
\(145\) 0 0
\(146\) −3.23596 −0.267810
\(147\) 5.95142 0.490865
\(148\) 10.7857 0.886577
\(149\) −11.6917 −0.957818 −0.478909 0.877864i \(-0.658968\pi\)
−0.478909 + 0.877864i \(0.658968\pi\)
\(150\) 0 0
\(151\) −23.0439 −1.87529 −0.937644 0.347596i \(-0.886998\pi\)
−0.937644 + 0.347596i \(0.886998\pi\)
\(152\) 1.17730 0.0954920
\(153\) −0.943222 −0.0762550
\(154\) 0.586251 0.0472415
\(155\) 0 0
\(156\) −13.3023 −1.06503
\(157\) 12.2578 0.978277 0.489138 0.872206i \(-0.337311\pi\)
0.489138 + 0.872206i \(0.337311\pi\)
\(158\) −3.25452 −0.258916
\(159\) −8.52066 −0.675732
\(160\) 0 0
\(161\) 7.85967 0.619429
\(162\) 3.04736 0.239423
\(163\) 8.00570 0.627055 0.313527 0.949579i \(-0.398489\pi\)
0.313527 + 0.949579i \(0.398489\pi\)
\(164\) 6.73843 0.526183
\(165\) 0 0
\(166\) −2.44715 −0.189936
\(167\) 5.00463 0.387270 0.193635 0.981074i \(-0.437972\pi\)
0.193635 + 0.981074i \(0.437972\pi\)
\(168\) −4.24855 −0.327782
\(169\) 1.12494 0.0865341
\(170\) 0 0
\(171\) 0.436480 0.0333784
\(172\) −5.23330 −0.399035
\(173\) 7.15833 0.544237 0.272119 0.962264i \(-0.412276\pi\)
0.272119 + 0.962264i \(0.412276\pi\)
\(174\) −1.22223 −0.0926567
\(175\) 0 0
\(176\) −3.46406 −0.261113
\(177\) −5.74855 −0.432087
\(178\) 3.34737 0.250896
\(179\) −4.12217 −0.308106 −0.154053 0.988063i \(-0.549233\pi\)
−0.154053 + 0.988063i \(0.549233\pi\)
\(180\) 0 0
\(181\) 17.5288 1.30291 0.651454 0.758688i \(-0.274159\pi\)
0.651454 + 0.758688i \(0.274159\pi\)
\(182\) 2.20332 0.163321
\(183\) 18.7178 1.38366
\(184\) 4.75334 0.350421
\(185\) 0 0
\(186\) 1.79501 0.131616
\(187\) 2.16098 0.158026
\(188\) 3.58390 0.261383
\(189\) 9.25099 0.672911
\(190\) 0 0
\(191\) −27.5080 −1.99041 −0.995205 0.0978155i \(-0.968814\pi\)
−0.995205 + 0.0978155i \(0.968814\pi\)
\(192\) 10.9463 0.789979
\(193\) 13.1923 0.949603 0.474802 0.880093i \(-0.342520\pi\)
0.474802 + 0.880093i \(0.342520\pi\)
\(194\) −3.44798 −0.247551
\(195\) 0 0
\(196\) 6.12970 0.437836
\(197\) −11.4532 −0.816006 −0.408003 0.912981i \(-0.633775\pi\)
−0.408003 + 0.912981i \(0.633775\pi\)
\(198\) 0.131448 0.00934159
\(199\) −19.7562 −1.40048 −0.700240 0.713907i \(-0.746924\pi\)
−0.700240 + 0.713907i \(0.746924\pi\)
\(200\) 0 0
\(201\) −22.3605 −1.57719
\(202\) −2.06737 −0.145460
\(203\) −4.26187 −0.299124
\(204\) −7.64861 −0.535510
\(205\) 0 0
\(206\) 4.13358 0.288000
\(207\) 1.76228 0.122487
\(208\) −13.0190 −0.902708
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −22.4594 −1.54617 −0.773085 0.634303i \(-0.781287\pi\)
−0.773085 + 0.634303i \(0.781287\pi\)
\(212\) −8.77591 −0.602732
\(213\) −11.9071 −0.815862
\(214\) −4.47349 −0.305801
\(215\) 0 0
\(216\) 5.59477 0.380676
\(217\) 6.25914 0.424898
\(218\) 2.14938 0.145574
\(219\) −19.9191 −1.34601
\(220\) 0 0
\(221\) 8.12163 0.546320
\(222\) −3.15368 −0.211661
\(223\) 10.3113 0.690499 0.345249 0.938511i \(-0.387794\pi\)
0.345249 + 0.938511i \(0.387794\pi\)
\(224\) −6.61448 −0.441949
\(225\) 0 0
\(226\) 0.368986 0.0245446
\(227\) −19.9756 −1.32583 −0.662914 0.748696i \(-0.730680\pi\)
−0.662914 + 0.748696i \(0.730680\pi\)
\(228\) 3.53942 0.234404
\(229\) −15.7027 −1.03767 −0.518833 0.854876i \(-0.673633\pi\)
−0.518833 + 0.854876i \(0.673633\pi\)
\(230\) 0 0
\(231\) 3.60871 0.237435
\(232\) −2.57747 −0.169219
\(233\) 21.1823 1.38770 0.693850 0.720120i \(-0.255913\pi\)
0.693850 + 0.720120i \(0.255913\pi\)
\(234\) 0.494022 0.0322953
\(235\) 0 0
\(236\) −5.92075 −0.385408
\(237\) −20.0334 −1.30131
\(238\) 1.26687 0.0821193
\(239\) −0.429031 −0.0277517 −0.0138759 0.999904i \(-0.504417\pi\)
−0.0138759 + 0.999904i \(0.504417\pi\)
\(240\) 0 0
\(241\) 11.9657 0.770781 0.385391 0.922753i \(-0.374067\pi\)
0.385391 + 0.922753i \(0.374067\pi\)
\(242\) −0.301154 −0.0193589
\(243\) 4.50164 0.288780
\(244\) 19.2785 1.23418
\(245\) 0 0
\(246\) −1.97029 −0.125621
\(247\) −3.75832 −0.239136
\(248\) 3.78537 0.240371
\(249\) −15.0636 −0.954617
\(250\) 0 0
\(251\) 7.13854 0.450581 0.225290 0.974292i \(-0.427667\pi\)
0.225290 + 0.974292i \(0.427667\pi\)
\(252\) −1.62231 −0.102196
\(253\) −4.03747 −0.253834
\(254\) 0.536602 0.0336694
\(255\) 0 0
\(256\) 9.22763 0.576727
\(257\) −6.86772 −0.428397 −0.214198 0.976790i \(-0.568714\pi\)
−0.214198 + 0.976790i \(0.568714\pi\)
\(258\) 1.53019 0.0952657
\(259\) −10.9968 −0.683308
\(260\) 0 0
\(261\) −0.955585 −0.0591492
\(262\) −5.54737 −0.342718
\(263\) −0.473180 −0.0291775 −0.0145888 0.999894i \(-0.504644\pi\)
−0.0145888 + 0.999894i \(0.504644\pi\)
\(264\) 2.18246 0.134321
\(265\) 0 0
\(266\) −0.586251 −0.0359454
\(267\) 20.6049 1.26100
\(268\) −23.0304 −1.40680
\(269\) 2.26859 0.138318 0.0691591 0.997606i \(-0.477968\pi\)
0.0691591 + 0.997606i \(0.477968\pi\)
\(270\) 0 0
\(271\) −13.6384 −0.828474 −0.414237 0.910169i \(-0.635952\pi\)
−0.414237 + 0.910169i \(0.635952\pi\)
\(272\) −7.48575 −0.453891
\(273\) 13.5627 0.820850
\(274\) 4.18382 0.252754
\(275\) 0 0
\(276\) 14.2903 0.860177
\(277\) −1.64693 −0.0989544 −0.0494772 0.998775i \(-0.515756\pi\)
−0.0494772 + 0.998775i \(0.515756\pi\)
\(278\) 1.31052 0.0785997
\(279\) 1.40341 0.0840198
\(280\) 0 0
\(281\) 16.7390 0.998565 0.499282 0.866439i \(-0.333597\pi\)
0.499282 + 0.866439i \(0.333597\pi\)
\(282\) −1.04792 −0.0624025
\(283\) −9.91741 −0.589529 −0.294764 0.955570i \(-0.595241\pi\)
−0.294764 + 0.955570i \(0.595241\pi\)
\(284\) −12.2638 −0.727723
\(285\) 0 0
\(286\) −1.13183 −0.0669267
\(287\) −6.87034 −0.405543
\(288\) −1.48308 −0.0873915
\(289\) −12.3302 −0.725305
\(290\) 0 0
\(291\) −21.2243 −1.24419
\(292\) −20.5158 −1.20060
\(293\) −14.1780 −0.828289 −0.414144 0.910211i \(-0.635919\pi\)
−0.414144 + 0.910211i \(0.635919\pi\)
\(294\) −1.79230 −0.104529
\(295\) 0 0
\(296\) −6.65059 −0.386558
\(297\) −4.75219 −0.275750
\(298\) 3.52100 0.203966
\(299\) −15.1741 −0.877541
\(300\) 0 0
\(301\) 5.33574 0.307547
\(302\) 6.93978 0.399340
\(303\) −12.7258 −0.731079
\(304\) 3.46406 0.198678
\(305\) 0 0
\(306\) 0.284056 0.0162384
\(307\) −7.49870 −0.427974 −0.213987 0.976837i \(-0.568645\pi\)
−0.213987 + 0.976837i \(0.568645\pi\)
\(308\) 3.71681 0.211785
\(309\) 25.4445 1.44749
\(310\) 0 0
\(311\) −22.6904 −1.28666 −0.643328 0.765591i \(-0.722447\pi\)
−0.643328 + 0.765591i \(0.722447\pi\)
\(312\) 8.20236 0.464367
\(313\) −16.3070 −0.921723 −0.460862 0.887472i \(-0.652460\pi\)
−0.460862 + 0.887472i \(0.652460\pi\)
\(314\) −3.69148 −0.208322
\(315\) 0 0
\(316\) −20.6336 −1.16073
\(317\) −26.5837 −1.49309 −0.746544 0.665336i \(-0.768288\pi\)
−0.746544 + 0.665336i \(0.768288\pi\)
\(318\) 2.56603 0.143896
\(319\) 2.18930 0.122577
\(320\) 0 0
\(321\) −27.5368 −1.53696
\(322\) −2.36697 −0.131906
\(323\) −2.16098 −0.120240
\(324\) 19.3201 1.07334
\(325\) 0 0
\(326\) −2.41095 −0.133530
\(327\) 13.2306 0.731655
\(328\) −4.15501 −0.229422
\(329\) −3.65406 −0.201455
\(330\) 0 0
\(331\) 12.6060 0.692887 0.346444 0.938071i \(-0.387389\pi\)
0.346444 + 0.938071i \(0.387389\pi\)
\(332\) −15.5148 −0.851487
\(333\) −2.46567 −0.135118
\(334\) −1.50717 −0.0824685
\(335\) 0 0
\(336\) −12.5008 −0.681974
\(337\) −6.18984 −0.337182 −0.168591 0.985686i \(-0.553922\pi\)
−0.168591 + 0.985686i \(0.553922\pi\)
\(338\) −0.338782 −0.0184273
\(339\) 2.27131 0.123361
\(340\) 0 0
\(341\) −3.21529 −0.174118
\(342\) −0.131448 −0.00710788
\(343\) −19.8765 −1.07323
\(344\) 3.22693 0.173984
\(345\) 0 0
\(346\) −2.15576 −0.115894
\(347\) −12.5724 −0.674921 −0.337460 0.941340i \(-0.609568\pi\)
−0.337460 + 0.941340i \(0.609568\pi\)
\(348\) −7.74886 −0.415382
\(349\) 17.0337 0.911793 0.455897 0.890033i \(-0.349319\pi\)
0.455897 + 0.890033i \(0.349319\pi\)
\(350\) 0 0
\(351\) −17.8602 −0.953308
\(352\) 3.39783 0.181105
\(353\) −8.51838 −0.453387 −0.226694 0.973966i \(-0.572792\pi\)
−0.226694 + 0.973966i \(0.572792\pi\)
\(354\) 1.73120 0.0920123
\(355\) 0 0
\(356\) 21.2222 1.12477
\(357\) 7.79833 0.412731
\(358\) 1.24141 0.0656106
\(359\) −6.43010 −0.339367 −0.169684 0.985499i \(-0.554275\pi\)
−0.169684 + 0.985499i \(0.554275\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −5.27889 −0.277452
\(363\) −1.85377 −0.0972979
\(364\) 13.9689 0.732172
\(365\) 0 0
\(366\) −5.63696 −0.294648
\(367\) −29.9633 −1.56407 −0.782037 0.623233i \(-0.785819\pi\)
−0.782037 + 0.623233i \(0.785819\pi\)
\(368\) 13.9861 0.729074
\(369\) −1.54045 −0.0801926
\(370\) 0 0
\(371\) 8.94769 0.464541
\(372\) 11.3803 0.590039
\(373\) −34.6611 −1.79469 −0.897343 0.441334i \(-0.854505\pi\)
−0.897343 + 0.441334i \(0.854505\pi\)
\(374\) −0.650787 −0.0336514
\(375\) 0 0
\(376\) −2.20988 −0.113966
\(377\) 8.22808 0.423768
\(378\) −2.78598 −0.143295
\(379\) 1.82041 0.0935083 0.0467542 0.998906i \(-0.485112\pi\)
0.0467542 + 0.998906i \(0.485112\pi\)
\(380\) 0 0
\(381\) 3.30309 0.169222
\(382\) 8.28415 0.423854
\(383\) 17.0353 0.870463 0.435232 0.900319i \(-0.356666\pi\)
0.435232 + 0.900319i \(0.356666\pi\)
\(384\) −15.8941 −0.811094
\(385\) 0 0
\(386\) −3.97292 −0.202216
\(387\) 1.19637 0.0608147
\(388\) −21.8601 −1.10978
\(389\) −21.7778 −1.10418 −0.552090 0.833784i \(-0.686170\pi\)
−0.552090 + 0.833784i \(0.686170\pi\)
\(390\) 0 0
\(391\) −8.72488 −0.441236
\(392\) −3.77966 −0.190902
\(393\) −34.1472 −1.72250
\(394\) 3.44918 0.173767
\(395\) 0 0
\(396\) 0.833374 0.0418786
\(397\) 8.81816 0.442571 0.221285 0.975209i \(-0.428975\pi\)
0.221285 + 0.975209i \(0.428975\pi\)
\(398\) 5.94967 0.298230
\(399\) −3.60871 −0.180661
\(400\) 0 0
\(401\) −11.0380 −0.551210 −0.275605 0.961271i \(-0.588878\pi\)
−0.275605 + 0.961271i \(0.588878\pi\)
\(402\) 6.73397 0.335860
\(403\) −12.0841 −0.601950
\(404\) −13.1070 −0.652099
\(405\) 0 0
\(406\) 1.28348 0.0636980
\(407\) 5.64900 0.280011
\(408\) 4.71624 0.233489
\(409\) −25.9583 −1.28355 −0.641777 0.766891i \(-0.721803\pi\)
−0.641777 + 0.766891i \(0.721803\pi\)
\(410\) 0 0
\(411\) 25.7538 1.27034
\(412\) 26.2067 1.29111
\(413\) 6.03665 0.297044
\(414\) −0.530717 −0.0260833
\(415\) 0 0
\(416\) 12.7701 0.626106
\(417\) 8.06698 0.395042
\(418\) 0.301154 0.0147299
\(419\) 8.14823 0.398067 0.199034 0.979993i \(-0.436220\pi\)
0.199034 + 0.979993i \(0.436220\pi\)
\(420\) 0 0
\(421\) −14.7467 −0.718711 −0.359356 0.933201i \(-0.617003\pi\)
−0.359356 + 0.933201i \(0.617003\pi\)
\(422\) 6.76375 0.329254
\(423\) −0.819304 −0.0398359
\(424\) 5.41134 0.262798
\(425\) 0 0
\(426\) 3.58588 0.173736
\(427\) −19.6559 −0.951216
\(428\) −28.3617 −1.37092
\(429\) −6.96707 −0.336373
\(430\) 0 0
\(431\) −28.0178 −1.34957 −0.674785 0.738014i \(-0.735764\pi\)
−0.674785 + 0.738014i \(0.735764\pi\)
\(432\) 16.4619 0.792022
\(433\) −2.14430 −0.103049 −0.0515243 0.998672i \(-0.516408\pi\)
−0.0515243 + 0.998672i \(0.516408\pi\)
\(434\) −1.88497 −0.0904813
\(435\) 0 0
\(436\) 13.6270 0.652613
\(437\) 4.03747 0.193139
\(438\) 5.99873 0.286631
\(439\) 4.98343 0.237846 0.118923 0.992903i \(-0.462056\pi\)
0.118923 + 0.992903i \(0.462056\pi\)
\(440\) 0 0
\(441\) −1.40129 −0.0667281
\(442\) −2.44586 −0.116338
\(443\) −10.3754 −0.492952 −0.246476 0.969149i \(-0.579273\pi\)
−0.246476 + 0.969149i \(0.579273\pi\)
\(444\) −19.9942 −0.948883
\(445\) 0 0
\(446\) −3.10531 −0.147041
\(447\) 21.6737 1.02513
\(448\) −11.4949 −0.543081
\(449\) 35.9683 1.69745 0.848725 0.528835i \(-0.177371\pi\)
0.848725 + 0.528835i \(0.177371\pi\)
\(450\) 0 0
\(451\) 3.52926 0.166186
\(452\) 2.33935 0.110034
\(453\) 42.7183 2.00708
\(454\) 6.01574 0.282333
\(455\) 0 0
\(456\) −2.18246 −0.102203
\(457\) −27.3842 −1.28098 −0.640490 0.767966i \(-0.721269\pi\)
−0.640490 + 0.767966i \(0.721269\pi\)
\(458\) 4.72895 0.220969
\(459\) −10.2694 −0.479333
\(460\) 0 0
\(461\) −9.58866 −0.446589 −0.223294 0.974751i \(-0.571681\pi\)
−0.223294 + 0.974751i \(0.571681\pi\)
\(462\) −1.08678 −0.0505615
\(463\) −27.4154 −1.27410 −0.637050 0.770822i \(-0.719846\pi\)
−0.637050 + 0.770822i \(0.719846\pi\)
\(464\) −7.58387 −0.352072
\(465\) 0 0
\(466\) −6.37915 −0.295508
\(467\) 30.6386 1.41778 0.708892 0.705317i \(-0.249195\pi\)
0.708892 + 0.705317i \(0.249195\pi\)
\(468\) 3.13208 0.144781
\(469\) 23.4812 1.08426
\(470\) 0 0
\(471\) −22.7231 −1.04703
\(472\) 3.65082 0.168043
\(473\) −2.74094 −0.126029
\(474\) 6.03315 0.277112
\(475\) 0 0
\(476\) 8.03193 0.368143
\(477\) 2.00623 0.0918589
\(478\) 0.129205 0.00590968
\(479\) 5.55489 0.253809 0.126905 0.991915i \(-0.459496\pi\)
0.126905 + 0.991915i \(0.459496\pi\)
\(480\) 0 0
\(481\) 21.2307 0.968038
\(482\) −3.60354 −0.164137
\(483\) −14.5701 −0.662961
\(484\) −1.90931 −0.0867866
\(485\) 0 0
\(486\) −1.35569 −0.0614953
\(487\) 15.4839 0.701642 0.350821 0.936442i \(-0.385902\pi\)
0.350821 + 0.936442i \(0.385902\pi\)
\(488\) −11.8874 −0.538118
\(489\) −14.8408 −0.671122
\(490\) 0 0
\(491\) −25.8915 −1.16847 −0.584233 0.811586i \(-0.698605\pi\)
−0.584233 + 0.811586i \(0.698605\pi\)
\(492\) −12.4915 −0.563162
\(493\) 4.73103 0.213075
\(494\) 1.13183 0.0509236
\(495\) 0 0
\(496\) 11.1380 0.500109
\(497\) 12.5039 0.560875
\(498\) 4.53647 0.203284
\(499\) −33.9769 −1.52102 −0.760508 0.649329i \(-0.775050\pi\)
−0.760508 + 0.649329i \(0.775050\pi\)
\(500\) 0 0
\(501\) −9.27746 −0.414486
\(502\) −2.14980 −0.0959504
\(503\) −11.0065 −0.490756 −0.245378 0.969428i \(-0.578912\pi\)
−0.245378 + 0.969428i \(0.578912\pi\)
\(504\) 1.00034 0.0445587
\(505\) 0 0
\(506\) 1.21590 0.0540535
\(507\) −2.08539 −0.0926155
\(508\) 3.40204 0.150941
\(509\) −20.0893 −0.890443 −0.445221 0.895420i \(-0.646875\pi\)
−0.445221 + 0.895420i \(0.646875\pi\)
\(510\) 0 0
\(511\) 20.9174 0.925332
\(512\) −19.9268 −0.880648
\(513\) 4.75219 0.209814
\(514\) 2.06824 0.0912264
\(515\) 0 0
\(516\) 9.70136 0.427079
\(517\) 1.87707 0.0825535
\(518\) 3.31173 0.145509
\(519\) −13.2699 −0.582485
\(520\) 0 0
\(521\) −5.13542 −0.224987 −0.112493 0.993652i \(-0.535884\pi\)
−0.112493 + 0.993652i \(0.535884\pi\)
\(522\) 0.287779 0.0125957
\(523\) 11.4021 0.498578 0.249289 0.968429i \(-0.419803\pi\)
0.249289 + 0.968429i \(0.419803\pi\)
\(524\) −35.1701 −1.53641
\(525\) 0 0
\(526\) 0.142500 0.00621330
\(527\) −6.94816 −0.302667
\(528\) 6.42159 0.279464
\(529\) −6.69880 −0.291252
\(530\) 0 0
\(531\) 1.35352 0.0587379
\(532\) −3.71681 −0.161144
\(533\) 13.2641 0.574530
\(534\) −6.20527 −0.268528
\(535\) 0 0
\(536\) 14.2008 0.613383
\(537\) 7.64158 0.329758
\(538\) −0.683195 −0.0294546
\(539\) 3.21044 0.138283
\(540\) 0 0
\(541\) 31.1731 1.34024 0.670118 0.742254i \(-0.266243\pi\)
0.670118 + 0.742254i \(0.266243\pi\)
\(542\) 4.10726 0.176422
\(543\) −32.4945 −1.39447
\(544\) 7.34262 0.314812
\(545\) 0 0
\(546\) −4.08445 −0.174798
\(547\) 36.3618 1.55472 0.777360 0.629056i \(-0.216558\pi\)
0.777360 + 0.629056i \(0.216558\pi\)
\(548\) 26.5252 1.13310
\(549\) −4.40720 −0.188095
\(550\) 0 0
\(551\) −2.18930 −0.0932673
\(552\) −8.81161 −0.375047
\(553\) 21.0374 0.894603
\(554\) 0.495980 0.0210722
\(555\) 0 0
\(556\) 8.30863 0.352364
\(557\) 0.847753 0.0359204 0.0179602 0.999839i \(-0.494283\pi\)
0.0179602 + 0.999839i \(0.494283\pi\)
\(558\) −0.422642 −0.0178919
\(559\) −10.3013 −0.435700
\(560\) 0 0
\(561\) −4.00596 −0.169132
\(562\) −5.04102 −0.212643
\(563\) −39.5533 −1.66697 −0.833486 0.552541i \(-0.813658\pi\)
−0.833486 + 0.552541i \(0.813658\pi\)
\(564\) −6.64375 −0.279752
\(565\) 0 0
\(566\) 2.98667 0.125539
\(567\) −19.6983 −0.827251
\(568\) 7.56203 0.317296
\(569\) 18.3932 0.771082 0.385541 0.922691i \(-0.374015\pi\)
0.385541 + 0.922691i \(0.374015\pi\)
\(570\) 0 0
\(571\) 33.4442 1.39959 0.699797 0.714341i \(-0.253274\pi\)
0.699797 + 0.714341i \(0.253274\pi\)
\(572\) −7.17578 −0.300034
\(573\) 50.9936 2.13029
\(574\) 2.06903 0.0863597
\(575\) 0 0
\(576\) −2.57735 −0.107390
\(577\) −32.8041 −1.36565 −0.682826 0.730581i \(-0.739249\pi\)
−0.682826 + 0.730581i \(0.739249\pi\)
\(578\) 3.71329 0.154452
\(579\) −24.4556 −1.01634
\(580\) 0 0
\(581\) 15.8185 0.656264
\(582\) 6.39178 0.264948
\(583\) −4.59639 −0.190363
\(584\) 12.6503 0.523475
\(585\) 0 0
\(586\) 4.26977 0.176383
\(587\) 26.2752 1.08449 0.542246 0.840220i \(-0.317574\pi\)
0.542246 + 0.840220i \(0.317574\pi\)
\(588\) −11.3631 −0.468606
\(589\) 3.21529 0.132484
\(590\) 0 0
\(591\) 21.2316 0.873353
\(592\) −19.5685 −0.804260
\(593\) −19.5147 −0.801371 −0.400686 0.916216i \(-0.631228\pi\)
−0.400686 + 0.916216i \(0.631228\pi\)
\(594\) 1.43114 0.0587205
\(595\) 0 0
\(596\) 22.3230 0.914384
\(597\) 36.6236 1.49890
\(598\) 4.56975 0.186871
\(599\) 15.9690 0.652476 0.326238 0.945288i \(-0.394219\pi\)
0.326238 + 0.945288i \(0.394219\pi\)
\(600\) 0 0
\(601\) −18.0313 −0.735510 −0.367755 0.929923i \(-0.619874\pi\)
−0.367755 + 0.929923i \(0.619874\pi\)
\(602\) −1.60688 −0.0654916
\(603\) 5.26489 0.214403
\(604\) 43.9979 1.79025
\(605\) 0 0
\(606\) 3.83244 0.155682
\(607\) 9.02312 0.366237 0.183119 0.983091i \(-0.441381\pi\)
0.183119 + 0.983091i \(0.441381\pi\)
\(608\) −3.39783 −0.137800
\(609\) 7.90054 0.320146
\(610\) 0 0
\(611\) 7.05463 0.285400
\(612\) 1.80090 0.0727971
\(613\) −35.9969 −1.45390 −0.726950 0.686690i \(-0.759063\pi\)
−0.726950 + 0.686690i \(0.759063\pi\)
\(614\) 2.25827 0.0911362
\(615\) 0 0
\(616\) −2.29184 −0.0923407
\(617\) −44.8507 −1.80562 −0.902811 0.430037i \(-0.858501\pi\)
−0.902811 + 0.430037i \(0.858501\pi\)
\(618\) −7.66272 −0.308240
\(619\) 36.6485 1.47303 0.736514 0.676422i \(-0.236470\pi\)
0.736514 + 0.676422i \(0.236470\pi\)
\(620\) 0 0
\(621\) 19.1868 0.769941
\(622\) 6.83331 0.273991
\(623\) −21.6376 −0.866892
\(624\) 24.1344 0.966148
\(625\) 0 0
\(626\) 4.91091 0.196279
\(627\) 1.85377 0.0740326
\(628\) −23.4038 −0.933915
\(629\) 12.2074 0.486739
\(630\) 0 0
\(631\) −39.4249 −1.56948 −0.784739 0.619826i \(-0.787203\pi\)
−0.784739 + 0.619826i \(0.787203\pi\)
\(632\) 12.7229 0.506091
\(633\) 41.6347 1.65483
\(634\) 8.00579 0.317950
\(635\) 0 0
\(636\) 16.2686 0.645090
\(637\) 12.0658 0.478066
\(638\) −0.659317 −0.0261026
\(639\) 2.80359 0.110908
\(640\) 0 0
\(641\) −6.91478 −0.273117 −0.136559 0.990632i \(-0.543604\pi\)
−0.136559 + 0.990632i \(0.543604\pi\)
\(642\) 8.29283 0.327292
\(643\) 38.8033 1.53025 0.765127 0.643879i \(-0.222676\pi\)
0.765127 + 0.643879i \(0.222676\pi\)
\(644\) −15.0065 −0.591340
\(645\) 0 0
\(646\) 0.650787 0.0256049
\(647\) −18.4537 −0.725490 −0.362745 0.931888i \(-0.618160\pi\)
−0.362745 + 0.931888i \(0.618160\pi\)
\(648\) −11.9131 −0.467989
\(649\) −3.10100 −0.121725
\(650\) 0 0
\(651\) −11.6030 −0.454758
\(652\) −15.2853 −0.598620
\(653\) 12.6249 0.494052 0.247026 0.969009i \(-0.420547\pi\)
0.247026 + 0.969009i \(0.420547\pi\)
\(654\) −3.98446 −0.155805
\(655\) 0 0
\(656\) −12.2256 −0.477328
\(657\) 4.69005 0.182976
\(658\) 1.10044 0.0428994
\(659\) 37.4031 1.45702 0.728509 0.685036i \(-0.240214\pi\)
0.728509 + 0.685036i \(0.240214\pi\)
\(660\) 0 0
\(661\) −12.6089 −0.490431 −0.245215 0.969469i \(-0.578859\pi\)
−0.245215 + 0.969469i \(0.578859\pi\)
\(662\) −3.79634 −0.147549
\(663\) −15.0557 −0.584714
\(664\) 9.56666 0.371259
\(665\) 0 0
\(666\) 0.742549 0.0287732
\(667\) −8.83924 −0.342257
\(668\) −9.55538 −0.369709
\(669\) −19.1149 −0.739025
\(670\) 0 0
\(671\) 10.0971 0.389796
\(672\) 12.2618 0.473008
\(673\) −6.19541 −0.238815 −0.119408 0.992845i \(-0.538100\pi\)
−0.119408 + 0.992845i \(0.538100\pi\)
\(674\) 1.86410 0.0718023
\(675\) 0 0
\(676\) −2.14786 −0.0826101
\(677\) 26.5020 1.01855 0.509276 0.860603i \(-0.329913\pi\)
0.509276 + 0.860603i \(0.329913\pi\)
\(678\) −0.684016 −0.0262695
\(679\) 22.2880 0.855334
\(680\) 0 0
\(681\) 37.0303 1.41900
\(682\) 0.968298 0.0370780
\(683\) 44.3957 1.69876 0.849378 0.527786i \(-0.176978\pi\)
0.849378 + 0.527786i \(0.176978\pi\)
\(684\) −0.833374 −0.0318648
\(685\) 0 0
\(686\) 5.98588 0.228542
\(687\) 29.1093 1.11059
\(688\) 9.49480 0.361986
\(689\) −17.2747 −0.658113
\(690\) 0 0
\(691\) −15.0873 −0.573948 −0.286974 0.957938i \(-0.592649\pi\)
−0.286974 + 0.957938i \(0.592649\pi\)
\(692\) −13.6674 −0.519558
\(693\) −0.849687 −0.0322769
\(694\) 3.78623 0.143723
\(695\) 0 0
\(696\) 4.77805 0.181112
\(697\) 7.62664 0.288880
\(698\) −5.12978 −0.194165
\(699\) −39.2672 −1.48522
\(700\) 0 0
\(701\) 8.66946 0.327441 0.163720 0.986507i \(-0.447650\pi\)
0.163720 + 0.986507i \(0.447650\pi\)
\(702\) 5.37869 0.203005
\(703\) −5.64900 −0.213056
\(704\) 5.90485 0.222548
\(705\) 0 0
\(706\) 2.56535 0.0965481
\(707\) 13.3636 0.502590
\(708\) 10.9757 0.412494
\(709\) 42.7639 1.60603 0.803015 0.595958i \(-0.203228\pi\)
0.803015 + 0.595958i \(0.203228\pi\)
\(710\) 0 0
\(711\) 4.71696 0.176900
\(712\) −13.0859 −0.490415
\(713\) 12.9816 0.486166
\(714\) −2.34850 −0.0878904
\(715\) 0 0
\(716\) 7.87049 0.294134
\(717\) 0.795327 0.0297020
\(718\) 1.93645 0.0722677
\(719\) 27.6655 1.03175 0.515875 0.856664i \(-0.327467\pi\)
0.515875 + 0.856664i \(0.327467\pi\)
\(720\) 0 0
\(721\) −26.7197 −0.995093
\(722\) −0.301154 −0.0112078
\(723\) −22.1818 −0.824950
\(724\) −33.4679 −1.24383
\(725\) 0 0
\(726\) 0.558272 0.0207194
\(727\) 20.9256 0.776088 0.388044 0.921641i \(-0.373151\pi\)
0.388044 + 0.921641i \(0.373151\pi\)
\(728\) −8.61344 −0.319235
\(729\) 22.0118 0.815250
\(730\) 0 0
\(731\) −5.92311 −0.219074
\(732\) −35.7381 −1.32092
\(733\) 27.8171 1.02745 0.513723 0.857956i \(-0.328266\pi\)
0.513723 + 0.857956i \(0.328266\pi\)
\(734\) 9.02359 0.333067
\(735\) 0 0
\(736\) −13.7186 −0.505676
\(737\) −12.0622 −0.444316
\(738\) 0.463913 0.0170769
\(739\) 9.45503 0.347809 0.173904 0.984763i \(-0.444362\pi\)
0.173904 + 0.984763i \(0.444362\pi\)
\(740\) 0 0
\(741\) 6.96707 0.255942
\(742\) −2.69464 −0.0989232
\(743\) −5.01586 −0.184014 −0.0920070 0.995758i \(-0.529328\pi\)
−0.0920070 + 0.995758i \(0.529328\pi\)
\(744\) −7.01723 −0.257264
\(745\) 0 0
\(746\) 10.4384 0.382175
\(747\) 3.54679 0.129770
\(748\) −4.12596 −0.150860
\(749\) 28.9169 1.05660
\(750\) 0 0
\(751\) −42.5515 −1.55273 −0.776363 0.630286i \(-0.782938\pi\)
−0.776363 + 0.630286i \(0.782938\pi\)
\(752\) −6.50229 −0.237114
\(753\) −13.2332 −0.482246
\(754\) −2.47792 −0.0902406
\(755\) 0 0
\(756\) −17.6630 −0.642396
\(757\) 6.77125 0.246105 0.123053 0.992400i \(-0.460732\pi\)
0.123053 + 0.992400i \(0.460732\pi\)
\(758\) −0.548225 −0.0199124
\(759\) 7.48457 0.271673
\(760\) 0 0
\(761\) 31.5134 1.14236 0.571180 0.820825i \(-0.306486\pi\)
0.571180 + 0.820825i \(0.306486\pi\)
\(762\) −0.994739 −0.0360356
\(763\) −13.8937 −0.502986
\(764\) 52.5212 1.90015
\(765\) 0 0
\(766\) −5.13025 −0.185364
\(767\) −11.6545 −0.420821
\(768\) −17.1059 −0.617258
\(769\) 10.0557 0.362617 0.181308 0.983426i \(-0.441967\pi\)
0.181308 + 0.983426i \(0.441967\pi\)
\(770\) 0 0
\(771\) 12.7312 0.458503
\(772\) −25.1882 −0.906541
\(773\) 4.14250 0.148995 0.0744977 0.997221i \(-0.476265\pi\)
0.0744977 + 0.997221i \(0.476265\pi\)
\(774\) −0.360291 −0.0129504
\(775\) 0 0
\(776\) 13.4792 0.483876
\(777\) 20.3856 0.731329
\(778\) 6.55849 0.235133
\(779\) −3.52926 −0.126449
\(780\) 0 0
\(781\) −6.42317 −0.229839
\(782\) 2.62754 0.0939605
\(783\) −10.4040 −0.371807
\(784\) −11.1211 −0.397184
\(785\) 0 0
\(786\) 10.2836 0.366803
\(787\) −6.39855 −0.228084 −0.114042 0.993476i \(-0.536380\pi\)
−0.114042 + 0.993476i \(0.536380\pi\)
\(788\) 21.8677 0.779003
\(789\) 0.877168 0.0312280
\(790\) 0 0
\(791\) −2.38515 −0.0848060
\(792\) −0.513870 −0.0182596
\(793\) 37.9483 1.34758
\(794\) −2.65563 −0.0942447
\(795\) 0 0
\(796\) 37.7207 1.33697
\(797\) 31.6848 1.12233 0.561166 0.827703i \(-0.310353\pi\)
0.561166 + 0.827703i \(0.310353\pi\)
\(798\) 1.08678 0.0384715
\(799\) 4.05631 0.143502
\(800\) 0 0
\(801\) −4.85153 −0.171420
\(802\) 3.32413 0.117379
\(803\) −10.7452 −0.379189
\(804\) 42.6931 1.50567
\(805\) 0 0
\(806\) 3.63917 0.128184
\(807\) −4.20545 −0.148039
\(808\) 8.08198 0.284323
\(809\) 34.9827 1.22993 0.614963 0.788556i \(-0.289171\pi\)
0.614963 + 0.788556i \(0.289171\pi\)
\(810\) 0 0
\(811\) −27.4687 −0.964556 −0.482278 0.876018i \(-0.660190\pi\)
−0.482278 + 0.876018i \(0.660190\pi\)
\(812\) 8.13721 0.285560
\(813\) 25.2825 0.886697
\(814\) −1.70122 −0.0596278
\(815\) 0 0
\(816\) 13.8769 0.485789
\(817\) 2.74094 0.0958935
\(818\) 7.81745 0.273331
\(819\) −3.19339 −0.111586
\(820\) 0 0
\(821\) 17.9999 0.628200 0.314100 0.949390i \(-0.398297\pi\)
0.314100 + 0.949390i \(0.398297\pi\)
\(822\) −7.75586 −0.270517
\(823\) 8.49995 0.296290 0.148145 0.988966i \(-0.452670\pi\)
0.148145 + 0.988966i \(0.452670\pi\)
\(824\) −16.1594 −0.562940
\(825\) 0 0
\(826\) −1.81796 −0.0632550
\(827\) 10.1919 0.354406 0.177203 0.984174i \(-0.443295\pi\)
0.177203 + 0.984174i \(0.443295\pi\)
\(828\) −3.36472 −0.116932
\(829\) −4.33629 −0.150605 −0.0753027 0.997161i \(-0.523992\pi\)
−0.0753027 + 0.997161i \(0.523992\pi\)
\(830\) 0 0
\(831\) 3.05304 0.105909
\(832\) 22.1923 0.769380
\(833\) 6.93768 0.240376
\(834\) −2.42941 −0.0841234
\(835\) 0 0
\(836\) 1.90931 0.0660347
\(837\) 15.2796 0.528142
\(838\) −2.45388 −0.0847677
\(839\) −42.6097 −1.47105 −0.735525 0.677498i \(-0.763064\pi\)
−0.735525 + 0.677498i \(0.763064\pi\)
\(840\) 0 0
\(841\) −24.2070 −0.834723
\(842\) 4.44104 0.153048
\(843\) −31.0303 −1.06874
\(844\) 42.8819 1.47606
\(845\) 0 0
\(846\) 0.246737 0.00848299
\(847\) 1.94668 0.0668887
\(848\) 15.9222 0.546769
\(849\) 18.3846 0.630959
\(850\) 0 0
\(851\) −22.8077 −0.781838
\(852\) 22.7343 0.778865
\(853\) 26.8673 0.919917 0.459959 0.887940i \(-0.347864\pi\)
0.459959 + 0.887940i \(0.347864\pi\)
\(854\) 5.91946 0.202560
\(855\) 0 0
\(856\) 17.4882 0.597735
\(857\) −56.6619 −1.93553 −0.967767 0.251846i \(-0.918962\pi\)
−0.967767 + 0.251846i \(0.918962\pi\)
\(858\) 2.09816 0.0716301
\(859\) 51.0188 1.74074 0.870370 0.492398i \(-0.163880\pi\)
0.870370 + 0.492398i \(0.163880\pi\)
\(860\) 0 0
\(861\) 12.7361 0.434043
\(862\) 8.43768 0.287389
\(863\) 33.1838 1.12959 0.564795 0.825231i \(-0.308955\pi\)
0.564795 + 0.825231i \(0.308955\pi\)
\(864\) −16.1471 −0.549336
\(865\) 0 0
\(866\) 0.645766 0.0219440
\(867\) 22.8574 0.776277
\(868\) −11.9506 −0.405630
\(869\) −10.8068 −0.366597
\(870\) 0 0
\(871\) −45.3334 −1.53606
\(872\) −8.40257 −0.284547
\(873\) 4.99736 0.169135
\(874\) −1.21590 −0.0411285
\(875\) 0 0
\(876\) 38.0317 1.28497
\(877\) −43.4040 −1.46565 −0.732825 0.680418i \(-0.761798\pi\)
−0.732825 + 0.680418i \(0.761798\pi\)
\(878\) −1.50078 −0.0506490
\(879\) 26.2828 0.886499
\(880\) 0 0
\(881\) −34.1080 −1.14913 −0.574565 0.818459i \(-0.694829\pi\)
−0.574565 + 0.818459i \(0.694829\pi\)
\(882\) 0.422005 0.0142096
\(883\) 31.9448 1.07503 0.537514 0.843255i \(-0.319364\pi\)
0.537514 + 0.843255i \(0.319364\pi\)
\(884\) −15.5067 −0.521546
\(885\) 0 0
\(886\) 3.12461 0.104973
\(887\) 15.6331 0.524907 0.262453 0.964945i \(-0.415468\pi\)
0.262453 + 0.964945i \(0.415468\pi\)
\(888\) 12.3287 0.413724
\(889\) −3.46863 −0.116334
\(890\) 0 0
\(891\) 10.1189 0.338997
\(892\) −19.6875 −0.659187
\(893\) −1.87707 −0.0628138
\(894\) −6.52713 −0.218300
\(895\) 0 0
\(896\) 16.6907 0.557597
\(897\) 28.1294 0.939212
\(898\) −10.8320 −0.361469
\(899\) −7.03923 −0.234771
\(900\) 0 0
\(901\) −9.93268 −0.330905
\(902\) −1.06285 −0.0353891
\(903\) −9.89126 −0.329161
\(904\) −1.44248 −0.0479761
\(905\) 0 0
\(906\) −12.8648 −0.427404
\(907\) −15.2429 −0.506132 −0.253066 0.967449i \(-0.581439\pi\)
−0.253066 + 0.967449i \(0.581439\pi\)
\(908\) 38.1395 1.26571
\(909\) 2.99635 0.0993828
\(910\) 0 0
\(911\) 10.7020 0.354574 0.177287 0.984159i \(-0.443268\pi\)
0.177287 + 0.984159i \(0.443268\pi\)
\(912\) −6.42159 −0.212640
\(913\) −8.12591 −0.268928
\(914\) 8.24688 0.272783
\(915\) 0 0
\(916\) 29.9813 0.990611
\(917\) 35.8586 1.18415
\(918\) 3.09266 0.102073
\(919\) −59.1251 −1.95036 −0.975178 0.221421i \(-0.928931\pi\)
−0.975178 + 0.221421i \(0.928931\pi\)
\(920\) 0 0
\(921\) 13.9009 0.458050
\(922\) 2.88767 0.0951003
\(923\) −24.1403 −0.794588
\(924\) −6.89012 −0.226668
\(925\) 0 0
\(926\) 8.25626 0.271318
\(927\) −5.99102 −0.196771
\(928\) 7.43886 0.244193
\(929\) 41.4480 1.35986 0.679932 0.733276i \(-0.262009\pi\)
0.679932 + 0.733276i \(0.262009\pi\)
\(930\) 0 0
\(931\) −3.21044 −0.105218
\(932\) −40.4435 −1.32477
\(933\) 42.0629 1.37708
\(934\) −9.22694 −0.301915
\(935\) 0 0
\(936\) −1.93128 −0.0631260
\(937\) −2.64414 −0.0863803 −0.0431901 0.999067i \(-0.513752\pi\)
−0.0431901 + 0.999067i \(0.513752\pi\)
\(938\) −7.07146 −0.230891
\(939\) 30.2294 0.986500
\(940\) 0 0
\(941\) −55.5911 −1.81222 −0.906109 0.423045i \(-0.860961\pi\)
−0.906109 + 0.423045i \(0.860961\pi\)
\(942\) 6.84317 0.222963
\(943\) −14.2493 −0.464021
\(944\) 10.7420 0.349624
\(945\) 0 0
\(946\) 0.825447 0.0268376
\(947\) −30.2967 −0.984509 −0.492255 0.870451i \(-0.663827\pi\)
−0.492255 + 0.870451i \(0.663827\pi\)
\(948\) 38.2500 1.24230
\(949\) −40.3838 −1.31091
\(950\) 0 0
\(951\) 49.2801 1.59802
\(952\) −4.95260 −0.160515
\(953\) 0.456421 0.0147849 0.00739247 0.999973i \(-0.497647\pi\)
0.00739247 + 0.999973i \(0.497647\pi\)
\(954\) −0.604185 −0.0195612
\(955\) 0 0
\(956\) 0.819152 0.0264933
\(957\) −4.05847 −0.131192
\(958\) −1.67288 −0.0540482
\(959\) −27.0445 −0.873312
\(960\) 0 0
\(961\) −20.6619 −0.666514
\(962\) −6.39373 −0.206142
\(963\) 6.48367 0.208933
\(964\) −22.8463 −0.735829
\(965\) 0 0
\(966\) 4.38784 0.141176
\(967\) 34.6037 1.11278 0.556389 0.830922i \(-0.312186\pi\)
0.556389 + 0.830922i \(0.312186\pi\)
\(968\) 1.17730 0.0378400
\(969\) 4.00596 0.128690
\(970\) 0 0
\(971\) −20.0747 −0.644228 −0.322114 0.946701i \(-0.604393\pi\)
−0.322114 + 0.946701i \(0.604393\pi\)
\(972\) −8.59501 −0.275685
\(973\) −8.47127 −0.271576
\(974\) −4.66304 −0.149414
\(975\) 0 0
\(976\) −34.9771 −1.11959
\(977\) 8.02308 0.256681 0.128341 0.991730i \(-0.459035\pi\)
0.128341 + 0.991730i \(0.459035\pi\)
\(978\) 4.46936 0.142914
\(979\) 11.1151 0.355241
\(980\) 0 0
\(981\) −3.11521 −0.0994611
\(982\) 7.79733 0.248823
\(983\) 10.1782 0.324633 0.162317 0.986739i \(-0.448103\pi\)
0.162317 + 0.986739i \(0.448103\pi\)
\(984\) 7.70245 0.245545
\(985\) 0 0
\(986\) −1.42477 −0.0453739
\(987\) 6.77380 0.215612
\(988\) 7.17578 0.228292
\(989\) 11.0665 0.351894
\(990\) 0 0
\(991\) 25.7709 0.818641 0.409321 0.912391i \(-0.365766\pi\)
0.409321 + 0.912391i \(0.365766\pi\)
\(992\) −10.9250 −0.346869
\(993\) −23.3686 −0.741581
\(994\) −3.76559 −0.119437
\(995\) 0 0
\(996\) 28.7610 0.911328
\(997\) −33.3107 −1.05496 −0.527481 0.849567i \(-0.676863\pi\)
−0.527481 + 0.849567i \(0.676863\pi\)
\(998\) 10.2323 0.323898
\(999\) −26.8451 −0.849342
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 5225.2.a.z.1.8 16
5.2 odd 4 1045.2.b.b.419.8 16
5.3 odd 4 1045.2.b.b.419.9 yes 16
5.4 even 2 inner 5225.2.a.z.1.9 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.b.419.8 16 5.2 odd 4
1045.2.b.b.419.9 yes 16 5.3 odd 4
5225.2.a.z.1.8 16 1.1 even 1 trivial
5225.2.a.z.1.9 16 5.4 even 2 inner