Properties

Label 4024.2.a.g.1.18
Level $4024$
Weight $2$
Character 4024.1
Self dual yes
Analytic conductor $32.132$
Analytic rank $0$
Dimension $33$
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] = [4024,2,Mod(1,4024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4024, 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("4024.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4024 = 2^{3} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4024.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: \(32.1318017734\)
Analytic rank: \(0\)
Dimension: \(33\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
Character \(\chi\) \(=\) 4024.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.649284 q^{3} -2.92377 q^{5} -1.90965 q^{7} -2.57843 q^{9} +O(q^{10})\) \(q+0.649284 q^{3} -2.92377 q^{5} -1.90965 q^{7} -2.57843 q^{9} +3.79600 q^{11} +1.26781 q^{13} -1.89835 q^{15} -3.64607 q^{17} +4.46389 q^{19} -1.23991 q^{21} -9.12945 q^{23} +3.54841 q^{25} -3.62199 q^{27} +2.77231 q^{29} -4.74950 q^{31} +2.46468 q^{33} +5.58338 q^{35} +1.24878 q^{37} +0.823169 q^{39} +8.26383 q^{41} -6.09603 q^{43} +7.53873 q^{45} -1.58222 q^{47} -3.35322 q^{49} -2.36733 q^{51} +9.70315 q^{53} -11.0986 q^{55} +2.89833 q^{57} +7.05146 q^{59} +1.40971 q^{61} +4.92391 q^{63} -3.70678 q^{65} +6.44588 q^{67} -5.92761 q^{69} -10.2738 q^{71} -4.81030 q^{73} +2.30393 q^{75} -7.24905 q^{77} +16.6520 q^{79} +5.38359 q^{81} +4.24583 q^{83} +10.6603 q^{85} +1.80001 q^{87} +14.4821 q^{89} -2.42108 q^{91} -3.08378 q^{93} -13.0514 q^{95} -7.34573 q^{97} -9.78772 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) \) 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 0
\(3\) 0.649284 0.374864 0.187432 0.982278i \(-0.439984\pi\)
0.187432 + 0.982278i \(0.439984\pi\)
\(4\) 0 0
\(5\) −2.92377 −1.30755 −0.653774 0.756690i \(-0.726815\pi\)
−0.653774 + 0.756690i \(0.726815\pi\)
\(6\) 0 0
\(7\) −1.90965 −0.721782 −0.360891 0.932608i \(-0.617527\pi\)
−0.360891 + 0.932608i \(0.617527\pi\)
\(8\) 0 0
\(9\) −2.57843 −0.859477
\(10\) 0 0
\(11\) 3.79600 1.14454 0.572269 0.820066i \(-0.306063\pi\)
0.572269 + 0.820066i \(0.306063\pi\)
\(12\) 0 0
\(13\) 1.26781 0.351627 0.175814 0.984423i \(-0.443744\pi\)
0.175814 + 0.984423i \(0.443744\pi\)
\(14\) 0 0
\(15\) −1.89835 −0.490153
\(16\) 0 0
\(17\) −3.64607 −0.884301 −0.442151 0.896941i \(-0.645784\pi\)
−0.442151 + 0.896941i \(0.645784\pi\)
\(18\) 0 0
\(19\) 4.46389 1.02409 0.512044 0.858959i \(-0.328889\pi\)
0.512044 + 0.858959i \(0.328889\pi\)
\(20\) 0 0
\(21\) −1.23991 −0.270570
\(22\) 0 0
\(23\) −9.12945 −1.90362 −0.951811 0.306684i \(-0.900781\pi\)
−0.951811 + 0.306684i \(0.900781\pi\)
\(24\) 0 0
\(25\) 3.54841 0.709682
\(26\) 0 0
\(27\) −3.62199 −0.697052
\(28\) 0 0
\(29\) 2.77231 0.514805 0.257402 0.966304i \(-0.417133\pi\)
0.257402 + 0.966304i \(0.417133\pi\)
\(30\) 0 0
\(31\) −4.74950 −0.853036 −0.426518 0.904479i \(-0.640260\pi\)
−0.426518 + 0.904479i \(0.640260\pi\)
\(32\) 0 0
\(33\) 2.46468 0.429046
\(34\) 0 0
\(35\) 5.58338 0.943764
\(36\) 0 0
\(37\) 1.24878 0.205298 0.102649 0.994718i \(-0.467268\pi\)
0.102649 + 0.994718i \(0.467268\pi\)
\(38\) 0 0
\(39\) 0.823169 0.131812
\(40\) 0 0
\(41\) 8.26383 1.29059 0.645296 0.763932i \(-0.276734\pi\)
0.645296 + 0.763932i \(0.276734\pi\)
\(42\) 0 0
\(43\) −6.09603 −0.929637 −0.464818 0.885406i \(-0.653880\pi\)
−0.464818 + 0.885406i \(0.653880\pi\)
\(44\) 0 0
\(45\) 7.53873 1.12381
\(46\) 0 0
\(47\) −1.58222 −0.230790 −0.115395 0.993320i \(-0.536813\pi\)
−0.115395 + 0.993320i \(0.536813\pi\)
\(48\) 0 0
\(49\) −3.35322 −0.479031
\(50\) 0 0
\(51\) −2.36733 −0.331493
\(52\) 0 0
\(53\) 9.70315 1.33283 0.666415 0.745581i \(-0.267828\pi\)
0.666415 + 0.745581i \(0.267828\pi\)
\(54\) 0 0
\(55\) −11.0986 −1.49654
\(56\) 0 0
\(57\) 2.89833 0.383894
\(58\) 0 0
\(59\) 7.05146 0.918022 0.459011 0.888431i \(-0.348204\pi\)
0.459011 + 0.888431i \(0.348204\pi\)
\(60\) 0 0
\(61\) 1.40971 0.180495 0.0902474 0.995919i \(-0.471234\pi\)
0.0902474 + 0.995919i \(0.471234\pi\)
\(62\) 0 0
\(63\) 4.92391 0.620355
\(64\) 0 0
\(65\) −3.70678 −0.459769
\(66\) 0 0
\(67\) 6.44588 0.787489 0.393745 0.919220i \(-0.371179\pi\)
0.393745 + 0.919220i \(0.371179\pi\)
\(68\) 0 0
\(69\) −5.92761 −0.713600
\(70\) 0 0
\(71\) −10.2738 −1.21928 −0.609639 0.792679i \(-0.708685\pi\)
−0.609639 + 0.792679i \(0.708685\pi\)
\(72\) 0 0
\(73\) −4.81030 −0.563003 −0.281502 0.959561i \(-0.590832\pi\)
−0.281502 + 0.959561i \(0.590832\pi\)
\(74\) 0 0
\(75\) 2.30393 0.266034
\(76\) 0 0
\(77\) −7.24905 −0.826106
\(78\) 0 0
\(79\) 16.6520 1.87350 0.936748 0.350005i \(-0.113820\pi\)
0.936748 + 0.350005i \(0.113820\pi\)
\(80\) 0 0
\(81\) 5.38359 0.598177
\(82\) 0 0
\(83\) 4.24583 0.466040 0.233020 0.972472i \(-0.425139\pi\)
0.233020 + 0.972472i \(0.425139\pi\)
\(84\) 0 0
\(85\) 10.6603 1.15627
\(86\) 0 0
\(87\) 1.80001 0.192982
\(88\) 0 0
\(89\) 14.4821 1.53510 0.767551 0.640988i \(-0.221475\pi\)
0.767551 + 0.640988i \(0.221475\pi\)
\(90\) 0 0
\(91\) −2.42108 −0.253798
\(92\) 0 0
\(93\) −3.08378 −0.319773
\(94\) 0 0
\(95\) −13.0514 −1.33904
\(96\) 0 0
\(97\) −7.34573 −0.745846 −0.372923 0.927862i \(-0.621644\pi\)
−0.372923 + 0.927862i \(0.621644\pi\)
\(98\) 0 0
\(99\) −9.78772 −0.983703
\(100\) 0 0
\(101\) 4.70072 0.467739 0.233869 0.972268i \(-0.424861\pi\)
0.233869 + 0.972268i \(0.424861\pi\)
\(102\) 0 0
\(103\) −2.26044 −0.222728 −0.111364 0.993780i \(-0.535522\pi\)
−0.111364 + 0.993780i \(0.535522\pi\)
\(104\) 0 0
\(105\) 3.62520 0.353784
\(106\) 0 0
\(107\) 13.4572 1.30096 0.650478 0.759526i \(-0.274569\pi\)
0.650478 + 0.759526i \(0.274569\pi\)
\(108\) 0 0
\(109\) −15.4456 −1.47942 −0.739709 0.672927i \(-0.765037\pi\)
−0.739709 + 0.672927i \(0.765037\pi\)
\(110\) 0 0
\(111\) 0.810814 0.0769591
\(112\) 0 0
\(113\) −6.64048 −0.624683 −0.312342 0.949970i \(-0.601113\pi\)
−0.312342 + 0.949970i \(0.601113\pi\)
\(114\) 0 0
\(115\) 26.6924 2.48908
\(116\) 0 0
\(117\) −3.26896 −0.302215
\(118\) 0 0
\(119\) 6.96273 0.638273
\(120\) 0 0
\(121\) 3.40963 0.309966
\(122\) 0 0
\(123\) 5.36557 0.483797
\(124\) 0 0
\(125\) 4.24411 0.379605
\(126\) 0 0
\(127\) −3.11813 −0.276690 −0.138345 0.990384i \(-0.544178\pi\)
−0.138345 + 0.990384i \(0.544178\pi\)
\(128\) 0 0
\(129\) −3.95806 −0.348488
\(130\) 0 0
\(131\) 17.6238 1.53980 0.769901 0.638163i \(-0.220306\pi\)
0.769901 + 0.638163i \(0.220306\pi\)
\(132\) 0 0
\(133\) −8.52449 −0.739167
\(134\) 0 0
\(135\) 10.5898 0.911428
\(136\) 0 0
\(137\) 19.5644 1.67150 0.835750 0.549110i \(-0.185033\pi\)
0.835750 + 0.549110i \(0.185033\pi\)
\(138\) 0 0
\(139\) 7.60405 0.644967 0.322483 0.946575i \(-0.395482\pi\)
0.322483 + 0.946575i \(0.395482\pi\)
\(140\) 0 0
\(141\) −1.02731 −0.0865151
\(142\) 0 0
\(143\) 4.81261 0.402450
\(144\) 0 0
\(145\) −8.10558 −0.673132
\(146\) 0 0
\(147\) −2.17719 −0.179572
\(148\) 0 0
\(149\) 11.5856 0.949127 0.474564 0.880221i \(-0.342606\pi\)
0.474564 + 0.880221i \(0.342606\pi\)
\(150\) 0 0
\(151\) 5.87029 0.477718 0.238859 0.971054i \(-0.423227\pi\)
0.238859 + 0.971054i \(0.423227\pi\)
\(152\) 0 0
\(153\) 9.40113 0.760037
\(154\) 0 0
\(155\) 13.8864 1.11539
\(156\) 0 0
\(157\) 11.0186 0.879382 0.439691 0.898149i \(-0.355088\pi\)
0.439691 + 0.898149i \(0.355088\pi\)
\(158\) 0 0
\(159\) 6.30010 0.499630
\(160\) 0 0
\(161\) 17.4341 1.37400
\(162\) 0 0
\(163\) 10.2705 0.804449 0.402225 0.915541i \(-0.368237\pi\)
0.402225 + 0.915541i \(0.368237\pi\)
\(164\) 0 0
\(165\) −7.20616 −0.560999
\(166\) 0 0
\(167\) −0.169266 −0.0130982 −0.00654909 0.999979i \(-0.502085\pi\)
−0.00654909 + 0.999979i \(0.502085\pi\)
\(168\) 0 0
\(169\) −11.3927 −0.876358
\(170\) 0 0
\(171\) −11.5098 −0.880179
\(172\) 0 0
\(173\) 2.30188 0.175009 0.0875043 0.996164i \(-0.472111\pi\)
0.0875043 + 0.996164i \(0.472111\pi\)
\(174\) 0 0
\(175\) −6.77624 −0.512235
\(176\) 0 0
\(177\) 4.57840 0.344134
\(178\) 0 0
\(179\) 23.0819 1.72522 0.862612 0.505866i \(-0.168827\pi\)
0.862612 + 0.505866i \(0.168827\pi\)
\(180\) 0 0
\(181\) 19.4781 1.44780 0.723898 0.689907i \(-0.242349\pi\)
0.723898 + 0.689907i \(0.242349\pi\)
\(182\) 0 0
\(183\) 0.915302 0.0676611
\(184\) 0 0
\(185\) −3.65115 −0.268438
\(186\) 0 0
\(187\) −13.8405 −1.01212
\(188\) 0 0
\(189\) 6.91674 0.503119
\(190\) 0 0
\(191\) −23.3292 −1.68804 −0.844021 0.536310i \(-0.819818\pi\)
−0.844021 + 0.536310i \(0.819818\pi\)
\(192\) 0 0
\(193\) 2.48731 0.179040 0.0895202 0.995985i \(-0.471467\pi\)
0.0895202 + 0.995985i \(0.471467\pi\)
\(194\) 0 0
\(195\) −2.40675 −0.172351
\(196\) 0 0
\(197\) −11.8559 −0.844699 −0.422350 0.906433i \(-0.638795\pi\)
−0.422350 + 0.906433i \(0.638795\pi\)
\(198\) 0 0
\(199\) 25.3537 1.79728 0.898638 0.438691i \(-0.144558\pi\)
0.898638 + 0.438691i \(0.144558\pi\)
\(200\) 0 0
\(201\) 4.18521 0.295202
\(202\) 0 0
\(203\) −5.29415 −0.371576
\(204\) 0 0
\(205\) −24.1615 −1.68751
\(206\) 0 0
\(207\) 23.5397 1.63612
\(208\) 0 0
\(209\) 16.9449 1.17211
\(210\) 0 0
\(211\) 15.8161 1.08882 0.544412 0.838818i \(-0.316753\pi\)
0.544412 + 0.838818i \(0.316753\pi\)
\(212\) 0 0
\(213\) −6.67062 −0.457064
\(214\) 0 0
\(215\) 17.8234 1.21554
\(216\) 0 0
\(217\) 9.06991 0.615706
\(218\) 0 0
\(219\) −3.12325 −0.211050
\(220\) 0 0
\(221\) −4.62252 −0.310944
\(222\) 0 0
\(223\) −20.9037 −1.39981 −0.699907 0.714234i \(-0.746775\pi\)
−0.699907 + 0.714234i \(0.746775\pi\)
\(224\) 0 0
\(225\) −9.14933 −0.609955
\(226\) 0 0
\(227\) −24.4607 −1.62352 −0.811758 0.583995i \(-0.801489\pi\)
−0.811758 + 0.583995i \(0.801489\pi\)
\(228\) 0 0
\(229\) −5.07631 −0.335452 −0.167726 0.985834i \(-0.553642\pi\)
−0.167726 + 0.985834i \(0.553642\pi\)
\(230\) 0 0
\(231\) −4.70669 −0.309678
\(232\) 0 0
\(233\) 11.0189 0.721873 0.360936 0.932590i \(-0.382457\pi\)
0.360936 + 0.932590i \(0.382457\pi\)
\(234\) 0 0
\(235\) 4.62604 0.301769
\(236\) 0 0
\(237\) 10.8119 0.702307
\(238\) 0 0
\(239\) 2.50016 0.161722 0.0808611 0.996725i \(-0.474233\pi\)
0.0808611 + 0.996725i \(0.474233\pi\)
\(240\) 0 0
\(241\) −16.3262 −1.05166 −0.525831 0.850589i \(-0.676246\pi\)
−0.525831 + 0.850589i \(0.676246\pi\)
\(242\) 0 0
\(243\) 14.3614 0.921287
\(244\) 0 0
\(245\) 9.80403 0.626356
\(246\) 0 0
\(247\) 5.65937 0.360097
\(248\) 0 0
\(249\) 2.75675 0.174702
\(250\) 0 0
\(251\) 19.9630 1.26005 0.630026 0.776574i \(-0.283044\pi\)
0.630026 + 0.776574i \(0.283044\pi\)
\(252\) 0 0
\(253\) −34.6554 −2.17877
\(254\) 0 0
\(255\) 6.92153 0.433443
\(256\) 0 0
\(257\) −8.64384 −0.539188 −0.269594 0.962974i \(-0.586889\pi\)
−0.269594 + 0.962974i \(0.586889\pi\)
\(258\) 0 0
\(259\) −2.38474 −0.148181
\(260\) 0 0
\(261\) −7.14820 −0.442463
\(262\) 0 0
\(263\) −0.118392 −0.00730038 −0.00365019 0.999993i \(-0.501162\pi\)
−0.00365019 + 0.999993i \(0.501162\pi\)
\(264\) 0 0
\(265\) −28.3697 −1.74274
\(266\) 0 0
\(267\) 9.40301 0.575455
\(268\) 0 0
\(269\) −4.67830 −0.285241 −0.142621 0.989777i \(-0.545553\pi\)
−0.142621 + 0.989777i \(0.545553\pi\)
\(270\) 0 0
\(271\) 20.1728 1.22541 0.612705 0.790312i \(-0.290081\pi\)
0.612705 + 0.790312i \(0.290081\pi\)
\(272\) 0 0
\(273\) −1.57197 −0.0951398
\(274\) 0 0
\(275\) 13.4698 0.812258
\(276\) 0 0
\(277\) −3.39604 −0.204049 −0.102024 0.994782i \(-0.532532\pi\)
−0.102024 + 0.994782i \(0.532532\pi\)
\(278\) 0 0
\(279\) 12.2463 0.733164
\(280\) 0 0
\(281\) 2.89693 0.172816 0.0864081 0.996260i \(-0.472461\pi\)
0.0864081 + 0.996260i \(0.472461\pi\)
\(282\) 0 0
\(283\) −21.8412 −1.29832 −0.649162 0.760650i \(-0.724880\pi\)
−0.649162 + 0.760650i \(0.724880\pi\)
\(284\) 0 0
\(285\) −8.47405 −0.501960
\(286\) 0 0
\(287\) −15.7811 −0.931526
\(288\) 0 0
\(289\) −3.70619 −0.218011
\(290\) 0 0
\(291\) −4.76947 −0.279591
\(292\) 0 0
\(293\) 3.00629 0.175629 0.0878147 0.996137i \(-0.472012\pi\)
0.0878147 + 0.996137i \(0.472012\pi\)
\(294\) 0 0
\(295\) −20.6168 −1.20036
\(296\) 0 0
\(297\) −13.7491 −0.797802
\(298\) 0 0
\(299\) −11.5744 −0.669365
\(300\) 0 0
\(301\) 11.6413 0.670995
\(302\) 0 0
\(303\) 3.05210 0.175339
\(304\) 0 0
\(305\) −4.12166 −0.236006
\(306\) 0 0
\(307\) −30.7196 −1.75326 −0.876631 0.481163i \(-0.840214\pi\)
−0.876631 + 0.481163i \(0.840214\pi\)
\(308\) 0 0
\(309\) −1.46767 −0.0834928
\(310\) 0 0
\(311\) −8.78128 −0.497941 −0.248970 0.968511i \(-0.580092\pi\)
−0.248970 + 0.968511i \(0.580092\pi\)
\(312\) 0 0
\(313\) 13.1310 0.742209 0.371104 0.928591i \(-0.378979\pi\)
0.371104 + 0.928591i \(0.378979\pi\)
\(314\) 0 0
\(315\) −14.3964 −0.811143
\(316\) 0 0
\(317\) 18.2280 1.02378 0.511892 0.859050i \(-0.328945\pi\)
0.511892 + 0.859050i \(0.328945\pi\)
\(318\) 0 0
\(319\) 10.5237 0.589213
\(320\) 0 0
\(321\) 8.73754 0.487682
\(322\) 0 0
\(323\) −16.2757 −0.905602
\(324\) 0 0
\(325\) 4.49871 0.249543
\(326\) 0 0
\(327\) −10.0286 −0.554581
\(328\) 0 0
\(329\) 3.02149 0.166580
\(330\) 0 0
\(331\) 18.4672 1.01505 0.507526 0.861637i \(-0.330560\pi\)
0.507526 + 0.861637i \(0.330560\pi\)
\(332\) 0 0
\(333\) −3.21990 −0.176449
\(334\) 0 0
\(335\) −18.8462 −1.02968
\(336\) 0 0
\(337\) 29.6384 1.61451 0.807254 0.590204i \(-0.200953\pi\)
0.807254 + 0.590204i \(0.200953\pi\)
\(338\) 0 0
\(339\) −4.31156 −0.234172
\(340\) 0 0
\(341\) −18.0291 −0.976331
\(342\) 0 0
\(343\) 19.7711 1.06754
\(344\) 0 0
\(345\) 17.3309 0.933067
\(346\) 0 0
\(347\) 22.3202 1.19821 0.599105 0.800670i \(-0.295523\pi\)
0.599105 + 0.800670i \(0.295523\pi\)
\(348\) 0 0
\(349\) −3.58420 −0.191858 −0.0959289 0.995388i \(-0.530582\pi\)
−0.0959289 + 0.995388i \(0.530582\pi\)
\(350\) 0 0
\(351\) −4.59199 −0.245102
\(352\) 0 0
\(353\) −20.3245 −1.08176 −0.540881 0.841099i \(-0.681909\pi\)
−0.540881 + 0.841099i \(0.681909\pi\)
\(354\) 0 0
\(355\) 30.0382 1.59426
\(356\) 0 0
\(357\) 4.52079 0.239266
\(358\) 0 0
\(359\) 20.8194 1.09880 0.549402 0.835558i \(-0.314856\pi\)
0.549402 + 0.835558i \(0.314856\pi\)
\(360\) 0 0
\(361\) 0.926334 0.0487544
\(362\) 0 0
\(363\) 2.21382 0.116195
\(364\) 0 0
\(365\) 14.0642 0.736154
\(366\) 0 0
\(367\) −16.3899 −0.855548 −0.427774 0.903886i \(-0.640702\pi\)
−0.427774 + 0.903886i \(0.640702\pi\)
\(368\) 0 0
\(369\) −21.3077 −1.10923
\(370\) 0 0
\(371\) −18.5297 −0.962012
\(372\) 0 0
\(373\) −32.2454 −1.66961 −0.834803 0.550549i \(-0.814418\pi\)
−0.834803 + 0.550549i \(0.814418\pi\)
\(374\) 0 0
\(375\) 2.75563 0.142300
\(376\) 0 0
\(377\) 3.51476 0.181019
\(378\) 0 0
\(379\) 13.6498 0.701143 0.350571 0.936536i \(-0.385987\pi\)
0.350571 + 0.936536i \(0.385987\pi\)
\(380\) 0 0
\(381\) −2.02455 −0.103721
\(382\) 0 0
\(383\) −28.3953 −1.45093 −0.725465 0.688259i \(-0.758375\pi\)
−0.725465 + 0.688259i \(0.758375\pi\)
\(384\) 0 0
\(385\) 21.1945 1.08017
\(386\) 0 0
\(387\) 15.7182 0.799001
\(388\) 0 0
\(389\) −25.9741 −1.31694 −0.658469 0.752607i \(-0.728796\pi\)
−0.658469 + 0.752607i \(0.728796\pi\)
\(390\) 0 0
\(391\) 33.2866 1.68338
\(392\) 0 0
\(393\) 11.4429 0.577217
\(394\) 0 0
\(395\) −48.6865 −2.44969
\(396\) 0 0
\(397\) 21.7097 1.08958 0.544790 0.838572i \(-0.316609\pi\)
0.544790 + 0.838572i \(0.316609\pi\)
\(398\) 0 0
\(399\) −5.53482 −0.277087
\(400\) 0 0
\(401\) −30.2178 −1.50901 −0.754503 0.656296i \(-0.772122\pi\)
−0.754503 + 0.656296i \(0.772122\pi\)
\(402\) 0 0
\(403\) −6.02146 −0.299951
\(404\) 0 0
\(405\) −15.7404 −0.782145
\(406\) 0 0
\(407\) 4.74038 0.234972
\(408\) 0 0
\(409\) −29.7393 −1.47051 −0.735257 0.677788i \(-0.762939\pi\)
−0.735257 + 0.677788i \(0.762939\pi\)
\(410\) 0 0
\(411\) 12.7029 0.626586
\(412\) 0 0
\(413\) −13.4659 −0.662612
\(414\) 0 0
\(415\) −12.4138 −0.609370
\(416\) 0 0
\(417\) 4.93719 0.241775
\(418\) 0 0
\(419\) −8.56794 −0.418571 −0.209286 0.977855i \(-0.567114\pi\)
−0.209286 + 0.977855i \(0.567114\pi\)
\(420\) 0 0
\(421\) −38.3028 −1.86677 −0.933383 0.358883i \(-0.883158\pi\)
−0.933383 + 0.358883i \(0.883158\pi\)
\(422\) 0 0
\(423\) 4.07964 0.198359
\(424\) 0 0
\(425\) −12.9377 −0.627573
\(426\) 0 0
\(427\) −2.69206 −0.130278
\(428\) 0 0
\(429\) 3.12475 0.150864
\(430\) 0 0
\(431\) 17.6208 0.848763 0.424381 0.905484i \(-0.360492\pi\)
0.424381 + 0.905484i \(0.360492\pi\)
\(432\) 0 0
\(433\) 23.8451 1.14592 0.572960 0.819583i \(-0.305795\pi\)
0.572960 + 0.819583i \(0.305795\pi\)
\(434\) 0 0
\(435\) −5.26282 −0.252333
\(436\) 0 0
\(437\) −40.7529 −1.94948
\(438\) 0 0
\(439\) 22.3019 1.06441 0.532207 0.846614i \(-0.321363\pi\)
0.532207 + 0.846614i \(0.321363\pi\)
\(440\) 0 0
\(441\) 8.64604 0.411716
\(442\) 0 0
\(443\) −14.8788 −0.706911 −0.353456 0.935451i \(-0.614993\pi\)
−0.353456 + 0.935451i \(0.614993\pi\)
\(444\) 0 0
\(445\) −42.3423 −2.00722
\(446\) 0 0
\(447\) 7.52233 0.355794
\(448\) 0 0
\(449\) −12.4180 −0.586043 −0.293021 0.956106i \(-0.594661\pi\)
−0.293021 + 0.956106i \(0.594661\pi\)
\(450\) 0 0
\(451\) 31.3695 1.47713
\(452\) 0 0
\(453\) 3.81149 0.179079
\(454\) 0 0
\(455\) 7.07867 0.331853
\(456\) 0 0
\(457\) 18.9508 0.886480 0.443240 0.896403i \(-0.353829\pi\)
0.443240 + 0.896403i \(0.353829\pi\)
\(458\) 0 0
\(459\) 13.2060 0.616404
\(460\) 0 0
\(461\) −2.48209 −0.115602 −0.0578012 0.998328i \(-0.518409\pi\)
−0.0578012 + 0.998328i \(0.518409\pi\)
\(462\) 0 0
\(463\) −9.11042 −0.423397 −0.211699 0.977335i \(-0.567899\pi\)
−0.211699 + 0.977335i \(0.567899\pi\)
\(464\) 0 0
\(465\) 9.01624 0.418118
\(466\) 0 0
\(467\) 37.2288 1.72274 0.861372 0.507974i \(-0.169605\pi\)
0.861372 + 0.507974i \(0.169605\pi\)
\(468\) 0 0
\(469\) −12.3094 −0.568395
\(470\) 0 0
\(471\) 7.15422 0.329649
\(472\) 0 0
\(473\) −23.1406 −1.06400
\(474\) 0 0
\(475\) 15.8397 0.726776
\(476\) 0 0
\(477\) −25.0189 −1.14554
\(478\) 0 0
\(479\) 2.96384 0.135421 0.0677106 0.997705i \(-0.478431\pi\)
0.0677106 + 0.997705i \(0.478431\pi\)
\(480\) 0 0
\(481\) 1.58322 0.0721885
\(482\) 0 0
\(483\) 11.3197 0.515064
\(484\) 0 0
\(485\) 21.4772 0.975229
\(486\) 0 0
\(487\) 24.4702 1.10885 0.554424 0.832234i \(-0.312938\pi\)
0.554424 + 0.832234i \(0.312938\pi\)
\(488\) 0 0
\(489\) 6.66848 0.301559
\(490\) 0 0
\(491\) −17.0876 −0.771151 −0.385576 0.922676i \(-0.625997\pi\)
−0.385576 + 0.922676i \(0.625997\pi\)
\(492\) 0 0
\(493\) −10.1080 −0.455242
\(494\) 0 0
\(495\) 28.6170 1.28624
\(496\) 0 0
\(497\) 19.6194 0.880052
\(498\) 0 0
\(499\) 9.05669 0.405433 0.202716 0.979237i \(-0.435023\pi\)
0.202716 + 0.979237i \(0.435023\pi\)
\(500\) 0 0
\(501\) −0.109902 −0.00491004
\(502\) 0 0
\(503\) 1.00000 0.0445878
\(504\) 0 0
\(505\) −13.7438 −0.611591
\(506\) 0 0
\(507\) −7.39707 −0.328515
\(508\) 0 0
\(509\) 10.4770 0.464385 0.232193 0.972670i \(-0.425410\pi\)
0.232193 + 0.972670i \(0.425410\pi\)
\(510\) 0 0
\(511\) 9.18601 0.406365
\(512\) 0 0
\(513\) −16.1682 −0.713841
\(514\) 0 0
\(515\) 6.60901 0.291228
\(516\) 0 0
\(517\) −6.00610 −0.264148
\(518\) 0 0
\(519\) 1.49457 0.0656045
\(520\) 0 0
\(521\) 42.0358 1.84162 0.920812 0.390007i \(-0.127527\pi\)
0.920812 + 0.390007i \(0.127527\pi\)
\(522\) 0 0
\(523\) −21.3746 −0.934647 −0.467323 0.884086i \(-0.654782\pi\)
−0.467323 + 0.884086i \(0.654782\pi\)
\(524\) 0 0
\(525\) −4.39970 −0.192019
\(526\) 0 0
\(527\) 17.3170 0.754341
\(528\) 0 0
\(529\) 60.3469 2.62378
\(530\) 0 0
\(531\) −18.1817 −0.789019
\(532\) 0 0
\(533\) 10.4770 0.453808
\(534\) 0 0
\(535\) −39.3457 −1.70106
\(536\) 0 0
\(537\) 14.9867 0.646725
\(538\) 0 0
\(539\) −12.7288 −0.548269
\(540\) 0 0
\(541\) −9.32108 −0.400744 −0.200372 0.979720i \(-0.564215\pi\)
−0.200372 + 0.979720i \(0.564215\pi\)
\(542\) 0 0
\(543\) 12.6468 0.542727
\(544\) 0 0
\(545\) 45.1592 1.93441
\(546\) 0 0
\(547\) −24.7564 −1.05851 −0.529253 0.848464i \(-0.677528\pi\)
−0.529253 + 0.848464i \(0.677528\pi\)
\(548\) 0 0
\(549\) −3.63484 −0.155131
\(550\) 0 0
\(551\) 12.3753 0.527205
\(552\) 0 0
\(553\) −31.7996 −1.35225
\(554\) 0 0
\(555\) −2.37063 −0.100628
\(556\) 0 0
\(557\) −7.60566 −0.322262 −0.161131 0.986933i \(-0.551514\pi\)
−0.161131 + 0.986933i \(0.551514\pi\)
\(558\) 0 0
\(559\) −7.72861 −0.326885
\(560\) 0 0
\(561\) −8.98640 −0.379406
\(562\) 0 0
\(563\) 1.31552 0.0554427 0.0277213 0.999616i \(-0.491175\pi\)
0.0277213 + 0.999616i \(0.491175\pi\)
\(564\) 0 0
\(565\) 19.4152 0.816804
\(566\) 0 0
\(567\) −10.2808 −0.431753
\(568\) 0 0
\(569\) −34.3068 −1.43822 −0.719108 0.694899i \(-0.755449\pi\)
−0.719108 + 0.694899i \(0.755449\pi\)
\(570\) 0 0
\(571\) 13.8306 0.578794 0.289397 0.957209i \(-0.406545\pi\)
0.289397 + 0.957209i \(0.406545\pi\)
\(572\) 0 0
\(573\) −15.1473 −0.632787
\(574\) 0 0
\(575\) −32.3950 −1.35097
\(576\) 0 0
\(577\) −36.8822 −1.53542 −0.767712 0.640795i \(-0.778605\pi\)
−0.767712 + 0.640795i \(0.778605\pi\)
\(578\) 0 0
\(579\) 1.61497 0.0671159
\(580\) 0 0
\(581\) −8.10806 −0.336379
\(582\) 0 0
\(583\) 36.8332 1.52547
\(584\) 0 0
\(585\) 9.55767 0.395161
\(586\) 0 0
\(587\) 15.7123 0.648516 0.324258 0.945969i \(-0.394885\pi\)
0.324258 + 0.945969i \(0.394885\pi\)
\(588\) 0 0
\(589\) −21.2013 −0.873583
\(590\) 0 0
\(591\) −7.69786 −0.316648
\(592\) 0 0
\(593\) 23.6896 0.972814 0.486407 0.873732i \(-0.338307\pi\)
0.486407 + 0.873732i \(0.338307\pi\)
\(594\) 0 0
\(595\) −20.3574 −0.834572
\(596\) 0 0
\(597\) 16.4618 0.673735
\(598\) 0 0
\(599\) 12.9584 0.529468 0.264734 0.964322i \(-0.414716\pi\)
0.264734 + 0.964322i \(0.414716\pi\)
\(600\) 0 0
\(601\) −34.6005 −1.41138 −0.705692 0.708519i \(-0.749364\pi\)
−0.705692 + 0.708519i \(0.749364\pi\)
\(602\) 0 0
\(603\) −16.6202 −0.676829
\(604\) 0 0
\(605\) −9.96895 −0.405295
\(606\) 0 0
\(607\) 20.2430 0.821640 0.410820 0.911717i \(-0.365243\pi\)
0.410820 + 0.911717i \(0.365243\pi\)
\(608\) 0 0
\(609\) −3.43741 −0.139291
\(610\) 0 0
\(611\) −2.00595 −0.0811521
\(612\) 0 0
\(613\) −39.9202 −1.61236 −0.806181 0.591669i \(-0.798469\pi\)
−0.806181 + 0.591669i \(0.798469\pi\)
\(614\) 0 0
\(615\) −15.6877 −0.632588
\(616\) 0 0
\(617\) 49.2647 1.98332 0.991662 0.128870i \(-0.0411349\pi\)
0.991662 + 0.128870i \(0.0411349\pi\)
\(618\) 0 0
\(619\) 15.1002 0.606926 0.303463 0.952843i \(-0.401857\pi\)
0.303463 + 0.952843i \(0.401857\pi\)
\(620\) 0 0
\(621\) 33.0668 1.32692
\(622\) 0 0
\(623\) −27.6558 −1.10801
\(624\) 0 0
\(625\) −30.1508 −1.20603
\(626\) 0 0
\(627\) 11.0021 0.439381
\(628\) 0 0
\(629\) −4.55314 −0.181546
\(630\) 0 0
\(631\) 15.5662 0.619680 0.309840 0.950789i \(-0.399724\pi\)
0.309840 + 0.950789i \(0.399724\pi\)
\(632\) 0 0
\(633\) 10.2691 0.408161
\(634\) 0 0
\(635\) 9.11669 0.361785
\(636\) 0 0
\(637\) −4.25124 −0.168440
\(638\) 0 0
\(639\) 26.4903 1.04794
\(640\) 0 0
\(641\) 5.14200 0.203097 0.101548 0.994831i \(-0.467620\pi\)
0.101548 + 0.994831i \(0.467620\pi\)
\(642\) 0 0
\(643\) 6.79831 0.268099 0.134050 0.990975i \(-0.457202\pi\)
0.134050 + 0.990975i \(0.457202\pi\)
\(644\) 0 0
\(645\) 11.5724 0.455664
\(646\) 0 0
\(647\) 12.4590 0.489812 0.244906 0.969547i \(-0.421243\pi\)
0.244906 + 0.969547i \(0.421243\pi\)
\(648\) 0 0
\(649\) 26.7674 1.05071
\(650\) 0 0
\(651\) 5.88895 0.230806
\(652\) 0 0
\(653\) −1.09746 −0.0429471 −0.0214735 0.999769i \(-0.506836\pi\)
−0.0214735 + 0.999769i \(0.506836\pi\)
\(654\) 0 0
\(655\) −51.5280 −2.01337
\(656\) 0 0
\(657\) 12.4030 0.483888
\(658\) 0 0
\(659\) 27.0878 1.05519 0.527596 0.849496i \(-0.323094\pi\)
0.527596 + 0.849496i \(0.323094\pi\)
\(660\) 0 0
\(661\) −16.6931 −0.649287 −0.324644 0.945836i \(-0.605244\pi\)
−0.324644 + 0.945836i \(0.605244\pi\)
\(662\) 0 0
\(663\) −3.00133 −0.116562
\(664\) 0 0
\(665\) 24.9236 0.966497
\(666\) 0 0
\(667\) −25.3097 −0.979994
\(668\) 0 0
\(669\) −13.5724 −0.524740
\(670\) 0 0
\(671\) 5.35126 0.206583
\(672\) 0 0
\(673\) 42.7644 1.64845 0.824224 0.566264i \(-0.191612\pi\)
0.824224 + 0.566264i \(0.191612\pi\)
\(674\) 0 0
\(675\) −12.8523 −0.494685
\(676\) 0 0
\(677\) 6.94186 0.266797 0.133399 0.991062i \(-0.457411\pi\)
0.133399 + 0.991062i \(0.457411\pi\)
\(678\) 0 0
\(679\) 14.0278 0.538338
\(680\) 0 0
\(681\) −15.8820 −0.608598
\(682\) 0 0
\(683\) 16.6251 0.636143 0.318072 0.948067i \(-0.396965\pi\)
0.318072 + 0.948067i \(0.396965\pi\)
\(684\) 0 0
\(685\) −57.2018 −2.18557
\(686\) 0 0
\(687\) −3.29597 −0.125749
\(688\) 0 0
\(689\) 12.3017 0.468659
\(690\) 0 0
\(691\) 2.92763 0.111372 0.0556861 0.998448i \(-0.482265\pi\)
0.0556861 + 0.998448i \(0.482265\pi\)
\(692\) 0 0
\(693\) 18.6912 0.710019
\(694\) 0 0
\(695\) −22.2325 −0.843325
\(696\) 0 0
\(697\) −30.1305 −1.14127
\(698\) 0 0
\(699\) 7.15440 0.270604
\(700\) 0 0
\(701\) 0.193799 0.00731970 0.00365985 0.999993i \(-0.498835\pi\)
0.00365985 + 0.999993i \(0.498835\pi\)
\(702\) 0 0
\(703\) 5.57443 0.210243
\(704\) 0 0
\(705\) 3.00361 0.113123
\(706\) 0 0
\(707\) −8.97674 −0.337605
\(708\) 0 0
\(709\) 38.5430 1.44751 0.723756 0.690056i \(-0.242414\pi\)
0.723756 + 0.690056i \(0.242414\pi\)
\(710\) 0 0
\(711\) −42.9360 −1.61023
\(712\) 0 0
\(713\) 43.3604 1.62386
\(714\) 0 0
\(715\) −14.0709 −0.526223
\(716\) 0 0
\(717\) 1.62332 0.0606239
\(718\) 0 0
\(719\) −29.6071 −1.10416 −0.552078 0.833792i \(-0.686165\pi\)
−0.552078 + 0.833792i \(0.686165\pi\)
\(720\) 0 0
\(721\) 4.31667 0.160761
\(722\) 0 0
\(723\) −10.6003 −0.394231
\(724\) 0 0
\(725\) 9.83728 0.365347
\(726\) 0 0
\(727\) 13.7172 0.508743 0.254371 0.967107i \(-0.418131\pi\)
0.254371 + 0.967107i \(0.418131\pi\)
\(728\) 0 0
\(729\) −6.82612 −0.252819
\(730\) 0 0
\(731\) 22.2266 0.822079
\(732\) 0 0
\(733\) −51.6865 −1.90908 −0.954541 0.298079i \(-0.903654\pi\)
−0.954541 + 0.298079i \(0.903654\pi\)
\(734\) 0 0
\(735\) 6.36560 0.234799
\(736\) 0 0
\(737\) 24.4686 0.901311
\(738\) 0 0
\(739\) −38.4545 −1.41457 −0.707285 0.706929i \(-0.750080\pi\)
−0.707285 + 0.706929i \(0.750080\pi\)
\(740\) 0 0
\(741\) 3.67454 0.134987
\(742\) 0 0
\(743\) −25.5027 −0.935604 −0.467802 0.883833i \(-0.654954\pi\)
−0.467802 + 0.883833i \(0.654954\pi\)
\(744\) 0 0
\(745\) −33.8735 −1.24103
\(746\) 0 0
\(747\) −10.9476 −0.400550
\(748\) 0 0
\(749\) −25.6986 −0.939005
\(750\) 0 0
\(751\) 20.1189 0.734150 0.367075 0.930191i \(-0.380359\pi\)
0.367075 + 0.930191i \(0.380359\pi\)
\(752\) 0 0
\(753\) 12.9617 0.472349
\(754\) 0 0
\(755\) −17.1634 −0.624639
\(756\) 0 0
\(757\) 32.3402 1.17543 0.587713 0.809070i \(-0.300029\pi\)
0.587713 + 0.809070i \(0.300029\pi\)
\(758\) 0 0
\(759\) −22.5012 −0.816742
\(760\) 0 0
\(761\) 16.1196 0.584334 0.292167 0.956367i \(-0.405624\pi\)
0.292167 + 0.956367i \(0.405624\pi\)
\(762\) 0 0
\(763\) 29.4957 1.06782
\(764\) 0 0
\(765\) −27.4867 −0.993784
\(766\) 0 0
\(767\) 8.93991 0.322802
\(768\) 0 0
\(769\) −27.3213 −0.985231 −0.492616 0.870247i \(-0.663959\pi\)
−0.492616 + 0.870247i \(0.663959\pi\)
\(770\) 0 0
\(771\) −5.61231 −0.202122
\(772\) 0 0
\(773\) −9.13547 −0.328580 −0.164290 0.986412i \(-0.552533\pi\)
−0.164290 + 0.986412i \(0.552533\pi\)
\(774\) 0 0
\(775\) −16.8532 −0.605384
\(776\) 0 0
\(777\) −1.54837 −0.0555476
\(778\) 0 0
\(779\) 36.8888 1.32168
\(780\) 0 0
\(781\) −38.9994 −1.39551
\(782\) 0 0
\(783\) −10.0413 −0.358845
\(784\) 0 0
\(785\) −32.2159 −1.14983
\(786\) 0 0
\(787\) −9.33974 −0.332926 −0.166463 0.986048i \(-0.553235\pi\)
−0.166463 + 0.986048i \(0.553235\pi\)
\(788\) 0 0
\(789\) −0.0768702 −0.00273665
\(790\) 0 0
\(791\) 12.6810 0.450885
\(792\) 0 0
\(793\) 1.78724 0.0634669
\(794\) 0 0
\(795\) −18.4200 −0.653291
\(796\) 0 0
\(797\) 7.62797 0.270197 0.135098 0.990832i \(-0.456865\pi\)
0.135098 + 0.990832i \(0.456865\pi\)
\(798\) 0 0
\(799\) 5.76888 0.204088
\(800\) 0 0
\(801\) −37.3411 −1.31938
\(802\) 0 0
\(803\) −18.2599 −0.644378
\(804\) 0 0
\(805\) −50.9733 −1.79657
\(806\) 0 0
\(807\) −3.03755 −0.106927
\(808\) 0 0
\(809\) 24.0887 0.846913 0.423457 0.905916i \(-0.360817\pi\)
0.423457 + 0.905916i \(0.360817\pi\)
\(810\) 0 0
\(811\) −11.1940 −0.393076 −0.196538 0.980496i \(-0.562970\pi\)
−0.196538 + 0.980496i \(0.562970\pi\)
\(812\) 0 0
\(813\) 13.0979 0.459363
\(814\) 0 0
\(815\) −30.0286 −1.05186
\(816\) 0 0
\(817\) −27.2120 −0.952029
\(818\) 0 0
\(819\) 6.24258 0.218133
\(820\) 0 0
\(821\) −7.64165 −0.266695 −0.133348 0.991069i \(-0.542573\pi\)
−0.133348 + 0.991069i \(0.542573\pi\)
\(822\) 0 0
\(823\) −54.9967 −1.91707 −0.958533 0.284983i \(-0.908012\pi\)
−0.958533 + 0.284983i \(0.908012\pi\)
\(824\) 0 0
\(825\) 8.74571 0.304486
\(826\) 0 0
\(827\) 31.9183 1.10991 0.554954 0.831881i \(-0.312736\pi\)
0.554954 + 0.831881i \(0.312736\pi\)
\(828\) 0 0
\(829\) −12.0931 −0.420012 −0.210006 0.977700i \(-0.567348\pi\)
−0.210006 + 0.977700i \(0.567348\pi\)
\(830\) 0 0
\(831\) −2.20500 −0.0764905
\(832\) 0 0
\(833\) 12.2261 0.423608
\(834\) 0 0
\(835\) 0.494893 0.0171265
\(836\) 0 0
\(837\) 17.2026 0.594610
\(838\) 0 0
\(839\) 41.0022 1.41555 0.707776 0.706437i \(-0.249699\pi\)
0.707776 + 0.706437i \(0.249699\pi\)
\(840\) 0 0
\(841\) −21.3143 −0.734976
\(842\) 0 0
\(843\) 1.88093 0.0647826
\(844\) 0 0
\(845\) 33.3095 1.14588
\(846\) 0 0
\(847\) −6.51121 −0.223728
\(848\) 0 0
\(849\) −14.1811 −0.486695
\(850\) 0 0
\(851\) −11.4007 −0.390811
\(852\) 0 0
\(853\) 19.1215 0.654709 0.327355 0.944902i \(-0.393843\pi\)
0.327355 + 0.944902i \(0.393843\pi\)
\(854\) 0 0
\(855\) 33.6521 1.15088
\(856\) 0 0
\(857\) 26.7002 0.912062 0.456031 0.889964i \(-0.349271\pi\)
0.456031 + 0.889964i \(0.349271\pi\)
\(858\) 0 0
\(859\) −18.3963 −0.627672 −0.313836 0.949477i \(-0.601614\pi\)
−0.313836 + 0.949477i \(0.601614\pi\)
\(860\) 0 0
\(861\) −10.2464 −0.349196
\(862\) 0 0
\(863\) 54.9965 1.87210 0.936052 0.351862i \(-0.114452\pi\)
0.936052 + 0.351862i \(0.114452\pi\)
\(864\) 0 0
\(865\) −6.73016 −0.228832
\(866\) 0 0
\(867\) −2.40637 −0.0817245
\(868\) 0 0
\(869\) 63.2110 2.14429
\(870\) 0 0
\(871\) 8.17215 0.276903
\(872\) 0 0
\(873\) 18.9405 0.641037
\(874\) 0 0
\(875\) −8.10479 −0.273992
\(876\) 0 0
\(877\) 9.83069 0.331959 0.165979 0.986129i \(-0.446921\pi\)
0.165979 + 0.986129i \(0.446921\pi\)
\(878\) 0 0
\(879\) 1.95194 0.0658372
\(880\) 0 0
\(881\) 2.95845 0.0996726 0.0498363 0.998757i \(-0.484130\pi\)
0.0498363 + 0.998757i \(0.484130\pi\)
\(882\) 0 0
\(883\) 29.7455 1.00102 0.500508 0.865732i \(-0.333147\pi\)
0.500508 + 0.865732i \(0.333147\pi\)
\(884\) 0 0
\(885\) −13.3862 −0.449971
\(886\) 0 0
\(887\) −4.04360 −0.135771 −0.0678854 0.997693i \(-0.521625\pi\)
−0.0678854 + 0.997693i \(0.521625\pi\)
\(888\) 0 0
\(889\) 5.95456 0.199710
\(890\) 0 0
\(891\) 20.4361 0.684636
\(892\) 0 0
\(893\) −7.06285 −0.236349
\(894\) 0 0
\(895\) −67.4862 −2.25581
\(896\) 0 0
\(897\) −7.51508 −0.250921
\(898\) 0 0
\(899\) −13.1671 −0.439147
\(900\) 0 0
\(901\) −35.3783 −1.17862
\(902\) 0 0
\(903\) 7.55852 0.251532
\(904\) 0 0
\(905\) −56.9494 −1.89306
\(906\) 0 0
\(907\) 41.1144 1.36518 0.682591 0.730801i \(-0.260853\pi\)
0.682591 + 0.730801i \(0.260853\pi\)
\(908\) 0 0
\(909\) −12.1205 −0.402011
\(910\) 0 0
\(911\) −9.69984 −0.321370 −0.160685 0.987006i \(-0.551370\pi\)
−0.160685 + 0.987006i \(0.551370\pi\)
\(912\) 0 0
\(913\) 16.1172 0.533400
\(914\) 0 0
\(915\) −2.67613 −0.0884701
\(916\) 0 0
\(917\) −33.6555 −1.11140
\(918\) 0 0
\(919\) −53.1616 −1.75364 −0.876820 0.480819i \(-0.840339\pi\)
−0.876820 + 0.480819i \(0.840339\pi\)
\(920\) 0 0
\(921\) −19.9458 −0.657235
\(922\) 0 0
\(923\) −13.0252 −0.428731
\(924\) 0 0
\(925\) 4.43119 0.145697
\(926\) 0 0
\(927\) 5.82839 0.191430
\(928\) 0 0
\(929\) −26.3532 −0.864621 −0.432311 0.901725i \(-0.642302\pi\)
−0.432311 + 0.901725i \(0.642302\pi\)
\(930\) 0 0
\(931\) −14.9684 −0.490570
\(932\) 0 0
\(933\) −5.70154 −0.186660
\(934\) 0 0
\(935\) 40.4663 1.32339
\(936\) 0 0
\(937\) −53.1013 −1.73474 −0.867372 0.497660i \(-0.834193\pi\)
−0.867372 + 0.497660i \(0.834193\pi\)
\(938\) 0 0
\(939\) 8.52576 0.278228
\(940\) 0 0
\(941\) 29.2627 0.953936 0.476968 0.878921i \(-0.341736\pi\)
0.476968 + 0.878921i \(0.341736\pi\)
\(942\) 0 0
\(943\) −75.4442 −2.45680
\(944\) 0 0
\(945\) −20.2229 −0.657852
\(946\) 0 0
\(947\) 49.5823 1.61121 0.805604 0.592454i \(-0.201841\pi\)
0.805604 + 0.592454i \(0.201841\pi\)
\(948\) 0 0
\(949\) −6.09855 −0.197967
\(950\) 0 0
\(951\) 11.8351 0.383780
\(952\) 0 0
\(953\) −26.7995 −0.868121 −0.434060 0.900884i \(-0.642920\pi\)
−0.434060 + 0.900884i \(0.642920\pi\)
\(954\) 0 0
\(955\) 68.2092 2.20720
\(956\) 0 0
\(957\) 6.83286 0.220875
\(958\) 0 0
\(959\) −37.3613 −1.20646
\(960\) 0 0
\(961\) −8.44223 −0.272330
\(962\) 0 0
\(963\) −34.6984 −1.11814
\(964\) 0 0
\(965\) −7.27231 −0.234104
\(966\) 0 0
\(967\) 10.4565 0.336258 0.168129 0.985765i \(-0.446227\pi\)
0.168129 + 0.985765i \(0.446227\pi\)
\(968\) 0 0
\(969\) −10.5675 −0.339478
\(970\) 0 0
\(971\) 4.10178 0.131632 0.0658162 0.997832i \(-0.479035\pi\)
0.0658162 + 0.997832i \(0.479035\pi\)
\(972\) 0 0
\(973\) −14.5211 −0.465525
\(974\) 0 0
\(975\) 2.92094 0.0935449
\(976\) 0 0
\(977\) 40.0544 1.28145 0.640726 0.767769i \(-0.278633\pi\)
0.640726 + 0.767769i \(0.278633\pi\)
\(978\) 0 0
\(979\) 54.9741 1.75698
\(980\) 0 0
\(981\) 39.8253 1.27152
\(982\) 0 0
\(983\) 29.2160 0.931845 0.465923 0.884826i \(-0.345723\pi\)
0.465923 + 0.884826i \(0.345723\pi\)
\(984\) 0 0
\(985\) 34.6639 1.10448
\(986\) 0 0
\(987\) 1.96181 0.0624450
\(988\) 0 0
\(989\) 55.6535 1.76968
\(990\) 0 0
\(991\) −16.2779 −0.517084 −0.258542 0.966000i \(-0.583242\pi\)
−0.258542 + 0.966000i \(0.583242\pi\)
\(992\) 0 0
\(993\) 11.9905 0.380506
\(994\) 0 0
\(995\) −74.1283 −2.35003
\(996\) 0 0
\(997\) 42.5090 1.34627 0.673137 0.739518i \(-0.264947\pi\)
0.673137 + 0.739518i \(0.264947\pi\)
\(998\) 0 0
\(999\) −4.52307 −0.143104
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 4024.2.a.g.1.18 33
4.3 odd 2 8048.2.a.x.1.16 33
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4024.2.a.g.1.18 33 1.1 even 1 trivial
8048.2.a.x.1.16 33 4.3 odd 2