Properties

Label 6039.2.a.n.1.2
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $1$
Dimension $25$
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] = [6039,2,Mod(1,6039)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, 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("6039.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.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: \(48.2216577807\)
Analytic rank: \(1\)
Dimension: \(25\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Character \(\chi\) \(=\) 6039.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.62068 q^{2} +4.86797 q^{4} -4.12800 q^{5} -0.639358 q^{7} -7.51602 q^{8} +O(q^{10})\) \(q-2.62068 q^{2} +4.86797 q^{4} -4.12800 q^{5} -0.639358 q^{7} -7.51602 q^{8} +10.8182 q^{10} -1.00000 q^{11} +3.39765 q^{13} +1.67555 q^{14} +9.96116 q^{16} -2.27541 q^{17} -3.83711 q^{19} -20.0950 q^{20} +2.62068 q^{22} -0.231954 q^{23} +12.0404 q^{25} -8.90417 q^{26} -3.11237 q^{28} +5.21505 q^{29} -6.79146 q^{31} -11.0730 q^{32} +5.96313 q^{34} +2.63927 q^{35} +0.00835933 q^{37} +10.0558 q^{38} +31.0262 q^{40} +1.25345 q^{41} -5.91520 q^{43} -4.86797 q^{44} +0.607877 q^{46} +2.04677 q^{47} -6.59122 q^{49} -31.5541 q^{50} +16.5397 q^{52} +5.78491 q^{53} +4.12800 q^{55} +4.80543 q^{56} -13.6670 q^{58} +6.98726 q^{59} -1.00000 q^{61} +17.7983 q^{62} +9.09640 q^{64} -14.0255 q^{65} -4.12479 q^{67} -11.0766 q^{68} -6.91669 q^{70} -2.69731 q^{71} +6.26305 q^{73} -0.0219071 q^{74} -18.6789 q^{76} +0.639358 q^{77} +3.96236 q^{79} -41.1197 q^{80} -3.28489 q^{82} -2.54007 q^{83} +9.39291 q^{85} +15.5018 q^{86} +7.51602 q^{88} +13.7324 q^{89} -2.17232 q^{91} -1.12914 q^{92} -5.36393 q^{94} +15.8396 q^{95} -5.19208 q^{97} +17.2735 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8} - 25 q^{11} + 4 q^{13} - 18 q^{14} + 21 q^{16} - 20 q^{17} + 14 q^{19} - 12 q^{20} + 5 q^{22} - 20 q^{23} + 13 q^{25} - 16 q^{26} - 14 q^{28} - 28 q^{29} - 12 q^{31} - 35 q^{32} + 6 q^{34} - 10 q^{35} - 8 q^{37} - 32 q^{38} + 24 q^{40} - 26 q^{41} + 18 q^{43} - 25 q^{44} + 4 q^{46} - 12 q^{47} + 23 q^{49} - 43 q^{50} + 22 q^{52} - 36 q^{53} + 4 q^{55} - 26 q^{56} - 20 q^{58} - 46 q^{59} - 25 q^{61} + 14 q^{62} - 13 q^{64} - 60 q^{65} - 20 q^{67} - 44 q^{68} - 20 q^{70} - 52 q^{71} + 6 q^{73} - 32 q^{74} - 4 q^{77} + 26 q^{79} - 52 q^{80} + 6 q^{82} - 38 q^{83} - 4 q^{85} - 34 q^{86} + 15 q^{88} - 82 q^{89} - 58 q^{91} - 36 q^{92} + 16 q^{94} - 30 q^{95} - 35 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.62068 −1.85310 −0.926550 0.376171i \(-0.877241\pi\)
−0.926550 + 0.376171i \(0.877241\pi\)
\(3\) 0 0
\(4\) 4.86797 2.43398
\(5\) −4.12800 −1.84610 −0.923050 0.384681i \(-0.874312\pi\)
−0.923050 + 0.384681i \(0.874312\pi\)
\(6\) 0 0
\(7\) −0.639358 −0.241655 −0.120827 0.992674i \(-0.538555\pi\)
−0.120827 + 0.992674i \(0.538555\pi\)
\(8\) −7.51602 −2.65731
\(9\) 0 0
\(10\) 10.8182 3.42101
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 3.39765 0.942340 0.471170 0.882042i \(-0.343832\pi\)
0.471170 + 0.882042i \(0.343832\pi\)
\(14\) 1.67555 0.447810
\(15\) 0 0
\(16\) 9.96116 2.49029
\(17\) −2.27541 −0.551868 −0.275934 0.961177i \(-0.588987\pi\)
−0.275934 + 0.961177i \(0.588987\pi\)
\(18\) 0 0
\(19\) −3.83711 −0.880294 −0.440147 0.897926i \(-0.645074\pi\)
−0.440147 + 0.897926i \(0.645074\pi\)
\(20\) −20.0950 −4.49337
\(21\) 0 0
\(22\) 2.62068 0.558731
\(23\) −0.231954 −0.0483657 −0.0241829 0.999708i \(-0.507698\pi\)
−0.0241829 + 0.999708i \(0.507698\pi\)
\(24\) 0 0
\(25\) 12.0404 2.40808
\(26\) −8.90417 −1.74625
\(27\) 0 0
\(28\) −3.11237 −0.588183
\(29\) 5.21505 0.968410 0.484205 0.874955i \(-0.339109\pi\)
0.484205 + 0.874955i \(0.339109\pi\)
\(30\) 0 0
\(31\) −6.79146 −1.21978 −0.609891 0.792485i \(-0.708787\pi\)
−0.609891 + 0.792485i \(0.708787\pi\)
\(32\) −11.0730 −1.95744
\(33\) 0 0
\(34\) 5.96313 1.02267
\(35\) 2.63927 0.446118
\(36\) 0 0
\(37\) 0.00835933 0.00137427 0.000687133 1.00000i \(-0.499781\pi\)
0.000687133 1.00000i \(0.499781\pi\)
\(38\) 10.0558 1.63127
\(39\) 0 0
\(40\) 31.0262 4.90567
\(41\) 1.25345 0.195756 0.0978778 0.995198i \(-0.468795\pi\)
0.0978778 + 0.995198i \(0.468795\pi\)
\(42\) 0 0
\(43\) −5.91520 −0.902059 −0.451029 0.892509i \(-0.648943\pi\)
−0.451029 + 0.892509i \(0.648943\pi\)
\(44\) −4.86797 −0.733873
\(45\) 0 0
\(46\) 0.607877 0.0896266
\(47\) 2.04677 0.298552 0.149276 0.988796i \(-0.452306\pi\)
0.149276 + 0.988796i \(0.452306\pi\)
\(48\) 0 0
\(49\) −6.59122 −0.941603
\(50\) −31.5541 −4.46242
\(51\) 0 0
\(52\) 16.5397 2.29364
\(53\) 5.78491 0.794618 0.397309 0.917685i \(-0.369944\pi\)
0.397309 + 0.917685i \(0.369944\pi\)
\(54\) 0 0
\(55\) 4.12800 0.556620
\(56\) 4.80543 0.642152
\(57\) 0 0
\(58\) −13.6670 −1.79456
\(59\) 6.98726 0.909664 0.454832 0.890577i \(-0.349699\pi\)
0.454832 + 0.890577i \(0.349699\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) 17.7983 2.26038
\(63\) 0 0
\(64\) 9.09640 1.13705
\(65\) −14.0255 −1.73965
\(66\) 0 0
\(67\) −4.12479 −0.503923 −0.251962 0.967737i \(-0.581076\pi\)
−0.251962 + 0.967737i \(0.581076\pi\)
\(68\) −11.0766 −1.34324
\(69\) 0 0
\(70\) −6.91669 −0.826702
\(71\) −2.69731 −0.320111 −0.160056 0.987108i \(-0.551167\pi\)
−0.160056 + 0.987108i \(0.551167\pi\)
\(72\) 0 0
\(73\) 6.26305 0.733035 0.366517 0.930411i \(-0.380550\pi\)
0.366517 + 0.930411i \(0.380550\pi\)
\(74\) −0.0219071 −0.00254665
\(75\) 0 0
\(76\) −18.6789 −2.14262
\(77\) 0.639358 0.0728616
\(78\) 0 0
\(79\) 3.96236 0.445800 0.222900 0.974841i \(-0.428448\pi\)
0.222900 + 0.974841i \(0.428448\pi\)
\(80\) −41.1197 −4.59732
\(81\) 0 0
\(82\) −3.28489 −0.362755
\(83\) −2.54007 −0.278809 −0.139404 0.990236i \(-0.544519\pi\)
−0.139404 + 0.990236i \(0.544519\pi\)
\(84\) 0 0
\(85\) 9.39291 1.01880
\(86\) 15.5018 1.67161
\(87\) 0 0
\(88\) 7.51602 0.801211
\(89\) 13.7324 1.45564 0.727818 0.685770i \(-0.240534\pi\)
0.727818 + 0.685770i \(0.240534\pi\)
\(90\) 0 0
\(91\) −2.17232 −0.227721
\(92\) −1.12914 −0.117721
\(93\) 0 0
\(94\) −5.36393 −0.553248
\(95\) 15.8396 1.62511
\(96\) 0 0
\(97\) −5.19208 −0.527175 −0.263588 0.964635i \(-0.584906\pi\)
−0.263588 + 0.964635i \(0.584906\pi\)
\(98\) 17.2735 1.74489
\(99\) 0 0
\(100\) 58.6123 5.86123
\(101\) −12.9663 −1.29019 −0.645097 0.764101i \(-0.723183\pi\)
−0.645097 + 0.764101i \(0.723183\pi\)
\(102\) 0 0
\(103\) 18.0846 1.78193 0.890964 0.454075i \(-0.150030\pi\)
0.890964 + 0.454075i \(0.150030\pi\)
\(104\) −25.5368 −2.50409
\(105\) 0 0
\(106\) −15.1604 −1.47251
\(107\) 15.0119 1.45126 0.725628 0.688088i \(-0.241550\pi\)
0.725628 + 0.688088i \(0.241550\pi\)
\(108\) 0 0
\(109\) −0.845703 −0.0810037 −0.0405018 0.999179i \(-0.512896\pi\)
−0.0405018 + 0.999179i \(0.512896\pi\)
\(110\) −10.8182 −1.03147
\(111\) 0 0
\(112\) −6.36875 −0.601790
\(113\) −5.27747 −0.496462 −0.248231 0.968701i \(-0.579849\pi\)
−0.248231 + 0.968701i \(0.579849\pi\)
\(114\) 0 0
\(115\) 0.957506 0.0892879
\(116\) 25.3867 2.35709
\(117\) 0 0
\(118\) −18.3114 −1.68570
\(119\) 1.45480 0.133362
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.62068 0.237265
\(123\) 0 0
\(124\) −33.0606 −2.96893
\(125\) −29.0629 −2.59946
\(126\) 0 0
\(127\) 2.53585 0.225021 0.112510 0.993651i \(-0.464111\pi\)
0.112510 + 0.993651i \(0.464111\pi\)
\(128\) −1.69281 −0.149625
\(129\) 0 0
\(130\) 36.7564 3.22375
\(131\) 20.6328 1.80270 0.901350 0.433092i \(-0.142578\pi\)
0.901350 + 0.433092i \(0.142578\pi\)
\(132\) 0 0
\(133\) 2.45329 0.212727
\(134\) 10.8098 0.933821
\(135\) 0 0
\(136\) 17.1020 1.46649
\(137\) 2.22550 0.190137 0.0950686 0.995471i \(-0.469693\pi\)
0.0950686 + 0.995471i \(0.469693\pi\)
\(138\) 0 0
\(139\) 2.85310 0.241996 0.120998 0.992653i \(-0.461390\pi\)
0.120998 + 0.992653i \(0.461390\pi\)
\(140\) 12.8479 1.08584
\(141\) 0 0
\(142\) 7.06878 0.593199
\(143\) −3.39765 −0.284126
\(144\) 0 0
\(145\) −21.5277 −1.78778
\(146\) −16.4135 −1.35839
\(147\) 0 0
\(148\) 0.0406929 0.00334494
\(149\) 4.97733 0.407759 0.203879 0.978996i \(-0.434645\pi\)
0.203879 + 0.978996i \(0.434645\pi\)
\(150\) 0 0
\(151\) 23.9900 1.95228 0.976138 0.217149i \(-0.0696759\pi\)
0.976138 + 0.217149i \(0.0696759\pi\)
\(152\) 28.8398 2.33922
\(153\) 0 0
\(154\) −1.67555 −0.135020
\(155\) 28.0352 2.25184
\(156\) 0 0
\(157\) 21.6861 1.73074 0.865371 0.501131i \(-0.167083\pi\)
0.865371 + 0.501131i \(0.167083\pi\)
\(158\) −10.3841 −0.826112
\(159\) 0 0
\(160\) 45.7093 3.61363
\(161\) 0.148302 0.0116878
\(162\) 0 0
\(163\) 1.46311 0.114599 0.0572996 0.998357i \(-0.481751\pi\)
0.0572996 + 0.998357i \(0.481751\pi\)
\(164\) 6.10174 0.476466
\(165\) 0 0
\(166\) 6.65670 0.516660
\(167\) −1.34205 −0.103851 −0.0519253 0.998651i \(-0.516536\pi\)
−0.0519253 + 0.998651i \(0.516536\pi\)
\(168\) 0 0
\(169\) −1.45594 −0.111996
\(170\) −24.6158 −1.88795
\(171\) 0 0
\(172\) −28.7950 −2.19560
\(173\) −20.7606 −1.57840 −0.789199 0.614137i \(-0.789504\pi\)
−0.789199 + 0.614137i \(0.789504\pi\)
\(174\) 0 0
\(175\) −7.69813 −0.581924
\(176\) −9.96116 −0.750851
\(177\) 0 0
\(178\) −35.9884 −2.69744
\(179\) −1.18299 −0.0884206 −0.0442103 0.999022i \(-0.514077\pi\)
−0.0442103 + 0.999022i \(0.514077\pi\)
\(180\) 0 0
\(181\) −8.98508 −0.667856 −0.333928 0.942599i \(-0.608374\pi\)
−0.333928 + 0.942599i \(0.608374\pi\)
\(182\) 5.69295 0.421990
\(183\) 0 0
\(184\) 1.74337 0.128523
\(185\) −0.0345073 −0.00253703
\(186\) 0 0
\(187\) 2.27541 0.166395
\(188\) 9.96361 0.726671
\(189\) 0 0
\(190\) −41.5106 −3.01149
\(191\) −25.1907 −1.82274 −0.911369 0.411591i \(-0.864973\pi\)
−0.911369 + 0.411591i \(0.864973\pi\)
\(192\) 0 0
\(193\) 6.01947 0.433291 0.216645 0.976250i \(-0.430488\pi\)
0.216645 + 0.976250i \(0.430488\pi\)
\(194\) 13.6068 0.976909
\(195\) 0 0
\(196\) −32.0858 −2.29185
\(197\) −22.1592 −1.57878 −0.789389 0.613894i \(-0.789602\pi\)
−0.789389 + 0.613894i \(0.789602\pi\)
\(198\) 0 0
\(199\) 4.21905 0.299081 0.149540 0.988756i \(-0.452221\pi\)
0.149540 + 0.988756i \(0.452221\pi\)
\(200\) −90.4960 −6.39903
\(201\) 0 0
\(202\) 33.9805 2.39086
\(203\) −3.33428 −0.234021
\(204\) 0 0
\(205\) −5.17424 −0.361384
\(206\) −47.3939 −3.30209
\(207\) 0 0
\(208\) 33.8446 2.34670
\(209\) 3.83711 0.265419
\(210\) 0 0
\(211\) −4.98072 −0.342887 −0.171443 0.985194i \(-0.554843\pi\)
−0.171443 + 0.985194i \(0.554843\pi\)
\(212\) 28.1607 1.93409
\(213\) 0 0
\(214\) −39.3414 −2.68932
\(215\) 24.4179 1.66529
\(216\) 0 0
\(217\) 4.34218 0.294766
\(218\) 2.21632 0.150108
\(219\) 0 0
\(220\) 20.0950 1.35480
\(221\) −7.73106 −0.520048
\(222\) 0 0
\(223\) −3.93611 −0.263581 −0.131791 0.991278i \(-0.542073\pi\)
−0.131791 + 0.991278i \(0.542073\pi\)
\(224\) 7.07959 0.473025
\(225\) 0 0
\(226\) 13.8306 0.919995
\(227\) 19.4854 1.29329 0.646647 0.762789i \(-0.276171\pi\)
0.646647 + 0.762789i \(0.276171\pi\)
\(228\) 0 0
\(229\) −9.52720 −0.629575 −0.314787 0.949162i \(-0.601933\pi\)
−0.314787 + 0.949162i \(0.601933\pi\)
\(230\) −2.50932 −0.165460
\(231\) 0 0
\(232\) −39.1964 −2.57337
\(233\) −25.2717 −1.65561 −0.827803 0.561019i \(-0.810409\pi\)
−0.827803 + 0.561019i \(0.810409\pi\)
\(234\) 0 0
\(235\) −8.44908 −0.551157
\(236\) 34.0137 2.21411
\(237\) 0 0
\(238\) −3.81257 −0.247132
\(239\) 1.36331 0.0881853 0.0440927 0.999027i \(-0.485960\pi\)
0.0440927 + 0.999027i \(0.485960\pi\)
\(240\) 0 0
\(241\) 7.47697 0.481634 0.240817 0.970571i \(-0.422585\pi\)
0.240817 + 0.970571i \(0.422585\pi\)
\(242\) −2.62068 −0.168464
\(243\) 0 0
\(244\) −4.86797 −0.311640
\(245\) 27.2086 1.73829
\(246\) 0 0
\(247\) −13.0372 −0.829536
\(248\) 51.0448 3.24135
\(249\) 0 0
\(250\) 76.1644 4.81706
\(251\) −3.90296 −0.246353 −0.123176 0.992385i \(-0.539308\pi\)
−0.123176 + 0.992385i \(0.539308\pi\)
\(252\) 0 0
\(253\) 0.231954 0.0145828
\(254\) −6.64566 −0.416986
\(255\) 0 0
\(256\) −13.7565 −0.859780
\(257\) −8.69767 −0.542546 −0.271273 0.962502i \(-0.587445\pi\)
−0.271273 + 0.962502i \(0.587445\pi\)
\(258\) 0 0
\(259\) −0.00534460 −0.000332098 0
\(260\) −68.2758 −4.23429
\(261\) 0 0
\(262\) −54.0721 −3.34058
\(263\) 24.6129 1.51770 0.758849 0.651266i \(-0.225762\pi\)
0.758849 + 0.651266i \(0.225762\pi\)
\(264\) 0 0
\(265\) −23.8801 −1.46694
\(266\) −6.42929 −0.394205
\(267\) 0 0
\(268\) −20.0793 −1.22654
\(269\) 9.87079 0.601833 0.300916 0.953651i \(-0.402707\pi\)
0.300916 + 0.953651i \(0.402707\pi\)
\(270\) 0 0
\(271\) −16.8200 −1.02174 −0.510871 0.859658i \(-0.670677\pi\)
−0.510871 + 0.859658i \(0.670677\pi\)
\(272\) −22.6657 −1.37431
\(273\) 0 0
\(274\) −5.83232 −0.352344
\(275\) −12.0404 −0.726064
\(276\) 0 0
\(277\) −4.27328 −0.256757 −0.128378 0.991725i \(-0.540977\pi\)
−0.128378 + 0.991725i \(0.540977\pi\)
\(278\) −7.47705 −0.448444
\(279\) 0 0
\(280\) −19.8368 −1.18548
\(281\) −8.03268 −0.479190 −0.239595 0.970873i \(-0.577015\pi\)
−0.239595 + 0.970873i \(0.577015\pi\)
\(282\) 0 0
\(283\) 15.2077 0.904006 0.452003 0.892016i \(-0.350710\pi\)
0.452003 + 0.892016i \(0.350710\pi\)
\(284\) −13.1304 −0.779146
\(285\) 0 0
\(286\) 8.90417 0.526514
\(287\) −0.801402 −0.0473053
\(288\) 0 0
\(289\) −11.8225 −0.695441
\(290\) 56.4173 3.31294
\(291\) 0 0
\(292\) 30.4883 1.78419
\(293\) 20.1569 1.17758 0.588788 0.808287i \(-0.299605\pi\)
0.588788 + 0.808287i \(0.299605\pi\)
\(294\) 0 0
\(295\) −28.8434 −1.67933
\(296\) −0.0628289 −0.00365186
\(297\) 0 0
\(298\) −13.0440 −0.755618
\(299\) −0.788099 −0.0455770
\(300\) 0 0
\(301\) 3.78193 0.217987
\(302\) −62.8701 −3.61777
\(303\) 0 0
\(304\) −38.2221 −2.19219
\(305\) 4.12800 0.236369
\(306\) 0 0
\(307\) 7.07947 0.404047 0.202023 0.979381i \(-0.435248\pi\)
0.202023 + 0.979381i \(0.435248\pi\)
\(308\) 3.11237 0.177344
\(309\) 0 0
\(310\) −73.4712 −4.17289
\(311\) 10.2641 0.582026 0.291013 0.956719i \(-0.406008\pi\)
0.291013 + 0.956719i \(0.406008\pi\)
\(312\) 0 0
\(313\) 13.0568 0.738012 0.369006 0.929427i \(-0.379698\pi\)
0.369006 + 0.929427i \(0.379698\pi\)
\(314\) −56.8325 −3.20724
\(315\) 0 0
\(316\) 19.2886 1.08507
\(317\) 16.3206 0.916656 0.458328 0.888783i \(-0.348449\pi\)
0.458328 + 0.888783i \(0.348449\pi\)
\(318\) 0 0
\(319\) −5.21505 −0.291987
\(320\) −37.5500 −2.09911
\(321\) 0 0
\(322\) −0.388651 −0.0216587
\(323\) 8.73101 0.485807
\(324\) 0 0
\(325\) 40.9092 2.26923
\(326\) −3.83433 −0.212364
\(327\) 0 0
\(328\) −9.42094 −0.520184
\(329\) −1.30862 −0.0721466
\(330\) 0 0
\(331\) −15.8397 −0.870631 −0.435315 0.900278i \(-0.643363\pi\)
−0.435315 + 0.900278i \(0.643363\pi\)
\(332\) −12.3650 −0.678615
\(333\) 0 0
\(334\) 3.51707 0.192446
\(335\) 17.0272 0.930293
\(336\) 0 0
\(337\) 33.3236 1.81525 0.907626 0.419780i \(-0.137893\pi\)
0.907626 + 0.419780i \(0.137893\pi\)
\(338\) 3.81556 0.207539
\(339\) 0 0
\(340\) 45.7243 2.47975
\(341\) 6.79146 0.367778
\(342\) 0 0
\(343\) 8.68966 0.469197
\(344\) 44.4587 2.39705
\(345\) 0 0
\(346\) 54.4069 2.92493
\(347\) 20.0513 1.07641 0.538206 0.842814i \(-0.319102\pi\)
0.538206 + 0.842814i \(0.319102\pi\)
\(348\) 0 0
\(349\) −30.1440 −1.61357 −0.806786 0.590844i \(-0.798795\pi\)
−0.806786 + 0.590844i \(0.798795\pi\)
\(350\) 20.1744 1.07836
\(351\) 0 0
\(352\) 11.0730 0.590191
\(353\) 6.43535 0.342519 0.171259 0.985226i \(-0.445216\pi\)
0.171259 + 0.985226i \(0.445216\pi\)
\(354\) 0 0
\(355\) 11.1345 0.590958
\(356\) 66.8491 3.54299
\(357\) 0 0
\(358\) 3.10023 0.163852
\(359\) −19.5047 −1.02942 −0.514708 0.857365i \(-0.672100\pi\)
−0.514708 + 0.857365i \(0.672100\pi\)
\(360\) 0 0
\(361\) −4.27656 −0.225082
\(362\) 23.5470 1.23760
\(363\) 0 0
\(364\) −10.5748 −0.554268
\(365\) −25.8539 −1.35325
\(366\) 0 0
\(367\) −27.5866 −1.44001 −0.720004 0.693969i \(-0.755860\pi\)
−0.720004 + 0.693969i \(0.755860\pi\)
\(368\) −2.31053 −0.120445
\(369\) 0 0
\(370\) 0.0904327 0.00470137
\(371\) −3.69863 −0.192023
\(372\) 0 0
\(373\) −4.80244 −0.248661 −0.124331 0.992241i \(-0.539678\pi\)
−0.124331 + 0.992241i \(0.539678\pi\)
\(374\) −5.96313 −0.308346
\(375\) 0 0
\(376\) −15.3836 −0.793348
\(377\) 17.7189 0.912571
\(378\) 0 0
\(379\) −24.3813 −1.25239 −0.626193 0.779668i \(-0.715388\pi\)
−0.626193 + 0.779668i \(0.715388\pi\)
\(380\) 77.1067 3.95549
\(381\) 0 0
\(382\) 66.0169 3.37772
\(383\) −36.5592 −1.86809 −0.934044 0.357159i \(-0.883745\pi\)
−0.934044 + 0.357159i \(0.883745\pi\)
\(384\) 0 0
\(385\) −2.63927 −0.134510
\(386\) −15.7751 −0.802932
\(387\) 0 0
\(388\) −25.2748 −1.28314
\(389\) −18.4912 −0.937539 −0.468770 0.883320i \(-0.655303\pi\)
−0.468770 + 0.883320i \(0.655303\pi\)
\(390\) 0 0
\(391\) 0.527791 0.0266915
\(392\) 49.5398 2.50214
\(393\) 0 0
\(394\) 58.0722 2.92563
\(395\) −16.3566 −0.822991
\(396\) 0 0
\(397\) 35.3234 1.77283 0.886414 0.462893i \(-0.153189\pi\)
0.886414 + 0.462893i \(0.153189\pi\)
\(398\) −11.0568 −0.554226
\(399\) 0 0
\(400\) 119.936 5.99682
\(401\) −30.7487 −1.53552 −0.767759 0.640739i \(-0.778628\pi\)
−0.767759 + 0.640739i \(0.778628\pi\)
\(402\) 0 0
\(403\) −23.0750 −1.14945
\(404\) −63.1194 −3.14031
\(405\) 0 0
\(406\) 8.73809 0.433664
\(407\) −0.00835933 −0.000414357 0
\(408\) 0 0
\(409\) 26.4879 1.30974 0.654872 0.755740i \(-0.272723\pi\)
0.654872 + 0.755740i \(0.272723\pi\)
\(410\) 13.5600 0.669682
\(411\) 0 0
\(412\) 88.0351 4.33718
\(413\) −4.46736 −0.219824
\(414\) 0 0
\(415\) 10.4854 0.514708
\(416\) −37.6221 −1.84458
\(417\) 0 0
\(418\) −10.0558 −0.491848
\(419\) 32.9507 1.60975 0.804875 0.593445i \(-0.202232\pi\)
0.804875 + 0.593445i \(0.202232\pi\)
\(420\) 0 0
\(421\) 0.537173 0.0261802 0.0130901 0.999914i \(-0.495833\pi\)
0.0130901 + 0.999914i \(0.495833\pi\)
\(422\) 13.0529 0.635404
\(423\) 0 0
\(424\) −43.4795 −2.11155
\(425\) −27.3969 −1.32894
\(426\) 0 0
\(427\) 0.639358 0.0309407
\(428\) 73.0774 3.53233
\(429\) 0 0
\(430\) −63.9916 −3.08595
\(431\) −24.2267 −1.16696 −0.583480 0.812127i \(-0.698309\pi\)
−0.583480 + 0.812127i \(0.698309\pi\)
\(432\) 0 0
\(433\) 16.0674 0.772151 0.386076 0.922467i \(-0.373830\pi\)
0.386076 + 0.922467i \(0.373830\pi\)
\(434\) −11.3795 −0.546231
\(435\) 0 0
\(436\) −4.11685 −0.197162
\(437\) 0.890034 0.0425761
\(438\) 0 0
\(439\) 26.3777 1.25894 0.629468 0.777026i \(-0.283273\pi\)
0.629468 + 0.777026i \(0.283273\pi\)
\(440\) −31.0262 −1.47911
\(441\) 0 0
\(442\) 20.2606 0.963700
\(443\) 27.7002 1.31608 0.658039 0.752984i \(-0.271386\pi\)
0.658039 + 0.752984i \(0.271386\pi\)
\(444\) 0 0
\(445\) −56.6876 −2.68725
\(446\) 10.3153 0.488443
\(447\) 0 0
\(448\) −5.81586 −0.274773
\(449\) −28.8938 −1.36358 −0.681792 0.731546i \(-0.738799\pi\)
−0.681792 + 0.731546i \(0.738799\pi\)
\(450\) 0 0
\(451\) −1.25345 −0.0590225
\(452\) −25.6905 −1.20838
\(453\) 0 0
\(454\) −51.0651 −2.39660
\(455\) 8.96734 0.420395
\(456\) 0 0
\(457\) −5.17075 −0.241877 −0.120939 0.992660i \(-0.538590\pi\)
−0.120939 + 0.992660i \(0.538590\pi\)
\(458\) 24.9677 1.16667
\(459\) 0 0
\(460\) 4.66111 0.217325
\(461\) −0.443270 −0.0206451 −0.0103226 0.999947i \(-0.503286\pi\)
−0.0103226 + 0.999947i \(0.503286\pi\)
\(462\) 0 0
\(463\) −0.752394 −0.0349667 −0.0174834 0.999847i \(-0.505565\pi\)
−0.0174834 + 0.999847i \(0.505565\pi\)
\(464\) 51.9479 2.41162
\(465\) 0 0
\(466\) 66.2291 3.06800
\(467\) −18.1513 −0.839942 −0.419971 0.907538i \(-0.637960\pi\)
−0.419971 + 0.907538i \(0.637960\pi\)
\(468\) 0 0
\(469\) 2.63722 0.121775
\(470\) 22.1423 1.02135
\(471\) 0 0
\(472\) −52.5164 −2.41726
\(473\) 5.91520 0.271981
\(474\) 0 0
\(475\) −46.2004 −2.11982
\(476\) 7.08193 0.324600
\(477\) 0 0
\(478\) −3.57281 −0.163416
\(479\) −27.0401 −1.23549 −0.617747 0.786377i \(-0.711954\pi\)
−0.617747 + 0.786377i \(0.711954\pi\)
\(480\) 0 0
\(481\) 0.0284021 0.00129502
\(482\) −19.5948 −0.892516
\(483\) 0 0
\(484\) 4.86797 0.221271
\(485\) 21.4329 0.973218
\(486\) 0 0
\(487\) 12.5533 0.568843 0.284421 0.958699i \(-0.408199\pi\)
0.284421 + 0.958699i \(0.408199\pi\)
\(488\) 7.51602 0.340234
\(489\) 0 0
\(490\) −71.3050 −3.22123
\(491\) 17.7127 0.799361 0.399681 0.916654i \(-0.369121\pi\)
0.399681 + 0.916654i \(0.369121\pi\)
\(492\) 0 0
\(493\) −11.8664 −0.534435
\(494\) 34.1663 1.53721
\(495\) 0 0
\(496\) −67.6508 −3.03761
\(497\) 1.72455 0.0773564
\(498\) 0 0
\(499\) −27.9820 −1.25265 −0.626323 0.779563i \(-0.715441\pi\)
−0.626323 + 0.779563i \(0.715441\pi\)
\(500\) −141.477 −6.32704
\(501\) 0 0
\(502\) 10.2284 0.456517
\(503\) −8.96909 −0.399912 −0.199956 0.979805i \(-0.564080\pi\)
−0.199956 + 0.979805i \(0.564080\pi\)
\(504\) 0 0
\(505\) 53.5249 2.38182
\(506\) −0.607877 −0.0270234
\(507\) 0 0
\(508\) 12.3444 0.547696
\(509\) 6.64241 0.294420 0.147210 0.989105i \(-0.452971\pi\)
0.147210 + 0.989105i \(0.452971\pi\)
\(510\) 0 0
\(511\) −4.00433 −0.177141
\(512\) 39.4369 1.74288
\(513\) 0 0
\(514\) 22.7938 1.00539
\(515\) −74.6532 −3.28961
\(516\) 0 0
\(517\) −2.04677 −0.0900169
\(518\) 0.0140065 0.000615410 0
\(519\) 0 0
\(520\) 105.416 4.62281
\(521\) −0.260058 −0.0113933 −0.00569667 0.999984i \(-0.501813\pi\)
−0.00569667 + 0.999984i \(0.501813\pi\)
\(522\) 0 0
\(523\) 20.7386 0.906838 0.453419 0.891298i \(-0.350204\pi\)
0.453419 + 0.891298i \(0.350204\pi\)
\(524\) 100.440 4.38774
\(525\) 0 0
\(526\) −64.5026 −2.81245
\(527\) 15.4534 0.673159
\(528\) 0 0
\(529\) −22.9462 −0.997661
\(530\) 62.5822 2.71840
\(531\) 0 0
\(532\) 11.9425 0.517774
\(533\) 4.25878 0.184468
\(534\) 0 0
\(535\) −61.9692 −2.67916
\(536\) 31.0020 1.33908
\(537\) 0 0
\(538\) −25.8682 −1.11526
\(539\) 6.59122 0.283904
\(540\) 0 0
\(541\) −20.3924 −0.876738 −0.438369 0.898795i \(-0.644444\pi\)
−0.438369 + 0.898795i \(0.644444\pi\)
\(542\) 44.0798 1.89339
\(543\) 0 0
\(544\) 25.1956 1.08025
\(545\) 3.49107 0.149541
\(546\) 0 0
\(547\) 15.7406 0.673019 0.336509 0.941680i \(-0.390754\pi\)
0.336509 + 0.941680i \(0.390754\pi\)
\(548\) 10.8337 0.462791
\(549\) 0 0
\(550\) 31.5541 1.34547
\(551\) −20.0107 −0.852486
\(552\) 0 0
\(553\) −2.53337 −0.107730
\(554\) 11.1989 0.475796
\(555\) 0 0
\(556\) 13.8888 0.589015
\(557\) −31.8995 −1.35162 −0.675812 0.737074i \(-0.736207\pi\)
−0.675812 + 0.737074i \(0.736207\pi\)
\(558\) 0 0
\(559\) −20.0978 −0.850046
\(560\) 26.2902 1.11096
\(561\) 0 0
\(562\) 21.0511 0.887987
\(563\) 22.4287 0.945258 0.472629 0.881261i \(-0.343305\pi\)
0.472629 + 0.881261i \(0.343305\pi\)
\(564\) 0 0
\(565\) 21.7854 0.916519
\(566\) −39.8546 −1.67522
\(567\) 0 0
\(568\) 20.2730 0.850637
\(569\) −1.24170 −0.0520547 −0.0260274 0.999661i \(-0.508286\pi\)
−0.0260274 + 0.999661i \(0.508286\pi\)
\(570\) 0 0
\(571\) −1.51920 −0.0635766 −0.0317883 0.999495i \(-0.510120\pi\)
−0.0317883 + 0.999495i \(0.510120\pi\)
\(572\) −16.5397 −0.691558
\(573\) 0 0
\(574\) 2.10022 0.0876614
\(575\) −2.79282 −0.116469
\(576\) 0 0
\(577\) −28.7053 −1.19502 −0.597509 0.801862i \(-0.703843\pi\)
−0.597509 + 0.801862i \(0.703843\pi\)
\(578\) 30.9830 1.28872
\(579\) 0 0
\(580\) −104.796 −4.35143
\(581\) 1.62401 0.0673754
\(582\) 0 0
\(583\) −5.78491 −0.239586
\(584\) −47.0732 −1.94790
\(585\) 0 0
\(586\) −52.8247 −2.18217
\(587\) −40.9540 −1.69035 −0.845176 0.534488i \(-0.820504\pi\)
−0.845176 + 0.534488i \(0.820504\pi\)
\(588\) 0 0
\(589\) 26.0596 1.07377
\(590\) 75.5894 3.11197
\(591\) 0 0
\(592\) 0.0832686 0.00342232
\(593\) −47.1502 −1.93623 −0.968113 0.250515i \(-0.919400\pi\)
−0.968113 + 0.250515i \(0.919400\pi\)
\(594\) 0 0
\(595\) −6.00543 −0.246199
\(596\) 24.2295 0.992478
\(597\) 0 0
\(598\) 2.06536 0.0844587
\(599\) 17.6382 0.720676 0.360338 0.932822i \(-0.382661\pi\)
0.360338 + 0.932822i \(0.382661\pi\)
\(600\) 0 0
\(601\) −22.2149 −0.906163 −0.453082 0.891469i \(-0.649675\pi\)
−0.453082 + 0.891469i \(0.649675\pi\)
\(602\) −9.91122 −0.403951
\(603\) 0 0
\(604\) 116.782 4.75181
\(605\) −4.12800 −0.167827
\(606\) 0 0
\(607\) 30.1450 1.22355 0.611775 0.791032i \(-0.290456\pi\)
0.611775 + 0.791032i \(0.290456\pi\)
\(608\) 42.4883 1.72313
\(609\) 0 0
\(610\) −10.8182 −0.438015
\(611\) 6.95422 0.281338
\(612\) 0 0
\(613\) −20.8524 −0.842219 −0.421110 0.907010i \(-0.638359\pi\)
−0.421110 + 0.907010i \(0.638359\pi\)
\(614\) −18.5530 −0.748739
\(615\) 0 0
\(616\) −4.80543 −0.193616
\(617\) −15.5474 −0.625913 −0.312956 0.949768i \(-0.601319\pi\)
−0.312956 + 0.949768i \(0.601319\pi\)
\(618\) 0 0
\(619\) 32.1684 1.29296 0.646479 0.762932i \(-0.276241\pi\)
0.646479 + 0.762932i \(0.276241\pi\)
\(620\) 136.474 5.48094
\(621\) 0 0
\(622\) −26.8990 −1.07855
\(623\) −8.77995 −0.351761
\(624\) 0 0
\(625\) 59.7695 2.39078
\(626\) −34.2176 −1.36761
\(627\) 0 0
\(628\) 105.567 4.21260
\(629\) −0.0190209 −0.000758413 0
\(630\) 0 0
\(631\) −9.52156 −0.379047 −0.189524 0.981876i \(-0.560694\pi\)
−0.189524 + 0.981876i \(0.560694\pi\)
\(632\) −29.7812 −1.18463
\(633\) 0 0
\(634\) −42.7710 −1.69866
\(635\) −10.4680 −0.415410
\(636\) 0 0
\(637\) −22.3947 −0.887310
\(638\) 13.6670 0.541080
\(639\) 0 0
\(640\) 6.98794 0.276223
\(641\) −32.6251 −1.28861 −0.644306 0.764767i \(-0.722854\pi\)
−0.644306 + 0.764767i \(0.722854\pi\)
\(642\) 0 0
\(643\) −48.8116 −1.92494 −0.962472 0.271382i \(-0.912519\pi\)
−0.962472 + 0.271382i \(0.912519\pi\)
\(644\) 0.721927 0.0284479
\(645\) 0 0
\(646\) −22.8812 −0.900249
\(647\) 3.82743 0.150472 0.0752359 0.997166i \(-0.476029\pi\)
0.0752359 + 0.997166i \(0.476029\pi\)
\(648\) 0 0
\(649\) −6.98726 −0.274274
\(650\) −107.210 −4.20512
\(651\) 0 0
\(652\) 7.12235 0.278933
\(653\) 11.7107 0.458275 0.229138 0.973394i \(-0.426409\pi\)
0.229138 + 0.973394i \(0.426409\pi\)
\(654\) 0 0
\(655\) −85.1724 −3.32796
\(656\) 12.4858 0.487488
\(657\) 0 0
\(658\) 3.42947 0.133695
\(659\) −48.5716 −1.89208 −0.946040 0.324050i \(-0.894956\pi\)
−0.946040 + 0.324050i \(0.894956\pi\)
\(660\) 0 0
\(661\) −37.4239 −1.45562 −0.727811 0.685778i \(-0.759462\pi\)
−0.727811 + 0.685778i \(0.759462\pi\)
\(662\) 41.5109 1.61337
\(663\) 0 0
\(664\) 19.0912 0.740882
\(665\) −10.1272 −0.392716
\(666\) 0 0
\(667\) −1.20965 −0.0468378
\(668\) −6.53303 −0.252771
\(669\) 0 0
\(670\) −44.6227 −1.72393
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) 9.73519 0.375264 0.187632 0.982239i \(-0.439919\pi\)
0.187632 + 0.982239i \(0.439919\pi\)
\(674\) −87.3305 −3.36384
\(675\) 0 0
\(676\) −7.08748 −0.272596
\(677\) −1.47674 −0.0567556 −0.0283778 0.999597i \(-0.509034\pi\)
−0.0283778 + 0.999597i \(0.509034\pi\)
\(678\) 0 0
\(679\) 3.31960 0.127394
\(680\) −70.5973 −2.70728
\(681\) 0 0
\(682\) −17.7983 −0.681530
\(683\) −31.8263 −1.21780 −0.608900 0.793247i \(-0.708389\pi\)
−0.608900 + 0.793247i \(0.708389\pi\)
\(684\) 0 0
\(685\) −9.18687 −0.351012
\(686\) −22.7728 −0.869470
\(687\) 0 0
\(688\) −58.9222 −2.24639
\(689\) 19.6551 0.748800
\(690\) 0 0
\(691\) −41.9701 −1.59662 −0.798308 0.602249i \(-0.794271\pi\)
−0.798308 + 0.602249i \(0.794271\pi\)
\(692\) −101.062 −3.84179
\(693\) 0 0
\(694\) −52.5481 −1.99470
\(695\) −11.7776 −0.446750
\(696\) 0 0
\(697\) −2.85211 −0.108031
\(698\) 78.9978 2.99011
\(699\) 0 0
\(700\) −37.4743 −1.41639
\(701\) −11.3844 −0.429984 −0.214992 0.976616i \(-0.568973\pi\)
−0.214992 + 0.976616i \(0.568973\pi\)
\(702\) 0 0
\(703\) −0.0320757 −0.00120976
\(704\) −9.09640 −0.342833
\(705\) 0 0
\(706\) −16.8650 −0.634722
\(707\) 8.29010 0.311781
\(708\) 0 0
\(709\) −35.6077 −1.33727 −0.668637 0.743589i \(-0.733122\pi\)
−0.668637 + 0.743589i \(0.733122\pi\)
\(710\) −29.1800 −1.09510
\(711\) 0 0
\(712\) −103.213 −3.86808
\(713\) 1.57531 0.0589957
\(714\) 0 0
\(715\) 14.0255 0.524525
\(716\) −5.75874 −0.215214
\(717\) 0 0
\(718\) 51.1155 1.90761
\(719\) 21.7668 0.811764 0.405882 0.913925i \(-0.366964\pi\)
0.405882 + 0.913925i \(0.366964\pi\)
\(720\) 0 0
\(721\) −11.5625 −0.430611
\(722\) 11.2075 0.417099
\(723\) 0 0
\(724\) −43.7391 −1.62555
\(725\) 62.7913 2.33201
\(726\) 0 0
\(727\) −24.0400 −0.891594 −0.445797 0.895134i \(-0.647080\pi\)
−0.445797 + 0.895134i \(0.647080\pi\)
\(728\) 16.3272 0.605126
\(729\) 0 0
\(730\) 67.7548 2.50772
\(731\) 13.4595 0.497818
\(732\) 0 0
\(733\) 24.5940 0.908398 0.454199 0.890900i \(-0.349925\pi\)
0.454199 + 0.890900i \(0.349925\pi\)
\(734\) 72.2957 2.66848
\(735\) 0 0
\(736\) 2.56842 0.0946732
\(737\) 4.12479 0.151939
\(738\) 0 0
\(739\) 9.40759 0.346064 0.173032 0.984916i \(-0.444644\pi\)
0.173032 + 0.984916i \(0.444644\pi\)
\(740\) −0.167981 −0.00617509
\(741\) 0 0
\(742\) 9.69292 0.355838
\(743\) 25.2236 0.925364 0.462682 0.886524i \(-0.346887\pi\)
0.462682 + 0.886524i \(0.346887\pi\)
\(744\) 0 0
\(745\) −20.5464 −0.752763
\(746\) 12.5857 0.460794
\(747\) 0 0
\(748\) 11.0766 0.405002
\(749\) −9.59798 −0.350703
\(750\) 0 0
\(751\) −38.5585 −1.40702 −0.703509 0.710686i \(-0.748385\pi\)
−0.703509 + 0.710686i \(0.748385\pi\)
\(752\) 20.3882 0.743482
\(753\) 0 0
\(754\) −46.4356 −1.69109
\(755\) −99.0307 −3.60410
\(756\) 0 0
\(757\) −45.4064 −1.65032 −0.825162 0.564896i \(-0.808916\pi\)
−0.825162 + 0.564896i \(0.808916\pi\)
\(758\) 63.8957 2.32080
\(759\) 0 0
\(760\) −119.051 −4.31843
\(761\) 2.88067 0.104424 0.0522121 0.998636i \(-0.483373\pi\)
0.0522121 + 0.998636i \(0.483373\pi\)
\(762\) 0 0
\(763\) 0.540707 0.0195749
\(764\) −122.628 −4.43651
\(765\) 0 0
\(766\) 95.8099 3.46175
\(767\) 23.7403 0.857212
\(768\) 0 0
\(769\) −50.4764 −1.82023 −0.910113 0.414361i \(-0.864005\pi\)
−0.910113 + 0.414361i \(0.864005\pi\)
\(770\) 6.91669 0.249260
\(771\) 0 0
\(772\) 29.3026 1.05462
\(773\) −46.6675 −1.67851 −0.839257 0.543735i \(-0.817010\pi\)
−0.839257 + 0.543735i \(0.817010\pi\)
\(774\) 0 0
\(775\) −81.7720 −2.93734
\(776\) 39.0238 1.40087
\(777\) 0 0
\(778\) 48.4594 1.73735
\(779\) −4.80962 −0.172323
\(780\) 0 0
\(781\) 2.69731 0.0965172
\(782\) −1.38317 −0.0494621
\(783\) 0 0
\(784\) −65.6562 −2.34486
\(785\) −89.5205 −3.19512
\(786\) 0 0
\(787\) −15.1302 −0.539332 −0.269666 0.962954i \(-0.586913\pi\)
−0.269666 + 0.962954i \(0.586913\pi\)
\(788\) −107.870 −3.84272
\(789\) 0 0
\(790\) 42.8655 1.52509
\(791\) 3.37419 0.119972
\(792\) 0 0
\(793\) −3.39765 −0.120654
\(794\) −92.5712 −3.28523
\(795\) 0 0
\(796\) 20.5382 0.727957
\(797\) 0.575451 0.0203835 0.0101918 0.999948i \(-0.496756\pi\)
0.0101918 + 0.999948i \(0.496756\pi\)
\(798\) 0 0
\(799\) −4.65725 −0.164762
\(800\) −133.323 −4.71368
\(801\) 0 0
\(802\) 80.5826 2.84547
\(803\) −6.26305 −0.221018
\(804\) 0 0
\(805\) −0.612189 −0.0215768
\(806\) 60.4723 2.13005
\(807\) 0 0
\(808\) 97.4549 3.42845
\(809\) 17.2764 0.607406 0.303703 0.952767i \(-0.401777\pi\)
0.303703 + 0.952767i \(0.401777\pi\)
\(810\) 0 0
\(811\) 5.50916 0.193453 0.0967264 0.995311i \(-0.469163\pi\)
0.0967264 + 0.995311i \(0.469163\pi\)
\(812\) −16.2312 −0.569602
\(813\) 0 0
\(814\) 0.0219071 0.000767845 0
\(815\) −6.03971 −0.211562
\(816\) 0 0
\(817\) 22.6973 0.794077
\(818\) −69.4164 −2.42709
\(819\) 0 0
\(820\) −25.1880 −0.879603
\(821\) 20.9755 0.732049 0.366025 0.930605i \(-0.380719\pi\)
0.366025 + 0.930605i \(0.380719\pi\)
\(822\) 0 0
\(823\) 25.3390 0.883263 0.441631 0.897197i \(-0.354400\pi\)
0.441631 + 0.897197i \(0.354400\pi\)
\(824\) −135.924 −4.73514
\(825\) 0 0
\(826\) 11.7075 0.407357
\(827\) 44.1075 1.53377 0.766884 0.641786i \(-0.221806\pi\)
0.766884 + 0.641786i \(0.221806\pi\)
\(828\) 0 0
\(829\) 21.3771 0.742456 0.371228 0.928542i \(-0.378937\pi\)
0.371228 + 0.928542i \(0.378937\pi\)
\(830\) −27.4789 −0.953806
\(831\) 0 0
\(832\) 30.9064 1.07149
\(833\) 14.9977 0.519641
\(834\) 0 0
\(835\) 5.53997 0.191719
\(836\) 18.6789 0.646025
\(837\) 0 0
\(838\) −86.3534 −2.98303
\(839\) −14.8834 −0.513833 −0.256917 0.966434i \(-0.582707\pi\)
−0.256917 + 0.966434i \(0.582707\pi\)
\(840\) 0 0
\(841\) −1.80329 −0.0621826
\(842\) −1.40776 −0.0485146
\(843\) 0 0
\(844\) −24.2460 −0.834580
\(845\) 6.01014 0.206755
\(846\) 0 0
\(847\) −0.639358 −0.0219686
\(848\) 57.6244 1.97883
\(849\) 0 0
\(850\) 71.7985 2.46267
\(851\) −0.00193898 −6.64673e−5 0
\(852\) 0 0
\(853\) −41.8057 −1.43140 −0.715700 0.698408i \(-0.753892\pi\)
−0.715700 + 0.698408i \(0.753892\pi\)
\(854\) −1.67555 −0.0573362
\(855\) 0 0
\(856\) −112.830 −3.85644
\(857\) 21.9535 0.749918 0.374959 0.927041i \(-0.377657\pi\)
0.374959 + 0.927041i \(0.377657\pi\)
\(858\) 0 0
\(859\) 12.9211 0.440863 0.220432 0.975402i \(-0.429253\pi\)
0.220432 + 0.975402i \(0.429253\pi\)
\(860\) 118.866 4.05329
\(861\) 0 0
\(862\) 63.4905 2.16250
\(863\) −20.4337 −0.695573 −0.347786 0.937574i \(-0.613067\pi\)
−0.347786 + 0.937574i \(0.613067\pi\)
\(864\) 0 0
\(865\) 85.6998 2.91388
\(866\) −42.1076 −1.43087
\(867\) 0 0
\(868\) 21.1376 0.717456
\(869\) −3.96236 −0.134414
\(870\) 0 0
\(871\) −14.0146 −0.474867
\(872\) 6.35632 0.215252
\(873\) 0 0
\(874\) −2.33249 −0.0788978
\(875\) 18.5816 0.628172
\(876\) 0 0
\(877\) 41.8719 1.41391 0.706957 0.707257i \(-0.250068\pi\)
0.706957 + 0.707257i \(0.250068\pi\)
\(878\) −69.1274 −2.33294
\(879\) 0 0
\(880\) 41.1197 1.38614
\(881\) 4.45061 0.149945 0.0749724 0.997186i \(-0.476113\pi\)
0.0749724 + 0.997186i \(0.476113\pi\)
\(882\) 0 0
\(883\) 19.8793 0.668990 0.334495 0.942398i \(-0.391434\pi\)
0.334495 + 0.942398i \(0.391434\pi\)
\(884\) −37.6345 −1.26579
\(885\) 0 0
\(886\) −72.5935 −2.43883
\(887\) 29.2454 0.981966 0.490983 0.871169i \(-0.336638\pi\)
0.490983 + 0.871169i \(0.336638\pi\)
\(888\) 0 0
\(889\) −1.62132 −0.0543773
\(890\) 148.560 4.97974
\(891\) 0 0
\(892\) −19.1608 −0.641552
\(893\) −7.85370 −0.262814
\(894\) 0 0
\(895\) 4.88338 0.163233
\(896\) 1.08231 0.0361576
\(897\) 0 0
\(898\) 75.7215 2.52686
\(899\) −35.4178 −1.18125
\(900\) 0 0
\(901\) −13.1630 −0.438525
\(902\) 3.28489 0.109375
\(903\) 0 0
\(904\) 39.6656 1.31926
\(905\) 37.0904 1.23293
\(906\) 0 0
\(907\) −31.0733 −1.03177 −0.515886 0.856657i \(-0.672537\pi\)
−0.515886 + 0.856657i \(0.672537\pi\)
\(908\) 94.8544 3.14786
\(909\) 0 0
\(910\) −23.5005 −0.779035
\(911\) 42.3077 1.40172 0.700859 0.713300i \(-0.252800\pi\)
0.700859 + 0.713300i \(0.252800\pi\)
\(912\) 0 0
\(913\) 2.54007 0.0840640
\(914\) 13.5509 0.448223
\(915\) 0 0
\(916\) −46.3781 −1.53237
\(917\) −13.1918 −0.435631
\(918\) 0 0
\(919\) 47.8295 1.57775 0.788874 0.614555i \(-0.210664\pi\)
0.788874 + 0.614555i \(0.210664\pi\)
\(920\) −7.19664 −0.237266
\(921\) 0 0
\(922\) 1.16167 0.0382575
\(923\) −9.16452 −0.301654
\(924\) 0 0
\(925\) 0.100650 0.00330934
\(926\) 1.97178 0.0647968
\(927\) 0 0
\(928\) −57.7461 −1.89561
\(929\) 45.8569 1.50452 0.752258 0.658869i \(-0.228965\pi\)
0.752258 + 0.658869i \(0.228965\pi\)
\(930\) 0 0
\(931\) 25.2913 0.828888
\(932\) −123.022 −4.02972
\(933\) 0 0
\(934\) 47.5687 1.55650
\(935\) −9.39291 −0.307181
\(936\) 0 0
\(937\) −56.1627 −1.83475 −0.917377 0.398019i \(-0.869698\pi\)
−0.917377 + 0.398019i \(0.869698\pi\)
\(938\) −6.91131 −0.225662
\(939\) 0 0
\(940\) −41.1298 −1.34151
\(941\) 60.4814 1.97164 0.985819 0.167814i \(-0.0536709\pi\)
0.985819 + 0.167814i \(0.0536709\pi\)
\(942\) 0 0
\(943\) −0.290742 −0.00946786
\(944\) 69.6012 2.26533
\(945\) 0 0
\(946\) −15.5018 −0.504008
\(947\) −50.3674 −1.63672 −0.818360 0.574706i \(-0.805116\pi\)
−0.818360 + 0.574706i \(0.805116\pi\)
\(948\) 0 0
\(949\) 21.2797 0.690768
\(950\) 121.077 3.92824
\(951\) 0 0
\(952\) −10.9343 −0.354384
\(953\) −37.2162 −1.20555 −0.602775 0.797911i \(-0.705938\pi\)
−0.602775 + 0.797911i \(0.705938\pi\)
\(954\) 0 0
\(955\) 103.987 3.36495
\(956\) 6.63656 0.214642
\(957\) 0 0
\(958\) 70.8635 2.28950
\(959\) −1.42289 −0.0459475
\(960\) 0 0
\(961\) 15.1240 0.487869
\(962\) −0.0744329 −0.00239981
\(963\) 0 0
\(964\) 36.3976 1.17229
\(965\) −24.8484 −0.799898
\(966\) 0 0
\(967\) −5.27617 −0.169670 −0.0848352 0.996395i \(-0.527036\pi\)
−0.0848352 + 0.996395i \(0.527036\pi\)
\(968\) −7.51602 −0.241574
\(969\) 0 0
\(970\) −56.1688 −1.80347
\(971\) 29.9077 0.959785 0.479892 0.877327i \(-0.340676\pi\)
0.479892 + 0.877327i \(0.340676\pi\)
\(972\) 0 0
\(973\) −1.82415 −0.0584796
\(974\) −32.8981 −1.05412
\(975\) 0 0
\(976\) −9.96116 −0.318849
\(977\) −33.7657 −1.08026 −0.540130 0.841581i \(-0.681625\pi\)
−0.540130 + 0.841581i \(0.681625\pi\)
\(978\) 0 0
\(979\) −13.7324 −0.438891
\(980\) 132.450 4.23097
\(981\) 0 0
\(982\) −46.4192 −1.48130
\(983\) 20.0762 0.640331 0.320165 0.947362i \(-0.396261\pi\)
0.320165 + 0.947362i \(0.396261\pi\)
\(984\) 0 0
\(985\) 91.4732 2.91458
\(986\) 31.0980 0.990361
\(987\) 0 0
\(988\) −63.4646 −2.01908
\(989\) 1.37205 0.0436287
\(990\) 0 0
\(991\) 15.9157 0.505579 0.252790 0.967521i \(-0.418652\pi\)
0.252790 + 0.967521i \(0.418652\pi\)
\(992\) 75.2017 2.38765
\(993\) 0 0
\(994\) −4.51948 −0.143349
\(995\) −17.4163 −0.552132
\(996\) 0 0
\(997\) 38.6749 1.22485 0.612424 0.790530i \(-0.290195\pi\)
0.612424 + 0.790530i \(0.290195\pi\)
\(998\) 73.3319 2.32128
\(999\) 0 0
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 6039.2.a.n.1.2 25
3.2 odd 2 6039.2.a.o.1.24 yes 25
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6039.2.a.n.1.2 25 1.1 even 1 trivial
6039.2.a.o.1.24 yes 25 3.2 odd 2