Properties

Label 5265.2.a.bf.1.6
Level $5265$
Weight $2$
Character 5265.1
Self dual yes
Analytic conductor $42.041$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5265,2,Mod(1,5265)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5265, 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("5265.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5265.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: \(42.0412366642\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 8x^{6} + 31x^{5} - x^{4} - 70x^{3} + 66x^{2} - 19x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 585)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(1.63404\) of defining polynomial
Character \(\chi\) \(=\) 5265.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.63404 q^{2} +0.670078 q^{4} -1.00000 q^{5} +2.13012 q^{7} -2.17314 q^{8} +O(q^{10})\) \(q+1.63404 q^{2} +0.670078 q^{4} -1.00000 q^{5} +2.13012 q^{7} -2.17314 q^{8} -1.63404 q^{10} -0.0526457 q^{11} -1.00000 q^{13} +3.48069 q^{14} -4.89115 q^{16} -2.48047 q^{17} +2.13105 q^{19} -0.670078 q^{20} -0.0860250 q^{22} +4.93083 q^{23} +1.00000 q^{25} -1.63404 q^{26} +1.42734 q^{28} -2.48694 q^{29} -8.16059 q^{31} -3.64604 q^{32} -4.05318 q^{34} -2.13012 q^{35} -1.11963 q^{37} +3.48221 q^{38} +2.17314 q^{40} -5.47417 q^{41} -9.47722 q^{43} -0.0352767 q^{44} +8.05716 q^{46} +9.76560 q^{47} -2.46260 q^{49} +1.63404 q^{50} -0.670078 q^{52} -3.64091 q^{53} +0.0526457 q^{55} -4.62905 q^{56} -4.06374 q^{58} +7.49087 q^{59} -5.78404 q^{61} -13.3347 q^{62} +3.82454 q^{64} +1.00000 q^{65} -6.23591 q^{67} -1.66211 q^{68} -3.48069 q^{70} +2.50050 q^{71} -1.10245 q^{73} -1.82952 q^{74} +1.42797 q^{76} -0.112141 q^{77} -15.6156 q^{79} +4.89115 q^{80} -8.94500 q^{82} +0.489152 q^{83} +2.48047 q^{85} -15.4861 q^{86} +0.114407 q^{88} -4.64901 q^{89} -2.13012 q^{91} +3.30404 q^{92} +15.9574 q^{94} -2.13105 q^{95} +7.34097 q^{97} -4.02399 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{2} + 9 q^{4} - 8 q^{5} - 11 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{2} + 9 q^{4} - 8 q^{5} - 11 q^{7} - 6 q^{8} - 3 q^{10} + 6 q^{11} - 8 q^{13} + 10 q^{14} + 11 q^{16} - 2 q^{17} - 10 q^{19} - 9 q^{20} + 3 q^{22} + 6 q^{23} + 8 q^{25} - 3 q^{26} - 34 q^{28} + 14 q^{29} - 31 q^{31} + q^{32} - 7 q^{34} + 11 q^{35} + q^{37} + 9 q^{38} + 6 q^{40} - 12 q^{41} + 15 q^{43} + 16 q^{44} - 32 q^{46} - 18 q^{47} + 17 q^{49} + 3 q^{50} - 9 q^{52} + 2 q^{53} - 6 q^{55} + 16 q^{56} - 42 q^{58} + 24 q^{59} - 9 q^{61} - 20 q^{62} - 30 q^{64} + 8 q^{65} - 18 q^{67} - 14 q^{68} - 10 q^{70} + 10 q^{71} + 6 q^{73} - 37 q^{74} - 53 q^{76} - 34 q^{77} - 3 q^{79} - 11 q^{80} - 34 q^{82} - 10 q^{83} + 2 q^{85} + 60 q^{86} - 14 q^{88} - 13 q^{89} + 11 q^{91} + 5 q^{92} + 17 q^{94} + 10 q^{95} - 34 q^{97} - 30 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.63404 1.15544 0.577719 0.816235i \(-0.303943\pi\)
0.577719 + 0.816235i \(0.303943\pi\)
\(3\) 0 0
\(4\) 0.670078 0.335039
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.13012 0.805108 0.402554 0.915396i \(-0.368123\pi\)
0.402554 + 0.915396i \(0.368123\pi\)
\(8\) −2.17314 −0.768322
\(9\) 0 0
\(10\) −1.63404 −0.516728
\(11\) −0.0526457 −0.0158733 −0.00793664 0.999969i \(-0.502526\pi\)
−0.00793664 + 0.999969i \(0.502526\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 3.48069 0.930253
\(15\) 0 0
\(16\) −4.89115 −1.22279
\(17\) −2.48047 −0.601603 −0.300801 0.953687i \(-0.597254\pi\)
−0.300801 + 0.953687i \(0.597254\pi\)
\(18\) 0 0
\(19\) 2.13105 0.488895 0.244448 0.969662i \(-0.421393\pi\)
0.244448 + 0.969662i \(0.421393\pi\)
\(20\) −0.670078 −0.149834
\(21\) 0 0
\(22\) −0.0860250 −0.0183406
\(23\) 4.93083 1.02815 0.514075 0.857745i \(-0.328135\pi\)
0.514075 + 0.857745i \(0.328135\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.63404 −0.320461
\(27\) 0 0
\(28\) 1.42734 0.269743
\(29\) −2.48694 −0.461812 −0.230906 0.972976i \(-0.574169\pi\)
−0.230906 + 0.972976i \(0.574169\pi\)
\(30\) 0 0
\(31\) −8.16059 −1.46569 −0.732843 0.680398i \(-0.761807\pi\)
−0.732843 + 0.680398i \(0.761807\pi\)
\(32\) −3.64604 −0.644535
\(33\) 0 0
\(34\) −4.05318 −0.695115
\(35\) −2.13012 −0.360055
\(36\) 0 0
\(37\) −1.11963 −0.184066 −0.0920331 0.995756i \(-0.529337\pi\)
−0.0920331 + 0.995756i \(0.529337\pi\)
\(38\) 3.48221 0.564889
\(39\) 0 0
\(40\) 2.17314 0.343604
\(41\) −5.47417 −0.854922 −0.427461 0.904034i \(-0.640592\pi\)
−0.427461 + 0.904034i \(0.640592\pi\)
\(42\) 0 0
\(43\) −9.47722 −1.44526 −0.722632 0.691233i \(-0.757068\pi\)
−0.722632 + 0.691233i \(0.757068\pi\)
\(44\) −0.0352767 −0.00531816
\(45\) 0 0
\(46\) 8.05716 1.18796
\(47\) 9.76560 1.42446 0.712230 0.701946i \(-0.247685\pi\)
0.712230 + 0.701946i \(0.247685\pi\)
\(48\) 0 0
\(49\) −2.46260 −0.351801
\(50\) 1.63404 0.231088
\(51\) 0 0
\(52\) −0.670078 −0.0929231
\(53\) −3.64091 −0.500118 −0.250059 0.968231i \(-0.580450\pi\)
−0.250059 + 0.968231i \(0.580450\pi\)
\(54\) 0 0
\(55\) 0.0526457 0.00709875
\(56\) −4.62905 −0.618582
\(57\) 0 0
\(58\) −4.06374 −0.533596
\(59\) 7.49087 0.975228 0.487614 0.873059i \(-0.337867\pi\)
0.487614 + 0.873059i \(0.337867\pi\)
\(60\) 0 0
\(61\) −5.78404 −0.740570 −0.370285 0.928918i \(-0.620740\pi\)
−0.370285 + 0.928918i \(0.620740\pi\)
\(62\) −13.3347 −1.69351
\(63\) 0 0
\(64\) 3.82454 0.478067
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −6.23591 −0.761837 −0.380919 0.924609i \(-0.624392\pi\)
−0.380919 + 0.924609i \(0.624392\pi\)
\(68\) −1.66211 −0.201560
\(69\) 0 0
\(70\) −3.48069 −0.416022
\(71\) 2.50050 0.296755 0.148377 0.988931i \(-0.452595\pi\)
0.148377 + 0.988931i \(0.452595\pi\)
\(72\) 0 0
\(73\) −1.10245 −0.129032 −0.0645159 0.997917i \(-0.520550\pi\)
−0.0645159 + 0.997917i \(0.520550\pi\)
\(74\) −1.82952 −0.212677
\(75\) 0 0
\(76\) 1.42797 0.163799
\(77\) −0.112141 −0.0127797
\(78\) 0 0
\(79\) −15.6156 −1.75689 −0.878445 0.477844i \(-0.841418\pi\)
−0.878445 + 0.477844i \(0.841418\pi\)
\(80\) 4.89115 0.546847
\(81\) 0 0
\(82\) −8.94500 −0.987810
\(83\) 0.489152 0.0536914 0.0268457 0.999640i \(-0.491454\pi\)
0.0268457 + 0.999640i \(0.491454\pi\)
\(84\) 0 0
\(85\) 2.48047 0.269045
\(86\) −15.4861 −1.66991
\(87\) 0 0
\(88\) 0.114407 0.0121958
\(89\) −4.64901 −0.492794 −0.246397 0.969169i \(-0.579247\pi\)
−0.246397 + 0.969169i \(0.579247\pi\)
\(90\) 0 0
\(91\) −2.13012 −0.223297
\(92\) 3.30404 0.344470
\(93\) 0 0
\(94\) 15.9574 1.64588
\(95\) −2.13105 −0.218641
\(96\) 0 0
\(97\) 7.34097 0.745363 0.372682 0.927959i \(-0.378438\pi\)
0.372682 + 0.927959i \(0.378438\pi\)
\(98\) −4.02399 −0.406484
\(99\) 0 0
\(100\) 0.670078 0.0670078
\(101\) 0.552859 0.0550115 0.0275058 0.999622i \(-0.491244\pi\)
0.0275058 + 0.999622i \(0.491244\pi\)
\(102\) 0 0
\(103\) −7.83795 −0.772296 −0.386148 0.922437i \(-0.626195\pi\)
−0.386148 + 0.922437i \(0.626195\pi\)
\(104\) 2.17314 0.213094
\(105\) 0 0
\(106\) −5.94939 −0.577855
\(107\) 19.4173 1.87714 0.938572 0.345084i \(-0.112149\pi\)
0.938572 + 0.345084i \(0.112149\pi\)
\(108\) 0 0
\(109\) −2.13502 −0.204498 −0.102249 0.994759i \(-0.532604\pi\)
−0.102249 + 0.994759i \(0.532604\pi\)
\(110\) 0.0860250 0.00820217
\(111\) 0 0
\(112\) −10.4187 −0.984477
\(113\) −6.23529 −0.586567 −0.293283 0.956026i \(-0.594748\pi\)
−0.293283 + 0.956026i \(0.594748\pi\)
\(114\) 0 0
\(115\) −4.93083 −0.459802
\(116\) −1.66644 −0.154725
\(117\) 0 0
\(118\) 12.2404 1.12682
\(119\) −5.28369 −0.484355
\(120\) 0 0
\(121\) −10.9972 −0.999748
\(122\) −9.45133 −0.855683
\(123\) 0 0
\(124\) −5.46823 −0.491062
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −10.6036 −0.940918 −0.470459 0.882422i \(-0.655912\pi\)
−0.470459 + 0.882422i \(0.655912\pi\)
\(128\) 13.5415 1.19691
\(129\) 0 0
\(130\) 1.63404 0.143315
\(131\) −16.1923 −1.41473 −0.707365 0.706849i \(-0.750116\pi\)
−0.707365 + 0.706849i \(0.750116\pi\)
\(132\) 0 0
\(133\) 4.53938 0.393614
\(134\) −10.1897 −0.880256
\(135\) 0 0
\(136\) 5.39042 0.462224
\(137\) −20.7686 −1.77438 −0.887189 0.461406i \(-0.847345\pi\)
−0.887189 + 0.461406i \(0.847345\pi\)
\(138\) 0 0
\(139\) −2.50075 −0.212111 −0.106055 0.994360i \(-0.533822\pi\)
−0.106055 + 0.994360i \(0.533822\pi\)
\(140\) −1.42734 −0.120633
\(141\) 0 0
\(142\) 4.08591 0.342882
\(143\) 0.0526457 0.00440245
\(144\) 0 0
\(145\) 2.48694 0.206529
\(146\) −1.80144 −0.149088
\(147\) 0 0
\(148\) −0.750240 −0.0616693
\(149\) −1.89025 −0.154856 −0.0774279 0.996998i \(-0.524671\pi\)
−0.0774279 + 0.996998i \(0.524671\pi\)
\(150\) 0 0
\(151\) −4.88685 −0.397686 −0.198843 0.980031i \(-0.563718\pi\)
−0.198843 + 0.980031i \(0.563718\pi\)
\(152\) −4.63107 −0.375629
\(153\) 0 0
\(154\) −0.183243 −0.0147662
\(155\) 8.16059 0.655475
\(156\) 0 0
\(157\) −17.9998 −1.43654 −0.718269 0.695765i \(-0.755065\pi\)
−0.718269 + 0.695765i \(0.755065\pi\)
\(158\) −25.5164 −2.02998
\(159\) 0 0
\(160\) 3.64604 0.288245
\(161\) 10.5032 0.827772
\(162\) 0 0
\(163\) 9.41748 0.737634 0.368817 0.929502i \(-0.379763\pi\)
0.368817 + 0.929502i \(0.379763\pi\)
\(164\) −3.66812 −0.286432
\(165\) 0 0
\(166\) 0.799292 0.0620371
\(167\) −15.4905 −1.19869 −0.599344 0.800492i \(-0.704572\pi\)
−0.599344 + 0.800492i \(0.704572\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 4.05318 0.310865
\(171\) 0 0
\(172\) −6.35048 −0.484219
\(173\) −9.67663 −0.735700 −0.367850 0.929885i \(-0.619906\pi\)
−0.367850 + 0.929885i \(0.619906\pi\)
\(174\) 0 0
\(175\) 2.13012 0.161022
\(176\) 0.257498 0.0194096
\(177\) 0 0
\(178\) −7.59665 −0.569393
\(179\) −3.31630 −0.247872 −0.123936 0.992290i \(-0.539552\pi\)
−0.123936 + 0.992290i \(0.539552\pi\)
\(180\) 0 0
\(181\) −22.6189 −1.68125 −0.840623 0.541621i \(-0.817811\pi\)
−0.840623 + 0.541621i \(0.817811\pi\)
\(182\) −3.48069 −0.258006
\(183\) 0 0
\(184\) −10.7154 −0.789950
\(185\) 1.11963 0.0823169
\(186\) 0 0
\(187\) 0.130586 0.00954940
\(188\) 6.54371 0.477249
\(189\) 0 0
\(190\) −3.48221 −0.252626
\(191\) 5.88211 0.425614 0.212807 0.977094i \(-0.431739\pi\)
0.212807 + 0.977094i \(0.431739\pi\)
\(192\) 0 0
\(193\) −0.459539 −0.0330783 −0.0165392 0.999863i \(-0.505265\pi\)
−0.0165392 + 0.999863i \(0.505265\pi\)
\(194\) 11.9954 0.861221
\(195\) 0 0
\(196\) −1.65014 −0.117867
\(197\) −2.31119 −0.164665 −0.0823326 0.996605i \(-0.526237\pi\)
−0.0823326 + 0.996605i \(0.526237\pi\)
\(198\) 0 0
\(199\) 24.3734 1.72779 0.863893 0.503675i \(-0.168019\pi\)
0.863893 + 0.503675i \(0.168019\pi\)
\(200\) −2.17314 −0.153664
\(201\) 0 0
\(202\) 0.903392 0.0635624
\(203\) −5.29746 −0.371809
\(204\) 0 0
\(205\) 5.47417 0.382333
\(206\) −12.8075 −0.892341
\(207\) 0 0
\(208\) 4.89115 0.339140
\(209\) −0.112190 −0.00776037
\(210\) 0 0
\(211\) 27.3938 1.88587 0.942934 0.332980i \(-0.108054\pi\)
0.942934 + 0.332980i \(0.108054\pi\)
\(212\) −2.43969 −0.167559
\(213\) 0 0
\(214\) 31.7286 2.16893
\(215\) 9.47722 0.646341
\(216\) 0 0
\(217\) −17.3830 −1.18004
\(218\) −3.48870 −0.236285
\(219\) 0 0
\(220\) 0.0352767 0.00237836
\(221\) 2.48047 0.166855
\(222\) 0 0
\(223\) 25.3359 1.69661 0.848307 0.529504i \(-0.177622\pi\)
0.848307 + 0.529504i \(0.177622\pi\)
\(224\) −7.76649 −0.518920
\(225\) 0 0
\(226\) −10.1887 −0.677742
\(227\) −21.2417 −1.40986 −0.704931 0.709276i \(-0.749022\pi\)
−0.704931 + 0.709276i \(0.749022\pi\)
\(228\) 0 0
\(229\) 12.2337 0.808427 0.404214 0.914665i \(-0.367545\pi\)
0.404214 + 0.914665i \(0.367545\pi\)
\(230\) −8.05716 −0.531274
\(231\) 0 0
\(232\) 5.40446 0.354820
\(233\) 11.4048 0.747154 0.373577 0.927599i \(-0.378131\pi\)
0.373577 + 0.927599i \(0.378131\pi\)
\(234\) 0 0
\(235\) −9.76560 −0.637038
\(236\) 5.01947 0.326739
\(237\) 0 0
\(238\) −8.63375 −0.559643
\(239\) 14.8571 0.961025 0.480512 0.876988i \(-0.340451\pi\)
0.480512 + 0.876988i \(0.340451\pi\)
\(240\) 0 0
\(241\) −18.5908 −1.19754 −0.598768 0.800922i \(-0.704343\pi\)
−0.598768 + 0.800922i \(0.704343\pi\)
\(242\) −17.9699 −1.15515
\(243\) 0 0
\(244\) −3.87575 −0.248120
\(245\) 2.46260 0.157330
\(246\) 0 0
\(247\) −2.13105 −0.135595
\(248\) 17.7341 1.12612
\(249\) 0 0
\(250\) −1.63404 −0.103346
\(251\) 30.0723 1.89815 0.949073 0.315058i \(-0.102024\pi\)
0.949073 + 0.315058i \(0.102024\pi\)
\(252\) 0 0
\(253\) −0.259587 −0.0163201
\(254\) −17.3267 −1.08717
\(255\) 0 0
\(256\) 14.4783 0.904892
\(257\) 3.04064 0.189670 0.0948348 0.995493i \(-0.469768\pi\)
0.0948348 + 0.995493i \(0.469768\pi\)
\(258\) 0 0
\(259\) −2.38494 −0.148193
\(260\) 0.670078 0.0415565
\(261\) 0 0
\(262\) −26.4588 −1.63463
\(263\) −4.80102 −0.296044 −0.148022 0.988984i \(-0.547291\pi\)
−0.148022 + 0.988984i \(0.547291\pi\)
\(264\) 0 0
\(265\) 3.64091 0.223659
\(266\) 7.41751 0.454797
\(267\) 0 0
\(268\) −4.17854 −0.255245
\(269\) −16.8723 −1.02873 −0.514363 0.857573i \(-0.671971\pi\)
−0.514363 + 0.857573i \(0.671971\pi\)
\(270\) 0 0
\(271\) −2.34099 −0.142205 −0.0711026 0.997469i \(-0.522652\pi\)
−0.0711026 + 0.997469i \(0.522652\pi\)
\(272\) 12.1324 0.735632
\(273\) 0 0
\(274\) −33.9366 −2.05019
\(275\) −0.0526457 −0.00317466
\(276\) 0 0
\(277\) −3.77233 −0.226657 −0.113329 0.993558i \(-0.536151\pi\)
−0.113329 + 0.993558i \(0.536151\pi\)
\(278\) −4.08632 −0.245081
\(279\) 0 0
\(280\) 4.62905 0.276638
\(281\) 10.9560 0.653577 0.326789 0.945097i \(-0.394033\pi\)
0.326789 + 0.945097i \(0.394033\pi\)
\(282\) 0 0
\(283\) 17.5697 1.04441 0.522204 0.852821i \(-0.325110\pi\)
0.522204 + 0.852821i \(0.325110\pi\)
\(284\) 1.67553 0.0994243
\(285\) 0 0
\(286\) 0.0860250 0.00508677
\(287\) −11.6606 −0.688305
\(288\) 0 0
\(289\) −10.8473 −0.638074
\(290\) 4.06374 0.238631
\(291\) 0 0
\(292\) −0.738726 −0.0432307
\(293\) 24.5445 1.43391 0.716953 0.697122i \(-0.245536\pi\)
0.716953 + 0.697122i \(0.245536\pi\)
\(294\) 0 0
\(295\) −7.49087 −0.436135
\(296\) 2.43312 0.141422
\(297\) 0 0
\(298\) −3.08875 −0.178926
\(299\) −4.93083 −0.285157
\(300\) 0 0
\(301\) −20.1876 −1.16359
\(302\) −7.98530 −0.459502
\(303\) 0 0
\(304\) −10.4233 −0.597815
\(305\) 5.78404 0.331193
\(306\) 0 0
\(307\) −11.0689 −0.631735 −0.315868 0.948803i \(-0.602296\pi\)
−0.315868 + 0.948803i \(0.602296\pi\)
\(308\) −0.0751435 −0.00428170
\(309\) 0 0
\(310\) 13.3347 0.757361
\(311\) 14.7933 0.838853 0.419426 0.907789i \(-0.362231\pi\)
0.419426 + 0.907789i \(0.362231\pi\)
\(312\) 0 0
\(313\) 27.0525 1.52910 0.764548 0.644567i \(-0.222962\pi\)
0.764548 + 0.644567i \(0.222962\pi\)
\(314\) −29.4123 −1.65983
\(315\) 0 0
\(316\) −10.4636 −0.588626
\(317\) 0.577998 0.0324636 0.0162318 0.999868i \(-0.494833\pi\)
0.0162318 + 0.999868i \(0.494833\pi\)
\(318\) 0 0
\(319\) 0.130926 0.00733047
\(320\) −3.82454 −0.213798
\(321\) 0 0
\(322\) 17.1627 0.956440
\(323\) −5.28600 −0.294121
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 15.3885 0.852291
\(327\) 0 0
\(328\) 11.8962 0.656855
\(329\) 20.8019 1.14684
\(330\) 0 0
\(331\) −24.1140 −1.32542 −0.662712 0.748874i \(-0.730595\pi\)
−0.662712 + 0.748874i \(0.730595\pi\)
\(332\) 0.327770 0.0179887
\(333\) 0 0
\(334\) −25.3120 −1.38501
\(335\) 6.23591 0.340704
\(336\) 0 0
\(337\) 6.13491 0.334190 0.167095 0.985941i \(-0.446561\pi\)
0.167095 + 0.985941i \(0.446561\pi\)
\(338\) 1.63404 0.0888799
\(339\) 0 0
\(340\) 1.66211 0.0901405
\(341\) 0.429620 0.0232652
\(342\) 0 0
\(343\) −20.1564 −1.08835
\(344\) 20.5954 1.11043
\(345\) 0 0
\(346\) −15.8120 −0.850057
\(347\) −3.72399 −0.199914 −0.0999572 0.994992i \(-0.531871\pi\)
−0.0999572 + 0.994992i \(0.531871\pi\)
\(348\) 0 0
\(349\) −3.34865 −0.179249 −0.0896246 0.995976i \(-0.528567\pi\)
−0.0896246 + 0.995976i \(0.528567\pi\)
\(350\) 3.48069 0.186051
\(351\) 0 0
\(352\) 0.191948 0.0102309
\(353\) −35.5626 −1.89281 −0.946403 0.322988i \(-0.895313\pi\)
−0.946403 + 0.322988i \(0.895313\pi\)
\(354\) 0 0
\(355\) −2.50050 −0.132713
\(356\) −3.11520 −0.165105
\(357\) 0 0
\(358\) −5.41896 −0.286401
\(359\) 9.06902 0.478645 0.239322 0.970940i \(-0.423075\pi\)
0.239322 + 0.970940i \(0.423075\pi\)
\(360\) 0 0
\(361\) −14.4586 −0.760981
\(362\) −36.9600 −1.94258
\(363\) 0 0
\(364\) −1.42734 −0.0748131
\(365\) 1.10245 0.0577048
\(366\) 0 0
\(367\) −24.6186 −1.28508 −0.642541 0.766252i \(-0.722120\pi\)
−0.642541 + 0.766252i \(0.722120\pi\)
\(368\) −24.1174 −1.25721
\(369\) 0 0
\(370\) 1.82952 0.0951121
\(371\) −7.75557 −0.402649
\(372\) 0 0
\(373\) 26.9758 1.39675 0.698376 0.715731i \(-0.253906\pi\)
0.698376 + 0.715731i \(0.253906\pi\)
\(374\) 0.213383 0.0110338
\(375\) 0 0
\(376\) −21.2220 −1.09444
\(377\) 2.48694 0.128084
\(378\) 0 0
\(379\) 14.2429 0.731611 0.365805 0.930691i \(-0.380794\pi\)
0.365805 + 0.930691i \(0.380794\pi\)
\(380\) −1.42797 −0.0732531
\(381\) 0 0
\(382\) 9.61158 0.491771
\(383\) 25.9736 1.32719 0.663595 0.748092i \(-0.269030\pi\)
0.663595 + 0.748092i \(0.269030\pi\)
\(384\) 0 0
\(385\) 0.112141 0.00571526
\(386\) −0.750903 −0.0382200
\(387\) 0 0
\(388\) 4.91902 0.249726
\(389\) 25.7906 1.30764 0.653819 0.756651i \(-0.273166\pi\)
0.653819 + 0.756651i \(0.273166\pi\)
\(390\) 0 0
\(391\) −12.2308 −0.618537
\(392\) 5.35159 0.270296
\(393\) 0 0
\(394\) −3.77657 −0.190261
\(395\) 15.6156 0.785705
\(396\) 0 0
\(397\) −19.2890 −0.968086 −0.484043 0.875044i \(-0.660832\pi\)
−0.484043 + 0.875044i \(0.660832\pi\)
\(398\) 39.8271 1.99635
\(399\) 0 0
\(400\) −4.89115 −0.244558
\(401\) −16.4631 −0.822128 −0.411064 0.911607i \(-0.634843\pi\)
−0.411064 + 0.911607i \(0.634843\pi\)
\(402\) 0 0
\(403\) 8.16059 0.406508
\(404\) 0.370458 0.0184310
\(405\) 0 0
\(406\) −8.65625 −0.429602
\(407\) 0.0589438 0.00292173
\(408\) 0 0
\(409\) −19.7526 −0.976704 −0.488352 0.872647i \(-0.662402\pi\)
−0.488352 + 0.872647i \(0.662402\pi\)
\(410\) 8.94500 0.441762
\(411\) 0 0
\(412\) −5.25204 −0.258749
\(413\) 15.9564 0.785164
\(414\) 0 0
\(415\) −0.489152 −0.0240115
\(416\) 3.64604 0.178762
\(417\) 0 0
\(418\) −0.183323 −0.00896664
\(419\) −17.4583 −0.852894 −0.426447 0.904513i \(-0.640235\pi\)
−0.426447 + 0.904513i \(0.640235\pi\)
\(420\) 0 0
\(421\) −7.41596 −0.361432 −0.180716 0.983535i \(-0.557841\pi\)
−0.180716 + 0.983535i \(0.557841\pi\)
\(422\) 44.7625 2.17900
\(423\) 0 0
\(424\) 7.91222 0.384251
\(425\) −2.48047 −0.120321
\(426\) 0 0
\(427\) −12.3207 −0.596239
\(428\) 13.0111 0.628916
\(429\) 0 0
\(430\) 15.4861 0.746808
\(431\) 33.6097 1.61892 0.809461 0.587174i \(-0.199760\pi\)
0.809461 + 0.587174i \(0.199760\pi\)
\(432\) 0 0
\(433\) 30.2826 1.45529 0.727644 0.685955i \(-0.240615\pi\)
0.727644 + 0.685955i \(0.240615\pi\)
\(434\) −28.4045 −1.36346
\(435\) 0 0
\(436\) −1.43063 −0.0685147
\(437\) 10.5078 0.502658
\(438\) 0 0
\(439\) −28.7370 −1.37154 −0.685772 0.727817i \(-0.740535\pi\)
−0.685772 + 0.727817i \(0.740535\pi\)
\(440\) −0.114407 −0.00545412
\(441\) 0 0
\(442\) 4.05318 0.192790
\(443\) −22.4218 −1.06529 −0.532646 0.846338i \(-0.678802\pi\)
−0.532646 + 0.846338i \(0.678802\pi\)
\(444\) 0 0
\(445\) 4.64901 0.220384
\(446\) 41.3997 1.96033
\(447\) 0 0
\(448\) 8.14671 0.384896
\(449\) 16.0149 0.755791 0.377895 0.925848i \(-0.376648\pi\)
0.377895 + 0.925848i \(0.376648\pi\)
\(450\) 0 0
\(451\) 0.288192 0.0135704
\(452\) −4.17813 −0.196523
\(453\) 0 0
\(454\) −34.7097 −1.62901
\(455\) 2.13012 0.0998614
\(456\) 0 0
\(457\) 1.06556 0.0498448 0.0249224 0.999689i \(-0.492066\pi\)
0.0249224 + 0.999689i \(0.492066\pi\)
\(458\) 19.9904 0.934089
\(459\) 0 0
\(460\) −3.30404 −0.154052
\(461\) 24.6387 1.14754 0.573769 0.819017i \(-0.305481\pi\)
0.573769 + 0.819017i \(0.305481\pi\)
\(462\) 0 0
\(463\) 7.12328 0.331047 0.165524 0.986206i \(-0.447069\pi\)
0.165524 + 0.986206i \(0.447069\pi\)
\(464\) 12.1640 0.564698
\(465\) 0 0
\(466\) 18.6359 0.863290
\(467\) 20.4662 0.947063 0.473531 0.880777i \(-0.342979\pi\)
0.473531 + 0.880777i \(0.342979\pi\)
\(468\) 0 0
\(469\) −13.2832 −0.613361
\(470\) −15.9574 −0.736058
\(471\) 0 0
\(472\) −16.2787 −0.749289
\(473\) 0.498935 0.0229411
\(474\) 0 0
\(475\) 2.13105 0.0977791
\(476\) −3.54048 −0.162278
\(477\) 0 0
\(478\) 24.2770 1.11041
\(479\) −14.4699 −0.661145 −0.330573 0.943781i \(-0.607242\pi\)
−0.330573 + 0.943781i \(0.607242\pi\)
\(480\) 0 0
\(481\) 1.11963 0.0510508
\(482\) −30.3780 −1.38368
\(483\) 0 0
\(484\) −7.36900 −0.334954
\(485\) −7.34097 −0.333336
\(486\) 0 0
\(487\) 29.8595 1.35306 0.676531 0.736414i \(-0.263483\pi\)
0.676531 + 0.736414i \(0.263483\pi\)
\(488\) 12.5695 0.568996
\(489\) 0 0
\(490\) 4.02399 0.181785
\(491\) 22.0470 0.994965 0.497483 0.867474i \(-0.334258\pi\)
0.497483 + 0.867474i \(0.334258\pi\)
\(492\) 0 0
\(493\) 6.16877 0.277827
\(494\) −3.48221 −0.156672
\(495\) 0 0
\(496\) 39.9147 1.79222
\(497\) 5.32635 0.238920
\(498\) 0 0
\(499\) 21.1723 0.947802 0.473901 0.880578i \(-0.342845\pi\)
0.473901 + 0.880578i \(0.342845\pi\)
\(500\) −0.670078 −0.0299668
\(501\) 0 0
\(502\) 49.1392 2.19319
\(503\) −40.3143 −1.79752 −0.898762 0.438436i \(-0.855533\pi\)
−0.898762 + 0.438436i \(0.855533\pi\)
\(504\) 0 0
\(505\) −0.552859 −0.0246019
\(506\) −0.424175 −0.0188569
\(507\) 0 0
\(508\) −7.10524 −0.315244
\(509\) 19.0167 0.842900 0.421450 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(510\) 0 0
\(511\) −2.34834 −0.103885
\(512\) −3.42501 −0.151366
\(513\) 0 0
\(514\) 4.96851 0.219152
\(515\) 7.83795 0.345381
\(516\) 0 0
\(517\) −0.514117 −0.0226108
\(518\) −3.89709 −0.171228
\(519\) 0 0
\(520\) −2.17314 −0.0952986
\(521\) −33.4407 −1.46507 −0.732533 0.680732i \(-0.761662\pi\)
−0.732533 + 0.680732i \(0.761662\pi\)
\(522\) 0 0
\(523\) −6.40250 −0.279962 −0.139981 0.990154i \(-0.544704\pi\)
−0.139981 + 0.990154i \(0.544704\pi\)
\(524\) −10.8501 −0.473989
\(525\) 0 0
\(526\) −7.84504 −0.342060
\(527\) 20.2421 0.881761
\(528\) 0 0
\(529\) 1.31310 0.0570915
\(530\) 5.94939 0.258425
\(531\) 0 0
\(532\) 3.04173 0.131876
\(533\) 5.47417 0.237113
\(534\) 0 0
\(535\) −19.4173 −0.839484
\(536\) 13.5515 0.585336
\(537\) 0 0
\(538\) −27.5700 −1.18863
\(539\) 0.129646 0.00558423
\(540\) 0 0
\(541\) −17.4745 −0.751286 −0.375643 0.926764i \(-0.622578\pi\)
−0.375643 + 0.926764i \(0.622578\pi\)
\(542\) −3.82527 −0.164309
\(543\) 0 0
\(544\) 9.04389 0.387754
\(545\) 2.13502 0.0914542
\(546\) 0 0
\(547\) −10.0747 −0.430761 −0.215381 0.976530i \(-0.569099\pi\)
−0.215381 + 0.976530i \(0.569099\pi\)
\(548\) −13.9166 −0.594486
\(549\) 0 0
\(550\) −0.0860250 −0.00366812
\(551\) −5.29977 −0.225778
\(552\) 0 0
\(553\) −33.2630 −1.41449
\(554\) −6.16413 −0.261889
\(555\) 0 0
\(556\) −1.67570 −0.0710654
\(557\) 21.2315 0.899607 0.449804 0.893127i \(-0.351494\pi\)
0.449804 + 0.893127i \(0.351494\pi\)
\(558\) 0 0
\(559\) 9.47722 0.400844
\(560\) 10.4187 0.440271
\(561\) 0 0
\(562\) 17.9024 0.755169
\(563\) 12.3449 0.520276 0.260138 0.965571i \(-0.416232\pi\)
0.260138 + 0.965571i \(0.416232\pi\)
\(564\) 0 0
\(565\) 6.23529 0.262321
\(566\) 28.7095 1.20675
\(567\) 0 0
\(568\) −5.43394 −0.228003
\(569\) 7.75747 0.325210 0.162605 0.986691i \(-0.448010\pi\)
0.162605 + 0.986691i \(0.448010\pi\)
\(570\) 0 0
\(571\) −10.2139 −0.427440 −0.213720 0.976895i \(-0.568558\pi\)
−0.213720 + 0.976895i \(0.568558\pi\)
\(572\) 0.0352767 0.00147499
\(573\) 0 0
\(574\) −19.0539 −0.795294
\(575\) 4.93083 0.205630
\(576\) 0 0
\(577\) −44.5539 −1.85480 −0.927402 0.374065i \(-0.877964\pi\)
−0.927402 + 0.374065i \(0.877964\pi\)
\(578\) −17.7248 −0.737256
\(579\) 0 0
\(580\) 1.66644 0.0691952
\(581\) 1.04195 0.0432274
\(582\) 0 0
\(583\) 0.191678 0.00793851
\(584\) 2.39578 0.0991380
\(585\) 0 0
\(586\) 40.1066 1.65679
\(587\) 20.9069 0.862921 0.431461 0.902132i \(-0.357998\pi\)
0.431461 + 0.902132i \(0.357998\pi\)
\(588\) 0 0
\(589\) −17.3906 −0.716567
\(590\) −12.2404 −0.503928
\(591\) 0 0
\(592\) 5.47628 0.225074
\(593\) −7.48843 −0.307513 −0.153756 0.988109i \(-0.549137\pi\)
−0.153756 + 0.988109i \(0.549137\pi\)
\(594\) 0 0
\(595\) 5.28369 0.216610
\(596\) −1.26662 −0.0518827
\(597\) 0 0
\(598\) −8.05716 −0.329482
\(599\) −27.9314 −1.14125 −0.570623 0.821212i \(-0.693298\pi\)
−0.570623 + 0.821212i \(0.693298\pi\)
\(600\) 0 0
\(601\) 32.7780 1.33704 0.668521 0.743693i \(-0.266927\pi\)
0.668521 + 0.743693i \(0.266927\pi\)
\(602\) −32.9873 −1.34446
\(603\) 0 0
\(604\) −3.27457 −0.133240
\(605\) 10.9972 0.447101
\(606\) 0 0
\(607\) 39.1704 1.58988 0.794940 0.606689i \(-0.207503\pi\)
0.794940 + 0.606689i \(0.207503\pi\)
\(608\) −7.76988 −0.315110
\(609\) 0 0
\(610\) 9.45133 0.382673
\(611\) −9.76560 −0.395074
\(612\) 0 0
\(613\) −10.2019 −0.412051 −0.206026 0.978547i \(-0.566053\pi\)
−0.206026 + 0.978547i \(0.566053\pi\)
\(614\) −18.0870 −0.729932
\(615\) 0 0
\(616\) 0.243699 0.00981893
\(617\) −9.68947 −0.390084 −0.195042 0.980795i \(-0.562484\pi\)
−0.195042 + 0.980795i \(0.562484\pi\)
\(618\) 0 0
\(619\) −19.8750 −0.798842 −0.399421 0.916768i \(-0.630789\pi\)
−0.399421 + 0.916768i \(0.630789\pi\)
\(620\) 5.46823 0.219610
\(621\) 0 0
\(622\) 24.1728 0.969243
\(623\) −9.90292 −0.396752
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 44.2047 1.76678
\(627\) 0 0
\(628\) −12.0612 −0.481296
\(629\) 2.77721 0.110735
\(630\) 0 0
\(631\) −38.9270 −1.54966 −0.774830 0.632170i \(-0.782165\pi\)
−0.774830 + 0.632170i \(0.782165\pi\)
\(632\) 33.9349 1.34986
\(633\) 0 0
\(634\) 0.944470 0.0375097
\(635\) 10.6036 0.420791
\(636\) 0 0
\(637\) 2.46260 0.0975719
\(638\) 0.213939 0.00846991
\(639\) 0 0
\(640\) −13.5415 −0.535275
\(641\) −7.89412 −0.311799 −0.155899 0.987773i \(-0.549828\pi\)
−0.155899 + 0.987773i \(0.549828\pi\)
\(642\) 0 0
\(643\) 35.7751 1.41083 0.705415 0.708794i \(-0.250760\pi\)
0.705415 + 0.708794i \(0.250760\pi\)
\(644\) 7.03799 0.277336
\(645\) 0 0
\(646\) −8.63752 −0.339839
\(647\) −1.47889 −0.0581412 −0.0290706 0.999577i \(-0.509255\pi\)
−0.0290706 + 0.999577i \(0.509255\pi\)
\(648\) 0 0
\(649\) −0.394362 −0.0154801
\(650\) −1.63404 −0.0640922
\(651\) 0 0
\(652\) 6.31044 0.247136
\(653\) −6.95046 −0.271992 −0.135996 0.990709i \(-0.543423\pi\)
−0.135996 + 0.990709i \(0.543423\pi\)
\(654\) 0 0
\(655\) 16.1923 0.632686
\(656\) 26.7750 1.04539
\(657\) 0 0
\(658\) 33.9910 1.32511
\(659\) 30.9546 1.20582 0.602910 0.797810i \(-0.294008\pi\)
0.602910 + 0.797810i \(0.294008\pi\)
\(660\) 0 0
\(661\) 3.57678 0.139121 0.0695603 0.997578i \(-0.477840\pi\)
0.0695603 + 0.997578i \(0.477840\pi\)
\(662\) −39.4032 −1.53145
\(663\) 0 0
\(664\) −1.06300 −0.0412522
\(665\) −4.53938 −0.176029
\(666\) 0 0
\(667\) −12.2627 −0.474812
\(668\) −10.3798 −0.401607
\(669\) 0 0
\(670\) 10.1897 0.393663
\(671\) 0.304505 0.0117553
\(672\) 0 0
\(673\) −28.0993 −1.08315 −0.541575 0.840653i \(-0.682172\pi\)
−0.541575 + 0.840653i \(0.682172\pi\)
\(674\) 10.0247 0.386136
\(675\) 0 0
\(676\) 0.670078 0.0257722
\(677\) 44.2934 1.70233 0.851167 0.524895i \(-0.175895\pi\)
0.851167 + 0.524895i \(0.175895\pi\)
\(678\) 0 0
\(679\) 15.6371 0.600098
\(680\) −5.39042 −0.206713
\(681\) 0 0
\(682\) 0.702016 0.0268816
\(683\) −1.69041 −0.0646816 −0.0323408 0.999477i \(-0.510296\pi\)
−0.0323408 + 0.999477i \(0.510296\pi\)
\(684\) 0 0
\(685\) 20.7686 0.793526
\(686\) −32.9364 −1.25752
\(687\) 0 0
\(688\) 46.3545 1.76725
\(689\) 3.64091 0.138708
\(690\) 0 0
\(691\) −10.8144 −0.411398 −0.205699 0.978615i \(-0.565947\pi\)
−0.205699 + 0.978615i \(0.565947\pi\)
\(692\) −6.48409 −0.246488
\(693\) 0 0
\(694\) −6.08514 −0.230989
\(695\) 2.50075 0.0948589
\(696\) 0 0
\(697\) 13.5785 0.514323
\(698\) −5.47182 −0.207111
\(699\) 0 0
\(700\) 1.42734 0.0539485
\(701\) −32.1755 −1.21525 −0.607626 0.794224i \(-0.707878\pi\)
−0.607626 + 0.794224i \(0.707878\pi\)
\(702\) 0 0
\(703\) −2.38598 −0.0899891
\(704\) −0.201346 −0.00758850
\(705\) 0 0
\(706\) −58.1106 −2.18702
\(707\) 1.17765 0.0442902
\(708\) 0 0
\(709\) 31.9120 1.19848 0.599240 0.800570i \(-0.295470\pi\)
0.599240 + 0.800570i \(0.295470\pi\)
\(710\) −4.08591 −0.153341
\(711\) 0 0
\(712\) 10.1030 0.378624
\(713\) −40.2385 −1.50694
\(714\) 0 0
\(715\) −0.0526457 −0.00196884
\(716\) −2.22218 −0.0830467
\(717\) 0 0
\(718\) 14.8191 0.553045
\(719\) 20.6891 0.771572 0.385786 0.922588i \(-0.373930\pi\)
0.385786 + 0.922588i \(0.373930\pi\)
\(720\) 0 0
\(721\) −16.6958 −0.621782
\(722\) −23.6260 −0.879267
\(723\) 0 0
\(724\) −15.1564 −0.563283
\(725\) −2.48694 −0.0923625
\(726\) 0 0
\(727\) −24.6791 −0.915299 −0.457649 0.889133i \(-0.651308\pi\)
−0.457649 + 0.889133i \(0.651308\pi\)
\(728\) 4.62905 0.171564
\(729\) 0 0
\(730\) 1.80144 0.0666744
\(731\) 23.5080 0.869474
\(732\) 0 0
\(733\) −21.7459 −0.803203 −0.401602 0.915814i \(-0.631546\pi\)
−0.401602 + 0.915814i \(0.631546\pi\)
\(734\) −40.2277 −1.48483
\(735\) 0 0
\(736\) −17.9780 −0.662678
\(737\) 0.328294 0.0120929
\(738\) 0 0
\(739\) 43.0460 1.58347 0.791736 0.610864i \(-0.209178\pi\)
0.791736 + 0.610864i \(0.209178\pi\)
\(740\) 0.750240 0.0275794
\(741\) 0 0
\(742\) −12.6729 −0.465236
\(743\) 7.89816 0.289755 0.144878 0.989450i \(-0.453721\pi\)
0.144878 + 0.989450i \(0.453721\pi\)
\(744\) 0 0
\(745\) 1.89025 0.0692536
\(746\) 44.0794 1.61386
\(747\) 0 0
\(748\) 0.0875029 0.00319942
\(749\) 41.3612 1.51130
\(750\) 0 0
\(751\) 38.6450 1.41018 0.705089 0.709119i \(-0.250907\pi\)
0.705089 + 0.709119i \(0.250907\pi\)
\(752\) −47.7650 −1.74181
\(753\) 0 0
\(754\) 4.06374 0.147993
\(755\) 4.88685 0.177851
\(756\) 0 0
\(757\) −15.3437 −0.557678 −0.278839 0.960338i \(-0.589950\pi\)
−0.278839 + 0.960338i \(0.589950\pi\)
\(758\) 23.2735 0.845331
\(759\) 0 0
\(760\) 4.63107 0.167986
\(761\) 48.7170 1.76599 0.882995 0.469382i \(-0.155523\pi\)
0.882995 + 0.469382i \(0.155523\pi\)
\(762\) 0 0
\(763\) −4.54784 −0.164643
\(764\) 3.94147 0.142597
\(765\) 0 0
\(766\) 42.4418 1.53349
\(767\) −7.49087 −0.270480
\(768\) 0 0
\(769\) 6.37109 0.229747 0.114874 0.993380i \(-0.463354\pi\)
0.114874 + 0.993380i \(0.463354\pi\)
\(770\) 0.183243 0.00660363
\(771\) 0 0
\(772\) −0.307927 −0.0110825
\(773\) 29.8732 1.07446 0.537232 0.843435i \(-0.319470\pi\)
0.537232 + 0.843435i \(0.319470\pi\)
\(774\) 0 0
\(775\) −8.16059 −0.293137
\(776\) −15.9530 −0.572679
\(777\) 0 0
\(778\) 42.1429 1.51090
\(779\) −11.6657 −0.417967
\(780\) 0 0
\(781\) −0.131641 −0.00471047
\(782\) −19.9856 −0.714682
\(783\) 0 0
\(784\) 12.0450 0.430178
\(785\) 17.9998 0.642439
\(786\) 0 0
\(787\) −23.6556 −0.843232 −0.421616 0.906774i \(-0.638537\pi\)
−0.421616 + 0.906774i \(0.638537\pi\)
\(788\) −1.54868 −0.0551693
\(789\) 0 0
\(790\) 25.5164 0.907834
\(791\) −13.2819 −0.472250
\(792\) 0 0
\(793\) 5.78404 0.205397
\(794\) −31.5189 −1.11856
\(795\) 0 0
\(796\) 16.3321 0.578876
\(797\) −30.0006 −1.06268 −0.531339 0.847160i \(-0.678311\pi\)
−0.531339 + 0.847160i \(0.678311\pi\)
\(798\) 0 0
\(799\) −24.2233 −0.856959
\(800\) −3.64604 −0.128907
\(801\) 0 0
\(802\) −26.9013 −0.949918
\(803\) 0.0580392 0.00204816
\(804\) 0 0
\(805\) −10.5032 −0.370191
\(806\) 13.3347 0.469695
\(807\) 0 0
\(808\) −1.20144 −0.0422665
\(809\) 37.0787 1.30362 0.651808 0.758384i \(-0.274011\pi\)
0.651808 + 0.758384i \(0.274011\pi\)
\(810\) 0 0
\(811\) −42.7582 −1.50144 −0.750722 0.660618i \(-0.770294\pi\)
−0.750722 + 0.660618i \(0.770294\pi\)
\(812\) −3.54971 −0.124570
\(813\) 0 0
\(814\) 0.0963163 0.00337588
\(815\) −9.41748 −0.329880
\(816\) 0 0
\(817\) −20.1964 −0.706583
\(818\) −32.2765 −1.12852
\(819\) 0 0
\(820\) 3.66812 0.128096
\(821\) 28.5669 0.996992 0.498496 0.866892i \(-0.333886\pi\)
0.498496 + 0.866892i \(0.333886\pi\)
\(822\) 0 0
\(823\) 32.0041 1.11559 0.557797 0.829977i \(-0.311647\pi\)
0.557797 + 0.829977i \(0.311647\pi\)
\(824\) 17.0330 0.593372
\(825\) 0 0
\(826\) 26.0734 0.907209
\(827\) 31.4986 1.09531 0.547657 0.836703i \(-0.315520\pi\)
0.547657 + 0.836703i \(0.315520\pi\)
\(828\) 0 0
\(829\) 32.3851 1.12478 0.562390 0.826872i \(-0.309882\pi\)
0.562390 + 0.826872i \(0.309882\pi\)
\(830\) −0.799292 −0.0277438
\(831\) 0 0
\(832\) −3.82454 −0.132592
\(833\) 6.10842 0.211644
\(834\) 0 0
\(835\) 15.4905 0.536070
\(836\) −0.0751763 −0.00260003
\(837\) 0 0
\(838\) −28.5275 −0.985467
\(839\) 46.0763 1.59073 0.795365 0.606131i \(-0.207279\pi\)
0.795365 + 0.606131i \(0.207279\pi\)
\(840\) 0 0
\(841\) −22.8152 −0.786729
\(842\) −12.1180 −0.417612
\(843\) 0 0
\(844\) 18.3560 0.631839
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −23.4254 −0.804905
\(848\) 17.8083 0.611538
\(849\) 0 0
\(850\) −4.05318 −0.139023
\(851\) −5.52071 −0.189248
\(852\) 0 0
\(853\) 48.7573 1.66942 0.834709 0.550692i \(-0.185636\pi\)
0.834709 + 0.550692i \(0.185636\pi\)
\(854\) −20.1324 −0.688918
\(855\) 0 0
\(856\) −42.1966 −1.44225
\(857\) 31.9479 1.09132 0.545660 0.838007i \(-0.316279\pi\)
0.545660 + 0.838007i \(0.316279\pi\)
\(858\) 0 0
\(859\) −7.33499 −0.250267 −0.125133 0.992140i \(-0.539936\pi\)
−0.125133 + 0.992140i \(0.539936\pi\)
\(860\) 6.35048 0.216549
\(861\) 0 0
\(862\) 54.9195 1.87056
\(863\) 28.4742 0.969273 0.484637 0.874716i \(-0.338952\pi\)
0.484637 + 0.874716i \(0.338952\pi\)
\(864\) 0 0
\(865\) 9.67663 0.329015
\(866\) 49.4829 1.68150
\(867\) 0 0
\(868\) −11.6480 −0.395358
\(869\) 0.822093 0.0278876
\(870\) 0 0
\(871\) 6.23591 0.211296
\(872\) 4.63970 0.157120
\(873\) 0 0
\(874\) 17.1702 0.580790
\(875\) −2.13012 −0.0720111
\(876\) 0 0
\(877\) 45.8529 1.54834 0.774171 0.632977i \(-0.218167\pi\)
0.774171 + 0.632977i \(0.218167\pi\)
\(878\) −46.9574 −1.58473
\(879\) 0 0
\(880\) −0.257498 −0.00868026
\(881\) −37.4454 −1.26157 −0.630784 0.775959i \(-0.717267\pi\)
−0.630784 + 0.775959i \(0.717267\pi\)
\(882\) 0 0
\(883\) 50.8068 1.70978 0.854892 0.518805i \(-0.173623\pi\)
0.854892 + 0.518805i \(0.173623\pi\)
\(884\) 1.66211 0.0559028
\(885\) 0 0
\(886\) −36.6381 −1.23088
\(887\) −45.2968 −1.52092 −0.760459 0.649386i \(-0.775026\pi\)
−0.760459 + 0.649386i \(0.775026\pi\)
\(888\) 0 0
\(889\) −22.5869 −0.757541
\(890\) 7.59665 0.254640
\(891\) 0 0
\(892\) 16.9770 0.568432
\(893\) 20.8109 0.696412
\(894\) 0 0
\(895\) 3.31630 0.110852
\(896\) 28.8450 0.963644
\(897\) 0 0
\(898\) 26.1690 0.873270
\(899\) 20.2949 0.676872
\(900\) 0 0
\(901\) 9.03118 0.300872
\(902\) 0.470916 0.0156798
\(903\) 0 0
\(904\) 13.5502 0.450672
\(905\) 22.6189 0.751876
\(906\) 0 0
\(907\) 14.3690 0.477116 0.238558 0.971128i \(-0.423325\pi\)
0.238558 + 0.971128i \(0.423325\pi\)
\(908\) −14.2336 −0.472359
\(909\) 0 0
\(910\) 3.48069 0.115384
\(911\) 53.9063 1.78600 0.892998 0.450060i \(-0.148597\pi\)
0.892998 + 0.450060i \(0.148597\pi\)
\(912\) 0 0
\(913\) −0.0257517 −0.000852258 0
\(914\) 1.74117 0.0575926
\(915\) 0 0
\(916\) 8.19755 0.270855
\(917\) −34.4915 −1.13901
\(918\) 0 0
\(919\) −15.6573 −0.516487 −0.258244 0.966080i \(-0.583144\pi\)
−0.258244 + 0.966080i \(0.583144\pi\)
\(920\) 10.7154 0.353276
\(921\) 0 0
\(922\) 40.2605 1.32591
\(923\) −2.50050 −0.0823049
\(924\) 0 0
\(925\) −1.11963 −0.0368132
\(926\) 11.6397 0.382505
\(927\) 0 0
\(928\) 9.06746 0.297654
\(929\) −23.1439 −0.759327 −0.379664 0.925125i \(-0.623960\pi\)
−0.379664 + 0.925125i \(0.623960\pi\)
\(930\) 0 0
\(931\) −5.24792 −0.171994
\(932\) 7.64211 0.250326
\(933\) 0 0
\(934\) 33.4425 1.09427
\(935\) −0.130586 −0.00427062
\(936\) 0 0
\(937\) 14.6354 0.478118 0.239059 0.971005i \(-0.423161\pi\)
0.239059 + 0.971005i \(0.423161\pi\)
\(938\) −21.7052 −0.708702
\(939\) 0 0
\(940\) −6.54371 −0.213432
\(941\) −22.0393 −0.718462 −0.359231 0.933249i \(-0.616961\pi\)
−0.359231 + 0.933249i \(0.616961\pi\)
\(942\) 0 0
\(943\) −26.9922 −0.878987
\(944\) −36.6390 −1.19250
\(945\) 0 0
\(946\) 0.815279 0.0265070
\(947\) −50.3906 −1.63747 −0.818737 0.574168i \(-0.805326\pi\)
−0.818737 + 0.574168i \(0.805326\pi\)
\(948\) 0 0
\(949\) 1.10245 0.0357870
\(950\) 3.48221 0.112978
\(951\) 0 0
\(952\) 11.4822 0.372141
\(953\) −7.28058 −0.235841 −0.117921 0.993023i \(-0.537623\pi\)
−0.117921 + 0.993023i \(0.537623\pi\)
\(954\) 0 0
\(955\) −5.88211 −0.190340
\(956\) 9.95540 0.321981
\(957\) 0 0
\(958\) −23.6443 −0.763913
\(959\) −44.2395 −1.42857
\(960\) 0 0
\(961\) 35.5953 1.14824
\(962\) 1.82952 0.0589861
\(963\) 0 0
\(964\) −12.4573 −0.401221
\(965\) 0.459539 0.0147931
\(966\) 0 0
\(967\) 1.64327 0.0528441 0.0264220 0.999651i \(-0.491589\pi\)
0.0264220 + 0.999651i \(0.491589\pi\)
\(968\) 23.8985 0.768128
\(969\) 0 0
\(970\) −11.9954 −0.385150
\(971\) 59.5135 1.90988 0.954939 0.296803i \(-0.0959204\pi\)
0.954939 + 0.296803i \(0.0959204\pi\)
\(972\) 0 0
\(973\) −5.32689 −0.170772
\(974\) 48.7915 1.56338
\(975\) 0 0
\(976\) 28.2906 0.905560
\(977\) −15.3273 −0.490363 −0.245181 0.969477i \(-0.578847\pi\)
−0.245181 + 0.969477i \(0.578847\pi\)
\(978\) 0 0
\(979\) 0.244750 0.00782225
\(980\) 1.65014 0.0527117
\(981\) 0 0
\(982\) 36.0255 1.14962
\(983\) −3.26954 −0.104282 −0.0521411 0.998640i \(-0.516605\pi\)
−0.0521411 + 0.998640i \(0.516605\pi\)
\(984\) 0 0
\(985\) 2.31119 0.0736405
\(986\) 10.0800 0.321013
\(987\) 0 0
\(988\) −1.42797 −0.0454297
\(989\) −46.7306 −1.48595
\(990\) 0 0
\(991\) −39.2309 −1.24621 −0.623106 0.782138i \(-0.714129\pi\)
−0.623106 + 0.782138i \(0.714129\pi\)
\(992\) 29.7538 0.944685
\(993\) 0 0
\(994\) 8.70346 0.276057
\(995\) −24.3734 −0.772690
\(996\) 0 0
\(997\) −32.8254 −1.03959 −0.519795 0.854291i \(-0.673992\pi\)
−0.519795 + 0.854291i \(0.673992\pi\)
\(998\) 34.5963 1.09513
\(999\) 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 5265.2.a.bf.1.6 8
3.2 odd 2 5265.2.a.ba.1.3 8
9.2 odd 6 1755.2.i.f.1171.6 16
9.4 even 3 585.2.i.e.196.3 16
9.5 odd 6 1755.2.i.f.586.6 16
9.7 even 3 585.2.i.e.391.3 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.i.e.196.3 16 9.4 even 3
585.2.i.e.391.3 yes 16 9.7 even 3
1755.2.i.f.586.6 16 9.5 odd 6
1755.2.i.f.1171.6 16 9.2 odd 6
5265.2.a.ba.1.3 8 3.2 odd 2
5265.2.a.bf.1.6 8 1.1 even 1 trivial