Properties

Label 9680.2.a.cu.1.4
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("9680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.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: \(77.2951891566\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.35567\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.19353 q^{3} +1.00000 q^{5} +0.544113 q^{7} +1.81156 q^{9} +O(q^{10})\) \(q+2.19353 q^{3} +1.00000 q^{5} +0.544113 q^{7} +1.81156 q^{9} +1.63743 q^{13} +2.19353 q^{15} +0.169045 q^{17} +4.72333 q^{19} +1.19353 q^{21} +2.26236 q^{23} +1.00000 q^{25} -2.60687 q^{27} +9.43468 q^{29} +3.01116 q^{31} +0.544113 q^{35} -0.0537019 q^{37} +3.59174 q^{39} -2.54230 q^{41} +7.46097 q^{43} +1.81156 q^{45} -9.07211 q^{47} -6.70394 q^{49} +0.370805 q^{51} -2.03452 q^{53} +10.3608 q^{57} +12.6782 q^{59} +2.80781 q^{61} +0.985694 q^{63} +1.63743 q^{65} +1.26369 q^{67} +4.96255 q^{69} -13.8311 q^{71} +7.97504 q^{73} +2.19353 q^{75} -3.13271 q^{79} -11.1529 q^{81} -2.73764 q^{83} +0.169045 q^{85} +20.6952 q^{87} -10.9251 q^{89} +0.890946 q^{91} +6.60506 q^{93} +4.72333 q^{95} +7.89798 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 4 q^{5} + 7 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 4 q^{5} + 7 q^{7} - 4 q^{9} + 3 q^{13} + 2 q^{15} + 11 q^{17} - 2 q^{19} - 2 q^{21} + 11 q^{23} + 4 q^{25} - q^{27} + 4 q^{29} + 17 q^{31} + 7 q^{35} - 3 q^{37} - q^{39} + 13 q^{41} + 7 q^{43} - 4 q^{45} + q^{47} - 5 q^{49} + q^{51} - 15 q^{53} + 17 q^{57} + 17 q^{59} + 4 q^{61} - 15 q^{63} + 3 q^{65} - 7 q^{67} + 4 q^{69} + 15 q^{71} + 7 q^{73} + 2 q^{75} + 12 q^{79} - 8 q^{81} - 9 q^{83} + 11 q^{85} + 23 q^{87} - 12 q^{89} + 24 q^{91} - 11 q^{93} - 2 q^{95} + 2 q^{97}+O(q^{100}) \) 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
\(3\) 2.19353 1.26643 0.633217 0.773975i \(-0.281734\pi\)
0.633217 + 0.773975i \(0.281734\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.544113 0.205655 0.102828 0.994699i \(-0.467211\pi\)
0.102828 + 0.994699i \(0.467211\pi\)
\(8\) 0 0
\(9\) 1.81156 0.603854
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 1.63743 0.454141 0.227070 0.973878i \(-0.427085\pi\)
0.227070 + 0.973878i \(0.427085\pi\)
\(14\) 0 0
\(15\) 2.19353 0.566366
\(16\) 0 0
\(17\) 0.169045 0.0409995 0.0204997 0.999790i \(-0.493474\pi\)
0.0204997 + 0.999790i \(0.493474\pi\)
\(18\) 0 0
\(19\) 4.72333 1.08361 0.541804 0.840505i \(-0.317742\pi\)
0.541804 + 0.840505i \(0.317742\pi\)
\(20\) 0 0
\(21\) 1.19353 0.260449
\(22\) 0 0
\(23\) 2.26236 0.471735 0.235867 0.971785i \(-0.424207\pi\)
0.235867 + 0.971785i \(0.424207\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −2.60687 −0.501693
\(28\) 0 0
\(29\) 9.43468 1.75198 0.875988 0.482332i \(-0.160210\pi\)
0.875988 + 0.482332i \(0.160210\pi\)
\(30\) 0 0
\(31\) 3.01116 0.540820 0.270410 0.962745i \(-0.412841\pi\)
0.270410 + 0.962745i \(0.412841\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.544113 0.0919719
\(36\) 0 0
\(37\) −0.0537019 −0.00882854 −0.00441427 0.999990i \(-0.501405\pi\)
−0.00441427 + 0.999990i \(0.501405\pi\)
\(38\) 0 0
\(39\) 3.59174 0.575139
\(40\) 0 0
\(41\) −2.54230 −0.397041 −0.198521 0.980097i \(-0.563614\pi\)
−0.198521 + 0.980097i \(0.563614\pi\)
\(42\) 0 0
\(43\) 7.46097 1.13779 0.568894 0.822411i \(-0.307371\pi\)
0.568894 + 0.822411i \(0.307371\pi\)
\(44\) 0 0
\(45\) 1.81156 0.270052
\(46\) 0 0
\(47\) −9.07211 −1.32330 −0.661652 0.749811i \(-0.730144\pi\)
−0.661652 + 0.749811i \(0.730144\pi\)
\(48\) 0 0
\(49\) −6.70394 −0.957706
\(50\) 0 0
\(51\) 0.370805 0.0519231
\(52\) 0 0
\(53\) −2.03452 −0.279463 −0.139732 0.990189i \(-0.544624\pi\)
−0.139732 + 0.990189i \(0.544624\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 10.3608 1.37232
\(58\) 0 0
\(59\) 12.6782 1.65056 0.825278 0.564727i \(-0.191018\pi\)
0.825278 + 0.564727i \(0.191018\pi\)
\(60\) 0 0
\(61\) 2.80781 0.359503 0.179751 0.983712i \(-0.442471\pi\)
0.179751 + 0.983712i \(0.442471\pi\)
\(62\) 0 0
\(63\) 0.985694 0.124186
\(64\) 0 0
\(65\) 1.63743 0.203098
\(66\) 0 0
\(67\) 1.26369 0.154385 0.0771924 0.997016i \(-0.475404\pi\)
0.0771924 + 0.997016i \(0.475404\pi\)
\(68\) 0 0
\(69\) 4.96255 0.597420
\(70\) 0 0
\(71\) −13.8311 −1.64145 −0.820724 0.571325i \(-0.806430\pi\)
−0.820724 + 0.571325i \(0.806430\pi\)
\(72\) 0 0
\(73\) 7.97504 0.933408 0.466704 0.884414i \(-0.345441\pi\)
0.466704 + 0.884414i \(0.345441\pi\)
\(74\) 0 0
\(75\) 2.19353 0.253287
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.13271 −0.352458 −0.176229 0.984349i \(-0.556390\pi\)
−0.176229 + 0.984349i \(0.556390\pi\)
\(80\) 0 0
\(81\) −11.1529 −1.23921
\(82\) 0 0
\(83\) −2.73764 −0.300495 −0.150248 0.988648i \(-0.548007\pi\)
−0.150248 + 0.988648i \(0.548007\pi\)
\(84\) 0 0
\(85\) 0.169045 0.0183355
\(86\) 0 0
\(87\) 20.6952 2.21876
\(88\) 0 0
\(89\) −10.9251 −1.15806 −0.579029 0.815307i \(-0.696568\pi\)
−0.579029 + 0.815307i \(0.696568\pi\)
\(90\) 0 0
\(91\) 0.890946 0.0933965
\(92\) 0 0
\(93\) 6.60506 0.684913
\(94\) 0 0
\(95\) 4.72333 0.484604
\(96\) 0 0
\(97\) 7.89798 0.801918 0.400959 0.916096i \(-0.368677\pi\)
0.400959 + 0.916096i \(0.368677\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.54545 −0.452289 −0.226144 0.974094i \(-0.572612\pi\)
−0.226144 + 0.974094i \(0.572612\pi\)
\(102\) 0 0
\(103\) −0.475281 −0.0468308 −0.0234154 0.999726i \(-0.507454\pi\)
−0.0234154 + 0.999726i \(0.507454\pi\)
\(104\) 0 0
\(105\) 1.19353 0.116476
\(106\) 0 0
\(107\) −3.45671 −0.334173 −0.167086 0.985942i \(-0.553436\pi\)
−0.167086 + 0.985942i \(0.553436\pi\)
\(108\) 0 0
\(109\) −12.7327 −1.21957 −0.609785 0.792567i \(-0.708744\pi\)
−0.609785 + 0.792567i \(0.708744\pi\)
\(110\) 0 0
\(111\) −0.117797 −0.0111808
\(112\) 0 0
\(113\) 11.8988 1.11935 0.559673 0.828714i \(-0.310927\pi\)
0.559673 + 0.828714i \(0.310927\pi\)
\(114\) 0 0
\(115\) 2.26236 0.210966
\(116\) 0 0
\(117\) 2.96630 0.274235
\(118\) 0 0
\(119\) 0.0919797 0.00843176
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −5.57661 −0.502826
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 18.7284 1.66188 0.830939 0.556363i \(-0.187804\pi\)
0.830939 + 0.556363i \(0.187804\pi\)
\(128\) 0 0
\(129\) 16.3659 1.44093
\(130\) 0 0
\(131\) 5.25037 0.458727 0.229364 0.973341i \(-0.426336\pi\)
0.229364 + 0.973341i \(0.426336\pi\)
\(132\) 0 0
\(133\) 2.57003 0.222850
\(134\) 0 0
\(135\) −2.60687 −0.224364
\(136\) 0 0
\(137\) −18.3401 −1.56690 −0.783449 0.621456i \(-0.786541\pi\)
−0.783449 + 0.621456i \(0.786541\pi\)
\(138\) 0 0
\(139\) −2.07720 −0.176186 −0.0880929 0.996112i \(-0.528077\pi\)
−0.0880929 + 0.996112i \(0.528077\pi\)
\(140\) 0 0
\(141\) −19.8999 −1.67588
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 9.43468 0.783508
\(146\) 0 0
\(147\) −14.7053 −1.21287
\(148\) 0 0
\(149\) 23.4562 1.92161 0.960805 0.277225i \(-0.0894149\pi\)
0.960805 + 0.277225i \(0.0894149\pi\)
\(150\) 0 0
\(151\) 5.92720 0.482349 0.241174 0.970482i \(-0.422467\pi\)
0.241174 + 0.970482i \(0.422467\pi\)
\(152\) 0 0
\(153\) 0.306236 0.0247577
\(154\) 0 0
\(155\) 3.01116 0.241862
\(156\) 0 0
\(157\) 13.1554 1.04991 0.524957 0.851129i \(-0.324081\pi\)
0.524957 + 0.851129i \(0.324081\pi\)
\(158\) 0 0
\(159\) −4.46278 −0.353922
\(160\) 0 0
\(161\) 1.23098 0.0970148
\(162\) 0 0
\(163\) 6.12253 0.479554 0.239777 0.970828i \(-0.422926\pi\)
0.239777 + 0.970828i \(0.422926\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −19.6820 −1.52304 −0.761521 0.648140i \(-0.775547\pi\)
−0.761521 + 0.648140i \(0.775547\pi\)
\(168\) 0 0
\(169\) −10.3188 −0.793756
\(170\) 0 0
\(171\) 8.55661 0.654340
\(172\) 0 0
\(173\) 9.62920 0.732094 0.366047 0.930596i \(-0.380711\pi\)
0.366047 + 0.930596i \(0.380711\pi\)
\(174\) 0 0
\(175\) 0.544113 0.0411311
\(176\) 0 0
\(177\) 27.8099 2.09032
\(178\) 0 0
\(179\) 14.6636 1.09601 0.548003 0.836476i \(-0.315388\pi\)
0.548003 + 0.836476i \(0.315388\pi\)
\(180\) 0 0
\(181\) −11.8766 −0.882784 −0.441392 0.897314i \(-0.645515\pi\)
−0.441392 + 0.897314i \(0.645515\pi\)
\(182\) 0 0
\(183\) 6.15900 0.455287
\(184\) 0 0
\(185\) −0.0537019 −0.00394824
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.41843 −0.103176
\(190\) 0 0
\(191\) 19.6970 1.42523 0.712614 0.701556i \(-0.247511\pi\)
0.712614 + 0.701556i \(0.247511\pi\)
\(192\) 0 0
\(193\) −12.1991 −0.878112 −0.439056 0.898460i \(-0.644687\pi\)
−0.439056 + 0.898460i \(0.644687\pi\)
\(194\) 0 0
\(195\) 3.59174 0.257210
\(196\) 0 0
\(197\) 6.60821 0.470815 0.235408 0.971897i \(-0.424357\pi\)
0.235408 + 0.971897i \(0.424357\pi\)
\(198\) 0 0
\(199\) 0.680185 0.0482170 0.0241085 0.999709i \(-0.492325\pi\)
0.0241085 + 0.999709i \(0.492325\pi\)
\(200\) 0 0
\(201\) 2.77195 0.195518
\(202\) 0 0
\(203\) 5.13354 0.360304
\(204\) 0 0
\(205\) −2.54230 −0.177562
\(206\) 0 0
\(207\) 4.09840 0.284859
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −16.5424 −1.13883 −0.569414 0.822051i \(-0.692830\pi\)
−0.569414 + 0.822051i \(0.692830\pi\)
\(212\) 0 0
\(213\) −30.3389 −2.07878
\(214\) 0 0
\(215\) 7.46097 0.508834
\(216\) 0 0
\(217\) 1.63841 0.111223
\(218\) 0 0
\(219\) 17.4935 1.18210
\(220\) 0 0
\(221\) 0.276799 0.0186195
\(222\) 0 0
\(223\) 25.6110 1.71504 0.857519 0.514452i \(-0.172005\pi\)
0.857519 + 0.514452i \(0.172005\pi\)
\(224\) 0 0
\(225\) 1.81156 0.120771
\(226\) 0 0
\(227\) 25.5668 1.69693 0.848464 0.529254i \(-0.177528\pi\)
0.848464 + 0.529254i \(0.177528\pi\)
\(228\) 0 0
\(229\) 0.0811121 0.00536004 0.00268002 0.999996i \(-0.499147\pi\)
0.00268002 + 0.999996i \(0.499147\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.5895 −0.955787 −0.477894 0.878418i \(-0.658600\pi\)
−0.477894 + 0.878418i \(0.658600\pi\)
\(234\) 0 0
\(235\) −9.07211 −0.591799
\(236\) 0 0
\(237\) −6.87169 −0.446364
\(238\) 0 0
\(239\) 24.6474 1.59431 0.797155 0.603774i \(-0.206337\pi\)
0.797155 + 0.603774i \(0.206337\pi\)
\(240\) 0 0
\(241\) 20.8968 1.34608 0.673040 0.739606i \(-0.264988\pi\)
0.673040 + 0.739606i \(0.264988\pi\)
\(242\) 0 0
\(243\) −16.6436 −1.06769
\(244\) 0 0
\(245\) −6.70394 −0.428299
\(246\) 0 0
\(247\) 7.73412 0.492110
\(248\) 0 0
\(249\) −6.00509 −0.380557
\(250\) 0 0
\(251\) 19.6300 1.23904 0.619518 0.784982i \(-0.287328\pi\)
0.619518 + 0.784982i \(0.287328\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0.370805 0.0232207
\(256\) 0 0
\(257\) −5.51761 −0.344179 −0.172089 0.985081i \(-0.555052\pi\)
−0.172089 + 0.985081i \(0.555052\pi\)
\(258\) 0 0
\(259\) −0.0292199 −0.00181564
\(260\) 0 0
\(261\) 17.0915 1.05794
\(262\) 0 0
\(263\) 17.7620 1.09525 0.547627 0.836722i \(-0.315531\pi\)
0.547627 + 0.836722i \(0.315531\pi\)
\(264\) 0 0
\(265\) −2.03452 −0.124980
\(266\) 0 0
\(267\) −23.9645 −1.46660
\(268\) 0 0
\(269\) −10.8735 −0.662969 −0.331484 0.943461i \(-0.607549\pi\)
−0.331484 + 0.943461i \(0.607549\pi\)
\(270\) 0 0
\(271\) 9.75306 0.592456 0.296228 0.955117i \(-0.404271\pi\)
0.296228 + 0.955117i \(0.404271\pi\)
\(272\) 0 0
\(273\) 1.95431 0.118280
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 26.2082 1.57470 0.787349 0.616508i \(-0.211453\pi\)
0.787349 + 0.616508i \(0.211453\pi\)
\(278\) 0 0
\(279\) 5.45490 0.326576
\(280\) 0 0
\(281\) −30.0840 −1.79466 −0.897331 0.441358i \(-0.854497\pi\)
−0.897331 + 0.441358i \(0.854497\pi\)
\(282\) 0 0
\(283\) −7.32137 −0.435210 −0.217605 0.976037i \(-0.569824\pi\)
−0.217605 + 0.976037i \(0.569824\pi\)
\(284\) 0 0
\(285\) 10.3608 0.613719
\(286\) 0 0
\(287\) −1.38330 −0.0816537
\(288\) 0 0
\(289\) −16.9714 −0.998319
\(290\) 0 0
\(291\) 17.3244 1.01558
\(292\) 0 0
\(293\) 12.9860 0.758650 0.379325 0.925264i \(-0.376156\pi\)
0.379325 + 0.925264i \(0.376156\pi\)
\(294\) 0 0
\(295\) 12.6782 0.738151
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.70445 0.214234
\(300\) 0 0
\(301\) 4.05962 0.233992
\(302\) 0 0
\(303\) −9.97056 −0.572794
\(304\) 0 0
\(305\) 2.80781 0.160775
\(306\) 0 0
\(307\) −2.30624 −0.131624 −0.0658119 0.997832i \(-0.520964\pi\)
−0.0658119 + 0.997832i \(0.520964\pi\)
\(308\) 0 0
\(309\) −1.04254 −0.0593081
\(310\) 0 0
\(311\) 22.3608 1.26796 0.633981 0.773348i \(-0.281420\pi\)
0.633981 + 0.773348i \(0.281420\pi\)
\(312\) 0 0
\(313\) −31.0185 −1.75327 −0.876636 0.481154i \(-0.840218\pi\)
−0.876636 + 0.481154i \(0.840218\pi\)
\(314\) 0 0
\(315\) 0.985694 0.0555376
\(316\) 0 0
\(317\) −15.7100 −0.882364 −0.441182 0.897418i \(-0.645441\pi\)
−0.441182 + 0.897418i \(0.645441\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −7.58239 −0.423208
\(322\) 0 0
\(323\) 0.798457 0.0444273
\(324\) 0 0
\(325\) 1.63743 0.0908282
\(326\) 0 0
\(327\) −27.9295 −1.54450
\(328\) 0 0
\(329\) −4.93626 −0.272145
\(330\) 0 0
\(331\) 30.0204 1.65007 0.825034 0.565083i \(-0.191156\pi\)
0.825034 + 0.565083i \(0.191156\pi\)
\(332\) 0 0
\(333\) −0.0972843 −0.00533115
\(334\) 0 0
\(335\) 1.26369 0.0690430
\(336\) 0 0
\(337\) −29.2748 −1.59470 −0.797351 0.603516i \(-0.793766\pi\)
−0.797351 + 0.603516i \(0.793766\pi\)
\(338\) 0 0
\(339\) 26.1003 1.41758
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −7.45650 −0.402613
\(344\) 0 0
\(345\) 4.96255 0.267175
\(346\) 0 0
\(347\) −32.6474 −1.75261 −0.876303 0.481761i \(-0.839997\pi\)
−0.876303 + 0.481761i \(0.839997\pi\)
\(348\) 0 0
\(349\) 23.6430 1.26558 0.632789 0.774324i \(-0.281910\pi\)
0.632789 + 0.774324i \(0.281910\pi\)
\(350\) 0 0
\(351\) −4.26857 −0.227839
\(352\) 0 0
\(353\) −31.4013 −1.67132 −0.835662 0.549244i \(-0.814916\pi\)
−0.835662 + 0.549244i \(0.814916\pi\)
\(354\) 0 0
\(355\) −13.8311 −0.734078
\(356\) 0 0
\(357\) 0.201760 0.0106783
\(358\) 0 0
\(359\) −0.530101 −0.0279777 −0.0139888 0.999902i \(-0.504453\pi\)
−0.0139888 + 0.999902i \(0.504453\pi\)
\(360\) 0 0
\(361\) 3.30989 0.174205
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 7.97504 0.417433
\(366\) 0 0
\(367\) 4.33628 0.226352 0.113176 0.993575i \(-0.463898\pi\)
0.113176 + 0.993575i \(0.463898\pi\)
\(368\) 0 0
\(369\) −4.60554 −0.239755
\(370\) 0 0
\(371\) −1.10701 −0.0574732
\(372\) 0 0
\(373\) 10.4807 0.542669 0.271335 0.962485i \(-0.412535\pi\)
0.271335 + 0.962485i \(0.412535\pi\)
\(374\) 0 0
\(375\) 2.19353 0.113273
\(376\) 0 0
\(377\) 15.4486 0.795644
\(378\) 0 0
\(379\) 11.6117 0.596455 0.298228 0.954495i \(-0.403605\pi\)
0.298228 + 0.954495i \(0.403605\pi\)
\(380\) 0 0
\(381\) 41.0813 2.10466
\(382\) 0 0
\(383\) 2.38279 0.121755 0.0608775 0.998145i \(-0.480610\pi\)
0.0608775 + 0.998145i \(0.480610\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 13.5160 0.687057
\(388\) 0 0
\(389\) −19.2329 −0.975148 −0.487574 0.873082i \(-0.662118\pi\)
−0.487574 + 0.873082i \(0.662118\pi\)
\(390\) 0 0
\(391\) 0.382441 0.0193409
\(392\) 0 0
\(393\) 11.5168 0.580948
\(394\) 0 0
\(395\) −3.13271 −0.157624
\(396\) 0 0
\(397\) 34.1748 1.71518 0.857591 0.514332i \(-0.171960\pi\)
0.857591 + 0.514332i \(0.171960\pi\)
\(398\) 0 0
\(399\) 5.63743 0.282224
\(400\) 0 0
\(401\) −4.11422 −0.205455 −0.102727 0.994710i \(-0.532757\pi\)
−0.102727 + 0.994710i \(0.532757\pi\)
\(402\) 0 0
\(403\) 4.93056 0.245609
\(404\) 0 0
\(405\) −11.1529 −0.554194
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −19.5039 −0.964404 −0.482202 0.876060i \(-0.660163\pi\)
−0.482202 + 0.876060i \(0.660163\pi\)
\(410\) 0 0
\(411\) −40.2294 −1.98437
\(412\) 0 0
\(413\) 6.89835 0.339446
\(414\) 0 0
\(415\) −2.73764 −0.134385
\(416\) 0 0
\(417\) −4.55639 −0.223128
\(418\) 0 0
\(419\) −21.6184 −1.05613 −0.528064 0.849205i \(-0.677082\pi\)
−0.528064 + 0.849205i \(0.677082\pi\)
\(420\) 0 0
\(421\) −15.0594 −0.733950 −0.366975 0.930231i \(-0.619607\pi\)
−0.366975 + 0.930231i \(0.619607\pi\)
\(422\) 0 0
\(423\) −16.4347 −0.799082
\(424\) 0 0
\(425\) 0.169045 0.00819989
\(426\) 0 0
\(427\) 1.52777 0.0739337
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.08620 −0.0523204 −0.0261602 0.999658i \(-0.508328\pi\)
−0.0261602 + 0.999658i \(0.508328\pi\)
\(432\) 0 0
\(433\) −8.19203 −0.393684 −0.196842 0.980435i \(-0.563069\pi\)
−0.196842 + 0.980435i \(0.563069\pi\)
\(434\) 0 0
\(435\) 20.6952 0.992260
\(436\) 0 0
\(437\) 10.6859 0.511175
\(438\) 0 0
\(439\) 17.4476 0.832727 0.416364 0.909198i \(-0.363304\pi\)
0.416364 + 0.909198i \(0.363304\pi\)
\(440\) 0 0
\(441\) −12.1446 −0.578314
\(442\) 0 0
\(443\) 10.8052 0.513369 0.256685 0.966495i \(-0.417370\pi\)
0.256685 + 0.966495i \(0.417370\pi\)
\(444\) 0 0
\(445\) −10.9251 −0.517899
\(446\) 0 0
\(447\) 51.4519 2.43359
\(448\) 0 0
\(449\) 30.0753 1.41934 0.709671 0.704533i \(-0.248844\pi\)
0.709671 + 0.704533i \(0.248844\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 13.0015 0.610862
\(454\) 0 0
\(455\) 0.890946 0.0417682
\(456\) 0 0
\(457\) 29.5182 1.38081 0.690403 0.723425i \(-0.257433\pi\)
0.690403 + 0.723425i \(0.257433\pi\)
\(458\) 0 0
\(459\) −0.440679 −0.0205691
\(460\) 0 0
\(461\) −27.4603 −1.27896 −0.639478 0.768810i \(-0.720849\pi\)
−0.639478 + 0.768810i \(0.720849\pi\)
\(462\) 0 0
\(463\) 8.13077 0.377869 0.188934 0.981990i \(-0.439497\pi\)
0.188934 + 0.981990i \(0.439497\pi\)
\(464\) 0 0
\(465\) 6.60506 0.306302
\(466\) 0 0
\(467\) 3.22107 0.149053 0.0745267 0.997219i \(-0.476255\pi\)
0.0745267 + 0.997219i \(0.476255\pi\)
\(468\) 0 0
\(469\) 0.687593 0.0317501
\(470\) 0 0
\(471\) 28.8567 1.32965
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.72333 0.216721
\(476\) 0 0
\(477\) −3.68567 −0.168755
\(478\) 0 0
\(479\) 7.50962 0.343123 0.171562 0.985173i \(-0.445119\pi\)
0.171562 + 0.985173i \(0.445119\pi\)
\(480\) 0 0
\(481\) −0.0879330 −0.00400940
\(482\) 0 0
\(483\) 2.70019 0.122863
\(484\) 0 0
\(485\) 7.89798 0.358629
\(486\) 0 0
\(487\) −28.3021 −1.28249 −0.641245 0.767336i \(-0.721582\pi\)
−0.641245 + 0.767336i \(0.721582\pi\)
\(488\) 0 0
\(489\) 13.4299 0.607323
\(490\) 0 0
\(491\) −13.1296 −0.592533 −0.296266 0.955105i \(-0.595742\pi\)
−0.296266 + 0.955105i \(0.595742\pi\)
\(492\) 0 0
\(493\) 1.59489 0.0718301
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7.52568 −0.337573
\(498\) 0 0
\(499\) 39.5888 1.77224 0.886119 0.463458i \(-0.153392\pi\)
0.886119 + 0.463458i \(0.153392\pi\)
\(500\) 0 0
\(501\) −43.1731 −1.92883
\(502\) 0 0
\(503\) −9.83455 −0.438501 −0.219250 0.975669i \(-0.570361\pi\)
−0.219250 + 0.975669i \(0.570361\pi\)
\(504\) 0 0
\(505\) −4.54545 −0.202270
\(506\) 0 0
\(507\) −22.6346 −1.00524
\(508\) 0 0
\(509\) −30.0107 −1.33020 −0.665100 0.746755i \(-0.731611\pi\)
−0.665100 + 0.746755i \(0.731611\pi\)
\(510\) 0 0
\(511\) 4.33933 0.191961
\(512\) 0 0
\(513\) −12.3131 −0.543638
\(514\) 0 0
\(515\) −0.475281 −0.0209434
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 21.1219 0.927149
\(520\) 0 0
\(521\) −22.3793 −0.980456 −0.490228 0.871594i \(-0.663087\pi\)
−0.490228 + 0.871594i \(0.663087\pi\)
\(522\) 0 0
\(523\) −32.4955 −1.42093 −0.710464 0.703734i \(-0.751515\pi\)
−0.710464 + 0.703734i \(0.751515\pi\)
\(524\) 0 0
\(525\) 1.19353 0.0520898
\(526\) 0 0
\(527\) 0.509022 0.0221733
\(528\) 0 0
\(529\) −17.8817 −0.777466
\(530\) 0 0
\(531\) 22.9673 0.996694
\(532\) 0 0
\(533\) −4.16284 −0.180313
\(534\) 0 0
\(535\) −3.45671 −0.149447
\(536\) 0 0
\(537\) 32.1649 1.38802
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −0.492040 −0.0211544 −0.0105772 0.999944i \(-0.503367\pi\)
−0.0105772 + 0.999944i \(0.503367\pi\)
\(542\) 0 0
\(543\) −26.0517 −1.11799
\(544\) 0 0
\(545\) −12.7327 −0.545408
\(546\) 0 0
\(547\) 24.7570 1.05853 0.529267 0.848456i \(-0.322467\pi\)
0.529267 + 0.848456i \(0.322467\pi\)
\(548\) 0 0
\(549\) 5.08652 0.217087
\(550\) 0 0
\(551\) 44.5632 1.89845
\(552\) 0 0
\(553\) −1.70455 −0.0724848
\(554\) 0 0
\(555\) −0.117797 −0.00500019
\(556\) 0 0
\(557\) 27.8342 1.17937 0.589685 0.807633i \(-0.299252\pi\)
0.589685 + 0.807633i \(0.299252\pi\)
\(558\) 0 0
\(559\) 12.2168 0.516716
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.69669 0.366522 0.183261 0.983064i \(-0.441335\pi\)
0.183261 + 0.983064i \(0.441335\pi\)
\(564\) 0 0
\(565\) 11.8988 0.500586
\(566\) 0 0
\(567\) −6.06846 −0.254851
\(568\) 0 0
\(569\) 9.71276 0.407180 0.203590 0.979056i \(-0.434739\pi\)
0.203590 + 0.979056i \(0.434739\pi\)
\(570\) 0 0
\(571\) −25.4484 −1.06498 −0.532491 0.846436i \(-0.678744\pi\)
−0.532491 + 0.846436i \(0.678744\pi\)
\(572\) 0 0
\(573\) 43.2060 1.80496
\(574\) 0 0
\(575\) 2.26236 0.0943469
\(576\) 0 0
\(577\) 2.06822 0.0861013 0.0430506 0.999073i \(-0.486292\pi\)
0.0430506 + 0.999073i \(0.486292\pi\)
\(578\) 0 0
\(579\) −26.7591 −1.11207
\(580\) 0 0
\(581\) −1.48959 −0.0617985
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 2.96630 0.122641
\(586\) 0 0
\(587\) −17.8303 −0.735934 −0.367967 0.929839i \(-0.619946\pi\)
−0.367967 + 0.929839i \(0.619946\pi\)
\(588\) 0 0
\(589\) 14.2227 0.586037
\(590\) 0 0
\(591\) 14.4953 0.596256
\(592\) 0 0
\(593\) −21.8042 −0.895392 −0.447696 0.894186i \(-0.647755\pi\)
−0.447696 + 0.894186i \(0.647755\pi\)
\(594\) 0 0
\(595\) 0.0919797 0.00377080
\(596\) 0 0
\(597\) 1.49200 0.0610637
\(598\) 0 0
\(599\) −32.9202 −1.34508 −0.672542 0.740059i \(-0.734797\pi\)
−0.672542 + 0.740059i \(0.734797\pi\)
\(600\) 0 0
\(601\) 0.208174 0.00849159 0.00424579 0.999991i \(-0.498649\pi\)
0.00424579 + 0.999991i \(0.498649\pi\)
\(602\) 0 0
\(603\) 2.28926 0.0932259
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −8.17531 −0.331826 −0.165913 0.986140i \(-0.553057\pi\)
−0.165913 + 0.986140i \(0.553057\pi\)
\(608\) 0 0
\(609\) 11.2605 0.456301
\(610\) 0 0
\(611\) −14.8549 −0.600966
\(612\) 0 0
\(613\) −39.0929 −1.57895 −0.789474 0.613785i \(-0.789646\pi\)
−0.789474 + 0.613785i \(0.789646\pi\)
\(614\) 0 0
\(615\) −5.57661 −0.224871
\(616\) 0 0
\(617\) 30.1150 1.21238 0.606191 0.795319i \(-0.292697\pi\)
0.606191 + 0.795319i \(0.292697\pi\)
\(618\) 0 0
\(619\) −40.2004 −1.61579 −0.807894 0.589327i \(-0.799393\pi\)
−0.807894 + 0.589327i \(0.799393\pi\)
\(620\) 0 0
\(621\) −5.89768 −0.236666
\(622\) 0 0
\(623\) −5.94449 −0.238161
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −0.00907804 −0.000361965 0
\(630\) 0 0
\(631\) −32.2741 −1.28481 −0.642406 0.766365i \(-0.722064\pi\)
−0.642406 + 0.766365i \(0.722064\pi\)
\(632\) 0 0
\(633\) −36.2863 −1.44225
\(634\) 0 0
\(635\) 18.7284 0.743215
\(636\) 0 0
\(637\) −10.9772 −0.434933
\(638\) 0 0
\(639\) −25.0559 −0.991195
\(640\) 0 0
\(641\) −2.99104 −0.118139 −0.0590695 0.998254i \(-0.518813\pi\)
−0.0590695 + 0.998254i \(0.518813\pi\)
\(642\) 0 0
\(643\) 2.60901 0.102889 0.0514447 0.998676i \(-0.483617\pi\)
0.0514447 + 0.998676i \(0.483617\pi\)
\(644\) 0 0
\(645\) 16.3659 0.644405
\(646\) 0 0
\(647\) −49.8554 −1.96002 −0.980009 0.198954i \(-0.936246\pi\)
−0.980009 + 0.198954i \(0.936246\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 3.59390 0.140856
\(652\) 0 0
\(653\) −46.7093 −1.82788 −0.913938 0.405854i \(-0.866974\pi\)
−0.913938 + 0.405854i \(0.866974\pi\)
\(654\) 0 0
\(655\) 5.25037 0.205149
\(656\) 0 0
\(657\) 14.4473 0.563642
\(658\) 0 0
\(659\) −36.0483 −1.40424 −0.702121 0.712058i \(-0.747763\pi\)
−0.702121 + 0.712058i \(0.747763\pi\)
\(660\) 0 0
\(661\) −14.5536 −0.566071 −0.283036 0.959109i \(-0.591341\pi\)
−0.283036 + 0.959109i \(0.591341\pi\)
\(662\) 0 0
\(663\) 0.607166 0.0235804
\(664\) 0 0
\(665\) 2.57003 0.0996614
\(666\) 0 0
\(667\) 21.3446 0.826468
\(668\) 0 0
\(669\) 56.1784 2.17198
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −34.4049 −1.32621 −0.663105 0.748526i \(-0.730762\pi\)
−0.663105 + 0.748526i \(0.730762\pi\)
\(674\) 0 0
\(675\) −2.60687 −0.100339
\(676\) 0 0
\(677\) 9.88151 0.379777 0.189889 0.981806i \(-0.439187\pi\)
0.189889 + 0.981806i \(0.439187\pi\)
\(678\) 0 0
\(679\) 4.29739 0.164919
\(680\) 0 0
\(681\) 56.0814 2.14905
\(682\) 0 0
\(683\) 48.2825 1.84748 0.923739 0.383022i \(-0.125117\pi\)
0.923739 + 0.383022i \(0.125117\pi\)
\(684\) 0 0
\(685\) −18.3401 −0.700738
\(686\) 0 0
\(687\) 0.177922 0.00678813
\(688\) 0 0
\(689\) −3.33139 −0.126916
\(690\) 0 0
\(691\) 13.2798 0.505188 0.252594 0.967572i \(-0.418716\pi\)
0.252594 + 0.967572i \(0.418716\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2.07720 −0.0787927
\(696\) 0 0
\(697\) −0.429764 −0.0162785
\(698\) 0 0
\(699\) −32.0024 −1.21044
\(700\) 0 0
\(701\) −51.7261 −1.95367 −0.976833 0.214003i \(-0.931350\pi\)
−0.976833 + 0.214003i \(0.931350\pi\)
\(702\) 0 0
\(703\) −0.253652 −0.00956667
\(704\) 0 0
\(705\) −19.8999 −0.749474
\(706\) 0 0
\(707\) −2.47324 −0.0930157
\(708\) 0 0
\(709\) 27.7606 1.04257 0.521286 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(710\) 0 0
\(711\) −5.67510 −0.212833
\(712\) 0 0
\(713\) 6.81233 0.255124
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 54.0648 2.01909
\(718\) 0 0
\(719\) 4.36929 0.162947 0.0814735 0.996676i \(-0.474037\pi\)
0.0814735 + 0.996676i \(0.474037\pi\)
\(720\) 0 0
\(721\) −0.258606 −0.00963101
\(722\) 0 0
\(723\) 45.8376 1.70472
\(724\) 0 0
\(725\) 9.43468 0.350395
\(726\) 0 0
\(727\) −8.78087 −0.325665 −0.162832 0.986654i \(-0.552063\pi\)
−0.162832 + 0.986654i \(0.552063\pi\)
\(728\) 0 0
\(729\) −3.04947 −0.112943
\(730\) 0 0
\(731\) 1.26124 0.0466487
\(732\) 0 0
\(733\) −33.0190 −1.21958 −0.609792 0.792561i \(-0.708747\pi\)
−0.609792 + 0.792561i \(0.708747\pi\)
\(734\) 0 0
\(735\) −14.7053 −0.542412
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 5.38791 0.198198 0.0990988 0.995078i \(-0.468404\pi\)
0.0990988 + 0.995078i \(0.468404\pi\)
\(740\) 0 0
\(741\) 16.9650 0.623225
\(742\) 0 0
\(743\) −30.1554 −1.10629 −0.553147 0.833083i \(-0.686573\pi\)
−0.553147 + 0.833083i \(0.686573\pi\)
\(744\) 0 0
\(745\) 23.4562 0.859370
\(746\) 0 0
\(747\) −4.95940 −0.181455
\(748\) 0 0
\(749\) −1.88084 −0.0687245
\(750\) 0 0
\(751\) −0.565451 −0.0206336 −0.0103168 0.999947i \(-0.503284\pi\)
−0.0103168 + 0.999947i \(0.503284\pi\)
\(752\) 0 0
\(753\) 43.0590 1.56916
\(754\) 0 0
\(755\) 5.92720 0.215713
\(756\) 0 0
\(757\) 5.64054 0.205009 0.102504 0.994733i \(-0.467314\pi\)
0.102504 + 0.994733i \(0.467314\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.400570 0.0145206 0.00726032 0.999974i \(-0.497689\pi\)
0.00726032 + 0.999974i \(0.497689\pi\)
\(762\) 0 0
\(763\) −6.92802 −0.250811
\(764\) 0 0
\(765\) 0.306236 0.0110720
\(766\) 0 0
\(767\) 20.7596 0.749585
\(768\) 0 0
\(769\) −19.0463 −0.686827 −0.343413 0.939184i \(-0.611583\pi\)
−0.343413 + 0.939184i \(0.611583\pi\)
\(770\) 0 0
\(771\) −12.1030 −0.435879
\(772\) 0 0
\(773\) −3.81849 −0.137342 −0.0686708 0.997639i \(-0.521876\pi\)
−0.0686708 + 0.997639i \(0.521876\pi\)
\(774\) 0 0
\(775\) 3.01116 0.108164
\(776\) 0 0
\(777\) −0.0640947 −0.00229938
\(778\) 0 0
\(779\) −12.0082 −0.430237
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −24.5950 −0.878954
\(784\) 0 0
\(785\) 13.1554 0.469536
\(786\) 0 0
\(787\) 13.6634 0.487047 0.243523 0.969895i \(-0.421697\pi\)
0.243523 + 0.969895i \(0.421697\pi\)
\(788\) 0 0
\(789\) 38.9615 1.38707
\(790\) 0 0
\(791\) 6.47430 0.230199
\(792\) 0 0
\(793\) 4.59758 0.163265
\(794\) 0 0
\(795\) −4.46278 −0.158279
\(796\) 0 0
\(797\) 37.1730 1.31673 0.658367 0.752697i \(-0.271247\pi\)
0.658367 + 0.752697i \(0.271247\pi\)
\(798\) 0 0
\(799\) −1.53360 −0.0542547
\(800\) 0 0
\(801\) −19.7915 −0.699297
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 1.23098 0.0433863
\(806\) 0 0
\(807\) −23.8513 −0.839606
\(808\) 0 0
\(809\) 29.0463 1.02121 0.510607 0.859814i \(-0.329421\pi\)
0.510607 + 0.859814i \(0.329421\pi\)
\(810\) 0 0
\(811\) 9.77746 0.343333 0.171667 0.985155i \(-0.445085\pi\)
0.171667 + 0.985155i \(0.445085\pi\)
\(812\) 0 0
\(813\) 21.3936 0.750307
\(814\) 0 0
\(815\) 6.12253 0.214463
\(816\) 0 0
\(817\) 35.2407 1.23292
\(818\) 0 0
\(819\) 1.61400 0.0563978
\(820\) 0 0
\(821\) 24.0003 0.837617 0.418809 0.908075i \(-0.362448\pi\)
0.418809 + 0.908075i \(0.362448\pi\)
\(822\) 0 0
\(823\) −3.22709 −0.112489 −0.0562447 0.998417i \(-0.517913\pi\)
−0.0562447 + 0.998417i \(0.517913\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 4.52002 0.157177 0.0785883 0.996907i \(-0.474959\pi\)
0.0785883 + 0.996907i \(0.474959\pi\)
\(828\) 0 0
\(829\) 36.7844 1.27757 0.638787 0.769383i \(-0.279436\pi\)
0.638787 + 0.769383i \(0.279436\pi\)
\(830\) 0 0
\(831\) 57.4884 1.99425
\(832\) 0 0
\(833\) −1.13327 −0.0392654
\(834\) 0 0
\(835\) −19.6820 −0.681125
\(836\) 0 0
\(837\) −7.84971 −0.271326
\(838\) 0 0
\(839\) 15.1279 0.522274 0.261137 0.965302i \(-0.415903\pi\)
0.261137 + 0.965302i \(0.415903\pi\)
\(840\) 0 0
\(841\) 60.0132 2.06942
\(842\) 0 0
\(843\) −65.9901 −2.27282
\(844\) 0 0
\(845\) −10.3188 −0.354979
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −16.0596 −0.551165
\(850\) 0 0
\(851\) −0.121493 −0.00416473
\(852\) 0 0
\(853\) −13.4410 −0.460210 −0.230105 0.973166i \(-0.573907\pi\)
−0.230105 + 0.973166i \(0.573907\pi\)
\(854\) 0 0
\(855\) 8.55661 0.292630
\(856\) 0 0
\(857\) −53.6470 −1.83255 −0.916274 0.400553i \(-0.868818\pi\)
−0.916274 + 0.400553i \(0.868818\pi\)
\(858\) 0 0
\(859\) −18.4797 −0.630520 −0.315260 0.949005i \(-0.602092\pi\)
−0.315260 + 0.949005i \(0.602092\pi\)
\(860\) 0 0
\(861\) −3.03431 −0.103409
\(862\) 0 0
\(863\) −10.1120 −0.344217 −0.172109 0.985078i \(-0.555058\pi\)
−0.172109 + 0.985078i \(0.555058\pi\)
\(864\) 0 0
\(865\) 9.62920 0.327402
\(866\) 0 0
\(867\) −37.2273 −1.26430
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 2.06921 0.0701125
\(872\) 0 0
\(873\) 14.3077 0.484241
\(874\) 0 0
\(875\) 0.544113 0.0183944
\(876\) 0 0
\(877\) 9.11534 0.307803 0.153902 0.988086i \(-0.450816\pi\)
0.153902 + 0.988086i \(0.450816\pi\)
\(878\) 0 0
\(879\) 28.4851 0.960779
\(880\) 0 0
\(881\) −54.4607 −1.83483 −0.917414 0.397934i \(-0.869727\pi\)
−0.917414 + 0.397934i \(0.869727\pi\)
\(882\) 0 0
\(883\) −53.0001 −1.78359 −0.891797 0.452436i \(-0.850555\pi\)
−0.891797 + 0.452436i \(0.850555\pi\)
\(884\) 0 0
\(885\) 27.8099 0.934819
\(886\) 0 0
\(887\) −6.70248 −0.225047 −0.112524 0.993649i \(-0.535893\pi\)
−0.112524 + 0.993649i \(0.535893\pi\)
\(888\) 0 0
\(889\) 10.1904 0.341774
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −42.8506 −1.43394
\(894\) 0 0
\(895\) 14.6636 0.490149
\(896\) 0 0
\(897\) 8.12581 0.271313
\(898\) 0 0
\(899\) 28.4094 0.947505
\(900\) 0 0
\(901\) −0.343926 −0.0114578
\(902\) 0 0
\(903\) 8.90488 0.296336
\(904\) 0 0
\(905\) −11.8766 −0.394793
\(906\) 0 0
\(907\) 9.71846 0.322696 0.161348 0.986898i \(-0.448416\pi\)
0.161348 + 0.986898i \(0.448416\pi\)
\(908\) 0 0
\(909\) −8.23436 −0.273116
\(910\) 0 0
\(911\) 8.21748 0.272257 0.136129 0.990691i \(-0.456534\pi\)
0.136129 + 0.990691i \(0.456534\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 6.15900 0.203610
\(916\) 0 0
\(917\) 2.85680 0.0943398
\(918\) 0 0
\(919\) −2.56256 −0.0845310 −0.0422655 0.999106i \(-0.513458\pi\)
−0.0422655 + 0.999106i \(0.513458\pi\)
\(920\) 0 0
\(921\) −5.05879 −0.166693
\(922\) 0 0
\(923\) −22.6474 −0.745449
\(924\) 0 0
\(925\) −0.0537019 −0.00176571
\(926\) 0 0
\(927\) −0.861000 −0.0282789
\(928\) 0 0
\(929\) −34.4098 −1.12895 −0.564474 0.825451i \(-0.690921\pi\)
−0.564474 + 0.825451i \(0.690921\pi\)
\(930\) 0 0
\(931\) −31.6650 −1.03778
\(932\) 0 0
\(933\) 49.0489 1.60579
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −35.4691 −1.15873 −0.579363 0.815070i \(-0.696699\pi\)
−0.579363 + 0.815070i \(0.696699\pi\)
\(938\) 0 0
\(939\) −68.0400 −2.22040
\(940\) 0 0
\(941\) 10.1919 0.332245 0.166122 0.986105i \(-0.446875\pi\)
0.166122 + 0.986105i \(0.446875\pi\)
\(942\) 0 0
\(943\) −5.75160 −0.187298
\(944\) 0 0
\(945\) −1.41843 −0.0461417
\(946\) 0 0
\(947\) 27.7310 0.901137 0.450568 0.892742i \(-0.351221\pi\)
0.450568 + 0.892742i \(0.351221\pi\)
\(948\) 0 0
\(949\) 13.0586 0.423899
\(950\) 0 0
\(951\) −34.4604 −1.11746
\(952\) 0 0
\(953\) 26.3962 0.855056 0.427528 0.904002i \(-0.359384\pi\)
0.427528 + 0.904002i \(0.359384\pi\)
\(954\) 0 0
\(955\) 19.6970 0.637381
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −9.97907 −0.322241
\(960\) 0 0
\(961\) −21.9329 −0.707513
\(962\) 0 0
\(963\) −6.26204 −0.201792
\(964\) 0 0
\(965\) −12.1991 −0.392704
\(966\) 0 0
\(967\) −8.50127 −0.273383 −0.136691 0.990614i \(-0.543647\pi\)
−0.136691 + 0.990614i \(0.543647\pi\)
\(968\) 0 0
\(969\) 1.75144 0.0562642
\(970\) 0 0
\(971\) −18.7537 −0.601833 −0.300917 0.953650i \(-0.597293\pi\)
−0.300917 + 0.953650i \(0.597293\pi\)
\(972\) 0 0
\(973\) −1.13023 −0.0362336
\(974\) 0 0
\(975\) 3.59174 0.115028
\(976\) 0 0
\(977\) 14.6354 0.468229 0.234115 0.972209i \(-0.424781\pi\)
0.234115 + 0.972209i \(0.424781\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −23.0660 −0.736442
\(982\) 0 0
\(983\) −42.2727 −1.34829 −0.674145 0.738599i \(-0.735488\pi\)
−0.674145 + 0.738599i \(0.735488\pi\)
\(984\) 0 0
\(985\) 6.60821 0.210555
\(986\) 0 0
\(987\) −10.8278 −0.344653
\(988\) 0 0
\(989\) 16.8794 0.536734
\(990\) 0 0
\(991\) −2.99694 −0.0952009 −0.0476004 0.998866i \(-0.515157\pi\)
−0.0476004 + 0.998866i \(0.515157\pi\)
\(992\) 0 0
\(993\) 65.8505 2.08970
\(994\) 0 0
\(995\) 0.680185 0.0215633
\(996\) 0 0
\(997\) 24.1939 0.766228 0.383114 0.923701i \(-0.374852\pi\)
0.383114 + 0.923701i \(0.374852\pi\)
\(998\) 0 0
\(999\) 0.139994 0.00442922
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 9680.2.a.cu.1.4 4
4.3 odd 2 4840.2.a.y.1.1 4
11.3 even 5 880.2.bo.d.801.1 8
11.4 even 5 880.2.bo.d.401.1 8
11.10 odd 2 9680.2.a.ct.1.4 4
44.3 odd 10 440.2.y.a.361.2 8
44.15 odd 10 440.2.y.a.401.2 yes 8
44.43 even 2 4840.2.a.z.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.a.361.2 8 44.3 odd 10
440.2.y.a.401.2 yes 8 44.15 odd 10
880.2.bo.d.401.1 8 11.4 even 5
880.2.bo.d.801.1 8 11.3 even 5
4840.2.a.y.1.1 4 4.3 odd 2
4840.2.a.z.1.1 4 44.43 even 2
9680.2.a.ct.1.4 4 11.10 odd 2
9680.2.a.cu.1.4 4 1.1 even 1 trivial