Properties

Label 4840.2.a.z.1.1
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $1$
Dimension $4$
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] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(1\)
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 1.1
Root \(-1.35567\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-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 \(-0.718266\pi\)
−0.633217 + 0.773975i \(0.718266\pi\)
\(4\) 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 \(-0.572915\pi\)
−0.227070 + 0.973878i \(0.572915\pi\)
\(14\) 0 0
\(15\) −2.19353 −0.566366
\(16\) 0 0
\(17\) −0.169045 −0.0409995 −0.0204997 0.999790i \(-0.506526\pi\)
−0.0204997 + 0.999790i \(0.506526\pi\)
\(18\) 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 \(-0.575793\pi\)
−0.235867 + 0.971785i \(0.575793\pi\)
\(24\) 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 \(-0.839790\pi\)
−0.875988 + 0.482332i \(0.839790\pi\)
\(30\) 0 0
\(31\) −3.01116 −0.540820 −0.270410 0.962745i \(-0.587159\pi\)
−0.270410 + 0.962745i \(0.587159\pi\)
\(32\) 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 \(-0.436386\pi\)
0.198521 + 0.980097i \(0.436386\pi\)
\(42\) 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 \(-0.269856\pi\)
0.661652 + 0.749811i \(0.269856\pi\)
\(48\) 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 \(-0.808982\pi\)
−0.825278 + 0.564727i \(0.808982\pi\)
\(60\) 0 0
\(61\) −2.80781 −0.359503 −0.179751 0.983712i \(-0.557529\pi\)
−0.179751 + 0.983712i \(0.557529\pi\)
\(62\) 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 \(-0.524596\pi\)
−0.0771924 + 0.997016i \(0.524596\pi\)
\(68\) 0 0
\(69\) 4.96255 0.597420
\(70\) 0 0
\(71\) 13.8311 1.64145 0.820724 0.571325i \(-0.193570\pi\)
0.820724 + 0.571325i \(0.193570\pi\)
\(72\) 0 0
\(73\) −7.97504 −0.933408 −0.466704 0.884414i \(-0.654559\pi\)
−0.466704 + 0.884414i \(0.654559\pi\)
\(74\) 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 \(-0.427388\pi\)
0.226144 + 0.974094i \(0.427388\pi\)
\(102\) 0 0
\(103\) 0.475281 0.0468308 0.0234154 0.999726i \(-0.492546\pi\)
0.0234154 + 0.999726i \(0.492546\pi\)
\(104\) 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 \(-0.291256\pi\)
0.609785 + 0.792567i \(0.291256\pi\)
\(110\) 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 \(-0.910585\pi\)
−0.960805 + 0.277225i \(0.910585\pi\)
\(150\) 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 \(-0.577074\pi\)
−0.239777 + 0.970828i \(0.577074\pi\)
\(164\) 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 \(-0.619289\pi\)
−0.366047 + 0.930596i \(0.619289\pi\)
\(174\) 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 \(-0.684612\pi\)
−0.548003 + 0.836476i \(0.684612\pi\)
\(180\) 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 \(-0.752489\pi\)
−0.712614 + 0.701556i \(0.752489\pi\)
\(192\) 0 0
\(193\) 12.1991 0.878112 0.439056 0.898460i \(-0.355313\pi\)
0.439056 + 0.898460i \(0.355313\pi\)
\(194\) 0 0
\(195\) 3.59174 0.257210
\(196\) 0 0
\(197\) −6.60821 −0.470815 −0.235408 0.971897i \(-0.575643\pi\)
−0.235408 + 0.971897i \(0.575643\pi\)
\(198\) 0 0
\(199\) −0.680185 −0.0482170 −0.0241085 0.999709i \(-0.507675\pi\)
−0.0241085 + 0.999709i \(0.507675\pi\)
\(200\) 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 \(-0.827995\pi\)
−0.857519 + 0.514452i \(0.827995\pi\)
\(224\) 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 \(-0.341400\pi\)
0.477894 + 0.878418i \(0.341400\pi\)
\(234\) 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 \(-0.735012\pi\)
−0.673040 + 0.739606i \(0.735012\pi\)
\(242\) 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 \(-0.712672\pi\)
−0.619518 + 0.784982i \(0.712672\pi\)
\(252\) 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 \(-0.788547\pi\)
−0.787349 + 0.616508i \(0.788547\pi\)
\(278\) 0 0
\(279\) −5.45490 −0.326576
\(280\) 0 0
\(281\) 30.0840 1.79466 0.897331 0.441358i \(-0.145503\pi\)
0.897331 + 0.441358i \(0.145503\pi\)
\(282\) 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 \(-0.623844\pi\)
−0.379325 + 0.925264i \(0.623844\pi\)
\(294\) 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 \(-0.718580\pi\)
−0.633981 + 0.773348i \(0.718580\pi\)
\(312\) 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 \(-0.808844\pi\)
−0.825034 + 0.565083i \(0.808844\pi\)
\(332\) 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 \(-0.206234\pi\)
0.797351 + 0.603516i \(0.206234\pi\)
\(338\) 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 \(-0.718090\pi\)
−0.632789 + 0.774324i \(0.718090\pi\)
\(350\) 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 \(-0.536102\pi\)
−0.113176 + 0.993575i \(0.536102\pi\)
\(368\) 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 \(-0.587465\pi\)
−0.271335 + 0.962485i \(0.587465\pi\)
\(374\) 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 \(-0.596395\pi\)
−0.298228 + 0.954495i \(0.596395\pi\)
\(380\) 0 0
\(381\) −41.0813 −2.10466
\(382\) 0 0
\(383\) −2.38279 −0.121755 −0.0608775 0.998145i \(-0.519390\pi\)
−0.0608775 + 0.998145i \(0.519390\pi\)
\(384\) 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 \(-0.339837\pi\)
0.482202 + 0.876060i \(0.339837\pi\)
\(410\) 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 \(-0.322918\pi\)
0.528064 + 0.849205i \(0.322918\pi\)
\(420\) 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 \(-0.582630\pi\)
−0.256685 + 0.966495i \(0.582630\pi\)
\(444\) 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 \(-0.742567\pi\)
−0.690403 + 0.723425i \(0.742567\pi\)
\(458\) 0 0
\(459\) −0.440679 −0.0205691
\(460\) 0 0
\(461\) 27.4603 1.27896 0.639478 0.768810i \(-0.279151\pi\)
0.639478 + 0.768810i \(0.279151\pi\)
\(462\) 0 0
\(463\) −8.13077 −0.377869 −0.188934 0.981990i \(-0.560503\pi\)
−0.188934 + 0.981990i \(0.560503\pi\)
\(464\) 0 0
\(465\) 6.60506 0.306302
\(466\) 0 0
\(467\) −3.22107 −0.149053 −0.0745267 0.997219i \(-0.523745\pi\)
−0.0745267 + 0.997219i \(0.523745\pi\)
\(468\) 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 \(-0.278418\pi\)
0.641245 + 0.767336i \(0.278418\pi\)
\(488\) 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 \(-0.846608\pi\)
−0.886119 + 0.463458i \(0.846608\pi\)
\(500\) 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 \(-0.496633\pi\)
0.0105772 + 0.999944i \(0.496633\pi\)
\(542\) 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 \(-0.700748\pi\)
−0.589685 + 0.807633i \(0.700748\pi\)
\(558\) 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 \(-0.565261\pi\)
−0.203590 + 0.979056i \(0.565261\pi\)
\(570\) 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 \(-0.380054\pi\)
0.367967 + 0.929839i \(0.380054\pi\)
\(588\) 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 \(-0.352245\pi\)
0.447696 + 0.894186i \(0.352245\pi\)
\(594\) 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 \(-0.265203\pi\)
0.672542 + 0.740059i \(0.265203\pi\)
\(600\) 0 0
\(601\) −0.208174 −0.00849159 −0.00424579 0.999991i \(-0.501351\pi\)
−0.00424579 + 0.999991i \(0.501351\pi\)
\(602\) 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 \(-0.210354\pi\)
0.789474 + 0.613785i \(0.210354\pi\)
\(614\) 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 \(-0.200607\pi\)
0.807894 + 0.589327i \(0.200607\pi\)
\(620\) 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 \(-0.277936\pi\)
0.642406 + 0.766365i \(0.277936\pi\)
\(632\) 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 \(-0.516383\pi\)
−0.0514447 + 0.998676i \(0.516383\pi\)
\(644\) 0 0
\(645\) −16.3659 −0.644405
\(646\) 0 0
\(647\) 49.8554 1.96002 0.980009 0.198954i \(-0.0637545\pi\)
0.980009 + 0.198954i \(0.0637545\pi\)
\(648\) 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 \(-0.269238\pi\)
0.663105 + 0.748526i \(0.269238\pi\)
\(674\) 0 0
\(675\) 2.60687 0.100339
\(676\) 0 0
\(677\) −9.88151 −0.379777 −0.189889 0.981806i \(-0.560813\pi\)
−0.189889 + 0.981806i \(0.560813\pi\)
\(678\) 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 \(-0.874883\pi\)
−0.923739 + 0.383022i \(0.874883\pi\)
\(684\) 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 \(-0.581284\pi\)
−0.252594 + 0.967572i \(0.581284\pi\)
\(692\) 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 \(-0.0686501\pi\)
0.976833 + 0.214003i \(0.0686501\pi\)
\(702\) 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 \(-0.525963\pi\)
−0.0814735 + 0.996676i \(0.525963\pi\)
\(720\) 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 \(-0.447937\pi\)
0.162832 + 0.986654i \(0.447937\pi\)
\(728\) 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 \(-0.291253\pi\)
0.609792 + 0.792561i \(0.291253\pi\)
\(734\) 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 \(-0.496716\pi\)
0.0103168 + 0.999947i \(0.496716\pi\)
\(752\) 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 \(-0.502311\pi\)
−0.00726032 + 0.999974i \(0.502311\pi\)
\(762\) 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 \(-0.388417\pi\)
0.343413 + 0.939184i \(0.388417\pi\)
\(770\) 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 \(-0.670579\pi\)
−0.510607 + 0.859814i \(0.670579\pi\)
\(810\) 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 \(-0.637552\pi\)
−0.418809 + 0.908075i \(0.637552\pi\)
\(822\) 0 0
\(823\) 3.22709 0.112489 0.0562447 0.998417i \(-0.482087\pi\)
0.0562447 + 0.998417i \(0.482087\pi\)
\(824\) 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 \(-0.584097\pi\)
−0.261137 + 0.965302i \(0.584097\pi\)
\(840\) 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 \(-0.426093\pi\)
0.230105 + 0.973166i \(0.426093\pi\)
\(854\) 0 0
\(855\) 8.55661 0.292630
\(856\) 0 0
\(857\) 53.6470 1.83255 0.916274 0.400553i \(-0.131182\pi\)
0.916274 + 0.400553i \(0.131182\pi\)
\(858\) 0 0
\(859\) 18.4797 0.630520 0.315260 0.949005i \(-0.397908\pi\)
0.315260 + 0.949005i \(0.397908\pi\)
\(860\) 0 0
\(861\) −3.03431 −0.103409
\(862\) 0 0
\(863\) 10.1120 0.344217 0.172109 0.985078i \(-0.444942\pi\)
0.172109 + 0.985078i \(0.444942\pi\)
\(864\) 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 \(-0.549184\pi\)
−0.153902 + 0.988086i \(0.549184\pi\)
\(878\) 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 \(-0.149445\pi\)
0.891797 + 0.452436i \(0.149445\pi\)
\(884\) 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 \(-0.551584\pi\)
−0.161348 + 0.986898i \(0.551584\pi\)
\(908\) 0 0
\(909\) 8.23436 0.273116
\(910\) 0 0
\(911\) −8.21748 −0.272257 −0.136129 0.990691i \(-0.543466\pi\)
−0.136129 + 0.990691i \(0.543466\pi\)
\(912\) 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 \(-0.303301\pi\)
0.579363 + 0.815070i \(0.303301\pi\)
\(938\) 0 0
\(939\) 68.0400 2.22040
\(940\) 0 0
\(941\) −10.1919 −0.332245 −0.166122 0.986105i \(-0.553125\pi\)
−0.166122 + 0.986105i \(0.553125\pi\)
\(942\) 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 \(-0.648779\pi\)
−0.450568 + 0.892742i \(0.648779\pi\)
\(948\) 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 \(-0.640616\pi\)
−0.427528 + 0.904002i \(0.640616\pi\)
\(954\) 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 \(-0.402707\pi\)
0.300917 + 0.953650i \(0.402707\pi\)
\(972\) 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 \(-0.264512\pi\)
0.674145 + 0.738599i \(0.264512\pi\)
\(984\) 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 \(-0.484843\pi\)
0.0476004 + 0.998866i \(0.484843\pi\)
\(992\) 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 \(-0.625148\pi\)
−0.383114 + 0.923701i \(0.625148\pi\)
\(998\) 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 4840.2.a.z.1.1 4
4.3 odd 2 9680.2.a.ct.1.4 4
11.7 odd 10 440.2.y.a.401.2 yes 8
11.8 odd 10 440.2.y.a.361.2 8
11.10 odd 2 4840.2.a.y.1.1 4
44.7 even 10 880.2.bo.d.401.1 8
44.19 even 10 880.2.bo.d.801.1 8
44.43 even 2 9680.2.a.cu.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.a.361.2 8 11.8 odd 10
440.2.y.a.401.2 yes 8 11.7 odd 10
880.2.bo.d.401.1 8 44.7 even 10
880.2.bo.d.801.1 8 44.19 even 10
4840.2.a.y.1.1 4 11.10 odd 2
4840.2.a.z.1.1 4 1.1 even 1 trivial
9680.2.a.ct.1.4 4 4.3 odd 2
9680.2.a.cu.1.4 4 44.43 even 2