Properties

Label 201.4.j.a.161.2
Level $201$
Weight $4$
Character 201.161
Analytic conductor $11.859$
Analytic rank $0$
Dimension $20$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [201,4,Mod(5,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 201.j (of order \(22\), degree \(10\), minimal)

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: no
Analytic conductor: \(11.8593839112\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: \(\Q(\zeta_{33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{22}]$

Embedding invariants

Embedding label 161.2
Root \(-0.786053 + 0.618159i\) of defining polynomial
Character \(\chi\) \(=\) 201.161
Dual form 201.4.j.a.5.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.72659 + 2.15856i) q^{3} +(1.13852 - 7.91857i) q^{4} +(11.6079 - 10.0583i) q^{7} +(17.6812 + 20.4052i) q^{9} +O(q^{10})\) \(q+(4.72659 + 2.15856i) q^{3} +(1.13852 - 7.91857i) q^{4} +(11.6079 - 10.0583i) q^{7} +(17.6812 + 20.4052i) q^{9} +(22.4740 - 34.9703i) q^{12} +(6.52521 - 10.1534i) q^{13} +(-61.4076 - 18.0309i) q^{16} +(108.092 - 124.745i) q^{19} +(76.5774 - 22.4851i) q^{21} +(-105.157 - 67.5801i) q^{25} +(39.5260 + 134.613i) q^{27} +(-66.4317 - 103.370i) q^{28} +(127.777 + 198.825i) q^{31} +(181.711 - 116.778i) q^{36} +143.439 q^{37} +(52.7587 - 33.9060i) q^{39} +(96.5466 - 13.8813i) q^{43} +(-251.327 - 217.776i) q^{48} +(-15.2399 + 105.996i) q^{49} +(-72.9715 - 63.2302i) q^{52} +(780.174 - 356.294i) q^{57} +(-156.106 - 531.649i) q^{61} +(410.485 + 59.0189i) q^{63} +(-212.692 + 465.732i) q^{64} +(193.169 + 513.273i) q^{67} +(-1196.68 + 351.377i) q^{73} +(-351.157 - 546.410i) q^{75} +(-864.735 - 997.957i) q^{76} +(-754.678 + 1174.30i) q^{79} +(-103.748 + 721.580i) q^{81} +(-90.8655 - 631.983i) q^{84} +(-26.3823 - 183.493i) q^{91} +(174.774 + 1215.58i) q^{93} +1779.67i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 16 q^{4} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 16 q^{4} + 54 q^{9} - 128 q^{16} + 112 q^{19} + 324 q^{21} + 250 q^{25} - 432 q^{36} + 220 q^{37} - 648 q^{39} + 1258 q^{49} - 6160 q^{52} + 8910 q^{57} + 5940 q^{63} + 1024 q^{64} - 880 q^{67} - 10710 q^{73} - 896 q^{76} - 9724 q^{79} - 1458 q^{81} + 11664 q^{84} - 3888 q^{91} - 1620 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\) \(68\) \(136\)
\(\chi(n)\) \(-1\) \(e\left(\frac{17}{22}\right)\)

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 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(3\) 4.72659 + 2.15856i 0.909632 + 0.415415i
\(4\) 1.13852 7.91857i 0.142315 0.989821i
\(5\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(6\) 0 0
\(7\) 11.6079 10.0583i 0.626769 0.543099i −0.282521 0.959261i \(-0.591171\pi\)
0.909290 + 0.416162i \(0.136625\pi\)
\(8\) 0 0
\(9\) 17.6812 + 20.4052i 0.654861 + 0.755750i
\(10\) 0 0
\(11\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(12\) 22.4740 34.9703i 0.540641 0.841254i
\(13\) 6.52521 10.1534i 0.139213 0.216619i −0.764646 0.644451i \(-0.777086\pi\)
0.903858 + 0.427832i \(0.140722\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −61.4076 18.0309i −0.959493 0.281733i
\(17\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(18\) 0 0
\(19\) 108.092 124.745i 1.30516 1.50623i 0.589534 0.807743i \(-0.299311\pi\)
0.715622 0.698487i \(-0.246143\pi\)
\(20\) 0 0
\(21\) 76.5774 22.4851i 0.795741 0.233651i
\(22\) 0 0
\(23\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(24\) 0 0
\(25\) −105.157 67.5801i −0.841254 0.540641i
\(26\) 0 0
\(27\) 39.5260 + 134.613i 0.281733 + 0.959493i
\(28\) −66.4317 103.370i −0.448372 0.697681i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 127.777 + 198.825i 0.740305 + 1.15194i 0.983315 + 0.181913i \(0.0582288\pi\)
−0.243010 + 0.970024i \(0.578135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 181.711 116.778i 0.841254 0.540641i
\(37\) 143.439 0.637332 0.318666 0.947867i \(-0.396765\pi\)
0.318666 + 0.947867i \(0.396765\pi\)
\(38\) 0 0
\(39\) 52.7587 33.9060i 0.216619 0.139213i
\(40\) 0 0
\(41\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(42\) 0 0
\(43\) 96.5466 13.8813i 0.342400 0.0492297i 0.0310302 0.999518i \(-0.490121\pi\)
0.311370 + 0.950289i \(0.399212\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(48\) −251.327 217.776i −0.755750 0.654861i
\(49\) −15.2399 + 105.996i −0.0444313 + 0.309026i
\(50\) 0 0
\(51\) 0 0
\(52\) −72.9715 63.2302i −0.194603 0.168624i
\(53\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 780.174 356.294i 1.81292 0.827934i
\(58\) 0 0
\(59\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(60\) 0 0
\(61\) −156.106 531.649i −0.327662 1.11591i −0.944415 0.328756i \(-0.893371\pi\)
0.616753 0.787157i \(-0.288448\pi\)
\(62\) 0 0
\(63\) 410.485 + 59.0189i 0.820893 + 0.118027i
\(64\) −212.692 + 465.732i −0.415415 + 0.909632i
\(65\) 0 0
\(66\) 0 0
\(67\) 193.169 + 513.273i 0.352228 + 0.935914i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(72\) 0 0
\(73\) −1196.68 + 351.377i −1.91864 + 0.563363i −0.953966 + 0.299916i \(0.903041\pi\)
−0.964674 + 0.263447i \(0.915141\pi\)
\(74\) 0 0
\(75\) −351.157 546.410i −0.540641 0.841254i
\(76\) −864.735 997.957i −1.30516 1.50623i
\(77\) 0 0
\(78\) 0 0
\(79\) −754.678 + 1174.30i −1.07478 + 1.67240i −0.445304 + 0.895379i \(0.646904\pi\)
−0.629480 + 0.777017i \(0.716732\pi\)
\(80\) 0 0
\(81\) −103.748 + 721.580i −0.142315 + 0.989821i
\(82\) 0 0
\(83\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(84\) −90.8655 631.983i −0.118027 0.820893i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(90\) 0 0
\(91\) −26.3823 183.493i −0.0303914 0.211377i
\(92\) 0 0
\(93\) 174.774 + 1215.58i 0.194873 + 1.35537i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1779.67i 1.86287i 0.363911 + 0.931434i \(0.381441\pi\)
−0.363911 + 0.931434i \(0.618559\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −654.861 + 755.750i −0.654861 + 0.755750i
\(101\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(102\) 0 0
\(103\) 348.533 223.989i 0.333418 0.214275i −0.363210 0.931707i \(-0.618319\pi\)
0.696628 + 0.717433i \(0.254683\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(108\) 1110.94 159.730i 0.989821 0.142315i
\(109\) −1033.70 + 1608.46i −0.908350 + 1.41342i 0.00218975 + 0.999998i \(0.499303\pi\)
−0.910539 + 0.413422i \(0.864333\pi\)
\(110\) 0 0
\(111\) 677.978 + 309.622i 0.579737 + 0.264757i
\(112\) −894.175 + 408.356i −0.754389 + 0.344518i
\(113\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 322.557 46.3767i 0.254875 0.0366455i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1119.71 719.593i −0.841254 0.540641i
\(122\) 0 0
\(123\) 0 0
\(124\) 1719.89 785.446i 1.24557 0.568832i
\(125\) 0 0
\(126\) 0 0
\(127\) −795.116 917.612i −0.555552 0.641141i 0.406616 0.913599i \(-0.366709\pi\)
−0.962168 + 0.272458i \(0.912163\pi\)
\(128\) 0 0
\(129\) 486.299 + 142.790i 0.331909 + 0.0974573i
\(130\) 0 0
\(131\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(132\) 0 0
\(133\) 2535.25i 1.65289i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(138\) 0 0
\(139\) 422.986 1440.56i 0.258109 0.879040i −0.723852 0.689956i \(-0.757630\pi\)
0.981961 0.189084i \(-0.0605518\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −717.837 1571.84i −0.415415 0.909632i
\(145\) 0 0
\(146\) 0 0
\(147\) −300.832 + 468.103i −0.168790 + 0.262643i
\(148\) 163.308 1135.83i 0.0907018 0.630845i
\(149\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(150\) 0 0
\(151\) 107.436 + 747.237i 0.0579010 + 0.402710i 0.998075 + 0.0620161i \(0.0197530\pi\)
−0.940174 + 0.340694i \(0.889338\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −208.420 456.376i −0.106968 0.234227i
\(157\) 792.863 1736.13i 0.403040 0.882535i −0.593913 0.804529i \(-0.702418\pi\)
0.996953 0.0780055i \(-0.0248552\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2860.06 1.37434 0.687169 0.726498i \(-0.258853\pi\)
0.687169 + 0.726498i \(0.258853\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(168\) 0 0
\(169\) 852.153 + 1865.96i 0.387871 + 0.849320i
\(170\) 0 0
\(171\) 4456.64 1.99303
\(172\) 780.315i 0.345921i
\(173\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(174\) 0 0
\(175\) −1900.39 + 273.235i −0.820893 + 0.118027i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(180\) 0 0
\(181\) −699.161 + 1530.95i −0.287117 + 0.628699i −0.997148 0.0754717i \(-0.975954\pi\)
0.710031 + 0.704171i \(0.248681\pi\)
\(182\) 0 0
\(183\) 409.747 2849.85i 0.165516 1.15119i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1812.80 + 1165.01i 0.697681 + 0.448372i
\(190\) 0 0
\(191\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(192\) −2010.62 + 1742.21i −0.755750 + 0.654861i
\(193\) 3963.56 2547.23i 1.47826 0.950018i 0.480944 0.876751i \(-0.340294\pi\)
0.997312 0.0732663i \(-0.0233423\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 821.986 + 241.357i 0.299558 + 0.0879581i
\(197\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(198\) 0 0
\(199\) −825.591 952.783i −0.294093 0.339402i 0.589404 0.807839i \(-0.299363\pi\)
−0.883497 + 0.468437i \(0.844817\pi\)
\(200\) 0 0
\(201\) −194.901 + 2842.99i −0.0683944 + 0.997658i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −583.772 + 505.842i −0.194603 + 0.168624i
\(209\) 0 0
\(210\) 0 0
\(211\) 2276.57 + 4985.00i 0.742776 + 1.62645i 0.778934 + 0.627106i \(0.215761\pi\)
−0.0361575 + 0.999346i \(0.511512\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 3483.07 + 1022.72i 1.08962 + 0.319940i
\(218\) 0 0
\(219\) −6414.67 922.291i −1.97929 0.284578i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −2758.72 6040.74i −0.828418 1.81398i −0.483469 0.875362i \(-0.660623\pi\)
−0.344949 0.938621i \(-0.612104\pi\)
\(224\) 0 0
\(225\) −480.313 3340.65i −0.142315 0.989821i
\(226\) 0 0
\(227\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(228\) −1933.09 6583.51i −0.561501 1.91230i
\(229\) 3715.94 + 5782.11i 1.07230 + 1.66853i 0.644370 + 0.764714i \(0.277120\pi\)
0.427927 + 0.903813i \(0.359244\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −6101.85 + 3921.42i −1.67240 + 1.07478i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 5308.44 1558.70i 1.41887 0.416617i 0.519747 0.854320i \(-0.326026\pi\)
0.899119 + 0.437704i \(0.144208\pi\)
\(242\) 0 0
\(243\) −2047.94 + 3186.66i −0.540641 + 0.841254i
\(244\) −4387.63 + 630.846i −1.15119 + 0.165516i
\(245\) 0 0
\(246\) 0 0
\(247\) −561.263 1911.49i −0.144584 0.492409i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(252\) 934.690 3183.26i 0.233651 0.795741i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 3445.77 + 2214.46i 0.841254 + 0.540641i
\(257\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(258\) 0 0
\(259\) 1665.03 1442.76i 0.399460 0.346134i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 4284.31 945.249i 0.976515 0.215449i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −5851.10 2672.11i −1.31155 0.598963i −0.367883 0.929872i \(-0.619917\pi\)
−0.943663 + 0.330909i \(0.892645\pi\)
\(272\) 0 0
\(273\) 271.382 924.243i 0.0601641 0.204900i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4999.94 5770.24i −1.08454 1.25162i −0.965963 0.258680i \(-0.916713\pi\)
−0.118576 0.992945i \(-0.537833\pi\)
\(278\) 0 0
\(279\) −1797.81 + 6122.80i −0.385779 + 1.31384i
\(280\) 0 0
\(281\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(282\) 0 0
\(283\) 2097.93 2421.14i 0.440667 0.508557i −0.491354 0.870960i \(-0.663498\pi\)
0.932022 + 0.362403i \(0.118043\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4713.99 + 1384.15i −0.959493 + 0.281733i
\(290\) 0 0
\(291\) −3841.52 + 8411.77i −0.773863 + 1.69452i
\(292\) 1419.96 + 9876.04i 0.284578 + 1.97929i
\(293\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −4726.59 + 2158.56i −0.909632 + 0.415415i
\(301\) 981.083 1132.23i 0.187869 0.216813i
\(302\) 0 0
\(303\) 0 0
\(304\) −8886.91 + 5711.27i −1.67664 + 1.07751i
\(305\) 0 0
\(306\) 0 0
\(307\) −2597.85 + 1669.54i −0.482955 + 0.310376i −0.759366 0.650663i \(-0.774491\pi\)
0.276412 + 0.961039i \(0.410855\pi\)
\(308\) 0 0
\(309\) 2130.87 306.372i 0.392300 0.0564043i
\(310\) 0 0
\(311\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(312\) 0 0
\(313\) 4058.54 1853.47i 0.732915 0.334711i −0.0137555 0.999905i \(-0.504379\pi\)
0.746670 + 0.665194i \(0.231651\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 8439.58 + 7312.94i 1.50242 + 1.30185i
\(317\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 5595.76 + 1643.06i 0.959493 + 0.281733i
\(325\) −1372.34 + 626.726i −0.234227 + 0.106968i
\(326\) 0 0
\(327\) −8357.82 + 5371.24i −1.41342 + 0.908350i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4042.35 581.202i −0.671262 0.0965129i −0.201747 0.979438i \(-0.564662\pi\)
−0.469515 + 0.882925i \(0.655571\pi\)
\(332\) 0 0
\(333\) 2536.18 + 2926.91i 0.417364 + 0.481663i
\(334\) 0 0
\(335\) 0 0
\(336\) −5107.86 −0.829335
\(337\) 5200.71 4506.44i 0.840654 0.728431i −0.123906 0.992294i \(-0.539542\pi\)
0.964560 + 0.263863i \(0.0849967\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 3737.50 + 5815.66i 0.588356 + 0.915499i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(348\) 0 0
\(349\) 1680.39 11687.4i 0.257734 1.79258i −0.291148 0.956678i \(-0.594037\pi\)
0.548881 0.835900i \(-0.315054\pi\)
\(350\) 0 0
\(351\) 1624.70 + 477.055i 0.247066 + 0.0725450i
\(352\) 0 0
\(353\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(360\) 0 0
\(361\) −2901.24 20178.6i −0.422983 2.94191i
\(362\) 0 0
\(363\) −3739.11 5818.18i −0.540641 0.841254i
\(364\) −1483.04 −0.213550
\(365\) 0 0
\(366\) 0 0
\(367\) −10561.2 + 4823.13i −1.50215 + 0.686010i −0.985420 0.170138i \(-0.945579\pi\)
−0.516731 + 0.856148i \(0.672851\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 9824.63 1.36931
\(373\) 4474.37i 0.621110i 0.950555 + 0.310555i \(0.100515\pi\)
−0.950555 + 0.310555i \(0.899485\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1248.94 + 570.370i 0.169271 + 0.0773033i 0.498248 0.867034i \(-0.333977\pi\)
−0.328978 + 0.944338i \(0.606704\pi\)
\(380\) 0 0
\(381\) −1777.46 6053.48i −0.239008 0.813987i
\(382\) 0 0
\(383\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1990.31 + 1724.62i 0.261430 + 0.226530i
\(388\) 14092.4 + 2026.19i 1.84391 + 0.265114i
\(389\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −15144.9 4446.95i −1.91461 0.562182i −0.976312 0.216366i \(-0.930580\pi\)
−0.938302 0.345816i \(-0.887602\pi\)
\(398\) 0 0
\(399\) 5472.49 11983.1i 0.686634 1.50352i
\(400\) 5238.89 + 6046.00i 0.654861 + 0.755750i
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 2852.53 0.352592
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 8643.73 7489.83i 1.04500 0.905497i 0.0493597 0.998781i \(-0.484282\pi\)
0.995640 + 0.0932841i \(0.0297365\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1376.86 3014.90i −0.164643 0.360518i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5108.81 5895.88i 0.599951 0.692380i
\(418\) 0 0
\(419\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(420\) 0 0
\(421\) 10214.1 11787.6i 1.18243 1.36460i 0.266211 0.963915i \(-0.414228\pi\)
0.916218 0.400680i \(-0.131226\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −7159.57 4601.18i −0.811419 0.521467i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 8978.95i 1.00000i
\(433\) 8867.86 + 13798.7i 0.984208 + 1.53146i 0.840481 + 0.541840i \(0.182272\pi\)
0.143727 + 0.989617i \(0.454091\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 11559.8 + 10016.7i 1.26976 + 1.10025i
\(437\) 0 0
\(438\) 0 0
\(439\) 12770.5 1.38838 0.694192 0.719790i \(-0.255762\pi\)
0.694192 + 0.719790i \(0.255762\pi\)
\(440\) 0 0
\(441\) −2432.34 + 1563.17i −0.262643 + 0.168790i
\(442\) 0 0
\(443\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(444\) 3223.66 5016.11i 0.344568 0.536158i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 2215.56 + 7545.51i 0.233651 + 0.795741i
\(449\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −1105.15 + 3763.79i −0.114623 + 0.390371i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2433.08 + 1563.64i 0.249047 + 0.160053i 0.659209 0.751960i \(-0.270891\pi\)
−0.410162 + 0.912013i \(0.634528\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(462\) 0 0
\(463\) −3768.64 12834.8i −0.378280 1.28830i −0.900263 0.435347i \(-0.856626\pi\)
0.521983 0.852956i \(-0.325192\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(468\) 2606.99i 0.257496i
\(469\) 7404.95 + 4015.08i 0.729060 + 0.395307i
\(470\) 0 0
\(471\) 7495.07 6494.51i 0.733237 0.635353i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −19796.8 + 5812.88i −1.91230 + 0.561501i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(480\) 0 0
\(481\) 935.971 1456.40i 0.0887248 0.138058i
\(482\) 0 0
\(483\) 0 0
\(484\) −6972.96 + 8047.22i −0.654861 + 0.755750i
\(485\) 0 0
\(486\) 0 0
\(487\) −21183.8 3045.77i −1.97111 0.283402i −0.998755 0.0498761i \(-0.984117\pi\)
−0.972351 0.233526i \(-0.924974\pi\)
\(488\) 0 0
\(489\) 13518.3 + 6173.61i 1.25014 + 0.570920i
\(490\) 0 0
\(491\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −4261.49 14513.3i −0.385779 1.31384i
\(497\) 0 0
\(498\) 0 0
\(499\) 21952.7i 1.96941i 0.174223 + 0.984706i \(0.444259\pi\)
−0.174223 + 0.984706i \(0.555741\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 10659.0i 0.933696i
\(508\) −8171.43 + 5251.46i −0.713679 + 0.458653i
\(509\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(510\) 0 0
\(511\) −10356.7 + 16115.3i −0.896582 + 1.39511i
\(512\) 0 0
\(513\) 21064.7 + 9619.93i 1.81292 + 0.827934i
\(514\) 0 0
\(515\) 0 0
\(516\) 1684.36 3688.23i 0.143701 0.314661i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(522\) 0 0
\(523\) −19774.6 12708.4i −1.65331 1.06252i −0.926935 0.375222i \(-0.877566\pi\)
−0.726377 0.687297i \(-0.758797\pi\)
\(524\) 0 0
\(525\) −9572.17 2810.64i −0.795741 0.233651i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7967.69 9195.21i −0.654861 0.755750i
\(530\) 0 0
\(531\) 0 0
\(532\) −20075.6 2886.43i −1.63606 0.235230i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 5886.59 20047.9i 0.467808 1.59321i −0.300931 0.953646i \(-0.597297\pi\)
0.768739 0.639562i \(-0.220884\pi\)
\(542\) 0 0
\(543\) −6609.29 + 5726.98i −0.522342 + 0.452612i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −566.436 + 1929.10i −0.0442761 + 0.150791i −0.978664 0.205466i \(-0.934129\pi\)
0.934388 + 0.356257i \(0.115947\pi\)
\(548\) 0 0
\(549\) 8088.27 12585.6i 0.628778 0.978398i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 3051.27 + 21222.0i 0.234635 + 1.63192i
\(554\) 0 0
\(555\) 0 0
\(556\) −10925.6 4989.55i −0.833359 0.380582i
\(557\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(558\) 0 0
\(559\) 489.044 1070.86i 0.0370024 0.0810240i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 6053.59 + 9419.57i 0.448372 + 0.697681i
\(568\) 0 0
\(569\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(570\) 0 0
\(571\) −1337.40 2928.49i −0.0980182 0.214630i 0.854270 0.519829i \(-0.174004\pi\)
−0.952289 + 0.305199i \(0.901277\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −13264.0 + 3894.67i −0.959493 + 0.281733i
\(577\) −15314.2 + 2201.85i −1.10492 + 0.158864i −0.670546 0.741868i \(-0.733940\pi\)
−0.434376 + 0.900732i \(0.643031\pi\)
\(578\) 0 0
\(579\) 24232.5 3484.10i 1.73932 0.250077i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(588\) 3364.21 + 2915.10i 0.235948 + 0.204450i
\(589\) 38614.0 + 5551.86i 2.70130 + 0.388388i
\(590\) 0 0
\(591\) 0 0
\(592\) −8808.26 2586.34i −0.611515 0.179557i
\(593\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1845.59 6285.50i −0.126524 0.430902i
\(598\) 0 0
\(599\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(600\) 0 0
\(601\) 18986.7 + 21911.8i 1.28866 + 1.48719i 0.779331 + 0.626612i \(0.215559\pi\)
0.509324 + 0.860575i \(0.329896\pi\)
\(602\) 0 0
\(603\) −7057.99 + 13017.0i −0.476656 + 0.879090i
\(604\) 6039.37 0.406852
\(605\) 0 0
\(606\) 0 0
\(607\) −3651.25 + 25395.0i −0.244151 + 1.69811i 0.386701 + 0.922205i \(0.373615\pi\)
−0.630852 + 0.775903i \(0.717294\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −11258.1 24651.8i −0.741777 1.62427i −0.780609 0.625020i \(-0.785091\pi\)
0.0388314 0.999246i \(-0.487636\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(618\) 0 0
\(619\) −20368.5 5980.73i −1.32258 0.388345i −0.457158 0.889385i \(-0.651133\pi\)
−0.865424 + 0.501040i \(0.832951\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −3851.14 + 1130.80i −0.247066 + 0.0725450i
\(625\) 6490.86 + 14213.0i 0.415415 + 0.909632i
\(626\) 0 0
\(627\) 0 0
\(628\) −12845.0 8254.95i −0.816193 0.524536i
\(629\) 0 0
\(630\) 0 0
\(631\) −8037.52 12506.6i −0.507082 0.789035i 0.489469 0.872021i \(-0.337191\pi\)
−0.996551 + 0.0829861i \(0.973554\pi\)
\(632\) 0 0
\(633\) 28476.1i 1.78803i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 976.779 + 846.384i 0.0607557 + 0.0526451i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −30133.3 + 8847.95i −1.84812 + 0.542658i −0.848207 + 0.529665i \(0.822318\pi\)
−0.999916 + 0.0129929i \(0.995864\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 14255.4 + 12352.4i 0.858241 + 0.743670i
\(652\) 3256.23 22647.6i 0.195589 1.36035i
\(653\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −28328.7 18205.7i −1.68220 1.08109i
\(658\) 0 0
\(659\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(660\) 0 0
\(661\) −12404.4 + 10748.5i −0.729918 + 0.632477i −0.938401 0.345549i \(-0.887693\pi\)
0.208483 + 0.978026i \(0.433147\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 34507.0i 1.99419i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 28258.5 + 12905.2i 1.61855 + 0.739168i 0.998953 0.0457511i \(-0.0145681\pi\)
0.619599 + 0.784919i \(0.287295\pi\)
\(674\) 0 0
\(675\) 4940.75 16826.6i 0.281733 0.959493i
\(676\) 15745.9 4623.41i 0.895875 0.263053i
\(677\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(678\) 0 0
\(679\) 17900.5 + 20658.3i 1.01172 + 1.16759i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(684\) 5073.97 35290.2i 0.283638 1.97274i
\(685\) 0 0
\(686\) 0 0
\(687\) 5082.67 + 35350.7i 0.282265 + 1.96319i
\(688\) −6178.98 888.403i −0.342400 0.0492297i
\(689\) 0 0
\(690\) 0 0
\(691\) −29873.5 + 8771.64i −1.64463 + 0.482907i −0.967482 0.252939i \(-0.918603\pi\)
−0.677149 + 0.735846i \(0.736785\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 15359.5i 0.829335i
\(701\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(702\) 0 0
\(703\) 15504.6 17893.3i 0.831818 0.959969i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −4281.78 + 2751.73i −0.226806 + 0.145759i −0.649109 0.760695i \(-0.724858\pi\)
0.422303 + 0.906455i \(0.361222\pi\)
\(710\) 0 0
\(711\) −37305.6 + 5363.73i −1.96775 + 0.282919i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(720\) 0 0
\(721\) 1792.80 6105.71i 0.0926037 0.315379i
\(722\) 0 0
\(723\) 28455.4 + 4091.26i 1.46371 + 0.210450i
\(724\) 11326.9 + 7279.37i 0.581439 + 0.373668i
\(725\) 0 0
\(726\) 0 0
\(727\) 4826.52 2204.20i 0.246225 0.112447i −0.288480 0.957486i \(-0.593150\pi\)
0.534705 + 0.845039i \(0.320423\pi\)
\(728\) 0 0
\(729\) −16558.4 + 10641.4i −0.841254 + 0.540641i
\(730\) 0 0
\(731\) 0 0
\(732\) −22100.2 6489.22i −1.11591 0.327662i
\(733\) 35064.1 + 5041.46i 1.76688 + 0.254039i 0.947624 0.319388i \(-0.103477\pi\)
0.819257 + 0.573427i \(0.194386\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 20771.7 17998.8i 1.03396 0.895935i 0.0393130 0.999227i \(-0.487483\pi\)
0.994651 + 0.103292i \(0.0329376\pi\)
\(740\) 0 0
\(741\) 1473.20 10246.3i 0.0730356 0.507973i
\(742\) 0 0
\(743\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 4290.09 29838.2i 0.208452 1.44982i −0.569757 0.821813i \(-0.692963\pi\)
0.778210 0.628005i \(-0.216128\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 11289.2 13028.4i 0.543099 0.626769i
\(757\) 7199.08 + 3287.71i 0.345647 + 0.157852i 0.580672 0.814137i \(-0.302790\pi\)
−0.235025 + 0.971989i \(0.575517\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(762\) 0 0
\(763\) 4179.37 + 29068.2i 0.198301 + 1.37921i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 11506.7 + 17904.8i 0.540641 + 0.841254i
\(769\) 30181.1 13783.2i 1.41529 0.646342i 0.446627 0.894720i \(-0.352625\pi\)
0.968663 + 0.248378i \(0.0798977\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −15657.8 34285.8i −0.729970 1.59841i
\(773\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(774\) 0 0
\(775\) 29543.0i 1.36931i
\(776\) 0 0
\(777\) 10984.2 3225.25i 0.507151 0.148913i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 2847.05 6234.17i 0.129694 0.283991i
\(785\) 0 0
\(786\) 0 0
\(787\) 30000.7 4313.45i 1.35884 0.195372i 0.575929 0.817500i \(-0.304641\pi\)
0.782916 + 0.622128i \(0.213731\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −6416.68 1884.11i −0.287343 0.0843716i
\(794\) 0 0
\(795\) 0 0
\(796\) −8484.63 + 5452.74i −0.377801 + 0.242798i
\(797\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 22290.6 + 4780.14i 0.977770 + 0.209680i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(810\) 0 0
\(811\) −31447.0 + 27249.0i −1.36160 + 1.17983i −0.396470 + 0.918048i \(0.629765\pi\)
−0.965127 + 0.261783i \(0.915690\pi\)
\(812\) 0 0
\(813\) −21887.8 25259.9i −0.944206 1.08967i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 8704.28 13544.1i 0.372735 0.579987i
\(818\) 0 0
\(819\) 3277.74 3782.72i 0.139846 0.161391i
\(820\) 0 0
\(821\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(822\) 0 0
\(823\) 27814.2 32099.3i 1.17806 1.35955i 0.258786 0.965935i \(-0.416677\pi\)
0.919274 0.393619i \(-0.128777\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(828\) 0 0
\(829\) 39167.0 + 25171.1i 1.64093 + 1.05456i 0.939915 + 0.341408i \(0.110904\pi\)
0.701010 + 0.713151i \(0.252733\pi\)
\(830\) 0 0
\(831\) −11177.2 38066.2i −0.466588 1.58905i
\(832\) 3340.91 + 5198.55i 0.139213 + 0.216619i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −21713.9 + 25059.2i −0.896707 + 1.03486i
\(838\) 0 0
\(839\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(840\) 0 0
\(841\) 24389.0 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 42066.0 12351.7i 1.71561 0.503747i
\(845\) 0 0
\(846\) 0 0
\(847\) −20235.4 + 2909.41i −0.820893 + 0.118027i
\(848\) 0 0
\(849\) 15142.2 6915.21i 0.612107 0.279540i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 4251.34 29568.7i 0.170648 1.18688i −0.706871 0.707343i \(-0.749894\pi\)
0.877519 0.479542i \(-0.159197\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(858\) 0 0
\(859\) 40091.5 + 25765.2i 1.59244 + 1.02340i 0.970738 + 0.240141i \(0.0771936\pi\)
0.621699 + 0.783256i \(0.286443\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −25268.9 3633.11i −0.989821 0.142315i
\(868\) 12064.1 26416.6i 0.471752 1.03299i
\(869\) 0 0
\(870\) 0 0
\(871\) 6471.94 + 1387.89i 0.251772 + 0.0539917i
\(872\) 0 0
\(873\) −36314.6 + 31466.8i −1.40786 + 1.21992i
\(874\) 0 0
\(875\) 0 0
\(876\) −14606.5 + 49745.0i −0.563363 + 1.91864i
\(877\) −41322.6 + 12133.4i −1.59107 + 0.467179i −0.953043 0.302836i \(-0.902066\pi\)
−0.638025 + 0.770016i \(0.720248\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(882\) 0 0
\(883\) 662.048 1030.17i 0.0252318 0.0392615i −0.828409 0.560124i \(-0.810753\pi\)
0.853640 + 0.520863i \(0.174390\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(888\) 0 0
\(889\) −18459.3 2654.05i −0.696406 0.100128i
\(890\) 0 0
\(891\) 0 0
\(892\) −50974.9 + 14967.6i −1.91342 + 0.561829i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −27000.0 −1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) 7081.16 3233.86i 0.260959 0.119176i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −28797.2 + 18506.9i −1.05424 + 0.677519i −0.948469 0.316871i \(-0.897368\pi\)
−0.105772 + 0.994390i \(0.533731\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(912\) −54332.9 + 7811.89i −1.97274 + 0.283638i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 50016.7 22841.9i 1.80415 0.823927i
\(917\) 0 0
\(918\) 0 0
\(919\) −40941.2 35475.8i −1.46956 1.27338i −0.887971 0.459899i \(-0.847886\pi\)
−0.581590 0.813482i \(-0.697569\pi\)
\(920\) 0 0
\(921\) −15882.7 + 2283.59i −0.568246 + 0.0817014i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −15083.6 9693.64i −0.536158 0.344568i
\(926\) 0 0
\(927\) 10733.0 + 3151.51i 0.380280 + 0.111660i
\(928\) 0 0
\(929\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(930\) 0 0
\(931\) 11575.1 + 13358.4i 0.407475 + 0.470252i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 52886.2i 1.84388i 0.387329 + 0.921942i \(0.373398\pi\)
−0.387329 + 0.921942i \(0.626602\pi\)
\(938\) 0 0
\(939\) 23183.9 0.805727
\(940\) 0 0
\(941\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(948\) 24105.0 + 52782.6i 0.825837 + 1.80833i
\(949\) −4240.90 + 14443.2i −0.145064 + 0.494042i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −10828.8 + 23711.7i −0.363492 + 0.795936i
\(962\) 0 0
\(963\) 0 0
\(964\) −6298.91 43809.9i −0.210450 1.46371i
\(965\) 0 0
\(966\) 0 0
\(967\) −9593.99 −0.319050 −0.159525 0.987194i \(-0.550996\pi\)
−0.159525 + 0.987194i \(0.550996\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(972\) 22902.2 + 19844.9i 0.755750 + 0.654861i
\(973\) −9579.61 20976.4i −0.315630 0.691134i
\(974\) 0 0
\(975\) −7839.30 −0.257496
\(976\) 35462.0i 1.16302i
\(977\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −51098.1 + 7346.80i −1.66303 + 0.239108i
\(982\) 0 0
\(983\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −15775.3 + 2268.14i −0.507973 + 0.0730356i
\(989\) 0 0
\(990\) 0 0
\(991\) 27693.0 + 3981.65i 0.887686 + 0.127630i 0.571048 0.820917i \(-0.306537\pi\)
0.316638 + 0.948547i \(0.397446\pi\)
\(992\) 0 0
\(993\) −17852.0 11472.8i −0.570508 0.366643i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −52903.6 + 33999.1i −1.68052 + 1.08000i −0.813958 + 0.580923i \(0.802692\pi\)
−0.866557 + 0.499078i \(0.833672\pi\)
\(998\) 0 0
\(999\) 5669.58 + 19308.8i 0.179557 + 0.611515i
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 201.4.j.a.161.2 yes 20
3.2 odd 2 CM 201.4.j.a.161.2 yes 20
67.5 odd 22 inner 201.4.j.a.5.2 20
201.5 even 22 inner 201.4.j.a.5.2 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
201.4.j.a.5.2 20 67.5 odd 22 inner
201.4.j.a.5.2 20 201.5 even 22 inner
201.4.j.a.161.2 yes 20 1.1 even 1 trivial
201.4.j.a.161.2 yes 20 3.2 odd 2 CM