Properties

Label 400.4.c.g.49.2
Level $400$
Weight $4$
Character 400.49
Analytic conductor $23.601$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,4,Mod(49,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not 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: \(23.6007640023\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 400.49
Dual form 400.4.c.g.49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q+5.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} -39.0000 q^{11} -84.0000i q^{13} -61.0000i q^{17} +151.000 q^{19} +10.0000 q^{21} -58.0000i q^{23} +145.000i q^{27} -192.000 q^{29} +18.0000 q^{31} -195.000i q^{33} -138.000i q^{37} +420.000 q^{39} +229.000 q^{41} -164.000i q^{43} +212.000i q^{47} +339.000 q^{49} +305.000 q^{51} -578.000i q^{53} +755.000i q^{57} -336.000 q^{59} +858.000 q^{61} -4.00000i q^{63} +209.000i q^{67} +290.000 q^{69} +780.000 q^{71} +403.000i q^{73} +78.0000i q^{77} -230.000 q^{79} -671.000 q^{81} -1293.00i q^{83} -960.000i q^{87} +1369.00 q^{89} -168.000 q^{91} +90.0000i q^{93} +382.000i q^{97} -78.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{9} - 78 q^{11} + 302 q^{19} + 20 q^{21} - 384 q^{29} + 36 q^{31} + 840 q^{39} + 458 q^{41} + 678 q^{49} + 610 q^{51} - 672 q^{59} + 1716 q^{61} + 580 q^{69} + 1560 q^{71} - 460 q^{79} - 1342 q^{81} + 2738 q^{89} - 336 q^{91} - 156 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 5.00000i 0.962250i 0.876652 + 0.481125i \(0.159772\pi\)
−0.876652 + 0.481125i \(0.840228\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.00000i − 0.107990i −0.998541 0.0539949i \(-0.982805\pi\)
0.998541 0.0539949i \(-0.0171955\pi\)
\(8\) 0 0
\(9\) 2.00000 0.0740741
\(10\) 0 0
\(11\) −39.0000 −1.06899 −0.534497 0.845170i \(-0.679499\pi\)
−0.534497 + 0.845170i \(0.679499\pi\)
\(12\) 0 0
\(13\) − 84.0000i − 1.79211i −0.443945 0.896054i \(-0.646421\pi\)
0.443945 0.896054i \(-0.353579\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 61.0000i − 0.870275i −0.900364 0.435137i \(-0.856700\pi\)
0.900364 0.435137i \(-0.143300\pi\)
\(18\) 0 0
\(19\) 151.000 1.82325 0.911626 0.411021i \(-0.134828\pi\)
0.911626 + 0.411021i \(0.134828\pi\)
\(20\) 0 0
\(21\) 10.0000 0.103913
\(22\) 0 0
\(23\) − 58.0000i − 0.525819i −0.964821 0.262909i \(-0.915318\pi\)
0.964821 0.262909i \(-0.0846821\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 145.000i 1.03353i
\(28\) 0 0
\(29\) −192.000 −1.22943 −0.614716 0.788749i \(-0.710729\pi\)
−0.614716 + 0.788749i \(0.710729\pi\)
\(30\) 0 0
\(31\) 18.0000 0.104287 0.0521435 0.998640i \(-0.483395\pi\)
0.0521435 + 0.998640i \(0.483395\pi\)
\(32\) 0 0
\(33\) − 195.000i − 1.02864i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 138.000i − 0.613164i −0.951844 0.306582i \(-0.900815\pi\)
0.951844 0.306582i \(-0.0991853\pi\)
\(38\) 0 0
\(39\) 420.000 1.72446
\(40\) 0 0
\(41\) 229.000 0.872288 0.436144 0.899877i \(-0.356344\pi\)
0.436144 + 0.899877i \(0.356344\pi\)
\(42\) 0 0
\(43\) − 164.000i − 0.581622i −0.956780 0.290811i \(-0.906075\pi\)
0.956780 0.290811i \(-0.0939252\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 212.000i 0.657944i 0.944340 + 0.328972i \(0.106702\pi\)
−0.944340 + 0.328972i \(0.893298\pi\)
\(48\) 0 0
\(49\) 339.000 0.988338
\(50\) 0 0
\(51\) 305.000 0.837422
\(52\) 0 0
\(53\) − 578.000i − 1.49801i −0.662566 0.749004i \(-0.730532\pi\)
0.662566 0.749004i \(-0.269468\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 755.000i 1.75442i
\(58\) 0 0
\(59\) −336.000 −0.741415 −0.370707 0.928750i \(-0.620885\pi\)
−0.370707 + 0.928750i \(0.620885\pi\)
\(60\) 0 0
\(61\) 858.000 1.80091 0.900456 0.434947i \(-0.143233\pi\)
0.900456 + 0.434947i \(0.143233\pi\)
\(62\) 0 0
\(63\) − 4.00000i − 0.00799925i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 209.000i 0.381096i 0.981678 + 0.190548i \(0.0610264\pi\)
−0.981678 + 0.190548i \(0.938974\pi\)
\(68\) 0 0
\(69\) 290.000 0.505970
\(70\) 0 0
\(71\) 780.000 1.30379 0.651894 0.758310i \(-0.273975\pi\)
0.651894 + 0.758310i \(0.273975\pi\)
\(72\) 0 0
\(73\) 403.000i 0.646131i 0.946377 + 0.323066i \(0.104713\pi\)
−0.946377 + 0.323066i \(0.895287\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 78.0000i 0.115441i
\(78\) 0 0
\(79\) −230.000 −0.327557 −0.163779 0.986497i \(-0.552368\pi\)
−0.163779 + 0.986497i \(0.552368\pi\)
\(80\) 0 0
\(81\) −671.000 −0.920439
\(82\) 0 0
\(83\) − 1293.00i − 1.70994i −0.518676 0.854971i \(-0.673575\pi\)
0.518676 0.854971i \(-0.326425\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 960.000i − 1.18302i
\(88\) 0 0
\(89\) 1369.00 1.63049 0.815246 0.579115i \(-0.196602\pi\)
0.815246 + 0.579115i \(0.196602\pi\)
\(90\) 0 0
\(91\) −168.000 −0.193530
\(92\) 0 0
\(93\) 90.0000i 0.100350i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 382.000i 0.399858i 0.979810 + 0.199929i \(0.0640711\pi\)
−0.979810 + 0.199929i \(0.935929\pi\)
\(98\) 0 0
\(99\) −78.0000 −0.0791848
\(100\) 0 0
\(101\) −794.000 −0.782237 −0.391119 0.920340i \(-0.627912\pi\)
−0.391119 + 0.920340i \(0.627912\pi\)
\(102\) 0 0
\(103\) − 1348.00i − 1.28954i −0.764378 0.644769i \(-0.776954\pi\)
0.764378 0.644769i \(-0.223046\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 775.000i 0.700206i 0.936711 + 0.350103i \(0.113853\pi\)
−0.936711 + 0.350103i \(0.886147\pi\)
\(108\) 0 0
\(109\) −446.000 −0.391918 −0.195959 0.980612i \(-0.562782\pi\)
−0.195959 + 0.980612i \(0.562782\pi\)
\(110\) 0 0
\(111\) 690.000 0.590017
\(112\) 0 0
\(113\) 231.000i 0.192307i 0.995367 + 0.0961533i \(0.0306539\pi\)
−0.995367 + 0.0961533i \(0.969346\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 168.000i − 0.132749i
\(118\) 0 0
\(119\) −122.000 −0.0939809
\(120\) 0 0
\(121\) 190.000 0.142750
\(122\) 0 0
\(123\) 1145.00i 0.839359i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 2386.00i − 1.66711i −0.552435 0.833556i \(-0.686301\pi\)
0.552435 0.833556i \(-0.313699\pi\)
\(128\) 0 0
\(129\) 820.000 0.559666
\(130\) 0 0
\(131\) −2452.00 −1.63536 −0.817680 0.575673i \(-0.804740\pi\)
−0.817680 + 0.575673i \(0.804740\pi\)
\(132\) 0 0
\(133\) − 302.000i − 0.196893i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 1125.00i − 0.701571i −0.936456 0.350786i \(-0.885915\pi\)
0.936456 0.350786i \(-0.114085\pi\)
\(138\) 0 0
\(139\) −1377.00 −0.840256 −0.420128 0.907465i \(-0.638015\pi\)
−0.420128 + 0.907465i \(0.638015\pi\)
\(140\) 0 0
\(141\) −1060.00 −0.633107
\(142\) 0 0
\(143\) 3276.00i 1.91575i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1695.00i 0.951029i
\(148\) 0 0
\(149\) −1920.00 −1.05565 −0.527827 0.849352i \(-0.676993\pi\)
−0.527827 + 0.849352i \(0.676993\pi\)
\(150\) 0 0
\(151\) −1854.00 −0.999181 −0.499591 0.866262i \(-0.666516\pi\)
−0.499591 + 0.866262i \(0.666516\pi\)
\(152\) 0 0
\(153\) − 122.000i − 0.0644648i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 634.000i − 0.322285i −0.986931 0.161142i \(-0.948482\pi\)
0.986931 0.161142i \(-0.0515178\pi\)
\(158\) 0 0
\(159\) 2890.00 1.44146
\(160\) 0 0
\(161\) −116.000 −0.0567831
\(162\) 0 0
\(163\) 103.000i 0.0494944i 0.999694 + 0.0247472i \(0.00787808\pi\)
−0.999694 + 0.0247472i \(0.992122\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 44.0000i − 0.0203882i −0.999948 0.0101941i \(-0.996755\pi\)
0.999948 0.0101941i \(-0.00324493\pi\)
\(168\) 0 0
\(169\) −4859.00 −2.21165
\(170\) 0 0
\(171\) 302.000 0.135056
\(172\) 0 0
\(173\) 1128.00i 0.495724i 0.968795 + 0.247862i \(0.0797280\pi\)
−0.968795 + 0.247862i \(0.920272\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 1680.00i − 0.713427i
\(178\) 0 0
\(179\) −2245.00 −0.937426 −0.468713 0.883351i \(-0.655282\pi\)
−0.468713 + 0.883351i \(0.655282\pi\)
\(180\) 0 0
\(181\) 3050.00 1.25251 0.626256 0.779617i \(-0.284586\pi\)
0.626256 + 0.779617i \(0.284586\pi\)
\(182\) 0 0
\(183\) 4290.00i 1.73293i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2379.00i 0.930319i
\(188\) 0 0
\(189\) 290.000 0.111611
\(190\) 0 0
\(191\) 4222.00 1.59944 0.799720 0.600373i \(-0.204981\pi\)
0.799720 + 0.600373i \(0.204981\pi\)
\(192\) 0 0
\(193\) 3357.00i 1.25203i 0.779810 + 0.626016i \(0.215316\pi\)
−0.779810 + 0.626016i \(0.784684\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 166.000i − 0.0600356i −0.999549 0.0300178i \(-0.990444\pi\)
0.999549 0.0300178i \(-0.00955640\pi\)
\(198\) 0 0
\(199\) 3372.00 1.20118 0.600590 0.799557i \(-0.294932\pi\)
0.600590 + 0.799557i \(0.294932\pi\)
\(200\) 0 0
\(201\) −1045.00 −0.366710
\(202\) 0 0
\(203\) 384.000i 0.132766i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 116.000i − 0.0389496i
\(208\) 0 0
\(209\) −5889.00 −1.94905
\(210\) 0 0
\(211\) −5601.00 −1.82743 −0.913717 0.406350i \(-0.866801\pi\)
−0.913717 + 0.406350i \(0.866801\pi\)
\(212\) 0 0
\(213\) 3900.00i 1.25457i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 36.0000i − 0.0112619i
\(218\) 0 0
\(219\) −2015.00 −0.621740
\(220\) 0 0
\(221\) −5124.00 −1.55963
\(222\) 0 0
\(223\) 828.000i 0.248641i 0.992242 + 0.124321i \(0.0396751\pi\)
−0.992242 + 0.124321i \(0.960325\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2388.00i 0.698225i 0.937081 + 0.349113i \(0.113517\pi\)
−0.937081 + 0.349113i \(0.886483\pi\)
\(228\) 0 0
\(229\) 2844.00 0.820685 0.410342 0.911932i \(-0.365409\pi\)
0.410342 + 0.911932i \(0.365409\pi\)
\(230\) 0 0
\(231\) −390.000 −0.111083
\(232\) 0 0
\(233\) − 5962.00i − 1.67632i −0.545421 0.838162i \(-0.683630\pi\)
0.545421 0.838162i \(-0.316370\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 1150.00i − 0.315192i
\(238\) 0 0
\(239\) −4320.00 −1.16919 −0.584597 0.811324i \(-0.698748\pi\)
−0.584597 + 0.811324i \(0.698748\pi\)
\(240\) 0 0
\(241\) 3857.00 1.03092 0.515459 0.856914i \(-0.327621\pi\)
0.515459 + 0.856914i \(0.327621\pi\)
\(242\) 0 0
\(243\) 560.000i 0.147835i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 12684.0i − 3.26746i
\(248\) 0 0
\(249\) 6465.00 1.64539
\(250\) 0 0
\(251\) 287.000 0.0721724 0.0360862 0.999349i \(-0.488511\pi\)
0.0360862 + 0.999349i \(0.488511\pi\)
\(252\) 0 0
\(253\) 2262.00i 0.562098i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2130.00i 0.516987i 0.966013 + 0.258494i \(0.0832261\pi\)
−0.966013 + 0.258494i \(0.916774\pi\)
\(258\) 0 0
\(259\) −276.000 −0.0662155
\(260\) 0 0
\(261\) −384.000 −0.0910690
\(262\) 0 0
\(263\) − 3066.00i − 0.718850i −0.933174 0.359425i \(-0.882973\pi\)
0.933174 0.359425i \(-0.117027\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6845.00i 1.56894i
\(268\) 0 0
\(269\) 3744.00 0.848609 0.424304 0.905520i \(-0.360519\pi\)
0.424304 + 0.905520i \(0.360519\pi\)
\(270\) 0 0
\(271\) 3346.00 0.750019 0.375009 0.927021i \(-0.377640\pi\)
0.375009 + 0.927021i \(0.377640\pi\)
\(272\) 0 0
\(273\) − 840.000i − 0.186224i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7040.00i 1.52705i 0.645779 + 0.763525i \(0.276533\pi\)
−0.645779 + 0.763525i \(0.723467\pi\)
\(278\) 0 0
\(279\) 36.0000 0.00772496
\(280\) 0 0
\(281\) −3010.00 −0.639009 −0.319505 0.947585i \(-0.603516\pi\)
−0.319505 + 0.947585i \(0.603516\pi\)
\(282\) 0 0
\(283\) − 6001.00i − 1.26050i −0.776391 0.630252i \(-0.782952\pi\)
0.776391 0.630252i \(-0.217048\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 458.000i − 0.0941982i
\(288\) 0 0
\(289\) 1192.00 0.242622
\(290\) 0 0
\(291\) −1910.00 −0.384764
\(292\) 0 0
\(293\) − 4802.00i − 0.957460i −0.877962 0.478730i \(-0.841097\pi\)
0.877962 0.478730i \(-0.158903\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 5655.00i − 1.10484i
\(298\) 0 0
\(299\) −4872.00 −0.942325
\(300\) 0 0
\(301\) −328.000 −0.0628093
\(302\) 0 0
\(303\) − 3970.00i − 0.752708i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 6149.00i − 1.14313i −0.820556 0.571567i \(-0.806336\pi\)
0.820556 0.571567i \(-0.193664\pi\)
\(308\) 0 0
\(309\) 6740.00 1.24086
\(310\) 0 0
\(311\) 878.000 0.160086 0.0800431 0.996791i \(-0.474494\pi\)
0.0800431 + 0.996791i \(0.474494\pi\)
\(312\) 0 0
\(313\) 4042.00i 0.729928i 0.931022 + 0.364964i \(0.118919\pi\)
−0.931022 + 0.364964i \(0.881081\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3844.00i 0.681074i 0.940231 + 0.340537i \(0.110609\pi\)
−0.940231 + 0.340537i \(0.889391\pi\)
\(318\) 0 0
\(319\) 7488.00 1.31426
\(320\) 0 0
\(321\) −3875.00 −0.673774
\(322\) 0 0
\(323\) − 9211.00i − 1.58673i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 2230.00i − 0.377123i
\(328\) 0 0
\(329\) 424.000 0.0710513
\(330\) 0 0
\(331\) 2717.00 0.451178 0.225589 0.974223i \(-0.427569\pi\)
0.225589 + 0.974223i \(0.427569\pi\)
\(332\) 0 0
\(333\) − 276.000i − 0.0454195i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 1603.00i − 0.259113i −0.991572 0.129556i \(-0.958645\pi\)
0.991572 0.129556i \(-0.0413553\pi\)
\(338\) 0 0
\(339\) −1155.00 −0.185047
\(340\) 0 0
\(341\) −702.000 −0.111482
\(342\) 0 0
\(343\) − 1364.00i − 0.214720i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11607.0i 1.79567i 0.440335 + 0.897833i \(0.354860\pi\)
−0.440335 + 0.897833i \(0.645140\pi\)
\(348\) 0 0
\(349\) −4030.00 −0.618112 −0.309056 0.951044i \(-0.600013\pi\)
−0.309056 + 0.951044i \(0.600013\pi\)
\(350\) 0 0
\(351\) 12180.0 1.85219
\(352\) 0 0
\(353\) − 2106.00i − 0.317538i −0.987316 0.158769i \(-0.949247\pi\)
0.987316 0.158769i \(-0.0507526\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 610.000i − 0.0904331i
\(358\) 0 0
\(359\) 7394.00 1.08702 0.543510 0.839402i \(-0.317095\pi\)
0.543510 + 0.839402i \(0.317095\pi\)
\(360\) 0 0
\(361\) 15942.0 2.32425
\(362\) 0 0
\(363\) 950.000i 0.137361i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 6940.00i 0.987098i 0.869718 + 0.493549i \(0.164301\pi\)
−0.869718 + 0.493549i \(0.835699\pi\)
\(368\) 0 0
\(369\) 458.000 0.0646139
\(370\) 0 0
\(371\) −1156.00 −0.161770
\(372\) 0 0
\(373\) − 7486.00i − 1.03917i −0.854419 0.519585i \(-0.826087\pi\)
0.854419 0.519585i \(-0.173913\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16128.0i 2.20327i
\(378\) 0 0
\(379\) 1285.00 0.174158 0.0870792 0.996201i \(-0.472247\pi\)
0.0870792 + 0.996201i \(0.472247\pi\)
\(380\) 0 0
\(381\) 11930.0 1.60418
\(382\) 0 0
\(383\) 9622.00i 1.28371i 0.766826 + 0.641855i \(0.221835\pi\)
−0.766826 + 0.641855i \(0.778165\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 328.000i − 0.0430831i
\(388\) 0 0
\(389\) −1974.00 −0.257290 −0.128645 0.991691i \(-0.541063\pi\)
−0.128645 + 0.991691i \(0.541063\pi\)
\(390\) 0 0
\(391\) −3538.00 −0.457607
\(392\) 0 0
\(393\) − 12260.0i − 1.57363i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 8084.00i − 1.02198i −0.859588 0.510988i \(-0.829280\pi\)
0.859588 0.510988i \(-0.170720\pi\)
\(398\) 0 0
\(399\) 1510.00 0.189460
\(400\) 0 0
\(401\) −5667.00 −0.705727 −0.352863 0.935675i \(-0.614792\pi\)
−0.352863 + 0.935675i \(0.614792\pi\)
\(402\) 0 0
\(403\) − 1512.00i − 0.186894i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5382.00i 0.655469i
\(408\) 0 0
\(409\) 4835.00 0.584536 0.292268 0.956336i \(-0.405590\pi\)
0.292268 + 0.956336i \(0.405590\pi\)
\(410\) 0 0
\(411\) 5625.00 0.675087
\(412\) 0 0
\(413\) 672.000i 0.0800653i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 6885.00i − 0.808537i
\(418\) 0 0
\(419\) 4619.00 0.538551 0.269276 0.963063i \(-0.413216\pi\)
0.269276 + 0.963063i \(0.413216\pi\)
\(420\) 0 0
\(421\) 7476.00 0.865458 0.432729 0.901524i \(-0.357551\pi\)
0.432729 + 0.901524i \(0.357551\pi\)
\(422\) 0 0
\(423\) 424.000i 0.0487366i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 1716.00i − 0.194480i
\(428\) 0 0
\(429\) −16380.0 −1.84344
\(430\) 0 0
\(431\) −7810.00 −0.872841 −0.436420 0.899743i \(-0.643754\pi\)
−0.436420 + 0.899743i \(0.643754\pi\)
\(432\) 0 0
\(433\) 2029.00i 0.225191i 0.993641 + 0.112595i \(0.0359164\pi\)
−0.993641 + 0.112595i \(0.964084\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 8758.00i − 0.958700i
\(438\) 0 0
\(439\) 3208.00 0.348769 0.174384 0.984678i \(-0.444206\pi\)
0.174384 + 0.984678i \(0.444206\pi\)
\(440\) 0 0
\(441\) 678.000 0.0732102
\(442\) 0 0
\(443\) 13227.0i 1.41859i 0.704914 + 0.709293i \(0.250986\pi\)
−0.704914 + 0.709293i \(0.749014\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 9600.00i − 1.01580i
\(448\) 0 0
\(449\) −3617.00 −0.380171 −0.190086 0.981768i \(-0.560877\pi\)
−0.190086 + 0.981768i \(0.560877\pi\)
\(450\) 0 0
\(451\) −8931.00 −0.932471
\(452\) 0 0
\(453\) − 9270.00i − 0.961463i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6215.00i 0.636161i 0.948064 + 0.318080i \(0.103038\pi\)
−0.948064 + 0.318080i \(0.896962\pi\)
\(458\) 0 0
\(459\) 8845.00 0.899454
\(460\) 0 0
\(461\) −7108.00 −0.718118 −0.359059 0.933315i \(-0.616902\pi\)
−0.359059 + 0.933315i \(0.616902\pi\)
\(462\) 0 0
\(463\) − 3364.00i − 0.337664i −0.985645 0.168832i \(-0.946000\pi\)
0.985645 0.168832i \(-0.0539995\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18964.0i 1.87912i 0.342384 + 0.939560i \(0.388766\pi\)
−0.342384 + 0.939560i \(0.611234\pi\)
\(468\) 0 0
\(469\) 418.000 0.0411545
\(470\) 0 0
\(471\) 3170.00 0.310119
\(472\) 0 0
\(473\) 6396.00i 0.621751i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 1156.00i − 0.110964i
\(478\) 0 0
\(479\) −10926.0 −1.04222 −0.521108 0.853491i \(-0.674481\pi\)
−0.521108 + 0.853491i \(0.674481\pi\)
\(480\) 0 0
\(481\) −11592.0 −1.09886
\(482\) 0 0
\(483\) − 580.000i − 0.0546396i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 4350.00i 0.404758i 0.979307 + 0.202379i \(0.0648673\pi\)
−0.979307 + 0.202379i \(0.935133\pi\)
\(488\) 0 0
\(489\) −515.000 −0.0476260
\(490\) 0 0
\(491\) 1324.00 0.121693 0.0608465 0.998147i \(-0.480620\pi\)
0.0608465 + 0.998147i \(0.480620\pi\)
\(492\) 0 0
\(493\) 11712.0i 1.06994i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 1560.00i − 0.140796i
\(498\) 0 0
\(499\) 9068.00 0.813506 0.406753 0.913538i \(-0.366661\pi\)
0.406753 + 0.913538i \(0.366661\pi\)
\(500\) 0 0
\(501\) 220.000 0.0196185
\(502\) 0 0
\(503\) − 19836.0i − 1.75834i −0.476511 0.879169i \(-0.658099\pi\)
0.476511 0.879169i \(-0.341901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 24295.0i − 2.12816i
\(508\) 0 0
\(509\) −2682.00 −0.233551 −0.116776 0.993158i \(-0.537256\pi\)
−0.116776 + 0.993158i \(0.537256\pi\)
\(510\) 0 0
\(511\) 806.000 0.0697756
\(512\) 0 0
\(513\) 21895.0i 1.88438i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 8268.00i − 0.703339i
\(518\) 0 0
\(519\) −5640.00 −0.477011
\(520\) 0 0
\(521\) −3035.00 −0.255213 −0.127606 0.991825i \(-0.540729\pi\)
−0.127606 + 0.991825i \(0.540729\pi\)
\(522\) 0 0
\(523\) 7701.00i 0.643865i 0.946763 + 0.321932i \(0.104332\pi\)
−0.946763 + 0.321932i \(0.895668\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 1098.00i − 0.0907583i
\(528\) 0 0
\(529\) 8803.00 0.723514
\(530\) 0 0
\(531\) −672.000 −0.0549196
\(532\) 0 0
\(533\) − 19236.0i − 1.56323i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 11225.0i − 0.902038i
\(538\) 0 0
\(539\) −13221.0 −1.05653
\(540\) 0 0
\(541\) −18112.0 −1.43936 −0.719682 0.694304i \(-0.755712\pi\)
−0.719682 + 0.694304i \(0.755712\pi\)
\(542\) 0 0
\(543\) 15250.0i 1.20523i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 19541.0i − 1.52745i −0.645544 0.763723i \(-0.723369\pi\)
0.645544 0.763723i \(-0.276631\pi\)
\(548\) 0 0
\(549\) 1716.00 0.133401
\(550\) 0 0
\(551\) −28992.0 −2.24156
\(552\) 0 0
\(553\) 460.000i 0.0353729i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 13508.0i 1.02756i 0.857921 + 0.513781i \(0.171756\pi\)
−0.857921 + 0.513781i \(0.828244\pi\)
\(558\) 0 0
\(559\) −13776.0 −1.04233
\(560\) 0 0
\(561\) −11895.0 −0.895200
\(562\) 0 0
\(563\) − 8712.00i − 0.652162i −0.945342 0.326081i \(-0.894272\pi\)
0.945342 0.326081i \(-0.105728\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1342.00i 0.0993981i
\(568\) 0 0
\(569\) 9623.00 0.708993 0.354497 0.935057i \(-0.384652\pi\)
0.354497 + 0.935057i \(0.384652\pi\)
\(570\) 0 0
\(571\) −604.000 −0.0442673 −0.0221336 0.999755i \(-0.507046\pi\)
−0.0221336 + 0.999755i \(0.507046\pi\)
\(572\) 0 0
\(573\) 21110.0i 1.53906i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 3629.00i 0.261832i 0.991393 + 0.130916i \(0.0417919\pi\)
−0.991393 + 0.130916i \(0.958208\pi\)
\(578\) 0 0
\(579\) −16785.0 −1.20477
\(580\) 0 0
\(581\) −2586.00 −0.184656
\(582\) 0 0
\(583\) 22542.0i 1.60136i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 9219.00i − 0.648226i −0.946018 0.324113i \(-0.894934\pi\)
0.946018 0.324113i \(-0.105066\pi\)
\(588\) 0 0
\(589\) 2718.00 0.190141
\(590\) 0 0
\(591\) 830.000 0.0577693
\(592\) 0 0
\(593\) 19111.0i 1.32343i 0.749755 + 0.661716i \(0.230171\pi\)
−0.749755 + 0.661716i \(0.769829\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 16860.0i 1.15584i
\(598\) 0 0
\(599\) −17086.0 −1.16547 −0.582734 0.812663i \(-0.698017\pi\)
−0.582734 + 0.812663i \(0.698017\pi\)
\(600\) 0 0
\(601\) 9035.00 0.613220 0.306610 0.951835i \(-0.400805\pi\)
0.306610 + 0.951835i \(0.400805\pi\)
\(602\) 0 0
\(603\) 418.000i 0.0282293i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 14784.0i 0.988573i 0.869299 + 0.494287i \(0.164571\pi\)
−0.869299 + 0.494287i \(0.835429\pi\)
\(608\) 0 0
\(609\) −1920.00 −0.127754
\(610\) 0 0
\(611\) 17808.0 1.17911
\(612\) 0 0
\(613\) 17846.0i 1.17585i 0.808917 + 0.587923i \(0.200054\pi\)
−0.808917 + 0.587923i \(0.799946\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 11618.0i 0.758060i 0.925384 + 0.379030i \(0.123742\pi\)
−0.925384 + 0.379030i \(0.876258\pi\)
\(618\) 0 0
\(619\) −9556.00 −0.620498 −0.310249 0.950655i \(-0.600412\pi\)
−0.310249 + 0.950655i \(0.600412\pi\)
\(620\) 0 0
\(621\) 8410.00 0.543449
\(622\) 0 0
\(623\) − 2738.00i − 0.176076i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 29445.0i − 1.87547i
\(628\) 0 0
\(629\) −8418.00 −0.533621
\(630\) 0 0
\(631\) 19394.0 1.22355 0.611777 0.791030i \(-0.290455\pi\)
0.611777 + 0.791030i \(0.290455\pi\)
\(632\) 0 0
\(633\) − 28005.0i − 1.75845i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 28476.0i − 1.77121i
\(638\) 0 0
\(639\) 1560.00 0.0965769
\(640\) 0 0
\(641\) 12138.0 0.747929 0.373964 0.927443i \(-0.377998\pi\)
0.373964 + 0.927443i \(0.377998\pi\)
\(642\) 0 0
\(643\) 27036.0i 1.65816i 0.559131 + 0.829079i \(0.311135\pi\)
−0.559131 + 0.829079i \(0.688865\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17556.0i 1.06677i 0.845874 + 0.533383i \(0.179080\pi\)
−0.845874 + 0.533383i \(0.820920\pi\)
\(648\) 0 0
\(649\) 13104.0 0.792569
\(650\) 0 0
\(651\) 180.000 0.0108368
\(652\) 0 0
\(653\) 17262.0i 1.03448i 0.855841 + 0.517239i \(0.173040\pi\)
−0.855841 + 0.517239i \(0.826960\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 806.000i 0.0478616i
\(658\) 0 0
\(659\) 10517.0 0.621675 0.310838 0.950463i \(-0.399390\pi\)
0.310838 + 0.950463i \(0.399390\pi\)
\(660\) 0 0
\(661\) 1408.00 0.0828515 0.0414258 0.999142i \(-0.486810\pi\)
0.0414258 + 0.999142i \(0.486810\pi\)
\(662\) 0 0
\(663\) − 25620.0i − 1.50075i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 11136.0i 0.646458i
\(668\) 0 0
\(669\) −4140.00 −0.239255
\(670\) 0 0
\(671\) −33462.0 −1.92517
\(672\) 0 0
\(673\) 9626.00i 0.551345i 0.961252 + 0.275672i \(0.0889005\pi\)
−0.961252 + 0.275672i \(0.911100\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 28464.0i 1.61589i 0.589255 + 0.807947i \(0.299421\pi\)
−0.589255 + 0.807947i \(0.700579\pi\)
\(678\) 0 0
\(679\) 764.000 0.0431806
\(680\) 0 0
\(681\) −11940.0 −0.671868
\(682\) 0 0
\(683\) − 3963.00i − 0.222020i −0.993819 0.111010i \(-0.964591\pi\)
0.993819 0.111010i \(-0.0354086\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 14220.0i 0.789704i
\(688\) 0 0
\(689\) −48552.0 −2.68459
\(690\) 0 0
\(691\) 31781.0 1.74965 0.874824 0.484442i \(-0.160977\pi\)
0.874824 + 0.484442i \(0.160977\pi\)
\(692\) 0 0
\(693\) 156.000i 0.00855115i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 13969.0i − 0.759130i
\(698\) 0 0
\(699\) 29810.0 1.61304
\(700\) 0 0
\(701\) −28004.0 −1.50884 −0.754420 0.656392i \(-0.772082\pi\)
−0.754420 + 0.656392i \(0.772082\pi\)
\(702\) 0 0
\(703\) − 20838.0i − 1.11795i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1588.00i 0.0844737i
\(708\) 0 0
\(709\) 35228.0 1.86603 0.933015 0.359837i \(-0.117168\pi\)
0.933015 + 0.359837i \(0.117168\pi\)
\(710\) 0 0
\(711\) −460.000 −0.0242635
\(712\) 0 0
\(713\) − 1044.00i − 0.0548361i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 21600.0i − 1.12506i
\(718\) 0 0
\(719\) −8658.00 −0.449081 −0.224540 0.974465i \(-0.572088\pi\)
−0.224540 + 0.974465i \(0.572088\pi\)
\(720\) 0 0
\(721\) −2696.00 −0.139257
\(722\) 0 0
\(723\) 19285.0i 0.992001i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 5728.00i − 0.292214i −0.989269 0.146107i \(-0.953326\pi\)
0.989269 0.146107i \(-0.0466744\pi\)
\(728\) 0 0
\(729\) −20917.0 −1.06269
\(730\) 0 0
\(731\) −10004.0 −0.506171
\(732\) 0 0
\(733\) − 21460.0i − 1.08137i −0.841226 0.540684i \(-0.818165\pi\)
0.841226 0.540684i \(-0.181835\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 8151.00i − 0.407389i
\(738\) 0 0
\(739\) 29164.0 1.45171 0.725856 0.687847i \(-0.241444\pi\)
0.725856 + 0.687847i \(0.241444\pi\)
\(740\) 0 0
\(741\) 63420.0 3.14412
\(742\) 0 0
\(743\) 29478.0i 1.45551i 0.685838 + 0.727754i \(0.259436\pi\)
−0.685838 + 0.727754i \(0.740564\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 2586.00i − 0.126662i
\(748\) 0 0
\(749\) 1550.00 0.0756152
\(750\) 0 0
\(751\) −576.000 −0.0279874 −0.0139937 0.999902i \(-0.504454\pi\)
−0.0139937 + 0.999902i \(0.504454\pi\)
\(752\) 0 0
\(753\) 1435.00i 0.0694480i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2880.00i 0.138277i 0.997607 + 0.0691383i \(0.0220250\pi\)
−0.997607 + 0.0691383i \(0.977975\pi\)
\(758\) 0 0
\(759\) −11310.0 −0.540879
\(760\) 0 0
\(761\) 20789.0 0.990277 0.495138 0.868814i \(-0.335117\pi\)
0.495138 + 0.868814i \(0.335117\pi\)
\(762\) 0 0
\(763\) 892.000i 0.0423232i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 28224.0i 1.32870i
\(768\) 0 0
\(769\) −26421.0 −1.23897 −0.619484 0.785010i \(-0.712658\pi\)
−0.619484 + 0.785010i \(0.712658\pi\)
\(770\) 0 0
\(771\) −10650.0 −0.497471
\(772\) 0 0
\(773\) − 32504.0i − 1.51240i −0.654339 0.756202i \(-0.727053\pi\)
0.654339 0.756202i \(-0.272947\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 1380.00i − 0.0637159i
\(778\) 0 0
\(779\) 34579.0 1.59040
\(780\) 0 0
\(781\) −30420.0 −1.39374
\(782\) 0 0
\(783\) − 27840.0i − 1.27065i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 996.000i − 0.0451125i −0.999746 0.0225563i \(-0.992820\pi\)
0.999746 0.0225563i \(-0.00718049\pi\)
\(788\) 0 0
\(789\) 15330.0 0.691714
\(790\) 0 0
\(791\) 462.000 0.0207672
\(792\) 0 0
\(793\) − 72072.0i − 3.22743i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15134.0i 0.672615i 0.941752 + 0.336307i \(0.109178\pi\)
−0.941752 + 0.336307i \(0.890822\pi\)
\(798\) 0 0
\(799\) 12932.0 0.572592
\(800\) 0 0
\(801\) 2738.00 0.120777
\(802\) 0 0
\(803\) − 15717.0i − 0.690711i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 18720.0i 0.816574i
\(808\) 0 0
\(809\) 36942.0 1.60545 0.802727 0.596347i \(-0.203382\pi\)
0.802727 + 0.596347i \(0.203382\pi\)
\(810\) 0 0
\(811\) 11748.0 0.508666 0.254333 0.967117i \(-0.418144\pi\)
0.254333 + 0.967117i \(0.418144\pi\)
\(812\) 0 0
\(813\) 16730.0i 0.721706i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 24764.0i − 1.06044i
\(818\) 0 0
\(819\) −336.000 −0.0143355
\(820\) 0 0
\(821\) −1198.00 −0.0509263 −0.0254631 0.999676i \(-0.508106\pi\)
−0.0254631 + 0.999676i \(0.508106\pi\)
\(822\) 0 0
\(823\) − 6788.00i − 0.287503i −0.989614 0.143751i \(-0.954083\pi\)
0.989614 0.143751i \(-0.0459166\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 33011.0i − 1.38803i −0.719958 0.694017i \(-0.755839\pi\)
0.719958 0.694017i \(-0.244161\pi\)
\(828\) 0 0
\(829\) −17732.0 −0.742892 −0.371446 0.928454i \(-0.621138\pi\)
−0.371446 + 0.928454i \(0.621138\pi\)
\(830\) 0 0
\(831\) −35200.0 −1.46940
\(832\) 0 0
\(833\) − 20679.0i − 0.860126i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 2610.00i 0.107784i
\(838\) 0 0
\(839\) 8480.00 0.348942 0.174471 0.984662i \(-0.444179\pi\)
0.174471 + 0.984662i \(0.444179\pi\)
\(840\) 0 0
\(841\) 12475.0 0.511501
\(842\) 0 0
\(843\) − 15050.0i − 0.614887i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 380.000i − 0.0154155i
\(848\) 0 0
\(849\) 30005.0 1.21292
\(850\) 0 0
\(851\) −8004.00 −0.322413
\(852\) 0 0
\(853\) − 30014.0i − 1.20476i −0.798210 0.602380i \(-0.794219\pi\)
0.798210 0.602380i \(-0.205781\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 21643.0i 0.862673i 0.902191 + 0.431337i \(0.141958\pi\)
−0.902191 + 0.431337i \(0.858042\pi\)
\(858\) 0 0
\(859\) 2799.00 0.111177 0.0555883 0.998454i \(-0.482297\pi\)
0.0555883 + 0.998454i \(0.482297\pi\)
\(860\) 0 0
\(861\) 2290.00 0.0906423
\(862\) 0 0
\(863\) − 19384.0i − 0.764588i −0.924041 0.382294i \(-0.875134\pi\)
0.924041 0.382294i \(-0.124866\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 5960.00i 0.233463i
\(868\) 0 0
\(869\) 8970.00 0.350157
\(870\) 0 0
\(871\) 17556.0 0.682965
\(872\) 0 0
\(873\) 764.000i 0.0296191i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 5132.00i 0.197600i 0.995107 + 0.0988001i \(0.0315004\pi\)
−0.995107 + 0.0988001i \(0.968500\pi\)
\(878\) 0 0
\(879\) 24010.0 0.921316
\(880\) 0 0
\(881\) 4430.00 0.169410 0.0847052 0.996406i \(-0.473005\pi\)
0.0847052 + 0.996406i \(0.473005\pi\)
\(882\) 0 0
\(883\) 24317.0i 0.926764i 0.886159 + 0.463382i \(0.153364\pi\)
−0.886159 + 0.463382i \(0.846636\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 26100.0i 0.987996i 0.869463 + 0.493998i \(0.164465\pi\)
−0.869463 + 0.493998i \(0.835535\pi\)
\(888\) 0 0
\(889\) −4772.00 −0.180031
\(890\) 0 0
\(891\) 26169.0 0.983944
\(892\) 0 0
\(893\) 32012.0i 1.19960i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 24360.0i − 0.906752i
\(898\) 0 0
\(899\) −3456.00 −0.128214
\(900\) 0 0
\(901\) −35258.0 −1.30368
\(902\) 0 0
\(903\) − 1640.00i − 0.0604383i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 24356.0i − 0.891651i −0.895120 0.445826i \(-0.852910\pi\)
0.895120 0.445826i \(-0.147090\pi\)
\(908\) 0 0
\(909\) −1588.00 −0.0579435
\(910\) 0 0
\(911\) 29900.0 1.08741 0.543705 0.839276i \(-0.317021\pi\)
0.543705 + 0.839276i \(0.317021\pi\)
\(912\) 0 0
\(913\) 50427.0i 1.82792i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4904.00i 0.176602i
\(918\) 0 0
\(919\) −34838.0 −1.25049 −0.625245 0.780429i \(-0.715001\pi\)
−0.625245 + 0.780429i \(0.715001\pi\)
\(920\) 0 0
\(921\) 30745.0 1.09998
\(922\) 0 0
\(923\) − 65520.0i − 2.33653i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 2696.00i − 0.0955213i
\(928\) 0 0
\(929\) −26334.0 −0.930022 −0.465011 0.885305i \(-0.653950\pi\)
−0.465011 + 0.885305i \(0.653950\pi\)
\(930\) 0 0
\(931\) 51189.0 1.80199
\(932\) 0 0
\(933\) 4390.00i 0.154043i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 30949.0i 1.07904i 0.841973 + 0.539520i \(0.181394\pi\)
−0.841973 + 0.539520i \(0.818606\pi\)
\(938\) 0 0
\(939\) −20210.0 −0.702373
\(940\) 0 0
\(941\) 25276.0 0.875637 0.437818 0.899063i \(-0.355751\pi\)
0.437818 + 0.899063i \(0.355751\pi\)
\(942\) 0 0
\(943\) − 13282.0i − 0.458665i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1216.00i 0.0417262i 0.999782 + 0.0208631i \(0.00664141\pi\)
−0.999782 + 0.0208631i \(0.993359\pi\)
\(948\) 0 0
\(949\) 33852.0 1.15794
\(950\) 0 0
\(951\) −19220.0 −0.655364
\(952\) 0 0
\(953\) 6033.00i 0.205066i 0.994730 + 0.102533i \(0.0326947\pi\)
−0.994730 + 0.102533i \(0.967305\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 37440.0i 1.26464i
\(958\) 0 0
\(959\) −2250.00 −0.0757626
\(960\) 0 0
\(961\) −29467.0 −0.989124
\(962\) 0 0
\(963\) 1550.00i 0.0518671i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 41792.0i 1.38980i 0.719105 + 0.694902i \(0.244552\pi\)
−0.719105 + 0.694902i \(0.755448\pi\)
\(968\) 0 0
\(969\) 46055.0 1.52683
\(970\) 0 0
\(971\) 2105.00 0.0695702 0.0347851 0.999395i \(-0.488925\pi\)
0.0347851 + 0.999395i \(0.488925\pi\)
\(972\) 0 0
\(973\) 2754.00i 0.0907391i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 30119.0i 0.986277i 0.869951 + 0.493138i \(0.164150\pi\)
−0.869951 + 0.493138i \(0.835850\pi\)
\(978\) 0 0
\(979\) −53391.0 −1.74299
\(980\) 0 0
\(981\) −892.000 −0.0290310
\(982\) 0 0
\(983\) 18438.0i 0.598251i 0.954214 + 0.299126i \(0.0966949\pi\)
−0.954214 + 0.299126i \(0.903305\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 2120.00i 0.0683691i
\(988\) 0 0
\(989\) −9512.00 −0.305828
\(990\) 0 0
\(991\) 2230.00 0.0714816 0.0357408 0.999361i \(-0.488621\pi\)
0.0357408 + 0.999361i \(0.488621\pi\)
\(992\) 0 0
\(993\) 13585.0i 0.434146i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 6804.00i − 0.216133i −0.994144 0.108067i \(-0.965534\pi\)
0.994144 0.108067i \(-0.0344660\pi\)
\(998\) 0 0
\(999\) 20010.0 0.633722
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 400.4.c.g.49.2 2
4.3 odd 2 200.4.c.d.49.1 2
5.2 odd 4 400.4.a.q.1.1 1
5.3 odd 4 400.4.a.f.1.1 1
5.4 even 2 inner 400.4.c.g.49.1 2
12.11 even 2 1800.4.f.c.649.2 2
20.3 even 4 200.4.a.h.1.1 yes 1
20.7 even 4 200.4.a.c.1.1 1
20.19 odd 2 200.4.c.d.49.2 2
40.3 even 4 1600.4.a.m.1.1 1
40.13 odd 4 1600.4.a.bo.1.1 1
40.27 even 4 1600.4.a.bn.1.1 1
40.37 odd 4 1600.4.a.n.1.1 1
60.23 odd 4 1800.4.a.t.1.1 1
60.47 odd 4 1800.4.a.p.1.1 1
60.59 even 2 1800.4.f.c.649.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.a.c.1.1 1 20.7 even 4
200.4.a.h.1.1 yes 1 20.3 even 4
200.4.c.d.49.1 2 4.3 odd 2
200.4.c.d.49.2 2 20.19 odd 2
400.4.a.f.1.1 1 5.3 odd 4
400.4.a.q.1.1 1 5.2 odd 4
400.4.c.g.49.1 2 5.4 even 2 inner
400.4.c.g.49.2 2 1.1 even 1 trivial
1600.4.a.m.1.1 1 40.3 even 4
1600.4.a.n.1.1 1 40.37 odd 4
1600.4.a.bn.1.1 1 40.27 even 4
1600.4.a.bo.1.1 1 40.13 odd 4
1800.4.a.p.1.1 1 60.47 odd 4
1800.4.a.t.1.1 1 60.23 odd 4
1800.4.f.c.649.1 2 60.59 even 2
1800.4.f.c.649.2 2 12.11 even 2