Properties

Label 196.3.b.b.97.3
Level $196$
Weight $3$
Character 196.97
Analytic conductor $5.341$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [196,3,Mod(97,196)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(196, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.97");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 196.b (of order \(2\), degree \(1\), 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: \(5.34061318146\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2048.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 97.3
Root \(1.84776i\) of defining polynomial
Character \(\chi\) \(=\) 196.97
Dual form 196.3.b.b.97.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.317025i q^{3} -5.67459i q^{5} +8.89949 q^{9} +O(q^{10})\) \(q+0.317025i q^{3} -5.67459i q^{5} +8.89949 q^{9} -15.8995 q^{11} -20.1940i q^{13} +1.79899 q^{15} +0.951076i q^{17} -31.8602i q^{19} +26.0000 q^{23} -7.20101 q^{25} +5.67459i q^{27} +27.7990 q^{29} +15.1535i q^{31} -5.04054i q^{33} -32.0000 q^{37} +6.40202 q^{39} -17.3408i q^{41} -59.2965 q^{43} -50.5010i q^{45} +76.3378i q^{47} -0.301515 q^{51} +25.7990 q^{53} +90.2232i q^{55} +10.1005 q^{57} +68.4440i q^{59} +54.2416i q^{61} -114.593 q^{65} +87.3970 q^{67} +8.24266i q^{69} +16.4020 q^{71} -70.3462i q^{73} -2.28290i q^{75} +40.2010 q^{79} +78.2965 q^{81} +71.5505i q^{83} +5.39697 q^{85} +8.81298i q^{87} +77.3207i q^{89} -4.80404 q^{93} -180.794 q^{95} -128.328i q^{97} -141.497 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 24 q^{11} - 72 q^{15} + 104 q^{23} - 108 q^{25} + 32 q^{29} - 128 q^{37} + 184 q^{39} + 40 q^{43} - 120 q^{51} + 24 q^{53} + 80 q^{57} + 96 q^{65} + 112 q^{67} + 224 q^{71} + 240 q^{79} + 36 q^{81} - 216 q^{85} - 336 q^{93} - 248 q^{95} - 368 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).

\(n\) \(99\) \(101\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) 0.317025i 0.105675i 0.998603 + 0.0528376i \(0.0168266\pi\)
−0.998603 + 0.0528376i \(0.983173\pi\)
\(4\) 0 0
\(5\) − 5.67459i − 1.13492i −0.823401 0.567459i \(-0.807926\pi\)
0.823401 0.567459i \(-0.192074\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 8.89949 0.988833
\(10\) 0 0
\(11\) −15.8995 −1.44541 −0.722704 0.691157i \(-0.757101\pi\)
−0.722704 + 0.691157i \(0.757101\pi\)
\(12\) 0 0
\(13\) − 20.1940i − 1.55339i −0.629879 0.776694i \(-0.716895\pi\)
0.629879 0.776694i \(-0.283105\pi\)
\(14\) 0 0
\(15\) 1.79899 0.119933
\(16\) 0 0
\(17\) 0.951076i 0.0559456i 0.999609 + 0.0279728i \(0.00890519\pi\)
−0.999609 + 0.0279728i \(0.991095\pi\)
\(18\) 0 0
\(19\) − 31.8602i − 1.67686i −0.545013 0.838428i \(-0.683475\pi\)
0.545013 0.838428i \(-0.316525\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 26.0000 1.13043 0.565217 0.824942i \(-0.308792\pi\)
0.565217 + 0.824942i \(0.308792\pi\)
\(24\) 0 0
\(25\) −7.20101 −0.288040
\(26\) 0 0
\(27\) 5.67459i 0.210170i
\(28\) 0 0
\(29\) 27.7990 0.958586 0.479293 0.877655i \(-0.340893\pi\)
0.479293 + 0.877655i \(0.340893\pi\)
\(30\) 0 0
\(31\) 15.1535i 0.488822i 0.969672 + 0.244411i \(0.0785946\pi\)
−0.969672 + 0.244411i \(0.921405\pi\)
\(32\) 0 0
\(33\) − 5.04054i − 0.152744i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −32.0000 −0.864865 −0.432432 0.901666i \(-0.642345\pi\)
−0.432432 + 0.901666i \(0.642345\pi\)
\(38\) 0 0
\(39\) 6.40202 0.164154
\(40\) 0 0
\(41\) − 17.3408i − 0.422946i −0.977384 0.211473i \(-0.932174\pi\)
0.977384 0.211473i \(-0.0678261\pi\)
\(42\) 0 0
\(43\) −59.2965 −1.37899 −0.689494 0.724292i \(-0.742167\pi\)
−0.689494 + 0.724292i \(0.742167\pi\)
\(44\) 0 0
\(45\) − 50.5010i − 1.12224i
\(46\) 0 0
\(47\) 76.3378i 1.62421i 0.583513 + 0.812104i \(0.301678\pi\)
−0.583513 + 0.812104i \(0.698322\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −0.301515 −0.00591206
\(52\) 0 0
\(53\) 25.7990 0.486773 0.243387 0.969929i \(-0.421742\pi\)
0.243387 + 0.969929i \(0.421742\pi\)
\(54\) 0 0
\(55\) 90.2232i 1.64042i
\(56\) 0 0
\(57\) 10.1005 0.177202
\(58\) 0 0
\(59\) 68.4440i 1.16007i 0.814592 + 0.580034i \(0.196961\pi\)
−0.814592 + 0.580034i \(0.803039\pi\)
\(60\) 0 0
\(61\) 54.2416i 0.889206i 0.895728 + 0.444603i \(0.146655\pi\)
−0.895728 + 0.444603i \(0.853345\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −114.593 −1.76297
\(66\) 0 0
\(67\) 87.3970 1.30443 0.652216 0.758033i \(-0.273839\pi\)
0.652216 + 0.758033i \(0.273839\pi\)
\(68\) 0 0
\(69\) 8.24266i 0.119459i
\(70\) 0 0
\(71\) 16.4020 0.231014 0.115507 0.993307i \(-0.463151\pi\)
0.115507 + 0.993307i \(0.463151\pi\)
\(72\) 0 0
\(73\) − 70.3462i − 0.963646i −0.876269 0.481823i \(-0.839975\pi\)
0.876269 0.481823i \(-0.160025\pi\)
\(74\) 0 0
\(75\) − 2.28290i − 0.0304387i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 40.2010 0.508874 0.254437 0.967089i \(-0.418110\pi\)
0.254437 + 0.967089i \(0.418110\pi\)
\(80\) 0 0
\(81\) 78.2965 0.966623
\(82\) 0 0
\(83\) 71.5505i 0.862055i 0.902339 + 0.431027i \(0.141849\pi\)
−0.902339 + 0.431027i \(0.858151\pi\)
\(84\) 0 0
\(85\) 5.39697 0.0634938
\(86\) 0 0
\(87\) 8.81298i 0.101299i
\(88\) 0 0
\(89\) 77.3207i 0.868772i 0.900727 + 0.434386i \(0.143035\pi\)
−0.900727 + 0.434386i \(0.856965\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.80404 −0.0516563
\(94\) 0 0
\(95\) −180.794 −1.90309
\(96\) 0 0
\(97\) − 128.328i − 1.32297i −0.749957 0.661486i \(-0.769926\pi\)
0.749957 0.661486i \(-0.230074\pi\)
\(98\) 0 0
\(99\) −141.497 −1.42927
\(100\) 0 0
\(101\) 100.272i 0.992796i 0.868095 + 0.496398i \(0.165344\pi\)
−0.868095 + 0.496398i \(0.834656\pi\)
\(102\) 0 0
\(103\) − 49.2011i − 0.477680i −0.971059 0.238840i \(-0.923233\pi\)
0.971059 0.238840i \(-0.0767672\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 29.5980 0.276617 0.138308 0.990389i \(-0.455834\pi\)
0.138308 + 0.990389i \(0.455834\pi\)
\(108\) 0 0
\(109\) −11.1960 −0.102715 −0.0513576 0.998680i \(-0.516355\pi\)
−0.0513576 + 0.998680i \(0.516355\pi\)
\(110\) 0 0
\(111\) − 10.1448i − 0.0913947i
\(112\) 0 0
\(113\) 133.698 1.18317 0.591586 0.806242i \(-0.298502\pi\)
0.591586 + 0.806242i \(0.298502\pi\)
\(114\) 0 0
\(115\) − 147.539i − 1.28295i
\(116\) 0 0
\(117\) − 179.717i − 1.53604i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 131.794 1.08921
\(122\) 0 0
\(123\) 5.49747 0.0446949
\(124\) 0 0
\(125\) − 101.002i − 0.808016i
\(126\) 0 0
\(127\) 120.995 0.952716 0.476358 0.879251i \(-0.341957\pi\)
0.476358 + 0.879251i \(0.341957\pi\)
\(128\) 0 0
\(129\) − 18.7985i − 0.145725i
\(130\) 0 0
\(131\) − 127.060i − 0.969925i −0.874535 0.484963i \(-0.838833\pi\)
0.874535 0.484963i \(-0.161167\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 32.2010 0.238526
\(136\) 0 0
\(137\) −149.899 −1.09416 −0.547078 0.837081i \(-0.684260\pi\)
−0.547078 + 0.837081i \(0.684260\pi\)
\(138\) 0 0
\(139\) 14.2661i 0.102634i 0.998682 + 0.0513171i \(0.0163419\pi\)
−0.998682 + 0.0513171i \(0.983658\pi\)
\(140\) 0 0
\(141\) −24.2010 −0.171638
\(142\) 0 0
\(143\) 321.075i 2.24528i
\(144\) 0 0
\(145\) − 157.748i − 1.08792i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 50.6030 0.339618 0.169809 0.985477i \(-0.445685\pi\)
0.169809 + 0.985477i \(0.445685\pi\)
\(150\) 0 0
\(151\) −41.7990 −0.276815 −0.138407 0.990375i \(-0.544198\pi\)
−0.138407 + 0.990375i \(0.544198\pi\)
\(152\) 0 0
\(153\) 8.46410i 0.0553209i
\(154\) 0 0
\(155\) 85.9899 0.554774
\(156\) 0 0
\(157\) − 143.260i − 0.912487i −0.889855 0.456243i \(-0.849195\pi\)
0.889855 0.456243i \(-0.150805\pi\)
\(158\) 0 0
\(159\) 8.17893i 0.0514398i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 114.291 0.701174 0.350587 0.936530i \(-0.385982\pi\)
0.350587 + 0.936530i \(0.385982\pi\)
\(164\) 0 0
\(165\) −28.6030 −0.173352
\(166\) 0 0
\(167\) − 290.895i − 1.74189i −0.491382 0.870944i \(-0.663508\pi\)
0.491382 0.870944i \(-0.336492\pi\)
\(168\) 0 0
\(169\) −238.799 −1.41301
\(170\) 0 0
\(171\) − 283.540i − 1.65813i
\(172\) 0 0
\(173\) 212.496i 1.22830i 0.789189 + 0.614151i \(0.210501\pi\)
−0.789189 + 0.614151i \(0.789499\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −21.6985 −0.122590
\(178\) 0 0
\(179\) 23.6081 0.131889 0.0659444 0.997823i \(-0.478994\pi\)
0.0659444 + 0.997823i \(0.478994\pi\)
\(180\) 0 0
\(181\) 156.384i 0.864002i 0.901873 + 0.432001i \(0.142192\pi\)
−0.901873 + 0.432001i \(0.857808\pi\)
\(182\) 0 0
\(183\) −17.1960 −0.0939670
\(184\) 0 0
\(185\) 181.587i 0.981551i
\(186\) 0 0
\(187\) − 15.1216i − 0.0808643i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −245.588 −1.28580 −0.642900 0.765950i \(-0.722269\pi\)
−0.642900 + 0.765950i \(0.722269\pi\)
\(192\) 0 0
\(193\) 57.4975 0.297914 0.148957 0.988844i \(-0.452408\pi\)
0.148957 + 0.988844i \(0.452408\pi\)
\(194\) 0 0
\(195\) − 36.3289i − 0.186302i
\(196\) 0 0
\(197\) 174.402 0.885289 0.442645 0.896697i \(-0.354040\pi\)
0.442645 + 0.896697i \(0.354040\pi\)
\(198\) 0 0
\(199\) 167.765i 0.843042i 0.906819 + 0.421521i \(0.138504\pi\)
−0.906819 + 0.421521i \(0.861496\pi\)
\(200\) 0 0
\(201\) 27.7071i 0.137846i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −98.4020 −0.480010
\(206\) 0 0
\(207\) 231.387 1.11781
\(208\) 0 0
\(209\) 506.562i 2.42374i
\(210\) 0 0
\(211\) −81.7889 −0.387625 −0.193813 0.981039i \(-0.562085\pi\)
−0.193813 + 0.981039i \(0.562085\pi\)
\(212\) 0 0
\(213\) 5.19986i 0.0244125i
\(214\) 0 0
\(215\) 336.483i 1.56504i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 22.3015 0.101833
\(220\) 0 0
\(221\) 19.2061 0.0869053
\(222\) 0 0
\(223\) 91.4275i 0.409989i 0.978763 + 0.204994i \(0.0657176\pi\)
−0.978763 + 0.204994i \(0.934282\pi\)
\(224\) 0 0
\(225\) −64.0854 −0.284824
\(226\) 0 0
\(227\) − 367.550i − 1.61916i −0.587007 0.809582i \(-0.699694\pi\)
0.587007 0.809582i \(-0.300306\pi\)
\(228\) 0 0
\(229\) − 12.0788i − 0.0527460i −0.999652 0.0263730i \(-0.991604\pi\)
0.999652 0.0263730i \(-0.00839575\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.69848 −0.00728963 −0.00364482 0.999993i \(-0.501160\pi\)
−0.00364482 + 0.999993i \(0.501160\pi\)
\(234\) 0 0
\(235\) 433.186 1.84334
\(236\) 0 0
\(237\) 12.7447i 0.0537753i
\(238\) 0 0
\(239\) 201.397 0.842665 0.421333 0.906906i \(-0.361563\pi\)
0.421333 + 0.906906i \(0.361563\pi\)
\(240\) 0 0
\(241\) 97.4192i 0.404229i 0.979362 + 0.202114i \(0.0647813\pi\)
−0.979362 + 0.202114i \(0.935219\pi\)
\(242\) 0 0
\(243\) 75.8933i 0.312318i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −643.387 −2.60481
\(248\) 0 0
\(249\) −22.6833 −0.0910977
\(250\) 0 0
\(251\) − 77.2251i − 0.307670i −0.988097 0.153835i \(-0.950838\pi\)
0.988097 0.153835i \(-0.0491624\pi\)
\(252\) 0 0
\(253\) −413.387 −1.63394
\(254\) 0 0
\(255\) 1.71098i 0.00670971i
\(256\) 0 0
\(257\) 363.682i 1.41511i 0.706660 + 0.707553i \(0.250201\pi\)
−0.706660 + 0.707553i \(0.749799\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 247.397 0.947881
\(262\) 0 0
\(263\) −410.593 −1.56119 −0.780595 0.625037i \(-0.785084\pi\)
−0.780595 + 0.625037i \(0.785084\pi\)
\(264\) 0 0
\(265\) − 146.399i − 0.552448i
\(266\) 0 0
\(267\) −24.5126 −0.0918076
\(268\) 0 0
\(269\) 307.855i 1.14444i 0.820099 + 0.572222i \(0.193918\pi\)
−0.820099 + 0.572222i \(0.806082\pi\)
\(270\) 0 0
\(271\) 178.417i 0.658364i 0.944266 + 0.329182i \(0.106773\pi\)
−0.944266 + 0.329182i \(0.893227\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 114.492 0.416336
\(276\) 0 0
\(277\) 96.1909 0.347260 0.173630 0.984811i \(-0.444450\pi\)
0.173630 + 0.984811i \(0.444450\pi\)
\(278\) 0 0
\(279\) 134.858i 0.483363i
\(280\) 0 0
\(281\) −155.106 −0.551977 −0.275989 0.961161i \(-0.589005\pi\)
−0.275989 + 0.961161i \(0.589005\pi\)
\(282\) 0 0
\(283\) − 313.245i − 1.10687i −0.832892 0.553436i \(-0.813316\pi\)
0.832892 0.553436i \(-0.186684\pi\)
\(284\) 0 0
\(285\) − 57.3163i − 0.201110i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 288.095 0.996870
\(290\) 0 0
\(291\) 40.6833 0.139805
\(292\) 0 0
\(293\) 202.543i 0.691271i 0.938369 + 0.345636i \(0.112337\pi\)
−0.938369 + 0.345636i \(0.887663\pi\)
\(294\) 0 0
\(295\) 388.392 1.31658
\(296\) 0 0
\(297\) − 90.2232i − 0.303782i
\(298\) 0 0
\(299\) − 525.045i − 1.75600i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −31.7889 −0.104914
\(304\) 0 0
\(305\) 307.799 1.00918
\(306\) 0 0
\(307\) − 378.772i − 1.23378i −0.787047 0.616892i \(-0.788391\pi\)
0.787047 0.616892i \(-0.211609\pi\)
\(308\) 0 0
\(309\) 15.5980 0.0504789
\(310\) 0 0
\(311\) 205.808i 0.661763i 0.943672 + 0.330882i \(0.107346\pi\)
−0.943672 + 0.330882i \(0.892654\pi\)
\(312\) 0 0
\(313\) − 451.971i − 1.44400i −0.691894 0.721999i \(-0.743224\pi\)
0.691894 0.721999i \(-0.256776\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −83.3869 −0.263050 −0.131525 0.991313i \(-0.541987\pi\)
−0.131525 + 0.991313i \(0.541987\pi\)
\(318\) 0 0
\(319\) −441.990 −1.38555
\(320\) 0 0
\(321\) 9.38331i 0.0292315i
\(322\) 0 0
\(323\) 30.3015 0.0938127
\(324\) 0 0
\(325\) 145.417i 0.447438i
\(326\) 0 0
\(327\) − 3.54940i − 0.0108544i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 153.487 0.463708 0.231854 0.972751i \(-0.425521\pi\)
0.231854 + 0.972751i \(0.425521\pi\)
\(332\) 0 0
\(333\) −284.784 −0.855207
\(334\) 0 0
\(335\) − 495.942i − 1.48042i
\(336\) 0 0
\(337\) 519.377 1.54118 0.770589 0.637333i \(-0.219962\pi\)
0.770589 + 0.637333i \(0.219962\pi\)
\(338\) 0 0
\(339\) 42.3858i 0.125032i
\(340\) 0 0
\(341\) − 240.933i − 0.706548i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 46.7737 0.135576
\(346\) 0 0
\(347\) 84.4823 0.243465 0.121732 0.992563i \(-0.461155\pi\)
0.121732 + 0.992563i \(0.461155\pi\)
\(348\) 0 0
\(349\) 187.959i 0.538565i 0.963061 + 0.269283i \(0.0867866\pi\)
−0.963061 + 0.269283i \(0.913213\pi\)
\(350\) 0 0
\(351\) 114.593 0.326476
\(352\) 0 0
\(353\) − 113.270i − 0.320879i −0.987046 0.160440i \(-0.948709\pi\)
0.987046 0.160440i \(-0.0512912\pi\)
\(354\) 0 0
\(355\) − 93.0748i − 0.262183i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 661.176 1.84172 0.920858 0.389899i \(-0.127490\pi\)
0.920858 + 0.389899i \(0.127490\pi\)
\(360\) 0 0
\(361\) −654.075 −1.81184
\(362\) 0 0
\(363\) 41.7820i 0.115102i
\(364\) 0 0
\(365\) −399.186 −1.09366
\(366\) 0 0
\(367\) − 87.8781i − 0.239450i −0.992807 0.119725i \(-0.961799\pi\)
0.992807 0.119725i \(-0.0382013\pi\)
\(368\) 0 0
\(369\) − 154.324i − 0.418223i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 426.201 1.14263 0.571315 0.820731i \(-0.306433\pi\)
0.571315 + 0.820731i \(0.306433\pi\)
\(374\) 0 0
\(375\) 32.0202 0.0853872
\(376\) 0 0
\(377\) − 561.374i − 1.48905i
\(378\) 0 0
\(379\) −719.879 −1.89942 −0.949709 0.313134i \(-0.898621\pi\)
−0.949709 + 0.313134i \(0.898621\pi\)
\(380\) 0 0
\(381\) 38.3585i 0.100678i
\(382\) 0 0
\(383\) − 241.503i − 0.630557i −0.948999 0.315278i \(-0.897902\pi\)
0.948999 0.315278i \(-0.102098\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −527.709 −1.36359
\(388\) 0 0
\(389\) −265.407 −0.682280 −0.341140 0.940012i \(-0.610813\pi\)
−0.341140 + 0.940012i \(0.610813\pi\)
\(390\) 0 0
\(391\) 24.7280i 0.0632429i
\(392\) 0 0
\(393\) 40.2813 0.102497
\(394\) 0 0
\(395\) − 228.124i − 0.577530i
\(396\) 0 0
\(397\) 181.112i 0.456202i 0.973637 + 0.228101i \(0.0732517\pi\)
−0.973637 + 0.228101i \(0.926748\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −420.593 −1.04886 −0.524430 0.851454i \(-0.675722\pi\)
−0.524430 + 0.851454i \(0.675722\pi\)
\(402\) 0 0
\(403\) 306.010 0.759330
\(404\) 0 0
\(405\) − 444.301i − 1.09704i
\(406\) 0 0
\(407\) 508.784 1.25008
\(408\) 0 0
\(409\) 146.145i 0.357324i 0.983911 + 0.178662i \(0.0571768\pi\)
−0.983911 + 0.178662i \(0.942823\pi\)
\(410\) 0 0
\(411\) − 47.5219i − 0.115625i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 406.020 0.978362
\(416\) 0 0
\(417\) −4.52273 −0.0108459
\(418\) 0 0
\(419\) 521.905i 1.24560i 0.782383 + 0.622798i \(0.214004\pi\)
−0.782383 + 0.622798i \(0.785996\pi\)
\(420\) 0 0
\(421\) 746.181 1.77240 0.886200 0.463302i \(-0.153335\pi\)
0.886200 + 0.463302i \(0.153335\pi\)
\(422\) 0 0
\(423\) 679.368i 1.60607i
\(424\) 0 0
\(425\) − 6.84871i − 0.0161146i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −101.789 −0.237270
\(430\) 0 0
\(431\) 248.593 0.576782 0.288391 0.957513i \(-0.406880\pi\)
0.288391 + 0.957513i \(0.406880\pi\)
\(432\) 0 0
\(433\) 494.357i 1.14170i 0.821054 + 0.570851i \(0.193387\pi\)
−0.821054 + 0.570851i \(0.806613\pi\)
\(434\) 0 0
\(435\) 50.0101 0.114966
\(436\) 0 0
\(437\) − 828.366i − 1.89558i
\(438\) 0 0
\(439\) 655.274i 1.49265i 0.665581 + 0.746325i \(0.268184\pi\)
−0.665581 + 0.746325i \(0.731816\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −39.3970 −0.0889322 −0.0444661 0.999011i \(-0.514159\pi\)
−0.0444661 + 0.999011i \(0.514159\pi\)
\(444\) 0 0
\(445\) 438.764 0.985986
\(446\) 0 0
\(447\) 16.0424i 0.0358891i
\(448\) 0 0
\(449\) −700.362 −1.55983 −0.779913 0.625888i \(-0.784737\pi\)
−0.779913 + 0.625888i \(0.784737\pi\)
\(450\) 0 0
\(451\) 275.710i 0.611330i
\(452\) 0 0
\(453\) − 13.2513i − 0.0292524i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −353.477 −0.773473 −0.386737 0.922190i \(-0.626398\pi\)
−0.386737 + 0.922190i \(0.626398\pi\)
\(458\) 0 0
\(459\) −5.39697 −0.0117581
\(460\) 0 0
\(461\) 382.482i 0.829680i 0.909894 + 0.414840i \(0.136162\pi\)
−0.909894 + 0.414840i \(0.863838\pi\)
\(462\) 0 0
\(463\) −309.005 −0.667398 −0.333699 0.942680i \(-0.608297\pi\)
−0.333699 + 0.942680i \(0.608297\pi\)
\(464\) 0 0
\(465\) 27.2610i 0.0586258i
\(466\) 0 0
\(467\) − 460.278i − 0.985605i −0.870141 0.492803i \(-0.835972\pi\)
0.870141 0.492803i \(-0.164028\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 45.4172 0.0964271
\(472\) 0 0
\(473\) 942.784 1.99320
\(474\) 0 0
\(475\) 229.426i 0.483002i
\(476\) 0 0
\(477\) 229.598 0.481337
\(478\) 0 0
\(479\) − 14.9623i − 0.0312366i −0.999878 0.0156183i \(-0.995028\pi\)
0.999878 0.0156183i \(-0.00497166\pi\)
\(480\) 0 0
\(481\) 646.209i 1.34347i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −728.211 −1.50147
\(486\) 0 0
\(487\) −437.377 −0.898104 −0.449052 0.893506i \(-0.648238\pi\)
−0.449052 + 0.893506i \(0.648238\pi\)
\(488\) 0 0
\(489\) 36.2333i 0.0740967i
\(490\) 0 0
\(491\) −816.583 −1.66310 −0.831551 0.555449i \(-0.812546\pi\)
−0.831551 + 0.555449i \(0.812546\pi\)
\(492\) 0 0
\(493\) 26.4390i 0.0536287i
\(494\) 0 0
\(495\) 802.941i 1.62210i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −249.005 −0.499008 −0.249504 0.968374i \(-0.580268\pi\)
−0.249504 + 0.968374i \(0.580268\pi\)
\(500\) 0 0
\(501\) 92.2212 0.184074
\(502\) 0 0
\(503\) 423.536i 0.842020i 0.907056 + 0.421010i \(0.138324\pi\)
−0.907056 + 0.421010i \(0.861676\pi\)
\(504\) 0 0
\(505\) 569.005 1.12674
\(506\) 0 0
\(507\) − 75.7053i − 0.149320i
\(508\) 0 0
\(509\) 775.138i 1.52286i 0.648245 + 0.761432i \(0.275503\pi\)
−0.648245 + 0.761432i \(0.724497\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 180.794 0.352425
\(514\) 0 0
\(515\) −279.196 −0.542128
\(516\) 0 0
\(517\) − 1213.73i − 2.34764i
\(518\) 0 0
\(519\) −67.3667 −0.129801
\(520\) 0 0
\(521\) − 509.447i − 0.977825i −0.872333 0.488913i \(-0.837394\pi\)
0.872333 0.488913i \(-0.162606\pi\)
\(522\) 0 0
\(523\) 528.150i 1.00985i 0.863164 + 0.504923i \(0.168479\pi\)
−0.863164 + 0.504923i \(0.831521\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −14.4121 −0.0273475
\(528\) 0 0
\(529\) 147.000 0.277883
\(530\) 0 0
\(531\) 609.117i 1.14711i
\(532\) 0 0
\(533\) −350.181 −0.657000
\(534\) 0 0
\(535\) − 167.957i − 0.313937i
\(536\) 0 0
\(537\) 7.48436i 0.0139374i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −529.598 −0.978924 −0.489462 0.872025i \(-0.662807\pi\)
−0.489462 + 0.872025i \(0.662807\pi\)
\(542\) 0 0
\(543\) −49.5778 −0.0913035
\(544\) 0 0
\(545\) 63.5325i 0.116573i
\(546\) 0 0
\(547\) 16.9045 0.0309041 0.0154521 0.999881i \(-0.495081\pi\)
0.0154521 + 0.999881i \(0.495081\pi\)
\(548\) 0 0
\(549\) 482.723i 0.879276i
\(550\) 0 0
\(551\) − 885.683i − 1.60741i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −57.5677 −0.103726
\(556\) 0 0
\(557\) 893.960 1.60495 0.802477 0.596683i \(-0.203515\pi\)
0.802477 + 0.596683i \(0.203515\pi\)
\(558\) 0 0
\(559\) 1197.43i 2.14210i
\(560\) 0 0
\(561\) 4.79394 0.00854535
\(562\) 0 0
\(563\) 97.1356i 0.172532i 0.996272 + 0.0862661i \(0.0274935\pi\)
−0.996272 + 0.0862661i \(0.972506\pi\)
\(564\) 0 0
\(565\) − 758.685i − 1.34280i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 760.181 1.33599 0.667997 0.744164i \(-0.267152\pi\)
0.667997 + 0.744164i \(0.267152\pi\)
\(570\) 0 0
\(571\) 338.291 0.592454 0.296227 0.955118i \(-0.404271\pi\)
0.296227 + 0.955118i \(0.404271\pi\)
\(572\) 0 0
\(573\) − 77.8576i − 0.135877i
\(574\) 0 0
\(575\) −187.226 −0.325611
\(576\) 0 0
\(577\) 241.377i 0.418332i 0.977880 + 0.209166i \(0.0670748\pi\)
−0.977880 + 0.209166i \(0.932925\pi\)
\(578\) 0 0
\(579\) 18.2282i 0.0314821i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −410.191 −0.703586
\(584\) 0 0
\(585\) −1019.82 −1.74328
\(586\) 0 0
\(587\) − 405.338i − 0.690525i −0.938506 0.345263i \(-0.887790\pi\)
0.938506 0.345263i \(-0.112210\pi\)
\(588\) 0 0
\(589\) 482.794 0.819684
\(590\) 0 0
\(591\) 55.2899i 0.0935531i
\(592\) 0 0
\(593\) 148.331i 0.250137i 0.992148 + 0.125068i \(0.0399151\pi\)
−0.992148 + 0.125068i \(0.960085\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −53.1859 −0.0890885
\(598\) 0 0
\(599\) 401.789 0.670766 0.335383 0.942082i \(-0.391134\pi\)
0.335383 + 0.942082i \(0.391134\pi\)
\(600\) 0 0
\(601\) − 782.716i − 1.30236i −0.758925 0.651178i \(-0.774275\pi\)
0.758925 0.651178i \(-0.225725\pi\)
\(602\) 0 0
\(603\) 777.789 1.28987
\(604\) 0 0
\(605\) − 747.877i − 1.23616i
\(606\) 0 0
\(607\) − 131.373i − 0.216430i −0.994128 0.108215i \(-0.965487\pi\)
0.994128 0.108215i \(-0.0345134\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1541.57 2.52302
\(612\) 0 0
\(613\) −1000.76 −1.63257 −0.816284 0.577651i \(-0.803969\pi\)
−0.816284 + 0.577651i \(0.803969\pi\)
\(614\) 0 0
\(615\) − 31.1959i − 0.0507251i
\(616\) 0 0
\(617\) 738.563 1.19702 0.598511 0.801115i \(-0.295759\pi\)
0.598511 + 0.801115i \(0.295759\pi\)
\(618\) 0 0
\(619\) 953.904i 1.54104i 0.637416 + 0.770520i \(0.280003\pi\)
−0.637416 + 0.770520i \(0.719997\pi\)
\(620\) 0 0
\(621\) 147.539i 0.237584i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −753.171 −1.20507
\(626\) 0 0
\(627\) −160.593 −0.256129
\(628\) 0 0
\(629\) − 30.4344i − 0.0483854i
\(630\) 0 0
\(631\) −743.176 −1.17777 −0.588887 0.808215i \(-0.700434\pi\)
−0.588887 + 0.808215i \(0.700434\pi\)
\(632\) 0 0
\(633\) − 25.9291i − 0.0409623i
\(634\) 0 0
\(635\) − 686.597i − 1.08126i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 145.970 0.228435
\(640\) 0 0
\(641\) 239.417 0.373506 0.186753 0.982407i \(-0.440204\pi\)
0.186753 + 0.982407i \(0.440204\pi\)
\(642\) 0 0
\(643\) − 812.453i − 1.26353i −0.775158 0.631767i \(-0.782330\pi\)
0.775158 0.631767i \(-0.217670\pi\)
\(644\) 0 0
\(645\) −106.674 −0.165386
\(646\) 0 0
\(647\) − 712.845i − 1.10177i −0.834581 0.550885i \(-0.814290\pi\)
0.834581 0.550885i \(-0.185710\pi\)
\(648\) 0 0
\(649\) − 1088.23i − 1.67677i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 296.583 0.454185 0.227093 0.973873i \(-0.427078\pi\)
0.227093 + 0.973873i \(0.427078\pi\)
\(654\) 0 0
\(655\) −721.015 −1.10079
\(656\) 0 0
\(657\) − 626.045i − 0.952885i
\(658\) 0 0
\(659\) −617.709 −0.937342 −0.468671 0.883373i \(-0.655267\pi\)
−0.468671 + 0.883373i \(0.655267\pi\)
\(660\) 0 0
\(661\) 325.099i 0.491829i 0.969292 + 0.245915i \(0.0790883\pi\)
−0.969292 + 0.245915i \(0.920912\pi\)
\(662\) 0 0
\(663\) 6.08881i 0.00918372i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 722.774 1.08362
\(668\) 0 0
\(669\) −28.9848 −0.0433256
\(670\) 0 0
\(671\) − 862.414i − 1.28527i
\(672\) 0 0
\(673\) −793.899 −1.17964 −0.589821 0.807534i \(-0.700802\pi\)
−0.589821 + 0.807534i \(0.700802\pi\)
\(674\) 0 0
\(675\) − 40.8628i − 0.0605375i
\(676\) 0 0
\(677\) − 342.980i − 0.506617i −0.967386 0.253309i \(-0.918481\pi\)
0.967386 0.253309i \(-0.0815188\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 116.523 0.171105
\(682\) 0 0
\(683\) −1192.16 −1.74548 −0.872738 0.488188i \(-0.837658\pi\)
−0.872738 + 0.488188i \(0.837658\pi\)
\(684\) 0 0
\(685\) 850.619i 1.24178i
\(686\) 0 0
\(687\) 3.82929 0.00557394
\(688\) 0 0
\(689\) − 520.986i − 0.756148i
\(690\) 0 0
\(691\) 312.961i 0.452911i 0.974022 + 0.226455i \(0.0727137\pi\)
−0.974022 + 0.226455i \(0.927286\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 80.9545 0.116481
\(696\) 0 0
\(697\) 16.4924 0.0236620
\(698\) 0 0
\(699\) − 0.538463i 0 0.000770333i
\(700\) 0 0
\(701\) 414.010 0.590599 0.295300 0.955405i \(-0.404581\pi\)
0.295300 + 0.955405i \(0.404581\pi\)
\(702\) 0 0
\(703\) 1019.53i 1.45025i
\(704\) 0 0
\(705\) 137.331i 0.194796i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1123.96 1.58527 0.792637 0.609694i \(-0.208707\pi\)
0.792637 + 0.609694i \(0.208707\pi\)
\(710\) 0 0
\(711\) 357.769 0.503191
\(712\) 0 0
\(713\) 393.991i 0.552582i
\(714\) 0 0
\(715\) 1821.97 2.54821
\(716\) 0 0
\(717\) 63.8479i 0.0890487i
\(718\) 0 0
\(719\) − 1085.72i − 1.51004i −0.655700 0.755021i \(-0.727626\pi\)
0.655700 0.755021i \(-0.272374\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −30.8843 −0.0427169
\(724\) 0 0
\(725\) −200.181 −0.276111
\(726\) 0 0
\(727\) − 270.606i − 0.372223i −0.982529 0.186111i \(-0.940412\pi\)
0.982529 0.186111i \(-0.0595885\pi\)
\(728\) 0 0
\(729\) 680.608 0.933619
\(730\) 0 0
\(731\) − 56.3954i − 0.0771484i
\(732\) 0 0
\(733\) − 308.298i − 0.420598i −0.977637 0.210299i \(-0.932556\pi\)
0.977637 0.210299i \(-0.0674437\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1389.57 −1.88544
\(738\) 0 0
\(739\) 272.523 0.368772 0.184386 0.982854i \(-0.440970\pi\)
0.184386 + 0.982854i \(0.440970\pi\)
\(740\) 0 0
\(741\) − 203.970i − 0.275263i
\(742\) 0 0
\(743\) −225.196 −0.303090 −0.151545 0.988450i \(-0.548425\pi\)
−0.151545 + 0.988450i \(0.548425\pi\)
\(744\) 0 0
\(745\) − 287.152i − 0.385438i
\(746\) 0 0
\(747\) 636.764i 0.852428i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −511.387 −0.680941 −0.340471 0.940255i \(-0.610586\pi\)
−0.340471 + 0.940255i \(0.610586\pi\)
\(752\) 0 0
\(753\) 24.4823 0.0325130
\(754\) 0 0
\(755\) 237.192i 0.314162i
\(756\) 0 0
\(757\) −779.598 −1.02985 −0.514926 0.857235i \(-0.672181\pi\)
−0.514926 + 0.857235i \(0.672181\pi\)
\(758\) 0 0
\(759\) − 131.054i − 0.172667i
\(760\) 0 0
\(761\) − 195.028i − 0.256278i −0.991756 0.128139i \(-0.959100\pi\)
0.991756 0.128139i \(-0.0409004\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 48.0303 0.0627847
\(766\) 0 0
\(767\) 1382.16 1.80203
\(768\) 0 0
\(769\) − 73.8956i − 0.0960931i −0.998845 0.0480465i \(-0.984700\pi\)
0.998845 0.0480465i \(-0.0152996\pi\)
\(770\) 0 0
\(771\) −115.296 −0.149541
\(772\) 0 0
\(773\) 185.930i 0.240530i 0.992742 + 0.120265i \(0.0383744\pi\)
−0.992742 + 0.120265i \(0.961626\pi\)
\(774\) 0 0
\(775\) − 109.120i − 0.140801i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −552.482 −0.709220
\(780\) 0 0
\(781\) −260.784 −0.333910
\(782\) 0 0
\(783\) 157.748i 0.201466i
\(784\) 0 0
\(785\) −812.944 −1.03560
\(786\) 0 0
\(787\) − 713.704i − 0.906866i −0.891290 0.453433i \(-0.850199\pi\)
0.891290 0.453433i \(-0.149801\pi\)
\(788\) 0 0
\(789\) − 130.168i − 0.164979i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1095.36 1.38128
\(794\) 0 0
\(795\) 46.4121 0.0583800
\(796\) 0 0
\(797\) 908.540i 1.13995i 0.821662 + 0.569975i \(0.193047\pi\)
−0.821662 + 0.569975i \(0.806953\pi\)
\(798\) 0 0
\(799\) −72.6030 −0.0908674
\(800\) 0 0
\(801\) 688.115i 0.859070i
\(802\) 0 0
\(803\) 1118.47i 1.39286i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −97.5980 −0.120939
\(808\) 0 0
\(809\) −320.111 −0.395687 −0.197843 0.980234i \(-0.563394\pi\)
−0.197843 + 0.980234i \(0.563394\pi\)
\(810\) 0 0
\(811\) 839.749i 1.03545i 0.855547 + 0.517724i \(0.173221\pi\)
−0.855547 + 0.517724i \(0.826779\pi\)
\(812\) 0 0
\(813\) −56.5626 −0.0695727
\(814\) 0 0
\(815\) − 648.557i − 0.795776i
\(816\) 0 0
\(817\) 1889.20i 2.31236i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 129.960 0.158294 0.0791471 0.996863i \(-0.474780\pi\)
0.0791471 + 0.996863i \(0.474780\pi\)
\(822\) 0 0
\(823\) 800.965 0.973226 0.486613 0.873618i \(-0.338232\pi\)
0.486613 + 0.873618i \(0.338232\pi\)
\(824\) 0 0
\(825\) 36.2970i 0.0439964i
\(826\) 0 0
\(827\) 289.156 0.349644 0.174822 0.984600i \(-0.444065\pi\)
0.174822 + 0.984600i \(0.444065\pi\)
\(828\) 0 0
\(829\) 316.541i 0.381835i 0.981606 + 0.190917i \(0.0611463\pi\)
−0.981606 + 0.190917i \(0.938854\pi\)
\(830\) 0 0
\(831\) 30.4950i 0.0366967i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −1650.71 −1.97690
\(836\) 0 0
\(837\) −85.9899 −0.102736
\(838\) 0 0
\(839\) − 802.370i − 0.956341i −0.878267 0.478171i \(-0.841300\pi\)
0.878267 0.478171i \(-0.158700\pi\)
\(840\) 0 0
\(841\) −68.2162 −0.0811132
\(842\) 0 0
\(843\) − 49.1724i − 0.0583302i
\(844\) 0 0
\(845\) 1355.09i 1.60365i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 99.3066 0.116969
\(850\) 0 0
\(851\) −832.000 −0.977673
\(852\) 0 0
\(853\) − 235.386i − 0.275950i −0.990436 0.137975i \(-0.955941\pi\)
0.990436 0.137975i \(-0.0440594\pi\)
\(854\) 0 0
\(855\) −1608.97 −1.88184
\(856\) 0 0
\(857\) 1416.28i 1.65260i 0.563232 + 0.826299i \(0.309558\pi\)
−0.563232 + 0.826299i \(0.690442\pi\)
\(858\) 0 0
\(859\) 1290.10i 1.50187i 0.660378 + 0.750933i \(0.270396\pi\)
−0.660378 + 0.750933i \(0.729604\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −372.161 −0.431241 −0.215620 0.976477i \(-0.569177\pi\)
−0.215620 + 0.976477i \(0.569177\pi\)
\(864\) 0 0
\(865\) 1205.83 1.39402
\(866\) 0 0
\(867\) 91.3336i 0.105344i
\(868\) 0 0
\(869\) −639.176 −0.735530
\(870\) 0 0
\(871\) − 1764.90i − 2.02629i
\(872\) 0 0
\(873\) − 1142.06i − 1.30820i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −778.593 −0.887791 −0.443896 0.896078i \(-0.646404\pi\)
−0.443896 + 0.896078i \(0.646404\pi\)
\(878\) 0 0
\(879\) −64.2111 −0.0730502
\(880\) 0 0
\(881\) 1206.76i 1.36976i 0.728657 + 0.684879i \(0.240145\pi\)
−0.728657 + 0.684879i \(0.759855\pi\)
\(882\) 0 0
\(883\) 1132.77 1.28287 0.641435 0.767178i \(-0.278340\pi\)
0.641435 + 0.767178i \(0.278340\pi\)
\(884\) 0 0
\(885\) 123.130i 0.139130i
\(886\) 0 0
\(887\) − 728.824i − 0.821673i −0.911709 0.410836i \(-0.865237\pi\)
0.911709 0.410836i \(-0.134763\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1244.87 −1.39717
\(892\) 0 0
\(893\) 2432.14 2.72356
\(894\) 0 0
\(895\) − 133.966i − 0.149683i
\(896\) 0 0
\(897\) 166.453 0.185566
\(898\) 0 0
\(899\) 421.252i 0.468578i
\(900\) 0 0
\(901\) 24.5368i 0.0272329i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 887.417 0.980571
\(906\) 0 0
\(907\) −9.61818 −0.0106044 −0.00530220 0.999986i \(-0.501688\pi\)
−0.00530220 + 0.999986i \(0.501688\pi\)
\(908\) 0 0
\(909\) 892.374i 0.981709i
\(910\) 0 0
\(911\) 1015.41 1.11461 0.557304 0.830309i \(-0.311836\pi\)
0.557304 + 0.830309i \(0.311836\pi\)
\(912\) 0 0
\(913\) − 1137.62i − 1.24602i
\(914\) 0 0
\(915\) 97.5801i 0.106645i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 483.015 0.525588 0.262794 0.964852i \(-0.415356\pi\)
0.262794 + 0.964852i \(0.415356\pi\)
\(920\) 0 0
\(921\) 120.080 0.130380
\(922\) 0 0
\(923\) − 331.223i − 0.358855i
\(924\) 0 0
\(925\) 230.432 0.249116
\(926\) 0 0
\(927\) − 437.865i − 0.472346i
\(928\) 0 0
\(929\) 942.621i 1.01466i 0.861751 + 0.507331i \(0.169368\pi\)
−0.861751 + 0.507331i \(0.830632\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −65.2465 −0.0699319
\(934\) 0 0
\(935\) −85.8091 −0.0917744
\(936\) 0 0
\(937\) 1489.44i 1.58958i 0.606882 + 0.794792i \(0.292420\pi\)
−0.606882 + 0.794792i \(0.707580\pi\)
\(938\) 0 0
\(939\) 143.286 0.152595
\(940\) 0 0
\(941\) 1098.56i 1.16744i 0.811955 + 0.583720i \(0.198403\pi\)
−0.811955 + 0.583720i \(0.801597\pi\)
\(942\) 0 0
\(943\) − 450.861i − 0.478113i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 304.905 0.321969 0.160984 0.986957i \(-0.448533\pi\)
0.160984 + 0.986957i \(0.448533\pi\)
\(948\) 0 0
\(949\) −1420.57 −1.49692
\(950\) 0 0
\(951\) − 26.4357i − 0.0277978i
\(952\) 0 0
\(953\) −803.578 −0.843209 −0.421604 0.906780i \(-0.638533\pi\)
−0.421604 + 0.906780i \(0.638533\pi\)
\(954\) 0 0
\(955\) 1393.61i 1.45928i
\(956\) 0 0
\(957\) − 140.122i − 0.146418i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 731.372 0.761053
\(962\) 0 0
\(963\) 263.407 0.273528
\(964\) 0 0
\(965\) − 326.275i − 0.338109i
\(966\) 0 0
\(967\) −93.6182 −0.0968130 −0.0484065 0.998828i \(-0.515414\pi\)
−0.0484065 + 0.998828i \(0.515414\pi\)
\(968\) 0 0
\(969\) 9.60635i 0.00991367i
\(970\) 0 0
\(971\) 419.447i 0.431974i 0.976396 + 0.215987i \(0.0692969\pi\)
−0.976396 + 0.215987i \(0.930703\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −46.1010 −0.0472831
\(976\) 0 0
\(977\) −1679.65 −1.71919 −0.859595 0.510977i \(-0.829284\pi\)
−0.859595 + 0.510977i \(0.829284\pi\)
\(978\) 0 0
\(979\) − 1229.36i − 1.25573i
\(980\) 0 0
\(981\) −99.6384 −0.101568
\(982\) 0 0
\(983\) − 266.359i − 0.270965i −0.990780 0.135483i \(-0.956742\pi\)
0.990780 0.135483i \(-0.0432585\pi\)
\(984\) 0 0
\(985\) − 989.661i − 1.00473i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1541.71 −1.55886
\(990\) 0 0
\(991\) 894.613 0.902738 0.451369 0.892337i \(-0.350936\pi\)
0.451369 + 0.892337i \(0.350936\pi\)
\(992\) 0 0
\(993\) 48.6594i 0.0490024i
\(994\) 0 0
\(995\) 952.000 0.956784
\(996\) 0 0
\(997\) − 638.884i − 0.640806i −0.947281 0.320403i \(-0.896182\pi\)
0.947281 0.320403i \(-0.103818\pi\)
\(998\) 0 0
\(999\) − 181.587i − 0.181769i
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 196.3.b.b.97.3 yes 4
3.2 odd 2 1764.3.d.e.685.3 4
4.3 odd 2 784.3.c.d.97.2 4
7.2 even 3 196.3.h.c.129.2 8
7.3 odd 6 196.3.h.c.117.2 8
7.4 even 3 196.3.h.c.117.3 8
7.5 odd 6 196.3.h.c.129.3 8
7.6 odd 2 inner 196.3.b.b.97.2 4
21.2 odd 6 1764.3.z.k.325.3 8
21.5 even 6 1764.3.z.k.325.2 8
21.11 odd 6 1764.3.z.k.901.2 8
21.17 even 6 1764.3.z.k.901.3 8
21.20 even 2 1764.3.d.e.685.2 4
28.3 even 6 784.3.s.g.705.3 8
28.11 odd 6 784.3.s.g.705.2 8
28.19 even 6 784.3.s.g.129.2 8
28.23 odd 6 784.3.s.g.129.3 8
28.27 even 2 784.3.c.d.97.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
196.3.b.b.97.2 4 7.6 odd 2 inner
196.3.b.b.97.3 yes 4 1.1 even 1 trivial
196.3.h.c.117.2 8 7.3 odd 6
196.3.h.c.117.3 8 7.4 even 3
196.3.h.c.129.2 8 7.2 even 3
196.3.h.c.129.3 8 7.5 odd 6
784.3.c.d.97.2 4 4.3 odd 2
784.3.c.d.97.3 4 28.27 even 2
784.3.s.g.129.2 8 28.19 even 6
784.3.s.g.129.3 8 28.23 odd 6
784.3.s.g.705.2 8 28.11 odd 6
784.3.s.g.705.3 8 28.3 even 6
1764.3.d.e.685.2 4 21.20 even 2
1764.3.d.e.685.3 4 3.2 odd 2
1764.3.z.k.325.2 8 21.5 even 6
1764.3.z.k.325.3 8 21.2 odd 6
1764.3.z.k.901.2 8 21.11 odd 6
1764.3.z.k.901.3 8 21.17 even 6