Properties

Label 4368.2.h.o.337.6
Level $4368$
Weight $2$
Character 4368.337
Analytic conductor $34.879$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4368,2,Mod(337,4368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4368.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.h (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: \(34.8786556029\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 273)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.6
Root \(0.403032 - 0.403032i\) of defining polynomial
Character \(\chi\) \(=\) 4368.337
Dual form 4368.2.h.o.337.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +4.15633i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +4.15633i q^{5} -1.00000i q^{7} +1.00000 q^{9} -3.19394i q^{11} +(1.48119 - 3.28726i) q^{13} -4.15633i q^{15} +3.35026 q^{17} +2.38787i q^{19} +1.00000i q^{21} -0.387873 q^{23} -12.2750 q^{25} -1.00000 q^{27} -7.92478 q^{29} +10.7005i q^{31} +3.19394i q^{33} +4.15633 q^{35} +1.61213i q^{37} +(-1.48119 + 3.28726i) q^{39} -1.45580i q^{41} -1.92478 q^{43} +4.15633i q^{45} -3.76845i q^{47} -1.00000 q^{49} -3.35026 q^{51} -6.00000 q^{53} +13.2750 q^{55} -2.38787i q^{57} -6.15633i q^{59} +14.4387 q^{61} -1.00000i q^{63} +(13.6629 + 6.15633i) q^{65} +5.61213i q^{67} +0.387873 q^{69} +11.8192i q^{71} +15.6629i q^{73} +12.2750 q^{75} -3.19394 q^{77} +8.96239 q^{79} +1.00000 q^{81} +6.99271i q^{83} +13.9248i q^{85} +7.92478 q^{87} -0.932071i q^{89} +(-3.28726 - 1.48119i) q^{91} -10.7005i q^{93} -9.92478 q^{95} +3.35026i q^{97} -3.19394i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{3} + 6 q^{9} - 2 q^{13} - 4 q^{23} - 10 q^{25} - 6 q^{27} - 4 q^{29} + 4 q^{35} + 2 q^{39} + 32 q^{43} - 6 q^{49} - 36 q^{53} + 16 q^{55} + 28 q^{61} + 20 q^{65} + 4 q^{69} + 10 q^{75} - 20 q^{77} + 32 q^{79} + 6 q^{81} + 4 q^{87} - 8 q^{91} - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times\).

\(n\) \(1093\) \(1249\) \(1457\) \(2017\) \(3823\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 4.15633i 1.85877i 0.369118 + 0.929383i \(0.379660\pi\)
−0.369118 + 0.929383i \(0.620340\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.19394i 0.963008i −0.876444 0.481504i \(-0.840091\pi\)
0.876444 0.481504i \(-0.159909\pi\)
\(12\) 0 0
\(13\) 1.48119 3.28726i 0.410809 0.911721i
\(14\) 0 0
\(15\) 4.15633i 1.07316i
\(16\) 0 0
\(17\) 3.35026 0.812558 0.406279 0.913749i \(-0.366826\pi\)
0.406279 + 0.913749i \(0.366826\pi\)
\(18\) 0 0
\(19\) 2.38787i 0.547816i 0.961756 + 0.273908i \(0.0883163\pi\)
−0.961756 + 0.273908i \(0.911684\pi\)
\(20\) 0 0
\(21\) 1.00000i 0.218218i
\(22\) 0 0
\(23\) −0.387873 −0.0808771 −0.0404386 0.999182i \(-0.512876\pi\)
−0.0404386 + 0.999182i \(0.512876\pi\)
\(24\) 0 0
\(25\) −12.2750 −2.45501
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −7.92478 −1.47159 −0.735797 0.677202i \(-0.763192\pi\)
−0.735797 + 0.677202i \(0.763192\pi\)
\(30\) 0 0
\(31\) 10.7005i 1.92187i 0.276773 + 0.960935i \(0.410735\pi\)
−0.276773 + 0.960935i \(0.589265\pi\)
\(32\) 0 0
\(33\) 3.19394i 0.555993i
\(34\) 0 0
\(35\) 4.15633 0.702547
\(36\) 0 0
\(37\) 1.61213i 0.265032i 0.991181 + 0.132516i \(0.0423056\pi\)
−0.991181 + 0.132516i \(0.957694\pi\)
\(38\) 0 0
\(39\) −1.48119 + 3.28726i −0.237181 + 0.526383i
\(40\) 0 0
\(41\) 1.45580i 0.227358i −0.993518 0.113679i \(-0.963736\pi\)
0.993518 0.113679i \(-0.0362635\pi\)
\(42\) 0 0
\(43\) −1.92478 −0.293526 −0.146763 0.989172i \(-0.546885\pi\)
−0.146763 + 0.989172i \(0.546885\pi\)
\(44\) 0 0
\(45\) 4.15633i 0.619588i
\(46\) 0 0
\(47\) 3.76845i 0.549685i −0.961489 0.274843i \(-0.911374\pi\)
0.961489 0.274843i \(-0.0886258\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −3.35026 −0.469130
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 13.2750 1.79001
\(56\) 0 0
\(57\) 2.38787i 0.316282i
\(58\) 0 0
\(59\) 6.15633i 0.801485i −0.916191 0.400743i \(-0.868752\pi\)
0.916191 0.400743i \(-0.131248\pi\)
\(60\) 0 0
\(61\) 14.4387 1.84868 0.924340 0.381569i \(-0.124616\pi\)
0.924340 + 0.381569i \(0.124616\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 13.6629 + 6.15633i 1.69468 + 0.763598i
\(66\) 0 0
\(67\) 5.61213i 0.685630i 0.939403 + 0.342815i \(0.111380\pi\)
−0.939403 + 0.342815i \(0.888620\pi\)
\(68\) 0 0
\(69\) 0.387873 0.0466944
\(70\) 0 0
\(71\) 11.8192i 1.40269i 0.712824 + 0.701343i \(0.247416\pi\)
−0.712824 + 0.701343i \(0.752584\pi\)
\(72\) 0 0
\(73\) 15.6629i 1.83321i 0.399800 + 0.916603i \(0.369080\pi\)
−0.399800 + 0.916603i \(0.630920\pi\)
\(74\) 0 0
\(75\) 12.2750 1.41740
\(76\) 0 0
\(77\) −3.19394 −0.363983
\(78\) 0 0
\(79\) 8.96239 1.00835 0.504174 0.863602i \(-0.331797\pi\)
0.504174 + 0.863602i \(0.331797\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.99271i 0.767549i 0.923427 + 0.383775i \(0.125376\pi\)
−0.923427 + 0.383775i \(0.874624\pi\)
\(84\) 0 0
\(85\) 13.9248i 1.51035i
\(86\) 0 0
\(87\) 7.92478 0.849625
\(88\) 0 0
\(89\) 0.932071i 0.0987994i −0.998779 0.0493997i \(-0.984269\pi\)
0.998779 0.0493997i \(-0.0157308\pi\)
\(90\) 0 0
\(91\) −3.28726 1.48119i −0.344598 0.155271i
\(92\) 0 0
\(93\) 10.7005i 1.10959i
\(94\) 0 0
\(95\) −9.92478 −1.01826
\(96\) 0 0
\(97\) 3.35026i 0.340168i 0.985430 + 0.170084i \(0.0544038\pi\)
−0.985430 + 0.170084i \(0.945596\pi\)
\(98\) 0 0
\(99\) 3.19394i 0.321003i
\(100\) 0 0
\(101\) −14.0508 −1.39811 −0.699053 0.715070i \(-0.746395\pi\)
−0.699053 + 0.715070i \(0.746395\pi\)
\(102\) 0 0
\(103\) −1.42548 −0.140457 −0.0702286 0.997531i \(-0.522373\pi\)
−0.0702286 + 0.997531i \(0.522373\pi\)
\(104\) 0 0
\(105\) −4.15633 −0.405616
\(106\) 0 0
\(107\) −9.53690 −0.921967 −0.460984 0.887409i \(-0.652503\pi\)
−0.460984 + 0.887409i \(0.652503\pi\)
\(108\) 0 0
\(109\) 6.23743i 0.597437i 0.954341 + 0.298719i \(0.0965592\pi\)
−0.954341 + 0.298719i \(0.903441\pi\)
\(110\) 0 0
\(111\) 1.61213i 0.153016i
\(112\) 0 0
\(113\) −3.92478 −0.369212 −0.184606 0.982813i \(-0.559101\pi\)
−0.184606 + 0.982813i \(0.559101\pi\)
\(114\) 0 0
\(115\) 1.61213i 0.150332i
\(116\) 0 0
\(117\) 1.48119 3.28726i 0.136936 0.303907i
\(118\) 0 0
\(119\) 3.35026i 0.307118i
\(120\) 0 0
\(121\) 0.798769 0.0726154
\(122\) 0 0
\(123\) 1.45580i 0.131265i
\(124\) 0 0
\(125\) 30.2374i 2.70452i
\(126\) 0 0
\(127\) 0.962389 0.0853982 0.0426991 0.999088i \(-0.486404\pi\)
0.0426991 + 0.999088i \(0.486404\pi\)
\(128\) 0 0
\(129\) 1.92478 0.169467
\(130\) 0 0
\(131\) −17.1490 −1.49832 −0.749159 0.662390i \(-0.769542\pi\)
−0.749159 + 0.662390i \(0.769542\pi\)
\(132\) 0 0
\(133\) 2.38787 0.207055
\(134\) 0 0
\(135\) 4.15633i 0.357720i
\(136\) 0 0
\(137\) 4.85685i 0.414949i −0.978240 0.207474i \(-0.933476\pi\)
0.978240 0.207474i \(-0.0665243\pi\)
\(138\) 0 0
\(139\) −2.57452 −0.218368 −0.109184 0.994022i \(-0.534824\pi\)
−0.109184 + 0.994022i \(0.534824\pi\)
\(140\) 0 0
\(141\) 3.76845i 0.317361i
\(142\) 0 0
\(143\) −10.4993 4.73084i −0.877995 0.395613i
\(144\) 0 0
\(145\) 32.9380i 2.73535i
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 13.3806i 1.09618i 0.836419 + 0.548090i \(0.184645\pi\)
−0.836419 + 0.548090i \(0.815355\pi\)
\(150\) 0 0
\(151\) 15.8496i 1.28982i 0.764259 + 0.644909i \(0.223105\pi\)
−0.764259 + 0.644909i \(0.776895\pi\)
\(152\) 0 0
\(153\) 3.35026 0.270853
\(154\) 0 0
\(155\) −44.4749 −3.57231
\(156\) 0 0
\(157\) −12.7005 −1.01361 −0.506806 0.862060i \(-0.669174\pi\)
−0.506806 + 0.862060i \(0.669174\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0.387873i 0.0305687i
\(162\) 0 0
\(163\) 9.61213i 0.752880i −0.926441 0.376440i \(-0.877148\pi\)
0.926441 0.376440i \(-0.122852\pi\)
\(164\) 0 0
\(165\) −13.2750 −1.03346
\(166\) 0 0
\(167\) 12.0811i 0.934864i 0.884029 + 0.467432i \(0.154821\pi\)
−0.884029 + 0.467432i \(0.845179\pi\)
\(168\) 0 0
\(169\) −8.61213 9.73813i −0.662471 0.749087i
\(170\) 0 0
\(171\) 2.38787i 0.182605i
\(172\) 0 0
\(173\) −9.27504 −0.705168 −0.352584 0.935780i \(-0.614697\pi\)
−0.352584 + 0.935780i \(0.614697\pi\)
\(174\) 0 0
\(175\) 12.2750i 0.927906i
\(176\) 0 0
\(177\) 6.15633i 0.462738i
\(178\) 0 0
\(179\) 5.01317 0.374702 0.187351 0.982293i \(-0.440010\pi\)
0.187351 + 0.982293i \(0.440010\pi\)
\(180\) 0 0
\(181\) −13.8496 −1.02943 −0.514715 0.857362i \(-0.672102\pi\)
−0.514715 + 0.857362i \(0.672102\pi\)
\(182\) 0 0
\(183\) −14.4387 −1.06734
\(184\) 0 0
\(185\) −6.70052 −0.492632
\(186\) 0 0
\(187\) 10.7005i 0.782500i
\(188\) 0 0
\(189\) 1.00000i 0.0727393i
\(190\) 0 0
\(191\) 18.5647 1.34329 0.671646 0.740872i \(-0.265588\pi\)
0.671646 + 0.740872i \(0.265588\pi\)
\(192\) 0 0
\(193\) 13.0884i 0.942123i 0.882100 + 0.471062i \(0.156129\pi\)
−0.882100 + 0.471062i \(0.843871\pi\)
\(194\) 0 0
\(195\) −13.6629 6.15633i −0.978421 0.440864i
\(196\) 0 0
\(197\) 7.24472i 0.516165i 0.966123 + 0.258083i \(0.0830906\pi\)
−0.966123 + 0.258083i \(0.916909\pi\)
\(198\) 0 0
\(199\) −8.62530 −0.611431 −0.305716 0.952123i \(-0.598896\pi\)
−0.305716 + 0.952123i \(0.598896\pi\)
\(200\) 0 0
\(201\) 5.61213i 0.395849i
\(202\) 0 0
\(203\) 7.92478i 0.556210i
\(204\) 0 0
\(205\) 6.05079 0.422605
\(206\) 0 0
\(207\) −0.387873 −0.0269590
\(208\) 0 0
\(209\) 7.62672 0.527551
\(210\) 0 0
\(211\) −18.8872 −1.30025 −0.650123 0.759829i \(-0.725283\pi\)
−0.650123 + 0.759829i \(0.725283\pi\)
\(212\) 0 0
\(213\) 11.8192i 0.809841i
\(214\) 0 0
\(215\) 8.00000i 0.545595i
\(216\) 0 0
\(217\) 10.7005 0.726399
\(218\) 0 0
\(219\) 15.6629i 1.05840i
\(220\) 0 0
\(221\) 4.96239 11.0132i 0.333806 0.740826i
\(222\) 0 0
\(223\) 16.1622i 1.08230i −0.840926 0.541151i \(-0.817989\pi\)
0.840926 0.541151i \(-0.182011\pi\)
\(224\) 0 0
\(225\) −12.2750 −0.818336
\(226\) 0 0
\(227\) 27.2447i 1.80830i −0.427220 0.904148i \(-0.640507\pi\)
0.427220 0.904148i \(-0.359493\pi\)
\(228\) 0 0
\(229\) 19.5125i 1.28942i 0.764427 + 0.644710i \(0.223022\pi\)
−0.764427 + 0.644710i \(0.776978\pi\)
\(230\) 0 0
\(231\) 3.19394 0.210146
\(232\) 0 0
\(233\) −21.8496 −1.43141 −0.715706 0.698402i \(-0.753895\pi\)
−0.715706 + 0.698402i \(0.753895\pi\)
\(234\) 0 0
\(235\) 15.6629 1.02174
\(236\) 0 0
\(237\) −8.96239 −0.582170
\(238\) 0 0
\(239\) 8.49341i 0.549393i 0.961531 + 0.274697i \(0.0885774\pi\)
−0.961531 + 0.274697i \(0.911423\pi\)
\(240\) 0 0
\(241\) 5.48612i 0.353392i 0.984265 + 0.176696i \(0.0565409\pi\)
−0.984265 + 0.176696i \(0.943459\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 4.15633i 0.265538i
\(246\) 0 0
\(247\) 7.84955 + 3.53690i 0.499455 + 0.225048i
\(248\) 0 0
\(249\) 6.99271i 0.443145i
\(250\) 0 0
\(251\) −6.44851 −0.407026 −0.203513 0.979072i \(-0.565236\pi\)
−0.203513 + 0.979072i \(0.565236\pi\)
\(252\) 0 0
\(253\) 1.23884i 0.0778853i
\(254\) 0 0
\(255\) 13.9248i 0.872003i
\(256\) 0 0
\(257\) 9.90034 0.617566 0.308783 0.951132i \(-0.400078\pi\)
0.308783 + 0.951132i \(0.400078\pi\)
\(258\) 0 0
\(259\) 1.61213 0.100173
\(260\) 0 0
\(261\) −7.92478 −0.490531
\(262\) 0 0
\(263\) −3.46168 −0.213456 −0.106728 0.994288i \(-0.534037\pi\)
−0.106728 + 0.994288i \(0.534037\pi\)
\(264\) 0 0
\(265\) 24.9380i 1.53193i
\(266\) 0 0
\(267\) 0.932071i 0.0570418i
\(268\) 0 0
\(269\) 16.7513 1.02135 0.510673 0.859775i \(-0.329396\pi\)
0.510673 + 0.859775i \(0.329396\pi\)
\(270\) 0 0
\(271\) 6.38787i 0.388036i 0.980998 + 0.194018i \(0.0621520\pi\)
−0.980998 + 0.194018i \(0.937848\pi\)
\(272\) 0 0
\(273\) 3.28726 + 1.48119i 0.198954 + 0.0896460i
\(274\) 0 0
\(275\) 39.2057i 2.36419i
\(276\) 0 0
\(277\) 12.1768 0.731633 0.365816 0.930687i \(-0.380790\pi\)
0.365816 + 0.930687i \(0.380790\pi\)
\(278\) 0 0
\(279\) 10.7005i 0.640624i
\(280\) 0 0
\(281\) 27.3054i 1.62890i −0.580233 0.814450i \(-0.697039\pi\)
0.580233 0.814450i \(-0.302961\pi\)
\(282\) 0 0
\(283\) −27.0738 −1.60937 −0.804685 0.593701i \(-0.797666\pi\)
−0.804685 + 0.593701i \(0.797666\pi\)
\(284\) 0 0
\(285\) 9.92478 0.587893
\(286\) 0 0
\(287\) −1.45580 −0.0859333
\(288\) 0 0
\(289\) −5.77575 −0.339750
\(290\) 0 0
\(291\) 3.35026i 0.196396i
\(292\) 0 0
\(293\) 16.7816i 0.980393i 0.871612 + 0.490197i \(0.163075\pi\)
−0.871612 + 0.490197i \(0.836925\pi\)
\(294\) 0 0
\(295\) 25.5877 1.48977
\(296\) 0 0
\(297\) 3.19394i 0.185331i
\(298\) 0 0
\(299\) −0.574515 + 1.27504i −0.0332251 + 0.0737374i
\(300\) 0 0
\(301\) 1.92478i 0.110942i
\(302\) 0 0
\(303\) 14.0508 0.807197
\(304\) 0 0
\(305\) 60.0118i 3.43626i
\(306\) 0 0
\(307\) 17.2995i 0.987333i −0.869651 0.493667i \(-0.835656\pi\)
0.869651 0.493667i \(-0.164344\pi\)
\(308\) 0 0
\(309\) 1.42548 0.0810930
\(310\) 0 0
\(311\) −17.1490 −0.972432 −0.486216 0.873839i \(-0.661623\pi\)
−0.486216 + 0.873839i \(0.661623\pi\)
\(312\) 0 0
\(313\) −10.8119 −0.611127 −0.305564 0.952172i \(-0.598845\pi\)
−0.305564 + 0.952172i \(0.598845\pi\)
\(314\) 0 0
\(315\) 4.15633 0.234182
\(316\) 0 0
\(317\) 14.1563i 0.795098i −0.917581 0.397549i \(-0.869861\pi\)
0.917581 0.397549i \(-0.130139\pi\)
\(318\) 0 0
\(319\) 25.3112i 1.41716i
\(320\) 0 0
\(321\) 9.53690 0.532298
\(322\) 0 0
\(323\) 8.00000i 0.445132i
\(324\) 0 0
\(325\) −18.1817 + 40.3512i −1.00854 + 2.23828i
\(326\) 0 0
\(327\) 6.23743i 0.344931i
\(328\) 0 0
\(329\) −3.76845 −0.207761
\(330\) 0 0
\(331\) 17.2506i 0.948179i −0.880477 0.474089i \(-0.842777\pi\)
0.880477 0.474089i \(-0.157223\pi\)
\(332\) 0 0
\(333\) 1.61213i 0.0883440i
\(334\) 0 0
\(335\) −23.3258 −1.27443
\(336\) 0 0
\(337\) −5.97556 −0.325510 −0.162755 0.986667i \(-0.552038\pi\)
−0.162755 + 0.986667i \(0.552038\pi\)
\(338\) 0 0
\(339\) 3.92478 0.213165
\(340\) 0 0
\(341\) 34.1768 1.85078
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 1.61213i 0.0867940i
\(346\) 0 0
\(347\) −25.7889 −1.38442 −0.692211 0.721695i \(-0.743363\pi\)
−0.692211 + 0.721695i \(0.743363\pi\)
\(348\) 0 0
\(349\) 12.9624i 0.693861i 0.937891 + 0.346930i \(0.112776\pi\)
−0.937891 + 0.346930i \(0.887224\pi\)
\(350\) 0 0
\(351\) −1.48119 + 3.28726i −0.0790603 + 0.175461i
\(352\) 0 0
\(353\) 3.78304i 0.201351i −0.994919 0.100675i \(-0.967900\pi\)
0.994919 0.100675i \(-0.0321004\pi\)
\(354\) 0 0
\(355\) −49.1246 −2.60726
\(356\) 0 0
\(357\) 3.35026i 0.177315i
\(358\) 0 0
\(359\) 10.6702i 0.563152i −0.959539 0.281576i \(-0.909143\pi\)
0.959539 0.281576i \(-0.0908571\pi\)
\(360\) 0 0
\(361\) 13.2981 0.699898
\(362\) 0 0
\(363\) −0.798769 −0.0419245
\(364\) 0 0
\(365\) −65.1002 −3.40750
\(366\) 0 0
\(367\) −6.82653 −0.356342 −0.178171 0.984000i \(-0.557018\pi\)
−0.178171 + 0.984000i \(0.557018\pi\)
\(368\) 0 0
\(369\) 1.45580i 0.0757860i
\(370\) 0 0
\(371\) 6.00000i 0.311504i
\(372\) 0 0
\(373\) 16.5745 0.858196 0.429098 0.903258i \(-0.358832\pi\)
0.429098 + 0.903258i \(0.358832\pi\)
\(374\) 0 0
\(375\) 30.2374i 1.56145i
\(376\) 0 0
\(377\) −11.7381 + 26.0508i −0.604545 + 1.34168i
\(378\) 0 0
\(379\) 19.0738i 0.979756i 0.871791 + 0.489878i \(0.162959\pi\)
−0.871791 + 0.489878i \(0.837041\pi\)
\(380\) 0 0
\(381\) −0.962389 −0.0493047
\(382\) 0 0
\(383\) 8.29218i 0.423711i −0.977301 0.211855i \(-0.932049\pi\)
0.977301 0.211855i \(-0.0679506\pi\)
\(384\) 0 0
\(385\) 13.2750i 0.676559i
\(386\) 0 0
\(387\) −1.92478 −0.0978419
\(388\) 0 0
\(389\) 0.700523 0.0355180 0.0177590 0.999842i \(-0.494347\pi\)
0.0177590 + 0.999842i \(0.494347\pi\)
\(390\) 0 0
\(391\) −1.29948 −0.0657174
\(392\) 0 0
\(393\) 17.1490 0.865054
\(394\) 0 0
\(395\) 37.2506i 1.87428i
\(396\) 0 0
\(397\) 12.4993i 0.627322i −0.949535 0.313661i \(-0.898445\pi\)
0.949535 0.313661i \(-0.101555\pi\)
\(398\) 0 0
\(399\) −2.38787 −0.119543
\(400\) 0 0
\(401\) 28.7064i 1.43353i 0.697315 + 0.716765i \(0.254378\pi\)
−0.697315 + 0.716765i \(0.745622\pi\)
\(402\) 0 0
\(403\) 35.1754 + 15.8496i 1.75221 + 0.789523i
\(404\) 0 0
\(405\) 4.15633i 0.206529i
\(406\) 0 0
\(407\) 5.14903 0.255228
\(408\) 0 0
\(409\) 23.4518i 1.15962i −0.814752 0.579809i \(-0.803127\pi\)
0.814752 0.579809i \(-0.196873\pi\)
\(410\) 0 0
\(411\) 4.85685i 0.239571i
\(412\) 0 0
\(413\) −6.15633 −0.302933
\(414\) 0 0
\(415\) −29.0640 −1.42669
\(416\) 0 0
\(417\) 2.57452 0.126075
\(418\) 0 0
\(419\) 34.1768 1.66965 0.834823 0.550519i \(-0.185570\pi\)
0.834823 + 0.550519i \(0.185570\pi\)
\(420\) 0 0
\(421\) 5.55149i 0.270563i 0.990807 + 0.135282i \(0.0431939\pi\)
−0.990807 + 0.135282i \(0.956806\pi\)
\(422\) 0 0
\(423\) 3.76845i 0.183228i
\(424\) 0 0
\(425\) −41.1246 −1.99484
\(426\) 0 0
\(427\) 14.4387i 0.698736i
\(428\) 0 0
\(429\) 10.4993 + 4.73084i 0.506911 + 0.228407i
\(430\) 0 0
\(431\) 5.43136i 0.261620i 0.991407 + 0.130810i \(0.0417577\pi\)
−0.991407 + 0.130810i \(0.958242\pi\)
\(432\) 0 0
\(433\) 40.2130 1.93251 0.966256 0.257582i \(-0.0829257\pi\)
0.966256 + 0.257582i \(0.0829257\pi\)
\(434\) 0 0
\(435\) 32.9380i 1.57925i
\(436\) 0 0
\(437\) 0.926192i 0.0443058i
\(438\) 0 0
\(439\) 22.5745 1.07742 0.538711 0.842490i \(-0.318911\pi\)
0.538711 + 0.842490i \(0.318911\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) 8.23743 0.391372 0.195686 0.980667i \(-0.437307\pi\)
0.195686 + 0.980667i \(0.437307\pi\)
\(444\) 0 0
\(445\) 3.87399 0.183645
\(446\) 0 0
\(447\) 13.3806i 0.632880i
\(448\) 0 0
\(449\) 2.40834i 0.113657i 0.998384 + 0.0568283i \(0.0180988\pi\)
−0.998384 + 0.0568283i \(0.981901\pi\)
\(450\) 0 0
\(451\) −4.64974 −0.218948
\(452\) 0 0
\(453\) 15.8496i 0.744677i
\(454\) 0 0
\(455\) 6.15633 13.6629i 0.288613 0.640527i
\(456\) 0 0
\(457\) 29.5633i 1.38291i 0.722419 + 0.691455i \(0.243030\pi\)
−0.722419 + 0.691455i \(0.756970\pi\)
\(458\) 0 0
\(459\) −3.35026 −0.156377
\(460\) 0 0
\(461\) 33.7196i 1.57048i −0.619193 0.785239i \(-0.712540\pi\)
0.619193 0.785239i \(-0.287460\pi\)
\(462\) 0 0
\(463\) 16.8364i 0.782453i −0.920294 0.391226i \(-0.872051\pi\)
0.920294 0.391226i \(-0.127949\pi\)
\(464\) 0 0
\(465\) 44.4749 2.06247
\(466\) 0 0
\(467\) 22.7005 1.05045 0.525227 0.850962i \(-0.323980\pi\)
0.525227 + 0.850962i \(0.323980\pi\)
\(468\) 0 0
\(469\) 5.61213 0.259144
\(470\) 0 0
\(471\) 12.7005 0.585209
\(472\) 0 0
\(473\) 6.14762i 0.282668i
\(474\) 0 0
\(475\) 29.3112i 1.34489i
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 10.0957i 0.461284i 0.973039 + 0.230642i \(0.0740826\pi\)
−0.973039 + 0.230642i \(0.925917\pi\)
\(480\) 0 0
\(481\) 5.29948 + 2.38787i 0.241635 + 0.108878i
\(482\) 0 0
\(483\) 0.387873i 0.0176488i
\(484\) 0 0
\(485\) −13.9248 −0.632292
\(486\) 0 0
\(487\) 9.40105i 0.426002i −0.977052 0.213001i \(-0.931676\pi\)
0.977052 0.213001i \(-0.0683238\pi\)
\(488\) 0 0
\(489\) 9.61213i 0.434675i
\(490\) 0 0
\(491\) −20.7612 −0.936938 −0.468469 0.883480i \(-0.655194\pi\)
−0.468469 + 0.883480i \(0.655194\pi\)
\(492\) 0 0
\(493\) −26.5501 −1.19576
\(494\) 0 0
\(495\) 13.2750 0.596669
\(496\) 0 0
\(497\) 11.8192 0.530165
\(498\) 0 0
\(499\) 22.9525i 1.02750i 0.857941 + 0.513748i \(0.171744\pi\)
−0.857941 + 0.513748i \(0.828256\pi\)
\(500\) 0 0
\(501\) 12.0811i 0.539744i
\(502\) 0 0
\(503\) 34.9234 1.55716 0.778578 0.627548i \(-0.215941\pi\)
0.778578 + 0.627548i \(0.215941\pi\)
\(504\) 0 0
\(505\) 58.3996i 2.59875i
\(506\) 0 0
\(507\) 8.61213 + 9.73813i 0.382478 + 0.432486i
\(508\) 0 0
\(509\) 19.2301i 0.852361i 0.904638 + 0.426180i \(0.140141\pi\)
−0.904638 + 0.426180i \(0.859859\pi\)
\(510\) 0 0
\(511\) 15.6629 0.692886
\(512\) 0 0
\(513\) 2.38787i 0.105427i
\(514\) 0 0
\(515\) 5.92478i 0.261077i
\(516\) 0 0
\(517\) −12.0362 −0.529351
\(518\) 0 0
\(519\) 9.27504 0.407129
\(520\) 0 0
\(521\) 21.5271 0.943117 0.471559 0.881835i \(-0.343692\pi\)
0.471559 + 0.881835i \(0.343692\pi\)
\(522\) 0 0
\(523\) −39.6747 −1.73485 −0.867426 0.497566i \(-0.834227\pi\)
−0.867426 + 0.497566i \(0.834227\pi\)
\(524\) 0 0
\(525\) 12.2750i 0.535727i
\(526\) 0 0
\(527\) 35.8496i 1.56163i
\(528\) 0 0
\(529\) −22.8496 −0.993459
\(530\) 0 0
\(531\) 6.15633i 0.267162i
\(532\) 0 0
\(533\) −4.78560 2.15633i −0.207287 0.0934008i
\(534\) 0 0
\(535\) 39.6385i 1.71372i
\(536\) 0 0
\(537\) −5.01317 −0.216334
\(538\) 0 0
\(539\) 3.19394i 0.137573i
\(540\) 0 0
\(541\) 10.0263i 0.431066i −0.976497 0.215533i \(-0.930851\pi\)
0.976497 0.215533i \(-0.0691489\pi\)
\(542\) 0 0
\(543\) 13.8496 0.594341
\(544\) 0 0
\(545\) −25.9248 −1.11050
\(546\) 0 0
\(547\) 26.4485 1.13086 0.565428 0.824797i \(-0.308711\pi\)
0.565428 + 0.824797i \(0.308711\pi\)
\(548\) 0 0
\(549\) 14.4387 0.616227
\(550\) 0 0
\(551\) 18.9234i 0.806162i
\(552\) 0 0
\(553\) 8.96239i 0.381120i
\(554\) 0 0
\(555\) 6.70052 0.284421
\(556\) 0 0
\(557\) 5.84367i 0.247604i 0.992307 + 0.123802i \(0.0395088\pi\)
−0.992307 + 0.123802i \(0.960491\pi\)
\(558\) 0 0
\(559\) −2.85097 + 6.32724i −0.120583 + 0.267614i
\(560\) 0 0
\(561\) 10.7005i 0.451776i
\(562\) 0 0
\(563\) −1.25060 −0.0527066 −0.0263533 0.999653i \(-0.508389\pi\)
−0.0263533 + 0.999653i \(0.508389\pi\)
\(564\) 0 0
\(565\) 16.3127i 0.686278i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) −38.2228 −1.60238 −0.801192 0.598407i \(-0.795801\pi\)
−0.801192 + 0.598407i \(0.795801\pi\)
\(570\) 0 0
\(571\) 9.73813 0.407528 0.203764 0.979020i \(-0.434682\pi\)
0.203764 + 0.979020i \(0.434682\pi\)
\(572\) 0 0
\(573\) −18.5647 −0.775550
\(574\) 0 0
\(575\) 4.76116 0.198554
\(576\) 0 0
\(577\) 3.87399i 0.161276i −0.996743 0.0806382i \(-0.974304\pi\)
0.996743 0.0806382i \(-0.0256958\pi\)
\(578\) 0 0
\(579\) 13.0884i 0.543935i
\(580\) 0 0
\(581\) 6.99271 0.290106
\(582\) 0 0
\(583\) 19.1636i 0.793676i
\(584\) 0 0
\(585\) 13.6629 + 6.15633i 0.564892 + 0.254533i
\(586\) 0 0
\(587\) 11.2447i 0.464119i 0.972702 + 0.232060i \(0.0745465\pi\)
−0.972702 + 0.232060i \(0.925454\pi\)
\(588\) 0 0
\(589\) −25.5515 −1.05283
\(590\) 0 0
\(591\) 7.24472i 0.298008i
\(592\) 0 0
\(593\) 28.7210i 1.17943i 0.807612 + 0.589715i \(0.200760\pi\)
−0.807612 + 0.589715i \(0.799240\pi\)
\(594\) 0 0
\(595\) 13.9248 0.570860
\(596\) 0 0
\(597\) 8.62530 0.353010
\(598\) 0 0
\(599\) 21.1344 0.863530 0.431765 0.901986i \(-0.357891\pi\)
0.431765 + 0.901986i \(0.357891\pi\)
\(600\) 0 0
\(601\) 18.9135 0.771498 0.385749 0.922604i \(-0.373943\pi\)
0.385749 + 0.922604i \(0.373943\pi\)
\(602\) 0 0
\(603\) 5.61213i 0.228543i
\(604\) 0 0
\(605\) 3.31994i 0.134975i
\(606\) 0 0
\(607\) 41.1246 1.66920 0.834598 0.550860i \(-0.185700\pi\)
0.834598 + 0.550860i \(0.185700\pi\)
\(608\) 0 0
\(609\) 7.92478i 0.321128i
\(610\) 0 0
\(611\) −12.3879 5.58181i −0.501160 0.225816i
\(612\) 0 0
\(613\) 23.3258i 0.942121i −0.882101 0.471061i \(-0.843871\pi\)
0.882101 0.471061i \(-0.156129\pi\)
\(614\) 0 0
\(615\) −6.05079 −0.243991
\(616\) 0 0
\(617\) 17.7830i 0.715918i 0.933737 + 0.357959i \(0.116527\pi\)
−0.933737 + 0.357959i \(0.883473\pi\)
\(618\) 0 0
\(619\) 8.93795i 0.359247i 0.983735 + 0.179623i \(0.0574879\pi\)
−0.983735 + 0.179623i \(0.942512\pi\)
\(620\) 0 0
\(621\) 0.387873 0.0155648
\(622\) 0 0
\(623\) −0.932071 −0.0373427
\(624\) 0 0
\(625\) 64.3014 2.57206
\(626\) 0 0
\(627\) −7.62672 −0.304582
\(628\) 0 0
\(629\) 5.40105i 0.215354i
\(630\) 0 0
\(631\) 32.2638i 1.28440i −0.766537 0.642200i \(-0.778022\pi\)
0.766537 0.642200i \(-0.221978\pi\)
\(632\) 0 0
\(633\) 18.8872 0.750697
\(634\) 0 0
\(635\) 4.00000i 0.158735i
\(636\) 0 0
\(637\) −1.48119 + 3.28726i −0.0586871 + 0.130246i
\(638\) 0 0
\(639\) 11.8192i 0.467562i
\(640\) 0 0
\(641\) −49.4274 −1.95226 −0.976132 0.217176i \(-0.930315\pi\)
−0.976132 + 0.217176i \(0.930315\pi\)
\(642\) 0 0
\(643\) 6.07522i 0.239583i 0.992799 + 0.119792i \(0.0382227\pi\)
−0.992799 + 0.119792i \(0.961777\pi\)
\(644\) 0 0
\(645\) 8.00000i 0.315000i
\(646\) 0 0
\(647\) 12.9986 0.511027 0.255514 0.966805i \(-0.417755\pi\)
0.255514 + 0.966805i \(0.417755\pi\)
\(648\) 0 0
\(649\) −19.6629 −0.771837
\(650\) 0 0
\(651\) −10.7005 −0.419387
\(652\) 0 0
\(653\) 15.4010 0.602690 0.301345 0.953515i \(-0.402564\pi\)
0.301345 + 0.953515i \(0.402564\pi\)
\(654\) 0 0
\(655\) 71.2769i 2.78502i
\(656\) 0 0
\(657\) 15.6629i 0.611068i
\(658\) 0 0
\(659\) −8.76116 −0.341286 −0.170643 0.985333i \(-0.554585\pi\)
−0.170643 + 0.985333i \(0.554585\pi\)
\(660\) 0 0
\(661\) 33.4372i 1.30056i 0.759695 + 0.650279i \(0.225348\pi\)
−0.759695 + 0.650279i \(0.774652\pi\)
\(662\) 0 0
\(663\) −4.96239 + 11.0132i −0.192723 + 0.427716i
\(664\) 0 0
\(665\) 9.92478i 0.384866i
\(666\) 0 0
\(667\) 3.07381 0.119018
\(668\) 0 0
\(669\) 16.1622i 0.624867i
\(670\) 0 0
\(671\) 46.1162i 1.78029i
\(672\) 0 0
\(673\) −0.176793 −0.00681488 −0.00340744 0.999994i \(-0.501085\pi\)
−0.00340744 + 0.999994i \(0.501085\pi\)
\(674\) 0 0
\(675\) 12.2750 0.472466
\(676\) 0 0
\(677\) −5.67750 −0.218204 −0.109102 0.994031i \(-0.534798\pi\)
−0.109102 + 0.994031i \(0.534798\pi\)
\(678\) 0 0
\(679\) 3.35026 0.128571
\(680\) 0 0
\(681\) 27.2447i 1.04402i
\(682\) 0 0
\(683\) 50.4299i 1.92965i 0.262896 + 0.964824i \(0.415322\pi\)
−0.262896 + 0.964824i \(0.584678\pi\)
\(684\) 0 0
\(685\) 20.1866 0.771292
\(686\) 0 0
\(687\) 19.5125i 0.744447i
\(688\) 0 0
\(689\) −8.88717 + 19.7235i −0.338574 + 0.751407i
\(690\) 0 0
\(691\) 41.5633i 1.58114i 0.612371 + 0.790570i \(0.290216\pi\)
−0.612371 + 0.790570i \(0.709784\pi\)
\(692\) 0 0
\(693\) −3.19394 −0.121328
\(694\) 0 0
\(695\) 10.7005i 0.405894i
\(696\) 0 0
\(697\) 4.87732i 0.184742i
\(698\) 0 0
\(699\) 21.8496 0.826426
\(700\) 0 0
\(701\) −2.47486 −0.0934740 −0.0467370 0.998907i \(-0.514882\pi\)
−0.0467370 + 0.998907i \(0.514882\pi\)
\(702\) 0 0
\(703\) −3.84955 −0.145189
\(704\) 0 0
\(705\) −15.6629 −0.589899
\(706\) 0 0
\(707\) 14.0508i 0.528434i
\(708\) 0 0
\(709\) 7.91019i 0.297073i 0.988907 + 0.148537i \(0.0474563\pi\)
−0.988907 + 0.148537i \(0.952544\pi\)
\(710\) 0 0
\(711\) 8.96239 0.336116
\(712\) 0 0
\(713\) 4.15045i 0.155435i
\(714\) 0 0
\(715\) 19.6629 43.6385i 0.735351 1.63199i
\(716\) 0 0
\(717\) 8.49341i 0.317192i
\(718\) 0 0
\(719\) −26.5501 −0.990151 −0.495075 0.868850i \(-0.664860\pi\)
−0.495075 + 0.868850i \(0.664860\pi\)
\(720\) 0 0
\(721\) 1.42548i 0.0530878i
\(722\) 0 0
\(723\) 5.48612i 0.204031i
\(724\) 0 0
\(725\) 97.2769 3.61278
\(726\) 0 0
\(727\) 6.44851 0.239162 0.119581 0.992824i \(-0.461845\pi\)
0.119581 + 0.992824i \(0.461845\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.44851 −0.238507
\(732\) 0 0
\(733\) 28.8119i 1.06419i −0.846684 0.532097i \(-0.821404\pi\)
0.846684 0.532097i \(-0.178596\pi\)
\(734\) 0 0
\(735\) 4.15633i 0.153308i
\(736\) 0 0
\(737\) 17.9248 0.660268
\(738\) 0 0
\(739\) 29.6121i 1.08930i 0.838664 + 0.544650i \(0.183337\pi\)
−0.838664 + 0.544650i \(0.816663\pi\)
\(740\) 0 0
\(741\) −7.84955 3.53690i −0.288361 0.129931i
\(742\) 0 0
\(743\) 3.77830i 0.138612i −0.997595 0.0693062i \(-0.977921\pi\)
0.997595 0.0693062i \(-0.0220786\pi\)
\(744\) 0 0
\(745\) −55.6140 −2.03754
\(746\) 0 0
\(747\) 6.99271i 0.255850i
\(748\) 0 0
\(749\) 9.53690i 0.348471i
\(750\) 0 0
\(751\) 10.0752 0.367650 0.183825 0.982959i \(-0.441152\pi\)
0.183825 + 0.982959i \(0.441152\pi\)
\(752\) 0 0
\(753\) 6.44851 0.234997
\(754\) 0 0
\(755\) −65.8759 −2.39747
\(756\) 0 0
\(757\) 39.4979 1.43557 0.717787 0.696262i \(-0.245155\pi\)
0.717787 + 0.696262i \(0.245155\pi\)
\(758\) 0 0
\(759\) 1.23884i 0.0449671i
\(760\) 0 0
\(761\) 25.7685i 0.934106i −0.884229 0.467053i \(-0.845316\pi\)
0.884229 0.467053i \(-0.154684\pi\)
\(762\) 0 0
\(763\) 6.23743 0.225810
\(764\) 0 0
\(765\) 13.9248i 0.503451i
\(766\) 0 0
\(767\) −20.2374 9.11871i −0.730731 0.329258i
\(768\) 0 0
\(769\) 49.4372i 1.78275i −0.453264 0.891376i \(-0.649741\pi\)
0.453264 0.891376i \(-0.350259\pi\)
\(770\) 0 0
\(771\) −9.90034 −0.356552
\(772\) 0 0
\(773\) 15.2184i 0.547367i 0.961820 + 0.273683i \(0.0882421\pi\)
−0.961820 + 0.273683i \(0.911758\pi\)
\(774\) 0 0
\(775\) 131.349i 4.71821i
\(776\) 0 0
\(777\) −1.61213 −0.0578347
\(778\) 0 0
\(779\) 3.47627 0.124550
\(780\) 0 0
\(781\) 37.7499 1.35080
\(782\) 0 0
\(783\) 7.92478 0.283208
\(784\) 0 0
\(785\) 52.7875i 1.88407i
\(786\) 0 0
\(787\) 26.6107i 0.948569i 0.880372 + 0.474285i \(0.157293\pi\)
−0.880372 + 0.474285i \(0.842707\pi\)
\(788\) 0 0
\(789\) 3.46168 0.123239
\(790\) 0 0
\(791\) 3.92478i 0.139549i
\(792\) 0 0
\(793\) 21.3865 47.4636i 0.759455 1.68548i
\(794\) 0 0
\(795\) 24.9380i 0.884458i
\(796\) 0 0
\(797\) 12.2473 0.433821 0.216910 0.976192i \(-0.430402\pi\)
0.216910 + 0.976192i \(0.430402\pi\)
\(798\) 0 0
\(799\) 12.6253i 0.446651i
\(800\) 0 0
\(801\) 0.932071i 0.0329331i
\(802\) 0 0
\(803\) 50.0263 1.76539
\(804\) 0 0
\(805\) −1.61213 −0.0568200
\(806\) 0 0
\(807\) −16.7513 −0.589674
\(808\) 0 0
\(809\) −28.9234 −1.01689 −0.508446 0.861094i \(-0.669780\pi\)
−0.508446 + 0.861094i \(0.669780\pi\)
\(810\) 0 0
\(811\) 33.1002i 1.16230i 0.813795 + 0.581152i \(0.197398\pi\)
−0.813795 + 0.581152i \(0.802602\pi\)
\(812\) 0 0
\(813\) 6.38787i 0.224032i
\(814\) 0 0
\(815\) 39.9511 1.39943
\(816\) 0 0
\(817\) 4.59612i 0.160798i
\(818\) 0 0
\(819\) −3.28726 1.48119i −0.114866 0.0517571i
\(820\) 0 0
\(821\) 6.46898i 0.225769i −0.993608 0.112884i \(-0.963991\pi\)
0.993608 0.112884i \(-0.0360090\pi\)
\(822\) 0 0
\(823\) −9.55149 −0.332944 −0.166472 0.986046i \(-0.553238\pi\)
−0.166472 + 0.986046i \(0.553238\pi\)
\(824\) 0 0
\(825\) 39.2057i 1.36497i
\(826\) 0 0
\(827\) 36.3430i 1.26377i −0.775063 0.631884i \(-0.782282\pi\)
0.775063 0.631884i \(-0.217718\pi\)
\(828\) 0 0
\(829\) −21.7645 −0.755912 −0.377956 0.925824i \(-0.623373\pi\)
−0.377956 + 0.925824i \(0.623373\pi\)
\(830\) 0 0
\(831\) −12.1768 −0.422408
\(832\) 0 0
\(833\) −3.35026 −0.116080
\(834\) 0 0
\(835\) −50.2130 −1.73769
\(836\) 0 0
\(837\) 10.7005i 0.369864i
\(838\) 0 0
\(839\) 0.0693432i 0.00239399i 0.999999 + 0.00119700i \(0.000381016\pi\)
−0.999999 + 0.00119700i \(0.999619\pi\)
\(840\) 0 0
\(841\) 33.8021 1.16559
\(842\) 0 0
\(843\) 27.3054i 0.940446i
\(844\) 0 0
\(845\) 40.4749 35.7948i 1.39238 1.23138i
\(846\) 0 0
\(847\) 0.798769i 0.0274460i
\(848\) 0 0
\(849\) 27.0738 0.929171
\(850\) 0 0
\(851\) 0.625301i 0.0214350i
\(852\) 0 0
\(853\) 5.17347i 0.177136i −0.996070 0.0885681i \(-0.971771\pi\)
0.996070 0.0885681i \(-0.0282291\pi\)
\(854\) 0 0
\(855\) −9.92478 −0.339420
\(856\) 0 0
\(857\) 41.2750 1.40993 0.704964 0.709243i \(-0.250963\pi\)
0.704964 + 0.709243i \(0.250963\pi\)
\(858\) 0 0
\(859\) 14.1768 0.483706 0.241853 0.970313i \(-0.422245\pi\)
0.241853 + 0.970313i \(0.422245\pi\)
\(860\) 0 0
\(861\) 1.45580 0.0496136
\(862\) 0 0
\(863\) 7.62786i 0.259655i 0.991537 + 0.129828i \(0.0414424\pi\)
−0.991537 + 0.129828i \(0.958558\pi\)
\(864\) 0 0
\(865\) 38.5501i 1.31074i
\(866\) 0 0
\(867\) 5.77575 0.196155
\(868\) 0 0
\(869\) 28.6253i 0.971047i
\(870\) 0 0
\(871\) 18.4485 + 8.31265i 0.625104 + 0.281663i
\(872\) 0 0
\(873\) 3.35026i 0.113389i
\(874\) 0 0
\(875\) −30.2374 −1.02221
\(876\) 0 0
\(877\) 43.1754i 1.45793i 0.684552 + 0.728964i \(0.259998\pi\)
−0.684552 + 0.728964i \(0.740002\pi\)
\(878\) 0 0
\(879\) 16.7816i 0.566030i
\(880\) 0 0
\(881\) −29.9003 −1.00737 −0.503684 0.863888i \(-0.668022\pi\)
−0.503684 + 0.863888i \(0.668022\pi\)
\(882\) 0 0
\(883\) 9.83971 0.331132 0.165566 0.986199i \(-0.447055\pi\)
0.165566 + 0.986199i \(0.447055\pi\)
\(884\) 0 0
\(885\) −25.5877 −0.860121
\(886\) 0 0
\(887\) 43.0249 1.44464 0.722318 0.691561i \(-0.243077\pi\)
0.722318 + 0.691561i \(0.243077\pi\)
\(888\) 0 0
\(889\) 0.962389i 0.0322775i
\(890\) 0 0
\(891\) 3.19394i 0.107001i
\(892\) 0 0
\(893\) 8.99859 0.301126
\(894\) 0 0
\(895\) 20.8364i 0.696483i
\(896\) 0 0
\(897\) 0.574515 1.27504i 0.0191825 0.0425723i
\(898\) 0 0
\(899\) 84.7993i 2.82821i
\(900\) 0 0
\(901\) −20.1016 −0.669680
\(902\) 0 0
\(903\) 1.92478i 0.0640526i
\(904\) 0 0
\(905\) 57.5633i 1.91347i
\(906\) 0 0
\(907\) 11.2605 0.373897 0.186949 0.982370i \(-0.440140\pi\)
0.186949 + 0.982370i \(0.440140\pi\)
\(908\) 0 0
\(909\) −14.0508 −0.466035
\(910\) 0 0
\(911\) −23.5633 −0.780685 −0.390343 0.920670i \(-0.627643\pi\)
−0.390343 + 0.920670i \(0.627643\pi\)
\(912\) 0 0
\(913\) 22.3343 0.739156
\(914\) 0 0
\(915\) 60.0118i 1.98393i
\(916\) 0 0
\(917\) 17.1490i 0.566311i
\(918\) 0 0
\(919\) −12.8411 −0.423589 −0.211795 0.977314i \(-0.567931\pi\)
−0.211795 + 0.977314i \(0.567931\pi\)
\(920\) 0 0
\(921\) 17.2995i 0.570037i
\(922\) 0 0
\(923\) 38.8529 + 17.5066i 1.27886 + 0.576236i
\(924\) 0 0
\(925\) 19.7889i 0.650656i
\(926\) 0 0
\(927\) −1.42548 −0.0468191
\(928\) 0 0
\(929\) 24.6194i 0.807737i 0.914817 + 0.403869i \(0.132335\pi\)
−0.914817 + 0.403869i \(0.867665\pi\)
\(930\) 0 0
\(931\) 2.38787i 0.0782594i
\(932\) 0 0
\(933\) 17.1490 0.561434
\(934\) 0 0
\(935\) 44.4749 1.45448
\(936\) 0 0
\(937\) 44.4847 1.45325 0.726626 0.687033i \(-0.241087\pi\)
0.726626 + 0.687033i \(0.241087\pi\)
\(938\) 0 0
\(939\) 10.8119 0.352834
\(940\) 0 0
\(941\) 2.69464i 0.0878429i 0.999035 + 0.0439214i \(0.0139851\pi\)
−0.999035 + 0.0439214i \(0.986015\pi\)
\(942\) 0 0
\(943\) 0.564666i 0.0183881i
\(944\) 0 0
\(945\) −4.15633 −0.135205
\(946\) 0 0
\(947\) 3.09237i 0.100488i 0.998737 + 0.0502442i \(0.0160000\pi\)
−0.998737 + 0.0502442i \(0.984000\pi\)
\(948\) 0 0
\(949\) 51.4880 + 23.1998i 1.67137 + 0.753098i
\(950\) 0 0
\(951\) 14.1563i 0.459050i
\(952\) 0 0
\(953\) 8.57925 0.277909 0.138955 0.990299i \(-0.455626\pi\)
0.138955 + 0.990299i \(0.455626\pi\)
\(954\) 0 0
\(955\) 77.1608i 2.49686i
\(956\) 0 0
\(957\) 25.3112i 0.818196i
\(958\) 0 0
\(959\) −4.85685 −0.156836
\(960\) 0 0
\(961\) −83.5012 −2.69359
\(962\) 0 0
\(963\) −9.53690 −0.307322
\(964\) 0 0
\(965\) −54.3996 −1.75119
\(966\) 0 0
\(967\) 7.43533i 0.239104i −0.992828 0.119552i \(-0.961854\pi\)
0.992828 0.119552i \(-0.0381459\pi\)
\(968\) 0 0
\(969\) 8.00000i 0.256997i
\(970\) 0 0
\(971\) −52.3244 −1.67917 −0.839585 0.543228i \(-0.817202\pi\)
−0.839585 + 0.543228i \(0.817202\pi\)
\(972\) 0 0
\(973\) 2.57452i 0.0825352i
\(974\) 0 0
\(975\) 18.1817 40.3512i 0.582281 1.29227i
\(976\) 0 0
\(977\) 12.0811i 0.386509i −0.981149 0.193254i \(-0.938096\pi\)
0.981149 0.193254i \(-0.0619043\pi\)
\(978\) 0 0
\(979\) −2.97698 −0.0951446
\(980\) 0 0
\(981\) 6.23743i 0.199146i
\(982\) 0 0
\(983\) 16.7553i 0.534410i −0.963640 0.267205i \(-0.913900\pi\)
0.963640 0.267205i \(-0.0861001\pi\)
\(984\) 0 0
\(985\) −30.1114 −0.959430
\(986\) 0 0
\(987\) 3.76845 0.119951
\(988\) 0 0
\(989\) 0.746569 0.0237395
\(990\) 0 0
\(991\) 1.77433 0.0563635 0.0281818 0.999603i \(-0.491028\pi\)
0.0281818 + 0.999603i \(0.491028\pi\)
\(992\) 0 0
\(993\) 17.2506i 0.547431i
\(994\) 0 0
\(995\) 35.8496i 1.13651i
\(996\) 0 0
\(997\) 24.5501 0.777509 0.388754 0.921341i \(-0.372905\pi\)
0.388754 + 0.921341i \(0.372905\pi\)
\(998\) 0 0
\(999\) 1.61213i 0.0510054i
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 4368.2.h.o.337.6 6
4.3 odd 2 273.2.c.b.64.2 6
12.11 even 2 819.2.c.c.64.5 6
13.12 even 2 inner 4368.2.h.o.337.1 6
28.27 even 2 1911.2.c.h.883.2 6
52.31 even 4 3549.2.a.q.1.1 3
52.47 even 4 3549.2.a.k.1.3 3
52.51 odd 2 273.2.c.b.64.5 yes 6
156.155 even 2 819.2.c.c.64.2 6
364.363 even 2 1911.2.c.h.883.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.c.b.64.2 6 4.3 odd 2
273.2.c.b.64.5 yes 6 52.51 odd 2
819.2.c.c.64.2 6 156.155 even 2
819.2.c.c.64.5 6 12.11 even 2
1911.2.c.h.883.2 6 28.27 even 2
1911.2.c.h.883.5 6 364.363 even 2
3549.2.a.k.1.3 3 52.47 even 4
3549.2.a.q.1.1 3 52.31 even 4
4368.2.h.o.337.1 6 13.12 even 2 inner
4368.2.h.o.337.6 6 1.1 even 1 trivial