Properties

Label 4400.2.b.bb.4049.4
Level $4400$
Weight $2$
Character 4400.4049
Analytic conductor $35.134$
Analytic rank $0$
Dimension $6$
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] = [4400,2,Mod(4049,4400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4400.4049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.b (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: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.96668224.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 15x^{4} + 61x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2200)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.4
Root \(0.841083i\) of defining polynomial
Character \(\chi\) \(=\) 4400.4049
Dual form 4400.2.b.bb.4049.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.841083i q^{3} +3.29258i q^{7} +2.29258 q^{9} +O(q^{10})\) \(q+0.841083i q^{3} +3.29258i q^{7} +2.29258 q^{9} -1.00000 q^{11} -1.00000i q^{13} -1.15892i q^{17} +1.31783 q^{19} -2.76933 q^{21} +3.84108i q^{23} +4.45150i q^{27} -6.61041 q^{29} -7.58516 q^{31} -0.841083i q^{33} +2.13366i q^{37} +0.841083 q^{39} +6.13366 q^{41} +8.81583i q^{43} -2.76933i q^{47} -3.84108 q^{49} +0.974745 q^{51} +10.1956i q^{53} +1.10841i q^{57} +2.76933 q^{59} -3.61041 q^{61} +7.54850i q^{63} -2.90299i q^{67} -3.23067 q^{69} -0.866337 q^{71} -5.87774i q^{73} -3.29258i q^{77} +11.7441 q^{79} +3.13366 q^{81} +8.92825i q^{83} -5.55991i q^{87} -14.2926 q^{89} +3.29258 q^{91} -6.37975i q^{93} -12.4262i q^{97} -2.29258 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 12 q^{9} - 6 q^{11} + 14 q^{19} - 20 q^{29} + 6 q^{31} + 2 q^{39} + 8 q^{41} - 20 q^{49} - 26 q^{51} - 2 q^{61} - 36 q^{69} - 34 q^{71} + 22 q^{79} - 10 q^{81} - 60 q^{89} - 6 q^{91} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(1201\) \(2751\) \(3301\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 0.841083i 0.485599i 0.970076 + 0.242800i \(0.0780658\pi\)
−0.970076 + 0.242800i \(0.921934\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.29258i 1.24448i 0.782827 + 0.622239i \(0.213777\pi\)
−0.782827 + 0.622239i \(0.786223\pi\)
\(8\) 0 0
\(9\) 2.29258 0.764193
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i −0.990338 0.138675i \(-0.955716\pi\)
0.990338 0.138675i \(-0.0442844\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.15892i − 0.281079i −0.990075 0.140539i \(-0.955116\pi\)
0.990075 0.140539i \(-0.0448837\pi\)
\(18\) 0 0
\(19\) 1.31783 0.302332 0.151166 0.988508i \(-0.451697\pi\)
0.151166 + 0.988508i \(0.451697\pi\)
\(20\) 0 0
\(21\) −2.76933 −0.604318
\(22\) 0 0
\(23\) 3.84108i 0.800921i 0.916314 + 0.400461i \(0.131150\pi\)
−0.916314 + 0.400461i \(0.868850\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.45150i 0.856691i
\(28\) 0 0
\(29\) −6.61041 −1.22752 −0.613762 0.789491i \(-0.710344\pi\)
−0.613762 + 0.789491i \(0.710344\pi\)
\(30\) 0 0
\(31\) −7.58516 −1.36233 −0.681167 0.732128i \(-0.738527\pi\)
−0.681167 + 0.732128i \(0.738527\pi\)
\(32\) 0 0
\(33\) − 0.841083i − 0.146414i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.13366i 0.350772i 0.984500 + 0.175386i \(0.0561173\pi\)
−0.984500 + 0.175386i \(0.943883\pi\)
\(38\) 0 0
\(39\) 0.841083 0.134681
\(40\) 0 0
\(41\) 6.13366 0.957917 0.478959 0.877838i \(-0.341014\pi\)
0.478959 + 0.877838i \(0.341014\pi\)
\(42\) 0 0
\(43\) 8.81583i 1.34440i 0.740369 + 0.672201i \(0.234651\pi\)
−0.740369 + 0.672201i \(0.765349\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 2.76933i − 0.403949i −0.979391 0.201974i \(-0.935264\pi\)
0.979391 0.201974i \(-0.0647357\pi\)
\(48\) 0 0
\(49\) −3.84108 −0.548726
\(50\) 0 0
\(51\) 0.974745 0.136492
\(52\) 0 0
\(53\) 10.1956i 1.40047i 0.713912 + 0.700235i \(0.246921\pi\)
−0.713912 + 0.700235i \(0.753079\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.10841i 0.146812i
\(58\) 0 0
\(59\) 2.76933 0.360536 0.180268 0.983618i \(-0.442303\pi\)
0.180268 + 0.983618i \(0.442303\pi\)
\(60\) 0 0
\(61\) −3.61041 −0.462266 −0.231133 0.972922i \(-0.574243\pi\)
−0.231133 + 0.972922i \(0.574243\pi\)
\(62\) 0 0
\(63\) 7.54850i 0.951022i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 2.90299i − 0.354657i −0.984152 0.177329i \(-0.943254\pi\)
0.984152 0.177329i \(-0.0567455\pi\)
\(68\) 0 0
\(69\) −3.23067 −0.388927
\(70\) 0 0
\(71\) −0.866337 −0.102815 −0.0514077 0.998678i \(-0.516371\pi\)
−0.0514077 + 0.998678i \(0.516371\pi\)
\(72\) 0 0
\(73\) − 5.87774i − 0.687937i −0.938981 0.343969i \(-0.888229\pi\)
0.938981 0.343969i \(-0.111771\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 3.29258i − 0.375224i
\(78\) 0 0
\(79\) 11.7441 1.32131 0.660656 0.750689i \(-0.270278\pi\)
0.660656 + 0.750689i \(0.270278\pi\)
\(80\) 0 0
\(81\) 3.13366 0.348185
\(82\) 0 0
\(83\) 8.92825i 0.980003i 0.871722 + 0.490001i \(0.163004\pi\)
−0.871722 + 0.490001i \(0.836996\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 5.55991i − 0.596084i
\(88\) 0 0
\(89\) −14.2926 −1.51501 −0.757505 0.652829i \(-0.773582\pi\)
−0.757505 + 0.652829i \(0.773582\pi\)
\(90\) 0 0
\(91\) 3.29258 0.345156
\(92\) 0 0
\(93\) − 6.37975i − 0.661549i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 12.4262i − 1.26169i −0.775907 0.630847i \(-0.782708\pi\)
0.775907 0.630847i \(-0.217292\pi\)
\(98\) 0 0
\(99\) −2.29258 −0.230413
\(100\) 0 0
\(101\) 2.15892 0.214820 0.107410 0.994215i \(-0.465744\pi\)
0.107410 + 0.994215i \(0.465744\pi\)
\(102\) 0 0
\(103\) 3.07175i 0.302669i 0.988483 + 0.151334i \(0.0483570\pi\)
−0.988483 + 0.151334i \(0.951643\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.4882i 1.59397i 0.603999 + 0.796985i \(0.293573\pi\)
−0.603999 + 0.796985i \(0.706427\pi\)
\(108\) 0 0
\(109\) −10.2926 −0.985850 −0.492925 0.870072i \(-0.664072\pi\)
−0.492925 + 0.870072i \(0.664072\pi\)
\(110\) 0 0
\(111\) −1.79459 −0.170335
\(112\) 0 0
\(113\) 11.4980i 1.08164i 0.841138 + 0.540820i \(0.181886\pi\)
−0.841138 + 0.540820i \(0.818114\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 2.29258i − 0.211949i
\(118\) 0 0
\(119\) 3.81583 0.349796
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 5.15892i 0.465164i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 4.37975i − 0.388640i −0.980938 0.194320i \(-0.937750\pi\)
0.980938 0.194320i \(-0.0622500\pi\)
\(128\) 0 0
\(129\) −7.41484 −0.652840
\(130\) 0 0
\(131\) 5.01140 0.437848 0.218924 0.975742i \(-0.429745\pi\)
0.218924 + 0.975742i \(0.429745\pi\)
\(132\) 0 0
\(133\) 4.33908i 0.376246i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 16.2421i − 1.38765i −0.720142 0.693827i \(-0.755923\pi\)
0.720142 0.693827i \(-0.244077\pi\)
\(138\) 0 0
\(139\) −22.7555 −1.93009 −0.965047 0.262076i \(-0.915593\pi\)
−0.965047 + 0.262076i \(0.915593\pi\)
\(140\) 0 0
\(141\) 2.32924 0.196157
\(142\) 0 0
\(143\) 1.00000i 0.0836242i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 3.23067i − 0.266461i
\(148\) 0 0
\(149\) −18.2673 −1.49652 −0.748259 0.663407i \(-0.769110\pi\)
−0.748259 + 0.663407i \(0.769110\pi\)
\(150\) 0 0
\(151\) −16.0832 −1.30883 −0.654414 0.756136i \(-0.727085\pi\)
−0.654414 + 0.756136i \(0.727085\pi\)
\(152\) 0 0
\(153\) − 2.65691i − 0.214798i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 15.1703i − 1.21072i −0.795951 0.605362i \(-0.793028\pi\)
0.795951 0.605362i \(-0.206972\pi\)
\(158\) 0 0
\(159\) −8.57532 −0.680067
\(160\) 0 0
\(161\) −12.6471 −0.996729
\(162\) 0 0
\(163\) 6.24608i 0.489231i 0.969620 + 0.244616i \(0.0786617\pi\)
−0.969620 + 0.244616i \(0.921338\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.594999i 0.0460424i 0.999735 + 0.0230212i \(0.00732852\pi\)
−0.999735 + 0.0230212i \(0.992671\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 3.02124 0.231040
\(172\) 0 0
\(173\) 1.31783i 0.100193i 0.998744 + 0.0500966i \(0.0159529\pi\)
−0.998744 + 0.0500966i \(0.984047\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2.32924i 0.175076i
\(178\) 0 0
\(179\) −4.70742 −0.351849 −0.175925 0.984404i \(-0.556291\pi\)
−0.175925 + 0.984404i \(0.556291\pi\)
\(180\) 0 0
\(181\) −10.0212 −0.744873 −0.372437 0.928058i \(-0.621478\pi\)
−0.372437 + 0.928058i \(0.621478\pi\)
\(182\) 0 0
\(183\) − 3.03666i − 0.224476i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.15892i 0.0847484i
\(188\) 0 0
\(189\) −14.6569 −1.06613
\(190\) 0 0
\(191\) −16.2421 −1.17523 −0.587617 0.809139i \(-0.699934\pi\)
−0.587617 + 0.809139i \(0.699934\pi\)
\(192\) 0 0
\(193\) 15.3643i 1.10595i 0.833198 + 0.552974i \(0.186507\pi\)
−0.833198 + 0.552974i \(0.813493\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.52325i 0.536009i 0.963418 + 0.268005i \(0.0863642\pi\)
−0.963418 + 0.268005i \(0.913636\pi\)
\(198\) 0 0
\(199\) 24.5094 1.73742 0.868712 0.495317i \(-0.164948\pi\)
0.868712 + 0.495317i \(0.164948\pi\)
\(200\) 0 0
\(201\) 2.44166 0.172221
\(202\) 0 0
\(203\) − 21.7653i − 1.52763i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 8.80599i 0.612059i
\(208\) 0 0
\(209\) −1.31783 −0.0911565
\(210\) 0 0
\(211\) −5.35449 −0.368618 −0.184309 0.982868i \(-0.559005\pi\)
−0.184309 + 0.982868i \(0.559005\pi\)
\(212\) 0 0
\(213\) − 0.728661i − 0.0499271i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 24.9747i − 1.69540i
\(218\) 0 0
\(219\) 4.94367 0.334062
\(220\) 0 0
\(221\) −1.15892 −0.0779572
\(222\) 0 0
\(223\) 7.40500i 0.495876i 0.968776 + 0.247938i \(0.0797529\pi\)
−0.968776 + 0.247938i \(0.920247\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 16.2926i − 1.08138i −0.841223 0.540688i \(-0.818164\pi\)
0.841223 0.540688i \(-0.181836\pi\)
\(228\) 0 0
\(229\) 19.0579 1.25938 0.629691 0.776846i \(-0.283182\pi\)
0.629691 + 0.776846i \(0.283182\pi\)
\(230\) 0 0
\(231\) 2.76933 0.182209
\(232\) 0 0
\(233\) 6.97475i 0.456931i 0.973552 + 0.228465i \(0.0733708\pi\)
−0.973552 + 0.228465i \(0.926629\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 9.87774i 0.641628i
\(238\) 0 0
\(239\) −0.513409 −0.0332097 −0.0166048 0.999862i \(-0.505286\pi\)
−0.0166048 + 0.999862i \(0.505286\pi\)
\(240\) 0 0
\(241\) 12.4727 0.803440 0.401720 0.915763i \(-0.368413\pi\)
0.401720 + 0.915763i \(0.368413\pi\)
\(242\) 0 0
\(243\) 15.9902i 1.02577i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 1.31783i − 0.0838518i
\(248\) 0 0
\(249\) −7.50940 −0.475889
\(250\) 0 0
\(251\) −14.4727 −0.913511 −0.456756 0.889592i \(-0.650989\pi\)
−0.456756 + 0.889592i \(0.650989\pi\)
\(252\) 0 0
\(253\) − 3.84108i − 0.241487i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.6218i 1.41111i 0.708655 + 0.705555i \(0.249302\pi\)
−0.708655 + 0.705555i \(0.750698\pi\)
\(258\) 0 0
\(259\) −7.02525 −0.436528
\(260\) 0 0
\(261\) −15.1549 −0.938065
\(262\) 0 0
\(263\) 14.5445i 0.896852i 0.893820 + 0.448426i \(0.148015\pi\)
−0.893820 + 0.448426i \(0.851985\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 12.0212i − 0.735688i
\(268\) 0 0
\(269\) −7.42624 −0.452786 −0.226393 0.974036i \(-0.572693\pi\)
−0.226393 + 0.974036i \(0.572693\pi\)
\(270\) 0 0
\(271\) 1.40500 0.0853477 0.0426739 0.999089i \(-0.486412\pi\)
0.0426739 + 0.999089i \(0.486412\pi\)
\(272\) 0 0
\(273\) 2.76933i 0.167608i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 17.9397i 1.07789i 0.842341 + 0.538945i \(0.181177\pi\)
−0.842341 + 0.538945i \(0.818823\pi\)
\(278\) 0 0
\(279\) −17.3896 −1.04109
\(280\) 0 0
\(281\) −5.49799 −0.327983 −0.163991 0.986462i \(-0.552437\pi\)
−0.163991 + 0.986462i \(0.552437\pi\)
\(282\) 0 0
\(283\) 8.25592i 0.490764i 0.969427 + 0.245382i \(0.0789133\pi\)
−0.969427 + 0.245382i \(0.921087\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 20.1956i 1.19211i
\(288\) 0 0
\(289\) 15.6569 0.920995
\(290\) 0 0
\(291\) 10.4515 0.612678
\(292\) 0 0
\(293\) 14.6357i 0.855025i 0.904009 + 0.427512i \(0.140610\pi\)
−0.904009 + 0.427512i \(0.859390\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 4.45150i − 0.258302i
\(298\) 0 0
\(299\) 3.84108 0.222136
\(300\) 0 0
\(301\) −29.0268 −1.67308
\(302\) 0 0
\(303\) 1.81583i 0.104317i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 18.3292i − 1.04610i −0.852301 0.523052i \(-0.824793\pi\)
0.852301 0.523052i \(-0.175207\pi\)
\(308\) 0 0
\(309\) −2.58360 −0.146976
\(310\) 0 0
\(311\) −6.09701 −0.345729 −0.172865 0.984946i \(-0.555302\pi\)
−0.172865 + 0.984946i \(0.555302\pi\)
\(312\) 0 0
\(313\) − 0.317835i − 0.0179651i −0.999960 0.00898254i \(-0.997141\pi\)
0.999960 0.00898254i \(-0.00285927\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.3797i 1.36930i 0.728871 + 0.684651i \(0.240046\pi\)
−0.728871 + 0.684651i \(0.759954\pi\)
\(318\) 0 0
\(319\) 6.61041 0.370112
\(320\) 0 0
\(321\) −13.8679 −0.774031
\(322\) 0 0
\(323\) − 1.52726i − 0.0849791i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 8.65691i − 0.478728i
\(328\) 0 0
\(329\) 9.11825 0.502705
\(330\) 0 0
\(331\) 3.54850 0.195043 0.0975217 0.995233i \(-0.468908\pi\)
0.0975217 + 0.995233i \(0.468908\pi\)
\(332\) 0 0
\(333\) 4.89159i 0.268058i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 10.0000i − 0.544735i −0.962193 0.272367i \(-0.912193\pi\)
0.962193 0.272367i \(-0.0878066\pi\)
\(338\) 0 0
\(339\) −9.67076 −0.525244
\(340\) 0 0
\(341\) 7.58516 0.410759
\(342\) 0 0
\(343\) 10.4010i 0.561601i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 8.70742i − 0.467439i −0.972304 0.233719i \(-0.924910\pi\)
0.972304 0.233719i \(-0.0750897\pi\)
\(348\) 0 0
\(349\) −16.8565 −0.902308 −0.451154 0.892446i \(-0.648987\pi\)
−0.451154 + 0.892446i \(0.648987\pi\)
\(350\) 0 0
\(351\) 4.45150 0.237603
\(352\) 0 0
\(353\) 10.3643i 0.551638i 0.961210 + 0.275819i \(0.0889490\pi\)
−0.961210 + 0.275819i \(0.911051\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 3.20943i 0.169861i
\(358\) 0 0
\(359\) −24.9030 −1.31433 −0.657165 0.753747i \(-0.728244\pi\)
−0.657165 + 0.753747i \(0.728244\pi\)
\(360\) 0 0
\(361\) −17.2633 −0.908595
\(362\) 0 0
\(363\) 0.841083i 0.0441454i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 2.98860i − 0.156004i −0.996953 0.0780018i \(-0.975146\pi\)
0.996953 0.0780018i \(-0.0248540\pi\)
\(368\) 0 0
\(369\) 14.0619 0.732034
\(370\) 0 0
\(371\) −33.5697 −1.74285
\(372\) 0 0
\(373\) − 21.4475i − 1.11051i −0.831681 0.555254i \(-0.812621\pi\)
0.831681 0.555254i \(-0.187379\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.61041i 0.340454i
\(378\) 0 0
\(379\) 6.45150 0.331391 0.165696 0.986177i \(-0.447013\pi\)
0.165696 + 0.986177i \(0.447013\pi\)
\(380\) 0 0
\(381\) 3.68373 0.188723
\(382\) 0 0
\(383\) − 22.0872i − 1.12860i −0.825569 0.564301i \(-0.809146\pi\)
0.825569 0.564301i \(-0.190854\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 20.2110i 1.02738i
\(388\) 0 0
\(389\) 31.2633 1.58511 0.792556 0.609799i \(-0.208750\pi\)
0.792556 + 0.609799i \(0.208750\pi\)
\(390\) 0 0
\(391\) 4.45150 0.225122
\(392\) 0 0
\(393\) 4.21500i 0.212619i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 5.74408i − 0.288287i −0.989557 0.144143i \(-0.953957\pi\)
0.989557 0.144143i \(-0.0460427\pi\)
\(398\) 0 0
\(399\) −3.64952 −0.182705
\(400\) 0 0
\(401\) 34.1605 1.70589 0.852947 0.521998i \(-0.174813\pi\)
0.852947 + 0.521998i \(0.174813\pi\)
\(402\) 0 0
\(403\) 7.58516i 0.377844i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 2.13366i − 0.105762i
\(408\) 0 0
\(409\) 20.7188 1.02448 0.512240 0.858842i \(-0.328816\pi\)
0.512240 + 0.858842i \(0.328816\pi\)
\(410\) 0 0
\(411\) 13.6609 0.673844
\(412\) 0 0
\(413\) 9.11825i 0.448680i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 19.1392i − 0.937253i
\(418\) 0 0
\(419\) 6.81984 0.333171 0.166586 0.986027i \(-0.446726\pi\)
0.166586 + 0.986027i \(0.446726\pi\)
\(420\) 0 0
\(421\) −36.5134 −1.77955 −0.889777 0.456395i \(-0.849140\pi\)
−0.889777 + 0.456395i \(0.849140\pi\)
\(422\) 0 0
\(423\) − 6.34891i − 0.308695i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 11.8876i − 0.575280i
\(428\) 0 0
\(429\) −0.841083 −0.0406079
\(430\) 0 0
\(431\) 10.4010 0.500998 0.250499 0.968117i \(-0.419405\pi\)
0.250499 + 0.968117i \(0.419405\pi\)
\(432\) 0 0
\(433\) 29.7090i 1.42772i 0.700287 + 0.713861i \(0.253055\pi\)
−0.700287 + 0.713861i \(0.746945\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5.06191i 0.242144i
\(438\) 0 0
\(439\) −30.3292 −1.44754 −0.723768 0.690044i \(-0.757591\pi\)
−0.723768 + 0.690044i \(0.757591\pi\)
\(440\) 0 0
\(441\) −8.80599 −0.419333
\(442\) 0 0
\(443\) − 25.0465i − 1.18999i −0.803728 0.594997i \(-0.797153\pi\)
0.803728 0.594997i \(-0.202847\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 15.3643i − 0.726708i
\(448\) 0 0
\(449\) 38.2322 1.80429 0.902145 0.431432i \(-0.141992\pi\)
0.902145 + 0.431432i \(0.141992\pi\)
\(450\) 0 0
\(451\) −6.13366 −0.288823
\(452\) 0 0
\(453\) − 13.5273i − 0.635566i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 32.7302i − 1.53106i −0.643403 0.765528i \(-0.722478\pi\)
0.643403 0.765528i \(-0.277522\pi\)
\(458\) 0 0
\(459\) 5.15892 0.240798
\(460\) 0 0
\(461\) 20.8118 0.969303 0.484651 0.874707i \(-0.338946\pi\)
0.484651 + 0.874707i \(0.338946\pi\)
\(462\) 0 0
\(463\) 1.51185i 0.0702614i 0.999383 + 0.0351307i \(0.0111848\pi\)
−0.999383 + 0.0351307i \(0.988815\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 23.4783i − 1.08645i −0.839588 0.543223i \(-0.817204\pi\)
0.839588 0.543223i \(-0.182796\pi\)
\(468\) 0 0
\(469\) 9.55834 0.441363
\(470\) 0 0
\(471\) 12.7595 0.587926
\(472\) 0 0
\(473\) − 8.81583i − 0.405352i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 23.3742i 1.07023i
\(478\) 0 0
\(479\) −14.7188 −0.672520 −0.336260 0.941769i \(-0.609162\pi\)
−0.336260 + 0.941769i \(0.609162\pi\)
\(480\) 0 0
\(481\) 2.13366 0.0972866
\(482\) 0 0
\(483\) − 10.6372i − 0.484011i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 23.7595i − 1.07665i −0.842739 0.538323i \(-0.819058\pi\)
0.842739 0.538323i \(-0.180942\pi\)
\(488\) 0 0
\(489\) −5.25347 −0.237570
\(490\) 0 0
\(491\) −18.7555 −0.846423 −0.423211 0.906031i \(-0.639097\pi\)
−0.423211 + 0.906031i \(0.639097\pi\)
\(492\) 0 0
\(493\) 7.66092i 0.345031i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 2.85249i − 0.127951i
\(498\) 0 0
\(499\) −26.9747 −1.20756 −0.603778 0.797153i \(-0.706339\pi\)
−0.603778 + 0.797153i \(0.706339\pi\)
\(500\) 0 0
\(501\) −0.500443 −0.0223582
\(502\) 0 0
\(503\) 43.2941i 1.93039i 0.261530 + 0.965195i \(0.415773\pi\)
−0.261530 + 0.965195i \(0.584227\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 10.0930i 0.448246i
\(508\) 0 0
\(509\) 10.8312 0.480086 0.240043 0.970762i \(-0.422838\pi\)
0.240043 + 0.970762i \(0.422838\pi\)
\(510\) 0 0
\(511\) 19.3529 0.856123
\(512\) 0 0
\(513\) 5.86634i 0.259005i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.76933i 0.121795i
\(518\) 0 0
\(519\) −1.10841 −0.0486537
\(520\) 0 0
\(521\) 21.1337 0.925883 0.462941 0.886389i \(-0.346794\pi\)
0.462941 + 0.886389i \(0.346794\pi\)
\(522\) 0 0
\(523\) − 31.7921i − 1.39017i −0.718926 0.695087i \(-0.755366\pi\)
0.718926 0.695087i \(-0.244634\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.79057i 0.382923i
\(528\) 0 0
\(529\) 8.24608 0.358525
\(530\) 0 0
\(531\) 6.34891 0.275519
\(532\) 0 0
\(533\) − 6.13366i − 0.265678i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 3.95933i − 0.170858i
\(538\) 0 0
\(539\) 3.84108 0.165447
\(540\) 0 0
\(541\) 30.3757 1.30595 0.652977 0.757377i \(-0.273520\pi\)
0.652977 + 0.757377i \(0.273520\pi\)
\(542\) 0 0
\(543\) − 8.42869i − 0.361710i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 4.83707i − 0.206818i −0.994639 0.103409i \(-0.967025\pi\)
0.994639 0.103409i \(-0.0329751\pi\)
\(548\) 0 0
\(549\) −8.27716 −0.353261
\(550\) 0 0
\(551\) −8.71143 −0.371120
\(552\) 0 0
\(553\) 38.6683i 1.64434i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 30.6317i 1.29790i 0.760829 + 0.648952i \(0.224793\pi\)
−0.760829 + 0.648952i \(0.775207\pi\)
\(558\) 0 0
\(559\) 8.81583 0.372870
\(560\) 0 0
\(561\) −0.974745 −0.0411538
\(562\) 0 0
\(563\) − 14.2461i − 0.600401i −0.953876 0.300200i \(-0.902946\pi\)
0.953876 0.300200i \(-0.0970536\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 10.3178i 0.433308i
\(568\) 0 0
\(569\) −40.3194 −1.69028 −0.845139 0.534547i \(-0.820482\pi\)
−0.845139 + 0.534547i \(0.820482\pi\)
\(570\) 0 0
\(571\) 8.98860 0.376161 0.188081 0.982154i \(-0.439773\pi\)
0.188081 + 0.982154i \(0.439773\pi\)
\(572\) 0 0
\(573\) − 13.6609i − 0.570693i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 10.2787i 0.427909i 0.976844 + 0.213955i \(0.0686344\pi\)
−0.976844 + 0.213955i \(0.931366\pi\)
\(578\) 0 0
\(579\) −12.9227 −0.537048
\(580\) 0 0
\(581\) −29.3970 −1.21959
\(582\) 0 0
\(583\) − 10.1956i − 0.422258i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 4.52325i − 0.186694i −0.995634 0.0933472i \(-0.970243\pi\)
0.995634 0.0933472i \(-0.0297567\pi\)
\(588\) 0 0
\(589\) −9.99599 −0.411877
\(590\) 0 0
\(591\) −6.32767 −0.260286
\(592\) 0 0
\(593\) 28.0326i 1.15116i 0.817745 + 0.575581i \(0.195224\pi\)
−0.817745 + 0.575581i \(0.804776\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 20.6144i 0.843692i
\(598\) 0 0
\(599\) 15.8876 0.649149 0.324574 0.945860i \(-0.394779\pi\)
0.324574 + 0.945860i \(0.394779\pi\)
\(600\) 0 0
\(601\) 4.83124 0.197071 0.0985353 0.995134i \(-0.468584\pi\)
0.0985353 + 0.995134i \(0.468584\pi\)
\(602\) 0 0
\(603\) − 6.65535i − 0.271027i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 37.2633i 1.51247i 0.654299 + 0.756236i \(0.272964\pi\)
−0.654299 + 0.756236i \(0.727036\pi\)
\(608\) 0 0
\(609\) 18.3064 0.741814
\(610\) 0 0
\(611\) −2.76933 −0.112035
\(612\) 0 0
\(613\) − 44.1857i − 1.78465i −0.451398 0.892323i \(-0.649075\pi\)
0.451398 0.892323i \(-0.350925\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 13.3080i − 0.535760i −0.963452 0.267880i \(-0.913677\pi\)
0.963452 0.267880i \(-0.0863230\pi\)
\(618\) 0 0
\(619\) −33.4783 −1.34561 −0.672804 0.739821i \(-0.734910\pi\)
−0.672804 + 0.739821i \(0.734910\pi\)
\(620\) 0 0
\(621\) −17.0986 −0.686142
\(622\) 0 0
\(623\) − 47.0595i − 1.88540i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 1.10841i − 0.0442655i
\(628\) 0 0
\(629\) 2.47274 0.0985945
\(630\) 0 0
\(631\) 49.1411 1.95627 0.978137 0.207961i \(-0.0666826\pi\)
0.978137 + 0.207961i \(0.0666826\pi\)
\(632\) 0 0
\(633\) − 4.50357i − 0.179001i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3.84108i 0.152189i
\(638\) 0 0
\(639\) −1.98615 −0.0785708
\(640\) 0 0
\(641\) 8.36433 0.330371 0.165186 0.986262i \(-0.447178\pi\)
0.165186 + 0.986262i \(0.447178\pi\)
\(642\) 0 0
\(643\) − 40.8020i − 1.60907i −0.593903 0.804536i \(-0.702414\pi\)
0.593903 0.804536i \(-0.297586\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 40.0832i − 1.57583i −0.615783 0.787916i \(-0.711160\pi\)
0.615783 0.787916i \(-0.288840\pi\)
\(648\) 0 0
\(649\) −2.76933 −0.108706
\(650\) 0 0
\(651\) 21.0058 0.823283
\(652\) 0 0
\(653\) − 14.1352i − 0.553154i −0.960992 0.276577i \(-0.910800\pi\)
0.960992 0.276577i \(-0.0892001\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 13.4752i − 0.525717i
\(658\) 0 0
\(659\) −1.51341 −0.0589541 −0.0294770 0.999565i \(-0.509384\pi\)
−0.0294770 + 0.999565i \(0.509384\pi\)
\(660\) 0 0
\(661\) 15.4050 0.599185 0.299593 0.954067i \(-0.403149\pi\)
0.299593 + 0.954067i \(0.403149\pi\)
\(662\) 0 0
\(663\) − 0.974745i − 0.0378560i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 25.3911i − 0.983149i
\(668\) 0 0
\(669\) −6.22822 −0.240797
\(670\) 0 0
\(671\) 3.61041 0.139379
\(672\) 0 0
\(673\) − 36.8426i − 1.42018i −0.704111 0.710090i \(-0.748654\pi\)
0.704111 0.710090i \(-0.251346\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 28.0367i 1.07754i 0.842454 + 0.538768i \(0.181110\pi\)
−0.842454 + 0.538768i \(0.818890\pi\)
\(678\) 0 0
\(679\) 40.9144 1.57015
\(680\) 0 0
\(681\) 13.7034 0.525116
\(682\) 0 0
\(683\) 28.7416i 1.09977i 0.835241 + 0.549884i \(0.185328\pi\)
−0.835241 + 0.549884i \(0.814672\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 16.0293i 0.611555i
\(688\) 0 0
\(689\) 10.1956 0.388420
\(690\) 0 0
\(691\) 23.2729 0.885343 0.442671 0.896684i \(-0.354031\pi\)
0.442671 + 0.896684i \(0.354031\pi\)
\(692\) 0 0
\(693\) − 7.54850i − 0.286744i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 7.10841i − 0.269250i
\(698\) 0 0
\(699\) −5.86634 −0.221885
\(700\) 0 0
\(701\) −26.8891 −1.01559 −0.507794 0.861478i \(-0.669539\pi\)
−0.507794 + 0.861478i \(0.669539\pi\)
\(702\) 0 0
\(703\) 2.81181i 0.106050i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7.10841i 0.267339i
\(708\) 0 0
\(709\) 16.6357 0.624766 0.312383 0.949956i \(-0.398873\pi\)
0.312383 + 0.949956i \(0.398873\pi\)
\(710\) 0 0
\(711\) 26.9242 1.00974
\(712\) 0 0
\(713\) − 29.1352i − 1.09112i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 0.431820i − 0.0161266i
\(718\) 0 0
\(719\) −11.8158 −0.440656 −0.220328 0.975426i \(-0.570713\pi\)
−0.220328 + 0.975426i \(0.570713\pi\)
\(720\) 0 0
\(721\) −10.1140 −0.376664
\(722\) 0 0
\(723\) 10.4906i 0.390150i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 30.7767i 1.14145i 0.821143 + 0.570723i \(0.193337\pi\)
−0.821143 + 0.570723i \(0.806663\pi\)
\(728\) 0 0
\(729\) −4.04806 −0.149928
\(730\) 0 0
\(731\) 10.2168 0.377883
\(732\) 0 0
\(733\) 23.6782i 0.874572i 0.899322 + 0.437286i \(0.144060\pi\)
−0.899322 + 0.437286i \(0.855940\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.90299i 0.106933i
\(738\) 0 0
\(739\) 30.6569 1.12773 0.563866 0.825866i \(-0.309313\pi\)
0.563866 + 0.825866i \(0.309313\pi\)
\(740\) 0 0
\(741\) 1.10841 0.0407184
\(742\) 0 0
\(743\) 17.4164i 0.638946i 0.947595 + 0.319473i \(0.103506\pi\)
−0.947595 + 0.319473i \(0.896494\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 20.4687i 0.748911i
\(748\) 0 0
\(749\) −54.2886 −1.98366
\(750\) 0 0
\(751\) 24.2926 0.886449 0.443224 0.896411i \(-0.353834\pi\)
0.443224 + 0.896411i \(0.353834\pi\)
\(752\) 0 0
\(753\) − 12.1728i − 0.443600i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 34.1565i 1.24144i 0.784033 + 0.620719i \(0.213159\pi\)
−0.784033 + 0.620719i \(0.786841\pi\)
\(758\) 0 0
\(759\) 3.23067 0.117266
\(760\) 0 0
\(761\) 52.5347 1.90438 0.952190 0.305507i \(-0.0988260\pi\)
0.952190 + 0.305507i \(0.0988260\pi\)
\(762\) 0 0
\(763\) − 33.8891i − 1.22687i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 2.76933i − 0.0999948i
\(768\) 0 0
\(769\) 24.4010 0.879922 0.439961 0.898017i \(-0.354992\pi\)
0.439961 + 0.898017i \(0.354992\pi\)
\(770\) 0 0
\(771\) −19.0268 −0.685234
\(772\) 0 0
\(773\) 12.2966i 0.442278i 0.975242 + 0.221139i \(0.0709774\pi\)
−0.975242 + 0.221139i \(0.929023\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 5.90882i − 0.211978i
\(778\) 0 0
\(779\) 8.08315 0.289609
\(780\) 0 0
\(781\) 0.866337 0.0310000
\(782\) 0 0
\(783\) − 29.4262i − 1.05161i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 2.66831i 0.0951151i 0.998868 + 0.0475575i \(0.0151437\pi\)
−0.998868 + 0.0475575i \(0.984856\pi\)
\(788\) 0 0
\(789\) −12.2331 −0.435511
\(790\) 0 0
\(791\) −37.8581 −1.34608
\(792\) 0 0
\(793\) 3.61041i 0.128210i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 26.2054i 0.928243i 0.885771 + 0.464122i \(0.153630\pi\)
−0.885771 + 0.464122i \(0.846370\pi\)
\(798\) 0 0
\(799\) −3.20943 −0.113541
\(800\) 0 0
\(801\) −32.7669 −1.15776
\(802\) 0 0
\(803\) 5.87774i 0.207421i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 6.24608i − 0.219873i
\(808\) 0 0
\(809\) 20.0733 0.705740 0.352870 0.935672i \(-0.385206\pi\)
0.352870 + 0.935672i \(0.385206\pi\)
\(810\) 0 0
\(811\) 14.2575 0.500648 0.250324 0.968162i \(-0.419463\pi\)
0.250324 + 0.968162i \(0.419463\pi\)
\(812\) 0 0
\(813\) 1.18172i 0.0414448i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 11.6178i 0.406455i
\(818\) 0 0
\(819\) 7.54850 0.263766
\(820\) 0 0
\(821\) −16.0872 −0.561446 −0.280723 0.959789i \(-0.590574\pi\)
−0.280723 + 0.959789i \(0.590574\pi\)
\(822\) 0 0
\(823\) 27.6120i 0.962493i 0.876585 + 0.481247i \(0.159816\pi\)
−0.876585 + 0.481247i \(0.840184\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 20.4980i 0.712785i 0.934336 + 0.356393i \(0.115993\pi\)
−0.934336 + 0.356393i \(0.884007\pi\)
\(828\) 0 0
\(829\) 30.5232 1.06012 0.530058 0.847961i \(-0.322170\pi\)
0.530058 + 0.847961i \(0.322170\pi\)
\(830\) 0 0
\(831\) −15.0887 −0.523422
\(832\) 0 0
\(833\) 4.45150i 0.154235i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 33.7653i − 1.16710i
\(838\) 0 0
\(839\) −28.8737 −0.996832 −0.498416 0.866938i \(-0.666085\pi\)
−0.498416 + 0.866938i \(0.666085\pi\)
\(840\) 0 0
\(841\) 14.6976 0.506813
\(842\) 0 0
\(843\) − 4.62427i − 0.159268i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3.29258i 0.113134i
\(848\) 0 0
\(849\) −6.94391 −0.238314
\(850\) 0 0
\(851\) −8.19557 −0.280941
\(852\) 0 0
\(853\) 47.5729i 1.62886i 0.580259 + 0.814432i \(0.302951\pi\)
−0.580259 + 0.814432i \(0.697049\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.7148i 0.673445i 0.941604 + 0.336723i \(0.109318\pi\)
−0.941604 + 0.336723i \(0.890682\pi\)
\(858\) 0 0
\(859\) 47.6822 1.62689 0.813447 0.581639i \(-0.197588\pi\)
0.813447 + 0.581639i \(0.197588\pi\)
\(860\) 0 0
\(861\) −16.9861 −0.578886
\(862\) 0 0
\(863\) − 15.8297i − 0.538849i −0.963022 0.269424i \(-0.913167\pi\)
0.963022 0.269424i \(-0.0868334\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13.1688i 0.447234i
\(868\) 0 0
\(869\) −11.7441 −0.398391
\(870\) 0 0
\(871\) −2.90299 −0.0983642
\(872\) 0 0
\(873\) − 28.4882i − 0.964178i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 43.5248i − 1.46973i −0.678214 0.734864i \(-0.737246\pi\)
0.678214 0.734864i \(-0.262754\pi\)
\(878\) 0 0
\(879\) −12.3098 −0.415200
\(880\) 0 0
\(881\) 53.5461 1.80401 0.902006 0.431723i \(-0.142094\pi\)
0.902006 + 0.431723i \(0.142094\pi\)
\(882\) 0 0
\(883\) 45.2535i 1.52290i 0.648223 + 0.761450i \(0.275512\pi\)
−0.648223 + 0.761450i \(0.724488\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.54850i 0.0519936i 0.999662 + 0.0259968i \(0.00827598\pi\)
−0.999662 + 0.0259968i \(0.991724\pi\)
\(888\) 0 0
\(889\) 14.4207 0.483654
\(890\) 0 0
\(891\) −3.13366 −0.104982
\(892\) 0 0
\(893\) − 3.64952i − 0.122127i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3.23067i 0.107869i
\(898\) 0 0
\(899\) 50.1411 1.67230
\(900\) 0 0
\(901\) 11.8158 0.393642
\(902\) 0 0
\(903\) − 24.4140i − 0.812446i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 40.7090i 1.35172i 0.737030 + 0.675860i \(0.236228\pi\)
−0.737030 + 0.675860i \(0.763772\pi\)
\(908\) 0 0
\(909\) 4.94949 0.164164
\(910\) 0 0
\(911\) 24.9030 0.825073 0.412537 0.910941i \(-0.364643\pi\)
0.412537 + 0.910941i \(0.364643\pi\)
\(912\) 0 0
\(913\) − 8.92825i − 0.295482i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16.5004i 0.544893i
\(918\) 0 0
\(919\) −2.09299 −0.0690414 −0.0345207 0.999404i \(-0.510990\pi\)
−0.0345207 + 0.999404i \(0.510990\pi\)
\(920\) 0 0
\(921\) 15.4164 0.507988
\(922\) 0 0
\(923\) 0.866337i 0.0285158i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 7.04223i 0.231297i
\(928\) 0 0
\(929\) 36.7555 1.20591 0.602954 0.797776i \(-0.293990\pi\)
0.602954 + 0.797776i \(0.293990\pi\)
\(930\) 0 0
\(931\) −5.06191 −0.165897
\(932\) 0 0
\(933\) − 5.12809i − 0.167886i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 50.2673i − 1.64216i −0.570812 0.821081i \(-0.693371\pi\)
0.570812 0.821081i \(-0.306629\pi\)
\(938\) 0 0
\(939\) 0.267325 0.00872383
\(940\) 0 0
\(941\) 40.3000 1.31374 0.656871 0.754003i \(-0.271880\pi\)
0.656871 + 0.754003i \(0.271880\pi\)
\(942\) 0 0
\(943\) 23.5599i 0.767216i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 0.133663i − 0.00434345i −0.999998 0.00217173i \(-0.999309\pi\)
0.999998 0.00217173i \(-0.000691282\pi\)
\(948\) 0 0
\(949\) −5.87774 −0.190800
\(950\) 0 0
\(951\) −20.5054 −0.664933
\(952\) 0 0
\(953\) − 0.728661i − 0.0236037i −0.999930 0.0118018i \(-0.996243\pi\)
0.999930 0.0118018i \(-0.00375673\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 5.55991i 0.179726i
\(958\) 0 0
\(959\) 53.4783 1.72690
\(960\) 0 0
\(961\) 26.5347 0.855956
\(962\) 0 0
\(963\) 37.8004i 1.21810i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 56.5769i 1.81939i 0.415277 + 0.909695i \(0.363685\pi\)
−0.415277 + 0.909695i \(0.636315\pi\)
\(968\) 0 0
\(969\) 1.28455 0.0412658
\(970\) 0 0
\(971\) −48.8232 −1.56681 −0.783406 0.621511i \(-0.786519\pi\)
−0.783406 + 0.621511i \(0.786519\pi\)
\(972\) 0 0
\(973\) − 74.9242i − 2.40196i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 7.21099i − 0.230700i −0.993325 0.115350i \(-0.963201\pi\)
0.993325 0.115350i \(-0.0367990\pi\)
\(978\) 0 0
\(979\) 14.2926 0.456793
\(980\) 0 0
\(981\) −23.5966 −0.753380
\(982\) 0 0
\(983\) − 58.4278i − 1.86356i −0.363027 0.931779i \(-0.618257\pi\)
0.363027 0.931779i \(-0.381743\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 7.66920i 0.244113i
\(988\) 0 0
\(989\) −33.8623 −1.07676
\(990\) 0 0
\(991\) 24.0114 0.762747 0.381374 0.924421i \(-0.375451\pi\)
0.381374 + 0.924421i \(0.375451\pi\)
\(992\) 0 0
\(993\) 2.98458i 0.0947129i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 38.4222i 1.21684i 0.793614 + 0.608422i \(0.208197\pi\)
−0.793614 + 0.608422i \(0.791803\pi\)
\(998\) 0 0
\(999\) −9.49799 −0.300503
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 4400.2.b.bb.4049.4 6
4.3 odd 2 2200.2.b.m.1849.3 6
5.2 odd 4 4400.2.a.bz.1.2 3
5.3 odd 4 4400.2.a.by.1.2 3
5.4 even 2 inner 4400.2.b.bb.4049.3 6
20.3 even 4 2200.2.a.v.1.2 yes 3
20.7 even 4 2200.2.a.u.1.2 3
20.19 odd 2 2200.2.b.m.1849.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2200.2.a.u.1.2 3 20.7 even 4
2200.2.a.v.1.2 yes 3 20.3 even 4
2200.2.b.m.1849.3 6 4.3 odd 2
2200.2.b.m.1849.4 6 20.19 odd 2
4400.2.a.by.1.2 3 5.3 odd 4
4400.2.a.bz.1.2 3 5.2 odd 4
4400.2.b.bb.4049.3 6 5.4 even 2 inner
4400.2.b.bb.4049.4 6 1.1 even 1 trivial