Properties

Label 5700.2.f.n.3649.4
Level $5700$
Weight $2$
Character 5700.3649
Analytic conductor $45.515$
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] = [5700,2,Mod(3649,5700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5700, 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("5700.3649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5700.f (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: \(45.5147291521\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.4
Root \(2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 5700.3649
Dual form 5700.2.f.n.3649.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +4.60555i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +4.60555i q^{7} -1.00000 q^{9} +2.60555 q^{11} +2.60555i q^{13} +2.00000i q^{17} +1.00000 q^{19} -4.60555 q^{21} +2.00000i q^{23} -1.00000i q^{27} +4.60555 q^{29} +4.00000 q^{31} +2.60555i q^{33} +10.6056i q^{37} -2.60555 q^{39} -0.605551 q^{41} -3.39445i q^{43} +6.00000i q^{47} -14.2111 q^{49} -2.00000 q^{51} +1.00000i q^{57} +9.21110 q^{59} +7.21110 q^{61} -4.60555i q^{63} +4.00000i q^{67} -2.00000 q^{69} +5.21110 q^{71} -6.00000i q^{73} +12.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} -3.21110i q^{83} +4.60555i q^{87} +0.605551 q^{89} -12.0000 q^{91} +4.00000i q^{93} +9.39445i q^{97} -2.60555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 4 q^{11} + 4 q^{19} - 4 q^{21} + 4 q^{29} + 16 q^{31} + 4 q^{39} + 12 q^{41} - 28 q^{49} - 8 q^{51} + 8 q^{59} - 8 q^{69} - 8 q^{71} - 32 q^{79} + 4 q^{81} - 12 q^{89} - 48 q^{91} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5700\mathbb{Z}\right)^\times\).

\(n\) \(1901\) \(2851\) \(3877\) \(4201\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.60555i 1.74073i 0.492403 + 0.870367i \(0.336119\pi\)
−0.492403 + 0.870367i \(0.663881\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.60555 0.785603 0.392802 0.919623i \(-0.371506\pi\)
0.392802 + 0.919623i \(0.371506\pi\)
\(12\) 0 0
\(13\) 2.60555i 0.722650i 0.932440 + 0.361325i \(0.117675\pi\)
−0.932440 + 0.361325i \(0.882325\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −4.60555 −1.00501
\(22\) 0 0
\(23\) 2.00000i 0.417029i 0.978019 + 0.208514i \(0.0668628\pi\)
−0.978019 + 0.208514i \(0.933137\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 4.60555 0.855229 0.427615 0.903961i \(-0.359354\pi\)
0.427615 + 0.903961i \(0.359354\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 2.60555i 0.453568i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.6056i 1.74354i 0.489914 + 0.871771i \(0.337028\pi\)
−0.489914 + 0.871771i \(0.662972\pi\)
\(38\) 0 0
\(39\) −2.60555 −0.417222
\(40\) 0 0
\(41\) −0.605551 −0.0945712 −0.0472856 0.998881i \(-0.515057\pi\)
−0.0472856 + 0.998881i \(0.515057\pi\)
\(42\) 0 0
\(43\) − 3.39445i − 0.517649i −0.965924 0.258824i \(-0.916665\pi\)
0.965924 0.258824i \(-0.0833351\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) 0 0
\(49\) −14.2111 −2.03016
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000i 0.132453i
\(58\) 0 0
\(59\) 9.21110 1.19918 0.599592 0.800306i \(-0.295330\pi\)
0.599592 + 0.800306i \(0.295330\pi\)
\(60\) 0 0
\(61\) 7.21110 0.923287 0.461644 0.887066i \(-0.347260\pi\)
0.461644 + 0.887066i \(0.347260\pi\)
\(62\) 0 0
\(63\) − 4.60555i − 0.580245i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) 5.21110 0.618444 0.309222 0.950990i \(-0.399931\pi\)
0.309222 + 0.950990i \(0.399931\pi\)
\(72\) 0 0
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.0000i 1.36753i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 3.21110i − 0.352464i −0.984349 0.176232i \(-0.943609\pi\)
0.984349 0.176232i \(-0.0563909\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.60555i 0.493767i
\(88\) 0 0
\(89\) 0.605551 0.0641883 0.0320942 0.999485i \(-0.489782\pi\)
0.0320942 + 0.999485i \(0.489782\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) 0 0
\(93\) 4.00000i 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.39445i 0.953862i 0.878941 + 0.476931i \(0.158251\pi\)
−0.878941 + 0.476931i \(0.841749\pi\)
\(98\) 0 0
\(99\) −2.60555 −0.261868
\(100\) 0 0
\(101\) 7.21110 0.717532 0.358766 0.933428i \(-0.383198\pi\)
0.358766 + 0.933428i \(0.383198\pi\)
\(102\) 0 0
\(103\) 10.4222i 1.02693i 0.858110 + 0.513465i \(0.171638\pi\)
−0.858110 + 0.513465i \(0.828362\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 10.4222i − 1.00755i −0.863834 0.503776i \(-0.831944\pi\)
0.863834 0.503776i \(-0.168056\pi\)
\(108\) 0 0
\(109\) −3.21110 −0.307568 −0.153784 0.988105i \(-0.549146\pi\)
−0.153784 + 0.988105i \(0.549146\pi\)
\(110\) 0 0
\(111\) −10.6056 −1.00663
\(112\) 0 0
\(113\) − 13.2111i − 1.24280i −0.783495 0.621398i \(-0.786565\pi\)
0.783495 0.621398i \(-0.213435\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 2.60555i − 0.240883i
\(118\) 0 0
\(119\) −9.21110 −0.844380
\(120\) 0 0
\(121\) −4.21110 −0.382828
\(122\) 0 0
\(123\) − 0.605551i − 0.0546007i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 10.4222i − 0.924821i −0.886666 0.462411i \(-0.846985\pi\)
0.886666 0.462411i \(-0.153015\pi\)
\(128\) 0 0
\(129\) 3.39445 0.298865
\(130\) 0 0
\(131\) −22.6056 −1.97506 −0.987528 0.157443i \(-0.949675\pi\)
−0.987528 + 0.157443i \(0.949675\pi\)
\(132\) 0 0
\(133\) 4.60555i 0.399352i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.4222i 1.40304i 0.712648 + 0.701522i \(0.247496\pi\)
−0.712648 + 0.701522i \(0.752504\pi\)
\(138\) 0 0
\(139\) 5.21110 0.442000 0.221000 0.975274i \(-0.429068\pi\)
0.221000 + 0.975274i \(0.429068\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 6.78890i 0.567716i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 14.2111i − 1.17211i
\(148\) 0 0
\(149\) 12.4222 1.01767 0.508833 0.860865i \(-0.330077\pi\)
0.508833 + 0.860865i \(0.330077\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) − 2.00000i − 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.788897i 0.0629609i 0.999504 + 0.0314804i \(0.0100222\pi\)
−0.999504 + 0.0314804i \(0.989978\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −9.21110 −0.725937
\(162\) 0 0
\(163\) − 19.0278i − 1.49037i −0.666858 0.745184i \(-0.732361\pi\)
0.666858 0.745184i \(-0.267639\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 21.2111i − 1.64136i −0.571385 0.820682i \(-0.693594\pi\)
0.571385 0.820682i \(-0.306406\pi\)
\(168\) 0 0
\(169\) 6.21110 0.477777
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) − 5.21110i − 0.396193i −0.980183 0.198096i \(-0.936524\pi\)
0.980183 0.198096i \(-0.0634759\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 9.21110i 0.692349i
\(178\) 0 0
\(179\) 6.78890 0.507426 0.253713 0.967280i \(-0.418348\pi\)
0.253713 + 0.967280i \(0.418348\pi\)
\(180\) 0 0
\(181\) −15.2111 −1.13063 −0.565316 0.824874i \(-0.691246\pi\)
−0.565316 + 0.824874i \(0.691246\pi\)
\(182\) 0 0
\(183\) 7.21110i 0.533060i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 5.21110i 0.381074i
\(188\) 0 0
\(189\) 4.60555 0.335005
\(190\) 0 0
\(191\) 15.8167 1.14445 0.572226 0.820096i \(-0.306080\pi\)
0.572226 + 0.820096i \(0.306080\pi\)
\(192\) 0 0
\(193\) − 6.60555i − 0.475478i −0.971329 0.237739i \(-0.923594\pi\)
0.971329 0.237739i \(-0.0764063\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.0000i 0.997459i 0.866758 + 0.498729i \(0.166200\pi\)
−0.866758 + 0.498729i \(0.833800\pi\)
\(198\) 0 0
\(199\) −2.78890 −0.197700 −0.0988498 0.995102i \(-0.531516\pi\)
−0.0988498 + 0.995102i \(0.531516\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 21.2111i 1.48873i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 2.00000i − 0.139010i
\(208\) 0 0
\(209\) 2.60555 0.180230
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 5.21110i 0.357059i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 18.4222i 1.25058i
\(218\) 0 0
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) −5.21110 −0.350537
\(222\) 0 0
\(223\) − 18.4222i − 1.23364i −0.787103 0.616821i \(-0.788420\pi\)
0.787103 0.616821i \(-0.211580\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.21110i 0.0803837i 0.999192 + 0.0401918i \(0.0127969\pi\)
−0.999192 + 0.0401918i \(0.987203\pi\)
\(228\) 0 0
\(229\) −11.2111 −0.740851 −0.370425 0.928862i \(-0.620788\pi\)
−0.370425 + 0.928862i \(0.620788\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 0 0
\(233\) − 12.4222i − 0.813806i −0.913471 0.406903i \(-0.866609\pi\)
0.913471 0.406903i \(-0.133391\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 8.00000i − 0.519656i
\(238\) 0 0
\(239\) −4.18335 −0.270598 −0.135299 0.990805i \(-0.543200\pi\)
−0.135299 + 0.990805i \(0.543200\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.60555i 0.165787i
\(248\) 0 0
\(249\) 3.21110 0.203495
\(250\) 0 0
\(251\) −27.8167 −1.75577 −0.877886 0.478870i \(-0.841047\pi\)
−0.877886 + 0.478870i \(0.841047\pi\)
\(252\) 0 0
\(253\) 5.21110i 0.327619i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 17.2111i 1.07360i 0.843710 + 0.536800i \(0.180367\pi\)
−0.843710 + 0.536800i \(0.819633\pi\)
\(258\) 0 0
\(259\) −48.8444 −3.03504
\(260\) 0 0
\(261\) −4.60555 −0.285076
\(262\) 0 0
\(263\) 7.21110i 0.444656i 0.974972 + 0.222328i \(0.0713655\pi\)
−0.974972 + 0.222328i \(0.928634\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0.605551i 0.0370591i
\(268\) 0 0
\(269\) 24.6056 1.50023 0.750113 0.661309i \(-0.229999\pi\)
0.750113 + 0.661309i \(0.229999\pi\)
\(270\) 0 0
\(271\) −31.6333 −1.92159 −0.960793 0.277266i \(-0.910572\pi\)
−0.960793 + 0.277266i \(0.910572\pi\)
\(272\) 0 0
\(273\) − 12.0000i − 0.726273i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −32.2389 −1.92321 −0.961605 0.274439i \(-0.911508\pi\)
−0.961605 + 0.274439i \(0.911508\pi\)
\(282\) 0 0
\(283\) 11.3944i 0.677330i 0.940907 + 0.338665i \(0.109975\pi\)
−0.940907 + 0.338665i \(0.890025\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 2.78890i − 0.164623i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −9.39445 −0.550712
\(292\) 0 0
\(293\) 23.6333i 1.38067i 0.723489 + 0.690336i \(0.242537\pi\)
−0.723489 + 0.690336i \(0.757463\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 2.60555i − 0.151189i
\(298\) 0 0
\(299\) −5.21110 −0.301366
\(300\) 0 0
\(301\) 15.6333 0.901089
\(302\) 0 0
\(303\) 7.21110i 0.414267i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 31.6333i 1.80541i 0.430262 + 0.902704i \(0.358421\pi\)
−0.430262 + 0.902704i \(0.641579\pi\)
\(308\) 0 0
\(309\) −10.4222 −0.592899
\(310\) 0 0
\(311\) 9.39445 0.532710 0.266355 0.963875i \(-0.414181\pi\)
0.266355 + 0.963875i \(0.414181\pi\)
\(312\) 0 0
\(313\) − 15.2111i − 0.859782i −0.902881 0.429891i \(-0.858552\pi\)
0.902881 0.429891i \(-0.141448\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.4222i 1.25936i 0.776856 + 0.629678i \(0.216813\pi\)
−0.776856 + 0.629678i \(0.783187\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) 10.4222 0.581711
\(322\) 0 0
\(323\) 2.00000i 0.111283i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 3.21110i − 0.177574i
\(328\) 0 0
\(329\) −27.6333 −1.52347
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) − 10.6056i − 0.581181i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 33.0278i − 1.79914i −0.436780 0.899568i \(-0.643881\pi\)
0.436780 0.899568i \(-0.356119\pi\)
\(338\) 0 0
\(339\) 13.2111 0.717529
\(340\) 0 0
\(341\) 10.4222 0.564394
\(342\) 0 0
\(343\) − 33.2111i − 1.79323i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 21.6333i − 1.16134i −0.814140 0.580668i \(-0.802791\pi\)
0.814140 0.580668i \(-0.197209\pi\)
\(348\) 0 0
\(349\) 22.8444 1.22283 0.611417 0.791309i \(-0.290600\pi\)
0.611417 + 0.791309i \(0.290600\pi\)
\(350\) 0 0
\(351\) 2.60555 0.139074
\(352\) 0 0
\(353\) 28.4222i 1.51276i 0.654132 + 0.756381i \(0.273034\pi\)
−0.654132 + 0.756381i \(0.726966\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 9.21110i − 0.487503i
\(358\) 0 0
\(359\) −7.81665 −0.412547 −0.206274 0.978494i \(-0.566134\pi\)
−0.206274 + 0.978494i \(0.566134\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) − 4.21110i − 0.221026i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 11.0278i − 0.575644i −0.957684 0.287822i \(-0.907069\pi\)
0.957684 0.287822i \(-0.0929312\pi\)
\(368\) 0 0
\(369\) 0.605551 0.0315237
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 9.39445i − 0.486426i −0.969973 0.243213i \(-0.921799\pi\)
0.969973 0.243213i \(-0.0782014\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) 2.42221 0.124420 0.0622102 0.998063i \(-0.480185\pi\)
0.0622102 + 0.998063i \(0.480185\pi\)
\(380\) 0 0
\(381\) 10.4222 0.533946
\(382\) 0 0
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.39445i 0.172550i
\(388\) 0 0
\(389\) −7.57779 −0.384209 −0.192105 0.981374i \(-0.561531\pi\)
−0.192105 + 0.981374i \(0.561531\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) − 22.6056i − 1.14030i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 36.4222i − 1.82798i −0.405739 0.913989i \(-0.632986\pi\)
0.405739 0.913989i \(-0.367014\pi\)
\(398\) 0 0
\(399\) −4.60555 −0.230566
\(400\) 0 0
\(401\) 32.6056 1.62824 0.814122 0.580694i \(-0.197219\pi\)
0.814122 + 0.580694i \(0.197219\pi\)
\(402\) 0 0
\(403\) 10.4222i 0.519167i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 27.6333i 1.36973i
\(408\) 0 0
\(409\) 27.2111 1.34550 0.672751 0.739869i \(-0.265112\pi\)
0.672751 + 0.739869i \(0.265112\pi\)
\(410\) 0 0
\(411\) −16.4222 −0.810048
\(412\) 0 0
\(413\) 42.4222i 2.08746i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5.21110i 0.255189i
\(418\) 0 0
\(419\) −29.0278 −1.41810 −0.709049 0.705159i \(-0.750876\pi\)
−0.709049 + 0.705159i \(0.750876\pi\)
\(420\) 0 0
\(421\) 20.4222 0.995317 0.497659 0.867373i \(-0.334193\pi\)
0.497659 + 0.867373i \(0.334193\pi\)
\(422\) 0 0
\(423\) − 6.00000i − 0.291730i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 33.2111i 1.60720i
\(428\) 0 0
\(429\) −6.78890 −0.327771
\(430\) 0 0
\(431\) 5.21110 0.251010 0.125505 0.992093i \(-0.459945\pi\)
0.125505 + 0.992093i \(0.459945\pi\)
\(432\) 0 0
\(433\) 9.39445i 0.451468i 0.974189 + 0.225734i \(0.0724781\pi\)
−0.974189 + 0.225734i \(0.927522\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.00000i 0.0956730i
\(438\) 0 0
\(439\) −5.57779 −0.266214 −0.133107 0.991102i \(-0.542495\pi\)
−0.133107 + 0.991102i \(0.542495\pi\)
\(440\) 0 0
\(441\) 14.2111 0.676719
\(442\) 0 0
\(443\) 38.8444i 1.84555i 0.385335 + 0.922777i \(0.374086\pi\)
−0.385335 + 0.922777i \(0.625914\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 12.4222i 0.587550i
\(448\) 0 0
\(449\) −32.2389 −1.52145 −0.760723 0.649077i \(-0.775155\pi\)
−0.760723 + 0.649077i \(0.775155\pi\)
\(450\) 0 0
\(451\) −1.57779 −0.0742955
\(452\) 0 0
\(453\) − 12.0000i − 0.563809i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 40.0555i − 1.87372i −0.349708 0.936859i \(-0.613719\pi\)
0.349708 0.936859i \(-0.386281\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 34.8444 1.62287 0.811433 0.584445i \(-0.198688\pi\)
0.811433 + 0.584445i \(0.198688\pi\)
\(462\) 0 0
\(463\) 3.02776i 0.140712i 0.997522 + 0.0703559i \(0.0224135\pi\)
−0.997522 + 0.0703559i \(0.977587\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 34.8444i − 1.61241i −0.591638 0.806204i \(-0.701519\pi\)
0.591638 0.806204i \(-0.298481\pi\)
\(468\) 0 0
\(469\) −18.4222 −0.850658
\(470\) 0 0
\(471\) −0.788897 −0.0363505
\(472\) 0 0
\(473\) − 8.84441i − 0.406666i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −21.3944 −0.977537 −0.488769 0.872413i \(-0.662554\pi\)
−0.488769 + 0.872413i \(0.662554\pi\)
\(480\) 0 0
\(481\) −27.6333 −1.25997
\(482\) 0 0
\(483\) − 9.21110i − 0.419120i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 43.6333i 1.97721i 0.150520 + 0.988607i \(0.451905\pi\)
−0.150520 + 0.988607i \(0.548095\pi\)
\(488\) 0 0
\(489\) 19.0278 0.860465
\(490\) 0 0
\(491\) 38.2389 1.72570 0.862848 0.505464i \(-0.168679\pi\)
0.862848 + 0.505464i \(0.168679\pi\)
\(492\) 0 0
\(493\) 9.21110i 0.414847i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.0000i 1.07655i
\(498\) 0 0
\(499\) −15.6333 −0.699843 −0.349921 0.936779i \(-0.613792\pi\)
−0.349921 + 0.936779i \(0.613792\pi\)
\(500\) 0 0
\(501\) 21.2111 0.947642
\(502\) 0 0
\(503\) − 27.2111i − 1.21328i −0.794975 0.606642i \(-0.792516\pi\)
0.794975 0.606642i \(-0.207484\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6.21110i 0.275845i
\(508\) 0 0
\(509\) −29.4500 −1.30535 −0.652673 0.757639i \(-0.726353\pi\)
−0.652673 + 0.757639i \(0.726353\pi\)
\(510\) 0 0
\(511\) 27.6333 1.22243
\(512\) 0 0
\(513\) − 1.00000i − 0.0441511i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 15.6333i 0.687552i
\(518\) 0 0
\(519\) 5.21110 0.228742
\(520\) 0 0
\(521\) −25.8167 −1.13105 −0.565524 0.824732i \(-0.691326\pi\)
−0.565524 + 0.824732i \(0.691326\pi\)
\(522\) 0 0
\(523\) 10.7889i 0.471766i 0.971782 + 0.235883i \(0.0757981\pi\)
−0.971782 + 0.235883i \(0.924202\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.00000i 0.348485i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) −9.21110 −0.399728
\(532\) 0 0
\(533\) − 1.57779i − 0.0683419i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 6.78890i 0.292963i
\(538\) 0 0
\(539\) −37.0278 −1.59490
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 0 0
\(543\) − 15.2111i − 0.652771i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 37.2111i − 1.59103i −0.605933 0.795516i \(-0.707200\pi\)
0.605933 0.795516i \(-0.292800\pi\)
\(548\) 0 0
\(549\) −7.21110 −0.307762
\(550\) 0 0
\(551\) 4.60555 0.196203
\(552\) 0 0
\(553\) − 36.8444i − 1.56678i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 24.4222i 1.03480i 0.855743 + 0.517401i \(0.173100\pi\)
−0.855743 + 0.517401i \(0.826900\pi\)
\(558\) 0 0
\(559\) 8.84441 0.374079
\(560\) 0 0
\(561\) −5.21110 −0.220013
\(562\) 0 0
\(563\) 11.6333i 0.490285i 0.969487 + 0.245143i \(0.0788348\pi\)
−0.969487 + 0.245143i \(0.921165\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.60555i 0.193415i
\(568\) 0 0
\(569\) 25.4500 1.06692 0.533459 0.845826i \(-0.320892\pi\)
0.533459 + 0.845826i \(0.320892\pi\)
\(570\) 0 0
\(571\) −35.6333 −1.49121 −0.745604 0.666390i \(-0.767839\pi\)
−0.745604 + 0.666390i \(0.767839\pi\)
\(572\) 0 0
\(573\) 15.8167i 0.660750i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 8.78890i − 0.365887i −0.983123 0.182943i \(-0.941438\pi\)
0.983123 0.182943i \(-0.0585625\pi\)
\(578\) 0 0
\(579\) 6.60555 0.274517
\(580\) 0 0
\(581\) 14.7889 0.613547
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 16.4222i − 0.677817i −0.940819 0.338908i \(-0.889942\pi\)
0.940819 0.338908i \(-0.110058\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) −14.0000 −0.575883
\(592\) 0 0
\(593\) 32.4222i 1.33142i 0.746210 + 0.665710i \(0.231871\pi\)
−0.746210 + 0.665710i \(0.768129\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 2.78890i − 0.114142i
\(598\) 0 0
\(599\) 20.8444 0.851680 0.425840 0.904799i \(-0.359979\pi\)
0.425840 + 0.904799i \(0.359979\pi\)
\(600\) 0 0
\(601\) 19.2111 0.783637 0.391819 0.920042i \(-0.371846\pi\)
0.391819 + 0.920042i \(0.371846\pi\)
\(602\) 0 0
\(603\) − 4.00000i − 0.162893i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 19.6333i 0.796891i 0.917192 + 0.398446i \(0.130450\pi\)
−0.917192 + 0.398446i \(0.869550\pi\)
\(608\) 0 0
\(609\) −21.2111 −0.859517
\(610\) 0 0
\(611\) −15.6333 −0.632456
\(612\) 0 0
\(613\) − 19.2111i − 0.775929i −0.921674 0.387965i \(-0.873178\pi\)
0.921674 0.387965i \(-0.126822\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 34.0000i 1.36879i 0.729112 + 0.684394i \(0.239933\pi\)
−0.729112 + 0.684394i \(0.760067\pi\)
\(618\) 0 0
\(619\) 47.6333 1.91454 0.957272 0.289189i \(-0.0933855\pi\)
0.957272 + 0.289189i \(0.0933855\pi\)
\(620\) 0 0
\(621\) 2.00000 0.0802572
\(622\) 0 0
\(623\) 2.78890i 0.111735i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 2.60555i 0.104056i
\(628\) 0 0
\(629\) −21.2111 −0.845742
\(630\) 0 0
\(631\) −20.8444 −0.829803 −0.414901 0.909866i \(-0.636184\pi\)
−0.414901 + 0.909866i \(0.636184\pi\)
\(632\) 0 0
\(633\) − 4.00000i − 0.158986i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 37.0278i − 1.46709i
\(638\) 0 0
\(639\) −5.21110 −0.206148
\(640\) 0 0
\(641\) −29.4500 −1.16320 −0.581602 0.813474i \(-0.697574\pi\)
−0.581602 + 0.813474i \(0.697574\pi\)
\(642\) 0 0
\(643\) − 12.6056i − 0.497114i −0.968617 0.248557i \(-0.920044\pi\)
0.968617 0.248557i \(-0.0799564\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 45.6333i − 1.79403i −0.442000 0.897015i \(-0.645731\pi\)
0.442000 0.897015i \(-0.354269\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) −18.4222 −0.722023
\(652\) 0 0
\(653\) − 42.0000i − 1.64359i −0.569785 0.821794i \(-0.692974\pi\)
0.569785 0.821794i \(-0.307026\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −16.7889 −0.653012 −0.326506 0.945195i \(-0.605871\pi\)
−0.326506 + 0.945195i \(0.605871\pi\)
\(662\) 0 0
\(663\) − 5.21110i − 0.202382i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9.21110i 0.356655i
\(668\) 0 0
\(669\) 18.4222 0.712244
\(670\) 0 0
\(671\) 18.7889 0.725337
\(672\) 0 0
\(673\) − 29.3944i − 1.13307i −0.824037 0.566536i \(-0.808283\pi\)
0.824037 0.566536i \(-0.191717\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 18.4222i − 0.708023i −0.935241 0.354011i \(-0.884817\pi\)
0.935241 0.354011i \(-0.115183\pi\)
\(678\) 0 0
\(679\) −43.2666 −1.66042
\(680\) 0 0
\(681\) −1.21110 −0.0464096
\(682\) 0 0
\(683\) 48.0000i 1.83667i 0.395805 + 0.918334i \(0.370466\pi\)
−0.395805 + 0.918334i \(0.629534\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 11.2111i − 0.427730i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 30.0555 1.14337 0.571683 0.820475i \(-0.306291\pi\)
0.571683 + 0.820475i \(0.306291\pi\)
\(692\) 0 0
\(693\) − 12.0000i − 0.455842i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 1.21110i − 0.0458738i
\(698\) 0 0
\(699\) 12.4222 0.469851
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 10.6056i 0.399996i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 33.2111i 1.24903i
\(708\) 0 0
\(709\) −17.6333 −0.662233 −0.331116 0.943590i \(-0.607425\pi\)
−0.331116 + 0.943590i \(0.607425\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 8.00000i 0.299602i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 4.18335i − 0.156230i
\(718\) 0 0
\(719\) 34.2389 1.27689 0.638447 0.769666i \(-0.279577\pi\)
0.638447 + 0.769666i \(0.279577\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) 0 0
\(723\) − 10.0000i − 0.371904i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 24.6056i − 0.912569i −0.889834 0.456285i \(-0.849180\pi\)
0.889834 0.456285i \(-0.150820\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 6.78890 0.251096
\(732\) 0 0
\(733\) − 4.42221i − 0.163338i −0.996660 0.0816689i \(-0.973975\pi\)
0.996660 0.0816689i \(-0.0260250\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.4222i 0.383907i
\(738\) 0 0
\(739\) −9.57779 −0.352325 −0.176162 0.984361i \(-0.556368\pi\)
−0.176162 + 0.984361i \(0.556368\pi\)
\(740\) 0 0
\(741\) −2.60555 −0.0957173
\(742\) 0 0
\(743\) 20.0000i 0.733729i 0.930274 + 0.366864i \(0.119569\pi\)
−0.930274 + 0.366864i \(0.880431\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 3.21110i 0.117488i
\(748\) 0 0
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) −16.8444 −0.614661 −0.307331 0.951603i \(-0.599436\pi\)
−0.307331 + 0.951603i \(0.599436\pi\)
\(752\) 0 0
\(753\) − 27.8167i − 1.01370i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 36.0555i 1.31046i 0.755429 + 0.655230i \(0.227428\pi\)
−0.755429 + 0.655230i \(0.772572\pi\)
\(758\) 0 0
\(759\) −5.21110 −0.189151
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) − 14.7889i − 0.535394i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000i 0.866590i
\(768\) 0 0
\(769\) −8.42221 −0.303712 −0.151856 0.988403i \(-0.548525\pi\)
−0.151856 + 0.988403i \(0.548525\pi\)
\(770\) 0 0
\(771\) −17.2111 −0.619843
\(772\) 0 0
\(773\) − 43.2666i − 1.55619i −0.628145 0.778096i \(-0.716186\pi\)
0.628145 0.778096i \(-0.283814\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 48.8444i − 1.75228i
\(778\) 0 0
\(779\) −0.605551 −0.0216961
\(780\) 0 0
\(781\) 13.5778 0.485852
\(782\) 0 0
\(783\) − 4.60555i − 0.164589i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 23.6333i 0.842436i 0.906959 + 0.421218i \(0.138397\pi\)
−0.906959 + 0.421218i \(0.861603\pi\)
\(788\) 0 0
\(789\) −7.21110 −0.256722
\(790\) 0 0
\(791\) 60.8444 2.16338
\(792\) 0 0
\(793\) 18.7889i 0.667213i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.78890i 0.0987878i 0.998779 + 0.0493939i \(0.0157290\pi\)
−0.998779 + 0.0493939i \(0.984271\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) −0.605551 −0.0213961
\(802\) 0 0
\(803\) − 15.6333i − 0.551687i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 24.6056i 0.866156i
\(808\) 0 0
\(809\) 19.5778 0.688319 0.344159 0.938911i \(-0.388164\pi\)
0.344159 + 0.938911i \(0.388164\pi\)
\(810\) 0 0
\(811\) 16.8444 0.591487 0.295744 0.955267i \(-0.404433\pi\)
0.295744 + 0.955267i \(0.404433\pi\)
\(812\) 0 0
\(813\) − 31.6333i − 1.10943i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 3.39445i − 0.118757i
\(818\) 0 0
\(819\) 12.0000 0.419314
\(820\) 0 0
\(821\) 28.7889 1.00474 0.502370 0.864653i \(-0.332462\pi\)
0.502370 + 0.864653i \(0.332462\pi\)
\(822\) 0 0
\(823\) 26.1833i 0.912694i 0.889802 + 0.456347i \(0.150842\pi\)
−0.889802 + 0.456347i \(0.849158\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 15.6333i 0.543623i 0.962350 + 0.271812i \(0.0876228\pi\)
−0.962350 + 0.271812i \(0.912377\pi\)
\(828\) 0 0
\(829\) −0.788897 −0.0273995 −0.0136998 0.999906i \(-0.504361\pi\)
−0.0136998 + 0.999906i \(0.504361\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) − 28.4222i − 0.984771i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 4.00000i − 0.138260i
\(838\) 0 0
\(839\) 15.6333 0.539722 0.269861 0.962899i \(-0.413022\pi\)
0.269861 + 0.962899i \(0.413022\pi\)
\(840\) 0 0
\(841\) −7.78890 −0.268583
\(842\) 0 0
\(843\) − 32.2389i − 1.11037i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 19.3944i − 0.666401i
\(848\) 0 0
\(849\) −11.3944 −0.391056
\(850\) 0 0
\(851\) −21.2111 −0.727107
\(852\) 0 0
\(853\) 13.6333i 0.466796i 0.972381 + 0.233398i \(0.0749844\pi\)
−0.972381 + 0.233398i \(0.925016\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 22.4222i − 0.765928i −0.923763 0.382964i \(-0.874903\pi\)
0.923763 0.382964i \(-0.125097\pi\)
\(858\) 0 0
\(859\) 48.8444 1.66655 0.833275 0.552859i \(-0.186463\pi\)
0.833275 + 0.552859i \(0.186463\pi\)
\(860\) 0 0
\(861\) 2.78890 0.0950454
\(862\) 0 0
\(863\) 22.4222i 0.763261i 0.924315 + 0.381630i \(0.124637\pi\)
−0.924315 + 0.381630i \(0.875363\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13.0000i 0.441503i
\(868\) 0 0
\(869\) −20.8444 −0.707098
\(870\) 0 0
\(871\) −10.4222 −0.353143
\(872\) 0 0
\(873\) − 9.39445i − 0.317954i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 9.02776i − 0.304846i −0.988315 0.152423i \(-0.951292\pi\)
0.988315 0.152423i \(-0.0487076\pi\)
\(878\) 0 0
\(879\) −23.6333 −0.797132
\(880\) 0 0
\(881\) 44.7889 1.50898 0.754488 0.656314i \(-0.227885\pi\)
0.754488 + 0.656314i \(0.227885\pi\)
\(882\) 0 0
\(883\) − 6.18335i − 0.208086i −0.994573 0.104043i \(-0.966822\pi\)
0.994573 0.104043i \(-0.0331780\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 36.8444i 1.23711i 0.785740 + 0.618557i \(0.212282\pi\)
−0.785740 + 0.618557i \(0.787718\pi\)
\(888\) 0 0
\(889\) 48.0000 1.60987
\(890\) 0 0
\(891\) 2.60555 0.0872893
\(892\) 0 0
\(893\) 6.00000i 0.200782i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 5.21110i − 0.173994i
\(898\) 0 0
\(899\) 18.4222 0.614415
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 15.6333i 0.520244i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 42.0555i − 1.39643i −0.715888 0.698215i \(-0.753978\pi\)
0.715888 0.698215i \(-0.246022\pi\)
\(908\) 0 0
\(909\) −7.21110 −0.239177
\(910\) 0 0
\(911\) −18.4222 −0.610355 −0.305177 0.952296i \(-0.598716\pi\)
−0.305177 + 0.952296i \(0.598716\pi\)
\(912\) 0 0
\(913\) − 8.36669i − 0.276897i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 104.111i − 3.43805i
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −31.6333 −1.04235
\(922\) 0 0
\(923\) 13.5778i 0.446919i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 10.4222i − 0.342310i
\(928\) 0 0
\(929\) 2.36669 0.0776487 0.0388243 0.999246i \(-0.487639\pi\)
0.0388243 + 0.999246i \(0.487639\pi\)
\(930\) 0 0
\(931\) −14.2111 −0.465750
\(932\) 0 0
\(933\) 9.39445i 0.307560i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 4.78890i − 0.156446i −0.996936 0.0782232i \(-0.975075\pi\)
0.996936 0.0782232i \(-0.0249247\pi\)
\(938\) 0 0
\(939\) 15.2111 0.496396
\(940\) 0 0
\(941\) 20.2389 0.659768 0.329884 0.944021i \(-0.392990\pi\)
0.329884 + 0.944021i \(0.392990\pi\)
\(942\) 0 0
\(943\) − 1.21110i − 0.0394389i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 7.21110i 0.234329i 0.993113 + 0.117165i \(0.0373805\pi\)
−0.993113 + 0.117165i \(0.962619\pi\)
\(948\) 0 0
\(949\) 15.6333 0.507479
\(950\) 0 0
\(951\) −22.4222 −0.727090
\(952\) 0 0
\(953\) 9.21110i 0.298377i 0.988809 + 0.149188i \(0.0476661\pi\)
−0.988809 + 0.149188i \(0.952334\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 12.0000i 0.387905i
\(958\) 0 0
\(959\) −75.6333 −2.44233
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 10.4222i 0.335851i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 24.2389i 0.779469i 0.920927 + 0.389735i \(0.127433\pi\)
−0.920927 + 0.389735i \(0.872567\pi\)
\(968\) 0 0
\(969\) −2.00000 −0.0642493
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 24.0000i 0.769405i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 22.4222i − 0.717350i −0.933463 0.358675i \(-0.883229\pi\)
0.933463 0.358675i \(-0.116771\pi\)
\(978\) 0 0
\(979\) 1.57779 0.0504265
\(980\) 0 0
\(981\) 3.21110 0.102523
\(982\) 0 0
\(983\) − 5.21110i − 0.166208i −0.996541 0.0831042i \(-0.973517\pi\)
0.996541 0.0831042i \(-0.0264834\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 27.6333i − 0.879578i
\(988\) 0 0
\(989\) 6.78890 0.215874
\(990\) 0 0
\(991\) 21.5778 0.685441 0.342721 0.939437i \(-0.388652\pi\)
0.342721 + 0.939437i \(0.388652\pi\)
\(992\) 0 0
\(993\) − 8.00000i − 0.253872i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 18.3667i − 0.581679i −0.956772 0.290839i \(-0.906065\pi\)
0.956772 0.290839i \(-0.0939346\pi\)
\(998\) 0 0
\(999\) 10.6056 0.335545
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 5700.2.f.n.3649.4 4
5.2 odd 4 5700.2.a.u.1.1 2
5.3 odd 4 1140.2.a.e.1.2 2
5.4 even 2 inner 5700.2.f.n.3649.1 4
15.8 even 4 3420.2.a.i.1.2 2
20.3 even 4 4560.2.a.bl.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.a.e.1.2 2 5.3 odd 4
3420.2.a.i.1.2 2 15.8 even 4
4560.2.a.bl.1.1 2 20.3 even 4
5700.2.a.u.1.1 2 5.2 odd 4
5700.2.f.n.3649.1 4 5.4 even 2 inner
5700.2.f.n.3649.4 4 1.1 even 1 trivial