Properties

Label 3800.2.d.n.3649.5
Level $3800$
Weight $2$
Character 3800.3649
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,2,Mod(3649,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, 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("3800.3649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.d (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: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.5
Root \(1.40680 - 0.144584i\) of defining polynomial
Character \(\chi\) \(=\) 3800.3649
Dual form 3800.2.d.n.3649.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.81361i q^{3} +4.91638i q^{7} -0.289169 q^{9} +O(q^{10})\) \(q+1.81361i q^{3} +4.91638i q^{7} -0.289169 q^{9} +0.578337 q^{11} -6.39194i q^{13} +0.710831i q^{17} -1.00000 q^{19} -8.91638 q^{21} -2.71083i q^{23} +4.91638i q^{27} -6.54359 q^{29} +1.42166 q^{31} +1.04888i q^{33} +9.10278i q^{37} +11.5925 q^{39} -11.0489 q^{41} +5.83276i q^{43} +1.15667i q^{47} -17.1708 q^{49} -1.28917 q^{51} +13.2736i q^{53} -1.81361i q^{57} -11.3869 q^{59} -9.04888 q^{61} -1.42166i q^{63} -2.97028i q^{67} +4.91638 q^{69} -9.38692i q^{73} +2.84333i q^{77} -4.37279 q^{79} -9.78389 q^{81} -0.372787i q^{83} -11.8675i q^{87} +16.6167 q^{89} +31.4252 q^{91} +2.57834i q^{93} -3.94610i q^{97} -0.167237 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{19} - 26 q^{21} + 14 q^{29} + 12 q^{31} - 6 q^{39} - 44 q^{41} - 24 q^{49} - 6 q^{51} - 22 q^{59} - 32 q^{61} + 2 q^{69} - 52 q^{79} - 26 q^{81} + 12 q^{89} + 58 q^{91} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.81361i 1.04709i 0.851999 + 0.523543i \(0.175390\pi\)
−0.851999 + 0.523543i \(0.824610\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.91638i 1.85822i 0.369807 + 0.929109i \(0.379424\pi\)
−0.369807 + 0.929109i \(0.620576\pi\)
\(8\) 0 0
\(9\) −0.289169 −0.0963895
\(10\) 0 0
\(11\) 0.578337 0.174375 0.0871876 0.996192i \(-0.472212\pi\)
0.0871876 + 0.996192i \(0.472212\pi\)
\(12\) 0 0
\(13\) − 6.39194i − 1.77281i −0.462914 0.886403i \(-0.653196\pi\)
0.462914 0.886403i \(-0.346804\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.710831i 0.172402i 0.996278 + 0.0862010i \(0.0274727\pi\)
−0.996278 + 0.0862010i \(0.972527\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −8.91638 −1.94571
\(22\) 0 0
\(23\) − 2.71083i − 0.565247i −0.959231 0.282624i \(-0.908795\pi\)
0.959231 0.282624i \(-0.0912048\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.91638i 0.946158i
\(28\) 0 0
\(29\) −6.54359 −1.21512 −0.607558 0.794276i \(-0.707851\pi\)
−0.607558 + 0.794276i \(0.707851\pi\)
\(30\) 0 0
\(31\) 1.42166 0.255338 0.127669 0.991817i \(-0.459250\pi\)
0.127669 + 0.991817i \(0.459250\pi\)
\(32\) 0 0
\(33\) 1.04888i 0.182586i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.10278i 1.49649i 0.663424 + 0.748243i \(0.269103\pi\)
−0.663424 + 0.748243i \(0.730897\pi\)
\(38\) 0 0
\(39\) 11.5925 1.85628
\(40\) 0 0
\(41\) −11.0489 −1.72554 −0.862772 0.505593i \(-0.831274\pi\)
−0.862772 + 0.505593i \(0.831274\pi\)
\(42\) 0 0
\(43\) 5.83276i 0.889488i 0.895658 + 0.444744i \(0.146705\pi\)
−0.895658 + 0.444744i \(0.853295\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.15667i 0.168718i 0.996435 + 0.0843591i \(0.0268843\pi\)
−0.996435 + 0.0843591i \(0.973116\pi\)
\(48\) 0 0
\(49\) −17.1708 −2.45297
\(50\) 0 0
\(51\) −1.28917 −0.180520
\(52\) 0 0
\(53\) 13.2736i 1.82327i 0.411004 + 0.911633i \(0.365178\pi\)
−0.411004 + 0.911633i \(0.634822\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.81361i − 0.240218i
\(58\) 0 0
\(59\) −11.3869 −1.48245 −0.741225 0.671256i \(-0.765755\pi\)
−0.741225 + 0.671256i \(0.765755\pi\)
\(60\) 0 0
\(61\) −9.04888 −1.15859 −0.579295 0.815118i \(-0.696672\pi\)
−0.579295 + 0.815118i \(0.696672\pi\)
\(62\) 0 0
\(63\) − 1.42166i − 0.179113i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 2.97028i − 0.362878i −0.983402 0.181439i \(-0.941925\pi\)
0.983402 0.181439i \(-0.0580754\pi\)
\(68\) 0 0
\(69\) 4.91638 0.591863
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 9.38692i − 1.09866i −0.835607 0.549328i \(-0.814884\pi\)
0.835607 0.549328i \(-0.185116\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.84333i 0.324027i
\(78\) 0 0
\(79\) −4.37279 −0.491977 −0.245988 0.969273i \(-0.579113\pi\)
−0.245988 + 0.969273i \(0.579113\pi\)
\(80\) 0 0
\(81\) −9.78389 −1.08710
\(82\) 0 0
\(83\) − 0.372787i − 0.0409187i −0.999791 0.0204593i \(-0.993487\pi\)
0.999791 0.0204593i \(-0.00651287\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 11.8675i − 1.27233i
\(88\) 0 0
\(89\) 16.6167 1.76136 0.880681 0.473710i \(-0.157086\pi\)
0.880681 + 0.473710i \(0.157086\pi\)
\(90\) 0 0
\(91\) 31.4252 3.29426
\(92\) 0 0
\(93\) 2.57834i 0.267361i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 3.94610i − 0.400666i −0.979728 0.200333i \(-0.935798\pi\)
0.979728 0.200333i \(-0.0642024\pi\)
\(98\) 0 0
\(99\) −0.167237 −0.0168079
\(100\) 0 0
\(101\) −6.20555 −0.617475 −0.308738 0.951147i \(-0.599907\pi\)
−0.308738 + 0.951147i \(0.599907\pi\)
\(102\) 0 0
\(103\) 8.15165i 0.803206i 0.915814 + 0.401603i \(0.131547\pi\)
−0.915814 + 0.401603i \(0.868453\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.0680i 1.26333i 0.775240 + 0.631667i \(0.217629\pi\)
−0.775240 + 0.631667i \(0.782371\pi\)
\(108\) 0 0
\(109\) 11.5925 1.11036 0.555179 0.831731i \(-0.312650\pi\)
0.555179 + 0.831731i \(0.312650\pi\)
\(110\) 0 0
\(111\) −16.5089 −1.56695
\(112\) 0 0
\(113\) − 14.9355i − 1.40502i −0.711675 0.702509i \(-0.752063\pi\)
0.711675 0.702509i \(-0.247937\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.84835i 0.170880i
\(118\) 0 0
\(119\) −3.49472 −0.320360
\(120\) 0 0
\(121\) −10.6655 −0.969593
\(122\) 0 0
\(123\) − 20.0383i − 1.80679i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 11.8867i − 1.05477i −0.849626 0.527385i \(-0.823172\pi\)
0.849626 0.527385i \(-0.176828\pi\)
\(128\) 0 0
\(129\) −10.5783 −0.931371
\(130\) 0 0
\(131\) −9.15667 −0.800022 −0.400011 0.916510i \(-0.630994\pi\)
−0.400011 + 0.916510i \(0.630994\pi\)
\(132\) 0 0
\(133\) − 4.91638i − 0.426304i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 13.3869i − 1.14372i −0.820351 0.571861i \(-0.806222\pi\)
0.820351 0.571861i \(-0.193778\pi\)
\(138\) 0 0
\(139\) −3.42166 −0.290222 −0.145111 0.989415i \(-0.546354\pi\)
−0.145111 + 0.989415i \(0.546354\pi\)
\(140\) 0 0
\(141\) −2.09775 −0.176663
\(142\) 0 0
\(143\) − 3.69670i − 0.309133i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 31.1411i − 2.56847i
\(148\) 0 0
\(149\) 3.36222 0.275444 0.137722 0.990471i \(-0.456022\pi\)
0.137722 + 0.990471i \(0.456022\pi\)
\(150\) 0 0
\(151\) 21.1950 1.72482 0.862412 0.506207i \(-0.168953\pi\)
0.862412 + 0.506207i \(0.168953\pi\)
\(152\) 0 0
\(153\) − 0.205550i − 0.0166177i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 11.7944i 0.941300i 0.882320 + 0.470650i \(0.155980\pi\)
−0.882320 + 0.470650i \(0.844020\pi\)
\(158\) 0 0
\(159\) −24.0731 −1.90912
\(160\) 0 0
\(161\) 13.3275 1.05035
\(162\) 0 0
\(163\) − 14.2439i − 1.11567i −0.829953 0.557833i \(-0.811633\pi\)
0.829953 0.557833i \(-0.188367\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.47556i 0.578476i 0.957257 + 0.289238i \(0.0934020\pi\)
−0.957257 + 0.289238i \(0.906598\pi\)
\(168\) 0 0
\(169\) −27.8569 −2.14284
\(170\) 0 0
\(171\) 0.289169 0.0221133
\(172\) 0 0
\(173\) 4.05390i 0.308212i 0.988054 + 0.154106i \(0.0492498\pi\)
−0.988054 + 0.154106i \(0.950750\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 20.6514i − 1.55225i
\(178\) 0 0
\(179\) −2.95112 −0.220577 −0.110289 0.993900i \(-0.535178\pi\)
−0.110289 + 0.993900i \(0.535178\pi\)
\(180\) 0 0
\(181\) 9.66553 0.718433 0.359216 0.933254i \(-0.383044\pi\)
0.359216 + 0.933254i \(0.383044\pi\)
\(182\) 0 0
\(183\) − 16.4111i − 1.21314i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.411100i 0.0300626i
\(188\) 0 0
\(189\) −24.1708 −1.75817
\(190\) 0 0
\(191\) −15.9058 −1.15090 −0.575452 0.817835i \(-0.695174\pi\)
−0.575452 + 0.817835i \(0.695174\pi\)
\(192\) 0 0
\(193\) 15.6116i 1.12375i 0.827222 + 0.561875i \(0.189920\pi\)
−0.827222 + 0.561875i \(0.810080\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.6655i 0.973628i 0.873506 + 0.486814i \(0.161841\pi\)
−0.873506 + 0.486814i \(0.838159\pi\)
\(198\) 0 0
\(199\) −8.91638 −0.632066 −0.316033 0.948748i \(-0.602351\pi\)
−0.316033 + 0.948748i \(0.602351\pi\)
\(200\) 0 0
\(201\) 5.38692 0.379964
\(202\) 0 0
\(203\) − 32.1708i − 2.25795i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.783887i 0.0544839i
\(208\) 0 0
\(209\) −0.578337 −0.0400044
\(210\) 0 0
\(211\) 19.7980 1.36295 0.681476 0.731840i \(-0.261338\pi\)
0.681476 + 0.731840i \(0.261338\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6.98944i 0.474474i
\(218\) 0 0
\(219\) 17.0242 1.15039
\(220\) 0 0
\(221\) 4.54359 0.305635
\(222\) 0 0
\(223\) − 1.57331i − 0.105357i −0.998612 0.0526784i \(-0.983224\pi\)
0.998612 0.0526784i \(-0.0167758\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 28.0575i 1.86224i 0.364713 + 0.931120i \(0.381167\pi\)
−0.364713 + 0.931120i \(0.618833\pi\)
\(228\) 0 0
\(229\) −2.47054 −0.163258 −0.0816289 0.996663i \(-0.526012\pi\)
−0.0816289 + 0.996663i \(0.526012\pi\)
\(230\) 0 0
\(231\) −5.15667 −0.339284
\(232\) 0 0
\(233\) 3.15667i 0.206801i 0.994640 + 0.103400i \(0.0329723\pi\)
−0.994640 + 0.103400i \(0.967028\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 7.93051i − 0.515142i
\(238\) 0 0
\(239\) 2.74914 0.177827 0.0889137 0.996039i \(-0.471660\pi\)
0.0889137 + 0.996039i \(0.471660\pi\)
\(240\) 0 0
\(241\) 8.88164 0.572117 0.286058 0.958212i \(-0.407655\pi\)
0.286058 + 0.958212i \(0.407655\pi\)
\(242\) 0 0
\(243\) − 2.99498i − 0.192128i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.39194i 0.406710i
\(248\) 0 0
\(249\) 0.676089 0.0428454
\(250\) 0 0
\(251\) 6.31335 0.398495 0.199248 0.979949i \(-0.436150\pi\)
0.199248 + 0.979949i \(0.436150\pi\)
\(252\) 0 0
\(253\) − 1.56777i − 0.0985651i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 7.83830i − 0.488940i −0.969657 0.244470i \(-0.921386\pi\)
0.969657 0.244470i \(-0.0786140\pi\)
\(258\) 0 0
\(259\) −44.7527 −2.78080
\(260\) 0 0
\(261\) 1.89220 0.117124
\(262\) 0 0
\(263\) 13.8711i 0.855327i 0.903938 + 0.427664i \(0.140663\pi\)
−0.903938 + 0.427664i \(0.859337\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 30.1361i 1.84430i
\(268\) 0 0
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) −4.07306 −0.247421 −0.123710 0.992318i \(-0.539479\pi\)
−0.123710 + 0.992318i \(0.539479\pi\)
\(272\) 0 0
\(273\) 56.9930i 3.44937i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 16.1361i − 0.969522i −0.874647 0.484761i \(-0.838907\pi\)
0.874647 0.484761i \(-0.161093\pi\)
\(278\) 0 0
\(279\) −0.411100 −0.0246119
\(280\) 0 0
\(281\) −11.4600 −0.683645 −0.341822 0.939765i \(-0.611044\pi\)
−0.341822 + 0.939765i \(0.611044\pi\)
\(282\) 0 0
\(283\) − 21.7250i − 1.29142i −0.763585 0.645708i \(-0.776563\pi\)
0.763585 0.645708i \(-0.223437\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 54.3205i − 3.20644i
\(288\) 0 0
\(289\) 16.4947 0.970278
\(290\) 0 0
\(291\) 7.15667 0.419532
\(292\) 0 0
\(293\) − 5.91136i − 0.345345i −0.984979 0.172673i \(-0.944760\pi\)
0.984979 0.172673i \(-0.0552403\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.84333i 0.164986i
\(298\) 0 0
\(299\) −17.3275 −1.00207
\(300\) 0 0
\(301\) −28.6761 −1.65286
\(302\) 0 0
\(303\) − 11.2544i − 0.646550i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 29.7194i 1.69618i 0.529854 + 0.848089i \(0.322247\pi\)
−0.529854 + 0.848089i \(0.677753\pi\)
\(308\) 0 0
\(309\) −14.7839 −0.841026
\(310\) 0 0
\(311\) −0.407530 −0.0231089 −0.0115544 0.999933i \(-0.503678\pi\)
−0.0115544 + 0.999933i \(0.503678\pi\)
\(312\) 0 0
\(313\) − 0.338044i − 0.0191074i −0.999954 0.00955370i \(-0.996959\pi\)
0.999954 0.00955370i \(-0.00304108\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.75468i 0.491712i 0.969306 + 0.245856i \(0.0790690\pi\)
−0.969306 + 0.245856i \(0.920931\pi\)
\(318\) 0 0
\(319\) −3.78440 −0.211886
\(320\) 0 0
\(321\) −23.7003 −1.32282
\(322\) 0 0
\(323\) − 0.710831i − 0.0395517i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 21.0242i 1.16264i
\(328\) 0 0
\(329\) −5.68665 −0.313515
\(330\) 0 0
\(331\) 23.8675 1.31188 0.655938 0.754814i \(-0.272273\pi\)
0.655938 + 0.754814i \(0.272273\pi\)
\(332\) 0 0
\(333\) − 2.63224i − 0.144246i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 10.0439i 0.547124i 0.961854 + 0.273562i \(0.0882018\pi\)
−0.961854 + 0.273562i \(0.911798\pi\)
\(338\) 0 0
\(339\) 27.0872 1.47117
\(340\) 0 0
\(341\) 0.822200 0.0445246
\(342\) 0 0
\(343\) − 50.0036i − 2.69994i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.6272i 0.838913i 0.907775 + 0.419456i \(0.137779\pi\)
−0.907775 + 0.419456i \(0.862221\pi\)
\(348\) 0 0
\(349\) −17.2544 −0.923608 −0.461804 0.886982i \(-0.652798\pi\)
−0.461804 + 0.886982i \(0.652798\pi\)
\(350\) 0 0
\(351\) 31.4252 1.67735
\(352\) 0 0
\(353\) 27.2197i 1.44876i 0.689402 + 0.724379i \(0.257873\pi\)
−0.689402 + 0.724379i \(0.742127\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 6.33804i − 0.335445i
\(358\) 0 0
\(359\) 7.18137 0.379018 0.189509 0.981879i \(-0.439310\pi\)
0.189509 + 0.981879i \(0.439310\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) − 19.3431i − 1.01525i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 9.15667i − 0.477975i −0.971023 0.238987i \(-0.923185\pi\)
0.971023 0.238987i \(-0.0768154\pi\)
\(368\) 0 0
\(369\) 3.19499 0.166324
\(370\) 0 0
\(371\) −65.2580 −3.38803
\(372\) 0 0
\(373\) − 8.75468i − 0.453300i −0.973976 0.226650i \(-0.927223\pi\)
0.973976 0.226650i \(-0.0727774\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 41.8263i 2.15416i
\(378\) 0 0
\(379\) 12.9164 0.663470 0.331735 0.943373i \(-0.392366\pi\)
0.331735 + 0.943373i \(0.392366\pi\)
\(380\) 0 0
\(381\) 21.5577 1.10444
\(382\) 0 0
\(383\) − 1.64280i − 0.0839431i −0.999119 0.0419716i \(-0.986636\pi\)
0.999119 0.0419716i \(-0.0133639\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 1.68665i − 0.0857373i
\(388\) 0 0
\(389\) 11.8328 0.599945 0.299972 0.953948i \(-0.403023\pi\)
0.299972 + 0.953948i \(0.403023\pi\)
\(390\) 0 0
\(391\) 1.92694 0.0974498
\(392\) 0 0
\(393\) − 16.6066i − 0.837692i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 24.1361i 1.21135i 0.795710 + 0.605677i \(0.207098\pi\)
−0.795710 + 0.605677i \(0.792902\pi\)
\(398\) 0 0
\(399\) 8.91638 0.446377
\(400\) 0 0
\(401\) 19.9789 0.997697 0.498849 0.866689i \(-0.333756\pi\)
0.498849 + 0.866689i \(0.333756\pi\)
\(402\) 0 0
\(403\) − 9.08719i − 0.452665i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.26447i 0.260950i
\(408\) 0 0
\(409\) −5.66553 −0.280142 −0.140071 0.990141i \(-0.544733\pi\)
−0.140071 + 0.990141i \(0.544733\pi\)
\(410\) 0 0
\(411\) 24.2786 1.19758
\(412\) 0 0
\(413\) − 55.9824i − 2.75472i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 6.20555i − 0.303887i
\(418\) 0 0
\(419\) −22.6550 −1.10677 −0.553384 0.832926i \(-0.686664\pi\)
−0.553384 + 0.832926i \(0.686664\pi\)
\(420\) 0 0
\(421\) −25.5330 −1.24440 −0.622202 0.782857i \(-0.713762\pi\)
−0.622202 + 0.782857i \(0.713762\pi\)
\(422\) 0 0
\(423\) − 0.334474i − 0.0162627i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 44.4877i − 2.15291i
\(428\) 0 0
\(429\) 6.70436 0.323689
\(430\) 0 0
\(431\) −13.7944 −0.664455 −0.332228 0.943199i \(-0.607800\pi\)
−0.332228 + 0.943199i \(0.607800\pi\)
\(432\) 0 0
\(433\) − 29.4444i − 1.41501i −0.706710 0.707504i \(-0.749821\pi\)
0.706710 0.707504i \(-0.250179\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.71083i 0.129677i
\(438\) 0 0
\(439\) 34.5472 1.64885 0.824423 0.565974i \(-0.191500\pi\)
0.824423 + 0.565974i \(0.191500\pi\)
\(440\) 0 0
\(441\) 4.96526 0.236441
\(442\) 0 0
\(443\) − 23.5577i − 1.11926i −0.828742 0.559631i \(-0.810943\pi\)
0.828742 0.559631i \(-0.189057\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.09775i 0.288414i
\(448\) 0 0
\(449\) 7.49115 0.353529 0.176765 0.984253i \(-0.443437\pi\)
0.176765 + 0.984253i \(0.443437\pi\)
\(450\) 0 0
\(451\) −6.38997 −0.300892
\(452\) 0 0
\(453\) 38.4394i 1.80604i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 38.5819i − 1.80479i −0.430914 0.902393i \(-0.641809\pi\)
0.430914 0.902393i \(-0.358191\pi\)
\(458\) 0 0
\(459\) −3.49472 −0.163119
\(460\) 0 0
\(461\) −31.9789 −1.48940 −0.744702 0.667397i \(-0.767409\pi\)
−0.744702 + 0.667397i \(0.767409\pi\)
\(462\) 0 0
\(463\) 30.8122i 1.43196i 0.698120 + 0.715981i \(0.254020\pi\)
−0.698120 + 0.715981i \(0.745980\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 41.4600i − 1.91854i −0.282493 0.959269i \(-0.591161\pi\)
0.282493 0.959269i \(-0.408839\pi\)
\(468\) 0 0
\(469\) 14.6030 0.674305
\(470\) 0 0
\(471\) −21.3905 −0.985622
\(472\) 0 0
\(473\) 3.37330i 0.155105i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 3.83830i − 0.175744i
\(478\) 0 0
\(479\) 9.93051 0.453737 0.226868 0.973925i \(-0.427151\pi\)
0.226868 + 0.973925i \(0.427151\pi\)
\(480\) 0 0
\(481\) 58.1844 2.65298
\(482\) 0 0
\(483\) 24.1708i 1.09981i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 19.7789i 0.896266i 0.893967 + 0.448133i \(0.147911\pi\)
−0.893967 + 0.448133i \(0.852089\pi\)
\(488\) 0 0
\(489\) 25.8328 1.16820
\(490\) 0 0
\(491\) −16.1461 −0.728664 −0.364332 0.931269i \(-0.618703\pi\)
−0.364332 + 0.931269i \(0.618703\pi\)
\(492\) 0 0
\(493\) − 4.65139i − 0.209488i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 38.8222 1.73792 0.868960 0.494882i \(-0.164789\pi\)
0.868960 + 0.494882i \(0.164789\pi\)
\(500\) 0 0
\(501\) −13.5577 −0.605715
\(502\) 0 0
\(503\) 12.5436i 0.559291i 0.960103 + 0.279646i \(0.0902170\pi\)
−0.960103 + 0.279646i \(0.909783\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 50.5215i − 2.24374i
\(508\) 0 0
\(509\) −25.0278 −1.10934 −0.554668 0.832072i \(-0.687155\pi\)
−0.554668 + 0.832072i \(0.687155\pi\)
\(510\) 0 0
\(511\) 46.1497 2.04154
\(512\) 0 0
\(513\) − 4.91638i − 0.217064i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0.668948i 0.0294203i
\(518\) 0 0
\(519\) −7.35218 −0.322725
\(520\) 0 0
\(521\) −16.6550 −0.729667 −0.364834 0.931073i \(-0.618874\pi\)
−0.364834 + 0.931073i \(0.618874\pi\)
\(522\) 0 0
\(523\) 24.3608i 1.06522i 0.846360 + 0.532611i \(0.178789\pi\)
−0.846360 + 0.532611i \(0.821211\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.01056i 0.0440208i
\(528\) 0 0
\(529\) 15.6514 0.680495
\(530\) 0 0
\(531\) 3.29274 0.142893
\(532\) 0 0
\(533\) 70.6238i 3.05906i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 5.35218i − 0.230964i
\(538\) 0 0
\(539\) −9.93051 −0.427738
\(540\) 0 0
\(541\) 15.3622 0.660474 0.330237 0.943898i \(-0.392871\pi\)
0.330237 + 0.943898i \(0.392871\pi\)
\(542\) 0 0
\(543\) 17.5295i 0.752261i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 26.5527i 1.13531i 0.823266 + 0.567656i \(0.192150\pi\)
−0.823266 + 0.567656i \(0.807850\pi\)
\(548\) 0 0
\(549\) 2.61665 0.111676
\(550\) 0 0
\(551\) 6.54359 0.278767
\(552\) 0 0
\(553\) − 21.4983i − 0.914200i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 28.9583i 1.22700i 0.789694 + 0.613501i \(0.210239\pi\)
−0.789694 + 0.613501i \(0.789761\pi\)
\(558\) 0 0
\(559\) 37.2827 1.57689
\(560\) 0 0
\(561\) −0.745574 −0.0314782
\(562\) 0 0
\(563\) − 1.64280i − 0.0692357i −0.999401 0.0346179i \(-0.988979\pi\)
0.999401 0.0346179i \(-0.0110214\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 48.1013i − 2.02007i
\(568\) 0 0
\(569\) −14.3416 −0.601232 −0.300616 0.953745i \(-0.597192\pi\)
−0.300616 + 0.953745i \(0.597192\pi\)
\(570\) 0 0
\(571\) −39.0177 −1.63284 −0.816420 0.577458i \(-0.804045\pi\)
−0.816420 + 0.577458i \(0.804045\pi\)
\(572\) 0 0
\(573\) − 28.8469i − 1.20510i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 25.4947i − 1.06136i −0.847573 0.530680i \(-0.821937\pi\)
0.847573 0.530680i \(-0.178063\pi\)
\(578\) 0 0
\(579\) −28.3133 −1.17666
\(580\) 0 0
\(581\) 1.83276 0.0760358
\(582\) 0 0
\(583\) 7.67661i 0.317933i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9.72496i 0.401392i 0.979654 + 0.200696i \(0.0643204\pi\)
−0.979654 + 0.200696i \(0.935680\pi\)
\(588\) 0 0
\(589\) −1.42166 −0.0585786
\(590\) 0 0
\(591\) −24.7839 −1.01947
\(592\) 0 0
\(593\) − 13.0388i − 0.535441i −0.963497 0.267720i \(-0.913730\pi\)
0.963497 0.267720i \(-0.0862703\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 16.1708i − 0.661827i
\(598\) 0 0
\(599\) 18.6861 0.763495 0.381747 0.924267i \(-0.375322\pi\)
0.381747 + 0.924267i \(0.375322\pi\)
\(600\) 0 0
\(601\) 12.7355 0.519493 0.259747 0.965677i \(-0.416361\pi\)
0.259747 + 0.965677i \(0.416361\pi\)
\(602\) 0 0
\(603\) 0.858912i 0.0349776i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 23.0645i − 0.936158i −0.883687 0.468079i \(-0.844946\pi\)
0.883687 0.468079i \(-0.155054\pi\)
\(608\) 0 0
\(609\) 58.3452 2.36427
\(610\) 0 0
\(611\) 7.39340 0.299105
\(612\) 0 0
\(613\) 25.7038i 1.03817i 0.854723 + 0.519084i \(0.173727\pi\)
−0.854723 + 0.519084i \(0.826273\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.205550i 0.00827514i 0.999991 + 0.00413757i \(0.00131703\pi\)
−0.999991 + 0.00413757i \(0.998683\pi\)
\(618\) 0 0
\(619\) 11.0872 0.445632 0.222816 0.974861i \(-0.428475\pi\)
0.222816 + 0.974861i \(0.428475\pi\)
\(620\) 0 0
\(621\) 13.3275 0.534813
\(622\) 0 0
\(623\) 81.6938i 3.27299i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 1.04888i − 0.0418881i
\(628\) 0 0
\(629\) −6.47054 −0.257997
\(630\) 0 0
\(631\) 10.4806 0.417226 0.208613 0.977998i \(-0.433105\pi\)
0.208613 + 0.977998i \(0.433105\pi\)
\(632\) 0 0
\(633\) 35.9058i 1.42713i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 109.755i 4.34864i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 37.3694 1.47600 0.738001 0.674800i \(-0.235770\pi\)
0.738001 + 0.674800i \(0.235770\pi\)
\(642\) 0 0
\(643\) 42.0172i 1.65700i 0.559992 + 0.828498i \(0.310804\pi\)
−0.559992 + 0.828498i \(0.689196\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 23.4635i − 0.922447i −0.887284 0.461224i \(-0.847411\pi\)
0.887284 0.461224i \(-0.152589\pi\)
\(648\) 0 0
\(649\) −6.58548 −0.258503
\(650\) 0 0
\(651\) −12.6761 −0.496815
\(652\) 0 0
\(653\) 33.6272i 1.31593i 0.753047 + 0.657967i \(0.228583\pi\)
−0.753047 + 0.657967i \(0.771417\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.71440i 0.105899i
\(658\) 0 0
\(659\) −17.9653 −0.699827 −0.349914 0.936782i \(-0.613789\pi\)
−0.349914 + 0.936782i \(0.613789\pi\)
\(660\) 0 0
\(661\) 15.5542 0.604987 0.302493 0.953152i \(-0.402181\pi\)
0.302493 + 0.953152i \(0.402181\pi\)
\(662\) 0 0
\(663\) 8.24029i 0.320026i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 17.7386i 0.686841i
\(668\) 0 0
\(669\) 2.85337 0.110318
\(670\) 0 0
\(671\) −5.23330 −0.202029
\(672\) 0 0
\(673\) 12.2111i 0.470703i 0.971910 + 0.235351i \(0.0756241\pi\)
−0.971910 + 0.235351i \(0.924376\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 43.0680i 1.65524i 0.561290 + 0.827619i \(0.310305\pi\)
−0.561290 + 0.827619i \(0.689695\pi\)
\(678\) 0 0
\(679\) 19.4005 0.744524
\(680\) 0 0
\(681\) −50.8852 −1.94993
\(682\) 0 0
\(683\) − 45.3850i − 1.73661i −0.496033 0.868303i \(-0.665211\pi\)
0.496033 0.868303i \(-0.334789\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 4.48059i − 0.170945i
\(688\) 0 0
\(689\) 84.8440 3.23230
\(690\) 0 0
\(691\) −0.508852 −0.0193576 −0.00967882 0.999953i \(-0.503081\pi\)
−0.00967882 + 0.999953i \(0.503081\pi\)
\(692\) 0 0
\(693\) − 0.822200i − 0.0312328i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 7.85389i − 0.297487i
\(698\) 0 0
\(699\) −5.72496 −0.216538
\(700\) 0 0
\(701\) −40.1844 −1.51774 −0.758872 0.651239i \(-0.774249\pi\)
−0.758872 + 0.651239i \(0.774249\pi\)
\(702\) 0 0
\(703\) − 9.10278i − 0.343318i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 30.5089i − 1.14740i
\(708\) 0 0
\(709\) −20.0978 −0.754787 −0.377393 0.926053i \(-0.623180\pi\)
−0.377393 + 0.926053i \(0.623180\pi\)
\(710\) 0 0
\(711\) 1.26447 0.0474214
\(712\) 0 0
\(713\) − 3.85389i − 0.144329i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 4.98587i 0.186201i
\(718\) 0 0
\(719\) −35.5925 −1.32738 −0.663688 0.748010i \(-0.731010\pi\)
−0.663688 + 0.748010i \(0.731010\pi\)
\(720\) 0 0
\(721\) −40.0766 −1.49253
\(722\) 0 0
\(723\) 16.1078i 0.599055i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 20.1708i 0.748094i 0.927410 + 0.374047i \(0.122030\pi\)
−0.927410 + 0.374047i \(0.877970\pi\)
\(728\) 0 0
\(729\) −23.9200 −0.885924
\(730\) 0 0
\(731\) −4.14611 −0.153349
\(732\) 0 0
\(733\) 30.1461i 1.11347i 0.830689 + 0.556736i \(0.187947\pi\)
−0.830689 + 0.556736i \(0.812053\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 1.71782i − 0.0632768i
\(738\) 0 0
\(739\) −26.0283 −0.957465 −0.478733 0.877961i \(-0.658904\pi\)
−0.478733 + 0.877961i \(0.658904\pi\)
\(740\) 0 0
\(741\) −11.5925 −0.425860
\(742\) 0 0
\(743\) 0.221136i 0.00811269i 0.999992 + 0.00405635i \(0.00129118\pi\)
−0.999992 + 0.00405635i \(0.998709\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0.107798i 0.00394413i
\(748\) 0 0
\(749\) −64.2474 −2.34755
\(750\) 0 0
\(751\) 41.9406 1.53043 0.765216 0.643773i \(-0.222632\pi\)
0.765216 + 0.643773i \(0.222632\pi\)
\(752\) 0 0
\(753\) 11.4499i 0.417259i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 26.7456i 0.972084i 0.873935 + 0.486042i \(0.161560\pi\)
−0.873935 + 0.486042i \(0.838440\pi\)
\(758\) 0 0
\(759\) 2.84333 0.103206
\(760\) 0 0
\(761\) −11.6519 −0.422381 −0.211191 0.977445i \(-0.567734\pi\)
−0.211191 + 0.977445i \(0.567734\pi\)
\(762\) 0 0
\(763\) 56.9930i 2.06329i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 72.7846i 2.62810i
\(768\) 0 0
\(769\) 9.28917 0.334976 0.167488 0.985874i \(-0.446434\pi\)
0.167488 + 0.985874i \(0.446434\pi\)
\(770\) 0 0
\(771\) 14.2156 0.511962
\(772\) 0 0
\(773\) − 10.3225i − 0.371273i −0.982618 0.185637i \(-0.940565\pi\)
0.982618 0.185637i \(-0.0594347\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 81.1638i − 2.91174i
\(778\) 0 0
\(779\) 11.0489 0.395867
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 32.1708i − 1.14969i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 49.8902i − 1.77839i −0.457524 0.889197i \(-0.651264\pi\)
0.457524 0.889197i \(-0.348736\pi\)
\(788\) 0 0
\(789\) −25.1567 −0.895601
\(790\) 0 0
\(791\) 73.4288 2.61083
\(792\) 0 0
\(793\) 57.8399i 2.05396i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 28.7436i − 1.01815i −0.860722 0.509075i \(-0.829987\pi\)
0.860722 0.509075i \(-0.170013\pi\)
\(798\) 0 0
\(799\) −0.822200 −0.0290874
\(800\) 0 0
\(801\) −4.80501 −0.169777
\(802\) 0 0
\(803\) − 5.42880i − 0.191578i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 25.3905i 0.893788i
\(808\) 0 0
\(809\) −38.5124 −1.35402 −0.677012 0.735972i \(-0.736726\pi\)
−0.677012 + 0.735972i \(0.736726\pi\)
\(810\) 0 0
\(811\) −28.1013 −0.986771 −0.493385 0.869811i \(-0.664241\pi\)
−0.493385 + 0.869811i \(0.664241\pi\)
\(812\) 0 0
\(813\) − 7.38692i − 0.259071i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 5.83276i − 0.204063i
\(818\) 0 0
\(819\) −9.08719 −0.317532
\(820\) 0 0
\(821\) 22.2650 0.777053 0.388527 0.921437i \(-0.372984\pi\)
0.388527 + 0.921437i \(0.372984\pi\)
\(822\) 0 0
\(823\) 39.8363i 1.38861i 0.719682 + 0.694304i \(0.244288\pi\)
−0.719682 + 0.694304i \(0.755712\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 48.5874i 1.68955i 0.535121 + 0.844776i \(0.320266\pi\)
−0.535121 + 0.844776i \(0.679734\pi\)
\(828\) 0 0
\(829\) 45.9058 1.59437 0.797187 0.603732i \(-0.206320\pi\)
0.797187 + 0.603732i \(0.206320\pi\)
\(830\) 0 0
\(831\) 29.2645 1.01517
\(832\) 0 0
\(833\) − 12.2056i − 0.422897i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 6.98944i 0.241590i
\(838\) 0 0
\(839\) 27.9688 0.965591 0.482796 0.875733i \(-0.339621\pi\)
0.482796 + 0.875733i \(0.339621\pi\)
\(840\) 0 0
\(841\) 13.8186 0.476504
\(842\) 0 0
\(843\) − 20.7839i − 0.715835i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 52.4358i − 1.80172i
\(848\) 0 0
\(849\) 39.4005 1.35222
\(850\) 0 0
\(851\) 24.6761 0.845885
\(852\) 0 0
\(853\) 7.56777i 0.259116i 0.991572 + 0.129558i \(0.0413558\pi\)
−0.991572 + 0.129558i \(0.958644\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 21.4756i − 0.733591i −0.930302 0.366796i \(-0.880455\pi\)
0.930302 0.366796i \(-0.119545\pi\)
\(858\) 0 0
\(859\) 33.0177 1.12655 0.563275 0.826270i \(-0.309541\pi\)
0.563275 + 0.826270i \(0.309541\pi\)
\(860\) 0 0
\(861\) 98.5160 3.35742
\(862\) 0 0
\(863\) 18.6605i 0.635211i 0.948223 + 0.317605i \(0.102879\pi\)
−0.948223 + 0.317605i \(0.897121\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 29.9149i 1.01596i
\(868\) 0 0
\(869\) −2.52894 −0.0857886
\(870\) 0 0
\(871\) −18.9859 −0.643312
\(872\) 0 0
\(873\) 1.14109i 0.0386200i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 57.7925i 1.95151i 0.218858 + 0.975757i \(0.429767\pi\)
−0.218858 + 0.975757i \(0.570233\pi\)
\(878\) 0 0
\(879\) 10.7209 0.361606
\(880\) 0 0
\(881\) 36.2338 1.22075 0.610374 0.792113i \(-0.291019\pi\)
0.610374 + 0.792113i \(0.291019\pi\)
\(882\) 0 0
\(883\) − 2.98944i − 0.100603i −0.998734 0.0503013i \(-0.983982\pi\)
0.998734 0.0503013i \(-0.0160182\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 42.2494i 1.41860i 0.704909 + 0.709298i \(0.250988\pi\)
−0.704909 + 0.709298i \(0.749012\pi\)
\(888\) 0 0
\(889\) 58.4394 1.95999
\(890\) 0 0
\(891\) −5.65838 −0.189563
\(892\) 0 0
\(893\) − 1.15667i − 0.0387066i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 31.4252i − 1.04926i
\(898\) 0 0
\(899\) −9.30279 −0.310265
\(900\) 0 0
\(901\) −9.43528 −0.314335
\(902\) 0 0
\(903\) − 52.0071i − 1.73069i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 1.97080i − 0.0654392i −0.999465 0.0327196i \(-0.989583\pi\)
0.999465 0.0327196i \(-0.0104168\pi\)
\(908\) 0 0
\(909\) 1.79445 0.0595181
\(910\) 0 0
\(911\) 9.98995 0.330982 0.165491 0.986211i \(-0.447079\pi\)
0.165491 + 0.986211i \(0.447079\pi\)
\(912\) 0 0
\(913\) − 0.215597i − 0.00713520i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 45.0177i − 1.48662i
\(918\) 0 0
\(919\) −4.24029 −0.139874 −0.0699372 0.997551i \(-0.522280\pi\)
−0.0699372 + 0.997551i \(0.522280\pi\)
\(920\) 0 0
\(921\) −53.8993 −1.77604
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 2.35720i − 0.0774206i
\(928\) 0 0
\(929\) 13.1531 0.431539 0.215770 0.976444i \(-0.430774\pi\)
0.215770 + 0.976444i \(0.430774\pi\)
\(930\) 0 0
\(931\) 17.1708 0.562750
\(932\) 0 0
\(933\) − 0.739098i − 0.0241970i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 38.5436i 1.25916i 0.776934 + 0.629582i \(0.216774\pi\)
−0.776934 + 0.629582i \(0.783226\pi\)
\(938\) 0 0
\(939\) 0.613080 0.0200071
\(940\) 0 0
\(941\) −2.91638 −0.0950713 −0.0475357 0.998870i \(-0.515137\pi\)
−0.0475357 + 0.998870i \(0.515137\pi\)
\(942\) 0 0
\(943\) 29.9516i 0.975360i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 58.4011i − 1.89778i −0.315608 0.948890i \(-0.602208\pi\)
0.315608 0.948890i \(-0.397792\pi\)
\(948\) 0 0
\(949\) −60.0007 −1.94770
\(950\) 0 0
\(951\) −15.8776 −0.514865
\(952\) 0 0
\(953\) − 36.8378i − 1.19329i −0.802504 0.596646i \(-0.796499\pi\)
0.802504 0.596646i \(-0.203501\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 6.86342i − 0.221863i
\(958\) 0 0
\(959\) 65.8152 2.12528
\(960\) 0 0
\(961\) −28.9789 −0.934802
\(962\) 0 0
\(963\) − 3.77886i − 0.121772i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 43.0278i 1.38368i 0.722051 + 0.691840i \(0.243199\pi\)
−0.722051 + 0.691840i \(0.756801\pi\)
\(968\) 0 0
\(969\) 1.28917 0.0414141
\(970\) 0 0
\(971\) −49.0177 −1.57305 −0.786526 0.617557i \(-0.788123\pi\)
−0.786526 + 0.617557i \(0.788123\pi\)
\(972\) 0 0
\(973\) − 16.8222i − 0.539295i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 20.2383i − 0.647481i −0.946146 0.323741i \(-0.895059\pi\)
0.946146 0.323741i \(-0.104941\pi\)
\(978\) 0 0
\(979\) 9.61003 0.307138
\(980\) 0 0
\(981\) −3.35218 −0.107027
\(982\) 0 0
\(983\) 25.6811i 0.819100i 0.912288 + 0.409550i \(0.134314\pi\)
−0.912288 + 0.409550i \(0.865686\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 10.3133i − 0.328277i
\(988\) 0 0
\(989\) 15.8116 0.502781
\(990\) 0 0
\(991\) 8.56829 0.272181 0.136090 0.990696i \(-0.456546\pi\)
0.136090 + 0.990696i \(0.456546\pi\)
\(992\) 0 0
\(993\) 43.2863i 1.37365i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 29.9688i − 0.949122i −0.880223 0.474561i \(-0.842607\pi\)
0.880223 0.474561i \(-0.157393\pi\)
\(998\) 0 0
\(999\) −44.7527 −1.41591
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 3800.2.d.n.3649.5 6
5.2 odd 4 760.2.a.i.1.3 3
5.3 odd 4 3800.2.a.w.1.1 3
5.4 even 2 inner 3800.2.d.n.3649.2 6
15.2 even 4 6840.2.a.bm.1.1 3
20.3 even 4 7600.2.a.bp.1.3 3
20.7 even 4 1520.2.a.q.1.1 3
40.27 even 4 6080.2.a.br.1.3 3
40.37 odd 4 6080.2.a.bx.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.i.1.3 3 5.2 odd 4
1520.2.a.q.1.1 3 20.7 even 4
3800.2.a.w.1.1 3 5.3 odd 4
3800.2.d.n.3649.2 6 5.4 even 2 inner
3800.2.d.n.3649.5 6 1.1 even 1 trivial
6080.2.a.br.1.3 3 40.27 even 4
6080.2.a.bx.1.1 3 40.37 odd 4
6840.2.a.bm.1.1 3 15.2 even 4
7600.2.a.bp.1.3 3 20.3 even 4