Properties

Label 2850.2.d.w.799.3
Level $2850$
Weight $2$
Character 2850.799
Analytic conductor $22.757$
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] = [2850,2,Mod(799,2850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2850, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2850.799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.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: \(22.7573645761\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.3
Root \(1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 2850.799
Dual form 2850.2.d.w.799.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -0.449490i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -0.449490i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.44949 q^{11} +1.00000i q^{12} -2.44949i q^{13} +0.449490 q^{14} +1.00000 q^{16} -0.449490i q^{17} -1.00000i q^{18} -1.00000 q^{19} -0.449490 q^{21} -1.44949i q^{22} +1.00000i q^{23} -1.00000 q^{24} +2.44949 q^{26} +1.00000i q^{27} +0.449490i q^{28} -10.3485 q^{29} -3.00000 q^{31} +1.00000i q^{32} +1.44949i q^{33} +0.449490 q^{34} +1.00000 q^{36} +11.7980i q^{37} -1.00000i q^{38} -2.44949 q^{39} +8.89898 q^{41} -0.449490i q^{42} -2.44949i q^{43} +1.44949 q^{44} -1.00000 q^{46} +11.7980i q^{47} -1.00000i q^{48} +6.79796 q^{49} -0.449490 q^{51} +2.44949i q^{52} +2.55051i q^{53} -1.00000 q^{54} -0.449490 q^{56} +1.00000i q^{57} -10.3485i q^{58} -1.55051 q^{59} -4.55051 q^{61} -3.00000i q^{62} +0.449490i q^{63} -1.00000 q^{64} -1.44949 q^{66} -9.24745i q^{67} +0.449490i q^{68} +1.00000 q^{69} -6.44949 q^{71} +1.00000i q^{72} +1.00000i q^{73} -11.7980 q^{74} +1.00000 q^{76} +0.651531i q^{77} -2.44949i q^{78} +5.00000 q^{79} +1.00000 q^{81} +8.89898i q^{82} +6.34847i q^{83} +0.449490 q^{84} +2.44949 q^{86} +10.3485i q^{87} +1.44949i q^{88} -11.8990 q^{89} -1.10102 q^{91} -1.00000i q^{92} +3.00000i q^{93} -11.7980 q^{94} +1.00000 q^{96} +6.44949i q^{97} +6.79796i q^{98} +1.44949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 4 q^{11} - 8 q^{14} + 4 q^{16} - 4 q^{19} + 8 q^{21} - 4 q^{24} - 12 q^{29} - 12 q^{31} - 8 q^{34} + 4 q^{36} + 16 q^{41} - 4 q^{44} - 4 q^{46} - 12 q^{49} + 8 q^{51} - 4 q^{54} + 8 q^{56} - 16 q^{59} - 28 q^{61} - 4 q^{64} + 4 q^{66} + 4 q^{69} - 16 q^{71} - 8 q^{74} + 4 q^{76} + 20 q^{79} + 4 q^{81} - 8 q^{84} - 28 q^{89} - 24 q^{91} - 8 q^{94} + 4 q^{96} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times\).

\(n\) \(1027\) \(1351\) \(1901\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) − 0.449490i − 0.169891i −0.996386 0.0849456i \(-0.972928\pi\)
0.996386 0.0849456i \(-0.0270716\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.44949 −0.437038 −0.218519 0.975833i \(-0.570122\pi\)
−0.218519 + 0.975833i \(0.570122\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 2.44949i − 0.679366i −0.940540 0.339683i \(-0.889680\pi\)
0.940540 0.339683i \(-0.110320\pi\)
\(14\) 0.449490 0.120131
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 0.449490i − 0.109017i −0.998513 0.0545086i \(-0.982641\pi\)
0.998513 0.0545086i \(-0.0173592\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −0.449490 −0.0980867
\(22\) − 1.44949i − 0.309032i
\(23\) 1.00000i 0.208514i 0.994550 + 0.104257i \(0.0332465\pi\)
−0.994550 + 0.104257i \(0.966753\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 2.44949 0.480384
\(27\) 1.00000i 0.192450i
\(28\) 0.449490i 0.0849456i
\(29\) −10.3485 −1.92166 −0.960831 0.277134i \(-0.910615\pi\)
−0.960831 + 0.277134i \(0.910615\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 1.44949i 0.252324i
\(34\) 0.449490 0.0770869
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 11.7980i 1.93957i 0.243956 + 0.969786i \(0.421555\pi\)
−0.243956 + 0.969786i \(0.578445\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) −2.44949 −0.392232
\(40\) 0 0
\(41\) 8.89898 1.38979 0.694894 0.719113i \(-0.255451\pi\)
0.694894 + 0.719113i \(0.255451\pi\)
\(42\) − 0.449490i − 0.0693578i
\(43\) − 2.44949i − 0.373544i −0.982403 0.186772i \(-0.940197\pi\)
0.982403 0.186772i \(-0.0598025\pi\)
\(44\) 1.44949 0.218519
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 11.7980i 1.72091i 0.509527 + 0.860455i \(0.329820\pi\)
−0.509527 + 0.860455i \(0.670180\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 6.79796 0.971137
\(50\) 0 0
\(51\) −0.449490 −0.0629412
\(52\) 2.44949i 0.339683i
\(53\) 2.55051i 0.350340i 0.984538 + 0.175170i \(0.0560474\pi\)
−0.984538 + 0.175170i \(0.943953\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −0.449490 −0.0600656
\(57\) 1.00000i 0.132453i
\(58\) − 10.3485i − 1.35882i
\(59\) −1.55051 −0.201859 −0.100930 0.994894i \(-0.532182\pi\)
−0.100930 + 0.994894i \(0.532182\pi\)
\(60\) 0 0
\(61\) −4.55051 −0.582633 −0.291317 0.956627i \(-0.594093\pi\)
−0.291317 + 0.956627i \(0.594093\pi\)
\(62\) − 3.00000i − 0.381000i
\(63\) 0.449490i 0.0566304i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −1.44949 −0.178420
\(67\) − 9.24745i − 1.12976i −0.825175 0.564878i \(-0.808923\pi\)
0.825175 0.564878i \(-0.191077\pi\)
\(68\) 0.449490i 0.0545086i
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −6.44949 −0.765414 −0.382707 0.923870i \(-0.625008\pi\)
−0.382707 + 0.923870i \(0.625008\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 1.00000i 0.117041i 0.998286 + 0.0585206i \(0.0186383\pi\)
−0.998286 + 0.0585206i \(0.981362\pi\)
\(74\) −11.7980 −1.37148
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0.651531i 0.0742488i
\(78\) − 2.44949i − 0.277350i
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 8.89898i 0.982728i
\(83\) 6.34847i 0.696835i 0.937339 + 0.348418i \(0.113281\pi\)
−0.937339 + 0.348418i \(0.886719\pi\)
\(84\) 0.449490 0.0490434
\(85\) 0 0
\(86\) 2.44949 0.264135
\(87\) 10.3485i 1.10947i
\(88\) 1.44949i 0.154516i
\(89\) −11.8990 −1.26129 −0.630645 0.776072i \(-0.717209\pi\)
−0.630645 + 0.776072i \(0.717209\pi\)
\(90\) 0 0
\(91\) −1.10102 −0.115418
\(92\) − 1.00000i − 0.104257i
\(93\) 3.00000i 0.311086i
\(94\) −11.7980 −1.21687
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 6.44949i 0.654846i 0.944878 + 0.327423i \(0.106180\pi\)
−0.944878 + 0.327423i \(0.893820\pi\)
\(98\) 6.79796i 0.686698i
\(99\) 1.44949 0.145679
\(100\) 0 0
\(101\) −6.44949 −0.641748 −0.320874 0.947122i \(-0.603977\pi\)
−0.320874 + 0.947122i \(0.603977\pi\)
\(102\) − 0.449490i − 0.0445061i
\(103\) 7.89898i 0.778310i 0.921172 + 0.389155i \(0.127233\pi\)
−0.921172 + 0.389155i \(0.872767\pi\)
\(104\) −2.44949 −0.240192
\(105\) 0 0
\(106\) −2.55051 −0.247727
\(107\) 2.65153i 0.256333i 0.991753 + 0.128167i \(0.0409092\pi\)
−0.991753 + 0.128167i \(0.959091\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −6.89898 −0.660802 −0.330401 0.943841i \(-0.607184\pi\)
−0.330401 + 0.943841i \(0.607184\pi\)
\(110\) 0 0
\(111\) 11.7980 1.11981
\(112\) − 0.449490i − 0.0424728i
\(113\) − 2.79796i − 0.263210i −0.991302 0.131605i \(-0.957987\pi\)
0.991302 0.131605i \(-0.0420130\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) 10.3485 0.960831
\(117\) 2.44949i 0.226455i
\(118\) − 1.55051i − 0.142736i
\(119\) −0.202041 −0.0185211
\(120\) 0 0
\(121\) −8.89898 −0.808998
\(122\) − 4.55051i − 0.411984i
\(123\) − 8.89898i − 0.802394i
\(124\) 3.00000 0.269408
\(125\) 0 0
\(126\) −0.449490 −0.0400437
\(127\) 9.89898i 0.878392i 0.898391 + 0.439196i \(0.144737\pi\)
−0.898391 + 0.439196i \(0.855263\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −2.44949 −0.215666
\(130\) 0 0
\(131\) −15.2474 −1.33218 −0.666088 0.745873i \(-0.732032\pi\)
−0.666088 + 0.745873i \(0.732032\pi\)
\(132\) − 1.44949i − 0.126162i
\(133\) 0.449490i 0.0389757i
\(134\) 9.24745 0.798858
\(135\) 0 0
\(136\) −0.449490 −0.0385434
\(137\) 4.89898i 0.418548i 0.977857 + 0.209274i \(0.0671101\pi\)
−0.977857 + 0.209274i \(0.932890\pi\)
\(138\) 1.00000i 0.0851257i
\(139\) −2.24745 −0.190626 −0.0953131 0.995447i \(-0.530385\pi\)
−0.0953131 + 0.995447i \(0.530385\pi\)
\(140\) 0 0
\(141\) 11.7980 0.993567
\(142\) − 6.44949i − 0.541229i
\(143\) 3.55051i 0.296909i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −1.00000 −0.0827606
\(147\) − 6.79796i − 0.560686i
\(148\) − 11.7980i − 0.969786i
\(149\) 13.7980 1.13037 0.565186 0.824963i \(-0.308804\pi\)
0.565186 + 0.824963i \(0.308804\pi\)
\(150\) 0 0
\(151\) −4.20204 −0.341957 −0.170979 0.985275i \(-0.554693\pi\)
−0.170979 + 0.985275i \(0.554693\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) 0.449490i 0.0363391i
\(154\) −0.651531 −0.0525018
\(155\) 0 0
\(156\) 2.44949 0.196116
\(157\) 4.89898i 0.390981i 0.980706 + 0.195491i \(0.0626299\pi\)
−0.980706 + 0.195491i \(0.937370\pi\)
\(158\) 5.00000i 0.397779i
\(159\) 2.55051 0.202269
\(160\) 0 0
\(161\) 0.449490 0.0354248
\(162\) 1.00000i 0.0785674i
\(163\) − 0.202041i − 0.0158251i −0.999969 0.00791254i \(-0.997481\pi\)
0.999969 0.00791254i \(-0.00251867\pi\)
\(164\) −8.89898 −0.694894
\(165\) 0 0
\(166\) −6.34847 −0.492737
\(167\) 18.0000i 1.39288i 0.717614 + 0.696441i \(0.245234\pi\)
−0.717614 + 0.696441i \(0.754766\pi\)
\(168\) 0.449490i 0.0346789i
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 2.44949i 0.186772i
\(173\) 16.3485i 1.24295i 0.783434 + 0.621476i \(0.213466\pi\)
−0.783434 + 0.621476i \(0.786534\pi\)
\(174\) −10.3485 −0.784515
\(175\) 0 0
\(176\) −1.44949 −0.109259
\(177\) 1.55051i 0.116543i
\(178\) − 11.8990i − 0.891866i
\(179\) −6.20204 −0.463562 −0.231781 0.972768i \(-0.574455\pi\)
−0.231781 + 0.972768i \(0.574455\pi\)
\(180\) 0 0
\(181\) −9.55051 −0.709884 −0.354942 0.934888i \(-0.615499\pi\)
−0.354942 + 0.934888i \(0.615499\pi\)
\(182\) − 1.10102i − 0.0816131i
\(183\) 4.55051i 0.336383i
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −3.00000 −0.219971
\(187\) 0.651531i 0.0476446i
\(188\) − 11.7980i − 0.860455i
\(189\) 0.449490 0.0326956
\(190\) 0 0
\(191\) −9.89898 −0.716265 −0.358133 0.933671i \(-0.616586\pi\)
−0.358133 + 0.933671i \(0.616586\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 0.651531i 0.0468982i 0.999725 + 0.0234491i \(0.00746477\pi\)
−0.999725 + 0.0234491i \(0.992535\pi\)
\(194\) −6.44949 −0.463046
\(195\) 0 0
\(196\) −6.79796 −0.485568
\(197\) 12.6515i 0.901384i 0.892679 + 0.450692i \(0.148823\pi\)
−0.892679 + 0.450692i \(0.851177\pi\)
\(198\) 1.44949i 0.103011i
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) −9.24745 −0.652265
\(202\) − 6.44949i − 0.453785i
\(203\) 4.65153i 0.326473i
\(204\) 0.449490 0.0314706
\(205\) 0 0
\(206\) −7.89898 −0.550348
\(207\) − 1.00000i − 0.0695048i
\(208\) − 2.44949i − 0.169842i
\(209\) 1.44949 0.100263
\(210\) 0 0
\(211\) −18.3485 −1.26316 −0.631580 0.775310i \(-0.717593\pi\)
−0.631580 + 0.775310i \(0.717593\pi\)
\(212\) − 2.55051i − 0.175170i
\(213\) 6.44949i 0.441912i
\(214\) −2.65153 −0.181255
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 1.34847i 0.0915401i
\(218\) − 6.89898i − 0.467258i
\(219\) 1.00000 0.0675737
\(220\) 0 0
\(221\) −1.10102 −0.0740627
\(222\) 11.7980i 0.791827i
\(223\) − 25.8990i − 1.73432i −0.498026 0.867162i \(-0.665942\pi\)
0.498026 0.867162i \(-0.334058\pi\)
\(224\) 0.449490 0.0300328
\(225\) 0 0
\(226\) 2.79796 0.186117
\(227\) − 9.59592i − 0.636903i −0.947939 0.318452i \(-0.896837\pi\)
0.947939 0.318452i \(-0.103163\pi\)
\(228\) − 1.00000i − 0.0662266i
\(229\) 17.2474 1.13974 0.569872 0.821734i \(-0.306993\pi\)
0.569872 + 0.821734i \(0.306993\pi\)
\(230\) 0 0
\(231\) 0.651531 0.0428676
\(232\) 10.3485i 0.679410i
\(233\) 18.2474i 1.19543i 0.801709 + 0.597715i \(0.203925\pi\)
−0.801709 + 0.597715i \(0.796075\pi\)
\(234\) −2.44949 −0.160128
\(235\) 0 0
\(236\) 1.55051 0.100930
\(237\) − 5.00000i − 0.324785i
\(238\) − 0.202041i − 0.0130964i
\(239\) −20.6969 −1.33877 −0.669387 0.742914i \(-0.733443\pi\)
−0.669387 + 0.742914i \(0.733443\pi\)
\(240\) 0 0
\(241\) 0.449490 0.0289542 0.0144771 0.999895i \(-0.495392\pi\)
0.0144771 + 0.999895i \(0.495392\pi\)
\(242\) − 8.89898i − 0.572048i
\(243\) − 1.00000i − 0.0641500i
\(244\) 4.55051 0.291317
\(245\) 0 0
\(246\) 8.89898 0.567378
\(247\) 2.44949i 0.155857i
\(248\) 3.00000i 0.190500i
\(249\) 6.34847 0.402318
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) − 0.449490i − 0.0283152i
\(253\) − 1.44949i − 0.0911286i
\(254\) −9.89898 −0.621117
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 27.4949i 1.71508i 0.514413 + 0.857542i \(0.328010\pi\)
−0.514413 + 0.857542i \(0.671990\pi\)
\(258\) − 2.44949i − 0.152499i
\(259\) 5.30306 0.329516
\(260\) 0 0
\(261\) 10.3485 0.640554
\(262\) − 15.2474i − 0.941991i
\(263\) − 12.7980i − 0.789156i −0.918862 0.394578i \(-0.870891\pi\)
0.918862 0.394578i \(-0.129109\pi\)
\(264\) 1.44949 0.0892099
\(265\) 0 0
\(266\) −0.449490 −0.0275600
\(267\) 11.8990i 0.728206i
\(268\) 9.24745i 0.564878i
\(269\) −6.89898 −0.420638 −0.210319 0.977633i \(-0.567450\pi\)
−0.210319 + 0.977633i \(0.567450\pi\)
\(270\) 0 0
\(271\) −30.9444 −1.87974 −0.939869 0.341536i \(-0.889053\pi\)
−0.939869 + 0.341536i \(0.889053\pi\)
\(272\) − 0.449490i − 0.0272543i
\(273\) 1.10102i 0.0666368i
\(274\) −4.89898 −0.295958
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) − 23.0454i − 1.38466i −0.721579 0.692332i \(-0.756583\pi\)
0.721579 0.692332i \(-0.243417\pi\)
\(278\) − 2.24745i − 0.134793i
\(279\) 3.00000 0.179605
\(280\) 0 0
\(281\) 7.00000 0.417585 0.208792 0.977960i \(-0.433047\pi\)
0.208792 + 0.977960i \(0.433047\pi\)
\(282\) 11.7980i 0.702558i
\(283\) − 10.2020i − 0.606448i −0.952919 0.303224i \(-0.901937\pi\)
0.952919 0.303224i \(-0.0980631\pi\)
\(284\) 6.44949 0.382707
\(285\) 0 0
\(286\) −3.55051 −0.209946
\(287\) − 4.00000i − 0.236113i
\(288\) − 1.00000i − 0.0589256i
\(289\) 16.7980 0.988115
\(290\) 0 0
\(291\) 6.44949 0.378076
\(292\) − 1.00000i − 0.0585206i
\(293\) − 1.24745i − 0.0728767i −0.999336 0.0364384i \(-0.988399\pi\)
0.999336 0.0364384i \(-0.0116013\pi\)
\(294\) 6.79796 0.396465
\(295\) 0 0
\(296\) 11.7980 0.685742
\(297\) − 1.44949i − 0.0841079i
\(298\) 13.7980i 0.799294i
\(299\) 2.44949 0.141658
\(300\) 0 0
\(301\) −1.10102 −0.0634618
\(302\) − 4.20204i − 0.241800i
\(303\) 6.44949i 0.370514i
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) −0.449490 −0.0256956
\(307\) − 1.65153i − 0.0942578i −0.998889 0.0471289i \(-0.984993\pi\)
0.998889 0.0471289i \(-0.0150072\pi\)
\(308\) − 0.651531i − 0.0371244i
\(309\) 7.89898 0.449357
\(310\) 0 0
\(311\) −4.20204 −0.238276 −0.119138 0.992878i \(-0.538013\pi\)
−0.119138 + 0.992878i \(0.538013\pi\)
\(312\) 2.44949i 0.138675i
\(313\) − 22.1010i − 1.24922i −0.780935 0.624612i \(-0.785257\pi\)
0.780935 0.624612i \(-0.214743\pi\)
\(314\) −4.89898 −0.276465
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) − 9.24745i − 0.519388i −0.965691 0.259694i \(-0.916378\pi\)
0.965691 0.259694i \(-0.0836218\pi\)
\(318\) 2.55051i 0.143026i
\(319\) 15.0000 0.839839
\(320\) 0 0
\(321\) 2.65153 0.147994
\(322\) 0.449490i 0.0250491i
\(323\) 0.449490i 0.0250103i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 0.202041 0.0111900
\(327\) 6.89898i 0.381514i
\(328\) − 8.89898i − 0.491364i
\(329\) 5.30306 0.292367
\(330\) 0 0
\(331\) −7.65153 −0.420566 −0.210283 0.977641i \(-0.567439\pi\)
−0.210283 + 0.977641i \(0.567439\pi\)
\(332\) − 6.34847i − 0.348418i
\(333\) − 11.7980i − 0.646524i
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) −0.449490 −0.0245217
\(337\) − 15.1010i − 0.822605i −0.911499 0.411303i \(-0.865074\pi\)
0.911499 0.411303i \(-0.134926\pi\)
\(338\) 7.00000i 0.380750i
\(339\) −2.79796 −0.151964
\(340\) 0 0
\(341\) 4.34847 0.235483
\(342\) 1.00000i 0.0540738i
\(343\) − 6.20204i − 0.334879i
\(344\) −2.44949 −0.132068
\(345\) 0 0
\(346\) −16.3485 −0.878899
\(347\) 15.5959i 0.837233i 0.908163 + 0.418616i \(0.137485\pi\)
−0.908163 + 0.418616i \(0.862515\pi\)
\(348\) − 10.3485i − 0.554736i
\(349\) −27.9444 −1.49583 −0.747914 0.663795i \(-0.768945\pi\)
−0.747914 + 0.663795i \(0.768945\pi\)
\(350\) 0 0
\(351\) 2.44949 0.130744
\(352\) − 1.44949i − 0.0772581i
\(353\) − 6.24745i − 0.332518i −0.986082 0.166259i \(-0.946831\pi\)
0.986082 0.166259i \(-0.0531688\pi\)
\(354\) −1.55051 −0.0824087
\(355\) 0 0
\(356\) 11.8990 0.630645
\(357\) 0.202041i 0.0106931i
\(358\) − 6.20204i − 0.327788i
\(359\) 13.1010 0.691445 0.345723 0.938337i \(-0.387634\pi\)
0.345723 + 0.938337i \(0.387634\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 9.55051i − 0.501964i
\(363\) 8.89898i 0.467075i
\(364\) 1.10102 0.0577092
\(365\) 0 0
\(366\) −4.55051 −0.237859
\(367\) − 10.4495i − 0.545459i −0.962091 0.272729i \(-0.912074\pi\)
0.962091 0.272729i \(-0.0879264\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) −8.89898 −0.463262
\(370\) 0 0
\(371\) 1.14643 0.0595196
\(372\) − 3.00000i − 0.155543i
\(373\) 17.5505i 0.908731i 0.890815 + 0.454365i \(0.150134\pi\)
−0.890815 + 0.454365i \(0.849866\pi\)
\(374\) −0.651531 −0.0336899
\(375\) 0 0
\(376\) 11.7980 0.608433
\(377\) 25.3485i 1.30551i
\(378\) 0.449490i 0.0231193i
\(379\) 30.6969 1.57680 0.788398 0.615166i \(-0.210911\pi\)
0.788398 + 0.615166i \(0.210911\pi\)
\(380\) 0 0
\(381\) 9.89898 0.507140
\(382\) − 9.89898i − 0.506476i
\(383\) 8.24745i 0.421425i 0.977548 + 0.210712i \(0.0675784\pi\)
−0.977548 + 0.210712i \(0.932422\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −0.651531 −0.0331620
\(387\) 2.44949i 0.124515i
\(388\) − 6.44949i − 0.327423i
\(389\) 17.5959 0.892148 0.446074 0.894996i \(-0.352822\pi\)
0.446074 + 0.894996i \(0.352822\pi\)
\(390\) 0 0
\(391\) 0.449490 0.0227317
\(392\) − 6.79796i − 0.343349i
\(393\) 15.2474i 0.769132i
\(394\) −12.6515 −0.637375
\(395\) 0 0
\(396\) −1.44949 −0.0728396
\(397\) − 29.9444i − 1.50287i −0.659810 0.751433i \(-0.729363\pi\)
0.659810 0.751433i \(-0.270637\pi\)
\(398\) 10.0000i 0.501255i
\(399\) 0.449490 0.0225026
\(400\) 0 0
\(401\) 30.7980 1.53798 0.768988 0.639263i \(-0.220760\pi\)
0.768988 + 0.639263i \(0.220760\pi\)
\(402\) − 9.24745i − 0.461221i
\(403\) 7.34847i 0.366053i
\(404\) 6.44949 0.320874
\(405\) 0 0
\(406\) −4.65153 −0.230852
\(407\) − 17.1010i − 0.847666i
\(408\) 0.449490i 0.0222531i
\(409\) 13.1010 0.647804 0.323902 0.946091i \(-0.395005\pi\)
0.323902 + 0.946091i \(0.395005\pi\)
\(410\) 0 0
\(411\) 4.89898 0.241649
\(412\) − 7.89898i − 0.389155i
\(413\) 0.696938i 0.0342941i
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 2.44949 0.120096
\(417\) 2.24745i 0.110058i
\(418\) 1.44949i 0.0708969i
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) − 18.3485i − 0.893190i
\(423\) − 11.7980i − 0.573636i
\(424\) 2.55051 0.123864
\(425\) 0 0
\(426\) −6.44949 −0.312479
\(427\) 2.04541i 0.0989842i
\(428\) − 2.65153i − 0.128167i
\(429\) 3.55051 0.171420
\(430\) 0 0
\(431\) −14.8990 −0.717659 −0.358829 0.933403i \(-0.616824\pi\)
−0.358829 + 0.933403i \(0.616824\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 19.3485i − 0.929828i −0.885356 0.464914i \(-0.846085\pi\)
0.885356 0.464914i \(-0.153915\pi\)
\(434\) −1.34847 −0.0647286
\(435\) 0 0
\(436\) 6.89898 0.330401
\(437\) − 1.00000i − 0.0478365i
\(438\) 1.00000i 0.0477818i
\(439\) 21.8990 1.04518 0.522591 0.852584i \(-0.324966\pi\)
0.522591 + 0.852584i \(0.324966\pi\)
\(440\) 0 0
\(441\) −6.79796 −0.323712
\(442\) − 1.10102i − 0.0523702i
\(443\) 3.24745i 0.154291i 0.997020 + 0.0771455i \(0.0245806\pi\)
−0.997020 + 0.0771455i \(0.975419\pi\)
\(444\) −11.7980 −0.559906
\(445\) 0 0
\(446\) 25.8990 1.22635
\(447\) − 13.7980i − 0.652621i
\(448\) 0.449490i 0.0212364i
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 0 0
\(451\) −12.8990 −0.607389
\(452\) 2.79796i 0.131605i
\(453\) 4.20204i 0.197429i
\(454\) 9.59592 0.450359
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) − 15.1010i − 0.706396i −0.935549 0.353198i \(-0.885094\pi\)
0.935549 0.353198i \(-0.114906\pi\)
\(458\) 17.2474i 0.805920i
\(459\) 0.449490 0.0209804
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0.651531i 0.0303120i
\(463\) − 14.6969i − 0.683025i −0.939877 0.341512i \(-0.889061\pi\)
0.939877 0.341512i \(-0.110939\pi\)
\(464\) −10.3485 −0.480416
\(465\) 0 0
\(466\) −18.2474 −0.845297
\(467\) 8.34847i 0.386321i 0.981167 + 0.193161i \(0.0618738\pi\)
−0.981167 + 0.193161i \(0.938126\pi\)
\(468\) − 2.44949i − 0.113228i
\(469\) −4.15663 −0.191935
\(470\) 0 0
\(471\) 4.89898 0.225733
\(472\) 1.55051i 0.0713680i
\(473\) 3.55051i 0.163253i
\(474\) 5.00000 0.229658
\(475\) 0 0
\(476\) 0.202041 0.00926054
\(477\) − 2.55051i − 0.116780i
\(478\) − 20.6969i − 0.946656i
\(479\) 22.5959 1.03243 0.516217 0.856458i \(-0.327340\pi\)
0.516217 + 0.856458i \(0.327340\pi\)
\(480\) 0 0
\(481\) 28.8990 1.31768
\(482\) 0.449490i 0.0204737i
\(483\) − 0.449490i − 0.0204525i
\(484\) 8.89898 0.404499
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 32.0000i − 1.45006i −0.688718 0.725029i \(-0.741826\pi\)
0.688718 0.725029i \(-0.258174\pi\)
\(488\) 4.55051i 0.205992i
\(489\) −0.202041 −0.00913661
\(490\) 0 0
\(491\) 4.40408 0.198753 0.0993767 0.995050i \(-0.468315\pi\)
0.0993767 + 0.995050i \(0.468315\pi\)
\(492\) 8.89898i 0.401197i
\(493\) 4.65153i 0.209494i
\(494\) −2.44949 −0.110208
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) 2.89898i 0.130037i
\(498\) 6.34847i 0.284482i
\(499\) −39.8434 −1.78363 −0.891817 0.452396i \(-0.850569\pi\)
−0.891817 + 0.452396i \(0.850569\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) − 18.0000i − 0.803379i
\(503\) − 10.2020i − 0.454887i −0.973791 0.227443i \(-0.926963\pi\)
0.973791 0.227443i \(-0.0730366\pi\)
\(504\) 0.449490 0.0200219
\(505\) 0 0
\(506\) 1.44949 0.0644377
\(507\) − 7.00000i − 0.310881i
\(508\) − 9.89898i − 0.439196i
\(509\) −4.14643 −0.183787 −0.0918936 0.995769i \(-0.529292\pi\)
−0.0918936 + 0.995769i \(0.529292\pi\)
\(510\) 0 0
\(511\) 0.449490 0.0198843
\(512\) 1.00000i 0.0441942i
\(513\) − 1.00000i − 0.0441511i
\(514\) −27.4949 −1.21275
\(515\) 0 0
\(516\) 2.44949 0.107833
\(517\) − 17.1010i − 0.752102i
\(518\) 5.30306i 0.233003i
\(519\) 16.3485 0.717618
\(520\) 0 0
\(521\) −13.6969 −0.600074 −0.300037 0.953928i \(-0.596999\pi\)
−0.300037 + 0.953928i \(0.596999\pi\)
\(522\) 10.3485i 0.452940i
\(523\) − 25.3939i − 1.11040i −0.831718 0.555198i \(-0.812642\pi\)
0.831718 0.555198i \(-0.187358\pi\)
\(524\) 15.2474 0.666088
\(525\) 0 0
\(526\) 12.7980 0.558018
\(527\) 1.34847i 0.0587402i
\(528\) 1.44949i 0.0630809i
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) 1.55051 0.0672864
\(532\) − 0.449490i − 0.0194879i
\(533\) − 21.7980i − 0.944174i
\(534\) −11.8990 −0.514919
\(535\) 0 0
\(536\) −9.24745 −0.399429
\(537\) 6.20204i 0.267638i
\(538\) − 6.89898i − 0.297436i
\(539\) −9.85357 −0.424423
\(540\) 0 0
\(541\) −12.1464 −0.522216 −0.261108 0.965310i \(-0.584088\pi\)
−0.261108 + 0.965310i \(0.584088\pi\)
\(542\) − 30.9444i − 1.32918i
\(543\) 9.55051i 0.409852i
\(544\) 0.449490 0.0192717
\(545\) 0 0
\(546\) −1.10102 −0.0471193
\(547\) 36.6413i 1.56667i 0.621600 + 0.783335i \(0.286483\pi\)
−0.621600 + 0.783335i \(0.713517\pi\)
\(548\) − 4.89898i − 0.209274i
\(549\) 4.55051 0.194211
\(550\) 0 0
\(551\) 10.3485 0.440860
\(552\) − 1.00000i − 0.0425628i
\(553\) − 2.24745i − 0.0955712i
\(554\) 23.0454 0.979106
\(555\) 0 0
\(556\) 2.24745 0.0953131
\(557\) 12.6515i 0.536063i 0.963410 + 0.268031i \(0.0863731\pi\)
−0.963410 + 0.268031i \(0.913627\pi\)
\(558\) 3.00000i 0.127000i
\(559\) −6.00000 −0.253773
\(560\) 0 0
\(561\) 0.651531 0.0275077
\(562\) 7.00000i 0.295277i
\(563\) 18.2474i 0.769038i 0.923117 + 0.384519i \(0.125633\pi\)
−0.923117 + 0.384519i \(0.874367\pi\)
\(564\) −11.7980 −0.496784
\(565\) 0 0
\(566\) 10.2020 0.428824
\(567\) − 0.449490i − 0.0188768i
\(568\) 6.44949i 0.270615i
\(569\) −45.1918 −1.89454 −0.947270 0.320436i \(-0.896171\pi\)
−0.947270 + 0.320436i \(0.896171\pi\)
\(570\) 0 0
\(571\) 14.2474 0.596237 0.298119 0.954529i \(-0.403641\pi\)
0.298119 + 0.954529i \(0.403641\pi\)
\(572\) − 3.55051i − 0.148454i
\(573\) 9.89898i 0.413536i
\(574\) 4.00000 0.166957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 9.89898i 0.412100i 0.978541 + 0.206050i \(0.0660609\pi\)
−0.978541 + 0.206050i \(0.933939\pi\)
\(578\) 16.7980i 0.698703i
\(579\) 0.651531 0.0270767
\(580\) 0 0
\(581\) 2.85357 0.118386
\(582\) 6.44949i 0.267340i
\(583\) − 3.69694i − 0.153112i
\(584\) 1.00000 0.0413803
\(585\) 0 0
\(586\) 1.24745 0.0515316
\(587\) − 28.5505i − 1.17841i −0.807985 0.589203i \(-0.799442\pi\)
0.807985 0.589203i \(-0.200558\pi\)
\(588\) 6.79796i 0.280343i
\(589\) 3.00000 0.123613
\(590\) 0 0
\(591\) 12.6515 0.520414
\(592\) 11.7980i 0.484893i
\(593\) 40.4949i 1.66293i 0.555580 + 0.831463i \(0.312496\pi\)
−0.555580 + 0.831463i \(0.687504\pi\)
\(594\) 1.44949 0.0594733
\(595\) 0 0
\(596\) −13.7980 −0.565186
\(597\) − 10.0000i − 0.409273i
\(598\) 2.44949i 0.100167i
\(599\) −11.5505 −0.471941 −0.235971 0.971760i \(-0.575827\pi\)
−0.235971 + 0.971760i \(0.575827\pi\)
\(600\) 0 0
\(601\) −17.1464 −0.699417 −0.349709 0.936858i \(-0.613719\pi\)
−0.349709 + 0.936858i \(0.613719\pi\)
\(602\) − 1.10102i − 0.0448742i
\(603\) 9.24745i 0.376585i
\(604\) 4.20204 0.170979
\(605\) 0 0
\(606\) −6.44949 −0.261993
\(607\) − 32.1918i − 1.30663i −0.757088 0.653313i \(-0.773379\pi\)
0.757088 0.653313i \(-0.226621\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) 4.65153 0.188490
\(610\) 0 0
\(611\) 28.8990 1.16913
\(612\) − 0.449490i − 0.0181695i
\(613\) 41.1918i 1.66372i 0.554984 + 0.831861i \(0.312725\pi\)
−0.554984 + 0.831861i \(0.687275\pi\)
\(614\) 1.65153 0.0666504
\(615\) 0 0
\(616\) 0.651531 0.0262509
\(617\) − 40.2929i − 1.62213i −0.584957 0.811065i \(-0.698888\pi\)
0.584957 0.811065i \(-0.301112\pi\)
\(618\) 7.89898i 0.317744i
\(619\) −10.8536 −0.436242 −0.218121 0.975922i \(-0.569993\pi\)
−0.218121 + 0.975922i \(0.569993\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) − 4.20204i − 0.168486i
\(623\) 5.34847i 0.214282i
\(624\) −2.44949 −0.0980581
\(625\) 0 0
\(626\) 22.1010 0.883334
\(627\) − 1.44949i − 0.0578870i
\(628\) − 4.89898i − 0.195491i
\(629\) 5.30306 0.211447
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) − 5.00000i − 0.198889i
\(633\) 18.3485i 0.729286i
\(634\) 9.24745 0.367263
\(635\) 0 0
\(636\) −2.55051 −0.101134
\(637\) − 16.6515i − 0.659758i
\(638\) 15.0000i 0.593856i
\(639\) 6.44949 0.255138
\(640\) 0 0
\(641\) 18.2020 0.718937 0.359469 0.933157i \(-0.382958\pi\)
0.359469 + 0.933157i \(0.382958\pi\)
\(642\) 2.65153i 0.104648i
\(643\) 19.7980i 0.780755i 0.920655 + 0.390378i \(0.127656\pi\)
−0.920655 + 0.390378i \(0.872344\pi\)
\(644\) −0.449490 −0.0177124
\(645\) 0 0
\(646\) −0.449490 −0.0176849
\(647\) 16.1010i 0.632996i 0.948593 + 0.316498i \(0.102507\pi\)
−0.948593 + 0.316498i \(0.897493\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 2.24745 0.0882201
\(650\) 0 0
\(651\) 1.34847 0.0528507
\(652\) 0.202041i 0.00791254i
\(653\) − 14.6969i − 0.575136i −0.957760 0.287568i \(-0.907153\pi\)
0.957760 0.287568i \(-0.0928467\pi\)
\(654\) −6.89898 −0.269771
\(655\) 0 0
\(656\) 8.89898 0.347447
\(657\) − 1.00000i − 0.0390137i
\(658\) 5.30306i 0.206735i
\(659\) −26.0454 −1.01459 −0.507293 0.861774i \(-0.669354\pi\)
−0.507293 + 0.861774i \(0.669354\pi\)
\(660\) 0 0
\(661\) −15.5959 −0.606611 −0.303305 0.952893i \(-0.598090\pi\)
−0.303305 + 0.952893i \(0.598090\pi\)
\(662\) − 7.65153i − 0.297385i
\(663\) 1.10102i 0.0427601i
\(664\) 6.34847 0.246368
\(665\) 0 0
\(666\) 11.7980 0.457162
\(667\) − 10.3485i − 0.400694i
\(668\) − 18.0000i − 0.696441i
\(669\) −25.8990 −1.00131
\(670\) 0 0
\(671\) 6.59592 0.254633
\(672\) − 0.449490i − 0.0173394i
\(673\) 30.6515i 1.18153i 0.806844 + 0.590765i \(0.201174\pi\)
−0.806844 + 0.590765i \(0.798826\pi\)
\(674\) 15.1010 0.581670
\(675\) 0 0
\(676\) −7.00000 −0.269231
\(677\) 7.65153i 0.294072i 0.989131 + 0.147036i \(0.0469734\pi\)
−0.989131 + 0.147036i \(0.953027\pi\)
\(678\) − 2.79796i − 0.107455i
\(679\) 2.89898 0.111253
\(680\) 0 0
\(681\) −9.59592 −0.367716
\(682\) 4.34847i 0.166511i
\(683\) − 47.6413i − 1.82294i −0.411361 0.911472i \(-0.634947\pi\)
0.411361 0.911472i \(-0.365053\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) 6.20204 0.236795
\(687\) − 17.2474i − 0.658031i
\(688\) − 2.44949i − 0.0933859i
\(689\) 6.24745 0.238009
\(690\) 0 0
\(691\) 34.2474 1.30283 0.651417 0.758720i \(-0.274175\pi\)
0.651417 + 0.758720i \(0.274175\pi\)
\(692\) − 16.3485i − 0.621476i
\(693\) − 0.651531i − 0.0247496i
\(694\) −15.5959 −0.592013
\(695\) 0 0
\(696\) 10.3485 0.392258
\(697\) − 4.00000i − 0.151511i
\(698\) − 27.9444i − 1.05771i
\(699\) 18.2474 0.690182
\(700\) 0 0
\(701\) 16.4949 0.623004 0.311502 0.950246i \(-0.399168\pi\)
0.311502 + 0.950246i \(0.399168\pi\)
\(702\) 2.44949i 0.0924500i
\(703\) − 11.7980i − 0.444968i
\(704\) 1.44949 0.0546297
\(705\) 0 0
\(706\) 6.24745 0.235126
\(707\) 2.89898i 0.109027i
\(708\) − 1.55051i − 0.0582717i
\(709\) 37.2474 1.39886 0.699429 0.714702i \(-0.253438\pi\)
0.699429 + 0.714702i \(0.253438\pi\)
\(710\) 0 0
\(711\) −5.00000 −0.187515
\(712\) 11.8990i 0.445933i
\(713\) − 3.00000i − 0.112351i
\(714\) −0.202041 −0.00756120
\(715\) 0 0
\(716\) 6.20204 0.231781
\(717\) 20.6969i 0.772941i
\(718\) 13.1010i 0.488926i
\(719\) 38.7980 1.44692 0.723460 0.690366i \(-0.242551\pi\)
0.723460 + 0.690366i \(0.242551\pi\)
\(720\) 0 0
\(721\) 3.55051 0.132228
\(722\) 1.00000i 0.0372161i
\(723\) − 0.449490i − 0.0167167i
\(724\) 9.55051 0.354942
\(725\) 0 0
\(726\) −8.89898 −0.330272
\(727\) − 26.4949i − 0.982641i −0.870979 0.491321i \(-0.836514\pi\)
0.870979 0.491321i \(-0.163486\pi\)
\(728\) 1.10102i 0.0408065i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −1.10102 −0.0407227
\(732\) − 4.55051i − 0.168192i
\(733\) 30.1464i 1.11348i 0.830686 + 0.556742i \(0.187949\pi\)
−0.830686 + 0.556742i \(0.812051\pi\)
\(734\) 10.4495 0.385698
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 13.4041i 0.493746i
\(738\) − 8.89898i − 0.327576i
\(739\) −23.1010 −0.849785 −0.424892 0.905244i \(-0.639688\pi\)
−0.424892 + 0.905244i \(0.639688\pi\)
\(740\) 0 0
\(741\) 2.44949 0.0899843
\(742\) 1.14643i 0.0420867i
\(743\) − 5.39388i − 0.197882i −0.995093 0.0989411i \(-0.968454\pi\)
0.995093 0.0989411i \(-0.0315455\pi\)
\(744\) 3.00000 0.109985
\(745\) 0 0
\(746\) −17.5505 −0.642570
\(747\) − 6.34847i − 0.232278i
\(748\) − 0.651531i − 0.0238223i
\(749\) 1.19184 0.0435487
\(750\) 0 0
\(751\) −4.20204 −0.153335 −0.0766673 0.997057i \(-0.524428\pi\)
−0.0766673 + 0.997057i \(0.524428\pi\)
\(752\) 11.7980i 0.430227i
\(753\) 18.0000i 0.655956i
\(754\) −25.3485 −0.923137
\(755\) 0 0
\(756\) −0.449490 −0.0163478
\(757\) 13.8536i 0.503517i 0.967790 + 0.251758i \(0.0810088\pi\)
−0.967790 + 0.251758i \(0.918991\pi\)
\(758\) 30.6969i 1.11496i
\(759\) −1.44949 −0.0526131
\(760\) 0 0
\(761\) 36.6515 1.32862 0.664308 0.747459i \(-0.268726\pi\)
0.664308 + 0.747459i \(0.268726\pi\)
\(762\) 9.89898i 0.358602i
\(763\) 3.10102i 0.112264i
\(764\) 9.89898 0.358133
\(765\) 0 0
\(766\) −8.24745 −0.297992
\(767\) 3.79796i 0.137136i
\(768\) − 1.00000i − 0.0360844i
\(769\) −1.89898 −0.0684790 −0.0342395 0.999414i \(-0.510901\pi\)
−0.0342395 + 0.999414i \(0.510901\pi\)
\(770\) 0 0
\(771\) 27.4949 0.990205
\(772\) − 0.651531i − 0.0234491i
\(773\) 30.4949i 1.09683i 0.836208 + 0.548413i \(0.184768\pi\)
−0.836208 + 0.548413i \(0.815232\pi\)
\(774\) −2.44949 −0.0880451
\(775\) 0 0
\(776\) 6.44949 0.231523
\(777\) − 5.30306i − 0.190246i
\(778\) 17.5959i 0.630844i
\(779\) −8.89898 −0.318839
\(780\) 0 0
\(781\) 9.34847 0.334515
\(782\) 0.449490i 0.0160737i
\(783\) − 10.3485i − 0.369824i
\(784\) 6.79796 0.242784
\(785\) 0 0
\(786\) −15.2474 −0.543858
\(787\) − 28.5505i − 1.01772i −0.860851 0.508858i \(-0.830068\pi\)
0.860851 0.508858i \(-0.169932\pi\)
\(788\) − 12.6515i − 0.450692i
\(789\) −12.7980 −0.455619
\(790\) 0 0
\(791\) −1.25765 −0.0447170
\(792\) − 1.44949i − 0.0515054i
\(793\) 11.1464i 0.395821i
\(794\) 29.9444 1.06269
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) 52.4949i 1.85946i 0.368236 + 0.929732i \(0.379962\pi\)
−0.368236 + 0.929732i \(0.620038\pi\)
\(798\) 0.449490i 0.0159118i
\(799\) 5.30306 0.187609
\(800\) 0 0
\(801\) 11.8990 0.420430
\(802\) 30.7980i 1.08751i
\(803\) − 1.44949i − 0.0511514i
\(804\) 9.24745 0.326132
\(805\) 0 0
\(806\) −7.34847 −0.258839
\(807\) 6.89898i 0.242856i
\(808\) 6.44949i 0.226892i
\(809\) 8.44949 0.297068 0.148534 0.988907i \(-0.452545\pi\)
0.148534 + 0.988907i \(0.452545\pi\)
\(810\) 0 0
\(811\) −14.5505 −0.510938 −0.255469 0.966817i \(-0.582230\pi\)
−0.255469 + 0.966817i \(0.582230\pi\)
\(812\) − 4.65153i − 0.163237i
\(813\) 30.9444i 1.08527i
\(814\) 17.1010 0.599390
\(815\) 0 0
\(816\) −0.449490 −0.0157353
\(817\) 2.44949i 0.0856968i
\(818\) 13.1010i 0.458066i
\(819\) 1.10102 0.0384728
\(820\) 0 0
\(821\) 11.1464 0.389013 0.194507 0.980901i \(-0.437689\pi\)
0.194507 + 0.980901i \(0.437689\pi\)
\(822\) 4.89898i 0.170872i
\(823\) 12.2020i 0.425336i 0.977124 + 0.212668i \(0.0682153\pi\)
−0.977124 + 0.212668i \(0.931785\pi\)
\(824\) 7.89898 0.275174
\(825\) 0 0
\(826\) −0.696938 −0.0242496
\(827\) − 24.2474i − 0.843166i −0.906790 0.421583i \(-0.861475\pi\)
0.906790 0.421583i \(-0.138525\pi\)
\(828\) 1.00000i 0.0347524i
\(829\) −43.6413 −1.51573 −0.757863 0.652414i \(-0.773756\pi\)
−0.757863 + 0.652414i \(0.773756\pi\)
\(830\) 0 0
\(831\) −23.0454 −0.799436
\(832\) 2.44949i 0.0849208i
\(833\) − 3.05561i − 0.105871i
\(834\) −2.24745 −0.0778228
\(835\) 0 0
\(836\) −1.44949 −0.0501317
\(837\) − 3.00000i − 0.103695i
\(838\) 0 0
\(839\) −25.3485 −0.875126 −0.437563 0.899188i \(-0.644158\pi\)
−0.437563 + 0.899188i \(0.644158\pi\)
\(840\) 0 0
\(841\) 78.0908 2.69279
\(842\) 22.0000i 0.758170i
\(843\) − 7.00000i − 0.241093i
\(844\) 18.3485 0.631580
\(845\) 0 0
\(846\) 11.7980 0.405622
\(847\) 4.00000i 0.137442i
\(848\) 2.55051i 0.0875849i
\(849\) −10.2020 −0.350133
\(850\) 0 0
\(851\) −11.7980 −0.404429
\(852\) − 6.44949i − 0.220956i
\(853\) − 10.8990i − 0.373174i −0.982438 0.186587i \(-0.940257\pi\)
0.982438 0.186587i \(-0.0597426\pi\)
\(854\) −2.04541 −0.0699924
\(855\) 0 0
\(856\) 2.65153 0.0906275
\(857\) 12.4949i 0.426818i 0.976963 + 0.213409i \(0.0684566\pi\)
−0.976963 + 0.213409i \(0.931543\pi\)
\(858\) 3.55051i 0.121212i
\(859\) 23.7980 0.811976 0.405988 0.913878i \(-0.366927\pi\)
0.405988 + 0.913878i \(0.366927\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) − 14.8990i − 0.507461i
\(863\) 46.6969i 1.58958i 0.606883 + 0.794791i \(0.292420\pi\)
−0.606883 + 0.794791i \(0.707580\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 19.3485 0.657488
\(867\) − 16.7980i − 0.570489i
\(868\) − 1.34847i − 0.0457700i
\(869\) −7.24745 −0.245853
\(870\) 0 0
\(871\) −22.6515 −0.767518
\(872\) 6.89898i 0.233629i
\(873\) − 6.44949i − 0.218282i
\(874\) 1.00000 0.0338255
\(875\) 0 0
\(876\) −1.00000 −0.0337869
\(877\) − 18.8990i − 0.638173i −0.947726 0.319087i \(-0.896624\pi\)
0.947726 0.319087i \(-0.103376\pi\)
\(878\) 21.8990i 0.739055i
\(879\) −1.24745 −0.0420754
\(880\) 0 0
\(881\) 21.3031 0.717718 0.358859 0.933392i \(-0.383166\pi\)
0.358859 + 0.933392i \(0.383166\pi\)
\(882\) − 6.79796i − 0.228899i
\(883\) − 49.3485i − 1.66071i −0.557236 0.830354i \(-0.688138\pi\)
0.557236 0.830354i \(-0.311862\pi\)
\(884\) 1.10102 0.0370313
\(885\) 0 0
\(886\) −3.24745 −0.109100
\(887\) 4.89898i 0.164492i 0.996612 + 0.0822458i \(0.0262093\pi\)
−0.996612 + 0.0822458i \(0.973791\pi\)
\(888\) − 11.7980i − 0.395914i
\(889\) 4.44949 0.149231
\(890\) 0 0
\(891\) −1.44949 −0.0485597
\(892\) 25.8990i 0.867162i
\(893\) − 11.7980i − 0.394804i
\(894\) 13.7980 0.461473
\(895\) 0 0
\(896\) −0.449490 −0.0150164
\(897\) − 2.44949i − 0.0817861i
\(898\) 15.0000i 0.500556i
\(899\) 31.0454 1.03542
\(900\) 0 0
\(901\) 1.14643 0.0381931
\(902\) − 12.8990i − 0.429489i
\(903\) 1.10102i 0.0366397i
\(904\) −2.79796 −0.0930587
\(905\) 0 0
\(906\) −4.20204 −0.139603
\(907\) − 22.0000i − 0.730498i −0.930910 0.365249i \(-0.880984\pi\)
0.930910 0.365249i \(-0.119016\pi\)
\(908\) 9.59592i 0.318452i
\(909\) 6.44949 0.213916
\(910\) 0 0
\(911\) −42.4949 −1.40792 −0.703959 0.710240i \(-0.748586\pi\)
−0.703959 + 0.710240i \(0.748586\pi\)
\(912\) 1.00000i 0.0331133i
\(913\) − 9.20204i − 0.304543i
\(914\) 15.1010 0.499497
\(915\) 0 0
\(916\) −17.2474 −0.569872
\(917\) 6.85357i 0.226325i
\(918\) 0.449490i 0.0148354i
\(919\) 39.8434 1.31431 0.657156 0.753755i \(-0.271759\pi\)
0.657156 + 0.753755i \(0.271759\pi\)
\(920\) 0 0
\(921\) −1.65153 −0.0544198
\(922\) 12.0000i 0.395199i
\(923\) 15.7980i 0.519996i
\(924\) −0.651531 −0.0214338
\(925\) 0 0
\(926\) 14.6969 0.482971
\(927\) − 7.89898i − 0.259437i
\(928\) − 10.3485i − 0.339705i
\(929\) −7.59592 −0.249214 −0.124607 0.992206i \(-0.539767\pi\)
−0.124607 + 0.992206i \(0.539767\pi\)
\(930\) 0 0
\(931\) −6.79796 −0.222794
\(932\) − 18.2474i − 0.597715i
\(933\) 4.20204i 0.137569i
\(934\) −8.34847 −0.273170
\(935\) 0 0
\(936\) 2.44949 0.0800641
\(937\) 39.3939i 1.28694i 0.765471 + 0.643471i \(0.222506\pi\)
−0.765471 + 0.643471i \(0.777494\pi\)
\(938\) − 4.15663i − 0.135719i
\(939\) −22.1010 −0.721240
\(940\) 0 0
\(941\) −46.6413 −1.52046 −0.760232 0.649652i \(-0.774915\pi\)
−0.760232 + 0.649652i \(0.774915\pi\)
\(942\) 4.89898i 0.159617i
\(943\) 8.89898i 0.289791i
\(944\) −1.55051 −0.0504648
\(945\) 0 0
\(946\) −3.55051 −0.115437
\(947\) 8.00000i 0.259965i 0.991516 + 0.129983i \(0.0414921\pi\)
−0.991516 + 0.129983i \(0.958508\pi\)
\(948\) 5.00000i 0.162392i
\(949\) 2.44949 0.0795138
\(950\) 0 0
\(951\) −9.24745 −0.299869
\(952\) 0.202041i 0.00654819i
\(953\) − 33.4949i − 1.08501i −0.840054 0.542503i \(-0.817477\pi\)
0.840054 0.542503i \(-0.182523\pi\)
\(954\) 2.55051 0.0825758
\(955\) 0 0
\(956\) 20.6969 0.669387
\(957\) − 15.0000i − 0.484881i
\(958\) 22.5959i 0.730041i
\(959\) 2.20204 0.0711076
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 28.8990i 0.931740i
\(963\) − 2.65153i − 0.0854444i
\(964\) −0.449490 −0.0144771
\(965\) 0 0
\(966\) 0.449490 0.0144621
\(967\) − 25.1010i − 0.807194i −0.914937 0.403597i \(-0.867760\pi\)
0.914937 0.403597i \(-0.132240\pi\)
\(968\) 8.89898i 0.286024i
\(969\) 0.449490 0.0144397
\(970\) 0 0
\(971\) −41.6413 −1.33633 −0.668167 0.744011i \(-0.732921\pi\)
−0.668167 + 0.744011i \(0.732921\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 1.01021i 0.0323857i
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) −4.55051 −0.145658
\(977\) − 19.5959i − 0.626929i −0.949600 0.313464i \(-0.898510\pi\)
0.949600 0.313464i \(-0.101490\pi\)
\(978\) − 0.202041i − 0.00646056i
\(979\) 17.2474 0.551231
\(980\) 0 0
\(981\) 6.89898 0.220267
\(982\) 4.40408i 0.140540i
\(983\) − 45.5505i − 1.45284i −0.687253 0.726418i \(-0.741184\pi\)
0.687253 0.726418i \(-0.258816\pi\)
\(984\) −8.89898 −0.283689
\(985\) 0 0
\(986\) −4.65153 −0.148135
\(987\) − 5.30306i − 0.168798i
\(988\) − 2.44949i − 0.0779287i
\(989\) 2.44949 0.0778892
\(990\) 0 0
\(991\) 53.8990 1.71216 0.856079 0.516845i \(-0.172894\pi\)
0.856079 + 0.516845i \(0.172894\pi\)
\(992\) − 3.00000i − 0.0952501i
\(993\) 7.65153i 0.242814i
\(994\) −2.89898 −0.0919500
\(995\) 0 0
\(996\) −6.34847 −0.201159
\(997\) 34.5505i 1.09423i 0.837059 + 0.547113i \(0.184273\pi\)
−0.837059 + 0.547113i \(0.815727\pi\)
\(998\) − 39.8434i − 1.26122i
\(999\) −11.7980 −0.373271
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 2850.2.d.w.799.3 4
5.2 odd 4 2850.2.a.bc.1.2 2
5.3 odd 4 2850.2.a.bj.1.1 yes 2
5.4 even 2 inner 2850.2.d.w.799.2 4
15.2 even 4 8550.2.a.bv.1.2 2
15.8 even 4 8550.2.a.bu.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.bc.1.2 2 5.2 odd 4
2850.2.a.bj.1.1 yes 2 5.3 odd 4
2850.2.d.w.799.2 4 5.4 even 2 inner
2850.2.d.w.799.3 4 1.1 even 1 trivial
8550.2.a.bu.1.1 2 15.8 even 4
8550.2.a.bv.1.2 2 15.2 even 4