Properties

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

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +0.449490i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +0.449490i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.44949 q^{11} -1.00000i q^{12} +2.44949i q^{13} +0.449490 q^{14} +1.00000 q^{16} +0.449490i q^{17} +1.00000i q^{18} -1.00000 q^{19} -0.449490 q^{21} +1.44949i q^{22} -1.00000i q^{23} -1.00000 q^{24} +2.44949 q^{26} -1.00000i q^{27} -0.449490i q^{28} -10.3485 q^{29} -3.00000 q^{31} -1.00000i q^{32} -1.44949i q^{33} +0.449490 q^{34} +1.00000 q^{36} -11.7980i q^{37} +1.00000i q^{38} -2.44949 q^{39} +8.89898 q^{41} +0.449490i q^{42} +2.44949i q^{43} +1.44949 q^{44} -1.00000 q^{46} -11.7980i q^{47} +1.00000i q^{48} +6.79796 q^{49} -0.449490 q^{51} -2.44949i q^{52} -2.55051i q^{53} -1.00000 q^{54} -0.449490 q^{56} -1.00000i q^{57} +10.3485i q^{58} -1.55051 q^{59} -4.55051 q^{61} +3.00000i q^{62} -0.449490i q^{63} -1.00000 q^{64} -1.44949 q^{66} +9.24745i q^{67} -0.449490i q^{68} +1.00000 q^{69} -6.44949 q^{71} -1.00000i q^{72} -1.00000i q^{73} -11.7980 q^{74} +1.00000 q^{76} -0.651531i q^{77} +2.44949i q^{78} +5.00000 q^{79} +1.00000 q^{81} -8.89898i q^{82} -6.34847i q^{83} +0.449490 q^{84} +2.44949 q^{86} -10.3485i q^{87} -1.44949i q^{88} -11.8990 q^{89} -1.10102 q^{91} +1.00000i q^{92} -3.00000i q^{93} -11.7980 q^{94} +1.00000 q^{96} -6.44949i q^{97} -6.79796i q^{98} +1.44949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) 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 \(0.0270716\pi\)
−0.996386 + 0.0849456i \(0.972928\pi\)
\(8\) 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 \(0.110320\pi\)
−0.940540 + 0.339683i \(0.889680\pi\)
\(14\) 0.449490 0.120131
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.449490i 0.109017i 0.998513 + 0.0545086i \(0.0173592\pi\)
−0.998513 + 0.0545086i \(0.982641\pi\)
\(18\) 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 \(-0.966753\pi\)
0.994550 0.104257i \(-0.0332465\pi\)
\(24\) −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 \(-0.578445\pi\)
0.243956 0.969786i \(-0.421555\pi\)
\(38\) 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 \(0.0598025\pi\)
−0.982403 + 0.186772i \(0.940197\pi\)
\(44\) 1.44949 0.218519
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) − 11.7980i − 1.72091i −0.509527 0.860455i \(-0.670180\pi\)
0.509527 0.860455i \(-0.329820\pi\)
\(48\) 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 \(-0.943953\pi\)
0.984538 0.175170i \(-0.0560474\pi\)
\(54\) −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 \(0.191077\pi\)
−0.825175 + 0.564878i \(0.808923\pi\)
\(68\) − 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 \(-0.981362\pi\)
0.998286 0.0585206i \(-0.0186383\pi\)
\(74\) −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 \(-0.886719\pi\)
0.937339 0.348418i \(-0.113281\pi\)
\(84\) 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 \(-0.893820\pi\)
0.944878 0.327423i \(-0.106180\pi\)
\(98\) − 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 \(-0.872767\pi\)
0.921172 0.389155i \(-0.127233\pi\)
\(104\) −2.44949 −0.240192
\(105\) 0 0
\(106\) −2.55051 −0.247727
\(107\) − 2.65153i − 0.256333i −0.991753 0.128167i \(-0.959091\pi\)
0.991753 0.128167i \(-0.0409092\pi\)
\(108\) 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 \(0.0420130\pi\)
−0.991302 + 0.131605i \(0.957987\pi\)
\(114\) −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 \(-0.855263\pi\)
0.898391 0.439196i \(-0.144737\pi\)
\(128\) 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 \(-0.932890\pi\)
0.977857 0.209274i \(-0.0671101\pi\)
\(138\) − 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 \(-0.937370\pi\)
0.980706 0.195491i \(-0.0626299\pi\)
\(158\) − 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 \(0.00251867\pi\)
−0.999969 + 0.00791254i \(0.997481\pi\)
\(164\) −8.89898 −0.694894
\(165\) 0 0
\(166\) −6.34847 −0.492737
\(167\) − 18.0000i − 1.39288i −0.717614 0.696441i \(-0.754766\pi\)
0.717614 0.696441i \(-0.245234\pi\)
\(168\) − 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 \(-0.786534\pi\)
0.783434 0.621476i \(-0.213466\pi\)
\(174\) −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 \(-0.992535\pi\)
0.999725 0.0234491i \(-0.00746477\pi\)
\(194\) −6.44949 −0.463046
\(195\) 0 0
\(196\) −6.79796 −0.485568
\(197\) − 12.6515i − 0.901384i −0.892679 0.450692i \(-0.851177\pi\)
0.892679 0.450692i \(-0.148823\pi\)
\(198\) − 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 \(0.334058\pi\)
−0.498026 + 0.867162i \(0.665942\pi\)
\(224\) 0.449490 0.0300328
\(225\) 0 0
\(226\) 2.79796 0.186117
\(227\) 9.59592i 0.636903i 0.947939 + 0.318452i \(0.103163\pi\)
−0.947939 + 0.318452i \(0.896837\pi\)
\(228\) 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 \(-0.796075\pi\)
0.801709 0.597715i \(-0.203925\pi\)
\(234\) −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 \(-0.671990\pi\)
0.514413 0.857542i \(-0.328010\pi\)
\(258\) 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 \(0.129109\pi\)
−0.918862 + 0.394578i \(0.870891\pi\)
\(264\) 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 \(0.243417\pi\)
−0.721579 + 0.692332i \(0.756583\pi\)
\(278\) 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 \(0.0980631\pi\)
−0.952919 + 0.303224i \(0.901937\pi\)
\(284\) 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 \(0.0116013\pi\)
−0.999336 + 0.0364384i \(0.988399\pi\)
\(294\) 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 \(0.0150072\pi\)
−0.998889 + 0.0471289i \(0.984993\pi\)
\(308\) 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 \(0.214743\pi\)
−0.780935 + 0.624612i \(0.785257\pi\)
\(314\) −4.89898 −0.276465
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) 9.24745i 0.519388i 0.965691 + 0.259694i \(0.0836218\pi\)
−0.965691 + 0.259694i \(0.916378\pi\)
\(318\) − 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 \(0.134926\pi\)
−0.911499 + 0.411303i \(0.865074\pi\)
\(338\) − 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 \(-0.862515\pi\)
0.908163 0.418616i \(-0.137485\pi\)
\(348\) 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 \(0.0531688\pi\)
−0.986082 + 0.166259i \(0.946831\pi\)
\(354\) −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 \(0.0879264\pi\)
−0.962091 + 0.272729i \(0.912074\pi\)
\(368\) − 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 \(-0.849866\pi\)
0.890815 0.454365i \(-0.150134\pi\)
\(374\) −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 \(-0.932422\pi\)
0.977548 0.210712i \(-0.0675784\pi\)
\(384\) −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 \(0.270637\pi\)
−0.659810 + 0.751433i \(0.729363\pi\)
\(398\) − 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 \(0.153915\pi\)
−0.885356 + 0.464914i \(0.846085\pi\)
\(434\) −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 \(-0.975419\pi\)
0.997020 0.0771455i \(-0.0245806\pi\)
\(444\) −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 \(0.114906\pi\)
−0.935549 + 0.353198i \(0.885094\pi\)
\(458\) − 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 \(0.110939\pi\)
−0.939877 + 0.341512i \(0.889061\pi\)
\(464\) −10.3485 −0.480416
\(465\) 0 0
\(466\) −18.2474 −0.845297
\(467\) − 8.34847i − 0.386321i −0.981167 0.193161i \(-0.938126\pi\)
0.981167 0.193161i \(-0.0618738\pi\)
\(468\) 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 \(0.258174\pi\)
−0.688718 + 0.725029i \(0.741826\pi\)
\(488\) − 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 \(0.0730366\pi\)
−0.973791 + 0.227443i \(0.926963\pi\)
\(504\) 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 \(0.187358\pi\)
−0.831718 + 0.555198i \(0.812642\pi\)
\(524\) 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 \(-0.713517\pi\)
0.621600 0.783335i \(-0.286483\pi\)
\(548\) 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 \(-0.913627\pi\)
0.963410 0.268031i \(-0.0863731\pi\)
\(558\) − 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 \(-0.874367\pi\)
0.923117 0.384519i \(-0.125633\pi\)
\(564\) −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 \(-0.933939\pi\)
0.978541 0.206050i \(-0.0660609\pi\)
\(578\) − 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 \(0.200558\pi\)
−0.807985 + 0.589203i \(0.799442\pi\)
\(588\) − 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 \(-0.687504\pi\)
0.555580 0.831463i \(-0.312496\pi\)
\(594\) 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 \(0.226621\pi\)
−0.757088 + 0.653313i \(0.773379\pi\)
\(608\) 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 \(-0.687275\pi\)
0.554984 0.831861i \(-0.312725\pi\)
\(614\) 1.65153 0.0666504
\(615\) 0 0
\(616\) 0.651531 0.0262509
\(617\) 40.2929i 1.62213i 0.584957 + 0.811065i \(0.301112\pi\)
−0.584957 + 0.811065i \(0.698888\pi\)
\(618\) − 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 \(-0.872344\pi\)
0.920655 0.390378i \(-0.127656\pi\)
\(644\) −0.449490 −0.0177124
\(645\) 0 0
\(646\) −0.449490 −0.0176849
\(647\) − 16.1010i − 0.632996i −0.948593 0.316498i \(-0.897493\pi\)
0.948593 0.316498i \(-0.102507\pi\)
\(648\) 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 \(0.0928467\pi\)
−0.957760 + 0.287568i \(0.907153\pi\)
\(654\) −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 \(-0.798826\pi\)
0.806844 0.590765i \(-0.201174\pi\)
\(674\) 15.1010 0.581670
\(675\) 0 0
\(676\) −7.00000 −0.269231
\(677\) − 7.65153i − 0.294072i −0.989131 0.147036i \(-0.953027\pi\)
0.989131 0.147036i \(-0.0469734\pi\)
\(678\) 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 \(0.365053\pi\)
−0.411361 + 0.911472i \(0.634947\pi\)
\(684\) −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 \(0.163486\pi\)
−0.870979 + 0.491321i \(0.836514\pi\)
\(728\) − 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 \(-0.812051\pi\)
0.830686 0.556742i \(-0.187949\pi\)
\(734\) 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 \(0.0315455\pi\)
−0.995093 + 0.0989411i \(0.968454\pi\)
\(744\) 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 \(-0.918991\pi\)
0.967790 0.251758i \(-0.0810088\pi\)
\(758\) − 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 \(-0.815232\pi\)
0.836208 0.548413i \(-0.184768\pi\)
\(774\) −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 \(0.169932\pi\)
−0.860851 + 0.508858i \(0.830068\pi\)
\(788\) 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 \(-0.620038\pi\)
0.368236 0.929732i \(-0.379962\pi\)
\(798\) − 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 \(-0.931785\pi\)
0.977124 0.212668i \(-0.0682153\pi\)
\(824\) 7.89898 0.275174
\(825\) 0 0
\(826\) −0.696938 −0.0242496
\(827\) 24.2474i 0.843166i 0.906790 + 0.421583i \(0.138525\pi\)
−0.906790 + 0.421583i \(0.861475\pi\)
\(828\) − 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 \(0.0597426\pi\)
−0.982438 + 0.186587i \(0.940257\pi\)
\(854\) −2.04541 −0.0699924
\(855\) 0 0
\(856\) 2.65153 0.0906275
\(857\) − 12.4949i − 0.426818i −0.976963 0.213409i \(-0.931543\pi\)
0.976963 0.213409i \(-0.0684566\pi\)
\(858\) − 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 \(-0.707580\pi\)
0.606883 0.794791i \(-0.292420\pi\)
\(864\) −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 \(0.103376\pi\)
−0.947726 + 0.319087i \(0.896624\pi\)
\(878\) − 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 \(0.311862\pi\)
−0.557236 + 0.830354i \(0.688138\pi\)
\(884\) 1.10102 0.0370313
\(885\) 0 0
\(886\) −3.24745 −0.109100
\(887\) − 4.89898i − 0.164492i −0.996612 0.0822458i \(-0.973791\pi\)
0.996612 0.0822458i \(-0.0262093\pi\)
\(888\) 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 \(0.119016\pi\)
−0.930910 + 0.365249i \(0.880984\pi\)
\(908\) − 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 \(-0.777494\pi\)
0.765471 0.643471i \(-0.222506\pi\)
\(938\) 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 \(-0.958508\pi\)
0.991516 0.129983i \(-0.0414921\pi\)
\(948\) − 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 \(0.182523\pi\)
−0.840054 + 0.542503i \(0.817477\pi\)
\(954\) 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 \(0.132240\pi\)
−0.914937 + 0.403597i \(0.867760\pi\)
\(968\) − 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 \(0.101490\pi\)
−0.949600 + 0.313464i \(0.898510\pi\)
\(978\) 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 \(0.258816\pi\)
−0.687253 + 0.726418i \(0.741184\pi\)
\(984\) −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 \(-0.815727\pi\)
0.837059 0.547113i \(-0.184273\pi\)
\(998\) 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.799.2 4
5.2 odd 4 2850.2.a.bj.1.1 yes 2
5.3 odd 4 2850.2.a.bc.1.2 2
5.4 even 2 inner 2850.2.d.w.799.3 4
15.2 even 4 8550.2.a.bu.1.1 2
15.8 even 4 8550.2.a.bv.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.bc.1.2 2 5.3 odd 4
2850.2.a.bj.1.1 yes 2 5.2 odd 4
2850.2.d.w.799.2 4 1.1 even 1 trivial
2850.2.d.w.799.3 4 5.4 even 2 inner
8550.2.a.bu.1.1 2 15.2 even 4
8550.2.a.bv.1.2 2 15.8 even 4