Properties

Label 2550.2.d.s.2449.2
Level $2550$
Weight $2$
Character 2550.2449
Analytic conductor $20.362$
Analytic rank $0$
Dimension $2$
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] = [2550,2,Mod(2449,2550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2550, 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("2550.2449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2550.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: \(20.3618525154\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 510)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2449.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2550.2449
Dual form 2550.2.d.s.2449.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} +4.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} -2.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} -4.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} +4.00000i q^{38} -2.00000 q^{39} -10.0000 q^{41} +8.00000i q^{43} -4.00000 q^{44} +4.00000 q^{46} -1.00000i q^{48} +7.00000 q^{49} +1.00000 q^{51} +2.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} -4.00000i q^{57} -2.00000i q^{58} +8.00000 q^{59} +10.0000 q^{61} -4.00000i q^{62} -1.00000 q^{64} +4.00000 q^{66} -8.00000i q^{67} -1.00000i q^{68} -4.00000 q^{69} +8.00000 q^{71} +1.00000i q^{72} +2.00000i q^{73} +6.00000 q^{74} -4.00000 q^{76} -2.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} -4.00000i q^{83} -8.00000 q^{86} +2.00000i q^{87} -4.00000i q^{88} +14.0000 q^{89} +4.00000i q^{92} +4.00000i q^{93} +1.00000 q^{96} -10.0000i q^{97} +7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{11} + 2 q^{16} + 8 q^{19} - 2 q^{24} + 4 q^{26} - 4 q^{29} - 8 q^{31} - 2 q^{34} + 2 q^{36} - 4 q^{39} - 20 q^{41} - 8 q^{44} + 8 q^{46} + 14 q^{49} + 2 q^{51} - 2 q^{54} + 16 q^{59} + 20 q^{61} - 2 q^{64} + 8 q^{66} - 8 q^{69} + 16 q^{71} + 12 q^{74} - 8 q^{76} - 8 q^{79} + 2 q^{81} - 16 q^{86} + 28 q^{89} + 2 q^{96} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times\).

\(n\) \(751\) \(851\) \(1327\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000i 0.242536i
\(18\) − 1.00000i − 0.235702i
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.00000i 0.852803i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 4.00000i − 0.696311i
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 6.00000i − 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) 4.00000i 0.648886i
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 2.00000i 0.277350i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.00000i − 0.529813i
\(58\) − 2.00000i − 0.262613i
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) − 1.00000i − 0.121268i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) − 2.00000i − 0.226455i
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 10.0000i − 1.10432i
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 2.00000i 0.214423i
\(88\) − 4.00000i − 0.426401i
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000i 0.417029i
\(93\) 4.00000i 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 7.00000i 0.707107i
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 1.00000i 0.0990148i
\(103\) − 16.0000i − 1.57653i −0.615338 0.788263i \(-0.710980\pi\)
0.615338 0.788263i \(-0.289020\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) − 18.0000i − 1.69330i −0.532152 0.846649i \(-0.678617\pi\)
0.532152 0.846649i \(-0.321383\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 2.00000i 0.184900i
\(118\) 8.00000i 0.736460i
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000i 0.905357i
\(123\) 10.0000i 0.901670i
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) − 16.0000i − 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 4.00000i 0.348155i
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 1.00000 0.0857493
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) − 4.00000i − 0.340503i
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000i 0.671345i
\(143\) − 8.00000i − 0.668994i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) − 7.00000i − 0.577350i
\(148\) 6.00000i 0.493197i
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) − 4.00000i − 0.324443i
\(153\) − 1.00000i − 0.0808452i
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) − 6.00000i − 0.478852i −0.970915 0.239426i \(-0.923041\pi\)
0.970915 0.239426i \(-0.0769593\pi\)
\(158\) − 4.00000i − 0.318223i
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) − 12.0000i − 0.939913i −0.882690 0.469956i \(-0.844270\pi\)
0.882690 0.469956i \(-0.155730\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) − 8.00000i − 0.609994i
\(173\) − 2.00000i − 0.152057i −0.997106 0.0760286i \(-0.975776\pi\)
0.997106 0.0760286i \(-0.0242240\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) − 8.00000i − 0.601317i
\(178\) 14.0000i 1.04934i
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) − 10.0000i − 0.739221i
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 4.00000i 0.292509i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 6.00000i − 0.431889i −0.976406 0.215945i \(-0.930717\pi\)
0.976406 0.215945i \(-0.0692831\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) − 4.00000i − 0.284268i
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) − 6.00000i − 0.422159i
\(203\) 0 0
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) 4.00000i 0.278019i
\(208\) − 2.00000i − 0.138675i
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 6.00000i 0.412082i
\(213\) − 8.00000i − 0.548151i
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 14.0000i 0.948200i
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 2.00000 0.134535
\(222\) − 6.00000i − 0.402694i
\(223\) − 24.0000i − 1.60716i −0.595198 0.803579i \(-0.702926\pi\)
0.595198 0.803579i \(-0.297074\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 4.00000i 0.264906i
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.00000i 0.131306i
\(233\) 30.0000i 1.96537i 0.185296 + 0.982683i \(0.440675\pi\)
−0.185296 + 0.982683i \(0.559325\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 4.00000i 0.259828i
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 5.00000i 0.321412i
\(243\) − 1.00000i − 0.0641500i
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) − 8.00000i − 0.509028i
\(248\) 4.00000i 0.254000i
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) − 16.0000i − 1.00591i
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 8.00000i 0.498058i
\(259\) 0 0
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 4.00000i 0.247121i
\(263\) − 16.0000i − 0.986602i −0.869859 0.493301i \(-0.835790\pi\)
0.869859 0.493301i \(-0.164210\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) − 14.0000i − 0.856786i
\(268\) 8.00000i 0.488678i
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 1.00000i 0.0606339i
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) 10.0000i 0.600842i 0.953807 + 0.300421i \(0.0971271\pi\)
−0.953807 + 0.300421i \(0.902873\pi\)
\(278\) 4.00000i 0.239904i
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) − 20.0000i − 1.18888i −0.804141 0.594438i \(-0.797374\pi\)
0.804141 0.594438i \(-0.202626\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) 0 0
\(288\) − 1.00000i − 0.0589256i
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) − 2.00000i − 0.117041i
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 4.00000i 0.232104i
\(298\) − 2.00000i − 0.115857i
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 6.00000i 0.344691i
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 1.00000 0.0571662
\(307\) 16.0000i 0.913168i 0.889680 + 0.456584i \(0.150927\pi\)
−0.889680 + 0.456584i \(0.849073\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 2.00000i 0.113228i
\(313\) 2.00000i 0.113047i 0.998401 + 0.0565233i \(0.0180015\pi\)
−0.998401 + 0.0565233i \(0.981998\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 26.0000i 1.46031i 0.683284 + 0.730153i \(0.260551\pi\)
−0.683284 + 0.730153i \(0.739449\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 4.00000i 0.222566i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) − 14.0000i − 0.774202i
\(328\) 10.0000i 0.552158i
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 4.00000i 0.219529i
\(333\) 6.00000i 0.328798i
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) 6.00000i 0.326841i 0.986557 + 0.163420i \(0.0522527\pi\)
−0.986557 + 0.163420i \(0.947747\pi\)
\(338\) 9.00000i 0.489535i
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) − 4.00000i − 0.216295i
\(343\) 0 0
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) 4.00000i 0.214731i 0.994220 + 0.107366i \(0.0342415\pi\)
−0.994220 + 0.107366i \(0.965758\pi\)
\(348\) − 2.00000i − 0.107211i
\(349\) 34.0000 1.81998 0.909989 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 4.00000i 0.213201i
\(353\) 30.0000i 1.59674i 0.602168 + 0.798369i \(0.294304\pi\)
−0.602168 + 0.798369i \(0.705696\pi\)
\(354\) 8.00000 0.425195
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) 0 0
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 14.0000i − 0.735824i
\(363\) − 5.00000i − 0.262432i
\(364\) 0 0
\(365\) 0 0
\(366\) 10.0000 0.522708
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) − 4.00000i − 0.207390i
\(373\) − 18.0000i − 0.932005i −0.884783 0.466002i \(-0.845694\pi\)
0.884783 0.466002i \(-0.154306\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) 0 0
\(377\) 4.00000i 0.206010i
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) 24.0000i 1.22634i 0.789950 + 0.613171i \(0.210106\pi\)
−0.789950 + 0.613171i \(0.789894\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) − 8.00000i − 0.406663i
\(388\) 10.0000i 0.507673i
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) − 7.00000i − 0.353553i
\(393\) − 4.00000i − 0.201773i
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) 18.0000i 0.903394i 0.892171 + 0.451697i \(0.149181\pi\)
−0.892171 + 0.451697i \(0.850819\pi\)
\(398\) 4.00000i 0.200502i
\(399\) 0 0
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) − 8.00000i − 0.399004i
\(403\) 8.00000i 0.398508i
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) − 24.0000i − 1.18964i
\(408\) − 1.00000i − 0.0495074i
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) 16.0000i 0.788263i
\(413\) 0 0
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) − 4.00000i − 0.195881i
\(418\) 16.0000i 0.782586i
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 12.0000i 0.584151i
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 0 0
\(428\) − 12.0000i − 0.580042i
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 6.00000i 0.288342i 0.989553 + 0.144171i \(0.0460515\pi\)
−0.989553 + 0.144171i \(0.953949\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) − 16.0000i − 0.765384i
\(438\) 2.00000i 0.0955637i
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 2.00000i 0.0951303i
\(443\) 28.0000i 1.33032i 0.746701 + 0.665160i \(0.231637\pi\)
−0.746701 + 0.665160i \(0.768363\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) 24.0000 1.13643
\(447\) 2.00000i 0.0945968i
\(448\) 0 0
\(449\) 26.0000 1.22702 0.613508 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) 18.0000i 0.846649i
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 2.00000i 0.0935561i 0.998905 + 0.0467780i \(0.0148953\pi\)
−0.998905 + 0.0467780i \(0.985105\pi\)
\(458\) − 6.00000i − 0.280362i
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) −38.0000 −1.76984 −0.884918 0.465746i \(-0.845786\pi\)
−0.884918 + 0.465746i \(0.845786\pi\)
\(462\) 0 0
\(463\) 24.0000i 1.11537i 0.830051 + 0.557687i \(0.188311\pi\)
−0.830051 + 0.557687i \(0.811689\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −30.0000 −1.38972
\(467\) − 12.0000i − 0.555294i −0.960683 0.277647i \(-0.910445\pi\)
0.960683 0.277647i \(-0.0895545\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) 0 0
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) − 8.00000i − 0.368230i
\(473\) 32.0000i 1.47136i
\(474\) −4.00000 −0.183726
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000i 0.274721i
\(478\) − 8.00000i − 0.365911i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 10.0000i 0.455488i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 40.0000i − 1.81257i −0.422664 0.906287i \(-0.638905\pi\)
0.422664 0.906287i \(-0.361095\pi\)
\(488\) − 10.0000i − 0.452679i
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) − 10.0000i − 0.450835i
\(493\) − 2.00000i − 0.0900755i
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) − 4.00000i − 0.179244i
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 28.0000i 1.24846i 0.781241 + 0.624229i \(0.214587\pi\)
−0.781241 + 0.624229i \(0.785413\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 16.0000 0.711287
\(507\) − 9.00000i − 0.399704i
\(508\) 16.0000i 0.709885i
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) 0 0
\(518\) 0 0
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 2.00000i 0.0875376i
\(523\) 40.0000i 1.74908i 0.484955 + 0.874539i \(0.338836\pi\)
−0.484955 + 0.874539i \(0.661164\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) − 4.00000i − 0.174243i
\(528\) − 4.00000i − 0.174078i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 20.0000i 0.866296i
\(534\) 14.0000 0.605839
\(535\) 0 0
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) 14.0000i 0.603583i
\(539\) 28.0000 1.20605
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) − 8.00000i − 0.343629i
\(543\) 14.0000i 0.600798i
\(544\) −1.00000 −0.0428746
\(545\) 0 0
\(546\) 0 0
\(547\) − 20.0000i − 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) 6.00000i 0.256307i
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 4.00000i 0.170251i
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) 4.00000i 0.169334i
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) − 30.0000i − 1.26547i
\(563\) − 20.0000i − 0.842900i −0.906852 0.421450i \(-0.861521\pi\)
0.906852 0.421450i \(-0.138479\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) 0 0
\(568\) − 8.00000i − 0.335673i
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) −44.0000 −1.84134 −0.920671 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) 8.00000i 0.334497i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 6.00000i − 0.249783i −0.992170 0.124892i \(-0.960142\pi\)
0.992170 0.124892i \(-0.0398583\pi\)
\(578\) − 1.00000i − 0.0415945i
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) 0 0
\(582\) − 10.0000i − 0.414513i
\(583\) − 24.0000i − 0.993978i
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 7.00000i 0.288675i
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) − 6.00000i − 0.246598i
\(593\) 30.0000i 1.23195i 0.787765 + 0.615976i \(0.211238\pi\)
−0.787765 + 0.615976i \(0.788762\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 2.00000 0.0819232
\(597\) − 4.00000i − 0.163709i
\(598\) − 8.00000i − 0.327144i
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 34.0000 1.38689 0.693444 0.720510i \(-0.256092\pi\)
0.693444 + 0.720510i \(0.256092\pi\)
\(602\) 0 0
\(603\) 8.00000i 0.325785i
\(604\) 0 0
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) − 32.0000i − 1.29884i −0.760430 0.649420i \(-0.775012\pi\)
0.760430 0.649420i \(-0.224988\pi\)
\(608\) 4.00000i 0.162221i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 1.00000i 0.0404226i
\(613\) − 10.0000i − 0.403896i −0.979396 0.201948i \(-0.935273\pi\)
0.979396 0.201948i \(-0.0647272\pi\)
\(614\) −16.0000 −0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) 2.00000i 0.0805170i 0.999189 + 0.0402585i \(0.0128181\pi\)
−0.999189 + 0.0402585i \(0.987182\pi\)
\(618\) − 16.0000i − 0.643614i
\(619\) −44.0000 −1.76851 −0.884255 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) − 8.00000i − 0.320771i
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) −2.00000 −0.0799361
\(627\) − 16.0000i − 0.638978i
\(628\) 6.00000i 0.239426i
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 4.00000i 0.159111i
\(633\) − 12.0000i − 0.476957i
\(634\) −26.0000 −1.03259
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) − 14.0000i − 0.554700i
\(638\) − 8.00000i − 0.316723i
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 20.0000i 0.788723i 0.918955 + 0.394362i \(0.129034\pi\)
−0.918955 + 0.394362i \(0.870966\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) − 32.0000i − 1.25805i −0.777385 0.629025i \(-0.783454\pi\)
0.777385 0.629025i \(-0.216546\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 32.0000 1.25611
\(650\) 0 0
\(651\) 0 0
\(652\) 12.0000i 0.469956i
\(653\) − 34.0000i − 1.33052i −0.746611 0.665261i \(-0.768320\pi\)
0.746611 0.665261i \(-0.231680\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) − 2.00000i − 0.0780274i
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 20.0000i 0.777322i
\(663\) − 2.00000i − 0.0776736i
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 8.00000i 0.309761i
\(668\) − 12.0000i − 0.464294i
\(669\) −24.0000 −0.927894
\(670\) 0 0
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) 10.0000i 0.385472i 0.981251 + 0.192736i \(0.0617360\pi\)
−0.981251 + 0.192736i \(0.938264\pi\)
\(674\) −6.00000 −0.231111
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 18.0000i 0.691796i 0.938272 + 0.345898i \(0.112426\pi\)
−0.938272 + 0.345898i \(0.887574\pi\)
\(678\) − 18.0000i − 0.691286i
\(679\) 0 0
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) − 16.0000i − 0.612672i
\(683\) − 44.0000i − 1.68361i −0.539779 0.841807i \(-0.681492\pi\)
0.539779 0.841807i \(-0.318508\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) 6.00000i 0.228914i
\(688\) 8.00000i 0.304997i
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −36.0000 −1.36950 −0.684752 0.728776i \(-0.740090\pi\)
−0.684752 + 0.728776i \(0.740090\pi\)
\(692\) 2.00000i 0.0760286i
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) 2.00000 0.0758098
\(697\) − 10.0000i − 0.378777i
\(698\) 34.0000i 1.28692i
\(699\) 30.0000 1.13470
\(700\) 0 0
\(701\) 10.0000 0.377695 0.188847 0.982006i \(-0.439525\pi\)
0.188847 + 0.982006i \(0.439525\pi\)
\(702\) 2.00000i 0.0754851i
\(703\) − 24.0000i − 0.905177i
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) 0 0
\(708\) 8.00000i 0.300658i
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) − 14.0000i − 0.524672i
\(713\) 16.0000i 0.599205i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 8.00000i 0.298765i
\(718\) − 32.0000i − 1.19423i
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 3.00000i − 0.111648i
\(723\) − 10.0000i − 0.371904i
\(724\) 14.0000 0.520306
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) 40.0000i 1.48352i 0.670667 + 0.741759i \(0.266008\pi\)
−0.670667 + 0.741759i \(0.733992\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 10.0000i 0.369611i
\(733\) − 26.0000i − 0.960332i −0.877178 0.480166i \(-0.840576\pi\)
0.877178 0.480166i \(-0.159424\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) − 32.0000i − 1.17874i
\(738\) 10.0000i 0.368105i
\(739\) 44.0000 1.61857 0.809283 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) − 28.0000i − 1.02722i −0.858024 0.513610i \(-0.828308\pi\)
0.858024 0.513610i \(-0.171692\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) 18.0000 0.659027
\(747\) 4.00000i 0.146352i
\(748\) − 4.00000i − 0.146254i
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) 0 0
\(757\) − 14.0000i − 0.508839i −0.967094 0.254419i \(-0.918116\pi\)
0.967094 0.254419i \(-0.0818843\pi\)
\(758\) − 20.0000i − 0.726433i
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) − 16.0000i − 0.579619i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) − 16.0000i − 0.577727i
\(768\) − 1.00000i − 0.0360844i
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 6.00000i 0.215945i
\(773\) − 14.0000i − 0.503545i −0.967786 0.251773i \(-0.918987\pi\)
0.967786 0.251773i \(-0.0810135\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) 30.0000i 1.07555i
\(779\) −40.0000 −1.43315
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 4.00000i 0.143040i
\(783\) − 2.00000i − 0.0714742i
\(784\) 7.00000 0.250000
\(785\) 0 0
\(786\) 4.00000 0.142675
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) − 18.0000i − 0.641223i
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) 0 0
\(792\) 4.00000i 0.142134i
\(793\) − 20.0000i − 0.710221i
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) 6.00000i 0.212531i 0.994338 + 0.106265i \(0.0338893\pi\)
−0.994338 + 0.106265i \(0.966111\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) 6.00000i 0.211867i
\(803\) 8.00000i 0.282314i
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) − 14.0000i − 0.492823i
\(808\) 6.00000i 0.211079i
\(809\) −54.0000 −1.89854 −0.949269 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 8.00000i 0.280572i
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) 1.00000 0.0350070
\(817\) 32.0000i 1.11954i
\(818\) − 10.0000i − 0.349642i
\(819\) 0 0
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) − 6.00000i − 0.209274i
\(823\) − 16.0000i − 0.557725i −0.960331 0.278862i \(-0.910043\pi\)
0.960331 0.278862i \(-0.0899574\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0000i 1.25184i 0.779886 + 0.625921i \(0.215277\pi\)
−0.779886 + 0.625921i \(0.784723\pi\)
\(828\) − 4.00000i − 0.139010i
\(829\) −54.0000 −1.87550 −0.937749 0.347314i \(-0.887094\pi\)
−0.937749 + 0.347314i \(0.887094\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 2.00000i 0.0693375i
\(833\) 7.00000i 0.242536i
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) − 4.00000i − 0.138260i
\(838\) − 28.0000i − 0.967244i
\(839\) −16.0000 −0.552381 −0.276191 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 30.0000i 1.03387i
\(843\) 30.0000i 1.03325i
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) − 6.00000i − 0.206041i
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) 8.00000i 0.274075i
\(853\) 54.0000i 1.84892i 0.381273 + 0.924462i \(0.375486\pi\)
−0.381273 + 0.924462i \(0.624514\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 26.0000i 0.888143i 0.895991 + 0.444072i \(0.146466\pi\)
−0.895991 + 0.444072i \(0.853534\pi\)
\(858\) − 8.00000i − 0.273115i
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 16.0000i − 0.544962i
\(863\) 8.00000i 0.272323i 0.990687 + 0.136162i \(0.0434766\pi\)
−0.990687 + 0.136162i \(0.956523\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −6.00000 −0.203888
\(867\) 1.00000i 0.0339618i
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) − 14.0000i − 0.474100i
\(873\) 10.0000i 0.338449i
\(874\) 16.0000 0.541208
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) 50.0000i 1.68838i 0.536044 + 0.844190i \(0.319918\pi\)
−0.536044 + 0.844190i \(0.680082\pi\)
\(878\) − 20.0000i − 0.674967i
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) − 7.00000i − 0.235702i
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) −28.0000 −0.940678
\(887\) 12.0000i 0.402921i 0.979497 + 0.201460i \(0.0645687\pi\)
−0.979497 + 0.201460i \(0.935431\pi\)
\(888\) 6.00000i 0.201347i
\(889\) 0 0
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 24.0000i 0.803579i
\(893\) 0 0
\(894\) −2.00000 −0.0668900
\(895\) 0 0
\(896\) 0 0
\(897\) 8.00000i 0.267112i
\(898\) 26.0000i 0.867631i
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 6.00000 0.199889
\(902\) − 40.0000i − 1.33185i
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) 0 0
\(907\) − 36.0000i − 1.19536i −0.801735 0.597680i \(-0.796089\pi\)
0.801735 0.597680i \(-0.203911\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) − 16.0000i − 0.529523i
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 0 0
\(918\) − 1.00000i − 0.0330049i
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 16.0000 0.527218
\(922\) − 38.0000i − 1.25146i
\(923\) − 16.0000i − 0.526646i
\(924\) 0 0
\(925\) 0 0
\(926\) −24.0000 −0.788689
\(927\) 16.0000i 0.525509i
\(928\) − 2.00000i − 0.0656532i
\(929\) 50.0000 1.64045 0.820223 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) 0 0
\(931\) 28.0000 0.917663
\(932\) − 30.0000i − 0.982683i
\(933\) 8.00000i 0.261908i
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 2.00000i 0.0653372i 0.999466 + 0.0326686i \(0.0104006\pi\)
−0.999466 + 0.0326686i \(0.989599\pi\)
\(938\) 0 0
\(939\) 2.00000 0.0652675
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) − 6.00000i − 0.195491i
\(943\) 40.0000i 1.30258i
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) −32.0000 −1.04041
\(947\) − 28.0000i − 0.909878i −0.890523 0.454939i \(-0.849661\pi\)
0.890523 0.454939i \(-0.150339\pi\)
\(948\) − 4.00000i − 0.129914i
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 26.0000 0.843108
\(952\) 0 0
\(953\) 6.00000i 0.194359i 0.995267 + 0.0971795i \(0.0309821\pi\)
−0.995267 + 0.0971795i \(0.969018\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) 8.00000i 0.258603i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) − 12.0000i − 0.386896i
\(963\) − 12.0000i − 0.386695i
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) 24.0000i 0.771788i 0.922543 + 0.385894i \(0.126107\pi\)
−0.922543 + 0.385894i \(0.873893\pi\)
\(968\) − 5.00000i − 0.160706i
\(969\) 4.00000 0.128499
\(970\) 0 0
\(971\) 32.0000 1.02693 0.513464 0.858111i \(-0.328362\pi\)
0.513464 + 0.858111i \(0.328362\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 0 0
\(974\) 40.0000 1.28168
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) − 12.0000i − 0.383718i
\(979\) 56.0000 1.78977
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) 24.0000i 0.765871i
\(983\) 44.0000i 1.40338i 0.712481 + 0.701691i \(0.247571\pi\)
−0.712481 + 0.701691i \(0.752429\pi\)
\(984\) 10.0000 0.318788
\(985\) 0 0
\(986\) 2.00000 0.0636930
\(987\) 0 0
\(988\) 8.00000i 0.254514i
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) −44.0000 −1.39771 −0.698853 0.715265i \(-0.746306\pi\)
−0.698853 + 0.715265i \(0.746306\pi\)
\(992\) − 4.00000i − 0.127000i
\(993\) − 20.0000i − 0.634681i
\(994\) 0 0
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) 10.0000i 0.316703i 0.987383 + 0.158352i \(0.0506179\pi\)
−0.987383 + 0.158352i \(0.949382\pi\)
\(998\) − 28.0000i − 0.886325i
\(999\) 6.00000 0.189832
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 2550.2.d.s.2449.2 2
5.2 odd 4 2550.2.a.d.1.1 1
5.3 odd 4 510.2.a.f.1.1 1
5.4 even 2 inner 2550.2.d.s.2449.1 2
15.2 even 4 7650.2.a.bw.1.1 1
15.8 even 4 1530.2.a.f.1.1 1
20.3 even 4 4080.2.a.c.1.1 1
85.33 odd 4 8670.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.f.1.1 1 5.3 odd 4
1530.2.a.f.1.1 1 15.8 even 4
2550.2.a.d.1.1 1 5.2 odd 4
2550.2.d.s.2449.1 2 5.4 even 2 inner
2550.2.d.s.2449.2 2 1.1 even 1 trivial
4080.2.a.c.1.1 1 20.3 even 4
7650.2.a.bw.1.1 1 15.2 even 4
8670.2.a.r.1.1 1 85.33 odd 4