Properties

Label 2550.2.d.n.2449.2
Level $2550$
Weight $2$
Character 2550.2449
Analytic conductor $20.362$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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.n.2449.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} -2.00000 q^{21} -4.00000i q^{22} +8.00000i q^{23} -1.00000 q^{24} -4.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} -2.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} +4.00000i q^{33} +1.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} +4.00000i q^{38} +4.00000 q^{39} +8.00000 q^{41} -2.00000i q^{42} +6.00000i q^{43} +4.00000 q^{44} -8.00000 q^{46} -8.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -1.00000 q^{51} -4.00000i q^{52} -2.00000i q^{53} -1.00000 q^{54} -2.00000 q^{56} -4.00000i q^{57} -2.00000i q^{58} +6.00000 q^{59} +14.0000 q^{61} +4.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} +2.00000i q^{67} +1.00000i q^{68} +8.00000 q^{69} +2.00000 q^{71} +1.00000i q^{72} +4.00000i q^{73} +6.00000 q^{74} -4.00000 q^{76} +8.00000i q^{77} +4.00000i q^{78} +1.00000 q^{81} +8.00000i q^{82} -16.0000i q^{83} +2.00000 q^{84} -6.00000 q^{86} +2.00000i q^{87} +4.00000i q^{88} +2.00000 q^{89} +8.00000 q^{91} -8.00000i q^{92} -4.00000i q^{93} +8.00000 q^{94} +1.00000 q^{96} +3.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 4 q^{14} + 2 q^{16} + 8 q^{19} - 4 q^{21} - 2 q^{24} - 8 q^{26} - 4 q^{29} + 8 q^{31} + 2 q^{34} + 2 q^{36} + 8 q^{39} + 16 q^{41} + 8 q^{44} - 16 q^{46} + 6 q^{49} - 2 q^{51} - 2 q^{54} - 4 q^{56} + 12 q^{59} + 28 q^{61} - 2 q^{64} - 8 q^{66} + 16 q^{69} + 4 q^{71} + 12 q^{74} - 8 q^{76} + 2 q^{81} + 4 q^{84} - 12 q^{86} + 4 q^{89} + 16 q^{91} + 16 q^{94} + 2 q^{96} + 8 q^{99}+O(q^{100}) \) 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
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 2.00000 0.534522
\(15\) 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
\(21\) −2.00000 −0.436436
\(22\) − 4.00000i − 0.852803i
\(23\) 8.00000i 1.66812i 0.551677 + 0.834058i \(0.313988\pi\)
−0.551677 + 0.834058i \(0.686012\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 1.00000i 0.192450i
\(28\) 2.00000i 0.377964i
\(29\) −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 \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 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
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) − 8.00000i − 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) − 4.00000i − 0.554700i
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) − 4.00000i − 0.529813i
\(58\) − 2.00000i − 0.262613i
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 4.00000i 0.508001i
\(63\) 2.00000i 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) 2.00000i 0.244339i 0.992509 + 0.122169i \(0.0389851\pi\)
−0.992509 + 0.122169i \(0.961015\pi\)
\(68\) 1.00000i 0.121268i
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 8.00000i 0.911685i
\(78\) 4.00000i 0.452911i
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 8.00000i 0.883452i
\(83\) − 16.0000i − 1.75623i −0.478451 0.878114i \(-0.658802\pi\)
0.478451 0.878114i \(-0.341198\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −6.00000 −0.646997
\(87\) 2.00000i 0.214423i
\(88\) 4.00000i 0.426401i
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) − 8.00000i − 0.834058i
\(93\) − 4.00000i − 0.414781i
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) − 1.00000i − 0.0990148i
\(103\) − 12.0000i − 1.18240i −0.806527 0.591198i \(-0.798655\pi\)
0.806527 0.591198i \(-0.201345\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) − 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
\(112\) − 2.00000i − 0.188982i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) − 4.00000i − 0.369800i
\(118\) 6.00000i 0.552345i
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 14.0000i 1.26750i
\(123\) − 8.00000i − 0.721336i
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 4.00000i 0.354943i 0.984126 + 0.177471i \(0.0567917\pi\)
−0.984126 + 0.177471i \(0.943208\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) −16.0000 −1.39793 −0.698963 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) − 4.00000i − 0.348155i
\(133\) − 8.00000i − 0.693688i
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 8.00000i 0.681005i
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 2.00000i 0.167836i
\(143\) − 16.0000i − 1.33799i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) − 3.00000i − 0.247436i
\(148\) 6.00000i 0.493197i
\(149\) −16.0000 −1.31077 −0.655386 0.755295i \(-0.727494\pi\)
−0.655386 + 0.755295i \(0.727494\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) − 4.00000i − 0.324443i
\(153\) 1.00000i 0.0808452i
\(154\) −8.00000 −0.644658
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 4.00000i 0.319235i 0.987179 + 0.159617i \(0.0510260\pi\)
−0.987179 + 0.159617i \(0.948974\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 1.00000i 0.0785674i
\(163\) − 8.00000i − 0.626608i −0.949653 0.313304i \(-0.898564\pi\)
0.949653 0.313304i \(-0.101436\pi\)
\(164\) −8.00000 −0.624695
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 2.00000i 0.154303i
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) − 6.00000i − 0.457496i
\(173\) 22.0000i 1.67263i 0.548250 + 0.836315i \(0.315294\pi\)
−0.548250 + 0.836315i \(0.684706\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) − 6.00000i − 0.450988i
\(178\) 2.00000i 0.149906i
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) 26.0000 1.93256 0.966282 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 8.00000i 0.592999i
\(183\) − 14.0000i − 1.03491i
\(184\) 8.00000 0.589768
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) 4.00000i 0.292509i
\(188\) 8.00000i 0.583460i
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 24.0000i 1.72756i 0.503871 + 0.863779i \(0.331909\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 26.0000i 1.85242i 0.377004 + 0.926212i \(0.376954\pi\)
−0.377004 + 0.926212i \(0.623046\pi\)
\(198\) 4.00000i 0.284268i
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 2.00000 0.141069
\(202\) 8.00000i 0.562878i
\(203\) 4.00000i 0.280745i
\(204\) 1.00000 0.0700140
\(205\) 0 0
\(206\) 12.0000 0.836080
\(207\) − 8.00000i − 0.556038i
\(208\) 4.00000i 0.277350i
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 2.00000i 0.137361i
\(213\) − 2.00000i − 0.137038i
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) − 8.00000i − 0.543075i
\(218\) 14.0000i 0.948200i
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) − 6.00000i − 0.402694i
\(223\) 28.0000i 1.87502i 0.347960 + 0.937509i \(0.386874\pi\)
−0.347960 + 0.937509i \(0.613126\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) − 28.0000i − 1.85843i −0.369546 0.929213i \(-0.620487\pi\)
0.369546 0.929213i \(-0.379513\pi\)
\(228\) 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
\(231\) 8.00000 0.526361
\(232\) 2.00000i 0.131306i
\(233\) 18.0000i 1.17922i 0.807688 + 0.589610i \(0.200718\pi\)
−0.807688 + 0.589610i \(0.799282\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 0 0
\(238\) − 2.00000i − 0.129641i
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 5.00000i 0.321412i
\(243\) − 1.00000i − 0.0641500i
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) 8.00000 0.510061
\(247\) 16.0000i 1.01806i
\(248\) − 4.00000i − 0.254000i
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) 26.0000 1.64111 0.820553 0.571571i \(-0.193666\pi\)
0.820553 + 0.571571i \(0.193666\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) − 32.0000i − 2.01182i
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 6.00000i 0.373544i
\(259\) −12.0000 −0.745644
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) − 16.0000i − 0.988483i
\(263\) 8.00000i 0.493301i 0.969104 + 0.246651i \(0.0793300\pi\)
−0.969104 + 0.246651i \(0.920670\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) 8.00000 0.490511
\(267\) − 2.00000i − 0.122398i
\(268\) − 2.00000i − 0.122169i
\(269\) 30.0000 1.82913 0.914566 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) − 1.00000i − 0.0606339i
\(273\) − 8.00000i − 0.484182i
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 20.0000i 1.19952i
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) − 8.00000i − 0.476393i
\(283\) − 20.0000i − 1.18888i −0.804141 0.594438i \(-0.797374\pi\)
0.804141 0.594438i \(-0.202626\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 16.0000 0.946100
\(287\) − 16.0000i − 0.944450i
\(288\) − 1.00000i − 0.0589256i
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) − 4.00000i − 0.234082i
\(293\) − 14.0000i − 0.817889i −0.912559 0.408944i \(-0.865897\pi\)
0.912559 0.408944i \(-0.134103\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) − 4.00000i − 0.232104i
\(298\) − 16.0000i − 0.926855i
\(299\) −32.0000 −1.85061
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) − 8.00000i − 0.460348i
\(303\) − 8.00000i − 0.459588i
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) −1.00000 −0.0571662
\(307\) 10.0000i 0.570730i 0.958419 + 0.285365i \(0.0921148\pi\)
−0.958419 + 0.285365i \(0.907885\pi\)
\(308\) − 8.00000i − 0.455842i
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) 14.0000 0.793867 0.396934 0.917847i \(-0.370074\pi\)
0.396934 + 0.917847i \(0.370074\pi\)
\(312\) − 4.00000i − 0.226455i
\(313\) 16.0000i 0.904373i 0.891923 + 0.452187i \(0.149356\pi\)
−0.891923 + 0.452187i \(0.850644\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) 0 0
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) − 2.00000i − 0.112154i
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 16.0000i 0.891645i
\(323\) − 4.00000i − 0.222566i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 8.00000 0.443079
\(327\) − 14.0000i − 0.774202i
\(328\) − 8.00000i − 0.441726i
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 16.0000i 0.878114i
\(333\) 6.00000i 0.328798i
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) − 20.0000i − 1.08947i −0.838608 0.544735i \(-0.816630\pi\)
0.838608 0.544735i \(-0.183370\pi\)
\(338\) − 3.00000i − 0.163178i
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) − 4.00000i − 0.216295i
\(343\) − 20.0000i − 1.07990i
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) −22.0000 −1.18273
\(347\) 28.0000i 1.50312i 0.659665 + 0.751559i \(0.270698\pi\)
−0.659665 + 0.751559i \(0.729302\pi\)
\(348\) − 2.00000i − 0.107211i
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) − 4.00000i − 0.213201i
\(353\) − 10.0000i − 0.532246i −0.963939 0.266123i \(-0.914257\pi\)
0.963939 0.266123i \(-0.0857428\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 2.00000i 0.105851i
\(358\) − 6.00000i − 0.317110i
\(359\) −36.0000 −1.90001 −0.950004 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 26.0000i 1.36653i
\(363\) − 5.00000i − 0.262432i
\(364\) −8.00000 −0.419314
\(365\) 0 0
\(366\) 14.0000 0.731792
\(367\) − 22.0000i − 1.14839i −0.818718 0.574195i \(-0.805315\pi\)
0.818718 0.574195i \(-0.194685\pi\)
\(368\) 8.00000i 0.417029i
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 4.00000i 0.207390i
\(373\) − 16.0000i − 0.828449i −0.910175 0.414224i \(-0.864053\pi\)
0.910175 0.414224i \(-0.135947\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) − 8.00000i − 0.412021i
\(378\) 2.00000i 0.102869i
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 4.00000i 0.204658i
\(383\) 8.00000i 0.408781i 0.978889 + 0.204390i \(0.0655212\pi\)
−0.978889 + 0.204390i \(0.934479\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −24.0000 −1.22157
\(387\) − 6.00000i − 0.304997i
\(388\) 0 0
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) − 3.00000i − 0.151523i
\(393\) 16.0000i 0.807093i
\(394\) −26.0000 −1.30986
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) − 6.00000i − 0.301131i −0.988600 0.150566i \(-0.951890\pi\)
0.988600 0.150566i \(-0.0481095\pi\)
\(398\) 20.0000i 1.00251i
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 2.00000i 0.0997509i
\(403\) 16.0000i 0.797017i
\(404\) −8.00000 −0.398015
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) 24.0000i 1.18964i
\(408\) 1.00000i 0.0495074i
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 12.0000i 0.591198i
\(413\) − 12.0000i − 0.590481i
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) − 20.0000i − 0.979404i
\(418\) − 16.0000i − 0.782586i
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) 4.00000i 0.194717i
\(423\) 8.00000i 0.388973i
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) 2.00000 0.0969003
\(427\) − 28.0000i − 1.35501i
\(428\) 12.0000i 0.580042i
\(429\) −16.0000 −0.772487
\(430\) 0 0
\(431\) 14.0000 0.674356 0.337178 0.941441i \(-0.390528\pi\)
0.337178 + 0.941441i \(0.390528\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 26.0000i − 1.24948i −0.780833 0.624740i \(-0.785205\pi\)
0.780833 0.624740i \(-0.214795\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) 32.0000i 1.53077i
\(438\) 4.00000i 0.191127i
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 4.00000i 0.190261i
\(443\) 24.0000i 1.14027i 0.821549 + 0.570137i \(0.193110\pi\)
−0.821549 + 0.570137i \(0.806890\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) −28.0000 −1.32584
\(447\) 16.0000i 0.756774i
\(448\) 2.00000i 0.0944911i
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) −32.0000 −1.50682
\(452\) − 6.00000i − 0.282216i
\(453\) 8.00000i 0.375873i
\(454\) 28.0000 1.31411
\(455\) 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
\(461\) −28.0000 −1.30409 −0.652045 0.758180i \(-0.726089\pi\)
−0.652045 + 0.758180i \(0.726089\pi\)
\(462\) 8.00000i 0.372194i
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 4.00000i 0.184900i
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) − 6.00000i − 0.276172i
\(473\) − 24.0000i − 1.10352i
\(474\) 0 0
\(475\) 0 0
\(476\) 2.00000 0.0916698
\(477\) 2.00000i 0.0915737i
\(478\) 0 0
\(479\) 22.0000 1.00521 0.502603 0.864517i \(-0.332376\pi\)
0.502603 + 0.864517i \(0.332376\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) − 14.0000i − 0.637683i
\(483\) − 16.0000i − 0.728025i
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 22.0000i 0.996915i 0.866914 + 0.498458i \(0.166100\pi\)
−0.866914 + 0.498458i \(0.833900\pi\)
\(488\) − 14.0000i − 0.633750i
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) −18.0000 −0.812329 −0.406164 0.913800i \(-0.633134\pi\)
−0.406164 + 0.913800i \(0.633134\pi\)
\(492\) 8.00000i 0.360668i
\(493\) 2.00000i 0.0900755i
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) − 4.00000i − 0.179425i
\(498\) − 16.0000i − 0.716977i
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 26.0000i 1.16044i
\(503\) − 16.0000i − 0.713405i −0.934218 0.356702i \(-0.883901\pi\)
0.934218 0.356702i \(-0.116099\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 32.0000 1.42257
\(507\) 3.00000i 0.133235i
\(508\) − 4.00000i − 0.177471i
\(509\) −8.00000 −0.354594 −0.177297 0.984157i \(-0.556735\pi\)
−0.177297 + 0.984157i \(0.556735\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 1.00000i 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) −6.00000 −0.264135
\(517\) 32.0000i 1.40736i
\(518\) − 12.0000i − 0.527250i
\(519\) 22.0000 0.965693
\(520\) 0 0
\(521\) −36.0000 −1.57719 −0.788594 0.614914i \(-0.789191\pi\)
−0.788594 + 0.614914i \(0.789191\pi\)
\(522\) 2.00000i 0.0875376i
\(523\) − 14.0000i − 0.612177i −0.952003 0.306089i \(-0.900980\pi\)
0.952003 0.306089i \(-0.0990204\pi\)
\(524\) 16.0000 0.698963
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) − 4.00000i − 0.174243i
\(528\) 4.00000i 0.174078i
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 8.00000i 0.346844i
\(533\) 32.0000i 1.38607i
\(534\) 2.00000 0.0865485
\(535\) 0 0
\(536\) 2.00000 0.0863868
\(537\) 6.00000i 0.258919i
\(538\) 30.0000i 1.29339i
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) − 24.0000i − 1.03089i
\(543\) − 26.0000i − 1.11577i
\(544\) 1.00000 0.0428746
\(545\) 0 0
\(546\) 8.00000 0.342368
\(547\) − 44.0000i − 1.88130i −0.339372 0.940652i \(-0.610215\pi\)
0.339372 0.940652i \(-0.389785\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) − 8.00000i − 0.340503i
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 2.00000i 0.0847427i 0.999102 + 0.0423714i \(0.0134913\pi\)
−0.999102 + 0.0423714i \(0.986509\pi\)
\(558\) − 4.00000i − 0.169334i
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) − 2.00000i − 0.0843649i
\(563\) − 24.0000i − 1.01148i −0.862686 0.505740i \(-0.831220\pi\)
0.862686 0.505740i \(-0.168780\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) − 2.00000i − 0.0839921i
\(568\) − 2.00000i − 0.0839181i
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 16.0000i 0.668994i
\(573\) − 4.00000i − 0.167102i
\(574\) 16.0000 0.667827
\(575\) 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
\(579\) 24.0000 0.997406
\(580\) 0 0
\(581\) −32.0000 −1.32758
\(582\) 0 0
\(583\) 8.00000i 0.331326i
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) − 8.00000i − 0.330195i −0.986277 0.165098i \(-0.947206\pi\)
0.986277 0.165098i \(-0.0527939\pi\)
\(588\) 3.00000i 0.123718i
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 26.0000 1.06950
\(592\) − 6.00000i − 0.246598i
\(593\) − 18.0000i − 0.739171i −0.929197 0.369586i \(-0.879500\pi\)
0.929197 0.369586i \(-0.120500\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 16.0000 0.655386
\(597\) − 20.0000i − 0.818546i
\(598\) − 32.0000i − 1.30858i
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) 30.0000 1.22373 0.611863 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) 12.0000i 0.489083i
\(603\) − 2.00000i − 0.0814463i
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 8.00000 0.324978
\(607\) − 34.0000i − 1.38002i −0.723801 0.690009i \(-0.757607\pi\)
0.723801 0.690009i \(-0.242393\pi\)
\(608\) 4.00000i 0.162221i
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) 32.0000 1.29458
\(612\) − 1.00000i − 0.0404226i
\(613\) 16.0000i 0.646234i 0.946359 + 0.323117i \(0.104731\pi\)
−0.946359 + 0.323117i \(0.895269\pi\)
\(614\) −10.0000 −0.403567
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) − 12.0000i − 0.482711i
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 14.0000i 0.561349i
\(623\) − 4.00000i − 0.160257i
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −16.0000 −0.639489
\(627\) 16.0000i 0.638978i
\(628\) − 4.00000i − 0.159617i
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) − 4.00000i − 0.158986i
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) 12.0000i 0.475457i
\(638\) 8.00000i 0.316723i
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) 12.0000 0.473972 0.236986 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) − 12.0000i − 0.473602i
\(643\) 16.0000i 0.630978i 0.948929 + 0.315489i \(0.102169\pi\)
−0.948929 + 0.315489i \(0.897831\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) 16.0000i 0.629025i 0.949253 + 0.314512i \(0.101841\pi\)
−0.949253 + 0.314512i \(0.898159\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) 8.00000i 0.313304i
\(653\) − 30.0000i − 1.17399i −0.809590 0.586995i \(-0.800311\pi\)
0.809590 0.586995i \(-0.199689\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) 8.00000 0.312348
\(657\) − 4.00000i − 0.156055i
\(658\) − 16.0000i − 0.623745i
\(659\) −14.0000 −0.545363 −0.272681 0.962104i \(-0.587910\pi\)
−0.272681 + 0.962104i \(0.587910\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) − 8.00000i − 0.310929i
\(663\) − 4.00000i − 0.155347i
\(664\) −16.0000 −0.620920
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) − 16.0000i − 0.619522i
\(668\) − 12.0000i − 0.464294i
\(669\) 28.0000 1.08254
\(670\) 0 0
\(671\) −56.0000 −2.16186
\(672\) − 2.00000i − 0.0771517i
\(673\) − 8.00000i − 0.308377i −0.988041 0.154189i \(-0.950724\pi\)
0.988041 0.154189i \(-0.0492764\pi\)
\(674\) 20.0000 0.770371
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) − 6.00000i − 0.230599i −0.993331 0.115299i \(-0.963217\pi\)
0.993331 0.115299i \(-0.0367827\pi\)
\(678\) 6.00000i 0.230429i
\(679\) 0 0
\(680\) 0 0
\(681\) −28.0000 −1.07296
\(682\) − 16.0000i − 0.612672i
\(683\) 36.0000i 1.37750i 0.724998 + 0.688751i \(0.241841\pi\)
−0.724998 + 0.688751i \(0.758159\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 6.00000i 0.228914i
\(688\) 6.00000i 0.228748i
\(689\) 8.00000 0.304776
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) − 22.0000i − 0.836315i
\(693\) − 8.00000i − 0.303895i
\(694\) −28.0000 −1.06287
\(695\) 0 0
\(696\) 2.00000 0.0758098
\(697\) − 8.00000i − 0.303022i
\(698\) 14.0000i 0.529908i
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) − 4.00000i − 0.150970i
\(703\) − 24.0000i − 0.905177i
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) − 16.0000i − 0.601742i
\(708\) 6.00000i 0.225494i
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 2.00000i − 0.0749532i
\(713\) 32.0000i 1.19841i
\(714\) −2.00000 −0.0748481
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) 0 0
\(718\) − 36.0000i − 1.34351i
\(719\) −10.0000 −0.372937 −0.186469 0.982461i \(-0.559704\pi\)
−0.186469 + 0.982461i \(0.559704\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) − 3.00000i − 0.111648i
\(723\) 14.0000i 0.520666i
\(724\) −26.0000 −0.966282
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) − 44.0000i − 1.63187i −0.578144 0.815935i \(-0.696223\pi\)
0.578144 0.815935i \(-0.303777\pi\)
\(728\) − 8.00000i − 0.296500i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 6.00000 0.221918
\(732\) 14.0000i 0.517455i
\(733\) 48.0000i 1.77292i 0.462805 + 0.886460i \(0.346843\pi\)
−0.462805 + 0.886460i \(0.653157\pi\)
\(734\) 22.0000 0.812035
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) − 8.00000i − 0.294684i
\(738\) − 8.00000i − 0.294484i
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) − 4.00000i − 0.146845i
\(743\) − 20.0000i − 0.733729i −0.930274 0.366864i \(-0.880431\pi\)
0.930274 0.366864i \(-0.119569\pi\)
\(744\) −4.00000 −0.146647
\(745\) 0 0
\(746\) 16.0000 0.585802
\(747\) 16.0000i 0.585409i
\(748\) − 4.00000i − 0.146254i
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) −24.0000 −0.875772 −0.437886 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) − 8.00000i − 0.291730i
\(753\) − 26.0000i − 0.947493i
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) 28.0000i 1.01768i 0.860862 + 0.508839i \(0.169925\pi\)
−0.860862 + 0.508839i \(0.830075\pi\)
\(758\) 4.00000i 0.145287i
\(759\) −32.0000 −1.16153
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 4.00000i 0.144905i
\(763\) − 28.0000i − 1.01367i
\(764\) −4.00000 −0.144715
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) 24.0000i 0.866590i
\(768\) − 1.00000i − 0.0360844i
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) − 24.0000i − 0.863779i
\(773\) − 42.0000i − 1.51064i −0.655359 0.755318i \(-0.727483\pi\)
0.655359 0.755318i \(-0.272517\pi\)
\(774\) 6.00000 0.215666
\(775\) 0 0
\(776\) 0 0
\(777\) 12.0000i 0.430498i
\(778\) − 24.0000i − 0.860442i
\(779\) 32.0000 1.14652
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 8.00000i 0.286079i
\(783\) − 2.00000i − 0.0714742i
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) −16.0000 −0.570701
\(787\) − 24.0000i − 0.855508i −0.903895 0.427754i \(-0.859305\pi\)
0.903895 0.427754i \(-0.140695\pi\)
\(788\) − 26.0000i − 0.926212i
\(789\) 8.00000 0.284808
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) − 4.00000i − 0.142134i
\(793\) 56.0000i 1.98862i
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) −20.0000 −0.708881
\(797\) 14.0000i 0.495905i 0.968772 + 0.247953i \(0.0797578\pi\)
−0.968772 + 0.247953i \(0.920242\pi\)
\(798\) − 8.00000i − 0.283197i
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) − 12.0000i − 0.423735i
\(803\) − 16.0000i − 0.564628i
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) − 30.0000i − 1.05605i
\(808\) − 8.00000i − 0.281439i
\(809\) 48.0000 1.68759 0.843795 0.536666i \(-0.180316\pi\)
0.843795 + 0.536666i \(0.180316\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) − 4.00000i − 0.140372i
\(813\) 24.0000i 0.841717i
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) −1.00000 −0.0350070
\(817\) 24.0000i 0.839654i
\(818\) 26.0000i 0.909069i
\(819\) −8.00000 −0.279543
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 6.00000i 0.209274i
\(823\) 14.0000i 0.488009i 0.969774 + 0.244005i \(0.0784612\pi\)
−0.969774 + 0.244005i \(0.921539\pi\)
\(824\) −12.0000 −0.418040
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) − 44.0000i − 1.53003i −0.644013 0.765015i \(-0.722732\pi\)
0.644013 0.765015i \(-0.277268\pi\)
\(828\) 8.00000i 0.278019i
\(829\) 46.0000 1.59765 0.798823 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) − 4.00000i − 0.138675i
\(833\) − 3.00000i − 0.103944i
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 4.00000i 0.138260i
\(838\) 0 0
\(839\) −6.00000 −0.207143 −0.103572 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) − 18.0000i − 0.620321i
\(843\) 2.00000i 0.0688837i
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) − 10.0000i − 0.343604i
\(848\) − 2.00000i − 0.0686803i
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 48.0000 1.64542
\(852\) 2.00000i 0.0685189i
\(853\) 2.00000i 0.0684787i 0.999414 + 0.0342393i \(0.0109009\pi\)
−0.999414 + 0.0342393i \(0.989099\pi\)
\(854\) 28.0000 0.958140
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 22.0000i 0.751506i 0.926720 + 0.375753i \(0.122616\pi\)
−0.926720 + 0.375753i \(0.877384\pi\)
\(858\) − 16.0000i − 0.546231i
\(859\) 32.0000 1.09183 0.545913 0.837842i \(-0.316183\pi\)
0.545913 + 0.837842i \(0.316183\pi\)
\(860\) 0 0
\(861\) −16.0000 −0.545279
\(862\) 14.0000i 0.476842i
\(863\) − 16.0000i − 0.544646i −0.962206 0.272323i \(-0.912208\pi\)
0.962206 0.272323i \(-0.0877920\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 26.0000 0.883516
\(867\) 1.00000i 0.0339618i
\(868\) 8.00000i 0.271538i
\(869\) 0 0
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) − 14.0000i − 0.474100i
\(873\) 0 0
\(874\) −32.0000 −1.08242
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) − 50.0000i − 1.68838i −0.536044 0.844190i \(-0.680082\pi\)
0.536044 0.844190i \(-0.319918\pi\)
\(878\) 24.0000i 0.809961i
\(879\) −14.0000 −0.472208
\(880\) 0 0
\(881\) 36.0000 1.21287 0.606435 0.795133i \(-0.292599\pi\)
0.606435 + 0.795133i \(0.292599\pi\)
\(882\) − 3.00000i − 0.101015i
\(883\) − 46.0000i − 1.54802i −0.633171 0.774012i \(-0.718247\pi\)
0.633171 0.774012i \(-0.281753\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 6.00000i 0.201347i
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) − 28.0000i − 0.937509i
\(893\) − 32.0000i − 1.07084i
\(894\) −16.0000 −0.535120
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 32.0000i 1.06845i
\(898\) 12.0000i 0.400445i
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) −2.00000 −0.0666297
\(902\) − 32.0000i − 1.06548i
\(903\) − 12.0000i − 0.399335i
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) 32.0000i 1.06254i 0.847202 + 0.531271i \(0.178286\pi\)
−0.847202 + 0.531271i \(0.821714\pi\)
\(908\) 28.0000i 0.929213i
\(909\) −8.00000 −0.265343
\(910\) 0 0
\(911\) 50.0000 1.65657 0.828287 0.560304i \(-0.189316\pi\)
0.828287 + 0.560304i \(0.189316\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) 64.0000i 2.11809i
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 32.0000i 1.05673i
\(918\) 1.00000i 0.0330049i
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 10.0000 0.329511
\(922\) − 28.0000i − 0.922131i
\(923\) 8.00000i 0.263323i
\(924\) −8.00000 −0.263181
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) 12.0000i 0.394132i
\(928\) − 2.00000i − 0.0656532i
\(929\) 48.0000 1.57483 0.787414 0.616424i \(-0.211419\pi\)
0.787414 + 0.616424i \(0.211419\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) − 18.0000i − 0.589610i
\(933\) − 14.0000i − 0.458339i
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) 34.0000i 1.11073i 0.831606 + 0.555366i \(0.187422\pi\)
−0.831606 + 0.555366i \(0.812578\pi\)
\(938\) 4.00000i 0.130605i
\(939\) 16.0000 0.522140
\(940\) 0 0
\(941\) −2.00000 −0.0651981 −0.0325991 0.999469i \(-0.510378\pi\)
−0.0325991 + 0.999469i \(0.510378\pi\)
\(942\) 4.00000i 0.130327i
\(943\) 64.0000i 2.08413i
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 24.0000 0.780307
\(947\) 4.00000i 0.129983i 0.997886 + 0.0649913i \(0.0207020\pi\)
−0.997886 + 0.0649913i \(0.979298\pi\)
\(948\) 0 0
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) 2.00000i 0.0648204i
\(953\) 42.0000i 1.36051i 0.732974 + 0.680257i \(0.238132\pi\)
−0.732974 + 0.680257i \(0.761868\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) 0 0
\(957\) − 8.00000i − 0.258603i
\(958\) 22.0000i 0.710788i
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 24.0000i 0.773791i
\(963\) 12.0000i 0.386695i
\(964\) 14.0000 0.450910
\(965\) 0 0
\(966\) 16.0000 0.514792
\(967\) − 56.0000i − 1.80084i −0.435023 0.900419i \(-0.643260\pi\)
0.435023 0.900419i \(-0.356740\pi\)
\(968\) − 5.00000i − 0.160706i
\(969\) −4.00000 −0.128499
\(970\) 0 0
\(971\) −50.0000 −1.60458 −0.802288 0.596937i \(-0.796384\pi\)
−0.802288 + 0.596937i \(0.796384\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) − 40.0000i − 1.28234i
\(974\) −22.0000 −0.704925
\(975\) 0 0
\(976\) 14.0000 0.448129
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) − 8.00000i − 0.255812i
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) − 18.0000i − 0.574403i
\(983\) − 52.0000i − 1.65854i −0.558846 0.829271i \(-0.688756\pi\)
0.558846 0.829271i \(-0.311244\pi\)
\(984\) −8.00000 −0.255031
\(985\) 0 0
\(986\) −2.00000 −0.0636930
\(987\) 16.0000i 0.509286i
\(988\) − 16.0000i − 0.509028i
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 4.00000i 0.127000i
\(993\) 8.00000i 0.253872i
\(994\) 4.00000 0.126872
\(995\) 0 0
\(996\) 16.0000 0.506979
\(997\) 50.0000i 1.58352i 0.610835 + 0.791758i \(0.290834\pi\)
−0.610835 + 0.791758i \(0.709166\pi\)
\(998\) − 4.00000i − 0.126618i
\(999\) 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.n.2449.2 2
5.2 odd 4 510.2.a.a.1.1 1
5.3 odd 4 2550.2.a.bc.1.1 1
5.4 even 2 inner 2550.2.d.n.2449.1 2
15.2 even 4 1530.2.a.p.1.1 1
15.8 even 4 7650.2.a.k.1.1 1
20.7 even 4 4080.2.a.s.1.1 1
85.67 odd 4 8670.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.a.1.1 1 5.2 odd 4
1530.2.a.p.1.1 1 15.2 even 4
2550.2.a.bc.1.1 1 5.3 odd 4
2550.2.d.n.2449.1 2 5.4 even 2 inner
2550.2.d.n.2449.2 2 1.1 even 1 trivial
4080.2.a.s.1.1 1 20.7 even 4
7650.2.a.k.1.1 1 15.8 even 4
8670.2.a.k.1.1 1 85.67 odd 4