Properties

Label 1014.2.b.b.337.2
Level $1014$
Weight $2$
Character 1014.337
Analytic conductor $8.097$
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] = [1014,2,Mod(337,1014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1014.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.b (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: \(8.09683076496\)
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 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1014.337
Dual form 1014.2.b.b.337.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000i q^{5} -1.00000i q^{6} +4.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000i q^{5} -1.00000i q^{6} +4.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +2.00000 q^{10} -4.00000i q^{11} +1.00000 q^{12} -4.00000 q^{14} +2.00000i q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000i q^{18} +8.00000i q^{19} +2.00000i q^{20} -4.00000i q^{21} +4.00000 q^{22} +1.00000i q^{24} +1.00000 q^{25} -1.00000 q^{27} -4.00000i q^{28} +6.00000 q^{29} -2.00000 q^{30} +4.00000i q^{31} +1.00000i q^{32} +4.00000i q^{33} -2.00000i q^{34} +8.00000 q^{35} -1.00000 q^{36} -2.00000i q^{37} -8.00000 q^{38} -2.00000 q^{40} +10.0000i q^{41} +4.00000 q^{42} -4.00000 q^{43} +4.00000i q^{44} -2.00000i q^{45} +8.00000i q^{47} -1.00000 q^{48} -9.00000 q^{49} +1.00000i q^{50} +2.00000 q^{51} -10.0000 q^{53} -1.00000i q^{54} -8.00000 q^{55} +4.00000 q^{56} -8.00000i q^{57} +6.00000i q^{58} +4.00000i q^{59} -2.00000i q^{60} -2.00000 q^{61} -4.00000 q^{62} +4.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} +16.0000i q^{67} +2.00000 q^{68} +8.00000i q^{70} +8.00000i q^{71} -1.00000i q^{72} +2.00000i q^{73} +2.00000 q^{74} -1.00000 q^{75} -8.00000i q^{76} +16.0000 q^{77} +8.00000 q^{79} -2.00000i q^{80} +1.00000 q^{81} -10.0000 q^{82} -12.0000i q^{83} +4.00000i q^{84} +4.00000i q^{85} -4.00000i q^{86} -6.00000 q^{87} -4.00000 q^{88} +14.0000i q^{89} +2.00000 q^{90} -4.00000i q^{93} -8.00000 q^{94} +16.0000 q^{95} -1.00000i q^{96} -10.0000i q^{97} -9.00000i q^{98} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 4 q^{10} + 2 q^{12} - 8 q^{14} + 2 q^{16} - 4 q^{17} + 8 q^{22} + 2 q^{25} - 2 q^{27} + 12 q^{29} - 4 q^{30} + 16 q^{35} - 2 q^{36} - 16 q^{38} - 4 q^{40} + 8 q^{42} - 8 q^{43} - 2 q^{48} - 18 q^{49} + 4 q^{51} - 20 q^{53} - 16 q^{55} + 8 q^{56} - 4 q^{61} - 8 q^{62} - 2 q^{64} - 8 q^{66} + 4 q^{68} + 4 q^{74} - 2 q^{75} + 32 q^{77} + 16 q^{79} + 2 q^{81} - 20 q^{82} - 12 q^{87} - 8 q^{88} + 4 q^{90} - 16 q^{94} + 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) − 2.00000i − 0.894427i −0.894427 0.447214i \(-0.852416\pi\)
0.894427 0.447214i \(-0.147584\pi\)
\(6\) − 1.00000i − 0.408248i
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) − 4.00000i − 1.20605i −0.797724 0.603023i \(-0.793963\pi\)
0.797724 0.603023i \(-0.206037\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −4.00000 −1.06904
\(15\) 2.00000i 0.516398i
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 8.00000i 1.83533i 0.397360 + 0.917663i \(0.369927\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 2.00000i 0.447214i
\(21\) − 4.00000i − 0.872872i
\(22\) 4.00000 0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000i 0.204124i
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) − 4.00000i − 0.755929i
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −2.00000 −0.365148
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 4.00000i 0.696311i
\(34\) − 2.00000i − 0.342997i
\(35\) 8.00000 1.35225
\(36\) −1.00000 −0.166667
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) −8.00000 −1.29777
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) 10.0000i 1.56174i 0.624695 + 0.780869i \(0.285223\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 4.00000 0.617213
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 4.00000i 0.603023i
\(45\) − 2.00000i − 0.298142i
\(46\) 0 0
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) −1.00000 −0.144338
\(49\) −9.00000 −1.28571
\(50\) 1.00000i 0.141421i
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) −8.00000 −1.07872
\(56\) 4.00000 0.534522
\(57\) − 8.00000i − 1.05963i
\(58\) 6.00000i 0.787839i
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) − 2.00000i − 0.258199i
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −4.00000 −0.508001
\(63\) 4.00000i 0.503953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) 16.0000i 1.95471i 0.211604 + 0.977356i \(0.432131\pi\)
−0.211604 + 0.977356i \(0.567869\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 8.00000i 0.956183i
\(71\) 8.00000i 0.949425i 0.880141 + 0.474713i \(0.157448\pi\)
−0.880141 + 0.474713i \(0.842552\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) − 8.00000i − 0.917663i
\(77\) 16.0000 1.82337
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) − 2.00000i − 0.223607i
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 4.00000i 0.436436i
\(85\) 4.00000i 0.433861i
\(86\) − 4.00000i − 0.431331i
\(87\) −6.00000 −0.643268
\(88\) −4.00000 −0.426401
\(89\) 14.0000i 1.48400i 0.670402 + 0.741999i \(0.266122\pi\)
−0.670402 + 0.741999i \(0.733878\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 0 0
\(93\) − 4.00000i − 0.414781i
\(94\) −8.00000 −0.825137
\(95\) 16.0000 1.64157
\(96\) − 1.00000i − 0.102062i
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) − 9.00000i − 0.909137i
\(99\) − 4.00000i − 0.402015i
\(100\) −1.00000 −0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 2.00000i 0.198030i
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) −8.00000 −0.780720
\(106\) − 10.0000i − 0.971286i
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000i 0.191565i 0.995402 + 0.0957826i \(0.0305354\pi\)
−0.995402 + 0.0957826i \(0.969465\pi\)
\(110\) − 8.00000i − 0.762770i
\(111\) 2.00000i 0.189832i
\(112\) 4.00000i 0.377964i
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) − 8.00000i − 0.733359i
\(120\) 2.00000 0.182574
\(121\) −5.00000 −0.454545
\(122\) − 2.00000i − 0.181071i
\(123\) − 10.0000i − 0.901670i
\(124\) − 4.00000i − 0.359211i
\(125\) − 12.0000i − 1.07331i
\(126\) −4.00000 −0.356348
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 4.00000 0.352180
\(130\) 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
\(133\) −32.0000 −2.77475
\(134\) −16.0000 −1.38219
\(135\) 2.00000i 0.172133i
\(136\) 2.00000i 0.171499i
\(137\) − 10.0000i − 0.854358i −0.904167 0.427179i \(-0.859507\pi\)
0.904167 0.427179i \(-0.140493\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) −8.00000 −0.676123
\(141\) − 8.00000i − 0.673722i
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 12.0000i − 0.996546i
\(146\) −2.00000 −0.165521
\(147\) 9.00000 0.742307
\(148\) 2.00000i 0.164399i
\(149\) 6.00000i 0.491539i 0.969328 + 0.245770i \(0.0790407\pi\)
−0.969328 + 0.245770i \(0.920959\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) 12.0000i 0.976546i 0.872691 + 0.488273i \(0.162373\pi\)
−0.872691 + 0.488273i \(0.837627\pi\)
\(152\) 8.00000 0.648886
\(153\) −2.00000 −0.161690
\(154\) 16.0000i 1.28932i
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 8.00000i 0.636446i
\(159\) 10.0000 0.793052
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) − 16.0000i − 1.25322i −0.779334 0.626608i \(-0.784443\pi\)
0.779334 0.626608i \(-0.215557\pi\)
\(164\) − 10.0000i − 0.780869i
\(165\) 8.00000 0.622799
\(166\) 12.0000 0.931381
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) −4.00000 −0.308607
\(169\) 0 0
\(170\) −4.00000 −0.306786
\(171\) 8.00000i 0.611775i
\(172\) 4.00000 0.304997
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) − 6.00000i − 0.454859i
\(175\) 4.00000i 0.302372i
\(176\) − 4.00000i − 0.301511i
\(177\) − 4.00000i − 0.300658i
\(178\) −14.0000 −1.04934
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 2.00000i 0.149071i
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 4.00000 0.293294
\(187\) 8.00000i 0.585018i
\(188\) − 8.00000i − 0.583460i
\(189\) − 4.00000i − 0.290957i
\(190\) 16.0000i 1.16076i
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 1.00000 0.0721688
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 4.00000 0.284268
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) − 16.0000i − 1.12855i
\(202\) 2.00000i 0.140720i
\(203\) 24.0000i 1.68447i
\(204\) −2.00000 −0.140028
\(205\) 20.0000 1.39686
\(206\) − 16.0000i − 1.11477i
\(207\) 0 0
\(208\) 0 0
\(209\) 32.0000 2.21349
\(210\) − 8.00000i − 0.552052i
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 10.0000 0.686803
\(213\) − 8.00000i − 0.548151i
\(214\) 12.0000i 0.820303i
\(215\) 8.00000i 0.545595i
\(216\) 1.00000i 0.0680414i
\(217\) −16.0000 −1.08615
\(218\) −2.00000 −0.135457
\(219\) − 2.00000i − 0.135147i
\(220\) 8.00000 0.539360
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) −4.00000 −0.267261
\(225\) 1.00000 0.0666667
\(226\) − 6.00000i − 0.399114i
\(227\) − 20.0000i − 1.32745i −0.747978 0.663723i \(-0.768975\pi\)
0.747978 0.663723i \(-0.231025\pi\)
\(228\) 8.00000i 0.529813i
\(229\) 22.0000i 1.45380i 0.686743 + 0.726900i \(0.259040\pi\)
−0.686743 + 0.726900i \(0.740960\pi\)
\(230\) 0 0
\(231\) −16.0000 −1.05272
\(232\) − 6.00000i − 0.393919i
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 16.0000 1.04372
\(236\) − 4.00000i − 0.260378i
\(237\) −8.00000 −0.519656
\(238\) 8.00000 0.518563
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 2.00000i 0.129099i
\(241\) 10.0000i 0.644157i 0.946713 + 0.322078i \(0.104381\pi\)
−0.946713 + 0.322078i \(0.895619\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 18.0000i 1.14998i
\(246\) 10.0000 0.637577
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) 12.0000i 0.760469i
\(250\) 12.0000 0.758947
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) − 4.00000i − 0.251976i
\(253\) 0 0
\(254\) 0 0
\(255\) − 4.00000i − 0.250490i
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 4.00000i 0.249029i
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 4.00000i 0.247121i
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 4.00000 0.246183
\(265\) 20.0000i 1.22859i
\(266\) − 32.0000i − 1.96205i
\(267\) − 14.0000i − 0.856786i
\(268\) − 16.0000i − 0.977356i
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) −2.00000 −0.121716
\(271\) − 4.00000i − 0.242983i −0.992592 0.121491i \(-0.961232\pi\)
0.992592 0.121491i \(-0.0387677\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) − 4.00000i − 0.241209i
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 12.0000i 0.719712i
\(279\) 4.00000i 0.239474i
\(280\) − 8.00000i − 0.478091i
\(281\) − 26.0000i − 1.55103i −0.631329 0.775515i \(-0.717490\pi\)
0.631329 0.775515i \(-0.282510\pi\)
\(282\) 8.00000 0.476393
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) − 8.00000i − 0.474713i
\(285\) −16.0000 −0.947758
\(286\) 0 0
\(287\) −40.0000 −2.36113
\(288\) 1.00000i 0.0589256i
\(289\) −13.0000 −0.764706
\(290\) 12.0000 0.704664
\(291\) 10.0000i 0.586210i
\(292\) − 2.00000i − 0.117041i
\(293\) 26.0000i 1.51894i 0.650545 + 0.759468i \(0.274541\pi\)
−0.650545 + 0.759468i \(0.725459\pi\)
\(294\) 9.00000i 0.524891i
\(295\) 8.00000 0.465778
\(296\) −2.00000 −0.116248
\(297\) 4.00000i 0.232104i
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) − 16.0000i − 0.922225i
\(302\) −12.0000 −0.690522
\(303\) −2.00000 −0.114897
\(304\) 8.00000i 0.458831i
\(305\) 4.00000i 0.229039i
\(306\) − 2.00000i − 0.114332i
\(307\) − 8.00000i − 0.456584i −0.973593 0.228292i \(-0.926686\pi\)
0.973593 0.228292i \(-0.0733141\pi\)
\(308\) −16.0000 −0.911685
\(309\) 16.0000 0.910208
\(310\) 8.00000i 0.454369i
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 14.0000i 0.790066i
\(315\) 8.00000 0.450749
\(316\) −8.00000 −0.450035
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) 10.0000i 0.560772i
\(319\) − 24.0000i − 1.34374i
\(320\) 2.00000i 0.111803i
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) − 16.0000i − 0.890264i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) − 2.00000i − 0.110600i
\(328\) 10.0000 0.552158
\(329\) −32.0000 −1.76422
\(330\) 8.00000i 0.440386i
\(331\) − 8.00000i − 0.439720i −0.975531 0.219860i \(-0.929440\pi\)
0.975531 0.219860i \(-0.0705600\pi\)
\(332\) 12.0000i 0.658586i
\(333\) − 2.00000i − 0.109599i
\(334\) 0 0
\(335\) 32.0000 1.74835
\(336\) − 4.00000i − 0.218218i
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) − 4.00000i − 0.216930i
\(341\) 16.0000 0.866449
\(342\) −8.00000 −0.432590
\(343\) − 8.00000i − 0.431959i
\(344\) 4.00000i 0.215666i
\(345\) 0 0
\(346\) 10.0000i 0.537603i
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 6.00000 0.321634
\(349\) 6.00000i 0.321173i 0.987022 + 0.160586i \(0.0513385\pi\)
−0.987022 + 0.160586i \(0.948662\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) − 14.0000i − 0.745145i −0.928003 0.372572i \(-0.878476\pi\)
0.928003 0.372572i \(-0.121524\pi\)
\(354\) 4.00000 0.212598
\(355\) 16.0000 0.849192
\(356\) − 14.0000i − 0.741999i
\(357\) 8.00000i 0.423405i
\(358\) 12.0000i 0.634220i
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −2.00000 −0.105409
\(361\) −45.0000 −2.36842
\(362\) 10.0000i 0.525588i
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) 2.00000i 0.104542i
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 0 0
\(369\) 10.0000i 0.520579i
\(370\) − 4.00000i − 0.207950i
\(371\) − 40.0000i − 2.07670i
\(372\) 4.00000i 0.207390i
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) −8.00000 −0.413670
\(375\) 12.0000i 0.619677i
\(376\) 8.00000 0.412568
\(377\) 0 0
\(378\) 4.00000 0.205738
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) −16.0000 −0.820783
\(381\) 0 0
\(382\) − 8.00000i − 0.409316i
\(383\) 24.0000i 1.22634i 0.789950 + 0.613171i \(0.210106\pi\)
−0.789950 + 0.613171i \(0.789894\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) − 32.0000i − 1.63087i
\(386\) 14.0000 0.712581
\(387\) −4.00000 −0.203331
\(388\) 10.0000i 0.507673i
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 9.00000i 0.454569i
\(393\) −4.00000 −0.201773
\(394\) 18.0000 0.906827
\(395\) − 16.0000i − 0.805047i
\(396\) 4.00000i 0.201008i
\(397\) 6.00000i 0.301131i 0.988600 + 0.150566i \(0.0481095\pi\)
−0.988600 + 0.150566i \(0.951890\pi\)
\(398\) 8.00000i 0.401004i
\(399\) 32.0000 1.60200
\(400\) 1.00000 0.0500000
\(401\) 6.00000i 0.299626i 0.988714 + 0.149813i \(0.0478671\pi\)
−0.988714 + 0.149813i \(0.952133\pi\)
\(402\) 16.0000 0.798007
\(403\) 0 0
\(404\) −2.00000 −0.0995037
\(405\) − 2.00000i − 0.0993808i
\(406\) −24.0000 −1.19110
\(407\) −8.00000 −0.396545
\(408\) − 2.00000i − 0.0990148i
\(409\) − 2.00000i − 0.0988936i −0.998777 0.0494468i \(-0.984254\pi\)
0.998777 0.0494468i \(-0.0157458\pi\)
\(410\) 20.0000i 0.987730i
\(411\) 10.0000i 0.493264i
\(412\) 16.0000 0.788263
\(413\) −16.0000 −0.787309
\(414\) 0 0
\(415\) −24.0000 −1.17811
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 32.0000i 1.56517i
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 8.00000 0.390360
\(421\) − 22.0000i − 1.07221i −0.844150 0.536107i \(-0.819894\pi\)
0.844150 0.536107i \(-0.180106\pi\)
\(422\) 12.0000i 0.584151i
\(423\) 8.00000i 0.388973i
\(424\) 10.0000i 0.485643i
\(425\) −2.00000 −0.0970143
\(426\) 8.00000 0.387601
\(427\) − 8.00000i − 0.387147i
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 8.00000i 0.385346i 0.981263 + 0.192673i \(0.0617157\pi\)
−0.981263 + 0.192673i \(0.938284\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 30.0000 1.44171 0.720854 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) − 16.0000i − 0.768025i
\(435\) 12.0000i 0.575356i
\(436\) − 2.00000i − 0.0957826i
\(437\) 0 0
\(438\) 2.00000 0.0955637
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 8.00000i 0.381385i
\(441\) −9.00000 −0.428571
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) − 2.00000i − 0.0949158i
\(445\) 28.0000 1.32733
\(446\) −4.00000 −0.189405
\(447\) − 6.00000i − 0.283790i
\(448\) − 4.00000i − 0.188982i
\(449\) 6.00000i 0.283158i 0.989927 + 0.141579i \(0.0452178\pi\)
−0.989927 + 0.141579i \(0.954782\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) 40.0000 1.88353
\(452\) 6.00000 0.282216
\(453\) − 12.0000i − 0.563809i
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) −8.00000 −0.374634
\(457\) 30.0000i 1.40334i 0.712502 + 0.701670i \(0.247562\pi\)
−0.712502 + 0.701670i \(0.752438\pi\)
\(458\) −22.0000 −1.02799
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 6.00000i 0.279448i 0.990190 + 0.139724i \(0.0446215\pi\)
−0.990190 + 0.139724i \(0.955378\pi\)
\(462\) − 16.0000i − 0.744387i
\(463\) − 20.0000i − 0.929479i −0.885448 0.464739i \(-0.846148\pi\)
0.885448 0.464739i \(-0.153852\pi\)
\(464\) 6.00000 0.278543
\(465\) −8.00000 −0.370991
\(466\) − 18.0000i − 0.833834i
\(467\) 4.00000 0.185098 0.0925490 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) 0 0
\(469\) −64.0000 −2.95525
\(470\) 16.0000i 0.738025i
\(471\) −14.0000 −0.645086
\(472\) 4.00000 0.184115
\(473\) 16.0000i 0.735681i
\(474\) − 8.00000i − 0.367452i
\(475\) 8.00000i 0.367065i
\(476\) 8.00000i 0.366679i
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) − 16.0000i − 0.731059i −0.930800 0.365529i \(-0.880888\pi\)
0.930800 0.365529i \(-0.119112\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 0 0
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −20.0000 −0.908153
\(486\) − 1.00000i − 0.0453609i
\(487\) − 4.00000i − 0.181257i −0.995885 0.0906287i \(-0.971112\pi\)
0.995885 0.0906287i \(-0.0288876\pi\)
\(488\) 2.00000i 0.0905357i
\(489\) 16.0000i 0.723545i
\(490\) −18.0000 −0.813157
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 10.0000i 0.450835i
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) −8.00000 −0.359573
\(496\) 4.00000i 0.179605i
\(497\) −32.0000 −1.43540
\(498\) −12.0000 −0.537733
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 12.0000i 0.536656i
\(501\) 0 0
\(502\) − 4.00000i − 0.178529i
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) 4.00000 0.178174
\(505\) − 4.00000i − 0.177998i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 42.0000i − 1.86162i −0.365507 0.930809i \(-0.619104\pi\)
0.365507 0.930809i \(-0.380896\pi\)
\(510\) 4.00000 0.177123
\(511\) −8.00000 −0.353899
\(512\) 1.00000i 0.0441942i
\(513\) − 8.00000i − 0.353209i
\(514\) 6.00000i 0.264649i
\(515\) 32.0000i 1.41009i
\(516\) −4.00000 −0.176090
\(517\) 32.0000 1.40736
\(518\) 8.00000i 0.351500i
\(519\) −10.0000 −0.438951
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 6.00000i 0.262613i
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −4.00000 −0.174741
\(525\) − 4.00000i − 0.174574i
\(526\) 8.00000i 0.348817i
\(527\) − 8.00000i − 0.348485i
\(528\) 4.00000i 0.174078i
\(529\) −23.0000 −1.00000
\(530\) −20.0000 −0.868744
\(531\) 4.00000i 0.173585i
\(532\) 32.0000 1.38738
\(533\) 0 0
\(534\) 14.0000 0.605839
\(535\) − 24.0000i − 1.03761i
\(536\) 16.0000 0.691095
\(537\) −12.0000 −0.517838
\(538\) − 26.0000i − 1.12094i
\(539\) 36.0000i 1.55063i
\(540\) − 2.00000i − 0.0860663i
\(541\) − 34.0000i − 1.46177i −0.682498 0.730887i \(-0.739107\pi\)
0.682498 0.730887i \(-0.260893\pi\)
\(542\) 4.00000 0.171815
\(543\) −10.0000 −0.429141
\(544\) − 2.00000i − 0.0857493i
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 10.0000i 0.427179i
\(549\) −2.00000 −0.0853579
\(550\) 4.00000 0.170561
\(551\) 48.0000i 2.04487i
\(552\) 0 0
\(553\) 32.0000i 1.36078i
\(554\) − 22.0000i − 0.934690i
\(555\) 4.00000 0.169791
\(556\) −12.0000 −0.508913
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) −4.00000 −0.169334
\(559\) 0 0
\(560\) 8.00000 0.338062
\(561\) − 8.00000i − 0.337760i
\(562\) 26.0000 1.09674
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 8.00000i 0.336861i
\(565\) 12.0000i 0.504844i
\(566\) 4.00000i 0.168133i
\(567\) 4.00000i 0.167984i
\(568\) 8.00000 0.335673
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) − 16.0000i − 0.670166i
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 8.00000 0.334205
\(574\) − 40.0000i − 1.66957i
\(575\) 0 0
\(576\) −1.00000 −0.0416667
\(577\) − 18.0000i − 0.749350i −0.927156 0.374675i \(-0.877754\pi\)
0.927156 0.374675i \(-0.122246\pi\)
\(578\) − 13.0000i − 0.540729i
\(579\) 14.0000i 0.581820i
\(580\) 12.0000i 0.498273i
\(581\) 48.0000 1.99138
\(582\) −10.0000 −0.414513
\(583\) 40.0000i 1.65663i
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) − 4.00000i − 0.165098i −0.996587 0.0825488i \(-0.973694\pi\)
0.996587 0.0825488i \(-0.0263060\pi\)
\(588\) −9.00000 −0.371154
\(589\) −32.0000 −1.31854
\(590\) 8.00000i 0.329355i
\(591\) 18.0000i 0.740421i
\(592\) − 2.00000i − 0.0821995i
\(593\) − 42.0000i − 1.72473i −0.506284 0.862367i \(-0.668981\pi\)
0.506284 0.862367i \(-0.331019\pi\)
\(594\) −4.00000 −0.164122
\(595\) −16.0000 −0.655936
\(596\) − 6.00000i − 0.245770i
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 16.0000 0.652111
\(603\) 16.0000i 0.651570i
\(604\) − 12.0000i − 0.488273i
\(605\) 10.0000i 0.406558i
\(606\) − 2.00000i − 0.0812444i
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) −8.00000 −0.324443
\(609\) − 24.0000i − 0.972529i
\(610\) −4.00000 −0.161955
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) 2.00000i 0.0807792i 0.999184 + 0.0403896i \(0.0128599\pi\)
−0.999184 + 0.0403896i \(0.987140\pi\)
\(614\) 8.00000 0.322854
\(615\) −20.0000 −0.806478
\(616\) − 16.0000i − 0.644658i
\(617\) − 6.00000i − 0.241551i −0.992680 0.120775i \(-0.961462\pi\)
0.992680 0.120775i \(-0.0385381\pi\)
\(618\) 16.0000i 0.643614i
\(619\) 32.0000i 1.28619i 0.765787 + 0.643094i \(0.222350\pi\)
−0.765787 + 0.643094i \(0.777650\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) 0 0
\(623\) −56.0000 −2.24359
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) − 6.00000i − 0.239808i
\(627\) −32.0000 −1.27796
\(628\) −14.0000 −0.558661
\(629\) 4.00000i 0.159490i
\(630\) 8.00000i 0.318728i
\(631\) − 36.0000i − 1.43314i −0.697517 0.716569i \(-0.745712\pi\)
0.697517 0.716569i \(-0.254288\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) −12.0000 −0.476957
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) −10.0000 −0.396526
\(637\) 0 0
\(638\) 24.0000 0.950169
\(639\) 8.00000i 0.316475i
\(640\) −2.00000 −0.0790569
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) − 12.0000i − 0.473602i
\(643\) 16.0000i 0.630978i 0.948929 + 0.315489i \(0.102169\pi\)
−0.948929 + 0.315489i \(0.897831\pi\)
\(644\) 0 0
\(645\) − 8.00000i − 0.315000i
\(646\) 16.0000 0.629512
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 16.0000i 0.626608i
\(653\) −10.0000 −0.391330 −0.195665 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 2.00000 0.0782062
\(655\) − 8.00000i − 0.312586i
\(656\) 10.0000i 0.390434i
\(657\) 2.00000i 0.0780274i
\(658\) − 32.0000i − 1.24749i
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) −8.00000 −0.311400
\(661\) − 2.00000i − 0.0777910i −0.999243 0.0388955i \(-0.987616\pi\)
0.999243 0.0388955i \(-0.0123839\pi\)
\(662\) 8.00000 0.310929
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 64.0000i 2.48181i
\(666\) 2.00000 0.0774984
\(667\) 0 0
\(668\) 0 0
\(669\) − 4.00000i − 0.154649i
\(670\) 32.0000i 1.23627i
\(671\) 8.00000i 0.308837i
\(672\) 4.00000 0.154303
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) − 18.0000i − 0.693334i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 6.00000i 0.230429i
\(679\) 40.0000 1.53506
\(680\) 4.00000 0.153393
\(681\) 20.0000i 0.766402i
\(682\) 16.0000i 0.612672i
\(683\) 44.0000i 1.68361i 0.539779 + 0.841807i \(0.318508\pi\)
−0.539779 + 0.841807i \(0.681492\pi\)
\(684\) − 8.00000i − 0.305888i
\(685\) −20.0000 −0.764161
\(686\) 8.00000 0.305441
\(687\) − 22.0000i − 0.839352i
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) 32.0000i 1.21734i 0.793424 + 0.608669i \(0.208296\pi\)
−0.793424 + 0.608669i \(0.791704\pi\)
\(692\) −10.0000 −0.380143
\(693\) 16.0000 0.607790
\(694\) − 12.0000i − 0.455514i
\(695\) − 24.0000i − 0.910372i
\(696\) 6.00000i 0.227429i
\(697\) − 20.0000i − 0.757554i
\(698\) −6.00000 −0.227103
\(699\) 18.0000 0.680823
\(700\) − 4.00000i − 0.151186i
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 0 0
\(703\) 16.0000 0.603451
\(704\) 4.00000i 0.150756i
\(705\) −16.0000 −0.602595
\(706\) 14.0000 0.526897
\(707\) 8.00000i 0.300871i
\(708\) 4.00000i 0.150329i
\(709\) 6.00000i 0.225335i 0.993633 + 0.112667i \(0.0359394\pi\)
−0.993633 + 0.112667i \(0.964061\pi\)
\(710\) 16.0000i 0.600469i
\(711\) 8.00000 0.300023
\(712\) 14.0000 0.524672
\(713\) 0 0
\(714\) −8.00000 −0.299392
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) − 2.00000i − 0.0745356i
\(721\) − 64.0000i − 2.38348i
\(722\) − 45.0000i − 1.67473i
\(723\) − 10.0000i − 0.371904i
\(724\) −10.0000 −0.371647
\(725\) 6.00000 0.222834
\(726\) 5.00000i 0.185567i
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 4.00000i 0.148047i
\(731\) 8.00000 0.295891
\(732\) −2.00000 −0.0739221
\(733\) 2.00000i 0.0738717i 0.999318 + 0.0369358i \(0.0117597\pi\)
−0.999318 + 0.0369358i \(0.988240\pi\)
\(734\) 16.0000i 0.590571i
\(735\) − 18.0000i − 0.663940i
\(736\) 0 0
\(737\) 64.0000 2.35747
\(738\) −10.0000 −0.368105
\(739\) 40.0000i 1.47142i 0.677295 + 0.735712i \(0.263152\pi\)
−0.677295 + 0.735712i \(0.736848\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) 40.0000 1.46845
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) −4.00000 −0.146647
\(745\) 12.0000 0.439646
\(746\) 6.00000i 0.219676i
\(747\) − 12.0000i − 0.439057i
\(748\) − 8.00000i − 0.292509i
\(749\) 48.0000i 1.75388i
\(750\) −12.0000 −0.438178
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 8.00000i 0.291730i
\(753\) 4.00000 0.145768
\(754\) 0 0
\(755\) 24.0000 0.873449
\(756\) 4.00000i 0.145479i
\(757\) 54.0000 1.96266 0.981332 0.192323i \(-0.0616021\pi\)
0.981332 + 0.192323i \(0.0616021\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) − 16.0000i − 0.580381i
\(761\) − 26.0000i − 0.942499i −0.882000 0.471250i \(-0.843803\pi\)
0.882000 0.471250i \(-0.156197\pi\)
\(762\) 0 0
\(763\) −8.00000 −0.289619
\(764\) 8.00000 0.289430
\(765\) 4.00000i 0.144620i
\(766\) −24.0000 −0.867155
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) − 2.00000i − 0.0721218i −0.999350 0.0360609i \(-0.988519\pi\)
0.999350 0.0360609i \(-0.0114810\pi\)
\(770\) 32.0000 1.15320
\(771\) −6.00000 −0.216085
\(772\) 14.0000i 0.503871i
\(773\) 54.0000i 1.94225i 0.238581 + 0.971123i \(0.423318\pi\)
−0.238581 + 0.971123i \(0.576682\pi\)
\(774\) − 4.00000i − 0.143777i
\(775\) 4.00000i 0.143684i
\(776\) −10.0000 −0.358979
\(777\) −8.00000 −0.286998
\(778\) 26.0000i 0.932145i
\(779\) −80.0000 −2.86630
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) −9.00000 −0.321429
\(785\) − 28.0000i − 0.999363i
\(786\) − 4.00000i − 0.142675i
\(787\) 40.0000i 1.42585i 0.701242 + 0.712923i \(0.252629\pi\)
−0.701242 + 0.712923i \(0.747371\pi\)
\(788\) 18.0000i 0.641223i
\(789\) −8.00000 −0.284808
\(790\) 16.0000 0.569254
\(791\) − 24.0000i − 0.853342i
\(792\) −4.00000 −0.142134
\(793\) 0 0
\(794\) −6.00000 −0.212932
\(795\) − 20.0000i − 0.709327i
\(796\) −8.00000 −0.283552
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 32.0000i 1.13279i
\(799\) − 16.0000i − 0.566039i
\(800\) 1.00000i 0.0353553i
\(801\) 14.0000i 0.494666i
\(802\) −6.00000 −0.211867
\(803\) 8.00000 0.282314
\(804\) 16.0000i 0.564276i
\(805\) 0 0
\(806\) 0 0
\(807\) 26.0000 0.915243
\(808\) − 2.00000i − 0.0703598i
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 2.00000 0.0702728
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) − 24.0000i − 0.842235i
\(813\) 4.00000i 0.140286i
\(814\) − 8.00000i − 0.280400i
\(815\) −32.0000 −1.12091
\(816\) 2.00000 0.0700140
\(817\) − 32.0000i − 1.11954i
\(818\) 2.00000 0.0699284
\(819\) 0 0
\(820\) −20.0000 −0.698430
\(821\) − 42.0000i − 1.46581i −0.680331 0.732905i \(-0.738164\pi\)
0.680331 0.732905i \(-0.261836\pi\)
\(822\) −10.0000 −0.348790
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 16.0000i 0.557386i
\(825\) 4.00000i 0.139262i
\(826\) − 16.0000i − 0.556711i
\(827\) 28.0000i 0.973655i 0.873498 + 0.486828i \(0.161846\pi\)
−0.873498 + 0.486828i \(0.838154\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) − 24.0000i − 0.833052i
\(831\) 22.0000 0.763172
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) − 12.0000i − 0.415526i
\(835\) 0 0
\(836\) −32.0000 −1.10674
\(837\) − 4.00000i − 0.138260i
\(838\) 4.00000i 0.138178i
\(839\) − 40.0000i − 1.38095i −0.723355 0.690477i \(-0.757401\pi\)
0.723355 0.690477i \(-0.242599\pi\)
\(840\) 8.00000i 0.276026i
\(841\) 7.00000 0.241379
\(842\) 22.0000 0.758170
\(843\) 26.0000i 0.895488i
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) − 20.0000i − 0.687208i
\(848\) −10.0000 −0.343401
\(849\) −4.00000 −0.137280
\(850\) − 2.00000i − 0.0685994i
\(851\) 0 0
\(852\) 8.00000i 0.274075i
\(853\) − 2.00000i − 0.0684787i −0.999414 0.0342393i \(-0.989099\pi\)
0.999414 0.0342393i \(-0.0109009\pi\)
\(854\) 8.00000 0.273754
\(855\) 16.0000 0.547188
\(856\) − 12.0000i − 0.410152i
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) − 8.00000i − 0.272798i
\(861\) 40.0000 1.36320
\(862\) −8.00000 −0.272481
\(863\) − 40.0000i − 1.36162i −0.732462 0.680808i \(-0.761629\pi\)
0.732462 0.680808i \(-0.238371\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) − 20.0000i − 0.680020i
\(866\) 30.0000i 1.01944i
\(867\) 13.0000 0.441503
\(868\) 16.0000 0.543075
\(869\) − 32.0000i − 1.08553i
\(870\) −12.0000 −0.406838
\(871\) 0 0
\(872\) 2.00000 0.0677285
\(873\) − 10.0000i − 0.338449i
\(874\) 0 0
\(875\) 48.0000 1.62270
\(876\) 2.00000i 0.0675737i
\(877\) − 22.0000i − 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) − 16.0000i − 0.539974i
\(879\) − 26.0000i − 0.876958i
\(880\) −8.00000 −0.269680
\(881\) −26.0000 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) − 9.00000i − 0.303046i
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) −8.00000 −0.268917
\(886\) − 4.00000i − 0.134383i
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) 28.0000i 0.938562i
\(891\) − 4.00000i − 0.134005i
\(892\) − 4.00000i − 0.133930i
\(893\) −64.0000 −2.14168
\(894\) 6.00000 0.200670
\(895\) − 24.0000i − 0.802232i
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) 24.0000i 0.800445i
\(900\) −1.00000 −0.0333333
\(901\) 20.0000 0.666297
\(902\) 40.0000i 1.33185i
\(903\) 16.0000i 0.532447i
\(904\) 6.00000i 0.199557i
\(905\) − 20.0000i − 0.664822i
\(906\) 12.0000 0.398673
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 20.0000i 0.663723i
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) − 8.00000i − 0.264906i
\(913\) −48.0000 −1.58857
\(914\) −30.0000 −0.992312
\(915\) − 4.00000i − 0.132236i
\(916\) − 22.0000i − 0.726900i
\(917\) 16.0000i 0.528367i
\(918\) 2.00000i 0.0660098i
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 8.00000i 0.263609i
\(922\) −6.00000 −0.197599
\(923\) 0 0
\(924\) 16.0000 0.526361
\(925\) − 2.00000i − 0.0657596i
\(926\) 20.0000 0.657241
\(927\) −16.0000 −0.525509
\(928\) 6.00000i 0.196960i
\(929\) − 46.0000i − 1.50921i −0.656179 0.754606i \(-0.727828\pi\)
0.656179 0.754606i \(-0.272172\pi\)
\(930\) − 8.00000i − 0.262330i
\(931\) − 72.0000i − 2.35970i
\(932\) 18.0000 0.589610
\(933\) 0 0
\(934\) 4.00000i 0.130884i
\(935\) 16.0000 0.523256
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) − 64.0000i − 2.08967i
\(939\) 6.00000 0.195803
\(940\) −16.0000 −0.521862
\(941\) 46.0000i 1.49956i 0.661689 + 0.749779i \(0.269840\pi\)
−0.661689 + 0.749779i \(0.730160\pi\)
\(942\) − 14.0000i − 0.456145i
\(943\) 0 0
\(944\) 4.00000i 0.130189i
\(945\) −8.00000 −0.260240
\(946\) −16.0000 −0.520205
\(947\) − 4.00000i − 0.129983i −0.997886 0.0649913i \(-0.979298\pi\)
0.997886 0.0649913i \(-0.0207020\pi\)
\(948\) 8.00000 0.259828
\(949\) 0 0
\(950\) −8.00000 −0.259554
\(951\) − 6.00000i − 0.194563i
\(952\) −8.00000 −0.259281
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) − 10.0000i − 0.323762i
\(955\) 16.0000i 0.517748i
\(956\) 0 0
\(957\) 24.0000i 0.775810i
\(958\) 16.0000 0.516937
\(959\) 40.0000 1.29167
\(960\) − 2.00000i − 0.0645497i
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) − 10.0000i − 0.322078i
\(965\) −28.0000 −0.901352
\(966\) 0 0
\(967\) − 4.00000i − 0.128631i −0.997930 0.0643157i \(-0.979514\pi\)
0.997930 0.0643157i \(-0.0204865\pi\)
\(968\) 5.00000i 0.160706i
\(969\) 16.0000i 0.513994i
\(970\) − 20.0000i − 0.642161i
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 1.00000 0.0320750
\(973\) 48.0000i 1.53881i
\(974\) 4.00000 0.128168
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) − 6.00000i − 0.191957i −0.995383 0.0959785i \(-0.969402\pi\)
0.995383 0.0959785i \(-0.0305980\pi\)
\(978\) −16.0000 −0.511624
\(979\) 56.0000 1.78977
\(980\) − 18.0000i − 0.574989i
\(981\) 2.00000i 0.0638551i
\(982\) − 36.0000i − 1.14881i
\(983\) 24.0000i 0.765481i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) −10.0000 −0.318788
\(985\) −36.0000 −1.14706
\(986\) − 12.0000i − 0.382158i
\(987\) 32.0000 1.01857
\(988\) 0 0
\(989\) 0 0
\(990\) − 8.00000i − 0.254257i
\(991\) −48.0000 −1.52477 −0.762385 0.647124i \(-0.775972\pi\)
−0.762385 + 0.647124i \(0.775972\pi\)
\(992\) −4.00000 −0.127000
\(993\) 8.00000i 0.253872i
\(994\) − 32.0000i − 1.01498i
\(995\) − 16.0000i − 0.507234i
\(996\) − 12.0000i − 0.380235i
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 0 0
\(999\) 2.00000i 0.0632772i
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 1014.2.b.b.337.2 2
3.2 odd 2 3042.2.b.g.1351.1 2
13.2 odd 12 1014.2.e.c.529.1 2
13.3 even 3 1014.2.i.d.823.1 4
13.4 even 6 1014.2.i.d.361.1 4
13.5 odd 4 1014.2.a.d.1.1 1
13.6 odd 12 1014.2.e.c.991.1 2
13.7 odd 12 1014.2.e.f.991.1 2
13.8 odd 4 78.2.a.a.1.1 1
13.9 even 3 1014.2.i.d.361.2 4
13.10 even 6 1014.2.i.d.823.2 4
13.11 odd 12 1014.2.e.f.529.1 2
13.12 even 2 inner 1014.2.b.b.337.1 2
39.5 even 4 3042.2.a.f.1.1 1
39.8 even 4 234.2.a.c.1.1 1
39.38 odd 2 3042.2.b.g.1351.2 2
52.31 even 4 8112.2.a.v.1.1 1
52.47 even 4 624.2.a.h.1.1 1
65.8 even 4 1950.2.e.i.1249.2 2
65.34 odd 4 1950.2.a.w.1.1 1
65.47 even 4 1950.2.e.i.1249.1 2
91.34 even 4 3822.2.a.j.1.1 1
104.21 odd 4 2496.2.a.t.1.1 1
104.99 even 4 2496.2.a.b.1.1 1
117.34 odd 12 2106.2.e.q.1405.1 2
117.47 even 12 2106.2.e.j.1405.1 2
117.86 even 12 2106.2.e.j.703.1 2
117.112 odd 12 2106.2.e.q.703.1 2
143.21 even 4 9438.2.a.t.1.1 1
156.47 odd 4 1872.2.a.c.1.1 1
195.8 odd 4 5850.2.e.bb.5149.1 2
195.47 odd 4 5850.2.e.bb.5149.2 2
195.164 even 4 5850.2.a.d.1.1 1
312.125 even 4 7488.2.a.bz.1.1 1
312.203 odd 4 7488.2.a.bk.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.a.a.1.1 1 13.8 odd 4
234.2.a.c.1.1 1 39.8 even 4
624.2.a.h.1.1 1 52.47 even 4
1014.2.a.d.1.1 1 13.5 odd 4
1014.2.b.b.337.1 2 13.12 even 2 inner
1014.2.b.b.337.2 2 1.1 even 1 trivial
1014.2.e.c.529.1 2 13.2 odd 12
1014.2.e.c.991.1 2 13.6 odd 12
1014.2.e.f.529.1 2 13.11 odd 12
1014.2.e.f.991.1 2 13.7 odd 12
1014.2.i.d.361.1 4 13.4 even 6
1014.2.i.d.361.2 4 13.9 even 3
1014.2.i.d.823.1 4 13.3 even 3
1014.2.i.d.823.2 4 13.10 even 6
1872.2.a.c.1.1 1 156.47 odd 4
1950.2.a.w.1.1 1 65.34 odd 4
1950.2.e.i.1249.1 2 65.47 even 4
1950.2.e.i.1249.2 2 65.8 even 4
2106.2.e.j.703.1 2 117.86 even 12
2106.2.e.j.1405.1 2 117.47 even 12
2106.2.e.q.703.1 2 117.112 odd 12
2106.2.e.q.1405.1 2 117.34 odd 12
2496.2.a.b.1.1 1 104.99 even 4
2496.2.a.t.1.1 1 104.21 odd 4
3042.2.a.f.1.1 1 39.5 even 4
3042.2.b.g.1351.1 2 3.2 odd 2
3042.2.b.g.1351.2 2 39.38 odd 2
3822.2.a.j.1.1 1 91.34 even 4
5850.2.a.d.1.1 1 195.164 even 4
5850.2.e.bb.5149.1 2 195.8 odd 4
5850.2.e.bb.5149.2 2 195.47 odd 4
7488.2.a.bk.1.1 1 312.203 odd 4
7488.2.a.bz.1.1 1 312.125 even 4
8112.2.a.v.1.1 1 52.31 even 4
9438.2.a.t.1.1 1 143.21 even 4