Properties

Label 169.2.b.a.168.1
Level $169$
Weight $2$
Character 169.168
Analytic conductor $1.349$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,2,Mod(168,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.168");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.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: \(1.34947179416\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 168.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 169.168
Dual form 169.2.b.a.168.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205i q^{2} +2.00000 q^{3} -1.00000 q^{4} +1.73205i q^{5} -3.46410i q^{6} -1.73205i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{2} +2.00000 q^{3} -1.00000 q^{4} +1.73205i q^{5} -3.46410i q^{6} -1.73205i q^{8} +1.00000 q^{9} +3.00000 q^{10} -2.00000 q^{12} +3.46410i q^{15} -5.00000 q^{16} -3.00000 q^{17} -1.73205i q^{18} +3.46410i q^{19} -1.73205i q^{20} -6.00000 q^{23} -3.46410i q^{24} +2.00000 q^{25} -4.00000 q^{27} +3.00000 q^{29} +6.00000 q^{30} -3.46410i q^{31} +5.19615i q^{32} +5.19615i q^{34} -1.00000 q^{36} +8.66025i q^{37} +6.00000 q^{38} +3.00000 q^{40} -5.19615i q^{41} +8.00000 q^{43} +1.73205i q^{45} +10.3923i q^{46} +3.46410i q^{47} -10.0000 q^{48} +7.00000 q^{49} -3.46410i q^{50} -6.00000 q^{51} -3.00000 q^{53} +6.92820i q^{54} +6.92820i q^{57} -5.19615i q^{58} -6.92820i q^{59} -3.46410i q^{60} +1.00000 q^{61} -6.00000 q^{62} -1.00000 q^{64} +3.46410i q^{67} +3.00000 q^{68} -12.0000 q^{69} -3.46410i q^{71} -1.73205i q^{72} -1.73205i q^{73} +15.0000 q^{74} +4.00000 q^{75} -3.46410i q^{76} +4.00000 q^{79} -8.66025i q^{80} -11.0000 q^{81} -9.00000 q^{82} -13.8564i q^{83} -5.19615i q^{85} -13.8564i q^{86} +6.00000 q^{87} -6.92820i q^{89} +3.00000 q^{90} +6.00000 q^{92} -6.92820i q^{93} +6.00000 q^{94} -6.00000 q^{95} +10.3923i q^{96} -6.92820i q^{97} -12.1244i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{3} - 2 q^{4} + 2 q^{9} + 6 q^{10} - 4 q^{12} - 10 q^{16} - 6 q^{17} - 12 q^{23} + 4 q^{25} - 8 q^{27} + 6 q^{29} + 12 q^{30} - 2 q^{36} + 12 q^{38} + 6 q^{40} + 16 q^{43} - 20 q^{48} + 14 q^{49} - 12 q^{51} - 6 q^{53} + 2 q^{61} - 12 q^{62} - 2 q^{64} + 6 q^{68} - 24 q^{69} + 30 q^{74} + 8 q^{75} + 8 q^{79} - 22 q^{81} - 18 q^{82} + 12 q^{87} + 6 q^{90} + 12 q^{92} + 12 q^{94} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.73205i − 1.22474i −0.790569 0.612372i \(-0.790215\pi\)
0.790569 0.612372i \(-0.209785\pi\)
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) −1.00000 −0.500000
\(5\) 1.73205i 0.774597i 0.921954 + 0.387298i \(0.126592\pi\)
−0.921954 + 0.387298i \(0.873408\pi\)
\(6\) − 3.46410i − 1.41421i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) − 1.73205i − 0.612372i
\(9\) 1.00000 0.333333
\(10\) 3.00000 0.948683
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −2.00000 −0.577350
\(13\) 0 0
\(14\) 0 0
\(15\) 3.46410i 0.894427i
\(16\) −5.00000 −1.25000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) − 1.73205i − 0.408248i
\(19\) 3.46410i 0.794719i 0.917663 + 0.397360i \(0.130073\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) − 1.73205i − 0.387298i
\(21\) 0 0
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) − 3.46410i − 0.707107i
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 6.00000 1.09545
\(31\) − 3.46410i − 0.622171i −0.950382 0.311086i \(-0.899307\pi\)
0.950382 0.311086i \(-0.100693\pi\)
\(32\) 5.19615i 0.918559i
\(33\) 0 0
\(34\) 5.19615i 0.891133i
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 8.66025i 1.42374i 0.702313 + 0.711868i \(0.252151\pi\)
−0.702313 + 0.711868i \(0.747849\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) − 5.19615i − 0.811503i −0.913984 0.405751i \(-0.867010\pi\)
0.913984 0.405751i \(-0.132990\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 1.73205i 0.258199i
\(46\) 10.3923i 1.53226i
\(47\) 3.46410i 0.505291i 0.967559 + 0.252646i \(0.0813007\pi\)
−0.967559 + 0.252646i \(0.918699\pi\)
\(48\) −10.0000 −1.44338
\(49\) 7.00000 1.00000
\(50\) − 3.46410i − 0.489898i
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 6.92820i 0.942809i
\(55\) 0 0
\(56\) 0 0
\(57\) 6.92820i 0.917663i
\(58\) − 5.19615i − 0.682288i
\(59\) − 6.92820i − 0.901975i −0.892530 0.450988i \(-0.851072\pi\)
0.892530 0.450988i \(-0.148928\pi\)
\(60\) − 3.46410i − 0.447214i
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) −6.00000 −0.762001
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 3.46410i 0.423207i 0.977356 + 0.211604i \(0.0678686\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 3.00000 0.363803
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) − 3.46410i − 0.411113i −0.978645 0.205557i \(-0.934100\pi\)
0.978645 0.205557i \(-0.0659005\pi\)
\(72\) − 1.73205i − 0.204124i
\(73\) − 1.73205i − 0.202721i −0.994850 0.101361i \(-0.967680\pi\)
0.994850 0.101361i \(-0.0323196\pi\)
\(74\) 15.0000 1.74371
\(75\) 4.00000 0.461880
\(76\) − 3.46410i − 0.397360i
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) − 8.66025i − 0.968246i
\(81\) −11.0000 −1.22222
\(82\) −9.00000 −0.993884
\(83\) − 13.8564i − 1.52094i −0.649374 0.760469i \(-0.724969\pi\)
0.649374 0.760469i \(-0.275031\pi\)
\(84\) 0 0
\(85\) − 5.19615i − 0.563602i
\(86\) − 13.8564i − 1.49417i
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) − 6.92820i − 0.734388i −0.930144 0.367194i \(-0.880318\pi\)
0.930144 0.367194i \(-0.119682\pi\)
\(90\) 3.00000 0.316228
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) − 6.92820i − 0.718421i
\(94\) 6.00000 0.618853
\(95\) −6.00000 −0.615587
\(96\) 10.3923i 1.06066i
\(97\) − 6.92820i − 0.703452i −0.936103 0.351726i \(-0.885595\pi\)
0.936103 0.351726i \(-0.114405\pi\)
\(98\) − 12.1244i − 1.22474i
\(99\) 0 0
\(100\) −2.00000 −0.200000
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 10.3923i 1.02899i
\(103\) −10.0000 −0.985329 −0.492665 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 5.19615i 0.504695i
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 4.00000 0.384900
\(109\) 13.8564i 1.32720i 0.748086 + 0.663602i \(0.230973\pi\)
−0.748086 + 0.663602i \(0.769027\pi\)
\(110\) 0 0
\(111\) 17.3205i 1.64399i
\(112\) 0 0
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 12.0000 1.12390
\(115\) − 10.3923i − 0.969087i
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) 6.00000 0.547723
\(121\) 11.0000 1.00000
\(122\) − 1.73205i − 0.156813i
\(123\) − 10.3923i − 0.937043i
\(124\) 3.46410i 0.311086i
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 12.1244i 1.07165i
\(129\) 16.0000 1.40872
\(130\) 0 0
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 6.00000 0.518321
\(135\) − 6.92820i − 0.596285i
\(136\) 5.19615i 0.445566i
\(137\) 15.5885i 1.33181i 0.746036 + 0.665906i \(0.231955\pi\)
−0.746036 + 0.665906i \(0.768045\pi\)
\(138\) 20.7846i 1.76930i
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 6.92820i 0.583460i
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) −5.00000 −0.416667
\(145\) 5.19615i 0.431517i
\(146\) −3.00000 −0.248282
\(147\) 14.0000 1.15470
\(148\) − 8.66025i − 0.711868i
\(149\) 19.0526i 1.56085i 0.625252 + 0.780423i \(0.284996\pi\)
−0.625252 + 0.780423i \(0.715004\pi\)
\(150\) − 6.92820i − 0.565685i
\(151\) − 17.3205i − 1.40952i −0.709444 0.704761i \(-0.751054\pi\)
0.709444 0.704761i \(-0.248946\pi\)
\(152\) 6.00000 0.486664
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) − 6.92820i − 0.551178i
\(159\) −6.00000 −0.475831
\(160\) −9.00000 −0.711512
\(161\) 0 0
\(162\) 19.0526i 1.49691i
\(163\) − 20.7846i − 1.62798i −0.580881 0.813988i \(-0.697292\pi\)
0.580881 0.813988i \(-0.302708\pi\)
\(164\) 5.19615i 0.405751i
\(165\) 0 0
\(166\) −24.0000 −1.86276
\(167\) − 13.8564i − 1.07224i −0.844141 0.536120i \(-0.819889\pi\)
0.844141 0.536120i \(-0.180111\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −9.00000 −0.690268
\(171\) 3.46410i 0.264906i
\(172\) −8.00000 −0.609994
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) − 10.3923i − 0.787839i
\(175\) 0 0
\(176\) 0 0
\(177\) − 13.8564i − 1.04151i
\(178\) −12.0000 −0.899438
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) − 1.73205i − 0.129099i
\(181\) 11.0000 0.817624 0.408812 0.912619i \(-0.365943\pi\)
0.408812 + 0.912619i \(0.365943\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 10.3923i 0.766131i
\(185\) −15.0000 −1.10282
\(186\) −12.0000 −0.879883
\(187\) 0 0
\(188\) − 3.46410i − 0.252646i
\(189\) 0 0
\(190\) 10.3923i 0.753937i
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) −2.00000 −0.144338
\(193\) − 5.19615i − 0.374027i −0.982357 0.187014i \(-0.940119\pi\)
0.982357 0.187014i \(-0.0598809\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 13.8564i 0.987228i 0.869681 + 0.493614i \(0.164324\pi\)
−0.869681 + 0.493614i \(0.835676\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) − 3.46410i − 0.244949i
\(201\) 6.92820i 0.488678i
\(202\) 5.19615i 0.365600i
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 9.00000 0.628587
\(206\) 17.3205i 1.20678i
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) 3.00000 0.206041
\(213\) − 6.92820i − 0.474713i
\(214\) − 10.3923i − 0.710403i
\(215\) 13.8564i 0.944999i
\(216\) 6.92820i 0.471405i
\(217\) 0 0
\(218\) 24.0000 1.62549
\(219\) − 3.46410i − 0.234082i
\(220\) 0 0
\(221\) 0 0
\(222\) 30.0000 2.01347
\(223\) − 10.3923i − 0.695920i −0.937509 0.347960i \(-0.886874\pi\)
0.937509 0.347960i \(-0.113126\pi\)
\(224\) 0 0
\(225\) 2.00000 0.133333
\(226\) 25.9808i 1.72821i
\(227\) − 24.2487i − 1.60944i −0.593652 0.804722i \(-0.702314\pi\)
0.593652 0.804722i \(-0.297686\pi\)
\(228\) − 6.92820i − 0.458831i
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) −18.0000 −1.18688
\(231\) 0 0
\(232\) − 5.19615i − 0.341144i
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 6.92820i 0.450988i
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 20.7846i 1.34444i 0.740349 + 0.672222i \(0.234660\pi\)
−0.740349 + 0.672222i \(0.765340\pi\)
\(240\) − 17.3205i − 1.11803i
\(241\) 1.73205i 0.111571i 0.998443 + 0.0557856i \(0.0177663\pi\)
−0.998443 + 0.0557856i \(0.982234\pi\)
\(242\) − 19.0526i − 1.22474i
\(243\) −10.0000 −0.641500
\(244\) −1.00000 −0.0640184
\(245\) 12.1244i 0.774597i
\(246\) −18.0000 −1.14764
\(247\) 0 0
\(248\) −6.00000 −0.381000
\(249\) − 27.7128i − 1.75623i
\(250\) 21.0000 1.32816
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 3.46410i 0.217357i
\(255\) − 10.3923i − 0.650791i
\(256\) 19.0000 1.18750
\(257\) 3.00000 0.187135 0.0935674 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) − 27.7128i − 1.72532i
\(259\) 0 0
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) − 31.1769i − 1.92612i
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) − 5.19615i − 0.319197i
\(266\) 0 0
\(267\) − 13.8564i − 0.847998i
\(268\) − 3.46410i − 0.211604i
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) −12.0000 −0.730297
\(271\) 20.7846i 1.26258i 0.775549 + 0.631288i \(0.217473\pi\)
−0.775549 + 0.631288i \(0.782527\pi\)
\(272\) 15.0000 0.909509
\(273\) 0 0
\(274\) 27.0000 1.63113
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) −7.00000 −0.420589 −0.210295 0.977638i \(-0.567442\pi\)
−0.210295 + 0.977638i \(0.567442\pi\)
\(278\) 6.92820i 0.415526i
\(279\) − 3.46410i − 0.207390i
\(280\) 0 0
\(281\) 22.5167i 1.34323i 0.740900 + 0.671616i \(0.234399\pi\)
−0.740900 + 0.671616i \(0.765601\pi\)
\(282\) 12.0000 0.714590
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 3.46410i 0.205557i
\(285\) −12.0000 −0.710819
\(286\) 0 0
\(287\) 0 0
\(288\) 5.19615i 0.306186i
\(289\) −8.00000 −0.470588
\(290\) 9.00000 0.528498
\(291\) − 13.8564i − 0.812277i
\(292\) 1.73205i 0.101361i
\(293\) − 5.19615i − 0.303562i −0.988414 0.151781i \(-0.951499\pi\)
0.988414 0.151781i \(-0.0485009\pi\)
\(294\) − 24.2487i − 1.41421i
\(295\) 12.0000 0.698667
\(296\) 15.0000 0.871857
\(297\) 0 0
\(298\) 33.0000 1.91164
\(299\) 0 0
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) −30.0000 −1.72631
\(303\) −6.00000 −0.344691
\(304\) − 17.3205i − 0.993399i
\(305\) 1.73205i 0.0991769i
\(306\) 5.19615i 0.297044i
\(307\) 17.3205i 0.988534i 0.869310 + 0.494267i \(0.164563\pi\)
−0.869310 + 0.494267i \(0.835437\pi\)
\(308\) 0 0
\(309\) −20.0000 −1.13776
\(310\) − 10.3923i − 0.590243i
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 22.5167i 1.27069i
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 5.19615i 0.291845i 0.989296 + 0.145922i \(0.0466150\pi\)
−0.989296 + 0.145922i \(0.953385\pi\)
\(318\) 10.3923i 0.582772i
\(319\) 0 0
\(320\) − 1.73205i − 0.0968246i
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) − 10.3923i − 0.578243i
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) −36.0000 −1.99386
\(327\) 27.7128i 1.53252i
\(328\) −9.00000 −0.496942
\(329\) 0 0
\(330\) 0 0
\(331\) 27.7128i 1.52323i 0.648027 + 0.761617i \(0.275594\pi\)
−0.648027 + 0.761617i \(0.724406\pi\)
\(332\) 13.8564i 0.760469i
\(333\) 8.66025i 0.474579i
\(334\) −24.0000 −1.31322
\(335\) −6.00000 −0.327815
\(336\) 0 0
\(337\) −23.0000 −1.25289 −0.626445 0.779466i \(-0.715491\pi\)
−0.626445 + 0.779466i \(0.715491\pi\)
\(338\) 0 0
\(339\) −30.0000 −1.62938
\(340\) 5.19615i 0.281801i
\(341\) 0 0
\(342\) 6.00000 0.324443
\(343\) 0 0
\(344\) − 13.8564i − 0.747087i
\(345\) − 20.7846i − 1.11901i
\(346\) − 10.3923i − 0.558694i
\(347\) −30.0000 −1.61048 −0.805242 0.592946i \(-0.797965\pi\)
−0.805242 + 0.592946i \(0.797965\pi\)
\(348\) −6.00000 −0.321634
\(349\) 13.8564i 0.741716i 0.928689 + 0.370858i \(0.120936\pi\)
−0.928689 + 0.370858i \(0.879064\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 32.9090i 1.75157i 0.482704 + 0.875784i \(0.339655\pi\)
−0.482704 + 0.875784i \(0.660345\pi\)
\(354\) −24.0000 −1.27559
\(355\) 6.00000 0.318447
\(356\) 6.92820i 0.367194i
\(357\) 0 0
\(358\) 0 0
\(359\) 6.92820i 0.365657i 0.983145 + 0.182828i \(0.0585252\pi\)
−0.983145 + 0.182828i \(0.941475\pi\)
\(360\) 3.00000 0.158114
\(361\) 7.00000 0.368421
\(362\) − 19.0526i − 1.00138i
\(363\) 22.0000 1.15470
\(364\) 0 0
\(365\) 3.00000 0.157027
\(366\) − 3.46410i − 0.181071i
\(367\) −22.0000 −1.14839 −0.574195 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) 30.0000 1.56386
\(369\) − 5.19615i − 0.270501i
\(370\) 25.9808i 1.35068i
\(371\) 0 0
\(372\) 6.92820i 0.359211i
\(373\) 19.0000 0.983783 0.491891 0.870657i \(-0.336306\pi\)
0.491891 + 0.870657i \(0.336306\pi\)
\(374\) 0 0
\(375\) 24.2487i 1.25220i
\(376\) 6.00000 0.309426
\(377\) 0 0
\(378\) 0 0
\(379\) − 24.2487i − 1.24557i −0.782392 0.622786i \(-0.786001\pi\)
0.782392 0.622786i \(-0.213999\pi\)
\(380\) 6.00000 0.307794
\(381\) −4.00000 −0.204926
\(382\) − 31.1769i − 1.59515i
\(383\) 20.7846i 1.06204i 0.847358 + 0.531022i \(0.178192\pi\)
−0.847358 + 0.531022i \(0.821808\pi\)
\(384\) 24.2487i 1.23744i
\(385\) 0 0
\(386\) −9.00000 −0.458088
\(387\) 8.00000 0.406663
\(388\) 6.92820i 0.351726i
\(389\) −9.00000 −0.456318 −0.228159 0.973624i \(-0.573271\pi\)
−0.228159 + 0.973624i \(0.573271\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) − 12.1244i − 0.612372i
\(393\) 36.0000 1.81596
\(394\) 24.0000 1.20910
\(395\) 6.92820i 0.348596i
\(396\) 0 0
\(397\) 13.8564i 0.695433i 0.937600 + 0.347717i \(0.113043\pi\)
−0.937600 + 0.347717i \(0.886957\pi\)
\(398\) 3.46410i 0.173640i
\(399\) 0 0
\(400\) −10.0000 −0.500000
\(401\) − 1.73205i − 0.0864945i −0.999064 0.0432472i \(-0.986230\pi\)
0.999064 0.0432472i \(-0.0137703\pi\)
\(402\) 12.0000 0.598506
\(403\) 0 0
\(404\) 3.00000 0.149256
\(405\) − 19.0526i − 0.946729i
\(406\) 0 0
\(407\) 0 0
\(408\) 10.3923i 0.514496i
\(409\) − 15.5885i − 0.770800i −0.922750 0.385400i \(-0.874064\pi\)
0.922750 0.385400i \(-0.125936\pi\)
\(410\) − 15.5885i − 0.769859i
\(411\) 31.1769i 1.53784i
\(412\) 10.0000 0.492665
\(413\) 0 0
\(414\) 10.3923i 0.510754i
\(415\) 24.0000 1.17811
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) 18.0000 0.879358 0.439679 0.898155i \(-0.355092\pi\)
0.439679 + 0.898155i \(0.355092\pi\)
\(420\) 0 0
\(421\) − 15.5885i − 0.759735i −0.925041 0.379867i \(-0.875970\pi\)
0.925041 0.379867i \(-0.124030\pi\)
\(422\) − 17.3205i − 0.843149i
\(423\) 3.46410i 0.168430i
\(424\) 5.19615i 0.252347i
\(425\) −6.00000 −0.291043
\(426\) −12.0000 −0.581402
\(427\) 0 0
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) 24.0000 1.15738
\(431\) − 6.92820i − 0.333720i −0.985981 0.166860i \(-0.946637\pi\)
0.985981 0.166860i \(-0.0533628\pi\)
\(432\) 20.0000 0.962250
\(433\) −17.0000 −0.816968 −0.408484 0.912766i \(-0.633942\pi\)
−0.408484 + 0.912766i \(0.633942\pi\)
\(434\) 0 0
\(435\) 10.3923i 0.498273i
\(436\) − 13.8564i − 0.663602i
\(437\) − 20.7846i − 0.994263i
\(438\) −6.00000 −0.286691
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 7.00000 0.333333
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) − 17.3205i − 0.821995i
\(445\) 12.0000 0.568855
\(446\) −18.0000 −0.852325
\(447\) 38.1051i 1.80231i
\(448\) 0 0
\(449\) 6.92820i 0.326962i 0.986546 + 0.163481i \(0.0522723\pi\)
−0.986546 + 0.163481i \(0.947728\pi\)
\(450\) − 3.46410i − 0.163299i
\(451\) 0 0
\(452\) 15.0000 0.705541
\(453\) − 34.6410i − 1.62758i
\(454\) −42.0000 −1.97116
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) 1.73205i 0.0810219i 0.999179 + 0.0405110i \(0.0128986\pi\)
−0.999179 + 0.0405110i \(0.987101\pi\)
\(458\) 0 0
\(459\) 12.0000 0.560112
\(460\) 10.3923i 0.484544i
\(461\) − 22.5167i − 1.04871i −0.851501 0.524353i \(-0.824307\pi\)
0.851501 0.524353i \(-0.175693\pi\)
\(462\) 0 0
\(463\) − 13.8564i − 0.643962i −0.946746 0.321981i \(-0.895651\pi\)
0.946746 0.321981i \(-0.104349\pi\)
\(464\) −15.0000 −0.696358
\(465\) 12.0000 0.556487
\(466\) − 10.3923i − 0.481414i
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 10.3923i 0.479361i
\(471\) −26.0000 −1.19802
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) − 13.8564i − 0.636446i
\(475\) 6.92820i 0.317888i
\(476\) 0 0
\(477\) −3.00000 −0.137361
\(478\) 36.0000 1.64660
\(479\) 24.2487i 1.10795i 0.832533 + 0.553976i \(0.186890\pi\)
−0.832533 + 0.553976i \(0.813110\pi\)
\(480\) −18.0000 −0.821584
\(481\) 0 0
\(482\) 3.00000 0.136646
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 12.0000 0.544892
\(486\) 17.3205i 0.785674i
\(487\) 6.92820i 0.313947i 0.987603 + 0.156973i \(0.0501737\pi\)
−0.987603 + 0.156973i \(0.949826\pi\)
\(488\) − 1.73205i − 0.0784063i
\(489\) − 41.5692i − 1.87983i
\(490\) 21.0000 0.948683
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 10.3923i 0.468521i
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) 0 0
\(496\) 17.3205i 0.777714i
\(497\) 0 0
\(498\) −48.0000 −2.15093
\(499\) − 31.1769i − 1.39567i −0.716258 0.697835i \(-0.754147\pi\)
0.716258 0.697835i \(-0.245853\pi\)
\(500\) − 12.1244i − 0.542218i
\(501\) − 27.7128i − 1.23812i
\(502\) 31.1769i 1.39149i
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 0 0
\(505\) − 5.19615i − 0.231226i
\(506\) 0 0
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) 19.0526i 0.844490i 0.906482 + 0.422245i \(0.138758\pi\)
−0.906482 + 0.422245i \(0.861242\pi\)
\(510\) −18.0000 −0.797053
\(511\) 0 0
\(512\) − 8.66025i − 0.382733i
\(513\) − 13.8564i − 0.611775i
\(514\) − 5.19615i − 0.229192i
\(515\) − 17.3205i − 0.763233i
\(516\) −16.0000 −0.704361
\(517\) 0 0
\(518\) 0 0
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) 9.00000 0.394297 0.197149 0.980374i \(-0.436832\pi\)
0.197149 + 0.980374i \(0.436832\pi\)
\(522\) − 5.19615i − 0.227429i
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −18.0000 −0.786334
\(525\) 0 0
\(526\) − 20.7846i − 0.906252i
\(527\) 10.3923i 0.452696i
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −9.00000 −0.390935
\(531\) − 6.92820i − 0.300658i
\(532\) 0 0
\(533\) 0 0
\(534\) −24.0000 −1.03858
\(535\) 10.3923i 0.449299i
\(536\) 6.00000 0.259161
\(537\) 0 0
\(538\) 10.3923i 0.448044i
\(539\) 0 0
\(540\) 6.92820i 0.298142i
\(541\) − 29.4449i − 1.26593i −0.774179 0.632967i \(-0.781837\pi\)
0.774179 0.632967i \(-0.218163\pi\)
\(542\) 36.0000 1.54633
\(543\) 22.0000 0.944110
\(544\) − 15.5885i − 0.668350i
\(545\) −24.0000 −1.02805
\(546\) 0 0
\(547\) −22.0000 −0.940652 −0.470326 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) − 15.5885i − 0.665906i
\(549\) 1.00000 0.0426790
\(550\) 0 0
\(551\) 10.3923i 0.442727i
\(552\) 20.7846i 0.884652i
\(553\) 0 0
\(554\) 12.1244i 0.515115i
\(555\) −30.0000 −1.27343
\(556\) 4.00000 0.169638
\(557\) − 15.5885i − 0.660504i −0.943893 0.330252i \(-0.892866\pi\)
0.943893 0.330252i \(-0.107134\pi\)
\(558\) −6.00000 −0.254000
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 39.0000 1.64512
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) − 6.92820i − 0.291730i
\(565\) − 25.9808i − 1.09302i
\(566\) − 6.92820i − 0.291214i
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 20.7846i 0.870572i
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 0 0
\(573\) 36.0000 1.50392
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) −1.00000 −0.0416667
\(577\) 19.0526i 0.793168i 0.917998 + 0.396584i \(0.129805\pi\)
−0.917998 + 0.396584i \(0.870195\pi\)
\(578\) 13.8564i 0.576351i
\(579\) − 10.3923i − 0.431889i
\(580\) − 5.19615i − 0.215758i
\(581\) 0 0
\(582\) −24.0000 −0.994832
\(583\) 0 0
\(584\) −3.00000 −0.124141
\(585\) 0 0
\(586\) −9.00000 −0.371787
\(587\) − 20.7846i − 0.857873i −0.903335 0.428936i \(-0.858888\pi\)
0.903335 0.428936i \(-0.141112\pi\)
\(588\) −14.0000 −0.577350
\(589\) 12.0000 0.494451
\(590\) − 20.7846i − 0.855689i
\(591\) 27.7128i 1.13995i
\(592\) − 43.3013i − 1.77967i
\(593\) 25.9808i 1.06690i 0.845831 + 0.533451i \(0.179105\pi\)
−0.845831 + 0.533451i \(0.820895\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 19.0526i − 0.780423i
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) − 6.92820i − 0.282843i
\(601\) 25.0000 1.01977 0.509886 0.860242i \(-0.329688\pi\)
0.509886 + 0.860242i \(0.329688\pi\)
\(602\) 0 0
\(603\) 3.46410i 0.141069i
\(604\) 17.3205i 0.704761i
\(605\) 19.0526i 0.774597i
\(606\) 10.3923i 0.422159i
\(607\) −34.0000 −1.38002 −0.690009 0.723801i \(-0.742393\pi\)
−0.690009 + 0.723801i \(0.742393\pi\)
\(608\) −18.0000 −0.729996
\(609\) 0 0
\(610\) 3.00000 0.121466
\(611\) 0 0
\(612\) 3.00000 0.121268
\(613\) − 12.1244i − 0.489698i −0.969561 0.244849i \(-0.921262\pi\)
0.969561 0.244849i \(-0.0787384\pi\)
\(614\) 30.0000 1.21070
\(615\) 18.0000 0.725830
\(616\) 0 0
\(617\) 22.5167i 0.906487i 0.891387 + 0.453243i \(0.149733\pi\)
−0.891387 + 0.453243i \(0.850267\pi\)
\(618\) 34.6410i 1.39347i
\(619\) 20.7846i 0.835404i 0.908584 + 0.417702i \(0.137164\pi\)
−0.908584 + 0.417702i \(0.862836\pi\)
\(620\) −6.00000 −0.240966
\(621\) 24.0000 0.963087
\(622\) 51.9615i 2.08347i
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) − 17.3205i − 0.692267i
\(627\) 0 0
\(628\) 13.0000 0.518756
\(629\) − 25.9808i − 1.03592i
\(630\) 0 0
\(631\) 48.4974i 1.93065i 0.261048 + 0.965326i \(0.415932\pi\)
−0.261048 + 0.965326i \(0.584068\pi\)
\(632\) − 6.92820i − 0.275589i
\(633\) 20.0000 0.794929
\(634\) 9.00000 0.357436
\(635\) − 3.46410i − 0.137469i
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) − 3.46410i − 0.137038i
\(640\) −21.0000 −0.830098
\(641\) 33.0000 1.30342 0.651711 0.758468i \(-0.274052\pi\)
0.651711 + 0.758468i \(0.274052\pi\)
\(642\) − 20.7846i − 0.820303i
\(643\) − 13.8564i − 0.546443i −0.961951 0.273222i \(-0.911911\pi\)
0.961951 0.273222i \(-0.0880892\pi\)
\(644\) 0 0
\(645\) 27.7128i 1.09119i
\(646\) −18.0000 −0.708201
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 19.0526i 0.748455i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 20.7846i 0.813988i
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 48.0000 1.87695
\(655\) 31.1769i 1.21818i
\(656\) 25.9808i 1.01438i
\(657\) − 1.73205i − 0.0675737i
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) − 46.7654i − 1.81896i −0.415745 0.909481i \(-0.636479\pi\)
0.415745 0.909481i \(-0.363521\pi\)
\(662\) 48.0000 1.86557
\(663\) 0 0
\(664\) −24.0000 −0.931381
\(665\) 0 0
\(666\) 15.0000 0.581238
\(667\) −18.0000 −0.696963
\(668\) 13.8564i 0.536120i
\(669\) − 20.7846i − 0.803579i
\(670\) 10.3923i 0.401490i
\(671\) 0 0
\(672\) 0 0
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) 39.8372i 1.53447i
\(675\) −8.00000 −0.307920
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 51.9615i 1.99557i
\(679\) 0 0
\(680\) −9.00000 −0.345134
\(681\) − 48.4974i − 1.85843i
\(682\) 0 0
\(683\) − 24.2487i − 0.927851i −0.885874 0.463926i \(-0.846441\pi\)
0.885874 0.463926i \(-0.153559\pi\)
\(684\) − 3.46410i − 0.132453i
\(685\) −27.0000 −1.03162
\(686\) 0 0
\(687\) 0 0
\(688\) −40.0000 −1.52499
\(689\) 0 0
\(690\) −36.0000 −1.37050
\(691\) 13.8564i 0.527123i 0.964643 + 0.263561i \(0.0848971\pi\)
−0.964643 + 0.263561i \(0.915103\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 51.9615i 1.97243i
\(695\) − 6.92820i − 0.262802i
\(696\) − 10.3923i − 0.393919i
\(697\) 15.5885i 0.590455i
\(698\) 24.0000 0.908413
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) −30.0000 −1.13147
\(704\) 0 0
\(705\) −12.0000 −0.451946
\(706\) 57.0000 2.14522
\(707\) 0 0
\(708\) 13.8564i 0.520756i
\(709\) 5.19615i 0.195146i 0.995228 + 0.0975728i \(0.0311079\pi\)
−0.995228 + 0.0975728i \(0.968892\pi\)
\(710\) − 10.3923i − 0.390016i
\(711\) 4.00000 0.150012
\(712\) −12.0000 −0.449719
\(713\) 20.7846i 0.778390i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 41.5692i 1.55243i
\(718\) 12.0000 0.447836
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) − 8.66025i − 0.322749i
\(721\) 0 0
\(722\) − 12.1244i − 0.451222i
\(723\) 3.46410i 0.128831i
\(724\) −11.0000 −0.408812
\(725\) 6.00000 0.222834
\(726\) − 38.1051i − 1.41421i
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) − 5.19615i − 0.192318i
\(731\) −24.0000 −0.887672
\(732\) −2.00000 −0.0739221
\(733\) − 12.1244i − 0.447823i −0.974609 0.223912i \(-0.928117\pi\)
0.974609 0.223912i \(-0.0718827\pi\)
\(734\) 38.1051i 1.40649i
\(735\) 24.2487i 0.894427i
\(736\) − 31.1769i − 1.14920i
\(737\) 0 0
\(738\) −9.00000 −0.331295
\(739\) − 20.7846i − 0.764574i −0.924044 0.382287i \(-0.875137\pi\)
0.924044 0.382287i \(-0.124863\pi\)
\(740\) 15.0000 0.551411
\(741\) 0 0
\(742\) 0 0
\(743\) − 34.6410i − 1.27086i −0.772160 0.635428i \(-0.780824\pi\)
0.772160 0.635428i \(-0.219176\pi\)
\(744\) −12.0000 −0.439941
\(745\) −33.0000 −1.20903
\(746\) − 32.9090i − 1.20488i
\(747\) − 13.8564i − 0.506979i
\(748\) 0 0
\(749\) 0 0
\(750\) 42.0000 1.53362
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) − 17.3205i − 0.631614i
\(753\) −36.0000 −1.31191
\(754\) 0 0
\(755\) 30.0000 1.09181
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) −42.0000 −1.52551
\(759\) 0 0
\(760\) 10.3923i 0.376969i
\(761\) − 34.6410i − 1.25574i −0.778320 0.627868i \(-0.783928\pi\)
0.778320 0.627868i \(-0.216072\pi\)
\(762\) 6.92820i 0.250982i
\(763\) 0 0
\(764\) −18.0000 −0.651217
\(765\) − 5.19615i − 0.187867i
\(766\) 36.0000 1.30073
\(767\) 0 0
\(768\) 38.0000 1.37121
\(769\) 6.92820i 0.249837i 0.992167 + 0.124919i \(0.0398670\pi\)
−0.992167 + 0.124919i \(0.960133\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 5.19615i 0.187014i
\(773\) 34.6410i 1.24595i 0.782241 + 0.622975i \(0.214076\pi\)
−0.782241 + 0.622975i \(0.785924\pi\)
\(774\) − 13.8564i − 0.498058i
\(775\) − 6.92820i − 0.248868i
\(776\) −12.0000 −0.430775
\(777\) 0 0
\(778\) 15.5885i 0.558873i
\(779\) 18.0000 0.644917
\(780\) 0 0
\(781\) 0 0
\(782\) − 31.1769i − 1.11488i
\(783\) −12.0000 −0.428845
\(784\) −35.0000 −1.25000
\(785\) − 22.5167i − 0.803654i
\(786\) − 62.3538i − 2.22409i
\(787\) − 38.1051i − 1.35830i −0.733999 0.679150i \(-0.762348\pi\)
0.733999 0.679150i \(-0.237652\pi\)
\(788\) − 13.8564i − 0.493614i
\(789\) 24.0000 0.854423
\(790\) 12.0000 0.426941
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 24.0000 0.851728
\(795\) − 10.3923i − 0.368577i
\(796\) 2.00000 0.0708881
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) − 10.3923i − 0.367653i
\(800\) 10.3923i 0.367423i
\(801\) − 6.92820i − 0.244796i
\(802\) −3.00000 −0.105934
\(803\) 0 0
\(804\) − 6.92820i − 0.244339i
\(805\) 0 0
\(806\) 0 0
\(807\) −12.0000 −0.422420
\(808\) 5.19615i 0.182800i
\(809\) 33.0000 1.16022 0.580109 0.814539i \(-0.303010\pi\)
0.580109 + 0.814539i \(0.303010\pi\)
\(810\) −33.0000 −1.15950
\(811\) 38.1051i 1.33805i 0.743239 + 0.669026i \(0.233288\pi\)
−0.743239 + 0.669026i \(0.766712\pi\)
\(812\) 0 0
\(813\) 41.5692i 1.45790i
\(814\) 0 0
\(815\) 36.0000 1.26102
\(816\) 30.0000 1.05021
\(817\) 27.7128i 0.969549i
\(818\) −27.0000 −0.944033
\(819\) 0 0
\(820\) −9.00000 −0.314294
\(821\) − 41.5692i − 1.45078i −0.688340 0.725388i \(-0.741660\pi\)
0.688340 0.725388i \(-0.258340\pi\)
\(822\) 54.0000 1.88347
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 17.3205i 0.603388i
\(825\) 0 0
\(826\) 0 0
\(827\) − 20.7846i − 0.722752i −0.932420 0.361376i \(-0.882307\pi\)
0.932420 0.361376i \(-0.117693\pi\)
\(828\) 6.00000 0.208514
\(829\) 25.0000 0.868286 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) − 41.5692i − 1.44289i
\(831\) −14.0000 −0.485655
\(832\) 0 0
\(833\) −21.0000 −0.727607
\(834\) 13.8564i 0.479808i
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) 13.8564i 0.478947i
\(838\) − 31.1769i − 1.07699i
\(839\) − 45.0333i − 1.55472i −0.629054 0.777361i \(-0.716558\pi\)
0.629054 0.777361i \(-0.283442\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −27.0000 −0.930481
\(843\) 45.0333i 1.55103i
\(844\) −10.0000 −0.344214
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) 0 0
\(848\) 15.0000 0.515102
\(849\) 8.00000 0.274559
\(850\) 10.3923i 0.356453i
\(851\) − 51.9615i − 1.78122i
\(852\) 6.92820i 0.237356i
\(853\) − 25.9808i − 0.889564i −0.895639 0.444782i \(-0.853281\pi\)
0.895639 0.444782i \(-0.146719\pi\)
\(854\) 0 0
\(855\) −6.00000 −0.205196
\(856\) − 10.3923i − 0.355202i
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) −14.0000 −0.477674 −0.238837 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) − 13.8564i − 0.472500i
\(861\) 0 0
\(862\) −12.0000 −0.408722
\(863\) 27.7128i 0.943355i 0.881771 + 0.471678i \(0.156351\pi\)
−0.881771 + 0.471678i \(0.843649\pi\)
\(864\) − 20.7846i − 0.707107i
\(865\) 10.3923i 0.353349i
\(866\) 29.4449i 1.00058i
\(867\) −16.0000 −0.543388
\(868\) 0 0
\(869\) 0 0
\(870\) 18.0000 0.610257
\(871\) 0 0
\(872\) 24.0000 0.812743
\(873\) − 6.92820i − 0.234484i
\(874\) −36.0000 −1.21772
\(875\) 0 0
\(876\) 3.46410i 0.117041i
\(877\) − 12.1244i − 0.409410i −0.978824 0.204705i \(-0.934376\pi\)
0.978824 0.204705i \(-0.0656236\pi\)
\(878\) 48.4974i 1.63671i
\(879\) − 10.3923i − 0.350524i
\(880\) 0 0
\(881\) 27.0000 0.909653 0.454827 0.890580i \(-0.349701\pi\)
0.454827 + 0.890580i \(0.349701\pi\)
\(882\) − 12.1244i − 0.408248i
\(883\) 10.0000 0.336527 0.168263 0.985742i \(-0.446184\pi\)
0.168263 + 0.985742i \(0.446184\pi\)
\(884\) 0 0
\(885\) 24.0000 0.806751
\(886\) 20.7846i 0.698273i
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 30.0000 1.00673
\(889\) 0 0
\(890\) − 20.7846i − 0.696702i
\(891\) 0 0
\(892\) 10.3923i 0.347960i
\(893\) −12.0000 −0.401565
\(894\) 66.0000 2.20737
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 12.0000 0.400445
\(899\) − 10.3923i − 0.346603i
\(900\) −2.00000 −0.0666667
\(901\) 9.00000 0.299833
\(902\) 0 0
\(903\) 0 0
\(904\) 25.9808i 0.864107i
\(905\) 19.0526i 0.633328i
\(906\) −60.0000 −1.99337
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 24.2487i 0.804722i
\(909\) −3.00000 −0.0995037
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) − 34.6410i − 1.14708i
\(913\) 0 0
\(914\) 3.00000 0.0992312
\(915\) 3.46410i 0.114520i
\(916\) 0 0
\(917\) 0 0
\(918\) − 20.7846i − 0.685994i
\(919\) 22.0000 0.725713 0.362857 0.931845i \(-0.381802\pi\)
0.362857 + 0.931845i \(0.381802\pi\)
\(920\) −18.0000 −0.593442
\(921\) 34.6410i 1.14146i
\(922\) −39.0000 −1.28440
\(923\) 0 0
\(924\) 0 0
\(925\) 17.3205i 0.569495i
\(926\) −24.0000 −0.788689
\(927\) −10.0000 −0.328443
\(928\) 15.5885i 0.511716i
\(929\) 46.7654i 1.53432i 0.641455 + 0.767161i \(0.278331\pi\)
−0.641455 + 0.767161i \(0.721669\pi\)
\(930\) − 20.7846i − 0.681554i
\(931\) 24.2487i 0.794719i
\(932\) −6.00000 −0.196537
\(933\) −60.0000 −1.96431
\(934\) − 20.7846i − 0.680093i
\(935\) 0 0
\(936\) 0 0
\(937\) 7.00000 0.228680 0.114340 0.993442i \(-0.463525\pi\)
0.114340 + 0.993442i \(0.463525\pi\)
\(938\) 0 0
\(939\) 20.0000 0.652675
\(940\) 6.00000 0.195698
\(941\) − 20.7846i − 0.677559i −0.940866 0.338779i \(-0.889986\pi\)
0.940866 0.338779i \(-0.110014\pi\)
\(942\) 45.0333i 1.46726i
\(943\) 31.1769i 1.01526i
\(944\) 34.6410i 1.12747i
\(945\) 0 0
\(946\) 0 0
\(947\) − 17.3205i − 0.562841i −0.959585 0.281420i \(-0.909194\pi\)
0.959585 0.281420i \(-0.0908056\pi\)
\(948\) −8.00000 −0.259828
\(949\) 0 0
\(950\) 12.0000 0.389331
\(951\) 10.3923i 0.336994i
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 5.19615i 0.168232i
\(955\) 31.1769i 1.00886i
\(956\) − 20.7846i − 0.672222i
\(957\) 0 0
\(958\) 42.0000 1.35696
\(959\) 0 0
\(960\) − 3.46410i − 0.111803i
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) − 1.73205i − 0.0557856i
\(965\) 9.00000 0.289720
\(966\) 0 0
\(967\) − 58.8897i − 1.89377i −0.321578 0.946883i \(-0.604213\pi\)
0.321578 0.946883i \(-0.395787\pi\)
\(968\) − 19.0526i − 0.612372i
\(969\) − 20.7846i − 0.667698i
\(970\) − 20.7846i − 0.667354i
\(971\) 6.00000 0.192549 0.0962746 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) 10.0000 0.320750
\(973\) 0 0
\(974\) 12.0000 0.384505
\(975\) 0 0
\(976\) −5.00000 −0.160046
\(977\) 43.3013i 1.38533i 0.721259 + 0.692665i \(0.243564\pi\)
−0.721259 + 0.692665i \(0.756436\pi\)
\(978\) −72.0000 −2.30231
\(979\) 0 0
\(980\) − 12.1244i − 0.387298i
\(981\) 13.8564i 0.442401i
\(982\) − 20.7846i − 0.663264i
\(983\) − 51.9615i − 1.65732i −0.559756 0.828658i \(-0.689105\pi\)
0.559756 0.828658i \(-0.310895\pi\)
\(984\) −18.0000 −0.573819
\(985\) −24.0000 −0.764704
\(986\) 15.5885i 0.496438i
\(987\) 0 0
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) 18.0000 0.571501
\(993\) 55.4256i 1.75888i
\(994\) 0 0
\(995\) − 3.46410i − 0.109819i
\(996\) 27.7128i 0.878114i
\(997\) 17.0000 0.538395 0.269198 0.963085i \(-0.413241\pi\)
0.269198 + 0.963085i \(0.413241\pi\)
\(998\) −54.0000 −1.70934
\(999\) − 34.6410i − 1.09599i
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 169.2.b.a.168.1 2
3.2 odd 2 1521.2.b.a.1351.2 2
4.3 odd 2 2704.2.f.b.337.2 2
13.2 odd 12 169.2.c.a.22.2 4
13.3 even 3 169.2.e.a.147.1 2
13.4 even 6 169.2.e.a.23.1 2
13.5 odd 4 169.2.a.a.1.1 2
13.6 odd 12 169.2.c.a.146.2 4
13.7 odd 12 169.2.c.a.146.1 4
13.8 odd 4 169.2.a.a.1.2 2
13.9 even 3 13.2.e.a.10.1 yes 2
13.10 even 6 13.2.e.a.4.1 2
13.11 odd 12 169.2.c.a.22.1 4
13.12 even 2 inner 169.2.b.a.168.2 2
39.5 even 4 1521.2.a.k.1.2 2
39.8 even 4 1521.2.a.k.1.1 2
39.23 odd 6 117.2.q.c.82.1 2
39.35 odd 6 117.2.q.c.10.1 2
39.38 odd 2 1521.2.b.a.1351.1 2
52.23 odd 6 208.2.w.b.17.1 2
52.31 even 4 2704.2.a.o.1.2 2
52.35 odd 6 208.2.w.b.49.1 2
52.47 even 4 2704.2.a.o.1.1 2
52.51 odd 2 2704.2.f.b.337.1 2
65.9 even 6 325.2.n.a.101.1 2
65.22 odd 12 325.2.m.a.49.1 4
65.23 odd 12 325.2.m.a.199.1 4
65.34 odd 4 4225.2.a.v.1.1 2
65.44 odd 4 4225.2.a.v.1.2 2
65.48 odd 12 325.2.m.a.49.2 4
65.49 even 6 325.2.n.a.251.1 2
65.62 odd 12 325.2.m.a.199.2 4
91.9 even 3 637.2.u.c.361.1 2
91.10 odd 6 637.2.u.b.30.1 2
91.23 even 6 637.2.k.a.459.1 2
91.34 even 4 8281.2.a.q.1.2 2
91.48 odd 6 637.2.q.a.491.1 2
91.61 odd 6 637.2.u.b.361.1 2
91.62 odd 6 637.2.q.a.589.1 2
91.74 even 3 637.2.k.a.569.1 2
91.75 odd 6 637.2.k.c.459.1 2
91.83 even 4 8281.2.a.q.1.1 2
91.87 odd 6 637.2.k.c.569.1 2
91.88 even 6 637.2.u.c.30.1 2
104.35 odd 6 832.2.w.a.257.1 2
104.61 even 6 832.2.w.d.257.1 2
104.75 odd 6 832.2.w.a.641.1 2
104.101 even 6 832.2.w.d.641.1 2
156.23 even 6 1872.2.by.d.433.1 2
156.35 even 6 1872.2.by.d.1297.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.2.e.a.4.1 2 13.10 even 6
13.2.e.a.10.1 yes 2 13.9 even 3
117.2.q.c.10.1 2 39.35 odd 6
117.2.q.c.82.1 2 39.23 odd 6
169.2.a.a.1.1 2 13.5 odd 4
169.2.a.a.1.2 2 13.8 odd 4
169.2.b.a.168.1 2 1.1 even 1 trivial
169.2.b.a.168.2 2 13.12 even 2 inner
169.2.c.a.22.1 4 13.11 odd 12
169.2.c.a.22.2 4 13.2 odd 12
169.2.c.a.146.1 4 13.7 odd 12
169.2.c.a.146.2 4 13.6 odd 12
169.2.e.a.23.1 2 13.4 even 6
169.2.e.a.147.1 2 13.3 even 3
208.2.w.b.17.1 2 52.23 odd 6
208.2.w.b.49.1 2 52.35 odd 6
325.2.m.a.49.1 4 65.22 odd 12
325.2.m.a.49.2 4 65.48 odd 12
325.2.m.a.199.1 4 65.23 odd 12
325.2.m.a.199.2 4 65.62 odd 12
325.2.n.a.101.1 2 65.9 even 6
325.2.n.a.251.1 2 65.49 even 6
637.2.k.a.459.1 2 91.23 even 6
637.2.k.a.569.1 2 91.74 even 3
637.2.k.c.459.1 2 91.75 odd 6
637.2.k.c.569.1 2 91.87 odd 6
637.2.q.a.491.1 2 91.48 odd 6
637.2.q.a.589.1 2 91.62 odd 6
637.2.u.b.30.1 2 91.10 odd 6
637.2.u.b.361.1 2 91.61 odd 6
637.2.u.c.30.1 2 91.88 even 6
637.2.u.c.361.1 2 91.9 even 3
832.2.w.a.257.1 2 104.35 odd 6
832.2.w.a.641.1 2 104.75 odd 6
832.2.w.d.257.1 2 104.61 even 6
832.2.w.d.641.1 2 104.101 even 6
1521.2.a.k.1.1 2 39.8 even 4
1521.2.a.k.1.2 2 39.5 even 4
1521.2.b.a.1351.1 2 39.38 odd 2
1521.2.b.a.1351.2 2 3.2 odd 2
1872.2.by.d.433.1 2 156.23 even 6
1872.2.by.d.1297.1 2 156.35 even 6
2704.2.a.o.1.1 2 52.47 even 4
2704.2.a.o.1.2 2 52.31 even 4
2704.2.f.b.337.1 2 52.51 odd 2
2704.2.f.b.337.2 2 4.3 odd 2
4225.2.a.v.1.1 2 65.34 odd 4
4225.2.a.v.1.2 2 65.44 odd 4
8281.2.a.q.1.1 2 91.83 even 4
8281.2.a.q.1.2 2 91.34 even 4