Properties

Label 189.3.d.a.55.2
Level $189$
Weight $3$
Character 189.55
Analytic conductor $5.150$
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] = [189,3,Mod(55,189)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(189, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.55");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 189.d (of order \(2\), degree \(1\), 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: \(5.14987699641\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 55.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 189.55
Dual form 189.3.d.a.55.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -3.00000 q^{4} +1.73205i q^{5} +(-3.50000 - 6.06218i) q^{7} +7.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -3.00000 q^{4} +1.73205i q^{5} +(-3.50000 - 6.06218i) q^{7} +7.00000 q^{8} -1.73205i q^{10} +17.0000 q^{11} +17.3205i q^{13} +(3.50000 + 6.06218i) q^{14} +5.00000 q^{16} +31.1769i q^{17} -20.7846i q^{19} -5.19615i q^{20} -17.0000 q^{22} +32.0000 q^{23} +22.0000 q^{25} -17.3205i q^{26} +(10.5000 + 18.1865i) q^{28} +2.00000 q^{29} +22.5167i q^{31} -33.0000 q^{32} -31.1769i q^{34} +(10.5000 - 6.06218i) q^{35} +8.00000 q^{37} +20.7846i q^{38} +12.1244i q^{40} -13.8564i q^{41} +26.0000 q^{43} -51.0000 q^{44} -32.0000 q^{46} -38.1051i q^{47} +(-24.5000 + 42.4352i) q^{49} -22.0000 q^{50} -51.9615i q^{52} -1.00000 q^{53} +29.4449i q^{55} +(-24.5000 - 42.4352i) q^{56} -2.00000 q^{58} +90.0666i q^{59} -69.2820i q^{61} -22.5167i q^{62} +13.0000 q^{64} -30.0000 q^{65} -34.0000 q^{67} -93.5307i q^{68} +(-10.5000 + 6.06218i) q^{70} -10.0000 q^{71} -57.1577i q^{73} -8.00000 q^{74} +62.3538i q^{76} +(-59.5000 - 103.057i) q^{77} -46.0000 q^{79} +8.66025i q^{80} +13.8564i q^{82} +102.191i q^{83} -54.0000 q^{85} -26.0000 q^{86} +119.000 q^{88} -51.9615i q^{89} +(105.000 - 60.6218i) q^{91} -96.0000 q^{92} +38.1051i q^{94} +36.0000 q^{95} +91.7987i q^{97} +(24.5000 - 42.4352i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 6 q^{4} - 7 q^{7} + 14 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 6 q^{4} - 7 q^{7} + 14 q^{8} + 34 q^{11} + 7 q^{14} + 10 q^{16} - 34 q^{22} + 64 q^{23} + 44 q^{25} + 21 q^{28} + 4 q^{29} - 66 q^{32} + 21 q^{35} + 16 q^{37} + 52 q^{43} - 102 q^{44} - 64 q^{46} - 49 q^{49} - 44 q^{50} - 2 q^{53} - 49 q^{56} - 4 q^{58} + 26 q^{64} - 60 q^{65} - 68 q^{67} - 21 q^{70} - 20 q^{71} - 16 q^{74} - 119 q^{77} - 92 q^{79} - 108 q^{85} - 52 q^{86} + 238 q^{88} + 210 q^{91} - 192 q^{92} + 72 q^{95} + 49 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.500000 −0.250000 0.968246i \(-0.580431\pi\)
−0.250000 + 0.968246i \(0.580431\pi\)
\(3\) 0 0
\(4\) −3.00000 −0.750000
\(5\) 1.73205i 0.346410i 0.984886 + 0.173205i \(0.0554123\pi\)
−0.984886 + 0.173205i \(0.944588\pi\)
\(6\) 0 0
\(7\) −3.50000 6.06218i −0.500000 0.866025i
\(8\) 7.00000 0.875000
\(9\) 0 0
\(10\) 1.73205i 0.173205i
\(11\) 17.0000 1.54545 0.772727 0.634738i \(-0.218892\pi\)
0.772727 + 0.634738i \(0.218892\pi\)
\(12\) 0 0
\(13\) 17.3205i 1.33235i 0.745797 + 0.666173i \(0.232069\pi\)
−0.745797 + 0.666173i \(0.767931\pi\)
\(14\) 3.50000 + 6.06218i 0.250000 + 0.433013i
\(15\) 0 0
\(16\) 5.00000 0.312500
\(17\) 31.1769i 1.83394i 0.398961 + 0.916968i \(0.369371\pi\)
−0.398961 + 0.916968i \(0.630629\pi\)
\(18\) 0 0
\(19\) 20.7846i 1.09393i −0.837157 0.546963i \(-0.815784\pi\)
0.837157 0.546963i \(-0.184216\pi\)
\(20\) 5.19615i 0.259808i
\(21\) 0 0
\(22\) −17.0000 −0.772727
\(23\) 32.0000 1.39130 0.695652 0.718379i \(-0.255116\pi\)
0.695652 + 0.718379i \(0.255116\pi\)
\(24\) 0 0
\(25\) 22.0000 0.880000
\(26\) 17.3205i 0.666173i
\(27\) 0 0
\(28\) 10.5000 + 18.1865i 0.375000 + 0.649519i
\(29\) 2.00000 0.0689655 0.0344828 0.999405i \(-0.489022\pi\)
0.0344828 + 0.999405i \(0.489022\pi\)
\(30\) 0 0
\(31\) 22.5167i 0.726344i 0.931722 + 0.363172i \(0.118306\pi\)
−0.931722 + 0.363172i \(0.881694\pi\)
\(32\) −33.0000 −1.03125
\(33\) 0 0
\(34\) 31.1769i 0.916968i
\(35\) 10.5000 6.06218i 0.300000 0.173205i
\(36\) 0 0
\(37\) 8.00000 0.216216 0.108108 0.994139i \(-0.465521\pi\)
0.108108 + 0.994139i \(0.465521\pi\)
\(38\) 20.7846i 0.546963i
\(39\) 0 0
\(40\) 12.1244i 0.303109i
\(41\) 13.8564i 0.337961i −0.985619 0.168981i \(-0.945952\pi\)
0.985619 0.168981i \(-0.0540475\pi\)
\(42\) 0 0
\(43\) 26.0000 0.604651 0.302326 0.953205i \(-0.402237\pi\)
0.302326 + 0.953205i \(0.402237\pi\)
\(44\) −51.0000 −1.15909
\(45\) 0 0
\(46\) −32.0000 −0.695652
\(47\) 38.1051i 0.810747i −0.914151 0.405374i \(-0.867141\pi\)
0.914151 0.405374i \(-0.132859\pi\)
\(48\) 0 0
\(49\) −24.5000 + 42.4352i −0.500000 + 0.866025i
\(50\) −22.0000 −0.440000
\(51\) 0 0
\(52\) 51.9615i 0.999260i
\(53\) −1.00000 −0.0188679 −0.00943396 0.999955i \(-0.503003\pi\)
−0.00943396 + 0.999955i \(0.503003\pi\)
\(54\) 0 0
\(55\) 29.4449i 0.535361i
\(56\) −24.5000 42.4352i −0.437500 0.757772i
\(57\) 0 0
\(58\) −2.00000 −0.0344828
\(59\) 90.0666i 1.52655i 0.646072 + 0.763277i \(0.276411\pi\)
−0.646072 + 0.763277i \(0.723589\pi\)
\(60\) 0 0
\(61\) 69.2820i 1.13577i −0.823108 0.567886i \(-0.807762\pi\)
0.823108 0.567886i \(-0.192238\pi\)
\(62\) 22.5167i 0.363172i
\(63\) 0 0
\(64\) 13.0000 0.203125
\(65\) −30.0000 −0.461538
\(66\) 0 0
\(67\) −34.0000 −0.507463 −0.253731 0.967275i \(-0.581658\pi\)
−0.253731 + 0.967275i \(0.581658\pi\)
\(68\) 93.5307i 1.37545i
\(69\) 0 0
\(70\) −10.5000 + 6.06218i −0.150000 + 0.0866025i
\(71\) −10.0000 −0.140845 −0.0704225 0.997517i \(-0.522435\pi\)
−0.0704225 + 0.997517i \(0.522435\pi\)
\(72\) 0 0
\(73\) 57.1577i 0.782982i −0.920182 0.391491i \(-0.871959\pi\)
0.920182 0.391491i \(-0.128041\pi\)
\(74\) −8.00000 −0.108108
\(75\) 0 0
\(76\) 62.3538i 0.820445i
\(77\) −59.5000 103.057i −0.772727 1.33840i
\(78\) 0 0
\(79\) −46.0000 −0.582278 −0.291139 0.956681i \(-0.594034\pi\)
−0.291139 + 0.956681i \(0.594034\pi\)
\(80\) 8.66025i 0.108253i
\(81\) 0 0
\(82\) 13.8564i 0.168981i
\(83\) 102.191i 1.23122i 0.788052 + 0.615608i \(0.211090\pi\)
−0.788052 + 0.615608i \(0.788910\pi\)
\(84\) 0 0
\(85\) −54.0000 −0.635294
\(86\) −26.0000 −0.302326
\(87\) 0 0
\(88\) 119.000 1.35227
\(89\) 51.9615i 0.583837i −0.956443 0.291919i \(-0.905706\pi\)
0.956443 0.291919i \(-0.0942937\pi\)
\(90\) 0 0
\(91\) 105.000 60.6218i 1.15385 0.666173i
\(92\) −96.0000 −1.04348
\(93\) 0 0
\(94\) 38.1051i 0.405374i
\(95\) 36.0000 0.378947
\(96\) 0 0
\(97\) 91.7987i 0.946378i 0.880961 + 0.473189i \(0.156897\pi\)
−0.880961 + 0.473189i \(0.843103\pi\)
\(98\) 24.5000 42.4352i 0.250000 0.433013i
\(99\) 0 0
\(100\) −66.0000 −0.660000
\(101\) 122.976i 1.21758i 0.793331 + 0.608790i \(0.208345\pi\)
−0.793331 + 0.608790i \(0.791655\pi\)
\(102\) 0 0
\(103\) 6.92820i 0.0672641i 0.999434 + 0.0336321i \(0.0107074\pi\)
−0.999434 + 0.0336321i \(0.989293\pi\)
\(104\) 121.244i 1.16580i
\(105\) 0 0
\(106\) 1.00000 0.00943396
\(107\) 107.000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(108\) 0 0
\(109\) −88.0000 −0.807339 −0.403670 0.914905i \(-0.632266\pi\)
−0.403670 + 0.914905i \(0.632266\pi\)
\(110\) 29.4449i 0.267681i
\(111\) 0 0
\(112\) −17.5000 30.3109i −0.156250 0.270633i
\(113\) 200.000 1.76991 0.884956 0.465675i \(-0.154188\pi\)
0.884956 + 0.465675i \(0.154188\pi\)
\(114\) 0 0
\(115\) 55.4256i 0.481962i
\(116\) −6.00000 −0.0517241
\(117\) 0 0
\(118\) 90.0666i 0.763277i
\(119\) 189.000 109.119i 1.58824 0.916968i
\(120\) 0 0
\(121\) 168.000 1.38843
\(122\) 69.2820i 0.567886i
\(123\) 0 0
\(124\) 67.5500i 0.544758i
\(125\) 81.4064i 0.651251i
\(126\) 0 0
\(127\) 143.000 1.12598 0.562992 0.826462i \(-0.309650\pi\)
0.562992 + 0.826462i \(0.309650\pi\)
\(128\) 119.000 0.929688
\(129\) 0 0
\(130\) 30.0000 0.230769
\(131\) 12.1244i 0.0925523i 0.998929 + 0.0462762i \(0.0147354\pi\)
−0.998929 + 0.0462762i \(0.985265\pi\)
\(132\) 0 0
\(133\) −126.000 + 72.7461i −0.947368 + 0.546963i
\(134\) 34.0000 0.253731
\(135\) 0 0
\(136\) 218.238i 1.60469i
\(137\) −10.0000 −0.0729927 −0.0364964 0.999334i \(-0.511620\pi\)
−0.0364964 + 0.999334i \(0.511620\pi\)
\(138\) 0 0
\(139\) 138.564i 0.996864i −0.866929 0.498432i \(-0.833909\pi\)
0.866929 0.498432i \(-0.166091\pi\)
\(140\) −31.5000 + 18.1865i −0.225000 + 0.129904i
\(141\) 0 0
\(142\) 10.0000 0.0704225
\(143\) 294.449i 2.05908i
\(144\) 0 0
\(145\) 3.46410i 0.0238904i
\(146\) 57.1577i 0.391491i
\(147\) 0 0
\(148\) −24.0000 −0.162162
\(149\) −79.0000 −0.530201 −0.265101 0.964221i \(-0.585405\pi\)
−0.265101 + 0.964221i \(0.585405\pi\)
\(150\) 0 0
\(151\) −73.0000 −0.483444 −0.241722 0.970346i \(-0.577712\pi\)
−0.241722 + 0.970346i \(0.577712\pi\)
\(152\) 145.492i 0.957186i
\(153\) 0 0
\(154\) 59.5000 + 103.057i 0.386364 + 0.669201i
\(155\) −39.0000 −0.251613
\(156\) 0 0
\(157\) 200.918i 1.27973i −0.768487 0.639866i \(-0.778990\pi\)
0.768487 0.639866i \(-0.221010\pi\)
\(158\) 46.0000 0.291139
\(159\) 0 0
\(160\) 57.1577i 0.357235i
\(161\) −112.000 193.990i −0.695652 1.20490i
\(162\) 0 0
\(163\) −226.000 −1.38650 −0.693252 0.720696i \(-0.743823\pi\)
−0.693252 + 0.720696i \(0.743823\pi\)
\(164\) 41.5692i 0.253471i
\(165\) 0 0
\(166\) 102.191i 0.615608i
\(167\) 86.6025i 0.518578i −0.965800 0.259289i \(-0.916512\pi\)
0.965800 0.259289i \(-0.0834882\pi\)
\(168\) 0 0
\(169\) −131.000 −0.775148
\(170\) 54.0000 0.317647
\(171\) 0 0
\(172\) −78.0000 −0.453488
\(173\) 74.4782i 0.430510i −0.976558 0.215255i \(-0.930942\pi\)
0.976558 0.215255i \(-0.0690582\pi\)
\(174\) 0 0
\(175\) −77.0000 133.368i −0.440000 0.762102i
\(176\) 85.0000 0.482955
\(177\) 0 0
\(178\) 51.9615i 0.291919i
\(179\) 119.000 0.664804 0.332402 0.943138i \(-0.392141\pi\)
0.332402 + 0.943138i \(0.392141\pi\)
\(180\) 0 0
\(181\) 62.3538i 0.344496i 0.985054 + 0.172248i \(0.0551031\pi\)
−0.985054 + 0.172248i \(0.944897\pi\)
\(182\) −105.000 + 60.6218i −0.576923 + 0.333087i
\(183\) 0 0
\(184\) 224.000 1.21739
\(185\) 13.8564i 0.0748995i
\(186\) 0 0
\(187\) 530.008i 2.83426i
\(188\) 114.315i 0.608060i
\(189\) 0 0
\(190\) −36.0000 −0.189474
\(191\) −190.000 −0.994764 −0.497382 0.867532i \(-0.665705\pi\)
−0.497382 + 0.867532i \(0.665705\pi\)
\(192\) 0 0
\(193\) 311.000 1.61140 0.805699 0.592325i \(-0.201790\pi\)
0.805699 + 0.592325i \(0.201790\pi\)
\(194\) 91.7987i 0.473189i
\(195\) 0 0
\(196\) 73.5000 127.306i 0.375000 0.649519i
\(197\) −307.000 −1.55838 −0.779188 0.626791i \(-0.784368\pi\)
−0.779188 + 0.626791i \(0.784368\pi\)
\(198\) 0 0
\(199\) 254.611i 1.27945i −0.768602 0.639727i \(-0.779047\pi\)
0.768602 0.639727i \(-0.220953\pi\)
\(200\) 154.000 0.770000
\(201\) 0 0
\(202\) 122.976i 0.608790i
\(203\) −7.00000 12.1244i −0.0344828 0.0597259i
\(204\) 0 0
\(205\) 24.0000 0.117073
\(206\) 6.92820i 0.0336321i
\(207\) 0 0
\(208\) 86.6025i 0.416358i
\(209\) 353.338i 1.69061i
\(210\) 0 0
\(211\) −196.000 −0.928910 −0.464455 0.885597i \(-0.653750\pi\)
−0.464455 + 0.885597i \(0.653750\pi\)
\(212\) 3.00000 0.0141509
\(213\) 0 0
\(214\) −107.000 −0.500000
\(215\) 45.0333i 0.209457i
\(216\) 0 0
\(217\) 136.500 78.8083i 0.629032 0.363172i
\(218\) 88.0000 0.403670
\(219\) 0 0
\(220\) 88.3346i 0.401521i
\(221\) −540.000 −2.44344
\(222\) 0 0
\(223\) 76.2102i 0.341750i 0.985293 + 0.170875i \(0.0546594\pi\)
−0.985293 + 0.170875i \(0.945341\pi\)
\(224\) 115.500 + 200.052i 0.515625 + 0.893089i
\(225\) 0 0
\(226\) −200.000 −0.884956
\(227\) 76.2102i 0.335728i 0.985810 + 0.167864i \(0.0536869\pi\)
−0.985810 + 0.167864i \(0.946313\pi\)
\(228\) 0 0
\(229\) 24.2487i 0.105890i −0.998597 0.0529448i \(-0.983139\pi\)
0.998597 0.0529448i \(-0.0168607\pi\)
\(230\) 55.4256i 0.240981i
\(231\) 0 0
\(232\) 14.0000 0.0603448
\(233\) −178.000 −0.763948 −0.381974 0.924173i \(-0.624756\pi\)
−0.381974 + 0.924173i \(0.624756\pi\)
\(234\) 0 0
\(235\) 66.0000 0.280851
\(236\) 270.200i 1.14491i
\(237\) 0 0
\(238\) −189.000 + 109.119i −0.794118 + 0.458484i
\(239\) −88.0000 −0.368201 −0.184100 0.982907i \(-0.558937\pi\)
−0.184100 + 0.982907i \(0.558937\pi\)
\(240\) 0 0
\(241\) 200.918i 0.833684i 0.908979 + 0.416842i \(0.136863\pi\)
−0.908979 + 0.416842i \(0.863137\pi\)
\(242\) −168.000 −0.694215
\(243\) 0 0
\(244\) 207.846i 0.851828i
\(245\) −73.5000 42.4352i −0.300000 0.173205i
\(246\) 0 0
\(247\) 360.000 1.45749
\(248\) 157.617i 0.635551i
\(249\) 0 0
\(250\) 81.4064i 0.325626i
\(251\) 311.769i 1.24211i −0.783768 0.621054i \(-0.786705\pi\)
0.783768 0.621054i \(-0.213295\pi\)
\(252\) 0 0
\(253\) 544.000 2.15020
\(254\) −143.000 −0.562992
\(255\) 0 0
\(256\) −171.000 −0.667969
\(257\) 3.46410i 0.0134790i −0.999977 0.00673950i \(-0.997855\pi\)
0.999977 0.00673950i \(-0.00214526\pi\)
\(258\) 0 0
\(259\) −28.0000 48.4974i −0.108108 0.187249i
\(260\) 90.0000 0.346154
\(261\) 0 0
\(262\) 12.1244i 0.0462762i
\(263\) 116.000 0.441065 0.220532 0.975380i \(-0.429221\pi\)
0.220532 + 0.975380i \(0.429221\pi\)
\(264\) 0 0
\(265\) 1.73205i 0.00653604i
\(266\) 126.000 72.7461i 0.473684 0.273482i
\(267\) 0 0
\(268\) 102.000 0.380597
\(269\) 353.338i 1.31353i −0.754097 0.656763i \(-0.771925\pi\)
0.754097 0.656763i \(-0.228075\pi\)
\(270\) 0 0
\(271\) 67.5500i 0.249262i −0.992203 0.124631i \(-0.960225\pi\)
0.992203 0.124631i \(-0.0397747\pi\)
\(272\) 155.885i 0.573105i
\(273\) 0 0
\(274\) 10.0000 0.0364964
\(275\) 374.000 1.36000
\(276\) 0 0
\(277\) −136.000 −0.490975 −0.245487 0.969400i \(-0.578948\pi\)
−0.245487 + 0.969400i \(0.578948\pi\)
\(278\) 138.564i 0.498432i
\(279\) 0 0
\(280\) 73.5000 42.4352i 0.262500 0.151554i
\(281\) −370.000 −1.31673 −0.658363 0.752701i \(-0.728751\pi\)
−0.658363 + 0.752701i \(0.728751\pi\)
\(282\) 0 0
\(283\) 401.836i 1.41991i 0.704245 + 0.709957i \(0.251286\pi\)
−0.704245 + 0.709957i \(0.748714\pi\)
\(284\) 30.0000 0.105634
\(285\) 0 0
\(286\) 294.449i 1.02954i
\(287\) −84.0000 + 48.4974i −0.292683 + 0.168981i
\(288\) 0 0
\(289\) −683.000 −2.36332
\(290\) 3.46410i 0.0119452i
\(291\) 0 0
\(292\) 171.473i 0.587236i
\(293\) 48.4974i 0.165520i 0.996569 + 0.0827601i \(0.0263735\pi\)
−0.996569 + 0.0827601i \(0.973626\pi\)
\(294\) 0 0
\(295\) −156.000 −0.528814
\(296\) 56.0000 0.189189
\(297\) 0 0
\(298\) 79.0000 0.265101
\(299\) 554.256i 1.85370i
\(300\) 0 0
\(301\) −91.0000 157.617i −0.302326 0.523643i
\(302\) 73.0000 0.241722
\(303\) 0 0
\(304\) 103.923i 0.341852i
\(305\) 120.000 0.393443
\(306\) 0 0
\(307\) 426.084i 1.38790i −0.720024 0.693949i \(-0.755869\pi\)
0.720024 0.693949i \(-0.244131\pi\)
\(308\) 178.500 + 309.171i 0.579545 + 1.00380i
\(309\) 0 0
\(310\) 39.0000 0.125806
\(311\) 159.349i 0.512375i −0.966627 0.256188i \(-0.917534\pi\)
0.966627 0.256188i \(-0.0824665\pi\)
\(312\) 0 0
\(313\) 386.247i 1.23402i −0.786956 0.617009i \(-0.788344\pi\)
0.786956 0.617009i \(-0.211656\pi\)
\(314\) 200.918i 0.639866i
\(315\) 0 0
\(316\) 138.000 0.436709
\(317\) 227.000 0.716088 0.358044 0.933705i \(-0.383444\pi\)
0.358044 + 0.933705i \(0.383444\pi\)
\(318\) 0 0
\(319\) 34.0000 0.106583
\(320\) 22.5167i 0.0703646i
\(321\) 0 0
\(322\) 112.000 + 193.990i 0.347826 + 0.602452i
\(323\) 648.000 2.00619
\(324\) 0 0
\(325\) 381.051i 1.17247i
\(326\) 226.000 0.693252
\(327\) 0 0
\(328\) 96.9948i 0.295716i
\(329\) −231.000 + 133.368i −0.702128 + 0.405374i
\(330\) 0 0
\(331\) 398.000 1.20242 0.601208 0.799092i \(-0.294686\pi\)
0.601208 + 0.799092i \(0.294686\pi\)
\(332\) 306.573i 0.923413i
\(333\) 0 0
\(334\) 86.6025i 0.259289i
\(335\) 58.8897i 0.175790i
\(336\) 0 0
\(337\) −610.000 −1.81009 −0.905045 0.425317i \(-0.860163\pi\)
−0.905045 + 0.425317i \(0.860163\pi\)
\(338\) 131.000 0.387574
\(339\) 0 0
\(340\) 162.000 0.476471
\(341\) 382.783i 1.12253i
\(342\) 0 0
\(343\) 343.000 1.00000
\(344\) 182.000 0.529070
\(345\) 0 0
\(346\) 74.4782i 0.215255i
\(347\) 515.000 1.48415 0.742075 0.670317i \(-0.233842\pi\)
0.742075 + 0.670317i \(0.233842\pi\)
\(348\) 0 0
\(349\) 325.626i 0.933025i 0.884515 + 0.466512i \(0.154490\pi\)
−0.884515 + 0.466512i \(0.845510\pi\)
\(350\) 77.0000 + 133.368i 0.220000 + 0.381051i
\(351\) 0 0
\(352\) −561.000 −1.59375
\(353\) 235.559i 0.667306i −0.942696 0.333653i \(-0.891719\pi\)
0.942696 0.333653i \(-0.108281\pi\)
\(354\) 0 0
\(355\) 17.3205i 0.0487902i
\(356\) 155.885i 0.437878i
\(357\) 0 0
\(358\) −119.000 −0.332402
\(359\) −226.000 −0.629526 −0.314763 0.949170i \(-0.601925\pi\)
−0.314763 + 0.949170i \(0.601925\pi\)
\(360\) 0 0
\(361\) −71.0000 −0.196676
\(362\) 62.3538i 0.172248i
\(363\) 0 0
\(364\) −315.000 + 181.865i −0.865385 + 0.499630i
\(365\) 99.0000 0.271233
\(366\) 0 0
\(367\) 531.740i 1.44888i −0.689337 0.724441i \(-0.742098\pi\)
0.689337 0.724441i \(-0.257902\pi\)
\(368\) 160.000 0.434783
\(369\) 0 0
\(370\) 13.8564i 0.0374497i
\(371\) 3.50000 + 6.06218i 0.00943396 + 0.0163401i
\(372\) 0 0
\(373\) 218.000 0.584450 0.292225 0.956350i \(-0.405604\pi\)
0.292225 + 0.956350i \(0.405604\pi\)
\(374\) 530.008i 1.41713i
\(375\) 0 0
\(376\) 266.736i 0.709404i
\(377\) 34.6410i 0.0918860i
\(378\) 0 0
\(379\) 296.000 0.781003 0.390501 0.920602i \(-0.372302\pi\)
0.390501 + 0.920602i \(0.372302\pi\)
\(380\) −108.000 −0.284211
\(381\) 0 0
\(382\) 190.000 0.497382
\(383\) 640.859i 1.67326i 0.547768 + 0.836630i \(0.315478\pi\)
−0.547768 + 0.836630i \(0.684522\pi\)
\(384\) 0 0
\(385\) 178.500 103.057i 0.463636 0.267681i
\(386\) −311.000 −0.805699
\(387\) 0 0
\(388\) 275.396i 0.709784i
\(389\) −457.000 −1.17481 −0.587404 0.809294i \(-0.699850\pi\)
−0.587404 + 0.809294i \(0.699850\pi\)
\(390\) 0 0
\(391\) 997.661i 2.55156i
\(392\) −171.500 + 297.047i −0.437500 + 0.757772i
\(393\) 0 0
\(394\) 307.000 0.779188
\(395\) 79.6743i 0.201707i
\(396\) 0 0
\(397\) 114.315i 0.287948i 0.989581 + 0.143974i \(0.0459882\pi\)
−0.989581 + 0.143974i \(0.954012\pi\)
\(398\) 254.611i 0.639727i
\(399\) 0 0
\(400\) 110.000 0.275000
\(401\) −448.000 −1.11721 −0.558603 0.829435i \(-0.688663\pi\)
−0.558603 + 0.829435i \(0.688663\pi\)
\(402\) 0 0
\(403\) −390.000 −0.967742
\(404\) 368.927i 0.913185i
\(405\) 0 0
\(406\) 7.00000 + 12.1244i 0.0172414 + 0.0298629i
\(407\) 136.000 0.334152
\(408\) 0 0
\(409\) 476.314i 1.16458i −0.812980 0.582291i \(-0.802156\pi\)
0.812980 0.582291i \(-0.197844\pi\)
\(410\) −24.0000 −0.0585366
\(411\) 0 0
\(412\) 20.7846i 0.0504481i
\(413\) 546.000 315.233i 1.32203 0.763277i
\(414\) 0 0
\(415\) −177.000 −0.426506
\(416\) 571.577i 1.37398i
\(417\) 0 0
\(418\) 353.338i 0.845307i
\(419\) 242.487i 0.578728i −0.957219 0.289364i \(-0.906556\pi\)
0.957219 0.289364i \(-0.0934438\pi\)
\(420\) 0 0
\(421\) 548.000 1.30166 0.650831 0.759222i \(-0.274420\pi\)
0.650831 + 0.759222i \(0.274420\pi\)
\(422\) 196.000 0.464455
\(423\) 0 0
\(424\) −7.00000 −0.0165094
\(425\) 685.892i 1.61386i
\(426\) 0 0
\(427\) −420.000 + 242.487i −0.983607 + 0.567886i
\(428\) −321.000 −0.750000
\(429\) 0 0
\(430\) 45.0333i 0.104729i
\(431\) −358.000 −0.830626 −0.415313 0.909678i \(-0.636328\pi\)
−0.415313 + 0.909678i \(0.636328\pi\)
\(432\) 0 0
\(433\) 161.081i 0.372011i 0.982549 + 0.186005i \(0.0595542\pi\)
−0.982549 + 0.186005i \(0.940446\pi\)
\(434\) −136.500 + 78.8083i −0.314516 + 0.181586i
\(435\) 0 0
\(436\) 264.000 0.605505
\(437\) 665.108i 1.52199i
\(438\) 0 0
\(439\) 334.286i 0.761471i −0.924684 0.380736i \(-0.875671\pi\)
0.924684 0.380736i \(-0.124329\pi\)
\(440\) 206.114i 0.468441i
\(441\) 0 0
\(442\) 540.000 1.22172
\(443\) 650.000 1.46727 0.733634 0.679544i \(-0.237823\pi\)
0.733634 + 0.679544i \(0.237823\pi\)
\(444\) 0 0
\(445\) 90.0000 0.202247
\(446\) 76.2102i 0.170875i
\(447\) 0 0
\(448\) −45.5000 78.8083i −0.101562 0.175911i
\(449\) −316.000 −0.703786 −0.351893 0.936040i \(-0.614462\pi\)
−0.351893 + 0.936040i \(0.614462\pi\)
\(450\) 0 0
\(451\) 235.559i 0.522304i
\(452\) −600.000 −1.32743
\(453\) 0 0
\(454\) 76.2102i 0.167864i
\(455\) 105.000 + 181.865i 0.230769 + 0.399704i
\(456\) 0 0
\(457\) −19.0000 −0.0415755 −0.0207877 0.999784i \(-0.506617\pi\)
−0.0207877 + 0.999784i \(0.506617\pi\)
\(458\) 24.2487i 0.0529448i
\(459\) 0 0
\(460\) 166.277i 0.361471i
\(461\) 355.070i 0.770218i −0.922871 0.385109i \(-0.874164\pi\)
0.922871 0.385109i \(-0.125836\pi\)
\(462\) 0 0
\(463\) −835.000 −1.80346 −0.901728 0.432304i \(-0.857701\pi\)
−0.901728 + 0.432304i \(0.857701\pi\)
\(464\) 10.0000 0.0215517
\(465\) 0 0
\(466\) 178.000 0.381974
\(467\) 431.281i 0.923513i 0.887007 + 0.461757i \(0.152781\pi\)
−0.887007 + 0.461757i \(0.847219\pi\)
\(468\) 0 0
\(469\) 119.000 + 206.114i 0.253731 + 0.439476i
\(470\) −66.0000 −0.140426
\(471\) 0 0
\(472\) 630.466i 1.33573i
\(473\) 442.000 0.934461
\(474\) 0 0
\(475\) 457.261i 0.962656i
\(476\) −567.000 + 327.358i −1.19118 + 0.687726i
\(477\) 0 0
\(478\) 88.0000 0.184100
\(479\) 620.074i 1.29452i −0.762270 0.647259i \(-0.775915\pi\)
0.762270 0.647259i \(-0.224085\pi\)
\(480\) 0 0
\(481\) 138.564i 0.288075i
\(482\) 200.918i 0.416842i
\(483\) 0 0
\(484\) −504.000 −1.04132
\(485\) −159.000 −0.327835
\(486\) 0 0
\(487\) 98.0000 0.201232 0.100616 0.994925i \(-0.467919\pi\)
0.100616 + 0.994925i \(0.467919\pi\)
\(488\) 484.974i 0.993800i
\(489\) 0 0
\(490\) 73.5000 + 42.4352i 0.150000 + 0.0866025i
\(491\) −283.000 −0.576375 −0.288187 0.957574i \(-0.593053\pi\)
−0.288187 + 0.957574i \(0.593053\pi\)
\(492\) 0 0
\(493\) 62.3538i 0.126478i
\(494\) −360.000 −0.728745
\(495\) 0 0
\(496\) 112.583i 0.226982i
\(497\) 35.0000 + 60.6218i 0.0704225 + 0.121975i
\(498\) 0 0
\(499\) 302.000 0.605210 0.302605 0.953116i \(-0.402144\pi\)
0.302605 + 0.953116i \(0.402144\pi\)
\(500\) 244.219i 0.488438i
\(501\) 0 0
\(502\) 311.769i 0.621054i
\(503\) 665.108i 1.32228i −0.750262 0.661141i \(-0.770073\pi\)
0.750262 0.661141i \(-0.229927\pi\)
\(504\) 0 0
\(505\) −213.000 −0.421782
\(506\) −544.000 −1.07510
\(507\) 0 0
\(508\) −429.000 −0.844488
\(509\) 583.701i 1.14676i 0.819289 + 0.573380i \(0.194368\pi\)
−0.819289 + 0.573380i \(0.805632\pi\)
\(510\) 0 0
\(511\) −346.500 + 200.052i −0.678082 + 0.391491i
\(512\) −305.000 −0.595703
\(513\) 0 0
\(514\) 3.46410i 0.00673950i
\(515\) −12.0000 −0.0233010
\(516\) 0 0
\(517\) 647.787i 1.25297i
\(518\) 28.0000 + 48.4974i 0.0540541 + 0.0936244i
\(519\) 0 0
\(520\) −210.000 −0.403846
\(521\) 280.592i 0.538565i −0.963061 0.269282i \(-0.913214\pi\)
0.963061 0.269282i \(-0.0867865\pi\)
\(522\) 0 0
\(523\) 966.484i 1.84796i −0.382438 0.923981i \(-0.624915\pi\)
0.382438 0.923981i \(-0.375085\pi\)
\(524\) 36.3731i 0.0694142i
\(525\) 0 0
\(526\) −116.000 −0.220532
\(527\) −702.000 −1.33207
\(528\) 0 0
\(529\) 495.000 0.935728
\(530\) 1.73205i 0.00326802i
\(531\) 0 0
\(532\) 378.000 218.238i 0.710526 0.410223i
\(533\) 240.000 0.450281
\(534\) 0 0
\(535\) 185.329i 0.346410i
\(536\) −238.000 −0.444030
\(537\) 0 0
\(538\) 353.338i 0.656763i
\(539\) −416.500 + 721.399i −0.772727 + 1.33840i
\(540\) 0 0
\(541\) −784.000 −1.44917 −0.724584 0.689186i \(-0.757968\pi\)
−0.724584 + 0.689186i \(0.757968\pi\)
\(542\) 67.5500i 0.124631i
\(543\) 0 0
\(544\) 1028.84i 1.89125i
\(545\) 152.420i 0.279671i
\(546\) 0 0
\(547\) 26.0000 0.0475320 0.0237660 0.999718i \(-0.492434\pi\)
0.0237660 + 0.999718i \(0.492434\pi\)
\(548\) 30.0000 0.0547445
\(549\) 0 0
\(550\) −374.000 −0.680000
\(551\) 41.5692i 0.0754432i
\(552\) 0 0
\(553\) 161.000 + 278.860i 0.291139 + 0.504268i
\(554\) 136.000 0.245487
\(555\) 0 0
\(556\) 415.692i 0.747648i
\(557\) 629.000 1.12926 0.564632 0.825343i \(-0.309018\pi\)
0.564632 + 0.825343i \(0.309018\pi\)
\(558\) 0 0
\(559\) 450.333i 0.805605i
\(560\) 52.5000 30.3109i 0.0937500 0.0541266i
\(561\) 0 0
\(562\) 370.000 0.658363
\(563\) 874.686i 1.55362i 0.629738 + 0.776808i \(0.283162\pi\)
−0.629738 + 0.776808i \(0.716838\pi\)
\(564\) 0 0
\(565\) 346.410i 0.613115i
\(566\) 401.836i 0.709957i
\(567\) 0 0
\(568\) −70.0000 −0.123239
\(569\) 116.000 0.203866 0.101933 0.994791i \(-0.467497\pi\)
0.101933 + 0.994791i \(0.467497\pi\)
\(570\) 0 0
\(571\) 590.000 1.03327 0.516637 0.856204i \(-0.327184\pi\)
0.516637 + 0.856204i \(0.327184\pi\)
\(572\) 883.346i 1.54431i
\(573\) 0 0
\(574\) 84.0000 48.4974i 0.146341 0.0844903i
\(575\) 704.000 1.22435
\(576\) 0 0
\(577\) 270.200i 0.468284i 0.972202 + 0.234142i \(0.0752281\pi\)
−0.972202 + 0.234142i \(0.924772\pi\)
\(578\) 683.000 1.18166
\(579\) 0 0
\(580\) 10.3923i 0.0179178i
\(581\) 619.500 357.668i 1.06627 0.615608i
\(582\) 0 0
\(583\) −17.0000 −0.0291595
\(584\) 400.104i 0.685109i
\(585\) 0 0
\(586\) 48.4974i 0.0827601i
\(587\) 174.937i 0.298019i 0.988836 + 0.149009i \(0.0476085\pi\)
−0.988836 + 0.149009i \(0.952392\pi\)
\(588\) 0 0
\(589\) 468.000 0.794567
\(590\) 156.000 0.264407
\(591\) 0 0
\(592\) 40.0000 0.0675676
\(593\) 592.361i 0.998923i 0.866336 + 0.499462i \(0.166469\pi\)
−0.866336 + 0.499462i \(0.833531\pi\)
\(594\) 0 0
\(595\) 189.000 + 327.358i 0.317647 + 0.550181i
\(596\) 237.000 0.397651
\(597\) 0 0
\(598\) 554.256i 0.926850i
\(599\) −106.000 −0.176962 −0.0884808 0.996078i \(-0.528201\pi\)
−0.0884808 + 0.996078i \(0.528201\pi\)
\(600\) 0 0
\(601\) 365.463i 0.608091i −0.952658 0.304046i \(-0.901663\pi\)
0.952658 0.304046i \(-0.0983375\pi\)
\(602\) 91.0000 + 157.617i 0.151163 + 0.261822i
\(603\) 0 0
\(604\) 219.000 0.362583
\(605\) 290.985i 0.480966i
\(606\) 0 0
\(607\) 96.9948i 0.159794i −0.996803 0.0798969i \(-0.974541\pi\)
0.996803 0.0798969i \(-0.0254591\pi\)
\(608\) 685.892i 1.12811i
\(609\) 0 0
\(610\) −120.000 −0.196721
\(611\) 660.000 1.08020
\(612\) 0 0
\(613\) −802.000 −1.30832 −0.654160 0.756356i \(-0.726978\pi\)
−0.654160 + 0.756356i \(0.726978\pi\)
\(614\) 426.084i 0.693949i
\(615\) 0 0
\(616\) −416.500 721.399i −0.676136 1.17110i
\(617\) 104.000 0.168558 0.0842788 0.996442i \(-0.473141\pi\)
0.0842788 + 0.996442i \(0.473141\pi\)
\(618\) 0 0
\(619\) 523.079i 0.845039i 0.906354 + 0.422520i \(0.138854\pi\)
−0.906354 + 0.422520i \(0.861146\pi\)
\(620\) 117.000 0.188710
\(621\) 0 0
\(622\) 159.349i 0.256188i
\(623\) −315.000 + 181.865i −0.505618 + 0.291919i
\(624\) 0 0
\(625\) 409.000 0.654400
\(626\) 386.247i 0.617009i
\(627\) 0 0
\(628\) 602.754i 0.959799i
\(629\) 249.415i 0.396527i
\(630\) 0 0
\(631\) 767.000 1.21553 0.607765 0.794117i \(-0.292066\pi\)
0.607765 + 0.794117i \(0.292066\pi\)
\(632\) −322.000 −0.509494
\(633\) 0 0
\(634\) −227.000 −0.358044
\(635\) 247.683i 0.390052i
\(636\) 0 0
\(637\) −735.000 424.352i −1.15385 0.666173i
\(638\) −34.0000 −0.0532915
\(639\) 0 0
\(640\) 206.114i 0.322053i
\(641\) 260.000 0.405616 0.202808 0.979218i \(-0.434993\pi\)
0.202808 + 0.979218i \(0.434993\pi\)
\(642\) 0 0
\(643\) 159.349i 0.247821i −0.992293 0.123910i \(-0.960456\pi\)
0.992293 0.123910i \(-0.0395435\pi\)
\(644\) 336.000 + 581.969i 0.521739 + 0.903679i
\(645\) 0 0
\(646\) −648.000 −1.00310
\(647\) 519.615i 0.803115i 0.915834 + 0.401557i \(0.131531\pi\)
−0.915834 + 0.401557i \(0.868469\pi\)
\(648\) 0 0
\(649\) 1531.13i 2.35922i
\(650\) 381.051i 0.586233i
\(651\) 0 0
\(652\) 678.000 1.03988
\(653\) 1073.00 1.64319 0.821593 0.570075i \(-0.193086\pi\)
0.821593 + 0.570075i \(0.193086\pi\)
\(654\) 0 0
\(655\) −21.0000 −0.0320611
\(656\) 69.2820i 0.105613i
\(657\) 0 0
\(658\) 231.000 133.368i 0.351064 0.202687i
\(659\) 503.000 0.763278 0.381639 0.924312i \(-0.375360\pi\)
0.381639 + 0.924312i \(0.375360\pi\)
\(660\) 0 0
\(661\) 990.733i 1.49884i −0.662095 0.749420i \(-0.730333\pi\)
0.662095 0.749420i \(-0.269667\pi\)
\(662\) −398.000 −0.601208
\(663\) 0 0
\(664\) 715.337i 1.07731i
\(665\) −126.000 218.238i −0.189474 0.328178i
\(666\) 0 0
\(667\) 64.0000 0.0959520
\(668\) 259.808i 0.388934i
\(669\) 0 0
\(670\) 58.8897i 0.0878951i
\(671\) 1177.79i 1.75528i
\(672\) 0 0
\(673\) 305.000 0.453195 0.226597 0.973989i \(-0.427240\pi\)
0.226597 + 0.973989i \(0.427240\pi\)
\(674\) 610.000 0.905045
\(675\) 0 0
\(676\) 393.000 0.581361
\(677\) 6.92820i 0.0102337i −0.999987 0.00511684i \(-0.998371\pi\)
0.999987 0.00511684i \(-0.00162875\pi\)
\(678\) 0 0
\(679\) 556.500 321.295i 0.819588 0.473189i
\(680\) −378.000 −0.555882
\(681\) 0 0
\(682\) 382.783i 0.561266i
\(683\) −154.000 −0.225476 −0.112738 0.993625i \(-0.535962\pi\)
−0.112738 + 0.993625i \(0.535962\pi\)
\(684\) 0 0
\(685\) 17.3205i 0.0252854i
\(686\) −343.000 −0.500000
\(687\) 0 0
\(688\) 130.000 0.188953
\(689\) 17.3205i 0.0251386i
\(690\) 0 0
\(691\) 13.8564i 0.0200527i 0.999950 + 0.0100263i \(0.00319154\pi\)
−0.999950 + 0.0100263i \(0.996808\pi\)
\(692\) 223.435i 0.322882i
\(693\) 0 0
\(694\) −515.000 −0.742075
\(695\) 240.000 0.345324
\(696\) 0 0
\(697\) 432.000 0.619799
\(698\) 325.626i 0.466512i
\(699\) 0 0
\(700\) 231.000 + 400.104i 0.330000 + 0.571577i
\(701\) −469.000 −0.669044 −0.334522 0.942388i \(-0.608575\pi\)
−0.334522 + 0.942388i \(0.608575\pi\)
\(702\) 0 0
\(703\) 166.277i 0.236525i
\(704\) 221.000 0.313920
\(705\) 0 0
\(706\) 235.559i 0.333653i
\(707\) 745.500 430.415i 1.05446 0.608790i
\(708\) 0 0
\(709\) 80.0000 0.112835 0.0564175 0.998407i \(-0.482032\pi\)
0.0564175 + 0.998407i \(0.482032\pi\)
\(710\) 17.3205i 0.0243951i
\(711\) 0 0
\(712\) 363.731i 0.510858i
\(713\) 720.533i 1.01057i
\(714\) 0 0
\(715\) −510.000 −0.713287
\(716\) −357.000 −0.498603
\(717\) 0 0
\(718\) 226.000 0.314763
\(719\) 769.031i 1.06958i −0.844984 0.534792i \(-0.820390\pi\)
0.844984 0.534792i \(-0.179610\pi\)
\(720\) 0 0
\(721\) 42.0000 24.2487i 0.0582524 0.0336321i
\(722\) 71.0000 0.0983380
\(723\) 0 0
\(724\) 187.061i 0.258372i
\(725\) 44.0000 0.0606897
\(726\) 0 0
\(727\) 549.060i 0.755241i 0.925960 + 0.377620i \(0.123258\pi\)
−0.925960 + 0.377620i \(0.876742\pi\)
\(728\) 735.000 424.352i 1.00962 0.582902i
\(729\) 0 0
\(730\) −99.0000 −0.135616
\(731\) 810.600i 1.10889i
\(732\) 0 0
\(733\) 27.7128i 0.0378074i 0.999821 + 0.0189037i \(0.00601759\pi\)
−0.999821 + 0.0189037i \(0.993982\pi\)
\(734\) 531.740i 0.724441i
\(735\) 0 0
\(736\) −1056.00 −1.43478
\(737\) −578.000 −0.784261
\(738\) 0 0
\(739\) −730.000 −0.987821 −0.493911 0.869513i \(-0.664433\pi\)
−0.493911 + 0.869513i \(0.664433\pi\)
\(740\) 41.5692i 0.0561746i
\(741\) 0 0
\(742\) −3.50000 6.06218i −0.00471698 0.00817005i
\(743\) −934.000 −1.25707 −0.628533 0.777783i \(-0.716344\pi\)
−0.628533 + 0.777783i \(0.716344\pi\)
\(744\) 0 0
\(745\) 136.832i 0.183667i
\(746\) −218.000 −0.292225
\(747\) 0 0
\(748\) 1590.02i 2.12570i
\(749\) −374.500 648.653i −0.500000 0.866025i
\(750\) 0 0
\(751\) −763.000 −1.01598 −0.507989 0.861363i \(-0.669611\pi\)
−0.507989 + 0.861363i \(0.669611\pi\)
\(752\) 190.526i 0.253358i
\(753\) 0 0
\(754\) 34.6410i 0.0459430i
\(755\) 126.440i 0.167470i
\(756\) 0 0
\(757\) −682.000 −0.900925 −0.450462 0.892795i \(-0.648741\pi\)
−0.450462 + 0.892795i \(0.648741\pi\)
\(758\) −296.000 −0.390501
\(759\) 0 0
\(760\) 252.000 0.331579
\(761\) 193.990i 0.254914i 0.991844 + 0.127457i \(0.0406815\pi\)
−0.991844 + 0.127457i \(0.959318\pi\)
\(762\) 0 0
\(763\) 308.000 + 533.472i 0.403670 + 0.699176i
\(764\) 570.000 0.746073
\(765\) 0 0
\(766\) 640.859i 0.836630i
\(767\) −1560.00 −2.03390
\(768\) 0 0
\(769\) 1255.74i 1.63295i −0.577382 0.816474i \(-0.695926\pi\)
0.577382 0.816474i \(-0.304074\pi\)
\(770\) −178.500 + 103.057i −0.231818 + 0.133840i
\(771\) 0 0
\(772\) −933.000 −1.20855
\(773\) 665.108i 0.860424i 0.902728 + 0.430212i \(0.141561\pi\)
−0.902728 + 0.430212i \(0.858439\pi\)
\(774\) 0 0
\(775\) 495.367i 0.639183i
\(776\) 642.591i 0.828081i
\(777\) 0 0
\(778\) 457.000 0.587404
\(779\) −288.000 −0.369705
\(780\) 0 0
\(781\) −170.000 −0.217670
\(782\) 997.661i 1.27578i
\(783\) 0 0
\(784\) −122.500 + 212.176i −0.156250 + 0.270633i
\(785\) 348.000 0.443312
\(786\) 0 0
\(787\) 1150.08i 1.46135i 0.682726 + 0.730675i \(0.260794\pi\)
−0.682726 + 0.730675i \(0.739206\pi\)
\(788\) 921.000 1.16878
\(789\) 0 0
\(790\) 79.6743i 0.100854i
\(791\) −700.000 1212.44i −0.884956 1.53279i
\(792\) 0 0
\(793\) 1200.00 1.51324
\(794\) 114.315i 0.143974i
\(795\) 0 0
\(796\) 763.834i 0.959591i
\(797\) 968.216i 1.21483i 0.794386 + 0.607413i \(0.207793\pi\)
−0.794386 + 0.607413i \(0.792207\pi\)
\(798\) 0 0
\(799\) 1188.00 1.48686
\(800\) −726.000 −0.907500
\(801\) 0 0
\(802\) 448.000 0.558603
\(803\) 971.681i 1.21006i
\(804\) 0 0
\(805\) 336.000 193.990i 0.417391 0.240981i
\(806\) 390.000 0.483871
\(807\) 0 0
\(808\) 860.829i 1.06538i
\(809\) 638.000 0.788628 0.394314 0.918976i \(-0.370982\pi\)
0.394314 + 0.918976i \(0.370982\pi\)
\(810\) 0 0
\(811\) 581.969i 0.717594i −0.933416 0.358797i \(-0.883187\pi\)
0.933416 0.358797i \(-0.116813\pi\)
\(812\) 21.0000 + 36.3731i 0.0258621 + 0.0447944i
\(813\) 0 0
\(814\) −136.000 −0.167076
\(815\) 391.443i 0.480299i
\(816\) 0 0
\(817\) 540.400i 0.661444i
\(818\) 476.314i 0.582291i
\(819\) 0 0
\(820\) −72.0000 −0.0878049
\(821\) −262.000 −0.319123 −0.159562 0.987188i \(-0.551008\pi\)
−0.159562 + 0.987188i \(0.551008\pi\)
\(822\) 0 0
\(823\) −769.000 −0.934386 −0.467193 0.884155i \(-0.654735\pi\)
−0.467193 + 0.884155i \(0.654735\pi\)
\(824\) 48.4974i 0.0588561i
\(825\) 0 0
\(826\) −546.000 + 315.233i −0.661017 + 0.381638i
\(827\) −442.000 −0.534462 −0.267231 0.963633i \(-0.586109\pi\)
−0.267231 + 0.963633i \(0.586109\pi\)
\(828\) 0 0
\(829\) 1340.61i 1.61714i 0.588401 + 0.808569i \(0.299757\pi\)
−0.588401 + 0.808569i \(0.700243\pi\)
\(830\) 177.000 0.213253
\(831\) 0 0
\(832\) 225.167i 0.270633i
\(833\) −1323.00 763.834i −1.58824 0.916968i
\(834\) 0 0
\(835\) 150.000 0.179641
\(836\) 1060.02i 1.26796i
\(837\) 0 0
\(838\) 242.487i 0.289364i
\(839\) 1118.90i 1.33362i −0.745229 0.666809i \(-0.767660\pi\)
0.745229 0.666809i \(-0.232340\pi\)
\(840\) 0 0
\(841\) −837.000 −0.995244
\(842\) −548.000 −0.650831
\(843\) 0 0
\(844\) 588.000 0.696682
\(845\) 226.899i 0.268519i
\(846\) 0 0
\(847\) −588.000 1018.45i −0.694215 1.20242i
\(848\) −5.00000 −0.00589623
\(849\) 0 0
\(850\) 685.892i 0.806932i
\(851\) 256.000 0.300823
\(852\) 0 0
\(853\) 1077.34i 1.26300i −0.775377 0.631498i \(-0.782440\pi\)
0.775377 0.631498i \(-0.217560\pi\)
\(854\) 420.000 242.487i 0.491803 0.283943i
\(855\) 0 0
\(856\) 749.000 0.875000
\(857\) 100.459i 0.117222i −0.998281 0.0586108i \(-0.981333\pi\)
0.998281 0.0586108i \(-0.0186671\pi\)
\(858\) 0 0
\(859\) 96.9948i 0.112916i −0.998405 0.0564580i \(-0.982019\pi\)
0.998405 0.0564580i \(-0.0179807\pi\)
\(860\) 135.100i 0.157093i
\(861\) 0 0
\(862\) 358.000 0.415313
\(863\) −1216.00 −1.40904 −0.704519 0.709685i \(-0.748837\pi\)
−0.704519 + 0.709685i \(0.748837\pi\)
\(864\) 0 0
\(865\) 129.000 0.149133
\(866\) 161.081i 0.186005i
\(867\) 0 0
\(868\) −409.500 + 236.425i −0.471774 + 0.272379i
\(869\) −782.000 −0.899885
\(870\) 0 0
\(871\) 588.897i 0.676116i
\(872\) −616.000 −0.706422
\(873\) 0 0
\(874\) 665.108i 0.760993i
\(875\) 493.500 284.922i 0.564000 0.325626i
\(876\) 0 0
\(877\) 254.000 0.289624 0.144812 0.989459i \(-0.453742\pi\)
0.144812 + 0.989459i \(0.453742\pi\)
\(878\) 334.286i 0.380736i
\(879\) 0 0
\(880\) 147.224i 0.167300i
\(881\) 654.715i 0.743150i 0.928403 + 0.371575i \(0.121182\pi\)
−0.928403 + 0.371575i \(0.878818\pi\)
\(882\) 0 0
\(883\) 80.0000 0.0906002 0.0453001 0.998973i \(-0.485576\pi\)
0.0453001 + 0.998973i \(0.485576\pi\)
\(884\) 1620.00 1.83258
\(885\) 0 0
\(886\) −650.000 −0.733634
\(887\) 1063.48i 1.19896i −0.800389 0.599481i \(-0.795374\pi\)
0.800389 0.599481i \(-0.204626\pi\)
\(888\) 0 0
\(889\) −500.500 866.891i −0.562992 0.975131i
\(890\) −90.0000 −0.101124
\(891\) 0 0
\(892\) 228.631i 0.256312i
\(893\) −792.000 −0.886898
\(894\) 0 0
\(895\) 206.114i 0.230295i
\(896\) −416.500 721.399i −0.464844 0.805133i
\(897\) 0 0
\(898\) 316.000 0.351893
\(899\) 45.0333i 0.0500927i
\(900\) 0 0
\(901\) 31.1769i 0.0346026i
\(902\) 235.559i 0.261152i
\(903\) 0 0
\(904\) 1400.00 1.54867
\(905\) −108.000 −0.119337
\(906\) 0 0
\(907\) −208.000 −0.229327 −0.114664 0.993404i \(-0.536579\pi\)
−0.114664 + 0.993404i \(0.536579\pi\)
\(908\) 228.631i 0.251796i
\(909\) 0 0
\(910\) −105.000 181.865i −0.115385 0.199852i
\(911\) −1654.00 −1.81559 −0.907794 0.419417i \(-0.862234\pi\)
−0.907794 + 0.419417i \(0.862234\pi\)
\(912\) 0 0
\(913\) 1737.25i 1.90279i
\(914\) 19.0000 0.0207877
\(915\) 0 0
\(916\) 72.7461i 0.0794172i
\(917\) 73.5000 42.4352i 0.0801527 0.0462762i
\(918\) 0 0
\(919\) 53.0000 0.0576714 0.0288357 0.999584i \(-0.490820\pi\)
0.0288357 + 0.999584i \(0.490820\pi\)
\(920\) 387.979i 0.421717i
\(921\) 0 0
\(922\) 355.070i 0.385109i
\(923\) 173.205i 0.187654i
\(924\) 0 0
\(925\) 176.000 0.190270
\(926\) 835.000 0.901728
\(927\) 0 0
\(928\) −66.0000 −0.0711207
\(929\) 297.913i 0.320681i −0.987062 0.160341i \(-0.948741\pi\)
0.987062 0.160341i \(-0.0512592\pi\)
\(930\) 0 0
\(931\) 882.000 + 509.223i 0.947368 + 0.546963i
\(932\) 534.000 0.572961
\(933\) 0 0
\(934\) 431.281i 0.461757i
\(935\) −918.000 −0.981818
\(936\) 0 0
\(937\) 1834.24i 1.95757i 0.204893 + 0.978784i \(0.434315\pi\)
−0.204893 + 0.978784i \(0.565685\pi\)
\(938\) −119.000 206.114i −0.126866 0.219738i
\(939\) 0 0
\(940\) −198.000 −0.210638
\(941\) 1373.52i 1.45963i 0.683642 + 0.729817i \(0.260395\pi\)
−0.683642 + 0.729817i \(0.739605\pi\)
\(942\) 0 0
\(943\) 443.405i 0.470207i
\(944\) 450.333i 0.477048i
\(945\) 0 0
\(946\) −442.000 −0.467230
\(947\) −385.000 −0.406547 −0.203273 0.979122i \(-0.565158\pi\)
−0.203273 + 0.979122i \(0.565158\pi\)
\(948\) 0 0
\(949\) 990.000 1.04320
\(950\) 457.261i 0.481328i
\(951\) 0 0
\(952\) 1323.00 763.834i 1.38971 0.802347i
\(953\) 1244.00 1.30535 0.652676 0.757637i \(-0.273646\pi\)
0.652676 + 0.757637i \(0.273646\pi\)
\(954\) 0 0
\(955\) 329.090i 0.344596i
\(956\) 264.000 0.276151
\(957\) 0 0
\(958\) 620.074i 0.647259i
\(959\) 35.0000 + 60.6218i 0.0364964 + 0.0632135i
\(960\) 0 0
\(961\) 454.000 0.472425
\(962\) 138.564i 0.144037i
\(963\) 0 0
\(964\) 602.754i 0.625263i
\(965\) 538.668i 0.558205i
\(966\) 0 0
\(967\) −535.000 −0.553257 −0.276629 0.960977i \(-0.589217\pi\)
−0.276629 + 0.960977i \(0.589217\pi\)
\(968\) 1176.00 1.21488
\(969\) 0 0
\(970\) 159.000 0.163918
\(971\) 743.050i 0.765242i 0.923905 + 0.382621i \(0.124978\pi\)
−0.923905 + 0.382621i \(0.875022\pi\)
\(972\) 0 0
\(973\) −840.000 + 484.974i −0.863309 + 0.498432i
\(974\) −98.0000 −0.100616
\(975\) 0 0
\(976\) 346.410i 0.354928i
\(977\) 950.000 0.972364 0.486182 0.873857i \(-0.338389\pi\)
0.486182 + 0.873857i \(0.338389\pi\)
\(978\) 0 0
\(979\) 883.346i 0.902294i
\(980\) 220.500 + 127.306i 0.225000 + 0.129904i
\(981\) 0 0
\(982\) 283.000 0.288187
\(983\) 1603.88i 1.63162i 0.578322 + 0.815808i \(0.303708\pi\)
−0.578322 + 0.815808i \(0.696292\pi\)
\(984\) 0 0
\(985\) 531.740i 0.539837i
\(986\) 62.3538i 0.0632392i
\(987\) 0 0
\(988\) −1080.00 −1.09312
\(989\) 832.000 0.841254
\(990\) 0 0
\(991\) 1151.00 1.16145 0.580727 0.814099i \(-0.302769\pi\)
0.580727 + 0.814099i \(0.302769\pi\)
\(992\) 743.050i 0.749042i
\(993\) 0 0
\(994\) −35.0000 60.6218i −0.0352113 0.0609877i
\(995\) 441.000 0.443216
\(996\) 0 0
\(997\) 370.659i 0.371774i −0.982571 0.185887i \(-0.940484\pi\)
0.982571 0.185887i \(-0.0595159\pi\)
\(998\) −302.000 −0.302605
\(999\) 0 0
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 189.3.d.a.55.2 yes 2
3.2 odd 2 189.3.d.c.55.1 yes 2
7.6 odd 2 inner 189.3.d.a.55.1 2
21.20 even 2 189.3.d.c.55.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.3.d.a.55.1 2 7.6 odd 2 inner
189.3.d.a.55.2 yes 2 1.1 even 1 trivial
189.3.d.c.55.1 yes 2 3.2 odd 2
189.3.d.c.55.2 yes 2 21.20 even 2