Properties

Label 189.3.d.e.55.4
Level $189$
Weight $3$
Character 189.55
Analytic conductor $5.150$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Coefficient field: 8.0.2355463701504.14
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 79x^{4} + 18x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 55.4
Root \(1.48088 - 2.56496i\) of defining polynomial
Character \(\chi\) \(=\) 189.55
Dual form 189.3.d.e.55.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.28648 q^{2} +1.22800 q^{4} +6.53330i q^{5} +(5.77200 - 3.96030i) q^{7} +6.33813 q^{8} +O(q^{10})\) \(q-2.28648 q^{2} +1.22800 q^{4} +6.53330i q^{5} +(5.77200 - 3.96030i) q^{7} +6.33813 q^{8} -14.9383i q^{10} -13.1976 q^{11} -3.96030i q^{13} +(-13.1976 + 9.05516i) q^{14} -19.4040 q^{16} +15.5885i q^{17} +30.7795i q^{19} +8.02288i q^{20} +30.1760 q^{22} +1.76517 q^{23} -17.6840 q^{25} +9.05516i q^{26} +(7.08801 - 4.86324i) q^{28} -37.8276 q^{29} +57.5987i q^{31} +19.0144 q^{32} -35.6427i q^{34} +(25.8738 + 37.7102i) q^{35} -41.7720 q^{37} -70.3767i q^{38} +41.4089i q^{40} -24.6436i q^{41} -38.0880 q^{43} -16.2066 q^{44} -4.03602 q^{46} +32.6665i q^{47} +(17.6320 - 45.7177i) q^{49} +40.4341 q^{50} -4.86324i q^{52} +7.38076 q^{53} -86.2237i q^{55} +(36.5837 - 25.1009i) q^{56} +86.4920 q^{58} -59.8320i q^{59} +32.5854i q^{61} -131.698i q^{62} +34.1400 q^{64} +25.8738 q^{65} +91.5800 q^{67} +19.1426i q^{68} +(-59.1601 - 86.2237i) q^{70} +9.86847 q^{71} +23.7618i q^{73} +95.5109 q^{74} +37.7971i q^{76} +(-76.1764 + 52.2664i) q^{77} +74.6840 q^{79} -126.772i q^{80} +56.3472i q^{82} +126.197i q^{83} -101.844 q^{85} +87.0875 q^{86} -83.6480 q^{88} +27.1655i q^{89} +(-15.6840 - 22.8589i) q^{91} +2.16762 q^{92} -74.6913i q^{94} -201.092 q^{95} -48.4266i q^{97} +(-40.3153 + 104.533i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 44 q^{4} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 44 q^{4} + 12 q^{7} + 84 q^{16} - 32 q^{22} - 244 q^{25} - 80 q^{28} - 300 q^{37} - 168 q^{43} + 412 q^{46} - 64 q^{49} + 316 q^{58} + 444 q^{64} + 220 q^{67} + 552 q^{70} + 700 q^{79} + 108 q^{85} - 1216 q^{88} - 228 q^{91}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.28648 −1.14324 −0.571620 0.820518i \(-0.693685\pi\)
−0.571620 + 0.820518i \(0.693685\pi\)
\(3\) 0 0
\(4\) 1.22800 0.307000
\(5\) 6.53330i 1.30666i 0.757073 + 0.653330i \(0.226629\pi\)
−0.757073 + 0.653330i \(0.773371\pi\)
\(6\) 0 0
\(7\) 5.77200 3.96030i 0.824572 0.565757i
\(8\) 6.33813 0.792266
\(9\) 0 0
\(10\) 14.9383i 1.49383i
\(11\) −13.1976 −1.19978 −0.599890 0.800083i \(-0.704789\pi\)
−0.599890 + 0.800083i \(0.704789\pi\)
\(12\) 0 0
\(13\) 3.96030i 0.304639i −0.988331 0.152319i \(-0.951326\pi\)
0.988331 0.152319i \(-0.0486742\pi\)
\(14\) −13.1976 + 9.05516i −0.942684 + 0.646797i
\(15\) 0 0
\(16\) −19.4040 −1.21275
\(17\) 15.5885i 0.916968i 0.888703 + 0.458484i \(0.151607\pi\)
−0.888703 + 0.458484i \(0.848393\pi\)
\(18\) 0 0
\(19\) 30.7795i 1.61997i 0.586449 + 0.809986i \(0.300525\pi\)
−0.586449 + 0.809986i \(0.699475\pi\)
\(20\) 8.02288i 0.401144i
\(21\) 0 0
\(22\) 30.1760 1.37164
\(23\) 1.76517 0.0767464 0.0383732 0.999263i \(-0.487782\pi\)
0.0383732 + 0.999263i \(0.487782\pi\)
\(24\) 0 0
\(25\) −17.6840 −0.707360
\(26\) 9.05516i 0.348275i
\(27\) 0 0
\(28\) 7.08801 4.86324i 0.253143 0.173687i
\(29\) −37.8276 −1.30440 −0.652199 0.758048i \(-0.726153\pi\)
−0.652199 + 0.758048i \(0.726153\pi\)
\(30\) 0 0
\(31\) 57.5987i 1.85802i 0.370053 + 0.929011i \(0.379339\pi\)
−0.370053 + 0.929011i \(0.620661\pi\)
\(32\) 19.0144 0.594200
\(33\) 0 0
\(34\) 35.6427i 1.04832i
\(35\) 25.8738 + 37.7102i 0.739253 + 1.07743i
\(36\) 0 0
\(37\) −41.7720 −1.12897 −0.564487 0.825442i \(-0.690926\pi\)
−0.564487 + 0.825442i \(0.690926\pi\)
\(38\) 70.3767i 1.85202i
\(39\) 0 0
\(40\) 41.4089i 1.03522i
\(41\) 24.6436i 0.601064i −0.953772 0.300532i \(-0.902836\pi\)
0.953772 0.300532i \(-0.0971642\pi\)
\(42\) 0 0
\(43\) −38.0880 −0.885768 −0.442884 0.896579i \(-0.646045\pi\)
−0.442884 + 0.896579i \(0.646045\pi\)
\(44\) −16.2066 −0.368332
\(45\) 0 0
\(46\) −4.03602 −0.0877397
\(47\) 32.6665i 0.695032i 0.937674 + 0.347516i \(0.112975\pi\)
−0.937674 + 0.347516i \(0.887025\pi\)
\(48\) 0 0
\(49\) 17.6320 45.7177i 0.359837 0.933015i
\(50\) 40.4341 0.808683
\(51\) 0 0
\(52\) 4.86324i 0.0935239i
\(53\) 7.38076 0.139260 0.0696298 0.997573i \(-0.477818\pi\)
0.0696298 + 0.997573i \(0.477818\pi\)
\(54\) 0 0
\(55\) 86.2237i 1.56770i
\(56\) 36.5837 25.1009i 0.653280 0.448231i
\(57\) 0 0
\(58\) 86.4920 1.49124
\(59\) 59.8320i 1.01410i −0.861916 0.507051i \(-0.830736\pi\)
0.861916 0.507051i \(-0.169264\pi\)
\(60\) 0 0
\(61\) 32.5854i 0.534186i 0.963671 + 0.267093i \(0.0860631\pi\)
−0.963671 + 0.267093i \(0.913937\pi\)
\(62\) 131.698i 2.12417i
\(63\) 0 0
\(64\) 34.1400 0.533437
\(65\) 25.8738 0.398059
\(66\) 0 0
\(67\) 91.5800 1.36687 0.683433 0.730013i \(-0.260486\pi\)
0.683433 + 0.730013i \(0.260486\pi\)
\(68\) 19.1426i 0.281509i
\(69\) 0 0
\(70\) −59.1601 86.2237i −0.845144 1.23177i
\(71\) 9.86847 0.138992 0.0694962 0.997582i \(-0.477861\pi\)
0.0694962 + 0.997582i \(0.477861\pi\)
\(72\) 0 0
\(73\) 23.7618i 0.325504i 0.986667 + 0.162752i \(0.0520371\pi\)
−0.986667 + 0.162752i \(0.947963\pi\)
\(74\) 95.5109 1.29069
\(75\) 0 0
\(76\) 37.7971i 0.497331i
\(77\) −76.1764 + 52.2664i −0.989304 + 0.678784i
\(78\) 0 0
\(79\) 74.6840 0.945367 0.472684 0.881232i \(-0.343285\pi\)
0.472684 + 0.881232i \(0.343285\pi\)
\(80\) 126.772i 1.58465i
\(81\) 0 0
\(82\) 56.3472i 0.687161i
\(83\) 126.197i 1.52045i 0.649661 + 0.760224i \(0.274911\pi\)
−0.649661 + 0.760224i \(0.725089\pi\)
\(84\) 0 0
\(85\) −101.844 −1.19817
\(86\) 87.0875 1.01265
\(87\) 0 0
\(88\) −83.6480 −0.950545
\(89\) 27.1655i 0.305230i 0.988286 + 0.152615i \(0.0487695\pi\)
−0.988286 + 0.152615i \(0.951231\pi\)
\(90\) 0 0
\(91\) −15.6840 22.8589i −0.172352 0.251196i
\(92\) 2.16762 0.0235611
\(93\) 0 0
\(94\) 74.6913i 0.794589i
\(95\) −201.092 −2.11675
\(96\) 0 0
\(97\) 48.4266i 0.499243i −0.968343 0.249621i \(-0.919694\pi\)
0.968343 0.249621i \(-0.0803062\pi\)
\(98\) −40.3153 + 104.533i −0.411380 + 1.06666i
\(99\) 0 0
\(100\) −21.7159 −0.217159
\(101\) 158.864i 1.57291i −0.617649 0.786454i \(-0.711915\pi\)
0.617649 0.786454i \(-0.288085\pi\)
\(102\) 0 0
\(103\) 116.449i 1.13057i 0.824895 + 0.565285i \(0.191234\pi\)
−0.824895 + 0.565285i \(0.808766\pi\)
\(104\) 25.1009i 0.241355i
\(105\) 0 0
\(106\) −16.8760 −0.159207
\(107\) −64.0215 −0.598332 −0.299166 0.954201i \(-0.596708\pi\)
−0.299166 + 0.954201i \(0.596708\pi\)
\(108\) 0 0
\(109\) 20.9640 0.192330 0.0961650 0.995365i \(-0.469342\pi\)
0.0961650 + 0.995365i \(0.469342\pi\)
\(110\) 197.149i 1.79226i
\(111\) 0 0
\(112\) −112.000 + 76.8458i −1.00000 + 0.686123i
\(113\) 162.020 1.43381 0.716903 0.697173i \(-0.245559\pi\)
0.716903 + 0.697173i \(0.245559\pi\)
\(114\) 0 0
\(115\) 11.5324i 0.100281i
\(116\) −46.4522 −0.400450
\(117\) 0 0
\(118\) 136.805i 1.15936i
\(119\) 61.7350 + 89.9766i 0.518782 + 0.756106i
\(120\) 0 0
\(121\) 53.1760 0.439471
\(122\) 74.5058i 0.610703i
\(123\) 0 0
\(124\) 70.7310i 0.570412i
\(125\) 47.7977i 0.382381i
\(126\) 0 0
\(127\) 17.4200 0.137165 0.0685826 0.997645i \(-0.478152\pi\)
0.0685826 + 0.997645i \(0.478152\pi\)
\(128\) −154.118 −1.20405
\(129\) 0 0
\(130\) −59.1601 −0.455077
\(131\) 147.744i 1.12782i 0.825837 + 0.563908i \(0.190703\pi\)
−0.825837 + 0.563908i \(0.809297\pi\)
\(132\) 0 0
\(133\) 121.896 + 177.659i 0.916512 + 1.33578i
\(134\) −209.396 −1.56266
\(135\) 0 0
\(136\) 98.8017i 0.726483i
\(137\) 202.537 1.47837 0.739185 0.673503i \(-0.235211\pi\)
0.739185 + 0.673503i \(0.235211\pi\)
\(138\) 0 0
\(139\) 245.333i 1.76498i −0.470327 0.882492i \(-0.655864\pi\)
0.470327 0.882492i \(-0.344136\pi\)
\(140\) 31.7730 + 46.3081i 0.226950 + 0.330772i
\(141\) 0 0
\(142\) −22.5641 −0.158902
\(143\) 52.2664i 0.365499i
\(144\) 0 0
\(145\) 247.139i 1.70441i
\(146\) 54.3310i 0.372130i
\(147\) 0 0
\(148\) −51.2959 −0.346594
\(149\) −64.6252 −0.433726 −0.216863 0.976202i \(-0.569582\pi\)
−0.216863 + 0.976202i \(0.569582\pi\)
\(150\) 0 0
\(151\) 158.652 1.05068 0.525338 0.850894i \(-0.323939\pi\)
0.525338 + 0.850894i \(0.323939\pi\)
\(152\) 195.084i 1.28345i
\(153\) 0 0
\(154\) 174.176 119.506i 1.13101 0.776014i
\(155\) −376.309 −2.42780
\(156\) 0 0
\(157\) 79.5546i 0.506717i −0.967372 0.253359i \(-0.918465\pi\)
0.967372 0.253359i \(-0.0815352\pi\)
\(158\) −170.764 −1.08078
\(159\) 0 0
\(160\) 124.227i 0.776417i
\(161\) 10.1886 6.99060i 0.0632829 0.0434199i
\(162\) 0 0
\(163\) −323.320 −1.98356 −0.991779 0.127959i \(-0.959157\pi\)
−0.991779 + 0.127959i \(0.959157\pi\)
\(164\) 30.2623i 0.184526i
\(165\) 0 0
\(166\) 288.548i 1.73824i
\(167\) 1.48958i 0.00891964i −0.999990 0.00445982i \(-0.998580\pi\)
0.999990 0.00445982i \(-0.00141961\pi\)
\(168\) 0 0
\(169\) 153.316 0.907195
\(170\) 232.865 1.36979
\(171\) 0 0
\(172\) −46.7720 −0.271930
\(173\) 142.818i 0.825537i −0.910836 0.412769i \(-0.864562\pi\)
0.910836 0.412769i \(-0.135438\pi\)
\(174\) 0 0
\(175\) −102.072 + 70.0340i −0.583269 + 0.400194i
\(176\) 256.086 1.45503
\(177\) 0 0
\(178\) 62.1134i 0.348951i
\(179\) 273.619 1.52860 0.764298 0.644863i \(-0.223086\pi\)
0.764298 + 0.644863i \(0.223086\pi\)
\(180\) 0 0
\(181\) 61.0045i 0.337042i 0.985698 + 0.168521i \(0.0538991\pi\)
−0.985698 + 0.168521i \(0.946101\pi\)
\(182\) 35.8612 + 52.2664i 0.197039 + 0.287178i
\(183\) 0 0
\(184\) 11.1879 0.0608036
\(185\) 272.909i 1.47518i
\(186\) 0 0
\(187\) 205.730i 1.10016i
\(188\) 40.1144i 0.213374i
\(189\) 0 0
\(190\) 459.792 2.41996
\(191\) −6.01805 −0.0315081 −0.0157540 0.999876i \(-0.505015\pi\)
−0.0157540 + 0.999876i \(0.505015\pi\)
\(192\) 0 0
\(193\) −221.880 −1.14964 −0.574819 0.818281i \(-0.694927\pi\)
−0.574819 + 0.818281i \(0.694927\pi\)
\(194\) 110.726i 0.570755i
\(195\) 0 0
\(196\) 21.6521 56.1413i 0.110470 0.286435i
\(197\) 175.739 0.892076 0.446038 0.895014i \(-0.352835\pi\)
0.446038 + 0.895014i \(0.352835\pi\)
\(198\) 0 0
\(199\) 42.6604i 0.214374i −0.994239 0.107187i \(-0.965816\pi\)
0.994239 0.107187i \(-0.0341843\pi\)
\(200\) −112.083 −0.560417
\(201\) 0 0
\(202\) 363.239i 1.79821i
\(203\) −218.341 + 149.809i −1.07557 + 0.737973i
\(204\) 0 0
\(205\) 161.004 0.785386
\(206\) 266.258i 1.29251i
\(207\) 0 0
\(208\) 76.8458i 0.369451i
\(209\) 406.214i 1.94361i
\(210\) 0 0
\(211\) −186.876 −0.885668 −0.442834 0.896604i \(-0.646027\pi\)
−0.442834 + 0.896604i \(0.646027\pi\)
\(212\) 9.06356 0.0427526
\(213\) 0 0
\(214\) 146.384 0.684037
\(215\) 248.840i 1.15740i
\(216\) 0 0
\(217\) 228.108 + 332.460i 1.05119 + 1.53207i
\(218\) −47.9337 −0.219880
\(219\) 0 0
\(220\) 105.883i 0.481284i
\(221\) 61.7350 0.279344
\(222\) 0 0
\(223\) 320.927i 1.43914i 0.694422 + 0.719568i \(0.255660\pi\)
−0.694422 + 0.719568i \(0.744340\pi\)
\(224\) 109.751 75.3027i 0.489960 0.336173i
\(225\) 0 0
\(226\) −370.456 −1.63919
\(227\) 9.05516i 0.0398906i 0.999801 + 0.0199453i \(0.00634920\pi\)
−0.999801 + 0.0199453i \(0.993651\pi\)
\(228\) 0 0
\(229\) 229.698i 1.00305i −0.865144 0.501523i \(-0.832773\pi\)
0.865144 0.501523i \(-0.167227\pi\)
\(230\) 26.3686i 0.114646i
\(231\) 0 0
\(232\) −239.756 −1.03343
\(233\) 11.7525 0.0504399 0.0252199 0.999682i \(-0.491971\pi\)
0.0252199 + 0.999682i \(0.491971\pi\)
\(234\) 0 0
\(235\) −213.420 −0.908170
\(236\) 73.4735i 0.311329i
\(237\) 0 0
\(238\) −141.156 205.730i −0.593092 0.864411i
\(239\) 440.733 1.84407 0.922036 0.387104i \(-0.126525\pi\)
0.922036 + 0.387104i \(0.126525\pi\)
\(240\) 0 0
\(241\) 350.107i 1.45272i 0.687312 + 0.726362i \(0.258791\pi\)
−0.687312 + 0.726362i \(0.741209\pi\)
\(242\) −121.586 −0.502421
\(243\) 0 0
\(244\) 40.0148i 0.163995i
\(245\) 298.688 + 115.195i 1.21913 + 0.470184i
\(246\) 0 0
\(247\) 121.896 0.493506
\(248\) 365.068i 1.47205i
\(249\) 0 0
\(250\) 109.288i 0.437154i
\(251\) 207.576i 0.826996i 0.910505 + 0.413498i \(0.135693\pi\)
−0.910505 + 0.413498i \(0.864307\pi\)
\(252\) 0 0
\(253\) −23.2959 −0.0920788
\(254\) −39.8304 −0.156813
\(255\) 0 0
\(256\) 215.828 0.843078
\(257\) 442.435i 1.72154i −0.508996 0.860769i \(-0.669983\pi\)
0.508996 0.860769i \(-0.330017\pi\)
\(258\) 0 0
\(259\) −241.108 + 165.430i −0.930919 + 0.638725i
\(260\) 31.7730 0.122204
\(261\) 0 0
\(262\) 337.814i 1.28937i
\(263\) −128.244 −0.487621 −0.243810 0.969823i \(-0.578397\pi\)
−0.243810 + 0.969823i \(0.578397\pi\)
\(264\) 0 0
\(265\) 48.2207i 0.181965i
\(266\) −278.713 406.214i −1.04779 1.52712i
\(267\) 0 0
\(268\) 112.460 0.419627
\(269\) 81.0391i 0.301261i −0.988590 0.150630i \(-0.951870\pi\)
0.988590 0.150630i \(-0.0481303\pi\)
\(270\) 0 0
\(271\) 3.75443i 0.0138540i 0.999976 + 0.00692700i \(0.00220495\pi\)
−0.999976 + 0.00692700i \(0.997795\pi\)
\(272\) 302.479i 1.11205i
\(273\) 0 0
\(274\) −463.096 −1.69013
\(275\) 233.386 0.848676
\(276\) 0 0
\(277\) −111.004 −0.400737 −0.200368 0.979721i \(-0.564214\pi\)
−0.200368 + 0.979721i \(0.564214\pi\)
\(278\) 560.949i 2.01780i
\(279\) 0 0
\(280\) 163.992 + 239.012i 0.585685 + 0.853615i
\(281\) −47.1747 −0.167882 −0.0839408 0.996471i \(-0.526751\pi\)
−0.0839408 + 0.996471i \(0.526751\pi\)
\(282\) 0 0
\(283\) 209.342i 0.739723i 0.929087 + 0.369862i \(0.120595\pi\)
−0.929087 + 0.369862i \(0.879405\pi\)
\(284\) 12.1185 0.0426706
\(285\) 0 0
\(286\) 119.506i 0.417854i
\(287\) −97.5962 142.243i −0.340056 0.495620i
\(288\) 0 0
\(289\) 46.0000 0.159170
\(290\) 565.078i 1.94855i
\(291\) 0 0
\(292\) 29.1795i 0.0999297i
\(293\) 138.349i 0.472182i 0.971731 + 0.236091i \(0.0758663\pi\)
−0.971731 + 0.236091i \(0.924134\pi\)
\(294\) 0 0
\(295\) 390.900 1.32509
\(296\) −264.756 −0.894447
\(297\) 0 0
\(298\) 147.764 0.495853
\(299\) 6.99060i 0.0233799i
\(300\) 0 0
\(301\) −219.844 + 150.840i −0.730379 + 0.501130i
\(302\) −362.755 −1.20118
\(303\) 0 0
\(304\) 597.245i 1.96462i
\(305\) −212.890 −0.698000
\(306\) 0 0
\(307\) 422.295i 1.37555i −0.725922 0.687777i \(-0.758587\pi\)
0.725922 0.687777i \(-0.241413\pi\)
\(308\) −93.5445 + 64.1830i −0.303716 + 0.208386i
\(309\) 0 0
\(310\) 860.424 2.77556
\(311\) 71.8663i 0.231081i −0.993303 0.115541i \(-0.963140\pi\)
0.993303 0.115541i \(-0.0368601\pi\)
\(312\) 0 0
\(313\) 7.92060i 0.0253054i −0.999920 0.0126527i \(-0.995972\pi\)
0.999920 0.0126527i \(-0.00402759\pi\)
\(314\) 181.900i 0.579300i
\(315\) 0 0
\(316\) 91.7118 0.290227
\(317\) 360.624 1.13761 0.568807 0.822471i \(-0.307405\pi\)
0.568807 + 0.822471i \(0.307405\pi\)
\(318\) 0 0
\(319\) 499.232 1.56499
\(320\) 223.047i 0.697021i
\(321\) 0 0
\(322\) −23.2959 + 15.9839i −0.0723476 + 0.0496394i
\(323\) −479.805 −1.48546
\(324\) 0 0
\(325\) 70.0340i 0.215489i
\(326\) 739.265 2.26769
\(327\) 0 0
\(328\) 156.194i 0.476203i
\(329\) 129.369 + 188.551i 0.393219 + 0.573104i
\(330\) 0 0
\(331\) −69.4560 −0.209837 −0.104918 0.994481i \(-0.533458\pi\)
−0.104918 + 0.994481i \(0.533458\pi\)
\(332\) 154.970i 0.466777i
\(333\) 0 0
\(334\) 3.40590i 0.0101973i
\(335\) 598.320i 1.78603i
\(336\) 0 0
\(337\) −162.720 −0.482849 −0.241424 0.970420i \(-0.577615\pi\)
−0.241424 + 0.970420i \(0.577615\pi\)
\(338\) −350.554 −1.03714
\(339\) 0 0
\(340\) −125.064 −0.367836
\(341\) 760.163i 2.22922i
\(342\) 0 0
\(343\) −79.2841 333.711i −0.231149 0.972918i
\(344\) −241.407 −0.701764
\(345\) 0 0
\(346\) 326.551i 0.943788i
\(347\) −89.8953 −0.259064 −0.129532 0.991575i \(-0.541348\pi\)
−0.129532 + 0.991575i \(0.541348\pi\)
\(348\) 0 0
\(349\) 495.529i 1.41985i −0.704275 0.709927i \(-0.748728\pi\)
0.704275 0.709927i \(-0.251272\pi\)
\(350\) 233.386 160.131i 0.666817 0.457518i
\(351\) 0 0
\(352\) −250.944 −0.712909
\(353\) 441.978i 1.25206i 0.779798 + 0.626031i \(0.215322\pi\)
−0.779798 + 0.626031i \(0.784678\pi\)
\(354\) 0 0
\(355\) 64.4736i 0.181616i
\(356\) 33.3592i 0.0937055i
\(357\) 0 0
\(358\) −625.624 −1.74755
\(359\) −518.272 −1.44366 −0.721828 0.692073i \(-0.756698\pi\)
−0.721828 + 0.692073i \(0.756698\pi\)
\(360\) 0 0
\(361\) −586.376 −1.62431
\(362\) 139.486i 0.385320i
\(363\) 0 0
\(364\) −19.2599 28.0707i −0.0529119 0.0771172i
\(365\) −155.243 −0.425323
\(366\) 0 0
\(367\) 134.096i 0.365384i 0.983170 + 0.182692i \(0.0584811\pi\)
−0.983170 + 0.182692i \(0.941519\pi\)
\(368\) −34.2513 −0.0930743
\(369\) 0 0
\(370\) 624.001i 1.68649i
\(371\) 42.6018 29.2300i 0.114830 0.0787872i
\(372\) 0 0
\(373\) 530.060 1.42107 0.710536 0.703660i \(-0.248452\pi\)
0.710536 + 0.703660i \(0.248452\pi\)
\(374\) 470.398i 1.25775i
\(375\) 0 0
\(376\) 207.045i 0.550650i
\(377\) 149.809i 0.397370i
\(378\) 0 0
\(379\) 503.916 1.32959 0.664797 0.747024i \(-0.268518\pi\)
0.664797 + 0.747024i \(0.268518\pi\)
\(380\) −246.940 −0.649842
\(381\) 0 0
\(382\) 13.7601 0.0360213
\(383\) 85.8475i 0.224145i 0.993700 + 0.112072i \(0.0357489\pi\)
−0.993700 + 0.112072i \(0.964251\pi\)
\(384\) 0 0
\(385\) −341.472 497.683i −0.886940 1.29268i
\(386\) 507.325 1.31431
\(387\) 0 0
\(388\) 59.4677i 0.153267i
\(389\) −513.901 −1.32108 −0.660541 0.750790i \(-0.729673\pi\)
−0.660541 + 0.750790i \(0.729673\pi\)
\(390\) 0 0
\(391\) 27.5162i 0.0703740i
\(392\) 111.754 289.765i 0.285087 0.739197i
\(393\) 0 0
\(394\) −401.824 −1.01986
\(395\) 487.933i 1.23527i
\(396\) 0 0
\(397\) 233.309i 0.587681i 0.955854 + 0.293840i \(0.0949334\pi\)
−0.955854 + 0.293840i \(0.905067\pi\)
\(398\) 97.5422i 0.245081i
\(399\) 0 0
\(400\) 343.140 0.857851
\(401\) 329.418 0.821491 0.410746 0.911750i \(-0.365268\pi\)
0.410746 + 0.911750i \(0.365268\pi\)
\(402\) 0 0
\(403\) 228.108 0.566025
\(404\) 195.084i 0.482882i
\(405\) 0 0
\(406\) 499.232 342.535i 1.22964 0.843681i
\(407\) 551.289 1.35452
\(408\) 0 0
\(409\) 55.4442i 0.135560i 0.997700 + 0.0677802i \(0.0215917\pi\)
−0.997700 + 0.0677802i \(0.978408\pi\)
\(410\) −368.133 −0.897885
\(411\) 0 0
\(412\) 142.999i 0.347085i
\(413\) −236.953 345.350i −0.573735 0.836199i
\(414\) 0 0
\(415\) −824.484 −1.98671
\(416\) 75.3027i 0.181016i
\(417\) 0 0
\(418\) 928.802i 2.22201i
\(419\) 454.130i 1.08384i 0.840429 + 0.541921i \(0.182303\pi\)
−0.840429 + 0.541921i \(0.817697\pi\)
\(420\) 0 0
\(421\) −275.056 −0.653340 −0.326670 0.945138i \(-0.605927\pi\)
−0.326670 + 0.945138i \(0.605927\pi\)
\(422\) 427.288 1.01253
\(423\) 0 0
\(424\) 46.7802 0.110331
\(425\) 275.666i 0.648626i
\(426\) 0 0
\(427\) 129.048 + 188.083i 0.302220 + 0.440475i
\(428\) −78.6183 −0.183688
\(429\) 0 0
\(430\) 568.969i 1.32318i
\(431\) 312.004 0.723907 0.361954 0.932196i \(-0.382110\pi\)
0.361954 + 0.932196i \(0.382110\pi\)
\(432\) 0 0
\(433\) 77.1943i 0.178278i −0.996019 0.0891389i \(-0.971588\pi\)
0.996019 0.0891389i \(-0.0284115\pi\)
\(434\) −521.565 760.163i −1.20176 1.75153i
\(435\) 0 0
\(436\) 25.7437 0.0590452
\(437\) 54.3310i 0.124327i
\(438\) 0 0
\(439\) 337.117i 0.767920i 0.923350 + 0.383960i \(0.125440\pi\)
−0.923350 + 0.383960i \(0.874560\pi\)
\(440\) 546.497i 1.24204i
\(441\) 0 0
\(442\) −141.156 −0.319357
\(443\) −136.027 −0.307060 −0.153530 0.988144i \(-0.549064\pi\)
−0.153530 + 0.988144i \(0.549064\pi\)
\(444\) 0 0
\(445\) −177.480 −0.398832
\(446\) 733.794i 1.64528i
\(447\) 0 0
\(448\) 197.056 135.205i 0.439857 0.301796i
\(449\) 178.227 0.396941 0.198471 0.980107i \(-0.436403\pi\)
0.198471 + 0.980107i \(0.436403\pi\)
\(450\) 0 0
\(451\) 325.236i 0.721144i
\(452\) 198.960 0.440178
\(453\) 0 0
\(454\) 20.7045i 0.0456045i
\(455\) 149.344 102.468i 0.328228 0.225205i
\(456\) 0 0
\(457\) 112.068 0.245225 0.122613 0.992455i \(-0.460873\pi\)
0.122613 + 0.992455i \(0.460873\pi\)
\(458\) 525.199i 1.14672i
\(459\) 0 0
\(460\) 14.1617i 0.0307864i
\(461\) 38.6248i 0.0837848i −0.999122 0.0418924i \(-0.986661\pi\)
0.999122 0.0418924i \(-0.0133387\pi\)
\(462\) 0 0
\(463\) 100.580 0.217235 0.108618 0.994084i \(-0.465358\pi\)
0.108618 + 0.994084i \(0.465358\pi\)
\(464\) 734.006 1.58191
\(465\) 0 0
\(466\) −26.8719 −0.0576649
\(467\) 92.6161i 0.198322i −0.995071 0.0991608i \(-0.968384\pi\)
0.995071 0.0991608i \(-0.0316158\pi\)
\(468\) 0 0
\(469\) 528.600 362.685i 1.12708 0.773315i
\(470\) 487.981 1.03826
\(471\) 0 0
\(472\) 379.223i 0.803438i
\(473\) 502.669 1.06273
\(474\) 0 0
\(475\) 544.304i 1.14590i
\(476\) 75.8105 + 110.491i 0.159266 + 0.232124i
\(477\) 0 0
\(478\) −1007.73 −2.10822
\(479\) 551.672i 1.15172i −0.817550 0.575858i \(-0.804668\pi\)
0.817550 0.575858i \(-0.195332\pi\)
\(480\) 0 0
\(481\) 165.430i 0.343929i
\(482\) 800.512i 1.66081i
\(483\) 0 0
\(484\) 65.3000 0.134917
\(485\) 316.385 0.652341
\(486\) 0 0
\(487\) 172.176 0.353544 0.176772 0.984252i \(-0.443434\pi\)
0.176772 + 0.984252i \(0.443434\pi\)
\(488\) 206.530i 0.423218i
\(489\) 0 0
\(490\) −682.944 263.392i −1.39376 0.537534i
\(491\) −602.470 −1.22703 −0.613513 0.789685i \(-0.710244\pi\)
−0.613513 + 0.789685i \(0.710244\pi\)
\(492\) 0 0
\(493\) 589.673i 1.19609i
\(494\) −278.713 −0.564196
\(495\) 0 0
\(496\) 1117.64i 2.25332i
\(497\) 56.9608 39.0821i 0.114609 0.0786360i
\(498\) 0 0
\(499\) 226.860 0.454629 0.227315 0.973821i \(-0.427005\pi\)
0.227315 + 0.973821i \(0.427005\pi\)
\(500\) 58.6954i 0.117391i
\(501\) 0 0
\(502\) 474.619i 0.945456i
\(503\) 269.015i 0.534821i 0.963583 + 0.267411i \(0.0861681\pi\)
−0.963583 + 0.267411i \(0.913832\pi\)
\(504\) 0 0
\(505\) 1037.90 2.05526
\(506\) 53.2657 0.105268
\(507\) 0 0
\(508\) 21.3917 0.0421096
\(509\) 229.580i 0.451041i −0.974238 0.225521i \(-0.927592\pi\)
0.974238 0.225521i \(-0.0724083\pi\)
\(510\) 0 0
\(511\) 94.1040 + 137.153i 0.184156 + 0.268402i
\(512\) 122.985 0.240205
\(513\) 0 0
\(514\) 1011.62i 1.96813i
\(515\) −760.795 −1.47727
\(516\) 0 0
\(517\) 431.119i 0.833885i
\(518\) 551.289 378.252i 1.06426 0.730216i
\(519\) 0 0
\(520\) 163.992 0.315369
\(521\) 531.379i 1.01992i 0.860197 + 0.509961i \(0.170340\pi\)
−0.860197 + 0.509961i \(0.829660\pi\)
\(522\) 0 0
\(523\) 21.0530i 0.0402543i 0.999797 + 0.0201271i \(0.00640710\pi\)
−0.999797 + 0.0201271i \(0.993593\pi\)
\(524\) 181.429i 0.346239i
\(525\) 0 0
\(526\) 293.228 0.557468
\(527\) −897.874 −1.70375
\(528\) 0 0
\(529\) −525.884 −0.994110
\(530\) 110.256i 0.208030i
\(531\) 0 0
\(532\) 149.688 + 218.165i 0.281369 + 0.410085i
\(533\) −97.5962 −0.183107
\(534\) 0 0
\(535\) 418.271i 0.781816i
\(536\) 580.446 1.08292
\(537\) 0 0
\(538\) 185.294i 0.344414i
\(539\) −232.700 + 603.363i −0.431725 + 1.11941i
\(540\) 0 0
\(541\) 430.484 0.795720 0.397860 0.917446i \(-0.369753\pi\)
0.397860 + 0.917446i \(0.369753\pi\)
\(542\) 8.58444i 0.0158385i
\(543\) 0 0
\(544\) 296.405i 0.544862i
\(545\) 136.964i 0.251310i
\(546\) 0 0
\(547\) 212.996 0.389389 0.194695 0.980864i \(-0.437628\pi\)
0.194695 + 0.980864i \(0.437628\pi\)
\(548\) 248.715 0.453859
\(549\) 0 0
\(550\) −533.632 −0.970241
\(551\) 1164.31i 2.11309i
\(552\) 0 0
\(553\) 431.076 295.771i 0.779523 0.534848i
\(554\) 253.809 0.458139
\(555\) 0 0
\(556\) 301.268i 0.541849i
\(557\) 613.427 1.10130 0.550652 0.834735i \(-0.314379\pi\)
0.550652 + 0.834735i \(0.314379\pi\)
\(558\) 0 0
\(559\) 150.840i 0.269839i
\(560\) −502.056 731.729i −0.896529 1.30666i
\(561\) 0 0
\(562\) 107.864 0.191929
\(563\) 749.043i 1.33045i 0.746643 + 0.665225i \(0.231664\pi\)
−0.746643 + 0.665225i \(0.768336\pi\)
\(564\) 0 0
\(565\) 1058.53i 1.87350i
\(566\) 478.656i 0.845682i
\(567\) 0 0
\(568\) 62.5476 0.110119
\(569\) −277.835 −0.488287 −0.244143 0.969739i \(-0.578507\pi\)
−0.244143 + 0.969739i \(0.578507\pi\)
\(570\) 0 0
\(571\) 637.328 1.11616 0.558081 0.829787i \(-0.311538\pi\)
0.558081 + 0.829787i \(0.311538\pi\)
\(572\) 64.1830i 0.112208i
\(573\) 0 0
\(574\) 223.152 + 325.236i 0.388766 + 0.566613i
\(575\) −31.2152 −0.0542873
\(576\) 0 0
\(577\) 89.6296i 0.155337i −0.996979 0.0776686i \(-0.975252\pi\)
0.996979 0.0776686i \(-0.0247476\pi\)
\(578\) −105.178 −0.181969
\(579\) 0 0
\(580\) 303.486i 0.523252i
\(581\) 499.779 + 728.411i 0.860205 + 1.25372i
\(582\) 0 0
\(583\) −97.4081 −0.167081
\(584\) 150.605i 0.257886i
\(585\) 0 0
\(586\) 316.333i 0.539817i
\(587\) 1080.75i 1.84114i 0.390573 + 0.920572i \(0.372277\pi\)
−0.390573 + 0.920572i \(0.627723\pi\)
\(588\) 0 0
\(589\) −1772.86 −3.00994
\(590\) −893.786 −1.51489
\(591\) 0 0
\(592\) 810.544 1.36916
\(593\) 97.0849i 0.163718i −0.996644 0.0818591i \(-0.973914\pi\)
0.996644 0.0818591i \(-0.0260857\pi\)
\(594\) 0 0
\(595\) −587.844 + 403.333i −0.987973 + 0.677871i
\(596\) −79.3596 −0.133154
\(597\) 0 0
\(598\) 15.9839i 0.0267289i
\(599\) 567.651 0.947665 0.473832 0.880615i \(-0.342870\pi\)
0.473832 + 0.880615i \(0.342870\pi\)
\(600\) 0 0
\(601\) 108.180i 0.179999i 0.995942 + 0.0899997i \(0.0286866\pi\)
−0.995942 + 0.0899997i \(0.971313\pi\)
\(602\) 502.669 344.893i 0.834999 0.572912i
\(603\) 0 0
\(604\) 194.824 0.322557
\(605\) 347.415i 0.574239i
\(606\) 0 0
\(607\) 343.026i 0.565117i −0.959250 0.282558i \(-0.908817\pi\)
0.959250 0.282558i \(-0.0911831\pi\)
\(608\) 585.253i 0.962587i
\(609\) 0 0
\(610\) 486.769 0.797982
\(611\) 129.369 0.211734
\(612\) 0 0
\(613\) 526.640 0.859119 0.429560 0.903038i \(-0.358669\pi\)
0.429560 + 0.903038i \(0.358669\pi\)
\(614\) 965.570i 1.57259i
\(615\) 0 0
\(616\) −482.816 + 331.271i −0.783793 + 0.537778i
\(617\) −1123.43 −1.82080 −0.910398 0.413734i \(-0.864224\pi\)
−0.910398 + 0.413734i \(0.864224\pi\)
\(618\) 0 0
\(619\) 74.6913i 0.120665i 0.998178 + 0.0603323i \(0.0192160\pi\)
−0.998178 + 0.0603323i \(0.980784\pi\)
\(620\) −462.107 −0.745334
\(621\) 0 0
\(622\) 164.321i 0.264182i
\(623\) 107.583 + 156.799i 0.172686 + 0.251684i
\(624\) 0 0
\(625\) −754.376 −1.20700
\(626\) 18.1103i 0.0289302i
\(627\) 0 0
\(628\) 97.6929i 0.155562i
\(629\) 651.161i 1.03523i
\(630\) 0 0
\(631\) −905.740 −1.43540 −0.717702 0.696350i \(-0.754806\pi\)
−0.717702 + 0.696350i \(0.754806\pi\)
\(632\) 473.357 0.748983
\(633\) 0 0
\(634\) −824.560 −1.30057
\(635\) 113.810i 0.179228i
\(636\) 0 0
\(637\) −181.056 69.8281i −0.284232 0.109620i
\(638\) −1141.49 −1.78916
\(639\) 0 0
\(640\) 1006.90i 1.57328i
\(641\) −800.150 −1.24828 −0.624142 0.781311i \(-0.714551\pi\)
−0.624142 + 0.781311i \(0.714551\pi\)
\(642\) 0 0
\(643\) 1207.83i 1.87843i 0.343333 + 0.939214i \(0.388444\pi\)
−0.343333 + 0.939214i \(0.611556\pi\)
\(644\) 12.5115 8.58444i 0.0194278 0.0133299i
\(645\) 0 0
\(646\) 1097.06 1.69824
\(647\) 377.781i 0.583897i −0.956434 0.291949i \(-0.905696\pi\)
0.956434 0.291949i \(-0.0943036\pi\)
\(648\) 0 0
\(649\) 789.637i 1.21670i
\(650\) 160.131i 0.246356i
\(651\) 0 0
\(652\) −397.036 −0.608952
\(653\) −728.674 −1.11589 −0.557944 0.829879i \(-0.688410\pi\)
−0.557944 + 0.829879i \(0.688410\pi\)
\(654\) 0 0
\(655\) −965.256 −1.47367
\(656\) 478.185i 0.728941i
\(657\) 0 0
\(658\) −295.800 431.119i −0.449544 0.655195i
\(659\) 813.274 1.23410 0.617052 0.786923i \(-0.288327\pi\)
0.617052 + 0.786923i \(0.288327\pi\)
\(660\) 0 0
\(661\) 505.192i 0.764285i 0.924103 + 0.382142i \(0.124814\pi\)
−0.924103 + 0.382142i \(0.875186\pi\)
\(662\) 158.810 0.239894
\(663\) 0 0
\(664\) 799.855i 1.20460i
\(665\) −1160.70 + 796.383i −1.74541 + 1.19757i
\(666\) 0 0
\(667\) −66.7720 −0.100108
\(668\) 1.82920i 0.00273833i
\(669\) 0 0
\(670\) 1368.05i 2.04186i
\(671\) 430.048i 0.640906i
\(672\) 0 0
\(673\) 112.068 0.166520 0.0832600 0.996528i \(-0.473467\pi\)
0.0832600 + 0.996528i \(0.473467\pi\)
\(674\) 372.056 0.552012
\(675\) 0 0
\(676\) 188.272 0.278509
\(677\) 1053.59i 1.55626i 0.628105 + 0.778129i \(0.283831\pi\)
−0.628105 + 0.778129i \(0.716169\pi\)
\(678\) 0 0
\(679\) −191.784 279.518i −0.282450 0.411662i
\(680\) −645.501 −0.949266
\(681\) 0 0
\(682\) 1738.10i 2.54853i
\(683\) −272.576 −0.399087 −0.199543 0.979889i \(-0.563946\pi\)
−0.199543 + 0.979889i \(0.563946\pi\)
\(684\) 0 0
\(685\) 1323.23i 1.93173i
\(686\) 181.282 + 763.024i 0.264259 + 1.11228i
\(687\) 0 0
\(688\) 739.060 1.07422
\(689\) 29.2300i 0.0424239i
\(690\) 0 0
\(691\) 891.908i 1.29075i −0.763866 0.645375i \(-0.776701\pi\)
0.763866 0.645375i \(-0.223299\pi\)
\(692\) 175.380i 0.253440i
\(693\) 0 0
\(694\) 205.544 0.296173
\(695\) 1602.83 2.30623
\(696\) 0 0
\(697\) 384.156 0.551156
\(698\) 1133.02i 1.62323i
\(699\) 0 0
\(700\) −125.344 + 86.0016i −0.179063 + 0.122859i
\(701\) 1053.60 1.50300 0.751499 0.659734i \(-0.229331\pi\)
0.751499 + 0.659734i \(0.229331\pi\)
\(702\) 0 0
\(703\) 1285.72i 1.82891i
\(704\) −450.565 −0.640007
\(705\) 0 0
\(706\) 1010.57i 1.43141i
\(707\) −629.148 916.962i −0.889885 1.29698i
\(708\) 0 0
\(709\) 488.340 0.688773 0.344387 0.938828i \(-0.388087\pi\)
0.344387 + 0.938828i \(0.388087\pi\)
\(710\) 147.418i 0.207631i
\(711\) 0 0
\(712\) 172.178i 0.241824i
\(713\) 101.671i 0.142596i
\(714\) 0 0
\(715\) −341.472 −0.477583
\(716\) 336.003 0.469278
\(717\) 0 0
\(718\) 1185.02 1.65045
\(719\) 310.044i 0.431216i −0.976480 0.215608i \(-0.930827\pi\)
0.976480 0.215608i \(-0.0691733\pi\)
\(720\) 0 0
\(721\) 461.172 + 672.143i 0.639629 + 0.932237i
\(722\) 1340.74 1.85698
\(723\) 0 0
\(724\) 74.9135i 0.103472i
\(725\) 668.942 0.922679
\(726\) 0 0
\(727\) 606.623i 0.834420i 0.908810 + 0.417210i \(0.136992\pi\)
−0.908810 + 0.417210i \(0.863008\pi\)
\(728\) −99.4072 144.883i −0.136548 0.199014i
\(729\) 0 0
\(730\) 354.960 0.486247
\(731\) 593.733i 0.812221i
\(732\) 0 0
\(733\) 1186.02i 1.61803i −0.587788 0.809015i \(-0.700001\pi\)
0.587788 0.809015i \(-0.299999\pi\)
\(734\) 306.608i 0.417722i
\(735\) 0 0
\(736\) 33.5636 0.0456027
\(737\) −1208.63 −1.63994
\(738\) 0 0
\(739\) 559.472 0.757066 0.378533 0.925588i \(-0.376429\pi\)
0.378533 + 0.925588i \(0.376429\pi\)
\(740\) 335.132i 0.452881i
\(741\) 0 0
\(742\) −97.4081 + 66.8339i −0.131278 + 0.0900727i
\(743\) 272.091 0.366206 0.183103 0.983094i \(-0.441386\pi\)
0.183103 + 0.983094i \(0.441386\pi\)
\(744\) 0 0
\(745\) 422.216i 0.566732i
\(746\) −1211.97 −1.62463
\(747\) 0 0
\(748\) 252.636i 0.337748i
\(749\) −369.532 + 253.544i −0.493367 + 0.338511i
\(750\) 0 0
\(751\) 37.6562 0.0501414 0.0250707 0.999686i \(-0.492019\pi\)
0.0250707 + 0.999686i \(0.492019\pi\)
\(752\) 633.861i 0.842900i
\(753\) 0 0
\(754\) 342.535i 0.454290i
\(755\) 1036.52i 1.37288i
\(756\) 0 0
\(757\) 369.588 0.488227 0.244113 0.969747i \(-0.421503\pi\)
0.244113 + 0.969747i \(0.421503\pi\)
\(758\) −1152.19 −1.52005
\(759\) 0 0
\(760\) −1274.54 −1.67703
\(761\) 1232.17i 1.61914i −0.587022 0.809571i \(-0.699700\pi\)
0.587022 0.809571i \(-0.300300\pi\)
\(762\) 0 0
\(763\) 121.004 83.0237i 0.158590 0.108812i
\(764\) −7.39015 −0.00967297
\(765\) 0 0
\(766\) 196.289i 0.256252i
\(767\) −236.953 −0.308934
\(768\) 0 0
\(769\) 80.3149i 0.104441i 0.998636 + 0.0522203i \(0.0166298\pi\)
−0.998636 + 0.0522203i \(0.983370\pi\)
\(770\) 780.769 + 1137.94i 1.01399 + 1.47785i
\(771\) 0 0
\(772\) −272.468 −0.352938
\(773\) 638.774i 0.826357i −0.910650 0.413178i \(-0.864419\pi\)
0.910650 0.413178i \(-0.135581\pi\)
\(774\) 0 0
\(775\) 1018.57i 1.31429i
\(776\) 306.934i 0.395533i
\(777\) 0 0
\(778\) 1175.02 1.51031
\(779\) 758.518 0.973707
\(780\) 0 0
\(781\) −130.240 −0.166760
\(782\) 62.9154i 0.0804545i
\(783\) 0 0
\(784\) −342.132 + 887.108i −0.436393 + 1.13151i
\(785\) 519.754 0.662107
\(786\) 0 0
\(787\) 25.1560i 0.0319644i 0.999872 + 0.0159822i \(0.00508750\pi\)
−0.999872 + 0.0159822i \(0.994912\pi\)
\(788\) 215.807 0.273867
\(789\) 0 0
\(790\) 1115.65i 1.41221i
\(791\) 935.180 641.649i 1.18228 0.811187i
\(792\) 0 0
\(793\) 129.048 0.162734
\(794\) 533.457i 0.671861i
\(795\) 0 0
\(796\) 52.3869i 0.0658127i
\(797\) 92.1454i 0.115615i 0.998328 + 0.0578077i \(0.0184110\pi\)
−0.998328 + 0.0578077i \(0.981589\pi\)
\(798\) 0 0
\(799\) −509.220 −0.637322
\(800\) −336.250 −0.420313
\(801\) 0 0
\(802\) −753.208 −0.939163
\(803\) 313.598i 0.390533i
\(804\) 0 0
\(805\) 45.6717 + 66.5649i 0.0567350 + 0.0826893i
\(806\) −521.565 −0.647103
\(807\) 0 0
\(808\) 1006.90i 1.24616i
\(809\) 906.892 1.12100 0.560502 0.828153i \(-0.310608\pi\)
0.560502 + 0.828153i \(0.310608\pi\)
\(810\) 0 0
\(811\) 177.659i 0.219062i 0.993983 + 0.109531i \(0.0349349\pi\)
−0.993983 + 0.109531i \(0.965065\pi\)
\(812\) −268.122 + 183.965i −0.330200 + 0.226557i
\(813\) 0 0
\(814\) −1260.51 −1.54854
\(815\) 2112.35i 2.59184i
\(816\) 0 0
\(817\) 1172.33i 1.43492i
\(818\) 126.772i 0.154978i
\(819\) 0 0
\(820\) 197.713 0.241113
\(821\) 147.698 0.179900 0.0899498 0.995946i \(-0.471329\pi\)
0.0899498 + 0.995946i \(0.471329\pi\)
\(822\) 0 0
\(823\) 467.732 0.568325 0.284163 0.958776i \(-0.408284\pi\)
0.284163 + 0.958776i \(0.408284\pi\)
\(824\) 738.068i 0.895713i
\(825\) 0 0
\(826\) 541.788 + 789.637i 0.655918 + 0.955977i
\(827\) 52.2690 0.0632031 0.0316016 0.999501i \(-0.489939\pi\)
0.0316016 + 0.999501i \(0.489939\pi\)
\(828\) 0 0
\(829\) 1267.38i 1.52880i 0.644742 + 0.764401i \(0.276965\pi\)
−0.644742 + 0.764401i \(0.723035\pi\)
\(830\) 1885.17 2.27129
\(831\) 0 0
\(832\) 135.205i 0.162506i
\(833\) 712.669 + 274.856i 0.855545 + 0.329959i
\(834\) 0 0
\(835\) 9.73187 0.0116549
\(836\) 498.831i 0.596687i
\(837\) 0 0
\(838\) 1038.36i 1.23909i
\(839\) 1298.30i 1.54743i −0.633531 0.773717i \(-0.718395\pi\)
0.633531 0.773717i \(-0.281605\pi\)
\(840\) 0 0
\(841\) 589.924 0.701456
\(842\) 628.911 0.746925
\(843\) 0 0
\(844\) −229.483 −0.271900
\(845\) 1001.66i 1.18540i
\(846\) 0 0
\(847\) 306.932 210.593i 0.362376 0.248634i
\(848\) −143.216 −0.168887
\(849\) 0 0
\(850\) 630.306i 0.741536i
\(851\) −73.7346 −0.0866447
\(852\) 0 0
\(853\) 1200.05i 1.40686i 0.710765 + 0.703430i \(0.248349\pi\)
−0.710765 + 0.703430i \(0.751651\pi\)
\(854\) −295.066 430.048i −0.345510 0.503569i
\(855\) 0 0
\(856\) −405.777 −0.474038
\(857\) 186.369i 0.217467i 0.994071 + 0.108733i \(0.0346794\pi\)
−0.994071 + 0.108733i \(0.965321\pi\)
\(858\) 0 0
\(859\) 683.960i 0.796229i 0.917336 + 0.398114i \(0.130335\pi\)
−0.917336 + 0.398114i \(0.869665\pi\)
\(860\) 305.575i 0.355320i
\(861\) 0 0
\(862\) −713.392 −0.827601
\(863\) 886.597 1.02734 0.513672 0.857987i \(-0.328285\pi\)
0.513672 + 0.857987i \(0.328285\pi\)
\(864\) 0 0
\(865\) 933.073 1.07870
\(866\) 176.503i 0.203814i
\(867\) 0 0
\(868\) 280.116 + 408.260i 0.322715 + 0.470345i
\(869\) −985.648 −1.13423
\(870\) 0 0
\(871\) 362.685i 0.416400i
\(872\) 132.872 0.152377
\(873\) 0 0
\(874\) 124.227i 0.142136i
\(875\) 189.293 + 275.888i 0.216335 + 0.315301i
\(876\) 0 0
\(877\) −855.004 −0.974919 −0.487460 0.873146i \(-0.662076\pi\)
−0.487460 + 0.873146i \(0.662076\pi\)
\(878\) 770.812i 0.877918i
\(879\) 0 0
\(880\) 1673.09i 1.90123i
\(881\) 14.7781i 0.0167743i −0.999965 0.00838713i \(-0.997330\pi\)
0.999965 0.00838713i \(-0.00266974\pi\)
\(882\) 0 0
\(883\) −623.784 −0.706438 −0.353219 0.935541i \(-0.614913\pi\)
−0.353219 + 0.935541i \(0.614913\pi\)
\(884\) 75.8105 0.0857584
\(885\) 0 0
\(886\) 311.024 0.351043
\(887\) 810.939i 0.914250i 0.889403 + 0.457125i \(0.151121\pi\)
−0.889403 + 0.457125i \(0.848879\pi\)
\(888\) 0 0
\(889\) 100.548 68.9884i 0.113102 0.0776022i
\(890\) 405.805 0.455961
\(891\) 0 0
\(892\) 394.098i 0.441814i
\(893\) −1005.46 −1.12593
\(894\) 0 0
\(895\) 1787.63i 1.99736i
\(896\) −889.570 + 610.354i −0.992823 + 0.681199i
\(897\) 0 0
\(898\) −407.512 −0.453800
\(899\) 2178.82i 2.42360i
\(900\) 0 0
\(901\) 115.055i 0.127697i
\(902\) 743.646i 0.824441i
\(903\) 0 0
\(904\) 1026.90 1.13596
\(905\) −398.561 −0.440399
\(906\) 0 0
\(907\) −440.100 −0.485226 −0.242613 0.970123i \(-0.578005\pi\)
−0.242613 + 0.970123i \(0.578005\pi\)
\(908\) 11.1197i 0.0122464i
\(909\) 0 0
\(910\) −341.472 + 234.292i −0.375244 + 0.257463i
\(911\) −1396.74 −1.53319 −0.766597 0.642129i \(-0.778051\pi\)
−0.766597 + 0.642129i \(0.778051\pi\)
\(912\) 0 0
\(913\) 1665.50i 1.82420i
\(914\) −256.241 −0.280352
\(915\) 0 0
\(916\) 282.068i 0.307935i
\(917\) 585.111 + 852.779i 0.638071 + 0.929966i
\(918\) 0 0
\(919\) −77.7720 −0.0846268 −0.0423134 0.999104i \(-0.513473\pi\)
−0.0423134 + 0.999104i \(0.513473\pi\)
\(920\) 73.0937i 0.0794497i
\(921\) 0 0
\(922\) 88.3149i 0.0957863i
\(923\) 39.0821i 0.0423425i
\(924\) 0 0
\(925\) 738.696 0.798590
\(926\) −229.974 −0.248352
\(927\) 0 0
\(928\) −719.268 −0.775073
\(929\) 293.763i 0.316214i 0.987422 + 0.158107i \(0.0505391\pi\)
−0.987422 + 0.158107i \(0.949461\pi\)
\(930\) 0 0
\(931\) 1407.17 + 542.704i 1.51146 + 0.582926i
\(932\) 14.4320 0.0154850
\(933\) 0 0
\(934\) 211.765i 0.226729i
\(935\) 1344.09 1.43753
\(936\) 0 0
\(937\) 800.758i 0.854597i −0.904111 0.427299i \(-0.859465\pi\)
0.904111 0.427299i \(-0.140535\pi\)
\(938\) −1208.63 + 829.272i −1.28852 + 0.884085i
\(939\) 0 0
\(940\) −262.079 −0.278808
\(941\) 934.732i 0.993339i 0.867940 + 0.496670i \(0.165444\pi\)
−0.867940 + 0.496670i \(0.834556\pi\)
\(942\) 0 0
\(943\) 43.5001i 0.0461295i
\(944\) 1160.98i 1.22985i
\(945\) 0 0
\(946\) −1149.34 −1.21495
\(947\) −43.4796 −0.0459130 −0.0229565 0.999736i \(-0.507308\pi\)
−0.0229565 + 0.999736i \(0.507308\pi\)
\(948\) 0 0
\(949\) 94.1040 0.0991612
\(950\) 1244.54i 1.31004i
\(951\) 0 0
\(952\) 391.285 + 570.284i 0.411013 + 0.599037i
\(953\) 467.247 0.490291 0.245145 0.969486i \(-0.421164\pi\)
0.245145 + 0.969486i \(0.421164\pi\)
\(954\) 0 0
\(955\) 39.3177i 0.0411704i
\(956\) 541.219 0.566129
\(957\) 0 0
\(958\) 1261.39i 1.31669i
\(959\) 1169.04 802.106i 1.21902 0.836399i
\(960\) 0 0
\(961\) −2356.60 −2.45224
\(962\) 378.252i 0.393193i
\(963\) 0 0
\(964\) 429.930i 0.445986i
\(965\) 1449.61i 1.50219i
\(966\) 0 0
\(967\) 550.640 0.569431 0.284716 0.958612i \(-0.408101\pi\)
0.284716 + 0.958612i \(0.408101\pi\)
\(968\) 337.037 0.348178
\(969\) 0 0
\(970\) −723.409 −0.745783
\(971\) 1028.38i 1.05909i 0.848280 + 0.529547i \(0.177638\pi\)
−0.848280 + 0.529547i \(0.822362\pi\)
\(972\) 0 0
\(973\) −971.592 1416.06i −0.998553 1.45536i
\(974\) −393.677 −0.404186
\(975\) 0 0
\(976\) 632.287i 0.647835i
\(977\) −1121.51 −1.14791 −0.573956 0.818886i \(-0.694592\pi\)
−0.573956 + 0.818886i \(0.694592\pi\)
\(978\) 0 0
\(979\) 358.518i 0.366209i
\(980\) 366.788 + 141.459i 0.374273 + 0.144346i
\(981\) 0 0
\(982\) 1377.54 1.40279
\(983\) 884.975i 0.900279i −0.892958 0.450140i \(-0.851374\pi\)
0.892958 0.450140i \(-0.148626\pi\)
\(984\) 0 0
\(985\) 1148.16i 1.16564i
\(986\) 1348.28i 1.36742i
\(987\) 0 0
\(988\) 149.688 0.151506
\(989\) −67.2317 −0.0679795
\(990\) 0 0
\(991\) −886.836 −0.894890 −0.447445 0.894311i \(-0.647666\pi\)
−0.447445 + 0.894311i \(0.647666\pi\)
\(992\) 1095.20i 1.10404i
\(993\) 0 0
\(994\) −130.240 + 89.3605i −0.131026 + 0.0898999i
\(995\) 278.713 0.280114
\(996\) 0 0
\(997\) 656.935i 0.658912i −0.944171 0.329456i \(-0.893135\pi\)
0.944171 0.329456i \(-0.106865\pi\)
\(998\) −518.711 −0.519751
\(999\) 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.e.55.4 yes 8
3.2 odd 2 inner 189.3.d.e.55.5 yes 8
7.6 odd 2 inner 189.3.d.e.55.3 8
21.20 even 2 inner 189.3.d.e.55.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.3.d.e.55.3 8 7.6 odd 2 inner
189.3.d.e.55.4 yes 8 1.1 even 1 trivial
189.3.d.e.55.5 yes 8 3.2 odd 2 inner
189.3.d.e.55.6 yes 8 21.20 even 2 inner