Properties

Label 189.3.b.c.134.5
Level $189$
Weight $3$
Character 189.134
Analytic conductor $5.150$
Analytic rank $0$
Dimension $8$
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(134,189)]
 
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([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.134");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 189.b (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: 8.0.1997017344.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 134.5
Root \(1.27733i\) of defining polynomial
Character \(\chi\) \(=\) 189.134
Dual form 189.3.b.c.134.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.722666i q^{2} +3.47775 q^{4} +7.94527i q^{5} +2.64575 q^{7} +5.40392i q^{8} +O(q^{10})\) \(q+0.722666i q^{2} +3.47775 q^{4} +7.94527i q^{5} +2.64575 q^{7} +5.40392i q^{8} -5.74177 q^{10} -9.25195i q^{11} -22.1456 q^{13} +1.91199i q^{14} +10.0058 q^{16} +27.2692i q^{17} +31.8633 q^{19} +27.6317i q^{20} +6.68607 q^{22} -13.5294i q^{23} -38.1273 q^{25} -16.0038i q^{26} +9.20127 q^{28} +6.73870i q^{29} +11.9351 q^{31} +28.8465i q^{32} -19.7065 q^{34} +21.0212i q^{35} +26.8988 q^{37} +23.0265i q^{38} -42.9356 q^{40} -63.9635i q^{41} -16.2185 q^{43} -32.1760i q^{44} +9.77724 q^{46} -0.518396i q^{47} +7.00000 q^{49} -27.5533i q^{50} -77.0168 q^{52} -71.8912i q^{53} +73.5092 q^{55} +14.2974i q^{56} -4.86983 q^{58} -38.5540i q^{59} +27.9222 q^{61} +8.62506i q^{62} +19.1768 q^{64} -175.953i q^{65} +29.5649 q^{67} +94.8356i q^{68} -15.1913 q^{70} -39.4892i q^{71} +67.1187 q^{73} +19.4388i q^{74} +110.813 q^{76} -24.4784i q^{77} -18.7689 q^{79} +79.4987i q^{80} +46.2242 q^{82} -80.0893i q^{83} -216.661 q^{85} -11.7205i q^{86} +49.9968 q^{88} +62.9293i q^{89} -58.5917 q^{91} -47.0520i q^{92} +0.374627 q^{94} +253.162i q^{95} -46.7811 q^{97} +5.05866i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{4} - 52 q^{10} + 36 q^{13} + 132 q^{16} + 12 q^{19} - 136 q^{22} - 108 q^{25} + 56 q^{28} - 28 q^{31} - 12 q^{34} - 4 q^{37} + 336 q^{40} - 152 q^{43} + 108 q^{46} + 56 q^{49} - 272 q^{52} + 196 q^{55} - 220 q^{58} + 180 q^{61} - 700 q^{64} - 132 q^{67} + 196 q^{70} + 272 q^{73} + 544 q^{76} + 316 q^{79} + 28 q^{82} - 228 q^{85} - 56 q^{91} - 348 q^{94} - 364 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.722666i 0.361333i 0.983544 + 0.180666i \(0.0578254\pi\)
−0.983544 + 0.180666i \(0.942175\pi\)
\(3\) 0 0
\(4\) 3.47775 0.869439
\(5\) 7.94527i 1.58905i 0.607229 + 0.794527i \(0.292281\pi\)
−0.607229 + 0.794527i \(0.707719\pi\)
\(6\) 0 0
\(7\) 2.64575 0.377964
\(8\) 5.40392i 0.675490i
\(9\) 0 0
\(10\) −5.74177 −0.574177
\(11\) − 9.25195i − 0.841086i −0.907273 0.420543i \(-0.861840\pi\)
0.907273 0.420543i \(-0.138160\pi\)
\(12\) 0 0
\(13\) −22.1456 −1.70351 −0.851753 0.523944i \(-0.824460\pi\)
−0.851753 + 0.523944i \(0.824460\pi\)
\(14\) 1.91199i 0.136571i
\(15\) 0 0
\(16\) 10.0058 0.625362
\(17\) 27.2692i 1.60407i 0.597277 + 0.802035i \(0.296249\pi\)
−0.597277 + 0.802035i \(0.703751\pi\)
\(18\) 0 0
\(19\) 31.8633 1.67702 0.838508 0.544890i \(-0.183428\pi\)
0.838508 + 0.544890i \(0.183428\pi\)
\(20\) 27.6317i 1.38158i
\(21\) 0 0
\(22\) 6.68607 0.303912
\(23\) − 13.5294i − 0.588235i −0.955769 0.294118i \(-0.904974\pi\)
0.955769 0.294118i \(-0.0950257\pi\)
\(24\) 0 0
\(25\) −38.1273 −1.52509
\(26\) − 16.0038i − 0.615532i
\(27\) 0 0
\(28\) 9.20127 0.328617
\(29\) 6.73870i 0.232369i 0.993228 + 0.116184i \(0.0370664\pi\)
−0.993228 + 0.116184i \(0.962934\pi\)
\(30\) 0 0
\(31\) 11.9351 0.385002 0.192501 0.981297i \(-0.438340\pi\)
0.192501 + 0.981297i \(0.438340\pi\)
\(32\) 28.8465i 0.901453i
\(33\) 0 0
\(34\) −19.7065 −0.579603
\(35\) 21.0212i 0.600606i
\(36\) 0 0
\(37\) 26.8988 0.726993 0.363497 0.931595i \(-0.381583\pi\)
0.363497 + 0.931595i \(0.381583\pi\)
\(38\) 23.0265i 0.605961i
\(39\) 0 0
\(40\) −42.9356 −1.07339
\(41\) − 63.9635i − 1.56008i −0.625727 0.780042i \(-0.715197\pi\)
0.625727 0.780042i \(-0.284803\pi\)
\(42\) 0 0
\(43\) −16.2185 −0.377174 −0.188587 0.982056i \(-0.560391\pi\)
−0.188587 + 0.982056i \(0.560391\pi\)
\(44\) − 32.1760i − 0.731273i
\(45\) 0 0
\(46\) 9.77724 0.212549
\(47\) − 0.518396i − 0.0110297i −0.999985 0.00551485i \(-0.998245\pi\)
0.999985 0.00551485i \(-0.00175544\pi\)
\(48\) 0 0
\(49\) 7.00000 0.142857
\(50\) − 27.5533i − 0.551066i
\(51\) 0 0
\(52\) −77.0168 −1.48109
\(53\) − 71.8912i − 1.35644i −0.734860 0.678219i \(-0.762752\pi\)
0.734860 0.678219i \(-0.237248\pi\)
\(54\) 0 0
\(55\) 73.5092 1.33653
\(56\) 14.2974i 0.255311i
\(57\) 0 0
\(58\) −4.86983 −0.0839625
\(59\) − 38.5540i − 0.653457i −0.945118 0.326729i \(-0.894054\pi\)
0.945118 0.326729i \(-0.105946\pi\)
\(60\) 0 0
\(61\) 27.9222 0.457742 0.228871 0.973457i \(-0.426497\pi\)
0.228871 + 0.973457i \(0.426497\pi\)
\(62\) 8.62506i 0.139114i
\(63\) 0 0
\(64\) 19.1768 0.299637
\(65\) − 175.953i − 2.70696i
\(66\) 0 0
\(67\) 29.5649 0.441266 0.220633 0.975357i \(-0.429188\pi\)
0.220633 + 0.975357i \(0.429188\pi\)
\(68\) 94.8356i 1.39464i
\(69\) 0 0
\(70\) −15.1913 −0.217019
\(71\) − 39.4892i − 0.556186i −0.960554 0.278093i \(-0.910298\pi\)
0.960554 0.278093i \(-0.0897023\pi\)
\(72\) 0 0
\(73\) 67.1187 0.919434 0.459717 0.888066i \(-0.347951\pi\)
0.459717 + 0.888066i \(0.347951\pi\)
\(74\) 19.4388i 0.262687i
\(75\) 0 0
\(76\) 110.813 1.45806
\(77\) − 24.4784i − 0.317901i
\(78\) 0 0
\(79\) −18.7689 −0.237581 −0.118790 0.992919i \(-0.537902\pi\)
−0.118790 + 0.992919i \(0.537902\pi\)
\(80\) 79.4987i 0.993734i
\(81\) 0 0
\(82\) 46.2242 0.563710
\(83\) − 80.0893i − 0.964931i −0.875915 0.482466i \(-0.839741\pi\)
0.875915 0.482466i \(-0.160259\pi\)
\(84\) 0 0
\(85\) −216.661 −2.54895
\(86\) − 11.7205i − 0.136285i
\(87\) 0 0
\(88\) 49.9968 0.568145
\(89\) 62.9293i 0.707071i 0.935421 + 0.353535i \(0.115021\pi\)
−0.935421 + 0.353535i \(0.884979\pi\)
\(90\) 0 0
\(91\) −58.5917 −0.643864
\(92\) − 47.0520i − 0.511434i
\(93\) 0 0
\(94\) 0.374627 0.00398539
\(95\) 253.162i 2.66487i
\(96\) 0 0
\(97\) −46.7811 −0.482280 −0.241140 0.970490i \(-0.577521\pi\)
−0.241140 + 0.970490i \(0.577521\pi\)
\(98\) 5.05866i 0.0516190i
\(99\) 0 0
\(100\) −132.597 −1.32597
\(101\) 104.056i 1.03026i 0.857113 + 0.515128i \(0.172255\pi\)
−0.857113 + 0.515128i \(0.827745\pi\)
\(102\) 0 0
\(103\) −103.860 −1.00835 −0.504174 0.863602i \(-0.668203\pi\)
−0.504174 + 0.863602i \(0.668203\pi\)
\(104\) − 119.673i − 1.15070i
\(105\) 0 0
\(106\) 51.9533 0.490126
\(107\) − 116.531i − 1.08907i −0.838738 0.544536i \(-0.816706\pi\)
0.838738 0.544536i \(-0.183294\pi\)
\(108\) 0 0
\(109\) −13.9837 −0.128291 −0.0641456 0.997941i \(-0.520432\pi\)
−0.0641456 + 0.997941i \(0.520432\pi\)
\(110\) 53.1226i 0.482933i
\(111\) 0 0
\(112\) 26.4728 0.236365
\(113\) − 52.5714i − 0.465233i −0.972569 0.232617i \(-0.925271\pi\)
0.972569 0.232617i \(-0.0747287\pi\)
\(114\) 0 0
\(115\) 107.495 0.934737
\(116\) 23.4355i 0.202030i
\(117\) 0 0
\(118\) 27.8616 0.236116
\(119\) 72.1475i 0.606282i
\(120\) 0 0
\(121\) 35.4014 0.292574
\(122\) 20.1784i 0.165397i
\(123\) 0 0
\(124\) 41.5072 0.334736
\(125\) − 104.300i − 0.834400i
\(126\) 0 0
\(127\) 23.1204 0.182051 0.0910254 0.995849i \(-0.470986\pi\)
0.0910254 + 0.995849i \(0.470986\pi\)
\(128\) 129.244i 1.00972i
\(129\) 0 0
\(130\) 127.155 0.978114
\(131\) 52.3026i 0.399256i 0.979872 + 0.199628i \(0.0639734\pi\)
−0.979872 + 0.199628i \(0.936027\pi\)
\(132\) 0 0
\(133\) 84.3023 0.633852
\(134\) 21.3655i 0.159444i
\(135\) 0 0
\(136\) −147.360 −1.08353
\(137\) 145.292i 1.06052i 0.847834 + 0.530261i \(0.177906\pi\)
−0.847834 + 0.530261i \(0.822094\pi\)
\(138\) 0 0
\(139\) 48.7826 0.350954 0.175477 0.984484i \(-0.443853\pi\)
0.175477 + 0.984484i \(0.443853\pi\)
\(140\) 73.1066i 0.522190i
\(141\) 0 0
\(142\) 28.5375 0.200968
\(143\) 204.890i 1.43280i
\(144\) 0 0
\(145\) −53.5408 −0.369247
\(146\) 48.5043i 0.332222i
\(147\) 0 0
\(148\) 93.5473 0.632076
\(149\) − 221.450i − 1.48624i −0.669157 0.743121i \(-0.733345\pi\)
0.669157 0.743121i \(-0.266655\pi\)
\(150\) 0 0
\(151\) −282.901 −1.87352 −0.936759 0.349975i \(-0.886190\pi\)
−0.936759 + 0.349975i \(0.886190\pi\)
\(152\) 172.187i 1.13281i
\(153\) 0 0
\(154\) 17.6897 0.114868
\(155\) 94.8273i 0.611789i
\(156\) 0 0
\(157\) 34.3984 0.219098 0.109549 0.993981i \(-0.465059\pi\)
0.109549 + 0.993981i \(0.465059\pi\)
\(158\) − 13.5636i − 0.0858458i
\(159\) 0 0
\(160\) −229.193 −1.43246
\(161\) − 35.7954i − 0.222332i
\(162\) 0 0
\(163\) −0.514328 −0.00315539 −0.00157769 0.999999i \(-0.500502\pi\)
−0.00157769 + 0.999999i \(0.500502\pi\)
\(164\) − 222.449i − 1.35640i
\(165\) 0 0
\(166\) 57.8778 0.348661
\(167\) 156.191i 0.935275i 0.883920 + 0.467638i \(0.154895\pi\)
−0.883920 + 0.467638i \(0.845105\pi\)
\(168\) 0 0
\(169\) 321.426 1.90193
\(170\) − 156.574i − 0.921021i
\(171\) 0 0
\(172\) −56.4039 −0.327930
\(173\) 76.3720i 0.441457i 0.975335 + 0.220728i \(0.0708434\pi\)
−0.975335 + 0.220728i \(0.929157\pi\)
\(174\) 0 0
\(175\) −100.875 −0.576431
\(176\) − 92.5731i − 0.525984i
\(177\) 0 0
\(178\) −45.4768 −0.255488
\(179\) 92.6544i 0.517622i 0.965928 + 0.258811i \(0.0833307\pi\)
−0.965928 + 0.258811i \(0.916669\pi\)
\(180\) 0 0
\(181\) −150.103 −0.829296 −0.414648 0.909982i \(-0.636095\pi\)
−0.414648 + 0.909982i \(0.636095\pi\)
\(182\) − 42.3422i − 0.232649i
\(183\) 0 0
\(184\) 73.1118 0.397347
\(185\) 213.718i 1.15523i
\(186\) 0 0
\(187\) 252.293 1.34916
\(188\) − 1.80285i − 0.00958964i
\(189\) 0 0
\(190\) −182.952 −0.962904
\(191\) − 226.023i − 1.18337i −0.806171 0.591683i \(-0.798464\pi\)
0.806171 0.591683i \(-0.201536\pi\)
\(192\) 0 0
\(193\) 35.4187 0.183517 0.0917583 0.995781i \(-0.470751\pi\)
0.0917583 + 0.995781i \(0.470751\pi\)
\(194\) − 33.8071i − 0.174264i
\(195\) 0 0
\(196\) 24.3443 0.124206
\(197\) − 93.8848i − 0.476572i −0.971195 0.238286i \(-0.923414\pi\)
0.971195 0.238286i \(-0.0765856\pi\)
\(198\) 0 0
\(199\) −20.0037 −0.100521 −0.0502607 0.998736i \(-0.516005\pi\)
−0.0502607 + 0.998736i \(0.516005\pi\)
\(200\) − 206.037i − 1.03018i
\(201\) 0 0
\(202\) −75.1975 −0.372265
\(203\) 17.8289i 0.0878272i
\(204\) 0 0
\(205\) 508.207 2.47906
\(206\) − 75.0559i − 0.364349i
\(207\) 0 0
\(208\) −221.584 −1.06531
\(209\) − 294.798i − 1.41051i
\(210\) 0 0
\(211\) 241.002 1.14219 0.571095 0.820884i \(-0.306519\pi\)
0.571095 + 0.820884i \(0.306519\pi\)
\(212\) − 250.020i − 1.17934i
\(213\) 0 0
\(214\) 84.2127 0.393517
\(215\) − 128.860i − 0.599350i
\(216\) 0 0
\(217\) 31.5772 0.145517
\(218\) − 10.1056i − 0.0463558i
\(219\) 0 0
\(220\) 255.647 1.16203
\(221\) − 603.892i − 2.73254i
\(222\) 0 0
\(223\) −358.666 −1.60837 −0.804185 0.594380i \(-0.797398\pi\)
−0.804185 + 0.594380i \(0.797398\pi\)
\(224\) 76.3207i 0.340717i
\(225\) 0 0
\(226\) 37.9915 0.168104
\(227\) 126.918i 0.559111i 0.960129 + 0.279556i \(0.0901872\pi\)
−0.960129 + 0.279556i \(0.909813\pi\)
\(228\) 0 0
\(229\) −240.003 −1.04805 −0.524024 0.851703i \(-0.675570\pi\)
−0.524024 + 0.851703i \(0.675570\pi\)
\(230\) 77.6828i 0.337751i
\(231\) 0 0
\(232\) −36.4154 −0.156963
\(233\) 353.670i 1.51790i 0.651150 + 0.758949i \(0.274287\pi\)
−0.651150 + 0.758949i \(0.725713\pi\)
\(234\) 0 0
\(235\) 4.11879 0.0175268
\(236\) − 134.081i − 0.568141i
\(237\) 0 0
\(238\) −52.1385 −0.219069
\(239\) 460.988i 1.92882i 0.264413 + 0.964410i \(0.414822\pi\)
−0.264413 + 0.964410i \(0.585178\pi\)
\(240\) 0 0
\(241\) −286.281 −1.18789 −0.593944 0.804507i \(-0.702430\pi\)
−0.593944 + 0.804507i \(0.702430\pi\)
\(242\) 25.5834i 0.105716i
\(243\) 0 0
\(244\) 97.1067 0.397978
\(245\) 55.6169i 0.227008i
\(246\) 0 0
\(247\) −705.631 −2.85680
\(248\) 64.4961i 0.260065i
\(249\) 0 0
\(250\) 75.3741 0.301496
\(251\) − 171.893i − 0.684834i −0.939548 0.342417i \(-0.888754\pi\)
0.939548 0.342417i \(-0.111246\pi\)
\(252\) 0 0
\(253\) −125.173 −0.494757
\(254\) 16.7083i 0.0657809i
\(255\) 0 0
\(256\) −16.6934 −0.0652084
\(257\) − 477.109i − 1.85645i −0.372015 0.928227i \(-0.621333\pi\)
0.372015 0.928227i \(-0.378667\pi\)
\(258\) 0 0
\(259\) 71.1674 0.274778
\(260\) − 611.920i − 2.35354i
\(261\) 0 0
\(262\) −37.7973 −0.144264
\(263\) 309.722i 1.17765i 0.808260 + 0.588826i \(0.200410\pi\)
−0.808260 + 0.588826i \(0.799590\pi\)
\(264\) 0 0
\(265\) 571.195 2.15545
\(266\) 60.9224i 0.229032i
\(267\) 0 0
\(268\) 102.819 0.383654
\(269\) − 2.34739i − 0.00872637i −0.999990 0.00436318i \(-0.998611\pi\)
0.999990 0.00436318i \(-0.00138885\pi\)
\(270\) 0 0
\(271\) 13.0921 0.0483105 0.0241553 0.999708i \(-0.492310\pi\)
0.0241553 + 0.999708i \(0.492310\pi\)
\(272\) 272.850i 1.00312i
\(273\) 0 0
\(274\) −104.997 −0.383202
\(275\) 352.752i 1.28273i
\(276\) 0 0
\(277\) 223.245 0.805937 0.402968 0.915214i \(-0.367978\pi\)
0.402968 + 0.915214i \(0.367978\pi\)
\(278\) 35.2535i 0.126811i
\(279\) 0 0
\(280\) −113.597 −0.405703
\(281\) 42.6462i 0.151766i 0.997117 + 0.0758830i \(0.0241775\pi\)
−0.997117 + 0.0758830i \(0.975822\pi\)
\(282\) 0 0
\(283\) −103.687 −0.366386 −0.183193 0.983077i \(-0.558643\pi\)
−0.183193 + 0.983077i \(0.558643\pi\)
\(284\) − 137.334i − 0.483569i
\(285\) 0 0
\(286\) −148.067 −0.517716
\(287\) − 169.231i − 0.589657i
\(288\) 0 0
\(289\) −454.609 −1.57304
\(290\) − 38.6921i − 0.133421i
\(291\) 0 0
\(292\) 233.422 0.799391
\(293\) 322.920i 1.10212i 0.834467 + 0.551058i \(0.185776\pi\)
−0.834467 + 0.551058i \(0.814224\pi\)
\(294\) 0 0
\(295\) 306.322 1.03838
\(296\) 145.359i 0.491076i
\(297\) 0 0
\(298\) 160.034 0.537028
\(299\) 299.616i 1.00206i
\(300\) 0 0
\(301\) −42.9101 −0.142558
\(302\) − 204.443i − 0.676964i
\(303\) 0 0
\(304\) 318.818 1.04874
\(305\) 221.850i 0.727376i
\(306\) 0 0
\(307\) 553.767 1.80380 0.901901 0.431943i \(-0.142172\pi\)
0.901901 + 0.431943i \(0.142172\pi\)
\(308\) − 85.1297i − 0.276395i
\(309\) 0 0
\(310\) −68.5284 −0.221059
\(311\) − 281.881i − 0.906370i −0.891416 0.453185i \(-0.850288\pi\)
0.891416 0.453185i \(-0.149712\pi\)
\(312\) 0 0
\(313\) −423.175 −1.35200 −0.675999 0.736903i \(-0.736288\pi\)
−0.675999 + 0.736903i \(0.736288\pi\)
\(314\) 24.8586i 0.0791674i
\(315\) 0 0
\(316\) −65.2736 −0.206562
\(317\) − 163.024i − 0.514272i −0.966375 0.257136i \(-0.917221\pi\)
0.966375 0.257136i \(-0.0827788\pi\)
\(318\) 0 0
\(319\) 62.3461 0.195442
\(320\) 152.365i 0.476140i
\(321\) 0 0
\(322\) 25.8681 0.0803358
\(323\) 868.886i 2.69005i
\(324\) 0 0
\(325\) 844.351 2.59800
\(326\) − 0.371687i − 0.00114015i
\(327\) 0 0
\(328\) 345.653 1.05382
\(329\) − 1.37155i − 0.00416883i
\(330\) 0 0
\(331\) 5.69024 0.0171911 0.00859553 0.999963i \(-0.497264\pi\)
0.00859553 + 0.999963i \(0.497264\pi\)
\(332\) − 278.531i − 0.838948i
\(333\) 0 0
\(334\) −112.874 −0.337946
\(335\) 234.901i 0.701196i
\(336\) 0 0
\(337\) −642.277 −1.90587 −0.952933 0.303182i \(-0.901951\pi\)
−0.952933 + 0.303182i \(0.901951\pi\)
\(338\) 232.284i 0.687230i
\(339\) 0 0
\(340\) −753.494 −2.21616
\(341\) − 110.423i − 0.323820i
\(342\) 0 0
\(343\) 18.5203 0.0539949
\(344\) − 87.6433i − 0.254777i
\(345\) 0 0
\(346\) −55.1914 −0.159513
\(347\) − 30.2586i − 0.0872005i −0.999049 0.0436002i \(-0.986117\pi\)
0.999049 0.0436002i \(-0.0138828\pi\)
\(348\) 0 0
\(349\) −440.357 −1.26177 −0.630884 0.775877i \(-0.717308\pi\)
−0.630884 + 0.775877i \(0.717308\pi\)
\(350\) − 72.8992i − 0.208283i
\(351\) 0 0
\(352\) 266.886 0.758200
\(353\) − 268.129i − 0.759573i −0.925074 0.379787i \(-0.875997\pi\)
0.925074 0.379787i \(-0.124003\pi\)
\(354\) 0 0
\(355\) 313.752 0.883809
\(356\) 218.853i 0.614755i
\(357\) 0 0
\(358\) −66.9581 −0.187034
\(359\) 226.280i 0.630307i 0.949041 + 0.315153i \(0.102056\pi\)
−0.949041 + 0.315153i \(0.897944\pi\)
\(360\) 0 0
\(361\) 654.269 1.81238
\(362\) − 108.474i − 0.299652i
\(363\) 0 0
\(364\) −203.767 −0.559801
\(365\) 533.276i 1.46103i
\(366\) 0 0
\(367\) −546.177 −1.48822 −0.744110 0.668057i \(-0.767126\pi\)
−0.744110 + 0.668057i \(0.767126\pi\)
\(368\) − 135.372i − 0.367860i
\(369\) 0 0
\(370\) −154.447 −0.417423
\(371\) − 190.206i − 0.512685i
\(372\) 0 0
\(373\) −105.024 −0.281566 −0.140783 0.990040i \(-0.544962\pi\)
−0.140783 + 0.990040i \(0.544962\pi\)
\(374\) 182.324i 0.487496i
\(375\) 0 0
\(376\) 2.80137 0.00745044
\(377\) − 149.232i − 0.395842i
\(378\) 0 0
\(379\) 101.586 0.268037 0.134019 0.990979i \(-0.457212\pi\)
0.134019 + 0.990979i \(0.457212\pi\)
\(380\) 880.437i 2.31694i
\(381\) 0 0
\(382\) 163.339 0.427589
\(383\) 233.716i 0.610223i 0.952317 + 0.305112i \(0.0986938\pi\)
−0.952317 + 0.305112i \(0.901306\pi\)
\(384\) 0 0
\(385\) 194.487 0.505162
\(386\) 25.5959i 0.0663106i
\(387\) 0 0
\(388\) −162.693 −0.419313
\(389\) 72.3016i 0.185865i 0.995672 + 0.0929326i \(0.0296241\pi\)
−0.995672 + 0.0929326i \(0.970376\pi\)
\(390\) 0 0
\(391\) 368.936 0.943571
\(392\) 37.8274i 0.0964985i
\(393\) 0 0
\(394\) 67.8473 0.172201
\(395\) − 149.124i − 0.377529i
\(396\) 0 0
\(397\) −423.076 −1.06568 −0.532842 0.846215i \(-0.678876\pi\)
−0.532842 + 0.846215i \(0.678876\pi\)
\(398\) − 14.4560i − 0.0363217i
\(399\) 0 0
\(400\) −381.494 −0.953735
\(401\) 363.447i 0.906351i 0.891421 + 0.453176i \(0.149709\pi\)
−0.891421 + 0.453176i \(0.850291\pi\)
\(402\) 0 0
\(403\) −264.309 −0.655853
\(404\) 361.880i 0.895744i
\(405\) 0 0
\(406\) −12.8843 −0.0317348
\(407\) − 248.866i − 0.611464i
\(408\) 0 0
\(409\) −568.164 −1.38915 −0.694577 0.719418i \(-0.744408\pi\)
−0.694577 + 0.719418i \(0.744408\pi\)
\(410\) 367.264i 0.895765i
\(411\) 0 0
\(412\) −361.199 −0.876696
\(413\) − 102.004i − 0.246984i
\(414\) 0 0
\(415\) 636.331 1.53333
\(416\) − 638.822i − 1.53563i
\(417\) 0 0
\(418\) 213.040 0.509665
\(419\) − 638.204i − 1.52316i −0.648071 0.761580i \(-0.724424\pi\)
0.648071 0.761580i \(-0.275576\pi\)
\(420\) 0 0
\(421\) 246.608 0.585767 0.292883 0.956148i \(-0.405385\pi\)
0.292883 + 0.956148i \(0.405385\pi\)
\(422\) 174.164i 0.412711i
\(423\) 0 0
\(424\) 388.494 0.916260
\(425\) − 1039.70i − 2.44636i
\(426\) 0 0
\(427\) 73.8753 0.173010
\(428\) − 405.265i − 0.946880i
\(429\) 0 0
\(430\) 93.1229 0.216565
\(431\) − 138.307i − 0.320898i −0.987044 0.160449i \(-0.948706\pi\)
0.987044 0.160449i \(-0.0512942\pi\)
\(432\) 0 0
\(433\) 209.792 0.484507 0.242254 0.970213i \(-0.422113\pi\)
0.242254 + 0.970213i \(0.422113\pi\)
\(434\) 22.8198i 0.0525801i
\(435\) 0 0
\(436\) −48.6320 −0.111541
\(437\) − 431.091i − 0.986479i
\(438\) 0 0
\(439\) 714.425 1.62739 0.813696 0.581291i \(-0.197452\pi\)
0.813696 + 0.581291i \(0.197452\pi\)
\(440\) 397.238i 0.902813i
\(441\) 0 0
\(442\) 436.412 0.987357
\(443\) 523.302i 1.18127i 0.806940 + 0.590634i \(0.201122\pi\)
−0.806940 + 0.590634i \(0.798878\pi\)
\(444\) 0 0
\(445\) −499.990 −1.12357
\(446\) − 259.196i − 0.581157i
\(447\) 0 0
\(448\) 50.7370 0.113252
\(449\) − 453.325i − 1.00963i −0.863227 0.504816i \(-0.831560\pi\)
0.863227 0.504816i \(-0.168440\pi\)
\(450\) 0 0
\(451\) −591.787 −1.31217
\(452\) − 182.830i − 0.404492i
\(453\) 0 0
\(454\) −91.7195 −0.202025
\(455\) − 465.527i − 1.02314i
\(456\) 0 0
\(457\) 340.386 0.744826 0.372413 0.928067i \(-0.378530\pi\)
0.372413 + 0.928067i \(0.378530\pi\)
\(458\) − 173.442i − 0.378694i
\(459\) 0 0
\(460\) 373.841 0.812697
\(461\) 198.359i 0.430280i 0.976583 + 0.215140i \(0.0690208\pi\)
−0.976583 + 0.215140i \(0.930979\pi\)
\(462\) 0 0
\(463\) 578.341 1.24912 0.624559 0.780978i \(-0.285279\pi\)
0.624559 + 0.780978i \(0.285279\pi\)
\(464\) 67.4260i 0.145315i
\(465\) 0 0
\(466\) −255.585 −0.548467
\(467\) − 867.984i − 1.85864i −0.369277 0.929319i \(-0.620395\pi\)
0.369277 0.929319i \(-0.379605\pi\)
\(468\) 0 0
\(469\) 78.2212 0.166783
\(470\) 2.97651i 0.00633300i
\(471\) 0 0
\(472\) 208.343 0.441404
\(473\) 150.053i 0.317236i
\(474\) 0 0
\(475\) −1214.86 −2.55760
\(476\) 250.911i 0.527125i
\(477\) 0 0
\(478\) −333.140 −0.696946
\(479\) 723.856i 1.51118i 0.655043 + 0.755591i \(0.272650\pi\)
−0.655043 + 0.755591i \(0.727350\pi\)
\(480\) 0 0
\(481\) −595.688 −1.23844
\(482\) − 206.885i − 0.429223i
\(483\) 0 0
\(484\) 123.117 0.254375
\(485\) − 371.689i − 0.766369i
\(486\) 0 0
\(487\) −881.932 −1.81095 −0.905474 0.424401i \(-0.860485\pi\)
−0.905474 + 0.424401i \(0.860485\pi\)
\(488\) 150.889i 0.309200i
\(489\) 0 0
\(490\) −40.1924 −0.0820253
\(491\) − 751.995i − 1.53156i −0.643103 0.765779i \(-0.722353\pi\)
0.643103 0.765779i \(-0.277647\pi\)
\(492\) 0 0
\(493\) −183.759 −0.372736
\(494\) − 509.935i − 1.03226i
\(495\) 0 0
\(496\) 119.420 0.240766
\(497\) − 104.479i − 0.210218i
\(498\) 0 0
\(499\) 27.5359 0.0551822 0.0275911 0.999619i \(-0.491216\pi\)
0.0275911 + 0.999619i \(0.491216\pi\)
\(500\) − 362.730i − 0.725460i
\(501\) 0 0
\(502\) 124.221 0.247453
\(503\) 820.917i 1.63204i 0.578022 + 0.816021i \(0.303825\pi\)
−0.578022 + 0.816021i \(0.696175\pi\)
\(504\) 0 0
\(505\) −826.751 −1.63713
\(506\) − 90.4585i − 0.178772i
\(507\) 0 0
\(508\) 80.4072 0.158282
\(509\) − 490.271i − 0.963203i −0.876390 0.481602i \(-0.840055\pi\)
0.876390 0.481602i \(-0.159945\pi\)
\(510\) 0 0
\(511\) 177.579 0.347513
\(512\) 504.914i 0.986160i
\(513\) 0 0
\(514\) 344.790 0.670798
\(515\) − 825.194i − 1.60232i
\(516\) 0 0
\(517\) −4.79617 −0.00927693
\(518\) 51.4303i 0.0992862i
\(519\) 0 0
\(520\) 950.833 1.82852
\(521\) − 424.710i − 0.815182i −0.913164 0.407591i \(-0.866369\pi\)
0.913164 0.407591i \(-0.133631\pi\)
\(522\) 0 0
\(523\) 745.325 1.42510 0.712548 0.701624i \(-0.247541\pi\)
0.712548 + 0.701624i \(0.247541\pi\)
\(524\) 181.896i 0.347129i
\(525\) 0 0
\(526\) −223.826 −0.425524
\(527\) 325.460i 0.617570i
\(528\) 0 0
\(529\) 345.955 0.653979
\(530\) 412.783i 0.778836i
\(531\) 0 0
\(532\) 293.183 0.551096
\(533\) 1416.51i 2.65761i
\(534\) 0 0
\(535\) 925.867 1.73059
\(536\) 159.766i 0.298071i
\(537\) 0 0
\(538\) 1.69638 0.00315312
\(539\) − 64.7637i − 0.120155i
\(540\) 0 0
\(541\) −2.11927 −0.00391732 −0.00195866 0.999998i \(-0.500623\pi\)
−0.00195866 + 0.999998i \(0.500623\pi\)
\(542\) 9.46125i 0.0174562i
\(543\) 0 0
\(544\) −786.621 −1.44599
\(545\) − 111.105i − 0.203862i
\(546\) 0 0
\(547\) −172.013 −0.314466 −0.157233 0.987562i \(-0.550257\pi\)
−0.157233 + 0.987562i \(0.550257\pi\)
\(548\) 505.289i 0.922059i
\(549\) 0 0
\(550\) −254.922 −0.463494
\(551\) 214.717i 0.389686i
\(552\) 0 0
\(553\) −49.6578 −0.0897971
\(554\) 161.331i 0.291211i
\(555\) 0 0
\(556\) 169.654 0.305133
\(557\) − 866.241i − 1.55519i −0.628765 0.777595i \(-0.716439\pi\)
0.628765 0.777595i \(-0.283561\pi\)
\(558\) 0 0
\(559\) 359.168 0.642518
\(560\) 210.334i 0.375596i
\(561\) 0 0
\(562\) −30.8190 −0.0548380
\(563\) − 494.890i − 0.879023i −0.898237 0.439511i \(-0.855152\pi\)
0.898237 0.439511i \(-0.144848\pi\)
\(564\) 0 0
\(565\) 417.694 0.739281
\(566\) − 74.9312i − 0.132387i
\(567\) 0 0
\(568\) 213.396 0.375697
\(569\) − 476.783i − 0.837932i −0.908002 0.418966i \(-0.862393\pi\)
0.908002 0.418966i \(-0.137607\pi\)
\(570\) 0 0
\(571\) 724.636 1.26906 0.634532 0.772896i \(-0.281193\pi\)
0.634532 + 0.772896i \(0.281193\pi\)
\(572\) 712.556i 1.24573i
\(573\) 0 0
\(574\) 122.298 0.213062
\(575\) 515.840i 0.897113i
\(576\) 0 0
\(577\) 74.4585 0.129044 0.0645221 0.997916i \(-0.479448\pi\)
0.0645221 + 0.997916i \(0.479448\pi\)
\(578\) − 328.530i − 0.568392i
\(579\) 0 0
\(580\) −186.202 −0.321037
\(581\) − 211.896i − 0.364710i
\(582\) 0 0
\(583\) −665.134 −1.14088
\(584\) 362.704i 0.621068i
\(585\) 0 0
\(586\) −233.363 −0.398230
\(587\) − 750.028i − 1.27773i −0.769319 0.638865i \(-0.779404\pi\)
0.769319 0.638865i \(-0.220596\pi\)
\(588\) 0 0
\(589\) 380.290 0.645654
\(590\) 221.368i 0.375200i
\(591\) 0 0
\(592\) 269.143 0.454634
\(593\) − 830.097i − 1.39983i −0.714228 0.699913i \(-0.753222\pi\)
0.714228 0.699913i \(-0.246778\pi\)
\(594\) 0 0
\(595\) −573.231 −0.963414
\(596\) − 770.149i − 1.29220i
\(597\) 0 0
\(598\) −216.522 −0.362078
\(599\) 631.380i 1.05406i 0.849848 + 0.527029i \(0.176694\pi\)
−0.849848 + 0.527029i \(0.823306\pi\)
\(600\) 0 0
\(601\) 545.316 0.907347 0.453674 0.891168i \(-0.350113\pi\)
0.453674 + 0.891168i \(0.350113\pi\)
\(602\) − 31.0096i − 0.0515110i
\(603\) 0 0
\(604\) −983.861 −1.62891
\(605\) 281.274i 0.464915i
\(606\) 0 0
\(607\) 5.26052 0.00866643 0.00433321 0.999991i \(-0.498621\pi\)
0.00433321 + 0.999991i \(0.498621\pi\)
\(608\) 919.145i 1.51175i
\(609\) 0 0
\(610\) −160.323 −0.262825
\(611\) 11.4802i 0.0187891i
\(612\) 0 0
\(613\) −428.462 −0.698959 −0.349480 0.936944i \(-0.613642\pi\)
−0.349480 + 0.936944i \(0.613642\pi\)
\(614\) 400.189i 0.651773i
\(615\) 0 0
\(616\) 132.279 0.214739
\(617\) 1045.71i 1.69483i 0.530931 + 0.847415i \(0.321842\pi\)
−0.530931 + 0.847415i \(0.678158\pi\)
\(618\) 0 0
\(619\) −699.304 −1.12973 −0.564866 0.825182i \(-0.691072\pi\)
−0.564866 + 0.825182i \(0.691072\pi\)
\(620\) 329.786i 0.531913i
\(621\) 0 0
\(622\) 203.706 0.327501
\(623\) 166.495i 0.267248i
\(624\) 0 0
\(625\) −124.491 −0.199185
\(626\) − 305.814i − 0.488521i
\(627\) 0 0
\(628\) 119.629 0.190493
\(629\) 733.507i 1.16615i
\(630\) 0 0
\(631\) −748.642 −1.18644 −0.593218 0.805042i \(-0.702143\pi\)
−0.593218 + 0.805042i \(0.702143\pi\)
\(632\) − 101.425i − 0.160483i
\(633\) 0 0
\(634\) 117.812 0.185823
\(635\) 183.698i 0.289288i
\(636\) 0 0
\(637\) −155.019 −0.243358
\(638\) 45.0554i 0.0706197i
\(639\) 0 0
\(640\) −1026.88 −1.60450
\(641\) − 66.0855i − 0.103097i −0.998670 0.0515487i \(-0.983584\pi\)
0.998670 0.0515487i \(-0.0164158\pi\)
\(642\) 0 0
\(643\) 763.206 1.18695 0.593473 0.804854i \(-0.297757\pi\)
0.593473 + 0.804854i \(0.297757\pi\)
\(644\) − 124.488i − 0.193304i
\(645\) 0 0
\(646\) −627.914 −0.972004
\(647\) 522.507i 0.807585i 0.914851 + 0.403792i \(0.132308\pi\)
−0.914851 + 0.403792i \(0.867692\pi\)
\(648\) 0 0
\(649\) −356.700 −0.549614
\(650\) 610.183i 0.938744i
\(651\) 0 0
\(652\) −1.78871 −0.00274342
\(653\) − 38.5504i − 0.0590359i −0.999564 0.0295179i \(-0.990603\pi\)
0.999564 0.0295179i \(-0.00939721\pi\)
\(654\) 0 0
\(655\) −415.558 −0.634440
\(656\) − 640.005i − 0.975618i
\(657\) 0 0
\(658\) 0.991169 0.00150634
\(659\) − 749.071i − 1.13668i −0.822794 0.568339i \(-0.807586\pi\)
0.822794 0.568339i \(-0.192414\pi\)
\(660\) 0 0
\(661\) 605.366 0.915834 0.457917 0.888995i \(-0.348596\pi\)
0.457917 + 0.888995i \(0.348596\pi\)
\(662\) 4.11214i 0.00621169i
\(663\) 0 0
\(664\) 432.796 0.651801
\(665\) 669.805i 1.00723i
\(666\) 0 0
\(667\) 91.1706 0.136688
\(668\) 543.194i 0.813164i
\(669\) 0 0
\(670\) −169.755 −0.253365
\(671\) − 258.335i − 0.385000i
\(672\) 0 0
\(673\) 1042.10 1.54844 0.774222 0.632914i \(-0.218141\pi\)
0.774222 + 0.632914i \(0.218141\pi\)
\(674\) − 464.151i − 0.688652i
\(675\) 0 0
\(676\) 1117.84 1.65361
\(677\) − 42.8646i − 0.0633155i −0.999499 0.0316577i \(-0.989921\pi\)
0.999499 0.0316577i \(-0.0100787\pi\)
\(678\) 0 0
\(679\) −123.771 −0.182285
\(680\) − 1170.82i − 1.72179i
\(681\) 0 0
\(682\) 79.7986 0.117007
\(683\) 1041.21i 1.52446i 0.647305 + 0.762231i \(0.275896\pi\)
−0.647305 + 0.762231i \(0.724104\pi\)
\(684\) 0 0
\(685\) −1154.38 −1.68523
\(686\) 13.3840i 0.0195101i
\(687\) 0 0
\(688\) −162.279 −0.235870
\(689\) 1592.07i 2.31070i
\(690\) 0 0
\(691\) −521.023 −0.754014 −0.377007 0.926210i \(-0.623047\pi\)
−0.377007 + 0.926210i \(0.623047\pi\)
\(692\) 265.603i 0.383819i
\(693\) 0 0
\(694\) 21.8668 0.0315084
\(695\) 387.591i 0.557685i
\(696\) 0 0
\(697\) 1744.23 2.50249
\(698\) − 318.231i − 0.455918i
\(699\) 0 0
\(700\) −350.820 −0.501171
\(701\) 249.669i 0.356161i 0.984016 + 0.178080i \(0.0569887\pi\)
−0.984016 + 0.178080i \(0.943011\pi\)
\(702\) 0 0
\(703\) 857.083 1.21918
\(704\) − 177.423i − 0.252021i
\(705\) 0 0
\(706\) 193.768 0.274459
\(707\) 275.306i 0.389400i
\(708\) 0 0
\(709\) 1050.17 1.48120 0.740601 0.671945i \(-0.234541\pi\)
0.740601 + 0.671945i \(0.234541\pi\)
\(710\) 226.738i 0.319349i
\(711\) 0 0
\(712\) −340.065 −0.477619
\(713\) − 161.474i − 0.226472i
\(714\) 0 0
\(715\) −1627.90 −2.27679
\(716\) 322.229i 0.450041i
\(717\) 0 0
\(718\) −163.525 −0.227750
\(719\) 121.143i 0.168488i 0.996445 + 0.0842441i \(0.0268476\pi\)
−0.996445 + 0.0842441i \(0.973152\pi\)
\(720\) 0 0
\(721\) −274.787 −0.381120
\(722\) 472.818i 0.654873i
\(723\) 0 0
\(724\) −522.020 −0.721022
\(725\) − 256.928i − 0.354384i
\(726\) 0 0
\(727\) 961.438 1.32247 0.661236 0.750178i \(-0.270032\pi\)
0.661236 + 0.750178i \(0.270032\pi\)
\(728\) − 316.624i − 0.434924i
\(729\) 0 0
\(730\) −385.380 −0.527918
\(731\) − 442.265i − 0.605014i
\(732\) 0 0
\(733\) 161.183 0.219895 0.109948 0.993937i \(-0.464932\pi\)
0.109948 + 0.993937i \(0.464932\pi\)
\(734\) − 394.703i − 0.537743i
\(735\) 0 0
\(736\) 390.276 0.530267
\(737\) − 273.533i − 0.371143i
\(738\) 0 0
\(739\) −643.023 −0.870126 −0.435063 0.900400i \(-0.643274\pi\)
−0.435063 + 0.900400i \(0.643274\pi\)
\(740\) 743.258i 1.00440i
\(741\) 0 0
\(742\) 137.456 0.185250
\(743\) − 49.7283i − 0.0669291i −0.999440 0.0334646i \(-0.989346\pi\)
0.999440 0.0334646i \(-0.0106541\pi\)
\(744\) 0 0
\(745\) 1759.48 2.36172
\(746\) − 75.8973i − 0.101739i
\(747\) 0 0
\(748\) 877.414 1.17301
\(749\) − 308.311i − 0.411630i
\(750\) 0 0
\(751\) 966.138 1.28647 0.643235 0.765669i \(-0.277592\pi\)
0.643235 + 0.765669i \(0.277592\pi\)
\(752\) − 5.18696i − 0.00689755i
\(753\) 0 0
\(754\) 107.845 0.143031
\(755\) − 2247.73i − 2.97712i
\(756\) 0 0
\(757\) 136.110 0.179802 0.0899009 0.995951i \(-0.471345\pi\)
0.0899009 + 0.995951i \(0.471345\pi\)
\(758\) 73.4127i 0.0968506i
\(759\) 0 0
\(760\) −1368.07 −1.80009
\(761\) 1340.82i 1.76191i 0.473196 + 0.880957i \(0.343100\pi\)
−0.473196 + 0.880957i \(0.656900\pi\)
\(762\) 0 0
\(763\) −36.9975 −0.0484895
\(764\) − 786.052i − 1.02886i
\(765\) 0 0
\(766\) −168.898 −0.220494
\(767\) 853.800i 1.11317i
\(768\) 0 0
\(769\) −572.189 −0.744069 −0.372035 0.928219i \(-0.621340\pi\)
−0.372035 + 0.928219i \(0.621340\pi\)
\(770\) 140.549i 0.182531i
\(771\) 0 0
\(772\) 123.178 0.159556
\(773\) − 628.925i − 0.813616i −0.913514 0.406808i \(-0.866642\pi\)
0.913514 0.406808i \(-0.133358\pi\)
\(774\) 0 0
\(775\) −455.052 −0.587164
\(776\) − 252.801i − 0.325775i
\(777\) 0 0
\(778\) −52.2499 −0.0671592
\(779\) − 2038.09i − 2.61629i
\(780\) 0 0
\(781\) −365.352 −0.467800
\(782\) 266.617i 0.340943i
\(783\) 0 0
\(784\) 70.0405 0.0893374
\(785\) 273.305i 0.348159i
\(786\) 0 0
\(787\) −266.123 −0.338149 −0.169074 0.985603i \(-0.554078\pi\)
−0.169074 + 0.985603i \(0.554078\pi\)
\(788\) − 326.508i − 0.414350i
\(789\) 0 0
\(790\) 107.767 0.136414
\(791\) − 139.091i − 0.175842i
\(792\) 0 0
\(793\) −618.354 −0.779765
\(794\) − 305.743i − 0.385066i
\(795\) 0 0
\(796\) −69.5681 −0.0873971
\(797\) − 716.744i − 0.899303i −0.893204 0.449651i \(-0.851548\pi\)
0.893204 0.449651i \(-0.148452\pi\)
\(798\) 0 0
\(799\) 14.1362 0.0176924
\(800\) − 1099.84i − 1.37480i
\(801\) 0 0
\(802\) −262.651 −0.327494
\(803\) − 620.979i − 0.773323i
\(804\) 0 0
\(805\) 284.405 0.353298
\(806\) − 191.007i − 0.236981i
\(807\) 0 0
\(808\) −562.309 −0.695927
\(809\) 591.782i 0.731498i 0.930713 + 0.365749i \(0.119187\pi\)
−0.930713 + 0.365749i \(0.880813\pi\)
\(810\) 0 0
\(811\) −1308.23 −1.61311 −0.806555 0.591159i \(-0.798670\pi\)
−0.806555 + 0.591159i \(0.798670\pi\)
\(812\) 62.0046i 0.0763603i
\(813\) 0 0
\(814\) 179.847 0.220942
\(815\) − 4.08648i − 0.00501408i
\(816\) 0 0
\(817\) −516.774 −0.632527
\(818\) − 410.593i − 0.501947i
\(819\) 0 0
\(820\) 1767.42 2.15539
\(821\) − 142.567i − 0.173650i −0.996224 0.0868252i \(-0.972328\pi\)
0.996224 0.0868252i \(-0.0276722\pi\)
\(822\) 0 0
\(823\) 113.006 0.137310 0.0686550 0.997640i \(-0.478129\pi\)
0.0686550 + 0.997640i \(0.478129\pi\)
\(824\) − 561.250i − 0.681128i
\(825\) 0 0
\(826\) 73.7150 0.0892433
\(827\) − 69.0940i − 0.0835477i −0.999127 0.0417739i \(-0.986699\pi\)
0.999127 0.0417739i \(-0.0133009\pi\)
\(828\) 0 0
\(829\) −107.675 −0.129885 −0.0649426 0.997889i \(-0.520686\pi\)
−0.0649426 + 0.997889i \(0.520686\pi\)
\(830\) 459.855i 0.554042i
\(831\) 0 0
\(832\) −424.681 −0.510434
\(833\) 190.884i 0.229153i
\(834\) 0 0
\(835\) −1240.98 −1.48620
\(836\) − 1025.23i − 1.22636i
\(837\) 0 0
\(838\) 461.208 0.550368
\(839\) − 462.914i − 0.551745i −0.961194 0.275873i \(-0.911033\pi\)
0.961194 0.275873i \(-0.0889668\pi\)
\(840\) 0 0
\(841\) 795.590 0.946005
\(842\) 178.215i 0.211657i
\(843\) 0 0
\(844\) 838.147 0.993065
\(845\) 2553.82i 3.02227i
\(846\) 0 0
\(847\) 93.6633 0.110582
\(848\) − 719.329i − 0.848265i
\(849\) 0 0
\(850\) 751.356 0.883949
\(851\) − 363.924i − 0.427643i
\(852\) 0 0
\(853\) −890.954 −1.04449 −0.522247 0.852794i \(-0.674906\pi\)
−0.522247 + 0.852794i \(0.674906\pi\)
\(854\) 53.3871i 0.0625142i
\(855\) 0 0
\(856\) 629.722 0.735656
\(857\) 615.003i 0.717623i 0.933410 + 0.358811i \(0.116818\pi\)
−0.933410 + 0.358811i \(0.883182\pi\)
\(858\) 0 0
\(859\) 34.4978 0.0401604 0.0200802 0.999798i \(-0.493608\pi\)
0.0200802 + 0.999798i \(0.493608\pi\)
\(860\) − 448.144i − 0.521098i
\(861\) 0 0
\(862\) 99.9496 0.115951
\(863\) − 588.277i − 0.681665i −0.940124 0.340832i \(-0.889291\pi\)
0.940124 0.340832i \(-0.110709\pi\)
\(864\) 0 0
\(865\) −606.796 −0.701498
\(866\) 151.609i 0.175068i
\(867\) 0 0
\(868\) 109.818 0.126518
\(869\) 173.649i 0.199826i
\(870\) 0 0
\(871\) −654.730 −0.751700
\(872\) − 75.5669i − 0.0866593i
\(873\) 0 0
\(874\) 311.535 0.356447
\(875\) − 275.952i − 0.315374i
\(876\) 0 0
\(877\) 66.4285 0.0757452 0.0378726 0.999283i \(-0.487942\pi\)
0.0378726 + 0.999283i \(0.487942\pi\)
\(878\) 516.290i 0.588030i
\(879\) 0 0
\(880\) 735.518 0.835816
\(881\) 1060.27i 1.20349i 0.798689 + 0.601744i \(0.205527\pi\)
−0.798689 + 0.601744i \(0.794473\pi\)
\(882\) 0 0
\(883\) 828.431 0.938201 0.469100 0.883145i \(-0.344578\pi\)
0.469100 + 0.883145i \(0.344578\pi\)
\(884\) − 2100.19i − 2.37578i
\(885\) 0 0
\(886\) −378.172 −0.426831
\(887\) − 285.577i − 0.321958i −0.986958 0.160979i \(-0.948535\pi\)
0.986958 0.160979i \(-0.0514651\pi\)
\(888\) 0 0
\(889\) 61.1709 0.0688087
\(890\) − 361.326i − 0.405984i
\(891\) 0 0
\(892\) −1247.35 −1.39838
\(893\) − 16.5178i − 0.0184970i
\(894\) 0 0
\(895\) −736.164 −0.822530
\(896\) 341.949i 0.381639i
\(897\) 0 0
\(898\) 327.603 0.364814
\(899\) 80.4268i 0.0894625i
\(900\) 0 0
\(901\) 1960.42 2.17582
\(902\) − 427.664i − 0.474129i
\(903\) 0 0
\(904\) 284.091 0.314260
\(905\) − 1192.61i − 1.31780i
\(906\) 0 0
\(907\) 766.603 0.845207 0.422604 0.906315i \(-0.361116\pi\)
0.422604 + 0.906315i \(0.361116\pi\)
\(908\) 441.391i 0.486113i
\(909\) 0 0
\(910\) 336.420 0.369692
\(911\) − 1099.95i − 1.20741i −0.797207 0.603706i \(-0.793690\pi\)
0.797207 0.603706i \(-0.206310\pi\)
\(912\) 0 0
\(913\) −740.982 −0.811591
\(914\) 245.985i 0.269130i
\(915\) 0 0
\(916\) −834.672 −0.911214
\(917\) 138.380i 0.150905i
\(918\) 0 0
\(919\) 411.489 0.447757 0.223879 0.974617i \(-0.428128\pi\)
0.223879 + 0.974617i \(0.428128\pi\)
\(920\) 580.893i 0.631405i
\(921\) 0 0
\(922\) −143.347 −0.155474
\(923\) 874.510i 0.947465i
\(924\) 0 0
\(925\) −1025.58 −1.10873
\(926\) 417.947i 0.451347i
\(927\) 0 0
\(928\) −194.388 −0.209470
\(929\) − 217.553i − 0.234180i −0.993121 0.117090i \(-0.962643\pi\)
0.993121 0.117090i \(-0.0373566\pi\)
\(930\) 0 0
\(931\) 223.043 0.239574
\(932\) 1229.98i 1.31972i
\(933\) 0 0
\(934\) 627.262 0.671587
\(935\) 2004.54i 2.14389i
\(936\) 0 0
\(937\) −674.025 −0.719344 −0.359672 0.933079i \(-0.617111\pi\)
−0.359672 + 0.933079i \(0.617111\pi\)
\(938\) 56.5278i 0.0602642i
\(939\) 0 0
\(940\) 14.3242 0.0152385
\(941\) 1535.74i 1.63203i 0.578030 + 0.816015i \(0.303822\pi\)
−0.578030 + 0.816015i \(0.696178\pi\)
\(942\) 0 0
\(943\) −865.388 −0.917697
\(944\) − 385.763i − 0.408647i
\(945\) 0 0
\(946\) −108.438 −0.114628
\(947\) 312.347i 0.329828i 0.986308 + 0.164914i \(0.0527346\pi\)
−0.986308 + 0.164914i \(0.947265\pi\)
\(948\) 0 0
\(949\) −1486.38 −1.56626
\(950\) − 877.939i − 0.924146i
\(951\) 0 0
\(952\) −389.879 −0.409537
\(953\) − 842.068i − 0.883597i −0.897114 0.441798i \(-0.854341\pi\)
0.897114 0.441798i \(-0.145659\pi\)
\(954\) 0 0
\(955\) 1795.81 1.88043
\(956\) 1603.20i 1.67699i
\(957\) 0 0
\(958\) −523.106 −0.546040
\(959\) 384.406i 0.400840i
\(960\) 0 0
\(961\) −818.554 −0.851773
\(962\) − 430.483i − 0.447488i
\(963\) 0 0
\(964\) −995.615 −1.03280
\(965\) 281.411i 0.291618i
\(966\) 0 0
\(967\) 1270.48 1.31383 0.656916 0.753964i \(-0.271861\pi\)
0.656916 + 0.753964i \(0.271861\pi\)
\(968\) 191.306i 0.197630i
\(969\) 0 0
\(970\) 268.607 0.276914
\(971\) 140.610i 0.144810i 0.997375 + 0.0724048i \(0.0230673\pi\)
−0.997375 + 0.0724048i \(0.976933\pi\)
\(972\) 0 0
\(973\) 129.067 0.132648
\(974\) − 637.342i − 0.654355i
\(975\) 0 0
\(976\) 279.384 0.286254
\(977\) − 824.093i − 0.843493i −0.906714 0.421747i \(-0.861417\pi\)
0.906714 0.421747i \(-0.138583\pi\)
\(978\) 0 0
\(979\) 582.219 0.594708
\(980\) 193.422i 0.197369i
\(981\) 0 0
\(982\) 543.441 0.553403
\(983\) − 191.958i − 0.195277i −0.995222 0.0976387i \(-0.968871\pi\)
0.995222 0.0976387i \(-0.0311290\pi\)
\(984\) 0 0
\(985\) 745.940 0.757299
\(986\) − 132.796i − 0.134682i
\(987\) 0 0
\(988\) −2454.01 −2.48382
\(989\) 219.426i 0.221867i
\(990\) 0 0
\(991\) 1913.96 1.93135 0.965673 0.259761i \(-0.0836438\pi\)
0.965673 + 0.259761i \(0.0836438\pi\)
\(992\) 344.285i 0.347061i
\(993\) 0 0
\(994\) 75.5030 0.0759588
\(995\) − 158.935i − 0.159734i
\(996\) 0 0
\(997\) −1686.26 −1.69134 −0.845669 0.533708i \(-0.820798\pi\)
−0.845669 + 0.533708i \(0.820798\pi\)
\(998\) 19.8993i 0.0199391i
\(999\) 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.b.c.134.5 yes 8
3.2 odd 2 inner 189.3.b.c.134.4 8
4.3 odd 2 3024.3.d.j.1457.8 8
9.2 odd 6 567.3.r.e.134.4 16
9.4 even 3 567.3.r.e.512.4 16
9.5 odd 6 567.3.r.e.512.5 16
9.7 even 3 567.3.r.e.134.5 16
12.11 even 2 3024.3.d.j.1457.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.3.b.c.134.4 8 3.2 odd 2 inner
189.3.b.c.134.5 yes 8 1.1 even 1 trivial
567.3.r.e.134.4 16 9.2 odd 6
567.3.r.e.134.5 16 9.7 even 3
567.3.r.e.512.4 16 9.4 even 3
567.3.r.e.512.5 16 9.5 odd 6
3024.3.d.j.1457.1 8 12.11 even 2
3024.3.d.j.1457.8 8 4.3 odd 2