Properties

Label 2070.3.c.b.91.14
Level $2070$
Weight $3$
Character 2070.91
Analytic conductor $56.403$
Analytic rank $0$
Dimension $32$
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] = [2070,3,Mod(91,2070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2070, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2070.91");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2070.c (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: \(56.4034147226\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 690)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 91.14
Character \(\chi\) \(=\) 2070.91
Dual form 2070.3.c.b.91.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +3.80775i q^{7} -2.82843 q^{8} +O(q^{10})\) \(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +3.80775i q^{7} -2.82843 q^{8} -3.16228i q^{10} +16.1827i q^{11} +12.9618 q^{13} -5.38497i q^{14} +4.00000 q^{16} +10.8189i q^{17} +17.1074i q^{19} +4.47214i q^{20} -22.8858i q^{22} +(-11.6478 + 19.8325i) q^{23} -5.00000 q^{25} -18.3308 q^{26} +7.61550i q^{28} +10.1177 q^{29} +29.2531 q^{31} -5.65685 q^{32} -15.3003i q^{34} -8.51438 q^{35} +3.06540i q^{37} -24.1935i q^{38} -6.32456i q^{40} -55.3042 q^{41} +34.1448i q^{43} +32.3653i q^{44} +(16.4724 - 28.0475i) q^{46} +49.2975 q^{47} +34.5011 q^{49} +7.07107 q^{50} +25.9237 q^{52} -28.6856i q^{53} -36.1856 q^{55} -10.7699i q^{56} -14.3086 q^{58} -24.2529 q^{59} -52.9773i q^{61} -41.3702 q^{62} +8.00000 q^{64} +28.9835i q^{65} -7.48607i q^{67} +21.6379i q^{68} +12.0412 q^{70} +121.879 q^{71} -132.268 q^{73} -4.33513i q^{74} +34.2148i q^{76} -61.6195 q^{77} +18.6463i q^{79} +8.94427i q^{80} +78.2120 q^{82} +46.3088i q^{83} -24.1919 q^{85} -48.2881i q^{86} -45.7715i q^{88} +17.2346i q^{89} +49.3554i q^{91} +(-23.2955 + 39.6651i) q^{92} -69.7172 q^{94} -38.2533 q^{95} +31.4987i q^{97} -48.7919 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 64 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 64 q^{4} - 48 q^{13} + 128 q^{16} + 80 q^{23} - 160 q^{25} - 120 q^{29} + 248 q^{31} + 120 q^{35} - 72 q^{41} + 160 q^{46} - 400 q^{47} - 344 q^{49} - 96 q^{52} - 256 q^{58} - 120 q^{59} - 160 q^{62} + 256 q^{64} - 104 q^{71} + 16 q^{73} - 240 q^{77} + 64 q^{82} - 120 q^{85} + 160 q^{92} + 96 q^{94} + 160 q^{95} - 64 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(1891\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −0.707107
\(3\) 0 0
\(4\) 2.00000 0.500000
\(5\) 2.23607i 0.447214i
\(6\) 0 0
\(7\) 3.80775i 0.543964i 0.962302 + 0.271982i \(0.0876791\pi\)
−0.962302 + 0.271982i \(0.912321\pi\)
\(8\) −2.82843 −0.353553
\(9\) 0 0
\(10\) 3.16228i 0.316228i
\(11\) 16.1827i 1.47115i 0.677442 + 0.735576i \(0.263088\pi\)
−0.677442 + 0.735576i \(0.736912\pi\)
\(12\) 0 0
\(13\) 12.9618 0.997064 0.498532 0.866871i \(-0.333873\pi\)
0.498532 + 0.866871i \(0.333873\pi\)
\(14\) 5.38497i 0.384641i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 10.8189i 0.636408i 0.948022 + 0.318204i \(0.103080\pi\)
−0.948022 + 0.318204i \(0.896920\pi\)
\(18\) 0 0
\(19\) 17.1074i 0.900390i 0.892930 + 0.450195i \(0.148646\pi\)
−0.892930 + 0.450195i \(0.851354\pi\)
\(20\) 4.47214i 0.223607i
\(21\) 0 0
\(22\) 22.8858i 1.04026i
\(23\) −11.6478 + 19.8325i −0.506424 + 0.862285i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) −18.3308 −0.705031
\(27\) 0 0
\(28\) 7.61550i 0.271982i
\(29\) 10.1177 0.348887 0.174443 0.984667i \(-0.444187\pi\)
0.174443 + 0.984667i \(0.444187\pi\)
\(30\) 0 0
\(31\) 29.2531 0.943650 0.471825 0.881692i \(-0.343595\pi\)
0.471825 + 0.881692i \(0.343595\pi\)
\(32\) −5.65685 −0.176777
\(33\) 0 0
\(34\) 15.3003i 0.450008i
\(35\) −8.51438 −0.243268
\(36\) 0 0
\(37\) 3.06540i 0.0828486i 0.999142 + 0.0414243i \(0.0131895\pi\)
−0.999142 + 0.0414243i \(0.986810\pi\)
\(38\) 24.1935i 0.636672i
\(39\) 0 0
\(40\) 6.32456i 0.158114i
\(41\) −55.3042 −1.34888 −0.674441 0.738328i \(-0.735616\pi\)
−0.674441 + 0.738328i \(0.735616\pi\)
\(42\) 0 0
\(43\) 34.1448i 0.794065i 0.917804 + 0.397033i \(0.129960\pi\)
−0.917804 + 0.397033i \(0.870040\pi\)
\(44\) 32.3653i 0.735576i
\(45\) 0 0
\(46\) 16.4724 28.0475i 0.358096 0.609727i
\(47\) 49.2975 1.04888 0.524442 0.851446i \(-0.324274\pi\)
0.524442 + 0.851446i \(0.324274\pi\)
\(48\) 0 0
\(49\) 34.5011 0.704103
\(50\) 7.07107 0.141421
\(51\) 0 0
\(52\) 25.9237 0.498532
\(53\) 28.6856i 0.541238i −0.962687 0.270619i \(-0.912772\pi\)
0.962687 0.270619i \(-0.0872283\pi\)
\(54\) 0 0
\(55\) −36.1856 −0.657919
\(56\) 10.7699i 0.192320i
\(57\) 0 0
\(58\) −14.3086 −0.246700
\(59\) −24.2529 −0.411066 −0.205533 0.978650i \(-0.565893\pi\)
−0.205533 + 0.978650i \(0.565893\pi\)
\(60\) 0 0
\(61\) 52.9773i 0.868481i −0.900797 0.434240i \(-0.857017\pi\)
0.900797 0.434240i \(-0.142983\pi\)
\(62\) −41.3702 −0.667261
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) 28.9835i 0.445900i
\(66\) 0 0
\(67\) 7.48607i 0.111732i −0.998438 0.0558662i \(-0.982208\pi\)
0.998438 0.0558662i \(-0.0177920\pi\)
\(68\) 21.6379i 0.318204i
\(69\) 0 0
\(70\) 12.0412 0.172017
\(71\) 121.879 1.71661 0.858305 0.513140i \(-0.171518\pi\)
0.858305 + 0.513140i \(0.171518\pi\)
\(72\) 0 0
\(73\) −132.268 −1.81189 −0.905947 0.423391i \(-0.860840\pi\)
−0.905947 + 0.423391i \(0.860840\pi\)
\(74\) 4.33513i 0.0585828i
\(75\) 0 0
\(76\) 34.2148i 0.450195i
\(77\) −61.6195 −0.800254
\(78\) 0 0
\(79\) 18.6463i 0.236030i 0.993012 + 0.118015i \(0.0376530\pi\)
−0.993012 + 0.118015i \(0.962347\pi\)
\(80\) 8.94427i 0.111803i
\(81\) 0 0
\(82\) 78.2120 0.953804
\(83\) 46.3088i 0.557938i 0.960300 + 0.278969i \(0.0899926\pi\)
−0.960300 + 0.278969i \(0.910007\pi\)
\(84\) 0 0
\(85\) −24.1919 −0.284610
\(86\) 48.2881i 0.561489i
\(87\) 0 0
\(88\) 45.7715i 0.520131i
\(89\) 17.2346i 0.193647i 0.995302 + 0.0968234i \(0.0308682\pi\)
−0.995302 + 0.0968234i \(0.969132\pi\)
\(90\) 0 0
\(91\) 49.3554i 0.542367i
\(92\) −23.2955 + 39.6651i −0.253212 + 0.431142i
\(93\) 0 0
\(94\) −69.7172 −0.741672
\(95\) −38.2533 −0.402667
\(96\) 0 0
\(97\) 31.4987i 0.324728i 0.986731 + 0.162364i \(0.0519120\pi\)
−0.986731 + 0.162364i \(0.948088\pi\)
\(98\) −48.7919 −0.497876
\(99\) 0 0
\(100\) −10.0000 −0.100000
\(101\) −11.8823 −0.117646 −0.0588232 0.998268i \(-0.518735\pi\)
−0.0588232 + 0.998268i \(0.518735\pi\)
\(102\) 0 0
\(103\) 61.0381i 0.592603i −0.955094 0.296301i \(-0.904247\pi\)
0.955094 0.296301i \(-0.0957533\pi\)
\(104\) −36.6616 −0.352515
\(105\) 0 0
\(106\) 40.5676i 0.382713i
\(107\) 144.705i 1.35238i −0.736728 0.676190i \(-0.763630\pi\)
0.736728 0.676190i \(-0.236370\pi\)
\(108\) 0 0
\(109\) 162.558i 1.49135i 0.666308 + 0.745677i \(0.267874\pi\)
−0.666308 + 0.745677i \(0.732126\pi\)
\(110\) 51.1741 0.465219
\(111\) 0 0
\(112\) 15.2310i 0.135991i
\(113\) 141.903i 1.25578i −0.778304 0.627888i \(-0.783920\pi\)
0.778304 0.627888i \(-0.216080\pi\)
\(114\) 0 0
\(115\) −44.3469 26.0452i −0.385625 0.226480i
\(116\) 20.2354 0.174443
\(117\) 0 0
\(118\) 34.2988 0.290668
\(119\) −41.1958 −0.346183
\(120\) 0 0
\(121\) −140.879 −1.16429
\(122\) 74.9213i 0.614109i
\(123\) 0 0
\(124\) 58.5063 0.471825
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) 36.2207 0.285202 0.142601 0.989780i \(-0.454453\pi\)
0.142601 + 0.989780i \(0.454453\pi\)
\(128\) −11.3137 −0.0883883
\(129\) 0 0
\(130\) 40.9889i 0.315299i
\(131\) 0.512699 0.00391373 0.00195687 0.999998i \(-0.499377\pi\)
0.00195687 + 0.999998i \(0.499377\pi\)
\(132\) 0 0
\(133\) −65.1407 −0.489780
\(134\) 10.5869i 0.0790067i
\(135\) 0 0
\(136\) 30.6006i 0.225004i
\(137\) 64.7952i 0.472958i −0.971637 0.236479i \(-0.924007\pi\)
0.971637 0.236479i \(-0.0759934\pi\)
\(138\) 0 0
\(139\) −170.051 −1.22339 −0.611693 0.791095i \(-0.709511\pi\)
−0.611693 + 0.791095i \(0.709511\pi\)
\(140\) −17.0288 −0.121634
\(141\) 0 0
\(142\) −172.363 −1.21383
\(143\) 209.757i 1.46683i
\(144\) 0 0
\(145\) 22.6239i 0.156027i
\(146\) 187.056 1.28120
\(147\) 0 0
\(148\) 6.13079i 0.0414243i
\(149\) 54.6548i 0.366810i 0.983037 + 0.183405i \(0.0587120\pi\)
−0.983037 + 0.183405i \(0.941288\pi\)
\(150\) 0 0
\(151\) 131.579 0.871382 0.435691 0.900096i \(-0.356504\pi\)
0.435691 + 0.900096i \(0.356504\pi\)
\(152\) 48.3871i 0.318336i
\(153\) 0 0
\(154\) 87.1432 0.565865
\(155\) 65.4120i 0.422013i
\(156\) 0 0
\(157\) 62.7071i 0.399409i 0.979856 + 0.199704i \(0.0639982\pi\)
−0.979856 + 0.199704i \(0.936002\pi\)
\(158\) 26.3699i 0.166898i
\(159\) 0 0
\(160\) 12.6491i 0.0790569i
\(161\) −75.5173 44.3517i −0.469052 0.275477i
\(162\) 0 0
\(163\) 65.2642 0.400394 0.200197 0.979756i \(-0.435842\pi\)
0.200197 + 0.979756i \(0.435842\pi\)
\(164\) −110.608 −0.674441
\(165\) 0 0
\(166\) 65.4906i 0.394521i
\(167\) 89.9666 0.538722 0.269361 0.963039i \(-0.413187\pi\)
0.269361 + 0.963039i \(0.413187\pi\)
\(168\) 0 0
\(169\) −0.991006 −0.00586394
\(170\) 34.2125 0.201250
\(171\) 0 0
\(172\) 68.2896i 0.397033i
\(173\) 177.669 1.02699 0.513494 0.858093i \(-0.328351\pi\)
0.513494 + 0.858093i \(0.328351\pi\)
\(174\) 0 0
\(175\) 19.0387i 0.108793i
\(176\) 64.7307i 0.367788i
\(177\) 0 0
\(178\) 24.3734i 0.136929i
\(179\) 193.456 1.08076 0.540379 0.841421i \(-0.318281\pi\)
0.540379 + 0.841421i \(0.318281\pi\)
\(180\) 0 0
\(181\) 186.354i 1.02958i −0.857316 0.514790i \(-0.827870\pi\)
0.857316 0.514790i \(-0.172130\pi\)
\(182\) 69.7991i 0.383511i
\(183\) 0 0
\(184\) 32.9448 56.0949i 0.179048 0.304864i
\(185\) −6.85443 −0.0370510
\(186\) 0 0
\(187\) −175.079 −0.936253
\(188\) 98.5950 0.524442
\(189\) 0 0
\(190\) 54.0984 0.284728
\(191\) 18.7678i 0.0982605i −0.998792 0.0491303i \(-0.984355\pi\)
0.998792 0.0491303i \(-0.0156449\pi\)
\(192\) 0 0
\(193\) −161.771 −0.838193 −0.419097 0.907942i \(-0.637653\pi\)
−0.419097 + 0.907942i \(0.637653\pi\)
\(194\) 44.5458i 0.229618i
\(195\) 0 0
\(196\) 69.0021 0.352052
\(197\) −62.3846 −0.316673 −0.158337 0.987385i \(-0.550613\pi\)
−0.158337 + 0.987385i \(0.550613\pi\)
\(198\) 0 0
\(199\) 280.675i 1.41043i 0.708994 + 0.705214i \(0.249149\pi\)
−0.708994 + 0.705214i \(0.750851\pi\)
\(200\) 14.1421 0.0707107
\(201\) 0 0
\(202\) 16.8041 0.0831885
\(203\) 38.5257i 0.189782i
\(204\) 0 0
\(205\) 123.664i 0.603239i
\(206\) 86.3209i 0.419033i
\(207\) 0 0
\(208\) 51.8473 0.249266
\(209\) −276.844 −1.32461
\(210\) 0 0
\(211\) −252.159 −1.19507 −0.597533 0.801844i \(-0.703852\pi\)
−0.597533 + 0.801844i \(0.703852\pi\)
\(212\) 57.3712i 0.270619i
\(213\) 0 0
\(214\) 204.643i 0.956276i
\(215\) −76.3501 −0.355117
\(216\) 0 0
\(217\) 111.389i 0.513312i
\(218\) 229.891i 1.05455i
\(219\) 0 0
\(220\) −72.3711 −0.328960
\(221\) 140.233i 0.634539i
\(222\) 0 0
\(223\) −225.671 −1.01198 −0.505989 0.862540i \(-0.668872\pi\)
−0.505989 + 0.862540i \(0.668872\pi\)
\(224\) 21.5399i 0.0961602i
\(225\) 0 0
\(226\) 200.681i 0.887967i
\(227\) 397.011i 1.74895i −0.485074 0.874473i \(-0.661207\pi\)
0.485074 0.874473i \(-0.338793\pi\)
\(228\) 0 0
\(229\) 269.388i 1.17637i 0.808728 + 0.588183i \(0.200156\pi\)
−0.808728 + 0.588183i \(0.799844\pi\)
\(230\) 62.7160 + 36.8334i 0.272678 + 0.160145i
\(231\) 0 0
\(232\) −28.6172 −0.123350
\(233\) −350.279 −1.50334 −0.751672 0.659537i \(-0.770752\pi\)
−0.751672 + 0.659537i \(0.770752\pi\)
\(234\) 0 0
\(235\) 110.233i 0.469075i
\(236\) −48.5058 −0.205533
\(237\) 0 0
\(238\) 58.2596 0.244788
\(239\) −34.7609 −0.145443 −0.0727215 0.997352i \(-0.523168\pi\)
−0.0727215 + 0.997352i \(0.523168\pi\)
\(240\) 0 0
\(241\) 343.644i 1.42591i 0.701211 + 0.712954i \(0.252643\pi\)
−0.701211 + 0.712954i \(0.747357\pi\)
\(242\) 199.233 0.823276
\(243\) 0 0
\(244\) 105.955i 0.434240i
\(245\) 77.1467i 0.314884i
\(246\) 0 0
\(247\) 221.743i 0.897746i
\(248\) −82.7404 −0.333631
\(249\) 0 0
\(250\) 15.8114i 0.0632456i
\(251\) 442.427i 1.76266i 0.472505 + 0.881328i \(0.343350\pi\)
−0.472505 + 0.881328i \(0.656650\pi\)
\(252\) 0 0
\(253\) −320.944 188.492i −1.26855 0.745027i
\(254\) −51.2238 −0.201669
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) −377.957 −1.47065 −0.735325 0.677715i \(-0.762971\pi\)
−0.735325 + 0.677715i \(0.762971\pi\)
\(258\) 0 0
\(259\) −11.6723 −0.0450666
\(260\) 57.9671i 0.222950i
\(261\) 0 0
\(262\) −0.725066 −0.00276743
\(263\) 233.084i 0.886251i −0.896459 0.443126i \(-0.853870\pi\)
0.896459 0.443126i \(-0.146130\pi\)
\(264\) 0 0
\(265\) 64.1430 0.242049
\(266\) 92.1229 0.346327
\(267\) 0 0
\(268\) 14.9721i 0.0558662i
\(269\) 278.353 1.03477 0.517385 0.855753i \(-0.326906\pi\)
0.517385 + 0.855753i \(0.326906\pi\)
\(270\) 0 0
\(271\) −508.787 −1.87744 −0.938722 0.344676i \(-0.887989\pi\)
−0.938722 + 0.344676i \(0.887989\pi\)
\(272\) 43.2757i 0.159102i
\(273\) 0 0
\(274\) 91.6343i 0.334432i
\(275\) 80.9134i 0.294230i
\(276\) 0 0
\(277\) −19.7229 −0.0712019 −0.0356009 0.999366i \(-0.511335\pi\)
−0.0356009 + 0.999366i \(0.511335\pi\)
\(278\) 240.488 0.865065
\(279\) 0 0
\(280\) 24.0823 0.0860083
\(281\) 343.453i 1.22225i 0.791533 + 0.611126i \(0.209283\pi\)
−0.791533 + 0.611126i \(0.790717\pi\)
\(282\) 0 0
\(283\) 149.531i 0.528378i −0.964471 0.264189i \(-0.914896\pi\)
0.964471 0.264189i \(-0.0851043\pi\)
\(284\) 243.759 0.858305
\(285\) 0 0
\(286\) 296.641i 1.03721i
\(287\) 210.584i 0.733744i
\(288\) 0 0
\(289\) 171.951 0.594985
\(290\) 31.9950i 0.110328i
\(291\) 0 0
\(292\) −264.537 −0.905947
\(293\) 80.2344i 0.273838i 0.990582 + 0.136919i \(0.0437199\pi\)
−0.990582 + 0.136919i \(0.956280\pi\)
\(294\) 0 0
\(295\) 54.2312i 0.183834i
\(296\) 8.67025i 0.0292914i
\(297\) 0 0
\(298\) 77.2935i 0.259374i
\(299\) −150.976 + 257.066i −0.504937 + 0.859753i
\(300\) 0 0
\(301\) −130.015 −0.431943
\(302\) −186.080 −0.616160
\(303\) 0 0
\(304\) 68.4296i 0.225097i
\(305\) 118.461 0.388396
\(306\) 0 0
\(307\) 153.780 0.500911 0.250456 0.968128i \(-0.419420\pi\)
0.250456 + 0.968128i \(0.419420\pi\)
\(308\) −123.239 −0.400127
\(309\) 0 0
\(310\) 92.5066i 0.298408i
\(311\) 124.567 0.400536 0.200268 0.979741i \(-0.435819\pi\)
0.200268 + 0.979741i \(0.435819\pi\)
\(312\) 0 0
\(313\) 222.424i 0.710619i −0.934749 0.355310i \(-0.884375\pi\)
0.934749 0.355310i \(-0.115625\pi\)
\(314\) 88.6813i 0.282425i
\(315\) 0 0
\(316\) 37.2927i 0.118015i
\(317\) −303.301 −0.956785 −0.478393 0.878146i \(-0.658780\pi\)
−0.478393 + 0.878146i \(0.658780\pi\)
\(318\) 0 0
\(319\) 163.732i 0.513265i
\(320\) 17.8885i 0.0559017i
\(321\) 0 0
\(322\) 106.798 + 62.7228i 0.331670 + 0.194791i
\(323\) −185.084 −0.573015
\(324\) 0 0
\(325\) −64.8091 −0.199413
\(326\) −92.2975 −0.283121
\(327\) 0 0
\(328\) 156.424 0.476902
\(329\) 187.713i 0.570555i
\(330\) 0 0
\(331\) −281.157 −0.849416 −0.424708 0.905330i \(-0.639623\pi\)
−0.424708 + 0.905330i \(0.639623\pi\)
\(332\) 92.6176i 0.278969i
\(333\) 0 0
\(334\) −127.232 −0.380934
\(335\) 16.7394 0.0499683
\(336\) 0 0
\(337\) 258.510i 0.767091i 0.923522 + 0.383546i \(0.125297\pi\)
−0.923522 + 0.383546i \(0.874703\pi\)
\(338\) 1.40149 0.00414643
\(339\) 0 0
\(340\) −48.3837 −0.142305
\(341\) 473.394i 1.38825i
\(342\) 0 0
\(343\) 317.951i 0.926971i
\(344\) 96.5761i 0.280744i
\(345\) 0 0
\(346\) −251.262 −0.726191
\(347\) −613.312 −1.76747 −0.883735 0.467988i \(-0.844979\pi\)
−0.883735 + 0.467988i \(0.844979\pi\)
\(348\) 0 0
\(349\) −75.5131 −0.216370 −0.108185 0.994131i \(-0.534504\pi\)
−0.108185 + 0.994131i \(0.534504\pi\)
\(350\) 26.9248i 0.0769281i
\(351\) 0 0
\(352\) 91.5430i 0.260065i
\(353\) 31.5279 0.0893142 0.0446571 0.999002i \(-0.485780\pi\)
0.0446571 + 0.999002i \(0.485780\pi\)
\(354\) 0 0
\(355\) 272.530i 0.767691i
\(356\) 34.4691i 0.0968234i
\(357\) 0 0
\(358\) −273.588 −0.764212
\(359\) 192.855i 0.537199i −0.963252 0.268600i \(-0.913439\pi\)
0.963252 0.268600i \(-0.0865608\pi\)
\(360\) 0 0
\(361\) 68.3365 0.189298
\(362\) 263.544i 0.728023i
\(363\) 0 0
\(364\) 98.7108i 0.271183i
\(365\) 295.761i 0.810304i
\(366\) 0 0
\(367\) 512.314i 1.39595i 0.716121 + 0.697976i \(0.245916\pi\)
−0.716121 + 0.697976i \(0.754084\pi\)
\(368\) −46.5910 + 79.3302i −0.126606 + 0.215571i
\(369\) 0 0
\(370\) 9.69363 0.0261990
\(371\) 109.228 0.294414
\(372\) 0 0
\(373\) 308.198i 0.826267i −0.910670 0.413134i \(-0.864434\pi\)
0.910670 0.413134i \(-0.135566\pi\)
\(374\) 247.599 0.662031
\(375\) 0 0
\(376\) −139.434 −0.370836
\(377\) 131.144 0.347862
\(378\) 0 0
\(379\) 274.002i 0.722959i −0.932380 0.361480i \(-0.882272\pi\)
0.932380 0.361480i \(-0.117728\pi\)
\(380\) −76.5067 −0.201333
\(381\) 0 0
\(382\) 26.5416i 0.0694807i
\(383\) 184.563i 0.481889i 0.970539 + 0.240944i \(0.0774571\pi\)
−0.970539 + 0.240944i \(0.922543\pi\)
\(384\) 0 0
\(385\) 137.785i 0.357884i
\(386\) 228.779 0.592692
\(387\) 0 0
\(388\) 62.9973i 0.162364i
\(389\) 384.497i 0.988425i 0.869341 + 0.494212i \(0.164543\pi\)
−0.869341 + 0.494212i \(0.835457\pi\)
\(390\) 0 0
\(391\) −214.567 126.016i −0.548765 0.322292i
\(392\) −97.5837 −0.248938
\(393\) 0 0
\(394\) 88.2252 0.223922
\(395\) −41.6945 −0.105556
\(396\) 0 0
\(397\) 340.530 0.857757 0.428879 0.903362i \(-0.358909\pi\)
0.428879 + 0.903362i \(0.358909\pi\)
\(398\) 396.935i 0.997324i
\(399\) 0 0
\(400\) −20.0000 −0.0500000
\(401\) 742.537i 1.85171i 0.377875 + 0.925857i \(0.376655\pi\)
−0.377875 + 0.925857i \(0.623345\pi\)
\(402\) 0 0
\(403\) 379.174 0.940879
\(404\) −23.7646 −0.0588232
\(405\) 0 0
\(406\) 54.4836i 0.134196i
\(407\) −49.6063 −0.121883
\(408\) 0 0
\(409\) 757.373 1.85177 0.925884 0.377807i \(-0.123322\pi\)
0.925884 + 0.377807i \(0.123322\pi\)
\(410\) 174.887i 0.426554i
\(411\) 0 0
\(412\) 122.076i 0.296301i
\(413\) 92.3490i 0.223605i
\(414\) 0 0
\(415\) −103.550 −0.249517
\(416\) −73.3232 −0.176258
\(417\) 0 0
\(418\) 391.516 0.936641
\(419\) 213.737i 0.510113i −0.966926 0.255056i \(-0.917906\pi\)
0.966926 0.255056i \(-0.0820940\pi\)
\(420\) 0 0
\(421\) 675.986i 1.60567i −0.596202 0.802834i \(-0.703324\pi\)
0.596202 0.802834i \(-0.296676\pi\)
\(422\) 356.607 0.845040
\(423\) 0 0
\(424\) 81.1351i 0.191356i
\(425\) 54.0947i 0.127282i
\(426\) 0 0
\(427\) 201.724 0.472422
\(428\) 289.409i 0.676190i
\(429\) 0 0
\(430\) 107.975 0.251105
\(431\) 8.04660i 0.0186696i 0.999956 + 0.00933480i \(0.00297140\pi\)
−0.999956 + 0.00933480i \(0.997029\pi\)
\(432\) 0 0
\(433\) 369.369i 0.853046i −0.904477 0.426523i \(-0.859738\pi\)
0.904477 0.426523i \(-0.140262\pi\)
\(434\) 157.527i 0.362966i
\(435\) 0 0
\(436\) 325.115i 0.745677i
\(437\) −339.283 199.263i −0.776392 0.455979i
\(438\) 0 0
\(439\) −54.3077 −0.123708 −0.0618539 0.998085i \(-0.519701\pi\)
−0.0618539 + 0.998085i \(0.519701\pi\)
\(440\) 102.348 0.232610
\(441\) 0 0
\(442\) 198.320i 0.448687i
\(443\) −263.570 −0.594966 −0.297483 0.954727i \(-0.596147\pi\)
−0.297483 + 0.954727i \(0.596147\pi\)
\(444\) 0 0
\(445\) −38.5377 −0.0866015
\(446\) 319.147 0.715576
\(447\) 0 0
\(448\) 30.4620i 0.0679955i
\(449\) −578.427 −1.28826 −0.644128 0.764917i \(-0.722780\pi\)
−0.644128 + 0.764917i \(0.722780\pi\)
\(450\) 0 0
\(451\) 894.970i 1.98441i
\(452\) 283.805i 0.627888i
\(453\) 0 0
\(454\) 561.458i 1.23669i
\(455\) −110.362 −0.242554
\(456\) 0 0
\(457\) 514.251i 1.12528i 0.826703 + 0.562638i \(0.190214\pi\)
−0.826703 + 0.562638i \(0.809786\pi\)
\(458\) 380.972i 0.831816i
\(459\) 0 0
\(460\) −88.6938 52.0903i −0.192813 0.113240i
\(461\) −545.697 −1.18372 −0.591862 0.806039i \(-0.701607\pi\)
−0.591862 + 0.806039i \(0.701607\pi\)
\(462\) 0 0
\(463\) −481.885 −1.04079 −0.520395 0.853926i \(-0.674215\pi\)
−0.520395 + 0.853926i \(0.674215\pi\)
\(464\) 40.4708 0.0872217
\(465\) 0 0
\(466\) 495.370 1.06302
\(467\) 6.72759i 0.0144060i −0.999974 0.00720299i \(-0.997707\pi\)
0.999974 0.00720299i \(-0.00229280\pi\)
\(468\) 0 0
\(469\) 28.5051 0.0607784
\(470\) 155.892i 0.331686i
\(471\) 0 0
\(472\) 68.5976 0.145334
\(473\) −552.554 −1.16819
\(474\) 0 0
\(475\) 85.5370i 0.180078i
\(476\) −82.3916 −0.173092
\(477\) 0 0
\(478\) 49.1593 0.102844
\(479\) 366.353i 0.764828i 0.923991 + 0.382414i \(0.124907\pi\)
−0.923991 + 0.382414i \(0.875093\pi\)
\(480\) 0 0
\(481\) 39.7331i 0.0826053i
\(482\) 485.985i 1.00827i
\(483\) 0 0
\(484\) −281.758 −0.582144
\(485\) −70.4331 −0.145223
\(486\) 0 0
\(487\) 212.080 0.435483 0.217742 0.976006i \(-0.430131\pi\)
0.217742 + 0.976006i \(0.430131\pi\)
\(488\) 149.843i 0.307054i
\(489\) 0 0
\(490\) 109.102i 0.222657i
\(491\) 323.833 0.659537 0.329768 0.944062i \(-0.393029\pi\)
0.329768 + 0.944062i \(0.393029\pi\)
\(492\) 0 0
\(493\) 109.463i 0.222034i
\(494\) 313.592i 0.634802i
\(495\) 0 0
\(496\) 117.013 0.235912
\(497\) 464.086i 0.933774i
\(498\) 0 0
\(499\) 113.014 0.226481 0.113241 0.993568i \(-0.463877\pi\)
0.113241 + 0.993568i \(0.463877\pi\)
\(500\) 22.3607i 0.0447214i
\(501\) 0 0
\(502\) 625.686i 1.24639i
\(503\) 841.141i 1.67225i −0.548541 0.836124i \(-0.684816\pi\)
0.548541 0.836124i \(-0.315184\pi\)
\(504\) 0 0
\(505\) 26.5696i 0.0526131i
\(506\) 453.883 + 266.568i 0.897001 + 0.526813i
\(507\) 0 0
\(508\) 72.4414 0.142601
\(509\) 570.869 1.12155 0.560775 0.827968i \(-0.310503\pi\)
0.560775 + 0.827968i \(0.310503\pi\)
\(510\) 0 0
\(511\) 503.644i 0.985605i
\(512\) −22.6274 −0.0441942
\(513\) 0 0
\(514\) 534.512 1.03991
\(515\) 136.485 0.265020
\(516\) 0 0
\(517\) 797.765i 1.54307i
\(518\) 16.5071 0.0318669
\(519\) 0 0
\(520\) 81.9778i 0.157650i
\(521\) 823.426i 1.58047i −0.612802 0.790236i \(-0.709958\pi\)
0.612802 0.790236i \(-0.290042\pi\)
\(522\) 0 0
\(523\) 692.210i 1.32354i −0.749709 0.661768i \(-0.769806\pi\)
0.749709 0.661768i \(-0.230194\pi\)
\(524\) 1.02540 0.00195687
\(525\) 0 0
\(526\) 329.631i 0.626674i
\(527\) 316.488i 0.600546i
\(528\) 0 0
\(529\) −257.660 462.009i −0.487069 0.873363i
\(530\) −90.7118 −0.171154
\(531\) 0 0
\(532\) −130.281 −0.244890
\(533\) −716.844 −1.34492
\(534\) 0 0
\(535\) 323.569 0.604802
\(536\) 21.1738i 0.0395034i
\(537\) 0 0
\(538\) −393.650 −0.731692
\(539\) 558.319i 1.03584i
\(540\) 0 0
\(541\) 915.849 1.69288 0.846440 0.532483i \(-0.178741\pi\)
0.846440 + 0.532483i \(0.178741\pi\)
\(542\) 719.534 1.32755
\(543\) 0 0
\(544\) 61.2011i 0.112502i
\(545\) −363.490 −0.666954
\(546\) 0 0
\(547\) 203.592 0.372197 0.186099 0.982531i \(-0.440416\pi\)
0.186099 + 0.982531i \(0.440416\pi\)
\(548\) 129.590i 0.236479i
\(549\) 0 0
\(550\) 114.429i 0.208052i
\(551\) 173.088i 0.314134i
\(552\) 0 0
\(553\) −71.0006 −0.128392
\(554\) 27.8924 0.0503473
\(555\) 0 0
\(556\) −340.102 −0.611693
\(557\) 624.298i 1.12082i 0.828215 + 0.560411i \(0.189357\pi\)
−0.828215 + 0.560411i \(0.810643\pi\)
\(558\) 0 0
\(559\) 442.579i 0.791734i
\(560\) −34.0575 −0.0608170
\(561\) 0 0
\(562\) 485.716i 0.864263i
\(563\) 961.054i 1.70702i −0.521074 0.853511i \(-0.674469\pi\)
0.521074 0.853511i \(-0.325531\pi\)
\(564\) 0 0
\(565\) 317.304 0.561600
\(566\) 211.469i 0.373620i
\(567\) 0 0
\(568\) −344.727 −0.606913
\(569\) 576.159i 1.01258i 0.862363 + 0.506290i \(0.168984\pi\)
−0.862363 + 0.506290i \(0.831016\pi\)
\(570\) 0 0
\(571\) 346.570i 0.606952i −0.952839 0.303476i \(-0.901853\pi\)
0.952839 0.303476i \(-0.0981472\pi\)
\(572\) 419.514i 0.733416i
\(573\) 0 0
\(574\) 297.811i 0.518835i
\(575\) 58.2388 99.1627i 0.101285 0.172457i
\(576\) 0 0
\(577\) 675.477 1.17067 0.585335 0.810791i \(-0.300963\pi\)
0.585335 + 0.810791i \(0.300963\pi\)
\(578\) −243.175 −0.420718
\(579\) 0 0
\(580\) 45.2478i 0.0780134i
\(581\) −176.332 −0.303498
\(582\) 0 0
\(583\) 464.210 0.796243
\(584\) 374.111 0.640601
\(585\) 0 0
\(586\) 113.469i 0.193632i
\(587\) 820.083 1.39708 0.698538 0.715573i \(-0.253835\pi\)
0.698538 + 0.715573i \(0.253835\pi\)
\(588\) 0 0
\(589\) 500.446i 0.849653i
\(590\) 76.6944i 0.129991i
\(591\) 0 0
\(592\) 12.2616i 0.0207121i
\(593\) −768.200 −1.29545 −0.647723 0.761876i \(-0.724279\pi\)
−0.647723 + 0.761876i \(0.724279\pi\)
\(594\) 0 0
\(595\) 92.1166i 0.154818i
\(596\) 109.310i 0.183405i
\(597\) 0 0
\(598\) 213.513 363.546i 0.357044 0.607937i
\(599\) 192.340 0.321102 0.160551 0.987028i \(-0.448673\pi\)
0.160551 + 0.987028i \(0.448673\pi\)
\(600\) 0 0
\(601\) 829.949 1.38095 0.690474 0.723358i \(-0.257402\pi\)
0.690474 + 0.723358i \(0.257402\pi\)
\(602\) 183.869 0.305430
\(603\) 0 0
\(604\) 263.157 0.435691
\(605\) 315.015i 0.520685i
\(606\) 0 0
\(607\) 503.567 0.829600 0.414800 0.909913i \(-0.363851\pi\)
0.414800 + 0.909913i \(0.363851\pi\)
\(608\) 96.7741i 0.159168i
\(609\) 0 0
\(610\) −167.529 −0.274638
\(611\) 638.986 1.04580
\(612\) 0 0
\(613\) 117.049i 0.190944i −0.995432 0.0954721i \(-0.969564\pi\)
0.995432 0.0954721i \(-0.0304361\pi\)
\(614\) −217.477 −0.354198
\(615\) 0 0
\(616\) 174.286 0.282932
\(617\) 13.5947i 0.0220336i −0.999939 0.0110168i \(-0.996493\pi\)
0.999939 0.0110168i \(-0.00350682\pi\)
\(618\) 0 0
\(619\) 351.787i 0.568315i −0.958778 0.284158i \(-0.908286\pi\)
0.958778 0.284158i \(-0.0917139\pi\)
\(620\) 130.824i 0.211007i
\(621\) 0 0
\(622\) −176.164 −0.283222
\(623\) −65.6249 −0.105337
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 314.555i 0.502484i
\(627\) 0 0
\(628\) 125.414i 0.199704i
\(629\) −33.1643 −0.0527255
\(630\) 0 0
\(631\) 719.196i 1.13977i −0.821724 0.569886i \(-0.806987\pi\)
0.821724 0.569886i \(-0.193013\pi\)
\(632\) 52.7398i 0.0834491i
\(633\) 0 0
\(634\) 428.932 0.676549
\(635\) 80.9920i 0.127546i
\(636\) 0 0
\(637\) 447.197 0.702036
\(638\) 231.551i 0.362933i
\(639\) 0 0
\(640\) 25.2982i 0.0395285i
\(641\) 745.251i 1.16264i 0.813676 + 0.581319i \(0.197463\pi\)
−0.813676 + 0.581319i \(0.802537\pi\)
\(642\) 0 0
\(643\) 371.944i 0.578450i 0.957261 + 0.289225i \(0.0933976\pi\)
−0.957261 + 0.289225i \(0.906602\pi\)
\(644\) −151.035 88.7034i −0.234526 0.137738i
\(645\) 0 0
\(646\) 261.748 0.405183
\(647\) 772.723 1.19432 0.597159 0.802123i \(-0.296296\pi\)
0.597159 + 0.802123i \(0.296296\pi\)
\(648\) 0 0
\(649\) 392.477i 0.604741i
\(650\) 91.6540 0.141006
\(651\) 0 0
\(652\) 130.528 0.200197
\(653\) −42.1242 −0.0645088 −0.0322544 0.999480i \(-0.510269\pi\)
−0.0322544 + 0.999480i \(0.510269\pi\)
\(654\) 0 0
\(655\) 1.14643i 0.00175027i
\(656\) −221.217 −0.337221
\(657\) 0 0
\(658\) 265.466i 0.403443i
\(659\) 1033.80i 1.56873i 0.620297 + 0.784367i \(0.287012\pi\)
−0.620297 + 0.784367i \(0.712988\pi\)
\(660\) 0 0
\(661\) 512.065i 0.774683i 0.921936 + 0.387341i \(0.126607\pi\)
−0.921936 + 0.387341i \(0.873393\pi\)
\(662\) 397.615 0.600628
\(663\) 0 0
\(664\) 130.981i 0.197261i
\(665\) 145.659i 0.219036i
\(666\) 0 0
\(667\) −117.849 + 200.660i −0.176685 + 0.300840i
\(668\) 179.933 0.269361
\(669\) 0 0
\(670\) −23.6730 −0.0353329
\(671\) 857.315 1.27767
\(672\) 0 0
\(673\) 611.507 0.908628 0.454314 0.890842i \(-0.349884\pi\)
0.454314 + 0.890842i \(0.349884\pi\)
\(674\) 365.588i 0.542415i
\(675\) 0 0
\(676\) −1.98201 −0.00293197
\(677\) 109.122i 0.161184i −0.996747 0.0805920i \(-0.974319\pi\)
0.996747 0.0805920i \(-0.0256811\pi\)
\(678\) 0 0
\(679\) −119.939 −0.176641
\(680\) 68.4250 0.100625
\(681\) 0 0
\(682\) 669.480i 0.981643i
\(683\) 426.220 0.624042 0.312021 0.950075i \(-0.398994\pi\)
0.312021 + 0.950075i \(0.398994\pi\)
\(684\) 0 0
\(685\) 144.886 0.211513
\(686\) 449.651i 0.655467i
\(687\) 0 0
\(688\) 136.579i 0.198516i
\(689\) 371.818i 0.539649i
\(690\) 0 0
\(691\) 152.559 0.220780 0.110390 0.993888i \(-0.464790\pi\)
0.110390 + 0.993888i \(0.464790\pi\)
\(692\) 355.338 0.513494
\(693\) 0 0
\(694\) 867.354 1.24979
\(695\) 380.245i 0.547115i
\(696\) 0 0
\(697\) 598.333i 0.858440i
\(698\) 106.792 0.152997
\(699\) 0 0
\(700\) 38.0775i 0.0543964i
\(701\) 828.197i 1.18145i −0.806873 0.590725i \(-0.798842\pi\)
0.806873 0.590725i \(-0.201158\pi\)
\(702\) 0 0
\(703\) −52.4410 −0.0745960
\(704\) 129.461i 0.183894i
\(705\) 0 0
\(706\) −44.5872 −0.0631547
\(707\) 45.2447i 0.0639954i
\(708\) 0 0
\(709\) 319.241i 0.450269i −0.974328 0.225134i \(-0.927718\pi\)
0.974328 0.225134i \(-0.0722821\pi\)
\(710\) 385.416i 0.542840i
\(711\) 0 0
\(712\) 48.7467i 0.0684645i
\(713\) −340.734 + 580.164i −0.477887 + 0.813695i
\(714\) 0 0
\(715\) −469.031 −0.655987
\(716\) 386.912 0.540379
\(717\) 0 0
\(718\) 272.737i 0.379857i
\(719\) 894.454 1.24402 0.622012 0.783008i \(-0.286315\pi\)
0.622012 + 0.783008i \(0.286315\pi\)
\(720\) 0 0
\(721\) 232.418 0.322355
\(722\) −96.6425 −0.133854
\(723\) 0 0
\(724\) 372.708i 0.514790i
\(725\) −50.5886 −0.0697773
\(726\) 0 0
\(727\) 1349.88i 1.85679i 0.371601 + 0.928393i \(0.378809\pi\)
−0.371601 + 0.928393i \(0.621191\pi\)
\(728\) 139.598i 0.191756i
\(729\) 0 0
\(730\) 418.269i 0.572971i
\(731\) −369.410 −0.505349
\(732\) 0 0
\(733\) 584.473i 0.797371i 0.917088 + 0.398686i \(0.130534\pi\)
−0.917088 + 0.398686i \(0.869466\pi\)
\(734\) 724.522i 0.987087i
\(735\) 0 0
\(736\) 65.8896 112.190i 0.0895240 0.152432i
\(737\) 121.145 0.164375
\(738\) 0 0
\(739\) 100.840 0.136454 0.0682272 0.997670i \(-0.478266\pi\)
0.0682272 + 0.997670i \(0.478266\pi\)
\(740\) −13.7089 −0.0185255
\(741\) 0 0
\(742\) −154.471 −0.208182
\(743\) 1214.22i 1.63422i −0.576482 0.817110i \(-0.695575\pi\)
0.576482 0.817110i \(-0.304425\pi\)
\(744\) 0 0
\(745\) −122.212 −0.164043
\(746\) 435.857i 0.584259i
\(747\) 0 0
\(748\) −350.159 −0.468126
\(749\) 550.999 0.735646
\(750\) 0 0
\(751\) 209.572i 0.279057i 0.990218 + 0.139529i \(0.0445587\pi\)
−0.990218 + 0.139529i \(0.955441\pi\)
\(752\) 197.190 0.262221
\(753\) 0 0
\(754\) −185.466 −0.245976
\(755\) 294.219i 0.389694i
\(756\) 0 0
\(757\) 1291.76i 1.70643i −0.521563 0.853213i \(-0.674651\pi\)
0.521563 0.853213i \(-0.325349\pi\)
\(758\) 387.497i 0.511209i
\(759\) 0 0
\(760\) 108.197 0.142364
\(761\) 1137.06 1.49417 0.747083 0.664731i \(-0.231454\pi\)
0.747083 + 0.664731i \(0.231454\pi\)
\(762\) 0 0
\(763\) −618.978 −0.811243
\(764\) 37.5355i 0.0491303i
\(765\) 0 0
\(766\) 261.012i 0.340747i
\(767\) −314.362 −0.409859
\(768\) 0 0
\(769\) 1211.18i 1.57501i −0.616310 0.787504i \(-0.711373\pi\)
0.616310 0.787504i \(-0.288627\pi\)
\(770\) 194.858i 0.253062i
\(771\) 0 0
\(772\) −323.543 −0.419097
\(773\) 1006.14i 1.30160i −0.759247 0.650802i \(-0.774433\pi\)
0.759247 0.650802i \(-0.225567\pi\)
\(774\) 0 0
\(775\) −146.266 −0.188730
\(776\) 89.0917i 0.114809i
\(777\) 0 0
\(778\) 543.761i 0.698922i
\(779\) 946.112i 1.21452i
\(780\) 0 0
\(781\) 1972.33i 2.52539i
\(782\) 303.444 + 178.214i 0.388035 + 0.227895i
\(783\) 0 0
\(784\) 138.004 0.176026
\(785\) −140.217 −0.178621
\(786\) 0 0
\(787\) 837.892i 1.06467i −0.846535 0.532333i \(-0.821315\pi\)
0.846535 0.532333i \(-0.178685\pi\)
\(788\) −124.769 −0.158337
\(789\) 0 0
\(790\) 58.9649 0.0746391
\(791\) 540.330 0.683097
\(792\) 0 0
\(793\) 686.683i 0.865931i
\(794\) −481.582 −0.606526
\(795\) 0 0
\(796\) 561.351i 0.705214i
\(797\) 866.601i 1.08733i 0.839303 + 0.543665i \(0.182964\pi\)
−0.839303 + 0.543665i \(0.817036\pi\)
\(798\) 0 0
\(799\) 533.346i 0.667518i
\(800\) 28.2843 0.0353553
\(801\) 0 0
\(802\) 1050.11i 1.30936i
\(803\) 2140.45i 2.66557i
\(804\) 0 0
\(805\) 99.1735 168.862i 0.123197 0.209766i
\(806\) −536.233 −0.665302
\(807\) 0 0
\(808\) 33.6082 0.0415943
\(809\) 1290.76 1.59550 0.797749 0.602990i \(-0.206024\pi\)
0.797749 + 0.602990i \(0.206024\pi\)
\(810\) 0 0
\(811\) −526.205 −0.648835 −0.324418 0.945914i \(-0.605168\pi\)
−0.324418 + 0.945914i \(0.605168\pi\)
\(812\) 77.0514i 0.0948909i
\(813\) 0 0
\(814\) 70.1539 0.0861842
\(815\) 145.935i 0.179061i
\(816\) 0 0
\(817\) −584.129 −0.714968
\(818\) −1071.09 −1.30940
\(819\) 0 0
\(820\) 247.328i 0.301619i
\(821\) 445.751 0.542937 0.271468 0.962447i \(-0.412491\pi\)
0.271468 + 0.962447i \(0.412491\pi\)
\(822\) 0 0
\(823\) −1114.24 −1.35388 −0.676938 0.736040i \(-0.736693\pi\)
−0.676938 + 0.736040i \(0.736693\pi\)
\(824\) 172.642i 0.209517i
\(825\) 0 0
\(826\) 130.601i 0.158113i
\(827\) 426.354i 0.515543i −0.966206 0.257771i \(-0.917012\pi\)
0.966206 0.257771i \(-0.0829882\pi\)
\(828\) 0 0
\(829\) 413.046 0.498246 0.249123 0.968472i \(-0.419858\pi\)
0.249123 + 0.968472i \(0.419858\pi\)
\(830\) 146.441 0.176435
\(831\) 0 0
\(832\) 103.695 0.124633
\(833\) 373.265i 0.448097i
\(834\) 0 0
\(835\) 201.172i 0.240924i
\(836\) −553.687 −0.662305
\(837\) 0 0
\(838\) 302.270i 0.360704i
\(839\) 1356.51i 1.61682i −0.588619 0.808411i \(-0.700328\pi\)
0.588619 0.808411i \(-0.299672\pi\)
\(840\) 0 0
\(841\) −738.632 −0.878278
\(842\) 955.989i 1.13538i
\(843\) 0 0
\(844\) −504.318 −0.597533
\(845\) 2.21596i 0.00262243i
\(846\) 0 0
\(847\) 536.431i 0.633331i
\(848\) 114.742i 0.135309i
\(849\) 0 0
\(850\) 76.5014i 0.0900017i
\(851\) −60.7946 35.7050i −0.0714390 0.0419565i
\(852\) 0 0
\(853\) −1215.79 −1.42531 −0.712657 0.701512i \(-0.752509\pi\)
−0.712657 + 0.701512i \(0.752509\pi\)
\(854\) −285.281 −0.334053
\(855\) 0 0
\(856\) 409.286i 0.478138i
\(857\) 321.853 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(858\) 0 0
\(859\) 217.358 0.253036 0.126518 0.991964i \(-0.459620\pi\)
0.126518 + 0.991964i \(0.459620\pi\)
\(860\) −152.700 −0.177558
\(861\) 0 0
\(862\) 11.3796i 0.0132014i
\(863\) 1162.38 1.34690 0.673451 0.739232i \(-0.264811\pi\)
0.673451 + 0.739232i \(0.264811\pi\)
\(864\) 0 0
\(865\) 397.280i 0.459283i
\(866\) 522.366i 0.603194i
\(867\) 0 0
\(868\) 222.777i 0.256656i
\(869\) −301.748 −0.347235
\(870\) 0 0
\(871\) 97.0332i 0.111404i
\(872\) 459.782i 0.527273i
\(873\) 0 0
\(874\) 479.819 + 281.800i 0.548992 + 0.322426i
\(875\) 42.5719 0.0486536
\(876\) 0 0
\(877\) −143.677 −0.163828 −0.0819139 0.996639i \(-0.526103\pi\)
−0.0819139 + 0.996639i \(0.526103\pi\)
\(878\) 76.8027 0.0874746
\(879\) 0 0
\(880\) −144.742 −0.164480
\(881\) 26.5389i 0.0301237i −0.999887 0.0150618i \(-0.995205\pi\)
0.999887 0.0150618i \(-0.00479451\pi\)
\(882\) 0 0
\(883\) 1442.75 1.63391 0.816957 0.576698i \(-0.195659\pi\)
0.816957 + 0.576698i \(0.195659\pi\)
\(884\) 280.466i 0.317270i
\(885\) 0 0
\(886\) 372.744 0.420704
\(887\) 373.153 0.420691 0.210345 0.977627i \(-0.432541\pi\)
0.210345 + 0.977627i \(0.432541\pi\)
\(888\) 0 0
\(889\) 137.919i 0.155140i
\(890\) 54.5005 0.0612365
\(891\) 0 0
\(892\) −451.342 −0.505989
\(893\) 843.353i 0.944404i
\(894\) 0 0
\(895\) 432.580i 0.483330i
\(896\) 43.0798i 0.0480801i
\(897\) 0 0
\(898\) 818.020 0.910935
\(899\) 295.975 0.329227
\(900\) 0 0
\(901\) 310.348 0.344448
\(902\) 1265.68i 1.40319i
\(903\) 0 0
\(904\) 401.361i 0.443984i
\(905\) 416.700 0.460442
\(906\) 0 0
\(907\) 488.681i 0.538788i 0.963030 + 0.269394i \(0.0868234\pi\)
−0.963030 + 0.269394i \(0.913177\pi\)
\(908\) 794.022i 0.874473i
\(909\) 0 0
\(910\) 156.075 0.171511
\(911\) 978.988i 1.07463i −0.843382 0.537315i \(-0.819439\pi\)
0.843382 0.537315i \(-0.180561\pi\)
\(912\) 0 0
\(913\) −749.400 −0.820811
\(914\) 727.261i 0.795690i
\(915\) 0 0
\(916\) 538.775i 0.588183i
\(917\) 1.95223i 0.00212893i
\(918\) 0 0
\(919\) 346.891i 0.377466i 0.982028 + 0.188733i \(0.0604381\pi\)
−0.982028 + 0.188733i \(0.939562\pi\)
\(920\) 125.432 + 73.6669i 0.136339 + 0.0800727i
\(921\) 0 0
\(922\) 771.732 0.837019
\(923\) 1579.78 1.71157
\(924\) 0 0
\(925\) 15.3270i 0.0165697i
\(926\) 681.489 0.735949
\(927\) 0 0
\(928\) −57.2344 −0.0616750
\(929\) −700.599 −0.754143 −0.377072 0.926184i \(-0.623069\pi\)
−0.377072 + 0.926184i \(0.623069\pi\)
\(930\) 0 0
\(931\) 590.224i 0.633967i
\(932\) −700.558 −0.751672
\(933\) 0 0
\(934\) 9.51426i 0.0101866i
\(935\) 391.489i 0.418705i
\(936\) 0 0
\(937\) 1140.57i 1.21726i 0.793455 + 0.608628i \(0.208280\pi\)
−0.793455 + 0.608628i \(0.791720\pi\)
\(938\) −40.3123 −0.0429768
\(939\) 0 0
\(940\) 220.465i 0.234537i
\(941\) 561.160i 0.596344i −0.954512 0.298172i \(-0.903623\pi\)
0.954512 0.298172i \(-0.0963769\pi\)
\(942\) 0 0
\(943\) 644.170 1096.82i 0.683107 1.16312i
\(944\) −97.0116 −0.102767
\(945\) 0 0
\(946\) 781.430 0.826036
\(947\) 974.501 1.02904 0.514520 0.857478i \(-0.327970\pi\)
0.514520 + 0.857478i \(0.327970\pi\)
\(948\) 0 0
\(949\) −1714.44 −1.80657
\(950\) 120.968i 0.127334i
\(951\) 0 0
\(952\) 116.519 0.122394
\(953\) 245.849i 0.257974i −0.991646 0.128987i \(-0.958827\pi\)
0.991646 0.128987i \(-0.0411725\pi\)
\(954\) 0 0
\(955\) 41.9660 0.0439434
\(956\) −69.5218 −0.0727215
\(957\) 0 0
\(958\) 518.101i 0.540815i
\(959\) 246.724 0.257272
\(960\) 0 0
\(961\) −105.253 −0.109525
\(962\) 56.1911i 0.0584108i
\(963\) 0 0
\(964\) 687.287i 0.712954i
\(965\) 361.732i 0.374852i
\(966\) 0 0
\(967\) −958.537 −0.991248 −0.495624 0.868537i \(-0.665061\pi\)
−0.495624 + 0.868537i \(0.665061\pi\)
\(968\) 398.465 0.411638
\(969\) 0 0
\(970\) 99.6075 0.102688
\(971\) 797.683i 0.821506i 0.911747 + 0.410753i \(0.134734\pi\)
−0.911747 + 0.410753i \(0.865266\pi\)
\(972\) 0 0
\(973\) 647.511i 0.665479i
\(974\) −299.927 −0.307933
\(975\) 0 0
\(976\) 211.909i 0.217120i
\(977\) 968.304i 0.991099i 0.868580 + 0.495550i \(0.165033\pi\)
−0.868580 + 0.495550i \(0.834967\pi\)
\(978\) 0 0
\(979\) −278.901 −0.284884
\(980\) 154.293i 0.157442i
\(981\) 0 0
\(982\) −457.968 −0.466363
\(983\) 242.603i 0.246799i 0.992357 + 0.123399i \(0.0393796\pi\)
−0.992357 + 0.123399i \(0.960620\pi\)
\(984\) 0 0
\(985\) 139.496i 0.141621i
\(986\) 154.804i 0.157002i
\(987\) 0 0
\(988\) 443.487i 0.448873i
\(989\) −677.178 397.710i −0.684710 0.402134i
\(990\) 0 0
\(991\) 1231.78 1.24297 0.621483 0.783428i \(-0.286530\pi\)
0.621483 + 0.783428i \(0.286530\pi\)
\(992\) −165.481 −0.166815
\(993\) 0 0
\(994\) 656.316i 0.660278i
\(995\) −627.609 −0.630763
\(996\) 0 0
\(997\) −689.979 −0.692055 −0.346027 0.938224i \(-0.612470\pi\)
−0.346027 + 0.938224i \(0.612470\pi\)
\(998\) −159.826 −0.160146
\(999\) 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 2070.3.c.b.91.14 32
3.2 odd 2 690.3.c.a.91.27 32
23.22 odd 2 inner 2070.3.c.b.91.3 32
69.68 even 2 690.3.c.a.91.30 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.3.c.a.91.27 32 3.2 odd 2
690.3.c.a.91.30 yes 32 69.68 even 2
2070.3.c.b.91.3 32 23.22 odd 2 inner
2070.3.c.b.91.14 32 1.1 even 1 trivial