Properties

Label 684.5.y.e.145.1
Level $684$
Weight $5$
Character 684.145
Analytic conductor $70.705$
Analytic rank $0$
Dimension $14$
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] = [684,5,Mod(145,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.145");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 684.y (of order \(6\), degree \(2\), 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: \(70.7050547493\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + 2358 x^{12} + 15572 x^{11} + 4050518 x^{10} + 21628620 x^{9} + 2974230644 x^{8} + 5952397856 x^{7} + 1565987836020 x^{6} + \cdots + 96\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.1
Root \(-15.2075 + 26.3401i\) of defining polynomial
Character \(\chi\) \(=\) 684.145
Dual form 684.5.y.e.217.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-13.2075 - 22.8760i) q^{5} +3.95127 q^{7} +O(q^{10})\) \(q+(-13.2075 - 22.8760i) q^{5} +3.95127 q^{7} +66.0544 q^{11} +(251.823 + 145.390i) q^{13} +(-203.993 - 353.326i) q^{17} +(-248.401 - 261.951i) q^{19} +(-62.6209 + 108.463i) q^{23} +(-36.3749 + 63.0032i) q^{25} +(1371.33 + 791.736i) q^{29} +457.962i q^{31} +(-52.1863 - 90.3894i) q^{35} +2191.32i q^{37} +(1957.61 - 1130.22i) q^{41} +(1261.53 + 2185.04i) q^{43} +(1323.10 - 2291.68i) q^{47} -2385.39 q^{49} +(-3853.60 - 2224.88i) q^{53} +(-872.412 - 1511.06i) q^{55} +(3689.16 - 2129.93i) q^{59} +(123.741 - 214.326i) q^{61} -7680.93i q^{65} +(-6640.45 - 3833.86i) q^{67} +(-2609.52 + 1506.61i) q^{71} +(-1797.57 - 3113.47i) q^{73} +260.999 q^{77} +(6435.38 - 3715.47i) q^{79} +1653.76 q^{83} +(-5388.46 + 9333.08i) q^{85} +(1852.60 + 1069.60i) q^{89} +(995.019 + 574.474i) q^{91} +(-2711.65 + 9142.12i) q^{95} +(12756.3 - 7364.84i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 30 q^{5} + 106 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 30 q^{5} + 106 q^{7} + 264 q^{11} + 57 q^{13} - 282 q^{17} + 2 q^{19} - 96 q^{23} - 465 q^{25} + 630 q^{29} - 1434 q^{35} + 228 q^{41} - 2093 q^{43} + 4710 q^{47} + 4440 q^{49} - 2364 q^{53} + 6368 q^{55} - 11838 q^{59} - 1661 q^{61} - 20319 q^{67} - 624 q^{71} + 5851 q^{73} + 1080 q^{77} - 13299 q^{79} - 12252 q^{83} - 5740 q^{85} + 20010 q^{89} + 15951 q^{91} - 7770 q^{95} + 44904 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) −13.2075 22.8760i −0.528299 0.915041i −0.999456 0.0329913i \(-0.989497\pi\)
0.471157 0.882050i \(-0.343837\pi\)
\(6\) 0 0
\(7\) 3.95127 0.0806382 0.0403191 0.999187i \(-0.487163\pi\)
0.0403191 + 0.999187i \(0.487163\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 66.0544 0.545904 0.272952 0.962028i \(-0.412000\pi\)
0.272952 + 0.962028i \(0.412000\pi\)
\(12\) 0 0
\(13\) 251.823 + 145.390i 1.49007 + 0.860295i 0.999936 0.0113498i \(-0.00361285\pi\)
0.490139 + 0.871645i \(0.336946\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −203.993 353.326i −0.705857 1.22258i −0.966381 0.257113i \(-0.917229\pi\)
0.260524 0.965467i \(-0.416105\pi\)
\(18\) 0 0
\(19\) −248.401 261.951i −0.688090 0.725625i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −62.6209 + 108.463i −0.118376 + 0.205033i −0.919124 0.393968i \(-0.871102\pi\)
0.800748 + 0.599001i \(0.204435\pi\)
\(24\) 0 0
\(25\) −36.3749 + 63.0032i −0.0581999 + 0.100805i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1371.33 + 791.736i 1.63059 + 0.941422i 0.983911 + 0.178661i \(0.0571766\pi\)
0.646681 + 0.762761i \(0.276157\pi\)
\(30\) 0 0
\(31\) 457.962i 0.476548i 0.971198 + 0.238274i \(0.0765816\pi\)
−0.971198 + 0.238274i \(0.923418\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −52.1863 90.3894i −0.0426011 0.0737872i
\(36\) 0 0
\(37\) 2191.32i 1.60068i 0.599550 + 0.800338i \(0.295346\pi\)
−0.599550 + 0.800338i \(0.704654\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1957.61 1130.22i 1.16455 0.672352i 0.212159 0.977235i \(-0.431951\pi\)
0.952390 + 0.304883i \(0.0986173\pi\)
\(42\) 0 0
\(43\) 1261.53 + 2185.04i 0.682278 + 1.18174i 0.974284 + 0.225324i \(0.0723441\pi\)
−0.292006 + 0.956417i \(0.594323\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1323.10 2291.68i 0.598960 1.03743i −0.394015 0.919104i \(-0.628914\pi\)
0.992975 0.118325i \(-0.0377526\pi\)
\(48\) 0 0
\(49\) −2385.39 −0.993497
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3853.60 2224.88i −1.37188 0.792053i −0.380712 0.924694i \(-0.624321\pi\)
−0.991164 + 0.132640i \(0.957654\pi\)
\(54\) 0 0
\(55\) −872.412 1511.06i −0.288401 0.499525i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3689.16 2129.93i 1.05980 0.611874i 0.134421 0.990924i \(-0.457082\pi\)
0.925376 + 0.379050i \(0.123749\pi\)
\(60\) 0 0
\(61\) 123.741 214.326i 0.0332548 0.0575989i −0.848919 0.528523i \(-0.822746\pi\)
0.882174 + 0.470924i \(0.156079\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7680.93i 1.81797i
\(66\) 0 0
\(67\) −6640.45 3833.86i −1.47927 0.854057i −0.479546 0.877517i \(-0.659199\pi\)
−0.999725 + 0.0234595i \(0.992532\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2609.52 + 1506.61i −0.517660 + 0.298871i −0.735977 0.677007i \(-0.763277\pi\)
0.218317 + 0.975878i \(0.429943\pi\)
\(72\) 0 0
\(73\) −1797.57 3113.47i −0.337318 0.584251i 0.646610 0.762821i \(-0.276186\pi\)
−0.983927 + 0.178570i \(0.942853\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 260.999 0.0440207
\(78\) 0 0
\(79\) 6435.38 3715.47i 1.03115 0.595332i 0.113833 0.993500i \(-0.463687\pi\)
0.917312 + 0.398168i \(0.130354\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1653.76 0.240059 0.120029 0.992770i \(-0.461701\pi\)
0.120029 + 0.992770i \(0.461701\pi\)
\(84\) 0 0
\(85\) −5388.46 + 9333.08i −0.745807 + 1.29178i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1852.60 + 1069.60i 0.233884 + 0.135033i 0.612363 0.790577i \(-0.290219\pi\)
−0.378478 + 0.925610i \(0.623553\pi\)
\(90\) 0 0
\(91\) 995.019 + 574.474i 0.120157 + 0.0693726i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2711.65 + 9142.12i −0.300459 + 1.01298i
\(96\) 0 0
\(97\) 12756.3 7364.84i 1.35575 0.782744i 0.366704 0.930338i \(-0.380486\pi\)
0.989048 + 0.147594i \(0.0471529\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −190.130 + 329.315i −0.0186384 + 0.0322826i −0.875194 0.483772i \(-0.839266\pi\)
0.856556 + 0.516054i \(0.172600\pi\)
\(102\) 0 0
\(103\) 15522.1i 1.46310i −0.681787 0.731551i \(-0.738797\pi\)
0.681787 0.731551i \(-0.261203\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 19304.7i 1.68615i −0.537796 0.843075i \(-0.680743\pi\)
0.537796 0.843075i \(-0.319257\pi\)
\(108\) 0 0
\(109\) −5069.64 + 2926.96i −0.426701 + 0.246356i −0.697940 0.716156i \(-0.745900\pi\)
0.271239 + 0.962512i \(0.412567\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 20207.0i 1.58251i −0.611488 0.791254i \(-0.709429\pi\)
0.611488 0.791254i \(-0.290571\pi\)
\(114\) 0 0
\(115\) 3308.25 0.250152
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −806.030 1396.09i −0.0569190 0.0985867i
\(120\) 0 0
\(121\) −10277.8 −0.701989
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −14587.7 −0.933610
\(126\) 0 0
\(127\) 13379.6 + 7724.74i 0.829539 + 0.478935i 0.853695 0.520773i \(-0.174356\pi\)
−0.0241556 + 0.999708i \(0.507690\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1431.61 2479.63i −0.0834225 0.144492i 0.821295 0.570503i \(-0.193252\pi\)
−0.904718 + 0.426011i \(0.859918\pi\)
\(132\) 0 0
\(133\) −981.498 1035.04i −0.0554863 0.0585131i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1152.70 + 1996.53i −0.0614149 + 0.106374i −0.895098 0.445869i \(-0.852895\pi\)
0.833683 + 0.552243i \(0.186228\pi\)
\(138\) 0 0
\(139\) 16171.7 28010.3i 0.837003 1.44973i −0.0553863 0.998465i \(-0.517639\pi\)
0.892389 0.451267i \(-0.149028\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 16634.0 + 9603.64i 0.813438 + 0.469638i
\(144\) 0 0
\(145\) 41827.3i 1.98941i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6161.61 10672.2i −0.277538 0.480709i 0.693235 0.720712i \(-0.256185\pi\)
−0.970772 + 0.240003i \(0.922852\pi\)
\(150\) 0 0
\(151\) 26041.0i 1.14210i −0.820916 0.571050i \(-0.806536\pi\)
0.820916 0.571050i \(-0.193464\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 10476.4 6048.53i 0.436061 0.251760i
\(156\) 0 0
\(157\) 9055.18 + 15684.0i 0.367365 + 0.636295i 0.989153 0.146891i \(-0.0469267\pi\)
−0.621788 + 0.783186i \(0.713593\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −247.432 + 428.565i −0.00954562 + 0.0165335i
\(162\) 0 0
\(163\) 1816.65 0.0683748 0.0341874 0.999415i \(-0.489116\pi\)
0.0341874 + 0.999415i \(0.489116\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 26066.6 + 15049.5i 0.934655 + 0.539623i 0.888281 0.459301i \(-0.151900\pi\)
0.0463740 + 0.998924i \(0.485233\pi\)
\(168\) 0 0
\(169\) 27995.9 + 48490.3i 0.980214 + 1.69778i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −27606.0 + 15938.3i −0.922383 + 0.532538i −0.884395 0.466740i \(-0.845428\pi\)
−0.0379886 + 0.999278i \(0.512095\pi\)
\(174\) 0 0
\(175\) −143.727 + 248.943i −0.00469313 + 0.00812874i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 25052.3i 0.781884i −0.920415 0.390942i \(-0.872149\pi\)
0.920415 0.390942i \(-0.127851\pi\)
\(180\) 0 0
\(181\) 30483.9 + 17599.9i 0.930494 + 0.537221i 0.886968 0.461831i \(-0.152807\pi\)
0.0435262 + 0.999052i \(0.486141\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 50128.8 28941.9i 1.46468 0.845635i
\(186\) 0 0
\(187\) −13474.6 23338.7i −0.385330 0.667412i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −69403.7 −1.90246 −0.951231 0.308479i \(-0.900180\pi\)
−0.951231 + 0.308479i \(0.900180\pi\)
\(192\) 0 0
\(193\) 572.006 330.248i 0.0153563 0.00886595i −0.492302 0.870424i \(-0.663845\pi\)
0.507658 + 0.861558i \(0.330511\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8731.63 −0.224990 −0.112495 0.993652i \(-0.535884\pi\)
−0.112495 + 0.993652i \(0.535884\pi\)
\(198\) 0 0
\(199\) 18442.2 31942.8i 0.465700 0.806616i −0.533533 0.845779i \(-0.679136\pi\)
0.999233 + 0.0391632i \(0.0124692\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 5418.48 + 3128.36i 0.131488 + 0.0759146i
\(204\) 0 0
\(205\) −51710.1 29854.8i −1.23046 0.710406i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −16407.9 17303.0i −0.375631 0.396122i
\(210\) 0 0
\(211\) −28075.1 + 16209.2i −0.630604 + 0.364079i −0.780986 0.624549i \(-0.785283\pi\)
0.150382 + 0.988628i \(0.451950\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 33323.3 57717.7i 0.720894 1.24863i
\(216\) 0 0
\(217\) 1809.53i 0.0384279i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 118634.i 2.42898i
\(222\) 0 0
\(223\) −73819.4 + 42619.7i −1.48443 + 0.857039i −0.999843 0.0177024i \(-0.994365\pi\)
−0.484591 + 0.874741i \(0.661032\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 27523.0i 0.534126i −0.963679 0.267063i \(-0.913947\pi\)
0.963679 0.267063i \(-0.0860532\pi\)
\(228\) 0 0
\(229\) 53550.6 1.02116 0.510580 0.859830i \(-0.329431\pi\)
0.510580 + 0.859830i \(0.329431\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −20198.3 34984.4i −0.372051 0.644411i 0.617830 0.786312i \(-0.288012\pi\)
−0.989881 + 0.141901i \(0.954679\pi\)
\(234\) 0 0
\(235\) −69899.4 −1.26572
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 54594.4 0.955768 0.477884 0.878423i \(-0.341404\pi\)
0.477884 + 0.878423i \(0.341404\pi\)
\(240\) 0 0
\(241\) −41863.1 24169.7i −0.720771 0.416137i 0.0942654 0.995547i \(-0.469950\pi\)
−0.815036 + 0.579410i \(0.803283\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 31505.0 + 54568.2i 0.524864 + 0.909091i
\(246\) 0 0
\(247\) −24467.9 102080.i −0.401054 1.67320i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −37943.6 + 65720.3i −0.602270 + 1.04316i 0.390206 + 0.920728i \(0.372404\pi\)
−0.992476 + 0.122435i \(0.960930\pi\)
\(252\) 0 0
\(253\) −4136.38 + 7164.43i −0.0646219 + 0.111928i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 107798. + 62237.0i 1.63209 + 0.942286i 0.983448 + 0.181189i \(0.0579947\pi\)
0.648639 + 0.761096i \(0.275339\pi\)
\(258\) 0 0
\(259\) 8658.52i 0.129076i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −34739.9 60171.2i −0.502246 0.869916i −0.999997 0.00259575i \(-0.999174\pi\)
0.497750 0.867320i \(-0.334160\pi\)
\(264\) 0 0
\(265\) 117540.i 1.67376i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 54316.7 31359.8i 0.750635 0.433379i −0.0752883 0.997162i \(-0.523988\pi\)
0.825923 + 0.563782i \(0.190654\pi\)
\(270\) 0 0
\(271\) 24492.5 + 42422.3i 0.333499 + 0.577638i 0.983195 0.182556i \(-0.0584372\pi\)
−0.649696 + 0.760194i \(0.725104\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2402.72 + 4161.64i −0.0317716 + 0.0550299i
\(276\) 0 0
\(277\) 83714.4 1.09104 0.545520 0.838098i \(-0.316332\pi\)
0.545520 + 0.838098i \(0.316332\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −32897.2 18993.2i −0.416626 0.240539i 0.277007 0.960868i \(-0.410658\pi\)
−0.693633 + 0.720329i \(0.743991\pi\)
\(282\) 0 0
\(283\) 35732.7 + 61890.8i 0.446162 + 0.772775i 0.998132 0.0610884i \(-0.0194571\pi\)
−0.551970 + 0.833864i \(0.686124\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7735.03 4465.82i 0.0939071 0.0542173i
\(288\) 0 0
\(289\) −41465.6 + 71820.4i −0.496469 + 0.859909i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8987.73i 0.104692i −0.998629 0.0523461i \(-0.983330\pi\)
0.998629 0.0523461i \(-0.0166699\pi\)
\(294\) 0 0
\(295\) −97448.9 56262.1i −1.11978 0.646505i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −31538.7 + 18208.9i −0.352778 + 0.203676i
\(300\) 0 0
\(301\) 4984.66 + 8633.68i 0.0550177 + 0.0952934i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −6537.22 −0.0702738
\(306\) 0 0
\(307\) 7608.34 4392.68i 0.0807260 0.0466072i −0.459094 0.888388i \(-0.651826\pi\)
0.539820 + 0.841781i \(0.318492\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −59943.3 −0.619755 −0.309878 0.950776i \(-0.600288\pi\)
−0.309878 + 0.950776i \(0.600288\pi\)
\(312\) 0 0
\(313\) 20206.9 34999.3i 0.206258 0.357249i −0.744275 0.667873i \(-0.767205\pi\)
0.950533 + 0.310624i \(0.100538\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 43676.0 + 25216.4i 0.434635 + 0.250937i 0.701319 0.712847i \(-0.252595\pi\)
−0.266684 + 0.963784i \(0.585928\pi\)
\(318\) 0 0
\(319\) 90582.2 + 52297.7i 0.890147 + 0.513926i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −41882.0 + 141202.i −0.401442 + 1.35343i
\(324\) 0 0
\(325\) −18320.0 + 10577.1i −0.173444 + 0.100138i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 5227.94 9055.05i 0.0482991 0.0836564i
\(330\) 0 0
\(331\) 27515.7i 0.251145i 0.992084 + 0.125573i \(0.0400768\pi\)
−0.992084 + 0.125573i \(0.959923\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 202543.i 1.80479i
\(336\) 0 0
\(337\) 108897. 62872.0i 0.958866 0.553602i 0.0630421 0.998011i \(-0.479920\pi\)
0.895824 + 0.444409i \(0.146586\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 30250.4i 0.260149i
\(342\) 0 0
\(343\) −18912.3 −0.160752
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7145.75 + 12376.8i 0.0593457 + 0.102790i 0.894172 0.447724i \(-0.147765\pi\)
−0.834826 + 0.550514i \(0.814432\pi\)
\(348\) 0 0
\(349\) 66467.6 0.545706 0.272853 0.962056i \(-0.412033\pi\)
0.272853 + 0.962056i \(0.412033\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −107847. −0.865487 −0.432743 0.901517i \(-0.642454\pi\)
−0.432743 + 0.901517i \(0.642454\pi\)
\(354\) 0 0
\(355\) 68930.4 + 39797.0i 0.546959 + 0.315787i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −32192.7 55759.4i −0.249786 0.432643i 0.713680 0.700472i \(-0.247027\pi\)
−0.963466 + 0.267829i \(0.913694\pi\)
\(360\) 0 0
\(361\) −6915.34 + 130137.i −0.0530639 + 0.998591i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −47482.6 + 82242.3i −0.356409 + 0.617319i
\(366\) 0 0
\(367\) −8645.43 + 14974.3i −0.0641881 + 0.111177i −0.896334 0.443380i \(-0.853779\pi\)
0.832145 + 0.554557i \(0.187112\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −15226.6 8791.09i −0.110626 0.0638697i
\(372\) 0 0
\(373\) 83855.1i 0.602715i −0.953511 0.301358i \(-0.902560\pi\)
0.953511 0.301358i \(-0.0974398\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 230221. + 398754.i 1.61980 + 2.80558i
\(378\) 0 0
\(379\) 237159.i 1.65105i −0.564363 0.825526i \(-0.690878\pi\)
0.564363 0.825526i \(-0.309122\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −647.503 + 373.836i −0.00441412 + 0.00254849i −0.502205 0.864748i \(-0.667478\pi\)
0.497791 + 0.867297i \(0.334145\pi\)
\(384\) 0 0
\(385\) −3447.14 5970.61i −0.0232561 0.0402808i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −25400.9 + 43995.7i −0.167861 + 0.290744i −0.937668 0.347533i \(-0.887019\pi\)
0.769806 + 0.638277i \(0.220353\pi\)
\(390\) 0 0
\(391\) 51096.8 0.334226
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −169990. 98143.9i −1.08951 0.629027i
\(396\) 0 0
\(397\) 58870.5 + 101967.i 0.373522 + 0.646959i 0.990105 0.140331i \(-0.0448167\pi\)
−0.616582 + 0.787290i \(0.711483\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −143708. + 82970.0i −0.893702 + 0.515979i −0.875152 0.483849i \(-0.839238\pi\)
−0.0185504 + 0.999828i \(0.505905\pi\)
\(402\) 0 0
\(403\) −66583.0 + 115325.i −0.409971 + 0.710091i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 144747.i 0.873815i
\(408\) 0 0
\(409\) −184594. 106575.i −1.10349 0.637103i −0.166358 0.986066i \(-0.553201\pi\)
−0.937137 + 0.348963i \(0.886534\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 14576.9 8415.95i 0.0854601 0.0493404i
\(414\) 0 0
\(415\) −21842.1 37831.6i −0.126823 0.219664i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −59218.3 −0.337309 −0.168654 0.985675i \(-0.553942\pi\)
−0.168654 + 0.985675i \(0.553942\pi\)
\(420\) 0 0
\(421\) 57179.4 33012.5i 0.322608 0.186258i −0.329946 0.944000i \(-0.607031\pi\)
0.652555 + 0.757742i \(0.273697\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 29680.9 0.164323
\(426\) 0 0
\(427\) 488.934 846.859i 0.00268160 0.00464467i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 101448. + 58570.8i 0.546119 + 0.315302i 0.747555 0.664200i \(-0.231228\pi\)
−0.201436 + 0.979502i \(0.564561\pi\)
\(432\) 0 0
\(433\) 185685. + 107205.i 0.990379 + 0.571796i 0.905388 0.424586i \(-0.139580\pi\)
0.0849918 + 0.996382i \(0.472914\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 43966.9 10538.6i 0.230230 0.0551847i
\(438\) 0 0
\(439\) −228113. + 131701.i −1.18364 + 0.683376i −0.956854 0.290568i \(-0.906156\pi\)
−0.226788 + 0.973944i \(0.572822\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 25906.1 44870.8i 0.132006 0.228642i −0.792443 0.609945i \(-0.791191\pi\)
0.924450 + 0.381303i \(0.124525\pi\)
\(444\) 0 0
\(445\) 56506.8i 0.285352i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 300762.i 1.49187i 0.666019 + 0.745935i \(0.267997\pi\)
−0.666019 + 0.745935i \(0.732003\pi\)
\(450\) 0 0
\(451\) 129308. 74656.3i 0.635732 0.367040i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 30349.4i 0.146598i
\(456\) 0 0
\(457\) 143035. 0.684872 0.342436 0.939541i \(-0.388748\pi\)
0.342436 + 0.939541i \(0.388748\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 62369.0 + 108026.i 0.293472 + 0.508309i 0.974628 0.223830i \(-0.0718559\pi\)
−0.681156 + 0.732138i \(0.738523\pi\)
\(462\) 0 0
\(463\) −303815. −1.41725 −0.708626 0.705585i \(-0.750684\pi\)
−0.708626 + 0.705585i \(0.750684\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −407608. −1.86900 −0.934499 0.355966i \(-0.884152\pi\)
−0.934499 + 0.355966i \(0.884152\pi\)
\(468\) 0 0
\(469\) −26238.2 15148.6i −0.119286 0.0688696i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 83329.8 + 144331.i 0.372459 + 0.645117i
\(474\) 0 0
\(475\) 25539.3 6121.60i 0.113194 0.0271317i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 91751.1 158918.i 0.399890 0.692630i −0.593822 0.804596i \(-0.702382\pi\)
0.993712 + 0.111967i \(0.0357150\pi\)
\(480\) 0 0
\(481\) −318596. + 551825.i −1.37705 + 2.38512i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −336956. 194542.i −1.43248 0.827046i
\(486\) 0 0
\(487\) 74156.9i 0.312676i 0.987704 + 0.156338i \(0.0499688\pi\)
−0.987704 + 0.156338i \(0.950031\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −62825.2 108816.i −0.260598 0.451369i 0.705803 0.708408i \(-0.250586\pi\)
−0.966401 + 0.257039i \(0.917253\pi\)
\(492\) 0 0
\(493\) 646034.i 2.65804i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −10310.9 + 5953.02i −0.0417432 + 0.0241004i
\(498\) 0 0
\(499\) −51694.7 89537.8i −0.207608 0.359588i 0.743352 0.668900i \(-0.233235\pi\)
−0.950961 + 0.309312i \(0.899901\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 147700. 255825.i 0.583775 1.01113i −0.411252 0.911522i \(-0.634908\pi\)
0.995027 0.0996066i \(-0.0317584\pi\)
\(504\) 0 0
\(505\) 10044.6 0.0393866
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 311158. + 179647.i 1.20101 + 0.693401i 0.960779 0.277316i \(-0.0894448\pi\)
0.240227 + 0.970717i \(0.422778\pi\)
\(510\) 0 0
\(511\) −7102.67 12302.2i −0.0272007 0.0471129i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −355083. + 205007.i −1.33880 + 0.772956i
\(516\) 0 0
\(517\) 87396.8 151376.i 0.326975 0.566337i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 195107.i 0.718782i 0.933187 + 0.359391i \(0.117016\pi\)
−0.933187 + 0.359391i \(0.882984\pi\)
\(522\) 0 0
\(523\) 259323. + 149720.i 0.948063 + 0.547364i 0.892479 0.451090i \(-0.148965\pi\)
0.0555841 + 0.998454i \(0.482298\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 161810. 93421.0i 0.582618 0.336375i
\(528\) 0 0
\(529\) 132078. + 228765.i 0.471974 + 0.817483i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 657292. 2.31368
\(534\) 0 0
\(535\) −441616. + 254967.i −1.54290 + 0.890792i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −157565. −0.542354
\(540\) 0 0
\(541\) 83074.6 143889.i 0.283840 0.491625i −0.688487 0.725249i \(-0.741725\pi\)
0.972327 + 0.233623i \(0.0750582\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 133914. + 77315.4i 0.450852 + 0.260299i
\(546\) 0 0
\(547\) 345204. + 199304.i 1.15372 + 0.666102i 0.949792 0.312883i \(-0.101295\pi\)
0.203931 + 0.978985i \(0.434628\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −133243. 555888.i −0.438874 1.83098i
\(552\) 0 0
\(553\) 25427.9 14680.8i 0.0831497 0.0480065i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 157159. 272208.i 0.506558 0.877384i −0.493413 0.869795i \(-0.664251\pi\)
0.999971 0.00758920i \(-0.00241574\pi\)
\(558\) 0 0
\(559\) 733656.i 2.34784i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 39467.4i 0.124515i 0.998060 + 0.0622574i \(0.0198300\pi\)
−0.998060 + 0.0622574i \(0.980170\pi\)
\(564\) 0 0
\(565\) −462257. + 266884.i −1.44806 + 0.836037i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 376002.i 1.16136i 0.814133 + 0.580679i \(0.197213\pi\)
−0.814133 + 0.580679i \(0.802787\pi\)
\(570\) 0 0
\(571\) 452184. 1.38689 0.693446 0.720508i \(-0.256091\pi\)
0.693446 + 0.720508i \(0.256091\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4555.66 7890.63i −0.0137789 0.0238658i
\(576\) 0 0
\(577\) −394141. −1.18386 −0.591929 0.805990i \(-0.701633\pi\)
−0.591929 + 0.805990i \(0.701633\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 6534.47 0.0193579
\(582\) 0 0
\(583\) −254547. 146963.i −0.748913 0.432385i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −248196. 429888.i −0.720308 1.24761i −0.960876 0.276978i \(-0.910667\pi\)
0.240568 0.970632i \(-0.422666\pi\)
\(588\) 0 0
\(589\) 119964. 113758.i 0.345795 0.327908i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 89679.7 155330.i 0.255026 0.441718i −0.709876 0.704326i \(-0.751249\pi\)
0.964903 + 0.262608i \(0.0845826\pi\)
\(594\) 0 0
\(595\) −21291.3 + 36877.5i −0.0601405 + 0.104166i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −57674.8 33298.5i −0.160743 0.0928050i 0.417471 0.908690i \(-0.362917\pi\)
−0.578214 + 0.815885i \(0.696250\pi\)
\(600\) 0 0
\(601\) 117101.i 0.324200i −0.986774 0.162100i \(-0.948173\pi\)
0.986774 0.162100i \(-0.0518267\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 135744. + 235116.i 0.370860 + 0.642348i
\(606\) 0 0
\(607\) 480656.i 1.30454i −0.757988 0.652269i \(-0.773817\pi\)
0.757988 0.652269i \(-0.226183\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 666374. 384731.i 1.78499 1.03056i
\(612\) 0 0
\(613\) 274075. + 474712.i 0.729372 + 1.26331i 0.957149 + 0.289596i \(0.0935209\pi\)
−0.227777 + 0.973713i \(0.573146\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −346802. + 600678.i −0.910985 + 1.57787i −0.0983085 + 0.995156i \(0.531343\pi\)
−0.812676 + 0.582716i \(0.801990\pi\)
\(618\) 0 0
\(619\) −89967.6 −0.234804 −0.117402 0.993084i \(-0.537457\pi\)
−0.117402 + 0.993084i \(0.537457\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7320.11 + 4226.27i 0.0188600 + 0.0108888i
\(624\) 0 0
\(625\) 215401. + 373085.i 0.551425 + 0.955097i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 774251. 447014.i 1.95695 1.12985i
\(630\) 0 0
\(631\) 266143. 460974.i 0.668431 1.15776i −0.309912 0.950765i \(-0.600299\pi\)
0.978343 0.206991i \(-0.0663672\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 408097.i 1.01208i
\(636\) 0 0
\(637\) −600694. 346811.i −1.48038 0.854701i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 346114. 199829.i 0.842371 0.486343i −0.0156986 0.999877i \(-0.504997\pi\)
0.858069 + 0.513534i \(0.171664\pi\)
\(642\) 0 0
\(643\) 59063.7 + 102301.i 0.142856 + 0.247434i 0.928571 0.371155i \(-0.121038\pi\)
−0.785715 + 0.618589i \(0.787705\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −343903. −0.821538 −0.410769 0.911740i \(-0.634740\pi\)
−0.410769 + 0.911740i \(0.634740\pi\)
\(648\) 0 0
\(649\) 243685. 140692.i 0.578548 0.334025i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −327275. −0.767515 −0.383757 0.923434i \(-0.625370\pi\)
−0.383757 + 0.923434i \(0.625370\pi\)
\(654\) 0 0
\(655\) −37816.0 + 65499.3i −0.0881441 + 0.152670i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −112125. 64735.4i −0.258185 0.149063i 0.365321 0.930882i \(-0.380959\pi\)
−0.623506 + 0.781818i \(0.714292\pi\)
\(660\) 0 0
\(661\) 121616. + 70215.1i 0.278348 + 0.160704i 0.632675 0.774417i \(-0.281957\pi\)
−0.354327 + 0.935121i \(0.615290\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −10714.4 + 36123.0i −0.0242285 + 0.0816847i
\(666\) 0 0
\(667\) −171747. + 99158.4i −0.386045 + 0.222883i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 8173.64 14157.2i 0.0181539 0.0314435i
\(672\) 0 0
\(673\) 245792.i 0.542673i −0.962485 0.271336i \(-0.912534\pi\)
0.962485 0.271336i \(-0.0874656\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 429443.i 0.936974i 0.883470 + 0.468487i \(0.155201\pi\)
−0.883470 + 0.468487i \(0.844799\pi\)
\(678\) 0 0
\(679\) 50403.5 29100.5i 0.109325 0.0631190i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 110387.i 0.236634i −0.992976 0.118317i \(-0.962250\pi\)
0.992976 0.118317i \(-0.0377499\pi\)
\(684\) 0 0
\(685\) 60896.8 0.129782
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −646949. 1.12055e6i −1.36280 2.36044i
\(690\) 0 0
\(691\) 202528. 0.424159 0.212080 0.977252i \(-0.431976\pi\)
0.212080 + 0.977252i \(0.431976\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −854351. −1.76875
\(696\) 0 0
\(697\) −798675. 461115.i −1.64401 0.949169i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 121102. + 209755.i 0.246443 + 0.426851i 0.962536 0.271153i \(-0.0874050\pi\)
−0.716094 + 0.698004i \(0.754072\pi\)
\(702\) 0 0
\(703\) 574019. 544326.i 1.16149 1.10141i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −751.256 + 1301.21i −0.00150297 + 0.00260321i
\(708\) 0 0
\(709\) 187900. 325452.i 0.373795 0.647432i −0.616351 0.787472i \(-0.711390\pi\)
0.990146 + 0.140040i \(0.0447230\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −49671.7 28678.0i −0.0977080 0.0564118i
\(714\) 0 0
\(715\) 507359.i 0.992438i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 408662. + 707823.i 0.790508 + 1.36920i 0.925653 + 0.378374i \(0.123517\pi\)
−0.135144 + 0.990826i \(0.543150\pi\)
\(720\) 0 0
\(721\) 61331.8i 0.117982i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −99763.8 + 57598.7i −0.189800 + 0.109581i
\(726\) 0 0
\(727\) −423400. 733351.i −0.801092 1.38753i −0.918898 0.394495i \(-0.870920\pi\)
0.117806 0.993037i \(-0.462414\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 514687. 891464.i 0.963182 1.66828i
\(732\) 0 0
\(733\) 562446. 1.04682 0.523411 0.852080i \(-0.324659\pi\)
0.523411 + 0.852080i \(0.324659\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −438631. 253244.i −0.807540 0.466233i
\(738\) 0 0
\(739\) −138927. 240628.i −0.254388 0.440613i 0.710341 0.703858i \(-0.248541\pi\)
−0.964729 + 0.263245i \(0.915207\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −220105. + 127078.i −0.398705 + 0.230193i −0.685925 0.727672i \(-0.740602\pi\)
0.287220 + 0.957865i \(0.407269\pi\)
\(744\) 0 0
\(745\) −162759. + 281906.i −0.293246 + 0.507917i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 76278.2i 0.135968i
\(750\) 0 0
\(751\) 671263. + 387554.i 1.19018 + 0.687151i 0.958347 0.285605i \(-0.0921945\pi\)
0.231833 + 0.972756i \(0.425528\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −595715. + 343936.i −1.04507 + 0.603370i
\(756\) 0 0
\(757\) 82777.3 + 143374.i 0.144451 + 0.250196i 0.929168 0.369658i \(-0.120525\pi\)
−0.784717 + 0.619854i \(0.787192\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −88512.4 −0.152839 −0.0764196 0.997076i \(-0.524349\pi\)
−0.0764196 + 0.997076i \(0.524349\pi\)
\(762\) 0 0
\(763\) −20031.5 + 11565.2i −0.0344084 + 0.0198657i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.23868e6 2.10557
\(768\) 0 0
\(769\) 266238. 461138.i 0.450212 0.779790i −0.548187 0.836356i \(-0.684682\pi\)
0.998399 + 0.0565656i \(0.0180150\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −372719. 215189.i −0.623767 0.360132i 0.154567 0.987982i \(-0.450602\pi\)
−0.778334 + 0.627850i \(0.783935\pi\)
\(774\) 0 0
\(775\) −28853.1 16658.3i −0.0480384 0.0277350i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −782333. 232048.i −1.28919 0.382387i
\(780\) 0 0
\(781\) −172371. + 99518.2i −0.282593 + 0.163155i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 239192. 414293.i 0.388157 0.672308i
\(786\) 0 0
\(787\) 399690.i 0.645318i −0.946515 0.322659i \(-0.895423\pi\)
0.946515 0.322659i \(-0.104577\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 79843.5i 0.127610i
\(792\) 0 0
\(793\) 62321.5 35981.3i 0.0991041 0.0572178i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 993609.i 1.56422i 0.623138 + 0.782112i \(0.285858\pi\)
−0.623138 + 0.782112i \(0.714142\pi\)
\(798\) 0 0
\(799\) −1.07961e6 −1.69112
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −118737. 205659.i −0.184143 0.318945i
\(804\) 0 0
\(805\) 13071.8 0.0201718
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −162243. −0.247896 −0.123948 0.992289i \(-0.539556\pi\)
−0.123948 + 0.992289i \(0.539556\pi\)
\(810\) 0 0
\(811\) 208547. + 120405.i 0.317075 + 0.183064i 0.650088 0.759859i \(-0.274732\pi\)
−0.333013 + 0.942922i \(0.608065\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −23993.4 41557.7i −0.0361224 0.0625658i
\(816\) 0 0
\(817\) 259007. 873224.i 0.388032 1.30822i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −148698. + 257553.i −0.220607 + 0.382102i −0.954992 0.296630i \(-0.904137\pi\)
0.734386 + 0.678733i \(0.237470\pi\)
\(822\) 0 0
\(823\) −168468. + 291795.i −0.248724 + 0.430803i −0.963172 0.268886i \(-0.913345\pi\)
0.714448 + 0.699689i \(0.246678\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −346639. 200132.i −0.506835 0.292621i 0.224697 0.974429i \(-0.427861\pi\)
−0.731532 + 0.681807i \(0.761194\pi\)
\(828\) 0 0
\(829\) 450567.i 0.655617i −0.944744 0.327808i \(-0.893690\pi\)
0.944744 0.327808i \(-0.106310\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 486602. + 842819.i 0.701267 + 1.21463i
\(834\) 0 0
\(835\) 795066.i 1.14033i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 373845. 215840.i 0.531089 0.306625i −0.210371 0.977622i \(-0.567467\pi\)
0.741460 + 0.670997i \(0.234134\pi\)
\(840\) 0 0
\(841\) 900052. + 1.55894e6i 1.27255 + 2.20412i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 739510. 1.28087e6i 1.03569 1.79387i
\(846\) 0 0
\(847\) −40610.4 −0.0566071
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −237677. 137223.i −0.328191 0.189481i
\(852\) 0 0
\(853\) 190271. + 329558.i 0.261501 + 0.452933i 0.966641 0.256135i \(-0.0824492\pi\)
−0.705140 + 0.709068i \(0.749116\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 943560. 544765.i 1.28472 0.741732i 0.307011 0.951706i \(-0.400671\pi\)
0.977707 + 0.209973i \(0.0673378\pi\)
\(858\) 0 0
\(859\) 65215.6 112957.i 0.0883823 0.153083i −0.818445 0.574585i \(-0.805164\pi\)
0.906827 + 0.421502i \(0.138497\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 253989.i 0.341031i −0.985355 0.170515i \(-0.945457\pi\)
0.985355 0.170515i \(-0.0545432\pi\)
\(864\) 0 0
\(865\) 729211. + 421010.i 0.974588 + 0.562679i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 425085. 245423.i 0.562906 0.324994i
\(870\) 0 0
\(871\) −1.11481e6 1.93091e6i −1.46948 2.54522i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −57639.8 −0.0752846
\(876\) 0 0
\(877\) 759687. 438606.i 0.987724 0.570263i 0.0831309 0.996539i \(-0.473508\pi\)
0.904593 + 0.426276i \(0.140175\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 143314. 0.184645 0.0923223 0.995729i \(-0.470571\pi\)
0.0923223 + 0.995729i \(0.470571\pi\)
\(882\) 0 0
\(883\) −140425. + 243224.i −0.180104 + 0.311950i −0.941916 0.335849i \(-0.890977\pi\)
0.761812 + 0.647799i \(0.224310\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 214235. + 123689.i 0.272297 + 0.157211i 0.629931 0.776651i \(-0.283083\pi\)
−0.357634 + 0.933862i \(0.616416\pi\)
\(888\) 0 0
\(889\) 52866.6 + 30522.5i 0.0668925 + 0.0386204i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −928967. + 222667.i −1.16492 + 0.279224i
\(894\) 0 0
\(895\) −573098. + 330878.i −0.715456 + 0.413069i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −362585. + 628016.i −0.448632 + 0.777054i
\(900\) 0 0
\(901\) 1.81544e6i 2.23631i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 929801.i 1.13525i
\(906\) 0 0
\(907\) −967269. + 558453.i −1.17580 + 0.678847i −0.955039 0.296481i \(-0.904187\pi\)
−0.220759 + 0.975328i \(0.570853\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.08226e6i 1.30405i −0.758198 0.652025i \(-0.773920\pi\)
0.758198 0.652025i \(-0.226080\pi\)
\(912\) 0 0
\(913\) 109238. 0.131049
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −5656.69 9797.68i −0.00672704 0.0116516i
\(918\) 0 0
\(919\) −920659. −1.09010 −0.545052 0.838402i \(-0.683490\pi\)
−0.545052 + 0.838402i \(0.683490\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −876182. −1.02847
\(924\) 0 0
\(925\) −138060. 79709.2i −0.161356 0.0931591i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 282674. + 489605.i 0.327532 + 0.567302i 0.982022 0.188769i \(-0.0604497\pi\)
−0.654489 + 0.756071i \(0.727116\pi\)
\(930\) 0 0
\(931\) 592532. + 624854.i 0.683616 + 0.720907i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −355931. + 616491.i −0.407139 + 0.705186i
\(936\) 0 0
\(937\) 140444. 243257.i 0.159965 0.277068i −0.774891 0.632095i \(-0.782195\pi\)
0.934856 + 0.355027i \(0.115528\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.21593e6 + 702015.i 1.37318 + 0.792807i 0.991327 0.131416i \(-0.0419523\pi\)
0.381854 + 0.924223i \(0.375286\pi\)
\(942\) 0 0
\(943\) 283102.i 0.318361i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 458463. + 794081.i 0.511215 + 0.885451i 0.999916 + 0.0129993i \(0.00413793\pi\)
−0.488700 + 0.872452i \(0.662529\pi\)
\(948\) 0 0
\(949\) 1.04539e6i 1.16077i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.31224e6 757621.i 1.44486 0.834192i 0.446694 0.894687i \(-0.352601\pi\)
0.998169 + 0.0604947i \(0.0192678\pi\)
\(954\) 0 0
\(955\) 916648. + 1.58768e6i 1.00507 + 1.74083i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4554.61 + 7888.82i −0.00495238 + 0.00857778i
\(960\) 0 0
\(961\) 713792. 0.772902
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −15109.5 8723.48i −0.0162254 0.00936775i
\(966\) 0 0
\(967\) 660547. + 1.14410e6i 0.706400 + 1.22352i 0.966184 + 0.257854i \(0.0830155\pi\)
−0.259784 + 0.965667i \(0.583651\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −902679. + 521162.i −0.957403 + 0.552757i −0.895373 0.445317i \(-0.853091\pi\)
−0.0620301 + 0.998074i \(0.519757\pi\)
\(972\) 0 0
\(973\) 63898.9 110676.i 0.0674944 0.116904i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.19511e6i 1.25204i −0.779808 0.626019i \(-0.784683\pi\)
0.779808 0.626019i \(-0.215317\pi\)
\(978\) 0 0
\(979\) 122372. + 70651.6i 0.127678 + 0.0737152i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 904865. 522424.i 0.936433 0.540650i 0.0475925 0.998867i \(-0.484845\pi\)
0.888840 + 0.458217i \(0.151512\pi\)
\(984\) 0 0
\(985\) 115323. + 199745.i 0.118862 + 0.205875i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −315993. −0.323061
\(990\) 0 0
\(991\) −1.06594e6 + 615421.i −1.08539 + 0.626650i −0.932345 0.361569i \(-0.882241\pi\)
−0.153045 + 0.988219i \(0.548908\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −974299. −0.984116
\(996\) 0 0
\(997\) −729969. + 1.26434e6i −0.734369 + 1.27196i 0.220631 + 0.975357i \(0.429188\pi\)
−0.955000 + 0.296607i \(0.904145\pi\)
\(998\) 0 0
\(999\) 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 684.5.y.e.145.1 14
3.2 odd 2 228.5.l.a.145.7 14
19.8 odd 6 inner 684.5.y.e.217.1 14
57.8 even 6 228.5.l.a.217.7 yes 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
228.5.l.a.145.7 14 3.2 odd 2
228.5.l.a.217.7 yes 14 57.8 even 6
684.5.y.e.145.1 14 1.1 even 1 trivial
684.5.y.e.217.1 14 19.8 odd 6 inner