Properties

Label 684.5.y.c.145.6
Level $684$
Weight $5$
Character 684.145
Analytic conductor $70.705$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 631 x^{10} - 3100 x^{9} + 142264 x^{8} - 550522 x^{7} + 14083117 x^{6} - 40335478 x^{5} + 638031136 x^{4} - 1209472584 x^{3} + \cdots + 90728724573 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.6
Root \(0.500000 + 6.42649i\) of defining polynomial
Character \(\chi\) \(=\) 684.145
Dual form 684.5.y.c.217.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(13.2594 + 22.9660i) q^{5} -84.4930 q^{7} +O(q^{10})\) \(q+(13.2594 + 22.9660i) q^{5} -84.4930 q^{7} -33.3991 q^{11} +(-32.2160 - 18.5999i) q^{13} +(62.9369 + 109.010i) q^{17} +(-354.290 - 69.2806i) q^{19} +(-199.345 + 345.275i) q^{23} +(-39.1233 + 67.7635i) q^{25} +(480.756 + 277.565i) q^{29} -1323.28i q^{31} +(-1120.33 - 1940.46i) q^{35} -630.480i q^{37} +(-2615.46 + 1510.04i) q^{41} +(953.243 + 1651.07i) q^{43} +(1710.41 - 2962.52i) q^{47} +4738.07 q^{49} +(527.826 + 304.741i) q^{53} +(-442.852 - 767.041i) q^{55} +(4229.93 - 2442.15i) q^{59} +(2120.39 - 3672.62i) q^{61} -986.495i q^{65} +(-3198.52 - 1846.67i) q^{67} +(-588.629 + 339.845i) q^{71} +(2278.44 + 3946.38i) q^{73} +2821.99 q^{77} +(-6779.56 + 3914.18i) q^{79} +3820.26 q^{83} +(-1669.01 + 2890.81i) q^{85} +(12667.5 + 7313.61i) q^{89} +(2722.03 + 1571.56i) q^{91} +(-3106.57 - 9055.22i) q^{95} +(8157.18 - 4709.55i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 9 q^{5} - 52 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 9 q^{5} - 52 q^{7} - 6 q^{11} - 93 q^{13} + 483 q^{17} - 533 q^{19} - 531 q^{23} - 217 q^{25} - 2025 q^{29} + 1128 q^{35} + 1692 q^{41} - 63 q^{43} + 3471 q^{47} + 420 q^{49} + 3771 q^{53} - 2014 q^{55} + 9594 q^{59} + 1229 q^{61} + 7590 q^{67} - 963 q^{71} - 2838 q^{73} + 15408 q^{77} + 11073 q^{79} + 14202 q^{83} + 9455 q^{85} - 6525 q^{89} - 7686 q^{91} - 1521 q^{95} - 34110 q^{97}+O(q^{100}) \) 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
\(5\) 13.2594 + 22.9660i 0.530376 + 0.918638i 0.999372 + 0.0354377i \(0.0112825\pi\)
−0.468996 + 0.883200i \(0.655384\pi\)
\(6\) 0 0
\(7\) −84.4930 −1.72435 −0.862174 0.506613i \(-0.830897\pi\)
−0.862174 + 0.506613i \(0.830897\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −33.3991 −0.276025 −0.138013 0.990430i \(-0.544071\pi\)
−0.138013 + 0.990430i \(0.544071\pi\)
\(12\) 0 0
\(13\) −32.2160 18.5999i −0.190627 0.110059i 0.401649 0.915794i \(-0.368437\pi\)
−0.592276 + 0.805735i \(0.701771\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 62.9369 + 109.010i 0.217775 + 0.377197i 0.954127 0.299401i \(-0.0967869\pi\)
−0.736353 + 0.676598i \(0.763454\pi\)
\(18\) 0 0
\(19\) −354.290 69.2806i −0.981412 0.191913i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −199.345 + 345.275i −0.376833 + 0.652695i −0.990600 0.136793i \(-0.956320\pi\)
0.613766 + 0.789488i \(0.289654\pi\)
\(24\) 0 0
\(25\) −39.1233 + 67.7635i −0.0625972 + 0.108422i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 480.756 + 277.565i 0.571648 + 0.330041i 0.757807 0.652478i \(-0.226271\pi\)
−0.186159 + 0.982520i \(0.559604\pi\)
\(30\) 0 0
\(31\) 1323.28i 1.37698i −0.725244 0.688492i \(-0.758273\pi\)
0.725244 0.688492i \(-0.241727\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1120.33 1940.46i −0.914552 1.58405i
\(36\) 0 0
\(37\) 630.480i 0.460541i −0.973127 0.230270i \(-0.926039\pi\)
0.973127 0.230270i \(-0.0739610\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2615.46 + 1510.04i −1.55589 + 0.898296i −0.558251 + 0.829672i \(0.688527\pi\)
−0.997643 + 0.0686238i \(0.978139\pi\)
\(42\) 0 0
\(43\) 953.243 + 1651.07i 0.515545 + 0.892950i 0.999837 + 0.0180440i \(0.00574389\pi\)
−0.484292 + 0.874906i \(0.660923\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1710.41 2962.52i 0.774292 1.34111i −0.160900 0.986971i \(-0.551440\pi\)
0.935192 0.354142i \(-0.115227\pi\)
\(48\) 0 0
\(49\) 4738.07 1.97337
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 527.826 + 304.741i 0.187905 + 0.108487i 0.591002 0.806670i \(-0.298733\pi\)
−0.403096 + 0.915158i \(0.632066\pi\)
\(54\) 0 0
\(55\) −442.852 767.041i −0.146397 0.253567i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4229.93 2442.15i 1.21515 0.701566i 0.251271 0.967917i \(-0.419151\pi\)
0.963876 + 0.266351i \(0.0858180\pi\)
\(60\) 0 0
\(61\) 2120.39 3672.62i 0.569844 0.986998i −0.426737 0.904376i \(-0.640337\pi\)
0.996581 0.0826224i \(-0.0263295\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 986.495i 0.233490i
\(66\) 0 0
\(67\) −3198.52 1846.67i −0.712524 0.411376i 0.0994711 0.995040i \(-0.468285\pi\)
−0.811995 + 0.583665i \(0.801618\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −588.629 + 339.845i −0.116768 + 0.0674162i −0.557247 0.830347i \(-0.688142\pi\)
0.440478 + 0.897763i \(0.354809\pi\)
\(72\) 0 0
\(73\) 2278.44 + 3946.38i 0.427555 + 0.740548i 0.996655 0.0817208i \(-0.0260416\pi\)
−0.569100 + 0.822268i \(0.692708\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2821.99 0.475964
\(78\) 0 0
\(79\) −6779.56 + 3914.18i −1.08629 + 0.627172i −0.932587 0.360944i \(-0.882455\pi\)
−0.153707 + 0.988116i \(0.549121\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3820.26 0.554545 0.277272 0.960791i \(-0.410570\pi\)
0.277272 + 0.960791i \(0.410570\pi\)
\(84\) 0 0
\(85\) −1669.01 + 2890.81i −0.231005 + 0.400112i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12667.5 + 7313.61i 1.59924 + 0.923319i 0.991635 + 0.129077i \(0.0412014\pi\)
0.607601 + 0.794242i \(0.292132\pi\)
\(90\) 0 0
\(91\) 2722.03 + 1571.56i 0.328707 + 0.189779i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3106.57 9055.22i −0.344219 1.00335i
\(96\) 0 0
\(97\) 8157.18 4709.55i 0.866955 0.500536i 0.000619586 1.00000i \(-0.499803\pi\)
0.866335 + 0.499463i \(0.166469\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5659.50 + 9802.55i −0.554799 + 0.960940i 0.443120 + 0.896462i \(0.353871\pi\)
−0.997919 + 0.0644777i \(0.979462\pi\)
\(102\) 0 0
\(103\) 6757.10i 0.636922i −0.947936 0.318461i \(-0.896834\pi\)
0.947936 0.318461i \(-0.103166\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7055.61i 0.616264i −0.951344 0.308132i \(-0.900296\pi\)
0.951344 0.308132i \(-0.0997039\pi\)
\(108\) 0 0
\(109\) 2141.22 1236.23i 0.180222 0.104051i −0.407175 0.913350i \(-0.633486\pi\)
0.587397 + 0.809299i \(0.300153\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4623.82i 0.362113i −0.983473 0.181056i \(-0.942048\pi\)
0.983473 0.181056i \(-0.0579516\pi\)
\(114\) 0 0
\(115\) −10572.8 −0.799454
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5317.73 9210.57i −0.375519 0.650418i
\(120\) 0 0
\(121\) −13525.5 −0.923810
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 14499.2 0.927952
\(126\) 0 0
\(127\) 14711.8 + 8493.88i 0.912136 + 0.526622i 0.881118 0.472897i \(-0.156792\pi\)
0.0310181 + 0.999519i \(0.490125\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −9571.65 16578.6i −0.557756 0.966062i −0.997683 0.0680283i \(-0.978329\pi\)
0.439927 0.898033i \(-0.355004\pi\)
\(132\) 0 0
\(133\) 29935.0 + 5853.73i 1.69230 + 0.330925i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6048.91 10477.0i 0.322282 0.558208i −0.658677 0.752426i \(-0.728883\pi\)
0.980958 + 0.194218i \(0.0622168\pi\)
\(138\) 0 0
\(139\) 14680.6 25427.5i 0.759826 1.31606i −0.183114 0.983092i \(-0.558618\pi\)
0.942939 0.332965i \(-0.108049\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1075.98 + 621.220i 0.0526179 + 0.0303790i
\(144\) 0 0
\(145\) 14721.4i 0.700184i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2080.34 3603.26i −0.0937049 0.162302i 0.815362 0.578951i \(-0.196538\pi\)
−0.909067 + 0.416649i \(0.863204\pi\)
\(150\) 0 0
\(151\) 23274.8i 1.02078i −0.859943 0.510390i \(-0.829501\pi\)
0.859943 0.510390i \(-0.170499\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 30390.4 17545.9i 1.26495 0.730319i
\(156\) 0 0
\(157\) −20323.4 35201.2i −0.824513 1.42810i −0.902291 0.431127i \(-0.858116\pi\)
0.0777786 0.996971i \(-0.475217\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 16843.3 29173.4i 0.649792 1.12547i
\(162\) 0 0
\(163\) −25586.5 −0.963023 −0.481511 0.876440i \(-0.659912\pi\)
−0.481511 + 0.876440i \(0.659912\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16030.7 + 9255.32i 0.574803 + 0.331863i 0.759065 0.651014i \(-0.225656\pi\)
−0.184262 + 0.982877i \(0.558990\pi\)
\(168\) 0 0
\(169\) −13588.6 23536.1i −0.475774 0.824065i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 30408.0 17556.0i 1.01600 0.586590i 0.103059 0.994675i \(-0.467137\pi\)
0.912944 + 0.408086i \(0.133804\pi\)
\(174\) 0 0
\(175\) 3305.64 5725.54i 0.107939 0.186956i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 20859.6i 0.651028i −0.945537 0.325514i \(-0.894463\pi\)
0.945537 0.325514i \(-0.105537\pi\)
\(180\) 0 0
\(181\) 41594.9 + 24014.8i 1.26965 + 0.733030i 0.974921 0.222552i \(-0.0714387\pi\)
0.294725 + 0.955582i \(0.404772\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 14479.6 8359.79i 0.423070 0.244260i
\(186\) 0 0
\(187\) −2102.03 3640.83i −0.0601113 0.104116i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 52465.4 1.43816 0.719079 0.694929i \(-0.244564\pi\)
0.719079 + 0.694929i \(0.244564\pi\)
\(192\) 0 0
\(193\) 46145.2 26642.0i 1.23883 0.715240i 0.269976 0.962867i \(-0.412984\pi\)
0.968855 + 0.247627i \(0.0796508\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10872.8 0.280161 0.140081 0.990140i \(-0.455264\pi\)
0.140081 + 0.990140i \(0.455264\pi\)
\(198\) 0 0
\(199\) −10950.4 + 18966.6i −0.276517 + 0.478942i −0.970517 0.241033i \(-0.922514\pi\)
0.694000 + 0.719975i \(0.255847\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −40620.5 23452.3i −0.985720 0.569106i
\(204\) 0 0
\(205\) −69358.8 40044.3i −1.65042 0.952869i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 11832.9 + 2313.91i 0.270895 + 0.0529729i
\(210\) 0 0
\(211\) −19016.9 + 10979.4i −0.427143 + 0.246611i −0.698129 0.715972i \(-0.745984\pi\)
0.270985 + 0.962583i \(0.412650\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −25278.9 + 43784.3i −0.546865 + 0.947199i
\(216\) 0 0
\(217\) 111808.i 2.37440i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4682.48i 0.0958719i
\(222\) 0 0
\(223\) 36180.1 20888.6i 0.727545 0.420048i −0.0899783 0.995944i \(-0.528680\pi\)
0.817523 + 0.575895i \(0.195346\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 51321.1i 0.995966i 0.867187 + 0.497983i \(0.165926\pi\)
−0.867187 + 0.497983i \(0.834074\pi\)
\(228\) 0 0
\(229\) 9344.76 0.178196 0.0890978 0.996023i \(-0.471602\pi\)
0.0890978 + 0.996023i \(0.471602\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −37637.2 65189.6i −0.693275 1.20079i −0.970759 0.240057i \(-0.922834\pi\)
0.277484 0.960730i \(-0.410500\pi\)
\(234\) 0 0
\(235\) 90716.0 1.64266
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −32265.1 −0.564855 −0.282427 0.959289i \(-0.591140\pi\)
−0.282427 + 0.959289i \(0.591140\pi\)
\(240\) 0 0
\(241\) 13807.1 + 7971.52i 0.237721 + 0.137248i 0.614129 0.789206i \(-0.289507\pi\)
−0.376408 + 0.926454i \(0.622841\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 62824.0 + 108814.i 1.04663 + 1.81282i
\(246\) 0 0
\(247\) 10125.2 + 8821.70i 0.165962 + 0.144597i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −58283.1 + 100949.i −0.925114 + 1.60234i −0.133736 + 0.991017i \(0.542697\pi\)
−0.791378 + 0.611327i \(0.790636\pi\)
\(252\) 0 0
\(253\) 6657.93 11531.9i 0.104016 0.180160i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −86480.5 49929.6i −1.30934 0.755947i −0.327353 0.944902i \(-0.606157\pi\)
−0.981986 + 0.188955i \(0.939490\pi\)
\(258\) 0 0
\(259\) 53271.2i 0.794132i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −27926.3 48369.8i −0.403740 0.699299i 0.590434 0.807086i \(-0.298957\pi\)
−0.994174 + 0.107788i \(0.965623\pi\)
\(264\) 0 0
\(265\) 16162.7i 0.230156i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −82433.0 + 47592.7i −1.13919 + 0.657712i −0.946230 0.323495i \(-0.895142\pi\)
−0.192961 + 0.981207i \(0.561809\pi\)
\(270\) 0 0
\(271\) 19687.4 + 34099.6i 0.268071 + 0.464313i 0.968364 0.249543i \(-0.0802804\pi\)
−0.700293 + 0.713856i \(0.746947\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1306.68 2263.24i 0.0172784 0.0299271i
\(276\) 0 0
\(277\) −52790.5 −0.688013 −0.344006 0.938967i \(-0.611784\pi\)
−0.344006 + 0.938967i \(0.611784\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −112961. 65217.8i −1.43059 0.825950i −0.433421 0.901191i \(-0.642694\pi\)
−0.997165 + 0.0752419i \(0.976027\pi\)
\(282\) 0 0
\(283\) −37625.8 65169.8i −0.469800 0.813718i 0.529604 0.848245i \(-0.322341\pi\)
−0.999404 + 0.0345275i \(0.989007\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 220988. 127587.i 2.68290 1.54897i
\(288\) 0 0
\(289\) 33838.4 58609.8i 0.405148 0.701738i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 26421.2i 0.307763i 0.988089 + 0.153882i \(0.0491775\pi\)
−0.988089 + 0.153882i \(0.950823\pi\)
\(294\) 0 0
\(295\) 112173. + 64762.9i 1.28897 + 0.744187i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12844.2 7415.60i 0.143669 0.0829476i
\(300\) 0 0
\(301\) −80542.4 139503.i −0.888979 1.53976i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 112460. 1.20893
\(306\) 0 0
\(307\) −113735. + 65665.0i −1.20675 + 0.696718i −0.962048 0.272880i \(-0.912024\pi\)
−0.244703 + 0.969598i \(0.578690\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 66871.7 0.691388 0.345694 0.938347i \(-0.387644\pi\)
0.345694 + 0.938347i \(0.387644\pi\)
\(312\) 0 0
\(313\) 12163.3 21067.4i 0.124154 0.215042i −0.797248 0.603652i \(-0.793712\pi\)
0.921402 + 0.388611i \(0.127045\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 74607.7 + 43074.8i 0.742447 + 0.428652i 0.822958 0.568102i \(-0.192322\pi\)
−0.0805116 + 0.996754i \(0.525655\pi\)
\(318\) 0 0
\(319\) −16056.8 9270.40i −0.157789 0.0910998i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −14745.6 42981.4i −0.141338 0.411979i
\(324\) 0 0
\(325\) 2520.79 1455.38i 0.0238655 0.0137787i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −144518. + 250312.i −1.33515 + 2.31254i
\(330\) 0 0
\(331\) 101411.i 0.925616i −0.886459 0.462808i \(-0.846842\pi\)
0.886459 0.462808i \(-0.153158\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 97942.7i 0.872735i
\(336\) 0 0
\(337\) −30062.6 + 17356.6i −0.264708 + 0.152829i −0.626480 0.779437i \(-0.715505\pi\)
0.361773 + 0.932266i \(0.382172\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 44196.4i 0.380082i
\(342\) 0 0
\(343\) −197466. −1.67843
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −107740. 186612.i −0.894786 1.54981i −0.834069 0.551660i \(-0.813995\pi\)
−0.0607169 0.998155i \(-0.519339\pi\)
\(348\) 0 0
\(349\) −61536.9 −0.505225 −0.252612 0.967568i \(-0.581290\pi\)
−0.252612 + 0.967568i \(0.581290\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 184748. 1.48262 0.741311 0.671162i \(-0.234204\pi\)
0.741311 + 0.671162i \(0.234204\pi\)
\(354\) 0 0
\(355\) −15609.7 9012.29i −0.123862 0.0715119i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 40117.9 + 69486.2i 0.311279 + 0.539150i 0.978639 0.205584i \(-0.0659095\pi\)
−0.667361 + 0.744734i \(0.732576\pi\)
\(360\) 0 0
\(361\) 120721. + 49090.8i 0.926339 + 0.376692i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −60421.6 + 104653.i −0.453530 + 0.785537i
\(366\) 0 0
\(367\) 10611.8 18380.2i 0.0787877 0.136464i −0.823939 0.566678i \(-0.808228\pi\)
0.902727 + 0.430214i \(0.141562\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −44597.6 25748.4i −0.324014 0.187070i
\(372\) 0 0
\(373\) 136406.i 0.980428i 0.871602 + 0.490214i \(0.163081\pi\)
−0.871602 + 0.490214i \(0.836919\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −10325.4 17884.0i −0.0726478 0.125830i
\(378\) 0 0
\(379\) 120694.i 0.840244i −0.907468 0.420122i \(-0.861987\pi\)
0.907468 0.420122i \(-0.138013\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −199020. + 114904.i −1.35675 + 0.783320i −0.989184 0.146677i \(-0.953142\pi\)
−0.367566 + 0.929998i \(0.619809\pi\)
\(384\) 0 0
\(385\) 37417.9 + 64809.6i 0.252440 + 0.437238i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −125565. + 217485.i −0.829793 + 1.43724i 0.0684079 + 0.997657i \(0.478208\pi\)
−0.898201 + 0.439586i \(0.855125\pi\)
\(390\) 0 0
\(391\) −50184.6 −0.328259
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −179786. 103799.i −1.15229 0.665274i
\(396\) 0 0
\(397\) −37440.2 64848.4i −0.237551 0.411451i 0.722460 0.691413i \(-0.243012\pi\)
−0.960011 + 0.279962i \(0.909678\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 102868. 59390.9i 0.639723 0.369344i −0.144785 0.989463i \(-0.546249\pi\)
0.784508 + 0.620119i \(0.212916\pi\)
\(402\) 0 0
\(403\) −24612.9 + 42630.8i −0.151549 + 0.262491i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 21057.4i 0.127121i
\(408\) 0 0
\(409\) 282457. + 163076.i 1.68852 + 0.974865i 0.955655 + 0.294487i \(0.0951488\pi\)
0.732861 + 0.680378i \(0.238185\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −357399. + 206345.i −2.09534 + 1.20974i
\(414\) 0 0
\(415\) 50654.3 + 87735.9i 0.294117 + 0.509426i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −10223.8 −0.0582350 −0.0291175 0.999576i \(-0.509270\pi\)
−0.0291175 + 0.999576i \(0.509270\pi\)
\(420\) 0 0
\(421\) −221712. + 128006.i −1.25091 + 0.722213i −0.971290 0.237898i \(-0.923541\pi\)
−0.279619 + 0.960111i \(0.590208\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −9849.18 −0.0545283
\(426\) 0 0
\(427\) −179158. + 310311.i −0.982608 + 1.70193i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 75203.7 + 43418.8i 0.404841 + 0.233735i 0.688571 0.725169i \(-0.258239\pi\)
−0.283730 + 0.958904i \(0.591572\pi\)
\(432\) 0 0
\(433\) −282022. 162825.i −1.50420 0.868453i −0.999988 0.00487606i \(-0.998448\pi\)
−0.504217 0.863577i \(-0.668219\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 94546.8 108517.i 0.495090 0.568243i
\(438\) 0 0
\(439\) 200497. 115757.i 1.04035 0.600647i 0.120418 0.992723i \(-0.461576\pi\)
0.919933 + 0.392076i \(0.128243\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −103566. + 179381.i −0.527726 + 0.914048i 0.471752 + 0.881731i \(0.343622\pi\)
−0.999478 + 0.0323166i \(0.989712\pi\)
\(444\) 0 0
\(445\) 387896.i 1.95882i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 320126.i 1.58792i 0.607971 + 0.793959i \(0.291984\pi\)
−0.607971 + 0.793959i \(0.708016\pi\)
\(450\) 0 0
\(451\) 87353.8 50433.8i 0.429466 0.247952i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 83351.9i 0.402618i
\(456\) 0 0
\(457\) −186438. −0.892695 −0.446347 0.894860i \(-0.647275\pi\)
−0.446347 + 0.894860i \(0.647275\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −8745.73 15148.0i −0.0411523 0.0712779i 0.844716 0.535215i \(-0.179770\pi\)
−0.885868 + 0.463938i \(0.846436\pi\)
\(462\) 0 0
\(463\) 168691. 0.786920 0.393460 0.919342i \(-0.371278\pi\)
0.393460 + 0.919342i \(0.371278\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −183184. −0.839950 −0.419975 0.907536i \(-0.637961\pi\)
−0.419975 + 0.907536i \(0.637961\pi\)
\(468\) 0 0
\(469\) 270253. + 156030.i 1.22864 + 0.709355i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −31837.4 55144.0i −0.142304 0.246477i
\(474\) 0 0
\(475\) 18555.7 21297.4i 0.0822412 0.0943930i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −159926. + 276999.i −0.697022 + 1.20728i 0.272472 + 0.962164i \(0.412159\pi\)
−0.969494 + 0.245114i \(0.921174\pi\)
\(480\) 0 0
\(481\) −11726.9 + 20311.5i −0.0506865 + 0.0877916i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 216318. + 124892.i 0.919624 + 0.530945i
\(486\) 0 0
\(487\) 22667.1i 0.0955736i −0.998858 0.0477868i \(-0.984783\pi\)
0.998858 0.0477868i \(-0.0152168\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −102131. 176895.i −0.423636 0.733759i 0.572656 0.819796i \(-0.305913\pi\)
−0.996292 + 0.0860370i \(0.972580\pi\)
\(492\) 0 0
\(493\) 69876.2i 0.287498i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 49735.1 28714.6i 0.201349 0.116249i
\(498\) 0 0
\(499\) −207921. 360129.i −0.835020 1.44630i −0.894015 0.448038i \(-0.852123\pi\)
0.0589950 0.998258i \(-0.481210\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 217012. 375875.i 0.857724 1.48562i −0.0163716 0.999866i \(-0.505211\pi\)
0.874095 0.485755i \(-0.161455\pi\)
\(504\) 0 0
\(505\) −300166. −1.17701
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 315207. + 181985.i 1.21663 + 0.702424i 0.964196 0.265189i \(-0.0854345\pi\)
0.252437 + 0.967613i \(0.418768\pi\)
\(510\) 0 0
\(511\) −192513. 333441.i −0.737254 1.27696i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 155183. 89595.1i 0.585101 0.337808i
\(516\) 0 0
\(517\) −57126.1 + 98945.3i −0.213724 + 0.370181i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 100484.i 0.370187i −0.982721 0.185094i \(-0.940741\pi\)
0.982721 0.185094i \(-0.0592589\pi\)
\(522\) 0 0
\(523\) 5151.62 + 2974.29i 0.0188339 + 0.0108738i 0.509387 0.860537i \(-0.329872\pi\)
−0.490553 + 0.871411i \(0.663205\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 144251. 83283.2i 0.519394 0.299872i
\(528\) 0 0
\(529\) 60443.7 + 104692.i 0.215993 + 0.374111i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 112346. 0.395461
\(534\) 0 0
\(535\) 162039. 93553.1i 0.566124 0.326852i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −158247. −0.544701
\(540\) 0 0
\(541\) −25692.0 + 44499.9i −0.0877817 + 0.152042i −0.906573 0.422049i \(-0.861311\pi\)
0.818791 + 0.574091i \(0.194644\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 56782.5 + 32783.4i 0.191171 + 0.110373i
\(546\) 0 0
\(547\) −253560. 146393.i −0.847434 0.489266i 0.0123501 0.999924i \(-0.496069\pi\)
−0.859784 + 0.510657i \(0.829402\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −151097. 131645.i −0.497683 0.433613i
\(552\) 0 0
\(553\) 572826. 330721.i 1.87315 1.08146i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 52344.2 90662.9i 0.168717 0.292226i −0.769252 0.638945i \(-0.779371\pi\)
0.937969 + 0.346719i \(0.112704\pi\)
\(558\) 0 0
\(559\) 70920.9i 0.226961i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3123.45i 0.00985411i 0.999988 + 0.00492706i \(0.00156834\pi\)
−0.999988 + 0.00492706i \(0.998432\pi\)
\(564\) 0 0
\(565\) 106190. 61309.0i 0.332650 0.192056i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 35963.3i 0.111080i 0.998456 + 0.0555399i \(0.0176880\pi\)
−0.998456 + 0.0555399i \(0.982312\pi\)
\(570\) 0 0
\(571\) 62530.8 0.191788 0.0958941 0.995392i \(-0.469429\pi\)
0.0958941 + 0.995392i \(0.469429\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −15598.0 27016.6i −0.0471774 0.0817137i
\(576\) 0 0
\(577\) 242182. 0.727429 0.363715 0.931510i \(-0.381508\pi\)
0.363715 + 0.931510i \(0.381508\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −322785. −0.956228
\(582\) 0 0
\(583\) −17628.9 10178.0i −0.0518666 0.0299452i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −10340.3 17910.0i −0.0300095 0.0519779i 0.850631 0.525764i \(-0.176220\pi\)
−0.880640 + 0.473786i \(0.842887\pi\)
\(588\) 0 0
\(589\) −91677.8 + 468825.i −0.264261 + 1.35139i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 73184.7 126760.i 0.208118 0.360472i −0.743003 0.669288i \(-0.766599\pi\)
0.951122 + 0.308816i \(0.0999327\pi\)
\(594\) 0 0
\(595\) 141020. 244253.i 0.398333 0.689932i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 28855.0 + 16659.4i 0.0804206 + 0.0464308i 0.539671 0.841876i \(-0.318549\pi\)
−0.459250 + 0.888307i \(0.651882\pi\)
\(600\) 0 0
\(601\) 301579.i 0.834933i −0.908692 0.417467i \(-0.862918\pi\)
0.908692 0.417467i \(-0.137082\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −179340. 310626.i −0.489967 0.848647i
\(606\) 0 0
\(607\) 671101.i 1.82142i 0.413045 + 0.910711i \(0.364465\pi\)
−0.413045 + 0.910711i \(0.635535\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −110205. + 63627.0i −0.295202 + 0.170435i
\(612\) 0 0
\(613\) −116030. 200969.i −0.308779 0.534822i 0.669316 0.742978i \(-0.266587\pi\)
−0.978096 + 0.208156i \(0.933254\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −61529.0 + 106571.i −0.161625 + 0.279943i −0.935452 0.353455i \(-0.885007\pi\)
0.773826 + 0.633398i \(0.218340\pi\)
\(618\) 0 0
\(619\) 159886. 0.417283 0.208641 0.977992i \(-0.433096\pi\)
0.208641 + 0.977992i \(0.433096\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.07032e6 617949.i −2.75764 1.59212i
\(624\) 0 0
\(625\) 216703. + 375341.i 0.554760 + 0.960873i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 68728.5 39680.4i 0.173714 0.100294i
\(630\) 0 0
\(631\) 92668.8 160507.i 0.232742 0.403121i −0.725872 0.687830i \(-0.758564\pi\)
0.958614 + 0.284709i \(0.0918969\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 450495.i 1.11723i
\(636\) 0 0
\(637\) −152642. 88127.7i −0.376179 0.217187i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −142059. + 82017.6i −0.345742 + 0.199614i −0.662808 0.748789i \(-0.730636\pi\)
0.317066 + 0.948403i \(0.397302\pi\)
\(642\) 0 0
\(643\) −252661. 437622.i −0.611106 1.05847i −0.991054 0.133459i \(-0.957391\pi\)
0.379948 0.925008i \(-0.375942\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −95116.5 −0.227220 −0.113610 0.993525i \(-0.536241\pi\)
−0.113610 + 0.993525i \(0.536241\pi\)
\(648\) 0 0
\(649\) −141276. + 81565.6i −0.335412 + 0.193650i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 586580. 1.37563 0.687813 0.725888i \(-0.258571\pi\)
0.687813 + 0.725888i \(0.258571\pi\)
\(654\) 0 0
\(655\) 253829. 439644.i 0.591641 1.02475i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 619966. + 357938.i 1.42757 + 0.824207i 0.996929 0.0783166i \(-0.0249545\pi\)
0.430640 + 0.902524i \(0.358288\pi\)
\(660\) 0 0
\(661\) 316498. + 182730.i 0.724383 + 0.418223i 0.816364 0.577538i \(-0.195987\pi\)
−0.0919808 + 0.995761i \(0.529320\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 262484. + 765103.i 0.593552 + 1.73012i
\(666\) 0 0
\(667\) −191673. + 110662.i −0.430832 + 0.248741i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −70819.0 + 122662.i −0.157291 + 0.272436i
\(672\) 0 0
\(673\) 769776.i 1.69955i −0.527144 0.849776i \(-0.676737\pi\)
0.527144 0.849776i \(-0.323263\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 789698.i 1.72299i −0.507762 0.861497i \(-0.669527\pi\)
0.507762 0.861497i \(-0.330473\pi\)
\(678\) 0 0
\(679\) −689224. + 397924.i −1.49493 + 0.863099i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 644105.i 1.38075i −0.723451 0.690376i \(-0.757445\pi\)
0.723451 0.690376i \(-0.242555\pi\)
\(684\) 0 0
\(685\) 320819. 0.683722
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −11336.3 19635.0i −0.0238799 0.0413612i
\(690\) 0 0
\(691\) 338350. 0.708616 0.354308 0.935129i \(-0.384717\pi\)
0.354308 + 0.935129i \(0.384717\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 778623. 1.61197
\(696\) 0 0
\(697\) −329217. 190074.i −0.677668 0.391252i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 150371. + 260451.i 0.306006 + 0.530017i 0.977485 0.211006i \(-0.0676741\pi\)
−0.671479 + 0.741023i \(0.734341\pi\)
\(702\) 0 0
\(703\) −43680.1 + 223373.i −0.0883838 + 0.451980i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 478189. 828247.i 0.956666 1.65699i
\(708\) 0 0
\(709\) −207160. + 358811.i −0.412110 + 0.713795i −0.995120 0.0986695i \(-0.968541\pi\)
0.583010 + 0.812465i \(0.301875\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 456897. + 263789.i 0.898750 + 0.518894i
\(714\) 0 0
\(715\) 32948.0i 0.0644491i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −135419. 234553.i −0.261952 0.453715i 0.704808 0.709398i \(-0.251033\pi\)
−0.966761 + 0.255683i \(0.917700\pi\)
\(720\) 0 0
\(721\) 570928.i 1.09827i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −37617.5 + 21718.5i −0.0715672 + 0.0413193i
\(726\) 0 0
\(727\) 4386.18 + 7597.09i 0.00829885 + 0.0143740i 0.870145 0.492796i \(-0.164025\pi\)
−0.861846 + 0.507170i \(0.830692\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −119988. + 207826.i −0.224545 + 0.388924i
\(732\) 0 0
\(733\) 653473. 1.21624 0.608121 0.793844i \(-0.291924\pi\)
0.608121 + 0.793844i \(0.291924\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 106828. + 61676.9i 0.196675 + 0.113550i
\(738\) 0 0
\(739\) 189565. + 328336.i 0.347112 + 0.601215i 0.985735 0.168304i \(-0.0538292\pi\)
−0.638623 + 0.769519i \(0.720496\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 643576. 371569.i 1.16580 0.673072i 0.213109 0.977028i \(-0.431641\pi\)
0.952686 + 0.303956i \(0.0983076\pi\)
\(744\) 0 0
\(745\) 55168.2 95554.1i 0.0993977 0.172162i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 596150.i 1.06265i
\(750\) 0 0
\(751\) −832933. 480894.i −1.47683 0.852648i −0.477172 0.878810i \(-0.658338\pi\)
−0.999658 + 0.0261621i \(0.991671\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 534528. 308610.i 0.937728 0.541398i
\(756\) 0 0
\(757\) −68686.5 118968.i −0.119861 0.207606i 0.799851 0.600198i \(-0.204912\pi\)
−0.919713 + 0.392592i \(0.871578\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 275441. 0.475619 0.237809 0.971312i \(-0.423571\pi\)
0.237809 + 0.971312i \(0.423571\pi\)
\(762\) 0 0
\(763\) −180918. + 104453.i −0.310766 + 0.179421i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −181695. −0.308854
\(768\) 0 0
\(769\) 303928. 526419.i 0.513947 0.890182i −0.485922 0.874002i \(-0.661516\pi\)
0.999869 0.0161801i \(-0.00515050\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −955688. 551767.i −1.59940 0.923414i −0.991603 0.129321i \(-0.958720\pi\)
−0.607797 0.794092i \(-0.707947\pi\)
\(774\) 0 0
\(775\) 89670.1 + 51771.1i 0.149295 + 0.0861953i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.03125e6 353789.i 1.69937 0.583002i
\(780\) 0 0
\(781\) 19659.7 11350.5i 0.0322310 0.0186086i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 538952. 933493.i 0.874603 1.51486i
\(786\) 0 0
\(787\) 619317.i 0.999916i −0.866050 0.499958i \(-0.833349\pi\)
0.866050 0.499958i \(-0.166651\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 390680.i 0.624408i
\(792\) 0 0
\(793\) −136621. + 78878.1i −0.217255 + 0.125432i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 22197.8i 0.0349457i 0.999847 + 0.0174728i \(0.00556206\pi\)
−0.999847 + 0.0174728i \(0.994438\pi\)
\(798\) 0 0
\(799\) 430591. 0.674484
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −76097.9 131805.i −0.118016 0.204410i
\(804\) 0 0
\(805\) 893326. 1.37854
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −110260. −0.168469 −0.0842345 0.996446i \(-0.526844\pi\)
−0.0842345 + 0.996446i \(0.526844\pi\)
\(810\) 0 0
\(811\) 274722. + 158611.i 0.417688 + 0.241152i 0.694088 0.719890i \(-0.255808\pi\)
−0.276399 + 0.961043i \(0.589141\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −339262. 587619.i −0.510764 0.884669i
\(816\) 0 0
\(817\) −223337. 650997.i −0.334593 0.975292i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 423971. 734339.i 0.628999 1.08946i −0.358754 0.933432i \(-0.616798\pi\)
0.987753 0.156026i \(-0.0498683\pi\)
\(822\) 0 0
\(823\) −295443. + 511723.i −0.436189 + 0.755502i −0.997392 0.0721768i \(-0.977005\pi\)
0.561203 + 0.827678i \(0.310339\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 615143. + 355153.i 0.899425 + 0.519283i 0.877014 0.480466i \(-0.159532\pi\)
0.0224113 + 0.999749i \(0.492866\pi\)
\(828\) 0 0
\(829\) 986161.i 1.43496i 0.696581 + 0.717479i \(0.254704\pi\)
−0.696581 + 0.717479i \(0.745296\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 298199. + 516496.i 0.429751 + 0.744350i
\(834\) 0 0
\(835\) 490880.i 0.704048i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −296225. + 171026.i −0.420822 + 0.242962i −0.695429 0.718595i \(-0.744786\pi\)
0.274607 + 0.961557i \(0.411452\pi\)
\(840\) 0 0
\(841\) −199556. 345641.i −0.282146 0.488690i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 360353. 624150.i 0.504678 0.874129i
\(846\) 0 0
\(847\) 1.14281e6 1.59297
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 217689. + 125683.i 0.300592 + 0.173547i
\(852\) 0 0
\(853\) −478069. 828040.i −0.657041 1.13803i −0.981378 0.192087i \(-0.938474\pi\)
0.324337 0.945942i \(-0.394859\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.22953e6 + 709870.i −1.67409 + 0.966535i −0.708776 + 0.705434i \(0.750752\pi\)
−0.965312 + 0.261101i \(0.915914\pi\)
\(858\) 0 0
\(859\) 557394. 965435.i 0.755399 1.30839i −0.189777 0.981827i \(-0.560777\pi\)
0.945176 0.326562i \(-0.105890\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.04519e6i 1.40338i −0.712483 0.701690i \(-0.752429\pi\)
0.712483 0.701690i \(-0.247571\pi\)
\(864\) 0 0
\(865\) 806382. + 465565.i 1.07773 + 0.622226i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 226431. 130730.i 0.299845 0.173116i
\(870\) 0 0
\(871\) 68695.6 + 118984.i 0.0905509 + 0.156839i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.22508e6 −1.60011
\(876\) 0 0
\(877\) 121311. 70038.7i 0.157725 0.0910624i −0.419060 0.907958i \(-0.637640\pi\)
0.576785 + 0.816896i \(0.304307\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −517628. −0.666908 −0.333454 0.942766i \(-0.608214\pi\)
−0.333454 + 0.942766i \(0.608214\pi\)
\(882\) 0 0
\(883\) −199591. + 345702.i −0.255988 + 0.443384i −0.965163 0.261648i \(-0.915734\pi\)
0.709176 + 0.705032i \(0.249067\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −902167. 520867.i −1.14667 0.662032i −0.198599 0.980081i \(-0.563639\pi\)
−0.948074 + 0.318048i \(0.896973\pi\)
\(888\) 0 0
\(889\) −1.24305e6 717674.i −1.57284 0.908079i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −811226. + 931091.i −1.01728 + 1.16759i
\(894\) 0 0
\(895\) 479061. 276586.i 0.598059 0.345290i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 367296. 636176.i 0.454461 0.787150i
\(900\) 0 0
\(901\) 76717.7i 0.0945030i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.27369e6i 1.55513i
\(906\) 0 0
\(907\) 694514. 400978.i 0.844241 0.487422i −0.0144629 0.999895i \(-0.504604\pi\)
0.858703 + 0.512473i \(0.171271\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.38700e6i 1.67125i −0.549301 0.835625i \(-0.685106\pi\)
0.549301 0.835625i \(-0.314894\pi\)
\(912\) 0 0
\(913\) −127593. −0.153068
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 808738. + 1.40077e6i 0.961765 + 1.66583i
\(918\) 0 0
\(919\) −751701. −0.890049 −0.445025 0.895518i \(-0.646805\pi\)
−0.445025 + 0.895518i \(0.646805\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 25284.4 0.0296790
\(924\) 0 0
\(925\) 42723.5 + 24666.4i 0.0499325 + 0.0288286i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −570771. 988605.i −0.661349 1.14549i −0.980261 0.197706i \(-0.936651\pi\)
0.318912 0.947784i \(-0.396683\pi\)
\(930\) 0 0
\(931\) −1.67865e6 328257.i −1.93669 0.378716i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 55743.4 96550.4i 0.0637632 0.110441i
\(936\) 0 0
\(937\) −281582. + 487714.i −0.320719 + 0.555502i −0.980637 0.195836i \(-0.937258\pi\)
0.659917 + 0.751338i \(0.270591\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 927824. + 535679.i 1.04782 + 0.604959i 0.922038 0.387100i \(-0.126523\pi\)
0.125781 + 0.992058i \(0.459856\pi\)
\(942\) 0 0
\(943\) 1.20407e6i 1.35403i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 721456. + 1.24960e6i 0.804470 + 1.39338i 0.916648 + 0.399695i \(0.130884\pi\)
−0.112178 + 0.993688i \(0.535783\pi\)
\(948\) 0 0
\(949\) 169515.i 0.188225i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −239343. + 138185.i −0.263533 + 0.152151i −0.625945 0.779867i \(-0.715287\pi\)
0.362412 + 0.932018i \(0.381953\pi\)
\(954\) 0 0
\(955\) 695660. + 1.20492e6i 0.762764 + 1.32115i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −511090. + 885234.i −0.555726 + 0.962545i
\(960\) 0 0
\(961\) −827553. −0.896084
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.22372e6 + 706513.i 1.31409 + 0.758692i
\(966\) 0 0
\(967\) −43685.9 75666.3i −0.0467185 0.0809188i 0.841721 0.539913i \(-0.181543\pi\)
−0.888439 + 0.458995i \(0.848210\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −483492. + 279144.i −0.512803 + 0.296067i −0.733985 0.679166i \(-0.762342\pi\)
0.221182 + 0.975233i \(0.429008\pi\)
\(972\) 0 0
\(973\) −1.24041e6 + 2.14845e6i −1.31020 + 2.26934i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 36446.8i 0.0381830i 0.999818 + 0.0190915i \(0.00607738\pi\)
−0.999818 + 0.0190915i \(0.993923\pi\)
\(978\) 0 0
\(979\) −423084. 244268.i −0.441430 0.254859i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −819884. + 473360.i −0.848487 + 0.489874i −0.860140 0.510058i \(-0.829624\pi\)
0.0116529 + 0.999932i \(0.496291\pi\)
\(984\) 0 0
\(985\) 144167. + 249704.i 0.148591 + 0.257367i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −760097. −0.777099
\(990\) 0 0
\(991\) 361483. 208702.i 0.368078 0.212510i −0.304540 0.952499i \(-0.598503\pi\)
0.672619 + 0.739989i \(0.265169\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −580781. −0.586632
\(996\) 0 0
\(997\) 247061. 427922.i 0.248550 0.430501i −0.714574 0.699560i \(-0.753379\pi\)
0.963124 + 0.269059i \(0.0867126\pi\)
\(998\) 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.c.145.6 12
3.2 odd 2 76.5.h.a.69.5 yes 12
12.11 even 2 304.5.r.b.145.2 12
19.8 odd 6 inner 684.5.y.c.217.6 12
57.8 even 6 76.5.h.a.65.5 12
228.179 odd 6 304.5.r.b.65.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.5.h.a.65.5 12 57.8 even 6
76.5.h.a.69.5 yes 12 3.2 odd 2
304.5.r.b.65.2 12 228.179 odd 6
304.5.r.b.145.2 12 12.11 even 2
684.5.y.c.145.6 12 1.1 even 1 trivial
684.5.y.c.217.6 12 19.8 odd 6 inner