Properties

Label 1620.4.i.g.541.1
Level $1620$
Weight $4$
Character 1620.541
Analytic conductor $95.583$
Analytic rank $0$
Dimension $2$
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] = [1620,4,Mod(541,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.541");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1620.i (of order \(3\), degree \(2\), not 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: \(95.5830942093\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 541.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1620.541
Dual form 1620.4.i.g.1081.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.50000 + 4.33013i) q^{5} +(-16.0000 + 27.7128i) q^{7} +O(q^{10})\) \(q+(2.50000 + 4.33013i) q^{5} +(-16.0000 + 27.7128i) q^{7} +(18.0000 - 31.1769i) q^{11} +(5.00000 + 8.66025i) q^{13} +78.0000 q^{17} +140.000 q^{19} +(-96.0000 - 166.277i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(3.00000 - 5.19615i) q^{29} +(8.00000 + 13.8564i) q^{31} -160.000 q^{35} -34.0000 q^{37} +(-195.000 - 337.750i) q^{41} +(26.0000 - 45.0333i) q^{43} +(204.000 - 353.338i) q^{47} +(-340.500 - 589.763i) q^{49} +114.000 q^{53} +180.000 q^{55} +(258.000 + 446.869i) q^{59} +(29.0000 - 50.2295i) q^{61} +(-25.0000 + 43.3013i) q^{65} +(446.000 + 772.495i) q^{67} +120.000 q^{71} -646.000 q^{73} +(576.000 + 997.661i) q^{77} +(584.000 - 1011.52i) q^{79} +(-366.000 + 633.931i) q^{83} +(195.000 + 337.750i) q^{85} +1590.00 q^{89} -320.000 q^{91} +(350.000 + 606.218i) q^{95} +(-97.0000 + 168.009i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{5} - 32 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{5} - 32 q^{7} + 36 q^{11} + 10 q^{13} + 156 q^{17} + 280 q^{19} - 192 q^{23} - 25 q^{25} + 6 q^{29} + 16 q^{31} - 320 q^{35} - 68 q^{37} - 390 q^{41} + 52 q^{43} + 408 q^{47} - 681 q^{49} + 228 q^{53} + 360 q^{55} + 516 q^{59} + 58 q^{61} - 50 q^{65} + 892 q^{67} + 240 q^{71} - 1292 q^{73} + 1152 q^{77} + 1168 q^{79} - 732 q^{83} + 390 q^{85} + 3180 q^{89} - 640 q^{91} + 700 q^{95} - 194 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) 2.50000 + 4.33013i 0.223607 + 0.387298i
\(6\) 0 0
\(7\) −16.0000 + 27.7128i −0.863919 + 1.49635i 0.00419795 + 0.999991i \(0.498664\pi\)
−0.868117 + 0.496360i \(0.834670\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 18.0000 31.1769i 0.493382 0.854563i −0.506589 0.862188i \(-0.669094\pi\)
0.999971 + 0.00762479i \(0.00242707\pi\)
\(12\) 0 0
\(13\) 5.00000 + 8.66025i 0.106673 + 0.184763i 0.914421 0.404765i \(-0.132647\pi\)
−0.807747 + 0.589529i \(0.799313\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 78.0000 1.11281 0.556405 0.830911i \(-0.312180\pi\)
0.556405 + 0.830911i \(0.312180\pi\)
\(18\) 0 0
\(19\) 140.000 1.69043 0.845216 0.534425i \(-0.179472\pi\)
0.845216 + 0.534425i \(0.179472\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −96.0000 166.277i −0.870321 1.50744i −0.861665 0.507478i \(-0.830578\pi\)
−0.00865615 0.999963i \(-0.502755\pi\)
\(24\) 0 0
\(25\) −12.5000 + 21.6506i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00000 5.19615i 0.0192099 0.0332725i −0.856261 0.516544i \(-0.827218\pi\)
0.875471 + 0.483272i \(0.160552\pi\)
\(30\) 0 0
\(31\) 8.00000 + 13.8564i 0.0463498 + 0.0802801i 0.888270 0.459323i \(-0.151908\pi\)
−0.841920 + 0.539603i \(0.818574\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −160.000 −0.772712
\(36\) 0 0
\(37\) −34.0000 −0.151069 −0.0755347 0.997143i \(-0.524066\pi\)
−0.0755347 + 0.997143i \(0.524066\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −195.000 337.750i −0.742778 1.28653i −0.951226 0.308495i \(-0.900175\pi\)
0.208448 0.978033i \(-0.433159\pi\)
\(42\) 0 0
\(43\) 26.0000 45.0333i 0.0922084 0.159710i −0.816232 0.577725i \(-0.803941\pi\)
0.908440 + 0.418015i \(0.137274\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 204.000 353.338i 0.633116 1.09659i −0.353795 0.935323i \(-0.615109\pi\)
0.986911 0.161266i \(-0.0515578\pi\)
\(48\) 0 0
\(49\) −340.500 589.763i −0.992711 1.71943i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 114.000 0.295455 0.147727 0.989028i \(-0.452804\pi\)
0.147727 + 0.989028i \(0.452804\pi\)
\(54\) 0 0
\(55\) 180.000 0.441294
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 258.000 + 446.869i 0.569301 + 0.986058i 0.996635 + 0.0819641i \(0.0261193\pi\)
−0.427335 + 0.904094i \(0.640547\pi\)
\(60\) 0 0
\(61\) 29.0000 50.2295i 0.0608700 0.105430i −0.833985 0.551788i \(-0.813946\pi\)
0.894855 + 0.446358i \(0.147279\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −25.0000 + 43.3013i −0.0477057 + 0.0826286i
\(66\) 0 0
\(67\) 446.000 + 772.495i 0.813247 + 1.40859i 0.910580 + 0.413334i \(0.135636\pi\)
−0.0973322 + 0.995252i \(0.531031\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 120.000 0.200583 0.100291 0.994958i \(-0.468022\pi\)
0.100291 + 0.994958i \(0.468022\pi\)
\(72\) 0 0
\(73\) −646.000 −1.03573 −0.517867 0.855461i \(-0.673274\pi\)
−0.517867 + 0.855461i \(0.673274\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 576.000 + 997.661i 0.852484 + 1.47655i
\(78\) 0 0
\(79\) 584.000 1011.52i 0.831711 1.44056i −0.0649702 0.997887i \(-0.520695\pi\)
0.896681 0.442678i \(-0.145971\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −366.000 + 633.931i −0.484021 + 0.838348i −0.999832 0.0183540i \(-0.994157\pi\)
0.515811 + 0.856703i \(0.327491\pi\)
\(84\) 0 0
\(85\) 195.000 + 337.750i 0.248832 + 0.430990i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1590.00 1.89370 0.946852 0.321669i \(-0.104244\pi\)
0.946852 + 0.321669i \(0.104244\pi\)
\(90\) 0 0
\(91\) −320.000 −0.368628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 350.000 + 606.218i 0.377992 + 0.654701i
\(96\) 0 0
\(97\) −97.0000 + 168.009i −0.101535 + 0.175863i −0.912317 0.409484i \(-0.865709\pi\)
0.810782 + 0.585348i \(0.199042\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 399.000 691.088i 0.393089 0.680850i −0.599766 0.800175i \(-0.704740\pi\)
0.992855 + 0.119325i \(0.0380731\pi\)
\(102\) 0 0
\(103\) −136.000 235.559i −0.130102 0.225343i 0.793614 0.608422i \(-0.208197\pi\)
−0.923716 + 0.383079i \(0.874864\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −156.000 −0.140945 −0.0704724 0.997514i \(-0.522451\pi\)
−0.0704724 + 0.997514i \(0.522451\pi\)
\(108\) 0 0
\(109\) 1622.00 1.42532 0.712658 0.701512i \(-0.247491\pi\)
0.712658 + 0.701512i \(0.247491\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 537.000 + 930.111i 0.447051 + 0.774314i 0.998193 0.0600972i \(-0.0191411\pi\)
−0.551142 + 0.834411i \(0.685808\pi\)
\(114\) 0 0
\(115\) 480.000 831.384i 0.389219 0.674148i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1248.00 + 2161.60i −0.961378 + 1.66516i
\(120\) 0 0
\(121\) 17.5000 + 30.3109i 0.0131480 + 0.0227730i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −1528.00 −1.06762 −0.533811 0.845604i \(-0.679241\pi\)
−0.533811 + 0.845604i \(0.679241\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1206.00 + 2088.85i 0.804341 + 1.39316i 0.916735 + 0.399496i \(0.130815\pi\)
−0.112394 + 0.993664i \(0.535852\pi\)
\(132\) 0 0
\(133\) −2240.00 + 3879.79i −1.46040 + 2.52948i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1053.00 1823.85i 0.656671 1.13739i −0.324802 0.945782i \(-0.605298\pi\)
0.981472 0.191605i \(-0.0613691\pi\)
\(138\) 0 0
\(139\) 278.000 + 481.510i 0.169638 + 0.293821i 0.938293 0.345843i \(-0.112407\pi\)
−0.768655 + 0.639664i \(0.779074\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 360.000 0.210522
\(144\) 0 0
\(145\) 30.0000 0.0171818
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1209.00 2094.05i −0.664732 1.15135i −0.979358 0.202134i \(-0.935212\pi\)
0.314625 0.949216i \(-0.398121\pi\)
\(150\) 0 0
\(151\) −1420.00 + 2459.51i −0.765285 + 1.32551i 0.174812 + 0.984602i \(0.444068\pi\)
−0.940096 + 0.340910i \(0.889265\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −40.0000 + 69.2820i −0.0207282 + 0.0359024i
\(156\) 0 0
\(157\) −1027.00 1778.82i −0.522061 0.904236i −0.999671 0.0256636i \(-0.991830\pi\)
0.477610 0.878572i \(-0.341503\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6144.00 3.00755
\(162\) 0 0
\(163\) −460.000 −0.221043 −0.110521 0.993874i \(-0.535252\pi\)
−0.110521 + 0.993874i \(0.535252\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1008.00 + 1745.91i 0.467074 + 0.808996i 0.999292 0.0376110i \(-0.0119748\pi\)
−0.532218 + 0.846607i \(0.678641\pi\)
\(168\) 0 0
\(169\) 1048.50 1816.06i 0.477242 0.826607i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −309.000 + 535.204i −0.135797 + 0.235207i −0.925902 0.377765i \(-0.876693\pi\)
0.790105 + 0.612972i \(0.210026\pi\)
\(174\) 0 0
\(175\) −400.000 692.820i −0.172784 0.299270i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2964.00 1.23765 0.618826 0.785528i \(-0.287609\pi\)
0.618826 + 0.785528i \(0.287609\pi\)
\(180\) 0 0
\(181\) −370.000 −0.151944 −0.0759721 0.997110i \(-0.524206\pi\)
−0.0759721 + 0.997110i \(0.524206\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −85.0000 147.224i −0.0337801 0.0585089i
\(186\) 0 0
\(187\) 1404.00 2431.80i 0.549041 0.950967i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 552.000 956.092i 0.209117 0.362201i −0.742320 0.670046i \(-0.766274\pi\)
0.951437 + 0.307845i \(0.0996078\pi\)
\(192\) 0 0
\(193\) 1199.00 + 2076.73i 0.447181 + 0.774540i 0.998201 0.0599518i \(-0.0190947\pi\)
−0.551020 + 0.834492i \(0.685761\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1278.00 −0.462202 −0.231101 0.972930i \(-0.574233\pi\)
−0.231101 + 0.972930i \(0.574233\pi\)
\(198\) 0 0
\(199\) 4472.00 1.59302 0.796512 0.604623i \(-0.206676\pi\)
0.796512 + 0.604623i \(0.206676\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 96.0000 + 166.277i 0.0331915 + 0.0574894i
\(204\) 0 0
\(205\) 975.000 1688.75i 0.332180 0.575353i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2520.00 4364.77i 0.834029 1.44458i
\(210\) 0 0
\(211\) −670.000 1160.47i −0.218600 0.378627i 0.735780 0.677221i \(-0.236816\pi\)
−0.954380 + 0.298594i \(0.903482\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 260.000 0.0824737
\(216\) 0 0
\(217\) −512.000 −0.160170
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 390.000 + 675.500i 0.118707 + 0.205606i
\(222\) 0 0
\(223\) −1180.00 + 2043.82i −0.354344 + 0.613741i −0.987005 0.160687i \(-0.948629\pi\)
0.632662 + 0.774428i \(0.281962\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 690.000 1195.12i 0.201748 0.349439i −0.747343 0.664438i \(-0.768671\pi\)
0.949092 + 0.314999i \(0.102004\pi\)
\(228\) 0 0
\(229\) −847.000 1467.05i −0.244416 0.423341i 0.717551 0.696506i \(-0.245263\pi\)
−0.961967 + 0.273164i \(0.911930\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5190.00 1.45926 0.729631 0.683841i \(-0.239692\pi\)
0.729631 + 0.683841i \(0.239692\pi\)
\(234\) 0 0
\(235\) 2040.00 0.566276
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1176.00 + 2036.89i 0.318281 + 0.551279i 0.980129 0.198359i \(-0.0635612\pi\)
−0.661849 + 0.749638i \(0.730228\pi\)
\(240\) 0 0
\(241\) 1751.00 3032.82i 0.468016 0.810627i −0.531316 0.847174i \(-0.678302\pi\)
0.999332 + 0.0365464i \(0.0116357\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1702.50 2948.82i 0.443954 0.768951i
\(246\) 0 0
\(247\) 700.000 + 1212.44i 0.180324 + 0.312330i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4788.00 −1.20405 −0.602024 0.798478i \(-0.705639\pi\)
−0.602024 + 0.798478i \(0.705639\pi\)
\(252\) 0 0
\(253\) −6912.00 −1.71760
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 753.000 + 1304.23i 0.182766 + 0.316560i 0.942821 0.333298i \(-0.108162\pi\)
−0.760055 + 0.649858i \(0.774828\pi\)
\(258\) 0 0
\(259\) 544.000 942.236i 0.130512 0.226053i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −216.000 + 374.123i −0.0506431 + 0.0877164i −0.890236 0.455500i \(-0.849460\pi\)
0.839593 + 0.543217i \(0.182794\pi\)
\(264\) 0 0
\(265\) 285.000 + 493.634i 0.0660657 + 0.114429i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −54.0000 −0.0122395 −0.00611977 0.999981i \(-0.501948\pi\)
−0.00611977 + 0.999981i \(0.501948\pi\)
\(270\) 0 0
\(271\) −6496.00 −1.45610 −0.728051 0.685522i \(-0.759574\pi\)
−0.728051 + 0.685522i \(0.759574\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 450.000 + 779.423i 0.0986764 + 0.170913i
\(276\) 0 0
\(277\) 233.000 403.568i 0.0505401 0.0875381i −0.839649 0.543130i \(-0.817239\pi\)
0.890189 + 0.455592i \(0.150572\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2427.00 + 4203.69i −0.515241 + 0.892423i 0.484603 + 0.874734i \(0.338964\pi\)
−0.999844 + 0.0176890i \(0.994369\pi\)
\(282\) 0 0
\(283\) 2258.00 + 3910.97i 0.474290 + 0.821495i 0.999567 0.0294368i \(-0.00937136\pi\)
−0.525276 + 0.850932i \(0.676038\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12480.0 2.56680
\(288\) 0 0
\(289\) 1171.00 0.238347
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4287.00 + 7425.30i 0.854775 + 1.48051i 0.876853 + 0.480758i \(0.159639\pi\)
−0.0220777 + 0.999756i \(0.507028\pi\)
\(294\) 0 0
\(295\) −1290.00 + 2234.35i −0.254599 + 0.440978i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 960.000 1662.77i 0.185680 0.321607i
\(300\) 0 0
\(301\) 832.000 + 1441.07i 0.159321 + 0.275952i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 290.000 0.0544438
\(306\) 0 0
\(307\) 3476.00 0.646208 0.323104 0.946363i \(-0.395274\pi\)
0.323104 + 0.946363i \(0.395274\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1212.00 + 2099.25i 0.220985 + 0.382757i 0.955107 0.296260i \(-0.0957396\pi\)
−0.734123 + 0.679017i \(0.762406\pi\)
\(312\) 0 0
\(313\) 779.000 1349.27i 0.140676 0.243659i −0.787075 0.616857i \(-0.788406\pi\)
0.927751 + 0.373199i \(0.121739\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4269.00 + 7394.12i −0.756375 + 1.31008i 0.188313 + 0.982109i \(0.439698\pi\)
−0.944688 + 0.327971i \(0.893635\pi\)
\(318\) 0 0
\(319\) −108.000 187.061i −0.0189556 0.0328321i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 10920.0 1.88113
\(324\) 0 0
\(325\) −250.000 −0.0426692
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6528.00 + 11306.8i 1.09392 + 1.89473i
\(330\) 0 0
\(331\) 494.000 855.633i 0.0820323 0.142084i −0.822090 0.569357i \(-0.807192\pi\)
0.904123 + 0.427273i \(0.140526\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2230.00 + 3862.47i −0.363695 + 0.629939i
\(336\) 0 0
\(337\) −1273.00 2204.90i −0.205771 0.356405i 0.744607 0.667503i \(-0.232637\pi\)
−0.950378 + 0.311097i \(0.899303\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 576.000 0.0914726
\(342\) 0 0
\(343\) 10816.0 1.70265
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4278.00 + 7409.71i 0.661830 + 1.14632i 0.980134 + 0.198335i \(0.0635533\pi\)
−0.318304 + 0.947989i \(0.603113\pi\)
\(348\) 0 0
\(349\) 1853.00 3209.49i 0.284209 0.492264i −0.688208 0.725513i \(-0.741602\pi\)
0.972417 + 0.233249i \(0.0749358\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5697.00 9867.49i 0.858982 1.48780i −0.0139186 0.999903i \(-0.504431\pi\)
0.872901 0.487898i \(-0.162236\pi\)
\(354\) 0 0
\(355\) 300.000 + 519.615i 0.0448517 + 0.0776854i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 264.000 0.0388117 0.0194058 0.999812i \(-0.493823\pi\)
0.0194058 + 0.999812i \(0.493823\pi\)
\(360\) 0 0
\(361\) 12741.0 1.85756
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1615.00 2797.26i −0.231597 0.401138i
\(366\) 0 0
\(367\) −5116.00 + 8861.17i −0.727665 + 1.26035i 0.230203 + 0.973143i \(0.426061\pi\)
−0.957868 + 0.287210i \(0.907272\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1824.00 + 3159.26i −0.255249 + 0.442104i
\(372\) 0 0
\(373\) 281.000 + 486.706i 0.0390070 + 0.0675622i 0.884870 0.465838i \(-0.154247\pi\)
−0.845863 + 0.533401i \(0.820914\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 60.0000 0.00819670
\(378\) 0 0
\(379\) −7228.00 −0.979624 −0.489812 0.871828i \(-0.662935\pi\)
−0.489812 + 0.871828i \(0.662935\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2868.00 4967.52i −0.382632 0.662738i 0.608806 0.793319i \(-0.291649\pi\)
−0.991438 + 0.130582i \(0.958316\pi\)
\(384\) 0 0
\(385\) −2880.00 + 4988.31i −0.381243 + 0.660332i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −4593.00 + 7955.31i −0.598649 + 1.03689i 0.394372 + 0.918951i \(0.370962\pi\)
−0.993021 + 0.117939i \(0.962371\pi\)
\(390\) 0 0
\(391\) −7488.00 12969.6i −0.968502 1.67750i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5840.00 0.743905
\(396\) 0 0
\(397\) −394.000 −0.0498093 −0.0249047 0.999690i \(-0.507928\pi\)
−0.0249047 + 0.999690i \(0.507928\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −807.000 1397.77i −0.100498 0.174067i 0.811392 0.584502i \(-0.198710\pi\)
−0.911890 + 0.410435i \(0.865377\pi\)
\(402\) 0 0
\(403\) −80.0000 + 138.564i −0.00988855 + 0.0171275i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −612.000 + 1060.02i −0.0745349 + 0.129098i
\(408\) 0 0
\(409\) −517.000 895.470i −0.0625037 0.108260i 0.833080 0.553152i \(-0.186575\pi\)
−0.895584 + 0.444893i \(0.853242\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −16512.0 −1.96732
\(414\) 0 0
\(415\) −3660.00 −0.432921
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1854.00 + 3211.22i 0.216167 + 0.374412i 0.953633 0.300972i \(-0.0973112\pi\)
−0.737466 + 0.675384i \(0.763978\pi\)
\(420\) 0 0
\(421\) 2465.00 4269.51i 0.285360 0.494259i −0.687336 0.726340i \(-0.741220\pi\)
0.972697 + 0.232081i \(0.0745534\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −975.000 + 1688.75i −0.111281 + 0.192744i
\(426\) 0 0
\(427\) 928.000 + 1607.34i 0.105173 + 0.182166i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2592.00 0.289680 0.144840 0.989455i \(-0.453733\pi\)
0.144840 + 0.989455i \(0.453733\pi\)
\(432\) 0 0
\(433\) 2162.00 0.239952 0.119976 0.992777i \(-0.461718\pi\)
0.119976 + 0.992777i \(0.461718\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −13440.0 23278.8i −1.47122 2.54822i
\(438\) 0 0
\(439\) −676.000 + 1170.87i −0.0734937 + 0.127295i −0.900430 0.435000i \(-0.856748\pi\)
0.826937 + 0.562295i \(0.190082\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2766.00 4790.85i 0.296652 0.513816i −0.678716 0.734401i \(-0.737463\pi\)
0.975368 + 0.220585i \(0.0707967\pi\)
\(444\) 0 0
\(445\) 3975.00 + 6884.90i 0.423445 + 0.733428i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3198.00 0.336131 0.168066 0.985776i \(-0.446248\pi\)
0.168066 + 0.985776i \(0.446248\pi\)
\(450\) 0 0
\(451\) −14040.0 −1.46589
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −800.000 1385.64i −0.0824276 0.142769i
\(456\) 0 0
\(457\) 755.000 1307.70i 0.0772810 0.133855i −0.824795 0.565432i \(-0.808709\pi\)
0.902076 + 0.431577i \(0.142043\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8043.00 13930.9i 0.812581 1.40743i −0.0984709 0.995140i \(-0.531395\pi\)
0.911052 0.412292i \(-0.135272\pi\)
\(462\) 0 0
\(463\) −2692.00 4662.68i −0.270211 0.468020i 0.698704 0.715410i \(-0.253760\pi\)
−0.968916 + 0.247391i \(0.920427\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2604.00 0.258027 0.129014 0.991643i \(-0.458819\pi\)
0.129014 + 0.991643i \(0.458819\pi\)
\(468\) 0 0
\(469\) −28544.0 −2.81032
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −936.000 1621.20i −0.0909880 0.157596i
\(474\) 0 0
\(475\) −1750.00 + 3031.09i −0.169043 + 0.292791i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5568.00 9644.06i 0.531124 0.919934i −0.468216 0.883614i \(-0.655103\pi\)
0.999340 0.0363199i \(-0.0115635\pi\)
\(480\) 0 0
\(481\) −170.000 294.449i −0.0161150 0.0279121i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −970.000 −0.0908153
\(486\) 0 0
\(487\) 14624.0 1.36073 0.680366 0.732872i \(-0.261821\pi\)
0.680366 + 0.732872i \(0.261821\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5922.00 + 10257.2i 0.544310 + 0.942772i 0.998650 + 0.0519440i \(0.0165417\pi\)
−0.454340 + 0.890828i \(0.650125\pi\)
\(492\) 0 0
\(493\) 234.000 405.300i 0.0213769 0.0370259i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1920.00 + 3325.54i −0.173287 + 0.300142i
\(498\) 0 0
\(499\) 5642.00 + 9772.23i 0.506154 + 0.876684i 0.999975 + 0.00712011i \(0.00226642\pi\)
−0.493821 + 0.869564i \(0.664400\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4032.00 −0.357412 −0.178706 0.983903i \(-0.557191\pi\)
−0.178706 + 0.983903i \(0.557191\pi\)
\(504\) 0 0
\(505\) 3990.00 0.351589
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −8781.00 15209.1i −0.764658 1.32443i −0.940427 0.339995i \(-0.889575\pi\)
0.175769 0.984431i \(-0.443759\pi\)
\(510\) 0 0
\(511\) 10336.0 17902.5i 0.894790 1.54982i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 680.000 1177.79i 0.0581833 0.100776i
\(516\) 0 0
\(517\) −7344.00 12720.2i −0.624736 1.08208i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3162.00 −0.265892 −0.132946 0.991123i \(-0.542444\pi\)
−0.132946 + 0.991123i \(0.542444\pi\)
\(522\) 0 0
\(523\) 6764.00 0.565524 0.282762 0.959190i \(-0.408749\pi\)
0.282762 + 0.959190i \(0.408749\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 624.000 + 1080.80i 0.0515785 + 0.0893366i
\(528\) 0 0
\(529\) −12348.5 + 21388.2i −1.01492 + 1.75789i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1950.00 3377.50i 0.158469 0.274476i
\(534\) 0 0
\(535\) −390.000 675.500i −0.0315162 0.0545877i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −24516.0 −1.95914
\(540\) 0 0
\(541\) 17798.0 1.41441 0.707205 0.707009i \(-0.249956\pi\)
0.707205 + 0.707009i \(0.249956\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4055.00 + 7023.47i 0.318710 + 0.552022i
\(546\) 0 0
\(547\) 9998.00 17317.0i 0.781506 1.35361i −0.149559 0.988753i \(-0.547785\pi\)
0.931064 0.364855i \(-0.118881\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 420.000 727.461i 0.0324730 0.0562448i
\(552\) 0 0
\(553\) 18688.0 + 32368.6i 1.43706 + 2.48906i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −11094.0 −0.843928 −0.421964 0.906613i \(-0.638659\pi\)
−0.421964 + 0.906613i \(0.638659\pi\)
\(558\) 0 0
\(559\) 520.000 0.0393446
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 450.000 + 779.423i 0.0336860 + 0.0583459i 0.882377 0.470543i \(-0.155942\pi\)
−0.848691 + 0.528889i \(0.822609\pi\)
\(564\) 0 0
\(565\) −2685.00 + 4650.56i −0.199927 + 0.346284i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3957.00 6853.73i 0.291540 0.504962i −0.682634 0.730760i \(-0.739166\pi\)
0.974174 + 0.225799i \(0.0724991\pi\)
\(570\) 0 0
\(571\) 1190.00 + 2061.14i 0.0872153 + 0.151061i 0.906333 0.422564i \(-0.138870\pi\)
−0.819118 + 0.573625i \(0.805537\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4800.00 0.348128
\(576\) 0 0
\(577\) −25726.0 −1.85613 −0.928065 0.372417i \(-0.878529\pi\)
−0.928065 + 0.372417i \(0.878529\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −11712.0 20285.8i −0.836309 1.44853i
\(582\) 0 0
\(583\) 2052.00 3554.17i 0.145772 0.252485i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1806.00 3128.08i 0.126987 0.219949i −0.795521 0.605926i \(-0.792803\pi\)
0.922508 + 0.385978i \(0.126136\pi\)
\(588\) 0 0
\(589\) 1120.00 + 1939.90i 0.0783511 + 0.135708i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −2898.00 −0.200686 −0.100343 0.994953i \(-0.531994\pi\)
−0.100343 + 0.994953i \(0.531994\pi\)
\(594\) 0 0
\(595\) −12480.0 −0.859883
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1332.00 + 2307.09i 0.0908582 + 0.157371i 0.907872 0.419246i \(-0.137706\pi\)
−0.817014 + 0.576617i \(0.804372\pi\)
\(600\) 0 0
\(601\) 251.000 434.745i 0.0170358 0.0295068i −0.857382 0.514681i \(-0.827910\pi\)
0.874418 + 0.485174i \(0.161244\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −87.5000 + 151.554i −0.00587997 + 0.0101844i
\(606\) 0 0
\(607\) −3988.00 6907.42i −0.266669 0.461884i 0.701331 0.712836i \(-0.252590\pi\)
−0.967999 + 0.250952i \(0.919256\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4080.00 0.270146
\(612\) 0 0
\(613\) 20414.0 1.34505 0.672523 0.740076i \(-0.265210\pi\)
0.672523 + 0.740076i \(0.265210\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3171.00 5492.33i −0.206904 0.358368i 0.743834 0.668365i \(-0.233005\pi\)
−0.950738 + 0.309997i \(0.899672\pi\)
\(618\) 0 0
\(619\) −11338.0 + 19638.0i −0.736208 + 1.27515i 0.217983 + 0.975952i \(0.430052\pi\)
−0.954191 + 0.299197i \(0.903281\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −25440.0 + 44063.4i −1.63601 + 2.83365i
\(624\) 0 0
\(625\) −312.500 541.266i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2652.00 −0.168112
\(630\) 0 0
\(631\) −7048.00 −0.444654 −0.222327 0.974972i \(-0.571365\pi\)
−0.222327 + 0.974972i \(0.571365\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −3820.00 6616.43i −0.238728 0.413488i
\(636\) 0 0
\(637\) 3405.00 5897.63i 0.211791 0.366833i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −10143.0 + 17568.2i −0.624999 + 1.08253i 0.363542 + 0.931578i \(0.381567\pi\)
−0.988541 + 0.150952i \(0.951766\pi\)
\(642\) 0 0
\(643\) 8054.00 + 13949.9i 0.493964 + 0.855570i 0.999976 0.00695598i \(-0.00221417\pi\)
−0.506012 + 0.862526i \(0.668881\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −27456.0 −1.66833 −0.834163 0.551518i \(-0.814049\pi\)
−0.834163 + 0.551518i \(0.814049\pi\)
\(648\) 0 0
\(649\) 18576.0 1.12353
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6261.00 10844.4i −0.375210 0.649882i 0.615149 0.788411i \(-0.289096\pi\)
−0.990358 + 0.138529i \(0.955763\pi\)
\(654\) 0 0
\(655\) −6030.00 + 10444.3i −0.359712 + 0.623040i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −8154.00 + 14123.1i −0.481995 + 0.834840i −0.999786 0.0206670i \(-0.993421\pi\)
0.517791 + 0.855507i \(0.326754\pi\)
\(660\) 0 0
\(661\) −16039.0 27780.4i −0.943789 1.63469i −0.758157 0.652072i \(-0.773900\pi\)
−0.185632 0.982619i \(-0.559433\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −22400.0 −1.30622
\(666\) 0 0
\(667\) −1152.00 −0.0668750
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1044.00 1808.26i −0.0600643 0.104034i
\(672\) 0 0
\(673\) −2305.00 + 3992.38i −0.132023 + 0.228670i −0.924456 0.381288i \(-0.875480\pi\)
0.792434 + 0.609958i \(0.208814\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5391.00 9337.49i 0.306046 0.530087i −0.671448 0.741052i \(-0.734327\pi\)
0.977494 + 0.210965i \(0.0676606\pi\)
\(678\) 0 0
\(679\) −3104.00 5376.29i −0.175435 0.303863i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −2892.00 −0.162019 −0.0810097 0.996713i \(-0.525814\pi\)
−0.0810097 + 0.996713i \(0.525814\pi\)
\(684\) 0 0
\(685\) 10530.0 0.587344
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 570.000 + 987.269i 0.0315171 + 0.0545892i
\(690\) 0 0
\(691\) 14786.0 25610.1i 0.814017 1.40992i −0.0960141 0.995380i \(-0.530609\pi\)
0.910031 0.414539i \(-0.136057\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1390.00 + 2407.55i −0.0758643 + 0.131401i
\(696\) 0 0
\(697\) −15210.0 26344.5i −0.826571 1.43166i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −5766.00 −0.310669 −0.155334 0.987862i \(-0.549646\pi\)
−0.155334 + 0.987862i \(0.549646\pi\)
\(702\) 0 0
\(703\) −4760.00 −0.255372
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12768.0 + 22114.8i 0.679194 + 1.17640i
\(708\) 0 0
\(709\) −1663.00 + 2880.40i −0.0880892 + 0.152575i −0.906703 0.421769i \(-0.861409\pi\)
0.818614 + 0.574344i \(0.194743\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1536.00 2660.43i 0.0806783 0.139739i
\(714\) 0 0
\(715\) 900.000 + 1558.85i 0.0470743 + 0.0815350i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 7728.00 0.400843 0.200421 0.979710i \(-0.435769\pi\)
0.200421 + 0.979710i \(0.435769\pi\)
\(720\) 0 0
\(721\) 8704.00 0.449589
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 75.0000 + 129.904i 0.00384197 + 0.00665449i
\(726\) 0 0
\(727\) 10808.0 18720.0i 0.551371 0.955002i −0.446805 0.894631i \(-0.647438\pi\)
0.998176 0.0603709i \(-0.0192283\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2028.00 3512.60i 0.102611 0.177727i
\(732\) 0 0
\(733\) −5059.00 8762.45i −0.254923 0.441539i 0.709952 0.704250i \(-0.248717\pi\)
−0.964875 + 0.262711i \(0.915383\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 32112.0 1.60497
\(738\) 0 0
\(739\) 10460.0 0.520673 0.260336 0.965518i \(-0.416167\pi\)
0.260336 + 0.965518i \(0.416167\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 8616.00 + 14923.3i 0.425424 + 0.736857i 0.996460 0.0840686i \(-0.0267915\pi\)
−0.571035 + 0.820925i \(0.693458\pi\)
\(744\) 0 0
\(745\) 6045.00 10470.2i 0.297277 0.514900i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2496.00 4323.20i 0.121765 0.210903i
\(750\) 0 0
\(751\) −13456.0 23306.5i −0.653817 1.13244i −0.982189 0.187896i \(-0.939833\pi\)
0.328372 0.944548i \(-0.393500\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −14200.0 −0.684491
\(756\) 0 0
\(757\) 13838.0 0.664400 0.332200 0.943209i \(-0.392209\pi\)
0.332200 + 0.943209i \(0.392209\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −8619.00 14928.5i −0.410563 0.711116i 0.584388 0.811474i \(-0.301335\pi\)
−0.994951 + 0.100358i \(0.968001\pi\)
\(762\) 0 0
\(763\) −25952.0 + 44950.2i −1.23136 + 2.13277i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2580.00 + 4468.69i −0.121458 + 0.210372i
\(768\) 0 0
\(769\) −10849.0 18791.0i −0.508745 0.881172i −0.999949 0.0101275i \(-0.996776\pi\)
0.491204 0.871045i \(-0.336557\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −18366.0 −0.854565 −0.427283 0.904118i \(-0.640529\pi\)
−0.427283 + 0.904118i \(0.640529\pi\)
\(774\) 0 0
\(775\) −400.000 −0.0185399
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −27300.0 47285.0i −1.25561 2.17479i
\(780\) 0 0
\(781\) 2160.00 3741.23i 0.0989640 0.171411i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 5135.00 8894.08i 0.233473 0.404386i
\(786\) 0 0
\(787\) 15158.0 + 26254.4i 0.686562 + 1.18916i 0.972943 + 0.231045i \(0.0742143\pi\)
−0.286381 + 0.958116i \(0.592452\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −34368.0 −1.54486
\(792\) 0 0
\(793\) 580.000 0.0259728
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3747.00 + 6489.99i 0.166531 + 0.288441i 0.937198 0.348798i \(-0.113410\pi\)
−0.770667 + 0.637239i \(0.780077\pi\)
\(798\) 0 0
\(799\) 15912.0 27560.4i 0.704538 1.22030i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11628.0 + 20140.3i −0.511013 + 0.885100i
\(804\) 0 0
\(805\) 15360.0 + 26604.3i 0.672508 + 1.16482i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 11526.0 0.500906 0.250453 0.968129i \(-0.419421\pi\)
0.250453 + 0.968129i \(0.419421\pi\)
\(810\) 0 0
\(811\) −33820.0 −1.46434 −0.732171 0.681121i \(-0.761493\pi\)
−0.732171 + 0.681121i \(0.761493\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1150.00 1991.86i −0.0494267 0.0856095i
\(816\) 0 0
\(817\) 3640.00 6304.66i 0.155872 0.269978i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 9783.00 16944.7i 0.415870 0.720308i −0.579650 0.814866i \(-0.696811\pi\)
0.995519 + 0.0945583i \(0.0301439\pi\)
\(822\) 0 0
\(823\) 20048.0 + 34724.2i 0.849124 + 1.47073i 0.881991 + 0.471267i \(0.156203\pi\)
−0.0328664 + 0.999460i \(0.510464\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −31884.0 −1.34065 −0.670324 0.742069i \(-0.733845\pi\)
−0.670324 + 0.742069i \(0.733845\pi\)
\(828\) 0 0
\(829\) −24442.0 −1.02401 −0.512006 0.858982i \(-0.671097\pi\)
−0.512006 + 0.858982i \(0.671097\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −26559.0 46001.5i −1.10470 1.91340i
\(834\) 0 0
\(835\) −5040.00 + 8729.54i −0.208882 + 0.361794i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −15972.0 + 27664.3i −0.657228 + 1.13835i 0.324102 + 0.946022i \(0.394938\pi\)
−0.981330 + 0.192331i \(0.938395\pi\)
\(840\) 0 0
\(841\) 12176.5 + 21090.3i 0.499262 + 0.864747i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 10485.0 0.426858
\(846\) 0 0
\(847\) −1120.00 −0.0454352
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3264.00 + 5653.41i 0.131479 + 0.227728i
\(852\) 0 0
\(853\) −8743.00 + 15143.3i −0.350943 + 0.607852i −0.986415 0.164273i \(-0.947472\pi\)
0.635472 + 0.772124i \(0.280806\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 21717.0 37614.9i 0.865623 1.49930i −0.000804999 1.00000i \(-0.500256\pi\)
0.866428 0.499303i \(-0.166410\pi\)
\(858\) 0 0
\(859\) −5410.00 9370.39i −0.214886 0.372193i 0.738352 0.674416i \(-0.235605\pi\)
−0.953237 + 0.302223i \(0.902271\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −29976.0 −1.18238 −0.591191 0.806532i \(-0.701342\pi\)
−0.591191 + 0.806532i \(0.701342\pi\)
\(864\) 0 0
\(865\) −3090.00 −0.121460
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −21024.0 36414.6i −0.820702 1.42150i
\(870\) 0 0
\(871\) −4460.00 + 7724.95i −0.173503 + 0.300516i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2000.00 3464.10i 0.0772712 0.133838i
\(876\) 0 0
\(877\) 20261.0 + 35093.1i 0.780120 + 1.35121i 0.931871 + 0.362789i \(0.118175\pi\)
−0.151751 + 0.988419i \(0.548491\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −15570.0 −0.595422 −0.297711 0.954656i \(-0.596223\pi\)
−0.297711 + 0.954656i \(0.596223\pi\)
\(882\) 0 0
\(883\) −1084.00 −0.0413131 −0.0206566 0.999787i \(-0.506576\pi\)
−0.0206566 + 0.999787i \(0.506576\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −4104.00 7108.34i −0.155354 0.269081i 0.777834 0.628470i \(-0.216318\pi\)
−0.933188 + 0.359389i \(0.882985\pi\)
\(888\) 0 0
\(889\) 24448.0 42345.2i 0.922339 1.59754i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 28560.0 49467.4i 1.07024 1.85371i
\(894\) 0 0
\(895\) 7410.00 + 12834.5i 0.276747 + 0.479341i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 96.0000 0.00356149
\(900\) 0 0
\(901\) 8892.00 0.328785
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −925.000 1602.15i −0.0339757 0.0588477i
\(906\) 0 0
\(907\) −17038.0 + 29510.7i −0.623746 + 1.08036i 0.365036 + 0.930993i \(0.381057\pi\)
−0.988782 + 0.149366i \(0.952277\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −7536.00 + 13052.7i −0.274071 + 0.474705i −0.969900 0.243502i \(-0.921704\pi\)
0.695829 + 0.718207i \(0.255037\pi\)
\(912\) 0 0
\(913\) 13176.0 + 22821.5i 0.477614 + 0.827252i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −77184.0 −2.77954
\(918\) 0 0
\(919\) 24392.0 0.875536 0.437768 0.899088i \(-0.355769\pi\)
0.437768 + 0.899088i \(0.355769\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 600.000 + 1039.23i 0.0213968 + 0.0370603i
\(924\) 0 0
\(925\) 425.000 736.122i 0.0151069 0.0261660i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 6801.00 11779.7i 0.240187 0.416016i −0.720581 0.693371i \(-0.756125\pi\)
0.960767 + 0.277355i \(0.0894580\pi\)
\(930\) 0 0
\(931\) −47670.0 82566.9i −1.67811 2.90657i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 14040.0 0.491077
\(936\) 0 0
\(937\) −47974.0 −1.67262 −0.836309 0.548259i \(-0.815291\pi\)
−0.836309 + 0.548259i \(0.815291\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −24165.0 41855.0i −0.837148 1.44998i −0.892269 0.451504i \(-0.850888\pi\)
0.0551211 0.998480i \(-0.482446\pi\)
\(942\) 0 0
\(943\) −37440.0 + 64848.0i −1.29291 + 2.23939i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4662.00 + 8074.82i −0.159973 + 0.277082i −0.934859 0.355020i \(-0.884474\pi\)
0.774886 + 0.632102i \(0.217807\pi\)
\(948\) 0 0
\(949\) −3230.00 5594.52i −0.110485 0.191366i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 14838.0 0.504355 0.252177 0.967681i \(-0.418853\pi\)
0.252177 + 0.967681i \(0.418853\pi\)
\(954\) 0 0
\(955\) 5520.00 0.187040
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 33696.0 + 58363.2i 1.13462 + 1.96522i
\(960\) 0 0
\(961\) 14767.5 25578.1i 0.495703 0.858583i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −5995.00 + 10383.6i −0.199985 + 0.346385i
\(966\) 0 0
\(967\) −5680.00 9838.05i −0.188890 0.327167i 0.755991 0.654583i \(-0.227156\pi\)
−0.944880 + 0.327416i \(0.893822\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 6972.00 0.230424 0.115212 0.993341i \(-0.463245\pi\)
0.115212 + 0.993341i \(0.463245\pi\)
\(972\) 0 0
\(973\) −17792.0 −0.586213
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −20583.0 35650.8i −0.674011 1.16742i −0.976757 0.214350i \(-0.931237\pi\)
0.302746 0.953071i \(-0.402097\pi\)
\(978\) 0 0
\(979\) 28620.0 49571.3i 0.934320 1.61829i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 11232.0 19454.4i 0.364441 0.631230i −0.624246 0.781228i \(-0.714594\pi\)
0.988686 + 0.149998i \(0.0479269\pi\)
\(984\) 0 0
\(985\) −3195.00 5533.90i −0.103351 0.179010i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −9984.00 −0.321004
\(990\) 0 0
\(991\) −10192.0 −0.326700 −0.163350 0.986568i \(-0.552230\pi\)
−0.163350 + 0.986568i \(0.552230\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 11180.0 + 19364.3i 0.356211 + 0.616975i
\(996\) 0 0
\(997\) 161.000 278.860i 0.00511426 0.00885817i −0.863457 0.504423i \(-0.831705\pi\)
0.868571 + 0.495564i \(0.165039\pi\)
\(998\) 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 1620.4.i.g.541.1 2
3.2 odd 2 1620.4.i.a.541.1 2
9.2 odd 6 60.4.a.b.1.1 1
9.4 even 3 inner 1620.4.i.g.1081.1 2
9.5 odd 6 1620.4.i.a.1081.1 2
9.7 even 3 180.4.a.c.1.1 1
36.7 odd 6 720.4.a.c.1.1 1
36.11 even 6 240.4.a.j.1.1 1
45.2 even 12 300.4.d.d.49.2 2
45.7 odd 12 900.4.d.b.649.2 2
45.29 odd 6 300.4.a.e.1.1 1
45.34 even 6 900.4.a.b.1.1 1
45.38 even 12 300.4.d.d.49.1 2
45.43 odd 12 900.4.d.b.649.1 2
72.11 even 6 960.4.a.a.1.1 1
72.29 odd 6 960.4.a.bb.1.1 1
180.47 odd 12 1200.4.f.e.49.1 2
180.83 odd 12 1200.4.f.e.49.2 2
180.119 even 6 1200.4.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.4.a.b.1.1 1 9.2 odd 6
180.4.a.c.1.1 1 9.7 even 3
240.4.a.j.1.1 1 36.11 even 6
300.4.a.e.1.1 1 45.29 odd 6
300.4.d.d.49.1 2 45.38 even 12
300.4.d.d.49.2 2 45.2 even 12
720.4.a.c.1.1 1 36.7 odd 6
900.4.a.b.1.1 1 45.34 even 6
900.4.d.b.649.1 2 45.43 odd 12
900.4.d.b.649.2 2 45.7 odd 12
960.4.a.a.1.1 1 72.11 even 6
960.4.a.bb.1.1 1 72.29 odd 6
1200.4.a.s.1.1 1 180.119 even 6
1200.4.f.e.49.1 2 180.47 odd 12
1200.4.f.e.49.2 2 180.83 odd 12
1620.4.i.a.541.1 2 3.2 odd 2
1620.4.i.a.1081.1 2 9.5 odd 6
1620.4.i.g.541.1 2 1.1 even 1 trivial
1620.4.i.g.1081.1 2 9.4 even 3 inner