Properties

Label 2880.2.w.p.2177.5
Level $2880$
Weight $2$
Character 2880.2177
Analytic conductor $22.997$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2880,2,Mod(2177,2880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2880, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2880.2177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.w (of order \(4\), 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: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 27x^{8} + 107x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 1440)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 2177.5
Root \(-0.219986 - 0.219986i\) of defining polynomial
Character \(\chi\) \(=\) 2880.2177
Dual form 2880.2.w.p.2753.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.34577 + 1.78575i) q^{5} +(-2.90321 - 2.90321i) q^{7} +O(q^{10})\) \(q+(1.34577 + 1.78575i) q^{5} +(-2.90321 - 2.90321i) q^{7} -2.15728i q^{11} +(-1.90321 + 1.90321i) q^{13} +(0.974241 - 0.974241i) q^{17} +1.80642i q^{19} +(0.879946 + 0.879946i) q^{23} +(-1.37778 + 4.80642i) q^{25} -0.743067 q^{29} -7.05086 q^{31} +(1.27733 - 9.09147i) q^{35} +(8.33185 + 8.33185i) q^{37} +7.67726i q^{41} +(3.05086 - 3.05086i) q^{43} +(-7.54038 + 7.54038i) q^{47} +9.85728i q^{49} +(-5.08000 - 5.08000i) q^{53} +(3.85236 - 2.90321i) q^{55} +14.2658 q^{59} -4.56199 q^{61} +(-5.95995 - 0.837362i) q^{65} +(7.05086 + 7.05086i) q^{67} +12.5261i q^{71} +(3.62222 - 3.62222i) q^{73} +(-6.26304 + 6.26304i) q^{77} +15.0509i q^{79} +(-7.81413 - 7.81413i) q^{83} +(3.05086 + 0.428639i) q^{85} -7.40350 q^{89} +11.0509 q^{91} +(-3.22582 + 2.43104i) q^{95} +(6.67307 + 6.67307i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{7} + 4 q^{13} - 16 q^{25} - 32 q^{31} + 20 q^{37} - 16 q^{43} + 72 q^{55} + 32 q^{67} + 44 q^{73} - 16 q^{85} + 80 q^{91} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.34577 + 1.78575i 0.601848 + 0.798610i
\(6\) 0 0
\(7\) −2.90321 2.90321i −1.09731 1.09731i −0.994724 0.102587i \(-0.967288\pi\)
−0.102587 0.994724i \(-0.532712\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.15728i 0.650444i −0.945638 0.325222i \(-0.894561\pi\)
0.945638 0.325222i \(-0.105439\pi\)
\(12\) 0 0
\(13\) −1.90321 + 1.90321i −0.527856 + 0.527856i −0.919933 0.392077i \(-0.871757\pi\)
0.392077 + 0.919933i \(0.371757\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.974241 0.974241i 0.236288 0.236288i −0.579023 0.815311i \(-0.696566\pi\)
0.815311 + 0.579023i \(0.196566\pi\)
\(18\) 0 0
\(19\) 1.80642i 0.414422i 0.978296 + 0.207211i \(0.0664387\pi\)
−0.978296 + 0.207211i \(0.933561\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.879946 + 0.879946i 0.183481 + 0.183481i 0.792871 0.609390i \(-0.208585\pi\)
−0.609390 + 0.792871i \(0.708585\pi\)
\(24\) 0 0
\(25\) −1.37778 + 4.80642i −0.275557 + 0.961285i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.743067 −0.137984 −0.0689920 0.997617i \(-0.521978\pi\)
−0.0689920 + 0.997617i \(0.521978\pi\)
\(30\) 0 0
\(31\) −7.05086 −1.26637 −0.633185 0.774000i \(-0.718253\pi\)
−0.633185 + 0.774000i \(0.718253\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.27733 9.09147i 0.215909 1.53674i
\(36\) 0 0
\(37\) 8.33185 + 8.33185i 1.36975 + 1.36975i 0.860793 + 0.508955i \(0.169968\pi\)
0.508955 + 0.860793i \(0.330032\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.67726i 1.19899i 0.800380 + 0.599493i \(0.204631\pi\)
−0.800380 + 0.599493i \(0.795369\pi\)
\(42\) 0 0
\(43\) 3.05086 3.05086i 0.465251 0.465251i −0.435121 0.900372i \(-0.643294\pi\)
0.900372 + 0.435121i \(0.143294\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.54038 + 7.54038i −1.09988 + 1.09988i −0.105453 + 0.994424i \(0.533629\pi\)
−0.994424 + 0.105453i \(0.966371\pi\)
\(48\) 0 0
\(49\) 9.85728i 1.40818i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.08000 5.08000i −0.697792 0.697792i 0.266142 0.963934i \(-0.414251\pi\)
−0.963934 + 0.266142i \(0.914251\pi\)
\(54\) 0 0
\(55\) 3.85236 2.90321i 0.519452 0.391469i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 14.2658 1.85724 0.928622 0.371026i \(-0.120994\pi\)
0.928622 + 0.371026i \(0.120994\pi\)
\(60\) 0 0
\(61\) −4.56199 −0.584103 −0.292052 0.956403i \(-0.594338\pi\)
−0.292052 + 0.956403i \(0.594338\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.95995 0.837362i −0.739241 0.103862i
\(66\) 0 0
\(67\) 7.05086 + 7.05086i 0.861399 + 0.861399i 0.991501 0.130102i \(-0.0415304\pi\)
−0.130102 + 0.991501i \(0.541530\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.5261i 1.48657i 0.668973 + 0.743286i \(0.266734\pi\)
−0.668973 + 0.743286i \(0.733266\pi\)
\(72\) 0 0
\(73\) 3.62222 3.62222i 0.423948 0.423948i −0.462612 0.886561i \(-0.653088\pi\)
0.886561 + 0.462612i \(0.153088\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.26304 + 6.26304i −0.713740 + 0.713740i
\(78\) 0 0
\(79\) 15.0509i 1.69335i 0.532108 + 0.846677i \(0.321400\pi\)
−0.532108 + 0.846677i \(0.678600\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.81413 7.81413i −0.857713 0.857713i 0.133356 0.991068i \(-0.457425\pi\)
−0.991068 + 0.133356i \(0.957425\pi\)
\(84\) 0 0
\(85\) 3.05086 + 0.428639i 0.330912 + 0.0464925i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.40350 −0.784769 −0.392385 0.919801i \(-0.628350\pi\)
−0.392385 + 0.919801i \(0.628350\pi\)
\(90\) 0 0
\(91\) 11.0509 1.15844
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.22582 + 2.43104i −0.330962 + 0.249419i
\(96\) 0 0
\(97\) 6.67307 + 6.67307i 0.677548 + 0.677548i 0.959445 0.281897i \(-0.0909636\pi\)
−0.281897 + 0.959445i \(0.590964\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.27987i 0.724374i 0.932105 + 0.362187i \(0.117970\pi\)
−0.932105 + 0.362187i \(0.882030\pi\)
\(102\) 0 0
\(103\) 8.70964 8.70964i 0.858186 0.858186i −0.132938 0.991124i \(-0.542441\pi\)
0.991124 + 0.132938i \(0.0424412\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.49957 + 3.49957i −0.338317 + 0.338317i −0.855733 0.517417i \(-0.826894\pi\)
0.517417 + 0.855733i \(0.326894\pi\)
\(108\) 0 0
\(109\) 15.9081i 1.52372i 0.647740 + 0.761861i \(0.275714\pi\)
−0.647740 + 0.761861i \(0.724286\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −10.6000 10.6000i −0.997162 0.997162i 0.00283397 0.999996i \(-0.499098\pi\)
−0.999996 + 0.00283397i \(0.999098\pi\)
\(114\) 0 0
\(115\) −0.387152 + 2.75557i −0.0361021 + 0.256958i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.65685 −0.518563
\(120\) 0 0
\(121\) 6.34614 0.576922
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −10.4372 + 4.00799i −0.933535 + 0.358485i
\(126\) 0 0
\(127\) −10.9032 10.9032i −0.967504 0.967504i 0.0319847 0.999488i \(-0.489817\pi\)
−0.999488 + 0.0319847i \(0.989817\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 11.7111i 1.02320i 0.859223 + 0.511602i \(0.170948\pi\)
−0.859223 + 0.511602i \(0.829052\pi\)
\(132\) 0 0
\(133\) 5.24443 5.24443i 0.454750 0.454750i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.82307 + 5.82307i −0.497498 + 0.497498i −0.910658 0.413160i \(-0.864425\pi\)
0.413160 + 0.910658i \(0.364425\pi\)
\(138\) 0 0
\(139\) 3.14272i 0.266562i −0.991078 0.133281i \(-0.957449\pi\)
0.991078 0.133281i \(-0.0425513\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.10576 + 4.10576i 0.343341 + 0.343341i
\(144\) 0 0
\(145\) −1.00000 1.32693i −0.0830455 0.110195i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.31475 −0.517325 −0.258662 0.965968i \(-0.583282\pi\)
−0.258662 + 0.965968i \(0.583282\pi\)
\(150\) 0 0
\(151\) −18.3684 −1.49480 −0.747400 0.664374i \(-0.768698\pi\)
−0.747400 + 0.664374i \(0.768698\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −9.48886 12.5910i −0.762163 1.01134i
\(156\) 0 0
\(157\) −0.0365650 0.0365650i −0.00291821 0.00291821i 0.705646 0.708564i \(-0.250657\pi\)
−0.708564 + 0.705646i \(0.750657\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.10934i 0.402672i
\(162\) 0 0
\(163\) −7.05086 + 7.05086i −0.552266 + 0.552266i −0.927094 0.374829i \(-0.877702\pi\)
0.374829 + 0.927094i \(0.377702\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.09147 9.09147i 0.703519 0.703519i −0.261645 0.965164i \(-0.584265\pi\)
0.965164 + 0.261645i \(0.0842650\pi\)
\(168\) 0 0
\(169\) 5.75557i 0.442736i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −12.3599 12.3599i −0.939703 0.939703i 0.0585793 0.998283i \(-0.481343\pi\)
−0.998283 + 0.0585793i \(0.981343\pi\)
\(174\) 0 0
\(175\) 17.9541 9.95407i 1.35720 0.752457i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10.9163 0.815924 0.407962 0.912999i \(-0.366240\pi\)
0.407962 + 0.912999i \(0.366240\pi\)
\(180\) 0 0
\(181\) 19.5210 1.45098 0.725492 0.688231i \(-0.241612\pi\)
0.725492 + 0.688231i \(0.241612\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −3.66579 + 26.0914i −0.269514 + 1.91828i
\(186\) 0 0
\(187\) −2.10171 2.10171i −0.153692 0.153692i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 23.8398i 1.72499i −0.506068 0.862493i \(-0.668902\pi\)
0.506068 0.862493i \(-0.331098\pi\)
\(192\) 0 0
\(193\) −9.66370 + 9.66370i −0.695609 + 0.695609i −0.963460 0.267852i \(-0.913686\pi\)
0.267852 + 0.963460i \(0.413686\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 9.99379 9.99379i 0.712028 0.712028i −0.254931 0.966959i \(-0.582053\pi\)
0.966959 + 0.254931i \(0.0820528\pi\)
\(198\) 0 0
\(199\) 14.7556i 1.04599i 0.852334 + 0.522997i \(0.175186\pi\)
−0.852334 + 0.522997i \(0.824814\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.15728 + 2.15728i 0.151411 + 0.151411i
\(204\) 0 0
\(205\) −13.7096 + 10.3319i −0.957522 + 0.721608i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.89696 0.269559
\(210\) 0 0
\(211\) −8.47013 −0.583108 −0.291554 0.956554i \(-0.594172\pi\)
−0.291554 + 0.956554i \(0.594172\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 9.55382 + 1.34229i 0.651565 + 0.0915437i
\(216\) 0 0
\(217\) 20.4701 + 20.4701i 1.38960 + 1.38960i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.70837i 0.249452i
\(222\) 0 0
\(223\) −5.56691 + 5.56691i −0.372788 + 0.372788i −0.868492 0.495704i \(-0.834910\pi\)
0.495704 + 0.868492i \(0.334910\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.97141 + 9.97141i −0.661826 + 0.661826i −0.955810 0.293984i \(-0.905019\pi\)
0.293984 + 0.955810i \(0.405019\pi\)
\(228\) 0 0
\(229\) 15.1240i 0.999421i 0.866192 + 0.499711i \(0.166560\pi\)
−0.866192 + 0.499711i \(0.833440\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4.07643 4.07643i −0.267056 0.267056i 0.560857 0.827913i \(-0.310472\pi\)
−0.827913 + 0.560857i \(0.810472\pi\)
\(234\) 0 0
\(235\) −23.6128 3.31756i −1.54033 0.216414i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −0.547516 −0.0354159 −0.0177079 0.999843i \(-0.505637\pi\)
−0.0177079 + 0.999843i \(0.505637\pi\)
\(240\) 0 0
\(241\) −22.4701 −1.44743 −0.723714 0.690100i \(-0.757567\pi\)
−0.723714 + 0.690100i \(0.757567\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −17.6026 + 13.2657i −1.12459 + 0.847513i
\(246\) 0 0
\(247\) −3.43801 3.43801i −0.218755 0.218755i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 8.23173i 0.519582i 0.965665 + 0.259791i \(0.0836537\pi\)
−0.965665 + 0.259791i \(0.916346\pi\)
\(252\) 0 0
\(253\) 1.89829 1.89829i 0.119344 0.119344i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.61069 4.61069i 0.287607 0.287607i −0.548526 0.836133i \(-0.684811\pi\)
0.836133 + 0.548526i \(0.184811\pi\)
\(258\) 0 0
\(259\) 48.3783i 3.00608i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7.74918 7.74918i −0.477835 0.477835i 0.426604 0.904439i \(-0.359710\pi\)
−0.904439 + 0.426604i \(0.859710\pi\)
\(264\) 0 0
\(265\) 2.23506 15.9081i 0.137299 0.977229i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 13.3139 0.811763 0.405881 0.913926i \(-0.366965\pi\)
0.405881 + 0.913926i \(0.366965\pi\)
\(270\) 0 0
\(271\) 20.0830 1.21995 0.609977 0.792419i \(-0.291179\pi\)
0.609977 + 0.792419i \(0.291179\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.3688 + 2.97227i 0.625262 + 0.179234i
\(276\) 0 0
\(277\) 3.90321 + 3.90321i 0.234521 + 0.234521i 0.814577 0.580056i \(-0.196969\pi\)
−0.580056 + 0.814577i \(0.696969\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.60280i 0.0956152i −0.998857 0.0478076i \(-0.984777\pi\)
0.998857 0.0478076i \(-0.0152234\pi\)
\(282\) 0 0
\(283\) −18.1017 + 18.1017i −1.07603 + 1.07603i −0.0791742 + 0.996861i \(0.525228\pi\)
−0.996861 + 0.0791742i \(0.974772\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 22.2887 22.2887i 1.31566 1.31566i
\(288\) 0 0
\(289\) 15.1017i 0.888336i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 18.9553 + 18.9553i 1.10738 + 1.10738i 0.993493 + 0.113889i \(0.0363310\pi\)
0.113889 + 0.993493i \(0.463669\pi\)
\(294\) 0 0
\(295\) 19.1985 + 25.4750i 1.11778 + 1.48321i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.34945 −0.193704
\(300\) 0 0
\(301\) −17.7146 −1.02105
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −6.13941 8.14656i −0.351542 0.466471i
\(306\) 0 0
\(307\) 6.19358 + 6.19358i 0.353486 + 0.353486i 0.861405 0.507919i \(-0.169585\pi\)
−0.507919 + 0.861405i \(0.669585\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 17.3882i 0.985992i −0.870032 0.492996i \(-0.835902\pi\)
0.870032 0.492996i \(-0.164098\pi\)
\(312\) 0 0
\(313\) 15.2953 15.2953i 0.864541 0.864541i −0.127321 0.991862i \(-0.540638\pi\)
0.991862 + 0.127321i \(0.0406378\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.3430 11.3430i 0.637089 0.637089i −0.312747 0.949836i \(-0.601249\pi\)
0.949836 + 0.312747i \(0.101249\pi\)
\(318\) 0 0
\(319\) 1.60300i 0.0897510i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.75989 + 1.75989i 0.0979230 + 0.0979230i
\(324\) 0 0
\(325\) −6.52543 11.7699i −0.361966 0.652874i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 43.7826 2.41381
\(330\) 0 0
\(331\) 18.9304 1.04051 0.520255 0.854011i \(-0.325837\pi\)
0.520255 + 0.854011i \(0.325837\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3.10219 + 22.0799i −0.169490 + 1.20635i
\(336\) 0 0
\(337\) −4.09234 4.09234i −0.222924 0.222924i 0.586805 0.809729i \(-0.300386\pi\)
−0.809729 + 0.586805i \(0.800386\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 15.2107i 0.823704i
\(342\) 0 0
\(343\) 8.29529 8.29529i 0.447903 0.447903i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −16.4230 + 16.4230i −0.881635 + 0.881635i −0.993701 0.112065i \(-0.964253\pi\)
0.112065 + 0.993701i \(0.464253\pi\)
\(348\) 0 0
\(349\) 11.4380i 0.612262i 0.951989 + 0.306131i \(0.0990346\pi\)
−0.951989 + 0.306131i \(0.900965\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12.7055 12.7055i −0.676248 0.676248i 0.282901 0.959149i \(-0.408703\pi\)
−0.959149 + 0.282901i \(0.908703\pi\)
\(354\) 0 0
\(355\) −22.3684 + 16.8573i −1.18719 + 0.894691i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −30.8389 −1.62762 −0.813809 0.581133i \(-0.802610\pi\)
−0.813809 + 0.581133i \(0.802610\pi\)
\(360\) 0 0
\(361\) 15.7368 0.828254
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 11.3430 + 1.59368i 0.593722 + 0.0834169i
\(366\) 0 0
\(367\) 10.2494 + 10.2494i 0.535012 + 0.535012i 0.922060 0.387048i \(-0.126505\pi\)
−0.387048 + 0.922060i \(0.626505\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 29.4966i 1.53139i
\(372\) 0 0
\(373\) 16.4336 16.4336i 0.850898 0.850898i −0.139346 0.990244i \(-0.544500\pi\)
0.990244 + 0.139346i \(0.0445001\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.41421 1.41421i 0.0728357 0.0728357i
\(378\) 0 0
\(379\) 3.14272i 0.161431i −0.996737 0.0807154i \(-0.974280\pi\)
0.996737 0.0807154i \(-0.0257205\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.7796 + 12.7796i 0.653009 + 0.653009i 0.953716 0.300708i \(-0.0972228\pi\)
−0.300708 + 0.953716i \(0.597223\pi\)
\(384\) 0 0
\(385\) −19.6128 2.75557i −0.999563 0.140437i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 25.0452 1.26984 0.634921 0.772577i \(-0.281033\pi\)
0.634921 + 0.772577i \(0.281033\pi\)
\(390\) 0 0
\(391\) 1.71456 0.0867089
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −26.8770 + 20.2551i −1.35233 + 1.01914i
\(396\) 0 0
\(397\) −1.57628 1.57628i −0.0791114 0.0791114i 0.666444 0.745555i \(-0.267816\pi\)
−0.745555 + 0.666444i \(0.767816\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 32.7742i 1.63666i 0.574746 + 0.818332i \(0.305101\pi\)
−0.574746 + 0.818332i \(0.694899\pi\)
\(402\) 0 0
\(403\) 13.4193 13.4193i 0.668462 0.668462i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 17.9741 17.9741i 0.890945 0.890945i
\(408\) 0 0
\(409\) 2.75557i 0.136254i 0.997677 + 0.0681271i \(0.0217023\pi\)
−0.997677 + 0.0681271i \(0.978298\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −41.4165 41.4165i −2.03797 2.03797i
\(414\) 0 0
\(415\) 3.43801 24.4701i 0.168765 1.20119i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −16.8204 −0.821732 −0.410866 0.911696i \(-0.634774\pi\)
−0.410866 + 0.911696i \(0.634774\pi\)
\(420\) 0 0
\(421\) −2.38715 −0.116343 −0.0581713 0.998307i \(-0.518527\pi\)
−0.0581713 + 0.998307i \(0.518527\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.34032 + 6.02491i 0.162029 + 0.292251i
\(426\) 0 0
\(427\) 13.2444 + 13.2444i 0.640943 + 0.640943i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 21.5324i 1.03718i −0.855024 0.518589i \(-0.826457\pi\)
0.855024 0.518589i \(-0.173543\pi\)
\(432\) 0 0
\(433\) 6.31756 6.31756i 0.303603 0.303603i −0.538819 0.842422i \(-0.681129\pi\)
0.842422 + 0.538819i \(0.181129\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.58956 + 1.58956i −0.0760387 + 0.0760387i
\(438\) 0 0
\(439\) 17.8350i 0.851218i 0.904907 + 0.425609i \(0.139940\pi\)
−0.904907 + 0.425609i \(0.860060\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.55467 + 2.55467i 0.121376 + 0.121376i 0.765186 0.643810i \(-0.222647\pi\)
−0.643810 + 0.765186i \(0.722647\pi\)
\(444\) 0 0
\(445\) −9.96343 13.2208i −0.472312 0.626725i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −4.66024 −0.219930 −0.109965 0.993935i \(-0.535074\pi\)
−0.109965 + 0.993935i \(0.535074\pi\)
\(450\) 0 0
\(451\) 16.5620 0.779874
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 14.8720 + 19.7340i 0.697208 + 0.925146i
\(456\) 0 0
\(457\) −22.7748 22.7748i −1.06536 1.06536i −0.997709 0.0676502i \(-0.978450\pi\)
−0.0676502 0.997709i \(-0.521550\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 28.0175i 1.30490i −0.757830 0.652452i \(-0.773740\pi\)
0.757830 0.652452i \(-0.226260\pi\)
\(462\) 0 0
\(463\) 13.5669 13.5669i 0.630508 0.630508i −0.317687 0.948195i \(-0.602906\pi\)
0.948195 + 0.317687i \(0.102906\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.71195 4.71195i 0.218043 0.218043i −0.589630 0.807673i \(-0.700727\pi\)
0.807673 + 0.589630i \(0.200727\pi\)
\(468\) 0 0
\(469\) 40.9403i 1.89045i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6.58155 6.58155i −0.302620 0.302620i
\(474\) 0 0
\(475\) −8.68244 2.48886i −0.398378 0.114197i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.89696 −0.178057 −0.0890284 0.996029i \(-0.528376\pi\)
−0.0890284 + 0.996029i \(0.528376\pi\)
\(480\) 0 0
\(481\) −31.7146 −1.44606
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.93597 + 20.8969i −0.133316 + 0.948878i
\(486\) 0 0
\(487\) −5.39207 5.39207i −0.244338 0.244338i 0.574304 0.818642i \(-0.305273\pi\)
−0.818642 + 0.574304i \(0.805273\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 23.6897i 1.06910i −0.845137 0.534550i \(-0.820481\pi\)
0.845137 0.534550i \(-0.179519\pi\)
\(492\) 0 0
\(493\) −0.723926 + 0.723926i −0.0326040 + 0.0326040i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 36.3659 36.3659i 1.63123 1.63123i
\(498\) 0 0
\(499\) 9.21585i 0.412558i −0.978493 0.206279i \(-0.933865\pi\)
0.978493 0.206279i \(-0.0661355\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −2.01344 2.01344i −0.0897749 0.0897749i 0.660793 0.750568i \(-0.270220\pi\)
−0.750568 + 0.660793i \(0.770220\pi\)
\(504\) 0 0
\(505\) −13.0000 + 9.79706i −0.578492 + 0.435963i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 15.5766 0.690419 0.345209 0.938526i \(-0.387808\pi\)
0.345209 + 0.938526i \(0.387808\pi\)
\(510\) 0 0
\(511\) −21.0321 −0.930406
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 27.2744 + 3.83200i 1.20185 + 0.168858i
\(516\) 0 0
\(517\) 16.2667 + 16.2667i 0.715409 + 0.715409i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.42013i 0.281271i −0.990061 0.140636i \(-0.955085\pi\)
0.990061 0.140636i \(-0.0449146\pi\)
\(522\) 0 0
\(523\) −25.9813 + 25.9813i −1.13608 + 1.13608i −0.146935 + 0.989146i \(0.546941\pi\)
−0.989146 + 0.146935i \(0.953059\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6.86923 + 6.86923i −0.299228 + 0.299228i
\(528\) 0 0
\(529\) 21.4514i 0.932669i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −14.6114 14.6114i −0.632892 0.632892i
\(534\) 0 0
\(535\) −10.9590 1.53972i −0.473798 0.0665678i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 21.2649 0.915945
\(540\) 0 0
\(541\) −30.0098 −1.29022 −0.645112 0.764088i \(-0.723189\pi\)
−0.645112 + 0.764088i \(0.723189\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −28.4079 + 21.4088i −1.21686 + 0.917050i
\(546\) 0 0
\(547\) 9.12399 + 9.12399i 0.390113 + 0.390113i 0.874728 0.484614i \(-0.161040\pi\)
−0.484614 + 0.874728i \(0.661040\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.34229i 0.0571836i
\(552\) 0 0
\(553\) 43.6958 43.6958i 1.85814 1.85814i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.5039 14.5039i 0.614550 0.614550i −0.329578 0.944128i \(-0.606907\pi\)
0.944128 + 0.329578i \(0.106907\pi\)
\(558\) 0 0
\(559\) 11.6128i 0.491171i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −18.1829 18.1829i −0.766319 0.766319i 0.211137 0.977456i \(-0.432283\pi\)
−0.977456 + 0.211137i \(0.932283\pi\)
\(564\) 0 0
\(565\) 4.66370 33.1941i 0.196203 1.39648i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3.42137 −0.143431 −0.0717156 0.997425i \(-0.522847\pi\)
−0.0717156 + 0.997425i \(0.522847\pi\)
\(570\) 0 0
\(571\) 3.17130 0.132715 0.0663574 0.997796i \(-0.478862\pi\)
0.0663574 + 0.997796i \(0.478862\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5.44177 + 3.01702i −0.226937 + 0.125818i
\(576\) 0 0
\(577\) −8.22570 8.22570i −0.342440 0.342440i 0.514844 0.857284i \(-0.327850\pi\)
−0.857284 + 0.514844i \(0.827850\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 45.3722i 1.88236i
\(582\) 0 0
\(583\) −10.9590 + 10.9590i −0.453875 + 0.453875i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.4773 22.4773i 0.927737 0.927737i −0.0698224 0.997559i \(-0.522243\pi\)
0.997559 + 0.0698224i \(0.0222433\pi\)
\(588\) 0 0
\(589\) 12.7368i 0.524812i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −10.2228 10.2228i −0.419800 0.419800i 0.465335 0.885135i \(-0.345934\pi\)
−0.885135 + 0.465335i \(0.845934\pi\)
\(594\) 0 0
\(595\) −7.61285 10.1017i −0.312096 0.414130i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −30.1615 −1.23237 −0.616183 0.787603i \(-0.711322\pi\)
−0.616183 + 0.787603i \(0.711322\pi\)
\(600\) 0 0
\(601\) 40.0830 1.63502 0.817509 0.575915i \(-0.195354\pi\)
0.817509 + 0.575915i \(0.195354\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 8.54047 + 11.3326i 0.347220 + 0.460736i
\(606\) 0 0
\(607\) −13.3921 13.3921i −0.543568 0.543568i 0.381005 0.924573i \(-0.375578\pi\)
−0.924573 + 0.381005i \(0.875578\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 28.7019i 1.16115i
\(612\) 0 0
\(613\) 17.1891 17.1891i 0.694262 0.694262i −0.268904 0.963167i \(-0.586662\pi\)
0.963167 + 0.268904i \(0.0866616\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23.2831 23.2831i 0.937344 0.937344i −0.0608053 0.998150i \(-0.519367\pi\)
0.998150 + 0.0608053i \(0.0193669\pi\)
\(618\) 0 0
\(619\) 44.6735i 1.79558i −0.440422 0.897791i \(-0.645171\pi\)
0.440422 0.897791i \(-0.354829\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 21.4939 + 21.4939i 0.861136 + 0.861136i
\(624\) 0 0
\(625\) −21.2034 13.2444i −0.848137 0.529777i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 16.2345 0.647310
\(630\) 0 0
\(631\) −7.64143 −0.304200 −0.152100 0.988365i \(-0.548604\pi\)
−0.152100 + 0.988365i \(0.548604\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.79712 34.1436i 0.190368 1.35495i
\(636\) 0 0
\(637\) −18.7605 18.7605i −0.743318 0.743318i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.9304i 0.510721i −0.966846 0.255361i \(-0.917806\pi\)
0.966846 0.255361i \(-0.0821942\pi\)
\(642\) 0 0
\(643\) −1.51114 + 1.51114i −0.0595934 + 0.0595934i −0.736275 0.676682i \(-0.763417\pi\)
0.676682 + 0.736275i \(0.263417\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −5.78049 + 5.78049i −0.227254 + 0.227254i −0.811545 0.584290i \(-0.801373\pi\)
0.584290 + 0.811545i \(0.301373\pi\)
\(648\) 0 0
\(649\) 30.7753i 1.20803i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.34390 + 4.34390i 0.169990 + 0.169990i 0.786975 0.616985i \(-0.211646\pi\)
−0.616985 + 0.786975i \(0.711646\pi\)
\(654\) 0 0
\(655\) −20.9131 + 15.7605i −0.817141 + 0.615813i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3.74684 0.145956 0.0729780 0.997334i \(-0.476750\pi\)
0.0729780 + 0.997334i \(0.476750\pi\)
\(660\) 0 0
\(661\) −34.6637 −1.34826 −0.674131 0.738612i \(-0.735482\pi\)
−0.674131 + 0.738612i \(0.735482\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 16.4230 + 2.30741i 0.636858 + 0.0894774i
\(666\) 0 0
\(667\) −0.653858 0.653858i −0.0253175 0.0253175i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 9.84150i 0.379927i
\(672\) 0 0
\(673\) −4.15257 + 4.15257i −0.160070 + 0.160070i −0.782598 0.622528i \(-0.786106\pi\)
0.622528 + 0.782598i \(0.286106\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19.4959 19.4959i 0.749288 0.749288i −0.225057 0.974346i \(-0.572257\pi\)
0.974346 + 0.225057i \(0.0722570\pi\)
\(678\) 0 0
\(679\) 38.7467i 1.48696i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 25.0320 + 25.0320i 0.957821 + 0.957821i 0.999146 0.0413245i \(-0.0131577\pi\)
−0.0413245 + 0.999146i \(0.513158\pi\)
\(684\) 0 0
\(685\) −18.2351 2.56199i −0.696726 0.0978887i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 19.3366 0.736667
\(690\) 0 0
\(691\) −3.70471 −0.140934 −0.0704670 0.997514i \(-0.522449\pi\)
−0.0704670 + 0.997514i \(0.522449\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.61210 4.22939i 0.212879 0.160430i
\(696\) 0 0
\(697\) 7.47949 + 7.47949i 0.283306 + 0.283306i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 36.8352i 1.39125i −0.718407 0.695623i \(-0.755128\pi\)
0.718407 0.695623i \(-0.244872\pi\)
\(702\) 0 0
\(703\) −15.0509 + 15.0509i −0.567654 + 0.567654i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 21.1350 21.1350i 0.794863 0.794863i
\(708\) 0 0
\(709\) 1.40943i 0.0529322i 0.999650 + 0.0264661i \(0.00842540\pi\)
−0.999650 + 0.0264661i \(0.991575\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6.20437 6.20437i −0.232355 0.232355i
\(714\) 0 0
\(715\) −1.80642 + 12.8573i −0.0675564 + 0.480835i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −18.1829 −0.678109 −0.339055 0.940767i \(-0.610107\pi\)
−0.339055 + 0.940767i \(0.610107\pi\)
\(720\) 0 0
\(721\) −50.5718 −1.88339
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.02379 3.57149i 0.0380224 0.132642i
\(726\) 0 0
\(727\) −0.119063 0.119063i −0.00441580 0.00441580i 0.704895 0.709311i \(-0.250994\pi\)
−0.709311 + 0.704895i \(0.750994\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 5.94453i 0.219867i
\(732\) 0 0
\(733\) 13.4143 13.4143i 0.495470 0.495470i −0.414554 0.910025i \(-0.636063\pi\)
0.910025 + 0.414554i \(0.136063\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 15.2107 15.2107i 0.560292 0.560292i
\(738\) 0 0
\(739\) 48.8484i 1.79692i 0.439058 + 0.898458i \(0.355312\pi\)
−0.439058 + 0.898458i \(0.644688\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 6.74560 + 6.74560i 0.247472 + 0.247472i 0.819932 0.572460i \(-0.194011\pi\)
−0.572460 + 0.819932i \(0.694011\pi\)
\(744\) 0 0
\(745\) −8.49823 11.2766i −0.311351 0.413141i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 20.3200 0.742477
\(750\) 0 0
\(751\) −13.1526 −0.479944 −0.239972 0.970780i \(-0.577138\pi\)
−0.239972 + 0.970780i \(0.577138\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −24.7197 32.8013i −0.899643 1.19376i
\(756\) 0 0
\(757\) 6.85236 + 6.85236i 0.249053 + 0.249053i 0.820582 0.571529i \(-0.193649\pi\)
−0.571529 + 0.820582i \(0.693649\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 52.3133i 1.89636i 0.317739 + 0.948178i \(0.397077\pi\)
−0.317739 + 0.948178i \(0.602923\pi\)
\(762\) 0 0
\(763\) 46.1847 46.1847i 1.67200 1.67200i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −27.1508 + 27.1508i −0.980358 + 0.980358i
\(768\) 0 0
\(769\) 20.4701i 0.738172i 0.929395 + 0.369086i \(0.120329\pi\)
−0.929395 + 0.369086i \(0.879671\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −8.59979 8.59979i −0.309313 0.309313i 0.535330 0.844643i \(-0.320187\pi\)
−0.844643 + 0.535330i \(0.820187\pi\)
\(774\) 0 0
\(775\) 9.71456 33.8894i 0.348957 1.21734i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −13.8684 −0.496886
\(780\) 0 0
\(781\) 27.0223 0.966933
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.0160876 0.114504i 0.000574192 0.00408683i
\(786\) 0 0
\(787\) 15.5210 + 15.5210i 0.553263 + 0.553263i 0.927381 0.374118i \(-0.122054\pi\)
−0.374118 + 0.927381i \(0.622054\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 61.5480i 2.18839i
\(792\) 0 0
\(793\) 8.68244 8.68244i 0.308322 0.308322i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.04616 + 1.04616i −0.0370569 + 0.0370569i −0.725392 0.688336i \(-0.758342\pi\)
0.688336 + 0.725392i \(0.258342\pi\)
\(798\) 0 0
\(799\) 14.6923i 0.519776i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −7.81413 7.81413i −0.275755 0.275755i
\(804\) 0 0
\(805\) 9.12399 6.87601i 0.321578 0.242348i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 50.2210 1.76568 0.882838 0.469677i \(-0.155630\pi\)
0.882838 + 0.469677i \(0.155630\pi\)
\(810\) 0 0
\(811\) 29.9813 1.05278 0.526392 0.850242i \(-0.323544\pi\)
0.526392 + 0.850242i \(0.323544\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −22.0799 3.10219i −0.773425 0.108665i
\(816\) 0 0
\(817\) 5.51114 + 5.51114i 0.192810 + 0.192810i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 28.7536i 1.00351i 0.865011 + 0.501753i \(0.167312\pi\)
−0.865011 + 0.501753i \(0.832688\pi\)
\(822\) 0 0
\(823\) 21.8622 21.8622i 0.762068 0.762068i −0.214628 0.976696i \(-0.568854\pi\)
0.976696 + 0.214628i \(0.0688538\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −23.4222 + 23.4222i −0.814470 + 0.814470i −0.985300 0.170831i \(-0.945355\pi\)
0.170831 + 0.985300i \(0.445355\pi\)
\(828\) 0 0
\(829\) 39.3176i 1.36556i 0.730626 + 0.682778i \(0.239228\pi\)
−0.730626 + 0.682778i \(0.760772\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 9.60336 + 9.60336i 0.332737 + 0.332737i
\(834\) 0 0
\(835\) 28.4701 + 4.00000i 0.985249 + 0.138426i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 39.4681 1.36259 0.681294 0.732010i \(-0.261417\pi\)
0.681294 + 0.732010i \(0.261417\pi\)
\(840\) 0 0
\(841\) −28.4479 −0.980960
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −10.2780 + 7.74570i −0.353574 + 0.266460i
\(846\) 0 0
\(847\) −18.4242 18.4242i −0.633063 0.633063i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 14.6632i 0.502646i
\(852\) 0 0
\(853\) 22.8938 22.8938i 0.783870 0.783870i −0.196611 0.980482i \(-0.562994\pi\)
0.980482 + 0.196611i \(0.0629937\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −31.4947 + 31.4947i −1.07584 + 1.07584i −0.0789597 + 0.996878i \(0.525160\pi\)
−0.996878 + 0.0789597i \(0.974840\pi\)
\(858\) 0 0
\(859\) 55.6958i 1.90032i 0.311767 + 0.950158i \(0.399079\pi\)
−0.311767 + 0.950158i \(0.600921\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −28.1991 28.1991i −0.959909 0.959909i 0.0393181 0.999227i \(-0.487481\pi\)
−0.999227 + 0.0393181i \(0.987481\pi\)
\(864\) 0 0
\(865\) 5.43801 38.7052i 0.184898 1.31602i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 32.4689 1.10143
\(870\) 0 0
\(871\) −26.8385 −0.909389
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 41.9376 + 18.6655i 1.41775 + 0.631009i
\(876\) 0 0
\(877\) −20.7003 20.7003i −0.698998 0.698998i 0.265196 0.964194i \(-0.414563\pi\)
−0.964194 + 0.265196i \(0.914563\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 37.4212i 1.26075i −0.776290 0.630376i \(-0.782901\pi\)
0.776290 0.630376i \(-0.217099\pi\)
\(882\) 0 0
\(883\) −12.5620 + 12.5620i −0.422745 + 0.422745i −0.886148 0.463403i \(-0.846628\pi\)
0.463403 + 0.886148i \(0.346628\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −23.3774 + 23.3774i −0.784938 + 0.784938i −0.980659 0.195722i \(-0.937295\pi\)
0.195722 + 0.980659i \(0.437295\pi\)
\(888\) 0 0
\(889\) 63.3087i 2.12330i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −13.6211 13.6211i −0.455813 0.455813i
\(894\) 0 0
\(895\) 14.6909 + 19.4938i 0.491063 + 0.651605i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5.23926 0.174739
\(900\) 0 0
\(901\) −9.89829 −0.329760
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 26.2708 + 34.8595i 0.873272 + 1.15877i
\(906\) 0 0
\(907\) 13.4193 + 13.4193i 0.445580 + 0.445580i 0.893882 0.448302i \(-0.147971\pi\)
−0.448302 + 0.893882i \(0.647971\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 11.5610i 0.383032i 0.981489 + 0.191516i \(0.0613404\pi\)
−0.981489 + 0.191516i \(0.938660\pi\)
\(912\) 0 0
\(913\) −16.8573 + 16.8573i −0.557895 + 0.557895i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 33.9998 33.9998i 1.12277 1.12277i
\(918\) 0 0
\(919\) 2.07313i 0.0683862i 0.999415 + 0.0341931i \(0.0108861\pi\)
−0.999415 + 0.0341931i \(0.989114\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −23.8398 23.8398i −0.784696 0.784696i
\(924\) 0 0
\(925\) −51.5259 + 28.5669i −1.69416 + 0.939274i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 24.2889 0.796893 0.398446 0.917192i \(-0.369550\pi\)
0.398446 + 0.917192i \(0.369550\pi\)
\(930\) 0 0
\(931\) −17.8064 −0.583582
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0.924696 6.58155i 0.0302408 0.215240i
\(936\) 0 0
\(937\) −12.4479 12.4479i −0.406654 0.406654i 0.473916 0.880570i \(-0.342840\pi\)
−0.880570 + 0.473916i \(0.842840\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 20.6007i 0.671565i 0.941940 + 0.335782i \(0.109001\pi\)
−0.941940 + 0.335782i \(0.890999\pi\)
\(942\) 0 0
\(943\) −6.75557 + 6.75557i −0.219992 + 0.219992i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −11.5610 + 11.5610i −0.375681 + 0.375681i −0.869541 0.493860i \(-0.835585\pi\)
0.493860 + 0.869541i \(0.335585\pi\)
\(948\) 0 0
\(949\) 13.7877i 0.447567i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −42.1889 42.1889i −1.36663 1.36663i −0.865190 0.501444i \(-0.832802\pi\)
−0.501444 0.865190i \(-0.667198\pi\)
\(954\) 0 0
\(955\) 42.5718 32.0830i 1.37759 1.03818i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 33.8112 1.09182
\(960\) 0 0
\(961\) 18.7146 0.603695
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −30.2621 4.25177i −0.974171 0.136869i
\(966\) 0 0
\(967\) 8.23951 + 8.23951i 0.264965 + 0.264965i 0.827067 0.562103i \(-0.190007\pi\)
−0.562103 + 0.827067i \(0.690007\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 29.2292i 0.938009i −0.883196 0.469004i \(-0.844613\pi\)
0.883196 0.469004i \(-0.155387\pi\)
\(972\) 0 0
\(973\) −9.12399 + 9.12399i −0.292502 + 0.292502i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 7.17861 7.17861i 0.229664 0.229664i −0.582888 0.812552i \(-0.698077\pi\)
0.812552 + 0.582888i \(0.198077\pi\)
\(978\) 0 0
\(979\) 15.9714i 0.510449i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 22.7510 + 22.7510i 0.725646 + 0.725646i 0.969749 0.244103i \(-0.0784937\pi\)
−0.244103 + 0.969749i \(0.578494\pi\)
\(984\) 0 0
\(985\) 31.2958 + 4.39700i 0.997166 + 0.140100i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 5.36917 0.170730
\(990\) 0 0
\(991\) 45.7975 1.45481 0.727403 0.686210i \(-0.240727\pi\)
0.727403 + 0.686210i \(0.240727\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −26.3497 + 19.8577i −0.835342 + 0.629530i
\(996\) 0 0
\(997\) −5.08742 5.08742i −0.161120 0.161120i 0.621943 0.783063i \(-0.286344\pi\)
−0.783063 + 0.621943i \(0.786344\pi\)
\(998\) 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 2880.2.w.p.2177.5 12
3.2 odd 2 inner 2880.2.w.p.2177.2 12
4.3 odd 2 2880.2.w.q.2177.5 12
5.3 odd 4 inner 2880.2.w.p.2753.2 12
8.3 odd 2 1440.2.w.g.737.2 yes 12
8.5 even 2 1440.2.w.f.737.2 12
12.11 even 2 2880.2.w.q.2177.2 12
15.8 even 4 inner 2880.2.w.p.2753.5 12
20.3 even 4 2880.2.w.q.2753.2 12
24.5 odd 2 1440.2.w.f.737.5 yes 12
24.11 even 2 1440.2.w.g.737.5 yes 12
40.3 even 4 1440.2.w.g.1313.5 yes 12
40.13 odd 4 1440.2.w.f.1313.5 yes 12
60.23 odd 4 2880.2.w.q.2753.5 12
120.53 even 4 1440.2.w.f.1313.2 yes 12
120.83 odd 4 1440.2.w.g.1313.2 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1440.2.w.f.737.2 12 8.5 even 2
1440.2.w.f.737.5 yes 12 24.5 odd 2
1440.2.w.f.1313.2 yes 12 120.53 even 4
1440.2.w.f.1313.5 yes 12 40.13 odd 4
1440.2.w.g.737.2 yes 12 8.3 odd 2
1440.2.w.g.737.5 yes 12 24.11 even 2
1440.2.w.g.1313.2 yes 12 120.83 odd 4
1440.2.w.g.1313.5 yes 12 40.3 even 4
2880.2.w.p.2177.2 12 3.2 odd 2 inner
2880.2.w.p.2177.5 12 1.1 even 1 trivial
2880.2.w.p.2753.2 12 5.3 odd 4 inner
2880.2.w.p.2753.5 12 15.8 even 4 inner
2880.2.w.q.2177.2 12 12.11 even 2
2880.2.w.q.2177.5 12 4.3 odd 2
2880.2.w.q.2753.2 12 20.3 even 4
2880.2.w.q.2753.5 12 60.23 odd 4