Properties

Label 1216.2.i.p.961.2
Level $1216$
Weight $2$
Character 1216.961
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
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] = [1216,2,Mod(577,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.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: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.39075800976.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 12x^{6} - 13x^{5} + 125x^{4} - 116x^{3} + 232x^{2} + 96x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 608)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.2
Root \(1.51772 - 2.62877i\) of defining polynomial
Character \(\chi\) \(=\) 1216.961
Dual form 1216.2.i.p.577.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.684999 + 1.18645i) q^{3} +(0.780776 - 1.35234i) q^{5} +(0.561553 + 0.972638i) q^{9} +O(q^{10})\) \(q+(-0.684999 + 1.18645i) q^{3} +(0.780776 - 1.35234i) q^{5} +(0.561553 + 0.972638i) q^{9} -3.50932 q^{11} +(-0.219224 - 0.379706i) q^{13} +(1.06966 + 1.85271i) q^{15} +(0.219224 - 0.379706i) q^{17} +(-3.12466 + 3.03916i) q^{19} +(2.43966 + 4.22562i) q^{23} +(1.28078 + 2.21837i) q^{25} -5.64865 q^{27} +(2.34233 + 4.05703i) q^{29} +2.74000 q^{31} +(2.40388 - 4.16365i) q^{33} +1.12311 q^{37} +0.600672 q^{39} +(-0.500000 + 0.866025i) q^{41} +(-3.80966 + 6.59852i) q^{43} +1.75379 q^{45} +(1.06966 + 1.85271i) q^{47} -7.00000 q^{49} +(0.300336 + 0.520197i) q^{51} +(0.219224 + 0.379706i) q^{53} +(-2.74000 + 4.74581i) q^{55} +(-1.46543 - 5.78908i) q^{57} +(0.684999 - 1.18645i) q^{59} +(-1.21922 - 2.11176i) q^{61} -0.684658 q^{65} +(6.93432 + 12.0106i) q^{67} -6.68466 q^{69} +(-2.43966 + 4.22562i) q^{71} +(-0.623106 + 1.07925i) q^{73} -3.50932 q^{75} +(1.06966 - 1.85271i) q^{79} +(2.18466 - 3.78394i) q^{81} +6.24932 q^{83} +(-0.342329 - 0.592932i) q^{85} -6.41797 q^{87} +(5.34233 + 9.25319i) q^{89} +(-1.87689 + 3.25088i) q^{93} +(1.67033 + 6.59852i) q^{95} +(-5.62311 + 9.73950i) q^{97} +(-1.97067 - 3.41330i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{5} - 12 q^{9} - 10 q^{13} + 10 q^{17} + 2 q^{25} - 6 q^{29} - 22 q^{33} - 24 q^{37} - 4 q^{41} + 80 q^{45} - 56 q^{49} + 10 q^{53} + 46 q^{57} - 18 q^{61} + 44 q^{65} - 4 q^{69} + 28 q^{73} - 32 q^{81} + 22 q^{85} + 18 q^{89} - 48 q^{93} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) −0.684999 + 1.18645i −0.395484 + 0.684999i −0.993163 0.116737i \(-0.962757\pi\)
0.597679 + 0.801736i \(0.296090\pi\)
\(4\) 0 0
\(5\) 0.780776 1.35234i 0.349174 0.604787i −0.636929 0.770922i \(-0.719796\pi\)
0.986103 + 0.166136i \(0.0531289\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0.561553 + 0.972638i 0.187184 + 0.324213i
\(10\) 0 0
\(11\) −3.50932 −1.05810 −0.529050 0.848591i \(-0.677452\pi\)
−0.529050 + 0.848591i \(0.677452\pi\)
\(12\) 0 0
\(13\) −0.219224 0.379706i −0.0608017 0.105312i 0.834022 0.551731i \(-0.186032\pi\)
−0.894824 + 0.446419i \(0.852699\pi\)
\(14\) 0 0
\(15\) 1.06966 + 1.85271i 0.276186 + 0.478367i
\(16\) 0 0
\(17\) 0.219224 0.379706i 0.0531695 0.0920923i −0.838216 0.545339i \(-0.816401\pi\)
0.891385 + 0.453247i \(0.149734\pi\)
\(18\) 0 0
\(19\) −3.12466 + 3.03916i −0.716846 + 0.697232i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.43966 + 4.22562i 0.508704 + 0.881102i 0.999949 + 0.0100802i \(0.00320867\pi\)
−0.491245 + 0.871021i \(0.663458\pi\)
\(24\) 0 0
\(25\) 1.28078 + 2.21837i 0.256155 + 0.443674i
\(26\) 0 0
\(27\) −5.64865 −1.08708
\(28\) 0 0
\(29\) 2.34233 + 4.05703i 0.434960 + 0.753372i 0.997292 0.0735389i \(-0.0234293\pi\)
−0.562333 + 0.826911i \(0.690096\pi\)
\(30\) 0 0
\(31\) 2.74000 0.492118 0.246059 0.969255i \(-0.420864\pi\)
0.246059 + 0.969255i \(0.420864\pi\)
\(32\) 0 0
\(33\) 2.40388 4.16365i 0.418462 0.724798i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.12311 0.184637 0.0923187 0.995730i \(-0.470572\pi\)
0.0923187 + 0.995730i \(0.470572\pi\)
\(38\) 0 0
\(39\) 0.600672 0.0961845
\(40\) 0 0
\(41\) −0.500000 + 0.866025i −0.0780869 + 0.135250i −0.902424 0.430848i \(-0.858214\pi\)
0.824338 + 0.566099i \(0.191548\pi\)
\(42\) 0 0
\(43\) −3.80966 + 6.59852i −0.580967 + 1.00627i 0.414398 + 0.910096i \(0.363992\pi\)
−0.995365 + 0.0961691i \(0.969341\pi\)
\(44\) 0 0
\(45\) 1.75379 0.261439
\(46\) 0 0
\(47\) 1.06966 + 1.85271i 0.156026 + 0.270245i 0.933432 0.358754i \(-0.116798\pi\)
−0.777406 + 0.628999i \(0.783465\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0.300336 + 0.520197i 0.0420554 + 0.0728421i
\(52\) 0 0
\(53\) 0.219224 + 0.379706i 0.0301127 + 0.0521567i 0.880689 0.473695i \(-0.157080\pi\)
−0.850576 + 0.525852i \(0.823747\pi\)
\(54\) 0 0
\(55\) −2.74000 + 4.74581i −0.369461 + 0.639925i
\(56\) 0 0
\(57\) −1.46543 5.78908i −0.194102 0.766783i
\(58\) 0 0
\(59\) 0.684999 1.18645i 0.0891793 0.154463i −0.817985 0.575239i \(-0.804909\pi\)
0.907164 + 0.420776i \(0.138242\pi\)
\(60\) 0 0
\(61\) −1.21922 2.11176i −0.156106 0.270383i 0.777355 0.629062i \(-0.216561\pi\)
−0.933461 + 0.358679i \(0.883227\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.684658 −0.0849214
\(66\) 0 0
\(67\) 6.93432 + 12.0106i 0.847162 + 1.46733i 0.883731 + 0.467996i \(0.155024\pi\)
−0.0365690 + 0.999331i \(0.511643\pi\)
\(68\) 0 0
\(69\) −6.68466 −0.804738
\(70\) 0 0
\(71\) −2.43966 + 4.22562i −0.289534 + 0.501488i −0.973699 0.227840i \(-0.926834\pi\)
0.684164 + 0.729328i \(0.260167\pi\)
\(72\) 0 0
\(73\) −0.623106 + 1.07925i −0.0729290 + 0.126317i −0.900184 0.435510i \(-0.856568\pi\)
0.827255 + 0.561827i \(0.189901\pi\)
\(74\) 0 0
\(75\) −3.50932 −0.405222
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.06966 1.85271i 0.120346 0.208446i −0.799558 0.600589i \(-0.794933\pi\)
0.919904 + 0.392143i \(0.128266\pi\)
\(80\) 0 0
\(81\) 2.18466 3.78394i 0.242740 0.420438i
\(82\) 0 0
\(83\) 6.24932 0.685952 0.342976 0.939344i \(-0.388565\pi\)
0.342976 + 0.939344i \(0.388565\pi\)
\(84\) 0 0
\(85\) −0.342329 0.592932i −0.0371308 0.0643125i
\(86\) 0 0
\(87\) −6.41797 −0.688079
\(88\) 0 0
\(89\) 5.34233 + 9.25319i 0.566286 + 0.980836i 0.996929 + 0.0783134i \(0.0249535\pi\)
−0.430643 + 0.902522i \(0.641713\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.87689 + 3.25088i −0.194625 + 0.337100i
\(94\) 0 0
\(95\) 1.67033 + 6.59852i 0.171373 + 0.676994i
\(96\) 0 0
\(97\) −5.62311 + 9.73950i −0.570940 + 0.988897i 0.425530 + 0.904944i \(0.360088\pi\)
−0.996470 + 0.0839525i \(0.973246\pi\)
\(98\) 0 0
\(99\) −1.97067 3.41330i −0.198060 0.343050i
\(100\) 0 0
\(101\) −7.90388 13.6899i −0.786466 1.36220i −0.928120 0.372282i \(-0.878575\pi\)
0.141654 0.989916i \(-0.454758\pi\)
\(102\) 0 0
\(103\) −17.9786 −1.77149 −0.885743 0.464175i \(-0.846351\pi\)
−0.885743 + 0.464175i \(0.846351\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.47999 −0.529771 −0.264885 0.964280i \(-0.585334\pi\)
−0.264885 + 0.964280i \(0.585334\pi\)
\(108\) 0 0
\(109\) −5.90388 + 10.2258i −0.565489 + 0.979456i 0.431515 + 0.902106i \(0.357979\pi\)
−0.997004 + 0.0773503i \(0.975354\pi\)
\(110\) 0 0
\(111\) −0.769326 + 1.33251i −0.0730212 + 0.126476i
\(112\) 0 0
\(113\) −6.56155 −0.617259 −0.308629 0.951182i \(-0.599870\pi\)
−0.308629 + 0.951182i \(0.599870\pi\)
\(114\) 0 0
\(115\) 7.61932 0.710505
\(116\) 0 0
\(117\) 0.246211 0.426450i 0.0227622 0.0394254i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.31534 0.119577
\(122\) 0 0
\(123\) −0.684999 1.18645i −0.0617643 0.106979i
\(124\) 0 0
\(125\) 11.8078 1.05612
\(126\) 0 0
\(127\) −4.57898 7.93103i −0.406319 0.703765i 0.588155 0.808748i \(-0.299855\pi\)
−0.994474 + 0.104983i \(0.966521\pi\)
\(128\) 0 0
\(129\) −5.21922 9.03996i −0.459527 0.795924i
\(130\) 0 0
\(131\) 4.79499 8.30517i 0.418940 0.725626i −0.576893 0.816820i \(-0.695735\pi\)
0.995833 + 0.0911938i \(0.0290683\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −4.41033 + 7.63892i −0.379581 + 0.657453i
\(136\) 0 0
\(137\) 3.50000 + 6.06218i 0.299025 + 0.517927i 0.975913 0.218159i \(-0.0700052\pi\)
−0.676888 + 0.736086i \(0.736672\pi\)
\(138\) 0 0
\(139\) 5.56432 + 9.63768i 0.471959 + 0.817458i 0.999485 0.0320813i \(-0.0102136\pi\)
−0.527526 + 0.849539i \(0.676880\pi\)
\(140\) 0 0
\(141\) −2.93087 −0.246824
\(142\) 0 0
\(143\) 0.769326 + 1.33251i 0.0643343 + 0.111430i
\(144\) 0 0
\(145\) 7.31534 0.607506
\(146\) 0 0
\(147\) 4.79499 8.30517i 0.395484 0.684999i
\(148\) 0 0
\(149\) −4.46543 + 7.73436i −0.365823 + 0.633623i −0.988908 0.148531i \(-0.952546\pi\)
0.623085 + 0.782154i \(0.285879\pi\)
\(150\) 0 0
\(151\) −7.01864 −0.571169 −0.285585 0.958354i \(-0.592188\pi\)
−0.285585 + 0.958354i \(0.592188\pi\)
\(152\) 0 0
\(153\) 0.492423 0.0398100
\(154\) 0 0
\(155\) 2.13932 3.70542i 0.171835 0.297626i
\(156\) 0 0
\(157\) 9.90388 17.1540i 0.790416 1.36904i −0.135294 0.990806i \(-0.543198\pi\)
0.925710 0.378235i \(-0.123469\pi\)
\(158\) 0 0
\(159\) −0.600672 −0.0476364
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3.50932 0.274871 0.137436 0.990511i \(-0.456114\pi\)
0.137436 + 0.990511i \(0.456114\pi\)
\(164\) 0 0
\(165\) −3.75379 6.50175i −0.292232 0.506161i
\(166\) 0 0
\(167\) −11.4290 19.7956i −0.884401 1.53183i −0.846399 0.532549i \(-0.821234\pi\)
−0.0380014 0.999278i \(-0.512099\pi\)
\(168\) 0 0
\(169\) 6.40388 11.0918i 0.492606 0.853219i
\(170\) 0 0
\(171\) −4.71067 1.33251i −0.360234 0.101900i
\(172\) 0 0
\(173\) −2.90388 + 5.02967i −0.220778 + 0.382399i −0.955044 0.296463i \(-0.904193\pi\)
0.734266 + 0.678861i \(0.237526\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.938447 + 1.62544i 0.0705380 + 0.122175i
\(178\) 0 0
\(179\) 17.5466 1.31150 0.655748 0.754980i \(-0.272354\pi\)
0.655748 + 0.754980i \(0.272354\pi\)
\(180\) 0 0
\(181\) 10.4654 + 18.1267i 0.777890 + 1.34734i 0.933156 + 0.359472i \(0.117043\pi\)
−0.155266 + 0.987873i \(0.549624\pi\)
\(182\) 0 0
\(183\) 3.34067 0.246949
\(184\) 0 0
\(185\) 0.876894 1.51883i 0.0644706 0.111666i
\(186\) 0 0
\(187\) −0.769326 + 1.33251i −0.0562587 + 0.0974429i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 24.9973 1.80874 0.904370 0.426750i \(-0.140342\pi\)
0.904370 + 0.426750i \(0.140342\pi\)
\(192\) 0 0
\(193\) 4.90388 8.49377i 0.352989 0.611395i −0.633783 0.773511i \(-0.718499\pi\)
0.986772 + 0.162116i \(0.0518319\pi\)
\(194\) 0 0
\(195\) 0.468990 0.812315i 0.0335851 0.0581711i
\(196\) 0 0
\(197\) 19.3693 1.38001 0.690003 0.723806i \(-0.257609\pi\)
0.690003 + 0.723806i \(0.257609\pi\)
\(198\) 0 0
\(199\) −9.45830 16.3823i −0.670481 1.16131i −0.977768 0.209691i \(-0.932754\pi\)
0.307286 0.951617i \(-0.400579\pi\)
\(200\) 0 0
\(201\) −19.0000 −1.34016
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0.780776 + 1.35234i 0.0545318 + 0.0944518i
\(206\) 0 0
\(207\) −2.74000 + 4.74581i −0.190443 + 0.329857i
\(208\) 0 0
\(209\) 10.9654 10.6654i 0.758495 0.737741i
\(210\) 0 0
\(211\) 3.80966 6.59852i 0.262268 0.454261i −0.704576 0.709628i \(-0.748863\pi\)
0.966844 + 0.255367i \(0.0821963\pi\)
\(212\) 0 0
\(213\) −3.34233 5.78908i −0.229013 0.396662i
\(214\) 0 0
\(215\) 5.94898 + 10.3039i 0.405717 + 0.702723i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −0.853653 1.47857i −0.0576846 0.0999126i
\(220\) 0 0
\(221\) −0.192236 −0.0129312
\(222\) 0 0
\(223\) 9.45830 16.3823i 0.633375 1.09704i −0.353482 0.935441i \(-0.615002\pi\)
0.986857 0.161596i \(-0.0516642\pi\)
\(224\) 0 0
\(225\) −1.43845 + 2.49146i −0.0958965 + 0.166098i
\(226\) 0 0
\(227\) 10.5280 0.698766 0.349383 0.936980i \(-0.386391\pi\)
0.349383 + 0.936980i \(0.386391\pi\)
\(228\) 0 0
\(229\) 9.75379 0.644549 0.322274 0.946646i \(-0.395553\pi\)
0.322274 + 0.946646i \(0.395553\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.93845 + 3.35749i −0.126992 + 0.219956i −0.922510 0.385974i \(-0.873866\pi\)
0.795518 + 0.605930i \(0.207199\pi\)
\(234\) 0 0
\(235\) 3.34067 0.217921
\(236\) 0 0
\(237\) 1.46543 + 2.53821i 0.0951902 + 0.164874i
\(238\) 0 0
\(239\) 29.2759 1.89370 0.946851 0.321673i \(-0.104245\pi\)
0.946851 + 0.321673i \(0.104245\pi\)
\(240\) 0 0
\(241\) 9.74621 + 16.8809i 0.627809 + 1.08740i 0.987991 + 0.154514i \(0.0493813\pi\)
−0.360182 + 0.932882i \(0.617285\pi\)
\(242\) 0 0
\(243\) −5.47999 9.49162i −0.351542 0.608888i
\(244\) 0 0
\(245\) −5.46543 + 9.46641i −0.349174 + 0.604787i
\(246\) 0 0
\(247\) 1.83899 + 0.520197i 0.117012 + 0.0330993i
\(248\) 0 0
\(249\) −4.28078 + 7.41452i −0.271283 + 0.469876i
\(250\) 0 0
\(251\) −10.4436 18.0889i −0.659197 1.14176i −0.980824 0.194896i \(-0.937563\pi\)
0.321627 0.946866i \(-0.395770\pi\)
\(252\) 0 0
\(253\) −8.56155 14.8290i −0.538260 0.932294i
\(254\) 0 0
\(255\) 0.937981 0.0587386
\(256\) 0 0
\(257\) 5.62311 + 9.73950i 0.350760 + 0.607534i 0.986383 0.164466i \(-0.0525901\pi\)
−0.635623 + 0.772000i \(0.719257\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.63068 + 4.55648i −0.162835 + 0.282039i
\(262\) 0 0
\(263\) 11.4290 19.7956i 0.704741 1.22065i −0.262044 0.965056i \(-0.584397\pi\)
0.966785 0.255591i \(-0.0822700\pi\)
\(264\) 0 0
\(265\) 0.684658 0.0420582
\(266\) 0 0
\(267\) −14.6380 −0.895829
\(268\) 0 0
\(269\) −8.78078 + 15.2088i −0.535373 + 0.927294i 0.463772 + 0.885955i \(0.346496\pi\)
−0.999145 + 0.0413392i \(0.986838\pi\)
\(270\) 0 0
\(271\) −1.06966 + 1.85271i −0.0649773 + 0.112544i −0.896684 0.442671i \(-0.854031\pi\)
0.831707 + 0.555215i \(0.187364\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.49466 7.78497i −0.271038 0.469452i
\(276\) 0 0
\(277\) 4.00000 0.240337 0.120168 0.992754i \(-0.461657\pi\)
0.120168 + 0.992754i \(0.461657\pi\)
\(278\) 0 0
\(279\) 1.53865 + 2.66502i 0.0921167 + 0.159551i
\(280\) 0 0
\(281\) −11.5000 19.9186i −0.686032 1.18824i −0.973111 0.230336i \(-0.926017\pi\)
0.287079 0.957907i \(-0.407316\pi\)
\(282\) 0 0
\(283\) 4.79499 8.30517i 0.285033 0.493691i −0.687584 0.726104i \(-0.741329\pi\)
0.972617 + 0.232413i \(0.0746622\pi\)
\(284\) 0 0
\(285\) −8.97301 2.53821i −0.531515 0.150350i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.40388 + 14.5560i 0.494346 + 0.856232i
\(290\) 0 0
\(291\) −7.70364 13.3431i −0.451596 0.782186i
\(292\) 0 0
\(293\) 4.00000 0.233682 0.116841 0.993151i \(-0.462723\pi\)
0.116841 + 0.993151i \(0.462723\pi\)
\(294\) 0 0
\(295\) −1.06966 1.85271i −0.0622781 0.107869i
\(296\) 0 0
\(297\) 19.8229 1.15024
\(298\) 0 0
\(299\) 1.06966 1.85271i 0.0618602 0.107145i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 21.6566 1.24414
\(304\) 0 0
\(305\) −3.80776 −0.218032
\(306\) 0 0
\(307\) −8.30431 + 14.3835i −0.473952 + 0.820909i −0.999555 0.0298205i \(-0.990506\pi\)
0.525603 + 0.850730i \(0.323840\pi\)
\(308\) 0 0
\(309\) 12.3153 21.3308i 0.700595 1.21347i
\(310\) 0 0
\(311\) 13.7000 0.776855 0.388427 0.921479i \(-0.373018\pi\)
0.388427 + 0.921479i \(0.373018\pi\)
\(312\) 0 0
\(313\) −5.62311 9.73950i −0.317837 0.550509i 0.662200 0.749327i \(-0.269623\pi\)
−0.980036 + 0.198818i \(0.936290\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −15.5885 27.0001i −0.875540 1.51648i −0.856187 0.516666i \(-0.827173\pi\)
−0.0193529 0.999813i \(-0.506161\pi\)
\(318\) 0 0
\(319\) −8.21999 14.2374i −0.460231 0.797143i
\(320\) 0 0
\(321\) 3.75379 6.50175i 0.209516 0.362892i
\(322\) 0 0
\(323\) 0.468990 + 1.85271i 0.0260953 + 0.103087i
\(324\) 0 0
\(325\) 0.561553 0.972638i 0.0311493 0.0539522i
\(326\) 0 0
\(327\) −8.08831 14.0094i −0.447284 0.774719i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −21.4879 −1.18108 −0.590542 0.807007i \(-0.701086\pi\)
−0.590542 + 0.807007i \(0.701086\pi\)
\(332\) 0 0
\(333\) 0.630683 + 1.09238i 0.0345612 + 0.0598618i
\(334\) 0 0
\(335\) 21.6566 1.18323
\(336\) 0 0
\(337\) −3.06155 + 5.30277i −0.166773 + 0.288860i −0.937284 0.348568i \(-0.886668\pi\)
0.770510 + 0.637428i \(0.220002\pi\)
\(338\) 0 0
\(339\) 4.49466 7.78497i 0.244116 0.422822i
\(340\) 0 0
\(341\) −9.61553 −0.520710
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −5.21922 + 9.03996i −0.280994 + 0.486695i
\(346\) 0 0
\(347\) −4.79499 + 8.30517i −0.257409 + 0.445845i −0.965547 0.260229i \(-0.916202\pi\)
0.708138 + 0.706074i \(0.249535\pi\)
\(348\) 0 0
\(349\) 12.0000 0.642345 0.321173 0.947021i \(-0.395923\pi\)
0.321173 + 0.947021i \(0.395923\pi\)
\(350\) 0 0
\(351\) 1.23832 + 2.14483i 0.0660964 + 0.114482i
\(352\) 0 0
\(353\) −24.8078 −1.32038 −0.660192 0.751097i \(-0.729525\pi\)
−0.660192 + 0.751097i \(0.729525\pi\)
\(354\) 0 0
\(355\) 3.80966 + 6.59852i 0.202196 + 0.350213i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.43966 4.22562i 0.128760 0.223019i −0.794436 0.607348i \(-0.792234\pi\)
0.923197 + 0.384328i \(0.125567\pi\)
\(360\) 0 0
\(361\) 0.526988 18.9927i 0.0277362 0.999615i
\(362\) 0 0
\(363\) −0.901008 + 1.56059i −0.0472906 + 0.0819098i
\(364\) 0 0
\(365\) 0.973012 + 1.68531i 0.0509298 + 0.0882130i
\(366\) 0 0
\(367\) −9.45830 16.3823i −0.493719 0.855147i 0.506254 0.862384i \(-0.331030\pi\)
−0.999974 + 0.00723702i \(0.997696\pi\)
\(368\) 0 0
\(369\) −1.12311 −0.0584665
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −27.6155 −1.42988 −0.714939 0.699187i \(-0.753546\pi\)
−0.714939 + 0.699187i \(0.753546\pi\)
\(374\) 0 0
\(375\) −8.08831 + 14.0094i −0.417678 + 0.723440i
\(376\) 0 0
\(377\) 1.02699 1.77879i 0.0528926 0.0916126i
\(378\) 0 0
\(379\) −8.55730 −0.439559 −0.219779 0.975550i \(-0.570534\pi\)
−0.219779 + 0.975550i \(0.570534\pi\)
\(380\) 0 0
\(381\) 12.5464 0.642771
\(382\) 0 0
\(383\) −5.94898 + 10.3039i −0.303979 + 0.526507i −0.977033 0.213086i \(-0.931649\pi\)
0.673055 + 0.739593i \(0.264982\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.55730 −0.434992
\(388\) 0 0
\(389\) 14.1501 + 24.5087i 0.717438 + 1.24264i 0.962012 + 0.273009i \(0.0880188\pi\)
−0.244573 + 0.969631i \(0.578648\pi\)
\(390\) 0 0
\(391\) 2.13932 0.108190
\(392\) 0 0
\(393\) 6.56913 + 11.3781i 0.331369 + 0.573948i
\(394\) 0 0
\(395\) −1.67033 2.89310i −0.0840436 0.145568i
\(396\) 0 0
\(397\) −13.7808 + 23.8690i −0.691637 + 1.19795i 0.279664 + 0.960098i \(0.409777\pi\)
−0.971301 + 0.237853i \(0.923556\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8.50000 + 14.7224i −0.424470 + 0.735203i −0.996371 0.0851195i \(-0.972873\pi\)
0.571901 + 0.820323i \(0.306206\pi\)
\(402\) 0 0
\(403\) −0.600672 1.04039i −0.0299216 0.0518257i
\(404\) 0 0
\(405\) −3.41146 5.90882i −0.169517 0.293612i
\(406\) 0 0
\(407\) −3.94134 −0.195365
\(408\) 0 0
\(409\) 9.30776 + 16.1215i 0.460239 + 0.797158i 0.998973 0.0453185i \(-0.0144303\pi\)
−0.538733 + 0.842476i \(0.681097\pi\)
\(410\) 0 0
\(411\) −9.58999 −0.473039
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4.87932 8.45123i 0.239516 0.414855i
\(416\) 0 0
\(417\) −15.2462 −0.746610
\(418\) 0 0
\(419\) 30.4773 1.48891 0.744456 0.667672i \(-0.232709\pi\)
0.744456 + 0.667672i \(0.232709\pi\)
\(420\) 0 0
\(421\) −16.0270 + 27.7596i −0.781108 + 1.35292i 0.150189 + 0.988657i \(0.452012\pi\)
−0.931297 + 0.364261i \(0.881322\pi\)
\(422\) 0 0
\(423\) −1.20134 + 2.08079i −0.0584113 + 0.101171i
\(424\) 0 0
\(425\) 1.12311 0.0544786
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −2.10795 −0.101773
\(430\) 0 0
\(431\) 14.9383 + 25.8739i 0.719552 + 1.24630i 0.961177 + 0.275932i \(0.0889863\pi\)
−0.241625 + 0.970370i \(0.577680\pi\)
\(432\) 0 0
\(433\) −15.5885 27.0001i −0.749137 1.29754i −0.948237 0.317565i \(-0.897135\pi\)
0.199099 0.979979i \(-0.436198\pi\)
\(434\) 0 0
\(435\) −5.01100 + 8.67931i −0.240259 + 0.416141i
\(436\) 0 0
\(437\) −20.4654 5.78908i −0.978995 0.276929i
\(438\) 0 0
\(439\) −2.43966 + 4.22562i −0.116439 + 0.201678i −0.918354 0.395760i \(-0.870481\pi\)
0.801915 + 0.597438i \(0.203814\pi\)
\(440\) 0 0
\(441\) −3.93087 6.80847i −0.187184 0.324213i
\(442\) 0 0
\(443\) 2.05500 + 3.55936i 0.0976359 + 0.169110i 0.910706 0.413056i \(-0.135539\pi\)
−0.813070 + 0.582166i \(0.802205\pi\)
\(444\) 0 0
\(445\) 16.6847 0.790929
\(446\) 0 0
\(447\) −6.11764 10.5961i −0.289354 0.501176i
\(448\) 0 0
\(449\) 24.8078 1.17075 0.585375 0.810762i \(-0.300947\pi\)
0.585375 + 0.810762i \(0.300947\pi\)
\(450\) 0 0
\(451\) 1.75466 3.03916i 0.0826238 0.143109i
\(452\) 0 0
\(453\) 4.80776 8.32729i 0.225888 0.391250i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −32.5616 −1.52317 −0.761583 0.648068i \(-0.775577\pi\)
−0.761583 + 0.648068i \(0.775577\pi\)
\(458\) 0 0
\(459\) −1.23832 + 2.14483i −0.0577997 + 0.100112i
\(460\) 0 0
\(461\) 13.7808 23.8690i 0.641835 1.11169i −0.343188 0.939267i \(-0.611507\pi\)
0.985023 0.172424i \(-0.0551598\pi\)
\(462\) 0 0
\(463\) −16.7773 −0.779707 −0.389853 0.920877i \(-0.627474\pi\)
−0.389853 + 0.920877i \(0.627474\pi\)
\(464\) 0 0
\(465\) 2.93087 + 5.07642i 0.135916 + 0.235413i
\(466\) 0 0
\(467\) 14.4693 0.669560 0.334780 0.942296i \(-0.391338\pi\)
0.334780 + 0.942296i \(0.391338\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 13.5683 + 23.5010i 0.625194 + 1.08287i
\(472\) 0 0
\(473\) 13.3693 23.1563i 0.614722 1.06473i
\(474\) 0 0
\(475\) −10.7440 3.03916i −0.492967 0.139446i
\(476\) 0 0
\(477\) −0.246211 + 0.426450i −0.0112732 + 0.0195258i
\(478\) 0 0
\(479\) −4.41033 7.63892i −0.201513 0.349031i 0.747503 0.664258i \(-0.231253\pi\)
−0.949016 + 0.315227i \(0.897919\pi\)
\(480\) 0 0
\(481\) −0.246211 0.426450i −0.0112263 0.0194445i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.78078 + 15.2088i 0.398715 + 0.690594i
\(486\) 0 0
\(487\) −32.0159 −1.45078 −0.725390 0.688338i \(-0.758340\pi\)
−0.725390 + 0.688338i \(0.758340\pi\)
\(488\) 0 0
\(489\) −2.40388 + 4.16365i −0.108707 + 0.188287i
\(490\) 0 0
\(491\) 16.3083 28.2468i 0.735983 1.27476i −0.218308 0.975880i \(-0.570054\pi\)
0.954291 0.298880i \(-0.0966130\pi\)
\(492\) 0 0
\(493\) 2.05398 0.0925064
\(494\) 0 0
\(495\) −6.15461 −0.276629
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −6.33365 + 10.9702i −0.283533 + 0.491093i −0.972252 0.233935i \(-0.924840\pi\)
0.688719 + 0.725028i \(0.258173\pi\)
\(500\) 0 0
\(501\) 31.3153 1.39907
\(502\) 0 0
\(503\) 14.9383 + 25.8739i 0.666066 + 1.15366i 0.978995 + 0.203883i \(0.0653563\pi\)
−0.312930 + 0.949776i \(0.601310\pi\)
\(504\) 0 0
\(505\) −24.6847 −1.09845
\(506\) 0 0
\(507\) 8.77331 + 15.1958i 0.389636 + 0.674870i
\(508\) 0 0
\(509\) −0.534565 0.925894i −0.0236942 0.0410395i 0.853935 0.520379i \(-0.174209\pi\)
−0.877629 + 0.479340i \(0.840876\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 17.6501 17.1672i 0.779271 0.757948i
\(514\) 0 0
\(515\) −14.0373 + 24.3133i −0.618557 + 1.07137i
\(516\) 0 0
\(517\) −3.75379 6.50175i −0.165091 0.285947i
\(518\) 0 0
\(519\) −3.97831 6.89064i −0.174629 0.302465i
\(520\) 0 0
\(521\) 28.4233 1.24525 0.622624 0.782522i \(-0.286067\pi\)
0.622624 + 0.782522i \(0.286067\pi\)
\(522\) 0 0
\(523\) 5.94898 + 10.3039i 0.260131 + 0.450560i 0.966276 0.257507i \(-0.0829010\pi\)
−0.706146 + 0.708067i \(0.749568\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.600672 1.04039i 0.0261657 0.0453203i
\(528\) 0 0
\(529\) −0.403882 + 0.699544i −0.0175601 + 0.0304150i
\(530\) 0 0
\(531\) 1.53865 0.0667718
\(532\) 0 0
\(533\) 0.438447 0.0189913
\(534\) 0 0
\(535\) −4.27865 + 7.41084i −0.184982 + 0.320398i
\(536\) 0 0
\(537\) −12.0194 + 20.8182i −0.518676 + 0.898373i
\(538\) 0 0
\(539\) 24.5653 1.05810
\(540\) 0 0
\(541\) −14.1501 24.5087i −0.608360 1.05371i −0.991511 0.130025i \(-0.958494\pi\)
0.383151 0.923686i \(-0.374839\pi\)
\(542\) 0 0
\(543\) −28.6752 −1.23057
\(544\) 0 0
\(545\) 9.21922 + 15.9682i 0.394908 + 0.684001i
\(546\) 0 0
\(547\) −5.94898 10.3039i −0.254360 0.440565i 0.710361 0.703837i \(-0.248532\pi\)
−0.964722 + 0.263272i \(0.915198\pi\)
\(548\) 0 0
\(549\) 1.36932 2.37173i 0.0584410 0.101223i
\(550\) 0 0
\(551\) −19.6490 5.55813i −0.837074 0.236784i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 1.20134 + 2.08079i 0.0509942 + 0.0883245i
\(556\) 0 0
\(557\) −5.21922 9.03996i −0.221146 0.383035i 0.734011 0.679138i \(-0.237646\pi\)
−0.955156 + 0.296103i \(0.904313\pi\)
\(558\) 0 0
\(559\) 3.34067 0.141295
\(560\) 0 0
\(561\) −1.05398 1.82554i −0.0444989 0.0770743i
\(562\) 0 0
\(563\) 32.4479 1.36752 0.683759 0.729708i \(-0.260344\pi\)
0.683759 + 0.729708i \(0.260344\pi\)
\(564\) 0 0
\(565\) −5.12311 + 8.87348i −0.215531 + 0.373310i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0.630683 0.0264396 0.0132198 0.999913i \(-0.495792\pi\)
0.0132198 + 0.999913i \(0.495792\pi\)
\(570\) 0 0
\(571\) 14.4693 0.605522 0.302761 0.953067i \(-0.402092\pi\)
0.302761 + 0.953067i \(0.402092\pi\)
\(572\) 0 0
\(573\) −17.1231 + 29.6581i −0.715328 + 1.23898i
\(574\) 0 0
\(575\) −6.24932 + 10.8241i −0.260615 + 0.451398i
\(576\) 0 0
\(577\) 24.1771 1.00651 0.503253 0.864139i \(-0.332137\pi\)
0.503253 + 0.864139i \(0.332137\pi\)
\(578\) 0 0
\(579\) 6.71831 + 11.6365i 0.279203 + 0.483594i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −0.769326 1.33251i −0.0318622 0.0551870i
\(584\) 0 0
\(585\) −0.384472 0.665925i −0.0158960 0.0275326i
\(586\) 0 0
\(587\) −10.2276 + 17.7148i −0.422139 + 0.731167i −0.996149 0.0876819i \(-0.972054\pi\)
0.574009 + 0.818849i \(0.305387\pi\)
\(588\) 0 0
\(589\) −8.56155 + 8.32729i −0.352773 + 0.343120i
\(590\) 0 0
\(591\) −13.2680 + 22.9808i −0.545771 + 0.945303i
\(592\) 0 0
\(593\) 11.1847 + 19.3724i 0.459299 + 0.795529i 0.998924 0.0463764i \(-0.0147674\pi\)
−0.539625 + 0.841905i \(0.681434\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 25.9157 1.06066
\(598\) 0 0
\(599\) 7.91965 + 13.7172i 0.323588 + 0.560471i 0.981226 0.192863i \(-0.0617774\pi\)
−0.657637 + 0.753335i \(0.728444\pi\)
\(600\) 0 0
\(601\) 42.4233 1.73048 0.865241 0.501356i \(-0.167165\pi\)
0.865241 + 0.501356i \(0.167165\pi\)
\(602\) 0 0
\(603\) −7.78797 + 13.4892i −0.317151 + 0.549321i
\(604\) 0 0
\(605\) 1.02699 1.77879i 0.0417530 0.0723183i
\(606\) 0 0
\(607\) −39.0346 −1.58436 −0.792182 0.610285i \(-0.791055\pi\)
−0.792182 + 0.610285i \(0.791055\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0.468990 0.812315i 0.0189733 0.0328628i
\(612\) 0 0
\(613\) 22.9039 39.6707i 0.925079 1.60228i 0.133645 0.991029i \(-0.457332\pi\)
0.791434 0.611255i \(-0.209335\pi\)
\(614\) 0 0
\(615\) −2.13932 −0.0862659
\(616\) 0 0
\(617\) −16.1847 28.0327i −0.651570 1.12855i −0.982742 0.184982i \(-0.940777\pi\)
0.331172 0.943570i \(-0.392556\pi\)
\(618\) 0 0
\(619\) 19.5173 0.784466 0.392233 0.919866i \(-0.371703\pi\)
0.392233 + 0.919866i \(0.371703\pi\)
\(620\) 0 0
\(621\) −13.7808 23.8690i −0.553004 0.957830i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.81534 4.87631i 0.112614 0.195053i
\(626\) 0 0
\(627\) 5.14268 + 20.3158i 0.205379 + 0.811333i
\(628\) 0 0
\(629\) 0.246211 0.426450i 0.00981709 0.0170037i
\(630\) 0 0
\(631\) 8.08831 + 14.0094i 0.321990 + 0.557704i 0.980899 0.194520i \(-0.0623148\pi\)
−0.658908 + 0.752223i \(0.728981\pi\)
\(632\) 0 0
\(633\) 5.21922 + 9.03996i 0.207446 + 0.359306i
\(634\) 0 0
\(635\) −14.3007 −0.567504
\(636\) 0 0
\(637\) 1.53457 + 2.65794i 0.0608017 + 0.105312i
\(638\) 0 0
\(639\) −5.47999 −0.216785
\(640\) 0 0
\(641\) −7.18466 + 12.4442i −0.283777 + 0.491516i −0.972312 0.233687i \(-0.924921\pi\)
0.688535 + 0.725203i \(0.258254\pi\)
\(642\) 0 0
\(643\) −1.28567 + 2.22685i −0.0507019 + 0.0878183i −0.890263 0.455448i \(-0.849479\pi\)
0.839561 + 0.543266i \(0.182813\pi\)
\(644\) 0 0
\(645\) −16.3002 −0.641819
\(646\) 0 0
\(647\) −28.9386 −1.13769 −0.568847 0.822443i \(-0.692610\pi\)
−0.568847 + 0.822443i \(0.692610\pi\)
\(648\) 0 0
\(649\) −2.40388 + 4.16365i −0.0943606 + 0.163437i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 40.4924 1.58459 0.792295 0.610138i \(-0.208886\pi\)
0.792295 + 0.610138i \(0.208886\pi\)
\(654\) 0 0
\(655\) −7.48763 12.9690i −0.292566 0.506739i
\(656\) 0 0
\(657\) −1.39963 −0.0546046
\(658\) 0 0
\(659\) 5.94898 + 10.3039i 0.231739 + 0.401384i 0.958320 0.285697i \(-0.0922250\pi\)
−0.726581 + 0.687081i \(0.758892\pi\)
\(660\) 0 0
\(661\) −2.15009 3.72407i −0.0836289 0.144850i 0.821177 0.570673i \(-0.193318\pi\)
−0.904806 + 0.425824i \(0.859984\pi\)
\(662\) 0 0
\(663\) 0.131681 0.228079i 0.00511408 0.00885785i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −11.4290 + 19.7956i −0.442532 + 0.766487i
\(668\) 0 0
\(669\) 12.9579 + 22.4437i 0.500980 + 0.867722i
\(670\) 0 0
\(671\) 4.27865 + 7.41084i 0.165175 + 0.286092i
\(672\) 0 0
\(673\) −25.6155 −0.987406 −0.493703 0.869631i \(-0.664357\pi\)
−0.493703 + 0.869631i \(0.664357\pi\)
\(674\) 0 0
\(675\) −7.23465 12.5308i −0.278462 0.482310i
\(676\) 0 0
\(677\) −1.12311 −0.0431645 −0.0215822 0.999767i \(-0.506870\pi\)
−0.0215822 + 0.999767i \(0.506870\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −7.21165 + 12.4909i −0.276351 + 0.478654i
\(682\) 0 0
\(683\) 23.7959 0.910526 0.455263 0.890357i \(-0.349545\pi\)
0.455263 + 0.890357i \(0.349545\pi\)
\(684\) 0 0
\(685\) 10.9309 0.417647
\(686\) 0 0
\(687\) −6.68134 + 11.5724i −0.254909 + 0.441515i
\(688\) 0 0
\(689\) 0.0961180 0.166481i 0.00366180 0.00634243i
\(690\) 0 0
\(691\) −41.4372 −1.57635 −0.788174 0.615453i \(-0.788973\pi\)
−0.788174 + 0.615453i \(0.788973\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 17.3780 0.659183
\(696\) 0 0
\(697\) 0.219224 + 0.379706i 0.00830369 + 0.0143824i
\(698\) 0 0
\(699\) −2.65567 4.59975i −0.100447 0.173979i
\(700\) 0 0
\(701\) −12.7808 + 22.1370i −0.482723 + 0.836101i −0.999803 0.0198359i \(-0.993686\pi\)
0.517080 + 0.855937i \(0.327019\pi\)
\(702\) 0 0
\(703\) −3.50932 + 3.41330i −0.132357 + 0.128735i
\(704\) 0 0
\(705\) −2.28835 + 3.96355i −0.0861844 + 0.149276i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −12.9039 22.3502i −0.484615 0.839379i 0.515228 0.857053i \(-0.327707\pi\)
−0.999844 + 0.0176744i \(0.994374\pi\)
\(710\) 0 0
\(711\) 2.40269 0.0901078
\(712\) 0 0
\(713\) 6.68466 + 11.5782i 0.250342 + 0.433606i
\(714\) 0 0
\(715\) 2.40269 0.0898554
\(716\) 0 0
\(717\) −20.0540 + 34.7345i −0.748929 + 1.29718i
\(718\) 0 0
\(719\) −17.0776 + 29.5793i −0.636888 + 1.10312i 0.349224 + 0.937039i \(0.386445\pi\)
−0.986112 + 0.166083i \(0.946888\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −26.7046 −0.993154
\(724\) 0 0
\(725\) −6.00000 + 10.3923i −0.222834 + 0.385961i
\(726\) 0 0
\(727\) −13.5683 + 23.5010i −0.503220 + 0.871603i 0.496773 + 0.867881i \(0.334518\pi\)
−0.999993 + 0.00372251i \(0.998815\pi\)
\(728\) 0 0
\(729\) 28.1231 1.04160
\(730\) 0 0
\(731\) 1.67033 + 2.89310i 0.0617795 + 0.107005i
\(732\) 0 0
\(733\) 3.36932 0.124449 0.0622243 0.998062i \(-0.480181\pi\)
0.0622243 + 0.998062i \(0.480181\pi\)
\(734\) 0 0
\(735\) −7.48763 12.9690i −0.276186 0.478367i
\(736\) 0 0
\(737\) −24.3348 42.1490i −0.896382 1.55258i
\(738\) 0 0
\(739\) −4.79499 + 8.30517i −0.176387 + 0.305511i −0.940640 0.339405i \(-0.889774\pi\)
0.764254 + 0.644916i \(0.223108\pi\)
\(740\) 0 0
\(741\) −1.87689 + 1.82554i −0.0689494 + 0.0670628i
\(742\) 0 0
\(743\) 14.9383 25.8739i 0.548033 0.949221i −0.450376 0.892839i \(-0.648710\pi\)
0.998409 0.0563821i \(-0.0179565\pi\)
\(744\) 0 0
\(745\) 6.97301 + 12.0776i 0.255471 + 0.442489i
\(746\) 0 0
\(747\) 3.50932 + 6.07832i 0.128399 + 0.222394i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 21.9569 + 38.0305i 0.801220 + 1.38775i 0.918813 + 0.394692i \(0.129149\pi\)
−0.117593 + 0.993062i \(0.537518\pi\)
\(752\) 0 0
\(753\) 28.6155 1.04281
\(754\) 0 0
\(755\) −5.47999 + 9.49162i −0.199437 + 0.345436i
\(756\) 0 0
\(757\) −4.65767 + 8.06732i −0.169286 + 0.293212i −0.938169 0.346178i \(-0.887479\pi\)
0.768883 + 0.639389i \(0.220813\pi\)
\(758\) 0 0
\(759\) 23.4586 0.851494
\(760\) 0 0
\(761\) −21.9309 −0.794993 −0.397497 0.917604i \(-0.630121\pi\)
−0.397497 + 0.917604i \(0.630121\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.384472 0.665925i 0.0139006 0.0240766i
\(766\) 0 0
\(767\) −0.600672 −0.0216890
\(768\) 0 0
\(769\) −5.34233 9.25319i −0.192649 0.333678i 0.753478 0.657473i \(-0.228375\pi\)
−0.946127 + 0.323795i \(0.895041\pi\)
\(770\) 0 0
\(771\) −15.4073 −0.554880
\(772\) 0 0
\(773\) −14.3423 24.8416i −0.515858 0.893492i −0.999831 0.0184088i \(-0.994140\pi\)
0.483973 0.875083i \(-0.339193\pi\)
\(774\) 0 0
\(775\) 3.50932 + 6.07832i 0.126059 + 0.218340i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.06966 4.22562i −0.0383246 0.151398i
\(780\) 0 0
\(781\) 8.56155 14.8290i 0.306356 0.530625i
\(782\) 0 0
\(783\) −13.2310 22.9167i −0.472837 0.818978i
\(784\) 0 0
\(785\) −15.4654 26.7869i −0.551985 0.956066i
\(786\) 0 0
\(787\) −21.4879 −0.765963 −0.382981 0.923756i \(-0.625103\pi\)
−0.382981 + 0.923756i \(0.625103\pi\)
\(788\) 0 0
\(789\) 15.6577 + 27.1199i 0.557428 + 0.965493i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.534565 + 0.925894i −0.0189830 + 0.0328795i
\(794\) 0 0
\(795\) −0.468990 + 0.812315i −0.0166334 + 0.0288098i
\(796\) 0 0
\(797\) 12.6307 0.447402 0.223701 0.974658i \(-0.428186\pi\)
0.223701 + 0.974658i \(0.428186\pi\)
\(798\) 0 0
\(799\) 0.937981 0.0331834
\(800\) 0 0
\(801\) −6.00000 + 10.3923i −0.212000 + 0.367194i
\(802\) 0 0
\(803\) 2.18668 3.78744i 0.0771662 0.133656i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −12.0296 20.8360i −0.423464 0.733460i
\(808\) 0 0
\(809\) −40.1771 −1.41255 −0.706275 0.707937i \(-0.749626\pi\)
−0.706275 + 0.707937i \(0.749626\pi\)
\(810\) 0 0
\(811\) −26.0669 45.1493i −0.915334 1.58540i −0.806412 0.591354i \(-0.798593\pi\)
−0.108922 0.994050i \(-0.534740\pi\)
\(812\) 0 0
\(813\) −1.46543 2.53821i −0.0513950 0.0890188i
\(814\) 0 0
\(815\) 2.74000 4.74581i 0.0959779 0.166239i
\(816\) 0 0
\(817\) −8.15009 32.1963i −0.285136 1.12641i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −8.71165 15.0890i −0.304039 0.526610i 0.673008 0.739635i \(-0.265002\pi\)
−0.977047 + 0.213025i \(0.931668\pi\)
\(822\) 0 0
\(823\) 24.0963 + 41.7360i 0.839943 + 1.45482i 0.889942 + 0.456075i \(0.150745\pi\)
−0.0499986 + 0.998749i \(0.515922\pi\)
\(824\) 0 0
\(825\) 12.3153 0.428765
\(826\) 0 0
\(827\) −3.42499 5.93227i −0.119099 0.206285i 0.800312 0.599584i \(-0.204667\pi\)
−0.919411 + 0.393299i \(0.871334\pi\)
\(828\) 0 0
\(829\) 31.1231 1.08095 0.540475 0.841360i \(-0.318245\pi\)
0.540475 + 0.841360i \(0.318245\pi\)
\(830\) 0 0
\(831\) −2.74000 + 4.74581i −0.0950494 + 0.164630i
\(832\) 0 0
\(833\) −1.53457 + 2.65794i −0.0531695 + 0.0920923i
\(834\) 0 0
\(835\) −35.6939 −1.23524
\(836\) 0 0
\(837\) −15.4773 −0.534973
\(838\) 0 0
\(839\) 25.4663 44.1089i 0.879193 1.52281i 0.0269650 0.999636i \(-0.491416\pi\)
0.852228 0.523171i \(-0.175251\pi\)
\(840\) 0 0
\(841\) 3.52699 6.10892i 0.121620 0.210652i
\(842\) 0 0
\(843\) 31.5100 1.08526
\(844\) 0 0
\(845\) −10.0000 17.3205i −0.344010 0.595844i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 6.56913 + 11.3781i 0.225452 + 0.390494i
\(850\) 0 0
\(851\) 2.74000 + 4.74581i 0.0939258 + 0.162684i
\(852\) 0 0
\(853\) −1.78078 + 3.08440i −0.0609726 + 0.105608i −0.894900 0.446266i \(-0.852754\pi\)
0.833928 + 0.551874i \(0.186087\pi\)
\(854\) 0 0
\(855\) −5.47999 + 5.33005i −0.187412 + 0.182284i
\(856\) 0 0
\(857\) 0.815342 1.41221i 0.0278515 0.0482403i −0.851764 0.523926i \(-0.824467\pi\)
0.879615 + 0.475686i \(0.157800\pi\)
\(858\) 0 0
\(859\) 26.4516 + 45.8155i 0.902517 + 1.56321i 0.824216 + 0.566275i \(0.191616\pi\)
0.0783006 + 0.996930i \(0.475051\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 33.2173 1.13073 0.565364 0.824841i \(-0.308736\pi\)
0.565364 + 0.824841i \(0.308736\pi\)
\(864\) 0 0
\(865\) 4.53457 + 7.85410i 0.154180 + 0.267047i
\(866\) 0 0
\(867\) −23.0266 −0.782024
\(868\) 0 0
\(869\) −3.75379 + 6.50175i −0.127339 + 0.220557i
\(870\) 0 0
\(871\) 3.04033 5.26601i 0.103018 0.178432i
\(872\) 0 0
\(873\) −12.6307 −0.427484
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −19.6577 + 34.0481i −0.663792 + 1.14972i 0.315819 + 0.948820i \(0.397721\pi\)
−0.979611 + 0.200903i \(0.935613\pi\)
\(878\) 0 0
\(879\) −2.74000 + 4.74581i −0.0924178 + 0.160072i
\(880\) 0 0
\(881\) −11.0540 −0.372418 −0.186209 0.982510i \(-0.559620\pi\)
−0.186209 + 0.982510i \(0.559620\pi\)
\(882\) 0 0
\(883\) −17.4623 30.2456i −0.587653 1.01784i −0.994539 0.104365i \(-0.966719\pi\)
0.406886 0.913479i \(-0.366614\pi\)
\(884\) 0 0
\(885\) 2.93087 0.0985201
\(886\) 0 0
\(887\) −5.94898 10.3039i −0.199747 0.345972i 0.748699 0.662910i \(-0.230679\pi\)
−0.948446 + 0.316938i \(0.897345\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −7.66667 + 13.2791i −0.256843 + 0.444865i
\(892\) 0 0
\(893\) −8.97301 2.53821i −0.300270 0.0849379i
\(894\) 0 0
\(895\) 13.7000 23.7291i 0.457940 0.793175i
\(896\) 0 0
\(897\) 1.46543 + 2.53821i 0.0489294 + 0.0847483i
\(898\) 0 0
\(899\) 6.41797 + 11.1163i 0.214051 + 0.370748i
\(900\) 0 0
\(901\) 0.192236 0.00640431
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 32.6847 1.08647
\(906\) 0 0
\(907\) −6.76566 + 11.7185i −0.224650 + 0.389105i −0.956214 0.292667i \(-0.905457\pi\)
0.731564 + 0.681772i \(0.238791\pi\)
\(908\) 0 0
\(909\) 8.87689 15.3752i 0.294428 0.509964i
\(910\) 0 0
\(911\) 10.0959 0.334494 0.167247 0.985915i \(-0.446512\pi\)
0.167247 + 0.985915i \(0.446512\pi\)
\(912\) 0 0
\(913\) −21.9309 −0.725806
\(914\) 0 0
\(915\) 2.60831 4.51773i 0.0862282 0.149352i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 57.0132 1.88069 0.940346 0.340220i \(-0.110502\pi\)
0.940346 + 0.340220i \(0.110502\pi\)
\(920\) 0 0
\(921\) −11.3769 19.7054i −0.374881 0.649314i
\(922\) 0 0
\(923\) 2.13932 0.0704167
\(924\) 0 0
\(925\) 1.43845 + 2.49146i 0.0472959 + 0.0819188i
\(926\) 0 0
\(927\) −10.0959 17.4867i −0.331594 0.574338i
\(928\) 0 0
\(929\) −6.06155 + 10.4989i −0.198873 + 0.344458i −0.948163 0.317783i \(-0.897061\pi\)
0.749290 + 0.662242i \(0.230395\pi\)
\(930\) 0 0
\(931\) 21.8726 21.2741i 0.716846 0.697232i
\(932\) 0 0
\(933\) −9.38447 + 16.2544i −0.307234 + 0.532145i
\(934\) 0 0
\(935\) 1.20134 + 2.08079i 0.0392881 + 0.0680490i
\(936\) 0 0
\(937\) 20.9924 + 36.3599i 0.685793 + 1.18783i 0.973187 + 0.230015i \(0.0738776\pi\)
−0.287394 + 0.957812i \(0.592789\pi\)
\(938\) 0 0
\(939\) 15.4073 0.502798
\(940\) 0 0
\(941\) −3.78078 6.54850i −0.123250 0.213475i 0.797798 0.602925i \(-0.205998\pi\)
−0.921047 + 0.389450i \(0.872665\pi\)
\(942\) 0 0
\(943\) −4.87932 −0.158893
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −23.3269 + 40.4034i −0.758024 + 1.31294i 0.185833 + 0.982581i \(0.440502\pi\)
−0.943857 + 0.330354i \(0.892832\pi\)
\(948\) 0 0
\(949\) 0.546398 0.0177368
\(950\) 0 0
\(951\) 42.7125 1.38505
\(952\) 0 0
\(953\) −28.3078 + 49.0305i −0.916978 + 1.58825i −0.113000 + 0.993595i \(0.536046\pi\)
−0.803979 + 0.594658i \(0.797287\pi\)
\(954\) 0 0
\(955\) 19.5173 33.8049i 0.631564 1.09390i
\(956\) 0 0
\(957\) 22.5227 0.728057
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −23.4924 −0.757820
\(962\) 0 0
\(963\) −3.07730 5.33005i −0.0991648 0.171758i
\(964\) 0 0
\(965\) −7.65767 13.2635i −0.246509 0.426966i
\(966\) 0 0
\(967\) 15.5390 26.9143i 0.499700 0.865505i −0.500300 0.865852i \(-0.666777\pi\)
1.00000 0.000346935i \(0.000110433\pi\)
\(968\) 0 0
\(969\) −2.51941 0.712669i −0.0809351 0.0228942i
\(970\) 0 0
\(971\) 5.73297 9.92980i 0.183980 0.318662i −0.759252 0.650796i \(-0.774435\pi\)
0.943232 + 0.332134i \(0.107769\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0.769326 + 1.33251i 0.0246382 + 0.0426745i
\(976\) 0 0
\(977\) 37.3002 1.19334 0.596669 0.802487i \(-0.296490\pi\)
0.596669 + 0.802487i \(0.296490\pi\)
\(978\) 0 0
\(979\) −18.7480 32.4724i −0.599187 1.03782i
\(980\) 0 0
\(981\) −13.2614 −0.423403
\(982\) 0 0
\(983\) 19.0483 32.9926i 0.607546 1.05230i −0.384097 0.923293i \(-0.625487\pi\)
0.991644 0.129008i \(-0.0411793\pi\)
\(984\) 0 0
\(985\) 15.1231 26.1940i 0.481862 0.834610i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −37.1771 −1.18216
\(990\) 0 0
\(991\) −16.9090 + 29.2872i −0.537131 + 0.930338i 0.461926 + 0.886919i \(0.347159\pi\)
−0.999057 + 0.0434197i \(0.986175\pi\)
\(992\) 0 0
\(993\) 14.7192 25.4944i 0.467100 0.809042i
\(994\) 0 0
\(995\) −29.5393 −0.936458
\(996\) 0 0
\(997\) −11.4654 19.8587i −0.363114 0.628932i 0.625358 0.780338i \(-0.284953\pi\)
−0.988472 + 0.151406i \(0.951620\pi\)
\(998\) 0 0
\(999\) −6.34403 −0.200716
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 1216.2.i.p.961.2 8
4.3 odd 2 inner 1216.2.i.p.961.3 8
8.3 odd 2 608.2.i.d.353.2 8
8.5 even 2 608.2.i.d.353.3 yes 8
19.7 even 3 inner 1216.2.i.p.577.2 8
76.7 odd 6 inner 1216.2.i.p.577.3 8
152.45 even 6 608.2.i.d.577.3 yes 8
152.83 odd 6 608.2.i.d.577.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.i.d.353.2 8 8.3 odd 2
608.2.i.d.353.3 yes 8 8.5 even 2
608.2.i.d.577.2 yes 8 152.83 odd 6
608.2.i.d.577.3 yes 8 152.45 even 6
1216.2.i.p.577.2 8 19.7 even 3 inner
1216.2.i.p.577.3 8 76.7 odd 6 inner
1216.2.i.p.961.2 8 1.1 even 1 trivial
1216.2.i.p.961.3 8 4.3 odd 2 inner