Properties

Label 960.3.l.j.641.1
Level $960$
Weight $3$
Character 960.641
Analytic conductor $26.158$
Analytic rank $0$
Dimension $16$
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] = [960,3,Mod(641,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.641");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.l (of order \(2\), degree \(1\), 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: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 56 x^{14} - 252 x^{13} + 1094 x^{12} - 3652 x^{11} + 5452 x^{10} + 1164 x^{9} - 5879 x^{8} - 15060 x^{7} + 42772 x^{6} - 39536 x^{5} + 64024 x^{4} + \cdots + 20736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 480)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 641.1
Root \(2.45733 - 4.45426i\) of defining polynomial
Character \(\chi\) \(=\) 960.641
Dual form 960.3.l.j.641.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.99655 - 0.143843i) q^{3} +2.23607i q^{5} +2.24307 q^{7} +(8.95862 + 0.862068i) q^{9} +O(q^{10})\) \(q+(-2.99655 - 0.143843i) q^{3} +2.23607i q^{5} +2.24307 q^{7} +(8.95862 + 0.862068i) q^{9} -9.03137i q^{11} -8.03098 q^{13} +(0.321644 - 6.70049i) q^{15} +7.34437i q^{17} -14.0089 q^{19} +(-6.72147 - 0.322651i) q^{21} -12.6041i q^{23} -5.00000 q^{25} +(-26.7209 - 3.87187i) q^{27} +38.9853i q^{29} +44.0396 q^{31} +(-1.29910 + 27.0629i) q^{33} +5.01566i q^{35} +25.3292 q^{37} +(24.0652 + 1.15520i) q^{39} +8.51074i q^{41} -45.9971 q^{43} +(-1.92764 + 20.0321i) q^{45} -51.7389i q^{47} -43.9686 q^{49} +(1.05644 - 22.0078i) q^{51} -60.6457i q^{53} +20.1947 q^{55} +(41.9783 + 2.01508i) q^{57} -86.0177i q^{59} +67.0499 q^{61} +(20.0948 + 1.93368i) q^{63} -17.9578i q^{65} -15.5703 q^{67} +(-1.81302 + 37.7689i) q^{69} -68.6484i q^{71} -90.6777 q^{73} +(14.9827 + 0.719217i) q^{75} -20.2580i q^{77} +31.3181 q^{79} +(79.5137 + 15.4459i) q^{81} -143.046i q^{83} -16.4225 q^{85} +(5.60778 - 116.821i) q^{87} -140.581i q^{89} -18.0140 q^{91} +(-131.967 - 6.33480i) q^{93} -31.3248i q^{95} +0.825684 q^{97} +(7.78565 - 80.9086i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{9} + 16 q^{13} + 104 q^{21} - 80 q^{25} + 192 q^{33} - 144 q^{37} - 40 q^{45} - 128 q^{49} - 80 q^{57} - 144 q^{61} + 280 q^{69} + 192 q^{81} + 96 q^{93} - 160 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) −2.99655 0.143843i −0.998850 0.0479478i
\(4\) 0 0
\(5\) 2.23607i 0.447214i
\(6\) 0 0
\(7\) 2.24307 0.320439 0.160219 0.987081i \(-0.448780\pi\)
0.160219 + 0.987081i \(0.448780\pi\)
\(8\) 0 0
\(9\) 8.95862 + 0.862068i 0.995402 + 0.0957853i
\(10\) 0 0
\(11\) 9.03137i 0.821033i −0.911853 0.410517i \(-0.865348\pi\)
0.911853 0.410517i \(-0.134652\pi\)
\(12\) 0 0
\(13\) −8.03098 −0.617767 −0.308884 0.951100i \(-0.599955\pi\)
−0.308884 + 0.951100i \(0.599955\pi\)
\(14\) 0 0
\(15\) 0.321644 6.70049i 0.0214429 0.446699i
\(16\) 0 0
\(17\) 7.34437i 0.432022i 0.976391 + 0.216011i \(0.0693047\pi\)
−0.976391 + 0.216011i \(0.930695\pi\)
\(18\) 0 0
\(19\) −14.0089 −0.737310 −0.368655 0.929566i \(-0.620182\pi\)
−0.368655 + 0.929566i \(0.620182\pi\)
\(20\) 0 0
\(21\) −6.72147 0.322651i −0.320070 0.0153643i
\(22\) 0 0
\(23\) 12.6041i 0.548006i −0.961729 0.274003i \(-0.911652\pi\)
0.961729 0.274003i \(-0.0883478\pi\)
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −26.7209 3.87187i −0.989664 0.143402i
\(28\) 0 0
\(29\) 38.9853i 1.34432i 0.740405 + 0.672161i \(0.234634\pi\)
−0.740405 + 0.672161i \(0.765366\pi\)
\(30\) 0 0
\(31\) 44.0396 1.42063 0.710316 0.703883i \(-0.248552\pi\)
0.710316 + 0.703883i \(0.248552\pi\)
\(32\) 0 0
\(33\) −1.29910 + 27.0629i −0.0393667 + 0.820089i
\(34\) 0 0
\(35\) 5.01566i 0.143304i
\(36\) 0 0
\(37\) 25.3292 0.684573 0.342286 0.939596i \(-0.388799\pi\)
0.342286 + 0.939596i \(0.388799\pi\)
\(38\) 0 0
\(39\) 24.0652 + 1.15520i 0.617057 + 0.0296206i
\(40\) 0 0
\(41\) 8.51074i 0.207579i 0.994599 + 0.103790i \(0.0330968\pi\)
−0.994599 + 0.103790i \(0.966903\pi\)
\(42\) 0 0
\(43\) −45.9971 −1.06970 −0.534850 0.844947i \(-0.679632\pi\)
−0.534850 + 0.844947i \(0.679632\pi\)
\(44\) 0 0
\(45\) −1.92764 + 20.0321i −0.0428365 + 0.445157i
\(46\) 0 0
\(47\) 51.7389i 1.10083i −0.834892 0.550414i \(-0.814470\pi\)
0.834892 0.550414i \(-0.185530\pi\)
\(48\) 0 0
\(49\) −43.9686 −0.897319
\(50\) 0 0
\(51\) 1.05644 22.0078i 0.0207145 0.431525i
\(52\) 0 0
\(53\) 60.6457i 1.14426i −0.820163 0.572130i \(-0.806117\pi\)
0.820163 0.572130i \(-0.193883\pi\)
\(54\) 0 0
\(55\) 20.1947 0.367177
\(56\) 0 0
\(57\) 41.9783 + 2.01508i 0.736462 + 0.0353524i
\(58\) 0 0
\(59\) 86.0177i 1.45793i −0.684552 0.728964i \(-0.740002\pi\)
0.684552 0.728964i \(-0.259998\pi\)
\(60\) 0 0
\(61\) 67.0499 1.09918 0.549590 0.835435i \(-0.314784\pi\)
0.549590 + 0.835435i \(0.314784\pi\)
\(62\) 0 0
\(63\) 20.0948 + 1.93368i 0.318965 + 0.0306933i
\(64\) 0 0
\(65\) 17.9578i 0.276274i
\(66\) 0 0
\(67\) −15.5703 −0.232392 −0.116196 0.993226i \(-0.537070\pi\)
−0.116196 + 0.993226i \(0.537070\pi\)
\(68\) 0 0
\(69\) −1.81302 + 37.7689i −0.0262757 + 0.547375i
\(70\) 0 0
\(71\) 68.6484i 0.966879i −0.875378 0.483440i \(-0.839387\pi\)
0.875378 0.483440i \(-0.160613\pi\)
\(72\) 0 0
\(73\) −90.6777 −1.24216 −0.621080 0.783747i \(-0.713306\pi\)
−0.621080 + 0.783747i \(0.713306\pi\)
\(74\) 0 0
\(75\) 14.9827 + 0.719217i 0.199770 + 0.00958956i
\(76\) 0 0
\(77\) 20.2580i 0.263091i
\(78\) 0 0
\(79\) 31.3181 0.396431 0.198216 0.980158i \(-0.436485\pi\)
0.198216 + 0.980158i \(0.436485\pi\)
\(80\) 0 0
\(81\) 79.5137 + 15.4459i 0.981650 + 0.190690i
\(82\) 0 0
\(83\) 143.046i 1.72344i −0.507383 0.861720i \(-0.669387\pi\)
0.507383 0.861720i \(-0.330613\pi\)
\(84\) 0 0
\(85\) −16.4225 −0.193206
\(86\) 0 0
\(87\) 5.60778 116.821i 0.0644572 1.34278i
\(88\) 0 0
\(89\) 140.581i 1.57956i −0.613392 0.789778i \(-0.710196\pi\)
0.613392 0.789778i \(-0.289804\pi\)
\(90\) 0 0
\(91\) −18.0140 −0.197956
\(92\) 0 0
\(93\) −131.967 6.33480i −1.41900 0.0681161i
\(94\) 0 0
\(95\) 31.3248i 0.329735i
\(96\) 0 0
\(97\) 0.825684 0.00851221 0.00425610 0.999991i \(-0.498645\pi\)
0.00425610 + 0.999991i \(0.498645\pi\)
\(98\) 0 0
\(99\) 7.78565 80.9086i 0.0786429 0.817258i
\(100\) 0 0
\(101\) 149.256i 1.47778i −0.673824 0.738892i \(-0.735349\pi\)
0.673824 0.738892i \(-0.264651\pi\)
\(102\) 0 0
\(103\) −86.3106 −0.837967 −0.418984 0.907994i \(-0.637613\pi\)
−0.418984 + 0.907994i \(0.637613\pi\)
\(104\) 0 0
\(105\) 0.721469 15.0297i 0.00687113 0.143140i
\(106\) 0 0
\(107\) 156.269i 1.46046i 0.683202 + 0.730230i \(0.260587\pi\)
−0.683202 + 0.730230i \(0.739413\pi\)
\(108\) 0 0
\(109\) 26.0810 0.239275 0.119638 0.992818i \(-0.461827\pi\)
0.119638 + 0.992818i \(0.461827\pi\)
\(110\) 0 0
\(111\) −75.9002 3.64344i −0.683786 0.0328238i
\(112\) 0 0
\(113\) 42.8246i 0.378978i 0.981883 + 0.189489i \(0.0606832\pi\)
−0.981883 + 0.189489i \(0.939317\pi\)
\(114\) 0 0
\(115\) 28.1837 0.245076
\(116\) 0 0
\(117\) −71.9465 6.92324i −0.614927 0.0591730i
\(118\) 0 0
\(119\) 16.4739i 0.138436i
\(120\) 0 0
\(121\) 39.4344 0.325904
\(122\) 0 0
\(123\) 1.22421 25.5029i 0.00995296 0.207340i
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −98.4145 −0.774918 −0.387459 0.921887i \(-0.626647\pi\)
−0.387459 + 0.921887i \(0.626647\pi\)
\(128\) 0 0
\(129\) 137.832 + 6.61637i 1.06847 + 0.0512897i
\(130\) 0 0
\(131\) 98.5827i 0.752539i −0.926510 0.376270i \(-0.877207\pi\)
0.926510 0.376270i \(-0.122793\pi\)
\(132\) 0 0
\(133\) −31.4229 −0.236262
\(134\) 0 0
\(135\) 8.65776 59.7498i 0.0641315 0.442591i
\(136\) 0 0
\(137\) 154.756i 1.12961i −0.825225 0.564804i \(-0.808952\pi\)
0.825225 0.564804i \(-0.191048\pi\)
\(138\) 0 0
\(139\) 227.186 1.63443 0.817215 0.576333i \(-0.195517\pi\)
0.817215 + 0.576333i \(0.195517\pi\)
\(140\) 0 0
\(141\) −7.44230 + 155.038i −0.0527822 + 1.09956i
\(142\) 0 0
\(143\) 72.5307i 0.507208i
\(144\) 0 0
\(145\) −87.1738 −0.601199
\(146\) 0 0
\(147\) 131.754 + 6.32460i 0.896287 + 0.0430245i
\(148\) 0 0
\(149\) 104.398i 0.700659i −0.936627 0.350330i \(-0.886070\pi\)
0.936627 0.350330i \(-0.113930\pi\)
\(150\) 0 0
\(151\) −216.256 −1.43216 −0.716079 0.698020i \(-0.754065\pi\)
−0.716079 + 0.698020i \(0.754065\pi\)
\(152\) 0 0
\(153\) −6.33134 + 65.7954i −0.0413813 + 0.430035i
\(154\) 0 0
\(155\) 98.4755i 0.635326i
\(156\) 0 0
\(157\) 239.367 1.52463 0.762316 0.647205i \(-0.224062\pi\)
0.762316 + 0.647205i \(0.224062\pi\)
\(158\) 0 0
\(159\) −8.72349 + 181.728i −0.0548647 + 1.14294i
\(160\) 0 0
\(161\) 28.2719i 0.175602i
\(162\) 0 0
\(163\) 19.8765 0.121942 0.0609708 0.998140i \(-0.480580\pi\)
0.0609708 + 0.998140i \(0.480580\pi\)
\(164\) 0 0
\(165\) −60.5146 2.90488i −0.366755 0.0176053i
\(166\) 0 0
\(167\) 293.172i 1.75552i −0.479099 0.877761i \(-0.659036\pi\)
0.479099 0.877761i \(-0.340964\pi\)
\(168\) 0 0
\(169\) −104.503 −0.618363
\(170\) 0 0
\(171\) −125.500 12.0766i −0.733919 0.0706234i
\(172\) 0 0
\(173\) 90.1213i 0.520932i 0.965483 + 0.260466i \(0.0838762\pi\)
−0.965483 + 0.260466i \(0.916124\pi\)
\(174\) 0 0
\(175\) −11.2153 −0.0640877
\(176\) 0 0
\(177\) −12.3731 + 257.756i −0.0699044 + 1.45625i
\(178\) 0 0
\(179\) 328.689i 1.83625i −0.396292 0.918124i \(-0.629703\pi\)
0.396292 0.918124i \(-0.370297\pi\)
\(180\) 0 0
\(181\) 200.100 1.10552 0.552762 0.833339i \(-0.313574\pi\)
0.552762 + 0.833339i \(0.313574\pi\)
\(182\) 0 0
\(183\) −200.918 9.64469i −1.09791 0.0527032i
\(184\) 0 0
\(185\) 56.6378i 0.306150i
\(186\) 0 0
\(187\) 66.3297 0.354704
\(188\) 0 0
\(189\) −59.9369 8.68486i −0.317127 0.0459517i
\(190\) 0 0
\(191\) 197.777i 1.03548i 0.855537 + 0.517742i \(0.173227\pi\)
−0.855537 + 0.517742i \(0.826773\pi\)
\(192\) 0 0
\(193\) −212.677 −1.10195 −0.550977 0.834521i \(-0.685745\pi\)
−0.550977 + 0.834521i \(0.685745\pi\)
\(194\) 0 0
\(195\) −2.58311 + 53.8115i −0.0132467 + 0.275956i
\(196\) 0 0
\(197\) 68.5241i 0.347838i 0.984760 + 0.173919i \(0.0556431\pi\)
−0.984760 + 0.173919i \(0.944357\pi\)
\(198\) 0 0
\(199\) 252.252 1.26760 0.633798 0.773498i \(-0.281495\pi\)
0.633798 + 0.773498i \(0.281495\pi\)
\(200\) 0 0
\(201\) 46.6572 + 2.23968i 0.232125 + 0.0111427i
\(202\) 0 0
\(203\) 87.4468i 0.430772i
\(204\) 0 0
\(205\) −19.0306 −0.0928322
\(206\) 0 0
\(207\) 10.8656 112.916i 0.0524909 0.545486i
\(208\) 0 0
\(209\) 126.519i 0.605356i
\(210\) 0 0
\(211\) 17.9653 0.0851435 0.0425717 0.999093i \(-0.486445\pi\)
0.0425717 + 0.999093i \(0.486445\pi\)
\(212\) 0 0
\(213\) −9.87462 + 205.708i −0.0463597 + 0.965767i
\(214\) 0 0
\(215\) 102.853i 0.478384i
\(216\) 0 0
\(217\) 98.7838 0.455225
\(218\) 0 0
\(219\) 271.720 + 13.0434i 1.24073 + 0.0595589i
\(220\) 0 0
\(221\) 58.9825i 0.266889i
\(222\) 0 0
\(223\) −142.384 −0.638493 −0.319246 0.947672i \(-0.603430\pi\)
−0.319246 + 0.947672i \(0.603430\pi\)
\(224\) 0 0
\(225\) −44.7931 4.31034i −0.199080 0.0191571i
\(226\) 0 0
\(227\) 321.954i 1.41830i 0.705059 + 0.709149i \(0.250921\pi\)
−0.705059 + 0.709149i \(0.749079\pi\)
\(228\) 0 0
\(229\) 107.541 0.469613 0.234806 0.972042i \(-0.424554\pi\)
0.234806 + 0.972042i \(0.424554\pi\)
\(230\) 0 0
\(231\) −2.91398 + 60.7040i −0.0126146 + 0.262788i
\(232\) 0 0
\(233\) 291.098i 1.24935i −0.780885 0.624675i \(-0.785232\pi\)
0.780885 0.624675i \(-0.214768\pi\)
\(234\) 0 0
\(235\) 115.692 0.492305
\(236\) 0 0
\(237\) −93.8461 4.50490i −0.395975 0.0190080i
\(238\) 0 0
\(239\) 452.505i 1.89333i 0.322223 + 0.946664i \(0.395570\pi\)
−0.322223 + 0.946664i \(0.604430\pi\)
\(240\) 0 0
\(241\) 62.6968 0.260153 0.130076 0.991504i \(-0.458478\pi\)
0.130076 + 0.991504i \(0.458478\pi\)
\(242\) 0 0
\(243\) −236.045 57.7218i −0.971378 0.237538i
\(244\) 0 0
\(245\) 98.3169i 0.401293i
\(246\) 0 0
\(247\) 112.505 0.455486
\(248\) 0 0
\(249\) −20.5762 + 428.643i −0.0826352 + 1.72146i
\(250\) 0 0
\(251\) 242.908i 0.967759i 0.875135 + 0.483880i \(0.160773\pi\)
−0.875135 + 0.483880i \(0.839227\pi\)
\(252\) 0 0
\(253\) −113.833 −0.449931
\(254\) 0 0
\(255\) 49.2109 + 2.36227i 0.192984 + 0.00926380i
\(256\) 0 0
\(257\) 258.831i 1.00712i −0.863959 0.503562i \(-0.832022\pi\)
0.863959 0.503562i \(-0.167978\pi\)
\(258\) 0 0
\(259\) 56.8152 0.219364
\(260\) 0 0
\(261\) −33.6080 + 349.255i −0.128766 + 1.33814i
\(262\) 0 0
\(263\) 205.895i 0.782869i 0.920206 + 0.391435i \(0.128021\pi\)
−0.920206 + 0.391435i \(0.871979\pi\)
\(264\) 0 0
\(265\) 135.608 0.511728
\(266\) 0 0
\(267\) −20.2216 + 421.257i −0.0757363 + 1.57774i
\(268\) 0 0
\(269\) 200.612i 0.745768i −0.927878 0.372884i \(-0.878369\pi\)
0.927878 0.372884i \(-0.121631\pi\)
\(270\) 0 0
\(271\) 46.0058 0.169763 0.0848816 0.996391i \(-0.472949\pi\)
0.0848816 + 0.996391i \(0.472949\pi\)
\(272\) 0 0
\(273\) 53.9800 + 2.59120i 0.197729 + 0.00949158i
\(274\) 0 0
\(275\) 45.1568i 0.164207i
\(276\) 0 0
\(277\) −256.932 −0.927551 −0.463776 0.885953i \(-0.653506\pi\)
−0.463776 + 0.885953i \(0.653506\pi\)
\(278\) 0 0
\(279\) 394.534 + 37.9651i 1.41410 + 0.136076i
\(280\) 0 0
\(281\) 42.9617i 0.152889i 0.997074 + 0.0764444i \(0.0243568\pi\)
−0.997074 + 0.0764444i \(0.975643\pi\)
\(282\) 0 0
\(283\) −519.170 −1.83452 −0.917261 0.398287i \(-0.869605\pi\)
−0.917261 + 0.398287i \(0.869605\pi\)
\(284\) 0 0
\(285\) −4.50587 + 93.8664i −0.0158101 + 0.329356i
\(286\) 0 0
\(287\) 19.0902i 0.0665163i
\(288\) 0 0
\(289\) 235.060 0.813357
\(290\) 0 0
\(291\) −2.47420 0.118769i −0.00850242 0.000408142i
\(292\) 0 0
\(293\) 88.7773i 0.302994i −0.988458 0.151497i \(-0.951591\pi\)
0.988458 0.151497i \(-0.0484094\pi\)
\(294\) 0 0
\(295\) 192.341 0.652005
\(296\) 0 0
\(297\) −34.9682 + 241.327i −0.117738 + 0.812547i
\(298\) 0 0
\(299\) 101.223i 0.338540i
\(300\) 0 0
\(301\) −103.175 −0.342773
\(302\) 0 0
\(303\) −21.4695 + 447.254i −0.0708565 + 1.47608i
\(304\) 0 0
\(305\) 149.928i 0.491568i
\(306\) 0 0
\(307\) −345.304 −1.12477 −0.562385 0.826876i \(-0.690116\pi\)
−0.562385 + 0.826876i \(0.690116\pi\)
\(308\) 0 0
\(309\) 258.634 + 12.4152i 0.837003 + 0.0401787i
\(310\) 0 0
\(311\) 65.2492i 0.209804i −0.994483 0.104902i \(-0.966547\pi\)
0.994483 0.104902i \(-0.0334529\pi\)
\(312\) 0 0
\(313\) 114.858 0.366959 0.183480 0.983023i \(-0.441264\pi\)
0.183480 + 0.983023i \(0.441264\pi\)
\(314\) 0 0
\(315\) −4.32383 + 44.9333i −0.0137265 + 0.142646i
\(316\) 0 0
\(317\) 10.0703i 0.0317675i −0.999874 0.0158837i \(-0.994944\pi\)
0.999874 0.0158837i \(-0.00505616\pi\)
\(318\) 0 0
\(319\) 352.091 1.10373
\(320\) 0 0
\(321\) 22.4783 468.268i 0.0700258 1.45878i
\(322\) 0 0
\(323\) 102.886i 0.318534i
\(324\) 0 0
\(325\) 40.1549 0.123553
\(326\) 0 0
\(327\) −78.1530 3.75158i −0.239000 0.0114727i
\(328\) 0 0
\(329\) 116.054i 0.352748i
\(330\) 0 0
\(331\) −365.387 −1.10389 −0.551944 0.833881i \(-0.686114\pi\)
−0.551944 + 0.833881i \(0.686114\pi\)
\(332\) 0 0
\(333\) 226.915 + 21.8355i 0.681425 + 0.0655720i
\(334\) 0 0
\(335\) 34.8162i 0.103929i
\(336\) 0 0
\(337\) −103.132 −0.306028 −0.153014 0.988224i \(-0.548898\pi\)
−0.153014 + 0.988224i \(0.548898\pi\)
\(338\) 0 0
\(339\) 6.16003 128.326i 0.0181712 0.378543i
\(340\) 0 0
\(341\) 397.738i 1.16639i
\(342\) 0 0
\(343\) −208.535 −0.607974
\(344\) 0 0
\(345\) −84.4538 4.05404i −0.244794 0.0117508i
\(346\) 0 0
\(347\) 411.888i 1.18700i 0.804836 + 0.593498i \(0.202253\pi\)
−0.804836 + 0.593498i \(0.797747\pi\)
\(348\) 0 0
\(349\) −492.648 −1.41160 −0.705800 0.708411i \(-0.749412\pi\)
−0.705800 + 0.708411i \(0.749412\pi\)
\(350\) 0 0
\(351\) 214.595 + 31.0949i 0.611382 + 0.0885894i
\(352\) 0 0
\(353\) 444.692i 1.25975i −0.776697 0.629875i \(-0.783106\pi\)
0.776697 0.629875i \(-0.216894\pi\)
\(354\) 0 0
\(355\) 153.503 0.432402
\(356\) 0 0
\(357\) 2.36967 49.3650i 0.00663772 0.138277i
\(358\) 0 0
\(359\) 137.546i 0.383136i 0.981479 + 0.191568i \(0.0613572\pi\)
−0.981479 + 0.191568i \(0.938643\pi\)
\(360\) 0 0
\(361\) −164.751 −0.456375
\(362\) 0 0
\(363\) −118.167 5.67238i −0.325530 0.0156264i
\(364\) 0 0
\(365\) 202.762i 0.555511i
\(366\) 0 0
\(367\) −122.989 −0.335119 −0.167559 0.985862i \(-0.553589\pi\)
−0.167559 + 0.985862i \(0.553589\pi\)
\(368\) 0 0
\(369\) −7.33683 + 76.2445i −0.0198830 + 0.206625i
\(370\) 0 0
\(371\) 136.033i 0.366665i
\(372\) 0 0
\(373\) 17.5537 0.0470609 0.0235305 0.999723i \(-0.492509\pi\)
0.0235305 + 0.999723i \(0.492509\pi\)
\(374\) 0 0
\(375\) −1.60822 + 33.5024i −0.00428858 + 0.0893398i
\(376\) 0 0
\(377\) 313.090i 0.830478i
\(378\) 0 0
\(379\) 295.437 0.779517 0.389759 0.920917i \(-0.372558\pi\)
0.389759 + 0.920917i \(0.372558\pi\)
\(380\) 0 0
\(381\) 294.904 + 14.1563i 0.774026 + 0.0371556i
\(382\) 0 0
\(383\) 249.327i 0.650984i 0.945545 + 0.325492i \(0.105530\pi\)
−0.945545 + 0.325492i \(0.894470\pi\)
\(384\) 0 0
\(385\) 45.2982 0.117658
\(386\) 0 0
\(387\) −412.070 39.6526i −1.06478 0.102461i
\(388\) 0 0
\(389\) 100.365i 0.258008i 0.991644 + 0.129004i \(0.0411780\pi\)
−0.991644 + 0.129004i \(0.958822\pi\)
\(390\) 0 0
\(391\) 92.5694 0.236750
\(392\) 0 0
\(393\) −14.1805 + 295.408i −0.0360826 + 0.751674i
\(394\) 0 0
\(395\) 70.0293i 0.177289i
\(396\) 0 0
\(397\) 420.402 1.05895 0.529473 0.848327i \(-0.322390\pi\)
0.529473 + 0.848327i \(0.322390\pi\)
\(398\) 0 0
\(399\) 94.1603 + 4.51998i 0.235991 + 0.0113283i
\(400\) 0 0
\(401\) 465.360i 1.16050i 0.814439 + 0.580249i \(0.197045\pi\)
−0.814439 + 0.580249i \(0.802955\pi\)
\(402\) 0 0
\(403\) −353.681 −0.877620
\(404\) 0 0
\(405\) −34.5380 + 177.798i −0.0852790 + 0.439007i
\(406\) 0 0
\(407\) 228.757i 0.562057i
\(408\) 0 0
\(409\) −441.627 −1.07977 −0.539886 0.841738i \(-0.681533\pi\)
−0.539886 + 0.841738i \(0.681533\pi\)
\(410\) 0 0
\(411\) −22.2607 + 463.735i −0.0541622 + 1.12831i
\(412\) 0 0
\(413\) 192.944i 0.467176i
\(414\) 0 0
\(415\) 319.860 0.770746
\(416\) 0 0
\(417\) −680.773 32.6792i −1.63255 0.0783673i
\(418\) 0 0
\(419\) 133.888i 0.319542i 0.987154 + 0.159771i \(0.0510756\pi\)
−0.987154 + 0.159771i \(0.948924\pi\)
\(420\) 0 0
\(421\) −659.449 −1.56639 −0.783193 0.621778i \(-0.786411\pi\)
−0.783193 + 0.621778i \(0.786411\pi\)
\(422\) 0 0
\(423\) 44.6024 463.509i 0.105443 1.09577i
\(424\) 0 0
\(425\) 36.7219i 0.0864044i
\(426\) 0 0
\(427\) 150.398 0.352219
\(428\) 0 0
\(429\) 10.4331 217.342i 0.0243195 0.506624i
\(430\) 0 0
\(431\) 464.823i 1.07848i 0.842153 + 0.539238i \(0.181288\pi\)
−0.842153 + 0.539238i \(0.818712\pi\)
\(432\) 0 0
\(433\) 393.188 0.908055 0.454028 0.890988i \(-0.349987\pi\)
0.454028 + 0.890988i \(0.349987\pi\)
\(434\) 0 0
\(435\) 261.221 + 12.5394i 0.600507 + 0.0288262i
\(436\) 0 0
\(437\) 176.570i 0.404050i
\(438\) 0 0
\(439\) 406.911 0.926904 0.463452 0.886122i \(-0.346611\pi\)
0.463452 + 0.886122i \(0.346611\pi\)
\(440\) 0 0
\(441\) −393.898 37.9039i −0.893193 0.0859500i
\(442\) 0 0
\(443\) 531.885i 1.20064i 0.799758 + 0.600322i \(0.204961\pi\)
−0.799758 + 0.600322i \(0.795039\pi\)
\(444\) 0 0
\(445\) 314.348 0.706399
\(446\) 0 0
\(447\) −15.0170 + 312.834i −0.0335951 + 0.699853i
\(448\) 0 0
\(449\) 690.916i 1.53879i −0.638774 0.769394i \(-0.720558\pi\)
0.638774 0.769394i \(-0.279442\pi\)
\(450\) 0 0
\(451\) 76.8636 0.170429
\(452\) 0 0
\(453\) 648.021 + 31.1070i 1.43051 + 0.0686688i
\(454\) 0 0
\(455\) 40.2806i 0.0885288i
\(456\) 0 0
\(457\) 885.218 1.93702 0.968510 0.248974i \(-0.0800934\pi\)
0.968510 + 0.248974i \(0.0800934\pi\)
\(458\) 0 0
\(459\) 28.4364 196.248i 0.0619530 0.427557i
\(460\) 0 0
\(461\) 516.543i 1.12048i −0.828329 0.560242i \(-0.810708\pi\)
0.828329 0.560242i \(-0.189292\pi\)
\(462\) 0 0
\(463\) −465.526 −1.00546 −0.502728 0.864445i \(-0.667670\pi\)
−0.502728 + 0.864445i \(0.667670\pi\)
\(464\) 0 0
\(465\) 14.1650 295.087i 0.0304625 0.634595i
\(466\) 0 0
\(467\) 646.351i 1.38405i −0.721874 0.692025i \(-0.756719\pi\)
0.721874 0.692025i \(-0.243281\pi\)
\(468\) 0 0
\(469\) −34.9253 −0.0744675
\(470\) 0 0
\(471\) −717.275 34.4314i −1.52288 0.0731027i
\(472\) 0 0
\(473\) 415.416i 0.878258i
\(474\) 0 0
\(475\) 70.0444 0.147462
\(476\) 0 0
\(477\) 52.2807 543.302i 0.109603 1.13900i
\(478\) 0 0
\(479\) 55.5045i 0.115876i 0.998320 + 0.0579379i \(0.0184526\pi\)
−0.998320 + 0.0579379i \(0.981547\pi\)
\(480\) 0 0
\(481\) −203.418 −0.422907
\(482\) 0 0
\(483\) −4.06673 + 84.7183i −0.00841973 + 0.175400i
\(484\) 0 0
\(485\) 1.84629i 0.00380678i
\(486\) 0 0
\(487\) −188.411 −0.386880 −0.193440 0.981112i \(-0.561964\pi\)
−0.193440 + 0.981112i \(0.561964\pi\)
\(488\) 0 0
\(489\) −59.5608 2.85910i −0.121801 0.00584683i
\(490\) 0 0
\(491\) 154.621i 0.314910i −0.987526 0.157455i \(-0.949671\pi\)
0.987526 0.157455i \(-0.0503290\pi\)
\(492\) 0 0
\(493\) −286.323 −0.580776
\(494\) 0 0
\(495\) 180.917 + 17.4092i 0.365489 + 0.0351702i
\(496\) 0 0
\(497\) 153.983i 0.309825i
\(498\) 0 0
\(499\) 961.012 1.92588 0.962938 0.269722i \(-0.0869318\pi\)
0.962938 + 0.269722i \(0.0869318\pi\)
\(500\) 0 0
\(501\) −42.1709 + 878.505i −0.0841734 + 1.75350i
\(502\) 0 0
\(503\) 435.850i 0.866500i −0.901274 0.433250i \(-0.857367\pi\)
0.901274 0.433250i \(-0.142633\pi\)
\(504\) 0 0
\(505\) 333.747 0.660885
\(506\) 0 0
\(507\) 313.150 + 15.0321i 0.617652 + 0.0296492i
\(508\) 0 0
\(509\) 748.589i 1.47070i 0.677685 + 0.735352i \(0.262983\pi\)
−0.677685 + 0.735352i \(0.737017\pi\)
\(510\) 0 0
\(511\) −203.396 −0.398036
\(512\) 0 0
\(513\) 374.330 + 54.2405i 0.729689 + 0.105732i
\(514\) 0 0
\(515\) 192.996i 0.374750i
\(516\) 0 0
\(517\) −467.273 −0.903816
\(518\) 0 0
\(519\) 12.9633 270.053i 0.0249775 0.520333i
\(520\) 0 0
\(521\) 331.457i 0.636193i 0.948058 + 0.318097i \(0.103044\pi\)
−0.948058 + 0.318097i \(0.896956\pi\)
\(522\) 0 0
\(523\) 223.173 0.426717 0.213359 0.976974i \(-0.431560\pi\)
0.213359 + 0.976974i \(0.431560\pi\)
\(524\) 0 0
\(525\) 33.6073 + 1.61325i 0.0640140 + 0.00307286i
\(526\) 0 0
\(527\) 323.443i 0.613744i
\(528\) 0 0
\(529\) 370.136 0.699690
\(530\) 0 0
\(531\) 74.1531 770.600i 0.139648 1.45122i
\(532\) 0 0
\(533\) 68.3495i 0.128236i
\(534\) 0 0
\(535\) −349.428 −0.653137
\(536\) 0 0
\(537\) −47.2797 + 984.931i −0.0880441 + 1.83414i
\(538\) 0 0
\(539\) 397.097i 0.736729i
\(540\) 0 0
\(541\) −674.035 −1.24591 −0.622953 0.782259i \(-0.714067\pi\)
−0.622953 + 0.782259i \(0.714067\pi\)
\(542\) 0 0
\(543\) −599.609 28.7830i −1.10425 0.0530074i
\(544\) 0 0
\(545\) 58.3189i 0.107007i
\(546\) 0 0
\(547\) 205.195 0.375128 0.187564 0.982252i \(-0.439941\pi\)
0.187564 + 0.982252i \(0.439941\pi\)
\(548\) 0 0
\(549\) 600.675 + 57.8016i 1.09413 + 0.105285i
\(550\) 0 0
\(551\) 546.141i 0.991181i
\(552\) 0 0
\(553\) 70.2486 0.127032
\(554\) 0 0
\(555\) 8.14697 169.718i 0.0146792 0.305798i
\(556\) 0 0
\(557\) 886.612i 1.59176i 0.605453 + 0.795881i \(0.292992\pi\)
−0.605453 + 0.795881i \(0.707008\pi\)
\(558\) 0 0
\(559\) 369.401 0.660825
\(560\) 0 0
\(561\) −198.760 9.54109i −0.354296 0.0170073i
\(562\) 0 0
\(563\) 453.686i 0.805836i −0.915236 0.402918i \(-0.867996\pi\)
0.915236 0.402918i \(-0.132004\pi\)
\(564\) 0 0
\(565\) −95.7586 −0.169484
\(566\) 0 0
\(567\) 178.355 + 34.6462i 0.314559 + 0.0611043i
\(568\) 0 0
\(569\) 313.662i 0.551251i −0.961265 0.275625i \(-0.911115\pi\)
0.961265 0.275625i \(-0.0888849\pi\)
\(570\) 0 0
\(571\) −418.596 −0.733092 −0.366546 0.930400i \(-0.619460\pi\)
−0.366546 + 0.930400i \(0.619460\pi\)
\(572\) 0 0
\(573\) 28.4490 592.650i 0.0496492 1.03429i
\(574\) 0 0
\(575\) 63.0207i 0.109601i
\(576\) 0 0
\(577\) −18.7107 −0.0324276 −0.0162138 0.999869i \(-0.505161\pi\)
−0.0162138 + 0.999869i \(0.505161\pi\)
\(578\) 0 0
\(579\) 637.297 + 30.5922i 1.10069 + 0.0528362i
\(580\) 0 0
\(581\) 320.861i 0.552257i
\(582\) 0 0
\(583\) −547.714 −0.939475
\(584\) 0 0
\(585\) 15.4808 160.877i 0.0264630 0.275004i
\(586\) 0 0
\(587\) 937.505i 1.59711i −0.601920 0.798556i \(-0.705597\pi\)
0.601920 0.798556i \(-0.294403\pi\)
\(588\) 0 0
\(589\) −616.945 −1.04745
\(590\) 0 0
\(591\) 9.85674 205.336i 0.0166781 0.347438i
\(592\) 0 0
\(593\) 902.817i 1.52246i 0.648484 + 0.761228i \(0.275403\pi\)
−0.648484 + 0.761228i \(0.724597\pi\)
\(594\) 0 0
\(595\) −36.8368 −0.0619106
\(596\) 0 0
\(597\) −755.885 36.2847i −1.26614 0.0607784i
\(598\) 0 0
\(599\) 991.033i 1.65448i −0.561849 0.827240i \(-0.689910\pi\)
0.561849 0.827240i \(-0.310090\pi\)
\(600\) 0 0
\(601\) 218.484 0.363534 0.181767 0.983342i \(-0.441818\pi\)
0.181767 + 0.983342i \(0.441818\pi\)
\(602\) 0 0
\(603\) −139.488 13.4226i −0.231324 0.0222598i
\(604\) 0 0
\(605\) 88.1781i 0.145749i
\(606\) 0 0
\(607\) −438.702 −0.722738 −0.361369 0.932423i \(-0.617690\pi\)
−0.361369 + 0.932423i \(0.617690\pi\)
\(608\) 0 0
\(609\) 12.5786 262.039i 0.0206546 0.430277i
\(610\) 0 0
\(611\) 415.514i 0.680055i
\(612\) 0 0
\(613\) 597.090 0.974046 0.487023 0.873389i \(-0.338083\pi\)
0.487023 + 0.873389i \(0.338083\pi\)
\(614\) 0 0
\(615\) 57.0261 + 2.73742i 0.0927254 + 0.00445110i
\(616\) 0 0
\(617\) 321.463i 0.521010i −0.965473 0.260505i \(-0.916111\pi\)
0.965473 0.260505i \(-0.0838890\pi\)
\(618\) 0 0
\(619\) 511.960 0.827076 0.413538 0.910487i \(-0.364293\pi\)
0.413538 + 0.910487i \(0.364293\pi\)
\(620\) 0 0
\(621\) −48.8015 + 336.794i −0.0785854 + 0.542342i
\(622\) 0 0
\(623\) 315.332i 0.506151i
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) 18.1990 379.121i 0.0290255 0.604659i
\(628\) 0 0
\(629\) 186.027i 0.295750i
\(630\) 0 0
\(631\) −1070.42 −1.69639 −0.848195 0.529684i \(-0.822311\pi\)
−0.848195 + 0.529684i \(0.822311\pi\)
\(632\) 0 0
\(633\) −53.8338 2.58418i −0.0850455 0.00408244i
\(634\) 0 0
\(635\) 220.062i 0.346554i
\(636\) 0 0
\(637\) 353.111 0.554335
\(638\) 0 0
\(639\) 59.1796 614.995i 0.0926128 0.962434i
\(640\) 0 0
\(641\) 865.824i 1.35074i 0.737479 + 0.675370i \(0.236016\pi\)
−0.737479 + 0.675370i \(0.763984\pi\)
\(642\) 0 0
\(643\) −731.670 −1.13790 −0.568950 0.822372i \(-0.692650\pi\)
−0.568950 + 0.822372i \(0.692650\pi\)
\(644\) 0 0
\(645\) −14.7947 + 308.203i −0.0229375 + 0.477834i
\(646\) 0 0
\(647\) 230.583i 0.356388i 0.983995 + 0.178194i \(0.0570254\pi\)
−0.983995 + 0.178194i \(0.942975\pi\)
\(648\) 0 0
\(649\) −776.857 −1.19701
\(650\) 0 0
\(651\) −296.011 14.2094i −0.454701 0.0218270i
\(652\) 0 0
\(653\) 617.718i 0.945970i 0.881070 + 0.472985i \(0.156824\pi\)
−0.881070 + 0.472985i \(0.843176\pi\)
\(654\) 0 0
\(655\) 220.438 0.336546
\(656\) 0 0
\(657\) −812.347 78.1703i −1.23645 0.118981i
\(658\) 0 0
\(659\) 265.710i 0.403202i −0.979468 0.201601i \(-0.935386\pi\)
0.979468 0.201601i \(-0.0646145\pi\)
\(660\) 0 0
\(661\) 987.246 1.49356 0.746782 0.665069i \(-0.231598\pi\)
0.746782 + 0.665069i \(0.231598\pi\)
\(662\) 0 0
\(663\) −8.48424 + 176.744i −0.0127967 + 0.266582i
\(664\) 0 0
\(665\) 70.2637i 0.105660i
\(666\) 0 0
\(667\) 491.376 0.736696
\(668\) 0 0
\(669\) 426.660 + 20.4810i 0.637758 + 0.0306143i
\(670\) 0 0
\(671\) 605.552i 0.902463i
\(672\) 0 0
\(673\) 587.447 0.872878 0.436439 0.899734i \(-0.356240\pi\)
0.436439 + 0.899734i \(0.356240\pi\)
\(674\) 0 0
\(675\) 133.605 + 19.3593i 0.197933 + 0.0286805i
\(676\) 0 0
\(677\) 168.689i 0.249172i −0.992209 0.124586i \(-0.960240\pi\)
0.992209 0.124586i \(-0.0397602\pi\)
\(678\) 0 0
\(679\) 1.85207 0.00272764
\(680\) 0 0
\(681\) 46.3109 964.750i 0.0680043 1.41667i
\(682\) 0 0
\(683\) 363.184i 0.531749i −0.964008 0.265874i \(-0.914339\pi\)
0.964008 0.265874i \(-0.0856606\pi\)
\(684\) 0 0
\(685\) 346.045 0.505176
\(686\) 0 0
\(687\) −322.253 15.4691i −0.469073 0.0225169i
\(688\) 0 0
\(689\) 487.044i 0.706886i
\(690\) 0 0
\(691\) 395.111 0.571795 0.285898 0.958260i \(-0.407708\pi\)
0.285898 + 0.958260i \(0.407708\pi\)
\(692\) 0 0
\(693\) 17.4637 181.484i 0.0252002 0.261881i
\(694\) 0 0
\(695\) 508.003i 0.730939i
\(696\) 0 0
\(697\) −62.5060 −0.0896787
\(698\) 0 0
\(699\) −41.8726 + 872.291i −0.0599035 + 1.24791i
\(700\) 0 0
\(701\) 473.115i 0.674914i −0.941341 0.337457i \(-0.890433\pi\)
0.941341 0.337457i \(-0.109567\pi\)
\(702\) 0 0
\(703\) −354.834 −0.504742
\(704\) 0 0
\(705\) −346.676 16.6415i −0.491739 0.0236049i
\(706\) 0 0
\(707\) 334.792i 0.473539i
\(708\) 0 0
\(709\) −441.603 −0.622853 −0.311427 0.950270i \(-0.600807\pi\)
−0.311427 + 0.950270i \(0.600807\pi\)
\(710\) 0 0
\(711\) 280.567 + 26.9983i 0.394608 + 0.0379723i
\(712\) 0 0
\(713\) 555.081i 0.778514i
\(714\) 0 0
\(715\) −162.184 −0.226830
\(716\) 0 0
\(717\) 65.0899 1355.95i 0.0907809 1.89115i
\(718\) 0 0
\(719\) 1096.09i 1.52446i −0.647307 0.762229i \(-0.724105\pi\)
0.647307 0.762229i \(-0.275895\pi\)
\(720\) 0 0
\(721\) −193.601 −0.268517
\(722\) 0 0
\(723\) −187.874 9.01852i −0.259853 0.0124737i
\(724\) 0 0
\(725\) 194.927i 0.268864i
\(726\) 0 0
\(727\) 1331.47 1.83146 0.915731 0.401791i \(-0.131612\pi\)
0.915731 + 0.401791i \(0.131612\pi\)
\(728\) 0 0
\(729\) 699.017 + 206.920i 0.958871 + 0.283841i
\(730\) 0 0
\(731\) 337.819i 0.462133i
\(732\) 0 0
\(733\) 295.464 0.403089 0.201545 0.979479i \(-0.435404\pi\)
0.201545 + 0.979479i \(0.435404\pi\)
\(734\) 0 0
\(735\) −14.1422 + 294.611i −0.0192411 + 0.400832i
\(736\) 0 0
\(737\) 140.621i 0.190802i
\(738\) 0 0
\(739\) −512.085 −0.692943 −0.346471 0.938061i \(-0.612620\pi\)
−0.346471 + 0.938061i \(0.612620\pi\)
\(740\) 0 0
\(741\) −337.127 16.1831i −0.454962 0.0218395i
\(742\) 0 0
\(743\) 117.627i 0.158313i −0.996862 0.0791567i \(-0.974777\pi\)
0.996862 0.0791567i \(-0.0252227\pi\)
\(744\) 0 0
\(745\) 233.441 0.313344
\(746\) 0 0
\(747\) 123.315 1281.49i 0.165080 1.71552i
\(748\) 0 0
\(749\) 350.523i 0.467987i
\(750\) 0 0
\(751\) 313.995 0.418103 0.209051 0.977905i \(-0.432962\pi\)
0.209051 + 0.977905i \(0.432962\pi\)
\(752\) 0 0
\(753\) 34.9406 727.884i 0.0464019 0.966646i
\(754\) 0 0
\(755\) 483.563i 0.640480i
\(756\) 0 0
\(757\) 880.421 1.16304 0.581520 0.813532i \(-0.302458\pi\)
0.581520 + 0.813532i \(0.302458\pi\)
\(758\) 0 0
\(759\) 341.105 + 16.3741i 0.449413 + 0.0215732i
\(760\) 0 0
\(761\) 177.647i 0.233439i 0.993165 + 0.116720i \(0.0372379\pi\)
−0.993165 + 0.116720i \(0.962762\pi\)
\(762\) 0 0
\(763\) 58.5015 0.0766730
\(764\) 0 0
\(765\) −147.123 14.1573i −0.192318 0.0185063i
\(766\) 0 0
\(767\) 690.806i 0.900660i
\(768\) 0 0
\(769\) −539.556 −0.701633 −0.350816 0.936444i \(-0.614096\pi\)
−0.350816 + 0.936444i \(0.614096\pi\)
\(770\) 0 0
\(771\) −37.2311 + 775.600i −0.0482894 + 1.00597i
\(772\) 0 0
\(773\) 648.160i 0.838499i −0.907871 0.419250i \(-0.862293\pi\)
0.907871 0.419250i \(-0.137707\pi\)
\(774\) 0 0
\(775\) −220.198 −0.284126
\(776\) 0 0
\(777\) −170.249 8.17248i −0.219111 0.0105180i
\(778\) 0 0
\(779\) 119.226i 0.153050i
\(780\) 0 0
\(781\) −619.989 −0.793840
\(782\) 0 0
\(783\) 150.946 1041.72i 0.192779 1.33043i
\(784\) 0 0
\(785\) 535.241i 0.681836i
\(786\) 0 0
\(787\) −1149.99 −1.46123 −0.730615 0.682789i \(-0.760767\pi\)
−0.730615 + 0.682789i \(0.760767\pi\)
\(788\) 0 0
\(789\) 29.6166 616.973i 0.0375368 0.781969i
\(790\) 0 0
\(791\) 96.0585i 0.121439i
\(792\) 0 0
\(793\) −538.476 −0.679037
\(794\) 0 0
\(795\) −406.356 19.5063i −0.511140 0.0245362i
\(796\) 0 0
\(797\) 1436.14i 1.80194i 0.433884 + 0.900969i \(0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(798\) 0 0
\(799\) 379.990 0.475581
\(800\) 0 0
\(801\) 121.190 1259.41i 0.151298 1.57229i
\(802\) 0 0
\(803\) 818.944i 1.01986i
\(804\) 0 0
\(805\) 63.2180 0.0785317
\(806\) 0 0
\(807\) −28.8567 + 601.143i −0.0357579 + 0.744910i
\(808\) 0 0
\(809\) 9.50158i 0.0117449i −0.999983 0.00587243i \(-0.998131\pi\)
0.999983 0.00587243i \(-0.00186926\pi\)
\(810\) 0 0
\(811\) −938.529 −1.15725 −0.578624 0.815594i \(-0.696410\pi\)
−0.578624 + 0.815594i \(0.696410\pi\)
\(812\) 0 0
\(813\) −137.859 6.61763i −0.169568 0.00813977i
\(814\) 0 0
\(815\) 44.4451i 0.0545339i
\(816\) 0 0
\(817\) 644.367 0.788699
\(818\) 0 0
\(819\) −161.381 15.5293i −0.197046 0.0189613i
\(820\) 0 0
\(821\) 469.104i 0.571381i −0.958322 0.285691i \(-0.907777\pi\)
0.958322 0.285691i \(-0.0922230\pi\)
\(822\) 0 0
\(823\) 1016.98 1.23570 0.617852 0.786294i \(-0.288003\pi\)
0.617852 + 0.786294i \(0.288003\pi\)
\(824\) 0 0
\(825\) 6.49551 135.315i 0.00787335 0.164018i
\(826\) 0 0
\(827\) 460.051i 0.556289i −0.960539 0.278145i \(-0.910281\pi\)
0.960539 0.278145i \(-0.0897194\pi\)
\(828\) 0 0
\(829\) −18.7992 −0.0226770 −0.0113385 0.999936i \(-0.503609\pi\)
−0.0113385 + 0.999936i \(0.503609\pi\)
\(830\) 0 0
\(831\) 769.909 + 36.9579i 0.926484 + 0.0444740i
\(832\) 0 0
\(833\) 322.922i 0.387661i
\(834\) 0 0
\(835\) 655.553 0.785093
\(836\) 0 0
\(837\) −1176.78 170.515i −1.40595 0.203722i
\(838\) 0 0
\(839\) 496.989i 0.592359i −0.955132 0.296179i \(-0.904287\pi\)
0.955132 0.296179i \(-0.0957126\pi\)
\(840\) 0 0
\(841\) −678.856 −0.807200
\(842\) 0 0
\(843\) 6.17976 128.737i 0.00733068 0.152713i
\(844\) 0 0
\(845\) 233.677i 0.276541i
\(846\) 0 0
\(847\) 88.4542 0.104432
\(848\) 0 0
\(849\) 1555.72 + 74.6791i 1.83241 + 0.0879613i
\(850\) 0 0
\(851\) 319.253i 0.375150i
\(852\) 0 0
\(853\) −1432.68 −1.67958 −0.839791 0.542911i \(-0.817322\pi\)
−0.839791 + 0.542911i \(0.817322\pi\)
\(854\) 0 0
\(855\) 27.0041 280.627i 0.0315837 0.328219i
\(856\) 0 0
\(857\) 1211.90i 1.41412i −0.707154 0.707060i \(-0.750021\pi\)
0.707154 0.707060i \(-0.249979\pi\)
\(858\) 0 0
\(859\) −18.5883 −0.0216394 −0.0108197 0.999941i \(-0.503444\pi\)
−0.0108197 + 0.999941i \(0.503444\pi\)
\(860\) 0 0
\(861\) 2.74600 57.2047i 0.00318931 0.0664398i
\(862\) 0 0
\(863\) 936.142i 1.08475i −0.840135 0.542377i \(-0.817525\pi\)
0.840135 0.542377i \(-0.182475\pi\)
\(864\) 0 0
\(865\) −201.517 −0.232968
\(866\) 0 0
\(867\) −704.370 33.8119i −0.812422 0.0389987i
\(868\) 0 0
\(869\) 282.845i 0.325483i
\(870\) 0 0
\(871\) 125.045 0.143564
\(872\) 0 0
\(873\) 7.39699 + 0.711796i 0.00847307 + 0.000815344i
\(874\) 0 0
\(875\) 25.0783i 0.0286609i
\(876\) 0 0
\(877\) 873.860 0.996419 0.498210 0.867057i \(-0.333991\pi\)
0.498210 + 0.867057i \(0.333991\pi\)
\(878\) 0 0
\(879\) −12.7700 + 266.025i −0.0145279 + 0.302646i
\(880\) 0 0
\(881\) 808.376i 0.917566i 0.888548 + 0.458783i \(0.151714\pi\)
−0.888548 + 0.458783i \(0.848286\pi\)
\(882\) 0 0
\(883\) −177.913 −0.201487 −0.100743 0.994912i \(-0.532122\pi\)
−0.100743 + 0.994912i \(0.532122\pi\)
\(884\) 0 0
\(885\) −576.361 27.6670i −0.651255 0.0312622i
\(886\) 0 0
\(887\) 1157.53i 1.30500i 0.757789 + 0.652499i \(0.226279\pi\)
−0.757789 + 0.652499i \(0.773721\pi\)
\(888\) 0 0
\(889\) −220.751 −0.248313
\(890\) 0 0
\(891\) 139.497 718.117i 0.156563 0.805968i
\(892\) 0 0
\(893\) 724.804i 0.811651i
\(894\) 0 0
\(895\) 734.970 0.821195
\(896\) 0 0
\(897\) 14.5603 303.321i 0.0162322 0.338151i
\(898\) 0 0
\(899\) 1716.90i 1.90979i
\(900\) 0 0
\(901\) 445.405 0.494345
\(902\) 0 0
\(903\) 309.168 + 14.8410i 0.342379 + 0.0164352i
\(904\) 0 0
\(905\) 447.437i 0.494405i
\(906\) 0 0
\(907\) −89.1205 −0.0982586 −0.0491293 0.998792i \(-0.515645\pi\)
−0.0491293 + 0.998792i \(0.515645\pi\)
\(908\) 0 0
\(909\) 128.669 1337.13i 0.141550 1.47099i
\(910\) 0 0
\(911\) 280.274i 0.307655i −0.988098 0.153827i \(-0.950840\pi\)
0.988098 0.153827i \(-0.0491600\pi\)
\(912\) 0 0
\(913\) −1291.90 −1.41500
\(914\) 0 0
\(915\) 21.5662 449.267i 0.0235696 0.491002i
\(916\) 0 0
\(917\) 221.128i 0.241143i
\(918\) 0 0
\(919\) 240.880 0.262111 0.131056 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(920\) 0 0
\(921\) 1034.72 + 49.6697i 1.12348 + 0.0539302i
\(922\) 0 0
\(923\) 551.314i 0.597307i
\(924\) 0 0
\(925\) −126.646 −0.136915
\(926\) 0 0
\(927\) −773.224 74.4056i −0.834114 0.0802649i
\(928\) 0 0
\(929\) 1119.74i 1.20531i 0.798001 + 0.602657i \(0.205891\pi\)
−0.798001 + 0.602657i \(0.794109\pi\)
\(930\) 0 0
\(931\) 615.951 0.661602
\(932\) 0 0
\(933\) −9.38566 + 195.522i −0.0100597 + 0.209563i
\(934\) 0 0
\(935\) 148.318i 0.158629i
\(936\) 0 0
\(937\) 771.901 0.823800 0.411900 0.911229i \(-0.364865\pi\)
0.411900 + 0.911229i \(0.364865\pi\)
\(938\) 0 0
\(939\) −344.179 16.5216i −0.366537 0.0175949i
\(940\) 0 0
\(941\) 1423.21i 1.51245i 0.654313 + 0.756223i \(0.272958\pi\)
−0.654313 + 0.756223i \(0.727042\pi\)
\(942\) 0 0
\(943\) 107.270 0.113754
\(944\) 0 0
\(945\) 19.4199 134.023i 0.0205502 0.141823i
\(946\) 0 0
\(947\) 492.999i 0.520590i −0.965529 0.260295i \(-0.916180\pi\)
0.965529 0.260295i \(-0.0838199\pi\)
\(948\) 0 0
\(949\) 728.231 0.767366
\(950\) 0 0
\(951\) −1.44854 + 30.1761i −0.00152318 + 0.0317309i
\(952\) 0 0
\(953\) 1076.66i 1.12976i −0.825173 0.564880i \(-0.808922\pi\)
0.825173 0.564880i \(-0.191078\pi\)
\(954\) 0 0
\(955\) −442.244 −0.463082
\(956\) 0 0
\(957\) −1055.06 50.6459i −1.10246 0.0529215i
\(958\) 0 0
\(959\) 347.129i 0.361970i
\(960\) 0 0
\(961\) 978.485 1.01819
\(962\) 0 0
\(963\) −134.715 + 1399.96i −0.139891 + 1.45374i
\(964\) 0 0
\(965\) 475.560i 0.492808i
\(966\) 0 0
\(967\) −7.16082 −0.00740519 −0.00370260 0.999993i \(-0.501179\pi\)
−0.00370260 + 0.999993i \(0.501179\pi\)
\(968\) 0 0
\(969\) −14.7995 + 308.304i −0.0152730 + 0.318167i
\(970\) 0 0
\(971\) 161.099i 0.165911i 0.996553 + 0.0829554i \(0.0264359\pi\)
−0.996553 + 0.0829554i \(0.973564\pi\)
\(972\) 0 0
\(973\) 509.593 0.523734
\(974\) 0 0
\(975\) −120.326 5.77601i −0.123411 0.00592412i
\(976\) 0 0
\(977\) 1311.58i 1.34246i 0.741248 + 0.671231i \(0.234234\pi\)
−0.741248 + 0.671231i \(0.765766\pi\)
\(978\) 0 0
\(979\) −1269.63 −1.29687
\(980\) 0 0
\(981\) 233.650 + 22.4836i 0.238175 + 0.0229190i
\(982\) 0 0
\(983\) 674.305i 0.685966i 0.939342 + 0.342983i \(0.111437\pi\)
−0.939342 + 0.342983i \(0.888563\pi\)
\(984\) 0 0
\(985\) −153.225 −0.155558
\(986\) 0 0
\(987\) −16.6936 + 347.761i −0.0169135 + 0.352342i
\(988\) 0 0
\(989\) 579.753i 0.586201i
\(990\) 0 0
\(991\) 1291.05 1.30277 0.651386 0.758746i \(-0.274188\pi\)
0.651386 + 0.758746i \(0.274188\pi\)
\(992\) 0 0
\(993\) 1094.90 + 52.5584i 1.10262 + 0.0529289i
\(994\) 0 0
\(995\) 564.052i 0.566886i
\(996\) 0 0
\(997\) 409.552 0.410784 0.205392 0.978680i \(-0.434153\pi\)
0.205392 + 0.978680i \(0.434153\pi\)
\(998\) 0 0
\(999\) −676.820 98.0713i −0.677497 0.0981694i
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 960.3.l.j.641.1 16
3.2 odd 2 inner 960.3.l.j.641.2 16
4.3 odd 2 inner 960.3.l.j.641.16 16
8.3 odd 2 480.3.l.b.161.1 16
8.5 even 2 480.3.l.b.161.16 yes 16
12.11 even 2 inner 960.3.l.j.641.15 16
24.5 odd 2 480.3.l.b.161.15 yes 16
24.11 even 2 480.3.l.b.161.2 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
480.3.l.b.161.1 16 8.3 odd 2
480.3.l.b.161.2 yes 16 24.11 even 2
480.3.l.b.161.15 yes 16 24.5 odd 2
480.3.l.b.161.16 yes 16 8.5 even 2
960.3.l.j.641.1 16 1.1 even 1 trivial
960.3.l.j.641.2 16 3.2 odd 2 inner
960.3.l.j.641.15 16 12.11 even 2 inner
960.3.l.j.641.16 16 4.3 odd 2 inner