Properties

Label 1008.3.cg.q.145.6
Level $1008$
Weight $3$
Character 1008.145
Analytic conductor $27.466$
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] = [1008,3,Mod(145,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.145");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.cg (of order \(6\), 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: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 81 x^{14} - 118 x^{13} + 1960 x^{12} - 366 x^{11} + 37625 x^{10} - 83714 x^{9} + 623931 x^{8} + 289492 x^{7} + 241286 x^{6} - 4777608 x^{5} + \cdots + 1148023744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.6
Root \(1.75486 - 2.33310i\) of defining polynomial
Character \(\chi\) \(=\) 1008.145
Dual form 1008.3.cg.q.577.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.18217 + 1.25988i) q^{5} +(3.00972 - 6.31993i) q^{7} +O(q^{10})\) \(q+(2.18217 + 1.25988i) q^{5} +(3.00972 - 6.31993i) q^{7} +(-2.78132 - 4.81739i) q^{11} -6.50184i q^{13} +(-17.6019 + 10.1625i) q^{17} +(-9.98781 - 5.76646i) q^{19} +(16.3094 - 28.2488i) q^{23} +(-9.32541 - 16.1521i) q^{25} -19.7142 q^{29} +(5.23235 - 3.02090i) q^{31} +(14.5301 - 9.99931i) q^{35} +(-5.71348 + 9.89604i) q^{37} +69.9985i q^{41} -40.6641 q^{43} +(-33.5140 - 19.3493i) q^{47} +(-30.8831 - 38.0425i) q^{49} +(16.4735 + 28.5330i) q^{53} -14.0165i q^{55} +(61.4324 - 35.4680i) q^{59} +(-35.8685 - 20.7087i) q^{61} +(8.19153 - 14.1881i) q^{65} +(-47.9001 - 82.9655i) q^{67} +57.6429 q^{71} +(28.2999 - 16.3389i) q^{73} +(-38.8166 + 3.07876i) q^{77} +(31.1447 - 53.9442i) q^{79} -35.3256i q^{83} -51.2140 q^{85} +(-6.60075 - 3.81095i) q^{89} +(-41.0912 - 19.5687i) q^{91} +(-14.5301 - 25.1669i) q^{95} -130.272i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{7} + 24 q^{19} + 36 q^{25} + 84 q^{31} - 68 q^{37} - 80 q^{43} - 184 q^{49} + 216 q^{61} - 56 q^{67} + 156 q^{73} - 28 q^{79} + 448 q^{85} - 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.18217 + 1.25988i 0.436435 + 0.251976i 0.702084 0.712094i \(-0.252253\pi\)
−0.265649 + 0.964070i \(0.585586\pi\)
\(6\) 0 0
\(7\) 3.00972 6.31993i 0.429960 0.902848i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.78132 4.81739i −0.252848 0.437945i 0.711461 0.702726i \(-0.248034\pi\)
−0.964309 + 0.264781i \(0.914700\pi\)
\(12\) 0 0
\(13\) 6.50184i 0.500142i −0.968228 0.250071i \(-0.919546\pi\)
0.968228 0.250071i \(-0.0804539\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −17.6019 + 10.1625i −1.03541 + 0.597793i −0.918529 0.395353i \(-0.870622\pi\)
−0.116879 + 0.993146i \(0.537289\pi\)
\(18\) 0 0
\(19\) −9.98781 5.76646i −0.525674 0.303498i 0.213579 0.976926i \(-0.431488\pi\)
−0.739253 + 0.673428i \(0.764821\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 16.3094 28.2488i 0.709106 1.22821i −0.256084 0.966655i \(-0.582432\pi\)
0.965189 0.261552i \(-0.0842344\pi\)
\(24\) 0 0
\(25\) −9.32541 16.1521i −0.373016 0.646083i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −19.7142 −0.679801 −0.339900 0.940461i \(-0.610393\pi\)
−0.339900 + 0.940461i \(0.610393\pi\)
\(30\) 0 0
\(31\) 5.23235 3.02090i 0.168785 0.0974483i −0.413228 0.910628i \(-0.635599\pi\)
0.582013 + 0.813179i \(0.302265\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 14.5301 9.99931i 0.415145 0.285695i
\(36\) 0 0
\(37\) −5.71348 + 9.89604i −0.154418 + 0.267461i −0.932847 0.360272i \(-0.882684\pi\)
0.778429 + 0.627733i \(0.216017\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 69.9985i 1.70728i 0.520862 + 0.853641i \(0.325610\pi\)
−0.520862 + 0.853641i \(0.674390\pi\)
\(42\) 0 0
\(43\) −40.6641 −0.945677 −0.472838 0.881149i \(-0.656771\pi\)
−0.472838 + 0.881149i \(0.656771\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −33.5140 19.3493i −0.713063 0.411687i 0.0991311 0.995074i \(-0.468394\pi\)
−0.812194 + 0.583387i \(0.801727\pi\)
\(48\) 0 0
\(49\) −30.8831 38.0425i −0.630268 0.776378i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 16.4735 + 28.5330i 0.310821 + 0.538358i 0.978540 0.206055i \(-0.0660627\pi\)
−0.667719 + 0.744413i \(0.732729\pi\)
\(54\) 0 0
\(55\) 14.0165i 0.254846i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 61.4324 35.4680i 1.04123 0.601153i 0.121046 0.992647i \(-0.461375\pi\)
0.920181 + 0.391494i \(0.128042\pi\)
\(60\) 0 0
\(61\) −35.8685 20.7087i −0.588008 0.339487i 0.176301 0.984336i \(-0.443587\pi\)
−0.764310 + 0.644849i \(0.776920\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.19153 14.1881i 0.126024 0.218279i
\(66\) 0 0
\(67\) −47.9001 82.9655i −0.714928 1.23829i −0.962987 0.269546i \(-0.913126\pi\)
0.248060 0.968745i \(-0.420207\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 57.6429 0.811871 0.405936 0.913902i \(-0.366946\pi\)
0.405936 + 0.913902i \(0.366946\pi\)
\(72\) 0 0
\(73\) 28.2999 16.3389i 0.387669 0.223821i −0.293480 0.955965i \(-0.594814\pi\)
0.681150 + 0.732144i \(0.261480\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −38.8166 + 3.07876i −0.504112 + 0.0399839i
\(78\) 0 0
\(79\) 31.1447 53.9442i 0.394237 0.682838i −0.598767 0.800923i \(-0.704342\pi\)
0.993003 + 0.118086i \(0.0376757\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 35.3256i 0.425610i −0.977095 0.212805i \(-0.931740\pi\)
0.977095 0.212805i \(-0.0682599\pi\)
\(84\) 0 0
\(85\) −51.2140 −0.602517
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.60075 3.81095i −0.0741658 0.0428196i 0.462459 0.886641i \(-0.346967\pi\)
−0.536624 + 0.843821i \(0.680301\pi\)
\(90\) 0 0
\(91\) −41.0912 19.5687i −0.451552 0.215041i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −14.5301 25.1669i −0.152948 0.264914i
\(96\) 0 0
\(97\) 130.272i 1.34301i −0.740999 0.671506i \(-0.765648\pi\)
0.740999 0.671506i \(-0.234352\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −36.2733 + 20.9424i −0.359141 + 0.207350i −0.668704 0.743529i \(-0.733151\pi\)
0.309563 + 0.950879i \(0.399817\pi\)
\(102\) 0 0
\(103\) −74.4377 42.9766i −0.722696 0.417249i 0.0930481 0.995662i \(-0.470339\pi\)
−0.815744 + 0.578413i \(0.803672\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −80.8938 + 140.112i −0.756016 + 1.30946i 0.188851 + 0.982006i \(0.439524\pi\)
−0.944867 + 0.327453i \(0.893810\pi\)
\(108\) 0 0
\(109\) −43.4215 75.2082i −0.398362 0.689984i 0.595162 0.803606i \(-0.297088\pi\)
−0.993524 + 0.113622i \(0.963755\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 86.5429 0.765866 0.382933 0.923776i \(-0.374914\pi\)
0.382933 + 0.923776i \(0.374914\pi\)
\(114\) 0 0
\(115\) 71.1800 41.0958i 0.618957 0.357355i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 11.2493 + 141.829i 0.0945316 + 1.19184i
\(120\) 0 0
\(121\) 45.0285 77.9916i 0.372136 0.644559i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 109.989i 0.879916i
\(126\) 0 0
\(127\) −29.4865 −0.232177 −0.116089 0.993239i \(-0.537036\pi\)
−0.116089 + 0.993239i \(0.537036\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −169.021 97.5843i −1.29024 0.744918i −0.311540 0.950233i \(-0.600845\pi\)
−0.978696 + 0.205315i \(0.934178\pi\)
\(132\) 0 0
\(133\) −66.5042 + 45.7668i −0.500032 + 0.344112i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.821473 1.42283i −0.00599615 0.0103856i 0.863012 0.505184i \(-0.168575\pi\)
−0.869008 + 0.494798i \(0.835242\pi\)
\(138\) 0 0
\(139\) 9.93842i 0.0714994i −0.999361 0.0357497i \(-0.988618\pi\)
0.999361 0.0357497i \(-0.0113819\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −31.3219 + 18.0837i −0.219034 + 0.126460i
\(144\) 0 0
\(145\) −43.0199 24.8375i −0.296689 0.171293i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 145.960 252.810i 0.979598 1.69671i 0.315758 0.948840i \(-0.397741\pi\)
0.663840 0.747874i \(-0.268925\pi\)
\(150\) 0 0
\(151\) −40.9781 70.9761i −0.271378 0.470041i 0.697837 0.716257i \(-0.254146\pi\)
−0.969215 + 0.246216i \(0.920813\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 15.2239 0.0982185
\(156\) 0 0
\(157\) −44.0946 + 25.4580i −0.280857 + 0.162153i −0.633812 0.773488i \(-0.718511\pi\)
0.352954 + 0.935641i \(0.385177\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −129.443 188.095i −0.803996 1.16829i
\(162\) 0 0
\(163\) −40.6690 + 70.4408i −0.249503 + 0.432152i −0.963388 0.268111i \(-0.913601\pi\)
0.713885 + 0.700263i \(0.246934\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.97762i 0.0178301i 0.999960 + 0.00891503i \(0.00283778\pi\)
−0.999960 + 0.00891503i \(0.997162\pi\)
\(168\) 0 0
\(169\) 126.726 0.749858
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 190.687 + 110.093i 1.10224 + 0.636378i 0.936808 0.349843i \(-0.113765\pi\)
0.165431 + 0.986221i \(0.447099\pi\)
\(174\) 0 0
\(175\) −130.147 + 10.3227i −0.743697 + 0.0589867i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 129.984 + 225.139i 0.726167 + 1.25776i 0.958492 + 0.285119i \(0.0920332\pi\)
−0.232325 + 0.972638i \(0.574633\pi\)
\(180\) 0 0
\(181\) 337.053i 1.86217i 0.364799 + 0.931086i \(0.381138\pi\)
−0.364799 + 0.931086i \(0.618862\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −24.9356 + 14.3966i −0.134787 + 0.0778194i
\(186\) 0 0
\(187\) 97.9134 + 56.5303i 0.523601 + 0.302301i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −102.491 + 177.520i −0.536602 + 0.929422i 0.462482 + 0.886629i \(0.346959\pi\)
−0.999084 + 0.0427932i \(0.986374\pi\)
\(192\) 0 0
\(193\) −131.261 227.351i −0.680111 1.17799i −0.974947 0.222438i \(-0.928598\pi\)
0.294836 0.955548i \(-0.404735\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −142.697 −0.724349 −0.362175 0.932110i \(-0.617966\pi\)
−0.362175 + 0.932110i \(0.617966\pi\)
\(198\) 0 0
\(199\) −24.0813 + 13.9033i −0.121011 + 0.0698659i −0.559284 0.828976i \(-0.688924\pi\)
0.438273 + 0.898842i \(0.355590\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −59.3344 + 124.593i −0.292287 + 0.613757i
\(204\) 0 0
\(205\) −88.1897 + 152.749i −0.430194 + 0.745117i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 64.1536i 0.306955i
\(210\) 0 0
\(211\) 214.007 1.01425 0.507125 0.861872i \(-0.330708\pi\)
0.507125 + 0.861872i \(0.330708\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −88.7361 51.2318i −0.412726 0.238288i
\(216\) 0 0
\(217\) −3.34396 42.1602i −0.0154099 0.194287i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 66.0748 + 114.445i 0.298981 + 0.517851i
\(222\) 0 0
\(223\) 125.714i 0.563738i 0.959453 + 0.281869i \(0.0909543\pi\)
−0.959453 + 0.281869i \(0.909046\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 193.764 111.870i 0.853585 0.492818i −0.00827359 0.999966i \(-0.502634\pi\)
0.861859 + 0.507148i \(0.169300\pi\)
\(228\) 0 0
\(229\) 199.180 + 114.997i 0.869782 + 0.502169i 0.867276 0.497828i \(-0.165869\pi\)
0.00250654 + 0.999997i \(0.499202\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 201.147 348.396i 0.863291 1.49526i −0.00544355 0.999985i \(-0.501733\pi\)
0.868734 0.495278i \(-0.164934\pi\)
\(234\) 0 0
\(235\) −48.7555 84.4471i −0.207470 0.359349i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −52.4038 −0.219263 −0.109631 0.993972i \(-0.534967\pi\)
−0.109631 + 0.993972i \(0.534967\pi\)
\(240\) 0 0
\(241\) 414.018 239.034i 1.71792 0.991840i 0.795216 0.606327i \(-0.207358\pi\)
0.922702 0.385513i \(-0.125976\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −19.4634 121.924i −0.0794426 0.497651i
\(246\) 0 0
\(247\) −37.4926 + 64.9391i −0.151792 + 0.262912i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 98.1715i 0.391121i −0.980692 0.195561i \(-0.937347\pi\)
0.980692 0.195561i \(-0.0626527\pi\)
\(252\) 0 0
\(253\) −181.447 −0.717183
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 348.332 + 201.110i 1.35538 + 0.782527i 0.988997 0.147938i \(-0.0472636\pi\)
0.366380 + 0.930465i \(0.380597\pi\)
\(258\) 0 0
\(259\) 45.3463 + 65.8932i 0.175082 + 0.254414i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −183.196 317.305i −0.696564 1.20648i −0.969651 0.244494i \(-0.921378\pi\)
0.273087 0.961989i \(-0.411955\pi\)
\(264\) 0 0
\(265\) 83.0186i 0.313278i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −308.338 + 178.019i −1.14624 + 0.661780i −0.947967 0.318368i \(-0.896865\pi\)
−0.198269 + 0.980148i \(0.563532\pi\)
\(270\) 0 0
\(271\) 275.573 + 159.102i 1.01687 + 0.587092i 0.913197 0.407519i \(-0.133606\pi\)
0.103677 + 0.994611i \(0.466939\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −51.8740 + 89.8484i −0.188633 + 0.326721i
\(276\) 0 0
\(277\) 113.444 + 196.491i 0.409546 + 0.709354i 0.994839 0.101468i \(-0.0323540\pi\)
−0.585293 + 0.810822i \(0.699021\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −274.978 −0.978570 −0.489285 0.872124i \(-0.662742\pi\)
−0.489285 + 0.872124i \(0.662742\pi\)
\(282\) 0 0
\(283\) 134.926 77.8995i 0.476770 0.275263i −0.242299 0.970202i \(-0.577902\pi\)
0.719069 + 0.694938i \(0.244568\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 442.386 + 210.676i 1.54142 + 0.734063i
\(288\) 0 0
\(289\) 62.0521 107.477i 0.214713 0.371894i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 484.307i 1.65293i 0.562992 + 0.826463i \(0.309650\pi\)
−0.562992 + 0.826463i \(0.690350\pi\)
\(294\) 0 0
\(295\) 178.742 0.605904
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −183.669 106.041i −0.614277 0.354653i
\(300\) 0 0
\(301\) −122.388 + 256.994i −0.406604 + 0.853802i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −52.1809 90.3800i −0.171085 0.296328i
\(306\) 0 0
\(307\) 258.696i 0.842658i 0.906908 + 0.421329i \(0.138436\pi\)
−0.906908 + 0.421329i \(0.861564\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 451.891 260.899i 1.45303 0.838905i 0.454374 0.890811i \(-0.349863\pi\)
0.998652 + 0.0519066i \(0.0165298\pi\)
\(312\) 0 0
\(313\) 306.163 + 176.763i 0.978155 + 0.564738i 0.901713 0.432336i \(-0.142310\pi\)
0.0764426 + 0.997074i \(0.475644\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −37.9439 + 65.7208i −0.119697 + 0.207321i −0.919648 0.392745i \(-0.871526\pi\)
0.799951 + 0.600066i \(0.204859\pi\)
\(318\) 0 0
\(319\) 54.8316 + 94.9712i 0.171886 + 0.297715i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 234.406 0.725716
\(324\) 0 0
\(325\) −105.018 + 60.6323i −0.323133 + 0.186561i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −223.154 + 153.570i −0.678280 + 0.466778i
\(330\) 0 0
\(331\) −200.896 + 347.963i −0.606937 + 1.05125i 0.384805 + 0.922998i \(0.374269\pi\)
−0.991742 + 0.128248i \(0.959065\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 241.394i 0.720578i
\(336\) 0 0
\(337\) 292.693 0.868525 0.434262 0.900786i \(-0.357009\pi\)
0.434262 + 0.900786i \(0.357009\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −29.1057 16.8042i −0.0853540 0.0492792i
\(342\) 0 0
\(343\) −333.376 + 80.6820i −0.971941 + 0.235224i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −227.335 393.755i −0.655143 1.13474i −0.981858 0.189618i \(-0.939275\pi\)
0.326715 0.945123i \(-0.394058\pi\)
\(348\) 0 0
\(349\) 583.961i 1.67324i 0.547782 + 0.836621i \(0.315472\pi\)
−0.547782 + 0.836621i \(0.684528\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −263.295 + 152.013i −0.745878 + 0.430633i −0.824203 0.566295i \(-0.808376\pi\)
0.0783245 + 0.996928i \(0.475043\pi\)
\(354\) 0 0
\(355\) 125.787 + 72.6230i 0.354329 + 0.204572i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −85.5609 + 148.196i −0.238331 + 0.412802i −0.960236 0.279191i \(-0.909934\pi\)
0.721904 + 0.691993i \(0.243267\pi\)
\(360\) 0 0
\(361\) −113.996 197.447i −0.315778 0.546943i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 82.3403 0.225590
\(366\) 0 0
\(367\) 596.765 344.543i 1.62606 0.938808i 0.640811 0.767698i \(-0.278598\pi\)
0.985252 0.171110i \(-0.0547353\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 229.907 18.2352i 0.619696 0.0491515i
\(372\) 0 0
\(373\) −360.676 + 624.709i −0.966960 + 1.67482i −0.262707 + 0.964876i \(0.584615\pi\)
−0.704254 + 0.709948i \(0.748718\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 128.179i 0.339997i
\(378\) 0 0
\(379\) −282.500 −0.745383 −0.372692 0.927955i \(-0.621565\pi\)
−0.372692 + 0.927955i \(0.621565\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −588.573 339.813i −1.53674 0.887240i −0.999026 0.0441150i \(-0.985953\pi\)
−0.537718 0.843125i \(-0.680713\pi\)
\(384\) 0 0
\(385\) −88.5835 42.1859i −0.230087 0.109574i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 105.767 + 183.194i 0.271895 + 0.470935i 0.969347 0.245696i \(-0.0790164\pi\)
−0.697452 + 0.716631i \(0.745683\pi\)
\(390\) 0 0
\(391\) 662.977i 1.69559i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 135.926 78.4771i 0.344117 0.198676i
\(396\) 0 0
\(397\) 46.7416 + 26.9863i 0.117737 + 0.0679755i 0.557712 0.830035i \(-0.311679\pi\)
−0.439975 + 0.898010i \(0.645013\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −88.7998 + 153.806i −0.221446 + 0.383556i −0.955247 0.295808i \(-0.904411\pi\)
0.733801 + 0.679364i \(0.237744\pi\)
\(402\) 0 0
\(403\) −19.6414 34.0199i −0.0487380 0.0844166i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 63.5642 0.156177
\(408\) 0 0
\(409\) 429.670 248.070i 1.05054 0.606528i 0.127738 0.991808i \(-0.459228\pi\)
0.922800 + 0.385280i \(0.125895\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −39.2610 494.998i −0.0950629 1.19854i
\(414\) 0 0
\(415\) 44.5060 77.0866i 0.107243 0.185751i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 629.543i 1.50249i −0.660023 0.751245i \(-0.729454\pi\)
0.660023 0.751245i \(-0.270546\pi\)
\(420\) 0 0
\(421\) 342.355 0.813196 0.406598 0.913607i \(-0.366715\pi\)
0.406598 + 0.913607i \(0.366715\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 328.291 + 189.539i 0.772448 + 0.445973i
\(426\) 0 0
\(427\) −238.832 + 164.359i −0.559325 + 0.384916i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 239.388 + 414.633i 0.555425 + 0.962025i 0.997870 + 0.0652291i \(0.0207778\pi\)
−0.442445 + 0.896796i \(0.645889\pi\)
\(432\) 0 0
\(433\) 27.1130i 0.0626166i −0.999510 0.0313083i \(-0.990033\pi\)
0.999510 0.0313083i \(-0.00996737\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −325.791 + 188.095i −0.745517 + 0.430424i
\(438\) 0 0
\(439\) 406.217 + 234.530i 0.925325 + 0.534236i 0.885330 0.464964i \(-0.153933\pi\)
0.0399947 + 0.999200i \(0.487266\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −362.893 + 628.550i −0.819172 + 1.41885i 0.0871206 + 0.996198i \(0.472233\pi\)
−0.906293 + 0.422650i \(0.861100\pi\)
\(444\) 0 0
\(445\) −9.60266 16.6323i −0.0215790 0.0373760i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −333.343 −0.742412 −0.371206 0.928551i \(-0.621056\pi\)
−0.371206 + 0.928551i \(0.621056\pi\)
\(450\) 0 0
\(451\) 337.211 194.689i 0.747695 0.431682i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −65.0139 94.4723i −0.142888 0.207632i
\(456\) 0 0
\(457\) 284.818 493.319i 0.623234 1.07947i −0.365646 0.930754i \(-0.619152\pi\)
0.988880 0.148719i \(-0.0475148\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 137.485i 0.298233i −0.988820 0.149116i \(-0.952357\pi\)
0.988820 0.149116i \(-0.0476429\pi\)
\(462\) 0 0
\(463\) 216.686 0.468004 0.234002 0.972236i \(-0.424818\pi\)
0.234002 + 0.972236i \(0.424818\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 249.762 + 144.200i 0.534821 + 0.308779i 0.742978 0.669316i \(-0.233413\pi\)
−0.208156 + 0.978096i \(0.566746\pi\)
\(468\) 0 0
\(469\) −668.503 + 53.0226i −1.42538 + 0.113055i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 113.100 + 195.895i 0.239112 + 0.414154i
\(474\) 0 0
\(475\) 215.099i 0.452839i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 202.257 116.773i 0.422248 0.243785i −0.273791 0.961789i \(-0.588278\pi\)
0.696039 + 0.718004i \(0.254944\pi\)
\(480\) 0 0
\(481\) 64.3425 + 37.1482i 0.133768 + 0.0772311i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 164.127 284.277i 0.338407 0.586137i
\(486\) 0 0
\(487\) −411.600 712.913i −0.845175 1.46389i −0.885469 0.464699i \(-0.846163\pi\)
0.0402936 0.999188i \(-0.487171\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 506.517 1.03160 0.515801 0.856708i \(-0.327494\pi\)
0.515801 + 0.856708i \(0.327494\pi\)
\(492\) 0 0
\(493\) 347.008 200.345i 0.703871 0.406380i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 173.489 364.299i 0.349073 0.732996i
\(498\) 0 0
\(499\) 75.3022 130.427i 0.150906 0.261377i −0.780655 0.624963i \(-0.785114\pi\)
0.931561 + 0.363585i \(0.118448\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 562.598i 1.11848i 0.829004 + 0.559242i \(0.188908\pi\)
−0.829004 + 0.559242i \(0.811092\pi\)
\(504\) 0 0
\(505\) −105.539 −0.208989
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 511.698 + 295.429i 1.00530 + 0.580411i 0.909813 0.415019i \(-0.136225\pi\)
0.0954889 + 0.995430i \(0.469559\pi\)
\(510\) 0 0
\(511\) −18.0862 228.029i −0.0353938 0.446241i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −108.291 187.565i −0.210273 0.364204i
\(516\) 0 0
\(517\) 215.267i 0.416376i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 70.6996 40.8184i 0.135700 0.0783463i −0.430613 0.902537i \(-0.641703\pi\)
0.566313 + 0.824190i \(0.308369\pi\)
\(522\) 0 0
\(523\) −623.961 360.244i −1.19304 0.688803i −0.234047 0.972225i \(-0.575197\pi\)
−0.958995 + 0.283422i \(0.908530\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −61.3997 + 106.347i −0.116508 + 0.201798i
\(528\) 0 0
\(529\) −267.495 463.315i −0.505661 0.875831i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 455.119 0.853882
\(534\) 0 0
\(535\) −353.049 + 203.833i −0.659904 + 0.380996i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −97.3697 + 254.585i −0.180649 + 0.472328i
\(540\) 0 0
\(541\) −302.518 + 523.977i −0.559183 + 0.968534i 0.438381 + 0.898789i \(0.355552\pi\)
−0.997565 + 0.0697451i \(0.977781\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 218.823i 0.401511i
\(546\) 0 0
\(547\) 570.576 1.04310 0.521550 0.853221i \(-0.325354\pi\)
0.521550 + 0.853221i \(0.325354\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 196.902 + 113.681i 0.357354 + 0.206318i
\(552\) 0 0
\(553\) −247.187 359.189i −0.446992 0.649529i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −449.675 778.859i −0.807315 1.39831i −0.914717 0.404095i \(-0.867586\pi\)
0.107402 0.994216i \(-0.465747\pi\)
\(558\) 0 0
\(559\) 264.392i 0.472972i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −939.781 + 542.583i −1.66924 + 0.963735i −0.701188 + 0.712976i \(0.747347\pi\)
−0.968050 + 0.250759i \(0.919320\pi\)
\(564\) 0 0
\(565\) 188.852 + 109.034i 0.334251 + 0.192980i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 269.018 465.953i 0.472791 0.818899i −0.526724 0.850037i \(-0.676580\pi\)
0.999515 + 0.0311379i \(0.00991310\pi\)
\(570\) 0 0
\(571\) −373.747 647.348i −0.654548 1.13371i −0.982007 0.188845i \(-0.939526\pi\)
0.327459 0.944865i \(-0.393808\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −608.368 −1.05803
\(576\) 0 0
\(577\) 373.623 215.711i 0.647527 0.373850i −0.139981 0.990154i \(-0.544704\pi\)
0.787508 + 0.616304i \(0.211371\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −223.256 106.320i −0.384261 0.182995i
\(582\) 0 0
\(583\) 91.6364 158.719i 0.157181 0.272245i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 249.815i 0.425579i 0.977098 + 0.212790i \(0.0682549\pi\)
−0.977098 + 0.212790i \(0.931745\pi\)
\(588\) 0 0
\(589\) −69.6796 −0.118302
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 175.770 + 101.481i 0.296408 + 0.171131i 0.640828 0.767684i \(-0.278591\pi\)
−0.344420 + 0.938816i \(0.611925\pi\)
\(594\) 0 0
\(595\) −154.140 + 323.669i −0.259059 + 0.543981i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 10.5681 + 18.3044i 0.0176428 + 0.0305583i 0.874712 0.484643i \(-0.161050\pi\)
−0.857069 + 0.515201i \(0.827717\pi\)
\(600\) 0 0
\(601\) 112.712i 0.187540i 0.995594 + 0.0937701i \(0.0298919\pi\)
−0.995594 + 0.0937701i \(0.970108\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 196.520 113.461i 0.324826 0.187539i
\(606\) 0 0
\(607\) 178.155 + 102.858i 0.293501 + 0.169453i 0.639520 0.768775i \(-0.279133\pi\)
−0.346019 + 0.938228i \(0.612467\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −125.806 + 217.902i −0.205902 + 0.356633i
\(612\) 0 0
\(613\) −542.469 939.584i −0.884941 1.53276i −0.845781 0.533530i \(-0.820865\pi\)
−0.0391596 0.999233i \(-0.512468\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 454.282 0.736276 0.368138 0.929771i \(-0.379995\pi\)
0.368138 + 0.929771i \(0.379995\pi\)
\(618\) 0 0
\(619\) 699.389 403.793i 1.12987 0.652330i 0.185967 0.982556i \(-0.440458\pi\)
0.943902 + 0.330226i \(0.107125\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −43.9514 + 30.2464i −0.0705479 + 0.0485496i
\(624\) 0 0
\(625\) −94.5618 + 163.786i −0.151299 + 0.262057i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 232.253i 0.369241i
\(630\) 0 0
\(631\) −1139.48 −1.80583 −0.902915 0.429819i \(-0.858577\pi\)
−0.902915 + 0.429819i \(0.858577\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −64.3448 37.1495i −0.101330 0.0585031i
\(636\) 0 0
\(637\) −247.346 + 200.797i −0.388299 + 0.315223i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 43.6700 + 75.6387i 0.0681280 + 0.118001i 0.898077 0.439838i \(-0.144964\pi\)
−0.829949 + 0.557839i \(0.811631\pi\)
\(642\) 0 0
\(643\) 258.078i 0.401366i −0.979656 0.200683i \(-0.935684\pi\)
0.979656 0.200683i \(-0.0643161\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10.3257 5.96152i 0.0159593 0.00921410i −0.491999 0.870596i \(-0.663734\pi\)
0.507958 + 0.861382i \(0.330400\pi\)
\(648\) 0 0
\(649\) −341.727 197.296i −0.526544 0.304000i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 153.685 266.190i 0.235352 0.407641i −0.724023 0.689776i \(-0.757709\pi\)
0.959375 + 0.282135i \(0.0910425\pi\)
\(654\) 0 0
\(655\) −245.889 425.892i −0.375403 0.650217i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −135.643 −0.205832 −0.102916 0.994690i \(-0.532817\pi\)
−0.102916 + 0.994690i \(0.532817\pi\)
\(660\) 0 0
\(661\) −177.211 + 102.313i −0.268095 + 0.154785i −0.628022 0.778196i \(-0.716135\pi\)
0.359926 + 0.932981i \(0.382802\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −202.784 + 16.0840i −0.304939 + 0.0241864i
\(666\) 0 0
\(667\) −321.528 + 556.902i −0.482051 + 0.834936i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 230.390i 0.343354i
\(672\) 0 0
\(673\) 243.179 0.361336 0.180668 0.983544i \(-0.442174\pi\)
0.180668 + 0.983544i \(0.442174\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −571.368 329.880i −0.843971 0.487267i 0.0146410 0.999893i \(-0.495339\pi\)
−0.858612 + 0.512626i \(0.828673\pi\)
\(678\) 0 0
\(679\) −823.312 392.083i −1.21254 0.577442i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 162.302 + 281.116i 0.237631 + 0.411589i 0.960034 0.279883i \(-0.0902957\pi\)
−0.722403 + 0.691472i \(0.756962\pi\)
\(684\) 0 0
\(685\) 4.13982i 0.00604354i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 185.517 107.108i 0.269255 0.155455i
\(690\) 0 0
\(691\) 506.305 + 292.315i 0.732714 + 0.423033i 0.819414 0.573202i \(-0.194299\pi\)
−0.0867003 + 0.996234i \(0.527632\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.5212 21.6874i 0.0180161 0.0312048i
\(696\) 0 0
\(697\) −711.359 1232.11i −1.02060 1.76773i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −870.839 −1.24228 −0.621141 0.783699i \(-0.713330\pi\)
−0.621141 + 0.783699i \(0.713330\pi\)
\(702\) 0 0
\(703\) 114.130 65.8932i 0.162348 0.0937314i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 23.1820 + 292.276i 0.0327892 + 0.413402i
\(708\) 0 0
\(709\) 388.383 672.700i 0.547790 0.948801i −0.450635 0.892708i \(-0.648803\pi\)
0.998426 0.0560925i \(-0.0178642\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 197.077i 0.276405i
\(714\) 0 0
\(715\) −91.1332 −0.127459
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 188.524 + 108.844i 0.262202 + 0.151383i 0.625339 0.780353i \(-0.284961\pi\)
−0.363136 + 0.931736i \(0.618294\pi\)
\(720\) 0 0
\(721\) −495.646 + 341.094i −0.687443 + 0.473084i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 183.843 + 318.426i 0.253577 + 0.439208i
\(726\) 0 0
\(727\) 500.204i 0.688038i −0.938963 0.344019i \(-0.888212\pi\)
0.938963 0.344019i \(-0.111788\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 715.767 413.248i 0.979161 0.565319i
\(732\) 0 0
\(733\) 79.7682 + 46.0542i 0.108824 + 0.0628297i 0.553424 0.832899i \(-0.313321\pi\)
−0.444600 + 0.895729i \(0.646654\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −266.452 + 461.508i −0.361535 + 0.626198i
\(738\) 0 0
\(739\) −467.042 808.941i −0.631992 1.09464i −0.987144 0.159834i \(-0.948904\pi\)
0.355152 0.934809i \(-0.384429\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −779.457 −1.04907 −0.524534 0.851390i \(-0.675760\pi\)
−0.524534 + 0.851390i \(0.675760\pi\)
\(744\) 0 0
\(745\) 637.021 367.784i 0.855062 0.493670i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 642.031 + 932.942i 0.857185 + 1.24558i
\(750\) 0 0
\(751\) 99.7934 172.847i 0.132881 0.230156i −0.791905 0.610644i \(-0.790911\pi\)
0.924786 + 0.380488i \(0.124244\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 206.510i 0.273523i
\(756\) 0 0
\(757\) −541.529 −0.715361 −0.357681 0.933844i \(-0.616432\pi\)
−0.357681 + 0.933844i \(0.616432\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −929.152 536.446i −1.22096 0.704922i −0.255838 0.966720i \(-0.582351\pi\)
−0.965123 + 0.261797i \(0.915685\pi\)
\(762\) 0 0
\(763\) −605.998 + 48.0650i −0.794230 + 0.0629948i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −230.607 399.424i −0.300661 0.520761i
\(768\) 0 0
\(769\) 13.9429i 0.0181312i 0.999959 + 0.00906558i \(0.00288570\pi\)
−0.999959 + 0.00906558i \(0.997114\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1243.17 717.746i 1.60824 0.928520i 0.618481 0.785800i \(-0.287749\pi\)
0.989763 0.142720i \(-0.0455848\pi\)
\(774\) 0 0
\(775\) −97.5876 56.3422i −0.125920 0.0726997i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 403.644 699.132i 0.518157 0.897474i
\(780\) 0 0
\(781\) −160.323 277.688i −0.205280 0.355555i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −128.296 −0.163435
\(786\) 0 0
\(787\) 1217.14 702.715i 1.54655 0.892904i 0.548154 0.836378i \(-0.315331\pi\)
0.998401 0.0565261i \(-0.0180024\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 260.470 546.945i 0.329292 0.691460i
\(792\) 0 0
\(793\) −134.645 + 233.211i −0.169791 + 0.294087i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 269.295i 0.337885i 0.985626 + 0.168943i \(0.0540353\pi\)
−0.985626 + 0.168943i \(0.945965\pi\)
\(798\) 0 0
\(799\) 786.547 0.984415
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −157.422 90.8878i −0.196043 0.113185i
\(804\) 0 0
\(805\) −45.4906 573.540i −0.0565101 0.712472i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 10.9829 + 19.0229i 0.0135759 + 0.0235141i 0.872734 0.488197i \(-0.162345\pi\)
−0.859158 + 0.511711i \(0.829012\pi\)
\(810\) 0 0
\(811\) 453.011i 0.558583i 0.960206 + 0.279292i \(0.0900996\pi\)
−0.960206 + 0.279292i \(0.909900\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −177.494 + 102.476i −0.217784 + 0.125738i
\(816\) 0 0
\(817\) 406.145 + 234.488i 0.497118 + 0.287011i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −433.447 + 750.752i −0.527950 + 0.914436i 0.471519 + 0.881856i \(0.343706\pi\)
−0.999469 + 0.0325803i \(0.989628\pi\)
\(822\) 0 0
\(823\) −265.315 459.539i −0.322375 0.558371i 0.658602 0.752491i \(-0.271148\pi\)
−0.980978 + 0.194121i \(0.937815\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 215.336 0.260382 0.130191 0.991489i \(-0.458441\pi\)
0.130191 + 0.991489i \(0.458441\pi\)
\(828\) 0 0
\(829\) 473.322 273.272i 0.570955 0.329641i −0.186576 0.982441i \(-0.559739\pi\)
0.757531 + 0.652800i \(0.226406\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 930.209 + 355.772i 1.11670 + 0.427098i
\(834\) 0 0
\(835\) −3.75144 + 6.49769i −0.00449274 + 0.00778166i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 35.6266i 0.0424631i −0.999775 0.0212316i \(-0.993241\pi\)
0.999775 0.0212316i \(-0.00675873\pi\)
\(840\) 0 0
\(841\) −452.349 −0.537871
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 276.538 + 159.659i 0.327264 + 0.188946i
\(846\) 0 0
\(847\) −357.379 519.310i −0.421935 0.613117i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 186.367 + 322.798i 0.218998 + 0.379316i
\(852\) 0 0
\(853\) 3.01022i 0.00352898i −0.999998 0.00176449i \(-0.999438\pi\)
0.999998 0.00176449i \(-0.000561654\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −997.825 + 576.095i −1.16432 + 0.672223i −0.952336 0.305050i \(-0.901327\pi\)
−0.211987 + 0.977272i \(0.567994\pi\)
\(858\) 0 0
\(859\) −1319.48 761.802i −1.53606 0.886847i −0.999064 0.0432653i \(-0.986224\pi\)
−0.537001 0.843582i \(-0.680443\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 582.886 1009.59i 0.675418 1.16986i −0.300929 0.953647i \(-0.597297\pi\)
0.976347 0.216211i \(-0.0693700\pi\)
\(864\) 0 0
\(865\) 277.409 + 480.486i 0.320704 + 0.555475i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −346.494 −0.398727
\(870\) 0 0
\(871\) −539.428 + 311.439i −0.619321 + 0.357565i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −695.126 331.038i −0.794430 0.378329i
\(876\) 0 0
\(877\) 286.492 496.218i 0.326672 0.565813i −0.655177 0.755475i \(-0.727406\pi\)
0.981849 + 0.189662i \(0.0607393\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 741.161i 0.841272i 0.907230 + 0.420636i \(0.138193\pi\)
−0.907230 + 0.420636i \(0.861807\pi\)
\(882\) 0 0
\(883\) 902.855 1.02249 0.511243 0.859436i \(-0.329185\pi\)
0.511243 + 0.859436i \(0.329185\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 759.358 + 438.416i 0.856097 + 0.494268i 0.862703 0.505710i \(-0.168769\pi\)
−0.00660618 + 0.999978i \(0.502103\pi\)
\(888\) 0 0
\(889\) −88.7463 + 186.353i −0.0998271 + 0.209621i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 223.154 + 386.514i 0.249893 + 0.432827i
\(894\) 0 0
\(895\) 655.055i 0.731906i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −103.152 + 59.5547i −0.114741 + 0.0662455i
\(900\) 0 0
\(901\) −579.932 334.824i −0.643653 0.371613i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −424.646 + 735.509i −0.469222 + 0.812717i
\(906\) 0 0
\(907\) −149.904 259.642i −0.165275 0.286264i 0.771478 0.636256i \(-0.219518\pi\)
−0.936753 + 0.349992i \(0.886184\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1143.71 1.25544 0.627720 0.778440i \(-0.283988\pi\)
0.627720 + 0.778440i \(0.283988\pi\)
\(912\) 0 0
\(913\) −170.177 + 98.2520i −0.186394 + 0.107614i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1125.43 + 774.500i −1.22730 + 0.844601i
\(918\) 0 0
\(919\) −19.3081 + 33.4427i −0.0210099 + 0.0363903i −0.876339 0.481694i \(-0.840021\pi\)
0.855329 + 0.518085i \(0.173355\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 374.785i 0.406051i
\(924\) 0 0
\(925\) 213.122 0.230402
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −893.518 515.873i −0.961806 0.555299i −0.0650776 0.997880i \(-0.520730\pi\)
−0.896728 + 0.442581i \(0.854063\pi\)
\(930\) 0 0
\(931\) 89.0841 + 558.048i 0.0956865 + 0.599407i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 142.443 + 246.718i 0.152345 + 0.263869i
\(936\) 0 0
\(937\) 1219.23i 1.30120i 0.759420 + 0.650601i \(0.225483\pi\)
−0.759420 + 0.650601i \(0.774517\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −381.358 + 220.177i −0.405269 + 0.233982i −0.688755 0.724994i \(-0.741842\pi\)
0.283486 + 0.958976i \(0.408509\pi\)
\(942\) 0 0
\(943\) 1977.37 + 1141.64i 2.09689 + 1.21064i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 864.937 1498.12i 0.913345 1.58196i 0.104038 0.994573i \(-0.466824\pi\)
0.809307 0.587386i \(-0.199843\pi\)
\(948\) 0 0
\(949\) −106.233 184.001i −0.111942 0.193890i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1436.26 1.50709 0.753545 0.657397i \(-0.228342\pi\)
0.753545 + 0.657397i \(0.228342\pi\)
\(954\) 0 0
\(955\) −447.306 + 258.252i −0.468384 + 0.270421i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −11.4646 + 0.909321i −0.0119548 + 0.000948198i
\(960\) 0 0
\(961\) −462.248 + 800.638i −0.481008 + 0.833130i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 661.494i 0.685485i
\(966\) 0 0
\(967\) −446.032 −0.461254 −0.230627 0.973042i \(-0.574078\pi\)
−0.230627 + 0.973042i \(0.574078\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 166.691 + 96.2390i 0.171669 + 0.0991133i 0.583373 0.812205i \(-0.301733\pi\)
−0.411703 + 0.911318i \(0.635066\pi\)
\(972\) 0 0
\(973\) −62.8101 29.9119i −0.0645531 0.0307419i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −218.928 379.194i −0.224082 0.388121i 0.731962 0.681346i \(-0.238605\pi\)
−0.956044 + 0.293225i \(0.905272\pi\)
\(978\) 0 0
\(979\) 42.3979i 0.0433074i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −792.929 + 457.798i −0.806642 + 0.465715i −0.845788 0.533518i \(-0.820870\pi\)
0.0391462 + 0.999233i \(0.487536\pi\)
\(984\) 0 0
\(985\) −311.389 179.781i −0.316131 0.182518i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −663.208 + 1148.71i −0.670585 + 1.16149i
\(990\) 0 0
\(991\) 436.297 + 755.688i 0.440259 + 0.762551i 0.997708 0.0676598i \(-0.0215532\pi\)
−0.557449 + 0.830211i \(0.688220\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −70.0660 −0.0704181
\(996\) 0 0
\(997\) 1505.96 869.465i 1.51049 0.872081i 0.510564 0.859840i \(-0.329437\pi\)
0.999925 0.0122411i \(-0.00389656\pi\)
\(998\) 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 1008.3.cg.q.145.6 16
3.2 odd 2 inner 1008.3.cg.q.145.3 16
4.3 odd 2 504.3.by.d.145.6 yes 16
7.3 odd 6 inner 1008.3.cg.q.577.6 16
12.11 even 2 504.3.by.d.145.3 yes 16
21.17 even 6 inner 1008.3.cg.q.577.3 16
28.3 even 6 504.3.by.d.73.6 yes 16
28.19 even 6 3528.3.f.i.2449.11 16
28.23 odd 6 3528.3.f.i.2449.5 16
84.23 even 6 3528.3.f.i.2449.12 16
84.47 odd 6 3528.3.f.i.2449.6 16
84.59 odd 6 504.3.by.d.73.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.3.by.d.73.3 16 84.59 odd 6
504.3.by.d.73.6 yes 16 28.3 even 6
504.3.by.d.145.3 yes 16 12.11 even 2
504.3.by.d.145.6 yes 16 4.3 odd 2
1008.3.cg.q.145.3 16 3.2 odd 2 inner
1008.3.cg.q.145.6 16 1.1 even 1 trivial
1008.3.cg.q.577.3 16 21.17 even 6 inner
1008.3.cg.q.577.6 16 7.3 odd 6 inner
3528.3.f.i.2449.5 16 28.23 odd 6
3528.3.f.i.2449.6 16 84.47 odd 6
3528.3.f.i.2449.11 16 28.19 even 6
3528.3.f.i.2449.12 16 84.23 even 6