Properties

Label 1008.2.t.l.193.7
Level $1008$
Weight $2$
Character 1008.193
Analytic conductor $8.049$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,2,Mod(193,1008)]
 
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, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.t (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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 193.7
Character \(\chi\) \(=\) 1008.193
Dual form 1008.2.t.l.961.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.816621 - 1.52746i) q^{3} -1.78355 q^{5} +(-1.90167 + 1.83948i) q^{7} +(-1.66626 - 2.49471i) q^{9} +O(q^{10})\) \(q+(0.816621 - 1.52746i) q^{3} -1.78355 q^{5} +(-1.90167 + 1.83948i) q^{7} +(-1.66626 - 2.49471i) q^{9} +5.61411 q^{11} +(3.14009 - 5.43879i) q^{13} +(-1.45648 + 2.72430i) q^{15} +(0.646279 - 1.11939i) q^{17} +(-0.559062 - 0.968324i) q^{19} +(1.25678 + 4.40687i) q^{21} -7.61715 q^{23} -1.81896 q^{25} +(-5.17127 + 0.507916i) q^{27} +(-1.57496 - 2.72791i) q^{29} +(0.501553 + 0.868716i) q^{31} +(4.58460 - 8.57533i) q^{33} +(3.39171 - 3.28079i) q^{35} +(-5.96542 - 10.3324i) q^{37} +(-5.74327 - 9.23778i) q^{39} +(4.14160 - 7.17347i) q^{41} +(-2.34804 - 4.06693i) q^{43} +(2.97186 + 4.44943i) q^{45} +(-0.972001 + 1.68356i) q^{47} +(0.232662 - 6.99613i) q^{49} +(-1.18205 - 1.90128i) q^{51} +(-4.45992 + 7.72481i) q^{53} -10.0130 q^{55} +(-1.93562 + 0.0631911i) q^{57} +(4.19339 + 7.26317i) q^{59} +(-2.41288 + 4.17923i) q^{61} +(7.75763 + 1.67905i) q^{63} +(-5.60050 + 9.70034i) q^{65} +(-1.27814 - 2.21380i) q^{67} +(-6.22032 + 11.6349i) q^{69} +8.86178 q^{71} +(5.67598 - 9.83109i) q^{73} +(-1.48540 + 2.77838i) q^{75} +(-10.6762 + 10.3270i) q^{77} +(6.72883 - 11.6547i) q^{79} +(-3.44714 + 8.31368i) q^{81} +(1.60203 + 2.77479i) q^{83} +(-1.15267 + 1.99648i) q^{85} +(-5.45291 + 0.178018i) q^{87} +(-0.404646 - 0.700867i) q^{89} +(4.03312 + 16.1189i) q^{91} +(1.73651 - 0.0566909i) q^{93} +(0.997114 + 1.72705i) q^{95} +(1.10781 + 1.91879i) q^{97} +(-9.35458 - 14.0056i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q + 2 q^{3} - 2 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 22 q + 2 q^{3} - 2 q^{5} + q^{7} + 6 q^{11} + 7 q^{13} + q^{15} - q^{17} - 13 q^{19} + 33 q^{21} + 44 q^{25} + 2 q^{27} - 7 q^{29} - 6 q^{31} + 9 q^{33} - 2 q^{35} + 6 q^{37} + 4 q^{39} + 4 q^{41} - 2 q^{43} - 17 q^{47} + 29 q^{49} + 25 q^{51} + q^{53} - 2 q^{55} - 21 q^{57} + 21 q^{59} + 31 q^{61} + 7 q^{63} - 3 q^{65} + 26 q^{67} - 40 q^{69} + 32 q^{71} + 17 q^{73} + 16 q^{75} - 4 q^{77} + 16 q^{79} + 36 q^{83} + 28 q^{85} - 7 q^{87} - 2 q^{89} - 15 q^{91} - 56 q^{93} + 24 q^{95} + 19 q^{97} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.816621 1.52746i 0.471476 0.881879i
\(4\) 0 0
\(5\) −1.78355 −0.797627 −0.398814 0.917032i \(-0.630578\pi\)
−0.398814 + 0.917032i \(0.630578\pi\)
\(6\) 0 0
\(7\) −1.90167 + 1.83948i −0.718762 + 0.695256i
\(8\) 0 0
\(9\) −1.66626 2.49471i −0.555421 0.831570i
\(10\) 0 0
\(11\) 5.61411 1.69272 0.846359 0.532612i \(-0.178790\pi\)
0.846359 + 0.532612i \(0.178790\pi\)
\(12\) 0 0
\(13\) 3.14009 5.43879i 0.870903 1.50845i 0.00983976 0.999952i \(-0.496868\pi\)
0.861064 0.508497i \(-0.169799\pi\)
\(14\) 0 0
\(15\) −1.45648 + 2.72430i −0.376062 + 0.703410i
\(16\) 0 0
\(17\) 0.646279 1.11939i 0.156746 0.271491i −0.776948 0.629565i \(-0.783233\pi\)
0.933693 + 0.358074i \(0.116566\pi\)
\(18\) 0 0
\(19\) −0.559062 0.968324i −0.128258 0.222149i 0.794744 0.606945i \(-0.207605\pi\)
−0.923002 + 0.384796i \(0.874272\pi\)
\(20\) 0 0
\(21\) 1.25678 + 4.40687i 0.274253 + 0.961658i
\(22\) 0 0
\(23\) −7.61715 −1.58828 −0.794142 0.607732i \(-0.792080\pi\)
−0.794142 + 0.607732i \(0.792080\pi\)
\(24\) 0 0
\(25\) −1.81896 −0.363791
\(26\) 0 0
\(27\) −5.17127 + 0.507916i −0.995211 + 0.0977485i
\(28\) 0 0
\(29\) −1.57496 2.72791i −0.292462 0.506560i 0.681929 0.731418i \(-0.261141\pi\)
−0.974391 + 0.224859i \(0.927808\pi\)
\(30\) 0 0
\(31\) 0.501553 + 0.868716i 0.0900816 + 0.156026i 0.907545 0.419954i \(-0.137954\pi\)
−0.817464 + 0.575980i \(0.804621\pi\)
\(32\) 0 0
\(33\) 4.58460 8.57533i 0.798076 1.49277i
\(34\) 0 0
\(35\) 3.39171 3.28079i 0.573304 0.554555i
\(36\) 0 0
\(37\) −5.96542 10.3324i −0.980708 1.69864i −0.659642 0.751580i \(-0.729292\pi\)
−0.321067 0.947057i \(-0.604041\pi\)
\(38\) 0 0
\(39\) −5.74327 9.23778i −0.919659 1.47923i
\(40\) 0 0
\(41\) 4.14160 7.17347i 0.646810 1.12031i −0.337071 0.941479i \(-0.609436\pi\)
0.983880 0.178828i \(-0.0572306\pi\)
\(42\) 0 0
\(43\) −2.34804 4.06693i −0.358073 0.620200i 0.629566 0.776947i \(-0.283233\pi\)
−0.987639 + 0.156747i \(0.949899\pi\)
\(44\) 0 0
\(45\) 2.97186 + 4.44943i 0.443018 + 0.663282i
\(46\) 0 0
\(47\) −0.972001 + 1.68356i −0.141781 + 0.245572i −0.928167 0.372163i \(-0.878616\pi\)
0.786386 + 0.617735i \(0.211950\pi\)
\(48\) 0 0
\(49\) 0.232662 6.99613i 0.0332374 0.999447i
\(50\) 0 0
\(51\) −1.18205 1.90128i −0.165521 0.266232i
\(52\) 0 0
\(53\) −4.45992 + 7.72481i −0.612617 + 1.06108i 0.378180 + 0.925732i \(0.376550\pi\)
−0.990798 + 0.135352i \(0.956783\pi\)
\(54\) 0 0
\(55\) −10.0130 −1.35016
\(56\) 0 0
\(57\) −1.93562 + 0.0631911i −0.256379 + 0.00836987i
\(58\) 0 0
\(59\) 4.19339 + 7.26317i 0.545933 + 0.945584i 0.998548 + 0.0538778i \(0.0171581\pi\)
−0.452614 + 0.891706i \(0.649509\pi\)
\(60\) 0 0
\(61\) −2.41288 + 4.17923i −0.308937 + 0.535095i −0.978130 0.207994i \(-0.933307\pi\)
0.669193 + 0.743089i \(0.266640\pi\)
\(62\) 0 0
\(63\) 7.75763 + 1.67905i 0.977369 + 0.211541i
\(64\) 0 0
\(65\) −5.60050 + 9.70034i −0.694656 + 1.20318i
\(66\) 0 0
\(67\) −1.27814 2.21380i −0.156150 0.270459i 0.777327 0.629096i \(-0.216575\pi\)
−0.933477 + 0.358637i \(0.883242\pi\)
\(68\) 0 0
\(69\) −6.22032 + 11.6349i −0.748838 + 1.40067i
\(70\) 0 0
\(71\) 8.86178 1.05170 0.525850 0.850577i \(-0.323747\pi\)
0.525850 + 0.850577i \(0.323747\pi\)
\(72\) 0 0
\(73\) 5.67598 9.83109i 0.664323 1.15064i −0.315145 0.949044i \(-0.602053\pi\)
0.979468 0.201598i \(-0.0646135\pi\)
\(74\) 0 0
\(75\) −1.48540 + 2.77838i −0.171519 + 0.320820i
\(76\) 0 0
\(77\) −10.6762 + 10.3270i −1.21666 + 1.17687i
\(78\) 0 0
\(79\) 6.72883 11.6547i 0.757052 1.31125i −0.187295 0.982304i \(-0.559972\pi\)
0.944348 0.328949i \(-0.106695\pi\)
\(80\) 0 0
\(81\) −3.44714 + 8.31368i −0.383016 + 0.923742i
\(82\) 0 0
\(83\) 1.60203 + 2.77479i 0.175845 + 0.304573i 0.940453 0.339922i \(-0.110401\pi\)
−0.764608 + 0.644495i \(0.777067\pi\)
\(84\) 0 0
\(85\) −1.15267 + 1.99648i −0.125025 + 0.216549i
\(86\) 0 0
\(87\) −5.45291 + 0.178018i −0.584613 + 0.0190856i
\(88\) 0 0
\(89\) −0.404646 0.700867i −0.0428924 0.0742917i 0.843782 0.536686i \(-0.180324\pi\)
−0.886675 + 0.462394i \(0.846991\pi\)
\(90\) 0 0
\(91\) 4.03312 + 16.1189i 0.422786 + 1.68972i
\(92\) 0 0
\(93\) 1.73651 0.0566909i 0.180067 0.00587857i
\(94\) 0 0
\(95\) 0.997114 + 1.72705i 0.102302 + 0.177192i
\(96\) 0 0
\(97\) 1.10781 + 1.91879i 0.112481 + 0.194823i 0.916770 0.399415i \(-0.130787\pi\)
−0.804289 + 0.594238i \(0.797454\pi\)
\(98\) 0 0
\(99\) −9.35458 14.0056i −0.940171 1.40761i
\(100\) 0 0
\(101\) 9.40267 0.935601 0.467801 0.883834i \(-0.345047\pi\)
0.467801 + 0.883834i \(0.345047\pi\)
\(102\) 0 0
\(103\) 3.52698 0.347524 0.173762 0.984788i \(-0.444408\pi\)
0.173762 + 0.984788i \(0.444408\pi\)
\(104\) 0 0
\(105\) −2.24153 7.85986i −0.218751 0.767044i
\(106\) 0 0
\(107\) 3.39276 + 5.87644i 0.327991 + 0.568097i 0.982113 0.188292i \(-0.0602951\pi\)
−0.654122 + 0.756389i \(0.726962\pi\)
\(108\) 0 0
\(109\) 0.681848 1.18099i 0.0653092 0.113119i −0.831522 0.555492i \(-0.812530\pi\)
0.896831 + 0.442373i \(0.145863\pi\)
\(110\) 0 0
\(111\) −20.6538 + 0.674275i −1.96037 + 0.0639993i
\(112\) 0 0
\(113\) −2.76458 + 4.78840i −0.260070 + 0.450455i −0.966260 0.257568i \(-0.917079\pi\)
0.706190 + 0.708022i \(0.250412\pi\)
\(114\) 0 0
\(115\) 13.5855 1.26686
\(116\) 0 0
\(117\) −18.8004 + 1.22885i −1.73810 + 0.113607i
\(118\) 0 0
\(119\) 0.830080 + 3.31752i 0.0760933 + 0.304116i
\(120\) 0 0
\(121\) 20.5183 1.86530
\(122\) 0 0
\(123\) −7.57506 12.1841i −0.683020 1.09861i
\(124\) 0 0
\(125\) 12.1619 1.08780
\(126\) 0 0
\(127\) 12.8209 1.13767 0.568837 0.822450i \(-0.307394\pi\)
0.568837 + 0.822450i \(0.307394\pi\)
\(128\) 0 0
\(129\) −8.12952 + 0.265401i −0.715764 + 0.0233672i
\(130\) 0 0
\(131\) 5.81910 0.508417 0.254208 0.967149i \(-0.418185\pi\)
0.254208 + 0.967149i \(0.418185\pi\)
\(132\) 0 0
\(133\) 2.84436 + 0.813047i 0.246637 + 0.0705001i
\(134\) 0 0
\(135\) 9.22321 0.905892i 0.793807 0.0779668i
\(136\) 0 0
\(137\) −15.3409 −1.31066 −0.655332 0.755341i \(-0.727471\pi\)
−0.655332 + 0.755341i \(0.727471\pi\)
\(138\) 0 0
\(139\) 6.05803 10.4928i 0.513835 0.889988i −0.486036 0.873939i \(-0.661558\pi\)
0.999871 0.0160496i \(-0.00510897\pi\)
\(140\) 0 0
\(141\) 1.77781 + 2.85952i 0.149718 + 0.240815i
\(142\) 0 0
\(143\) 17.6288 30.5340i 1.47419 2.55338i
\(144\) 0 0
\(145\) 2.80901 + 4.86535i 0.233276 + 0.404046i
\(146\) 0 0
\(147\) −10.4963 6.06857i −0.865721 0.500527i
\(148\) 0 0
\(149\) 4.73699 0.388069 0.194035 0.980995i \(-0.437843\pi\)
0.194035 + 0.980995i \(0.437843\pi\)
\(150\) 0 0
\(151\) −24.3690 −1.98312 −0.991559 0.129657i \(-0.958612\pi\)
−0.991559 + 0.129657i \(0.958612\pi\)
\(152\) 0 0
\(153\) −3.86942 + 0.252916i −0.312824 + 0.0204470i
\(154\) 0 0
\(155\) −0.894545 1.54940i −0.0718516 0.124451i
\(156\) 0 0
\(157\) 3.15229 + 5.45993i 0.251580 + 0.435750i 0.963961 0.266043i \(-0.0857165\pi\)
−0.712381 + 0.701793i \(0.752383\pi\)
\(158\) 0 0
\(159\) 8.15727 + 13.1206i 0.646913 + 1.04053i
\(160\) 0 0
\(161\) 14.4853 14.0116i 1.14160 1.10426i
\(162\) 0 0
\(163\) −0.350678 0.607392i −0.0274672 0.0475746i 0.851965 0.523599i \(-0.175411\pi\)
−0.879432 + 0.476024i \(0.842078\pi\)
\(164\) 0 0
\(165\) −8.17686 + 15.2945i −0.636567 + 1.19068i
\(166\) 0 0
\(167\) −3.53822 + 6.12839i −0.273796 + 0.474229i −0.969831 0.243780i \(-0.921613\pi\)
0.696035 + 0.718008i \(0.254946\pi\)
\(168\) 0 0
\(169\) −13.2203 22.8982i −1.01695 1.76140i
\(170\) 0 0
\(171\) −1.48414 + 3.00818i −0.113495 + 0.230041i
\(172\) 0 0
\(173\) −3.31767 + 5.74638i −0.252238 + 0.436889i −0.964142 0.265388i \(-0.914500\pi\)
0.711904 + 0.702277i \(0.247833\pi\)
\(174\) 0 0
\(175\) 3.45904 3.34592i 0.261479 0.252928i
\(176\) 0 0
\(177\) 14.5186 0.473982i 1.09129 0.0356267i
\(178\) 0 0
\(179\) −5.63527 + 9.76057i −0.421200 + 0.729539i −0.996057 0.0887145i \(-0.971724\pi\)
0.574858 + 0.818253i \(0.305057\pi\)
\(180\) 0 0
\(181\) 21.1800 1.57430 0.787149 0.616762i \(-0.211556\pi\)
0.787149 + 0.616762i \(0.211556\pi\)
\(182\) 0 0
\(183\) 4.41319 + 7.09841i 0.326233 + 0.524730i
\(184\) 0 0
\(185\) 10.6396 + 18.4283i 0.782240 + 1.35488i
\(186\) 0 0
\(187\) 3.62828 6.28437i 0.265326 0.459559i
\(188\) 0 0
\(189\) 8.89972 10.4783i 0.647360 0.762185i
\(190\) 0 0
\(191\) −4.72044 + 8.17604i −0.341559 + 0.591598i −0.984722 0.174131i \(-0.944288\pi\)
0.643163 + 0.765729i \(0.277622\pi\)
\(192\) 0 0
\(193\) 3.14021 + 5.43900i 0.226037 + 0.391508i 0.956630 0.291305i \(-0.0940896\pi\)
−0.730593 + 0.682813i \(0.760756\pi\)
\(194\) 0 0
\(195\) 10.2434 + 16.4760i 0.733545 + 1.17987i
\(196\) 0 0
\(197\) 15.4780 1.10276 0.551382 0.834253i \(-0.314101\pi\)
0.551382 + 0.834253i \(0.314101\pi\)
\(198\) 0 0
\(199\) −2.19477 + 3.80145i −0.155583 + 0.269478i −0.933271 0.359173i \(-0.883059\pi\)
0.777688 + 0.628650i \(0.216392\pi\)
\(200\) 0 0
\(201\) −4.42525 + 0.144469i −0.312133 + 0.0101901i
\(202\) 0 0
\(203\) 8.01296 + 2.29047i 0.562400 + 0.160759i
\(204\) 0 0
\(205\) −7.38675 + 12.7942i −0.515913 + 0.893587i
\(206\) 0 0
\(207\) 12.6922 + 19.0026i 0.882166 + 1.32077i
\(208\) 0 0
\(209\) −3.13864 5.43628i −0.217104 0.376035i
\(210\) 0 0
\(211\) −7.93101 + 13.7369i −0.545993 + 0.945688i 0.452550 + 0.891739i \(0.350514\pi\)
−0.998544 + 0.0539495i \(0.982819\pi\)
\(212\) 0 0
\(213\) 7.23671 13.5360i 0.495851 0.927472i
\(214\) 0 0
\(215\) 4.18784 + 7.25356i 0.285609 + 0.494689i
\(216\) 0 0
\(217\) −2.55177 0.729412i −0.173225 0.0495157i
\(218\) 0 0
\(219\) −10.3815 16.6981i −0.701514 1.12835i
\(220\) 0 0
\(221\) −4.05874 7.02995i −0.273021 0.472886i
\(222\) 0 0
\(223\) −6.99253 12.1114i −0.468254 0.811040i 0.531088 0.847317i \(-0.321784\pi\)
−0.999342 + 0.0362769i \(0.988450\pi\)
\(224\) 0 0
\(225\) 3.03086 + 4.53776i 0.202057 + 0.302518i
\(226\) 0 0
\(227\) −10.7673 −0.714650 −0.357325 0.933980i \(-0.616311\pi\)
−0.357325 + 0.933980i \(0.616311\pi\)
\(228\) 0 0
\(229\) −1.61003 −0.106394 −0.0531969 0.998584i \(-0.516941\pi\)
−0.0531969 + 0.998584i \(0.516941\pi\)
\(230\) 0 0
\(231\) 7.05573 + 24.7407i 0.464233 + 1.62782i
\(232\) 0 0
\(233\) −0.510606 0.884395i −0.0334509 0.0579387i 0.848815 0.528690i \(-0.177316\pi\)
−0.882266 + 0.470751i \(0.843983\pi\)
\(234\) 0 0
\(235\) 1.73361 3.00270i 0.113088 0.195875i
\(236\) 0 0
\(237\) −12.3071 19.7955i −0.799434 1.28585i
\(238\) 0 0
\(239\) 6.96428 12.0625i 0.450482 0.780258i −0.547934 0.836522i \(-0.684586\pi\)
0.998416 + 0.0562640i \(0.0179189\pi\)
\(240\) 0 0
\(241\) −14.5758 −0.938908 −0.469454 0.882957i \(-0.655549\pi\)
−0.469454 + 0.882957i \(0.655549\pi\)
\(242\) 0 0
\(243\) 9.88379 + 12.0545i 0.634045 + 0.773296i
\(244\) 0 0
\(245\) −0.414964 + 12.4779i −0.0265111 + 0.797186i
\(246\) 0 0
\(247\) −7.02201 −0.446800
\(248\) 0 0
\(249\) 5.54663 0.181078i 0.351503 0.0114754i
\(250\) 0 0
\(251\) −18.0287 −1.13796 −0.568981 0.822350i \(-0.692662\pi\)
−0.568981 + 0.822350i \(0.692662\pi\)
\(252\) 0 0
\(253\) −42.7635 −2.68852
\(254\) 0 0
\(255\) 2.10825 + 3.39102i 0.132024 + 0.212354i
\(256\) 0 0
\(257\) 26.9266 1.67964 0.839818 0.542869i \(-0.182662\pi\)
0.839818 + 0.542869i \(0.182662\pi\)
\(258\) 0 0
\(259\) 30.3504 + 8.67554i 1.88588 + 0.539072i
\(260\) 0 0
\(261\) −4.18104 + 8.47447i −0.258800 + 0.524556i
\(262\) 0 0
\(263\) −1.53884 −0.0948888 −0.0474444 0.998874i \(-0.515108\pi\)
−0.0474444 + 0.998874i \(0.515108\pi\)
\(264\) 0 0
\(265\) 7.95448 13.7776i 0.488640 0.846349i
\(266\) 0 0
\(267\) −1.40099 + 0.0457374i −0.0857390 + 0.00279908i
\(268\) 0 0
\(269\) 3.26461 5.65446i 0.199047 0.344759i −0.749173 0.662374i \(-0.769549\pi\)
0.948220 + 0.317616i \(0.102882\pi\)
\(270\) 0 0
\(271\) 5.64494 + 9.77733i 0.342906 + 0.593930i 0.984971 0.172720i \(-0.0552555\pi\)
−0.642065 + 0.766650i \(0.721922\pi\)
\(272\) 0 0
\(273\) 27.9144 + 7.00257i 1.68946 + 0.423815i
\(274\) 0 0
\(275\) −10.2118 −0.615796
\(276\) 0 0
\(277\) 1.81188 0.108865 0.0544325 0.998517i \(-0.482665\pi\)
0.0544325 + 0.998517i \(0.482665\pi\)
\(278\) 0 0
\(279\) 1.33147 2.69874i 0.0797133 0.161569i
\(280\) 0 0
\(281\) −2.98798 5.17533i −0.178248 0.308734i 0.763033 0.646360i \(-0.223710\pi\)
−0.941280 + 0.337626i \(0.890376\pi\)
\(282\) 0 0
\(283\) −9.99760 17.3163i −0.594295 1.02935i −0.993646 0.112552i \(-0.964098\pi\)
0.399350 0.916798i \(-0.369236\pi\)
\(284\) 0 0
\(285\) 3.45227 0.112704i 0.204495 0.00667604i
\(286\) 0 0
\(287\) 5.31947 + 21.2599i 0.313998 + 1.25493i
\(288\) 0 0
\(289\) 7.66465 + 13.2756i 0.450862 + 0.780915i
\(290\) 0 0
\(291\) 3.83553 0.125217i 0.224843 0.00734033i
\(292\) 0 0
\(293\) −4.95166 + 8.57652i −0.289279 + 0.501046i −0.973638 0.228099i \(-0.926749\pi\)
0.684359 + 0.729145i \(0.260082\pi\)
\(294\) 0 0
\(295\) −7.47912 12.9542i −0.435451 0.754224i
\(296\) 0 0
\(297\) −29.0321 + 2.85150i −1.68461 + 0.165461i
\(298\) 0 0
\(299\) −23.9185 + 41.4281i −1.38324 + 2.39585i
\(300\) 0 0
\(301\) 11.9462 + 3.41477i 0.688567 + 0.196824i
\(302\) 0 0
\(303\) 7.67842 14.3622i 0.441114 0.825087i
\(304\) 0 0
\(305\) 4.30348 7.45385i 0.246417 0.426806i
\(306\) 0 0
\(307\) 23.7122 1.35332 0.676662 0.736293i \(-0.263426\pi\)
0.676662 + 0.736293i \(0.263426\pi\)
\(308\) 0 0
\(309\) 2.88021 5.38732i 0.163849 0.306474i
\(310\) 0 0
\(311\) 10.1724 + 17.6192i 0.576826 + 0.999092i 0.995841 + 0.0911122i \(0.0290422\pi\)
−0.419015 + 0.907979i \(0.637624\pi\)
\(312\) 0 0
\(313\) −4.85936 + 8.41665i −0.274667 + 0.475737i −0.970051 0.242901i \(-0.921901\pi\)
0.695384 + 0.718638i \(0.255234\pi\)
\(314\) 0 0
\(315\) −13.8361 2.99467i −0.779576 0.168731i
\(316\) 0 0
\(317\) 1.66419 2.88246i 0.0934703 0.161895i −0.815499 0.578759i \(-0.803537\pi\)
0.908969 + 0.416863i \(0.136871\pi\)
\(318\) 0 0
\(319\) −8.84199 15.3148i −0.495056 0.857463i
\(320\) 0 0
\(321\) 11.7466 0.383486i 0.655633 0.0214041i
\(322\) 0 0
\(323\) −1.44524 −0.0804153
\(324\) 0 0
\(325\) −5.71168 + 9.89292i −0.316827 + 0.548760i
\(326\) 0 0
\(327\) −1.24711 2.00592i −0.0689654 0.110928i
\(328\) 0 0
\(329\) −1.24844 4.98953i −0.0688286 0.275082i
\(330\) 0 0
\(331\) −0.717346 + 1.24248i −0.0394289 + 0.0682929i −0.885066 0.465465i \(-0.845887\pi\)
0.845638 + 0.533758i \(0.179221\pi\)
\(332\) 0 0
\(333\) −15.8364 + 32.0985i −0.867829 + 1.75899i
\(334\) 0 0
\(335\) 2.27962 + 3.94843i 0.124549 + 0.215726i
\(336\) 0 0
\(337\) 0.00257316 0.00445685i 0.000140169 0.000242780i −0.865955 0.500121i \(-0.833289\pi\)
0.866095 + 0.499879i \(0.166622\pi\)
\(338\) 0 0
\(339\) 5.05647 + 8.13309i 0.274630 + 0.441729i
\(340\) 0 0
\(341\) 2.81578 + 4.87707i 0.152483 + 0.264108i
\(342\) 0 0
\(343\) 12.4268 + 13.7323i 0.670982 + 0.741473i
\(344\) 0 0
\(345\) 11.0942 20.7514i 0.597294 1.11722i
\(346\) 0 0
\(347\) 11.5536 + 20.0114i 0.620229 + 1.07427i 0.989443 + 0.144924i \(0.0462936\pi\)
−0.369214 + 0.929344i \(0.620373\pi\)
\(348\) 0 0
\(349\) 6.09723 + 10.5607i 0.326377 + 0.565302i 0.981790 0.189969i \(-0.0608387\pi\)
−0.655413 + 0.755271i \(0.727505\pi\)
\(350\) 0 0
\(351\) −13.4758 + 29.7203i −0.719284 + 1.58635i
\(352\) 0 0
\(353\) 24.7072 1.31503 0.657515 0.753441i \(-0.271608\pi\)
0.657515 + 0.753441i \(0.271608\pi\)
\(354\) 0 0
\(355\) −15.8054 −0.838864
\(356\) 0 0
\(357\) 5.74523 + 1.44124i 0.304070 + 0.0762784i
\(358\) 0 0
\(359\) −8.20362 14.2091i −0.432970 0.749927i 0.564157 0.825667i \(-0.309201\pi\)
−0.997128 + 0.0757407i \(0.975868\pi\)
\(360\) 0 0
\(361\) 8.87490 15.3718i 0.467100 0.809041i
\(362\) 0 0
\(363\) 16.7556 31.3408i 0.879443 1.64497i
\(364\) 0 0
\(365\) −10.1234 + 17.5342i −0.529882 + 0.917783i
\(366\) 0 0
\(367\) 5.81801 0.303698 0.151849 0.988404i \(-0.451477\pi\)
0.151849 + 0.988404i \(0.451477\pi\)
\(368\) 0 0
\(369\) −24.7967 + 1.62078i −1.29086 + 0.0843744i
\(370\) 0 0
\(371\) −5.72832 22.8939i −0.297399 1.18859i
\(372\) 0 0
\(373\) 8.84966 0.458218 0.229109 0.973401i \(-0.426419\pi\)
0.229109 + 0.973401i \(0.426419\pi\)
\(374\) 0 0
\(375\) 9.93169 18.5769i 0.512870 0.959305i
\(376\) 0 0
\(377\) −19.7820 −1.01883
\(378\) 0 0
\(379\) −17.3300 −0.890181 −0.445091 0.895486i \(-0.646829\pi\)
−0.445091 + 0.895486i \(0.646829\pi\)
\(380\) 0 0
\(381\) 10.4698 19.5834i 0.536386 1.00329i
\(382\) 0 0
\(383\) −7.09090 −0.362328 −0.181164 0.983453i \(-0.557986\pi\)
−0.181164 + 0.983453i \(0.557986\pi\)
\(384\) 0 0
\(385\) 19.0415 18.4187i 0.970442 0.938706i
\(386\) 0 0
\(387\) −6.23335 + 12.6342i −0.316859 + 0.642235i
\(388\) 0 0
\(389\) 6.64240 0.336783 0.168392 0.985720i \(-0.446143\pi\)
0.168392 + 0.985720i \(0.446143\pi\)
\(390\) 0 0
\(391\) −4.92280 + 8.52654i −0.248957 + 0.431206i
\(392\) 0 0
\(393\) 4.75199 8.88843i 0.239706 0.448362i
\(394\) 0 0
\(395\) −12.0012 + 20.7867i −0.603845 + 1.04589i
\(396\) 0 0
\(397\) 7.86340 + 13.6198i 0.394653 + 0.683559i 0.993057 0.117636i \(-0.0375315\pi\)
−0.598404 + 0.801195i \(0.704198\pi\)
\(398\) 0 0
\(399\) 3.56466 3.68069i 0.178456 0.184265i
\(400\) 0 0
\(401\) 5.96559 0.297907 0.148954 0.988844i \(-0.452409\pi\)
0.148954 + 0.988844i \(0.452409\pi\)
\(402\) 0 0
\(403\) 6.29968 0.313810
\(404\) 0 0
\(405\) 6.14815 14.8278i 0.305504 0.736801i
\(406\) 0 0
\(407\) −33.4905 58.0073i −1.66006 2.87531i
\(408\) 0 0
\(409\) 8.80943 + 15.2584i 0.435598 + 0.754478i 0.997344 0.0728314i \(-0.0232035\pi\)
−0.561746 + 0.827310i \(0.689870\pi\)
\(410\) 0 0
\(411\) −12.5277 + 23.4326i −0.617947 + 1.15585i
\(412\) 0 0
\(413\) −21.3349 6.09848i −1.04982 0.300086i
\(414\) 0 0
\(415\) −2.85729 4.94897i −0.140259 0.242936i
\(416\) 0 0
\(417\) −11.0802 17.8220i −0.542601 0.872748i
\(418\) 0 0
\(419\) −9.62164 + 16.6652i −0.470048 + 0.814147i −0.999413 0.0342470i \(-0.989097\pi\)
0.529365 + 0.848394i \(0.322430\pi\)
\(420\) 0 0
\(421\) 7.77999 + 13.4753i 0.379174 + 0.656748i 0.990942 0.134289i \(-0.0428750\pi\)
−0.611769 + 0.791037i \(0.709542\pi\)
\(422\) 0 0
\(423\) 5.81959 0.380384i 0.282958 0.0184949i
\(424\) 0 0
\(425\) −1.17555 + 2.03612i −0.0570227 + 0.0987662i
\(426\) 0 0
\(427\) −3.09910 12.3859i −0.149976 0.599397i
\(428\) 0 0
\(429\) −32.2434 51.8620i −1.55672 2.50392i
\(430\) 0 0
\(431\) 16.9779 29.4067i 0.817799 1.41647i −0.0895020 0.995987i \(-0.528528\pi\)
0.907301 0.420482i \(-0.138139\pi\)
\(432\) 0 0
\(433\) −34.8338 −1.67401 −0.837004 0.547197i \(-0.815695\pi\)
−0.837004 + 0.547197i \(0.815695\pi\)
\(434\) 0 0
\(435\) 9.72553 0.317505i 0.466303 0.0152232i
\(436\) 0 0
\(437\) 4.25846 + 7.37587i 0.203710 + 0.352835i
\(438\) 0 0
\(439\) 7.77938 13.4743i 0.371290 0.643093i −0.618475 0.785805i \(-0.712249\pi\)
0.989764 + 0.142712i \(0.0455824\pi\)
\(440\) 0 0
\(441\) −17.8410 + 11.0770i −0.849571 + 0.527474i
\(442\) 0 0
\(443\) 9.16212 15.8693i 0.435305 0.753971i −0.562015 0.827127i \(-0.689974\pi\)
0.997321 + 0.0731560i \(0.0233071\pi\)
\(444\) 0 0
\(445\) 0.721705 + 1.25003i 0.0342121 + 0.0592571i
\(446\) 0 0
\(447\) 3.86832 7.23555i 0.182965 0.342230i
\(448\) 0 0
\(449\) −2.75072 −0.129815 −0.0649073 0.997891i \(-0.520675\pi\)
−0.0649073 + 0.997891i \(0.520675\pi\)
\(450\) 0 0
\(451\) 23.2514 40.2727i 1.09487 1.89637i
\(452\) 0 0
\(453\) −19.9002 + 37.2226i −0.934993 + 1.74887i
\(454\) 0 0
\(455\) −7.19327 28.7488i −0.337226 1.34776i
\(456\) 0 0
\(457\) −10.3407 + 17.9106i −0.483716 + 0.837821i −0.999825 0.0187018i \(-0.994047\pi\)
0.516109 + 0.856523i \(0.327380\pi\)
\(458\) 0 0
\(459\) −2.77353 + 6.11691i −0.129457 + 0.285513i
\(460\) 0 0
\(461\) 6.40670 + 11.0967i 0.298390 + 0.516826i 0.975768 0.218809i \(-0.0702172\pi\)
−0.677378 + 0.735635i \(0.736884\pi\)
\(462\) 0 0
\(463\) −5.54704 + 9.60775i −0.257793 + 0.446510i −0.965650 0.259845i \(-0.916328\pi\)
0.707858 + 0.706355i \(0.249662\pi\)
\(464\) 0 0
\(465\) −3.09714 + 0.101111i −0.143627 + 0.00468891i
\(466\) 0 0
\(467\) −5.36754 9.29686i −0.248380 0.430207i 0.714696 0.699435i \(-0.246565\pi\)
−0.963077 + 0.269228i \(0.913232\pi\)
\(468\) 0 0
\(469\) 6.50283 + 1.85881i 0.300273 + 0.0858317i
\(470\) 0 0
\(471\) 10.9140 0.356305i 0.502892 0.0164177i
\(472\) 0 0
\(473\) −13.1822 22.8322i −0.606117 1.04982i
\(474\) 0 0
\(475\) 1.01691 + 1.76134i 0.0466590 + 0.0808157i
\(476\) 0 0
\(477\) 26.7025 1.74535i 1.22263 0.0799141i
\(478\) 0 0
\(479\) 7.79508 0.356166 0.178083 0.984015i \(-0.443010\pi\)
0.178083 + 0.984015i \(0.443010\pi\)
\(480\) 0 0
\(481\) −74.9277 −3.41641
\(482\) 0 0
\(483\) −9.57311 33.5678i −0.435591 1.52739i
\(484\) 0 0
\(485\) −1.97583 3.42225i −0.0897180 0.155396i
\(486\) 0 0
\(487\) −13.9984 + 24.2459i −0.634326 + 1.09868i 0.352331 + 0.935875i \(0.385389\pi\)
−0.986657 + 0.162810i \(0.947944\pi\)
\(488\) 0 0
\(489\) −1.21414 + 0.0396373i −0.0549052 + 0.00179246i
\(490\) 0 0
\(491\) 12.1227 20.9971i 0.547089 0.947586i −0.451383 0.892330i \(-0.649069\pi\)
0.998472 0.0552556i \(-0.0175974\pi\)
\(492\) 0 0
\(493\) −4.07145 −0.183369
\(494\) 0 0
\(495\) 16.6843 + 24.9796i 0.749906 + 1.12275i
\(496\) 0 0
\(497\) −16.8521 + 16.3010i −0.755922 + 0.731201i
\(498\) 0 0
\(499\) 33.7748 1.51197 0.755984 0.654590i \(-0.227159\pi\)
0.755984 + 0.654590i \(0.227159\pi\)
\(500\) 0 0
\(501\) 6.47147 + 10.4091i 0.289124 + 0.465042i
\(502\) 0 0
\(503\) 1.09819 0.0489661 0.0244830 0.999700i \(-0.492206\pi\)
0.0244830 + 0.999700i \(0.492206\pi\)
\(504\) 0 0
\(505\) −16.7701 −0.746261
\(506\) 0 0
\(507\) −45.7720 + 1.49430i −2.03281 + 0.0663641i
\(508\) 0 0
\(509\) 16.2134 0.718648 0.359324 0.933213i \(-0.383007\pi\)
0.359324 + 0.933213i \(0.383007\pi\)
\(510\) 0 0
\(511\) 7.29023 + 29.1363i 0.322501 + 1.28891i
\(512\) 0 0
\(513\) 3.38289 + 4.72351i 0.149358 + 0.208548i
\(514\) 0 0
\(515\) −6.29054 −0.277194
\(516\) 0 0
\(517\) −5.45692 + 9.45167i −0.239995 + 0.415684i
\(518\) 0 0
\(519\) 6.06807 + 9.76021i 0.266359 + 0.428426i
\(520\) 0 0
\(521\) 19.1738 33.2099i 0.840017 1.45495i −0.0498617 0.998756i \(-0.515878\pi\)
0.889879 0.456197i \(-0.150789\pi\)
\(522\) 0 0
\(523\) −20.6021 35.6838i −0.900865 1.56034i −0.826374 0.563122i \(-0.809600\pi\)
−0.0744911 0.997222i \(-0.523733\pi\)
\(524\) 0 0
\(525\) −2.28603 8.01590i −0.0997707 0.349843i
\(526\) 0 0
\(527\) 1.29657 0.0564796
\(528\) 0 0
\(529\) 35.0209 1.52265
\(530\) 0 0
\(531\) 11.1322 22.5636i 0.483097 0.979178i
\(532\) 0 0
\(533\) −26.0100 45.0506i −1.12662 1.95136i
\(534\) 0 0
\(535\) −6.05116 10.4809i −0.261614 0.453129i
\(536\) 0 0
\(537\) 10.3070 + 16.5783i 0.444779 + 0.715407i
\(538\) 0 0
\(539\) 1.30619 39.2771i 0.0562616 1.69178i
\(540\) 0 0
\(541\) −9.09371 15.7508i −0.390969 0.677178i 0.601609 0.798791i \(-0.294527\pi\)
−0.992578 + 0.121613i \(0.961193\pi\)
\(542\) 0 0
\(543\) 17.2960 32.3516i 0.742244 1.38834i
\(544\) 0 0
\(545\) −1.21611 + 2.10636i −0.0520924 + 0.0902266i
\(546\) 0 0
\(547\) −0.338699 0.586644i −0.0144817 0.0250831i 0.858694 0.512489i \(-0.171277\pi\)
−0.873175 + 0.487406i \(0.837943\pi\)
\(548\) 0 0
\(549\) 14.4464 0.944258i 0.616559 0.0403000i
\(550\) 0 0
\(551\) −1.76100 + 3.05014i −0.0750211 + 0.129940i
\(552\) 0 0
\(553\) 8.64250 + 34.5408i 0.367516 + 1.46882i
\(554\) 0 0
\(555\) 36.8371 1.20260i 1.56365 0.0510476i
\(556\) 0 0
\(557\) 14.8659 25.7484i 0.629887 1.09100i −0.357687 0.933842i \(-0.616435\pi\)
0.987574 0.157155i \(-0.0502322\pi\)
\(558\) 0 0
\(559\) −29.4922 −1.24739
\(560\) 0 0
\(561\) −6.63619 10.6740i −0.280180 0.450657i
\(562\) 0 0
\(563\) 10.6739 + 18.4878i 0.449852 + 0.779167i 0.998376 0.0569680i \(-0.0181433\pi\)
−0.548524 + 0.836135i \(0.684810\pi\)
\(564\) 0 0
\(565\) 4.93077 8.54034i 0.207439 0.359295i
\(566\) 0 0
\(567\) −8.73749 22.1508i −0.366940 0.930245i
\(568\) 0 0
\(569\) 7.56212 13.0980i 0.317021 0.549096i −0.662844 0.748757i \(-0.730651\pi\)
0.979865 + 0.199661i \(0.0639842\pi\)
\(570\) 0 0
\(571\) 9.94314 + 17.2220i 0.416107 + 0.720719i 0.995544 0.0942981i \(-0.0300607\pi\)
−0.579437 + 0.815017i \(0.696727\pi\)
\(572\) 0 0
\(573\) 8.63376 + 13.8870i 0.360681 + 0.580138i
\(574\) 0 0
\(575\) 13.8553 0.577804
\(576\) 0 0
\(577\) 19.8090 34.3102i 0.824661 1.42835i −0.0775179 0.996991i \(-0.524700\pi\)
0.902178 0.431363i \(-0.141967\pi\)
\(578\) 0 0
\(579\) 10.8722 0.354940i 0.451834 0.0147508i
\(580\) 0 0
\(581\) −8.15068 2.32984i −0.338147 0.0966579i
\(582\) 0 0
\(583\) −25.0385 + 43.3679i −1.03699 + 1.79612i
\(584\) 0 0
\(585\) 33.5314 2.19170i 1.38635 0.0906158i
\(586\) 0 0
\(587\) −1.13275 1.96199i −0.0467537 0.0809798i 0.841701 0.539943i \(-0.181554\pi\)
−0.888455 + 0.458963i \(0.848221\pi\)
\(588\) 0 0
\(589\) 0.560799 0.971332i 0.0231073 0.0400230i
\(590\) 0 0
\(591\) 12.6397 23.6421i 0.519927 0.972504i
\(592\) 0 0
\(593\) 5.97295 + 10.3454i 0.245280 + 0.424837i 0.962210 0.272308i \(-0.0877869\pi\)
−0.716931 + 0.697145i \(0.754454\pi\)
\(594\) 0 0
\(595\) −1.48049 5.91695i −0.0606941 0.242571i
\(596\) 0 0
\(597\) 4.01426 + 6.45676i 0.164293 + 0.264257i
\(598\) 0 0
\(599\) −1.47636 2.55713i −0.0603224 0.104481i 0.834287 0.551330i \(-0.185880\pi\)
−0.894610 + 0.446849i \(0.852546\pi\)
\(600\) 0 0
\(601\) 15.9751 + 27.6697i 0.651638 + 1.12867i 0.982725 + 0.185071i \(0.0592514\pi\)
−0.331087 + 0.943600i \(0.607415\pi\)
\(602\) 0 0
\(603\) −3.39308 + 6.87736i −0.138177 + 0.280068i
\(604\) 0 0
\(605\) −36.5953 −1.48781
\(606\) 0 0
\(607\) −10.4014 −0.422179 −0.211089 0.977467i \(-0.567701\pi\)
−0.211089 + 0.977467i \(0.567701\pi\)
\(608\) 0 0
\(609\) 10.0421 10.3690i 0.406928 0.420174i
\(610\) 0 0
\(611\) 6.10433 + 10.5730i 0.246955 + 0.427739i
\(612\) 0 0
\(613\) 6.22441 10.7810i 0.251402 0.435441i −0.712510 0.701662i \(-0.752442\pi\)
0.963912 + 0.266221i \(0.0857751\pi\)
\(614\) 0 0
\(615\) 13.5105 + 21.7310i 0.544795 + 0.876278i
\(616\) 0 0
\(617\) −4.70100 + 8.14237i −0.189255 + 0.327799i −0.945002 0.327064i \(-0.893941\pi\)
0.755747 + 0.654864i \(0.227274\pi\)
\(618\) 0 0
\(619\) 22.1196 0.889062 0.444531 0.895763i \(-0.353370\pi\)
0.444531 + 0.895763i \(0.353370\pi\)
\(620\) 0 0
\(621\) 39.3903 3.86887i 1.58068 0.155252i
\(622\) 0 0
\(623\) 2.05873 + 0.588479i 0.0824812 + 0.0235769i
\(624\) 0 0
\(625\) −12.5966 −0.503865
\(626\) 0 0
\(627\) −10.8668 + 0.354762i −0.433977 + 0.0141678i
\(628\) 0 0
\(629\) −15.4213 −0.614887
\(630\) 0 0
\(631\) −18.3705 −0.731316 −0.365658 0.930749i \(-0.619156\pi\)
−0.365658 + 0.930749i \(0.619156\pi\)
\(632\) 0 0
\(633\) 14.5059 + 23.3321i 0.576560 + 0.927370i
\(634\) 0 0
\(635\) −22.8668 −0.907439
\(636\) 0 0
\(637\) −37.3199 23.2339i −1.47867 0.920559i
\(638\) 0 0
\(639\) −14.7660 22.1076i −0.584136 0.874562i
\(640\) 0 0
\(641\) −33.7233 −1.33199 −0.665995 0.745956i \(-0.731993\pi\)
−0.665995 + 0.745956i \(0.731993\pi\)
\(642\) 0 0
\(643\) −10.0635 + 17.4306i −0.396867 + 0.687394i −0.993338 0.115241i \(-0.963236\pi\)
0.596470 + 0.802635i \(0.296569\pi\)
\(644\) 0 0
\(645\) 14.4994 0.473355i 0.570913 0.0186383i
\(646\) 0 0
\(647\) 11.1891 19.3800i 0.439887 0.761907i −0.557793 0.829980i \(-0.688352\pi\)
0.997680 + 0.0680731i \(0.0216851\pi\)
\(648\) 0 0
\(649\) 23.5422 + 40.7763i 0.924112 + 1.60061i
\(650\) 0 0
\(651\) −3.19797 + 3.30207i −0.125338 + 0.129418i
\(652\) 0 0
\(653\) −40.6754 −1.59175 −0.795875 0.605460i \(-0.792989\pi\)
−0.795875 + 0.605460i \(0.792989\pi\)
\(654\) 0 0
\(655\) −10.3786 −0.405527
\(656\) 0 0
\(657\) −33.9834 + 2.22124i −1.32582 + 0.0866590i
\(658\) 0 0
\(659\) −7.69321 13.3250i −0.299685 0.519070i 0.676379 0.736554i \(-0.263548\pi\)
−0.976064 + 0.217484i \(0.930215\pi\)
\(660\) 0 0
\(661\) −24.5736 42.5628i −0.955804 1.65550i −0.732518 0.680747i \(-0.761655\pi\)
−0.223285 0.974753i \(-0.571678\pi\)
\(662\) 0 0
\(663\) −14.0524 + 0.458762i −0.545751 + 0.0178169i
\(664\) 0 0
\(665\) −5.07305 1.45011i −0.196724 0.0562328i
\(666\) 0 0
\(667\) 11.9967 + 20.7789i 0.464513 + 0.804561i
\(668\) 0 0
\(669\) −24.2099 + 0.790370i −0.936010 + 0.0305575i
\(670\) 0 0
\(671\) −13.5462 + 23.4627i −0.522944 + 0.905766i
\(672\) 0 0
\(673\) −6.99961 12.1237i −0.269815 0.467334i 0.698999 0.715123i \(-0.253629\pi\)
−0.968814 + 0.247789i \(0.920296\pi\)
\(674\) 0 0
\(675\) 9.40631 0.923876i 0.362049 0.0355600i
\(676\) 0 0
\(677\) 23.3870 40.5075i 0.898836 1.55683i 0.0698517 0.997557i \(-0.477747\pi\)
0.828984 0.559272i \(-0.188919\pi\)
\(678\) 0 0
\(679\) −5.63624 1.61110i −0.216299 0.0618282i
\(680\) 0 0
\(681\) −8.79279 + 16.4466i −0.336940 + 0.630235i
\(682\) 0 0
\(683\) −21.6222 + 37.4508i −0.827352 + 1.43302i 0.0727571 + 0.997350i \(0.476820\pi\)
−0.900109 + 0.435665i \(0.856513\pi\)
\(684\) 0 0
\(685\) 27.3613 1.04542
\(686\) 0 0
\(687\) −1.31478 + 2.45925i −0.0501621 + 0.0938264i
\(688\) 0 0
\(689\) 28.0091 + 48.5131i 1.06706 + 1.84820i
\(690\) 0 0
\(691\) −5.86072 + 10.1511i −0.222952 + 0.386165i −0.955703 0.294332i \(-0.904903\pi\)
0.732751 + 0.680497i \(0.238236\pi\)
\(692\) 0 0
\(693\) 43.5522 + 9.42640i 1.65441 + 0.358079i
\(694\) 0 0
\(695\) −10.8048 + 18.7144i −0.409849 + 0.709879i
\(696\) 0 0
\(697\) −5.35326 9.27212i −0.202769 0.351207i
\(698\) 0 0
\(699\) −1.76785 + 0.0577141i −0.0668662 + 0.00218295i
\(700\) 0 0
\(701\) 0.0120975 0.000456915 0.000228458 1.00000i \(-0.499927\pi\)
0.000228458 1.00000i \(0.499927\pi\)
\(702\) 0 0
\(703\) −6.67008 + 11.5529i −0.251567 + 0.435726i
\(704\) 0 0
\(705\) −3.17080 5.10009i −0.119419 0.192080i
\(706\) 0 0
\(707\) −17.8807 + 17.2960i −0.672474 + 0.650483i
\(708\) 0 0
\(709\) −0.537388 + 0.930783i −0.0201820 + 0.0349563i −0.875940 0.482420i \(-0.839758\pi\)
0.855758 + 0.517376i \(0.173091\pi\)
\(710\) 0 0
\(711\) −40.2870 + 2.63327i −1.51088 + 0.0987552i
\(712\) 0 0
\(713\) −3.82041 6.61714i −0.143075 0.247814i
\(714\) 0 0
\(715\) −31.4418 + 54.4588i −1.17586 + 2.03664i
\(716\) 0 0
\(717\) −12.7378 20.4881i −0.475701 0.765143i
\(718\) 0 0
\(719\) −12.7777 22.1317i −0.476529 0.825372i 0.523109 0.852266i \(-0.324772\pi\)
−0.999638 + 0.0268932i \(0.991439\pi\)
\(720\) 0 0
\(721\) −6.70714 + 6.48779i −0.249787 + 0.241618i
\(722\) 0 0
\(723\) −11.9029 + 22.2639i −0.442672 + 0.828003i
\(724\) 0 0
\(725\) 2.86478 + 4.96194i 0.106395 + 0.184282i
\(726\) 0 0
\(727\) −6.20522 10.7478i −0.230139 0.398612i 0.727710 0.685885i \(-0.240585\pi\)
−0.957849 + 0.287273i \(0.907251\pi\)
\(728\) 0 0
\(729\) 26.4840 5.25314i 0.980890 0.194561i
\(730\) 0 0
\(731\) −6.06996 −0.224505
\(732\) 0 0
\(733\) 29.4189 1.08661 0.543307 0.839534i \(-0.317172\pi\)
0.543307 + 0.839534i \(0.317172\pi\)
\(734\) 0 0
\(735\) 18.7207 + 10.8236i 0.690522 + 0.399234i
\(736\) 0 0
\(737\) −7.17562 12.4285i −0.264317 0.457811i
\(738\) 0 0
\(739\) −7.75910 + 13.4392i −0.285423 + 0.494368i −0.972712 0.232017i \(-0.925468\pi\)
0.687288 + 0.726385i \(0.258801\pi\)
\(740\) 0 0
\(741\) −5.73432 + 10.7258i −0.210656 + 0.394024i
\(742\) 0 0
\(743\) 13.6333 23.6136i 0.500159 0.866301i −0.499841 0.866117i \(-0.666608\pi\)
1.00000 0.000183414i \(-5.83824e-5\pi\)
\(744\) 0 0
\(745\) −8.44865 −0.309534
\(746\) 0 0
\(747\) 4.25290 8.62012i 0.155606 0.315394i
\(748\) 0 0
\(749\) −17.2615 4.93412i −0.630720 0.180289i
\(750\) 0 0
\(751\) 9.14353 0.333652 0.166826 0.985986i \(-0.446648\pi\)
0.166826 + 0.985986i \(0.446648\pi\)
\(752\) 0 0
\(753\) −14.7226 + 27.5381i −0.536522 + 1.00355i
\(754\) 0 0
\(755\) 43.4632 1.58179
\(756\) 0 0
\(757\) −20.6307 −0.749834 −0.374917 0.927058i \(-0.622329\pi\)
−0.374917 + 0.927058i \(0.622329\pi\)
\(758\) 0 0
\(759\) −34.9216 + 65.3195i −1.26757 + 2.37095i
\(760\) 0 0
\(761\) −0.478417 −0.0173426 −0.00867130 0.999962i \(-0.502760\pi\)
−0.00867130 + 0.999962i \(0.502760\pi\)
\(762\) 0 0
\(763\) 0.875765 + 3.50010i 0.0317048 + 0.126712i
\(764\) 0 0
\(765\) 6.90129 0.451087i 0.249517 0.0163091i
\(766\) 0 0
\(767\) 52.6705 1.90182
\(768\) 0 0
\(769\) −13.3518 + 23.1261i −0.481480 + 0.833948i −0.999774 0.0212548i \(-0.993234\pi\)
0.518294 + 0.855202i \(0.326567\pi\)
\(770\) 0 0
\(771\) 21.9888 41.1293i 0.791908 1.48123i
\(772\) 0 0
\(773\) −13.5143 + 23.4074i −0.486074 + 0.841905i −0.999872 0.0160062i \(-0.994905\pi\)
0.513798 + 0.857911i \(0.328238\pi\)
\(774\) 0 0
\(775\) −0.912303 1.58016i −0.0327709 0.0567609i
\(776\) 0 0
\(777\) 38.0363 39.2744i 1.36455 1.40896i
\(778\) 0 0
\(779\) −9.26165 −0.331833
\(780\) 0 0
\(781\) 49.7510 1.78023
\(782\) 0 0
\(783\) 9.53008 + 13.3068i 0.340577 + 0.475546i
\(784\) 0 0
\(785\) −5.62226 9.73804i −0.200667 0.347566i
\(786\) 0 0
\(787\) 24.5915 + 42.5937i 0.876593 + 1.51830i 0.855056 + 0.518535i \(0.173522\pi\)
0.0215363 + 0.999768i \(0.493144\pi\)
\(788\) 0 0
\(789\) −1.25665 + 2.35051i −0.0447378 + 0.0836804i
\(790\) 0 0
\(791\) −3.55083 14.1913i −0.126253 0.504585i
\(792\) 0 0
\(793\) 15.1533 + 26.2463i 0.538109 + 0.932032i
\(794\) 0 0
\(795\) −14.5489 23.4012i −0.515995 0.829955i
\(796\) 0 0
\(797\) 22.1538 38.3715i 0.784728 1.35919i −0.144434 0.989514i \(-0.546136\pi\)
0.929162 0.369674i \(-0.120531\pi\)
\(798\) 0 0
\(799\) 1.25637 + 2.17609i 0.0444471 + 0.0769846i
\(800\) 0 0
\(801\) −1.07421 + 2.17730i −0.0379555 + 0.0769311i
\(802\) 0 0
\(803\) 31.8656 55.1928i 1.12451 1.94771i
\(804\) 0 0
\(805\) −25.8352 + 24.9903i −0.910570 + 0.880792i
\(806\) 0 0
\(807\) −5.97102 9.60410i −0.210190 0.338080i
\(808\) 0 0
\(809\) −16.7359 + 28.9874i −0.588402 + 1.01914i 0.406040 + 0.913855i \(0.366909\pi\)
−0.994442 + 0.105286i \(0.966424\pi\)
\(810\) 0 0
\(811\) 43.7383 1.53586 0.767929 0.640535i \(-0.221287\pi\)
0.767929 + 0.640535i \(0.221287\pi\)
\(812\) 0 0
\(813\) 19.5442 0.638051i 0.685446 0.0223774i
\(814\) 0 0
\(815\) 0.625451 + 1.08331i 0.0219086 + 0.0379468i
\(816\) 0 0
\(817\) −2.62540 + 4.54733i −0.0918512 + 0.159091i
\(818\) 0 0
\(819\) 33.4916 36.9197i 1.17029 1.29008i
\(820\) 0 0
\(821\) 11.2566 19.4970i 0.392857 0.680449i −0.599968 0.800024i \(-0.704820\pi\)
0.992825 + 0.119575i \(0.0381533\pi\)
\(822\) 0 0
\(823\) 7.33674 + 12.7076i 0.255743 + 0.442959i 0.965097 0.261893i \(-0.0843467\pi\)
−0.709354 + 0.704852i \(0.751013\pi\)
\(824\) 0 0
\(825\) −8.33919 + 15.5981i −0.290333 + 0.543058i
\(826\) 0 0
\(827\) −45.0520 −1.56661 −0.783306 0.621636i \(-0.786468\pi\)
−0.783306 + 0.621636i \(0.786468\pi\)
\(828\) 0 0
\(829\) 20.6688 35.7993i 0.717856 1.24336i −0.243992 0.969777i \(-0.578457\pi\)
0.961848 0.273585i \(-0.0882095\pi\)
\(830\) 0 0
\(831\) 1.47961 2.76756i 0.0513273 0.0960058i
\(832\) 0 0
\(833\) −7.68102 4.78189i −0.266132 0.165683i
\(834\) 0 0
\(835\) 6.31059 10.9303i 0.218387 0.378258i
\(836\) 0 0
\(837\) −3.03490 4.23762i −0.104902 0.146473i
\(838\) 0 0
\(839\) −2.04477 3.54164i −0.0705932 0.122271i 0.828568 0.559888i \(-0.189156\pi\)
−0.899161 + 0.437617i \(0.855823\pi\)
\(840\) 0 0
\(841\) 9.53902 16.5221i 0.328932 0.569726i
\(842\) 0 0
\(843\) −10.3451 + 0.337733i −0.356306 + 0.0116321i
\(844\) 0 0
\(845\) 23.5790 + 40.8401i 0.811143 + 1.40494i
\(846\) 0 0
\(847\) −39.0189 + 37.7428i −1.34070 + 1.29686i
\(848\) 0 0
\(849\) −34.6143 + 1.13003i −1.18796 + 0.0387827i
\(850\) 0 0
\(851\) 45.4394 + 78.7034i 1.55764 + 2.69792i
\(852\) 0 0
\(853\) −21.1012 36.5484i −0.722491 1.25139i −0.959998 0.280006i \(-0.909664\pi\)
0.237507 0.971386i \(-0.423670\pi\)
\(854\) 0 0
\(855\) 2.64704 5.36523i 0.0905269 0.183487i
\(856\) 0 0
\(857\) −55.1245 −1.88302 −0.941509 0.336987i \(-0.890592\pi\)
−0.941509 + 0.336987i \(0.890592\pi\)
\(858\) 0 0
\(859\) 37.9534 1.29495 0.647476 0.762086i \(-0.275824\pi\)
0.647476 + 0.762086i \(0.275824\pi\)
\(860\) 0 0
\(861\) 36.8176 + 9.23600i 1.25474 + 0.314762i
\(862\) 0 0
\(863\) −6.27205 10.8635i −0.213503 0.369798i 0.739305 0.673370i \(-0.235154\pi\)
−0.952809 + 0.303572i \(0.901821\pi\)
\(864\) 0 0
\(865\) 5.91723 10.2489i 0.201192 0.348474i
\(866\) 0 0
\(867\) 26.5370 0.866340i 0.901243 0.0294224i
\(868\) 0 0
\(869\) 37.7764 65.4306i 1.28148 2.21958i
\(870\) 0 0
\(871\) −16.0539 −0.543965
\(872\) 0 0
\(873\) 2.94091 5.96086i 0.0995346 0.201745i
\(874\) 0 0
\(875\) −23.1279 + 22.3716i −0.781867 + 0.756297i
\(876\) 0 0
\(877\) 9.70673 0.327773 0.163887 0.986479i \(-0.447597\pi\)
0.163887 + 0.986479i \(0.447597\pi\)
\(878\) 0 0
\(879\) 9.05666 + 14.5672i 0.305474 + 0.491340i
\(880\) 0 0
\(881\) −11.6652 −0.393012 −0.196506 0.980503i \(-0.562959\pi\)
−0.196506 + 0.980503i \(0.562959\pi\)
\(882\) 0 0
\(883\) 13.1758 0.443401 0.221701 0.975115i \(-0.428839\pi\)
0.221701 + 0.975115i \(0.428839\pi\)
\(884\) 0 0
\(885\) −25.8946 + 0.845370i −0.870439 + 0.0284168i
\(886\) 0 0
\(887\) 0.830721 0.0278929 0.0139464 0.999903i \(-0.495561\pi\)
0.0139464 + 0.999903i \(0.495561\pi\)
\(888\) 0 0
\(889\) −24.3811 + 23.5838i −0.817717 + 0.790975i
\(890\) 0 0
\(891\) −19.3527 + 46.6739i −0.648338 + 1.56364i
\(892\) 0 0
\(893\) 2.17364 0.0727379
\(894\) 0 0
\(895\) 10.0508 17.4084i 0.335960 0.581900i
\(896\) 0 0
\(897\) 43.7473 + 70.3655i 1.46068 + 2.34944i
\(898\) 0 0
\(899\) 1.57985 2.73638i 0.0526910 0.0912634i
\(900\) 0 0
\(901\) 5.76470 + 9.98476i 0.192050 + 0.332641i
\(902\) 0 0
\(903\) 14.9714 15.4588i 0.498218 0.514435i
\(904\) 0 0
\(905\) −37.7756 −1.25570
\(906\) 0 0
\(907\) −33.3176 −1.10629 −0.553146 0.833084i \(-0.686573\pi\)
−0.553146 + 0.833084i \(0.686573\pi\)
\(908\) 0 0
\(909\) −15.6673 23.4569i −0.519652 0.778017i
\(910\) 0 0
\(911\) 11.1353 + 19.2870i 0.368930 + 0.639006i 0.989399 0.145226i \(-0.0463909\pi\)
−0.620469 + 0.784231i \(0.713058\pi\)
\(912\) 0 0
\(913\) 8.99396 + 15.5780i 0.297657 + 0.515556i
\(914\) 0 0
\(915\) −7.87114 12.6604i −0.260212 0.418539i
\(916\) 0 0
\(917\) −11.0660 + 10.7041i −0.365431 + 0.353480i
\(918\) 0 0
\(919\) −10.7906 18.6899i −0.355949 0.616522i 0.631331 0.775514i \(-0.282509\pi\)
−0.987280 + 0.158992i \(0.949176\pi\)
\(920\) 0 0
\(921\) 19.3638 36.2194i 0.638060 1.19347i
\(922\) 0 0
\(923\) 27.8268 48.1974i 0.915929 1.58644i
\(924\) 0 0
\(925\) 10.8508 + 18.7942i 0.356773 + 0.617949i
\(926\) 0 0
\(927\) −5.87687 8.79879i −0.193022 0.288990i
\(928\) 0 0
\(929\) 18.5780 32.1780i 0.609523 1.05572i −0.381796 0.924247i \(-0.624694\pi\)
0.991319 0.131478i \(-0.0419724\pi\)
\(930\) 0 0
\(931\) −6.90460 + 3.68598i −0.226289 + 0.120803i
\(932\) 0 0
\(933\) 35.2196 1.14980i 1.15304 0.0376427i
\(934\) 0 0
\(935\) −6.47122 + 11.2085i −0.211631 + 0.366556i
\(936\) 0 0
\(937\) 21.5238 0.703152 0.351576 0.936159i \(-0.385646\pi\)
0.351576 + 0.936159i \(0.385646\pi\)
\(938\) 0 0
\(939\) 8.88784 + 14.2957i 0.290044 + 0.466522i
\(940\) 0 0
\(941\) 23.7032 + 41.0551i 0.772702 + 1.33836i 0.936077 + 0.351795i \(0.114429\pi\)
−0.163375 + 0.986564i \(0.552238\pi\)
\(942\) 0 0
\(943\) −31.5472 + 54.6413i −1.02732 + 1.77937i
\(944\) 0 0
\(945\) −15.8731 + 18.6886i −0.516352 + 0.607939i
\(946\) 0 0
\(947\) −1.25059 + 2.16608i −0.0406386 + 0.0703881i −0.885629 0.464393i \(-0.846273\pi\)
0.844991 + 0.534781i \(0.179606\pi\)
\(948\) 0 0
\(949\) −35.6461 61.7409i −1.15712 2.00420i
\(950\) 0 0
\(951\) −3.04383 4.89586i −0.0987031 0.158759i
\(952\) 0 0
\(953\) −7.89914 −0.255878 −0.127939 0.991782i \(-0.540836\pi\)
−0.127939 + 0.991782i \(0.540836\pi\)
\(954\) 0 0
\(955\) 8.41913 14.5824i 0.272437 0.471874i
\(956\) 0 0
\(957\) −30.6132 + 0.999416i −0.989586 + 0.0323065i
\(958\) 0 0
\(959\) 29.1733 28.2193i 0.942055 0.911247i
\(960\) 0 0
\(961\) 14.9969 25.9754i 0.483771 0.837915i
\(962\) 0 0
\(963\) 9.00677 18.2556i 0.290239 0.588280i
\(964\) 0 0
\(965\) −5.60071 9.70072i −0.180293 0.312277i
\(966\) 0 0
\(967\) −5.76591 + 9.98684i −0.185419 + 0.321155i −0.943718 0.330752i \(-0.892698\pi\)
0.758299 + 0.651907i \(0.226031\pi\)
\(968\) 0 0
\(969\) −1.18021 + 2.20755i −0.0379139 + 0.0709166i
\(970\) 0 0
\(971\) 9.14669 + 15.8425i 0.293531 + 0.508411i 0.974642 0.223769i \(-0.0718362\pi\)
−0.681111 + 0.732180i \(0.738503\pi\)
\(972\) 0 0
\(973\) 7.78092 + 31.0974i 0.249445 + 0.996937i
\(974\) 0 0
\(975\) 10.4468 + 16.8031i 0.334564 + 0.538130i
\(976\) 0 0
\(977\) 14.9541 + 25.9013i 0.478425 + 0.828656i 0.999694 0.0247361i \(-0.00787454\pi\)
−0.521269 + 0.853392i \(0.674541\pi\)
\(978\) 0 0
\(979\) −2.27173 3.93475i −0.0726047 0.125755i
\(980\) 0 0
\(981\) −4.08238 + 0.266835i −0.130340 + 0.00851939i
\(982\) 0 0
\(983\) −14.5445 −0.463897 −0.231948 0.972728i \(-0.574510\pi\)
−0.231948 + 0.972728i \(0.574510\pi\)
\(984\) 0 0
\(985\) −27.6058 −0.879594
\(986\) 0 0
\(987\) −8.64080 2.16762i −0.275040 0.0689960i
\(988\) 0 0
\(989\) 17.8854 + 30.9784i 0.568722 + 0.985055i
\(990\) 0 0
\(991\) 14.3753 24.8987i 0.456646 0.790935i −0.542135 0.840292i \(-0.682384\pi\)
0.998781 + 0.0493567i \(0.0157171\pi\)
\(992\) 0 0
\(993\) 1.31204 + 2.11035i 0.0416363 + 0.0669700i
\(994\) 0 0
\(995\) 3.91447 6.78007i 0.124097 0.214943i
\(996\) 0 0
\(997\) 35.8938 1.13677 0.568384 0.822763i \(-0.307569\pi\)
0.568384 + 0.822763i \(0.307569\pi\)
\(998\) 0 0
\(999\) 36.0968 + 50.4017i 1.14205 + 1.59464i
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 1008.2.t.l.193.7 22
3.2 odd 2 3024.2.t.k.1873.8 22
4.3 odd 2 504.2.t.c.193.5 yes 22
7.2 even 3 1008.2.q.l.625.8 22
9.2 odd 6 3024.2.q.l.2881.4 22
9.7 even 3 1008.2.q.l.529.8 22
12.11 even 2 1512.2.t.c.361.8 22
21.2 odd 6 3024.2.q.l.2305.4 22
28.23 odd 6 504.2.q.c.121.4 yes 22
36.7 odd 6 504.2.q.c.25.4 22
36.11 even 6 1512.2.q.d.1369.4 22
63.2 odd 6 3024.2.t.k.289.8 22
63.16 even 3 inner 1008.2.t.l.961.7 22
84.23 even 6 1512.2.q.d.793.4 22
252.79 odd 6 504.2.t.c.457.5 yes 22
252.191 even 6 1512.2.t.c.289.8 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.q.c.25.4 22 36.7 odd 6
504.2.q.c.121.4 yes 22 28.23 odd 6
504.2.t.c.193.5 yes 22 4.3 odd 2
504.2.t.c.457.5 yes 22 252.79 odd 6
1008.2.q.l.529.8 22 9.7 even 3
1008.2.q.l.625.8 22 7.2 even 3
1008.2.t.l.193.7 22 1.1 even 1 trivial
1008.2.t.l.961.7 22 63.16 even 3 inner
1512.2.q.d.793.4 22 84.23 even 6
1512.2.q.d.1369.4 22 36.11 even 6
1512.2.t.c.289.8 22 252.191 even 6
1512.2.t.c.361.8 22 12.11 even 2
3024.2.q.l.2305.4 22 21.2 odd 6
3024.2.q.l.2881.4 22 9.2 odd 6
3024.2.t.k.289.8 22 63.2 odd 6
3024.2.t.k.1873.8 22 3.2 odd 2