Properties

Label 4020.2.g.c.1609.26
Level $4020$
Weight $2$
Character 4020.1609
Analytic conductor $32.100$
Analytic rank $0$
Dimension $38$
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] = [4020,2,Mod(1609,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4020.1609");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.g (of order \(2\), degree \(1\), 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: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(38\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1609.26
Character \(\chi\) \(=\) 4020.1609
Dual form 4020.2.g.c.1609.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +(-0.847850 - 2.06909i) q^{5} -3.67313i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +(-0.847850 - 2.06909i) q^{5} -3.67313i q^{7} -1.00000 q^{9} -3.98590 q^{11} +1.24921i q^{13} +(2.06909 - 0.847850i) q^{15} +0.354171i q^{17} -1.69097 q^{19} +3.67313 q^{21} +1.36335i q^{23} +(-3.56230 + 3.50856i) q^{25} -1.00000i q^{27} +0.677093 q^{29} +0.344129 q^{31} -3.98590i q^{33} +(-7.60004 + 3.11426i) q^{35} -8.16175i q^{37} -1.24921 q^{39} -5.45089 q^{41} +10.6444i q^{43} +(0.847850 + 2.06909i) q^{45} -10.1918i q^{47} -6.49186 q^{49} -0.354171 q^{51} -4.74012i q^{53} +(3.37945 + 8.24720i) q^{55} -1.69097i q^{57} +5.13578 q^{59} +5.19082 q^{61} +3.67313i q^{63} +(2.58472 - 1.05914i) q^{65} +1.00000i q^{67} -1.36335 q^{69} -3.56237 q^{71} +4.89398i q^{73} +(-3.50856 - 3.56230i) q^{75} +14.6407i q^{77} +5.19345 q^{79} +1.00000 q^{81} +7.49034i q^{83} +(0.732812 - 0.300284i) q^{85} +0.677093i q^{87} -11.5662 q^{89} +4.58849 q^{91} +0.344129i q^{93} +(1.43369 + 3.49878i) q^{95} +17.6632i q^{97} +3.98590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 38 q - 2 q^{5} - 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 38 q - 2 q^{5} - 38 q^{9} + 24 q^{11} + 2 q^{15} - 16 q^{19} + 12 q^{21} + 4 q^{25} - 56 q^{29} - 4 q^{31} - 2 q^{35} + 60 q^{41} + 2 q^{45} - 70 q^{49} + 12 q^{55} - 52 q^{59} + 48 q^{61} + 16 q^{65} - 12 q^{69} + 12 q^{75} - 24 q^{79} + 38 q^{81} + 16 q^{85} - 76 q^{89} + 44 q^{91} + 36 q^{95} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) −0.847850 2.06909i −0.379170 0.925327i
\(6\) 0 0
\(7\) 3.67313i 1.38831i −0.719825 0.694156i \(-0.755778\pi\)
0.719825 0.694156i \(-0.244222\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.98590 −1.20179 −0.600897 0.799327i \(-0.705190\pi\)
−0.600897 + 0.799327i \(0.705190\pi\)
\(12\) 0 0
\(13\) 1.24921i 0.346467i 0.984881 + 0.173234i \(0.0554216\pi\)
−0.984881 + 0.173234i \(0.944578\pi\)
\(14\) 0 0
\(15\) 2.06909 0.847850i 0.534238 0.218914i
\(16\) 0 0
\(17\) 0.354171i 0.0858990i 0.999077 + 0.0429495i \(0.0136755\pi\)
−0.999077 + 0.0429495i \(0.986325\pi\)
\(18\) 0 0
\(19\) −1.69097 −0.387936 −0.193968 0.981008i \(-0.562136\pi\)
−0.193968 + 0.981008i \(0.562136\pi\)
\(20\) 0 0
\(21\) 3.67313 0.801542
\(22\) 0 0
\(23\) 1.36335i 0.284279i 0.989847 + 0.142139i \(0.0453981\pi\)
−0.989847 + 0.142139i \(0.954602\pi\)
\(24\) 0 0
\(25\) −3.56230 + 3.50856i −0.712460 + 0.701713i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 0.677093 0.125733 0.0628665 0.998022i \(-0.479976\pi\)
0.0628665 + 0.998022i \(0.479976\pi\)
\(30\) 0 0
\(31\) 0.344129 0.0618074 0.0309037 0.999522i \(-0.490161\pi\)
0.0309037 + 0.999522i \(0.490161\pi\)
\(32\) 0 0
\(33\) 3.98590i 0.693856i
\(34\) 0 0
\(35\) −7.60004 + 3.11426i −1.28464 + 0.526406i
\(36\) 0 0
\(37\) 8.16175i 1.34178i −0.741555 0.670892i \(-0.765912\pi\)
0.741555 0.670892i \(-0.234088\pi\)
\(38\) 0 0
\(39\) −1.24921 −0.200033
\(40\) 0 0
\(41\) −5.45089 −0.851287 −0.425643 0.904891i \(-0.639952\pi\)
−0.425643 + 0.904891i \(0.639952\pi\)
\(42\) 0 0
\(43\) 10.6444i 1.62325i 0.584177 + 0.811626i \(0.301417\pi\)
−0.584177 + 0.811626i \(0.698583\pi\)
\(44\) 0 0
\(45\) 0.847850 + 2.06909i 0.126390 + 0.308442i
\(46\) 0 0
\(47\) 10.1918i 1.48663i −0.668944 0.743313i \(-0.733253\pi\)
0.668944 0.743313i \(-0.266747\pi\)
\(48\) 0 0
\(49\) −6.49186 −0.927408
\(50\) 0 0
\(51\) −0.354171 −0.0495938
\(52\) 0 0
\(53\) 4.74012i 0.651106i −0.945524 0.325553i \(-0.894450\pi\)
0.945524 0.325553i \(-0.105550\pi\)
\(54\) 0 0
\(55\) 3.37945 + 8.24720i 0.455684 + 1.11205i
\(56\) 0 0
\(57\) 1.69097i 0.223975i
\(58\) 0 0
\(59\) 5.13578 0.668621 0.334311 0.942463i \(-0.391497\pi\)
0.334311 + 0.942463i \(0.391497\pi\)
\(60\) 0 0
\(61\) 5.19082 0.664616 0.332308 0.943171i \(-0.392173\pi\)
0.332308 + 0.943171i \(0.392173\pi\)
\(62\) 0 0
\(63\) 3.67313i 0.462770i
\(64\) 0 0
\(65\) 2.58472 1.05914i 0.320595 0.131370i
\(66\) 0 0
\(67\) 1.00000i 0.122169i
\(68\) 0 0
\(69\) −1.36335 −0.164128
\(70\) 0 0
\(71\) −3.56237 −0.422776 −0.211388 0.977402i \(-0.567798\pi\)
−0.211388 + 0.977402i \(0.567798\pi\)
\(72\) 0 0
\(73\) 4.89398i 0.572797i 0.958111 + 0.286398i \(0.0924581\pi\)
−0.958111 + 0.286398i \(0.907542\pi\)
\(74\) 0 0
\(75\) −3.50856 3.56230i −0.405134 0.411339i
\(76\) 0 0
\(77\) 14.6407i 1.66846i
\(78\) 0 0
\(79\) 5.19345 0.584308 0.292154 0.956371i \(-0.405628\pi\)
0.292154 + 0.956371i \(0.405628\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.49034i 0.822171i 0.911597 + 0.411086i \(0.134850\pi\)
−0.911597 + 0.411086i \(0.865150\pi\)
\(84\) 0 0
\(85\) 0.732812 0.300284i 0.0794847 0.0325704i
\(86\) 0 0
\(87\) 0.677093i 0.0725920i
\(88\) 0 0
\(89\) −11.5662 −1.22601 −0.613007 0.790078i \(-0.710040\pi\)
−0.613007 + 0.790078i \(0.710040\pi\)
\(90\) 0 0
\(91\) 4.58849 0.481004
\(92\) 0 0
\(93\) 0.344129i 0.0356845i
\(94\) 0 0
\(95\) 1.43369 + 3.49878i 0.147094 + 0.358967i
\(96\) 0 0
\(97\) 17.6632i 1.79342i 0.442617 + 0.896711i \(0.354050\pi\)
−0.442617 + 0.896711i \(0.645950\pi\)
\(98\) 0 0
\(99\) 3.98590 0.400598
\(100\) 0 0
\(101\) −5.52695 −0.549952 −0.274976 0.961451i \(-0.588670\pi\)
−0.274976 + 0.961451i \(0.588670\pi\)
\(102\) 0 0
\(103\) 18.0243i 1.77599i 0.459853 + 0.887995i \(0.347902\pi\)
−0.459853 + 0.887995i \(0.652098\pi\)
\(104\) 0 0
\(105\) −3.11426 7.60004i −0.303921 0.741688i
\(106\) 0 0
\(107\) 18.2547i 1.76474i 0.470552 + 0.882372i \(0.344055\pi\)
−0.470552 + 0.882372i \(0.655945\pi\)
\(108\) 0 0
\(109\) 11.4396 1.09572 0.547859 0.836571i \(-0.315443\pi\)
0.547859 + 0.836571i \(0.315443\pi\)
\(110\) 0 0
\(111\) 8.16175 0.774679
\(112\) 0 0
\(113\) 12.1514i 1.14311i 0.820565 + 0.571553i \(0.193659\pi\)
−0.820565 + 0.571553i \(0.806341\pi\)
\(114\) 0 0
\(115\) 2.82091 1.15592i 0.263051 0.107790i
\(116\) 0 0
\(117\) 1.24921i 0.115489i
\(118\) 0 0
\(119\) 1.30091 0.119255
\(120\) 0 0
\(121\) 4.88739 0.444308
\(122\) 0 0
\(123\) 5.45089i 0.491491i
\(124\) 0 0
\(125\) 10.2798 + 4.39599i 0.919457 + 0.393190i
\(126\) 0 0
\(127\) 2.07566i 0.184185i 0.995750 + 0.0920927i \(0.0293556\pi\)
−0.995750 + 0.0920927i \(0.970644\pi\)
\(128\) 0 0
\(129\) −10.6444 −0.937185
\(130\) 0 0
\(131\) −7.87477 −0.688022 −0.344011 0.938966i \(-0.611786\pi\)
−0.344011 + 0.938966i \(0.611786\pi\)
\(132\) 0 0
\(133\) 6.21115i 0.538575i
\(134\) 0 0
\(135\) −2.06909 + 0.847850i −0.178079 + 0.0729714i
\(136\) 0 0
\(137\) 9.02338i 0.770920i 0.922725 + 0.385460i \(0.125957\pi\)
−0.922725 + 0.385460i \(0.874043\pi\)
\(138\) 0 0
\(139\) −3.33603 −0.282958 −0.141479 0.989941i \(-0.545186\pi\)
−0.141479 + 0.989941i \(0.545186\pi\)
\(140\) 0 0
\(141\) 10.1918 0.858304
\(142\) 0 0
\(143\) 4.97920i 0.416382i
\(144\) 0 0
\(145\) −0.574074 1.40097i −0.0476742 0.116344i
\(146\) 0 0
\(147\) 6.49186i 0.535439i
\(148\) 0 0
\(149\) 10.9548 0.897452 0.448726 0.893670i \(-0.351878\pi\)
0.448726 + 0.893670i \(0.351878\pi\)
\(150\) 0 0
\(151\) −5.79175 −0.471326 −0.235663 0.971835i \(-0.575726\pi\)
−0.235663 + 0.971835i \(0.575726\pi\)
\(152\) 0 0
\(153\) 0.354171i 0.0286330i
\(154\) 0 0
\(155\) −0.291770 0.712036i −0.0234355 0.0571921i
\(156\) 0 0
\(157\) 20.0431i 1.59961i −0.600258 0.799806i \(-0.704936\pi\)
0.600258 0.799806i \(-0.295064\pi\)
\(158\) 0 0
\(159\) 4.74012 0.375916
\(160\) 0 0
\(161\) 5.00777 0.394668
\(162\) 0 0
\(163\) 16.4495i 1.28843i 0.764846 + 0.644213i \(0.222815\pi\)
−0.764846 + 0.644213i \(0.777185\pi\)
\(164\) 0 0
\(165\) −8.24720 + 3.37945i −0.642043 + 0.263089i
\(166\) 0 0
\(167\) 10.2670i 0.794488i −0.917713 0.397244i \(-0.869967\pi\)
0.917713 0.397244i \(-0.130033\pi\)
\(168\) 0 0
\(169\) 11.4395 0.879960
\(170\) 0 0
\(171\) 1.69097 0.129312
\(172\) 0 0
\(173\) 9.03279i 0.686750i 0.939198 + 0.343375i \(0.111570\pi\)
−0.939198 + 0.343375i \(0.888430\pi\)
\(174\) 0 0
\(175\) 12.8874 + 13.0848i 0.974196 + 0.989116i
\(176\) 0 0
\(177\) 5.13578i 0.386029i
\(178\) 0 0
\(179\) −16.4474 −1.22933 −0.614666 0.788787i \(-0.710709\pi\)
−0.614666 + 0.788787i \(0.710709\pi\)
\(180\) 0 0
\(181\) 19.0006 1.41230 0.706152 0.708060i \(-0.250429\pi\)
0.706152 + 0.708060i \(0.250429\pi\)
\(182\) 0 0
\(183\) 5.19082i 0.383716i
\(184\) 0 0
\(185\) −16.8874 + 6.91994i −1.24159 + 0.508764i
\(186\) 0 0
\(187\) 1.41169i 0.103233i
\(188\) 0 0
\(189\) −3.67313 −0.267181
\(190\) 0 0
\(191\) −23.1050 −1.67182 −0.835910 0.548866i \(-0.815060\pi\)
−0.835910 + 0.548866i \(0.815060\pi\)
\(192\) 0 0
\(193\) 2.97961i 0.214477i 0.994233 + 0.107239i \(0.0342009\pi\)
−0.994233 + 0.107239i \(0.965799\pi\)
\(194\) 0 0
\(195\) 1.05914 + 2.58472i 0.0758465 + 0.185096i
\(196\) 0 0
\(197\) 7.15240i 0.509587i −0.966995 0.254794i \(-0.917992\pi\)
0.966995 0.254794i \(-0.0820075\pi\)
\(198\) 0 0
\(199\) −13.7887 −0.977452 −0.488726 0.872437i \(-0.662538\pi\)
−0.488726 + 0.872437i \(0.662538\pi\)
\(200\) 0 0
\(201\) −1.00000 −0.0705346
\(202\) 0 0
\(203\) 2.48705i 0.174557i
\(204\) 0 0
\(205\) 4.62154 + 11.2784i 0.322783 + 0.787718i
\(206\) 0 0
\(207\) 1.36335i 0.0947596i
\(208\) 0 0
\(209\) 6.74004 0.466218
\(210\) 0 0
\(211\) −0.00933814 −0.000642864 −0.000321432 1.00000i \(-0.500102\pi\)
−0.000321432 1.00000i \(0.500102\pi\)
\(212\) 0 0
\(213\) 3.56237i 0.244090i
\(214\) 0 0
\(215\) 22.0242 9.02484i 1.50204 0.615489i
\(216\) 0 0
\(217\) 1.26403i 0.0858080i
\(218\) 0 0
\(219\) −4.89398 −0.330704
\(220\) 0 0
\(221\) −0.442432 −0.0297612
\(222\) 0 0
\(223\) 8.79157i 0.588727i 0.955694 + 0.294364i \(0.0951077\pi\)
−0.955694 + 0.294364i \(0.904892\pi\)
\(224\) 0 0
\(225\) 3.56230 3.50856i 0.237487 0.233904i
\(226\) 0 0
\(227\) 25.8704i 1.71708i −0.512747 0.858540i \(-0.671372\pi\)
0.512747 0.858540i \(-0.328628\pi\)
\(228\) 0 0
\(229\) 7.38276 0.487867 0.243933 0.969792i \(-0.421562\pi\)
0.243933 + 0.969792i \(0.421562\pi\)
\(230\) 0 0
\(231\) −14.6407 −0.963288
\(232\) 0 0
\(233\) 4.02124i 0.263440i 0.991287 + 0.131720i \(0.0420500\pi\)
−0.991287 + 0.131720i \(0.957950\pi\)
\(234\) 0 0
\(235\) −21.0878 + 8.64111i −1.37561 + 0.563684i
\(236\) 0 0
\(237\) 5.19345i 0.337351i
\(238\) 0 0
\(239\) −4.65242 −0.300940 −0.150470 0.988615i \(-0.548079\pi\)
−0.150470 + 0.988615i \(0.548079\pi\)
\(240\) 0 0
\(241\) 5.31972 0.342674 0.171337 0.985213i \(-0.445191\pi\)
0.171337 + 0.985213i \(0.445191\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 5.50412 + 13.4323i 0.351646 + 0.858156i
\(246\) 0 0
\(247\) 2.11237i 0.134407i
\(248\) 0 0
\(249\) −7.49034 −0.474681
\(250\) 0 0
\(251\) 15.4986 0.978261 0.489131 0.872211i \(-0.337314\pi\)
0.489131 + 0.872211i \(0.337314\pi\)
\(252\) 0 0
\(253\) 5.43419i 0.341645i
\(254\) 0 0
\(255\) 0.300284 + 0.732812i 0.0188045 + 0.0458905i
\(256\) 0 0
\(257\) 11.0848i 0.691448i 0.938336 + 0.345724i \(0.112367\pi\)
−0.938336 + 0.345724i \(0.887633\pi\)
\(258\) 0 0
\(259\) −29.9791 −1.86281
\(260\) 0 0
\(261\) −0.677093 −0.0419110
\(262\) 0 0
\(263\) 18.3141i 1.12930i 0.825331 + 0.564649i \(0.190989\pi\)
−0.825331 + 0.564649i \(0.809011\pi\)
\(264\) 0 0
\(265\) −9.80775 + 4.01891i −0.602486 + 0.246880i
\(266\) 0 0
\(267\) 11.5662i 0.707839i
\(268\) 0 0
\(269\) −3.87552 −0.236294 −0.118147 0.992996i \(-0.537695\pi\)
−0.118147 + 0.992996i \(0.537695\pi\)
\(270\) 0 0
\(271\) −6.86979 −0.417310 −0.208655 0.977989i \(-0.566909\pi\)
−0.208655 + 0.977989i \(0.566909\pi\)
\(272\) 0 0
\(273\) 4.58849i 0.277708i
\(274\) 0 0
\(275\) 14.1990 13.9848i 0.856230 0.843314i
\(276\) 0 0
\(277\) 29.2508i 1.75751i −0.477274 0.878755i \(-0.658375\pi\)
0.477274 0.878755i \(-0.341625\pi\)
\(278\) 0 0
\(279\) −0.344129 −0.0206025
\(280\) 0 0
\(281\) −9.06296 −0.540651 −0.270325 0.962769i \(-0.587131\pi\)
−0.270325 + 0.962769i \(0.587131\pi\)
\(282\) 0 0
\(283\) 2.74959i 0.163446i −0.996655 0.0817232i \(-0.973958\pi\)
0.996655 0.0817232i \(-0.0260423\pi\)
\(284\) 0 0
\(285\) −3.49878 + 1.43369i −0.207250 + 0.0849245i
\(286\) 0 0
\(287\) 20.0218i 1.18185i
\(288\) 0 0
\(289\) 16.8746 0.992621
\(290\) 0 0
\(291\) −17.6632 −1.03543
\(292\) 0 0
\(293\) 5.28044i 0.308487i 0.988033 + 0.154243i \(0.0492940\pi\)
−0.988033 + 0.154243i \(0.950706\pi\)
\(294\) 0 0
\(295\) −4.35437 10.6264i −0.253521 0.618693i
\(296\) 0 0
\(297\) 3.98590i 0.231285i
\(298\) 0 0
\(299\) −1.70311 −0.0984933
\(300\) 0 0
\(301\) 39.0981 2.25358
\(302\) 0 0
\(303\) 5.52695i 0.317515i
\(304\) 0 0
\(305\) −4.40104 10.7403i −0.252003 0.614987i
\(306\) 0 0
\(307\) 31.1784i 1.77944i 0.456502 + 0.889722i \(0.349102\pi\)
−0.456502 + 0.889722i \(0.650898\pi\)
\(308\) 0 0
\(309\) −18.0243 −1.02537
\(310\) 0 0
\(311\) −17.4565 −0.989869 −0.494934 0.868930i \(-0.664808\pi\)
−0.494934 + 0.868930i \(0.664808\pi\)
\(312\) 0 0
\(313\) 10.3566i 0.585388i −0.956206 0.292694i \(-0.905448\pi\)
0.956206 0.292694i \(-0.0945518\pi\)
\(314\) 0 0
\(315\) 7.60004 3.11426i 0.428214 0.175469i
\(316\) 0 0
\(317\) 2.19437i 0.123248i −0.998099 0.0616240i \(-0.980372\pi\)
0.998099 0.0616240i \(-0.0196280\pi\)
\(318\) 0 0
\(319\) −2.69882 −0.151105
\(320\) 0 0
\(321\) −18.2547 −1.01888
\(322\) 0 0
\(323\) 0.598893i 0.0333233i
\(324\) 0 0
\(325\) −4.38292 4.45004i −0.243120 0.246844i
\(326\) 0 0
\(327\) 11.4396i 0.632613i
\(328\) 0 0
\(329\) −37.4357 −2.06390
\(330\) 0 0
\(331\) −9.64862 −0.530336 −0.265168 0.964202i \(-0.585427\pi\)
−0.265168 + 0.964202i \(0.585427\pi\)
\(332\) 0 0
\(333\) 8.16175i 0.447261i
\(334\) 0 0
\(335\) 2.06909 0.847850i 0.113047 0.0463230i
\(336\) 0 0
\(337\) 26.6003i 1.44901i −0.689269 0.724505i \(-0.742068\pi\)
0.689269 0.724505i \(-0.257932\pi\)
\(338\) 0 0
\(339\) −12.1514 −0.659972
\(340\) 0 0
\(341\) −1.37166 −0.0742798
\(342\) 0 0
\(343\) 1.86648i 0.100780i
\(344\) 0 0
\(345\) 1.15592 + 2.82091i 0.0622326 + 0.151873i
\(346\) 0 0
\(347\) 19.0663i 1.02353i 0.859125 + 0.511766i \(0.171008\pi\)
−0.859125 + 0.511766i \(0.828992\pi\)
\(348\) 0 0
\(349\) −34.9836 −1.87263 −0.936314 0.351164i \(-0.885786\pi\)
−0.936314 + 0.351164i \(0.885786\pi\)
\(350\) 0 0
\(351\) 1.24921 0.0666776
\(352\) 0 0
\(353\) 14.8946i 0.792758i 0.918087 + 0.396379i \(0.129733\pi\)
−0.918087 + 0.396379i \(0.870267\pi\)
\(354\) 0 0
\(355\) 3.02036 + 7.37089i 0.160304 + 0.391206i
\(356\) 0 0
\(357\) 1.30091i 0.0688517i
\(358\) 0 0
\(359\) −20.8058 −1.09809 −0.549043 0.835794i \(-0.685008\pi\)
−0.549043 + 0.835794i \(0.685008\pi\)
\(360\) 0 0
\(361\) −16.1406 −0.849506
\(362\) 0 0
\(363\) 4.88739i 0.256521i
\(364\) 0 0
\(365\) 10.1261 4.14936i 0.530024 0.217187i
\(366\) 0 0
\(367\) 9.13253i 0.476714i −0.971178 0.238357i \(-0.923391\pi\)
0.971178 0.238357i \(-0.0766089\pi\)
\(368\) 0 0
\(369\) 5.45089 0.283762
\(370\) 0 0
\(371\) −17.4111 −0.903937
\(372\) 0 0
\(373\) 18.9166i 0.979464i −0.871873 0.489732i \(-0.837095\pi\)
0.871873 0.489732i \(-0.162905\pi\)
\(374\) 0 0
\(375\) −4.39599 + 10.2798i −0.227008 + 0.530849i
\(376\) 0 0
\(377\) 0.845828i 0.0435624i
\(378\) 0 0
\(379\) 4.68889 0.240852 0.120426 0.992722i \(-0.461574\pi\)
0.120426 + 0.992722i \(0.461574\pi\)
\(380\) 0 0
\(381\) −2.07566 −0.106339
\(382\) 0 0
\(383\) 11.1860i 0.571575i 0.958293 + 0.285788i \(0.0922552\pi\)
−0.958293 + 0.285788i \(0.907745\pi\)
\(384\) 0 0
\(385\) 30.2930 12.4131i 1.54387 0.632632i
\(386\) 0 0
\(387\) 10.6444i 0.541084i
\(388\) 0 0
\(389\) −32.5925 −1.65250 −0.826252 0.563301i \(-0.809531\pi\)
−0.826252 + 0.563301i \(0.809531\pi\)
\(390\) 0 0
\(391\) −0.482860 −0.0244193
\(392\) 0 0
\(393\) 7.87477i 0.397230i
\(394\) 0 0
\(395\) −4.40327 10.7457i −0.221552 0.540676i
\(396\) 0 0
\(397\) 16.4542i 0.825811i 0.910774 + 0.412906i \(0.135486\pi\)
−0.910774 + 0.412906i \(0.864514\pi\)
\(398\) 0 0
\(399\) −6.21115 −0.310947
\(400\) 0 0
\(401\) 18.9972 0.948674 0.474337 0.880343i \(-0.342688\pi\)
0.474337 + 0.880343i \(0.342688\pi\)
\(402\) 0 0
\(403\) 0.429888i 0.0214142i
\(404\) 0 0
\(405\) −0.847850 2.06909i −0.0421300 0.102814i
\(406\) 0 0
\(407\) 32.5319i 1.61255i
\(408\) 0 0
\(409\) 2.04519 0.101128 0.0505641 0.998721i \(-0.483898\pi\)
0.0505641 + 0.998721i \(0.483898\pi\)
\(410\) 0 0
\(411\) −9.02338 −0.445091
\(412\) 0 0
\(413\) 18.8644i 0.928254i
\(414\) 0 0
\(415\) 15.4982 6.35069i 0.760777 0.311743i
\(416\) 0 0
\(417\) 3.33603i 0.163366i
\(418\) 0 0
\(419\) 11.8865 0.580695 0.290348 0.956921i \(-0.406229\pi\)
0.290348 + 0.956921i \(0.406229\pi\)
\(420\) 0 0
\(421\) 20.8761 1.01744 0.508719 0.860933i \(-0.330119\pi\)
0.508719 + 0.860933i \(0.330119\pi\)
\(422\) 0 0
\(423\) 10.1918i 0.495542i
\(424\) 0 0
\(425\) −1.24263 1.26166i −0.0602764 0.0611996i
\(426\) 0 0
\(427\) 19.0665i 0.922694i
\(428\) 0 0
\(429\) 4.97920 0.240398
\(430\) 0 0
\(431\) 1.98614 0.0956692 0.0478346 0.998855i \(-0.484768\pi\)
0.0478346 + 0.998855i \(0.484768\pi\)
\(432\) 0 0
\(433\) 8.40462i 0.403900i 0.979396 + 0.201950i \(0.0647279\pi\)
−0.979396 + 0.201950i \(0.935272\pi\)
\(434\) 0 0
\(435\) 1.40097 0.574074i 0.0671713 0.0275247i
\(436\) 0 0
\(437\) 2.30539i 0.110282i
\(438\) 0 0
\(439\) −26.1890 −1.24993 −0.624966 0.780652i \(-0.714887\pi\)
−0.624966 + 0.780652i \(0.714887\pi\)
\(440\) 0 0
\(441\) 6.49186 0.309136
\(442\) 0 0
\(443\) 31.3510i 1.48953i −0.667327 0.744765i \(-0.732561\pi\)
0.667327 0.744765i \(-0.267439\pi\)
\(444\) 0 0
\(445\) 9.80640 + 23.9315i 0.464868 + 1.13446i
\(446\) 0 0
\(447\) 10.9548i 0.518144i
\(448\) 0 0
\(449\) −0.482287 −0.0227605 −0.0113803 0.999935i \(-0.503623\pi\)
−0.0113803 + 0.999935i \(0.503623\pi\)
\(450\) 0 0
\(451\) 21.7267 1.02307
\(452\) 0 0
\(453\) 5.79175i 0.272120i
\(454\) 0 0
\(455\) −3.89035 9.49401i −0.182383 0.445086i
\(456\) 0 0
\(457\) 22.1593i 1.03657i 0.855208 + 0.518285i \(0.173429\pi\)
−0.855208 + 0.518285i \(0.826571\pi\)
\(458\) 0 0
\(459\) 0.354171 0.0165313
\(460\) 0 0
\(461\) −9.56263 −0.445376 −0.222688 0.974890i \(-0.571483\pi\)
−0.222688 + 0.974890i \(0.571483\pi\)
\(462\) 0 0
\(463\) 6.00166i 0.278921i 0.990228 + 0.139460i \(0.0445368\pi\)
−0.990228 + 0.139460i \(0.955463\pi\)
\(464\) 0 0
\(465\) 0.712036 0.291770i 0.0330199 0.0135305i
\(466\) 0 0
\(467\) 25.2288i 1.16745i −0.811951 0.583726i \(-0.801594\pi\)
0.811951 0.583726i \(-0.198406\pi\)
\(468\) 0 0
\(469\) 3.67313 0.169609
\(470\) 0 0
\(471\) 20.0431 0.923537
\(472\) 0 0
\(473\) 42.4274i 1.95081i
\(474\) 0 0
\(475\) 6.02375 5.93288i 0.276389 0.272219i
\(476\) 0 0
\(477\) 4.74012i 0.217035i
\(478\) 0 0
\(479\) −36.5193 −1.66861 −0.834305 0.551303i \(-0.814131\pi\)
−0.834305 + 0.551303i \(0.814131\pi\)
\(480\) 0 0
\(481\) 10.1957 0.464884
\(482\) 0 0
\(483\) 5.00777i 0.227861i
\(484\) 0 0
\(485\) 36.5467 14.9757i 1.65950 0.680012i
\(486\) 0 0
\(487\) 23.0782i 1.04578i 0.852402 + 0.522888i \(0.175145\pi\)
−0.852402 + 0.522888i \(0.824855\pi\)
\(488\) 0 0
\(489\) −16.4495 −0.743873
\(490\) 0 0
\(491\) −22.6298 −1.02127 −0.510635 0.859798i \(-0.670590\pi\)
−0.510635 + 0.859798i \(0.670590\pi\)
\(492\) 0 0
\(493\) 0.239806i 0.0108003i
\(494\) 0 0
\(495\) −3.37945 8.24720i −0.151895 0.370684i
\(496\) 0 0
\(497\) 13.0851i 0.586945i
\(498\) 0 0
\(499\) 19.3444 0.865972 0.432986 0.901401i \(-0.357460\pi\)
0.432986 + 0.901401i \(0.357460\pi\)
\(500\) 0 0
\(501\) 10.2670 0.458698
\(502\) 0 0
\(503\) 0.211980i 0.00945172i 0.999989 + 0.00472586i \(0.00150429\pi\)
−0.999989 + 0.00472586i \(0.998496\pi\)
\(504\) 0 0
\(505\) 4.68603 + 11.4358i 0.208525 + 0.508885i
\(506\) 0 0
\(507\) 11.4395i 0.508045i
\(508\) 0 0
\(509\) 12.3733 0.548435 0.274218 0.961668i \(-0.411581\pi\)
0.274218 + 0.961668i \(0.411581\pi\)
\(510\) 0 0
\(511\) 17.9762 0.795220
\(512\) 0 0
\(513\) 1.69097i 0.0746582i
\(514\) 0 0
\(515\) 37.2940 15.2819i 1.64337 0.673403i
\(516\) 0 0
\(517\) 40.6234i 1.78662i
\(518\) 0 0
\(519\) −9.03279 −0.396495
\(520\) 0 0
\(521\) 3.82228 0.167457 0.0837285 0.996489i \(-0.473317\pi\)
0.0837285 + 0.996489i \(0.473317\pi\)
\(522\) 0 0
\(523\) 22.8462i 0.998994i 0.866316 + 0.499497i \(0.166482\pi\)
−0.866316 + 0.499497i \(0.833518\pi\)
\(524\) 0 0
\(525\) −13.0848 + 12.8874i −0.571066 + 0.562452i
\(526\) 0 0
\(527\) 0.121880i 0.00530920i
\(528\) 0 0
\(529\) 21.1413 0.919186
\(530\) 0 0
\(531\) −5.13578 −0.222874
\(532\) 0 0
\(533\) 6.80929i 0.294943i
\(534\) 0 0
\(535\) 37.7706 15.4772i 1.63297 0.669139i
\(536\) 0 0
\(537\) 16.4474i 0.709756i
\(538\) 0 0
\(539\) 25.8759 1.11455
\(540\) 0 0
\(541\) 42.2690 1.81729 0.908643 0.417573i \(-0.137119\pi\)
0.908643 + 0.417573i \(0.137119\pi\)
\(542\) 0 0
\(543\) 19.0006i 0.815394i
\(544\) 0 0
\(545\) −9.69909 23.6697i −0.415464 1.01390i
\(546\) 0 0
\(547\) 22.6352i 0.967811i −0.875120 0.483906i \(-0.839218\pi\)
0.875120 0.483906i \(-0.160782\pi\)
\(548\) 0 0
\(549\) −5.19082 −0.221539
\(550\) 0 0
\(551\) −1.14495 −0.0487763
\(552\) 0 0
\(553\) 19.0762i 0.811202i
\(554\) 0 0
\(555\) −6.91994 16.8874i −0.293735 0.716831i
\(556\) 0 0
\(557\) 23.9334i 1.01409i −0.861920 0.507045i \(-0.830738\pi\)
0.861920 0.507045i \(-0.169262\pi\)
\(558\) 0 0
\(559\) −13.2970 −0.562404
\(560\) 0 0
\(561\) 1.41169 0.0596015
\(562\) 0 0
\(563\) 1.50398i 0.0633850i 0.999498 + 0.0316925i \(0.0100897\pi\)
−0.999498 + 0.0316925i \(0.989910\pi\)
\(564\) 0 0
\(565\) 25.1423 10.3025i 1.05775 0.433432i
\(566\) 0 0
\(567\) 3.67313i 0.154257i
\(568\) 0 0
\(569\) −46.5849 −1.95294 −0.976469 0.215657i \(-0.930811\pi\)
−0.976469 + 0.215657i \(0.930811\pi\)
\(570\) 0 0
\(571\) 2.19735 0.0919561 0.0459780 0.998942i \(-0.485360\pi\)
0.0459780 + 0.998942i \(0.485360\pi\)
\(572\) 0 0
\(573\) 23.1050i 0.965226i
\(574\) 0 0
\(575\) −4.78341 4.85667i −0.199482 0.202537i
\(576\) 0 0
\(577\) 31.1751i 1.29784i −0.760857 0.648919i \(-0.775221\pi\)
0.760857 0.648919i \(-0.224779\pi\)
\(578\) 0 0
\(579\) −2.97961 −0.123829
\(580\) 0 0
\(581\) 27.5130 1.14143
\(582\) 0 0
\(583\) 18.8936i 0.782495i
\(584\) 0 0
\(585\) −2.58472 + 1.05914i −0.106865 + 0.0437900i
\(586\) 0 0
\(587\) 19.7574i 0.815477i 0.913099 + 0.407738i \(0.133682\pi\)
−0.913099 + 0.407738i \(0.866318\pi\)
\(588\) 0 0
\(589\) −0.581913 −0.0239773
\(590\) 0 0
\(591\) 7.15240 0.294210
\(592\) 0 0
\(593\) 44.5399i 1.82903i −0.404547 0.914517i \(-0.632571\pi\)
0.404547 0.914517i \(-0.367429\pi\)
\(594\) 0 0
\(595\) −1.10298 2.69171i −0.0452178 0.110349i
\(596\) 0 0
\(597\) 13.7887i 0.564332i
\(598\) 0 0
\(599\) 25.4327 1.03915 0.519577 0.854424i \(-0.326090\pi\)
0.519577 + 0.854424i \(0.326090\pi\)
\(600\) 0 0
\(601\) −8.66513 −0.353458 −0.176729 0.984260i \(-0.556552\pi\)
−0.176729 + 0.984260i \(0.556552\pi\)
\(602\) 0 0
\(603\) 1.00000i 0.0407231i
\(604\) 0 0
\(605\) −4.14377 10.1125i −0.168468 0.411130i
\(606\) 0 0
\(607\) 13.3352i 0.541258i −0.962684 0.270629i \(-0.912768\pi\)
0.962684 0.270629i \(-0.0872316\pi\)
\(608\) 0 0
\(609\) 2.48705 0.100780
\(610\) 0 0
\(611\) 12.7316 0.515067
\(612\) 0 0
\(613\) 16.7704i 0.677350i 0.940903 + 0.338675i \(0.109979\pi\)
−0.940903 + 0.338675i \(0.890021\pi\)
\(614\) 0 0
\(615\) −11.2784 + 4.62154i −0.454789 + 0.186359i
\(616\) 0 0
\(617\) 25.6884i 1.03418i 0.855932 + 0.517088i \(0.172984\pi\)
−0.855932 + 0.517088i \(0.827016\pi\)
\(618\) 0 0
\(619\) −41.3538 −1.66215 −0.831075 0.556160i \(-0.812274\pi\)
−0.831075 + 0.556160i \(0.812274\pi\)
\(620\) 0 0
\(621\) 1.36335 0.0547095
\(622\) 0 0
\(623\) 42.4841i 1.70209i
\(624\) 0 0
\(625\) 0.379950 24.9971i 0.0151980 0.999885i
\(626\) 0 0
\(627\) 6.74004i 0.269171i
\(628\) 0 0
\(629\) 2.89065 0.115258
\(630\) 0 0
\(631\) 7.06053 0.281075 0.140538 0.990075i \(-0.455117\pi\)
0.140538 + 0.990075i \(0.455117\pi\)
\(632\) 0 0
\(633\) 0.00933814i 0.000371158i
\(634\) 0 0
\(635\) 4.29474 1.75985i 0.170432 0.0698376i
\(636\) 0 0
\(637\) 8.10966i 0.321316i
\(638\) 0 0
\(639\) 3.56237 0.140925
\(640\) 0 0
\(641\) −4.92465 −0.194512 −0.0972560 0.995259i \(-0.531007\pi\)
−0.0972560 + 0.995259i \(0.531007\pi\)
\(642\) 0 0
\(643\) 17.0877i 0.673871i 0.941528 + 0.336936i \(0.109390\pi\)
−0.941528 + 0.336936i \(0.890610\pi\)
\(644\) 0 0
\(645\) 9.02484 + 22.0242i 0.355353 + 0.867202i
\(646\) 0 0
\(647\) 26.7562i 1.05190i −0.850517 0.525948i \(-0.823711\pi\)
0.850517 0.525948i \(-0.176289\pi\)
\(648\) 0 0
\(649\) −20.4707 −0.803545
\(650\) 0 0
\(651\) 1.26403 0.0495412
\(652\) 0 0
\(653\) 21.9516i 0.859034i −0.903059 0.429517i \(-0.858684\pi\)
0.903059 0.429517i \(-0.141316\pi\)
\(654\) 0 0
\(655\) 6.67663 + 16.2936i 0.260877 + 0.636645i
\(656\) 0 0
\(657\) 4.89398i 0.190932i
\(658\) 0 0
\(659\) 18.6068 0.724819 0.362409 0.932019i \(-0.381954\pi\)
0.362409 + 0.932019i \(0.381954\pi\)
\(660\) 0 0
\(661\) 25.4378 0.989416 0.494708 0.869059i \(-0.335275\pi\)
0.494708 + 0.869059i \(0.335275\pi\)
\(662\) 0 0
\(663\) 0.442432i 0.0171826i
\(664\) 0 0
\(665\) 12.8515 5.26613i 0.498358 0.204212i
\(666\) 0 0
\(667\) 0.923117i 0.0357432i
\(668\) 0 0
\(669\) −8.79157 −0.339902
\(670\) 0 0
\(671\) −20.6901 −0.798731
\(672\) 0 0
\(673\) 12.5740i 0.484693i −0.970190 0.242346i \(-0.922083\pi\)
0.970190 0.242346i \(-0.0779170\pi\)
\(674\) 0 0
\(675\) 3.50856 + 3.56230i 0.135045 + 0.137113i
\(676\) 0 0
\(677\) 21.8260i 0.838843i 0.907792 + 0.419422i \(0.137767\pi\)
−0.907792 + 0.419422i \(0.862233\pi\)
\(678\) 0 0
\(679\) 64.8790 2.48983
\(680\) 0 0
\(681\) 25.8704 0.991356
\(682\) 0 0
\(683\) 9.52151i 0.364331i −0.983268 0.182165i \(-0.941689\pi\)
0.983268 0.182165i \(-0.0583106\pi\)
\(684\) 0 0
\(685\) 18.6702 7.65048i 0.713353 0.292310i
\(686\) 0 0
\(687\) 7.38276i 0.281670i
\(688\) 0 0
\(689\) 5.92138 0.225587
\(690\) 0 0
\(691\) 0.564108 0.0214597 0.0107298 0.999942i \(-0.496585\pi\)
0.0107298 + 0.999942i \(0.496585\pi\)
\(692\) 0 0
\(693\) 14.6407i 0.556154i
\(694\) 0 0
\(695\) 2.82845 + 6.90256i 0.107289 + 0.261829i
\(696\) 0 0
\(697\) 1.93055i 0.0731247i
\(698\) 0 0
\(699\) −4.02124 −0.152097
\(700\) 0 0
\(701\) −27.9780 −1.05671 −0.528356 0.849023i \(-0.677192\pi\)
−0.528356 + 0.849023i \(0.677192\pi\)
\(702\) 0 0
\(703\) 13.8013i 0.520526i
\(704\) 0 0
\(705\) −8.64111 21.0878i −0.325443 0.794211i
\(706\) 0 0
\(707\) 20.3012i 0.763504i
\(708\) 0 0
\(709\) 18.8221 0.706881 0.353440 0.935457i \(-0.385012\pi\)
0.353440 + 0.935457i \(0.385012\pi\)
\(710\) 0 0
\(711\) −5.19345 −0.194769
\(712\) 0 0
\(713\) 0.469170i 0.0175705i
\(714\) 0 0
\(715\) −10.3024 + 4.22162i −0.385290 + 0.157880i
\(716\) 0 0
\(717\) 4.65242i 0.173748i
\(718\) 0 0
\(719\) −22.8755 −0.853112 −0.426556 0.904461i \(-0.640273\pi\)
−0.426556 + 0.904461i \(0.640273\pi\)
\(720\) 0 0
\(721\) 66.2057 2.46563
\(722\) 0 0
\(723\) 5.31972i 0.197843i
\(724\) 0 0
\(725\) −2.41201 + 2.37562i −0.0895797 + 0.0882285i
\(726\) 0 0
\(727\) 37.5873i 1.39404i 0.717053 + 0.697018i \(0.245490\pi\)
−0.717053 + 0.697018i \(0.754510\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −3.76993 −0.139436
\(732\) 0 0
\(733\) 23.6054i 0.871886i 0.899974 + 0.435943i \(0.143585\pi\)
−0.899974 + 0.435943i \(0.856415\pi\)
\(734\) 0 0
\(735\) −13.4323 + 5.50412i −0.495456 + 0.203023i
\(736\) 0 0
\(737\) 3.98590i 0.146822i
\(738\) 0 0
\(739\) −9.12097 −0.335520 −0.167760 0.985828i \(-0.553653\pi\)
−0.167760 + 0.985828i \(0.553653\pi\)
\(740\) 0 0
\(741\) 2.11237 0.0775999
\(742\) 0 0
\(743\) 6.75655i 0.247874i 0.992290 + 0.123937i \(0.0395520\pi\)
−0.992290 + 0.123937i \(0.960448\pi\)
\(744\) 0 0
\(745\) −9.28803 22.6665i −0.340287 0.830436i
\(746\) 0 0
\(747\) 7.49034i 0.274057i
\(748\) 0 0
\(749\) 67.0517 2.45001
\(750\) 0 0
\(751\) −36.9238 −1.34737 −0.673683 0.739020i \(-0.735289\pi\)
−0.673683 + 0.739020i \(0.735289\pi\)
\(752\) 0 0
\(753\) 15.4986i 0.564799i
\(754\) 0 0
\(755\) 4.91054 + 11.9837i 0.178713 + 0.436131i
\(756\) 0 0
\(757\) 17.3715i 0.631378i 0.948863 + 0.315689i \(0.102236\pi\)
−0.948863 + 0.315689i \(0.897764\pi\)
\(758\) 0 0
\(759\) 5.43419 0.197249
\(760\) 0 0
\(761\) 36.6221 1.32755 0.663775 0.747933i \(-0.268953\pi\)
0.663775 + 0.747933i \(0.268953\pi\)
\(762\) 0 0
\(763\) 42.0192i 1.52120i
\(764\) 0 0
\(765\) −0.732812 + 0.300284i −0.0264949 + 0.0108568i
\(766\) 0 0
\(767\) 6.41564i 0.231655i
\(768\) 0 0
\(769\) −35.7791 −1.29023 −0.645114 0.764086i \(-0.723190\pi\)
−0.645114 + 0.764086i \(0.723190\pi\)
\(770\) 0 0
\(771\) −11.0848 −0.399208
\(772\) 0 0
\(773\) 22.5349i 0.810524i −0.914201 0.405262i \(-0.867180\pi\)
0.914201 0.405262i \(-0.132820\pi\)
\(774\) 0 0
\(775\) −1.22589 + 1.20740i −0.0440353 + 0.0433711i
\(776\) 0 0
\(777\) 29.9791i 1.07550i
\(778\) 0 0
\(779\) 9.21731 0.330244
\(780\) 0 0
\(781\) 14.1993 0.508090
\(782\) 0 0
\(783\) 0.677093i 0.0241973i
\(784\) 0 0
\(785\) −41.4710 + 16.9935i −1.48016 + 0.606525i
\(786\) 0 0
\(787\) 5.35297i 0.190813i 0.995438 + 0.0954064i \(0.0304151\pi\)
−0.995438 + 0.0954064i \(0.969585\pi\)
\(788\) 0 0
\(789\) −18.3141 −0.652001
\(790\) 0 0
\(791\) 44.6335 1.58699
\(792\) 0 0
\(793\) 6.48440i 0.230268i
\(794\) 0 0
\(795\) −4.01891 9.80775i −0.142536 0.347845i
\(796\) 0 0
\(797\) 40.7721i 1.44422i −0.691777 0.722111i \(-0.743172\pi\)
0.691777 0.722111i \(-0.256828\pi\)
\(798\) 0 0
\(799\) 3.60963 0.127700
\(800\) 0 0
\(801\) 11.5662 0.408671
\(802\) 0 0
\(803\) 19.5069i 0.688383i
\(804\) 0 0
\(805\) −4.24584 10.3615i −0.149646 0.365196i
\(806\) 0 0
\(807\) 3.87552i 0.136425i
\(808\) 0 0
\(809\) −19.3161 −0.679118 −0.339559 0.940585i \(-0.610278\pi\)
−0.339559 + 0.940585i \(0.610278\pi\)
\(810\) 0 0
\(811\) −40.7873 −1.43224 −0.716118 0.697979i \(-0.754083\pi\)
−0.716118 + 0.697979i \(0.754083\pi\)
\(812\) 0 0
\(813\) 6.86979i 0.240934i
\(814\) 0 0
\(815\) 34.0356 13.9467i 1.19222 0.488533i
\(816\) 0 0
\(817\) 17.9993i 0.629717i
\(818\) 0 0
\(819\) −4.58849 −0.160335
\(820\) 0 0
\(821\) −3.46470 −0.120919 −0.0604594 0.998171i \(-0.519257\pi\)
−0.0604594 + 0.998171i \(0.519257\pi\)
\(822\) 0 0
\(823\) 5.13737i 0.179077i 0.995983 + 0.0895387i \(0.0285393\pi\)
−0.995983 + 0.0895387i \(0.971461\pi\)
\(824\) 0 0
\(825\) 13.9848 + 14.1990i 0.486888 + 0.494344i
\(826\) 0 0
\(827\) 15.8139i 0.549904i −0.961458 0.274952i \(-0.911338\pi\)
0.961458 0.274952i \(-0.0886619\pi\)
\(828\) 0 0
\(829\) −0.764528 −0.0265531 −0.0132766 0.999912i \(-0.504226\pi\)
−0.0132766 + 0.999912i \(0.504226\pi\)
\(830\) 0 0
\(831\) 29.2508 1.01470
\(832\) 0 0
\(833\) 2.29923i 0.0796634i
\(834\) 0 0
\(835\) −21.2435 + 8.70492i −0.735161 + 0.301246i
\(836\) 0 0
\(837\) 0.344129i 0.0118948i
\(838\) 0 0
\(839\) −27.2097 −0.939383 −0.469692 0.882831i \(-0.655635\pi\)
−0.469692 + 0.882831i \(0.655635\pi\)
\(840\) 0 0
\(841\) −28.5415 −0.984191
\(842\) 0 0
\(843\) 9.06296i 0.312145i
\(844\) 0 0
\(845\) −9.69897 23.6694i −0.333655 0.814251i
\(846\) 0 0
\(847\) 17.9520i 0.616837i
\(848\) 0 0
\(849\) 2.74959 0.0943658
\(850\) 0 0
\(851\) 11.1274 0.381441
\(852\) 0 0
\(853\) 13.2637i 0.454142i −0.973878 0.227071i \(-0.927085\pi\)
0.973878 0.227071i \(-0.0729149\pi\)
\(854\) 0 0
\(855\) −1.43369 3.49878i −0.0490312 0.119656i
\(856\) 0 0
\(857\) 30.1619i 1.03031i 0.857096 + 0.515156i \(0.172266\pi\)
−0.857096 + 0.515156i \(0.827734\pi\)
\(858\) 0 0
\(859\) 26.6052 0.907758 0.453879 0.891063i \(-0.350040\pi\)
0.453879 + 0.891063i \(0.350040\pi\)
\(860\) 0 0
\(861\) −20.0218 −0.682342
\(862\) 0 0
\(863\) 19.7440i 0.672092i 0.941846 + 0.336046i \(0.109090\pi\)
−0.941846 + 0.336046i \(0.890910\pi\)
\(864\) 0 0
\(865\) 18.6897 7.65845i 0.635468 0.260395i
\(866\) 0 0
\(867\) 16.8746i 0.573090i
\(868\) 0 0
\(869\) −20.7005 −0.702218
\(870\) 0 0
\(871\) −1.24921 −0.0423277
\(872\) 0 0
\(873\) 17.6632i 0.597807i
\(874\) 0 0
\(875\) 16.1470 37.7592i 0.545870 1.27649i
\(876\) 0 0
\(877\) 25.9792i 0.877254i 0.898669 + 0.438627i \(0.144535\pi\)
−0.898669 + 0.438627i \(0.855465\pi\)
\(878\) 0 0
\(879\) −5.28044 −0.178105
\(880\) 0 0
\(881\) −55.9999 −1.88669 −0.943343 0.331820i \(-0.892337\pi\)
−0.943343 + 0.331820i \(0.892337\pi\)
\(882\) 0 0
\(883\) 10.0916i 0.339609i 0.985478 + 0.169804i \(0.0543136\pi\)
−0.985478 + 0.169804i \(0.945686\pi\)
\(884\) 0 0
\(885\) 10.6264 4.35437i 0.357203 0.146371i
\(886\) 0 0
\(887\) 0.754278i 0.0253262i −0.999920 0.0126631i \(-0.995969\pi\)
0.999920 0.0126631i \(-0.00403089\pi\)
\(888\) 0 0
\(889\) 7.62418 0.255707
\(890\) 0 0
\(891\) −3.98590 −0.133533
\(892\) 0 0
\(893\) 17.2340i 0.576715i
\(894\) 0 0
\(895\) 13.9449 + 34.0311i 0.466127 + 1.13753i
\(896\) 0 0
\(897\) 1.70311i 0.0568651i
\(898\) 0 0
\(899\) 0.233007 0.00777123
\(900\) 0 0
\(901\) 1.67881 0.0559293
\(902\) 0 0
\(903\) 39.0981i 1.30110i
\(904\) 0 0
\(905\) −16.1097 39.3140i −0.535504 1.30684i
\(906\) 0 0
\(907\) 0.545502i 0.0181131i −0.999959 0.00905655i \(-0.997117\pi\)
0.999959 0.00905655i \(-0.00288283\pi\)
\(908\) 0 0
\(909\) 5.52695 0.183317
\(910\) 0 0
\(911\) −2.04782 −0.0678471 −0.0339236 0.999424i \(-0.510800\pi\)
−0.0339236 + 0.999424i \(0.510800\pi\)
\(912\) 0 0
\(913\) 29.8557i 0.988080i
\(914\) 0 0
\(915\) 10.7403 4.40104i 0.355063 0.145494i
\(916\) 0 0
\(917\) 28.9250i 0.955189i
\(918\) 0 0
\(919\) −11.4017 −0.376107 −0.188053 0.982159i \(-0.560218\pi\)
−0.188053 + 0.982159i \(0.560218\pi\)
\(920\) 0 0
\(921\) −31.1784 −1.02736
\(922\) 0 0
\(923\) 4.45014i 0.146478i
\(924\) 0 0
\(925\) 28.6360 + 29.0746i 0.941547 + 0.955967i
\(926\) 0 0
\(927\) 18.0243i 0.591997i
\(928\) 0 0
\(929\) 2.37804 0.0780211 0.0390105 0.999239i \(-0.487579\pi\)
0.0390105 + 0.999239i \(0.487579\pi\)
\(930\) 0 0
\(931\) 10.9775 0.359775
\(932\) 0 0
\(933\) 17.4565i 0.571501i
\(934\) 0 0
\(935\) −2.92092 + 1.19690i −0.0955242 + 0.0391428i
\(936\) 0 0
\(937\) 4.00582i 0.130865i 0.997857 + 0.0654323i \(0.0208426\pi\)
−0.997857 + 0.0654323i \(0.979157\pi\)
\(938\) 0 0
\(939\) 10.3566 0.337974
\(940\) 0 0
\(941\) −9.21785 −0.300493 −0.150247 0.988649i \(-0.548007\pi\)
−0.150247 + 0.988649i \(0.548007\pi\)
\(942\) 0 0
\(943\) 7.43150i 0.242003i
\(944\) 0 0
\(945\) 3.11426 + 7.60004i 0.101307 + 0.247229i
\(946\) 0 0
\(947\) 40.5049i 1.31623i 0.752916 + 0.658116i \(0.228647\pi\)
−0.752916 + 0.658116i \(0.771353\pi\)
\(948\) 0 0
\(949\) −6.11358 −0.198455
\(950\) 0 0
\(951\) 2.19437 0.0711573
\(952\) 0 0
\(953\) 44.4728i 1.44061i −0.693655 0.720307i \(-0.744001\pi\)
0.693655 0.720307i \(-0.255999\pi\)
\(954\) 0 0
\(955\) 19.5896 + 47.8064i 0.633904 + 1.54698i
\(956\) 0 0
\(957\) 2.69882i 0.0872406i
\(958\) 0 0
\(959\) 33.1440 1.07028
\(960\) 0 0
\(961\) −30.8816 −0.996180
\(962\) 0 0
\(963\) 18.2547i 0.588248i
\(964\) 0 0
\(965\) 6.16510 2.52627i 0.198462 0.0813234i
\(966\) 0 0
\(967\) 10.0992i 0.324768i −0.986728 0.162384i \(-0.948082\pi\)
0.986728 0.162384i \(-0.0519184\pi\)
\(968\) 0 0
\(969\) 0.598893 0.0192392
\(970\) 0 0
\(971\) 3.85147 0.123599 0.0617997 0.998089i \(-0.480316\pi\)
0.0617997 + 0.998089i \(0.480316\pi\)
\(972\) 0 0
\(973\) 12.2537i 0.392834i
\(974\) 0 0
\(975\) 4.45004 4.38292i 0.142515 0.140366i
\(976\) 0 0
\(977\) 41.9997i 1.34369i −0.740692 0.671844i \(-0.765502\pi\)
0.740692 0.671844i \(-0.234498\pi\)
\(978\) 0 0
\(979\) 46.1016 1.47341
\(980\) 0 0
\(981\) −11.4396 −0.365239
\(982\) 0 0
\(983\) 17.9968i 0.574010i 0.957929 + 0.287005i \(0.0926596\pi\)
−0.957929 + 0.287005i \(0.907340\pi\)
\(984\) 0 0
\(985\) −14.7990 + 6.06416i −0.471535 + 0.193220i
\(986\) 0 0
\(987\) 37.4357i 1.19159i
\(988\) 0 0
\(989\) −14.5120 −0.461456
\(990\) 0 0
\(991\) 5.39621 0.171416 0.0857082 0.996320i \(-0.472685\pi\)
0.0857082 + 0.996320i \(0.472685\pi\)
\(992\) 0 0
\(993\) 9.64862i 0.306190i
\(994\) 0 0
\(995\) 11.6907 + 28.5300i 0.370621 + 0.904463i
\(996\) 0 0
\(997\) 4.36110i 0.138118i −0.997613 0.0690588i \(-0.978000\pi\)
0.997613 0.0690588i \(-0.0219996\pi\)
\(998\) 0 0
\(999\) −8.16175 −0.258226
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 4020.2.g.c.1609.26 yes 38
5.4 even 2 inner 4020.2.g.c.1609.7 38
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4020.2.g.c.1609.7 38 5.4 even 2 inner
4020.2.g.c.1609.26 yes 38 1.1 even 1 trivial