Properties

Label 4020.2.q.h.3781.1
Level $4020$
Weight $2$
Character 4020.3781
Analytic conductor $32.100$
Analytic rank $1$
Dimension $2$
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(841,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4020.841");
 
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.q (of order \(3\), degree \(2\), 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 3781.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 4020.3781
Dual form 4020.2.q.h.841.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +(0.500000 + 0.866025i) q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +(0.500000 + 0.866025i) q^{7} +1.00000 q^{9} +(-1.50000 - 2.59808i) q^{11} +(-2.50000 + 4.33013i) q^{13} -1.00000 q^{15} +(-1.50000 + 2.59808i) q^{17} +(0.500000 - 0.866025i) q^{19} +(0.500000 + 0.866025i) q^{21} +(-1.50000 + 2.59808i) q^{23} +1.00000 q^{25} +1.00000 q^{27} +(1.50000 + 2.59808i) q^{29} +(-2.50000 - 4.33013i) q^{31} +(-1.50000 - 2.59808i) q^{33} +(-0.500000 - 0.866025i) q^{35} +(3.50000 - 6.06218i) q^{37} +(-2.50000 + 4.33013i) q^{39} +(-4.50000 - 7.79423i) q^{41} -4.00000 q^{43} -1.00000 q^{45} +(1.50000 + 2.59808i) q^{47} +(3.00000 - 5.19615i) q^{49} +(-1.50000 + 2.59808i) q^{51} -6.00000 q^{53} +(1.50000 + 2.59808i) q^{55} +(0.500000 - 0.866025i) q^{57} -12.0000 q^{59} +(0.500000 - 0.866025i) q^{61} +(0.500000 + 0.866025i) q^{63} +(2.50000 - 4.33013i) q^{65} +(-8.00000 - 1.73205i) q^{67} +(-1.50000 + 2.59808i) q^{69} +(-1.50000 - 2.59808i) q^{71} +(3.50000 - 6.06218i) q^{73} +1.00000 q^{75} +(1.50000 - 2.59808i) q^{77} +(-8.50000 - 14.7224i) q^{79} +1.00000 q^{81} +(-7.50000 + 12.9904i) q^{83} +(1.50000 - 2.59808i) q^{85} +(1.50000 + 2.59808i) q^{87} -6.00000 q^{89} -5.00000 q^{91} +(-2.50000 - 4.33013i) q^{93} +(-0.500000 + 0.866025i) q^{95} +(-8.50000 + 14.7224i) q^{97} +(-1.50000 - 2.59808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9} - 3 q^{11} - 5 q^{13} - 2 q^{15} - 3 q^{17} + q^{19} + q^{21} - 3 q^{23} + 2 q^{25} + 2 q^{27} + 3 q^{29} - 5 q^{31} - 3 q^{33} - q^{35} + 7 q^{37} - 5 q^{39} - 9 q^{41} - 8 q^{43} - 2 q^{45} + 3 q^{47} + 6 q^{49} - 3 q^{51} - 12 q^{53} + 3 q^{55} + q^{57} - 24 q^{59} + q^{61} + q^{63} + 5 q^{65} - 16 q^{67} - 3 q^{69} - 3 q^{71} + 7 q^{73} + 2 q^{75} + 3 q^{77} - 17 q^{79} + 2 q^{81} - 15 q^{83} + 3 q^{85} + 3 q^{87} - 12 q^{89} - 10 q^{91} - 5 q^{93} - q^{95} - 17 q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(1\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i 0.944911 0.327327i \(-0.106148\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.50000 2.59808i −0.452267 0.783349i 0.546259 0.837616i \(-0.316051\pi\)
−0.998526 + 0.0542666i \(0.982718\pi\)
\(12\) 0 0
\(13\) −2.50000 + 4.33013i −0.693375 + 1.20096i 0.277350 + 0.960769i \(0.410544\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −1.50000 + 2.59808i −0.363803 + 0.630126i −0.988583 0.150675i \(-0.951855\pi\)
0.624780 + 0.780801i \(0.285189\pi\)
\(18\) 0 0
\(19\) 0.500000 0.866025i 0.114708 0.198680i −0.802955 0.596040i \(-0.796740\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0.500000 + 0.866025i 0.109109 + 0.188982i
\(22\) 0 0
\(23\) −1.50000 + 2.59808i −0.312772 + 0.541736i −0.978961 0.204046i \(-0.934591\pi\)
0.666190 + 0.745782i \(0.267924\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.50000 + 2.59808i 0.278543 + 0.482451i 0.971023 0.238987i \(-0.0768152\pi\)
−0.692480 + 0.721437i \(0.743482\pi\)
\(30\) 0 0
\(31\) −2.50000 4.33013i −0.449013 0.777714i 0.549309 0.835619i \(-0.314891\pi\)
−0.998322 + 0.0579057i \(0.981558\pi\)
\(32\) 0 0
\(33\) −1.50000 2.59808i −0.261116 0.452267i
\(34\) 0 0
\(35\) −0.500000 0.866025i −0.0845154 0.146385i
\(36\) 0 0
\(37\) 3.50000 6.06218i 0.575396 0.996616i −0.420602 0.907245i \(-0.638181\pi\)
0.995998 0.0893706i \(-0.0284856\pi\)
\(38\) 0 0
\(39\) −2.50000 + 4.33013i −0.400320 + 0.693375i
\(40\) 0 0
\(41\) −4.50000 7.79423i −0.702782 1.21725i −0.967486 0.252924i \(-0.918608\pi\)
0.264704 0.964330i \(-0.414726\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 1.50000 + 2.59808i 0.218797 + 0.378968i 0.954441 0.298401i \(-0.0964533\pi\)
−0.735643 + 0.677369i \(0.763120\pi\)
\(48\) 0 0
\(49\) 3.00000 5.19615i 0.428571 0.742307i
\(50\) 0 0
\(51\) −1.50000 + 2.59808i −0.210042 + 0.363803i
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 1.50000 + 2.59808i 0.202260 + 0.350325i
\(56\) 0 0
\(57\) 0.500000 0.866025i 0.0662266 0.114708i
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 0.500000 0.866025i 0.0640184 0.110883i −0.832240 0.554416i \(-0.812942\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 0.500000 + 0.866025i 0.0629941 + 0.109109i
\(64\) 0 0
\(65\) 2.50000 4.33013i 0.310087 0.537086i
\(66\) 0 0
\(67\) −8.00000 1.73205i −0.977356 0.211604i
\(68\) 0 0
\(69\) −1.50000 + 2.59808i −0.180579 + 0.312772i
\(70\) 0 0
\(71\) −1.50000 2.59808i −0.178017 0.308335i 0.763184 0.646181i \(-0.223635\pi\)
−0.941201 + 0.337846i \(0.890302\pi\)
\(72\) 0 0
\(73\) 3.50000 6.06218i 0.409644 0.709524i −0.585206 0.810885i \(-0.698986\pi\)
0.994850 + 0.101361i \(0.0323196\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 1.50000 2.59808i 0.170941 0.296078i
\(78\) 0 0
\(79\) −8.50000 14.7224i −0.956325 1.65640i −0.731307 0.682048i \(-0.761089\pi\)
−0.225018 0.974355i \(-0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −7.50000 + 12.9904i −0.823232 + 1.42588i 0.0800311 + 0.996792i \(0.474498\pi\)
−0.903263 + 0.429087i \(0.858835\pi\)
\(84\) 0 0
\(85\) 1.50000 2.59808i 0.162698 0.281801i
\(86\) 0 0
\(87\) 1.50000 + 2.59808i 0.160817 + 0.278543i
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −5.00000 −0.524142
\(92\) 0 0
\(93\) −2.50000 4.33013i −0.259238 0.449013i
\(94\) 0 0
\(95\) −0.500000 + 0.866025i −0.0512989 + 0.0888523i
\(96\) 0 0
\(97\) −8.50000 + 14.7224i −0.863044 + 1.49484i 0.00593185 + 0.999982i \(0.498112\pi\)
−0.868976 + 0.494854i \(0.835222\pi\)
\(98\) 0 0
\(99\) −1.50000 2.59808i −0.150756 0.261116i
\(100\) 0 0
\(101\) 1.50000 + 2.59808i 0.149256 + 0.258518i 0.930953 0.365140i \(-0.118979\pi\)
−0.781697 + 0.623658i \(0.785646\pi\)
\(102\) 0 0
\(103\) 6.50000 + 11.2583i 0.640464 + 1.10932i 0.985329 + 0.170664i \(0.0545913\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) 0 0
\(105\) −0.500000 0.866025i −0.0487950 0.0845154i
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 3.50000 6.06218i 0.332205 0.575396i
\(112\) 0 0
\(113\) 4.50000 + 7.79423i 0.423324 + 0.733219i 0.996262 0.0863794i \(-0.0275297\pi\)
−0.572938 + 0.819599i \(0.694196\pi\)
\(114\) 0 0
\(115\) 1.50000 2.59808i 0.139876 0.242272i
\(116\) 0 0
\(117\) −2.50000 + 4.33013i −0.231125 + 0.400320i
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 1.00000 1.73205i 0.0909091 0.157459i
\(122\) 0 0
\(123\) −4.50000 7.79423i −0.405751 0.702782i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 6.50000 + 11.2583i 0.576782 + 0.999015i 0.995846 + 0.0910585i \(0.0290250\pi\)
−0.419064 + 0.907957i \(0.637642\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 1.50000 + 2.59808i 0.126323 + 0.218797i
\(142\) 0 0
\(143\) 15.0000 1.25436
\(144\) 0 0
\(145\) −1.50000 2.59808i −0.124568 0.215758i
\(146\) 0 0
\(147\) 3.00000 5.19615i 0.247436 0.428571i
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 0.500000 0.866025i 0.0406894 0.0704761i −0.844963 0.534824i \(-0.820378\pi\)
0.885653 + 0.464348i \(0.153711\pi\)
\(152\) 0 0
\(153\) −1.50000 + 2.59808i −0.121268 + 0.210042i
\(154\) 0 0
\(155\) 2.50000 + 4.33013i 0.200805 + 0.347804i
\(156\) 0 0
\(157\) 3.50000 6.06218i 0.279330 0.483814i −0.691888 0.722005i \(-0.743221\pi\)
0.971219 + 0.238190i \(0.0765542\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) −5.50000 9.52628i −0.430793 0.746156i 0.566149 0.824303i \(-0.308433\pi\)
−0.996942 + 0.0781474i \(0.975100\pi\)
\(164\) 0 0
\(165\) 1.50000 + 2.59808i 0.116775 + 0.202260i
\(166\) 0 0
\(167\) −10.5000 18.1865i −0.812514 1.40732i −0.911099 0.412188i \(-0.864765\pi\)
0.0985846 0.995129i \(-0.468568\pi\)
\(168\) 0 0
\(169\) −6.00000 10.3923i −0.461538 0.799408i
\(170\) 0 0
\(171\) 0.500000 0.866025i 0.0382360 0.0662266i
\(172\) 0 0
\(173\) 10.5000 18.1865i 0.798300 1.38270i −0.122422 0.992478i \(-0.539066\pi\)
0.920722 0.390218i \(-0.127601\pi\)
\(174\) 0 0
\(175\) 0.500000 + 0.866025i 0.0377964 + 0.0654654i
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 6.50000 + 11.2583i 0.483141 + 0.836825i 0.999813 0.0193587i \(-0.00616244\pi\)
−0.516671 + 0.856184i \(0.672829\pi\)
\(182\) 0 0
\(183\) 0.500000 0.866025i 0.0369611 0.0640184i
\(184\) 0 0
\(185\) −3.50000 + 6.06218i −0.257325 + 0.445700i
\(186\) 0 0
\(187\) 9.00000 0.658145
\(188\) 0 0
\(189\) 0.500000 + 0.866025i 0.0363696 + 0.0629941i
\(190\) 0 0
\(191\) 1.50000 2.59808i 0.108536 0.187990i −0.806641 0.591041i \(-0.798717\pi\)
0.915177 + 0.403051i \(0.132050\pi\)
\(192\) 0 0
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 0 0
\(195\) 2.50000 4.33013i 0.179029 0.310087i
\(196\) 0 0
\(197\) −1.50000 2.59808i −0.106871 0.185105i 0.807630 0.589689i \(-0.200750\pi\)
−0.914501 + 0.404584i \(0.867416\pi\)
\(198\) 0 0
\(199\) −5.50000 + 9.52628i −0.389885 + 0.675300i −0.992434 0.122782i \(-0.960818\pi\)
0.602549 + 0.798082i \(0.294152\pi\)
\(200\) 0 0
\(201\) −8.00000 1.73205i −0.564276 0.122169i
\(202\) 0 0
\(203\) −1.50000 + 2.59808i −0.105279 + 0.182349i
\(204\) 0 0
\(205\) 4.50000 + 7.79423i 0.314294 + 0.544373i
\(206\) 0 0
\(207\) −1.50000 + 2.59808i −0.104257 + 0.180579i
\(208\) 0 0
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) −11.5000 + 19.9186i −0.791693 + 1.37125i 0.133226 + 0.991086i \(0.457467\pi\)
−0.924918 + 0.380166i \(0.875867\pi\)
\(212\) 0 0
\(213\) −1.50000 2.59808i −0.102778 0.178017i
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 2.50000 4.33013i 0.169711 0.293948i
\(218\) 0 0
\(219\) 3.50000 6.06218i 0.236508 0.409644i
\(220\) 0 0
\(221\) −7.50000 12.9904i −0.504505 0.873828i
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 1.50000 + 2.59808i 0.0995585 + 0.172440i 0.911502 0.411296i \(-0.134924\pi\)
−0.811943 + 0.583736i \(0.801590\pi\)
\(228\) 0 0
\(229\) −11.5000 + 19.9186i −0.759941 + 1.31626i 0.182939 + 0.983124i \(0.441439\pi\)
−0.942880 + 0.333133i \(0.891894\pi\)
\(230\) 0 0
\(231\) 1.50000 2.59808i 0.0986928 0.170941i
\(232\) 0 0
\(233\) 10.5000 + 18.1865i 0.687878 + 1.19144i 0.972523 + 0.232806i \(0.0747909\pi\)
−0.284645 + 0.958633i \(0.591876\pi\)
\(234\) 0 0
\(235\) −1.50000 2.59808i −0.0978492 0.169480i
\(236\) 0 0
\(237\) −8.50000 14.7224i −0.552134 0.956325i
\(238\) 0 0
\(239\) −13.5000 23.3827i −0.873242 1.51250i −0.858623 0.512607i \(-0.828680\pi\)
−0.0146191 0.999893i \(-0.504654\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −3.00000 + 5.19615i −0.191663 + 0.331970i
\(246\) 0 0
\(247\) 2.50000 + 4.33013i 0.159071 + 0.275519i
\(248\) 0 0
\(249\) −7.50000 + 12.9904i −0.475293 + 0.823232i
\(250\) 0 0
\(251\) 7.50000 12.9904i 0.473396 0.819946i −0.526140 0.850398i \(-0.676361\pi\)
0.999536 + 0.0304521i \(0.00969471\pi\)
\(252\) 0 0
\(253\) 9.00000 0.565825
\(254\) 0 0
\(255\) 1.50000 2.59808i 0.0939336 0.162698i
\(256\) 0 0
\(257\) −13.5000 23.3827i −0.842107 1.45857i −0.888110 0.459631i \(-0.847982\pi\)
0.0460033 0.998941i \(-0.485352\pi\)
\(258\) 0 0
\(259\) 7.00000 0.434959
\(260\) 0 0
\(261\) 1.50000 + 2.59808i 0.0928477 + 0.160817i
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) −5.00000 −0.302614
\(274\) 0 0
\(275\) −1.50000 2.59808i −0.0904534 0.156670i
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) −2.50000 4.33013i −0.149671 0.259238i
\(280\) 0 0
\(281\) −4.50000 + 7.79423i −0.268447 + 0.464965i −0.968461 0.249165i \(-0.919844\pi\)
0.700014 + 0.714130i \(0.253177\pi\)
\(282\) 0 0
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 0 0
\(285\) −0.500000 + 0.866025i −0.0296174 + 0.0512989i
\(286\) 0 0
\(287\) 4.50000 7.79423i 0.265627 0.460079i
\(288\) 0 0
\(289\) 4.00000 + 6.92820i 0.235294 + 0.407541i
\(290\) 0 0
\(291\) −8.50000 + 14.7224i −0.498279 + 0.863044i
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) −1.50000 2.59808i −0.0870388 0.150756i
\(298\) 0 0
\(299\) −7.50000 12.9904i −0.433736 0.751253i
\(300\) 0 0
\(301\) −2.00000 3.46410i −0.115278 0.199667i
\(302\) 0 0
\(303\) 1.50000 + 2.59808i 0.0861727 + 0.149256i
\(304\) 0 0
\(305\) −0.500000 + 0.866025i −0.0286299 + 0.0495885i
\(306\) 0 0
\(307\) −2.50000 + 4.33013i −0.142683 + 0.247133i −0.928506 0.371318i \(-0.878906\pi\)
0.785823 + 0.618451i \(0.212239\pi\)
\(308\) 0 0
\(309\) 6.50000 + 11.2583i 0.369772 + 0.640464i
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 0 0
\(315\) −0.500000 0.866025i −0.0281718 0.0487950i
\(316\) 0 0
\(317\) −7.50000 + 12.9904i −0.421242 + 0.729612i −0.996061 0.0886679i \(-0.971739\pi\)
0.574819 + 0.818280i \(0.305072\pi\)
\(318\) 0 0
\(319\) 4.50000 7.79423i 0.251952 0.436393i
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 1.50000 + 2.59808i 0.0834622 + 0.144561i
\(324\) 0 0
\(325\) −2.50000 + 4.33013i −0.138675 + 0.240192i
\(326\) 0 0
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) −1.50000 + 2.59808i −0.0826977 + 0.143237i
\(330\) 0 0
\(331\) −8.50000 14.7224i −0.467202 0.809218i 0.532096 0.846684i \(-0.321405\pi\)
−0.999298 + 0.0374662i \(0.988071\pi\)
\(332\) 0 0
\(333\) 3.50000 6.06218i 0.191799 0.332205i
\(334\) 0 0
\(335\) 8.00000 + 1.73205i 0.437087 + 0.0946320i
\(336\) 0 0
\(337\) 15.5000 26.8468i 0.844339 1.46244i −0.0418554 0.999124i \(-0.513327\pi\)
0.886194 0.463314i \(-0.153340\pi\)
\(338\) 0 0
\(339\) 4.50000 + 7.79423i 0.244406 + 0.423324i
\(340\) 0 0
\(341\) −7.50000 + 12.9904i −0.406148 + 0.703469i
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) 1.50000 2.59808i 0.0807573 0.139876i
\(346\) 0 0
\(347\) 13.5000 + 23.3827i 0.724718 + 1.25525i 0.959090 + 0.283101i \(0.0913633\pi\)
−0.234372 + 0.972147i \(0.575303\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) −2.50000 + 4.33013i −0.133440 + 0.231125i
\(352\) 0 0
\(353\) 10.5000 18.1865i 0.558859 0.967972i −0.438733 0.898617i \(-0.644573\pi\)
0.997592 0.0693543i \(-0.0220939\pi\)
\(354\) 0 0
\(355\) 1.50000 + 2.59808i 0.0796117 + 0.137892i
\(356\) 0 0
\(357\) −3.00000 −0.158777
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 9.00000 + 15.5885i 0.473684 + 0.820445i
\(362\) 0 0
\(363\) 1.00000 1.73205i 0.0524864 0.0909091i
\(364\) 0 0
\(365\) −3.50000 + 6.06218i −0.183198 + 0.317309i
\(366\) 0 0
\(367\) −11.5000 19.9186i −0.600295 1.03974i −0.992776 0.119982i \(-0.961716\pi\)
0.392481 0.919760i \(-0.371617\pi\)
\(368\) 0 0
\(369\) −4.50000 7.79423i −0.234261 0.405751i
\(370\) 0 0
\(371\) −3.00000 5.19615i −0.155752 0.269771i
\(372\) 0 0
\(373\) 15.5000 + 26.8468i 0.802560 + 1.39007i 0.917926 + 0.396751i \(0.129862\pi\)
−0.115367 + 0.993323i \(0.536804\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −15.0000 −0.772539
\(378\) 0 0
\(379\) −11.5000 + 19.9186i −0.590715 + 1.02315i 0.403421 + 0.915014i \(0.367821\pi\)
−0.994136 + 0.108134i \(0.965512\pi\)
\(380\) 0 0
\(381\) 6.50000 + 11.2583i 0.333005 + 0.576782i
\(382\) 0 0
\(383\) 16.5000 28.5788i 0.843111 1.46031i −0.0441413 0.999025i \(-0.514055\pi\)
0.887252 0.461285i \(-0.152611\pi\)
\(384\) 0 0
\(385\) −1.50000 + 2.59808i −0.0764471 + 0.132410i
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 1.50000 2.59808i 0.0760530 0.131728i −0.825491 0.564416i \(-0.809102\pi\)
0.901544 + 0.432688i \(0.142435\pi\)
\(390\) 0 0
\(391\) −4.50000 7.79423i −0.227575 0.394171i
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) 8.50000 + 14.7224i 0.427681 + 0.740766i
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 25.0000 1.24534
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −21.0000 −1.04093
\(408\) 0 0
\(409\) 6.50000 + 11.2583i 0.321404 + 0.556689i 0.980778 0.195127i \(-0.0625118\pi\)
−0.659374 + 0.751815i \(0.729178\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 0 0
\(413\) −6.00000 10.3923i −0.295241 0.511372i
\(414\) 0 0
\(415\) 7.50000 12.9904i 0.368161 0.637673i
\(416\) 0 0
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) −10.5000 + 18.1865i −0.512959 + 0.888470i 0.486928 + 0.873442i \(0.338117\pi\)
−0.999887 + 0.0150285i \(0.995216\pi\)
\(420\) 0 0
\(421\) 6.50000 11.2583i 0.316791 0.548697i −0.663026 0.748596i \(-0.730728\pi\)
0.979817 + 0.199899i \(0.0640614\pi\)
\(422\) 0 0
\(423\) 1.50000 + 2.59808i 0.0729325 + 0.126323i
\(424\) 0 0
\(425\) −1.50000 + 2.59808i −0.0727607 + 0.126025i
\(426\) 0 0
\(427\) 1.00000 0.0483934
\(428\) 0 0
\(429\) 15.0000 0.724207
\(430\) 0 0
\(431\) 10.5000 + 18.1865i 0.505767 + 0.876014i 0.999978 + 0.00667224i \(0.00212386\pi\)
−0.494211 + 0.869342i \(0.664543\pi\)
\(432\) 0 0
\(433\) 9.50000 + 16.4545i 0.456541 + 0.790752i 0.998775 0.0494752i \(-0.0157549\pi\)
−0.542234 + 0.840227i \(0.682422\pi\)
\(434\) 0 0
\(435\) −1.50000 2.59808i −0.0719195 0.124568i
\(436\) 0 0
\(437\) 1.50000 + 2.59808i 0.0717547 + 0.124283i
\(438\) 0 0
\(439\) 12.5000 21.6506i 0.596592 1.03333i −0.396728 0.917936i \(-0.629854\pi\)
0.993320 0.115392i \(-0.0368124\pi\)
\(440\) 0 0
\(441\) 3.00000 5.19615i 0.142857 0.247436i
\(442\) 0 0
\(443\) 7.50000 + 12.9904i 0.356336 + 0.617192i 0.987346 0.158583i \(-0.0506926\pi\)
−0.631010 + 0.775775i \(0.717359\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) 1.50000 + 2.59808i 0.0707894 + 0.122611i 0.899247 0.437440i \(-0.144115\pi\)
−0.828458 + 0.560051i \(0.810782\pi\)
\(450\) 0 0
\(451\) −13.5000 + 23.3827i −0.635690 + 1.10105i
\(452\) 0 0
\(453\) 0.500000 0.866025i 0.0234920 0.0406894i
\(454\) 0 0
\(455\) 5.00000 0.234404
\(456\) 0 0
\(457\) 9.50000 + 16.4545i 0.444391 + 0.769708i 0.998010 0.0630623i \(-0.0200867\pi\)
−0.553618 + 0.832771i \(0.686753\pi\)
\(458\) 0 0
\(459\) −1.50000 + 2.59808i −0.0700140 + 0.121268i
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 15.5000 26.8468i 0.720346 1.24768i −0.240515 0.970645i \(-0.577316\pi\)
0.960861 0.277031i \(-0.0893503\pi\)
\(464\) 0 0
\(465\) 2.50000 + 4.33013i 0.115935 + 0.200805i
\(466\) 0 0
\(467\) 4.50000 7.79423i 0.208235 0.360674i −0.742923 0.669376i \(-0.766561\pi\)
0.951159 + 0.308702i \(0.0998947\pi\)
\(468\) 0 0
\(469\) −2.50000 7.79423i −0.115439 0.359904i
\(470\) 0 0
\(471\) 3.50000 6.06218i 0.161271 0.279330i
\(472\) 0 0
\(473\) 6.00000 + 10.3923i 0.275880 + 0.477839i
\(474\) 0 0
\(475\) 0.500000 0.866025i 0.0229416 0.0397360i
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) −4.50000 + 7.79423i −0.205610 + 0.356127i −0.950327 0.311253i \(-0.899251\pi\)
0.744717 + 0.667381i \(0.232585\pi\)
\(480\) 0 0
\(481\) 17.5000 + 30.3109i 0.797931 + 1.38206i
\(482\) 0 0
\(483\) −3.00000 −0.136505
\(484\) 0 0
\(485\) 8.50000 14.7224i 0.385965 0.668511i
\(486\) 0 0
\(487\) −14.5000 + 25.1147i −0.657058 + 1.13806i 0.324316 + 0.945949i \(0.394866\pi\)
−0.981374 + 0.192109i \(0.938467\pi\)
\(488\) 0 0
\(489\) −5.50000 9.52628i −0.248719 0.430793i
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) 1.50000 + 2.59808i 0.0674200 + 0.116775i
\(496\) 0 0
\(497\) 1.50000 2.59808i 0.0672842 0.116540i
\(498\) 0 0
\(499\) 6.50000 11.2583i 0.290980 0.503992i −0.683062 0.730361i \(-0.739352\pi\)
0.974042 + 0.226369i \(0.0726854\pi\)
\(500\) 0 0
\(501\) −10.5000 18.1865i −0.469105 0.812514i
\(502\) 0 0
\(503\) 19.5000 + 33.7750i 0.869462 + 1.50595i 0.862547 + 0.505976i \(0.168868\pi\)
0.00691465 + 0.999976i \(0.497799\pi\)
\(504\) 0 0
\(505\) −1.50000 2.59808i −0.0667491 0.115613i
\(506\) 0 0
\(507\) −6.00000 10.3923i −0.266469 0.461538i
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 7.00000 0.309662
\(512\) 0 0
\(513\) 0.500000 0.866025i 0.0220755 0.0382360i
\(514\) 0 0
\(515\) −6.50000 11.2583i −0.286424 0.496101i
\(516\) 0 0
\(517\) 4.50000 7.79423i 0.197910 0.342790i
\(518\) 0 0
\(519\) 10.5000 18.1865i 0.460899 0.798300i
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) 15.5000 26.8468i 0.677768 1.17393i −0.297884 0.954602i \(-0.596281\pi\)
0.975652 0.219326i \(-0.0703858\pi\)
\(524\) 0 0
\(525\) 0.500000 + 0.866025i 0.0218218 + 0.0377964i
\(526\) 0 0
\(527\) 15.0000 0.653410
\(528\) 0 0
\(529\) 7.00000 + 12.1244i 0.304348 + 0.527146i
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 45.0000 1.94917
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 0 0
\(543\) 6.50000 + 11.2583i 0.278942 + 0.483141i
\(544\) 0 0
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) 0.500000 + 0.866025i 0.0213785 + 0.0370286i 0.876517 0.481371i \(-0.159861\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) 0 0
\(549\) 0.500000 0.866025i 0.0213395 0.0369611i
\(550\) 0 0
\(551\) 3.00000 0.127804
\(552\) 0 0
\(553\) 8.50000 14.7224i 0.361457 0.626061i
\(554\) 0 0
\(555\) −3.50000 + 6.06218i −0.148567 + 0.257325i
\(556\) 0 0
\(557\) 10.5000 + 18.1865i 0.444899 + 0.770588i 0.998045 0.0624962i \(-0.0199061\pi\)
−0.553146 + 0.833084i \(0.686573\pi\)
\(558\) 0 0
\(559\) 10.0000 17.3205i 0.422955 0.732579i
\(560\) 0 0
\(561\) 9.00000 0.379980
\(562\) 0 0
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) −4.50000 7.79423i −0.189316 0.327906i
\(566\) 0 0
\(567\) 0.500000 + 0.866025i 0.0209980 + 0.0363696i
\(568\) 0 0
\(569\) 7.50000 + 12.9904i 0.314416 + 0.544585i 0.979313 0.202350i \(-0.0648579\pi\)
−0.664897 + 0.746935i \(0.731525\pi\)
\(570\) 0 0
\(571\) −8.50000 14.7224i −0.355714 0.616115i 0.631526 0.775355i \(-0.282429\pi\)
−0.987240 + 0.159240i \(0.949096\pi\)
\(572\) 0 0
\(573\) 1.50000 2.59808i 0.0626634 0.108536i
\(574\) 0 0
\(575\) −1.50000 + 2.59808i −0.0625543 + 0.108347i
\(576\) 0 0
\(577\) −14.5000 25.1147i −0.603643 1.04554i −0.992264 0.124143i \(-0.960382\pi\)
0.388621 0.921397i \(-0.372951\pi\)
\(578\) 0 0
\(579\) −22.0000 −0.914289
\(580\) 0 0
\(581\) −15.0000 −0.622305
\(582\) 0 0
\(583\) 9.00000 + 15.5885i 0.372742 + 0.645608i
\(584\) 0 0
\(585\) 2.50000 4.33013i 0.103362 0.179029i
\(586\) 0 0
\(587\) −19.5000 + 33.7750i −0.804851 + 1.39404i 0.111540 + 0.993760i \(0.464422\pi\)
−0.916392 + 0.400283i \(0.868912\pi\)
\(588\) 0 0
\(589\) −5.00000 −0.206021
\(590\) 0 0
\(591\) −1.50000 2.59808i −0.0617018 0.106871i
\(592\) 0 0
\(593\) 16.5000 28.5788i 0.677574 1.17359i −0.298136 0.954524i \(-0.596365\pi\)
0.975709 0.219069i \(-0.0703019\pi\)
\(594\) 0 0
\(595\) 3.00000 0.122988
\(596\) 0 0
\(597\) −5.50000 + 9.52628i −0.225100 + 0.389885i
\(598\) 0 0
\(599\) −19.5000 33.7750i −0.796748 1.38001i −0.921723 0.387849i \(-0.873218\pi\)
0.124975 0.992160i \(-0.460115\pi\)
\(600\) 0 0
\(601\) 6.50000 11.2583i 0.265141 0.459237i −0.702460 0.711723i \(-0.747915\pi\)
0.967600 + 0.252486i \(0.0812483\pi\)
\(602\) 0 0
\(603\) −8.00000 1.73205i −0.325785 0.0705346i
\(604\) 0 0
\(605\) −1.00000 + 1.73205i −0.0406558 + 0.0704179i
\(606\) 0 0
\(607\) −5.50000 9.52628i −0.223238 0.386660i 0.732551 0.680712i \(-0.238329\pi\)
−0.955789 + 0.294052i \(0.904996\pi\)
\(608\) 0 0
\(609\) −1.50000 + 2.59808i −0.0607831 + 0.105279i
\(610\) 0 0
\(611\) −15.0000 −0.606835
\(612\) 0 0
\(613\) −2.50000 + 4.33013i −0.100974 + 0.174892i −0.912086 0.409998i \(-0.865529\pi\)
0.811112 + 0.584891i \(0.198863\pi\)
\(614\) 0 0
\(615\) 4.50000 + 7.79423i 0.181458 + 0.314294i
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) 18.5000 32.0429i 0.743578 1.28791i −0.207279 0.978282i \(-0.566461\pi\)
0.950856 0.309633i \(-0.100206\pi\)
\(620\) 0 0
\(621\) −1.50000 + 2.59808i −0.0601929 + 0.104257i
\(622\) 0 0
\(623\) −3.00000 5.19615i −0.120192 0.208179i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −3.00000 −0.119808
\(628\) 0 0
\(629\) 10.5000 + 18.1865i 0.418662 + 0.725145i
\(630\) 0 0
\(631\) 12.5000 21.6506i 0.497617 0.861898i −0.502379 0.864647i \(-0.667542\pi\)
0.999996 + 0.00274930i \(0.000875132\pi\)
\(632\) 0 0
\(633\) −11.5000 + 19.9186i −0.457084 + 0.791693i
\(634\) 0 0
\(635\) −6.50000 11.2583i −0.257945 0.446773i
\(636\) 0 0
\(637\) 15.0000 + 25.9808i 0.594322 + 1.02940i
\(638\) 0 0
\(639\) −1.50000 2.59808i −0.0593391 0.102778i
\(640\) 0 0
\(641\) −10.5000 18.1865i −0.414725 0.718325i 0.580674 0.814136i \(-0.302789\pi\)
−0.995400 + 0.0958109i \(0.969456\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) 4.00000 0.157500
\(646\) 0 0
\(647\) 16.5000 28.5788i 0.648682 1.12355i −0.334756 0.942305i \(-0.608654\pi\)
0.983438 0.181245i \(-0.0580128\pi\)
\(648\) 0 0
\(649\) 18.0000 + 31.1769i 0.706562 + 1.22380i
\(650\) 0 0
\(651\) 2.50000 4.33013i 0.0979827 0.169711i
\(652\) 0 0
\(653\) 4.50000 7.79423i 0.176099 0.305012i −0.764442 0.644692i \(-0.776986\pi\)
0.940541 + 0.339680i \(0.110319\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) 3.50000 6.06218i 0.136548 0.236508i
\(658\) 0 0
\(659\) −7.50000 12.9904i −0.292159 0.506033i 0.682161 0.731202i \(-0.261040\pi\)
−0.974320 + 0.225168i \(0.927707\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 0 0
\(663\) −7.50000 12.9904i −0.291276 0.504505i
\(664\) 0 0
\(665\) −1.00000 −0.0387783
\(666\) 0 0
\(667\) −9.00000 −0.348481
\(668\) 0 0
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) −3.00000 −0.115814
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 16.5000 + 28.5788i 0.634147 + 1.09837i 0.986695 + 0.162581i \(0.0519817\pi\)
−0.352549 + 0.935793i \(0.614685\pi\)
\(678\) 0 0
\(679\) −17.0000 −0.652400
\(680\) 0 0
\(681\) 1.50000 + 2.59808i 0.0574801 + 0.0995585i
\(682\) 0 0
\(683\) −19.5000 + 33.7750i −0.746147 + 1.29236i 0.203510 + 0.979073i \(0.434765\pi\)
−0.949657 + 0.313291i \(0.898568\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) −11.5000 + 19.9186i −0.438752 + 0.759941i
\(688\) 0 0
\(689\) 15.0000 25.9808i 0.571454 0.989788i
\(690\) 0 0
\(691\) −8.50000 14.7224i −0.323355 0.560068i 0.657823 0.753173i \(-0.271478\pi\)
−0.981178 + 0.193105i \(0.938144\pi\)
\(692\) 0 0
\(693\) 1.50000 2.59808i 0.0569803 0.0986928i
\(694\) 0 0
\(695\) −20.0000 −0.758643
\(696\) 0 0
\(697\) 27.0000 1.02270
\(698\) 0 0
\(699\) 10.5000 + 18.1865i 0.397146 + 0.687878i
\(700\) 0 0
\(701\) 1.50000 + 2.59808i 0.0566542 + 0.0981280i 0.892962 0.450133i \(-0.148623\pi\)
−0.836307 + 0.548261i \(0.815290\pi\)
\(702\) 0 0
\(703\) −3.50000 6.06218i −0.132005 0.228639i
\(704\) 0 0
\(705\) −1.50000 2.59808i −0.0564933 0.0978492i
\(706\) 0 0
\(707\) −1.50000 + 2.59808i −0.0564133 + 0.0977107i
\(708\) 0 0
\(709\) −11.5000 + 19.9186i −0.431892 + 0.748058i −0.997036 0.0769337i \(-0.975487\pi\)
0.565145 + 0.824992i \(0.308820\pi\)
\(710\) 0 0
\(711\) −8.50000 14.7224i −0.318775 0.552134i
\(712\) 0 0
\(713\) 15.0000 0.561754
\(714\) 0 0
\(715\) −15.0000 −0.560968
\(716\) 0 0
\(717\) −13.5000 23.3827i −0.504167 0.873242i
\(718\) 0 0
\(719\) −4.50000 + 7.79423i −0.167822 + 0.290676i −0.937654 0.347571i \(-0.887007\pi\)
0.769832 + 0.638247i \(0.220340\pi\)
\(720\) 0 0
\(721\) −6.50000 + 11.2583i −0.242073 + 0.419282i
\(722\) 0 0
\(723\) −10.0000 −0.371904
\(724\) 0 0
\(725\) 1.50000 + 2.59808i 0.0557086 + 0.0964901i
\(726\) 0 0
\(727\) 3.50000 6.06218i 0.129808 0.224834i −0.793794 0.608186i \(-0.791897\pi\)
0.923602 + 0.383353i \(0.125231\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.00000 10.3923i 0.221918 0.384373i
\(732\) 0 0
\(733\) 15.5000 + 26.8468i 0.572506 + 0.991609i 0.996308 + 0.0858539i \(0.0273618\pi\)
−0.423802 + 0.905755i \(0.639305\pi\)
\(734\) 0 0
\(735\) −3.00000 + 5.19615i −0.110657 + 0.191663i
\(736\) 0 0
\(737\) 7.50000 + 23.3827i 0.276266 + 0.861312i
\(738\) 0 0
\(739\) 12.5000 21.6506i 0.459820 0.796431i −0.539131 0.842222i \(-0.681247\pi\)
0.998951 + 0.0457903i \(0.0145806\pi\)
\(740\) 0 0
\(741\) 2.50000 + 4.33013i 0.0918398 + 0.159071i
\(742\) 0 0
\(743\) 4.50000 7.79423i 0.165089 0.285943i −0.771598 0.636111i \(-0.780542\pi\)
0.936687 + 0.350168i \(0.113876\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) −7.50000 + 12.9904i −0.274411 + 0.475293i
\(748\) 0 0
\(749\) −6.00000 10.3923i −0.219235 0.379727i
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) 7.50000 12.9904i 0.273315 0.473396i
\(754\) 0 0
\(755\) −0.500000 + 0.866025i −0.0181969 + 0.0315179i
\(756\) 0 0
\(757\) 21.5000 + 37.2391i 0.781431 + 1.35348i 0.931108 + 0.364743i \(0.118843\pi\)
−0.149677 + 0.988735i \(0.547824\pi\)
\(758\) 0 0
\(759\) 9.00000 0.326679
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 1.00000 + 1.73205i 0.0362024 + 0.0627044i
\(764\) 0 0
\(765\) 1.50000 2.59808i 0.0542326 0.0939336i
\(766\) 0 0
\(767\) 30.0000 51.9615i 1.08324 1.87622i
\(768\) 0 0
\(769\) 24.5000 + 42.4352i 0.883493 + 1.53025i 0.847432 + 0.530904i \(0.178148\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) −13.5000 23.3827i −0.486191 0.842107i
\(772\) 0 0
\(773\) 10.5000 + 18.1865i 0.377659 + 0.654124i 0.990721 0.135910i \(-0.0433959\pi\)
−0.613062 + 0.790034i \(0.710063\pi\)
\(774\) 0 0
\(775\) −2.50000 4.33013i −0.0898027 0.155543i
\(776\) 0 0
\(777\) 7.00000 0.251124
\(778\) 0 0
\(779\) −9.00000 −0.322458
\(780\) 0 0
\(781\) −4.50000 + 7.79423i −0.161023 + 0.278899i
\(782\) 0 0
\(783\) 1.50000 + 2.59808i 0.0536056 + 0.0928477i
\(784\) 0 0
\(785\) −3.50000 + 6.06218i −0.124920 + 0.216368i
\(786\) 0 0
\(787\) −8.50000 + 14.7224i −0.302992 + 0.524798i −0.976812 0.214097i \(-0.931319\pi\)
0.673820 + 0.738896i \(0.264652\pi\)
\(788\) 0 0
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −4.50000 + 7.79423i −0.160002 + 0.277131i
\(792\) 0 0
\(793\) 2.50000 + 4.33013i 0.0887776 + 0.153767i
\(794\) 0 0
\(795\) 6.00000 0.212798
\(796\) 0 0
\(797\) −19.5000 33.7750i −0.690725 1.19637i −0.971601 0.236627i \(-0.923958\pi\)
0.280875 0.959744i \(-0.409375\pi\)
\(798\) 0 0
\(799\) −9.00000 −0.318397
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) −21.0000 −0.741074
\(804\) 0 0
\(805\) 3.00000 0.105736
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) 3.50000 + 6.06218i 0.122902 + 0.212872i 0.920911 0.389774i \(-0.127447\pi\)
−0.798009 + 0.602645i \(0.794113\pi\)
\(812\) 0 0
\(813\) 20.0000 0.701431
\(814\) 0 0
\(815\) 5.50000 + 9.52628i 0.192657 + 0.333691i
\(816\) 0 0
\(817\) −2.00000 + 3.46410i −0.0699711 + 0.121194i
\(818\) 0 0
\(819\) −5.00000 −0.174714
\(820\) 0 0
\(821\) 7.50000 12.9904i 0.261752 0.453367i −0.704956 0.709251i \(-0.749033\pi\)
0.966708 + 0.255884i \(0.0823665\pi\)
\(822\) 0 0
\(823\) −2.50000 + 4.33013i −0.0871445 + 0.150939i −0.906303 0.422628i \(-0.861108\pi\)
0.819159 + 0.573567i \(0.194441\pi\)
\(824\) 0 0
\(825\) −1.50000 2.59808i −0.0522233 0.0904534i
\(826\) 0 0
\(827\) 4.50000 7.79423i 0.156480 0.271032i −0.777117 0.629356i \(-0.783319\pi\)
0.933597 + 0.358325i \(0.116652\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) 9.00000 + 15.5885i 0.311832 + 0.540108i
\(834\) 0 0
\(835\) 10.5000 + 18.1865i 0.363367 + 0.629371i
\(836\) 0 0
\(837\) −2.50000 4.33013i −0.0864126 0.149671i
\(838\) 0 0
\(839\) 22.5000 + 38.9711i 0.776786 + 1.34543i 0.933785 + 0.357834i \(0.116485\pi\)
−0.156999 + 0.987599i \(0.550182\pi\)
\(840\) 0 0
\(841\) 10.0000 17.3205i 0.344828 0.597259i
\(842\) 0 0
\(843\) −4.50000 + 7.79423i −0.154988 + 0.268447i
\(844\) 0 0
\(845\) 6.00000 + 10.3923i 0.206406 + 0.357506i
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 0 0
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) 10.5000 + 18.1865i 0.359935 + 0.623426i
\(852\) 0 0
\(853\) −8.50000 + 14.7224i −0.291034 + 0.504086i −0.974055 0.226313i \(-0.927333\pi\)
0.683020 + 0.730400i \(0.260666\pi\)
\(854\) 0 0
\(855\) −0.500000 + 0.866025i −0.0170996 + 0.0296174i
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 3.50000 + 6.06218i 0.119418 + 0.206839i 0.919537 0.393003i \(-0.128564\pi\)
−0.800119 + 0.599841i \(0.795230\pi\)
\(860\) 0 0
\(861\) 4.50000 7.79423i 0.153360 0.265627i
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −10.5000 + 18.1865i −0.357011 + 0.618361i
\(866\) 0 0
\(867\) 4.00000 + 6.92820i 0.135847 + 0.235294i
\(868\) 0 0
\(869\) −25.5000 + 44.1673i −0.865028 + 1.49827i
\(870\) 0 0
\(871\) 27.5000 30.3109i 0.931802 1.02705i
\(872\) 0 0
\(873\) −8.50000 + 14.7224i −0.287681 + 0.498279i
\(874\) 0 0
\(875\) −0.500000 0.866025i −0.0169031 0.0292770i
\(876\) 0 0
\(877\) 9.50000 16.4545i 0.320792 0.555628i −0.659860 0.751389i \(-0.729384\pi\)
0.980652 + 0.195761i \(0.0627176\pi\)
\(878\) 0 0
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −4.50000 + 7.79423i −0.151609 + 0.262594i −0.931819 0.362923i \(-0.881779\pi\)
0.780210 + 0.625517i \(0.215112\pi\)
\(882\) 0 0
\(883\) 18.5000 + 32.0429i 0.622575 + 1.07833i 0.989005 + 0.147885i \(0.0472466\pi\)
−0.366430 + 0.930446i \(0.619420\pi\)
\(884\) 0 0
\(885\) 12.0000 0.403376
\(886\) 0 0
\(887\) −7.50000 + 12.9904i −0.251825 + 0.436174i −0.964028 0.265799i \(-0.914364\pi\)
0.712203 + 0.701974i \(0.247698\pi\)
\(888\) 0 0
\(889\) −6.50000 + 11.2583i −0.218003 + 0.377592i
\(890\) 0 0
\(891\) −1.50000 2.59808i −0.0502519 0.0870388i
\(892\) 0 0
\(893\) 3.00000 0.100391
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) −7.50000 12.9904i −0.250418 0.433736i
\(898\) 0 0
\(899\) 7.50000 12.9904i 0.250139 0.433253i
\(900\) 0 0
\(901\) 9.00000 15.5885i 0.299833 0.519327i
\(902\) 0 0
\(903\) −2.00000 3.46410i −0.0665558 0.115278i
\(904\) 0 0
\(905\) −6.50000 11.2583i −0.216067 0.374240i
\(906\) 0 0
\(907\) −17.5000 30.3109i −0.581078 1.00646i −0.995352 0.0963043i \(-0.969298\pi\)
0.414274 0.910152i \(-0.364036\pi\)
\(908\) 0 0
\(909\) 1.50000 + 2.59808i 0.0497519 + 0.0861727i
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) 45.0000 1.48928
\(914\) 0 0
\(915\) −0.500000 + 0.866025i −0.0165295 + 0.0286299i
\(916\) 0 0
\(917\) −6.00000 10.3923i −0.198137 0.343184i
\(918\) 0 0
\(919\) 12.5000 21.6506i 0.412337 0.714189i −0.582808 0.812610i \(-0.698046\pi\)
0.995145 + 0.0984214i \(0.0313793\pi\)
\(920\) 0 0
\(921\) −2.50000 + 4.33013i −0.0823778 + 0.142683i
\(922\) 0 0
\(923\) 15.0000 0.493731
\(924\) 0 0
\(925\) 3.50000 6.06218i 0.115079 0.199323i
\(926\) 0 0
\(927\) 6.50000 + 11.2583i 0.213488 + 0.369772i
\(928\) 0 0
\(929\) −54.0000 −1.77168 −0.885841 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) 0 0
\(931\) −3.00000 5.19615i −0.0983210 0.170297i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −9.00000 −0.294331
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 0 0
\(939\) 26.0000 0.848478
\(940\) 0 0
\(941\) 54.0000 1.76035 0.880175 0.474650i \(-0.157425\pi\)
0.880175 + 0.474650i \(0.157425\pi\)
\(942\) 0 0
\(943\) 27.0000 0.879241
\(944\) 0 0
\(945\) −0.500000 0.866025i −0.0162650 0.0281718i
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 17.5000 + 30.3109i 0.568074 + 0.983933i
\(950\) 0 0
\(951\) −7.50000 + 12.9904i −0.243204 + 0.421242i
\(952\) 0 0
\(953\) 54.0000 1.74923 0.874616 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 0 0
\(955\) −1.50000 + 2.59808i −0.0485389 + 0.0840718i
\(956\) 0 0
\(957\) 4.50000 7.79423i 0.145464 0.251952i
\(958\) 0 0
\(959\) −9.00000 15.5885i −0.290625 0.503378i
\(960\) 0 0
\(961\) 3.00000 5.19615i 0.0967742 0.167618i
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) 22.0000 0.708205
\(966\) 0 0
\(967\) −17.5000 30.3109i −0.562762 0.974732i −0.997254 0.0740568i \(-0.976405\pi\)
0.434492 0.900676i \(-0.356928\pi\)
\(968\) 0 0
\(969\) 1.50000 + 2.59808i 0.0481869 + 0.0834622i
\(970\) 0 0
\(971\) −7.50000 12.9904i −0.240686 0.416881i 0.720224 0.693742i \(-0.244039\pi\)
−0.960910 + 0.276861i \(0.910706\pi\)
\(972\) 0 0
\(973\) 10.0000 + 17.3205i 0.320585 + 0.555270i
\(974\) 0 0
\(975\) −2.50000 + 4.33013i −0.0800641 + 0.138675i
\(976\) 0 0
\(977\) −19.5000 + 33.7750i −0.623860 + 1.08056i 0.364900 + 0.931047i \(0.381103\pi\)
−0.988760 + 0.149511i \(0.952230\pi\)
\(978\) 0 0
\(979\) 9.00000 + 15.5885i 0.287641 + 0.498209i
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) 1.50000 + 2.59808i 0.0477940 + 0.0827816i
\(986\) 0 0
\(987\) −1.50000 + 2.59808i −0.0477455 + 0.0826977i
\(988\) 0 0
\(989\) 6.00000 10.3923i 0.190789 0.330456i
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) −8.50000 14.7224i −0.269739 0.467202i
\(994\) 0 0
\(995\) 5.50000 9.52628i 0.174362 0.302003i
\(996\) 0 0
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) 0 0
\(999\) 3.50000 6.06218i 0.110735 0.191799i
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.q.h.3781.1 yes 2
67.37 even 3 inner 4020.2.q.h.841.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4020.2.q.h.841.1 2 67.37 even 3 inner
4020.2.q.h.3781.1 yes 2 1.1 even 1 trivial