Properties

Label 840.2.bg.i.361.3
Level $840$
Weight $2$
Character 840.361
Analytic conductor $6.707$
Analytic rank $0$
Dimension $6$
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] = [840,2,Mod(121,840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(840, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("840.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.bg (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: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.38363328.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 3x^{4} - 2x^{3} - 21x^{2} - 49x + 343 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.3
Root \(-2.33916 + 1.23625i\) of defining polynomial
Character \(\chi\) \(=\) 840.361
Dual form 840.2.bg.i.121.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(2.24021 + 1.40765i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(2.24021 + 1.40765i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(2.74021 - 4.74618i) q^{11} +0.197906 q^{13} +1.00000 q^{15} +(-1.40105 + 2.42668i) q^{17} +(0.302094 + 0.523243i) q^{19} +(-2.33916 + 1.23625i) q^{21} +(-3.93811 - 6.82101i) q^{23} +(-0.500000 + 0.866025i) q^{25} +1.00000 q^{27} +10.1587 q^{29} +(1.83916 - 3.18552i) q^{31} +(2.74021 + 4.74618i) q^{33} +(0.0989528 - 2.64390i) q^{35} +(2.57937 + 4.46760i) q^{37} +(-0.0989528 + 0.171391i) q^{39} +7.08461 q^{41} +4.48042 q^{43} +(-0.500000 + 0.866025i) q^{45} +(4.33916 + 7.51565i) q^{47} +(3.03707 + 6.30684i) q^{49} +(-1.40105 - 2.42668i) q^{51} +(2.53707 - 4.39433i) q^{53} -5.48042 q^{55} -0.604189 q^{57} +(3.07937 - 5.33362i) q^{59} +(-0.802094 - 1.38927i) q^{61} +(0.0989528 - 2.64390i) q^{63} +(-0.0989528 - 0.171391i) q^{65} +(-1.04230 + 1.80532i) q^{67} +7.87623 q^{69} -10.5546 q^{71} +(-2.43811 + 4.22294i) q^{73} +(-0.500000 - 0.866025i) q^{75} +(12.8196 - 6.77519i) q^{77} +(-5.64126 - 9.77094i) q^{79} +(-0.500000 + 0.866025i) q^{81} +11.7629 q^{83} +2.80209 q^{85} +(-5.07937 + 8.79773i) q^{87} +(-3.59895 - 6.23357i) q^{89} +(0.443350 + 0.278581i) q^{91} +(1.83916 + 3.18552i) q^{93} +(0.302094 - 0.523243i) q^{95} +2.39581 q^{97} -5.48042 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} - 3 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} - 3 q^{5} - 2 q^{7} - 3 q^{9} + q^{11} + 2 q^{13} + 6 q^{15} - 8 q^{17} + q^{19} + q^{21} - 9 q^{23} - 3 q^{25} + 6 q^{27} - 4 q^{31} + q^{33} + q^{35} - 15 q^{37} - q^{39} + 10 q^{41} - 4 q^{43} - 3 q^{45} + 11 q^{47} + 4 q^{49} - 8 q^{51} + q^{53} - 2 q^{55} - 2 q^{57} - 12 q^{59} - 4 q^{61} + q^{63} - q^{65} + 10 q^{67} + 18 q^{69} - 4 q^{71} - 3 q^{75} + 31 q^{77} - 18 q^{79} - 3 q^{81} + 8 q^{83} + 16 q^{85} - 22 q^{89} - 14 q^{91} - 4 q^{93} + q^{95} + 16 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(281\) \(337\) \(421\) \(631\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(1\) \(1\)

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
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) 0 0
\(5\) −0.500000 0.866025i −0.223607 0.387298i
\(6\) 0 0
\(7\) 2.24021 + 1.40765i 0.846719 + 0.532040i
\(8\) 0 0
\(9\) −0.500000 0.866025i −0.166667 0.288675i
\(10\) 0 0
\(11\) 2.74021 4.74618i 0.826204 1.43103i −0.0747918 0.997199i \(-0.523829\pi\)
0.900996 0.433828i \(-0.142837\pi\)
\(12\) 0 0
\(13\) 0.197906 0.0548891 0.0274446 0.999623i \(-0.491263\pi\)
0.0274446 + 0.999623i \(0.491263\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −1.40105 + 2.42668i −0.339804 + 0.588558i −0.984396 0.175969i \(-0.943694\pi\)
0.644592 + 0.764527i \(0.277027\pi\)
\(18\) 0 0
\(19\) 0.302094 + 0.523243i 0.0693052 + 0.120040i 0.898596 0.438778i \(-0.144588\pi\)
−0.829290 + 0.558818i \(0.811255\pi\)
\(20\) 0 0
\(21\) −2.33916 + 1.23625i −0.510447 + 0.269773i
\(22\) 0 0
\(23\) −3.93811 6.82101i −0.821154 1.42228i −0.904824 0.425786i \(-0.859998\pi\)
0.0836703 0.996493i \(-0.473336\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 10.1587 1.88643 0.943215 0.332182i \(-0.107785\pi\)
0.943215 + 0.332182i \(0.107785\pi\)
\(30\) 0 0
\(31\) 1.83916 3.18552i 0.330323 0.572136i −0.652252 0.758002i \(-0.726176\pi\)
0.982575 + 0.185866i \(0.0595090\pi\)
\(32\) 0 0
\(33\) 2.74021 + 4.74618i 0.477009 + 0.826204i
\(34\) 0 0
\(35\) 0.0989528 2.64390i 0.0167261 0.446901i
\(36\) 0 0
\(37\) 2.57937 + 4.46760i 0.424046 + 0.734469i 0.996331 0.0855860i \(-0.0272762\pi\)
−0.572285 + 0.820055i \(0.693943\pi\)
\(38\) 0 0
\(39\) −0.0989528 + 0.171391i −0.0158451 + 0.0274446i
\(40\) 0 0
\(41\) 7.08461 1.10643 0.553215 0.833039i \(-0.313401\pi\)
0.553215 + 0.833039i \(0.313401\pi\)
\(42\) 0 0
\(43\) 4.48042 0.683257 0.341629 0.939835i \(-0.389022\pi\)
0.341629 + 0.939835i \(0.389022\pi\)
\(44\) 0 0
\(45\) −0.500000 + 0.866025i −0.0745356 + 0.129099i
\(46\) 0 0
\(47\) 4.33916 + 7.51565i 0.632932 + 1.09627i 0.986949 + 0.161031i \(0.0514820\pi\)
−0.354018 + 0.935239i \(0.615185\pi\)
\(48\) 0 0
\(49\) 3.03707 + 6.30684i 0.433867 + 0.900977i
\(50\) 0 0
\(51\) −1.40105 2.42668i −0.196186 0.339804i
\(52\) 0 0
\(53\) 2.53707 4.39433i 0.348493 0.603607i −0.637489 0.770459i \(-0.720027\pi\)
0.985982 + 0.166852i \(0.0533602\pi\)
\(54\) 0 0
\(55\) −5.48042 −0.738979
\(56\) 0 0
\(57\) −0.604189 −0.0800267
\(58\) 0 0
\(59\) 3.07937 5.33362i 0.400900 0.694379i −0.592935 0.805250i \(-0.702031\pi\)
0.993835 + 0.110872i \(0.0353642\pi\)
\(60\) 0 0
\(61\) −0.802094 1.38927i −0.102698 0.177878i 0.810098 0.586295i \(-0.199414\pi\)
−0.912795 + 0.408418i \(0.866081\pi\)
\(62\) 0 0
\(63\) 0.0989528 2.64390i 0.0124669 0.333100i
\(64\) 0 0
\(65\) −0.0989528 0.171391i −0.0122736 0.0212585i
\(66\) 0 0
\(67\) −1.04230 + 1.80532i −0.127338 + 0.220555i −0.922644 0.385652i \(-0.873976\pi\)
0.795307 + 0.606207i \(0.207310\pi\)
\(68\) 0 0
\(69\) 7.87623 0.948186
\(70\) 0 0
\(71\) −10.5546 −1.25259 −0.626297 0.779584i \(-0.715430\pi\)
−0.626297 + 0.779584i \(0.715430\pi\)
\(72\) 0 0
\(73\) −2.43811 + 4.22294i −0.285360 + 0.494257i −0.972696 0.232082i \(-0.925446\pi\)
0.687337 + 0.726339i \(0.258780\pi\)
\(74\) 0 0
\(75\) −0.500000 0.866025i −0.0577350 0.100000i
\(76\) 0 0
\(77\) 12.8196 6.77519i 1.46093 0.772105i
\(78\) 0 0
\(79\) −5.64126 9.77094i −0.634691 1.09932i −0.986581 0.163275i \(-0.947794\pi\)
0.351890 0.936041i \(-0.385539\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 11.7629 1.29115 0.645575 0.763697i \(-0.276618\pi\)
0.645575 + 0.763697i \(0.276618\pi\)
\(84\) 0 0
\(85\) 2.80209 0.303930
\(86\) 0 0
\(87\) −5.07937 + 8.79773i −0.544566 + 0.943215i
\(88\) 0 0
\(89\) −3.59895 6.23357i −0.381488 0.660757i 0.609787 0.792565i \(-0.291255\pi\)
−0.991275 + 0.131808i \(0.957922\pi\)
\(90\) 0 0
\(91\) 0.443350 + 0.278581i 0.0464757 + 0.0292032i
\(92\) 0 0
\(93\) 1.83916 + 3.18552i 0.190712 + 0.330323i
\(94\) 0 0
\(95\) 0.302094 0.523243i 0.0309942 0.0536836i
\(96\) 0 0
\(97\) 2.39581 0.243258 0.121629 0.992576i \(-0.461188\pi\)
0.121629 + 0.992576i \(0.461188\pi\)
\(98\) 0 0
\(99\) −5.48042 −0.550803
\(100\) 0 0
\(101\) −7.40105 + 12.8190i −0.736432 + 1.27554i 0.217661 + 0.976025i \(0.430157\pi\)
−0.954092 + 0.299513i \(0.903176\pi\)
\(102\) 0 0
\(103\) 0.957697 + 1.65878i 0.0943647 + 0.163444i 0.909343 0.416047i \(-0.136585\pi\)
−0.814979 + 0.579491i \(0.803251\pi\)
\(104\) 0 0
\(105\) 2.24021 + 1.40765i 0.218622 + 0.137372i
\(106\) 0 0
\(107\) 2.07937 + 3.60157i 0.201020 + 0.348177i 0.948857 0.315705i \(-0.102241\pi\)
−0.747837 + 0.663882i \(0.768908\pi\)
\(108\) 0 0
\(109\) −8.31958 + 14.4099i −0.796871 + 1.38022i 0.124773 + 0.992185i \(0.460180\pi\)
−0.921644 + 0.388036i \(0.873154\pi\)
\(110\) 0 0
\(111\) −5.15874 −0.489646
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −3.93811 + 6.82101i −0.367231 + 0.636063i
\(116\) 0 0
\(117\) −0.0989528 0.171391i −0.00914819 0.0158451i
\(118\) 0 0
\(119\) −6.55455 + 3.46410i −0.600855 + 0.317554i
\(120\) 0 0
\(121\) −9.51748 16.4848i −0.865226 1.49861i
\(122\) 0 0
\(123\) −3.54230 + 6.13545i −0.319399 + 0.553215i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.59372 0.407626 0.203813 0.979010i \(-0.434666\pi\)
0.203813 + 0.979010i \(0.434666\pi\)
\(128\) 0 0
\(129\) −2.24021 + 3.88015i −0.197239 + 0.341629i
\(130\) 0 0
\(131\) −10.4979 18.1829i −0.917206 1.58865i −0.803639 0.595117i \(-0.797106\pi\)
−0.113566 0.993530i \(-0.536227\pi\)
\(132\) 0 0
\(133\) −0.0597862 + 1.59741i −0.00518412 + 0.138513i
\(134\) 0 0
\(135\) −0.500000 0.866025i −0.0430331 0.0745356i
\(136\) 0 0
\(137\) 4.88146 8.45494i 0.417052 0.722355i −0.578590 0.815619i \(-0.696397\pi\)
0.995641 + 0.0932641i \(0.0297301\pi\)
\(138\) 0 0
\(139\) 20.2433 1.71702 0.858509 0.512798i \(-0.171391\pi\)
0.858509 + 0.512798i \(0.171391\pi\)
\(140\) 0 0
\(141\) −8.67832 −0.730847
\(142\) 0 0
\(143\) 0.542303 0.939296i 0.0453496 0.0785479i
\(144\) 0 0
\(145\) −5.07937 8.79773i −0.421819 0.730611i
\(146\) 0 0
\(147\) −6.98042 0.523243i −0.575735 0.0431563i
\(148\) 0 0
\(149\) 2.00000 + 3.46410i 0.163846 + 0.283790i 0.936245 0.351348i \(-0.114277\pi\)
−0.772399 + 0.635138i \(0.780943\pi\)
\(150\) 0 0
\(151\) 3.80209 6.58542i 0.309410 0.535914i −0.668823 0.743421i \(-0.733202\pi\)
0.978233 + 0.207507i \(0.0665351\pi\)
\(152\) 0 0
\(153\) 2.80209 0.226536
\(154\) 0 0
\(155\) −3.67832 −0.295450
\(156\) 0 0
\(157\) −11.1413 + 19.2972i −0.889169 + 1.54009i −0.0483095 + 0.998832i \(0.515383\pi\)
−0.840860 + 0.541253i \(0.817950\pi\)
\(158\) 0 0
\(159\) 2.53707 + 4.39433i 0.201202 + 0.348493i
\(160\) 0 0
\(161\) 0.779375 20.8240i 0.0614234 1.64116i
\(162\) 0 0
\(163\) −8.00000 13.8564i −0.626608 1.08532i −0.988227 0.152992i \(-0.951109\pi\)
0.361619 0.932326i \(-0.382224\pi\)
\(164\) 0 0
\(165\) 2.74021 4.74618i 0.213325 0.369490i
\(166\) 0 0
\(167\) −24.0454 −1.86069 −0.930346 0.366683i \(-0.880493\pi\)
−0.930346 + 0.366683i \(0.880493\pi\)
\(168\) 0 0
\(169\) −12.9608 −0.996987
\(170\) 0 0
\(171\) 0.302094 0.523243i 0.0231017 0.0400134i
\(172\) 0 0
\(173\) −1.85874 3.21944i −0.141318 0.244769i 0.786675 0.617367i \(-0.211801\pi\)
−0.927993 + 0.372597i \(0.878467\pi\)
\(174\) 0 0
\(175\) −2.33916 + 1.23625i −0.176824 + 0.0934521i
\(176\) 0 0
\(177\) 3.07937 + 5.33362i 0.231460 + 0.400900i
\(178\) 0 0
\(179\) 8.33916 14.4439i 0.623298 1.07958i −0.365569 0.930784i \(-0.619126\pi\)
0.988867 0.148800i \(-0.0475410\pi\)
\(180\) 0 0
\(181\) −9.03497 −0.671564 −0.335782 0.941940i \(-0.609001\pi\)
−0.335782 + 0.941940i \(0.609001\pi\)
\(182\) 0 0
\(183\) 1.60419 0.118585
\(184\) 0 0
\(185\) 2.57937 4.46760i 0.189639 0.328464i
\(186\) 0 0
\(187\) 7.67832 + 13.2992i 0.561495 + 0.972537i
\(188\) 0 0
\(189\) 2.24021 + 1.40765i 0.162951 + 0.102391i
\(190\) 0 0
\(191\) −11.9608 20.7168i −0.865456 1.49901i −0.866594 0.499014i \(-0.833696\pi\)
0.00113863 0.999999i \(-0.499638\pi\)
\(192\) 0 0
\(193\) −9.31434 + 16.1329i −0.670461 + 1.16127i 0.307313 + 0.951609i \(0.400570\pi\)
−0.977774 + 0.209664i \(0.932763\pi\)
\(194\) 0 0
\(195\) 0.197906 0.0141723
\(196\) 0 0
\(197\) 3.08461 0.219769 0.109885 0.993944i \(-0.464952\pi\)
0.109885 + 0.993944i \(0.464952\pi\)
\(198\) 0 0
\(199\) −10.4804 + 18.1526i −0.742937 + 1.28680i 0.208215 + 0.978083i \(0.433235\pi\)
−0.951152 + 0.308722i \(0.900099\pi\)
\(200\) 0 0
\(201\) −1.04230 1.80532i −0.0735184 0.127338i
\(202\) 0 0
\(203\) 22.7577 + 14.2999i 1.59728 + 1.00366i
\(204\) 0 0
\(205\) −3.54230 6.13545i −0.247405 0.428518i
\(206\) 0 0
\(207\) −3.93811 + 6.82101i −0.273718 + 0.474093i
\(208\) 0 0
\(209\) 3.31121 0.229041
\(210\) 0 0
\(211\) 7.63916 0.525901 0.262951 0.964809i \(-0.415304\pi\)
0.262951 + 0.964809i \(0.415304\pi\)
\(212\) 0 0
\(213\) 5.27728 9.14051i 0.361593 0.626297i
\(214\) 0 0
\(215\) −2.24021 3.88015i −0.152781 0.264624i
\(216\) 0 0
\(217\) 8.60419 4.54734i 0.584090 0.308694i
\(218\) 0 0
\(219\) −2.43811 4.22294i −0.164752 0.285360i
\(220\) 0 0
\(221\) −0.277275 + 0.480255i −0.0186515 + 0.0323054i
\(222\) 0 0
\(223\) −16.3175 −1.09270 −0.546350 0.837557i \(-0.683983\pi\)
−0.546350 + 0.837557i \(0.683983\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −9.88146 + 17.1152i −0.655856 + 1.13598i 0.325823 + 0.945431i \(0.394359\pi\)
−0.981679 + 0.190545i \(0.938975\pi\)
\(228\) 0 0
\(229\) 1.51748 + 2.62836i 0.100278 + 0.173687i 0.911799 0.410636i \(-0.134693\pi\)
−0.811521 + 0.584323i \(0.801360\pi\)
\(230\) 0 0
\(231\) −0.542303 + 14.4897i −0.0356809 + 0.953351i
\(232\) 0 0
\(233\) 8.19791 + 14.1992i 0.537063 + 0.930220i 0.999060 + 0.0433387i \(0.0137995\pi\)
−0.461998 + 0.886881i \(0.652867\pi\)
\(234\) 0 0
\(235\) 4.33916 7.51565i 0.283056 0.490267i
\(236\) 0 0
\(237\) 11.2825 0.732878
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 2.94335 5.09803i 0.189598 0.328393i −0.755518 0.655127i \(-0.772615\pi\)
0.945116 + 0.326734i \(0.105948\pi\)
\(242\) 0 0
\(243\) −0.500000 0.866025i −0.0320750 0.0555556i
\(244\) 0 0
\(245\) 3.94335 5.78360i 0.251931 0.369500i
\(246\) 0 0
\(247\) 0.0597862 + 0.103553i 0.00380410 + 0.00658890i
\(248\) 0 0
\(249\) −5.88146 + 10.1870i −0.372723 + 0.645575i
\(250\) 0 0
\(251\) −9.87623 −0.623382 −0.311691 0.950184i \(-0.600895\pi\)
−0.311691 + 0.950184i \(0.600895\pi\)
\(252\) 0 0
\(253\) −43.1650 −2.71376
\(254\) 0 0
\(255\) −1.40105 + 2.42668i −0.0877370 + 0.151965i
\(256\) 0 0
\(257\) 3.88146 + 6.72289i 0.242119 + 0.419363i 0.961318 0.275442i \(-0.0888242\pi\)
−0.719199 + 0.694805i \(0.755491\pi\)
\(258\) 0 0
\(259\) −0.510472 + 13.6392i −0.0317192 + 0.847498i
\(260\) 0 0
\(261\) −5.07937 8.79773i −0.314405 0.544566i
\(262\) 0 0
\(263\) 2.80209 4.85337i 0.172785 0.299272i −0.766608 0.642116i \(-0.778057\pi\)
0.939392 + 0.342844i \(0.111390\pi\)
\(264\) 0 0
\(265\) −5.07413 −0.311702
\(266\) 0 0
\(267\) 7.19791 0.440505
\(268\) 0 0
\(269\) −10.8762 + 18.8382i −0.663135 + 1.14858i 0.316652 + 0.948542i \(0.397441\pi\)
−0.979787 + 0.200042i \(0.935892\pi\)
\(270\) 0 0
\(271\) 10.0846 + 17.4670i 0.612596 + 1.06105i 0.990801 + 0.135326i \(0.0432081\pi\)
−0.378205 + 0.925722i \(0.623459\pi\)
\(272\) 0 0
\(273\) −0.462933 + 0.244662i −0.0280180 + 0.0148076i
\(274\) 0 0
\(275\) 2.74021 + 4.74618i 0.165241 + 0.286205i
\(276\) 0 0
\(277\) 8.11644 14.0581i 0.487669 0.844668i −0.512230 0.858848i \(-0.671180\pi\)
0.999899 + 0.0141801i \(0.00451382\pi\)
\(278\) 0 0
\(279\) −3.67832 −0.220215
\(280\) 0 0
\(281\) −14.6783 −0.875635 −0.437818 0.899064i \(-0.644248\pi\)
−0.437818 + 0.899064i \(0.644248\pi\)
\(282\) 0 0
\(283\) 2.03183 3.51923i 0.120780 0.209197i −0.799296 0.600938i \(-0.794794\pi\)
0.920075 + 0.391741i \(0.128127\pi\)
\(284\) 0 0
\(285\) 0.302094 + 0.523243i 0.0178945 + 0.0309942i
\(286\) 0 0
\(287\) 15.8710 + 9.97261i 0.936835 + 0.588665i
\(288\) 0 0
\(289\) 4.57413 + 7.92263i 0.269067 + 0.466037i
\(290\) 0 0
\(291\) −1.19791 + 2.07483i −0.0702225 + 0.121629i
\(292\) 0 0
\(293\) 2.67832 0.156469 0.0782346 0.996935i \(-0.475072\pi\)
0.0782346 + 0.996935i \(0.475072\pi\)
\(294\) 0 0
\(295\) −6.15874 −0.358576
\(296\) 0 0
\(297\) 2.74021 4.74618i 0.159003 0.275401i
\(298\) 0 0
\(299\) −0.779375 1.34992i −0.0450724 0.0780677i
\(300\) 0 0
\(301\) 10.0371 + 6.30684i 0.578527 + 0.363520i
\(302\) 0 0
\(303\) −7.40105 12.8190i −0.425179 0.736432i
\(304\) 0 0
\(305\) −0.802094 + 1.38927i −0.0459278 + 0.0795493i
\(306\) 0 0
\(307\) 31.1937 1.78032 0.890159 0.455649i \(-0.150593\pi\)
0.890159 + 0.455649i \(0.150593\pi\)
\(308\) 0 0
\(309\) −1.91539 −0.108963
\(310\) 0 0
\(311\) −0.192670 + 0.333714i −0.0109253 + 0.0189232i −0.871436 0.490509i \(-0.836811\pi\)
0.860511 + 0.509432i \(0.170144\pi\)
\(312\) 0 0
\(313\) −4.24021 7.34426i −0.239671 0.415122i 0.720949 0.692988i \(-0.243706\pi\)
−0.960620 + 0.277866i \(0.910373\pi\)
\(314\) 0 0
\(315\) −2.33916 + 1.23625i −0.131797 + 0.0696550i
\(316\) 0 0
\(317\) −13.4360 23.2719i −0.754642 1.30708i −0.945552 0.325470i \(-0.894477\pi\)
0.190910 0.981607i \(-0.438856\pi\)
\(318\) 0 0
\(319\) 27.8371 48.2152i 1.55858 2.69953i
\(320\) 0 0
\(321\) −4.15874 −0.232118
\(322\) 0 0
\(323\) −1.69299 −0.0942007
\(324\) 0 0
\(325\) −0.0989528 + 0.171391i −0.00548891 + 0.00950708i
\(326\) 0 0
\(327\) −8.31958 14.4099i −0.460074 0.796871i
\(328\) 0 0
\(329\) −0.858744 + 22.9446i −0.0473441 + 1.26498i
\(330\) 0 0
\(331\) 4.58461 + 7.94077i 0.251993 + 0.436464i 0.964074 0.265632i \(-0.0855808\pi\)
−0.712082 + 0.702097i \(0.752247\pi\)
\(332\) 0 0
\(333\) 2.57937 4.46760i 0.141349 0.244823i
\(334\) 0 0
\(335\) 2.08461 0.113894
\(336\) 0 0
\(337\) 24.0063 1.30770 0.653852 0.756622i \(-0.273152\pi\)
0.653852 + 0.756622i \(0.273152\pi\)
\(338\) 0 0
\(339\) 3.00000 5.19615i 0.162938 0.282216i
\(340\) 0 0
\(341\) −10.0794 17.4580i −0.545828 0.945403i
\(342\) 0 0
\(343\) −2.07413 + 18.4037i −0.111993 + 0.993709i
\(344\) 0 0
\(345\) −3.93811 6.82101i −0.212021 0.367231i
\(346\) 0 0
\(347\) 1.71749 2.97478i 0.0921996 0.159694i −0.816237 0.577718i \(-0.803944\pi\)
0.908436 + 0.418023i \(0.137277\pi\)
\(348\) 0 0
\(349\) 16.6434 0.890898 0.445449 0.895307i \(-0.353044\pi\)
0.445449 + 0.895307i \(0.353044\pi\)
\(350\) 0 0
\(351\) 0.197906 0.0105634
\(352\) 0 0
\(353\) −3.88146 + 6.72289i −0.206589 + 0.357823i −0.950638 0.310302i \(-0.899570\pi\)
0.744049 + 0.668126i \(0.232903\pi\)
\(354\) 0 0
\(355\) 5.27728 + 9.14051i 0.280089 + 0.485128i
\(356\) 0 0
\(357\) 0.277275 7.40846i 0.0146750 0.392097i
\(358\) 0 0
\(359\) 15.0794 + 26.1182i 0.795859 + 1.37847i 0.922292 + 0.386493i \(0.126314\pi\)
−0.126434 + 0.991975i \(0.540353\pi\)
\(360\) 0 0
\(361\) 9.31748 16.1383i 0.490394 0.849387i
\(362\) 0 0
\(363\) 19.0350 0.999077
\(364\) 0 0
\(365\) 4.87623 0.255233
\(366\) 0 0
\(367\) −8.09895 + 14.0278i −0.422762 + 0.732245i −0.996208 0.0869979i \(-0.972273\pi\)
0.573447 + 0.819243i \(0.305606\pi\)
\(368\) 0 0
\(369\) −3.54230 6.13545i −0.184405 0.319399i
\(370\) 0 0
\(371\) 11.8692 6.27292i 0.616219 0.325674i
\(372\) 0 0
\(373\) −1.24021 2.14810i −0.0642156 0.111225i 0.832130 0.554580i \(-0.187121\pi\)
−0.896346 + 0.443356i \(0.853788\pi\)
\(374\) 0 0
\(375\) −0.500000 + 0.866025i −0.0258199 + 0.0447214i
\(376\) 0 0
\(377\) 2.01047 0.103545
\(378\) 0 0
\(379\) 6.92167 0.355542 0.177771 0.984072i \(-0.443111\pi\)
0.177771 + 0.984072i \(0.443111\pi\)
\(380\) 0 0
\(381\) −2.29686 + 3.97828i −0.117672 + 0.203813i
\(382\) 0 0
\(383\) −2.53707 4.39433i −0.129638 0.224540i 0.793898 0.608051i \(-0.208048\pi\)
−0.923536 + 0.383511i \(0.874715\pi\)
\(384\) 0 0
\(385\) −12.2773 7.71449i −0.625708 0.393167i
\(386\) 0 0
\(387\) −2.24021 3.88015i −0.113876 0.197239i
\(388\) 0 0
\(389\) 3.11854 5.40146i 0.158116 0.273865i −0.776073 0.630643i \(-0.782791\pi\)
0.934189 + 0.356778i \(0.116125\pi\)
\(390\) 0 0
\(391\) 22.0699 1.11612
\(392\) 0 0
\(393\) 20.9958 1.05910
\(394\) 0 0
\(395\) −5.64126 + 9.77094i −0.283842 + 0.491629i
\(396\) 0 0
\(397\) −13.7206 23.7648i −0.688618 1.19272i −0.972285 0.233799i \(-0.924884\pi\)
0.283666 0.958923i \(-0.408449\pi\)
\(398\) 0 0
\(399\) −1.35351 0.850484i −0.0677602 0.0425774i
\(400\) 0 0
\(401\) 15.4979 + 26.8432i 0.773928 + 1.34048i 0.935395 + 0.353605i \(0.115044\pi\)
−0.161467 + 0.986878i \(0.551622\pi\)
\(402\) 0 0
\(403\) 0.363980 0.630432i 0.0181312 0.0314041i
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 28.2720 1.40139
\(408\) 0 0
\(409\) 13.5175 23.4130i 0.668397 1.15770i −0.309956 0.950751i \(-0.600314\pi\)
0.978352 0.206946i \(-0.0663524\pi\)
\(410\) 0 0
\(411\) 4.88146 + 8.45494i 0.240785 + 0.417052i
\(412\) 0 0
\(413\) 14.4063 7.61377i 0.708887 0.374649i
\(414\) 0 0
\(415\) −5.88146 10.1870i −0.288710 0.500060i
\(416\) 0 0
\(417\) −10.1217 + 17.5313i −0.495660 + 0.858509i
\(418\) 0 0
\(419\) −12.9259 −0.631470 −0.315735 0.948847i \(-0.602251\pi\)
−0.315735 + 0.948847i \(0.602251\pi\)
\(420\) 0 0
\(421\) −18.0741 −0.880879 −0.440440 0.897782i \(-0.645177\pi\)
−0.440440 + 0.897782i \(0.645177\pi\)
\(422\) 0 0
\(423\) 4.33916 7.51565i 0.210977 0.365423i
\(424\) 0 0
\(425\) −1.40105 2.42668i −0.0679608 0.117712i
\(426\) 0 0
\(427\) 0.158739 4.24131i 0.00768192 0.205252i
\(428\) 0 0
\(429\) 0.542303 + 0.939296i 0.0261826 + 0.0453496i
\(430\) 0 0
\(431\) −17.0454 + 29.5236i −0.821050 + 1.42210i 0.0838512 + 0.996478i \(0.473278\pi\)
−0.904901 + 0.425622i \(0.860055\pi\)
\(432\) 0 0
\(433\) 7.51958 0.361368 0.180684 0.983541i \(-0.442169\pi\)
0.180684 + 0.983541i \(0.442169\pi\)
\(434\) 0 0
\(435\) 10.1587 0.487074
\(436\) 0 0
\(437\) 2.37936 4.12118i 0.113820 0.197143i
\(438\) 0 0
\(439\) −9.67832 16.7633i −0.461921 0.800071i 0.537136 0.843496i \(-0.319506\pi\)
−0.999057 + 0.0434251i \(0.986173\pi\)
\(440\) 0 0
\(441\) 3.94335 5.78360i 0.187779 0.275409i
\(442\) 0 0
\(443\) 8.15874 + 14.1314i 0.387633 + 0.671401i 0.992131 0.125206i \(-0.0399593\pi\)
−0.604497 + 0.796607i \(0.706626\pi\)
\(444\) 0 0
\(445\) −3.59895 + 6.23357i −0.170607 + 0.295500i
\(446\) 0 0
\(447\) −4.00000 −0.189194
\(448\) 0 0
\(449\) −25.2433 −1.19131 −0.595654 0.803241i \(-0.703107\pi\)
−0.595654 + 0.803241i \(0.703107\pi\)
\(450\) 0 0
\(451\) 19.4133 33.6248i 0.914136 1.58333i
\(452\) 0 0
\(453\) 3.80209 + 6.58542i 0.178638 + 0.309410i
\(454\) 0 0
\(455\) 0.0195833 0.523243i 0.000918080 0.0245300i
\(456\) 0 0
\(457\) −14.5122 25.1360i −0.678854 1.17581i −0.975326 0.220769i \(-0.929143\pi\)
0.296472 0.955042i \(-0.404190\pi\)
\(458\) 0 0
\(459\) −1.40105 + 2.42668i −0.0653953 + 0.113268i
\(460\) 0 0
\(461\) 21.3462 0.994190 0.497095 0.867696i \(-0.334400\pi\)
0.497095 + 0.867696i \(0.334400\pi\)
\(462\) 0 0
\(463\) −20.3462 −0.945567 −0.472783 0.881179i \(-0.656751\pi\)
−0.472783 + 0.881179i \(0.656751\pi\)
\(464\) 0 0
\(465\) 1.83916 3.18552i 0.0852891 0.147725i
\(466\) 0 0
\(467\) 19.0350 + 32.9695i 0.880833 + 1.52565i 0.850416 + 0.526111i \(0.176350\pi\)
0.0304171 + 0.999537i \(0.490316\pi\)
\(468\) 0 0
\(469\) −4.87623 + 2.57710i −0.225163 + 0.119000i
\(470\) 0 0
\(471\) −11.1413 19.2972i −0.513362 0.889169i
\(472\) 0 0
\(473\) 12.2773 21.2649i 0.564510 0.977760i
\(474\) 0 0
\(475\) −0.604189 −0.0277221
\(476\) 0 0
\(477\) −5.07413 −0.232329
\(478\) 0 0
\(479\) −5.88670 + 10.1961i −0.268970 + 0.465870i −0.968596 0.248639i \(-0.920017\pi\)
0.699626 + 0.714509i \(0.253350\pi\)
\(480\) 0 0
\(481\) 0.510472 + 0.884163i 0.0232755 + 0.0403144i
\(482\) 0 0
\(483\) 17.6444 + 11.0869i 0.802848 + 0.504473i
\(484\) 0 0
\(485\) −1.19791 2.07483i −0.0543941 0.0942133i
\(486\) 0 0
\(487\) −13.2010 + 22.8649i −0.598196 + 1.03611i 0.394891 + 0.918728i \(0.370782\pi\)
−0.993087 + 0.117378i \(0.962551\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) −24.3175 −1.09743 −0.548716 0.836009i \(-0.684883\pi\)
−0.548716 + 0.836009i \(0.684883\pi\)
\(492\) 0 0
\(493\) −14.2329 + 24.6521i −0.641016 + 1.11027i
\(494\) 0 0
\(495\) 2.74021 + 4.74618i 0.123163 + 0.213325i
\(496\) 0 0
\(497\) −23.6444 14.8571i −1.06060 0.666431i
\(498\) 0 0
\(499\) 11.3300 + 19.6242i 0.507203 + 0.878501i 0.999965 + 0.00833700i \(0.00265378\pi\)
−0.492763 + 0.870164i \(0.664013\pi\)
\(500\) 0 0
\(501\) 12.0227 20.8240i 0.537135 0.930346i
\(502\) 0 0
\(503\) −12.9608 −0.577895 −0.288947 0.957345i \(-0.593305\pi\)
−0.288947 + 0.957345i \(0.593305\pi\)
\(504\) 0 0
\(505\) 14.8021 0.658685
\(506\) 0 0
\(507\) 6.48042 11.2244i 0.287805 0.498494i
\(508\) 0 0
\(509\) −16.3671 28.3487i −0.725460 1.25653i −0.958785 0.284134i \(-0.908294\pi\)
0.233325 0.972399i \(-0.425039\pi\)
\(510\) 0 0
\(511\) −11.4063 + 6.02826i −0.504584 + 0.266674i
\(512\) 0 0
\(513\) 0.302094 + 0.523243i 0.0133378 + 0.0231017i
\(514\) 0 0
\(515\) 0.957697 1.65878i 0.0422012 0.0730946i
\(516\) 0 0
\(517\) 47.5608 2.09172
\(518\) 0 0
\(519\) 3.71749 0.163180
\(520\) 0 0
\(521\) −12.4238 + 21.5186i −0.544295 + 0.942747i 0.454356 + 0.890820i \(0.349869\pi\)
−0.998651 + 0.0519265i \(0.983464\pi\)
\(522\) 0 0
\(523\) 7.24021 + 12.5404i 0.316592 + 0.548354i 0.979775 0.200104i \(-0.0641280\pi\)
−0.663182 + 0.748458i \(0.730795\pi\)
\(524\) 0 0
\(525\) 0.0989528 2.64390i 0.00431866 0.115389i
\(526\) 0 0
\(527\) 5.15350 + 8.92613i 0.224490 + 0.388828i
\(528\) 0 0
\(529\) −19.5175 + 33.8053i −0.848586 + 1.46979i
\(530\) 0 0
\(531\) −6.15874 −0.267267
\(532\) 0 0
\(533\) 1.40208 0.0607310
\(534\) 0 0
\(535\) 2.07937 3.60157i 0.0898990 0.155710i
\(536\) 0 0
\(537\) 8.33916 + 14.4439i 0.359861 + 0.623298i
\(538\) 0 0
\(539\) 38.2556 + 2.86759i 1.64778 + 0.123516i
\(540\) 0 0
\(541\) 5.51748 + 9.55656i 0.237215 + 0.410869i 0.959914 0.280294i \(-0.0904320\pi\)
−0.722699 + 0.691163i \(0.757099\pi\)
\(542\) 0 0
\(543\) 4.51748 7.82451i 0.193864 0.335782i
\(544\) 0 0
\(545\) 16.6392 0.712743
\(546\) 0 0
\(547\) −13.1091 −0.560505 −0.280252 0.959926i \(-0.590418\pi\)
−0.280252 + 0.959926i \(0.590418\pi\)
\(548\) 0 0
\(549\) −0.802094 + 1.38927i −0.0342326 + 0.0592925i
\(550\) 0 0
\(551\) 3.06890 + 5.31549i 0.130739 + 0.226447i
\(552\) 0 0
\(553\) 1.11644 29.8298i 0.0474757 1.26849i
\(554\) 0 0
\(555\) 2.57937 + 4.46760i 0.109488 + 0.189639i
\(556\) 0 0
\(557\) 1.54754 2.68042i 0.0655713 0.113573i −0.831376 0.555710i \(-0.812446\pi\)
0.896947 + 0.442138i \(0.145780\pi\)
\(558\) 0 0
\(559\) 0.886700 0.0375034
\(560\) 0 0
\(561\) −15.3566 −0.648358
\(562\) 0 0
\(563\) 7.39581 12.8099i 0.311696 0.539874i −0.667033 0.745028i \(-0.732436\pi\)
0.978730 + 0.205154i \(0.0657695\pi\)
\(564\) 0 0
\(565\) 3.00000 + 5.19615i 0.126211 + 0.218604i
\(566\) 0 0
\(567\) −2.33916 + 1.23625i −0.0982355 + 0.0519178i
\(568\) 0 0
\(569\) 9.54230 + 16.5278i 0.400034 + 0.692879i 0.993730 0.111810i \(-0.0356649\pi\)
−0.593695 + 0.804690i \(0.702332\pi\)
\(570\) 0 0
\(571\) 13.3937 23.1986i 0.560509 0.970831i −0.436943 0.899489i \(-0.643939\pi\)
0.997452 0.0713413i \(-0.0227280\pi\)
\(572\) 0 0
\(573\) 23.9217 0.999342
\(574\) 0 0
\(575\) 7.87623 0.328461
\(576\) 0 0
\(577\) 6.11644 10.5940i 0.254631 0.441033i −0.710165 0.704036i \(-0.751379\pi\)
0.964795 + 0.263003i \(0.0847128\pi\)
\(578\) 0 0
\(579\) −9.31434 16.1329i −0.387091 0.670461i
\(580\) 0 0
\(581\) 26.3514 + 16.5580i 1.09324 + 0.686943i
\(582\) 0 0
\(583\) −13.9042 24.0828i −0.575852 0.997406i
\(584\) 0 0
\(585\) −0.0989528 + 0.171391i −0.00409120 + 0.00708616i
\(586\) 0 0
\(587\) −45.7629 −1.88884 −0.944419 0.328744i \(-0.893375\pi\)
−0.944419 + 0.328744i \(0.893375\pi\)
\(588\) 0 0
\(589\) 2.22240 0.0915724
\(590\) 0 0
\(591\) −1.54230 + 2.67135i −0.0634419 + 0.109885i
\(592\) 0 0
\(593\) 6.67309 + 11.5581i 0.274031 + 0.474635i 0.969890 0.243543i \(-0.0783096\pi\)
−0.695859 + 0.718178i \(0.744976\pi\)
\(594\) 0 0
\(595\) 6.27728 + 3.94436i 0.257343 + 0.161703i
\(596\) 0 0
\(597\) −10.4804 18.1526i −0.428935 0.742937i
\(598\) 0 0
\(599\) 9.48042 16.4206i 0.387359 0.670926i −0.604734 0.796427i \(-0.706721\pi\)
0.992093 + 0.125501i \(0.0400540\pi\)
\(600\) 0 0
\(601\) 27.9958 1.14197 0.570986 0.820960i \(-0.306561\pi\)
0.570986 + 0.820960i \(0.306561\pi\)
\(602\) 0 0
\(603\) 2.08461 0.0848917
\(604\) 0 0
\(605\) −9.51748 + 16.4848i −0.386941 + 0.670201i
\(606\) 0 0
\(607\) 20.1836 + 34.9589i 0.819225 + 1.41894i 0.906254 + 0.422734i \(0.138930\pi\)
−0.0870284 + 0.996206i \(0.527737\pi\)
\(608\) 0 0
\(609\) −23.7629 + 12.5588i −0.962922 + 0.508908i
\(610\) 0 0
\(611\) 0.858744 + 1.48739i 0.0347411 + 0.0601733i
\(612\) 0 0
\(613\) −10.5371 + 18.2507i −0.425588 + 0.737140i −0.996475 0.0838884i \(-0.973266\pi\)
0.570887 + 0.821029i \(0.306599\pi\)
\(614\) 0 0
\(615\) 7.08461 0.285679
\(616\) 0 0
\(617\) −18.8825 −0.760181 −0.380090 0.924949i \(-0.624107\pi\)
−0.380090 + 0.924949i \(0.624107\pi\)
\(618\) 0 0
\(619\) 14.8671 25.7506i 0.597560 1.03500i −0.395620 0.918414i \(-0.629470\pi\)
0.993180 0.116590i \(-0.0371963\pi\)
\(620\) 0 0
\(621\) −3.93811 6.82101i −0.158031 0.273718i
\(622\) 0 0
\(623\) 0.712253 19.0305i 0.0285358 0.762443i
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) −1.65560 + 2.86759i −0.0661184 + 0.114520i
\(628\) 0 0
\(629\) −14.4553 −0.576369
\(630\) 0 0
\(631\) 36.8616 1.46744 0.733718 0.679454i \(-0.237783\pi\)
0.733718 + 0.679454i \(0.237783\pi\)
\(632\) 0 0
\(633\) −3.81958 + 6.61570i −0.151815 + 0.262951i
\(634\) 0 0
\(635\) −2.29686 3.97828i −0.0911480 0.157873i
\(636\) 0 0
\(637\) 0.601053 + 1.24816i 0.0238146 + 0.0494539i
\(638\) 0 0
\(639\) 5.27728 + 9.14051i 0.208766 + 0.361593i
\(640\) 0 0
\(641\) −11.3794 + 19.7096i −0.449458 + 0.778484i −0.998351 0.0574088i \(-0.981716\pi\)
0.548893 + 0.835893i \(0.315049\pi\)
\(642\) 0 0
\(643\) 8.55875 0.337524 0.168762 0.985657i \(-0.446023\pi\)
0.168762 + 0.985657i \(0.446023\pi\)
\(644\) 0 0
\(645\) 4.48042 0.176416
\(646\) 0 0
\(647\) −17.3794 + 30.1019i −0.683253 + 1.18343i 0.290729 + 0.956805i \(0.406102\pi\)
−0.973982 + 0.226624i \(0.927231\pi\)
\(648\) 0 0
\(649\) −16.8762 29.2305i −0.662450 1.14740i
\(650\) 0 0
\(651\) −0.363980 + 9.72512i −0.0142655 + 0.381157i
\(652\) 0 0
\(653\) 23.8989 + 41.3942i 0.935238 + 1.61988i 0.774209 + 0.632930i \(0.218148\pi\)
0.161029 + 0.986950i \(0.448519\pi\)
\(654\) 0 0
\(655\) −10.4979 + 18.1829i −0.410187 + 0.710465i
\(656\) 0 0
\(657\) 4.87623 0.190240
\(658\) 0 0
\(659\) −8.79162 −0.342473 −0.171236 0.985230i \(-0.554776\pi\)
−0.171236 + 0.985230i \(0.554776\pi\)
\(660\) 0 0
\(661\) 7.23497 12.5313i 0.281408 0.487413i −0.690324 0.723500i \(-0.742532\pi\)
0.971732 + 0.236088i \(0.0758653\pi\)
\(662\) 0 0
\(663\) −0.277275 0.480255i −0.0107685 0.0186515i
\(664\) 0 0
\(665\) 1.41329 0.746931i 0.0548052 0.0289647i
\(666\) 0 0
\(667\) −40.0063 69.2929i −1.54905 2.68303i
\(668\) 0 0
\(669\) 8.15874 14.1314i 0.315435 0.546350i
\(670\) 0 0
\(671\) −8.79162 −0.339397
\(672\) 0 0
\(673\) −6.87623 −0.265059 −0.132530 0.991179i \(-0.542310\pi\)
−0.132530 + 0.991179i \(0.542310\pi\)
\(674\) 0 0
\(675\) −0.500000 + 0.866025i −0.0192450 + 0.0333333i
\(676\) 0 0
\(677\) −11.6906 20.2487i −0.449305 0.778219i 0.549036 0.835799i \(-0.314995\pi\)
−0.998341 + 0.0575796i \(0.981662\pi\)
\(678\) 0 0
\(679\) 5.36712 + 3.37245i 0.205971 + 0.129423i
\(680\) 0 0
\(681\) −9.88146 17.1152i −0.378659 0.655856i
\(682\) 0 0
\(683\) 2.59895 4.50152i 0.0994462 0.172246i −0.812009 0.583644i \(-0.801626\pi\)
0.911456 + 0.411399i \(0.134960\pi\)
\(684\) 0 0
\(685\) −9.76293 −0.373022
\(686\) 0 0
\(687\) −3.03497 −0.115791
\(688\) 0 0
\(689\) 0.502100 0.869662i 0.0191285 0.0331315i
\(690\) 0 0
\(691\) −0.923767 1.60001i −0.0351417 0.0608673i 0.847920 0.530125i \(-0.177855\pi\)
−0.883061 + 0.469258i \(0.844522\pi\)
\(692\) 0 0
\(693\) −12.2773 7.71449i −0.466375 0.293049i
\(694\) 0 0
\(695\) −10.1217 17.5313i −0.383937 0.664998i
\(696\) 0 0
\(697\) −9.92587 + 17.1921i −0.375969 + 0.651197i
\(698\) 0 0
\(699\) −16.3958 −0.620147
\(700\) 0 0
\(701\) −20.0105 −0.755785 −0.377893 0.925849i \(-0.623351\pi\)
−0.377893 + 0.925849i \(0.623351\pi\)
\(702\) 0 0
\(703\) −1.55843 + 2.69927i −0.0587771 + 0.101805i
\(704\) 0 0
\(705\) 4.33916 + 7.51565i 0.163422 + 0.283056i
\(706\) 0 0
\(707\) −34.6245 + 18.2992i −1.30219 + 0.688211i
\(708\) 0 0
\(709\) −7.19791 12.4671i −0.270323 0.468213i 0.698621 0.715491i \(-0.253797\pi\)
−0.968945 + 0.247278i \(0.920464\pi\)
\(710\) 0 0
\(711\) −5.64126 + 9.77094i −0.211564 + 0.366439i
\(712\) 0 0
\(713\) −28.9713 −1.08498
\(714\) 0 0
\(715\) −1.08461 −0.0405619
\(716\) 0 0
\(717\) 3.00000 5.19615i 0.112037 0.194054i
\(718\) 0 0
\(719\) −6.19791 10.7351i −0.231143 0.400351i 0.727002 0.686636i \(-0.240913\pi\)
−0.958145 + 0.286284i \(0.907580\pi\)
\(720\) 0 0
\(721\) −0.189534 + 5.06411i −0.00705860 + 0.188597i
\(722\) 0 0
\(723\) 2.94335 + 5.09803i 0.109464 + 0.189598i
\(724\) 0 0
\(725\) −5.07937 + 8.79773i −0.188643 + 0.326739i
\(726\) 0 0
\(727\) −39.2287 −1.45491 −0.727455 0.686155i \(-0.759297\pi\)
−0.727455 + 0.686155i \(0.759297\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.27728 + 10.8726i −0.232173 + 0.402136i
\(732\) 0 0
\(733\) −0.222725 0.385771i −0.00822653 0.0142488i 0.861883 0.507107i \(-0.169285\pi\)
−0.870109 + 0.492859i \(0.835952\pi\)
\(734\) 0 0
\(735\) 3.03707 + 6.30684i 0.112024 + 0.232631i
\(736\) 0 0
\(737\) 5.71225 + 9.89391i 0.210414 + 0.364447i
\(738\) 0 0
\(739\) 3.10419 5.37661i 0.114189 0.197782i −0.803266 0.595620i \(-0.796906\pi\)
0.917455 + 0.397839i \(0.130240\pi\)
\(740\) 0 0
\(741\) −0.119572 −0.00439260
\(742\) 0 0
\(743\) −33.9462 −1.24536 −0.622682 0.782475i \(-0.713957\pi\)
−0.622682 + 0.782475i \(0.713957\pi\)
\(744\) 0 0
\(745\) 2.00000 3.46410i 0.0732743 0.126915i
\(746\) 0 0
\(747\) −5.88146 10.1870i −0.215192 0.372723i
\(748\) 0 0
\(749\) −0.411519 + 10.9953i −0.0150366 + 0.401759i
\(750\) 0 0
\(751\) 17.6021 + 30.4877i 0.642309 + 1.11251i 0.984916 + 0.173033i \(0.0553569\pi\)
−0.342607 + 0.939479i \(0.611310\pi\)
\(752\) 0 0
\(753\) 4.93811 8.55306i 0.179955 0.311691i
\(754\) 0 0
\(755\) −7.60419 −0.276745
\(756\) 0 0
\(757\) 52.7133 1.91590 0.957949 0.286940i \(-0.0926381\pi\)
0.957949 + 0.286940i \(0.0926381\pi\)
\(758\) 0 0
\(759\) 21.5825 37.3820i 0.783395 1.35688i
\(760\) 0 0
\(761\) 16.5773 + 28.7127i 0.600926 + 1.04083i 0.992681 + 0.120764i \(0.0385345\pi\)
−0.391756 + 0.920069i \(0.628132\pi\)
\(762\) 0 0
\(763\) −38.9217 + 20.5702i −1.40906 + 0.744692i
\(764\) 0 0
\(765\) −1.40105 2.42668i −0.0506550 0.0877370i
\(766\) 0 0
\(767\) 0.609425 1.05555i 0.0220050 0.0381139i
\(768\) 0 0
\(769\) 12.4350 0.448417 0.224208 0.974541i \(-0.428020\pi\)
0.224208 + 0.974541i \(0.428020\pi\)
\(770\) 0 0
\(771\) −7.76293 −0.279575
\(772\) 0 0
\(773\) 17.3794 30.1019i 0.625092 1.08269i −0.363431 0.931621i \(-0.618395\pi\)
0.988523 0.151070i \(-0.0482720\pi\)
\(774\) 0 0
\(775\) 1.83916 + 3.18552i 0.0660646 + 0.114427i
\(776\) 0 0
\(777\) −11.5567 7.26168i −0.414593 0.260511i
\(778\) 0 0
\(779\) 2.14022 + 3.70697i 0.0766813 + 0.132816i
\(780\) 0 0
\(781\) −28.9217 + 50.0938i −1.03490 + 1.79250i
\(782\) 0 0
\(783\) 10.1587 0.363044
\(784\) 0 0
\(785\) 22.2825 0.795297
\(786\) 0 0
\(787\) −6.91539 + 11.9778i −0.246507 + 0.426963i −0.962554 0.271089i \(-0.912616\pi\)
0.716047 + 0.698052i \(0.245950\pi\)
\(788\) 0 0
\(789\) 2.80209 + 4.85337i 0.0997572 + 0.172785i
\(790\) 0 0
\(791\) −13.4413 8.44587i −0.477916 0.300301i
\(792\) 0 0
\(793\) −0.158739 0.274944i −0.00563699 0.00976355i
\(794\) 0 0
\(795\) 2.53707 4.39433i 0.0899805 0.155851i
\(796\) 0 0
\(797\) −31.6846 −1.12233 −0.561163 0.827705i \(-0.689646\pi\)
−0.561163 + 0.827705i \(0.689646\pi\)
\(798\) 0 0
\(799\) −24.3175 −0.860291
\(800\) 0 0
\(801\) −3.59895 + 6.23357i −0.127163 + 0.220252i
\(802\) 0 0
\(803\) 13.3619 + 23.1435i 0.471531 + 0.816715i
\(804\) 0 0
\(805\) −18.4238 + 9.73702i −0.649352 + 0.343185i
\(806\) 0 0
\(807\) −10.8762 18.8382i −0.382861 0.663135i
\(808\) 0 0
\(809\) 10.5084 18.2010i 0.369455 0.639914i −0.620026 0.784582i \(-0.712878\pi\)
0.989480 + 0.144667i \(0.0462112\pi\)
\(810\) 0 0
\(811\) −39.3916 −1.38323 −0.691613 0.722268i \(-0.743100\pi\)
−0.691613 + 0.722268i \(0.743100\pi\)
\(812\) 0 0
\(813\) −20.1692 −0.707365
\(814\) 0 0
\(815\) −8.00000 + 13.8564i −0.280228 + 0.485369i
\(816\) 0 0
\(817\) 1.35351 + 2.34435i 0.0473533 + 0.0820183i
\(818\) 0 0
\(819\) 0.0195833 0.523243i 0.000684297 0.0182836i
\(820\) 0 0
\(821\) 23.4360 + 40.5924i 0.817923 + 1.41668i 0.907210 + 0.420678i \(0.138208\pi\)
−0.0892877 + 0.996006i \(0.528459\pi\)
\(822\) 0 0
\(823\) 13.6783 23.6915i 0.476796 0.825835i −0.522850 0.852425i \(-0.675131\pi\)
0.999646 + 0.0265892i \(0.00846459\pi\)
\(824\) 0 0
\(825\) −5.48042 −0.190804
\(826\) 0 0
\(827\) 54.3175 1.88880 0.944402 0.328793i \(-0.106642\pi\)
0.944402 + 0.328793i \(0.106642\pi\)
\(828\) 0 0
\(829\) −11.0084 + 19.0671i −0.382337 + 0.662226i −0.991396 0.130898i \(-0.958214\pi\)
0.609059 + 0.793125i \(0.291547\pi\)
\(830\) 0 0
\(831\) 8.11644 + 14.0581i 0.281556 + 0.487669i
\(832\) 0 0
\(833\) −19.5598 1.46618i −0.677706 0.0508000i
\(834\) 0 0
\(835\) 12.0227 + 20.8240i 0.416063 + 0.720643i
\(836\) 0 0
\(837\) 1.83916 3.18552i 0.0635707 0.110108i
\(838\) 0 0
\(839\) −38.1587 −1.31739 −0.658693 0.752412i \(-0.728890\pi\)
−0.658693 + 0.752412i \(0.728890\pi\)
\(840\) 0 0
\(841\) 74.2000 2.55862
\(842\) 0 0
\(843\) 7.33916 12.7118i 0.252774 0.437818i
\(844\) 0 0
\(845\) 6.48042 + 11.2244i 0.222933 + 0.386131i
\(846\) 0 0
\(847\) 1.88356 50.3266i 0.0647200 1.72924i
\(848\) 0 0
\(849\) 2.03183 + 3.51923i 0.0697323 + 0.120780i
\(850\) 0 0
\(851\) 20.3157 35.1878i 0.696413 1.20622i
\(852\) 0 0
\(853\) −49.7937 −1.70490 −0.852452 0.522806i \(-0.824885\pi\)
−0.852452 + 0.522806i \(0.824885\pi\)
\(854\) 0 0
\(855\) −0.604189 −0.0206628
\(856\) 0 0
\(857\) −5.26680 + 9.12237i −0.179911 + 0.311614i −0.941850 0.336034i \(-0.890914\pi\)
0.761939 + 0.647649i \(0.224248\pi\)
\(858\) 0 0
\(859\) −21.6000 37.4123i −0.736982 1.27649i −0.953848 0.300289i \(-0.902917\pi\)
0.216866 0.976201i \(-0.430416\pi\)
\(860\) 0 0
\(861\) −16.5720 + 8.75837i −0.564773 + 0.298485i
\(862\) 0 0
\(863\) −9.00701 15.6006i −0.306602 0.531051i 0.671015 0.741444i \(-0.265859\pi\)
−0.977617 + 0.210394i \(0.932525\pi\)
\(864\) 0 0
\(865\) −1.85874 + 3.21944i −0.0631992 + 0.109464i
\(866\) 0 0
\(867\) −9.14827 −0.310691
\(868\) 0 0
\(869\) −61.8329 −2.09754
\(870\) 0 0
\(871\) −0.206278 + 0.357283i −0.00698945 + 0.0121061i
\(872\) 0 0
\(873\) −1.19791 2.07483i −0.0405430 0.0702225i
\(874\) 0 0
\(875\) 2.24021 + 1.40765i 0.0757329 + 0.0475871i
\(876\) 0 0
\(877\) −12.9825 22.4864i −0.438388 0.759311i 0.559177 0.829048i \(-0.311117\pi\)
−0.997565 + 0.0697373i \(0.977784\pi\)
\(878\) 0 0
\(879\) −1.33916 + 2.31950i −0.0451688 + 0.0782346i
\(880\) 0 0
\(881\) −25.0951 −0.845475 −0.422737 0.906252i \(-0.638931\pi\)
−0.422737 + 0.906252i \(0.638931\pi\)
\(882\) 0 0
\(883\) −50.9671 −1.71518 −0.857590 0.514334i \(-0.828039\pi\)
−0.857590 + 0.514334i \(0.828039\pi\)
\(884\) 0 0
\(885\) 3.07937 5.33362i 0.103512 0.179288i
\(886\) 0 0
\(887\) −12.0402 20.8542i −0.404270 0.700217i 0.589966 0.807428i \(-0.299141\pi\)
−0.994236 + 0.107211i \(0.965808\pi\)
\(888\) 0 0
\(889\) 10.2909 + 6.46633i 0.345145 + 0.216874i
\(890\) 0 0
\(891\) 2.74021 + 4.74618i 0.0918004 + 0.159003i
\(892\) 0 0
\(893\) −2.62167 + 4.54087i −0.0877309 + 0.151954i
\(894\) 0 0
\(895\) −16.6783 −0.557495
\(896\) 0 0
\(897\) 1.55875 0.0520451
\(898\) 0 0
\(899\) 18.6836 32.3609i 0.623132 1.07930i
\(900\) 0 0
\(901\) 7.10910 + 12.3133i 0.236838 + 0.410216i
\(902\) 0 0
\(903\) −10.4804 + 5.53894i −0.348766 + 0.184324i
\(904\) 0 0
\(905\) 4.51748 + 7.82451i 0.150166 + 0.260096i
\(906\) 0 0
\(907\) 10.1269 17.5403i 0.336258 0.582417i −0.647467 0.762093i \(-0.724172\pi\)
0.983726 + 0.179676i \(0.0575051\pi\)
\(908\) 0 0
\(909\) 14.8021 0.490954
\(910\) 0 0
\(911\) 38.7238 1.28298 0.641488 0.767133i \(-0.278318\pi\)
0.641488 + 0.767133i \(0.278318\pi\)
\(912\) 0 0
\(913\) 32.2329 55.8290i 1.06675 1.84767i
\(914\) 0 0
\(915\) −0.802094 1.38927i −0.0265164 0.0459278i
\(916\) 0 0
\(917\) 2.07759 55.5108i 0.0686082 1.83313i
\(918\) 0 0
\(919\) 8.68042 + 15.0349i 0.286341 + 0.495957i 0.972933 0.231086i \(-0.0742277\pi\)
−0.686593 + 0.727042i \(0.740894\pi\)
\(920\) 0 0
\(921\) −15.5969 + 27.0145i −0.513934 + 0.890159i
\(922\) 0 0
\(923\) −2.08881 −0.0687539
\(924\) 0 0
\(925\) −5.15874 −0.169618
\(926\) 0 0
\(927\) 0.957697 1.65878i 0.0314549 0.0544815i
\(928\) 0 0
\(929\) 16.0969 + 27.8806i 0.528121 + 0.914732i 0.999463 + 0.0327812i \(0.0104364\pi\)
−0.471342 + 0.881951i \(0.656230\pi\)
\(930\) 0 0
\(931\) −2.38253 + 3.49438i −0.0780842 + 0.114524i
\(932\) 0 0
\(933\) −0.192670 0.333714i −0.00630772 0.0109253i
\(934\) 0 0
\(935\) 7.67832 13.2992i 0.251108 0.434932i
\(936\) 0 0
\(937\) −14.7070 −0.480457 −0.240229 0.970716i \(-0.577222\pi\)
−0.240229 + 0.970716i \(0.577222\pi\)
\(938\) 0 0
\(939\) 8.48042 0.276748
\(940\) 0 0
\(941\) −14.9661 + 25.9220i −0.487880 + 0.845033i −0.999903 0.0139390i \(-0.995563\pi\)
0.512023 + 0.858972i \(0.328896\pi\)
\(942\) 0 0
\(943\) −27.9000 48.3242i −0.908548 1.57365i
\(944\) 0 0
\(945\) 0.0989528 2.64390i 0.00321894 0.0860061i
\(946\) 0 0
\(947\) −7.91643 13.7117i −0.257249 0.445569i 0.708255 0.705957i \(-0.249483\pi\)
−0.965504 + 0.260388i \(0.916150\pi\)
\(948\) 0 0
\(949\) −0.482517 + 0.835743i −0.0156631 + 0.0271294i
\(950\) 0 0
\(951\) 26.8720 0.871385
\(952\) 0 0
\(953\) −9.90072 −0.320716 −0.160358 0.987059i \(-0.551265\pi\)
−0.160358 + 0.987059i \(0.551265\pi\)
\(954\) 0 0
\(955\) −11.9608 + 20.7168i −0.387043 + 0.670379i
\(956\) 0 0
\(957\) 27.8371 + 48.2152i 0.899844 + 1.55858i
\(958\) 0 0
\(959\) 22.8371 12.0695i 0.737447 0.389743i
\(960\) 0 0
\(961\) 8.73497 + 15.1294i 0.281773 + 0.488046i
\(962\) 0 0
\(963\) 2.07937 3.60157i 0.0670067 0.116059i
\(964\) 0 0
\(965\) 18.6287 0.599679
\(966\) 0 0
\(967\) −4.48042 −0.144080 −0.0720402 0.997402i \(-0.522951\pi\)
−0.0720402 + 0.997402i \(0.522951\pi\)
\(968\) 0 0
\(969\) 0.846497 1.46618i 0.0271934 0.0471003i
\(970\) 0 0
\(971\) −10.0175 17.3508i −0.321476 0.556813i 0.659317 0.751865i \(-0.270846\pi\)
−0.980793 + 0.195052i \(0.937512\pi\)
\(972\) 0 0
\(973\) 45.3493 + 28.4955i 1.45383 + 0.913522i
\(974\) 0 0
\(975\) −0.0989528 0.171391i −0.00316903 0.00548891i
\(976\) 0 0
\(977\) −5.21361 + 9.03024i −0.166798 + 0.288903i −0.937292 0.348544i \(-0.886676\pi\)
0.770494 + 0.637447i \(0.220010\pi\)
\(978\) 0 0
\(979\) −39.4475 −1.26075
\(980\) 0 0
\(981\) 16.6392 0.531247
\(982\) 0 0
\(983\) 3.29476 5.70669i 0.105086 0.182015i −0.808687 0.588239i \(-0.799821\pi\)
0.913774 + 0.406224i \(0.133155\pi\)
\(984\) 0 0
\(985\) −1.54230 2.67135i −0.0491419 0.0851162i
\(986\) 0 0
\(987\) −19.4413 12.2160i −0.618822 0.388840i
\(988\) 0 0
\(989\) −17.6444 30.5610i −0.561059 0.971783i
\(990\) 0 0
\(991\) −17.6021 + 30.4877i −0.559149 + 0.968474i 0.438419 + 0.898771i \(0.355539\pi\)
−0.997568 + 0.0697034i \(0.977795\pi\)
\(992\) 0 0
\(993\) −9.16921 −0.290976
\(994\) 0 0
\(995\) 20.9608 0.664503
\(996\) 0 0
\(997\) −28.1164 + 48.6991i −0.890456 + 1.54232i −0.0511274 + 0.998692i \(0.516281\pi\)
−0.839329 + 0.543624i \(0.817052\pi\)
\(998\) 0 0
\(999\) 2.57937 + 4.46760i 0.0816076 + 0.141349i
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 840.2.bg.i.361.3 yes 6
3.2 odd 2 2520.2.bi.o.361.3 6
4.3 odd 2 1680.2.bg.u.1201.1 6
7.2 even 3 inner 840.2.bg.i.121.3 6
7.3 odd 6 5880.2.a.bt.1.1 3
7.4 even 3 5880.2.a.bw.1.1 3
21.2 odd 6 2520.2.bi.o.1801.3 6
28.23 odd 6 1680.2.bg.u.961.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.bg.i.121.3 6 7.2 even 3 inner
840.2.bg.i.361.3 yes 6 1.1 even 1 trivial
1680.2.bg.u.961.1 6 28.23 odd 6
1680.2.bg.u.1201.1 6 4.3 odd 2
2520.2.bi.o.361.3 6 3.2 odd 2
2520.2.bi.o.1801.3 6 21.2 odd 6
5880.2.a.bt.1.1 3 7.3 odd 6
5880.2.a.bw.1.1 3 7.4 even 3