Properties

Label 768.3.l.d.703.2
Level $768$
Weight $3$
Character 768.703
Analytic conductor $20.926$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,3,Mod(319,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.319");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 768.l (of order \(4\), 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: \(20.9264843029\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 703.2
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 768.703
Dual form 768.3.l.d.319.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.22474 + 1.22474i) q^{3} +(2.00000 - 2.00000i) q^{5} +0.378937 q^{7} -3.00000i q^{9} +O(q^{10})\) \(q+(-1.22474 + 1.22474i) q^{3} +(2.00000 - 2.00000i) q^{5} +0.378937 q^{7} -3.00000i q^{9} +(10.5558 + 10.5558i) q^{11} +(-6.46410 - 6.46410i) q^{13} +4.89898i q^{15} -11.8564 q^{17} +(18.6622 - 18.6622i) q^{19} +(-0.464102 + 0.464102i) q^{21} +12.0716 q^{23} +17.0000i q^{25} +(3.67423 + 3.67423i) q^{27} +(-1.07180 - 1.07180i) q^{29} +6.38752i q^{31} -25.8564 q^{33} +(0.757875 - 0.757875i) q^{35} +(35.3923 - 35.3923i) q^{37} +15.8338 q^{39} -3.56922i q^{41} +(29.9759 + 29.9759i) q^{43} +(-6.00000 - 6.00000i) q^{45} -31.6675i q^{47} -48.8564 q^{49} +(14.5211 - 14.5211i) q^{51} +(68.7846 - 68.7846i) q^{53} +42.2233 q^{55} +45.7128i q^{57} +(27.8781 + 27.8781i) q^{59} +(62.8179 + 62.8179i) q^{61} -1.13681i q^{63} -25.8564 q^{65} +(30.5579 - 30.5579i) q^{67} +(-14.7846 + 14.7846i) q^{69} +131.975 q^{71} +50.1436i q^{73} +(-20.8207 - 20.8207i) q^{75} +(4.00000 + 4.00000i) q^{77} +114.219i q^{79} -9.00000 q^{81} +(1.92189 - 1.92189i) q^{83} +(-23.7128 + 23.7128i) q^{85} +2.62536 q^{87} +74.0000i q^{89} +(-2.44949 - 2.44949i) q^{91} +(-7.82309 - 7.82309i) q^{93} -74.6487i q^{95} -89.2820 q^{97} +(31.6675 - 31.6675i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{5} - 24 q^{13} + 16 q^{17} + 24 q^{21} - 64 q^{29} - 96 q^{33} + 200 q^{37} - 48 q^{45} - 280 q^{49} + 384 q^{53} - 24 q^{61} - 96 q^{65} + 48 q^{69} + 32 q^{77} - 72 q^{81} + 32 q^{85} - 312 q^{93} - 160 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.22474 + 1.22474i −0.408248 + 0.408248i
\(4\) 0 0
\(5\) 2.00000 2.00000i 0.400000 0.400000i −0.478233 0.878233i \(-0.658723\pi\)
0.878233 + 0.478233i \(0.158723\pi\)
\(6\) 0 0
\(7\) 0.378937 0.0541339 0.0270670 0.999634i \(-0.491383\pi\)
0.0270670 + 0.999634i \(0.491383\pi\)
\(8\) 0 0
\(9\) 3.00000i 0.333333i
\(10\) 0 0
\(11\) 10.5558 + 10.5558i 0.959621 + 0.959621i 0.999216 0.0395946i \(-0.0126066\pi\)
−0.0395946 + 0.999216i \(0.512607\pi\)
\(12\) 0 0
\(13\) −6.46410 6.46410i −0.497239 0.497239i 0.413339 0.910577i \(-0.364363\pi\)
−0.910577 + 0.413339i \(0.864363\pi\)
\(14\) 0 0
\(15\) 4.89898i 0.326599i
\(16\) 0 0
\(17\) −11.8564 −0.697436 −0.348718 0.937228i \(-0.613383\pi\)
−0.348718 + 0.937228i \(0.613383\pi\)
\(18\) 0 0
\(19\) 18.6622 18.6622i 0.982220 0.982220i −0.0176248 0.999845i \(-0.505610\pi\)
0.999845 + 0.0176248i \(0.00561044\pi\)
\(20\) 0 0
\(21\) −0.464102 + 0.464102i −0.0221001 + 0.0221001i
\(22\) 0 0
\(23\) 12.0716 0.524851 0.262426 0.964952i \(-0.415478\pi\)
0.262426 + 0.964952i \(0.415478\pi\)
\(24\) 0 0
\(25\) 17.0000i 0.680000i
\(26\) 0 0
\(27\) 3.67423 + 3.67423i 0.136083 + 0.136083i
\(28\) 0 0
\(29\) −1.07180 1.07180i −0.0369585 0.0369585i 0.688386 0.725345i \(-0.258320\pi\)
−0.725345 + 0.688386i \(0.758320\pi\)
\(30\) 0 0
\(31\) 6.38752i 0.206049i 0.994679 + 0.103025i \(0.0328520\pi\)
−0.994679 + 0.103025i \(0.967148\pi\)
\(32\) 0 0
\(33\) −25.8564 −0.783527
\(34\) 0 0
\(35\) 0.757875 0.757875i 0.0216536 0.0216536i
\(36\) 0 0
\(37\) 35.3923 35.3923i 0.956549 0.956549i −0.0425457 0.999095i \(-0.513547\pi\)
0.999095 + 0.0425457i \(0.0135468\pi\)
\(38\) 0 0
\(39\) 15.8338 0.405994
\(40\) 0 0
\(41\) 3.56922i 0.0870541i −0.999052 0.0435271i \(-0.986141\pi\)
0.999052 0.0435271i \(-0.0138595\pi\)
\(42\) 0 0
\(43\) 29.9759 + 29.9759i 0.697114 + 0.697114i 0.963787 0.266673i \(-0.0859245\pi\)
−0.266673 + 0.963787i \(0.585924\pi\)
\(44\) 0 0
\(45\) −6.00000 6.00000i −0.133333 0.133333i
\(46\) 0 0
\(47\) 31.6675i 0.673777i −0.941545 0.336888i \(-0.890626\pi\)
0.941545 0.336888i \(-0.109374\pi\)
\(48\) 0 0
\(49\) −48.8564 −0.997070
\(50\) 0 0
\(51\) 14.5211 14.5211i 0.284727 0.284727i
\(52\) 0 0
\(53\) 68.7846 68.7846i 1.29782 1.29782i 0.367995 0.929828i \(-0.380044\pi\)
0.929828 0.367995i \(-0.119956\pi\)
\(54\) 0 0
\(55\) 42.2233 0.767697
\(56\) 0 0
\(57\) 45.7128i 0.801979i
\(58\) 0 0
\(59\) 27.8781 + 27.8781i 0.472511 + 0.472511i 0.902726 0.430216i \(-0.141562\pi\)
−0.430216 + 0.902726i \(0.641562\pi\)
\(60\) 0 0
\(61\) 62.8179 + 62.8179i 1.02980 + 1.02980i 0.999542 + 0.0302601i \(0.00963355\pi\)
0.0302601 + 0.999542i \(0.490366\pi\)
\(62\) 0 0
\(63\) 1.13681i 0.0180446i
\(64\) 0 0
\(65\) −25.8564 −0.397791
\(66\) 0 0
\(67\) 30.5579 30.5579i 0.456088 0.456088i −0.441281 0.897369i \(-0.645476\pi\)
0.897369 + 0.441281i \(0.145476\pi\)
\(68\) 0 0
\(69\) −14.7846 + 14.7846i −0.214270 + 0.214270i
\(70\) 0 0
\(71\) 131.975 1.85880 0.929402 0.369068i \(-0.120323\pi\)
0.929402 + 0.369068i \(0.120323\pi\)
\(72\) 0 0
\(73\) 50.1436i 0.686899i 0.939171 + 0.343449i \(0.111595\pi\)
−0.939171 + 0.343449i \(0.888405\pi\)
\(74\) 0 0
\(75\) −20.8207 20.8207i −0.277609 0.277609i
\(76\) 0 0
\(77\) 4.00000 + 4.00000i 0.0519481 + 0.0519481i
\(78\) 0 0
\(79\) 114.219i 1.44582i 0.690944 + 0.722908i \(0.257195\pi\)
−0.690944 + 0.722908i \(0.742805\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) 0 0
\(83\) 1.92189 1.92189i 0.0231553 0.0231553i −0.695434 0.718590i \(-0.744788\pi\)
0.718590 + 0.695434i \(0.244788\pi\)
\(84\) 0 0
\(85\) −23.7128 + 23.7128i −0.278974 + 0.278974i
\(86\) 0 0
\(87\) 2.62536 0.0301765
\(88\) 0 0
\(89\) 74.0000i 0.831461i 0.909488 + 0.415730i \(0.136474\pi\)
−0.909488 + 0.415730i \(0.863526\pi\)
\(90\) 0 0
\(91\) −2.44949 2.44949i −0.0269175 0.0269175i
\(92\) 0 0
\(93\) −7.82309 7.82309i −0.0841192 0.0841192i
\(94\) 0 0
\(95\) 74.6487i 0.785776i
\(96\) 0 0
\(97\) −89.2820 −0.920433 −0.460217 0.887807i \(-0.652228\pi\)
−0.460217 + 0.887807i \(0.652228\pi\)
\(98\) 0 0
\(99\) 31.6675 31.6675i 0.319874 0.319874i
\(100\) 0 0
\(101\) 123.923 123.923i 1.22696 1.22696i 0.261853 0.965108i \(-0.415666\pi\)
0.965108 0.261853i \(-0.0843336\pi\)
\(102\) 0 0
\(103\) 96.1393 0.933391 0.466696 0.884418i \(-0.345444\pi\)
0.466696 + 0.884418i \(0.345444\pi\)
\(104\) 0 0
\(105\) 1.85641i 0.0176801i
\(106\) 0 0
\(107\) 115.708 + 115.708i 1.08138 + 1.08138i 0.996381 + 0.0850027i \(0.0270899\pi\)
0.0850027 + 0.996381i \(0.472910\pi\)
\(108\) 0 0
\(109\) −81.2487 81.2487i −0.745401 0.745401i 0.228211 0.973612i \(-0.426712\pi\)
−0.973612 + 0.228211i \(0.926712\pi\)
\(110\) 0 0
\(111\) 86.6931i 0.781019i
\(112\) 0 0
\(113\) 164.851 1.45886 0.729430 0.684055i \(-0.239785\pi\)
0.729430 + 0.684055i \(0.239785\pi\)
\(114\) 0 0
\(115\) 24.1432 24.1432i 0.209941 0.209941i
\(116\) 0 0
\(117\) −19.3923 + 19.3923i −0.165746 + 0.165746i
\(118\) 0 0
\(119\) −4.49284 −0.0377549
\(120\) 0 0
\(121\) 101.851i 0.841746i
\(122\) 0 0
\(123\) 4.37138 + 4.37138i 0.0355397 + 0.0355397i
\(124\) 0 0
\(125\) 84.0000 + 84.0000i 0.672000 + 0.672000i
\(126\) 0 0
\(127\) 66.7454i 0.525555i −0.964857 0.262777i \(-0.915362\pi\)
0.964857 0.262777i \(-0.0846385\pi\)
\(128\) 0 0
\(129\) −73.4256 −0.569191
\(130\) 0 0
\(131\) −136.116 + 136.116i −1.03906 + 1.03906i −0.0398495 + 0.999206i \(0.512688\pi\)
−0.999206 + 0.0398495i \(0.987312\pi\)
\(132\) 0 0
\(133\) 7.07180 7.07180i 0.0531714 0.0531714i
\(134\) 0 0
\(135\) 14.6969 0.108866
\(136\) 0 0
\(137\) 225.559i 1.64642i −0.567740 0.823208i \(-0.692182\pi\)
0.567740 0.823208i \(-0.307818\pi\)
\(138\) 0 0
\(139\) −41.0593 41.0593i −0.295391 0.295391i 0.543815 0.839205i \(-0.316979\pi\)
−0.839205 + 0.543815i \(0.816979\pi\)
\(140\) 0 0
\(141\) 38.7846 + 38.7846i 0.275068 + 0.275068i
\(142\) 0 0
\(143\) 136.468i 0.954321i
\(144\) 0 0
\(145\) −4.28719 −0.0295668
\(146\) 0 0
\(147\) 59.8366 59.8366i 0.407052 0.407052i
\(148\) 0 0
\(149\) −31.8564 + 31.8564i −0.213801 + 0.213801i −0.805880 0.592079i \(-0.798307\pi\)
0.592079 + 0.805880i \(0.298307\pi\)
\(150\) 0 0
\(151\) −27.5536 −0.182474 −0.0912371 0.995829i \(-0.529082\pi\)
−0.0912371 + 0.995829i \(0.529082\pi\)
\(152\) 0 0
\(153\) 35.5692i 0.232479i
\(154\) 0 0
\(155\) 12.7750 + 12.7750i 0.0824196 + 0.0824196i
\(156\) 0 0
\(157\) −65.5359 65.5359i −0.417426 0.417426i 0.466890 0.884316i \(-0.345375\pi\)
−0.884316 + 0.466890i \(0.845375\pi\)
\(158\) 0 0
\(159\) 168.487i 1.05967i
\(160\) 0 0
\(161\) 4.57437 0.0284123
\(162\) 0 0
\(163\) 80.8332 80.8332i 0.495909 0.495909i −0.414253 0.910162i \(-0.635957\pi\)
0.910162 + 0.414253i \(0.135957\pi\)
\(164\) 0 0
\(165\) −51.7128 + 51.7128i −0.313411 + 0.313411i
\(166\) 0 0
\(167\) −252.528 −1.51214 −0.756071 0.654490i \(-0.772884\pi\)
−0.756071 + 0.654490i \(0.772884\pi\)
\(168\) 0 0
\(169\) 85.4308i 0.505508i
\(170\) 0 0
\(171\) −55.9865 55.9865i −0.327407 0.327407i
\(172\) 0 0
\(173\) −90.4205 90.4205i −0.522662 0.522662i 0.395713 0.918374i \(-0.370498\pi\)
−0.918374 + 0.395713i \(0.870498\pi\)
\(174\) 0 0
\(175\) 6.44194i 0.0368111i
\(176\) 0 0
\(177\) −68.2872 −0.385803
\(178\) 0 0
\(179\) −9.09450 + 9.09450i −0.0508072 + 0.0508072i −0.732054 0.681247i \(-0.761438\pi\)
0.681247 + 0.732054i \(0.261438\pi\)
\(180\) 0 0
\(181\) −214.310 + 214.310i −1.18403 + 1.18403i −0.205344 + 0.978690i \(0.565831\pi\)
−0.978690 + 0.205344i \(0.934169\pi\)
\(182\) 0 0
\(183\) −153.872 −0.840830
\(184\) 0 0
\(185\) 141.569i 0.765239i
\(186\) 0 0
\(187\) −125.154 125.154i −0.669274 0.669274i
\(188\) 0 0
\(189\) 1.39230 + 1.39230i 0.00736669 + 0.00736669i
\(190\) 0 0
\(191\) 91.1043i 0.476986i −0.971144 0.238493i \(-0.923347\pi\)
0.971144 0.238493i \(-0.0766534\pi\)
\(192\) 0 0
\(193\) −191.990 −0.994765 −0.497383 0.867531i \(-0.665705\pi\)
−0.497383 + 0.867531i \(0.665705\pi\)
\(194\) 0 0
\(195\) 31.6675 31.6675i 0.162397 0.162397i
\(196\) 0 0
\(197\) 252.631 252.631i 1.28239 1.28239i 0.343085 0.939304i \(-0.388528\pi\)
0.939304 0.343085i \(-0.111472\pi\)
\(198\) 0 0
\(199\) −268.876 −1.35114 −0.675569 0.737297i \(-0.736102\pi\)
−0.675569 + 0.737297i \(0.736102\pi\)
\(200\) 0 0
\(201\) 74.8513i 0.372394i
\(202\) 0 0
\(203\) −0.406144 0.406144i −0.00200071 0.00200071i
\(204\) 0 0
\(205\) −7.13844 7.13844i −0.0348217 0.0348217i
\(206\) 0 0
\(207\) 36.2147i 0.174950i
\(208\) 0 0
\(209\) 393.990 1.88512
\(210\) 0 0
\(211\) −140.663 + 140.663i −0.666652 + 0.666652i −0.956939 0.290288i \(-0.906249\pi\)
0.290288 + 0.956939i \(0.406249\pi\)
\(212\) 0 0
\(213\) −161.636 + 161.636i −0.758854 + 0.758854i
\(214\) 0 0
\(215\) 119.904 0.557691
\(216\) 0 0
\(217\) 2.42047i 0.0111542i
\(218\) 0 0
\(219\) −61.4131 61.4131i −0.280425 0.280425i
\(220\) 0 0
\(221\) 76.6410 + 76.6410i 0.346792 + 0.346792i
\(222\) 0 0
\(223\) 135.331i 0.606866i 0.952853 + 0.303433i \(0.0981329\pi\)
−0.952853 + 0.303433i \(0.901867\pi\)
\(224\) 0 0
\(225\) 51.0000 0.226667
\(226\) 0 0
\(227\) −219.102 + 219.102i −0.965205 + 0.965205i −0.999415 0.0342095i \(-0.989109\pi\)
0.0342095 + 0.999415i \(0.489109\pi\)
\(228\) 0 0
\(229\) −160.895 + 160.895i −0.702598 + 0.702598i −0.964967 0.262370i \(-0.915496\pi\)
0.262370 + 0.964967i \(0.415496\pi\)
\(230\) 0 0
\(231\) −9.79796 −0.0424154
\(232\) 0 0
\(233\) 231.990i 0.995664i −0.867274 0.497832i \(-0.834130\pi\)
0.867274 0.497832i \(-0.165870\pi\)
\(234\) 0 0
\(235\) −63.3350 63.3350i −0.269511 0.269511i
\(236\) 0 0
\(237\) −139.890 139.890i −0.590252 0.590252i
\(238\) 0 0
\(239\) 454.550i 1.90188i −0.309372 0.950941i \(-0.600119\pi\)
0.309372 0.950941i \(-0.399881\pi\)
\(240\) 0 0
\(241\) 330.554 1.37159 0.685796 0.727794i \(-0.259454\pi\)
0.685796 + 0.727794i \(0.259454\pi\)
\(242\) 0 0
\(243\) 11.0227 11.0227i 0.0453609 0.0453609i
\(244\) 0 0
\(245\) −97.7128 + 97.7128i −0.398828 + 0.398828i
\(246\) 0 0
\(247\) −241.268 −0.976795
\(248\) 0 0
\(249\) 4.70766i 0.0189063i
\(250\) 0 0
\(251\) −24.9010 24.9010i −0.0992073 0.0992073i 0.655761 0.754968i \(-0.272348\pi\)
−0.754968 + 0.655761i \(0.772348\pi\)
\(252\) 0 0
\(253\) 127.426 + 127.426i 0.503659 + 0.503659i
\(254\) 0 0
\(255\) 58.0843i 0.227782i
\(256\) 0 0
\(257\) 277.138 1.07836 0.539180 0.842191i \(-0.318734\pi\)
0.539180 + 0.842191i \(0.318734\pi\)
\(258\) 0 0
\(259\) 13.4115 13.4115i 0.0517817 0.0517817i
\(260\) 0 0
\(261\) −3.21539 + 3.21539i −0.0123195 + 0.0123195i
\(262\) 0 0
\(263\) −119.200 −0.453232 −0.226616 0.973984i \(-0.572766\pi\)
−0.226616 + 0.973984i \(0.572766\pi\)
\(264\) 0 0
\(265\) 275.138i 1.03826i
\(266\) 0 0
\(267\) −90.6311 90.6311i −0.339442 0.339442i
\(268\) 0 0
\(269\) −151.061 151.061i −0.561567 0.561567i 0.368185 0.929752i \(-0.379979\pi\)
−0.929752 + 0.368185i \(0.879979\pi\)
\(270\) 0 0
\(271\) 430.137i 1.58722i 0.608427 + 0.793610i \(0.291801\pi\)
−0.608427 + 0.793610i \(0.708199\pi\)
\(272\) 0 0
\(273\) 6.00000 0.0219780
\(274\) 0 0
\(275\) −179.449 + 179.449i −0.652542 + 0.652542i
\(276\) 0 0
\(277\) −213.249 + 213.249i −0.769851 + 0.769851i −0.978080 0.208229i \(-0.933230\pi\)
0.208229 + 0.978080i \(0.433230\pi\)
\(278\) 0 0
\(279\) 19.1626 0.0686830
\(280\) 0 0
\(281\) 78.0000i 0.277580i −0.990322 0.138790i \(-0.955679\pi\)
0.990322 0.138790i \(-0.0443213\pi\)
\(282\) 0 0
\(283\) −41.7628 41.7628i −0.147572 0.147572i 0.629461 0.777032i \(-0.283276\pi\)
−0.777032 + 0.629461i \(0.783276\pi\)
\(284\) 0 0
\(285\) 91.4256 + 91.4256i 0.320792 + 0.320792i
\(286\) 0 0
\(287\) 1.35251i 0.00471258i
\(288\) 0 0
\(289\) −148.426 −0.513583
\(290\) 0 0
\(291\) 109.348 109.348i 0.375765 0.375765i
\(292\) 0 0
\(293\) −141.933 + 141.933i −0.484414 + 0.484414i −0.906538 0.422124i \(-0.861285\pi\)
0.422124 + 0.906538i \(0.361285\pi\)
\(294\) 0 0
\(295\) 111.513 0.378009
\(296\) 0 0
\(297\) 77.5692i 0.261176i
\(298\) 0 0
\(299\) −78.0319 78.0319i −0.260976 0.260976i
\(300\) 0 0
\(301\) 11.3590 + 11.3590i 0.0377375 + 0.0377375i
\(302\) 0 0
\(303\) 303.548i 1.00181i
\(304\) 0 0
\(305\) 251.272 0.823842
\(306\) 0 0
\(307\) −16.3215 + 16.3215i −0.0531645 + 0.0531645i −0.733189 0.680025i \(-0.761969\pi\)
0.680025 + 0.733189i \(0.261969\pi\)
\(308\) 0 0
\(309\) −117.746 + 117.746i −0.381055 + 0.381055i
\(310\) 0 0
\(311\) −111.404 −0.358211 −0.179106 0.983830i \(-0.557320\pi\)
−0.179106 + 0.983830i \(0.557320\pi\)
\(312\) 0 0
\(313\) 502.267i 1.60469i 0.596864 + 0.802343i \(0.296413\pi\)
−0.596864 + 0.802343i \(0.703587\pi\)
\(314\) 0 0
\(315\) −2.27362 2.27362i −0.00721785 0.00721785i
\(316\) 0 0
\(317\) 140.785 + 140.785i 0.444115 + 0.444115i 0.893393 0.449277i \(-0.148318\pi\)
−0.449277 + 0.893393i \(0.648318\pi\)
\(318\) 0 0
\(319\) 22.6274i 0.0709323i
\(320\) 0 0
\(321\) −283.426 −0.882946
\(322\) 0 0
\(323\) −221.266 + 221.266i −0.685035 + 0.685035i
\(324\) 0 0
\(325\) 109.890 109.890i 0.338122 0.338122i
\(326\) 0 0
\(327\) 199.018 0.608617
\(328\) 0 0
\(329\) 12.0000i 0.0364742i
\(330\) 0 0
\(331\) −302.682 302.682i −0.914446 0.914446i 0.0821724 0.996618i \(-0.473814\pi\)
−0.996618 + 0.0821724i \(0.973814\pi\)
\(332\) 0 0
\(333\) −106.177 106.177i −0.318850 0.318850i
\(334\) 0 0
\(335\) 122.232i 0.364870i
\(336\) 0 0
\(337\) 435.559 1.29246 0.646230 0.763143i \(-0.276345\pi\)
0.646230 + 0.763143i \(0.276345\pi\)
\(338\) 0 0
\(339\) −201.901 + 201.901i −0.595577 + 0.595577i
\(340\) 0 0
\(341\) −67.4256 + 67.4256i −0.197729 + 0.197729i
\(342\) 0 0
\(343\) −37.0815 −0.108109
\(344\) 0 0
\(345\) 59.1384i 0.171416i
\(346\) 0 0
\(347\) 59.7885 + 59.7885i 0.172301 + 0.172301i 0.787990 0.615688i \(-0.211122\pi\)
−0.615688 + 0.787990i \(0.711122\pi\)
\(348\) 0 0
\(349\) 56.4538 + 56.4538i 0.161759 + 0.161759i 0.783345 0.621587i \(-0.213512\pi\)
−0.621587 + 0.783345i \(0.713512\pi\)
\(350\) 0 0
\(351\) 47.5013i 0.135331i
\(352\) 0 0
\(353\) −87.9897 −0.249263 −0.124631 0.992203i \(-0.539775\pi\)
−0.124631 + 0.992203i \(0.539775\pi\)
\(354\) 0 0
\(355\) 263.950 263.950i 0.743522 0.743522i
\(356\) 0 0
\(357\) 5.50258 5.50258i 0.0154134 0.0154134i
\(358\) 0 0
\(359\) −213.499 −0.594705 −0.297353 0.954768i \(-0.596104\pi\)
−0.297353 + 0.954768i \(0.596104\pi\)
\(360\) 0 0
\(361\) 335.554i 0.929512i
\(362\) 0 0
\(363\) −124.742 124.742i −0.343641 0.343641i
\(364\) 0 0
\(365\) 100.287 + 100.287i 0.274759 + 0.274759i
\(366\) 0 0
\(367\) 537.102i 1.46349i −0.681577 0.731746i \(-0.738706\pi\)
0.681577 0.731746i \(-0.261294\pi\)
\(368\) 0 0
\(369\) −10.7077 −0.0290180
\(370\) 0 0
\(371\) 26.0651 26.0651i 0.0702562 0.0702562i
\(372\) 0 0
\(373\) 120.895 120.895i 0.324115 0.324115i −0.526228 0.850343i \(-0.676394\pi\)
0.850343 + 0.526228i \(0.176394\pi\)
\(374\) 0 0
\(375\) −205.757 −0.548686
\(376\) 0 0
\(377\) 13.8564i 0.0367544i
\(378\) 0 0
\(379\) 178.922 + 178.922i 0.472089 + 0.472089i 0.902590 0.430501i \(-0.141663\pi\)
−0.430501 + 0.902590i \(0.641663\pi\)
\(380\) 0 0
\(381\) 81.7461 + 81.7461i 0.214557 + 0.214557i
\(382\) 0 0
\(383\) 61.1158i 0.159571i 0.996812 + 0.0797856i \(0.0254236\pi\)
−0.996812 + 0.0797856i \(0.974576\pi\)
\(384\) 0 0
\(385\) 16.0000 0.0415584
\(386\) 0 0
\(387\) 89.9277 89.9277i 0.232371 0.232371i
\(388\) 0 0
\(389\) 399.569 399.569i 1.02717 1.02717i 0.0275498 0.999620i \(-0.491230\pi\)
0.999620 0.0275498i \(-0.00877048\pi\)
\(390\) 0 0
\(391\) −143.126 −0.366050
\(392\) 0 0
\(393\) 333.415i 0.848385i
\(394\) 0 0
\(395\) 228.439 + 228.439i 0.578327 + 0.578327i
\(396\) 0 0
\(397\) −310.597 310.597i −0.782361 0.782361i 0.197868 0.980229i \(-0.436598\pi\)
−0.980229 + 0.197868i \(0.936598\pi\)
\(398\) 0 0
\(399\) 17.3223i 0.0434143i
\(400\) 0 0
\(401\) −447.856 −1.11685 −0.558424 0.829555i \(-0.688594\pi\)
−0.558424 + 0.829555i \(0.688594\pi\)
\(402\) 0 0
\(403\) 41.2896 41.2896i 0.102456 0.102456i
\(404\) 0 0
\(405\) −18.0000 + 18.0000i −0.0444444 + 0.0444444i
\(406\) 0 0
\(407\) 747.191 1.83585
\(408\) 0 0
\(409\) 208.133i 0.508883i 0.967088 + 0.254442i \(0.0818917\pi\)
−0.967088 + 0.254442i \(0.918108\pi\)
\(410\) 0 0
\(411\) 276.252 + 276.252i 0.672146 + 0.672146i
\(412\) 0 0
\(413\) 10.5641 + 10.5641i 0.0255788 + 0.0255788i
\(414\) 0 0
\(415\) 7.68757i 0.0185243i
\(416\) 0 0
\(417\) 100.574 0.241186
\(418\) 0 0
\(419\) 169.191 169.191i 0.403796 0.403796i −0.475772 0.879569i \(-0.657831\pi\)
0.879569 + 0.475772i \(0.157831\pi\)
\(420\) 0 0
\(421\) 64.9615 64.9615i 0.154303 0.154303i −0.625734 0.780037i \(-0.715200\pi\)
0.780037 + 0.625734i \(0.215200\pi\)
\(422\) 0 0
\(423\) −95.0025 −0.224592
\(424\) 0 0
\(425\) 201.559i 0.474256i
\(426\) 0 0
\(427\) 23.8041 + 23.8041i 0.0557472 + 0.0557472i
\(428\) 0 0
\(429\) 167.138 + 167.138i 0.389600 + 0.389600i
\(430\) 0 0
\(431\) 104.206i 0.241777i −0.992666 0.120888i \(-0.961426\pi\)
0.992666 0.120888i \(-0.0385743\pi\)
\(432\) 0 0
\(433\) −533.549 −1.23221 −0.616107 0.787663i \(-0.711291\pi\)
−0.616107 + 0.787663i \(0.711291\pi\)
\(434\) 0 0
\(435\) 5.25071 5.25071i 0.0120706 0.0120706i
\(436\) 0 0
\(437\) 225.282 225.282i 0.515520 0.515520i
\(438\) 0 0
\(439\) 163.318 0.372023 0.186012 0.982548i \(-0.440444\pi\)
0.186012 + 0.982548i \(0.440444\pi\)
\(440\) 0 0
\(441\) 146.569i 0.332357i
\(442\) 0 0
\(443\) −238.589 238.589i −0.538575 0.538575i 0.384535 0.923110i \(-0.374362\pi\)
−0.923110 + 0.384535i \(0.874362\pi\)
\(444\) 0 0
\(445\) 148.000 + 148.000i 0.332584 + 0.332584i
\(446\) 0 0
\(447\) 78.0319i 0.174568i
\(448\) 0 0
\(449\) −333.559 −0.742893 −0.371446 0.928454i \(-0.621138\pi\)
−0.371446 + 0.928454i \(0.621138\pi\)
\(450\) 0 0
\(451\) 37.6761 37.6761i 0.0835390 0.0835390i
\(452\) 0 0
\(453\) 33.7461 33.7461i 0.0744948 0.0744948i
\(454\) 0 0
\(455\) −9.79796 −0.0215340
\(456\) 0 0
\(457\) 431.292i 0.943747i 0.881666 + 0.471873i \(0.156422\pi\)
−0.881666 + 0.471873i \(0.843578\pi\)
\(458\) 0 0
\(459\) −43.5632 43.5632i −0.0949090 0.0949090i
\(460\) 0 0
\(461\) −232.277 232.277i −0.503854 0.503854i 0.408779 0.912633i \(-0.365955\pi\)
−0.912633 + 0.408779i \(0.865955\pi\)
\(462\) 0 0
\(463\) 470.681i 1.01659i 0.861183 + 0.508295i \(0.169724\pi\)
−0.861183 + 0.508295i \(0.830276\pi\)
\(464\) 0 0
\(465\) −31.2923 −0.0672954
\(466\) 0 0
\(467\) 412.950 412.950i 0.884262 0.884262i −0.109702 0.993964i \(-0.534990\pi\)
0.993964 + 0.109702i \(0.0349898\pi\)
\(468\) 0 0
\(469\) 11.5795 11.5795i 0.0246898 0.0246898i
\(470\) 0 0
\(471\) 160.530 0.340827
\(472\) 0 0
\(473\) 632.841i 1.33793i
\(474\) 0 0
\(475\) 317.257 + 317.257i 0.667910 + 0.667910i
\(476\) 0 0
\(477\) −206.354 206.354i −0.432608 0.432608i
\(478\) 0 0
\(479\) 802.784i 1.67596i 0.545703 + 0.837979i \(0.316263\pi\)
−0.545703 + 0.837979i \(0.683737\pi\)
\(480\) 0 0
\(481\) −457.559 −0.951266
\(482\) 0 0
\(483\) −5.60244 + 5.60244i −0.0115993 + 0.0115993i
\(484\) 0 0
\(485\) −178.564 + 178.564i −0.368173 + 0.368173i
\(486\) 0 0
\(487\) 317.326 0.651593 0.325797 0.945440i \(-0.394367\pi\)
0.325797 + 0.945440i \(0.394367\pi\)
\(488\) 0 0
\(489\) 198.000i 0.404908i
\(490\) 0 0
\(491\) 651.648 + 651.648i 1.32719 + 1.32719i 0.907816 + 0.419369i \(0.137749\pi\)
0.419369 + 0.907816i \(0.362251\pi\)
\(492\) 0 0
\(493\) 12.7077 + 12.7077i 0.0257762 + 0.0257762i
\(494\) 0 0
\(495\) 126.670i 0.255899i
\(496\) 0 0
\(497\) 50.0103 0.100624
\(498\) 0 0
\(499\) 38.9489 38.9489i 0.0780540 0.0780540i −0.667002 0.745056i \(-0.732423\pi\)
0.745056 + 0.667002i \(0.232423\pi\)
\(500\) 0 0
\(501\) 309.282 309.282i 0.617329 0.617329i
\(502\) 0 0
\(503\) −205.594 −0.408735 −0.204368 0.978894i \(-0.565514\pi\)
−0.204368 + 0.978894i \(0.565514\pi\)
\(504\) 0 0
\(505\) 495.692i 0.981569i
\(506\) 0 0
\(507\) 104.631 + 104.631i 0.206373 + 0.206373i
\(508\) 0 0
\(509\) −91.6565 91.6565i −0.180072 0.180072i 0.611315 0.791387i \(-0.290641\pi\)
−0.791387 + 0.611315i \(0.790641\pi\)
\(510\) 0 0
\(511\) 19.0013i 0.0371845i
\(512\) 0 0
\(513\) 137.138 0.267326
\(514\) 0 0
\(515\) 192.279 192.279i 0.373357 0.373357i
\(516\) 0 0
\(517\) 334.277 334.277i 0.646570 0.646570i
\(518\) 0 0
\(519\) 221.484 0.426751
\(520\) 0 0
\(521\) 155.282i 0.298046i −0.988834 0.149023i \(-0.952387\pi\)
0.988834 0.149023i \(-0.0476129\pi\)
\(522\) 0 0
\(523\) 501.090 + 501.090i 0.958107 + 0.958107i 0.999157 0.0410496i \(-0.0130702\pi\)
−0.0410496 + 0.999157i \(0.513070\pi\)
\(524\) 0 0
\(525\) −7.88973 7.88973i −0.0150281 0.0150281i
\(526\) 0 0
\(527\) 75.7331i 0.143706i
\(528\) 0 0
\(529\) −383.277 −0.724531
\(530\) 0 0
\(531\) 83.6344 83.6344i 0.157504 0.157504i
\(532\) 0 0
\(533\) −23.0718 + 23.0718i −0.0432867 + 0.0432867i
\(534\) 0 0
\(535\) 462.832 0.865107
\(536\) 0 0
\(537\) 22.2769i 0.0414839i
\(538\) 0 0
\(539\) −515.720 515.720i −0.956809 0.956809i
\(540\) 0 0
\(541\) −1.53590 1.53590i −0.00283900 0.00283900i 0.705686 0.708525i \(-0.250639\pi\)
−0.708525 + 0.705686i \(0.750639\pi\)
\(542\) 0 0
\(543\) 524.951i 0.966760i
\(544\) 0 0
\(545\) −324.995 −0.596321
\(546\) 0 0
\(547\) −456.891 + 456.891i −0.835266 + 0.835266i −0.988232 0.152965i \(-0.951118\pi\)
0.152965 + 0.988232i \(0.451118\pi\)
\(548\) 0 0
\(549\) 188.454 188.454i 0.343267 0.343267i
\(550\) 0 0
\(551\) −40.0041 −0.0726028
\(552\) 0 0
\(553\) 43.2820i 0.0782677i
\(554\) 0 0
\(555\) 173.386 + 173.386i 0.312408 + 0.312408i
\(556\) 0 0
\(557\) −9.86672 9.86672i −0.0177140 0.0177140i 0.698194 0.715908i \(-0.253987\pi\)
−0.715908 + 0.698194i \(0.753987\pi\)
\(558\) 0 0
\(559\) 387.534i 0.693264i
\(560\) 0 0
\(561\) 306.564 0.546460
\(562\) 0 0
\(563\) 533.909 533.909i 0.948329 0.948329i −0.0504003 0.998729i \(-0.516050\pi\)
0.998729 + 0.0504003i \(0.0160497\pi\)
\(564\) 0 0
\(565\) 329.703 329.703i 0.583544 0.583544i
\(566\) 0 0
\(567\) −3.41044 −0.00601488
\(568\) 0 0
\(569\) 673.846i 1.18426i 0.805841 + 0.592132i \(0.201714\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(570\) 0 0
\(571\) 490.467 + 490.467i 0.858962 + 0.858962i 0.991216 0.132254i \(-0.0422214\pi\)
−0.132254 + 0.991216i \(0.542221\pi\)
\(572\) 0 0
\(573\) 111.580 + 111.580i 0.194729 + 0.194729i
\(574\) 0 0
\(575\) 205.217i 0.356899i
\(576\) 0 0
\(577\) 403.005 0.698449 0.349225 0.937039i \(-0.386445\pi\)
0.349225 + 0.937039i \(0.386445\pi\)
\(578\) 0 0
\(579\) 235.138 235.138i 0.406111 0.406111i
\(580\) 0 0
\(581\) 0.728277 0.728277i 0.00125349 0.00125349i
\(582\) 0 0
\(583\) 1452.16 2.49084
\(584\) 0 0
\(585\) 77.5692i 0.132597i
\(586\) 0 0
\(587\) 157.688 + 157.688i 0.268635 + 0.268635i 0.828550 0.559915i \(-0.189166\pi\)
−0.559915 + 0.828550i \(0.689166\pi\)
\(588\) 0 0
\(589\) 119.205 + 119.205i 0.202386 + 0.202386i
\(590\) 0 0
\(591\) 618.816i 1.04707i
\(592\) 0 0
\(593\) 30.0000 0.0505902 0.0252951 0.999680i \(-0.491947\pi\)
0.0252951 + 0.999680i \(0.491947\pi\)
\(594\) 0 0
\(595\) −8.98567 + 8.98567i −0.0151020 + 0.0151020i
\(596\) 0 0
\(597\) 329.305 329.305i 0.551600 0.551600i
\(598\) 0 0
\(599\) 1139.76 1.90277 0.951384 0.308007i \(-0.0996622\pi\)
0.951384 + 0.308007i \(0.0996622\pi\)
\(600\) 0 0
\(601\) 327.559i 0.545023i −0.962153 0.272512i \(-0.912146\pi\)
0.962153 0.272512i \(-0.0878543\pi\)
\(602\) 0 0
\(603\) −91.6737 91.6737i −0.152029 0.152029i
\(604\) 0 0
\(605\) 203.703 + 203.703i 0.336698 + 0.336698i
\(606\) 0 0
\(607\) 484.540i 0.798254i −0.916896 0.399127i \(-0.869313\pi\)
0.916896 0.399127i \(-0.130687\pi\)
\(608\) 0 0
\(609\) 0.994845 0.00163357
\(610\) 0 0
\(611\) −204.702 + 204.702i −0.335028 + 0.335028i
\(612\) 0 0
\(613\) 463.105 463.105i 0.755473 0.755473i −0.220022 0.975495i \(-0.570613\pi\)
0.975495 + 0.220022i \(0.0706128\pi\)
\(614\) 0 0
\(615\) 17.4855 0.0284318
\(616\) 0 0
\(617\) 1115.39i 1.80777i 0.427774 + 0.903885i \(0.359298\pi\)
−0.427774 + 0.903885i \(0.640702\pi\)
\(618\) 0 0
\(619\) −590.884 590.884i −0.954578 0.954578i 0.0444343 0.999012i \(-0.485851\pi\)
−0.999012 + 0.0444343i \(0.985851\pi\)
\(620\) 0 0
\(621\) 44.3538 + 44.3538i 0.0714232 + 0.0714232i
\(622\) 0 0
\(623\) 28.0414i 0.0450102i
\(624\) 0 0
\(625\) −89.0000 −0.142400
\(626\) 0 0
\(627\) −482.537 + 482.537i −0.769596 + 0.769596i
\(628\) 0 0
\(629\) −419.626 + 419.626i −0.667131 + 0.667131i
\(630\) 0 0
\(631\) −1221.55 −1.93590 −0.967949 0.251145i \(-0.919193\pi\)
−0.967949 + 0.251145i \(0.919193\pi\)
\(632\) 0 0
\(633\) 344.554i 0.544319i
\(634\) 0 0
\(635\) −133.491 133.491i −0.210222 0.210222i
\(636\) 0 0
\(637\) 315.813 + 315.813i 0.495781 + 0.495781i
\(638\) 0 0
\(639\) 395.925i 0.619602i
\(640\) 0 0
\(641\) −1152.41 −1.79783 −0.898916 0.438121i \(-0.855644\pi\)
−0.898916 + 0.438121i \(0.855644\pi\)
\(642\) 0 0
\(643\) 177.649 177.649i 0.276281 0.276281i −0.555341 0.831622i \(-0.687412\pi\)
0.831622 + 0.555341i \(0.187412\pi\)
\(644\) 0 0
\(645\) −146.851 + 146.851i −0.227676 + 0.227676i
\(646\) 0 0
\(647\) −390.892 −0.604161 −0.302081 0.953282i \(-0.597681\pi\)
−0.302081 + 0.953282i \(0.597681\pi\)
\(648\) 0 0
\(649\) 588.554i 0.906862i
\(650\) 0 0
\(651\) −2.96446 2.96446i −0.00455370 0.00455370i
\(652\) 0 0
\(653\) 441.538 + 441.538i 0.676169 + 0.676169i 0.959131 0.282962i \(-0.0913171\pi\)
−0.282962 + 0.959131i \(0.591317\pi\)
\(654\) 0 0
\(655\) 544.465i 0.831244i
\(656\) 0 0
\(657\) 150.431 0.228966
\(658\) 0 0
\(659\) 671.947 671.947i 1.01965 1.01965i 0.0198437 0.999803i \(-0.493683\pi\)
0.999803 0.0198437i \(-0.00631685\pi\)
\(660\) 0 0
\(661\) −16.0333 + 16.0333i −0.0242562 + 0.0242562i −0.719131 0.694875i \(-0.755460\pi\)
0.694875 + 0.719131i \(0.255460\pi\)
\(662\) 0 0
\(663\) −187.731 −0.283154
\(664\) 0 0
\(665\) 28.2872i 0.0425371i
\(666\) 0 0
\(667\) −12.9383 12.9383i −0.0193977 0.0193977i
\(668\) 0 0
\(669\) −165.746 165.746i −0.247752 0.247752i
\(670\) 0 0
\(671\) 1326.19i 1.97644i
\(672\) 0 0
\(673\) 512.954 0.762190 0.381095 0.924536i \(-0.375547\pi\)
0.381095 + 0.924536i \(0.375547\pi\)
\(674\) 0 0
\(675\) −62.4620 + 62.4620i −0.0925363 + 0.0925363i
\(676\) 0 0
\(677\) −282.000 + 282.000i −0.416544 + 0.416544i −0.884011 0.467467i \(-0.845167\pi\)
0.467467 + 0.884011i \(0.345167\pi\)
\(678\) 0 0
\(679\) −33.8323 −0.0498267
\(680\) 0 0
\(681\) 536.687i 0.788087i
\(682\) 0 0
\(683\) −307.660 307.660i −0.450454 0.450454i 0.445051 0.895505i \(-0.353186\pi\)
−0.895505 + 0.445051i \(0.853186\pi\)
\(684\) 0 0
\(685\) −451.118 451.118i −0.658566 0.658566i
\(686\) 0 0
\(687\) 394.110i 0.573669i
\(688\) 0 0
\(689\) −889.261 −1.29066
\(690\) 0 0
\(691\) −599.422 + 599.422i −0.867470 + 0.867470i −0.992192 0.124722i \(-0.960196\pi\)
0.124722 + 0.992192i \(0.460196\pi\)
\(692\) 0 0
\(693\) 12.0000 12.0000i 0.0173160 0.0173160i
\(694\) 0 0
\(695\) −164.237 −0.236313
\(696\) 0 0
\(697\) 42.3181i 0.0607147i
\(698\) 0 0
\(699\) 284.128 + 284.128i 0.406478 + 0.406478i
\(700\) 0 0
\(701\) −63.3487 63.3487i −0.0903690 0.0903690i 0.660477 0.750846i \(-0.270354\pi\)
−0.750846 + 0.660477i \(0.770354\pi\)
\(702\) 0 0
\(703\) 1320.99i 1.87908i
\(704\) 0 0
\(705\) 155.138 0.220055
\(706\) 0 0
\(707\) 46.9591 46.9591i 0.0664202 0.0664202i
\(708\) 0 0
\(709\) −247.259 + 247.259i −0.348743 + 0.348743i −0.859641 0.510898i \(-0.829313\pi\)
0.510898 + 0.859641i \(0.329313\pi\)
\(710\) 0 0
\(711\) 342.658 0.481939
\(712\) 0 0
\(713\) 77.1075i 0.108145i
\(714\) 0 0
\(715\) −272.936 272.936i −0.381729 0.381729i
\(716\) 0 0
\(717\) 556.708 + 556.708i 0.776440 + 0.776440i
\(718\) 0 0
\(719\) 794.987i 1.10568i 0.833286 + 0.552842i \(0.186457\pi\)
−0.833286 + 0.552842i \(0.813543\pi\)
\(720\) 0 0
\(721\) 36.4308 0.0505281
\(722\) 0 0
\(723\) −404.844 + 404.844i −0.559950 + 0.559950i
\(724\) 0 0
\(725\) 18.2205 18.2205i 0.0251318 0.0251318i
\(726\) 0 0
\(727\) 395.706 0.544300 0.272150 0.962255i \(-0.412265\pi\)
0.272150 + 0.962255i \(0.412265\pi\)
\(728\) 0 0
\(729\) 27.0000i 0.0370370i
\(730\) 0 0
\(731\) −355.406 355.406i −0.486192 0.486192i
\(732\) 0 0
\(733\) −486.443 486.443i −0.663634 0.663634i 0.292601 0.956235i \(-0.405479\pi\)
−0.956235 + 0.292601i \(0.905479\pi\)
\(734\) 0 0
\(735\) 239.347i 0.325642i
\(736\) 0 0
\(737\) 645.128 0.875343
\(738\) 0 0
\(739\) 503.079 503.079i 0.680757 0.680757i −0.279414 0.960171i \(-0.590140\pi\)
0.960171 + 0.279414i \(0.0901402\pi\)
\(740\) 0 0
\(741\) 295.492 295.492i 0.398775 0.398775i
\(742\) 0 0
\(743\) −270.934 −0.364649 −0.182325 0.983238i \(-0.558362\pi\)
−0.182325 + 0.983238i \(0.558362\pi\)
\(744\) 0 0
\(745\) 127.426i 0.171041i
\(746\) 0 0
\(747\) −5.76568 5.76568i −0.00771845 0.00771845i
\(748\) 0 0
\(749\) 43.8461 + 43.8461i 0.0585395 + 0.0585395i
\(750\) 0 0
\(751\) 348.232i 0.463691i −0.972753 0.231845i \(-0.925524\pi\)
0.972753 0.231845i \(-0.0744763\pi\)
\(752\) 0 0
\(753\) 60.9948 0.0810025
\(754\) 0 0
\(755\) −55.1072 + 55.1072i −0.0729897 + 0.0729897i
\(756\) 0 0
\(757\) 231.792 231.792i 0.306198 0.306198i −0.537235 0.843433i \(-0.680531\pi\)
0.843433 + 0.537235i \(0.180531\pi\)
\(758\) 0 0
\(759\) −312.128 −0.411236
\(760\) 0 0
\(761\) 1038.99i 1.36530i −0.730745 0.682651i \(-0.760827\pi\)
0.730745 0.682651i \(-0.239173\pi\)
\(762\) 0 0
\(763\) −30.7882 30.7882i −0.0403515 0.0403515i
\(764\) 0 0
\(765\) 71.1384 + 71.1384i 0.0929914 + 0.0929914i
\(766\) 0 0
\(767\) 360.414i 0.469901i
\(768\) 0 0
\(769\) 1047.42 1.36205 0.681024 0.732261i \(-0.261535\pi\)
0.681024 + 0.732261i \(0.261535\pi\)
\(770\) 0 0
\(771\) −339.424 + 339.424i −0.440238 + 0.440238i
\(772\) 0 0
\(773\) −863.041 + 863.041i −1.11648 + 1.11648i −0.124229 + 0.992254i \(0.539646\pi\)
−0.992254 + 0.124229i \(0.960354\pi\)
\(774\) 0 0
\(775\) −108.588 −0.140113
\(776\) 0 0
\(777\) 32.8513i 0.0422796i
\(778\) 0 0
\(779\) −66.6094 66.6094i −0.0855063 0.0855063i
\(780\) 0 0
\(781\) 1393.11 + 1393.11i 1.78375 + 1.78375i
\(782\) 0 0
\(783\) 7.87607i 0.0100588i
\(784\) 0 0
\(785\) −262.144 −0.333941
\(786\) 0 0
\(787\) 61.6142 61.6142i 0.0782900 0.0782900i −0.666877 0.745167i \(-0.732370\pi\)
0.745167 + 0.666877i \(0.232370\pi\)
\(788\) 0 0
\(789\) 145.990 145.990i 0.185031 0.185031i
\(790\) 0 0
\(791\) 62.4683 0.0789738
\(792\) 0 0
\(793\) 812.123i 1.02411i
\(794\) 0 0
\(795\) 336.974 + 336.974i 0.423867 + 0.423867i
\(796\) 0 0
\(797\) −998.379 998.379i −1.25267 1.25267i −0.954518 0.298154i \(-0.903629\pi\)
−0.298154 0.954518i \(-0.596371\pi\)
\(798\) 0 0
\(799\) 375.463i 0.469916i
\(800\) 0 0
\(801\) 222.000 0.277154
\(802\) 0 0
\(803\) −529.307 + 529.307i −0.659162 + 0.659162i
\(804\) 0 0
\(805\) 9.14875 9.14875i 0.0113649 0.0113649i
\(806\) 0 0
\(807\) 370.024 0.458517
\(808\) 0 0
\(809\) 400.390i 0.494919i 0.968898 + 0.247460i \(0.0795957\pi\)
−0.968898 + 0.247460i \(0.920404\pi\)
\(810\) 0 0
\(811\) 504.096 + 504.096i 0.621574 + 0.621574i 0.945934 0.324360i \(-0.105149\pi\)
−0.324360 + 0.945934i \(0.605149\pi\)
\(812\) 0 0
\(813\) −526.808 526.808i −0.647980 0.647980i
\(814\) 0 0
\(815\) 323.333i 0.396727i
\(816\) 0 0
\(817\) 1118.83 1.36944
\(818\) 0 0
\(819\) −7.34847 + 7.34847i −0.00897249 + 0.00897249i
\(820\) 0 0
\(821\) −596.477 + 596.477i −0.726525 + 0.726525i −0.969926 0.243401i \(-0.921737\pi\)
0.243401 + 0.969926i \(0.421737\pi\)
\(822\) 0 0
\(823\) −1615.20 −1.96258 −0.981290 0.192534i \(-0.938329\pi\)
−0.981290 + 0.192534i \(0.938329\pi\)
\(824\) 0 0
\(825\) 439.559i 0.532799i
\(826\) 0 0
\(827\) −1010.90 1010.90i −1.22237 1.22237i −0.966788 0.255579i \(-0.917734\pi\)
−0.255579 0.966788i \(-0.582266\pi\)
\(828\) 0 0
\(829\) −933.561 933.561i −1.12613 1.12613i −0.990801 0.135329i \(-0.956791\pi\)
−0.135329 0.990801i \(-0.543209\pi\)
\(830\) 0 0
\(831\) 522.351i 0.628581i
\(832\) 0 0
\(833\) 579.261 0.695392
\(834\) 0 0
\(835\) −505.055 + 505.055i −0.604857 + 0.604857i
\(836\) 0 0
\(837\) −23.4693 + 23.4693i −0.0280397 + 0.0280397i
\(838\) 0 0
\(839\) −1408.20 −1.67843 −0.839216 0.543798i \(-0.816986\pi\)
−0.839216 + 0.543798i \(0.816986\pi\)
\(840\) 0 0
\(841\) 838.703i 0.997268i
\(842\) 0 0
\(843\) 95.5301 + 95.5301i 0.113322 + 0.113322i
\(844\) 0 0
\(845\) −170.862 170.862i −0.202203 0.202203i
\(846\) 0 0
\(847\) 38.5952i 0.0455670i
\(848\) 0 0
\(849\) 102.297 0.120492
\(850\) 0 0
\(851\) 427.241 427.241i 0.502046 0.502046i
\(852\) 0 0
\(853\) 41.8691 41.8691i 0.0490845 0.0490845i −0.682138 0.731223i \(-0.738950\pi\)
0.731223 + 0.682138i \(0.238950\pi\)
\(854\) 0 0
\(855\) −223.946 −0.261925
\(856\) 0 0
\(857\) 418.441i 0.488263i −0.969742 0.244131i \(-0.921497\pi\)
0.969742 0.244131i \(-0.0785028\pi\)
\(858\) 0 0
\(859\) 272.814 + 272.814i 0.317595 + 0.317595i 0.847843 0.530247i \(-0.177901\pi\)
−0.530247 + 0.847843i \(0.677901\pi\)
\(860\) 0 0
\(861\) 1.65648 + 1.65648i 0.00192390 + 0.00192390i
\(862\) 0 0
\(863\) 988.405i 1.14531i 0.819796 + 0.572656i \(0.194087\pi\)
−0.819796 + 0.572656i \(0.805913\pi\)
\(864\) 0 0
\(865\) −361.682 −0.418129
\(866\) 0 0
\(867\) 181.784 181.784i 0.209670 0.209670i
\(868\) 0 0
\(869\) −1205.68 + 1205.68i −1.38744 + 1.38744i
\(870\) 0 0
\(871\) −395.059 −0.453569
\(872\) 0 0
\(873\) 267.846i 0.306811i
\(874\) 0 0
\(875\) 31.8307 + 31.8307i 0.0363780 + 0.0363780i
\(876\) 0 0
\(877\) −116.254 116.254i −0.132559 0.132559i 0.637714 0.770273i \(-0.279880\pi\)
−0.770273 + 0.637714i \(0.779880\pi\)
\(878\) 0 0
\(879\) 347.664i 0.395523i
\(880\) 0 0
\(881\) −727.395 −0.825647 −0.412823 0.910811i \(-0.635457\pi\)
−0.412823 + 0.910811i \(0.635457\pi\)
\(882\) 0 0
\(883\) 44.6981 44.6981i 0.0506207 0.0506207i −0.681343 0.731964i \(-0.738604\pi\)
0.731964 + 0.681343i \(0.238604\pi\)
\(884\) 0 0
\(885\) −136.574 + 136.574i −0.154321 + 0.154321i
\(886\) 0 0
\(887\) −566.766 −0.638969 −0.319485 0.947591i \(-0.603510\pi\)
−0.319485 + 0.947591i \(0.603510\pi\)
\(888\) 0 0
\(889\) 25.2923i 0.0284503i
\(890\) 0 0
\(891\) −95.0025 95.0025i −0.106625 0.106625i
\(892\) 0 0
\(893\) −590.985 590.985i −0.661797 0.661797i
\(894\) 0 0
\(895\) 36.3780i 0.0406458i
\(896\) 0 0
\(897\) 191.138 0.213086
\(898\) 0 0
\(899\) 6.84613 6.84613i 0.00761527 0.00761527i
\(900\) 0 0
\(901\) −815.538 + 815.538i −0.905148 + 0.905148i
\(902\) 0 0
\(903\) −27.8237 −0.0308125
\(904\) 0 0
\(905\) 857.241i 0.947227i
\(906\) 0 0
\(907\) −259.499 259.499i −0.286107 0.286107i 0.549432 0.835539i \(-0.314844\pi\)
−0.835539 + 0.549432i \(0.814844\pi\)
\(908\) 0 0
\(909\) −371.769 371.769i −0.408987 0.408987i
\(910\) 0 0
\(911\) 1020.02i 1.11967i 0.828605 + 0.559834i \(0.189135\pi\)
−0.828605 + 0.559834i \(0.810865\pi\)
\(912\) 0 0
\(913\) 40.5744 0.0444407
\(914\) 0 0
\(915\) −307.744 + 307.744i −0.336332 + 0.336332i
\(916\) 0 0
\(917\) −51.5795 + 51.5795i −0.0562481 + 0.0562481i
\(918\) 0 0
\(919\) −398.632 −0.433768 −0.216884 0.976197i \(-0.569589\pi\)
−0.216884 + 0.976197i \(0.569589\pi\)
\(920\) 0 0
\(921\) 39.9794i 0.0434087i
\(922\) 0 0
\(923\) −853.101 853.101i −0.924269 0.924269i
\(924\) 0 0
\(925\) 601.669 + 601.669i 0.650453 + 0.650453i
\(926\) 0 0
\(927\) 288.418i 0.311130i
\(928\) 0 0
\(929\) 645.251 0.694565 0.347283 0.937761i \(-0.387104\pi\)
0.347283 + 0.937761i \(0.387104\pi\)
\(930\) 0 0
\(931\) −911.767 + 911.767i −0.979342 + 0.979342i
\(932\) 0 0
\(933\) 136.441 136.441i 0.146239 0.146239i
\(934\) 0 0
\(935\) −500.617 −0.535419
\(936\) 0 0
\(937\) 1279.72i 1.36577i −0.730528 0.682883i \(-0.760726\pi\)
0.730528 0.682883i \(-0.239274\pi\)
\(938\) 0 0
\(939\) −615.148 615.148i −0.655110 0.655110i
\(940\) 0 0
\(941\) 796.144 + 796.144i 0.846061 + 0.846061i 0.989639 0.143578i \(-0.0458607\pi\)
−0.143578 + 0.989639i \(0.545861\pi\)
\(942\) 0 0
\(943\) 43.0861i 0.0456905i
\(944\) 0 0
\(945\) 5.56922 0.00589335
\(946\) 0 0
\(947\) 903.254 903.254i 0.953806 0.953806i −0.0451730 0.998979i \(-0.514384\pi\)
0.998979 + 0.0451730i \(0.0143839\pi\)
\(948\) 0 0
\(949\) 324.133 324.133i 0.341552 0.341552i
\(950\) 0 0
\(951\) −344.850 −0.362619
\(952\) 0 0
\(953\) 286.687i 0.300826i 0.988623 + 0.150413i \(0.0480603\pi\)
−0.988623 + 0.150413i \(0.951940\pi\)
\(954\) 0 0
\(955\) −182.209 182.209i −0.190794 0.190794i
\(956\) 0 0
\(957\) 27.7128 + 27.7128i 0.0289580 + 0.0289580i
\(958\) 0 0
\(959\) 85.4727i 0.0891269i
\(960\) 0 0
\(961\) 920.200 0.957544
\(962\) 0 0
\(963\) 347.124 347.124i 0.360461 0.360461i
\(964\) 0 0
\(965\) −383.979 + 383.979i −0.397906 + 0.397906i
\(966\) 0 0
\(967\) −153.089 −0.158313 −0.0791565 0.996862i \(-0.525223\pi\)
−0.0791565 + 0.996862i \(0.525223\pi\)
\(968\) 0 0
\(969\) 541.990i 0.559329i
\(970\) 0 0
\(971\) 704.184 + 704.184i 0.725215 + 0.725215i 0.969663 0.244447i \(-0.0786065\pi\)
−0.244447 + 0.969663i \(0.578606\pi\)
\(972\) 0 0
\(973\) −15.5589 15.5589i −0.0159907 0.0159907i
\(974\) 0 0
\(975\) 269.174i 0.276076i
\(976\) 0 0
\(977\) −521.230 −0.533501 −0.266751 0.963766i \(-0.585950\pi\)
−0.266751 + 0.963766i \(0.585950\pi\)
\(978\) 0 0
\(979\) −781.132 + 781.132i −0.797887 + 0.797887i
\(980\) 0 0
\(981\) −243.746 + 243.746i −0.248467 + 0.248467i
\(982\) 0 0
\(983\) 952.730 0.969207 0.484603 0.874734i \(-0.338964\pi\)
0.484603 + 0.874734i \(0.338964\pi\)
\(984\) 0 0
\(985\) 1010.52i 1.02591i
\(986\) 0 0
\(987\) 14.6969 + 14.6969i 0.0148905 + 0.0148905i
\(988\) 0 0
\(989\) 361.856 + 361.856i 0.365881 + 0.365881i
\(990\) 0 0
\(991\) 788.654i 0.795816i 0.917425 + 0.397908i \(0.130264\pi\)
−0.917425 + 0.397908i \(0.869736\pi\)
\(992\) 0 0
\(993\) 741.415 0.746642
\(994\) 0 0
\(995\) −537.753 + 537.753i −0.540455 + 0.540455i
\(996\) 0 0
\(997\) −370.177 + 370.177i −0.371291 + 0.371291i −0.867947 0.496656i \(-0.834561\pi\)
0.496656 + 0.867947i \(0.334561\pi\)
\(998\) 0 0
\(999\) 260.079 0.260340
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 768.3.l.d.703.2 yes 8
4.3 odd 2 inner 768.3.l.d.703.3 yes 8
8.3 odd 2 768.3.l.a.703.1 yes 8
8.5 even 2 768.3.l.a.703.4 yes 8
16.3 odd 4 768.3.l.a.319.4 yes 8
16.5 even 4 inner 768.3.l.d.319.3 yes 8
16.11 odd 4 inner 768.3.l.d.319.2 yes 8
16.13 even 4 768.3.l.a.319.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
768.3.l.a.319.1 8 16.13 even 4
768.3.l.a.319.4 yes 8 16.3 odd 4
768.3.l.a.703.1 yes 8 8.3 odd 2
768.3.l.a.703.4 yes 8 8.5 even 2
768.3.l.d.319.2 yes 8 16.11 odd 4 inner
768.3.l.d.319.3 yes 8 16.5 even 4 inner
768.3.l.d.703.2 yes 8 1.1 even 1 trivial
768.3.l.d.703.3 yes 8 4.3 odd 2 inner