Properties

Label 640.2.s.d.223.9
Level $640$
Weight $2$
Character 640.223
Analytic conductor $5.110$
Analytic rank $0$
Dimension $18$
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] = [640,2,Mod(223,640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(640, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("640.223");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.s (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.11042572936\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + 74 x^{8} + 24 x^{7} - 80 x^{6} - 224 x^{5} - 160 x^{4} - 256 x^{3} + 256 x^{2} + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 223.9
Root \(0.235136 + 1.39453i\) of defining polynomial
Character \(\chi\) \(=\) 640.223
Dual form 640.2.s.d.287.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.96561 q^{3} +(0.177336 - 2.22902i) q^{5} +(-0.115101 + 0.115101i) q^{7} +5.79486 q^{9} +O(q^{10})\) \(q+2.96561 q^{3} +(0.177336 - 2.22902i) q^{5} +(-0.115101 + 0.115101i) q^{7} +5.79486 q^{9} +(-2.95966 - 2.95966i) q^{11} -1.55822i q^{13} +(0.525911 - 6.61042i) q^{15} +(0.299668 - 0.299668i) q^{17} +(2.26261 + 2.26261i) q^{19} +(-0.341344 + 0.341344i) q^{21} +(4.14573 + 4.14573i) q^{23} +(-4.93710 - 0.790575i) q^{25} +8.28846 q^{27} +(-0.289656 + 0.289656i) q^{29} +4.18508i q^{31} +(-8.77721 - 8.77721i) q^{33} +(0.236151 + 0.276974i) q^{35} +1.63643i q^{37} -4.62107i q^{39} -7.61648i q^{41} +6.72651i q^{43} +(1.02764 - 12.9169i) q^{45} +(4.38366 + 4.38366i) q^{47} +6.97350i q^{49} +(0.888698 - 0.888698i) q^{51} -11.4324 q^{53} +(-7.12202 + 6.07231i) q^{55} +(6.71003 + 6.71003i) q^{57} +(-1.63497 + 1.63497i) q^{59} +(1.23034 + 1.23034i) q^{61} +(-0.666993 + 0.666993i) q^{63} +(-3.47331 - 0.276329i) q^{65} +2.49337i q^{67} +(12.2946 + 12.2946i) q^{69} +8.00096 q^{71} +(-1.12102 + 1.12102i) q^{73} +(-14.6415 - 2.34454i) q^{75} +0.681319 q^{77} +3.62218 q^{79} +7.19579 q^{81} -1.62629 q^{83} +(-0.614825 - 0.721109i) q^{85} +(-0.859007 + 0.859007i) q^{87} -15.7149 q^{89} +(0.179352 + 0.179352i) q^{91} +12.4113i q^{93} +(5.44467 - 4.64218i) q^{95} +(9.69217 - 9.69217i) q^{97} +(-17.1508 - 17.1508i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{5} + 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 2 q^{5} + 2 q^{7} + 10 q^{9} + 2 q^{11} - 20 q^{15} - 6 q^{17} + 2 q^{19} + 16 q^{21} - 2 q^{23} - 6 q^{25} + 24 q^{27} - 14 q^{29} - 8 q^{33} - 2 q^{35} + 14 q^{45} + 38 q^{47} - 8 q^{51} - 12 q^{53} - 6 q^{55} - 24 q^{57} - 10 q^{59} - 14 q^{61} - 6 q^{63} + 32 q^{69} + 24 q^{71} - 14 q^{73} - 16 q^{75} + 44 q^{77} - 16 q^{79} + 2 q^{81} - 40 q^{83} - 14 q^{85} + 24 q^{87} + 12 q^{89} + 34 q^{95} + 18 q^{97} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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
\(3\) 2.96561 1.71220 0.856099 0.516813i \(-0.172882\pi\)
0.856099 + 0.516813i \(0.172882\pi\)
\(4\) 0 0
\(5\) 0.177336 2.22902i 0.0793073 0.996850i
\(6\) 0 0
\(7\) −0.115101 + 0.115101i −0.0435040 + 0.0435040i −0.728524 0.685020i \(-0.759793\pi\)
0.685020 + 0.728524i \(0.259793\pi\)
\(8\) 0 0
\(9\) 5.79486 1.93162
\(10\) 0 0
\(11\) −2.95966 2.95966i −0.892372 0.892372i 0.102374 0.994746i \(-0.467356\pi\)
−0.994746 + 0.102374i \(0.967356\pi\)
\(12\) 0 0
\(13\) 1.55822i 0.432172i −0.976374 0.216086i \(-0.930671\pi\)
0.976374 0.216086i \(-0.0693292\pi\)
\(14\) 0 0
\(15\) 0.525911 6.61042i 0.135790 1.70680i
\(16\) 0 0
\(17\) 0.299668 0.299668i 0.0726801 0.0726801i −0.669832 0.742512i \(-0.733634\pi\)
0.742512 + 0.669832i \(0.233634\pi\)
\(18\) 0 0
\(19\) 2.26261 + 2.26261i 0.519079 + 0.519079i 0.917293 0.398214i \(-0.130370\pi\)
−0.398214 + 0.917293i \(0.630370\pi\)
\(20\) 0 0
\(21\) −0.341344 + 0.341344i −0.0744874 + 0.0744874i
\(22\) 0 0
\(23\) 4.14573 + 4.14573i 0.864444 + 0.864444i 0.991851 0.127406i \(-0.0406652\pi\)
−0.127406 + 0.991851i \(0.540665\pi\)
\(24\) 0 0
\(25\) −4.93710 0.790575i −0.987421 0.158115i
\(26\) 0 0
\(27\) 8.28846 1.59511
\(28\) 0 0
\(29\) −0.289656 + 0.289656i −0.0537878 + 0.0537878i −0.733489 0.679701i \(-0.762109\pi\)
0.679701 + 0.733489i \(0.262109\pi\)
\(30\) 0 0
\(31\) 4.18508i 0.751663i 0.926688 + 0.375832i \(0.122643\pi\)
−0.926688 + 0.375832i \(0.877357\pi\)
\(32\) 0 0
\(33\) −8.77721 8.77721i −1.52792 1.52792i
\(34\) 0 0
\(35\) 0.236151 + 0.276974i 0.0399168 + 0.0468172i
\(36\) 0 0
\(37\) 1.63643i 0.269027i 0.990912 + 0.134514i \(0.0429472\pi\)
−0.990912 + 0.134514i \(0.957053\pi\)
\(38\) 0 0
\(39\) 4.62107i 0.739964i
\(40\) 0 0
\(41\) 7.61648i 1.18949i −0.803913 0.594747i \(-0.797252\pi\)
0.803913 0.594747i \(-0.202748\pi\)
\(42\) 0 0
\(43\) 6.72651i 1.02578i 0.858453 + 0.512892i \(0.171426\pi\)
−0.858453 + 0.512892i \(0.828574\pi\)
\(44\) 0 0
\(45\) 1.02764 12.9169i 0.153191 1.92553i
\(46\) 0 0
\(47\) 4.38366 + 4.38366i 0.639423 + 0.639423i 0.950413 0.310990i \(-0.100661\pi\)
−0.310990 + 0.950413i \(0.600661\pi\)
\(48\) 0 0
\(49\) 6.97350i 0.996215i
\(50\) 0 0
\(51\) 0.888698 0.888698i 0.124443 0.124443i
\(52\) 0 0
\(53\) −11.4324 −1.57036 −0.785182 0.619265i \(-0.787431\pi\)
−0.785182 + 0.619265i \(0.787431\pi\)
\(54\) 0 0
\(55\) −7.12202 + 6.07231i −0.960333 + 0.818790i
\(56\) 0 0
\(57\) 6.71003 + 6.71003i 0.888766 + 0.888766i
\(58\) 0 0
\(59\) −1.63497 + 1.63497i −0.212855 + 0.212855i −0.805479 0.592624i \(-0.798092\pi\)
0.592624 + 0.805479i \(0.298092\pi\)
\(60\) 0 0
\(61\) 1.23034 + 1.23034i 0.157528 + 0.157528i 0.781471 0.623942i \(-0.214470\pi\)
−0.623942 + 0.781471i \(0.714470\pi\)
\(62\) 0 0
\(63\) −0.666993 + 0.666993i −0.0840332 + 0.0840332i
\(64\) 0 0
\(65\) −3.47331 0.276329i −0.430811 0.0342744i
\(66\) 0 0
\(67\) 2.49337i 0.304614i 0.988333 + 0.152307i \(0.0486702\pi\)
−0.988333 + 0.152307i \(0.951330\pi\)
\(68\) 0 0
\(69\) 12.2946 + 12.2946i 1.48010 + 1.48010i
\(70\) 0 0
\(71\) 8.00096 0.949540 0.474770 0.880110i \(-0.342531\pi\)
0.474770 + 0.880110i \(0.342531\pi\)
\(72\) 0 0
\(73\) −1.12102 + 1.12102i −0.131205 + 0.131205i −0.769660 0.638454i \(-0.779574\pi\)
0.638454 + 0.769660i \(0.279574\pi\)
\(74\) 0 0
\(75\) −14.6415 2.34454i −1.69066 0.270724i
\(76\) 0 0
\(77\) 0.681319 0.0776435
\(78\) 0 0
\(79\) 3.62218 0.407527 0.203763 0.979020i \(-0.434683\pi\)
0.203763 + 0.979020i \(0.434683\pi\)
\(80\) 0 0
\(81\) 7.19579 0.799532
\(82\) 0 0
\(83\) −1.62629 −0.178509 −0.0892545 0.996009i \(-0.528448\pi\)
−0.0892545 + 0.996009i \(0.528448\pi\)
\(84\) 0 0
\(85\) −0.614825 0.721109i −0.0666871 0.0782152i
\(86\) 0 0
\(87\) −0.859007 + 0.859007i −0.0920953 + 0.0920953i
\(88\) 0 0
\(89\) −15.7149 −1.66577 −0.832887 0.553443i \(-0.813314\pi\)
−0.832887 + 0.553443i \(0.813314\pi\)
\(90\) 0 0
\(91\) 0.179352 + 0.179352i 0.0188012 + 0.0188012i
\(92\) 0 0
\(93\) 12.4113i 1.28700i
\(94\) 0 0
\(95\) 5.44467 4.64218i 0.558611 0.476277i
\(96\) 0 0
\(97\) 9.69217 9.69217i 0.984091 0.984091i −0.0157848 0.999875i \(-0.505025\pi\)
0.999875 + 0.0157848i \(0.00502467\pi\)
\(98\) 0 0
\(99\) −17.1508 17.1508i −1.72372 1.72372i
\(100\) 0 0
\(101\) 12.8067 12.8067i 1.27432 1.27432i 0.330516 0.943800i \(-0.392777\pi\)
0.943800 0.330516i \(-0.107223\pi\)
\(102\) 0 0
\(103\) −4.33738 4.33738i −0.427375 0.427375i 0.460358 0.887733i \(-0.347721\pi\)
−0.887733 + 0.460358i \(0.847721\pi\)
\(104\) 0 0
\(105\) 0.700332 + 0.821398i 0.0683454 + 0.0801602i
\(106\) 0 0
\(107\) 11.9807 1.15822 0.579108 0.815251i \(-0.303401\pi\)
0.579108 + 0.815251i \(0.303401\pi\)
\(108\) 0 0
\(109\) −4.01503 + 4.01503i −0.384570 + 0.384570i −0.872746 0.488175i \(-0.837663\pi\)
0.488175 + 0.872746i \(0.337663\pi\)
\(110\) 0 0
\(111\) 4.85301i 0.460628i
\(112\) 0 0
\(113\) 6.47754 + 6.47754i 0.609356 + 0.609356i 0.942778 0.333422i \(-0.108203\pi\)
−0.333422 + 0.942778i \(0.608203\pi\)
\(114\) 0 0
\(115\) 9.97612 8.50575i 0.930278 0.793165i
\(116\) 0 0
\(117\) 9.02966i 0.834792i
\(118\) 0 0
\(119\) 0.0689840i 0.00632375i
\(120\) 0 0
\(121\) 6.51921i 0.592655i
\(122\) 0 0
\(123\) 22.5875i 2.03665i
\(124\) 0 0
\(125\) −2.63774 + 10.8647i −0.235927 + 0.971771i
\(126\) 0 0
\(127\) −12.2756 12.2756i −1.08928 1.08928i −0.995603 0.0936781i \(-0.970138\pi\)
−0.0936781 0.995603i \(-0.529862\pi\)
\(128\) 0 0
\(129\) 19.9482i 1.75634i
\(130\) 0 0
\(131\) −7.99562 + 7.99562i −0.698581 + 0.698581i −0.964104 0.265524i \(-0.914455\pi\)
0.265524 + 0.964104i \(0.414455\pi\)
\(132\) 0 0
\(133\) −0.520857 −0.0451641
\(134\) 0 0
\(135\) 1.46985 18.4752i 0.126504 1.59009i
\(136\) 0 0
\(137\) −3.08551 3.08551i −0.263613 0.263613i 0.562907 0.826520i \(-0.309683\pi\)
−0.826520 + 0.562907i \(0.809683\pi\)
\(138\) 0 0
\(139\) −12.2206 + 12.2206i −1.03654 + 1.03654i −0.0372284 + 0.999307i \(0.511853\pi\)
−0.999307 + 0.0372284i \(0.988147\pi\)
\(140\) 0 0
\(141\) 13.0002 + 13.0002i 1.09482 + 1.09482i
\(142\) 0 0
\(143\) −4.61180 + 4.61180i −0.385658 + 0.385658i
\(144\) 0 0
\(145\) 0.594284 + 0.697017i 0.0493526 + 0.0578841i
\(146\) 0 0
\(147\) 20.6807i 1.70572i
\(148\) 0 0
\(149\) −2.59172 2.59172i −0.212322 0.212322i 0.592931 0.805253i \(-0.297971\pi\)
−0.805253 + 0.592931i \(0.797971\pi\)
\(150\) 0 0
\(151\) −16.9594 −1.38014 −0.690068 0.723745i \(-0.742419\pi\)
−0.690068 + 0.723745i \(0.742419\pi\)
\(152\) 0 0
\(153\) 1.73653 1.73653i 0.140390 0.140390i
\(154\) 0 0
\(155\) 9.32865 + 0.742168i 0.749296 + 0.0596124i
\(156\) 0 0
\(157\) −8.55235 −0.682552 −0.341276 0.939963i \(-0.610859\pi\)
−0.341276 + 0.939963i \(0.610859\pi\)
\(158\) 0 0
\(159\) −33.9041 −2.68877
\(160\) 0 0
\(161\) −0.954354 −0.0752136
\(162\) 0 0
\(163\) 3.57797 0.280248 0.140124 0.990134i \(-0.455250\pi\)
0.140124 + 0.990134i \(0.455250\pi\)
\(164\) 0 0
\(165\) −21.1211 + 18.0081i −1.64428 + 1.40193i
\(166\) 0 0
\(167\) 0.482874 0.482874i 0.0373659 0.0373659i −0.688177 0.725543i \(-0.741589\pi\)
0.725543 + 0.688177i \(0.241589\pi\)
\(168\) 0 0
\(169\) 10.5720 0.813227
\(170\) 0 0
\(171\) 13.1115 + 13.1115i 1.00266 + 1.00266i
\(172\) 0 0
\(173\) 11.8189i 0.898576i −0.893387 0.449288i \(-0.851678\pi\)
0.893387 0.449288i \(-0.148322\pi\)
\(174\) 0 0
\(175\) 0.659260 0.477269i 0.0498354 0.0360781i
\(176\) 0 0
\(177\) −4.84870 + 4.84870i −0.364451 + 0.364451i
\(178\) 0 0
\(179\) 4.71524 + 4.71524i 0.352433 + 0.352433i 0.861014 0.508581i \(-0.169830\pi\)
−0.508581 + 0.861014i \(0.669830\pi\)
\(180\) 0 0
\(181\) −13.1843 + 13.1843i −0.979983 + 0.979983i −0.999804 0.0198205i \(-0.993691\pi\)
0.0198205 + 0.999804i \(0.493691\pi\)
\(182\) 0 0
\(183\) 3.64870 + 3.64870i 0.269720 + 0.269720i
\(184\) 0 0
\(185\) 3.64764 + 0.290199i 0.268180 + 0.0213358i
\(186\) 0 0
\(187\) −1.77383 −0.129715
\(188\) 0 0
\(189\) −0.954008 + 0.954008i −0.0693939 + 0.0693939i
\(190\) 0 0
\(191\) 13.9872i 1.01208i 0.862510 + 0.506040i \(0.168891\pi\)
−0.862510 + 0.506040i \(0.831109\pi\)
\(192\) 0 0
\(193\) 3.88875 + 3.88875i 0.279919 + 0.279919i 0.833076 0.553158i \(-0.186577\pi\)
−0.553158 + 0.833076i \(0.686577\pi\)
\(194\) 0 0
\(195\) −10.3005 0.819485i −0.737633 0.0586845i
\(196\) 0 0
\(197\) 22.3277i 1.59078i −0.606097 0.795391i \(-0.707266\pi\)
0.606097 0.795391i \(-0.292734\pi\)
\(198\) 0 0
\(199\) 9.83847i 0.697431i 0.937229 + 0.348715i \(0.113382\pi\)
−0.937229 + 0.348715i \(0.886618\pi\)
\(200\) 0 0
\(201\) 7.39437i 0.521559i
\(202\) 0 0
\(203\) 0.0666793i 0.00467997i
\(204\) 0 0
\(205\) −16.9773 1.35068i −1.18575 0.0943355i
\(206\) 0 0
\(207\) 24.0239 + 24.0239i 1.66978 + 1.66978i
\(208\) 0 0
\(209\) 13.3931i 0.926423i
\(210\) 0 0
\(211\) −11.0531 + 11.0531i −0.760925 + 0.760925i −0.976490 0.215565i \(-0.930841\pi\)
0.215565 + 0.976490i \(0.430841\pi\)
\(212\) 0 0
\(213\) 23.7278 1.62580
\(214\) 0 0
\(215\) 14.9936 + 1.19286i 1.02255 + 0.0813521i
\(216\) 0 0
\(217\) −0.481706 0.481706i −0.0327004 0.0327004i
\(218\) 0 0
\(219\) −3.32451 + 3.32451i −0.224650 + 0.224650i
\(220\) 0 0
\(221\) −0.466948 0.466948i −0.0314103 0.0314103i
\(222\) 0 0
\(223\) 5.93975 5.93975i 0.397755 0.397755i −0.479686 0.877440i \(-0.659249\pi\)
0.877440 + 0.479686i \(0.159249\pi\)
\(224\) 0 0
\(225\) −28.6098 4.58127i −1.90732 0.305418i
\(226\) 0 0
\(227\) 23.2105i 1.54054i −0.637720 0.770269i \(-0.720122\pi\)
0.637720 0.770269i \(-0.279878\pi\)
\(228\) 0 0
\(229\) −5.59944 5.59944i −0.370021 0.370021i 0.497464 0.867485i \(-0.334265\pi\)
−0.867485 + 0.497464i \(0.834265\pi\)
\(230\) 0 0
\(231\) 2.02053 0.132941
\(232\) 0 0
\(233\) 3.01998 3.01998i 0.197845 0.197845i −0.601230 0.799076i \(-0.705323\pi\)
0.799076 + 0.601230i \(0.205323\pi\)
\(234\) 0 0
\(235\) 10.5487 8.99391i 0.688120 0.586698i
\(236\) 0 0
\(237\) 10.7420 0.697766
\(238\) 0 0
\(239\) −0.00138865 −8.98241e−5 −4.49120e−5 1.00000i \(-0.500014\pi\)
−4.49120e−5 1.00000i \(0.500014\pi\)
\(240\) 0 0
\(241\) −12.8578 −0.828245 −0.414123 0.910221i \(-0.635912\pi\)
−0.414123 + 0.910221i \(0.635912\pi\)
\(242\) 0 0
\(243\) −3.52546 −0.226158
\(244\) 0 0
\(245\) 15.5441 + 1.23666i 0.993077 + 0.0790071i
\(246\) 0 0
\(247\) 3.52565 3.52565i 0.224332 0.224332i
\(248\) 0 0
\(249\) −4.82296 −0.305643
\(250\) 0 0
\(251\) 9.14111 + 9.14111i 0.576982 + 0.576982i 0.934071 0.357089i \(-0.116231\pi\)
−0.357089 + 0.934071i \(0.616231\pi\)
\(252\) 0 0
\(253\) 24.5399i 1.54281i
\(254\) 0 0
\(255\) −1.82333 2.13853i −0.114181 0.133920i
\(256\) 0 0
\(257\) 21.2733 21.2733i 1.32699 1.32699i 0.419013 0.907980i \(-0.362376\pi\)
0.907980 0.419013i \(-0.137624\pi\)
\(258\) 0 0
\(259\) −0.188354 0.188354i −0.0117038 0.0117038i
\(260\) 0 0
\(261\) −1.67851 + 1.67851i −0.103897 + 0.103897i
\(262\) 0 0
\(263\) −16.7214 16.7214i −1.03108 1.03108i −0.999501 0.0315818i \(-0.989946\pi\)
−0.0315818 0.999501i \(-0.510054\pi\)
\(264\) 0 0
\(265\) −2.02739 + 25.4832i −0.124541 + 1.56542i
\(266\) 0 0
\(267\) −46.6043 −2.85213
\(268\) 0 0
\(269\) −15.9096 + 15.9096i −0.970026 + 0.970026i −0.999564 0.0295378i \(-0.990596\pi\)
0.0295378 + 0.999564i \(0.490596\pi\)
\(270\) 0 0
\(271\) 12.3601i 0.750824i −0.926858 0.375412i \(-0.877501\pi\)
0.926858 0.375412i \(-0.122499\pi\)
\(272\) 0 0
\(273\) 0.531889 + 0.531889i 0.0321914 + 0.0321914i
\(274\) 0 0
\(275\) 12.2723 + 16.9520i 0.740049 + 1.02224i
\(276\) 0 0
\(277\) 21.0270i 1.26339i 0.775217 + 0.631695i \(0.217641\pi\)
−0.775217 + 0.631695i \(0.782359\pi\)
\(278\) 0 0
\(279\) 24.2520i 1.45193i
\(280\) 0 0
\(281\) 10.6807i 0.637158i −0.947896 0.318579i \(-0.896794\pi\)
0.947896 0.318579i \(-0.103206\pi\)
\(282\) 0 0
\(283\) 12.5946i 0.748673i −0.927293 0.374336i \(-0.877871\pi\)
0.927293 0.374336i \(-0.122129\pi\)
\(284\) 0 0
\(285\) 16.1468 13.7669i 0.956452 0.815481i
\(286\) 0 0
\(287\) 0.876663 + 0.876663i 0.0517478 + 0.0517478i
\(288\) 0 0
\(289\) 16.8204i 0.989435i
\(290\) 0 0
\(291\) 28.7432 28.7432i 1.68496 1.68496i
\(292\) 0 0
\(293\) −3.43132 −0.200460 −0.100230 0.994964i \(-0.531958\pi\)
−0.100230 + 0.994964i \(0.531958\pi\)
\(294\) 0 0
\(295\) 3.35446 + 3.93434i 0.195304 + 0.229066i
\(296\) 0 0
\(297\) −24.5310 24.5310i −1.42344 1.42344i
\(298\) 0 0
\(299\) 6.45996 6.45996i 0.373589 0.373589i
\(300\) 0 0
\(301\) −0.774227 0.774227i −0.0446257 0.0446257i
\(302\) 0 0
\(303\) 37.9798 37.9798i 2.18188 2.18188i
\(304\) 0 0
\(305\) 2.96063 2.52427i 0.169525 0.144539i
\(306\) 0 0
\(307\) 11.8104i 0.674053i −0.941495 0.337027i \(-0.890579\pi\)
0.941495 0.337027i \(-0.109421\pi\)
\(308\) 0 0
\(309\) −12.8630 12.8630i −0.731750 0.731750i
\(310\) 0 0
\(311\) 22.6262 1.28301 0.641506 0.767118i \(-0.278310\pi\)
0.641506 + 0.767118i \(0.278310\pi\)
\(312\) 0 0
\(313\) 7.08945 7.08945i 0.400719 0.400719i −0.477767 0.878486i \(-0.658554\pi\)
0.878486 + 0.477767i \(0.158554\pi\)
\(314\) 0 0
\(315\) 1.36846 + 1.60503i 0.0771040 + 0.0904329i
\(316\) 0 0
\(317\) 25.1265 1.41124 0.705621 0.708589i \(-0.250668\pi\)
0.705621 + 0.708589i \(0.250668\pi\)
\(318\) 0 0
\(319\) 1.71457 0.0959974
\(320\) 0 0
\(321\) 35.5300 1.98309
\(322\) 0 0
\(323\) 1.35606 0.0754535
\(324\) 0 0
\(325\) −1.23189 + 7.69309i −0.0683329 + 0.426736i
\(326\) 0 0
\(327\) −11.9070 + 11.9070i −0.658460 + 0.658460i
\(328\) 0 0
\(329\) −1.00913 −0.0556349
\(330\) 0 0
\(331\) −5.80829 5.80829i −0.319253 0.319253i 0.529227 0.848480i \(-0.322482\pi\)
−0.848480 + 0.529227i \(0.822482\pi\)
\(332\) 0 0
\(333\) 9.48287i 0.519658i
\(334\) 0 0
\(335\) 5.55778 + 0.442166i 0.303654 + 0.0241581i
\(336\) 0 0
\(337\) −7.41679 + 7.41679i −0.404019 + 0.404019i −0.879647 0.475628i \(-0.842221\pi\)
0.475628 + 0.879647i \(0.342221\pi\)
\(338\) 0 0
\(339\) 19.2099 + 19.2099i 1.04334 + 1.04334i
\(340\) 0 0
\(341\) 12.3864 12.3864i 0.670763 0.670763i
\(342\) 0 0
\(343\) −1.60836 1.60836i −0.0868434 0.0868434i
\(344\) 0 0
\(345\) 29.5853 25.2247i 1.59282 1.35805i
\(346\) 0 0
\(347\) 18.2493 0.979673 0.489837 0.871814i \(-0.337056\pi\)
0.489837 + 0.871814i \(0.337056\pi\)
\(348\) 0 0
\(349\) 19.4413 19.4413i 1.04067 1.04067i 0.0415330 0.999137i \(-0.486776\pi\)
0.999137 0.0415330i \(-0.0132242\pi\)
\(350\) 0 0
\(351\) 12.9152i 0.689364i
\(352\) 0 0
\(353\) −1.13598 1.13598i −0.0604622 0.0604622i 0.676229 0.736691i \(-0.263613\pi\)
−0.736691 + 0.676229i \(0.763613\pi\)
\(354\) 0 0
\(355\) 1.41886 17.8343i 0.0753054 0.946549i
\(356\) 0 0
\(357\) 0.204580i 0.0108275i
\(358\) 0 0
\(359\) 28.4140i 1.49963i −0.661645 0.749817i \(-0.730141\pi\)
0.661645 0.749817i \(-0.269859\pi\)
\(360\) 0 0
\(361\) 8.76116i 0.461114i
\(362\) 0 0
\(363\) 19.3334i 1.01474i
\(364\) 0 0
\(365\) 2.29998 + 2.69758i 0.120387 + 0.141198i
\(366\) 0 0
\(367\) −2.29692 2.29692i −0.119898 0.119898i 0.644612 0.764510i \(-0.277019\pi\)
−0.764510 + 0.644612i \(0.777019\pi\)
\(368\) 0 0
\(369\) 44.1364i 2.29765i
\(370\) 0 0
\(371\) 1.31588 1.31588i 0.0683172 0.0683172i
\(372\) 0 0
\(373\) 18.0787 0.936081 0.468040 0.883707i \(-0.344960\pi\)
0.468040 + 0.883707i \(0.344960\pi\)
\(374\) 0 0
\(375\) −7.82251 + 32.2206i −0.403953 + 1.66386i
\(376\) 0 0
\(377\) 0.451348 + 0.451348i 0.0232456 + 0.0232456i
\(378\) 0 0
\(379\) 2.79031 2.79031i 0.143328 0.143328i −0.631802 0.775130i \(-0.717684\pi\)
0.775130 + 0.631802i \(0.217684\pi\)
\(380\) 0 0
\(381\) −36.4046 36.4046i −1.86506 1.86506i
\(382\) 0 0
\(383\) −8.12206 + 8.12206i −0.415018 + 0.415018i −0.883482 0.468464i \(-0.844807\pi\)
0.468464 + 0.883482i \(0.344807\pi\)
\(384\) 0 0
\(385\) 0.120823 1.51868i 0.00615770 0.0773990i
\(386\) 0 0
\(387\) 38.9792i 1.98142i
\(388\) 0 0
\(389\) 14.4341 + 14.4341i 0.731839 + 0.731839i 0.970984 0.239145i \(-0.0768670\pi\)
−0.239145 + 0.970984i \(0.576867\pi\)
\(390\) 0 0
\(391\) 2.48468 0.125656
\(392\) 0 0
\(393\) −23.7119 + 23.7119i −1.19611 + 1.19611i
\(394\) 0 0
\(395\) 0.642344 8.07392i 0.0323198 0.406243i
\(396\) 0 0
\(397\) 35.1624 1.76475 0.882374 0.470549i \(-0.155944\pi\)
0.882374 + 0.470549i \(0.155944\pi\)
\(398\) 0 0
\(399\) −1.54466 −0.0773298
\(400\) 0 0
\(401\) −23.5164 −1.17435 −0.587176 0.809459i \(-0.699760\pi\)
−0.587176 + 0.809459i \(0.699760\pi\)
\(402\) 0 0
\(403\) 6.52128 0.324848
\(404\) 0 0
\(405\) 1.27608 16.0396i 0.0634087 0.797014i
\(406\) 0 0
\(407\) 4.84328 4.84328i 0.240072 0.240072i
\(408\) 0 0
\(409\) 23.2595 1.15011 0.575054 0.818115i \(-0.304981\pi\)
0.575054 + 0.818115i \(0.304981\pi\)
\(410\) 0 0
\(411\) −9.15043 9.15043i −0.451357 0.451357i
\(412\) 0 0
\(413\) 0.376374i 0.0185201i
\(414\) 0 0
\(415\) −0.288401 + 3.62505i −0.0141571 + 0.177947i
\(416\) 0 0
\(417\) −36.2415 + 36.2415i −1.77475 + 1.77475i
\(418\) 0 0
\(419\) −6.63975 6.63975i −0.324373 0.324373i 0.526069 0.850442i \(-0.323665\pi\)
−0.850442 + 0.526069i \(0.823665\pi\)
\(420\) 0 0
\(421\) −7.28216 + 7.28216i −0.354911 + 0.354911i −0.861933 0.507022i \(-0.830746\pi\)
0.507022 + 0.861933i \(0.330746\pi\)
\(422\) 0 0
\(423\) 25.4027 + 25.4027i 1.23512 + 1.23512i
\(424\) 0 0
\(425\) −1.71640 + 1.24258i −0.0832576 + 0.0602740i
\(426\) 0 0
\(427\) −0.283225 −0.0137062
\(428\) 0 0
\(429\) −13.6768 + 13.6768i −0.660323 + 0.660323i
\(430\) 0 0
\(431\) 11.7250i 0.564771i 0.959301 + 0.282386i \(0.0911258\pi\)
−0.959301 + 0.282386i \(0.908874\pi\)
\(432\) 0 0
\(433\) −20.8827 20.8827i −1.00356 1.00356i −0.999994 0.00356603i \(-0.998865\pi\)
−0.00356603 0.999994i \(-0.501135\pi\)
\(434\) 0 0
\(435\) 1.76242 + 2.06708i 0.0845014 + 0.0991090i
\(436\) 0 0
\(437\) 18.7604i 0.897430i
\(438\) 0 0
\(439\) 7.53661i 0.359703i 0.983694 + 0.179851i \(0.0575617\pi\)
−0.983694 + 0.179851i \(0.942438\pi\)
\(440\) 0 0
\(441\) 40.4105i 1.92431i
\(442\) 0 0
\(443\) 25.7280i 1.22237i 0.791486 + 0.611187i \(0.209308\pi\)
−0.791486 + 0.611187i \(0.790692\pi\)
\(444\) 0 0
\(445\) −2.78682 + 35.0289i −0.132108 + 1.66053i
\(446\) 0 0
\(447\) −7.68604 7.68604i −0.363537 0.363537i
\(448\) 0 0
\(449\) 2.33824i 0.110348i −0.998477 0.0551741i \(-0.982429\pi\)
0.998477 0.0551741i \(-0.0175714\pi\)
\(450\) 0 0
\(451\) −22.5422 + 22.5422i −1.06147 + 1.06147i
\(452\) 0 0
\(453\) −50.2950 −2.36306
\(454\) 0 0
\(455\) 0.431586 0.367975i 0.0202331 0.0172509i
\(456\) 0 0
\(457\) 10.4561 + 10.4561i 0.489115 + 0.489115i 0.908027 0.418912i \(-0.137588\pi\)
−0.418912 + 0.908027i \(0.637588\pi\)
\(458\) 0 0
\(459\) 2.48378 2.48378i 0.115933 0.115933i
\(460\) 0 0
\(461\) −15.6903 15.6903i −0.730769 0.730769i 0.240003 0.970772i \(-0.422852\pi\)
−0.970772 + 0.240003i \(0.922852\pi\)
\(462\) 0 0
\(463\) −19.6332 + 19.6332i −0.912434 + 0.912434i −0.996463 0.0840297i \(-0.973221\pi\)
0.0840297 + 0.996463i \(0.473221\pi\)
\(464\) 0 0
\(465\) 27.6652 + 2.20098i 1.28294 + 0.102068i
\(466\) 0 0
\(467\) 24.4862i 1.13309i −0.824032 0.566543i \(-0.808281\pi\)
0.824032 0.566543i \(-0.191719\pi\)
\(468\) 0 0
\(469\) −0.286989 0.286989i −0.0132519 0.0132519i
\(470\) 0 0
\(471\) −25.3630 −1.16866
\(472\) 0 0
\(473\) 19.9082 19.9082i 0.915380 0.915380i
\(474\) 0 0
\(475\) −9.38199 12.9595i −0.430475 0.594624i
\(476\) 0 0
\(477\) −66.2493 −3.03335
\(478\) 0 0
\(479\) 37.0609 1.69335 0.846677 0.532108i \(-0.178600\pi\)
0.846677 + 0.532108i \(0.178600\pi\)
\(480\) 0 0
\(481\) 2.54991 0.116266
\(482\) 0 0
\(483\) −2.83024 −0.128781
\(484\) 0 0
\(485\) −19.8853 23.3229i −0.902945 1.05904i
\(486\) 0 0
\(487\) 20.1912 20.1912i 0.914950 0.914950i −0.0817061 0.996656i \(-0.526037\pi\)
0.996656 + 0.0817061i \(0.0260369\pi\)
\(488\) 0 0
\(489\) 10.6109 0.479840
\(490\) 0 0
\(491\) 7.45822 + 7.45822i 0.336585 + 0.336585i 0.855080 0.518496i \(-0.173508\pi\)
−0.518496 + 0.855080i \(0.673508\pi\)
\(492\) 0 0
\(493\) 0.173601i 0.00781860i
\(494\) 0 0
\(495\) −41.2711 + 35.1881i −1.85500 + 1.58159i
\(496\) 0 0
\(497\) −0.920917 + 0.920917i −0.0413088 + 0.0413088i
\(498\) 0 0
\(499\) −8.17420 8.17420i −0.365927 0.365927i 0.500062 0.865990i \(-0.333311\pi\)
−0.865990 + 0.500062i \(0.833311\pi\)
\(500\) 0 0
\(501\) 1.43202 1.43202i 0.0639778 0.0639778i
\(502\) 0 0
\(503\) −29.2327 29.2327i −1.30342 1.30342i −0.926072 0.377348i \(-0.876836\pi\)
−0.377348 0.926072i \(-0.623164\pi\)
\(504\) 0 0
\(505\) −26.2754 30.8176i −1.16924 1.37136i
\(506\) 0 0
\(507\) 31.3523 1.39241
\(508\) 0 0
\(509\) 20.0340 20.0340i 0.887992 0.887992i −0.106338 0.994330i \(-0.533912\pi\)
0.994330 + 0.106338i \(0.0339125\pi\)
\(510\) 0 0
\(511\) 0.258061i 0.0114159i
\(512\) 0 0
\(513\) 18.7536 + 18.7536i 0.827991 + 0.827991i
\(514\) 0 0
\(515\) −10.4373 + 8.89895i −0.459922 + 0.392135i
\(516\) 0 0
\(517\) 25.9483i 1.14121i
\(518\) 0 0
\(519\) 35.0504i 1.53854i
\(520\) 0 0
\(521\) 5.89264i 0.258161i 0.991634 + 0.129081i \(0.0412026\pi\)
−0.991634 + 0.129081i \(0.958797\pi\)
\(522\) 0 0
\(523\) 24.6537i 1.07803i 0.842296 + 0.539015i \(0.181203\pi\)
−0.842296 + 0.539015i \(0.818797\pi\)
\(524\) 0 0
\(525\) 1.95511 1.41539i 0.0853280 0.0617729i
\(526\) 0 0
\(527\) 1.25413 + 1.25413i 0.0546309 + 0.0546309i
\(528\) 0 0
\(529\) 11.3742i 0.494528i
\(530\) 0 0
\(531\) −9.47444 + 9.47444i −0.411156 + 0.411156i
\(532\) 0 0
\(533\) −11.8681 −0.514066
\(534\) 0 0
\(535\) 2.12461 26.7052i 0.0918549 1.15457i
\(536\) 0 0
\(537\) 13.9836 + 13.9836i 0.603435 + 0.603435i
\(538\) 0 0
\(539\) 20.6392 20.6392i 0.888994 0.888994i
\(540\) 0 0
\(541\) 27.1762 + 27.1762i 1.16840 + 1.16840i 0.982585 + 0.185812i \(0.0594916\pi\)
0.185812 + 0.982585i \(0.440508\pi\)
\(542\) 0 0
\(543\) −39.0996 + 39.0996i −1.67792 + 1.67792i
\(544\) 0 0
\(545\) 8.23759 + 9.66162i 0.352860 + 0.413858i
\(546\) 0 0
\(547\) 3.69225i 0.157869i 0.996880 + 0.0789347i \(0.0251519\pi\)
−0.996880 + 0.0789347i \(0.974848\pi\)
\(548\) 0 0
\(549\) 7.12962 + 7.12962i 0.304285 + 0.304285i
\(550\) 0 0
\(551\) −1.31076 −0.0558402
\(552\) 0 0
\(553\) −0.416915 + 0.416915i −0.0177290 + 0.0177290i
\(554\) 0 0
\(555\) 10.8175 + 0.860616i 0.459177 + 0.0365311i
\(556\) 0 0
\(557\) −12.2117 −0.517426 −0.258713 0.965954i \(-0.583298\pi\)
−0.258713 + 0.965954i \(0.583298\pi\)
\(558\) 0 0
\(559\) 10.4814 0.443315
\(560\) 0 0
\(561\) −5.26049 −0.222098
\(562\) 0 0
\(563\) 12.2211 0.515057 0.257528 0.966271i \(-0.417092\pi\)
0.257528 + 0.966271i \(0.417092\pi\)
\(564\) 0 0
\(565\) 15.5873 13.2899i 0.655763 0.559110i
\(566\) 0 0
\(567\) −0.828241 + 0.828241i −0.0347829 + 0.0347829i
\(568\) 0 0
\(569\) −30.9592 −1.29788 −0.648938 0.760841i \(-0.724787\pi\)
−0.648938 + 0.760841i \(0.724787\pi\)
\(570\) 0 0
\(571\) −30.1508 30.1508i −1.26177 1.26177i −0.950233 0.311539i \(-0.899156\pi\)
−0.311539 0.950233i \(-0.600844\pi\)
\(572\) 0 0
\(573\) 41.4806i 1.73288i
\(574\) 0 0
\(575\) −17.1904 23.7454i −0.716889 0.990252i
\(576\) 0 0
\(577\) 1.98215 1.98215i 0.0825181 0.0825181i −0.664643 0.747161i \(-0.731416\pi\)
0.747161 + 0.664643i \(0.231416\pi\)
\(578\) 0 0
\(579\) 11.5325 + 11.5325i 0.479276 + 0.479276i
\(580\) 0 0
\(581\) 0.187188 0.187188i 0.00776586 0.00776586i
\(582\) 0 0
\(583\) 33.8361 + 33.8361i 1.40135 + 1.40135i
\(584\) 0 0
\(585\) −20.1273 1.60129i −0.832163 0.0662051i
\(586\) 0 0
\(587\) −26.9680 −1.11309 −0.556544 0.830818i \(-0.687873\pi\)
−0.556544 + 0.830818i \(0.687873\pi\)
\(588\) 0 0
\(589\) −9.46923 + 9.46923i −0.390173 + 0.390173i
\(590\) 0 0
\(591\) 66.2153i 2.72373i
\(592\) 0 0
\(593\) 16.6701 + 16.6701i 0.684560 + 0.684560i 0.961024 0.276464i \(-0.0891626\pi\)
−0.276464 + 0.961024i \(0.589163\pi\)
\(594\) 0 0
\(595\) 0.153767 + 0.0122334i 0.00630383 + 0.000501520i
\(596\) 0 0
\(597\) 29.1771i 1.19414i
\(598\) 0 0
\(599\) 28.8376i 1.17827i 0.808033 + 0.589137i \(0.200532\pi\)
−0.808033 + 0.589137i \(0.799468\pi\)
\(600\) 0 0
\(601\) 1.91377i 0.0780642i 0.999238 + 0.0390321i \(0.0124275\pi\)
−0.999238 + 0.0390321i \(0.987573\pi\)
\(602\) 0 0
\(603\) 14.4487i 0.588397i
\(604\) 0 0
\(605\) 14.5315 + 1.15609i 0.590789 + 0.0470019i
\(606\) 0 0
\(607\) −7.89049 7.89049i −0.320265 0.320265i 0.528604 0.848869i \(-0.322716\pi\)
−0.848869 + 0.528604i \(0.822716\pi\)
\(608\) 0 0
\(609\) 0.197745i 0.00801303i
\(610\) 0 0
\(611\) 6.83071 6.83071i 0.276341 0.276341i
\(612\) 0 0
\(613\) 40.1035 1.61976 0.809882 0.586592i \(-0.199531\pi\)
0.809882 + 0.586592i \(0.199531\pi\)
\(614\) 0 0
\(615\) −50.3481 4.00559i −2.03023 0.161521i
\(616\) 0 0
\(617\) 14.5821 + 14.5821i 0.587052 + 0.587052i 0.936832 0.349780i \(-0.113744\pi\)
−0.349780 + 0.936832i \(0.613744\pi\)
\(618\) 0 0
\(619\) 4.01752 4.01752i 0.161478 0.161478i −0.621743 0.783221i \(-0.713575\pi\)
0.783221 + 0.621743i \(0.213575\pi\)
\(620\) 0 0
\(621\) 34.3617 + 34.3617i 1.37889 + 1.37889i
\(622\) 0 0
\(623\) 1.80880 1.80880i 0.0724679 0.0724679i
\(624\) 0 0
\(625\) 23.7500 + 7.80630i 0.949999 + 0.312252i
\(626\) 0 0
\(627\) 39.7189i 1.58622i
\(628\) 0 0
\(629\) 0.490385 + 0.490385i 0.0195529 + 0.0195529i
\(630\) 0 0
\(631\) −26.9309 −1.07210 −0.536052 0.844185i \(-0.680085\pi\)
−0.536052 + 0.844185i \(0.680085\pi\)
\(632\) 0 0
\(633\) −32.7791 + 32.7791i −1.30285 + 1.30285i
\(634\) 0 0
\(635\) −29.5395 + 25.1856i −1.17224 + 0.999462i
\(636\) 0 0
\(637\) 10.8662 0.430536
\(638\) 0 0
\(639\) 46.3644 1.83415
\(640\) 0 0
\(641\) 18.6880 0.738131 0.369065 0.929403i \(-0.379678\pi\)
0.369065 + 0.929403i \(0.379678\pi\)
\(642\) 0 0
\(643\) −29.6249 −1.16829 −0.584146 0.811648i \(-0.698571\pi\)
−0.584146 + 0.811648i \(0.698571\pi\)
\(644\) 0 0
\(645\) 44.4651 + 3.53755i 1.75081 + 0.139291i
\(646\) 0 0
\(647\) −5.04426 + 5.04426i −0.198310 + 0.198310i −0.799275 0.600965i \(-0.794783\pi\)
0.600965 + 0.799275i \(0.294783\pi\)
\(648\) 0 0
\(649\) 9.67794 0.379893
\(650\) 0 0
\(651\) −1.42855 1.42855i −0.0559895 0.0559895i
\(652\) 0 0
\(653\) 3.04934i 0.119330i −0.998218 0.0596649i \(-0.980997\pi\)
0.998218 0.0596649i \(-0.0190032\pi\)
\(654\) 0 0
\(655\) 16.4045 + 19.2404i 0.640978 + 0.751783i
\(656\) 0 0
\(657\) −6.49615 + 6.49615i −0.253439 + 0.253439i
\(658\) 0 0
\(659\) −22.0441 22.0441i −0.858718 0.858718i 0.132469 0.991187i \(-0.457709\pi\)
−0.991187 + 0.132469i \(0.957709\pi\)
\(660\) 0 0
\(661\) −8.09788 + 8.09788i −0.314971 + 0.314971i −0.846832 0.531861i \(-0.821493\pi\)
0.531861 + 0.846832i \(0.321493\pi\)
\(662\) 0 0
\(663\) −1.38479 1.38479i −0.0537807 0.0537807i
\(664\) 0 0
\(665\) −0.0923670 + 1.16100i −0.00358184 + 0.0450218i
\(666\) 0 0
\(667\) −2.40167 −0.0929931
\(668\) 0 0
\(669\) 17.6150 17.6150i 0.681035 0.681035i
\(670\) 0 0
\(671\) 7.28276i 0.281148i
\(672\) 0 0
\(673\) −27.1768 27.1768i −1.04759 1.04759i −0.998810 0.0487786i \(-0.984467\pi\)
−0.0487786 0.998810i \(-0.515533\pi\)
\(674\) 0 0
\(675\) −40.9210 6.55265i −1.57505 0.252212i
\(676\) 0 0
\(677\) 28.6501i 1.10111i 0.834798 + 0.550557i \(0.185585\pi\)
−0.834798 + 0.550557i \(0.814415\pi\)
\(678\) 0 0
\(679\) 2.23115i 0.0856238i
\(680\) 0 0
\(681\) 68.8334i 2.63770i
\(682\) 0 0
\(683\) 30.8472i 1.18034i −0.807281 0.590168i \(-0.799062\pi\)
0.807281 0.590168i \(-0.200938\pi\)
\(684\) 0 0
\(685\) −7.42485 + 6.33051i −0.283689 + 0.241876i
\(686\) 0 0
\(687\) −16.6058 16.6058i −0.633549 0.633549i
\(688\) 0 0
\(689\) 17.8142i 0.678668i
\(690\) 0 0
\(691\) −0.253186 + 0.253186i −0.00963164 + 0.00963164i −0.711906 0.702275i \(-0.752168\pi\)
0.702275 + 0.711906i \(0.252168\pi\)
\(692\) 0 0
\(693\) 3.94815 0.149978
\(694\) 0 0
\(695\) 25.0728 + 29.4071i 0.951066 + 1.11548i
\(696\) 0 0
\(697\) −2.28241 2.28241i −0.0864525 0.0864525i
\(698\) 0 0
\(699\) 8.95608 8.95608i 0.338750 0.338750i
\(700\) 0 0
\(701\) −10.5238 10.5238i −0.397479 0.397479i 0.479864 0.877343i \(-0.340686\pi\)
−0.877343 + 0.479864i \(0.840686\pi\)
\(702\) 0 0
\(703\) −3.70261 + 3.70261i −0.139646 + 0.139646i
\(704\) 0 0
\(705\) 31.2833 26.6725i 1.17820 1.00454i
\(706\) 0 0
\(707\) 2.94813i 0.110876i
\(708\) 0 0
\(709\) −1.58968 1.58968i −0.0597015 0.0597015i 0.676626 0.736327i \(-0.263442\pi\)
−0.736327 + 0.676626i \(0.763442\pi\)
\(710\) 0 0
\(711\) 20.9900 0.787186
\(712\) 0 0
\(713\) −17.3502 + 17.3502i −0.649771 + 0.649771i
\(714\) 0 0
\(715\) 9.46198 + 11.0977i 0.353858 + 0.415029i
\(716\) 0 0
\(717\) −0.00411819 −0.000153797
\(718\) 0 0
\(719\) −22.8919 −0.853722 −0.426861 0.904317i \(-0.640381\pi\)
−0.426861 + 0.904317i \(0.640381\pi\)
\(720\) 0 0
\(721\) 0.998472 0.0371850
\(722\) 0 0
\(723\) −38.1313 −1.41812
\(724\) 0 0
\(725\) 1.65906 1.20107i 0.0616158 0.0446065i
\(726\) 0 0
\(727\) 20.1893 20.1893i 0.748780 0.748780i −0.225470 0.974250i \(-0.572392\pi\)
0.974250 + 0.225470i \(0.0723919\pi\)
\(728\) 0 0
\(729\) −32.0425 −1.18676
\(730\) 0 0
\(731\) 2.01572 + 2.01572i 0.0745540 + 0.0745540i
\(732\) 0 0
\(733\) 14.3253i 0.529118i 0.964370 + 0.264559i \(0.0852263\pi\)
−0.964370 + 0.264559i \(0.914774\pi\)
\(734\) 0 0
\(735\) 46.0978 + 3.66744i 1.70034 + 0.135276i
\(736\) 0 0
\(737\) 7.37954 7.37954i 0.271829 0.271829i
\(738\) 0 0
\(739\) 32.3401 + 32.3401i 1.18965 + 1.18965i 0.977164 + 0.212487i \(0.0681564\pi\)
0.212487 + 0.977164i \(0.431844\pi\)
\(740\) 0 0
\(741\) 10.4557 10.4557i 0.384100 0.384100i
\(742\) 0 0
\(743\) −6.06842 6.06842i −0.222629 0.222629i 0.586976 0.809605i \(-0.300318\pi\)
−0.809605 + 0.586976i \(0.800318\pi\)
\(744\) 0 0
\(745\) −6.23662 + 5.31741i −0.228492 + 0.194815i
\(746\) 0 0
\(747\) −9.42414 −0.344811
\(748\) 0 0
\(749\) −1.37898 + 1.37898i −0.0503870 + 0.0503870i
\(750\) 0 0
\(751\) 49.6431i 1.81150i 0.423810 + 0.905751i \(0.360692\pi\)
−0.423810 + 0.905751i \(0.639308\pi\)
\(752\) 0 0
\(753\) 27.1090 + 27.1090i 0.987907 + 0.987907i
\(754\) 0 0
\(755\) −3.00752 + 37.8029i −0.109455 + 1.37579i
\(756\) 0 0
\(757\) 9.18443i 0.333814i −0.985973 0.166907i \(-0.946622\pi\)
0.985973 0.166907i \(-0.0533779\pi\)
\(758\) 0 0
\(759\) 72.7759i 2.64160i
\(760\) 0 0
\(761\) 4.75310i 0.172300i −0.996282 0.0861499i \(-0.972544\pi\)
0.996282 0.0861499i \(-0.0274564\pi\)
\(762\) 0 0
\(763\) 0.924267i 0.0334607i
\(764\) 0 0
\(765\) −3.56282 4.17872i −0.128814 0.151082i
\(766\) 0 0
\(767\) 2.54765 + 2.54765i 0.0919902 + 0.0919902i
\(768\) 0 0
\(769\) 19.4153i 0.700135i −0.936724 0.350067i \(-0.886159\pi\)
0.936724 0.350067i \(-0.113841\pi\)
\(770\) 0 0
\(771\) 63.0884 63.0884i 2.27207 2.27207i
\(772\) 0 0
\(773\) −26.0890 −0.938356 −0.469178 0.883104i \(-0.655450\pi\)
−0.469178 + 0.883104i \(0.655450\pi\)
\(774\) 0 0
\(775\) 3.30862 20.6622i 0.118849 0.742208i
\(776\) 0 0
\(777\) −0.558586 0.558586i −0.0200392 0.0200392i
\(778\) 0 0
\(779\) 17.2331 17.2331i 0.617442 0.617442i
\(780\) 0 0
\(781\) −23.6802 23.6802i −0.847343 0.847343i
\(782\) 0 0
\(783\) −2.40080 + 2.40080i −0.0857977 + 0.0857977i
\(784\) 0 0
\(785\) −1.51664 + 19.0634i −0.0541313 + 0.680402i
\(786\) 0 0
\(787\) 14.2339i 0.507384i −0.967285 0.253692i \(-0.918355\pi\)
0.967285 0.253692i \(-0.0816449\pi\)
\(788\) 0 0
\(789\) −49.5891 49.5891i −1.76542 1.76542i
\(790\) 0 0
\(791\) −1.49114 −0.0530189
\(792\) 0 0
\(793\) 1.91713 1.91713i 0.0680794 0.0680794i
\(794\) 0 0
\(795\) −6.01244 + 75.5732i −0.213239 + 2.68030i
\(796\) 0 0
\(797\) −19.8283 −0.702353 −0.351176 0.936309i \(-0.614218\pi\)
−0.351176 + 0.936309i \(0.614218\pi\)
\(798\) 0 0
\(799\) 2.62729 0.0929467
\(800\) 0 0
\(801\) −91.0655 −3.21764
\(802\) 0 0
\(803\) 6.63568 0.234168
\(804\) 0 0
\(805\) −0.169242 + 2.12728i −0.00596499 + 0.0749767i
\(806\) 0 0
\(807\) −47.1817 + 47.1817i −1.66088 + 1.66088i
\(808\) 0 0
\(809\) 21.3864 0.751907 0.375954 0.926639i \(-0.377315\pi\)
0.375954 + 0.926639i \(0.377315\pi\)
\(810\) 0 0
\(811\) 9.90624 + 9.90624i 0.347855 + 0.347855i 0.859310 0.511455i \(-0.170893\pi\)
−0.511455 + 0.859310i \(0.670893\pi\)
\(812\) 0 0
\(813\) 36.6553i 1.28556i
\(814\) 0 0
\(815\) 0.634504 7.97538i 0.0222257 0.279365i
\(816\) 0 0
\(817\) −15.2195 + 15.2195i −0.532463 + 0.532463i
\(818\) 0 0
\(819\) 1.03932 + 1.03932i 0.0363168 + 0.0363168i
\(820\) 0 0
\(821\) −22.6209 + 22.6209i −0.789474 + 0.789474i −0.981408 0.191934i \(-0.938524\pi\)
0.191934 + 0.981408i \(0.438524\pi\)
\(822\) 0 0
\(823\) 4.89892 + 4.89892i 0.170766 + 0.170766i 0.787316 0.616550i \(-0.211470\pi\)
−0.616550 + 0.787316i \(0.711470\pi\)
\(824\) 0 0
\(825\) 36.3950 + 50.2731i 1.26711 + 1.75028i
\(826\) 0 0
\(827\) −1.05434 −0.0366630 −0.0183315 0.999832i \(-0.505835\pi\)
−0.0183315 + 0.999832i \(0.505835\pi\)
\(828\) 0 0
\(829\) −11.7754 + 11.7754i −0.408978 + 0.408978i −0.881382 0.472404i \(-0.843386\pi\)
0.472404 + 0.881382i \(0.343386\pi\)
\(830\) 0 0
\(831\) 62.3580i 2.16317i
\(832\) 0 0
\(833\) 2.08973 + 2.08973i 0.0724050 + 0.0724050i
\(834\) 0 0
\(835\) −0.990707 1.16197i −0.0342848 0.0402116i
\(836\) 0 0
\(837\) 34.6879i 1.19899i
\(838\) 0 0
\(839\) 41.1678i 1.42127i 0.703560 + 0.710636i \(0.251593\pi\)
−0.703560 + 0.710636i \(0.748407\pi\)
\(840\) 0 0
\(841\) 28.8322i 0.994214i
\(842\) 0 0
\(843\) 31.6749i 1.09094i
\(844\) 0 0
\(845\) 1.87479 23.5651i 0.0644948 0.810666i
\(846\) 0 0
\(847\) −0.750366 0.750366i −0.0257829 0.0257829i
\(848\) 0 0
\(849\) 37.3508i 1.28188i
\(850\) 0 0
\(851\) −6.78419 + 6.78419i −0.232559 + 0.232559i
\(852\) 0 0
\(853\) −11.7179 −0.401212 −0.200606 0.979672i \(-0.564291\pi\)
−0.200606 + 0.979672i \(0.564291\pi\)
\(854\) 0 0
\(855\) 31.5511 26.9008i 1.07902 0.919986i
\(856\) 0 0
\(857\) −12.2154 12.2154i −0.417270 0.417270i 0.466992 0.884262i \(-0.345338\pi\)
−0.884262 + 0.466992i \(0.845338\pi\)
\(858\) 0 0
\(859\) −17.2170 + 17.2170i −0.587436 + 0.587436i −0.936936 0.349500i \(-0.886351\pi\)
0.349500 + 0.936936i \(0.386351\pi\)
\(860\) 0 0
\(861\) 2.59984 + 2.59984i 0.0886024 + 0.0886024i
\(862\) 0 0
\(863\) −11.1929 + 11.1929i −0.381011 + 0.381011i −0.871466 0.490455i \(-0.836831\pi\)
0.490455 + 0.871466i \(0.336831\pi\)
\(864\) 0 0
\(865\) −26.3447 2.09593i −0.895746 0.0712637i
\(866\) 0 0
\(867\) 49.8828i 1.69411i
\(868\) 0 0
\(869\) −10.7204 10.7204i −0.363665 0.363665i
\(870\) 0 0
\(871\) 3.88522 0.131646
\(872\) 0 0
\(873\) 56.1647 56.1647i 1.90089 1.90089i
\(874\) 0 0
\(875\) −0.946933 1.55415i −0.0320122 0.0525397i
\(876\) 0 0
\(877\) −43.1739 −1.45788 −0.728940 0.684578i \(-0.759987\pi\)
−0.728940 + 0.684578i \(0.759987\pi\)
\(878\) 0 0
\(879\) −10.1760 −0.343227
\(880\) 0 0
\(881\) −33.4204 −1.12596 −0.562981 0.826470i \(-0.690346\pi\)
−0.562981 + 0.826470i \(0.690346\pi\)
\(882\) 0 0
\(883\) −2.00362 −0.0674270 −0.0337135 0.999432i \(-0.510733\pi\)
−0.0337135 + 0.999432i \(0.510733\pi\)
\(884\) 0 0
\(885\) 9.94802 + 11.6677i 0.334399 + 0.392206i
\(886\) 0 0
\(887\) −16.1765 + 16.1765i −0.543154 + 0.543154i −0.924452 0.381298i \(-0.875477\pi\)
0.381298 + 0.924452i \(0.375477\pi\)
\(888\) 0 0
\(889\) 2.82586 0.0947762
\(890\) 0 0
\(891\) −21.2971 21.2971i −0.713480 0.713480i
\(892\) 0 0
\(893\) 19.8371i 0.663822i
\(894\) 0 0
\(895\) 11.3466 9.67419i 0.379274 0.323373i
\(896\) 0 0
\(897\) 19.1577 19.1577i 0.639658 0.639658i
\(898\) 0 0
\(899\) −1.21223 1.21223i −0.0404303 0.0404303i
\(900\) 0 0
\(901\) −3.42593 + 3.42593i −0.114134 + 0.114134i
\(902\) 0 0
\(903\) −2.29606 2.29606i −0.0764080 0.0764080i
\(904\) 0 0
\(905\) 27.0501 + 31.7262i 0.899176 + 1.05462i
\(906\) 0 0
\(907\) 29.7116 0.986559 0.493279 0.869871i \(-0.335798\pi\)
0.493279 + 0.869871i \(0.335798\pi\)
\(908\) 0 0
\(909\) 74.2131 74.2131i 2.46149 2.46149i
\(910\) 0 0
\(911\) 44.6931i 1.48075i −0.672195 0.740374i \(-0.734648\pi\)
0.672195 0.740374i \(-0.265352\pi\)
\(912\) 0 0
\(913\) 4.81328 + 4.81328i 0.159296 + 0.159296i
\(914\) 0 0
\(915\) 8.78009 7.48599i 0.290261 0.247479i
\(916\) 0 0
\(917\) 1.84061i 0.0607821i
\(918\) 0 0
\(919\) 40.1278i 1.32369i 0.749639 + 0.661847i \(0.230227\pi\)
−0.749639 + 0.661847i \(0.769773\pi\)
\(920\) 0 0
\(921\) 35.0250i 1.15411i
\(922\) 0 0
\(923\) 12.4673i 0.410365i
\(924\) 0 0
\(925\) 1.29372 8.07922i 0.0425372 0.265643i
\(926\) 0 0
\(927\) −25.1345 25.1345i −0.825525 0.825525i
\(928\) 0 0
\(929\) 27.7519i 0.910512i −0.890361 0.455256i \(-0.849548\pi\)
0.890361 0.455256i \(-0.150452\pi\)
\(930\) 0 0
\(931\) −15.7783 + 15.7783i −0.517114 + 0.517114i
\(932\) 0 0
\(933\) 67.1004 2.19677
\(934\) 0 0
\(935\) −0.314565 + 3.95391i −0.0102874 + 0.129307i
\(936\) 0 0
\(937\) −17.2805 17.2805i −0.564531 0.564531i 0.366060 0.930591i \(-0.380706\pi\)
−0.930591 + 0.366060i \(0.880706\pi\)
\(938\) 0 0
\(939\) 21.0245 21.0245i 0.686110 0.686110i
\(940\) 0 0
\(941\) 4.81532 + 4.81532i 0.156975 + 0.156975i 0.781225 0.624250i \(-0.214595\pi\)
−0.624250 + 0.781225i \(0.714595\pi\)
\(942\) 0 0
\(943\) 31.5759 31.5759i 1.02825 1.02825i
\(944\) 0 0
\(945\) 1.95733 + 2.29569i 0.0636719 + 0.0746788i
\(946\) 0 0
\(947\) 3.37347i 0.109623i 0.998497 + 0.0548115i \(0.0174558\pi\)
−0.998497 + 0.0548115i \(0.982544\pi\)
\(948\) 0 0
\(949\) 1.74680 + 1.74680i 0.0567034 + 0.0567034i
\(950\) 0 0
\(951\) 74.5153 2.41632
\(952\) 0 0
\(953\) 14.3663 14.3663i 0.465369 0.465369i −0.435041 0.900410i \(-0.643266\pi\)
0.900410 + 0.435041i \(0.143266\pi\)
\(954\) 0 0
\(955\) 31.1778 + 2.48044i 1.00889 + 0.0802652i
\(956\) 0 0
\(957\) 5.08475 0.164366
\(958\) 0 0
\(959\) 0.710289 0.0229364
\(960\) 0 0
\(961\) 13.4851 0.435003
\(962\) 0 0
\(963\) 69.4263 2.23723
\(964\) 0 0
\(965\) 9.35775 7.97851i 0.301236 0.256837i
\(966\) 0 0
\(967\) −11.8576 + 11.8576i −0.381315 + 0.381315i −0.871576 0.490260i \(-0.836902\pi\)
0.490260 + 0.871576i \(0.336902\pi\)
\(968\) 0 0
\(969\) 4.02156 0.129191
\(970\) 0 0
\(971\) 14.6082 + 14.6082i 0.468799 + 0.468799i 0.901525 0.432726i \(-0.142448\pi\)
−0.432726 + 0.901525i \(0.642448\pi\)
\(972\) 0 0
\(973\) 2.81319i 0.0901869i
\(974\) 0 0
\(975\) −3.65330 + 22.8147i −0.116999 + 0.730656i
\(976\) 0 0
\(977\) 12.9249 12.9249i 0.413504 0.413504i −0.469454 0.882957i \(-0.655549\pi\)
0.882957 + 0.469454i \(0.155549\pi\)
\(978\) 0 0
\(979\) 46.5108 + 46.5108i 1.48649 + 1.48649i
\(980\) 0 0
\(981\) −23.2665 + 23.2665i −0.742843 + 0.742843i
\(982\) 0 0
\(983\) −0.133323 0.133323i −0.00425235 0.00425235i 0.704977 0.709230i \(-0.250957\pi\)
−0.709230 + 0.704977i \(0.750957\pi\)
\(984\) 0 0
\(985\) −49.7690 3.95951i −1.58577 0.126161i
\(986\) 0 0
\(987\) −2.99268 −0.0952580
\(988\) 0 0
\(989\) −27.8863 + 27.8863i −0.886733 + 0.886733i
\(990\) 0 0
\(991\) 47.9032i 1.52170i −0.648930 0.760848i \(-0.724783\pi\)
0.648930 0.760848i \(-0.275217\pi\)
\(992\) 0 0
\(993\) −17.2251 17.2251i −0.546624 0.546624i
\(994\) 0 0
\(995\) 21.9302 + 1.74472i 0.695234 + 0.0553114i
\(996\) 0 0
\(997\) 54.9379i 1.73990i −0.493138 0.869951i \(-0.664150\pi\)
0.493138 0.869951i \(-0.335850\pi\)
\(998\) 0 0
\(999\) 13.5635i 0.429129i
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 640.2.s.d.223.9 18
4.3 odd 2 640.2.s.c.223.1 18
5.2 odd 4 640.2.j.d.607.1 18
8.3 odd 2 320.2.s.b.303.9 18
8.5 even 2 80.2.s.b.3.5 yes 18
16.3 odd 4 80.2.j.b.43.1 18
16.5 even 4 640.2.j.c.543.1 18
16.11 odd 4 640.2.j.d.543.9 18
16.13 even 4 320.2.j.b.143.9 18
20.7 even 4 640.2.j.c.607.9 18
24.5 odd 2 720.2.z.g.163.5 18
40.3 even 4 1600.2.j.d.1007.9 18
40.13 odd 4 400.2.j.d.307.9 18
40.19 odd 2 1600.2.s.d.943.1 18
40.27 even 4 320.2.j.b.47.1 18
40.29 even 2 400.2.s.d.243.5 18
40.37 odd 4 80.2.j.b.67.1 yes 18
48.35 even 4 720.2.bd.g.523.9 18
80.3 even 4 400.2.s.d.107.5 18
80.13 odd 4 1600.2.s.d.207.1 18
80.19 odd 4 400.2.j.d.43.9 18
80.27 even 4 inner 640.2.s.d.287.9 18
80.29 even 4 1600.2.j.d.143.1 18
80.37 odd 4 640.2.s.c.287.1 18
80.67 even 4 80.2.s.b.27.5 yes 18
80.77 odd 4 320.2.s.b.207.9 18
120.77 even 4 720.2.bd.g.307.9 18
240.227 odd 4 720.2.z.g.667.5 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.j.b.43.1 18 16.3 odd 4
80.2.j.b.67.1 yes 18 40.37 odd 4
80.2.s.b.3.5 yes 18 8.5 even 2
80.2.s.b.27.5 yes 18 80.67 even 4
320.2.j.b.47.1 18 40.27 even 4
320.2.j.b.143.9 18 16.13 even 4
320.2.s.b.207.9 18 80.77 odd 4
320.2.s.b.303.9 18 8.3 odd 2
400.2.j.d.43.9 18 80.19 odd 4
400.2.j.d.307.9 18 40.13 odd 4
400.2.s.d.107.5 18 80.3 even 4
400.2.s.d.243.5 18 40.29 even 2
640.2.j.c.543.1 18 16.5 even 4
640.2.j.c.607.9 18 20.7 even 4
640.2.j.d.543.9 18 16.11 odd 4
640.2.j.d.607.1 18 5.2 odd 4
640.2.s.c.223.1 18 4.3 odd 2
640.2.s.c.287.1 18 80.37 odd 4
640.2.s.d.223.9 18 1.1 even 1 trivial
640.2.s.d.287.9 18 80.27 even 4 inner
720.2.z.g.163.5 18 24.5 odd 2
720.2.z.g.667.5 18 240.227 odd 4
720.2.bd.g.307.9 18 120.77 even 4
720.2.bd.g.523.9 18 48.35 even 4
1600.2.j.d.143.1 18 80.29 even 4
1600.2.j.d.1007.9 18 40.3 even 4
1600.2.s.d.207.1 18 80.13 odd 4
1600.2.s.d.943.1 18 40.19 odd 2