Properties

Label 672.2.bk.a.529.13
Level $672$
Weight $2$
Character 672.529
Analytic conductor $5.366$
Analytic rank $0$
Dimension $32$
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] = [672,2,Mod(529,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.bk (of order \(6\), 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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 529.13
Character \(\chi\) \(=\) 672.529
Dual form 672.2.bk.a.625.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.866025 + 0.500000i) q^{3} +(-0.0402223 + 0.0232224i) q^{5} +(1.97032 + 1.76574i) q^{7} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(0.866025 + 0.500000i) q^{3} +(-0.0402223 + 0.0232224i) q^{5} +(1.97032 + 1.76574i) q^{7} +(0.500000 + 0.866025i) q^{9} +(3.11586 + 1.79894i) q^{11} -6.29415i q^{13} -0.0464447 q^{15} +(0.258110 - 0.447060i) q^{17} +(-2.80834 + 1.62140i) q^{19} +(0.823477 + 2.51434i) q^{21} +(3.47322 + 6.01578i) q^{23} +(-2.49892 + 4.32826i) q^{25} +1.00000i q^{27} +2.29579i q^{29} +(1.05460 - 1.82662i) q^{31} +(1.79894 + 3.11586i) q^{33} +(-0.120255 - 0.0252667i) q^{35} +(1.12720 - 0.650790i) q^{37} +(3.14708 - 5.45089i) q^{39} +10.1883 q^{41} -9.12162i q^{43} +(-0.0402223 - 0.0232224i) q^{45} +(2.32465 + 4.02641i) q^{47} +(0.764321 + 6.95815i) q^{49} +(0.447060 - 0.258110i) q^{51} +(-4.91727 - 2.83899i) q^{53} -0.167103 q^{55} -3.24279 q^{57} +(-1.42910 - 0.825090i) q^{59} +(1.10386 - 0.637312i) q^{61} +(-0.544016 + 2.58922i) q^{63} +(0.146165 + 0.253165i) q^{65} +(-6.72366 - 3.88191i) q^{67} +6.94643i q^{69} -11.8320 q^{71} +(5.57028 - 9.64802i) q^{73} +(-4.32826 + 2.49892i) q^{75} +(2.96278 + 9.04630i) q^{77} +(2.75086 + 4.76463i) q^{79} +(-0.500000 + 0.866025i) q^{81} +8.25030i q^{83} +0.0239757i q^{85} +(-1.14789 + 1.98821i) q^{87} +(-7.38218 - 12.7863i) q^{89} +(11.1138 - 12.4015i) q^{91} +(1.82662 - 1.05460i) q^{93} +(0.0753053 - 0.130433i) q^{95} -6.16464 q^{97} +3.59789i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 16 q^{9} + 8 q^{23} + 16 q^{25} + 24 q^{31} + 24 q^{47} + 8 q^{49} + 64 q^{55} - 16 q^{57} + 80 q^{71} + 8 q^{73} - 8 q^{79} - 16 q^{81} - 24 q^{87} - 24 q^{95} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 0.866025 + 0.500000i 0.500000 + 0.288675i
\(4\) 0 0
\(5\) −0.0402223 + 0.0232224i −0.0179880 + 0.0103854i −0.508967 0.860786i \(-0.669972\pi\)
0.490979 + 0.871171i \(0.336639\pi\)
\(6\) 0 0
\(7\) 1.97032 + 1.76574i 0.744711 + 0.667387i
\(8\) 0 0
\(9\) 0.500000 + 0.866025i 0.166667 + 0.288675i
\(10\) 0 0
\(11\) 3.11586 + 1.79894i 0.939468 + 0.542402i 0.889793 0.456364i \(-0.150848\pi\)
0.0496743 + 0.998765i \(0.484182\pi\)
\(12\) 0 0
\(13\) 6.29415i 1.74568i −0.488004 0.872842i \(-0.662275\pi\)
0.488004 0.872842i \(-0.337725\pi\)
\(14\) 0 0
\(15\) −0.0464447 −0.0119920
\(16\) 0 0
\(17\) 0.258110 0.447060i 0.0626010 0.108428i −0.833026 0.553233i \(-0.813394\pi\)
0.895627 + 0.444805i \(0.146727\pi\)
\(18\) 0 0
\(19\) −2.80834 + 1.62140i −0.644278 + 0.371974i −0.786261 0.617895i \(-0.787986\pi\)
0.141983 + 0.989869i \(0.454652\pi\)
\(20\) 0 0
\(21\) 0.823477 + 2.51434i 0.179697 + 0.548673i
\(22\) 0 0
\(23\) 3.47322 + 6.01578i 0.724215 + 1.25438i 0.959296 + 0.282402i \(0.0911312\pi\)
−0.235081 + 0.971976i \(0.575535\pi\)
\(24\) 0 0
\(25\) −2.49892 + 4.32826i −0.499784 + 0.865652i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 2.29579i 0.426317i 0.977018 + 0.213159i \(0.0683751\pi\)
−0.977018 + 0.213159i \(0.931625\pi\)
\(30\) 0 0
\(31\) 1.05460 1.82662i 0.189411 0.328070i −0.755643 0.654984i \(-0.772675\pi\)
0.945054 + 0.326914i \(0.106009\pi\)
\(32\) 0 0
\(33\) 1.79894 + 3.11586i 0.313156 + 0.542402i
\(34\) 0 0
\(35\) −0.120255 0.0252667i −0.0203269 0.00427085i
\(36\) 0 0
\(37\) 1.12720 0.650790i 0.185311 0.106989i −0.404475 0.914549i \(-0.632546\pi\)
0.589785 + 0.807560i \(0.299212\pi\)
\(38\) 0 0
\(39\) 3.14708 5.45089i 0.503935 0.872842i
\(40\) 0 0
\(41\) 10.1883 1.59115 0.795576 0.605854i \(-0.207168\pi\)
0.795576 + 0.605854i \(0.207168\pi\)
\(42\) 0 0
\(43\) 9.12162i 1.39103i −0.718510 0.695517i \(-0.755176\pi\)
0.718510 0.695517i \(-0.244824\pi\)
\(44\) 0 0
\(45\) −0.0402223 0.0232224i −0.00599599 0.00346178i
\(46\) 0 0
\(47\) 2.32465 + 4.02641i 0.339085 + 0.587313i 0.984261 0.176722i \(-0.0565492\pi\)
−0.645176 + 0.764034i \(0.723216\pi\)
\(48\) 0 0
\(49\) 0.764321 + 6.95815i 0.109189 + 0.994021i
\(50\) 0 0
\(51\) 0.447060 0.258110i 0.0626010 0.0361427i
\(52\) 0 0
\(53\) −4.91727 2.83899i −0.675439 0.389965i 0.122696 0.992444i \(-0.460846\pi\)
−0.798134 + 0.602480i \(0.794179\pi\)
\(54\) 0 0
\(55\) −0.167103 −0.0225321
\(56\) 0 0
\(57\) −3.24279 −0.429519
\(58\) 0 0
\(59\) −1.42910 0.825090i −0.186053 0.107418i 0.404081 0.914723i \(-0.367591\pi\)
−0.590133 + 0.807306i \(0.700925\pi\)
\(60\) 0 0
\(61\) 1.10386 0.637312i 0.141334 0.0815995i −0.427665 0.903937i \(-0.640664\pi\)
0.569000 + 0.822338i \(0.307331\pi\)
\(62\) 0 0
\(63\) −0.544016 + 2.58922i −0.0685396 + 0.326211i
\(64\) 0 0
\(65\) 0.146165 + 0.253165i 0.0181295 + 0.0314013i
\(66\) 0 0
\(67\) −6.72366 3.88191i −0.821426 0.474251i 0.0294820 0.999565i \(-0.490614\pi\)
−0.850908 + 0.525315i \(0.823948\pi\)
\(68\) 0 0
\(69\) 6.94643i 0.836252i
\(70\) 0 0
\(71\) −11.8320 −1.40420 −0.702099 0.712080i \(-0.747753\pi\)
−0.702099 + 0.712080i \(0.747753\pi\)
\(72\) 0 0
\(73\) 5.57028 9.64802i 0.651953 1.12921i −0.330696 0.943737i \(-0.607283\pi\)
0.982649 0.185478i \(-0.0593832\pi\)
\(74\) 0 0
\(75\) −4.32826 + 2.49892i −0.499784 + 0.288551i
\(76\) 0 0
\(77\) 2.96278 + 9.04630i 0.337640 + 1.03092i
\(78\) 0 0
\(79\) 2.75086 + 4.76463i 0.309496 + 0.536063i 0.978252 0.207419i \(-0.0665063\pi\)
−0.668756 + 0.743482i \(0.733173\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 8.25030i 0.905588i 0.891615 + 0.452794i \(0.149573\pi\)
−0.891615 + 0.452794i \(0.850427\pi\)
\(84\) 0 0
\(85\) 0.0239757i 0.00260053i
\(86\) 0 0
\(87\) −1.14789 + 1.98821i −0.123067 + 0.213159i
\(88\) 0 0
\(89\) −7.38218 12.7863i −0.782510 1.35535i −0.930475 0.366354i \(-0.880606\pi\)
0.147966 0.988993i \(-0.452727\pi\)
\(90\) 0 0
\(91\) 11.1138 12.4015i 1.16505 1.30003i
\(92\) 0 0
\(93\) 1.82662 1.05460i 0.189411 0.109357i
\(94\) 0 0
\(95\) 0.0753053 0.130433i 0.00772616 0.0133821i
\(96\) 0 0
\(97\) −6.16464 −0.625925 −0.312962 0.949766i \(-0.601321\pi\)
−0.312962 + 0.949766i \(0.601321\pi\)
\(98\) 0 0
\(99\) 3.59789i 0.361601i
\(100\) 0 0
\(101\) 6.93470 + 4.00375i 0.690029 + 0.398388i 0.803623 0.595139i \(-0.202903\pi\)
−0.113594 + 0.993527i \(0.536236\pi\)
\(102\) 0 0
\(103\) −6.43821 11.1513i −0.634376 1.09877i −0.986647 0.162873i \(-0.947924\pi\)
0.352271 0.935898i \(-0.385409\pi\)
\(104\) 0 0
\(105\) −0.0915109 0.0820093i −0.00893055 0.00800329i
\(106\) 0 0
\(107\) 8.84815 5.10848i 0.855383 0.493856i −0.00708048 0.999975i \(-0.502254\pi\)
0.862463 + 0.506119i \(0.168920\pi\)
\(108\) 0 0
\(109\) −10.1275 5.84714i −0.970043 0.560054i −0.0707935 0.997491i \(-0.522553\pi\)
−0.899249 + 0.437437i \(0.855886\pi\)
\(110\) 0 0
\(111\) 1.30158 0.123540
\(112\) 0 0
\(113\) −9.08476 −0.854623 −0.427311 0.904105i \(-0.640539\pi\)
−0.427311 + 0.904105i \(0.640539\pi\)
\(114\) 0 0
\(115\) −0.279401 0.161312i −0.0260543 0.0150425i
\(116\) 0 0
\(117\) 5.45089 3.14708i 0.503935 0.290947i
\(118\) 0 0
\(119\) 1.29795 0.425096i 0.118983 0.0389685i
\(120\) 0 0
\(121\) 0.972397 + 1.68424i 0.0883997 + 0.153113i
\(122\) 0 0
\(123\) 8.82337 + 5.09417i 0.795576 + 0.459326i
\(124\) 0 0
\(125\) 0.464347i 0.0415324i
\(126\) 0 0
\(127\) −14.1059 −1.25170 −0.625849 0.779944i \(-0.715247\pi\)
−0.625849 + 0.779944i \(0.715247\pi\)
\(128\) 0 0
\(129\) 4.56081 7.89955i 0.401557 0.695517i
\(130\) 0 0
\(131\) −14.8183 + 8.55535i −1.29468 + 0.747484i −0.979480 0.201541i \(-0.935405\pi\)
−0.315200 + 0.949025i \(0.602072\pi\)
\(132\) 0 0
\(133\) −8.39630 1.76413i −0.728052 0.152970i
\(134\) 0 0
\(135\) −0.0232224 0.0402223i −0.00199866 0.00346178i
\(136\) 0 0
\(137\) 2.48594 4.30578i 0.212388 0.367867i −0.740073 0.672526i \(-0.765209\pi\)
0.952462 + 0.304659i \(0.0985425\pi\)
\(138\) 0 0
\(139\) 5.20468i 0.441455i −0.975335 0.220728i \(-0.929157\pi\)
0.975335 0.220728i \(-0.0708432\pi\)
\(140\) 0 0
\(141\) 4.64930i 0.391542i
\(142\) 0 0
\(143\) 11.3228 19.6117i 0.946862 1.64001i
\(144\) 0 0
\(145\) −0.0533136 0.0923419i −0.00442746 0.00766858i
\(146\) 0 0
\(147\) −2.81715 + 6.40809i −0.232355 + 0.528531i
\(148\) 0 0
\(149\) 0.725874 0.419084i 0.0594659 0.0343327i −0.469972 0.882681i \(-0.655736\pi\)
0.529438 + 0.848349i \(0.322403\pi\)
\(150\) 0 0
\(151\) 3.21803 5.57379i 0.261879 0.453588i −0.704862 0.709345i \(-0.748991\pi\)
0.966741 + 0.255756i \(0.0823244\pi\)
\(152\) 0 0
\(153\) 0.516221 0.0417340
\(154\) 0 0
\(155\) 0.0979610i 0.00786842i
\(156\) 0 0
\(157\) 4.73349 + 2.73288i 0.377774 + 0.218108i 0.676849 0.736122i \(-0.263345\pi\)
−0.299075 + 0.954229i \(0.596678\pi\)
\(158\) 0 0
\(159\) −2.83899 4.91727i −0.225146 0.389965i
\(160\) 0 0
\(161\) −3.77897 + 17.9858i −0.297825 + 1.41748i
\(162\) 0 0
\(163\) −13.2105 + 7.62708i −1.03473 + 0.597399i −0.918335 0.395805i \(-0.870466\pi\)
−0.116390 + 0.993204i \(0.537132\pi\)
\(164\) 0 0
\(165\) −0.144715 0.0835514i −0.0112661 0.00650447i
\(166\) 0 0
\(167\) 1.77463 0.137325 0.0686625 0.997640i \(-0.478127\pi\)
0.0686625 + 0.997640i \(0.478127\pi\)
\(168\) 0 0
\(169\) −26.6163 −2.04741
\(170\) 0 0
\(171\) −2.80834 1.62140i −0.214759 0.123991i
\(172\) 0 0
\(173\) 2.23631 1.29114i 0.170024 0.0981633i −0.412573 0.910924i \(-0.635370\pi\)
0.582597 + 0.812761i \(0.302037\pi\)
\(174\) 0 0
\(175\) −12.5663 + 4.11561i −0.949920 + 0.311111i
\(176\) 0 0
\(177\) −0.825090 1.42910i −0.0620175 0.107418i
\(178\) 0 0
\(179\) 17.0438 + 9.84027i 1.27392 + 0.735496i 0.975723 0.219009i \(-0.0702824\pi\)
0.298194 + 0.954505i \(0.403616\pi\)
\(180\) 0 0
\(181\) 25.5613i 1.89996i −0.312315 0.949979i \(-0.601104\pi\)
0.312315 0.949979i \(-0.398896\pi\)
\(182\) 0 0
\(183\) 1.27462 0.0942230
\(184\) 0 0
\(185\) −0.0302257 + 0.0523525i −0.00222224 + 0.00384903i
\(186\) 0 0
\(187\) 1.60847 0.928652i 0.117623 0.0679098i
\(188\) 0 0
\(189\) −1.76574 + 1.97032i −0.128439 + 0.143320i
\(190\) 0 0
\(191\) −6.98804 12.1036i −0.505637 0.875788i −0.999979 0.00652098i \(-0.997924\pi\)
0.494342 0.869267i \(-0.335409\pi\)
\(192\) 0 0
\(193\) 9.00706 15.6007i 0.648342 1.12296i −0.335176 0.942155i \(-0.608796\pi\)
0.983519 0.180806i \(-0.0578707\pi\)
\(194\) 0 0
\(195\) 0.292330i 0.0209342i
\(196\) 0 0
\(197\) 3.70082i 0.263672i −0.991272 0.131836i \(-0.957913\pi\)
0.991272 0.131836i \(-0.0420873\pi\)
\(198\) 0 0
\(199\) −0.882192 + 1.52800i −0.0625369 + 0.108317i −0.895599 0.444863i \(-0.853252\pi\)
0.833062 + 0.553180i \(0.186586\pi\)
\(200\) 0 0
\(201\) −3.88191 6.72366i −0.273809 0.474251i
\(202\) 0 0
\(203\) −4.05377 + 4.52344i −0.284519 + 0.317483i
\(204\) 0 0
\(205\) −0.409799 + 0.236597i −0.0286216 + 0.0165247i
\(206\) 0 0
\(207\) −3.47322 + 6.01578i −0.241405 + 0.418126i
\(208\) 0 0
\(209\) −11.6672 −0.807038
\(210\) 0 0
\(211\) 10.6757i 0.734945i 0.930034 + 0.367473i \(0.119777\pi\)
−0.930034 + 0.367473i \(0.880223\pi\)
\(212\) 0 0
\(213\) −10.2468 5.91599i −0.702099 0.405357i
\(214\) 0 0
\(215\) 0.211825 + 0.366892i 0.0144464 + 0.0250218i
\(216\) 0 0
\(217\) 5.30323 1.73688i 0.360007 0.117907i
\(218\) 0 0
\(219\) 9.64802 5.57028i 0.651953 0.376405i
\(220\) 0 0
\(221\) −2.81386 1.62459i −0.189281 0.109281i
\(222\) 0 0
\(223\) 22.2212 1.48804 0.744020 0.668158i \(-0.232917\pi\)
0.744020 + 0.668158i \(0.232917\pi\)
\(224\) 0 0
\(225\) −4.99784 −0.333190
\(226\) 0 0
\(227\) −17.0728 9.85697i −1.13316 0.654230i −0.188432 0.982086i \(-0.560340\pi\)
−0.944728 + 0.327856i \(0.893674\pi\)
\(228\) 0 0
\(229\) −6.06806 + 3.50340i −0.400989 + 0.231511i −0.686911 0.726742i \(-0.741034\pi\)
0.285922 + 0.958253i \(0.407700\pi\)
\(230\) 0 0
\(231\) −1.95731 + 9.31571i −0.128781 + 0.612929i
\(232\) 0 0
\(233\) 12.3717 + 21.4284i 0.810495 + 1.40382i 0.912518 + 0.409036i \(0.134135\pi\)
−0.102023 + 0.994782i \(0.532532\pi\)
\(234\) 0 0
\(235\) −0.187006 0.107968i −0.0121989 0.00704303i
\(236\) 0 0
\(237\) 5.50172i 0.357375i
\(238\) 0 0
\(239\) −21.9503 −1.41985 −0.709924 0.704278i \(-0.751271\pi\)
−0.709924 + 0.704278i \(0.751271\pi\)
\(240\) 0 0
\(241\) 7.39162 12.8027i 0.476136 0.824692i −0.523490 0.852032i \(-0.675370\pi\)
0.999626 + 0.0273397i \(0.00870359\pi\)
\(242\) 0 0
\(243\) −0.866025 + 0.500000i −0.0555556 + 0.0320750i
\(244\) 0 0
\(245\) −0.192327 0.262123i −0.0122873 0.0167464i
\(246\) 0 0
\(247\) 10.2053 + 17.6761i 0.649349 + 1.12471i
\(248\) 0 0
\(249\) −4.12515 + 7.14497i −0.261421 + 0.452794i
\(250\) 0 0
\(251\) 14.4785i 0.913876i −0.889499 0.456938i \(-0.848946\pi\)
0.889499 0.456938i \(-0.151054\pi\)
\(252\) 0 0
\(253\) 24.9925i 1.57126i
\(254\) 0 0
\(255\) −0.0119879 + 0.0207636i −0.000750709 + 0.00130027i
\(256\) 0 0
\(257\) −7.93731 13.7478i −0.495116 0.857566i 0.504868 0.863196i \(-0.331541\pi\)
−0.999984 + 0.00563049i \(0.998208\pi\)
\(258\) 0 0
\(259\) 3.37007 + 0.708080i 0.209406 + 0.0439980i
\(260\) 0 0
\(261\) −1.98821 + 1.14789i −0.123067 + 0.0710529i
\(262\) 0 0
\(263\) 0.162710 0.281822i 0.0100331 0.0173779i −0.860965 0.508664i \(-0.830140\pi\)
0.870998 + 0.491286i \(0.163473\pi\)
\(264\) 0 0
\(265\) 0.263712 0.0161997
\(266\) 0 0
\(267\) 14.7644i 0.903565i
\(268\) 0 0
\(269\) −20.3761 11.7642i −1.24235 0.717273i −0.272780 0.962076i \(-0.587943\pi\)
−0.969573 + 0.244803i \(0.921277\pi\)
\(270\) 0 0
\(271\) 0.751567 + 1.30175i 0.0456544 + 0.0790758i 0.887950 0.459941i \(-0.152129\pi\)
−0.842295 + 0.539017i \(0.818796\pi\)
\(272\) 0 0
\(273\) 15.8256 5.18309i 0.957809 0.313695i
\(274\) 0 0
\(275\) −15.5726 + 8.99084i −0.939062 + 0.542168i
\(276\) 0 0
\(277\) 12.6685 + 7.31418i 0.761178 + 0.439466i 0.829719 0.558182i \(-0.188501\pi\)
−0.0685405 + 0.997648i \(0.521834\pi\)
\(278\) 0 0
\(279\) 2.10920 0.126274
\(280\) 0 0
\(281\) 21.3285 1.27235 0.636175 0.771545i \(-0.280516\pi\)
0.636175 + 0.771545i \(0.280516\pi\)
\(282\) 0 0
\(283\) 15.0068 + 8.66416i 0.892059 + 0.515031i 0.874616 0.484817i \(-0.161114\pi\)
0.0174438 + 0.999848i \(0.494447\pi\)
\(284\) 0 0
\(285\) 0.130433 0.0753053i 0.00772616 0.00446070i
\(286\) 0 0
\(287\) 20.0743 + 17.9900i 1.18495 + 1.06191i
\(288\) 0 0
\(289\) 8.36676 + 14.4917i 0.492162 + 0.852450i
\(290\) 0 0
\(291\) −5.33874 3.08232i −0.312962 0.180689i
\(292\) 0 0
\(293\) 29.6927i 1.73466i 0.497730 + 0.867332i \(0.334167\pi\)
−0.497730 + 0.867332i \(0.665833\pi\)
\(294\) 0 0
\(295\) 0.0766421 0.00446228
\(296\) 0 0
\(297\) −1.79894 + 3.11586i −0.104385 + 0.180801i
\(298\) 0 0
\(299\) 37.8643 21.8609i 2.18975 1.26425i
\(300\) 0 0
\(301\) 16.1064 17.9725i 0.928358 1.03592i
\(302\) 0 0
\(303\) 4.00375 + 6.93470i 0.230010 + 0.398388i
\(304\) 0 0
\(305\) −0.0295998 + 0.0512683i −0.00169488 + 0.00293562i
\(306\) 0 0
\(307\) 4.88243i 0.278655i 0.990246 + 0.139327i \(0.0444940\pi\)
−0.990246 + 0.139327i \(0.955506\pi\)
\(308\) 0 0
\(309\) 12.8764i 0.732514i
\(310\) 0 0
\(311\) 9.08277 15.7318i 0.515036 0.892069i −0.484811 0.874619i \(-0.661112\pi\)
0.999848 0.0174504i \(-0.00555493\pi\)
\(312\) 0 0
\(313\) 6.80533 + 11.7872i 0.384660 + 0.666251i 0.991722 0.128404i \(-0.0409853\pi\)
−0.607062 + 0.794655i \(0.707652\pi\)
\(314\) 0 0
\(315\) −0.0382461 0.116778i −0.00215493 0.00657967i
\(316\) 0 0
\(317\) 21.6407 12.4943i 1.21546 0.701749i 0.251520 0.967852i \(-0.419070\pi\)
0.963944 + 0.266103i \(0.0857363\pi\)
\(318\) 0 0
\(319\) −4.13000 + 7.15336i −0.231235 + 0.400511i
\(320\) 0 0
\(321\) 10.2170 0.570255
\(322\) 0 0
\(323\) 1.67400i 0.0931437i
\(324\) 0 0
\(325\) 27.2427 + 15.7286i 1.51115 + 0.872465i
\(326\) 0 0
\(327\) −5.84714 10.1275i −0.323348 0.560054i
\(328\) 0 0
\(329\) −2.52929 + 12.0380i −0.139445 + 0.663679i
\(330\) 0 0
\(331\) −6.34783 + 3.66492i −0.348908 + 0.201442i −0.664204 0.747551i \(-0.731229\pi\)
0.315296 + 0.948993i \(0.397896\pi\)
\(332\) 0 0
\(333\) 1.12720 + 0.650790i 0.0617702 + 0.0356631i
\(334\) 0 0
\(335\) 0.360588 0.0197010
\(336\) 0 0
\(337\) 1.49107 0.0812237 0.0406118 0.999175i \(-0.487069\pi\)
0.0406118 + 0.999175i \(0.487069\pi\)
\(338\) 0 0
\(339\) −7.86764 4.54238i −0.427311 0.246708i
\(340\) 0 0
\(341\) 6.57197 3.79433i 0.355892 0.205474i
\(342\) 0 0
\(343\) −10.7803 + 15.0594i −0.582083 + 0.813129i
\(344\) 0 0
\(345\) −0.161312 0.279401i −0.00868477 0.0150425i
\(346\) 0 0
\(347\) −20.6494 11.9219i −1.10852 0.640003i −0.170073 0.985431i \(-0.554400\pi\)
−0.938445 + 0.345428i \(0.887734\pi\)
\(348\) 0 0
\(349\) 11.5733i 0.619504i 0.950817 + 0.309752i \(0.100246\pi\)
−0.950817 + 0.309752i \(0.899754\pi\)
\(350\) 0 0
\(351\) 6.29415 0.335957
\(352\) 0 0
\(353\) −5.01488 + 8.68603i −0.266915 + 0.462311i −0.968064 0.250705i \(-0.919338\pi\)
0.701148 + 0.713015i \(0.252671\pi\)
\(354\) 0 0
\(355\) 0.475909 0.274766i 0.0252586 0.0145831i
\(356\) 0 0
\(357\) 1.33661 + 0.280832i 0.0707408 + 0.0148632i
\(358\) 0 0
\(359\) 1.80639 + 3.12876i 0.0953375 + 0.165129i 0.909749 0.415158i \(-0.136274\pi\)
−0.814412 + 0.580287i \(0.802940\pi\)
\(360\) 0 0
\(361\) −4.24214 + 7.34760i −0.223271 + 0.386716i
\(362\) 0 0
\(363\) 1.94479i 0.102075i
\(364\) 0 0
\(365\) 0.517420i 0.0270830i
\(366\) 0 0
\(367\) 11.0055 19.0620i 0.574480 0.995029i −0.421617 0.906774i \(-0.638537\pi\)
0.996098 0.0882554i \(-0.0281292\pi\)
\(368\) 0 0
\(369\) 5.09417 + 8.82337i 0.265192 + 0.459326i
\(370\) 0 0
\(371\) −4.67568 14.2763i −0.242749 0.741190i
\(372\) 0 0
\(373\) −27.9915 + 16.1609i −1.44934 + 0.836780i −0.998442 0.0557935i \(-0.982231\pi\)
−0.450903 + 0.892573i \(0.648898\pi\)
\(374\) 0 0
\(375\) 0.232173 0.402136i 0.0119894 0.0207662i
\(376\) 0 0
\(377\) 14.4500 0.744215
\(378\) 0 0
\(379\) 16.6668i 0.856115i 0.903751 + 0.428058i \(0.140802\pi\)
−0.903751 + 0.428058i \(0.859198\pi\)
\(380\) 0 0
\(381\) −12.2161 7.05296i −0.625849 0.361334i
\(382\) 0 0
\(383\) 12.1998 + 21.1307i 0.623383 + 1.07973i 0.988851 + 0.148907i \(0.0475755\pi\)
−0.365468 + 0.930824i \(0.619091\pi\)
\(384\) 0 0
\(385\) −0.329246 0.295060i −0.0167799 0.0150377i
\(386\) 0 0
\(387\) 7.89955 4.56081i 0.401557 0.231839i
\(388\) 0 0
\(389\) 7.38356 + 4.26290i 0.374362 + 0.216138i 0.675362 0.737486i \(-0.263987\pi\)
−0.301001 + 0.953624i \(0.597321\pi\)
\(390\) 0 0
\(391\) 3.58589 0.181346
\(392\) 0 0
\(393\) −17.1107 −0.863120
\(394\) 0 0
\(395\) −0.221292 0.127763i −0.0111344 0.00642845i
\(396\) 0 0
\(397\) −5.91805 + 3.41679i −0.297019 + 0.171484i −0.641103 0.767455i \(-0.721523\pi\)
0.344084 + 0.938939i \(0.388189\pi\)
\(398\) 0 0
\(399\) −6.38934 5.72593i −0.319867 0.286655i
\(400\) 0 0
\(401\) 5.24408 + 9.08301i 0.261877 + 0.453584i 0.966741 0.255759i \(-0.0823253\pi\)
−0.704864 + 0.709343i \(0.748992\pi\)
\(402\) 0 0
\(403\) −11.4970 6.63780i −0.572707 0.330652i
\(404\) 0 0
\(405\) 0.0464447i 0.00230786i
\(406\) 0 0
\(407\) 4.68294 0.232125
\(408\) 0 0
\(409\) −4.01466 + 6.95360i −0.198512 + 0.343833i −0.948046 0.318133i \(-0.896944\pi\)
0.749534 + 0.661966i \(0.230278\pi\)
\(410\) 0 0
\(411\) 4.30578 2.48594i 0.212388 0.122622i
\(412\) 0 0
\(413\) −1.35888 4.14911i −0.0668663 0.204164i
\(414\) 0 0
\(415\) −0.191591 0.331846i −0.00940485 0.0162897i
\(416\) 0 0
\(417\) 2.60234 4.50739i 0.127437 0.220728i
\(418\) 0 0
\(419\) 16.7264i 0.817139i 0.912727 + 0.408570i \(0.133972\pi\)
−0.912727 + 0.408570i \(0.866028\pi\)
\(420\) 0 0
\(421\) 14.7157i 0.717199i 0.933492 + 0.358599i \(0.116746\pi\)
−0.933492 + 0.358599i \(0.883254\pi\)
\(422\) 0 0
\(423\) −2.32465 + 4.02641i −0.113028 + 0.195771i
\(424\) 0 0
\(425\) 1.28999 + 2.23434i 0.0625739 + 0.108381i
\(426\) 0 0
\(427\) 3.30028 + 0.693416i 0.159712 + 0.0335568i
\(428\) 0 0
\(429\) 19.6117 11.3228i 0.946862 0.546671i
\(430\) 0 0
\(431\) −1.57036 + 2.71995i −0.0756416 + 0.131015i −0.901365 0.433060i \(-0.857434\pi\)
0.825723 + 0.564075i \(0.190767\pi\)
\(432\) 0 0
\(433\) 33.6748 1.61831 0.809155 0.587595i \(-0.199925\pi\)
0.809155 + 0.587595i \(0.199925\pi\)
\(434\) 0 0
\(435\) 0.106627i 0.00511239i
\(436\) 0 0
\(437\) −19.5080 11.2629i −0.933192 0.538779i
\(438\) 0 0
\(439\) −3.68016 6.37423i −0.175645 0.304225i 0.764740 0.644340i \(-0.222868\pi\)
−0.940384 + 0.340114i \(0.889534\pi\)
\(440\) 0 0
\(441\) −5.64377 + 4.14100i −0.268751 + 0.197190i
\(442\) 0 0
\(443\) 22.3955 12.9300i 1.06404 0.614325i 0.137494 0.990503i \(-0.456095\pi\)
0.926547 + 0.376178i \(0.122762\pi\)
\(444\) 0 0
\(445\) 0.593857 + 0.342863i 0.0281515 + 0.0162533i
\(446\) 0 0
\(447\) 0.838167 0.0396439
\(448\) 0 0
\(449\) −19.7508 −0.932096 −0.466048 0.884759i \(-0.654323\pi\)
−0.466048 + 0.884759i \(0.654323\pi\)
\(450\) 0 0
\(451\) 31.7455 + 18.3283i 1.49484 + 0.863044i
\(452\) 0 0
\(453\) 5.57379 3.21803i 0.261879 0.151196i
\(454\) 0 0
\(455\) −0.159032 + 0.756906i −0.00745555 + 0.0354843i
\(456\) 0 0
\(457\) −1.28552 2.22658i −0.0601339 0.104155i 0.834391 0.551173i \(-0.185819\pi\)
−0.894525 + 0.447018i \(0.852486\pi\)
\(458\) 0 0
\(459\) 0.447060 + 0.258110i 0.0208670 + 0.0120476i
\(460\) 0 0
\(461\) 30.2081i 1.40693i −0.710728 0.703467i \(-0.751634\pi\)
0.710728 0.703467i \(-0.248366\pi\)
\(462\) 0 0
\(463\) 20.1707 0.937412 0.468706 0.883354i \(-0.344720\pi\)
0.468706 + 0.883354i \(0.344720\pi\)
\(464\) 0 0
\(465\) −0.0489805 + 0.0848367i −0.00227142 + 0.00393421i
\(466\) 0 0
\(467\) −7.99333 + 4.61495i −0.369887 + 0.213554i −0.673409 0.739270i \(-0.735171\pi\)
0.303522 + 0.952824i \(0.401837\pi\)
\(468\) 0 0
\(469\) −6.39332 19.5208i −0.295216 0.901389i
\(470\) 0 0
\(471\) 2.73288 + 4.73349i 0.125925 + 0.218108i
\(472\) 0 0
\(473\) 16.4093 28.4217i 0.754499 1.30683i
\(474\) 0 0
\(475\) 16.2070i 0.743627i
\(476\) 0 0
\(477\) 5.67797i 0.259976i
\(478\) 0 0
\(479\) −19.6055 + 33.9577i −0.895797 + 1.55157i −0.0629820 + 0.998015i \(0.520061\pi\)
−0.832815 + 0.553551i \(0.813272\pi\)
\(480\) 0 0
\(481\) −4.09617 7.09477i −0.186769 0.323494i
\(482\) 0 0
\(483\) −12.2656 + 13.6867i −0.558104 + 0.622766i
\(484\) 0 0
\(485\) 0.247956 0.143157i 0.0112591 0.00650045i
\(486\) 0 0
\(487\) 3.06454 5.30794i 0.138868 0.240526i −0.788201 0.615418i \(-0.788987\pi\)
0.927068 + 0.374893i \(0.122320\pi\)
\(488\) 0 0
\(489\) −15.2542 −0.689817
\(490\) 0 0
\(491\) 20.5291i 0.926463i 0.886237 + 0.463232i \(0.153310\pi\)
−0.886237 + 0.463232i \(0.846690\pi\)
\(492\) 0 0
\(493\) 1.02636 + 0.592567i 0.0462248 + 0.0266879i
\(494\) 0 0
\(495\) −0.0835514 0.144715i −0.00375536 0.00650447i
\(496\) 0 0
\(497\) −23.3128 20.8922i −1.04572 0.937143i
\(498\) 0 0
\(499\) −9.57940 + 5.53067i −0.428833 + 0.247587i −0.698849 0.715269i \(-0.746304\pi\)
0.270016 + 0.962856i \(0.412971\pi\)
\(500\) 0 0
\(501\) 1.53688 + 0.887315i 0.0686625 + 0.0396423i
\(502\) 0 0
\(503\) −2.64242 −0.117820 −0.0589099 0.998263i \(-0.518762\pi\)
−0.0589099 + 0.998263i \(0.518762\pi\)
\(504\) 0 0
\(505\) −0.371906 −0.0165496
\(506\) 0 0
\(507\) −23.0504 13.3082i −1.02371 0.591036i
\(508\) 0 0
\(509\) −28.9903 + 16.7376i −1.28497 + 0.741880i −0.977753 0.209758i \(-0.932732\pi\)
−0.307220 + 0.951638i \(0.599399\pi\)
\(510\) 0 0
\(511\) 28.0111 9.17400i 1.23914 0.405834i
\(512\) 0 0
\(513\) −1.62140 2.80834i −0.0715864 0.123991i
\(514\) 0 0
\(515\) 0.517919 + 0.299021i 0.0228222 + 0.0131764i
\(516\) 0 0
\(517\) 16.7277i 0.735682i
\(518\) 0 0
\(519\) 2.58227 0.113349
\(520\) 0 0
\(521\) 12.4133 21.5005i 0.543837 0.941954i −0.454842 0.890572i \(-0.650304\pi\)
0.998679 0.0513815i \(-0.0163625\pi\)
\(522\) 0 0
\(523\) −4.00205 + 2.31058i −0.174997 + 0.101035i −0.584940 0.811076i \(-0.698882\pi\)
0.409943 + 0.912111i \(0.365549\pi\)
\(524\) 0 0
\(525\) −12.9405 2.71891i −0.564770 0.118663i
\(526\) 0 0
\(527\) −0.544406 0.942938i −0.0237147 0.0410750i
\(528\) 0 0
\(529\) −12.6264 + 21.8696i −0.548976 + 0.950854i
\(530\) 0 0
\(531\) 1.65018i 0.0716117i
\(532\) 0 0
\(533\) 64.1270i 2.77765i
\(534\) 0 0
\(535\) −0.237262 + 0.410950i −0.0102577 + 0.0177669i
\(536\) 0 0
\(537\) 9.84027 + 17.0438i 0.424639 + 0.735496i
\(538\) 0 0
\(539\) −10.1358 + 23.0556i −0.436580 + 0.993075i
\(540\) 0 0
\(541\) −21.7241 + 12.5424i −0.933992 + 0.539241i −0.888072 0.459704i \(-0.847955\pi\)
−0.0459203 + 0.998945i \(0.514622\pi\)
\(542\) 0 0
\(543\) 12.7807 22.1367i 0.548470 0.949979i
\(544\) 0 0
\(545\) 0.543137 0.0232654
\(546\) 0 0
\(547\) 38.8014i 1.65903i 0.558487 + 0.829514i \(0.311382\pi\)
−0.558487 + 0.829514i \(0.688618\pi\)
\(548\) 0 0
\(549\) 1.10386 + 0.637312i 0.0471115 + 0.0271998i
\(550\) 0 0
\(551\) −3.72239 6.44736i −0.158579 0.274667i
\(552\) 0 0
\(553\) −2.99303 + 14.2452i −0.127276 + 0.605766i
\(554\) 0 0
\(555\) −0.0523525 + 0.0302257i −0.00222224 + 0.00128301i
\(556\) 0 0
\(557\) 27.2335 + 15.7233i 1.15392 + 0.666217i 0.949840 0.312737i \(-0.101246\pi\)
0.204082 + 0.978954i \(0.434579\pi\)
\(558\) 0 0
\(559\) −57.4128 −2.42830
\(560\) 0 0
\(561\) 1.85730 0.0784154
\(562\) 0 0
\(563\) −9.22107 5.32379i −0.388622 0.224371i 0.292941 0.956131i \(-0.405366\pi\)
−0.681563 + 0.731760i \(0.738699\pi\)
\(564\) 0 0
\(565\) 0.365410 0.210970i 0.0153729 0.00887556i
\(566\) 0 0
\(567\) −2.51434 + 0.823477i −0.105592 + 0.0345828i
\(568\) 0 0
\(569\) −13.1394 22.7581i −0.550832 0.954069i −0.998215 0.0597265i \(-0.980977\pi\)
0.447383 0.894343i \(-0.352356\pi\)
\(570\) 0 0
\(571\) 6.37133 + 3.67849i 0.266632 + 0.153940i 0.627356 0.778733i \(-0.284137\pi\)
−0.360724 + 0.932673i \(0.617470\pi\)
\(572\) 0 0
\(573\) 13.9761i 0.583859i
\(574\) 0 0
\(575\) −34.7172 −1.44781
\(576\) 0 0
\(577\) 6.42935 11.1360i 0.267658 0.463596i −0.700599 0.713555i \(-0.747084\pi\)
0.968256 + 0.249959i \(0.0804171\pi\)
\(578\) 0 0
\(579\) 15.6007 9.00706i 0.648342 0.374321i
\(580\) 0 0
\(581\) −14.5679 + 16.2557i −0.604378 + 0.674402i
\(582\) 0 0
\(583\) −10.2144 17.6918i −0.423035 0.732719i
\(584\) 0 0
\(585\) −0.146165 + 0.253165i −0.00604318 + 0.0104671i
\(586\) 0 0
\(587\) 7.51402i 0.310137i −0.987904 0.155068i \(-0.950440\pi\)
0.987904 0.155068i \(-0.0495598\pi\)
\(588\) 0 0
\(589\) 6.83969i 0.281825i
\(590\) 0 0
\(591\) 1.85041 3.20500i 0.0761157 0.131836i
\(592\) 0 0
\(593\) −9.63761 16.6928i −0.395769 0.685492i 0.597430 0.801921i \(-0.296189\pi\)
−0.993199 + 0.116429i \(0.962855\pi\)
\(594\) 0 0
\(595\) −0.0423349 + 0.0472398i −0.00173556 + 0.00193664i
\(596\) 0 0
\(597\) −1.52800 + 0.882192i −0.0625369 + 0.0361057i
\(598\) 0 0
\(599\) −7.76773 + 13.4541i −0.317381 + 0.549720i −0.979941 0.199289i \(-0.936137\pi\)
0.662560 + 0.749009i \(0.269470\pi\)
\(600\) 0 0
\(601\) −44.5011 −1.81524 −0.907620 0.419793i \(-0.862103\pi\)
−0.907620 + 0.419793i \(0.862103\pi\)
\(602\) 0 0
\(603\) 7.76382i 0.316167i
\(604\) 0 0
\(605\) −0.0782241 0.0451627i −0.00318026 0.00183612i
\(606\) 0 0
\(607\) −11.9183 20.6432i −0.483751 0.837881i 0.516075 0.856543i \(-0.327393\pi\)
−0.999826 + 0.0186624i \(0.994059\pi\)
\(608\) 0 0
\(609\) −5.77239 + 1.89053i −0.233909 + 0.0766081i
\(610\) 0 0
\(611\) 25.3428 14.6317i 1.02526 0.591935i
\(612\) 0 0
\(613\) 17.7261 + 10.2342i 0.715950 + 0.413354i 0.813260 0.581900i \(-0.197690\pi\)
−0.0973099 + 0.995254i \(0.531024\pi\)
\(614\) 0 0
\(615\) −0.473195 −0.0190811
\(616\) 0 0
\(617\) 19.3039 0.777146 0.388573 0.921418i \(-0.372968\pi\)
0.388573 + 0.921418i \(0.372968\pi\)
\(618\) 0 0
\(619\) 21.1475 + 12.2095i 0.849990 + 0.490742i 0.860647 0.509201i \(-0.170059\pi\)
−0.0106577 + 0.999943i \(0.503393\pi\)
\(620\) 0 0
\(621\) −6.01578 + 3.47322i −0.241405 + 0.139375i
\(622\) 0 0
\(623\) 8.03205 38.2282i 0.321797 1.53158i
\(624\) 0 0
\(625\) −12.4838 21.6226i −0.499353 0.864905i
\(626\) 0 0
\(627\) −10.1041 5.83361i −0.403519 0.232972i
\(628\) 0 0
\(629\) 0.671902i 0.0267905i
\(630\) 0 0
\(631\) 20.7613 0.826494 0.413247 0.910619i \(-0.364395\pi\)
0.413247 + 0.910619i \(0.364395\pi\)
\(632\) 0 0
\(633\) −5.33785 + 9.24542i −0.212160 + 0.367473i
\(634\) 0 0
\(635\) 0.567372 0.327573i 0.0225155 0.0129993i
\(636\) 0 0
\(637\) 43.7956 4.81075i 1.73525 0.190609i
\(638\) 0 0
\(639\) −5.91599 10.2468i −0.234033 0.405357i
\(640\) 0 0
\(641\) −11.0483 + 19.1363i −0.436383 + 0.755838i −0.997407 0.0719616i \(-0.977074\pi\)
0.561024 + 0.827799i \(0.310407\pi\)
\(642\) 0 0
\(643\) 31.9358i 1.25943i −0.776828 0.629713i \(-0.783173\pi\)
0.776828 0.629713i \(-0.216827\pi\)
\(644\) 0 0
\(645\) 0.423651i 0.0166812i
\(646\) 0 0
\(647\) 1.48741 2.57626i 0.0584760 0.101283i −0.835305 0.549786i \(-0.814709\pi\)
0.893781 + 0.448503i \(0.148043\pi\)
\(648\) 0 0
\(649\) −2.96858 5.14173i −0.116527 0.201831i
\(650\) 0 0
\(651\) 5.46117 + 1.14744i 0.214040 + 0.0449716i
\(652\) 0 0
\(653\) −14.9217 + 8.61507i −0.583933 + 0.337134i −0.762695 0.646758i \(-0.776124\pi\)
0.178762 + 0.983892i \(0.442791\pi\)
\(654\) 0 0
\(655\) 0.397351 0.688231i 0.0155258 0.0268914i
\(656\) 0 0
\(657\) 11.1406 0.434635
\(658\) 0 0
\(659\) 27.7588i 1.08133i −0.841238 0.540665i \(-0.818173\pi\)
0.841238 0.540665i \(-0.181827\pi\)
\(660\) 0 0
\(661\) 28.3232 + 16.3524i 1.10164 + 0.636034i 0.936652 0.350261i \(-0.113907\pi\)
0.164991 + 0.986295i \(0.447241\pi\)
\(662\) 0 0
\(663\) −1.62459 2.81386i −0.0630937 0.109281i
\(664\) 0 0
\(665\) 0.378686 0.124024i 0.0146848 0.00480946i
\(666\) 0 0
\(667\) −13.8110 + 7.97377i −0.534763 + 0.308746i
\(668\) 0 0
\(669\) 19.2441 + 11.1106i 0.744020 + 0.429560i
\(670\) 0 0
\(671\) 4.58596 0.177039
\(672\) 0 0
\(673\) 12.9387 0.498752 0.249376 0.968407i \(-0.419775\pi\)
0.249376 + 0.968407i \(0.419775\pi\)
\(674\) 0 0
\(675\) −4.32826 2.49892i −0.166595 0.0961835i
\(676\) 0 0
\(677\) 1.12548 0.649798i 0.0432558 0.0249738i −0.478216 0.878242i \(-0.658716\pi\)
0.521472 + 0.853268i \(0.325383\pi\)
\(678\) 0 0
\(679\) −12.1463 10.8852i −0.466133 0.417734i
\(680\) 0 0
\(681\) −9.85697 17.0728i −0.377720 0.654230i
\(682\) 0 0
\(683\) −16.8225 9.71249i −0.643696 0.371638i 0.142341 0.989818i \(-0.454537\pi\)
−0.786037 + 0.618179i \(0.787870\pi\)
\(684\) 0 0
\(685\) 0.230918i 0.00882291i
\(686\) 0 0
\(687\) −7.00679 −0.267326
\(688\) 0 0
\(689\) −17.8690 + 30.9500i −0.680755 + 1.17910i
\(690\) 0 0
\(691\) −11.1725 + 6.45042i −0.425020 + 0.245386i −0.697223 0.716854i \(-0.745581\pi\)
0.272203 + 0.962240i \(0.412248\pi\)
\(692\) 0 0
\(693\) −6.35294 + 7.08899i −0.241328 + 0.269288i
\(694\) 0 0
\(695\) 0.120865 + 0.209344i 0.00458467 + 0.00794088i
\(696\) 0 0
\(697\) 2.62972 4.55480i 0.0996076 0.172526i
\(698\) 0 0
\(699\) 24.7433i 0.935879i
\(700\) 0 0
\(701\) 34.3868i 1.29877i 0.760460 + 0.649385i \(0.224974\pi\)
−0.760460 + 0.649385i \(0.775026\pi\)
\(702\) 0 0
\(703\) −2.11038 + 3.65528i −0.0795944 + 0.137862i
\(704\) 0 0
\(705\) −0.107968 0.187006i −0.00406630 0.00704303i
\(706\) 0 0
\(707\) 6.59400 + 20.1336i 0.247993 + 0.757201i
\(708\) 0 0
\(709\) −19.7915 + 11.4266i −0.743284 + 0.429135i −0.823262 0.567661i \(-0.807848\pi\)
0.0799779 + 0.996797i \(0.474515\pi\)
\(710\) 0 0
\(711\) −2.75086 + 4.76463i −0.103165 + 0.178688i
\(712\) 0 0
\(713\) 14.6514 0.548699
\(714\) 0 0
\(715\) 1.05177i 0.0393340i
\(716\) 0 0
\(717\) −19.0095 10.9752i −0.709924 0.409875i
\(718\) 0 0
\(719\) −5.21620 9.03472i −0.194531 0.336938i 0.752215 0.658917i \(-0.228985\pi\)
−0.946747 + 0.321979i \(0.895652\pi\)
\(720\) 0 0
\(721\) 7.00498 33.3399i 0.260879 1.24164i
\(722\) 0 0
\(723\) 12.8027 7.39162i 0.476136 0.274897i
\(724\) 0 0
\(725\) −9.93677 5.73700i −0.369042 0.213067i
\(726\) 0 0
\(727\) 2.48753 0.0922572 0.0461286 0.998936i \(-0.485312\pi\)
0.0461286 + 0.998936i \(0.485312\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −4.07791 2.35438i −0.150827 0.0870800i
\(732\) 0 0
\(733\) −0.674033 + 0.389153i −0.0248960 + 0.0143737i −0.512396 0.858749i \(-0.671242\pi\)
0.487500 + 0.873123i \(0.337909\pi\)
\(734\) 0 0
\(735\) −0.0354987 0.323169i −0.00130939 0.0119203i
\(736\) 0 0
\(737\) −13.9667 24.1910i −0.514469 0.891086i
\(738\) 0 0
\(739\) −22.4331 12.9518i −0.825215 0.476438i 0.0269963 0.999636i \(-0.491406\pi\)
−0.852212 + 0.523197i \(0.824739\pi\)
\(740\) 0 0
\(741\) 20.4106i 0.749804i
\(742\) 0 0
\(743\) 27.7729 1.01889 0.509445 0.860503i \(-0.329851\pi\)
0.509445 + 0.860503i \(0.329851\pi\)
\(744\) 0 0
\(745\) −0.0194642 + 0.0337130i −0.000713113 + 0.00123515i
\(746\) 0 0
\(747\) −7.14497 + 4.12515i −0.261421 + 0.150931i
\(748\) 0 0
\(749\) 26.4539 + 5.55819i 0.966606 + 0.203092i
\(750\) 0 0
\(751\) 10.5549 + 18.2816i 0.385153 + 0.667105i 0.991790 0.127874i \(-0.0408153\pi\)
−0.606637 + 0.794979i \(0.707482\pi\)
\(752\) 0 0
\(753\) 7.23926 12.5388i 0.263813 0.456938i
\(754\) 0 0
\(755\) 0.298921i 0.0108788i
\(756\) 0 0
\(757\) 46.6419i 1.69523i −0.530612 0.847615i \(-0.678038\pi\)
0.530612 0.847615i \(-0.321962\pi\)
\(758\) 0 0
\(759\) −12.4962 + 21.6441i −0.453585 + 0.785632i
\(760\) 0 0
\(761\) 12.5114 + 21.6703i 0.453537 + 0.785550i 0.998603 0.0528439i \(-0.0168286\pi\)
−0.545066 + 0.838393i \(0.683495\pi\)
\(762\) 0 0
\(763\) −9.62997 29.4034i −0.348628 1.06447i
\(764\) 0 0
\(765\) −0.0207636 + 0.0119879i −0.000750709 + 0.000433422i
\(766\) 0 0
\(767\) −5.19324 + 8.99495i −0.187517 + 0.324789i
\(768\) 0 0
\(769\) 1.32800 0.0478891 0.0239445 0.999713i \(-0.492377\pi\)
0.0239445 + 0.999713i \(0.492377\pi\)
\(770\) 0 0
\(771\) 15.8746i 0.571711i
\(772\) 0 0
\(773\) 19.2488 + 11.1133i 0.692332 + 0.399718i 0.804485 0.593973i \(-0.202441\pi\)
−0.112153 + 0.993691i \(0.535775\pi\)
\(774\) 0 0
\(775\) 5.27072 + 9.12915i 0.189330 + 0.327929i
\(776\) 0 0
\(777\) 2.56453 + 2.29825i 0.0920020 + 0.0824493i
\(778\) 0 0
\(779\) −28.6124 + 16.5194i −1.02514 + 0.591867i
\(780\) 0 0
\(781\) −36.8668 21.2851i −1.31920 0.761639i
\(782\) 0 0
\(783\) −2.29579 −0.0820448
\(784\) 0 0
\(785\) −0.253856 −0.00906050
\(786\) 0 0
\(787\) 16.5930 + 9.57996i 0.591476 + 0.341489i 0.765681 0.643221i \(-0.222402\pi\)
−0.174205 + 0.984709i \(0.555736\pi\)
\(788\) 0 0
\(789\) 0.281822 0.162710i 0.0100331 0.00579263i
\(790\) 0 0
\(791\) −17.8999 16.0413i −0.636447 0.570364i
\(792\) 0 0
\(793\) −4.01134 6.94784i −0.142447 0.246725i
\(794\) 0 0
\(795\) 0.228381 + 0.131856i 0.00809984 + 0.00467645i
\(796\) 0 0
\(797\) 35.9779i 1.27440i 0.770697 + 0.637201i \(0.219908\pi\)
−0.770697 + 0.637201i \(0.780092\pi\)
\(798\) 0 0
\(799\) 2.40006 0.0849082
\(800\) 0 0
\(801\) 7.38218 12.7863i 0.260837 0.451782i
\(802\) 0 0
\(803\) 34.7125 20.0413i 1.22498 0.707241i
\(804\) 0 0
\(805\) −0.265674 0.811187i −0.00936378 0.0285906i
\(806\) 0 0
\(807\) −11.7642 20.3761i −0.414118 0.717273i
\(808\) 0 0
\(809\) 4.56264 7.90272i 0.160414 0.277845i −0.774603 0.632447i \(-0.782050\pi\)
0.935017 + 0.354603i \(0.115384\pi\)
\(810\) 0 0
\(811\) 9.64051i 0.338524i −0.985571 0.169262i \(-0.945862\pi\)
0.985571 0.169262i \(-0.0541384\pi\)
\(812\) 0 0
\(813\) 1.50313i 0.0527172i
\(814\) 0 0
\(815\) 0.354237 0.613557i 0.0124084 0.0214920i
\(816\) 0 0
\(817\) 14.7898 + 25.6166i 0.517428 + 0.896212i
\(818\) 0 0
\(819\) 16.2969 + 3.42412i 0.569461 + 0.119648i
\(820\) 0 0
\(821\) 16.6309 9.60187i 0.580423 0.335107i −0.180878 0.983505i \(-0.557894\pi\)
0.761301 + 0.648398i \(0.224561\pi\)
\(822\) 0 0
\(823\) −0.820157 + 1.42055i −0.0285889 + 0.0495174i −0.879966 0.475037i \(-0.842435\pi\)
0.851377 + 0.524554i \(0.175768\pi\)
\(824\) 0 0
\(825\) −17.9817 −0.626042
\(826\) 0 0
\(827\) 27.3891i 0.952412i −0.879334 0.476206i \(-0.842012\pi\)
0.879334 0.476206i \(-0.157988\pi\)
\(828\) 0 0
\(829\) −15.4922 8.94441i −0.538065 0.310652i 0.206229 0.978504i \(-0.433881\pi\)
−0.744294 + 0.667852i \(0.767214\pi\)
\(830\) 0 0
\(831\) 7.31418 + 12.6685i 0.253726 + 0.439466i
\(832\) 0 0
\(833\) 3.30799 + 1.45427i 0.114615 + 0.0503875i
\(834\) 0 0
\(835\) −0.0713797 + 0.0412111i −0.00247020 + 0.00142617i
\(836\) 0 0
\(837\) 1.82662 + 1.05460i 0.0631372 + 0.0364523i
\(838\) 0 0
\(839\) −11.0644 −0.381986 −0.190993 0.981591i \(-0.561171\pi\)
−0.190993 + 0.981591i \(0.561171\pi\)
\(840\) 0 0
\(841\) 23.7294 0.818253
\(842\) 0 0
\(843\) 18.4710 + 10.6642i 0.636175 + 0.367296i
\(844\) 0 0
\(845\) 1.07057 0.618094i 0.0368287 0.0212631i
\(846\) 0 0
\(847\) −1.05800 + 5.03550i −0.0363533 + 0.173022i
\(848\) 0 0
\(849\) 8.66416 + 15.0068i 0.297353 + 0.515031i
\(850\) 0 0
\(851\) 7.83002 + 4.52067i 0.268410 + 0.154966i
\(852\) 0 0
\(853\) 29.4621i 1.00876i −0.863480 0.504382i \(-0.831720\pi\)
0.863480 0.504382i \(-0.168280\pi\)
\(854\) 0 0
\(855\) 0.150611 0.00515078
\(856\) 0 0
\(857\) −8.88103 + 15.3824i −0.303370 + 0.525453i −0.976897 0.213710i \(-0.931445\pi\)
0.673527 + 0.739163i \(0.264779\pi\)
\(858\) 0 0
\(859\) −31.7456 + 18.3283i −1.08314 + 0.625354i −0.931743 0.363118i \(-0.881712\pi\)
−0.151402 + 0.988472i \(0.548379\pi\)
\(860\) 0 0
\(861\) 8.38987 + 25.6169i 0.285926 + 0.873022i
\(862\) 0 0
\(863\) −1.57129 2.72156i −0.0534874 0.0926429i 0.838042 0.545606i \(-0.183700\pi\)
−0.891529 + 0.452963i \(0.850367\pi\)
\(864\) 0 0
\(865\) −0.0599664 + 0.103865i −0.00203892 + 0.00353151i
\(866\) 0 0
\(867\) 16.7335i 0.568300i
\(868\) 0 0
\(869\) 19.7946i 0.671485i
\(870\) 0 0
\(871\) −24.4333 + 42.3197i −0.827891 + 1.43395i
\(872\) 0 0
\(873\) −3.08232 5.33874i −0.104321 0.180689i
\(874\) 0 0
\(875\) 0.819916 0.914912i 0.0277182 0.0309297i
\(876\) 0 0
\(877\) 12.4428 7.18384i 0.420162 0.242581i −0.274984 0.961449i \(-0.588673\pi\)
0.695147 + 0.718868i \(0.255339\pi\)
\(878\) 0 0
\(879\) −14.8463 + 25.7146i −0.500754 + 0.867332i
\(880\) 0 0
\(881\) −21.4314 −0.722043 −0.361022 0.932557i \(-0.617572\pi\)
−0.361022 + 0.932557i \(0.617572\pi\)
\(882\) 0 0
\(883\) 29.5761i 0.995316i −0.867373 0.497658i \(-0.834194\pi\)
0.867373 0.497658i \(-0.165806\pi\)
\(884\) 0 0
\(885\) 0.0663740 + 0.0383210i 0.00223114 + 0.00128815i
\(886\) 0 0
\(887\) −1.90744 3.30379i −0.0640456 0.110930i 0.832225 0.554439i \(-0.187067\pi\)
−0.896270 + 0.443508i \(0.853734\pi\)
\(888\) 0 0
\(889\) −27.7932 24.9074i −0.932153 0.835367i
\(890\) 0 0
\(891\) −3.11586 + 1.79894i −0.104385 + 0.0602669i
\(892\) 0 0
\(893\) −13.0568 7.53836i −0.436930 0.252262i
\(894\) 0 0
\(895\) −0.914057 −0.0305536
\(896\) 0 0
\(897\) 43.7219 1.45983
\(898\) 0 0
\(899\) 4.19353 + 2.42114i 0.139862 + 0.0807494i
\(900\) 0 0
\(901\) −2.53840 + 1.46554i −0.0845662 + 0.0488243i
\(902\) 0 0
\(903\) 22.9348 7.51144i 0.763223 0.249965i
\(904\) 0 0
\(905\) 0.593594 + 1.02813i 0.0197317 + 0.0341763i
\(906\) 0 0
\(907\) 4.23020 + 2.44231i 0.140462 + 0.0810955i 0.568584 0.822625i \(-0.307491\pi\)
−0.428122 + 0.903721i \(0.640825\pi\)
\(908\) 0 0
\(909\) 8.00751i 0.265592i
\(910\) 0 0
\(911\) 47.3325 1.56820 0.784098 0.620637i \(-0.213126\pi\)
0.784098 + 0.620637i \(0.213126\pi\)
\(912\) 0 0
\(913\) −14.8418 + 25.7068i −0.491193 + 0.850771i
\(914\) 0 0
\(915\) −0.0512683 + 0.0295998i −0.00169488 + 0.000978539i
\(916\) 0 0
\(917\) −44.3033 9.30849i −1.46302 0.307394i
\(918\) 0 0
\(919\) −15.5826 26.9898i −0.514022 0.890312i −0.999868 0.0162672i \(-0.994822\pi\)
0.485846 0.874044i \(-0.338512\pi\)
\(920\) 0 0
\(921\) −2.44121 + 4.22831i −0.0804407 + 0.139327i
\(922\) 0 0
\(923\) 74.4722i 2.45128i
\(924\) 0 0
\(925\) 6.50509i 0.213886i
\(926\) 0 0
\(927\) 6.43821 11.1513i 0.211459 0.366257i
\(928\) 0 0
\(929\) 14.3019 + 24.7717i 0.469231 + 0.812733i 0.999381 0.0351712i \(-0.0111977\pi\)
−0.530150 + 0.847904i \(0.677864\pi\)
\(930\) 0 0
\(931\) −13.4284 18.3016i −0.440098 0.599811i
\(932\) 0 0
\(933\) 15.7318 9.08277i 0.515036 0.297356i
\(934\) 0 0
\(935\) −0.0431310 + 0.0747050i −0.00141053 + 0.00244312i
\(936\) 0 0
\(937\) 5.62816 0.183864 0.0919320 0.995765i \(-0.470696\pi\)
0.0919320 + 0.995765i \(0.470696\pi\)
\(938\) 0 0
\(939\) 13.6107i 0.444167i
\(940\) 0 0
\(941\) −29.2442 16.8842i −0.953334 0.550408i −0.0592190 0.998245i \(-0.518861\pi\)
−0.894115 + 0.447837i \(0.852194\pi\)
\(942\) 0 0
\(943\) 35.3863 + 61.2909i 1.15234 + 1.99591i
\(944\) 0 0
\(945\) 0.0252667 0.120255i 0.000821925 0.00391191i
\(946\) 0 0
\(947\) 10.1554 5.86321i 0.330005 0.190529i −0.325838 0.945426i \(-0.605646\pi\)
0.655843 + 0.754897i \(0.272313\pi\)
\(948\) 0 0
\(949\) −60.7261 35.0602i −1.97125 1.13810i
\(950\) 0 0
\(951\) 24.9886 0.810310
\(952\) 0 0
\(953\) 37.7047 1.22137 0.610687 0.791872i \(-0.290893\pi\)
0.610687 + 0.791872i \(0.290893\pi\)
\(954\) 0 0
\(955\) 0.562150 + 0.324557i 0.0181907 + 0.0105024i
\(956\) 0 0
\(957\) −7.15336 + 4.13000i −0.231235 + 0.133504i
\(958\) 0 0
\(959\) 12.5010 4.09423i 0.403678 0.132210i
\(960\) 0 0
\(961\) 13.2756 + 22.9941i 0.428247 + 0.741745i
\(962\) 0 0
\(963\) 8.84815 + 5.10848i 0.285128 + 0.164619i
\(964\) 0 0
\(965\) 0.836660i 0.0269330i
\(966\) 0 0
\(967\) 8.89648 0.286092 0.143046 0.989716i \(-0.454310\pi\)
0.143046 + 0.989716i \(0.454310\pi\)
\(968\) 0 0
\(969\) −0.836999 + 1.44972i −0.0268883 + 0.0465719i
\(970\) 0 0
\(971\) −52.8744 + 30.5270i −1.69682 + 0.979659i −0.748074 + 0.663615i \(0.769021\pi\)
−0.948745 + 0.316044i \(0.897645\pi\)
\(972\) 0 0
\(973\) 9.19012 10.2549i 0.294622 0.328757i
\(974\) 0 0
\(975\) 15.7286 + 27.2427i 0.503718 + 0.872465i
\(976\) 0 0
\(977\) −23.8132 + 41.2456i −0.761851 + 1.31956i 0.180045 + 0.983658i \(0.442376\pi\)
−0.941896 + 0.335906i \(0.890958\pi\)
\(978\) 0 0
\(979\) 53.1205i 1.69774i
\(980\) 0 0
\(981\) 11.6943i 0.373370i
\(982\) 0 0
\(983\) 6.89129 11.9361i 0.219798 0.380701i −0.734948 0.678123i \(-0.762793\pi\)
0.954746 + 0.297422i \(0.0961268\pi\)
\(984\) 0 0
\(985\) 0.0859417 + 0.148855i 0.00273833 + 0.00474293i
\(986\) 0 0
\(987\) −8.20946 + 9.16061i −0.261310 + 0.291585i
\(988\) 0 0
\(989\) 54.8737 31.6813i 1.74488 1.00741i
\(990\) 0 0
\(991\) 4.32034 7.48305i 0.137240 0.237707i −0.789211 0.614122i \(-0.789510\pi\)
0.926451 + 0.376415i \(0.122843\pi\)
\(992\) 0 0
\(993\) −7.32984 −0.232605
\(994\) 0 0
\(995\) 0.0819463i 0.00259787i
\(996\) 0 0
\(997\) 38.0017 + 21.9403i 1.20353 + 0.694856i 0.961337 0.275374i \(-0.0888016\pi\)
0.242188 + 0.970229i \(0.422135\pi\)
\(998\) 0 0
\(999\) 0.650790 + 1.12720i 0.0205901 + 0.0356631i
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 672.2.bk.a.529.13 32
3.2 odd 2 2016.2.cr.e.1873.9 32
4.3 odd 2 168.2.bc.a.109.3 yes 32
7.2 even 3 inner 672.2.bk.a.625.4 32
7.3 odd 6 4704.2.c.f.2353.12 16
7.4 even 3 4704.2.c.e.2353.5 16
8.3 odd 2 168.2.bc.a.109.14 yes 32
8.5 even 2 inner 672.2.bk.a.529.4 32
12.11 even 2 504.2.cj.e.109.14 32
21.2 odd 6 2016.2.cr.e.1297.8 32
24.5 odd 2 2016.2.cr.e.1873.8 32
24.11 even 2 504.2.cj.e.109.3 32
28.3 even 6 1176.2.c.f.589.9 16
28.11 odd 6 1176.2.c.e.589.9 16
28.23 odd 6 168.2.bc.a.37.14 yes 32
56.3 even 6 1176.2.c.f.589.10 16
56.11 odd 6 1176.2.c.e.589.10 16
56.37 even 6 inner 672.2.bk.a.625.13 32
56.45 odd 6 4704.2.c.f.2353.5 16
56.51 odd 6 168.2.bc.a.37.3 32
56.53 even 6 4704.2.c.e.2353.12 16
84.23 even 6 504.2.cj.e.37.3 32
168.107 even 6 504.2.cj.e.37.14 32
168.149 odd 6 2016.2.cr.e.1297.9 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.bc.a.37.3 32 56.51 odd 6
168.2.bc.a.37.14 yes 32 28.23 odd 6
168.2.bc.a.109.3 yes 32 4.3 odd 2
168.2.bc.a.109.14 yes 32 8.3 odd 2
504.2.cj.e.37.3 32 84.23 even 6
504.2.cj.e.37.14 32 168.107 even 6
504.2.cj.e.109.3 32 24.11 even 2
504.2.cj.e.109.14 32 12.11 even 2
672.2.bk.a.529.4 32 8.5 even 2 inner
672.2.bk.a.529.13 32 1.1 even 1 trivial
672.2.bk.a.625.4 32 7.2 even 3 inner
672.2.bk.a.625.13 32 56.37 even 6 inner
1176.2.c.e.589.9 16 28.11 odd 6
1176.2.c.e.589.10 16 56.11 odd 6
1176.2.c.f.589.9 16 28.3 even 6
1176.2.c.f.589.10 16 56.3 even 6
2016.2.cr.e.1297.8 32 21.2 odd 6
2016.2.cr.e.1297.9 32 168.149 odd 6
2016.2.cr.e.1873.8 32 24.5 odd 2
2016.2.cr.e.1873.9 32 3.2 odd 2
4704.2.c.e.2353.5 16 7.4 even 3
4704.2.c.e.2353.12 16 56.53 even 6
4704.2.c.f.2353.5 16 56.45 odd 6
4704.2.c.f.2353.12 16 7.3 odd 6