Properties

Label 280.2.x.a.97.1
Level $280$
Weight $2$
Character 280.97
Analytic conductor $2.236$
Analytic rank $0$
Dimension $24$
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] = [280,2,Mod(97,280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(280, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("280.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.x (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: \(2.23581125660\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 97.1
Character \(\chi\) \(=\) 280.97
Dual form 280.2.x.a.153.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.16993 - 2.16993i) q^{3} +(0.272751 + 2.21937i) q^{5} +(-1.63589 + 2.07939i) q^{7} +6.41716i q^{9} +O(q^{10})\) \(q+(-2.16993 - 2.16993i) q^{3} +(0.272751 + 2.21937i) q^{5} +(-1.63589 + 2.07939i) q^{7} +6.41716i q^{9} +2.56986 q^{11} +(1.35447 + 1.35447i) q^{13} +(4.22402 - 5.40772i) q^{15} +(2.10000 - 2.10000i) q^{17} +4.43874 q^{19} +(8.06189 - 0.962346i) q^{21} +(-5.78997 + 5.78997i) q^{23} +(-4.85121 + 1.21067i) q^{25} +(7.41498 - 7.41498i) q^{27} +4.28527i q^{29} +9.70587i q^{31} +(-5.57640 - 5.57640i) q^{33} +(-5.06112 - 3.06350i) q^{35} +(0.183701 + 0.183701i) q^{37} -5.87819i q^{39} +3.81823i q^{41} +(5.86065 - 5.86065i) q^{43} +(-14.2421 + 1.75029i) q^{45} +(-1.83004 + 1.83004i) q^{47} +(-1.64770 - 6.80331i) q^{49} -9.11369 q^{51} +(-2.38616 + 2.38616i) q^{53} +(0.700932 + 5.70346i) q^{55} +(-9.63174 - 9.63174i) q^{57} -6.34515 q^{59} -8.62046i q^{61} +(-13.3438 - 10.4978i) q^{63} +(-2.63663 + 3.37550i) q^{65} +(6.68908 + 6.68908i) q^{67} +25.1276 q^{69} -6.04557 q^{71} +(1.88962 + 1.88962i) q^{73} +(13.1538 + 7.89970i) q^{75} +(-4.20401 + 5.34372i) q^{77} -12.2456i q^{79} -12.9285 q^{81} +(-2.08951 - 2.08951i) q^{83} +(5.23346 + 4.08790i) q^{85} +(9.29871 - 9.29871i) q^{87} +10.1525 q^{89} +(-5.03222 + 0.600696i) q^{91} +(21.0610 - 21.0610i) q^{93} +(1.21067 + 9.85121i) q^{95} +(7.15926 - 7.15926i) q^{97} +16.4912i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 4 q^{7} + 8 q^{11} - 8 q^{15} + 16 q^{21} - 32 q^{23} + 8 q^{25} + 12 q^{35} - 8 q^{37} + 16 q^{43} - 24 q^{51} - 16 q^{53} + 20 q^{63} - 48 q^{65} - 32 q^{67} - 32 q^{71} - 40 q^{77} - 72 q^{81} + 16 q^{85} - 64 q^{91} + 72 q^{93} - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.16993 2.16993i −1.25281 1.25281i −0.954456 0.298351i \(-0.903563\pi\)
−0.298351 0.954456i \(-0.596437\pi\)
\(4\) 0 0
\(5\) 0.272751 + 2.21937i 0.121978 + 0.992533i
\(6\) 0 0
\(7\) −1.63589 + 2.07939i −0.618310 + 0.785934i
\(8\) 0 0
\(9\) 6.41716i 2.13905i
\(10\) 0 0
\(11\) 2.56986 0.774841 0.387420 0.921903i \(-0.373366\pi\)
0.387420 + 0.921903i \(0.373366\pi\)
\(12\) 0 0
\(13\) 1.35447 + 1.35447i 0.375661 + 0.375661i 0.869534 0.493873i \(-0.164419\pi\)
−0.493873 + 0.869534i \(0.664419\pi\)
\(14\) 0 0
\(15\) 4.22402 5.40772i 1.09064 1.39627i
\(16\) 0 0
\(17\) 2.10000 2.10000i 0.509325 0.509325i −0.404994 0.914319i \(-0.632726\pi\)
0.914319 + 0.404994i \(0.132726\pi\)
\(18\) 0 0
\(19\) 4.43874 1.01832 0.509159 0.860673i \(-0.329957\pi\)
0.509159 + 0.860673i \(0.329957\pi\)
\(20\) 0 0
\(21\) 8.06189 0.962346i 1.75925 0.210001i
\(22\) 0 0
\(23\) −5.78997 + 5.78997i −1.20729 + 1.20729i −0.235391 + 0.971901i \(0.575637\pi\)
−0.971901 + 0.235391i \(0.924363\pi\)
\(24\) 0 0
\(25\) −4.85121 + 1.21067i −0.970243 + 0.242135i
\(26\) 0 0
\(27\) 7.41498 7.41498i 1.42701 1.42701i
\(28\) 0 0
\(29\) 4.28527i 0.795754i 0.917439 + 0.397877i \(0.130253\pi\)
−0.917439 + 0.397877i \(0.869747\pi\)
\(30\) 0 0
\(31\) 9.70587i 1.74323i 0.490194 + 0.871613i \(0.336926\pi\)
−0.490194 + 0.871613i \(0.663074\pi\)
\(32\) 0 0
\(33\) −5.57640 5.57640i −0.970726 0.970726i
\(34\) 0 0
\(35\) −5.06112 3.06350i −0.855486 0.517826i
\(36\) 0 0
\(37\) 0.183701 + 0.183701i 0.0302002 + 0.0302002i 0.722046 0.691845i \(-0.243202\pi\)
−0.691845 + 0.722046i \(0.743202\pi\)
\(38\) 0 0
\(39\) 5.87819i 0.941263i
\(40\) 0 0
\(41\) 3.81823i 0.596307i 0.954518 + 0.298153i \(0.0963707\pi\)
−0.954518 + 0.298153i \(0.903629\pi\)
\(42\) 0 0
\(43\) 5.86065 5.86065i 0.893741 0.893741i −0.101132 0.994873i \(-0.532246\pi\)
0.994873 + 0.101132i \(0.0322464\pi\)
\(44\) 0 0
\(45\) −14.2421 + 1.75029i −2.12308 + 0.260918i
\(46\) 0 0
\(47\) −1.83004 + 1.83004i −0.266939 + 0.266939i −0.827866 0.560926i \(-0.810445\pi\)
0.560926 + 0.827866i \(0.310445\pi\)
\(48\) 0 0
\(49\) −1.64770 6.80331i −0.235386 0.971902i
\(50\) 0 0
\(51\) −9.11369 −1.27617
\(52\) 0 0
\(53\) −2.38616 + 2.38616i −0.327764 + 0.327764i −0.851736 0.523972i \(-0.824450\pi\)
0.523972 + 0.851736i \(0.324450\pi\)
\(54\) 0 0
\(55\) 0.700932 + 5.70346i 0.0945136 + 0.769055i
\(56\) 0 0
\(57\) −9.63174 9.63174i −1.27576 1.27576i
\(58\) 0 0
\(59\) −6.34515 −0.826068 −0.413034 0.910716i \(-0.635531\pi\)
−0.413034 + 0.910716i \(0.635531\pi\)
\(60\) 0 0
\(61\) 8.62046i 1.10374i −0.833931 0.551868i \(-0.813915\pi\)
0.833931 0.551868i \(-0.186085\pi\)
\(62\) 0 0
\(63\) −13.3438 10.4978i −1.68116 1.32260i
\(64\) 0 0
\(65\) −2.63663 + 3.37550i −0.327034 + 0.418679i
\(66\) 0 0
\(67\) 6.68908 + 6.68908i 0.817201 + 0.817201i 0.985702 0.168501i \(-0.0538925\pi\)
−0.168501 + 0.985702i \(0.553892\pi\)
\(68\) 0 0
\(69\) 25.1276 3.02501
\(70\) 0 0
\(71\) −6.04557 −0.717477 −0.358739 0.933438i \(-0.616793\pi\)
−0.358739 + 0.933438i \(0.616793\pi\)
\(72\) 0 0
\(73\) 1.88962 + 1.88962i 0.221164 + 0.221164i 0.808988 0.587825i \(-0.200015\pi\)
−0.587825 + 0.808988i \(0.700015\pi\)
\(74\) 0 0
\(75\) 13.1538 + 7.89970i 1.51888 + 0.912179i
\(76\) 0 0
\(77\) −4.20401 + 5.34372i −0.479092 + 0.608974i
\(78\) 0 0
\(79\) 12.2456i 1.37774i −0.724887 0.688868i \(-0.758108\pi\)
0.724887 0.688868i \(-0.241892\pi\)
\(80\) 0 0
\(81\) −12.9285 −1.43650
\(82\) 0 0
\(83\) −2.08951 2.08951i −0.229354 0.229354i 0.583069 0.812423i \(-0.301852\pi\)
−0.812423 + 0.583069i \(0.801852\pi\)
\(84\) 0 0
\(85\) 5.23346 + 4.08790i 0.567648 + 0.443395i
\(86\) 0 0
\(87\) 9.29871 9.29871i 0.996927 0.996927i
\(88\) 0 0
\(89\) 10.1525 1.07616 0.538081 0.842893i \(-0.319150\pi\)
0.538081 + 0.842893i \(0.319150\pi\)
\(90\) 0 0
\(91\) −5.03222 + 0.600696i −0.527520 + 0.0629700i
\(92\) 0 0
\(93\) 21.0610 21.0610i 2.18393 2.18393i
\(94\) 0 0
\(95\) 1.21067 + 9.85121i 0.124212 + 1.01071i
\(96\) 0 0
\(97\) 7.15926 7.15926i 0.726913 0.726913i −0.243091 0.970004i \(-0.578161\pi\)
0.970004 + 0.243091i \(0.0781613\pi\)
\(98\) 0 0
\(99\) 16.4912i 1.65743i
\(100\) 0 0
\(101\) 1.12506i 0.111947i 0.998432 + 0.0559737i \(0.0178263\pi\)
−0.998432 + 0.0559737i \(0.982174\pi\)
\(102\) 0 0
\(103\) −1.56743 1.56743i −0.154443 0.154443i 0.625656 0.780099i \(-0.284831\pi\)
−0.780099 + 0.625656i \(0.784831\pi\)
\(104\) 0 0
\(105\) 4.33469 + 17.6298i 0.423023 + 1.72050i
\(106\) 0 0
\(107\) −0.145240 0.145240i −0.0140409 0.0140409i 0.700051 0.714092i \(-0.253160\pi\)
−0.714092 + 0.700051i \(0.753160\pi\)
\(108\) 0 0
\(109\) 6.51130i 0.623669i −0.950136 0.311835i \(-0.899057\pi\)
0.950136 0.311835i \(-0.100943\pi\)
\(110\) 0 0
\(111\) 0.797233i 0.0756700i
\(112\) 0 0
\(113\) −7.49203 + 7.49203i −0.704791 + 0.704791i −0.965435 0.260644i \(-0.916065\pi\)
0.260644 + 0.965435i \(0.416065\pi\)
\(114\) 0 0
\(115\) −14.4293 11.2709i −1.34554 1.05101i
\(116\) 0 0
\(117\) −8.69183 + 8.69183i −0.803560 + 0.803560i
\(118\) 0 0
\(119\) 0.931334 + 7.80209i 0.0853753 + 0.715217i
\(120\) 0 0
\(121\) −4.39584 −0.399622
\(122\) 0 0
\(123\) 8.28527 8.28527i 0.747057 0.747057i
\(124\) 0 0
\(125\) −4.01011 10.4364i −0.358675 0.933463i
\(126\) 0 0
\(127\) 6.78578 + 6.78578i 0.602141 + 0.602141i 0.940880 0.338740i \(-0.110001\pi\)
−0.338740 + 0.940880i \(0.610001\pi\)
\(128\) 0 0
\(129\) −25.4344 −2.23937
\(130\) 0 0
\(131\) 8.37853i 0.732036i 0.930608 + 0.366018i \(0.119279\pi\)
−0.930608 + 0.366018i \(0.880721\pi\)
\(132\) 0 0
\(133\) −7.26131 + 9.22986i −0.629636 + 0.800330i
\(134\) 0 0
\(135\) 18.4790 + 14.4342i 1.59042 + 1.24229i
\(136\) 0 0
\(137\) 7.56986 + 7.56986i 0.646736 + 0.646736i 0.952203 0.305466i \(-0.0988124\pi\)
−0.305466 + 0.952203i \(0.598812\pi\)
\(138\) 0 0
\(139\) −11.2432 −0.953636 −0.476818 0.879002i \(-0.658210\pi\)
−0.476818 + 0.879002i \(0.658210\pi\)
\(140\) 0 0
\(141\) 7.94212 0.668847
\(142\) 0 0
\(143\) 3.48078 + 3.48078i 0.291078 + 0.291078i
\(144\) 0 0
\(145\) −9.51060 + 1.16881i −0.789812 + 0.0970646i
\(146\) 0 0
\(147\) −11.1873 + 18.3381i −0.922713 + 1.51250i
\(148\) 0 0
\(149\) 18.0067i 1.47516i −0.675258 0.737581i \(-0.735968\pi\)
0.675258 0.737581i \(-0.264032\pi\)
\(150\) 0 0
\(151\) 12.1231 0.986563 0.493282 0.869870i \(-0.335797\pi\)
0.493282 + 0.869870i \(0.335797\pi\)
\(152\) 0 0
\(153\) 13.4760 + 13.4760i 1.08947 + 1.08947i
\(154\) 0 0
\(155\) −21.5409 + 2.64729i −1.73021 + 0.212635i
\(156\) 0 0
\(157\) 11.6764 11.6764i 0.931876 0.931876i −0.0659475 0.997823i \(-0.521007\pi\)
0.997823 + 0.0659475i \(0.0210070\pi\)
\(158\) 0 0
\(159\) 10.3556 0.821249
\(160\) 0 0
\(161\) −2.56781 21.5114i −0.202371 1.69533i
\(162\) 0 0
\(163\) 10.5217 10.5217i 0.824123 0.824123i −0.162574 0.986696i \(-0.551980\pi\)
0.986696 + 0.162574i \(0.0519796\pi\)
\(164\) 0 0
\(165\) 10.8551 13.8971i 0.845070 1.08188i
\(166\) 0 0
\(167\) 9.06283 9.06283i 0.701303 0.701303i −0.263387 0.964690i \(-0.584840\pi\)
0.964690 + 0.263387i \(0.0848398\pi\)
\(168\) 0 0
\(169\) 9.33084i 0.717757i
\(170\) 0 0
\(171\) 28.4841i 2.17823i
\(172\) 0 0
\(173\) −9.99321 9.99321i −0.759770 0.759770i 0.216511 0.976280i \(-0.430532\pi\)
−0.976280 + 0.216511i \(0.930532\pi\)
\(174\) 0 0
\(175\) 5.41862 12.0681i 0.409609 0.912261i
\(176\) 0 0
\(177\) 13.7685 + 13.7685i 1.03490 + 1.03490i
\(178\) 0 0
\(179\) 14.8694i 1.11139i 0.831386 + 0.555695i \(0.187548\pi\)
−0.831386 + 0.555695i \(0.812452\pi\)
\(180\) 0 0
\(181\) 14.2095i 1.05618i 0.849188 + 0.528090i \(0.177092\pi\)
−0.849188 + 0.528090i \(0.822908\pi\)
\(182\) 0 0
\(183\) −18.7058 + 18.7058i −1.38277 + 1.38277i
\(184\) 0 0
\(185\) −0.357595 + 0.457804i −0.0262909 + 0.0336584i
\(186\) 0 0
\(187\) 5.39670 5.39670i 0.394646 0.394646i
\(188\) 0 0
\(189\) 3.28849 + 27.5488i 0.239202 + 2.00388i
\(190\) 0 0
\(191\) −1.04143 −0.0753549 −0.0376775 0.999290i \(-0.511996\pi\)
−0.0376775 + 0.999290i \(0.511996\pi\)
\(192\) 0 0
\(193\) 0.0690260 0.0690260i 0.00496860 0.00496860i −0.704618 0.709587i \(-0.748882\pi\)
0.709587 + 0.704618i \(0.248882\pi\)
\(194\) 0 0
\(195\) 13.0459 1.60328i 0.934234 0.114813i
\(196\) 0 0
\(197\) 12.7945 + 12.7945i 0.911571 + 0.911571i 0.996396 0.0848244i \(-0.0270329\pi\)
−0.0848244 + 0.996396i \(0.527033\pi\)
\(198\) 0 0
\(199\) 4.09793 0.290494 0.145247 0.989395i \(-0.453602\pi\)
0.145247 + 0.989395i \(0.453602\pi\)
\(200\) 0 0
\(201\) 29.0296i 2.04759i
\(202\) 0 0
\(203\) −8.91073 7.01025i −0.625411 0.492023i
\(204\) 0 0
\(205\) −8.47406 + 1.04143i −0.591854 + 0.0727364i
\(206\) 0 0
\(207\) −37.1551 37.1551i −2.58246 2.58246i
\(208\) 0 0
\(209\) 11.4069 0.789033
\(210\) 0 0
\(211\) 28.7179 1.97702 0.988510 0.151154i \(-0.0482988\pi\)
0.988510 + 0.151154i \(0.0482988\pi\)
\(212\) 0 0
\(213\) 13.1184 + 13.1184i 0.898861 + 0.898861i
\(214\) 0 0
\(215\) 14.6055 + 11.4085i 0.996084 + 0.778050i
\(216\) 0 0
\(217\) −20.1823 15.8778i −1.37006 1.07785i
\(218\) 0 0
\(219\) 8.20069i 0.554151i
\(220\) 0 0
\(221\) 5.68876 0.382668
\(222\) 0 0
\(223\) 3.69411 + 3.69411i 0.247376 + 0.247376i 0.819893 0.572517i \(-0.194033\pi\)
−0.572517 + 0.819893i \(0.694033\pi\)
\(224\) 0 0
\(225\) −7.76908 31.1310i −0.517939 2.07540i
\(226\) 0 0
\(227\) −7.85781 + 7.85781i −0.521541 + 0.521541i −0.918037 0.396495i \(-0.870226\pi\)
0.396495 + 0.918037i \(0.370226\pi\)
\(228\) 0 0
\(229\) −9.48513 −0.626795 −0.313397 0.949622i \(-0.601467\pi\)
−0.313397 + 0.949622i \(0.601467\pi\)
\(230\) 0 0
\(231\) 20.7179 2.47309i 1.36314 0.162717i
\(232\) 0 0
\(233\) −13.6024 + 13.6024i −0.891126 + 0.891126i −0.994629 0.103503i \(-0.966995\pi\)
0.103503 + 0.994629i \(0.466995\pi\)
\(234\) 0 0
\(235\) −4.56069 3.56240i −0.297507 0.232385i
\(236\) 0 0
\(237\) −26.5720 + 26.5720i −1.72604 + 1.72604i
\(238\) 0 0
\(239\) 18.7581i 1.21336i −0.794946 0.606680i \(-0.792501\pi\)
0.794946 0.606680i \(-0.207499\pi\)
\(240\) 0 0
\(241\) 2.78395i 0.179330i −0.995972 0.0896649i \(-0.971420\pi\)
0.995972 0.0896649i \(-0.0285796\pi\)
\(242\) 0 0
\(243\) 5.80885 + 5.80885i 0.372638 + 0.372638i
\(244\) 0 0
\(245\) 14.6497 5.51247i 0.935933 0.352179i
\(246\) 0 0
\(247\) 6.01213 + 6.01213i 0.382543 + 0.382543i
\(248\) 0 0
\(249\) 9.06817i 0.574672i
\(250\) 0 0
\(251\) 28.3562i 1.78983i −0.446241 0.894913i \(-0.647237\pi\)
0.446241 0.894913i \(-0.352763\pi\)
\(252\) 0 0
\(253\) −14.8794 + 14.8794i −0.935458 + 0.935458i
\(254\) 0 0
\(255\) −2.48577 20.2267i −0.155665 1.26664i
\(256\) 0 0
\(257\) −17.9740 + 17.9740i −1.12119 + 1.12119i −0.129626 + 0.991563i \(0.541378\pi\)
−0.991563 + 0.129626i \(0.958622\pi\)
\(258\) 0 0
\(259\) −0.682499 + 0.0814698i −0.0424084 + 0.00506229i
\(260\) 0 0
\(261\) −27.4992 −1.70216
\(262\) 0 0
\(263\) 0.986652 0.986652i 0.0608396 0.0608396i −0.676032 0.736872i \(-0.736302\pi\)
0.736872 + 0.676032i \(0.236302\pi\)
\(264\) 0 0
\(265\) −5.94659 4.64494i −0.365296 0.285336i
\(266\) 0 0
\(267\) −22.0301 22.0301i −1.34822 1.34822i
\(268\) 0 0
\(269\) −2.65510 −0.161884 −0.0809422 0.996719i \(-0.525793\pi\)
−0.0809422 + 0.996719i \(0.525793\pi\)
\(270\) 0 0
\(271\) 0.573473i 0.0348360i −0.999848 0.0174180i \(-0.994455\pi\)
0.999848 0.0174180i \(-0.00554460\pi\)
\(272\) 0 0
\(273\) 12.2230 + 9.61609i 0.739771 + 0.581992i
\(274\) 0 0
\(275\) −12.4669 + 3.11125i −0.751783 + 0.187616i
\(276\) 0 0
\(277\) 6.98543 + 6.98543i 0.419714 + 0.419714i 0.885105 0.465391i \(-0.154086\pi\)
−0.465391 + 0.885105i \(0.654086\pi\)
\(278\) 0 0
\(279\) −62.2841 −3.72885
\(280\) 0 0
\(281\) 23.2936 1.38958 0.694789 0.719213i \(-0.255498\pi\)
0.694789 + 0.719213i \(0.255498\pi\)
\(282\) 0 0
\(283\) 9.27105 + 9.27105i 0.551107 + 0.551107i 0.926760 0.375653i \(-0.122582\pi\)
−0.375653 + 0.926760i \(0.622582\pi\)
\(284\) 0 0
\(285\) 18.7493 24.0035i 1.11061 1.42184i
\(286\) 0 0
\(287\) −7.93957 6.24621i −0.468658 0.368702i
\(288\) 0 0
\(289\) 8.17999i 0.481176i
\(290\) 0 0
\(291\) −31.0701 −1.82136
\(292\) 0 0
\(293\) −11.3951 11.3951i −0.665707 0.665707i 0.291012 0.956719i \(-0.406008\pi\)
−0.956719 + 0.291012i \(0.906008\pi\)
\(294\) 0 0
\(295\) −1.73065 14.0822i −0.100762 0.819899i
\(296\) 0 0
\(297\) 19.0554 19.0554i 1.10571 1.10571i
\(298\) 0 0
\(299\) −15.6846 −0.907066
\(300\) 0 0
\(301\) 2.59915 + 21.7740i 0.149813 + 1.25503i
\(302\) 0 0
\(303\) 2.44129 2.44129i 0.140249 0.140249i
\(304\) 0 0
\(305\) 19.1320 2.35124i 1.09549 0.134632i
\(306\) 0 0
\(307\) 12.7022 12.7022i 0.724953 0.724953i −0.244657 0.969610i \(-0.578675\pi\)
0.969610 + 0.244657i \(0.0786753\pi\)
\(308\) 0 0
\(309\) 6.80241i 0.386976i
\(310\) 0 0
\(311\) 12.2723i 0.695896i −0.937514 0.347948i \(-0.886879\pi\)
0.937514 0.347948i \(-0.113121\pi\)
\(312\) 0 0
\(313\) 11.9640 + 11.9640i 0.676246 + 0.676246i 0.959149 0.282903i \(-0.0912974\pi\)
−0.282903 + 0.959149i \(0.591297\pi\)
\(314\) 0 0
\(315\) 19.6590 32.4780i 1.10766 1.82993i
\(316\) 0 0
\(317\) −18.9944 18.9944i −1.06683 1.06683i −0.997600 0.0692338i \(-0.977945\pi\)
−0.0692338 0.997600i \(-0.522055\pi\)
\(318\) 0 0
\(319\) 11.0125i 0.616583i
\(320\) 0 0
\(321\) 0.630320i 0.0351811i
\(322\) 0 0
\(323\) 9.32136 9.32136i 0.518654 0.518654i
\(324\) 0 0
\(325\) −8.21062 4.93099i −0.455443 0.273522i
\(326\) 0 0
\(327\) −14.1290 + 14.1290i −0.781337 + 0.781337i
\(328\) 0 0
\(329\) −0.811611 6.79913i −0.0447455 0.374848i
\(330\) 0 0
\(331\) −23.8309 −1.30986 −0.654931 0.755689i \(-0.727302\pi\)
−0.654931 + 0.755689i \(0.727302\pi\)
\(332\) 0 0
\(333\) −1.17884 + 1.17884i −0.0645998 + 0.0645998i
\(334\) 0 0
\(335\) −13.0211 + 16.6700i −0.711418 + 0.910779i
\(336\) 0 0
\(337\) 8.26895 + 8.26895i 0.450439 + 0.450439i 0.895500 0.445061i \(-0.146818\pi\)
−0.445061 + 0.895500i \(0.646818\pi\)
\(338\) 0 0
\(339\) 32.5143 1.76593
\(340\) 0 0
\(341\) 24.9427i 1.35072i
\(342\) 0 0
\(343\) 16.8422 + 7.70330i 0.909392 + 0.415939i
\(344\) 0 0
\(345\) 6.85359 + 55.7675i 0.368985 + 3.00242i
\(346\) 0 0
\(347\) 12.0104 + 12.0104i 0.644754 + 0.644754i 0.951720 0.306966i \(-0.0993139\pi\)
−0.306966 + 0.951720i \(0.599314\pi\)
\(348\) 0 0
\(349\) 13.0227 0.697092 0.348546 0.937292i \(-0.386676\pi\)
0.348546 + 0.937292i \(0.386676\pi\)
\(350\) 0 0
\(351\) 20.0867 1.07215
\(352\) 0 0
\(353\) 9.48642 + 9.48642i 0.504911 + 0.504911i 0.912960 0.408049i \(-0.133791\pi\)
−0.408049 + 0.912960i \(0.633791\pi\)
\(354\) 0 0
\(355\) −1.64894 13.4174i −0.0875165 0.712120i
\(356\) 0 0
\(357\) 14.9090 18.9509i 0.789070 1.00299i
\(358\) 0 0
\(359\) 9.36637i 0.494338i −0.968972 0.247169i \(-0.920500\pi\)
0.968972 0.247169i \(-0.0795003\pi\)
\(360\) 0 0
\(361\) 0.702427 0.0369698
\(362\) 0 0
\(363\) 9.53865 + 9.53865i 0.500649 + 0.500649i
\(364\) 0 0
\(365\) −3.67838 + 4.70918i −0.192535 + 0.246490i
\(366\) 0 0
\(367\) 1.22951 1.22951i 0.0641798 0.0641798i −0.674288 0.738468i \(-0.735549\pi\)
0.738468 + 0.674288i \(0.235549\pi\)
\(368\) 0 0
\(369\) −24.5022 −1.27553
\(370\) 0 0
\(371\) −1.05824 8.86524i −0.0549412 0.460260i
\(372\) 0 0
\(373\) −6.68837 + 6.68837i −0.346311 + 0.346311i −0.858733 0.512423i \(-0.828748\pi\)
0.512423 + 0.858733i \(0.328748\pi\)
\(374\) 0 0
\(375\) −13.9446 + 31.3479i −0.720098 + 1.61880i
\(376\) 0 0
\(377\) −5.80425 + 5.80425i −0.298934 + 0.298934i
\(378\) 0 0
\(379\) 0.0350675i 0.00180130i 1.00000 0.000900649i \(0.000286685\pi\)
−1.00000 0.000900649i \(0.999713\pi\)
\(380\) 0 0
\(381\) 29.4493i 1.50873i
\(382\) 0 0
\(383\) 20.5450 + 20.5450i 1.04980 + 1.04980i 0.998693 + 0.0511048i \(0.0162743\pi\)
0.0511048 + 0.998693i \(0.483726\pi\)
\(384\) 0 0
\(385\) −13.0064 7.87275i −0.662865 0.401233i
\(386\) 0 0
\(387\) 37.6087 + 37.6087i 1.91176 + 1.91176i
\(388\) 0 0
\(389\) 11.6510i 0.590730i −0.955385 0.295365i \(-0.904559\pi\)
0.955385 0.295365i \(-0.0954412\pi\)
\(390\) 0 0
\(391\) 24.3179i 1.22981i
\(392\) 0 0
\(393\) 18.1808 18.1808i 0.917100 0.917100i
\(394\) 0 0
\(395\) 27.1775 3.34000i 1.36745 0.168054i
\(396\) 0 0
\(397\) 4.22472 4.22472i 0.212033 0.212033i −0.593098 0.805131i \(-0.702095\pi\)
0.805131 + 0.593098i \(0.202095\pi\)
\(398\) 0 0
\(399\) 35.7846 4.27160i 1.79147 0.213848i
\(400\) 0 0
\(401\) −4.26395 −0.212932 −0.106466 0.994316i \(-0.533953\pi\)
−0.106466 + 0.994316i \(0.533953\pi\)
\(402\) 0 0
\(403\) −13.1463 + 13.1463i −0.654863 + 0.654863i
\(404\) 0 0
\(405\) −3.52625 28.6930i −0.175221 1.42577i
\(406\) 0 0
\(407\) 0.472084 + 0.472084i 0.0234003 + 0.0234003i
\(408\) 0 0
\(409\) −25.3610 −1.25402 −0.627011 0.779010i \(-0.715722\pi\)
−0.627011 + 0.779010i \(0.715722\pi\)
\(410\) 0 0
\(411\) 32.8521i 1.62047i
\(412\) 0 0
\(413\) 10.3800 13.1940i 0.510766 0.649235i
\(414\) 0 0
\(415\) 4.06748 5.20732i 0.199665 0.255617i
\(416\) 0 0
\(417\) 24.3969 + 24.3969i 1.19472 + 1.19472i
\(418\) 0 0
\(419\) −0.597961 −0.0292123 −0.0146061 0.999893i \(-0.504649\pi\)
−0.0146061 + 0.999893i \(0.504649\pi\)
\(420\) 0 0
\(421\) −9.94725 −0.484799 −0.242400 0.970176i \(-0.577935\pi\)
−0.242400 + 0.970176i \(0.577935\pi\)
\(422\) 0 0
\(423\) −11.7437 11.7437i −0.570998 0.570998i
\(424\) 0 0
\(425\) −7.64514 + 12.7300i −0.370844 + 0.617494i
\(426\) 0 0
\(427\) 17.9253 + 14.1022i 0.867464 + 0.682451i
\(428\) 0 0
\(429\) 15.1061i 0.729329i
\(430\) 0 0
\(431\) −3.29543 −0.158735 −0.0793677 0.996845i \(-0.525290\pi\)
−0.0793677 + 0.996845i \(0.525290\pi\)
\(432\) 0 0
\(433\) −14.8212 14.8212i −0.712260 0.712260i 0.254747 0.967008i \(-0.418008\pi\)
−0.967008 + 0.254747i \(0.918008\pi\)
\(434\) 0 0
\(435\) 23.1735 + 18.1011i 1.11109 + 0.867879i
\(436\) 0 0
\(437\) −25.7002 + 25.7002i −1.22941 + 1.22941i
\(438\) 0 0
\(439\) −3.93710 −0.187907 −0.0939537 0.995577i \(-0.529951\pi\)
−0.0939537 + 0.995577i \(0.529951\pi\)
\(440\) 0 0
\(441\) 43.6580 10.5735i 2.07895 0.503502i
\(442\) 0 0
\(443\) 23.0246 23.0246i 1.09393 1.09393i 0.0988281 0.995105i \(-0.468491\pi\)
0.995105 0.0988281i \(-0.0315094\pi\)
\(444\) 0 0
\(445\) 2.76910 + 22.5321i 0.131268 + 1.06813i
\(446\) 0 0
\(447\) −39.0731 + 39.0731i −1.84810 + 1.84810i
\(448\) 0 0
\(449\) 9.44923i 0.445937i 0.974826 + 0.222968i \(0.0715747\pi\)
−0.974826 + 0.222968i \(0.928425\pi\)
\(450\) 0 0
\(451\) 9.81229i 0.462043i
\(452\) 0 0
\(453\) −26.3062 26.3062i −1.23597 1.23597i
\(454\) 0 0
\(455\) −2.70571 11.0045i −0.126846 0.515900i
\(456\) 0 0
\(457\) −17.8997 17.8997i −0.837313 0.837313i 0.151191 0.988505i \(-0.451689\pi\)
−0.988505 + 0.151191i \(0.951689\pi\)
\(458\) 0 0
\(459\) 31.1429i 1.45363i
\(460\) 0 0
\(461\) 8.91063i 0.415009i 0.978234 + 0.207505i \(0.0665342\pi\)
−0.978234 + 0.207505i \(0.933466\pi\)
\(462\) 0 0
\(463\) 1.48151 1.48151i 0.0688518 0.0688518i −0.671842 0.740694i \(-0.734497\pi\)
0.740694 + 0.671842i \(0.234497\pi\)
\(464\) 0 0
\(465\) 52.4867 + 40.9978i 2.43401 + 1.90123i
\(466\) 0 0
\(467\) −0.836995 + 0.836995i −0.0387315 + 0.0387315i −0.726207 0.687476i \(-0.758719\pi\)
0.687476 + 0.726207i \(0.258719\pi\)
\(468\) 0 0
\(469\) −24.8518 + 2.96656i −1.14755 + 0.136983i
\(470\) 0 0
\(471\) −50.6737 −2.33492
\(472\) 0 0
\(473\) 15.0610 15.0610i 0.692507 0.692507i
\(474\) 0 0
\(475\) −21.5333 + 5.37386i −0.988015 + 0.246570i
\(476\) 0 0
\(477\) −15.3123 15.3123i −0.701104 0.701104i
\(478\) 0 0
\(479\) 23.6142 1.07896 0.539479 0.841999i \(-0.318621\pi\)
0.539479 + 0.841999i \(0.318621\pi\)
\(480\) 0 0
\(481\) 0.497632i 0.0226901i
\(482\) 0 0
\(483\) −41.1061 + 52.2500i −1.87039 + 2.37746i
\(484\) 0 0
\(485\) 17.8417 + 13.9364i 0.810152 + 0.632817i
\(486\) 0 0
\(487\) 22.7931 + 22.7931i 1.03285 + 1.03285i 0.999442 + 0.0334131i \(0.0106377\pi\)
0.0334131 + 0.999442i \(0.489362\pi\)
\(488\) 0 0
\(489\) −45.6626 −2.06493
\(490\) 0 0
\(491\) 5.02524 0.226786 0.113393 0.993550i \(-0.463828\pi\)
0.113393 + 0.993550i \(0.463828\pi\)
\(492\) 0 0
\(493\) 8.99907 + 8.99907i 0.405298 + 0.405298i
\(494\) 0 0
\(495\) −36.6000 + 4.49799i −1.64505 + 0.202170i
\(496\) 0 0
\(497\) 9.88992 12.5711i 0.443623 0.563890i
\(498\) 0 0
\(499\) 10.9254i 0.489089i 0.969638 + 0.244544i \(0.0786384\pi\)
−0.969638 + 0.244544i \(0.921362\pi\)
\(500\) 0 0
\(501\) −39.3313 −1.75719
\(502\) 0 0
\(503\) −13.4112 13.4112i −0.597974 0.597974i 0.341799 0.939773i \(-0.388964\pi\)
−0.939773 + 0.341799i \(0.888964\pi\)
\(504\) 0 0
\(505\) −2.49692 + 0.306861i −0.111111 + 0.0136551i
\(506\) 0 0
\(507\) −20.2472 + 20.2472i −0.899211 + 0.899211i
\(508\) 0 0
\(509\) 40.1376 1.77907 0.889533 0.456871i \(-0.151030\pi\)
0.889533 + 0.456871i \(0.151030\pi\)
\(510\) 0 0
\(511\) −7.02049 + 0.838034i −0.310568 + 0.0370724i
\(512\) 0 0
\(513\) 32.9132 32.9132i 1.45315 1.45315i
\(514\) 0 0
\(515\) 3.05119 3.90622i 0.134451 0.172129i
\(516\) 0 0
\(517\) −4.70295 + 4.70295i −0.206835 + 0.206835i
\(518\) 0 0
\(519\) 43.3691i 1.90369i
\(520\) 0 0
\(521\) 19.2753i 0.844467i 0.906487 + 0.422234i \(0.138754\pi\)
−0.906487 + 0.422234i \(0.861246\pi\)
\(522\) 0 0
\(523\) −10.6573 10.6573i −0.466013 0.466013i 0.434607 0.900620i \(-0.356887\pi\)
−0.900620 + 0.434607i \(0.856887\pi\)
\(524\) 0 0
\(525\) −37.9448 + 14.4288i −1.65605 + 0.629727i
\(526\) 0 0
\(527\) 20.3823 + 20.3823i 0.887869 + 0.887869i
\(528\) 0 0
\(529\) 44.0474i 1.91511i
\(530\) 0 0
\(531\) 40.7178i 1.76700i
\(532\) 0 0
\(533\) −5.17166 + 5.17166i −0.224009 + 0.224009i
\(534\) 0 0
\(535\) 0.282727 0.361956i 0.0122234 0.0156487i
\(536\) 0 0
\(537\) 32.2655 32.2655i 1.39236 1.39236i
\(538\) 0 0
\(539\) −4.23435 17.4835i −0.182386 0.753069i
\(540\) 0 0
\(541\) 12.8503 0.552479 0.276240 0.961089i \(-0.410912\pi\)
0.276240 + 0.961089i \(0.410912\pi\)
\(542\) 0 0
\(543\) 30.8335 30.8335i 1.32319 1.32319i
\(544\) 0 0
\(545\) 14.4510 1.77597i 0.619012 0.0760740i
\(546\) 0 0
\(547\) −26.5300 26.5300i −1.13434 1.13434i −0.989447 0.144895i \(-0.953716\pi\)
−0.144895 0.989447i \(-0.546284\pi\)
\(548\) 0 0
\(549\) 55.3188 2.36095
\(550\) 0 0
\(551\) 19.0212i 0.810330i
\(552\) 0 0
\(553\) 25.4633 + 20.0325i 1.08281 + 0.851868i
\(554\) 0 0
\(555\) 1.76936 0.217446i 0.0751050 0.00923009i
\(556\) 0 0
\(557\) −25.9969 25.9969i −1.10152 1.10152i −0.994227 0.107298i \(-0.965780\pi\)
−0.107298 0.994227i \(-0.534220\pi\)
\(558\) 0 0
\(559\) 15.8761 0.671488
\(560\) 0 0
\(561\) −23.4209 −0.988830
\(562\) 0 0
\(563\) 1.11717 + 1.11717i 0.0470833 + 0.0470833i 0.730256 0.683173i \(-0.239401\pi\)
−0.683173 + 0.730256i \(0.739401\pi\)
\(564\) 0 0
\(565\) −18.6711 14.5841i −0.785497 0.613559i
\(566\) 0 0
\(567\) 21.1496 26.8833i 0.888199 1.12899i
\(568\) 0 0
\(569\) 22.9125i 0.960543i −0.877120 0.480271i \(-0.840538\pi\)
0.877120 0.480271i \(-0.159462\pi\)
\(570\) 0 0
\(571\) 33.1839 1.38870 0.694351 0.719636i \(-0.255692\pi\)
0.694351 + 0.719636i \(0.255692\pi\)
\(572\) 0 0
\(573\) 2.25982 + 2.25982i 0.0944052 + 0.0944052i
\(574\) 0 0
\(575\) 21.0786 35.0981i 0.879039 1.46369i
\(576\) 0 0
\(577\) 29.4941 29.4941i 1.22786 1.22786i 0.263084 0.964773i \(-0.415260\pi\)
0.964773 0.263084i \(-0.0847397\pi\)
\(578\) 0 0
\(579\) −0.299563 −0.0124494
\(580\) 0 0
\(581\) 7.76312 0.926682i 0.322069 0.0384452i
\(582\) 0 0
\(583\) −6.13207 + 6.13207i −0.253965 + 0.253965i
\(584\) 0 0
\(585\) −21.6611 16.9197i −0.895576 0.699543i
\(586\) 0 0
\(587\) 14.6733 14.6733i 0.605634 0.605634i −0.336168 0.941802i \(-0.609131\pi\)
0.941802 + 0.336168i \(0.109131\pi\)
\(588\) 0 0
\(589\) 43.0819i 1.77516i
\(590\) 0 0
\(591\) 55.5263i 2.28405i
\(592\) 0 0
\(593\) 21.3914 + 21.3914i 0.878439 + 0.878439i 0.993373 0.114934i \(-0.0366657\pi\)
−0.114934 + 0.993373i \(0.536666\pi\)
\(594\) 0 0
\(595\) −17.0617 + 4.19501i −0.699462 + 0.171979i
\(596\) 0 0
\(597\) −8.89220 8.89220i −0.363933 0.363933i
\(598\) 0 0
\(599\) 13.4975i 0.551494i 0.961230 + 0.275747i \(0.0889252\pi\)
−0.961230 + 0.275747i \(0.911075\pi\)
\(600\) 0 0
\(601\) 36.3870i 1.48426i 0.670258 + 0.742128i \(0.266183\pi\)
−0.670258 + 0.742128i \(0.733817\pi\)
\(602\) 0 0
\(603\) −42.9249 + 42.9249i −1.74804 + 1.74804i
\(604\) 0 0
\(605\) −1.19897 9.75600i −0.0487451 0.396638i
\(606\) 0 0
\(607\) 29.3687 29.3687i 1.19204 1.19204i 0.215545 0.976494i \(-0.430847\pi\)
0.976494 0.215545i \(-0.0691529\pi\)
\(608\) 0 0
\(609\) 4.12391 + 34.5473i 0.167109 + 1.39993i
\(610\) 0 0
\(611\) −4.95747 −0.200558
\(612\) 0 0
\(613\) −20.4722 + 20.4722i −0.826863 + 0.826863i −0.987082 0.160218i \(-0.948780\pi\)
0.160218 + 0.987082i \(0.448780\pi\)
\(614\) 0 0
\(615\) 20.6479 + 16.1283i 0.832604 + 0.650354i
\(616\) 0 0
\(617\) −5.88024 5.88024i −0.236729 0.236729i 0.578765 0.815494i \(-0.303535\pi\)
−0.815494 + 0.578765i \(0.803535\pi\)
\(618\) 0 0
\(619\) −22.3760 −0.899368 −0.449684 0.893188i \(-0.648463\pi\)
−0.449684 + 0.893188i \(0.648463\pi\)
\(620\) 0 0
\(621\) 85.8650i 3.44564i
\(622\) 0 0
\(623\) −16.6084 + 21.1109i −0.665401 + 0.845792i
\(624\) 0 0
\(625\) 22.0685 11.7465i 0.882742 0.469859i
\(626\) 0 0
\(627\) −24.7522 24.7522i −0.988507 0.988507i
\(628\) 0 0
\(629\) 0.771543 0.0307634
\(630\) 0 0
\(631\) −42.0404 −1.67360 −0.836802 0.547506i \(-0.815577\pi\)
−0.836802 + 0.547506i \(0.815577\pi\)
\(632\) 0 0
\(633\) −62.3157 62.3157i −2.47683 2.47683i
\(634\) 0 0
\(635\) −13.2093 + 16.9110i −0.524196 + 0.671092i
\(636\) 0 0
\(637\) 6.98311 11.4466i 0.276681 0.453531i
\(638\) 0 0
\(639\) 38.7954i 1.53472i
\(640\) 0 0
\(641\) −1.69064 −0.0667764 −0.0333882 0.999442i \(-0.510630\pi\)
−0.0333882 + 0.999442i \(0.510630\pi\)
\(642\) 0 0
\(643\) −0.492457 0.492457i −0.0194206 0.0194206i 0.697330 0.716750i \(-0.254371\pi\)
−0.716750 + 0.697330i \(0.754371\pi\)
\(644\) 0 0
\(645\) −6.93726 56.4483i −0.273154 2.22265i
\(646\) 0 0
\(647\) −6.32851 + 6.32851i −0.248799 + 0.248799i −0.820478 0.571678i \(-0.806293\pi\)
0.571678 + 0.820478i \(0.306293\pi\)
\(648\) 0 0
\(649\) −16.3061 −0.640071
\(650\) 0 0
\(651\) 9.34040 + 78.2476i 0.366079 + 3.06677i
\(652\) 0 0
\(653\) 24.3667 24.3667i 0.953543 0.953543i −0.0454247 0.998968i \(-0.514464\pi\)
0.998968 + 0.0454247i \(0.0144641\pi\)
\(654\) 0 0
\(655\) −18.5951 + 2.28526i −0.726569 + 0.0892923i
\(656\) 0 0
\(657\) −12.1260 + 12.1260i −0.473081 + 0.473081i
\(658\) 0 0
\(659\) 10.7004i 0.416829i 0.978041 + 0.208414i \(0.0668303\pi\)
−0.978041 + 0.208414i \(0.933170\pi\)
\(660\) 0 0
\(661\) 34.1408i 1.32792i −0.747766 0.663962i \(-0.768874\pi\)
0.747766 0.663962i \(-0.231126\pi\)
\(662\) 0 0
\(663\) −12.3442 12.3442i −0.479409 0.479409i
\(664\) 0 0
\(665\) −22.4650 13.5981i −0.871156 0.527311i
\(666\) 0 0
\(667\) −24.8116 24.8116i −0.960707 0.960707i
\(668\) 0 0
\(669\) 16.0319i 0.619829i
\(670\) 0 0
\(671\) 22.1533i 0.855220i
\(672\) 0 0
\(673\) −11.2267 + 11.2267i −0.432758 + 0.432758i −0.889566 0.456808i \(-0.848993\pi\)
0.456808 + 0.889566i \(0.348993\pi\)
\(674\) 0 0
\(675\) −26.9946 + 44.9488i −1.03902 + 1.73008i
\(676\) 0 0
\(677\) 2.84843 2.84843i 0.109474 0.109474i −0.650248 0.759722i \(-0.725335\pi\)
0.759722 + 0.650248i \(0.225335\pi\)
\(678\) 0 0
\(679\) 3.17508 + 26.5987i 0.121848 + 1.02076i
\(680\) 0 0
\(681\) 34.1017 1.30678
\(682\) 0 0
\(683\) 22.5165 22.5165i 0.861571 0.861571i −0.129950 0.991521i \(-0.541482\pi\)
0.991521 + 0.129950i \(0.0414816\pi\)
\(684\) 0 0
\(685\) −14.7356 + 18.8650i −0.563019 + 0.720795i
\(686\) 0 0
\(687\) 20.5820 + 20.5820i 0.785253 + 0.785253i
\(688\) 0 0
\(689\) −6.46394 −0.246256
\(690\) 0 0
\(691\) 7.28794i 0.277246i 0.990345 + 0.138623i \(0.0442677\pi\)
−0.990345 + 0.138623i \(0.955732\pi\)
\(692\) 0 0
\(693\) −34.2915 26.9778i −1.30263 1.02480i
\(694\) 0 0
\(695\) −3.06660 24.9528i −0.116323 0.946515i
\(696\) 0 0
\(697\) 8.01828 + 8.01828i 0.303714 + 0.303714i
\(698\) 0 0
\(699\) 59.0326 2.23282
\(700\) 0 0
\(701\) 8.29057 0.313130 0.156565 0.987668i \(-0.449958\pi\)
0.156565 + 0.987668i \(0.449958\pi\)
\(702\) 0 0
\(703\) 0.815399 + 0.815399i 0.0307534 + 0.0307534i
\(704\) 0 0
\(705\) 2.16622 + 17.6265i 0.0815847 + 0.663853i
\(706\) 0 0
\(707\) −2.33943 1.84048i −0.0879833 0.0692182i
\(708\) 0 0
\(709\) 15.8960i 0.596987i 0.954412 + 0.298493i \(0.0964841\pi\)
−0.954412 + 0.298493i \(0.903516\pi\)
\(710\) 0 0
\(711\) 78.5819 2.94705
\(712\) 0 0
\(713\) −56.1967 56.1967i −2.10458 2.10458i
\(714\) 0 0
\(715\) −6.77576 + 8.67454i −0.253399 + 0.324409i
\(716\) 0 0
\(717\) −40.7036 + 40.7036i −1.52011 + 1.52011i
\(718\) 0 0
\(719\) −41.3180 −1.54090 −0.770450 0.637500i \(-0.779969\pi\)
−0.770450 + 0.637500i \(0.779969\pi\)
\(720\) 0 0
\(721\) 5.82344 0.695143i 0.216876 0.0258885i
\(722\) 0 0
\(723\) −6.04096 + 6.04096i −0.224666 + 0.224666i
\(724\) 0 0
\(725\) −5.18806 20.7887i −0.192680 0.772075i
\(726\) 0 0
\(727\) −2.27196 + 2.27196i −0.0842625 + 0.0842625i −0.747982 0.663719i \(-0.768977\pi\)
0.663719 + 0.747982i \(0.268977\pi\)
\(728\) 0 0
\(729\) 13.5758i 0.502809i
\(730\) 0 0
\(731\) 24.6147i 0.910409i
\(732\) 0 0
\(733\) 4.64097 + 4.64097i 0.171418 + 0.171418i 0.787602 0.616184i \(-0.211322\pi\)
−0.616184 + 0.787602i \(0.711322\pi\)
\(734\) 0 0
\(735\) −43.7503 19.8270i −1.61376 0.731332i
\(736\) 0 0
\(737\) 17.1900 + 17.1900i 0.633201 + 0.633201i
\(738\) 0 0
\(739\) 2.48628i 0.0914592i 0.998954 + 0.0457296i \(0.0145613\pi\)
−0.998954 + 0.0457296i \(0.985439\pi\)
\(740\) 0 0
\(741\) 26.0917i 0.958504i
\(742\) 0 0
\(743\) 11.7230 11.7230i 0.430075 0.430075i −0.458579 0.888654i \(-0.651641\pi\)
0.888654 + 0.458579i \(0.151641\pi\)
\(744\) 0 0
\(745\) 39.9634 4.91134i 1.46415 0.179938i
\(746\) 0 0
\(747\) 13.4087 13.4087i 0.490600 0.490600i
\(748\) 0 0
\(749\) 0.539608 0.0644129i 0.0197168 0.00235359i
\(750\) 0 0
\(751\) −18.4627 −0.673714 −0.336857 0.941556i \(-0.609364\pi\)
−0.336857 + 0.941556i \(0.609364\pi\)
\(752\) 0 0
\(753\) −61.5308 + 61.5308i −2.24231 + 2.24231i
\(754\) 0 0
\(755\) 3.30659 + 26.9056i 0.120339 + 0.979196i
\(756\) 0 0
\(757\) −2.72346 2.72346i −0.0989857 0.0989857i 0.655880 0.754865i \(-0.272298\pi\)
−0.754865 + 0.655880i \(0.772298\pi\)
\(758\) 0 0
\(759\) 64.5743 2.34390
\(760\) 0 0
\(761\) 13.4733i 0.488406i −0.969724 0.244203i \(-0.921474\pi\)
0.969724 0.244203i \(-0.0785263\pi\)
\(762\) 0 0
\(763\) 13.5395 + 10.6518i 0.490163 + 0.385621i
\(764\) 0 0
\(765\) −26.2327 + 33.5839i −0.948446 + 1.21423i
\(766\) 0 0
\(767\) −8.59429 8.59429i −0.310322 0.310322i
\(768\) 0 0
\(769\) 13.2864 0.479121 0.239561 0.970881i \(-0.422997\pi\)
0.239561 + 0.970881i \(0.422997\pi\)
\(770\) 0 0
\(771\) 78.0046 2.80927
\(772\) 0 0
\(773\) 0.232304 + 0.232304i 0.00835541 + 0.00835541i 0.711272 0.702917i \(-0.248119\pi\)
−0.702917 + 0.711272i \(0.748119\pi\)
\(774\) 0 0
\(775\) −11.7506 47.0853i −0.422095 1.69135i
\(776\) 0 0
\(777\) 1.65776 + 1.30419i 0.0594717 + 0.0467875i
\(778\) 0 0
\(779\) 16.9481i 0.607229i
\(780\) 0 0
\(781\) −15.5362 −0.555931
\(782\) 0 0
\(783\) 31.7752 + 31.7752i 1.13555 + 1.13555i
\(784\) 0 0
\(785\) 29.0989 + 22.7294i 1.03859 + 0.811249i
\(786\) 0 0
\(787\) −32.0984 + 32.0984i −1.14419 + 1.14419i −0.156509 + 0.987676i \(0.550024\pi\)
−0.987676 + 0.156509i \(0.949976\pi\)
\(788\) 0 0
\(789\) −4.28192 −0.152441
\(790\) 0 0
\(791\) −3.32266 27.8350i −0.118140 0.989699i
\(792\) 0 0
\(793\) 11.6761 11.6761i 0.414631 0.414631i
\(794\) 0 0
\(795\) 2.82449 + 22.9828i 0.100174 + 0.815117i
\(796\) 0 0
\(797\) 31.3730 31.3730i 1.11129 1.11129i 0.118313 0.992976i \(-0.462251\pi\)
0.992976 0.118313i \(-0.0377485\pi\)
\(798\) 0 0
\(799\) 7.68619i 0.271918i
\(800\) 0 0
\(801\) 65.1501i 2.30197i
\(802\) 0 0
\(803\) 4.85606 + 4.85606i 0.171367 + 0.171367i
\(804\) 0 0
\(805\) 47.0413 11.5662i 1.65799 0.407654i
\(806\) 0 0
\(807\) 5.76138 + 5.76138i 0.202810 + 0.202810i
\(808\) 0 0
\(809\) 43.8681i 1.54232i 0.636642 + 0.771160i \(0.280323\pi\)
−0.636642 + 0.771160i \(0.719677\pi\)
\(810\) 0 0
\(811\) 20.3680i 0.715217i 0.933872 + 0.357609i \(0.116408\pi\)
−0.933872 + 0.357609i \(0.883592\pi\)
\(812\) 0 0
\(813\) −1.24439 + 1.24439i −0.0436428 + 0.0436428i
\(814\) 0 0
\(815\) 26.2213 + 20.4817i 0.918494 + 0.717444i
\(816\) 0 0
\(817\) 26.0139 26.0139i 0.910112 0.910112i
\(818\) 0 0
\(819\) −3.85476 32.2926i −0.134696 1.12839i
\(820\) 0 0
\(821\) 2.15818 0.0753209 0.0376604 0.999291i \(-0.488009\pi\)
0.0376604 + 0.999291i \(0.488009\pi\)
\(822\) 0 0
\(823\) 24.0905 24.0905i 0.839741 0.839741i −0.149083 0.988825i \(-0.547632\pi\)
0.988825 + 0.149083i \(0.0476322\pi\)
\(824\) 0 0
\(825\) 33.8035 + 20.3011i 1.17689 + 0.706794i
\(826\) 0 0
\(827\) −24.0992 24.0992i −0.838011 0.838011i 0.150586 0.988597i \(-0.451884\pi\)
−0.988597 + 0.150586i \(0.951884\pi\)
\(828\) 0 0
\(829\) 26.8499 0.932536 0.466268 0.884643i \(-0.345598\pi\)
0.466268 + 0.884643i \(0.345598\pi\)
\(830\) 0 0
\(831\) 30.3157i 1.05164i
\(832\) 0 0
\(833\) −17.7471 10.8268i −0.614902 0.375126i
\(834\) 0 0
\(835\) 22.5857 + 17.6419i 0.781610 + 0.610522i
\(836\) 0 0
\(837\) 71.9689 + 71.9689i 2.48761 + 2.48761i
\(838\) 0 0
\(839\) 16.1545 0.557717 0.278858 0.960332i \(-0.410044\pi\)
0.278858 + 0.960332i \(0.410044\pi\)
\(840\) 0 0
\(841\) 10.6365 0.366775
\(842\) 0 0
\(843\) −50.5454 50.5454i −1.74087 1.74087i
\(844\) 0 0
\(845\) 20.7086 2.54500i 0.712397 0.0875506i
\(846\) 0 0
\(847\) 7.19113 9.14066i 0.247090 0.314077i
\(848\) 0 0
\(849\) 40.2350i 1.38086i
\(850\) 0 0
\(851\) −2.12724 −0.0729208
\(852\) 0 0
\(853\) −2.00773 2.00773i −0.0687435 0.0687435i 0.671899 0.740643i \(-0.265479\pi\)
−0.740643 + 0.671899i \(0.765479\pi\)
\(854\) 0 0
\(855\) −63.2168 + 7.76908i −2.16197 + 0.265697i
\(856\) 0 0
\(857\) −23.9532 + 23.9532i −0.818225 + 0.818225i −0.985851 0.167626i \(-0.946390\pi\)
0.167626 + 0.985851i \(0.446390\pi\)
\(858\) 0 0
\(859\) 28.0499 0.957049 0.478525 0.878074i \(-0.341172\pi\)
0.478525 + 0.878074i \(0.341172\pi\)
\(860\) 0 0
\(861\) 3.67445 + 30.7821i 0.125225 + 1.04905i
\(862\) 0 0
\(863\) 0.191578 0.191578i 0.00652139 0.00652139i −0.703839 0.710360i \(-0.748532\pi\)
0.710360 + 0.703839i \(0.248532\pi\)
\(864\) 0 0
\(865\) 19.4530 24.9043i 0.661421 0.846772i
\(866\) 0 0
\(867\) 17.7500 17.7500i 0.602821 0.602821i
\(868\) 0 0
\(869\) 31.4694i 1.06753i
\(870\) 0 0
\(871\) 18.1203i 0.613982i
\(872\) 0 0
\(873\) 45.9421 + 45.9421i 1.55490 + 1.55490i
\(874\) 0 0
\(875\) 28.2615 + 8.73433i 0.955413 + 0.295274i
\(876\) 0 0
\(877\) 41.2975 + 41.2975i 1.39452 + 1.39452i 0.814863 + 0.579653i \(0.196812\pi\)
0.579653 + 0.814863i \(0.303188\pi\)
\(878\) 0 0
\(879\) 49.4529i 1.66801i
\(880\) 0 0
\(881\) 24.7259i 0.833037i −0.909127 0.416519i \(-0.863250\pi\)
0.909127 0.416519i \(-0.136750\pi\)
\(882\) 0 0
\(883\) 12.2632 12.2632i 0.412688 0.412688i −0.469986 0.882674i \(-0.655741\pi\)
0.882674 + 0.469986i \(0.155741\pi\)
\(884\) 0 0
\(885\) −26.8020 + 34.3128i −0.900940 + 1.15341i
\(886\) 0 0
\(887\) −14.0519 + 14.0519i −0.471816 + 0.471816i −0.902502 0.430686i \(-0.858272\pi\)
0.430686 + 0.902502i \(0.358272\pi\)
\(888\) 0 0
\(889\) −25.2111 + 3.00944i −0.845553 + 0.100933i
\(890\) 0 0
\(891\) −33.2243 −1.11305
\(892\) 0 0
\(893\) −8.12309 + 8.12309i −0.271829 + 0.271829i
\(894\) 0 0
\(895\) −33.0007 + 4.05564i −1.10309 + 0.135565i
\(896\) 0 0
\(897\) 34.0345 + 34.0345i 1.13638 + 1.13638i
\(898\) 0 0
\(899\) −41.5923 −1.38718
\(900\) 0 0
\(901\) 10.0219i 0.333876i
\(902\) 0 0
\(903\) 41.6079 52.8879i 1.38463 1.76000i
\(904\) 0 0
\(905\) −31.5360 + 3.87565i −1.04829 + 0.128831i
\(906\) 0 0
\(907\) −30.9592 30.9592i −1.02798 1.02798i −0.999597 0.0283860i \(-0.990963\pi\)
−0.0283860 0.999597i \(-0.509037\pi\)
\(908\) 0 0
\(909\) −7.21967 −0.239461
\(910\) 0 0
\(911\) 22.1629 0.734289 0.367145 0.930164i \(-0.380335\pi\)
0.367145 + 0.930164i \(0.380335\pi\)
\(912\) 0 0
\(913\) −5.36974 5.36974i −0.177712 0.177712i
\(914\) 0 0
\(915\) −46.6170 36.4130i −1.54111 1.20378i
\(916\) 0 0
\(917\) −17.4222 13.7064i −0.575332 0.452625i
\(918\) 0 0
\(919\) 5.60542i 0.184906i 0.995717 + 0.0924529i \(0.0294708\pi\)
−0.995717 + 0.0924529i \(0.970529\pi\)
\(920\) 0 0
\(921\) −55.1257 −1.81645
\(922\) 0 0
\(923\) −8.18853 8.18853i −0.269529 0.269529i
\(924\) 0 0
\(925\) −1.11357 0.668769i −0.0366140 0.0219890i
\(926\) 0 0
\(927\) 10.0584 10.0584i 0.330363 0.330363i
\(928\) 0 0
\(929\) 29.8751 0.980170 0.490085 0.871675i \(-0.336966\pi\)
0.490085 + 0.871675i \(0.336966\pi\)
\(930\) 0 0
\(931\) −7.31371 30.1982i −0.239697 0.989705i
\(932\) 0 0
\(933\) −26.6299 + 26.6299i −0.871824 + 0.871824i
\(934\) 0 0
\(935\) 13.4492 + 10.5053i 0.439837 + 0.343561i
\(936\) 0 0
\(937\) 34.6529 34.6529i 1.13206 1.13206i 0.142226 0.989834i \(-0.454574\pi\)
0.989834 0.142226i \(-0.0454258\pi\)
\(938\) 0 0
\(939\) 51.9220i 1.69441i
\(940\) 0 0
\(941\) 21.6170i 0.704693i −0.935870 0.352346i \(-0.885384\pi\)
0.935870 0.352346i \(-0.114616\pi\)
\(942\) 0 0
\(943\) −22.1074 22.1074i −0.719916 0.719916i
\(944\) 0 0
\(945\) −60.2440 + 14.8123i −1.95974 + 0.481845i
\(946\) 0 0
\(947\) 7.95185 + 7.95185i 0.258400 + 0.258400i 0.824403 0.566003i \(-0.191511\pi\)
−0.566003 + 0.824403i \(0.691511\pi\)
\(948\) 0 0
\(949\) 5.11887i 0.166165i
\(950\) 0 0
\(951\) 82.4331i 2.67308i
\(952\) 0 0
\(953\) −12.7397 + 12.7397i −0.412678 + 0.412678i −0.882671 0.469992i \(-0.844257\pi\)
0.469992 + 0.882671i \(0.344257\pi\)
\(954\) 0 0
\(955\) −0.284050 2.31131i −0.00919165 0.0747923i
\(956\) 0 0
\(957\) 23.8964 23.8964i 0.772459 0.772459i
\(958\) 0 0
\(959\) −28.1241 + 3.35717i −0.908176 + 0.108409i
\(960\) 0 0
\(961\) −63.2040 −2.03884
\(962\) 0 0
\(963\) 0.932029 0.932029i 0.0300342 0.0300342i
\(964\) 0 0
\(965\) 0.172021 + 0.134367i 0.00553756 + 0.00432544i
\(966\) 0 0
\(967\) 11.8492 + 11.8492i 0.381045 + 0.381045i 0.871478 0.490434i \(-0.163162\pi\)
−0.490434 + 0.871478i \(0.663162\pi\)
\(968\) 0 0
\(969\) −40.4533 −1.29955
\(970\) 0 0
\(971\) 48.5371i 1.55763i 0.627255 + 0.778814i \(0.284179\pi\)
−0.627255 + 0.778814i \(0.715821\pi\)
\(972\) 0 0
\(973\) 18.3927 23.3790i 0.589642 0.749495i
\(974\) 0 0
\(975\) 7.11656 + 28.5163i 0.227912 + 0.913254i
\(976\) 0 0
\(977\) −18.9303 18.9303i −0.605635 0.605635i 0.336167 0.941802i \(-0.390869\pi\)
−0.941802 + 0.336167i \(0.890869\pi\)
\(978\) 0 0
\(979\) 26.0904 0.833854
\(980\) 0 0
\(981\) 41.7840 1.33406
\(982\) 0 0
\(983\) −7.97683 7.97683i −0.254421 0.254421i 0.568359 0.822781i \(-0.307578\pi\)
−0.822781 + 0.568359i \(0.807578\pi\)
\(984\) 0 0
\(985\) −24.9061 + 31.8855i −0.793573 + 1.01596i
\(986\) 0 0
\(987\) −12.9925 + 16.5147i −0.413555 + 0.525670i
\(988\) 0 0
\(989\) 67.8660i 2.15801i
\(990\) 0 0
\(991\) −49.6706 −1.57784 −0.788920 0.614496i \(-0.789359\pi\)
−0.788920 + 0.614496i \(0.789359\pi\)
\(992\) 0 0
\(993\) 51.7112 + 51.7112i 1.64100 + 1.64100i
\(994\) 0 0
\(995\) 1.11771 + 9.09482i 0.0354339 + 0.288325i
\(996\) 0 0
\(997\) −12.4002 + 12.4002i −0.392718 + 0.392718i −0.875655 0.482937i \(-0.839570\pi\)
0.482937 + 0.875655i \(0.339570\pi\)
\(998\) 0 0
\(999\) 2.72427 0.0861922
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 280.2.x.a.97.1 24
4.3 odd 2 560.2.bj.d.97.12 24
5.2 odd 4 1400.2.x.b.993.1 24
5.3 odd 4 inner 280.2.x.a.153.12 yes 24
5.4 even 2 1400.2.x.b.657.12 24
7.6 odd 2 inner 280.2.x.a.97.12 yes 24
20.3 even 4 560.2.bj.d.433.1 24
28.27 even 2 560.2.bj.d.97.1 24
35.13 even 4 inner 280.2.x.a.153.1 yes 24
35.27 even 4 1400.2.x.b.993.12 24
35.34 odd 2 1400.2.x.b.657.1 24
140.83 odd 4 560.2.bj.d.433.12 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.x.a.97.1 24 1.1 even 1 trivial
280.2.x.a.97.12 yes 24 7.6 odd 2 inner
280.2.x.a.153.1 yes 24 35.13 even 4 inner
280.2.x.a.153.12 yes 24 5.3 odd 4 inner
560.2.bj.d.97.1 24 28.27 even 2
560.2.bj.d.97.12 24 4.3 odd 2
560.2.bj.d.433.1 24 20.3 even 4
560.2.bj.d.433.12 24 140.83 odd 4
1400.2.x.b.657.1 24 35.34 odd 2
1400.2.x.b.657.12 24 5.4 even 2
1400.2.x.b.993.1 24 5.2 odd 4
1400.2.x.b.993.12 24 35.27 even 4