Properties

Label 2352.2.bl.n.607.2
Level $2352$
Weight $2$
Character 2352.607
Analytic conductor $18.781$
Analytic rank $0$
Dimension $4$
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] = [2352,2,Mod(31,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.bl (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: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 607.2
Root \(0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 2352.607
Dual form 2352.2.bl.n.31.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(2.12132 + 1.22474i) q^{5} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{3} +(2.12132 + 1.22474i) q^{5} +(-0.500000 + 0.866025i) q^{9} +(-2.12132 + 1.22474i) q^{11} -4.89898i q^{13} +2.44949i q^{15} +(-2.12132 + 1.22474i) q^{17} +(-1.00000 + 1.73205i) q^{19} +(6.36396 + 3.67423i) q^{23} +(0.500000 + 0.866025i) q^{25} -1.00000 q^{27} +6.00000 q^{29} +(4.00000 + 6.92820i) q^{31} +(-2.12132 - 1.22474i) q^{33} +(-2.00000 + 3.46410i) q^{37} +(4.24264 - 2.44949i) q^{39} +7.34847i q^{41} +4.89898i q^{43} +(-2.12132 + 1.22474i) q^{45} +(-6.00000 + 10.3923i) q^{47} +(-2.12132 - 1.22474i) q^{51} +(3.00000 + 5.19615i) q^{53} -6.00000 q^{55} -2.00000 q^{57} +(6.00000 + 10.3923i) q^{59} +(6.00000 - 10.3923i) q^{65} +7.34847i q^{69} -12.2474i q^{71} +(12.7279 - 7.34847i) q^{73} +(-0.500000 + 0.866025i) q^{75} +(-8.48528 - 4.89898i) q^{79} +(-0.500000 - 0.866025i) q^{81} -6.00000 q^{85} +(3.00000 + 5.19615i) q^{87} +(-10.6066 - 6.12372i) q^{89} +(-4.00000 + 6.92820i) q^{93} +(-4.24264 + 2.44949i) q^{95} -4.89898i q^{97} -2.44949i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 2 q^{9} - 4 q^{19} + 2 q^{25} - 4 q^{27} + 24 q^{29} + 16 q^{31} - 8 q^{37} - 24 q^{47} + 12 q^{53} - 24 q^{55} - 8 q^{57} + 24 q^{59} + 24 q^{65} - 2 q^{75} - 2 q^{81} - 24 q^{85} + 12 q^{87} - 16 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) 2.12132 + 1.22474i 0.948683 + 0.547723i 0.892672 0.450708i \(-0.148828\pi\)
0.0560116 + 0.998430i \(0.482162\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −2.12132 + 1.22474i −0.639602 + 0.369274i −0.784461 0.620178i \(-0.787060\pi\)
0.144859 + 0.989452i \(0.453727\pi\)
\(12\) 0 0
\(13\) 4.89898i 1.35873i −0.733799 0.679366i \(-0.762255\pi\)
0.733799 0.679366i \(-0.237745\pi\)
\(14\) 0 0
\(15\) 2.44949i 0.632456i
\(16\) 0 0
\(17\) −2.12132 + 1.22474i −0.514496 + 0.297044i −0.734680 0.678414i \(-0.762668\pi\)
0.220184 + 0.975458i \(0.429334\pi\)
\(18\) 0 0
\(19\) −1.00000 + 1.73205i −0.229416 + 0.397360i −0.957635 0.287984i \(-0.907015\pi\)
0.728219 + 0.685344i \(0.240348\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.36396 + 3.67423i 1.32698 + 0.766131i 0.984831 0.173516i \(-0.0555127\pi\)
0.342147 + 0.939647i \(0.388846\pi\)
\(24\) 0 0
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 + 6.92820i 0.718421 + 1.24434i 0.961625 + 0.274367i \(0.0884683\pi\)
−0.243204 + 0.969975i \(0.578198\pi\)
\(32\) 0 0
\(33\) −2.12132 1.22474i −0.369274 0.213201i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 + 3.46410i −0.328798 + 0.569495i −0.982274 0.187453i \(-0.939977\pi\)
0.653476 + 0.756948i \(0.273310\pi\)
\(38\) 0 0
\(39\) 4.24264 2.44949i 0.679366 0.392232i
\(40\) 0 0
\(41\) 7.34847i 1.14764i 0.818982 + 0.573819i \(0.194539\pi\)
−0.818982 + 0.573819i \(0.805461\pi\)
\(42\) 0 0
\(43\) 4.89898i 0.747087i 0.927613 + 0.373544i \(0.121857\pi\)
−0.927613 + 0.373544i \(0.878143\pi\)
\(44\) 0 0
\(45\) −2.12132 + 1.22474i −0.316228 + 0.182574i
\(46\) 0 0
\(47\) −6.00000 + 10.3923i −0.875190 + 1.51587i −0.0186297 + 0.999826i \(0.505930\pi\)
−0.856560 + 0.516047i \(0.827403\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −2.12132 1.22474i −0.297044 0.171499i
\(52\) 0 0
\(53\) 3.00000 + 5.19615i 0.412082 + 0.713746i 0.995117 0.0987002i \(-0.0314685\pi\)
−0.583036 + 0.812447i \(0.698135\pi\)
\(54\) 0 0
\(55\) −6.00000 −0.809040
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) 6.00000 + 10.3923i 0.781133 + 1.35296i 0.931282 + 0.364299i \(0.118692\pi\)
−0.150148 + 0.988663i \(0.547975\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000 10.3923i 0.744208 1.28901i
\(66\) 0 0
\(67\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 0 0
\(69\) 7.34847i 0.884652i
\(70\) 0 0
\(71\) 12.2474i 1.45350i −0.686900 0.726752i \(-0.741029\pi\)
0.686900 0.726752i \(-0.258971\pi\)
\(72\) 0 0
\(73\) 12.7279 7.34847i 1.48969 0.860073i 0.489760 0.871857i \(-0.337084\pi\)
0.999931 + 0.0117840i \(0.00375105\pi\)
\(74\) 0 0
\(75\) −0.500000 + 0.866025i −0.0577350 + 0.100000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.48528 4.89898i −0.954669 0.551178i −0.0601406 0.998190i \(-0.519155\pi\)
−0.894528 + 0.447012i \(0.852488\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 3.00000 + 5.19615i 0.321634 + 0.557086i
\(88\) 0 0
\(89\) −10.6066 6.12372i −1.12430 0.649113i −0.181803 0.983335i \(-0.558193\pi\)
−0.942495 + 0.334221i \(0.891527\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.00000 + 6.92820i −0.414781 + 0.718421i
\(94\) 0 0
\(95\) −4.24264 + 2.44949i −0.435286 + 0.251312i
\(96\) 0 0
\(97\) 4.89898i 0.497416i −0.968579 0.248708i \(-0.919994\pi\)
0.968579 0.248708i \(-0.0800060\pi\)
\(98\) 0 0
\(99\) 2.44949i 0.246183i
\(100\) 0 0
\(101\) 2.12132 1.22474i 0.211079 0.121867i −0.390734 0.920504i \(-0.627779\pi\)
0.601813 + 0.798637i \(0.294445\pi\)
\(102\) 0 0
\(103\) −8.00000 + 13.8564i −0.788263 + 1.36531i 0.138767 + 0.990325i \(0.455686\pi\)
−0.927030 + 0.374987i \(0.877647\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.6066 6.12372i −1.02538 0.592003i −0.109721 0.993962i \(-0.534996\pi\)
−0.915657 + 0.401960i \(0.868329\pi\)
\(108\) 0 0
\(109\) −1.00000 1.73205i −0.0957826 0.165900i 0.814152 0.580651i \(-0.197202\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 9.00000 + 15.5885i 0.839254 + 1.45363i
\(116\) 0 0
\(117\) 4.24264 + 2.44949i 0.392232 + 0.226455i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.50000 + 4.33013i −0.227273 + 0.393648i
\(122\) 0 0
\(123\) −6.36396 + 3.67423i −0.573819 + 0.331295i
\(124\) 0 0
\(125\) 9.79796i 0.876356i
\(126\) 0 0
\(127\) 9.79796i 0.869428i −0.900568 0.434714i \(-0.856849\pi\)
0.900568 0.434714i \(-0.143151\pi\)
\(128\) 0 0
\(129\) −4.24264 + 2.44949i −0.373544 + 0.215666i
\(130\) 0 0
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −2.12132 1.22474i −0.182574 0.105409i
\(136\) 0 0
\(137\) −3.00000 5.19615i −0.256307 0.443937i 0.708942 0.705266i \(-0.249173\pi\)
−0.965250 + 0.261329i \(0.915839\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 0 0
\(143\) 6.00000 + 10.3923i 0.501745 + 0.869048i
\(144\) 0 0
\(145\) 12.7279 + 7.34847i 1.05700 + 0.610257i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.00000 15.5885i 0.737309 1.27706i −0.216394 0.976306i \(-0.569430\pi\)
0.953703 0.300750i \(-0.0972370\pi\)
\(150\) 0 0
\(151\) 12.7279 7.34847i 1.03578 0.598010i 0.117147 0.993115i \(-0.462625\pi\)
0.918636 + 0.395105i \(0.129292\pi\)
\(152\) 0 0
\(153\) 2.44949i 0.198030i
\(154\) 0 0
\(155\) 19.5959i 1.57398i
\(156\) 0 0
\(157\) −16.9706 + 9.79796i −1.35440 + 0.781962i −0.988862 0.148835i \(-0.952448\pi\)
−0.365536 + 0.930797i \(0.619114\pi\)
\(158\) 0 0
\(159\) −3.00000 + 5.19615i −0.237915 + 0.412082i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 16.9706 + 9.79796i 1.32924 + 0.767435i 0.985182 0.171514i \(-0.0548659\pi\)
0.344055 + 0.938949i \(0.388199\pi\)
\(164\) 0 0
\(165\) −3.00000 5.19615i −0.233550 0.404520i
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) −1.00000 1.73205i −0.0764719 0.132453i
\(172\) 0 0
\(173\) 6.36396 + 3.67423i 0.483843 + 0.279347i 0.722017 0.691876i \(-0.243215\pi\)
−0.238174 + 0.971223i \(0.576549\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −6.00000 + 10.3923i −0.450988 + 0.781133i
\(178\) 0 0
\(179\) 14.8492 8.57321i 1.10988 0.640792i 0.171085 0.985256i \(-0.445273\pi\)
0.938799 + 0.344464i \(0.111939\pi\)
\(180\) 0 0
\(181\) 4.89898i 0.364138i 0.983286 + 0.182069i \(0.0582795\pi\)
−0.983286 + 0.182069i \(0.941721\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.48528 + 4.89898i −0.623850 + 0.360180i
\(186\) 0 0
\(187\) 3.00000 5.19615i 0.219382 0.379980i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2.12132 1.22474i −0.153493 0.0886194i 0.421286 0.906928i \(-0.361579\pi\)
−0.574779 + 0.818308i \(0.694912\pi\)
\(192\) 0 0
\(193\) −8.00000 13.8564i −0.575853 0.997406i −0.995948 0.0899262i \(-0.971337\pi\)
0.420096 0.907480i \(-0.361996\pi\)
\(194\) 0 0
\(195\) 12.0000 0.859338
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −1.00000 1.73205i −0.0708881 0.122782i 0.828403 0.560133i \(-0.189250\pi\)
−0.899291 + 0.437351i \(0.855917\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −9.00000 + 15.5885i −0.628587 + 1.08875i
\(206\) 0 0
\(207\) −6.36396 + 3.67423i −0.442326 + 0.255377i
\(208\) 0 0
\(209\) 4.89898i 0.338869i
\(210\) 0 0
\(211\) 4.89898i 0.337260i −0.985679 0.168630i \(-0.946066\pi\)
0.985679 0.168630i \(-0.0539342\pi\)
\(212\) 0 0
\(213\) 10.6066 6.12372i 0.726752 0.419591i
\(214\) 0 0
\(215\) −6.00000 + 10.3923i −0.409197 + 0.708749i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 12.7279 + 7.34847i 0.860073 + 0.496564i
\(220\) 0 0
\(221\) 6.00000 + 10.3923i 0.403604 + 0.699062i
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) 6.00000 + 10.3923i 0.398234 + 0.689761i 0.993508 0.113761i \(-0.0362899\pi\)
−0.595274 + 0.803523i \(0.702957\pi\)
\(228\) 0 0
\(229\) −21.2132 12.2474i −1.40181 0.809334i −0.407230 0.913326i \(-0.633505\pi\)
−0.994578 + 0.103992i \(0.966839\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.0000 25.9808i 0.982683 1.70206i 0.330870 0.943676i \(-0.392658\pi\)
0.651813 0.758380i \(-0.274009\pi\)
\(234\) 0 0
\(235\) −25.4558 + 14.6969i −1.66056 + 0.958723i
\(236\) 0 0
\(237\) 9.79796i 0.636446i
\(238\) 0 0
\(239\) 12.2474i 0.792222i 0.918203 + 0.396111i \(0.129640\pi\)
−0.918203 + 0.396111i \(0.870360\pi\)
\(240\) 0 0
\(241\) −4.24264 + 2.44949i −0.273293 + 0.157786i −0.630383 0.776284i \(-0.717102\pi\)
0.357090 + 0.934070i \(0.383769\pi\)
\(242\) 0 0
\(243\) 0.500000 0.866025i 0.0320750 0.0555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.48528 + 4.89898i 0.539906 + 0.311715i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −18.0000 −1.13165
\(254\) 0 0
\(255\) −3.00000 5.19615i −0.187867 0.325396i
\(256\) 0 0
\(257\) 19.0919 + 11.0227i 1.19092 + 0.687577i 0.958515 0.285042i \(-0.0920075\pi\)
0.232404 + 0.972619i \(0.425341\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3.00000 + 5.19615i −0.185695 + 0.321634i
\(262\) 0 0
\(263\) −2.12132 + 1.22474i −0.130806 + 0.0755210i −0.563975 0.825792i \(-0.690729\pi\)
0.433169 + 0.901313i \(0.357395\pi\)
\(264\) 0 0
\(265\) 14.6969i 0.902826i
\(266\) 0 0
\(267\) 12.2474i 0.749532i
\(268\) 0 0
\(269\) 19.0919 11.0227i 1.16405 0.672066i 0.211781 0.977317i \(-0.432074\pi\)
0.952272 + 0.305251i \(0.0987404\pi\)
\(270\) 0 0
\(271\) −8.00000 + 13.8564i −0.485965 + 0.841717i −0.999870 0.0161307i \(-0.994865\pi\)
0.513905 + 0.857847i \(0.328199\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.12132 1.22474i −0.127920 0.0738549i
\(276\) 0 0
\(277\) 4.00000 + 6.92820i 0.240337 + 0.416275i 0.960810 0.277207i \(-0.0894088\pi\)
−0.720473 + 0.693482i \(0.756075\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −7.00000 12.1244i −0.416107 0.720718i 0.579437 0.815017i \(-0.303272\pi\)
−0.995544 + 0.0942988i \(0.969939\pi\)
\(284\) 0 0
\(285\) −4.24264 2.44949i −0.251312 0.145095i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −5.50000 + 9.52628i −0.323529 + 0.560369i
\(290\) 0 0
\(291\) 4.24264 2.44949i 0.248708 0.143592i
\(292\) 0 0
\(293\) 22.0454i 1.28791i 0.765065 + 0.643953i \(0.222707\pi\)
−0.765065 + 0.643953i \(0.777293\pi\)
\(294\) 0 0
\(295\) 29.3939i 1.71138i
\(296\) 0 0
\(297\) 2.12132 1.22474i 0.123091 0.0710669i
\(298\) 0 0
\(299\) 18.0000 31.1769i 1.04097 1.80301i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 2.12132 + 1.22474i 0.121867 + 0.0703598i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 26.0000 1.48390 0.741949 0.670456i \(-0.233902\pi\)
0.741949 + 0.670456i \(0.233902\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −12.0000 20.7846i −0.680458 1.17859i −0.974841 0.222900i \(-0.928448\pi\)
0.294384 0.955687i \(-0.404886\pi\)
\(312\) 0 0
\(313\) −8.48528 4.89898i −0.479616 0.276907i 0.240640 0.970614i \(-0.422643\pi\)
−0.720257 + 0.693708i \(0.755976\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.00000 15.5885i 0.505490 0.875535i −0.494489 0.869184i \(-0.664645\pi\)
0.999980 0.00635137i \(-0.00202172\pi\)
\(318\) 0 0
\(319\) −12.7279 + 7.34847i −0.712627 + 0.411435i
\(320\) 0 0
\(321\) 12.2474i 0.683586i
\(322\) 0 0
\(323\) 4.89898i 0.272587i
\(324\) 0 0
\(325\) 4.24264 2.44949i 0.235339 0.135873i
\(326\) 0 0
\(327\) 1.00000 1.73205i 0.0553001 0.0957826i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −21.2132 12.2474i −1.16598 0.673181i −0.213253 0.976997i \(-0.568406\pi\)
−0.952731 + 0.303816i \(0.901739\pi\)
\(332\) 0 0
\(333\) −2.00000 3.46410i −0.109599 0.189832i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 3.00000 + 5.19615i 0.162938 + 0.282216i
\(340\) 0 0
\(341\) −16.9706 9.79796i −0.919007 0.530589i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −9.00000 + 15.5885i −0.484544 + 0.839254i
\(346\) 0 0
\(347\) 14.8492 8.57321i 0.797149 0.460234i −0.0453242 0.998972i \(-0.514432\pi\)
0.842473 + 0.538738i \(0.181099\pi\)
\(348\) 0 0
\(349\) 19.5959i 1.04895i 0.851427 + 0.524473i \(0.175738\pi\)
−0.851427 + 0.524473i \(0.824262\pi\)
\(350\) 0 0
\(351\) 4.89898i 0.261488i
\(352\) 0 0
\(353\) 2.12132 1.22474i 0.112906 0.0651866i −0.442483 0.896777i \(-0.645902\pi\)
0.555390 + 0.831590i \(0.312569\pi\)
\(354\) 0 0
\(355\) 15.0000 25.9808i 0.796117 1.37892i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.6066 + 6.12372i 0.559795 + 0.323198i 0.753063 0.657948i \(-0.228575\pi\)
−0.193268 + 0.981146i \(0.561909\pi\)
\(360\) 0 0
\(361\) 7.50000 + 12.9904i 0.394737 + 0.683704i
\(362\) 0 0
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 36.0000 1.88433
\(366\) 0 0
\(367\) −7.00000 12.1244i −0.365397 0.632886i 0.623443 0.781869i \(-0.285733\pi\)
−0.988840 + 0.148983i \(0.952400\pi\)
\(368\) 0 0
\(369\) −6.36396 3.67423i −0.331295 0.191273i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 5.00000 8.66025i 0.258890 0.448411i −0.707055 0.707159i \(-0.749977\pi\)
0.965945 + 0.258748i \(0.0833099\pi\)
\(374\) 0 0
\(375\) 8.48528 4.89898i 0.438178 0.252982i
\(376\) 0 0
\(377\) 29.3939i 1.51386i
\(378\) 0 0
\(379\) 34.2929i 1.76151i −0.473576 0.880753i \(-0.657037\pi\)
0.473576 0.880753i \(-0.342963\pi\)
\(380\) 0 0
\(381\) 8.48528 4.89898i 0.434714 0.250982i
\(382\) 0 0
\(383\) −6.00000 + 10.3923i −0.306586 + 0.531022i −0.977613 0.210411i \(-0.932520\pi\)
0.671027 + 0.741433i \(0.265853\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.24264 2.44949i −0.215666 0.124515i
\(388\) 0 0
\(389\) 15.0000 + 25.9808i 0.760530 + 1.31728i 0.942578 + 0.333987i \(0.108394\pi\)
−0.182047 + 0.983290i \(0.558272\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −12.0000 20.7846i −0.603786 1.04579i
\(396\) 0 0
\(397\) 8.48528 + 4.89898i 0.425864 + 0.245873i 0.697583 0.716504i \(-0.254259\pi\)
−0.271719 + 0.962377i \(0.587592\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9.00000 15.5885i 0.449439 0.778450i −0.548911 0.835881i \(-0.684957\pi\)
0.998350 + 0.0574304i \(0.0182907\pi\)
\(402\) 0 0
\(403\) 33.9411 19.5959i 1.69073 0.976142i
\(404\) 0 0
\(405\) 2.44949i 0.121716i
\(406\) 0 0
\(407\) 9.79796i 0.485667i
\(408\) 0 0
\(409\) −4.24264 + 2.44949i −0.209785 + 0.121119i −0.601212 0.799090i \(-0.705315\pi\)
0.391426 + 0.920209i \(0.371982\pi\)
\(410\) 0 0
\(411\) 3.00000 5.19615i 0.147979 0.256307i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 2.00000 + 3.46410i 0.0979404 + 0.169638i
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) −8.00000 −0.389896 −0.194948 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) 0 0
\(423\) −6.00000 10.3923i −0.291730 0.505291i
\(424\) 0 0
\(425\) −2.12132 1.22474i −0.102899 0.0594089i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −6.00000 + 10.3923i −0.289683 + 0.501745i
\(430\) 0 0
\(431\) −14.8492 + 8.57321i −0.715263 + 0.412957i −0.813007 0.582254i \(-0.802171\pi\)
0.0977438 + 0.995212i \(0.468837\pi\)
\(432\) 0 0
\(433\) 9.79796i 0.470860i 0.971891 + 0.235430i \(0.0756498\pi\)
−0.971891 + 0.235430i \(0.924350\pi\)
\(434\) 0 0
\(435\) 14.6969i 0.704664i
\(436\) 0 0
\(437\) −12.7279 + 7.34847i −0.608859 + 0.351525i
\(438\) 0 0
\(439\) 11.0000 19.0526i 0.525001 0.909329i −0.474575 0.880215i \(-0.657398\pi\)
0.999576 0.0291138i \(-0.00926853\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.12132 1.22474i −0.100787 0.0581894i 0.448759 0.893653i \(-0.351866\pi\)
−0.549546 + 0.835463i \(0.685199\pi\)
\(444\) 0 0
\(445\) −15.0000 25.9808i −0.711068 1.23161i
\(446\) 0 0
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −9.00000 15.5885i −0.423793 0.734032i
\(452\) 0 0
\(453\) 12.7279 + 7.34847i 0.598010 + 0.345261i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −19.0000 + 32.9090i −0.888783 + 1.53942i −0.0474665 + 0.998873i \(0.515115\pi\)
−0.841316 + 0.540544i \(0.818219\pi\)
\(458\) 0 0
\(459\) 2.12132 1.22474i 0.0990148 0.0571662i
\(460\) 0 0
\(461\) 2.44949i 0.114084i −0.998372 0.0570421i \(-0.981833\pi\)
0.998372 0.0570421i \(-0.0181669\pi\)
\(462\) 0 0
\(463\) 9.79796i 0.455350i −0.973737 0.227675i \(-0.926888\pi\)
0.973737 0.227675i \(-0.0731123\pi\)
\(464\) 0 0
\(465\) −16.9706 + 9.79796i −0.786991 + 0.454369i
\(466\) 0 0
\(467\) 12.0000 20.7846i 0.555294 0.961797i −0.442587 0.896726i \(-0.645939\pi\)
0.997881 0.0650714i \(-0.0207275\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −16.9706 9.79796i −0.781962 0.451466i
\(472\) 0 0
\(473\) −6.00000 10.3923i −0.275880 0.477839i
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) −12.0000 20.7846i −0.548294 0.949673i −0.998392 0.0566937i \(-0.981944\pi\)
0.450098 0.892979i \(-0.351389\pi\)
\(480\) 0 0
\(481\) 16.9706 + 9.79796i 0.773791 + 0.446748i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.00000 10.3923i 0.272446 0.471890i
\(486\) 0 0
\(487\) −4.24264 + 2.44949i −0.192252 + 0.110997i −0.593037 0.805176i \(-0.702071\pi\)
0.400784 + 0.916172i \(0.368738\pi\)
\(488\) 0 0
\(489\) 19.5959i 0.886158i
\(490\) 0 0
\(491\) 36.7423i 1.65816i −0.559131 0.829079i \(-0.688865\pi\)
0.559131 0.829079i \(-0.311135\pi\)
\(492\) 0 0
\(493\) −12.7279 + 7.34847i −0.573237 + 0.330958i
\(494\) 0 0
\(495\) 3.00000 5.19615i 0.134840 0.233550i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −12.7279 7.34847i −0.569780 0.328963i 0.187281 0.982306i \(-0.440032\pi\)
−0.757061 + 0.653344i \(0.773366\pi\)
\(500\) 0 0
\(501\) −6.00000 10.3923i −0.268060 0.464294i
\(502\) 0 0
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) −5.50000 9.52628i −0.244264 0.423077i
\(508\) 0 0
\(509\) 14.8492 + 8.57321i 0.658181 + 0.380001i 0.791584 0.611061i \(-0.209257\pi\)
−0.133402 + 0.991062i \(0.542590\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 1.00000 1.73205i 0.0441511 0.0764719i
\(514\) 0 0
\(515\) −33.9411 + 19.5959i −1.49562 + 0.863499i
\(516\) 0 0
\(517\) 29.3939i 1.29274i
\(518\) 0 0
\(519\) 7.34847i 0.322562i
\(520\) 0 0
\(521\) 2.12132 1.22474i 0.0929367 0.0536570i −0.452811 0.891606i \(-0.649579\pi\)
0.545748 + 0.837949i \(0.316246\pi\)
\(522\) 0 0
\(523\) 10.0000 17.3205i 0.437269 0.757373i −0.560208 0.828352i \(-0.689279\pi\)
0.997478 + 0.0709788i \(0.0226123\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −16.9706 9.79796i −0.739249 0.426806i
\(528\) 0 0
\(529\) 15.5000 + 26.8468i 0.673913 + 1.16725i
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 36.0000 1.55933
\(534\) 0 0
\(535\) −15.0000 25.9808i −0.648507 1.12325i
\(536\) 0 0
\(537\) 14.8492 + 8.57321i 0.640792 + 0.369961i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −8.00000 + 13.8564i −0.343947 + 0.595733i −0.985162 0.171628i \(-0.945097\pi\)
0.641215 + 0.767361i \(0.278431\pi\)
\(542\) 0 0
\(543\) −4.24264 + 2.44949i −0.182069 + 0.105118i
\(544\) 0 0
\(545\) 4.89898i 0.209849i
\(546\) 0 0
\(547\) 29.3939i 1.25679i 0.777894 + 0.628396i \(0.216288\pi\)
−0.777894 + 0.628396i \(0.783712\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.00000 + 10.3923i −0.255609 + 0.442727i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −8.48528 4.89898i −0.360180 0.207950i
\(556\) 0 0
\(557\) −3.00000 5.19615i −0.127114 0.220168i 0.795443 0.606028i \(-0.207238\pi\)
−0.922557 + 0.385860i \(0.873905\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) 0 0
\(563\) −12.0000 20.7846i −0.505740 0.875967i −0.999978 0.00664037i \(-0.997886\pi\)
0.494238 0.869326i \(-0.335447\pi\)
\(564\) 0 0
\(565\) 12.7279 + 7.34847i 0.535468 + 0.309152i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.00000 5.19615i 0.125767 0.217834i −0.796266 0.604947i \(-0.793194\pi\)
0.922032 + 0.387113i \(0.126528\pi\)
\(570\) 0 0
\(571\) 16.9706 9.79796i 0.710196 0.410032i −0.100938 0.994893i \(-0.532184\pi\)
0.811134 + 0.584861i \(0.198851\pi\)
\(572\) 0 0
\(573\) 2.44949i 0.102329i
\(574\) 0 0
\(575\) 7.34847i 0.306452i
\(576\) 0 0
\(577\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(578\) 0 0
\(579\) 8.00000 13.8564i 0.332469 0.575853i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −12.7279 7.34847i −0.527137 0.304342i
\(584\) 0 0
\(585\) 6.00000 + 10.3923i 0.248069 + 0.429669i
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 9.00000 + 15.5885i 0.370211 + 0.641223i
\(592\) 0 0
\(593\) −40.3051 23.2702i −1.65513 0.955591i −0.974914 0.222580i \(-0.928552\pi\)
−0.680217 0.733011i \(-0.738114\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.00000 1.73205i 0.0409273 0.0708881i
\(598\) 0 0
\(599\) −2.12132 + 1.22474i −0.0866748 + 0.0500417i −0.542711 0.839920i \(-0.682602\pi\)
0.456036 + 0.889961i \(0.349269\pi\)
\(600\) 0 0
\(601\) 19.5959i 0.799334i −0.916660 0.399667i \(-0.869126\pi\)
0.916660 0.399667i \(-0.130874\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −10.6066 + 6.12372i −0.431220 + 0.248965i
\(606\) 0 0
\(607\) −7.00000 + 12.1244i −0.284121 + 0.492112i −0.972396 0.233338i \(-0.925035\pi\)
0.688274 + 0.725450i \(0.258368\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 50.9117 + 29.3939i 2.05967 + 1.18915i
\(612\) 0 0
\(613\) −13.0000 22.5167i −0.525065 0.909439i −0.999574 0.0291886i \(-0.990708\pi\)
0.474509 0.880251i \(-0.342626\pi\)
\(614\) 0 0
\(615\) −18.0000 −0.725830
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) −2.00000 3.46410i −0.0803868 0.139234i 0.823029 0.567999i \(-0.192282\pi\)
−0.903416 + 0.428765i \(0.858949\pi\)
\(620\) 0 0
\(621\) −6.36396 3.67423i −0.255377 0.147442i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 14.5000 25.1147i 0.580000 1.00459i
\(626\) 0 0
\(627\) 4.24264 2.44949i 0.169435 0.0978232i
\(628\) 0 0
\(629\) 9.79796i 0.390670i
\(630\) 0 0
\(631\) 19.5959i 0.780101i 0.920793 + 0.390051i \(0.127542\pi\)
−0.920793 + 0.390051i \(0.872458\pi\)
\(632\) 0 0
\(633\) 4.24264 2.44949i 0.168630 0.0973585i
\(634\) 0 0
\(635\) 12.0000 20.7846i 0.476205 0.824812i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 10.6066 + 6.12372i 0.419591 + 0.242251i
\(640\) 0 0
\(641\) 3.00000 + 5.19615i 0.118493 + 0.205236i 0.919171 0.393860i \(-0.128860\pi\)
−0.800678 + 0.599095i \(0.795527\pi\)
\(642\) 0 0
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 0 0
\(645\) −12.0000 −0.472500
\(646\) 0 0
\(647\) −12.0000 20.7846i −0.471769 0.817127i 0.527710 0.849425i \(-0.323051\pi\)
−0.999478 + 0.0322975i \(0.989718\pi\)
\(648\) 0 0
\(649\) −25.4558 14.6969i −0.999229 0.576905i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.00000 5.19615i 0.117399 0.203341i −0.801337 0.598213i \(-0.795878\pi\)
0.918736 + 0.394872i \(0.129211\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 14.6969i 0.573382i
\(658\) 0 0
\(659\) 7.34847i 0.286256i −0.989704 0.143128i \(-0.954284\pi\)
0.989704 0.143128i \(-0.0457160\pi\)
\(660\) 0 0
\(661\) 25.4558 14.6969i 0.990118 0.571645i 0.0848081 0.996397i \(-0.472972\pi\)
0.905309 + 0.424753i \(0.139639\pi\)
\(662\) 0 0
\(663\) −6.00000 + 10.3923i −0.233021 + 0.403604i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 38.1838 + 22.0454i 1.47848 + 0.853602i
\(668\) 0 0
\(669\) −5.00000 8.66025i −0.193311 0.334825i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −8.00000 −0.308377 −0.154189 0.988041i \(-0.549276\pi\)
−0.154189 + 0.988041i \(0.549276\pi\)
\(674\) 0 0
\(675\) −0.500000 0.866025i −0.0192450 0.0333333i
\(676\) 0 0
\(677\) 10.6066 + 6.12372i 0.407645 + 0.235354i 0.689777 0.724022i \(-0.257708\pi\)
−0.282132 + 0.959375i \(0.591042\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −6.00000 + 10.3923i −0.229920 + 0.398234i
\(682\) 0 0
\(683\) −6.36396 + 3.67423i −0.243510 + 0.140591i −0.616789 0.787129i \(-0.711567\pi\)
0.373279 + 0.927719i \(0.378233\pi\)
\(684\) 0 0
\(685\) 14.6969i 0.561541i
\(686\) 0 0
\(687\) 24.4949i 0.934539i
\(688\) 0 0
\(689\) 25.4558 14.6969i 0.969790 0.559909i
\(690\) 0 0
\(691\) 10.0000 17.3205i 0.380418 0.658903i −0.610704 0.791859i \(-0.709113\pi\)
0.991122 + 0.132956i \(0.0424468\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.48528 + 4.89898i 0.321865 + 0.185829i
\(696\) 0 0
\(697\) −9.00000 15.5885i −0.340899 0.590455i
\(698\) 0 0
\(699\) 30.0000 1.13470
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) −4.00000 6.92820i −0.150863 0.261302i
\(704\) 0 0
\(705\) −25.4558 14.6969i −0.958723 0.553519i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 5.00000 8.66025i 0.187779 0.325243i −0.756730 0.653727i \(-0.773204\pi\)
0.944509 + 0.328484i \(0.106538\pi\)
\(710\) 0 0
\(711\) 8.48528 4.89898i 0.318223 0.183726i
\(712\) 0 0
\(713\) 58.7878i 2.20162i
\(714\) 0 0
\(715\) 29.3939i 1.09927i
\(716\) 0 0
\(717\) −10.6066 + 6.12372i −0.396111 + 0.228695i
\(718\) 0 0
\(719\) 12.0000 20.7846i 0.447524 0.775135i −0.550700 0.834703i \(-0.685639\pi\)
0.998224 + 0.0595683i \(0.0189724\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −4.24264 2.44949i −0.157786 0.0910975i
\(724\) 0 0
\(725\) 3.00000 + 5.19615i 0.111417 + 0.192980i
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.00000 10.3923i −0.221918 0.384373i
\(732\) 0 0
\(733\) 4.24264 + 2.44949i 0.156706 + 0.0904740i 0.576302 0.817237i \(-0.304495\pi\)
−0.419597 + 0.907711i \(0.637828\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −8.48528 + 4.89898i −0.312136 + 0.180212i −0.647882 0.761741i \(-0.724345\pi\)
0.335746 + 0.941953i \(0.391012\pi\)
\(740\) 0 0
\(741\) 9.79796i 0.359937i
\(742\) 0 0
\(743\) 12.2474i 0.449315i 0.974438 + 0.224658i \(0.0721264\pi\)
−0.974438 + 0.224658i \(0.927874\pi\)
\(744\) 0 0
\(745\) 38.1838 22.0454i 1.39894 0.807681i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 29.6985 + 17.1464i 1.08371 + 0.625682i 0.931896 0.362726i \(-0.118154\pi\)
0.151817 + 0.988409i \(0.451487\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 36.0000 1.31017
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) −9.00000 15.5885i −0.326679 0.565825i
\(760\) 0 0
\(761\) 36.0624 + 20.8207i 1.30726 + 0.754748i 0.981638 0.190751i \(-0.0610923\pi\)
0.325624 + 0.945499i \(0.394426\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 3.00000 5.19615i 0.108465 0.187867i
\(766\) 0 0
\(767\) 50.9117 29.3939i 1.83831 1.06135i
\(768\) 0 0
\(769\) 9.79796i 0.353323i 0.984272 + 0.176662i \(0.0565299\pi\)
−0.984272 + 0.176662i \(0.943470\pi\)
\(770\) 0 0
\(771\) 22.0454i 0.793946i
\(772\) 0 0
\(773\) −19.0919 + 11.0227i −0.686687 + 0.396459i −0.802370 0.596827i \(-0.796428\pi\)
0.115683 + 0.993286i \(0.463094\pi\)
\(774\) 0 0
\(775\) −4.00000 + 6.92820i −0.143684 + 0.248868i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −12.7279 7.34847i −0.456025 0.263286i
\(780\) 0 0
\(781\) 15.0000 + 25.9808i 0.536742 + 0.929665i
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) 0 0
\(785\) −48.0000 −1.71319
\(786\) 0 0
\(787\) 10.0000 + 17.3205i 0.356462 + 0.617409i 0.987367 0.158450i \(-0.0506498\pi\)
−0.630905 + 0.775860i \(0.717316\pi\)
\(788\) 0 0
\(789\) −2.12132 1.22474i −0.0755210 0.0436021i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −12.7279 + 7.34847i −0.451413 + 0.260623i
\(796\) 0 0
\(797\) 36.7423i 1.30148i 0.759300 + 0.650740i \(0.225541\pi\)
−0.759300 + 0.650740i \(0.774459\pi\)
\(798\) 0 0
\(799\) 29.3939i 1.03988i
\(800\) 0 0
\(801\) 10.6066 6.12372i 0.374766 0.216371i
\(802\) 0 0
\(803\) −18.0000 + 31.1769i −0.635206 + 1.10021i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 19.0919 + 11.0227i 0.672066 + 0.388018i
\(808\) 0 0
\(809\) −9.00000 15.5885i −0.316423 0.548061i 0.663316 0.748340i \(-0.269149\pi\)
−0.979739 + 0.200279i \(0.935815\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) 24.0000 + 41.5692i 0.840683 + 1.45611i
\(816\) 0 0
\(817\) −8.48528 4.89898i −0.296862 0.171394i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −9.00000 + 15.5885i −0.314102 + 0.544041i −0.979246 0.202674i \(-0.935037\pi\)
0.665144 + 0.746715i \(0.268370\pi\)
\(822\) 0 0
\(823\) −16.9706 + 9.79796i −0.591557 + 0.341535i −0.765713 0.643183i \(-0.777614\pi\)
0.174156 + 0.984718i \(0.444280\pi\)
\(824\) 0 0
\(825\) 2.44949i 0.0852803i
\(826\) 0 0
\(827\) 46.5403i 1.61836i 0.587557 + 0.809182i \(0.300090\pi\)
−0.587557 + 0.809182i \(0.699910\pi\)
\(828\) 0 0
\(829\) −4.24264 + 2.44949i −0.147353 + 0.0850743i −0.571864 0.820349i \(-0.693779\pi\)
0.424511 + 0.905423i \(0.360446\pi\)
\(830\) 0 0
\(831\) −4.00000 + 6.92820i −0.138758 + 0.240337i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −25.4558 14.6969i −0.880936 0.508609i
\(836\) 0 0
\(837\) −4.00000 6.92820i −0.138260 0.239474i
\(838\) 0 0
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 3.00000 + 5.19615i 0.103325 + 0.178965i
\(844\) 0 0
\(845\) −23.3345 13.4722i −0.802732 0.463458i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 7.00000 12.1244i 0.240239 0.416107i
\(850\) 0 0
\(851\) −25.4558 + 14.6969i −0.872615 + 0.503805i
\(852\) 0 0
\(853\) 29.3939i 1.00643i −0.864162 0.503214i \(-0.832151\pi\)
0.864162 0.503214i \(-0.167849\pi\)
\(854\) 0 0
\(855\) 4.89898i 0.167542i
\(856\) 0 0
\(857\) −19.0919 + 11.0227i −0.652166 + 0.376528i −0.789286 0.614026i \(-0.789549\pi\)
0.137119 + 0.990555i \(0.456216\pi\)
\(858\) 0 0
\(859\) 11.0000 19.0526i 0.375315 0.650065i −0.615059 0.788481i \(-0.710868\pi\)
0.990374 + 0.138416i \(0.0442012\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −44.5477 25.7196i −1.51642 0.875507i −0.999814 0.0192849i \(-0.993861\pi\)
−0.516608 0.856222i \(-0.672806\pi\)
\(864\) 0 0
\(865\) 9.00000 + 15.5885i 0.306009 + 0.530023i
\(866\) 0 0
\(867\) −11.0000 −0.373580
\(868\) 0 0
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 4.24264 + 2.44949i 0.143592 + 0.0829027i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 11.0000 19.0526i 0.371444 0.643359i −0.618344 0.785907i \(-0.712196\pi\)
0.989788 + 0.142548i \(0.0455296\pi\)
\(878\) 0 0
\(879\) −19.0919 + 11.0227i −0.643953 + 0.371787i
\(880\) 0 0
\(881\) 2.44949i 0.0825254i 0.999148 + 0.0412627i \(0.0131381\pi\)
−0.999148 + 0.0412627i \(0.986862\pi\)
\(882\) 0 0
\(883\) 34.2929i 1.15405i −0.816728 0.577023i \(-0.804214\pi\)
0.816728 0.577023i \(-0.195786\pi\)
\(884\) 0 0
\(885\) −25.4558 + 14.6969i −0.855689 + 0.494032i
\(886\) 0 0
\(887\) 6.00000 10.3923i 0.201460 0.348939i −0.747539 0.664218i \(-0.768765\pi\)
0.948999 + 0.315279i \(0.102098\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2.12132 + 1.22474i 0.0710669 + 0.0410305i
\(892\) 0 0
\(893\) −12.0000 20.7846i −0.401565 0.695530i
\(894\) 0 0
\(895\) 42.0000 1.40391
\(896\) 0 0
\(897\) 36.0000 1.20201
\(898\) 0 0
\(899\) 24.0000 + 41.5692i 0.800445 + 1.38641i
\(900\) 0 0
\(901\) −12.7279 7.34847i −0.424029 0.244813i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −6.00000 + 10.3923i −0.199447 + 0.345452i
\(906\) 0 0
\(907\) −21.2132 + 12.2474i −0.704373 + 0.406670i −0.808974 0.587844i \(-0.799977\pi\)
0.104601 + 0.994514i \(0.466643\pi\)
\(908\) 0 0
\(909\) 2.44949i 0.0812444i
\(910\) 0 0
\(911\) 7.34847i 0.243466i 0.992563 + 0.121733i \(0.0388451\pi\)
−0.992563 + 0.121733i \(0.961155\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −12.7279 7.34847i −0.419855 0.242404i 0.275160 0.961398i \(-0.411269\pi\)
−0.695015 + 0.718995i \(0.744602\pi\)
\(920\) 0 0
\(921\) 13.0000 + 22.5167i 0.428365 + 0.741949i
\(922\) 0 0
\(923\) −60.0000 −1.97492
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) −8.00000 13.8564i −0.262754 0.455104i
\(928\) 0 0
\(929\) 10.6066 + 6.12372i 0.347991 + 0.200913i 0.663800 0.747910i \(-0.268943\pi\)
−0.315809 + 0.948823i \(0.602276\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 12.0000 20.7846i 0.392862 0.680458i
\(934\) 0 0
\(935\) 12.7279 7.34847i 0.416248 0.240321i
\(936\) 0 0
\(937\) 39.1918i 1.28034i −0.768233 0.640171i \(-0.778864\pi\)
0.768233 0.640171i \(-0.221136\pi\)
\(938\) 0 0
\(939\) 9.79796i 0.319744i
\(940\) 0 0
\(941\) −10.6066 + 6.12372i −0.345765 + 0.199628i −0.662819 0.748780i \(-0.730640\pi\)
0.317053 + 0.948408i \(0.397307\pi\)
\(942\) 0 0
\(943\) −27.0000 + 46.7654i −0.879241 + 1.52289i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −23.3345 13.4722i −0.758270 0.437787i 0.0704042 0.997519i \(-0.477571\pi\)
−0.828674 + 0.559731i \(0.810904\pi\)
\(948\) 0 0
\(949\) −36.0000 62.3538i −1.16861 2.02409i
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) −3.00000 5.19615i −0.0970777 0.168144i
\(956\) 0 0
\(957\) −12.7279 7.34847i −0.411435 0.237542i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −16.5000 + 28.5788i −0.532258 + 0.921898i
\(962\) 0 0
\(963\) 10.6066 6.12372i 0.341793 0.197334i
\(964\) 0 0
\(965\) 39.1918i 1.26163i
\(966\) 0 0
\(967\) 58.7878i 1.89049i −0.326366 0.945243i \(-0.605824\pi\)
0.326366 0.945243i \(-0.394176\pi\)
\(968\) 0 0
\(969\) 4.24264 2.44949i 0.136293 0.0786889i
\(970\) 0 0
\(971\) −18.0000 + 31.1769i −0.577647 + 1.00051i 0.418101 + 0.908401i \(0.362696\pi\)
−0.995748 + 0.0921142i \(0.970638\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 4.24264 + 2.44949i 0.135873 + 0.0784465i
\(976\) 0 0
\(977\) −3.00000 5.19615i −0.0959785 0.166240i 0.814038 0.580812i \(-0.197265\pi\)
−0.910017 + 0.414572i \(0.863931\pi\)
\(978\) 0 0
\(979\) 30.0000 0.958804
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) 6.00000 + 10.3923i 0.191370 + 0.331463i 0.945705 0.325027i \(-0.105374\pi\)
−0.754334 + 0.656490i \(0.772040\pi\)
\(984\) 0 0
\(985\) 38.1838 + 22.0454i 1.21664 + 0.702425i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −18.0000 + 31.1769i −0.572367 + 0.991368i
\(990\) 0 0
\(991\) 12.7279 7.34847i 0.404316 0.233432i −0.284029 0.958816i \(-0.591671\pi\)
0.688344 + 0.725384i \(0.258338\pi\)
\(992\) 0 0
\(993\) 24.4949i 0.777322i
\(994\) 0 0
\(995\) 4.89898i 0.155308i
\(996\) 0 0
\(997\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(998\) 0 0
\(999\) 2.00000 3.46410i 0.0632772 0.109599i
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 2352.2.bl.n.607.2 4
4.3 odd 2 2352.2.bl.m.607.2 4
7.2 even 3 336.2.b.b.223.1 2
7.3 odd 6 2352.2.bl.m.31.2 4
7.4 even 3 inner 2352.2.bl.n.31.1 4
7.5 odd 6 336.2.b.c.223.2 yes 2
7.6 odd 2 2352.2.bl.m.607.1 4
21.2 odd 6 1008.2.b.e.559.2 2
21.5 even 6 1008.2.b.d.559.1 2
28.3 even 6 inner 2352.2.bl.n.31.2 4
28.11 odd 6 2352.2.bl.m.31.1 4
28.19 even 6 336.2.b.b.223.2 yes 2
28.23 odd 6 336.2.b.c.223.1 yes 2
28.27 even 2 inner 2352.2.bl.n.607.1 4
56.5 odd 6 1344.2.b.a.895.1 2
56.19 even 6 1344.2.b.d.895.1 2
56.37 even 6 1344.2.b.d.895.2 2
56.51 odd 6 1344.2.b.a.895.2 2
84.23 even 6 1008.2.b.d.559.2 2
84.47 odd 6 1008.2.b.e.559.1 2
168.5 even 6 4032.2.b.d.3583.2 2
168.107 even 6 4032.2.b.d.3583.1 2
168.131 odd 6 4032.2.b.f.3583.2 2
168.149 odd 6 4032.2.b.f.3583.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.b.b.223.1 2 7.2 even 3
336.2.b.b.223.2 yes 2 28.19 even 6
336.2.b.c.223.1 yes 2 28.23 odd 6
336.2.b.c.223.2 yes 2 7.5 odd 6
1008.2.b.d.559.1 2 21.5 even 6
1008.2.b.d.559.2 2 84.23 even 6
1008.2.b.e.559.1 2 84.47 odd 6
1008.2.b.e.559.2 2 21.2 odd 6
1344.2.b.a.895.1 2 56.5 odd 6
1344.2.b.a.895.2 2 56.51 odd 6
1344.2.b.d.895.1 2 56.19 even 6
1344.2.b.d.895.2 2 56.37 even 6
2352.2.bl.m.31.1 4 28.11 odd 6
2352.2.bl.m.31.2 4 7.3 odd 6
2352.2.bl.m.607.1 4 7.6 odd 2
2352.2.bl.m.607.2 4 4.3 odd 2
2352.2.bl.n.31.1 4 7.4 even 3 inner
2352.2.bl.n.31.2 4 28.3 even 6 inner
2352.2.bl.n.607.1 4 28.27 even 2 inner
2352.2.bl.n.607.2 4 1.1 even 1 trivial
4032.2.b.d.3583.1 2 168.107 even 6
4032.2.b.d.3583.2 2 168.5 even 6
4032.2.b.f.3583.1 2 168.149 odd 6
4032.2.b.f.3583.2 2 168.131 odd 6