Properties

Label 363.2.f.c.233.1
Level $363$
Weight $2$
Character 363.233
Analytic conductor $2.899$
Analytic rank $0$
Dimension $8$
CM discriminant -11
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,2,Mod(161,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.161");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.f (of order \(10\), degree \(4\), 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: \(2.89856959337\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.228765625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2x^{6} + 5x^{5} + x^{4} + 15x^{3} - 18x^{2} - 27x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label 233.1
Root \(1.37924 + 1.04771i\) of defining polynomial
Character \(\chi\) \(=\) 363.233
Dual form 363.2.f.c.215.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.37924 - 1.04771i) q^{3} +(1.61803 + 1.17557i) q^{4} +(3.15430 - 1.02489i) q^{5} +(0.804606 + 2.89009i) q^{9} +O(q^{10})\) \(q+(-1.37924 - 1.04771i) q^{3} +(1.61803 + 1.17557i) q^{4} +(3.15430 - 1.02489i) q^{5} +(0.804606 + 2.89009i) q^{9} +(-1.00000 - 3.31662i) q^{12} +(-5.42433 - 1.89122i) q^{15} +(1.23607 + 3.80423i) q^{16} +(6.30860 + 2.04979i) q^{20} -3.31662i q^{23} +(4.85410 - 3.52671i) q^{25} +(1.91823 - 4.82912i) q^{27} +(1.54508 - 4.75528i) q^{31} +(-2.09562 + 5.62213i) q^{36} +(5.66312 + 4.11450i) q^{37} +(5.50000 + 8.29156i) q^{45} +(-3.89893 - 5.36641i) q^{47} +(2.28089 - 6.54198i) q^{48} +(2.16312 + 6.65740i) q^{49} +(-12.6172 - 4.09957i) q^{53} +(-1.94946 + 2.68321i) q^{59} +(-6.55348 - 9.43673i) q^{60} +(-2.47214 + 7.60845i) q^{64} -13.0000 q^{67} +(-3.47486 + 4.57442i) q^{69} +(-15.7715 + 5.12447i) q^{71} +(-10.3899 - 0.221511i) q^{75} +(7.79785 + 10.7328i) q^{80} +(-7.70522 + 4.65077i) q^{81} +16.5831i q^{89} +(3.89893 - 5.36641i) q^{92} +(-7.11320 + 4.93987i) q^{93} +(5.25329 - 16.1680i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{3} + 4 q^{4} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{3} + 4 q^{4} + 5 q^{9} - 8 q^{12} - 11 q^{15} - 8 q^{16} + 12 q^{25} + 8 q^{27} - 10 q^{31} - 10 q^{36} + 14 q^{37} + 44 q^{45} - 4 q^{48} - 14 q^{49} + 22 q^{60} + 16 q^{64} - 104 q^{67} - 11 q^{69} + 6 q^{75} - 7 q^{81} - 5 q^{93} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{10}\right)\)

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 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(3\) −1.37924 1.04771i −0.796305 0.604896i
\(4\) 1.61803 + 1.17557i 0.809017 + 0.587785i
\(5\) 3.15430 1.02489i 1.41064 0.458346i 0.498027 0.867161i \(-0.334058\pi\)
0.912617 + 0.408815i \(0.134058\pi\)
\(6\) 0 0
\(7\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(8\) 0 0
\(9\) 0.804606 + 2.89009i 0.268202 + 0.963363i
\(10\) 0 0
\(11\) 0 0
\(12\) −1.00000 3.31662i −0.288675 0.957427i
\(13\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(14\) 0 0
\(15\) −5.42433 1.89122i −1.40055 0.488310i
\(16\) 1.23607 + 3.80423i 0.309017 + 0.951057i
\(17\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(18\) 0 0
\(19\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(20\) 6.30860 + 2.04979i 1.41064 + 0.458346i
\(21\) 0 0
\(22\) 0 0
\(23\) 3.31662i 0.691564i −0.938315 0.345782i \(-0.887614\pi\)
0.938315 0.345782i \(-0.112386\pi\)
\(24\) 0 0
\(25\) 4.85410 3.52671i 0.970820 0.705342i
\(26\) 0 0
\(27\) 1.91823 4.82912i 0.369163 0.929364i
\(28\) 0 0
\(29\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(30\) 0 0
\(31\) 1.54508 4.75528i 0.277505 0.854074i −0.711040 0.703151i \(-0.751776\pi\)
0.988546 0.150923i \(-0.0482244\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −2.09562 + 5.62213i −0.349270 + 0.937022i
\(37\) 5.66312 + 4.11450i 0.931011 + 0.676419i 0.946240 0.323465i \(-0.104848\pi\)
−0.0152291 + 0.999884i \(0.504848\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 5.50000 + 8.29156i 0.819892 + 1.23603i
\(46\) 0 0
\(47\) −3.89893 5.36641i −0.568717 0.782772i 0.423685 0.905810i \(-0.360736\pi\)
−0.992402 + 0.123038i \(0.960736\pi\)
\(48\) 2.28089 6.54198i 0.329218 0.944254i
\(49\) 2.16312 + 6.65740i 0.309017 + 0.951057i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.6172 4.09957i −1.73310 0.563120i −0.739212 0.673473i \(-0.764802\pi\)
−0.993892 + 0.110353i \(0.964802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.94946 + 2.68321i −0.253798 + 0.349324i −0.916837 0.399262i \(-0.869267\pi\)
0.663039 + 0.748585i \(0.269267\pi\)
\(60\) −6.55348 9.43673i −0.846051 1.21828i
\(61\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −2.47214 + 7.60845i −0.309017 + 0.951057i
\(65\) 0 0
\(66\) 0 0
\(67\) −13.0000 −1.58820 −0.794101 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) 0 0
\(69\) −3.47486 + 4.57442i −0.418324 + 0.550696i
\(70\) 0 0
\(71\) −15.7715 + 5.12447i −1.87173 + 0.608162i −0.880855 + 0.473386i \(0.843032\pi\)
−0.990876 + 0.134777i \(0.956968\pi\)
\(72\) 0 0
\(73\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(74\) 0 0
\(75\) −10.3899 0.221511i −1.19973 0.0255779i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(80\) 7.79785 + 10.7328i 0.871826 + 1.19997i
\(81\) −7.70522 + 4.65077i −0.856135 + 0.516752i
\(82\) 0 0
\(83\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.5831i 1.75781i 0.476999 + 0.878904i \(0.341725\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 3.89893 5.36641i 0.406491 0.559487i
\(93\) −7.11320 + 4.93987i −0.737605 + 0.512241i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.25329 16.1680i 0.533391 1.64161i −0.213710 0.976897i \(-0.568555\pi\)
0.747101 0.664711i \(-0.231445\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 12.0000 1.20000
\(101\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(102\) 0 0
\(103\) 3.23607 + 2.35114i 0.318859 + 0.231665i 0.735689 0.677320i \(-0.236859\pi\)
−0.416829 + 0.908985i \(0.636859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(108\) 8.78073 5.55867i 0.844926 0.534883i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) −3.50000 11.6082i −0.332205 1.10180i
\(112\) 0 0
\(113\) 1.94946 + 2.68321i 0.183390 + 0.252415i 0.890807 0.454382i \(-0.150140\pi\)
−0.707417 + 0.706796i \(0.750140\pi\)
\(114\) 0 0
\(115\) −3.39919 10.4616i −0.316976 0.975551i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 8.09017 5.87785i 0.726519 0.527847i
\(125\) 1.94946 2.68321i 0.174365 0.239993i
\(126\) 0 0
\(127\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1.10133 17.1985i 0.0947878 1.48021i
\(136\) 0 0
\(137\) 22.0801 7.17425i 1.88643 0.612938i 0.903613 0.428350i \(-0.140905\pi\)
0.982816 0.184588i \(-0.0590949\pi\)
\(138\) 0 0
\(139\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(140\) 0 0
\(141\) −0.244889 + 11.4865i −0.0206234 + 0.967339i
\(142\) 0 0
\(143\) 0 0
\(144\) −10.0000 + 6.63325i −0.833333 + 0.552771i
\(145\) 0 0
\(146\) 0 0
\(147\) 3.99156 11.4485i 0.329218 0.944254i
\(148\) 4.32624 + 13.3148i 0.355615 + 1.09447i
\(149\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(150\) 0 0
\(151\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 16.5831i 1.33199i
\(156\) 0 0
\(157\) −18.6074 + 13.5191i −1.48503 + 1.07894i −0.509140 + 0.860684i \(0.670036\pi\)
−0.975892 + 0.218255i \(0.929964\pi\)
\(158\) 0 0
\(159\) 13.1070 + 18.8735i 1.03945 + 1.49676i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.94427 + 15.2169i −0.387265 + 1.19188i 0.547558 + 0.836768i \(0.315557\pi\)
−0.934824 + 0.355112i \(0.884443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(168\) 0 0
\(169\) −10.5172 7.64121i −0.809017 0.587785i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.50000 1.65831i 0.413405 0.124646i
\(178\) 0 0
\(179\) −9.74732 13.4160i −0.728549 1.00276i −0.999196 0.0400827i \(-0.987238\pi\)
0.270648 0.962678i \(-0.412762\pi\)
\(180\) −0.848129 + 19.8817i −0.0632158 + 1.48189i
\(181\) −7.72542 23.7764i −0.574226 1.76729i −0.638799 0.769374i \(-0.720568\pi\)
0.0645725 0.997913i \(-0.479432\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 22.0801 + 7.17425i 1.62336 + 0.527462i
\(186\) 0 0
\(187\) 0 0
\(188\) 13.2665i 0.967559i
\(189\) 0 0
\(190\) 0 0
\(191\) −13.6462 + 18.7824i −0.987407 + 1.35905i −0.0546656 + 0.998505i \(0.517409\pi\)
−0.932742 + 0.360545i \(0.882591\pi\)
\(192\) 11.3811 7.90380i 0.821362 0.570408i
\(193\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −4.32624 + 13.3148i −0.309017 + 0.951057i
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 17.9301 + 13.6202i 1.26469 + 0.960697i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 9.58534 2.66858i 0.666227 0.185479i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(212\) −15.5957 21.4656i −1.07112 1.47427i
\(213\) 27.1216 + 9.45608i 1.85834 + 0.647920i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0.809017 0.587785i 0.0541758 0.0393610i −0.560368 0.828244i \(-0.689340\pi\)
0.614544 + 0.788883i \(0.289340\pi\)
\(224\) 0 0
\(225\) 14.0981 + 11.1912i 0.939877 + 0.746078i
\(226\) 0 0
\(227\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(228\) 0 0
\(229\) 1.54508 4.75528i 0.102102 0.314238i −0.886937 0.461890i \(-0.847172\pi\)
0.989039 + 0.147652i \(0.0471715\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(234\) 0 0
\(235\) −17.7984 12.9313i −1.16104 0.843543i
\(236\) −6.30860 + 2.04979i −0.410655 + 0.133430i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(240\) 0.489779 22.9730i 0.0316151 1.48290i
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 15.5000 + 1.65831i 0.994325 + 0.106381i
\(244\) 0 0
\(245\) 13.6462 + 18.7824i 0.871826 + 1.19997i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 15.7715 + 5.12447i 0.995488 + 0.323453i 0.761061 0.648680i \(-0.224679\pi\)
0.234427 + 0.972134i \(0.424679\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −12.9443 + 9.40456i −0.809017 + 0.587785i
\(257\) 15.5957 21.4656i 0.972833 1.33899i 0.0322308 0.999480i \(-0.489739\pi\)
0.940603 0.339510i \(-0.110261\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −44.0000 −2.70290
\(266\) 0 0
\(267\) 17.3743 22.8721i 1.06329 1.39975i
\(268\) −21.0344 15.2824i −1.28488 0.933522i
\(269\) 12.6172 4.09957i 0.769284 0.249955i 0.102025 0.994782i \(-0.467468\pi\)
0.667258 + 0.744826i \(0.267468\pi\)
\(270\) 0 0
\(271\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −11.0000 + 3.31662i −0.662122 + 0.199637i
\(277\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(278\) 0 0
\(279\) 14.9864 + 0.639301i 0.897211 + 0.0382740i
\(280\) 0 0
\(281\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(282\) 0 0
\(283\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(284\) −31.5430 10.2489i −1.87173 0.608162i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 13.7533 9.99235i 0.809017 0.587785i
\(290\) 0 0
\(291\) −24.1849 + 16.7956i −1.41774 + 0.984574i
\(292\) 0 0
\(293\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(294\) 0 0
\(295\) −3.39919 + 10.4616i −0.197908 + 0.609099i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −16.5509 12.5725i −0.955566 0.725875i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) −2.00000 6.63325i −0.113776 0.377352i
\(310\) 0 0
\(311\) 19.4946 + 26.8321i 1.10544 + 1.52151i 0.827970 + 0.560772i \(0.189495\pi\)
0.277469 + 0.960735i \(0.410505\pi\)
\(312\) 0 0
\(313\) −5.87132 18.0701i −0.331867 1.02138i −0.968245 0.250004i \(-0.919568\pi\)
0.636378 0.771377i \(-0.280432\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.0801 7.17425i −1.24014 0.402946i −0.385763 0.922598i \(-0.626062\pi\)
−0.854378 + 0.519652i \(0.826062\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 26.5330i 1.48324i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −17.9346 1.53293i −0.996367 0.0851627i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 35.0000 1.92377 0.961887 0.273447i \(-0.0881639\pi\)
0.961887 + 0.273447i \(0.0881639\pi\)
\(332\) 0 0
\(333\) −7.33468 + 19.6775i −0.401938 + 1.07832i
\(334\) 0 0
\(335\) −41.0059 + 13.3236i −2.24039 + 0.727947i
\(336\) 0 0
\(337\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(338\) 0 0
\(339\) 0.122445 5.74326i 0.00665028 0.311931i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −6.27245 + 17.9905i −0.337698 + 0.968573i
\(346\) 0 0
\(347\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(348\) 0 0
\(349\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 36.4829i 1.94179i 0.239511 + 0.970894i \(0.423013\pi\)
−0.239511 + 0.970894i \(0.576987\pi\)
\(354\) 0 0
\(355\) −44.4959 + 32.3282i −2.36160 + 1.71580i
\(356\) −19.4946 + 26.8321i −1.03321 + 1.42210i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(360\) 0 0
\(361\) 5.87132 18.0701i 0.309017 0.951057i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 29.9336 + 21.7481i 1.56252 + 1.13524i 0.933903 + 0.357526i \(0.116380\pi\)
0.628620 + 0.777713i \(0.283620\pi\)
\(368\) 12.6172 4.09957i 0.657717 0.213705i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −17.3166 0.369185i −0.897822 0.0191413i
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −5.50000 + 1.65831i −0.284019 + 0.0856349i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −7.72542 23.7764i −0.396828 1.22131i −0.927528 0.373753i \(-0.878071\pi\)
0.530700 0.847560i \(-0.321929\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.15430 1.02489i −0.161177 0.0523696i 0.227317 0.973821i \(-0.427005\pi\)
−0.388494 + 0.921451i \(0.627005\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 27.5066 19.9847i 1.39643 1.01457i
\(389\) 21.4441 29.5153i 1.08726 1.49648i 0.235988 0.971756i \(-0.424167\pi\)
0.851270 0.524727i \(-0.175833\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 19.4164 + 14.1068i 0.970820 + 0.705342i
\(401\) −25.2344 + 8.19915i −1.26014 + 0.409446i −0.861546 0.507679i \(-0.830503\pi\)
−0.398599 + 0.917125i \(0.630503\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −19.5380 + 22.5669i −0.970851 + 1.12136i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(410\) 0 0
\(411\) −37.9703 13.2385i −1.87294 0.653008i
\(412\) 2.47214 + 7.60845i 0.121793 + 0.374842i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 33.1662i 1.62028i −0.586238 0.810139i \(-0.699392\pi\)
0.586238 0.810139i \(-0.300608\pi\)
\(420\) 0 0
\(421\) 8.09017 5.87785i 0.394291 0.286469i −0.372921 0.927863i \(-0.621644\pi\)
0.767211 + 0.641394i \(0.221644\pi\)
\(422\) 0 0
\(423\) 12.3723 15.5861i 0.601562 0.757822i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(432\) 20.7421 + 1.32826i 0.997956 + 0.0639059i
\(433\) −23.4615 17.0458i −1.12749 0.819168i −0.142160 0.989844i \(-0.545405\pi\)
−0.985327 + 0.170676i \(0.945405\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −17.5000 + 11.6082i −0.833333 + 0.552771i
\(442\) 0 0
\(443\) −21.4441 29.5153i −1.01884 1.40231i −0.913014 0.407928i \(-0.866252\pi\)
−0.105825 0.994385i \(-0.533748\pi\)
\(444\) 7.98312 22.8969i 0.378862 1.08664i
\(445\) 16.9959 + 52.3081i 0.805685 + 2.47964i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 15.7715 + 5.12447i 0.744303 + 0.241839i 0.656528 0.754302i \(-0.272024\pi\)
0.0877747 + 0.996140i \(0.472024\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.63325i 0.312002i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 6.79837 20.9232i 0.316976 0.975551i
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −31.0000 −1.44069 −0.720346 0.693615i \(-0.756017\pi\)
−0.720346 + 0.693615i \(0.756017\pi\)
\(464\) 0 0
\(465\) −17.3743 + 22.8721i −0.805714 + 1.06067i
\(466\) 0 0
\(467\) 41.0059 13.3236i 1.89753 0.616543i 0.927313 0.374286i \(-0.122112\pi\)
0.970212 0.242257i \(-0.0778878\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 39.8281 + 0.849124i 1.83518 + 0.0391256i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.69626 39.7633i 0.0776663 1.82064i
\(478\) 0 0
\(479\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 56.3826i 2.56020i
\(486\) 0 0
\(487\) 34.7877 25.2748i 1.57638 1.14531i 0.655683 0.755036i \(-0.272381\pi\)
0.920699 0.390273i \(-0.127619\pi\)
\(488\) 0 0
\(489\) 22.7622 15.8076i 1.02934 0.714844i
\(490\) 0 0
\(491\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 20.0000 0.898027
\(497\) 0 0
\(498\) 0 0
\(499\) 32.3607 + 23.5114i 1.44866 + 1.05252i 0.986141 + 0.165907i \(0.0530552\pi\)
0.462522 + 0.886608i \(0.346945\pi\)
\(500\) 6.30860 2.04979i 0.282129 0.0916693i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6.50000 + 21.5581i 0.288675 + 0.957427i
\(508\) 0 0
\(509\) 1.94946 + 2.68321i 0.0864084 + 0.118931i 0.850033 0.526730i \(-0.176582\pi\)
−0.763624 + 0.645661i \(0.776582\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 12.6172 + 4.09957i 0.555980 + 0.180649i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −25.3430 + 34.8817i −1.11030 + 1.52819i −0.289321 + 0.957232i \(0.593430\pi\)
−0.820977 + 0.570962i \(0.806570\pi\)
\(522\) 0 0
\(523\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 12.0000 0.521739
\(530\) 0 0
\(531\) −9.32325 3.47520i −0.404595 0.150811i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −0.612223 + 28.7163i −0.0264194 + 1.23920i
\(538\) 0 0
\(539\) 0 0
\(540\) 22.0000 26.5330i 0.946729 1.14180i
\(541\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(542\) 0 0
\(543\) −14.2556 + 40.8874i −0.611765 + 1.75465i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(548\) 44.1602 + 14.3485i 1.88643 + 0.612938i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −22.9372 33.0285i −0.973630 1.40198i
\(556\) 0 0
\(557\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(564\) −13.8994 + 18.2977i −0.585272 + 0.770472i
\(565\) 8.89919 + 6.46564i 0.374392 + 0.272011i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 38.5000 11.6082i 1.60836 0.484939i
\(574\) 0 0
\(575\) −11.6968 16.0992i −0.487789 0.671385i
\(576\) −23.9782 1.02288i −0.999091 0.0426201i
\(577\) 14.5238 + 44.6997i 0.604634 + 1.86087i 0.499290 + 0.866435i \(0.333594\pi\)
0.105344 + 0.994436i \(0.466406\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.89893 5.36641i 0.160926 0.221496i −0.720938 0.693000i \(-0.756289\pi\)
0.881864 + 0.471504i \(0.156289\pi\)
\(588\) 19.9170 13.8316i 0.821362 0.570408i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −8.65248 + 26.6296i −0.355615 + 1.09447i
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −27.5848 20.9542i −1.12897 0.857599i
\(598\) 0 0
\(599\) 31.5430 10.2489i 1.28881 0.418760i 0.417136 0.908844i \(-0.363034\pi\)
0.871675 + 0.490084i \(0.163034\pi\)
\(600\) 0 0
\(601\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(602\) 0 0
\(603\) −10.4599 37.5711i −0.425959 1.53002i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.5330i 1.06818i 0.845428 + 0.534089i \(0.179345\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) 0.809017 0.587785i 0.0325171 0.0236251i −0.571408 0.820666i \(-0.693603\pi\)
0.603925 + 0.797041i \(0.293603\pi\)
\(620\) 19.4946 26.8321i 0.782923 1.07760i
\(621\) −16.0164 6.36205i −0.642715 0.255300i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −5.87132 + 18.0701i −0.234853 + 0.722803i
\(626\) 0 0
\(627\) 0 0
\(628\) −46.0000 −1.83560
\(629\) 0 0
\(630\) 0 0
\(631\) 5.66312 + 4.11450i 0.225445 + 0.163796i 0.694774 0.719228i \(-0.255504\pi\)
−0.469329 + 0.883023i \(0.655504\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −0.979557 + 45.9461i −0.0388420 + 1.82188i
\(637\) 0 0
\(638\) 0 0
\(639\) −27.5000 41.4578i −1.08788 1.64005i
\(640\) 0 0
\(641\) 13.6462 + 18.7824i 0.538994 + 0.741862i 0.988468 0.151432i \(-0.0483884\pi\)
−0.449474 + 0.893294i \(0.648388\pi\)
\(642\) 0 0
\(643\) 12.6697 + 38.9933i 0.499644 + 1.53775i 0.809592 + 0.586993i \(0.199688\pi\)
−0.309948 + 0.950753i \(0.600312\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −41.0059 13.3236i −1.61211 0.523805i −0.642046 0.766666i \(-0.721914\pi\)
−0.970061 + 0.242861i \(0.921914\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −25.8885 + 18.8091i −1.01387 + 0.736622i
\(653\) −1.94946 + 2.68321i −0.0762884 + 0.105002i −0.845456 0.534045i \(-0.820671\pi\)
0.769167 + 0.639047i \(0.220671\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −1.73166 0.0369185i −0.0669497 0.00142735i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(674\) 0 0
\(675\) −7.71963 30.2061i −0.297129 1.16263i
\(676\) −8.03444 24.7275i −0.309017 0.951057i
\(677\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 46.4327i 1.77670i 0.459167 + 0.888350i \(0.348148\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 0 0
\(685\) 62.2943 45.2595i 2.38014 1.72928i
\(686\) 0 0
\(687\) −7.11320 + 4.93987i −0.271386 + 0.188468i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 5.25329 16.1680i 0.199845 0.615058i −0.800041 0.599945i \(-0.795189\pi\)
0.999886 0.0151132i \(-0.00481087\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 11.0000 + 36.4829i 0.414284 + 1.37402i
\(706\) 0 0
\(707\) 0 0
\(708\) 10.8487 + 3.78243i 0.407717 + 0.142152i
\(709\) −5.87132 18.0701i −0.220502 0.678636i −0.998717 0.0506378i \(-0.983875\pi\)
0.778215 0.627998i \(-0.216125\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −15.7715 5.12447i −0.590647 0.191913i
\(714\) 0 0
\(715\) 0 0
\(716\) 33.1662i 1.23948i
\(717\) 0 0
\(718\) 0 0
\(719\) 9.74732 13.4160i 0.363514 0.500333i −0.587610 0.809144i \(-0.699931\pi\)
0.951123 + 0.308811i \(0.0999310\pi\)
\(720\) −24.7446 + 31.1722i −0.922177 + 1.16172i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 15.4508 47.5528i 0.574226 1.76729i
\(725\) 0 0
\(726\) 0 0
\(727\) 53.0000 1.96566 0.982831 0.184510i \(-0.0590699\pi\)
0.982831 + 0.184510i \(0.0590699\pi\)
\(728\) 0 0
\(729\) −19.6408 18.5267i −0.727437 0.686175i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(734\) 0 0
\(735\) 0.857113 40.2028i 0.0316151 1.48290i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(740\) 27.2925 + 37.5649i 1.00329 + 1.38091i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −18.6074 + 13.5191i −0.678993 + 0.493318i −0.873024 0.487678i \(-0.837844\pi\)
0.194030 + 0.980996i \(0.437844\pi\)
\(752\) 15.5957 21.4656i 0.568717 0.782772i
\(753\) −16.3837 23.5918i −0.597056 0.859734i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 11.7426 36.1401i 0.426794 1.31354i −0.474473 0.880270i \(-0.657361\pi\)
0.901266 0.433266i \(-0.142639\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −44.1602 + 14.3485i −1.59766 + 0.519111i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 27.7065 + 0.590695i 0.999773 + 0.0213149i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −44.0000 + 13.2665i −1.58462 + 0.477781i
\(772\) 0 0
\(773\) 7.79785 + 10.7328i 0.280469 + 0.386033i 0.925889 0.377795i \(-0.123318\pi\)
−0.645420 + 0.763828i \(0.723318\pi\)
\(774\) 0 0
\(775\) −9.27051 28.5317i −0.333007 1.02489i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −22.6525 + 16.4580i −0.809017 + 0.587785i
\(785\) −44.8377 + 61.7137i −1.60032 + 2.20266i
\(786\) 0 0
\(787\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 60.6866 + 46.0993i 2.15233 + 1.63497i
\(796\) 32.3607 + 23.5114i 1.14699 + 0.833340i
\(797\) −53.6231 + 17.4232i −1.89943 + 0.617161i −0.933294 + 0.359113i \(0.883079\pi\)
−0.966132 + 0.258048i \(0.916921\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −47.9267 + 13.3429i −1.69341 + 0.471448i
\(802\) 0 0
\(803\) 0 0
\(804\) 13.0000 + 43.1161i 0.458475 + 1.52059i
\(805\) 0 0
\(806\) 0 0
\(807\) −21.6973 7.56486i −0.763781 0.266296i
\(808\) 0 0
\(809\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(810\) 0 0
\(811\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 53.0660i 1.85882i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(822\) 0 0
\(823\) −15.1418 + 46.6018i −0.527811 + 1.62444i 0.230878 + 0.972983i \(0.425840\pi\)
−0.758689 + 0.651453i \(0.774160\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(828\) 18.6465 + 6.95039i 0.648011 + 0.241543i
\(829\) −23.4615 17.0458i −0.814851 0.592024i 0.100382 0.994949i \(-0.467994\pi\)
−0.915233 + 0.402925i \(0.867994\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −20.0000 16.5831i −0.691301 0.573197i
\(838\) 0 0
\(839\) −21.4441 29.5153i −0.740332 1.01898i −0.998599 0.0529065i \(-0.983151\pi\)
0.258267 0.966074i \(-0.416849\pi\)
\(840\) 0 0
\(841\) −8.96149 27.5806i −0.309017 0.951057i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −41.0059 13.3236i −1.41064 0.458346i
\(846\) 0 0
\(847\) 0 0
\(848\) 53.0660i 1.82229i
\(849\) 0 0
\(850\) 0 0
\(851\) 13.6462 18.7824i 0.467787 0.643854i
\(852\) 32.7674 + 47.1836i 1.12259 + 1.61648i
\(853\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −31.0000 −1.05771 −0.528853 0.848713i \(-0.677378\pi\)
−0.528853 + 0.848713i \(0.677378\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −44.1602 + 14.3485i −1.50323 + 0.488429i −0.940958 0.338524i \(-0.890072\pi\)
−0.562272 + 0.826953i \(0.690072\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −29.4382 0.627614i −0.999773 0.0213149i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 50.9537 + 2.17362i 1.72452 + 0.0735660i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 16.5831i 0.558700i 0.960189 + 0.279350i \(0.0901189\pi\)
−0.960189 + 0.279350i \(0.909881\pi\)
\(882\) 0 0
\(883\) −45.3050 + 32.9160i −1.52463 + 1.10771i −0.565503 + 0.824747i \(0.691318\pi\)
−0.959130 + 0.282964i \(0.908682\pi\)
\(884\) 0 0
\(885\) 15.6490 10.8677i 0.526037 0.365315i
\(886\) 0 0
\(887\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 2.00000 0.0669650
\(893\) 0 0
\(894\) 0 0
\(895\) −44.4959 32.3282i −1.48734 1.08061i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 9.65528 + 34.6811i 0.321843 + 1.15604i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −48.7366 67.0801i −1.62006 2.22982i
\(906\) 0 0
\(907\) 2.47214 + 7.60845i 0.0820859 + 0.252635i 0.983674 0.179962i \(-0.0575975\pi\)
−0.901588 + 0.432597i \(0.857597\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 6.30860 + 2.04979i 0.209013 + 0.0679125i 0.411652 0.911341i \(-0.364952\pi\)
−0.202639 + 0.979253i \(0.564952\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 8.09017 5.87785i 0.267307 0.194210i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 42.0000 1.38095
\(926\) 0 0
\(927\) −4.19124 + 11.2443i −0.137659 + 0.369310i
\(928\) 0 0
\(929\) 50.4688 16.3983i 1.65583 0.538011i 0.675835 0.737053i \(-0.263783\pi\)
0.979991 + 0.199042i \(0.0637830\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 1.22445 57.4326i 0.0400866 1.88026i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(938\) 0 0
\(939\) −10.8342 + 31.0744i −0.353562 + 1.01408i
\(940\) −13.5967 41.8465i −0.443477 1.36488i
\(941\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −12.6172 4.09957i −0.410655 0.133430i
\(945\) 0 0
\(946\) 0 0
\(947\) 23.2164i 0.754431i −0.926126 0.377215i \(-0.876882\pi\)
0.926126 0.377215i \(-0.123118\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 22.9372 + 33.0285i 0.743790 + 1.07102i
\(952\) 0 0
\(953\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(954\) 0 0
\(955\) −23.7943 + 73.2314i −0.769966 + 2.36971i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 27.7989 36.5954i 0.897205 1.18111i
\(961\) 4.85410 + 3.52671i 0.156584 + 0.113765i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 25.3430 + 34.8817i 0.813296 + 1.11941i 0.990806 + 0.135287i \(0.0431957\pi\)
−0.177510 + 0.984119i \(0.556804\pi\)
\(972\) 23.1301 + 20.9045i 0.741897 + 0.670514i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 53.6231 + 17.4232i 1.71555 + 0.557417i 0.991242 0.132056i \(-0.0421578\pi\)
0.724311 + 0.689473i \(0.242158\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 46.4327i 1.48324i
\(981\) 0 0
\(982\) 0 0
\(983\) 21.4441 29.5153i 0.683960 0.941391i −0.316012 0.948755i \(-0.602344\pi\)
0.999973 + 0.00736431i \(0.00234415\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 0 0
\(993\) −48.2734 36.6699i −1.53191 1.16368i
\(994\) 0 0
\(995\) 63.0860 20.4979i 1.99996 0.649826i
\(996\) 0 0
\(997\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(998\) 0 0
\(999\) 30.7326 19.4553i 0.972335 0.615539i
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 363.2.f.c.233.1 8
3.2 odd 2 inner 363.2.f.c.233.2 8
11.2 odd 10 inner 363.2.f.c.239.1 8
11.3 even 5 inner 363.2.f.c.161.2 8
11.4 even 5 33.2.d.a.32.2 yes 2
11.5 even 5 inner 363.2.f.c.215.2 8
11.6 odd 10 inner 363.2.f.c.215.2 8
11.7 odd 10 33.2.d.a.32.2 yes 2
11.8 odd 10 inner 363.2.f.c.161.2 8
11.9 even 5 inner 363.2.f.c.239.1 8
11.10 odd 2 CM 363.2.f.c.233.1 8
33.2 even 10 inner 363.2.f.c.239.2 8
33.5 odd 10 inner 363.2.f.c.215.1 8
33.8 even 10 inner 363.2.f.c.161.1 8
33.14 odd 10 inner 363.2.f.c.161.1 8
33.17 even 10 inner 363.2.f.c.215.1 8
33.20 odd 10 inner 363.2.f.c.239.2 8
33.26 odd 10 33.2.d.a.32.1 2
33.29 even 10 33.2.d.a.32.1 2
33.32 even 2 inner 363.2.f.c.233.2 8
44.7 even 10 528.2.b.a.65.1 2
44.15 odd 10 528.2.b.a.65.1 2
55.4 even 10 825.2.f.a.626.1 2
55.7 even 20 825.2.d.a.824.3 4
55.18 even 20 825.2.d.a.824.2 4
55.29 odd 10 825.2.f.a.626.1 2
55.37 odd 20 825.2.d.a.824.3 4
55.48 odd 20 825.2.d.a.824.2 4
88.29 odd 10 2112.2.b.e.65.1 2
88.37 even 10 2112.2.b.e.65.1 2
88.51 even 10 2112.2.b.f.65.2 2
88.59 odd 10 2112.2.b.f.65.2 2
99.4 even 15 891.2.g.a.593.1 4
99.7 odd 30 891.2.g.a.296.2 4
99.29 even 30 891.2.g.a.296.1 4
99.40 odd 30 891.2.g.a.593.1 4
99.59 odd 30 891.2.g.a.593.2 4
99.70 even 15 891.2.g.a.296.2 4
99.92 odd 30 891.2.g.a.296.1 4
99.95 even 30 891.2.g.a.593.2 4
132.59 even 10 528.2.b.a.65.2 2
132.95 odd 10 528.2.b.a.65.2 2
165.29 even 10 825.2.f.a.626.2 2
165.59 odd 10 825.2.f.a.626.2 2
165.62 odd 20 825.2.d.a.824.1 4
165.92 even 20 825.2.d.a.824.1 4
165.128 odd 20 825.2.d.a.824.4 4
165.158 even 20 825.2.d.a.824.4 4
264.29 even 10 2112.2.b.e.65.2 2
264.59 even 10 2112.2.b.f.65.1 2
264.125 odd 10 2112.2.b.e.65.2 2
264.227 odd 10 2112.2.b.f.65.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.d.a.32.1 2 33.26 odd 10
33.2.d.a.32.1 2 33.29 even 10
33.2.d.a.32.2 yes 2 11.4 even 5
33.2.d.a.32.2 yes 2 11.7 odd 10
363.2.f.c.161.1 8 33.8 even 10 inner
363.2.f.c.161.1 8 33.14 odd 10 inner
363.2.f.c.161.2 8 11.3 even 5 inner
363.2.f.c.161.2 8 11.8 odd 10 inner
363.2.f.c.215.1 8 33.5 odd 10 inner
363.2.f.c.215.1 8 33.17 even 10 inner
363.2.f.c.215.2 8 11.5 even 5 inner
363.2.f.c.215.2 8 11.6 odd 10 inner
363.2.f.c.233.1 8 1.1 even 1 trivial
363.2.f.c.233.1 8 11.10 odd 2 CM
363.2.f.c.233.2 8 3.2 odd 2 inner
363.2.f.c.233.2 8 33.32 even 2 inner
363.2.f.c.239.1 8 11.2 odd 10 inner
363.2.f.c.239.1 8 11.9 even 5 inner
363.2.f.c.239.2 8 33.2 even 10 inner
363.2.f.c.239.2 8 33.20 odd 10 inner
528.2.b.a.65.1 2 44.7 even 10
528.2.b.a.65.1 2 44.15 odd 10
528.2.b.a.65.2 2 132.59 even 10
528.2.b.a.65.2 2 132.95 odd 10
825.2.d.a.824.1 4 165.62 odd 20
825.2.d.a.824.1 4 165.92 even 20
825.2.d.a.824.2 4 55.18 even 20
825.2.d.a.824.2 4 55.48 odd 20
825.2.d.a.824.3 4 55.7 even 20
825.2.d.a.824.3 4 55.37 odd 20
825.2.d.a.824.4 4 165.128 odd 20
825.2.d.a.824.4 4 165.158 even 20
825.2.f.a.626.1 2 55.4 even 10
825.2.f.a.626.1 2 55.29 odd 10
825.2.f.a.626.2 2 165.29 even 10
825.2.f.a.626.2 2 165.59 odd 10
891.2.g.a.296.1 4 99.29 even 30
891.2.g.a.296.1 4 99.92 odd 30
891.2.g.a.296.2 4 99.7 odd 30
891.2.g.a.296.2 4 99.70 even 15
891.2.g.a.593.1 4 99.4 even 15
891.2.g.a.593.1 4 99.40 odd 30
891.2.g.a.593.2 4 99.59 odd 30
891.2.g.a.593.2 4 99.95 even 30
2112.2.b.e.65.1 2 88.29 odd 10
2112.2.b.e.65.1 2 88.37 even 10
2112.2.b.e.65.2 2 264.29 even 10
2112.2.b.e.65.2 2 264.125 odd 10
2112.2.b.f.65.1 2 264.59 even 10
2112.2.b.f.65.1 2 264.227 odd 10
2112.2.b.f.65.2 2 88.51 even 10
2112.2.b.f.65.2 2 88.59 odd 10