Properties

Label 900.3.u.d.149.5
Level $900$
Weight $3$
Character 900.149
Analytic conductor $24.523$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,3,Mod(149,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.149");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.u (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: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 149.5
Character \(\chi\) \(=\) 900.149
Dual form 900.3.u.d.749.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.60876 + 2.53217i) q^{3} +(8.13249 - 4.69530i) q^{7} +(-3.82376 - 8.14732i) q^{9} +O(q^{10})\) \(q+(-1.60876 + 2.53217i) q^{3} +(8.13249 - 4.69530i) q^{7} +(-3.82376 - 8.14732i) q^{9} +(15.4241 - 8.90510i) q^{11} +(-11.9889 - 6.92181i) q^{13} -30.7147 q^{17} -28.5585 q^{19} +(-1.19398 + 28.1465i) q^{21} +(-6.16230 + 10.6734i) q^{23} +(26.7819 + 3.42472i) q^{27} +(5.02918 - 2.90360i) q^{29} +(-3.25379 + 5.63573i) q^{31} +(-2.26449 + 53.3826i) q^{33} +66.5560i q^{37} +(36.8146 - 19.2224i) q^{39} +(-33.0070 - 19.0566i) q^{41} +(-47.7660 + 27.5777i) q^{43} +(8.23479 + 14.2631i) q^{47} +(19.5916 - 33.9337i) q^{49} +(49.4126 - 77.7747i) q^{51} -69.8669 q^{53} +(45.9439 - 72.3150i) q^{57} +(-91.2592 - 52.6885i) q^{59} +(-33.5580 - 58.1242i) q^{61} +(-69.3508 - 48.3044i) q^{63} +(-39.8118 - 22.9853i) q^{67} +(-17.1132 - 32.7750i) q^{69} -31.1942i q^{71} +73.5877i q^{73} +(83.6241 - 144.841i) q^{77} +(47.3263 + 81.9715i) q^{79} +(-51.7578 + 62.3068i) q^{81} +(-7.73763 - 13.4020i) q^{83} +(-0.738362 + 17.4059i) q^{87} -52.7229i q^{89} -130.000 q^{91} +(-9.03604 - 17.3057i) q^{93} +(66.0130 - 38.1126i) q^{97} +(-131.531 - 91.6140i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q - 28 q^{9} - 4 q^{19} + 2 q^{21} - 18 q^{29} + 16 q^{31} - 38 q^{39} + 108 q^{41} + 90 q^{49} + 180 q^{51} - 18 q^{59} - 110 q^{61} + 294 q^{69} - 22 q^{79} - 260 q^{81} - 268 q^{91} - 504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\)

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
\(3\) −1.60876 + 2.53217i −0.536255 + 0.844056i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 8.13249 4.69530i 1.16178 0.670757i 0.210053 0.977690i \(-0.432636\pi\)
0.951731 + 0.306933i \(0.0993029\pi\)
\(8\) 0 0
\(9\) −3.82376 8.14732i −0.424862 0.905258i
\(10\) 0 0
\(11\) 15.4241 8.90510i 1.40219 0.809554i 0.407572 0.913173i \(-0.366376\pi\)
0.994617 + 0.103619i \(0.0330422\pi\)
\(12\) 0 0
\(13\) −11.9889 6.92181i −0.922226 0.532447i −0.0378812 0.999282i \(-0.512061\pi\)
−0.884344 + 0.466835i \(0.845394\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −30.7147 −1.80675 −0.903373 0.428856i \(-0.858917\pi\)
−0.903373 + 0.428856i \(0.858917\pi\)
\(18\) 0 0
\(19\) −28.5585 −1.50308 −0.751540 0.659687i \(-0.770689\pi\)
−0.751540 + 0.659687i \(0.770689\pi\)
\(20\) 0 0
\(21\) −1.19398 + 28.1465i −0.0568560 + 1.34031i
\(22\) 0 0
\(23\) −6.16230 + 10.6734i −0.267926 + 0.464062i −0.968326 0.249689i \(-0.919672\pi\)
0.700400 + 0.713751i \(0.253005\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 26.7819 + 3.42472i 0.991923 + 0.126841i
\(28\) 0 0
\(29\) 5.02918 2.90360i 0.173420 0.100124i −0.410777 0.911736i \(-0.634743\pi\)
0.584198 + 0.811612i \(0.301409\pi\)
\(30\) 0 0
\(31\) −3.25379 + 5.63573i −0.104961 + 0.181798i −0.913722 0.406339i \(-0.866805\pi\)
0.808761 + 0.588137i \(0.200138\pi\)
\(32\) 0 0
\(33\) −2.26449 + 53.3826i −0.0686210 + 1.61765i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 66.5560i 1.79881i 0.437116 + 0.899405i \(0.356000\pi\)
−0.437116 + 0.899405i \(0.644000\pi\)
\(38\) 0 0
\(39\) 36.8146 19.2224i 0.943963 0.492883i
\(40\) 0 0
\(41\) −33.0070 19.0566i −0.805049 0.464795i 0.0401848 0.999192i \(-0.487205\pi\)
−0.845233 + 0.534397i \(0.820539\pi\)
\(42\) 0 0
\(43\) −47.7660 + 27.5777i −1.11084 + 0.641342i −0.939046 0.343792i \(-0.888289\pi\)
−0.171791 + 0.985133i \(0.554955\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.23479 + 14.2631i 0.175208 + 0.303470i 0.940233 0.340531i \(-0.110607\pi\)
−0.765025 + 0.644001i \(0.777273\pi\)
\(48\) 0 0
\(49\) 19.5916 33.9337i 0.399829 0.692523i
\(50\) 0 0
\(51\) 49.4126 77.7747i 0.968875 1.52499i
\(52\) 0 0
\(53\) −69.8669 −1.31824 −0.659121 0.752036i \(-0.729072\pi\)
−0.659121 + 0.752036i \(0.729072\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 45.9439 72.3150i 0.806034 1.26868i
\(58\) 0 0
\(59\) −91.2592 52.6885i −1.54677 0.893026i −0.998386 0.0567963i \(-0.981911\pi\)
−0.548380 0.836229i \(-0.684755\pi\)
\(60\) 0 0
\(61\) −33.5580 58.1242i −0.550131 0.952855i −0.998265 0.0588885i \(-0.981244\pi\)
0.448133 0.893967i \(-0.352089\pi\)
\(62\) 0 0
\(63\) −69.3508 48.3044i −1.10081 0.766736i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −39.8118 22.9853i −0.594206 0.343065i 0.172553 0.985000i \(-0.444798\pi\)
−0.766759 + 0.641935i \(0.778132\pi\)
\(68\) 0 0
\(69\) −17.1132 32.7750i −0.248018 0.475000i
\(70\) 0 0
\(71\) 31.1942i 0.439354i −0.975573 0.219677i \(-0.929500\pi\)
0.975573 0.219677i \(-0.0705004\pi\)
\(72\) 0 0
\(73\) 73.5877i 1.00805i 0.863689 + 0.504026i \(0.168148\pi\)
−0.863689 + 0.504026i \(0.831852\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 83.6241 144.841i 1.08603 1.88105i
\(78\) 0 0
\(79\) 47.3263 + 81.9715i 0.599067 + 1.03761i 0.992959 + 0.118458i \(0.0377950\pi\)
−0.393892 + 0.919157i \(0.628872\pi\)
\(80\) 0 0
\(81\) −51.7578 + 62.3068i −0.638985 + 0.769220i
\(82\) 0 0
\(83\) −7.73763 13.4020i −0.0932245 0.161470i 0.815642 0.578557i \(-0.196384\pi\)
−0.908866 + 0.417088i \(0.863051\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.738362 + 17.4059i −0.00848692 + 0.200068i
\(88\) 0 0
\(89\) 52.7229i 0.592392i −0.955127 0.296196i \(-0.904282\pi\)
0.955127 0.296196i \(-0.0957181\pi\)
\(90\) 0 0
\(91\) −130.000 −1.42857
\(92\) 0 0
\(93\) −9.03604 17.3057i −0.0971617 0.186083i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 66.0130 38.1126i 0.680546 0.392913i −0.119515 0.992832i \(-0.538134\pi\)
0.800061 + 0.599919i \(0.204801\pi\)
\(98\) 0 0
\(99\) −131.531 91.6140i −1.32859 0.925394i
\(100\) 0 0
\(101\) 69.5637 40.1626i 0.688750 0.397650i −0.114394 0.993435i \(-0.536493\pi\)
0.803144 + 0.595786i \(0.203159\pi\)
\(102\) 0 0
\(103\) −26.0991 15.0683i −0.253390 0.146295i 0.367926 0.929855i \(-0.380068\pi\)
−0.621315 + 0.783561i \(0.713401\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.4620 0.0977760 0.0488880 0.998804i \(-0.484432\pi\)
0.0488880 + 0.998804i \(0.484432\pi\)
\(108\) 0 0
\(109\) 187.263 1.71801 0.859006 0.511965i \(-0.171082\pi\)
0.859006 + 0.511965i \(0.171082\pi\)
\(110\) 0 0
\(111\) −168.531 107.073i −1.51830 0.964620i
\(112\) 0 0
\(113\) 46.5365 80.6035i 0.411827 0.713305i −0.583263 0.812284i \(-0.698224\pi\)
0.995090 + 0.0989784i \(0.0315575\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −10.5515 + 124.145i −0.0901836 + 1.06107i
\(118\) 0 0
\(119\) −249.787 + 144.214i −2.09905 + 1.21189i
\(120\) 0 0
\(121\) 98.1015 169.917i 0.810756 1.40427i
\(122\) 0 0
\(123\) 101.355 52.9217i 0.824024 0.430258i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 48.4899i 0.381810i −0.981608 0.190905i \(-0.938858\pi\)
0.981608 0.190905i \(-0.0611423\pi\)
\(128\) 0 0
\(129\) 7.01278 165.318i 0.0543627 1.28153i
\(130\) 0 0
\(131\) −54.0490 31.2052i −0.412588 0.238208i 0.279313 0.960200i \(-0.409893\pi\)
−0.691901 + 0.721992i \(0.743227\pi\)
\(132\) 0 0
\(133\) −232.252 + 134.091i −1.74626 + 1.00820i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.07104 1.85510i −0.00781782 0.0135409i 0.862090 0.506755i \(-0.169155\pi\)
−0.869908 + 0.493214i \(0.835822\pi\)
\(138\) 0 0
\(139\) 7.40371 12.8236i 0.0532641 0.0922562i −0.838164 0.545418i \(-0.816371\pi\)
0.891428 + 0.453162i \(0.149704\pi\)
\(140\) 0 0
\(141\) −49.3644 2.09404i −0.350102 0.0148514i
\(142\) 0 0
\(143\) −246.558 −1.72418
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 54.4075 + 104.200i 0.370119 + 0.708847i
\(148\) 0 0
\(149\) 177.328 + 102.380i 1.19012 + 0.687116i 0.958334 0.285651i \(-0.0922099\pi\)
0.231785 + 0.972767i \(0.425543\pi\)
\(150\) 0 0
\(151\) 7.45266 + 12.9084i 0.0493553 + 0.0854860i 0.889648 0.456648i \(-0.150950\pi\)
−0.840292 + 0.542134i \(0.817617\pi\)
\(152\) 0 0
\(153\) 117.445 + 250.242i 0.767617 + 1.63557i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −15.8545 9.15362i −0.100984 0.0583033i 0.448657 0.893704i \(-0.351902\pi\)
−0.549642 + 0.835401i \(0.685236\pi\)
\(158\) 0 0
\(159\) 112.399 176.915i 0.706914 1.11267i
\(160\) 0 0
\(161\) 115.735i 0.718853i
\(162\) 0 0
\(163\) 43.2107i 0.265096i 0.991177 + 0.132548i \(0.0423159\pi\)
−0.991177 + 0.132548i \(0.957684\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 35.6591 61.7635i 0.213528 0.369841i −0.739288 0.673389i \(-0.764838\pi\)
0.952816 + 0.303548i \(0.0981713\pi\)
\(168\) 0 0
\(169\) 11.3230 + 19.6120i 0.0670000 + 0.116047i
\(170\) 0 0
\(171\) 109.201 + 232.676i 0.638602 + 1.36068i
\(172\) 0 0
\(173\) −83.9449 145.397i −0.485230 0.840444i 0.514626 0.857415i \(-0.327931\pi\)
−0.999856 + 0.0169713i \(0.994598\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 280.231 146.320i 1.58322 0.826668i
\(178\) 0 0
\(179\) 117.614i 0.657063i 0.944493 + 0.328531i \(0.106554\pi\)
−0.944493 + 0.328531i \(0.893446\pi\)
\(180\) 0 0
\(181\) 96.5280 0.533304 0.266652 0.963793i \(-0.414083\pi\)
0.266652 + 0.963793i \(0.414083\pi\)
\(182\) 0 0
\(183\) 201.167 + 8.53353i 1.09927 + 0.0466313i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −473.746 + 273.517i −2.53340 + 1.46266i
\(188\) 0 0
\(189\) 233.884 97.8975i 1.23748 0.517976i
\(190\) 0 0
\(191\) 51.9090 29.9697i 0.271775 0.156909i −0.357919 0.933753i \(-0.616514\pi\)
0.629694 + 0.776843i \(0.283180\pi\)
\(192\) 0 0
\(193\) 53.1675 + 30.6963i 0.275479 + 0.159048i 0.631375 0.775478i \(-0.282491\pi\)
−0.355896 + 0.934526i \(0.615824\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −289.758 −1.47085 −0.735427 0.677604i \(-0.763018\pi\)
−0.735427 + 0.677604i \(0.763018\pi\)
\(198\) 0 0
\(199\) 100.347 0.504254 0.252127 0.967694i \(-0.418870\pi\)
0.252127 + 0.967694i \(0.418870\pi\)
\(200\) 0 0
\(201\) 122.251 63.8322i 0.608212 0.317573i
\(202\) 0 0
\(203\) 27.2665 47.2270i 0.134318 0.232645i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 110.523 + 9.39369i 0.533927 + 0.0453801i
\(208\) 0 0
\(209\) −440.489 + 254.316i −2.10760 + 1.21683i
\(210\) 0 0
\(211\) −106.282 + 184.086i −0.503706 + 0.872444i 0.496285 + 0.868160i \(0.334697\pi\)
−0.999991 + 0.00428419i \(0.998636\pi\)
\(212\) 0 0
\(213\) 78.9889 + 50.1840i 0.370840 + 0.235606i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 61.1100i 0.281613i
\(218\) 0 0
\(219\) −186.337 118.385i −0.850852 0.540572i
\(220\) 0 0
\(221\) 368.236 + 212.601i 1.66623 + 0.961996i
\(222\) 0 0
\(223\) 192.880 111.359i 0.864932 0.499369i −0.000728529 1.00000i \(-0.500232\pi\)
0.865661 + 0.500631i \(0.166899\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −68.0187 117.812i −0.299642 0.518995i 0.676412 0.736523i \(-0.263534\pi\)
−0.976054 + 0.217528i \(0.930201\pi\)
\(228\) 0 0
\(229\) −197.762 + 342.533i −0.863589 + 1.49578i 0.00485333 + 0.999988i \(0.498455\pi\)
−0.868442 + 0.495791i \(0.834878\pi\)
\(230\) 0 0
\(231\) 232.231 + 444.766i 1.00533 + 1.92539i
\(232\) 0 0
\(233\) 403.347 1.73110 0.865551 0.500821i \(-0.166969\pi\)
0.865551 + 0.500821i \(0.166969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −283.703 12.0347i −1.19706 0.0507793i
\(238\) 0 0
\(239\) −370.505 213.911i −1.55023 0.895025i −0.998122 0.0612505i \(-0.980491\pi\)
−0.552106 0.833774i \(-0.686176\pi\)
\(240\) 0 0
\(241\) −27.4457 47.5373i −0.113882 0.197250i 0.803450 0.595372i \(-0.202995\pi\)
−0.917332 + 0.398122i \(0.869662\pi\)
\(242\) 0 0
\(243\) −74.5053 231.296i −0.306606 0.951836i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 342.386 + 197.677i 1.38618 + 0.800311i
\(248\) 0 0
\(249\) 46.3841 + 1.96762i 0.186281 + 0.00790208i
\(250\) 0 0
\(251\) 404.670i 1.61223i −0.591759 0.806115i \(-0.701566\pi\)
0.591759 0.806115i \(-0.298434\pi\)
\(252\) 0 0
\(253\) 219.504i 0.867603i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 44.6196 77.2834i 0.173617 0.300713i −0.766065 0.642763i \(-0.777788\pi\)
0.939682 + 0.342050i \(0.111121\pi\)
\(258\) 0 0
\(259\) 312.500 + 541.266i 1.20656 + 2.08983i
\(260\) 0 0
\(261\) −42.8869 29.8717i −0.164318 0.114451i
\(262\) 0 0
\(263\) −168.170 291.278i −0.639428 1.10752i −0.985558 0.169335i \(-0.945838\pi\)
0.346130 0.938186i \(-0.387495\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 133.503 + 84.8186i 0.500012 + 0.317673i
\(268\) 0 0
\(269\) 225.504i 0.838305i −0.907916 0.419153i \(-0.862327\pi\)
0.907916 0.419153i \(-0.137673\pi\)
\(270\) 0 0
\(271\) −196.656 −0.725669 −0.362834 0.931854i \(-0.618191\pi\)
−0.362834 + 0.931854i \(0.618191\pi\)
\(272\) 0 0
\(273\) 209.139 329.182i 0.766077 1.20579i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −154.236 + 89.0481i −0.556808 + 0.321473i −0.751863 0.659319i \(-0.770845\pi\)
0.195055 + 0.980792i \(0.437511\pi\)
\(278\) 0 0
\(279\) 58.3578 + 4.96001i 0.209168 + 0.0177778i
\(280\) 0 0
\(281\) 74.3930 42.9508i 0.264744 0.152850i −0.361753 0.932274i \(-0.617822\pi\)
0.626497 + 0.779424i \(0.284488\pi\)
\(282\) 0 0
\(283\) −105.635 60.9884i −0.373269 0.215507i 0.301617 0.953429i \(-0.402474\pi\)
−0.674886 + 0.737922i \(0.735807\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −357.905 −1.24706
\(288\) 0 0
\(289\) 654.391 2.26433
\(290\) 0 0
\(291\) −9.69173 + 228.470i −0.0333049 + 0.785121i
\(292\) 0 0
\(293\) −44.5599 + 77.1800i −0.152081 + 0.263413i −0.931992 0.362478i \(-0.881931\pi\)
0.779911 + 0.625890i \(0.215264\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 443.584 185.672i 1.49355 0.625160i
\(298\) 0 0
\(299\) 147.759 85.3086i 0.494177 0.285313i
\(300\) 0 0
\(301\) −258.971 + 448.551i −0.860368 + 1.49020i
\(302\) 0 0
\(303\) −10.2130 + 240.759i −0.0337064 + 0.794585i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 602.803i 1.96353i 0.190105 + 0.981764i \(0.439117\pi\)
−0.190105 + 0.981764i \(0.560883\pi\)
\(308\) 0 0
\(309\) 80.1430 41.8460i 0.259362 0.135424i
\(310\) 0 0
\(311\) 142.632 + 82.3487i 0.458624 + 0.264787i 0.711466 0.702721i \(-0.248032\pi\)
−0.252841 + 0.967508i \(0.581365\pi\)
\(312\) 0 0
\(313\) −12.7018 + 7.33340i −0.0405809 + 0.0234294i −0.520153 0.854073i \(-0.674125\pi\)
0.479572 + 0.877502i \(0.340792\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −236.060 408.868i −0.744670 1.28981i −0.950349 0.311186i \(-0.899274\pi\)
0.205679 0.978619i \(-0.434060\pi\)
\(318\) 0 0
\(319\) 51.7137 89.5707i 0.162112 0.280786i
\(320\) 0 0
\(321\) −16.8309 + 26.4916i −0.0524328 + 0.0825284i
\(322\) 0 0
\(323\) 877.166 2.71568
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −301.263 + 474.182i −0.921292 + 1.45010i
\(328\) 0 0
\(329\) 133.939 + 77.3296i 0.407109 + 0.235044i
\(330\) 0 0
\(331\) −241.860 418.914i −0.730695 1.26560i −0.956587 0.291448i \(-0.905863\pi\)
0.225892 0.974152i \(-0.427470\pi\)
\(332\) 0 0
\(333\) 542.253 254.494i 1.62839 0.764246i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −314.346 181.488i −0.932778 0.538540i −0.0450890 0.998983i \(-0.514357\pi\)
−0.887689 + 0.460443i \(0.847690\pi\)
\(338\) 0 0
\(339\) 129.236 + 247.510i 0.381226 + 0.730118i
\(340\) 0 0
\(341\) 115.901i 0.339886i
\(342\) 0 0
\(343\) 92.1855i 0.268762i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 184.089 318.851i 0.530515 0.918879i −0.468851 0.883277i \(-0.655332\pi\)
0.999366 0.0356015i \(-0.0113347\pi\)
\(348\) 0 0
\(349\) 199.861 + 346.170i 0.572668 + 0.991890i 0.996291 + 0.0860512i \(0.0274249\pi\)
−0.423623 + 0.905839i \(0.639242\pi\)
\(350\) 0 0
\(351\) −297.381 226.438i −0.847240 0.645123i
\(352\) 0 0
\(353\) −277.703 480.996i −0.786694 1.36259i −0.927982 0.372626i \(-0.878458\pi\)
0.141288 0.989969i \(-0.454876\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 36.6726 864.509i 0.102724 2.42159i
\(358\) 0 0
\(359\) 670.647i 1.86810i 0.357147 + 0.934048i \(0.383749\pi\)
−0.357147 + 0.934048i \(0.616251\pi\)
\(360\) 0 0
\(361\) 454.590 1.25925
\(362\) 0 0
\(363\) 272.436 + 521.766i 0.750512 + 1.43737i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 81.5985 47.1109i 0.222339 0.128368i −0.384694 0.923044i \(-0.625693\pi\)
0.607033 + 0.794677i \(0.292360\pi\)
\(368\) 0 0
\(369\) −29.0495 + 341.786i −0.0787249 + 0.926251i
\(370\) 0 0
\(371\) −568.192 + 328.046i −1.53151 + 0.884220i
\(372\) 0 0
\(373\) 166.777 + 96.2886i 0.447123 + 0.258146i 0.706614 0.707599i \(-0.250222\pi\)
−0.259492 + 0.965745i \(0.583555\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −80.3927 −0.213243
\(378\) 0 0
\(379\) −184.387 −0.486510 −0.243255 0.969962i \(-0.578215\pi\)
−0.243255 + 0.969962i \(0.578215\pi\)
\(380\) 0 0
\(381\) 122.785 + 78.0088i 0.322269 + 0.204748i
\(382\) 0 0
\(383\) 332.462 575.841i 0.868047 1.50350i 0.00405754 0.999992i \(-0.498708\pi\)
0.863990 0.503510i \(-0.167958\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 407.330 + 283.714i 1.05253 + 0.733112i
\(388\) 0 0
\(389\) −539.718 + 311.606i −1.38745 + 0.801044i −0.993027 0.117885i \(-0.962388\pi\)
−0.394422 + 0.918929i \(0.629055\pi\)
\(390\) 0 0
\(391\) 189.273 327.830i 0.484074 0.838441i
\(392\) 0 0
\(393\) 165.969 86.6594i 0.422313 0.220507i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 247.371i 0.623100i −0.950230 0.311550i \(-0.899152\pi\)
0.950230 0.311550i \(-0.100848\pi\)
\(398\) 0 0
\(399\) 34.0982 803.822i 0.0854591 2.01459i
\(400\) 0 0
\(401\) 369.248 + 213.185i 0.920818 + 0.531635i 0.883896 0.467684i \(-0.154911\pi\)
0.0369222 + 0.999318i \(0.488245\pi\)
\(402\) 0 0
\(403\) 78.0189 45.0443i 0.193595 0.111772i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 592.687 + 1026.56i 1.45623 + 2.52227i
\(408\) 0 0
\(409\) 87.0543 150.782i 0.212847 0.368661i −0.739758 0.672873i \(-0.765060\pi\)
0.952604 + 0.304212i \(0.0983932\pi\)
\(410\) 0 0
\(411\) 6.42047 + 0.272357i 0.0156216 + 0.000662669i
\(412\) 0 0
\(413\) −989.553 −2.39601
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 20.5607 + 39.3776i 0.0493063 + 0.0944307i
\(418\) 0 0
\(419\) 265.100 + 153.055i 0.632696 + 0.365287i 0.781796 0.623535i \(-0.214304\pi\)
−0.149099 + 0.988822i \(0.547637\pi\)
\(420\) 0 0
\(421\) 140.251 + 242.922i 0.333138 + 0.577012i 0.983125 0.182933i \(-0.0585591\pi\)
−0.649987 + 0.759945i \(0.725226\pi\)
\(422\) 0 0
\(423\) 84.7181 121.630i 0.200279 0.287542i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −545.820 315.130i −1.27827 0.738008i
\(428\) 0 0
\(429\) 396.653 624.326i 0.924599 1.45530i
\(430\) 0 0
\(431\) 675.034i 1.56620i −0.621893 0.783102i \(-0.713636\pi\)
0.621893 0.783102i \(-0.286364\pi\)
\(432\) 0 0
\(433\) 273.497i 0.631633i −0.948820 0.315817i \(-0.897722\pi\)
0.948820 0.315817i \(-0.102278\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 175.986 304.817i 0.402715 0.697522i
\(438\) 0 0
\(439\) −262.426 454.535i −0.597781 1.03539i −0.993148 0.116864i \(-0.962716\pi\)
0.395367 0.918523i \(-0.370618\pi\)
\(440\) 0 0
\(441\) −351.382 29.8651i −0.796784 0.0677212i
\(442\) 0 0
\(443\) 44.2748 + 76.6862i 0.0999431 + 0.173106i 0.911661 0.410943i \(-0.134801\pi\)
−0.811718 + 0.584050i \(0.801467\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −544.522 + 284.318i −1.21817 + 0.636059i
\(448\) 0 0
\(449\) 46.0938i 0.102659i 0.998682 + 0.0513294i \(0.0163458\pi\)
−0.998682 + 0.0513294i \(0.983654\pi\)
\(450\) 0 0
\(451\) −678.803 −1.50511
\(452\) 0 0
\(453\) −44.6758 1.89515i −0.0986220 0.00418355i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 163.904 94.6301i 0.358652 0.207068i −0.309837 0.950790i \(-0.600275\pi\)
0.668489 + 0.743722i \(0.266941\pi\)
\(458\) 0 0
\(459\) −822.598 105.189i −1.79215 0.229170i
\(460\) 0 0
\(461\) −362.663 + 209.384i −0.786689 + 0.454195i −0.838795 0.544447i \(-0.816740\pi\)
0.0521069 + 0.998642i \(0.483406\pi\)
\(462\) 0 0
\(463\) −271.090 156.514i −0.585507 0.338043i 0.177812 0.984064i \(-0.443098\pi\)
−0.763319 + 0.646022i \(0.776431\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −148.419 −0.317814 −0.158907 0.987294i \(-0.550797\pi\)
−0.158907 + 0.987294i \(0.550797\pi\)
\(468\) 0 0
\(469\) −431.692 −0.920452
\(470\) 0 0
\(471\) 48.6847 25.4203i 0.103365 0.0539710i
\(472\) 0 0
\(473\) −491.164 + 850.721i −1.03840 + 1.79856i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 267.154 + 569.228i 0.560071 + 1.19335i
\(478\) 0 0
\(479\) −273.219 + 157.743i −0.570395 + 0.329317i −0.757307 0.653059i \(-0.773485\pi\)
0.186912 + 0.982377i \(0.440152\pi\)
\(480\) 0 0
\(481\) 460.688 797.935i 0.957771 1.65891i
\(482\) 0 0
\(483\) −293.061 186.191i −0.606752 0.385488i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 492.691i 1.01169i −0.862626 0.505843i \(-0.831182\pi\)
0.862626 0.505843i \(-0.168818\pi\)
\(488\) 0 0
\(489\) −109.417 69.5158i −0.223756 0.142159i
\(490\) 0 0
\(491\) 719.318 + 415.298i 1.46501 + 0.845822i 0.999236 0.0390843i \(-0.0124441\pi\)
0.465770 + 0.884906i \(0.345777\pi\)
\(492\) 0 0
\(493\) −154.470 + 89.1831i −0.313326 + 0.180899i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −146.466 253.686i −0.294700 0.510435i
\(498\) 0 0
\(499\) 14.1379 24.4876i 0.0283325 0.0490733i −0.851512 0.524336i \(-0.824314\pi\)
0.879844 + 0.475263i \(0.157647\pi\)
\(500\) 0 0
\(501\) 99.0283 + 189.658i 0.197661 + 0.378558i
\(502\) 0 0
\(503\) 31.5709 0.0627652 0.0313826 0.999507i \(-0.490009\pi\)
0.0313826 + 0.999507i \(0.490009\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −67.8769 2.87935i −0.133880 0.00567918i
\(508\) 0 0
\(509\) −204.163 117.874i −0.401107 0.231579i 0.285855 0.958273i \(-0.407723\pi\)
−0.686961 + 0.726694i \(0.741056\pi\)
\(510\) 0 0
\(511\) 345.516 + 598.452i 0.676157 + 1.17114i
\(512\) 0 0
\(513\) −764.852 97.8049i −1.49094 0.190653i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 254.028 + 146.663i 0.491350 + 0.283681i
\(518\) 0 0
\(519\) 503.217 + 21.3465i 0.969589 + 0.0411301i
\(520\) 0 0
\(521\) 582.254i 1.11757i −0.829312 0.558785i \(-0.811268\pi\)
0.829312 0.558785i \(-0.188732\pi\)
\(522\) 0 0
\(523\) 201.726i 0.385709i 0.981227 + 0.192855i \(0.0617745\pi\)
−0.981227 + 0.192855i \(0.938225\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 99.9391 173.100i 0.189638 0.328462i
\(528\) 0 0
\(529\) 188.552 + 326.582i 0.356431 + 0.617357i
\(530\) 0 0
\(531\) −80.3173 + 944.986i −0.151257 + 1.77963i
\(532\) 0 0
\(533\) 263.812 + 456.937i 0.494958 + 0.857292i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −297.819 189.213i −0.554598 0.352353i
\(538\) 0 0
\(539\) 697.860i 1.29473i
\(540\) 0 0
\(541\) 390.540 0.721886 0.360943 0.932588i \(-0.382455\pi\)
0.360943 + 0.932588i \(0.382455\pi\)
\(542\) 0 0
\(543\) −155.291 + 244.425i −0.285987 + 0.450138i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 556.158 321.098i 1.01674 0.587017i 0.103584 0.994621i \(-0.466969\pi\)
0.913159 + 0.407604i \(0.133636\pi\)
\(548\) 0 0
\(549\) −345.239 + 495.661i −0.628850 + 0.902843i
\(550\) 0 0
\(551\) −143.626 + 82.9225i −0.260664 + 0.150495i
\(552\) 0 0
\(553\) 769.761 + 444.422i 1.39197 + 0.803656i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −145.052 −0.260417 −0.130208 0.991487i \(-0.541565\pi\)
−0.130208 + 0.991487i \(0.541565\pi\)
\(558\) 0 0
\(559\) 763.551 1.36592
\(560\) 0 0
\(561\) 69.5532 1639.63i 0.123981 2.92269i
\(562\) 0 0
\(563\) −422.949 + 732.570i −0.751242 + 1.30119i 0.195979 + 0.980608i \(0.437212\pi\)
−0.947221 + 0.320581i \(0.896122\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −128.371 + 749.727i −0.226403 + 1.32227i
\(568\) 0 0
\(569\) 725.474 418.853i 1.27500 0.736121i 0.299074 0.954230i \(-0.403322\pi\)
0.975924 + 0.218109i \(0.0699888\pi\)
\(570\) 0 0
\(571\) −348.675 + 603.922i −0.610639 + 1.05766i 0.380494 + 0.924783i \(0.375754\pi\)
−0.991133 + 0.132874i \(0.957579\pi\)
\(572\) 0 0
\(573\) −7.62104 + 179.656i −0.0133003 + 0.313537i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 713.999i 1.23743i 0.785614 + 0.618717i \(0.212347\pi\)
−0.785614 + 0.618717i \(0.787653\pi\)
\(578\) 0 0
\(579\) −163.262 + 85.2460i −0.281972 + 0.147230i
\(580\) 0 0
\(581\) −125.852 72.6610i −0.216614 0.125062i
\(582\) 0 0
\(583\) −1077.63 + 622.171i −1.84843 + 1.06719i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 408.180 + 706.988i 0.695366 + 1.20441i 0.970057 + 0.242876i \(0.0780909\pi\)
−0.274692 + 0.961532i \(0.588576\pi\)
\(588\) 0 0
\(589\) 92.9235 160.948i 0.157765 0.273257i
\(590\) 0 0
\(591\) 466.152 733.717i 0.788752 1.24148i
\(592\) 0 0
\(593\) 64.6676 0.109052 0.0545258 0.998512i \(-0.482635\pi\)
0.0545258 + 0.998512i \(0.482635\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −161.434 + 254.095i −0.270409 + 0.425619i
\(598\) 0 0
\(599\) 204.565 + 118.105i 0.341510 + 0.197171i 0.660940 0.750439i \(-0.270158\pi\)
−0.319430 + 0.947610i \(0.603491\pi\)
\(600\) 0 0
\(601\) 84.0789 + 145.629i 0.139898 + 0.242311i 0.927458 0.373927i \(-0.121989\pi\)
−0.787560 + 0.616238i \(0.788656\pi\)
\(602\) 0 0
\(603\) −35.0384 + 412.250i −0.0581068 + 0.683665i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 95.2911 + 55.0163i 0.156987 + 0.0906365i 0.576435 0.817143i \(-0.304443\pi\)
−0.419449 + 0.907779i \(0.637777\pi\)
\(608\) 0 0
\(609\) 75.7213 + 145.021i 0.124337 + 0.238129i
\(610\) 0 0
\(611\) 227.999i 0.373157i
\(612\) 0 0
\(613\) 387.013i 0.631343i 0.948869 + 0.315671i \(0.102230\pi\)
−0.948869 + 0.315671i \(0.897770\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 192.373 333.200i 0.311787 0.540032i −0.666962 0.745092i \(-0.732406\pi\)
0.978749 + 0.205060i \(0.0657390\pi\)
\(618\) 0 0
\(619\) −403.446 698.789i −0.651771 1.12890i −0.982693 0.185242i \(-0.940693\pi\)
0.330922 0.943658i \(-0.392640\pi\)
\(620\) 0 0
\(621\) −201.592 + 264.750i −0.324624 + 0.426329i
\(622\) 0 0
\(623\) −247.549 428.768i −0.397351 0.688232i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 64.6706 1524.53i 0.103143 2.43146i
\(628\) 0 0
\(629\) 2044.24i 3.24999i
\(630\) 0 0
\(631\) −753.299 −1.19382 −0.596909 0.802309i \(-0.703605\pi\)
−0.596909 + 0.802309i \(0.703605\pi\)
\(632\) 0 0
\(633\) −295.153 565.274i −0.466277 0.893008i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −469.765 + 271.219i −0.737464 + 0.425775i
\(638\) 0 0
\(639\) −254.149 + 119.279i −0.397729 + 0.186665i
\(640\) 0 0
\(641\) −570.820 + 329.563i −0.890515 + 0.514139i −0.874111 0.485726i \(-0.838555\pi\)
−0.0164041 + 0.999865i \(0.505222\pi\)
\(642\) 0 0
\(643\) −783.548 452.382i −1.21858 0.703549i −0.253968 0.967213i \(-0.581736\pi\)
−0.964615 + 0.263664i \(0.915069\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −360.764 −0.557594 −0.278797 0.960350i \(-0.589936\pi\)
−0.278797 + 0.960350i \(0.589936\pi\)
\(648\) 0 0
\(649\) −1876.79 −2.89181
\(650\) 0 0
\(651\) −154.741 98.3116i −0.237697 0.151016i
\(652\) 0 0
\(653\) 211.022 365.501i 0.323158 0.559726i −0.657980 0.753035i \(-0.728589\pi\)
0.981138 + 0.193310i \(0.0619222\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 599.543 281.382i 0.912547 0.428283i
\(658\) 0 0
\(659\) −520.928 + 300.758i −0.790483 + 0.456385i −0.840133 0.542381i \(-0.817523\pi\)
0.0496496 + 0.998767i \(0.484190\pi\)
\(660\) 0 0
\(661\) 465.379 806.060i 0.704053 1.21946i −0.262979 0.964802i \(-0.584705\pi\)
0.967032 0.254654i \(-0.0819617\pi\)
\(662\) 0 0
\(663\) −1130.75 + 590.411i −1.70550 + 0.890514i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 71.5714i 0.107303i
\(668\) 0 0
\(669\) −28.3178 + 667.555i −0.0423285 + 0.997840i
\(670\) 0 0
\(671\) −1035.20 597.675i −1.54278 0.890722i
\(672\) 0 0
\(673\) −50.6352 + 29.2343i −0.0752381 + 0.0434387i −0.537147 0.843489i \(-0.680498\pi\)
0.461909 + 0.886927i \(0.347165\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 317.363 + 549.689i 0.468778 + 0.811948i 0.999363 0.0356841i \(-0.0113610\pi\)
−0.530585 + 0.847632i \(0.678028\pi\)
\(678\) 0 0
\(679\) 357.900 619.901i 0.527099 0.912962i
\(680\) 0 0
\(681\) 407.746 + 17.2966i 0.598746 + 0.0253988i
\(682\) 0 0
\(683\) 387.682 0.567617 0.283808 0.958881i \(-0.408402\pi\)
0.283808 + 0.958881i \(0.408402\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −549.201 1051.82i −0.799419 1.53104i
\(688\) 0 0
\(689\) 837.629 + 483.605i 1.21572 + 0.701895i
\(690\) 0 0
\(691\) −483.665 837.732i −0.699949 1.21235i −0.968484 0.249077i \(-0.919873\pi\)
0.268535 0.963270i \(-0.413461\pi\)
\(692\) 0 0
\(693\) −1499.83 127.475i −2.16425 0.183947i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1013.80 + 585.317i 1.45452 + 0.839766i
\(698\) 0 0
\(699\) −648.889 + 1021.34i −0.928311 + 1.46115i
\(700\) 0 0
\(701\) 895.039i 1.27680i 0.769703 + 0.638402i \(0.220404\pi\)
−0.769703 + 0.638402i \(0.779596\pi\)
\(702\) 0 0
\(703\) 1900.74i 2.70376i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 377.151 653.245i 0.533453 0.923967i
\(708\) 0 0
\(709\) −624.531 1081.72i −0.880861 1.52570i −0.850385 0.526161i \(-0.823631\pi\)
−0.0304761 0.999535i \(-0.509702\pi\)
\(710\) 0 0
\(711\) 486.884 699.022i 0.684788 0.983153i
\(712\) 0 0
\(713\) −40.1017 69.4581i −0.0562436 0.0974167i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1137.71 594.048i 1.58677 0.828519i
\(718\) 0 0
\(719\) 418.228i 0.581681i −0.956772 0.290840i \(-0.906065\pi\)
0.956772 0.290840i \(-0.0939348\pi\)
\(720\) 0 0
\(721\) −283.001 −0.392512
\(722\) 0 0
\(723\) 164.526 + 6.97921i 0.227560 + 0.00965312i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 447.070 258.116i 0.614952 0.355043i −0.159949 0.987125i \(-0.551133\pi\)
0.774901 + 0.632082i \(0.217800\pi\)
\(728\) 0 0
\(729\) 705.543 + 183.441i 0.967823 + 0.251634i
\(730\) 0 0
\(731\) 1467.12 847.040i 2.00700 1.15874i
\(732\) 0 0
\(733\) −7.14580 4.12563i −0.00974871 0.00562842i 0.495118 0.868826i \(-0.335125\pi\)
−0.504866 + 0.863197i \(0.668458\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −818.747 −1.11092
\(738\) 0 0
\(739\) −294.686 −0.398763 −0.199381 0.979922i \(-0.563893\pi\)
−0.199381 + 0.979922i \(0.563893\pi\)
\(740\) 0 0
\(741\) −1051.37 + 548.965i −1.41885 + 0.740843i
\(742\) 0 0
\(743\) −253.191 + 438.540i −0.340769 + 0.590229i −0.984576 0.174959i \(-0.944021\pi\)
0.643807 + 0.765188i \(0.277354\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −79.6034 + 114.287i −0.106564 + 0.152995i
\(748\) 0 0
\(749\) 85.0824 49.1223i 0.113595 0.0655839i
\(750\) 0 0
\(751\) −395.768 + 685.491i −0.526989 + 0.912771i 0.472517 + 0.881322i \(0.343346\pi\)
−0.999505 + 0.0314493i \(0.989988\pi\)
\(752\) 0 0
\(753\) 1024.69 + 651.018i 1.36081 + 0.864566i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 957.457i 1.26480i −0.774640 0.632402i \(-0.782069\pi\)
0.774640 0.632402i \(-0.217931\pi\)
\(758\) 0 0
\(759\) −555.820 353.129i −0.732306 0.465256i
\(760\) 0 0
\(761\) −378.389 218.463i −0.497225 0.287073i 0.230342 0.973110i \(-0.426016\pi\)
−0.727567 + 0.686037i \(0.759349\pi\)
\(762\) 0 0
\(763\) 1522.92 879.257i 1.99596 1.15237i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 729.400 + 1263.36i 0.950978 + 1.64714i
\(768\) 0 0
\(769\) −302.360 + 523.703i −0.393186 + 0.681018i −0.992868 0.119221i \(-0.961960\pi\)
0.599682 + 0.800238i \(0.295294\pi\)
\(770\) 0 0
\(771\) 123.912 + 237.315i 0.160716 + 0.307802i
\(772\) 0 0
\(773\) 596.625 0.771831 0.385915 0.922534i \(-0.373886\pi\)
0.385915 + 0.922534i \(0.373886\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1873.32 79.4662i −2.41096 0.102273i
\(778\) 0 0
\(779\) 942.631 + 544.228i 1.21005 + 0.698625i
\(780\) 0 0
\(781\) −277.787 481.141i −0.355681 0.616058i
\(782\) 0 0
\(783\) 144.635 60.5404i 0.184719 0.0773186i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −471.499 272.220i −0.599109 0.345896i 0.169582 0.985516i \(-0.445758\pi\)
−0.768691 + 0.639620i \(0.779092\pi\)
\(788\) 0 0
\(789\) 1008.11 + 42.7642i 1.27771 + 0.0542005i
\(790\) 0 0
\(791\) 874.010i 1.10494i
\(792\) 0 0
\(793\) 929.129i 1.17166i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −115.508 + 200.065i −0.144928 + 0.251023i −0.929346 0.369210i \(-0.879628\pi\)
0.784418 + 0.620232i \(0.212962\pi\)
\(798\) 0 0
\(799\) −252.929 438.086i −0.316557 0.548293i
\(800\) 0 0
\(801\) −429.550 + 201.599i −0.536267 + 0.251685i
\(802\) 0 0
\(803\) 655.306 + 1135.02i 0.816072 + 1.41348i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 571.014 + 362.783i 0.707577 + 0.449545i
\(808\) 0 0
\(809\) 75.6693i 0.0935344i 0.998906 + 0.0467672i \(0.0148919\pi\)
−0.998906 + 0.0467672i \(0.985108\pi\)
\(810\) 0 0
\(811\) 1184.72 1.46081 0.730404 0.683015i \(-0.239332\pi\)
0.730404 + 0.683015i \(0.239332\pi\)
\(812\) 0 0
\(813\) 316.373 497.967i 0.389143 0.612505i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1364.13 787.579i 1.66968 0.963988i
\(818\) 0 0
\(819\) 497.088 + 1059.15i 0.606945 + 1.29322i
\(820\) 0 0
\(821\) −1282.37 + 740.377i −1.56196 + 0.901799i −0.564904 + 0.825157i \(0.691087\pi\)
−0.997059 + 0.0766424i \(0.975580\pi\)
\(822\) 0 0
\(823\) −659.371 380.688i −0.801180 0.462562i 0.0427034 0.999088i \(-0.486403\pi\)
−0.843884 + 0.536526i \(0.819736\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1019.60 −1.23290 −0.616448 0.787396i \(-0.711429\pi\)
−0.616448 + 0.787396i \(0.711429\pi\)
\(828\) 0 0
\(829\) −58.7988 −0.0709274 −0.0354637 0.999371i \(-0.511291\pi\)
−0.0354637 + 0.999371i \(0.511291\pi\)
\(830\) 0 0
\(831\) 22.6442 533.809i 0.0272494 0.642369i
\(832\) 0 0
\(833\) −601.750 + 1042.26i −0.722388 + 1.25121i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −106.444 + 139.792i −0.127173 + 0.167016i
\(838\) 0 0
\(839\) 1005.77 580.682i 1.19877 0.692112i 0.238492 0.971144i \(-0.423347\pi\)
0.960282 + 0.279032i \(0.0900136\pi\)
\(840\) 0 0
\(841\) −403.638 + 699.122i −0.479950 + 0.831298i
\(842\) 0 0
\(843\) −10.9221 + 257.473i −0.0129562 + 0.305425i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1842.46i 2.17528i
\(848\) 0 0
\(849\) 324.375 169.370i 0.382067 0.199493i
\(850\) 0 0
\(851\) −710.380 410.138i −0.834759 0.481948i
\(852\) 0 0
\(853\) 1204.25 695.272i 1.41178 0.815090i 0.416222 0.909263i \(-0.363354\pi\)
0.995556 + 0.0941731i \(0.0300207\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 528.220 + 914.904i 0.616359 + 1.06757i 0.990144 + 0.140050i \(0.0447264\pi\)
−0.373785 + 0.927515i \(0.621940\pi\)
\(858\) 0 0
\(859\) −537.750 + 931.410i −0.626018 + 1.08430i 0.362325 + 0.932052i \(0.381983\pi\)
−0.988343 + 0.152244i \(0.951350\pi\)
\(860\) 0 0
\(861\) 575.785 906.277i 0.668740 1.05259i
\(862\) 0 0
\(863\) 66.3453 0.0768776 0.0384388 0.999261i \(-0.487762\pi\)
0.0384388 + 0.999261i \(0.487762\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −1052.76 + 1657.03i −1.21426 + 1.91122i
\(868\) 0 0
\(869\) 1459.93 + 842.890i 1.68001 + 0.969954i
\(870\) 0 0
\(871\) 318.200 + 551.139i 0.365328 + 0.632766i
\(872\) 0 0
\(873\) −562.933 392.096i −0.644826 0.449136i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 366.507 + 211.603i 0.417909 + 0.241280i 0.694182 0.719799i \(-0.255766\pi\)
−0.276273 + 0.961079i \(0.589099\pi\)
\(878\) 0 0
\(879\) −123.746 236.997i −0.140781 0.269622i
\(880\) 0 0
\(881\) 525.156i 0.596090i 0.954552 + 0.298045i \(0.0963346\pi\)
−0.954552 + 0.298045i \(0.903665\pi\)
\(882\) 0 0
\(883\) 44.5609i 0.0504653i −0.999682 0.0252327i \(-0.991967\pi\)
0.999682 0.0252327i \(-0.00803266\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 706.516 1223.72i 0.796523 1.37962i −0.125345 0.992113i \(-0.540004\pi\)
0.921868 0.387505i \(-0.126663\pi\)
\(888\) 0 0
\(889\) −227.674 394.344i −0.256102 0.443581i
\(890\) 0 0
\(891\) −243.468 + 1421.93i −0.273252 + 1.59588i
\(892\) 0 0
\(893\) −235.174 407.333i −0.263352 0.456140i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −21.6933 + 511.392i −0.0241843 + 0.570113i
\(898\) 0 0
\(899\) 37.7908i 0.0420365i
\(900\) 0 0
\(901\) 2145.94 2.38173
\(902\) 0 0
\(903\) −719.183 1377.37i −0.796438 1.52533i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −261.298 + 150.861i −0.288090 + 0.166329i −0.637080 0.770797i \(-0.719858\pi\)
0.348990 + 0.937126i \(0.386525\pi\)
\(908\) 0 0
\(909\) −593.213 413.186i −0.652599 0.454550i
\(910\) 0 0
\(911\) 243.600 140.643i 0.267399 0.154383i −0.360306 0.932834i \(-0.617328\pi\)
0.627705 + 0.778451i \(0.283994\pi\)
\(912\) 0 0
\(913\) −238.692 137.809i −0.261437 0.150941i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −586.071 −0.639117
\(918\) 0 0
\(919\) −309.762 −0.337064 −0.168532 0.985696i \(-0.553903\pi\)
−0.168532 + 0.985696i \(0.553903\pi\)
\(920\) 0 0
\(921\) −1526.40 969.768i −1.65733 1.05295i
\(922\) 0 0
\(923\) −215.920 + 373.985i −0.233933 + 0.405184i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −22.9699 + 270.256i −0.0247787 + 0.291538i
\(928\) 0 0
\(929\) 416.036 240.198i 0.447832 0.258556i −0.259082 0.965855i \(-0.583420\pi\)
0.706914 + 0.707300i \(0.250087\pi\)
\(930\) 0 0
\(931\) −559.507 + 969.095i −0.600975 + 1.04092i
\(932\) 0 0
\(933\) −437.982 + 228.689i −0.469435 + 0.245112i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 918.055i 0.979781i 0.871784 + 0.489891i \(0.162963\pi\)
−0.871784 + 0.489891i \(0.837037\pi\)
\(938\) 0 0
\(939\) 1.86482 43.9609i 0.00198597 0.0468167i
\(940\) 0 0
\(941\) 1228.87 + 709.489i 1.30592 + 0.753973i 0.981413 0.191910i \(-0.0614682\pi\)
0.324507 + 0.945883i \(0.394802\pi\)
\(942\) 0 0
\(943\) 406.798 234.865i 0.431387 0.249061i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 134.189 + 232.422i 0.141699 + 0.245430i 0.928136 0.372240i \(-0.121410\pi\)
−0.786438 + 0.617670i \(0.788077\pi\)
\(948\) 0 0
\(949\) 509.361 882.238i 0.536734 0.929651i
\(950\) 0 0
\(951\) 1415.09 + 60.0282i 1.48800 + 0.0631212i
\(952\) 0 0
\(953\) −476.437 −0.499934 −0.249967 0.968254i \(-0.580420\pi\)
−0.249967 + 0.968254i \(0.580420\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 143.613 + 275.046i 0.150066 + 0.287404i
\(958\) 0 0
\(959\) −17.4205 10.0577i −0.0181652 0.0104877i
\(960\) 0 0
\(961\) 459.326 + 795.575i 0.477966 + 0.827862i
\(962\) 0 0
\(963\) −40.0043 85.2376i −0.0415413 0.0885125i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −876.486 506.040i −0.906398 0.523309i −0.0271272 0.999632i \(-0.508636\pi\)
−0.879270 + 0.476323i \(0.841969\pi\)
\(968\) 0 0
\(969\) −1411.15 + 2221.13i −1.45630 + 2.29219i
\(970\) 0 0
\(971\) 772.488i 0.795560i 0.917481 + 0.397780i \(0.130219\pi\)
−0.917481 + 0.397780i \(0.869781\pi\)
\(972\) 0 0
\(973\) 139.050i 0.142909i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −812.523 + 1407.33i −0.831651 + 1.44046i 0.0650777 + 0.997880i \(0.479270\pi\)
−0.896728 + 0.442581i \(0.854063\pi\)
\(978\) 0 0
\(979\) −469.502 813.202i −0.479573 0.830645i
\(980\) 0 0
\(981\) −716.050 1525.70i −0.729918 1.55524i
\(982\) 0 0
\(983\) 403.300 + 698.536i 0.410275 + 0.710616i 0.994920 0.100673i \(-0.0320995\pi\)
−0.584645 + 0.811289i \(0.698766\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −411.287 + 214.750i −0.416704 + 0.217579i
\(988\) 0 0
\(989\) 679.768i 0.687329i
\(990\) 0 0
\(991\) 434.414 0.438359 0.219180 0.975685i \(-0.429662\pi\)
0.219180 + 0.975685i \(0.429662\pi\)
\(992\) 0 0
\(993\) 1449.86 + 61.5030i 1.46008 + 0.0619366i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1530.07 + 883.384i −1.53467 + 0.886042i −0.535533 + 0.844514i \(0.679889\pi\)
−0.999137 + 0.0415279i \(0.986777\pi\)
\(998\) 0 0
\(999\) −227.935 + 1782.50i −0.228164 + 1.78428i
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 900.3.u.d.149.5 32
3.2 odd 2 2700.3.u.d.449.14 32
5.2 odd 4 900.3.p.e.401.7 yes 16
5.3 odd 4 900.3.p.d.401.2 yes 16
5.4 even 2 inner 900.3.u.d.149.12 32
9.2 odd 6 inner 900.3.u.d.749.12 32
9.7 even 3 2700.3.u.d.2249.3 32
15.2 even 4 2700.3.p.d.2501.7 16
15.8 even 4 2700.3.p.e.2501.2 16
15.14 odd 2 2700.3.u.d.449.3 32
45.2 even 12 900.3.p.e.101.7 yes 16
45.7 odd 12 2700.3.p.d.1601.7 16
45.29 odd 6 inner 900.3.u.d.749.5 32
45.34 even 6 2700.3.u.d.2249.14 32
45.38 even 12 900.3.p.d.101.2 16
45.43 odd 12 2700.3.p.e.1601.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.3.p.d.101.2 16 45.38 even 12
900.3.p.d.401.2 yes 16 5.3 odd 4
900.3.p.e.101.7 yes 16 45.2 even 12
900.3.p.e.401.7 yes 16 5.2 odd 4
900.3.u.d.149.5 32 1.1 even 1 trivial
900.3.u.d.149.12 32 5.4 even 2 inner
900.3.u.d.749.5 32 45.29 odd 6 inner
900.3.u.d.749.12 32 9.2 odd 6 inner
2700.3.p.d.1601.7 16 45.7 odd 12
2700.3.p.d.2501.7 16 15.2 even 4
2700.3.p.e.1601.2 16 45.43 odd 12
2700.3.p.e.2501.2 16 15.8 even 4
2700.3.u.d.449.3 32 15.14 odd 2
2700.3.u.d.449.14 32 3.2 odd 2
2700.3.u.d.2249.3 32 9.7 even 3
2700.3.u.d.2249.14 32 45.34 even 6