Properties

Label 900.3.p.f.401.1
Level $900$
Weight $3$
Character 900.401
Analytic conductor $24.523$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,3,Mod(101,900)]
 
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, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.101");
 
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.p (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 401.1
Character \(\chi\) \(=\) 900.401
Dual form 900.3.p.f.101.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.88925 - 0.807601i) q^{3} +(-6.19003 - 10.7214i) q^{7} +(7.69556 + 4.66673i) q^{9} +O(q^{10})\) \(q+(-2.88925 - 0.807601i) q^{3} +(-6.19003 - 10.7214i) q^{7} +(7.69556 + 4.66673i) q^{9} +(-4.00882 + 2.31449i) q^{11} +(-0.631764 + 1.09425i) q^{13} -22.9122i q^{17} -32.8729 q^{19} +(9.22592 + 35.9760i) q^{21} +(-0.690324 - 0.398559i) q^{23} +(-18.4656 - 19.6983i) q^{27} +(34.3302 - 19.8206i) q^{29} +(-0.730215 + 1.26477i) q^{31} +(13.4517 - 3.44963i) q^{33} +44.9884 q^{37} +(2.70904 - 2.65134i) q^{39} +(-33.8915 - 19.5673i) q^{41} +(18.1908 + 31.5074i) q^{43} +(-52.8629 + 30.5204i) q^{47} +(-52.1330 + 90.2970i) q^{49} +(-18.5039 + 66.1992i) q^{51} -13.9872i q^{53} +(94.9780 + 26.5482i) q^{57} +(50.2415 + 29.0069i) q^{59} +(27.5253 + 47.6752i) q^{61} +(2.39828 - 111.395i) q^{63} +(-33.4960 + 58.0167i) q^{67} +(1.67264 + 1.70904i) q^{69} +70.6560i q^{71} -23.1498 q^{73} +(49.6294 + 28.6536i) q^{77} +(-14.5725 - 25.2404i) q^{79} +(37.4434 + 71.8261i) q^{81} +(-112.958 + 65.2165i) q^{83} +(-115.196 + 29.5415i) q^{87} +48.0873i q^{89} +15.6425 q^{91} +(3.13120 - 3.06452i) q^{93} +(-9.68803 - 16.7802i) q^{97} +(-41.6512 - 0.896733i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 2 q^{9} - 18 q^{11} - 26 q^{21} - 36 q^{29} + 30 q^{31} - 6 q^{39} - 36 q^{41} - 108 q^{49} + 124 q^{51} + 306 q^{59} + 48 q^{61} + 268 q^{69} - 114 q^{79} - 14 q^{81} - 84 q^{91} - 418 q^{99}+O(q^{100}) \) 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
\(3\) −2.88925 0.807601i −0.963084 0.269200i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −6.19003 10.7214i −0.884290 1.53164i −0.846525 0.532349i \(-0.821309\pi\)
−0.0377652 0.999287i \(-0.512024\pi\)
\(8\) 0 0
\(9\) 7.69556 + 4.66673i 0.855062 + 0.518525i
\(10\) 0 0
\(11\) −4.00882 + 2.31449i −0.364438 + 0.210408i −0.671026 0.741434i \(-0.734146\pi\)
0.306588 + 0.951842i \(0.400813\pi\)
\(12\) 0 0
\(13\) −0.631764 + 1.09425i −0.0485972 + 0.0841728i −0.889301 0.457323i \(-0.848808\pi\)
0.840704 + 0.541496i \(0.182142\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 22.9122i 1.34778i −0.738833 0.673889i \(-0.764623\pi\)
0.738833 0.673889i \(-0.235377\pi\)
\(18\) 0 0
\(19\) −32.8729 −1.73015 −0.865075 0.501642i \(-0.832730\pi\)
−0.865075 + 0.501642i \(0.832730\pi\)
\(20\) 0 0
\(21\) 9.22592 + 35.9760i 0.439329 + 1.71315i
\(22\) 0 0
\(23\) −0.690324 0.398559i −0.0300141 0.0173286i 0.484918 0.874560i \(-0.338849\pi\)
−0.514932 + 0.857231i \(0.672183\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −18.4656 19.6983i −0.683910 0.729566i
\(28\) 0 0
\(29\) 34.3302 19.8206i 1.18380 0.683468i 0.226910 0.973916i \(-0.427138\pi\)
0.956891 + 0.290448i \(0.0938042\pi\)
\(30\) 0 0
\(31\) −0.730215 + 1.26477i −0.0235553 + 0.0407990i −0.877563 0.479462i \(-0.840832\pi\)
0.854007 + 0.520261i \(0.174165\pi\)
\(32\) 0 0
\(33\) 13.4517 3.44963i 0.407627 0.104534i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 44.9884 1.21590 0.607952 0.793974i \(-0.291991\pi\)
0.607952 + 0.793974i \(0.291991\pi\)
\(38\) 0 0
\(39\) 2.70904 2.65134i 0.0694626 0.0679831i
\(40\) 0 0
\(41\) −33.8915 19.5673i −0.826623 0.477251i 0.0260720 0.999660i \(-0.491700\pi\)
−0.852695 + 0.522409i \(0.825033\pi\)
\(42\) 0 0
\(43\) 18.1908 + 31.5074i 0.423043 + 0.732731i 0.996235 0.0866892i \(-0.0276287\pi\)
−0.573193 + 0.819421i \(0.694295\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −52.8629 + 30.5204i −1.12474 + 0.649370i −0.942607 0.333904i \(-0.891634\pi\)
−0.182134 + 0.983274i \(0.558301\pi\)
\(48\) 0 0
\(49\) −52.1330 + 90.2970i −1.06394 + 1.84280i
\(50\) 0 0
\(51\) −18.5039 + 66.1992i −0.362822 + 1.29802i
\(52\) 0 0
\(53\) 13.9872i 0.263909i −0.991256 0.131954i \(-0.957875\pi\)
0.991256 0.131954i \(-0.0421252\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 94.9780 + 26.5482i 1.66628 + 0.465757i
\(58\) 0 0
\(59\) 50.2415 + 29.0069i 0.851550 + 0.491643i 0.861174 0.508311i \(-0.169730\pi\)
−0.00962347 + 0.999954i \(0.503063\pi\)
\(60\) 0 0
\(61\) 27.5253 + 47.6752i 0.451234 + 0.781561i 0.998463 0.0554222i \(-0.0176505\pi\)
−0.547229 + 0.836983i \(0.684317\pi\)
\(62\) 0 0
\(63\) 2.39828 111.395i 0.0380680 1.76817i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −33.4960 + 58.0167i −0.499940 + 0.865921i −1.00000 6.94337e-5i \(-0.999978\pi\)
0.500060 + 0.865991i \(0.333311\pi\)
\(68\) 0 0
\(69\) 1.67264 + 1.70904i 0.0242412 + 0.0247687i
\(70\) 0 0
\(71\) 70.6560i 0.995156i 0.867419 + 0.497578i \(0.165777\pi\)
−0.867419 + 0.497578i \(0.834223\pi\)
\(72\) 0 0
\(73\) −23.1498 −0.317121 −0.158561 0.987349i \(-0.550685\pi\)
−0.158561 + 0.987349i \(0.550685\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 49.6294 + 28.6536i 0.644538 + 0.372124i
\(78\) 0 0
\(79\) −14.5725 25.2404i −0.184462 0.319498i 0.758933 0.651169i \(-0.225721\pi\)
−0.943395 + 0.331671i \(0.892388\pi\)
\(80\) 0 0
\(81\) 37.4434 + 71.8261i 0.462264 + 0.886743i
\(82\) 0 0
\(83\) −112.958 + 65.2165i −1.36094 + 0.785741i −0.989749 0.142815i \(-0.954385\pi\)
−0.371193 + 0.928556i \(0.621051\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −115.196 + 29.5415i −1.32409 + 0.339558i
\(88\) 0 0
\(89\) 48.0873i 0.540307i 0.962817 + 0.270153i \(0.0870744\pi\)
−0.962817 + 0.270153i \(0.912926\pi\)
\(90\) 0 0
\(91\) 15.6425 0.171896
\(92\) 0 0
\(93\) 3.13120 3.06452i 0.0336689 0.0329518i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.68803 16.7802i −0.0998766 0.172991i 0.811757 0.583996i \(-0.198511\pi\)
−0.911633 + 0.411004i \(0.865178\pi\)
\(98\) 0 0
\(99\) −41.6512 0.896733i −0.420719 0.00905791i
\(100\) 0 0
\(101\) −112.152 + 64.7508i −1.11041 + 0.641097i −0.938936 0.344092i \(-0.888187\pi\)
−0.171476 + 0.985188i \(0.554854\pi\)
\(102\) 0 0
\(103\) 4.20123 7.27674i 0.0407886 0.0706480i −0.844910 0.534908i \(-0.820346\pi\)
0.885699 + 0.464260i \(0.153680\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 61.1389i 0.571391i −0.958320 0.285696i \(-0.907775\pi\)
0.958320 0.285696i \(-0.0922246\pi\)
\(108\) 0 0
\(109\) 161.771 1.48414 0.742068 0.670324i \(-0.233845\pi\)
0.742068 + 0.670324i \(0.233845\pi\)
\(110\) 0 0
\(111\) −129.983 36.3327i −1.17102 0.327322i
\(112\) 0 0
\(113\) −29.6592 17.1238i −0.262471 0.151538i 0.362990 0.931793i \(-0.381756\pi\)
−0.625461 + 0.780255i \(0.715089\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −9.96833 + 5.47258i −0.0851994 + 0.0467742i
\(118\) 0 0
\(119\) −245.652 + 141.827i −2.06430 + 1.19183i
\(120\) 0 0
\(121\) −49.7862 + 86.2323i −0.411457 + 0.712664i
\(122\) 0 0
\(123\) 82.1187 + 83.9057i 0.667632 + 0.682160i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 104.320 0.821421 0.410710 0.911766i \(-0.365281\pi\)
0.410710 + 0.911766i \(0.365281\pi\)
\(128\) 0 0
\(129\) −27.1125 105.724i −0.210174 0.819565i
\(130\) 0 0
\(131\) 190.209 + 109.817i 1.45198 + 0.838299i 0.998594 0.0530175i \(-0.0168839\pi\)
0.453382 + 0.891316i \(0.350217\pi\)
\(132\) 0 0
\(133\) 203.484 + 352.445i 1.52996 + 2.64996i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −164.090 + 94.7374i −1.19774 + 0.691514i −0.960050 0.279828i \(-0.909722\pi\)
−0.237687 + 0.971342i \(0.576389\pi\)
\(138\) 0 0
\(139\) 110.020 190.560i 0.791511 1.37094i −0.133520 0.991046i \(-0.542628\pi\)
0.925031 0.379892i \(-0.124039\pi\)
\(140\) 0 0
\(141\) 177.382 45.4890i 1.25803 0.322617i
\(142\) 0 0
\(143\) 5.84885i 0.0409010i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 223.549 218.788i 1.52074 1.48835i
\(148\) 0 0
\(149\) 9.00402 + 5.19847i 0.0604297 + 0.0348891i 0.529910 0.848054i \(-0.322226\pi\)
−0.469481 + 0.882943i \(0.655559\pi\)
\(150\) 0 0
\(151\) 40.8811 + 70.8081i 0.270736 + 0.468928i 0.969050 0.246863i \(-0.0793998\pi\)
−0.698315 + 0.715791i \(0.746066\pi\)
\(152\) 0 0
\(153\) 106.925 176.322i 0.698856 1.15243i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −66.1558 + 114.585i −0.421375 + 0.729842i −0.996074 0.0885222i \(-0.971786\pi\)
0.574700 + 0.818364i \(0.305119\pi\)
\(158\) 0 0
\(159\) −11.2960 + 40.4124i −0.0710443 + 0.254166i
\(160\) 0 0
\(161\) 9.86836i 0.0612942i
\(162\) 0 0
\(163\) 81.7721 0.501669 0.250835 0.968030i \(-0.419295\pi\)
0.250835 + 0.968030i \(0.419295\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 23.9909 + 13.8512i 0.143658 + 0.0829411i 0.570106 0.821571i \(-0.306902\pi\)
−0.426448 + 0.904512i \(0.640235\pi\)
\(168\) 0 0
\(169\) 83.7017 + 144.976i 0.495277 + 0.857844i
\(170\) 0 0
\(171\) −252.975 153.409i −1.47939 0.897127i
\(172\) 0 0
\(173\) −241.288 + 139.308i −1.39473 + 0.805246i −0.993834 0.110878i \(-0.964634\pi\)
−0.400894 + 0.916125i \(0.631300\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −121.734 124.383i −0.687764 0.702731i
\(178\) 0 0
\(179\) 175.928i 0.982839i −0.870923 0.491419i \(-0.836478\pi\)
0.870923 0.491419i \(-0.163522\pi\)
\(180\) 0 0
\(181\) −180.887 −0.999377 −0.499689 0.866205i \(-0.666552\pi\)
−0.499689 + 0.866205i \(0.666552\pi\)
\(182\) 0 0
\(183\) −41.0250 159.975i −0.224180 0.874181i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 53.0302 + 91.8509i 0.283584 + 0.491181i
\(188\) 0 0
\(189\) −96.8917 + 319.911i −0.512655 + 1.69265i
\(190\) 0 0
\(191\) 120.178 69.3846i 0.629202 0.363270i −0.151241 0.988497i \(-0.548327\pi\)
0.780443 + 0.625227i \(0.214994\pi\)
\(192\) 0 0
\(193\) 89.4876 154.997i 0.463666 0.803093i −0.535474 0.844552i \(-0.679867\pi\)
0.999140 + 0.0414582i \(0.0132004\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 136.877i 0.694809i −0.937715 0.347405i \(-0.887063\pi\)
0.937715 0.347405i \(-0.112937\pi\)
\(198\) 0 0
\(199\) −11.4634 −0.0576050 −0.0288025 0.999585i \(-0.509169\pi\)
−0.0288025 + 0.999585i \(0.509169\pi\)
\(200\) 0 0
\(201\) 143.633 140.574i 0.714590 0.699371i
\(202\) 0 0
\(203\) −425.010 245.380i −2.09365 1.20877i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −3.45247 6.28869i −0.0166786 0.0303801i
\(208\) 0 0
\(209\) 131.781 76.0840i 0.630533 0.364038i
\(210\) 0 0
\(211\) 38.7228 67.0698i 0.183520 0.317867i −0.759557 0.650441i \(-0.774584\pi\)
0.943077 + 0.332575i \(0.107917\pi\)
\(212\) 0 0
\(213\) 57.0619 204.143i 0.267896 0.958419i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 18.0802 0.0833190
\(218\) 0 0
\(219\) 66.8857 + 18.6958i 0.305414 + 0.0853691i
\(220\) 0 0
\(221\) 25.0716 + 14.4751i 0.113446 + 0.0654982i
\(222\) 0 0
\(223\) 72.1175 + 124.911i 0.323397 + 0.560140i 0.981187 0.193062i \(-0.0618417\pi\)
−0.657790 + 0.753202i \(0.728508\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 42.5248 24.5517i 0.187334 0.108157i −0.403400 0.915024i \(-0.632172\pi\)
0.590734 + 0.806867i \(0.298838\pi\)
\(228\) 0 0
\(229\) −50.4902 + 87.4515i −0.220481 + 0.381884i −0.954954 0.296753i \(-0.904096\pi\)
0.734473 + 0.678638i \(0.237429\pi\)
\(230\) 0 0
\(231\) −120.251 122.868i −0.520568 0.531897i
\(232\) 0 0
\(233\) 120.466i 0.517021i −0.966009 0.258510i \(-0.916768\pi\)
0.966009 0.258510i \(-0.0832316\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 21.7196 + 84.6946i 0.0916438 + 0.357361i
\(238\) 0 0
\(239\) −157.093 90.6979i −0.657294 0.379489i 0.133951 0.990988i \(-0.457234\pi\)
−0.791245 + 0.611499i \(0.790567\pi\)
\(240\) 0 0
\(241\) −81.1538 140.562i −0.336738 0.583247i 0.647079 0.762423i \(-0.275990\pi\)
−0.983817 + 0.179176i \(0.942657\pi\)
\(242\) 0 0
\(243\) −50.1765 237.763i −0.206488 0.978449i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 20.7679 35.9710i 0.0840805 0.145632i
\(248\) 0 0
\(249\) 379.034 97.2017i 1.52222 0.390368i
\(250\) 0 0
\(251\) 445.457i 1.77473i −0.461069 0.887364i \(-0.652534\pi\)
0.461069 0.887364i \(-0.347466\pi\)
\(252\) 0 0
\(253\) 3.68984 0.0145844
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −214.270 123.709i −0.833734 0.481357i 0.0213954 0.999771i \(-0.493189\pi\)
−0.855129 + 0.518414i \(0.826522\pi\)
\(258\) 0 0
\(259\) −278.480 482.341i −1.07521 1.86232i
\(260\) 0 0
\(261\) 356.688 + 7.67933i 1.36662 + 0.0294227i
\(262\) 0 0
\(263\) −74.4992 + 43.0121i −0.283267 + 0.163544i −0.634901 0.772593i \(-0.718959\pi\)
0.351635 + 0.936137i \(0.385626\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 38.8353 138.936i 0.145451 0.520361i
\(268\) 0 0
\(269\) 368.618i 1.37033i −0.728389 0.685164i \(-0.759731\pi\)
0.728389 0.685164i \(-0.240269\pi\)
\(270\) 0 0
\(271\) −130.740 −0.482434 −0.241217 0.970471i \(-0.577546\pi\)
−0.241217 + 0.970471i \(0.577546\pi\)
\(272\) 0 0
\(273\) −45.1953 12.6329i −0.165550 0.0462745i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −108.635 188.162i −0.392185 0.679284i 0.600553 0.799585i \(-0.294947\pi\)
−0.992737 + 0.120301i \(0.961614\pi\)
\(278\) 0 0
\(279\) −11.5217 + 6.32540i −0.0412966 + 0.0226717i
\(280\) 0 0
\(281\) 149.240 86.1636i 0.531102 0.306632i −0.210363 0.977623i \(-0.567465\pi\)
0.741465 + 0.670991i \(0.234131\pi\)
\(282\) 0 0
\(283\) −218.079 + 377.723i −0.770596 + 1.33471i 0.166640 + 0.986018i \(0.446708\pi\)
−0.937237 + 0.348694i \(0.886625\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 484.489i 1.68811i
\(288\) 0 0
\(289\) −235.970 −0.816504
\(290\) 0 0
\(291\) 14.4395 + 56.3062i 0.0496203 + 0.193492i
\(292\) 0 0
\(293\) 25.1499 + 14.5203i 0.0858359 + 0.0495574i 0.542304 0.840183i \(-0.317552\pi\)
−0.456468 + 0.889740i \(0.650886\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 119.617 + 36.2284i 0.402750 + 0.121981i
\(298\) 0 0
\(299\) 0.872243 0.503590i 0.00291720 0.00168425i
\(300\) 0 0
\(301\) 225.204 390.064i 0.748185 1.29589i
\(302\) 0 0
\(303\) 376.327 96.5076i 1.24200 0.318507i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 306.163 0.997273 0.498637 0.866811i \(-0.333834\pi\)
0.498637 + 0.866811i \(0.333834\pi\)
\(308\) 0 0
\(309\) −18.0151 + 17.6314i −0.0583013 + 0.0570596i
\(310\) 0 0
\(311\) −344.003 198.610i −1.10612 0.638618i −0.168298 0.985736i \(-0.553827\pi\)
−0.937821 + 0.347118i \(0.887160\pi\)
\(312\) 0 0
\(313\) 195.881 + 339.276i 0.625818 + 1.08395i 0.988382 + 0.151990i \(0.0485681\pi\)
−0.362564 + 0.931959i \(0.618099\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 140.112 80.8940i 0.441995 0.255186i −0.262448 0.964946i \(-0.584530\pi\)
0.704444 + 0.709760i \(0.251197\pi\)
\(318\) 0 0
\(319\) −91.7491 + 158.914i −0.287615 + 0.498164i
\(320\) 0 0
\(321\) −49.3758 + 176.646i −0.153819 + 0.550298i
\(322\) 0 0
\(323\) 753.190i 2.33186i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −467.397 130.646i −1.42935 0.399530i
\(328\) 0 0
\(329\) 654.446 + 377.844i 1.98920 + 1.14846i
\(330\) 0 0
\(331\) −118.594 205.411i −0.358291 0.620578i 0.629385 0.777094i \(-0.283307\pi\)
−0.987675 + 0.156516i \(0.949974\pi\)
\(332\) 0 0
\(333\) 346.211 + 209.949i 1.03967 + 0.630476i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −211.024 + 365.505i −0.626185 + 1.08458i 0.362125 + 0.932129i \(0.382051\pi\)
−0.988310 + 0.152455i \(0.951282\pi\)
\(338\) 0 0
\(339\) 71.8638 + 73.4276i 0.211988 + 0.216601i
\(340\) 0 0
\(341\) 6.76031i 0.0198250i
\(342\) 0 0
\(343\) 684.196 1.99474
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −478.474 276.247i −1.37889 0.796102i −0.386863 0.922137i \(-0.626441\pi\)
−0.992026 + 0.126036i \(0.959775\pi\)
\(348\) 0 0
\(349\) 198.589 + 343.966i 0.569022 + 0.985575i 0.996663 + 0.0816263i \(0.0260114\pi\)
−0.427641 + 0.903949i \(0.640655\pi\)
\(350\) 0 0
\(351\) 33.2207 7.76123i 0.0946458 0.0221118i
\(352\) 0 0
\(353\) −267.320 + 154.337i −0.757281 + 0.437216i −0.828319 0.560257i \(-0.810702\pi\)
0.0710376 + 0.997474i \(0.477369\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 824.291 211.386i 2.30894 0.592118i
\(358\) 0 0
\(359\) 278.225i 0.775000i −0.921870 0.387500i \(-0.873339\pi\)
0.921870 0.387500i \(-0.126661\pi\)
\(360\) 0 0
\(361\) 719.626 1.99342
\(362\) 0 0
\(363\) 213.486 208.940i 0.588117 0.575591i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 304.122 + 526.754i 0.828669 + 1.43530i 0.899082 + 0.437780i \(0.144235\pi\)
−0.0704128 + 0.997518i \(0.522432\pi\)
\(368\) 0 0
\(369\) −169.499 308.744i −0.459348 0.836704i
\(370\) 0 0
\(371\) −149.963 + 86.5810i −0.404212 + 0.233372i
\(372\) 0 0
\(373\) −179.439 + 310.798i −0.481070 + 0.833238i −0.999764 0.0217222i \(-0.993085\pi\)
0.518694 + 0.854960i \(0.326418\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 50.0877i 0.132859i
\(378\) 0 0
\(379\) −353.208 −0.931948 −0.465974 0.884798i \(-0.654296\pi\)
−0.465974 + 0.884798i \(0.654296\pi\)
\(380\) 0 0
\(381\) −301.408 84.2493i −0.791097 0.221127i
\(382\) 0 0
\(383\) −235.029 135.694i −0.613653 0.354293i 0.160741 0.986997i \(-0.448612\pi\)
−0.774394 + 0.632704i \(0.781945\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −7.04790 + 327.359i −0.0182116 + 0.845889i
\(388\) 0 0
\(389\) −227.502 + 131.349i −0.584839 + 0.337657i −0.763054 0.646334i \(-0.776301\pi\)
0.178215 + 0.983992i \(0.442968\pi\)
\(390\) 0 0
\(391\) −9.13186 + 15.8169i −0.0233552 + 0.0404523i
\(392\) 0 0
\(393\) −460.873 470.902i −1.17270 1.19822i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −53.4516 −0.134639 −0.0673194 0.997731i \(-0.521445\pi\)
−0.0673194 + 0.997731i \(0.521445\pi\)
\(398\) 0 0
\(399\) −303.282 1182.64i −0.760106 2.96400i
\(400\) 0 0
\(401\) −160.143 92.4584i −0.399358 0.230570i 0.286849 0.957976i \(-0.407392\pi\)
−0.686207 + 0.727406i \(0.740726\pi\)
\(402\) 0 0
\(403\) −0.922647 1.59807i −0.00228945 0.00396544i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −180.350 + 104.125i −0.443122 + 0.255836i
\(408\) 0 0
\(409\) 116.052 201.008i 0.283746 0.491463i −0.688558 0.725181i \(-0.741756\pi\)
0.972304 + 0.233718i \(0.0750893\pi\)
\(410\) 0 0
\(411\) 550.607 141.201i 1.33968 0.343555i
\(412\) 0 0
\(413\) 718.215i 1.73902i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −471.772 + 461.725i −1.13135 + 1.10725i
\(418\) 0 0
\(419\) 336.098 + 194.046i 0.802143 + 0.463117i 0.844220 0.535997i \(-0.180064\pi\)
−0.0420770 + 0.999114i \(0.513397\pi\)
\(420\) 0 0
\(421\) −77.0016 133.371i −0.182902 0.316795i 0.759966 0.649963i \(-0.225216\pi\)
−0.942867 + 0.333168i \(0.891882\pi\)
\(422\) 0 0
\(423\) −549.240 11.8249i −1.29844 0.0279548i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 340.765 590.222i 0.798044 1.38225i
\(428\) 0 0
\(429\) −4.72354 + 16.8988i −0.0110106 + 0.0393912i
\(430\) 0 0
\(431\) 238.147i 0.552544i −0.961079 0.276272i \(-0.910901\pi\)
0.961079 0.276272i \(-0.0890991\pi\)
\(432\) 0 0
\(433\) −523.956 −1.21006 −0.605030 0.796203i \(-0.706839\pi\)
−0.605030 + 0.796203i \(0.706839\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 22.6929 + 13.1018i 0.0519289 + 0.0299812i
\(438\) 0 0
\(439\) 228.172 + 395.206i 0.519755 + 0.900242i 0.999736 + 0.0229630i \(0.00730999\pi\)
−0.479982 + 0.877279i \(0.659357\pi\)
\(440\) 0 0
\(441\) −822.584 + 451.596i −1.86527 + 1.02403i
\(442\) 0 0
\(443\) 235.368 135.890i 0.531305 0.306749i −0.210242 0.977649i \(-0.567425\pi\)
0.741548 + 0.670900i \(0.234092\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −21.8166 22.2914i −0.0488067 0.0498688i
\(448\) 0 0
\(449\) 386.192i 0.860116i 0.902801 + 0.430058i \(0.141507\pi\)
−0.902801 + 0.430058i \(0.858493\pi\)
\(450\) 0 0
\(451\) 181.153 0.401671
\(452\) 0 0
\(453\) −60.9311 237.598i −0.134506 0.524499i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.24416 + 3.88700i 0.00491063 + 0.00850546i 0.868470 0.495741i \(-0.165104\pi\)
−0.863560 + 0.504247i \(0.831770\pi\)
\(458\) 0 0
\(459\) −451.331 + 423.087i −0.983293 + 0.921759i
\(460\) 0 0
\(461\) −283.502 + 163.680i −0.614972 + 0.355054i −0.774909 0.632073i \(-0.782204\pi\)
0.159937 + 0.987127i \(0.448871\pi\)
\(462\) 0 0
\(463\) 106.071 183.720i 0.229094 0.396803i −0.728446 0.685103i \(-0.759757\pi\)
0.957540 + 0.288301i \(0.0930903\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 471.246i 1.00909i 0.863385 + 0.504546i \(0.168340\pi\)
−0.863385 + 0.504546i \(0.831660\pi\)
\(468\) 0 0
\(469\) 829.364 1.76837
\(470\) 0 0
\(471\) 283.680 277.638i 0.602293 0.589465i
\(472\) 0 0
\(473\) −145.847 84.2051i −0.308346 0.178023i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 65.2742 107.639i 0.136843 0.225658i
\(478\) 0 0
\(479\) −762.164 + 440.035i −1.59116 + 0.918654i −0.598047 + 0.801461i \(0.704056\pi\)
−0.993109 + 0.117193i \(0.962610\pi\)
\(480\) 0 0
\(481\) −28.4221 + 49.2285i −0.0590895 + 0.102346i
\(482\) 0 0
\(483\) 7.96970 28.5122i 0.0165004 0.0590315i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −606.421 −1.24522 −0.622609 0.782533i \(-0.713927\pi\)
−0.622609 + 0.782533i \(0.713927\pi\)
\(488\) 0 0
\(489\) −236.260 66.0392i −0.483150 0.135050i
\(490\) 0 0
\(491\) 488.016 + 281.756i 0.993922 + 0.573841i 0.906444 0.422325i \(-0.138786\pi\)
0.0874776 + 0.996166i \(0.472119\pi\)
\(492\) 0 0
\(493\) −454.133 786.582i −0.921163 1.59550i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 757.535 437.363i 1.52422 0.880006i
\(498\) 0 0
\(499\) 35.9407 62.2511i 0.0720254 0.124752i −0.827763 0.561077i \(-0.810387\pi\)
0.899789 + 0.436326i \(0.143720\pi\)
\(500\) 0 0
\(501\) −58.1296 59.3946i −0.116027 0.118552i
\(502\) 0 0
\(503\) 124.538i 0.247591i 0.992308 + 0.123795i \(0.0395067\pi\)
−0.992308 + 0.123795i \(0.960493\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −124.753 486.469i −0.246061 0.959505i
\(508\) 0 0
\(509\) 466.058 + 269.079i 0.915634 + 0.528642i 0.882240 0.470801i \(-0.156035\pi\)
0.0333945 + 0.999442i \(0.489368\pi\)
\(510\) 0 0
\(511\) 143.298 + 248.200i 0.280427 + 0.485714i
\(512\) 0 0
\(513\) 607.016 + 647.539i 1.18327 + 1.26226i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 141.278 244.701i 0.273266 0.473310i
\(518\) 0 0
\(519\) 809.647 207.631i 1.56001 0.400059i
\(520\) 0 0
\(521\) 147.767i 0.283622i 0.989894 + 0.141811i \(0.0452925\pi\)
−0.989894 + 0.141811i \(0.954707\pi\)
\(522\) 0 0
\(523\) 701.166 1.34066 0.670331 0.742063i \(-0.266152\pi\)
0.670331 + 0.742063i \(0.266152\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 28.9787 + 16.7308i 0.0549880 + 0.0317473i
\(528\) 0 0
\(529\) −264.182 457.577i −0.499399 0.864985i
\(530\) 0 0
\(531\) 251.269 + 457.688i 0.473200 + 0.861935i
\(532\) 0 0
\(533\) 42.8229 24.7238i 0.0803431 0.0463861i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −142.080 + 508.301i −0.264580 + 0.946556i
\(538\) 0 0
\(539\) 482.646i 0.895446i
\(540\) 0 0
\(541\) −573.411 −1.05991 −0.529955 0.848026i \(-0.677791\pi\)
−0.529955 + 0.848026i \(0.677791\pi\)
\(542\) 0 0
\(543\) 522.629 + 146.085i 0.962484 + 0.269033i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 152.510 + 264.155i 0.278812 + 0.482916i 0.971090 0.238715i \(-0.0767261\pi\)
−0.692278 + 0.721631i \(0.743393\pi\)
\(548\) 0 0
\(549\) −10.6645 + 495.341i −0.0194253 + 0.902260i
\(550\) 0 0
\(551\) −1128.53 + 651.559i −2.04815 + 1.18250i
\(552\) 0 0
\(553\) −180.409 + 312.477i −0.326237 + 0.565058i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 686.875i 1.23317i −0.787288 0.616585i \(-0.788516\pi\)
0.787288 0.616585i \(-0.211484\pi\)
\(558\) 0 0
\(559\) −45.9692 −0.0822348
\(560\) 0 0
\(561\) −79.0386 308.208i −0.140889 0.549390i
\(562\) 0 0
\(563\) 402.756 + 232.531i 0.715374 + 0.413021i 0.813048 0.582197i \(-0.197807\pi\)
−0.0976736 + 0.995219i \(0.531140\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 538.305 846.053i 0.949391 1.49216i
\(568\) 0 0
\(569\) −954.086 + 550.842i −1.67678 + 0.968088i −0.713084 + 0.701079i \(0.752702\pi\)
−0.963694 + 0.267009i \(0.913965\pi\)
\(570\) 0 0
\(571\) −381.027 + 659.958i −0.667298 + 1.15579i 0.311359 + 0.950292i \(0.399216\pi\)
−0.978657 + 0.205501i \(0.934118\pi\)
\(572\) 0 0
\(573\) −403.259 + 103.414i −0.703767 + 0.180478i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −673.918 −1.16797 −0.583985 0.811765i \(-0.698507\pi\)
−0.583985 + 0.811765i \(0.698507\pi\)
\(578\) 0 0
\(579\) −383.728 + 375.555i −0.662743 + 0.648628i
\(580\) 0 0
\(581\) 1398.43 + 807.384i 2.40694 + 1.38965i
\(582\) 0 0
\(583\) 32.3732 + 56.0720i 0.0555286 + 0.0961784i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −337.196 + 194.680i −0.574439 + 0.331653i −0.758920 0.651183i \(-0.774273\pi\)
0.184481 + 0.982836i \(0.440939\pi\)
\(588\) 0 0
\(589\) 24.0043 41.5766i 0.0407543 0.0705885i
\(590\) 0 0
\(591\) −110.542 + 395.473i −0.187043 + 0.669160i
\(592\) 0 0
\(593\) 640.797i 1.08060i 0.841472 + 0.540301i \(0.181690\pi\)
−0.841472 + 0.540301i \(0.818310\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 33.1206 + 9.25784i 0.0554785 + 0.0155073i
\(598\) 0 0
\(599\) −403.936 233.212i −0.674350 0.389336i 0.123373 0.992360i \(-0.460629\pi\)
−0.797723 + 0.603024i \(0.793962\pi\)
\(600\) 0 0
\(601\) 493.215 + 854.273i 0.820657 + 1.42142i 0.905194 + 0.425000i \(0.139726\pi\)
−0.0845364 + 0.996420i \(0.526941\pi\)
\(602\) 0 0
\(603\) −528.518 + 290.155i −0.876482 + 0.481185i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 434.982 753.411i 0.716610 1.24120i −0.245725 0.969339i \(-0.579026\pi\)
0.962335 0.271865i \(-0.0876405\pi\)
\(608\) 0 0
\(609\) 1029.79 + 1052.20i 1.69096 + 1.72776i
\(610\) 0 0
\(611\) 77.1267i 0.126230i
\(612\) 0 0
\(613\) 376.400 0.614029 0.307014 0.951705i \(-0.400670\pi\)
0.307014 + 0.951705i \(0.400670\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −100.567 58.0623i −0.162993 0.0941041i 0.416285 0.909234i \(-0.363332\pi\)
−0.579278 + 0.815130i \(0.696665\pi\)
\(618\) 0 0
\(619\) 119.765 + 207.439i 0.193482 + 0.335120i 0.946402 0.322992i \(-0.104689\pi\)
−0.752920 + 0.658112i \(0.771355\pi\)
\(620\) 0 0
\(621\) 4.89630 + 20.9578i 0.00788455 + 0.0337485i
\(622\) 0 0
\(623\) 515.566 297.662i 0.827553 0.477788i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −442.195 + 113.399i −0.705255 + 0.180860i
\(628\) 0 0
\(629\) 1030.78i 1.63877i
\(630\) 0 0
\(631\) −590.863 −0.936392 −0.468196 0.883625i \(-0.655096\pi\)
−0.468196 + 0.883625i \(0.655096\pi\)
\(632\) 0 0
\(633\) −166.046 + 162.509i −0.262315 + 0.256729i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −65.8715 114.093i −0.103409 0.179109i
\(638\) 0 0
\(639\) −329.732 + 543.738i −0.516013 + 0.850920i
\(640\) 0 0
\(641\) 663.399 383.014i 1.03494 0.597525i 0.116547 0.993185i \(-0.462817\pi\)
0.918397 + 0.395660i \(0.129484\pi\)
\(642\) 0 0
\(643\) −38.0263 + 65.8635i −0.0591389 + 0.102432i −0.894079 0.447909i \(-0.852169\pi\)
0.834940 + 0.550341i \(0.185502\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 287.130i 0.443786i −0.975071 0.221893i \(-0.928776\pi\)
0.975071 0.221893i \(-0.0712236\pi\)
\(648\) 0 0
\(649\) −268.545 −0.413783
\(650\) 0 0
\(651\) −52.2383 14.6016i −0.0802432 0.0224295i
\(652\) 0 0
\(653\) 194.364 + 112.216i 0.297647 + 0.171847i 0.641386 0.767219i \(-0.278360\pi\)
−0.343738 + 0.939066i \(0.611693\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −178.151 108.034i −0.271158 0.164435i
\(658\) 0 0
\(659\) 942.931 544.401i 1.43085 0.826102i 0.433665 0.901074i \(-0.357220\pi\)
0.997186 + 0.0749720i \(0.0238867\pi\)
\(660\) 0 0
\(661\) −439.376 + 761.021i −0.664714 + 1.15132i 0.314649 + 0.949208i \(0.398113\pi\)
−0.979363 + 0.202110i \(0.935220\pi\)
\(662\) 0 0
\(663\) −60.7481 62.0701i −0.0916262 0.0936201i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −31.5986 −0.0473743
\(668\) 0 0
\(669\) −107.487 419.142i −0.160669 0.626521i
\(670\) 0 0
\(671\) −220.688 127.414i −0.328894 0.189887i
\(672\) 0 0
\(673\) −278.142 481.757i −0.413287 0.715835i 0.581960 0.813218i \(-0.302286\pi\)
−0.995247 + 0.0973831i \(0.968953\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −486.720 + 281.008i −0.718937 + 0.415078i −0.814361 0.580358i \(-0.802913\pi\)
0.0954244 + 0.995437i \(0.469579\pi\)
\(678\) 0 0
\(679\) −119.938 + 207.739i −0.176640 + 0.305949i
\(680\) 0 0
\(681\) −142.693 + 36.5930i −0.209534 + 0.0537342i
\(682\) 0 0
\(683\) 500.931i 0.733427i −0.930334 0.366714i \(-0.880483\pi\)
0.930334 0.366714i \(-0.119517\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 216.505 211.894i 0.315145 0.308433i
\(688\) 0 0
\(689\) 15.3054 + 8.83658i 0.0222139 + 0.0128252i
\(690\) 0 0
\(691\) −361.250 625.703i −0.522793 0.905504i −0.999648 0.0265221i \(-0.991557\pi\)
0.476855 0.878982i \(-0.341777\pi\)
\(692\) 0 0
\(693\) 248.208 + 452.112i 0.358165 + 0.652399i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −448.330 + 776.530i −0.643228 + 1.11410i
\(698\) 0 0
\(699\) −97.2883 + 348.056i −0.139182 + 0.497934i
\(700\) 0 0
\(701\) 788.759i 1.12519i 0.826732 + 0.562595i \(0.190197\pi\)
−0.826732 + 0.562595i \(0.809803\pi\)
\(702\) 0 0
\(703\) −1478.90 −2.10370
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1388.44 + 801.618i 1.96385 + 1.13383i
\(708\) 0 0
\(709\) −199.643 345.792i −0.281584 0.487718i 0.690191 0.723627i \(-0.257526\pi\)
−0.971775 + 0.235910i \(0.924193\pi\)
\(710\) 0 0
\(711\) 5.64602 262.245i 0.00794095 0.368839i
\(712\) 0 0
\(713\) 1.00817 0.582067i 0.00141398 0.000816364i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 380.635 + 388.918i 0.530871 + 0.542424i
\(718\) 0 0
\(719\) 844.389i 1.17439i −0.809444 0.587197i \(-0.800232\pi\)
0.809444 0.587197i \(-0.199768\pi\)
\(720\) 0 0
\(721\) −104.023 −0.144276
\(722\) 0 0
\(723\) 120.955 + 471.660i 0.167297 + 0.652365i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 202.857 + 351.359i 0.279033 + 0.483300i 0.971145 0.238491i \(-0.0766526\pi\)
−0.692111 + 0.721791i \(0.743319\pi\)
\(728\) 0 0
\(729\) −47.0452 + 727.480i −0.0645339 + 0.997916i
\(730\) 0 0
\(731\) 721.905 416.792i 0.987559 0.570167i
\(732\) 0 0
\(733\) 602.495 1043.55i 0.821958 1.42367i −0.0822636 0.996611i \(-0.526215\pi\)
0.904222 0.427063i \(-0.140452\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 310.105i 0.420766i
\(738\) 0 0
\(739\) −539.145 −0.729561 −0.364780 0.931094i \(-0.618856\pi\)
−0.364780 + 0.931094i \(0.618856\pi\)
\(740\) 0 0
\(741\) −89.0539 + 87.1572i −0.120181 + 0.117621i
\(742\) 0 0
\(743\) 269.337 + 155.502i 0.362500 + 0.209289i 0.670177 0.742202i \(-0.266218\pi\)
−0.307677 + 0.951491i \(0.599552\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −1173.62 25.2676i −1.57112 0.0338255i
\(748\) 0 0
\(749\) −655.497 + 378.452i −0.875163 + 0.505276i
\(750\) 0 0
\(751\) −463.288 + 802.439i −0.616895 + 1.06849i 0.373153 + 0.927770i \(0.378276\pi\)
−0.990049 + 0.140724i \(0.955057\pi\)
\(752\) 0 0
\(753\) −359.751 + 1287.04i −0.477757 + 1.70921i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −450.364 −0.594933 −0.297467 0.954732i \(-0.596142\pi\)
−0.297467 + 0.954732i \(0.596142\pi\)
\(758\) 0 0
\(759\) −10.6609 2.97992i −0.0140460 0.00392612i
\(760\) 0 0
\(761\) −581.319 335.624i −0.763888 0.441031i 0.0668021 0.997766i \(-0.478720\pi\)
−0.830690 + 0.556735i \(0.812054\pi\)
\(762\) 0 0
\(763\) −1001.37 1734.42i −1.31241 2.27316i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −63.4815 + 36.6510i −0.0827659 + 0.0477849i
\(768\) 0 0
\(769\) −37.3520 + 64.6955i −0.0485721 + 0.0841294i −0.889289 0.457345i \(-0.848800\pi\)
0.840717 + 0.541475i \(0.182134\pi\)
\(770\) 0 0
\(771\) 519.172 + 530.470i 0.673375 + 0.688028i
\(772\) 0 0
\(773\) 1226.34i 1.58647i −0.608917 0.793234i \(-0.708396\pi\)
0.608917 0.793234i \(-0.291604\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 415.060 + 1618.51i 0.534182 + 2.08302i
\(778\) 0 0
\(779\) 1114.11 + 643.233i 1.43018 + 0.825716i
\(780\) 0 0
\(781\) −163.533 283.247i −0.209389 0.362673i
\(782\) 0 0
\(783\) −1024.36 310.249i −1.30825 0.396231i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 129.596 224.466i 0.164670 0.285218i −0.771868 0.635783i \(-0.780677\pi\)
0.936538 + 0.350566i \(0.114011\pi\)
\(788\) 0 0
\(789\) 249.984 64.1073i 0.316836 0.0812513i
\(790\) 0 0
\(791\) 423.986i 0.536013i
\(792\) 0 0
\(793\) −69.5579 −0.0877149
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1286.14 + 742.554i 1.61373 + 0.931686i 0.988496 + 0.151247i \(0.0483287\pi\)
0.625231 + 0.780439i \(0.285005\pi\)
\(798\) 0 0
\(799\) 699.290 + 1211.21i 0.875206 + 1.51590i
\(800\) 0 0
\(801\) −224.410 + 370.059i −0.280163 + 0.461996i
\(802\) 0 0
\(803\) 92.8035 53.5801i 0.115571 0.0667249i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −297.696 + 1065.03i −0.368893 + 1.31974i
\(808\) 0 0
\(809\) 10.2026i 0.0126114i −0.999980 0.00630569i \(-0.997993\pi\)
0.999980 0.00630569i \(-0.00200718\pi\)
\(810\) 0 0
\(811\) 1113.80 1.37337 0.686685 0.726955i \(-0.259065\pi\)
0.686685 + 0.726955i \(0.259065\pi\)
\(812\) 0 0
\(813\) 377.739 + 105.585i 0.464624 + 0.129871i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −597.985 1035.74i −0.731928 1.26774i
\(818\) 0 0
\(819\) 120.378 + 72.9995i 0.146982 + 0.0891324i
\(820\) 0 0
\(821\) −266.290 + 153.742i −0.324348 + 0.187262i −0.653329 0.757074i \(-0.726628\pi\)
0.328981 + 0.944337i \(0.393295\pi\)
\(822\) 0 0
\(823\) −297.749 + 515.717i −0.361785 + 0.626630i −0.988255 0.152816i \(-0.951166\pi\)
0.626469 + 0.779446i \(0.284499\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1110.81i 1.34318i 0.740921 + 0.671592i \(0.234389\pi\)
−0.740921 + 0.671592i \(0.765611\pi\)
\(828\) 0 0
\(829\) −390.454 −0.470994 −0.235497 0.971875i \(-0.575672\pi\)
−0.235497 + 0.971875i \(0.575672\pi\)
\(830\) 0 0
\(831\) 161.915 + 631.381i 0.194844 + 0.759784i
\(832\) 0 0
\(833\) 2068.90 + 1194.48i 2.48368 + 1.43395i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 38.3976 8.97071i 0.0458753 0.0107177i
\(838\) 0 0
\(839\) −928.946 + 536.327i −1.10721 + 0.639246i −0.938104 0.346353i \(-0.887420\pi\)
−0.169102 + 0.985599i \(0.554087\pi\)
\(840\) 0 0
\(841\) 365.210 632.562i 0.434257 0.752155i
\(842\) 0 0
\(843\) −500.777 + 128.422i −0.594041 + 0.152340i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1232.71 1.45539
\(848\) 0 0
\(849\) 935.134 915.218i 1.10145 1.07800i
\(850\) 0 0
\(851\) −31.0566 17.9305i −0.0364942 0.0210700i
\(852\) 0 0
\(853\) 454.724 + 787.604i 0.533087 + 0.923334i 0.999253 + 0.0386371i \(0.0123016\pi\)
−0.466166 + 0.884697i \(0.654365\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 564.185 325.732i 0.658325 0.380084i −0.133313 0.991074i \(-0.542562\pi\)
0.791639 + 0.610990i \(0.209228\pi\)
\(858\) 0 0
\(859\) 230.239 398.786i 0.268032 0.464244i −0.700322 0.713827i \(-0.746960\pi\)
0.968353 + 0.249583i \(0.0802935\pi\)
\(860\) 0 0
\(861\) 391.273 1399.81i 0.454441 1.62580i
\(862\) 0 0
\(863\) 359.492i 0.416560i 0.978069 + 0.208280i \(0.0667866\pi\)
−0.978069 + 0.208280i \(0.933213\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 681.776 + 190.569i 0.786362 + 0.219803i
\(868\) 0 0
\(869\) 116.837 + 67.4560i 0.134450 + 0.0776249i
\(870\) 0 0
\(871\) −42.3231 73.3057i −0.0485914 0.0841627i
\(872\) 0 0
\(873\) 3.75356 174.344i 0.00429960 0.199707i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −569.520 + 986.437i −0.649395 + 1.12479i 0.333872 + 0.942618i \(0.391645\pi\)
−0.983268 + 0.182167i \(0.941689\pi\)
\(878\) 0 0
\(879\) −60.9379 62.2639i −0.0693263 0.0708350i
\(880\) 0 0
\(881\) 1612.41i 1.83020i −0.403227 0.915100i \(-0.632112\pi\)
0.403227 0.915100i \(-0.367888\pi\)
\(882\) 0 0
\(883\) 688.542 0.779775 0.389888 0.920862i \(-0.372514\pi\)
0.389888 + 0.920862i \(0.372514\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −974.792 562.797i −1.09898 0.634494i −0.163025 0.986622i \(-0.552125\pi\)
−0.935952 + 0.352128i \(0.885458\pi\)
\(888\) 0 0
\(889\) −645.747 1118.47i −0.726374 1.25812i
\(890\) 0 0
\(891\) −316.345 201.276i −0.355045 0.225899i
\(892\) 0 0
\(893\) 1737.75 1003.29i 1.94597 1.12351i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −2.92683 + 0.750574i −0.00326291 + 0.000836761i
\(898\) 0 0
\(899\) 57.8931i 0.0643972i
\(900\) 0 0
\(901\) −320.477 −0.355690
\(902\) 0 0
\(903\) −965.686 + 945.119i −1.06942 + 1.04664i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −311.993 540.388i −0.343984 0.595798i 0.641185 0.767387i \(-0.278443\pi\)
−0.985169 + 0.171589i \(0.945110\pi\)
\(908\) 0 0
\(909\) −1165.24 25.0872i −1.28190 0.0275987i
\(910\) 0 0
\(911\) −1321.03 + 762.700i −1.45009 + 0.837212i −0.998486 0.0550052i \(-0.982482\pi\)
−0.451607 + 0.892217i \(0.649149\pi\)
\(912\) 0 0
\(913\) 301.886 522.882i 0.330653 0.572708i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2719.09i 2.96520i
\(918\) 0 0
\(919\) 875.498 0.952663 0.476332 0.879266i \(-0.341966\pi\)
0.476332 + 0.879266i \(0.341966\pi\)
\(920\) 0 0
\(921\) −884.582 247.257i −0.960458 0.268466i
\(922\) 0 0
\(923\) −77.3152 44.6379i −0.0837651 0.0483618i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 66.2894 36.3926i 0.0715096 0.0392585i
\(928\) 0 0
\(929\) −341.363 + 197.086i −0.367452 + 0.212148i −0.672345 0.740238i \(-0.734713\pi\)
0.304893 + 0.952387i \(0.401379\pi\)
\(930\) 0 0
\(931\) 1713.76 2968.32i 1.84077 3.18831i
\(932\) 0 0
\(933\) 833.514 + 851.652i 0.893370 + 0.912811i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 600.533 0.640911 0.320455 0.947264i \(-0.396164\pi\)
0.320455 + 0.947264i \(0.396164\pi\)
\(938\) 0 0
\(939\) −291.950 1138.45i −0.310916 1.21240i
\(940\) 0 0
\(941\) −572.208 330.364i −0.608085 0.351078i 0.164131 0.986439i \(-0.447518\pi\)
−0.772216 + 0.635361i \(0.780851\pi\)
\(942\) 0 0
\(943\) 15.5974 + 27.0155i 0.0165402 + 0.0286485i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1096.03 + 632.795i −1.15737 + 0.668210i −0.950673 0.310194i \(-0.899606\pi\)
−0.206701 + 0.978404i \(0.566273\pi\)
\(948\) 0 0
\(949\) 14.6252 25.3316i 0.0154112 0.0266930i
\(950\) 0 0
\(951\) −470.150 + 120.568i −0.494375 + 0.126780i
\(952\) 0 0
\(953\) 428.958i 0.450113i 0.974346 + 0.225057i \(0.0722567\pi\)
−0.974346 + 0.225057i \(0.927743\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 393.426 385.046i 0.411103 0.402347i
\(958\) 0 0
\(959\) 2031.44 + 1172.85i 2.11829 + 1.22300i
\(960\) 0 0
\(961\) 479.434 + 830.403i 0.498890 + 0.864103i
\(962\) 0 0
\(963\) 285.318 470.498i 0.296281 0.488575i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 221.642 383.896i 0.229206 0.396996i −0.728367 0.685187i \(-0.759720\pi\)
0.957573 + 0.288191i \(0.0930537\pi\)
\(968\) 0 0
\(969\) 608.277 2176.16i 0.627737 2.24578i
\(970\) 0 0
\(971\) 444.042i 0.457304i −0.973508 0.228652i \(-0.926568\pi\)
0.973508 0.228652i \(-0.0734317\pi\)
\(972\) 0 0
\(973\) −2724.11 −2.79970
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −200.009 115.475i −0.204717 0.118194i 0.394137 0.919052i \(-0.371044\pi\)
−0.598854 + 0.800858i \(0.704377\pi\)
\(978\) 0 0
\(979\) −111.298 192.773i −0.113685 0.196908i
\(980\) 0 0
\(981\) 1244.92 + 754.940i 1.26903 + 0.769562i
\(982\) 0 0
\(983\) −971.023 + 560.620i −0.987816 + 0.570316i −0.904621 0.426218i \(-0.859846\pi\)
−0.0831949 + 0.996533i \(0.526512\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −1585.71 1620.22i −1.60660 1.64156i
\(988\) 0 0
\(989\) 29.0005i 0.0293230i
\(990\) 0 0
\(991\) 182.717 0.184377 0.0921884 0.995742i \(-0.470614\pi\)
0.0921884 + 0.995742i \(0.470614\pi\)
\(992\) 0 0
\(993\) 176.758 + 689.262i 0.178005 + 0.694121i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −616.618 1068.01i −0.618473 1.07123i −0.989764 0.142711i \(-0.954418\pi\)
0.371291 0.928517i \(-0.378915\pi\)
\(998\) 0 0
\(999\) −830.737 886.195i −0.831569 0.887082i
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.p.f.401.1 24
3.2 odd 2 2700.3.p.f.2501.1 24
5.2 odd 4 180.3.t.a.149.6 yes 24
5.3 odd 4 180.3.t.a.149.7 yes 24
5.4 even 2 inner 900.3.p.f.401.12 24
9.2 odd 6 inner 900.3.p.f.101.1 24
9.7 even 3 2700.3.p.f.1601.1 24
15.2 even 4 540.3.t.a.449.5 24
15.8 even 4 540.3.t.a.449.10 24
15.14 odd 2 2700.3.p.f.2501.12 24
45.2 even 12 180.3.t.a.29.7 yes 24
45.7 odd 12 540.3.t.a.89.10 24
45.13 odd 12 1620.3.b.b.809.22 24
45.22 odd 12 1620.3.b.b.809.4 24
45.23 even 12 1620.3.b.b.809.3 24
45.29 odd 6 inner 900.3.p.f.101.12 24
45.32 even 12 1620.3.b.b.809.21 24
45.34 even 6 2700.3.p.f.1601.12 24
45.38 even 12 180.3.t.a.29.6 24
45.43 odd 12 540.3.t.a.89.5 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.3.t.a.29.6 24 45.38 even 12
180.3.t.a.29.7 yes 24 45.2 even 12
180.3.t.a.149.6 yes 24 5.2 odd 4
180.3.t.a.149.7 yes 24 5.3 odd 4
540.3.t.a.89.5 24 45.43 odd 12
540.3.t.a.89.10 24 45.7 odd 12
540.3.t.a.449.5 24 15.2 even 4
540.3.t.a.449.10 24 15.8 even 4
900.3.p.f.101.1 24 9.2 odd 6 inner
900.3.p.f.101.12 24 45.29 odd 6 inner
900.3.p.f.401.1 24 1.1 even 1 trivial
900.3.p.f.401.12 24 5.4 even 2 inner
1620.3.b.b.809.3 24 45.23 even 12
1620.3.b.b.809.4 24 45.22 odd 12
1620.3.b.b.809.21 24 45.32 even 12
1620.3.b.b.809.22 24 45.13 odd 12
2700.3.p.f.1601.1 24 9.7 even 3
2700.3.p.f.1601.12 24 45.34 even 6
2700.3.p.f.2501.1 24 3.2 odd 2
2700.3.p.f.2501.12 24 15.14 odd 2