Properties

Label 3420.2.bj.e.1189.5
Level $3420$
Weight $2$
Character 3420.1189
Analytic conductor $27.309$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3420,2,Mod(1189,3420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3420, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3420.1189");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.bj (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: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1189.5
Character \(\chi\) \(=\) 3420.1189
Dual form 3420.2.bj.e.2629.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.42631 + 1.72210i) q^{5} +2.19629i q^{7} +O(q^{10})\) \(q+(-1.42631 + 1.72210i) q^{5} +2.19629i q^{7} +0.851781 q^{11} +(0.197307 - 0.113916i) q^{13} +(6.57835 + 3.79801i) q^{17} +(3.93036 + 1.88474i) q^{19} +(1.46423 - 0.845372i) q^{23} +(-0.931273 - 4.91251i) q^{25} +(0.798103 + 1.38236i) q^{29} +5.76406 q^{31} +(-3.78224 - 3.13259i) q^{35} -8.94456i q^{37} +(-3.54377 + 6.13800i) q^{41} +(-6.55871 - 3.78668i) q^{43} +(5.06032 - 2.92158i) q^{47} +2.17631 q^{49} +(-4.14208 + 2.39143i) q^{53} +(-1.21490 + 1.46685i) q^{55} +(-5.33889 + 9.24723i) q^{59} +(-1.17130 - 2.02874i) q^{61} +(-0.0852477 + 0.502263i) q^{65} +(3.70127 - 2.13693i) q^{67} +(-3.14788 + 5.45228i) q^{71} +(7.74621 + 4.47228i) q^{73} +1.87076i q^{77} +(-0.332949 + 0.576685i) q^{79} +8.71844i q^{83} +(-15.9233 + 5.91144i) q^{85} +(5.33889 + 9.24723i) q^{89} +(0.250192 + 0.433345i) q^{91} +(-8.85164 + 4.08026i) q^{95} +(8.06614 + 4.65699i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q+O(q^{10}) \) Copy content Toggle raw display \( 40 q - 8 q^{19} - 8 q^{31} - 16 q^{49} - 8 q^{61} - 8 q^{79} + 28 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.42631 + 1.72210i −0.637866 + 0.770148i
\(6\) 0 0
\(7\) 2.19629i 0.830120i 0.909794 + 0.415060i \(0.136239\pi\)
−0.909794 + 0.415060i \(0.863761\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.851781 0.256821 0.128411 0.991721i \(-0.459012\pi\)
0.128411 + 0.991721i \(0.459012\pi\)
\(12\) 0 0
\(13\) 0.197307 0.113916i 0.0547233 0.0315945i −0.472389 0.881390i \(-0.656608\pi\)
0.527112 + 0.849796i \(0.323275\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.57835 + 3.79801i 1.59548 + 0.921153i 0.992342 + 0.123519i \(0.0394181\pi\)
0.603142 + 0.797634i \(0.293915\pi\)
\(18\) 0 0
\(19\) 3.93036 + 1.88474i 0.901687 + 0.432389i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.46423 0.845372i 0.305313 0.176272i −0.339514 0.940601i \(-0.610263\pi\)
0.644827 + 0.764329i \(0.276929\pi\)
\(24\) 0 0
\(25\) −0.931273 4.91251i −0.186255 0.982502i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.798103 + 1.38236i 0.148204 + 0.256697i 0.930564 0.366130i \(-0.119317\pi\)
−0.782360 + 0.622827i \(0.785984\pi\)
\(30\) 0 0
\(31\) 5.76406 1.03526 0.517628 0.855606i \(-0.326815\pi\)
0.517628 + 0.855606i \(0.326815\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.78224 3.13259i −0.639315 0.529505i
\(36\) 0 0
\(37\) 8.94456i 1.47048i −0.677809 0.735238i \(-0.737070\pi\)
0.677809 0.735238i \(-0.262930\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.54377 + 6.13800i −0.553445 + 0.958594i 0.444578 + 0.895740i \(0.353354\pi\)
−0.998023 + 0.0628541i \(0.979980\pi\)
\(42\) 0 0
\(43\) −6.55871 3.78668i −1.00019 0.577463i −0.0918887 0.995769i \(-0.529290\pi\)
−0.908306 + 0.418307i \(0.862624\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.06032 2.92158i 0.738124 0.426156i −0.0832628 0.996528i \(-0.526534\pi\)
0.821387 + 0.570372i \(0.193201\pi\)
\(48\) 0 0
\(49\) 2.17631 0.310901
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.14208 + 2.39143i −0.568959 + 0.328488i −0.756733 0.653724i \(-0.773206\pi\)
0.187775 + 0.982212i \(0.439873\pi\)
\(54\) 0 0
\(55\) −1.21490 + 1.46685i −0.163818 + 0.197790i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.33889 + 9.24723i −0.695065 + 1.20389i 0.275094 + 0.961417i \(0.411291\pi\)
−0.970159 + 0.242470i \(0.922042\pi\)
\(60\) 0 0
\(61\) −1.17130 2.02874i −0.149969 0.259754i 0.781247 0.624222i \(-0.214584\pi\)
−0.931216 + 0.364468i \(0.881251\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.0852477 + 0.502263i −0.0105737 + 0.0622980i
\(66\) 0 0
\(67\) 3.70127 2.13693i 0.452182 0.261067i −0.256569 0.966526i \(-0.582592\pi\)
0.708751 + 0.705459i \(0.249259\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.14788 + 5.45228i −0.373584 + 0.647067i −0.990114 0.140265i \(-0.955205\pi\)
0.616530 + 0.787332i \(0.288538\pi\)
\(72\) 0 0
\(73\) 7.74621 + 4.47228i 0.906626 + 0.523441i 0.879344 0.476187i \(-0.157982\pi\)
0.0272818 + 0.999628i \(0.491315\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.87076i 0.213193i
\(78\) 0 0
\(79\) −0.332949 + 0.576685i −0.0374597 + 0.0648822i −0.884147 0.467208i \(-0.845260\pi\)
0.846688 + 0.532090i \(0.178593\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.71844i 0.956973i 0.878095 + 0.478487i \(0.158814\pi\)
−0.878095 + 0.478487i \(0.841186\pi\)
\(84\) 0 0
\(85\) −15.9233 + 5.91144i −1.72713 + 0.641186i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.33889 + 9.24723i 0.565922 + 0.980205i 0.996963 + 0.0778729i \(0.0248128\pi\)
−0.431042 + 0.902332i \(0.641854\pi\)
\(90\) 0 0
\(91\) 0.250192 + 0.433345i 0.0262272 + 0.0454268i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.85164 + 4.08026i −0.908159 + 0.418626i
\(96\) 0 0
\(97\) 8.06614 + 4.65699i 0.818993 + 0.472846i 0.850069 0.526671i \(-0.176560\pi\)
−0.0310763 + 0.999517i \(0.509893\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.03264 + 5.25268i 0.301759 + 0.522661i 0.976534 0.215362i \(-0.0690930\pi\)
−0.674776 + 0.738023i \(0.735760\pi\)
\(102\) 0 0
\(103\) 10.2756i 1.01248i 0.862391 + 0.506242i \(0.168966\pi\)
−0.862391 + 0.506242i \(0.831034\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.21742i 0.601061i −0.953772 0.300531i \(-0.902836\pi\)
0.953772 0.300531i \(-0.0971637\pi\)
\(108\) 0 0
\(109\) −1.75405 + 3.03811i −0.168008 + 0.290998i −0.937719 0.347394i \(-0.887067\pi\)
0.769711 + 0.638392i \(0.220400\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 17.9659i 1.69009i −0.534698 0.845043i \(-0.679575\pi\)
0.534698 0.845043i \(-0.320425\pi\)
\(114\) 0 0
\(115\) −0.632627 + 3.72731i −0.0589927 + 0.347574i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −8.34154 + 14.4480i −0.764667 + 1.32444i
\(120\) 0 0
\(121\) −10.2745 −0.934043
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.78813 + 5.40302i 0.875477 + 0.483261i
\(126\) 0 0
\(127\) 12.6002 7.27472i 1.11809 0.645527i 0.177175 0.984179i \(-0.443304\pi\)
0.940912 + 0.338652i \(0.109971\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.18051 + 10.7050i −0.539994 + 0.935297i 0.458910 + 0.888483i \(0.348240\pi\)
−0.998904 + 0.0468142i \(0.985093\pi\)
\(132\) 0 0
\(133\) −4.13943 + 8.63222i −0.358935 + 0.748508i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.5283 + 7.23324i −1.07037 + 0.617978i −0.928282 0.371877i \(-0.878714\pi\)
−0.142086 + 0.989854i \(0.545381\pi\)
\(138\) 0 0
\(139\) 1.71963 + 2.97849i 0.145857 + 0.252632i 0.929692 0.368337i \(-0.120073\pi\)
−0.783835 + 0.620969i \(0.786739\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.168063 0.0970310i 0.0140541 0.00811414i
\(144\) 0 0
\(145\) −3.51890 0.597254i −0.292229 0.0495992i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0.0236837 0.0410214i 0.00194025 0.00336060i −0.865054 0.501679i \(-0.832716\pi\)
0.866994 + 0.498319i \(0.166049\pi\)
\(150\) 0 0
\(151\) −9.94886 −0.809627 −0.404813 0.914399i \(-0.632664\pi\)
−0.404813 + 0.914399i \(0.632664\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.22134 + 9.92630i −0.660354 + 0.797300i
\(156\) 0 0
\(157\) −15.6549 9.03839i −1.24940 0.721342i −0.278411 0.960462i \(-0.589808\pi\)
−0.970990 + 0.239120i \(0.923141\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.85668 + 3.21587i 0.146327 + 0.253446i
\(162\) 0 0
\(163\) 14.2125i 1.11321i −0.830777 0.556605i \(-0.812104\pi\)
0.830777 0.556605i \(-0.187896\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.18567 + 1.26190i −0.169132 + 0.0976484i −0.582177 0.813062i \(-0.697799\pi\)
0.413044 + 0.910711i \(0.364465\pi\)
\(168\) 0 0
\(169\) −6.47405 + 11.2134i −0.498004 + 0.862567i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.8605 + 6.84768i 0.901740 + 0.520620i 0.877764 0.479093i \(-0.159034\pi\)
0.0239756 + 0.999713i \(0.492368\pi\)
\(174\) 0 0
\(175\) 10.7893 2.04535i 0.815594 0.154614i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1.76670 −0.132050 −0.0660248 0.997818i \(-0.521032\pi\)
−0.0660248 + 0.997818i \(0.521032\pi\)
\(180\) 0 0
\(181\) −5.34722 9.26166i −0.397456 0.688414i 0.595955 0.803018i \(-0.296774\pi\)
−0.993411 + 0.114604i \(0.963440\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 15.4034 + 12.7577i 1.13248 + 0.937966i
\(186\) 0 0
\(187\) 5.60331 + 3.23507i 0.409755 + 0.236572i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −15.2844 −1.10594 −0.552971 0.833201i \(-0.686506\pi\)
−0.552971 + 0.833201i \(0.686506\pi\)
\(192\) 0 0
\(193\) 1.60192 + 0.924867i 0.115309 + 0.0665734i 0.556545 0.830817i \(-0.312127\pi\)
−0.441237 + 0.897391i \(0.645460\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.71391i 0.549594i −0.961502 0.274797i \(-0.911389\pi\)
0.961502 0.274797i \(-0.0886106\pi\)
\(198\) 0 0
\(199\) 5.35147 + 9.26901i 0.379355 + 0.657063i 0.990969 0.134094i \(-0.0428124\pi\)
−0.611613 + 0.791157i \(0.709479\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.03605 + 1.75287i −0.213089 + 0.123027i
\(204\) 0 0
\(205\) −5.51573 14.8574i −0.385236 1.03769i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.34781 + 1.60538i 0.231573 + 0.111047i
\(210\) 0 0
\(211\) −8.47981 + 14.6875i −0.583774 + 1.01113i 0.411253 + 0.911521i \(0.365091\pi\)
−0.995027 + 0.0996050i \(0.968242\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 15.8758 5.89380i 1.08272 0.401954i
\(216\) 0 0
\(217\) 12.6595i 0.859386i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.73061 0.116413
\(222\) 0 0
\(223\) −23.9444 13.8243i −1.60344 0.925745i −0.990793 0.135386i \(-0.956773\pi\)
−0.612644 0.790359i \(-0.709894\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.45083i 0.428157i 0.976816 + 0.214078i \(0.0686748\pi\)
−0.976816 + 0.214078i \(0.931325\pi\)
\(228\) 0 0
\(229\) 17.1452 1.13299 0.566494 0.824066i \(-0.308299\pi\)
0.566494 + 0.824066i \(0.308299\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.457945 0.264395i −0.0300010 0.0173211i 0.484924 0.874556i \(-0.338847\pi\)
−0.514925 + 0.857235i \(0.672180\pi\)
\(234\) 0 0
\(235\) −2.18634 + 12.8815i −0.142621 + 0.840295i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.81671 −0.440936 −0.220468 0.975394i \(-0.570758\pi\)
−0.220468 + 0.975394i \(0.570758\pi\)
\(240\) 0 0
\(241\) −8.93999 15.4845i −0.575875 0.997445i −0.995946 0.0899536i \(-0.971328\pi\)
0.420071 0.907491i \(-0.362005\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.10409 + 3.74783i −0.198313 + 0.239440i
\(246\) 0 0
\(247\) 0.990191 0.0758562i 0.0630044 0.00482662i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 8.21999 + 14.2374i 0.518841 + 0.898659i 0.999760 + 0.0218940i \(0.00696963\pi\)
−0.480919 + 0.876765i \(0.659697\pi\)
\(252\) 0 0
\(253\) 1.24720 0.720072i 0.0784108 0.0452705i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.57011 + 4.37061i −0.472211 + 0.272631i −0.717165 0.696904i \(-0.754560\pi\)
0.244954 + 0.969535i \(0.421227\pi\)
\(258\) 0 0
\(259\) 19.6448 1.22067
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 19.0332 + 10.9888i 1.17364 + 0.677599i 0.954534 0.298103i \(-0.0963540\pi\)
0.219102 + 0.975702i \(0.429687\pi\)
\(264\) 0 0
\(265\) 1.78961 10.5440i 0.109935 0.647714i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.17878 + 5.50580i −0.193813 + 0.335695i −0.946511 0.322672i \(-0.895419\pi\)
0.752697 + 0.658367i \(0.228752\pi\)
\(270\) 0 0
\(271\) 7.39590 12.8101i 0.449269 0.778156i −0.549070 0.835776i \(-0.685018\pi\)
0.998339 + 0.0576202i \(0.0183512\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.793240 4.18438i −0.0478342 0.252328i
\(276\) 0 0
\(277\) 8.26106i 0.496359i 0.968714 + 0.248180i \(0.0798323\pi\)
−0.968714 + 0.248180i \(0.920168\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 11.4531 + 19.8374i 0.683235 + 1.18340i 0.973988 + 0.226600i \(0.0727610\pi\)
−0.290753 + 0.956798i \(0.593906\pi\)
\(282\) 0 0
\(283\) 25.9233 + 14.9668i 1.54098 + 0.889684i 0.998777 + 0.0494347i \(0.0157420\pi\)
0.542200 + 0.840249i \(0.317591\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −13.4808 7.78316i −0.795748 0.459425i
\(288\) 0 0
\(289\) 20.3498 + 35.2469i 1.19705 + 2.07335i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 22.4761i 1.31307i −0.754296 0.656534i \(-0.772022\pi\)
0.754296 0.656534i \(-0.227978\pi\)
\(294\) 0 0
\(295\) −8.30976 22.3836i −0.483813 1.30322i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.192602 0.333597i 0.0111385 0.0192924i
\(300\) 0 0
\(301\) 8.31664 14.4048i 0.479363 0.830281i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.16434 + 0.876529i 0.295709 + 0.0501899i
\(306\) 0 0
\(307\) −20.6663 11.9317i −1.17949 0.680979i −0.223593 0.974682i \(-0.571779\pi\)
−0.955897 + 0.293704i \(0.905112\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.5315 −0.710598 −0.355299 0.934753i \(-0.615621\pi\)
−0.355299 + 0.934753i \(0.615621\pi\)
\(312\) 0 0
\(313\) 3.44426 1.98854i 0.194681 0.112399i −0.399491 0.916737i \(-0.630813\pi\)
0.594172 + 0.804338i \(0.297480\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.1149 10.4587i 1.01744 0.587417i 0.104076 0.994569i \(-0.466812\pi\)
0.913360 + 0.407153i \(0.133478\pi\)
\(318\) 0 0
\(319\) 0.679809 + 1.17746i 0.0380620 + 0.0659253i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 18.6970 + 27.3260i 1.04033 + 1.52046i
\(324\) 0 0
\(325\) −0.743358 0.863188i −0.0412341 0.0478811i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.41663 + 11.1139i 0.353761 + 0.612731i
\(330\) 0 0
\(331\) −13.5367 −0.744042 −0.372021 0.928224i \(-0.621335\pi\)
−0.372021 + 0.928224i \(0.621335\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.59915 + 9.42189i −0.0873709 + 0.514773i
\(336\) 0 0
\(337\) 11.3929 + 6.57769i 0.620610 + 0.358310i 0.777107 0.629369i \(-0.216687\pi\)
−0.156496 + 0.987679i \(0.550020\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4.90971 0.265876
\(342\) 0 0
\(343\) 20.1538i 1.08820i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.21363 0.700688i −0.0651509 0.0376149i 0.467071 0.884220i \(-0.345309\pi\)
−0.532222 + 0.846605i \(0.678643\pi\)
\(348\) 0 0
\(349\) 11.4686 0.613899 0.306949 0.951726i \(-0.400692\pi\)
0.306949 + 0.951726i \(0.400692\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5.17971i 0.275688i 0.990454 + 0.137844i \(0.0440173\pi\)
−0.990454 + 0.137844i \(0.955983\pi\)
\(354\) 0 0
\(355\) −4.89954 13.1976i −0.260040 0.700457i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.29572 2.24425i 0.0683853 0.118447i −0.829805 0.558053i \(-0.811549\pi\)
0.898191 + 0.439606i \(0.144882\pi\)
\(360\) 0 0
\(361\) 11.8955 + 14.8154i 0.626080 + 0.779759i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −18.7502 + 6.96091i −0.981432 + 0.364351i
\(366\) 0 0
\(367\) −11.3728 + 6.56609i −0.593655 + 0.342747i −0.766541 0.642195i \(-0.778024\pi\)
0.172886 + 0.984942i \(0.444691\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5.25228 9.09721i −0.272685 0.472304i
\(372\) 0 0
\(373\) 28.8123i 1.49184i −0.666035 0.745921i \(-0.732010\pi\)
0.666035 0.745921i \(-0.267990\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.314944 + 0.181833i 0.0162204 + 0.00936486i
\(378\) 0 0
\(379\) 25.2519 1.29710 0.648551 0.761171i \(-0.275375\pi\)
0.648551 + 0.761171i \(0.275375\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −18.4782 10.6684i −0.944190 0.545128i −0.0529189 0.998599i \(-0.516852\pi\)
−0.891271 + 0.453470i \(0.850186\pi\)
\(384\) 0 0
\(385\) −3.22164 2.66828i −0.164190 0.135988i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −15.2358 26.3892i −0.772486 1.33799i −0.936196 0.351477i \(-0.885680\pi\)
0.163710 0.986508i \(-0.447654\pi\)
\(390\) 0 0
\(391\) 12.8429 0.649495
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −0.518222 1.39591i −0.0260746 0.0702356i
\(396\) 0 0
\(397\) −4.28442 2.47361i −0.215029 0.124147i 0.388617 0.921399i \(-0.372953\pi\)
−0.603646 + 0.797252i \(0.706286\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 15.4584 26.7747i 0.771956 1.33707i −0.164533 0.986371i \(-0.552612\pi\)
0.936490 0.350696i \(-0.114055\pi\)
\(402\) 0 0
\(403\) 1.13729 0.656616i 0.0566526 0.0327084i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.61880i 0.377650i
\(408\) 0 0
\(409\) 2.31244 + 4.00526i 0.114343 + 0.198047i 0.917517 0.397697i \(-0.130191\pi\)
−0.803174 + 0.595744i \(0.796857\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −20.3096 11.7258i −0.999371 0.576987i
\(414\) 0 0
\(415\) −15.0140 12.4352i −0.737011 0.610420i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −26.8070 −1.30961 −0.654803 0.755800i \(-0.727248\pi\)
−0.654803 + 0.755800i \(0.727248\pi\)
\(420\) 0 0
\(421\) 11.1453 19.3043i 0.543191 0.940834i −0.455528 0.890222i \(-0.650549\pi\)
0.998718 0.0506123i \(-0.0161173\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 12.5315 35.8532i 0.607868 1.73913i
\(426\) 0 0
\(427\) 4.45571 2.57251i 0.215627 0.124492i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −14.5079 25.1283i −0.698819 1.21039i −0.968876 0.247545i \(-0.920376\pi\)
0.270058 0.962844i \(-0.412957\pi\)
\(432\) 0 0
\(433\) 6.47132 3.73622i 0.310992 0.179551i −0.336379 0.941727i \(-0.609202\pi\)
0.647370 + 0.762176i \(0.275869\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.34825 0.562932i 0.351515 0.0269287i
\(438\) 0 0
\(439\) −10.7675 + 18.6499i −0.513907 + 0.890112i 0.485963 + 0.873979i \(0.338469\pi\)
−0.999870 + 0.0161330i \(0.994864\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −29.4833 + 17.0222i −1.40079 + 0.808748i −0.994474 0.104983i \(-0.966521\pi\)
−0.406319 + 0.913731i \(0.633188\pi\)
\(444\) 0 0
\(445\) −23.5396 3.99531i −1.11588 0.189396i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 36.9505 1.74380 0.871901 0.489682i \(-0.162887\pi\)
0.871901 + 0.489682i \(0.162887\pi\)
\(450\) 0 0
\(451\) −3.01852 + 5.22823i −0.142136 + 0.246188i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.10311 0.187229i −0.0517148 0.00877741i
\(456\) 0 0
\(457\) 30.3030i 1.41752i −0.705452 0.708758i \(-0.749256\pi\)
0.705452 0.708758i \(-0.250744\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.18504 10.7128i 0.288066 0.498945i −0.685282 0.728278i \(-0.740321\pi\)
0.973348 + 0.229333i \(0.0736545\pi\)
\(462\) 0 0
\(463\) 25.3893i 1.17994i −0.807425 0.589971i \(-0.799139\pi\)
0.807425 0.589971i \(-0.200861\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 28.5919i 1.32307i 0.749912 + 0.661537i \(0.230096\pi\)
−0.749912 + 0.661537i \(0.769904\pi\)
\(468\) 0 0
\(469\) 4.69331 + 8.12906i 0.216717 + 0.375365i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5.58659 3.22542i −0.256871 0.148305i
\(474\) 0 0
\(475\) 5.59856 21.0631i 0.256879 0.966443i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 6.22157 + 10.7761i 0.284271 + 0.492372i 0.972432 0.233186i \(-0.0749152\pi\)
−0.688161 + 0.725558i \(0.741582\pi\)
\(480\) 0 0
\(481\) −1.01892 1.76483i −0.0464589 0.0804692i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −19.5246 + 7.24841i −0.886568 + 0.329133i
\(486\) 0 0
\(487\) 18.2489i 0.826937i 0.910518 + 0.413469i \(0.135683\pi\)
−0.910518 + 0.413469i \(0.864317\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −9.23347 + 15.9928i −0.416701 + 0.721747i −0.995605 0.0936485i \(-0.970147\pi\)
0.578905 + 0.815395i \(0.303480\pi\)
\(492\) 0 0
\(493\) 12.1248i 0.546075i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −11.9748 6.91365i −0.537143 0.310120i
\(498\) 0 0
\(499\) 0.746715 1.29335i 0.0334276 0.0578982i −0.848828 0.528670i \(-0.822691\pi\)
0.882255 + 0.470771i \(0.156024\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.34173 5.39345i 0.416527 0.240482i −0.277063 0.960852i \(-0.589361\pi\)
0.693590 + 0.720370i \(0.256028\pi\)
\(504\) 0 0
\(505\) −13.3711 2.26945i −0.595008 0.100989i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2.28888 3.96446i −0.101453 0.175722i 0.810831 0.585281i \(-0.199016\pi\)
−0.912283 + 0.409559i \(0.865682\pi\)
\(510\) 0 0
\(511\) −9.82242 + 17.0129i −0.434518 + 0.752608i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −17.6956 14.6562i −0.779762 0.645829i
\(516\) 0 0
\(517\) 4.31028 2.48854i 0.189566 0.109446i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 29.1613 1.27758 0.638791 0.769380i \(-0.279435\pi\)
0.638791 + 0.769380i \(0.279435\pi\)
\(522\) 0 0
\(523\) −27.7312 + 16.0106i −1.21260 + 0.700097i −0.963325 0.268336i \(-0.913526\pi\)
−0.249277 + 0.968432i \(0.580193\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 37.9180 + 21.8920i 1.65173 + 0.953629i
\(528\) 0 0
\(529\) −10.0707 + 17.4430i −0.437856 + 0.758389i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.61476i 0.0699432i
\(534\) 0 0
\(535\) 10.7070 + 8.86798i 0.462906 + 0.383396i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.85374 0.0798462
\(540\) 0 0
\(541\) −5.36073 9.28505i −0.230476 0.399196i 0.727473 0.686137i \(-0.240695\pi\)
−0.957948 + 0.286941i \(0.907362\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.73011 7.35395i −0.116945 0.315009i
\(546\) 0 0
\(547\) −14.7394 + 8.50982i −0.630213 + 0.363854i −0.780835 0.624738i \(-0.785206\pi\)
0.150622 + 0.988591i \(0.451873\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0.531456 + 6.93738i 0.0226408 + 0.295542i
\(552\) 0 0
\(553\) −1.26657 0.731253i −0.0538600 0.0310961i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 19.3042 11.1453i 0.817946 0.472241i −0.0317616 0.999495i \(-0.510112\pi\)
0.849708 + 0.527254i \(0.176778\pi\)
\(558\) 0 0
\(559\) −1.72544 −0.0729785
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 45.7030i 1.92615i −0.269234 0.963075i \(-0.586770\pi\)
0.269234 0.963075i \(-0.413230\pi\)
\(564\) 0 0
\(565\) 30.9391 + 25.6249i 1.30162 + 1.07805i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12.2807 0.514835 0.257417 0.966300i \(-0.417128\pi\)
0.257417 + 0.966300i \(0.417128\pi\)
\(570\) 0 0
\(571\) −16.5574 −0.692907 −0.346453 0.938067i \(-0.612614\pi\)
−0.346453 + 0.938067i \(0.612614\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5.51649 6.40576i −0.230054 0.267139i
\(576\) 0 0
\(577\) 29.0257i 1.20836i 0.796849 + 0.604179i \(0.206499\pi\)
−0.796849 + 0.604179i \(0.793501\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −19.1482 −0.794402
\(582\) 0 0
\(583\) −3.52814 + 2.03698i −0.146121 + 0.0843629i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.69083 + 5.01765i 0.358709 + 0.207101i 0.668514 0.743699i \(-0.266931\pi\)
−0.309805 + 0.950800i \(0.600264\pi\)
\(588\) 0 0
\(589\) 22.6548 + 10.8637i 0.933477 + 0.447633i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 13.2446 7.64676i 0.543890 0.314015i −0.202764 0.979228i \(-0.564993\pi\)
0.746654 + 0.665213i \(0.231659\pi\)
\(594\) 0 0
\(595\) −12.9832 34.9723i −0.532261 1.43372i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −13.0303 22.5691i −0.532402 0.922147i −0.999284 0.0378276i \(-0.987956\pi\)
0.466883 0.884319i \(-0.345377\pi\)
\(600\) 0 0
\(601\) 15.7129 0.640942 0.320471 0.947258i \(-0.396159\pi\)
0.320471 + 0.947258i \(0.396159\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 14.6546 17.6937i 0.595794 0.719351i
\(606\) 0 0
\(607\) 26.1236i 1.06032i 0.847896 + 0.530162i \(0.177869\pi\)
−0.847896 + 0.530162i \(0.822131\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0.665626 1.15290i 0.0269284 0.0466413i
\(612\) 0 0
\(613\) 33.1549 + 19.1420i 1.33912 + 0.773139i 0.986676 0.162696i \(-0.0520190\pi\)
0.352439 + 0.935835i \(0.385352\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −15.0154 + 8.66916i −0.604498 + 0.349007i −0.770809 0.637066i \(-0.780148\pi\)
0.166311 + 0.986073i \(0.446814\pi\)
\(618\) 0 0
\(619\) 32.1118 1.29068 0.645342 0.763894i \(-0.276715\pi\)
0.645342 + 0.763894i \(0.276715\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −20.3096 + 11.7258i −0.813687 + 0.469783i
\(624\) 0 0
\(625\) −23.2655 + 9.14977i −0.930618 + 0.365991i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 33.9715 58.8404i 1.35453 2.34612i
\(630\) 0 0
\(631\) 0.172064 + 0.298024i 0.00684978 + 0.0118642i 0.869430 0.494056i \(-0.164486\pi\)
−0.862580 + 0.505920i \(0.831153\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.44398 + 32.0748i −0.216038 + 1.27285i
\(636\) 0 0
\(637\) 0.429402 0.247915i 0.0170135 0.00982277i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 15.6410 27.0909i 0.617781 1.07003i −0.372109 0.928189i \(-0.621365\pi\)
0.989890 0.141839i \(-0.0453015\pi\)
\(642\) 0 0
\(643\) −5.11932 2.95564i −0.201886 0.116559i 0.395649 0.918402i \(-0.370520\pi\)
−0.597535 + 0.801843i \(0.703853\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19.1643i 0.753425i −0.926330 0.376712i \(-0.877055\pi\)
0.926330 0.376712i \(-0.122945\pi\)
\(648\) 0 0
\(649\) −4.54757 + 7.87661i −0.178508 + 0.309184i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 42.5209i 1.66397i 0.554799 + 0.831985i \(0.312795\pi\)
−0.554799 + 0.831985i \(0.687205\pi\)
\(654\) 0 0
\(655\) −9.61971 25.9121i −0.375873 1.01247i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −18.4989 32.0411i −0.720616 1.24814i −0.960753 0.277405i \(-0.910526\pi\)
0.240137 0.970739i \(-0.422808\pi\)
\(660\) 0 0
\(661\) −11.1793 19.3631i −0.434824 0.753137i 0.562457 0.826826i \(-0.309856\pi\)
−0.997281 + 0.0736893i \(0.976523\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.96144 19.4408i −0.347510 0.753880i
\(666\) 0 0
\(667\) 2.33721 + 1.34939i 0.0904971 + 0.0522485i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −0.997687 1.72804i −0.0385153 0.0667104i
\(672\) 0 0
\(673\) 4.94723i 0.190702i −0.995444 0.0953508i \(-0.969603\pi\)
0.995444 0.0953508i \(-0.0303973\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.58590i 0.176250i 0.996109 + 0.0881252i \(0.0280875\pi\)
−0.996109 + 0.0881252i \(0.971912\pi\)
\(678\) 0 0
\(679\) −10.2281 + 17.7156i −0.392518 + 0.679862i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 26.4117i 1.01062i −0.862939 0.505308i \(-0.831379\pi\)
0.862939 0.505308i \(-0.168621\pi\)
\(684\) 0 0
\(685\) 5.41293 31.8919i 0.206817 1.21853i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −0.544843 + 0.943695i −0.0207568 + 0.0359519i
\(690\) 0 0
\(691\) −43.1599 −1.64188 −0.820939 0.571016i \(-0.806550\pi\)
−0.820939 + 0.571016i \(0.806550\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7.58199 1.28687i −0.287601 0.0488137i
\(696\) 0 0
\(697\) −46.6244 + 26.9186i −1.76602 + 1.01961i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 13.5715 23.5065i 0.512588 0.887829i −0.487305 0.873232i \(-0.662020\pi\)
0.999893 0.0145974i \(-0.00464665\pi\)
\(702\) 0 0
\(703\) 16.8582 35.1554i 0.635818 1.32591i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −11.5364 + 6.66055i −0.433871 + 0.250496i
\(708\) 0 0
\(709\) 10.3352 + 17.9011i 0.388146 + 0.672288i 0.992200 0.124655i \(-0.0397824\pi\)
−0.604054 + 0.796943i \(0.706449\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8.43989 4.87277i 0.316076 0.182487i
\(714\) 0 0
\(715\) −0.0726123 + 0.427818i −0.00271555 + 0.0159995i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 15.5398 26.9157i 0.579537 1.00379i −0.415995 0.909367i \(-0.636567\pi\)
0.995532 0.0944208i \(-0.0300999\pi\)
\(720\) 0 0
\(721\) −22.5682 −0.840483
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6.04758 5.20804i 0.224601 0.193422i
\(726\) 0 0
\(727\) 20.0463 + 11.5737i 0.743475 + 0.429246i 0.823332 0.567561i \(-0.192113\pi\)
−0.0798562 + 0.996806i \(0.525446\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −28.7637 49.8202i −1.06386 1.84267i
\(732\) 0 0
\(733\) 17.8489i 0.659266i 0.944109 + 0.329633i \(0.106925\pi\)
−0.944109 + 0.329633i \(0.893075\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.15267 1.82019i 0.116130 0.0670477i
\(738\) 0 0
\(739\) −17.7079 + 30.6710i −0.651396 + 1.12825i 0.331388 + 0.943494i \(0.392483\pi\)
−0.982784 + 0.184756i \(0.940850\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 31.3859 + 18.1207i 1.15144 + 0.664782i 0.949237 0.314562i \(-0.101858\pi\)
0.202200 + 0.979344i \(0.435191\pi\)
\(744\) 0 0
\(745\) 0.0368627 + 0.0992951i 0.00135054 + 0.00363789i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 13.6553 0.498953
\(750\) 0 0
\(751\) −20.1202 34.8491i −0.734195 1.27166i −0.955076 0.296362i \(-0.904226\pi\)
0.220881 0.975301i \(-0.429107\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 14.1902 17.1330i 0.516433 0.623532i
\(756\) 0 0
\(757\) −26.1071 15.0730i −0.948880 0.547836i −0.0561469 0.998423i \(-0.517882\pi\)
−0.892733 + 0.450587i \(0.851215\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −15.6964 −0.568994 −0.284497 0.958677i \(-0.591827\pi\)
−0.284497 + 0.958677i \(0.591827\pi\)
\(762\) 0 0
\(763\) −6.67257 3.85241i −0.241563 0.139467i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.43273i 0.0878408i
\(768\) 0 0
\(769\) 9.71335 + 16.8240i 0.350272 + 0.606689i 0.986297 0.164979i \(-0.0527557\pi\)
−0.636025 + 0.771669i \(0.719422\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −42.5491 + 24.5657i −1.53039 + 0.883568i −0.531042 + 0.847346i \(0.678199\pi\)
−0.999344 + 0.0362225i \(0.988467\pi\)
\(774\) 0 0
\(775\) −5.36791 28.3160i −0.192821 1.01714i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −25.4968 + 17.4455i −0.913519 + 0.625049i
\(780\) 0 0
\(781\) −2.68130 + 4.64415i −0.0959445 + 0.166181i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 37.8939 14.0679i 1.35249 0.502104i
\(786\) 0 0
\(787\) 24.8250i 0.884917i −0.896789 0.442459i \(-0.854106\pi\)
0.896789 0.442459i \(-0.145894\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 39.4582 1.40297
\(792\) 0 0
\(793\) −0.462211 0.266858i −0.0164136 0.00947639i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 29.8866i 1.05864i −0.848423 0.529319i \(-0.822447\pi\)
0.848423 0.529319i \(-0.177553\pi\)
\(798\) 0 0
\(799\) 44.3848 1.57022
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.59807 + 3.80940i 0.232841 + 0.134431i
\(804\) 0 0
\(805\) −8.18626 1.38943i −0.288528 0.0489710i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 46.8929 1.64867 0.824333 0.566105i \(-0.191550\pi\)
0.824333 + 0.566105i \(0.191550\pi\)
\(810\) 0 0
\(811\) 0.623347 + 1.07967i 0.0218887 + 0.0379123i 0.876762 0.480924i \(-0.159699\pi\)
−0.854874 + 0.518836i \(0.826365\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 24.4754 + 20.2715i 0.857336 + 0.710078i
\(816\) 0 0
\(817\) −18.6412 27.2445i −0.652174 0.953164i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0.101045 + 0.175014i 0.00352648 + 0.00610805i 0.867783 0.496943i \(-0.165544\pi\)
−0.864257 + 0.503051i \(0.832211\pi\)
\(822\) 0 0
\(823\) −12.8967 + 7.44589i −0.449549 + 0.259548i −0.707640 0.706573i \(-0.750240\pi\)
0.258090 + 0.966121i \(0.416907\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.72867 + 1.57540i −0.0948851 + 0.0547819i −0.546692 0.837334i \(-0.684113\pi\)
0.451807 + 0.892116i \(0.350780\pi\)
\(828\) 0 0
\(829\) 40.7012 1.41361 0.706805 0.707409i \(-0.250136\pi\)
0.706805 + 0.707409i \(0.250136\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 14.3165 + 8.26565i 0.496038 + 0.286388i
\(834\) 0 0
\(835\) 0.944328 5.56380i 0.0326798 0.192543i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −13.1206 + 22.7255i −0.452973 + 0.784572i −0.998569 0.0534763i \(-0.982970\pi\)
0.545596 + 0.838048i \(0.316303\pi\)
\(840\) 0 0
\(841\) 13.2261 22.9082i 0.456071 0.789938i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −10.0766 27.1427i −0.346645 0.933739i
\(846\) 0 0
\(847\) 22.5657i 0.775367i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −7.56148 13.0969i −0.259204 0.448955i
\(852\) 0 0
\(853\) −2.85321 1.64730i −0.0976919 0.0564025i 0.450358 0.892848i \(-0.351296\pi\)
−0.548050 + 0.836446i \(0.684630\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −18.0460 10.4188i −0.616439 0.355901i 0.159042 0.987272i \(-0.449159\pi\)
−0.775481 + 0.631371i \(0.782493\pi\)
\(858\) 0 0
\(859\) 14.7172 + 25.4910i 0.502145 + 0.869740i 0.999997 + 0.00247823i \(0.000788846\pi\)
−0.497852 + 0.867262i \(0.665878\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 49.0615i 1.67007i −0.550195 0.835036i \(-0.685447\pi\)
0.550195 0.835036i \(-0.314553\pi\)
\(864\) 0 0
\(865\) −28.7092 + 10.6581i −0.976143 + 0.362387i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −0.283600 + 0.491209i −0.00962047 + 0.0166631i
\(870\) 0 0
\(871\) 0.486859 0.843264i 0.0164966 0.0285729i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −11.8666 + 21.4976i −0.401164 + 0.726750i
\(876\) 0 0
\(877\) 42.4480 + 24.5074i 1.43337 + 0.827556i 0.997376 0.0723944i \(-0.0230640\pi\)
0.435993 + 0.899950i \(0.356397\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 43.8746 1.47817 0.739086 0.673611i \(-0.235258\pi\)
0.739086 + 0.673611i \(0.235258\pi\)
\(882\) 0 0
\(883\) 48.3690 27.9259i 1.62775 0.939780i 0.642984 0.765880i \(-0.277696\pi\)
0.984763 0.173900i \(-0.0556370\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −20.5390 + 11.8582i −0.689633 + 0.398160i −0.803475 0.595339i \(-0.797018\pi\)
0.113841 + 0.993499i \(0.463684\pi\)
\(888\) 0 0
\(889\) 15.9774 + 27.6737i 0.535865 + 0.928145i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 25.3953 1.94548i 0.849822 0.0651029i
\(894\) 0 0
\(895\) 2.51987 3.04244i 0.0842299 0.101698i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 4.60031 + 7.96798i 0.153429 + 0.265747i
\(900\) 0 0
\(901\) −36.3308 −1.21035
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 23.5763 + 4.00155i 0.783704 + 0.133016i
\(906\) 0 0
\(907\) 21.7205 + 12.5403i 0.721216 + 0.416394i 0.815200 0.579179i \(-0.196627\pi\)
−0.0939839 + 0.995574i \(0.529960\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3.00341 0.0995074 0.0497537 0.998762i \(-0.484156\pi\)
0.0497537 + 0.998762i \(0.484156\pi\)
\(912\) 0 0
\(913\) 7.42620i 0.245771i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −23.5112 13.5742i −0.776409 0.448260i
\(918\) 0 0
\(919\) 18.5467 0.611799 0.305899 0.952064i \(-0.401043\pi\)
0.305899 + 0.952064i \(0.401043\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.43437i 0.0472128i
\(924\) 0 0
\(925\) −43.9402 + 8.32982i −1.44474 + 0.273883i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −4.47449 + 7.75005i −0.146803 + 0.254271i −0.930044 0.367447i \(-0.880232\pi\)
0.783241 + 0.621718i \(0.213565\pi\)
\(930\) 0 0
\(931\) 8.55369 + 4.10178i 0.280336 + 0.134430i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −13.5632 + 5.03525i −0.443564 + 0.164670i
\(936\) 0 0
\(937\) 27.4136 15.8272i 0.895563 0.517054i 0.0198050 0.999804i \(-0.493695\pi\)
0.875758 + 0.482750i \(0.160362\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −14.9011 25.8095i −0.485763 0.841366i 0.514103 0.857728i \(-0.328125\pi\)
−0.999866 + 0.0163621i \(0.994792\pi\)
\(942\) 0 0
\(943\) 11.9832i 0.390228i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −50.1907 28.9776i −1.63098 0.941646i −0.983792 0.179314i \(-0.942612\pi\)
−0.647187 0.762332i \(-0.724055\pi\)
\(948\) 0 0
\(949\) 2.03785 0.0661513
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 26.6329 + 15.3765i 0.862725 + 0.498094i 0.864924 0.501903i \(-0.167367\pi\)
−0.00219907 + 0.999998i \(0.500700\pi\)
\(954\) 0 0
\(955\) 21.8003 26.3213i 0.705442 0.851738i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −15.8863 27.5159i −0.512995 0.888534i
\(960\) 0 0
\(961\) 2.22438 0.0717541
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −3.87755 + 1.43952i −0.124823 + 0.0463397i
\(966\) 0 0
\(967\) −3.12068 1.80172i −0.100354 0.0579395i 0.448983 0.893540i \(-0.351786\pi\)
−0.549337 + 0.835601i \(0.685120\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 15.9543 27.6337i 0.511998 0.886807i −0.487905 0.872897i \(-0.662239\pi\)
0.999903 0.0139100i \(-0.00442783\pi\)
\(972\) 0 0
\(973\) −6.54162 + 3.77681i −0.209715 + 0.121079i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 40.9910i 1.31142i 0.755014 + 0.655709i \(0.227630\pi\)
−0.755014 + 0.655709i \(0.772370\pi\)
\(978\) 0 0
\(979\) 4.54757 + 7.87661i 0.145341 + 0.251738i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −39.9335 23.0556i −1.27368 0.735361i −0.298003 0.954565i \(-0.596321\pi\)
−0.975679 + 0.219204i \(0.929654\pi\)
\(984\) 0 0
\(985\) 13.2842 + 11.0024i 0.423268 + 0.350567i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −12.8046 −0.407163
\(990\) 0 0
\(991\) 1.26185 2.18559i 0.0400840 0.0694276i −0.845287 0.534312i \(-0.820571\pi\)
0.885371 + 0.464884i \(0.153904\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −23.5950 4.00472i −0.748013 0.126958i
\(996\) 0 0
\(997\) −0.194307 + 0.112183i −0.00615376 + 0.00355288i −0.503074 0.864244i \(-0.667798\pi\)
0.496920 + 0.867796i \(0.334464\pi\)
\(998\) 0 0
\(999\) 0 0
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 3420.2.bj.e.1189.5 40
3.2 odd 2 inner 3420.2.bj.e.1189.16 yes 40
5.4 even 2 inner 3420.2.bj.e.1189.9 yes 40
15.14 odd 2 inner 3420.2.bj.e.1189.12 yes 40
19.7 even 3 inner 3420.2.bj.e.2629.9 yes 40
57.26 odd 6 inner 3420.2.bj.e.2629.12 yes 40
95.64 even 6 inner 3420.2.bj.e.2629.5 yes 40
285.254 odd 6 inner 3420.2.bj.e.2629.16 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3420.2.bj.e.1189.5 40 1.1 even 1 trivial
3420.2.bj.e.1189.9 yes 40 5.4 even 2 inner
3420.2.bj.e.1189.12 yes 40 15.14 odd 2 inner
3420.2.bj.e.1189.16 yes 40 3.2 odd 2 inner
3420.2.bj.e.2629.5 yes 40 95.64 even 6 inner
3420.2.bj.e.2629.9 yes 40 19.7 even 3 inner
3420.2.bj.e.2629.12 yes 40 57.26 odd 6 inner
3420.2.bj.e.2629.16 yes 40 285.254 odd 6 inner