Properties

Label 252.2.k.a.37.1
Level $252$
Weight $2$
Character 252.37
Analytic conductor $2.012$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,2,Mod(37,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.k (of order \(3\), 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: \(2.01223013094\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 37.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 252.37
Dual form 252.2.k.a.109.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 1.73205i) q^{5} +(0.500000 + 2.59808i) q^{7} +O(q^{10})\) \(q+(-1.00000 + 1.73205i) q^{5} +(0.500000 + 2.59808i) q^{7} +(1.00000 + 1.73205i) q^{11} -3.00000 q^{13} +(4.00000 + 6.92820i) q^{17} +(0.500000 - 0.866025i) q^{19} +(4.00000 - 6.92820i) q^{23} +(0.500000 + 0.866025i) q^{25} -4.00000 q^{29} +(-1.50000 - 2.59808i) q^{31} +(-5.00000 - 1.73205i) q^{35} +(0.500000 - 0.866025i) q^{37} -6.00000 q^{41} +11.0000 q^{43} +(3.00000 - 5.19615i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(-6.00000 - 10.3923i) q^{53} -4.00000 q^{55} +(2.00000 + 3.46410i) q^{59} +(3.00000 - 5.19615i) q^{61} +(3.00000 - 5.19615i) q^{65} +(-6.50000 - 11.2583i) q^{67} +10.0000 q^{71} +(5.50000 + 9.52628i) q^{73} +(-4.00000 + 3.46410i) q^{77} +(1.50000 - 2.59808i) q^{79} -2.00000 q^{83} -16.0000 q^{85} +(-1.50000 - 7.79423i) q^{91} +(1.00000 + 1.73205i) q^{95} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + q^{7} + 2 q^{11} - 6 q^{13} + 8 q^{17} + q^{19} + 8 q^{23} + q^{25} - 8 q^{29} - 3 q^{31} - 10 q^{35} + q^{37} - 12 q^{41} + 22 q^{43} + 6 q^{47} - 13 q^{49} - 12 q^{53} - 8 q^{55} + 4 q^{59} + 6 q^{61} + 6 q^{65} - 13 q^{67} + 20 q^{71} + 11 q^{73} - 8 q^{77} + 3 q^{79} - 4 q^{83} - 32 q^{85} - 3 q^{91} + 2 q^{95} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 + 1.73205i −0.447214 + 0.774597i −0.998203 0.0599153i \(-0.980917\pi\)
0.550990 + 0.834512i \(0.314250\pi\)
\(6\) 0 0
\(7\) 0.500000 + 2.59808i 0.188982 + 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 + 1.73205i 0.301511 + 0.522233i 0.976478 0.215615i \(-0.0691756\pi\)
−0.674967 + 0.737848i \(0.735842\pi\)
\(12\) 0 0
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 + 6.92820i 0.970143 + 1.68034i 0.695113 + 0.718900i \(0.255354\pi\)
0.275029 + 0.961436i \(0.411312\pi\)
\(18\) 0 0
\(19\) 0.500000 0.866025i 0.114708 0.198680i −0.802955 0.596040i \(-0.796740\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 6.92820i 0.834058 1.44463i −0.0607377 0.998154i \(-0.519345\pi\)
0.894795 0.446476i \(-0.147321\pi\)
\(24\) 0 0
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −1.50000 2.59808i −0.269408 0.466628i 0.699301 0.714827i \(-0.253495\pi\)
−0.968709 + 0.248199i \(0.920161\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.00000 1.73205i −0.845154 0.292770i
\(36\) 0 0
\(37\) 0.500000 0.866025i 0.0821995 0.142374i −0.821995 0.569495i \(-0.807139\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 11.0000 1.67748 0.838742 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00000 5.19615i 0.437595 0.757937i −0.559908 0.828554i \(-0.689164\pi\)
0.997503 + 0.0706177i \(0.0224970\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 10.3923i −0.824163 1.42749i −0.902557 0.430570i \(-0.858312\pi\)
0.0783936 0.996922i \(-0.475021\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.00000 + 3.46410i 0.260378 + 0.450988i 0.966342 0.257260i \(-0.0828195\pi\)
−0.705965 + 0.708247i \(0.749486\pi\)
\(60\) 0 0
\(61\) 3.00000 5.19615i 0.384111 0.665299i −0.607535 0.794293i \(-0.707841\pi\)
0.991645 + 0.128994i \(0.0411748\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 5.19615i 0.372104 0.644503i
\(66\) 0 0
\(67\) −6.50000 11.2583i −0.794101 1.37542i −0.923408 0.383819i \(-0.874609\pi\)
0.129307 0.991605i \(-0.458725\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 0 0
\(73\) 5.50000 + 9.52628i 0.643726 + 1.11497i 0.984594 + 0.174855i \(0.0559458\pi\)
−0.340868 + 0.940111i \(0.610721\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.00000 + 3.46410i −0.455842 + 0.394771i
\(78\) 0 0
\(79\) 1.50000 2.59808i 0.168763 0.292306i −0.769222 0.638982i \(-0.779356\pi\)
0.937985 + 0.346675i \(0.112689\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.00000 −0.219529 −0.109764 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 0 0
\(85\) −16.0000 −1.73544
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) −1.50000 7.79423i −0.157243 0.817057i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 + 1.73205i 0.102598 + 0.177705i
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.00000 + 8.66025i 0.497519 + 0.861727i 0.999996 0.00286291i \(-0.000911295\pi\)
−0.502477 + 0.864590i \(0.667578\pi\)
\(102\) 0 0
\(103\) −5.50000 + 9.52628i −0.541931 + 0.938652i 0.456862 + 0.889538i \(0.348973\pi\)
−0.998793 + 0.0491146i \(0.984360\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(108\) 0 0
\(109\) 5.50000 + 9.52628i 0.526804 + 0.912452i 0.999512 + 0.0312328i \(0.00994332\pi\)
−0.472708 + 0.881219i \(0.656723\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 8.00000 + 13.8564i 0.746004 + 1.29212i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −16.0000 + 13.8564i −1.46672 + 1.27021i
\(120\) 0 0
\(121\) 3.50000 6.06218i 0.318182 0.551107i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 3.00000 0.266207 0.133103 0.991102i \(-0.457506\pi\)
0.133103 + 0.991102i \(0.457506\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.00000 + 1.73205i −0.0873704 + 0.151330i −0.906399 0.422423i \(-0.861180\pi\)
0.819028 + 0.573753i \(0.194513\pi\)
\(132\) 0 0
\(133\) 2.50000 + 0.866025i 0.216777 + 0.0750939i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.00000 + 3.46410i 0.170872 + 0.295958i 0.938725 0.344668i \(-0.112008\pi\)
−0.767853 + 0.640626i \(0.778675\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.00000 5.19615i −0.250873 0.434524i
\(144\) 0 0
\(145\) 4.00000 6.92820i 0.332182 0.575356i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 + 10.3923i −0.491539 + 0.851371i −0.999953 0.00974235i \(-0.996899\pi\)
0.508413 + 0.861113i \(0.330232\pi\)
\(150\) 0 0
\(151\) 4.00000 + 6.92820i 0.325515 + 0.563809i 0.981617 0.190864i \(-0.0611289\pi\)
−0.656101 + 0.754673i \(0.727796\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −1.00000 1.73205i −0.0798087 0.138233i 0.823359 0.567521i \(-0.192098\pi\)
−0.903167 + 0.429289i \(0.858764\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 20.0000 + 6.92820i 1.57622 + 0.546019i
\(162\) 0 0
\(163\) 2.00000 3.46410i 0.156652 0.271329i −0.777007 0.629492i \(-0.783263\pi\)
0.933659 + 0.358162i \(0.116597\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.00000 13.8564i 0.608229 1.05348i −0.383304 0.923622i \(-0.625214\pi\)
0.991532 0.129861i \(-0.0414530\pi\)
\(174\) 0 0
\(175\) −2.00000 + 1.73205i −0.151186 + 0.130931i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3.00000 + 5.19615i 0.224231 + 0.388379i 0.956088 0.293079i \(-0.0946798\pi\)
−0.731858 + 0.681457i \(0.761346\pi\)
\(180\) 0 0
\(181\) −15.0000 −1.11494 −0.557471 0.830197i \(-0.688228\pi\)
−0.557471 + 0.830197i \(0.688228\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.00000 + 1.73205i 0.0735215 + 0.127343i
\(186\) 0 0
\(187\) −8.00000 + 13.8564i −0.585018 + 1.01328i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.00000 5.19615i 0.217072 0.375980i −0.736839 0.676068i \(-0.763683\pi\)
0.953912 + 0.300088i \(0.0970159\pi\)
\(192\) 0 0
\(193\) −5.50000 9.52628i −0.395899 0.685717i 0.597317 0.802005i \(-0.296234\pi\)
−0.993215 + 0.116289i \(0.962900\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.00000 −0.569976 −0.284988 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) 0 0
\(199\) −4.00000 6.92820i −0.283552 0.491127i 0.688705 0.725042i \(-0.258180\pi\)
−0.972257 + 0.233915i \(0.924846\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.00000 10.3923i −0.140372 0.729397i
\(204\) 0 0
\(205\) 6.00000 10.3923i 0.419058 0.725830i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −11.0000 + 19.0526i −0.750194 + 1.29937i
\(216\) 0 0
\(217\) 6.00000 5.19615i 0.407307 0.352738i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −12.0000 20.7846i −0.807207 1.39812i
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.00000 15.5885i −0.597351 1.03464i −0.993210 0.116331i \(-0.962887\pi\)
0.395860 0.918311i \(-0.370447\pi\)
\(228\) 0 0
\(229\) −0.500000 + 0.866025i −0.0330409 + 0.0572286i −0.882073 0.471113i \(-0.843853\pi\)
0.849032 + 0.528341i \(0.177186\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.00000 12.1244i 0.458585 0.794293i −0.540301 0.841472i \(-0.681690\pi\)
0.998886 + 0.0471787i \(0.0150230\pi\)
\(234\) 0 0
\(235\) 6.00000 + 10.3923i 0.391397 + 0.677919i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −18.0000 −1.16432 −0.582162 0.813073i \(-0.697793\pi\)
−0.582162 + 0.813073i \(0.697793\pi\)
\(240\) 0 0
\(241\) −7.00000 12.1244i −0.450910 0.780998i 0.547533 0.836784i \(-0.315567\pi\)
−0.998443 + 0.0557856i \(0.982234\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.00000 13.8564i 0.127775 0.885253i
\(246\) 0 0
\(247\) −1.50000 + 2.59808i −0.0954427 + 0.165312i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.00000 15.5885i 0.561405 0.972381i −0.435970 0.899961i \(-0.643595\pi\)
0.997374 0.0724199i \(-0.0230722\pi\)
\(258\) 0 0
\(259\) 2.50000 + 0.866025i 0.155342 + 0.0538122i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.00000 + 10.3923i 0.369976 + 0.640817i 0.989561 0.144112i \(-0.0460326\pi\)
−0.619586 + 0.784929i \(0.712699\pi\)
\(264\) 0 0
\(265\) 24.0000 1.47431
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.00000 1.73205i −0.0609711 0.105605i 0.833929 0.551872i \(-0.186086\pi\)
−0.894900 + 0.446267i \(0.852753\pi\)
\(270\) 0 0
\(271\) 12.0000 20.7846i 0.728948 1.26258i −0.228380 0.973572i \(-0.573343\pi\)
0.957328 0.289003i \(-0.0933238\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.00000 + 1.73205i −0.0603023 + 0.104447i
\(276\) 0 0
\(277\) −8.50000 14.7224i −0.510716 0.884585i −0.999923 0.0124177i \(-0.996047\pi\)
0.489207 0.872167i \(-0.337286\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −20.0000 −1.19310 −0.596550 0.802576i \(-0.703462\pi\)
−0.596550 + 0.802576i \(0.703462\pi\)
\(282\) 0 0
\(283\) −9.50000 16.4545i −0.564716 0.978117i −0.997076 0.0764162i \(-0.975652\pi\)
0.432360 0.901701i \(-0.357681\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.00000 15.5885i −0.177084 0.920158i
\(288\) 0 0
\(289\) −23.5000 + 40.7032i −1.38235 + 2.39431i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −12.0000 + 20.7846i −0.693978 + 1.20201i
\(300\) 0 0
\(301\) 5.50000 + 28.5788i 0.317015 + 1.64726i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6.00000 + 10.3923i 0.343559 + 0.595062i
\(306\) 0 0
\(307\) −23.0000 −1.31268 −0.656340 0.754466i \(-0.727896\pi\)
−0.656340 + 0.754466i \(0.727896\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.00000 1.73205i −0.0567048 0.0982156i 0.836280 0.548303i \(-0.184726\pi\)
−0.892984 + 0.450088i \(0.851393\pi\)
\(312\) 0 0
\(313\) 8.50000 14.7224i 0.480448 0.832161i −0.519300 0.854592i \(-0.673807\pi\)
0.999748 + 0.0224310i \(0.00714060\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.0000 + 20.7846i −0.673987 + 1.16738i 0.302777 + 0.953062i \(0.402086\pi\)
−0.976764 + 0.214318i \(0.931247\pi\)
\(318\) 0 0
\(319\) −4.00000 6.92820i −0.223957 0.387905i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) −1.50000 2.59808i −0.0832050 0.144115i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 15.0000 + 5.19615i 0.826977 + 0.286473i
\(330\) 0 0
\(331\) −8.50000 + 14.7224i −0.467202 + 0.809218i −0.999298 0.0374662i \(-0.988071\pi\)
0.532096 + 0.846684i \(0.321405\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 26.0000 1.42053
\(336\) 0 0
\(337\) 21.0000 1.14394 0.571971 0.820274i \(-0.306179\pi\)
0.571971 + 0.820274i \(0.306179\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.00000 5.19615i 0.162459 0.281387i
\(342\) 0 0
\(343\) −10.0000 15.5885i −0.539949 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.0000 + 20.7846i 0.644194 + 1.11578i 0.984487 + 0.175457i \(0.0561403\pi\)
−0.340293 + 0.940319i \(0.610526\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3.00000 5.19615i −0.159674 0.276563i 0.775077 0.631867i \(-0.217711\pi\)
−0.934751 + 0.355303i \(0.884378\pi\)
\(354\) 0 0
\(355\) −10.0000 + 17.3205i −0.530745 + 0.919277i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10.0000 + 17.3205i −0.527780 + 0.914141i 0.471696 + 0.881761i \(0.343642\pi\)
−0.999476 + 0.0323801i \(0.989691\pi\)
\(360\) 0 0
\(361\) 9.00000 + 15.5885i 0.473684 + 0.820445i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −22.0000 −1.15153
\(366\) 0 0
\(367\) −2.50000 4.33013i −0.130499 0.226031i 0.793370 0.608740i \(-0.208325\pi\)
−0.923869 + 0.382709i \(0.874991\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 24.0000 20.7846i 1.24602 1.07908i
\(372\) 0 0
\(373\) 2.50000 4.33013i 0.129445 0.224205i −0.794017 0.607896i \(-0.792014\pi\)
0.923462 + 0.383691i \(0.125347\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 13.0000 0.667765 0.333883 0.942615i \(-0.391641\pi\)
0.333883 + 0.942615i \(0.391641\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −14.0000 + 24.2487i −0.715367 + 1.23905i 0.247451 + 0.968900i \(0.420407\pi\)
−0.962818 + 0.270151i \(0.912926\pi\)
\(384\) 0 0
\(385\) −2.00000 10.3923i −0.101929 0.529641i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.00000 + 8.66025i 0.253510 + 0.439092i 0.964490 0.264120i \(-0.0850816\pi\)
−0.710980 + 0.703213i \(0.751748\pi\)
\(390\) 0 0
\(391\) 64.0000 3.23662
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.00000 + 5.19615i 0.150946 + 0.261447i
\(396\) 0 0
\(397\) −1.50000 + 2.59808i −0.0752828 + 0.130394i −0.901209 0.433384i \(-0.857319\pi\)
0.825926 + 0.563778i \(0.190653\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 + 10.3923i −0.299626 + 0.518967i −0.976050 0.217545i \(-0.930195\pi\)
0.676425 + 0.736512i \(0.263528\pi\)
\(402\) 0 0
\(403\) 4.50000 + 7.79423i 0.224161 + 0.388258i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) 0 0
\(409\) 9.50000 + 16.4545i 0.469745 + 0.813622i 0.999402 0.0345902i \(-0.0110126\pi\)
−0.529657 + 0.848212i \(0.677679\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −8.00000 + 6.92820i −0.393654 + 0.340915i
\(414\) 0 0
\(415\) 2.00000 3.46410i 0.0981761 0.170046i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 0 0
\(421\) −27.0000 −1.31590 −0.657950 0.753062i \(-0.728576\pi\)
−0.657950 + 0.753062i \(0.728576\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4.00000 + 6.92820i −0.194029 + 0.336067i
\(426\) 0 0
\(427\) 15.0000 + 5.19615i 0.725901 + 0.251459i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −15.0000 25.9808i −0.722525 1.25145i −0.959985 0.280052i \(-0.909648\pi\)
0.237460 0.971397i \(-0.423685\pi\)
\(432\) 0 0
\(433\) −25.0000 −1.20142 −0.600712 0.799466i \(-0.705116\pi\)
−0.600712 + 0.799466i \(0.705116\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.00000 6.92820i −0.191346 0.331421i
\(438\) 0 0
\(439\) −12.0000 + 20.7846i −0.572729 + 0.991995i 0.423556 + 0.905870i \(0.360782\pi\)
−0.996284 + 0.0861252i \(0.972552\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.00000 + 3.46410i −0.0950229 + 0.164584i −0.909618 0.415445i \(-0.863626\pi\)
0.814595 + 0.580030i \(0.196959\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 0 0
\(451\) −6.00000 10.3923i −0.282529 0.489355i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 15.0000 + 5.19615i 0.703211 + 0.243599i
\(456\) 0 0
\(457\) −6.50000 + 11.2583i −0.304057 + 0.526642i −0.977051 0.213006i \(-0.931675\pi\)
0.672994 + 0.739648i \(0.265008\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −4.00000 −0.186299 −0.0931493 0.995652i \(-0.529693\pi\)
−0.0931493 + 0.995652i \(0.529693\pi\)
\(462\) 0 0
\(463\) −11.0000 −0.511213 −0.255607 0.966781i \(-0.582275\pi\)
−0.255607 + 0.966781i \(0.582275\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 17.0000 29.4449i 0.786666 1.36255i −0.141332 0.989962i \(-0.545139\pi\)
0.927999 0.372584i \(-0.121528\pi\)
\(468\) 0 0
\(469\) 26.0000 22.5167i 1.20057 1.03972i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 11.0000 + 19.0526i 0.505781 + 0.876038i
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −14.0000 24.2487i −0.639676 1.10795i −0.985504 0.169654i \(-0.945735\pi\)
0.345827 0.938298i \(-0.387598\pi\)
\(480\) 0 0
\(481\) −1.50000 + 2.59808i −0.0683941 + 0.118462i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −10.0000 + 17.3205i −0.454077 + 0.786484i
\(486\) 0 0
\(487\) 9.50000 + 16.4545i 0.430486 + 0.745624i 0.996915 0.0784867i \(-0.0250088\pi\)
−0.566429 + 0.824110i \(0.691675\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 0 0
\(493\) −16.0000 27.7128i −0.720604 1.24812i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.00000 + 25.9808i 0.224281 + 1.16540i
\(498\) 0 0
\(499\) 14.5000 25.1147i 0.649109 1.12429i −0.334227 0.942493i \(-0.608475\pi\)
0.983336 0.181797i \(-0.0581915\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) −20.0000 −0.889988
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 9.00000 15.5885i 0.398918 0.690946i −0.594675 0.803966i \(-0.702719\pi\)
0.993593 + 0.113020i \(0.0360525\pi\)
\(510\) 0 0
\(511\) −22.0000 + 19.0526i −0.973223 + 0.842836i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −11.0000 19.0526i −0.484718 0.839556i
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −18.0000 31.1769i −0.788594 1.36589i −0.926828 0.375486i \(-0.877476\pi\)
0.138234 0.990400i \(-0.455857\pi\)
\(522\) 0 0
\(523\) 15.5000 26.8468i 0.677768 1.17393i −0.297884 0.954602i \(-0.596281\pi\)
0.975652 0.219326i \(-0.0703858\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000 20.7846i 0.522728 0.905392i
\(528\) 0 0
\(529\) −20.5000 35.5070i −0.891304 1.54378i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 18.0000 0.779667
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −11.0000 8.66025i −0.473804 0.373024i
\(540\) 0 0
\(541\) 7.50000 12.9904i 0.322450 0.558500i −0.658543 0.752543i \(-0.728827\pi\)
0.980993 + 0.194043i \(0.0621602\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −22.0000 −0.942376
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.00000 + 3.46410i −0.0852029 + 0.147576i
\(552\) 0 0
\(553\) 7.50000 + 2.59808i 0.318932 + 0.110481i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11.0000 + 19.0526i 0.466085 + 0.807283i 0.999250 0.0387286i \(-0.0123308\pi\)
−0.533165 + 0.846011i \(0.678997\pi\)
\(558\) 0 0
\(559\) −33.0000 −1.39575
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −23.0000 39.8372i −0.969334 1.67894i −0.697489 0.716596i \(-0.745699\pi\)
−0.271846 0.962341i \(-0.587634\pi\)
\(564\) 0 0
\(565\) −14.0000 + 24.2487i −0.588984 + 1.02015i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.00000 5.19615i 0.125767 0.217834i −0.796266 0.604947i \(-0.793194\pi\)
0.922032 + 0.387113i \(0.126528\pi\)
\(570\) 0 0
\(571\) 10.5000 + 18.1865i 0.439411 + 0.761083i 0.997644 0.0686016i \(-0.0218537\pi\)
−0.558233 + 0.829684i \(0.688520\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 8.00000 0.333623
\(576\) 0 0
\(577\) 20.5000 + 35.5070i 0.853426 + 1.47818i 0.878097 + 0.478482i \(0.158813\pi\)
−0.0246713 + 0.999696i \(0.507854\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1.00000 5.19615i −0.0414870 0.215573i
\(582\) 0 0
\(583\) 12.0000 20.7846i 0.496989 0.860811i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −32.0000 −1.32078 −0.660391 0.750922i \(-0.729609\pi\)
−0.660391 + 0.750922i \(0.729609\pi\)
\(588\) 0 0
\(589\) −3.00000 −0.123613
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −3.00000 + 5.19615i −0.123195 + 0.213380i −0.921026 0.389501i \(-0.872647\pi\)
0.797831 + 0.602881i \(0.205981\pi\)
\(594\) 0 0
\(595\) −8.00000 41.5692i −0.327968 1.70417i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6.00000 10.3923i −0.245153 0.424618i 0.717021 0.697051i \(-0.245505\pi\)
−0.962175 + 0.272433i \(0.912172\pi\)
\(600\) 0 0
\(601\) −1.00000 −0.0407909 −0.0203954 0.999792i \(-0.506493\pi\)
−0.0203954 + 0.999792i \(0.506493\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.00000 + 12.1244i 0.284590 + 0.492925i
\(606\) 0 0
\(607\) 1.50000 2.59808i 0.0608831 0.105453i −0.833977 0.551799i \(-0.813942\pi\)
0.894860 + 0.446346i \(0.147275\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.00000 + 15.5885i −0.364101 + 0.630641i
\(612\) 0 0
\(613\) 15.0000 + 25.9808i 0.605844 + 1.04935i 0.991917 + 0.126885i \(0.0404979\pi\)
−0.386073 + 0.922468i \(0.626169\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −26.0000 −1.04672 −0.523360 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) 0 0
\(619\) 5.50000 + 9.52628i 0.221064 + 0.382893i 0.955131 0.296183i \(-0.0957138\pi\)
−0.734068 + 0.679076i \(0.762380\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9.50000 16.4545i 0.380000 0.658179i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −3.00000 + 5.19615i −0.119051 + 0.206203i
\(636\) 0 0
\(637\) 19.5000 7.79423i 0.772618 0.308819i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −20.0000 34.6410i −0.789953 1.36824i −0.925995 0.377535i \(-0.876772\pi\)
0.136043 0.990703i \(-0.456562\pi\)
\(642\) 0 0
\(643\) 35.0000 1.38027 0.690133 0.723683i \(-0.257552\pi\)
0.690133 + 0.723683i \(0.257552\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.00000 + 5.19615i 0.117942 + 0.204282i 0.918952 0.394369i \(-0.129037\pi\)
−0.801010 + 0.598651i \(0.795704\pi\)
\(648\) 0 0
\(649\) −4.00000 + 6.92820i −0.157014 + 0.271956i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3.00000 + 5.19615i −0.117399 + 0.203341i −0.918736 0.394872i \(-0.870789\pi\)
0.801337 + 0.598213i \(0.204122\pi\)
\(654\) 0 0
\(655\) −2.00000 3.46410i −0.0781465 0.135354i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 28.0000 1.09073 0.545363 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 0 0
\(661\) 14.5000 + 25.1147i 0.563985 + 0.976850i 0.997143 + 0.0755324i \(0.0240656\pi\)
−0.433159 + 0.901318i \(0.642601\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4.00000 + 3.46410i −0.155113 + 0.134332i
\(666\) 0 0
\(667\) −16.0000 + 27.7128i −0.619522 + 1.07304i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.00000 10.3923i 0.230599 0.399409i −0.727386 0.686229i \(-0.759265\pi\)
0.957984 + 0.286820i \(0.0925982\pi\)
\(678\) 0 0
\(679\) 5.00000 + 25.9808i 0.191882 + 0.997050i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 18.0000 + 31.1769i 0.688751 + 1.19295i 0.972242 + 0.233977i \(0.0751739\pi\)
−0.283491 + 0.958975i \(0.591493\pi\)
\(684\) 0 0
\(685\) −8.00000 −0.305664
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 18.0000 + 31.1769i 0.685745 + 1.18775i
\(690\) 0 0
\(691\) 21.5000 37.2391i 0.817899 1.41664i −0.0893292 0.996002i \(-0.528472\pi\)
0.907228 0.420640i \(-0.138194\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.00000 8.66025i 0.189661 0.328502i
\(696\) 0 0
\(697\) −24.0000 41.5692i −0.909065 1.57455i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 8.00000 0.302156 0.151078 0.988522i \(-0.451726\pi\)
0.151078 + 0.988522i \(0.451726\pi\)
\(702\) 0 0
\(703\) −0.500000 0.866025i −0.0188579 0.0326628i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −20.0000 + 17.3205i −0.752177 + 0.651405i
\(708\) 0 0
\(709\) −7.00000 + 12.1244i −0.262891 + 0.455340i −0.967009 0.254743i \(-0.918009\pi\)
0.704118 + 0.710083i \(0.251342\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 12.0000 0.448775
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −3.00000 + 5.19615i −0.111881 + 0.193784i −0.916529 0.399969i \(-0.869021\pi\)
0.804648 + 0.593753i \(0.202354\pi\)
\(720\) 0 0
\(721\) −27.5000 9.52628i −1.02415 0.354777i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2.00000 3.46410i −0.0742781 0.128654i
\(726\) 0 0
\(727\) −23.0000 −0.853023 −0.426511 0.904482i \(-0.640258\pi\)
−0.426511 + 0.904482i \(0.640258\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 44.0000 + 76.2102i 1.62740 + 2.81874i
\(732\) 0 0
\(733\) −22.5000 + 38.9711i −0.831056 + 1.43943i 0.0661448 + 0.997810i \(0.478930\pi\)
−0.897201 + 0.441622i \(0.854403\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 13.0000 22.5167i 0.478861 0.829412i
\(738\) 0 0
\(739\) 4.50000 + 7.79423i 0.165535 + 0.286715i 0.936845 0.349744i \(-0.113732\pi\)
−0.771310 + 0.636460i \(0.780398\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 18.0000 0.660356 0.330178 0.943919i \(-0.392891\pi\)
0.330178 + 0.943919i \(0.392891\pi\)
\(744\) 0 0
\(745\) −12.0000 20.7846i −0.439646 0.761489i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −7.50000 + 12.9904i −0.273679 + 0.474026i −0.969801 0.243898i \(-0.921574\pi\)
0.696122 + 0.717923i \(0.254907\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) 42.0000 1.52652 0.763258 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.00000 6.92820i 0.145000 0.251147i −0.784373 0.620289i \(-0.787015\pi\)
0.929373 + 0.369142i \(0.120348\pi\)
\(762\) 0 0
\(763\) −22.0000 + 19.0526i −0.796453 + 0.689749i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −6.00000 10.3923i −0.216647 0.375244i
\(768\) 0 0
\(769\) 31.0000 1.11789 0.558944 0.829205i \(-0.311207\pi\)
0.558944 + 0.829205i \(0.311207\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 11.0000 + 19.0526i 0.395643 + 0.685273i 0.993183 0.116566i \(-0.0371886\pi\)
−0.597540 + 0.801839i \(0.703855\pi\)
\(774\) 0 0
\(775\) 1.50000 2.59808i 0.0538816 0.0933257i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.00000 + 5.19615i −0.107486 + 0.186171i
\(780\) 0 0
\(781\) 10.0000 + 17.3205i 0.357828 + 0.619777i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) −12.0000 20.7846i −0.427754 0.740891i 0.568919 0.822393i \(-0.307362\pi\)
−0.996673 + 0.0815020i \(0.974028\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 7.00000 + 36.3731i 0.248891 + 1.29328i
\(792\) 0 0
\(793\) −9.00000 + 15.5885i −0.319599 + 0.553562i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 48.0000 1.70025 0.850124 0.526583i \(-0.176527\pi\)
0.850124 + 0.526583i \(0.176527\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11.0000 + 19.0526i −0.388182 + 0.672350i
\(804\) 0 0
\(805\) −32.0000 + 27.7128i −1.12785 + 0.976748i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 11.0000 + 19.0526i 0.386739 + 0.669852i 0.992009 0.126168i \(-0.0402680\pi\)
−0.605269 + 0.796021i \(0.706935\pi\)
\(810\) 0 0
\(811\) −32.0000 −1.12367 −0.561836 0.827249i \(-0.689905\pi\)
−0.561836 + 0.827249i \(0.689905\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 4.00000 + 6.92820i 0.140114 + 0.242684i
\(816\) 0 0
\(817\) 5.50000 9.52628i 0.192421 0.333282i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.00000 1.73205i 0.0349002 0.0604490i −0.848048 0.529920i \(-0.822222\pi\)
0.882948 + 0.469471i \(0.155555\pi\)
\(822\) 0 0
\(823\) −20.0000 34.6410i −0.697156 1.20751i −0.969448 0.245295i \(-0.921115\pi\)
0.272292 0.962215i \(-0.412218\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −54.0000 −1.87776 −0.938882 0.344239i \(-0.888137\pi\)
−0.938882 + 0.344239i \(0.888137\pi\)
\(828\) 0 0
\(829\) 5.50000 + 9.52628i 0.191023 + 0.330861i 0.945589 0.325362i \(-0.105486\pi\)
−0.754567 + 0.656223i \(0.772153\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −44.0000 34.6410i −1.52451 1.20024i
\(834\) 0 0
\(835\) −2.00000 + 3.46410i −0.0692129 + 0.119880i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 4.00000 0.138095 0.0690477 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4.00000 6.92820i 0.137604 0.238337i
\(846\) 0 0
\(847\) 17.5000 + 6.06218i 0.601307 + 0.208299i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −4.00000 6.92820i −0.137118 0.237496i
\(852\) 0 0
\(853\) 23.0000 0.787505 0.393753 0.919216i \(-0.371177\pi\)
0.393753 + 0.919216i \(0.371177\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(858\) 0 0
\(859\) −4.00000 + 6.92820i −0.136478 + 0.236387i −0.926161 0.377128i \(-0.876912\pi\)
0.789683 + 0.613515i \(0.210245\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 23.0000 39.8372i 0.782929 1.35607i −0.147299 0.989092i \(-0.547058\pi\)
0.930228 0.366981i \(-0.119609\pi\)
\(864\) 0 0
\(865\) 16.0000 + 27.7128i 0.544016 + 0.942264i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 6.00000 0.203536
\(870\) 0 0
\(871\) 19.5000 + 33.7750i 0.660732 + 1.14442i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −6.00000 31.1769i −0.202837 1.05397i
\(876\) 0 0
\(877\) 19.0000 32.9090i 0.641584 1.11126i −0.343495 0.939155i \(-0.611611\pi\)
0.985079 0.172102i \(-0.0550559\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −48.0000 −1.61716 −0.808581 0.588386i \(-0.799764\pi\)
−0.808581 + 0.588386i \(0.799764\pi\)
\(882\) 0 0
\(883\) 29.0000 0.975928 0.487964 0.872864i \(-0.337740\pi\)
0.487964 + 0.872864i \(0.337740\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.00000 + 5.19615i −0.100730 + 0.174470i −0.911986 0.410222i \(-0.865451\pi\)
0.811256 + 0.584692i \(0.198785\pi\)
\(888\) 0 0
\(889\) 1.50000 + 7.79423i 0.0503084 + 0.261410i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −3.00000 5.19615i −0.100391 0.173883i
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 6.00000 + 10.3923i 0.200111 + 0.346603i
\(900\) 0 0
\(901\) 48.0000 83.1384i 1.59911 2.76974i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 15.0000 25.9808i 0.498617 0.863630i
\(906\) 0 0
\(907\) −10.5000 18.1865i −0.348647 0.603874i 0.637363 0.770564i \(-0.280025\pi\)
−0.986009 + 0.166690i \(0.946692\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) −2.00000 3.46410i −0.0661903 0.114645i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −5.00000 1.73205i −0.165115 0.0571974i
\(918\) 0 0
\(919\) 5.50000 9.52628i 0.181428 0.314243i −0.760939 0.648824i \(-0.775261\pi\)
0.942367 + 0.334581i \(0.108595\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −30.0000 −0.987462
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 3.00000 5.19615i 0.0984268 0.170480i −0.812607 0.582812i \(-0.801952\pi\)
0.911034 + 0.412332i \(0.135286\pi\)
\(930\) 0 0
\(931\) −1.00000 + 6.92820i −0.0327737 + 0.227063i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −16.0000 27.7128i −0.523256 0.906306i
\(936\) 0 0
\(937\) −49.0000 −1.60076 −0.800380 0.599493i \(-0.795369\pi\)
−0.800380 + 0.599493i \(0.795369\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 26.0000 + 45.0333i 0.847576 + 1.46804i 0.883365 + 0.468685i \(0.155272\pi\)
−0.0357896 + 0.999359i \(0.511395\pi\)
\(942\) 0 0
\(943\) −24.0000 + 41.5692i −0.781548 + 1.35368i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 5.00000 8.66025i 0.162478 0.281420i −0.773279 0.634066i \(-0.781385\pi\)
0.935757 + 0.352646i \(0.114718\pi\)
\(948\) 0 0
\(949\) −16.5000 28.5788i −0.535613 0.927708i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 28.0000 0.907009 0.453504 0.891254i \(-0.350174\pi\)
0.453504 + 0.891254i \(0.350174\pi\)
\(954\) 0 0
\(955\) 6.00000 + 10.3923i 0.194155 + 0.336287i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.00000 + 6.92820i −0.258333 + 0.223723i
\(960\) 0 0
\(961\) 11.0000 19.0526i 0.354839 0.614599i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 22.0000 0.708205
\(966\) 0 0
\(967\) −31.0000 −0.996893 −0.498446 0.866921i \(-0.666096\pi\)
−0.498446 + 0.866921i \(0.666096\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 18.0000 31.1769i 0.577647 1.00051i −0.418101 0.908401i \(-0.637304\pi\)
0.995748 0.0921142i \(-0.0293625\pi\)
\(972\) 0 0
\(973\) −2.50000 12.9904i −0.0801463 0.416452i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −9.00000 15.5885i −0.287936 0.498719i 0.685381 0.728184i \(-0.259636\pi\)
−0.973317 + 0.229465i \(0.926302\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −6.00000 10.3923i −0.191370 0.331463i 0.754334 0.656490i \(-0.227960\pi\)
−0.945705 + 0.325027i \(0.894626\pi\)
\(984\) 0 0
\(985\) 8.00000 13.8564i 0.254901 0.441502i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 44.0000 76.2102i 1.39912 2.42334i
\(990\) 0 0
\(991\) −5.50000 9.52628i −0.174713 0.302612i 0.765349 0.643616i \(-0.222567\pi\)
−0.940062 + 0.341004i \(0.889233\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) 0 0
\(997\) 0.500000 + 0.866025i 0.0158352 + 0.0274273i 0.873834 0.486224i \(-0.161626\pi\)
−0.857999 + 0.513651i \(0.828293\pi\)
\(998\) 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 252.2.k.a.37.1 2
3.2 odd 2 84.2.i.a.37.1 yes 2
4.3 odd 2 1008.2.s.c.289.1 2
7.2 even 3 1764.2.a.h.1.1 1
7.3 odd 6 1764.2.k.j.361.1 2
7.4 even 3 inner 252.2.k.a.109.1 2
7.5 odd 6 1764.2.a.c.1.1 1
7.6 odd 2 1764.2.k.j.1549.1 2
9.2 odd 6 2268.2.i.g.2053.1 2
9.4 even 3 2268.2.l.g.541.1 2
9.5 odd 6 2268.2.l.b.541.1 2
9.7 even 3 2268.2.i.b.2053.1 2
12.11 even 2 336.2.q.c.289.1 2
15.2 even 4 2100.2.bc.a.1549.2 4
15.8 even 4 2100.2.bc.a.1549.1 4
15.14 odd 2 2100.2.q.b.1801.1 2
21.2 odd 6 588.2.a.a.1.1 1
21.5 even 6 588.2.a.f.1.1 1
21.11 odd 6 84.2.i.a.25.1 2
21.17 even 6 588.2.i.b.361.1 2
21.20 even 2 588.2.i.b.373.1 2
24.5 odd 2 1344.2.q.b.961.1 2
24.11 even 2 1344.2.q.n.961.1 2
28.11 odd 6 1008.2.s.c.865.1 2
28.19 even 6 7056.2.a.o.1.1 1
28.23 odd 6 7056.2.a.bs.1.1 1
63.4 even 3 2268.2.i.b.865.1 2
63.11 odd 6 2268.2.l.b.109.1 2
63.25 even 3 2268.2.l.g.109.1 2
63.32 odd 6 2268.2.i.g.865.1 2
84.11 even 6 336.2.q.c.193.1 2
84.23 even 6 2352.2.a.o.1.1 1
84.47 odd 6 2352.2.a.k.1.1 1
84.59 odd 6 2352.2.q.q.1537.1 2
84.83 odd 2 2352.2.q.q.961.1 2
105.32 even 12 2100.2.bc.a.949.1 4
105.53 even 12 2100.2.bc.a.949.2 4
105.74 odd 6 2100.2.q.b.1201.1 2
168.5 even 6 9408.2.a.i.1.1 1
168.11 even 6 1344.2.q.n.193.1 2
168.53 odd 6 1344.2.q.b.193.1 2
168.107 even 6 9408.2.a.bi.1.1 1
168.131 odd 6 9408.2.a.bx.1.1 1
168.149 odd 6 9408.2.a.cx.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.2.i.a.25.1 2 21.11 odd 6
84.2.i.a.37.1 yes 2 3.2 odd 2
252.2.k.a.37.1 2 1.1 even 1 trivial
252.2.k.a.109.1 2 7.4 even 3 inner
336.2.q.c.193.1 2 84.11 even 6
336.2.q.c.289.1 2 12.11 even 2
588.2.a.a.1.1 1 21.2 odd 6
588.2.a.f.1.1 1 21.5 even 6
588.2.i.b.361.1 2 21.17 even 6
588.2.i.b.373.1 2 21.20 even 2
1008.2.s.c.289.1 2 4.3 odd 2
1008.2.s.c.865.1 2 28.11 odd 6
1344.2.q.b.193.1 2 168.53 odd 6
1344.2.q.b.961.1 2 24.5 odd 2
1344.2.q.n.193.1 2 168.11 even 6
1344.2.q.n.961.1 2 24.11 even 2
1764.2.a.c.1.1 1 7.5 odd 6
1764.2.a.h.1.1 1 7.2 even 3
1764.2.k.j.361.1 2 7.3 odd 6
1764.2.k.j.1549.1 2 7.6 odd 2
2100.2.q.b.1201.1 2 105.74 odd 6
2100.2.q.b.1801.1 2 15.14 odd 2
2100.2.bc.a.949.1 4 105.32 even 12
2100.2.bc.a.949.2 4 105.53 even 12
2100.2.bc.a.1549.1 4 15.8 even 4
2100.2.bc.a.1549.2 4 15.2 even 4
2268.2.i.b.865.1 2 63.4 even 3
2268.2.i.b.2053.1 2 9.7 even 3
2268.2.i.g.865.1 2 63.32 odd 6
2268.2.i.g.2053.1 2 9.2 odd 6
2268.2.l.b.109.1 2 63.11 odd 6
2268.2.l.b.541.1 2 9.5 odd 6
2268.2.l.g.109.1 2 63.25 even 3
2268.2.l.g.541.1 2 9.4 even 3
2352.2.a.k.1.1 1 84.47 odd 6
2352.2.a.o.1.1 1 84.23 even 6
2352.2.q.q.961.1 2 84.83 odd 2
2352.2.q.q.1537.1 2 84.59 odd 6
7056.2.a.o.1.1 1 28.19 even 6
7056.2.a.bs.1.1 1 28.23 odd 6
9408.2.a.i.1.1 1 168.5 even 6
9408.2.a.bi.1.1 1 168.107 even 6
9408.2.a.bx.1.1 1 168.131 odd 6
9408.2.a.cx.1.1 1 168.149 odd 6