Properties

Label 196.4.e.e.165.1
Level $196$
Weight $4$
Character 196.165
Analytic conductor $11.564$
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] = [196,4,Mod(165,196)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(196, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.165");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 196.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5643743611\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 165.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 196.165
Dual form 196.4.e.e.177.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.00000 - 3.46410i) q^{3} +(10.0000 + 17.3205i) q^{5} +(5.50000 + 9.52628i) q^{9} +O(q^{10})\) \(q+(2.00000 - 3.46410i) q^{3} +(10.0000 + 17.3205i) q^{5} +(5.50000 + 9.52628i) q^{9} +(-22.0000 + 38.1051i) q^{11} -44.0000 q^{13} +80.0000 q^{15} +(-36.0000 + 62.3538i) q^{17} +(-50.0000 - 86.6025i) q^{19} +(60.0000 + 103.923i) q^{23} +(-137.500 + 238.157i) q^{25} +152.000 q^{27} +218.000 q^{29} +(140.000 - 242.487i) q^{31} +(88.0000 + 152.420i) q^{33} +(15.0000 + 25.9808i) q^{37} +(-88.0000 + 152.420i) q^{39} +120.000 q^{41} +220.000 q^{43} +(-110.000 + 190.526i) q^{45} +(-44.0000 - 76.2102i) q^{47} +(144.000 + 249.415i) q^{51} +(-55.0000 + 95.2628i) q^{53} -880.000 q^{55} -400.000 q^{57} +(-290.000 + 502.295i) q^{59} +(-190.000 - 329.090i) q^{61} +(-440.000 - 762.102i) q^{65} +(490.000 - 848.705i) q^{67} +480.000 q^{69} -112.000 q^{71} +(320.000 - 554.256i) q^{73} +(550.000 + 952.628i) q^{75} +(244.000 + 422.620i) q^{79} +(155.500 - 269.334i) q^{81} +660.000 q^{83} -1440.00 q^{85} +(436.000 - 755.174i) q^{87} +(-160.000 - 277.128i) q^{89} +(-560.000 - 969.948i) q^{93} +(1000.00 - 1732.05i) q^{95} +248.000 q^{97} -484.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} + 20 q^{5} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{3} + 20 q^{5} + 11 q^{9} - 44 q^{11} - 88 q^{13} + 160 q^{15} - 72 q^{17} - 100 q^{19} + 120 q^{23} - 275 q^{25} + 304 q^{27} + 436 q^{29} + 280 q^{31} + 176 q^{33} + 30 q^{37} - 176 q^{39} + 240 q^{41} + 440 q^{43} - 220 q^{45} - 88 q^{47} + 288 q^{51} - 110 q^{53} - 1760 q^{55} - 800 q^{57} - 580 q^{59} - 380 q^{61} - 880 q^{65} + 980 q^{67} + 960 q^{69} - 224 q^{71} + 640 q^{73} + 1100 q^{75} + 488 q^{79} + 311 q^{81} + 1320 q^{83} - 2880 q^{85} + 872 q^{87} - 320 q^{89} - 1120 q^{93} + 2000 q^{95} + 496 q^{97} - 968 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).

\(n\) \(99\) \(101\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 2.00000 3.46410i 0.384900 0.666667i −0.606855 0.794812i \(-0.707569\pi\)
0.991755 + 0.128146i \(0.0409025\pi\)
\(4\) 0 0
\(5\) 10.0000 + 17.3205i 0.894427 + 1.54919i 0.834512 + 0.550990i \(0.185750\pi\)
0.0599153 + 0.998203i \(0.480917\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 5.50000 + 9.52628i 0.203704 + 0.352825i
\(10\) 0 0
\(11\) −22.0000 + 38.1051i −0.603023 + 1.04447i 0.389338 + 0.921095i \(0.372704\pi\)
−0.992361 + 0.123371i \(0.960630\pi\)
\(12\) 0 0
\(13\) −44.0000 −0.938723 −0.469362 0.883006i \(-0.655516\pi\)
−0.469362 + 0.883006i \(0.655516\pi\)
\(14\) 0 0
\(15\) 80.0000 1.37706
\(16\) 0 0
\(17\) −36.0000 + 62.3538i −0.513605 + 0.889590i 0.486271 + 0.873808i \(0.338357\pi\)
−0.999875 + 0.0157814i \(0.994976\pi\)
\(18\) 0 0
\(19\) −50.0000 86.6025i −0.603726 1.04568i −0.992251 0.124246i \(-0.960349\pi\)
0.388526 0.921438i \(-0.372984\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 60.0000 + 103.923i 0.543951 + 0.942150i 0.998672 + 0.0515165i \(0.0164055\pi\)
−0.454721 + 0.890634i \(0.650261\pi\)
\(24\) 0 0
\(25\) −137.500 + 238.157i −1.10000 + 1.90526i
\(26\) 0 0
\(27\) 152.000 1.08342
\(28\) 0 0
\(29\) 218.000 1.39592 0.697958 0.716138i \(-0.254092\pi\)
0.697958 + 0.716138i \(0.254092\pi\)
\(30\) 0 0
\(31\) 140.000 242.487i 0.811121 1.40490i −0.100960 0.994891i \(-0.532191\pi\)
0.912080 0.410012i \(-0.134475\pi\)
\(32\) 0 0
\(33\) 88.0000 + 152.420i 0.464207 + 0.804030i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 15.0000 + 25.9808i 0.0666482 + 0.115438i 0.897424 0.441169i \(-0.145436\pi\)
−0.830776 + 0.556607i \(0.812103\pi\)
\(38\) 0 0
\(39\) −88.0000 + 152.420i −0.361315 + 0.625816i
\(40\) 0 0
\(41\) 120.000 0.457094 0.228547 0.973533i \(-0.426603\pi\)
0.228547 + 0.973533i \(0.426603\pi\)
\(42\) 0 0
\(43\) 220.000 0.780225 0.390113 0.920767i \(-0.372436\pi\)
0.390113 + 0.920767i \(0.372436\pi\)
\(44\) 0 0
\(45\) −110.000 + 190.526i −0.364396 + 0.631153i
\(46\) 0 0
\(47\) −44.0000 76.2102i −0.136554 0.236519i 0.789636 0.613576i \(-0.210270\pi\)
−0.926190 + 0.377057i \(0.876936\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 144.000 + 249.415i 0.395373 + 0.684806i
\(52\) 0 0
\(53\) −55.0000 + 95.2628i −0.142544 + 0.246893i −0.928454 0.371448i \(-0.878862\pi\)
0.785910 + 0.618341i \(0.212195\pi\)
\(54\) 0 0
\(55\) −880.000 −2.15744
\(56\) 0 0
\(57\) −400.000 −0.929496
\(58\) 0 0
\(59\) −290.000 + 502.295i −0.639912 + 1.10836i 0.345540 + 0.938404i \(0.387696\pi\)
−0.985452 + 0.169955i \(0.945638\pi\)
\(60\) 0 0
\(61\) −190.000 329.090i −0.398803 0.690748i 0.594775 0.803892i \(-0.297241\pi\)
−0.993579 + 0.113144i \(0.963908\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −440.000 762.102i −0.839620 1.45426i
\(66\) 0 0
\(67\) 490.000 848.705i 0.893478 1.54755i 0.0578010 0.998328i \(-0.481591\pi\)
0.835677 0.549221i \(-0.185076\pi\)
\(68\) 0 0
\(69\) 480.000 0.837467
\(70\) 0 0
\(71\) −112.000 −0.187211 −0.0936053 0.995609i \(-0.529839\pi\)
−0.0936053 + 0.995609i \(0.529839\pi\)
\(72\) 0 0
\(73\) 320.000 554.256i 0.513057 0.888641i −0.486828 0.873498i \(-0.661846\pi\)
0.999885 0.0151432i \(-0.00482042\pi\)
\(74\) 0 0
\(75\) 550.000 + 952.628i 0.846780 + 1.46667i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 244.000 + 422.620i 0.347496 + 0.601880i 0.985804 0.167901i \(-0.0536989\pi\)
−0.638308 + 0.769781i \(0.720366\pi\)
\(80\) 0 0
\(81\) 155.500 269.334i 0.213306 0.369457i
\(82\) 0 0
\(83\) 660.000 0.872824 0.436412 0.899747i \(-0.356249\pi\)
0.436412 + 0.899747i \(0.356249\pi\)
\(84\) 0 0
\(85\) −1440.00 −1.83753
\(86\) 0 0
\(87\) 436.000 755.174i 0.537289 0.930611i
\(88\) 0 0
\(89\) −160.000 277.128i −0.190561 0.330062i 0.754875 0.655869i \(-0.227697\pi\)
−0.945436 + 0.325807i \(0.894364\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −560.000 969.948i −0.624401 1.08149i
\(94\) 0 0
\(95\) 1000.00 1732.05i 1.07998 1.87058i
\(96\) 0 0
\(97\) 248.000 0.259594 0.129797 0.991541i \(-0.458567\pi\)
0.129797 + 0.991541i \(0.458567\pi\)
\(98\) 0 0
\(99\) −484.000 −0.491352
\(100\) 0 0
\(101\) −110.000 + 190.526i −0.108370 + 0.187703i −0.915110 0.403204i \(-0.867897\pi\)
0.806740 + 0.590907i \(0.201230\pi\)
\(102\) 0 0
\(103\) −668.000 1157.01i −0.639029 1.10683i −0.985646 0.168824i \(-0.946003\pi\)
0.346617 0.938007i \(-0.387330\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −50.0000 86.6025i −0.0451746 0.0782447i 0.842554 0.538612i \(-0.181051\pi\)
−0.887729 + 0.460367i \(0.847718\pi\)
\(108\) 0 0
\(109\) −341.000 + 590.629i −0.299650 + 0.519009i −0.976056 0.217520i \(-0.930203\pi\)
0.676406 + 0.736529i \(0.263537\pi\)
\(110\) 0 0
\(111\) 120.000 0.102612
\(112\) 0 0
\(113\) 370.000 0.308024 0.154012 0.988069i \(-0.450781\pi\)
0.154012 + 0.988069i \(0.450781\pi\)
\(114\) 0 0
\(115\) −1200.00 + 2078.46i −0.973048 + 1.68537i
\(116\) 0 0
\(117\) −242.000 419.156i −0.191221 0.331205i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −302.500 523.945i −0.227273 0.393648i
\(122\) 0 0
\(123\) 240.000 415.692i 0.175936 0.304729i
\(124\) 0 0
\(125\) −3000.00 −2.14663
\(126\) 0 0
\(127\) −160.000 −0.111793 −0.0558965 0.998437i \(-0.517802\pi\)
−0.0558965 + 0.998437i \(0.517802\pi\)
\(128\) 0 0
\(129\) 440.000 762.102i 0.300309 0.520150i
\(130\) 0 0
\(131\) 150.000 + 259.808i 0.100042 + 0.173279i 0.911702 0.410852i \(-0.134769\pi\)
−0.811659 + 0.584131i \(0.801435\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1520.00 + 2632.72i 0.969043 + 1.67843i
\(136\) 0 0
\(137\) −595.000 + 1030.57i −0.371053 + 0.642683i −0.989728 0.142964i \(-0.954337\pi\)
0.618675 + 0.785647i \(0.287670\pi\)
\(138\) 0 0
\(139\) 2220.00 1.35466 0.677331 0.735679i \(-0.263137\pi\)
0.677331 + 0.735679i \(0.263137\pi\)
\(140\) 0 0
\(141\) −352.000 −0.210239
\(142\) 0 0
\(143\) 968.000 1676.63i 0.566072 0.980465i
\(144\) 0 0
\(145\) 2180.00 + 3775.87i 1.24855 + 2.16254i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1207.00 2090.59i −0.663633 1.14945i −0.979654 0.200694i \(-0.935680\pi\)
0.316021 0.948752i \(-0.397653\pi\)
\(150\) 0 0
\(151\) −1028.00 + 1780.55i −0.554023 + 0.959596i 0.443956 + 0.896049i \(0.353575\pi\)
−0.997979 + 0.0635472i \(0.979759\pi\)
\(152\) 0 0
\(153\) −792.000 −0.418493
\(154\) 0 0
\(155\) 5600.00 2.90195
\(156\) 0 0
\(157\) −530.000 + 917.987i −0.269418 + 0.466645i −0.968712 0.248189i \(-0.920165\pi\)
0.699294 + 0.714834i \(0.253498\pi\)
\(158\) 0 0
\(159\) 220.000 + 381.051i 0.109730 + 0.190059i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 990.000 + 1714.73i 0.475723 + 0.823976i 0.999613 0.0278098i \(-0.00885328\pi\)
−0.523891 + 0.851786i \(0.675520\pi\)
\(164\) 0 0
\(165\) −1760.00 + 3048.41i −0.830399 + 1.43829i
\(166\) 0 0
\(167\) 488.000 0.226123 0.113062 0.993588i \(-0.463934\pi\)
0.113062 + 0.993588i \(0.463934\pi\)
\(168\) 0 0
\(169\) −261.000 −0.118798
\(170\) 0 0
\(171\) 550.000 952.628i 0.245962 0.426019i
\(172\) 0 0
\(173\) −1738.00 3010.30i −0.763802 1.32294i −0.940878 0.338746i \(-0.889997\pi\)
0.177076 0.984197i \(-0.443336\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1160.00 + 2009.18i 0.492604 + 0.853215i
\(178\) 0 0
\(179\) 42.0000 72.7461i 0.0175376 0.0303760i −0.857123 0.515111i \(-0.827751\pi\)
0.874661 + 0.484735i \(0.161084\pi\)
\(180\) 0 0
\(181\) 2180.00 0.895238 0.447619 0.894224i \(-0.352272\pi\)
0.447619 + 0.894224i \(0.352272\pi\)
\(182\) 0 0
\(183\) −1520.00 −0.613998
\(184\) 0 0
\(185\) −300.000 + 519.615i −0.119224 + 0.206502i
\(186\) 0 0
\(187\) −1584.00 2743.57i −0.619431 1.07289i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −348.000 602.754i −0.131835 0.228344i 0.792549 0.609808i \(-0.208753\pi\)
−0.924384 + 0.381464i \(0.875420\pi\)
\(192\) 0 0
\(193\) −1345.00 + 2329.61i −0.501633 + 0.868854i 0.498365 + 0.866967i \(0.333934\pi\)
−0.999998 + 0.00188695i \(0.999399\pi\)
\(194\) 0 0
\(195\) −3520.00 −1.29268
\(196\) 0 0
\(197\) 5310.00 1.92042 0.960208 0.279287i \(-0.0900980\pi\)
0.960208 + 0.279287i \(0.0900980\pi\)
\(198\) 0 0
\(199\) −1540.00 + 2667.36i −0.548581 + 0.950171i 0.449791 + 0.893134i \(0.351499\pi\)
−0.998372 + 0.0570369i \(0.981835\pi\)
\(200\) 0 0
\(201\) −1960.00 3394.82i −0.687800 1.19130i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1200.00 + 2078.46i 0.408837 + 0.708127i
\(206\) 0 0
\(207\) −660.000 + 1143.15i −0.221610 + 0.383839i
\(208\) 0 0
\(209\) 4400.00 1.45624
\(210\) 0 0
\(211\) −4.00000 −0.00130508 −0.000652539 1.00000i \(-0.500208\pi\)
−0.000652539 1.00000i \(0.500208\pi\)
\(212\) 0 0
\(213\) −224.000 + 387.979i −0.0720574 + 0.124807i
\(214\) 0 0
\(215\) 2200.00 + 3810.51i 0.697855 + 1.20872i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1280.00 2217.03i −0.394952 0.684076i
\(220\) 0 0
\(221\) 1584.00 2743.57i 0.482133 0.835079i
\(222\) 0 0
\(223\) −2416.00 −0.725504 −0.362752 0.931886i \(-0.618163\pi\)
−0.362752 + 0.931886i \(0.618163\pi\)
\(224\) 0 0
\(225\) −3025.00 −0.896296
\(226\) 0 0
\(227\) 2486.00 4305.88i 0.726879 1.25899i −0.231316 0.972879i \(-0.574303\pi\)
0.958196 0.286113i \(-0.0923634\pi\)
\(228\) 0 0
\(229\) 2230.00 + 3862.47i 0.643505 + 1.11458i 0.984645 + 0.174570i \(0.0558536\pi\)
−0.341140 + 0.940012i \(0.610813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2275.00 3940.42i −0.639658 1.10792i −0.985508 0.169630i \(-0.945743\pi\)
0.345850 0.938290i \(-0.387590\pi\)
\(234\) 0 0
\(235\) 880.000 1524.20i 0.244276 0.423099i
\(236\) 0 0
\(237\) 1952.00 0.535004
\(238\) 0 0
\(239\) −2112.00 −0.571606 −0.285803 0.958288i \(-0.592260\pi\)
−0.285803 + 0.958288i \(0.592260\pi\)
\(240\) 0 0
\(241\) 2420.00 4191.56i 0.646829 1.12034i −0.337046 0.941488i \(-0.609428\pi\)
0.983876 0.178853i \(-0.0572388\pi\)
\(242\) 0 0
\(243\) 1430.00 + 2476.83i 0.377508 + 0.653864i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2200.00 + 3810.51i 0.566731 + 0.981608i
\(248\) 0 0
\(249\) 1320.00 2286.31i 0.335950 0.581883i
\(250\) 0 0
\(251\) −4340.00 −1.09139 −0.545694 0.837985i \(-0.683734\pi\)
−0.545694 + 0.837985i \(0.683734\pi\)
\(252\) 0 0
\(253\) −5280.00 −1.31206
\(254\) 0 0
\(255\) −2880.00 + 4988.31i −0.707265 + 1.22502i
\(256\) 0 0
\(257\) 1760.00 + 3048.41i 0.427182 + 0.739901i 0.996621 0.0821323i \(-0.0261730\pi\)
−0.569439 + 0.822033i \(0.692840\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1199.00 + 2076.73i 0.284353 + 0.492515i
\(262\) 0 0
\(263\) 1280.00 2217.03i 0.300107 0.519801i −0.676053 0.736853i \(-0.736311\pi\)
0.976160 + 0.217052i \(0.0696442\pi\)
\(264\) 0 0
\(265\) −2200.00 −0.509981
\(266\) 0 0
\(267\) −1280.00 −0.293388
\(268\) 0 0
\(269\) −3410.00 + 5906.29i −0.772905 + 1.33871i 0.163060 + 0.986616i \(0.447864\pi\)
−0.935965 + 0.352094i \(0.885470\pi\)
\(270\) 0 0
\(271\) −1760.00 3048.41i −0.394511 0.683312i 0.598528 0.801102i \(-0.295753\pi\)
−0.993039 + 0.117789i \(0.962419\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6050.00 10478.9i −1.32665 2.29783i
\(276\) 0 0
\(277\) 4345.00 7525.76i 0.942476 1.63242i 0.181747 0.983345i \(-0.441825\pi\)
0.760728 0.649070i \(-0.224842\pi\)
\(278\) 0 0
\(279\) 3080.00 0.660913
\(280\) 0 0
\(281\) 3894.00 0.826678 0.413339 0.910577i \(-0.364362\pi\)
0.413339 + 0.910577i \(0.364362\pi\)
\(282\) 0 0
\(283\) 3278.00 5677.66i 0.688540 1.19259i −0.283770 0.958892i \(-0.591585\pi\)
0.972310 0.233694i \(-0.0750815\pi\)
\(284\) 0 0
\(285\) −4000.00 6928.20i −0.831367 1.43997i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −135.500 234.693i −0.0275799 0.0477698i
\(290\) 0 0
\(291\) 496.000 859.097i 0.0999176 0.173062i
\(292\) 0 0
\(293\) −2484.00 −0.495279 −0.247640 0.968852i \(-0.579655\pi\)
−0.247640 + 0.968852i \(0.579655\pi\)
\(294\) 0 0
\(295\) −11600.0 −2.28942
\(296\) 0 0
\(297\) −3344.00 + 5791.98i −0.653328 + 1.13160i
\(298\) 0 0
\(299\) −2640.00 4572.61i −0.510619 0.884418i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 440.000 + 762.102i 0.0834236 + 0.144494i
\(304\) 0 0
\(305\) 3800.00 6581.79i 0.713401 1.23565i
\(306\) 0 0
\(307\) 308.000 0.0572589 0.0286295 0.999590i \(-0.490886\pi\)
0.0286295 + 0.999590i \(0.490886\pi\)
\(308\) 0 0
\(309\) −5344.00 −0.983850
\(310\) 0 0
\(311\) −880.000 + 1524.20i −0.160451 + 0.277909i −0.935030 0.354567i \(-0.884628\pi\)
0.774580 + 0.632476i \(0.217961\pi\)
\(312\) 0 0
\(313\) −12.0000 20.7846i −0.00216703 0.00375340i 0.864940 0.501876i \(-0.167356\pi\)
−0.867107 + 0.498122i \(0.834023\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2445.00 + 4234.86i 0.433202 + 0.750327i 0.997147 0.0754850i \(-0.0240505\pi\)
−0.563945 + 0.825812i \(0.690717\pi\)
\(318\) 0 0
\(319\) −4796.00 + 8306.92i −0.841769 + 1.45799i
\(320\) 0 0
\(321\) −400.000 −0.0695509
\(322\) 0 0
\(323\) 7200.00 1.24031
\(324\) 0 0
\(325\) 6050.00 10478.9i 1.03260 1.78851i
\(326\) 0 0
\(327\) 1364.00 + 2362.52i 0.230671 + 0.399534i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4054.00 7021.73i −0.673196 1.16601i −0.976993 0.213273i \(-0.931588\pi\)
0.303796 0.952737i \(-0.401746\pi\)
\(332\) 0 0
\(333\) −165.000 + 285.788i −0.0271530 + 0.0470304i
\(334\) 0 0
\(335\) 19600.0 3.19660
\(336\) 0 0
\(337\) 2990.00 0.483311 0.241655 0.970362i \(-0.422310\pi\)
0.241655 + 0.970362i \(0.422310\pi\)
\(338\) 0 0
\(339\) 740.000 1281.72i 0.118558 0.205349i
\(340\) 0 0
\(341\) 6160.00 + 10669.4i 0.978248 + 1.69438i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 4800.00 + 8313.84i 0.749053 + 1.29740i
\(346\) 0 0
\(347\) 3930.00 6806.96i 0.607993 1.05307i −0.383578 0.923508i \(-0.625308\pi\)
0.991571 0.129566i \(-0.0413583\pi\)
\(348\) 0 0
\(349\) −6060.00 −0.929468 −0.464734 0.885450i \(-0.653850\pi\)
−0.464734 + 0.885450i \(0.653850\pi\)
\(350\) 0 0
\(351\) −6688.00 −1.01703
\(352\) 0 0
\(353\) 1512.00 2618.86i 0.227976 0.394867i −0.729232 0.684267i \(-0.760122\pi\)
0.957208 + 0.289400i \(0.0934558\pi\)
\(354\) 0 0
\(355\) −1120.00 1939.90i −0.167446 0.290025i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2948.00 5106.09i −0.433397 0.750665i 0.563766 0.825934i \(-0.309352\pi\)
−0.997163 + 0.0752688i \(0.976019\pi\)
\(360\) 0 0
\(361\) −1570.50 + 2720.19i −0.228969 + 0.396586i
\(362\) 0 0
\(363\) −2420.00 −0.349909
\(364\) 0 0
\(365\) 12800.0 1.83557
\(366\) 0 0
\(367\) −1096.00 + 1898.33i −0.155888 + 0.270005i −0.933382 0.358885i \(-0.883157\pi\)
0.777494 + 0.628890i \(0.216490\pi\)
\(368\) 0 0
\(369\) 660.000 + 1143.15i 0.0931117 + 0.161274i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1705.00 + 2953.15i 0.236680 + 0.409941i 0.959760 0.280823i \(-0.0906074\pi\)
−0.723080 + 0.690765i \(0.757274\pi\)
\(374\) 0 0
\(375\) −6000.00 + 10392.3i −0.826236 + 1.43108i
\(376\) 0 0
\(377\) −9592.00 −1.31038
\(378\) 0 0
\(379\) 5916.00 0.801806 0.400903 0.916120i \(-0.368696\pi\)
0.400903 + 0.916120i \(0.368696\pi\)
\(380\) 0 0
\(381\) −320.000 + 554.256i −0.0430291 + 0.0745286i
\(382\) 0 0
\(383\) −1020.00 1766.69i −0.136082 0.235702i 0.789928 0.613200i \(-0.210118\pi\)
−0.926010 + 0.377498i \(0.876785\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1210.00 + 2095.78i 0.158935 + 0.275283i
\(388\) 0 0
\(389\) −5889.00 + 10200.0i −0.767569 + 1.32947i 0.171309 + 0.985217i \(0.445200\pi\)
−0.938878 + 0.344250i \(0.888133\pi\)
\(390\) 0 0
\(391\) −8640.00 −1.11750
\(392\) 0 0
\(393\) 1200.00 0.154025
\(394\) 0 0
\(395\) −4880.00 + 8452.41i −0.621619 + 1.07668i
\(396\) 0 0
\(397\) −2354.00 4077.25i −0.297592 0.515444i 0.677993 0.735069i \(-0.262850\pi\)
−0.975584 + 0.219625i \(0.929517\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2937.00 + 5087.03i 0.365753 + 0.633502i 0.988897 0.148605i \(-0.0474783\pi\)
−0.623144 + 0.782107i \(0.714145\pi\)
\(402\) 0 0
\(403\) −6160.00 + 10669.4i −0.761418 + 1.31881i
\(404\) 0 0
\(405\) 6220.00 0.763146
\(406\) 0 0
\(407\) −1320.00 −0.160762
\(408\) 0 0
\(409\) −7020.00 + 12159.0i −0.848696 + 1.46998i 0.0336764 + 0.999433i \(0.489278\pi\)
−0.882372 + 0.470552i \(0.844055\pi\)
\(410\) 0 0
\(411\) 2380.00 + 4122.28i 0.285637 + 0.494738i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 6600.00 + 11431.5i 0.780678 + 1.35217i
\(416\) 0 0
\(417\) 4440.00 7690.31i 0.521409 0.903108i
\(418\) 0 0
\(419\) −7940.00 −0.925762 −0.462881 0.886420i \(-0.653184\pi\)
−0.462881 + 0.886420i \(0.653184\pi\)
\(420\) 0 0
\(421\) 5214.00 0.603598 0.301799 0.953372i \(-0.402413\pi\)
0.301799 + 0.953372i \(0.402413\pi\)
\(422\) 0 0
\(423\) 484.000 838.313i 0.0556333 0.0963597i
\(424\) 0 0
\(425\) −9900.00 17147.3i −1.12993 1.95710i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −3872.00 6706.50i −0.435762 0.754762i
\(430\) 0 0
\(431\) −5192.00 + 8992.81i −0.580255 + 1.00503i 0.415194 + 0.909733i \(0.363714\pi\)
−0.995449 + 0.0952980i \(0.969620\pi\)
\(432\) 0 0
\(433\) 6520.00 0.723629 0.361814 0.932250i \(-0.382157\pi\)
0.361814 + 0.932250i \(0.382157\pi\)
\(434\) 0 0
\(435\) 17440.0 1.92226
\(436\) 0 0
\(437\) 6000.00 10392.3i 0.656794 1.13760i
\(438\) 0 0
\(439\) 5760.00 + 9976.61i 0.626218 + 1.08464i 0.988304 + 0.152497i \(0.0487314\pi\)
−0.362086 + 0.932145i \(0.617935\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6690.00 11587.4i −0.717498 1.24274i −0.961988 0.273091i \(-0.911954\pi\)
0.244491 0.969652i \(-0.421379\pi\)
\(444\) 0 0
\(445\) 3200.00 5542.56i 0.340887 0.590433i
\(446\) 0 0
\(447\) −9656.00 −1.02173
\(448\) 0 0
\(449\) 4098.00 0.430727 0.215364 0.976534i \(-0.430906\pi\)
0.215364 + 0.976534i \(0.430906\pi\)
\(450\) 0 0
\(451\) −2640.00 + 4572.61i −0.275638 + 0.477419i
\(452\) 0 0
\(453\) 4112.00 + 7122.19i 0.426487 + 0.738697i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8195.00 + 14194.2i 0.838831 + 1.45290i 0.890872 + 0.454254i \(0.150094\pi\)
−0.0520411 + 0.998645i \(0.516573\pi\)
\(458\) 0 0
\(459\) −5472.00 + 9477.78i −0.556451 + 0.963802i
\(460\) 0 0
\(461\) 9540.00 0.963822 0.481911 0.876220i \(-0.339943\pi\)
0.481911 + 0.876220i \(0.339943\pi\)
\(462\) 0 0
\(463\) −8920.00 −0.895351 −0.447676 0.894196i \(-0.647748\pi\)
−0.447676 + 0.894196i \(0.647748\pi\)
\(464\) 0 0
\(465\) 11200.0 19399.0i 1.11696 1.93464i
\(466\) 0 0
\(467\) 4214.00 + 7298.86i 0.417560 + 0.723236i 0.995693 0.0927068i \(-0.0295519\pi\)
−0.578133 + 0.815942i \(0.696219\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2120.00 + 3671.95i 0.207398 + 0.359224i
\(472\) 0 0
\(473\) −4840.00 + 8383.13i −0.470494 + 0.814919i
\(474\) 0 0
\(475\) 27500.0 2.65639
\(476\) 0 0
\(477\) −1210.00 −0.116147
\(478\) 0 0
\(479\) 7260.00 12574.7i 0.692522 1.19948i −0.278487 0.960440i \(-0.589833\pi\)
0.971009 0.239043i \(-0.0768336\pi\)
\(480\) 0 0
\(481\) −660.000 1143.15i −0.0625643 0.108364i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2480.00 + 4295.49i 0.232188 + 0.402161i
\(486\) 0 0
\(487\) 5260.00 9110.59i 0.489432 0.847721i −0.510494 0.859881i \(-0.670537\pi\)
0.999926 + 0.0121603i \(0.00387083\pi\)
\(488\) 0 0
\(489\) 7920.00 0.732423
\(490\) 0 0
\(491\) −5436.00 −0.499640 −0.249820 0.968292i \(-0.580371\pi\)
−0.249820 + 0.968292i \(0.580371\pi\)
\(492\) 0 0
\(493\) −7848.00 + 13593.1i −0.716950 + 1.24179i
\(494\) 0 0
\(495\) −4840.00 8383.13i −0.439478 0.761199i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −9174.00 15889.8i −0.823015 1.42550i −0.903427 0.428743i \(-0.858957\pi\)
0.0804114 0.996762i \(-0.474377\pi\)
\(500\) 0 0
\(501\) 976.000 1690.48i 0.0870349 0.150749i
\(502\) 0 0
\(503\) −20944.0 −1.85655 −0.928277 0.371889i \(-0.878710\pi\)
−0.928277 + 0.371889i \(0.878710\pi\)
\(504\) 0 0
\(505\) −4400.00 −0.387718
\(506\) 0 0
\(507\) −522.000 + 904.131i −0.0457255 + 0.0791989i
\(508\) 0 0
\(509\) −130.000 225.167i −0.0113205 0.0196077i 0.860310 0.509772i \(-0.170270\pi\)
−0.871630 + 0.490164i \(0.836937\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −7600.00 13163.6i −0.654090 1.13292i
\(514\) 0 0
\(515\) 13360.0 23140.2i 1.14313 1.97996i
\(516\) 0 0
\(517\) 3872.00 0.329382
\(518\) 0 0
\(519\) −13904.0 −1.17595
\(520\) 0 0
\(521\) −7300.00 + 12644.0i −0.613856 + 1.06323i 0.376728 + 0.926324i \(0.377049\pi\)
−0.990584 + 0.136905i \(0.956284\pi\)
\(522\) 0 0
\(523\) −3410.00 5906.29i −0.285103 0.493813i 0.687531 0.726155i \(-0.258694\pi\)
−0.972634 + 0.232342i \(0.925361\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10080.0 + 17459.1i 0.833191 + 1.44313i
\(528\) 0 0
\(529\) −1116.50 + 1933.83i −0.0917646 + 0.158941i
\(530\) 0 0
\(531\) −6380.00 −0.521409
\(532\) 0 0
\(533\) −5280.00 −0.429085
\(534\) 0 0
\(535\) 1000.00 1732.05i 0.0808108 0.139968i
\(536\) 0 0
\(537\) −168.000 290.985i −0.0135004 0.0233834i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −6083.00 10536.1i −0.483417 0.837303i 0.516402 0.856347i \(-0.327271\pi\)
−0.999819 + 0.0190437i \(0.993938\pi\)
\(542\) 0 0
\(543\) 4360.00 7551.74i 0.344577 0.596826i
\(544\) 0 0
\(545\) −13640.0 −1.07206
\(546\) 0 0
\(547\) 22660.0 1.77125 0.885623 0.464405i \(-0.153732\pi\)
0.885623 + 0.464405i \(0.153732\pi\)
\(548\) 0 0
\(549\) 2090.00 3619.99i 0.162475 0.281416i
\(550\) 0 0
\(551\) −10900.0 18879.4i −0.842751 1.45969i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 1200.00 + 2078.46i 0.0917787 + 0.158965i
\(556\) 0 0
\(557\) 4245.00 7352.56i 0.322920 0.559314i −0.658169 0.752870i \(-0.728669\pi\)
0.981089 + 0.193556i \(0.0620022\pi\)
\(558\) 0 0
\(559\) −9680.00 −0.732416
\(560\) 0 0
\(561\) −12672.0 −0.953676
\(562\) 0 0
\(563\) −7150.00 + 12384.2i −0.535234 + 0.927052i 0.463918 + 0.885878i \(0.346443\pi\)
−0.999152 + 0.0411740i \(0.986890\pi\)
\(564\) 0 0
\(565\) 3700.00 + 6408.59i 0.275505 + 0.477188i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −9147.00 15843.1i −0.673923 1.16727i −0.976782 0.214234i \(-0.931275\pi\)
0.302859 0.953035i \(-0.402059\pi\)
\(570\) 0 0
\(571\) 7194.00 12460.4i 0.527250 0.913223i −0.472246 0.881467i \(-0.656557\pi\)
0.999496 0.0317563i \(-0.0101101\pi\)
\(572\) 0 0
\(573\) −2784.00 −0.202973
\(574\) 0 0
\(575\) −33000.0 −2.39338
\(576\) 0 0
\(577\) −12616.0 + 21851.6i −0.910244 + 1.57659i −0.0965256 + 0.995331i \(0.530773\pi\)
−0.813719 + 0.581259i \(0.802560\pi\)
\(578\) 0 0
\(579\) 5380.00 + 9318.43i 0.386157 + 0.668844i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −2420.00 4191.56i −0.171915 0.297765i
\(584\) 0 0
\(585\) 4840.00 8383.13i 0.342067 0.592478i
\(586\) 0 0
\(587\) 16060.0 1.12925 0.564623 0.825349i \(-0.309022\pi\)
0.564623 + 0.825349i \(0.309022\pi\)
\(588\) 0 0
\(589\) −28000.0 −1.95878
\(590\) 0 0
\(591\) 10620.0 18394.4i 0.739168 1.28028i
\(592\) 0 0
\(593\) 11928.0 + 20659.9i 0.826011 + 1.43069i 0.901145 + 0.433519i \(0.142728\pi\)
−0.0751340 + 0.997173i \(0.523938\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6160.00 + 10669.4i 0.422298 + 0.731442i
\(598\) 0 0
\(599\) −1584.00 + 2743.57i −0.108048 + 0.187144i −0.914979 0.403501i \(-0.867793\pi\)
0.806932 + 0.590645i \(0.201127\pi\)
\(600\) 0 0
\(601\) −12320.0 −0.836179 −0.418089 0.908406i \(-0.637300\pi\)
−0.418089 + 0.908406i \(0.637300\pi\)
\(602\) 0 0
\(603\) 10780.0 0.728019
\(604\) 0 0
\(605\) 6050.00 10478.9i 0.406558 0.704179i
\(606\) 0 0
\(607\) 4464.00 + 7731.87i 0.298498 + 0.517013i 0.975792 0.218698i \(-0.0701811\pi\)
−0.677295 + 0.735712i \(0.736848\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1936.00 + 3353.25i 0.128187 + 0.222026i
\(612\) 0 0
\(613\) 2255.00 3905.77i 0.148578 0.257345i −0.782124 0.623123i \(-0.785864\pi\)
0.930702 + 0.365778i \(0.119197\pi\)
\(614\) 0 0
\(615\) 9600.00 0.629446
\(616\) 0 0
\(617\) 5830.00 0.380400 0.190200 0.981745i \(-0.439086\pi\)
0.190200 + 0.981745i \(0.439086\pi\)
\(618\) 0 0
\(619\) 12230.0 21183.0i 0.794128 1.37547i −0.129263 0.991610i \(-0.541261\pi\)
0.923391 0.383860i \(-0.125405\pi\)
\(620\) 0 0
\(621\) 9120.00 + 15796.3i 0.589328 + 1.02075i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12812.5 22191.9i −0.820000 1.42028i
\(626\) 0 0
\(627\) 8800.00 15242.0i 0.560507 0.970827i
\(628\) 0 0
\(629\) −2160.00 −0.136923
\(630\) 0 0
\(631\) 15216.0 0.959967 0.479984 0.877277i \(-0.340643\pi\)
0.479984 + 0.877277i \(0.340643\pi\)
\(632\) 0 0
\(633\) −8.00000 + 13.8564i −0.000502325 + 0.000870052i
\(634\) 0 0
\(635\) −1600.00 2771.28i −0.0999907 0.173189i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −616.000 1066.94i −0.0381355 0.0660526i
\(640\) 0 0
\(641\) −6239.00 + 10806.3i −0.384439 + 0.665869i −0.991691 0.128641i \(-0.958939\pi\)
0.607252 + 0.794509i \(0.292272\pi\)
\(642\) 0 0
\(643\) 15996.0 0.981059 0.490529 0.871425i \(-0.336803\pi\)
0.490529 + 0.871425i \(0.336803\pi\)
\(644\) 0 0
\(645\) 17600.0 1.07442
\(646\) 0 0
\(647\) 2940.00 5092.23i 0.178645 0.309422i −0.762772 0.646668i \(-0.776162\pi\)
0.941417 + 0.337246i \(0.109495\pi\)
\(648\) 0 0
\(649\) −12760.0 22101.0i −0.771762 1.33673i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3755.00 + 6503.85i 0.225030 + 0.389763i 0.956328 0.292294i \(-0.0944187\pi\)
−0.731299 + 0.682058i \(0.761085\pi\)
\(654\) 0 0
\(655\) −3000.00 + 5196.15i −0.178961 + 0.309970i
\(656\) 0 0
\(657\) 7040.00 0.418047
\(658\) 0 0
\(659\) −16508.0 −0.975812 −0.487906 0.872896i \(-0.662239\pi\)
−0.487906 + 0.872896i \(0.662239\pi\)
\(660\) 0 0
\(661\) 610.000 1056.55i 0.0358945 0.0621711i −0.847520 0.530763i \(-0.821905\pi\)
0.883415 + 0.468592i \(0.155239\pi\)
\(662\) 0 0
\(663\) −6336.00 10974.3i −0.371146 0.642844i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 13080.0 + 22655.2i 0.759310 + 1.31516i
\(668\) 0 0
\(669\) −4832.00 + 8369.27i −0.279247 + 0.483669i
\(670\) 0 0
\(671\) 16720.0 0.961950
\(672\) 0 0
\(673\) −13090.0 −0.749751 −0.374875 0.927075i \(-0.622315\pi\)
−0.374875 + 0.927075i \(0.622315\pi\)
\(674\) 0 0
\(675\) −20900.0 + 36199.9i −1.19177 + 2.06420i
\(676\) 0 0
\(677\) −7554.00 13083.9i −0.428839 0.742770i 0.567932 0.823076i \(-0.307744\pi\)
−0.996770 + 0.0803053i \(0.974410\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −9944.00 17223.5i −0.559552 0.969172i
\(682\) 0 0
\(683\) −770.000 + 1333.68i −0.0431380 + 0.0747171i −0.886788 0.462176i \(-0.847069\pi\)
0.843650 + 0.536893i \(0.180402\pi\)
\(684\) 0 0
\(685\) −23800.0 −1.32752
\(686\) 0 0
\(687\) 17840.0 0.990740
\(688\) 0 0
\(689\) 2420.00 4191.56i 0.133809 0.231765i
\(690\) 0 0
\(691\) −2890.00 5005.63i −0.159104 0.275576i 0.775442 0.631419i \(-0.217527\pi\)
−0.934546 + 0.355843i \(0.884194\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 22200.0 + 38451.5i 1.21165 + 2.09863i
\(696\) 0 0
\(697\) −4320.00 + 7482.46i −0.234766 + 0.406626i
\(698\) 0 0
\(699\) −18200.0 −0.984817
\(700\) 0 0
\(701\) −10406.0 −0.560669 −0.280335 0.959902i \(-0.590445\pi\)
−0.280335 + 0.959902i \(0.590445\pi\)
\(702\) 0 0
\(703\) 1500.00 2598.08i 0.0804745 0.139386i
\(704\) 0 0
\(705\) −3520.00 6096.82i −0.188044 0.325701i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −14321.0 24804.7i −0.758585 1.31391i −0.943572 0.331166i \(-0.892558\pi\)
0.184988 0.982741i \(-0.440775\pi\)
\(710\) 0 0
\(711\) −2684.00 + 4648.82i −0.141572 + 0.245210i
\(712\) 0 0
\(713\) 33600.0 1.76484
\(714\) 0 0
\(715\) 38720.0 2.02524
\(716\) 0 0
\(717\) −4224.00 + 7316.18i −0.220011 + 0.381071i
\(718\) 0 0
\(719\) −3780.00 6547.15i −0.196064 0.339593i 0.751185 0.660092i \(-0.229483\pi\)
−0.947249 + 0.320499i \(0.896149\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −9680.00 16766.3i −0.497930 0.862439i
\(724\) 0 0
\(725\) −29975.0 + 51918.2i −1.53551 + 2.65958i
\(726\) 0 0
\(727\) −20360.0 −1.03867 −0.519333 0.854572i \(-0.673820\pi\)
−0.519333 + 0.854572i \(0.673820\pi\)
\(728\) 0 0
\(729\) 19837.0 1.00782
\(730\) 0 0
\(731\) −7920.00 + 13717.8i −0.400727 + 0.694080i
\(732\) 0 0
\(733\) −11638.0 20157.6i −0.586438 1.01574i −0.994694 0.102874i \(-0.967196\pi\)
0.408256 0.912867i \(-0.366137\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 21560.0 + 37343.0i 1.07758 + 1.86641i
\(738\) 0 0
\(739\) −1586.00 + 2747.03i −0.0789472 + 0.136740i −0.902796 0.430069i \(-0.858489\pi\)
0.823849 + 0.566810i \(0.191823\pi\)
\(740\) 0 0
\(741\) 17600.0 0.872540
\(742\) 0 0
\(743\) −6600.00 −0.325882 −0.162941 0.986636i \(-0.552098\pi\)
−0.162941 + 0.986636i \(0.552098\pi\)
\(744\) 0 0
\(745\) 24140.0 41811.7i 1.18714 2.05619i
\(746\) 0 0
\(747\) 3630.00 + 6287.34i 0.177798 + 0.307954i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 8208.00 + 14216.7i 0.398820 + 0.690777i 0.993581 0.113126i \(-0.0360863\pi\)
−0.594760 + 0.803903i \(0.702753\pi\)
\(752\) 0 0
\(753\) −8680.00 + 15034.2i −0.420075 + 0.727592i
\(754\) 0 0
\(755\) −41120.0 −1.98213
\(756\) 0 0
\(757\) 36850.0 1.76927 0.884634 0.466286i \(-0.154408\pi\)
0.884634 + 0.466286i \(0.154408\pi\)
\(758\) 0 0
\(759\) −10560.0 + 18290.5i −0.505011 + 0.874706i
\(760\) 0 0
\(761\) 18700.0 + 32389.4i 0.890768 + 1.54286i 0.838956 + 0.544199i \(0.183166\pi\)
0.0518116 + 0.998657i \(0.483500\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −7920.00 13717.8i −0.374311 0.648326i
\(766\) 0 0
\(767\) 12760.0 22101.0i 0.600700 1.04044i
\(768\) 0 0
\(769\) −5720.00 −0.268229 −0.134115 0.990966i \(-0.542819\pi\)
−0.134115 + 0.990966i \(0.542819\pi\)
\(770\) 0 0
\(771\) 14080.0 0.657690
\(772\) 0 0
\(773\) −3678.00 + 6370.48i −0.171136 + 0.296417i −0.938817 0.344415i \(-0.888077\pi\)
0.767681 + 0.640832i \(0.221411\pi\)
\(774\) 0 0
\(775\) 38500.0 + 66684.0i 1.78447 + 3.09079i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6000.00 10392.3i −0.275959 0.477976i
\(780\) 0 0
\(781\) 2464.00 4267.77i 0.112892 0.195535i
\(782\) 0 0
\(783\) 33136.0 1.51237
\(784\) 0 0
\(785\) −21200.0 −0.963899
\(786\) 0 0
\(787\) 19286.0 33404.3i 0.873535 1.51301i 0.0152188 0.999884i \(-0.495156\pi\)
0.858316 0.513122i \(-0.171511\pi\)
\(788\) 0 0
\(789\) −5120.00 8868.10i −0.231023 0.400143i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 8360.00 + 14479.9i 0.374366 + 0.648421i
\(794\) 0 0
\(795\) −4400.00 + 7621.02i −0.196292 + 0.339987i
\(796\) 0 0
\(797\) 7812.00 0.347196 0.173598 0.984817i \(-0.444461\pi\)
0.173598 + 0.984817i \(0.444461\pi\)
\(798\) 0 0
\(799\) 6336.00 0.280540
\(800\) 0 0
\(801\) 1760.00 3048.41i 0.0776361 0.134470i
\(802\) 0 0
\(803\) 14080.0 + 24387.3i 0.618770 + 1.07174i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 13640.0 + 23625.2i 0.594982 + 1.03054i
\(808\) 0 0
\(809\) −22509.0 + 38986.7i −0.978213 + 1.69431i −0.309316 + 0.950959i \(0.600100\pi\)
−0.668897 + 0.743355i \(0.733233\pi\)
\(810\) 0 0
\(811\) 2740.00 0.118637 0.0593184 0.998239i \(-0.481107\pi\)
0.0593184 + 0.998239i \(0.481107\pi\)
\(812\) 0 0
\(813\) −14080.0 −0.607389
\(814\) 0 0
\(815\) −19800.0 + 34294.6i −0.850998 + 1.47397i
\(816\) 0 0
\(817\) −11000.0 19052.6i −0.471042 0.815869i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −727.000 1259.20i −0.0309044 0.0535279i 0.850160 0.526525i \(-0.176505\pi\)
−0.881064 + 0.472997i \(0.843172\pi\)
\(822\) 0 0
\(823\) −18560.0 + 32146.9i −0.786101 + 1.36157i 0.142239 + 0.989832i \(0.454570\pi\)
−0.928339 + 0.371734i \(0.878763\pi\)
\(824\) 0 0
\(825\) −48400.0 −2.04251
\(826\) 0 0
\(827\) −7260.00 −0.305266 −0.152633 0.988283i \(-0.548775\pi\)
−0.152633 + 0.988283i \(0.548775\pi\)
\(828\) 0 0
\(829\) 5570.00 9647.52i 0.233358 0.404189i −0.725436 0.688290i \(-0.758362\pi\)
0.958794 + 0.284101i \(0.0916951\pi\)
\(830\) 0 0
\(831\) −17380.0 30103.0i −0.725518 1.25663i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 4880.00 + 8452.41i 0.202251 + 0.350309i
\(836\) 0 0
\(837\) 21280.0 36858.0i 0.878787 1.52210i
\(838\) 0 0
\(839\) 8760.00 0.360463 0.180232 0.983624i \(-0.442315\pi\)
0.180232 + 0.983624i \(0.442315\pi\)
\(840\) 0 0
\(841\) 23135.0 0.948583
\(842\) 0 0
\(843\) 7788.00 13489.2i 0.318189 0.551119i
\(844\) 0 0
\(845\) −2610.00 4520.65i −0.106256 0.184042i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −13112.0 22710.7i −0.530038 0.918054i
\(850\) 0 0
\(851\) −1800.00 + 3117.69i −0.0725067 + 0.125585i
\(852\) 0 0
\(853\) 820.000 0.0329147 0.0164574 0.999865i \(-0.494761\pi\)
0.0164574 + 0.999865i \(0.494761\pi\)
\(854\) 0 0
\(855\) 22000.0 0.879981
\(856\) 0 0
\(857\) 660.000 1143.15i 0.0263071 0.0455652i −0.852572 0.522610i \(-0.824959\pi\)
0.878879 + 0.477044i \(0.158292\pi\)
\(858\) 0 0
\(859\) −21390.0 37048.6i −0.849613 1.47157i −0.881554 0.472083i \(-0.843502\pi\)
0.0319414 0.999490i \(-0.489831\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −14940.0 25876.8i −0.589297 1.02069i −0.994325 0.106388i \(-0.966071\pi\)
0.405027 0.914305i \(-0.367262\pi\)
\(864\) 0 0
\(865\) 34760.0 60206.1i 1.36633 2.36655i
\(866\) 0 0
\(867\) −1084.00 −0.0424620
\(868\) 0 0
\(869\) −21472.0 −0.838191
\(870\) 0 0
\(871\) −21560.0 + 37343.0i −0.838729 + 1.45272i
\(872\) 0 0
\(873\) 1364.00 + 2362.52i 0.0528802 + 0.0915912i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −14085.0 24395.9i −0.542322 0.939330i −0.998770 0.0495796i \(-0.984212\pi\)
0.456448 0.889750i \(-0.349121\pi\)
\(878\) 0 0
\(879\) −4968.00 + 8604.83i −0.190633 + 0.330186i
\(880\) 0 0
\(881\) 20800.0 0.795425 0.397713 0.917510i \(-0.369804\pi\)
0.397713 + 0.917510i \(0.369804\pi\)
\(882\) 0 0
\(883\) −20900.0 −0.796536 −0.398268 0.917269i \(-0.630389\pi\)
−0.398268 + 0.917269i \(0.630389\pi\)
\(884\) 0 0
\(885\) −23200.0 + 40183.6i −0.881197 + 1.52628i
\(886\) 0 0
\(887\) −5820.00 10080.5i −0.220312 0.381591i 0.734591 0.678510i \(-0.237374\pi\)
−0.954903 + 0.296919i \(0.904041\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 6842.00 + 11850.7i 0.257257 + 0.445581i
\(892\) 0 0
\(893\) −4400.00 + 7621.02i −0.164883 + 0.285585i
\(894\) 0 0
\(895\) 1680.00 0.0627444
\(896\) 0 0
\(897\) −21120.0 −0.786150
\(898\) 0 0
\(899\) 30520.0 52862.2i 1.13226 1.96113i
\(900\) 0 0
\(901\) −3960.00 6858.92i −0.146423 0.253611i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 21800.0 + 37758.7i 0.800725 + 1.38690i
\(906\) 0 0
\(907\) 18790.0 32545.2i 0.687885 1.19145i −0.284636 0.958636i \(-0.591873\pi\)
0.972521 0.232816i \(-0.0747940\pi\)
\(908\) 0 0
\(909\) −2420.00 −0.0883018
\(910\) 0 0
\(911\) −40832.0 −1.48499 −0.742494 0.669852i \(-0.766357\pi\)
−0.742494 + 0.669852i \(0.766357\pi\)
\(912\) 0 0
\(913\) −14520.0 + 25149.4i −0.526333 + 0.911635i
\(914\) 0 0
\(915\) −15200.0 26327.2i −0.549177 0.951202i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −21384.0 37038.2i −0.767566 1.32946i −0.938879 0.344247i \(-0.888134\pi\)
0.171313 0.985217i \(-0.445199\pi\)
\(920\) 0 0
\(921\) 616.000 1066.94i 0.0220390 0.0381726i
\(922\) 0 0
\(923\) 4928.00 0.175739
\(924\) 0 0
\(925\) −8250.00 −0.293252
\(926\) 0 0
\(927\) 7348.00 12727.1i 0.260345 0.450931i
\(928\) 0 0
\(929\) 4380.00 + 7586.38i 0.154686 + 0.267924i 0.932945 0.360020i \(-0.117230\pi\)
−0.778259 + 0.627944i \(0.783897\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 3520.00 + 6096.82i 0.123515 + 0.213934i
\(934\) 0 0
\(935\) 31680.0 54871.4i 1.10807 1.91924i
\(936\) 0 0
\(937\) 49632.0 1.73042 0.865212 0.501407i \(-0.167184\pi\)
0.865212 + 0.501407i \(0.167184\pi\)
\(938\) 0 0
\(939\) −96.0000 −0.00333636
\(940\) 0 0
\(941\) 18450.0 31956.3i 0.639163 1.10706i −0.346453 0.938067i \(-0.612614\pi\)
0.985617 0.168996i \(-0.0540526\pi\)
\(942\) 0 0
\(943\) 7200.00 + 12470.8i 0.248637 + 0.430651i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 16670.0 + 28873.3i 0.572019 + 0.990766i 0.996358 + 0.0852631i \(0.0271731\pi\)
−0.424339 + 0.905503i \(0.639494\pi\)
\(948\) 0 0
\(949\) −14080.0 + 24387.3i −0.481619 + 0.834188i
\(950\) 0 0
\(951\) 19560.0 0.666957
\(952\) 0 0
\(953\) 5610.00 0.190688 0.0953440 0.995444i \(-0.469605\pi\)
0.0953440 + 0.995444i \(0.469605\pi\)
\(954\) 0 0
\(955\) 6960.00 12055.1i 0.235833 0.408474i
\(956\) 0 0
\(957\) 19184.0 + 33227.7i 0.647994 + 1.12236i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −24304.5 42096.6i −0.815834 1.41307i
\(962\) 0 0
\(963\) 550.000 952.628i 0.0184045 0.0318775i
\(964\) 0 0
\(965\) −53800.0 −1.79470
\(966\) 0 0
\(967\) −25160.0 −0.836702 −0.418351 0.908285i \(-0.637392\pi\)
−0.418351 + 0.908285i \(0.637392\pi\)
\(968\) 0 0
\(969\) 14400.0 24941.5i 0.477394 0.826870i
\(970\) 0 0
\(971\) −530.000 917.987i −0.0175165 0.0303394i 0.857134 0.515093i \(-0.172243\pi\)
−0.874651 + 0.484754i \(0.838909\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −24200.0 41915.6i −0.794893 1.37679i
\(976\) 0 0
\(977\) 19385.0 33575.8i 0.634781 1.09947i −0.351780 0.936083i \(-0.614424\pi\)
0.986561 0.163390i \(-0.0522430\pi\)
\(978\) 0 0
\(979\) 14080.0 0.459651
\(980\) 0 0
\(981\) −7502.00 −0.244159
\(982\) 0 0
\(983\) 12148.0 21041.0i 0.394162 0.682708i −0.598832 0.800875i \(-0.704368\pi\)
0.992994 + 0.118166i \(0.0377016\pi\)
\(984\) 0 0
\(985\) 53100.0 + 91971.9i 1.71767 + 2.97509i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 13200.0 + 22863.1i 0.424404 + 0.735089i
\(990\) 0 0
\(991\) 836.000 1447.99i 0.0267976 0.0464148i −0.852316 0.523028i \(-0.824802\pi\)
0.879113 + 0.476613i \(0.158136\pi\)
\(992\) 0 0
\(993\) −32432.0 −1.03645
\(994\) 0 0
\(995\) −61600.0 −1.96266
\(996\) 0 0
\(997\) −22814.0 + 39515.0i −0.724701 + 1.25522i 0.234396 + 0.972141i \(0.424689\pi\)
−0.959097 + 0.283077i \(0.908645\pi\)
\(998\) 0 0
\(999\) 2280.00 + 3949.08i 0.0722082 + 0.125068i
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 196.4.e.e.165.1 2
3.2 odd 2 1764.4.k.a.361.1 2
7.2 even 3 inner 196.4.e.e.177.1 2
7.3 odd 6 196.4.a.c.1.1 yes 1
7.4 even 3 196.4.a.a.1.1 1
7.5 odd 6 196.4.e.b.177.1 2
7.6 odd 2 196.4.e.b.165.1 2
21.2 odd 6 1764.4.k.a.1549.1 2
21.5 even 6 1764.4.k.p.1549.1 2
21.11 odd 6 1764.4.a.m.1.1 1
21.17 even 6 1764.4.a.a.1.1 1
21.20 even 2 1764.4.k.p.361.1 2
28.3 even 6 784.4.a.f.1.1 1
28.11 odd 6 784.4.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
196.4.a.a.1.1 1 7.4 even 3
196.4.a.c.1.1 yes 1 7.3 odd 6
196.4.e.b.165.1 2 7.6 odd 2
196.4.e.b.177.1 2 7.5 odd 6
196.4.e.e.165.1 2 1.1 even 1 trivial
196.4.e.e.177.1 2 7.2 even 3 inner
784.4.a.f.1.1 1 28.3 even 6
784.4.a.m.1.1 1 28.11 odd 6
1764.4.a.a.1.1 1 21.17 even 6
1764.4.a.m.1.1 1 21.11 odd 6
1764.4.k.a.361.1 2 3.2 odd 2
1764.4.k.a.1549.1 2 21.2 odd 6
1764.4.k.p.361.1 2 21.20 even 2
1764.4.k.p.1549.1 2 21.5 even 6