Properties

Label 2940.2.q.q.961.2
Level $2940$
Weight $2$
Character 2940.961
Analytic conductor $23.476$
Analytic rank $0$
Dimension $4$
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] = [2940,2,Mod(361,2940)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2940, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2940.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2940.q (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: \(23.4760181943\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.2
Root \(0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 2940.961
Dual form 2940.2.q.q.361.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{9} +(2.12132 + 3.67423i) q^{11} -3.24264 q^{13} -1.00000 q^{15} +(2.12132 + 3.67423i) q^{17} +(-3.50000 + 6.06218i) q^{19} +(2.12132 - 3.67423i) q^{23} +(-0.500000 - 0.866025i) q^{25} -1.00000 q^{27} -1.75736 q^{29} +(-4.74264 - 8.21449i) q^{31} +(-2.12132 + 3.67423i) q^{33} +(-1.62132 + 2.80821i) q^{37} +(-1.62132 - 2.80821i) q^{39} +4.24264 q^{41} +3.24264 q^{43} +(-0.500000 - 0.866025i) q^{45} +(-3.00000 + 5.19615i) q^{47} +(-2.12132 + 3.67423i) q^{51} +(-4.24264 - 7.34847i) q^{53} -4.24264 q^{55} -7.00000 q^{57} +(-5.12132 - 8.87039i) q^{59} +(2.24264 - 3.88437i) q^{61} +(1.62132 - 2.80821i) q^{65} +(2.62132 + 4.54026i) q^{67} +4.24264 q^{69} -12.7279 q^{71} +(4.62132 + 8.00436i) q^{73} +(0.500000 - 0.866025i) q^{75} +(-5.50000 + 9.52628i) q^{79} +(-0.500000 - 0.866025i) q^{81} +10.2426 q^{83} -4.24264 q^{85} +(-0.878680 - 1.52192i) q^{87} +(-5.12132 + 8.87039i) q^{89} +(4.74264 - 8.21449i) q^{93} +(-3.50000 - 6.06218i) q^{95} +0.485281 q^{97} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 2 q^{5} - 2 q^{9} + 4 q^{13} - 4 q^{15} - 14 q^{19} - 2 q^{25} - 4 q^{27} - 24 q^{29} - 2 q^{31} + 2 q^{37} + 2 q^{39} - 4 q^{43} - 2 q^{45} - 12 q^{47} - 28 q^{57} - 12 q^{59} - 8 q^{61} - 2 q^{65} + 2 q^{67} + 10 q^{73} + 2 q^{75} - 22 q^{79} - 2 q^{81} + 24 q^{83} - 12 q^{87} - 12 q^{89} + 2 q^{93} - 14 q^{95} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(1177\) \(1471\) \(1961\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 2.12132 + 3.67423i 0.639602 + 1.10782i 0.985520 + 0.169559i \(0.0542342\pi\)
−0.345918 + 0.938265i \(0.612432\pi\)
\(12\) 0 0
\(13\) −3.24264 −0.899347 −0.449673 0.893193i \(-0.648460\pi\)
−0.449673 + 0.893193i \(0.648460\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 2.12132 + 3.67423i 0.514496 + 0.891133i 0.999859 + 0.0168199i \(0.00535420\pi\)
−0.485363 + 0.874313i \(0.661312\pi\)
\(18\) 0 0
\(19\) −3.50000 + 6.06218i −0.802955 + 1.39076i 0.114708 + 0.993399i \(0.463407\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.12132 3.67423i 0.442326 0.766131i −0.555536 0.831493i \(-0.687487\pi\)
0.997862 + 0.0653618i \(0.0208201\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.75736 −0.326333 −0.163167 0.986599i \(-0.552171\pi\)
−0.163167 + 0.986599i \(0.552171\pi\)
\(30\) 0 0
\(31\) −4.74264 8.21449i −0.851803 1.47537i −0.879579 0.475753i \(-0.842176\pi\)
0.0277757 0.999614i \(-0.491158\pi\)
\(32\) 0 0
\(33\) −2.12132 + 3.67423i −0.369274 + 0.639602i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.62132 + 2.80821i −0.266543 + 0.461667i −0.967967 0.251078i \(-0.919215\pi\)
0.701423 + 0.712745i \(0.252548\pi\)
\(38\) 0 0
\(39\) −1.62132 2.80821i −0.259619 0.449673i
\(40\) 0 0
\(41\) 4.24264 0.662589 0.331295 0.943527i \(-0.392515\pi\)
0.331295 + 0.943527i \(0.392515\pi\)
\(42\) 0 0
\(43\) 3.24264 0.494498 0.247249 0.968952i \(-0.420473\pi\)
0.247249 + 0.968952i \(0.420473\pi\)
\(44\) 0 0
\(45\) −0.500000 0.866025i −0.0745356 0.129099i
\(46\) 0 0
\(47\) −3.00000 + 5.19615i −0.437595 + 0.757937i −0.997503 0.0706177i \(-0.977503\pi\)
0.559908 + 0.828554i \(0.310836\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −2.12132 + 3.67423i −0.297044 + 0.514496i
\(52\) 0 0
\(53\) −4.24264 7.34847i −0.582772 1.00939i −0.995149 0.0983769i \(-0.968635\pi\)
0.412378 0.911013i \(-0.364698\pi\)
\(54\) 0 0
\(55\) −4.24264 −0.572078
\(56\) 0 0
\(57\) −7.00000 −0.927173
\(58\) 0 0
\(59\) −5.12132 8.87039i −0.666739 1.15483i −0.978811 0.204767i \(-0.934356\pi\)
0.312072 0.950059i \(-0.398977\pi\)
\(60\) 0 0
\(61\) 2.24264 3.88437i 0.287141 0.497342i −0.685985 0.727615i \(-0.740629\pi\)
0.973126 + 0.230273i \(0.0739619\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.62132 2.80821i 0.201100 0.348315i
\(66\) 0 0
\(67\) 2.62132 + 4.54026i 0.320245 + 0.554681i 0.980539 0.196327i \(-0.0629013\pi\)
−0.660293 + 0.751008i \(0.729568\pi\)
\(68\) 0 0
\(69\) 4.24264 0.510754
\(70\) 0 0
\(71\) −12.7279 −1.51053 −0.755263 0.655422i \(-0.772491\pi\)
−0.755263 + 0.655422i \(0.772491\pi\)
\(72\) 0 0
\(73\) 4.62132 + 8.00436i 0.540885 + 0.936840i 0.998854 + 0.0478714i \(0.0152438\pi\)
−0.457969 + 0.888968i \(0.651423\pi\)
\(74\) 0 0
\(75\) 0.500000 0.866025i 0.0577350 0.100000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −5.50000 + 9.52628i −0.618798 + 1.07179i 0.370907 + 0.928670i \(0.379047\pi\)
−0.989705 + 0.143120i \(0.954286\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 10.2426 1.12428 0.562138 0.827043i \(-0.309979\pi\)
0.562138 + 0.827043i \(0.309979\pi\)
\(84\) 0 0
\(85\) −4.24264 −0.460179
\(86\) 0 0
\(87\) −0.878680 1.52192i −0.0942043 0.163167i
\(88\) 0 0
\(89\) −5.12132 + 8.87039i −0.542859 + 0.940259i 0.455879 + 0.890042i \(0.349325\pi\)
−0.998738 + 0.0502176i \(0.984009\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.74264 8.21449i 0.491789 0.851803i
\(94\) 0 0
\(95\) −3.50000 6.06218i −0.359092 0.621966i
\(96\) 0 0
\(97\) 0.485281 0.0492729 0.0246364 0.999696i \(-0.492157\pi\)
0.0246364 + 0.999696i \(0.492157\pi\)
\(98\) 0 0
\(99\) −4.24264 −0.426401
\(100\) 0 0
\(101\) −3.87868 6.71807i −0.385943 0.668473i 0.605956 0.795498i \(-0.292791\pi\)
−0.991900 + 0.127025i \(0.959457\pi\)
\(102\) 0 0
\(103\) −8.62132 + 14.9326i −0.849484 + 1.47135i 0.0321856 + 0.999482i \(0.489753\pi\)
−0.881670 + 0.471867i \(0.843580\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.36396 + 11.0227i −0.615227 + 1.06561i 0.375117 + 0.926977i \(0.377602\pi\)
−0.990345 + 0.138628i \(0.955731\pi\)
\(108\) 0 0
\(109\) 4.74264 + 8.21449i 0.454263 + 0.786806i 0.998645 0.0520310i \(-0.0165695\pi\)
−0.544383 + 0.838837i \(0.683236\pi\)
\(110\) 0 0
\(111\) −3.24264 −0.307778
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 2.12132 + 3.67423i 0.197814 + 0.342624i
\(116\) 0 0
\(117\) 1.62132 2.80821i 0.149891 0.259619i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −3.50000 + 6.06218i −0.318182 + 0.551107i
\(122\) 0 0
\(123\) 2.12132 + 3.67423i 0.191273 + 0.331295i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −2.75736 −0.244676 −0.122338 0.992488i \(-0.539039\pi\)
−0.122338 + 0.992488i \(0.539039\pi\)
\(128\) 0 0
\(129\) 1.62132 + 2.80821i 0.142749 + 0.247249i
\(130\) 0 0
\(131\) 7.24264 12.5446i 0.632792 1.09603i −0.354186 0.935175i \(-0.615242\pi\)
0.986978 0.160854i \(-0.0514247\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.500000 0.866025i 0.0430331 0.0745356i
\(136\) 0 0
\(137\) −2.12132 3.67423i −0.181237 0.313911i 0.761065 0.648675i \(-0.224677\pi\)
−0.942302 + 0.334764i \(0.891343\pi\)
\(138\) 0 0
\(139\) 15.4853 1.31344 0.656722 0.754133i \(-0.271942\pi\)
0.656722 + 0.754133i \(0.271942\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) −6.87868 11.9142i −0.575224 0.996317i
\(144\) 0 0
\(145\) 0.878680 1.52192i 0.0729704 0.126388i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.00000 10.3923i 0.491539 0.851371i −0.508413 0.861113i \(-0.669768\pi\)
0.999953 + 0.00974235i \(0.00310113\pi\)
\(150\) 0 0
\(151\) −11.2426 19.4728i −0.914913 1.58468i −0.807028 0.590513i \(-0.798926\pi\)
−0.107885 0.994163i \(-0.534408\pi\)
\(152\) 0 0
\(153\) −4.24264 −0.342997
\(154\) 0 0
\(155\) 9.48528 0.761876
\(156\) 0 0
\(157\) −3.24264 5.61642i −0.258791 0.448239i 0.707127 0.707086i \(-0.249991\pi\)
−0.965918 + 0.258847i \(0.916657\pi\)
\(158\) 0 0
\(159\) 4.24264 7.34847i 0.336463 0.582772i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 + 6.92820i −0.313304 + 0.542659i −0.979076 0.203497i \(-0.934769\pi\)
0.665771 + 0.746156i \(0.268103\pi\)
\(164\) 0 0
\(165\) −2.12132 3.67423i −0.165145 0.286039i
\(166\) 0 0
\(167\) −18.7279 −1.44921 −0.724605 0.689164i \(-0.757978\pi\)
−0.724605 + 0.689164i \(0.757978\pi\)
\(168\) 0 0
\(169\) −2.48528 −0.191175
\(170\) 0 0
\(171\) −3.50000 6.06218i −0.267652 0.463586i
\(172\) 0 0
\(173\) −10.2426 + 17.7408i −0.778734 + 1.34881i 0.153938 + 0.988080i \(0.450804\pi\)
−0.932672 + 0.360726i \(0.882529\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.12132 8.87039i 0.384942 0.666739i
\(178\) 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
\(181\) 13.0000 0.966282 0.483141 0.875542i \(-0.339496\pi\)
0.483141 + 0.875542i \(0.339496\pi\)
\(182\) 0 0
\(183\) 4.48528 0.331562
\(184\) 0 0
\(185\) −1.62132 2.80821i −0.119202 0.206464i
\(186\) 0 0
\(187\) −9.00000 + 15.5885i −0.658145 + 1.13994i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3.00000 + 5.19615i −0.217072 + 0.375980i −0.953912 0.300088i \(-0.902984\pi\)
0.736839 + 0.676068i \(0.236317\pi\)
\(192\) 0 0
\(193\) −3.37868 5.85204i −0.243203 0.421239i 0.718422 0.695607i \(-0.244865\pi\)
−0.961625 + 0.274368i \(0.911531\pi\)
\(194\) 0 0
\(195\) 3.24264 0.232210
\(196\) 0 0
\(197\) −16.2426 −1.15724 −0.578620 0.815597i \(-0.696409\pi\)
−0.578620 + 0.815597i \(0.696409\pi\)
\(198\) 0 0
\(199\) 5.24264 + 9.08052i 0.371641 + 0.643701i 0.989818 0.142338i \(-0.0454619\pi\)
−0.618177 + 0.786039i \(0.712129\pi\)
\(200\) 0 0
\(201\) −2.62132 + 4.54026i −0.184894 + 0.320245i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2.12132 + 3.67423i −0.148159 + 0.256620i
\(206\) 0 0
\(207\) 2.12132 + 3.67423i 0.147442 + 0.255377i
\(208\) 0 0
\(209\) −29.6985 −2.05429
\(210\) 0 0
\(211\) 22.4853 1.54795 0.773975 0.633216i \(-0.218265\pi\)
0.773975 + 0.633216i \(0.218265\pi\)
\(212\) 0 0
\(213\) −6.36396 11.0227i −0.436051 0.755263i
\(214\) 0 0
\(215\) −1.62132 + 2.80821i −0.110573 + 0.191518i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −4.62132 + 8.00436i −0.312280 + 0.540885i
\(220\) 0 0
\(221\) −6.87868 11.9142i −0.462710 0.801437i
\(222\) 0 0
\(223\) 7.51472 0.503223 0.251611 0.967828i \(-0.419040\pi\)
0.251611 + 0.967828i \(0.419040\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −7.60660 13.1750i −0.504868 0.874457i −0.999984 0.00563010i \(-0.998208\pi\)
0.495116 0.868827i \(-0.335125\pi\)
\(228\) 0 0
\(229\) −3.50000 + 6.06218i −0.231287 + 0.400600i −0.958187 0.286143i \(-0.907627\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −7.24264 + 12.5446i −0.474481 + 0.821825i −0.999573 0.0292201i \(-0.990698\pi\)
0.525092 + 0.851046i \(0.324031\pi\)
\(234\) 0 0
\(235\) −3.00000 5.19615i −0.195698 0.338960i
\(236\) 0 0
\(237\) −11.0000 −0.714527
\(238\) 0 0
\(239\) 10.9706 0.709627 0.354813 0.934937i \(-0.384544\pi\)
0.354813 + 0.934937i \(0.384544\pi\)
\(240\) 0 0
\(241\) −2.00000 3.46410i −0.128831 0.223142i 0.794393 0.607404i \(-0.207789\pi\)
−0.923224 + 0.384262i \(0.874456\pi\)
\(242\) 0 0
\(243\) 0.500000 0.866025i 0.0320750 0.0555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 11.3492 19.6575i 0.722135 1.25077i
\(248\) 0 0
\(249\) 5.12132 + 8.87039i 0.324550 + 0.562138i
\(250\) 0 0
\(251\) 18.7279 1.18210 0.591048 0.806636i \(-0.298714\pi\)
0.591048 + 0.806636i \(0.298714\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) −2.12132 3.67423i −0.132842 0.230089i
\(256\) 0 0
\(257\) 5.12132 8.87039i 0.319459 0.553320i −0.660916 0.750460i \(-0.729832\pi\)
0.980375 + 0.197140i \(0.0631654\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0.878680 1.52192i 0.0543889 0.0942043i
\(262\) 0 0
\(263\) 4.24264 + 7.34847i 0.261612 + 0.453126i 0.966671 0.256023i \(-0.0824124\pi\)
−0.705058 + 0.709150i \(0.749079\pi\)
\(264\) 0 0
\(265\) 8.48528 0.521247
\(266\) 0 0
\(267\) −10.2426 −0.626839
\(268\) 0 0
\(269\) −1.24264 2.15232i −0.0757651 0.131229i 0.825654 0.564177i \(-0.190807\pi\)
−0.901419 + 0.432948i \(0.857473\pi\)
\(270\) 0 0
\(271\) −11.7279 + 20.3134i −0.712421 + 1.23395i 0.251525 + 0.967851i \(0.419068\pi\)
−0.963946 + 0.266098i \(0.914266\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.12132 3.67423i 0.127920 0.221565i
\(276\) 0 0
\(277\) −8.86396 15.3528i −0.532584 0.922462i −0.999276 0.0380425i \(-0.987888\pi\)
0.466692 0.884420i \(-0.345446\pi\)
\(278\) 0 0
\(279\) 9.48528 0.567869
\(280\) 0 0
\(281\) 28.9706 1.72824 0.864119 0.503287i \(-0.167876\pi\)
0.864119 + 0.503287i \(0.167876\pi\)
\(282\) 0 0
\(283\) 11.8640 + 20.5490i 0.705239 + 1.22151i 0.966605 + 0.256270i \(0.0824938\pi\)
−0.261366 + 0.965240i \(0.584173\pi\)
\(284\) 0 0
\(285\) 3.50000 6.06218i 0.207322 0.359092i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.500000 + 0.866025i −0.0294118 + 0.0509427i
\(290\) 0 0
\(291\) 0.242641 + 0.420266i 0.0142238 + 0.0246364i
\(292\) 0 0
\(293\) 4.97056 0.290383 0.145192 0.989404i \(-0.453620\pi\)
0.145192 + 0.989404i \(0.453620\pi\)
\(294\) 0 0
\(295\) 10.2426 0.596350
\(296\) 0 0
\(297\) −2.12132 3.67423i −0.123091 0.213201i
\(298\) 0 0
\(299\) −6.87868 + 11.9142i −0.397804 + 0.689017i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 3.87868 6.71807i 0.222824 0.385943i
\(304\) 0 0
\(305\) 2.24264 + 3.88437i 0.128413 + 0.222418i
\(306\) 0 0
\(307\) −3.24264 −0.185067 −0.0925336 0.995710i \(-0.529497\pi\)
−0.0925336 + 0.995710i \(0.529497\pi\)
\(308\) 0 0
\(309\) −17.2426 −0.980900
\(310\) 0 0
\(311\) 10.6066 + 18.3712i 0.601445 + 1.04173i 0.992602 + 0.121410i \(0.0387415\pi\)
−0.391157 + 0.920324i \(0.627925\pi\)
\(312\) 0 0
\(313\) 11.8640 20.5490i 0.670591 1.16150i −0.307146 0.951662i \(-0.599374\pi\)
0.977737 0.209835i \(-0.0672926\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.3640 21.4150i 0.694429 1.20279i −0.275943 0.961174i \(-0.588990\pi\)
0.970373 0.241613i \(-0.0776764\pi\)
\(318\) 0 0
\(319\) −3.72792 6.45695i −0.208724 0.361520i
\(320\) 0 0
\(321\) −12.7279 −0.710403
\(322\) 0 0
\(323\) −29.6985 −1.65247
\(324\) 0 0
\(325\) 1.62132 + 2.80821i 0.0899347 + 0.155771i
\(326\) 0 0
\(327\) −4.74264 + 8.21449i −0.262269 + 0.454263i
\(328\) 0 0
\(329\) 0 0
\(330\) 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
\(333\) −1.62132 2.80821i −0.0888478 0.153889i
\(334\) 0 0
\(335\) −5.24264 −0.286436
\(336\) 0 0
\(337\) −13.7279 −0.747808 −0.373904 0.927467i \(-0.621981\pi\)
−0.373904 + 0.927467i \(0.621981\pi\)
\(338\) 0 0
\(339\) −9.00000 15.5885i −0.488813 0.846649i
\(340\) 0 0
\(341\) 20.1213 34.8511i 1.08963 1.88730i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −2.12132 + 3.67423i −0.114208 + 0.197814i
\(346\) 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
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 3.24264 0.173079
\(352\) 0 0
\(353\) −5.12132 8.87039i −0.272580 0.472123i 0.696941 0.717128i \(-0.254544\pi\)
−0.969522 + 0.245005i \(0.921210\pi\)
\(354\) 0 0
\(355\) 6.36396 11.0227i 0.337764 0.585024i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −0.878680 + 1.52192i −0.0463749 + 0.0803237i −0.888281 0.459300i \(-0.848100\pi\)
0.841906 + 0.539624i \(0.181434\pi\)
\(360\) 0 0
\(361\) −15.0000 25.9808i −0.789474 1.36741i
\(362\) 0 0
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) −9.24264 −0.483782
\(366\) 0 0
\(367\) 17.8640 + 30.9413i 0.932491 + 1.61512i 0.779048 + 0.626965i \(0.215703\pi\)
0.153443 + 0.988157i \(0.450964\pi\)
\(368\) 0 0
\(369\) −2.12132 + 3.67423i −0.110432 + 0.191273i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −8.86396 + 15.3528i −0.458959 + 0.794939i −0.998906 0.0467591i \(-0.985111\pi\)
0.539948 + 0.841699i \(0.318444\pi\)
\(374\) 0 0
\(375\) 0.500000 + 0.866025i 0.0258199 + 0.0447214i
\(376\) 0 0
\(377\) 5.69848 0.293487
\(378\) 0 0
\(379\) −20.4558 −1.05075 −0.525373 0.850872i \(-0.676074\pi\)
−0.525373 + 0.850872i \(0.676074\pi\)
\(380\) 0 0
\(381\) −1.37868 2.38794i −0.0706319 0.122338i
\(382\) 0 0
\(383\) 12.7279 22.0454i 0.650366 1.12647i −0.332668 0.943044i \(-0.607949\pi\)
0.983034 0.183424i \(-0.0587180\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.62132 + 2.80821i −0.0824163 + 0.142749i
\(388\) 0 0
\(389\) 10.6066 + 18.3712i 0.537776 + 0.931455i 0.999023 + 0.0441839i \(0.0140687\pi\)
−0.461247 + 0.887272i \(0.652598\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 0 0
\(393\) 14.4853 0.730686
\(394\) 0 0
\(395\) −5.50000 9.52628i −0.276735 0.479319i
\(396\) 0 0
\(397\) −4.37868 + 7.58410i −0.219760 + 0.380635i −0.954734 0.297460i \(-0.903861\pi\)
0.734975 + 0.678094i \(0.237194\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.75736 3.04384i 0.0877583 0.152002i −0.818805 0.574072i \(-0.805363\pi\)
0.906563 + 0.422070i \(0.138696\pi\)
\(402\) 0 0
\(403\) 15.3787 + 26.6367i 0.766067 + 1.32687i
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −13.7574 −0.681927
\(408\) 0 0
\(409\) 8.50000 + 14.7224i 0.420298 + 0.727977i 0.995968 0.0897044i \(-0.0285922\pi\)
−0.575670 + 0.817682i \(0.695259\pi\)
\(410\) 0 0
\(411\) 2.12132 3.67423i 0.104637 0.181237i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −5.12132 + 8.87039i −0.251396 + 0.435430i
\(416\) 0 0
\(417\) 7.74264 + 13.4106i 0.379159 + 0.656722i
\(418\) 0 0
\(419\) 14.4853 0.707652 0.353826 0.935311i \(-0.384880\pi\)
0.353826 + 0.935311i \(0.384880\pi\)
\(420\) 0 0
\(421\) 31.4853 1.53450 0.767249 0.641349i \(-0.221625\pi\)
0.767249 + 0.641349i \(0.221625\pi\)
\(422\) 0 0
\(423\) −3.00000 5.19615i −0.145865 0.252646i
\(424\) 0 0
\(425\) 2.12132 3.67423i 0.102899 0.178227i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 6.87868 11.9142i 0.332106 0.575224i
\(430\) 0 0
\(431\) −9.72792 16.8493i −0.468578 0.811600i 0.530777 0.847511i \(-0.321900\pi\)
−0.999355 + 0.0359112i \(0.988567\pi\)
\(432\) 0 0
\(433\) −33.2426 −1.59754 −0.798770 0.601637i \(-0.794515\pi\)
−0.798770 + 0.601637i \(0.794515\pi\)
\(434\) 0 0
\(435\) 1.75736 0.0842589
\(436\) 0 0
\(437\) 14.8492 + 25.7196i 0.710336 + 1.23034i
\(438\) 0 0
\(439\) −5.00000 + 8.66025i −0.238637 + 0.413331i −0.960323 0.278889i \(-0.910034\pi\)
0.721686 + 0.692220i \(0.243367\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.2426 + 17.7408i −0.486643 + 0.842890i −0.999882 0.0153558i \(-0.995112\pi\)
0.513240 + 0.858245i \(0.328445\pi\)
\(444\) 0 0
\(445\) −5.12132 8.87039i −0.242774 0.420497i
\(446\) 0 0
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 9.00000 + 15.5885i 0.423793 + 0.734032i
\(452\) 0 0
\(453\) 11.2426 19.4728i 0.528225 0.914913i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −16.1066 + 27.8975i −0.753435 + 1.30499i 0.192714 + 0.981255i \(0.438271\pi\)
−0.946149 + 0.323733i \(0.895062\pi\)
\(458\) 0 0
\(459\) −2.12132 3.67423i −0.0990148 0.171499i
\(460\) 0 0
\(461\) −18.7279 −0.872246 −0.436123 0.899887i \(-0.643649\pi\)
−0.436123 + 0.899887i \(0.643649\pi\)
\(462\) 0 0
\(463\) 10.2721 0.477384 0.238692 0.971095i \(-0.423281\pi\)
0.238692 + 0.971095i \(0.423281\pi\)
\(464\) 0 0
\(465\) 4.74264 + 8.21449i 0.219935 + 0.380938i
\(466\) 0 0
\(467\) −5.48528 + 9.50079i −0.253829 + 0.439644i −0.964577 0.263803i \(-0.915023\pi\)
0.710748 + 0.703447i \(0.248357\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 3.24264 5.61642i 0.149413 0.258791i
\(472\) 0 0
\(473\) 6.87868 + 11.9142i 0.316282 + 0.547817i
\(474\) 0 0
\(475\) 7.00000 0.321182
\(476\) 0 0
\(477\) 8.48528 0.388514
\(478\) 0 0
\(479\) 6.00000 + 10.3923i 0.274147 + 0.474837i 0.969920 0.243426i \(-0.0782712\pi\)
−0.695773 + 0.718262i \(0.744938\pi\)
\(480\) 0 0
\(481\) 5.25736 9.10601i 0.239715 0.415198i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −0.242641 + 0.420266i −0.0110177 + 0.0190833i
\(486\) 0 0
\(487\) 18.8640 + 32.6733i 0.854808 + 1.48057i 0.876823 + 0.480813i \(0.159658\pi\)
−0.0220157 + 0.999758i \(0.507008\pi\)
\(488\) 0 0
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) 15.5147 0.700169 0.350085 0.936718i \(-0.386153\pi\)
0.350085 + 0.936718i \(0.386153\pi\)
\(492\) 0 0
\(493\) −3.72792 6.45695i −0.167897 0.290806i
\(494\) 0 0
\(495\) 2.12132 3.67423i 0.0953463 0.165145i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −9.74264 + 16.8747i −0.436140 + 0.755417i −0.997388 0.0722305i \(-0.976988\pi\)
0.561247 + 0.827648i \(0.310322\pi\)
\(500\) 0 0
\(501\) −9.36396 16.2189i −0.418351 0.724605i
\(502\) 0 0
\(503\) 26.4853 1.18092 0.590460 0.807067i \(-0.298946\pi\)
0.590460 + 0.807067i \(0.298946\pi\)
\(504\) 0 0
\(505\) 7.75736 0.345198
\(506\) 0 0
\(507\) −1.24264 2.15232i −0.0551876 0.0955877i
\(508\) 0 0
\(509\) −9.72792 + 16.8493i −0.431183 + 0.746830i −0.996975 0.0777173i \(-0.975237\pi\)
0.565793 + 0.824547i \(0.308570\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3.50000 6.06218i 0.154529 0.267652i
\(514\) 0 0
\(515\) −8.62132 14.9326i −0.379901 0.658007i
\(516\) 0 0
\(517\) −25.4558 −1.11955
\(518\) 0 0
\(519\) −20.4853 −0.899204
\(520\) 0 0
\(521\) −6.72792 11.6531i −0.294756 0.510532i 0.680172 0.733052i \(-0.261905\pi\)
−0.974928 + 0.222520i \(0.928572\pi\)
\(522\) 0 0
\(523\) −2.62132 + 4.54026i −0.114622 + 0.198532i −0.917629 0.397439i \(-0.869899\pi\)
0.803006 + 0.595970i \(0.203232\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 20.1213 34.8511i 0.876498 1.51814i
\(528\) 0 0
\(529\) 2.50000 + 4.33013i 0.108696 + 0.188266i
\(530\) 0 0
\(531\) 10.2426 0.444493
\(532\) 0 0
\(533\) −13.7574 −0.595897
\(534\) 0 0
\(535\) −6.36396 11.0227i −0.275138 0.476553i
\(536\) 0 0
\(537\) −3.00000 + 5.19615i −0.129460 + 0.224231i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 20.4706 35.4561i 0.880098 1.52437i 0.0288675 0.999583i \(-0.490810\pi\)
0.851231 0.524792i \(-0.175857\pi\)
\(542\) 0 0
\(543\) 6.50000 + 11.2583i 0.278942 + 0.483141i
\(544\) 0 0
\(545\) −9.48528 −0.406305
\(546\) 0 0
\(547\) 33.4558 1.43047 0.715234 0.698885i \(-0.246320\pi\)
0.715234 + 0.698885i \(0.246320\pi\)
\(548\) 0 0
\(549\) 2.24264 + 3.88437i 0.0957136 + 0.165781i
\(550\) 0 0
\(551\) 6.15076 10.6534i 0.262031 0.453851i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 1.62132 2.80821i 0.0688212 0.119202i
\(556\) 0 0
\(557\) 15.0000 + 25.9808i 0.635570 + 1.10084i 0.986394 + 0.164399i \(0.0525683\pi\)
−0.350824 + 0.936442i \(0.614098\pi\)
\(558\) 0 0
\(559\) −10.5147 −0.444725
\(560\) 0 0
\(561\) −18.0000 −0.759961
\(562\) 0 0
\(563\) −3.00000 5.19615i −0.126435 0.218992i 0.795858 0.605483i \(-0.207020\pi\)
−0.922293 + 0.386492i \(0.873687\pi\)
\(564\) 0 0
\(565\) 9.00000 15.5885i 0.378633 0.655811i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 20.8492 36.1119i 0.874046 1.51389i 0.0162699 0.999868i \(-0.494821\pi\)
0.857776 0.514024i \(-0.171846\pi\)
\(570\) 0 0
\(571\) 14.4706 + 25.0637i 0.605574 + 1.04889i 0.991960 + 0.126548i \(0.0403898\pi\)
−0.386386 + 0.922337i \(0.626277\pi\)
\(572\) 0 0
\(573\) −6.00000 −0.250654
\(574\) 0 0
\(575\) −4.24264 −0.176930
\(576\) 0 0
\(577\) 6.37868 + 11.0482i 0.265548 + 0.459942i 0.967707 0.252078i \(-0.0811138\pi\)
−0.702159 + 0.712020i \(0.747780\pi\)
\(578\) 0 0
\(579\) 3.37868 5.85204i 0.140413 0.243203i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 18.0000 31.1769i 0.745484 1.29122i
\(584\) 0 0
\(585\) 1.62132 + 2.80821i 0.0670333 + 0.116105i
\(586\) 0 0
\(587\) −45.2132 −1.86615 −0.933074 0.359684i \(-0.882885\pi\)
−0.933074 + 0.359684i \(0.882885\pi\)
\(588\) 0 0
\(589\) 66.3970 2.73584
\(590\) 0 0
\(591\) −8.12132 14.0665i −0.334066 0.578620i
\(592\) 0 0
\(593\) 1.60660 2.78272i 0.0659752 0.114272i −0.831151 0.556047i \(-0.812318\pi\)
0.897126 + 0.441774i \(0.145651\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −5.24264 + 9.08052i −0.214567 + 0.371641i
\(598\) 0 0
\(599\) 16.2426 + 28.1331i 0.663656 + 1.14949i 0.979648 + 0.200724i \(0.0643296\pi\)
−0.315991 + 0.948762i \(0.602337\pi\)
\(600\) 0 0
\(601\) 3.48528 0.142168 0.0710838 0.997470i \(-0.477354\pi\)
0.0710838 + 0.997470i \(0.477354\pi\)
\(602\) 0 0
\(603\) −5.24264 −0.213497
\(604\) 0 0
\(605\) −3.50000 6.06218i −0.142295 0.246463i
\(606\) 0 0
\(607\) −14.6213 + 25.3249i −0.593461 + 1.02790i 0.400301 + 0.916384i \(0.368906\pi\)
−0.993762 + 0.111521i \(0.964428\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.72792 16.8493i 0.393550 0.681648i
\(612\) 0 0
\(613\) 2.72792 + 4.72490i 0.110180 + 0.190837i 0.915843 0.401537i \(-0.131524\pi\)
−0.805663 + 0.592374i \(0.798191\pi\)
\(614\) 0 0
\(615\) −4.24264 −0.171080
\(616\) 0 0
\(617\) −26.4853 −1.06626 −0.533129 0.846034i \(-0.678984\pi\)
−0.533129 + 0.846034i \(0.678984\pi\)
\(618\) 0 0
\(619\) −5.98528 10.3668i −0.240569 0.416677i 0.720308 0.693655i \(-0.244001\pi\)
−0.960876 + 0.276977i \(0.910667\pi\)
\(620\) 0 0
\(621\) −2.12132 + 3.67423i −0.0851257 + 0.147442i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) −14.8492 25.7196i −0.593022 1.02714i
\(628\) 0 0
\(629\) −13.7574 −0.548542
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 11.2426 + 19.4728i 0.446855 + 0.773975i
\(634\) 0 0
\(635\) 1.37868 2.38794i 0.0547112 0.0947626i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6.36396 11.0227i 0.251754 0.436051i
\(640\) 0 0
\(641\) −17.1213 29.6550i −0.676251 1.17130i −0.976101 0.217316i \(-0.930270\pi\)
0.299850 0.953986i \(-0.403063\pi\)
\(642\) 0 0
\(643\) 19.7279 0.777993 0.388997 0.921239i \(-0.372822\pi\)
0.388997 + 0.921239i \(0.372822\pi\)
\(644\) 0 0
\(645\) −3.24264 −0.127679
\(646\) 0 0
\(647\) 5.12132 + 8.87039i 0.201340 + 0.348731i 0.948960 0.315395i \(-0.102137\pi\)
−0.747621 + 0.664126i \(0.768804\pi\)
\(648\) 0 0
\(649\) 21.7279 37.6339i 0.852896 1.47726i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −5.12132 + 8.87039i −0.200413 + 0.347125i −0.948661 0.316293i \(-0.897562\pi\)
0.748249 + 0.663418i \(0.230895\pi\)
\(654\) 0 0
\(655\) 7.24264 + 12.5446i 0.282993 + 0.490159i
\(656\) 0 0
\(657\) −9.24264 −0.360590
\(658\) 0 0
\(659\) 40.9706 1.59599 0.797993 0.602666i \(-0.205895\pi\)
0.797993 + 0.602666i \(0.205895\pi\)
\(660\) 0 0
\(661\) −1.01472 1.75754i −0.0394680 0.0683605i 0.845617 0.533791i \(-0.179233\pi\)
−0.885085 + 0.465430i \(0.845900\pi\)
\(662\) 0 0
\(663\) 6.87868 11.9142i 0.267146 0.462710i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3.72792 + 6.45695i −0.144346 + 0.250014i
\(668\) 0 0
\(669\) 3.75736 + 6.50794i 0.145268 + 0.251611i
\(670\) 0 0
\(671\) 19.0294 0.734623
\(672\) 0 0
\(673\) 29.7279 1.14593 0.572964 0.819581i \(-0.305794\pi\)
0.572964 + 0.819581i \(0.305794\pi\)
\(674\) 0 0
\(675\) 0.500000 + 0.866025i 0.0192450 + 0.0333333i
\(676\) 0 0
\(677\) 6.36396 11.0227i 0.244587 0.423637i −0.717428 0.696632i \(-0.754681\pi\)
0.962015 + 0.272995i \(0.0880143\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 7.60660 13.1750i 0.291486 0.504868i
\(682\) 0 0
\(683\) −16.6066 28.7635i −0.635434 1.10060i −0.986423 0.164224i \(-0.947488\pi\)
0.350989 0.936380i \(-0.385845\pi\)
\(684\) 0 0
\(685\) 4.24264 0.162103
\(686\) 0 0
\(687\) −7.00000 −0.267067
\(688\) 0 0
\(689\) 13.7574 + 23.8284i 0.524114 + 0.907791i
\(690\) 0 0
\(691\) 13.4706 23.3317i 0.512444 0.887580i −0.487452 0.873150i \(-0.662073\pi\)
0.999896 0.0144296i \(-0.00459325\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7.74264 + 13.4106i −0.293695 + 0.508695i
\(696\) 0 0
\(697\) 9.00000 + 15.5885i 0.340899 + 0.590455i
\(698\) 0 0
\(699\) −14.4853 −0.547884
\(700\) 0 0
\(701\) 8.78680 0.331873 0.165936 0.986136i \(-0.446935\pi\)
0.165936 + 0.986136i \(0.446935\pi\)
\(702\) 0 0
\(703\) −11.3492 19.6575i −0.428045 0.741395i
\(704\) 0 0
\(705\) 3.00000 5.19615i 0.112987 0.195698i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 18.2426 31.5972i 0.685117 1.18666i −0.288283 0.957545i \(-0.593085\pi\)
0.973400 0.229112i \(-0.0735822\pi\)
\(710\) 0 0
\(711\) −5.50000 9.52628i −0.206266 0.357263i
\(712\) 0 0
\(713\) −40.2426 −1.50710
\(714\) 0 0
\(715\) 13.7574 0.514496
\(716\) 0 0
\(717\) 5.48528 + 9.50079i 0.204852 + 0.354813i
\(718\) 0 0
\(719\) 13.2426 22.9369i 0.493867 0.855403i −0.506108 0.862470i \(-0.668916\pi\)
0.999975 + 0.00706717i \(0.00224957\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 2.00000 3.46410i 0.0743808 0.128831i
\(724\) 0 0
\(725\) 0.878680 + 1.52192i 0.0326333 + 0.0565226i
\(726\) 0 0
\(727\) −0.757359 −0.0280889 −0.0140445 0.999901i \(-0.504471\pi\)
−0.0140445 + 0.999901i \(0.504471\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.87868 + 11.9142i 0.254417 + 0.440663i
\(732\) 0 0
\(733\) 0.893398 1.54741i 0.0329984 0.0571549i −0.849055 0.528305i \(-0.822828\pi\)
0.882053 + 0.471150i \(0.156161\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11.1213 + 19.2627i −0.409659 + 0.709550i
\(738\) 0 0
\(739\) 7.74264 + 13.4106i 0.284818 + 0.493319i 0.972565 0.232632i \(-0.0747336\pi\)
−0.687747 + 0.725950i \(0.741400\pi\)
\(740\) 0 0
\(741\) 22.6985 0.833850
\(742\) 0 0
\(743\) 15.2132 0.558118 0.279059 0.960274i \(-0.409977\pi\)
0.279059 + 0.960274i \(0.409977\pi\)
\(744\) 0 0
\(745\) 6.00000 + 10.3923i 0.219823 + 0.380745i
\(746\) 0 0
\(747\) −5.12132 + 8.87039i −0.187379 + 0.324550i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −22.4706 + 38.9202i −0.819962 + 1.42022i 0.0857467 + 0.996317i \(0.472672\pi\)
−0.905709 + 0.423900i \(0.860661\pi\)
\(752\) 0 0
\(753\) 9.36396 + 16.2189i 0.341242 + 0.591048i
\(754\) 0 0
\(755\) 22.4853 0.818323
\(756\) 0 0
\(757\) 9.02944 0.328180 0.164090 0.986445i \(-0.447531\pi\)
0.164090 + 0.986445i \(0.447531\pi\)
\(758\) 0 0
\(759\) 9.00000 + 15.5885i 0.326679 + 0.565825i
\(760\) 0 0
\(761\) 23.1213 40.0473i 0.838147 1.45171i −0.0532948 0.998579i \(-0.516972\pi\)
0.891442 0.453135i \(-0.149694\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2.12132 3.67423i 0.0766965 0.132842i
\(766\) 0 0
\(767\) 16.6066 + 28.7635i 0.599630 + 1.03859i
\(768\) 0 0
\(769\) −5.00000 −0.180305 −0.0901523 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) 10.2426 0.368880
\(772\) 0 0
\(773\) 5.84924 + 10.1312i 0.210383 + 0.364393i 0.951834 0.306613i \(-0.0991957\pi\)
−0.741452 + 0.671006i \(0.765862\pi\)
\(774\) 0 0
\(775\) −4.74264 + 8.21449i −0.170361 + 0.295073i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −14.8492 + 25.7196i −0.532029 + 0.921502i
\(780\) 0 0
\(781\) −27.0000 46.7654i −0.966136 1.67340i
\(782\) 0 0
\(783\) 1.75736 0.0628029
\(784\) 0 0
\(785\) 6.48528 0.231470
\(786\) 0 0
\(787\) 14.2426 + 24.6690i 0.507695 + 0.879354i 0.999960 + 0.00890869i \(0.00283576\pi\)
−0.492265 + 0.870445i \(0.663831\pi\)
\(788\) 0 0
\(789\) −4.24264 + 7.34847i −0.151042 + 0.261612i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −7.27208 + 12.5956i −0.258239 + 0.447283i
\(794\) 0 0
\(795\) 4.24264 + 7.34847i 0.150471 + 0.260623i
\(796\) 0 0
\(797\) 31.7574 1.12490 0.562452 0.826830i \(-0.309858\pi\)
0.562452 + 0.826830i \(0.309858\pi\)
\(798\) 0 0
\(799\) −25.4558 −0.900563
\(800\) 0 0
\(801\) −5.12132 8.87039i −0.180953 0.313420i
\(802\) 0 0
\(803\) −19.6066 + 33.9596i −0.691902 + 1.19841i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1.24264 2.15232i 0.0437430 0.0757651i
\(808\) 0 0
\(809\) −13.9706 24.1977i −0.491179 0.850747i 0.508770 0.860903i \(-0.330101\pi\)
−0.999948 + 0.0101560i \(0.996767\pi\)
\(810\) 0 0
\(811\) −11.9411 −0.419310 −0.209655 0.977775i \(-0.567234\pi\)
−0.209655 + 0.977775i \(0.567234\pi\)
\(812\) 0 0
\(813\) −23.4558 −0.822632
\(814\) 0 0
\(815\) −4.00000 6.92820i −0.140114 0.242684i
\(816\) 0 0
\(817\) −11.3492 + 19.6575i −0.397060 + 0.687728i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.84924 + 4.93503i −0.0994392 + 0.172234i −0.911453 0.411405i \(-0.865038\pi\)
0.812013 + 0.583639i \(0.198372\pi\)
\(822\) 0 0
\(823\) 19.4853 + 33.7495i 0.679214 + 1.17643i 0.975218 + 0.221247i \(0.0710126\pi\)
−0.296004 + 0.955187i \(0.595654\pi\)
\(824\) 0 0
\(825\) 4.24264 0.147710
\(826\) 0 0
\(827\) 43.4558 1.51111 0.755554 0.655087i \(-0.227368\pi\)
0.755554 + 0.655087i \(0.227368\pi\)
\(828\) 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
\(831\) 8.86396 15.3528i 0.307487 0.532584i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 9.36396 16.2189i 0.324053 0.561277i
\(836\) 0 0
\(837\) 4.74264 + 8.21449i 0.163930 + 0.283934i
\(838\) 0 0
\(839\) −25.7574 −0.889243 −0.444621 0.895719i \(-0.646662\pi\)
−0.444621 + 0.895719i \(0.646662\pi\)
\(840\) 0 0
\(841\) −25.9117 −0.893506
\(842\) 0 0
\(843\) 14.4853 + 25.0892i 0.498900 + 0.864119i
\(844\) 0 0
\(845\) 1.24264 2.15232i 0.0427481 0.0740419i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −11.8640 + 20.5490i −0.407170 + 0.705239i
\(850\) 0 0
\(851\) 6.87868 + 11.9142i 0.235798 + 0.408414i
\(852\) 0 0
\(853\) 25.7279 0.880907 0.440454 0.897775i \(-0.354818\pi\)
0.440454 + 0.897775i \(0.354818\pi\)
\(854\) 0 0
\(855\) 7.00000 0.239395
\(856\) 0 0
\(857\) 2.12132 + 3.67423i 0.0724629 + 0.125509i 0.899980 0.435931i \(-0.143581\pi\)
−0.827517 + 0.561440i \(0.810247\pi\)
\(858\) 0 0
\(859\) −11.0000 + 19.0526i −0.375315 + 0.650065i −0.990374 0.138416i \(-0.955799\pi\)
0.615059 + 0.788481i \(0.289132\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 10.7574 18.6323i 0.366185 0.634251i −0.622781 0.782396i \(-0.713997\pi\)
0.988966 + 0.148146i \(0.0473305\pi\)
\(864\) 0 0
\(865\) −10.2426 17.7408i −0.348260 0.603204i
\(866\) 0 0
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) −46.6690 −1.58314
\(870\) 0 0
\(871\) −8.50000 14.7224i −0.288012 0.498851i
\(872\) 0 0
\(873\) −0.242641 + 0.420266i −0.00821214 + 0.0142238i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 5.00000 8.66025i 0.168838 0.292436i −0.769174 0.639040i \(-0.779332\pi\)
0.938012 + 0.346604i \(0.112665\pi\)
\(878\) 0 0
\(879\) 2.48528 + 4.30463i 0.0838265 + 0.145192i
\(880\) 0 0
\(881\) −40.9706 −1.38033 −0.690167 0.723650i \(-0.742463\pi\)
−0.690167 + 0.723650i \(0.742463\pi\)
\(882\) 0 0
\(883\) 5.72792 0.192760 0.0963800 0.995345i \(-0.469274\pi\)
0.0963800 + 0.995345i \(0.469274\pi\)
\(884\) 0 0
\(885\) 5.12132 + 8.87039i 0.172151 + 0.298175i
\(886\) 0 0
\(887\) 7.60660 13.1750i 0.255405 0.442374i −0.709601 0.704604i \(-0.751125\pi\)
0.965005 + 0.262230i \(0.0844580\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2.12132 3.67423i 0.0710669 0.123091i
\(892\) 0 0
\(893\) −21.0000 36.3731i −0.702738 1.21718i
\(894\) 0 0
\(895\) −6.00000 −0.200558
\(896\) 0 0
\(897\) −13.7574 −0.459345
\(898\) 0 0
\(899\) 8.33452 + 14.4358i 0.277972 + 0.481462i
\(900\) 0 0
\(901\) 18.0000 31.1769i 0.599667 1.03865i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −6.50000 + 11.2583i −0.216067 + 0.374240i
\(906\) 0 0
\(907\) 15.1360 + 26.2164i 0.502584 + 0.870501i 0.999996 + 0.00298623i \(0.000950549\pi\)
−0.497412 + 0.867515i \(0.665716\pi\)
\(908\) 0 0
\(909\) 7.75736 0.257295
\(910\) 0 0
\(911\) 51.2132 1.69677 0.848385 0.529380i \(-0.177576\pi\)
0.848385 + 0.529380i \(0.177576\pi\)
\(912\) 0 0
\(913\) 21.7279 + 37.6339i 0.719089 + 1.24550i
\(914\) 0 0
\(915\) −2.24264 + 3.88437i −0.0741394 + 0.128413i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −16.9853 + 29.4194i −0.560293 + 0.970455i 0.437178 + 0.899375i \(0.355978\pi\)
−0.997471 + 0.0710804i \(0.977355\pi\)
\(920\) 0 0
\(921\) −1.62132 2.80821i −0.0534243 0.0925336i
\(922\) 0 0
\(923\) 41.2721 1.35849
\(924\) 0 0
\(925\) 3.24264 0.106617
\(926\) 0 0
\(927\) −8.62132 14.9326i −0.283161 0.490450i
\(928\) 0 0
\(929\) 18.3640 31.8073i 0.602502 1.04356i −0.389939 0.920841i \(-0.627504\pi\)
0.992441 0.122723i \(-0.0391628\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −10.6066 + 18.3712i −0.347245 + 0.601445i
\(934\) 0 0
\(935\) −9.00000 15.5885i −0.294331 0.509797i
\(936\) 0 0
\(937\) 18.6985 0.610853 0.305426 0.952216i \(-0.401201\pi\)
0.305426 + 0.952216i \(0.401201\pi\)
\(938\) 0 0
\(939\) 23.7279 0.774331
\(940\) 0 0
\(941\) −23.8492 41.3081i −0.777463 1.34661i −0.933400 0.358838i \(-0.883173\pi\)
0.155937 0.987767i \(-0.450160\pi\)
\(942\) 0 0
\(943\) 9.00000 15.5885i 0.293080 0.507630i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.60660 2.78272i 0.0522075 0.0904261i −0.838741 0.544531i \(-0.816708\pi\)
0.890948 + 0.454105i \(0.150041\pi\)
\(948\) 0 0
\(949\) −14.9853 25.9553i −0.486443 0.842544i
\(950\) 0 0
\(951\) 24.7279 0.801858
\(952\) 0 0
\(953\) −27.5147 −0.891289 −0.445645 0.895210i \(-0.647025\pi\)
−0.445645 + 0.895210i \(0.647025\pi\)
\(954\) 0 0
\(955\) −3.00000 5.19615i −0.0970777 0.168144i
\(956\) 0 0
\(957\) 3.72792 6.45695i 0.120507 0.208724i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −29.4853 + 51.0700i −0.951138 + 1.64742i
\(962\) 0 0
\(963\) −6.36396 11.0227i −0.205076 0.355202i
\(964\) 0 0
\(965\) 6.75736 0.217527
\(966\) 0 0
\(967\) −35.2426 −1.13333 −0.566663 0.823949i \(-0.691766\pi\)
−0.566663 + 0.823949i \(0.691766\pi\)
\(968\) 0 0
\(969\) −14.8492 25.7196i −0.477026 0.826234i
\(970\) 0 0
\(971\) 26.4853 45.8739i 0.849953 1.47216i −0.0312961 0.999510i \(-0.509963\pi\)
0.881249 0.472652i \(-0.156703\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −1.62132 + 2.80821i −0.0519238 + 0.0899347i
\(976\) 0 0
\(977\) 4.39340 + 7.60959i 0.140557 + 0.243452i 0.927707 0.373310i \(-0.121777\pi\)
−0.787149 + 0.616762i \(0.788444\pi\)
\(978\) 0 0
\(979\) −43.4558 −1.38885
\(980\) 0 0
\(981\) −9.48528 −0.302842
\(982\) 0 0
\(983\) −20.8492 36.1119i −0.664988 1.15179i −0.979289 0.202469i \(-0.935103\pi\)
0.314301 0.949323i \(-0.398230\pi\)
\(984\) 0 0
\(985\) 8.12132 14.0665i 0.258767 0.448197i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 6.87868 11.9142i 0.218729 0.378850i
\(990\) 0 0
\(991\) −7.47056 12.9394i −0.237310 0.411033i 0.722631 0.691234i \(-0.242932\pi\)
−0.959942 + 0.280200i \(0.909599\pi\)
\(992\) 0 0
\(993\) −17.0000 −0.539479
\(994\) 0 0
\(995\) −10.4853 −0.332406
\(996\) 0 0
\(997\) −12.8640 22.2810i −0.407406 0.705647i 0.587192 0.809447i \(-0.300233\pi\)
−0.994598 + 0.103800i \(0.966900\pi\)
\(998\) 0 0
\(999\) 1.62132 2.80821i 0.0512963 0.0888478i
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 2940.2.q.q.961.2 4
7.2 even 3 2940.2.a.p.1.1 2
7.3 odd 6 420.2.q.d.361.2 yes 4
7.4 even 3 inner 2940.2.q.q.361.2 4
7.5 odd 6 2940.2.a.r.1.1 2
7.6 odd 2 420.2.q.d.121.2 4
21.2 odd 6 8820.2.a.bf.1.2 2
21.5 even 6 8820.2.a.bk.1.2 2
21.17 even 6 1260.2.s.e.361.2 4
21.20 even 2 1260.2.s.e.541.2 4
28.3 even 6 1680.2.bg.t.1201.1 4
28.27 even 2 1680.2.bg.t.961.1 4
35.3 even 12 2100.2.bc.f.949.1 8
35.13 even 4 2100.2.bc.f.1549.4 8
35.17 even 12 2100.2.bc.f.949.4 8
35.24 odd 6 2100.2.q.k.1201.1 4
35.27 even 4 2100.2.bc.f.1549.1 8
35.34 odd 2 2100.2.q.k.1801.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.q.d.121.2 4 7.6 odd 2
420.2.q.d.361.2 yes 4 7.3 odd 6
1260.2.s.e.361.2 4 21.17 even 6
1260.2.s.e.541.2 4 21.20 even 2
1680.2.bg.t.961.1 4 28.27 even 2
1680.2.bg.t.1201.1 4 28.3 even 6
2100.2.q.k.1201.1 4 35.24 odd 6
2100.2.q.k.1801.1 4 35.34 odd 2
2100.2.bc.f.949.1 8 35.3 even 12
2100.2.bc.f.949.4 8 35.17 even 12
2100.2.bc.f.1549.1 8 35.27 even 4
2100.2.bc.f.1549.4 8 35.13 even 4
2940.2.a.p.1.1 2 7.2 even 3
2940.2.a.r.1.1 2 7.5 odd 6
2940.2.q.q.361.2 4 7.4 even 3 inner
2940.2.q.q.961.2 4 1.1 even 1 trivial
8820.2.a.bf.1.2 2 21.2 odd 6
8820.2.a.bk.1.2 2 21.5 even 6