Properties

Label 1944.2.i.k.649.2
Level $1944$
Weight $2$
Character 1944.649
Analytic conductor $15.523$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1944,2,Mod(649,1944)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1944, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("1944.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1944 = 2^{3} \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1944.i (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: \(15.5229181529\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 649.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1944.649
Dual form 1944.2.i.k.1297.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.366025 + 0.633975i) q^{5} +(-1.73205 + 3.00000i) q^{7} +O(q^{10})\) \(q+(0.366025 + 0.633975i) q^{5} +(-1.73205 + 3.00000i) q^{7} +(-3.09808 + 5.36603i) q^{11} +(1.23205 + 2.13397i) q^{13} +0.732051 q^{17} -4.46410 q^{19} +(-4.36603 - 7.56218i) q^{23} +(2.23205 - 3.86603i) q^{25} +(-1.26795 + 2.19615i) q^{29} +(-1.23205 - 2.13397i) q^{31} -2.53590 q^{35} +2.53590 q^{37} +(-4.73205 - 8.19615i) q^{41} +(2.96410 - 5.13397i) q^{43} +(3.46410 - 6.00000i) q^{47} +(-2.50000 - 4.33013i) q^{49} -7.66025 q^{53} -4.53590 q^{55} +(-0.901924 - 1.56218i) q^{59} +(0.767949 - 1.33013i) q^{61} +(-0.901924 + 1.56218i) q^{65} +(1.76795 + 3.06218i) q^{67} -6.19615 q^{71} +11.3923 q^{73} +(-10.7321 - 18.5885i) q^{77} +(-6.96410 + 12.0622i) q^{79} +(-6.36603 + 11.0263i) q^{83} +(0.267949 + 0.464102i) q^{85} -12.0000 q^{89} -8.53590 q^{91} +(-1.63397 - 2.83013i) q^{95} +(-5.96410 + 10.3301i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{5} - 2 q^{11} - 2 q^{13} - 4 q^{17} - 4 q^{19} - 14 q^{23} + 2 q^{25} - 12 q^{29} + 2 q^{31} - 24 q^{35} + 24 q^{37} - 12 q^{41} - 2 q^{43} - 10 q^{49} + 4 q^{53} - 32 q^{55} - 14 q^{59} + 10 q^{61} - 14 q^{65} + 14 q^{67} - 4 q^{71} + 4 q^{73} - 36 q^{77} - 14 q^{79} - 22 q^{83} + 8 q^{85} - 48 q^{89} - 48 q^{91} - 10 q^{95} - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1944\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(973\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0.366025 + 0.633975i 0.163692 + 0.283522i 0.936190 0.351495i \(-0.114326\pi\)
−0.772498 + 0.635017i \(0.780993\pi\)
\(6\) 0 0
\(7\) −1.73205 + 3.00000i −0.654654 + 1.13389i 0.327327 + 0.944911i \(0.393852\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.09808 + 5.36603i −0.934105 + 1.61792i −0.157883 + 0.987458i \(0.550467\pi\)
−0.776222 + 0.630460i \(0.782866\pi\)
\(12\) 0 0
\(13\) 1.23205 + 2.13397i 0.341709 + 0.591858i 0.984750 0.173974i \(-0.0556608\pi\)
−0.643041 + 0.765832i \(0.722327\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.732051 0.177548 0.0887742 0.996052i \(-0.471705\pi\)
0.0887742 + 0.996052i \(0.471705\pi\)
\(18\) 0 0
\(19\) −4.46410 −1.02414 −0.512068 0.858945i \(-0.671120\pi\)
−0.512068 + 0.858945i \(0.671120\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.36603 7.56218i −0.910379 1.57682i −0.813529 0.581524i \(-0.802457\pi\)
−0.0968500 0.995299i \(-0.530877\pi\)
\(24\) 0 0
\(25\) 2.23205 3.86603i 0.446410 0.773205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.26795 + 2.19615i −0.235452 + 0.407815i −0.959404 0.282035i \(-0.908990\pi\)
0.723952 + 0.689851i \(0.242324\pi\)
\(30\) 0 0
\(31\) −1.23205 2.13397i −0.221283 0.383273i 0.733915 0.679241i \(-0.237691\pi\)
−0.955198 + 0.295968i \(0.904358\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.53590 −0.428645
\(36\) 0 0
\(37\) 2.53590 0.416899 0.208450 0.978033i \(-0.433158\pi\)
0.208450 + 0.978033i \(0.433158\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.73205 8.19615i −0.739022 1.28002i −0.952936 0.303171i \(-0.901955\pi\)
0.213914 0.976853i \(-0.431379\pi\)
\(42\) 0 0
\(43\) 2.96410 5.13397i 0.452021 0.782924i −0.546490 0.837465i \(-0.684036\pi\)
0.998511 + 0.0545417i \(0.0173698\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.46410 6.00000i 0.505291 0.875190i −0.494690 0.869069i \(-0.664718\pi\)
0.999981 0.00612051i \(-0.00194823\pi\)
\(48\) 0 0
\(49\) −2.50000 4.33013i −0.357143 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.66025 −1.05222 −0.526108 0.850418i \(-0.676349\pi\)
−0.526108 + 0.850418i \(0.676349\pi\)
\(54\) 0 0
\(55\) −4.53590 −0.611620
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.901924 1.56218i −0.117420 0.203378i 0.801324 0.598230i \(-0.204129\pi\)
−0.918745 + 0.394852i \(0.870796\pi\)
\(60\) 0 0
\(61\) 0.767949 1.33013i 0.0983258 0.170305i −0.812666 0.582730i \(-0.801985\pi\)
0.910992 + 0.412424i \(0.135318\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.901924 + 1.56218i −0.111870 + 0.193764i
\(66\) 0 0
\(67\) 1.76795 + 3.06218i 0.215989 + 0.374105i 0.953578 0.301146i \(-0.0973690\pi\)
−0.737589 + 0.675250i \(0.764036\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.19615 −0.735348 −0.367674 0.929955i \(-0.619846\pi\)
−0.367674 + 0.929955i \(0.619846\pi\)
\(72\) 0 0
\(73\) 11.3923 1.33337 0.666684 0.745340i \(-0.267713\pi\)
0.666684 + 0.745340i \(0.267713\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −10.7321 18.5885i −1.22303 2.11835i
\(78\) 0 0
\(79\) −6.96410 + 12.0622i −0.783523 + 1.35710i 0.146355 + 0.989232i \(0.453246\pi\)
−0.929878 + 0.367869i \(0.880088\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.36603 + 11.0263i −0.698762 + 1.21029i 0.270134 + 0.962823i \(0.412932\pi\)
−0.968896 + 0.247469i \(0.920401\pi\)
\(84\) 0 0
\(85\) 0.267949 + 0.464102i 0.0290632 + 0.0503389i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) −8.53590 −0.894805
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.63397 2.83013i −0.167642 0.290365i
\(96\) 0 0
\(97\) −5.96410 + 10.3301i −0.605563 + 1.04887i 0.386400 + 0.922332i \(0.373719\pi\)
−0.991962 + 0.126534i \(0.959615\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.19615 + 3.80385i −0.218525 + 0.378497i −0.954357 0.298667i \(-0.903458\pi\)
0.735832 + 0.677164i \(0.236791\pi\)
\(102\) 0 0
\(103\) 9.42820 + 16.3301i 0.928988 + 1.60906i 0.785018 + 0.619473i \(0.212654\pi\)
0.143971 + 0.989582i \(0.454013\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.46410 0.914929 0.457465 0.889228i \(-0.348758\pi\)
0.457465 + 0.889228i \(0.348758\pi\)
\(108\) 0 0
\(109\) 16.4641 1.57697 0.788487 0.615051i \(-0.210865\pi\)
0.788487 + 0.615051i \(0.210865\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.46410 6.00000i −0.325875 0.564433i 0.655814 0.754923i \(-0.272326\pi\)
−0.981689 + 0.190490i \(0.938992\pi\)
\(114\) 0 0
\(115\) 3.19615 5.53590i 0.298043 0.516225i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.26795 + 2.19615i −0.116233 + 0.201321i
\(120\) 0 0
\(121\) −13.6962 23.7224i −1.24510 2.15658i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) 1.92820 0.171100 0.0855502 0.996334i \(-0.472735\pi\)
0.0855502 + 0.996334i \(0.472735\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −7.63397 13.2224i −0.666983 1.15525i −0.978743 0.205088i \(-0.934252\pi\)
0.311760 0.950161i \(-0.399082\pi\)
\(132\) 0 0
\(133\) 7.73205 13.3923i 0.670454 1.16126i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.46410 + 16.3923i −0.808573 + 1.40049i 0.105280 + 0.994443i \(0.466426\pi\)
−0.913852 + 0.406046i \(0.866907\pi\)
\(138\) 0 0
\(139\) 1.73205 + 3.00000i 0.146911 + 0.254457i 0.930084 0.367347i \(-0.119734\pi\)
−0.783174 + 0.621803i \(0.786400\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −15.2679 −1.27677
\(144\) 0 0
\(145\) −1.85641 −0.154166
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.0263 + 17.3660i 0.821385 + 1.42268i 0.904651 + 0.426153i \(0.140131\pi\)
−0.0832663 + 0.996527i \(0.526535\pi\)
\(150\) 0 0
\(151\) 1.76795 3.06218i 0.143874 0.249196i −0.785078 0.619396i \(-0.787377\pi\)
0.928952 + 0.370200i \(0.120711\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.901924 1.56218i 0.0724443 0.125477i
\(156\) 0 0
\(157\) −2.69615 4.66987i −0.215176 0.372696i 0.738151 0.674636i \(-0.235699\pi\)
−0.953327 + 0.301939i \(0.902366\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 30.2487 2.38393
\(162\) 0 0
\(163\) −22.3923 −1.75390 −0.876950 0.480581i \(-0.840426\pi\)
−0.876950 + 0.480581i \(0.840426\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.19615 3.80385i −0.169943 0.294351i 0.768456 0.639902i \(-0.221025\pi\)
−0.938400 + 0.345552i \(0.887692\pi\)
\(168\) 0 0
\(169\) 3.46410 6.00000i 0.266469 0.461538i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.46410 + 6.00000i −0.263371 + 0.456172i −0.967135 0.254262i \(-0.918168\pi\)
0.703765 + 0.710433i \(0.251501\pi\)
\(174\) 0 0
\(175\) 7.73205 + 13.3923i 0.584488 + 1.01236i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.53590 0.189542 0.0947710 0.995499i \(-0.469788\pi\)
0.0947710 + 0.995499i \(0.469788\pi\)
\(180\) 0 0
\(181\) −11.3205 −0.841447 −0.420723 0.907189i \(-0.638224\pi\)
−0.420723 + 0.907189i \(0.638224\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.928203 + 1.60770i 0.0682429 + 0.118200i
\(186\) 0 0
\(187\) −2.26795 + 3.92820i −0.165849 + 0.287259i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.46410 + 16.3923i −0.684798 + 1.18611i 0.288702 + 0.957419i \(0.406776\pi\)
−0.973500 + 0.228686i \(0.926557\pi\)
\(192\) 0 0
\(193\) 1.42820 + 2.47372i 0.102804 + 0.178062i 0.912839 0.408320i \(-0.133885\pi\)
−0.810035 + 0.586382i \(0.800552\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −23.3205 −1.66152 −0.830759 0.556633i \(-0.812093\pi\)
−0.830759 + 0.556633i \(0.812093\pi\)
\(198\) 0 0
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.39230 7.60770i −0.308279 0.533956i
\(204\) 0 0
\(205\) 3.46410 6.00000i 0.241943 0.419058i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 13.8301 23.9545i 0.956650 1.65697i
\(210\) 0 0
\(211\) −7.42820 12.8660i −0.511379 0.885734i −0.999913 0.0131891i \(-0.995802\pi\)
0.488534 0.872545i \(-0.337532\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.33975 0.295968
\(216\) 0 0
\(217\) 8.53590 0.579455
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.901924 + 1.56218i 0.0606700 + 0.105083i
\(222\) 0 0
\(223\) −6.96410 + 12.0622i −0.466351 + 0.807743i −0.999261 0.0384284i \(-0.987765\pi\)
0.532911 + 0.846172i \(0.321098\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.3923 18.0000i 0.689761 1.19470i −0.282153 0.959369i \(-0.591049\pi\)
0.971915 0.235333i \(-0.0756180\pi\)
\(228\) 0 0
\(229\) 8.16025 + 14.1340i 0.539245 + 0.933999i 0.998945 + 0.0459251i \(0.0146236\pi\)
−0.459700 + 0.888074i \(0.652043\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −17.1244 −1.12185 −0.560927 0.827865i \(-0.689555\pi\)
−0.560927 + 0.827865i \(0.689555\pi\)
\(234\) 0 0
\(235\) 5.07180 0.330848
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0.562178 + 0.973721i 0.0363643 + 0.0629847i 0.883635 0.468177i \(-0.155089\pi\)
−0.847270 + 0.531162i \(0.821756\pi\)
\(240\) 0 0
\(241\) −5.66025 + 9.80385i −0.364609 + 0.631521i −0.988713 0.149820i \(-0.952131\pi\)
0.624104 + 0.781341i \(0.285464\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.83013 3.16987i 0.116923 0.202516i
\(246\) 0 0
\(247\) −5.50000 9.52628i −0.349957 0.606143i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −22.5885 −1.42577 −0.712885 0.701281i \(-0.752612\pi\)
−0.712885 + 0.701281i \(0.752612\pi\)
\(252\) 0 0
\(253\) 54.1051 3.40156
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.16987 + 3.75833i 0.135353 + 0.234438i 0.925732 0.378180i \(-0.123450\pi\)
−0.790379 + 0.612618i \(0.790116\pi\)
\(258\) 0 0
\(259\) −4.39230 + 7.60770i −0.272925 + 0.472719i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.90192 5.02628i 0.178940 0.309934i −0.762578 0.646897i \(-0.776066\pi\)
0.941518 + 0.336963i \(0.109400\pi\)
\(264\) 0 0
\(265\) −2.80385 4.85641i −0.172239 0.298327i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.2679 0.930903 0.465452 0.885073i \(-0.345892\pi\)
0.465452 + 0.885073i \(0.345892\pi\)
\(270\) 0 0
\(271\) −10.4641 −0.635649 −0.317824 0.948150i \(-0.602952\pi\)
−0.317824 + 0.948150i \(0.602952\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 13.8301 + 23.9545i 0.833988 + 1.44451i
\(276\) 0 0
\(277\) 0.303848 0.526279i 0.0182564 0.0316211i −0.856753 0.515727i \(-0.827522\pi\)
0.875009 + 0.484106i \(0.160855\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.36603 7.56218i 0.260455 0.451122i −0.705908 0.708304i \(-0.749461\pi\)
0.966363 + 0.257182i \(0.0827940\pi\)
\(282\) 0 0
\(283\) 11.2321 + 19.4545i 0.667676 + 1.15645i 0.978552 + 0.205999i \(0.0660442\pi\)
−0.310876 + 0.950450i \(0.600622\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 32.7846 1.93521
\(288\) 0 0
\(289\) −16.4641 −0.968477
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −5.66025 9.80385i −0.330676 0.572747i 0.651969 0.758246i \(-0.273943\pi\)
−0.982644 + 0.185499i \(0.940610\pi\)
\(294\) 0 0
\(295\) 0.660254 1.14359i 0.0384415 0.0665826i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10.7583 18.6340i 0.622170 1.07763i
\(300\) 0 0
\(301\) 10.2679 + 17.7846i 0.591835 + 1.02509i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.12436 0.0643804
\(306\) 0 0
\(307\) −20.5359 −1.17205 −0.586023 0.810295i \(-0.699307\pi\)
−0.586023 + 0.810295i \(0.699307\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 14.1962 + 24.5885i 0.804990 + 1.39428i 0.916298 + 0.400498i \(0.131163\pi\)
−0.111308 + 0.993786i \(0.535504\pi\)
\(312\) 0 0
\(313\) −3.80385 + 6.58846i −0.215006 + 0.372402i −0.953274 0.302106i \(-0.902310\pi\)
0.738268 + 0.674507i \(0.235644\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.46410 16.3923i 0.531557 0.920684i −0.467765 0.883853i \(-0.654941\pi\)
0.999322 0.0368305i \(-0.0117262\pi\)
\(318\) 0 0
\(319\) −7.85641 13.6077i −0.439874 0.761885i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3.26795 −0.181834
\(324\) 0 0
\(325\) 11.0000 0.610170
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12.0000 + 20.7846i 0.661581 + 1.14589i
\(330\) 0 0
\(331\) −10.2321 + 17.7224i −0.562404 + 0.974113i 0.434882 + 0.900488i \(0.356790\pi\)
−0.997286 + 0.0736253i \(0.976543\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.29423 + 2.24167i −0.0707113 + 0.122476i
\(336\) 0 0
\(337\) −16.7321 28.9808i −0.911453 1.57868i −0.812013 0.583639i \(-0.801628\pi\)
−0.0994397 0.995044i \(-0.531705\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 15.2679 0.826806
\(342\) 0 0
\(343\) −6.92820 −0.374088
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.26795 + 2.19615i 0.0680671 + 0.117896i 0.898050 0.439893i \(-0.144983\pi\)
−0.829983 + 0.557788i \(0.811650\pi\)
\(348\) 0 0
\(349\) 14.1962 24.5885i 0.759903 1.31619i −0.182997 0.983113i \(-0.558580\pi\)
0.942900 0.333077i \(-0.108087\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −9.83013 + 17.0263i −0.523205 + 0.906217i 0.476430 + 0.879212i \(0.341930\pi\)
−0.999635 + 0.0270052i \(0.991403\pi\)
\(354\) 0 0
\(355\) −2.26795 3.92820i −0.120370 0.208487i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.7846 1.09697 0.548485 0.836160i \(-0.315205\pi\)
0.548485 + 0.836160i \(0.315205\pi\)
\(360\) 0 0
\(361\) 0.928203 0.0488528
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.16987 + 7.22243i 0.218261 + 0.378039i
\(366\) 0 0
\(367\) 15.4282 26.7224i 0.805346 1.39490i −0.110712 0.993853i \(-0.535313\pi\)
0.916057 0.401047i \(-0.131354\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 13.2679 22.9808i 0.688838 1.19310i
\(372\) 0 0
\(373\) 12.5000 + 21.6506i 0.647225 + 1.12103i 0.983783 + 0.179364i \(0.0574041\pi\)
−0.336557 + 0.941663i \(0.609263\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.24871 −0.321825
\(378\) 0 0
\(379\) 16.7846 0.862167 0.431084 0.902312i \(-0.358131\pi\)
0.431084 + 0.902312i \(0.358131\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 14.1962 + 24.5885i 0.725390 + 1.25641i 0.958813 + 0.284036i \(0.0916737\pi\)
−0.233424 + 0.972375i \(0.574993\pi\)
\(384\) 0 0
\(385\) 7.85641 13.6077i 0.400400 0.693512i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −7.83013 + 13.5622i −0.397003 + 0.687630i −0.993355 0.115094i \(-0.963283\pi\)
0.596351 + 0.802723i \(0.296616\pi\)
\(390\) 0 0
\(391\) −3.19615 5.53590i −0.161636 0.279962i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −10.1962 −0.513024
\(396\) 0 0
\(397\) −13.0000 −0.652451 −0.326226 0.945292i \(-0.605777\pi\)
−0.326226 + 0.945292i \(0.605777\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8.19615 14.1962i −0.409296 0.708922i 0.585515 0.810662i \(-0.300892\pi\)
−0.994811 + 0.101740i \(0.967559\pi\)
\(402\) 0 0
\(403\) 3.03590 5.25833i 0.151229 0.261936i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −7.85641 + 13.6077i −0.389428 + 0.674508i
\(408\) 0 0
\(409\) −11.6603 20.1962i −0.576562 0.998635i −0.995870 0.0907912i \(-0.971060\pi\)
0.419307 0.907844i \(-0.362273\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.24871 0.307479
\(414\) 0 0
\(415\) −9.32051 −0.457526
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 15.1244 + 26.1962i 0.738873 + 1.27977i 0.953003 + 0.302961i \(0.0979751\pi\)
−0.214130 + 0.976805i \(0.568692\pi\)
\(420\) 0 0
\(421\) −1.00000 + 1.73205i −0.0487370 + 0.0844150i −0.889365 0.457198i \(-0.848853\pi\)
0.840628 + 0.541613i \(0.182186\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.63397 2.83013i 0.0792594 0.137281i
\(426\) 0 0
\(427\) 2.66025 + 4.60770i 0.128739 + 0.222982i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 20.3397 0.979731 0.489866 0.871798i \(-0.337046\pi\)
0.489866 + 0.871798i \(0.337046\pi\)
\(432\) 0 0
\(433\) −15.3923 −0.739707 −0.369853 0.929090i \(-0.620592\pi\)
−0.369853 + 0.929090i \(0.620592\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 19.4904 + 33.7583i 0.932351 + 1.61488i
\(438\) 0 0
\(439\) 11.1962 19.3923i 0.534363 0.925544i −0.464831 0.885400i \(-0.653885\pi\)
0.999194 0.0401446i \(-0.0127819\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −14.2224 + 24.6340i −0.675728 + 1.17040i 0.300527 + 0.953773i \(0.402837\pi\)
−0.976255 + 0.216622i \(0.930496\pi\)
\(444\) 0 0
\(445\) −4.39230 7.60770i −0.208215 0.360639i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 13.8038 0.651444 0.325722 0.945466i \(-0.394393\pi\)
0.325722 + 0.945466i \(0.394393\pi\)
\(450\) 0 0
\(451\) 58.6410 2.76130
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3.12436 5.41154i −0.146472 0.253697i
\(456\) 0 0
\(457\) −9.42820 + 16.3301i −0.441033 + 0.763891i −0.997766 0.0667999i \(-0.978721\pi\)
0.556734 + 0.830691i \(0.312054\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 11.0981 19.2224i 0.516889 0.895278i −0.482919 0.875665i \(-0.660423\pi\)
0.999808 0.0196127i \(-0.00624332\pi\)
\(462\) 0 0
\(463\) −3.30385 5.72243i −0.153543 0.265944i 0.778985 0.627043i \(-0.215735\pi\)
−0.932527 + 0.361099i \(0.882402\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.46410 0.437946 0.218973 0.975731i \(-0.429729\pi\)
0.218973 + 0.975731i \(0.429729\pi\)
\(468\) 0 0
\(469\) −12.2487 −0.565593
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 18.3660 + 31.8109i 0.844471 + 1.46267i
\(474\) 0 0
\(475\) −9.96410 + 17.2583i −0.457184 + 0.791866i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 7.83013 13.5622i 0.357768 0.619672i −0.629820 0.776741i \(-0.716871\pi\)
0.987588 + 0.157069i \(0.0502047\pi\)
\(480\) 0 0
\(481\) 3.12436 + 5.41154i 0.142458 + 0.246745i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.73205 −0.396502
\(486\) 0 0
\(487\) −4.07180 −0.184511 −0.0922554 0.995735i \(-0.529408\pi\)
−0.0922554 + 0.995735i \(0.529408\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −0.928203 1.60770i −0.0418892 0.0725543i 0.844321 0.535838i \(-0.180004\pi\)
−0.886210 + 0.463284i \(0.846671\pi\)
\(492\) 0 0
\(493\) −0.928203 + 1.60770i −0.0418042 + 0.0724069i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 10.7321 18.5885i 0.481398 0.833806i
\(498\) 0 0
\(499\) 14.6603 + 25.3923i 0.656283 + 1.13672i 0.981570 + 0.191100i \(0.0612056\pi\)
−0.325287 + 0.945615i \(0.605461\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −21.4641 −0.957037 −0.478518 0.878077i \(-0.658826\pi\)
−0.478518 + 0.878077i \(0.658826\pi\)
\(504\) 0 0
\(505\) −3.21539 −0.143083
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 16.3923 + 28.3923i 0.726576 + 1.25847i 0.958322 + 0.285690i \(0.0922229\pi\)
−0.231746 + 0.972776i \(0.574444\pi\)
\(510\) 0 0
\(511\) −19.7321 + 34.1769i −0.872895 + 1.51190i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.90192 + 11.9545i −0.304135 + 0.526777i
\(516\) 0 0
\(517\) 21.4641 + 37.1769i 0.943990 + 1.63504i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 17.8038 0.780001 0.390000 0.920815i \(-0.372475\pi\)
0.390000 + 0.920815i \(0.372475\pi\)
\(522\) 0 0
\(523\) −12.8564 −0.562171 −0.281086 0.959683i \(-0.590695\pi\)
−0.281086 + 0.959683i \(0.590695\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −0.901924 1.56218i −0.0392884 0.0680495i
\(528\) 0 0
\(529\) −26.6244 + 46.1147i −1.15758 + 2.00499i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 11.6603 20.1962i 0.505062 0.874792i
\(534\) 0 0
\(535\) 3.46410 + 6.00000i 0.149766 + 0.259403i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 30.9808 1.33444
\(540\) 0 0
\(541\) −4.39230 −0.188840 −0.0944200 0.995532i \(-0.530100\pi\)
−0.0944200 + 0.995532i \(0.530100\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.02628 + 10.4378i 0.258137 + 0.447107i
\(546\) 0 0
\(547\) 3.83975 6.65064i 0.164176 0.284361i −0.772187 0.635396i \(-0.780837\pi\)
0.936362 + 0.351035i \(0.114170\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5.66025 9.80385i 0.241135 0.417658i
\(552\) 0 0
\(553\) −24.1244 41.7846i −1.02587 1.77686i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.19615 0.262539 0.131270 0.991347i \(-0.458095\pi\)
0.131270 + 0.991347i \(0.458095\pi\)
\(558\) 0 0
\(559\) 14.6077 0.617840
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −6.56218 11.3660i −0.276563 0.479021i 0.693965 0.720008i \(-0.255862\pi\)
−0.970528 + 0.240987i \(0.922529\pi\)
\(564\) 0 0
\(565\) 2.53590 4.39230i 0.106686 0.184786i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 13.2942 23.0263i 0.557323 0.965312i −0.440396 0.897804i \(-0.645162\pi\)
0.997719 0.0675081i \(-0.0215048\pi\)
\(570\) 0 0
\(571\) 1.73205 + 3.00000i 0.0724841 + 0.125546i 0.899989 0.435912i \(-0.143574\pi\)
−0.827505 + 0.561458i \(0.810241\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −38.9808 −1.62561
\(576\) 0 0
\(577\) −14.8564 −0.618480 −0.309240 0.950984i \(-0.600075\pi\)
−0.309240 + 0.950984i \(0.600075\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −22.0526 38.1962i −0.914894 1.58464i
\(582\) 0 0
\(583\) 23.7321 41.1051i 0.982881 1.70240i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.90192 5.02628i 0.119775 0.207457i −0.799903 0.600129i \(-0.795116\pi\)
0.919679 + 0.392672i \(0.128449\pi\)
\(588\) 0 0
\(589\) 5.50000 + 9.52628i 0.226624 + 0.392524i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −26.5885 −1.09186 −0.545929 0.837832i \(-0.683823\pi\)
−0.545929 + 0.837832i \(0.683823\pi\)
\(594\) 0 0
\(595\) −1.85641 −0.0761052
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −7.85641 13.6077i −0.321004 0.555995i 0.659691 0.751537i \(-0.270687\pi\)
−0.980695 + 0.195541i \(0.937354\pi\)
\(600\) 0 0
\(601\) −6.16025 + 10.6699i −0.251282 + 0.435233i −0.963879 0.266340i \(-0.914185\pi\)
0.712597 + 0.701574i \(0.247519\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10.0263 17.3660i 0.407626 0.706029i
\(606\) 0 0
\(607\) 6.39230 + 11.0718i 0.259456 + 0.449390i 0.966096 0.258182i \(-0.0831235\pi\)
−0.706641 + 0.707573i \(0.749790\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 17.0718 0.690651
\(612\) 0 0
\(613\) −19.9282 −0.804893 −0.402446 0.915444i \(-0.631840\pi\)
−0.402446 + 0.915444i \(0.631840\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.73205 + 8.19615i 0.190505 + 0.329965i 0.945418 0.325861i \(-0.105654\pi\)
−0.754913 + 0.655825i \(0.772321\pi\)
\(618\) 0 0
\(619\) −0.500000 + 0.866025i −0.0200967 + 0.0348085i −0.875899 0.482495i \(-0.839731\pi\)
0.855802 + 0.517303i \(0.173064\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 20.7846 36.0000i 0.832718 1.44231i
\(624\) 0 0
\(625\) −8.62436 14.9378i −0.344974 0.597513i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.85641 0.0740198
\(630\) 0 0
\(631\) 28.0000 1.11466 0.557331 0.830290i \(-0.311825\pi\)
0.557331 + 0.830290i \(0.311825\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0.705771 + 1.22243i 0.0280077 + 0.0485107i
\(636\) 0 0
\(637\) 6.16025 10.6699i 0.244078 0.422756i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −15.0981 + 26.1506i −0.596338 + 1.03289i 0.397018 + 0.917811i \(0.370045\pi\)
−0.993357 + 0.115077i \(0.963288\pi\)
\(642\) 0 0
\(643\) −12.8038 22.1769i −0.504934 0.874572i −0.999984 0.00570722i \(-0.998183\pi\)
0.495049 0.868865i \(-0.335150\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −25.1244 −0.987740 −0.493870 0.869536i \(-0.664418\pi\)
−0.493870 + 0.869536i \(0.664418\pi\)
\(648\) 0 0
\(649\) 11.1769 0.438732
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −4.39230 7.60770i −0.171884 0.297712i 0.767194 0.641415i \(-0.221652\pi\)
−0.939079 + 0.343703i \(0.888319\pi\)
\(654\) 0 0
\(655\) 5.58846 9.67949i 0.218359 0.378209i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6.00000 10.3923i 0.233727 0.404827i −0.725175 0.688565i \(-0.758241\pi\)
0.958902 + 0.283738i \(0.0915745\pi\)
\(660\) 0 0
\(661\) 14.3564 + 24.8660i 0.558399 + 0.967176i 0.997630 + 0.0688021i \(0.0219177\pi\)
−0.439231 + 0.898374i \(0.644749\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 11.3205 0.438990
\(666\) 0 0
\(667\) 22.1436 0.857403
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.75833 + 8.24167i 0.183693 + 0.318166i
\(672\) 0 0
\(673\) 8.62436 14.9378i 0.332444 0.575811i −0.650546 0.759467i \(-0.725460\pi\)
0.982991 + 0.183656i \(0.0587933\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −22.9545 + 39.7583i −0.882212 + 1.52804i −0.0333368 + 0.999444i \(0.510613\pi\)
−0.848876 + 0.528593i \(0.822720\pi\)
\(678\) 0 0
\(679\) −20.6603 35.7846i −0.792868 1.37329i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −0.0525589 −0.00201111 −0.00100555 0.999999i \(-0.500320\pi\)
−0.00100555 + 0.999999i \(0.500320\pi\)
\(684\) 0 0
\(685\) −13.8564 −0.529426
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −9.43782 16.3468i −0.359552 0.622763i
\(690\) 0 0
\(691\) 16.0885 27.8660i 0.612034 1.06007i −0.378863 0.925453i \(-0.623685\pi\)
0.990897 0.134621i \(-0.0429817\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.26795 + 2.19615i −0.0480961 + 0.0833048i
\(696\) 0 0
\(697\) −3.46410 6.00000i −0.131212 0.227266i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 23.3205 0.880803 0.440402 0.897801i \(-0.354836\pi\)
0.440402 + 0.897801i \(0.354836\pi\)
\(702\) 0 0
\(703\) −11.3205 −0.426961
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −7.60770 13.1769i −0.286117 0.495569i
\(708\) 0 0
\(709\) 0.0358984 0.0621778i 0.00134819 0.00233514i −0.865351 0.501167i \(-0.832904\pi\)
0.866699 + 0.498832i \(0.166238\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −10.7583 + 18.6340i −0.402903 + 0.697848i
\(714\) 0 0
\(715\) −5.58846 9.67949i −0.208996 0.361992i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −12.7321 −0.474825 −0.237413 0.971409i \(-0.576299\pi\)
−0.237413 + 0.971409i \(0.576299\pi\)
\(720\) 0 0
\(721\) −65.3205 −2.43266
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.66025 + 9.80385i 0.210217 + 0.364106i
\(726\) 0 0
\(727\) 11.1962 19.3923i 0.415242 0.719221i −0.580212 0.814466i \(-0.697030\pi\)
0.995454 + 0.0952450i \(0.0303635\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.16987 3.75833i 0.0802557 0.139007i
\(732\) 0 0
\(733\) 18.9641 + 32.8468i 0.700455 + 1.21322i 0.968307 + 0.249764i \(0.0803529\pi\)
−0.267852 + 0.963460i \(0.586314\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −21.9090 −0.807027
\(738\) 0 0
\(739\) −14.6077 −0.537353 −0.268676 0.963231i \(-0.586586\pi\)
−0.268676 + 0.963231i \(0.586586\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 6.02628 + 10.4378i 0.221083 + 0.382927i 0.955137 0.296164i \(-0.0957076\pi\)
−0.734054 + 0.679091i \(0.762374\pi\)
\(744\) 0 0
\(745\) −7.33975 + 12.7128i −0.268907 + 0.465761i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −16.3923 + 28.3923i −0.598962 + 1.03743i
\(750\) 0 0
\(751\) 10.2679 + 17.7846i 0.374683 + 0.648970i 0.990280 0.139092i \(-0.0444183\pi\)
−0.615597 + 0.788061i \(0.711085\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2.58846 0.0942036
\(756\) 0 0
\(757\) −0.0717968 −0.00260950 −0.00130475 0.999999i \(-0.500415\pi\)
−0.00130475 + 0.999999i \(0.500415\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −17.0981 29.6147i −0.619805 1.07353i −0.989521 0.144389i \(-0.953878\pi\)
0.369716 0.929145i \(-0.379455\pi\)
\(762\) 0 0
\(763\) −28.5167 + 49.3923i −1.03237 + 1.78812i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.22243 3.84936i 0.0802474 0.138993i
\(768\) 0 0
\(769\) 15.2846 + 26.4737i 0.551177 + 0.954667i 0.998190 + 0.0601393i \(0.0191545\pi\)
−0.447013 + 0.894528i \(0.647512\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −33.4641 −1.20362 −0.601810 0.798639i \(-0.705554\pi\)
−0.601810 + 0.798639i \(0.705554\pi\)
\(774\) 0 0
\(775\) −11.0000 −0.395132
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 21.1244 + 36.5885i 0.756859 + 1.31092i
\(780\) 0 0
\(781\) 19.1962 33.2487i 0.686892 1.18973i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.97372 3.41858i 0.0704451 0.122015i
\(786\) 0 0
\(787\) −23.8205 41.2583i −0.849109 1.47070i −0.882004 0.471242i \(-0.843806\pi\)
0.0328946 0.999459i \(-0.489527\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 24.0000 0.853342
\(792\) 0 0
\(793\) 3.78461 0.134395
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4.36603 7.56218i −0.154653 0.267866i 0.778280 0.627918i \(-0.216092\pi\)
−0.932932 + 0.360051i \(0.882759\pi\)
\(798\) 0 0
\(799\) 2.53590 4.39230i 0.0897136 0.155389i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −35.2942 + 61.1314i −1.24551 + 2.15728i
\(804\) 0 0
\(805\) 11.0718 + 19.1769i 0.390230 + 0.675897i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −4.33975 −0.152577 −0.0762887 0.997086i \(-0.524307\pi\)
−0.0762887 + 0.997086i \(0.524307\pi\)
\(810\) 0 0
\(811\) 45.7846 1.60772 0.803858 0.594822i \(-0.202777\pi\)
0.803858 + 0.594822i \(0.202777\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −8.19615 14.1962i −0.287099 0.497270i
\(816\) 0 0
\(817\) −13.2321 + 22.9186i −0.462931 + 0.801820i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.9282 22.3923i 0.451197 0.781497i −0.547263 0.836960i \(-0.684330\pi\)
0.998461 + 0.0554637i \(0.0176637\pi\)
\(822\) 0 0
\(823\) −21.0885 36.5263i −0.735097 1.27323i −0.954681 0.297632i \(-0.903803\pi\)
0.219583 0.975594i \(-0.429530\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 21.5167 0.748208 0.374104 0.927387i \(-0.377950\pi\)
0.374104 + 0.927387i \(0.377950\pi\)
\(828\) 0 0
\(829\) 13.2487 0.460147 0.230073 0.973173i \(-0.426103\pi\)
0.230073 + 0.973173i \(0.426103\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.83013 3.16987i −0.0634101 0.109830i
\(834\) 0 0
\(835\) 1.60770 2.78461i 0.0556366 0.0963654i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −5.66025 + 9.80385i −0.195414 + 0.338466i −0.947036 0.321127i \(-0.895938\pi\)
0.751622 + 0.659594i \(0.229272\pi\)
\(840\) 0 0
\(841\) 11.2846 + 19.5455i 0.389124 + 0.673983i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 5.07180 0.174475
\(846\) 0 0
\(847\) 94.8897 3.26045
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −11.0718 19.1769i −0.379536 0.657376i
\(852\) 0 0
\(853\) −15.8923 + 27.5263i −0.544142 + 0.942482i 0.454518 + 0.890737i \(0.349811\pi\)
−0.998660 + 0.0517444i \(0.983522\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −6.58846 + 11.4115i −0.225057 + 0.389811i −0.956337 0.292267i \(-0.905590\pi\)
0.731279 + 0.682078i \(0.238924\pi\)
\(858\) 0 0
\(859\) −16.2679 28.1769i −0.555055 0.961384i −0.997899 0.0647852i \(-0.979364\pi\)
0.442844 0.896599i \(-0.353970\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −15.9474 −0.542857 −0.271429 0.962459i \(-0.587496\pi\)
−0.271429 + 0.962459i \(0.587496\pi\)
\(864\) 0 0
\(865\) −5.07180 −0.172446
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −43.1506 74.7391i −1.46379 2.53535i
\(870\) 0 0
\(871\) −4.35641 + 7.54552i −0.147611 + 0.255670i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −12.0000 + 20.7846i −0.405674 + 0.702648i
\(876\) 0 0
\(877\) 22.7321 + 39.3731i 0.767607 + 1.32953i 0.938857 + 0.344306i \(0.111886\pi\)
−0.171250 + 0.985228i \(0.554781\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −15.6603 −0.527607 −0.263804 0.964576i \(-0.584977\pi\)
−0.263804 + 0.964576i \(0.584977\pi\)
\(882\) 0 0
\(883\) 11.6795 0.393046 0.196523 0.980499i \(-0.437035\pi\)
0.196523 + 0.980499i \(0.437035\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −14.5622 25.2224i −0.488950 0.846886i 0.510969 0.859599i \(-0.329287\pi\)
−0.999919 + 0.0127127i \(0.995953\pi\)
\(888\) 0 0
\(889\) −3.33975 + 5.78461i −0.112011 + 0.194010i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −15.4641 + 26.7846i −0.517486 + 0.896313i
\(894\) 0 0
\(895\) 0.928203 + 1.60770i 0.0310264 + 0.0537393i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 6.24871 0.208406
\(900\) 0 0
\(901\) −5.60770 −0.186819
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.14359 7.17691i −0.137738 0.238569i
\(906\) 0 0
\(907\) −3.50000 + 6.06218i −0.116216 + 0.201291i −0.918265 0.395966i \(-0.870410\pi\)
0.802049 + 0.597258i \(0.203743\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −6.90192 + 11.9545i −0.228671 + 0.396070i −0.957414 0.288717i \(-0.906771\pi\)
0.728744 + 0.684787i \(0.240105\pi\)
\(912\) 0 0
\(913\) −39.4449 68.3205i −1.30543 2.26108i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 52.8897 1.74657
\(918\) 0 0
\(919\) 4.00000 0.131948 0.0659739 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −7.63397 13.2224i −0.251275 0.435222i
\(924\) 0 0
\(925\) 5.66025 9.80385i 0.186108 0.322349i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −25.2942 + 43.8109i −0.829877 + 1.43739i 0.0682577 + 0.997668i \(0.478256\pi\)
−0.898134 + 0.439721i \(0.855077\pi\)
\(930\) 0 0
\(931\) 11.1603 + 19.3301i 0.365763 + 0.633519i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3.32051 −0.108592
\(936\) 0 0
\(937\) −16.3923 −0.535513 −0.267757 0.963487i \(-0.586282\pi\)
−0.267757 + 0.963487i \(0.586282\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 3.46410 + 6.00000i 0.112926 + 0.195594i 0.916949 0.399004i \(-0.130644\pi\)
−0.804022 + 0.594599i \(0.797311\pi\)
\(942\) 0 0
\(943\) −41.3205 + 71.5692i −1.34558 + 2.33061i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −9.12436 + 15.8038i −0.296502 + 0.513556i −0.975333 0.220738i \(-0.929153\pi\)
0.678831 + 0.734294i \(0.262487\pi\)
\(948\) 0 0
\(949\) 14.0359 + 24.3109i 0.455625 + 0.789165i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −3.21539 −0.104157 −0.0520784 0.998643i \(-0.516585\pi\)
−0.0520784 + 0.998643i \(0.516585\pi\)
\(954\) 0 0
\(955\) −13.8564 −0.448383
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −32.7846 56.7846i −1.05867 1.83367i
\(960\) 0 0
\(961\) 12.4641 21.5885i 0.402068 0.696402i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.04552 + 1.81089i −0.0336564 + 0.0582946i
\(966\) 0 0
\(967\) 21.6244 + 37.4545i 0.695392 + 1.20445i 0.970048 + 0.242912i \(0.0781027\pi\)
−0.274656 + 0.961543i \(0.588564\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −46.6410 −1.49678 −0.748391 0.663258i \(-0.769173\pi\)
−0.748391 + 0.663258i \(0.769173\pi\)
\(972\) 0 0
\(973\) −12.0000 −0.384702
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 5.66025 + 9.80385i 0.181088 + 0.313653i 0.942251 0.334907i \(-0.108705\pi\)
−0.761164 + 0.648560i \(0.775372\pi\)
\(978\) 0 0
\(979\) 37.1769 64.3923i 1.18818 2.05799i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 20.1962 34.9808i 0.644157 1.11571i −0.340338 0.940303i \(-0.610542\pi\)
0.984495 0.175410i \(-0.0561251\pi\)
\(984\) 0 0
\(985\) −8.53590 14.7846i −0.271976 0.471077i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −51.7654 −1.64604
\(990\) 0 0
\(991\) 13.9282 0.442444 0.221222 0.975223i \(-0.428995\pi\)
0.221222 + 0.975223i \(0.428995\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.02628 6.97372i −0.127642 0.221082i
\(996\) 0 0
\(997\) −5.00000 + 8.66025i −0.158352 + 0.274273i −0.934274 0.356555i \(-0.883951\pi\)
0.775923 + 0.630828i \(0.217285\pi\)
\(998\) 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 1944.2.i.k.649.2 4
3.2 odd 2 1944.2.i.n.649.1 4
9.2 odd 6 1944.2.a.k.1.2 2
9.4 even 3 inner 1944.2.i.k.1297.2 4
9.5 odd 6 1944.2.i.n.1297.1 4
9.7 even 3 1944.2.a.n.1.1 yes 2
36.7 odd 6 3888.2.a.bc.1.1 2
36.11 even 6 3888.2.a.w.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1944.2.a.k.1.2 2 9.2 odd 6
1944.2.a.n.1.1 yes 2 9.7 even 3
1944.2.i.k.649.2 4 1.1 even 1 trivial
1944.2.i.k.1297.2 4 9.4 even 3 inner
1944.2.i.n.649.1 4 3.2 odd 2
1944.2.i.n.1297.1 4 9.5 odd 6
3888.2.a.w.1.2 2 36.11 even 6
3888.2.a.bc.1.1 2 36.7 odd 6