Properties

Label 2736.2.bm.q.1855.4
Level $2736$
Weight $2$
Character 2736.1855
Analytic conductor $21.847$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,2,Mod(559,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.559");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.bm (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.897122304.10
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 51x^{4} - 104x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1855.4
Root \(-1.30421 + 0.752986i\) of defining polynomial
Character \(\chi\) \(=\) 2736.1855
Dual form 2736.2.bm.q.559.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.05719 + 3.56317i) q^{5} +1.00000i q^{7} +O(q^{10})\) \(q+(2.05719 + 3.56317i) q^{5} +1.00000i q^{7} +4.11439i q^{11} +(-1.50000 - 0.866025i) q^{13} +(1.50597 + 2.60842i) q^{17} +(-1.73205 - 4.00000i) q^{19} +(-0.954747 - 0.551224i) q^{23} +(-5.96410 + 10.3301i) q^{25} +(4.51791 + 2.60842i) q^{29} +4.26795 q^{31} +(-3.56317 + 2.05719i) q^{35} +4.26795i q^{37} +(-5.59808 + 3.23205i) q^{43} +(-5.21684 - 3.01194i) q^{47} +6.00000 q^{49} +(1.65367 + 0.954747i) q^{53} +(-14.6603 + 8.46410i) q^{55} +(0.954747 + 1.65367i) q^{59} +(-1.96410 + 3.40192i) q^{61} -7.12633i q^{65} +(5.59808 - 9.69615i) q^{67} +(-4.51791 - 7.82526i) q^{71} +(2.69615 + 4.66987i) q^{73} -4.11439 q^{77} +(-7.33013 - 12.6962i) q^{79} +11.2407i q^{83} +(-6.19615 + 10.7321i) q^{85} +(-10.6895 - 6.17158i) q^{89} +(0.866025 - 1.50000i) q^{91} +(10.6895 - 14.4004i) q^{95} +(-5.19615 + 3.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{13} - 20 q^{25} + 48 q^{31} - 24 q^{43} + 48 q^{49} - 48 q^{55} + 12 q^{61} + 24 q^{67} - 20 q^{73} - 24 q^{79} - 8 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\) \(1\)

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 0
\(4\) 0 0
\(5\) 2.05719 + 3.56317i 0.920006 + 1.59350i 0.799403 + 0.600795i \(0.205149\pi\)
0.120603 + 0.992701i \(0.461517\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.11439i 1.24054i 0.784390 + 0.620268i \(0.212976\pi\)
−0.784390 + 0.620268i \(0.787024\pi\)
\(12\) 0 0
\(13\) −1.50000 0.866025i −0.416025 0.240192i 0.277350 0.960769i \(-0.410544\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.50597 + 2.60842i 0.365252 + 0.632634i 0.988817 0.149137i \(-0.0476496\pi\)
−0.623565 + 0.781772i \(0.714316\pi\)
\(18\) 0 0
\(19\) −1.73205 4.00000i −0.397360 0.917663i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.954747 0.551224i −0.199079 0.114938i 0.397147 0.917755i \(-0.370000\pi\)
−0.596226 + 0.802817i \(0.703334\pi\)
\(24\) 0 0
\(25\) −5.96410 + 10.3301i −1.19282 + 2.06603i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.51791 + 2.60842i 0.838955 + 0.484371i 0.856909 0.515468i \(-0.172382\pi\)
−0.0179536 + 0.999839i \(0.505715\pi\)
\(30\) 0 0
\(31\) 4.26795 0.766546 0.383273 0.923635i \(-0.374797\pi\)
0.383273 + 0.923635i \(0.374797\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.56317 + 2.05719i −0.602285 + 0.347729i
\(36\) 0 0
\(37\) 4.26795i 0.701647i 0.936442 + 0.350823i \(0.114098\pi\)
−0.936442 + 0.350823i \(0.885902\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 0 0
\(43\) −5.59808 + 3.23205i −0.853699 + 0.492883i −0.861897 0.507083i \(-0.830724\pi\)
0.00819845 + 0.999966i \(0.497390\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.21684 3.01194i −0.760954 0.439337i 0.0686842 0.997638i \(-0.478120\pi\)
−0.829638 + 0.558302i \(0.811453\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.65367 + 0.954747i 0.227149 + 0.131145i 0.609256 0.792973i \(-0.291468\pi\)
−0.382107 + 0.924118i \(0.624801\pi\)
\(54\) 0 0
\(55\) −14.6603 + 8.46410i −1.97679 + 1.14130i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.954747 + 1.65367i 0.124298 + 0.215290i 0.921458 0.388478i \(-0.126999\pi\)
−0.797161 + 0.603767i \(0.793666\pi\)
\(60\) 0 0
\(61\) −1.96410 + 3.40192i −0.251477 + 0.435572i −0.963933 0.266146i \(-0.914250\pi\)
0.712455 + 0.701717i \(0.247583\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.12633i 0.883913i
\(66\) 0 0
\(67\) 5.59808 9.69615i 0.683914 1.18457i −0.289863 0.957068i \(-0.593610\pi\)
0.973777 0.227505i \(-0.0730569\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.51791 7.82526i −0.536178 0.928687i −0.999105 0.0422909i \(-0.986534\pi\)
0.462928 0.886396i \(-0.346799\pi\)
\(72\) 0 0
\(73\) 2.69615 + 4.66987i 0.315561 + 0.546567i 0.979557 0.201169i \(-0.0644741\pi\)
−0.663996 + 0.747736i \(0.731141\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.11439 −0.468878
\(78\) 0 0
\(79\) −7.33013 12.6962i −0.824704 1.42843i −0.902146 0.431432i \(-0.858009\pi\)
0.0774418 0.996997i \(-0.475325\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 11.2407i 1.23383i 0.787030 + 0.616915i \(0.211618\pi\)
−0.787030 + 0.616915i \(0.788382\pi\)
\(84\) 0 0
\(85\) −6.19615 + 10.7321i −0.672067 + 1.16405i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.6895 6.17158i −1.13308 0.654187i −0.188376 0.982097i \(-0.560322\pi\)
−0.944709 + 0.327910i \(0.893656\pi\)
\(90\) 0 0
\(91\) 0.866025 1.50000i 0.0907841 0.157243i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 10.6895 14.4004i 1.09672 1.47745i
\(96\) 0 0
\(97\) −5.19615 + 3.00000i −0.527589 + 0.304604i −0.740034 0.672569i \(-0.765191\pi\)
0.212445 + 0.977173i \(0.431857\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.62036 9.73475i 0.559247 0.968644i −0.438313 0.898823i \(-0.644424\pi\)
0.997560 0.0698213i \(-0.0222429\pi\)
\(102\) 0 0
\(103\) −14.6603 −1.44452 −0.722259 0.691623i \(-0.756896\pi\)
−0.722259 + 0.691623i \(0.756896\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 19.4695 1.88219 0.941094 0.338145i \(-0.109800\pi\)
0.941094 + 0.338145i \(0.109800\pi\)
\(108\) 0 0
\(109\) −11.1962 + 6.46410i −1.07240 + 0.619149i −0.928835 0.370493i \(-0.879189\pi\)
−0.143562 + 0.989641i \(0.545856\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.90949i 0.179630i 0.995958 + 0.0898151i \(0.0286276\pi\)
−0.995958 + 0.0898151i \(0.971372\pi\)
\(114\) 0 0
\(115\) 4.53590i 0.422975i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.60842 + 1.50597i −0.239113 + 0.138052i
\(120\) 0 0
\(121\) −5.92820 −0.538928
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −28.5053 −2.54959
\(126\) 0 0
\(127\) −6.92820 + 12.0000i −0.614779 + 1.06483i 0.375645 + 0.926764i \(0.377421\pi\)
−0.990423 + 0.138064i \(0.955912\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.60842 1.50597i 0.227899 0.131577i −0.381704 0.924285i \(-0.624663\pi\)
0.609602 + 0.792707i \(0.291329\pi\)
\(132\) 0 0
\(133\) 4.00000 1.73205i 0.346844 0.150188i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.403524 0.698924i 0.0344754 0.0597131i −0.848273 0.529559i \(-0.822357\pi\)
0.882748 + 0.469846i \(0.155691\pi\)
\(138\) 0 0
\(139\) 9.86603 + 5.69615i 0.836825 + 0.483141i 0.856184 0.516671i \(-0.172829\pi\)
−0.0193585 + 0.999813i \(0.506162\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.56317 6.17158i 0.297967 0.516094i
\(144\) 0 0
\(145\) 21.4641i 1.78250i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.15964 5.47266i −0.258848 0.448338i 0.707086 0.707128i \(-0.250010\pi\)
−0.965934 + 0.258790i \(0.916676\pi\)
\(150\) 0 0
\(151\) 5.07180 0.412737 0.206368 0.978474i \(-0.433835\pi\)
0.206368 + 0.978474i \(0.433835\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.78000 + 15.2074i 0.705227 + 1.22149i
\(156\) 0 0
\(157\) 6.69615 + 11.5981i 0.534411 + 0.925627i 0.999192 + 0.0402013i \(0.0127999\pi\)
−0.464780 + 0.885426i \(0.653867\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.551224 0.954747i 0.0434425 0.0752446i
\(162\) 0 0
\(163\) 5.00000i 0.391630i −0.980641 0.195815i \(-0.937265\pi\)
0.980641 0.195815i \(-0.0627352\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.3884 19.7253i 0.881263 1.52639i 0.0313242 0.999509i \(-0.490028\pi\)
0.849938 0.526882i \(-0.176639\pi\)
\(168\) 0 0
\(169\) −5.00000 8.66025i −0.384615 0.666173i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 16.8611 9.73475i 1.28192 0.740119i 0.304724 0.952441i \(-0.401436\pi\)
0.977200 + 0.212321i \(0.0681023\pi\)
\(174\) 0 0
\(175\) −10.3301 5.96410i −0.780884 0.450844i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −22.7768 −1.70242 −0.851211 0.524824i \(-0.824131\pi\)
−0.851211 + 0.524824i \(0.824131\pi\)
\(180\) 0 0
\(181\) 19.3923 + 11.1962i 1.44142 + 0.832203i 0.997945 0.0640837i \(-0.0204125\pi\)
0.443474 + 0.896287i \(0.353746\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −15.2074 + 8.78000i −1.11807 + 0.645519i
\(186\) 0 0
\(187\) −10.7321 + 6.19615i −0.784805 + 0.453108i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.31928i 0.457247i 0.973515 + 0.228624i \(0.0734225\pi\)
−0.973515 + 0.228624i \(0.926577\pi\)
\(192\) 0 0
\(193\) −11.8923 + 6.86603i −0.856027 + 0.494227i −0.862680 0.505751i \(-0.831216\pi\)
0.00665294 + 0.999978i \(0.497882\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.10245 −0.0785461 −0.0392731 0.999229i \(-0.512504\pi\)
−0.0392731 + 0.999229i \(0.512504\pi\)
\(198\) 0 0
\(199\) 0.401924 + 0.232051i 0.0284916 + 0.0164496i 0.514178 0.857683i \(-0.328097\pi\)
−0.485687 + 0.874133i \(0.661430\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.60842 + 4.51791i −0.183075 + 0.317095i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 16.4576 7.12633i 1.13839 0.492939i
\(210\) 0 0
\(211\) −7.33013 12.6962i −0.504627 0.874039i −0.999986 0.00535078i \(-0.998297\pi\)
0.495359 0.868688i \(-0.335037\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −23.0327 13.2979i −1.57081 0.906910i
\(216\) 0 0
\(217\) 4.26795i 0.289727i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.21684i 0.350922i
\(222\) 0 0
\(223\) 6.86603 + 11.8923i 0.459783 + 0.796368i 0.998949 0.0458318i \(-0.0145938\pi\)
−0.539166 + 0.842199i \(0.681260\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.30734 0.219516 0.109758 0.993958i \(-0.464992\pi\)
0.109758 + 0.993958i \(0.464992\pi\)
\(228\) 0 0
\(229\) −26.3205 −1.73931 −0.869654 0.493662i \(-0.835658\pi\)
−0.869654 + 0.493662i \(0.835658\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.52986 + 13.0421i 0.493297 + 0.854416i 0.999970 0.00772244i \(-0.00245815\pi\)
−0.506673 + 0.862138i \(0.669125\pi\)
\(234\) 0 0
\(235\) 24.7846i 1.61677i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 24.3909i 1.57772i 0.614574 + 0.788859i \(0.289328\pi\)
−0.614574 + 0.788859i \(0.710672\pi\)
\(240\) 0 0
\(241\) 9.69615 + 5.59808i 0.624584 + 0.360604i 0.778652 0.627457i \(-0.215904\pi\)
−0.154068 + 0.988060i \(0.549237\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 12.3432 + 21.3790i 0.788576 + 1.36585i
\(246\) 0 0
\(247\) −0.866025 + 7.50000i −0.0551039 + 0.477214i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 14.2527 + 8.22878i 0.899620 + 0.519396i 0.877077 0.480350i \(-0.159490\pi\)
0.0225432 + 0.999746i \(0.492824\pi\)
\(252\) 0 0
\(253\) 2.26795 3.92820i 0.142585 0.246964i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 23.0327 + 13.2979i 1.43674 + 0.829501i 0.997622 0.0689288i \(-0.0219581\pi\)
0.439117 + 0.898430i \(0.355291\pi\)
\(258\) 0 0
\(259\) −4.26795 −0.265197
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −22.0779 + 12.7467i −1.36138 + 0.785995i −0.989808 0.142408i \(-0.954515\pi\)
−0.371575 + 0.928403i \(0.621182\pi\)
\(264\) 0 0
\(265\) 7.85641i 0.482615i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −13.9968 + 8.08108i −0.853402 + 0.492712i −0.861797 0.507253i \(-0.830661\pi\)
0.00839497 + 0.999965i \(0.497328\pi\)
\(270\) 0 0
\(271\) −1.39230 + 0.803848i −0.0845765 + 0.0488303i −0.541692 0.840577i \(-0.682216\pi\)
0.457115 + 0.889407i \(0.348883\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −42.5022 24.5386i −2.56298 1.47974i
\(276\) 0 0
\(277\) −21.4641 −1.28965 −0.644826 0.764329i \(-0.723070\pi\)
−0.644826 + 0.764329i \(0.723070\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 27.5506 + 15.9063i 1.64353 + 0.948892i 0.979565 + 0.201129i \(0.0644610\pi\)
0.663965 + 0.747764i \(0.268872\pi\)
\(282\) 0 0
\(283\) 9.58846 5.53590i 0.569975 0.329075i −0.187165 0.982329i \(-0.559930\pi\)
0.757139 + 0.653254i \(0.226596\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 3.96410 6.86603i 0.233182 0.403884i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 14.2527i 0.832650i −0.909216 0.416325i \(-0.863318\pi\)
0.909216 0.416325i \(-0.136682\pi\)
\(294\) 0 0
\(295\) −3.92820 + 6.80385i −0.228709 + 0.396135i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.954747 + 1.65367i 0.0552145 + 0.0956343i
\(300\) 0 0
\(301\) −3.23205 5.59808i −0.186292 0.322668i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −16.1622 −0.925443
\(306\) 0 0
\(307\) 10.2679 + 17.7846i 0.586023 + 1.01502i 0.994747 + 0.102363i \(0.0326404\pi\)
−0.408724 + 0.912658i \(0.634026\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 27.4029i 1.55387i 0.629578 + 0.776937i \(0.283228\pi\)
−0.629578 + 0.776937i \(0.716772\pi\)
\(312\) 0 0
\(313\) 8.19615 14.1962i 0.463274 0.802414i −0.535848 0.844315i \(-0.680008\pi\)
0.999122 + 0.0419006i \(0.0133413\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.2074 + 8.78000i 0.854133 + 0.493134i 0.862043 0.506835i \(-0.169185\pi\)
−0.00790998 + 0.999969i \(0.502518\pi\)
\(318\) 0 0
\(319\) −10.7321 + 18.5885i −0.600879 + 1.04075i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 7.82526 10.5418i 0.435409 0.586561i
\(324\) 0 0
\(325\) 17.8923 10.3301i 0.992487 0.573012i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.01194 5.21684i 0.166054 0.287614i
\(330\) 0 0
\(331\) 25.9808 1.42803 0.714016 0.700129i \(-0.246874\pi\)
0.714016 + 0.700129i \(0.246874\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 46.0653 2.51682
\(336\) 0 0
\(337\) 26.0885 15.0622i 1.42113 0.820489i 0.424733 0.905319i \(-0.360368\pi\)
0.996395 + 0.0848294i \(0.0270345\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 17.5600i 0.950928i
\(342\) 0 0
\(343\) 13.0000i 0.701934i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5.47266 + 3.15964i −0.293788 + 0.169618i −0.639649 0.768667i \(-0.720920\pi\)
0.345861 + 0.938286i \(0.387587\pi\)
\(348\) 0 0
\(349\) 23.0000 1.23116 0.615581 0.788074i \(-0.288921\pi\)
0.615581 + 0.788074i \(0.288921\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −16.7530 −0.891670 −0.445835 0.895115i \(-0.647093\pi\)
−0.445835 + 0.895115i \(0.647093\pi\)
\(354\) 0 0
\(355\) 18.5885 32.1962i 0.986573 1.70879i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10.9453 + 6.31928i −0.577672 + 0.333519i −0.760208 0.649680i \(-0.774903\pi\)
0.182536 + 0.983199i \(0.441569\pi\)
\(360\) 0 0
\(361\) −13.0000 + 13.8564i −0.684211 + 0.729285i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −11.0930 + 19.2137i −0.580635 + 1.00569i
\(366\) 0 0
\(367\) −21.9904 12.6962i −1.14789 0.662734i −0.199517 0.979894i \(-0.563937\pi\)
−0.948372 + 0.317161i \(0.897271\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −0.954747 + 1.65367i −0.0495680 + 0.0858543i
\(372\) 0 0
\(373\) 20.5359i 1.06331i −0.846961 0.531654i \(-0.821571\pi\)
0.846961 0.531654i \(-0.178429\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.51791 7.82526i −0.232684 0.403021i
\(378\) 0 0
\(379\) 21.5885 1.10892 0.554462 0.832209i \(-0.312924\pi\)
0.554462 + 0.832209i \(0.312924\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 17.8158 + 30.8579i 0.910346 + 1.57677i 0.813575 + 0.581460i \(0.197518\pi\)
0.0967711 + 0.995307i \(0.469149\pi\)
\(384\) 0 0
\(385\) −8.46410 14.6603i −0.431371 0.747156i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 17.0088 29.4601i 0.862380 1.49369i −0.00724629 0.999974i \(-0.502307\pi\)
0.869626 0.493711i \(-0.164360\pi\)
\(390\) 0 0
\(391\) 3.32051i 0.167925i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 30.1590 52.2369i 1.51746 2.62832i
\(396\) 0 0
\(397\) 12.8923 + 22.3301i 0.647046 + 1.12072i 0.983825 + 0.179133i \(0.0573293\pi\)
−0.336779 + 0.941584i \(0.609337\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13.9968 8.08108i 0.698969 0.403550i −0.107994 0.994152i \(-0.534443\pi\)
0.806963 + 0.590602i \(0.201110\pi\)
\(402\) 0 0
\(403\) −6.40192 3.69615i −0.318903 0.184118i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −17.5600 −0.870417
\(408\) 0 0
\(409\) −31.9808 18.4641i −1.58135 0.912991i −0.994663 0.103175i \(-0.967100\pi\)
−0.586684 0.809816i \(-0.699567\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1.65367 + 0.954747i −0.0813718 + 0.0469801i
\(414\) 0 0
\(415\) −40.0526 + 23.1244i −1.96610 + 1.13513i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 16.7530i 0.818436i −0.912437 0.409218i \(-0.865802\pi\)
0.912437 0.409218i \(-0.134198\pi\)
\(420\) 0 0
\(421\) −9.00000 + 5.19615i −0.438633 + 0.253245i −0.703018 0.711172i \(-0.748165\pi\)
0.264385 + 0.964417i \(0.414831\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −35.9271 −1.74272
\(426\) 0 0
\(427\) −3.40192 1.96410i −0.164631 0.0950495i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −5.21684 + 9.03583i −0.251286 + 0.435240i −0.963880 0.266336i \(-0.914187\pi\)
0.712594 + 0.701577i \(0.247520\pi\)
\(432\) 0 0
\(433\) −19.5000 11.2583i −0.937110 0.541041i −0.0480569 0.998845i \(-0.515303\pi\)
−0.889053 + 0.457804i \(0.848636\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −0.551224 + 4.77374i −0.0263686 + 0.228359i
\(438\) 0 0
\(439\) 1.79423 + 3.10770i 0.0856339 + 0.148322i 0.905661 0.424002i \(-0.139375\pi\)
−0.820027 + 0.572324i \(0.806042\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.60842 1.50597i −0.123930 0.0715509i 0.436754 0.899581i \(-0.356128\pi\)
−0.560683 + 0.828030i \(0.689461\pi\)
\(444\) 0 0
\(445\) 50.7846i 2.40742i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 12.8548i 0.606656i 0.952886 + 0.303328i \(0.0980978\pi\)
−0.952886 + 0.303328i \(0.901902\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 7.12633 0.334088
\(456\) 0 0
\(457\) 17.3923 0.813578 0.406789 0.913522i \(-0.366648\pi\)
0.406789 + 0.913522i \(0.366648\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −4.37021 7.56943i −0.203541 0.352544i 0.746126 0.665805i \(-0.231912\pi\)
−0.949667 + 0.313261i \(0.898578\pi\)
\(462\) 0 0
\(463\) 20.3205i 0.944374i −0.881498 0.472187i \(-0.843465\pi\)
0.881498 0.472187i \(-0.156535\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.84287i 0.455474i 0.973723 + 0.227737i \(0.0731326\pi\)
−0.973723 + 0.227737i \(0.926867\pi\)
\(468\) 0 0
\(469\) 9.69615 + 5.59808i 0.447727 + 0.258495i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −13.2979 23.0327i −0.611439 1.05904i
\(474\) 0 0
\(475\) 51.6506 + 5.96410i 2.36989 + 0.273652i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 27.2948 + 15.7586i 1.24713 + 0.720030i 0.970536 0.240957i \(-0.0774611\pi\)
0.276593 + 0.960987i \(0.410794\pi\)
\(480\) 0 0
\(481\) 3.69615 6.40192i 0.168530 0.291903i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −21.3790 12.3432i −0.970770 0.560474i
\(486\) 0 0
\(487\) −4.39230 −0.199034 −0.0995172 0.995036i \(-0.531730\pi\)
−0.0995172 + 0.995036i \(0.531730\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −4.00627 + 2.31302i −0.180800 + 0.104385i −0.587669 0.809102i \(-0.699954\pi\)
0.406868 + 0.913487i \(0.366621\pi\)
\(492\) 0 0
\(493\) 15.7128i 0.707670i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.82526 4.51791i 0.351011 0.202656i
\(498\) 0 0
\(499\) 30.3109 17.5000i 1.35690 0.783408i 0.367697 0.929946i \(-0.380146\pi\)
0.989205 + 0.146538i \(0.0468131\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −27.2948 15.7586i −1.21701 0.702643i −0.252735 0.967535i \(-0.581330\pi\)
−0.964278 + 0.264893i \(0.914663\pi\)
\(504\) 0 0
\(505\) 46.2487 2.05804
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 12.3432 + 7.12633i 0.547101 + 0.315869i 0.747952 0.663753i \(-0.231037\pi\)
−0.200851 + 0.979622i \(0.564371\pi\)
\(510\) 0 0
\(511\) −4.66987 + 2.69615i −0.206583 + 0.119271i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −30.1590 52.2369i −1.32896 2.30183i
\(516\) 0 0
\(517\) 12.3923 21.4641i 0.545013 0.943990i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 35.6317i 1.56105i −0.625124 0.780526i \(-0.714951\pi\)
0.625124 0.780526i \(-0.285049\pi\)
\(522\) 0 0
\(523\) −0.866025 + 1.50000i −0.0378686 + 0.0655904i −0.884339 0.466846i \(-0.845390\pi\)
0.846470 + 0.532437i \(0.178724\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.42741 + 11.1326i 0.279982 + 0.484944i
\(528\) 0 0
\(529\) −10.8923 18.8660i −0.473578 0.820262i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 40.0526 + 69.3731i 1.73162 + 2.99926i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 24.6863i 1.06332i
\(540\) 0 0
\(541\) 7.89230 13.6699i 0.339317 0.587714i −0.644988 0.764193i \(-0.723138\pi\)
0.984304 + 0.176479i \(0.0564708\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −46.0653 26.5958i −1.97322 1.13924i
\(546\) 0 0
\(547\) 1.79423 3.10770i 0.0767157 0.132875i −0.825115 0.564964i \(-0.808890\pi\)
0.901831 + 0.432089i \(0.142223\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.60842 22.5896i 0.111122 0.962348i
\(552\) 0 0
\(553\) 12.6962 7.33013i 0.539895 0.311709i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 16.0540 27.8064i 0.680231 1.17820i −0.294679 0.955596i \(-0.595213\pi\)
0.974910 0.222599i \(-0.0714540\pi\)
\(558\) 0 0
\(559\) 11.1962 0.473547
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.39785 0.0589122 0.0294561 0.999566i \(-0.490622\pi\)
0.0294561 + 0.999566i \(0.490622\pi\)
\(564\) 0 0
\(565\) −6.80385 + 3.92820i −0.286240 + 0.165261i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 10.4337i 0.437402i 0.975792 + 0.218701i \(0.0701820\pi\)
−0.975792 + 0.218701i \(0.929818\pi\)
\(570\) 0 0
\(571\) 17.7846i 0.744263i 0.928180 + 0.372131i \(0.121373\pi\)
−0.928180 + 0.372131i \(0.878627\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 11.3884 6.57511i 0.474930 0.274201i
\(576\) 0 0
\(577\) 23.3205 0.970845 0.485423 0.874280i \(-0.338666\pi\)
0.485423 + 0.874280i \(0.338666\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −11.2407 −0.466344
\(582\) 0 0
\(583\) −3.92820 + 6.80385i −0.162690 + 0.281787i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 38.4959 22.2256i 1.58890 0.917350i 0.595407 0.803425i \(-0.296991\pi\)
0.993489 0.113925i \(-0.0363424\pi\)
\(588\) 0 0
\(589\) −7.39230 17.0718i −0.304595 0.703431i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0.551224 0.954747i 0.0226360 0.0392068i −0.854485 0.519475i \(-0.826127\pi\)
0.877122 + 0.480268i \(0.159461\pi\)
\(594\) 0 0
\(595\) −10.7321 6.19615i −0.439971 0.254017i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 10.6895 18.5148i 0.436761 0.756492i −0.560676 0.828035i \(-0.689459\pi\)
0.997438 + 0.0715426i \(0.0227922\pi\)
\(600\) 0 0
\(601\) 19.7321i 0.804887i −0.915445 0.402444i \(-0.868161\pi\)
0.915445 0.402444i \(-0.131839\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −12.1955 21.1232i −0.495816 0.858779i
\(606\) 0 0
\(607\) 21.3397 0.866154 0.433077 0.901357i \(-0.357428\pi\)
0.433077 + 0.901357i \(0.357428\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 5.21684 + 9.03583i 0.211051 + 0.365550i
\(612\) 0 0
\(613\) 3.46410 + 6.00000i 0.139914 + 0.242338i 0.927464 0.373913i \(-0.121984\pi\)
−0.787550 + 0.616251i \(0.788651\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.05719 3.56317i 0.0828195 0.143448i −0.821640 0.570006i \(-0.806941\pi\)
0.904460 + 0.426558i \(0.140274\pi\)
\(618\) 0 0
\(619\) 5.39230i 0.216735i −0.994111 0.108368i \(-0.965438\pi\)
0.994111 0.108368i \(-0.0345623\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.17158 10.6895i 0.247259 0.428266i
\(624\) 0 0
\(625\) −28.8205 49.9186i −1.15282 1.99674i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −11.1326 + 6.42741i −0.443886 + 0.256278i
\(630\) 0 0
\(631\) 1.91858 + 1.10770i 0.0763776 + 0.0440966i 0.537702 0.843135i \(-0.319292\pi\)
−0.461325 + 0.887231i \(0.652626\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −57.0107 −2.26240
\(636\) 0 0
\(637\) −9.00000 5.19615i −0.356593 0.205879i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −3.30734 + 1.90949i −0.130632 + 0.0754205i −0.563892 0.825849i \(-0.690697\pi\)
0.433260 + 0.901269i \(0.357363\pi\)
\(642\) 0 0
\(643\) 30.7750 17.7679i 1.21365 0.700700i 0.250095 0.968221i \(-0.419538\pi\)
0.963552 + 0.267522i \(0.0862047\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 25.4934i 1.00225i 0.865375 + 0.501124i \(0.167080\pi\)
−0.865375 + 0.501124i \(0.832920\pi\)
\(648\) 0 0
\(649\) −6.80385 + 3.92820i −0.267074 + 0.154195i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 25.4934 0.997633 0.498817 0.866708i \(-0.333768\pi\)
0.498817 + 0.866708i \(0.333768\pi\)
\(654\) 0 0
\(655\) 10.7321 + 6.19615i 0.419336 + 0.242104i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −5.47266 + 9.47893i −0.213185 + 0.369247i −0.952709 0.303883i \(-0.901717\pi\)
0.739525 + 0.673129i \(0.235050\pi\)
\(660\) 0 0
\(661\) −20.7846 12.0000i −0.808428 0.466746i 0.0379819 0.999278i \(-0.487907\pi\)
−0.846410 + 0.532533i \(0.821240\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 14.4004 + 10.6895i 0.558422 + 0.414521i
\(666\) 0 0
\(667\) −2.87564 4.98076i −0.111345 0.192856i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −13.9968 8.08108i −0.540342 0.311967i
\(672\) 0 0
\(673\) 24.3731i 0.939513i 0.882796 + 0.469756i \(0.155658\pi\)
−0.882796 + 0.469756i \(0.844342\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 20.8673i 0.801997i 0.916079 + 0.400999i \(0.131337\pi\)
−0.916079 + 0.400999i \(0.868663\pi\)
\(678\) 0 0
\(679\) −3.00000 5.19615i −0.115129 0.199410i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 49.8843 1.90877 0.954385 0.298578i \(-0.0965122\pi\)
0.954385 + 0.298578i \(0.0965122\pi\)
\(684\) 0 0
\(685\) 3.32051 0.126870
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.65367 2.86424i −0.0629999 0.109119i
\(690\) 0 0
\(691\) 36.2487i 1.37897i −0.724302 0.689483i \(-0.757838\pi\)
0.724302 0.689483i \(-0.242162\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 46.8724i 1.77797i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 17.4123 + 30.1590i 0.657654 + 1.13909i 0.981221 + 0.192885i \(0.0617843\pi\)
−0.323568 + 0.946205i \(0.604882\pi\)
\(702\) 0 0
\(703\) 17.0718 7.39230i 0.643875 0.278806i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.73475 + 5.62036i 0.366113 + 0.211375i
\(708\) 0 0
\(709\) 19.0885 33.0622i 0.716882 1.24168i −0.245347 0.969435i \(-0.578902\pi\)
0.962229 0.272241i \(-0.0877647\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −4.07481 2.35259i −0.152603 0.0881054i
\(714\) 0 0
\(715\) 29.3205 1.09652
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −17.1169 + 9.88245i −0.638353 + 0.368553i −0.783980 0.620786i \(-0.786813\pi\)
0.145627 + 0.989340i \(0.453480\pi\)
\(720\) 0 0
\(721\) 14.6603i 0.545976i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −53.8906 + 31.1137i −2.00145 + 1.15554i
\(726\) 0 0
\(727\) −28.7942 + 16.6244i −1.06792 + 0.616563i −0.927612 0.373545i \(-0.878142\pi\)
−0.140307 + 0.990108i \(0.544809\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −16.8611 9.73475i −0.623630 0.360053i
\(732\) 0 0
\(733\) 15.6077 0.576483 0.288242 0.957558i \(-0.406929\pi\)
0.288242 + 0.957558i \(0.406929\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 39.8938 + 23.0327i 1.46951 + 0.848419i
\(738\) 0 0
\(739\) −22.6699 + 13.0885i −0.833925 + 0.481467i −0.855195 0.518307i \(-0.826562\pi\)
0.0212698 + 0.999774i \(0.493229\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.56317 + 6.17158i 0.130720 + 0.226413i 0.923954 0.382503i \(-0.124938\pi\)
−0.793234 + 0.608916i \(0.791604\pi\)
\(744\) 0 0
\(745\) 13.0000 22.5167i 0.476283 0.824947i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 19.4695i 0.711400i
\(750\) 0 0
\(751\) 6.06218 10.5000i 0.221212 0.383150i −0.733964 0.679188i \(-0.762332\pi\)
0.955176 + 0.296038i \(0.0956654\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 10.4337 + 18.0717i 0.379720 + 0.657695i
\(756\) 0 0
\(757\) −1.37564 2.38269i −0.0499986 0.0866002i 0.839943 0.542675i \(-0.182588\pi\)
−0.889942 + 0.456075i \(0.849255\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 45.4745 1.64845 0.824225 0.566262i \(-0.191611\pi\)
0.824225 + 0.566262i \(0.191611\pi\)
\(762\) 0 0
\(763\) −6.46410 11.1962i −0.234016 0.405328i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.30734i 0.119421i
\(768\) 0 0
\(769\) −7.03590 + 12.1865i −0.253721 + 0.439458i −0.964547 0.263910i \(-0.914988\pi\)
0.710826 + 0.703368i \(0.248321\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 33.7222 + 19.4695i 1.21290 + 0.700269i 0.963390 0.268103i \(-0.0863966\pi\)
0.249512 + 0.968372i \(0.419730\pi\)
\(774\) 0 0
\(775\) −25.4545 + 44.0885i −0.914352 + 1.58370i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 32.1962 18.5885i 1.15207 0.665147i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −27.5506 + 47.7190i −0.983322 + 1.70316i
\(786\) 0 0
\(787\) −45.5885 −1.62505 −0.812526 0.582924i \(-0.801908\pi\)
−0.812526 + 0.582924i \(0.801908\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.90949 −0.0678938
\(792\) 0 0
\(793\) 5.89230 3.40192i 0.209242 0.120806i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15.6505i 0.554370i −0.960817 0.277185i \(-0.910599\pi\)
0.960817 0.277185i \(-0.0894014\pi\)
\(798\) 0 0
\(799\) 18.1436i 0.641874i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −19.2137 + 11.0930i −0.678036 + 0.391464i
\(804\) 0 0
\(805\) 4.53590 0.159869
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 43.0534 1.51368 0.756838 0.653602i \(-0.226743\pi\)
0.756838 + 0.653602i \(0.226743\pi\)
\(810\) 0 0
\(811\) −18.9282 + 32.7846i −0.664659 + 1.15122i 0.314719 + 0.949185i \(0.398090\pi\)
−0.979378 + 0.202038i \(0.935243\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 17.8158 10.2860i 0.624061 0.360302i
\(816\) 0 0
\(817\) 22.6244 + 16.7942i 0.791526 + 0.587556i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −3.71087 + 6.42741i −0.129510 + 0.224318i −0.923487 0.383630i \(-0.874674\pi\)
0.793977 + 0.607948i \(0.208007\pi\)
\(822\) 0 0
\(823\) 36.7128 + 21.1962i 1.27973 + 0.738851i 0.976798 0.214161i \(-0.0687019\pi\)
0.302930 + 0.953013i \(0.402035\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −20.6801 + 35.8189i −0.719117 + 1.24555i 0.242234 + 0.970218i \(0.422120\pi\)
−0.961350 + 0.275329i \(0.911213\pi\)
\(828\) 0 0
\(829\) 4.26795i 0.148232i 0.997250 + 0.0741160i \(0.0236135\pi\)
−0.997250 + 0.0741160i \(0.976386\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 9.03583 + 15.6505i 0.313073 + 0.542258i
\(834\) 0 0
\(835\) 93.7128 3.24307
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −0.255824 0.443100i −0.00883202 0.0152975i 0.861576 0.507629i \(-0.169478\pi\)
−0.870408 + 0.492332i \(0.836145\pi\)
\(840\) 0 0
\(841\) −0.892305 1.54552i −0.0307691 0.0532937i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 20.5719 35.6317i 0.707697 1.22577i
\(846\) 0 0
\(847\) 5.92820i 0.203695i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2.35259 4.07481i 0.0806459 0.139683i
\(852\) 0 0
\(853\) −16.8923 29.2583i −0.578382 1.00179i −0.995665 0.0930099i \(-0.970351\pi\)
0.417284 0.908776i \(-0.362982\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5.72848 + 3.30734i −0.195681 + 0.112977i −0.594639 0.803992i \(-0.702705\pi\)
0.398958 + 0.916969i \(0.369372\pi\)
\(858\) 0 0
\(859\) −9.86603 5.69615i −0.336624 0.194350i 0.322154 0.946687i \(-0.395593\pi\)
−0.658778 + 0.752337i \(0.728926\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −24.6863 −0.840333 −0.420166 0.907447i \(-0.638028\pi\)
−0.420166 + 0.907447i \(0.638028\pi\)
\(864\) 0 0
\(865\) 69.3731 + 40.0526i 2.35876 + 1.36183i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 52.2369 30.1590i 1.77202 1.02307i
\(870\) 0 0
\(871\) −16.7942 + 9.69615i −0.569051 + 0.328542i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 28.5053i 0.963656i
\(876\) 0 0
\(877\) −21.1077 + 12.1865i −0.712756 + 0.411510i −0.812081 0.583545i \(-0.801665\pi\)
0.0993245 + 0.995055i \(0.468332\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −28.8007 −0.970321 −0.485161 0.874425i \(-0.661239\pi\)
−0.485161 + 0.874425i \(0.661239\pi\)
\(882\) 0 0
\(883\) 31.6699 + 18.2846i 1.06578 + 0.615326i 0.927024 0.375001i \(-0.122358\pi\)
0.138752 + 0.990327i \(0.455691\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −0.954747 + 1.65367i −0.0320573 + 0.0555248i −0.881609 0.471981i \(-0.843539\pi\)
0.849552 + 0.527505i \(0.176873\pi\)
\(888\) 0 0
\(889\) −12.0000 6.92820i −0.402467 0.232364i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −3.01194 + 26.0842i −0.100791 + 0.872874i
\(894\) 0 0
\(895\) −46.8564 81.1577i −1.56624 2.71280i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 19.2822 + 11.1326i 0.643098 + 0.371293i
\(900\) 0 0
\(901\) 5.75129i 0.191603i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 92.1307i 3.06253i
\(906\) 0 0
\(907\) 24.5885 + 42.5885i 0.816446 + 1.41413i 0.908285 + 0.418353i \(0.137392\pi\)
−0.0918384 + 0.995774i \(0.529274\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −24.6863 −0.817895 −0.408947 0.912558i \(-0.634104\pi\)
−0.408947 + 0.912558i \(0.634104\pi\)
\(912\) 0 0
\(913\) −46.2487 −1.53061
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.50597 + 2.60842i 0.0497315 + 0.0861376i
\(918\) 0 0
\(919\) 8.17691i 0.269732i −0.990864 0.134866i \(-0.956940\pi\)
0.990864 0.134866i \(-0.0430603\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 15.6505i 0.515143i
\(924\) 0 0
\(925\) −44.0885 25.4545i −1.44962 0.836938i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −17.4123 30.1590i −0.571279 0.989485i −0.996435 0.0843645i \(-0.973114\pi\)
0.425156 0.905120i \(-0.360219\pi\)
\(930\) 0 0
\(931\) −10.3923 24.0000i −0.340594 0.786568i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −44.1558 25.4934i −1.44405 0.833723i
\(936\) 0 0
\(937\) −10.6962 + 18.5263i −0.349428 + 0.605227i −0.986148 0.165868i \(-0.946957\pi\)
0.636720 + 0.771095i \(0.280291\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −15.6505 9.03583i −0.510192 0.294560i 0.222721 0.974882i \(-0.428506\pi\)
−0.732913 + 0.680323i \(0.761840\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −31.1137 + 17.9635i −1.01106 + 0.583736i −0.911502 0.411295i \(-0.865076\pi\)
−0.0995588 + 0.995032i \(0.531743\pi\)
\(948\) 0 0
\(949\) 9.33975i 0.303181i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 39.8938 23.0327i 1.29229 0.746101i 0.313226 0.949678i \(-0.398590\pi\)
0.979059 + 0.203577i \(0.0652568\pi\)
\(954\) 0 0
\(955\) −22.5167 + 13.0000i −0.728622 + 0.420670i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0.698924 + 0.403524i 0.0225694 + 0.0130305i
\(960\) 0 0
\(961\) −12.7846 −0.412407
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −48.9296 28.2495i −1.57510 0.909384i
\(966\) 0 0
\(967\) −46.9186 + 27.0885i −1.50880 + 0.871106i −0.508853 + 0.860854i \(0.669930\pi\)
−0.999947 + 0.0102523i \(0.996737\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 12.3432 + 21.3790i 0.396111 + 0.686085i 0.993242 0.116059i \(-0.0370262\pi\)
−0.597131 + 0.802144i \(0.703693\pi\)
\(972\) 0 0
\(973\) −5.69615 + 9.86603i −0.182610 + 0.316290i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 45.5537i 1.45739i −0.684838 0.728696i \(-0.740127\pi\)
0.684838 0.728696i \(-0.259873\pi\)
\(978\) 0 0
\(979\) 25.3923 43.9808i 0.811542 1.40563i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −10.6895 18.5148i −0.340942 0.590529i 0.643666 0.765307i \(-0.277413\pi\)
−0.984608 + 0.174778i \(0.944079\pi\)
\(984\) 0 0
\(985\) −2.26795 3.92820i −0.0722629 0.125163i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7.12633 0.226604
\(990\) 0 0
\(991\) −6.99038 12.1077i −0.222057 0.384614i 0.733376 0.679824i \(-0.237944\pi\)
−0.955432 + 0.295210i \(0.904610\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.90949i 0.0605351i
\(996\) 0 0
\(997\) −13.1603 + 22.7942i −0.416789 + 0.721900i −0.995614 0.0935513i \(-0.970178\pi\)
0.578825 + 0.815452i \(0.303511\pi\)
\(998\) 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 2736.2.bm.q.1855.4 yes 8
3.2 odd 2 inner 2736.2.bm.q.1855.1 yes 8
4.3 odd 2 2736.2.bm.p.1855.4 yes 8
12.11 even 2 2736.2.bm.p.1855.1 yes 8
19.8 odd 6 2736.2.bm.p.559.4 yes 8
57.8 even 6 2736.2.bm.p.559.1 8
76.27 even 6 inner 2736.2.bm.q.559.4 yes 8
228.179 odd 6 inner 2736.2.bm.q.559.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2736.2.bm.p.559.1 8 57.8 even 6
2736.2.bm.p.559.4 yes 8 19.8 odd 6
2736.2.bm.p.1855.1 yes 8 12.11 even 2
2736.2.bm.p.1855.4 yes 8 4.3 odd 2
2736.2.bm.q.559.1 yes 8 228.179 odd 6 inner
2736.2.bm.q.559.4 yes 8 76.27 even 6 inner
2736.2.bm.q.1855.1 yes 8 3.2 odd 2 inner
2736.2.bm.q.1855.4 yes 8 1.1 even 1 trivial