Properties

Label 1620.2.x.b.917.1
Level $1620$
Weight $2$
Character 1620.917
Analytic conductor $12.936$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,2,Mod(53,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 10, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.x (of order \(12\), degree \(4\), 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: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 917.1
Root \(1.40126 + 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 1620.917
Dual form 1620.2.x.b.53.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.11803 - 1.93649i) q^{5} +(1.36603 - 0.366025i) q^{7} +O(q^{10})\) \(q+(-1.11803 - 1.93649i) q^{5} +(1.36603 - 0.366025i) q^{7} +(-3.87298 - 2.23607i) q^{11} +(4.09808 + 1.09808i) q^{13} +(2.23607 - 2.23607i) q^{17} -2.00000i q^{19} +(-0.818458 + 3.05453i) q^{23} +(-2.50000 + 4.33013i) q^{25} +(2.23607 - 3.87298i) q^{29} +(-2.00000 - 3.46410i) q^{31} +(-2.23607 - 2.23607i) q^{35} +(-3.00000 - 3.00000i) q^{37} +(-7.74597 + 4.47214i) q^{41} +(-1.09808 - 4.09808i) q^{43} +(-2.45537 - 9.16358i) q^{47} +(-4.33013 + 2.50000i) q^{49} +(2.23607 + 2.23607i) q^{53} +10.0000i q^{55} +(-4.47214 - 7.74597i) q^{59} +(3.00000 - 5.19615i) q^{61} +(-2.45537 - 9.16358i) q^{65} +(-0.366025 + 1.36603i) q^{67} -4.47214i q^{71} +(1.00000 - 1.00000i) q^{73} +(-6.10905 - 1.63692i) q^{77} +(5.19615 + 3.00000i) q^{79} +(-9.16358 + 2.45537i) q^{83} +(-6.83013 - 1.83013i) q^{85} +4.47214 q^{89} +6.00000 q^{91} +(-3.87298 + 2.23607i) q^{95} +(-12.2942 + 3.29423i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{7} + 12 q^{13} - 20 q^{25} - 16 q^{31} - 24 q^{37} + 12 q^{43} + 24 q^{61} + 4 q^{67} + 8 q^{73} - 20 q^{85} + 48 q^{91} - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{1}{6}\right)\)

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
\(5\) −1.11803 1.93649i −0.500000 0.866025i
\(6\) 0 0
\(7\) 1.36603 0.366025i 0.516309 0.138345i 0.00875026 0.999962i \(-0.497215\pi\)
0.507559 + 0.861617i \(0.330548\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.87298 2.23607i −1.16775 0.674200i −0.214600 0.976702i \(-0.568845\pi\)
−0.953149 + 0.302502i \(0.902178\pi\)
\(12\) 0 0
\(13\) 4.09808 + 1.09808i 1.13660 + 0.304552i 0.777584 0.628779i \(-0.216445\pi\)
0.359018 + 0.933331i \(0.383112\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.23607 2.23607i 0.542326 0.542326i −0.381884 0.924210i \(-0.624725\pi\)
0.924210 + 0.381884i \(0.124725\pi\)
\(18\) 0 0
\(19\) 2.00000i 0.458831i −0.973329 0.229416i \(-0.926318\pi\)
0.973329 0.229416i \(-0.0736815\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.818458 + 3.05453i −0.170660 + 0.636913i 0.826590 + 0.562805i \(0.190278\pi\)
−0.997250 + 0.0741081i \(0.976389\pi\)
\(24\) 0 0
\(25\) −2.50000 + 4.33013i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.23607 3.87298i 0.415227 0.719195i −0.580225 0.814456i \(-0.697035\pi\)
0.995452 + 0.0952614i \(0.0303687\pi\)
\(30\) 0 0
\(31\) −2.00000 3.46410i −0.359211 0.622171i 0.628619 0.777714i \(-0.283621\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.23607 2.23607i −0.377964 0.377964i
\(36\) 0 0
\(37\) −3.00000 3.00000i −0.493197 0.493197i 0.416115 0.909312i \(-0.363391\pi\)
−0.909312 + 0.416115i \(0.863391\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.74597 + 4.47214i −1.20972 + 0.698430i −0.962697 0.270580i \(-0.912784\pi\)
−0.247019 + 0.969011i \(0.579451\pi\)
\(42\) 0 0
\(43\) −1.09808 4.09808i −0.167455 0.624951i −0.997714 0.0675734i \(-0.978474\pi\)
0.830259 0.557377i \(-0.188192\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.45537 9.16358i −0.358153 1.33665i −0.876470 0.481457i \(-0.840108\pi\)
0.518317 0.855189i \(-0.326559\pi\)
\(48\) 0 0
\(49\) −4.33013 + 2.50000i −0.618590 + 0.357143i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.23607 + 2.23607i 0.307148 + 0.307148i 0.843802 0.536655i \(-0.180312\pi\)
−0.536655 + 0.843802i \(0.680312\pi\)
\(54\) 0 0
\(55\) 10.0000i 1.34840i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.47214 7.74597i −0.582223 1.00844i −0.995215 0.0977047i \(-0.968850\pi\)
0.412993 0.910734i \(-0.364483\pi\)
\(60\) 0 0
\(61\) 3.00000 5.19615i 0.384111 0.665299i −0.607535 0.794293i \(-0.707841\pi\)
0.991645 + 0.128994i \(0.0411748\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.45537 9.16358i −0.304552 1.13660i
\(66\) 0 0
\(67\) −0.366025 + 1.36603i −0.0447171 + 0.166887i −0.984673 0.174408i \(-0.944199\pi\)
0.939956 + 0.341295i \(0.110865\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.47214i 0.530745i −0.964146 0.265372i \(-0.914505\pi\)
0.964146 0.265372i \(-0.0854949\pi\)
\(72\) 0 0
\(73\) 1.00000 1.00000i 0.117041 0.117041i −0.646160 0.763202i \(-0.723626\pi\)
0.763202 + 0.646160i \(0.223626\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.10905 1.63692i −0.696191 0.186544i
\(78\) 0 0
\(79\) 5.19615 + 3.00000i 0.584613 + 0.337526i 0.762964 0.646440i \(-0.223743\pi\)
−0.178352 + 0.983967i \(0.557076\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.16358 + 2.45537i −1.00583 + 0.269512i −0.723887 0.689918i \(-0.757647\pi\)
−0.281946 + 0.959430i \(0.590980\pi\)
\(84\) 0 0
\(85\) −6.83013 1.83013i −0.740831 0.198505i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.47214 0.474045 0.237023 0.971504i \(-0.423828\pi\)
0.237023 + 0.971504i \(0.423828\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.87298 + 2.23607i −0.397360 + 0.229416i
\(96\) 0 0
\(97\) −12.2942 + 3.29423i −1.24829 + 0.334478i −0.821676 0.569955i \(-0.806960\pi\)
−0.426614 + 0.904434i \(0.640294\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.74597 4.47214i −0.770752 0.444994i 0.0623905 0.998052i \(-0.480128\pi\)
−0.833143 + 0.553058i \(0.813461\pi\)
\(102\) 0 0
\(103\) −1.36603 0.366025i −0.134598 0.0360656i 0.190891 0.981611i \(-0.438862\pi\)
−0.325489 + 0.945546i \(0.605529\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.23607 2.23607i 0.216169 0.216169i −0.590713 0.806882i \(-0.701153\pi\)
0.806882 + 0.590713i \(0.201153\pi\)
\(108\) 0 0
\(109\) 8.00000i 0.766261i 0.923694 + 0.383131i \(0.125154\pi\)
−0.923694 + 0.383131i \(0.874846\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.818458 3.05453i 0.0769940 0.287346i −0.916684 0.399613i \(-0.869145\pi\)
0.993678 + 0.112267i \(0.0358113\pi\)
\(114\) 0 0
\(115\) 6.83013 1.83013i 0.636913 0.170660i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.23607 3.87298i 0.204980 0.355036i
\(120\) 0 0
\(121\) 4.50000 + 7.79423i 0.409091 + 0.708566i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1803 1.00000
\(126\) 0 0
\(127\) −13.0000 13.0000i −1.15356 1.15356i −0.985833 0.167731i \(-0.946356\pi\)
−0.167731 0.985833i \(-0.553644\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.87298 2.23607i 0.338384 0.195366i −0.321173 0.947020i \(-0.604077\pi\)
0.659557 + 0.751654i \(0.270744\pi\)
\(132\) 0 0
\(133\) −0.732051 2.73205i −0.0634769 0.236899i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.45537 9.16358i −0.209777 0.782897i −0.987940 0.154836i \(-0.950515\pi\)
0.778164 0.628062i \(-0.216151\pi\)
\(138\) 0 0
\(139\) 12.1244 7.00000i 1.02837 0.593732i 0.111856 0.993724i \(-0.464321\pi\)
0.916519 + 0.399992i \(0.130987\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −13.4164 13.4164i −1.12194 1.12194i
\(144\) 0 0
\(145\) −10.0000 −0.830455
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.70820 + 11.6190i 0.549557 + 0.951861i 0.998305 + 0.0582028i \(0.0185370\pi\)
−0.448747 + 0.893659i \(0.648130\pi\)
\(150\) 0 0
\(151\) −2.00000 + 3.46410i −0.162758 + 0.281905i −0.935857 0.352381i \(-0.885372\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.47214 + 7.74597i −0.359211 + 0.622171i
\(156\) 0 0
\(157\) 3.29423 12.2942i 0.262908 0.981186i −0.700610 0.713544i \(-0.747089\pi\)
0.963518 0.267642i \(-0.0862445\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 4.47214i 0.352454i
\(162\) 0 0
\(163\) 17.0000 17.0000i 1.33154 1.33154i 0.427552 0.903991i \(-0.359376\pi\)
0.903991 0.427552i \(-0.140624\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.2726 4.09229i −1.18183 0.316671i −0.386179 0.922424i \(-0.626205\pi\)
−0.795653 + 0.605753i \(0.792872\pi\)
\(168\) 0 0
\(169\) 4.33013 + 2.50000i 0.333087 + 0.192308i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 21.3817 5.72920i 1.62562 0.435583i 0.672974 0.739667i \(-0.265017\pi\)
0.952645 + 0.304083i \(0.0983501\pi\)
\(174\) 0 0
\(175\) −1.83013 + 6.83013i −0.138345 + 0.516309i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −8.94427 −0.668526 −0.334263 0.942480i \(-0.608487\pi\)
−0.334263 + 0.942480i \(0.608487\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.45537 + 9.16358i −0.180523 + 0.673720i
\(186\) 0 0
\(187\) −13.6603 + 3.66025i −0.998937 + 0.267664i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.87298 + 2.23607i 0.280239 + 0.161796i 0.633532 0.773717i \(-0.281605\pi\)
−0.353292 + 0.935513i \(0.614938\pi\)
\(192\) 0 0
\(193\) 4.09808 + 1.09808i 0.294986 + 0.0790413i 0.403277 0.915078i \(-0.367871\pi\)
−0.108291 + 0.994119i \(0.534538\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.70820 + 6.70820i −0.477940 + 0.477940i −0.904472 0.426532i \(-0.859735\pi\)
0.426532 + 0.904472i \(0.359735\pi\)
\(198\) 0 0
\(199\) 18.0000i 1.27599i 0.770042 + 0.637993i \(0.220235\pi\)
−0.770042 + 0.637993i \(0.779765\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.63692 6.10905i 0.114889 0.428771i
\(204\) 0 0
\(205\) 17.3205 + 10.0000i 1.20972 + 0.698430i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.47214 + 7.74597i −0.309344 + 0.535800i
\(210\) 0 0
\(211\) 8.00000 + 13.8564i 0.550743 + 0.953914i 0.998221 + 0.0596196i \(0.0189888\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.70820 + 6.70820i −0.457496 + 0.457496i
\(216\) 0 0
\(217\) −4.00000 4.00000i −0.271538 0.271538i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 11.6190 6.70820i 0.781575 0.451243i
\(222\) 0 0
\(223\) −1.09808 4.09808i −0.0735326 0.274427i 0.919364 0.393408i \(-0.128704\pi\)
−0.992897 + 0.118981i \(0.962037\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.09229 + 15.2726i 0.271615 + 1.01368i 0.958076 + 0.286514i \(0.0924964\pi\)
−0.686461 + 0.727166i \(0.740837\pi\)
\(228\) 0 0
\(229\) −10.3923 + 6.00000i −0.686743 + 0.396491i −0.802391 0.596799i \(-0.796439\pi\)
0.115648 + 0.993290i \(0.463106\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.1246 + 20.1246i 1.31841 + 1.31841i 0.915037 + 0.403370i \(0.132161\pi\)
0.403370 + 0.915037i \(0.367839\pi\)
\(234\) 0 0
\(235\) −15.0000 + 15.0000i −0.978492 + 0.978492i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(240\) 0 0
\(241\) 13.0000 22.5167i 0.837404 1.45043i −0.0546547 0.998505i \(-0.517406\pi\)
0.892058 0.451920i \(-0.149261\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 9.68246 + 5.59017i 0.618590 + 0.357143i
\(246\) 0 0
\(247\) 2.19615 8.19615i 0.139738 0.521509i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 13.4164i 0.846836i −0.905934 0.423418i \(-0.860830\pi\)
0.905934 0.423418i \(-0.139170\pi\)
\(252\) 0 0
\(253\) 10.0000 10.0000i 0.628695 0.628695i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 21.3817 + 5.72920i 1.33375 + 0.357378i 0.854113 0.520088i \(-0.174101\pi\)
0.479640 + 0.877466i \(0.340767\pi\)
\(258\) 0 0
\(259\) −5.19615 3.00000i −0.322873 0.186411i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −21.3817 + 5.72920i −1.31845 + 0.353278i −0.848397 0.529360i \(-0.822432\pi\)
−0.470054 + 0.882638i \(0.655765\pi\)
\(264\) 0 0
\(265\) 1.83013 6.83013i 0.112424 0.419571i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 22.3607 1.36335 0.681677 0.731653i \(-0.261251\pi\)
0.681677 + 0.731653i \(0.261251\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 19.3649 11.1803i 1.16775 0.674200i
\(276\) 0 0
\(277\) −12.2942 + 3.29423i −0.738689 + 0.197931i −0.608495 0.793558i \(-0.708227\pi\)
−0.130193 + 0.991489i \(0.541560\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 23.2379 + 13.4164i 1.38626 + 0.800356i 0.992891 0.119026i \(-0.0379773\pi\)
0.393366 + 0.919382i \(0.371311\pi\)
\(282\) 0 0
\(283\) −23.2224 6.22243i −1.38043 0.369885i −0.509153 0.860676i \(-0.670041\pi\)
−0.871277 + 0.490791i \(0.836708\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.94427 + 8.94427i −0.527964 + 0.527964i
\(288\) 0 0
\(289\) 7.00000i 0.411765i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.45537 + 9.16358i −0.143444 + 0.535342i 0.856375 + 0.516354i \(0.172711\pi\)
−0.999820 + 0.0189880i \(0.993956\pi\)
\(294\) 0 0
\(295\) −10.0000 + 17.3205i −0.582223 + 1.00844i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.70820 + 11.6190i −0.387945 + 0.671941i
\(300\) 0 0
\(301\) −3.00000 5.19615i −0.172917 0.299501i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −13.4164 −0.768221
\(306\) 0 0
\(307\) −21.0000 21.0000i −1.19853 1.19853i −0.974606 0.223928i \(-0.928112\pi\)
−0.223928 0.974606i \(-0.571888\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.87298 + 2.23607i −0.219617 + 0.126796i −0.605773 0.795638i \(-0.707136\pi\)
0.386156 + 0.922433i \(0.373803\pi\)
\(312\) 0 0
\(313\) −1.09808 4.09808i −0.0620669 0.231637i 0.927924 0.372770i \(-0.121592\pi\)
−0.989991 + 0.141133i \(0.954925\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.36612 + 27.4907i 0.413722 + 1.54403i 0.787381 + 0.616467i \(0.211437\pi\)
−0.373658 + 0.927566i \(0.621897\pi\)
\(318\) 0 0
\(319\) −17.3205 + 10.0000i −0.969762 + 0.559893i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.47214 4.47214i −0.248836 0.248836i
\(324\) 0 0
\(325\) −15.0000 + 15.0000i −0.832050 + 0.832050i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −6.70820 11.6190i −0.369835 0.640573i
\(330\) 0 0
\(331\) 4.00000 6.92820i 0.219860 0.380808i −0.734905 0.678170i \(-0.762773\pi\)
0.954765 + 0.297361i \(0.0961066\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.05453 0.818458i 0.166887 0.0447171i
\(336\) 0 0
\(337\) 3.29423 12.2942i 0.179448 0.669709i −0.816303 0.577624i \(-0.803980\pi\)
0.995751 0.0920854i \(-0.0293533\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 17.8885i 0.968719i
\(342\) 0 0
\(343\) −12.0000 + 12.0000i −0.647939 + 0.647939i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 33.5998 + 9.00303i 1.80373 + 0.483308i 0.994551 0.104253i \(-0.0332451\pi\)
0.809180 + 0.587561i \(0.199912\pi\)
\(348\) 0 0
\(349\) −3.46410 2.00000i −0.185429 0.107058i 0.404412 0.914577i \(-0.367476\pi\)
−0.589841 + 0.807519i \(0.700810\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9.16358 2.45537i 0.487728 0.130686i −0.00657077 0.999978i \(-0.502092\pi\)
0.494299 + 0.869292i \(0.335425\pi\)
\(354\) 0 0
\(355\) −8.66025 + 5.00000i −0.459639 + 0.265372i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −35.7771 −1.88824 −0.944121 0.329598i \(-0.893087\pi\)
−0.944121 + 0.329598i \(0.893087\pi\)
\(360\) 0 0
\(361\) 15.0000 0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.05453 0.818458i −0.159881 0.0428400i
\(366\) 0 0
\(367\) 1.36603 0.366025i 0.0713059 0.0191064i −0.222990 0.974821i \(-0.571582\pi\)
0.294296 + 0.955714i \(0.404915\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.87298 + 2.23607i 0.201075 + 0.116091i
\(372\) 0 0
\(373\) 31.4186 + 8.41858i 1.62679 + 0.435898i 0.952987 0.303011i \(-0.0979919\pi\)
0.673806 + 0.738909i \(0.264659\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 13.4164 13.4164i 0.690980 0.690980i
\(378\) 0 0
\(379\) 2.00000i 0.102733i −0.998680 0.0513665i \(-0.983642\pi\)
0.998680 0.0513665i \(-0.0163577\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.45537 9.16358i 0.125464 0.468237i −0.874392 0.485220i \(-0.838739\pi\)
0.999856 + 0.0169831i \(0.00540616\pi\)
\(384\) 0 0
\(385\) 3.66025 + 13.6603i 0.186544 + 0.696191i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.70820 11.6190i 0.340119 0.589104i −0.644335 0.764743i \(-0.722866\pi\)
0.984455 + 0.175639i \(0.0561992\pi\)
\(390\) 0 0
\(391\) 5.00000 + 8.66025i 0.252861 + 0.437968i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 13.4164i 0.675053i
\(396\) 0 0
\(397\) −11.0000 11.0000i −0.552074 0.552074i 0.374965 0.927039i \(-0.377655\pi\)
−0.927039 + 0.374965i \(0.877655\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0 0
\(403\) −4.39230 16.3923i −0.218796 0.816559i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.91075 + 18.3272i 0.243417 + 0.908443i
\(408\) 0 0
\(409\) −10.3923 + 6.00000i −0.513866 + 0.296681i −0.734422 0.678694i \(-0.762546\pi\)
0.220555 + 0.975375i \(0.429213\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −8.94427 8.94427i −0.440119 0.440119i
\(414\) 0 0
\(415\) 15.0000 + 15.0000i 0.736321 + 0.736321i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 13.4164 + 23.2379i 0.655434 + 1.13525i 0.981785 + 0.189997i \(0.0608478\pi\)
−0.326350 + 0.945249i \(0.605819\pi\)
\(420\) 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
\(425\) 4.09229 + 15.2726i 0.198505 + 0.740831i
\(426\) 0 0
\(427\) 2.19615 8.19615i 0.106279 0.396640i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 31.3050i 1.50791i 0.656928 + 0.753953i \(0.271855\pi\)
−0.656928 + 0.753953i \(0.728145\pi\)
\(432\) 0 0
\(433\) 17.0000 17.0000i 0.816968 0.816968i −0.168700 0.985668i \(-0.553957\pi\)
0.985668 + 0.168700i \(0.0539568\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.10905 + 1.63692i 0.292236 + 0.0783043i
\(438\) 0 0
\(439\) 19.0526 + 11.0000i 0.909329 + 0.525001i 0.880215 0.474575i \(-0.157398\pi\)
0.0291138 + 0.999576i \(0.490731\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 27.4907 7.36612i 1.30612 0.349975i 0.462361 0.886692i \(-0.347002\pi\)
0.843763 + 0.536717i \(0.180336\pi\)
\(444\) 0 0
\(445\) −5.00000 8.66025i −0.237023 0.410535i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −4.47214 −0.211053 −0.105527 0.994416i \(-0.533653\pi\)
−0.105527 + 0.994416i \(0.533653\pi\)
\(450\) 0 0
\(451\) 40.0000 1.88353
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6.70820 11.6190i −0.314485 0.544705i
\(456\) 0 0
\(457\) 4.09808 1.09808i 0.191700 0.0513658i −0.161692 0.986841i \(-0.551695\pi\)
0.353392 + 0.935475i \(0.385028\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 7.74597 + 4.47214i 0.360766 + 0.208288i 0.669417 0.742887i \(-0.266544\pi\)
−0.308651 + 0.951175i \(0.599877\pi\)
\(462\) 0 0
\(463\) 25.9545 + 6.95448i 1.20621 + 0.323202i 0.805273 0.592905i \(-0.202019\pi\)
0.400934 + 0.916107i \(0.368686\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.23607 2.23607i 0.103473 0.103473i −0.653475 0.756948i \(-0.726690\pi\)
0.756948 + 0.653475i \(0.226690\pi\)
\(468\) 0 0
\(469\) 2.00000i 0.0923514i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4.91075 + 18.3272i −0.225796 + 0.842683i
\(474\) 0 0
\(475\) 8.66025 + 5.00000i 0.397360 + 0.229416i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −17.8885 + 30.9839i −0.817348 + 1.41569i 0.0902809 + 0.995916i \(0.471224\pi\)
−0.907629 + 0.419773i \(0.862110\pi\)
\(480\) 0 0
\(481\) −9.00000 15.5885i −0.410365 0.710772i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 20.1246 + 20.1246i 0.913812 + 0.913812i
\(486\) 0 0
\(487\) −21.0000 21.0000i −0.951601 0.951601i 0.0472808 0.998882i \(-0.484944\pi\)
−0.998882 + 0.0472808i \(0.984944\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 19.3649 11.1803i 0.873926 0.504562i 0.00527540 0.999986i \(-0.498321\pi\)
0.868651 + 0.495424i \(0.164987\pi\)
\(492\) 0 0
\(493\) −3.66025 13.6603i −0.164850 0.615227i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.63692 6.10905i −0.0734257 0.274028i
\(498\) 0 0
\(499\) 32.9090 19.0000i 1.47321 0.850557i 0.473662 0.880707i \(-0.342932\pi\)
0.999545 + 0.0301498i \(0.00959843\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.70820 + 6.70820i 0.299104 + 0.299104i 0.840663 0.541559i \(-0.182166\pi\)
−0.541559 + 0.840663i \(0.682166\pi\)
\(504\) 0 0
\(505\) 20.0000i 0.889988i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.23607 + 3.87298i 0.0991120 + 0.171667i 0.911317 0.411705i \(-0.135066\pi\)
−0.812205 + 0.583372i \(0.801733\pi\)
\(510\) 0 0
\(511\) 1.00000 1.73205i 0.0442374 0.0766214i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.818458 + 3.05453i 0.0360656 + 0.134598i
\(516\) 0 0
\(517\) −10.9808 + 40.9808i −0.482933 + 1.80233i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −3.00000 + 3.00000i −0.131181 + 0.131181i −0.769649 0.638468i \(-0.779569\pi\)
0.638468 + 0.769649i \(0.279569\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.2181 3.27383i −0.532229 0.142610i
\(528\) 0 0
\(529\) 11.2583 + 6.50000i 0.489493 + 0.282609i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −36.6543 + 9.82149i −1.58767 + 0.425416i
\(534\) 0 0
\(535\) −6.83013 1.83013i −0.295292 0.0791233i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 22.3607 0.963143
\(540\) 0 0
\(541\) −26.0000 −1.11783 −0.558914 0.829226i \(-0.688782\pi\)
−0.558914 + 0.829226i \(0.688782\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 15.4919 8.94427i 0.663602 0.383131i
\(546\) 0 0
\(547\) −36.8827 + 9.88269i −1.57699 + 0.422553i −0.937992 0.346656i \(-0.887317\pi\)
−0.638997 + 0.769209i \(0.720651\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −7.74597 4.47214i −0.329989 0.190519i
\(552\) 0 0
\(553\) 8.19615 + 2.19615i 0.348536 + 0.0933899i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −6.70820 + 6.70820i −0.284236 + 0.284236i −0.834796 0.550560i \(-0.814414\pi\)
0.550560 + 0.834796i \(0.314414\pi\)
\(558\) 0 0
\(559\) 18.0000i 0.761319i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.45537 9.16358i 0.103482 0.386199i −0.894687 0.446694i \(-0.852601\pi\)
0.998168 + 0.0604952i \(0.0192680\pi\)
\(564\) 0 0
\(565\) −6.83013 + 1.83013i −0.287346 + 0.0769940i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 11.1803 19.3649i 0.468704 0.811820i −0.530656 0.847587i \(-0.678054\pi\)
0.999360 + 0.0357678i \(0.0113877\pi\)
\(570\) 0 0
\(571\) 8.00000 + 13.8564i 0.334790 + 0.579873i 0.983444 0.181210i \(-0.0580014\pi\)
−0.648655 + 0.761083i \(0.724668\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −11.1803 11.1803i −0.466252 0.466252i
\(576\) 0 0
\(577\) −11.0000 11.0000i −0.457936 0.457936i 0.440041 0.897977i \(-0.354964\pi\)
−0.897977 + 0.440041i \(0.854964\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −11.6190 + 6.70820i −0.482035 + 0.278303i
\(582\) 0 0
\(583\) −3.66025 13.6603i −0.151592 0.565750i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2.45537 9.16358i −0.101344 0.378221i 0.896561 0.442921i \(-0.146058\pi\)
−0.997905 + 0.0646996i \(0.979391\pi\)
\(588\) 0 0
\(589\) −6.92820 + 4.00000i −0.285472 + 0.164817i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 11.1803 + 11.1803i 0.459122 + 0.459122i 0.898367 0.439246i \(-0.144754\pi\)
−0.439246 + 0.898367i \(0.644754\pi\)
\(594\) 0 0
\(595\) −10.0000 −0.409960
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −17.8885 30.9839i −0.730906 1.26597i −0.956496 0.291744i \(-0.905764\pi\)
0.225590 0.974222i \(-0.427569\pi\)
\(600\) 0 0
\(601\) −7.00000 + 12.1244i −0.285536 + 0.494563i −0.972739 0.231903i \(-0.925505\pi\)
0.687203 + 0.726465i \(0.258838\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10.0623 17.4284i 0.409091 0.708566i
\(606\) 0 0
\(607\) 9.88269 36.8827i 0.401126 1.49702i −0.409965 0.912101i \(-0.634459\pi\)
0.811091 0.584921i \(-0.198874\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 40.2492i 1.62831i
\(612\) 0 0
\(613\) 21.0000 21.0000i 0.848182 0.848182i −0.141724 0.989906i \(-0.545265\pi\)
0.989906 + 0.141724i \(0.0452646\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −27.4907 7.36612i −1.10674 0.296549i −0.341231 0.939980i \(-0.610844\pi\)
−0.765504 + 0.643431i \(0.777510\pi\)
\(618\) 0 0
\(619\) 36.3731 + 21.0000i 1.46196 + 0.844061i 0.999102 0.0423727i \(-0.0134917\pi\)
0.462855 + 0.886434i \(0.346825\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.10905 1.63692i 0.244754 0.0655816i
\(624\) 0 0
\(625\) −12.5000 21.6506i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −13.4164 −0.534947
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −10.6399 + 39.7088i −0.422234 + 1.57580i
\(636\) 0 0
\(637\) −20.4904 + 5.49038i −0.811858 + 0.217537i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −7.74597 4.47214i −0.305947 0.176639i 0.339164 0.940727i \(-0.389856\pi\)
−0.645112 + 0.764088i \(0.723189\pi\)
\(642\) 0 0
\(643\) −28.6865 7.68653i −1.13129 0.303127i −0.355844 0.934545i \(-0.615807\pi\)
−0.775443 + 0.631418i \(0.782473\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15.6525 + 15.6525i −0.615362 + 0.615362i −0.944338 0.328976i \(-0.893296\pi\)
0.328976 + 0.944338i \(0.393296\pi\)
\(648\) 0 0
\(649\) 40.0000i 1.57014i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7.36612 27.4907i 0.288259 1.07580i −0.658167 0.752872i \(-0.728668\pi\)
0.946425 0.322923i \(-0.104666\pi\)
\(654\) 0 0
\(655\) −8.66025 5.00000i −0.338384 0.195366i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4.47214 7.74597i 0.174210 0.301740i −0.765678 0.643224i \(-0.777596\pi\)
0.939887 + 0.341484i \(0.110930\pi\)
\(660\) 0 0
\(661\) −1.00000 1.73205i −0.0388955 0.0673690i 0.845922 0.533306i \(-0.179051\pi\)
−0.884818 + 0.465937i \(0.845717\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4.47214 + 4.47214i −0.173422 + 0.173422i
\(666\) 0 0
\(667\) 10.0000 + 10.0000i 0.387202 + 0.387202i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −23.2379 + 13.4164i −0.897089 + 0.517935i
\(672\) 0 0
\(673\) 0.366025 + 1.36603i 0.0141092 + 0.0526564i 0.972622 0.232395i \(-0.0746561\pi\)
−0.958512 + 0.285051i \(0.907989\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −9.00303 33.5998i −0.346015 1.29134i −0.891422 0.453174i \(-0.850292\pi\)
0.545408 0.838171i \(-0.316375\pi\)
\(678\) 0 0
\(679\) −15.5885 + 9.00000i −0.598230 + 0.345388i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −20.1246 20.1246i −0.770047 0.770047i 0.208068 0.978114i \(-0.433283\pi\)
−0.978114 + 0.208068i \(0.933283\pi\)
\(684\) 0 0
\(685\) −15.0000 + 15.0000i −0.573121 + 0.573121i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 6.70820 + 11.6190i 0.255562 + 0.442647i
\(690\) 0 0
\(691\) 24.0000 41.5692i 0.913003 1.58137i 0.103204 0.994660i \(-0.467091\pi\)
0.809799 0.586707i \(-0.199576\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −27.1109 15.6525i −1.02837 0.593732i
\(696\) 0 0
\(697\) −7.32051 + 27.3205i −0.277284 + 1.03484i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 35.7771i 1.35128i −0.737231 0.675641i \(-0.763867\pi\)
0.737231 0.675641i \(-0.236133\pi\)
\(702\) 0 0
\(703\) −6.00000 + 6.00000i −0.226294 + 0.226294i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.2181 3.27383i −0.459509 0.123125i
\(708\) 0 0
\(709\) −24.2487 14.0000i −0.910679 0.525781i −0.0300298 0.999549i \(-0.509560\pi\)
−0.880650 + 0.473768i \(0.842894\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 12.2181 3.27383i 0.457572 0.122606i
\(714\) 0 0
\(715\) −10.9808 + 40.9808i −0.410657 + 1.53259i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 35.7771 1.33426 0.667130 0.744941i \(-0.267522\pi\)
0.667130 + 0.744941i \(0.267522\pi\)
\(720\) 0 0
\(721\) −2.00000 −0.0744839
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 11.1803 + 19.3649i 0.415227 + 0.719195i
\(726\) 0 0
\(727\) 28.6865 7.68653i 1.06392 0.285078i 0.315930 0.948783i \(-0.397684\pi\)
0.747995 + 0.663705i \(0.231017\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −11.6190 6.70820i −0.429742 0.248112i
\(732\) 0 0
\(733\) −1.36603 0.366025i −0.0504553 0.0135195i 0.233503 0.972356i \(-0.424981\pi\)
−0.283958 + 0.958837i \(0.591648\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.47214 4.47214i 0.164733 0.164733i
\(738\) 0 0
\(739\) 14.0000i 0.514998i 0.966279 + 0.257499i \(0.0828985\pi\)
−0.966279 + 0.257499i \(0.917102\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −4.09229 + 15.2726i −0.150132 + 0.560298i 0.849342 + 0.527843i \(0.176999\pi\)
−0.999473 + 0.0324550i \(0.989667\pi\)
\(744\) 0 0
\(745\) 15.0000 25.9808i 0.549557 0.951861i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.23607 3.87298i 0.0817041 0.141516i
\(750\) 0 0
\(751\) −6.00000 10.3923i −0.218943 0.379221i 0.735542 0.677479i \(-0.236928\pi\)
−0.954485 + 0.298259i \(0.903594\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.94427 0.325515
\(756\) 0 0
\(757\) 9.00000 + 9.00000i 0.327111 + 0.327111i 0.851487 0.524376i \(-0.175701\pi\)
−0.524376 + 0.851487i \(0.675701\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15.4919 8.94427i 0.561582 0.324230i −0.192198 0.981356i \(-0.561562\pi\)
0.753780 + 0.657127i \(0.228228\pi\)
\(762\) 0 0
\(763\) 2.92820 + 10.9282i 0.106008 + 0.395628i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −9.82149 36.6543i −0.354634 1.32351i
\(768\) 0 0
\(769\) 6.92820 4.00000i 0.249837 0.144244i −0.369852 0.929091i \(-0.620592\pi\)
0.619690 + 0.784847i \(0.287258\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −6.70820 6.70820i −0.241277 0.241277i 0.576101 0.817378i \(-0.304573\pi\)
−0.817378 + 0.576101i \(0.804573\pi\)
\(774\) 0 0
\(775\) 20.0000 0.718421
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 8.94427 + 15.4919i 0.320462 + 0.555056i
\(780\) 0 0
\(781\) −10.0000 + 17.3205i −0.357828 + 0.619777i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −27.4907 + 7.36612i −0.981186 + 0.262908i
\(786\) 0 0
\(787\) 6.95448 25.9545i 0.247901 0.925177i −0.724003 0.689797i \(-0.757700\pi\)
0.971903 0.235380i \(-0.0756336\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 4.47214i 0.159011i
\(792\) 0 0
\(793\) 18.0000 18.0000i 0.639199 0.639199i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −27.4907 7.36612i −0.973772 0.260921i −0.263352 0.964700i \(-0.584828\pi\)
−0.710420 + 0.703778i \(0.751495\pi\)
\(798\) 0 0
\(799\) −25.9808 15.0000i −0.919133 0.530662i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.10905 + 1.63692i −0.215584 + 0.0577655i
\(804\) 0 0
\(805\) 8.66025 5.00000i 0.305234 0.176227i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 4.47214 0.157232 0.0786160 0.996905i \(-0.474950\pi\)
0.0786160 + 0.996905i \(0.474950\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −51.9269 13.9138i −1.81892 0.487378i
\(816\) 0 0
\(817\) −8.19615 + 2.19615i −0.286747 + 0.0768336i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −15.4919 8.94427i −0.540672 0.312157i 0.204679 0.978829i \(-0.434385\pi\)
−0.745351 + 0.666672i \(0.767718\pi\)
\(822\) 0 0
\(823\) −23.2224 6.22243i −0.809483 0.216900i −0.169740 0.985489i \(-0.554293\pi\)
−0.639743 + 0.768589i \(0.720959\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 20.1246 20.1246i 0.699801 0.699801i −0.264566 0.964368i \(-0.585229\pi\)
0.964368 + 0.264566i \(0.0852288\pi\)
\(828\) 0 0
\(829\) 16.0000i 0.555703i −0.960624 0.277851i \(-0.910378\pi\)
0.960624 0.277851i \(-0.0896223\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4.09229 + 15.2726i −0.141789 + 0.529165i
\(834\) 0 0
\(835\) 9.15064 + 34.1506i 0.316671 + 1.18183i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 17.8885 30.9839i 0.617581 1.06968i −0.372345 0.928095i \(-0.621446\pi\)
0.989926 0.141587i \(-0.0452206\pi\)
\(840\) 0 0
\(841\) 4.50000 + 7.79423i 0.155172 + 0.268767i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 11.1803i 0.384615i
\(846\) 0 0
\(847\) 9.00000 + 9.00000i 0.309244 + 0.309244i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 11.6190 6.70820i 0.398292 0.229954i
\(852\) 0 0
\(853\) 13.5429 + 50.5429i 0.463701 + 1.73056i 0.661159 + 0.750246i \(0.270065\pi\)
−0.197458 + 0.980311i \(0.563268\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.45537 9.16358i −0.0838739 0.313022i 0.911225 0.411910i \(-0.135138\pi\)
−0.995099 + 0.0988880i \(0.968471\pi\)
\(858\) 0 0
\(859\) 12.1244 7.00000i 0.413678 0.238837i −0.278691 0.960381i \(-0.589901\pi\)
0.692369 + 0.721544i \(0.256567\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.5967 + 24.5967i 0.837283 + 0.837283i 0.988501 0.151217i \(-0.0483194\pi\)
−0.151217 + 0.988501i \(0.548319\pi\)
\(864\) 0 0
\(865\) −35.0000 35.0000i −1.19004 1.19004i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −13.4164 23.2379i −0.455120 0.788292i
\(870\) 0 0
\(871\) −3.00000 + 5.19615i −0.101651 + 0.176065i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 15.2726 4.09229i 0.516309 0.138345i
\(876\) 0 0
\(877\) 3.29423 12.2942i 0.111238 0.415147i −0.887740 0.460346i \(-0.847726\pi\)
0.998978 + 0.0451990i \(0.0143922\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 44.7214i 1.50670i 0.657619 + 0.753350i \(0.271564\pi\)
−0.657619 + 0.753350i \(0.728436\pi\)
\(882\) 0 0
\(883\) −23.0000 + 23.0000i −0.774012 + 0.774012i −0.978805 0.204794i \(-0.934348\pi\)
0.204794 + 0.978805i \(0.434348\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −27.4907 7.36612i −0.923048 0.247330i −0.234160 0.972198i \(-0.575234\pi\)
−0.688888 + 0.724868i \(0.741901\pi\)
\(888\) 0 0
\(889\) −22.5167 13.0000i −0.755185 0.436006i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −18.3272 + 4.91075i −0.613295 + 0.164332i
\(894\) 0 0
\(895\) 10.0000 + 17.3205i 0.334263 + 0.578961i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −17.8885 −0.596616
\(900\) 0 0
\(901\) 10.0000 0.333148
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −15.6525 27.1109i −0.520306 0.901196i
\(906\) 0 0
\(907\) −9.56218 + 2.56218i −0.317507 + 0.0850757i −0.414053 0.910253i \(-0.635887\pi\)
0.0965460 + 0.995329i \(0.469220\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −42.6028 24.5967i −1.41149 0.814927i −0.415965 0.909381i \(-0.636556\pi\)
−0.995529 + 0.0944540i \(0.969889\pi\)
\(912\) 0 0
\(913\) 40.9808 + 10.9808i 1.35627 + 0.363410i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.47214 4.47214i 0.147683 0.147683i
\(918\) 0 0
\(919\) 22.0000i 0.725713i −0.931845 0.362857i \(-0.881802\pi\)
0.931845 0.362857i \(-0.118198\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 4.91075 18.3272i 0.161639 0.603246i
\(924\) 0 0
\(925\) 20.4904 5.49038i 0.673720 0.180523i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2.23607 3.87298i 0.0733630 0.127068i −0.827010 0.562187i \(-0.809960\pi\)
0.900373 + 0.435118i \(0.143293\pi\)
\(930\) 0 0
\(931\) 5.00000 + 8.66025i 0.163868 + 0.283828i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 22.3607 + 22.3607i 0.731272 + 0.731272i
\(936\) 0 0
\(937\) −11.0000 11.0000i −0.359354 0.359354i 0.504221 0.863575i \(-0.331780\pi\)
−0.863575 + 0.504221i \(0.831780\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −23.2379 + 13.4164i −0.757534 + 0.437362i −0.828410 0.560123i \(-0.810754\pi\)
0.0708757 + 0.997485i \(0.477421\pi\)
\(942\) 0 0
\(943\) −7.32051 27.3205i −0.238389 0.889678i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.09229 + 15.2726i 0.132982 + 0.496294i 0.999998 0.00198565i \(-0.000632053\pi\)
−0.867017 + 0.498279i \(0.833965\pi\)
\(948\) 0 0
\(949\) 5.19615 3.00000i 0.168674 0.0973841i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −33.5410 33.5410i −1.08650 1.08650i −0.995886 0.0906141i \(-0.971117\pi\)
−0.0906141 0.995886i \(-0.528883\pi\)
\(954\) 0 0
\(955\) 10.0000i 0.323592i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −6.70820 11.6190i −0.216619 0.375195i
\(960\) 0 0
\(961\) 7.50000 12.9904i 0.241935 0.419045i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2.45537 9.16358i −0.0790413 0.294986i
\(966\) 0 0
\(967\) 2.56218 9.56218i 0.0823941 0.307499i −0.912414 0.409269i \(-0.865784\pi\)
0.994808 + 0.101770i \(0.0324505\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 22.3607i 0.717588i 0.933417 + 0.358794i \(0.116812\pi\)
−0.933417 + 0.358794i \(0.883188\pi\)
\(972\) 0 0
\(973\) 14.0000 14.0000i 0.448819 0.448819i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9.16358 + 2.45537i 0.293169 + 0.0785543i 0.402406 0.915461i \(-0.368174\pi\)
−0.109237 + 0.994016i \(0.534841\pi\)
\(978\) 0 0
\(979\) −17.3205 10.0000i −0.553566 0.319601i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 15.2726 4.09229i 0.487121 0.130524i −0.00689536 0.999976i \(-0.502195\pi\)
0.494017 + 0.869452i \(0.335528\pi\)
\(984\) 0 0
\(985\) 20.4904 + 5.49038i 0.652878 + 0.174938i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 13.4164 0.426617
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 34.8569 20.1246i 1.10504 0.637993i
\(996\) 0 0
\(997\) 31.4186 8.41858i 0.995037 0.266619i 0.275672 0.961252i \(-0.411100\pi\)
0.719365 + 0.694633i \(0.244433\pi\)
\(998\) 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 1620.2.x.b.917.1 8
3.2 odd 2 inner 1620.2.x.b.917.2 8
5.3 odd 4 inner 1620.2.x.b.593.2 8
9.2 odd 6 60.2.i.a.17.2 yes 4
9.4 even 3 inner 1620.2.x.b.377.1 8
9.5 odd 6 inner 1620.2.x.b.377.2 8
9.7 even 3 60.2.i.a.17.1 4
15.8 even 4 inner 1620.2.x.b.593.1 8
36.7 odd 6 240.2.v.b.17.2 4
36.11 even 6 240.2.v.b.17.1 4
45.2 even 12 300.2.i.a.293.2 4
45.7 odd 12 300.2.i.a.293.1 4
45.13 odd 12 inner 1620.2.x.b.53.2 8
45.23 even 12 inner 1620.2.x.b.53.1 8
45.29 odd 6 300.2.i.a.257.1 4
45.34 even 6 300.2.i.a.257.2 4
45.38 even 12 60.2.i.a.53.1 yes 4
45.43 odd 12 60.2.i.a.53.2 yes 4
72.11 even 6 960.2.v.h.257.2 4
72.29 odd 6 960.2.v.e.257.1 4
72.43 odd 6 960.2.v.h.257.1 4
72.61 even 6 960.2.v.e.257.2 4
180.7 even 12 1200.2.v.i.593.2 4
180.43 even 12 240.2.v.b.113.1 4
180.47 odd 12 1200.2.v.i.593.1 4
180.79 odd 6 1200.2.v.i.257.1 4
180.83 odd 12 240.2.v.b.113.2 4
180.119 even 6 1200.2.v.i.257.2 4
360.43 even 12 960.2.v.h.833.2 4
360.83 odd 12 960.2.v.h.833.1 4
360.133 odd 12 960.2.v.e.833.1 4
360.173 even 12 960.2.v.e.833.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.2.i.a.17.1 4 9.7 even 3
60.2.i.a.17.2 yes 4 9.2 odd 6
60.2.i.a.53.1 yes 4 45.38 even 12
60.2.i.a.53.2 yes 4 45.43 odd 12
240.2.v.b.17.1 4 36.11 even 6
240.2.v.b.17.2 4 36.7 odd 6
240.2.v.b.113.1 4 180.43 even 12
240.2.v.b.113.2 4 180.83 odd 12
300.2.i.a.257.1 4 45.29 odd 6
300.2.i.a.257.2 4 45.34 even 6
300.2.i.a.293.1 4 45.7 odd 12
300.2.i.a.293.2 4 45.2 even 12
960.2.v.e.257.1 4 72.29 odd 6
960.2.v.e.257.2 4 72.61 even 6
960.2.v.e.833.1 4 360.133 odd 12
960.2.v.e.833.2 4 360.173 even 12
960.2.v.h.257.1 4 72.43 odd 6
960.2.v.h.257.2 4 72.11 even 6
960.2.v.h.833.1 4 360.83 odd 12
960.2.v.h.833.2 4 360.43 even 12
1200.2.v.i.257.1 4 180.79 odd 6
1200.2.v.i.257.2 4 180.119 even 6
1200.2.v.i.593.1 4 180.47 odd 12
1200.2.v.i.593.2 4 180.7 even 12
1620.2.x.b.53.1 8 45.23 even 12 inner
1620.2.x.b.53.2 8 45.13 odd 12 inner
1620.2.x.b.377.1 8 9.4 even 3 inner
1620.2.x.b.377.2 8 9.5 odd 6 inner
1620.2.x.b.593.1 8 15.8 even 4 inner
1620.2.x.b.593.2 8 5.3 odd 4 inner
1620.2.x.b.917.1 8 1.1 even 1 trivial
1620.2.x.b.917.2 8 3.2 odd 2 inner