Properties

Label 1620.2.x.a.593.1
Level $1620$
Weight $2$
Character 1620.593
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: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 593.1
Root \(-0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1620.593
Dual form 1620.2.x.a.377.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.48356 + 1.67303i) q^{5} +(0.732051 + 2.73205i) q^{7} +O(q^{10})\) \(q+(-1.48356 + 1.67303i) q^{5} +(0.732051 + 2.73205i) q^{7} +(2.44949 + 1.41421i) q^{11} +(1.09808 - 4.09808i) q^{13} +(1.41421 + 1.41421i) q^{17} -4.00000i q^{19} +(7.72741 + 2.07055i) q^{23} +(-0.598076 - 4.96410i) q^{25} +(-4.94975 + 8.57321i) q^{29} +(4.00000 + 6.92820i) q^{31} +(-5.65685 - 2.82843i) q^{35} +(-3.00000 + 3.00000i) q^{37} +(1.22474 - 0.707107i) q^{41} +(-11.5911 + 3.10583i) q^{47} +(-0.866025 + 0.500000i) q^{49} +(-7.07107 + 7.07107i) q^{53} +(-6.00000 + 2.00000i) q^{55} +(1.41421 + 2.44949i) q^{59} +(5.22715 + 7.91688i) q^{65} +(-10.9282 - 2.92820i) q^{67} -5.65685i q^{71} +(7.00000 + 7.00000i) q^{73} +(-2.07055 + 7.72741i) q^{77} +(3.10583 + 11.5911i) q^{83} +(-4.46410 + 0.267949i) q^{85} -1.41421 q^{89} +12.0000 q^{91} +(6.69213 + 5.93426i) q^{95} +(1.09808 + 4.09808i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} - 12 q^{13} + 16 q^{25} + 32 q^{31} - 24 q^{37} - 48 q^{55} - 32 q^{67} + 56 q^{73} - 8 q^{85} + 96 q^{91} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{1}{6}\right)\)

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
\(5\) −1.48356 + 1.67303i −0.663470 + 0.748203i
\(6\) 0 0
\(7\) 0.732051 + 2.73205i 0.276689 + 1.03262i 0.954701 + 0.297567i \(0.0961752\pi\)
−0.678012 + 0.735051i \(0.737158\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.44949 + 1.41421i 0.738549 + 0.426401i 0.821541 0.570149i \(-0.193114\pi\)
−0.0829925 + 0.996550i \(0.526448\pi\)
\(12\) 0 0
\(13\) 1.09808 4.09808i 0.304552 1.13660i −0.628779 0.777584i \(-0.716445\pi\)
0.933331 0.359018i \(-0.116888\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.41421 + 1.41421i 0.342997 + 0.342997i 0.857493 0.514496i \(-0.172021\pi\)
−0.514496 + 0.857493i \(0.672021\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i −0.888523 0.458831i \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.72741 + 2.07055i 1.61128 + 0.431740i 0.948422 0.317009i \(-0.102679\pi\)
0.662853 + 0.748749i \(0.269345\pi\)
\(24\) 0 0
\(25\) −0.598076 4.96410i −0.119615 0.992820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.94975 + 8.57321i −0.919145 + 1.59201i −0.118428 + 0.992963i \(0.537786\pi\)
−0.800717 + 0.599043i \(0.795548\pi\)
\(30\) 0 0
\(31\) 4.00000 + 6.92820i 0.718421 + 1.24434i 0.961625 + 0.274367i \(0.0884683\pi\)
−0.243204 + 0.969975i \(0.578198\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.65685 2.82843i −0.956183 0.478091i
\(36\) 0 0
\(37\) −3.00000 + 3.00000i −0.493197 + 0.493197i −0.909312 0.416115i \(-0.863391\pi\)
0.416115 + 0.909312i \(0.363391\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.22474 0.707107i 0.191273 0.110432i −0.401305 0.915944i \(-0.631443\pi\)
0.592578 + 0.805513i \(0.298110\pi\)
\(42\) 0 0
\(43\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.5911 + 3.10583i −1.69074 + 0.453032i −0.970580 0.240779i \(-0.922597\pi\)
−0.720157 + 0.693811i \(0.755930\pi\)
\(48\) 0 0
\(49\) −0.866025 + 0.500000i −0.123718 + 0.0714286i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.07107 + 7.07107i −0.971286 + 0.971286i −0.999599 0.0283132i \(-0.990986\pi\)
0.0283132 + 0.999599i \(0.490986\pi\)
\(54\) 0 0
\(55\) −6.00000 + 2.00000i −0.809040 + 0.269680i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.41421 + 2.44949i 0.184115 + 0.318896i 0.943278 0.332004i \(-0.107725\pi\)
−0.759163 + 0.650901i \(0.774391\pi\)
\(60\) 0 0
\(61\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.22715 + 7.91688i 0.648348 + 0.981968i
\(66\) 0 0
\(67\) −10.9282 2.92820i −1.33509 0.357737i −0.480481 0.877005i \(-0.659538\pi\)
−0.854611 + 0.519268i \(0.826205\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.65685i 0.671345i −0.941979 0.335673i \(-0.891036\pi\)
0.941979 0.335673i \(-0.108964\pi\)
\(72\) 0 0
\(73\) 7.00000 + 7.00000i 0.819288 + 0.819288i 0.986005 0.166717i \(-0.0533166\pi\)
−0.166717 + 0.986005i \(0.553317\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.07055 + 7.72741i −0.235961 + 0.880620i
\(78\) 0 0
\(79\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.10583 + 11.5911i 0.340909 + 1.27229i 0.897319 + 0.441382i \(0.145512\pi\)
−0.556410 + 0.830908i \(0.687822\pi\)
\(84\) 0 0
\(85\) −4.46410 + 0.267949i −0.484200 + 0.0290632i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.41421 −0.149906 −0.0749532 0.997187i \(-0.523881\pi\)
−0.0749532 + 0.997187i \(0.523881\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.69213 + 5.93426i 0.686598 + 0.608842i
\(96\) 0 0
\(97\) 1.09808 + 4.09808i 0.111493 + 0.416097i 0.999001 0.0446959i \(-0.0142319\pi\)
−0.887508 + 0.460793i \(0.847565\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.22474 + 0.707107i 0.121867 + 0.0703598i 0.559694 0.828699i \(-0.310919\pi\)
−0.437828 + 0.899059i \(0.644252\pi\)
\(102\) 0 0
\(103\) −0.732051 + 2.73205i −0.0721311 + 0.269197i −0.992567 0.121695i \(-0.961167\pi\)
0.920436 + 0.390892i \(0.127834\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.82843 2.82843i −0.273434 0.273434i 0.557047 0.830481i \(-0.311934\pi\)
−0.830481 + 0.557047i \(0.811934\pi\)
\(108\) 0 0
\(109\) 8.00000i 0.766261i −0.923694 0.383131i \(-0.874846\pi\)
0.923694 0.383131i \(-0.125154\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.86370 + 1.03528i 0.363467 + 0.0973906i 0.435930 0.899980i \(-0.356419\pi\)
−0.0724636 + 0.997371i \(0.523086\pi\)
\(114\) 0 0
\(115\) −14.9282 + 9.85641i −1.39206 + 0.919115i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.82843 + 4.89898i −0.259281 + 0.449089i
\(120\) 0 0
\(121\) −1.50000 2.59808i −0.136364 0.236189i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.19239 + 6.36396i 0.822192 + 0.569210i
\(126\) 0 0
\(127\) −10.0000 + 10.0000i −0.887357 + 0.887357i −0.994268 0.106912i \(-0.965904\pi\)
0.106912 + 0.994268i \(0.465904\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.44949 + 1.41421i −0.214013 + 0.123560i −0.603175 0.797609i \(-0.706098\pi\)
0.389162 + 0.921169i \(0.372765\pi\)
\(132\) 0 0
\(133\) 10.9282 2.92820i 0.947595 0.253907i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.5911 3.10583i 0.990295 0.265349i 0.272921 0.962037i \(-0.412010\pi\)
0.717375 + 0.696688i \(0.245344\pi\)
\(138\) 0 0
\(139\) 13.8564 8.00000i 1.17529 0.678551i 0.220366 0.975417i \(-0.429275\pi\)
0.954919 + 0.296866i \(0.0959415\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.48528 8.48528i 0.709575 0.709575i
\(144\) 0 0
\(145\) −7.00000 21.0000i −0.581318 1.74396i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.36396 + 11.0227i 0.521356 + 0.903015i 0.999691 + 0.0248379i \(0.00790696\pi\)
−0.478335 + 0.878177i \(0.658760\pi\)
\(150\) 0 0
\(151\) 10.0000 17.3205i 0.813788 1.40952i −0.0964061 0.995342i \(-0.530735\pi\)
0.910195 0.414181i \(-0.135932\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −17.5254 3.58630i −1.40767 0.288059i
\(156\) 0 0
\(157\) 12.2942 + 3.29423i 0.981186 + 0.262908i 0.713544 0.700610i \(-0.247089\pi\)
0.267642 + 0.963518i \(0.413756\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 22.6274i 1.78329i
\(162\) 0 0
\(163\) −4.00000 4.00000i −0.313304 0.313304i 0.532884 0.846188i \(-0.321108\pi\)
−0.846188 + 0.532884i \(0.821108\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.14110 15.4548i 0.320448 1.19593i −0.598361 0.801227i \(-0.704181\pi\)
0.918809 0.394702i \(-0.129152\pi\)
\(168\) 0 0
\(169\) −4.33013 2.50000i −0.333087 0.192308i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.03528 + 3.86370i 0.0787106 + 0.293752i 0.994049 0.108933i \(-0.0347435\pi\)
−0.915338 + 0.402685i \(0.868077\pi\)
\(174\) 0 0
\(175\) 13.1244 5.26795i 0.992108 0.398220i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −14.1421 −1.05703 −0.528516 0.848923i \(-0.677252\pi\)
−0.528516 + 0.848923i \(0.677252\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.568406 9.46979i −0.0417900 0.696233i
\(186\) 0 0
\(187\) 1.46410 + 5.46410i 0.107066 + 0.399575i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.79796 5.65685i −0.708955 0.409316i 0.101719 0.994813i \(-0.467566\pi\)
−0.810674 + 0.585498i \(0.800899\pi\)
\(192\) 0 0
\(193\) −5.49038 + 20.4904i −0.395206 + 1.47493i 0.426222 + 0.904618i \(0.359844\pi\)
−0.821429 + 0.570311i \(0.806823\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.48528 8.48528i −0.604551 0.604551i 0.336966 0.941517i \(-0.390599\pi\)
−0.941517 + 0.336966i \(0.890599\pi\)
\(198\) 0 0
\(199\) 12.0000i 0.850657i 0.905039 + 0.425329i \(0.139842\pi\)
−0.905039 + 0.425329i \(0.860158\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −27.0459 7.24693i −1.89825 0.508635i
\(204\) 0 0
\(205\) −0.633975 + 3.09808i −0.0442787 + 0.216379i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.65685 9.79796i 0.391293 0.677739i
\(210\) 0 0
\(211\) −4.00000 6.92820i −0.275371 0.476957i 0.694857 0.719148i \(-0.255467\pi\)
−0.970229 + 0.242190i \(0.922134\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −16.0000 + 16.0000i −1.08615 + 1.08615i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 7.34847 4.24264i 0.494312 0.285391i
\(222\) 0 0
\(223\) 8.19615 2.19615i 0.548855 0.147065i 0.0262738 0.999655i \(-0.491636\pi\)
0.522581 + 0.852590i \(0.324969\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.72741 2.07055i 0.512886 0.137427i 0.00691198 0.999976i \(-0.497800\pi\)
0.505974 + 0.862549i \(0.331133\pi\)
\(228\) 0 0
\(229\) −5.19615 + 3.00000i −0.343371 + 0.198246i −0.661762 0.749714i \(-0.730191\pi\)
0.318390 + 0.947960i \(0.396858\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.24264 4.24264i 0.277945 0.277945i −0.554343 0.832288i \(-0.687031\pi\)
0.832288 + 0.554343i \(0.187031\pi\)
\(234\) 0 0
\(235\) 12.0000 24.0000i 0.782794 1.56559i
\(236\) 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
\(241\) −5.00000 + 8.66025i −0.322078 + 0.557856i −0.980917 0.194429i \(-0.937715\pi\)
0.658838 + 0.752285i \(0.271048\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.448288 2.19067i 0.0286401 0.139957i
\(246\) 0 0
\(247\) −16.3923 4.39230i −1.04302 0.279476i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 25.4558i 1.60676i −0.595468 0.803379i \(-0.703033\pi\)
0.595468 0.803379i \(-0.296967\pi\)
\(252\) 0 0
\(253\) 16.0000 + 16.0000i 1.00591 + 1.00591i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.17638 + 19.3185i −0.322894 + 1.20506i 0.593519 + 0.804820i \(0.297738\pi\)
−0.916412 + 0.400236i \(0.868928\pi\)
\(258\) 0 0
\(259\) −10.3923 6.00000i −0.645746 0.372822i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −4.14110 15.4548i −0.255351 0.952985i −0.967895 0.251356i \(-0.919123\pi\)
0.712543 0.701628i \(-0.247543\pi\)
\(264\) 0 0
\(265\) −1.33975 22.3205i −0.0822999 1.37114i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 18.3848 1.12094 0.560470 0.828175i \(-0.310621\pi\)
0.560470 + 0.828175i \(0.310621\pi\)
\(270\) 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
\(275\) 5.55532 13.0053i 0.334998 0.784251i
\(276\) 0 0
\(277\) 5.49038 + 20.4904i 0.329885 + 1.23115i 0.909309 + 0.416121i \(0.136611\pi\)
−0.579424 + 0.815026i \(0.696722\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 11.0227 + 6.36396i 0.657559 + 0.379642i 0.791346 0.611368i \(-0.209380\pi\)
−0.133787 + 0.991010i \(0.542714\pi\)
\(282\) 0 0
\(283\) −1.46410 + 5.46410i −0.0870318 + 0.324807i −0.995691 0.0927310i \(-0.970440\pi\)
0.908659 + 0.417538i \(0.137107\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.82843 + 2.82843i 0.166957 + 0.166957i
\(288\) 0 0
\(289\) 13.0000i 0.764706i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −11.5911 3.10583i −0.677160 0.181444i −0.0961820 0.995364i \(-0.530663\pi\)
−0.580978 + 0.813919i \(0.697330\pi\)
\(294\) 0 0
\(295\) −6.19615 1.26795i −0.360754 0.0738229i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 16.9706 29.3939i 0.981433 1.69989i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −12.0000 + 12.0000i −0.684876 + 0.684876i −0.961095 0.276219i \(-0.910919\pi\)
0.276219 + 0.961095i \(0.410919\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.89898 + 2.82843i −0.277796 + 0.160385i −0.632425 0.774622i \(-0.717940\pi\)
0.354629 + 0.935007i \(0.384607\pi\)
\(312\) 0 0
\(313\) −4.09808 + 1.09808i −0.231637 + 0.0620669i −0.372770 0.927924i \(-0.621592\pi\)
0.141133 + 0.989991i \(0.454925\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.79555 1.55291i 0.325511 0.0872204i −0.0923631 0.995725i \(-0.529442\pi\)
0.417874 + 0.908505i \(0.362775\pi\)
\(318\) 0 0
\(319\) −24.2487 + 14.0000i −1.35767 + 0.783850i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5.65685 5.65685i 0.314756 0.314756i
\(324\) 0 0
\(325\) −21.0000 3.00000i −1.16487 0.166410i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −16.9706 29.3939i −0.935617 1.62054i
\(330\) 0 0
\(331\) 10.0000 17.3205i 0.549650 0.952021i −0.448649 0.893708i \(-0.648095\pi\)
0.998298 0.0583130i \(-0.0185721\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 21.1117 13.9391i 1.15345 0.761572i
\(336\) 0 0
\(337\) 4.09808 + 1.09808i 0.223236 + 0.0598160i 0.368703 0.929547i \(-0.379802\pi\)
−0.145467 + 0.989363i \(0.546468\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 22.6274i 1.22534i
\(342\) 0 0
\(343\) 12.0000 + 12.0000i 0.647939 + 0.647939i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.07055 7.72741i 0.111153 0.414829i −0.887817 0.460196i \(-0.847779\pi\)
0.998970 + 0.0453672i \(0.0144458\pi\)
\(348\) 0 0
\(349\) −27.7128 16.0000i −1.48343 0.856460i −0.483610 0.875284i \(-0.660675\pi\)
−0.999823 + 0.0188232i \(0.994008\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4.65874 17.3867i −0.247960 0.925399i −0.971873 0.235507i \(-0.924325\pi\)
0.723913 0.689892i \(-0.242342\pi\)
\(354\) 0 0
\(355\) 9.46410 + 8.39230i 0.502302 + 0.445417i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11.3137 0.597115 0.298557 0.954392i \(-0.403495\pi\)
0.298557 + 0.954392i \(0.403495\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −22.0962 + 1.32628i −1.15657 + 0.0694207i
\(366\) 0 0
\(367\) 5.12436 + 19.1244i 0.267489 + 0.998283i 0.960709 + 0.277557i \(0.0895248\pi\)
−0.693220 + 0.720726i \(0.743809\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −24.4949 14.1421i −1.27171 0.734223i
\(372\) 0 0
\(373\) −1.83013 + 6.83013i −0.0947604 + 0.353651i −0.996983 0.0776200i \(-0.975268\pi\)
0.902223 + 0.431271i \(0.141935\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 29.6985 + 29.6985i 1.52955 + 1.52955i
\(378\) 0 0
\(379\) 16.0000i 0.821865i −0.911666 0.410932i \(-0.865203\pi\)
0.911666 0.410932i \(-0.134797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 34.7733 + 9.31749i 1.77683 + 0.476101i 0.990000 0.141064i \(-0.0450523\pi\)
0.786834 + 0.617165i \(0.211719\pi\)
\(384\) 0 0
\(385\) −9.85641 14.9282i −0.502329 0.760812i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 10.6066 18.3712i 0.537776 0.931455i −0.461247 0.887272i \(-0.652598\pi\)
0.999023 0.0441839i \(-0.0140687\pi\)
\(390\) 0 0
\(391\) 8.00000 + 13.8564i 0.404577 + 0.700749i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −17.0000 + 17.0000i −0.853206 + 0.853206i −0.990527 0.137321i \(-0.956151\pi\)
0.137321 + 0.990527i \(0.456151\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.3712 + 10.6066i −0.917413 + 0.529668i −0.882809 0.469733i \(-0.844350\pi\)
−0.0346039 + 0.999401i \(0.511017\pi\)
\(402\) 0 0
\(403\) 32.7846 8.78461i 1.63312 0.437593i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −11.5911 + 3.10583i −0.574550 + 0.153950i
\(408\) 0 0
\(409\) 20.7846 12.0000i 1.02773 0.593362i 0.111398 0.993776i \(-0.464467\pi\)
0.916334 + 0.400414i \(0.131134\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −5.65685 + 5.65685i −0.278356 + 0.278356i
\(414\) 0 0
\(415\) −24.0000 12.0000i −1.17811 0.589057i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.7279 22.0454i −0.621800 1.07699i −0.989150 0.146906i \(-0.953068\pi\)
0.367351 0.930082i \(-0.380265\pi\)
\(420\) 0 0
\(421\) 11.0000 19.0526i 0.536107 0.928565i −0.463002 0.886357i \(-0.653228\pi\)
0.999109 0.0422075i \(-0.0134391\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.17449 7.86611i 0.299507 0.381562i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 28.2843i 1.36241i −0.732095 0.681203i \(-0.761457\pi\)
0.732095 0.681203i \(-0.238543\pi\)
\(432\) 0 0
\(433\) 23.0000 + 23.0000i 1.10531 + 1.10531i 0.993759 + 0.111551i \(0.0355818\pi\)
0.111551 + 0.993759i \(0.464418\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.28221 30.9096i 0.396192 1.47861i
\(438\) 0 0
\(439\) 17.3205 + 10.0000i 0.826663 + 0.477274i 0.852709 0.522387i \(-0.174958\pi\)
−0.0260459 + 0.999661i \(0.508292\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6.21166 23.1822i −0.295125 1.10142i −0.941118 0.338078i \(-0.890223\pi\)
0.645994 0.763343i \(-0.276443\pi\)
\(444\) 0 0
\(445\) 2.09808 2.36603i 0.0994584 0.112160i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.41421 0.0667409 0.0333704 0.999443i \(-0.489376\pi\)
0.0333704 + 0.999443i \(0.489376\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −17.8028 + 20.0764i −0.834607 + 0.941196i
\(456\) 0 0
\(457\) 5.49038 + 20.4904i 0.256829 + 0.958500i 0.967064 + 0.254534i \(0.0819222\pi\)
−0.710235 + 0.703965i \(0.751411\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 13.4722 + 7.77817i 0.627463 + 0.362266i 0.779769 0.626068i \(-0.215337\pi\)
−0.152306 + 0.988333i \(0.548670\pi\)
\(462\) 0 0
\(463\) 9.51666 35.5167i 0.442277 1.65060i −0.280751 0.959781i \(-0.590584\pi\)
0.723028 0.690819i \(-0.242750\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −19.7990 19.7990i −0.916188 0.916188i 0.0805616 0.996750i \(-0.474329\pi\)
−0.996750 + 0.0805616i \(0.974329\pi\)
\(468\) 0 0
\(469\) 32.0000i 1.47762i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −19.8564 + 2.39230i −0.911074 + 0.109766i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −19.7990 + 34.2929i −0.904639 + 1.56688i −0.0832378 + 0.996530i \(0.526526\pi\)
−0.821401 + 0.570351i \(0.806807\pi\)
\(480\) 0 0
\(481\) 9.00000 + 15.5885i 0.410365 + 0.710772i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.48528 4.24264i −0.385297 0.192648i
\(486\) 0 0
\(487\) 6.00000 6.00000i 0.271886 0.271886i −0.557973 0.829859i \(-0.688421\pi\)
0.829859 + 0.557973i \(0.188421\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 31.8434 18.3848i 1.43707 0.829693i 0.439426 0.898279i \(-0.355182\pi\)
0.997645 + 0.0685856i \(0.0218486\pi\)
\(492\) 0 0
\(493\) −19.1244 + 5.12436i −0.861318 + 0.230789i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 15.4548 4.14110i 0.693243 0.185754i
\(498\) 0 0
\(499\) 3.46410 2.00000i 0.155074 0.0895323i −0.420455 0.907314i \(-0.638129\pi\)
0.575529 + 0.817781i \(0.304796\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −8.48528 + 8.48528i −0.378340 + 0.378340i −0.870503 0.492163i \(-0.836206\pi\)
0.492163 + 0.870503i \(0.336206\pi\)
\(504\) 0 0
\(505\) −3.00000 + 1.00000i −0.133498 + 0.0444994i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.707107 1.22474i −0.0313420 0.0542859i 0.849929 0.526897i \(-0.176645\pi\)
−0.881271 + 0.472611i \(0.843311\pi\)
\(510\) 0 0
\(511\) −14.0000 + 24.2487i −0.619324 + 1.07270i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3.48477 5.27792i −0.153557 0.232573i
\(516\) 0 0
\(517\) −32.7846 8.78461i −1.44187 0.386347i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 29.6985i 1.30111i −0.759457 0.650557i \(-0.774535\pi\)
0.759457 0.650557i \(-0.225465\pi\)
\(522\) 0 0
\(523\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.14110 + 15.4548i −0.180389 + 0.673222i
\(528\) 0 0
\(529\) 35.5070 + 20.5000i 1.54378 + 0.891304i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.55291 5.79555i −0.0672642 0.251033i
\(534\) 0 0
\(535\) 8.92820 0.535898i 0.386000 0.0231689i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.82843 −0.121829
\(540\) 0 0
\(541\) 40.0000 1.71973 0.859867 0.510518i \(-0.170546\pi\)
0.859867 + 0.510518i \(0.170546\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 13.3843 + 11.8685i 0.573319 + 0.508391i
\(546\) 0 0
\(547\) −4.39230 16.3923i −0.187801 0.700884i −0.994013 0.109258i \(-0.965153\pi\)
0.806212 0.591627i \(-0.201514\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 34.2929 + 19.7990i 1.46092 + 0.843465i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 12.7279 + 12.7279i 0.539299 + 0.539299i 0.923323 0.384024i \(-0.125462\pi\)
−0.384024 + 0.923323i \(0.625462\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −23.1822 6.21166i −0.977014 0.261790i −0.265227 0.964186i \(-0.585447\pi\)
−0.711787 + 0.702396i \(0.752114\pi\)
\(564\) 0 0
\(565\) −7.46410 + 4.92820i −0.314017 + 0.207331i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −7.77817 + 13.4722i −0.326078 + 0.564784i −0.981730 0.190280i \(-0.939061\pi\)
0.655652 + 0.755063i \(0.272394\pi\)
\(570\) 0 0
\(571\) 2.00000 + 3.46410i 0.0836974 + 0.144968i 0.904835 0.425762i \(-0.139994\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 5.65685 39.5980i 0.235907 1.65135i
\(576\) 0 0
\(577\) 7.00000 7.00000i 0.291414 0.291414i −0.546225 0.837639i \(-0.683936\pi\)
0.837639 + 0.546225i \(0.183936\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −29.3939 + 16.9706i −1.21946 + 0.704058i
\(582\) 0 0
\(583\) −27.3205 + 7.32051i −1.13150 + 0.303184i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −11.5911 + 3.10583i −0.478416 + 0.128191i −0.489965 0.871742i \(-0.662990\pi\)
0.0115488 + 0.999933i \(0.496324\pi\)
\(588\) 0 0
\(589\) 27.7128 16.0000i 1.14189 0.659269i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −14.1421 + 14.1421i −0.580748 + 0.580748i −0.935109 0.354361i \(-0.884698\pi\)
0.354361 + 0.935109i \(0.384698\pi\)
\(594\) 0 0
\(595\) −4.00000 12.0000i −0.163984 0.491952i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 14.1421 + 24.4949i 0.577832 + 1.00083i 0.995728 + 0.0923393i \(0.0294344\pi\)
−0.417896 + 0.908495i \(0.637232\pi\)
\(600\) 0 0
\(601\) 20.0000 34.6410i 0.815817 1.41304i −0.0929227 0.995673i \(-0.529621\pi\)
0.908740 0.417363i \(-0.137046\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 6.57201 + 1.34486i 0.267190 + 0.0546765i
\(606\) 0 0
\(607\) −24.5885 6.58846i −0.998015 0.267417i −0.277401 0.960754i \(-0.589473\pi\)
−0.720614 + 0.693337i \(0.756140\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 50.9117i 2.05967i
\(612\) 0 0
\(613\) −27.0000 27.0000i −1.09052 1.09052i −0.995473 0.0950469i \(-0.969700\pi\)
−0.0950469 0.995473i \(-0.530300\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.55291 5.79555i 0.0625180 0.233320i −0.927596 0.373585i \(-0.878129\pi\)
0.990114 + 0.140264i \(0.0447952\pi\)
\(618\) 0 0
\(619\) −20.7846 12.0000i −0.835404 0.482321i 0.0202954 0.999794i \(-0.493539\pi\)
−0.855699 + 0.517473i \(0.826873\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.03528 3.86370i −0.0414775 0.154796i
\(624\) 0 0
\(625\) −24.2846 + 5.93782i −0.971384 + 0.237513i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −8.48528 −0.338330
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.89469 31.5660i −0.0751884 1.25266i
\(636\) 0 0
\(637\) 1.09808 + 4.09808i 0.0435074 + 0.162372i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −35.5176 20.5061i −1.40286 0.809942i −0.408176 0.912903i \(-0.633835\pi\)
−0.994685 + 0.102961i \(0.967168\pi\)
\(642\) 0 0
\(643\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.82843 + 2.82843i 0.111197 + 0.111197i 0.760516 0.649319i \(-0.224946\pi\)
−0.649319 + 0.760516i \(0.724946\pi\)
\(648\) 0 0
\(649\) 8.00000i 0.314027i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.79555 + 1.55291i 0.226798 + 0.0607702i 0.370428 0.928861i \(-0.379211\pi\)
−0.143630 + 0.989631i \(0.545878\pi\)
\(654\) 0 0
\(655\) 1.26795 6.19615i 0.0495429 0.242104i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.41421 + 2.44949i −0.0550899 + 0.0954186i −0.892255 0.451531i \(-0.850878\pi\)
0.837165 + 0.546950i \(0.184211\pi\)
\(660\) 0 0
\(661\) 8.00000 + 13.8564i 0.311164 + 0.538952i 0.978615 0.205702i \(-0.0659478\pi\)
−0.667451 + 0.744654i \(0.732615\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −11.3137 + 22.6274i −0.438727 + 0.877454i
\(666\) 0 0
\(667\) −56.0000 + 56.0000i −2.16833 + 2.16833i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 31.4186 8.41858i 1.21110 0.324513i 0.403905 0.914801i \(-0.367653\pi\)
0.807192 + 0.590288i \(0.200986\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −13.5230 + 3.62347i −0.519730 + 0.139261i −0.509142 0.860683i \(-0.670037\pi\)
−0.0105881 + 0.999944i \(0.503370\pi\)
\(678\) 0 0
\(679\) −10.3923 + 6.00000i −0.398820 + 0.230259i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 8.48528 8.48528i 0.324680 0.324680i −0.525879 0.850559i \(-0.676264\pi\)
0.850559 + 0.525879i \(0.176264\pi\)
\(684\) 0 0
\(685\) −12.0000 + 24.0000i −0.458496 + 0.916993i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 21.2132 + 36.7423i 0.808159 + 1.39977i
\(690\) 0 0
\(691\) −6.00000 + 10.3923i −0.228251 + 0.395342i −0.957290 0.289130i \(-0.906634\pi\)
0.729039 + 0.684472i \(0.239967\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7.17260 + 35.0507i −0.272072 + 1.32955i
\(696\) 0 0
\(697\) 2.73205 + 0.732051i 0.103484 + 0.0277284i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9.89949i 0.373899i 0.982370 + 0.186949i \(0.0598600\pi\)
−0.982370 + 0.186949i \(0.940140\pi\)
\(702\) 0 0
\(703\) 12.0000 + 12.0000i 0.452589 + 0.452589i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.03528 + 3.86370i −0.0389356 + 0.145310i
\(708\) 0 0
\(709\) 8.66025 + 5.00000i 0.325243 + 0.187779i 0.653727 0.756730i \(-0.273204\pi\)
−0.328484 + 0.944509i \(0.606538\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 16.5644 + 61.8193i 0.620342 + 2.31515i
\(714\) 0 0
\(715\) 1.60770 + 26.7846i 0.0601244 + 1.00169i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 39.5980 1.47676 0.738378 0.674387i \(-0.235592\pi\)
0.738378 + 0.674387i \(0.235592\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 45.5186 + 19.4436i 1.69052 + 0.722118i
\(726\) 0 0
\(727\) −2.19615 8.19615i −0.0814508 0.303978i 0.913168 0.407584i \(-0.133629\pi\)
−0.994618 + 0.103606i \(0.966962\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −10.6147 + 39.6147i −0.392064 + 1.46320i 0.434659 + 0.900595i \(0.356869\pi\)
−0.826723 + 0.562609i \(0.809798\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −22.6274 22.6274i −0.833492 0.833492i
\(738\) 0 0
\(739\) 28.0000i 1.03000i 0.857191 + 0.514998i \(0.172207\pi\)
−0.857191 + 0.514998i \(0.827793\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 27.0459 + 7.24693i 0.992219 + 0.265864i 0.718182 0.695855i \(-0.244974\pi\)
0.274036 + 0.961719i \(0.411641\pi\)
\(744\) 0 0
\(745\) −27.8827 5.70577i −1.02154 0.209043i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 5.65685 9.79796i 0.206697 0.358010i
\(750\) 0 0
\(751\) 12.0000 + 20.7846i 0.437886 + 0.758441i 0.997526 0.0702946i \(-0.0223939\pi\)
−0.559640 + 0.828736i \(0.689061\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 14.1421 + 42.4264i 0.514685 + 1.54406i
\(756\) 0 0
\(757\) 9.00000 9.00000i 0.327111 0.327111i −0.524376 0.851487i \(-0.675701\pi\)
0.851487 + 0.524376i \(0.175701\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 23.2702 13.4350i 0.843542 0.487019i −0.0149244 0.999889i \(-0.504751\pi\)
0.858467 + 0.512869i \(0.171417\pi\)
\(762\) 0 0
\(763\) 21.8564 5.85641i 0.791255 0.212016i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 11.5911 3.10583i 0.418531 0.112145i
\(768\) 0 0
\(769\) −6.92820 + 4.00000i −0.249837 + 0.144244i −0.619690 0.784847i \(-0.712742\pi\)
0.369852 + 0.929091i \(0.379408\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 12.7279 12.7279i 0.457792 0.457792i −0.440138 0.897930i \(-0.645071\pi\)
0.897930 + 0.440138i \(0.145071\pi\)
\(774\) 0 0
\(775\) 32.0000 24.0000i 1.14947 0.862105i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.82843 4.89898i −0.101339 0.175524i
\(780\) 0 0
\(781\) 8.00000 13.8564i 0.286263 0.495821i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −23.7506 + 15.6814i −0.847696 + 0.559695i
\(786\) 0 0
\(787\) 27.3205 + 7.32051i 0.973871 + 0.260948i 0.710461 0.703736i \(-0.248486\pi\)
0.263410 + 0.964684i \(0.415153\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 11.3137i 0.402269i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.10583 + 11.5911i −0.110014 + 0.410578i −0.998866 0.0476171i \(-0.984837\pi\)
0.888852 + 0.458195i \(0.151504\pi\)
\(798\) 0 0
\(799\) −20.7846 12.0000i −0.735307 0.424529i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7.24693 + 27.0459i 0.255739 + 0.954430i
\(804\) 0 0
\(805\) −37.8564 33.5692i −1.33426 1.18316i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −43.8406 −1.54135 −0.770677 0.637226i \(-0.780082\pi\)
−0.770677 + 0.637226i \(0.780082\pi\)
\(810\) 0 0
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 12.6264 0.757875i 0.442283 0.0265472i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 13.4722 + 7.77817i 0.470183 + 0.271460i 0.716316 0.697776i \(-0.245827\pi\)
−0.246133 + 0.969236i \(0.579160\pi\)
\(822\) 0 0
\(823\) 9.51666 35.5167i 0.331730 1.23803i −0.575641 0.817702i \(-0.695247\pi\)
0.907371 0.420331i \(-0.138086\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −33.9411 33.9411i −1.18025 1.18025i −0.979680 0.200569i \(-0.935721\pi\)
−0.200569 0.979680i \(-0.564279\pi\)
\(828\) 0 0
\(829\) 8.00000i 0.277851i −0.990303 0.138926i \(-0.955635\pi\)
0.990303 0.138926i \(-0.0443649\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.93185 0.517638i −0.0669347 0.0179351i
\(834\) 0 0
\(835\) 19.7128 + 29.8564i 0.682190 + 1.03322i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −5.65685 + 9.79796i −0.195296 + 0.338263i −0.946998 0.321241i \(-0.895900\pi\)
0.751701 + 0.659504i \(0.229234\pi\)
\(840\) 0 0
\(841\) −34.5000 59.7558i −1.18966 2.06054i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 10.6066 3.53553i 0.364878 0.121626i
\(846\) 0 0
\(847\) 6.00000 6.00000i 0.206162 0.206162i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −29.3939 + 16.9706i −1.00761 + 0.581743i
\(852\) 0 0
\(853\) 6.83013 1.83013i 0.233859 0.0626624i −0.139986 0.990153i \(-0.544706\pi\)
0.373845 + 0.927491i \(0.378039\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −40.5689 + 10.8704i −1.38581 + 0.371326i −0.873227 0.487314i \(-0.837977\pi\)
−0.512580 + 0.858640i \(0.671310\pi\)
\(858\) 0 0
\(859\) −6.92820 + 4.00000i −0.236387 + 0.136478i −0.613515 0.789683i \(-0.710245\pi\)
0.377128 + 0.926161i \(0.376912\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −5.65685 + 5.65685i −0.192562 + 0.192562i −0.796802 0.604240i \(-0.793477\pi\)
0.604240 + 0.796802i \(0.293477\pi\)
\(864\) 0 0
\(865\) −8.00000 4.00000i −0.272008 0.136004i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −24.0000 + 41.5692i −0.813209 + 1.40852i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −10.6574 + 29.7728i −0.360285 + 1.00650i
\(876\) 0 0
\(877\) 20.4904 + 5.49038i 0.691911 + 0.185397i 0.587605 0.809148i \(-0.300071\pi\)
0.104306 + 0.994545i \(0.466738\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 41.0122i 1.38174i −0.722981 0.690868i \(-0.757229\pi\)
0.722981 0.690868i \(-0.242771\pi\)
\(882\) 0 0
\(883\) −20.0000 20.0000i −0.673054 0.673054i 0.285365 0.958419i \(-0.407885\pi\)
−0.958419 + 0.285365i \(0.907885\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −12.4233 + 46.3644i −0.417134 + 1.55677i 0.363388 + 0.931638i \(0.381620\pi\)
−0.780522 + 0.625128i \(0.785047\pi\)
\(888\) 0 0
\(889\) −34.6410 20.0000i −1.16182 0.670778i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 12.4233 + 46.3644i 0.415730 + 1.55153i
\(894\) 0 0
\(895\) 20.9808 23.6603i 0.701310 0.790875i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −79.1960 −2.64133
\(900\) 0 0
\(901\) −20.0000 −0.666297
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −11.8685 + 13.3843i −0.394523 + 0.444908i
\(906\) 0 0
\(907\) 5.85641 + 21.8564i 0.194459 + 0.725730i 0.992406 + 0.123003i \(0.0392525\pi\)
−0.797948 + 0.602727i \(0.794081\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −24.4949 14.1421i −0.811552 0.468550i 0.0359424 0.999354i \(-0.488557\pi\)
−0.847495 + 0.530804i \(0.821890\pi\)
\(912\) 0 0
\(913\) −8.78461 + 32.7846i −0.290728 + 1.08501i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −5.65685 5.65685i −0.186806 0.186806i
\(918\) 0 0
\(919\) 32.0000i 1.05558i −0.849374 0.527791i \(-0.823020\pi\)
0.849374 0.527791i \(-0.176980\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −23.1822 6.21166i −0.763052 0.204459i
\(924\) 0 0
\(925\) 16.6865 + 13.0981i 0.548650 + 0.430662i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −17.6777 + 30.6186i −0.579986 + 1.00456i 0.415495 + 0.909596i \(0.363608\pi\)
−0.995480 + 0.0949688i \(0.969725\pi\)
\(930\) 0 0
\(931\) 2.00000 + 3.46410i 0.0655474 + 0.113531i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −11.3137 5.65685i −0.369998 0.184999i
\(936\) 0 0
\(937\) 37.0000 37.0000i 1.20874 1.20874i 0.237301 0.971436i \(-0.423737\pi\)
0.971436 0.237301i \(-0.0762628\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 33.0681 19.0919i 1.07799 0.622378i 0.147636 0.989042i \(-0.452834\pi\)
0.930353 + 0.366664i \(0.119500\pi\)
\(942\) 0 0
\(943\) 10.9282 2.92820i 0.355871 0.0953554i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 30.9096 8.28221i 1.00443 0.269136i 0.281128 0.959670i \(-0.409291\pi\)
0.723299 + 0.690535i \(0.242625\pi\)
\(948\) 0 0
\(949\) 36.3731 21.0000i 1.18072 0.681689i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 25.4558 25.4558i 0.824596 0.824596i −0.162168 0.986763i \(-0.551849\pi\)
0.986763 + 0.162168i \(0.0518485\pi\)
\(954\) 0 0
\(955\) 24.0000 8.00000i 0.776622 0.258874i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 16.9706 + 29.3939i 0.548008 + 0.949178i
\(960\) 0 0
\(961\) −16.5000 + 28.5788i −0.532258 + 0.921898i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −26.1357 39.5844i −0.841339 1.27427i
\(966\) 0 0
\(967\) 2.73205 + 0.732051i 0.0878568 + 0.0235412i 0.302480 0.953156i \(-0.402186\pi\)
−0.214623 + 0.976697i \(0.568852\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 53.7401i 1.72460i 0.506396 + 0.862301i \(0.330978\pi\)
−0.506396 + 0.862301i \(0.669022\pi\)
\(972\) 0 0
\(973\) 32.0000 + 32.0000i 1.02587 + 1.02587i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4.65874 17.3867i 0.149046 0.556249i −0.850495 0.525982i \(-0.823698\pi\)
0.999542 0.0302663i \(-0.00963555\pi\)
\(978\) 0 0
\(979\) −3.46410 2.00000i −0.110713 0.0639203i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 5.17638 + 19.3185i 0.165101 + 0.616165i 0.998027 + 0.0627812i \(0.0199970\pi\)
−0.832926 + 0.553384i \(0.813336\pi\)
\(984\) 0 0
\(985\) 26.7846 1.60770i 0.853429 0.0512254i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 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
\(995\) −20.0764 17.8028i −0.636464 0.564386i
\(996\) 0 0
\(997\) −4.02628 15.0263i −0.127514 0.475887i 0.872403 0.488787i \(-0.162560\pi\)
−0.999917 + 0.0128998i \(0.995894\pi\)
\(998\) 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.a.593.1 8
3.2 odd 2 inner 1620.2.x.a.593.2 8
5.2 odd 4 inner 1620.2.x.a.917.1 8
9.2 odd 6 180.2.j.a.53.2 yes 4
9.4 even 3 inner 1620.2.x.a.53.2 8
9.5 odd 6 inner 1620.2.x.a.53.1 8
9.7 even 3 180.2.j.a.53.1 yes 4
15.2 even 4 inner 1620.2.x.a.917.2 8
36.7 odd 6 720.2.w.b.593.1 4
36.11 even 6 720.2.w.b.593.2 4
45.2 even 12 180.2.j.a.17.1 4
45.7 odd 12 180.2.j.a.17.2 yes 4
45.22 odd 12 inner 1620.2.x.a.377.2 8
45.29 odd 6 900.2.j.a.593.2 4
45.32 even 12 inner 1620.2.x.a.377.1 8
45.34 even 6 900.2.j.a.593.1 4
45.38 even 12 900.2.j.a.557.2 4
45.43 odd 12 900.2.j.a.557.1 4
72.11 even 6 2880.2.w.a.2753.1 4
72.29 odd 6 2880.2.w.j.2753.1 4
72.43 odd 6 2880.2.w.a.2753.2 4
72.61 even 6 2880.2.w.j.2753.2 4
180.7 even 12 720.2.w.b.17.2 4
180.43 even 12 3600.2.w.f.1457.2 4
180.47 odd 12 720.2.w.b.17.1 4
180.79 odd 6 3600.2.w.f.593.2 4
180.83 odd 12 3600.2.w.f.1457.1 4
180.119 even 6 3600.2.w.f.593.1 4
360.187 even 12 2880.2.w.a.2177.1 4
360.227 odd 12 2880.2.w.a.2177.2 4
360.277 odd 12 2880.2.w.j.2177.1 4
360.317 even 12 2880.2.w.j.2177.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.2.j.a.17.1 4 45.2 even 12
180.2.j.a.17.2 yes 4 45.7 odd 12
180.2.j.a.53.1 yes 4 9.7 even 3
180.2.j.a.53.2 yes 4 9.2 odd 6
720.2.w.b.17.1 4 180.47 odd 12
720.2.w.b.17.2 4 180.7 even 12
720.2.w.b.593.1 4 36.7 odd 6
720.2.w.b.593.2 4 36.11 even 6
900.2.j.a.557.1 4 45.43 odd 12
900.2.j.a.557.2 4 45.38 even 12
900.2.j.a.593.1 4 45.34 even 6
900.2.j.a.593.2 4 45.29 odd 6
1620.2.x.a.53.1 8 9.5 odd 6 inner
1620.2.x.a.53.2 8 9.4 even 3 inner
1620.2.x.a.377.1 8 45.32 even 12 inner
1620.2.x.a.377.2 8 45.22 odd 12 inner
1620.2.x.a.593.1 8 1.1 even 1 trivial
1620.2.x.a.593.2 8 3.2 odd 2 inner
1620.2.x.a.917.1 8 5.2 odd 4 inner
1620.2.x.a.917.2 8 15.2 even 4 inner
2880.2.w.a.2177.1 4 360.187 even 12
2880.2.w.a.2177.2 4 360.227 odd 12
2880.2.w.a.2753.1 4 72.11 even 6
2880.2.w.a.2753.2 4 72.43 odd 6
2880.2.w.j.2177.1 4 360.277 odd 12
2880.2.w.j.2177.2 4 360.317 even 12
2880.2.w.j.2753.1 4 72.29 odd 6
2880.2.w.j.2753.2 4 72.61 even 6
3600.2.w.f.593.1 4 180.119 even 6
3600.2.w.f.593.2 4 180.79 odd 6
3600.2.w.f.1457.1 4 180.83 odd 12
3600.2.w.f.1457.2 4 180.43 even 12