Properties

Label 1620.2.x.a.917.2
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: \(\Q(\zeta_{24})\)
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 917.2
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1620.917
Dual form 1620.2.x.a.53.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.19067 - 0.448288i) q^{5} +(-2.73205 + 0.732051i) q^{7} +O(q^{10})\) \(q+(2.19067 - 0.448288i) q^{5} +(-2.73205 + 0.732051i) q^{7} +(-2.44949 - 1.41421i) q^{11} +(-4.09808 - 1.09808i) q^{13} +(1.41421 - 1.41421i) q^{17} +4.00000i q^{19} +(-2.07055 + 7.72741i) q^{23} +(4.59808 - 1.96410i) q^{25} +(-4.94975 + 8.57321i) q^{29} +(4.00000 + 6.92820i) q^{31} +(-5.65685 + 2.82843i) q^{35} +(-3.00000 - 3.00000i) q^{37} +(-1.22474 + 0.707107i) q^{41} +(3.10583 + 11.5911i) q^{47} +(0.866025 - 0.500000i) q^{49} +(-7.07107 - 7.07107i) q^{53} +(-6.00000 - 2.00000i) q^{55} +(1.41421 + 2.44949i) q^{59} +(-9.46979 - 0.568406i) q^{65} +(2.92820 - 10.9282i) q^{67} +5.65685i q^{71} +(7.00000 - 7.00000i) q^{73} +(7.72741 + 2.07055i) q^{77} +(-11.5911 + 3.10583i) q^{83} +(2.46410 - 3.73205i) q^{85} -1.41421 q^{89} +12.0000 q^{91} +(1.79315 + 8.76268i) q^{95} +(-4.09808 + 1.09808i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

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

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
\(5\) 2.19067 0.448288i 0.979698 0.200480i
\(6\) 0 0
\(7\) −2.73205 + 0.732051i −1.03262 + 0.276689i −0.735051 0.678012i \(-0.762842\pi\)
−0.297567 + 0.954701i \(0.596175\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.44949 1.41421i −0.738549 0.426401i 0.0829925 0.996550i \(-0.473552\pi\)
−0.821541 + 0.570149i \(0.806886\pi\)
\(12\) 0 0
\(13\) −4.09808 1.09808i −1.13660 0.304552i −0.359018 0.933331i \(-0.616888\pi\)
−0.777584 + 0.628779i \(0.783555\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.41421 1.41421i 0.342997 0.342997i −0.514496 0.857493i \(-0.672021\pi\)
0.857493 + 0.514496i \(0.172021\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.07055 + 7.72741i −0.431740 + 1.61128i 0.317009 + 0.948422i \(0.397321\pi\)
−0.748749 + 0.662853i \(0.769345\pi\)
\(24\) 0 0
\(25\) 4.59808 1.96410i 0.919615 0.392820i
\(26\) 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 0.416115 0.909312i \(-0.363391\pi\)
−0.909312 + 0.416115i \(0.863391\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.22474 + 0.707107i −0.191273 + 0.110432i −0.592578 0.805513i \(-0.701890\pi\)
0.401305 + 0.915944i \(0.368557\pi\)
\(42\) 0 0
\(43\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.10583 + 11.5911i 0.453032 + 1.69074i 0.693811 + 0.720157i \(0.255930\pi\)
−0.240779 + 0.970580i \(0.577403\pi\)
\(48\) 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 0.0283132 0.999599i \(-0.490986\pi\)
−0.999599 + 0.0283132i \(0.990986\pi\)
\(54\) 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
\(65\) −9.46979 0.568406i −1.17458 0.0705021i
\(66\) 0 0
\(67\) 2.92820 10.9282i 0.357737 1.33509i −0.519268 0.854611i \(-0.673795\pi\)
0.877005 0.480481i \(-0.159538\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.65685i 0.671345i 0.941979 + 0.335673i \(0.108964\pi\)
−0.941979 + 0.335673i \(0.891036\pi\)
\(72\) 0 0
\(73\) 7.00000 7.00000i 0.819288 0.819288i −0.166717 0.986005i \(-0.553317\pi\)
0.986005 + 0.166717i \(0.0533166\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.72741 + 2.07055i 0.880620 + 0.235961i
\(78\) 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
\(83\) −11.5911 + 3.10583i −1.27229 + 0.340909i −0.830908 0.556410i \(-0.812178\pi\)
−0.441382 + 0.897319i \(0.645512\pi\)
\(84\) 0 0
\(85\) 2.46410 3.73205i 0.267269 0.404798i
\(86\) 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
\(95\) 1.79315 + 8.76268i 0.183973 + 0.899032i
\(96\) 0 0
\(97\) −4.09808 + 1.09808i −0.416097 + 0.111493i −0.460793 0.887508i \(-0.652435\pi\)
0.0446959 + 0.999001i \(0.485768\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.22474 0.707107i −0.121867 0.0703598i 0.437828 0.899059i \(-0.355748\pi\)
−0.559694 + 0.828699i \(0.689081\pi\)
\(102\) 0 0
\(103\) 2.73205 + 0.732051i 0.269197 + 0.0721311i 0.390892 0.920436i \(-0.372166\pi\)
−0.121695 + 0.992567i \(0.538833\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.82843 + 2.82843i −0.273434 + 0.273434i −0.830481 0.557047i \(-0.811934\pi\)
0.557047 + 0.830481i \(0.311934\pi\)
\(108\) 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
\(113\) −1.03528 + 3.86370i −0.0973906 + 0.363467i −0.997371 0.0724636i \(-0.976914\pi\)
0.899980 + 0.435930i \(0.143581\pi\)
\(114\) 0 0
\(115\) −1.07180 + 17.8564i −0.0999456 + 1.66512i
\(116\) 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 0.106912 0.994268i \(-0.465904\pi\)
−0.994268 + 0.106912i \(0.965904\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.44949 1.41421i 0.214013 0.123560i −0.389162 0.921169i \(-0.627235\pi\)
0.603175 + 0.797609i \(0.293902\pi\)
\(132\) 0 0
\(133\) −2.92820 10.9282i −0.253907 0.947595i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −3.10583 11.5911i −0.265349 0.990295i −0.962037 0.272921i \(-0.912010\pi\)
0.696688 0.717375i \(-0.254656\pi\)
\(138\) 0 0
\(139\) −13.8564 + 8.00000i −1.17529 + 0.678551i −0.954919 0.296866i \(-0.904058\pi\)
−0.220366 + 0.975417i \(0.570725\pi\)
\(140\) 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
\(155\) 11.8685 + 13.3843i 0.953302 + 1.07505i
\(156\) 0 0
\(157\) −3.29423 + 12.2942i −0.262908 + 0.981186i 0.700610 + 0.713544i \(0.252911\pi\)
−0.963518 + 0.267642i \(0.913756\pi\)
\(158\) 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 −0.846188 0.532884i \(-0.821108\pi\)
0.532884 + 0.846188i \(0.321108\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.4548 4.14110i −1.19593 0.320448i −0.394702 0.918809i \(-0.629152\pi\)
−0.801227 + 0.598361i \(0.795819\pi\)
\(168\) 0 0
\(169\) 4.33013 + 2.50000i 0.333087 + 0.192308i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.86370 + 1.03528i −0.293752 + 0.0787106i −0.402685 0.915338i \(-0.631923\pi\)
0.108933 + 0.994049i \(0.465256\pi\)
\(174\) 0 0
\(175\) −11.1244 + 8.73205i −0.840922 + 0.660081i
\(176\) 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
\(185\) −7.91688 5.22715i −0.582060 0.384308i
\(186\) 0 0
\(187\) −5.46410 + 1.46410i −0.399575 + 0.107066i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 9.79796 + 5.65685i 0.708955 + 0.409316i 0.810674 0.585498i \(-0.199101\pi\)
−0.101719 + 0.994813i \(0.532434\pi\)
\(192\) 0 0
\(193\) 20.4904 + 5.49038i 1.47493 + 0.395206i 0.904618 0.426222i \(-0.140156\pi\)
0.570311 + 0.821429i \(0.306823\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.48528 + 8.48528i −0.604551 + 0.604551i −0.941517 0.336966i \(-0.890599\pi\)
0.336966 + 0.941517i \(0.390599\pi\)
\(198\) 0 0
\(199\) 12.0000i 0.850657i −0.905039 0.425329i \(-0.860158\pi\)
0.905039 0.425329i \(-0.139842\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 7.24693 27.0459i 0.508635 1.89825i
\(204\) 0 0
\(205\) −2.36603 + 2.09808i −0.165250 + 0.146536i
\(206\) 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
\(223\) −2.19615 8.19615i −0.147065 0.548855i −0.999655 0.0262738i \(-0.991636\pi\)
0.852590 0.522581i \(-0.175031\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.07055 7.72741i −0.137427 0.512886i −0.999976 0.00691198i \(-0.997800\pi\)
0.862549 0.505974i \(-0.168867\pi\)
\(228\) 0 0
\(229\) 5.19615 3.00000i 0.343371 0.198246i −0.318390 0.947960i \(-0.603142\pi\)
0.661762 + 0.749714i \(0.269809\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.24264 + 4.24264i 0.277945 + 0.277945i 0.832288 0.554343i \(-0.187031\pi\)
−0.554343 + 0.832288i \(0.687031\pi\)
\(234\) 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
\(245\) 1.67303 1.48356i 0.106886 0.0947814i
\(246\) 0 0
\(247\) 4.39230 16.3923i 0.279476 1.04302i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 25.4558i 1.60676i 0.595468 + 0.803379i \(0.296967\pi\)
−0.595468 + 0.803379i \(0.703033\pi\)
\(252\) 0 0
\(253\) 16.0000 16.0000i 1.00591 1.00591i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 19.3185 + 5.17638i 1.20506 + 0.322894i 0.804820 0.593519i \(-0.202262\pi\)
0.400236 + 0.916412i \(0.368928\pi\)
\(258\) 0 0
\(259\) 10.3923 + 6.00000i 0.645746 + 0.372822i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 15.4548 4.14110i 0.952985 0.255351i 0.251356 0.967895i \(-0.419123\pi\)
0.701628 + 0.712543i \(0.252457\pi\)
\(264\) 0 0
\(265\) −18.6603 12.3205i −1.14629 0.756843i
\(266\) 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
\(275\) −14.0406 1.69161i −0.846680 0.102008i
\(276\) 0 0
\(277\) −20.4904 + 5.49038i −1.23115 + 0.329885i −0.815026 0.579424i \(-0.803278\pi\)
−0.416121 + 0.909309i \(0.636611\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −11.0227 6.36396i −0.657559 0.379642i 0.133787 0.991010i \(-0.457286\pi\)
−0.791346 + 0.611368i \(0.790620\pi\)
\(282\) 0 0
\(283\) 5.46410 + 1.46410i 0.324807 + 0.0870318i 0.417538 0.908659i \(-0.362893\pi\)
−0.0927310 + 0.995691i \(0.529560\pi\)
\(284\) 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
\(293\) 3.10583 11.5911i 0.181444 0.677160i −0.813919 0.580978i \(-0.802670\pi\)
0.995364 0.0961820i \(-0.0306631\pi\)
\(294\) 0 0
\(295\) 4.19615 + 4.73205i 0.244309 + 0.275511i
\(296\) 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 0.276219 0.961095i \(-0.410919\pi\)
−0.961095 + 0.276219i \(0.910919\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.89898 2.82843i 0.277796 0.160385i −0.354629 0.935007i \(-0.615393\pi\)
0.632425 + 0.774622i \(0.282060\pi\)
\(312\) 0 0
\(313\) 1.09808 + 4.09808i 0.0620669 + 0.231637i 0.989991 0.141133i \(-0.0450746\pi\)
−0.927924 + 0.372770i \(0.878408\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.55291 5.79555i −0.0872204 0.325511i 0.908505 0.417874i \(-0.137225\pi\)
−0.995725 + 0.0923631i \(0.970558\pi\)
\(318\) 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
\(335\) 1.51575 25.2528i 0.0828142 1.37971i
\(336\) 0 0
\(337\) −1.09808 + 4.09808i −0.0598160 + 0.223236i −0.989363 0.145467i \(-0.953532\pi\)
0.929547 + 0.368703i \(0.120198\pi\)
\(338\) 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
\(347\) −7.72741 2.07055i −0.414829 0.111153i 0.0453672 0.998970i \(-0.485554\pi\)
−0.460196 + 0.887817i \(0.652221\pi\)
\(348\) 0 0
\(349\) 27.7128 + 16.0000i 1.48343 + 0.856460i 0.999823 0.0188232i \(-0.00599197\pi\)
0.483610 + 0.875284i \(0.339325\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 17.3867 4.65874i 0.925399 0.247960i 0.235507 0.971873i \(-0.424325\pi\)
0.689892 + 0.723913i \(0.257658\pi\)
\(354\) 0 0
\(355\) 2.53590 + 12.3923i 0.134592 + 0.657715i
\(356\) 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
\(365\) 12.1967 18.4727i 0.638403 0.966906i
\(366\) 0 0
\(367\) −19.1244 + 5.12436i −0.998283 + 0.267489i −0.720726 0.693220i \(-0.756191\pi\)
−0.277557 + 0.960709i \(0.589525\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 24.4949 + 14.1421i 1.27171 + 0.734223i
\(372\) 0 0
\(373\) 6.83013 + 1.83013i 0.353651 + 0.0947604i 0.431271 0.902223i \(-0.358065\pi\)
−0.0776200 + 0.996983i \(0.524732\pi\)
\(374\) 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 \(0.134797\pi\)
−0.911666 + 0.410932i \(0.865203\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −9.31749 + 34.7733i −0.476101 + 1.77683i 0.141064 + 0.990000i \(0.454948\pi\)
−0.617165 + 0.786834i \(0.711719\pi\)
\(384\) 0 0
\(385\) 17.8564 + 1.07180i 0.910047 + 0.0546238i
\(386\) 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 0.137321 0.990527i \(-0.456151\pi\)
−0.990527 + 0.137321i \(0.956151\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.3712 10.6066i 0.917413 0.529668i 0.0346039 0.999401i \(-0.488983\pi\)
0.882809 + 0.469733i \(0.155650\pi\)
\(402\) 0 0
\(403\) −8.78461 32.7846i −0.437593 1.63312i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.10583 + 11.5911i 0.153950 + 0.574550i
\(408\) 0 0
\(409\) −20.7846 + 12.0000i −1.02773 + 0.593362i −0.916334 0.400414i \(-0.868866\pi\)
−0.111398 + 0.993776i \(0.535533\pi\)
\(410\) 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
\(425\) 3.72500 9.28032i 0.180689 0.450162i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 28.2843i 1.36241i 0.732095 + 0.681203i \(0.238543\pi\)
−0.732095 + 0.681203i \(0.761457\pi\)
\(432\) 0 0
\(433\) 23.0000 23.0000i 1.10531 1.10531i 0.111551 0.993759i \(-0.464418\pi\)
0.993759 0.111551i \(-0.0355818\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −30.9096 8.28221i −1.47861 0.396192i
\(438\) 0 0
\(439\) −17.3205 10.0000i −0.826663 0.477274i 0.0260459 0.999661i \(-0.491708\pi\)
−0.852709 + 0.522387i \(0.825042\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 23.1822 6.21166i 1.10142 0.295125i 0.338078 0.941118i \(-0.390223\pi\)
0.763343 + 0.645994i \(0.223557\pi\)
\(444\) 0 0
\(445\) −3.09808 + 0.633975i −0.146863 + 0.0300533i
\(446\) 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
\(455\) 26.2880 5.37945i 1.23240 0.252193i
\(456\) 0 0
\(457\) −20.4904 + 5.49038i −0.958500 + 0.256829i −0.703965 0.710235i \(-0.748589\pi\)
−0.254534 + 0.967064i \(0.581922\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −13.4722 7.77817i −0.627463 0.362266i 0.152306 0.988333i \(-0.451330\pi\)
−0.779769 + 0.626068i \(0.784663\pi\)
\(462\) 0 0
\(463\) −35.5167 9.51666i −1.65060 0.442277i −0.690819 0.723028i \(-0.742750\pi\)
−0.959781 + 0.280751i \(0.909416\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −19.7990 + 19.7990i −0.916188 + 0.916188i −0.996750 0.0805616i \(-0.974329\pi\)
0.0805616 + 0.996750i \(0.474329\pi\)
\(468\) 0 0
\(469\) 32.0000i 1.47762i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 7.85641 + 18.3923i 0.360477 + 0.843897i
\(476\) 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 0.829859 0.557973i \(-0.188421\pi\)
−0.557973 + 0.829859i \(0.688421\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −31.8434 + 18.3848i −1.43707 + 0.829693i −0.997645 0.0685856i \(-0.978151\pi\)
−0.439426 + 0.898279i \(0.644818\pi\)
\(492\) 0 0
\(493\) 5.12436 + 19.1244i 0.230789 + 0.861318i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.14110 15.4548i −0.185754 0.693243i
\(498\) 0 0
\(499\) −3.46410 + 2.00000i −0.155074 + 0.0895323i −0.575529 0.817781i \(-0.695204\pi\)
0.420455 + 0.907314i \(0.361871\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −8.48528 8.48528i −0.378340 0.378340i 0.492163 0.870503i \(-0.336206\pi\)
−0.870503 + 0.492163i \(0.836206\pi\)
\(504\) 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
\(515\) 6.31319 + 0.378937i 0.278193 + 0.0166980i
\(516\) 0 0
\(517\) 8.78461 32.7846i 0.386347 1.44187i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 29.6985i 1.30111i 0.759457 + 0.650557i \(0.225465\pi\)
−0.759457 + 0.650557i \(0.774535\pi\)
\(522\) 0 0
\(523\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 15.4548 + 4.14110i 0.673222 + 0.180389i
\(528\) 0 0
\(529\) −35.5070 20.5000i −1.54378 0.891304i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 5.79555 1.55291i 0.251033 0.0672642i
\(534\) 0 0
\(535\) −4.92820 + 7.46410i −0.213065 + 0.322701i
\(536\) 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
\(545\) 3.58630 + 17.5254i 0.153620 + 0.750704i
\(546\) 0 0
\(547\) 16.3923 4.39230i 0.700884 0.187801i 0.109258 0.994013i \(-0.465153\pi\)
0.591627 + 0.806212i \(0.298486\pi\)
\(548\) 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 −0.384024 0.923323i \(-0.625462\pi\)
0.923323 + 0.384024i \(0.125462\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 6.21166 23.1822i 0.261790 0.977014i −0.702396 0.711787i \(-0.747886\pi\)
0.964186 0.265227i \(-0.0854470\pi\)
\(564\) 0 0
\(565\) −0.535898 + 8.92820i −0.0225454 + 0.375612i
\(566\) 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 0.837639 0.546225i \(-0.183936\pi\)
−0.546225 + 0.837639i \(0.683936\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 29.3939 16.9706i 1.21946 0.704058i
\(582\) 0 0
\(583\) 7.32051 + 27.3205i 0.303184 + 1.13150i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.10583 + 11.5911i 0.128191 + 0.478416i 0.999933 0.0115488i \(-0.00367618\pi\)
−0.871742 + 0.489965i \(0.837010\pi\)
\(588\) 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 0.354361 0.935109i \(-0.384698\pi\)
−0.935109 + 0.354361i \(0.884698\pi\)
\(594\) 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
\(605\) −4.45069 5.01910i −0.180946 0.204055i
\(606\) 0 0
\(607\) 6.58846 24.5885i 0.267417 0.998015i −0.693337 0.720614i \(-0.743860\pi\)
0.960754 0.277401i \(-0.0894731\pi\)
\(608\) 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 −0.0950469 + 0.995473i \(0.530300\pi\)
−0.995473 + 0.0950469i \(0.969700\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5.79555 1.55291i −0.233320 0.0625180i 0.140264 0.990114i \(-0.455205\pi\)
−0.373585 + 0.927596i \(0.621871\pi\)
\(618\) 0 0
\(619\) 20.7846 + 12.0000i 0.835404 + 0.482321i 0.855699 0.517473i \(-0.173127\pi\)
−0.0202954 + 0.999794i \(0.506461\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.86370 1.03528i 0.154796 0.0414775i
\(624\) 0 0
\(625\) 17.2846 18.0622i 0.691384 0.722487i
\(626\) 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
\(635\) −26.3896 17.4238i −1.04724 0.691444i
\(636\) 0 0
\(637\) −4.09808 + 1.09808i −0.162372 + 0.0435074i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 35.5176 + 20.5061i 1.40286 + 0.809942i 0.994685 0.102961i \(-0.0328318\pi\)
0.408176 + 0.912903i \(0.366165\pi\)
\(642\) 0 0
\(643\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.82843 2.82843i 0.111197 0.111197i −0.649319 0.760516i \(-0.724946\pi\)
0.760516 + 0.649319i \(0.224946\pi\)
\(648\) 0 0
\(649\) 8.00000i 0.314027i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.55291 + 5.79555i −0.0607702 + 0.226798i −0.989631 0.143630i \(-0.954122\pi\)
0.928861 + 0.370428i \(0.120789\pi\)
\(654\) 0 0
\(655\) 4.73205 4.19615i 0.184897 0.163957i
\(656\) 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
\(673\) −8.41858 31.4186i −0.324513 1.21110i −0.914801 0.403905i \(-0.867653\pi\)
0.590288 0.807192i \(-0.299014\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.62347 + 13.5230i 0.139261 + 0.519730i 0.999944 + 0.0105881i \(0.00337036\pi\)
−0.860683 + 0.509142i \(0.829963\pi\)
\(678\) 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 0.850559 0.525879i \(-0.176264\pi\)
−0.525879 + 0.850559i \(0.676264\pi\)
\(684\) 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
\(695\) −26.7685 + 23.7370i −1.01539 + 0.900397i
\(696\) 0 0
\(697\) −0.732051 + 2.73205i −0.0277284 + 0.103484i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9.89949i 0.373899i −0.982370 0.186949i \(-0.940140\pi\)
0.982370 0.186949i \(-0.0598600\pi\)
\(702\) 0 0
\(703\) 12.0000 12.0000i 0.452589 0.452589i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.86370 + 1.03528i 0.145310 + 0.0389356i
\(708\) 0 0
\(709\) −8.66025 5.00000i −0.325243 0.187779i 0.328484 0.944509i \(-0.393462\pi\)
−0.653727 + 0.756730i \(0.726796\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −61.8193 + 16.5644i −2.31515 + 0.620342i
\(714\) 0 0
\(715\) 22.3923 + 14.7846i 0.837425 + 0.552913i
\(716\) 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
\(725\) −5.92065 + 49.1421i −0.219888 + 1.82509i
\(726\) 0 0
\(727\) 8.19615 2.19615i 0.303978 0.0814508i −0.103606 0.994618i \(-0.533038\pi\)
0.407584 + 0.913168i \(0.366371\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 39.6147 + 10.6147i 1.46320 + 0.392064i 0.900595 0.434659i \(-0.143131\pi\)
0.562609 + 0.826723i \(0.309798\pi\)
\(734\) 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 \(-0.827793\pi\)
0.857191 0.514998i \(-0.172207\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −7.24693 + 27.0459i −0.265864 + 0.992219i 0.695855 + 0.718182i \(0.255026\pi\)
−0.961719 + 0.274036i \(0.911641\pi\)
\(744\) 0 0
\(745\) 18.8827 + 21.2942i 0.691808 + 0.780160i
\(746\) 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 0.851487 0.524376i \(-0.175701\pi\)
−0.524376 + 0.851487i \(0.675701\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −23.2702 + 13.4350i −0.843542 + 0.487019i −0.858467 0.512869i \(-0.828583\pi\)
0.0149244 + 0.999889i \(0.495249\pi\)
\(762\) 0 0
\(763\) −5.85641 21.8564i −0.212016 0.791255i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −3.10583 11.5911i −0.112145 0.418531i
\(768\) 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
\(773\) 12.7279 + 12.7279i 0.457792 + 0.457792i 0.897930 0.440138i \(-0.145071\pi\)
−0.440138 + 0.897930i \(0.645071\pi\)
\(774\) 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
\(785\) −1.70522 + 28.4094i −0.0608618 + 1.01397i
\(786\) 0 0
\(787\) −7.32051 + 27.3205i −0.260948 + 0.973871i 0.703736 + 0.710461i \(0.251514\pi\)
−0.964684 + 0.263410i \(0.915153\pi\)
\(788\) 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
\(797\) 11.5911 + 3.10583i 0.410578 + 0.110014i 0.458195 0.888852i \(-0.348496\pi\)
−0.0476171 + 0.998866i \(0.515163\pi\)
\(798\) 0 0
\(799\) 20.7846 + 12.0000i 0.735307 + 0.424529i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −27.0459 + 7.24693i −0.954430 + 0.255739i
\(804\) 0 0
\(805\) −10.1436 49.5692i −0.357515 1.74709i
\(806\) 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
\(815\) −6.96953 + 10.5558i −0.244132 + 0.369755i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −13.4722 7.77817i −0.470183 0.271460i 0.246133 0.969236i \(-0.420840\pi\)
−0.716316 + 0.697776i \(0.754173\pi\)
\(822\) 0 0
\(823\) −35.5167 9.51666i −1.23803 0.331730i −0.420331 0.907371i \(-0.638086\pi\)
−0.817702 + 0.575641i \(0.804753\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −33.9411 + 33.9411i −1.18025 + 1.18025i −0.200569 + 0.979680i \(0.564279\pi\)
−0.979680 + 0.200569i \(0.935721\pi\)
\(828\) 0 0
\(829\) 8.00000i 0.277851i 0.990303 + 0.138926i \(0.0443649\pi\)
−0.990303 + 0.138926i \(0.955635\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.517638 1.93185i 0.0179351 0.0669347i
\(834\) 0 0
\(835\) −35.7128 2.14359i −1.23589 0.0741821i
\(836\) 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
\(853\) −1.83013 6.83013i −0.0626624 0.233859i 0.927491 0.373845i \(-0.121961\pi\)
−0.990153 + 0.139986i \(0.955294\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 10.8704 + 40.5689i 0.371326 + 1.38581i 0.858640 + 0.512580i \(0.171310\pi\)
−0.487314 + 0.873227i \(0.662023\pi\)
\(858\) 0 0
\(859\) 6.92820 4.00000i 0.236387 0.136478i −0.377128 0.926161i \(-0.623088\pi\)
0.613515 + 0.789683i \(0.289755\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −5.65685 5.65685i −0.192562 0.192562i 0.604240 0.796802i \(-0.293477\pi\)
−0.796802 + 0.604240i \(0.793477\pi\)
\(864\) 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
\(875\) −20.4553 + 24.1160i −0.691516 + 0.815268i
\(876\) 0 0
\(877\) −5.49038 + 20.4904i −0.185397 + 0.691911i 0.809148 + 0.587605i \(0.199929\pi\)
−0.994545 + 0.104306i \(0.966738\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 41.0122i 1.38174i 0.722981 + 0.690868i \(0.242771\pi\)
−0.722981 + 0.690868i \(0.757229\pi\)
\(882\) 0 0
\(883\) −20.0000 + 20.0000i −0.673054 + 0.673054i −0.958419 0.285365i \(-0.907885\pi\)
0.285365 + 0.958419i \(0.407885\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 46.3644 + 12.4233i 1.55677 + 0.417134i 0.931638 0.363388i \(-0.118380\pi\)
0.625128 + 0.780522i \(0.285047\pi\)
\(888\) 0 0
\(889\) 34.6410 + 20.0000i 1.16182 + 0.670778i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −46.3644 + 12.4233i −1.55153 + 0.415730i
\(894\) 0 0
\(895\) −30.9808 + 6.33975i −1.03557 + 0.211914i
\(896\) 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
\(905\) 17.5254 3.58630i 0.582563 0.119213i
\(906\) 0 0
\(907\) −21.8564 + 5.85641i −0.725730 + 0.194459i −0.602727 0.797948i \(-0.705919\pi\)
−0.123003 + 0.992406i \(0.539253\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 24.4949 + 14.1421i 0.811552 + 0.468550i 0.847495 0.530804i \(-0.178110\pi\)
−0.0359424 + 0.999354i \(0.511443\pi\)
\(912\) 0 0
\(913\) 32.7846 + 8.78461i 1.08501 + 0.290728i
\(914\) 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 \(0.176980\pi\)
−0.849374 + 0.527791i \(0.823020\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 6.21166 23.1822i 0.204459 0.763052i
\(924\) 0 0
\(925\) −19.6865 7.90192i −0.647289 0.259814i
\(926\) 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 0.971436 + 0.237301i \(0.0762628\pi\)
0.237301 + 0.971436i \(0.423737\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −33.0681 + 19.0919i −1.07799 + 0.622378i −0.930353 0.366664i \(-0.880500\pi\)
−0.147636 + 0.989042i \(0.547166\pi\)
\(942\) 0 0
\(943\) −2.92820 10.9282i −0.0953554 0.355871i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −8.28221 30.9096i −0.269136 1.00443i −0.959670 0.281128i \(-0.909291\pi\)
0.690535 0.723299i \(-0.257375\pi\)
\(948\) 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 0.986763 0.162168i \(-0.0518485\pi\)
−0.162168 + 0.986763i \(0.551849\pi\)
\(954\) 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
\(965\) 47.3489 + 2.84203i 1.52422 + 0.0914882i
\(966\) 0 0
\(967\) −0.732051 + 2.73205i −0.0235412 + 0.0878568i −0.976697 0.214623i \(-0.931148\pi\)
0.953156 + 0.302480i \(0.0978144\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 53.7401i 1.72460i −0.506396 0.862301i \(-0.669022\pi\)
0.506396 0.862301i \(-0.330978\pi\)
\(972\) 0 0
\(973\) 32.0000 32.0000i 1.02587 1.02587i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −17.3867 4.65874i −0.556249 0.149046i −0.0302663 0.999542i \(-0.509636\pi\)
−0.525982 + 0.850495i \(0.676302\pi\)
\(978\) 0 0
\(979\) 3.46410 + 2.00000i 0.110713 + 0.0639203i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −19.3185 + 5.17638i −0.616165 + 0.165101i −0.553384 0.832926i \(-0.686664\pi\)
−0.0627812 + 0.998027i \(0.519997\pi\)
\(984\) 0 0
\(985\) −14.7846 + 22.3923i −0.471077 + 0.713478i
\(986\) 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
\(995\) −5.37945 26.2880i −0.170540 0.833387i
\(996\) 0 0
\(997\) 15.0263 4.02628i 0.475887 0.127514i −0.0128998 0.999917i \(-0.504106\pi\)
0.488787 + 0.872403i \(0.337440\pi\)
\(998\) 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.917.2 8
3.2 odd 2 inner 1620.2.x.a.917.1 8
5.3 odd 4 inner 1620.2.x.a.593.2 8
9.2 odd 6 180.2.j.a.17.2 yes 4
9.4 even 3 inner 1620.2.x.a.377.1 8
9.5 odd 6 inner 1620.2.x.a.377.2 8
9.7 even 3 180.2.j.a.17.1 4
15.8 even 4 inner 1620.2.x.a.593.1 8
36.7 odd 6 720.2.w.b.17.1 4
36.11 even 6 720.2.w.b.17.2 4
45.2 even 12 900.2.j.a.593.1 4
45.7 odd 12 900.2.j.a.593.2 4
45.13 odd 12 inner 1620.2.x.a.53.1 8
45.23 even 12 inner 1620.2.x.a.53.2 8
45.29 odd 6 900.2.j.a.557.1 4
45.34 even 6 900.2.j.a.557.2 4
45.38 even 12 180.2.j.a.53.1 yes 4
45.43 odd 12 180.2.j.a.53.2 yes 4
72.11 even 6 2880.2.w.a.2177.1 4
72.29 odd 6 2880.2.w.j.2177.1 4
72.43 odd 6 2880.2.w.a.2177.2 4
72.61 even 6 2880.2.w.j.2177.2 4
180.7 even 12 3600.2.w.f.593.1 4
180.43 even 12 720.2.w.b.593.2 4
180.47 odd 12 3600.2.w.f.593.2 4
180.79 odd 6 3600.2.w.f.1457.1 4
180.83 odd 12 720.2.w.b.593.1 4
180.119 even 6 3600.2.w.f.1457.2 4
360.43 even 12 2880.2.w.a.2753.1 4
360.83 odd 12 2880.2.w.a.2753.2 4
360.133 odd 12 2880.2.w.j.2753.1 4
360.173 even 12 2880.2.w.j.2753.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.2.j.a.17.1 4 9.7 even 3
180.2.j.a.17.2 yes 4 9.2 odd 6
180.2.j.a.53.1 yes 4 45.38 even 12
180.2.j.a.53.2 yes 4 45.43 odd 12
720.2.w.b.17.1 4 36.7 odd 6
720.2.w.b.17.2 4 36.11 even 6
720.2.w.b.593.1 4 180.83 odd 12
720.2.w.b.593.2 4 180.43 even 12
900.2.j.a.557.1 4 45.29 odd 6
900.2.j.a.557.2 4 45.34 even 6
900.2.j.a.593.1 4 45.2 even 12
900.2.j.a.593.2 4 45.7 odd 12
1620.2.x.a.53.1 8 45.13 odd 12 inner
1620.2.x.a.53.2 8 45.23 even 12 inner
1620.2.x.a.377.1 8 9.4 even 3 inner
1620.2.x.a.377.2 8 9.5 odd 6 inner
1620.2.x.a.593.1 8 15.8 even 4 inner
1620.2.x.a.593.2 8 5.3 odd 4 inner
1620.2.x.a.917.1 8 3.2 odd 2 inner
1620.2.x.a.917.2 8 1.1 even 1 trivial
2880.2.w.a.2177.1 4 72.11 even 6
2880.2.w.a.2177.2 4 72.43 odd 6
2880.2.w.a.2753.1 4 360.43 even 12
2880.2.w.a.2753.2 4 360.83 odd 12
2880.2.w.j.2177.1 4 72.29 odd 6
2880.2.w.j.2177.2 4 72.61 even 6
2880.2.w.j.2753.1 4 360.133 odd 12
2880.2.w.j.2753.2 4 360.173 even 12
3600.2.w.f.593.1 4 180.7 even 12
3600.2.w.f.593.2 4 180.47 odd 12
3600.2.w.f.1457.1 4 180.79 odd 6
3600.2.w.f.1457.2 4 180.119 even 6