Properties

Label 1960.2.q.u.361.2
Level $1960$
Weight $2$
Character 1960.361
Analytic conductor $15.651$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1960,2,Mod(361,1960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1960, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1960.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1960.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.6506787962\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.2
Root \(1.28078 - 2.21837i\) of defining polynomial
Character \(\chi\) \(=\) 1960.361
Dual form 1960.2.q.u.961.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.28078 - 2.21837i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-1.78078 - 3.08440i) q^{9} +O(q^{10})\) \(q+(1.28078 - 2.21837i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-1.78078 - 3.08440i) q^{9} +(1.28078 - 2.21837i) q^{11} +5.68466 q^{13} +2.56155 q^{15} +(1.71922 - 2.97778i) q^{17} +(0.561553 + 0.972638i) q^{19} +(2.56155 + 4.43674i) q^{23} +(-0.500000 + 0.866025i) q^{25} -1.43845 q^{27} +4.56155 q^{29} +(-5.12311 + 8.87348i) q^{31} +(-3.28078 - 5.68247i) q^{33} +(-4.12311 - 7.14143i) q^{37} +(7.28078 - 12.6107i) q^{39} -7.12311 q^{41} +1.12311 q^{43} +(1.78078 - 3.08440i) q^{45} +(3.28078 + 5.68247i) q^{47} +(-4.40388 - 7.62775i) q^{51} +(2.43845 - 4.22351i) q^{53} +2.56155 q^{55} +2.87689 q^{57} +(-2.00000 + 3.46410i) q^{59} +(-7.56155 - 13.0970i) q^{61} +(2.84233 + 4.92306i) q^{65} +(7.12311 - 12.3376i) q^{67} +13.1231 q^{69} +(6.12311 - 10.6055i) q^{73} +(1.28078 + 2.21837i) q^{75} +(5.84233 + 10.1192i) q^{79} +(3.50000 - 6.06218i) q^{81} -12.0000 q^{83} +3.43845 q^{85} +(5.84233 - 10.1192i) q^{87} +(-1.56155 - 2.70469i) q^{89} +(13.1231 + 22.7299i) q^{93} +(-0.561553 + 0.972638i) q^{95} -13.6847 q^{97} -9.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} + 2 q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} + 2 q^{5} - 3 q^{9} + q^{11} - 2 q^{13} + 2 q^{15} + 11 q^{17} - 6 q^{19} + 2 q^{23} - 2 q^{25} - 14 q^{27} + 10 q^{29} - 4 q^{31} - 9 q^{33} + 25 q^{39} - 12 q^{41} - 12 q^{43} + 3 q^{45} + 9 q^{47} + 3 q^{51} + 18 q^{53} + 2 q^{55} + 28 q^{57} - 8 q^{59} - 22 q^{61} - q^{65} + 12 q^{67} + 36 q^{69} + 8 q^{73} + q^{75} + 11 q^{79} + 14 q^{81} - 48 q^{83} + 22 q^{85} + 11 q^{87} + 2 q^{89} + 36 q^{93} + 6 q^{95} - 30 q^{97} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.28078 2.21837i 0.739457 1.28078i −0.213284 0.976990i \(-0.568416\pi\)
0.952740 0.303786i \(-0.0982508\pi\)
\(4\) 0 0
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.78078 3.08440i −0.593592 1.02813i
\(10\) 0 0
\(11\) 1.28078 2.21837i 0.386169 0.668864i −0.605762 0.795646i \(-0.707132\pi\)
0.991931 + 0.126782i \(0.0404650\pi\)
\(12\) 0 0
\(13\) 5.68466 1.57664 0.788320 0.615265i \(-0.210951\pi\)
0.788320 + 0.615265i \(0.210951\pi\)
\(14\) 0 0
\(15\) 2.56155 0.661390
\(16\) 0 0
\(17\) 1.71922 2.97778i 0.416973 0.722218i −0.578660 0.815569i \(-0.696424\pi\)
0.995633 + 0.0933503i \(0.0297576\pi\)
\(18\) 0 0
\(19\) 0.561553 + 0.972638i 0.128829 + 0.223138i 0.923223 0.384264i \(-0.125545\pi\)
−0.794394 + 0.607403i \(0.792211\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.56155 + 4.43674i 0.534121 + 0.925124i 0.999205 + 0.0398580i \(0.0126905\pi\)
−0.465085 + 0.885266i \(0.653976\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) −1.43845 −0.276829
\(28\) 0 0
\(29\) 4.56155 0.847059 0.423530 0.905882i \(-0.360791\pi\)
0.423530 + 0.905882i \(0.360791\pi\)
\(30\) 0 0
\(31\) −5.12311 + 8.87348i −0.920137 + 1.59372i −0.120936 + 0.992660i \(0.538589\pi\)
−0.799201 + 0.601064i \(0.794744\pi\)
\(32\) 0 0
\(33\) −3.28078 5.68247i −0.571110 0.989191i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.12311 7.14143i −0.677834 1.17404i −0.975632 0.219414i \(-0.929585\pi\)
0.297797 0.954629i \(-0.403748\pi\)
\(38\) 0 0
\(39\) 7.28078 12.6107i 1.16586 2.01932i
\(40\) 0 0
\(41\) −7.12311 −1.11244 −0.556221 0.831034i \(-0.687749\pi\)
−0.556221 + 0.831034i \(0.687749\pi\)
\(42\) 0 0
\(43\) 1.12311 0.171272 0.0856360 0.996326i \(-0.472708\pi\)
0.0856360 + 0.996326i \(0.472708\pi\)
\(44\) 0 0
\(45\) 1.78078 3.08440i 0.265462 0.459794i
\(46\) 0 0
\(47\) 3.28078 + 5.68247i 0.478550 + 0.828874i 0.999698 0.0245932i \(-0.00782905\pi\)
−0.521147 + 0.853467i \(0.674496\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −4.40388 7.62775i −0.616667 1.06810i
\(52\) 0 0
\(53\) 2.43845 4.22351i 0.334946 0.580144i −0.648528 0.761191i \(-0.724615\pi\)
0.983475 + 0.181046i \(0.0579484\pi\)
\(54\) 0 0
\(55\) 2.56155 0.345400
\(56\) 0 0
\(57\) 2.87689 0.381054
\(58\) 0 0
\(59\) −2.00000 + 3.46410i −0.260378 + 0.450988i −0.966342 0.257260i \(-0.917180\pi\)
0.705965 + 0.708247i \(0.250514\pi\)
\(60\) 0 0
\(61\) −7.56155 13.0970i −0.968158 1.67690i −0.700881 0.713278i \(-0.747210\pi\)
−0.267277 0.963620i \(-0.586124\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.84233 + 4.92306i 0.352548 + 0.610630i
\(66\) 0 0
\(67\) 7.12311 12.3376i 0.870226 1.50728i 0.00846293 0.999964i \(-0.497306\pi\)
0.861763 0.507311i \(-0.169361\pi\)
\(68\) 0 0
\(69\) 13.1231 1.57984
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 6.12311 10.6055i 0.716655 1.24128i −0.245662 0.969355i \(-0.579005\pi\)
0.962318 0.271928i \(-0.0876612\pi\)
\(74\) 0 0
\(75\) 1.28078 + 2.21837i 0.147891 + 0.256155i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.84233 + 10.1192i 0.657313 + 1.13850i 0.981308 + 0.192441i \(0.0616405\pi\)
−0.323995 + 0.946059i \(0.605026\pi\)
\(80\) 0 0
\(81\) 3.50000 6.06218i 0.388889 0.673575i
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 3.43845 0.372952
\(86\) 0 0
\(87\) 5.84233 10.1192i 0.626363 1.08489i
\(88\) 0 0
\(89\) −1.56155 2.70469i −0.165524 0.286696i 0.771317 0.636451i \(-0.219598\pi\)
−0.936841 + 0.349755i \(0.886265\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 13.1231 + 22.7299i 1.36080 + 2.35698i
\(94\) 0 0
\(95\) −0.561553 + 0.972638i −0.0576141 + 0.0997906i
\(96\) 0 0
\(97\) −13.6847 −1.38947 −0.694733 0.719267i \(-0.744478\pi\)
−0.694733 + 0.719267i \(0.744478\pi\)
\(98\) 0 0
\(99\) −9.12311 −0.916907
\(100\) 0 0
\(101\) 3.00000 5.19615i 0.298511 0.517036i −0.677284 0.735721i \(-0.736843\pi\)
0.975796 + 0.218685i \(0.0701767\pi\)
\(102\) 0 0
\(103\) 4.40388 + 7.62775i 0.433927 + 0.751584i 0.997207 0.0746818i \(-0.0237941\pi\)
−0.563280 + 0.826266i \(0.690461\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.43845 5.95557i −0.332407 0.575746i 0.650576 0.759441i \(-0.274528\pi\)
−0.982983 + 0.183695i \(0.941194\pi\)
\(108\) 0 0
\(109\) 4.28078 7.41452i 0.410024 0.710182i −0.584868 0.811129i \(-0.698854\pi\)
0.994892 + 0.100946i \(0.0321870\pi\)
\(110\) 0 0
\(111\) −21.1231 −2.00492
\(112\) 0 0
\(113\) −8.24621 −0.775738 −0.387869 0.921714i \(-0.626789\pi\)
−0.387869 + 0.921714i \(0.626789\pi\)
\(114\) 0 0
\(115\) −2.56155 + 4.43674i −0.238866 + 0.413728i
\(116\) 0 0
\(117\) −10.1231 17.5337i −0.935881 1.62099i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.21922 + 3.84381i 0.201748 + 0.349437i
\(122\) 0 0
\(123\) −9.12311 + 15.8017i −0.822603 + 1.42479i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 10.2462 0.909204 0.454602 0.890695i \(-0.349781\pi\)
0.454602 + 0.890695i \(0.349781\pi\)
\(128\) 0 0
\(129\) 1.43845 2.49146i 0.126648 0.219361i
\(130\) 0 0
\(131\) 3.43845 + 5.95557i 0.300419 + 0.520340i 0.976231 0.216734i \(-0.0695403\pi\)
−0.675812 + 0.737074i \(0.736207\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −0.719224 1.24573i −0.0619009 0.107216i
\(136\) 0 0
\(137\) −8.68466 + 15.0423i −0.741980 + 1.28515i 0.209612 + 0.977785i \(0.432780\pi\)
−0.951592 + 0.307363i \(0.900553\pi\)
\(138\) 0 0
\(139\) −3.36932 −0.285782 −0.142891 0.989738i \(-0.545640\pi\)
−0.142891 + 0.989738i \(0.545640\pi\)
\(140\) 0 0
\(141\) 16.8078 1.41547
\(142\) 0 0
\(143\) 7.28078 12.6107i 0.608849 1.05456i
\(144\) 0 0
\(145\) 2.28078 + 3.95042i 0.189408 + 0.328065i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.12311 14.0696i −0.665471 1.15263i −0.979157 0.203103i \(-0.934898\pi\)
0.313687 0.949527i \(-0.398436\pi\)
\(150\) 0 0
\(151\) −0.403882 + 0.699544i −0.0328675 + 0.0569281i −0.881991 0.471266i \(-0.843797\pi\)
0.849124 + 0.528194i \(0.177131\pi\)
\(152\) 0 0
\(153\) −12.2462 −0.990048
\(154\) 0 0
\(155\) −10.2462 −0.822995
\(156\) 0 0
\(157\) −6.12311 + 10.6055i −0.488677 + 0.846413i −0.999915 0.0130257i \(-0.995854\pi\)
0.511238 + 0.859439i \(0.329187\pi\)
\(158\) 0 0
\(159\) −6.24621 10.8188i −0.495357 0.857983i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.8078 + 18.7196i 0.846529 + 1.46623i 0.884286 + 0.466945i \(0.154645\pi\)
−0.0377569 + 0.999287i \(0.512021\pi\)
\(164\) 0 0
\(165\) 3.28078 5.68247i 0.255408 0.442380i
\(166\) 0 0
\(167\) −24.8078 −1.91968 −0.959841 0.280544i \(-0.909485\pi\)
−0.959841 + 0.280544i \(0.909485\pi\)
\(168\) 0 0
\(169\) 19.3153 1.48580
\(170\) 0 0
\(171\) 2.00000 3.46410i 0.152944 0.264906i
\(172\) 0 0
\(173\) 11.0885 + 19.2059i 0.843046 + 1.46020i 0.887307 + 0.461179i \(0.152573\pi\)
−0.0442611 + 0.999020i \(0.514093\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.12311 + 8.87348i 0.385076 + 0.666972i
\(178\) 0 0
\(179\) 2.00000 3.46410i 0.149487 0.258919i −0.781551 0.623841i \(-0.785571\pi\)
0.931038 + 0.364922i \(0.118904\pi\)
\(180\) 0 0
\(181\) −3.12311 −0.232139 −0.116069 0.993241i \(-0.537029\pi\)
−0.116069 + 0.993241i \(0.537029\pi\)
\(182\) 0 0
\(183\) −38.7386 −2.86364
\(184\) 0 0
\(185\) 4.12311 7.14143i 0.303137 0.525048i
\(186\) 0 0
\(187\) −4.40388 7.62775i −0.322044 0.557796i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.84233 10.1192i −0.422736 0.732200i 0.573470 0.819227i \(-0.305597\pi\)
−0.996206 + 0.0870262i \(0.972264\pi\)
\(192\) 0 0
\(193\) −2.43845 + 4.22351i −0.175523 + 0.304015i −0.940342 0.340230i \(-0.889495\pi\)
0.764819 + 0.644245i \(0.222828\pi\)
\(194\) 0 0
\(195\) 14.5616 1.04277
\(196\) 0 0
\(197\) −4.87689 −0.347464 −0.173732 0.984793i \(-0.555583\pi\)
−0.173732 + 0.984793i \(0.555583\pi\)
\(198\) 0 0
\(199\) 1.12311 1.94528i 0.0796148 0.137897i −0.823469 0.567361i \(-0.807964\pi\)
0.903084 + 0.429464i \(0.141298\pi\)
\(200\) 0 0
\(201\) −18.2462 31.6034i −1.28699 2.22913i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −3.56155 6.16879i −0.248750 0.430847i
\(206\) 0 0
\(207\) 9.12311 15.8017i 0.634100 1.09829i
\(208\) 0 0
\(209\) 2.87689 0.198999
\(210\) 0 0
\(211\) −13.4384 −0.925141 −0.462570 0.886583i \(-0.653073\pi\)
−0.462570 + 0.886583i \(0.653073\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.561553 + 0.972638i 0.0382976 + 0.0663334i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −15.6847 27.1666i −1.05987 1.83575i
\(220\) 0 0
\(221\) 9.77320 16.9277i 0.657417 1.13868i
\(222\) 0 0
\(223\) −12.3153 −0.824696 −0.412348 0.911026i \(-0.635291\pi\)
−0.412348 + 0.911026i \(0.635291\pi\)
\(224\) 0 0
\(225\) 3.56155 0.237437
\(226\) 0 0
\(227\) −11.8423 + 20.5115i −0.786003 + 1.36140i 0.142395 + 0.989810i \(0.454520\pi\)
−0.928398 + 0.371587i \(0.878814\pi\)
\(228\) 0 0
\(229\) 6.68466 + 11.5782i 0.441735 + 0.765107i 0.997818 0.0660194i \(-0.0210299\pi\)
−0.556084 + 0.831126i \(0.687697\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.56155 + 16.5611i 0.626398 + 1.08495i 0.988269 + 0.152725i \(0.0488049\pi\)
−0.361871 + 0.932228i \(0.617862\pi\)
\(234\) 0 0
\(235\) −3.28078 + 5.68247i −0.214014 + 0.370684i
\(236\) 0 0
\(237\) 29.9309 1.94422
\(238\) 0 0
\(239\) 19.0540 1.23250 0.616250 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(240\) 0 0
\(241\) 1.00000 1.73205i 0.0644157 0.111571i −0.832019 0.554747i \(-0.812815\pi\)
0.896435 + 0.443176i \(0.146148\pi\)
\(242\) 0 0
\(243\) −11.1231 19.2658i −0.713548 1.23590i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.19224 + 5.52911i 0.203117 + 0.351809i
\(248\) 0 0
\(249\) −15.3693 + 26.6204i −0.973991 + 1.68700i
\(250\) 0 0
\(251\) 11.3693 0.717625 0.358812 0.933410i \(-0.383182\pi\)
0.358812 + 0.933410i \(0.383182\pi\)
\(252\) 0 0
\(253\) 13.1231 0.825043
\(254\) 0 0
\(255\) 4.40388 7.62775i 0.275782 0.477668i
\(256\) 0 0
\(257\) 7.24621 + 12.5508i 0.452006 + 0.782898i 0.998511 0.0545585i \(-0.0173751\pi\)
−0.546504 + 0.837456i \(0.684042\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −8.12311 14.0696i −0.502808 0.870888i
\(262\) 0 0
\(263\) 7.68466 13.3102i 0.473856 0.820743i −0.525696 0.850673i \(-0.676195\pi\)
0.999552 + 0.0299295i \(0.00952826\pi\)
\(264\) 0 0
\(265\) 4.87689 0.299585
\(266\) 0 0
\(267\) −8.00000 −0.489592
\(268\) 0 0
\(269\) −6.12311 + 10.6055i −0.373332 + 0.646631i −0.990076 0.140534i \(-0.955118\pi\)
0.616744 + 0.787164i \(0.288452\pi\)
\(270\) 0 0
\(271\) 5.12311 + 8.87348i 0.311207 + 0.539025i 0.978624 0.205658i \(-0.0659336\pi\)
−0.667417 + 0.744684i \(0.732600\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.28078 + 2.21837i 0.0772337 + 0.133773i
\(276\) 0 0
\(277\) −4.12311 + 7.14143i −0.247733 + 0.429087i −0.962897 0.269871i \(-0.913019\pi\)
0.715163 + 0.698958i \(0.246352\pi\)
\(278\) 0 0
\(279\) 36.4924 2.18474
\(280\) 0 0
\(281\) −15.4384 −0.920981 −0.460490 0.887665i \(-0.652326\pi\)
−0.460490 + 0.887665i \(0.652326\pi\)
\(282\) 0 0
\(283\) 3.84233 6.65511i 0.228403 0.395605i −0.728932 0.684586i \(-0.759983\pi\)
0.957335 + 0.288981i \(0.0933164\pi\)
\(284\) 0 0
\(285\) 1.43845 + 2.49146i 0.0852063 + 0.147582i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.58854 + 4.48348i 0.152267 + 0.263734i
\(290\) 0 0
\(291\) −17.5270 + 30.3576i −1.02745 + 1.77960i
\(292\) 0 0
\(293\) 5.05398 0.295256 0.147628 0.989043i \(-0.452836\pi\)
0.147628 + 0.989043i \(0.452836\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) −1.84233 + 3.19101i −0.106903 + 0.185161i
\(298\) 0 0
\(299\) 14.5616 + 25.2213i 0.842116 + 1.45859i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −7.68466 13.3102i −0.441472 0.764652i
\(304\) 0 0
\(305\) 7.56155 13.0970i 0.432973 0.749932i
\(306\) 0 0
\(307\) 16.3153 0.931166 0.465583 0.885004i \(-0.345845\pi\)
0.465583 + 0.885004i \(0.345845\pi\)
\(308\) 0 0
\(309\) 22.5616 1.28348
\(310\) 0 0
\(311\) −10.5616 + 18.2931i −0.598891 + 1.03731i 0.394094 + 0.919070i \(0.371058\pi\)
−0.992985 + 0.118239i \(0.962275\pi\)
\(312\) 0 0
\(313\) 7.15767 + 12.3974i 0.404575 + 0.700745i 0.994272 0.106880i \(-0.0340860\pi\)
−0.589696 + 0.807625i \(0.700753\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.00000 + 8.66025i 0.280828 + 0.486408i 0.971589 0.236675i \(-0.0760576\pi\)
−0.690761 + 0.723083i \(0.742724\pi\)
\(318\) 0 0
\(319\) 5.84233 10.1192i 0.327108 0.566567i
\(320\) 0 0
\(321\) −17.6155 −0.983203
\(322\) 0 0
\(323\) 3.86174 0.214873
\(324\) 0 0
\(325\) −2.84233 + 4.92306i −0.157664 + 0.273082i
\(326\) 0 0
\(327\) −10.9654 18.9927i −0.606390 1.05030i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 6.00000 + 10.3923i 0.329790 + 0.571213i 0.982470 0.186421i \(-0.0596888\pi\)
−0.652680 + 0.757634i \(0.726355\pi\)
\(332\) 0 0
\(333\) −14.6847 + 25.4346i −0.804714 + 1.39381i
\(334\) 0 0
\(335\) 14.2462 0.778354
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) −10.5616 + 18.2931i −0.573625 + 0.993547i
\(340\) 0 0
\(341\) 13.1231 + 22.7299i 0.710656 + 1.23089i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 6.56155 + 11.3649i 0.353262 + 0.611868i
\(346\) 0 0
\(347\) −3.43845 + 5.95557i −0.184586 + 0.319711i −0.943437 0.331552i \(-0.892428\pi\)
0.758851 + 0.651264i \(0.225761\pi\)
\(348\) 0 0
\(349\) 28.2462 1.51199 0.755993 0.654580i \(-0.227155\pi\)
0.755993 + 0.654580i \(0.227155\pi\)
\(350\) 0 0
\(351\) −8.17708 −0.436460
\(352\) 0 0
\(353\) −11.4039 + 19.7521i −0.606967 + 1.05130i 0.384770 + 0.923013i \(0.374281\pi\)
−0.991737 + 0.128286i \(0.959053\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −16.0000 27.7128i −0.844448 1.46263i −0.886100 0.463494i \(-0.846596\pi\)
0.0416523 0.999132i \(-0.486738\pi\)
\(360\) 0 0
\(361\) 8.86932 15.3621i 0.466806 0.808532i
\(362\) 0 0
\(363\) 11.3693 0.596734
\(364\) 0 0
\(365\) 12.2462 0.640996
\(366\) 0 0
\(367\) −6.96543 + 12.0645i −0.363593 + 0.629761i −0.988549 0.150898i \(-0.951783\pi\)
0.624957 + 0.780660i \(0.285117\pi\)
\(368\) 0 0
\(369\) 12.6847 + 21.9705i 0.660337 + 1.14374i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 4.68466 + 8.11407i 0.242562 + 0.420130i 0.961443 0.275003i \(-0.0886787\pi\)
−0.718881 + 0.695133i \(0.755345\pi\)
\(374\) 0 0
\(375\) −1.28078 + 2.21837i −0.0661390 + 0.114556i
\(376\) 0 0
\(377\) 25.9309 1.33551
\(378\) 0 0
\(379\) −0.492423 −0.0252940 −0.0126470 0.999920i \(-0.504026\pi\)
−0.0126470 + 0.999920i \(0.504026\pi\)
\(380\) 0 0
\(381\) 13.1231 22.7299i 0.672317 1.16449i
\(382\) 0 0
\(383\) −13.1231 22.7299i −0.670559 1.16144i −0.977746 0.209794i \(-0.932721\pi\)
0.307186 0.951649i \(-0.400613\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.00000 3.46410i −0.101666 0.176090i
\(388\) 0 0
\(389\) 15.6501 27.1068i 0.793491 1.37437i −0.130302 0.991474i \(-0.541595\pi\)
0.923793 0.382893i \(-0.125072\pi\)
\(390\) 0 0
\(391\) 17.6155 0.890856
\(392\) 0 0
\(393\) 17.6155 0.888586
\(394\) 0 0
\(395\) −5.84233 + 10.1192i −0.293959 + 0.509153i
\(396\) 0 0
\(397\) 11.0885 + 19.2059i 0.556518 + 0.963917i 0.997784 + 0.0665408i \(0.0211963\pi\)
−0.441266 + 0.897376i \(0.645470\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7.96543 13.7965i −0.397775 0.688966i 0.595676 0.803225i \(-0.296884\pi\)
−0.993451 + 0.114258i \(0.963551\pi\)
\(402\) 0 0
\(403\) −29.1231 + 50.4427i −1.45073 + 2.51273i
\(404\) 0 0
\(405\) 7.00000 0.347833
\(406\) 0 0
\(407\) −21.1231 −1.04703
\(408\) 0 0
\(409\) 7.24621 12.5508i 0.358302 0.620597i −0.629375 0.777102i \(-0.716689\pi\)
0.987677 + 0.156504i \(0.0500225\pi\)
\(410\) 0 0
\(411\) 22.2462 + 38.5316i 1.09732 + 1.90062i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −6.00000 10.3923i −0.294528 0.510138i
\(416\) 0 0
\(417\) −4.31534 + 7.47439i −0.211323 + 0.366022i
\(418\) 0 0
\(419\) 1.75379 0.0856782 0.0428391 0.999082i \(-0.486360\pi\)
0.0428391 + 0.999082i \(0.486360\pi\)
\(420\) 0 0
\(421\) −0.561553 −0.0273684 −0.0136842 0.999906i \(-0.504356\pi\)
−0.0136842 + 0.999906i \(0.504356\pi\)
\(422\) 0 0
\(423\) 11.6847 20.2384i 0.568128 0.984026i
\(424\) 0 0
\(425\) 1.71922 + 2.97778i 0.0833946 + 0.144444i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −18.6501 32.3029i −0.900435 1.55960i
\(430\) 0 0
\(431\) −2.96543 + 5.13628i −0.142840 + 0.247406i −0.928565 0.371170i \(-0.878957\pi\)
0.785725 + 0.618576i \(0.212290\pi\)
\(432\) 0 0
\(433\) −36.2462 −1.74188 −0.870941 0.491388i \(-0.836490\pi\)
−0.870941 + 0.491388i \(0.836490\pi\)
\(434\) 0 0
\(435\) 11.6847 0.560236
\(436\) 0 0
\(437\) −2.87689 + 4.98293i −0.137621 + 0.238366i
\(438\) 0 0
\(439\) 5.43845 + 9.41967i 0.259563 + 0.449576i 0.966125 0.258075i \(-0.0830882\pi\)
−0.706562 + 0.707651i \(0.749755\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.8078 18.7196i −0.513492 0.889395i −0.999878 0.0156504i \(-0.995018\pi\)
0.486385 0.873745i \(-0.338315\pi\)
\(444\) 0 0
\(445\) 1.56155 2.70469i 0.0740247 0.128215i
\(446\) 0 0
\(447\) −41.6155 −1.96835
\(448\) 0 0
\(449\) −33.6847 −1.58968 −0.794839 0.606821i \(-0.792445\pi\)
−0.794839 + 0.606821i \(0.792445\pi\)
\(450\) 0 0
\(451\) −9.12311 + 15.8017i −0.429590 + 0.744072i
\(452\) 0 0
\(453\) 1.03457 + 1.79192i 0.0486081 + 0.0841917i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −16.6847 28.8987i −0.780475 1.35182i −0.931665 0.363318i \(-0.881644\pi\)
0.151190 0.988505i \(-0.451690\pi\)
\(458\) 0 0
\(459\) −2.47301 + 4.28338i −0.115430 + 0.199931i
\(460\) 0 0
\(461\) −27.1231 −1.26325 −0.631624 0.775274i \(-0.717612\pi\)
−0.631624 + 0.775274i \(0.717612\pi\)
\(462\) 0 0
\(463\) 10.2462 0.476182 0.238091 0.971243i \(-0.423478\pi\)
0.238091 + 0.971243i \(0.423478\pi\)
\(464\) 0 0
\(465\) −13.1231 + 22.7299i −0.608569 + 1.05407i
\(466\) 0 0
\(467\) 10.7192 + 18.5662i 0.496027 + 0.859143i 0.999990 0.00458213i \(-0.00145854\pi\)
−0.503963 + 0.863725i \(0.668125\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 15.6847 + 27.1666i 0.722711 + 1.25177i
\(472\) 0 0
\(473\) 1.43845 2.49146i 0.0661399 0.114558i
\(474\) 0 0
\(475\) −1.12311 −0.0515316
\(476\) 0 0
\(477\) −17.3693 −0.795286
\(478\) 0 0
\(479\) −3.68466 + 6.38202i −0.168356 + 0.291602i −0.937842 0.347062i \(-0.887179\pi\)
0.769486 + 0.638664i \(0.220513\pi\)
\(480\) 0 0
\(481\) −23.4384 40.5966i −1.06870 1.85104i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6.84233 11.8513i −0.310694 0.538138i
\(486\) 0 0
\(487\) 2.56155 4.43674i 0.116075 0.201048i −0.802134 0.597144i \(-0.796302\pi\)
0.918209 + 0.396096i \(0.129635\pi\)
\(488\) 0 0
\(489\) 55.3693 2.50389
\(490\) 0 0
\(491\) −15.6847 −0.707839 −0.353919 0.935276i \(-0.615151\pi\)
−0.353919 + 0.935276i \(0.615151\pi\)
\(492\) 0 0
\(493\) 7.84233 13.5833i 0.353201 0.611762i
\(494\) 0 0
\(495\) −4.56155 7.90084i −0.205027 0.355116i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 10.4039 + 18.0201i 0.465742 + 0.806688i 0.999235 0.0391162i \(-0.0124542\pi\)
−0.533493 + 0.845805i \(0.679121\pi\)
\(500\) 0 0
\(501\) −31.7732 + 55.0328i −1.41952 + 2.45868i
\(502\) 0 0
\(503\) −16.1771 −0.721300 −0.360650 0.932701i \(-0.617445\pi\)
−0.360650 + 0.932701i \(0.617445\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 24.7386 42.8486i 1.09868 1.90297i
\(508\) 0 0
\(509\) −8.36932 14.4961i −0.370963 0.642528i 0.618751 0.785588i \(-0.287639\pi\)
−0.989714 + 0.143060i \(0.954306\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −0.807764 1.39909i −0.0356637 0.0617713i
\(514\) 0 0
\(515\) −4.40388 + 7.62775i −0.194058 + 0.336119i
\(516\) 0 0
\(517\) 16.8078 0.739205
\(518\) 0 0
\(519\) 56.8078 2.49358
\(520\) 0 0
\(521\) 2.12311 3.67733i 0.0930149 0.161107i −0.815763 0.578386i \(-0.803683\pi\)
0.908778 + 0.417279i \(0.137016\pi\)
\(522\) 0 0
\(523\) −6.00000 10.3923i −0.262362 0.454424i 0.704507 0.709697i \(-0.251168\pi\)
−0.966869 + 0.255273i \(0.917835\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 17.6155 + 30.5110i 0.767344 + 1.32908i
\(528\) 0 0
\(529\) −1.62311 + 2.81130i −0.0705698 + 0.122230i
\(530\) 0 0
\(531\) 14.2462 0.618233
\(532\) 0 0
\(533\) −40.4924 −1.75392
\(534\) 0 0
\(535\) 3.43845 5.95557i 0.148657 0.257482i
\(536\) 0 0
\(537\) −5.12311 8.87348i −0.221078 0.382919i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 10.2116 + 17.6871i 0.439033 + 0.760427i 0.997615 0.0690218i \(-0.0219878\pi\)
−0.558582 + 0.829449i \(0.688654\pi\)
\(542\) 0 0
\(543\) −4.00000 + 6.92820i −0.171656 + 0.297318i
\(544\) 0 0
\(545\) 8.56155 0.366737
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 0 0
\(549\) −26.9309 + 46.6456i −1.14938 + 1.99079i
\(550\) 0 0
\(551\) 2.56155 + 4.43674i 0.109126 + 0.189011i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −10.5616 18.2931i −0.448313 0.776501i
\(556\) 0 0
\(557\) −11.8078 + 20.4516i −0.500311 + 0.866564i 0.499689 + 0.866205i \(0.333448\pi\)
−1.00000 0.000359173i \(0.999886\pi\)
\(558\) 0 0
\(559\) 6.38447 0.270034
\(560\) 0 0
\(561\) −22.5616 −0.952550
\(562\) 0 0
\(563\) −20.2462 + 35.0675i −0.853276 + 1.47792i 0.0249591 + 0.999688i \(0.492054\pi\)
−0.878235 + 0.478229i \(0.841279\pi\)
\(564\) 0 0
\(565\) −4.12311 7.14143i −0.173460 0.300442i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 23.4924 + 40.6901i 0.984854 + 1.70582i 0.642585 + 0.766214i \(0.277862\pi\)
0.342268 + 0.939602i \(0.388805\pi\)
\(570\) 0 0
\(571\) −10.0000 + 17.3205i −0.418487 + 0.724841i −0.995788 0.0916910i \(-0.970773\pi\)
0.577301 + 0.816532i \(0.304106\pi\)
\(572\) 0 0
\(573\) −29.9309 −1.25038
\(574\) 0 0
\(575\) −5.12311 −0.213648
\(576\) 0 0
\(577\) 0.280776 0.486319i 0.0116889 0.0202457i −0.860122 0.510089i \(-0.829613\pi\)
0.871811 + 0.489843i \(0.162946\pi\)
\(578\) 0 0
\(579\) 6.24621 + 10.8188i 0.259584 + 0.449612i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −6.24621 10.8188i −0.258692 0.448067i
\(584\) 0 0
\(585\) 10.1231 17.5337i 0.418539 0.724931i
\(586\) 0 0
\(587\) 6.24621 0.257809 0.128904 0.991657i \(-0.458854\pi\)
0.128904 + 0.991657i \(0.458854\pi\)
\(588\) 0 0
\(589\) −11.5076 −0.474161
\(590\) 0 0
\(591\) −6.24621 + 10.8188i −0.256935 + 0.445024i
\(592\) 0 0
\(593\) −12.2116 21.1512i −0.501472 0.868575i −0.999999 0.00170079i \(-0.999459\pi\)
0.498526 0.866875i \(-0.333875\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.87689 4.98293i −0.117743 0.203938i
\(598\) 0 0
\(599\) 9.03457 15.6483i 0.369142 0.639373i −0.620289 0.784373i \(-0.712985\pi\)
0.989432 + 0.145000i \(0.0463182\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) −50.7386 −2.06624
\(604\) 0 0
\(605\) −2.21922 + 3.84381i −0.0902243 + 0.156273i
\(606\) 0 0
\(607\) 1.84233 + 3.19101i 0.0747778 + 0.129519i 0.900990 0.433841i \(-0.142842\pi\)
−0.826212 + 0.563360i \(0.809509\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 18.6501 + 32.3029i 0.754502 + 1.30684i
\(612\) 0 0
\(613\) 16.3693 28.3525i 0.661150 1.14515i −0.319163 0.947700i \(-0.603402\pi\)
0.980314 0.197446i \(-0.0632649\pi\)
\(614\) 0 0
\(615\) −18.2462 −0.735758
\(616\) 0 0
\(617\) −32.2462 −1.29818 −0.649092 0.760710i \(-0.724851\pi\)
−0.649092 + 0.760710i \(0.724851\pi\)
\(618\) 0 0
\(619\) 17.6847 30.6307i 0.710806 1.23115i −0.253748 0.967270i \(-0.581664\pi\)
0.964555 0.263883i \(-0.0850032\pi\)
\(620\) 0 0
\(621\) −3.68466 6.38202i −0.147860 0.256101i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 3.68466 6.38202i 0.147151 0.254873i
\(628\) 0 0
\(629\) −28.3542 −1.13055
\(630\) 0 0
\(631\) 34.4233 1.37037 0.685185 0.728369i \(-0.259721\pi\)
0.685185 + 0.728369i \(0.259721\pi\)
\(632\) 0 0
\(633\) −17.2116 + 29.8114i −0.684102 + 1.18490i
\(634\) 0 0
\(635\) 5.12311 + 8.87348i 0.203304 + 0.352133i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −21.4924 + 37.2260i −0.848900 + 1.47034i 0.0332914 + 0.999446i \(0.489401\pi\)
−0.882191 + 0.470892i \(0.843932\pi\)
\(642\) 0 0
\(643\) −7.05398 −0.278182 −0.139091 0.990280i \(-0.544418\pi\)
−0.139091 + 0.990280i \(0.544418\pi\)
\(644\) 0 0
\(645\) 2.87689 0.113278
\(646\) 0 0
\(647\) 4.00000 6.92820i 0.157256 0.272376i −0.776622 0.629967i \(-0.783068\pi\)
0.933878 + 0.357591i \(0.116402\pi\)
\(648\) 0 0
\(649\) 5.12311 + 8.87348i 0.201099 + 0.348315i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.75379 + 4.76970i 0.107764 + 0.186653i 0.914864 0.403762i \(-0.132298\pi\)
−0.807100 + 0.590415i \(0.798964\pi\)
\(654\) 0 0
\(655\) −3.43845 + 5.95557i −0.134351 + 0.232703i
\(656\) 0 0
\(657\) −43.6155 −1.70160
\(658\) 0 0
\(659\) −20.8078 −0.810555 −0.405278 0.914194i \(-0.632825\pi\)
−0.405278 + 0.914194i \(0.632825\pi\)
\(660\) 0 0
\(661\) 11.8078 20.4516i 0.459269 0.795477i −0.539654 0.841887i \(-0.681445\pi\)
0.998922 + 0.0464102i \(0.0147781\pi\)
\(662\) 0 0
\(663\) −25.0346 43.3611i −0.972262 1.68401i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 11.6847 + 20.2384i 0.452432 + 0.783635i
\(668\) 0 0
\(669\) −15.7732 + 27.3200i −0.609827 + 1.05625i
\(670\) 0 0
\(671\) −38.7386 −1.49549
\(672\) 0 0
\(673\) 0.384472 0.0148203 0.00741015 0.999973i \(-0.497641\pi\)
0.00741015 + 0.999973i \(0.497641\pi\)
\(674\) 0 0
\(675\) 0.719224 1.24573i 0.0276829 0.0479482i
\(676\) 0 0
\(677\) −6.84233 11.8513i −0.262972 0.455481i 0.704058 0.710142i \(-0.251369\pi\)
−0.967030 + 0.254661i \(0.918036\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 30.3348 + 52.5413i 1.16243 + 2.01339i
\(682\) 0 0
\(683\) 10.4924 18.1734i 0.401481 0.695386i −0.592424 0.805627i \(-0.701829\pi\)
0.993905 + 0.110241i \(0.0351621\pi\)
\(684\) 0 0
\(685\) −17.3693 −0.663647
\(686\) 0 0
\(687\) 34.2462 1.30657
\(688\) 0 0
\(689\) 13.8617 24.0092i 0.528090 0.914679i
\(690\) 0 0
\(691\) −8.24621 14.2829i −0.313701 0.543345i 0.665460 0.746434i \(-0.268236\pi\)
−0.979160 + 0.203088i \(0.934902\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.68466 2.91791i −0.0639027 0.110683i
\(696\) 0 0
\(697\) −12.2462 + 21.2111i −0.463858 + 0.803426i
\(698\) 0 0
\(699\) 48.9848 1.85278
\(700\) 0 0
\(701\) −21.6847 −0.819018 −0.409509 0.912306i \(-0.634300\pi\)
−0.409509 + 0.912306i \(0.634300\pi\)
\(702\) 0 0
\(703\) 4.63068 8.02058i 0.174650 0.302502i
\(704\) 0 0
\(705\) 8.40388 + 14.5560i 0.316509 + 0.548209i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −16.5270 28.6256i −0.620684 1.07506i −0.989359 0.145498i \(-0.953522\pi\)
0.368675 0.929559i \(-0.379812\pi\)
\(710\) 0 0
\(711\) 20.8078 36.0401i 0.780352 1.35161i
\(712\) 0 0
\(713\) −52.4924 −1.96586
\(714\) 0 0
\(715\) 14.5616 0.544571
\(716\) 0 0
\(717\) 24.4039 42.2688i 0.911380 1.57856i
\(718\) 0 0
\(719\) 8.80776 + 15.2555i 0.328474 + 0.568934i 0.982209 0.187789i \(-0.0601323\pi\)
−0.653735 + 0.756723i \(0.726799\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −2.56155 4.43674i −0.0952652 0.165004i
\(724\) 0 0
\(725\) −2.28078 + 3.95042i −0.0847059 + 0.146715i
\(726\) 0 0
\(727\) 16.9848 0.629933 0.314967 0.949103i \(-0.398007\pi\)
0.314967 + 0.949103i \(0.398007\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) 1.93087 3.34436i 0.0714158 0.123696i
\(732\) 0 0
\(733\) 10.2808 + 17.8068i 0.379729 + 0.657710i 0.991023 0.133694i \(-0.0426839\pi\)
−0.611294 + 0.791404i \(0.709351\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −18.2462 31.6034i −0.672108 1.16412i
\(738\) 0 0
\(739\) 5.28078 9.14657i 0.194257 0.336462i −0.752400 0.658706i \(-0.771104\pi\)
0.946657 + 0.322244i \(0.104437\pi\)
\(740\) 0 0
\(741\) 16.3542 0.600785
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 8.12311 14.0696i 0.297608 0.515471i
\(746\) 0 0
\(747\) 21.3693 + 37.0127i 0.781862 + 1.35423i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 21.8423 + 37.8320i 0.797038 + 1.38051i 0.921537 + 0.388290i \(0.126934\pi\)
−0.124499 + 0.992220i \(0.539732\pi\)
\(752\) 0 0
\(753\) 14.5616 25.2213i 0.530652 0.919117i
\(754\) 0 0
\(755\) −0.807764 −0.0293975
\(756\) 0 0
\(757\) −9.36932 −0.340534 −0.170267 0.985398i \(-0.554463\pi\)
−0.170267 + 0.985398i \(0.554463\pi\)
\(758\) 0 0
\(759\) 16.8078 29.1119i 0.610083 1.05670i
\(760\) 0 0
\(761\) 13.0000 + 22.5167i 0.471250 + 0.816228i 0.999459 0.0328858i \(-0.0104698\pi\)
−0.528209 + 0.849114i \(0.677136\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −6.12311 10.6055i −0.221381 0.383444i
\(766\) 0 0
\(767\) −11.3693 + 19.6922i −0.410522 + 0.711045i
\(768\) 0 0
\(769\) 22.9848 0.828855 0.414427 0.910082i \(-0.363982\pi\)
0.414427 + 0.910082i \(0.363982\pi\)
\(770\) 0 0
\(771\) 37.1231 1.33696
\(772\) 0 0
\(773\) −4.59612 + 7.96071i −0.165311 + 0.286327i −0.936766 0.349957i \(-0.886196\pi\)
0.771455 + 0.636284i \(0.219529\pi\)
\(774\) 0 0
\(775\) −5.12311 8.87348i −0.184027 0.318745i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.00000 6.92820i −0.143315 0.248229i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −6.56155 −0.234491
\(784\) 0 0
\(785\) −12.2462 −0.437086
\(786\) 0 0
\(787\) −6.08854 + 10.5457i −0.217033 + 0.375912i −0.953900 0.300126i \(-0.902971\pi\)
0.736867 + 0.676038i \(0.236305\pi\)
\(788\) 0 0
\(789\) −19.6847 34.0948i −0.700792 1.21381i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −42.9848 74.4519i −1.52644 2.64387i
\(794\) 0 0
\(795\) 6.24621 10.8188i 0.221530 0.383702i
\(796\) 0 0
\(797\) 2.80776 0.0994561 0.0497281 0.998763i \(-0.484165\pi\)
0.0497281 + 0.998763i \(0.484165\pi\)
\(798\) 0 0
\(799\) 22.5616 0.798170
\(800\) 0 0
\(801\) −5.56155 + 9.63289i −0.196508 + 0.340362i
\(802\) 0 0
\(803\) −15.6847 27.1666i −0.553500 0.958689i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 15.6847 + 27.1666i 0.552126 + 0.956311i
\(808\) 0 0
\(809\) 16.5270 28.6256i 0.581058 1.00642i −0.414296 0.910142i \(-0.635972\pi\)
0.995354 0.0962798i \(-0.0306944\pi\)
\(810\) 0 0
\(811\) −13.6155 −0.478106 −0.239053 0.971007i \(-0.576837\pi\)
−0.239053 + 0.971007i \(0.576837\pi\)
\(812\) 0 0
\(813\) 26.2462 0.920495
\(814\) 0 0
\(815\) −10.8078 + 18.7196i −0.378579 + 0.655719i
\(816\) 0 0
\(817\) 0.630683 + 1.09238i 0.0220648 + 0.0382174i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −0.349907 0.606056i −0.0122118 0.0211515i 0.859855 0.510538i \(-0.170554\pi\)
−0.872067 + 0.489387i \(0.837221\pi\)
\(822\) 0 0
\(823\) 17.1231 29.6581i 0.596874 1.03382i −0.396406 0.918076i \(-0.629743\pi\)
0.993279 0.115740i \(-0.0369241\pi\)
\(824\) 0 0
\(825\) 6.56155 0.228444
\(826\) 0 0
\(827\) 33.1231 1.15180 0.575902 0.817519i \(-0.304651\pi\)
0.575902 + 0.817519i \(0.304651\pi\)
\(828\) 0 0
\(829\) −17.8078 + 30.8440i −0.618489 + 1.07125i 0.371272 + 0.928524i \(0.378922\pi\)
−0.989762 + 0.142731i \(0.954412\pi\)
\(830\) 0 0
\(831\) 10.5616 + 18.2931i 0.366376 + 0.634582i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −12.4039 21.4842i −0.429254 0.743490i
\(836\) 0 0
\(837\) 7.36932 12.7640i 0.254721 0.441189i
\(838\) 0 0
\(839\) −10.8769 −0.375512 −0.187756 0.982216i \(-0.560121\pi\)
−0.187756 + 0.982216i \(0.560121\pi\)
\(840\) 0 0
\(841\) −8.19224 −0.282491
\(842\) 0 0
\(843\) −19.7732 + 34.2482i −0.681025 + 1.17957i
\(844\) 0 0
\(845\) 9.65767 + 16.7276i 0.332234 + 0.575446i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −9.84233 17.0474i −0.337788 0.585066i
\(850\) 0 0
\(851\) 21.1231 36.5863i 0.724091 1.25416i
\(852\) 0 0
\(853\) −32.2462 −1.10409 −0.552045 0.833815i \(-0.686152\pi\)
−0.552045 + 0.833815i \(0.686152\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) 1.63068 2.82443i 0.0557031 0.0964806i −0.836829 0.547464i \(-0.815593\pi\)
0.892532 + 0.450983i \(0.148927\pi\)
\(858\) 0 0
\(859\) −14.4924 25.1016i −0.494475 0.856456i 0.505505 0.862824i \(-0.331306\pi\)
−0.999980 + 0.00636793i \(0.997973\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.24621 + 3.89055i 0.0764619 + 0.132436i 0.901721 0.432318i \(-0.142304\pi\)
−0.825259 + 0.564754i \(0.808971\pi\)
\(864\) 0 0
\(865\) −11.0885 + 19.2059i −0.377022 + 0.653021i
\(866\) 0 0
\(867\) 13.2614 0.450380
\(868\) 0 0
\(869\) 29.9309 1.01534
\(870\) 0 0
\(871\) 40.4924 70.1349i 1.37203 2.37643i
\(872\) 0 0
\(873\) 24.3693 + 42.2089i 0.824776 + 1.42855i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 23.2462 + 40.2636i 0.784969 + 1.35961i 0.929018 + 0.370035i \(0.120654\pi\)
−0.144049 + 0.989571i \(0.546012\pi\)
\(878\) 0 0
\(879\) 6.47301 11.2116i 0.218329 0.378157i
\(880\) 0 0
\(881\) 20.1080 0.677454 0.338727 0.940885i \(-0.390004\pi\)
0.338727 + 0.940885i \(0.390004\pi\)
\(882\) 0 0
\(883\) 38.2462 1.28709 0.643544 0.765409i \(-0.277463\pi\)
0.643544 + 0.765409i \(0.277463\pi\)
\(884\) 0 0
\(885\) −5.12311 + 8.87348i −0.172211 + 0.298279i
\(886\) 0 0
\(887\) −1.75379 3.03765i −0.0588865 0.101994i 0.835079 0.550130i \(-0.185422\pi\)
−0.893966 + 0.448135i \(0.852088\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −8.96543 15.5286i −0.300353 0.520227i
\(892\) 0 0
\(893\) −3.68466 + 6.38202i −0.123302 + 0.213566i
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) 0 0
\(897\) 74.6004 2.49083
\(898\) 0 0
\(899\) −23.3693 + 40.4768i −0.779410 + 1.34998i
\(900\) 0 0
\(901\) −8.38447 14.5223i −0.279327 0.483809i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.56155 2.70469i −0.0519078 0.0899069i
\(906\) 0 0
\(907\) −13.0540 + 22.6101i −0.433450 + 0.750758i −0.997168 0.0752099i \(-0.976037\pi\)
0.563718 + 0.825968i \(0.309371\pi\)
\(908\) 0 0
\(909\) −21.3693 −0.708776
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) −15.3693 + 26.6204i −0.508650 + 0.881008i
\(914\) 0 0
\(915\) −19.3693 33.5486i −0.640330 1.10908i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −3.28078 5.68247i −0.108223 0.187447i 0.806828 0.590787i \(-0.201183\pi\)
−0.915050 + 0.403340i \(0.867849\pi\)
\(920\) 0 0
\(921\) 20.8963 36.1935i 0.688557 1.19262i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 8.24621 0.271134
\(926\) 0 0
\(927\) 15.6847 27.1666i 0.515152 0.892269i
\(928\) 0 0
\(929\) −5.56155 9.63289i −0.182469 0.316045i 0.760252 0.649628i \(-0.225075\pi\)
−0.942721 + 0.333583i \(0.891742\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 27.0540 + 46.8589i 0.885707 + 1.53409i
\(934\) 0 0
\(935\) 4.40388 7.62775i 0.144022 0.249454i
\(936\) 0 0
\(937\) −17.1922 −0.561646 −0.280823 0.959760i \(-0.590607\pi\)
−0.280823 + 0.959760i \(0.590607\pi\)
\(938\) 0 0
\(939\) 36.6695 1.19666
\(940\) 0 0
\(941\) 26.6847 46.2192i 0.869895 1.50670i 0.00779178 0.999970i \(-0.497520\pi\)
0.862103 0.506733i \(-0.169147\pi\)
\(942\) 0 0
\(943\) −18.2462 31.6034i −0.594178 1.02915i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −23.6155 40.9033i −0.767402 1.32918i −0.938968 0.344006i \(-0.888216\pi\)
0.171566 0.985173i \(-0.445117\pi\)
\(948\) 0 0
\(949\) 34.8078 60.2888i 1.12991 1.95706i
\(950\) 0 0
\(951\) 25.6155 0.830640
\(952\) 0 0
\(953\) 5.86174 0.189880 0.0949402 0.995483i \(-0.469734\pi\)
0.0949402 + 0.995483i \(0.469734\pi\)
\(954\) 0 0
\(955\) 5.84233 10.1192i 0.189053 0.327450i
\(956\) 0 0
\(957\) −14.9654 25.9209i −0.483764 0.837903i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −36.9924 64.0728i −1.19330 2.06686i
\(962\) 0 0
\(963\) −12.2462 + 21.2111i −0.394629 + 0.683517i
\(964\) 0 0
\(965\) −4.87689 −0.156993
\(966\) 0 0
\(967\) −30.1080 −0.968206 −0.484103 0.875011i \(-0.660854\pi\)
−0.484103 + 0.875011i \(0.660854\pi\)
\(968\) 0 0
\(969\) 4.94602 8.56677i 0.158889 0.275204i
\(970\) 0 0
\(971\) 13.3693 + 23.1563i 0.429042 + 0.743122i 0.996788 0.0800812i \(-0.0255180\pi\)
−0.567747 + 0.823203i \(0.692185\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 7.28078 + 12.6107i 0.233171 + 0.403865i
\(976\) 0 0
\(977\) 27.4924 47.6183i 0.879561 1.52344i 0.0277366 0.999615i \(-0.491170\pi\)
0.851824 0.523828i \(-0.175497\pi\)
\(978\) 0 0
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) −30.4924 −0.973548
\(982\) 0 0
\(983\) 8.08854 14.0098i 0.257984 0.446842i −0.707717 0.706496i \(-0.750275\pi\)
0.965702 + 0.259654i \(0.0836084\pi\)
\(984\) 0 0
\(985\) −2.43845 4.22351i −0.0776954 0.134572i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.87689 + 4.98293i 0.0914799 + 0.158448i
\(990\) 0 0
\(991\) −6.24621 + 10.8188i −0.198417 + 0.343669i −0.948015 0.318224i \(-0.896914\pi\)
0.749598 + 0.661893i \(0.230247\pi\)
\(992\) 0 0
\(993\) 30.7386 0.975461
\(994\) 0 0
\(995\) 2.24621 0.0712097
\(996\) 0 0
\(997\) −14.0346 + 24.3086i −0.444479 + 0.769860i −0.998016 0.0629644i \(-0.979945\pi\)
0.553537 + 0.832825i \(0.313278\pi\)
\(998\) 0 0
\(999\) 5.93087 + 10.2726i 0.187644 + 0.325010i
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 1960.2.q.u.361.2 4
7.2 even 3 inner 1960.2.q.u.961.2 4
7.3 odd 6 280.2.a.d.1.2 2
7.4 even 3 1960.2.a.r.1.1 2
7.5 odd 6 1960.2.q.s.961.1 4
7.6 odd 2 1960.2.q.s.361.1 4
21.17 even 6 2520.2.a.w.1.2 2
28.3 even 6 560.2.a.g.1.1 2
28.11 odd 6 3920.2.a.bu.1.2 2
35.3 even 12 1400.2.g.k.449.4 4
35.4 even 6 9800.2.a.by.1.2 2
35.17 even 12 1400.2.g.k.449.1 4
35.24 odd 6 1400.2.a.p.1.1 2
56.3 even 6 2240.2.a.bi.1.2 2
56.45 odd 6 2240.2.a.be.1.1 2
84.59 odd 6 5040.2.a.bq.1.1 2
140.3 odd 12 2800.2.g.u.449.1 4
140.59 even 6 2800.2.a.bn.1.2 2
140.87 odd 12 2800.2.g.u.449.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.d.1.2 2 7.3 odd 6
560.2.a.g.1.1 2 28.3 even 6
1400.2.a.p.1.1 2 35.24 odd 6
1400.2.g.k.449.1 4 35.17 even 12
1400.2.g.k.449.4 4 35.3 even 12
1960.2.a.r.1.1 2 7.4 even 3
1960.2.q.s.361.1 4 7.6 odd 2
1960.2.q.s.961.1 4 7.5 odd 6
1960.2.q.u.361.2 4 1.1 even 1 trivial
1960.2.q.u.961.2 4 7.2 even 3 inner
2240.2.a.be.1.1 2 56.45 odd 6
2240.2.a.bi.1.2 2 56.3 even 6
2520.2.a.w.1.2 2 21.17 even 6
2800.2.a.bn.1.2 2 140.59 even 6
2800.2.g.u.449.1 4 140.3 odd 12
2800.2.g.u.449.4 4 140.87 odd 12
3920.2.a.bu.1.2 2 28.11 odd 6
5040.2.a.bq.1.1 2 84.59 odd 6
9800.2.a.by.1.2 2 35.4 even 6