Properties

Label 1296.3.o.a.703.1
Level $1296$
Weight $3$
Character 1296.703
Analytic conductor $35.313$
Analytic rank $0$
Dimension $2$
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] = [1296,3,Mod(271,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1296.271");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1296.o (of order \(6\), 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: \(35.3134422611\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 703.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1296.703
Dual form 1296.3.o.a.271.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.00000 - 5.19615i) q^{5} +(-6.00000 - 3.46410i) q^{7} +O(q^{10})\) \(q+(-3.00000 - 5.19615i) q^{5} +(-6.00000 - 3.46410i) q^{7} +(18.0000 + 10.3923i) q^{11} +(7.00000 + 12.1244i) q^{13} -6.00000 q^{17} -6.92820i q^{19} +(-5.50000 + 9.52628i) q^{25} +(-15.0000 + 25.9808i) q^{29} +(18.0000 - 10.3923i) q^{31} +41.5692i q^{35} +26.0000 q^{37} +(27.0000 + 46.7654i) q^{41} +(18.0000 + 10.3923i) q^{43} +(-36.0000 - 20.7846i) q^{47} +(-0.500000 - 0.866025i) q^{49} -18.0000 q^{53} -124.708i q^{55} +(18.0000 - 10.3923i) q^{59} +(35.0000 - 60.6218i) q^{61} +(42.0000 - 72.7461i) q^{65} +(102.000 - 58.8897i) q^{67} -83.1384i q^{71} +82.0000 q^{73} +(-72.0000 - 124.708i) q^{77} +(-66.0000 - 38.1051i) q^{79} +(-18.0000 - 10.3923i) q^{83} +(18.0000 + 31.1769i) q^{85} +114.000 q^{89} -96.9948i q^{91} +(-36.0000 + 20.7846i) q^{95} +(-17.0000 + 29.4449i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{5} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{5} - 12 q^{7} + 36 q^{11} + 14 q^{13} - 12 q^{17} - 11 q^{25} - 30 q^{29} + 36 q^{31} + 52 q^{37} + 54 q^{41} + 36 q^{43} - 72 q^{47} - q^{49} - 36 q^{53} + 36 q^{59} + 70 q^{61} + 84 q^{65} + 204 q^{67} + 164 q^{73} - 144 q^{77} - 132 q^{79} - 36 q^{83} + 36 q^{85} + 228 q^{89} - 72 q^{95} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) −3.00000 5.19615i −0.600000 1.03923i −0.992820 0.119615i \(-0.961834\pi\)
0.392820 0.919615i \(-0.371499\pi\)
\(6\) 0 0
\(7\) −6.00000 3.46410i −0.857143 0.494872i 0.00591161 0.999983i \(-0.498118\pi\)
−0.863054 + 0.505111i \(0.831452\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 18.0000 + 10.3923i 1.63636 + 0.944755i 0.982071 + 0.188514i \(0.0603671\pi\)
0.654293 + 0.756241i \(0.272966\pi\)
\(12\) 0 0
\(13\) 7.00000 + 12.1244i 0.538462 + 0.932643i 0.998987 + 0.0449963i \(0.0143276\pi\)
−0.460526 + 0.887646i \(0.652339\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −0.352941 −0.176471 0.984306i \(-0.556468\pi\)
−0.176471 + 0.984306i \(0.556468\pi\)
\(18\) 0 0
\(19\) 6.92820i 0.364642i −0.983239 0.182321i \(-0.941639\pi\)
0.983239 0.182321i \(-0.0583610\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) −5.50000 + 9.52628i −0.220000 + 0.381051i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −15.0000 + 25.9808i −0.517241 + 0.895888i 0.482558 + 0.875864i \(0.339708\pi\)
−0.999799 + 0.0200244i \(0.993626\pi\)
\(30\) 0 0
\(31\) 18.0000 10.3923i 0.580645 0.335236i −0.180745 0.983530i \(-0.557851\pi\)
0.761390 + 0.648294i \(0.224517\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 41.5692i 1.18769i
\(36\) 0 0
\(37\) 26.0000 0.702703 0.351351 0.936244i \(-0.385722\pi\)
0.351351 + 0.936244i \(0.385722\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 27.0000 + 46.7654i 0.658537 + 1.14062i 0.980995 + 0.194035i \(0.0621576\pi\)
−0.322458 + 0.946584i \(0.604509\pi\)
\(42\) 0 0
\(43\) 18.0000 + 10.3923i 0.418605 + 0.241682i 0.694480 0.719512i \(-0.255634\pi\)
−0.275875 + 0.961193i \(0.588968\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −36.0000 20.7846i −0.765957 0.442226i 0.0654732 0.997854i \(-0.479144\pi\)
−0.831431 + 0.555629i \(0.812478\pi\)
\(48\) 0 0
\(49\) −0.500000 0.866025i −0.0102041 0.0176740i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −18.0000 −0.339623 −0.169811 0.985477i \(-0.554316\pi\)
−0.169811 + 0.985477i \(0.554316\pi\)
\(54\) 0 0
\(55\) 124.708i 2.26741i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 18.0000 10.3923i 0.305085 0.176141i −0.339640 0.940555i \(-0.610305\pi\)
0.644725 + 0.764415i \(0.276972\pi\)
\(60\) 0 0
\(61\) 35.0000 60.6218i 0.573770 0.993800i −0.422404 0.906408i \(-0.638813\pi\)
0.996174 0.0873918i \(-0.0278532\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 42.0000 72.7461i 0.646154 1.11917i
\(66\) 0 0
\(67\) 102.000 58.8897i 1.52239 0.878951i 0.522738 0.852493i \(-0.324911\pi\)
0.999650 0.0264578i \(-0.00842278\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 83.1384i 1.17096i −0.810685 0.585482i \(-0.800905\pi\)
0.810685 0.585482i \(-0.199095\pi\)
\(72\) 0 0
\(73\) 82.0000 1.12329 0.561644 0.827379i \(-0.310169\pi\)
0.561644 + 0.827379i \(0.310169\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −72.0000 124.708i −0.935065 1.61958i
\(78\) 0 0
\(79\) −66.0000 38.1051i −0.835443 0.482343i 0.0202696 0.999795i \(-0.493548\pi\)
−0.855713 + 0.517451i \(0.826881\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −18.0000 10.3923i −0.216867 0.125208i 0.387631 0.921814i \(-0.373293\pi\)
−0.604499 + 0.796606i \(0.706627\pi\)
\(84\) 0 0
\(85\) 18.0000 + 31.1769i 0.211765 + 0.366787i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 114.000 1.28090 0.640449 0.768000i \(-0.278748\pi\)
0.640449 + 0.768000i \(0.278748\pi\)
\(90\) 0 0
\(91\) 96.9948i 1.06588i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −36.0000 + 20.7846i −0.378947 + 0.218785i
\(96\) 0 0
\(97\) −17.0000 + 29.4449i −0.175258 + 0.303555i −0.940250 0.340484i \(-0.889409\pi\)
0.764993 + 0.644039i \(0.222743\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.00000 15.5885i 0.0891089 0.154341i −0.818026 0.575182i \(-0.804931\pi\)
0.907135 + 0.420840i \(0.138265\pi\)
\(102\) 0 0
\(103\) 114.000 65.8179i 1.10680 0.639009i 0.168798 0.985651i \(-0.446011\pi\)
0.937998 + 0.346642i \(0.112678\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 145.492i 1.35974i −0.733332 0.679870i \(-0.762036\pi\)
0.733332 0.679870i \(-0.237964\pi\)
\(108\) 0 0
\(109\) 34.0000 0.311927 0.155963 0.987763i \(-0.450152\pi\)
0.155963 + 0.987763i \(0.450152\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 39.0000 + 67.5500i 0.345133 + 0.597787i 0.985378 0.170383i \(-0.0545006\pi\)
−0.640245 + 0.768171i \(0.721167\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 36.0000 + 20.7846i 0.302521 + 0.174661i
\(120\) 0 0
\(121\) 155.500 + 269.334i 1.28512 + 2.22590i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −84.0000 −0.672000
\(126\) 0 0
\(127\) 103.923i 0.818292i −0.912469 0.409146i \(-0.865827\pi\)
0.912469 0.409146i \(-0.134173\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −90.0000 + 51.9615i −0.687023 + 0.396653i −0.802496 0.596658i \(-0.796495\pi\)
0.115473 + 0.993311i \(0.463162\pi\)
\(132\) 0 0
\(133\) −24.0000 + 41.5692i −0.180451 + 0.312551i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −93.0000 + 161.081i −0.678832 + 1.17577i 0.296501 + 0.955033i \(0.404180\pi\)
−0.975333 + 0.220739i \(0.929153\pi\)
\(138\) 0 0
\(139\) 42.0000 24.2487i 0.302158 0.174451i −0.341254 0.939971i \(-0.610852\pi\)
0.643412 + 0.765520i \(0.277518\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 290.985i 2.03486i
\(144\) 0 0
\(145\) 180.000 1.24138
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 93.0000 + 161.081i 0.624161 + 1.08108i 0.988702 + 0.149892i \(0.0478924\pi\)
−0.364541 + 0.931187i \(0.618774\pi\)
\(150\) 0 0
\(151\) −30.0000 17.3205i −0.198675 0.114705i 0.397362 0.917662i \(-0.369926\pi\)
−0.596038 + 0.802957i \(0.703259\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −108.000 62.3538i −0.696774 0.402283i
\(156\) 0 0
\(157\) −85.0000 147.224i −0.541401 0.937735i −0.998824 0.0484851i \(-0.984561\pi\)
0.457423 0.889249i \(-0.348773\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 284.056i 1.74268i −0.490683 0.871338i \(-0.663253\pi\)
0.490683 0.871338i \(-0.336747\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 180.000 103.923i 1.07784 0.622294i 0.147530 0.989058i \(-0.452868\pi\)
0.930314 + 0.366764i \(0.119534\pi\)
\(168\) 0 0
\(169\) −13.5000 + 23.3827i −0.0798817 + 0.138359i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 21.0000 36.3731i 0.121387 0.210249i −0.798928 0.601427i \(-0.794599\pi\)
0.920315 + 0.391178i \(0.127932\pi\)
\(174\) 0 0
\(175\) 66.0000 38.1051i 0.377143 0.217744i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 145.492i 0.812806i 0.913694 + 0.406403i \(0.133217\pi\)
−0.913694 + 0.406403i \(0.866783\pi\)
\(180\) 0 0
\(181\) 82.0000 0.453039 0.226519 0.974007i \(-0.427265\pi\)
0.226519 + 0.974007i \(0.427265\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −78.0000 135.100i −0.421622 0.730270i
\(186\) 0 0
\(187\) −108.000 62.3538i −0.577540 0.333443i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 288.000 + 166.277i 1.50785 + 0.870560i 0.999958 + 0.00914069i \(0.00290961\pi\)
0.507895 + 0.861419i \(0.330424\pi\)
\(192\) 0 0
\(193\) 47.0000 + 81.4064i 0.243523 + 0.421795i 0.961715 0.274050i \(-0.0883634\pi\)
−0.718192 + 0.695845i \(0.755030\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −258.000 −1.30964 −0.654822 0.755783i \(-0.727257\pi\)
−0.654822 + 0.755783i \(0.727257\pi\)
\(198\) 0 0
\(199\) 117.779i 0.591857i −0.955210 0.295928i \(-0.904371\pi\)
0.955210 0.295928i \(-0.0956289\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 180.000 103.923i 0.886700 0.511936i
\(204\) 0 0
\(205\) 162.000 280.592i 0.790244 1.36874i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 72.0000 124.708i 0.344498 0.596687i
\(210\) 0 0
\(211\) 78.0000 45.0333i 0.369668 0.213428i −0.303645 0.952785i \(-0.598204\pi\)
0.673314 + 0.739357i \(0.264870\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 124.708i 0.580036i
\(216\) 0 0
\(217\) −144.000 −0.663594
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −42.0000 72.7461i −0.190045 0.329168i
\(222\) 0 0
\(223\) 306.000 + 176.669i 1.37220 + 0.792238i 0.991204 0.132339i \(-0.0422489\pi\)
0.380993 + 0.924578i \(0.375582\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 126.000 + 72.7461i 0.555066 + 0.320468i 0.751163 0.660117i \(-0.229493\pi\)
−0.196097 + 0.980585i \(0.562827\pi\)
\(228\) 0 0
\(229\) −113.000 195.722i −0.493450 0.854680i 0.506522 0.862227i \(-0.330931\pi\)
−0.999972 + 0.00754710i \(0.997598\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 114.000 0.489270 0.244635 0.969615i \(-0.421332\pi\)
0.244635 + 0.969615i \(0.421332\pi\)
\(234\) 0 0
\(235\) 249.415i 1.06134i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −288.000 + 166.277i −1.20502 + 0.695719i −0.961667 0.274219i \(-0.911581\pi\)
−0.243354 + 0.969938i \(0.578248\pi\)
\(240\) 0 0
\(241\) −89.0000 + 154.153i −0.369295 + 0.639637i −0.989455 0.144838i \(-0.953734\pi\)
0.620161 + 0.784475i \(0.287067\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.00000 + 5.19615i −0.0122449 + 0.0212088i
\(246\) 0 0
\(247\) 84.0000 48.4974i 0.340081 0.196346i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 103.923i 0.414036i −0.978337 0.207018i \(-0.933624\pi\)
0.978337 0.207018i \(-0.0663759\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −129.000 223.435i −0.501946 0.869395i −0.999997 0.00224796i \(-0.999284\pi\)
0.498052 0.867147i \(-0.334049\pi\)
\(258\) 0 0
\(259\) −156.000 90.0666i −0.602317 0.347748i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −324.000 187.061i −1.23194 0.711260i −0.264505 0.964384i \(-0.585209\pi\)
−0.967434 + 0.253124i \(0.918542\pi\)
\(264\) 0 0
\(265\) 54.0000 + 93.5307i 0.203774 + 0.352946i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 510.000 1.89591 0.947955 0.318403i \(-0.103147\pi\)
0.947955 + 0.318403i \(0.103147\pi\)
\(270\) 0 0
\(271\) 450.333i 1.66175i 0.556462 + 0.830873i \(0.312158\pi\)
−0.556462 + 0.830873i \(0.687842\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −198.000 + 114.315i −0.720000 + 0.415692i
\(276\) 0 0
\(277\) 7.00000 12.1244i 0.0252708 0.0437702i −0.853113 0.521725i \(-0.825289\pi\)
0.878384 + 0.477955i \(0.158622\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −177.000 + 306.573i −0.629893 + 1.09101i 0.357679 + 0.933844i \(0.383568\pi\)
−0.987573 + 0.157163i \(0.949765\pi\)
\(282\) 0 0
\(283\) −126.000 + 72.7461i −0.445230 + 0.257053i −0.705813 0.708398i \(-0.749418\pi\)
0.260584 + 0.965451i \(0.416085\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 374.123i 1.30356i
\(288\) 0 0
\(289\) −253.000 −0.875433
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 249.000 + 431.281i 0.849829 + 1.47195i 0.881360 + 0.472445i \(0.156628\pi\)
−0.0315309 + 0.999503i \(0.510038\pi\)
\(294\) 0 0
\(295\) −108.000 62.3538i −0.366102 0.211369i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −72.0000 124.708i −0.239203 0.414311i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −420.000 −1.37705
\(306\) 0 0
\(307\) 187.061i 0.609321i −0.952461 0.304660i \(-0.901457\pi\)
0.952461 0.304660i \(-0.0985430\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 36.0000 20.7846i 0.115756 0.0668315i −0.441004 0.897505i \(-0.645378\pi\)
0.556760 + 0.830673i \(0.312044\pi\)
\(312\) 0 0
\(313\) −145.000 + 251.147i −0.463259 + 0.802388i −0.999121 0.0419176i \(-0.986653\pi\)
0.535862 + 0.844305i \(0.319987\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 105.000 181.865i 0.331230 0.573708i −0.651523 0.758629i \(-0.725870\pi\)
0.982753 + 0.184921i \(0.0592030\pi\)
\(318\) 0 0
\(319\) −540.000 + 311.769i −1.69279 + 0.977333i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 41.5692i 0.128697i
\(324\) 0 0
\(325\) −154.000 −0.473846
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 144.000 + 249.415i 0.437690 + 0.758101i
\(330\) 0 0
\(331\) 174.000 + 100.459i 0.525680 + 0.303501i 0.739255 0.673425i \(-0.235178\pi\)
−0.213576 + 0.976927i \(0.568511\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −612.000 353.338i −1.82687 1.05474i
\(336\) 0 0
\(337\) 151.000 + 261.540i 0.448071 + 0.776082i 0.998260 0.0589579i \(-0.0187778\pi\)
−0.550189 + 0.835040i \(0.685444\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 432.000 1.26686
\(342\) 0 0
\(343\) 346.410i 1.00994i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 54.0000 31.1769i 0.155620 0.0898470i −0.420168 0.907446i \(-0.638029\pi\)
0.575788 + 0.817599i \(0.304696\pi\)
\(348\) 0 0
\(349\) 179.000 310.037i 0.512894 0.888358i −0.486994 0.873405i \(-0.661907\pi\)
0.999888 0.0149532i \(-0.00475994\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 279.000 483.242i 0.790368 1.36896i −0.135371 0.990795i \(-0.543223\pi\)
0.925739 0.378163i \(-0.123444\pi\)
\(354\) 0 0
\(355\) −432.000 + 249.415i −1.21690 + 0.702578i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 83.1384i 0.231583i 0.993274 + 0.115792i \(0.0369405\pi\)
−0.993274 + 0.115792i \(0.963059\pi\)
\(360\) 0 0
\(361\) 313.000 0.867036
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −246.000 426.084i −0.673973 1.16735i
\(366\) 0 0
\(367\) −186.000 107.387i −0.506812 0.292608i 0.224710 0.974426i \(-0.427856\pi\)
−0.731522 + 0.681818i \(0.761190\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 108.000 + 62.3538i 0.291105 + 0.168070i
\(372\) 0 0
\(373\) −277.000 479.778i −0.742627 1.28627i −0.951295 0.308282i \(-0.900246\pi\)
0.208668 0.977987i \(-0.433087\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −420.000 −1.11406
\(378\) 0 0
\(379\) 533.472i 1.40758i 0.710410 + 0.703788i \(0.248510\pi\)
−0.710410 + 0.703788i \(0.751490\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 432.000 249.415i 1.12794 0.651215i 0.184522 0.982828i \(-0.440926\pi\)
0.943415 + 0.331613i \(0.107593\pi\)
\(384\) 0 0
\(385\) −432.000 + 748.246i −1.12208 + 1.94350i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −99.0000 + 171.473i −0.254499 + 0.440805i −0.964759 0.263134i \(-0.915244\pi\)
0.710261 + 0.703939i \(0.248577\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 457.261i 1.15762i
\(396\) 0 0
\(397\) −646.000 −1.62720 −0.813602 0.581422i \(-0.802496\pi\)
−0.813602 + 0.581422i \(0.802496\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −165.000 285.788i −0.411471 0.712689i 0.583580 0.812056i \(-0.301652\pi\)
−0.995051 + 0.0993667i \(0.968318\pi\)
\(402\) 0 0
\(403\) 252.000 + 145.492i 0.625310 + 0.361023i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 468.000 + 270.200i 1.14988 + 0.663882i
\(408\) 0 0
\(409\) −65.0000 112.583i −0.158924 0.275265i 0.775557 0.631278i \(-0.217469\pi\)
−0.934481 + 0.356013i \(0.884136\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −144.000 −0.348668
\(414\) 0 0
\(415\) 124.708i 0.300500i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 306.000 176.669i 0.730310 0.421645i −0.0882254 0.996101i \(-0.528120\pi\)
0.818536 + 0.574456i \(0.194786\pi\)
\(420\) 0 0
\(421\) 199.000 344.678i 0.472684 0.818713i −0.526827 0.849972i \(-0.676619\pi\)
0.999511 + 0.0312596i \(0.00995185\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 33.0000 57.1577i 0.0776471 0.134489i
\(426\) 0 0
\(427\) −420.000 + 242.487i −0.983607 + 0.567886i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 124.708i 0.289345i 0.989480 + 0.144672i \(0.0462128\pi\)
−0.989480 + 0.144672i \(0.953787\pi\)
\(432\) 0 0
\(433\) −142.000 −0.327945 −0.163972 0.986465i \(-0.552431\pi\)
−0.163972 + 0.986465i \(0.552431\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 486.000 + 280.592i 1.10706 + 0.639162i 0.938067 0.346455i \(-0.112615\pi\)
0.168995 + 0.985617i \(0.445948\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 378.000 + 218.238i 0.853273 + 0.492637i 0.861754 0.507327i \(-0.169366\pi\)
−0.00848075 + 0.999964i \(0.502700\pi\)
\(444\) 0 0
\(445\) −342.000 592.361i −0.768539 1.33115i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −198.000 −0.440980 −0.220490 0.975389i \(-0.570766\pi\)
−0.220490 + 0.975389i \(0.570766\pi\)
\(450\) 0 0
\(451\) 1122.37i 2.48862i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −504.000 + 290.985i −1.10769 + 0.639526i
\(456\) 0 0
\(457\) 223.000 386.247i 0.487965 0.845180i −0.511939 0.859022i \(-0.671073\pi\)
0.999904 + 0.0138415i \(0.00440603\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −171.000 + 296.181i −0.370933 + 0.642474i −0.989709 0.143093i \(-0.954295\pi\)
0.618777 + 0.785567i \(0.287629\pi\)
\(462\) 0 0
\(463\) −138.000 + 79.6743i −0.298056 + 0.172083i −0.641569 0.767065i \(-0.721716\pi\)
0.343513 + 0.939148i \(0.388383\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 394.908i 0.845627i −0.906217 0.422813i \(-0.861043\pi\)
0.906217 0.422813i \(-0.138957\pi\)
\(468\) 0 0
\(469\) −816.000 −1.73987
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 216.000 + 374.123i 0.456660 + 0.790958i
\(474\) 0 0
\(475\) 66.0000 + 38.1051i 0.138947 + 0.0802213i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 684.000 + 394.908i 1.42797 + 0.824442i 0.996961 0.0779035i \(-0.0248226\pi\)
0.431014 + 0.902345i \(0.358156\pi\)
\(480\) 0 0
\(481\) 182.000 + 315.233i 0.378378 + 0.655371i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 204.000 0.420619
\(486\) 0 0
\(487\) 6.92820i 0.0142263i 0.999975 + 0.00711315i \(0.00226420\pi\)
−0.999975 + 0.00711315i \(0.997736\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −558.000 + 322.161i −1.13646 + 0.656133i −0.945551 0.325475i \(-0.894476\pi\)
−0.190906 + 0.981608i \(0.561142\pi\)
\(492\) 0 0
\(493\) 90.0000 155.885i 0.182556 0.316196i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −288.000 + 498.831i −0.579477 + 1.00368i
\(498\) 0 0
\(499\) 702.000 405.300i 1.40681 0.812224i 0.411734 0.911304i \(-0.364923\pi\)
0.995079 + 0.0990798i \(0.0315899\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 332.554i 0.661141i 0.943781 + 0.330570i \(0.107241\pi\)
−0.943781 + 0.330570i \(0.892759\pi\)
\(504\) 0 0
\(505\) −108.000 −0.213861
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 153.000 + 265.004i 0.300589 + 0.520636i 0.976270 0.216559i \(-0.0694833\pi\)
−0.675680 + 0.737195i \(0.736150\pi\)
\(510\) 0 0
\(511\) −492.000 284.056i −0.962818 0.555883i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −684.000 394.908i −1.32816 0.766811i
\(516\) 0 0
\(517\) −432.000 748.246i −0.835590 1.44728i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 522.000 1.00192 0.500960 0.865471i \(-0.332980\pi\)
0.500960 + 0.865471i \(0.332980\pi\)
\(522\) 0 0
\(523\) 48.4974i 0.0927293i −0.998925 0.0463646i \(-0.985236\pi\)
0.998925 0.0463646i \(-0.0147636\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −108.000 + 62.3538i −0.204934 + 0.118318i
\(528\) 0 0
\(529\) −264.500 + 458.127i −0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −378.000 + 654.715i −0.709193 + 1.22836i
\(534\) 0 0
\(535\) −756.000 + 436.477i −1.41308 + 0.815844i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 20.7846i 0.0385614i
\(540\) 0 0
\(541\) 802.000 1.48244 0.741220 0.671262i \(-0.234248\pi\)
0.741220 + 0.671262i \(0.234248\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −102.000 176.669i −0.187156 0.324164i
\(546\) 0 0
\(547\) 30.0000 + 17.3205i 0.0548446 + 0.0316645i 0.527172 0.849759i \(-0.323253\pi\)
−0.472327 + 0.881423i \(0.656586\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 180.000 + 103.923i 0.326679 + 0.188608i
\(552\) 0 0
\(553\) 264.000 + 457.261i 0.477396 + 0.826874i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −474.000 −0.850987 −0.425494 0.904961i \(-0.639900\pi\)
−0.425494 + 0.904961i \(0.639900\pi\)
\(558\) 0 0
\(559\) 290.985i 0.520545i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 594.000 342.946i 1.05506 0.609140i 0.131000 0.991382i \(-0.458181\pi\)
0.924062 + 0.382242i \(0.124848\pi\)
\(564\) 0 0
\(565\) 234.000 405.300i 0.414159 0.717345i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 75.0000 129.904i 0.131810 0.228302i −0.792564 0.609788i \(-0.791254\pi\)
0.924374 + 0.381487i \(0.124588\pi\)
\(570\) 0 0
\(571\) −582.000 + 336.018i −1.01926 + 0.588473i −0.913891 0.405960i \(-0.866937\pi\)
−0.105374 + 0.994433i \(0.533604\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −46.0000 −0.0797227 −0.0398614 0.999205i \(-0.512692\pi\)
−0.0398614 + 0.999205i \(0.512692\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 72.0000 + 124.708i 0.123924 + 0.214643i
\(582\) 0 0
\(583\) −324.000 187.061i −0.555746 0.320860i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −306.000 176.669i −0.521295 0.300970i 0.216170 0.976356i \(-0.430644\pi\)
−0.737464 + 0.675386i \(0.763977\pi\)
\(588\) 0 0
\(589\) −72.0000 124.708i −0.122241 0.211728i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 114.000 0.192243 0.0961214 0.995370i \(-0.469356\pi\)
0.0961214 + 0.995370i \(0.469356\pi\)
\(594\) 0 0
\(595\) 249.415i 0.419185i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 216.000 124.708i 0.360601 0.208193i −0.308743 0.951145i \(-0.599908\pi\)
0.669344 + 0.742952i \(0.266575\pi\)
\(600\) 0 0
\(601\) −313.000 + 542.132i −0.520799 + 0.902050i 0.478909 + 0.877865i \(0.341032\pi\)
−0.999707 + 0.0241851i \(0.992301\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 933.000 1616.00i 1.54215 2.67108i
\(606\) 0 0
\(607\) 582.000 336.018i 0.958814 0.553571i 0.0630061 0.998013i \(-0.479931\pi\)
0.895808 + 0.444442i \(0.146598\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 581.969i 0.952486i
\(612\) 0 0
\(613\) −694.000 −1.13214 −0.566069 0.824358i \(-0.691536\pi\)
−0.566069 + 0.824358i \(0.691536\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 15.0000 + 25.9808i 0.0243112 + 0.0421082i 0.877925 0.478798i \(-0.158927\pi\)
−0.853614 + 0.520906i \(0.825594\pi\)
\(618\) 0 0
\(619\) 294.000 + 169.741i 0.474960 + 0.274218i 0.718314 0.695719i \(-0.244914\pi\)
−0.243354 + 0.969938i \(0.578248\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −684.000 394.908i −1.09791 0.633881i
\(624\) 0 0
\(625\) 389.500 + 674.634i 0.623200 + 1.07941i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −156.000 −0.248013
\(630\) 0 0
\(631\) 464.190i 0.735641i −0.929897 0.367821i \(-0.880104\pi\)
0.929897 0.367821i \(-0.119896\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −540.000 + 311.769i −0.850394 + 0.490975i
\(636\) 0 0
\(637\) 7.00000 12.1244i 0.0109890 0.0190335i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 195.000 337.750i 0.304212 0.526911i −0.672873 0.739758i \(-0.734940\pi\)
0.977086 + 0.212847i \(0.0682735\pi\)
\(642\) 0 0
\(643\) −702.000 + 405.300i −1.09176 + 0.630326i −0.934044 0.357158i \(-0.883746\pi\)
−0.157714 + 0.987485i \(0.550412\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 581.969i 0.899489i 0.893157 + 0.449744i \(0.148485\pi\)
−0.893157 + 0.449744i \(0.851515\pi\)
\(648\) 0 0
\(649\) 432.000 0.665639
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −387.000 670.304i −0.592649 1.02650i −0.993874 0.110519i \(-0.964749\pi\)
0.401225 0.915980i \(-0.368585\pi\)
\(654\) 0 0
\(655\) 540.000 + 311.769i 0.824427 + 0.475983i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −198.000 114.315i −0.300455 0.173468i 0.342192 0.939630i \(-0.388831\pi\)
−0.642647 + 0.766162i \(0.722164\pi\)
\(660\) 0 0
\(661\) 227.000 + 393.176i 0.343419 + 0.594819i 0.985065 0.172182i \(-0.0550816\pi\)
−0.641646 + 0.767001i \(0.721748\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 288.000 0.433083
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1260.00 727.461i 1.87779 1.08415i
\(672\) 0 0
\(673\) −217.000 + 375.855i −0.322437 + 0.558477i −0.980990 0.194057i \(-0.937835\pi\)
0.658553 + 0.752534i \(0.271169\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 117.000 202.650i 0.172821 0.299335i −0.766584 0.642144i \(-0.778045\pi\)
0.939405 + 0.342809i \(0.111378\pi\)
\(678\) 0 0
\(679\) 204.000 117.779i 0.300442 0.173460i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 270.200i 0.395608i −0.980242 0.197804i \(-0.936619\pi\)
0.980242 0.197804i \(-0.0633809\pi\)
\(684\) 0 0
\(685\) 1116.00 1.62920
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −126.000 218.238i −0.182874 0.316747i
\(690\) 0 0
\(691\) −18.0000 10.3923i −0.0260492 0.0150395i 0.486919 0.873447i \(-0.338121\pi\)
−0.512968 + 0.858408i \(0.671454\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −252.000 145.492i −0.362590 0.209341i
\(696\) 0 0
\(697\) −162.000 280.592i −0.232425 0.402571i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1074.00 −1.53210 −0.766049 0.642783i \(-0.777780\pi\)
−0.766049 + 0.642783i \(0.777780\pi\)
\(702\) 0 0
\(703\) 180.133i 0.256235i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −108.000 + 62.3538i −0.152758 + 0.0881949i
\(708\) 0 0
\(709\) −449.000 + 777.691i −0.633286 + 1.09688i 0.353589 + 0.935401i \(0.384961\pi\)
−0.986875 + 0.161483i \(0.948372\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 1512.00 872.954i 2.11469 1.22091i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 956.092i 1.32975i −0.746954 0.664876i \(-0.768484\pi\)
0.746954 0.664876i \(-0.231516\pi\)
\(720\) 0 0
\(721\) −912.000 −1.26491
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −165.000 285.788i −0.227586 0.394191i
\(726\) 0 0
\(727\) −702.000 405.300i −0.965612 0.557496i −0.0677164 0.997705i \(-0.521571\pi\)
−0.897896 + 0.440208i \(0.854905\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −108.000 62.3538i −0.147743 0.0852994i
\(732\) 0 0
\(733\) −185.000 320.429i −0.252387 0.437148i 0.711795 0.702387i \(-0.247882\pi\)
−0.964183 + 0.265239i \(0.914549\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2448.00 3.32157
\(738\) 0 0
\(739\) 852.169i 1.15314i −0.817048 0.576569i \(-0.804391\pi\)
0.817048 0.576569i \(-0.195609\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1188.00 + 685.892i −1.59892 + 0.923139i −0.607228 + 0.794527i \(0.707719\pi\)
−0.991695 + 0.128611i \(0.958948\pi\)
\(744\) 0 0
\(745\) 558.000 966.484i 0.748993 1.29729i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −504.000 + 872.954i −0.672897 + 1.16549i
\(750\) 0 0
\(751\) 66.0000 38.1051i 0.0878828 0.0507392i −0.455415 0.890279i \(-0.650509\pi\)
0.543297 + 0.839540i \(0.317176\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 207.846i 0.275293i
\(756\) 0 0
\(757\) 514.000 0.678996 0.339498 0.940607i \(-0.389743\pi\)
0.339498 + 0.940607i \(0.389743\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 483.000 + 836.581i 0.634691 + 1.09932i 0.986580 + 0.163276i \(0.0522060\pi\)
−0.351889 + 0.936042i \(0.614461\pi\)
\(762\) 0 0
\(763\) −204.000 117.779i −0.267366 0.154364i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 252.000 + 145.492i 0.328553 + 0.189690i
\(768\) 0 0
\(769\) 479.000 + 829.652i 0.622887 + 1.07887i 0.988945 + 0.148280i \(0.0473736\pi\)
−0.366059 + 0.930592i \(0.619293\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −546.000 −0.706339 −0.353169 0.935559i \(-0.614896\pi\)
−0.353169 + 0.935559i \(0.614896\pi\)
\(774\) 0 0
\(775\) 228.631i 0.295007i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 324.000 187.061i 0.415918 0.240130i
\(780\) 0 0
\(781\) 864.000 1496.49i 1.10627 1.91612i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −510.000 + 883.346i −0.649682 + 1.12528i
\(786\) 0 0
\(787\) 210.000 121.244i 0.266836 0.154058i −0.360613 0.932716i \(-0.617432\pi\)
0.627449 + 0.778658i \(0.284099\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 540.400i 0.683186i
\(792\) 0 0
\(793\) 980.000 1.23581
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 669.000 + 1158.74i 0.839398 + 1.45388i 0.890399 + 0.455181i \(0.150425\pi\)
−0.0510012 + 0.998699i \(0.516241\pi\)
\(798\) 0 0
\(799\) 216.000 + 124.708i 0.270338 + 0.156080i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1476.00 + 852.169i 1.83811 + 1.06123i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −966.000 −1.19407 −0.597033 0.802216i \(-0.703654\pi\)
−0.597033 + 0.802216i \(0.703654\pi\)
\(810\) 0 0
\(811\) 1517.28i 1.87087i −0.353497 0.935436i \(-0.615008\pi\)
0.353497 0.935436i \(-0.384992\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1476.00 + 852.169i −1.81104 + 1.04561i
\(816\) 0 0
\(817\) 72.0000 124.708i 0.0881273 0.152641i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −111.000 + 192.258i −0.135201 + 0.234175i −0.925674 0.378322i \(-0.876501\pi\)
0.790473 + 0.612497i \(0.209835\pi\)
\(822\) 0 0
\(823\) 1110.00 640.859i 1.34872 0.778686i 0.360655 0.932699i \(-0.382553\pi\)
0.988069 + 0.154013i \(0.0492197\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1434.14i 1.73415i 0.498182 + 0.867073i \(0.334001\pi\)
−0.498182 + 0.867073i \(0.665999\pi\)
\(828\) 0 0
\(829\) 226.000 0.272618 0.136309 0.990666i \(-0.456476\pi\)
0.136309 + 0.990666i \(0.456476\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.00000 + 5.19615i 0.00360144 + 0.00623788i
\(834\) 0 0
\(835\) −1080.00 623.538i −1.29341 0.746752i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −432.000 249.415i −0.514899 0.297277i 0.219946 0.975512i \(-0.429412\pi\)
−0.734845 + 0.678235i \(0.762745\pi\)
\(840\) 0 0
\(841\) −29.5000 51.0955i −0.0350773 0.0607556i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 162.000 0.191716
\(846\) 0 0
\(847\) 2154.67i 2.54389i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 35.0000 60.6218i 0.0410317 0.0710689i −0.844780 0.535113i \(-0.820269\pi\)
0.885812 + 0.464044i \(0.153602\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −21.0000 + 36.3731i −0.0245041 + 0.0424423i −0.878017 0.478629i \(-0.841134\pi\)
0.853513 + 0.521071i \(0.174467\pi\)
\(858\) 0 0
\(859\) 798.000 460.726i 0.928987 0.536351i 0.0424961 0.999097i \(-0.486469\pi\)
0.886491 + 0.462746i \(0.153136\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 166.277i 0.192673i −0.995349 0.0963365i \(-0.969287\pi\)
0.995349 0.0963365i \(-0.0307125\pi\)
\(864\) 0 0
\(865\) −252.000 −0.291329
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −792.000 1371.78i −0.911392 1.57858i
\(870\) 0 0
\(871\) 1428.00 + 824.456i 1.63949 + 0.946563i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 504.000 + 290.985i 0.576000 + 0.332554i
\(876\) 0 0
\(877\) 83.0000 + 143.760i 0.0946408 + 0.163923i 0.909459 0.415794i \(-0.136496\pi\)
−0.814818 + 0.579717i \(0.803163\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −702.000 −0.796822 −0.398411 0.917207i \(-0.630438\pi\)
−0.398411 + 0.917207i \(0.630438\pi\)
\(882\) 0 0
\(883\) 630.466i 0.714005i 0.934103 + 0.357003i \(0.116201\pi\)
−0.934103 + 0.357003i \(0.883799\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1116.00 644.323i 1.25817 0.726407i 0.285454 0.958392i \(-0.407856\pi\)
0.972719 + 0.231985i \(0.0745222\pi\)
\(888\) 0 0
\(889\) −360.000 + 623.538i −0.404949 + 0.701393i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −144.000 + 249.415i −0.161254 + 0.279300i
\(894\) 0 0
\(895\) 756.000 436.477i 0.844693 0.487684i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 623.538i 0.693591i
\(900\) 0 0
\(901\) 108.000 0.119867
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −246.000 426.084i −0.271823 0.470812i
\(906\) 0 0
\(907\) −1254.00 723.997i −1.38258 0.798233i −0.390115 0.920766i \(-0.627565\pi\)
−0.992464 + 0.122533i \(0.960898\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 288.000 + 166.277i 0.316136 + 0.182521i 0.649669 0.760217i \(-0.274907\pi\)
−0.333533 + 0.942738i \(0.608241\pi\)
\(912\) 0 0
\(913\) −216.000 374.123i −0.236583 0.409773i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 720.000 0.785169
\(918\) 0 0
\(919\) 561.184i 0.610647i 0.952249 + 0.305323i \(0.0987646\pi\)
−0.952249 + 0.305323i \(0.901235\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1008.00 581.969i 1.09209 0.630519i
\(924\) 0 0
\(925\) −143.000 + 247.683i −0.154595 + 0.267766i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 219.000 379.319i 0.235737 0.408309i −0.723749 0.690063i \(-0.757583\pi\)
0.959487 + 0.281754i \(0.0909162\pi\)
\(930\) 0 0
\(931\) −6.00000 + 3.46410i −0.00644468 + 0.00372084i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 748.246i 0.800263i
\(936\) 0 0
\(937\) 1826.00 1.94877 0.974386 0.224881i \(-0.0721992\pi\)
0.974386 + 0.224881i \(0.0721992\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 165.000 + 285.788i 0.175345 + 0.303707i 0.940281 0.340400i \(-0.110562\pi\)
−0.764935 + 0.644107i \(0.777229\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 162.000 + 93.5307i 0.171067 + 0.0987653i 0.583089 0.812409i \(-0.301844\pi\)
−0.412022 + 0.911174i \(0.635177\pi\)
\(948\) 0 0
\(949\) 574.000 + 994.197i 0.604847 + 1.04763i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1110.00 −1.16474 −0.582371 0.812923i \(-0.697875\pi\)
−0.582371 + 0.812923i \(0.697875\pi\)
\(954\) 0 0
\(955\) 1995.32i 2.08934i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1116.00 644.323i 1.16371 0.671870i
\(960\) 0 0
\(961\) −264.500 + 458.127i −0.275234 + 0.476719i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 282.000 488.438i 0.292228 0.506154i
\(966\) 0 0
\(967\) −654.000 + 377.587i −0.676319 + 0.390473i −0.798467 0.602039i \(-0.794355\pi\)
0.122148 + 0.992512i \(0.461022\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 394.908i 0.406702i 0.979106 + 0.203351i \(0.0651832\pi\)
−0.979106 + 0.203351i \(0.934817\pi\)
\(972\) 0 0
\(973\) −336.000 −0.345324
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 459.000 + 795.011i 0.469806 + 0.813727i 0.999404 0.0345214i \(-0.0109907\pi\)
−0.529598 + 0.848248i \(0.677657\pi\)
\(978\) 0 0
\(979\) 2052.00 + 1184.72i 2.09602 + 1.21014i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 36.0000 + 20.7846i 0.0366226 + 0.0211441i 0.518200 0.855260i \(-0.326602\pi\)
−0.481577 + 0.876404i \(0.659936\pi\)
\(984\) 0 0
\(985\) 774.000 + 1340.61i 0.785787 + 1.36102i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 48.4974i 0.0489379i 0.999701 + 0.0244689i \(0.00778948\pi\)
−0.999701 + 0.0244689i \(0.992211\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −612.000 + 353.338i −0.615075 + 0.355114i
\(996\) 0 0
\(997\) −277.000 + 479.778i −0.277834 + 0.481222i −0.970846 0.239704i \(-0.922950\pi\)
0.693013 + 0.720925i \(0.256283\pi\)
\(998\) 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 1296.3.o.a.703.1 2
3.2 odd 2 1296.3.o.n.703.1 2
4.3 odd 2 1296.3.o.c.703.1 2
9.2 odd 6 144.3.g.b.127.2 2
9.4 even 3 1296.3.o.c.271.1 2
9.5 odd 6 1296.3.o.p.271.1 2
9.7 even 3 48.3.g.a.31.2 yes 2
12.11 even 2 1296.3.o.p.703.1 2
36.7 odd 6 48.3.g.a.31.1 2
36.11 even 6 144.3.g.b.127.1 2
36.23 even 6 1296.3.o.n.271.1 2
36.31 odd 6 inner 1296.3.o.a.271.1 2
45.2 even 12 3600.3.j.i.1999.2 4
45.7 odd 12 1200.3.j.a.799.3 4
45.29 odd 6 3600.3.e.t.3151.1 2
45.34 even 6 1200.3.e.h.751.1 2
45.38 even 12 3600.3.j.i.1999.4 4
45.43 odd 12 1200.3.j.a.799.1 4
63.34 odd 6 2352.3.m.a.1471.1 2
72.11 even 6 576.3.g.i.127.1 2
72.29 odd 6 576.3.g.i.127.2 2
72.43 odd 6 192.3.g.a.127.2 2
72.61 even 6 192.3.g.a.127.1 2
144.11 even 12 2304.3.b.n.127.2 4
144.29 odd 12 2304.3.b.n.127.3 4
144.43 odd 12 768.3.b.b.127.2 4
144.61 even 12 768.3.b.b.127.1 4
144.83 even 12 2304.3.b.n.127.4 4
144.101 odd 12 2304.3.b.n.127.1 4
144.115 odd 12 768.3.b.b.127.3 4
144.133 even 12 768.3.b.b.127.4 4
180.7 even 12 1200.3.j.a.799.2 4
180.43 even 12 1200.3.j.a.799.4 4
180.47 odd 12 3600.3.j.i.1999.3 4
180.79 odd 6 1200.3.e.h.751.2 2
180.83 odd 12 3600.3.j.i.1999.1 4
180.119 even 6 3600.3.e.t.3151.2 2
252.223 even 6 2352.3.m.a.1471.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.3.g.a.31.1 2 36.7 odd 6
48.3.g.a.31.2 yes 2 9.7 even 3
144.3.g.b.127.1 2 36.11 even 6
144.3.g.b.127.2 2 9.2 odd 6
192.3.g.a.127.1 2 72.61 even 6
192.3.g.a.127.2 2 72.43 odd 6
576.3.g.i.127.1 2 72.11 even 6
576.3.g.i.127.2 2 72.29 odd 6
768.3.b.b.127.1 4 144.61 even 12
768.3.b.b.127.2 4 144.43 odd 12
768.3.b.b.127.3 4 144.115 odd 12
768.3.b.b.127.4 4 144.133 even 12
1200.3.e.h.751.1 2 45.34 even 6
1200.3.e.h.751.2 2 180.79 odd 6
1200.3.j.a.799.1 4 45.43 odd 12
1200.3.j.a.799.2 4 180.7 even 12
1200.3.j.a.799.3 4 45.7 odd 12
1200.3.j.a.799.4 4 180.43 even 12
1296.3.o.a.271.1 2 36.31 odd 6 inner
1296.3.o.a.703.1 2 1.1 even 1 trivial
1296.3.o.c.271.1 2 9.4 even 3
1296.3.o.c.703.1 2 4.3 odd 2
1296.3.o.n.271.1 2 36.23 even 6
1296.3.o.n.703.1 2 3.2 odd 2
1296.3.o.p.271.1 2 9.5 odd 6
1296.3.o.p.703.1 2 12.11 even 2
2304.3.b.n.127.1 4 144.101 odd 12
2304.3.b.n.127.2 4 144.11 even 12
2304.3.b.n.127.3 4 144.29 odd 12
2304.3.b.n.127.4 4 144.83 even 12
2352.3.m.a.1471.1 2 63.34 odd 6
2352.3.m.a.1471.2 2 252.223 even 6
3600.3.e.t.3151.1 2 45.29 odd 6
3600.3.e.t.3151.2 2 180.119 even 6
3600.3.j.i.1999.1 4 180.83 odd 12
3600.3.j.i.1999.2 4 45.2 even 12
3600.3.j.i.1999.3 4 180.47 odd 12
3600.3.j.i.1999.4 4 45.38 even 12