Properties

Label 252.9.bk.a.53.4
Level $252$
Weight $9$
Character 252.53
Analytic conductor $102.659$
Analytic rank $0$
Dimension $44$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,9,Mod(53,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.53");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 252.bk (of order \(6\), degree \(2\), 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: \(102.659409735\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 53.4
Character \(\chi\) \(=\) 252.53
Dual form 252.9.bk.a.233.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-700.103 + 404.204i) q^{5} +(-641.390 + 2313.75i) q^{7} +O(q^{10})\) \(q+(-700.103 + 404.204i) q^{5} +(-641.390 + 2313.75i) q^{7} +(5978.45 + 3451.66i) q^{11} -33154.5 q^{13} +(61631.2 + 35582.8i) q^{17} +(91181.6 + 157931. i) q^{19} +(-439230. + 253590. i) q^{23} +(131450. - 227678. i) q^{25} +1.28004e6i q^{29} +(-220600. + 382090. i) q^{31} +(-486187. - 1.87911e6i) q^{35} +(1.41122e6 + 2.44431e6i) q^{37} +2.61027e6i q^{41} +5.88701e6 q^{43} +(3.57507e6 - 2.06406e6i) q^{47} +(-4.94204e6 - 2.96803e6i) q^{49} +(-1.19334e7 - 6.88974e6i) q^{53} -5.58070e6 q^{55} +(-2.24676e6 - 1.29717e6i) q^{59} +(-6.73438e6 - 1.16643e7i) q^{61} +(2.32115e7 - 1.34012e7i) q^{65} +(1.29779e7 - 2.24784e7i) q^{67} +2.46905e7i q^{71} +(-1.86854e7 + 3.23641e7i) q^{73} +(-1.18208e7 + 1.16187e7i) q^{77} +(-1.45367e7 - 2.51783e7i) q^{79} -4.80101e6i q^{83} -5.75309e7 q^{85} +(-1.67403e7 + 9.66503e6i) q^{89} +(2.12649e7 - 7.67110e7i) q^{91} +(-1.27673e8 - 7.37120e7i) q^{95} +1.62373e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q + 1230 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 44 q + 1230 q^{7} - 101380 q^{13} + 62770 q^{19} + 2247666 q^{25} + 1389254 q^{31} - 2136026 q^{37} + 11510140 q^{43} - 3824398 q^{49} - 42646528 q^{55} + 27346232 q^{61} + 14239194 q^{67} - 64344138 q^{73} + 7061786 q^{79} - 54198208 q^{85} - 45697066 q^{91} + 476543496 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −700.103 + 404.204i −1.12016 + 0.646727i −0.941443 0.337173i \(-0.890529\pi\)
−0.178721 + 0.983900i \(0.557196\pi\)
\(6\) 0 0
\(7\) −641.390 + 2313.75i −0.267135 + 0.963659i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5978.45 + 3451.66i 0.408336 + 0.235753i 0.690075 0.723738i \(-0.257578\pi\)
−0.281738 + 0.959491i \(0.590911\pi\)
\(12\) 0 0
\(13\) −33154.5 −1.16083 −0.580415 0.814321i \(-0.697110\pi\)
−0.580415 + 0.814321i \(0.697110\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 61631.2 + 35582.8i 0.737913 + 0.426034i 0.821310 0.570482i \(-0.193244\pi\)
−0.0833972 + 0.996516i \(0.526577\pi\)
\(18\) 0 0
\(19\) 91181.6 + 157931.i 0.699669 + 1.21186i 0.968581 + 0.248698i \(0.0800025\pi\)
−0.268912 + 0.963165i \(0.586664\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −439230. + 253590.i −1.56957 + 0.906192i −0.573351 + 0.819310i \(0.694357\pi\)
−0.996219 + 0.0868819i \(0.972310\pi\)
\(24\) 0 0
\(25\) 131450. 227678.i 0.336512 0.582856i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.28004e6i 1.80981i 0.425615 + 0.904904i \(0.360058\pi\)
−0.425615 + 0.904904i \(0.639942\pi\)
\(30\) 0 0
\(31\) −220600. + 382090.i −0.238868 + 0.413731i −0.960390 0.278660i \(-0.910110\pi\)
0.721522 + 0.692392i \(0.243443\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −486187. 1.87911e6i −0.323990 1.25222i
\(36\) 0 0
\(37\) 1.41122e6 + 2.44431e6i 0.752989 + 1.30421i 0.946368 + 0.323090i \(0.104722\pi\)
−0.193380 + 0.981124i \(0.561945\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.61027e6i 0.923739i 0.886948 + 0.461870i \(0.152821\pi\)
−0.886948 + 0.461870i \(0.847179\pi\)
\(42\) 0 0
\(43\) 5.88701e6 1.72195 0.860976 0.508645i \(-0.169854\pi\)
0.860976 + 0.508645i \(0.169854\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.57507e6 2.06406e6i 0.732643 0.422992i −0.0867452 0.996231i \(-0.527647\pi\)
0.819388 + 0.573239i \(0.194313\pi\)
\(48\) 0 0
\(49\) −4.94204e6 2.96803e6i −0.857278 0.514853i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.19334e7 6.88974e6i −1.51238 0.873171i −0.999895 0.0144717i \(-0.995393\pi\)
−0.512480 0.858699i \(-0.671273\pi\)
\(54\) 0 0
\(55\) −5.58070e6 −0.609871
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.24676e6 1.29717e6i −0.185416 0.107050i 0.404419 0.914574i \(-0.367474\pi\)
−0.589835 + 0.807524i \(0.700807\pi\)
\(60\) 0 0
\(61\) −6.73438e6 1.16643e7i −0.486383 0.842440i 0.513495 0.858093i \(-0.328351\pi\)
−0.999877 + 0.0156529i \(0.995017\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.32115e7 1.34012e7i 1.30032 0.750740i
\(66\) 0 0
\(67\) 1.29779e7 2.24784e7i 0.644030 1.11549i −0.340494 0.940247i \(-0.610594\pi\)
0.984525 0.175247i \(-0.0560723\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.46905e7i 0.971622i 0.874064 + 0.485811i \(0.161476\pi\)
−0.874064 + 0.485811i \(0.838524\pi\)
\(72\) 0 0
\(73\) −1.86854e7 + 3.23641e7i −0.657979 + 1.13965i 0.323159 + 0.946345i \(0.395255\pi\)
−0.981138 + 0.193308i \(0.938078\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.18208e7 + 1.16187e7i −0.336266 + 0.330519i
\(78\) 0 0
\(79\) −1.45367e7 2.51783e7i −0.373214 0.646426i 0.616844 0.787086i \(-0.288411\pi\)
−0.990058 + 0.140660i \(0.955078\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.80101e6i 0.101163i −0.998720 0.0505813i \(-0.983893\pi\)
0.998720 0.0505813i \(-0.0161074\pi\)
\(84\) 0 0
\(85\) −5.75309e7 −1.10211
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.67403e7 + 9.66503e6i −0.266811 + 0.154043i −0.627438 0.778667i \(-0.715896\pi\)
0.360626 + 0.932710i \(0.382563\pi\)
\(90\) 0 0
\(91\) 2.12649e7 7.67110e7i 0.310098 1.11864i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.27673e8 7.37120e7i −1.56749 0.904990i
\(96\) 0 0
\(97\) 1.62373e8 1.83412 0.917060 0.398749i \(-0.130556\pi\)
0.917060 + 0.398749i \(0.130556\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.50788e7 2.60263e7i −0.433199 0.250107i 0.267510 0.963555i \(-0.413799\pi\)
−0.700708 + 0.713448i \(0.747133\pi\)
\(102\) 0 0
\(103\) 8.26371e7 + 1.43132e8i 0.734220 + 1.27171i 0.955065 + 0.296397i \(0.0957851\pi\)
−0.220845 + 0.975309i \(0.570882\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.34074e8 + 7.74076e7i −1.02284 + 0.590539i −0.914926 0.403621i \(-0.867752\pi\)
−0.107917 + 0.994160i \(0.534418\pi\)
\(108\) 0 0
\(109\) −4.41466e7 + 7.64642e7i −0.312746 + 0.541692i −0.978956 0.204072i \(-0.934582\pi\)
0.666210 + 0.745764i \(0.267916\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.87851e7i 0.421872i −0.977500 0.210936i \(-0.932349\pi\)
0.977500 0.210936i \(-0.0676511\pi\)
\(114\) 0 0
\(115\) 2.05004e8 3.55077e8i 1.17212 2.03017i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.21859e8 + 1.19777e8i −0.607674 + 0.597288i
\(120\) 0 0
\(121\) −8.33515e7 1.44369e8i −0.388841 0.673493i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.03254e8i 0.422929i
\(126\) 0 0
\(127\) 1.10140e8 0.423380 0.211690 0.977337i \(-0.432103\pi\)
0.211690 + 0.977337i \(0.432103\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.29909e8 + 7.50031e7i −0.441118 + 0.254680i −0.704072 0.710129i \(-0.748637\pi\)
0.262954 + 0.964808i \(0.415303\pi\)
\(132\) 0 0
\(133\) −4.23895e8 + 1.09676e8i −1.35473 + 0.350512i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −3.52727e8 2.03647e8i −1.00128 0.578090i −0.0926542 0.995698i \(-0.529535\pi\)
−0.908627 + 0.417608i \(0.862868\pi\)
\(138\) 0 0
\(139\) 6.95491e7 0.186308 0.0931542 0.995652i \(-0.470305\pi\)
0.0931542 + 0.995652i \(0.470305\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.98212e8 1.14438e8i −0.474009 0.273669i
\(144\) 0 0
\(145\) −5.17399e8 8.96162e8i −1.17045 2.02728i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.52070e8 + 2.61003e8i −0.917193 + 0.529541i −0.882738 0.469865i \(-0.844303\pi\)
−0.0344543 + 0.999406i \(0.510969\pi\)
\(150\) 0 0
\(151\) 2.08313e8 3.60808e8i 0.400690 0.694015i −0.593120 0.805114i \(-0.702104\pi\)
0.993809 + 0.111099i \(0.0354372\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.56669e8i 0.617930i
\(156\) 0 0
\(157\) 1.54864e8 2.68232e8i 0.254889 0.441481i −0.709976 0.704226i \(-0.751294\pi\)
0.964865 + 0.262745i \(0.0846277\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.05024e8 1.17892e9i −0.453973 1.75461i
\(162\) 0 0
\(163\) −6.76515e8 1.17176e9i −0.958356 1.65992i −0.726494 0.687173i \(-0.758852\pi\)
−0.231862 0.972749i \(-0.574482\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.00867e9i 1.29683i 0.761288 + 0.648414i \(0.224568\pi\)
−0.761288 + 0.648414i \(0.775432\pi\)
\(168\) 0 0
\(169\) 2.83488e8 0.347526
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.04404e9 + 6.02778e8i −1.16556 + 0.672935i −0.952630 0.304133i \(-0.901633\pi\)
−0.212928 + 0.977068i \(0.568300\pi\)
\(174\) 0 0
\(175\) 4.42478e8 + 4.50172e8i 0.471780 + 0.479984i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.75263e8 3.89863e8i −0.657750 0.379752i 0.133669 0.991026i \(-0.457324\pi\)
−0.791419 + 0.611274i \(0.790657\pi\)
\(180\) 0 0
\(181\) 1.78614e9 1.66418 0.832091 0.554639i \(-0.187144\pi\)
0.832091 + 0.554639i \(0.187144\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.97600e9 1.14084e9i −1.68694 0.973956i
\(186\) 0 0
\(187\) 2.45639e8 + 4.25460e8i 0.200878 + 0.347930i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.65713e9 9.56742e8i 1.24515 0.718888i 0.275012 0.961441i \(-0.411318\pi\)
0.970138 + 0.242553i \(0.0779847\pi\)
\(192\) 0 0
\(193\) −2.74296e8 + 4.75094e8i −0.197692 + 0.342413i −0.947780 0.318925i \(-0.896678\pi\)
0.750087 + 0.661339i \(0.230011\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.76800e8i 0.582151i 0.956700 + 0.291076i \(0.0940131\pi\)
−0.956700 + 0.291076i \(0.905987\pi\)
\(198\) 0 0
\(199\) 7.36157e8 1.27506e9i 0.469416 0.813052i −0.529973 0.848015i \(-0.677798\pi\)
0.999389 + 0.0349623i \(0.0111311\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.96170e9 8.21007e8i −1.74404 0.483463i
\(204\) 0 0
\(205\) −1.05508e9 1.82745e9i −0.597407 1.03474i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.25891e9i 0.659796i
\(210\) 0 0
\(211\) 2.46116e9 1.24168 0.620840 0.783937i \(-0.286792\pi\)
0.620840 + 0.783937i \(0.286792\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.12151e9 + 2.37956e9i −1.92887 + 1.11363i
\(216\) 0 0
\(217\) −7.42568e8 7.55480e8i −0.334886 0.340709i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.04335e9 1.17973e9i −0.856591 0.494553i
\(222\) 0 0
\(223\) −2.61371e9 −1.05691 −0.528454 0.848962i \(-0.677228\pi\)
−0.528454 + 0.848962i \(0.677228\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.08436e9 + 2.35810e9i 1.53823 + 0.888095i 0.998943 + 0.0459712i \(0.0146382\pi\)
0.539284 + 0.842124i \(0.318695\pi\)
\(228\) 0 0
\(229\) 1.41923e9 + 2.45818e9i 0.516074 + 0.893866i 0.999826 + 0.0186609i \(0.00594029\pi\)
−0.483752 + 0.875205i \(0.660726\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.01012e9 5.83193e8i 0.342728 0.197874i −0.318750 0.947839i \(-0.603263\pi\)
0.661478 + 0.749965i \(0.269930\pi\)
\(234\) 0 0
\(235\) −1.66861e9 + 2.89011e9i −0.547120 + 0.947640i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 9.74981e8i 0.298816i −0.988776 0.149408i \(-0.952263\pi\)
0.988776 0.149408i \(-0.0477369\pi\)
\(240\) 0 0
\(241\) 1.87499e9 3.24757e9i 0.555814 0.962699i −0.442025 0.897003i \(-0.645740\pi\)
0.997840 0.0656961i \(-0.0209268\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.65962e9 + 8.03300e7i 1.29326 + 0.0222953i
\(246\) 0 0
\(247\) −3.02308e9 5.23612e9i −0.812197 1.40677i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.11453e9i 0.784689i −0.919818 0.392345i \(-0.871664\pi\)
0.919818 0.392345i \(-0.128336\pi\)
\(252\) 0 0
\(253\) −3.50122e9 −0.854549
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.73155e9 2.15441e9i 0.855377 0.493852i −0.00708477 0.999975i \(-0.502255\pi\)
0.862461 + 0.506123i \(0.168922\pi\)
\(258\) 0 0
\(259\) −6.56065e9 + 1.69745e9i −1.45797 + 0.377224i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2.97787e9 1.71927e9i −0.622419 0.359354i 0.155391 0.987853i \(-0.450336\pi\)
−0.777810 + 0.628499i \(0.783669\pi\)
\(264\) 0 0
\(265\) 1.11394e10 2.25881
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.04971e9 + 6.06049e8i 0.200474 + 0.115744i 0.596877 0.802333i \(-0.296408\pi\)
−0.396402 + 0.918077i \(0.629741\pi\)
\(270\) 0 0
\(271\) 5.97273e8 + 1.03451e9i 0.110738 + 0.191804i 0.916068 0.401023i \(-0.131345\pi\)
−0.805330 + 0.592827i \(0.798012\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.57173e9 9.07441e8i 0.274820 0.158667i
\(276\) 0 0
\(277\) 6.27297e8 1.08651e9i 0.106550 0.184550i −0.807820 0.589429i \(-0.799353\pi\)
0.914370 + 0.404878i \(0.132686\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3.32808e9i 0.533788i 0.963726 + 0.266894i \(0.0859974\pi\)
−0.963726 + 0.266894i \(0.914003\pi\)
\(282\) 0 0
\(283\) −2.33297e9 + 4.04082e9i −0.363717 + 0.629976i −0.988569 0.150767i \(-0.951826\pi\)
0.624852 + 0.780743i \(0.285159\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.03949e9 1.67420e9i −0.890170 0.246763i
\(288\) 0 0
\(289\) −9.55608e8 1.65516e9i −0.136990 0.237273i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.20087e9i 0.705676i 0.935684 + 0.352838i \(0.114783\pi\)
−0.935684 + 0.352838i \(0.885217\pi\)
\(294\) 0 0
\(295\) 2.09728e9 0.276929
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.45624e10 8.40763e9i 1.82200 1.05193i
\(300\) 0 0
\(301\) −3.77587e9 + 1.36211e10i −0.459993 + 1.65938i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 9.42951e9 + 5.44413e9i 1.08966 + 0.629114i
\(306\) 0 0
\(307\) 3.89434e9 0.438410 0.219205 0.975679i \(-0.429654\pi\)
0.219205 + 0.975679i \(0.429654\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.39232e9 + 8.03858e8i 0.148833 + 0.0859286i 0.572567 0.819858i \(-0.305948\pi\)
−0.423734 + 0.905787i \(0.639281\pi\)
\(312\) 0 0
\(313\) 4.77134e9 + 8.26421e9i 0.497123 + 0.861041i 0.999994 0.00331945i \(-0.00105662\pi\)
−0.502872 + 0.864361i \(0.667723\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.22083e9 4.74630e9i 0.814101 0.470022i −0.0342769 0.999412i \(-0.510913\pi\)
0.848378 + 0.529391i \(0.177579\pi\)
\(318\) 0 0
\(319\) −4.41827e9 + 7.65267e9i −0.426668 + 0.739010i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.29780e10i 1.19233i
\(324\) 0 0
\(325\) −4.35815e9 + 7.54854e9i −0.390633 + 0.676596i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.48271e9 + 9.59566e9i 0.211906 + 0.819014i
\(330\) 0 0
\(331\) 9.22096e9 + 1.59712e10i 0.768182 + 1.33053i 0.938548 + 0.345149i \(0.112172\pi\)
−0.170366 + 0.985381i \(0.554495\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.09830e10i 1.66605i
\(336\) 0 0
\(337\) −6.09585e9 −0.472623 −0.236311 0.971677i \(-0.575939\pi\)
−0.236311 + 0.971677i \(0.575939\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2.63769e9 + 1.52287e9i −0.195077 + 0.112628i
\(342\) 0 0
\(343\) 1.00370e10 9.53096e9i 0.725152 0.688589i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.20375e9 6.94985e8i −0.0830267 0.0479355i 0.457912 0.888998i \(-0.348598\pi\)
−0.540939 + 0.841062i \(0.681931\pi\)
\(348\) 0 0
\(349\) 7.89927e7 0.00532458 0.00266229 0.999996i \(-0.499153\pi\)
0.00266229 + 0.999996i \(0.499153\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.44156e10 + 1.40964e10i 1.57242 + 0.907838i 0.995871 + 0.0907756i \(0.0289346\pi\)
0.576550 + 0.817062i \(0.304399\pi\)
\(354\) 0 0
\(355\) −9.98002e9 1.72859e10i −0.628374 1.08838i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4.10750e9 2.37147e9i 0.247286 0.142771i −0.371235 0.928539i \(-0.621066\pi\)
0.618521 + 0.785768i \(0.287732\pi\)
\(360\) 0 0
\(361\) −8.13637e9 + 1.40926e10i −0.479073 + 0.829779i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.02110e10i 1.70213i
\(366\) 0 0
\(367\) −6.29244e9 + 1.08988e10i −0.346860 + 0.600780i −0.985690 0.168569i \(-0.946086\pi\)
0.638830 + 0.769348i \(0.279419\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.35950e10 2.31918e10i 1.24545 1.22416i
\(372\) 0 0
\(373\) 2.91240e9 + 5.04442e9i 0.150458 + 0.260601i 0.931396 0.364008i \(-0.118592\pi\)
−0.780938 + 0.624609i \(0.785258\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.24392e10i 2.10088i
\(378\) 0 0
\(379\) −5.27413e9 −0.255619 −0.127810 0.991799i \(-0.540795\pi\)
−0.127810 + 0.991799i \(0.540795\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.79667e10 + 1.03731e10i −0.834975 + 0.482073i −0.855553 0.517715i \(-0.826783\pi\)
0.0205779 + 0.999788i \(0.493449\pi\)
\(384\) 0 0
\(385\) 3.57941e9 1.29123e10i 0.162918 0.587708i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.88921e10 + 1.09074e10i 0.825053 + 0.476345i 0.852156 0.523288i \(-0.175295\pi\)
−0.0271026 + 0.999633i \(0.508628\pi\)
\(390\) 0 0
\(391\) −3.60937e10 −1.54427
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.03544e10 + 1.17516e10i 0.836122 + 0.482735i
\(396\) 0 0
\(397\) 1.25903e10 + 2.18071e10i 0.506844 + 0.877880i 0.999969 + 0.00792086i \(0.00252131\pi\)
−0.493125 + 0.869959i \(0.664145\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.80415e9 3.92838e9i 0.263146 0.151927i −0.362623 0.931936i \(-0.618119\pi\)
0.625769 + 0.780009i \(0.284785\pi\)
\(402\) 0 0
\(403\) 7.31386e9 1.26680e10i 0.277285 0.480272i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.94842e10i 0.710077i
\(408\) 0 0
\(409\) 5.66718e9 9.81584e9i 0.202523 0.350780i −0.746818 0.665029i \(-0.768419\pi\)
0.949341 + 0.314249i \(0.101753\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.44236e9 4.36643e9i 0.152691 0.150081i
\(414\) 0 0
\(415\) 1.94059e9 + 3.36120e9i 0.0654246 + 0.113319i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.52508e10i 0.819254i 0.912253 + 0.409627i \(0.134341\pi\)
−0.912253 + 0.409627i \(0.865659\pi\)
\(420\) 0 0
\(421\) 1.17703e10 0.374679 0.187339 0.982295i \(-0.440014\pi\)
0.187339 + 0.982295i \(0.440014\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.62028e10 9.35472e9i 0.496633 0.286731i
\(426\) 0 0
\(427\) 3.13076e10 8.10028e9i 0.941755 0.243663i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.91524e9 + 5.14722e9i 0.258359 + 0.149164i 0.623586 0.781755i \(-0.285675\pi\)
−0.365227 + 0.930919i \(0.619008\pi\)
\(432\) 0 0
\(433\) −3.82082e10 −1.08694 −0.543469 0.839429i \(-0.682889\pi\)
−0.543469 + 0.839429i \(0.682889\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −8.00994e10 4.62454e10i −2.19636 1.26807i
\(438\) 0 0
\(439\) −1.98026e9 3.42990e9i −0.0533167 0.0923472i 0.838135 0.545462i \(-0.183646\pi\)
−0.891452 + 0.453115i \(0.850313\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.51740e10 + 3.18547e10i −1.43258 + 0.827102i −0.997317 0.0731995i \(-0.976679\pi\)
−0.435266 + 0.900302i \(0.643346\pi\)
\(444\) 0 0
\(445\) 7.81330e9 1.35330e10i 0.199248 0.345108i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3.62898e9i 0.0892893i −0.999003 0.0446446i \(-0.985784\pi\)
0.999003 0.0446446i \(-0.0142156\pi\)
\(450\) 0 0
\(451\) −9.00975e9 + 1.56053e10i −0.217774 + 0.377196i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.61193e10 + 6.23010e10i 0.376097 + 1.45361i
\(456\) 0 0
\(457\) 3.40813e9 + 5.90306e9i 0.0781361 + 0.135336i 0.902446 0.430804i \(-0.141770\pi\)
−0.824310 + 0.566139i \(0.808436\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.72817e10i 0.825452i −0.910855 0.412726i \(-0.864577\pi\)
0.910855 0.412726i \(-0.135423\pi\)
\(462\) 0 0
\(463\) −4.90344e9 −0.106703 −0.0533516 0.998576i \(-0.516990\pi\)
−0.0533516 + 0.998576i \(0.516990\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.47483e10 + 1.42884e10i −0.520328 + 0.300412i −0.737069 0.675817i \(-0.763791\pi\)
0.216741 + 0.976229i \(0.430457\pi\)
\(468\) 0 0
\(469\) 4.36855e10 + 4.44451e10i 0.902913 + 0.918613i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.51952e10 + 2.03200e10i 0.703135 + 0.405955i
\(474\) 0 0
\(475\) 4.79433e10 0.941788
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4.25993e10 2.45947e10i −0.809208 0.467197i 0.0374726 0.999298i \(-0.488069\pi\)
−0.846681 + 0.532101i \(0.821403\pi\)
\(480\) 0 0
\(481\) −4.67883e10 8.10397e10i −0.874092 1.51397i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.13678e11 + 6.56320e10i −2.05452 + 1.18618i
\(486\) 0 0
\(487\) 4.22069e10 7.31045e10i 0.750357 1.29966i −0.197293 0.980345i \(-0.563215\pi\)
0.947650 0.319311i \(-0.103452\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.47411e10i 0.425690i 0.977086 + 0.212845i \(0.0682730\pi\)
−0.977086 + 0.212845i \(0.931727\pi\)
\(492\) 0 0
\(493\) −4.55475e10 + 7.88906e10i −0.771040 + 1.33548i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −5.71276e10 1.58363e10i −0.936312 0.259554i
\(498\) 0 0
\(499\) 6.13363e9 + 1.06238e10i 0.0989271 + 0.171347i 0.911241 0.411874i \(-0.135126\pi\)
−0.812314 + 0.583221i \(0.801792\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.16195e11i 1.81517i −0.419871 0.907584i \(-0.637925\pi\)
0.419871 0.907584i \(-0.362075\pi\)
\(504\) 0 0
\(505\) 4.20797e10 0.647005
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8.92103e10 5.15056e10i 1.32906 0.767332i 0.343903 0.939005i \(-0.388251\pi\)
0.985154 + 0.171673i \(0.0549174\pi\)
\(510\) 0 0
\(511\) −6.28977e10 6.39914e10i −0.922468 0.938508i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.15709e11 6.68045e10i −1.64489 0.949680i
\(516\) 0 0
\(517\) 2.84978e10 0.398886
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.00487e11 + 5.80160e10i 1.36382 + 0.787403i 0.990130 0.140150i \(-0.0447584\pi\)
0.373692 + 0.927553i \(0.378092\pi\)
\(522\) 0 0
\(523\) −1.86272e10 3.22633e10i −0.248967 0.431224i 0.714272 0.699868i \(-0.246758\pi\)
−0.963239 + 0.268644i \(0.913424\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.71916e10 + 1.56991e10i −0.352527 + 0.203532i
\(528\) 0 0
\(529\) 8.94598e10 1.54949e11i 1.14237 1.97864i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.65420e10i 1.07230i
\(534\) 0 0
\(535\) 6.25770e10 1.08387e11i 0.763835 1.32300i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.93011e10 3.48024e10i −0.228679 0.412339i
\(540\) 0 0
\(541\) 1.13051e10 + 1.95811e10i 0.131974 + 0.228585i 0.924437 0.381334i \(-0.124535\pi\)
−0.792464 + 0.609919i \(0.791202\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.13771e10i 0.809045i
\(546\) 0 0
\(547\) −1.58562e11 −1.77113 −0.885563 0.464519i \(-0.846227\pi\)
−0.885563 + 0.464519i \(0.846227\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.02159e11 + 1.16716e11i −2.19324 + 1.26627i
\(552\) 0 0
\(553\) 6.75800e10 1.74851e10i 0.722633 0.186969i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.00309e10 5.19793e10i −0.935342 0.540020i −0.0468452 0.998902i \(-0.514917\pi\)
−0.888497 + 0.458882i \(0.848250\pi\)
\(558\) 0 0
\(559\) −1.95181e11 −1.99889
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4.42375e10 2.55405e10i −0.440308 0.254212i 0.263420 0.964681i \(-0.415149\pi\)
−0.703728 + 0.710469i \(0.748483\pi\)
\(564\) 0 0
\(565\) 2.78032e10 + 4.81566e10i 0.272836 + 0.472566i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.62604e10 2.09350e10i 0.345926 0.199721i −0.316963 0.948438i \(-0.602663\pi\)
0.662890 + 0.748717i \(0.269330\pi\)
\(570\) 0 0
\(571\) −1.83659e10 + 3.18106e10i −0.172769 + 0.299245i −0.939387 0.342859i \(-0.888605\pi\)
0.766618 + 0.642104i \(0.221938\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.33337e11i 1.21978i
\(576\) 0 0
\(577\) 4.04732e10 7.01016e10i 0.365144 0.632448i −0.623655 0.781700i \(-0.714353\pi\)
0.988799 + 0.149252i \(0.0476864\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.11083e10 + 3.07932e9i 0.0974863 + 0.0270240i
\(582\) 0 0
\(583\) −4.75620e10 8.23799e10i −0.411705 0.713094i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.69569e10i 0.395500i 0.980252 + 0.197750i \(0.0633635\pi\)
−0.980252 + 0.197750i \(0.936637\pi\)
\(588\) 0 0
\(589\) −8.04585e10 −0.668514
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.02621e11 5.92484e10i 0.829886 0.479135i −0.0239277 0.999714i \(-0.507617\pi\)
0.853814 + 0.520579i \(0.174284\pi\)
\(594\) 0 0
\(595\) 3.68998e10 1.33112e11i 0.294412 1.06206i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.19459e10 + 6.89700e9i 0.0927926 + 0.0535738i 0.545678 0.837995i \(-0.316272\pi\)
−0.452886 + 0.891569i \(0.649605\pi\)
\(600\) 0 0
\(601\) 2.06073e11 1.57951 0.789757 0.613420i \(-0.210207\pi\)
0.789757 + 0.613420i \(0.210207\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.16709e11 + 6.73821e10i 0.871132 + 0.502948i
\(606\) 0 0
\(607\) 8.39703e9 + 1.45441e10i 0.0618544 + 0.107135i 0.895294 0.445475i \(-0.146965\pi\)
−0.833440 + 0.552610i \(0.813632\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.18529e11 + 6.84330e10i −0.850474 + 0.491021i
\(612\) 0 0
\(613\) −1.06152e11 + 1.83861e11i −0.751776 + 1.30211i 0.195186 + 0.980766i \(0.437469\pi\)
−0.946961 + 0.321347i \(0.895864\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.48225e10i 0.102278i −0.998692 0.0511388i \(-0.983715\pi\)
0.998692 0.0511388i \(-0.0162851\pi\)
\(618\) 0 0
\(619\) −1.02885e11 + 1.78203e11i −0.700795 + 1.21381i 0.267393 + 0.963588i \(0.413838\pi\)
−0.968188 + 0.250225i \(0.919495\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.16253e10 4.49319e10i −0.0771709 0.298265i
\(624\) 0 0
\(625\) 9.30834e10 + 1.61225e11i 0.610031 + 1.05661i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.00861e11i 1.28320i
\(630\) 0 0
\(631\) 1.67614e11 1.05729 0.528643 0.848844i \(-0.322701\pi\)
0.528643 + 0.848844i \(0.322701\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −7.71093e10 + 4.45191e10i −0.474255 + 0.273811i
\(636\) 0 0
\(637\) 1.63851e11 + 9.84034e10i 0.995154 + 0.597657i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2.04139e11 1.17860e11i −1.20919 0.698125i −0.246606 0.969116i \(-0.579315\pi\)
−0.962582 + 0.270991i \(0.912649\pi\)
\(642\) 0 0
\(643\) 1.02507e11 0.599666 0.299833 0.953992i \(-0.403069\pi\)
0.299833 + 0.953992i \(0.403069\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.26048e11 7.27741e10i −0.719317 0.415298i 0.0951843 0.995460i \(-0.469656\pi\)
−0.814501 + 0.580162i \(0.802989\pi\)
\(648\) 0 0
\(649\) −8.95475e9 1.55101e10i −0.0504748 0.0874249i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.29084e11 7.45267e10i 0.709937 0.409882i −0.101101 0.994876i \(-0.532236\pi\)
0.811038 + 0.584994i \(0.198903\pi\)
\(654\) 0 0
\(655\) 6.06332e10 1.05020e11i 0.329416 0.570566i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.29763e11i 0.688034i 0.938963 + 0.344017i \(0.111788\pi\)
−0.938963 + 0.344017i \(0.888212\pi\)
\(660\) 0 0
\(661\) −3.68060e10 + 6.37498e10i −0.192802 + 0.333943i −0.946178 0.323647i \(-0.895091\pi\)
0.753376 + 0.657591i \(0.228424\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.52439e11 2.48124e11i 1.29083 1.26877i
\(666\) 0 0
\(667\) −3.24606e11 5.62234e11i −1.64003 2.84062i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 9.29791e10i 0.458665i
\(672\) 0 0
\(673\) 6.98570e10 0.340525 0.170263 0.985399i \(-0.445538\pi\)
0.170263 + 0.985399i \(0.445538\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.08476e11 + 6.26284e10i −0.516389 + 0.298138i −0.735456 0.677572i \(-0.763032\pi\)
0.219067 + 0.975710i \(0.429699\pi\)
\(678\) 0 0
\(679\) −1.04145e11 + 3.75691e11i −0.489957 + 1.76747i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.80457e11 + 1.61922e11i 1.28880 + 0.744086i 0.978439 0.206534i \(-0.0662185\pi\)
0.310356 + 0.950620i \(0.399552\pi\)
\(684\) 0 0
\(685\) 3.29260e11 1.49547
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3.95645e11 + 2.28425e11i 1.75561 + 1.01360i
\(690\) 0 0
\(691\) 1.73309e10 + 3.00180e10i 0.0760167 + 0.131665i 0.901528 0.432721i \(-0.142446\pi\)
−0.825511 + 0.564386i \(0.809113\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.86915e10 + 2.81121e10i −0.208696 + 0.120491i
\(696\) 0 0
\(697\) −9.28806e10 + 1.60874e11i −0.393545 + 0.681639i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.09479e11i 0.867497i −0.901034 0.433748i \(-0.857191\pi\)
0.901034 0.433748i \(-0.142809\pi\)
\(702\) 0 0
\(703\) −2.57355e11 + 4.45752e11i −1.05369 + 1.82504i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.91313e10 8.76079e10i 0.356741 0.350644i
\(708\) 0 0
\(709\) −6.29890e10 1.09100e11i −0.249275 0.431758i 0.714049 0.700095i \(-0.246859\pi\)
−0.963325 + 0.268337i \(0.913526\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.23767e11i 0.865841i
\(714\) 0 0
\(715\) 1.85025e11 0.707957
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 2.21606e11 1.27944e11i 0.829212 0.478746i −0.0243706 0.999703i \(-0.507758\pi\)
0.853583 + 0.520957i \(0.174425\pi\)
\(720\) 0 0
\(721\) −3.84173e11 + 9.93980e10i −1.42163 + 0.367821i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.91438e11 + 1.68262e11i 1.05486 + 0.609022i
\(726\) 0 0
\(727\) −2.28182e10 −0.0816851 −0.0408426 0.999166i \(-0.513004\pi\)
−0.0408426 + 0.999166i \(0.513004\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.62824e11 + 2.09476e11i 1.27065 + 0.733610i
\(732\) 0 0
\(733\) −6.38696e10 1.10625e11i −0.221247 0.383212i 0.733940 0.679215i \(-0.237679\pi\)
−0.955187 + 0.296003i \(0.904346\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.55176e11 8.95908e10i 0.525962 0.303664i
\(738\) 0 0
\(739\) −2.93903e11 + 5.09056e11i −0.985432 + 1.70682i −0.345434 + 0.938443i \(0.612268\pi\)
−0.639999 + 0.768376i \(0.721065\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.81454e11i 1.25166i 0.779959 + 0.625830i \(0.215240\pi\)
−0.779959 + 0.625830i \(0.784760\pi\)
\(744\) 0 0
\(745\) 2.10997e11 3.65457e11i 0.684938 1.18635i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.31079e10 3.59862e11i −0.295842 1.14343i
\(750\) 0 0
\(751\) −7.92521e10 1.37269e11i −0.249144 0.431531i 0.714144 0.699999i \(-0.246816\pi\)
−0.963289 + 0.268468i \(0.913483\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 3.36804e11i 1.03655i
\(756\) 0 0
\(757\) −5.23086e11 −1.59290 −0.796452 0.604702i \(-0.793292\pi\)
−0.796452 + 0.604702i \(0.793292\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.89734e11 1.09543e11i 0.565727 0.326623i −0.189714 0.981839i \(-0.560756\pi\)
0.755441 + 0.655217i \(0.227423\pi\)
\(762\) 0 0
\(763\) −1.48604e11 1.51188e11i −0.438461 0.446085i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7.44900e10 + 4.30068e10i 0.215237 + 0.124267i
\(768\) 0 0
\(769\) 1.67374e11 0.478610 0.239305 0.970944i \(-0.423080\pi\)
0.239305 + 0.970944i \(0.423080\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2.36157e11 + 1.36345e11i 0.661428 + 0.381876i 0.792821 0.609455i \(-0.208612\pi\)
−0.131393 + 0.991330i \(0.541945\pi\)
\(774\) 0 0
\(775\) 5.79956e10 + 1.00451e11i 0.160764 + 0.278451i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.12242e11 + 2.38008e11i −1.11944 + 0.646312i
\(780\) 0 0
\(781\) −8.52233e10 + 1.47611e11i −0.229063 + 0.396748i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2.50387e11i 0.659375i
\(786\) 0 0
\(787\) 1.07075e11 1.85459e11i 0.279119 0.483447i −0.692047 0.721852i \(-0.743291\pi\)
0.971166 + 0.238405i \(0.0766244\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.59151e11 + 4.41181e10i 0.406541 + 0.112697i
\(792\) 0 0
\(793\) 2.23275e11 + 3.86723e11i 0.564608 + 0.977929i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 4.97596e11i 1.23323i −0.787265 0.616614i \(-0.788504\pi\)
0.787265 0.616614i \(-0.211496\pi\)
\(798\) 0 0
\(799\) 2.93781e11 0.720836
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2.23420e11 + 1.28992e11i −0.537353 + 0.310241i
\(804\) 0 0
\(805\) 6.90071e11 + 7.02070e11i 1.64328 + 1.67185i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −4.50290e11 2.59975e11i −1.05123 0.606929i −0.128238 0.991743i \(-0.540932\pi\)
−0.922994 + 0.384814i \(0.874265\pi\)
\(810\) 0 0
\(811\) 3.02055e11 0.698237 0.349118 0.937079i \(-0.386481\pi\)
0.349118 + 0.937079i \(0.386481\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 9.47260e11 + 5.46901e11i 2.14703 + 1.23959i
\(816\) 0 0
\(817\) 5.36787e11 + 9.29742e11i 1.20480 + 2.08677i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 6.46145e11 3.73052e11i 1.42219 0.821101i 0.425703 0.904863i \(-0.360027\pi\)
0.996486 + 0.0837617i \(0.0266934\pi\)
\(822\) 0 0
\(823\) 2.22067e11 3.84632e11i 0.484045 0.838390i −0.515787 0.856717i \(-0.672501\pi\)
0.999832 + 0.0183266i \(0.00583385\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.64292e11i 0.778802i 0.921068 + 0.389401i \(0.127318\pi\)
−0.921068 + 0.389401i \(0.872682\pi\)
\(828\) 0 0
\(829\) −3.65493e11 + 6.33053e11i −0.773858 + 1.34036i 0.161576 + 0.986860i \(0.448342\pi\)
−0.935434 + 0.353502i \(0.884991\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.98973e11 3.58775e11i −0.413251 0.745147i
\(834\) 0 0
\(835\) −4.07708e11 7.06172e11i −0.838694 1.45266i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.01006e11i 0.203845i −0.994792 0.101923i \(-0.967501\pi\)
0.994792 0.101923i \(-0.0324994\pi\)
\(840\) 0 0
\(841\) −1.13826e12 −2.27541
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.98470e11 + 1.14587e11i −0.389286 + 0.224755i
\(846\) 0 0
\(847\) 3.87494e11 1.00257e11i 0.752890 0.194797i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1.23970e12 7.15742e11i −2.36374 1.36470i
\(852\) 0 0
\(853\) 1.84535e11 0.348564 0.174282 0.984696i \(-0.444240\pi\)
0.174282 + 0.984696i \(0.444240\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 6.69695e11 + 3.86649e11i 1.24152 + 0.716792i 0.969404 0.245472i \(-0.0789431\pi\)
0.272117 + 0.962264i \(0.412276\pi\)
\(858\) 0 0
\(859\) 1.57615e10 + 2.72998e10i 0.0289485 + 0.0501403i 0.880137 0.474720i \(-0.157451\pi\)
−0.851188 + 0.524861i \(0.824117\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.68796e10 + 3.86129e10i −0.120573 + 0.0696129i −0.559074 0.829118i \(-0.688843\pi\)
0.438501 + 0.898731i \(0.355510\pi\)
\(864\) 0 0
\(865\) 4.87291e11 8.44014e11i 0.870411 1.50760i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.00703e11i 0.351945i
\(870\) 0 0
\(871\) −4.30276e11 + 7.45261e11i −0.747610 + 1.29490i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.38904e11 + 6.62261e10i 0.407559 + 0.112979i
\(876\) 0 0
\(877\) −1.42310e11 2.46488e11i −0.240568 0.416676i 0.720308 0.693654i \(-0.244000\pi\)
−0.960876 + 0.276978i \(0.910667\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 3.54602e10i 0.0588624i −0.999567 0.0294312i \(-0.990630\pi\)
0.999567 0.0294312i \(-0.00936959\pi\)
\(882\) 0 0
\(883\) 8.80065e10 0.144768 0.0723839 0.997377i \(-0.476939\pi\)
0.0723839 + 0.997377i \(0.476939\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −6.11155e11 + 3.52851e11i −0.987318 + 0.570028i −0.904472 0.426534i \(-0.859734\pi\)
−0.0828466 + 0.996562i \(0.526401\pi\)
\(888\) 0 0
\(889\) −7.06427e10 + 2.54836e11i −0.113099 + 0.407994i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.51960e11 + 3.76409e11i 1.02522 + 0.591908i
\(894\) 0 0
\(895\) 6.30338e11 0.982384
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −4.89091e11 2.82377e11i −0.748775 0.432305i
\(900\) 0 0
\(901\) −4.90312e11 8.49246e11i −0.744001 1.28865i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.25048e12 + 7.21965e11i −1.86416 + 1.07627i
\(906\) 0 0
\(907\) −5.90256e9 + 1.02235e10i −0.00872190 + 0.0151068i −0.870353 0.492428i \(-0.836110\pi\)
0.861632 + 0.507534i \(0.169443\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3.42842e11i 0.497761i 0.968534 + 0.248880i \(0.0800626\pi\)
−0.968534 + 0.248880i \(0.919937\pi\)
\(912\) 0 0
\(913\) 1.65714e10 2.87026e10i 0.0238494 0.0413083i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −9.02157e10 3.48683e11i −0.127587 0.493121i
\(918\) 0 0
\(919\) −4.28433e11 7.42068e11i −0.600649 1.04036i −0.992723 0.120421i \(-0.961575\pi\)
0.392074 0.919934i \(-0.371758\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 8.18601e11i 1.12789i
\(924\) 0 0
\(925\) 7.42020e11 1.01356
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 8.99296e11 5.19209e11i 1.20737 0.697074i 0.245185 0.969476i \(-0.421151\pi\)
0.962184 + 0.272402i \(0.0878180\pi\)
\(930\) 0 0
\(931\) 1.81211e10 1.05113e12i 0.0241204 1.39913i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3.43945e11 1.98577e11i −0.450032 0.259826i
\(936\) 0 0
\(937\) −9.49832e11 −1.23222 −0.616110 0.787660i \(-0.711292\pi\)
−0.616110 + 0.787660i \(0.711292\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.29832e11 + 1.32694e11i 0.293125 + 0.169236i 0.639350 0.768916i \(-0.279203\pi\)
−0.346226 + 0.938151i \(0.612537\pi\)
\(942\) 0 0
\(943\) −6.61936e11 1.14651e12i −0.837085 1.44987i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.46887e11 + 3.15745e11i −0.679982 + 0.392588i −0.799848 0.600202i \(-0.795087\pi\)
0.119866 + 0.992790i \(0.461753\pi\)
\(948\) 0 0
\(949\) 6.19506e11 1.07302e12i 0.763802 1.32294i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 2.74996e11i 0.333391i 0.986008 + 0.166696i \(0.0533097\pi\)
−0.986008 + 0.166696i \(0.946690\pi\)
\(954\) 0 0
\(955\) −7.73439e11 + 1.33964e12i −0.929849 + 1.61055i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 6.97423e11 6.85503e11i 0.824559 0.810466i
\(960\) 0 0
\(961\) 3.29117e11 + 5.70048e11i 0.385884 + 0.668371i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.43486e11i 0.511412i
\(966\) 0 0
\(967\) −7.42625e11 −0.849305 −0.424652 0.905356i \(-0.639604\pi\)
−0.424652 + 0.905356i \(0.639604\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.98214e10 1.14439e10i 0.0222976 0.0128735i −0.488810 0.872390i \(-0.662569\pi\)
0.511107 + 0.859517i \(0.329235\pi\)
\(972\) 0 0
\(973\) −4.46081e10 + 1.60919e11i −0.0497694 + 0.179538i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −8.25487e11 4.76595e11i −0.906007 0.523084i −0.0268630 0.999639i \(-0.508552\pi\)
−0.879144 + 0.476556i \(0.841885\pi\)
\(978\) 0 0
\(979\) −1.33442e11 −0.145265
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −9.00006e11 5.19619e11i −0.963898 0.556507i −0.0665277 0.997785i \(-0.521192\pi\)
−0.897371 + 0.441278i \(0.854525\pi\)
\(984\) 0 0
\(985\) −3.54407e11 6.13850e11i −0.376493 0.652105i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.58575e12 + 1.49288e12i −2.70272 + 1.56042i
\(990\) 0 0
\(991\) −3.06924e11 + 5.31608e11i −0.318227 + 0.551185i −0.980118 0.198415i \(-0.936421\pi\)
0.661891 + 0.749600i \(0.269754\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.19023e12i 1.21434i
\(996\) 0 0
\(997\) 6.94291e11 1.20255e12i 0.702685 1.21709i −0.264835 0.964294i \(-0.585318\pi\)
0.967520 0.252793i \(-0.0813491\pi\)
\(998\) 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 252.9.bk.a.53.4 44
3.2 odd 2 inner 252.9.bk.a.53.19 yes 44
7.2 even 3 inner 252.9.bk.a.233.19 yes 44
21.2 odd 6 inner 252.9.bk.a.233.4 yes 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.9.bk.a.53.4 44 1.1 even 1 trivial
252.9.bk.a.53.19 yes 44 3.2 odd 2 inner
252.9.bk.a.233.4 yes 44 21.2 odd 6 inner
252.9.bk.a.233.19 yes 44 7.2 even 3 inner