Properties

Label 20.11.d.c.19.2
Level $20$
Weight $11$
Character 20.19
Analytic conductor $12.707$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [20,11,Mod(19,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.19");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 20.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.7071450535\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 19.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 20.19
Dual form 20.11.d.c.19.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+32.0000i q^{2} -1024.00 q^{4} +(-237.000 + 3116.00i) q^{5} -32768.0i q^{8} -59049.0 q^{9} +O(q^{10})\) \(q+32.0000i q^{2} -1024.00 q^{4} +(-237.000 + 3116.00i) q^{5} -32768.0i q^{8} -59049.0 q^{9} +(-99712.0 - 7584.00i) q^{10} -291336. i q^{13} +1.04858e6 q^{16} -1.81154e6i q^{17} -1.88957e6i q^{18} +(242688. - 3.19078e6i) q^{20} +(-9.65329e6 - 1.47698e6i) q^{25} +9.32275e6 q^{26} -3.23198e7 q^{29} +3.35544e7i q^{32} +5.79692e7 q^{34} +6.04662e7 q^{36} +1.38237e8i q^{37} +(1.02105e8 + 7.76602e6i) q^{40} -2.07190e8 q^{41} +(1.39946e7 - 1.83997e8i) q^{45} -2.82475e8 q^{49} +(4.72635e7 - 3.08905e8i) q^{50} +2.98328e8i q^{52} -2.93540e8i q^{53} -1.03423e9i q^{58} +1.33009e9 q^{61} -1.07374e9 q^{64} +(9.07803e8 + 6.90466e7i) q^{65} +1.85501e9i q^{68} +1.93492e9i q^{72} +1.78891e9i q^{73} -4.42357e9 q^{74} +(-2.48513e8 + 3.26736e9i) q^{80} +3.48678e9 q^{81} -6.63008e9i q^{82} +(5.64475e9 + 4.29334e8i) q^{85} -8.56202e9 q^{89} +(5.88789e9 + 4.47828e8i) q^{90} -1.48172e10i q^{97} -9.03921e9i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2048 q^{4} - 474 q^{5} - 118098 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2048 q^{4} - 474 q^{5} - 118098 q^{9} - 199424 q^{10} + 2097152 q^{16} + 485376 q^{20} - 19306574 q^{25} + 18645504 q^{26} - 64639596 q^{29} + 115938304 q^{34} + 120932352 q^{36} + 204210176 q^{40} - 414380196 q^{41} + 27989226 q^{45} - 564950498 q^{49} + 94526976 q^{50} + 2660180204 q^{61} - 2147483648 q^{64} + 1815605952 q^{65} - 8847146496 q^{74} - 497025024 q^{80} + 6973568802 q^{81} + 11289492352 q^{85} - 17124032796 q^{89} + 11775787776 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 32.0000i 1.00000i
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −1024.00 −1.00000
\(5\) −237.000 + 3116.00i −0.0758400 + 0.997120i
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 32768.0i 1.00000i
\(9\) −59049.0 −1.00000
\(10\) −99712.0 7584.00i −0.997120 0.0758400i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 291336.i 0.784653i −0.919826 0.392326i \(-0.871670\pi\)
0.919826 0.392326i \(-0.128330\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.04858e6 1.00000
\(17\) 1.81154e6i 1.27586i −0.770095 0.637929i \(-0.779791\pi\)
0.770095 0.637929i \(-0.220209\pi\)
\(18\) 1.88957e6i 1.00000i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 242688. 3.19078e6i 0.0758400 0.997120i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −9.65329e6 1.47698e6i −0.988497 0.151243i
\(26\) 9.32275e6 0.784653
\(27\) 0 0
\(28\) 0 0
\(29\) −3.23198e7 −1.57572 −0.787859 0.615855i \(-0.788811\pi\)
−0.787859 + 0.615855i \(0.788811\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 3.35544e7i 1.00000i
\(33\) 0 0
\(34\) 5.79692e7 1.27586
\(35\) 0 0
\(36\) 6.04662e7 1.00000
\(37\) 1.38237e8i 1.99349i 0.0806029 + 0.996746i \(0.474315\pi\)
−0.0806029 + 0.996746i \(0.525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.02105e8 + 7.76602e6i 0.997120 + 0.0758400i
\(41\) −2.07190e8 −1.78834 −0.894169 0.447729i \(-0.852233\pi\)
−0.894169 + 0.447729i \(0.852233\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 1.39946e7 1.83997e8i 0.0758400 0.997120i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −2.82475e8 −1.00000
\(50\) 4.72635e7 3.08905e8i 0.151243 0.988497i
\(51\) 0 0
\(52\) 2.98328e8i 0.784653i
\(53\) 2.93540e8i 0.701920i −0.936390 0.350960i \(-0.885855\pi\)
0.936390 0.350960i \(-0.114145\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 1.03423e9i 1.57572i
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 1.33009e9 1.57482 0.787412 0.616427i \(-0.211421\pi\)
0.787412 + 0.616427i \(0.211421\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.07374e9 −1.00000
\(65\) 9.07803e8 + 6.90466e7i 0.782393 + 0.0595080i
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 1.85501e9i 1.27586i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 1.93492e9i 1.00000i
\(73\) 1.78891e9i 0.862926i 0.902131 + 0.431463i \(0.142003\pi\)
−0.902131 + 0.431463i \(0.857997\pi\)
\(74\) −4.42357e9 −1.99349
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −2.48513e8 + 3.26736e9i −0.0758400 + 0.997120i
\(81\) 3.48678e9 1.00000
\(82\) 6.63008e9i 1.78834i
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 5.64475e9 + 4.29334e8i 1.27218 + 0.0967611i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.56202e9 −1.53330 −0.766648 0.642068i \(-0.778077\pi\)
−0.766648 + 0.642068i \(0.778077\pi\)
\(90\) 5.88789e9 + 4.47828e8i 0.997120 + 0.0758400i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.48172e10i 1.72547i −0.505659 0.862733i \(-0.668751\pi\)
0.505659 0.862733i \(-0.331249\pi\)
\(98\) 9.03921e9i 1.00000i
\(99\) 0 0
\(100\) 9.88497e9 + 1.51243e9i 0.988497 + 0.151243i
\(101\) −1.14158e10 −1.08618 −0.543088 0.839676i \(-0.682745\pi\)
−0.543088 + 0.839676i \(0.682745\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −9.54650e9 −0.784653
\(105\) 0 0
\(106\) 9.39327e9 0.701920
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −2.99829e10 −1.94868 −0.974340 0.225080i \(-0.927736\pi\)
−0.974340 + 0.225080i \(0.927736\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.89814e10i 1.57299i 0.617595 + 0.786496i \(0.288107\pi\)
−0.617595 + 0.786496i \(0.711893\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 3.30955e10 1.57572
\(117\) 1.72031e10i 0.784653i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.59374e10 1.00000
\(122\) 4.25629e10i 1.57482i
\(123\) 0 0
\(124\) 0 0
\(125\) 6.89011e9 2.97296e10i 0.225775 0.974179i
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 3.43597e10i 1.00000i
\(129\) 0 0
\(130\) −2.20949e9 + 2.90497e10i −0.0595080 + 0.782393i
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −5.93604e10 −1.27586
\(137\) 3.27454e10i 0.678496i 0.940697 + 0.339248i \(0.110173\pi\)
−0.940697 + 0.339248i \(0.889827\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −6.19174e10 −1.00000
\(145\) 7.65979e9 1.00708e11i 0.119502 1.57118i
\(146\) −5.72451e10 −0.862926
\(147\) 0 0
\(148\) 1.41554e11i 1.99349i
\(149\) 1.44612e11 1.96913 0.984565 0.175020i \(-0.0559989\pi\)
0.984565 + 0.175020i \(0.0559989\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 1.06969e11i 1.27586i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.83291e11i 1.92151i −0.277392 0.960757i \(-0.589470\pi\)
0.277392 0.960757i \(-0.410530\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −1.04556e11 7.95240e9i −0.997120 0.0758400i
\(161\) 0 0
\(162\) 1.11577e11i 1.00000i
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 2.12163e11 1.78834
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 5.29818e10 0.384320
\(170\) −1.37387e10 + 1.80632e11i −0.0967611 + 1.27218i
\(171\) 0 0
\(172\) 0 0
\(173\) 3.09624e11i 1.99804i 0.0442886 + 0.999019i \(0.485898\pi\)
−0.0442886 + 0.999019i \(0.514102\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 2.73985e11i 1.53330i
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −1.43305e10 + 1.88413e11i −0.0758400 + 0.997120i
\(181\) −1.95015e11 −1.00386 −0.501932 0.864907i \(-0.667377\pi\)
−0.501932 + 0.864907i \(0.667377\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.30745e11 3.27621e10i −1.98775 0.151186i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 4.51372e11i 1.68557i 0.538247 + 0.842787i \(0.319087\pi\)
−0.538247 + 0.842787i \(0.680913\pi\)
\(194\) 4.74149e11 1.72547
\(195\) 0 0
\(196\) 2.89255e11 1.00000
\(197\) 3.88192e11i 1.30832i −0.756354 0.654162i \(-0.773021\pi\)
0.756354 0.654162i \(-0.226979\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −4.83978e10 + 3.16319e11i −0.151243 + 0.988497i
\(201\) 0 0
\(202\) 3.65306e11i 1.08618i
\(203\) 0 0
\(204\) 0 0
\(205\) 4.91041e10 6.45604e11i 0.135628 1.78319i
\(206\) 0 0
\(207\) 0 0
\(208\) 3.05488e11i 0.784653i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 3.00585e11i 0.701920i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 9.59452e11i 1.94868i
\(219\) 0 0
\(220\) 0 0
\(221\) −5.27766e11 −1.00111
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 5.70017e11 + 8.72144e10i 0.988497 + 0.151243i
\(226\) −9.27403e11 −1.57299
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 3.05850e11 0.485658 0.242829 0.970069i \(-0.421925\pi\)
0.242829 + 0.970069i \(0.421925\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.05906e12i 1.57572i
\(233\) 9.52777e11i 1.38743i 0.720249 + 0.693716i \(0.244028\pi\)
−0.720249 + 0.693716i \(0.755972\pi\)
\(234\) −5.50499e11 −0.784653
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 1.39831e12 1.71996 0.859979 0.510330i \(-0.170477\pi\)
0.859979 + 0.510330i \(0.170477\pi\)
\(242\) 8.29998e11i 1.00000i
\(243\) 0 0
\(244\) −1.36201e12 −1.57482
\(245\) 6.69466e10 8.80193e11i 0.0758400 0.997120i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 9.51347e11 + 2.20484e11i 0.974179 + 0.225775i
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.09951e12 1.00000
\(257\) 1.31049e12i 1.16888i 0.811438 + 0.584439i \(0.198685\pi\)
−0.811438 + 0.584439i \(0.801315\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −9.29590e11 7.07038e10i −0.782393 0.0595080i
\(261\) 1.90845e12 1.57572
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 9.14670e11 + 6.95689e10i 0.699898 + 0.0532336i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.71212e12 −1.92552 −0.962758 0.270364i \(-0.912856\pi\)
−0.962758 + 0.270364i \(0.912856\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 1.89953e12i 1.27586i
\(273\) 0 0
\(274\) −1.04785e12 −0.678496
\(275\) 0 0
\(276\) 0 0
\(277\) 1.75954e12i 1.07895i −0.842003 0.539473i \(-0.818624\pi\)
0.842003 0.539473i \(-0.181376\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3.48173e12 −1.98730 −0.993651 0.112505i \(-0.964112\pi\)
−0.993651 + 0.112505i \(0.964112\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.98136e12i 1.00000i
\(289\) −1.26567e12 −0.627814
\(290\) 3.22267e12 + 2.45113e11i 1.57118 + 0.119502i
\(291\) 0 0
\(292\) 1.83184e12i 0.862926i
\(293\) 3.97841e12i 1.84235i −0.389150 0.921174i \(-0.627231\pi\)
0.389150 0.921174i \(-0.372769\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 4.52974e12 1.99349
\(297\) 0 0
\(298\) 4.62760e12i 1.96913i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3.15231e11 + 4.14456e12i −0.119435 + 1.57029i
\(306\) −3.42302e12 −1.27586
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 5.53451e12i 1.84229i 0.389222 + 0.921144i \(0.372744\pi\)
−0.389222 + 0.921144i \(0.627256\pi\)
\(314\) 5.86532e12 1.92151
\(315\) 0 0
\(316\) 0 0
\(317\) 2.35391e12i 0.735349i −0.929955 0.367674i \(-0.880154\pi\)
0.929955 0.367674i \(-0.119846\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 2.54477e11 3.34578e12i 0.0758400 0.997120i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −3.57047e12 −1.00000
\(325\) −4.30299e11 + 2.81235e12i −0.118673 + 0.775626i
\(326\) 0 0
\(327\) 0 0
\(328\) 6.78921e12i 1.78834i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 8.16274e12i 1.99349i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 7.96749e12i 1.83304i 0.399990 + 0.916520i \(0.369014\pi\)
−0.399990 + 0.916520i \(0.630986\pi\)
\(338\) 1.69542e12i 0.384320i
\(339\) 0 0
\(340\) −5.78022e12 4.39638e11i −1.27218 0.0967611i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −9.90796e12 −1.99804
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 9.40330e12 1.81616 0.908078 0.418801i \(-0.137549\pi\)
0.908078 + 0.418801i \(0.137549\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.04265e13i 1.90223i −0.308830 0.951117i \(-0.599938\pi\)
0.308830 0.951117i \(-0.400062\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 8.76750e12 1.53330
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −6.02920e12 4.58575e11i −0.997120 0.0758400i
\(361\) 6.13107e12 1.00000
\(362\) 6.24047e12i 1.00386i
\(363\) 0 0
\(364\) 0 0
\(365\) −5.57424e12 4.23971e11i −0.860441 0.0654443i
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 1.22344e13 1.78834
\(370\) 1.04839e12 1.37839e13i 0.151186 1.98775i
\(371\) 0 0
\(372\) 0 0
\(373\) 7.75981e12i 1.07475i −0.843344 0.537374i \(-0.819416\pi\)
0.843344 0.537374i \(-0.180584\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 9.41592e12i 1.23639i
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.44439e13 −1.68557
\(387\) 0 0
\(388\) 1.51728e13i 1.72547i
\(389\) −1.01301e13 −1.13728 −0.568640 0.822586i \(-0.692530\pi\)
−0.568640 + 0.822586i \(0.692530\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 9.25615e12i 1.00000i
\(393\) 0 0
\(394\) 1.24221e13 1.30832
\(395\) 0 0
\(396\) 0 0
\(397\) 1.63131e12i 0.165419i −0.996574 0.0827093i \(-0.973643\pi\)
0.996574 0.0827093i \(-0.0263573\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.01222e13 1.54873e12i −0.988497 0.151243i
\(401\) −1.82028e13 −1.75556 −0.877782 0.479060i \(-0.840977\pi\)
−0.877782 + 0.479060i \(0.840977\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 1.16898e13 1.08618
\(405\) −8.26368e11 + 1.08648e13i −0.0758400 + 0.997120i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.87253e12 0.163611 0.0818053 0.996648i \(-0.473931\pi\)
0.0818053 + 0.996648i \(0.473931\pi\)
\(410\) 2.06593e13 + 1.57133e12i 1.78319 + 0.135628i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 9.77561e12 0.784653
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −8.93792e12 −0.675812 −0.337906 0.941180i \(-0.609719\pi\)
−0.337906 + 0.941180i \(0.609719\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −9.61871e12 −0.701920
\(425\) −2.67561e12 + 1.74873e13i −0.192965 + 1.26118i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 4.14409e12i 0.272264i 0.990691 + 0.136132i \(0.0434671\pi\)
−0.990691 + 0.136132i \(0.956533\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 3.07025e13 1.94868
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.66799e13 1.00000
\(442\) 1.68885e13i 1.00111i
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 2.02920e12 2.66792e13i 0.116285 1.52888i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3.55762e13 −1.94952 −0.974759 0.223258i \(-0.928331\pi\)
−0.974759 + 0.223258i \(0.928331\pi\)
\(450\) −2.79086e12 + 1.82405e13i −0.151243 + 0.988497i
\(451\) 0 0
\(452\) 2.96769e13i 1.57299i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.79492e13i 1.90380i −0.306409 0.951900i \(-0.599128\pi\)
0.306409 0.951900i \(-0.400872\pi\)
\(458\) 9.78719e12i 0.485658i
\(459\) 0 0
\(460\) 0 0
\(461\) 5.49695e12 0.264008 0.132004 0.991249i \(-0.457859\pi\)
0.132004 + 0.991249i \(0.457859\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) −3.38898e13 −1.57572
\(465\) 0 0
\(466\) −3.04889e13 −1.38743
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 1.76160e13i 0.784653i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.73332e13i 0.701920i
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 4.02733e13 1.56420
\(482\) 4.47459e13i 1.71996i
\(483\) 0 0
\(484\) −2.65599e13 −1.00000
\(485\) 4.61703e13 + 3.51167e12i 1.72050 + 0.130859i
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 4.35844e13i 1.57482i
\(489\) 0 0
\(490\) 2.81662e13 + 2.14229e12i 0.997120 + 0.0758400i
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 5.85485e13i 2.01039i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −7.05547e12 + 3.04431e13i −0.225775 + 0.974179i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 2.70555e12 3.55717e13i 0.0823755 1.08305i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4.21090e13 1.23250 0.616248 0.787552i \(-0.288652\pi\)
0.616248 + 0.787552i \(0.288652\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 3.51844e13i 1.00000i
\(513\) 0 0
\(514\) −4.19358e13 −1.16888
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 2.26252e12 2.97469e13i 0.0595080 0.782393i
\(521\) −2.38622e13 −0.621616 −0.310808 0.950473i \(-0.600600\pi\)
−0.310808 + 0.950473i \(0.600600\pi\)
\(522\) 6.10705e13i 1.57572i
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −4.14265e13 −1.00000
\(530\) −2.22621e12 + 2.92694e13i −0.0532336 + 0.699898i
\(531\) 0 0
\(532\) 0 0
\(533\) 6.03619e13i 1.40322i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 8.67878e13i 1.92552i
\(539\) 0 0
\(540\) 0 0
\(541\) −2.45613e13 −0.529986 −0.264993 0.964250i \(-0.585370\pi\)
−0.264993 + 0.964250i \(0.585370\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 6.07851e13 1.27586
\(545\) 7.10594e12 9.34266e13i 0.147788 1.94307i
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 3.35313e13i 0.678496i
\(549\) −7.85405e13 −1.57482
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 5.63052e13 1.07895
\(555\) 0 0
\(556\) 0 0
\(557\) 7.18764e12i 0.134063i 0.997751 + 0.0670317i \(0.0213529\pi\)
−0.997751 + 0.0670317i \(0.978647\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 1.11415e14i 1.98730i
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) −9.03059e13 6.86858e12i −1.56846 0.119296i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.03553e14 1.73620 0.868101 0.496388i \(-0.165341\pi\)
0.868101 + 0.496388i \(0.165341\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 6.34034e13 1.00000
\(577\) 5.17394e13i 0.808989i −0.914541 0.404494i \(-0.867448\pi\)
0.914541 0.404494i \(-0.132552\pi\)
\(578\) 4.05014e13i 0.627814i
\(579\) 0 0
\(580\) −7.84363e12 + 1.03125e14i −0.119502 + 1.57118i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 5.86189e13 0.862926
\(585\) −5.36049e13 4.07713e12i −0.782393 0.0595080i
\(586\) 1.27309e14 1.84235
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 1.44952e14i 1.99349i
\(593\) 2.99660e13i 0.408653i 0.978903 + 0.204327i \(0.0655004\pi\)
−0.978903 + 0.204327i \(0.934500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.48083e14 −1.96913
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −7.28552e13 −0.929155 −0.464578 0.885532i \(-0.653794\pi\)
−0.464578 + 0.885532i \(0.653794\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −6.14717e12 + 8.08210e13i −0.0758400 + 0.997120i
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −1.32626e14 1.00874e13i −1.57029 0.119435i
\(611\) 0 0
\(612\) 1.09537e14i 1.27586i
\(613\) 1.66100e14i 1.91896i 0.281768 + 0.959482i \(0.409079\pi\)
−0.281768 + 0.959482i \(0.590921\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.17349e14i 1.31236i −0.754606 0.656179i \(-0.772172\pi\)
0.754606 0.656179i \(-0.227828\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9.10045e13 + 2.85155e13i 0.954251 + 0.299007i
\(626\) −1.77104e14 −1.84229
\(627\) 0 0
\(628\) 1.87690e14i 1.92151i
\(629\) 2.50421e14 2.54341
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 7.53251e13 0.735349
\(635\) 0 0
\(636\) 0 0
\(637\) 8.22952e13i 0.784653i
\(638\) 0 0
\(639\) 0 0
\(640\) 1.07065e14 + 8.14326e12i 0.997120 + 0.0758400i
\(641\) 3.41001e12 0.0315113 0.0157556 0.999876i \(-0.494985\pi\)
0.0157556 + 0.999876i \(0.494985\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 1.14255e14i 1.00000i
\(649\) 0 0
\(650\) −8.99952e13 1.37696e13i −0.775626 0.118673i
\(651\) 0 0
\(652\) 0 0
\(653\) 1.92616e14i 1.62229i −0.584848 0.811143i \(-0.698846\pi\)
0.584848 0.811143i \(-0.301154\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2.17255e14 −1.78834
\(657\) 1.05633e14i 0.862926i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 1.78319e14 1.41316 0.706580 0.707633i \(-0.250237\pi\)
0.706580 + 0.707633i \(0.250237\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 2.61208e14 1.99349
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 2.74840e14i 1.99070i −0.0963342 0.995349i \(-0.530712\pi\)
0.0963342 0.995349i \(-0.469288\pi\)
\(674\) −2.54960e14 −1.83304
\(675\) 0 0
\(676\) −5.42534e13 −0.384320
\(677\) 1.06668e14i 0.750054i 0.927014 + 0.375027i \(0.122367\pi\)
−0.927014 + 0.375027i \(0.877633\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 1.40684e13 1.84967e14i 0.0967611 1.27218i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −1.02035e14 7.76066e12i −0.676542 0.0514571i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −8.55187e13 −0.550763
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 3.17055e14i 1.99804i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 3.75332e14i 2.28167i
\(698\) 3.00906e14i 1.81616i
\(699\) 0 0
\(700\) 0 0
\(701\) −1.09411e14 −0.646357 −0.323179 0.946338i \(-0.604752\pi\)
−0.323179 + 0.946338i \(0.604752\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 3.33647e14 1.90223
\(707\) 0 0
\(708\) 0 0
\(709\) −3.42414e14 −1.91127 −0.955633 0.294561i \(-0.904826\pi\)
−0.955633 + 0.294561i \(0.904826\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.80560e14i 1.53330i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 1.46744e13 1.92935e14i 0.0758400 0.997120i
\(721\) 0 0
\(722\) 1.96194e14i 1.00000i
\(723\) 0 0
\(724\) 1.99695e14 1.00386
\(725\) 3.11992e14 + 4.77358e13i 1.55759 + 0.238317i
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −2.05891e14 −1.00000
\(730\) 1.35671e13 1.78376e14i 0.0654443 0.860441i
\(731\) 0 0
\(732\) 0 0
\(733\) 2.85171e14i 1.34768i −0.738879 0.673838i \(-0.764645\pi\)
0.738879 0.673838i \(-0.235355\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 3.91500e14i 1.78834i
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 4.41083e14 + 3.35484e13i 1.98775 + 0.151186i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −3.42732e13 + 4.50612e14i −0.149339 + 1.96346i
\(746\) 2.48314e14 1.07475
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −3.01309e14 −1.23639
\(755\) 0 0
\(756\) 0 0
\(757\) 9.43111e13i 0.379388i 0.981843 + 0.189694i \(0.0607496\pi\)
−0.981843 + 0.189694i \(0.939250\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.29427e14 −0.507111 −0.253555 0.967321i \(-0.581600\pi\)
−0.253555 + 0.967321i \(0.581600\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −3.33317e14 2.53517e13i −1.27218 0.0967611i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1.26380e14 −0.469943 −0.234971 0.972002i \(-0.575500\pi\)
−0.234971 + 0.972002i \(0.575500\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.62205e14i 1.68557i
\(773\) 1.60854e14i 0.582821i −0.956598 0.291410i \(-0.905876\pi\)
0.956598 0.291410i \(-0.0941245\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −4.85529e14 −1.72547
\(777\) 0 0
\(778\) 3.24165e14i 1.13728i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −2.96197e14 −1.00000
\(785\) 5.71136e14 + 4.34400e13i 1.91598 + 0.145728i
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 3.97509e14i 1.30832i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 3.87503e14i 1.23569i
\(794\) 5.22020e13 0.165419
\(795\) 0 0
\(796\) 0 0
\(797\) 4.87758e14i 1.51675i −0.651820 0.758373i \(-0.725994\pi\)
0.651820 0.758373i \(-0.274006\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 4.95594e13 3.23911e14i 0.151243 0.988497i
\(801\) 5.05579e14 1.53330
\(802\) 5.82490e14i 1.75556i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 3.74073e14i 1.08618i
\(809\) 1.35171e14 0.390068 0.195034 0.980796i \(-0.437518\pi\)
0.195034 + 0.980796i \(0.437518\pi\)
\(810\) −3.47674e14 2.64438e13i −0.997120 0.0758400i
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 5.99209e13i 0.163611i
\(819\) 0 0
\(820\) −5.02826e13 + 6.61099e14i −0.135628 + 1.78319i
\(821\) −2.85345e14 −0.764988 −0.382494 0.923958i \(-0.624935\pi\)
−0.382494 + 0.923958i \(0.624935\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 7.19582e14 1.83784 0.918920 0.394445i \(-0.129063\pi\)
0.918920 + 0.394445i \(0.129063\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3.12820e14i 0.784653i
\(833\) 5.11714e14i 1.27586i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 6.23862e14 1.48289
\(842\) 2.86013e14i 0.675812i
\(843\) 0 0
\(844\) 0 0
\(845\) −1.25567e13 + 1.65091e14i −0.0291469 + 0.383214i
\(846\) 0 0
\(847\) 0 0
\(848\) 3.07799e14i 0.701920i
\(849\) 0 0
\(850\) −5.59593e14 8.56195e13i −1.26118 0.192965i
\(851\) 0 0
\(852\) 0 0
\(853\) 3.15864e14i 0.699448i −0.936853 0.349724i \(-0.886275\pi\)
0.936853 0.349724i \(-0.113725\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 9.06104e14i 1.96008i 0.198801 + 0.980040i \(0.436295\pi\)
−0.198801 + 0.980040i \(0.563705\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −9.64787e14 7.33808e13i −1.99228 0.151531i
\(866\) −1.32611e14 −0.272264
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 9.82479e14i 1.94868i
\(873\) 8.74939e14i 1.72547i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 9.25377e14i 1.78369i −0.452337 0.891847i \(-0.649409\pi\)
0.452337 0.891847i \(-0.350591\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −8.81991e14 −1.66182 −0.830911 0.556405i \(-0.812180\pi\)
−0.830911 + 0.556405i \(0.812180\pi\)
\(882\) 5.33756e14i 1.00000i
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 5.40432e14 1.00111
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 8.53736e14 + 6.49343e13i 1.52888 + 0.116285i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 1.13844e15i 1.94952i
\(899\) 0 0
\(900\) −5.83697e14 8.93076e13i −0.988497 0.151243i
\(901\) −5.31758e14 −0.895550
\(902\) 0 0
\(903\) 0 0
\(904\) 9.49661e14 1.57299
\(905\) 4.62185e13 6.07666e14i 0.0761330 1.00097i
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 6.74092e14 1.08618
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 1.21437e15 1.90380
\(915\) 0 0
\(916\) −3.13190e14 −0.485658
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.75902e14i 0.264008i
\(923\) 0 0
\(924\) 0 0
\(925\) 2.04173e14 1.33444e15i 0.301502 1.97056i
\(926\) 0 0
\(927\) 0 0
\(928\) 1.08447e15i 1.57572i
\(929\) −8.87879e14 −1.28314 −0.641572 0.767063i \(-0.721717\pi\)
−0.641572 + 0.767063i \(0.721717\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 9.75644e14i 1.38743i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 5.63711e14 0.784653
\(937\) 5.80047e14i 0.803092i 0.915839 + 0.401546i \(0.131527\pi\)
−0.915839 + 0.401546i \(0.868473\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.45205e15 −1.96804 −0.984022 0.178048i \(-0.943022\pi\)
−0.984022 + 0.178048i \(0.943022\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 5.21173e14 0.677097
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.46820e15i 1.86776i 0.357591 + 0.933878i \(0.383598\pi\)
−0.357591 + 0.933878i \(0.616402\pi\)
\(954\) −5.54663e14 −0.701920
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 8.19628e14 1.00000
\(962\) 1.28875e15i 1.56420i
\(963\) 0 0
\(964\) −1.43187e15 −1.71996
\(965\) −1.40647e15 1.06975e14i −1.68072 0.127834i
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 8.49918e14i 1.00000i
\(969\) 0 0
\(970\) −1.12373e14 + 1.47745e15i −0.130859 + 1.72050i
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 1.39470e15 1.57482
\(977\) 1.70723e15i 1.91788i −0.283617 0.958938i \(-0.591534\pi\)
0.283617 0.958938i \(-0.408466\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −6.85534e13 + 9.01318e14i −0.0758400 + 0.997120i
\(981\) 1.77046e15 1.94868
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 1.20961e15 + 9.20015e13i 1.30456 + 0.0992234i
\(986\) −1.87355e15 −2.01039
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.85671e15i 1.88481i 0.334475 + 0.942405i \(0.391441\pi\)
−0.334475 + 0.942405i \(0.608559\pi\)
\(998\) 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 20.11.d.c.19.2 yes 2
4.3 odd 2 CM 20.11.d.c.19.2 yes 2
5.2 odd 4 100.11.b.a.51.1 1
5.3 odd 4 100.11.b.b.51.1 1
5.4 even 2 inner 20.11.d.c.19.1 2
20.3 even 4 100.11.b.b.51.1 1
20.7 even 4 100.11.b.a.51.1 1
20.19 odd 2 inner 20.11.d.c.19.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.11.d.c.19.1 2 5.4 even 2 inner
20.11.d.c.19.1 2 20.19 odd 2 inner
20.11.d.c.19.2 yes 2 1.1 even 1 trivial
20.11.d.c.19.2 yes 2 4.3 odd 2 CM
100.11.b.a.51.1 1 5.2 odd 4
100.11.b.a.51.1 1 20.7 even 4
100.11.b.b.51.1 1 5.3 odd 4
100.11.b.b.51.1 1 20.3 even 4