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Properties

Label 20.7.d.b.19.1
Level $20$
Weight $7$
Character 20.19
Self dual yes
Analytic conductor $4.601$
Analytic rank $0$
Dimension $1$
CM discriminant -20
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [20,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.19");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
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: yes
Analytic conductor: \(4.60108167240\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 19.1
Character \(\chi\) \(=\) 20.19

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.00000 q^{2} +44.0000 q^{3} +64.0000 q^{4} -125.000 q^{5} +352.000 q^{6} -524.000 q^{7} +512.000 q^{8} +1207.00 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +44.0000 q^{3} +64.0000 q^{4} -125.000 q^{5} +352.000 q^{6} -524.000 q^{7} +512.000 q^{8} +1207.00 q^{9} -1000.00 q^{10} +2816.00 q^{12} -4192.00 q^{14} -5500.00 q^{15} +4096.00 q^{16} +9656.00 q^{18} -8000.00 q^{20} -23056.0 q^{21} -15356.0 q^{23} +22528.0 q^{24} +15625.0 q^{25} +21032.0 q^{27} -33536.0 q^{28} +44858.0 q^{29} -44000.0 q^{30} +32768.0 q^{32} +65500.0 q^{35} +77248.0 q^{36} -64000.0 q^{40} -74338.0 q^{41} -184448. q^{42} +17404.0 q^{43} -150875. q^{45} -122848. q^{46} -26444.0 q^{47} +180224. q^{48} +156927. q^{49} +125000. q^{50} +168256. q^{54} -268288. q^{56} +358864. q^{58} -352000. q^{60} +452342. q^{61} -632468. q^{63} +262144. q^{64} +1276.00 q^{67} -675664. q^{69} +524000. q^{70} +617984. q^{72} +687500. q^{75} -512000. q^{80} +45505.0 q^{81} -594704. q^{82} -1.13172e6 q^{83} -1.47558e6 q^{84} +139232. q^{86} +1.97375e6 q^{87} +511058. q^{89} -1.20700e6 q^{90} -982784. q^{92} -211552. q^{94} +1.44179e6 q^{96} +1.25542e6 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 1.00000
\(3\) 44.0000 1.62963 0.814815 − 0.579721i \(-0.196839\pi\)
0.814815 + 0.579721i \(0.196839\pi\)
\(4\) 64.0000 1.00000
\(5\) −125.000 −1.00000
\(6\) 352.000 1.62963
\(7\) −524.000 −1.52770 −0.763848 − 0.645396i \(-0.776692\pi\)
−0.763848 + 0.645396i \(0.776692\pi\)
\(8\) 512.000 1.00000
\(9\) 1207.00 1.65569
\(10\) −1000.00 −1.00000
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 2816.00 1.62963
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) −4192.00 −1.52770
\(15\) −5500.00 −1.62963
\(16\) 4096.00 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 9656.00 1.65569
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −8000.00 −1.00000
\(21\) −23056.0 −2.48958
\(22\) 0 0
\(23\) −15356.0 −1.26210 −0.631051 − 0.775741i \(-0.717376\pi\)
−0.631051 + 0.775741i \(0.717376\pi\)
\(24\) 22528.0 1.62963
\(25\) 15625.0 1.00000
\(26\) 0 0
\(27\) 21032.0 1.06854
\(28\) −33536.0 −1.52770
\(29\) 44858.0 1.83927 0.919636 − 0.392772i \(-0.128484\pi\)
0.919636 + 0.392772i \(0.128484\pi\)
\(30\) −44000.0 −1.62963
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 32768.0 1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) 65500.0 1.52770
\(36\) 77248.0 1.65569
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −64000.0 −1.00000
\(41\) −74338.0 −1.07860 −0.539299 − 0.842115i \(-0.681311\pi\)
−0.539299 + 0.842115i \(0.681311\pi\)
\(42\) −184448. −2.48958
\(43\) 17404.0 0.218899 0.109449 − 0.993992i \(-0.465091\pi\)
0.109449 + 0.993992i \(0.465091\pi\)
\(44\) 0 0
\(45\) −150875. −1.65569
\(46\) −122848. −1.26210
\(47\) −26444.0 −0.254703 −0.127351 − 0.991858i \(-0.540648\pi\)
−0.127351 + 0.991858i \(0.540648\pi\)
\(48\) 180224. 1.62963
\(49\) 156927. 1.33386
\(50\) 125000. 1.00000
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 168256. 1.06854
\(55\) 0 0
\(56\) −268288. −1.52770
\(57\) 0 0
\(58\) 358864. 1.83927
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) −352000. −1.62963
\(61\) 452342. 1.99286 0.996431 − 0.0844063i \(-0.0268994\pi\)
0.996431 + 0.0844063i \(0.0268994\pi\)
\(62\) 0 0
\(63\) −632468. −2.52940
\(64\) 262144. 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 1276.00 0.00424254 0.00212127 − 0.999998i \(-0.499325\pi\)
0.00212127 + 0.999998i \(0.499325\pi\)
\(68\) 0 0
\(69\) −675664. −2.05676
\(70\) 524000. 1.52770
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 617984. 1.65569
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 687500. 1.62963
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −512000. −1.00000
\(81\) 45505.0 0.0856257
\(82\) −594704. −1.07860
\(83\) −1.13172e6 −1.97926 −0.989631 − 0.143635i \(-0.954121\pi\)
−0.989631 + 0.143635i \(0.954121\pi\)
\(84\) −1.47558e6 −2.48958
\(85\) 0 0
\(86\) 139232. 0.218899
\(87\) 1.97375e6 2.99733
\(88\) 0 0
\(89\) 511058. 0.724937 0.362468 − 0.931996i \(-0.381934\pi\)
0.362468 + 0.931996i \(0.381934\pi\)
\(90\) −1.20700e6 −1.65569
\(91\) 0 0
\(92\) −982784. −1.26210
\(93\) 0 0
\(94\) −211552. −0.254703
\(95\) 0 0
\(96\) 1.44179e6 1.62963
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 1.25542e6 1.33386
\(99\) 0 0
\(100\) 1.00000e6 1.00000
\(101\) −1.91772e6 −1.86132 −0.930659 − 0.365887i \(-0.880766\pi\)
−0.930659 + 0.365887i \(0.880766\pi\)
\(102\) 0 0
\(103\) 1.31520e6 1.20360 0.601799 − 0.798648i \(-0.294451\pi\)
0.601799 + 0.798648i \(0.294451\pi\)
\(104\) 0 0
\(105\) 2.88200e6 2.48958
\(106\) 0 0
\(107\) −2.35240e6 −1.92026 −0.960131 − 0.279550i \(-0.909815\pi\)
−0.960131 + 0.279550i \(0.909815\pi\)
\(108\) 1.34605e6 1.06854
\(109\) −1.29956e6 −1.00350 −0.501750 − 0.865013i \(-0.667310\pi\)
−0.501750 + 0.865013i \(0.667310\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.14630e6 −1.52770
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 1.91950e6 1.26210
\(116\) 2.87091e6 1.83927
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) −2.81600e6 −1.62963
\(121\) 1.77156e6 1.00000
\(122\) 3.61874e6 1.99286
\(123\) −3.27087e6 −1.75771
\(124\) 0 0
\(125\) −1.95312e6 −1.00000
\(126\) −5.05974e6 −2.52940
\(127\) −1.72492e6 −0.842091 −0.421045 − 0.907040i \(-0.638337\pi\)
−0.421045 + 0.907040i \(0.638337\pi\)
\(128\) 2.09715e6 1.00000
\(129\) 765776. 0.356724
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 10208.0 0.00424254
\(135\) −2.62900e6 −1.06854
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) −5.40531e6 −2.05676
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 4.19200e6 1.52770
\(141\) −1.16354e6 −0.415071
\(142\) 0 0
\(143\) 0 0
\(144\) 4.94387e6 1.65569
\(145\) −5.60725e6 −1.83927
\(146\) 0 0
\(147\) 6.90479e6 2.17369
\(148\) 0 0
\(149\) 2.96932e6 0.897631 0.448816 − 0.893624i \(-0.351846\pi\)
0.448816 + 0.893624i \(0.351846\pi\)
\(150\) 5.50000e6 1.62963
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −4.09600e6 −1.00000
\(161\) 8.04654e6 1.92811
\(162\) 364040. 0.0856257
\(163\) 8.66100e6 1.99989 0.999943 − 0.0106368i \(-0.00338586\pi\)
0.999943 + 0.0106368i \(0.00338586\pi\)
\(164\) −4.75763e6 −1.07860
\(165\) 0 0
\(166\) −9.05373e6 −1.97926
\(167\) −5.88796e6 −1.26420 −0.632100 − 0.774887i \(-0.717807\pi\)
−0.632100 + 0.774887i \(0.717807\pi\)
\(168\) −1.18047e7 −2.48958
\(169\) 4.82681e6 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 1.11386e6 0.218899
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 1.57900e7 2.99733
\(175\) −8.18750e6 −1.52770
\(176\) 0 0
\(177\) 0 0
\(178\) 4.08846e6 0.724937
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −9.65600e6 −1.65569
\(181\) −1.06974e7 −1.80402 −0.902012 − 0.431710i \(-0.857910\pi\)
−0.902012 + 0.431710i \(0.857910\pi\)
\(182\) 0 0
\(183\) 1.99030e7 3.24763
\(184\) −7.86227e6 −1.26210
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −1.69242e6 −0.254703
\(189\) −1.10208e7 −1.63240
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 1.15343e7 1.62963
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.00433e7 1.33386
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 8.00000e6 1.00000
\(201\) 56144.0 0.00691377
\(202\) −1.53417e7 −1.86132
\(203\) −2.35056e7 −2.80985
\(204\) 0 0
\(205\) 9.29225e6 1.07860
\(206\) 1.05216e7 1.20360
\(207\) −1.85347e7 −2.08965
\(208\) 0 0
\(209\) 0 0
\(210\) 2.30560e7 2.48958
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −1.88192e7 −1.92026
\(215\) −2.17550e6 −0.218899
\(216\) 1.07684e7 1.06854
\(217\) 0 0
\(218\) −1.03965e7 −1.00350
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.78363e7 1.60839 0.804194 − 0.594367i \(-0.202597\pi\)
0.804194 + 0.594367i \(0.202597\pi\)
\(224\) −1.71704e7 −1.52770
\(225\) 1.88594e7 1.65569
\(226\) 0 0
\(227\) 9.91496e6 0.847643 0.423822 − 0.905746i \(-0.360688\pi\)
0.423822 + 0.905746i \(0.360688\pi\)
\(228\) 0 0
\(229\) 2.32339e7 1.93471 0.967354 − 0.253427i \(-0.0815578\pi\)
0.967354 + 0.253427i \(0.0815578\pi\)
\(230\) 1.53560e7 1.26210
\(231\) 0 0
\(232\) 2.29673e7 1.83927
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 3.30550e6 0.254703
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) −2.25280e7 −1.62963
\(241\) −2.50778e7 −1.79159 −0.895794 − 0.444470i \(-0.853392\pi\)
−0.895794 + 0.444470i \(0.853392\pi\)
\(242\) 1.41725e7 1.00000
\(243\) −1.33301e7 −0.928998
\(244\) 2.89499e7 1.99286
\(245\) −1.96159e7 −1.33386
\(246\) −2.61670e7 −1.75771
\(247\) 0 0
\(248\) 0 0
\(249\) −4.97955e7 −3.22546
\(250\) −1.56250e7 −1.00000
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −4.04780e7 −2.52940
\(253\) 0 0
\(254\) −1.37994e7 −0.842091
\(255\) 0 0
\(256\) 1.67772e7 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 6.12621e6 0.356724
\(259\) 0 0
\(260\) 0 0
\(261\) 5.41436e7 3.04527
\(262\) 0 0
\(263\) 3.60257e7 1.98036 0.990182 − 0.139785i \(-0.0446410\pi\)
0.990182 + 0.139785i \(0.0446410\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2.24866e7 1.18138
\(268\) 81664.0 0.00424254
\(269\) −8.19428e6 −0.420973 −0.210486 − 0.977597i \(-0.567505\pi\)
−0.210486 + 0.977597i \(0.567505\pi\)
\(270\) −2.10320e7 −1.06854
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −4.32425e7 −2.05676
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 3.35360e7 1.52770
\(281\) 2.59825e7 1.17101 0.585506 − 0.810668i \(-0.300896\pi\)
0.585506 + 0.810668i \(0.300896\pi\)
\(282\) −9.30829e6 −0.415071
\(283\) −4.43699e7 −1.95762 −0.978811 − 0.204765i \(-0.934357\pi\)
−0.978811 + 0.204765i \(0.934357\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.89531e7 1.64777
\(288\) 3.95510e7 1.65569
\(289\) 2.41376e7 1.00000
\(290\) −4.48580e7 −1.83927
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 5.52383e7 2.17369
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 2.37545e7 0.897631
\(299\) 0 0
\(300\) 4.40000e7 1.62963
\(301\) −9.11970e6 −0.334411
\(302\) 0 0
\(303\) −8.43796e7 −3.03326
\(304\) 0 0
\(305\) −5.65428e7 −1.99286
\(306\) 0 0
\(307\) −4.31915e7 −1.49274 −0.746369 − 0.665533i \(-0.768204\pi\)
−0.746369 + 0.665533i \(0.768204\pi\)
\(308\) 0 0
\(309\) 5.78690e7 1.96142
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 7.90585e7 2.52940
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −3.27680e7 −1.00000
\(321\) −1.03506e8 −3.12932
\(322\) 6.43724e7 1.92811
\(323\) 0 0
\(324\) 2.91232e6 0.0856257
\(325\) 0 0
\(326\) 6.92880e7 1.99989
\(327\) −5.71807e7 −1.63533
\(328\) −3.80611e7 −1.07860
\(329\) 1.38567e7 0.389109
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −7.24298e7 −1.97926
\(333\) 0 0
\(334\) −4.71037e7 −1.26420
\(335\) −159500. −0.00424254
\(336\) −9.44374e7 −2.48958
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 3.86145e7 1.00000
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −2.05817e7 −0.510033
\(344\) 8.91085e6 0.218899
\(345\) 8.44580e7 2.05676
\(346\) 0 0
\(347\) 4.03414e7 0.965524 0.482762 − 0.875752i \(-0.339634\pi\)
0.482762 + 0.875752i \(0.339634\pi\)
\(348\) 1.26320e8 2.99733
\(349\) 8.02822e6 0.188861 0.0944306 − 0.995531i \(-0.469897\pi\)
0.0944306 + 0.995531i \(0.469897\pi\)
\(350\) −6.55000e7 −1.52770
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 3.27077e7 0.724937
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −7.72480e7 −1.65569
\(361\) 4.70459e7 1.00000
\(362\) −8.55792e7 −1.80402
\(363\) 7.79487e7 1.62963
\(364\) 0 0
\(365\) 0 0
\(366\) 1.59224e8 3.24763
\(367\) 8.69589e7 1.75920 0.879601 − 0.475711i \(-0.157809\pi\)
0.879601 + 0.475711i \(0.157809\pi\)
\(368\) −6.28982e7 −1.26210
\(369\) −8.97260e7 −1.78583
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −8.59375e7 −1.62963
\(376\) −1.35393e7 −0.254703
\(377\) 0 0
\(378\) −8.81661e7 −1.63240
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −7.58967e7 −1.37230
\(382\) 0 0
\(383\) 1.92778e7 0.343131 0.171566 − 0.985173i \(-0.445117\pi\)
0.171566 + 0.985173i \(0.445117\pi\)
\(384\) 9.22747e7 1.62963
\(385\) 0 0
\(386\) 0 0
\(387\) 2.10066e7 0.362429
\(388\) 0 0
\(389\) 8.34581e7 1.41782 0.708908 − 0.705301i \(-0.249188\pi\)
0.708908 + 0.705301i \(0.249188\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 8.03466e7 1.33386
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 6.40000e7 1.00000
\(401\) 1.21373e8 1.88230 0.941152 − 0.337982i \(-0.109744\pi\)
0.941152 + 0.337982i \(0.109744\pi\)
\(402\) 449152. 0.00691377
\(403\) 0 0
\(404\) −1.22734e8 −1.86132
\(405\) −5.68812e6 −0.0856257
\(406\) −1.88045e8 −2.80985
\(407\) 0 0
\(408\) 0 0
\(409\) −1.13372e8 −1.65704 −0.828522 − 0.559956i \(-0.810818\pi\)
−0.828522 + 0.559956i \(0.810818\pi\)
\(410\) 7.43380e7 1.07860
\(411\) 0 0
\(412\) 8.41731e7 1.20360
\(413\) 0 0
\(414\) −1.48278e8 −2.08965
\(415\) 1.41464e8 1.97926
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 1.84448e8 2.48958
\(421\) −5.72305e7 −0.766975 −0.383487 − 0.923546i \(-0.625277\pi\)
−0.383487 + 0.923546i \(0.625277\pi\)
\(422\) 0 0
\(423\) −3.19179e7 −0.421709
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −2.37027e8 −3.04449
\(428\) −1.50554e8 −1.92026
\(429\) 0 0
\(430\) −1.74040e7 −0.218899
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 8.61471e7 1.06854
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) −2.46719e8 −2.99733
\(436\) −8.31720e7 −1.00350
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.89411e8 2.20846
\(442\) 0 0
\(443\) 3.57157e7 0.410817 0.205408 − 0.978676i \(-0.434148\pi\)
0.205408 + 0.978676i \(0.434148\pi\)
\(444\) 0 0
\(445\) −6.38822e7 −0.724937
\(446\) 1.42691e8 1.60839
\(447\) 1.30650e8 1.46281
\(448\) −1.37363e8 −1.52770
\(449\) −1.77667e8 −1.96276 −0.981380 − 0.192076i \(-0.938478\pi\)
−0.981380 + 0.192076i \(0.938478\pi\)
\(450\) 1.50875e8 1.65569
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 7.93196e7 0.847643
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 1.85871e8 1.93471
\(459\) 0 0
\(460\) 1.22848e8 1.26210
\(461\) 6.01196e7 0.613640 0.306820 − 0.951768i \(-0.400735\pi\)
0.306820 + 0.951768i \(0.400735\pi\)
\(462\) 0 0
\(463\) 4.53900e7 0.457317 0.228658 − 0.973507i \(-0.426566\pi\)
0.228658 + 0.973507i \(0.426566\pi\)
\(464\) 1.83738e8 1.83927
\(465\) 0 0
\(466\) 0 0
\(467\) −7.92225e7 −0.777854 −0.388927 − 0.921269i \(-0.627154\pi\)
−0.388927 + 0.921269i \(0.627154\pi\)
\(468\) 0 0
\(469\) −668624. −0.00648132
\(470\) 2.64440e7 0.254703
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) −1.80224e8 −1.62963
\(481\) 0 0
\(482\) −2.00622e8 −1.79159
\(483\) 3.54048e8 3.14210
\(484\) 1.13380e8 1.00000
\(485\) 0 0
\(486\) −1.06641e8 −0.928998
\(487\) −2.30989e8 −1.99989 −0.999943 − 0.0106586i \(-0.996607\pi\)
−0.999943 + 0.0106586i \(0.996607\pi\)
\(488\) 2.31599e8 1.99286
\(489\) 3.81084e8 3.25907
\(490\) −1.56927e8 −1.33386
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) −2.09336e8 −1.75771
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −3.98364e8 −3.22546
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −1.25000e8 −1.00000
\(501\) −2.59070e8 −2.06018
\(502\) 0 0
\(503\) 7.33066e7 0.576022 0.288011 − 0.957627i \(-0.407006\pi\)
0.288011 + 0.957627i \(0.407006\pi\)
\(504\) −3.23824e8 −2.52940
\(505\) 2.39715e8 1.86132
\(506\) 0 0
\(507\) 2.12380e8 1.62963
\(508\) −1.10395e8 −0.842091
\(509\) −1.83714e8 −1.39312 −0.696559 − 0.717500i \(-0.745286\pi\)
−0.696559 + 0.717500i \(0.745286\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.34218e8 1.00000
\(513\) 0 0
\(514\) 0 0
\(515\) −1.64400e8 −1.20360
\(516\) 4.90097e7 0.356724
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −2.11574e8 −1.49606 −0.748031 − 0.663664i \(-0.769000\pi\)
−0.748031 + 0.663664i \(0.769000\pi\)
\(522\) 4.33149e8 3.04527
\(523\) 1.30165e8 0.909893 0.454946 − 0.890519i \(-0.349658\pi\)
0.454946 + 0.890519i \(0.349658\pi\)
\(524\) 0 0
\(525\) −3.60250e8 −2.48958
\(526\) 2.88205e8 1.98036
\(527\) 0 0
\(528\) 0 0
\(529\) 8.77708e7 0.592902
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 1.79892e8 1.18138
\(535\) 2.94050e8 1.92026
\(536\) 653312. 0.00424254
\(537\) 0 0
\(538\) −6.55543e7 −0.420973
\(539\) 0 0
\(540\) −1.68256e8 −1.06854
\(541\) −2.70414e8 −1.70780 −0.853900 − 0.520438i \(-0.825769\pi\)
−0.853900 + 0.520438i \(0.825769\pi\)
\(542\) 0 0
\(543\) −4.70686e8 −2.93989
\(544\) 0 0
\(545\) 1.62445e8 1.00350
\(546\) 0 0
\(547\) 3.00733e8 1.83747 0.918733 − 0.394880i \(-0.129214\pi\)
0.918733 + 0.394880i \(0.129214\pi\)
\(548\) 0 0
\(549\) 5.45977e8 3.29957
\(550\) 0 0
\(551\) 0 0
\(552\) −3.45940e8 −2.05676
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 2.68288e8 1.52770
\(561\) 0 0
\(562\) 2.07860e8 1.17101
\(563\) −3.51215e8 −1.96810 −0.984052 − 0.177881i \(-0.943076\pi\)
−0.984052 + 0.177881i \(0.943076\pi\)
\(564\) −7.44663e7 −0.415071
\(565\) 0 0
\(566\) −3.54959e8 −1.95762
\(567\) −2.38446e7 −0.130810
\(568\) 0 0
\(569\) −1.49518e8 −0.811630 −0.405815 − 0.913955i \(-0.633012\pi\)
−0.405815 + 0.913955i \(0.633012\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 3.11625e8 1.64777
\(575\) −2.39938e8 −1.26210
\(576\) 3.16408e8 1.65569
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 1.93101e8 1.00000
\(579\) 0 0
\(580\) −3.58864e8 −1.83927
\(581\) 5.93019e8 3.02371
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.33498e8 −0.660026 −0.330013 − 0.943976i \(-0.607053\pi\)
−0.330013 + 0.943976i \(0.607053\pi\)
\(588\) 4.41906e8 2.17369
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.90036e8 0.897631
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 3.52000e8 1.62963
\(601\) 3.79911e8 1.75008 0.875042 − 0.484047i \(-0.160834\pi\)
0.875042 + 0.484047i \(0.160834\pi\)
\(602\) −7.29576e7 −0.334411
\(603\) 1.54013e6 0.00702435
\(604\) 0 0
\(605\) −2.21445e8 −1.00000
\(606\) −6.75037e8 −3.03326
\(607\) −1.69713e8 −0.758839 −0.379419 − 0.925225i \(-0.623876\pi\)
−0.379419 + 0.925225i \(0.623876\pi\)
\(608\) 0 0
\(609\) −1.03425e9 −4.57901
\(610\) −4.52342e8 −1.99286
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −3.45532e8 −1.49274
\(615\) 4.08859e8 1.75771
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 4.62952e8 1.96142
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) −3.22967e8 −1.34860
\(622\) 0 0
\(623\) −2.67794e8 −1.10748
\(624\) 0 0
\(625\) 2.44141e8 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 6.32468e8 2.52940
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.15616e8 0.842091
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −2.62144e8 −1.00000
\(641\) −7.10123e7 −0.269625 −0.134812 − 0.990871i \(-0.543043\pi\)
−0.134812 + 0.990871i \(0.543043\pi\)
\(642\) −8.28046e8 −3.12932
\(643\) 4.35161e8 1.63688 0.818440 − 0.574592i \(-0.194839\pi\)
0.818440 + 0.574592i \(0.194839\pi\)
\(644\) 5.14979e8 1.92811
\(645\) −9.57220e7 −0.356724
\(646\) 0 0
\(647\) 3.26848e8 1.20679 0.603396 − 0.797441i \(-0.293814\pi\)
0.603396 + 0.797441i \(0.293814\pi\)
\(648\) 2.32986e7 0.0856257
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 5.54304e8 1.99989
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) −4.57446e8 −1.63533
\(655\) 0 0
\(656\) −3.04488e8 −1.07860
\(657\) 0 0
\(658\) 1.10853e8 0.389109
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 3.64144e8 1.26086 0.630432 − 0.776244i \(-0.282878\pi\)
0.630432 + 0.776244i \(0.282878\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −5.79439e8 −1.97926
\(665\) 0 0
\(666\) 0 0
\(667\) −6.88839e8 −2.32135
\(668\) −3.76830e8 −1.26420
\(669\) 7.84798e8 2.62108
\(670\) −1.27600e6 −0.00424254
\(671\) 0 0
\(672\) −7.55499e8 −2.48958
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 3.28625e8 1.06854
\(676\) 3.08916e8 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 4.36258e8 1.38134
\(682\) 0 0
\(683\) −6.06224e8 −1.90270 −0.951352 − 0.308107i \(-0.900304\pi\)
−0.951352 + 0.308107i \(0.900304\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.64653e8 −0.510033
\(687\) 1.02229e9 3.15286
\(688\) 7.12868e7 0.218899
\(689\) 0 0
\(690\) 6.75664e8 2.05676
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 3.22731e8 0.965524
\(695\) 0 0
\(696\) 1.01056e9 2.99733
\(697\) 0 0
\(698\) 6.42257e7 0.188861
\(699\) 0 0
\(700\) −5.24000e8 −1.52770
\(701\) −5.95860e8 −1.72978 −0.864889 − 0.501963i \(-0.832611\pi\)
−0.864889 + 0.501963i \(0.832611\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 1.45442e8 0.415071
\(706\) 0 0
\(707\) 1.00488e9 2.84353
\(708\) 0 0
\(709\) −7.12546e8 −1.99928 −0.999641 − 0.0268005i \(-0.991468\pi\)
−0.999641 + 0.0268005i \(0.991468\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.61662e8 0.724937
\(713\) 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 \(\pi\)
\(720\) −6.17984e8 −1.65569
\(721\) −6.89167e8 −1.83873
\(722\) 3.76367e8 1.00000
\(723\) −1.10342e9 −2.91962
\(724\) −6.84633e8 −1.80402
\(725\) 7.00906e8 1.83927
\(726\) 6.23589e8 1.62963
\(727\) −6.84267e8 −1.78083 −0.890415 − 0.455150i \(-0.849586\pi\)
−0.890415 + 0.455150i \(0.849586\pi\)
\(728\) 0 0
\(729\) −6.19698e8 −1.59955
\(730\) 0 0
\(731\) 0 0
\(732\) 1.27380e9 3.24763
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 6.95671e8 1.75920
\(735\) −8.63098e8 −2.17369
\(736\) −5.03185e8 −1.26210
\(737\) 0 0
\(738\) −7.17808e8 −1.78583
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 8.19353e8 1.99758 0.998790 − 0.0491695i \(-0.0156575\pi\)
0.998790 + 0.0491695i \(0.0156575\pi\)
\(744\) 0 0
\(745\) −3.71165e8 −0.897631
\(746\) 0 0
\(747\) −1.36598e9 −3.27705
\(748\) 0 0
\(749\) 1.23266e9 2.93358
\(750\) −6.87500e8 −1.62963
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) −1.08315e8 −0.254703
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −7.05329e8 −1.63240
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4.06269e8 −0.921850 −0.460925 − 0.887439i \(-0.652482\pi\)
−0.460925 + 0.887439i \(0.652482\pi\)
\(762\) −6.07173e8 −1.37230
\(763\) 6.80970e8 1.53304
\(764\) 0 0
\(765\) 0 0
\(766\) 1.54222e8 0.343131
\(767\) 0 0
\(768\) 7.38198e8 1.62963
\(769\) −3.60743e7 −0.0793266 −0.0396633 − 0.999213i \(-0.512629\pi\)
−0.0396633 + 0.999213i \(0.512629\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 1.68053e8 0.362429
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 6.67665e8 1.41782
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 9.43453e8 1.96533
\(784\) 6.42773e8 1.33386
\(785\) 0 0
\(786\) 0 0
\(787\) 2.13980e8 0.438983 0.219492 − 0.975614i \(-0.429560\pi\)
0.219492 + 0.975614i \(0.429560\pi\)
\(788\) 0 0
\(789\) 1.58513e9 3.22726
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 5.12000e8 1.00000
\(801\) 6.16847e8 1.20027
\(802\) 9.70986e8 1.88230
\(803\) 0 0
\(804\) 3.59322e6 0.00691377
\(805\) −1.00582e9 −1.92811
\(806\) 0 0
\(807\) −3.60548e8 −0.686030
\(808\) −9.81872e8 −1.86132
\(809\) −3.61673e8 −0.683079 −0.341540 − 0.939867i \(-0.610948\pi\)
−0.341540 + 0.939867i \(0.610948\pi\)
\(810\) −4.55050e7 −0.0856257
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) −1.50436e9 −2.80985
\(813\) 0 0
\(814\) 0 0
\(815\) −1.08263e9 −1.99989
\(816\) 0 0
\(817\) 0 0
\(818\) −9.06972e8 −1.65704
\(819\) 0 0
\(820\) 5.94704e8 1.07860
\(821\) −1.04853e9 −1.89474 −0.947372 − 0.320135i \(-0.896272\pi\)
−0.947372 + 0.320135i \(0.896272\pi\)
\(822\) 0 0
\(823\) −1.16107e8 −0.208285 −0.104142 − 0.994562i \(-0.533210\pi\)
−0.104142 + 0.994562i \(0.533210\pi\)
\(824\) 6.73384e8 1.20360
\(825\) 0 0
\(826\) 0 0
\(827\) 1.01116e9 1.78773 0.893865 − 0.448337i \(-0.147983\pi\)
0.893865 + 0.448337i \(0.147983\pi\)
\(828\) −1.18622e9 −2.08965
\(829\) 1.81441e8 0.318472 0.159236 − 0.987241i \(-0.449097\pi\)
0.159236 + 0.987241i \(0.449097\pi\)
\(830\) 1.13172e9 1.97926
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 7.35996e8 1.26420
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 1.47558e9 2.48958
\(841\) 1.41742e9 2.38292
\(842\) −4.57844e8 −0.766975
\(843\) 1.14323e9 1.90832
\(844\) 0 0
\(845\) −6.03351e8 −1.00000
\(846\) −2.55343e8 −0.421709
\(847\) −9.28298e8 −1.52770
\(848\) 0 0
\(849\) −1.95227e9 −3.19020
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) −1.89622e9 −3.04449
\(855\) 0 0
\(856\) −1.20443e9 −1.92026
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) −1.39232e8 −0.218899
\(861\) 1.71394e9 2.68525
\(862\) 0 0
\(863\) 7.23421e8 1.12553 0.562767 − 0.826615i \(-0.309737\pi\)
0.562767 + 0.826615i \(0.309737\pi\)
\(864\) 6.89177e8 1.06854
\(865\) 0 0
\(866\) 0 0
\(867\) 1.06205e9 1.62963
\(868\) 0 0
\(869\) 0 0
\(870\) −1.97375e9 −2.99733
\(871\) 0 0
\(872\) −6.65376e8 −1.00350
\(873\) 0 0
\(874\) 0 0
\(875\) 1.02344e9 1.52770
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −4.68316e8 −0.684874 −0.342437 − 0.939541i \(-0.611252\pi\)
−0.342437 + 0.939541i \(0.611252\pi\)
\(882\) 1.51529e9 2.20846
\(883\) −9.07097e8 −1.31756 −0.658782 − 0.752334i \(-0.728928\pi\)
−0.658782 + 0.752334i \(0.728928\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 2.85726e8 0.410817
\(887\) −1.37949e9 −1.97674 −0.988369 − 0.152074i \(-0.951405\pi\)
−0.988369 + 0.152074i \(0.951405\pi\)
\(888\) 0 0
\(889\) 9.03860e8 1.28646
\(890\) −5.11058e8 −0.724937
\(891\) 0 0
\(892\) 1.14152e9 1.60839
\(893\) 0 0
\(894\) 1.04520e9 1.46281
\(895\) 0 0
\(896\) −1.09891e9 −1.52770
\(897\) 0 0
\(898\) −1.42133e9 −1.96276
\(899\) 0 0
\(900\) 1.20700e9 1.65569
\(901\) 0 0
\(902\) 0 0
\(903\) −4.01267e8 −0.544966
\(904\) 0 0
\(905\) 1.33717e9 1.80402
\(906\) 0 0
\(907\) −1.36077e9 −1.82374 −0.911872 − 0.410474i \(-0.865363\pi\)
−0.911872 + 0.410474i \(0.865363\pi\)
\(908\) 6.34557e8 0.847643
\(909\) −2.31469e9 −3.08177
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −2.48788e9 −3.24763
\(916\) 1.48697e9 1.93471
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 9.82784e8 1.26210
\(921\) −1.90043e9 −2.43261
\(922\) 4.80957e8 0.613640
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 3.63120e8 0.457317
\(927\) 1.58745e9 1.99279
\(928\) 1.46991e9 1.83927
\(929\) 1.27758e9 1.59346 0.796731 − 0.604334i \(-0.206561\pi\)
0.796731 + 0.604334i \(0.206561\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) −6.33780e8 −0.777854
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) −5.34899e6 −0.00648132
\(939\) 0 0
\(940\) 2.11552e8 0.254703
\(941\) 3.11817e8 0.374224 0.187112 − 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(942\) 0 0
\(943\) 1.14153e9 1.36130
\(944\) 0 0
\(945\) 1.37760e9 1.63240
\(946\) 0 0
\(947\) 1.01744e9 1.19800 0.599001 − 0.800749i \(-0.295565\pi\)
0.599001 + 0.800749i \(0.295565\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −1.44179e9 −1.62963
\(961\) 8.87504e8 1.00000
\(962\) 0 0
\(963\) −2.83935e9 −3.17936
\(964\) −1.60498e9 −1.79159
\(965\) 0 0
\(966\) 2.83238e9 3.14210
\(967\) −6.69958e8 −0.740915 −0.370458 − 0.928849i \(-0.620799\pi\)
−0.370458 + 0.928849i \(0.620799\pi\)
\(968\) 9.07039e8 1.00000
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −8.53127e8 −0.928998
\(973\) 0 0
\(974\) −1.84792e9 −1.99989
\(975\) 0 0
\(976\) 1.85279e9 1.99286
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 3.04867e9 3.25907
\(979\) 0 0
\(980\) −1.25542e9 −1.33386
\(981\) −1.56857e9 −1.66149
\(982\) 0 0
\(983\) 8.00372e8 0.842619 0.421310 − 0.906917i \(-0.361571\pi\)
0.421310 + 0.906917i \(0.361571\pi\)
\(984\) −1.67469e9 −1.75771
\(985\) 0 0
\(986\) 0 0
\(987\) 6.09693e8 0.634103
\(988\) 0 0
\(989\) −2.67256e8 −0.276273
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) −3.18691e9 −3.22546
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 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 20.7.d.b.19.1 yes 1
3.2 odd 2 180.7.f.a.19.1 1
4.3 odd 2 20.7.d.a.19.1 ✓ 1
5.2 odd 4 100.7.b.d.51.2 2
5.3 odd 4 100.7.b.d.51.1 2
5.4 even 2 20.7.d.a.19.1 ✓ 1
8.3 odd 2 320.7.h.b.319.1 1
8.5 even 2 320.7.h.a.319.1 1
12.11 even 2 180.7.f.b.19.1 1
15.14 odd 2 180.7.f.b.19.1 1
20.3 even 4 100.7.b.d.51.2 2
20.7 even 4 100.7.b.d.51.1 2
20.19 odd 2 CM 20.7.d.b.19.1 yes 1
40.19 odd 2 320.7.h.a.319.1 1
40.29 even 2 320.7.h.b.319.1 1
60.59 even 2 180.7.f.a.19.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.7.d.a.19.1 ✓ 1 4.3 odd 2
20.7.d.a.19.1 ✓ 1 5.4 even 2
20.7.d.b.19.1 yes 1 1.1 even 1 trivial
20.7.d.b.19.1 yes 1 20.19 odd 2 CM
100.7.b.d.51.1 2 5.3 odd 4
100.7.b.d.51.1 2 20.7 even 4
100.7.b.d.51.2 2 5.2 odd 4
100.7.b.d.51.2 2 20.3 even 4
180.7.f.a.19.1 1 3.2 odd 2
180.7.f.a.19.1 1 60.59 even 2
180.7.f.b.19.1 1 12.11 even 2
180.7.f.b.19.1 1 15.14 odd 2
320.7.h.a.319.1 1 8.5 even 2
320.7.h.a.319.1 1 40.19 odd 2
320.7.h.b.319.1 1 8.3 odd 2
320.7.h.b.319.1 1 40.29 even 2