Properties

Label 1344.4.b.f.895.2
Level $1344$
Weight $4$
Character 1344.895
Analytic conductor $79.299$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,4,Mod(895,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.895");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 158x^{6} + 8461x^{4} + 180672x^{2} + 1232100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 895.2
Root \(-5.92762i\) of defining polynomial
Character \(\chi\) \(=\) 1344.895
Dual form 1344.4.b.f.895.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -17.7376i q^{5} +(2.09706 - 18.4011i) q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -17.7376i q^{5} +(2.09706 - 18.4011i) q^{7} +9.00000 q^{9} +32.9212i q^{11} +53.7866i q^{13} -53.2127i q^{15} +48.1047i q^{17} -110.800 q^{19} +(6.29119 - 55.2034i) q^{21} +157.820i q^{23} -189.621 q^{25} +27.0000 q^{27} +72.0082 q^{29} +184.189 q^{31} +98.7635i q^{33} +(-326.391 - 37.1968i) q^{35} -422.967 q^{37} +161.360i q^{39} +346.448i q^{41} +198.168i q^{43} -159.638i q^{45} -29.0729 q^{47} +(-334.205 - 77.1768i) q^{49} +144.314i q^{51} -682.553 q^{53} +583.941 q^{55} -332.400 q^{57} -305.320 q^{59} -172.135i q^{61} +(18.8736 - 165.610i) q^{63} +954.042 q^{65} +109.033i q^{67} +473.459i q^{69} -741.453i q^{71} +752.602i q^{73} -568.863 q^{75} +(605.787 + 69.0378i) q^{77} +449.132i q^{79} +81.0000 q^{81} -262.988 q^{83} +853.261 q^{85} +216.025 q^{87} -674.294i q^{89} +(989.735 + 112.794i) q^{91} +552.567 q^{93} +1965.32i q^{95} +1459.49i q^{97} +296.290i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{3} - 4 q^{7} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{3} - 4 q^{7} + 72 q^{9} + 56 q^{19} - 12 q^{21} - 656 q^{25} + 216 q^{27} - 240 q^{29} + 320 q^{31} - 600 q^{35} - 392 q^{37} - 816 q^{47} - 16 q^{49} - 288 q^{53} - 456 q^{55} + 168 q^{57} + 1824 q^{59} - 36 q^{63} - 816 q^{65} - 1968 q^{75} + 2064 q^{77} + 648 q^{81} - 1680 q^{83} - 2568 q^{85} - 720 q^{87} + 864 q^{91} + 960 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 17.7376i 1.58650i −0.608899 0.793248i \(-0.708389\pi\)
0.608899 0.793248i \(-0.291611\pi\)
\(6\) 0 0
\(7\) 2.09706 18.4011i 0.113231 0.993569i
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 32.9212i 0.902373i 0.892430 + 0.451186i \(0.148999\pi\)
−0.892430 + 0.451186i \(0.851001\pi\)
\(12\) 0 0
\(13\) 53.7866i 1.14752i 0.819025 + 0.573758i \(0.194515\pi\)
−0.819025 + 0.573758i \(0.805485\pi\)
\(14\) 0 0
\(15\) 53.2127i 0.915963i
\(16\) 0 0
\(17\) 48.1047i 0.686301i 0.939280 + 0.343150i \(0.111494\pi\)
−0.939280 + 0.343150i \(0.888506\pi\)
\(18\) 0 0
\(19\) −110.800 −1.33785 −0.668927 0.743328i \(-0.733246\pi\)
−0.668927 + 0.743328i \(0.733246\pi\)
\(20\) 0 0
\(21\) 6.29119 55.2034i 0.0653739 0.573637i
\(22\) 0 0
\(23\) 157.820i 1.43077i 0.698732 + 0.715384i \(0.253748\pi\)
−0.698732 + 0.715384i \(0.746252\pi\)
\(24\) 0 0
\(25\) −189.621 −1.51697
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 72.0082 0.461089 0.230545 0.973062i \(-0.425949\pi\)
0.230545 + 0.973062i \(0.425949\pi\)
\(30\) 0 0
\(31\) 184.189 1.06714 0.533570 0.845756i \(-0.320850\pi\)
0.533570 + 0.845756i \(0.320850\pi\)
\(32\) 0 0
\(33\) 98.7635i 0.520985i
\(34\) 0 0
\(35\) −326.391 37.1968i −1.57629 0.179640i
\(36\) 0 0
\(37\) −422.967 −1.87933 −0.939667 0.342091i \(-0.888865\pi\)
−0.939667 + 0.342091i \(0.888865\pi\)
\(38\) 0 0
\(39\) 161.360i 0.662519i
\(40\) 0 0
\(41\) 346.448i 1.31966i 0.751415 + 0.659830i \(0.229372\pi\)
−0.751415 + 0.659830i \(0.770628\pi\)
\(42\) 0 0
\(43\) 198.168i 0.702800i 0.936225 + 0.351400i \(0.114294\pi\)
−0.936225 + 0.351400i \(0.885706\pi\)
\(44\) 0 0
\(45\) 159.638i 0.528832i
\(46\) 0 0
\(47\) −29.0729 −0.0902281 −0.0451141 0.998982i \(-0.514365\pi\)
−0.0451141 + 0.998982i \(0.514365\pi\)
\(48\) 0 0
\(49\) −334.205 77.1768i −0.974358 0.225005i
\(50\) 0 0
\(51\) 144.314i 0.396236i
\(52\) 0 0
\(53\) −682.553 −1.76898 −0.884490 0.466560i \(-0.845493\pi\)
−0.884490 + 0.466560i \(0.845493\pi\)
\(54\) 0 0
\(55\) 583.941 1.43161
\(56\) 0 0
\(57\) −332.400 −0.772411
\(58\) 0 0
\(59\) −305.320 −0.673716 −0.336858 0.941555i \(-0.609364\pi\)
−0.336858 + 0.941555i \(0.609364\pi\)
\(60\) 0 0
\(61\) 172.135i 0.361306i −0.983547 0.180653i \(-0.942179\pi\)
0.983547 0.180653i \(-0.0578211\pi\)
\(62\) 0 0
\(63\) 18.8736 165.610i 0.0377436 0.331190i
\(64\) 0 0
\(65\) 954.042 1.82053
\(66\) 0 0
\(67\) 109.033i 0.198814i 0.995047 + 0.0994071i \(0.0316946\pi\)
−0.995047 + 0.0994071i \(0.968305\pi\)
\(68\) 0 0
\(69\) 473.459i 0.826054i
\(70\) 0 0
\(71\) 741.453i 1.23936i −0.784856 0.619678i \(-0.787263\pi\)
0.784856 0.619678i \(-0.212737\pi\)
\(72\) 0 0
\(73\) 752.602i 1.20665i 0.797496 + 0.603325i \(0.206158\pi\)
−0.797496 + 0.603325i \(0.793842\pi\)
\(74\) 0 0
\(75\) −568.863 −0.875821
\(76\) 0 0
\(77\) 605.787 + 69.0378i 0.896569 + 0.102176i
\(78\) 0 0
\(79\) 449.132i 0.639637i 0.947479 + 0.319818i \(0.103622\pi\)
−0.947479 + 0.319818i \(0.896378\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −262.988 −0.347791 −0.173895 0.984764i \(-0.555635\pi\)
−0.173895 + 0.984764i \(0.555635\pi\)
\(84\) 0 0
\(85\) 853.261 1.08881
\(86\) 0 0
\(87\) 216.025 0.266210
\(88\) 0 0
\(89\) 674.294i 0.803090i −0.915839 0.401545i \(-0.868473\pi\)
0.915839 0.401545i \(-0.131527\pi\)
\(90\) 0 0
\(91\) 989.735 + 112.794i 1.14014 + 0.129934i
\(92\) 0 0
\(93\) 552.567 0.616113
\(94\) 0 0
\(95\) 1965.32i 2.12250i
\(96\) 0 0
\(97\) 1459.49i 1.52771i 0.645385 + 0.763857i \(0.276697\pi\)
−0.645385 + 0.763857i \(0.723303\pi\)
\(98\) 0 0
\(99\) 296.290i 0.300791i
\(100\) 0 0
\(101\) 226.672i 0.223314i 0.993747 + 0.111657i \(0.0356158\pi\)
−0.993747 + 0.111657i \(0.964384\pi\)
\(102\) 0 0
\(103\) 712.681 0.681772 0.340886 0.940105i \(-0.389273\pi\)
0.340886 + 0.940105i \(0.389273\pi\)
\(104\) 0 0
\(105\) −979.174 111.590i −0.910073 0.103715i
\(106\) 0 0
\(107\) 759.426i 0.686135i −0.939311 0.343068i \(-0.888534\pi\)
0.939311 0.343068i \(-0.111466\pi\)
\(108\) 0 0
\(109\) 1081.36 0.950231 0.475115 0.879923i \(-0.342406\pi\)
0.475115 + 0.879923i \(0.342406\pi\)
\(110\) 0 0
\(111\) −1268.90 −1.08503
\(112\) 0 0
\(113\) 1934.09 1.61013 0.805063 0.593189i \(-0.202131\pi\)
0.805063 + 0.593189i \(0.202131\pi\)
\(114\) 0 0
\(115\) 2799.33 2.26991
\(116\) 0 0
\(117\) 484.079i 0.382505i
\(118\) 0 0
\(119\) 885.183 + 100.879i 0.681887 + 0.0777104i
\(120\) 0 0
\(121\) 247.198 0.185723
\(122\) 0 0
\(123\) 1039.34i 0.761906i
\(124\) 0 0
\(125\) 1146.22i 0.820166i
\(126\) 0 0
\(127\) 2518.01i 1.75935i −0.475574 0.879676i \(-0.657760\pi\)
0.475574 0.879676i \(-0.342240\pi\)
\(128\) 0 0
\(129\) 594.505i 0.405762i
\(130\) 0 0
\(131\) 1114.33 0.743204 0.371602 0.928392i \(-0.378809\pi\)
0.371602 + 0.928392i \(0.378809\pi\)
\(132\) 0 0
\(133\) −232.355 + 2038.85i −0.151486 + 1.32925i
\(134\) 0 0
\(135\) 478.914i 0.305321i
\(136\) 0 0
\(137\) 260.914 0.162711 0.0813555 0.996685i \(-0.474075\pi\)
0.0813555 + 0.996685i \(0.474075\pi\)
\(138\) 0 0
\(139\) −505.973 −0.308749 −0.154374 0.988012i \(-0.549336\pi\)
−0.154374 + 0.988012i \(0.549336\pi\)
\(140\) 0 0
\(141\) −87.2188 −0.0520932
\(142\) 0 0
\(143\) −1770.72 −1.03549
\(144\) 0 0
\(145\) 1277.25i 0.731516i
\(146\) 0 0
\(147\) −1002.61 231.530i −0.562546 0.129907i
\(148\) 0 0
\(149\) −1397.28 −0.768254 −0.384127 0.923280i \(-0.625497\pi\)
−0.384127 + 0.923280i \(0.625497\pi\)
\(150\) 0 0
\(151\) 344.187i 0.185493i −0.995690 0.0927467i \(-0.970435\pi\)
0.995690 0.0927467i \(-0.0295647\pi\)
\(152\) 0 0
\(153\) 432.943i 0.228767i
\(154\) 0 0
\(155\) 3267.06i 1.69301i
\(156\) 0 0
\(157\) 2450.31i 1.24558i 0.782390 + 0.622789i \(0.214000\pi\)
−0.782390 + 0.622789i \(0.786000\pi\)
\(158\) 0 0
\(159\) −2047.66 −1.02132
\(160\) 0 0
\(161\) 2904.06 + 330.958i 1.42157 + 0.162007i
\(162\) 0 0
\(163\) 1749.52i 0.840693i 0.907364 + 0.420346i \(0.138091\pi\)
−0.907364 + 0.420346i \(0.861909\pi\)
\(164\) 0 0
\(165\) 1751.82 0.826540
\(166\) 0 0
\(167\) −4072.62 −1.88712 −0.943560 0.331202i \(-0.892546\pi\)
−0.943560 + 0.331202i \(0.892546\pi\)
\(168\) 0 0
\(169\) −695.995 −0.316793
\(170\) 0 0
\(171\) −997.199 −0.445952
\(172\) 0 0
\(173\) 3988.51i 1.75284i 0.481550 + 0.876419i \(0.340074\pi\)
−0.481550 + 0.876419i \(0.659926\pi\)
\(174\) 0 0
\(175\) −397.647 + 3489.24i −0.171767 + 1.50721i
\(176\) 0 0
\(177\) −915.960 −0.388970
\(178\) 0 0
\(179\) 32.2713i 0.0134753i 0.999977 + 0.00673763i \(0.00214467\pi\)
−0.999977 + 0.00673763i \(0.997855\pi\)
\(180\) 0 0
\(181\) 233.112i 0.0957299i −0.998854 0.0478650i \(-0.984758\pi\)
0.998854 0.0478650i \(-0.0152417\pi\)
\(182\) 0 0
\(183\) 516.405i 0.208600i
\(184\) 0 0
\(185\) 7502.40i 2.98155i
\(186\) 0 0
\(187\) −1583.66 −0.619299
\(188\) 0 0
\(189\) 56.6208 496.831i 0.0217913 0.191212i
\(190\) 0 0
\(191\) 111.172i 0.0421158i −0.999778 0.0210579i \(-0.993297\pi\)
0.999778 0.0210579i \(-0.00670344\pi\)
\(192\) 0 0
\(193\) −1713.45 −0.639052 −0.319526 0.947577i \(-0.603524\pi\)
−0.319526 + 0.947577i \(0.603524\pi\)
\(194\) 0 0
\(195\) 2862.13 1.05108
\(196\) 0 0
\(197\) −4045.10 −1.46295 −0.731475 0.681868i \(-0.761168\pi\)
−0.731475 + 0.681868i \(0.761168\pi\)
\(198\) 0 0
\(199\) −4480.63 −1.59610 −0.798048 0.602594i \(-0.794134\pi\)
−0.798048 + 0.602594i \(0.794134\pi\)
\(200\) 0 0
\(201\) 327.100i 0.114785i
\(202\) 0 0
\(203\) 151.006 1325.03i 0.0522096 0.458124i
\(204\) 0 0
\(205\) 6145.14 2.09363
\(206\) 0 0
\(207\) 1420.38i 0.476922i
\(208\) 0 0
\(209\) 3647.66i 1.20724i
\(210\) 0 0
\(211\) 3937.96i 1.28483i −0.766355 0.642417i \(-0.777932\pi\)
0.766355 0.642417i \(-0.222068\pi\)
\(212\) 0 0
\(213\) 2224.36i 0.715543i
\(214\) 0 0
\(215\) 3515.02 1.11499
\(216\) 0 0
\(217\) 386.256 3389.29i 0.120833 1.06028i
\(218\) 0 0
\(219\) 2257.81i 0.696659i
\(220\) 0 0
\(221\) −2587.39 −0.787541
\(222\) 0 0
\(223\) 6231.07 1.87114 0.935568 0.353147i \(-0.114888\pi\)
0.935568 + 0.353147i \(0.114888\pi\)
\(224\) 0 0
\(225\) −1706.59 −0.505656
\(226\) 0 0
\(227\) −2627.92 −0.768377 −0.384188 0.923255i \(-0.625519\pi\)
−0.384188 + 0.923255i \(0.625519\pi\)
\(228\) 0 0
\(229\) 1930.43i 0.557059i 0.960428 + 0.278530i \(0.0898471\pi\)
−0.960428 + 0.278530i \(0.910153\pi\)
\(230\) 0 0
\(231\) 1817.36 + 207.113i 0.517635 + 0.0589916i
\(232\) 0 0
\(233\) −5138.78 −1.44486 −0.722431 0.691443i \(-0.756975\pi\)
−0.722431 + 0.691443i \(0.756975\pi\)
\(234\) 0 0
\(235\) 515.683i 0.143147i
\(236\) 0 0
\(237\) 1347.40i 0.369294i
\(238\) 0 0
\(239\) 5724.04i 1.54919i −0.632455 0.774597i \(-0.717953\pi\)
0.632455 0.774597i \(-0.282047\pi\)
\(240\) 0 0
\(241\) 5731.84i 1.53204i 0.642820 + 0.766018i \(0.277764\pi\)
−0.642820 + 0.766018i \(0.722236\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −1368.93 + 5927.97i −0.356970 + 1.54581i
\(246\) 0 0
\(247\) 5959.55i 1.53521i
\(248\) 0 0
\(249\) −788.963 −0.200797
\(250\) 0 0
\(251\) 2433.43 0.611940 0.305970 0.952041i \(-0.401019\pi\)
0.305970 + 0.952041i \(0.401019\pi\)
\(252\) 0 0
\(253\) −5195.60 −1.29109
\(254\) 0 0
\(255\) 2559.78 0.628626
\(256\) 0 0
\(257\) 2495.76i 0.605762i 0.953028 + 0.302881i \(0.0979485\pi\)
−0.953028 + 0.302881i \(0.902051\pi\)
\(258\) 0 0
\(259\) −886.989 + 7783.08i −0.212799 + 1.86725i
\(260\) 0 0
\(261\) 648.074 0.153696
\(262\) 0 0
\(263\) 1255.35i 0.294328i −0.989112 0.147164i \(-0.952985\pi\)
0.989112 0.147164i \(-0.0470146\pi\)
\(264\) 0 0
\(265\) 12106.8i 2.80648i
\(266\) 0 0
\(267\) 2022.88i 0.463664i
\(268\) 0 0
\(269\) 2843.99i 0.644614i −0.946635 0.322307i \(-0.895542\pi\)
0.946635 0.322307i \(-0.104458\pi\)
\(270\) 0 0
\(271\) 688.100 0.154240 0.0771201 0.997022i \(-0.475428\pi\)
0.0771201 + 0.997022i \(0.475428\pi\)
\(272\) 0 0
\(273\) 2969.20 + 338.382i 0.658258 + 0.0750176i
\(274\) 0 0
\(275\) 6242.54i 1.36887i
\(276\) 0 0
\(277\) −3641.63 −0.789906 −0.394953 0.918701i \(-0.629239\pi\)
−0.394953 + 0.918701i \(0.629239\pi\)
\(278\) 0 0
\(279\) 1657.70 0.355713
\(280\) 0 0
\(281\) 471.011 0.0999935 0.0499967 0.998749i \(-0.484079\pi\)
0.0499967 + 0.998749i \(0.484079\pi\)
\(282\) 0 0
\(283\) 6928.51 1.45533 0.727663 0.685935i \(-0.240607\pi\)
0.727663 + 0.685935i \(0.240607\pi\)
\(284\) 0 0
\(285\) 5895.96i 1.22543i
\(286\) 0 0
\(287\) 6375.04 + 726.523i 1.31117 + 0.149426i
\(288\) 0 0
\(289\) 2598.93 0.528991
\(290\) 0 0
\(291\) 4378.46i 0.882026i
\(292\) 0 0
\(293\) 6739.54i 1.34378i −0.740650 0.671891i \(-0.765482\pi\)
0.740650 0.671891i \(-0.234518\pi\)
\(294\) 0 0
\(295\) 5415.63i 1.06885i
\(296\) 0 0
\(297\) 888.871i 0.173662i
\(298\) 0 0
\(299\) −8488.57 −1.64183
\(300\) 0 0
\(301\) 3646.53 + 415.572i 0.698280 + 0.0795787i
\(302\) 0 0
\(303\) 680.016i 0.128930i
\(304\) 0 0
\(305\) −3053.26 −0.573210
\(306\) 0 0
\(307\) −7974.24 −1.48246 −0.741228 0.671253i \(-0.765756\pi\)
−0.741228 + 0.671253i \(0.765756\pi\)
\(308\) 0 0
\(309\) 2138.04 0.393621
\(310\) 0 0
\(311\) −461.001 −0.0840546 −0.0420273 0.999116i \(-0.513382\pi\)
−0.0420273 + 0.999116i \(0.513382\pi\)
\(312\) 0 0
\(313\) 1977.23i 0.357059i −0.983935 0.178530i \(-0.942866\pi\)
0.983935 0.178530i \(-0.0571340\pi\)
\(314\) 0 0
\(315\) −2937.52 334.771i −0.525431 0.0598801i
\(316\) 0 0
\(317\) 1570.62 0.278280 0.139140 0.990273i \(-0.455566\pi\)
0.139140 + 0.990273i \(0.455566\pi\)
\(318\) 0 0
\(319\) 2370.59i 0.416075i
\(320\) 0 0
\(321\) 2278.28i 0.396140i
\(322\) 0 0
\(323\) 5330.00i 0.918171i
\(324\) 0 0
\(325\) 10199.1i 1.74074i
\(326\) 0 0
\(327\) 3244.07 0.548616
\(328\) 0 0
\(329\) −60.9678 + 534.975i −0.0102166 + 0.0896479i
\(330\) 0 0
\(331\) 5222.32i 0.867204i −0.901104 0.433602i \(-0.857242\pi\)
0.901104 0.433602i \(-0.142758\pi\)
\(332\) 0 0
\(333\) −3806.70 −0.626444
\(334\) 0 0
\(335\) 1933.99 0.315418
\(336\) 0 0
\(337\) −3136.18 −0.506940 −0.253470 0.967343i \(-0.581572\pi\)
−0.253470 + 0.967343i \(0.581572\pi\)
\(338\) 0 0
\(339\) 5802.28 0.929607
\(340\) 0 0
\(341\) 6063.72i 0.962958i
\(342\) 0 0
\(343\) −2120.99 + 5987.90i −0.333886 + 0.942614i
\(344\) 0 0
\(345\) 8398.00 1.31053
\(346\) 0 0
\(347\) 2160.06i 0.334173i 0.985942 + 0.167087i \(0.0534359\pi\)
−0.985942 + 0.167087i \(0.946564\pi\)
\(348\) 0 0
\(349\) 4717.67i 0.723584i 0.932259 + 0.361792i \(0.117835\pi\)
−0.932259 + 0.361792i \(0.882165\pi\)
\(350\) 0 0
\(351\) 1452.24i 0.220840i
\(352\) 0 0
\(353\) 2083.75i 0.314184i 0.987584 + 0.157092i \(0.0502119\pi\)
−0.987584 + 0.157092i \(0.949788\pi\)
\(354\) 0 0
\(355\) −13151.6 −1.96623
\(356\) 0 0
\(357\) 2655.55 + 302.636i 0.393688 + 0.0448661i
\(358\) 0 0
\(359\) 2144.44i 0.315263i −0.987498 0.157631i \(-0.949614\pi\)
0.987498 0.157631i \(-0.0503858\pi\)
\(360\) 0 0
\(361\) 5417.62 0.789855
\(362\) 0 0
\(363\) 741.594 0.107227
\(364\) 0 0
\(365\) 13349.3 1.91434
\(366\) 0 0
\(367\) −2435.16 −0.346361 −0.173181 0.984890i \(-0.555404\pi\)
−0.173181 + 0.984890i \(0.555404\pi\)
\(368\) 0 0
\(369\) 3118.03i 0.439887i
\(370\) 0 0
\(371\) −1431.36 + 12559.8i −0.200303 + 1.75760i
\(372\) 0 0
\(373\) −9190.90 −1.27584 −0.637918 0.770105i \(-0.720204\pi\)
−0.637918 + 0.770105i \(0.720204\pi\)
\(374\) 0 0
\(375\) 3438.65i 0.473523i
\(376\) 0 0
\(377\) 3873.08i 0.529108i
\(378\) 0 0
\(379\) 3771.98i 0.511223i 0.966780 + 0.255612i \(0.0822769\pi\)
−0.966780 + 0.255612i \(0.917723\pi\)
\(380\) 0 0
\(381\) 7554.04i 1.01576i
\(382\) 0 0
\(383\) −3974.64 −0.530273 −0.265136 0.964211i \(-0.585417\pi\)
−0.265136 + 0.964211i \(0.585417\pi\)
\(384\) 0 0
\(385\) 1224.56 10745.2i 0.162102 1.42240i
\(386\) 0 0
\(387\) 1783.52i 0.234267i
\(388\) 0 0
\(389\) 8337.10 1.08665 0.543326 0.839522i \(-0.317165\pi\)
0.543326 + 0.839522i \(0.317165\pi\)
\(390\) 0 0
\(391\) −7591.87 −0.981937
\(392\) 0 0
\(393\) 3343.00 0.429089
\(394\) 0 0
\(395\) 7966.51 1.01478
\(396\) 0 0
\(397\) 8116.13i 1.02604i 0.858378 + 0.513019i \(0.171473\pi\)
−0.858378 + 0.513019i \(0.828527\pi\)
\(398\) 0 0
\(399\) −697.064 + 6116.54i −0.0874607 + 0.767443i
\(400\) 0 0
\(401\) 8543.46 1.06394 0.531970 0.846763i \(-0.321452\pi\)
0.531970 + 0.846763i \(0.321452\pi\)
\(402\) 0 0
\(403\) 9906.90i 1.22456i
\(404\) 0 0
\(405\) 1436.74i 0.176277i
\(406\) 0 0
\(407\) 13924.6i 1.69586i
\(408\) 0 0
\(409\) 4970.55i 0.600923i 0.953794 + 0.300462i \(0.0971408\pi\)
−0.953794 + 0.300462i \(0.902859\pi\)
\(410\) 0 0
\(411\) 782.743 0.0939412
\(412\) 0 0
\(413\) −640.276 + 5618.24i −0.0762855 + 0.669383i
\(414\) 0 0
\(415\) 4664.76i 0.551768i
\(416\) 0 0
\(417\) −1517.92 −0.178256
\(418\) 0 0
\(419\) 11967.9 1.39540 0.697699 0.716391i \(-0.254207\pi\)
0.697699 + 0.716391i \(0.254207\pi\)
\(420\) 0 0
\(421\) 15762.7 1.82476 0.912381 0.409341i \(-0.134241\pi\)
0.912381 + 0.409341i \(0.134241\pi\)
\(422\) 0 0
\(423\) −261.656 −0.0300760
\(424\) 0 0
\(425\) 9121.66i 1.04110i
\(426\) 0 0
\(427\) −3167.48 360.979i −0.358982 0.0409110i
\(428\) 0 0
\(429\) −5312.15 −0.597839
\(430\) 0 0
\(431\) 3980.94i 0.444908i −0.974943 0.222454i \(-0.928593\pi\)
0.974943 0.222454i \(-0.0714067\pi\)
\(432\) 0 0
\(433\) 7828.52i 0.868856i −0.900707 0.434428i \(-0.856951\pi\)
0.900707 0.434428i \(-0.143049\pi\)
\(434\) 0 0
\(435\) 3831.75i 0.422341i
\(436\) 0 0
\(437\) 17486.4i 1.91416i
\(438\) 0 0
\(439\) −8174.69 −0.888739 −0.444370 0.895844i \(-0.646572\pi\)
−0.444370 + 0.895844i \(0.646572\pi\)
\(440\) 0 0
\(441\) −3007.84 694.591i −0.324786 0.0750018i
\(442\) 0 0
\(443\) 13439.8i 1.44141i −0.693243 0.720704i \(-0.743819\pi\)
0.693243 0.720704i \(-0.256181\pi\)
\(444\) 0 0
\(445\) −11960.3 −1.27410
\(446\) 0 0
\(447\) −4191.84 −0.443551
\(448\) 0 0
\(449\) −8966.31 −0.942419 −0.471210 0.882021i \(-0.656182\pi\)
−0.471210 + 0.882021i \(0.656182\pi\)
\(450\) 0 0
\(451\) −11405.5 −1.19082
\(452\) 0 0
\(453\) 1032.56i 0.107095i
\(454\) 0 0
\(455\) 2000.69 17555.5i 0.206140 1.80882i
\(456\) 0 0
\(457\) 7444.29 0.761990 0.380995 0.924577i \(-0.375582\pi\)
0.380995 + 0.924577i \(0.375582\pi\)
\(458\) 0 0
\(459\) 1298.83i 0.132079i
\(460\) 0 0
\(461\) 5552.31i 0.560947i −0.959862 0.280474i \(-0.909508\pi\)
0.959862 0.280474i \(-0.0904915\pi\)
\(462\) 0 0
\(463\) 3923.67i 0.393841i 0.980419 + 0.196921i \(0.0630941\pi\)
−0.980419 + 0.196921i \(0.936906\pi\)
\(464\) 0 0
\(465\) 9801.19i 0.977461i
\(466\) 0 0
\(467\) 8843.37 0.876279 0.438140 0.898907i \(-0.355638\pi\)
0.438140 + 0.898907i \(0.355638\pi\)
\(468\) 0 0
\(469\) 2006.34 + 228.650i 0.197535 + 0.0225119i
\(470\) 0 0
\(471\) 7350.92i 0.719134i
\(472\) 0 0
\(473\) −6523.93 −0.634188
\(474\) 0 0
\(475\) 21010.0 2.02948
\(476\) 0 0
\(477\) −6142.98 −0.589660
\(478\) 0 0
\(479\) −19687.6 −1.87798 −0.938988 0.343951i \(-0.888235\pi\)
−0.938988 + 0.343951i \(0.888235\pi\)
\(480\) 0 0
\(481\) 22749.9i 2.15657i
\(482\) 0 0
\(483\) 8712.18 + 992.874i 0.820741 + 0.0935348i
\(484\) 0 0
\(485\) 25887.7 2.42371
\(486\) 0 0
\(487\) 17969.6i 1.67203i 0.548706 + 0.836015i \(0.315121\pi\)
−0.548706 + 0.836015i \(0.684879\pi\)
\(488\) 0 0
\(489\) 5248.56i 0.485374i
\(490\) 0 0
\(491\) 2351.82i 0.216163i 0.994142 + 0.108082i \(0.0344708\pi\)
−0.994142 + 0.108082i \(0.965529\pi\)
\(492\) 0 0
\(493\) 3463.94i 0.316446i
\(494\) 0 0
\(495\) 5255.47 0.477203
\(496\) 0 0
\(497\) −13643.6 1554.88i −1.23139 0.140333i
\(498\) 0 0
\(499\) 1548.25i 0.138896i 0.997586 + 0.0694479i \(0.0221238\pi\)
−0.997586 + 0.0694479i \(0.977876\pi\)
\(500\) 0 0
\(501\) −12217.9 −1.08953
\(502\) 0 0
\(503\) 17661.5 1.56558 0.782791 0.622284i \(-0.213795\pi\)
0.782791 + 0.622284i \(0.213795\pi\)
\(504\) 0 0
\(505\) 4020.61 0.354287
\(506\) 0 0
\(507\) −2087.98 −0.182901
\(508\) 0 0
\(509\) 7607.03i 0.662427i 0.943556 + 0.331214i \(0.107458\pi\)
−0.943556 + 0.331214i \(0.892542\pi\)
\(510\) 0 0
\(511\) 13848.7 + 1578.25i 1.19889 + 0.136630i
\(512\) 0 0
\(513\) −2991.60 −0.257470
\(514\) 0 0
\(515\) 12641.2i 1.08163i
\(516\) 0 0
\(517\) 957.114i 0.0814194i
\(518\) 0 0
\(519\) 11965.5i 1.01200i
\(520\) 0 0
\(521\) 1121.16i 0.0942781i −0.998888 0.0471391i \(-0.984990\pi\)
0.998888 0.0471391i \(-0.0150104\pi\)
\(522\) 0 0
\(523\) 10129.6 0.846918 0.423459 0.905915i \(-0.360816\pi\)
0.423459 + 0.905915i \(0.360816\pi\)
\(524\) 0 0
\(525\) −1192.94 + 10467.7i −0.0991700 + 0.870189i
\(526\) 0 0
\(527\) 8860.37i 0.732379i
\(528\) 0 0
\(529\) −12740.0 −1.04710
\(530\) 0 0
\(531\) −2747.88 −0.224572
\(532\) 0 0
\(533\) −18634.2 −1.51433
\(534\) 0 0
\(535\) −13470.4 −1.08855
\(536\) 0 0
\(537\) 96.8139i 0.00777994i
\(538\) 0 0
\(539\) 2540.75 11002.4i 0.203039 0.879234i
\(540\) 0 0
\(541\) −14846.0 −1.17981 −0.589907 0.807471i \(-0.700836\pi\)
−0.589907 + 0.807471i \(0.700836\pi\)
\(542\) 0 0
\(543\) 699.337i 0.0552697i
\(544\) 0 0
\(545\) 19180.6i 1.50754i
\(546\) 0 0
\(547\) 5857.85i 0.457886i −0.973440 0.228943i \(-0.926473\pi\)
0.973440 0.228943i \(-0.0735269\pi\)
\(548\) 0 0
\(549\) 1549.22i 0.120435i
\(550\) 0 0
\(551\) −7978.51 −0.616871
\(552\) 0 0
\(553\) 8264.55 + 941.859i 0.635523 + 0.0724266i
\(554\) 0 0
\(555\) 22507.2i 1.72140i
\(556\) 0 0
\(557\) −8589.49 −0.653408 −0.326704 0.945127i \(-0.605938\pi\)
−0.326704 + 0.945127i \(0.605938\pi\)
\(558\) 0 0
\(559\) −10658.8 −0.806474
\(560\) 0 0
\(561\) −4750.99 −0.357553
\(562\) 0 0
\(563\) 6159.17 0.461062 0.230531 0.973065i \(-0.425954\pi\)
0.230531 + 0.973065i \(0.425954\pi\)
\(564\) 0 0
\(565\) 34306.1i 2.55446i
\(566\) 0 0
\(567\) 169.862 1490.49i 0.0125812 0.110397i
\(568\) 0 0
\(569\) −14756.9 −1.08724 −0.543620 0.839331i \(-0.682947\pi\)
−0.543620 + 0.839331i \(0.682947\pi\)
\(570\) 0 0
\(571\) 11216.5i 0.822059i 0.911622 + 0.411030i \(0.134831\pi\)
−0.911622 + 0.411030i \(0.865169\pi\)
\(572\) 0 0
\(573\) 333.516i 0.0243156i
\(574\) 0 0
\(575\) 29925.9i 2.17043i
\(576\) 0 0
\(577\) 25252.3i 1.82195i 0.412461 + 0.910975i \(0.364669\pi\)
−0.412461 + 0.910975i \(0.635331\pi\)
\(578\) 0 0
\(579\) −5140.36 −0.368957
\(580\) 0 0
\(581\) −551.502 + 4839.27i −0.0393807 + 0.345554i
\(582\) 0 0
\(583\) 22470.4i 1.59628i
\(584\) 0 0
\(585\) 8586.38 0.606843
\(586\) 0 0
\(587\) −16488.5 −1.15938 −0.579688 0.814839i \(-0.696826\pi\)
−0.579688 + 0.814839i \(0.696826\pi\)
\(588\) 0 0
\(589\) −20408.1 −1.42768
\(590\) 0 0
\(591\) −12135.3 −0.844634
\(592\) 0 0
\(593\) 8899.79i 0.616308i −0.951336 0.308154i \(-0.900289\pi\)
0.951336 0.308154i \(-0.0997112\pi\)
\(594\) 0 0
\(595\) 1789.34 15701.0i 0.123287 1.08181i
\(596\) 0 0
\(597\) −13441.9 −0.921507
\(598\) 0 0
\(599\) 10099.6i 0.688911i −0.938803 0.344456i \(-0.888064\pi\)
0.938803 0.344456i \(-0.111936\pi\)
\(600\) 0 0
\(601\) 8719.01i 0.591773i −0.955223 0.295887i \(-0.904385\pi\)
0.955223 0.295887i \(-0.0956150\pi\)
\(602\) 0 0
\(603\) 981.300i 0.0662714i
\(604\) 0 0
\(605\) 4384.69i 0.294649i
\(606\) 0 0
\(607\) 15841.9 1.05931 0.529655 0.848213i \(-0.322321\pi\)
0.529655 + 0.848213i \(0.322321\pi\)
\(608\) 0 0
\(609\) 453.018 3975.10i 0.0301432 0.264498i
\(610\) 0 0
\(611\) 1563.73i 0.103538i
\(612\) 0 0
\(613\) 6841.48 0.450775 0.225387 0.974269i \(-0.427635\pi\)
0.225387 + 0.974269i \(0.427635\pi\)
\(614\) 0 0
\(615\) 18435.4 1.20876
\(616\) 0 0
\(617\) 7676.90 0.500908 0.250454 0.968129i \(-0.419420\pi\)
0.250454 + 0.968129i \(0.419420\pi\)
\(618\) 0 0
\(619\) −1213.39 −0.0787886 −0.0393943 0.999224i \(-0.512543\pi\)
−0.0393943 + 0.999224i \(0.512543\pi\)
\(620\) 0 0
\(621\) 4261.13i 0.275351i
\(622\) 0 0
\(623\) −12407.8 1414.04i −0.797925 0.0909346i
\(624\) 0 0
\(625\) −3371.53 −0.215778
\(626\) 0 0
\(627\) 10943.0i 0.697002i
\(628\) 0 0
\(629\) 20346.7i 1.28979i
\(630\) 0 0
\(631\) 26695.3i 1.68419i 0.539328 + 0.842096i \(0.318678\pi\)
−0.539328 + 0.842096i \(0.681322\pi\)
\(632\) 0 0
\(633\) 11813.9i 0.741800i
\(634\) 0 0
\(635\) −44663.4 −2.79120
\(636\) 0 0
\(637\) 4151.08 17975.7i 0.258197 1.11809i
\(638\) 0 0
\(639\) 6673.08i 0.413119i
\(640\) 0 0
\(641\) −17576.8 −1.08306 −0.541530 0.840681i \(-0.682155\pi\)
−0.541530 + 0.840681i \(0.682155\pi\)
\(642\) 0 0
\(643\) 26186.4 1.60605 0.803025 0.595945i \(-0.203223\pi\)
0.803025 + 0.595945i \(0.203223\pi\)
\(644\) 0 0
\(645\) 10545.1 0.643739
\(646\) 0 0
\(647\) 9990.55 0.607062 0.303531 0.952822i \(-0.401834\pi\)
0.303531 + 0.952822i \(0.401834\pi\)
\(648\) 0 0
\(649\) 10051.5i 0.607943i
\(650\) 0 0
\(651\) 1158.77 10167.9i 0.0697630 0.612151i
\(652\) 0 0
\(653\) 28277.7 1.69463 0.847315 0.531091i \(-0.178218\pi\)
0.847315 + 0.531091i \(0.178218\pi\)
\(654\) 0 0
\(655\) 19765.6i 1.17909i
\(656\) 0 0
\(657\) 6773.42i 0.402216i
\(658\) 0 0
\(659\) 3090.10i 0.182661i 0.995821 + 0.0913303i \(0.0291119\pi\)
−0.995821 + 0.0913303i \(0.970888\pi\)
\(660\) 0 0
\(661\) 11860.9i 0.697937i 0.937134 + 0.348969i \(0.113468\pi\)
−0.937134 + 0.348969i \(0.886532\pi\)
\(662\) 0 0
\(663\) −7762.17 −0.454687
\(664\) 0 0
\(665\) 36164.1 + 4121.40i 2.10885 + 0.240332i
\(666\) 0 0
\(667\) 11364.3i 0.659712i
\(668\) 0 0
\(669\) 18693.2 1.08030
\(670\) 0 0
\(671\) 5666.89 0.326032
\(672\) 0 0
\(673\) −6269.15 −0.359076 −0.179538 0.983751i \(-0.557460\pi\)
−0.179538 + 0.983751i \(0.557460\pi\)
\(674\) 0 0
\(675\) −5119.76 −0.291940
\(676\) 0 0
\(677\) 30107.8i 1.70921i −0.519275 0.854607i \(-0.673798\pi\)
0.519275 0.854607i \(-0.326202\pi\)
\(678\) 0 0
\(679\) 26856.2 + 3060.64i 1.51789 + 0.172984i
\(680\) 0 0
\(681\) −7883.77 −0.443622
\(682\) 0 0
\(683\) 21336.3i 1.19533i −0.801746 0.597665i \(-0.796095\pi\)
0.801746 0.597665i \(-0.203905\pi\)
\(684\) 0 0
\(685\) 4627.98i 0.258140i
\(686\) 0 0
\(687\) 5791.30i 0.321618i
\(688\) 0 0
\(689\) 36712.2i 2.02993i
\(690\) 0 0
\(691\) −8597.07 −0.473297 −0.236648 0.971595i \(-0.576049\pi\)
−0.236648 + 0.971595i \(0.576049\pi\)
\(692\) 0 0
\(693\) 5452.08 + 621.340i 0.298856 + 0.0340588i
\(694\) 0 0
\(695\) 8974.73i 0.489829i
\(696\) 0 0
\(697\) −16665.8 −0.905683
\(698\) 0 0
\(699\) −15416.4 −0.834192
\(700\) 0 0
\(701\) −14251.7 −0.767872 −0.383936 0.923360i \(-0.625432\pi\)
−0.383936 + 0.923360i \(0.625432\pi\)
\(702\) 0 0
\(703\) 46864.7 2.51427
\(704\) 0 0
\(705\) 1547.05i 0.0826457i
\(706\) 0 0
\(707\) 4171.03 + 475.346i 0.221878 + 0.0252860i
\(708\) 0 0
\(709\) 21610.0 1.14468 0.572342 0.820015i \(-0.306035\pi\)
0.572342 + 0.820015i \(0.306035\pi\)
\(710\) 0 0
\(711\) 4042.19i 0.213212i
\(712\) 0 0
\(713\) 29068.6i 1.52683i
\(714\) 0 0
\(715\) 31408.2i 1.64280i
\(716\) 0 0
\(717\) 17172.1i 0.894427i
\(718\) 0 0
\(719\) −32462.5 −1.68379 −0.841896 0.539639i \(-0.818561\pi\)
−0.841896 + 0.539639i \(0.818561\pi\)
\(720\) 0 0
\(721\) 1494.54 13114.1i 0.0771977 0.677388i
\(722\) 0 0
\(723\) 17195.5i 0.884521i
\(724\) 0 0
\(725\) −13654.3 −0.699458
\(726\) 0 0
\(727\) 27608.0 1.40842 0.704211 0.709991i \(-0.251301\pi\)
0.704211 + 0.709991i \(0.251301\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −9532.84 −0.482332
\(732\) 0 0
\(733\) 6821.15i 0.343718i 0.985122 + 0.171859i \(0.0549773\pi\)
−0.985122 + 0.171859i \(0.945023\pi\)
\(734\) 0 0
\(735\) −4106.78 + 17783.9i −0.206097 + 0.892476i
\(736\) 0 0
\(737\) −3589.50 −0.179404
\(738\) 0 0
\(739\) 17123.2i 0.852351i −0.904641 0.426175i \(-0.859861\pi\)
0.904641 0.426175i \(-0.140139\pi\)
\(740\) 0 0
\(741\) 17878.6i 0.886354i
\(742\) 0 0
\(743\) 2411.20i 0.119055i −0.998227 0.0595277i \(-0.981041\pi\)
0.998227 0.0595277i \(-0.0189595\pi\)
\(744\) 0 0
\(745\) 24784.4i 1.21883i
\(746\) 0 0
\(747\) −2366.89 −0.115930
\(748\) 0 0
\(749\) −13974.3 1592.57i −0.681723 0.0776917i
\(750\) 0 0
\(751\) 9873.14i 0.479728i 0.970806 + 0.239864i \(0.0771029\pi\)
−0.970806 + 0.239864i \(0.922897\pi\)
\(752\) 0 0
\(753\) 7300.30 0.353304
\(754\) 0 0
\(755\) −6105.03 −0.294284
\(756\) 0 0
\(757\) −17357.4 −0.833376 −0.416688 0.909050i \(-0.636809\pi\)
−0.416688 + 0.909050i \(0.636809\pi\)
\(758\) 0 0
\(759\) −15586.8 −0.745409
\(760\) 0 0
\(761\) 26432.1i 1.25909i 0.776966 + 0.629543i \(0.216758\pi\)
−0.776966 + 0.629543i \(0.783242\pi\)
\(762\) 0 0
\(763\) 2267.67 19898.2i 0.107595 0.944120i
\(764\) 0 0
\(765\) 7679.35 0.362938
\(766\) 0 0
\(767\) 16422.1i 0.773100i
\(768\) 0 0
\(769\) 28143.2i 1.31973i −0.751385 0.659864i \(-0.770614\pi\)
0.751385 0.659864i \(-0.229386\pi\)
\(770\) 0 0
\(771\) 7487.27i 0.349737i
\(772\) 0 0
\(773\) 15072.7i 0.701328i −0.936501 0.350664i \(-0.885956\pi\)
0.936501 0.350664i \(-0.114044\pi\)
\(774\) 0 0
\(775\) −34926.1 −1.61882
\(776\) 0 0
\(777\) −2660.97 + 23349.2i −0.122859 + 1.07806i
\(778\) 0 0
\(779\) 38386.4i 1.76551i
\(780\) 0 0
\(781\) 24409.5 1.11836
\(782\) 0 0
\(783\) 1944.22 0.0887367
\(784\) 0 0
\(785\) 43462.4 1.97610
\(786\) 0 0
\(787\) −22371.5 −1.01329 −0.506645 0.862155i \(-0.669114\pi\)
−0.506645 + 0.862155i \(0.669114\pi\)
\(788\) 0 0
\(789\) 3766.06i 0.169931i
\(790\) 0 0
\(791\) 4055.92 35589.6i 0.182316 1.59977i
\(792\) 0 0
\(793\) 9258.56 0.414604
\(794\) 0 0
\(795\) 36320.5i 1.62032i
\(796\) 0 0
\(797\) 28223.2i 1.25435i 0.778878 + 0.627175i \(0.215789\pi\)
−0.778878 + 0.627175i \(0.784211\pi\)
\(798\) 0 0
\(799\) 1398.55i 0.0619236i
\(800\) 0 0
\(801\) 6068.64i 0.267697i
\(802\) 0 0
\(803\) −24776.5 −1.08885
\(804\) 0 0
\(805\) 5870.38 51510.9i 0.257023 2.25531i
\(806\) 0 0
\(807\) 8531.97i 0.372168i
\(808\) 0 0
\(809\) 750.987 0.0326370 0.0163185 0.999867i \(-0.494805\pi\)
0.0163185 + 0.999867i \(0.494805\pi\)
\(810\) 0 0
\(811\) 5362.03 0.232166 0.116083 0.993240i \(-0.462966\pi\)
0.116083 + 0.993240i \(0.462966\pi\)
\(812\) 0 0
\(813\) 2064.30 0.0890506
\(814\) 0 0
\(815\) 31032.2 1.33376
\(816\) 0 0
\(817\) 21957.0i 0.940244i
\(818\) 0 0
\(819\) 8907.61 + 1015.15i 0.380045 + 0.0433114i
\(820\) 0 0
\(821\) −34039.9 −1.44702 −0.723508 0.690316i \(-0.757471\pi\)
−0.723508 + 0.690316i \(0.757471\pi\)
\(822\) 0 0
\(823\) 3953.11i 0.167432i 0.996490 + 0.0837161i \(0.0266789\pi\)
−0.996490 + 0.0837161i \(0.973321\pi\)
\(824\) 0 0
\(825\) 18727.6i 0.790317i
\(826\) 0 0
\(827\) 22122.3i 0.930190i 0.885261 + 0.465095i \(0.153980\pi\)
−0.885261 + 0.465095i \(0.846020\pi\)
\(828\) 0 0
\(829\) 45342.1i 1.89963i 0.312805 + 0.949817i \(0.398731\pi\)
−0.312805 + 0.949817i \(0.601269\pi\)
\(830\) 0 0
\(831\) −10924.9 −0.456053
\(832\) 0 0
\(833\) 3712.57 16076.8i 0.154421 0.668702i
\(834\) 0 0
\(835\) 72238.4i 2.99391i
\(836\) 0 0
\(837\) 4973.10 0.205371
\(838\) 0 0
\(839\) −29527.4 −1.21502 −0.607508 0.794313i \(-0.707831\pi\)
−0.607508 + 0.794313i \(0.707831\pi\)
\(840\) 0 0
\(841\) −19203.8 −0.787396
\(842\) 0 0
\(843\) 1413.03 0.0577313
\(844\) 0 0
\(845\) 12345.2i 0.502591i
\(846\) 0 0
\(847\) 518.390 4548.72i 0.0210296 0.184529i
\(848\) 0 0
\(849\) 20785.5 0.840233
\(850\) 0 0
\(851\) 66752.5i 2.68889i
\(852\) 0 0
\(853\) 18339.4i 0.736140i 0.929798 + 0.368070i \(0.119981\pi\)
−0.929798 + 0.368070i \(0.880019\pi\)
\(854\) 0 0
\(855\) 17687.9i 0.707500i
\(856\) 0 0
\(857\) 14576.0i 0.580988i −0.956877 0.290494i \(-0.906180\pi\)
0.956877 0.290494i \(-0.0938196\pi\)
\(858\) 0 0
\(859\) 3972.38 0.157783 0.0788917 0.996883i \(-0.474862\pi\)
0.0788917 + 0.996883i \(0.474862\pi\)
\(860\) 0 0
\(861\) 19125.1 + 2179.57i 0.757006 + 0.0862712i
\(862\) 0 0
\(863\) 40644.4i 1.60319i 0.597869 + 0.801594i \(0.296014\pi\)
−0.597869 + 0.801594i \(0.703986\pi\)
\(864\) 0 0
\(865\) 70746.4 2.78087
\(866\) 0 0
\(867\) 7796.80 0.305413
\(868\) 0 0
\(869\) −14785.9 −0.577191
\(870\) 0 0
\(871\) −5864.53 −0.228142
\(872\) 0 0
\(873\) 13135.4i 0.509238i
\(874\) 0 0
\(875\) 21091.7 + 2403.69i 0.814891 + 0.0928681i
\(876\) 0 0
\(877\) 9602.49 0.369730 0.184865 0.982764i \(-0.440815\pi\)
0.184865 + 0.982764i \(0.440815\pi\)
\(878\) 0 0
\(879\) 20218.6i 0.775833i
\(880\) 0 0
\(881\) 5204.78i 0.199039i 0.995036 + 0.0995196i \(0.0317306\pi\)
−0.995036 + 0.0995196i \(0.968269\pi\)
\(882\) 0 0
\(883\) 22108.6i 0.842596i −0.906922 0.421298i \(-0.861575\pi\)
0.906922 0.421298i \(-0.138425\pi\)
\(884\) 0 0
\(885\) 16246.9i 0.617099i
\(886\) 0 0
\(887\) 21305.3 0.806497 0.403248 0.915091i \(-0.367881\pi\)
0.403248 + 0.915091i \(0.367881\pi\)
\(888\) 0 0
\(889\) −46334.4 5280.44i −1.74804 0.199213i
\(890\) 0 0
\(891\) 2666.61i 0.100264i
\(892\) 0 0
\(893\) 3221.28 0.120712
\(894\) 0 0
\(895\) 572.414 0.0213784
\(896\) 0 0
\(897\) −25465.7 −0.947910
\(898\) 0 0
\(899\) 13263.1 0.492047
\(900\) 0 0
\(901\) 32834.1i 1.21405i
\(902\) 0 0
\(903\) 10939.6 + 1246.72i 0.403152 + 0.0459448i
\(904\) 0 0
\(905\) −4134.85 −0.151875
\(906\) 0 0
\(907\) 5073.71i 0.185744i 0.995678 + 0.0928720i \(0.0296047\pi\)
−0.995678 + 0.0928720i \(0.970395\pi\)
\(908\) 0 0
\(909\) 2040.05i 0.0744380i
\(910\) 0 0
\(911\) 22711.5i 0.825977i 0.910736 + 0.412989i \(0.135515\pi\)
−0.910736 + 0.412989i \(0.864485\pi\)
\(912\) 0 0
\(913\) 8657.85i 0.313837i
\(914\) 0 0
\(915\) −9159.77 −0.330943
\(916\) 0 0
\(917\) 2336.83 20505.0i 0.0841537 0.738425i
\(918\) 0 0
\(919\) 8104.63i 0.290911i 0.989365 + 0.145455i \(0.0464647\pi\)
−0.989365 + 0.145455i \(0.953535\pi\)
\(920\) 0 0
\(921\) −23922.7 −0.855897
\(922\) 0 0
\(923\) 39880.2 1.42218
\(924\) 0 0
\(925\) 80203.4 2.85089
\(926\) 0 0
\(927\) 6414.13 0.227257
\(928\) 0 0
\(929\) 28055.3i 0.990814i 0.868661 + 0.495407i \(0.164981\pi\)
−0.868661 + 0.495407i \(0.835019\pi\)
\(930\) 0 0
\(931\) 37029.8 + 8551.18i 1.30355 + 0.301024i
\(932\) 0 0
\(933\) −1383.00 −0.0485290
\(934\) 0 0
\(935\) 28090.3i 0.982515i
\(936\) 0 0
\(937\) 26420.7i 0.921160i 0.887618 + 0.460580i \(0.152359\pi\)
−0.887618 + 0.460580i \(0.847641\pi\)
\(938\) 0 0
\(939\) 5931.69i 0.206148i
\(940\) 0 0
\(941\) 48331.7i 1.67435i 0.546931 + 0.837177i \(0.315796\pi\)
−0.546931 + 0.837177i \(0.684204\pi\)
\(942\) 0 0
\(943\) −54676.2 −1.88813
\(944\) 0 0
\(945\) −8812.57 1004.31i −0.303358 0.0345718i
\(946\) 0 0
\(947\) 56278.5i 1.93116i −0.260115 0.965578i \(-0.583760\pi\)
0.260115 0.965578i \(-0.416240\pi\)
\(948\) 0 0
\(949\) −40479.9 −1.38465
\(950\) 0 0
\(951\) 4711.85 0.160665
\(952\) 0 0
\(953\) 6765.72 0.229972 0.114986 0.993367i \(-0.463318\pi\)
0.114986 + 0.993367i \(0.463318\pi\)
\(954\) 0 0
\(955\) −1971.92 −0.0668166
\(956\) 0 0
\(957\) 7111.78i 0.240221i
\(958\) 0 0
\(959\) 547.154 4801.12i 0.0184239 0.161665i
\(960\) 0 0
\(961\) 4134.61 0.138787
\(962\) 0 0
\(963\) 6834.83i 0.228712i
\(964\) 0 0
\(965\) 30392.5i 1.01385i
\(966\) 0 0
\(967\) 30443.1i 1.01239i 0.862418 + 0.506196i \(0.168949\pi\)
−0.862418 + 0.506196i \(0.831051\pi\)
\(968\) 0 0
\(969\) 15990.0i 0.530106i
\(970\) 0 0
\(971\) 50571.9 1.67140 0.835701 0.549185i \(-0.185062\pi\)
0.835701 + 0.549185i \(0.185062\pi\)
\(972\) 0 0
\(973\) −1061.06 + 9310.49i −0.0349599 + 0.306763i
\(974\) 0 0
\(975\) 30597.2i 1.00502i
\(976\) 0 0
\(977\) 3249.34 0.106403 0.0532014 0.998584i \(-0.483057\pi\)
0.0532014 + 0.998584i \(0.483057\pi\)
\(978\) 0 0
\(979\) 22198.5 0.724686
\(980\) 0 0
\(981\) 9732.21 0.316744
\(982\) 0 0
\(983\) −29605.3 −0.960592 −0.480296 0.877106i \(-0.659471\pi\)
−0.480296 + 0.877106i \(0.659471\pi\)
\(984\) 0 0
\(985\) 71750.1i 2.32096i
\(986\) 0 0
\(987\) −182.903 + 1604.93i −0.00589856 + 0.0517582i
\(988\) 0 0
\(989\) −31274.9 −1.00554
\(990\) 0 0
\(991\) 40111.2i 1.28575i −0.765972 0.642873i \(-0.777742\pi\)
0.765972 0.642873i \(-0.222258\pi\)
\(992\) 0 0
\(993\) 15667.0i 0.500681i
\(994\) 0 0
\(995\) 79475.4i 2.53220i
\(996\) 0 0
\(997\) 50086.1i 1.59102i 0.605943 + 0.795508i \(0.292796\pi\)
−0.605943 + 0.795508i \(0.707204\pi\)
\(998\) 0 0
\(999\) −11420.1 −0.361678
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 1344.4.b.f.895.2 8
4.3 odd 2 1344.4.b.e.895.2 8
7.6 odd 2 1344.4.b.e.895.7 8
8.3 odd 2 336.4.b.f.223.7 yes 8
8.5 even 2 336.4.b.e.223.7 yes 8
24.5 odd 2 1008.4.b.i.559.2 8
24.11 even 2 1008.4.b.k.559.2 8
28.27 even 2 inner 1344.4.b.f.895.7 8
56.13 odd 2 336.4.b.f.223.2 yes 8
56.27 even 2 336.4.b.e.223.2 8
168.83 odd 2 1008.4.b.i.559.7 8
168.125 even 2 1008.4.b.k.559.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.4.b.e.223.2 8 56.27 even 2
336.4.b.e.223.7 yes 8 8.5 even 2
336.4.b.f.223.2 yes 8 56.13 odd 2
336.4.b.f.223.7 yes 8 8.3 odd 2
1008.4.b.i.559.2 8 24.5 odd 2
1008.4.b.i.559.7 8 168.83 odd 2
1008.4.b.k.559.2 8 24.11 even 2
1008.4.b.k.559.7 8 168.125 even 2
1344.4.b.e.895.2 8 4.3 odd 2
1344.4.b.e.895.7 8 7.6 odd 2
1344.4.b.f.895.2 8 1.1 even 1 trivial
1344.4.b.f.895.7 8 28.27 even 2 inner