Properties

Label 1344.4.b.g.895.12
Level $1344$
Weight $4$
Character 1344.895
Analytic conductor $79.299$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 2x^{10} - 6x^{9} + 56x^{7} - 448x^{6} + 448x^{5} - 3072x^{3} - 8192x^{2} - 32768x + 262144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{30} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 895.12
Root \(0.965027 - 2.65871i\) of defining polynomial
Character \(\chi\) \(=\) 1344.895
Dual form 1344.4.b.g.895.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +19.4608i q^{5} +(-1.95109 + 18.4172i) q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +19.4608i q^{5} +(-1.95109 + 18.4172i) q^{7} +9.00000 q^{9} +24.6621i q^{11} +5.25549i q^{13} -58.3823i q^{15} -80.8832i q^{17} -86.5920 q^{19} +(5.85328 - 55.2516i) q^{21} +108.350i q^{23} -253.721 q^{25} -27.0000 q^{27} -278.744 q^{29} +116.511 q^{31} -73.9863i q^{33} +(-358.413 - 37.9698i) q^{35} -53.5203 q^{37} -15.7665i q^{39} +303.555i q^{41} +176.565i q^{43} +175.147i q^{45} +102.276 q^{47} +(-335.386 - 71.8674i) q^{49} +242.650i q^{51} -185.933 q^{53} -479.943 q^{55} +259.776 q^{57} +732.951 q^{59} -443.155i q^{61} +(-17.5599 + 165.755i) q^{63} -102.276 q^{65} -166.969i q^{67} -325.050i q^{69} +378.024i q^{71} -664.824i q^{73} +761.164 q^{75} +(-454.207 - 48.1181i) q^{77} +737.826i q^{79} +81.0000 q^{81} +913.229 q^{83} +1574.05 q^{85} +836.233 q^{87} -1497.53i q^{89} +(-96.7913 - 10.2540i) q^{91} -349.533 q^{93} -1685.15i q^{95} +1445.52i q^{97} +221.959i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 36 q^{3} - 10 q^{7} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 36 q^{3} - 10 q^{7} + 108 q^{9} - 84 q^{19} + 30 q^{21} - 216 q^{25} - 324 q^{27} - 200 q^{29} + 384 q^{31} + 84 q^{35} + 244 q^{37} - 280 q^{47} - 424 q^{49} + 16 q^{53} - 212 q^{55} + 252 q^{57} + 1168 q^{59} - 90 q^{63} + 280 q^{65} + 648 q^{75} - 968 q^{77} + 972 q^{81} - 968 q^{83} + 852 q^{85} + 600 q^{87} + 1648 q^{91} - 1152 q^{93}+O(q^{100}) \) 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
\(5\) 19.4608i 1.74062i 0.492501 + 0.870312i \(0.336083\pi\)
−0.492501 + 0.870312i \(0.663917\pi\)
\(6\) 0 0
\(7\) −1.95109 + 18.4172i −0.105349 + 0.994435i
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 24.6621i 0.675991i 0.941148 + 0.337996i \(0.109749\pi\)
−0.941148 + 0.337996i \(0.890251\pi\)
\(12\) 0 0
\(13\) 5.25549i 0.112124i 0.998427 + 0.0560619i \(0.0178544\pi\)
−0.998427 + 0.0560619i \(0.982146\pi\)
\(14\) 0 0
\(15\) 58.3823i 1.00495i
\(16\) 0 0
\(17\) 80.8832i 1.15394i −0.816764 0.576972i \(-0.804234\pi\)
0.816764 0.576972i \(-0.195766\pi\)
\(18\) 0 0
\(19\) −86.5920 −1.04556 −0.522778 0.852469i \(-0.675104\pi\)
−0.522778 + 0.852469i \(0.675104\pi\)
\(20\) 0 0
\(21\) 5.85328 55.2516i 0.0608234 0.574137i
\(22\) 0 0
\(23\) 108.350i 0.982285i 0.871079 + 0.491143i \(0.163421\pi\)
−0.871079 + 0.491143i \(0.836579\pi\)
\(24\) 0 0
\(25\) −253.721 −2.02977
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −278.744 −1.78488 −0.892440 0.451167i \(-0.851008\pi\)
−0.892440 + 0.451167i \(0.851008\pi\)
\(30\) 0 0
\(31\) 116.511 0.675032 0.337516 0.941320i \(-0.390413\pi\)
0.337516 + 0.941320i \(0.390413\pi\)
\(32\) 0 0
\(33\) 73.9863i 0.390284i
\(34\) 0 0
\(35\) −358.413 37.9698i −1.73094 0.183373i
\(36\) 0 0
\(37\) −53.5203 −0.237802 −0.118901 0.992906i \(-0.537937\pi\)
−0.118901 + 0.992906i \(0.537937\pi\)
\(38\) 0 0
\(39\) 15.7665i 0.0647347i
\(40\) 0 0
\(41\) 303.555i 1.15628i 0.815938 + 0.578139i \(0.196221\pi\)
−0.815938 + 0.578139i \(0.803779\pi\)
\(42\) 0 0
\(43\) 176.565i 0.626183i 0.949723 + 0.313092i \(0.101365\pi\)
−0.949723 + 0.313092i \(0.898635\pi\)
\(44\) 0 0
\(45\) 175.147i 0.580208i
\(46\) 0 0
\(47\) 102.276 0.317414 0.158707 0.987326i \(-0.449268\pi\)
0.158707 + 0.987326i \(0.449268\pi\)
\(48\) 0 0
\(49\) −335.386 71.8674i −0.977803 0.209526i
\(50\) 0 0
\(51\) 242.650i 0.666230i
\(52\) 0 0
\(53\) −185.933 −0.481885 −0.240942 0.970539i \(-0.577457\pi\)
−0.240942 + 0.970539i \(0.577457\pi\)
\(54\) 0 0
\(55\) −479.943 −1.17665
\(56\) 0 0
\(57\) 259.776 0.603652
\(58\) 0 0
\(59\) 732.951 1.61732 0.808662 0.588274i \(-0.200192\pi\)
0.808662 + 0.588274i \(0.200192\pi\)
\(60\) 0 0
\(61\) 443.155i 0.930167i −0.885267 0.465083i \(-0.846024\pi\)
0.885267 0.465083i \(-0.153976\pi\)
\(62\) 0 0
\(63\) −17.5599 + 165.755i −0.0351164 + 0.331478i
\(64\) 0 0
\(65\) −102.276 −0.195165
\(66\) 0 0
\(67\) 166.969i 0.304455i −0.988345 0.152227i \(-0.951355\pi\)
0.988345 0.152227i \(-0.0486446\pi\)
\(68\) 0 0
\(69\) 325.050i 0.567123i
\(70\) 0 0
\(71\) 378.024i 0.631877i 0.948780 + 0.315938i \(0.102319\pi\)
−0.948780 + 0.315938i \(0.897681\pi\)
\(72\) 0 0
\(73\) 664.824i 1.06592i −0.846142 0.532958i \(-0.821081\pi\)
0.846142 0.532958i \(-0.178919\pi\)
\(74\) 0 0
\(75\) 761.164 1.17189
\(76\) 0 0
\(77\) −454.207 48.1181i −0.672229 0.0712151i
\(78\) 0 0
\(79\) 737.826i 1.05078i 0.850860 + 0.525392i \(0.176081\pi\)
−0.850860 + 0.525392i \(0.823919\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 913.229 1.20771 0.603855 0.797094i \(-0.293631\pi\)
0.603855 + 0.797094i \(0.293631\pi\)
\(84\) 0 0
\(85\) 1574.05 2.00858
\(86\) 0 0
\(87\) 836.233 1.03050
\(88\) 0 0
\(89\) 1497.53i 1.78358i −0.452453 0.891788i \(-0.649451\pi\)
0.452453 0.891788i \(-0.350549\pi\)
\(90\) 0 0
\(91\) −96.7913 10.2540i −0.111500 0.0118122i
\(92\) 0 0
\(93\) −349.533 −0.389730
\(94\) 0 0
\(95\) 1685.15i 1.81992i
\(96\) 0 0
\(97\) 1445.52i 1.51309i 0.653940 + 0.756547i \(0.273115\pi\)
−0.653940 + 0.756547i \(0.726885\pi\)
\(98\) 0 0
\(99\) 221.959i 0.225330i
\(100\) 0 0
\(101\) 240.349i 0.236788i −0.992967 0.118394i \(-0.962225\pi\)
0.992967 0.118394i \(-0.0377746\pi\)
\(102\) 0 0
\(103\) −1698.72 −1.62505 −0.812524 0.582927i \(-0.801907\pi\)
−0.812524 + 0.582927i \(0.801907\pi\)
\(104\) 0 0
\(105\) 1075.24 + 113.909i 0.999357 + 0.105871i
\(106\) 0 0
\(107\) 287.075i 0.259370i 0.991555 + 0.129685i \(0.0413966\pi\)
−0.991555 + 0.129685i \(0.958603\pi\)
\(108\) 0 0
\(109\) 14.9647 0.0131500 0.00657502 0.999978i \(-0.497907\pi\)
0.00657502 + 0.999978i \(0.497907\pi\)
\(110\) 0 0
\(111\) 160.561 0.137295
\(112\) 0 0
\(113\) −737.215 −0.613729 −0.306864 0.951753i \(-0.599280\pi\)
−0.306864 + 0.951753i \(0.599280\pi\)
\(114\) 0 0
\(115\) −2108.58 −1.70979
\(116\) 0 0
\(117\) 47.2994i 0.0373746i
\(118\) 0 0
\(119\) 1489.64 + 157.811i 1.14752 + 0.121567i
\(120\) 0 0
\(121\) 722.781 0.543036
\(122\) 0 0
\(123\) 910.666i 0.667577i
\(124\) 0 0
\(125\) 2505.01i 1.79244i
\(126\) 0 0
\(127\) 1572.28i 1.09856i 0.835637 + 0.549282i \(0.185099\pi\)
−0.835637 + 0.549282i \(0.814901\pi\)
\(128\) 0 0
\(129\) 529.694i 0.361527i
\(130\) 0 0
\(131\) −941.864 −0.628176 −0.314088 0.949394i \(-0.601699\pi\)
−0.314088 + 0.949394i \(0.601699\pi\)
\(132\) 0 0
\(133\) 168.949 1594.78i 0.110149 1.03974i
\(134\) 0 0
\(135\) 525.441i 0.334983i
\(136\) 0 0
\(137\) 2087.46 1.30178 0.650889 0.759173i \(-0.274396\pi\)
0.650889 + 0.759173i \(0.274396\pi\)
\(138\) 0 0
\(139\) −1894.46 −1.15602 −0.578008 0.816031i \(-0.696170\pi\)
−0.578008 + 0.816031i \(0.696170\pi\)
\(140\) 0 0
\(141\) −306.827 −0.183259
\(142\) 0 0
\(143\) −129.611 −0.0757947
\(144\) 0 0
\(145\) 5424.57i 3.10680i
\(146\) 0 0
\(147\) 1006.16 + 215.602i 0.564535 + 0.120970i
\(148\) 0 0
\(149\) 2506.20 1.37796 0.688981 0.724780i \(-0.258058\pi\)
0.688981 + 0.724780i \(0.258058\pi\)
\(150\) 0 0
\(151\) 465.035i 0.250623i 0.992117 + 0.125311i \(0.0399930\pi\)
−0.992117 + 0.125311i \(0.960007\pi\)
\(152\) 0 0
\(153\) 727.949i 0.384648i
\(154\) 0 0
\(155\) 2267.39i 1.17498i
\(156\) 0 0
\(157\) 441.217i 0.224286i 0.993692 + 0.112143i \(0.0357715\pi\)
−0.993692 + 0.112143i \(0.964228\pi\)
\(158\) 0 0
\(159\) 557.800 0.278216
\(160\) 0 0
\(161\) −1995.51 211.401i −0.976819 0.103483i
\(162\) 0 0
\(163\) 374.876i 0.180139i 0.995936 + 0.0900693i \(0.0287089\pi\)
−0.995936 + 0.0900693i \(0.971291\pi\)
\(164\) 0 0
\(165\) 1439.83 0.679337
\(166\) 0 0
\(167\) 986.292 0.457015 0.228508 0.973542i \(-0.426615\pi\)
0.228508 + 0.973542i \(0.426615\pi\)
\(168\) 0 0
\(169\) 2169.38 0.987428
\(170\) 0 0
\(171\) −779.328 −0.348519
\(172\) 0 0
\(173\) 1618.78i 0.711407i 0.934599 + 0.355704i \(0.115759\pi\)
−0.934599 + 0.355704i \(0.884241\pi\)
\(174\) 0 0
\(175\) 495.034 4672.83i 0.213835 2.01847i
\(176\) 0 0
\(177\) −2198.85 −0.933762
\(178\) 0 0
\(179\) 3693.28i 1.54217i −0.636730 0.771087i \(-0.719714\pi\)
0.636730 0.771087i \(-0.280286\pi\)
\(180\) 0 0
\(181\) 2447.09i 1.00492i −0.864600 0.502461i \(-0.832428\pi\)
0.864600 0.502461i \(-0.167572\pi\)
\(182\) 0 0
\(183\) 1329.46i 0.537032i
\(184\) 0 0
\(185\) 1041.55i 0.413924i
\(186\) 0 0
\(187\) 1994.75 0.780056
\(188\) 0 0
\(189\) 52.6796 497.264i 0.0202745 0.191379i
\(190\) 0 0
\(191\) 9.19972i 0.00348518i 0.999998 + 0.00174259i \(0.000554683\pi\)
−0.999998 + 0.00174259i \(0.999445\pi\)
\(192\) 0 0
\(193\) −1468.72 −0.547776 −0.273888 0.961762i \(-0.588310\pi\)
−0.273888 + 0.961762i \(0.588310\pi\)
\(194\) 0 0
\(195\) 306.827 0.112679
\(196\) 0 0
\(197\) −3914.64 −1.41577 −0.707884 0.706328i \(-0.750350\pi\)
−0.707884 + 0.706328i \(0.750350\pi\)
\(198\) 0 0
\(199\) −2960.22 −1.05449 −0.527247 0.849712i \(-0.676776\pi\)
−0.527247 + 0.849712i \(0.676776\pi\)
\(200\) 0 0
\(201\) 500.906i 0.175777i
\(202\) 0 0
\(203\) 543.856 5133.69i 0.188036 1.77495i
\(204\) 0 0
\(205\) −5907.42 −2.01264
\(206\) 0 0
\(207\) 975.151i 0.327428i
\(208\) 0 0
\(209\) 2135.54i 0.706787i
\(210\) 0 0
\(211\) 1329.13i 0.433654i 0.976210 + 0.216827i \(0.0695708\pi\)
−0.976210 + 0.216827i \(0.930429\pi\)
\(212\) 0 0
\(213\) 1134.07i 0.364814i
\(214\) 0 0
\(215\) −3436.08 −1.08995
\(216\) 0 0
\(217\) −227.324 + 2145.81i −0.0711141 + 0.671276i
\(218\) 0 0
\(219\) 1994.47i 0.615406i
\(220\) 0 0
\(221\) 425.080 0.129385
\(222\) 0 0
\(223\) 850.725 0.255465 0.127733 0.991809i \(-0.459230\pi\)
0.127733 + 0.991809i \(0.459230\pi\)
\(224\) 0 0
\(225\) −2283.49 −0.676590
\(226\) 0 0
\(227\) 2626.60 0.767988 0.383994 0.923336i \(-0.374548\pi\)
0.383994 + 0.923336i \(0.374548\pi\)
\(228\) 0 0
\(229\) 1681.55i 0.485239i −0.970122 0.242620i \(-0.921993\pi\)
0.970122 0.242620i \(-0.0780067\pi\)
\(230\) 0 0
\(231\) 1362.62 + 144.354i 0.388112 + 0.0411161i
\(232\) 0 0
\(233\) −1824.86 −0.513093 −0.256547 0.966532i \(-0.582585\pi\)
−0.256547 + 0.966532i \(0.582585\pi\)
\(234\) 0 0
\(235\) 1990.36i 0.552498i
\(236\) 0 0
\(237\) 2213.48i 0.606671i
\(238\) 0 0
\(239\) 776.076i 0.210043i −0.994470 0.105021i \(-0.966509\pi\)
0.994470 0.105021i \(-0.0334911\pi\)
\(240\) 0 0
\(241\) 40.5909i 0.0108493i −0.999985 0.00542466i \(-0.998273\pi\)
0.999985 0.00542466i \(-0.00172673\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 1398.59 6526.88i 0.364706 1.70199i
\(246\) 0 0
\(247\) 455.083i 0.117232i
\(248\) 0 0
\(249\) −2739.69 −0.697272
\(250\) 0 0
\(251\) −4231.68 −1.06415 −0.532074 0.846698i \(-0.678587\pi\)
−0.532074 + 0.846698i \(0.678587\pi\)
\(252\) 0 0
\(253\) −2672.14 −0.664016
\(254\) 0 0
\(255\) −4722.14 −1.15966
\(256\) 0 0
\(257\) 1872.50i 0.454489i −0.973838 0.227244i \(-0.927028\pi\)
0.973838 0.227244i \(-0.0729716\pi\)
\(258\) 0 0
\(259\) 104.423 985.694i 0.0250523 0.236479i
\(260\) 0 0
\(261\) −2508.70 −0.594960
\(262\) 0 0
\(263\) 5126.25i 1.20190i −0.799288 0.600948i \(-0.794790\pi\)
0.799288 0.600948i \(-0.205210\pi\)
\(264\) 0 0
\(265\) 3618.40i 0.838780i
\(266\) 0 0
\(267\) 4492.60i 1.02975i
\(268\) 0 0
\(269\) 2399.98i 0.543975i −0.962301 0.271988i \(-0.912319\pi\)
0.962301 0.271988i \(-0.0876810\pi\)
\(270\) 0 0
\(271\) 5666.29 1.27012 0.635060 0.772462i \(-0.280975\pi\)
0.635060 + 0.772462i \(0.280975\pi\)
\(272\) 0 0
\(273\) 290.374 + 30.7619i 0.0643745 + 0.00681975i
\(274\) 0 0
\(275\) 6257.30i 1.37211i
\(276\) 0 0
\(277\) 1089.92 0.236414 0.118207 0.992989i \(-0.462285\pi\)
0.118207 + 0.992989i \(0.462285\pi\)
\(278\) 0 0
\(279\) 1048.60 0.225011
\(280\) 0 0
\(281\) −3251.98 −0.690381 −0.345191 0.938533i \(-0.612186\pi\)
−0.345191 + 0.938533i \(0.612186\pi\)
\(282\) 0 0
\(283\) 8087.11 1.69869 0.849344 0.527840i \(-0.176998\pi\)
0.849344 + 0.527840i \(0.176998\pi\)
\(284\) 0 0
\(285\) 5055.44i 1.05073i
\(286\) 0 0
\(287\) −5590.64 592.265i −1.14984 0.121813i
\(288\) 0 0
\(289\) −1629.09 −0.331587
\(290\) 0 0
\(291\) 4336.55i 0.873585i
\(292\) 0 0
\(293\) 1615.35i 0.322080i −0.986948 0.161040i \(-0.948515\pi\)
0.986948 0.161040i \(-0.0514849\pi\)
\(294\) 0 0
\(295\) 14263.8i 2.81515i
\(296\) 0 0
\(297\) 665.877i 0.130095i
\(298\) 0 0
\(299\) −569.433 −0.110138
\(300\) 0 0
\(301\) −3251.83 344.495i −0.622699 0.0659679i
\(302\) 0 0
\(303\) 721.046i 0.136710i
\(304\) 0 0
\(305\) 8624.13 1.61907
\(306\) 0 0
\(307\) 2726.45 0.506863 0.253431 0.967353i \(-0.418441\pi\)
0.253431 + 0.967353i \(0.418441\pi\)
\(308\) 0 0
\(309\) 5096.17 0.938222
\(310\) 0 0
\(311\) 6495.92 1.18441 0.592203 0.805789i \(-0.298258\pi\)
0.592203 + 0.805789i \(0.298258\pi\)
\(312\) 0 0
\(313\) 2315.02i 0.418059i −0.977909 0.209030i \(-0.932970\pi\)
0.977909 0.209030i \(-0.0670305\pi\)
\(314\) 0 0
\(315\) −3225.71 341.728i −0.576979 0.0611244i
\(316\) 0 0
\(317\) 3545.18 0.628130 0.314065 0.949402i \(-0.398309\pi\)
0.314065 + 0.949402i \(0.398309\pi\)
\(318\) 0 0
\(319\) 6874.42i 1.20656i
\(320\) 0 0
\(321\) 861.225i 0.149747i
\(322\) 0 0
\(323\) 7003.83i 1.20651i
\(324\) 0 0
\(325\) 1333.43i 0.227585i
\(326\) 0 0
\(327\) −44.8940 −0.00759218
\(328\) 0 0
\(329\) −199.550 + 1883.63i −0.0334393 + 0.315648i
\(330\) 0 0
\(331\) 5480.40i 0.910060i 0.890476 + 0.455030i \(0.150371\pi\)
−0.890476 + 0.455030i \(0.849629\pi\)
\(332\) 0 0
\(333\) −481.683 −0.0792674
\(334\) 0 0
\(335\) 3249.34 0.529941
\(336\) 0 0
\(337\) −142.057 −0.0229624 −0.0114812 0.999934i \(-0.503655\pi\)
−0.0114812 + 0.999934i \(0.503655\pi\)
\(338\) 0 0
\(339\) 2211.64 0.354336
\(340\) 0 0
\(341\) 2873.41i 0.456316i
\(342\) 0 0
\(343\) 1977.97 6036.66i 0.311371 0.950288i
\(344\) 0 0
\(345\) 6325.73 0.987147
\(346\) 0 0
\(347\) 1106.02i 0.171108i 0.996334 + 0.0855540i \(0.0272660\pi\)
−0.996334 + 0.0855540i \(0.972734\pi\)
\(348\) 0 0
\(349\) 5314.57i 0.815136i 0.913175 + 0.407568i \(0.133623\pi\)
−0.913175 + 0.407568i \(0.866377\pi\)
\(350\) 0 0
\(351\) 141.898i 0.0215782i
\(352\) 0 0
\(353\) 10581.0i 1.59538i −0.603068 0.797690i \(-0.706055\pi\)
0.603068 0.797690i \(-0.293945\pi\)
\(354\) 0 0
\(355\) −7356.64 −1.09986
\(356\) 0 0
\(357\) −4468.92 473.432i −0.662523 0.0701868i
\(358\) 0 0
\(359\) 2661.07i 0.391214i −0.980682 0.195607i \(-0.937332\pi\)
0.980682 0.195607i \(-0.0626677\pi\)
\(360\) 0 0
\(361\) 639.174 0.0931876
\(362\) 0 0
\(363\) −2168.34 −0.313522
\(364\) 0 0
\(365\) 12938.0 1.85536
\(366\) 0 0
\(367\) 5895.68 0.838562 0.419281 0.907857i \(-0.362282\pi\)
0.419281 + 0.907857i \(0.362282\pi\)
\(368\) 0 0
\(369\) 2732.00i 0.385426i
\(370\) 0 0
\(371\) 362.774 3424.37i 0.0507662 0.479203i
\(372\) 0 0
\(373\) 6208.52 0.861837 0.430918 0.902391i \(-0.358190\pi\)
0.430918 + 0.902391i \(0.358190\pi\)
\(374\) 0 0
\(375\) 7515.04i 1.03487i
\(376\) 0 0
\(377\) 1464.94i 0.200127i
\(378\) 0 0
\(379\) 3532.73i 0.478797i −0.970921 0.239399i \(-0.923050\pi\)
0.970921 0.239399i \(-0.0769503\pi\)
\(380\) 0 0
\(381\) 4716.85i 0.634256i
\(382\) 0 0
\(383\) −9380.75 −1.25153 −0.625763 0.780014i \(-0.715212\pi\)
−0.625763 + 0.780014i \(0.715212\pi\)
\(384\) 0 0
\(385\) 936.414 8839.21i 0.123959 1.17010i
\(386\) 0 0
\(387\) 1589.08i 0.208728i
\(388\) 0 0
\(389\) −5267.14 −0.686516 −0.343258 0.939241i \(-0.611531\pi\)
−0.343258 + 0.939241i \(0.611531\pi\)
\(390\) 0 0
\(391\) 8763.70 1.13350
\(392\) 0 0
\(393\) 2825.59 0.362678
\(394\) 0 0
\(395\) −14358.7 −1.82902
\(396\) 0 0
\(397\) 8515.34i 1.07651i −0.842783 0.538253i \(-0.819085\pi\)
0.842783 0.538253i \(-0.180915\pi\)
\(398\) 0 0
\(399\) −506.848 + 4784.35i −0.0635943 + 0.600293i
\(400\) 0 0
\(401\) −2804.56 −0.349259 −0.174629 0.984634i \(-0.555873\pi\)
−0.174629 + 0.984634i \(0.555873\pi\)
\(402\) 0 0
\(403\) 612.322i 0.0756872i
\(404\) 0 0
\(405\) 1576.32i 0.193403i
\(406\) 0 0
\(407\) 1319.92i 0.160752i
\(408\) 0 0
\(409\) 13823.4i 1.67121i 0.549334 + 0.835603i \(0.314882\pi\)
−0.549334 + 0.835603i \(0.685118\pi\)
\(410\) 0 0
\(411\) −6262.38 −0.751582
\(412\) 0 0
\(413\) −1430.06 + 13498.9i −0.170384 + 1.60832i
\(414\) 0 0
\(415\) 17772.1i 2.10217i
\(416\) 0 0
\(417\) 5683.39 0.667426
\(418\) 0 0
\(419\) 5047.91 0.588560 0.294280 0.955719i \(-0.404920\pi\)
0.294280 + 0.955719i \(0.404920\pi\)
\(420\) 0 0
\(421\) −6467.30 −0.748686 −0.374343 0.927290i \(-0.622132\pi\)
−0.374343 + 0.927290i \(0.622132\pi\)
\(422\) 0 0
\(423\) 920.482 0.105805
\(424\) 0 0
\(425\) 20521.8i 2.34224i
\(426\) 0 0
\(427\) 8161.67 + 864.637i 0.924990 + 0.0979923i
\(428\) 0 0
\(429\) 388.834 0.0437601
\(430\) 0 0
\(431\) 12799.9i 1.43051i −0.698861 0.715257i \(-0.746309\pi\)
0.698861 0.715257i \(-0.253691\pi\)
\(432\) 0 0
\(433\) 7154.28i 0.794025i 0.917813 + 0.397013i \(0.129953\pi\)
−0.917813 + 0.397013i \(0.870047\pi\)
\(434\) 0 0
\(435\) 16273.7i 1.79371i
\(436\) 0 0
\(437\) 9382.25i 1.02703i
\(438\) 0 0
\(439\) −10688.8 −1.16207 −0.581034 0.813879i \(-0.697352\pi\)
−0.581034 + 0.813879i \(0.697352\pi\)
\(440\) 0 0
\(441\) −3018.48 646.807i −0.325934 0.0698420i
\(442\) 0 0
\(443\) 4652.95i 0.499025i −0.968372 0.249513i \(-0.919730\pi\)
0.968372 0.249513i \(-0.0802704\pi\)
\(444\) 0 0
\(445\) 29143.2 3.10453
\(446\) 0 0
\(447\) −7518.61 −0.795567
\(448\) 0 0
\(449\) 16721.0 1.75749 0.878743 0.477296i \(-0.158383\pi\)
0.878743 + 0.477296i \(0.158383\pi\)
\(450\) 0 0
\(451\) −7486.31 −0.781633
\(452\) 0 0
\(453\) 1395.10i 0.144697i
\(454\) 0 0
\(455\) 199.550 1883.63i 0.0205605 0.194079i
\(456\) 0 0
\(457\) −7571.11 −0.774971 −0.387485 0.921876i \(-0.626656\pi\)
−0.387485 + 0.921876i \(0.626656\pi\)
\(458\) 0 0
\(459\) 2183.85i 0.222077i
\(460\) 0 0
\(461\) 8410.07i 0.849666i 0.905272 + 0.424833i \(0.139667\pi\)
−0.905272 + 0.424833i \(0.860333\pi\)
\(462\) 0 0
\(463\) 10100.0i 1.01379i 0.862007 + 0.506897i \(0.169208\pi\)
−0.862007 + 0.506897i \(0.830792\pi\)
\(464\) 0 0
\(465\) 6802.18i 0.678373i
\(466\) 0 0
\(467\) −2012.75 −0.199441 −0.0997207 0.995015i \(-0.531795\pi\)
−0.0997207 + 0.995015i \(0.531795\pi\)
\(468\) 0 0
\(469\) 3075.10 + 325.772i 0.302761 + 0.0320741i
\(470\) 0 0
\(471\) 1323.65i 0.129492i
\(472\) 0 0
\(473\) −4354.46 −0.423294
\(474\) 0 0
\(475\) 21970.2 2.12224
\(476\) 0 0
\(477\) −1673.40 −0.160628
\(478\) 0 0
\(479\) −13299.9 −1.26866 −0.634330 0.773062i \(-0.718724\pi\)
−0.634330 + 0.773062i \(0.718724\pi\)
\(480\) 0 0
\(481\) 281.275i 0.0266633i
\(482\) 0 0
\(483\) 5986.52 + 634.204i 0.563967 + 0.0597459i
\(484\) 0 0
\(485\) −28130.9 −2.63372
\(486\) 0 0
\(487\) 2587.62i 0.240773i −0.992727 0.120386i \(-0.961587\pi\)
0.992727 0.120386i \(-0.0384134\pi\)
\(488\) 0 0
\(489\) 1124.63i 0.104003i
\(490\) 0 0
\(491\) 588.505i 0.0540913i 0.999634 + 0.0270457i \(0.00860996\pi\)
−0.999634 + 0.0270457i \(0.991390\pi\)
\(492\) 0 0
\(493\) 22545.7i 2.05965i
\(494\) 0 0
\(495\) −4319.49 −0.392215
\(496\) 0 0
\(497\) −6962.15 737.561i −0.628360 0.0665677i
\(498\) 0 0
\(499\) 16006.0i 1.43592i 0.696083 + 0.717961i \(0.254925\pi\)
−0.696083 + 0.717961i \(0.745075\pi\)
\(500\) 0 0
\(501\) −2958.88 −0.263858
\(502\) 0 0
\(503\) −10633.0 −0.942551 −0.471276 0.881986i \(-0.656206\pi\)
−0.471276 + 0.881986i \(0.656206\pi\)
\(504\) 0 0
\(505\) 4677.36 0.412158
\(506\) 0 0
\(507\) −6508.14 −0.570092
\(508\) 0 0
\(509\) 2697.13i 0.234869i −0.993081 0.117434i \(-0.962533\pi\)
0.993081 0.117434i \(-0.0374670\pi\)
\(510\) 0 0
\(511\) 12244.2 + 1297.14i 1.05998 + 0.112293i
\(512\) 0 0
\(513\) 2337.98 0.201217
\(514\) 0 0
\(515\) 33058.4i 2.82860i
\(516\) 0 0
\(517\) 2522.33i 0.214569i
\(518\) 0 0
\(519\) 4856.33i 0.410731i
\(520\) 0 0
\(521\) 11342.2i 0.953766i 0.878967 + 0.476883i \(0.158233\pi\)
−0.878967 + 0.476883i \(0.841767\pi\)
\(522\) 0 0
\(523\) 4722.12 0.394807 0.197403 0.980322i \(-0.436749\pi\)
0.197403 + 0.980322i \(0.436749\pi\)
\(524\) 0 0
\(525\) −1485.10 + 14018.5i −0.123457 + 1.16537i
\(526\) 0 0
\(527\) 9423.78i 0.778950i
\(528\) 0 0
\(529\) 427.252 0.0351156
\(530\) 0 0
\(531\) 6596.56 0.539108
\(532\) 0 0
\(533\) −1595.33 −0.129646
\(534\) 0 0
\(535\) −5586.70 −0.451465
\(536\) 0 0
\(537\) 11079.9i 0.890374i
\(538\) 0 0
\(539\) 1772.40 8271.33i 0.141638 0.660986i
\(540\) 0 0
\(541\) 6125.28 0.486777 0.243389 0.969929i \(-0.421741\pi\)
0.243389 + 0.969929i \(0.421741\pi\)
\(542\) 0 0
\(543\) 7341.27i 0.580192i
\(544\) 0 0
\(545\) 291.224i 0.0228893i
\(546\) 0 0
\(547\) 25304.7i 1.97797i −0.148015 0.988985i \(-0.547288\pi\)
0.148015 0.988985i \(-0.452712\pi\)
\(548\) 0 0
\(549\) 3988.39i 0.310056i
\(550\) 0 0
\(551\) 24137.0 1.86619
\(552\) 0 0
\(553\) −13588.7 1439.57i −1.04494 0.110699i
\(554\) 0 0
\(555\) 3124.64i 0.238979i
\(556\) 0 0
\(557\) −9686.90 −0.736889 −0.368444 0.929650i \(-0.620109\pi\)
−0.368444 + 0.929650i \(0.620109\pi\)
\(558\) 0 0
\(559\) −927.934 −0.0702100
\(560\) 0 0
\(561\) −5984.24 −0.450365
\(562\) 0 0
\(563\) 6110.70 0.457434 0.228717 0.973493i \(-0.426547\pi\)
0.228717 + 0.973493i \(0.426547\pi\)
\(564\) 0 0
\(565\) 14346.8i 1.06827i
\(566\) 0 0
\(567\) −158.039 + 1491.79i −0.0117055 + 0.110493i
\(568\) 0 0
\(569\) −16843.7 −1.24099 −0.620497 0.784209i \(-0.713069\pi\)
−0.620497 + 0.784209i \(0.713069\pi\)
\(570\) 0 0
\(571\) 15796.9i 1.15775i 0.815415 + 0.578877i \(0.196509\pi\)
−0.815415 + 0.578877i \(0.803491\pi\)
\(572\) 0 0
\(573\) 27.5992i 0.00201217i
\(574\) 0 0
\(575\) 27490.7i 1.99381i
\(576\) 0 0
\(577\) 6756.48i 0.487480i 0.969841 + 0.243740i \(0.0783743\pi\)
−0.969841 + 0.243740i \(0.921626\pi\)
\(578\) 0 0
\(579\) 4406.16 0.316259
\(580\) 0 0
\(581\) −1781.80 + 16819.1i −0.127231 + 1.20099i
\(582\) 0 0
\(583\) 4585.50i 0.325750i
\(584\) 0 0
\(585\) −920.482 −0.0650551
\(586\) 0 0
\(587\) 20504.0 1.44172 0.720861 0.693080i \(-0.243747\pi\)
0.720861 + 0.693080i \(0.243747\pi\)
\(588\) 0 0
\(589\) −10088.9 −0.705784
\(590\) 0 0
\(591\) 11743.9 0.817394
\(592\) 0 0
\(593\) 1432.90i 0.0992281i 0.998768 + 0.0496140i \(0.0157991\pi\)
−0.998768 + 0.0496140i \(0.984201\pi\)
\(594\) 0 0
\(595\) −3071.12 + 28989.6i −0.211603 + 1.99740i
\(596\) 0 0
\(597\) 8880.65 0.608812
\(598\) 0 0
\(599\) 21643.1i 1.47632i 0.674628 + 0.738158i \(0.264304\pi\)
−0.674628 + 0.738158i \(0.735696\pi\)
\(600\) 0 0
\(601\) 9462.88i 0.642261i 0.947035 + 0.321131i \(0.104063\pi\)
−0.947035 + 0.321131i \(0.895937\pi\)
\(602\) 0 0
\(603\) 1502.72i 0.101485i
\(604\) 0 0
\(605\) 14065.9i 0.945221i
\(606\) 0 0
\(607\) −523.640 −0.0350146 −0.0175073 0.999847i \(-0.505573\pi\)
−0.0175073 + 0.999847i \(0.505573\pi\)
\(608\) 0 0
\(609\) −1631.57 + 15401.1i −0.108562 + 1.02477i
\(610\) 0 0
\(611\) 537.509i 0.0355897i
\(612\) 0 0
\(613\) −25130.7 −1.65582 −0.827912 0.560858i \(-0.810471\pi\)
−0.827912 + 0.560858i \(0.810471\pi\)
\(614\) 0 0
\(615\) 17722.3 1.16200
\(616\) 0 0
\(617\) −22315.2 −1.45604 −0.728021 0.685555i \(-0.759560\pi\)
−0.728021 + 0.685555i \(0.759560\pi\)
\(618\) 0 0
\(619\) −20000.9 −1.29871 −0.649357 0.760484i \(-0.724962\pi\)
−0.649357 + 0.760484i \(0.724962\pi\)
\(620\) 0 0
\(621\) 2925.45i 0.189041i
\(622\) 0 0
\(623\) 27580.4 + 2921.83i 1.77365 + 0.187898i
\(624\) 0 0
\(625\) 17034.3 1.09019
\(626\) 0 0
\(627\) 6406.62i 0.408063i
\(628\) 0 0
\(629\) 4328.89i 0.274411i
\(630\) 0 0
\(631\) 6793.62i 0.428605i 0.976767 + 0.214302i \(0.0687478\pi\)
−0.976767 + 0.214302i \(0.931252\pi\)
\(632\) 0 0
\(633\) 3987.39i 0.250370i
\(634\) 0 0
\(635\) −30597.8 −1.91218
\(636\) 0 0
\(637\) 377.698 1762.62i 0.0234928 0.109635i
\(638\) 0 0
\(639\) 3402.22i 0.210626i
\(640\) 0 0
\(641\) −1369.18 −0.0843675 −0.0421837 0.999110i \(-0.513431\pi\)
−0.0421837 + 0.999110i \(0.513431\pi\)
\(642\) 0 0
\(643\) −24036.2 −1.47417 −0.737087 0.675797i \(-0.763799\pi\)
−0.737087 + 0.675797i \(0.763799\pi\)
\(644\) 0 0
\(645\) 10308.3 0.629282
\(646\) 0 0
\(647\) −13781.3 −0.837403 −0.418702 0.908124i \(-0.637515\pi\)
−0.418702 + 0.908124i \(0.637515\pi\)
\(648\) 0 0
\(649\) 18076.1i 1.09330i
\(650\) 0 0
\(651\) 681.972 6437.42i 0.0410578 0.387561i
\(652\) 0 0
\(653\) 10312.2 0.617991 0.308995 0.951064i \(-0.400007\pi\)
0.308995 + 0.951064i \(0.400007\pi\)
\(654\) 0 0
\(655\) 18329.4i 1.09342i
\(656\) 0 0
\(657\) 5983.42i 0.355305i
\(658\) 0 0
\(659\) 18371.4i 1.08596i −0.839745 0.542980i \(-0.817296\pi\)
0.839745 0.542980i \(-0.182704\pi\)
\(660\) 0 0
\(661\) 19068.1i 1.12203i 0.827806 + 0.561015i \(0.189589\pi\)
−0.827806 + 0.561015i \(0.810411\pi\)
\(662\) 0 0
\(663\) −1275.24 −0.0747002
\(664\) 0 0
\(665\) 31035.7 + 3287.88i 1.80979 + 0.191727i
\(666\) 0 0
\(667\) 30202.0i 1.75326i
\(668\) 0 0
\(669\) −2552.17 −0.147493
\(670\) 0 0
\(671\) 10929.1 0.628784
\(672\) 0 0
\(673\) −5215.69 −0.298737 −0.149368 0.988782i \(-0.547724\pi\)
−0.149368 + 0.988782i \(0.547724\pi\)
\(674\) 0 0
\(675\) 6850.47 0.390629
\(676\) 0 0
\(677\) 5346.63i 0.303527i −0.988417 0.151764i \(-0.951505\pi\)
0.988417 0.151764i \(-0.0484952\pi\)
\(678\) 0 0
\(679\) −26622.4 2820.34i −1.50467 0.159403i
\(680\) 0 0
\(681\) −7879.79 −0.443398
\(682\) 0 0
\(683\) 7087.40i 0.397060i 0.980095 + 0.198530i \(0.0636167\pi\)
−0.980095 + 0.198530i \(0.936383\pi\)
\(684\) 0 0
\(685\) 40623.5i 2.26591i
\(686\) 0 0
\(687\) 5044.64i 0.280153i
\(688\) 0 0
\(689\) 977.170i 0.0540308i
\(690\) 0 0
\(691\) −29981.3 −1.65057 −0.825284 0.564718i \(-0.808985\pi\)
−0.825284 + 0.564718i \(0.808985\pi\)
\(692\) 0 0
\(693\) −4087.86 433.063i −0.224076 0.0237384i
\(694\) 0 0
\(695\) 36867.7i 2.01219i
\(696\) 0 0
\(697\) 24552.5 1.33428
\(698\) 0 0
\(699\) 5474.59 0.296235
\(700\) 0 0
\(701\) 4928.54 0.265547 0.132773 0.991146i \(-0.457612\pi\)
0.132773 + 0.991146i \(0.457612\pi\)
\(702\) 0 0
\(703\) 4634.43 0.248636
\(704\) 0 0
\(705\) 5971.09i 0.318985i
\(706\) 0 0
\(707\) 4426.55 + 468.943i 0.235470 + 0.0249454i
\(708\) 0 0
\(709\) −22659.0 −1.20025 −0.600125 0.799906i \(-0.704883\pi\)
−0.600125 + 0.799906i \(0.704883\pi\)
\(710\) 0 0
\(711\) 6640.44i 0.350261i
\(712\) 0 0
\(713\) 12624.0i 0.663074i
\(714\) 0 0
\(715\) 2522.33i 0.131930i
\(716\) 0 0
\(717\) 2328.23i 0.121268i
\(718\) 0 0
\(719\) −23078.3 −1.19705 −0.598523 0.801106i \(-0.704246\pi\)
−0.598523 + 0.801106i \(0.704246\pi\)
\(720\) 0 0
\(721\) 3314.37 31285.7i 0.171198 1.61601i
\(722\) 0 0
\(723\) 121.773i 0.00626386i
\(724\) 0 0
\(725\) 70723.3 3.62289
\(726\) 0 0
\(727\) −24974.6 −1.27408 −0.637039 0.770831i \(-0.719841\pi\)
−0.637039 + 0.770831i \(0.719841\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 14281.1 0.722580
\(732\) 0 0
\(733\) 642.353i 0.0323682i 0.999869 + 0.0161841i \(0.00515178\pi\)
−0.999869 + 0.0161841i \(0.994848\pi\)
\(734\) 0 0
\(735\) −4195.78 + 19580.6i −0.210563 + 0.982643i
\(736\) 0 0
\(737\) 4117.80 0.205809
\(738\) 0 0
\(739\) 32931.9i 1.63927i 0.572888 + 0.819633i \(0.305823\pi\)
−0.572888 + 0.819633i \(0.694177\pi\)
\(740\) 0 0
\(741\) 1365.25i 0.0676838i
\(742\) 0 0
\(743\) 24636.6i 1.21646i −0.793762 0.608229i \(-0.791880\pi\)
0.793762 0.608229i \(-0.208120\pi\)
\(744\) 0 0
\(745\) 48772.6i 2.39851i
\(746\) 0 0
\(747\) 8219.07 0.402570
\(748\) 0 0
\(749\) −5287.12 560.110i −0.257927 0.0273244i
\(750\) 0 0
\(751\) 32225.0i 1.56579i 0.622156 + 0.782893i \(0.286257\pi\)
−0.622156 + 0.782893i \(0.713743\pi\)
\(752\) 0 0
\(753\) 12695.0 0.614386
\(754\) 0 0
\(755\) −9049.93 −0.436240
\(756\) 0 0
\(757\) 16949.5 0.813792 0.406896 0.913474i \(-0.366611\pi\)
0.406896 + 0.913474i \(0.366611\pi\)
\(758\) 0 0
\(759\) 8016.42 0.383370
\(760\) 0 0
\(761\) 3896.80i 0.185623i 0.995684 + 0.0928114i \(0.0295854\pi\)
−0.995684 + 0.0928114i \(0.970415\pi\)
\(762\) 0 0
\(763\) −29.1975 + 275.607i −0.00138535 + 0.0130769i
\(764\) 0 0
\(765\) 14166.4 0.669527
\(766\) 0 0
\(767\) 3852.01i 0.181340i
\(768\) 0 0
\(769\) 18651.6i 0.874634i −0.899307 0.437317i \(-0.855929\pi\)
0.899307 0.437317i \(-0.144071\pi\)
\(770\) 0 0
\(771\) 5617.51i 0.262399i
\(772\) 0 0
\(773\) 30262.1i 1.40809i −0.710156 0.704045i \(-0.751376\pi\)
0.710156 0.704045i \(-0.248624\pi\)
\(774\) 0 0
\(775\) −29561.3 −1.37016
\(776\) 0 0
\(777\) −313.270 + 2957.08i −0.0144639 + 0.136531i
\(778\) 0 0
\(779\) 26285.5i 1.20895i
\(780\) 0 0
\(781\) −9322.87 −0.427143
\(782\) 0 0
\(783\) 7526.09 0.343500
\(784\) 0 0
\(785\) −8586.42 −0.390398
\(786\) 0 0
\(787\) 25673.9 1.16286 0.581432 0.813595i \(-0.302493\pi\)
0.581432 + 0.813595i \(0.302493\pi\)
\(788\) 0 0
\(789\) 15378.8i 0.693914i
\(790\) 0 0
\(791\) 1438.38 13577.4i 0.0646558 0.610314i
\(792\) 0 0
\(793\) 2328.99 0.104294
\(794\) 0 0
\(795\) 10855.2i 0.484270i
\(796\) 0 0
\(797\) 9725.85i 0.432255i 0.976365 + 0.216127i \(0.0693427\pi\)
−0.976365 + 0.216127i \(0.930657\pi\)
\(798\) 0 0
\(799\) 8272.39i 0.366278i
\(800\) 0 0
\(801\) 13477.8i 0.594525i
\(802\) 0 0
\(803\) 16396.0 0.720549
\(804\) 0 0
\(805\) 4114.03 38834.1i 0.180125 1.70027i
\(806\) 0 0
\(807\) 7199.94i 0.314064i
\(808\) 0 0
\(809\) 22322.4 0.970106 0.485053 0.874485i \(-0.338800\pi\)
0.485053 + 0.874485i \(0.338800\pi\)
\(810\) 0 0
\(811\) 8693.62 0.376417 0.188209 0.982129i \(-0.439732\pi\)
0.188209 + 0.982129i \(0.439732\pi\)
\(812\) 0 0
\(813\) −16998.9 −0.733305
\(814\) 0 0
\(815\) −7295.38 −0.313553
\(816\) 0 0
\(817\) 15289.1i 0.654710i
\(818\) 0 0
\(819\) −871.122 92.2856i −0.0371666 0.00393739i
\(820\) 0 0
\(821\) 15530.6 0.660196 0.330098 0.943947i \(-0.392918\pi\)
0.330098 + 0.943947i \(0.392918\pi\)
\(822\) 0 0
\(823\) 37619.9i 1.59337i 0.604392 + 0.796687i \(0.293416\pi\)
−0.604392 + 0.796687i \(0.706584\pi\)
\(824\) 0 0
\(825\) 18771.9i 0.792186i
\(826\) 0 0
\(827\) 22298.2i 0.937587i −0.883308 0.468794i \(-0.844689\pi\)
0.883308 0.468794i \(-0.155311\pi\)
\(828\) 0 0
\(829\) 26664.3i 1.11712i 0.829465 + 0.558559i \(0.188646\pi\)
−0.829465 + 0.558559i \(0.811354\pi\)
\(830\) 0 0
\(831\) −3269.75 −0.136494
\(832\) 0 0
\(833\) −5812.86 + 27127.1i −0.241781 + 1.12833i
\(834\) 0 0
\(835\) 19194.0i 0.795491i
\(836\) 0 0
\(837\) −3145.80 −0.129910
\(838\) 0 0
\(839\) 33449.9 1.37642 0.688212 0.725510i \(-0.258396\pi\)
0.688212 + 0.725510i \(0.258396\pi\)
\(840\) 0 0
\(841\) 53309.3 2.18579
\(842\) 0 0
\(843\) 9755.95 0.398592
\(844\) 0 0
\(845\) 42217.8i 1.71874i
\(846\) 0 0
\(847\) −1410.21 + 13311.6i −0.0572084 + 0.540014i
\(848\) 0 0
\(849\) −24261.3 −0.980738
\(850\) 0 0
\(851\) 5798.93i 0.233590i
\(852\) 0 0
\(853\) 47864.2i 1.92127i −0.277822 0.960633i \(-0.589613\pi\)
0.277822 0.960633i \(-0.410387\pi\)
\(854\) 0 0
\(855\) 15166.3i 0.606640i
\(856\) 0 0
\(857\) 2792.59i 0.111310i −0.998450 0.0556552i \(-0.982275\pi\)
0.998450 0.0556552i \(-0.0177248\pi\)
\(858\) 0 0
\(859\) −24841.7 −0.986715 −0.493357 0.869827i \(-0.664231\pi\)
−0.493357 + 0.869827i \(0.664231\pi\)
\(860\) 0 0
\(861\) 16771.9 + 1776.80i 0.663862 + 0.0703287i
\(862\) 0 0
\(863\) 12119.8i 0.478058i −0.971012 0.239029i \(-0.923171\pi\)
0.971012 0.239029i \(-0.0768291\pi\)
\(864\) 0 0
\(865\) −31502.7 −1.23829
\(866\) 0 0
\(867\) 4887.26 0.191442
\(868\) 0 0
\(869\) −18196.3 −0.710321
\(870\) 0 0
\(871\) 877.502 0.0341366
\(872\) 0 0
\(873\) 13009.7i 0.504364i
\(874\) 0 0
\(875\) 46135.3 + 4887.52i 1.78247 + 0.188832i
\(876\) 0 0
\(877\) −14849.8 −0.571770 −0.285885 0.958264i \(-0.592288\pi\)
−0.285885 + 0.958264i \(0.592288\pi\)
\(878\) 0 0
\(879\) 4846.04i 0.185953i
\(880\) 0 0
\(881\) 20213.3i 0.772990i −0.922291 0.386495i \(-0.873686\pi\)
0.922291 0.386495i \(-0.126314\pi\)
\(882\) 0 0
\(883\) 9218.17i 0.351321i −0.984451 0.175660i \(-0.943794\pi\)
0.984451 0.175660i \(-0.0562061\pi\)
\(884\) 0 0
\(885\) 42791.3i 1.62533i
\(886\) 0 0
\(887\) −42349.9 −1.60312 −0.801561 0.597913i \(-0.795997\pi\)
−0.801561 + 0.597913i \(0.795997\pi\)
\(888\) 0 0
\(889\) −28957.0 3067.67i −1.09245 0.115733i
\(890\) 0 0
\(891\) 1997.63i 0.0751101i
\(892\) 0 0
\(893\) −8856.26 −0.331874
\(894\) 0 0
\(895\) 71874.1 2.68434
\(896\) 0 0
\(897\) 1708.30 0.0635880
\(898\) 0 0
\(899\) −32476.8 −1.20485
\(900\) 0 0
\(901\) 15038.9i 0.556068i
\(902\) 0 0
\(903\) 9755.49 + 1033.48i 0.359515 + 0.0380866i
\(904\) 0 0
\(905\) 47622.2 1.74919
\(906\) 0 0
\(907\) 40185.9i 1.47117i 0.677432 + 0.735586i \(0.263093\pi\)
−0.677432 + 0.735586i \(0.736907\pi\)
\(908\) 0 0
\(909\) 2163.14i 0.0789293i
\(910\) 0 0
\(911\) 11410.6i 0.414984i −0.978237 0.207492i \(-0.933470\pi\)
0.978237 0.207492i \(-0.0665301\pi\)
\(912\) 0 0
\(913\) 22522.2i 0.816401i
\(914\) 0 0
\(915\) −25872.4 −0.934770
\(916\) 0 0
\(917\) 1837.67 17346.5i 0.0661779 0.624680i
\(918\) 0 0
\(919\) 27571.5i 0.989662i −0.868989 0.494831i \(-0.835230\pi\)
0.868989 0.494831i \(-0.164770\pi\)
\(920\) 0 0
\(921\) −8179.36 −0.292637
\(922\) 0 0
\(923\) −1986.70 −0.0708484
\(924\) 0 0
\(925\) 13579.2 0.482684
\(926\) 0 0
\(927\) −15288.5 −0.541683
\(928\) 0 0
\(929\) 25071.2i 0.885424i −0.896664 0.442712i \(-0.854016\pi\)
0.896664 0.442712i \(-0.145984\pi\)
\(930\) 0 0
\(931\) 29041.8 + 6223.14i 1.02235 + 0.219071i
\(932\) 0 0
\(933\) −19487.8 −0.683817
\(934\) 0 0
\(935\) 38819.3i 1.35778i
\(936\) 0 0
\(937\) 19811.3i 0.690721i 0.938470 + 0.345360i \(0.112243\pi\)
−0.938470 + 0.345360i \(0.887757\pi\)
\(938\) 0 0
\(939\) 6945.05i 0.241367i
\(940\) 0 0
\(941\) 3182.72i 0.110259i −0.998479 0.0551295i \(-0.982443\pi\)
0.998479 0.0551295i \(-0.0175572\pi\)
\(942\) 0 0
\(943\) −32890.3 −1.13579
\(944\) 0 0
\(945\) 9677.14 + 1025.18i 0.333119 + 0.0352902i
\(946\) 0 0
\(947\) 25087.2i 0.860848i 0.902627 + 0.430424i \(0.141636\pi\)
−0.902627 + 0.430424i \(0.858364\pi\)
\(948\) 0 0
\(949\) 3493.97 0.119514
\(950\) 0 0
\(951\) −10635.5 −0.362651
\(952\) 0 0
\(953\) 247.587 0.00841568 0.00420784 0.999991i \(-0.498661\pi\)
0.00420784 + 0.999991i \(0.498661\pi\)
\(954\) 0 0
\(955\) −179.034 −0.00606638
\(956\) 0 0
\(957\) 20623.2i 0.696609i
\(958\) 0 0
\(959\) −4072.83 + 38445.1i −0.137141 + 1.29453i
\(960\) 0 0
\(961\) −16216.2 −0.544331
\(962\) 0 0
\(963\) 2583.67i 0.0864566i
\(964\) 0 0
\(965\) 28582.4i 0.953471i
\(966\) 0 0
\(967\) 31623.0i 1.05163i −0.850598 0.525816i \(-0.823760\pi\)
0.850598 0.525816i \(-0.176240\pi\)
\(968\) 0 0
\(969\) 21011.5i 0.696581i
\(970\) 0 0
\(971\) −33729.9 −1.11477 −0.557386 0.830253i \(-0.688196\pi\)
−0.557386 + 0.830253i \(0.688196\pi\)
\(972\) 0 0
\(973\) 3696.28 34890.7i 0.121785 1.14958i
\(974\) 0 0
\(975\) 4000.28i 0.131397i
\(976\) 0 0
\(977\) −45295.7 −1.48325 −0.741626 0.670813i \(-0.765945\pi\)
−0.741626 + 0.670813i \(0.765945\pi\)
\(978\) 0 0
\(979\) 36932.3 1.20568
\(980\) 0 0
\(981\) 134.682 0.00438335
\(982\) 0 0
\(983\) 43789.8 1.42083 0.710416 0.703782i \(-0.248507\pi\)
0.710416 + 0.703782i \(0.248507\pi\)
\(984\) 0 0
\(985\) 76181.8i 2.46432i
\(986\) 0 0
\(987\) 598.649 5650.90i 0.0193062 0.182239i
\(988\) 0 0
\(989\) −19130.8 −0.615090
\(990\) 0 0
\(991\) 398.825i 0.0127842i 0.999980 + 0.00639208i \(0.00203468\pi\)
−0.999980 + 0.00639208i \(0.997965\pi\)
\(992\) 0 0
\(993\) 16441.2i 0.525423i
\(994\) 0 0
\(995\) 57608.1i 1.83548i
\(996\) 0 0
\(997\) 40219.6i 1.27760i 0.769373 + 0.638800i \(0.220569\pi\)
−0.769373 + 0.638800i \(0.779431\pi\)
\(998\) 0 0
\(999\) 1445.05 0.0457651
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.g.895.12 12
4.3 odd 2 1344.4.b.h.895.12 12
7.6 odd 2 1344.4.b.h.895.1 12
8.3 odd 2 84.4.b.a.55.8 yes 12
8.5 even 2 84.4.b.b.55.7 yes 12
24.5 odd 2 252.4.b.e.55.6 12
24.11 even 2 252.4.b.f.55.5 12
28.27 even 2 inner 1344.4.b.g.895.1 12
56.13 odd 2 84.4.b.a.55.7 12
56.27 even 2 84.4.b.b.55.8 yes 12
168.83 odd 2 252.4.b.e.55.5 12
168.125 even 2 252.4.b.f.55.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.b.a.55.7 12 56.13 odd 2
84.4.b.a.55.8 yes 12 8.3 odd 2
84.4.b.b.55.7 yes 12 8.5 even 2
84.4.b.b.55.8 yes 12 56.27 even 2
252.4.b.e.55.5 12 168.83 odd 2
252.4.b.e.55.6 12 24.5 odd 2
252.4.b.f.55.5 12 24.11 even 2
252.4.b.f.55.6 12 168.125 even 2
1344.4.b.g.895.1 12 28.27 even 2 inner
1344.4.b.g.895.12 12 1.1 even 1 trivial
1344.4.b.h.895.1 12 7.6 odd 2
1344.4.b.h.895.12 12 4.3 odd 2