Properties

Label 192.5.g.b.127.1
Level $192$
Weight $5$
Character 192.127
Analytic conductor $19.847$
Analytic rank $0$
Dimension $2$
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] = [192,5,Mod(127,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.127");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 192.g (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: \(19.8470329121\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 192.127
Dual form 192.5.g.b.127.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.19615i q^{3} +42.0000 q^{5} -76.2102i q^{7} -27.0000 q^{9} +O(q^{10})\) \(q-5.19615i q^{3} +42.0000 q^{5} -76.2102i q^{7} -27.0000 q^{9} -20.7846i q^{11} +182.000 q^{13} -218.238i q^{15} -246.000 q^{17} -117.779i q^{19} -396.000 q^{21} +748.246i q^{23} +1139.00 q^{25} +140.296i q^{27} -78.0000 q^{29} -1475.71i q^{31} -108.000 q^{33} -3200.83i q^{35} -530.000 q^{37} -945.700i q^{39} -918.000 q^{41} -852.169i q^{43} -1134.00 q^{45} -3782.80i q^{47} -3407.00 q^{49} +1278.25i q^{51} +4626.00 q^{53} -872.954i q^{55} -612.000 q^{57} -228.631i q^{59} -1346.00 q^{61} +2057.68i q^{63} +7644.00 q^{65} +1087.73i q^{67} +3888.00 q^{69} +1829.05i q^{71} -926.000 q^{73} -5918.42i q^{75} -1584.00 q^{77} +4399.41i q^{79} +729.000 q^{81} +11992.7i q^{83} -10332.0 q^{85} +405.300i q^{87} +11586.0 q^{89} -13870.3i q^{91} -7668.00 q^{93} -4946.74i q^{95} -13118.0 q^{97} +561.184i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 84 q^{5} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 84 q^{5} - 54 q^{9} + 364 q^{13} - 492 q^{17} - 792 q^{21} + 2278 q^{25} - 156 q^{29} - 216 q^{33} - 1060 q^{37} - 1836 q^{41} - 2268 q^{45} - 6814 q^{49} + 9252 q^{53} - 1224 q^{57} - 2692 q^{61} + 15288 q^{65} + 7776 q^{69} - 1852 q^{73} - 3168 q^{77} + 1458 q^{81} - 20664 q^{85} + 23172 q^{89} - 15336 q^{93} - 26236 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 5.19615i − 0.577350i
\(4\) 0 0
\(5\) 42.0000 1.68000 0.840000 0.542586i \(-0.182555\pi\)
0.840000 + 0.542586i \(0.182555\pi\)
\(6\) 0 0
\(7\) − 76.2102i − 1.55531i −0.628691 0.777655i \(-0.716409\pi\)
0.628691 0.777655i \(-0.283591\pi\)
\(8\) 0 0
\(9\) −27.0000 −0.333333
\(10\) 0 0
\(11\) − 20.7846i − 0.171774i −0.996305 0.0858868i \(-0.972628\pi\)
0.996305 0.0858868i \(-0.0273723\pi\)
\(12\) 0 0
\(13\) 182.000 1.07692 0.538462 0.842650i \(-0.319006\pi\)
0.538462 + 0.842650i \(0.319006\pi\)
\(14\) 0 0
\(15\) − 218.238i − 0.969948i
\(16\) 0 0
\(17\) −246.000 −0.851211 −0.425606 0.904909i \(-0.639939\pi\)
−0.425606 + 0.904909i \(0.639939\pi\)
\(18\) 0 0
\(19\) − 117.779i − 0.326259i −0.986605 0.163129i \(-0.947841\pi\)
0.986605 0.163129i \(-0.0521588\pi\)
\(20\) 0 0
\(21\) −396.000 −0.897959
\(22\) 0 0
\(23\) 748.246i 1.41445i 0.706987 + 0.707227i \(0.250054\pi\)
−0.706987 + 0.707227i \(0.749946\pi\)
\(24\) 0 0
\(25\) 1139.00 1.82240
\(26\) 0 0
\(27\) 140.296i 0.192450i
\(28\) 0 0
\(29\) −78.0000 −0.0927467 −0.0463734 0.998924i \(-0.514766\pi\)
−0.0463734 + 0.998924i \(0.514766\pi\)
\(30\) 0 0
\(31\) − 1475.71i − 1.53560i −0.640692 0.767798i \(-0.721353\pi\)
0.640692 0.767798i \(-0.278647\pi\)
\(32\) 0 0
\(33\) −108.000 −0.0991736
\(34\) 0 0
\(35\) − 3200.83i − 2.61292i
\(36\) 0 0
\(37\) −530.000 −0.387144 −0.193572 0.981086i \(-0.562007\pi\)
−0.193572 + 0.981086i \(0.562007\pi\)
\(38\) 0 0
\(39\) − 945.700i − 0.621762i
\(40\) 0 0
\(41\) −918.000 −0.546104 −0.273052 0.961999i \(-0.588033\pi\)
−0.273052 + 0.961999i \(0.588033\pi\)
\(42\) 0 0
\(43\) − 852.169i − 0.460881i −0.973086 0.230441i \(-0.925983\pi\)
0.973086 0.230441i \(-0.0740167\pi\)
\(44\) 0 0
\(45\) −1134.00 −0.560000
\(46\) 0 0
\(47\) − 3782.80i − 1.71245i −0.516604 0.856224i \(-0.672804\pi\)
0.516604 0.856224i \(-0.327196\pi\)
\(48\) 0 0
\(49\) −3407.00 −1.41899
\(50\) 0 0
\(51\) 1278.25i 0.491447i
\(52\) 0 0
\(53\) 4626.00 1.64685 0.823425 0.567426i \(-0.192061\pi\)
0.823425 + 0.567426i \(0.192061\pi\)
\(54\) 0 0
\(55\) − 872.954i − 0.288580i
\(56\) 0 0
\(57\) −612.000 −0.188366
\(58\) 0 0
\(59\) − 228.631i − 0.0656796i −0.999461 0.0328398i \(-0.989545\pi\)
0.999461 0.0328398i \(-0.0104551\pi\)
\(60\) 0 0
\(61\) −1346.00 −0.361731 −0.180865 0.983508i \(-0.557890\pi\)
−0.180865 + 0.983508i \(0.557890\pi\)
\(62\) 0 0
\(63\) 2057.68i 0.518437i
\(64\) 0 0
\(65\) 7644.00 1.80923
\(66\) 0 0
\(67\) 1087.73i 0.242310i 0.992634 + 0.121155i \(0.0386597\pi\)
−0.992634 + 0.121155i \(0.961340\pi\)
\(68\) 0 0
\(69\) 3888.00 0.816635
\(70\) 0 0
\(71\) 1829.05i 0.362834i 0.983406 + 0.181417i \(0.0580684\pi\)
−0.983406 + 0.181417i \(0.941932\pi\)
\(72\) 0 0
\(73\) −926.000 −0.173766 −0.0868831 0.996219i \(-0.527691\pi\)
−0.0868831 + 0.996219i \(0.527691\pi\)
\(74\) 0 0
\(75\) − 5918.42i − 1.05216i
\(76\) 0 0
\(77\) −1584.00 −0.267161
\(78\) 0 0
\(79\) 4399.41i 0.704921i 0.935827 + 0.352460i \(0.114655\pi\)
−0.935827 + 0.352460i \(0.885345\pi\)
\(80\) 0 0
\(81\) 729.000 0.111111
\(82\) 0 0
\(83\) 11992.7i 1.74085i 0.492301 + 0.870425i \(0.336156\pi\)
−0.492301 + 0.870425i \(0.663844\pi\)
\(84\) 0 0
\(85\) −10332.0 −1.43003
\(86\) 0 0
\(87\) 405.300i 0.0535473i
\(88\) 0 0
\(89\) 11586.0 1.46269 0.731347 0.682005i \(-0.238892\pi\)
0.731347 + 0.682005i \(0.238892\pi\)
\(90\) 0 0
\(91\) − 13870.3i − 1.67495i
\(92\) 0 0
\(93\) −7668.00 −0.886576
\(94\) 0 0
\(95\) − 4946.74i − 0.548115i
\(96\) 0 0
\(97\) −13118.0 −1.39420 −0.697099 0.716975i \(-0.745526\pi\)
−0.697099 + 0.716975i \(0.745526\pi\)
\(98\) 0 0
\(99\) 561.184i 0.0572579i
\(100\) 0 0
\(101\) 5490.00 0.538183 0.269091 0.963115i \(-0.413277\pi\)
0.269091 + 0.963115i \(0.413277\pi\)
\(102\) 0 0
\(103\) 5701.91i 0.537460i 0.963216 + 0.268730i \(0.0866039\pi\)
−0.963216 + 0.268730i \(0.913396\pi\)
\(104\) 0 0
\(105\) −16632.0 −1.50857
\(106\) 0 0
\(107\) 10080.5i 0.880473i 0.897882 + 0.440237i \(0.145105\pi\)
−0.897882 + 0.440237i \(0.854895\pi\)
\(108\) 0 0
\(109\) 16166.0 1.36066 0.680330 0.732906i \(-0.261836\pi\)
0.680330 + 0.732906i \(0.261836\pi\)
\(110\) 0 0
\(111\) 2753.96i 0.223518i
\(112\) 0 0
\(113\) 1842.00 0.144256 0.0721278 0.997395i \(-0.477021\pi\)
0.0721278 + 0.997395i \(0.477021\pi\)
\(114\) 0 0
\(115\) 31426.3i 2.37628i
\(116\) 0 0
\(117\) −4914.00 −0.358974
\(118\) 0 0
\(119\) 18747.7i 1.32390i
\(120\) 0 0
\(121\) 14209.0 0.970494
\(122\) 0 0
\(123\) 4770.07i 0.315293i
\(124\) 0 0
\(125\) 21588.0 1.38163
\(126\) 0 0
\(127\) 394.908i 0.0244843i 0.999925 + 0.0122422i \(0.00389690\pi\)
−0.999925 + 0.0122422i \(0.996103\pi\)
\(128\) 0 0
\(129\) −4428.00 −0.266090
\(130\) 0 0
\(131\) − 353.338i − 0.0205896i −0.999947 0.0102948i \(-0.996723\pi\)
0.999947 0.0102948i \(-0.00327700\pi\)
\(132\) 0 0
\(133\) −8976.00 −0.507434
\(134\) 0 0
\(135\) 5892.44i 0.323316i
\(136\) 0 0
\(137\) −13254.0 −0.706164 −0.353082 0.935592i \(-0.614866\pi\)
−0.353082 + 0.935592i \(0.614866\pi\)
\(138\) 0 0
\(139\) 13212.1i 0.683820i 0.939733 + 0.341910i \(0.111074\pi\)
−0.939733 + 0.341910i \(0.888926\pi\)
\(140\) 0 0
\(141\) −19656.0 −0.988683
\(142\) 0 0
\(143\) − 3782.80i − 0.184987i
\(144\) 0 0
\(145\) −3276.00 −0.155815
\(146\) 0 0
\(147\) 17703.3i 0.819255i
\(148\) 0 0
\(149\) −438.000 −0.0197288 −0.00986442 0.999951i \(-0.503140\pi\)
−0.00986442 + 0.999951i \(0.503140\pi\)
\(150\) 0 0
\(151\) 28052.3i 1.23031i 0.788406 + 0.615155i \(0.210907\pi\)
−0.788406 + 0.615155i \(0.789093\pi\)
\(152\) 0 0
\(153\) 6642.00 0.283737
\(154\) 0 0
\(155\) − 61979.7i − 2.57980i
\(156\) 0 0
\(157\) −19346.0 −0.784859 −0.392430 0.919782i \(-0.628365\pi\)
−0.392430 + 0.919782i \(0.628365\pi\)
\(158\) 0 0
\(159\) − 24037.4i − 0.950809i
\(160\) 0 0
\(161\) 57024.0 2.19992
\(162\) 0 0
\(163\) − 36255.3i − 1.36457i −0.731086 0.682286i \(-0.760986\pi\)
0.731086 0.682286i \(-0.239014\pi\)
\(164\) 0 0
\(165\) −4536.00 −0.166612
\(166\) 0 0
\(167\) 18747.7i 0.672226i 0.941822 + 0.336113i \(0.109112\pi\)
−0.941822 + 0.336113i \(0.890888\pi\)
\(168\) 0 0
\(169\) 4563.00 0.159763
\(170\) 0 0
\(171\) 3180.05i 0.108753i
\(172\) 0 0
\(173\) 34410.0 1.14972 0.574861 0.818251i \(-0.305056\pi\)
0.574861 + 0.818251i \(0.305056\pi\)
\(174\) 0 0
\(175\) − 86803.5i − 2.83440i
\(176\) 0 0
\(177\) −1188.00 −0.0379201
\(178\) 0 0
\(179\) 16856.3i 0.526086i 0.964784 + 0.263043i \(0.0847261\pi\)
−0.964784 + 0.263043i \(0.915274\pi\)
\(180\) 0 0
\(181\) −15706.0 −0.479411 −0.239706 0.970846i \(-0.577051\pi\)
−0.239706 + 0.970846i \(0.577051\pi\)
\(182\) 0 0
\(183\) 6994.02i 0.208845i
\(184\) 0 0
\(185\) −22260.0 −0.650402
\(186\) 0 0
\(187\) 5113.01i 0.146216i
\(188\) 0 0
\(189\) 10692.0 0.299320
\(190\) 0 0
\(191\) − 2660.43i − 0.0729265i −0.999335 0.0364632i \(-0.988391\pi\)
0.999335 0.0364632i \(-0.0116092\pi\)
\(192\) 0 0
\(193\) −26782.0 −0.718999 −0.359500 0.933145i \(-0.617053\pi\)
−0.359500 + 0.933145i \(0.617053\pi\)
\(194\) 0 0
\(195\) − 39719.4i − 1.04456i
\(196\) 0 0
\(197\) 52482.0 1.35232 0.676158 0.736757i \(-0.263644\pi\)
0.676158 + 0.736757i \(0.263644\pi\)
\(198\) 0 0
\(199\) 23077.8i 0.582759i 0.956608 + 0.291380i \(0.0941143\pi\)
−0.956608 + 0.291380i \(0.905886\pi\)
\(200\) 0 0
\(201\) 5652.00 0.139898
\(202\) 0 0
\(203\) 5944.40i 0.144250i
\(204\) 0 0
\(205\) −38556.0 −0.917454
\(206\) 0 0
\(207\) − 20202.6i − 0.471485i
\(208\) 0 0
\(209\) −2448.00 −0.0560427
\(210\) 0 0
\(211\) 23895.4i 0.536721i 0.963319 + 0.268361i \(0.0864819\pi\)
−0.963319 + 0.268361i \(0.913518\pi\)
\(212\) 0 0
\(213\) 9504.00 0.209482
\(214\) 0 0
\(215\) − 35791.1i − 0.774280i
\(216\) 0 0
\(217\) −112464. −2.38833
\(218\) 0 0
\(219\) 4811.64i 0.100324i
\(220\) 0 0
\(221\) −44772.0 −0.916689
\(222\) 0 0
\(223\) − 852.169i − 0.0171363i −0.999963 0.00856813i \(-0.997273\pi\)
0.999963 0.00856813i \(-0.00272735\pi\)
\(224\) 0 0
\(225\) −30753.0 −0.607467
\(226\) 0 0
\(227\) 76175.6i 1.47831i 0.673538 + 0.739153i \(0.264774\pi\)
−0.673538 + 0.739153i \(0.735226\pi\)
\(228\) 0 0
\(229\) 48470.0 0.924277 0.462138 0.886808i \(-0.347082\pi\)
0.462138 + 0.886808i \(0.347082\pi\)
\(230\) 0 0
\(231\) 8230.71i 0.154246i
\(232\) 0 0
\(233\) 48738.0 0.897751 0.448875 0.893594i \(-0.351825\pi\)
0.448875 + 0.893594i \(0.351825\pi\)
\(234\) 0 0
\(235\) − 158878.i − 2.87691i
\(236\) 0 0
\(237\) 22860.0 0.406986
\(238\) 0 0
\(239\) − 71000.2i − 1.24298i −0.783422 0.621490i \(-0.786528\pi\)
0.783422 0.621490i \(-0.213472\pi\)
\(240\) 0 0
\(241\) 73138.0 1.25924 0.629621 0.776903i \(-0.283210\pi\)
0.629621 + 0.776903i \(0.283210\pi\)
\(242\) 0 0
\(243\) − 3788.00i − 0.0641500i
\(244\) 0 0
\(245\) −143094. −2.38391
\(246\) 0 0
\(247\) − 21435.9i − 0.351356i
\(248\) 0 0
\(249\) 62316.0 1.00508
\(250\) 0 0
\(251\) − 91888.8i − 1.45853i −0.684232 0.729264i \(-0.739862\pi\)
0.684232 0.729264i \(-0.260138\pi\)
\(252\) 0 0
\(253\) 15552.0 0.242966
\(254\) 0 0
\(255\) 53686.6i 0.825631i
\(256\) 0 0
\(257\) −48894.0 −0.740269 −0.370134 0.928978i \(-0.620688\pi\)
−0.370134 + 0.928978i \(0.620688\pi\)
\(258\) 0 0
\(259\) 40391.4i 0.602129i
\(260\) 0 0
\(261\) 2106.00 0.0309156
\(262\) 0 0
\(263\) − 78191.7i − 1.13044i −0.824939 0.565222i \(-0.808790\pi\)
0.824939 0.565222i \(-0.191210\pi\)
\(264\) 0 0
\(265\) 194292. 2.76671
\(266\) 0 0
\(267\) − 60202.6i − 0.844487i
\(268\) 0 0
\(269\) 71538.0 0.988626 0.494313 0.869284i \(-0.335420\pi\)
0.494313 + 0.869284i \(0.335420\pi\)
\(270\) 0 0
\(271\) 108198.i 1.47326i 0.676296 + 0.736630i \(0.263584\pi\)
−0.676296 + 0.736630i \(0.736416\pi\)
\(272\) 0 0
\(273\) −72072.0 −0.967033
\(274\) 0 0
\(275\) − 23673.7i − 0.313040i
\(276\) 0 0
\(277\) 120518. 1.57070 0.785348 0.619054i \(-0.212484\pi\)
0.785348 + 0.619054i \(0.212484\pi\)
\(278\) 0 0
\(279\) 39844.1i 0.511865i
\(280\) 0 0
\(281\) −3054.00 −0.0386773 −0.0193387 0.999813i \(-0.506156\pi\)
−0.0193387 + 0.999813i \(0.506156\pi\)
\(282\) 0 0
\(283\) 132959.i 1.66014i 0.557657 + 0.830071i \(0.311700\pi\)
−0.557657 + 0.830071i \(0.688300\pi\)
\(284\) 0 0
\(285\) −25704.0 −0.316454
\(286\) 0 0
\(287\) 69961.0i 0.849361i
\(288\) 0 0
\(289\) −23005.0 −0.275440
\(290\) 0 0
\(291\) 68163.1i 0.804940i
\(292\) 0 0
\(293\) −151662. −1.76661 −0.883307 0.468795i \(-0.844688\pi\)
−0.883307 + 0.468795i \(0.844688\pi\)
\(294\) 0 0
\(295\) − 9602.49i − 0.110342i
\(296\) 0 0
\(297\) 2916.00 0.0330579
\(298\) 0 0
\(299\) 136181.i 1.52326i
\(300\) 0 0
\(301\) −64944.0 −0.716813
\(302\) 0 0
\(303\) − 28526.9i − 0.310720i
\(304\) 0 0
\(305\) −56532.0 −0.607708
\(306\) 0 0
\(307\) − 5424.78i − 0.0575580i −0.999586 0.0287790i \(-0.990838\pi\)
0.999586 0.0287790i \(-0.00916190\pi\)
\(308\) 0 0
\(309\) 29628.0 0.310303
\(310\) 0 0
\(311\) 141127.i 1.45912i 0.683917 + 0.729560i \(0.260275\pi\)
−0.683917 + 0.729560i \(0.739725\pi\)
\(312\) 0 0
\(313\) −128686. −1.31354 −0.656769 0.754092i \(-0.728077\pi\)
−0.656769 + 0.754092i \(0.728077\pi\)
\(314\) 0 0
\(315\) 86422.4i 0.870974i
\(316\) 0 0
\(317\) 73986.0 0.736260 0.368130 0.929774i \(-0.379998\pi\)
0.368130 + 0.929774i \(0.379998\pi\)
\(318\) 0 0
\(319\) 1621.20i 0.0159314i
\(320\) 0 0
\(321\) 52380.0 0.508341
\(322\) 0 0
\(323\) 28973.7i 0.277715i
\(324\) 0 0
\(325\) 207298. 1.96258
\(326\) 0 0
\(327\) − 84001.0i − 0.785577i
\(328\) 0 0
\(329\) −288288. −2.66339
\(330\) 0 0
\(331\) 57026.0i 0.520496i 0.965542 + 0.260248i \(0.0838043\pi\)
−0.965542 + 0.260248i \(0.916196\pi\)
\(332\) 0 0
\(333\) 14310.0 0.129048
\(334\) 0 0
\(335\) 45684.6i 0.407080i
\(336\) 0 0
\(337\) 98674.0 0.868846 0.434423 0.900709i \(-0.356952\pi\)
0.434423 + 0.900709i \(0.356952\pi\)
\(338\) 0 0
\(339\) − 9571.31i − 0.0832860i
\(340\) 0 0
\(341\) −30672.0 −0.263775
\(342\) 0 0
\(343\) 76667.5i 0.651663i
\(344\) 0 0
\(345\) 163296. 1.37195
\(346\) 0 0
\(347\) 56929.0i 0.472797i 0.971656 + 0.236399i \(0.0759671\pi\)
−0.971656 + 0.236399i \(0.924033\pi\)
\(348\) 0 0
\(349\) −181346. −1.48887 −0.744436 0.667694i \(-0.767281\pi\)
−0.744436 + 0.667694i \(0.767281\pi\)
\(350\) 0 0
\(351\) 25533.9i 0.207254i
\(352\) 0 0
\(353\) −4302.00 −0.0345240 −0.0172620 0.999851i \(-0.505495\pi\)
−0.0172620 + 0.999851i \(0.505495\pi\)
\(354\) 0 0
\(355\) 76819.9i 0.609561i
\(356\) 0 0
\(357\) 97416.0 0.764353
\(358\) 0 0
\(359\) 185232.i 1.43724i 0.695405 + 0.718618i \(0.255225\pi\)
−0.695405 + 0.718618i \(0.744775\pi\)
\(360\) 0 0
\(361\) 116449. 0.893555
\(362\) 0 0
\(363\) − 73832.1i − 0.560315i
\(364\) 0 0
\(365\) −38892.0 −0.291927
\(366\) 0 0
\(367\) − 182690.i − 1.35638i −0.734885 0.678191i \(-0.762764\pi\)
0.734885 0.678191i \(-0.237236\pi\)
\(368\) 0 0
\(369\) 24786.0 0.182035
\(370\) 0 0
\(371\) − 352549.i − 2.56136i
\(372\) 0 0
\(373\) −151778. −1.09092 −0.545458 0.838138i \(-0.683644\pi\)
−0.545458 + 0.838138i \(0.683644\pi\)
\(374\) 0 0
\(375\) − 112175.i − 0.797686i
\(376\) 0 0
\(377\) −14196.0 −0.0998811
\(378\) 0 0
\(379\) 36005.9i 0.250666i 0.992115 + 0.125333i \(0.0399999\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(380\) 0 0
\(381\) 2052.00 0.0141360
\(382\) 0 0
\(383\) 65346.8i 0.445479i 0.974878 + 0.222739i \(0.0714999\pi\)
−0.974878 + 0.222739i \(0.928500\pi\)
\(384\) 0 0
\(385\) −66528.0 −0.448831
\(386\) 0 0
\(387\) 23008.6i 0.153627i
\(388\) 0 0
\(389\) −105750. −0.698846 −0.349423 0.936965i \(-0.613622\pi\)
−0.349423 + 0.936965i \(0.613622\pi\)
\(390\) 0 0
\(391\) − 184069.i − 1.20400i
\(392\) 0 0
\(393\) −1836.00 −0.0118874
\(394\) 0 0
\(395\) 184775.i 1.18427i
\(396\) 0 0
\(397\) 27934.0 0.177236 0.0886180 0.996066i \(-0.471755\pi\)
0.0886180 + 0.996066i \(0.471755\pi\)
\(398\) 0 0
\(399\) 46640.7i 0.292967i
\(400\) 0 0
\(401\) 237882. 1.47936 0.739678 0.672961i \(-0.234978\pi\)
0.739678 + 0.672961i \(0.234978\pi\)
\(402\) 0 0
\(403\) − 268579.i − 1.65372i
\(404\) 0 0
\(405\) 30618.0 0.186667
\(406\) 0 0
\(407\) 11015.8i 0.0665011i
\(408\) 0 0
\(409\) −20270.0 −0.121173 −0.0605867 0.998163i \(-0.519297\pi\)
−0.0605867 + 0.998163i \(0.519297\pi\)
\(410\) 0 0
\(411\) 68869.8i 0.407704i
\(412\) 0 0
\(413\) −17424.0 −0.102152
\(414\) 0 0
\(415\) 503694.i 2.92463i
\(416\) 0 0
\(417\) 68652.0 0.394804
\(418\) 0 0
\(419\) − 24089.4i − 0.137214i −0.997644 0.0686068i \(-0.978145\pi\)
0.997644 0.0686068i \(-0.0218554\pi\)
\(420\) 0 0
\(421\) −116698. −0.658414 −0.329207 0.944258i \(-0.606781\pi\)
−0.329207 + 0.944258i \(0.606781\pi\)
\(422\) 0 0
\(423\) 102136.i 0.570816i
\(424\) 0 0
\(425\) −280194. −1.55125
\(426\) 0 0
\(427\) 102579.i 0.562604i
\(428\) 0 0
\(429\) −19656.0 −0.106802
\(430\) 0 0
\(431\) − 355542.i − 1.91397i −0.290132 0.956986i \(-0.593699\pi\)
0.290132 0.956986i \(-0.406301\pi\)
\(432\) 0 0
\(433\) −199726. −1.06527 −0.532634 0.846346i \(-0.678798\pi\)
−0.532634 + 0.846346i \(0.678798\pi\)
\(434\) 0 0
\(435\) 17022.6i 0.0899595i
\(436\) 0 0
\(437\) 88128.0 0.461478
\(438\) 0 0
\(439\) − 146469.i − 0.760006i −0.924985 0.380003i \(-0.875923\pi\)
0.924985 0.380003i \(-0.124077\pi\)
\(440\) 0 0
\(441\) 91989.0 0.472997
\(442\) 0 0
\(443\) 50444.2i 0.257042i 0.991707 + 0.128521i \(0.0410230\pi\)
−0.991707 + 0.128521i \(0.958977\pi\)
\(444\) 0 0
\(445\) 486612. 2.45733
\(446\) 0 0
\(447\) 2275.91i 0.0113905i
\(448\) 0 0
\(449\) 149994. 0.744014 0.372007 0.928230i \(-0.378670\pi\)
0.372007 + 0.928230i \(0.378670\pi\)
\(450\) 0 0
\(451\) 19080.3i 0.0938062i
\(452\) 0 0
\(453\) 145764. 0.710320
\(454\) 0 0
\(455\) − 582551.i − 2.81392i
\(456\) 0 0
\(457\) 284338. 1.36145 0.680726 0.732538i \(-0.261664\pi\)
0.680726 + 0.732538i \(0.261664\pi\)
\(458\) 0 0
\(459\) − 34512.8i − 0.163816i
\(460\) 0 0
\(461\) 183402. 0.862983 0.431491 0.902117i \(-0.357987\pi\)
0.431491 + 0.902117i \(0.357987\pi\)
\(462\) 0 0
\(463\) − 172422.i − 0.804324i −0.915568 0.402162i \(-0.868259\pi\)
0.915568 0.402162i \(-0.131741\pi\)
\(464\) 0 0
\(465\) −322056. −1.48945
\(466\) 0 0
\(467\) − 68734.7i − 0.315168i −0.987506 0.157584i \(-0.949629\pi\)
0.987506 0.157584i \(-0.0503705\pi\)
\(468\) 0 0
\(469\) 82896.0 0.376867
\(470\) 0 0
\(471\) 100525.i 0.453139i
\(472\) 0 0
\(473\) −17712.0 −0.0791672
\(474\) 0 0
\(475\) − 134151.i − 0.594574i
\(476\) 0 0
\(477\) −124902. −0.548950
\(478\) 0 0
\(479\) 249956.i 1.08941i 0.838627 + 0.544706i \(0.183359\pi\)
−0.838627 + 0.544706i \(0.816641\pi\)
\(480\) 0 0
\(481\) −96460.0 −0.416924
\(482\) 0 0
\(483\) − 296305.i − 1.27012i
\(484\) 0 0
\(485\) −550956. −2.34225
\(486\) 0 0
\(487\) − 271108.i − 1.14310i −0.820568 0.571549i \(-0.806343\pi\)
0.820568 0.571549i \(-0.193657\pi\)
\(488\) 0 0
\(489\) −188388. −0.787835
\(490\) 0 0
\(491\) − 227862.i − 0.945166i −0.881286 0.472583i \(-0.843322\pi\)
0.881286 0.472583i \(-0.156678\pi\)
\(492\) 0 0
\(493\) 19188.0 0.0789470
\(494\) 0 0
\(495\) 23569.7i 0.0961932i
\(496\) 0 0
\(497\) 139392. 0.564320
\(498\) 0 0
\(499\) − 248854.i − 0.999410i −0.866196 0.499705i \(-0.833442\pi\)
0.866196 0.499705i \(-0.166558\pi\)
\(500\) 0 0
\(501\) 97416.0 0.388110
\(502\) 0 0
\(503\) − 446537.i − 1.76490i −0.470403 0.882452i \(-0.655891\pi\)
0.470403 0.882452i \(-0.344109\pi\)
\(504\) 0 0
\(505\) 230580. 0.904147
\(506\) 0 0
\(507\) − 23710.0i − 0.0922394i
\(508\) 0 0
\(509\) 39330.0 0.151806 0.0759029 0.997115i \(-0.475816\pi\)
0.0759029 + 0.997115i \(0.475816\pi\)
\(510\) 0 0
\(511\) 70570.7i 0.270260i
\(512\) 0 0
\(513\) 16524.0 0.0627886
\(514\) 0 0
\(515\) 239480.i 0.902933i
\(516\) 0 0
\(517\) −78624.0 −0.294154
\(518\) 0 0
\(519\) − 178800.i − 0.663792i
\(520\) 0 0
\(521\) −464598. −1.71160 −0.855799 0.517308i \(-0.826934\pi\)
−0.855799 + 0.517308i \(0.826934\pi\)
\(522\) 0 0
\(523\) − 135509.i − 0.495409i −0.968836 0.247704i \(-0.920324\pi\)
0.968836 0.247704i \(-0.0796762\pi\)
\(524\) 0 0
\(525\) −451044. −1.63644
\(526\) 0 0
\(527\) 363024.i 1.30712i
\(528\) 0 0
\(529\) −280031. −1.00068
\(530\) 0 0
\(531\) 6173.03i 0.0218932i
\(532\) 0 0
\(533\) −167076. −0.588111
\(534\) 0 0
\(535\) 423382.i 1.47919i
\(536\) 0 0
\(537\) 87588.0 0.303736
\(538\) 0 0
\(539\) 70813.2i 0.243745i
\(540\) 0 0
\(541\) −360442. −1.23152 −0.615759 0.787934i \(-0.711151\pi\)
−0.615759 + 0.787934i \(0.711151\pi\)
\(542\) 0 0
\(543\) 81610.8i 0.276788i
\(544\) 0 0
\(545\) 678972. 2.28591
\(546\) 0 0
\(547\) − 261644.i − 0.874451i −0.899352 0.437225i \(-0.855961\pi\)
0.899352 0.437225i \(-0.144039\pi\)
\(548\) 0 0
\(549\) 36342.0 0.120577
\(550\) 0 0
\(551\) 9186.80i 0.0302594i
\(552\) 0 0
\(553\) 335280. 1.09637
\(554\) 0 0
\(555\) 115666.i 0.375510i
\(556\) 0 0
\(557\) 233274. 0.751893 0.375946 0.926641i \(-0.377318\pi\)
0.375946 + 0.926641i \(0.377318\pi\)
\(558\) 0 0
\(559\) − 155095.i − 0.496333i
\(560\) 0 0
\(561\) 26568.0 0.0844176
\(562\) 0 0
\(563\) 419704.i 1.32412i 0.749453 + 0.662058i \(0.230317\pi\)
−0.749453 + 0.662058i \(0.769683\pi\)
\(564\) 0 0
\(565\) 77364.0 0.242349
\(566\) 0 0
\(567\) − 55557.3i − 0.172812i
\(568\) 0 0
\(569\) 470058. 1.45187 0.725934 0.687765i \(-0.241408\pi\)
0.725934 + 0.687765i \(0.241408\pi\)
\(570\) 0 0
\(571\) 320381.i 0.982640i 0.870979 + 0.491320i \(0.163485\pi\)
−0.870979 + 0.491320i \(0.836515\pi\)
\(572\) 0 0
\(573\) −13824.0 −0.0421041
\(574\) 0 0
\(575\) 852252.i 2.57770i
\(576\) 0 0
\(577\) −341038. −1.02436 −0.512178 0.858879i \(-0.671161\pi\)
−0.512178 + 0.858879i \(0.671161\pi\)
\(578\) 0 0
\(579\) 139163.i 0.415114i
\(580\) 0 0
\(581\) 913968. 2.70756
\(582\) 0 0
\(583\) − 96149.6i − 0.282885i
\(584\) 0 0
\(585\) −206388. −0.603077
\(586\) 0 0
\(587\) − 114128.i − 0.331220i −0.986191 0.165610i \(-0.947041\pi\)
0.986191 0.165610i \(-0.0529594\pi\)
\(588\) 0 0
\(589\) −173808. −0.501002
\(590\) 0 0
\(591\) − 272704.i − 0.780760i
\(592\) 0 0
\(593\) −96846.0 −0.275405 −0.137703 0.990474i \(-0.543972\pi\)
−0.137703 + 0.990474i \(0.543972\pi\)
\(594\) 0 0
\(595\) 787404.i 2.22415i
\(596\) 0 0
\(597\) 119916. 0.336456
\(598\) 0 0
\(599\) 519782.i 1.44866i 0.689452 + 0.724331i \(0.257851\pi\)
−0.689452 + 0.724331i \(0.742149\pi\)
\(600\) 0 0
\(601\) −627742. −1.73793 −0.868965 0.494874i \(-0.835214\pi\)
−0.868965 + 0.494874i \(0.835214\pi\)
\(602\) 0 0
\(603\) − 29368.7i − 0.0807699i
\(604\) 0 0
\(605\) 596778. 1.63043
\(606\) 0 0
\(607\) 133195.i 0.361501i 0.983529 + 0.180751i \(0.0578527\pi\)
−0.983529 + 0.180751i \(0.942147\pi\)
\(608\) 0 0
\(609\) 30888.0 0.0832828
\(610\) 0 0
\(611\) − 688469.i − 1.84418i
\(612\) 0 0
\(613\) −247202. −0.657856 −0.328928 0.944355i \(-0.606687\pi\)
−0.328928 + 0.944355i \(0.606687\pi\)
\(614\) 0 0
\(615\) 200343.i 0.529692i
\(616\) 0 0
\(617\) −31758.0 −0.0834224 −0.0417112 0.999130i \(-0.513281\pi\)
−0.0417112 + 0.999130i \(0.513281\pi\)
\(618\) 0 0
\(619\) 656094.i 1.71232i 0.516712 + 0.856160i \(0.327156\pi\)
−0.516712 + 0.856160i \(0.672844\pi\)
\(620\) 0 0
\(621\) −104976. −0.272212
\(622\) 0 0
\(623\) − 882972.i − 2.27494i
\(624\) 0 0
\(625\) 194821. 0.498742
\(626\) 0 0
\(627\) 12720.2i 0.0323563i
\(628\) 0 0
\(629\) 130380. 0.329541
\(630\) 0 0
\(631\) − 417736.i − 1.04916i −0.851360 0.524582i \(-0.824222\pi\)
0.851360 0.524582i \(-0.175778\pi\)
\(632\) 0 0
\(633\) 124164. 0.309876
\(634\) 0 0
\(635\) 16586.1i 0.0411337i
\(636\) 0 0
\(637\) −620074. −1.52815
\(638\) 0 0
\(639\) − 49384.2i − 0.120945i
\(640\) 0 0
\(641\) −152214. −0.370458 −0.185229 0.982695i \(-0.559303\pi\)
−0.185229 + 0.982695i \(0.559303\pi\)
\(642\) 0 0
\(643\) − 714138.i − 1.72727i −0.504117 0.863635i \(-0.668182\pi\)
0.504117 0.863635i \(-0.331818\pi\)
\(644\) 0 0
\(645\) −185976. −0.447031
\(646\) 0 0
\(647\) 259558.i 0.620049i 0.950729 + 0.310025i \(0.100337\pi\)
−0.950729 + 0.310025i \(0.899663\pi\)
\(648\) 0 0
\(649\) −4752.00 −0.0112820
\(650\) 0 0
\(651\) 584380.i 1.37890i
\(652\) 0 0
\(653\) 330714. 0.775579 0.387790 0.921748i \(-0.373239\pi\)
0.387790 + 0.921748i \(0.373239\pi\)
\(654\) 0 0
\(655\) − 14840.2i − 0.0345906i
\(656\) 0 0
\(657\) 25002.0 0.0579221
\(658\) 0 0
\(659\) 253884.i 0.584608i 0.956326 + 0.292304i \(0.0944219\pi\)
−0.956326 + 0.292304i \(0.905578\pi\)
\(660\) 0 0
\(661\) 722158. 1.65283 0.826417 0.563058i \(-0.190375\pi\)
0.826417 + 0.563058i \(0.190375\pi\)
\(662\) 0 0
\(663\) 232642.i 0.529251i
\(664\) 0 0
\(665\) −376992. −0.852489
\(666\) 0 0
\(667\) − 58363.2i − 0.131186i
\(668\) 0 0
\(669\) −4428.00 −0.00989362
\(670\) 0 0
\(671\) 27976.1i 0.0621358i
\(672\) 0 0
\(673\) −552910. −1.22074 −0.610372 0.792115i \(-0.708980\pi\)
−0.610372 + 0.792115i \(0.708980\pi\)
\(674\) 0 0
\(675\) 159797.i 0.350721i
\(676\) 0 0
\(677\) −609030. −1.32881 −0.664403 0.747375i \(-0.731314\pi\)
−0.664403 + 0.747375i \(0.731314\pi\)
\(678\) 0 0
\(679\) 999726.i 2.16841i
\(680\) 0 0
\(681\) 395820. 0.853500
\(682\) 0 0
\(683\) − 23715.2i − 0.0508377i −0.999677 0.0254189i \(-0.991908\pi\)
0.999677 0.0254189i \(-0.00809195\pi\)
\(684\) 0 0
\(685\) −556668. −1.18636
\(686\) 0 0
\(687\) − 251858.i − 0.533631i
\(688\) 0 0
\(689\) 841932. 1.77353
\(690\) 0 0
\(691\) 431842.i 0.904417i 0.891912 + 0.452208i \(0.149364\pi\)
−0.891912 + 0.452208i \(0.850636\pi\)
\(692\) 0 0
\(693\) 42768.0 0.0890538
\(694\) 0 0
\(695\) 554908.i 1.14882i
\(696\) 0 0
\(697\) 225828. 0.464849
\(698\) 0 0
\(699\) − 253250.i − 0.518317i
\(700\) 0 0
\(701\) −44958.0 −0.0914894 −0.0457447 0.998953i \(-0.514566\pi\)
−0.0457447 + 0.998953i \(0.514566\pi\)
\(702\) 0 0
\(703\) 62423.1i 0.126309i
\(704\) 0 0
\(705\) −825552. −1.66099
\(706\) 0 0
\(707\) − 418394.i − 0.837041i
\(708\) 0 0
\(709\) −533002. −1.06032 −0.530159 0.847898i \(-0.677868\pi\)
−0.530159 + 0.847898i \(0.677868\pi\)
\(710\) 0 0
\(711\) − 118784.i − 0.234974i
\(712\) 0 0
\(713\) 1.10419e6 2.17203
\(714\) 0 0
\(715\) − 158878.i − 0.310778i
\(716\) 0 0
\(717\) −368928. −0.717634
\(718\) 0 0
\(719\) − 292107.i − 0.565046i −0.959260 0.282523i \(-0.908829\pi\)
0.959260 0.282523i \(-0.0911714\pi\)
\(720\) 0 0
\(721\) 434544. 0.835917
\(722\) 0 0
\(723\) − 380036.i − 0.727023i
\(724\) 0 0
\(725\) −88842.0 −0.169022
\(726\) 0 0
\(727\) 755791.i 1.42999i 0.699130 + 0.714995i \(0.253571\pi\)
−0.699130 + 0.714995i \(0.746429\pi\)
\(728\) 0 0
\(729\) −19683.0 −0.0370370
\(730\) 0 0
\(731\) 209634.i 0.392307i
\(732\) 0 0
\(733\) 832982. 1.55034 0.775171 0.631751i \(-0.217664\pi\)
0.775171 + 0.631751i \(0.217664\pi\)
\(734\) 0 0
\(735\) 743538.i 1.37635i
\(736\) 0 0
\(737\) 22608.0 0.0416224
\(738\) 0 0
\(739\) 698093.i 1.27827i 0.769093 + 0.639137i \(0.220708\pi\)
−0.769093 + 0.639137i \(0.779292\pi\)
\(740\) 0 0
\(741\) −111384. −0.202855
\(742\) 0 0
\(743\) − 461044.i − 0.835151i −0.908642 0.417575i \(-0.862880\pi\)
0.908642 0.417575i \(-0.137120\pi\)
\(744\) 0 0
\(745\) −18396.0 −0.0331445
\(746\) 0 0
\(747\) − 323803.i − 0.580284i
\(748\) 0 0
\(749\) 768240. 1.36941
\(750\) 0 0
\(751\) − 937060.i − 1.66145i −0.556682 0.830726i \(-0.687926\pi\)
0.556682 0.830726i \(-0.312074\pi\)
\(752\) 0 0
\(753\) −477468. −0.842082
\(754\) 0 0
\(755\) 1.17820e6i 2.06692i
\(756\) 0 0
\(757\) −295786. −0.516162 −0.258081 0.966123i \(-0.583090\pi\)
−0.258081 + 0.966123i \(0.583090\pi\)
\(758\) 0 0
\(759\) − 80810.6i − 0.140276i
\(760\) 0 0
\(761\) −1.02615e6 −1.77191 −0.885955 0.463772i \(-0.846496\pi\)
−0.885955 + 0.463772i \(0.846496\pi\)
\(762\) 0 0
\(763\) − 1.23201e6i − 2.11625i
\(764\) 0 0
\(765\) 278964. 0.476678
\(766\) 0 0
\(767\) − 41610.8i − 0.0707319i
\(768\) 0 0
\(769\) 362306. 0.612665 0.306332 0.951925i \(-0.400898\pi\)
0.306332 + 0.951925i \(0.400898\pi\)
\(770\) 0 0
\(771\) 254061.i 0.427394i
\(772\) 0 0
\(773\) −1.02608e6 −1.71720 −0.858601 0.512644i \(-0.828666\pi\)
−0.858601 + 0.512644i \(0.828666\pi\)
\(774\) 0 0
\(775\) − 1.68083e6i − 2.79847i
\(776\) 0 0
\(777\) 209880. 0.347639
\(778\) 0 0
\(779\) 108122.i 0.178171i
\(780\) 0 0
\(781\) 38016.0 0.0623253
\(782\) 0 0
\(783\) − 10943.1i − 0.0178491i
\(784\) 0 0
\(785\) −812532. −1.31856
\(786\) 0 0
\(787\) − 850042.i − 1.37243i −0.727398 0.686216i \(-0.759270\pi\)
0.727398 0.686216i \(-0.240730\pi\)
\(788\) 0 0
\(789\) −406296. −0.652662
\(790\) 0 0
\(791\) − 140379.i − 0.224362i
\(792\) 0 0
\(793\) −244972. −0.389556
\(794\) 0 0
\(795\) − 1.00957e6i − 1.59736i
\(796\) 0 0
\(797\) −761478. −1.19878 −0.599392 0.800456i \(-0.704591\pi\)
−0.599392 + 0.800456i \(0.704591\pi\)
\(798\) 0 0
\(799\) 930569.i 1.45766i
\(800\) 0 0
\(801\) −312822. −0.487565
\(802\) 0 0
\(803\) 19246.5i 0.0298484i
\(804\) 0 0
\(805\) 2.39501e6 3.69586
\(806\) 0 0
\(807\) − 371722.i − 0.570784i
\(808\) 0 0
\(809\) 247674. 0.378428 0.189214 0.981936i \(-0.439406\pi\)
0.189214 + 0.981936i \(0.439406\pi\)
\(810\) 0 0
\(811\) − 920197.i − 1.39907i −0.714599 0.699534i \(-0.753391\pi\)
0.714599 0.699534i \(-0.246609\pi\)
\(812\) 0 0
\(813\) 562212. 0.850588
\(814\) 0 0
\(815\) − 1.52272e6i − 2.29248i
\(816\) 0 0
\(817\) −100368. −0.150367
\(818\) 0 0
\(819\) 374497.i 0.558317i
\(820\) 0 0
\(821\) 250242. 0.371256 0.185628 0.982620i \(-0.440568\pi\)
0.185628 + 0.982620i \(0.440568\pi\)
\(822\) 0 0
\(823\) 400762.i 0.591680i 0.955238 + 0.295840i \(0.0955995\pi\)
−0.955238 + 0.295840i \(0.904401\pi\)
\(824\) 0 0
\(825\) −123012. −0.180734
\(826\) 0 0
\(827\) − 17272.0i − 0.0252541i −0.999920 0.0126270i \(-0.995981\pi\)
0.999920 0.0126270i \(-0.00401942\pi\)
\(828\) 0 0
\(829\) 15686.0 0.0228246 0.0114123 0.999935i \(-0.496367\pi\)
0.0114123 + 0.999935i \(0.496367\pi\)
\(830\) 0 0
\(831\) − 626230.i − 0.906842i
\(832\) 0 0
\(833\) 838122. 1.20786
\(834\) 0 0
\(835\) 787404.i 1.12934i
\(836\) 0 0
\(837\) 207036. 0.295525
\(838\) 0 0
\(839\) 115479.i 0.164051i 0.996630 + 0.0820257i \(0.0261390\pi\)
−0.996630 + 0.0820257i \(0.973861\pi\)
\(840\) 0 0
\(841\) −701197. −0.991398
\(842\) 0 0
\(843\) 15869.0i 0.0223304i
\(844\) 0 0
\(845\) 191646. 0.268402
\(846\) 0 0
\(847\) − 1.08287e6i − 1.50942i
\(848\) 0 0
\(849\) 690876. 0.958484
\(850\) 0 0
\(851\) − 396570.i − 0.547597i
\(852\) 0 0
\(853\) −345938. −0.475445 −0.237722 0.971333i \(-0.576401\pi\)
−0.237722 + 0.971333i \(0.576401\pi\)
\(854\) 0 0
\(855\) 133562.i 0.182705i
\(856\) 0 0
\(857\) −267990. −0.364886 −0.182443 0.983216i \(-0.558400\pi\)
−0.182443 + 0.983216i \(0.558400\pi\)
\(858\) 0 0
\(859\) − 522407.i − 0.707983i −0.935249 0.353992i \(-0.884824\pi\)
0.935249 0.353992i \(-0.115176\pi\)
\(860\) 0 0
\(861\) 363528. 0.490379
\(862\) 0 0
\(863\) − 826895.i − 1.11027i −0.831760 0.555135i \(-0.812667\pi\)
0.831760 0.555135i \(-0.187333\pi\)
\(864\) 0 0
\(865\) 1.44522e6 1.93153
\(866\) 0 0
\(867\) 119537.i 0.159025i
\(868\) 0 0
\(869\) 91440.0 0.121087
\(870\) 0 0
\(871\) 197966.i 0.260949i
\(872\) 0 0
\(873\) 354186. 0.464732
\(874\) 0 0
\(875\) − 1.64523e6i − 2.14887i
\(876\) 0 0
\(877\) −1.11629e6 −1.45137 −0.725685 0.688028i \(-0.758477\pi\)
−0.725685 + 0.688028i \(0.758477\pi\)
\(878\) 0 0
\(879\) 788059.i 1.01995i
\(880\) 0 0
\(881\) 19170.0 0.0246985 0.0123492 0.999924i \(-0.496069\pi\)
0.0123492 + 0.999924i \(0.496069\pi\)
\(882\) 0 0
\(883\) 568909.i 0.729662i 0.931074 + 0.364831i \(0.118873\pi\)
−0.931074 + 0.364831i \(0.881127\pi\)
\(884\) 0 0
\(885\) −49896.0 −0.0637058
\(886\) 0 0
\(887\) 1.09015e6i 1.38561i 0.721126 + 0.692804i \(0.243625\pi\)
−0.721126 + 0.692804i \(0.756375\pi\)
\(888\) 0 0
\(889\) 30096.0 0.0380807
\(890\) 0 0
\(891\) − 15152.0i − 0.0190860i
\(892\) 0 0
\(893\) −445536. −0.558702
\(894\) 0 0
\(895\) 707965.i 0.883824i
\(896\) 0 0
\(897\) 707616. 0.879453
\(898\) 0 0
\(899\) 115105.i 0.142421i
\(900\) 0 0
\(901\) −1.13800e6 −1.40182
\(902\) 0 0
\(903\) 337459.i 0.413852i
\(904\) 0 0
\(905\) −659652. −0.805411
\(906\) 0 0
\(907\) 916193.i 1.11371i 0.830610 + 0.556855i \(0.187992\pi\)
−0.830610 + 0.556855i \(0.812008\pi\)
\(908\) 0 0
\(909\) −148230. −0.179394
\(910\) 0 0
\(911\) 995500.i 1.19951i 0.800183 + 0.599756i \(0.204736\pi\)
−0.800183 + 0.599756i \(0.795264\pi\)
\(912\) 0 0
\(913\) 249264. 0.299032
\(914\) 0 0
\(915\) 293749.i 0.350860i
\(916\) 0 0
\(917\) −26928.0 −0.0320233
\(918\) 0 0
\(919\) 97084.9i 0.114953i 0.998347 + 0.0574766i \(0.0183054\pi\)
−0.998347 + 0.0574766i \(0.981695\pi\)
\(920\) 0 0
\(921\) −28188.0 −0.0332311
\(922\) 0 0
\(923\) 332886.i 0.390744i
\(924\) 0 0
\(925\) −603670. −0.705531
\(926\) 0 0
\(927\) − 153952.i − 0.179153i
\(928\) 0 0
\(929\) −1.27882e6 −1.48176 −0.740881 0.671636i \(-0.765592\pi\)
−0.740881 + 0.671636i \(0.765592\pi\)
\(930\) 0 0
\(931\) 401275.i 0.462959i
\(932\) 0 0
\(933\) 733320. 0.842423
\(934\) 0 0
\(935\) 214747.i 0.245642i
\(936\) 0 0
\(937\) −981262. −1.11765 −0.558825 0.829286i \(-0.688748\pi\)
−0.558825 + 0.829286i \(0.688748\pi\)
\(938\) 0 0
\(939\) 668672.i 0.758371i
\(940\) 0 0
\(941\) −284406. −0.321188 −0.160594 0.987021i \(-0.551341\pi\)
−0.160594 + 0.987021i \(0.551341\pi\)
\(942\) 0 0
\(943\) − 686890.i − 0.772438i
\(944\) 0 0
\(945\) 449064. 0.502857
\(946\) 0 0
\(947\) − 993109.i − 1.10738i −0.832722 0.553691i \(-0.813219\pi\)
0.832722 0.553691i \(-0.186781\pi\)
\(948\) 0 0
\(949\) −168532. −0.187133
\(950\) 0 0
\(951\) − 384443.i − 0.425080i
\(952\) 0 0
\(953\) 602922. 0.663858 0.331929 0.943304i \(-0.392301\pi\)
0.331929 + 0.943304i \(0.392301\pi\)
\(954\) 0 0
\(955\) − 111738.i − 0.122516i
\(956\) 0 0
\(957\) 8424.00 0.00919802
\(958\) 0 0
\(959\) 1.01009e6i 1.09831i
\(960\) 0 0
\(961\) −1.25419e6 −1.35805
\(962\) 0 0
\(963\) − 272174.i − 0.293491i
\(964\) 0 0
\(965\) −1.12484e6 −1.20792
\(966\) 0 0
\(967\) 575810.i 0.615781i 0.951422 + 0.307890i \(0.0996230\pi\)
−0.951422 + 0.307890i \(0.900377\pi\)
\(968\) 0 0
\(969\) 150552. 0.160339
\(970\) 0 0
\(971\) − 1.23920e6i − 1.31432i −0.753749 0.657162i \(-0.771757\pi\)
0.753749 0.657162i \(-0.228243\pi\)
\(972\) 0 0
\(973\) 1.00690e6 1.06355
\(974\) 0 0
\(975\) − 1.07715e6i − 1.13310i
\(976\) 0 0
\(977\) −1.04074e6 −1.09032 −0.545160 0.838332i \(-0.683531\pi\)
−0.545160 + 0.838332i \(0.683531\pi\)
\(978\) 0 0
\(979\) − 240810.i − 0.251252i
\(980\) 0 0
\(981\) −436482. −0.453553
\(982\) 0 0
\(983\) − 948734.i − 0.981833i −0.871207 0.490916i \(-0.836662\pi\)
0.871207 0.490916i \(-0.163338\pi\)
\(984\) 0 0
\(985\) 2.20424e6 2.27189
\(986\) 0 0
\(987\) 1.49799e6i 1.53771i
\(988\) 0 0
\(989\) 637632. 0.651895
\(990\) 0 0
\(991\) − 616007.i − 0.627247i −0.949547 0.313623i \(-0.898457\pi\)
0.949547 0.313623i \(-0.101543\pi\)
\(992\) 0 0
\(993\) 296316. 0.300508
\(994\) 0 0
\(995\) 969269.i 0.979035i
\(996\) 0 0
\(997\) 535870. 0.539100 0.269550 0.962986i \(-0.413125\pi\)
0.269550 + 0.962986i \(0.413125\pi\)
\(998\) 0 0
\(999\) − 74356.9i − 0.0745059i
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 192.5.g.b.127.1 2
3.2 odd 2 576.5.g.d.127.1 2
4.3 odd 2 inner 192.5.g.b.127.2 2
8.3 odd 2 48.5.g.a.31.1 2
8.5 even 2 48.5.g.a.31.2 yes 2
12.11 even 2 576.5.g.d.127.2 2
16.3 odd 4 768.5.b.c.127.1 4
16.5 even 4 768.5.b.c.127.2 4
16.11 odd 4 768.5.b.c.127.4 4
16.13 even 4 768.5.b.c.127.3 4
24.5 odd 2 144.5.g.f.127.1 2
24.11 even 2 144.5.g.f.127.2 2
40.3 even 4 1200.5.j.b.799.3 4
40.13 odd 4 1200.5.j.b.799.2 4
40.19 odd 2 1200.5.e.b.751.2 2
40.27 even 4 1200.5.j.b.799.1 4
40.29 even 2 1200.5.e.b.751.1 2
40.37 odd 4 1200.5.j.b.799.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.5.g.a.31.1 2 8.3 odd 2
48.5.g.a.31.2 yes 2 8.5 even 2
144.5.g.f.127.1 2 24.5 odd 2
144.5.g.f.127.2 2 24.11 even 2
192.5.g.b.127.1 2 1.1 even 1 trivial
192.5.g.b.127.2 2 4.3 odd 2 inner
576.5.g.d.127.1 2 3.2 odd 2
576.5.g.d.127.2 2 12.11 even 2
768.5.b.c.127.1 4 16.3 odd 4
768.5.b.c.127.2 4 16.5 even 4
768.5.b.c.127.3 4 16.13 even 4
768.5.b.c.127.4 4 16.11 odd 4
1200.5.e.b.751.1 2 40.29 even 2
1200.5.e.b.751.2 2 40.19 odd 2
1200.5.j.b.799.1 4 40.27 even 4
1200.5.j.b.799.2 4 40.13 odd 4
1200.5.j.b.799.3 4 40.3 even 4
1200.5.j.b.799.4 4 40.37 odd 4