Properties

Label 1200.5.j.b.799.1
Level $1200$
Weight $5$
Character 1200.799
Analytic conductor $124.044$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,5,Mod(799,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.799");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1200.j (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: \(124.043955701\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1200.799
Dual form 1200.5.j.b.799.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.19615 q^{3} -76.2102 q^{7} +27.0000 q^{9} +O(q^{10})\) \(q-5.19615 q^{3} -76.2102 q^{7} +27.0000 q^{9} -20.7846i q^{11} +182.000i q^{13} -246.000i q^{17} +117.779i q^{19} +396.000 q^{21} -748.246 q^{23} -140.296 q^{27} -78.0000 q^{29} +1475.71i q^{31} +108.000i q^{33} +530.000i q^{37} -945.700i q^{39} -918.000 q^{41} -852.169 q^{43} -3782.80 q^{47} +3407.00 q^{49} +1278.25i q^{51} +4626.00i q^{53} -612.000i q^{57} +228.631i q^{59} +1346.00 q^{61} -2057.68 q^{63} -1087.73 q^{67} +3888.00 q^{69} -1829.05i q^{71} +926.000i q^{73} +1584.00i q^{77} +4399.41i q^{79} +729.000 q^{81} +11992.7 q^{83} +405.300 q^{87} -11586.0 q^{89} -13870.3i q^{91} -7668.00i q^{93} -13118.0i q^{97} -561.184i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 108 q^{9} + 1584 q^{21} - 312 q^{29} - 3672 q^{41} + 13628 q^{49} + 5384 q^{61} + 15552 q^{69} + 2916 q^{81} - 46344 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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
\(3\) −5.19615 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −76.2102 −1.55531 −0.777655 0.628691i \(-0.783591\pi\)
−0.777655 + 0.628691i \(0.783591\pi\)
\(8\) 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
\(13\) 182.000i 1.07692i 0.842650 + 0.538462i \(0.180994\pi\)
−0.842650 + 0.538462i \(0.819006\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 246.000i − 0.851211i −0.904909 0.425606i \(-0.860061\pi\)
0.904909 0.425606i \(-0.139939\pi\)
\(18\) 0 0
\(19\) 117.779i 0.326259i 0.986605 + 0.163129i \(0.0521588\pi\)
−0.986605 + 0.163129i \(0.947841\pi\)
\(20\) 0 0
\(21\) 396.000 0.897959
\(22\) 0 0
\(23\) −748.246 −1.41445 −0.707227 0.706987i \(-0.750054\pi\)
−0.707227 + 0.706987i \(0.750054\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −140.296 −0.192450
\(28\) 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 \(0.278647\pi\)
−0.640692 + 0.767798i \(0.721353\pi\)
\(32\) 0 0
\(33\) 108.000i 0.0991736i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 530.000i 0.387144i 0.981086 + 0.193572i \(0.0620073\pi\)
−0.981086 + 0.193572i \(0.937993\pi\)
\(38\) 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
\(43\) −852.169 −0.460881 −0.230441 0.973086i \(-0.574017\pi\)
−0.230441 + 0.973086i \(0.574017\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3782.80 −1.71245 −0.856224 0.516604i \(-0.827196\pi\)
−0.856224 + 0.516604i \(0.827196\pi\)
\(48\) 0 0
\(49\) 3407.00 1.41899
\(50\) 0 0
\(51\) 1278.25i 0.491447i
\(52\) 0 0
\(53\) 4626.00i 1.64685i 0.567426 + 0.823425i \(0.307939\pi\)
−0.567426 + 0.823425i \(0.692061\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 612.000i − 0.188366i
\(58\) 0 0
\(59\) 228.631i 0.0656796i 0.999461 + 0.0328398i \(0.0104551\pi\)
−0.999461 + 0.0328398i \(0.989545\pi\)
\(60\) 0 0
\(61\) 1346.00 0.361731 0.180865 0.983508i \(-0.442110\pi\)
0.180865 + 0.983508i \(0.442110\pi\)
\(62\) 0 0
\(63\) −2057.68 −0.518437
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1087.73 −0.242310 −0.121155 0.992634i \(-0.538660\pi\)
−0.121155 + 0.992634i \(0.538660\pi\)
\(68\) 0 0
\(69\) 3888.00 0.816635
\(70\) 0 0
\(71\) − 1829.05i − 0.362834i −0.983406 0.181417i \(-0.941932\pi\)
0.983406 0.181417i \(-0.0580684\pi\)
\(72\) 0 0
\(73\) 926.000i 0.173766i 0.996219 + 0.0868831i \(0.0276907\pi\)
−0.996219 + 0.0868831i \(0.972309\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1584.00i 0.267161i
\(78\) 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
\(83\) 11992.7 1.74085 0.870425 0.492301i \(-0.163844\pi\)
0.870425 + 0.492301i \(0.163844\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 405.300 0.0535473
\(88\) 0 0
\(89\) −11586.0 −1.46269 −0.731347 0.682005i \(-0.761108\pi\)
−0.731347 + 0.682005i \(0.761108\pi\)
\(90\) 0 0
\(91\) − 13870.3i − 1.67495i
\(92\) 0 0
\(93\) − 7668.00i − 0.886576i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 13118.0i − 1.39420i −0.716975 0.697099i \(-0.754474\pi\)
0.716975 0.697099i \(-0.245526\pi\)
\(98\) 0 0
\(99\) − 561.184i − 0.0572579i
\(100\) 0 0
\(101\) −5490.00 −0.538183 −0.269091 0.963115i \(-0.586723\pi\)
−0.269091 + 0.963115i \(0.586723\pi\)
\(102\) 0 0
\(103\) −5701.91 −0.537460 −0.268730 0.963216i \(-0.586604\pi\)
−0.268730 + 0.963216i \(0.586604\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10080.5 −0.880473 −0.440237 0.897882i \(-0.645105\pi\)
−0.440237 + 0.897882i \(0.645105\pi\)
\(108\) 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
\(113\) − 1842.00i − 0.144256i −0.997395 0.0721278i \(-0.977021\pi\)
0.997395 0.0721278i \(-0.0229789\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4914.00i 0.358974i
\(118\) 0 0
\(119\) 18747.7i 1.32390i
\(120\) 0 0
\(121\) 14209.0 0.970494
\(122\) 0 0
\(123\) 4770.07 0.315293
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 394.908 0.0244843 0.0122422 0.999925i \(-0.496103\pi\)
0.0122422 + 0.999925i \(0.496103\pi\)
\(128\) 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
\(133\) − 8976.00i − 0.507434i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 13254.0i − 0.706164i −0.935592 0.353082i \(-0.885134\pi\)
0.935592 0.353082i \(-0.114866\pi\)
\(138\) 0 0
\(139\) − 13212.1i − 0.683820i −0.939733 0.341910i \(-0.888926\pi\)
0.939733 0.341910i \(-0.111074\pi\)
\(140\) 0 0
\(141\) 19656.0 0.988683
\(142\) 0 0
\(143\) 3782.80 0.184987
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −17703.3 −0.819255
\(148\) 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 \(-0.789093\pi\)
0.788406 0.615155i \(-0.210907\pi\)
\(152\) 0 0
\(153\) − 6642.00i − 0.283737i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 19346.0i 0.784859i 0.919782 + 0.392430i \(0.128365\pi\)
−0.919782 + 0.392430i \(0.871635\pi\)
\(158\) 0 0
\(159\) − 24037.4i − 0.950809i
\(160\) 0 0
\(161\) 57024.0 2.19992
\(162\) 0 0
\(163\) −36255.3 −1.36457 −0.682286 0.731086i \(-0.739014\pi\)
−0.682286 + 0.731086i \(0.739014\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 18747.7 0.672226 0.336113 0.941822i \(-0.390888\pi\)
0.336113 + 0.941822i \(0.390888\pi\)
\(168\) 0 0
\(169\) −4563.00 −0.159763
\(170\) 0 0
\(171\) 3180.05i 0.108753i
\(172\) 0 0
\(173\) 34410.0i 1.14972i 0.818251 + 0.574861i \(0.194944\pi\)
−0.818251 + 0.574861i \(0.805056\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 1188.00i − 0.0379201i
\(178\) 0 0
\(179\) − 16856.3i − 0.526086i −0.964784 0.263043i \(-0.915274\pi\)
0.964784 0.263043i \(-0.0847261\pi\)
\(180\) 0 0
\(181\) 15706.0 0.479411 0.239706 0.970846i \(-0.422949\pi\)
0.239706 + 0.970846i \(0.422949\pi\)
\(182\) 0 0
\(183\) −6994.02 −0.208845
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −5113.01 −0.146216
\(188\) 0 0
\(189\) 10692.0 0.299320
\(190\) 0 0
\(191\) 2660.43i 0.0729265i 0.999335 + 0.0364632i \(0.0116092\pi\)
−0.999335 + 0.0364632i \(0.988391\pi\)
\(192\) 0 0
\(193\) 26782.0i 0.718999i 0.933145 + 0.359500i \(0.117053\pi\)
−0.933145 + 0.359500i \(0.882947\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 52482.0i − 1.35232i −0.736757 0.676158i \(-0.763644\pi\)
0.736757 0.676158i \(-0.236356\pi\)
\(198\) 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
\(203\) 5944.40 0.144250
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −20202.6 −0.471485
\(208\) 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
\(213\) 9504.00i 0.209482i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 112464.i − 2.38833i
\(218\) 0 0
\(219\) − 4811.64i − 0.100324i
\(220\) 0 0
\(221\) 44772.0 0.916689
\(222\) 0 0
\(223\) 852.169 0.0171363 0.00856813 0.999963i \(-0.497273\pi\)
0.00856813 + 0.999963i \(0.497273\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −76175.6 −1.47831 −0.739153 0.673538i \(-0.764774\pi\)
−0.739153 + 0.673538i \(0.764774\pi\)
\(228\) 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
\(233\) − 48738.0i − 0.897751i −0.893594 0.448875i \(-0.851825\pi\)
0.893594 0.448875i \(-0.148175\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 22860.0i − 0.406986i
\(238\) 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
\(243\) −3788.00 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −21435.9 −0.351356
\(248\) 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
\(253\) 15552.0i 0.242966i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 48894.0i − 0.740269i −0.928978 0.370134i \(-0.879312\pi\)
0.928978 0.370134i \(-0.120688\pi\)
\(258\) 0 0
\(259\) − 40391.4i − 0.602129i
\(260\) 0 0
\(261\) −2106.00 −0.0309156
\(262\) 0 0
\(263\) 78191.7 1.13044 0.565222 0.824939i \(-0.308790\pi\)
0.565222 + 0.824939i \(0.308790\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 60202.6 0.844487
\(268\) 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 \(-0.736416\pi\)
0.676296 0.736630i \(-0.263584\pi\)
\(272\) 0 0
\(273\) 72072.0i 0.967033i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 120518.i − 1.57070i −0.619054 0.785348i \(-0.712484\pi\)
0.619054 0.785348i \(-0.287516\pi\)
\(278\) 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. 1.66014 0.830071 0.557657i \(-0.188300\pi\)
0.830071 + 0.557657i \(0.188300\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 69961.0 0.849361
\(288\) 0 0
\(289\) 23005.0 0.275440
\(290\) 0 0
\(291\) 68163.1i 0.804940i
\(292\) 0 0
\(293\) − 151662.i − 1.76661i −0.468795 0.883307i \(-0.655312\pi\)
0.468795 0.883307i \(-0.344688\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2916.00i 0.0330579i
\(298\) 0 0
\(299\) − 136181.i − 1.52326i
\(300\) 0 0
\(301\) 64944.0 0.716813
\(302\) 0 0
\(303\) 28526.9 0.310720
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5424.78 0.0575580 0.0287790 0.999586i \(-0.490838\pi\)
0.0287790 + 0.999586i \(0.490838\pi\)
\(308\) 0 0
\(309\) 29628.0 0.310303
\(310\) 0 0
\(311\) − 141127.i − 1.45912i −0.683917 0.729560i \(-0.739725\pi\)
0.683917 0.729560i \(-0.260275\pi\)
\(312\) 0 0
\(313\) 128686.i 1.31354i 0.754092 + 0.656769i \(0.228077\pi\)
−0.754092 + 0.656769i \(0.771923\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 73986.0i − 0.736260i −0.929774 0.368130i \(-0.879998\pi\)
0.929774 0.368130i \(-0.120002\pi\)
\(318\) 0 0
\(319\) 1621.20i 0.0159314i
\(320\) 0 0
\(321\) 52380.0 0.508341
\(322\) 0 0
\(323\) 28973.7 0.277715
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −84001.0 −0.785577
\(328\) 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
\(333\) 14310.0i 0.129048i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 98674.0i 0.868846i 0.900709 + 0.434423i \(0.143048\pi\)
−0.900709 + 0.434423i \(0.856952\pi\)
\(338\) 0 0
\(339\) 9571.31i 0.0832860i
\(340\) 0 0
\(341\) 30672.0 0.263775
\(342\) 0 0
\(343\) −76667.5 −0.651663
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −56929.0 −0.472797 −0.236399 0.971656i \(-0.575967\pi\)
−0.236399 + 0.971656i \(0.575967\pi\)
\(348\) 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
\(353\) 4302.00i 0.0345240i 0.999851 + 0.0172620i \(0.00549494\pi\)
−0.999851 + 0.0172620i \(0.994505\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 97416.0i − 0.764353i
\(358\) 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
\(363\) −73832.1 −0.560315
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −182690. −1.35638 −0.678191 0.734885i \(-0.737236\pi\)
−0.678191 + 0.734885i \(0.737236\pi\)
\(368\) 0 0
\(369\) −24786.0 −0.182035
\(370\) 0 0
\(371\) − 352549.i − 2.56136i
\(372\) 0 0
\(373\) − 151778.i − 1.09092i −0.838138 0.545458i \(-0.816356\pi\)
0.838138 0.545458i \(-0.183644\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 14196.0i − 0.0998811i
\(378\) 0 0
\(379\) − 36005.9i − 0.250666i −0.992115 0.125333i \(-0.960000\pi\)
0.992115 0.125333i \(-0.0399999\pi\)
\(380\) 0 0
\(381\) −2052.00 −0.0141360
\(382\) 0 0
\(383\) −65346.8 −0.445479 −0.222739 0.974878i \(-0.571500\pi\)
−0.222739 + 0.974878i \(0.571500\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −23008.6 −0.153627
\(388\) 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
\(393\) 1836.00i 0.0118874i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 27934.0i − 0.177236i −0.996066 0.0886180i \(-0.971755\pi\)
0.996066 0.0886180i \(-0.0282450\pi\)
\(398\) 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. −1.65372
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11015.8 0.0665011
\(408\) 0 0
\(409\) 20270.0 0.121173 0.0605867 0.998163i \(-0.480703\pi\)
0.0605867 + 0.998163i \(0.480703\pi\)
\(410\) 0 0
\(411\) 68869.8i 0.407704i
\(412\) 0 0
\(413\) − 17424.0i − 0.102152i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 68652.0i 0.394804i
\(418\) 0 0
\(419\) 24089.4i 0.137214i 0.997644 + 0.0686068i \(0.0218554\pi\)
−0.997644 + 0.0686068i \(0.978145\pi\)
\(420\) 0 0
\(421\) 116698. 0.658414 0.329207 0.944258i \(-0.393219\pi\)
0.329207 + 0.944258i \(0.393219\pi\)
\(422\) 0 0
\(423\) −102136. −0.570816
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −102579. −0.562604
\(428\) 0 0
\(429\) −19656.0 −0.106802
\(430\) 0 0
\(431\) 355542.i 1.91397i 0.290132 + 0.956986i \(0.406301\pi\)
−0.290132 + 0.956986i \(0.593699\pi\)
\(432\) 0 0
\(433\) 199726.i 1.06527i 0.846346 + 0.532634i \(0.178798\pi\)
−0.846346 + 0.532634i \(0.821202\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 88128.0i − 0.461478i
\(438\) 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
\(443\) 50444.2 0.257042 0.128521 0.991707i \(-0.458977\pi\)
0.128521 + 0.991707i \(0.458977\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 2275.91 0.0113905
\(448\) 0 0
\(449\) −149994. −0.744014 −0.372007 0.928230i \(-0.621330\pi\)
−0.372007 + 0.928230i \(0.621330\pi\)
\(450\) 0 0
\(451\) 19080.3i 0.0938062i
\(452\) 0 0
\(453\) 145764.i 0.710320i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 284338.i 1.36145i 0.732538 + 0.680726i \(0.238336\pi\)
−0.732538 + 0.680726i \(0.761664\pi\)
\(458\) 0 0
\(459\) 34512.8i 0.163816i
\(460\) 0 0
\(461\) −183402. −0.862983 −0.431491 0.902117i \(-0.642013\pi\)
−0.431491 + 0.902117i \(0.642013\pi\)
\(462\) 0 0
\(463\) 172422. 0.804324 0.402162 0.915568i \(-0.368259\pi\)
0.402162 + 0.915568i \(0.368259\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 68734.7 0.315168 0.157584 0.987506i \(-0.449629\pi\)
0.157584 + 0.987506i \(0.449629\pi\)
\(468\) 0 0
\(469\) 82896.0 0.376867
\(470\) 0 0
\(471\) − 100525.i − 0.453139i
\(472\) 0 0
\(473\) 17712.0i 0.0791672i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 124902.i 0.548950i
\(478\) 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. −1.27012
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −271108. −1.14310 −0.571549 0.820568i \(-0.693657\pi\)
−0.571549 + 0.820568i \(0.693657\pi\)
\(488\) 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
\(493\) 19188.0i 0.0789470i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 139392.i 0.564320i
\(498\) 0 0
\(499\) 248854.i 0.999410i 0.866196 + 0.499705i \(0.166558\pi\)
−0.866196 + 0.499705i \(0.833442\pi\)
\(500\) 0 0
\(501\) −97416.0 −0.388110
\(502\) 0 0
\(503\) 446537. 1.76490 0.882452 0.470403i \(-0.155891\pi\)
0.882452 + 0.470403i \(0.155891\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 23710.0 0.0922394
\(508\) 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
\(513\) − 16524.0i − 0.0627886i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 78624.0i 0.294154i
\(518\) 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. −0.495409 −0.247704 0.968836i \(-0.579676\pi\)
−0.247704 + 0.968836i \(0.579676\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 363024. 1.30712
\(528\) 0 0
\(529\) 280031. 1.00068
\(530\) 0 0
\(531\) 6173.03i 0.0218932i
\(532\) 0 0
\(533\) − 167076.i − 0.588111i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 87588.0i 0.303736i
\(538\) 0 0
\(539\) − 70813.2i − 0.243745i
\(540\) 0 0
\(541\) 360442. 1.23152 0.615759 0.787934i \(-0.288849\pi\)
0.615759 + 0.787934i \(0.288849\pi\)
\(542\) 0 0
\(543\) −81610.8 −0.276788
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 261644. 0.874451 0.437225 0.899352i \(-0.355961\pi\)
0.437225 + 0.899352i \(0.355961\pi\)
\(548\) 0 0
\(549\) 36342.0 0.120577
\(550\) 0 0
\(551\) − 9186.80i − 0.0302594i
\(552\) 0 0
\(553\) − 335280.i − 1.09637i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 233274.i − 0.751893i −0.926641 0.375946i \(-0.877318\pi\)
0.926641 0.375946i \(-0.122682\pi\)
\(558\) 0 0
\(559\) − 155095.i − 0.496333i
\(560\) 0 0
\(561\) 26568.0 0.0844176
\(562\) 0 0
\(563\) 419704. 1.32412 0.662058 0.749453i \(-0.269683\pi\)
0.662058 + 0.749453i \(0.269683\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −55557.3 −0.172812
\(568\) 0 0
\(569\) −470058. −1.45187 −0.725934 0.687765i \(-0.758592\pi\)
−0.725934 + 0.687765i \(0.758592\pi\)
\(570\) 0 0
\(571\) 320381.i 0.982640i 0.870979 + 0.491320i \(0.163485\pi\)
−0.870979 + 0.491320i \(0.836515\pi\)
\(572\) 0 0
\(573\) − 13824.0i − 0.0421041i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 341038.i − 1.02436i −0.858879 0.512178i \(-0.828839\pi\)
0.858879 0.512178i \(-0.171161\pi\)
\(578\) 0 0
\(579\) − 139163.i − 0.415114i
\(580\) 0 0
\(581\) −913968. −2.70756
\(582\) 0 0
\(583\) 96149.6 0.282885
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 114128. 0.331220 0.165610 0.986191i \(-0.447041\pi\)
0.165610 + 0.986191i \(0.447041\pi\)
\(588\) 0 0
\(589\) −173808. −0.501002
\(590\) 0 0
\(591\) 272704.i 0.780760i
\(592\) 0 0
\(593\) 96846.0i 0.275405i 0.990474 + 0.137703i \(0.0439718\pi\)
−0.990474 + 0.137703i \(0.956028\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 119916.i − 0.336456i
\(598\) 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
\(603\) −29368.7 −0.0807699
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 133195. 0.361501 0.180751 0.983529i \(-0.442147\pi\)
0.180751 + 0.983529i \(0.442147\pi\)
\(608\) 0 0
\(609\) −30888.0 −0.0832828
\(610\) 0 0
\(611\) − 688469.i − 1.84418i
\(612\) 0 0
\(613\) − 247202.i − 0.657856i −0.944355 0.328928i \(-0.893313\pi\)
0.944355 0.328928i \(-0.106687\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 31758.0i − 0.0834224i −0.999130 0.0417112i \(-0.986719\pi\)
0.999130 0.0417112i \(-0.0132809\pi\)
\(618\) 0 0
\(619\) − 656094.i − 1.71232i −0.516712 0.856160i \(-0.672844\pi\)
0.516712 0.856160i \(-0.327156\pi\)
\(620\) 0 0
\(621\) 104976. 0.272212
\(622\) 0 0
\(623\) 882972. 2.27494
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −12720.2 −0.0323563
\(628\) 0 0
\(629\) 130380. 0.329541
\(630\) 0 0
\(631\) 417736.i 1.04916i 0.851360 + 0.524582i \(0.175778\pi\)
−0.851360 + 0.524582i \(0.824222\pi\)
\(632\) 0 0
\(633\) − 124164.i − 0.309876i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 620074.i 1.52815i
\(638\) 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. −1.72727 −0.863635 0.504117i \(-0.831818\pi\)
−0.863635 + 0.504117i \(0.831818\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 259558. 0.620049 0.310025 0.950729i \(-0.399663\pi\)
0.310025 + 0.950729i \(0.399663\pi\)
\(648\) 0 0
\(649\) 4752.00 0.0112820
\(650\) 0 0
\(651\) 584380.i 1.37890i
\(652\) 0 0
\(653\) 330714.i 0.775579i 0.921748 + 0.387790i \(0.126761\pi\)
−0.921748 + 0.387790i \(0.873239\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 25002.0i 0.0579221i
\(658\) 0 0
\(659\) − 253884.i − 0.584608i −0.956326 0.292304i \(-0.905578\pi\)
0.956326 0.292304i \(-0.0944219\pi\)
\(660\) 0 0
\(661\) −722158. −1.65283 −0.826417 0.563058i \(-0.809625\pi\)
−0.826417 + 0.563058i \(0.809625\pi\)
\(662\) 0 0
\(663\) −232642. −0.529251
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 58363.2 0.131186
\(668\) 0 0
\(669\) −4428.00 −0.00989362
\(670\) 0 0
\(671\) − 27976.1i − 0.0621358i
\(672\) 0 0
\(673\) 552910.i 1.22074i 0.792115 + 0.610372i \(0.208980\pi\)
−0.792115 + 0.610372i \(0.791020\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 609030.i 1.32881i 0.747375 + 0.664403i \(0.231314\pi\)
−0.747375 + 0.664403i \(0.768686\pi\)
\(678\) 0 0
\(679\) 999726.i 2.16841i
\(680\) 0 0
\(681\) 395820. 0.853500
\(682\) 0 0
\(683\) −23715.2 −0.0508377 −0.0254189 0.999677i \(-0.508092\pi\)
−0.0254189 + 0.999677i \(0.508092\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −251858. −0.533631
\(688\) 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
\(693\) 42768.0i 0.0890538i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 225828.i 0.464849i
\(698\) 0 0
\(699\) 253250.i 0.518317i
\(700\) 0 0
\(701\) 44958.0 0.0914894 0.0457447 0.998953i \(-0.485434\pi\)
0.0457447 + 0.998953i \(0.485434\pi\)
\(702\) 0 0
\(703\) −62423.1 −0.126309
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 418394. 0.837041
\(708\) 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
\(713\) − 1.10419e6i − 2.17203i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 368928.i 0.717634i
\(718\) 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. −0.727023
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 755791. 1.42999 0.714995 0.699130i \(-0.246429\pi\)
0.714995 + 0.699130i \(0.246429\pi\)
\(728\) 0 0
\(729\) 19683.0 0.0370370
\(730\) 0 0
\(731\) 209634.i 0.392307i
\(732\) 0 0
\(733\) 832982.i 1.55034i 0.631751 + 0.775171i \(0.282336\pi\)
−0.631751 + 0.775171i \(0.717664\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 22608.0i 0.0416224i
\(738\) 0 0
\(739\) − 698093.i − 1.27827i −0.769093 0.639137i \(-0.779292\pi\)
0.769093 0.639137i \(-0.220708\pi\)
\(740\) 0 0
\(741\) 111384. 0.202855
\(742\) 0 0
\(743\) 461044. 0.835151 0.417575 0.908642i \(-0.362880\pi\)
0.417575 + 0.908642i \(0.362880\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 323803. 0.580284
\(748\) 0 0
\(749\) 768240. 1.36941
\(750\) 0 0
\(751\) 937060.i 1.66145i 0.556682 + 0.830726i \(0.312074\pi\)
−0.556682 + 0.830726i \(0.687926\pi\)
\(752\) 0 0
\(753\) 477468.i 0.842082i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 295786.i 0.516162i 0.966123 + 0.258081i \(0.0830901\pi\)
−0.966123 + 0.258081i \(0.916910\pi\)
\(758\) 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
\(763\) −1.23201e6 −2.11625
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −41610.8 −0.0707319
\(768\) 0 0
\(769\) −362306. −0.612665 −0.306332 0.951925i \(-0.599102\pi\)
−0.306332 + 0.951925i \(0.599102\pi\)
\(770\) 0 0
\(771\) 254061.i 0.427394i
\(772\) 0 0
\(773\) − 1.02608e6i − 1.71720i −0.512644 0.858601i \(-0.671334\pi\)
0.512644 0.858601i \(-0.328666\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 209880.i 0.347639i
\(778\) 0 0
\(779\) − 108122.i − 0.178171i
\(780\) 0 0
\(781\) −38016.0 −0.0623253
\(782\) 0 0
\(783\) 10943.1 0.0178491
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 850042. 1.37243 0.686216 0.727398i \(-0.259270\pi\)
0.686216 + 0.727398i \(0.259270\pi\)
\(788\) 0 0
\(789\) −406296. −0.652662
\(790\) 0 0
\(791\) 140379.i 0.224362i
\(792\) 0 0
\(793\) 244972.i 0.389556i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 761478.i 1.19878i 0.800456 + 0.599392i \(0.204591\pi\)
−0.800456 + 0.599392i \(0.795409\pi\)
\(798\) 0 0
\(799\) 930569.i 1.45766i
\(800\) 0 0
\(801\) −312822. −0.487565
\(802\) 0 0
\(803\) 19246.5 0.0298484
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −371722. −0.570784
\(808\) 0 0
\(809\) −247674. −0.378428 −0.189214 0.981936i \(-0.560594\pi\)
−0.189214 + 0.981936i \(0.560594\pi\)
\(810\) 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.i 0.850588i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 100368.i − 0.150367i
\(818\) 0 0
\(819\) − 374497.i − 0.558317i
\(820\) 0 0
\(821\) −250242. −0.371256 −0.185628 0.982620i \(-0.559432\pi\)
−0.185628 + 0.982620i \(0.559432\pi\)
\(822\) 0 0
\(823\) −400762. −0.591680 −0.295840 0.955238i \(-0.595599\pi\)
−0.295840 + 0.955238i \(0.595599\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 17272.0 0.0252541 0.0126270 0.999920i \(-0.495981\pi\)
0.0126270 + 0.999920i \(0.495981\pi\)
\(828\) 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.i − 1.20786i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 207036.i − 0.295525i
\(838\) 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
\(843\) 15869.0 0.0223304
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1.08287e6 −1.50942
\(848\) 0 0
\(849\) −690876. −0.958484
\(850\) 0 0
\(851\) − 396570.i − 0.547597i
\(852\) 0 0
\(853\) − 345938.i − 0.475445i −0.971333 0.237722i \(-0.923599\pi\)
0.971333 0.237722i \(-0.0764009\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 267990.i − 0.364886i −0.983216 0.182443i \(-0.941600\pi\)
0.983216 0.182443i \(-0.0584005\pi\)
\(858\) 0 0
\(859\) 522407.i 0.707983i 0.935249 + 0.353992i \(0.115176\pi\)
−0.935249 + 0.353992i \(0.884824\pi\)
\(860\) 0 0
\(861\) −363528. −0.490379
\(862\) 0 0
\(863\) 826895. 1.11027 0.555135 0.831760i \(-0.312667\pi\)
0.555135 + 0.831760i \(0.312667\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −119537. −0.159025
\(868\) 0 0
\(869\) 91440.0 0.121087
\(870\) 0 0
\(871\) − 197966.i − 0.260949i
\(872\) 0 0
\(873\) − 354186.i − 0.464732i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.11629e6i 1.45137i 0.688028 + 0.725685i \(0.258477\pi\)
−0.688028 + 0.725685i \(0.741523\pi\)
\(878\) 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. 0.729662 0.364831 0.931074i \(-0.381127\pi\)
0.364831 + 0.931074i \(0.381127\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.09015e6 1.38561 0.692804 0.721126i \(-0.256375\pi\)
0.692804 + 0.721126i \(0.256375\pi\)
\(888\) 0 0
\(889\) −30096.0 −0.0380807
\(890\) 0 0
\(891\) − 15152.0i − 0.0190860i
\(892\) 0 0
\(893\) − 445536.i − 0.558702i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 707616.i 0.879453i
\(898\) 0 0
\(899\) − 115105.i − 0.142421i
\(900\) 0 0
\(901\) 1.13800e6 1.40182
\(902\) 0 0
\(903\) −337459. −0.413852
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −916193. −1.11371 −0.556855 0.830610i \(-0.687992\pi\)
−0.556855 + 0.830610i \(0.687992\pi\)
\(908\) 0 0
\(909\) −148230. −0.179394
\(910\) 0 0
\(911\) − 995500.i − 1.19951i −0.800183 0.599756i \(-0.795264\pi\)
0.800183 0.599756i \(-0.204736\pi\)
\(912\) 0 0
\(913\) − 249264.i − 0.299032i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 26928.0i 0.0320233i
\(918\) 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. 0.390744
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −153952. −0.179153
\(928\) 0 0
\(929\) 1.27882e6 1.48176 0.740881 0.671636i \(-0.234408\pi\)
0.740881 + 0.671636i \(0.234408\pi\)
\(930\) 0 0
\(931\) 401275.i 0.462959i
\(932\) 0 0
\(933\) 733320.i 0.842423i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 981262.i − 1.11765i −0.829286 0.558825i \(-0.811252\pi\)
0.829286 0.558825i \(-0.188748\pi\)
\(938\) 0 0
\(939\) − 668672.i − 0.758371i
\(940\) 0 0
\(941\) 284406. 0.321188 0.160594 0.987021i \(-0.448659\pi\)
0.160594 + 0.987021i \(0.448659\pi\)
\(942\) 0 0
\(943\) 686890. 0.772438
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 993109. 1.10738 0.553691 0.832722i \(-0.313219\pi\)
0.553691 + 0.832722i \(0.313219\pi\)
\(948\) 0 0
\(949\) −168532. −0.187133
\(950\) 0 0
\(951\) 384443.i 0.425080i
\(952\) 0 0
\(953\) − 602922.i − 0.663858i −0.943304 0.331929i \(-0.892301\pi\)
0.943304 0.331929i \(-0.107699\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 8424.00i − 0.00919802i
\(958\) 0 0
\(959\) 1.01009e6i 1.09831i
\(960\) 0 0
\(961\) −1.25419e6 −1.35805
\(962\) 0 0
\(963\) −272174. −0.293491
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 575810. 0.615781 0.307890 0.951422i \(-0.400377\pi\)
0.307890 + 0.951422i \(0.400377\pi\)
\(968\) 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
\(973\) 1.00690e6i 1.06355i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 1.04074e6i − 1.09032i −0.838332 0.545160i \(-0.816469\pi\)
0.838332 0.545160i \(-0.183531\pi\)
\(978\) 0 0
\(979\) 240810.i 0.251252i
\(980\) 0 0
\(981\) 436482. 0.453553
\(982\) 0 0
\(983\) 948734. 0.981833 0.490916 0.871207i \(-0.336662\pi\)
0.490916 + 0.871207i \(0.336662\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −1.49799e6 −1.53771
\(988\) 0 0
\(989\) 637632. 0.651895
\(990\) 0 0
\(991\) 616007.i 0.627247i 0.949547 + 0.313623i \(0.101543\pi\)
−0.949547 + 0.313623i \(0.898457\pi\)
\(992\) 0 0
\(993\) − 296316.i − 0.300508i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 535870.i − 0.539100i −0.962986 0.269550i \(-0.913125\pi\)
0.962986 0.269550i \(-0.0868749\pi\)
\(998\) 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 1200.5.j.b.799.1 4
4.3 odd 2 inner 1200.5.j.b.799.4 4
5.2 odd 4 1200.5.e.b.751.2 2
5.3 odd 4 48.5.g.a.31.1 2
5.4 even 2 inner 1200.5.j.b.799.3 4
15.8 even 4 144.5.g.f.127.2 2
20.3 even 4 48.5.g.a.31.2 yes 2
20.7 even 4 1200.5.e.b.751.1 2
20.19 odd 2 inner 1200.5.j.b.799.2 4
40.3 even 4 192.5.g.b.127.1 2
40.13 odd 4 192.5.g.b.127.2 2
60.23 odd 4 144.5.g.f.127.1 2
80.3 even 4 768.5.b.c.127.2 4
80.13 odd 4 768.5.b.c.127.4 4
80.43 even 4 768.5.b.c.127.3 4
80.53 odd 4 768.5.b.c.127.1 4
120.53 even 4 576.5.g.d.127.2 2
120.83 odd 4 576.5.g.d.127.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.5.g.a.31.1 2 5.3 odd 4
48.5.g.a.31.2 yes 2 20.3 even 4
144.5.g.f.127.1 2 60.23 odd 4
144.5.g.f.127.2 2 15.8 even 4
192.5.g.b.127.1 2 40.3 even 4
192.5.g.b.127.2 2 40.13 odd 4
576.5.g.d.127.1 2 120.83 odd 4
576.5.g.d.127.2 2 120.53 even 4
768.5.b.c.127.1 4 80.53 odd 4
768.5.b.c.127.2 4 80.3 even 4
768.5.b.c.127.3 4 80.43 even 4
768.5.b.c.127.4 4 80.13 odd 4
1200.5.e.b.751.1 2 20.7 even 4
1200.5.e.b.751.2 2 5.2 odd 4
1200.5.j.b.799.1 4 1.1 even 1 trivial
1200.5.j.b.799.2 4 20.19 odd 2 inner
1200.5.j.b.799.3 4 5.4 even 2 inner
1200.5.j.b.799.4 4 4.3 odd 2 inner