Properties

Label 192.4.d.c.97.3
Level $192$
Weight $4$
Character 192.97
Analytic conductor $11.328$
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] = [192,4,Mod(97,192)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.97");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 192.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.3283667211\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 97.3
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 192.97
Dual form 192.4.d.c.97.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} -3.46410i q^{5} +24.2487 q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} -3.46410i q^{5} +24.2487 q^{7} -9.00000 q^{9} -48.0000i q^{11} +41.5692i q^{13} +10.3923 q^{15} +54.0000 q^{17} -4.00000i q^{19} +72.7461i q^{21} +173.205 q^{23} +113.000 q^{25} -27.0000i q^{27} +162.813i q^{29} +58.8897 q^{31} +144.000 q^{33} -84.0000i q^{35} +325.626i q^{37} -124.708 q^{39} -294.000 q^{41} -188.000i q^{43} +31.1769i q^{45} +505.759 q^{47} +245.000 q^{49} +162.000i q^{51} -744.782i q^{53} -166.277 q^{55} +12.0000 q^{57} -252.000i q^{59} -90.0666i q^{61} -218.238 q^{63} +144.000 q^{65} +628.000i q^{67} +519.615i q^{69} -6.92820 q^{71} -1006.00 q^{73} +339.000i q^{75} -1163.94i q^{77} -1340.61 q^{79} +81.0000 q^{81} -720.000i q^{83} -187.061i q^{85} -488.438 q^{87} -1482.00 q^{89} +1008.00i q^{91} +176.669i q^{93} -13.8564 q^{95} +1822.00 q^{97} +432.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 36 q^{9} + 216 q^{17} + 452 q^{25} + 576 q^{33} - 1176 q^{41} + 980 q^{49} + 48 q^{57} + 576 q^{65} - 4024 q^{73} + 324 q^{81} - 5928 q^{89} + 7288 q^{97}+O(q^{100}) \) 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
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) − 3.46410i − 0.309839i −0.987927 0.154919i \(-0.950488\pi\)
0.987927 0.154919i \(-0.0495118\pi\)
\(6\) 0 0
\(7\) 24.2487 1.30931 0.654654 0.755929i \(-0.272814\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) − 48.0000i − 1.31569i −0.753155 0.657843i \(-0.771469\pi\)
0.753155 0.657843i \(-0.228531\pi\)
\(12\) 0 0
\(13\) 41.5692i 0.886864i 0.896308 + 0.443432i \(0.146239\pi\)
−0.896308 + 0.443432i \(0.853761\pi\)
\(14\) 0 0
\(15\) 10.3923 0.178885
\(16\) 0 0
\(17\) 54.0000 0.770407 0.385204 0.922832i \(-0.374131\pi\)
0.385204 + 0.922832i \(0.374131\pi\)
\(18\) 0 0
\(19\) − 4.00000i − 0.0482980i −0.999708 0.0241490i \(-0.992312\pi\)
0.999708 0.0241490i \(-0.00768762\pi\)
\(20\) 0 0
\(21\) 72.7461i 0.755929i
\(22\) 0 0
\(23\) 173.205 1.57025 0.785125 0.619337i \(-0.212599\pi\)
0.785125 + 0.619337i \(0.212599\pi\)
\(24\) 0 0
\(25\) 113.000 0.904000
\(26\) 0 0
\(27\) − 27.0000i − 0.192450i
\(28\) 0 0
\(29\) 162.813i 1.04254i 0.853393 + 0.521269i \(0.174541\pi\)
−0.853393 + 0.521269i \(0.825459\pi\)
\(30\) 0 0
\(31\) 58.8897 0.341191 0.170595 0.985341i \(-0.445431\pi\)
0.170595 + 0.985341i \(0.445431\pi\)
\(32\) 0 0
\(33\) 144.000 0.759612
\(34\) 0 0
\(35\) − 84.0000i − 0.405674i
\(36\) 0 0
\(37\) 325.626i 1.44682i 0.690416 + 0.723412i \(0.257427\pi\)
−0.690416 + 0.723412i \(0.742573\pi\)
\(38\) 0 0
\(39\) −124.708 −0.512031
\(40\) 0 0
\(41\) −294.000 −1.11988 −0.559940 0.828533i \(-0.689176\pi\)
−0.559940 + 0.828533i \(0.689176\pi\)
\(42\) 0 0
\(43\) − 188.000i − 0.666738i −0.942796 0.333369i \(-0.891815\pi\)
0.942796 0.333369i \(-0.108185\pi\)
\(44\) 0 0
\(45\) 31.1769i 0.103280i
\(46\) 0 0
\(47\) 505.759 1.56963 0.784814 0.619731i \(-0.212758\pi\)
0.784814 + 0.619731i \(0.212758\pi\)
\(48\) 0 0
\(49\) 245.000 0.714286
\(50\) 0 0
\(51\) 162.000i 0.444795i
\(52\) 0 0
\(53\) − 744.782i − 1.93026i −0.261775 0.965129i \(-0.584308\pi\)
0.261775 0.965129i \(-0.415692\pi\)
\(54\) 0 0
\(55\) −166.277 −0.407650
\(56\) 0 0
\(57\) 12.0000 0.0278849
\(58\) 0 0
\(59\) − 252.000i − 0.556061i −0.960572 0.278031i \(-0.910318\pi\)
0.960572 0.278031i \(-0.0896817\pi\)
\(60\) 0 0
\(61\) − 90.0666i − 0.189047i −0.995523 0.0945234i \(-0.969867\pi\)
0.995523 0.0945234i \(-0.0301327\pi\)
\(62\) 0 0
\(63\) −218.238 −0.436436
\(64\) 0 0
\(65\) 144.000 0.274785
\(66\) 0 0
\(67\) 628.000i 1.14511i 0.819866 + 0.572555i \(0.194048\pi\)
−0.819866 + 0.572555i \(0.805952\pi\)
\(68\) 0 0
\(69\) 519.615i 0.906584i
\(70\) 0 0
\(71\) −6.92820 −0.0115807 −0.00579033 0.999983i \(-0.501843\pi\)
−0.00579033 + 0.999983i \(0.501843\pi\)
\(72\) 0 0
\(73\) −1006.00 −1.61292 −0.806462 0.591286i \(-0.798620\pi\)
−0.806462 + 0.591286i \(0.798620\pi\)
\(74\) 0 0
\(75\) 339.000i 0.521925i
\(76\) 0 0
\(77\) − 1163.94i − 1.72264i
\(78\) 0 0
\(79\) −1340.61 −1.90924 −0.954621 0.297824i \(-0.903739\pi\)
−0.954621 + 0.297824i \(0.903739\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) − 720.000i − 0.952172i −0.879399 0.476086i \(-0.842055\pi\)
0.879399 0.476086i \(-0.157945\pi\)
\(84\) 0 0
\(85\) − 187.061i − 0.238702i
\(86\) 0 0
\(87\) −488.438 −0.601909
\(88\) 0 0
\(89\) −1482.00 −1.76508 −0.882538 0.470242i \(-0.844167\pi\)
−0.882538 + 0.470242i \(0.844167\pi\)
\(90\) 0 0
\(91\) 1008.00i 1.16118i
\(92\) 0 0
\(93\) 176.669i 0.196986i
\(94\) 0 0
\(95\) −13.8564 −0.0149646
\(96\) 0 0
\(97\) 1822.00 1.90718 0.953588 0.301114i \(-0.0973586\pi\)
0.953588 + 0.301114i \(0.0973586\pi\)
\(98\) 0 0
\(99\) 432.000i 0.438562i
\(100\) 0 0
\(101\) − 911.059i − 0.897562i −0.893642 0.448781i \(-0.851858\pi\)
0.893642 0.448781i \(-0.148142\pi\)
\(102\) 0 0
\(103\) −453.797 −0.434116 −0.217058 0.976159i \(-0.569646\pi\)
−0.217058 + 0.976159i \(0.569646\pi\)
\(104\) 0 0
\(105\) 252.000 0.234216
\(106\) 0 0
\(107\) 1188.00i 1.07335i 0.843790 + 0.536674i \(0.180320\pi\)
−0.843790 + 0.536674i \(0.819680\pi\)
\(108\) 0 0
\(109\) 471.118i 0.413990i 0.978342 + 0.206995i \(0.0663684\pi\)
−0.978342 + 0.206995i \(0.933632\pi\)
\(110\) 0 0
\(111\) −976.877 −0.835325
\(112\) 0 0
\(113\) −390.000 −0.324674 −0.162337 0.986735i \(-0.551903\pi\)
−0.162337 + 0.986735i \(0.551903\pi\)
\(114\) 0 0
\(115\) − 600.000i − 0.486524i
\(116\) 0 0
\(117\) − 374.123i − 0.295621i
\(118\) 0 0
\(119\) 1309.43 1.00870
\(120\) 0 0
\(121\) −973.000 −0.731029
\(122\) 0 0
\(123\) − 882.000i − 0.646563i
\(124\) 0 0
\(125\) − 824.456i − 0.589933i
\(126\) 0 0
\(127\) 606.218 0.423568 0.211784 0.977317i \(-0.432073\pi\)
0.211784 + 0.977317i \(0.432073\pi\)
\(128\) 0 0
\(129\) 564.000 0.384941
\(130\) 0 0
\(131\) − 1380.00i − 0.920391i −0.887818 0.460195i \(-0.847779\pi\)
0.887818 0.460195i \(-0.152221\pi\)
\(132\) 0 0
\(133\) − 96.9948i − 0.0632370i
\(134\) 0 0
\(135\) −93.5307 −0.0596285
\(136\) 0 0
\(137\) −1158.00 −0.722150 −0.361075 0.932537i \(-0.617590\pi\)
−0.361075 + 0.932537i \(0.617590\pi\)
\(138\) 0 0
\(139\) − 1180.00i − 0.720045i −0.932944 0.360023i \(-0.882769\pi\)
0.932944 0.360023i \(-0.117231\pi\)
\(140\) 0 0
\(141\) 1517.28i 0.906225i
\(142\) 0 0
\(143\) 1995.32 1.16683
\(144\) 0 0
\(145\) 564.000 0.323018
\(146\) 0 0
\(147\) 735.000i 0.412393i
\(148\) 0 0
\(149\) 2171.99i 1.19420i 0.802165 + 0.597102i \(0.203681\pi\)
−0.802165 + 0.597102i \(0.796319\pi\)
\(150\) 0 0
\(151\) −142.028 −0.0765436 −0.0382718 0.999267i \(-0.512185\pi\)
−0.0382718 + 0.999267i \(0.512185\pi\)
\(152\) 0 0
\(153\) −486.000 −0.256802
\(154\) 0 0
\(155\) − 204.000i − 0.105714i
\(156\) 0 0
\(157\) 1337.14i 0.679717i 0.940476 + 0.339859i \(0.110379\pi\)
−0.940476 + 0.339859i \(0.889621\pi\)
\(158\) 0 0
\(159\) 2234.35 1.11443
\(160\) 0 0
\(161\) 4200.00 2.05594
\(162\) 0 0
\(163\) 1748.00i 0.839963i 0.907533 + 0.419981i \(0.137963\pi\)
−0.907533 + 0.419981i \(0.862037\pi\)
\(164\) 0 0
\(165\) − 498.831i − 0.235357i
\(166\) 0 0
\(167\) 13.8564 0.00642060 0.00321030 0.999995i \(-0.498978\pi\)
0.00321030 + 0.999995i \(0.498978\pi\)
\(168\) 0 0
\(169\) 469.000 0.213473
\(170\) 0 0
\(171\) 36.0000i 0.0160993i
\(172\) 0 0
\(173\) 599.290i 0.263371i 0.991292 + 0.131685i \(0.0420389\pi\)
−0.991292 + 0.131685i \(0.957961\pi\)
\(174\) 0 0
\(175\) 2740.10 1.18361
\(176\) 0 0
\(177\) 756.000 0.321042
\(178\) 0 0
\(179\) − 3228.00i − 1.34789i −0.738782 0.673944i \(-0.764599\pi\)
0.738782 0.673944i \(-0.235401\pi\)
\(180\) 0 0
\(181\) 2023.04i 0.830779i 0.909644 + 0.415390i \(0.136355\pi\)
−0.909644 + 0.415390i \(0.863645\pi\)
\(182\) 0 0
\(183\) 270.200 0.109146
\(184\) 0 0
\(185\) 1128.00 0.448282
\(186\) 0 0
\(187\) − 2592.00i − 1.01361i
\(188\) 0 0
\(189\) − 654.715i − 0.251976i
\(190\) 0 0
\(191\) −3477.96 −1.31757 −0.658786 0.752330i \(-0.728930\pi\)
−0.658786 + 0.752330i \(0.728930\pi\)
\(192\) 0 0
\(193\) −766.000 −0.285689 −0.142844 0.989745i \(-0.545625\pi\)
−0.142844 + 0.989745i \(0.545625\pi\)
\(194\) 0 0
\(195\) 432.000i 0.158647i
\(196\) 0 0
\(197\) 2899.45i 1.04862i 0.851529 + 0.524308i \(0.175676\pi\)
−0.851529 + 0.524308i \(0.824324\pi\)
\(198\) 0 0
\(199\) 1735.51 0.618228 0.309114 0.951025i \(-0.399968\pi\)
0.309114 + 0.951025i \(0.399968\pi\)
\(200\) 0 0
\(201\) −1884.00 −0.661130
\(202\) 0 0
\(203\) 3948.00i 1.36500i
\(204\) 0 0
\(205\) 1018.45i 0.346982i
\(206\) 0 0
\(207\) −1558.85 −0.523417
\(208\) 0 0
\(209\) −192.000 −0.0635451
\(210\) 0 0
\(211\) − 1100.00i − 0.358896i −0.983767 0.179448i \(-0.942569\pi\)
0.983767 0.179448i \(-0.0574312\pi\)
\(212\) 0 0
\(213\) − 20.7846i − 0.00668609i
\(214\) 0 0
\(215\) −651.251 −0.206581
\(216\) 0 0
\(217\) 1428.00 0.446723
\(218\) 0 0
\(219\) − 3018.00i − 0.931222i
\(220\) 0 0
\(221\) 2244.74i 0.683246i
\(222\) 0 0
\(223\) −391.443 −0.117547 −0.0587735 0.998271i \(-0.518719\pi\)
−0.0587735 + 0.998271i \(0.518719\pi\)
\(224\) 0 0
\(225\) −1017.00 −0.301333
\(226\) 0 0
\(227\) 3336.00i 0.975410i 0.873008 + 0.487705i \(0.162166\pi\)
−0.873008 + 0.487705i \(0.837834\pi\)
\(228\) 0 0
\(229\) − 5999.82i − 1.73135i −0.500605 0.865676i \(-0.666889\pi\)
0.500605 0.865676i \(-0.333111\pi\)
\(230\) 0 0
\(231\) 3491.81 0.994565
\(232\) 0 0
\(233\) 318.000 0.0894115 0.0447057 0.999000i \(-0.485765\pi\)
0.0447057 + 0.999000i \(0.485765\pi\)
\(234\) 0 0
\(235\) − 1752.00i − 0.486331i
\(236\) 0 0
\(237\) − 4021.82i − 1.10230i
\(238\) 0 0
\(239\) −859.097 −0.232512 −0.116256 0.993219i \(-0.537089\pi\)
−0.116256 + 0.993219i \(0.537089\pi\)
\(240\) 0 0
\(241\) −2710.00 −0.724342 −0.362171 0.932112i \(-0.617964\pi\)
−0.362171 + 0.932112i \(0.617964\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) − 848.705i − 0.221313i
\(246\) 0 0
\(247\) 166.277 0.0428338
\(248\) 0 0
\(249\) 2160.00 0.549737
\(250\) 0 0
\(251\) 5136.00i 1.29156i 0.763524 + 0.645780i \(0.223468\pi\)
−0.763524 + 0.645780i \(0.776532\pi\)
\(252\) 0 0
\(253\) − 8313.84i − 2.06596i
\(254\) 0 0
\(255\) 561.184 0.137815
\(256\) 0 0
\(257\) −4398.00 −1.06747 −0.533735 0.845652i \(-0.679212\pi\)
−0.533735 + 0.845652i \(0.679212\pi\)
\(258\) 0 0
\(259\) 7896.00i 1.89434i
\(260\) 0 0
\(261\) − 1465.31i − 0.347512i
\(262\) 0 0
\(263\) −6817.35 −1.59839 −0.799194 0.601073i \(-0.794740\pi\)
−0.799194 + 0.601073i \(0.794740\pi\)
\(264\) 0 0
\(265\) −2580.00 −0.598068
\(266\) 0 0
\(267\) − 4446.00i − 1.01907i
\(268\) 0 0
\(269\) 4624.58i 1.04820i 0.851657 + 0.524099i \(0.175598\pi\)
−0.851657 + 0.524099i \(0.824402\pi\)
\(270\) 0 0
\(271\) −3883.26 −0.870447 −0.435223 0.900322i \(-0.643331\pi\)
−0.435223 + 0.900322i \(0.643331\pi\)
\(272\) 0 0
\(273\) −3024.00 −0.670406
\(274\) 0 0
\(275\) − 5424.00i − 1.18938i
\(276\) 0 0
\(277\) 1524.20i 0.330616i 0.986242 + 0.165308i \(0.0528618\pi\)
−0.986242 + 0.165308i \(0.947138\pi\)
\(278\) 0 0
\(279\) −530.008 −0.113730
\(280\) 0 0
\(281\) 4398.00 0.933675 0.466838 0.884343i \(-0.345393\pi\)
0.466838 + 0.884343i \(0.345393\pi\)
\(282\) 0 0
\(283\) − 4372.00i − 0.918334i −0.888350 0.459167i \(-0.848148\pi\)
0.888350 0.459167i \(-0.151852\pi\)
\(284\) 0 0
\(285\) − 41.5692i − 0.00863982i
\(286\) 0 0
\(287\) −7129.12 −1.46627
\(288\) 0 0
\(289\) −1997.00 −0.406473
\(290\) 0 0
\(291\) 5466.00i 1.10111i
\(292\) 0 0
\(293\) − 3571.49i − 0.712111i −0.934465 0.356056i \(-0.884121\pi\)
0.934465 0.356056i \(-0.115879\pi\)
\(294\) 0 0
\(295\) −872.954 −0.172289
\(296\) 0 0
\(297\) −1296.00 −0.253204
\(298\) 0 0
\(299\) 7200.00i 1.39260i
\(300\) 0 0
\(301\) − 4558.76i − 0.872965i
\(302\) 0 0
\(303\) 2733.18 0.518207
\(304\) 0 0
\(305\) −312.000 −0.0585740
\(306\) 0 0
\(307\) − 4172.00i − 0.775598i −0.921744 0.387799i \(-0.873235\pi\)
0.921744 0.387799i \(-0.126765\pi\)
\(308\) 0 0
\(309\) − 1361.39i − 0.250637i
\(310\) 0 0
\(311\) −6470.94 −1.17985 −0.589925 0.807458i \(-0.700843\pi\)
−0.589925 + 0.807458i \(0.700843\pi\)
\(312\) 0 0
\(313\) −74.0000 −0.0133633 −0.00668167 0.999978i \(-0.502127\pi\)
−0.00668167 + 0.999978i \(0.502127\pi\)
\(314\) 0 0
\(315\) 756.000i 0.135225i
\(316\) 0 0
\(317\) 1964.15i 0.348004i 0.984745 + 0.174002i \(0.0556700\pi\)
−0.984745 + 0.174002i \(0.944330\pi\)
\(318\) 0 0
\(319\) 7815.01 1.37165
\(320\) 0 0
\(321\) −3564.00 −0.619698
\(322\) 0 0
\(323\) − 216.000i − 0.0372092i
\(324\) 0 0
\(325\) 4697.32i 0.801725i
\(326\) 0 0
\(327\) −1413.35 −0.239017
\(328\) 0 0
\(329\) 12264.0 2.05513
\(330\) 0 0
\(331\) 7556.00i 1.25473i 0.778726 + 0.627365i \(0.215866\pi\)
−0.778726 + 0.627365i \(0.784134\pi\)
\(332\) 0 0
\(333\) − 2930.63i − 0.482275i
\(334\) 0 0
\(335\) 2175.46 0.354800
\(336\) 0 0
\(337\) −4106.00 −0.663703 −0.331852 0.943332i \(-0.607673\pi\)
−0.331852 + 0.943332i \(0.607673\pi\)
\(338\) 0 0
\(339\) − 1170.00i − 0.187450i
\(340\) 0 0
\(341\) − 2826.71i − 0.448900i
\(342\) 0 0
\(343\) −2376.37 −0.374088
\(344\) 0 0
\(345\) 1800.00 0.280895
\(346\) 0 0
\(347\) 5256.00i 0.813132i 0.913621 + 0.406566i \(0.133274\pi\)
−0.913621 + 0.406566i \(0.866726\pi\)
\(348\) 0 0
\(349\) 10385.4i 1.59288i 0.604715 + 0.796442i \(0.293287\pi\)
−0.604715 + 0.796442i \(0.706713\pi\)
\(350\) 0 0
\(351\) 1122.37 0.170677
\(352\) 0 0
\(353\) −3942.00 −0.594367 −0.297183 0.954820i \(-0.596047\pi\)
−0.297183 + 0.954820i \(0.596047\pi\)
\(354\) 0 0
\(355\) 24.0000i 0.00358813i
\(356\) 0 0
\(357\) 3928.29i 0.582373i
\(358\) 0 0
\(359\) 6644.15 0.976782 0.488391 0.872625i \(-0.337584\pi\)
0.488391 + 0.872625i \(0.337584\pi\)
\(360\) 0 0
\(361\) 6843.00 0.997667
\(362\) 0 0
\(363\) − 2919.00i − 0.422060i
\(364\) 0 0
\(365\) 3484.89i 0.499746i
\(366\) 0 0
\(367\) −2906.38 −0.413384 −0.206692 0.978406i \(-0.566270\pi\)
−0.206692 + 0.978406i \(0.566270\pi\)
\(368\) 0 0
\(369\) 2646.00 0.373293
\(370\) 0 0
\(371\) − 18060.0i − 2.52730i
\(372\) 0 0
\(373\) − 10246.8i − 1.42241i −0.702983 0.711206i \(-0.748149\pi\)
0.702983 0.711206i \(-0.251851\pi\)
\(374\) 0 0
\(375\) 2473.37 0.340598
\(376\) 0 0
\(377\) −6768.00 −0.924588
\(378\) 0 0
\(379\) − 13844.0i − 1.87630i −0.346226 0.938151i \(-0.612537\pi\)
0.346226 0.938151i \(-0.387463\pi\)
\(380\) 0 0
\(381\) 1818.65i 0.244547i
\(382\) 0 0
\(383\) 7163.76 0.955747 0.477874 0.878429i \(-0.341408\pi\)
0.477874 + 0.878429i \(0.341408\pi\)
\(384\) 0 0
\(385\) −4032.00 −0.533740
\(386\) 0 0
\(387\) 1692.00i 0.222246i
\(388\) 0 0
\(389\) 12993.8i 1.69361i 0.531904 + 0.846805i \(0.321477\pi\)
−0.531904 + 0.846805i \(0.678523\pi\)
\(390\) 0 0
\(391\) 9353.07 1.20973
\(392\) 0 0
\(393\) 4140.00 0.531388
\(394\) 0 0
\(395\) 4644.00i 0.591557i
\(396\) 0 0
\(397\) − 117.779i − 0.0148896i −0.999972 0.00744481i \(-0.997630\pi\)
0.999972 0.00744481i \(-0.00236978\pi\)
\(398\) 0 0
\(399\) 290.985 0.0365099
\(400\) 0 0
\(401\) −5418.00 −0.674718 −0.337359 0.941376i \(-0.609534\pi\)
−0.337359 + 0.941376i \(0.609534\pi\)
\(402\) 0 0
\(403\) 2448.00i 0.302589i
\(404\) 0 0
\(405\) − 280.592i − 0.0344265i
\(406\) 0 0
\(407\) 15630.0 1.90357
\(408\) 0 0
\(409\) 11450.0 1.38427 0.692135 0.721768i \(-0.256670\pi\)
0.692135 + 0.721768i \(0.256670\pi\)
\(410\) 0 0
\(411\) − 3474.00i − 0.416934i
\(412\) 0 0
\(413\) − 6110.68i − 0.728055i
\(414\) 0 0
\(415\) −2494.15 −0.295020
\(416\) 0 0
\(417\) 3540.00 0.415718
\(418\) 0 0
\(419\) 1176.00i 0.137115i 0.997647 + 0.0685577i \(0.0218397\pi\)
−0.997647 + 0.0685577i \(0.978160\pi\)
\(420\) 0 0
\(421\) 10032.0i 1.16136i 0.814133 + 0.580679i \(0.197213\pi\)
−0.814133 + 0.580679i \(0.802787\pi\)
\(422\) 0 0
\(423\) −4551.83 −0.523209
\(424\) 0 0
\(425\) 6102.00 0.696448
\(426\) 0 0
\(427\) − 2184.00i − 0.247520i
\(428\) 0 0
\(429\) 5985.97i 0.673672i
\(430\) 0 0
\(431\) −838.313 −0.0936893 −0.0468447 0.998902i \(-0.514917\pi\)
−0.0468447 + 0.998902i \(0.514917\pi\)
\(432\) 0 0
\(433\) 4318.00 0.479237 0.239619 0.970867i \(-0.422978\pi\)
0.239619 + 0.970867i \(0.422978\pi\)
\(434\) 0 0
\(435\) 1692.00i 0.186495i
\(436\) 0 0
\(437\) − 692.820i − 0.0758400i
\(438\) 0 0
\(439\) 1610.81 0.175124 0.0875622 0.996159i \(-0.472092\pi\)
0.0875622 + 0.996159i \(0.472092\pi\)
\(440\) 0 0
\(441\) −2205.00 −0.238095
\(442\) 0 0
\(443\) 1032.00i 0.110681i 0.998468 + 0.0553406i \(0.0176245\pi\)
−0.998468 + 0.0553406i \(0.982376\pi\)
\(444\) 0 0
\(445\) 5133.80i 0.546889i
\(446\) 0 0
\(447\) −6515.98 −0.689474
\(448\) 0 0
\(449\) 726.000 0.0763075 0.0381537 0.999272i \(-0.487852\pi\)
0.0381537 + 0.999272i \(0.487852\pi\)
\(450\) 0 0
\(451\) 14112.0i 1.47341i
\(452\) 0 0
\(453\) − 426.084i − 0.0441925i
\(454\) 0 0
\(455\) 3491.81 0.359778
\(456\) 0 0
\(457\) 8666.00 0.887042 0.443521 0.896264i \(-0.353729\pi\)
0.443521 + 0.896264i \(0.353729\pi\)
\(458\) 0 0
\(459\) − 1458.00i − 0.148265i
\(460\) 0 0
\(461\) − 14684.3i − 1.48355i −0.670648 0.741776i \(-0.733984\pi\)
0.670648 0.741776i \(-0.266016\pi\)
\(462\) 0 0
\(463\) 4998.70 0.501748 0.250874 0.968020i \(-0.419282\pi\)
0.250874 + 0.968020i \(0.419282\pi\)
\(464\) 0 0
\(465\) 612.000 0.0610340
\(466\) 0 0
\(467\) − 16824.0i − 1.66707i −0.552466 0.833535i \(-0.686313\pi\)
0.552466 0.833535i \(-0.313687\pi\)
\(468\) 0 0
\(469\) 15228.2i 1.49930i
\(470\) 0 0
\(471\) −4011.43 −0.392435
\(472\) 0 0
\(473\) −9024.00 −0.877218
\(474\) 0 0
\(475\) − 452.000i − 0.0436614i
\(476\) 0 0
\(477\) 6703.04i 0.643419i
\(478\) 0 0
\(479\) 10953.5 1.04484 0.522419 0.852689i \(-0.325030\pi\)
0.522419 + 0.852689i \(0.325030\pi\)
\(480\) 0 0
\(481\) −13536.0 −1.28314
\(482\) 0 0
\(483\) 12600.0i 1.18700i
\(484\) 0 0
\(485\) − 6311.59i − 0.590917i
\(486\) 0 0
\(487\) −10714.5 −0.996959 −0.498479 0.866902i \(-0.666108\pi\)
−0.498479 + 0.866902i \(0.666108\pi\)
\(488\) 0 0
\(489\) −5244.00 −0.484953
\(490\) 0 0
\(491\) − 852.000i − 0.0783100i −0.999233 0.0391550i \(-0.987533\pi\)
0.999233 0.0391550i \(-0.0124666\pi\)
\(492\) 0 0
\(493\) 8791.89i 0.803178i
\(494\) 0 0
\(495\) 1496.49 0.135883
\(496\) 0 0
\(497\) −168.000 −0.0151626
\(498\) 0 0
\(499\) − 11156.0i − 1.00082i −0.865787 0.500412i \(-0.833182\pi\)
0.865787 0.500412i \(-0.166818\pi\)
\(500\) 0 0
\(501\) 41.5692i 0.00370694i
\(502\) 0 0
\(503\) −14999.6 −1.32962 −0.664808 0.747014i \(-0.731487\pi\)
−0.664808 + 0.747014i \(0.731487\pi\)
\(504\) 0 0
\(505\) −3156.00 −0.278099
\(506\) 0 0
\(507\) 1407.00i 0.123249i
\(508\) 0 0
\(509\) − 9287.26i − 0.808743i −0.914595 0.404372i \(-0.867490\pi\)
0.914595 0.404372i \(-0.132510\pi\)
\(510\) 0 0
\(511\) −24394.2 −2.11181
\(512\) 0 0
\(513\) −108.000 −0.00929496
\(514\) 0 0
\(515\) 1572.00i 0.134506i
\(516\) 0 0
\(517\) − 24276.4i − 2.06514i
\(518\) 0 0
\(519\) −1797.87 −0.152057
\(520\) 0 0
\(521\) −2766.00 −0.232592 −0.116296 0.993215i \(-0.537102\pi\)
−0.116296 + 0.993215i \(0.537102\pi\)
\(522\) 0 0
\(523\) 18988.0i 1.58755i 0.608213 + 0.793774i \(0.291887\pi\)
−0.608213 + 0.793774i \(0.708113\pi\)
\(524\) 0 0
\(525\) 8220.31i 0.683360i
\(526\) 0 0
\(527\) 3180.05 0.262856
\(528\) 0 0
\(529\) 17833.0 1.46569
\(530\) 0 0
\(531\) 2268.00i 0.185354i
\(532\) 0 0
\(533\) − 12221.4i − 0.993181i
\(534\) 0 0
\(535\) 4115.35 0.332565
\(536\) 0 0
\(537\) 9684.00 0.778204
\(538\) 0 0
\(539\) − 11760.0i − 0.939776i
\(540\) 0 0
\(541\) − 12997.3i − 1.03290i −0.856318 0.516449i \(-0.827254\pi\)
0.856318 0.516449i \(-0.172746\pi\)
\(542\) 0 0
\(543\) −6069.11 −0.479651
\(544\) 0 0
\(545\) 1632.00 0.128270
\(546\) 0 0
\(547\) 21188.0i 1.65619i 0.560591 + 0.828093i \(0.310574\pi\)
−0.560591 + 0.828093i \(0.689426\pi\)
\(548\) 0 0
\(549\) 810.600i 0.0630156i
\(550\) 0 0
\(551\) 651.251 0.0503525
\(552\) 0 0
\(553\) −32508.0 −2.49978
\(554\) 0 0
\(555\) 3384.00i 0.258816i
\(556\) 0 0
\(557\) − 12231.7i − 0.930477i −0.885185 0.465238i \(-0.845969\pi\)
0.885185 0.465238i \(-0.154031\pi\)
\(558\) 0 0
\(559\) 7815.01 0.591306
\(560\) 0 0
\(561\) 7776.00 0.585210
\(562\) 0 0
\(563\) 504.000i 0.0377284i 0.999822 + 0.0188642i \(0.00600501\pi\)
−0.999822 + 0.0188642i \(0.993995\pi\)
\(564\) 0 0
\(565\) 1351.00i 0.100596i
\(566\) 0 0
\(567\) 1964.15 0.145479
\(568\) 0 0
\(569\) −20358.0 −1.49992 −0.749958 0.661486i \(-0.769926\pi\)
−0.749958 + 0.661486i \(0.769926\pi\)
\(570\) 0 0
\(571\) − 13300.0i − 0.974760i −0.873190 0.487380i \(-0.837953\pi\)
0.873190 0.487380i \(-0.162047\pi\)
\(572\) 0 0
\(573\) − 10433.9i − 0.760700i
\(574\) 0 0
\(575\) 19572.2 1.41951
\(576\) 0 0
\(577\) 4606.00 0.332323 0.166161 0.986099i \(-0.446863\pi\)
0.166161 + 0.986099i \(0.446863\pi\)
\(578\) 0 0
\(579\) − 2298.00i − 0.164942i
\(580\) 0 0
\(581\) − 17459.1i − 1.24669i
\(582\) 0 0
\(583\) −35749.5 −2.53961
\(584\) 0 0
\(585\) −1296.00 −0.0915949
\(586\) 0 0
\(587\) 13980.0i 0.982992i 0.870880 + 0.491496i \(0.163550\pi\)
−0.870880 + 0.491496i \(0.836450\pi\)
\(588\) 0 0
\(589\) − 235.559i − 0.0164788i
\(590\) 0 0
\(591\) −8698.36 −0.605419
\(592\) 0 0
\(593\) −12486.0 −0.864652 −0.432326 0.901717i \(-0.642307\pi\)
−0.432326 + 0.901717i \(0.642307\pi\)
\(594\) 0 0
\(595\) − 4536.00i − 0.312534i
\(596\) 0 0
\(597\) 5206.54i 0.356934i
\(598\) 0 0
\(599\) 8778.03 0.598766 0.299383 0.954133i \(-0.403219\pi\)
0.299383 + 0.954133i \(0.403219\pi\)
\(600\) 0 0
\(601\) 6986.00 0.474151 0.237076 0.971491i \(-0.423811\pi\)
0.237076 + 0.971491i \(0.423811\pi\)
\(602\) 0 0
\(603\) − 5652.00i − 0.381704i
\(604\) 0 0
\(605\) 3370.57i 0.226501i
\(606\) 0 0
\(607\) 4596.86 0.307382 0.153691 0.988119i \(-0.450884\pi\)
0.153691 + 0.988119i \(0.450884\pi\)
\(608\) 0 0
\(609\) −11844.0 −0.788084
\(610\) 0 0
\(611\) 21024.0i 1.39205i
\(612\) 0 0
\(613\) 11092.1i 0.730838i 0.930843 + 0.365419i \(0.119074\pi\)
−0.930843 + 0.365419i \(0.880926\pi\)
\(614\) 0 0
\(615\) −3055.34 −0.200330
\(616\) 0 0
\(617\) −2850.00 −0.185959 −0.0929795 0.995668i \(-0.529639\pi\)
−0.0929795 + 0.995668i \(0.529639\pi\)
\(618\) 0 0
\(619\) − 20116.0i − 1.30619i −0.757277 0.653094i \(-0.773471\pi\)
0.757277 0.653094i \(-0.226529\pi\)
\(620\) 0 0
\(621\) − 4676.54i − 0.302195i
\(622\) 0 0
\(623\) −35936.6 −2.31103
\(624\) 0 0
\(625\) 11269.0 0.721216
\(626\) 0 0
\(627\) − 576.000i − 0.0366878i
\(628\) 0 0
\(629\) 17583.8i 1.11464i
\(630\) 0 0
\(631\) 7271.15 0.458732 0.229366 0.973340i \(-0.426335\pi\)
0.229366 + 0.973340i \(0.426335\pi\)
\(632\) 0 0
\(633\) 3300.00 0.207209
\(634\) 0 0
\(635\) − 2100.00i − 0.131238i
\(636\) 0 0
\(637\) 10184.5i 0.633474i
\(638\) 0 0
\(639\) 62.3538 0.00386022
\(640\) 0 0
\(641\) 7230.00 0.445504 0.222752 0.974875i \(-0.428496\pi\)
0.222752 + 0.974875i \(0.428496\pi\)
\(642\) 0 0
\(643\) 2948.00i 0.180805i 0.995905 + 0.0904026i \(0.0288154\pi\)
−0.995905 + 0.0904026i \(0.971185\pi\)
\(644\) 0 0
\(645\) − 1953.75i − 0.119270i
\(646\) 0 0
\(647\) 17161.2 1.04277 0.521387 0.853320i \(-0.325415\pi\)
0.521387 + 0.853320i \(0.325415\pi\)
\(648\) 0 0
\(649\) −12096.0 −0.731602
\(650\) 0 0
\(651\) 4284.00i 0.257916i
\(652\) 0 0
\(653\) 10381.9i 0.622168i 0.950382 + 0.311084i \(0.100692\pi\)
−0.950382 + 0.311084i \(0.899308\pi\)
\(654\) 0 0
\(655\) −4780.46 −0.285173
\(656\) 0 0
\(657\) 9054.00 0.537641
\(658\) 0 0
\(659\) − 10308.0i − 0.609321i −0.952461 0.304661i \(-0.901457\pi\)
0.952461 0.304661i \(-0.0985430\pi\)
\(660\) 0 0
\(661\) − 15803.2i − 0.929916i −0.885333 0.464958i \(-0.846069\pi\)
0.885333 0.464958i \(-0.153931\pi\)
\(662\) 0 0
\(663\) −6734.21 −0.394472
\(664\) 0 0
\(665\) −336.000 −0.0195933
\(666\) 0 0
\(667\) 28200.0i 1.63704i
\(668\) 0 0
\(669\) − 1174.33i − 0.0678658i
\(670\) 0 0
\(671\) −4323.20 −0.248726
\(672\) 0 0
\(673\) 30910.0 1.77042 0.885210 0.465191i \(-0.154014\pi\)
0.885210 + 0.465191i \(0.154014\pi\)
\(674\) 0 0
\(675\) − 3051.00i − 0.173975i
\(676\) 0 0
\(677\) − 14802.1i − 0.840312i −0.907452 0.420156i \(-0.861975\pi\)
0.907452 0.420156i \(-0.138025\pi\)
\(678\) 0 0
\(679\) 44181.2 2.49708
\(680\) 0 0
\(681\) −10008.0 −0.563153
\(682\) 0 0
\(683\) − 528.000i − 0.0295803i −0.999891 0.0147902i \(-0.995292\pi\)
0.999891 0.0147902i \(-0.00470803\pi\)
\(684\) 0 0
\(685\) 4011.43i 0.223750i
\(686\) 0 0
\(687\) 17999.5 0.999596
\(688\) 0 0
\(689\) 30960.0 1.71188
\(690\) 0 0
\(691\) − 9052.00i − 0.498342i −0.968460 0.249171i \(-0.919842\pi\)
0.968460 0.249171i \(-0.0801581\pi\)
\(692\) 0 0
\(693\) 10475.4i 0.574212i
\(694\) 0 0
\(695\) −4087.64 −0.223098
\(696\) 0 0
\(697\) −15876.0 −0.862764
\(698\) 0 0
\(699\) 954.000i 0.0516217i
\(700\) 0 0
\(701\) 32600.7i 1.75650i 0.478197 + 0.878252i \(0.341290\pi\)
−0.478197 + 0.878252i \(0.658710\pi\)
\(702\) 0 0
\(703\) 1302.50 0.0698788
\(704\) 0 0
\(705\) 5256.00 0.280784
\(706\) 0 0
\(707\) − 22092.0i − 1.17518i
\(708\) 0 0
\(709\) 27227.8i 1.44226i 0.692799 + 0.721130i \(0.256377\pi\)
−0.692799 + 0.721130i \(0.743623\pi\)
\(710\) 0 0
\(711\) 12065.5 0.636414
\(712\) 0 0
\(713\) 10200.0 0.535755
\(714\) 0 0
\(715\) − 6912.00i − 0.361530i
\(716\) 0 0
\(717\) − 2577.29i − 0.134241i
\(718\) 0 0
\(719\) 685.892 0.0355764 0.0177882 0.999842i \(-0.494338\pi\)
0.0177882 + 0.999842i \(0.494338\pi\)
\(720\) 0 0
\(721\) −11004.0 −0.568392
\(722\) 0 0
\(723\) − 8130.00i − 0.418199i
\(724\) 0 0
\(725\) 18397.8i 0.942453i
\(726\) 0 0
\(727\) −20192.2 −1.03011 −0.515054 0.857158i \(-0.672228\pi\)
−0.515054 + 0.857158i \(0.672228\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) − 10152.0i − 0.513660i
\(732\) 0 0
\(733\) − 35236.8i − 1.77558i −0.460246 0.887792i \(-0.652239\pi\)
0.460246 0.887792i \(-0.347761\pi\)
\(734\) 0 0
\(735\) 2546.11 0.127775
\(736\) 0 0
\(737\) 30144.0 1.50661
\(738\) 0 0
\(739\) − 13940.0i − 0.693899i −0.937884 0.346949i \(-0.887218\pi\)
0.937884 0.346949i \(-0.112782\pi\)
\(740\) 0 0
\(741\) 498.831i 0.0247301i
\(742\) 0 0
\(743\) −11002.0 −0.543235 −0.271618 0.962405i \(-0.587559\pi\)
−0.271618 + 0.962405i \(0.587559\pi\)
\(744\) 0 0
\(745\) 7524.00 0.370011
\(746\) 0 0
\(747\) 6480.00i 0.317391i
\(748\) 0 0
\(749\) 28807.5i 1.40534i
\(750\) 0 0
\(751\) 33342.0 1.62006 0.810031 0.586388i \(-0.199450\pi\)
0.810031 + 0.586388i \(0.199450\pi\)
\(752\) 0 0
\(753\) −15408.0 −0.745682
\(754\) 0 0
\(755\) 492.000i 0.0237162i
\(756\) 0 0
\(757\) 17445.2i 0.837592i 0.908080 + 0.418796i \(0.137548\pi\)
−0.908080 + 0.418796i \(0.862452\pi\)
\(758\) 0 0
\(759\) 24941.5 1.19278
\(760\) 0 0
\(761\) −30222.0 −1.43961 −0.719807 0.694174i \(-0.755770\pi\)
−0.719807 + 0.694174i \(0.755770\pi\)
\(762\) 0 0
\(763\) 11424.0i 0.542040i
\(764\) 0 0
\(765\) 1683.55i 0.0795673i
\(766\) 0 0
\(767\) 10475.4 0.493150
\(768\) 0 0
\(769\) −11758.0 −0.551371 −0.275686 0.961248i \(-0.588905\pi\)
−0.275686 + 0.961248i \(0.588905\pi\)
\(770\) 0 0
\(771\) − 13194.0i − 0.616304i
\(772\) 0 0
\(773\) 1874.08i 0.0872004i 0.999049 + 0.0436002i \(0.0138828\pi\)
−0.999049 + 0.0436002i \(0.986117\pi\)
\(774\) 0 0
\(775\) 6654.54 0.308436
\(776\) 0 0
\(777\) −23688.0 −1.09370
\(778\) 0 0
\(779\) 1176.00i 0.0540880i
\(780\) 0 0
\(781\) 332.554i 0.0152365i
\(782\) 0 0
\(783\) 4395.94 0.200636
\(784\) 0 0
\(785\) 4632.00 0.210603
\(786\) 0 0
\(787\) − 31012.0i − 1.40465i −0.711857 0.702324i \(-0.752146\pi\)
0.711857 0.702324i \(-0.247854\pi\)
\(788\) 0 0
\(789\) − 20452.1i − 0.922829i
\(790\) 0 0
\(791\) −9457.00 −0.425097
\(792\) 0 0
\(793\) 3744.00 0.167659
\(794\) 0 0
\(795\) − 7740.00i − 0.345295i
\(796\) 0 0
\(797\) − 7091.02i − 0.315153i −0.987507 0.157576i \(-0.949632\pi\)
0.987507 0.157576i \(-0.0503680\pi\)
\(798\) 0 0
\(799\) 27311.0 1.20925
\(800\) 0 0
\(801\) 13338.0 0.588358
\(802\) 0 0
\(803\) 48288.0i 2.12210i
\(804\) 0 0
\(805\) − 14549.2i − 0.637010i
\(806\) 0 0
\(807\) −13873.7 −0.605178
\(808\) 0 0
\(809\) 40650.0 1.76660 0.883299 0.468810i \(-0.155317\pi\)
0.883299 + 0.468810i \(0.155317\pi\)
\(810\) 0 0
\(811\) − 8372.00i − 0.362492i −0.983438 0.181246i \(-0.941987\pi\)
0.983438 0.181246i \(-0.0580130\pi\)
\(812\) 0 0
\(813\) − 11649.8i − 0.502553i
\(814\) 0 0
\(815\) 6055.25 0.260253
\(816\) 0 0
\(817\) −752.000 −0.0322021
\(818\) 0 0
\(819\) − 9072.00i − 0.387059i
\(820\) 0 0
\(821\) 9370.39i 0.398330i 0.979966 + 0.199165i \(0.0638230\pi\)
−0.979966 + 0.199165i \(0.936177\pi\)
\(822\) 0 0
\(823\) 21668.0 0.917737 0.458868 0.888504i \(-0.348255\pi\)
0.458868 + 0.888504i \(0.348255\pi\)
\(824\) 0 0
\(825\) 16272.0 0.686689
\(826\) 0 0
\(827\) − 6684.00i − 0.281046i −0.990077 0.140523i \(-0.955122\pi\)
0.990077 0.140523i \(-0.0448785\pi\)
\(828\) 0 0
\(829\) 24359.6i 1.02056i 0.860009 + 0.510279i \(0.170458\pi\)
−0.860009 + 0.510279i \(0.829542\pi\)
\(830\) 0 0
\(831\) −4572.61 −0.190881
\(832\) 0 0
\(833\) 13230.0 0.550291
\(834\) 0 0
\(835\) − 48.0000i − 0.00198935i
\(836\) 0 0
\(837\) − 1590.02i − 0.0656622i
\(838\) 0 0
\(839\) −19613.7 −0.807082 −0.403541 0.914962i \(-0.632221\pi\)
−0.403541 + 0.914962i \(0.632221\pi\)
\(840\) 0 0
\(841\) −2119.00 −0.0868834
\(842\) 0 0
\(843\) 13194.0i 0.539058i
\(844\) 0 0
\(845\) − 1624.66i − 0.0661422i
\(846\) 0 0
\(847\) −23594.0 −0.957142
\(848\) 0 0
\(849\) 13116.0 0.530200
\(850\) 0 0
\(851\) 56400.0i 2.27188i
\(852\) 0 0
\(853\) − 20569.8i − 0.825671i −0.910806 0.412836i \(-0.864538\pi\)
0.910806 0.412836i \(-0.135462\pi\)
\(854\) 0 0
\(855\) 124.708 0.00498820
\(856\) 0 0
\(857\) −27222.0 −1.08505 −0.542524 0.840040i \(-0.682531\pi\)
−0.542524 + 0.840040i \(0.682531\pi\)
\(858\) 0 0
\(859\) − 3548.00i − 0.140927i −0.997514 0.0704634i \(-0.977552\pi\)
0.997514 0.0704634i \(-0.0224478\pi\)
\(860\) 0 0
\(861\) − 21387.4i − 0.846550i
\(862\) 0 0
\(863\) −45047.2 −1.77685 −0.888426 0.459019i \(-0.848201\pi\)
−0.888426 + 0.459019i \(0.848201\pi\)
\(864\) 0 0
\(865\) 2076.00 0.0816024
\(866\) 0 0
\(867\) − 5991.00i − 0.234677i
\(868\) 0 0
\(869\) 64349.2i 2.51196i
\(870\) 0 0
\(871\) −26105.5 −1.01556
\(872\) 0 0
\(873\) −16398.0 −0.635725
\(874\) 0 0
\(875\) − 19992.0i − 0.772403i
\(876\) 0 0
\(877\) − 29022.2i − 1.11746i −0.829350 0.558729i \(-0.811289\pi\)
0.829350 0.558729i \(-0.188711\pi\)
\(878\) 0 0
\(879\) 10714.5 0.411138
\(880\) 0 0
\(881\) −48318.0 −1.84776 −0.923879 0.382685i \(-0.875000\pi\)
−0.923879 + 0.382685i \(0.875000\pi\)
\(882\) 0 0
\(883\) − 14380.0i − 0.548047i −0.961723 0.274024i \(-0.911645\pi\)
0.961723 0.274024i \(-0.0883546\pi\)
\(884\) 0 0
\(885\) − 2618.86i − 0.0994712i
\(886\) 0 0
\(887\) −34086.8 −1.29033 −0.645164 0.764044i \(-0.723211\pi\)
−0.645164 + 0.764044i \(0.723211\pi\)
\(888\) 0 0
\(889\) 14700.0 0.554581
\(890\) 0 0
\(891\) − 3888.00i − 0.146187i
\(892\) 0 0
\(893\) − 2023.04i − 0.0758100i
\(894\) 0 0
\(895\) −11182.1 −0.417628
\(896\) 0 0
\(897\) −21600.0 −0.804017
\(898\) 0 0
\(899\) 9588.00i 0.355704i
\(900\) 0 0
\(901\) − 40218.2i − 1.48708i
\(902\) 0 0
\(903\) 13676.3 0.504007
\(904\) 0 0
\(905\) 7008.00 0.257408
\(906\) 0 0
\(907\) 31252.0i 1.14411i 0.820216 + 0.572054i \(0.193853\pi\)
−0.820216 + 0.572054i \(0.806147\pi\)
\(908\) 0 0
\(909\) 8199.53i 0.299187i
\(910\) 0 0
\(911\) 13080.4 0.475713 0.237857 0.971300i \(-0.423555\pi\)
0.237857 + 0.971300i \(0.423555\pi\)
\(912\) 0 0
\(913\) −34560.0 −1.25276
\(914\) 0 0
\(915\) − 936.000i − 0.0338177i
\(916\) 0 0
\(917\) − 33463.2i − 1.20507i
\(918\) 0 0
\(919\) −8843.85 −0.317445 −0.158722 0.987323i \(-0.550737\pi\)
−0.158722 + 0.987323i \(0.550737\pi\)
\(920\) 0 0
\(921\) 12516.0 0.447792
\(922\) 0 0
\(923\) − 288.000i − 0.0102705i
\(924\) 0 0
\(925\) 36795.7i 1.30793i
\(926\) 0 0
\(927\) 4084.18 0.144705
\(928\) 0 0
\(929\) 29622.0 1.04614 0.523071 0.852289i \(-0.324786\pi\)
0.523071 + 0.852289i \(0.324786\pi\)
\(930\) 0 0
\(931\) − 980.000i − 0.0344986i
\(932\) 0 0
\(933\) − 19412.8i − 0.681187i
\(934\) 0 0
\(935\) −8978.95 −0.314057
\(936\) 0 0
\(937\) −23210.0 −0.809218 −0.404609 0.914490i \(-0.632592\pi\)
−0.404609 + 0.914490i \(0.632592\pi\)
\(938\) 0 0
\(939\) − 222.000i − 0.00771533i
\(940\) 0 0
\(941\) − 19728.1i − 0.683439i −0.939802 0.341720i \(-0.888991\pi\)
0.939802 0.341720i \(-0.111009\pi\)
\(942\) 0 0
\(943\) −50922.3 −1.75849
\(944\) 0 0
\(945\) −2268.00 −0.0780720
\(946\) 0 0
\(947\) 1236.00i 0.0424125i 0.999775 + 0.0212062i \(0.00675066\pi\)
−0.999775 + 0.0212062i \(0.993249\pi\)
\(948\) 0 0
\(949\) − 41818.6i − 1.43044i
\(950\) 0 0
\(951\) −5892.44 −0.200920
\(952\) 0 0
\(953\) 18402.0 0.625498 0.312749 0.949836i \(-0.398750\pi\)
0.312749 + 0.949836i \(0.398750\pi\)
\(954\) 0 0
\(955\) 12048.0i 0.408235i
\(956\) 0 0
\(957\) 23445.0i 0.791923i
\(958\) 0 0
\(959\) −28080.0 −0.945517
\(960\) 0 0
\(961\) −26323.0 −0.883589
\(962\) 0 0
\(963\) − 10692.0i − 0.357783i
\(964\) 0 0
\(965\) 2653.50i 0.0885174i
\(966\) 0 0
\(967\) 41836.0 1.39127 0.695633 0.718398i \(-0.255124\pi\)
0.695633 + 0.718398i \(0.255124\pi\)
\(968\) 0 0
\(969\) 648.000 0.0214827
\(970\) 0 0
\(971\) 8832.00i 0.291897i 0.989292 + 0.145949i \(0.0466234\pi\)
−0.989292 + 0.145949i \(0.953377\pi\)
\(972\) 0 0
\(973\) − 28613.5i − 0.942761i
\(974\) 0 0
\(975\) −14092.0 −0.462876
\(976\) 0 0
\(977\) −23034.0 −0.754271 −0.377136 0.926158i \(-0.623091\pi\)
−0.377136 + 0.926158i \(0.623091\pi\)
\(978\) 0 0
\(979\) 71136.0i 2.32228i
\(980\) 0 0
\(981\) − 4240.06i − 0.137997i
\(982\) 0 0
\(983\) 17791.6 0.577278 0.288639 0.957438i \(-0.406797\pi\)
0.288639 + 0.957438i \(0.406797\pi\)
\(984\) 0 0
\(985\) 10044.0 0.324902
\(986\) 0 0
\(987\) 36792.0i 1.18653i
\(988\) 0 0
\(989\) − 32562.6i − 1.04695i
\(990\) 0 0
\(991\) −3072.66 −0.0984926 −0.0492463 0.998787i \(-0.515682\pi\)
−0.0492463 + 0.998787i \(0.515682\pi\)
\(992\) 0 0
\(993\) −22668.0 −0.724418
\(994\) 0 0
\(995\) − 6012.00i − 0.191551i
\(996\) 0 0
\(997\) 52273.3i 1.66049i 0.557396 + 0.830247i \(0.311800\pi\)
−0.557396 + 0.830247i \(0.688200\pi\)
\(998\) 0 0
\(999\) 8791.89 0.278442
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 192.4.d.c.97.3 yes 4
3.2 odd 2 576.4.d.c.289.4 4
4.3 odd 2 inner 192.4.d.c.97.1 4
8.3 odd 2 inner 192.4.d.c.97.4 yes 4
8.5 even 2 inner 192.4.d.c.97.2 yes 4
12.11 even 2 576.4.d.c.289.3 4
16.3 odd 4 768.4.a.o.1.1 2
16.5 even 4 768.4.a.o.1.2 2
16.11 odd 4 768.4.a.f.1.2 2
16.13 even 4 768.4.a.f.1.1 2
24.5 odd 2 576.4.d.c.289.2 4
24.11 even 2 576.4.d.c.289.1 4
48.5 odd 4 2304.4.a.x.1.1 2
48.11 even 4 2304.4.a.bk.1.1 2
48.29 odd 4 2304.4.a.bk.1.2 2
48.35 even 4 2304.4.a.x.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
192.4.d.c.97.1 4 4.3 odd 2 inner
192.4.d.c.97.2 yes 4 8.5 even 2 inner
192.4.d.c.97.3 yes 4 1.1 even 1 trivial
192.4.d.c.97.4 yes 4 8.3 odd 2 inner
576.4.d.c.289.1 4 24.11 even 2
576.4.d.c.289.2 4 24.5 odd 2
576.4.d.c.289.3 4 12.11 even 2
576.4.d.c.289.4 4 3.2 odd 2
768.4.a.f.1.1 2 16.13 even 4
768.4.a.f.1.2 2 16.11 odd 4
768.4.a.o.1.1 2 16.3 odd 4
768.4.a.o.1.2 2 16.5 even 4
2304.4.a.x.1.1 2 48.5 odd 4
2304.4.a.x.1.2 2 48.35 even 4
2304.4.a.bk.1.1 2 48.11 even 4
2304.4.a.bk.1.2 2 48.29 odd 4