Properties

Label 192.2.d.a.97.3
Level $192$
Weight $2$
Character 192.97
Analytic conductor $1.533$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(1.53312771881\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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 192.2.d.a.97.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} -3.46410i q^{5} +3.46410 q^{7} -1.00000 q^{9} +3.46410 q^{15} +6.00000 q^{17} +4.00000i q^{19} +3.46410i q^{21} -6.92820 q^{23} -7.00000 q^{25} -1.00000i q^{27} -3.46410i q^{29} -3.46410 q^{31} -12.0000i q^{35} +6.92820i q^{37} -6.00000 q^{41} -4.00000i q^{43} +3.46410i q^{45} -6.92820 q^{47} +5.00000 q^{49} +6.00000i q^{51} +3.46410i q^{53} -4.00000 q^{57} +12.0000i q^{59} +6.92820i q^{61} -3.46410 q^{63} -4.00000i q^{67} -6.92820i q^{69} +6.92820 q^{71} +2.00000 q^{73} -7.00000i q^{75} +10.3923 q^{79} +1.00000 q^{81} -20.7846i q^{85} +3.46410 q^{87} +6.00000 q^{89} -3.46410i q^{93} +13.8564 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} + 24 q^{17} - 28 q^{25} - 24 q^{41} + 20 q^{49} - 16 q^{57} + 8 q^{73} + 4 q^{81} + 24 q^{89} - 8 q^{97}+O(q^{100}) \) 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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) − 3.46410i − 1.54919i −0.632456 0.774597i \(-0.717953\pi\)
0.632456 0.774597i \(-0.282047\pi\)
\(6\) 0 0
\(7\) 3.46410 1.30931 0.654654 0.755929i \(-0.272814\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 3.46410 0.894427
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 3.46410i 0.755929i
\(22\) 0 0
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) − 3.46410i − 0.643268i −0.946864 0.321634i \(-0.895768\pi\)
0.946864 0.321634i \(-0.104232\pi\)
\(30\) 0 0
\(31\) −3.46410 −0.622171 −0.311086 0.950382i \(-0.600693\pi\)
−0.311086 + 0.950382i \(0.600693\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 12.0000i − 2.02837i
\(36\) 0 0
\(37\) 6.92820i 1.13899i 0.821995 + 0.569495i \(0.192861\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 3.46410i 0.516398i
\(46\) 0 0
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 6.00000i 0.840168i
\(52\) 0 0
\(53\) 3.46410i 0.475831i 0.971286 + 0.237915i \(0.0764641\pi\)
−0.971286 + 0.237915i \(0.923536\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 12.0000i 1.56227i 0.624364 + 0.781133i \(0.285358\pi\)
−0.624364 + 0.781133i \(0.714642\pi\)
\(60\) 0 0
\(61\) 6.92820i 0.887066i 0.896258 + 0.443533i \(0.146275\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) −3.46410 −0.436436
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) − 6.92820i − 0.834058i
\(70\) 0 0
\(71\) 6.92820 0.822226 0.411113 0.911584i \(-0.365140\pi\)
0.411113 + 0.911584i \(0.365140\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) − 7.00000i − 0.808290i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.3923 1.16923 0.584613 0.811312i \(-0.301246\pi\)
0.584613 + 0.811312i \(0.301246\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) − 20.7846i − 2.25441i
\(86\) 0 0
\(87\) 3.46410 0.371391
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 3.46410i − 0.359211i
\(94\) 0 0
\(95\) 13.8564 1.42164
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.46410i 0.344691i 0.985037 + 0.172345i \(0.0551346\pi\)
−0.985037 + 0.172345i \(0.944865\pi\)
\(102\) 0 0
\(103\) −17.3205 −1.70664 −0.853320 0.521387i \(-0.825415\pi\)
−0.853320 + 0.521387i \(0.825415\pi\)
\(104\) 0 0
\(105\) 12.0000 1.17108
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) − 13.8564i − 1.32720i −0.748086 0.663602i \(-0.769027\pi\)
0.748086 0.663602i \(-0.230973\pi\)
\(110\) 0 0
\(111\) −6.92820 −0.657596
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 24.0000i 2.23801i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 20.7846 1.90532
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) − 6.00000i − 0.541002i
\(124\) 0 0
\(125\) 6.92820i 0.619677i
\(126\) 0 0
\(127\) 3.46410 0.307389 0.153695 0.988118i \(-0.450883\pi\)
0.153695 + 0.988118i \(0.450883\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) − 12.0000i − 1.04844i −0.851581 0.524222i \(-0.824356\pi\)
0.851581 0.524222i \(-0.175644\pi\)
\(132\) 0 0
\(133\) 13.8564i 1.20150i
\(134\) 0 0
\(135\) −3.46410 −0.298142
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) − 20.0000i − 1.69638i −0.529694 0.848189i \(-0.677693\pi\)
0.529694 0.848189i \(-0.322307\pi\)
\(140\) 0 0
\(141\) − 6.92820i − 0.583460i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 0 0
\(147\) 5.00000i 0.412393i
\(148\) 0 0
\(149\) 10.3923i 0.851371i 0.904871 + 0.425685i \(0.139967\pi\)
−0.904871 + 0.425685i \(0.860033\pi\)
\(150\) 0 0
\(151\) 3.46410 0.281905 0.140952 0.990016i \(-0.454984\pi\)
0.140952 + 0.990016i \(0.454984\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 12.0000i 0.963863i
\(156\) 0 0
\(157\) − 6.92820i − 0.552931i −0.961024 0.276465i \(-0.910837\pi\)
0.961024 0.276465i \(-0.0891631\pi\)
\(158\) 0 0
\(159\) −3.46410 −0.274721
\(160\) 0 0
\(161\) −24.0000 −1.89146
\(162\) 0 0
\(163\) − 20.0000i − 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −13.8564 −1.07224 −0.536120 0.844141i \(-0.680111\pi\)
−0.536120 + 0.844141i \(0.680111\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) − 4.00000i − 0.305888i
\(172\) 0 0
\(173\) 17.3205i 1.31685i 0.752645 + 0.658427i \(0.228778\pi\)
−0.752645 + 0.658427i \(0.771222\pi\)
\(174\) 0 0
\(175\) −24.2487 −1.83303
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) 12.0000i 0.896922i 0.893802 + 0.448461i \(0.148028\pi\)
−0.893802 + 0.448461i \(0.851972\pi\)
\(180\) 0 0
\(181\) 13.8564i 1.02994i 0.857209 + 0.514969i \(0.172197\pi\)
−0.857209 + 0.514969i \(0.827803\pi\)
\(182\) 0 0
\(183\) −6.92820 −0.512148
\(184\) 0 0
\(185\) 24.0000 1.76452
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) − 3.46410i − 0.251976i
\(190\) 0 0
\(191\) −13.8564 −1.00261 −0.501307 0.865269i \(-0.667147\pi\)
−0.501307 + 0.865269i \(0.667147\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 10.3923i − 0.740421i −0.928948 0.370211i \(-0.879286\pi\)
0.928948 0.370211i \(-0.120714\pi\)
\(198\) 0 0
\(199\) 10.3923 0.736691 0.368345 0.929689i \(-0.379924\pi\)
0.368345 + 0.929689i \(0.379924\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) − 12.0000i − 0.842235i
\(204\) 0 0
\(205\) 20.7846i 1.45166i
\(206\) 0 0
\(207\) 6.92820 0.481543
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) − 4.00000i − 0.275371i −0.990476 0.137686i \(-0.956034\pi\)
0.990476 0.137686i \(-0.0439664\pi\)
\(212\) 0 0
\(213\) 6.92820i 0.474713i
\(214\) 0 0
\(215\) −13.8564 −0.944999
\(216\) 0 0
\(217\) −12.0000 −0.814613
\(218\) 0 0
\(219\) 2.00000i 0.135147i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 3.46410 0.231973 0.115987 0.993251i \(-0.462997\pi\)
0.115987 + 0.993251i \(0.462997\pi\)
\(224\) 0 0
\(225\) 7.00000 0.466667
\(226\) 0 0
\(227\) − 24.0000i − 1.59294i −0.604681 0.796468i \(-0.706699\pi\)
0.604681 0.796468i \(-0.293301\pi\)
\(228\) 0 0
\(229\) − 27.7128i − 1.83131i −0.401960 0.915657i \(-0.631671\pi\)
0.401960 0.915657i \(-0.368329\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 24.0000i 1.56559i
\(236\) 0 0
\(237\) 10.3923i 0.675053i
\(238\) 0 0
\(239\) 27.7128 1.79259 0.896296 0.443455i \(-0.146248\pi\)
0.896296 + 0.443455i \(0.146248\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) − 17.3205i − 1.10657i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 20.7846 1.30158
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 24.0000i 1.49129i
\(260\) 0 0
\(261\) 3.46410i 0.214423i
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) 0 0
\(269\) − 31.1769i − 1.90089i −0.310893 0.950445i \(-0.600628\pi\)
0.310893 0.950445i \(-0.399372\pi\)
\(270\) 0 0
\(271\) 3.46410 0.210429 0.105215 0.994450i \(-0.466447\pi\)
0.105215 + 0.994450i \(0.466447\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 13.8564i 0.832551i 0.909239 + 0.416275i \(0.136665\pi\)
−0.909239 + 0.416275i \(0.863335\pi\)
\(278\) 0 0
\(279\) 3.46410 0.207390
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 0 0
\(285\) 13.8564i 0.820783i
\(286\) 0 0
\(287\) −20.7846 −1.22688
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) − 2.00000i − 0.117242i
\(292\) 0 0
\(293\) 3.46410i 0.202375i 0.994867 + 0.101187i \(0.0322642\pi\)
−0.994867 + 0.101187i \(0.967736\pi\)
\(294\) 0 0
\(295\) 41.5692 2.42025
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 13.8564i − 0.798670i
\(302\) 0 0
\(303\) −3.46410 −0.199007
\(304\) 0 0
\(305\) 24.0000 1.37424
\(306\) 0 0
\(307\) − 4.00000i − 0.228292i −0.993464 0.114146i \(-0.963587\pi\)
0.993464 0.114146i \(-0.0364132\pi\)
\(308\) 0 0
\(309\) − 17.3205i − 0.985329i
\(310\) 0 0
\(311\) −13.8564 −0.785725 −0.392862 0.919597i \(-0.628515\pi\)
−0.392862 + 0.919597i \(0.628515\pi\)
\(312\) 0 0
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) 0 0
\(315\) 12.0000i 0.676123i
\(316\) 0 0
\(317\) − 31.1769i − 1.75107i −0.483155 0.875535i \(-0.660509\pi\)
0.483155 0.875535i \(-0.339491\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 24.0000i 1.33540i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 13.8564 0.766261
\(328\) 0 0
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) − 20.0000i − 1.09930i −0.835395 0.549650i \(-0.814761\pi\)
0.835395 0.549650i \(-0.185239\pi\)
\(332\) 0 0
\(333\) − 6.92820i − 0.379663i
\(334\) 0 0
\(335\) −13.8564 −0.757056
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) − 6.00000i − 0.325875i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −6.92820 −0.374088
\(344\) 0 0
\(345\) −24.0000 −1.29212
\(346\) 0 0
\(347\) − 24.0000i − 1.28839i −0.764862 0.644194i \(-0.777193\pi\)
0.764862 0.644194i \(-0.222807\pi\)
\(348\) 0 0
\(349\) 6.92820i 0.370858i 0.982658 + 0.185429i \(0.0593675\pi\)
−0.982658 + 0.185429i \(0.940632\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) − 24.0000i − 1.27379i
\(356\) 0 0
\(357\) 20.7846i 1.10004i
\(358\) 0 0
\(359\) 6.92820 0.365657 0.182828 0.983145i \(-0.441475\pi\)
0.182828 + 0.983145i \(0.441475\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 11.0000i 0.577350i
\(364\) 0 0
\(365\) − 6.92820i − 0.362639i
\(366\) 0 0
\(367\) 24.2487 1.26577 0.632886 0.774245i \(-0.281870\pi\)
0.632886 + 0.774245i \(0.281870\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 12.0000i 0.623009i
\(372\) 0 0
\(373\) 20.7846i 1.07619i 0.842885 + 0.538093i \(0.180855\pi\)
−0.842885 + 0.538093i \(0.819145\pi\)
\(374\) 0 0
\(375\) −6.92820 −0.357771
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 20.0000i 1.02733i 0.857991 + 0.513665i \(0.171713\pi\)
−0.857991 + 0.513665i \(0.828287\pi\)
\(380\) 0 0
\(381\) 3.46410i 0.177471i
\(382\) 0 0
\(383\) −13.8564 −0.708029 −0.354015 0.935240i \(-0.615184\pi\)
−0.354015 + 0.935240i \(0.615184\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.00000i 0.203331i
\(388\) 0 0
\(389\) 24.2487i 1.22946i 0.788738 + 0.614729i \(0.210735\pi\)
−0.788738 + 0.614729i \(0.789265\pi\)
\(390\) 0 0
\(391\) −41.5692 −2.10225
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) − 36.0000i − 1.81136i
\(396\) 0 0
\(397\) 34.6410i 1.73858i 0.494300 + 0.869291i \(0.335424\pi\)
−0.494300 + 0.869291i \(0.664576\pi\)
\(398\) 0 0
\(399\) −13.8564 −0.693688
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 3.46410i − 0.172133i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) − 6.00000i − 0.295958i
\(412\) 0 0
\(413\) 41.5692i 2.04549i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) 24.0000i 1.17248i 0.810139 + 0.586238i \(0.199392\pi\)
−0.810139 + 0.586238i \(0.800608\pi\)
\(420\) 0 0
\(421\) − 13.8564i − 0.675320i −0.941268 0.337660i \(-0.890365\pi\)
0.941268 0.337660i \(-0.109635\pi\)
\(422\) 0 0
\(423\) 6.92820 0.336861
\(424\) 0 0
\(425\) −42.0000 −2.03730
\(426\) 0 0
\(427\) 24.0000i 1.16144i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.92820 0.333720 0.166860 0.985981i \(-0.446637\pi\)
0.166860 + 0.985981i \(0.446637\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) − 12.0000i − 0.575356i
\(436\) 0 0
\(437\) − 27.7128i − 1.32568i
\(438\) 0 0
\(439\) −31.1769 −1.48799 −0.743996 0.668184i \(-0.767072\pi\)
−0.743996 + 0.668184i \(0.767072\pi\)
\(440\) 0 0
\(441\) −5.00000 −0.238095
\(442\) 0 0
\(443\) − 24.0000i − 1.14027i −0.821549 0.570137i \(-0.806890\pi\)
0.821549 0.570137i \(-0.193110\pi\)
\(444\) 0 0
\(445\) − 20.7846i − 0.985285i
\(446\) 0 0
\(447\) −10.3923 −0.491539
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 3.46410i 0.162758i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) − 6.00000i − 0.280056i
\(460\) 0 0
\(461\) 31.1769i 1.45205i 0.687666 + 0.726027i \(0.258635\pi\)
−0.687666 + 0.726027i \(0.741365\pi\)
\(462\) 0 0
\(463\) −10.3923 −0.482971 −0.241486 0.970404i \(-0.577635\pi\)
−0.241486 + 0.970404i \(0.577635\pi\)
\(464\) 0 0
\(465\) −12.0000 −0.556487
\(466\) 0 0
\(467\) − 24.0000i − 1.11059i −0.831654 0.555294i \(-0.812606\pi\)
0.831654 0.555294i \(-0.187394\pi\)
\(468\) 0 0
\(469\) − 13.8564i − 0.639829i
\(470\) 0 0
\(471\) 6.92820 0.319235
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) − 28.0000i − 1.28473i
\(476\) 0 0
\(477\) − 3.46410i − 0.158610i
\(478\) 0 0
\(479\) 20.7846 0.949673 0.474837 0.880074i \(-0.342507\pi\)
0.474837 + 0.880074i \(0.342507\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) − 24.0000i − 1.09204i
\(484\) 0 0
\(485\) 6.92820i 0.314594i
\(486\) 0 0
\(487\) −10.3923 −0.470920 −0.235460 0.971884i \(-0.575660\pi\)
−0.235460 + 0.971884i \(0.575660\pi\)
\(488\) 0 0
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) 36.0000i 1.62466i 0.583200 + 0.812329i \(0.301800\pi\)
−0.583200 + 0.812329i \(0.698200\pi\)
\(492\) 0 0
\(493\) − 20.7846i − 0.936092i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) − 28.0000i − 1.25345i −0.779240 0.626726i \(-0.784395\pi\)
0.779240 0.626726i \(-0.215605\pi\)
\(500\) 0 0
\(501\) − 13.8564i − 0.619059i
\(502\) 0 0
\(503\) 34.6410 1.54457 0.772283 0.635278i \(-0.219115\pi\)
0.772283 + 0.635278i \(0.219115\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 13.0000i 0.577350i
\(508\) 0 0
\(509\) 24.2487i 1.07481i 0.843326 + 0.537403i \(0.180594\pi\)
−0.843326 + 0.537403i \(0.819406\pi\)
\(510\) 0 0
\(511\) 6.92820 0.306486
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) 60.0000i 2.64392i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −17.3205 −0.760286
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) 0 0
\(525\) − 24.2487i − 1.05830i
\(526\) 0 0
\(527\) −20.7846 −0.905392
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) − 12.0000i − 0.520756i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 41.5692 1.79719
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) − 13.8564i − 0.595733i −0.954607 0.297867i \(-0.903725\pi\)
0.954607 0.297867i \(-0.0962751\pi\)
\(542\) 0 0
\(543\) −13.8564 −0.594635
\(544\) 0 0
\(545\) −48.0000 −2.05609
\(546\) 0 0
\(547\) 28.0000i 1.19719i 0.801050 + 0.598597i \(0.204275\pi\)
−0.801050 + 0.598597i \(0.795725\pi\)
\(548\) 0 0
\(549\) − 6.92820i − 0.295689i
\(550\) 0 0
\(551\) 13.8564 0.590303
\(552\) 0 0
\(553\) 36.0000 1.53088
\(554\) 0 0
\(555\) 24.0000i 1.01874i
\(556\) 0 0
\(557\) − 10.3923i − 0.440336i −0.975462 0.220168i \(-0.929339\pi\)
0.975462 0.220168i \(-0.0706606\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 24.0000i 1.01148i 0.862686 + 0.505740i \(0.168780\pi\)
−0.862686 + 0.505740i \(0.831220\pi\)
\(564\) 0 0
\(565\) 20.7846i 0.874415i
\(566\) 0 0
\(567\) 3.46410 0.145479
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 4.00000i 0.167395i 0.996491 + 0.0836974i \(0.0266729\pi\)
−0.996491 + 0.0836974i \(0.973327\pi\)
\(572\) 0 0
\(573\) − 13.8564i − 0.578860i
\(574\) 0 0
\(575\) 48.4974 2.02248
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 0 0
\(579\) 2.00000i 0.0831172i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 0 0
\(589\) − 13.8564i − 0.570943i
\(590\) 0 0
\(591\) 10.3923 0.427482
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) − 72.0000i − 2.95171i
\(596\) 0 0
\(597\) 10.3923i 0.425329i
\(598\) 0 0
\(599\) 34.6410 1.41539 0.707697 0.706516i \(-0.249734\pi\)
0.707697 + 0.706516i \(0.249734\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) − 38.1051i − 1.54919i
\(606\) 0 0
\(607\) 3.46410 0.140604 0.0703018 0.997526i \(-0.477604\pi\)
0.0703018 + 0.997526i \(0.477604\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 34.6410i − 1.39914i −0.714565 0.699569i \(-0.753375\pi\)
0.714565 0.699569i \(-0.246625\pi\)
\(614\) 0 0
\(615\) −20.7846 −0.838116
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) 4.00000i 0.160774i 0.996764 + 0.0803868i \(0.0256155\pi\)
−0.996764 + 0.0803868i \(0.974384\pi\)
\(620\) 0 0
\(621\) 6.92820i 0.278019i
\(622\) 0 0
\(623\) 20.7846 0.832718
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 41.5692i 1.65747i
\(630\) 0 0
\(631\) 17.3205 0.689519 0.344759 0.938691i \(-0.387961\pi\)
0.344759 + 0.938691i \(0.387961\pi\)
\(632\) 0 0
\(633\) 4.00000 0.158986
\(634\) 0 0
\(635\) − 12.0000i − 0.476205i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −6.92820 −0.274075
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 28.0000i 1.10421i 0.833774 + 0.552106i \(0.186176\pi\)
−0.833774 + 0.552106i \(0.813824\pi\)
\(644\) 0 0
\(645\) − 13.8564i − 0.545595i
\(646\) 0 0
\(647\) −34.6410 −1.36188 −0.680939 0.732340i \(-0.738428\pi\)
−0.680939 + 0.732340i \(0.738428\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) − 12.0000i − 0.470317i
\(652\) 0 0
\(653\) − 10.3923i − 0.406682i −0.979108 0.203341i \(-0.934820\pi\)
0.979108 0.203341i \(-0.0651801\pi\)
\(654\) 0 0
\(655\) −41.5692 −1.62424
\(656\) 0 0
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) − 12.0000i − 0.467454i −0.972302 0.233727i \(-0.924908\pi\)
0.972302 0.233727i \(-0.0750921\pi\)
\(660\) 0 0
\(661\) 6.92820i 0.269476i 0.990881 + 0.134738i \(0.0430193\pi\)
−0.990881 + 0.134738i \(0.956981\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 48.0000 1.86136
\(666\) 0 0
\(667\) 24.0000i 0.929284i
\(668\) 0 0
\(669\) 3.46410i 0.133930i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 0 0
\(675\) 7.00000i 0.269430i
\(676\) 0 0
\(677\) − 3.46410i − 0.133136i −0.997782 0.0665681i \(-0.978795\pi\)
0.997782 0.0665681i \(-0.0212050\pi\)
\(678\) 0 0
\(679\) −6.92820 −0.265880
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 20.7846i 0.794139i
\(686\) 0 0
\(687\) 27.7128 1.05731
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 28.0000i 1.06517i 0.846376 + 0.532585i \(0.178779\pi\)
−0.846376 + 0.532585i \(0.821221\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −69.2820 −2.62802
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 0 0
\(699\) − 18.0000i − 0.680823i
\(700\) 0 0
\(701\) 10.3923i 0.392512i 0.980553 + 0.196256i \(0.0628784\pi\)
−0.980553 + 0.196256i \(0.937122\pi\)
\(702\) 0 0
\(703\) −27.7128 −1.04521
\(704\) 0 0
\(705\) −24.0000 −0.903892
\(706\) 0 0
\(707\) 12.0000i 0.451306i
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) −10.3923 −0.389742
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 27.7128i 1.03495i
\(718\) 0 0
\(719\) −20.7846 −0.775135 −0.387568 0.921841i \(-0.626685\pi\)
−0.387568 + 0.921841i \(0.626685\pi\)
\(720\) 0 0
\(721\) −60.0000 −2.23452
\(722\) 0 0
\(723\) − 22.0000i − 0.818189i
\(724\) 0 0
\(725\) 24.2487i 0.900575i
\(726\) 0 0
\(727\) −10.3923 −0.385429 −0.192715 0.981255i \(-0.561729\pi\)
−0.192715 + 0.981255i \(0.561729\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 24.0000i − 0.887672i
\(732\) 0 0
\(733\) 27.7128i 1.02360i 0.859106 + 0.511798i \(0.171020\pi\)
−0.859106 + 0.511798i \(0.828980\pi\)
\(734\) 0 0
\(735\) 17.3205 0.638877
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 28.0000i − 1.03000i −0.857191 0.514998i \(-0.827793\pi\)
0.857191 0.514998i \(-0.172207\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 27.7128 1.01668 0.508342 0.861155i \(-0.330258\pi\)
0.508342 + 0.861155i \(0.330258\pi\)
\(744\) 0 0
\(745\) 36.0000 1.31894
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 41.5692i 1.51891i
\(750\) 0 0
\(751\) 24.2487 0.884848 0.442424 0.896806i \(-0.354119\pi\)
0.442424 + 0.896806i \(0.354119\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 12.0000i − 0.436725i
\(756\) 0 0
\(757\) − 27.7128i − 1.00724i −0.863925 0.503620i \(-0.832001\pi\)
0.863925 0.503620i \(-0.167999\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) − 48.0000i − 1.73772i
\(764\) 0 0
\(765\) 20.7846i 0.751469i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −46.0000 −1.65880 −0.829401 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 0 0
\(771\) 18.0000i 0.648254i
\(772\) 0 0
\(773\) − 38.1051i − 1.37055i −0.728286 0.685273i \(-0.759683\pi\)
0.728286 0.685273i \(-0.240317\pi\)
\(774\) 0 0
\(775\) 24.2487 0.871039
\(776\) 0 0
\(777\) −24.0000 −0.860995
\(778\) 0 0
\(779\) − 24.0000i − 0.859889i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −3.46410 −0.123797
\(784\) 0 0
\(785\) −24.0000 −0.856597
\(786\) 0 0
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −20.7846 −0.739016
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 12.0000i 0.425596i
\(796\) 0 0
\(797\) − 24.2487i − 0.858933i −0.903083 0.429467i \(-0.858702\pi\)
0.903083 0.429467i \(-0.141298\pi\)
\(798\) 0 0
\(799\) −41.5692 −1.47061
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 83.1384i 2.93024i
\(806\) 0 0
\(807\) 31.1769 1.09748
\(808\) 0 0
\(809\) −54.0000 −1.89854 −0.949269 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 0 0
\(811\) 20.0000i 0.702295i 0.936320 + 0.351147i \(0.114208\pi\)
−0.936320 + 0.351147i \(0.885792\pi\)
\(812\) 0 0
\(813\) 3.46410i 0.121491i
\(814\) 0 0
\(815\) −69.2820 −2.42684
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 24.2487i − 0.846286i −0.906063 0.423143i \(-0.860927\pi\)
0.906063 0.423143i \(-0.139073\pi\)
\(822\) 0 0
\(823\) 31.1769 1.08676 0.543379 0.839487i \(-0.317144\pi\)
0.543379 + 0.839487i \(0.317144\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 0 0
\(829\) 41.5692i 1.44376i 0.692019 + 0.721879i \(0.256721\pi\)
−0.692019 + 0.721879i \(0.743279\pi\)
\(830\) 0 0
\(831\) −13.8564 −0.480673
\(832\) 0 0
\(833\) 30.0000 1.03944
\(834\) 0 0
\(835\) 48.0000i 1.66111i
\(836\) 0 0
\(837\) 3.46410i 0.119737i
\(838\) 0 0
\(839\) −6.92820 −0.239188 −0.119594 0.992823i \(-0.538159\pi\)
−0.119594 + 0.992823i \(0.538159\pi\)
\(840\) 0 0
\(841\) 17.0000 0.586207
\(842\) 0 0
\(843\) 30.0000i 1.03325i
\(844\) 0 0
\(845\) − 45.0333i − 1.54919i
\(846\) 0 0
\(847\) 38.1051 1.30931
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) − 48.0000i − 1.64542i
\(852\) 0 0
\(853\) − 48.4974i − 1.66052i −0.557376 0.830260i \(-0.688192\pi\)
0.557376 0.830260i \(-0.311808\pi\)
\(854\) 0 0
\(855\) −13.8564 −0.473879
\(856\) 0 0
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) − 4.00000i − 0.136478i −0.997669 0.0682391i \(-0.978262\pi\)
0.997669 0.0682391i \(-0.0217381\pi\)
\(860\) 0 0
\(861\) − 20.7846i − 0.708338i
\(862\) 0 0
\(863\) −13.8564 −0.471678 −0.235839 0.971792i \(-0.575784\pi\)
−0.235839 + 0.971792i \(0.575784\pi\)
\(864\) 0 0
\(865\) 60.0000 2.04006
\(866\) 0 0
\(867\) 19.0000i 0.645274i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 24.0000i 0.811348i
\(876\) 0 0
\(877\) 6.92820i 0.233949i 0.993135 + 0.116974i \(0.0373195\pi\)
−0.993135 + 0.116974i \(0.962680\pi\)
\(878\) 0 0
\(879\) −3.46410 −0.116841
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) − 20.0000i − 0.673054i −0.941674 0.336527i \(-0.890748\pi\)
0.941674 0.336527i \(-0.109252\pi\)
\(884\) 0 0
\(885\) 41.5692i 1.39733i
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 12.0000 0.402467
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 27.7128i − 0.927374i
\(894\) 0 0
\(895\) 41.5692 1.38951
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 12.0000i 0.400222i
\(900\) 0 0
\(901\) 20.7846i 0.692436i
\(902\) 0 0
\(903\) 13.8564 0.461112
\(904\) 0 0
\(905\) 48.0000 1.59557
\(906\) 0 0
\(907\) − 52.0000i − 1.72663i −0.504664 0.863316i \(-0.668384\pi\)
0.504664 0.863316i \(-0.331616\pi\)
\(908\) 0 0
\(909\) − 3.46410i − 0.114897i
\(910\) 0 0
\(911\) 55.4256 1.83633 0.918166 0.396195i \(-0.129670\pi\)
0.918166 + 0.396195i \(0.129670\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 24.0000i 0.793416i
\(916\) 0 0
\(917\) − 41.5692i − 1.37274i
\(918\) 0 0
\(919\) −51.9615 −1.71405 −0.857026 0.515273i \(-0.827691\pi\)
−0.857026 + 0.515273i \(0.827691\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) − 48.4974i − 1.59459i
\(926\) 0 0
\(927\) 17.3205 0.568880
\(928\) 0 0
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) 20.0000i 0.655474i
\(932\) 0 0
\(933\) − 13.8564i − 0.453638i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) − 26.0000i − 0.848478i
\(940\) 0 0
\(941\) − 24.2487i − 0.790485i −0.918577 0.395243i \(-0.870660\pi\)
0.918577 0.395243i \(-0.129340\pi\)
\(942\) 0 0
\(943\) 41.5692 1.35368
\(944\) 0 0
\(945\) −12.0000 −0.390360
\(946\) 0 0
\(947\) − 36.0000i − 1.16984i −0.811090 0.584921i \(-0.801125\pi\)
0.811090 0.584921i \(-0.198875\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 31.1769 1.01098
\(952\) 0 0
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 0 0
\(955\) 48.0000i 1.55324i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −20.7846 −0.671170
\(960\) 0 0
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) − 12.0000i − 0.386695i
\(964\) 0 0
\(965\) − 6.92820i − 0.223027i
\(966\) 0 0
\(967\) 38.1051 1.22538 0.612689 0.790324i \(-0.290088\pi\)
0.612689 + 0.790324i \(0.290088\pi\)
\(968\) 0 0
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 48.0000i 1.54039i 0.637806 + 0.770197i \(0.279842\pi\)
−0.637806 + 0.770197i \(0.720158\pi\)
\(972\) 0 0
\(973\) − 69.2820i − 2.22108i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 54.0000 1.72761 0.863807 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 13.8564i 0.442401i
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −36.0000 −1.14706
\(986\) 0 0
\(987\) − 24.0000i − 0.763928i
\(988\) 0 0
\(989\) 27.7128i 0.881216i
\(990\) 0 0
\(991\) 24.2487 0.770286 0.385143 0.922857i \(-0.374152\pi\)
0.385143 + 0.922857i \(0.374152\pi\)
\(992\) 0 0
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) − 36.0000i − 1.14128i
\(996\) 0 0
\(997\) 20.7846i 0.658255i 0.944285 + 0.329128i \(0.106755\pi\)
−0.944285 + 0.329128i \(0.893245\pi\)
\(998\) 0 0
\(999\) 6.92820 0.219199
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 192.2.d.a.97.3 yes 4
3.2 odd 2 576.2.d.b.289.4 4
4.3 odd 2 inner 192.2.d.a.97.1 4
5.2 odd 4 4800.2.d.o.1249.4 4
5.3 odd 4 4800.2.d.j.1249.1 4
5.4 even 2 4800.2.k.j.2401.1 4
8.3 odd 2 inner 192.2.d.a.97.4 yes 4
8.5 even 2 inner 192.2.d.a.97.2 yes 4
12.11 even 2 576.2.d.b.289.3 4
16.3 odd 4 768.2.a.k.1.1 2
16.5 even 4 768.2.a.k.1.2 2
16.11 odd 4 768.2.a.j.1.2 2
16.13 even 4 768.2.a.j.1.1 2
20.3 even 4 4800.2.d.o.1249.3 4
20.7 even 4 4800.2.d.j.1249.2 4
20.19 odd 2 4800.2.k.j.2401.4 4
24.5 odd 2 576.2.d.b.289.2 4
24.11 even 2 576.2.d.b.289.1 4
40.3 even 4 4800.2.d.j.1249.3 4
40.13 odd 4 4800.2.d.o.1249.1 4
40.19 odd 2 4800.2.k.j.2401.2 4
40.27 even 4 4800.2.d.o.1249.2 4
40.29 even 2 4800.2.k.j.2401.3 4
40.37 odd 4 4800.2.d.j.1249.4 4
48.5 odd 4 2304.2.a.u.1.1 2
48.11 even 4 2304.2.a.s.1.1 2
48.29 odd 4 2304.2.a.s.1.2 2
48.35 even 4 2304.2.a.u.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
192.2.d.a.97.1 4 4.3 odd 2 inner
192.2.d.a.97.2 yes 4 8.5 even 2 inner
192.2.d.a.97.3 yes 4 1.1 even 1 trivial
192.2.d.a.97.4 yes 4 8.3 odd 2 inner
576.2.d.b.289.1 4 24.11 even 2
576.2.d.b.289.2 4 24.5 odd 2
576.2.d.b.289.3 4 12.11 even 2
576.2.d.b.289.4 4 3.2 odd 2
768.2.a.j.1.1 2 16.13 even 4
768.2.a.j.1.2 2 16.11 odd 4
768.2.a.k.1.1 2 16.3 odd 4
768.2.a.k.1.2 2 16.5 even 4
2304.2.a.s.1.1 2 48.11 even 4
2304.2.a.s.1.2 2 48.29 odd 4
2304.2.a.u.1.1 2 48.5 odd 4
2304.2.a.u.1.2 2 48.35 even 4
4800.2.d.j.1249.1 4 5.3 odd 4
4800.2.d.j.1249.2 4 20.7 even 4
4800.2.d.j.1249.3 4 40.3 even 4
4800.2.d.j.1249.4 4 40.37 odd 4
4800.2.d.o.1249.1 4 40.13 odd 4
4800.2.d.o.1249.2 4 40.27 even 4
4800.2.d.o.1249.3 4 20.3 even 4
4800.2.d.o.1249.4 4 5.2 odd 4
4800.2.k.j.2401.1 4 5.4 even 2
4800.2.k.j.2401.2 4 40.19 odd 2
4800.2.k.j.2401.3 4 40.29 even 2
4800.2.k.j.2401.4 4 20.19 odd 2