Properties

Label 672.2.b.b.223.6
Level $672$
Weight $2$
Character 672.223
Analytic conductor $5.366$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,2,Mod(223,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.223");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.b (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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.836829184.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 223.6
Root \(-0.222191i\) of defining polynomial
Character \(\chi\) \(=\) 672.223
Dual form 672.2.b.b.223.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.72844i q^{5} +(2.63640 - 0.222191i) q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.72844i q^{5} +(2.63640 - 0.222191i) q^{7} +1.00000 q^{9} +6.17282i q^{11} -2.82843i q^{13} +1.72844i q^{15} +1.72844i q^{17} -5.90126 q^{19} +(2.63640 - 0.222191i) q^{21} -1.54437i q^{23} +2.01250 q^{25} +1.00000 q^{27} +8.28531 q^{29} -4.62845 q^{31} +6.17282i q^{33} +(0.384044 + 4.55687i) q^{35} +2.24441 q^{37} -2.82843i q^{39} -3.92841i q^{41} +10.1012i q^{43} +1.72844i q^{45} +4.88877 q^{47} +(6.90126 - 1.17157i) q^{49} +1.72844i q^{51} -9.17407 q^{53} -10.6694 q^{55} -5.90126 q^{57} +9.65685 q^{59} -14.2853i q^{61} +(2.63640 - 0.222191i) q^{63} +4.88877 q^{65} +6.64436i q^{67} -1.54437i q^{69} -5.00125i q^{71} +2.19998i q^{73} +2.01250 q^{75} +(1.37155 + 16.2741i) q^{77} -9.55812i q^{79} +1.00000 q^{81} -12.2025 q^{83} -2.98750 q^{85} +8.28531 q^{87} -9.16282i q^{89} +(-0.628452 - 7.45688i) q^{91} -4.62845 q^{93} -10.2000i q^{95} -17.1137i q^{97} +6.17282i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 4 q^{7} + 8 q^{9} + 8 q^{19} + 4 q^{21} - 16 q^{25} + 8 q^{27} - 16 q^{31} - 8 q^{35} + 8 q^{37} + 16 q^{47} + 16 q^{53} - 8 q^{55} + 8 q^{57} + 32 q^{59} + 4 q^{63} + 16 q^{65} - 16 q^{75} + 32 q^{77} + 8 q^{81} + 16 q^{83} - 56 q^{85} + 16 q^{91} - 16 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.72844i 0.772982i 0.922293 + 0.386491i \(0.126313\pi\)
−0.922293 + 0.386491i \(0.873687\pi\)
\(6\) 0 0
\(7\) 2.63640 0.222191i 0.996467 0.0839804i
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.17282i 1.86118i 0.366068 + 0.930588i \(0.380704\pi\)
−0.366068 + 0.930588i \(0.619296\pi\)
\(12\) 0 0
\(13\) 2.82843i 0.784465i −0.919866 0.392232i \(-0.871703\pi\)
0.919866 0.392232i \(-0.128297\pi\)
\(14\) 0 0
\(15\) 1.72844i 0.446281i
\(16\) 0 0
\(17\) 1.72844i 0.419208i 0.977786 + 0.209604i \(0.0672175\pi\)
−0.977786 + 0.209604i \(0.932782\pi\)
\(18\) 0 0
\(19\) −5.90126 −1.35384 −0.676921 0.736056i \(-0.736686\pi\)
−0.676921 + 0.736056i \(0.736686\pi\)
\(20\) 0 0
\(21\) 2.63640 0.222191i 0.575311 0.0484861i
\(22\) 0 0
\(23\) 1.54437i 0.322023i −0.986952 0.161012i \(-0.948524\pi\)
0.986952 0.161012i \(-0.0514757\pi\)
\(24\) 0 0
\(25\) 2.01250 0.402499
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 8.28531 1.53854 0.769271 0.638922i \(-0.220619\pi\)
0.769271 + 0.638922i \(0.220619\pi\)
\(30\) 0 0
\(31\) −4.62845 −0.831295 −0.415647 0.909526i \(-0.636445\pi\)
−0.415647 + 0.909526i \(0.636445\pi\)
\(32\) 0 0
\(33\) 6.17282i 1.07455i
\(34\) 0 0
\(35\) 0.384044 + 4.55687i 0.0649153 + 0.770251i
\(36\) 0 0
\(37\) 2.24441 0.368978 0.184489 0.982835i \(-0.440937\pi\)
0.184489 + 0.982835i \(0.440937\pi\)
\(38\) 0 0
\(39\) 2.82843i 0.452911i
\(40\) 0 0
\(41\) 3.92841i 0.613515i −0.951788 0.306758i \(-0.900756\pi\)
0.951788 0.306758i \(-0.0992441\pi\)
\(42\) 0 0
\(43\) 10.1012i 1.54042i 0.637788 + 0.770212i \(0.279850\pi\)
−0.637788 + 0.770212i \(0.720150\pi\)
\(44\) 0 0
\(45\) 1.72844i 0.257661i
\(46\) 0 0
\(47\) 4.88877 0.713100 0.356550 0.934276i \(-0.383953\pi\)
0.356550 + 0.934276i \(0.383953\pi\)
\(48\) 0 0
\(49\) 6.90126 1.17157i 0.985895 0.167368i
\(50\) 0 0
\(51\) 1.72844i 0.242030i
\(52\) 0 0
\(53\) −9.17407 −1.26016 −0.630078 0.776532i \(-0.716977\pi\)
−0.630078 + 0.776532i \(0.716977\pi\)
\(54\) 0 0
\(55\) −10.6694 −1.43865
\(56\) 0 0
\(57\) −5.90126 −0.781641
\(58\) 0 0
\(59\) 9.65685 1.25722 0.628608 0.777723i \(-0.283625\pi\)
0.628608 + 0.777723i \(0.283625\pi\)
\(60\) 0 0
\(61\) 14.2853i 1.82905i −0.404534 0.914523i \(-0.632566\pi\)
0.404534 0.914523i \(-0.367434\pi\)
\(62\) 0 0
\(63\) 2.63640 0.222191i 0.332156 0.0279935i
\(64\) 0 0
\(65\) 4.88877 0.606377
\(66\) 0 0
\(67\) 6.64436i 0.811737i 0.913931 + 0.405869i \(0.133031\pi\)
−0.913931 + 0.405869i \(0.866969\pi\)
\(68\) 0 0
\(69\) 1.54437i 0.185920i
\(70\) 0 0
\(71\) 5.00125i 0.593539i −0.954949 0.296770i \(-0.904091\pi\)
0.954949 0.296770i \(-0.0959093\pi\)
\(72\) 0 0
\(73\) 2.19998i 0.257488i 0.991678 + 0.128744i \(0.0410945\pi\)
−0.991678 + 0.128744i \(0.958906\pi\)
\(74\) 0 0
\(75\) 2.01250 0.232383
\(76\) 0 0
\(77\) 1.37155 + 16.2741i 0.156302 + 1.85460i
\(78\) 0 0
\(79\) 9.55812i 1.07537i −0.843145 0.537686i \(-0.819299\pi\)
0.843145 0.537686i \(-0.180701\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.2025 −1.33940 −0.669698 0.742634i \(-0.733576\pi\)
−0.669698 + 0.742634i \(0.733576\pi\)
\(84\) 0 0
\(85\) −2.98750 −0.324040
\(86\) 0 0
\(87\) 8.28531 0.888278
\(88\) 0 0
\(89\) 9.16282i 0.971257i −0.874165 0.485629i \(-0.838591\pi\)
0.874165 0.485629i \(-0.161409\pi\)
\(90\) 0 0
\(91\) −0.628452 7.45688i −0.0658797 0.781693i
\(92\) 0 0
\(93\) −4.62845 −0.479948
\(94\) 0 0
\(95\) 10.2000i 1.04650i
\(96\) 0 0
\(97\) 17.1137i 1.73764i −0.495131 0.868818i \(-0.664880\pi\)
0.495131 0.868818i \(-0.335120\pi\)
\(98\) 0 0
\(99\) 6.17282i 0.620392i
\(100\) 0 0
\(101\) 9.58527i 0.953770i 0.878966 + 0.476885i \(0.158234\pi\)
−0.878966 + 0.476885i \(0.841766\pi\)
\(102\) 0 0
\(103\) −3.37155 −0.332208 −0.166104 0.986108i \(-0.553119\pi\)
−0.166104 + 0.986108i \(0.553119\pi\)
\(104\) 0 0
\(105\) 0.384044 + 4.55687i 0.0374789 + 0.444705i
\(106\) 0 0
\(107\) 5.28406i 0.510829i −0.966832 0.255415i \(-0.917788\pi\)
0.966832 0.255415i \(-0.0822119\pi\)
\(108\) 0 0
\(109\) 8.54562 0.818522 0.409261 0.912417i \(-0.365787\pi\)
0.409261 + 0.912417i \(0.365787\pi\)
\(110\) 0 0
\(111\) 2.24441 0.213030
\(112\) 0 0
\(113\) −2.36814 −0.222776 −0.111388 0.993777i \(-0.535530\pi\)
−0.111388 + 0.993777i \(0.535530\pi\)
\(114\) 0 0
\(115\) 2.66935 0.248918
\(116\) 0 0
\(117\) 2.82843i 0.261488i
\(118\) 0 0
\(119\) 0.384044 + 4.55687i 0.0352053 + 0.417727i
\(120\) 0 0
\(121\) −27.1037 −2.46398
\(122\) 0 0
\(123\) 3.92841i 0.354213i
\(124\) 0 0
\(125\) 12.1207i 1.08411i
\(126\) 0 0
\(127\) 6.24441i 0.554102i −0.960855 0.277051i \(-0.910643\pi\)
0.960855 0.277051i \(-0.0893570\pi\)
\(128\) 0 0
\(129\) 10.1012i 0.889364i
\(130\) 0 0
\(131\) 10.1457 0.886431 0.443216 0.896415i \(-0.353838\pi\)
0.443216 + 0.896415i \(0.353838\pi\)
\(132\) 0 0
\(133\) −15.5581 + 1.31121i −1.34906 + 0.113696i
\(134\) 0 0
\(135\) 1.72844i 0.148760i
\(136\) 0 0
\(137\) −6.76809 −0.578237 −0.289118 0.957293i \(-0.593362\pi\)
−0.289118 + 0.957293i \(0.593362\pi\)
\(138\) 0 0
\(139\) −15.3137 −1.29889 −0.649446 0.760408i \(-0.724999\pi\)
−0.649446 + 0.760408i \(0.724999\pi\)
\(140\) 0 0
\(141\) 4.88877 0.411708
\(142\) 0 0
\(143\) 17.4594 1.46003
\(144\) 0 0
\(145\) 14.3207i 1.18927i
\(146\) 0 0
\(147\) 6.90126 1.17157i 0.569206 0.0966297i
\(148\) 0 0
\(149\) −8.40598 −0.688645 −0.344322 0.938851i \(-0.611891\pi\)
−0.344322 + 0.938851i \(0.611891\pi\)
\(150\) 0 0
\(151\) 4.04443i 0.329131i −0.986366 0.164566i \(-0.947378\pi\)
0.986366 0.164566i \(-0.0526222\pi\)
\(152\) 0 0
\(153\) 1.72844i 0.139736i
\(154\) 0 0
\(155\) 8.00000i 0.642575i
\(156\) 0 0
\(157\) 2.97160i 0.237159i −0.992945 0.118580i \(-0.962166\pi\)
0.992945 0.118580i \(-0.0378341\pi\)
\(158\) 0 0
\(159\) −9.17407 −0.727551
\(160\) 0 0
\(161\) −0.343146 4.07159i −0.0270437 0.320886i
\(162\) 0 0
\(163\) 12.6694i 0.992340i −0.868225 0.496170i \(-0.834739\pi\)
0.868225 0.496170i \(-0.165261\pi\)
\(164\) 0 0
\(165\) −10.6694 −0.830608
\(166\) 0 0
\(167\) −9.28872 −0.718782 −0.359391 0.933187i \(-0.617016\pi\)
−0.359391 + 0.933187i \(0.617016\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) −5.90126 −0.451281
\(172\) 0 0
\(173\) 12.0716i 0.917786i −0.888492 0.458893i \(-0.848246\pi\)
0.888492 0.458893i \(-0.151754\pi\)
\(174\) 0 0
\(175\) 5.30576 0.447159i 0.401077 0.0338021i
\(176\) 0 0
\(177\) 9.65685 0.725854
\(178\) 0 0
\(179\) 8.17532i 0.611052i −0.952184 0.305526i \(-0.901168\pi\)
0.952184 0.305526i \(-0.0988323\pi\)
\(180\) 0 0
\(181\) 22.8334i 1.69720i −0.529039 0.848598i \(-0.677447\pi\)
0.529039 0.848598i \(-0.322553\pi\)
\(182\) 0 0
\(183\) 14.2853i 1.05600i
\(184\) 0 0
\(185\) 3.87932i 0.285214i
\(186\) 0 0
\(187\) −10.6694 −0.780220
\(188\) 0 0
\(189\) 2.63640 0.222191i 0.191770 0.0161620i
\(190\) 0 0
\(191\) 12.6331i 0.914100i 0.889441 + 0.457050i \(0.151094\pi\)
−0.889441 + 0.457050i \(0.848906\pi\)
\(192\) 0 0
\(193\) 8.30121 0.597534 0.298767 0.954326i \(-0.403425\pi\)
0.298767 + 0.954326i \(0.403425\pi\)
\(194\) 0 0
\(195\) 4.88877 0.350092
\(196\) 0 0
\(197\) −14.8309 −1.05666 −0.528330 0.849039i \(-0.677182\pi\)
−0.528330 + 0.849039i \(0.677182\pi\)
\(198\) 0 0
\(199\) 12.0159 0.851785 0.425892 0.904774i \(-0.359960\pi\)
0.425892 + 0.904774i \(0.359960\pi\)
\(200\) 0 0
\(201\) 6.64436i 0.468657i
\(202\) 0 0
\(203\) 21.8434 1.84092i 1.53311 0.129208i
\(204\) 0 0
\(205\) 6.79003 0.474236
\(206\) 0 0
\(207\) 1.54437i 0.107341i
\(208\) 0 0
\(209\) 36.4274i 2.51974i
\(210\) 0 0
\(211\) 17.9038i 1.23255i −0.787533 0.616273i \(-0.788642\pi\)
0.787533 0.616273i \(-0.211358\pi\)
\(212\) 0 0
\(213\) 5.00125i 0.342680i
\(214\) 0 0
\(215\) −17.4594 −1.19072
\(216\) 0 0
\(217\) −12.2025 + 1.02840i −0.828358 + 0.0698125i
\(218\) 0 0
\(219\) 2.19998i 0.148661i
\(220\) 0 0
\(221\) 4.88877 0.328854
\(222\) 0 0
\(223\) −8.58652 −0.574996 −0.287498 0.957781i \(-0.592823\pi\)
−0.287498 + 0.957781i \(0.592823\pi\)
\(224\) 0 0
\(225\) 2.01250 0.134166
\(226\) 0 0
\(227\) −13.2569 −0.879892 −0.439946 0.898024i \(-0.645002\pi\)
−0.439946 + 0.898024i \(0.645002\pi\)
\(228\) 0 0
\(229\) 3.91467i 0.258689i −0.991600 0.129344i \(-0.958713\pi\)
0.991600 0.129344i \(-0.0412872\pi\)
\(230\) 0 0
\(231\) 1.37155 + 16.2741i 0.0902412 + 1.07075i
\(232\) 0 0
\(233\) −0.343146 −0.0224802 −0.0112401 0.999937i \(-0.503578\pi\)
−0.0112401 + 0.999937i \(0.503578\pi\)
\(234\) 0 0
\(235\) 8.44994i 0.551213i
\(236\) 0 0
\(237\) 9.55812i 0.620866i
\(238\) 0 0
\(239\) 28.8038i 1.86316i 0.363535 + 0.931581i \(0.381570\pi\)
−0.363535 + 0.931581i \(0.618430\pi\)
\(240\) 0 0
\(241\) 24.5481i 1.58128i 0.612279 + 0.790642i \(0.290253\pi\)
−0.612279 + 0.790642i \(0.709747\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 2.02499 + 11.9284i 0.129372 + 0.762078i
\(246\) 0 0
\(247\) 16.6913i 1.06204i
\(248\) 0 0
\(249\) −12.2025 −0.773300
\(250\) 0 0
\(251\) −12.7681 −0.805915 −0.402957 0.915219i \(-0.632018\pi\)
−0.402957 + 0.915219i \(0.632018\pi\)
\(252\) 0 0
\(253\) 9.53312 0.599342
\(254\) 0 0
\(255\) −2.98750 −0.187085
\(256\) 0 0
\(257\) 18.9853i 1.18427i 0.805838 + 0.592137i \(0.201715\pi\)
−0.805838 + 0.592137i \(0.798285\pi\)
\(258\) 0 0
\(259\) 5.91717 0.498688i 0.367675 0.0309870i
\(260\) 0 0
\(261\) 8.28531 0.512848
\(262\) 0 0
\(263\) 19.9150i 1.22801i −0.789301 0.614006i \(-0.789557\pi\)
0.789301 0.614006i \(-0.210443\pi\)
\(264\) 0 0
\(265\) 15.8568i 0.974077i
\(266\) 0 0
\(267\) 9.16282i 0.560756i
\(268\) 0 0
\(269\) 12.6197i 0.769437i 0.923034 + 0.384718i \(0.125701\pi\)
−0.923034 + 0.384718i \(0.874299\pi\)
\(270\) 0 0
\(271\) −5.14908 −0.312784 −0.156392 0.987695i \(-0.549986\pi\)
−0.156392 + 0.987695i \(0.549986\pi\)
\(272\) 0 0
\(273\) −0.628452 7.45688i −0.0380357 0.451311i
\(274\) 0 0
\(275\) 12.4228i 0.749122i
\(276\) 0 0
\(277\) 14.3262 0.860778 0.430389 0.902643i \(-0.358376\pi\)
0.430389 + 0.902643i \(0.358376\pi\)
\(278\) 0 0
\(279\) −4.62845 −0.277098
\(280\) 0 0
\(281\) −12.9456 −0.772268 −0.386134 0.922443i \(-0.626190\pi\)
−0.386134 + 0.922443i \(0.626190\pi\)
\(282\) 0 0
\(283\) 26.4719 1.57359 0.786795 0.617215i \(-0.211739\pi\)
0.786795 + 0.617215i \(0.211739\pi\)
\(284\) 0 0
\(285\) 10.2000i 0.604194i
\(286\) 0 0
\(287\) −0.872860 10.3569i −0.0515233 0.611348i
\(288\) 0 0
\(289\) 14.0125 0.824264
\(290\) 0 0
\(291\) 17.1137i 1.00322i
\(292\) 0 0
\(293\) 15.5285i 0.907183i 0.891210 + 0.453591i \(0.149857\pi\)
−0.891210 + 0.453591i \(0.850143\pi\)
\(294\) 0 0
\(295\) 16.6913i 0.971804i
\(296\) 0 0
\(297\) 6.17282i 0.358183i
\(298\) 0 0
\(299\) −4.36814 −0.252616
\(300\) 0 0
\(301\) 2.24441 + 26.6310i 0.129366 + 1.53498i
\(302\) 0 0
\(303\) 9.58527i 0.550659i
\(304\) 0 0
\(305\) 24.6913 1.41382
\(306\) 0 0
\(307\) 13.1332 0.749550 0.374775 0.927116i \(-0.377720\pi\)
0.374775 + 0.927116i \(0.377720\pi\)
\(308\) 0 0
\(309\) −3.37155 −0.191801
\(310\) 0 0
\(311\) 3.87932 0.219976 0.109988 0.993933i \(-0.464919\pi\)
0.109988 + 0.993933i \(0.464919\pi\)
\(312\) 0 0
\(313\) 21.1362i 1.19469i 0.801984 + 0.597345i \(0.203778\pi\)
−0.801984 + 0.597345i \(0.796222\pi\)
\(314\) 0 0
\(315\) 0.384044 + 4.55687i 0.0216384 + 0.256750i
\(316\) 0 0
\(317\) 13.9422 0.783070 0.391535 0.920163i \(-0.371944\pi\)
0.391535 + 0.920163i \(0.371944\pi\)
\(318\) 0 0
\(319\) 51.1437i 2.86350i
\(320\) 0 0
\(321\) 5.28406i 0.294927i
\(322\) 0 0
\(323\) 10.2000i 0.567542i
\(324\) 0 0
\(325\) 5.69220i 0.315746i
\(326\) 0 0
\(327\) 8.54562 0.472574
\(328\) 0 0
\(329\) 12.8888 1.08624i 0.710581 0.0598864i
\(330\) 0 0
\(331\) 14.3037i 0.786203i −0.919495 0.393102i \(-0.871402\pi\)
0.919495 0.393102i \(-0.128598\pi\)
\(332\) 0 0
\(333\) 2.24441 0.122993
\(334\) 0 0
\(335\) −11.4844 −0.627458
\(336\) 0 0
\(337\) 17.1162 0.932381 0.466190 0.884684i \(-0.345626\pi\)
0.466190 + 0.884684i \(0.345626\pi\)
\(338\) 0 0
\(339\) −2.36814 −0.128620
\(340\) 0 0
\(341\) 28.5706i 1.54719i
\(342\) 0 0
\(343\) 17.9342 4.62214i 0.968356 0.249572i
\(344\) 0 0
\(345\) 2.66935 0.143713
\(346\) 0 0
\(347\) 4.11602i 0.220959i 0.993878 + 0.110480i \(0.0352387\pi\)
−0.993878 + 0.110480i \(0.964761\pi\)
\(348\) 0 0
\(349\) 34.5766i 1.85085i 0.378936 + 0.925423i \(0.376290\pi\)
−0.378936 + 0.925423i \(0.623710\pi\)
\(350\) 0 0
\(351\) 2.82843i 0.150970i
\(352\) 0 0
\(353\) 14.2991i 0.761062i −0.924768 0.380531i \(-0.875741\pi\)
0.924768 0.380531i \(-0.124259\pi\)
\(354\) 0 0
\(355\) 8.64436 0.458795
\(356\) 0 0
\(357\) 0.384044 + 4.55687i 0.0203258 + 0.241175i
\(358\) 0 0
\(359\) 26.8831i 1.41883i 0.704789 + 0.709417i \(0.251042\pi\)
−0.704789 + 0.709417i \(0.748958\pi\)
\(360\) 0 0
\(361\) 15.8249 0.832889
\(362\) 0 0
\(363\) −27.1037 −1.42258
\(364\) 0 0
\(365\) −3.80252 −0.199033
\(366\) 0 0
\(367\) −16.3072 −0.851231 −0.425616 0.904904i \(-0.639942\pi\)
−0.425616 + 0.904904i \(0.639942\pi\)
\(368\) 0 0
\(369\) 3.92841i 0.204505i
\(370\) 0 0
\(371\) −24.1866 + 2.03840i −1.25570 + 0.105828i
\(372\) 0 0
\(373\) −28.9188 −1.49736 −0.748678 0.662934i \(-0.769311\pi\)
−0.748678 + 0.662934i \(0.769311\pi\)
\(374\) 0 0
\(375\) 12.1207i 0.625909i
\(376\) 0 0
\(377\) 23.4344i 1.20693i
\(378\) 0 0
\(379\) 28.6150i 1.46986i 0.678145 + 0.734928i \(0.262784\pi\)
−0.678145 + 0.734928i \(0.737216\pi\)
\(380\) 0 0
\(381\) 6.24441i 0.319911i
\(382\) 0 0
\(383\) 25.4594 1.30091 0.650457 0.759543i \(-0.274578\pi\)
0.650457 + 0.759543i \(0.274578\pi\)
\(384\) 0 0
\(385\) −28.1287 + 2.37064i −1.43357 + 0.120819i
\(386\) 0 0
\(387\) 10.1012i 0.513475i
\(388\) 0 0
\(389\) 8.25350 0.418469 0.209234 0.977866i \(-0.432903\pi\)
0.209234 + 0.977866i \(0.432903\pi\)
\(390\) 0 0
\(391\) 2.66935 0.134995
\(392\) 0 0
\(393\) 10.1457 0.511781
\(394\) 0 0
\(395\) 16.5206 0.831243
\(396\) 0 0
\(397\) 27.9422i 1.40238i −0.712976 0.701188i \(-0.752653\pi\)
0.712976 0.701188i \(-0.247347\pi\)
\(398\) 0 0
\(399\) −15.5581 + 1.31121i −0.778880 + 0.0656426i
\(400\) 0 0
\(401\) 23.8593 1.19148 0.595739 0.803178i \(-0.296859\pi\)
0.595739 + 0.803178i \(0.296859\pi\)
\(402\) 0 0
\(403\) 13.0912i 0.652121i
\(404\) 0 0
\(405\) 1.72844i 0.0858869i
\(406\) 0 0
\(407\) 13.8543i 0.686734i
\(408\) 0 0
\(409\) 10.9363i 0.540763i 0.962753 + 0.270381i \(0.0871498\pi\)
−0.962753 + 0.270381i \(0.912850\pi\)
\(410\) 0 0
\(411\) −6.76809 −0.333845
\(412\) 0 0
\(413\) 25.4594 2.14567i 1.25277 0.105582i
\(414\) 0 0
\(415\) 21.0912i 1.03533i
\(416\) 0 0
\(417\) −15.3137 −0.749916
\(418\) 0 0
\(419\) 5.08624 0.248479 0.124240 0.992252i \(-0.460351\pi\)
0.124240 + 0.992252i \(0.460351\pi\)
\(420\) 0 0
\(421\) −10.8150 −0.527092 −0.263546 0.964647i \(-0.584892\pi\)
−0.263546 + 0.964647i \(0.584892\pi\)
\(422\) 0 0
\(423\) 4.88877 0.237700
\(424\) 0 0
\(425\) 3.47848i 0.168731i
\(426\) 0 0
\(427\) −3.17407 37.6619i −0.153604 1.82258i
\(428\) 0 0
\(429\) 17.4594 0.842947
\(430\) 0 0
\(431\) 2.10999i 0.101634i −0.998708 0.0508172i \(-0.983817\pi\)
0.998708 0.0508172i \(-0.0161826\pi\)
\(432\) 0 0
\(433\) 21.2619i 1.02178i 0.859646 + 0.510891i \(0.170684\pi\)
−0.859646 + 0.510891i \(0.829316\pi\)
\(434\) 0 0
\(435\) 14.3207i 0.686623i
\(436\) 0 0
\(437\) 9.11373i 0.435969i
\(438\) 0 0
\(439\) 35.6210 1.70010 0.850048 0.526706i \(-0.176573\pi\)
0.850048 + 0.526706i \(0.176573\pi\)
\(440\) 0 0
\(441\) 6.90126 1.17157i 0.328632 0.0557892i
\(442\) 0 0
\(443\) 2.14101i 0.101723i −0.998706 0.0508613i \(-0.983803\pi\)
0.998706 0.0508613i \(-0.0161966\pi\)
\(444\) 0 0
\(445\) 15.8374 0.750764
\(446\) 0 0
\(447\) −8.40598 −0.397589
\(448\) 0 0
\(449\) 9.40257 0.443735 0.221867 0.975077i \(-0.428785\pi\)
0.221867 + 0.975077i \(0.428785\pi\)
\(450\) 0 0
\(451\) 24.2494 1.14186
\(452\) 0 0
\(453\) 4.04443i 0.190024i
\(454\) 0 0
\(455\) 12.8888 1.08624i 0.604235 0.0509238i
\(456\) 0 0
\(457\) 15.5801 0.728804 0.364402 0.931242i \(-0.381273\pi\)
0.364402 + 0.931242i \(0.381273\pi\)
\(458\) 0 0
\(459\) 1.72844i 0.0806767i
\(460\) 0 0
\(461\) 29.8766i 1.39149i −0.718288 0.695746i \(-0.755074\pi\)
0.718288 0.695746i \(-0.244926\pi\)
\(462\) 0 0
\(463\) 35.3356i 1.64219i −0.570794 0.821093i \(-0.693364\pi\)
0.570794 0.821093i \(-0.306636\pi\)
\(464\) 0 0
\(465\) 8.00000i 0.370991i
\(466\) 0 0
\(467\) −24.3232 −1.12554 −0.562771 0.826613i \(-0.690265\pi\)
−0.562771 + 0.826613i \(0.690265\pi\)
\(468\) 0 0
\(469\) 1.47632 + 17.5172i 0.0681701 + 0.808870i
\(470\) 0 0
\(471\) 2.97160i 0.136924i
\(472\) 0 0
\(473\) −62.3531 −2.86700
\(474\) 0 0
\(475\) −11.8763 −0.544921
\(476\) 0 0
\(477\) −9.17407 −0.420052
\(478\) 0 0
\(479\) −19.6000 −0.895549 −0.447775 0.894146i \(-0.647783\pi\)
−0.447775 + 0.894146i \(0.647783\pi\)
\(480\) 0 0
\(481\) 6.34814i 0.289450i
\(482\) 0 0
\(483\) −0.343146 4.07159i −0.0156137 0.185264i
\(484\) 0 0
\(485\) 29.5801 1.34316
\(486\) 0 0
\(487\) 21.7331i 0.984821i 0.870363 + 0.492410i \(0.163884\pi\)
−0.870363 + 0.492410i \(0.836116\pi\)
\(488\) 0 0
\(489\) 12.6694i 0.572928i
\(490\) 0 0
\(491\) 34.9409i 1.57686i −0.615124 0.788431i \(-0.710894\pi\)
0.615124 0.788431i \(-0.289106\pi\)
\(492\) 0 0
\(493\) 14.3207i 0.644970i
\(494\) 0 0
\(495\) −10.6694 −0.479552
\(496\) 0 0
\(497\) −1.11123 13.1853i −0.0498457 0.591442i
\(498\) 0 0
\(499\) 19.5288i 0.874229i 0.899406 + 0.437115i \(0.144000\pi\)
−0.899406 + 0.437115i \(0.856000\pi\)
\(500\) 0 0
\(501\) −9.28872 −0.414989
\(502\) 0 0
\(503\) −30.4249 −1.35658 −0.678291 0.734794i \(-0.737279\pi\)
−0.678291 + 0.734794i \(0.737279\pi\)
\(504\) 0 0
\(505\) −16.5676 −0.737247
\(506\) 0 0
\(507\) 5.00000 0.222058
\(508\) 0 0
\(509\) 13.8766i 0.615070i 0.951537 + 0.307535i \(0.0995041\pi\)
−0.951537 + 0.307535i \(0.900496\pi\)
\(510\) 0 0
\(511\) 0.488815 + 5.80002i 0.0216239 + 0.256578i
\(512\) 0 0
\(513\) −5.90126 −0.260547
\(514\) 0 0
\(515\) 5.82752i 0.256791i
\(516\) 0 0
\(517\) 30.1775i 1.32720i
\(518\) 0 0
\(519\) 12.0716i 0.529884i
\(520\) 0 0
\(521\) 11.3628i 0.497813i 0.968527 + 0.248907i \(0.0800712\pi\)
−0.968527 + 0.248907i \(0.919929\pi\)
\(522\) 0 0
\(523\) −4.52063 −0.197673 −0.0988366 0.995104i \(-0.531512\pi\)
−0.0988366 + 0.995104i \(0.531512\pi\)
\(524\) 0 0
\(525\) 5.30576 0.447159i 0.231562 0.0195156i
\(526\) 0 0
\(527\) 8.00000i 0.348485i
\(528\) 0 0
\(529\) 20.6149 0.896301
\(530\) 0 0
\(531\) 9.65685 0.419072
\(532\) 0 0
\(533\) −11.1112 −0.481281
\(534\) 0 0
\(535\) 9.13317 0.394862
\(536\) 0 0
\(537\) 8.17532i 0.352791i
\(538\) 0 0
\(539\) 7.23191 + 42.6003i 0.311500 + 1.83492i
\(540\) 0 0
\(541\) −46.1287 −1.98323 −0.991614 0.129231i \(-0.958749\pi\)
−0.991614 + 0.129231i \(0.958749\pi\)
\(542\) 0 0
\(543\) 22.8334i 0.979876i
\(544\) 0 0
\(545\) 14.7706i 0.632702i
\(546\) 0 0
\(547\) 16.8718i 0.721387i 0.932684 + 0.360694i \(0.117460\pi\)
−0.932684 + 0.360694i \(0.882540\pi\)
\(548\) 0 0
\(549\) 14.2853i 0.609682i
\(550\) 0 0
\(551\) −48.8938 −2.08294
\(552\) 0 0
\(553\) −2.12373 25.1991i −0.0903102 1.07157i
\(554\) 0 0
\(555\) 3.87932i 0.164668i
\(556\) 0 0
\(557\) −43.7197 −1.85246 −0.926231 0.376955i \(-0.876971\pi\)
−0.926231 + 0.376955i \(0.876971\pi\)
\(558\) 0 0
\(559\) 28.5706 1.20841
\(560\) 0 0
\(561\) −10.6694 −0.450460
\(562\) 0 0
\(563\) −19.4662 −0.820402 −0.410201 0.911995i \(-0.634542\pi\)
−0.410201 + 0.911995i \(0.634542\pi\)
\(564\) 0 0
\(565\) 4.09318i 0.172202i
\(566\) 0 0
\(567\) 2.63640 0.222191i 0.110719 0.00933116i
\(568\) 0 0
\(569\) −12.1775 −0.510507 −0.255253 0.966874i \(-0.582159\pi\)
−0.255253 + 0.966874i \(0.582159\pi\)
\(570\) 0 0
\(571\) 6.04193i 0.252847i −0.991976 0.126424i \(-0.959650\pi\)
0.991976 0.126424i \(-0.0403498\pi\)
\(572\) 0 0
\(573\) 12.6331i 0.527756i
\(574\) 0 0
\(575\) 3.10804i 0.129614i
\(576\) 0 0
\(577\) 3.03444i 0.126325i −0.998003 0.0631626i \(-0.979881\pi\)
0.998003 0.0631626i \(-0.0201187\pi\)
\(578\) 0 0
\(579\) 8.30121 0.344987
\(580\) 0 0
\(581\) −32.1707 + 2.71128i −1.33466 + 0.112483i
\(582\) 0 0
\(583\) 56.6299i 2.34537i
\(584\) 0 0
\(585\) 4.88877 0.202126
\(586\) 0 0
\(587\) 32.2075 1.32934 0.664672 0.747135i \(-0.268571\pi\)
0.664672 + 0.747135i \(0.268571\pi\)
\(588\) 0 0
\(589\) 27.3137 1.12544
\(590\) 0 0
\(591\) −14.8309 −0.610063
\(592\) 0 0
\(593\) 35.2196i 1.44630i 0.690693 + 0.723148i \(0.257306\pi\)
−0.690693 + 0.723148i \(0.742694\pi\)
\(594\) 0 0
\(595\) −7.87627 + 0.663798i −0.322896 + 0.0272130i
\(596\) 0 0
\(597\) 12.0159 0.491778
\(598\) 0 0
\(599\) 20.7149i 0.846388i −0.906039 0.423194i \(-0.860909\pi\)
0.906039 0.423194i \(-0.139091\pi\)
\(600\) 0 0
\(601\) 24.8620i 1.01414i −0.861905 0.507070i \(-0.830729\pi\)
0.861905 0.507070i \(-0.169271\pi\)
\(602\) 0 0
\(603\) 6.64436i 0.270579i
\(604\) 0 0
\(605\) 46.8472i 1.90461i
\(606\) 0 0
\(607\) 1.19101 0.0483417 0.0241709 0.999708i \(-0.492305\pi\)
0.0241709 + 0.999708i \(0.492305\pi\)
\(608\) 0 0
\(609\) 21.8434 1.84092i 0.885140 0.0745980i
\(610\) 0 0
\(611\) 13.8275i 0.559401i
\(612\) 0 0
\(613\) 2.60242 0.105111 0.0525555 0.998618i \(-0.483263\pi\)
0.0525555 + 0.998618i \(0.483263\pi\)
\(614\) 0 0
\(615\) 6.79003 0.273800
\(616\) 0 0
\(617\) 33.9161 1.36541 0.682706 0.730693i \(-0.260803\pi\)
0.682706 + 0.730693i \(0.260803\pi\)
\(618\) 0 0
\(619\) −18.8688 −0.758400 −0.379200 0.925315i \(-0.623801\pi\)
−0.379200 + 0.925315i \(0.623801\pi\)
\(620\) 0 0
\(621\) 1.54437i 0.0619734i
\(622\) 0 0
\(623\) −2.03590 24.1569i −0.0815666 0.967826i
\(624\) 0 0
\(625\) −10.8874 −0.435495
\(626\) 0 0
\(627\) 36.4274i 1.45477i
\(628\) 0 0
\(629\) 3.87932i 0.154679i
\(630\) 0 0
\(631\) 2.44688i 0.0974088i 0.998813 + 0.0487044i \(0.0155092\pi\)
−0.998813 + 0.0487044i \(0.984491\pi\)
\(632\) 0 0
\(633\) 17.9038i 0.711611i
\(634\) 0 0
\(635\) 10.7931 0.428310
\(636\) 0 0
\(637\) −3.31371 19.5197i −0.131294 0.773399i
\(638\) 0 0
\(639\) 5.00125i 0.197846i
\(640\) 0 0
\(641\) −23.6619 −0.934587 −0.467293 0.884102i \(-0.654771\pi\)
−0.467293 + 0.884102i \(0.654771\pi\)
\(642\) 0 0
\(643\) 20.1237 0.793602 0.396801 0.917905i \(-0.370120\pi\)
0.396801 + 0.917905i \(0.370120\pi\)
\(644\) 0 0
\(645\) −17.4594 −0.687462
\(646\) 0 0
\(647\) 5.94320 0.233651 0.116826 0.993152i \(-0.462728\pi\)
0.116826 + 0.993152i \(0.462728\pi\)
\(648\) 0 0
\(649\) 59.6100i 2.33990i
\(650\) 0 0
\(651\) −12.2025 + 1.02840i −0.478253 + 0.0403063i
\(652\) 0 0
\(653\) −5.33974 −0.208960 −0.104480 0.994527i \(-0.533318\pi\)
−0.104480 + 0.994527i \(0.533318\pi\)
\(654\) 0 0
\(655\) 17.5362i 0.685195i
\(656\) 0 0
\(657\) 2.19998i 0.0858292i
\(658\) 0 0
\(659\) 25.5754i 0.996276i −0.867098 0.498138i \(-0.834017\pi\)
0.867098 0.498138i \(-0.165983\pi\)
\(660\) 0 0
\(661\) 21.7197i 0.844798i 0.906410 + 0.422399i \(0.138812\pi\)
−0.906410 + 0.422399i \(0.861188\pi\)
\(662\) 0 0
\(663\) 4.88877 0.189864
\(664\) 0 0
\(665\) −2.26635 26.8913i −0.0878851 1.04280i
\(666\) 0 0
\(667\) 12.7956i 0.495447i
\(668\) 0 0
\(669\) −8.58652 −0.331974
\(670\) 0 0
\(671\) 88.1807 3.40418
\(672\) 0 0
\(673\) −42.6175 −1.64279 −0.821393 0.570363i \(-0.806803\pi\)
−0.821393 + 0.570363i \(0.806803\pi\)
\(674\) 0 0
\(675\) 2.01250 0.0774610
\(676\) 0 0
\(677\) 33.7559i 1.29735i −0.761067 0.648673i \(-0.775324\pi\)
0.761067 0.648673i \(-0.224676\pi\)
\(678\) 0 0
\(679\) −3.80252 45.1187i −0.145927 1.73150i
\(680\) 0 0
\(681\) −13.2569 −0.508006
\(682\) 0 0
\(683\) 38.1553i 1.45997i 0.683462 + 0.729986i \(0.260474\pi\)
−0.683462 + 0.729986i \(0.739526\pi\)
\(684\) 0 0
\(685\) 11.6982i 0.446966i
\(686\) 0 0
\(687\) 3.91467i 0.149354i
\(688\) 0 0
\(689\) 25.9482i 0.988547i
\(690\) 0 0
\(691\) 32.8120 1.24823 0.624113 0.781334i \(-0.285460\pi\)
0.624113 + 0.781334i \(0.285460\pi\)
\(692\) 0 0
\(693\) 1.37155 + 16.2741i 0.0521008 + 0.618200i
\(694\) 0 0
\(695\) 26.4688i 1.00402i
\(696\) 0 0
\(697\) 6.79003 0.257191
\(698\) 0 0
\(699\) −0.343146 −0.0129790
\(700\) 0 0
\(701\) −14.9077 −0.563057 −0.281529 0.959553i \(-0.590841\pi\)
−0.281529 + 0.959553i \(0.590841\pi\)
\(702\) 0 0
\(703\) −13.2448 −0.499539
\(704\) 0 0
\(705\) 8.44994i 0.318243i
\(706\) 0 0
\(707\) 2.12976 + 25.2707i 0.0800980 + 0.950401i
\(708\) 0 0
\(709\) 5.47937 0.205782 0.102891 0.994693i \(-0.467191\pi\)
0.102891 + 0.994693i \(0.467191\pi\)
\(710\) 0 0
\(711\) 9.55812i 0.358457i
\(712\) 0 0
\(713\) 7.14804i 0.267696i
\(714\) 0 0
\(715\) 30.1775i 1.12857i
\(716\) 0 0
\(717\) 28.8038i 1.07570i
\(718\) 0 0
\(719\) −3.60005 −0.134259 −0.0671296 0.997744i \(-0.521384\pi\)
−0.0671296 + 0.997744i \(0.521384\pi\)
\(720\) 0 0
\(721\) −8.88877 + 0.749129i −0.331035 + 0.0278990i
\(722\) 0 0
\(723\) 24.5481i 0.912954i
\(724\) 0 0
\(725\) 16.6741 0.619262
\(726\) 0 0
\(727\) −17.3147 −0.642168 −0.321084 0.947051i \(-0.604047\pi\)
−0.321084 + 0.947051i \(0.604047\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −17.4594 −0.645758
\(732\) 0 0
\(733\) 15.7490i 0.581703i 0.956768 + 0.290851i \(0.0939385\pi\)
−0.956768 + 0.290851i \(0.906061\pi\)
\(734\) 0 0
\(735\) 2.02499 + 11.9284i 0.0746930 + 0.439986i
\(736\) 0 0
\(737\) −41.0144 −1.51079
\(738\) 0 0
\(739\) 8.87182i 0.326355i 0.986597 + 0.163178i \(0.0521744\pi\)
−0.986597 + 0.163178i \(0.947826\pi\)
\(740\) 0 0
\(741\) 16.6913i 0.613170i
\(742\) 0 0
\(743\) 2.90057i 0.106412i 0.998584 + 0.0532058i \(0.0169439\pi\)
−0.998584 + 0.0532058i \(0.983056\pi\)
\(744\) 0 0
\(745\) 14.5292i 0.532310i
\(746\) 0 0
\(747\) −12.2025 −0.446465
\(748\) 0 0
\(749\) −1.17407 13.9309i −0.0428997 0.509025i
\(750\) 0 0
\(751\) 9.98056i 0.364196i −0.983280 0.182098i \(-0.941711\pi\)
0.983280 0.182098i \(-0.0582888\pi\)
\(752\) 0 0
\(753\) −12.7681 −0.465295
\(754\) 0 0
\(755\) 6.99056 0.254412
\(756\) 0 0
\(757\) 4.34814 0.158036 0.0790180 0.996873i \(-0.474822\pi\)
0.0790180 + 0.996873i \(0.474822\pi\)
\(758\) 0 0
\(759\) 9.53312 0.346030
\(760\) 0 0
\(761\) 46.8197i 1.69721i 0.529025 + 0.848606i \(0.322558\pi\)
−0.529025 + 0.848606i \(0.677442\pi\)
\(762\) 0 0
\(763\) 22.5297 1.89876i 0.815630 0.0687398i
\(764\) 0 0
\(765\) −2.98750 −0.108013
\(766\) 0 0
\(767\) 27.3137i 0.986241i
\(768\) 0 0
\(769\) 2.86377i 0.103270i −0.998666 0.0516351i \(-0.983557\pi\)
0.998666 0.0516351i \(-0.0164433\pi\)
\(770\) 0 0
\(771\) 18.9853i 0.683740i
\(772\) 0 0
\(773\) 52.5040i 1.88844i −0.329317 0.944219i \(-0.606819\pi\)
0.329317 0.944219i \(-0.393181\pi\)
\(774\) 0 0
\(775\) −9.31474 −0.334595
\(776\) 0 0
\(777\) 5.91717 0.498688i 0.212277 0.0178903i
\(778\) 0 0
\(779\) 23.1826i 0.830603i
\(780\) 0 0
\(781\) 30.8718 1.10468
\(782\) 0 0
\(783\) 8.28531 0.296093
\(784\) 0 0
\(785\) 5.13623 0.183320
\(786\) 0 0
\(787\) −46.3981 −1.65391 −0.826957 0.562265i \(-0.809930\pi\)
−0.826957 + 0.562265i \(0.809930\pi\)
\(788\) 0 0
\(789\) 19.9150i 0.708993i
\(790\) 0 0
\(791\) −6.24337 + 0.526180i −0.221989 + 0.0187088i
\(792\) 0 0
\(793\) −40.4049 −1.43482
\(794\) 0 0
\(795\) 15.8568i 0.562384i
\(796\) 0 0
\(797\) 28.5215i 1.01028i 0.863036 + 0.505142i \(0.168560\pi\)
−0.863036 + 0.505142i \(0.831440\pi\)
\(798\) 0 0
\(799\) 8.44994i 0.298937i
\(800\) 0 0
\(801\) 9.16282i 0.323752i
\(802\) 0 0
\(803\) −13.5801 −0.479230
\(804\) 0 0
\(805\) 7.03749 0.593107i 0.248039 0.0209043i
\(806\) 0 0
\(807\) 12.6197i 0.444235i
\(808\) 0 0
\(809\) 53.6100 1.88483 0.942414 0.334447i \(-0.108550\pi\)
0.942414 + 0.334447i \(0.108550\pi\)
\(810\) 0 0
\(811\) −27.7207 −0.973406 −0.486703 0.873567i \(-0.661801\pi\)
−0.486703 + 0.873567i \(0.661801\pi\)
\(812\) 0 0
\(813\) −5.14908 −0.180586
\(814\) 0 0
\(815\) 21.8982 0.767061
\(816\) 0 0
\(817\) 59.6100i 2.08549i
\(818\) 0 0
\(819\) −0.628452 7.45688i −0.0219599 0.260564i
\(820\) 0 0
\(821\) −1.56902 −0.0547593 −0.0273797 0.999625i \(-0.508716\pi\)
−0.0273797 + 0.999625i \(0.508716\pi\)
\(822\) 0 0
\(823\) 24.7132i 0.861449i −0.902483 0.430724i \(-0.858258\pi\)
0.902483 0.430724i \(-0.141742\pi\)
\(824\) 0 0
\(825\) 12.4228i 0.432506i
\(826\) 0 0
\(827\) 31.4840i 1.09481i −0.836869 0.547404i \(-0.815616\pi\)
0.836869 0.547404i \(-0.184384\pi\)
\(828\) 0 0
\(829\) 40.4403i 1.40455i −0.711906 0.702275i \(-0.752168\pi\)
0.711906 0.702275i \(-0.247832\pi\)
\(830\) 0 0
\(831\) 14.3262 0.496971
\(832\) 0 0
\(833\) 2.02499 + 11.9284i 0.0701618 + 0.413295i
\(834\) 0 0
\(835\) 16.0550i 0.555606i
\(836\) 0 0
\(837\) −4.62845 −0.159983
\(838\) 0 0
\(839\) −39.5732 −1.36622 −0.683110 0.730316i \(-0.739373\pi\)
−0.683110 + 0.730316i \(0.739373\pi\)
\(840\) 0 0
\(841\) 39.6463 1.36711
\(842\) 0 0
\(843\) −12.9456 −0.445869
\(844\) 0 0
\(845\) 8.64220i 0.297301i
\(846\) 0 0
\(847\) −71.4564 + 6.02222i −2.45527 + 0.206926i
\(848\) 0 0
\(849\) 26.4719 0.908512
\(850\) 0 0
\(851\) 3.46620i 0.118820i
\(852\) 0 0
\(853\) 33.5540i 1.14887i 0.818551 + 0.574434i \(0.194778\pi\)
−0.818551 + 0.574434i \(0.805222\pi\)
\(854\) 0 0
\(855\) 10.2000i 0.348832i
\(856\) 0 0
\(857\) 19.5060i 0.666311i 0.942872 + 0.333156i \(0.108113\pi\)
−0.942872 + 0.333156i \(0.891887\pi\)
\(858\) 0 0
\(859\) 8.72616 0.297733 0.148866 0.988857i \(-0.452438\pi\)
0.148866 + 0.988857i \(0.452438\pi\)
\(860\) 0 0
\(861\) −0.872860 10.3569i −0.0297470 0.352962i
\(862\) 0 0
\(863\) 3.08055i 0.104863i −0.998625 0.0524315i \(-0.983303\pi\)
0.998625 0.0524315i \(-0.0166971\pi\)
\(864\) 0 0
\(865\) 20.8650 0.709432
\(866\) 0 0
\(867\) 14.0125 0.475889
\(868\) 0 0
\(869\) 59.0006 2.00146
\(870\) 0 0
\(871\) 18.7931 0.636779
\(872\) 0 0
\(873\) 17.1137i 0.579212i
\(874\) 0 0
\(875\) 2.69311 + 31.9550i 0.0910437 + 1.08028i
\(876\) 0 0
\(877\) −45.4712 −1.53545 −0.767727 0.640778i \(-0.778612\pi\)
−0.767727 + 0.640778i \(0.778612\pi\)
\(878\) 0 0
\(879\) 15.5285i 0.523762i
\(880\) 0 0
\(881\) 38.6765i 1.30304i 0.758629 + 0.651522i \(0.225869\pi\)
−0.758629 + 0.651522i \(0.774131\pi\)
\(882\) 0 0
\(883\) 15.5874i 0.524559i 0.964992 + 0.262279i \(0.0844742\pi\)
−0.964992 + 0.262279i \(0.915526\pi\)
\(884\) 0 0
\(885\) 16.6913i 0.561072i
\(886\) 0 0
\(887\) 31.8525 1.06950 0.534751 0.845009i \(-0.320405\pi\)
0.534751 + 0.845009i \(0.320405\pi\)
\(888\) 0 0
\(889\) −1.38745 16.4628i −0.0465337 0.552144i
\(890\) 0 0
\(891\) 6.17282i 0.206797i
\(892\) 0 0
\(893\) −28.8499 −0.965425
\(894\) 0 0
\(895\) 14.1305 0.472332
\(896\) 0 0
\(897\) −4.36814 −0.145848
\(898\) 0 0
\(899\) −38.3481 −1.27898
\(900\) 0 0
\(901\) 15.8568i 0.528268i
\(902\) 0 0
\(903\) 2.24441 + 26.6310i 0.0746892 + 0.886223i
\(904\) 0 0
\(905\) 39.4662 1.31190
\(906\) 0 0
\(907\) 5.05387i 0.167811i −0.996474 0.0839056i \(-0.973261\pi\)
0.996474 0.0839056i \(-0.0267394\pi\)
\(908\) 0 0
\(909\) 9.58527i 0.317923i
\(910\) 0 0
\(911\) 32.0900i 1.06319i 0.846999 + 0.531594i \(0.178407\pi\)
−0.846999 + 0.531594i \(0.821593\pi\)
\(912\) 0 0
\(913\) 75.3237i 2.49285i
\(914\) 0 0
\(915\) 24.6913 0.816269
\(916\) 0 0
\(917\) 26.7481 2.25428i 0.883300 0.0744429i
\(918\) 0 0
\(919\) 20.9332i 0.690522i 0.938507 + 0.345261i \(0.112210\pi\)
−0.938507 + 0.345261i \(0.887790\pi\)
\(920\) 0 0
\(921\) 13.1332 0.432753
\(922\) 0 0
\(923\) −14.1457 −0.465610
\(924\) 0 0
\(925\) 4.51686 0.148514
\(926\) 0 0
\(927\) −3.37155 −0.110736
\(928\) 0 0
\(929\) 21.7059i 0.712149i −0.934458 0.356074i \(-0.884115\pi\)
0.934458 0.356074i \(-0.115885\pi\)
\(930\) 0 0
\(931\) −40.7262 + 6.91376i −1.33475 + 0.226589i
\(932\) 0 0
\(933\) 3.87932 0.127003
\(934\) 0 0
\(935\) 18.4413i 0.603096i
\(936\) 0 0
\(937\) 18.1825i 0.593996i 0.954878 + 0.296998i \(0.0959854\pi\)
−0.954878 + 0.296998i \(0.904015\pi\)
\(938\) 0 0
\(939\) 21.1362i 0.689755i
\(940\) 0 0
\(941\) 12.8079i 0.417524i 0.977966 + 0.208762i \(0.0669435\pi\)
−0.977966 + 0.208762i \(0.933057\pi\)
\(942\) 0 0
\(943\) −6.06693 −0.197566
\(944\) 0 0
\(945\) 0.384044 + 4.55687i 0.0124930 + 0.148235i
\(946\) 0 0
\(947\) 23.8797i 0.775985i 0.921663 + 0.387992i \(0.126831\pi\)
−0.921663 + 0.387992i \(0.873169\pi\)
\(948\) 0 0
\(949\) 6.22247 0.201990
\(950\) 0 0
\(951\) 13.9422 0.452106
\(952\) 0 0
\(953\) −37.0006 −1.19857 −0.599283 0.800537i \(-0.704547\pi\)
−0.599283 + 0.800537i \(0.704547\pi\)
\(954\) 0 0
\(955\) −21.8356 −0.706582
\(956\) 0 0
\(957\) 51.1437i 1.65324i
\(958\) 0 0
\(959\) −17.8434 + 1.50381i −0.576194 + 0.0485606i
\(960\) 0 0
\(961\) −9.57743 −0.308949
\(962\) 0 0
\(963\) 5.28406i 0.170276i
\(964\) 0 0
\(965\) 14.3481i 0.461883i
\(966\) 0 0
\(967\) 7.93307i 0.255110i 0.991831 + 0.127555i \(0.0407130\pi\)
−0.991831 + 0.127555i \(0.959287\pi\)
\(968\) 0 0
\(969\) 10.2000i 0.327670i
\(970\) 0 0
\(971\) 49.2169 1.57945 0.789723 0.613464i \(-0.210224\pi\)
0.789723 + 0.613464i \(0.210224\pi\)
\(972\) 0 0
\(973\) −40.3731 + 3.40257i −1.29430 + 0.109082i
\(974\) 0 0
\(975\) 5.69220i 0.182296i
\(976\) 0 0
\(977\) −21.8075 −0.697684 −0.348842 0.937181i \(-0.613425\pi\)
−0.348842 + 0.937181i \(0.613425\pi\)
\(978\) 0 0
\(979\) 56.5605 1.80768
\(980\) 0 0
\(981\) 8.54562 0.272841
\(982\) 0 0
\(983\) −45.7436 −1.45900 −0.729498 0.683983i \(-0.760246\pi\)
−0.729498 + 0.683983i \(0.760246\pi\)
\(984\) 0 0
\(985\) 25.6344i 0.816779i
\(986\) 0 0
\(987\) 12.8888 1.08624i 0.410254 0.0345754i
\(988\) 0 0
\(989\) 15.6000 0.496053
\(990\) 0 0
\(991\) 41.0468i 1.30389i 0.758264 + 0.651947i \(0.226048\pi\)
−0.758264 + 0.651947i \(0.773952\pi\)
\(992\) 0 0
\(993\) 14.3037i 0.453915i
\(994\) 0 0
\(995\) 20.7688i 0.658414i
\(996\) 0 0
\(997\) 23.3009i 0.737946i −0.929440 0.368973i \(-0.879710\pi\)
0.929440 0.368973i \(-0.120290\pi\)
\(998\) 0 0
\(999\) 2.24441 0.0710099
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 672.2.b.b.223.6 yes 8
3.2 odd 2 2016.2.b.c.1567.3 8
4.3 odd 2 672.2.b.a.223.6 yes 8
7.6 odd 2 672.2.b.a.223.3 8
8.3 odd 2 1344.2.b.h.895.3 8
8.5 even 2 1344.2.b.g.895.3 8
12.11 even 2 2016.2.b.a.1567.3 8
21.20 even 2 2016.2.b.a.1567.6 8
24.5 odd 2 4032.2.b.q.3583.6 8
24.11 even 2 4032.2.b.o.3583.6 8
28.27 even 2 inner 672.2.b.b.223.3 yes 8
56.13 odd 2 1344.2.b.h.895.6 8
56.27 even 2 1344.2.b.g.895.6 8
84.83 odd 2 2016.2.b.c.1567.6 8
168.83 odd 2 4032.2.b.q.3583.3 8
168.125 even 2 4032.2.b.o.3583.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.b.a.223.3 8 7.6 odd 2
672.2.b.a.223.6 yes 8 4.3 odd 2
672.2.b.b.223.3 yes 8 28.27 even 2 inner
672.2.b.b.223.6 yes 8 1.1 even 1 trivial
1344.2.b.g.895.3 8 8.5 even 2
1344.2.b.g.895.6 8 56.27 even 2
1344.2.b.h.895.3 8 8.3 odd 2
1344.2.b.h.895.6 8 56.13 odd 2
2016.2.b.a.1567.3 8 12.11 even 2
2016.2.b.a.1567.6 8 21.20 even 2
2016.2.b.c.1567.3 8 3.2 odd 2
2016.2.b.c.1567.6 8 84.83 odd 2
4032.2.b.o.3583.3 8 168.125 even 2
4032.2.b.o.3583.6 8 24.11 even 2
4032.2.b.q.3583.3 8 168.83 odd 2
4032.2.b.q.3583.6 8 24.5 odd 2