Properties

Label 1600.2.o.g.607.4
Level $1600$
Weight $2$
Character 1600.607
Analytic conductor $12.776$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(543,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.543");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.o (of order \(4\), degree \(2\), 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: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 607.4
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1600.607
Dual form 1600.2.o.g.543.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.41421 - 1.41421i) q^{3} +(2.44949 - 2.44949i) q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(1.41421 - 1.41421i) q^{3} +(2.44949 - 2.44949i) q^{7} -1.00000i q^{9} -3.46410 q^{11} +(-2.82843 - 2.82843i) q^{13} +(-4.89898 - 4.89898i) q^{17} +3.46410i q^{19} -6.92820i q^{21} +(-2.44949 - 2.44949i) q^{23} +(2.82843 + 2.82843i) q^{27} -6.92820 q^{29} -8.00000i q^{31} +(-4.89898 + 4.89898i) q^{33} +(5.65685 - 5.65685i) q^{37} -8.00000 q^{39} +6.00000 q^{41} +(-1.41421 + 1.41421i) q^{43} +(2.44949 - 2.44949i) q^{47} -5.00000i q^{49} -13.8564 q^{51} +(8.48528 + 8.48528i) q^{53} +(4.89898 + 4.89898i) q^{57} +10.3923i q^{59} -13.8564i q^{61} +(-2.44949 - 2.44949i) q^{63} +(1.41421 + 1.41421i) q^{67} -6.92820 q^{69} +(4.89898 - 4.89898i) q^{73} +(-8.48528 + 8.48528i) q^{77} +8.00000 q^{79} +11.0000 q^{81} +(-4.24264 + 4.24264i) q^{83} +(-9.79796 + 9.79796i) q^{87} -6.00000i q^{89} -13.8564 q^{91} +(-11.3137 - 11.3137i) q^{93} +(4.89898 + 4.89898i) q^{97} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 64 q^{39} + 48 q^{41} + 64 q^{79} + 88 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(-1\)

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
\(3\) 1.41421 1.41421i 0.816497 0.816497i −0.169102 0.985599i \(-0.554087\pi\)
0.985599 + 0.169102i \(0.0540867\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.44949 2.44949i 0.925820 0.925820i −0.0716124 0.997433i \(-0.522814\pi\)
0.997433 + 0.0716124i \(0.0228145\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) −3.46410 −1.04447 −0.522233 0.852803i \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) −2.82843 2.82843i −0.784465 0.784465i 0.196116 0.980581i \(-0.437167\pi\)
−0.980581 + 0.196116i \(0.937167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.89898 4.89898i −1.18818 1.18818i −0.977571 0.210606i \(-0.932456\pi\)
−0.210606 0.977571i \(-0.567544\pi\)
\(18\) 0 0
\(19\) 3.46410i 0.794719i 0.917663 + 0.397360i \(0.130073\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 6.92820i 1.51186i
\(22\) 0 0
\(23\) −2.44949 2.44949i −0.510754 0.510754i 0.404004 0.914757i \(-0.367618\pi\)
−0.914757 + 0.404004i \(0.867618\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.82843 + 2.82843i 0.544331 + 0.544331i
\(28\) 0 0
\(29\) −6.92820 −1.28654 −0.643268 0.765641i \(-0.722422\pi\)
−0.643268 + 0.765641i \(0.722422\pi\)
\(30\) 0 0
\(31\) 8.00000i 1.43684i −0.695608 0.718421i \(-0.744865\pi\)
0.695608 0.718421i \(-0.255135\pi\)
\(32\) 0 0
\(33\) −4.89898 + 4.89898i −0.852803 + 0.852803i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.65685 5.65685i 0.929981 0.929981i −0.0677230 0.997704i \(-0.521573\pi\)
0.997704 + 0.0677230i \(0.0215734\pi\)
\(38\) 0 0
\(39\) −8.00000 −1.28103
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −1.41421 + 1.41421i −0.215666 + 0.215666i −0.806669 0.591003i \(-0.798732\pi\)
0.591003 + 0.806669i \(0.298732\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.44949 2.44949i 0.357295 0.357295i −0.505520 0.862815i \(-0.668699\pi\)
0.862815 + 0.505520i \(0.168699\pi\)
\(48\) 0 0
\(49\) 5.00000i 0.714286i
\(50\) 0 0
\(51\) −13.8564 −1.94029
\(52\) 0 0
\(53\) 8.48528 + 8.48528i 1.16554 + 1.16554i 0.983243 + 0.182300i \(0.0583542\pi\)
0.182300 + 0.983243i \(0.441646\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.89898 + 4.89898i 0.648886 + 0.648886i
\(58\) 0 0
\(59\) 10.3923i 1.35296i 0.736460 + 0.676481i \(0.236496\pi\)
−0.736460 + 0.676481i \(0.763504\pi\)
\(60\) 0 0
\(61\) 13.8564i 1.77413i −0.461644 0.887066i \(-0.652740\pi\)
0.461644 0.887066i \(-0.347260\pi\)
\(62\) 0 0
\(63\) −2.44949 2.44949i −0.308607 0.308607i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.41421 + 1.41421i 0.172774 + 0.172774i 0.788197 0.615423i \(-0.211015\pi\)
−0.615423 + 0.788197i \(0.711015\pi\)
\(68\) 0 0
\(69\) −6.92820 −0.834058
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 4.89898 4.89898i 0.573382 0.573382i −0.359690 0.933072i \(-0.617117\pi\)
0.933072 + 0.359690i \(0.117117\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.48528 + 8.48528i −0.966988 + 0.966988i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 11.0000 1.22222
\(82\) 0 0
\(83\) −4.24264 + 4.24264i −0.465690 + 0.465690i −0.900515 0.434825i \(-0.856810\pi\)
0.434825 + 0.900515i \(0.356810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −9.79796 + 9.79796i −1.05045 + 1.05045i
\(88\) 0 0
\(89\) 6.00000i 0.635999i −0.948091 0.317999i \(-0.896989\pi\)
0.948091 0.317999i \(-0.103011\pi\)
\(90\) 0 0
\(91\) −13.8564 −1.45255
\(92\) 0 0
\(93\) −11.3137 11.3137i −1.17318 1.17318i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.89898 + 4.89898i 0.497416 + 0.497416i 0.910633 0.413217i \(-0.135595\pi\)
−0.413217 + 0.910633i \(0.635595\pi\)
\(98\) 0 0
\(99\) 3.46410i 0.348155i
\(100\) 0 0
\(101\) 6.92820i 0.689382i −0.938716 0.344691i \(-0.887984\pi\)
0.938716 0.344691i \(-0.112016\pi\)
\(102\) 0 0
\(103\) 12.2474 + 12.2474i 1.20678 + 1.20678i 0.972065 + 0.234712i \(0.0754147\pi\)
0.234712 + 0.972065i \(0.424585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.7279 12.7279i −1.23045 1.23045i −0.963789 0.266666i \(-0.914078\pi\)
−0.266666 0.963789i \(-0.585922\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 16.0000i 1.51865i
\(112\) 0 0
\(113\) 9.79796 9.79796i 0.921714 0.921714i −0.0754362 0.997151i \(-0.524035\pi\)
0.997151 + 0.0754362i \(0.0240349\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.82843 + 2.82843i −0.261488 + 0.261488i
\(118\) 0 0
\(119\) −24.0000 −2.20008
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 8.48528 8.48528i 0.765092 0.765092i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −7.34847 + 7.34847i −0.652071 + 0.652071i −0.953491 0.301420i \(-0.902539\pi\)
0.301420 + 0.953491i \(0.402539\pi\)
\(128\) 0 0
\(129\) 4.00000i 0.352180i
\(130\) 0 0
\(131\) 3.46410 0.302660 0.151330 0.988483i \(-0.451644\pi\)
0.151330 + 0.988483i \(0.451644\pi\)
\(132\) 0 0
\(133\) 8.48528 + 8.48528i 0.735767 + 0.735767i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(138\) 0 0
\(139\) 3.46410i 0.293821i −0.989150 0.146911i \(-0.953067\pi\)
0.989150 0.146911i \(-0.0469330\pi\)
\(140\) 0 0
\(141\) 6.92820i 0.583460i
\(142\) 0 0
\(143\) 9.79796 + 9.79796i 0.819346 + 0.819346i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −7.07107 7.07107i −0.583212 0.583212i
\(148\) 0 0
\(149\) −13.8564 −1.13516 −0.567581 0.823318i \(-0.692120\pi\)
−0.567581 + 0.823318i \(0.692120\pi\)
\(150\) 0 0
\(151\) 8.00000i 0.651031i 0.945537 + 0.325515i \(0.105538\pi\)
−0.945537 + 0.325515i \(0.894462\pi\)
\(152\) 0 0
\(153\) −4.89898 + 4.89898i −0.396059 + 0.396059i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −5.65685 + 5.65685i −0.451466 + 0.451466i −0.895841 0.444375i \(-0.853426\pi\)
0.444375 + 0.895841i \(0.353426\pi\)
\(158\) 0 0
\(159\) 24.0000 1.90332
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) −1.41421 + 1.41421i −0.110770 + 0.110770i −0.760319 0.649550i \(-0.774958\pi\)
0.649550 + 0.760319i \(0.274958\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.34847 7.34847i 0.568642 0.568642i −0.363106 0.931748i \(-0.618284\pi\)
0.931748 + 0.363106i \(0.118284\pi\)
\(168\) 0 0
\(169\) 3.00000i 0.230769i
\(170\) 0 0
\(171\) 3.46410 0.264906
\(172\) 0 0
\(173\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 14.6969 + 14.6969i 1.10469 + 1.10469i
\(178\) 0 0
\(179\) 10.3923i 0.776757i 0.921500 + 0.388379i \(0.126965\pi\)
−0.921500 + 0.388379i \(0.873035\pi\)
\(180\) 0 0
\(181\) 6.92820i 0.514969i 0.966282 + 0.257485i \(0.0828937\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 0 0
\(183\) −19.5959 19.5959i −1.44857 1.44857i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 16.9706 + 16.9706i 1.24101 + 1.24101i
\(188\) 0 0
\(189\) 13.8564 1.00791
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 4.89898 4.89898i 0.352636 0.352636i −0.508453 0.861090i \(-0.669783\pi\)
0.861090 + 0.508453i \(0.169783\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.48528 8.48528i 0.604551 0.604551i −0.336966 0.941517i \(-0.609401\pi\)
0.941517 + 0.336966i \(0.109401\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) −16.9706 + 16.9706i −1.19110 + 1.19110i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2.44949 + 2.44949i −0.170251 + 0.170251i
\(208\) 0 0
\(209\) 12.0000i 0.830057i
\(210\) 0 0
\(211\) 3.46410 0.238479 0.119239 0.992866i \(-0.461954\pi\)
0.119239 + 0.992866i \(0.461954\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −19.5959 19.5959i −1.33026 1.33026i
\(218\) 0 0
\(219\) 13.8564i 0.936329i
\(220\) 0 0
\(221\) 27.7128i 1.86417i
\(222\) 0 0
\(223\) −7.34847 7.34847i −0.492090 0.492090i 0.416874 0.908964i \(-0.363126\pi\)
−0.908964 + 0.416874i \(0.863126\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.7279 + 12.7279i 0.844782 + 0.844782i 0.989476 0.144695i \(-0.0462199\pi\)
−0.144695 + 0.989476i \(0.546220\pi\)
\(228\) 0 0
\(229\) 20.7846 1.37349 0.686743 0.726900i \(-0.259040\pi\)
0.686743 + 0.726900i \(0.259040\pi\)
\(230\) 0 0
\(231\) 24.0000i 1.57908i
\(232\) 0 0
\(233\) −14.6969 + 14.6969i −0.962828 + 0.962828i −0.999333 0.0365050i \(-0.988378\pi\)
0.0365050 + 0.999333i \(0.488378\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 11.3137 11.3137i 0.734904 0.734904i
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 7.07107 7.07107i 0.453609 0.453609i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 9.79796 9.79796i 0.623429 0.623429i
\(248\) 0 0
\(249\) 12.0000i 0.760469i
\(250\) 0 0
\(251\) 24.2487 1.53057 0.765283 0.643695i \(-0.222599\pi\)
0.765283 + 0.643695i \(0.222599\pi\)
\(252\) 0 0
\(253\) 8.48528 + 8.48528i 0.533465 + 0.533465i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −9.79796 9.79796i −0.611180 0.611180i 0.332074 0.943253i \(-0.392252\pi\)
−0.943253 + 0.332074i \(0.892252\pi\)
\(258\) 0 0
\(259\) 27.7128i 1.72199i
\(260\) 0 0
\(261\) 6.92820i 0.428845i
\(262\) 0 0
\(263\) 7.34847 + 7.34847i 0.453126 + 0.453126i 0.896391 0.443265i \(-0.146180\pi\)
−0.443265 + 0.896391i \(0.646180\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −8.48528 8.48528i −0.519291 0.519291i
\(268\) 0 0
\(269\) −13.8564 −0.844840 −0.422420 0.906400i \(-0.638819\pi\)
−0.422420 + 0.906400i \(0.638819\pi\)
\(270\) 0 0
\(271\) 8.00000i 0.485965i −0.970031 0.242983i \(-0.921874\pi\)
0.970031 0.242983i \(-0.0781258\pi\)
\(272\) 0 0
\(273\) −19.5959 + 19.5959i −1.18600 + 1.18600i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.65685 5.65685i 0.339887 0.339887i −0.516437 0.856325i \(-0.672742\pi\)
0.856325 + 0.516437i \(0.172742\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 9.89949 9.89949i 0.588464 0.588464i −0.348751 0.937215i \(-0.613394\pi\)
0.937215 + 0.348751i \(0.113394\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 14.6969 14.6969i 0.867533 0.867533i
\(288\) 0 0
\(289\) 31.0000i 1.82353i
\(290\) 0 0
\(291\) 13.8564 0.812277
\(292\) 0 0
\(293\) −16.9706 16.9706i −0.991431 0.991431i 0.00853273 0.999964i \(-0.497284\pi\)
−0.999964 + 0.00853273i \(0.997284\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −9.79796 9.79796i −0.568535 0.568535i
\(298\) 0 0
\(299\) 13.8564i 0.801337i
\(300\) 0 0
\(301\) 6.92820i 0.399335i
\(302\) 0 0
\(303\) −9.79796 9.79796i −0.562878 0.562878i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −9.89949 9.89949i −0.564994 0.564994i 0.365728 0.930722i \(-0.380820\pi\)
−0.930722 + 0.365728i \(0.880820\pi\)
\(308\) 0 0
\(309\) 34.6410 1.97066
\(310\) 0 0
\(311\) 24.0000i 1.36092i 0.732787 + 0.680458i \(0.238219\pi\)
−0.732787 + 0.680458i \(0.761781\pi\)
\(312\) 0 0
\(313\) 9.79796 9.79796i 0.553813 0.553813i −0.373726 0.927539i \(-0.621920\pi\)
0.927539 + 0.373726i \(0.121920\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.48528 8.48528i 0.476581 0.476581i −0.427456 0.904036i \(-0.640590\pi\)
0.904036 + 0.427456i \(0.140590\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) −36.0000 −2.00932
\(322\) 0 0
\(323\) 16.9706 16.9706i 0.944267 0.944267i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12.0000i 0.661581i
\(330\) 0 0
\(331\) 17.3205 0.952021 0.476011 0.879440i \(-0.342082\pi\)
0.476011 + 0.879440i \(0.342082\pi\)
\(332\) 0 0
\(333\) −5.65685 5.65685i −0.309994 0.309994i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −9.79796 9.79796i −0.533729 0.533729i 0.387951 0.921680i \(-0.373183\pi\)
−0.921680 + 0.387951i \(0.873183\pi\)
\(338\) 0 0
\(339\) 27.7128i 1.50515i
\(340\) 0 0
\(341\) 27.7128i 1.50073i
\(342\) 0 0
\(343\) 4.89898 + 4.89898i 0.264520 + 0.264520i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 21.2132 + 21.2132i 1.13878 + 1.13878i 0.988668 + 0.150116i \(0.0479647\pi\)
0.150116 + 0.988668i \(0.452035\pi\)
\(348\) 0 0
\(349\) 6.92820 0.370858 0.185429 0.982658i \(-0.440632\pi\)
0.185429 + 0.982658i \(0.440632\pi\)
\(350\) 0 0
\(351\) 16.0000i 0.854017i
\(352\) 0 0
\(353\) 19.5959 19.5959i 1.04299 1.04299i 0.0439518 0.999034i \(-0.486005\pi\)
0.999034 0.0439518i \(-0.0139948\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −33.9411 + 33.9411i −1.79635 + 1.79635i
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) 0 0
\(363\) 1.41421 1.41421i 0.0742270 0.0742270i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 17.1464 17.1464i 0.895036 0.895036i −0.0999556 0.994992i \(-0.531870\pi\)
0.994992 + 0.0999556i \(0.0318701\pi\)
\(368\) 0 0
\(369\) 6.00000i 0.312348i
\(370\) 0 0
\(371\) 41.5692 2.15817
\(372\) 0 0
\(373\) −22.6274 22.6274i −1.17160 1.17160i −0.981827 0.189776i \(-0.939224\pi\)
−0.189776 0.981827i \(-0.560776\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 19.5959 + 19.5959i 1.00924 + 1.00924i
\(378\) 0 0
\(379\) 24.2487i 1.24557i −0.782392 0.622786i \(-0.786001\pi\)
0.782392 0.622786i \(-0.213999\pi\)
\(380\) 0 0
\(381\) 20.7846i 1.06483i
\(382\) 0 0
\(383\) 2.44949 + 2.44949i 0.125163 + 0.125163i 0.766914 0.641750i \(-0.221792\pi\)
−0.641750 + 0.766914i \(0.721792\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.41421 + 1.41421i 0.0718885 + 0.0718885i
\(388\) 0 0
\(389\) −13.8564 −0.702548 −0.351274 0.936273i \(-0.614251\pi\)
−0.351274 + 0.936273i \(0.614251\pi\)
\(390\) 0 0
\(391\) 24.0000i 1.21373i
\(392\) 0 0
\(393\) 4.89898 4.89898i 0.247121 0.247121i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 2.82843 2.82843i 0.141955 0.141955i −0.632558 0.774513i \(-0.717995\pi\)
0.774513 + 0.632558i \(0.217995\pi\)
\(398\) 0 0
\(399\) 24.0000 1.20150
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) −22.6274 + 22.6274i −1.12715 + 1.12715i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −19.5959 + 19.5959i −0.971334 + 0.971334i
\(408\) 0 0
\(409\) 22.0000i 1.08783i 0.839140 + 0.543915i \(0.183059\pi\)
−0.839140 + 0.543915i \(0.816941\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 25.4558 + 25.4558i 1.25260 + 1.25260i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −4.89898 4.89898i −0.239904 0.239904i
\(418\) 0 0
\(419\) 38.1051i 1.86156i 0.365584 + 0.930778i \(0.380869\pi\)
−0.365584 + 0.930778i \(0.619131\pi\)
\(420\) 0 0
\(421\) 13.8564i 0.675320i 0.941268 + 0.337660i \(0.109635\pi\)
−0.941268 + 0.337660i \(0.890365\pi\)
\(422\) 0 0
\(423\) −2.44949 2.44949i −0.119098 0.119098i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −33.9411 33.9411i −1.64253 1.64253i
\(428\) 0 0
\(429\) 27.7128 1.33799
\(430\) 0 0
\(431\) 24.0000i 1.15604i 0.816023 + 0.578020i \(0.196174\pi\)
−0.816023 + 0.578020i \(0.803826\pi\)
\(432\) 0 0
\(433\) −14.6969 + 14.6969i −0.706290 + 0.706290i −0.965753 0.259463i \(-0.916454\pi\)
0.259463 + 0.965753i \(0.416454\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.48528 8.48528i 0.405906 0.405906i
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −5.00000 −0.238095
\(442\) 0 0
\(443\) 12.7279 12.7279i 0.604722 0.604722i −0.336840 0.941562i \(-0.609358\pi\)
0.941562 + 0.336840i \(0.109358\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −19.5959 + 19.5959i −0.926855 + 0.926855i
\(448\) 0 0
\(449\) 6.00000i 0.283158i −0.989927 0.141579i \(-0.954782\pi\)
0.989927 0.141579i \(-0.0452178\pi\)
\(450\) 0 0
\(451\) −20.7846 −0.978709
\(452\) 0 0
\(453\) 11.3137 + 11.3137i 0.531564 + 0.531564i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 29.3939 + 29.3939i 1.37499 + 1.37499i 0.852879 + 0.522108i \(0.174854\pi\)
0.522108 + 0.852879i \(0.325146\pi\)
\(458\) 0 0
\(459\) 27.7128i 1.29352i
\(460\) 0 0
\(461\) 20.7846i 0.968036i 0.875058 + 0.484018i \(0.160823\pi\)
−0.875058 + 0.484018i \(0.839177\pi\)
\(462\) 0 0
\(463\) 12.2474 + 12.2474i 0.569187 + 0.569187i 0.931901 0.362713i \(-0.118150\pi\)
−0.362713 + 0.931901i \(0.618150\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.24264 + 4.24264i 0.196326 + 0.196326i 0.798423 0.602097i \(-0.205668\pi\)
−0.602097 + 0.798423i \(0.705668\pi\)
\(468\) 0 0
\(469\) 6.92820 0.319915
\(470\) 0 0
\(471\) 16.0000i 0.737241i
\(472\) 0 0
\(473\) 4.89898 4.89898i 0.225255 0.225255i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 8.48528 8.48528i 0.388514 0.388514i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −32.0000 −1.45907
\(482\) 0 0
\(483\) −16.9706 + 16.9706i −0.772187 + 0.772187i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −12.2474 + 12.2474i −0.554985 + 0.554985i −0.927875 0.372890i \(-0.878367\pi\)
0.372890 + 0.927875i \(0.378367\pi\)
\(488\) 0 0
\(489\) 4.00000i 0.180886i
\(490\) 0 0
\(491\) 10.3923 0.468998 0.234499 0.972116i \(-0.424655\pi\)
0.234499 + 0.972116i \(0.424655\pi\)
\(492\) 0 0
\(493\) 33.9411 + 33.9411i 1.52863 + 1.52863i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 10.3923i 0.465223i −0.972570 0.232612i \(-0.925273\pi\)
0.972570 0.232612i \(-0.0747271\pi\)
\(500\) 0 0
\(501\) 20.7846i 0.928588i
\(502\) 0 0
\(503\) 12.2474 + 12.2474i 0.546087 + 0.546087i 0.925307 0.379220i \(-0.123808\pi\)
−0.379220 + 0.925307i \(0.623808\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 4.24264 + 4.24264i 0.188422 + 0.188422i
\(508\) 0 0
\(509\) −6.92820 −0.307087 −0.153544 0.988142i \(-0.549069\pi\)
−0.153544 + 0.988142i \(0.549069\pi\)
\(510\) 0 0
\(511\) 24.0000i 1.06170i
\(512\) 0 0
\(513\) −9.79796 + 9.79796i −0.432590 + 0.432590i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −8.48528 + 8.48528i −0.373182 + 0.373182i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) −24.0416 + 24.0416i −1.05127 + 1.05127i −0.0526543 + 0.998613i \(0.516768\pi\)
−0.998613 + 0.0526543i \(0.983232\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −39.1918 + 39.1918i −1.70722 + 1.70722i
\(528\) 0 0
\(529\) 11.0000i 0.478261i
\(530\) 0 0
\(531\) 10.3923 0.450988
\(532\) 0 0
\(533\) −16.9706 16.9706i −0.735077 0.735077i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 14.6969 + 14.6969i 0.634220 + 0.634220i
\(538\) 0 0
\(539\) 17.3205i 0.746047i
\(540\) 0 0
\(541\) 34.6410i 1.48933i −0.667436 0.744667i \(-0.732608\pi\)
0.667436 0.744667i \(-0.267392\pi\)
\(542\) 0 0
\(543\) 9.79796 + 9.79796i 0.420471 + 0.420471i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 7.07107 + 7.07107i 0.302337 + 0.302337i 0.841928 0.539591i \(-0.181421\pi\)
−0.539591 + 0.841928i \(0.681421\pi\)
\(548\) 0 0
\(549\) −13.8564 −0.591377
\(550\) 0 0
\(551\) 24.0000i 1.02243i
\(552\) 0 0
\(553\) 19.5959 19.5959i 0.833303 0.833303i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 48.0000 2.02656
\(562\) 0 0
\(563\) −4.24264 + 4.24264i −0.178806 + 0.178806i −0.790835 0.612029i \(-0.790353\pi\)
0.612029 + 0.790835i \(0.290353\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 26.9444 26.9444i 1.13156 1.13156i
\(568\) 0 0
\(569\) 6.00000i 0.251533i −0.992060 0.125767i \(-0.959861\pi\)
0.992060 0.125767i \(-0.0401390\pi\)
\(570\) 0 0
\(571\) 3.46410 0.144968 0.0724841 0.997370i \(-0.476907\pi\)
0.0724841 + 0.997370i \(0.476907\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 9.79796 + 9.79796i 0.407894 + 0.407894i 0.881004 0.473109i \(-0.156868\pi\)
−0.473109 + 0.881004i \(0.656868\pi\)
\(578\) 0 0
\(579\) 13.8564i 0.575853i
\(580\) 0 0
\(581\) 20.7846i 0.862291i
\(582\) 0 0
\(583\) −29.3939 29.3939i −1.21737 1.21737i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.24264 4.24264i −0.175113 0.175113i 0.614109 0.789221i \(-0.289516\pi\)
−0.789221 + 0.614109i \(0.789516\pi\)
\(588\) 0 0
\(589\) 27.7128 1.14189
\(590\) 0 0
\(591\) 24.0000i 0.987228i
\(592\) 0 0
\(593\) 19.5959 19.5959i 0.804708 0.804708i −0.179119 0.983827i \(-0.557325\pi\)
0.983827 + 0.179119i \(0.0573249\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −22.6274 + 22.6274i −0.926079 + 0.926079i
\(598\) 0 0
\(599\) −48.0000 −1.96123 −0.980613 0.195952i \(-0.937220\pi\)
−0.980613 + 0.195952i \(0.937220\pi\)
\(600\) 0 0
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 0 0
\(603\) 1.41421 1.41421i 0.0575912 0.0575912i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 2.44949 2.44949i 0.0994217 0.0994217i −0.655646 0.755068i \(-0.727604\pi\)
0.755068 + 0.655646i \(0.227604\pi\)
\(608\) 0 0
\(609\) 48.0000i 1.94506i
\(610\) 0 0
\(611\) −13.8564 −0.560570
\(612\) 0 0
\(613\) −14.1421 14.1421i −0.571195 0.571195i 0.361267 0.932462i \(-0.382344\pi\)
−0.932462 + 0.361267i \(0.882344\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −4.89898 4.89898i −0.197225 0.197225i 0.601584 0.798810i \(-0.294536\pi\)
−0.798810 + 0.601584i \(0.794536\pi\)
\(618\) 0 0
\(619\) 17.3205i 0.696170i 0.937463 + 0.348085i \(0.113168\pi\)
−0.937463 + 0.348085i \(0.886832\pi\)
\(620\) 0 0
\(621\) 13.8564i 0.556038i
\(622\) 0 0
\(623\) −14.6969 14.6969i −0.588820 0.588820i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −16.9706 16.9706i −0.677739 0.677739i
\(628\) 0 0
\(629\) −55.4256 −2.20996
\(630\) 0 0
\(631\) 8.00000i 0.318475i −0.987240 0.159237i \(-0.949096\pi\)
0.987240 0.159237i \(-0.0509036\pi\)
\(632\) 0 0
\(633\) 4.89898 4.89898i 0.194717 0.194717i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −14.1421 + 14.1421i −0.560332 + 0.560332i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) 32.5269 32.5269i 1.28274 1.28274i 0.343632 0.939105i \(-0.388343\pi\)
0.939105 0.343632i \(-0.111657\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.34847 7.34847i 0.288898 0.288898i −0.547746 0.836644i \(-0.684514\pi\)
0.836644 + 0.547746i \(0.184514\pi\)
\(648\) 0 0
\(649\) 36.0000i 1.41312i
\(650\) 0 0
\(651\) −55.4256 −2.17230
\(652\) 0 0
\(653\) −8.48528 8.48528i −0.332055 0.332055i 0.521312 0.853366i \(-0.325443\pi\)
−0.853366 + 0.521312i \(0.825443\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −4.89898 4.89898i −0.191127 0.191127i
\(658\) 0 0
\(659\) 10.3923i 0.404827i 0.979300 + 0.202413i \(0.0648785\pi\)
−0.979300 + 0.202413i \(0.935122\pi\)
\(660\) 0 0
\(661\) 27.7128i 1.07790i −0.842337 0.538952i \(-0.818821\pi\)
0.842337 0.538952i \(-0.181179\pi\)
\(662\) 0 0
\(663\) 39.1918 + 39.1918i 1.52208 + 1.52208i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16.9706 + 16.9706i 0.657103 + 0.657103i
\(668\) 0 0
\(669\) −20.7846 −0.803579
\(670\) 0 0
\(671\) 48.0000i 1.85302i
\(672\) 0 0
\(673\) −24.4949 + 24.4949i −0.944209 + 0.944209i −0.998524 0.0543149i \(-0.982703\pi\)
0.0543149 + 0.998524i \(0.482703\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8.48528 8.48528i 0.326116 0.326116i −0.524992 0.851107i \(-0.675932\pi\)
0.851107 + 0.524992i \(0.175932\pi\)
\(678\) 0 0
\(679\) 24.0000 0.921035
\(680\) 0 0
\(681\) 36.0000 1.37952
\(682\) 0 0
\(683\) 21.2132 21.2132i 0.811701 0.811701i −0.173188 0.984889i \(-0.555407\pi\)
0.984889 + 0.173188i \(0.0554069\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 29.3939 29.3939i 1.12145 1.12145i
\(688\) 0 0
\(689\) 48.0000i 1.82865i
\(690\) 0 0
\(691\) −31.1769 −1.18603 −0.593013 0.805193i \(-0.702062\pi\)
−0.593013 + 0.805193i \(0.702062\pi\)
\(692\) 0 0
\(693\) 8.48528 + 8.48528i 0.322329 + 0.322329i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −29.3939 29.3939i −1.11337 1.11337i
\(698\) 0 0
\(699\) 41.5692i 1.57229i
\(700\) 0 0
\(701\) 27.7128i 1.04670i −0.852118 0.523349i \(-0.824682\pi\)
0.852118 0.523349i \(-0.175318\pi\)
\(702\) 0 0
\(703\) 19.5959 + 19.5959i 0.739074 + 0.739074i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −16.9706 16.9706i −0.638244 0.638244i
\(708\) 0 0
\(709\) 34.6410 1.30097 0.650485 0.759519i \(-0.274566\pi\)
0.650485 + 0.759519i \(0.274566\pi\)
\(710\) 0 0
\(711\) 8.00000i 0.300023i
\(712\) 0 0
\(713\) −19.5959 + 19.5959i −0.733873 + 0.733873i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 33.9411 33.9411i 1.26755 1.26755i
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 60.0000 2.23452
\(722\) 0 0
\(723\) 14.1421 14.1421i 0.525952 0.525952i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −7.34847 + 7.34847i −0.272540 + 0.272540i −0.830122 0.557582i \(-0.811729\pi\)
0.557582 + 0.830122i \(0.311729\pi\)
\(728\) 0 0
\(729\) 13.0000i 0.481481i
\(730\) 0 0
\(731\) 13.8564 0.512498
\(732\) 0 0
\(733\) −22.6274 22.6274i −0.835763 0.835763i 0.152535 0.988298i \(-0.451256\pi\)
−0.988298 + 0.152535i \(0.951256\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.89898 4.89898i −0.180456 0.180456i
\(738\) 0 0
\(739\) 10.3923i 0.382287i 0.981562 + 0.191144i \(0.0612196\pi\)
−0.981562 + 0.191144i \(0.938780\pi\)
\(740\) 0 0
\(741\) 27.7128i 1.01806i
\(742\) 0 0
\(743\) −26.9444 26.9444i −0.988494 0.988494i 0.0114409 0.999935i \(-0.496358\pi\)
−0.999935 + 0.0114409i \(0.996358\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 4.24264 + 4.24264i 0.155230 + 0.155230i
\(748\) 0 0
\(749\) −62.3538 −2.27836
\(750\) 0 0
\(751\) 32.0000i 1.16770i −0.811863 0.583848i \(-0.801546\pi\)
0.811863 0.583848i \(-0.198454\pi\)
\(752\) 0 0
\(753\) 34.2929 34.2929i 1.24970 1.24970i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 11.3137 11.3137i 0.411204 0.411204i −0.470954 0.882158i \(-0.656090\pi\)
0.882158 + 0.470954i \(0.156090\pi\)
\(758\) 0 0
\(759\) 24.0000 0.871145
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 29.3939 29.3939i 1.06135 1.06135i
\(768\) 0 0
\(769\) 14.0000i 0.504853i −0.967616 0.252426i \(-0.918771\pi\)
0.967616 0.252426i \(-0.0812286\pi\)
\(770\) 0 0
\(771\) −27.7128 −0.998053
\(772\) 0 0
\(773\) −8.48528 8.48528i −0.305194 0.305194i 0.537848 0.843042i \(-0.319238\pi\)
−0.843042 + 0.537848i \(0.819238\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −39.1918 39.1918i −1.40600 1.40600i
\(778\) 0 0
\(779\) 20.7846i 0.744686i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −19.5959 19.5959i −0.700301 0.700301i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 15.5563 + 15.5563i 0.554524 + 0.554524i 0.927743 0.373219i \(-0.121746\pi\)
−0.373219 + 0.927743i \(0.621746\pi\)
\(788\) 0 0
\(789\) 20.7846 0.739952
\(790\) 0 0
\(791\) 48.0000i 1.70668i
\(792\) 0 0
\(793\) −39.1918 + 39.1918i −1.39174 + 1.39174i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −8.48528 + 8.48528i −0.300564 + 0.300564i −0.841235 0.540670i \(-0.818171\pi\)
0.540670 + 0.841235i \(0.318171\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) −16.9706 + 16.9706i −0.598878 + 0.598878i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −19.5959 + 19.5959i −0.689809 + 0.689809i
\(808\) 0 0
\(809\) 6.00000i 0.210949i −0.994422 0.105474i \(-0.966364\pi\)
0.994422 0.105474i \(-0.0336361\pi\)
\(810\) 0 0
\(811\) −31.1769 −1.09477 −0.547385 0.836881i \(-0.684377\pi\)
−0.547385 + 0.836881i \(0.684377\pi\)
\(812\) 0 0
\(813\) −11.3137 11.3137i −0.396789 0.396789i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −4.89898 4.89898i −0.171394 0.171394i
\(818\) 0 0
\(819\) 13.8564i 0.484182i
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −12.2474 12.2474i −0.426919 0.426919i 0.460658 0.887578i \(-0.347613\pi\)
−0.887578 + 0.460658i \(0.847613\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 4.24264 + 4.24264i 0.147531 + 0.147531i 0.777014 0.629483i \(-0.216733\pi\)
−0.629483 + 0.777014i \(0.716733\pi\)
\(828\) 0 0
\(829\) −13.8564 −0.481253 −0.240626 0.970618i \(-0.577353\pi\)
−0.240626 + 0.970618i \(0.577353\pi\)
\(830\) 0 0
\(831\) 16.0000i 0.555034i
\(832\) 0 0
\(833\) −24.4949 + 24.4949i −0.848698 + 0.848698i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 22.6274 22.6274i 0.782118 0.782118i
\(838\) 0 0
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) 19.0000 0.655172
\(842\) 0 0
\(843\) −42.4264 + 42.4264i −1.46124 + 1.46124i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.44949 2.44949i 0.0841655 0.0841655i
\(848\) 0 0
\(849\) 28.0000i 0.960958i
\(850\) 0 0
\(851\) −27.7128 −0.949983
\(852\) 0 0
\(853\) 19.7990 + 19.7990i 0.677905 + 0.677905i 0.959526 0.281621i \(-0.0908721\pi\)
−0.281621 + 0.959526i \(0.590872\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(858\) 0 0
\(859\) 45.0333i 1.53652i −0.640140 0.768259i \(-0.721124\pi\)
0.640140 0.768259i \(-0.278876\pi\)
\(860\) 0 0
\(861\) 41.5692i 1.41668i
\(862\) 0 0
\(863\) −31.8434 31.8434i −1.08396 1.08396i −0.996136 0.0878249i \(-0.972008\pi\)
−0.0878249 0.996136i \(-0.527992\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 43.8406 + 43.8406i 1.48891 + 1.48891i
\(868\) 0 0
\(869\) −27.7128 −0.940093
\(870\) 0 0
\(871\) 8.00000i 0.271070i
\(872\) 0 0
\(873\) 4.89898 4.89898i 0.165805 0.165805i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −28.2843 + 28.2843i −0.955092 + 0.955092i −0.999034 0.0439421i \(-0.986008\pi\)
0.0439421 + 0.999034i \(0.486008\pi\)
\(878\) 0 0
\(879\) −48.0000 −1.61900
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 18.3848 18.3848i 0.618697 0.618697i −0.326500 0.945197i \(-0.605869\pi\)
0.945197 + 0.326500i \(0.105869\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 31.8434 31.8434i 1.06920 1.06920i 0.0717747 0.997421i \(-0.477134\pi\)
0.997421 0.0717747i \(-0.0228663\pi\)
\(888\) 0 0
\(889\) 36.0000i 1.20740i
\(890\) 0 0
\(891\) −38.1051 −1.27657
\(892\) 0 0
\(893\) 8.48528 + 8.48528i 0.283949 + 0.283949i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 19.5959 + 19.5959i 0.654289 + 0.654289i
\(898\) 0 0
\(899\) 55.4256i 1.84855i
\(900\) 0 0
\(901\) 83.1384i 2.76974i
\(902\) 0 0
\(903\) 9.79796 + 9.79796i 0.326056 + 0.326056i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 18.3848 + 18.3848i 0.610456 + 0.610456i 0.943065 0.332609i \(-0.107929\pi\)
−0.332609 + 0.943065i \(0.607929\pi\)
\(908\) 0 0
\(909\) −6.92820 −0.229794
\(910\) 0 0
\(911\) 24.0000i 0.795155i 0.917568 + 0.397578i \(0.130149\pi\)
−0.917568 + 0.397578i \(0.869851\pi\)
\(912\) 0 0
\(913\) 14.6969 14.6969i 0.486398 0.486398i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.48528 8.48528i 0.280209 0.280209i
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 12.2474 12.2474i 0.402259 0.402259i
\(928\) 0 0
\(929\) 42.0000i 1.37798i 0.724773 + 0.688988i \(0.241945\pi\)
−0.724773 + 0.688988i \(0.758055\pi\)
\(930\) 0 0
\(931\) 17.3205 0.567657
\(932\) 0 0
\(933\) 33.9411 + 33.9411i 1.11118 + 1.11118i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.89898 + 4.89898i 0.160043 + 0.160043i 0.782586 0.622543i \(-0.213900\pi\)
−0.622543 + 0.782586i \(0.713900\pi\)
\(938\) 0 0
\(939\) 27.7128i 0.904373i
\(940\) 0 0
\(941\) 6.92820i 0.225853i −0.993603 0.112926i \(-0.963978\pi\)
0.993603 0.112926i \(-0.0360225\pi\)
\(942\) 0 0
\(943\) −14.6969 14.6969i −0.478598 0.478598i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −21.2132 21.2132i −0.689336 0.689336i 0.272749 0.962085i \(-0.412067\pi\)
−0.962085 + 0.272749i \(0.912067\pi\)
\(948\) 0 0
\(949\) −27.7128 −0.899596
\(950\) 0 0
\(951\) 24.0000i 0.778253i
\(952\) 0 0
\(953\) −19.5959 + 19.5959i −0.634774 + 0.634774i −0.949262 0.314488i \(-0.898167\pi\)
0.314488 + 0.949262i \(0.398167\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 33.9411 33.9411i 1.09716 1.09716i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −33.0000 −1.06452
\(962\) 0 0
\(963\) −12.7279 + 12.7279i −0.410152 + 0.410152i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −17.1464 + 17.1464i −0.551392 + 0.551392i −0.926842 0.375450i \(-0.877488\pi\)
0.375450 + 0.926842i \(0.377488\pi\)
\(968\) 0 0
\(969\) 48.0000i 1.54198i
\(970\) 0 0
\(971\) 17.3205 0.555842 0.277921 0.960604i \(-0.410355\pi\)
0.277921 + 0.960604i \(0.410355\pi\)
\(972\) 0 0
\(973\) −8.48528 8.48528i −0.272026 0.272026i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 24.4949 + 24.4949i 0.783661 + 0.783661i 0.980447 0.196785i \(-0.0630502\pi\)
−0.196785 + 0.980447i \(0.563050\pi\)
\(978\) 0 0
\(979\) 20.7846i 0.664279i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 17.1464 + 17.1464i 0.546886 + 0.546886i 0.925539 0.378653i \(-0.123613\pi\)
−0.378653 + 0.925539i \(0.623613\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −16.9706 16.9706i −0.540179 0.540179i
\(988\) 0 0
\(989\) 6.92820 0.220304
\(990\) 0 0
\(991\) 16.0000i 0.508257i −0.967170 0.254128i \(-0.918211\pi\)
0.967170 0.254128i \(-0.0817886\pi\)
\(992\) 0 0
\(993\) 24.4949 24.4949i 0.777322 0.777322i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 36.7696 36.7696i 1.16450 1.16450i 0.181025 0.983479i \(-0.442059\pi\)
0.983479 0.181025i \(-0.0579415\pi\)
\(998\) 0 0
\(999\) 32.0000 1.01244
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 1600.2.o.g.607.4 yes 8
4.3 odd 2 1600.2.o.j.607.1 yes 8
5.2 odd 4 1600.2.o.j.543.2 yes 8
5.3 odd 4 1600.2.o.j.543.3 yes 8
5.4 even 2 inner 1600.2.o.g.607.1 yes 8
8.3 odd 2 1600.2.o.j.607.3 yes 8
8.5 even 2 inner 1600.2.o.g.607.2 yes 8
20.3 even 4 inner 1600.2.o.g.543.2 yes 8
20.7 even 4 inner 1600.2.o.g.543.3 yes 8
20.19 odd 2 1600.2.o.j.607.4 yes 8
40.3 even 4 inner 1600.2.o.g.543.4 yes 8
40.13 odd 4 1600.2.o.j.543.1 yes 8
40.19 odd 2 1600.2.o.j.607.2 yes 8
40.27 even 4 inner 1600.2.o.g.543.1 8
40.29 even 2 inner 1600.2.o.g.607.3 yes 8
40.37 odd 4 1600.2.o.j.543.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1600.2.o.g.543.1 8 40.27 even 4 inner
1600.2.o.g.543.2 yes 8 20.3 even 4 inner
1600.2.o.g.543.3 yes 8 20.7 even 4 inner
1600.2.o.g.543.4 yes 8 40.3 even 4 inner
1600.2.o.g.607.1 yes 8 5.4 even 2 inner
1600.2.o.g.607.2 yes 8 8.5 even 2 inner
1600.2.o.g.607.3 yes 8 40.29 even 2 inner
1600.2.o.g.607.4 yes 8 1.1 even 1 trivial
1600.2.o.j.543.1 yes 8 40.13 odd 4
1600.2.o.j.543.2 yes 8 5.2 odd 4
1600.2.o.j.543.3 yes 8 5.3 odd 4
1600.2.o.j.543.4 yes 8 40.37 odd 4
1600.2.o.j.607.1 yes 8 4.3 odd 2
1600.2.o.j.607.2 yes 8 40.19 odd 2
1600.2.o.j.607.3 yes 8 8.3 odd 2
1600.2.o.j.607.4 yes 8 20.19 odd 2