Properties

Label 200.2.k.e.43.2
Level $200$
Weight $2$
Character 200.43
Analytic conductor $1.597$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [200,2,Mod(43,200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(200, 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("200.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 200.k (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: \(1.59700804043\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 43.2
Root \(-1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 200.43
Dual form 200.2.k.e.107.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 1.00000i) q^{2} +(2.22474 + 2.22474i) q^{3} -2.00000i q^{4} -4.44949 q^{6} +(2.00000 + 2.00000i) q^{8} +6.89898i q^{9} +0.550510 q^{11} +(4.44949 - 4.44949i) q^{12} -4.00000 q^{16} +(-1.67423 + 1.67423i) q^{17} +(-6.89898 - 6.89898i) q^{18} -8.34847i q^{19} +(-0.550510 + 0.550510i) q^{22} +8.89898i q^{24} +(-8.67423 + 8.67423i) q^{27} +(4.00000 - 4.00000i) q^{32} +(1.22474 + 1.22474i) q^{33} -3.34847i q^{34} +13.7980 q^{36} +(8.34847 + 8.34847i) q^{38} +12.7980 q^{41} +(-6.00000 - 6.00000i) q^{43} -1.10102i q^{44} +(-8.89898 - 8.89898i) q^{48} -7.00000i q^{49} -7.44949 q^{51} -17.3485i q^{54} +(18.5732 - 18.5732i) q^{57} +6.00000i q^{59} +8.00000i q^{64} -2.44949 q^{66} +(5.57321 - 5.57321i) q^{67} +(3.34847 + 3.34847i) q^{68} +(-13.7980 + 13.7980i) q^{72} +(7.22474 + 7.22474i) q^{73} -16.6969 q^{76} -17.8990 q^{81} +(-12.7980 + 12.7980i) q^{82} +(-10.0227 - 10.0227i) q^{83} +12.0000 q^{86} +(1.10102 + 1.10102i) q^{88} +13.8990i q^{89} +17.7980 q^{96} +(-12.0000 + 12.0000i) q^{97} +(7.00000 + 7.00000i) q^{98} +3.79796i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{3} - 8 q^{6} + 8 q^{8} + 12 q^{11} + 8 q^{12} - 16 q^{16} + 8 q^{17} - 8 q^{18} - 12 q^{22} - 20 q^{27} + 16 q^{32} + 16 q^{36} + 4 q^{38} + 12 q^{41} - 24 q^{43} - 16 q^{48} - 20 q^{51}+ \cdots + 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 + 1.00000i −0.707107 + 0.707107i
\(3\) 2.22474 + 2.22474i 1.28446 + 1.28446i 0.938104 + 0.346353i \(0.112580\pi\)
0.346353 + 0.938104i \(0.387420\pi\)
\(4\) 2.00000i 1.00000i
\(5\) 0 0
\(6\) −4.44949 −1.81650
\(7\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 2.00000 + 2.00000i 0.707107 + 0.707107i
\(9\) 6.89898i 2.29966i
\(10\) 0 0
\(11\) 0.550510 0.165985 0.0829925 0.996550i \(-0.473552\pi\)
0.0829925 + 0.996550i \(0.473552\pi\)
\(12\) 4.44949 4.44949i 1.28446 1.28446i
\(13\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −1.67423 + 1.67423i −0.406062 + 0.406062i −0.880363 0.474301i \(-0.842701\pi\)
0.474301 + 0.880363i \(0.342701\pi\)
\(18\) −6.89898 6.89898i −1.62611 1.62611i
\(19\) 8.34847i 1.91527i −0.287984 0.957635i \(-0.592985\pi\)
0.287984 0.957635i \(-0.407015\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.550510 + 0.550510i −0.117369 + 0.117369i
\(23\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 8.89898i 1.81650i
\(25\) 0 0
\(26\) 0 0
\(27\) −8.67423 + 8.67423i −1.66936 + 1.66936i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 4.00000 4.00000i 0.707107 0.707107i
\(33\) 1.22474 + 1.22474i 0.213201 + 0.213201i
\(34\) 3.34847i 0.574258i
\(35\) 0 0
\(36\) 13.7980 2.29966
\(37\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 8.34847 + 8.34847i 1.35430 + 1.35430i
\(39\) 0 0
\(40\) 0 0
\(41\) 12.7980 1.99871 0.999353 0.0359748i \(-0.0114536\pi\)
0.999353 + 0.0359748i \(0.0114536\pi\)
\(42\) 0 0
\(43\) −6.00000 6.00000i −0.914991 0.914991i 0.0816682 0.996660i \(-0.473975\pi\)
−0.996660 + 0.0816682i \(0.973975\pi\)
\(44\) 1.10102i 0.165985i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) −8.89898 8.89898i −1.28446 1.28446i
\(49\) 7.00000i 1.00000i
\(50\) 0 0
\(51\) −7.44949 −1.04314
\(52\) 0 0
\(53\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) 17.3485i 2.36083i
\(55\) 0 0
\(56\) 0 0
\(57\) 18.5732 18.5732i 2.46008 2.46008i
\(58\) 0 0
\(59\) 6.00000i 0.781133i 0.920575 + 0.390567i \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) −2.44949 −0.301511
\(67\) 5.57321 5.57321i 0.680876 0.680876i −0.279321 0.960198i \(-0.590109\pi\)
0.960198 + 0.279321i \(0.0901094\pi\)
\(68\) 3.34847 + 3.34847i 0.406062 + 0.406062i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −13.7980 + 13.7980i −1.62611 + 1.62611i
\(73\) 7.22474 + 7.22474i 0.845592 + 0.845592i 0.989580 0.143987i \(-0.0459924\pi\)
−0.143987 + 0.989580i \(0.545992\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −16.6969 −1.91527
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −17.8990 −1.98878
\(82\) −12.7980 + 12.7980i −1.41330 + 1.41330i
\(83\) −10.0227 10.0227i −1.10013 1.10013i −0.994394 0.105741i \(-0.966279\pi\)
−0.105741 0.994394i \(-0.533721\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) 1.10102 + 1.10102i 0.117369 + 0.117369i
\(89\) 13.8990i 1.47329i 0.676280 + 0.736644i \(0.263591\pi\)
−0.676280 + 0.736644i \(0.736409\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 17.7980 1.81650
\(97\) −12.0000 + 12.0000i −1.21842 + 1.21842i −0.250229 + 0.968187i \(0.580506\pi\)
−0.968187 + 0.250229i \(0.919494\pi\)
\(98\) 7.00000 + 7.00000i 0.707107 + 0.707107i
\(99\) 3.79796i 0.381709i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 7.44949 7.44949i 0.737609 0.737609i
\(103\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.32577 3.32577i 0.321514 0.321514i −0.527834 0.849348i \(-0.676996\pi\)
0.849348 + 0.527834i \(0.176996\pi\)
\(108\) 17.3485 + 17.3485i 1.66936 + 1.66936i
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −15.0227 15.0227i −1.41322 1.41322i −0.733148 0.680069i \(-0.761950\pi\)
−0.680069 0.733148i \(-0.738050\pi\)
\(114\) 37.1464i 3.47908i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −6.00000 6.00000i −0.552345 0.552345i
\(119\) 0 0
\(120\) 0 0
\(121\) −10.6969 −0.972449
\(122\) 0 0
\(123\) 28.4722 + 28.4722i 2.56725 + 2.56725i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(128\) −8.00000 8.00000i −0.707107 0.707107i
\(129\) 26.6969i 2.35053i
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 2.44949 2.44949i 0.213201 0.213201i
\(133\) 0 0
\(134\) 11.1464i 0.962905i
\(135\) 0 0
\(136\) −6.69694 −0.574258
\(137\) −11.6742 + 11.6742i −0.997397 + 0.997397i −0.999997 0.00259945i \(-0.999173\pi\)
0.00259945 + 0.999997i \(0.499173\pi\)
\(138\) 0 0
\(139\) 18.3485i 1.55630i −0.628080 0.778148i \(-0.716159\pi\)
0.628080 0.778148i \(-0.283841\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 27.5959i 2.29966i
\(145\) 0 0
\(146\) −14.4495 −1.19585
\(147\) 15.5732 15.5732i 1.28446 1.28446i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 16.6969 16.6969i 1.35430 1.35430i
\(153\) −11.5505 11.5505i −0.933803 0.933803i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 17.8990 17.8990i 1.40628 1.40628i
\(163\) −7.77526 7.77526i −0.609005 0.609005i 0.333681 0.942686i \(-0.391709\pi\)
−0.942686 + 0.333681i \(0.891709\pi\)
\(164\) 25.5959i 1.99871i
\(165\) 0 0
\(166\) 20.0454 1.55583
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 13.0000i 1.00000i
\(170\) 0 0
\(171\) 57.5959 4.40447
\(172\) −12.0000 + 12.0000i −0.914991 + 0.914991i
\(173\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.20204 −0.165985
\(177\) −13.3485 + 13.3485i −1.00333 + 1.00333i
\(178\) −13.8990 13.8990i −1.04177 1.04177i
\(179\) 26.1464i 1.95428i 0.212607 + 0.977138i \(0.431805\pi\)
−0.212607 + 0.977138i \(0.568195\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −0.921683 + 0.921683i −0.0674002 + 0.0674002i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −17.7980 + 17.7980i −1.28446 + 1.28446i
\(193\) 19.4722 + 19.4722i 1.40164 + 1.40164i 0.794904 + 0.606735i \(0.207521\pi\)
0.606735 + 0.794904i \(0.292479\pi\)
\(194\) 24.0000i 1.72310i
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(198\) −3.79796 3.79796i −0.269909 0.269909i
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 24.7980 1.74911
\(202\) 0 0
\(203\) 0 0
\(204\) 14.8990i 1.04314i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.59592i 0.317906i
\(210\) 0 0
\(211\) 15.0454 1.03577 0.517884 0.855451i \(-0.326720\pi\)
0.517884 + 0.855451i \(0.326720\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 6.65153i 0.454689i
\(215\) 0 0
\(216\) −34.6969 −2.36083
\(217\) 0 0
\(218\) 0 0
\(219\) 32.1464i 2.17225i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 30.0454 1.99859
\(227\) −2.00000 + 2.00000i −0.132745 + 0.132745i −0.770357 0.637613i \(-0.779922\pi\)
0.637613 + 0.770357i \(0.279922\pi\)
\(228\) −37.1464 37.1464i −2.46008 2.46008i
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.00000 + 4.00000i 0.262049 + 0.262049i 0.825886 0.563837i \(-0.190675\pi\)
−0.563837 + 0.825886i \(0.690675\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −1.69694 −0.109309 −0.0546547 0.998505i \(-0.517406\pi\)
−0.0546547 + 0.998505i \(0.517406\pi\)
\(242\) 10.6969 10.6969i 0.687625 0.687625i
\(243\) −13.7980 13.7980i −0.885139 0.885139i
\(244\) 0 0
\(245\) 0 0
\(246\) −56.9444 −3.63064
\(247\) 0 0
\(248\) 0 0
\(249\) 44.5959i 2.82615i
\(250\) 0 0
\(251\) −23.9444 −1.51136 −0.755678 0.654943i \(-0.772693\pi\)
−0.755678 + 0.654943i \(0.772693\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 8.00000 8.00000i 0.499026 0.499026i −0.412108 0.911135i \(-0.635208\pi\)
0.911135 + 0.412108i \(0.135208\pi\)
\(258\) 26.6969 + 26.6969i 1.66208 + 1.66208i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 18.0000 18.0000i 1.11204 1.11204i
\(263\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(264\) 4.89898i 0.301511i
\(265\) 0 0
\(266\) 0 0
\(267\) −30.9217 + 30.9217i −1.89238 + 1.89238i
\(268\) −11.1464 11.1464i −0.680876 0.680876i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 6.69694 6.69694i 0.406062 0.406062i
\(273\) 0 0
\(274\) 23.3485i 1.41053i
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(278\) 18.3485 + 18.3485i 1.10047 + 1.10047i
\(279\) 0 0
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 4.47219 + 4.47219i 0.265844 + 0.265844i 0.827423 0.561579i \(-0.189806\pi\)
−0.561579 + 0.827423i \(0.689806\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 27.5959 + 27.5959i 1.62611 + 1.62611i
\(289\) 11.3939i 0.670228i
\(290\) 0 0
\(291\) −53.3939 −3.13000
\(292\) 14.4495 14.4495i 0.845592 0.845592i
\(293\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(294\) 31.1464i 1.81650i
\(295\) 0 0
\(296\) 0 0
\(297\) −4.77526 + 4.77526i −0.277088 + 0.277088i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 33.3939i 1.91527i
\(305\) 0 0
\(306\) 23.1010 1.32060
\(307\) 17.8207 17.8207i 1.01708 1.01708i 0.0172273 0.999852i \(-0.494516\pi\)
0.999852 0.0172273i \(-0.00548391\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 24.0000 + 24.0000i 1.35656 + 1.35656i 0.878120 + 0.478440i \(0.158798\pi\)
0.478440 + 0.878120i \(0.341202\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 14.7980 0.825942
\(322\) 0 0
\(323\) 13.9773 + 13.9773i 0.777718 + 0.777718i
\(324\) 35.7980i 1.98878i
\(325\) 0 0
\(326\) 15.5505 0.861263
\(327\) 0 0
\(328\) 25.5959 + 25.5959i 1.41330 + 1.41330i
\(329\) 0 0
\(330\) 0 0
\(331\) 35.0454 1.92627 0.963135 0.269019i \(-0.0866994\pi\)
0.963135 + 0.269019i \(0.0866994\pi\)
\(332\) −20.0454 + 20.0454i −1.10013 + 1.10013i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 20.5732 20.5732i 1.12069 1.12069i 0.129057 0.991637i \(-0.458805\pi\)
0.991637 0.129057i \(-0.0411951\pi\)
\(338\) −13.0000 13.0000i −0.707107 0.707107i
\(339\) 66.8434i 3.63043i
\(340\) 0 0
\(341\) 0 0
\(342\) −57.5959 + 57.5959i −3.11443 + 3.11443i
\(343\) 0 0
\(344\) 24.0000i 1.29399i
\(345\) 0 0
\(346\) 0 0
\(347\) −16.6742 + 16.6742i −0.895120 + 0.895120i −0.995000 0.0998797i \(-0.968154\pi\)
0.0998797 + 0.995000i \(0.468154\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.20204 2.20204i 0.117369 0.117369i
\(353\) −16.0000 16.0000i −0.851594 0.851594i 0.138735 0.990329i \(-0.455696\pi\)
−0.990329 + 0.138735i \(0.955696\pi\)
\(354\) 26.6969i 1.41893i
\(355\) 0 0
\(356\) 27.7980 1.47329
\(357\) 0 0
\(358\) −26.1464 26.1464i −1.38188 1.38188i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −50.6969 −2.66826
\(362\) 0 0
\(363\) −23.7980 23.7980i −1.24907 1.24907i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(368\) 0 0
\(369\) 88.2929i 4.59634i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(374\) 1.84337i 0.0953182i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 11.6515i 0.598499i 0.954175 + 0.299249i \(0.0967363\pi\)
−0.954175 + 0.299249i \(0.903264\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 35.5959i 1.81650i
\(385\) 0 0
\(386\) −38.9444 −1.98222
\(387\) 41.3939 41.3939i 2.10417 2.10417i
\(388\) 24.0000 + 24.0000i 1.21842 + 1.21842i
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 14.0000 14.0000i 0.707107 0.707107i
\(393\) −40.0454 40.0454i −2.02002 2.02002i
\(394\) 0 0
\(395\) 0 0
\(396\) 7.59592 0.381709
\(397\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 37.2929 1.86232 0.931158 0.364615i \(-0.118800\pi\)
0.931158 + 0.364615i \(0.118800\pi\)
\(402\) −24.7980 + 24.7980i −1.23681 + 1.23681i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −14.8990 14.8990i −0.737609 0.737609i
\(409\) 18.3939i 0.909519i 0.890614 + 0.454759i \(0.150275\pi\)
−0.890614 + 0.454759i \(0.849725\pi\)
\(410\) 0 0
\(411\) −51.9444 −2.56223
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 40.8207 40.8207i 1.99900 1.99900i
\(418\) 4.59592 + 4.59592i 0.224794 + 0.224794i
\(419\) 22.8434i 1.11597i −0.829851 0.557986i \(-0.811574\pi\)
0.829851 0.557986i \(-0.188426\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) −15.0454 + 15.0454i −0.732399 + 0.732399i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −6.65153 6.65153i −0.321514 0.321514i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 34.6969 34.6969i 1.66936 1.66936i
\(433\) −17.2702 17.2702i −0.829951 0.829951i 0.157559 0.987510i \(-0.449638\pi\)
−0.987510 + 0.157559i \(0.949638\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −32.1464 32.1464i −1.53602 1.53602i
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 48.2929 2.29966
\(442\) 0 0
\(443\) 26.7196 + 26.7196i 1.26949 + 1.26949i 0.946353 + 0.323136i \(0.104737\pi\)
0.323136 + 0.946353i \(0.395263\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 16.1010i 0.759854i −0.925016 0.379927i \(-0.875949\pi\)
0.925016 0.379927i \(-0.124051\pi\)
\(450\) 0 0
\(451\) 7.04541 0.331755
\(452\) −30.0454 + 30.0454i −1.41322 + 1.41322i
\(453\) 0 0
\(454\) 4.00000i 0.187729i
\(455\) 0 0
\(456\) 74.2929 3.47908
\(457\) −3.92168 + 3.92168i −0.183449 + 0.183449i −0.792857 0.609408i \(-0.791407\pi\)
0.609408 + 0.792857i \(0.291407\pi\)
\(458\) 0 0
\(459\) 29.0454i 1.35572i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −8.00000 −0.370593
\(467\) −22.0000 + 22.0000i −1.01804 + 1.01804i −0.0182043 + 0.999834i \(0.505795\pi\)
−0.999834 + 0.0182043i \(0.994205\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −12.0000 + 12.0000i −0.552345 + 0.552345i
\(473\) −3.30306 3.30306i −0.151875 0.151875i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 1.69694 1.69694i 0.0772934 0.0772934i
\(483\) 0 0
\(484\) 21.3939i 0.972449i
\(485\) 0 0
\(486\) 27.5959 1.25178
\(487\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(488\) 0 0
\(489\) 34.5959i 1.56448i
\(490\) 0 0
\(491\) 42.0000 1.89543 0.947717 0.319113i \(-0.103385\pi\)
0.947717 + 0.319113i \(0.103385\pi\)
\(492\) 56.9444 56.9444i 2.56725 2.56725i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 44.5959 + 44.5959i 1.99839 + 1.99839i
\(499\) 14.0000i 0.626726i −0.949633 0.313363i \(-0.898544\pi\)
0.949633 0.313363i \(-0.101456\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 23.9444 23.9444i 1.06869 1.06869i
\(503\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −28.9217 + 28.9217i −1.28446 + 1.28446i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −16.0000 + 16.0000i −0.707107 + 0.707107i
\(513\) 72.4166 + 72.4166i 3.19727 + 3.19727i
\(514\) 16.0000i 0.705730i
\(515\) 0 0
\(516\) −53.3939 −2.35053
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −36.1918 −1.58559 −0.792797 0.609486i \(-0.791376\pi\)
−0.792797 + 0.609486i \(0.791376\pi\)
\(522\) 0 0
\(523\) −32.2702 32.2702i −1.41108 1.41108i −0.752619 0.658456i \(-0.771210\pi\)
−0.658456 0.752619i \(-0.728790\pi\)
\(524\) 36.0000i 1.57267i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −4.89898 4.89898i −0.213201 0.213201i
\(529\) 23.0000i 1.00000i
\(530\) 0 0
\(531\) −41.3939 −1.79634
\(532\) 0 0
\(533\) 0 0
\(534\) 61.8434i 2.67622i
\(535\) 0 0
\(536\) 22.2929 0.962905
\(537\) −58.1691 + 58.1691i −2.51018 + 2.51018i
\(538\) 0 0
\(539\) 3.85357i 0.165985i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 13.3939i 0.574258i
\(545\) 0 0
\(546\) 0 0
\(547\) −31.1691 + 31.1691i −1.33270 + 1.33270i −0.429746 + 0.902950i \(0.641397\pi\)
−0.902950 + 0.429746i \(0.858603\pi\)
\(548\) 23.3485 + 23.3485i 0.997397 + 0.997397i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −36.6969 −1.55630
\(557\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −4.10102 −0.173145
\(562\) 18.0000 18.0000i 0.759284 0.759284i
\(563\) −26.0000 26.0000i −1.09577 1.09577i −0.994900 0.100870i \(-0.967837\pi\)
−0.100870 0.994900i \(-0.532163\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −8.94439 −0.375961
\(567\) 0 0
\(568\) 0 0
\(569\) 40.5959i 1.70187i −0.525271 0.850935i \(-0.676036\pi\)
0.525271 0.850935i \(-0.323964\pi\)
\(570\) 0 0
\(571\) 22.0000 0.920671 0.460336 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −55.1918 −2.29966
\(577\) 32.8207 32.8207i 1.36634 1.36634i 0.500750 0.865592i \(-0.333058\pi\)
0.865592 0.500750i \(-0.166942\pi\)
\(578\) −11.3939 11.3939i −0.473923 0.473923i
\(579\) 86.6413i 3.60069i
\(580\) 0 0
\(581\) 0 0
\(582\) 53.3939 53.3939i 2.21325 2.21325i
\(583\) 0 0
\(584\) 28.8990i 1.19585i
\(585\) 0 0
\(586\) 0 0
\(587\) 13.3258 13.3258i 0.550013 0.550013i −0.376431 0.926445i \(-0.622849\pi\)
0.926445 + 0.376431i \(0.122849\pi\)
\(588\) −31.1464 31.1464i −1.28446 1.28446i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.97730 + 4.97730i 0.204393 + 0.204393i 0.801879 0.597486i \(-0.203834\pi\)
−0.597486 + 0.801879i \(0.703834\pi\)
\(594\) 9.55051i 0.391862i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 8.30306 0.338689 0.169344 0.985557i \(-0.445835\pi\)
0.169344 + 0.985557i \(0.445835\pi\)
\(602\) 0 0
\(603\) 38.4495 + 38.4495i 1.56578 + 1.56578i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) −33.3939 33.3939i −1.35430 1.35430i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −23.1010 + 23.1010i −0.933803 + 0.933803i
\(613\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(614\) 35.6413i 1.43837i
\(615\) 0 0
\(616\) 0 0
\(617\) 28.0000 28.0000i 1.12724 1.12724i 0.136613 0.990624i \(-0.456378\pi\)
0.990624 0.136613i \(-0.0436217\pi\)
\(618\) 0 0
\(619\) 26.0000i 1.04503i 0.852631 + 0.522514i \(0.175006\pi\)
−0.852631 + 0.522514i \(0.824994\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −48.0000 −1.91847
\(627\) 10.2247 10.2247i 0.408337 0.408337i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 33.4722 + 33.4722i 1.33040 + 1.33040i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) −14.7980 + 14.7980i −0.584029 + 0.584029i
\(643\) −6.00000 6.00000i −0.236617 0.236617i 0.578831 0.815448i \(-0.303509\pi\)
−0.815448 + 0.578831i \(0.803509\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −27.9546 −1.09986
\(647\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) −35.7980 35.7980i −1.40628 1.40628i
\(649\) 3.30306i 0.129657i
\(650\) 0 0
\(651\) 0 0
\(652\) −15.5505 + 15.5505i −0.609005 + 0.609005i
\(653\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −51.1918 −1.99871
\(657\) −49.8434 + 49.8434i −1.94457 + 1.94457i
\(658\) 0 0
\(659\) 50.6413i 1.97271i 0.164644 + 0.986353i \(0.447352\pi\)
−0.164644 + 0.986353i \(0.552648\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −35.0454 + 35.0454i −1.36208 + 1.36208i
\(663\) 0 0
\(664\) 40.0908i 1.55583i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −36.0000 36.0000i −1.38770 1.38770i −0.830134 0.557564i \(-0.811736\pi\)
−0.557564 0.830134i \(-0.688264\pi\)
\(674\) 41.1464i 1.58490i
\(675\) 0 0
\(676\) 26.0000 1.00000
\(677\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 66.8434 + 66.8434i 2.56710 + 2.56710i
\(679\) 0 0
\(680\) 0 0
\(681\) −8.89898 −0.341010
\(682\) 0 0
\(683\) 36.7196 + 36.7196i 1.40504 + 1.40504i 0.782937 + 0.622101i \(0.213721\pi\)
0.622101 + 0.782937i \(0.286279\pi\)
\(684\) 115.192i 4.40447i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 24.0000 + 24.0000i 0.914991 + 0.914991i
\(689\) 0 0
\(690\) 0 0
\(691\) 45.0454 1.71361 0.856804 0.515642i \(-0.172447\pi\)
0.856804 + 0.515642i \(0.172447\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 33.3485i 1.26589i
\(695\) 0 0
\(696\) 0 0
\(697\) −21.4268 + 21.4268i −0.811597 + 0.811597i
\(698\) 0 0
\(699\) 17.7980i 0.673181i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 4.40408i 0.165985i
\(705\) 0 0
\(706\) 32.0000 1.20434
\(707\) 0 0
\(708\) 26.6969 + 26.6969i 1.00333 + 1.00333i
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −27.7980 + 27.7980i −1.04177 + 1.04177i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 52.2929 1.95428
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 50.6969 50.6969i 1.88674 1.88674i
\(723\) −3.77526 3.77526i −0.140403 0.140403i
\(724\) 0 0
\(725\) 0 0
\(726\) 47.5959 1.76645
\(727\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(728\) 0 0
\(729\) 7.69694i 0.285072i
\(730\) 0 0
\(731\) 20.0908 0.743086
\(732\) 0 0
\(733\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.06811 3.06811i 0.113015 0.113015i
\(738\) −88.2929 88.2929i −3.25010 3.25010i
\(739\) 34.0000i 1.25071i −0.780340 0.625355i \(-0.784954\pi\)
0.780340 0.625355i \(-0.215046\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 69.1464 69.1464i 2.52994 2.52994i
\(748\) 1.84337 + 1.84337i 0.0674002 + 0.0674002i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −53.2702 53.2702i −1.94127 1.94127i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(758\) −11.6515 11.6515i −0.423203 0.423203i
\(759\) 0 0
\(760\) 0 0
\(761\) −17.2020 −0.623574 −0.311787 0.950152i \(-0.600927\pi\)
−0.311787 + 0.950152i \(0.600927\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 35.5959 + 35.5959i 1.28446 + 1.28446i
\(769\) 55.0908i 1.98663i −0.115454 0.993313i \(-0.536832\pi\)
0.115454 0.993313i \(-0.463168\pi\)
\(770\) 0 0
\(771\) 35.5959 1.28196
\(772\) 38.9444 38.9444i 1.40164 1.40164i
\(773\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(774\) 82.7878i 2.97574i
\(775\) 0 0
\(776\) −48.0000 −1.72310
\(777\) 0 0
\(778\) 0 0
\(779\) 106.843i 3.82806i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 28.0000i 1.00000i
\(785\) 0 0
\(786\) 80.0908 2.85674
\(787\) 18.0000 18.0000i 0.641631 0.641631i −0.309326 0.950956i \(-0.600103\pi\)
0.950956 + 0.309326i \(0.100103\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −7.59592 + 7.59592i −0.269909 + 0.269909i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −95.8888 −3.38806
\(802\) −37.2929 + 37.2929i −1.31686 + 1.31686i
\(803\) 3.97730 + 3.97730i 0.140356 + 0.140356i
\(804\) 49.5959i 1.74911i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 6.00000i 0.210949i 0.994422 + 0.105474i \(0.0336361\pi\)
−0.994422 + 0.105474i \(0.966364\pi\)
\(810\) 0 0
\(811\) −38.0000 −1.33436 −0.667180 0.744896i \(-0.732499\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 29.7980 1.04314
\(817\) −50.0908 + 50.0908i −1.75246 + 1.75246i
\(818\) −18.3939 18.3939i −0.643127 0.643127i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 51.9444 51.9444i 1.81177 1.81177i
\(823\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 40.0681 40.0681i 1.39330 1.39330i 0.575510 0.817795i \(-0.304804\pi\)
0.817795 0.575510i \(-0.195196\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 11.7196 + 11.7196i 0.406062 + 0.406062i
\(834\) 81.6413i 2.82701i
\(835\) 0 0
\(836\) −9.19184 −0.317906
\(837\) 0 0
\(838\) 22.8434 + 22.8434i 0.789111 + 0.789111i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −40.0454 40.0454i −1.37924 1.37924i
\(844\) 30.0908i 1.03577i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 19.8990i 0.682931i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 13.3031 0.454689
\(857\) 25.0681 25.0681i 0.856310 0.856310i −0.134591 0.990901i \(-0.542972\pi\)
0.990901 + 0.134591i \(0.0429720\pi\)
\(858\) 0 0
\(859\) 21.6515i 0.738741i 0.929282 + 0.369370i \(0.120427\pi\)
−0.929282 + 0.369370i \(0.879573\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(864\) 69.3939i 2.36083i
\(865\) 0 0
\(866\) 34.5403 1.17373
\(867\) −25.3485 + 25.3485i −0.860879 + 0.860879i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −82.7878 82.7878i −2.80194 2.80194i
\(874\) 0 0
\(875\) 0 0
\(876\) 64.2929 2.17225
\(877\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) −48.2929 + 48.2929i −1.62611 + 1.62611i
\(883\) 22.2247 + 22.2247i 0.747922 + 0.747922i 0.974089 0.226166i \(-0.0726193\pi\)
−0.226166 + 0.974089i \(0.572619\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −53.4393 −1.79533
\(887\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −9.85357 −0.330107
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 16.1010 + 16.1010i 0.537298 + 0.537298i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) −7.04541 + 7.04541i −0.234586 + 0.234586i
\(903\) 0 0
\(904\) 60.0908i 1.99859i
\(905\) 0 0
\(906\) 0 0
\(907\) −42.0000 + 42.0000i −1.39459 + 1.39459i −0.579898 + 0.814689i \(0.696908\pi\)
−0.814689 + 0.579898i \(0.803092\pi\)
\(908\) 4.00000 + 4.00000i 0.132745 + 0.132745i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −74.2929 + 74.2929i −2.46008 + 2.46008i
\(913\) −5.51760 5.51760i −0.182606 0.182606i
\(914\) 7.84337i 0.259436i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 29.0454 + 29.0454i 0.958641 + 0.958641i
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 79.2929 2.61279
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 54.0000i 1.77168i −0.463988 0.885841i \(-0.653582\pi\)
0.463988 0.885841i \(-0.346418\pi\)
\(930\) 0 0
\(931\) −58.4393 −1.91527
\(932\) 8.00000 8.00000i 0.262049 0.262049i
\(933\) 0 0
\(934\) 44.0000i 1.43972i
\(935\) 0 0
\(936\) 0 0
\(937\) 2.82066 2.82066i 0.0921470 0.0921470i −0.659531 0.751678i \(-0.729245\pi\)
0.751678 + 0.659531i \(0.229245\pi\)
\(938\) 0 0
\(939\) 106.788i 3.48489i
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 24.0000i 0.781133i
\(945\) 0 0
\(946\) 6.60612 0.214784
\(947\) 38.0000 38.0000i 1.23483 1.23483i 0.272749 0.962085i \(-0.412067\pi\)
0.962085 0.272749i \(-0.0879328\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 41.7196 + 41.7196i 1.35143 + 1.35143i 0.884062 + 0.467370i \(0.154798\pi\)
0.467370 + 0.884062i \(0.345202\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 0 0
\(963\) 22.9444 + 22.9444i 0.739373 + 0.739373i
\(964\) 3.39388i 0.109309i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(968\) −21.3939 21.3939i −0.687625 0.687625i
\(969\) 62.1918i 1.99789i
\(970\) 0 0
\(971\) −53.9444 −1.73116 −0.865579 0.500773i \(-0.833049\pi\)
−0.865579 + 0.500773i \(0.833049\pi\)
\(972\) −27.5959 + 27.5959i −0.885139 + 0.885139i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.3258 18.3258i 0.586293 0.586293i −0.350332 0.936625i \(-0.613931\pi\)
0.936625 + 0.350332i \(0.113931\pi\)
\(978\) 34.5959 + 34.5959i 1.10626 + 1.10626i
\(979\) 7.65153i 0.244544i
\(980\) 0 0
\(981\) 0 0
\(982\) −42.0000 + 42.0000i −1.34027 + 1.34027i
\(983\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(984\) 113.889i 3.63064i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 77.9671 + 77.9671i 2.47421 + 2.47421i
\(994\) 0 0
\(995\) 0 0
\(996\) −89.1918 −2.82615
\(997\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(998\) 14.0000 + 14.0000i 0.443162 + 0.443162i
\(999\) 0 0
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 200.2.k.e.43.2 4
4.3 odd 2 800.2.o.e.143.1 4
5.2 odd 4 inner 200.2.k.e.107.2 yes 4
5.3 odd 4 200.2.k.f.107.1 yes 4
5.4 even 2 200.2.k.f.43.1 yes 4
8.3 odd 2 CM 200.2.k.e.43.2 4
8.5 even 2 800.2.o.e.143.1 4
20.3 even 4 800.2.o.f.207.2 4
20.7 even 4 800.2.o.e.207.1 4
20.19 odd 2 800.2.o.f.143.2 4
40.3 even 4 200.2.k.f.107.1 yes 4
40.13 odd 4 800.2.o.f.207.2 4
40.19 odd 2 200.2.k.f.43.1 yes 4
40.27 even 4 inner 200.2.k.e.107.2 yes 4
40.29 even 2 800.2.o.f.143.2 4
40.37 odd 4 800.2.o.e.207.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.2.k.e.43.2 4 1.1 even 1 trivial
200.2.k.e.43.2 4 8.3 odd 2 CM
200.2.k.e.107.2 yes 4 5.2 odd 4 inner
200.2.k.e.107.2 yes 4 40.27 even 4 inner
200.2.k.f.43.1 yes 4 5.4 even 2
200.2.k.f.43.1 yes 4 40.19 odd 2
200.2.k.f.107.1 yes 4 5.3 odd 4
200.2.k.f.107.1 yes 4 40.3 even 4
800.2.o.e.143.1 4 4.3 odd 2
800.2.o.e.143.1 4 8.5 even 2
800.2.o.e.207.1 4 20.7 even 4
800.2.o.e.207.1 4 40.37 odd 4
800.2.o.f.143.2 4 20.19 odd 2
800.2.o.f.143.2 4 40.29 even 2
800.2.o.f.207.2 4 20.3 even 4
800.2.o.f.207.2 4 40.13 odd 4