Properties

Label 950.2.b.a.799.2
Level $950$
Weight $2$
Character 950.799
Analytic conductor $7.586$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [950,2,Mod(799,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.b (of order \(2\), degree \(1\), not 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: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 950.799
Dual form 950.2.b.a.799.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} -5.00000i q^{7} -1.00000i q^{8} -6.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} -5.00000i q^{7} -1.00000i q^{8} -6.00000 q^{9} -4.00000 q^{11} -3.00000i q^{12} +1.00000i q^{13} +5.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} -6.00000i q^{18} -1.00000 q^{19} +15.0000 q^{21} -4.00000i q^{22} -7.00000i q^{23} +3.00000 q^{24} -1.00000 q^{26} -9.00000i q^{27} +5.00000i q^{28} +3.00000 q^{29} -2.00000 q^{31} +1.00000i q^{32} -12.0000i q^{33} +3.00000 q^{34} +6.00000 q^{36} -2.00000i q^{37} -1.00000i q^{38} -3.00000 q^{39} -6.00000 q^{41} +15.0000i q^{42} -6.00000i q^{43} +4.00000 q^{44} +7.00000 q^{46} +3.00000i q^{48} -18.0000 q^{49} +9.00000 q^{51} -1.00000i q^{52} +13.0000i q^{53} +9.00000 q^{54} -5.00000 q^{56} -3.00000i q^{57} +3.00000i q^{58} +9.00000 q^{59} -12.0000 q^{61} -2.00000i q^{62} +30.0000i q^{63} -1.00000 q^{64} +12.0000 q^{66} -3.00000i q^{67} +3.00000i q^{68} +21.0000 q^{69} +6.00000i q^{72} -11.0000i q^{73} +2.00000 q^{74} +1.00000 q^{76} +20.0000i q^{77} -3.00000i q^{78} +2.00000 q^{79} +9.00000 q^{81} -6.00000i q^{82} +10.0000i q^{83} -15.0000 q^{84} +6.00000 q^{86} +9.00000i q^{87} +4.00000i q^{88} -2.00000 q^{89} +5.00000 q^{91} +7.00000i q^{92} -6.00000i q^{93} -3.00000 q^{96} -2.00000i q^{97} -18.0000i q^{98} +24.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 6 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 6 q^{6} - 12 q^{9} - 8 q^{11} + 10 q^{14} + 2 q^{16} - 2 q^{19} + 30 q^{21} + 6 q^{24} - 2 q^{26} + 6 q^{29} - 4 q^{31} + 6 q^{34} + 12 q^{36} - 6 q^{39} - 12 q^{41} + 8 q^{44} + 14 q^{46} - 36 q^{49} + 18 q^{51} + 18 q^{54} - 10 q^{56} + 18 q^{59} - 24 q^{61} - 2 q^{64} + 24 q^{66} + 42 q^{69} + 4 q^{74} + 2 q^{76} + 4 q^{79} + 18 q^{81} - 30 q^{84} + 12 q^{86} - 4 q^{89} + 10 q^{91} - 6 q^{96} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 3.00000i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) − 5.00000i − 1.88982i −0.327327 0.944911i \(-0.606148\pi\)
0.327327 0.944911i \(-0.393852\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) − 3.00000i − 0.866025i
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 5.00000 1.33631
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 3.00000i − 0.727607i −0.931476 0.363803i \(-0.881478\pi\)
0.931476 0.363803i \(-0.118522\pi\)
\(18\) − 6.00000i − 1.41421i
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 15.0000 3.27327
\(22\) − 4.00000i − 0.852803i
\(23\) − 7.00000i − 1.45960i −0.683660 0.729800i \(-0.739613\pi\)
0.683660 0.729800i \(-0.260387\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) − 9.00000i − 1.73205i
\(28\) 5.00000i 0.944911i
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 12.0000i − 2.08893i
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) −3.00000 −0.480384
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 15.0000i 2.31455i
\(43\) − 6.00000i − 0.914991i −0.889212 0.457496i \(-0.848747\pi\)
0.889212 0.457496i \(-0.151253\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 7.00000 1.03209
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 3.00000i 0.433013i
\(49\) −18.0000 −2.57143
\(50\) 0 0
\(51\) 9.00000 1.26025
\(52\) − 1.00000i − 0.138675i
\(53\) 13.0000i 1.78569i 0.450367 + 0.892844i \(0.351293\pi\)
−0.450367 + 0.892844i \(0.648707\pi\)
\(54\) 9.00000 1.22474
\(55\) 0 0
\(56\) −5.00000 −0.668153
\(57\) − 3.00000i − 0.397360i
\(58\) 3.00000i 0.393919i
\(59\) 9.00000 1.17170 0.585850 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) − 2.00000i − 0.254000i
\(63\) 30.0000i 3.77964i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 12.0000 1.47710
\(67\) − 3.00000i − 0.366508i −0.983066 0.183254i \(-0.941337\pi\)
0.983066 0.183254i \(-0.0586631\pi\)
\(68\) 3.00000i 0.363803i
\(69\) 21.0000 2.52810
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 6.00000i 0.707107i
\(73\) − 11.0000i − 1.28745i −0.765256 0.643726i \(-0.777388\pi\)
0.765256 0.643726i \(-0.222612\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 20.0000i 2.27921i
\(78\) − 3.00000i − 0.339683i
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) − 6.00000i − 0.662589i
\(83\) 10.0000i 1.09764i 0.835940 + 0.548821i \(0.184923\pi\)
−0.835940 + 0.548821i \(0.815077\pi\)
\(84\) −15.0000 −1.63663
\(85\) 0 0
\(86\) 6.00000 0.646997
\(87\) 9.00000i 0.964901i
\(88\) 4.00000i 0.426401i
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 5.00000 0.524142
\(92\) 7.00000i 0.729800i
\(93\) − 6.00000i − 0.622171i
\(94\) 0 0
\(95\) 0 0
\(96\) −3.00000 −0.306186
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) − 18.0000i − 1.81827i
\(99\) 24.0000 2.41209
\(100\) 0 0
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) 9.00000i 0.891133i
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −13.0000 −1.26267
\(107\) − 13.0000i − 1.25676i −0.777908 0.628379i \(-0.783719\pi\)
0.777908 0.628379i \(-0.216281\pi\)
\(108\) 9.00000i 0.866025i
\(109\) −19.0000 −1.81987 −0.909935 0.414751i \(-0.863869\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) − 5.00000i − 0.472456i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 3.00000 0.280976
\(115\) 0 0
\(116\) −3.00000 −0.278543
\(117\) − 6.00000i − 0.554700i
\(118\) 9.00000i 0.828517i
\(119\) −15.0000 −1.37505
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) − 12.0000i − 1.08643i
\(123\) − 18.0000i − 1.62301i
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) −30.0000 −2.67261
\(127\) − 6.00000i − 0.532414i −0.963916 0.266207i \(-0.914230\pi\)
0.963916 0.266207i \(-0.0857705\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 18.0000 1.58481
\(130\) 0 0
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 12.0000i 1.04447i
\(133\) 5.00000i 0.433555i
\(134\) 3.00000 0.259161
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 9.00000i 0.768922i 0.923141 + 0.384461i \(0.125613\pi\)
−0.923141 + 0.384461i \(0.874387\pi\)
\(138\) 21.0000i 1.78764i
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 4.00000i − 0.334497i
\(144\) −6.00000 −0.500000
\(145\) 0 0
\(146\) 11.0000 0.910366
\(147\) − 54.0000i − 4.45384i
\(148\) 2.00000i 0.164399i
\(149\) 4.00000 0.327693 0.163846 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) 18.0000i 1.45521i
\(154\) −20.0000 −1.61165
\(155\) 0 0
\(156\) 3.00000 0.240192
\(157\) 6.00000i 0.478852i 0.970915 + 0.239426i \(0.0769593\pi\)
−0.970915 + 0.239426i \(0.923041\pi\)
\(158\) 2.00000i 0.159111i
\(159\) −39.0000 −3.09290
\(160\) 0 0
\(161\) −35.0000 −2.75839
\(162\) 9.00000i 0.707107i
\(163\) − 22.0000i − 1.72317i −0.507611 0.861586i \(-0.669471\pi\)
0.507611 0.861586i \(-0.330529\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −10.0000 −0.776151
\(167\) − 2.00000i − 0.154765i −0.997001 0.0773823i \(-0.975344\pi\)
0.997001 0.0773823i \(-0.0246562\pi\)
\(168\) − 15.0000i − 1.15728i
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 6.00000i 0.457496i
\(173\) 14.0000i 1.06440i 0.846619 + 0.532200i \(0.178635\pi\)
−0.846619 + 0.532200i \(0.821365\pi\)
\(174\) −9.00000 −0.682288
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 27.0000i 2.02944i
\(178\) − 2.00000i − 0.149906i
\(179\) 8.00000 0.597948 0.298974 0.954261i \(-0.403356\pi\)
0.298974 + 0.954261i \(0.403356\pi\)
\(180\) 0 0
\(181\) 26.0000 1.93256 0.966282 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 5.00000i 0.370625i
\(183\) − 36.0000i − 2.66120i
\(184\) −7.00000 −0.516047
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) 12.0000i 0.877527i
\(188\) 0 0
\(189\) −45.0000 −3.27327
\(190\) 0 0
\(191\) 9.00000 0.651217 0.325609 0.945505i \(-0.394431\pi\)
0.325609 + 0.945505i \(0.394431\pi\)
\(192\) − 3.00000i − 0.216506i
\(193\) − 10.0000i − 0.719816i −0.932988 0.359908i \(-0.882808\pi\)
0.932988 0.359908i \(-0.117192\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 18.0000 1.28571
\(197\) − 22.0000i − 1.56744i −0.621117 0.783718i \(-0.713321\pi\)
0.621117 0.783718i \(-0.286679\pi\)
\(198\) 24.0000i 1.70561i
\(199\) 15.0000 1.06332 0.531661 0.846957i \(-0.321568\pi\)
0.531661 + 0.846957i \(0.321568\pi\)
\(200\) 0 0
\(201\) 9.00000 0.634811
\(202\) − 8.00000i − 0.562878i
\(203\) − 15.0000i − 1.05279i
\(204\) −9.00000 −0.630126
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 42.0000i 2.91920i
\(208\) 1.00000i 0.0693375i
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) − 13.0000i − 0.892844i
\(213\) 0 0
\(214\) 13.0000 0.888662
\(215\) 0 0
\(216\) −9.00000 −0.612372
\(217\) 10.0000i 0.678844i
\(218\) − 19.0000i − 1.28684i
\(219\) 33.0000 2.22993
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) 6.00000i 0.402694i
\(223\) 2.00000i 0.133930i 0.997755 + 0.0669650i \(0.0213316\pi\)
−0.997755 + 0.0669650i \(0.978668\pi\)
\(224\) 5.00000 0.334077
\(225\) 0 0
\(226\) 0 0
\(227\) 5.00000i 0.331862i 0.986137 + 0.165931i \(0.0530628\pi\)
−0.986137 + 0.165931i \(0.946937\pi\)
\(228\) 3.00000i 0.198680i
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) −60.0000 −3.94771
\(232\) − 3.00000i − 0.196960i
\(233\) − 10.0000i − 0.655122i −0.944830 0.327561i \(-0.893773\pi\)
0.944830 0.327561i \(-0.106227\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) −9.00000 −0.585850
\(237\) 6.00000i 0.389742i
\(238\) − 15.0000i − 0.972306i
\(239\) 11.0000 0.711531 0.355765 0.934575i \(-0.384220\pi\)
0.355765 + 0.934575i \(0.384220\pi\)
\(240\) 0 0
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 5.00000i 0.321412i
\(243\) 0 0
\(244\) 12.0000 0.768221
\(245\) 0 0
\(246\) 18.0000 1.14764
\(247\) − 1.00000i − 0.0636285i
\(248\) 2.00000i 0.127000i
\(249\) −30.0000 −1.90117
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) − 30.0000i − 1.88982i
\(253\) 28.0000i 1.76034i
\(254\) 6.00000 0.376473
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 22.0000i 1.37232i 0.727450 + 0.686161i \(0.240706\pi\)
−0.727450 + 0.686161i \(0.759294\pi\)
\(258\) 18.0000i 1.12063i
\(259\) −10.0000 −0.621370
\(260\) 0 0
\(261\) −18.0000 −1.11417
\(262\) 16.0000i 0.988483i
\(263\) − 8.00000i − 0.493301i −0.969104 0.246651i \(-0.920670\pi\)
0.969104 0.246651i \(-0.0793300\pi\)
\(264\) −12.0000 −0.738549
\(265\) 0 0
\(266\) −5.00000 −0.306570
\(267\) − 6.00000i − 0.367194i
\(268\) 3.00000i 0.183254i
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) −27.0000 −1.64013 −0.820067 0.572268i \(-0.806064\pi\)
−0.820067 + 0.572268i \(0.806064\pi\)
\(272\) − 3.00000i − 0.181902i
\(273\) 15.0000i 0.907841i
\(274\) −9.00000 −0.543710
\(275\) 0 0
\(276\) −21.0000 −1.26405
\(277\) − 8.00000i − 0.480673i −0.970690 0.240337i \(-0.922742\pi\)
0.970690 0.240337i \(-0.0772579\pi\)
\(278\) − 16.0000i − 0.959616i
\(279\) 12.0000 0.718421
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 2.00000i 0.118888i 0.998232 + 0.0594438i \(0.0189327\pi\)
−0.998232 + 0.0594438i \(0.981067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 30.0000i 1.77084i
\(288\) − 6.00000i − 0.353553i
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) 11.0000i 0.643726i
\(293\) 27.0000i 1.57736i 0.614806 + 0.788678i \(0.289234\pi\)
−0.614806 + 0.788678i \(0.710766\pi\)
\(294\) 54.0000 3.14934
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 36.0000i 2.08893i
\(298\) 4.00000i 0.231714i
\(299\) 7.00000 0.404820
\(300\) 0 0
\(301\) −30.0000 −1.72917
\(302\) − 10.0000i − 0.575435i
\(303\) − 24.0000i − 1.37876i
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) −18.0000 −1.02899
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) − 20.0000i − 1.13961i
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) 25.0000 1.41762 0.708810 0.705399i \(-0.249232\pi\)
0.708810 + 0.705399i \(0.249232\pi\)
\(312\) 3.00000i 0.169842i
\(313\) 1.00000i 0.0565233i 0.999601 + 0.0282617i \(0.00899717\pi\)
−0.999601 + 0.0282617i \(0.991003\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) 9.00000i 0.505490i 0.967533 + 0.252745i \(0.0813334\pi\)
−0.967533 + 0.252745i \(0.918667\pi\)
\(318\) − 39.0000i − 2.18701i
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) 39.0000 2.17677
\(322\) − 35.0000i − 1.95047i
\(323\) 3.00000i 0.166924i
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) 22.0000 1.21847
\(327\) − 57.0000i − 3.15211i
\(328\) 6.00000i 0.331295i
\(329\) 0 0
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) − 10.0000i − 0.548821i
\(333\) 12.0000i 0.657596i
\(334\) 2.00000 0.109435
\(335\) 0 0
\(336\) 15.0000 0.818317
\(337\) 6.00000i 0.326841i 0.986557 + 0.163420i \(0.0522527\pi\)
−0.986557 + 0.163420i \(0.947747\pi\)
\(338\) 12.0000i 0.652714i
\(339\) 0 0
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 6.00000i 0.324443i
\(343\) 55.0000i 2.96972i
\(344\) −6.00000 −0.323498
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) − 6.00000i − 0.322097i −0.986947 0.161048i \(-0.948512\pi\)
0.986947 0.161048i \(-0.0514875\pi\)
\(348\) − 9.00000i − 0.482451i
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 9.00000 0.480384
\(352\) − 4.00000i − 0.213201i
\(353\) − 7.00000i − 0.372572i −0.982496 0.186286i \(-0.940355\pi\)
0.982496 0.186286i \(-0.0596452\pi\)
\(354\) −27.0000 −1.43503
\(355\) 0 0
\(356\) 2.00000 0.106000
\(357\) − 45.0000i − 2.38165i
\(358\) 8.00000i 0.422813i
\(359\) 5.00000 0.263890 0.131945 0.991257i \(-0.457878\pi\)
0.131945 + 0.991257i \(0.457878\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 26.0000i 1.36653i
\(363\) 15.0000i 0.787296i
\(364\) −5.00000 −0.262071
\(365\) 0 0
\(366\) 36.0000 1.88175
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) − 7.00000i − 0.364900i
\(369\) 36.0000 1.87409
\(370\) 0 0
\(371\) 65.0000 3.37463
\(372\) 6.00000i 0.311086i
\(373\) 23.0000i 1.19089i 0.803394 + 0.595447i \(0.203025\pi\)
−0.803394 + 0.595447i \(0.796975\pi\)
\(374\) −12.0000 −0.620505
\(375\) 0 0
\(376\) 0 0
\(377\) 3.00000i 0.154508i
\(378\) − 45.0000i − 2.31455i
\(379\) 33.0000 1.69510 0.847548 0.530719i \(-0.178078\pi\)
0.847548 + 0.530719i \(0.178078\pi\)
\(380\) 0 0
\(381\) 18.0000 0.922168
\(382\) 9.00000i 0.460480i
\(383\) 4.00000i 0.204390i 0.994764 + 0.102195i \(0.0325866\pi\)
−0.994764 + 0.102195i \(0.967413\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 36.0000i 1.82998i
\(388\) 2.00000i 0.101535i
\(389\) 4.00000 0.202808 0.101404 0.994845i \(-0.467667\pi\)
0.101404 + 0.994845i \(0.467667\pi\)
\(390\) 0 0
\(391\) −21.0000 −1.06202
\(392\) 18.0000i 0.909137i
\(393\) 48.0000i 2.42128i
\(394\) 22.0000 1.10834
\(395\) 0 0
\(396\) −24.0000 −1.20605
\(397\) − 16.0000i − 0.803017i −0.915855 0.401508i \(-0.868486\pi\)
0.915855 0.401508i \(-0.131514\pi\)
\(398\) 15.0000i 0.751882i
\(399\) −15.0000 −0.750939
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 9.00000i 0.448879i
\(403\) − 2.00000i − 0.0996271i
\(404\) 8.00000 0.398015
\(405\) 0 0
\(406\) 15.0000 0.744438
\(407\) 8.00000i 0.396545i
\(408\) − 9.00000i − 0.445566i
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) −27.0000 −1.33181
\(412\) 4.00000i 0.197066i
\(413\) − 45.0000i − 2.21431i
\(414\) −42.0000 −2.06419
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) − 48.0000i − 2.35057i
\(418\) 4.00000i 0.195646i
\(419\) −14.0000 −0.683945 −0.341972 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) 0 0
\(421\) 1.00000 0.0487370 0.0243685 0.999703i \(-0.492242\pi\)
0.0243685 + 0.999703i \(0.492242\pi\)
\(422\) − 5.00000i − 0.243396i
\(423\) 0 0
\(424\) 13.0000 0.631336
\(425\) 0 0
\(426\) 0 0
\(427\) 60.0000i 2.90360i
\(428\) 13.0000i 0.628379i
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) −36.0000 −1.73406 −0.867029 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) − 9.00000i − 0.433013i
\(433\) 16.0000i 0.768911i 0.923144 + 0.384455i \(0.125611\pi\)
−0.923144 + 0.384455i \(0.874389\pi\)
\(434\) −10.0000 −0.480015
\(435\) 0 0
\(436\) 19.0000 0.909935
\(437\) 7.00000i 0.334855i
\(438\) 33.0000i 1.57680i
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 108.000 5.14286
\(442\) 3.00000i 0.142695i
\(443\) − 36.0000i − 1.71041i −0.518289 0.855206i \(-0.673431\pi\)
0.518289 0.855206i \(-0.326569\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) 12.0000i 0.567581i
\(448\) 5.00000i 0.236228i
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 0 0
\(453\) − 30.0000i − 1.40952i
\(454\) −5.00000 −0.234662
\(455\) 0 0
\(456\) −3.00000 −0.140488
\(457\) − 29.0000i − 1.35656i −0.734802 0.678281i \(-0.762725\pi\)
0.734802 0.678281i \(-0.237275\pi\)
\(458\) 6.00000i 0.280362i
\(459\) −27.0000 −1.26025
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) − 60.0000i − 2.79145i
\(463\) 8.00000i 0.371792i 0.982569 + 0.185896i \(0.0595187\pi\)
−0.982569 + 0.185896i \(0.940481\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) − 8.00000i − 0.370196i −0.982720 0.185098i \(-0.940740\pi\)
0.982720 0.185098i \(-0.0592602\pi\)
\(468\) 6.00000i 0.277350i
\(469\) −15.0000 −0.692636
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) − 9.00000i − 0.414259i
\(473\) 24.0000i 1.10352i
\(474\) −6.00000 −0.275589
\(475\) 0 0
\(476\) 15.0000 0.687524
\(477\) − 78.0000i − 3.57137i
\(478\) 11.0000i 0.503128i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) − 12.0000i − 0.546585i
\(483\) − 105.000i − 4.77767i
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 0 0
\(487\) − 38.0000i − 1.72194i −0.508652 0.860972i \(-0.669856\pi\)
0.508652 0.860972i \(-0.330144\pi\)
\(488\) 12.0000i 0.543214i
\(489\) 66.0000 2.98462
\(490\) 0 0
\(491\) 18.0000 0.812329 0.406164 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(492\) 18.0000i 0.811503i
\(493\) − 9.00000i − 0.405340i
\(494\) 1.00000 0.0449921
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) − 30.0000i − 1.34433i
\(499\) −42.0000 −1.88018 −0.940089 0.340929i \(-0.889258\pi\)
−0.940089 + 0.340929i \(0.889258\pi\)
\(500\) 0 0
\(501\) 6.00000 0.268060
\(502\) − 12.0000i − 0.535586i
\(503\) 21.0000i 0.936344i 0.883637 + 0.468172i \(0.155087\pi\)
−0.883637 + 0.468172i \(0.844913\pi\)
\(504\) 30.0000 1.33631
\(505\) 0 0
\(506\) −28.0000 −1.24475
\(507\) 36.0000i 1.59882i
\(508\) 6.00000i 0.266207i
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) −55.0000 −2.43306
\(512\) 1.00000i 0.0441942i
\(513\) 9.00000i 0.397360i
\(514\) −22.0000 −0.970378
\(515\) 0 0
\(516\) −18.0000 −0.792406
\(517\) 0 0
\(518\) − 10.0000i − 0.439375i
\(519\) −42.0000 −1.84360
\(520\) 0 0
\(521\) 24.0000 1.05146 0.525730 0.850652i \(-0.323792\pi\)
0.525730 + 0.850652i \(0.323792\pi\)
\(522\) − 18.0000i − 0.787839i
\(523\) 9.00000i 0.393543i 0.980449 + 0.196771i \(0.0630456\pi\)
−0.980449 + 0.196771i \(0.936954\pi\)
\(524\) −16.0000 −0.698963
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) 6.00000i 0.261364i
\(528\) − 12.0000i − 0.522233i
\(529\) −26.0000 −1.13043
\(530\) 0 0
\(531\) −54.0000 −2.34340
\(532\) − 5.00000i − 0.216777i
\(533\) − 6.00000i − 0.259889i
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) −3.00000 −0.129580
\(537\) 24.0000i 1.03568i
\(538\) − 2.00000i − 0.0862261i
\(539\) 72.0000 3.10126
\(540\) 0 0
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) − 27.0000i − 1.15975i
\(543\) 78.0000i 3.34730i
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) −15.0000 −0.641941
\(547\) 20.0000i 0.855138i 0.903983 + 0.427569i \(0.140630\pi\)
−0.903983 + 0.427569i \(0.859370\pi\)
\(548\) − 9.00000i − 0.384461i
\(549\) 72.0000 3.07289
\(550\) 0 0
\(551\) −3.00000 −0.127804
\(552\) − 21.0000i − 0.893819i
\(553\) − 10.0000i − 0.425243i
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) 12.0000i 0.508456i 0.967144 + 0.254228i \(0.0818214\pi\)
−0.967144 + 0.254228i \(0.918179\pi\)
\(558\) 12.0000i 0.508001i
\(559\) 6.00000 0.253773
\(560\) 0 0
\(561\) −36.0000 −1.51992
\(562\) − 18.0000i − 0.759284i
\(563\) − 20.0000i − 0.842900i −0.906852 0.421450i \(-0.861521\pi\)
0.906852 0.421450i \(-0.138479\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −2.00000 −0.0840663
\(567\) − 45.0000i − 1.88982i
\(568\) 0 0
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 6.00000 0.251092 0.125546 0.992088i \(-0.459932\pi\)
0.125546 + 0.992088i \(0.459932\pi\)
\(572\) 4.00000i 0.167248i
\(573\) 27.0000i 1.12794i
\(574\) −30.0000 −1.25218
\(575\) 0 0
\(576\) 6.00000 0.250000
\(577\) − 7.00000i − 0.291414i −0.989328 0.145707i \(-0.953454\pi\)
0.989328 0.145707i \(-0.0465456\pi\)
\(578\) 8.00000i 0.332756i
\(579\) 30.0000 1.24676
\(580\) 0 0
\(581\) 50.0000 2.07435
\(582\) 6.00000i 0.248708i
\(583\) − 52.0000i − 2.15362i
\(584\) −11.0000 −0.455183
\(585\) 0 0
\(586\) −27.0000 −1.11536
\(587\) 18.0000i 0.742940i 0.928445 + 0.371470i \(0.121146\pi\)
−0.928445 + 0.371470i \(0.878854\pi\)
\(588\) 54.0000i 2.22692i
\(589\) 2.00000 0.0824086
\(590\) 0 0
\(591\) 66.0000 2.71488
\(592\) − 2.00000i − 0.0821995i
\(593\) − 30.0000i − 1.23195i −0.787765 0.615976i \(-0.788762\pi\)
0.787765 0.615976i \(-0.211238\pi\)
\(594\) −36.0000 −1.47710
\(595\) 0 0
\(596\) −4.00000 −0.163846
\(597\) 45.0000i 1.84173i
\(598\) 7.00000i 0.286251i
\(599\) 26.0000 1.06233 0.531166 0.847268i \(-0.321754\pi\)
0.531166 + 0.847268i \(0.321754\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) − 30.0000i − 1.22271i
\(603\) 18.0000i 0.733017i
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) 24.0000 0.974933
\(607\) − 26.0000i − 1.05531i −0.849460 0.527654i \(-0.823072\pi\)
0.849460 0.527654i \(-0.176928\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) 45.0000 1.82349
\(610\) 0 0
\(611\) 0 0
\(612\) − 18.0000i − 0.727607i
\(613\) 20.0000i 0.807792i 0.914805 + 0.403896i \(0.132344\pi\)
−0.914805 + 0.403896i \(0.867656\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 20.0000 0.805823
\(617\) 14.0000i 0.563619i 0.959470 + 0.281809i \(0.0909346\pi\)
−0.959470 + 0.281809i \(0.909065\pi\)
\(618\) 12.0000i 0.482711i
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) 0 0
\(621\) −63.0000 −2.52810
\(622\) 25.0000i 1.00241i
\(623\) 10.0000i 0.400642i
\(624\) −3.00000 −0.120096
\(625\) 0 0
\(626\) −1.00000 −0.0399680
\(627\) 12.0000i 0.479234i
\(628\) − 6.00000i − 0.239426i
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) − 2.00000i − 0.0795557i
\(633\) − 15.0000i − 0.596196i
\(634\) −9.00000 −0.357436
\(635\) 0 0
\(636\) 39.0000 1.54645
\(637\) − 18.0000i − 0.713186i
\(638\) − 12.0000i − 0.475085i
\(639\) 0 0
\(640\) 0 0
\(641\) 8.00000 0.315981 0.157991 0.987441i \(-0.449498\pi\)
0.157991 + 0.987441i \(0.449498\pi\)
\(642\) 39.0000i 1.53921i
\(643\) − 26.0000i − 1.02534i −0.858586 0.512670i \(-0.828656\pi\)
0.858586 0.512670i \(-0.171344\pi\)
\(644\) 35.0000 1.37919
\(645\) 0 0
\(646\) −3.00000 −0.118033
\(647\) − 21.0000i − 0.825595i −0.910823 0.412798i \(-0.864552\pi\)
0.910823 0.412798i \(-0.135448\pi\)
\(648\) − 9.00000i − 0.353553i
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) −30.0000 −1.17579
\(652\) 22.0000i 0.861586i
\(653\) 16.0000i 0.626128i 0.949732 + 0.313064i \(0.101356\pi\)
−0.949732 + 0.313064i \(0.898644\pi\)
\(654\) 57.0000 2.22888
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 66.0000i 2.57491i
\(658\) 0 0
\(659\) −33.0000 −1.28550 −0.642749 0.766077i \(-0.722206\pi\)
−0.642749 + 0.766077i \(0.722206\pi\)
\(660\) 0 0
\(661\) 15.0000 0.583432 0.291716 0.956505i \(-0.405774\pi\)
0.291716 + 0.956505i \(0.405774\pi\)
\(662\) − 7.00000i − 0.272063i
\(663\) 9.00000i 0.349531i
\(664\) 10.0000 0.388075
\(665\) 0 0
\(666\) −12.0000 −0.464991
\(667\) − 21.0000i − 0.813123i
\(668\) 2.00000i 0.0773823i
\(669\) −6.00000 −0.231973
\(670\) 0 0
\(671\) 48.0000 1.85302
\(672\) 15.0000i 0.578638i
\(673\) 44.0000i 1.69608i 0.529936 + 0.848038i \(0.322216\pi\)
−0.529936 + 0.848038i \(0.677784\pi\)
\(674\) −6.00000 −0.231111
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) − 39.0000i − 1.49889i −0.662066 0.749446i \(-0.730320\pi\)
0.662066 0.749446i \(-0.269680\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) −15.0000 −0.574801
\(682\) 8.00000i 0.306336i
\(683\) 44.0000i 1.68361i 0.539779 + 0.841807i \(0.318508\pi\)
−0.539779 + 0.841807i \(0.681492\pi\)
\(684\) −6.00000 −0.229416
\(685\) 0 0
\(686\) −55.0000 −2.09991
\(687\) 18.0000i 0.686743i
\(688\) − 6.00000i − 0.228748i
\(689\) −13.0000 −0.495261
\(690\) 0 0
\(691\) −42.0000 −1.59776 −0.798878 0.601494i \(-0.794573\pi\)
−0.798878 + 0.601494i \(0.794573\pi\)
\(692\) − 14.0000i − 0.532200i
\(693\) − 120.000i − 4.55842i
\(694\) 6.00000 0.227757
\(695\) 0 0
\(696\) 9.00000 0.341144
\(697\) 18.0000i 0.681799i
\(698\) − 14.0000i − 0.529908i
\(699\) 30.0000 1.13470
\(700\) 0 0
\(701\) 24.0000 0.906467 0.453234 0.891392i \(-0.350270\pi\)
0.453234 + 0.891392i \(0.350270\pi\)
\(702\) 9.00000i 0.339683i
\(703\) 2.00000i 0.0754314i
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 7.00000 0.263448
\(707\) 40.0000i 1.50435i
\(708\) − 27.0000i − 1.01472i
\(709\) −2.00000 −0.0751116 −0.0375558 0.999295i \(-0.511957\pi\)
−0.0375558 + 0.999295i \(0.511957\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) 2.00000i 0.0749532i
\(713\) 14.0000i 0.524304i
\(714\) 45.0000 1.68408
\(715\) 0 0
\(716\) −8.00000 −0.298974
\(717\) 33.0000i 1.23241i
\(718\) 5.00000i 0.186598i
\(719\) 27.0000 1.00693 0.503465 0.864016i \(-0.332058\pi\)
0.503465 + 0.864016i \(0.332058\pi\)
\(720\) 0 0
\(721\) −20.0000 −0.744839
\(722\) 1.00000i 0.0372161i
\(723\) − 36.0000i − 1.33885i
\(724\) −26.0000 −0.966282
\(725\) 0 0
\(726\) −15.0000 −0.556702
\(727\) 23.0000i 0.853023i 0.904482 + 0.426511i \(0.140258\pi\)
−0.904482 + 0.426511i \(0.859742\pi\)
\(728\) − 5.00000i − 0.185312i
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) −18.0000 −0.665754
\(732\) 36.0000i 1.33060i
\(733\) − 36.0000i − 1.32969i −0.746981 0.664845i \(-0.768498\pi\)
0.746981 0.664845i \(-0.231502\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) 7.00000 0.258023
\(737\) 12.0000i 0.442026i
\(738\) 36.0000i 1.32518i
\(739\) 10.0000 0.367856 0.183928 0.982940i \(-0.441119\pi\)
0.183928 + 0.982940i \(0.441119\pi\)
\(740\) 0 0
\(741\) 3.00000 0.110208
\(742\) 65.0000i 2.38623i
\(743\) 18.0000i 0.660356i 0.943919 + 0.330178i \(0.107109\pi\)
−0.943919 + 0.330178i \(0.892891\pi\)
\(744\) −6.00000 −0.219971
\(745\) 0 0
\(746\) −23.0000 −0.842090
\(747\) − 60.0000i − 2.19529i
\(748\) − 12.0000i − 0.438763i
\(749\) −65.0000 −2.37505
\(750\) 0 0
\(751\) −26.0000 −0.948753 −0.474377 0.880322i \(-0.657327\pi\)
−0.474377 + 0.880322i \(0.657327\pi\)
\(752\) 0 0
\(753\) − 36.0000i − 1.31191i
\(754\) −3.00000 −0.109254
\(755\) 0 0
\(756\) 45.0000 1.63663
\(757\) − 6.00000i − 0.218074i −0.994038 0.109037i \(-0.965223\pi\)
0.994038 0.109037i \(-0.0347767\pi\)
\(758\) 33.0000i 1.19861i
\(759\) −84.0000 −3.04901
\(760\) 0 0
\(761\) 11.0000 0.398750 0.199375 0.979923i \(-0.436109\pi\)
0.199375 + 0.979923i \(0.436109\pi\)
\(762\) 18.0000i 0.652071i
\(763\) 95.0000i 3.43923i
\(764\) −9.00000 −0.325609
\(765\) 0 0
\(766\) −4.00000 −0.144526
\(767\) 9.00000i 0.324971i
\(768\) 3.00000i 0.108253i
\(769\) 47.0000 1.69486 0.847432 0.530904i \(-0.178148\pi\)
0.847432 + 0.530904i \(0.178148\pi\)
\(770\) 0 0
\(771\) −66.0000 −2.37693
\(772\) 10.0000i 0.359908i
\(773\) 51.0000i 1.83434i 0.398493 + 0.917171i \(0.369533\pi\)
−0.398493 + 0.917171i \(0.630467\pi\)
\(774\) −36.0000 −1.29399
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) − 30.0000i − 1.07624i
\(778\) 4.00000i 0.143407i
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) − 21.0000i − 0.750958i
\(783\) − 27.0000i − 0.964901i
\(784\) −18.0000 −0.642857
\(785\) 0 0
\(786\) −48.0000 −1.71210
\(787\) − 39.0000i − 1.39020i −0.718913 0.695100i \(-0.755360\pi\)
0.718913 0.695100i \(-0.244640\pi\)
\(788\) 22.0000i 0.783718i
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 0 0
\(792\) − 24.0000i − 0.852803i
\(793\) − 12.0000i − 0.426132i
\(794\) 16.0000 0.567819
\(795\) 0 0
\(796\) −15.0000 −0.531661
\(797\) 31.0000i 1.09808i 0.835797 + 0.549038i \(0.185006\pi\)
−0.835797 + 0.549038i \(0.814994\pi\)
\(798\) − 15.0000i − 0.530994i
\(799\) 0 0
\(800\) 0 0
\(801\) 12.0000 0.423999
\(802\) − 6.00000i − 0.211867i
\(803\) 44.0000i 1.55273i
\(804\) −9.00000 −0.317406
\(805\) 0 0
\(806\) 2.00000 0.0704470
\(807\) − 6.00000i − 0.211210i
\(808\) 8.00000i 0.281439i
\(809\) −25.0000 −0.878953 −0.439477 0.898254i \(-0.644836\pi\)
−0.439477 + 0.898254i \(0.644836\pi\)
\(810\) 0 0
\(811\) 37.0000 1.29925 0.649623 0.760257i \(-0.274927\pi\)
0.649623 + 0.760257i \(0.274927\pi\)
\(812\) 15.0000i 0.526397i
\(813\) − 81.0000i − 2.84079i
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) 9.00000 0.315063
\(817\) 6.00000i 0.209913i
\(818\) − 22.0000i − 0.769212i
\(819\) −30.0000 −1.04828
\(820\) 0 0
\(821\) −52.0000 −1.81481 −0.907406 0.420255i \(-0.861941\pi\)
−0.907406 + 0.420255i \(0.861941\pi\)
\(822\) − 27.0000i − 0.941733i
\(823\) 43.0000i 1.49889i 0.662069 + 0.749443i \(0.269679\pi\)
−0.662069 + 0.749443i \(0.730321\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 45.0000 1.56575
\(827\) − 3.00000i − 0.104320i −0.998639 0.0521601i \(-0.983389\pi\)
0.998639 0.0521601i \(-0.0166106\pi\)
\(828\) − 42.0000i − 1.45960i
\(829\) −35.0000 −1.21560 −0.607800 0.794090i \(-0.707948\pi\)
−0.607800 + 0.794090i \(0.707948\pi\)
\(830\) 0 0
\(831\) 24.0000 0.832551
\(832\) − 1.00000i − 0.0346688i
\(833\) 54.0000i 1.87099i
\(834\) 48.0000 1.66210
\(835\) 0 0
\(836\) −4.00000 −0.138343
\(837\) 18.0000i 0.622171i
\(838\) − 14.0000i − 0.483622i
\(839\) 4.00000 0.138095 0.0690477 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 1.00000i 0.0344623i
\(843\) − 54.0000i − 1.85986i
\(844\) 5.00000 0.172107
\(845\) 0 0
\(846\) 0 0
\(847\) − 25.0000i − 0.859010i
\(848\) 13.0000i 0.446422i
\(849\) −6.00000 −0.205919
\(850\) 0 0
\(851\) −14.0000 −0.479914
\(852\) 0 0
\(853\) − 42.0000i − 1.43805i −0.694983 0.719026i \(-0.744588\pi\)
0.694983 0.719026i \(-0.255412\pi\)
\(854\) −60.0000 −2.05316
\(855\) 0 0
\(856\) −13.0000 −0.444331
\(857\) − 40.0000i − 1.36637i −0.730243 0.683187i \(-0.760593\pi\)
0.730243 0.683187i \(-0.239407\pi\)
\(858\) 12.0000i 0.409673i
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) −90.0000 −3.06719
\(862\) − 36.0000i − 1.22616i
\(863\) − 56.0000i − 1.90626i −0.302558 0.953131i \(-0.597840\pi\)
0.302558 0.953131i \(-0.402160\pi\)
\(864\) 9.00000 0.306186
\(865\) 0 0
\(866\) −16.0000 −0.543702
\(867\) 24.0000i 0.815083i
\(868\) − 10.0000i − 0.339422i
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) 3.00000 0.101651
\(872\) 19.0000i 0.643421i
\(873\) 12.0000i 0.406138i
\(874\) −7.00000 −0.236779
\(875\) 0 0
\(876\) −33.0000 −1.11497
\(877\) 33.0000i 1.11433i 0.830402 + 0.557165i \(0.188111\pi\)
−0.830402 + 0.557165i \(0.811889\pi\)
\(878\) − 26.0000i − 0.877457i
\(879\) −81.0000 −2.73206
\(880\) 0 0
\(881\) 10.0000 0.336909 0.168454 0.985709i \(-0.446122\pi\)
0.168454 + 0.985709i \(0.446122\pi\)
\(882\) 108.000i 3.63655i
\(883\) − 30.0000i − 1.00958i −0.863242 0.504790i \(-0.831570\pi\)
0.863242 0.504790i \(-0.168430\pi\)
\(884\) −3.00000 −0.100901
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) − 28.0000i − 0.940148i −0.882627 0.470074i \(-0.844227\pi\)
0.882627 0.470074i \(-0.155773\pi\)
\(888\) − 6.00000i − 0.201347i
\(889\) −30.0000 −1.00617
\(890\) 0 0
\(891\) −36.0000 −1.20605
\(892\) − 2.00000i − 0.0669650i
\(893\) 0 0
\(894\) −12.0000 −0.401340
\(895\) 0 0
\(896\) −5.00000 −0.167038
\(897\) 21.0000i 0.701170i
\(898\) 22.0000i 0.734150i
\(899\) −6.00000 −0.200111
\(900\) 0 0
\(901\) 39.0000 1.29928
\(902\) 24.0000i 0.799113i
\(903\) − 90.0000i − 2.99501i
\(904\) 0 0
\(905\) 0 0
\(906\) 30.0000 0.996683
\(907\) − 1.00000i − 0.0332045i −0.999862 0.0166022i \(-0.994715\pi\)
0.999862 0.0166022i \(-0.00528490\pi\)
\(908\) − 5.00000i − 0.165931i
\(909\) 48.0000 1.59206
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) − 3.00000i − 0.0993399i
\(913\) − 40.0000i − 1.32381i
\(914\) 29.0000 0.959235
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) − 80.0000i − 2.64183i
\(918\) − 27.0000i − 0.891133i
\(919\) 5.00000 0.164935 0.0824674 0.996594i \(-0.473720\pi\)
0.0824674 + 0.996594i \(0.473720\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) 18.0000i 0.592798i
\(923\) 0 0
\(924\) 60.0000 1.97386
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) 24.0000i 0.788263i
\(928\) 3.00000i 0.0984798i
\(929\) 3.00000 0.0984268 0.0492134 0.998788i \(-0.484329\pi\)
0.0492134 + 0.998788i \(0.484329\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) 10.0000i 0.327561i
\(933\) 75.0000i 2.45539i
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) 47.0000i 1.53542i 0.640796 + 0.767712i \(0.278605\pi\)
−0.640796 + 0.767712i \(0.721395\pi\)
\(938\) − 15.0000i − 0.489767i
\(939\) −3.00000 −0.0979013
\(940\) 0 0
\(941\) −51.0000 −1.66255 −0.831276 0.555860i \(-0.812389\pi\)
−0.831276 + 0.555860i \(0.812389\pi\)
\(942\) − 18.0000i − 0.586472i
\(943\) 42.0000i 1.36771i
\(944\) 9.00000 0.292925
\(945\) 0 0
\(946\) −24.0000 −0.780307
\(947\) − 24.0000i − 0.779895i −0.920837 0.389948i \(-0.872493\pi\)
0.920837 0.389948i \(-0.127507\pi\)
\(948\) − 6.00000i − 0.194871i
\(949\) 11.0000 0.357075
\(950\) 0 0
\(951\) −27.0000 −0.875535
\(952\) 15.0000i 0.486153i
\(953\) 24.0000i 0.777436i 0.921357 + 0.388718i \(0.127082\pi\)
−0.921357 + 0.388718i \(0.872918\pi\)
\(954\) 78.0000 2.52534
\(955\) 0 0
\(956\) −11.0000 −0.355765
\(957\) − 36.0000i − 1.16371i
\(958\) 0 0
\(959\) 45.0000 1.45313
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 2.00000i 0.0644826i
\(963\) 78.0000i 2.51351i
\(964\) 12.0000 0.386494
\(965\) 0 0
\(966\) 105.000 3.37832
\(967\) − 44.0000i − 1.41494i −0.706741 0.707472i \(-0.749835\pi\)
0.706741 0.707472i \(-0.250165\pi\)
\(968\) − 5.00000i − 0.160706i
\(969\) −9.00000 −0.289122
\(970\) 0 0
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 0 0
\(973\) 80.0000i 2.56468i
\(974\) 38.0000 1.21760
\(975\) 0 0
\(976\) −12.0000 −0.384111
\(977\) − 62.0000i − 1.98356i −0.127971 0.991778i \(-0.540847\pi\)
0.127971 0.991778i \(-0.459153\pi\)
\(978\) 66.0000i 2.11045i
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) 114.000 3.63974
\(982\) 18.0000i 0.574403i
\(983\) 42.0000i 1.33959i 0.742545 + 0.669796i \(0.233618\pi\)
−0.742545 + 0.669796i \(0.766382\pi\)
\(984\) −18.0000 −0.573819
\(985\) 0 0
\(986\) 9.00000 0.286618
\(987\) 0 0
\(988\) 1.00000i 0.0318142i
\(989\) −42.0000 −1.33552
\(990\) 0 0
\(991\) 30.0000 0.952981 0.476491 0.879180i \(-0.341909\pi\)
0.476491 + 0.879180i \(0.341909\pi\)
\(992\) − 2.00000i − 0.0635001i
\(993\) − 21.0000i − 0.666415i
\(994\) 0 0
\(995\) 0 0
\(996\) 30.0000 0.950586
\(997\) − 50.0000i − 1.58352i −0.610835 0.791758i \(-0.709166\pi\)
0.610835 0.791758i \(-0.290834\pi\)
\(998\) − 42.0000i − 1.32949i
\(999\) −18.0000 −0.569495
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 950.2.b.a.799.2 2
5.2 odd 4 950.2.a.c.1.1 1
5.3 odd 4 190.2.a.b.1.1 1
5.4 even 2 inner 950.2.b.a.799.1 2
15.2 even 4 8550.2.a.bm.1.1 1
15.8 even 4 1710.2.a.g.1.1 1
20.3 even 4 1520.2.a.j.1.1 1
20.7 even 4 7600.2.a.a.1.1 1
35.13 even 4 9310.2.a.u.1.1 1
40.3 even 4 6080.2.a.b.1.1 1
40.13 odd 4 6080.2.a.x.1.1 1
95.18 even 4 3610.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.a.b.1.1 1 5.3 odd 4
950.2.a.c.1.1 1 5.2 odd 4
950.2.b.a.799.1 2 5.4 even 2 inner
950.2.b.a.799.2 2 1.1 even 1 trivial
1520.2.a.j.1.1 1 20.3 even 4
1710.2.a.g.1.1 1 15.8 even 4
3610.2.a.e.1.1 1 95.18 even 4
6080.2.a.b.1.1 1 40.3 even 4
6080.2.a.x.1.1 1 40.13 odd 4
7600.2.a.a.1.1 1 20.7 even 4
8550.2.a.bm.1.1 1 15.2 even 4
9310.2.a.u.1.1 1 35.13 even 4