Properties

Label 6900.2.f.e.6349.2
Level $6900$
Weight $2$
Character 6900.6349
Analytic conductor $55.097$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6900,2,Mod(6349,6900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6900.6349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6900.f (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: \(55.0967773947\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 6349.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 6900.6349
Dual form 6900.2.f.e.6349.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} -4.00000i q^{13} +3.00000i q^{17} +4.00000 q^{19} -1.00000 q^{21} -1.00000i q^{23} -1.00000i q^{27} +3.00000 q^{29} -7.00000 q^{31} -11.0000i q^{37} +4.00000 q^{39} -9.00000 q^{41} -4.00000i q^{43} -6.00000i q^{47} +6.00000 q^{49} -3.00000 q^{51} +9.00000i q^{53} +4.00000i q^{57} -3.00000 q^{59} -10.0000 q^{61} -1.00000i q^{63} +13.0000i q^{67} +1.00000 q^{69} +9.00000 q^{71} -16.0000i q^{73} -8.00000 q^{79} +1.00000 q^{81} -15.0000i q^{83} +3.00000i q^{87} +4.00000 q^{91} -7.00000i q^{93} -2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 8 q^{19} - 2 q^{21} + 6 q^{29} - 14 q^{31} + 8 q^{39} - 18 q^{41} + 12 q^{49} - 6 q^{51} - 6 q^{59} - 20 q^{61} + 2 q^{69} + 18 q^{71} - 16 q^{79} + 2 q^{81} + 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6900\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1201\) \(3451\) \(4601\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) − 4.00000i − 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 11.0000i − 1.80839i −0.427121 0.904194i \(-0.640472\pi\)
0.427121 0.904194i \(-0.359528\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 6.00000i − 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 9.00000i 1.23625i 0.786082 + 0.618123i \(0.212106\pi\)
−0.786082 + 0.618123i \(0.787894\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) − 1.00000i − 0.125988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 13.0000i 1.58820i 0.607785 + 0.794101i \(0.292058\pi\)
−0.607785 + 0.794101i \(0.707942\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 9.00000 1.06810 0.534052 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) − 16.0000i − 1.87266i −0.351123 0.936329i \(-0.614200\pi\)
0.351123 0.936329i \(-0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 15.0000i − 1.64646i −0.567705 0.823232i \(-0.692169\pi\)
0.567705 0.823232i \(-0.307831\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.00000i 0.321634i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) − 7.00000i − 0.725866i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.00000i 0.290021i 0.989430 + 0.145010i \(0.0463216\pi\)
−0.989430 + 0.145010i \(0.953678\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 11.0000 1.04407
\(112\) 0 0
\(113\) 9.00000i 0.846649i 0.905978 + 0.423324i \(0.139137\pi\)
−0.905978 + 0.423324i \(0.860863\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.00000i 0.369800i
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) − 9.00000i − 0.811503i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 14.0000i − 1.24230i −0.783692 0.621150i \(-0.786666\pi\)
0.783692 0.621150i \(-0.213334\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 18.0000i − 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 0 0
\(139\) 13.0000 1.10265 0.551323 0.834292i \(-0.314123\pi\)
0.551323 + 0.834292i \(0.314123\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.00000i 0.494872i
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) − 3.00000i − 0.242536i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 17.0000i − 1.35675i −0.734717 0.678374i \(-0.762685\pi\)
0.734717 0.678374i \(-0.237315\pi\)
\(158\) 0 0
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 2.00000i 0.156652i 0.996928 + 0.0783260i \(0.0249575\pi\)
−0.996928 + 0.0783260i \(0.975042\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 18.0000i − 1.39288i −0.717614 0.696441i \(-0.754766\pi\)
0.717614 0.696441i \(-0.245234\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) − 24.0000i − 1.82469i −0.409426 0.912343i \(-0.634271\pi\)
0.409426 0.912343i \(-0.365729\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 3.00000i − 0.225494i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) − 10.0000i − 0.739221i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 0 0
\(193\) − 4.00000i − 0.287926i −0.989583 0.143963i \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 0 0
\(201\) −13.0000 −0.916949
\(202\) 0 0
\(203\) 3.00000i 0.210559i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.00000i 0.0695048i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 17.0000 1.17033 0.585164 0.810915i \(-0.301030\pi\)
0.585164 + 0.810915i \(0.301030\pi\)
\(212\) 0 0
\(213\) 9.00000i 0.616670i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 7.00000i − 0.475191i
\(218\) 0 0
\(219\) 16.0000 1.08118
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) − 10.0000i − 0.669650i −0.942280 0.334825i \(-0.891323\pi\)
0.942280 0.334825i \(-0.108677\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 8.00000i − 0.519656i
\(238\) 0 0
\(239\) 9.00000 0.582162 0.291081 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 16.0000i − 1.01806i
\(248\) 0 0
\(249\) 15.0000 0.950586
\(250\) 0 0
\(251\) 30.0000 1.89358 0.946792 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 0 0
\(259\) 11.0000 0.683507
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) 0 0
\(263\) 27.0000i 1.66489i 0.554107 + 0.832446i \(0.313060\pi\)
−0.554107 + 0.832446i \(0.686940\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 21.0000 1.28039 0.640196 0.768211i \(-0.278853\pi\)
0.640196 + 0.768211i \(0.278853\pi\)
\(270\) 0 0
\(271\) −13.0000 −0.789694 −0.394847 0.918747i \(-0.629202\pi\)
−0.394847 + 0.918747i \(0.629202\pi\)
\(272\) 0 0
\(273\) 4.00000i 0.242091i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 0 0
\(279\) 7.00000 0.419079
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) 5.00000i 0.297219i 0.988896 + 0.148610i \(0.0474798\pi\)
−0.988896 + 0.148610i \(0.952520\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 9.00000i − 0.531253i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 0 0
\(293\) − 9.00000i − 0.525786i −0.964825 0.262893i \(-0.915323\pi\)
0.964825 0.262893i \(-0.0846766\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) − 3.00000i − 0.172345i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 20.0000i − 1.14146i −0.821138 0.570730i \(-0.806660\pi\)
0.821138 0.570730i \(-0.193340\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 17.0000i 0.960897i 0.877023 + 0.480448i \(0.159526\pi\)
−0.877023 + 0.480448i \(0.840474\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) 12.0000i 0.667698i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 4.00000i 0.221201i
\(328\) 0 0
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) 0 0
\(333\) 11.0000i 0.602796i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 22.0000i 1.19842i 0.800593 + 0.599208i \(0.204518\pi\)
−0.800593 + 0.599208i \(0.795482\pi\)
\(338\) 0 0
\(339\) −9.00000 −0.488813
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 13.0000i 0.701934i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 24.0000i − 1.28839i −0.764862 0.644194i \(-0.777193\pi\)
0.764862 0.644194i \(-0.222807\pi\)
\(348\) 0 0
\(349\) −35.0000 −1.87351 −0.936754 0.349990i \(-0.886185\pi\)
−0.936754 + 0.349990i \(0.886185\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 3.00000i − 0.158777i
\(358\) 0 0
\(359\) −30.0000 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) − 11.0000i − 0.577350i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 29.0000i − 1.51379i −0.653538 0.756894i \(-0.726716\pi\)
0.653538 0.756894i \(-0.273284\pi\)
\(368\) 0 0
\(369\) 9.00000 0.468521
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) 0 0
\(373\) − 22.0000i − 1.13912i −0.821951 0.569558i \(-0.807114\pi\)
0.821951 0.569558i \(-0.192886\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 12.0000i − 0.618031i
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 14.0000 0.717242
\(382\) 0 0
\(383\) − 21.0000i − 1.07305i −0.843884 0.536525i \(-0.819737\pi\)
0.843884 0.536525i \(-0.180263\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.00000i 0.203331i
\(388\) 0 0
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) 12.0000i 0.605320i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 16.0000i 0.803017i 0.915855 + 0.401508i \(0.131514\pi\)
−0.915855 + 0.401508i \(0.868486\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 28.0000i 1.39478i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 31.0000 1.53285 0.766426 0.642333i \(-0.222033\pi\)
0.766426 + 0.642333i \(0.222033\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 0 0
\(413\) − 3.00000i − 0.147620i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 13.0000i 0.636613i
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 0 0
\(423\) 6.00000i 0.291730i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 10.0000i − 0.483934i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) 11.0000i 0.528626i 0.964437 + 0.264313i \(0.0851452\pi\)
−0.964437 + 0.264313i \(0.914855\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 4.00000i − 0.191346i
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 6.00000i 0.285069i 0.989790 + 0.142534i \(0.0455251\pi\)
−0.989790 + 0.142534i \(0.954475\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 6.00000i − 0.283790i
\(448\) 0 0
\(449\) −21.0000 −0.991051 −0.495526 0.868593i \(-0.665025\pi\)
−0.495526 + 0.868593i \(0.665025\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) − 16.0000i − 0.751746i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 29.0000i − 1.35656i −0.734802 0.678281i \(-0.762725\pi\)
0.734802 0.678281i \(-0.237275\pi\)
\(458\) 0 0
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) − 16.0000i − 0.743583i −0.928316 0.371792i \(-0.878744\pi\)
0.928316 0.371792i \(-0.121256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 21.0000i − 0.971764i −0.874024 0.485882i \(-0.838498\pi\)
0.874024 0.485882i \(-0.161502\pi\)
\(468\) 0 0
\(469\) −13.0000 −0.600284
\(470\) 0 0
\(471\) 17.0000 0.783319
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 9.00000i − 0.412082i
\(478\) 0 0
\(479\) −6.00000 −0.274147 −0.137073 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 0 0
\(481\) −44.0000 −2.00623
\(482\) 0 0
\(483\) 1.00000i 0.0455016i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 2.00000i − 0.0906287i −0.998973 0.0453143i \(-0.985571\pi\)
0.998973 0.0453143i \(-0.0144289\pi\)
\(488\) 0 0
\(489\) −2.00000 −0.0904431
\(490\) 0 0
\(491\) 21.0000 0.947717 0.473858 0.880601i \(-0.342861\pi\)
0.473858 + 0.880601i \(0.342861\pi\)
\(492\) 0 0
\(493\) 9.00000i 0.405340i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.00000i 0.403705i
\(498\) 0 0
\(499\) 13.0000 0.581960 0.290980 0.956729i \(-0.406019\pi\)
0.290980 + 0.956729i \(0.406019\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 0 0
\(503\) − 9.00000i − 0.401290i −0.979664 0.200645i \(-0.935696\pi\)
0.979664 0.200645i \(-0.0643038\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 3.00000i − 0.133235i
\(508\) 0 0
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 0 0
\(513\) − 4.00000i − 0.176604i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) − 28.0000i − 1.22435i −0.790721 0.612177i \(-0.790294\pi\)
0.790721 0.612177i \(-0.209706\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 21.0000i − 0.914774i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 3.00000 0.130189
\(532\) 0 0
\(533\) 36.0000i 1.55933i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) − 16.0000i − 0.686626i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 28.0000i 1.19719i 0.801050 + 0.598597i \(0.204275\pi\)
−0.801050 + 0.598597i \(0.795725\pi\)
\(548\) 0 0
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) − 8.00000i − 0.340195i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 39.0000i 1.65248i 0.563316 + 0.826242i \(0.309525\pi\)
−0.563316 + 0.826242i \(0.690475\pi\)
\(558\) 0 0
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 21.0000i − 0.885044i −0.896758 0.442522i \(-0.854084\pi\)
0.896758 0.442522i \(-0.145916\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −22.0000 −0.920671 −0.460336 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 0 0
\(573\) 6.00000i 0.250654i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 44.0000i − 1.83174i −0.401470 0.915872i \(-0.631501\pi\)
0.401470 0.915872i \(-0.368499\pi\)
\(578\) 0 0
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 15.0000 0.622305
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 30.0000i 1.23823i 0.785299 + 0.619116i \(0.212509\pi\)
−0.785299 + 0.619116i \(0.787491\pi\)
\(588\) 0 0
\(589\) −28.0000 −1.15372
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 6.00000i 0.246390i 0.992382 + 0.123195i \(0.0393141\pi\)
−0.992382 + 0.123195i \(0.960686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 2.00000i − 0.0818546i
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) 23.0000 0.938190 0.469095 0.883148i \(-0.344580\pi\)
0.469095 + 0.883148i \(0.344580\pi\)
\(602\) 0 0
\(603\) − 13.0000i − 0.529401i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 14.0000i − 0.568242i −0.958788 0.284121i \(-0.908298\pi\)
0.958788 0.284121i \(-0.0917018\pi\)
\(608\) 0 0
\(609\) −3.00000 −0.121566
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) 14.0000i 0.565455i 0.959200 + 0.282727i \(0.0912392\pi\)
−0.959200 + 0.282727i \(0.908761\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.00000i 0.362326i 0.983453 + 0.181163i \(0.0579862\pi\)
−0.983453 + 0.181163i \(0.942014\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 33.0000 1.31580
\(630\) 0 0
\(631\) −34.0000 −1.35352 −0.676759 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) 0 0
\(633\) 17.0000i 0.675689i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 24.0000i − 0.950915i
\(638\) 0 0
\(639\) −9.00000 −0.356034
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) − 13.0000i − 0.512670i −0.966588 0.256335i \(-0.917485\pi\)
0.966588 0.256335i \(-0.0825150\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 7.00000 0.274352
\(652\) 0 0
\(653\) − 36.0000i − 1.40879i −0.709809 0.704394i \(-0.751219\pi\)
0.709809 0.704394i \(-0.248781\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 16.0000i 0.624219i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 0 0
\(663\) 12.0000i 0.466041i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 3.00000i − 0.116160i
\(668\) 0 0
\(669\) 10.0000 0.386622
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) − 22.0000i − 0.848038i −0.905653 0.424019i \(-0.860619\pi\)
0.905653 0.424019i \(-0.139381\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 33.0000i 1.26829i 0.773213 + 0.634147i \(0.218648\pi\)
−0.773213 + 0.634147i \(0.781352\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) − 30.0000i − 1.14792i −0.818884 0.573959i \(-0.805407\pi\)
0.818884 0.573959i \(-0.194593\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 4.00000i 0.152610i
\(688\) 0 0
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 27.0000i − 1.02270i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) − 44.0000i − 1.65949i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 3.00000i − 0.112827i
\(708\) 0 0
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 7.00000i 0.262152i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 9.00000i 0.336111i
\(718\) 0 0
\(719\) 45.0000 1.67822 0.839108 0.543964i \(-0.183077\pi\)
0.839108 + 0.543964i \(0.183077\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 8.00000i 0.297523i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 17.0000i − 0.630495i −0.949009 0.315248i \(-0.897912\pi\)
0.949009 0.315248i \(-0.102088\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) 0 0
\(733\) − 1.00000i − 0.0369358i −0.999829 0.0184679i \(-0.994121\pi\)
0.999829 0.0184679i \(-0.00587886\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −5.00000 −0.183928 −0.0919640 0.995762i \(-0.529314\pi\)
−0.0919640 + 0.995762i \(0.529314\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 15.0000i 0.548821i
\(748\) 0 0
\(749\) −3.00000 −0.109618
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) 30.0000i 1.09326i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 17.0000i − 0.617876i −0.951082 0.308938i \(-0.900027\pi\)
0.951082 0.308938i \(-0.0999735\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15.0000 0.543750 0.271875 0.962333i \(-0.412356\pi\)
0.271875 + 0.962333i \(0.412356\pi\)
\(762\) 0 0
\(763\) 4.00000i 0.144810i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000i 0.433295i
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 0 0
\(773\) 42.0000i 1.51064i 0.655359 + 0.755318i \(0.272517\pi\)
−0.655359 + 0.755318i \(0.727483\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 11.0000i 0.394623i
\(778\) 0 0
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 3.00000i − 0.107211i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 11.0000i − 0.392108i −0.980593 0.196054i \(-0.937187\pi\)
0.980593 0.196054i \(-0.0628127\pi\)
\(788\) 0 0
\(789\) −27.0000 −0.961225
\(790\) 0 0
\(791\) −9.00000 −0.320003
\(792\) 0 0
\(793\) 40.0000i 1.42044i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 33.0000i 1.16892i 0.811423 + 0.584460i \(0.198694\pi\)
−0.811423 + 0.584460i \(0.801306\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 21.0000i 0.739235i
\(808\) 0 0
\(809\) 39.0000 1.37117 0.685583 0.727994i \(-0.259547\pi\)
0.685583 + 0.727994i \(0.259547\pi\)
\(810\) 0 0
\(811\) 41.0000 1.43970 0.719852 0.694127i \(-0.244209\pi\)
0.719852 + 0.694127i \(0.244209\pi\)
\(812\) 0 0
\(813\) − 13.0000i − 0.455930i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 16.0000i − 0.559769i
\(818\) 0 0
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) − 16.0000i − 0.557725i −0.960331 0.278862i \(-0.910043\pi\)
0.960331 0.278862i \(-0.0899574\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 45.0000i 1.56480i 0.622774 + 0.782402i \(0.286006\pi\)
−0.622774 + 0.782402i \(0.713994\pi\)
\(828\) 0 0
\(829\) 7.00000 0.243120 0.121560 0.992584i \(-0.461210\pi\)
0.121560 + 0.992584i \(0.461210\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) 18.0000i 0.623663i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 7.00000i 0.241955i
\(838\) 0 0
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) − 12.0000i − 0.413302i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 11.0000i − 0.377964i
\(848\) 0 0
\(849\) −5.00000 −0.171600
\(850\) 0 0
\(851\) −11.0000 −0.377075
\(852\) 0 0
\(853\) 26.0000i 0.890223i 0.895475 + 0.445112i \(0.146836\pi\)
−0.895475 + 0.445112i \(0.853164\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 6.00000i − 0.204956i −0.994735 0.102478i \(-0.967323\pi\)
0.994735 0.102478i \(-0.0326771\pi\)
\(858\) 0 0
\(859\) −5.00000 −0.170598 −0.0852989 0.996355i \(-0.527185\pi\)
−0.0852989 + 0.996355i \(0.527185\pi\)
\(860\) 0 0
\(861\) 9.00000 0.306719
\(862\) 0 0
\(863\) − 6.00000i − 0.204242i −0.994772 0.102121i \(-0.967437\pi\)
0.994772 0.102121i \(-0.0325630\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 8.00000i 0.271694i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 52.0000 1.76195
\(872\) 0 0
\(873\) 2.00000i 0.0676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) 0 0
\(879\) 9.00000 0.303562
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 14.0000i 0.471138i 0.971858 + 0.235569i \(0.0756953\pi\)
−0.971858 + 0.235569i \(0.924305\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 6.00000i − 0.201460i −0.994914 0.100730i \(-0.967882\pi\)
0.994914 0.100730i \(-0.0321179\pi\)
\(888\) 0 0
\(889\) 14.0000 0.469545
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 24.0000i − 0.803129i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 4.00000i − 0.133556i
\(898\) 0 0
\(899\) −21.0000 −0.700389
\(900\) 0 0
\(901\) −27.0000 −0.899500
\(902\) 0 0
\(903\) 4.00000i 0.133112i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 13.0000i 0.431658i 0.976431 + 0.215829i \(0.0692454\pi\)
−0.976431 + 0.215829i \(0.930755\pi\)
\(908\) 0 0
\(909\) 3.00000 0.0995037
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12.0000i 0.396275i
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 0 0
\(923\) − 36.0000i − 1.18495i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 8.00000i − 0.262754i
\(928\) 0 0
\(929\) 15.0000 0.492134 0.246067 0.969253i \(-0.420862\pi\)
0.246067 + 0.969253i \(0.420862\pi\)
\(930\) 0 0
\(931\) 24.0000 0.786568
\(932\) 0 0
\(933\) 24.0000i 0.785725i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 2.00000i − 0.0653372i −0.999466 0.0326686i \(-0.989599\pi\)
0.999466 0.0326686i \(-0.0104006\pi\)
\(938\) 0 0
\(939\) −17.0000 −0.554774
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) 9.00000i 0.293080i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 48.0000i 1.55979i 0.625910 + 0.779895i \(0.284728\pi\)
−0.625910 + 0.779895i \(0.715272\pi\)
\(948\) 0 0
\(949\) −64.0000 −2.07753
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) − 6.00000i − 0.194359i −0.995267 0.0971795i \(-0.969018\pi\)
0.995267 0.0971795i \(-0.0309821\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) − 3.00000i − 0.0966736i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 26.0000i − 0.836104i −0.908423 0.418052i \(-0.862713\pi\)
0.908423 0.418052i \(-0.137287\pi\)
\(968\) 0 0
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) 13.0000i 0.416761i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 33.0000i 1.05576i 0.849318 + 0.527882i \(0.177014\pi\)
−0.849318 + 0.527882i \(0.822986\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) − 27.0000i − 0.861166i −0.902551 0.430583i \(-0.858308\pi\)
0.902551 0.430583i \(-0.141692\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.00000i 0.190982i
\(988\) 0 0
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) 53.0000 1.68360 0.841800 0.539789i \(-0.181496\pi\)
0.841800 + 0.539789i \(0.181496\pi\)
\(992\) 0 0
\(993\) − 7.00000i − 0.222138i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 8.00000i − 0.253363i −0.991943 0.126681i \(-0.959567\pi\)
0.991943 0.126681i \(-0.0404325\pi\)
\(998\) 0 0
\(999\) −11.0000 −0.348025
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 6900.2.f.e.6349.2 2
5.2 odd 4 1380.2.a.c.1.1 1
5.3 odd 4 6900.2.a.b.1.1 1
5.4 even 2 inner 6900.2.f.e.6349.1 2
15.2 even 4 4140.2.a.i.1.1 1
20.7 even 4 5520.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.a.c.1.1 1 5.2 odd 4
4140.2.a.i.1.1 1 15.2 even 4
5520.2.a.g.1.1 1 20.7 even 4
6900.2.a.b.1.1 1 5.3 odd 4
6900.2.f.e.6349.1 2 5.4 even 2 inner
6900.2.f.e.6349.2 2 1.1 even 1 trivial