Properties

Label 912.3.o.a.721.2
Level $912$
Weight $3$
Character 912.721
Analytic conductor $24.850$
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] = [912,3,Mod(721,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.721");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.o (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: \(24.8502001097\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 721.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 912.721
Dual form 912.3.o.a.721.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} +4.00000 q^{5} +10.0000 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+1.73205i q^{3} +4.00000 q^{5} +10.0000 q^{7} -3.00000 q^{9} -10.0000 q^{11} +24.2487i q^{13} +6.92820i q^{15} +10.0000 q^{17} -19.0000 q^{19} +17.3205i q^{21} +20.0000 q^{23} -9.00000 q^{25} -5.19615i q^{27} +34.6410i q^{29} +17.3205i q^{31} -17.3205i q^{33} +40.0000 q^{35} +10.3923i q^{37} -42.0000 q^{39} -34.6410i q^{41} +10.0000 q^{43} -12.0000 q^{45} +80.0000 q^{47} +51.0000 q^{49} +17.3205i q^{51} -41.5692i q^{53} -40.0000 q^{55} -32.9090i q^{57} +34.6410i q^{59} -10.0000 q^{61} -30.0000 q^{63} +96.9948i q^{65} -76.2102i q^{67} +34.6410i q^{69} +103.923i q^{71} -10.0000 q^{73} -15.5885i q^{75} -100.000 q^{77} -17.3205i q^{79} +9.00000 q^{81} -70.0000 q^{83} +40.0000 q^{85} -60.0000 q^{87} +103.923i q^{89} +242.487i q^{91} -30.0000 q^{93} -76.0000 q^{95} -76.2102i q^{97} +30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{5} + 20 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{5} + 20 q^{7} - 6 q^{9} - 20 q^{11} + 20 q^{17} - 38 q^{19} + 40 q^{23} - 18 q^{25} + 80 q^{35} - 84 q^{39} + 20 q^{43} - 24 q^{45} + 160 q^{47} + 102 q^{49} - 80 q^{55} - 20 q^{61} - 60 q^{63} - 20 q^{73} - 200 q^{77} + 18 q^{81} - 140 q^{83} + 80 q^{85} - 120 q^{87} - 60 q^{93} - 152 q^{95} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 1.73205i 0.577350i
\(4\) 0 0
\(5\) 4.00000 0.800000 0.400000 0.916515i \(-0.369010\pi\)
0.400000 + 0.916515i \(0.369010\pi\)
\(6\) 0 0
\(7\) 10.0000 1.42857 0.714286 0.699854i \(-0.246752\pi\)
0.714286 + 0.699854i \(0.246752\pi\)
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) −10.0000 −0.909091 −0.454545 0.890724i \(-0.650198\pi\)
−0.454545 + 0.890724i \(0.650198\pi\)
\(12\) 0 0
\(13\) 24.2487i 1.86529i 0.360801 + 0.932643i \(0.382503\pi\)
−0.360801 + 0.932643i \(0.617497\pi\)
\(14\) 0 0
\(15\) 6.92820i 0.461880i
\(16\) 0 0
\(17\) 10.0000 0.588235 0.294118 0.955769i \(-0.404974\pi\)
0.294118 + 0.955769i \(0.404974\pi\)
\(18\) 0 0
\(19\) −19.0000 −1.00000
\(20\) 0 0
\(21\) 17.3205i 0.824786i
\(22\) 0 0
\(23\) 20.0000 0.869565 0.434783 0.900535i \(-0.356825\pi\)
0.434783 + 0.900535i \(0.356825\pi\)
\(24\) 0 0
\(25\) −9.00000 −0.360000
\(26\) 0 0
\(27\) − 5.19615i − 0.192450i
\(28\) 0 0
\(29\) 34.6410i 1.19452i 0.802049 + 0.597259i \(0.203744\pi\)
−0.802049 + 0.597259i \(0.796256\pi\)
\(30\) 0 0
\(31\) 17.3205i 0.558726i 0.960186 + 0.279363i \(0.0901233\pi\)
−0.960186 + 0.279363i \(0.909877\pi\)
\(32\) 0 0
\(33\) − 17.3205i − 0.524864i
\(34\) 0 0
\(35\) 40.0000 1.14286
\(36\) 0 0
\(37\) 10.3923i 0.280873i 0.990090 + 0.140437i \(0.0448506\pi\)
−0.990090 + 0.140437i \(0.955149\pi\)
\(38\) 0 0
\(39\) −42.0000 −1.07692
\(40\) 0 0
\(41\) − 34.6410i − 0.844903i −0.906386 0.422451i \(-0.861170\pi\)
0.906386 0.422451i \(-0.138830\pi\)
\(42\) 0 0
\(43\) 10.0000 0.232558 0.116279 0.993217i \(-0.462903\pi\)
0.116279 + 0.993217i \(0.462903\pi\)
\(44\) 0 0
\(45\) −12.0000 −0.266667
\(46\) 0 0
\(47\) 80.0000 1.70213 0.851064 0.525062i \(-0.175958\pi\)
0.851064 + 0.525062i \(0.175958\pi\)
\(48\) 0 0
\(49\) 51.0000 1.04082
\(50\) 0 0
\(51\) 17.3205i 0.339618i
\(52\) 0 0
\(53\) − 41.5692i − 0.784325i −0.919896 0.392162i \(-0.871727\pi\)
0.919896 0.392162i \(-0.128273\pi\)
\(54\) 0 0
\(55\) −40.0000 −0.727273
\(56\) 0 0
\(57\) − 32.9090i − 0.577350i
\(58\) 0 0
\(59\) 34.6410i 0.587136i 0.955938 + 0.293568i \(0.0948427\pi\)
−0.955938 + 0.293568i \(0.905157\pi\)
\(60\) 0 0
\(61\) −10.0000 −0.163934 −0.0819672 0.996635i \(-0.526120\pi\)
−0.0819672 + 0.996635i \(0.526120\pi\)
\(62\) 0 0
\(63\) −30.0000 −0.476190
\(64\) 0 0
\(65\) 96.9948i 1.49223i
\(66\) 0 0
\(67\) − 76.2102i − 1.13747i −0.822522 0.568733i \(-0.807434\pi\)
0.822522 0.568733i \(-0.192566\pi\)
\(68\) 0 0
\(69\) 34.6410i 0.502044i
\(70\) 0 0
\(71\) 103.923i 1.46370i 0.681463 + 0.731852i \(0.261344\pi\)
−0.681463 + 0.731852i \(0.738656\pi\)
\(72\) 0 0
\(73\) −10.0000 −0.136986 −0.0684932 0.997652i \(-0.521819\pi\)
−0.0684932 + 0.997652i \(0.521819\pi\)
\(74\) 0 0
\(75\) − 15.5885i − 0.207846i
\(76\) 0 0
\(77\) −100.000 −1.29870
\(78\) 0 0
\(79\) − 17.3205i − 0.219247i −0.993973 0.109623i \(-0.965035\pi\)
0.993973 0.109623i \(-0.0349645\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) −70.0000 −0.843373 −0.421687 0.906742i \(-0.638562\pi\)
−0.421687 + 0.906742i \(0.638562\pi\)
\(84\) 0 0
\(85\) 40.0000 0.470588
\(86\) 0 0
\(87\) −60.0000 −0.689655
\(88\) 0 0
\(89\) 103.923i 1.16767i 0.811871 + 0.583837i \(0.198449\pi\)
−0.811871 + 0.583837i \(0.801551\pi\)
\(90\) 0 0
\(91\) 242.487i 2.66469i
\(92\) 0 0
\(93\) −30.0000 −0.322581
\(94\) 0 0
\(95\) −76.0000 −0.800000
\(96\) 0 0
\(97\) − 76.2102i − 0.785673i −0.919608 0.392836i \(-0.871494\pi\)
0.919608 0.392836i \(-0.128506\pi\)
\(98\) 0 0
\(99\) 30.0000 0.303030
\(100\) 0 0
\(101\) 100.000 0.990099 0.495050 0.868865i \(-0.335150\pi\)
0.495050 + 0.868865i \(0.335150\pi\)
\(102\) 0 0
\(103\) 183.597i 1.78250i 0.453513 + 0.891249i \(0.350170\pi\)
−0.453513 + 0.891249i \(0.649830\pi\)
\(104\) 0 0
\(105\) 69.2820i 0.659829i
\(106\) 0 0
\(107\) 62.3538i 0.582746i 0.956610 + 0.291373i \(0.0941121\pi\)
−0.956610 + 0.291373i \(0.905888\pi\)
\(108\) 0 0
\(109\) 155.885i 1.43013i 0.699056 + 0.715067i \(0.253604\pi\)
−0.699056 + 0.715067i \(0.746396\pi\)
\(110\) 0 0
\(111\) −18.0000 −0.162162
\(112\) 0 0
\(113\) 6.92820i 0.0613115i 0.999530 + 0.0306558i \(0.00975956\pi\)
−0.999530 + 0.0306558i \(0.990240\pi\)
\(114\) 0 0
\(115\) 80.0000 0.695652
\(116\) 0 0
\(117\) − 72.7461i − 0.621762i
\(118\) 0 0
\(119\) 100.000 0.840336
\(120\) 0 0
\(121\) −21.0000 −0.173554
\(122\) 0 0
\(123\) 60.0000 0.487805
\(124\) 0 0
\(125\) −136.000 −1.08800
\(126\) 0 0
\(127\) − 114.315i − 0.900121i −0.892998 0.450060i \(-0.851402\pi\)
0.892998 0.450060i \(-0.148598\pi\)
\(128\) 0 0
\(129\) 17.3205i 0.134268i
\(130\) 0 0
\(131\) 38.0000 0.290076 0.145038 0.989426i \(-0.453670\pi\)
0.145038 + 0.989426i \(0.453670\pi\)
\(132\) 0 0
\(133\) −190.000 −1.42857
\(134\) 0 0
\(135\) − 20.7846i − 0.153960i
\(136\) 0 0
\(137\) 190.000 1.38686 0.693431 0.720523i \(-0.256098\pi\)
0.693431 + 0.720523i \(0.256098\pi\)
\(138\) 0 0
\(139\) −50.0000 −0.359712 −0.179856 0.983693i \(-0.557563\pi\)
−0.179856 + 0.983693i \(0.557563\pi\)
\(140\) 0 0
\(141\) 138.564i 0.982724i
\(142\) 0 0
\(143\) − 242.487i − 1.69571i
\(144\) 0 0
\(145\) 138.564i 0.955614i
\(146\) 0 0
\(147\) 88.3346i 0.600916i
\(148\) 0 0
\(149\) −20.0000 −0.134228 −0.0671141 0.997745i \(-0.521379\pi\)
−0.0671141 + 0.997745i \(0.521379\pi\)
\(150\) 0 0
\(151\) − 225.167i − 1.49117i −0.666411 0.745585i \(-0.732170\pi\)
0.666411 0.745585i \(-0.267830\pi\)
\(152\) 0 0
\(153\) −30.0000 −0.196078
\(154\) 0 0
\(155\) 69.2820i 0.446981i
\(156\) 0 0
\(157\) 230.000 1.46497 0.732484 0.680784i \(-0.238361\pi\)
0.732484 + 0.680784i \(0.238361\pi\)
\(158\) 0 0
\(159\) 72.0000 0.452830
\(160\) 0 0
\(161\) 200.000 1.24224
\(162\) 0 0
\(163\) −170.000 −1.04294 −0.521472 0.853268i \(-0.674617\pi\)
−0.521472 + 0.853268i \(0.674617\pi\)
\(164\) 0 0
\(165\) − 69.2820i − 0.419891i
\(166\) 0 0
\(167\) − 131.636i − 0.788239i −0.919059 0.394119i \(-0.871050\pi\)
0.919059 0.394119i \(-0.128950\pi\)
\(168\) 0 0
\(169\) −419.000 −2.47929
\(170\) 0 0
\(171\) 57.0000 0.333333
\(172\) 0 0
\(173\) − 235.559i − 1.36161i −0.732464 0.680806i \(-0.761630\pi\)
0.732464 0.680806i \(-0.238370\pi\)
\(174\) 0 0
\(175\) −90.0000 −0.514286
\(176\) 0 0
\(177\) −60.0000 −0.338983
\(178\) 0 0
\(179\) − 103.923i − 0.580576i −0.956939 0.290288i \(-0.906249\pi\)
0.956939 0.290288i \(-0.0937511\pi\)
\(180\) 0 0
\(181\) − 259.808i − 1.43540i −0.696352 0.717701i \(-0.745195\pi\)
0.696352 0.717701i \(-0.254805\pi\)
\(182\) 0 0
\(183\) − 17.3205i − 0.0946476i
\(184\) 0 0
\(185\) 41.5692i 0.224698i
\(186\) 0 0
\(187\) −100.000 −0.534759
\(188\) 0 0
\(189\) − 51.9615i − 0.274929i
\(190\) 0 0
\(191\) 332.000 1.73822 0.869110 0.494619i \(-0.164692\pi\)
0.869110 + 0.494619i \(0.164692\pi\)
\(192\) 0 0
\(193\) 96.9948i 0.502564i 0.967914 + 0.251282i \(0.0808521\pi\)
−0.967914 + 0.251282i \(0.919148\pi\)
\(194\) 0 0
\(195\) −168.000 −0.861538
\(196\) 0 0
\(197\) 160.000 0.812183 0.406091 0.913832i \(-0.366891\pi\)
0.406091 + 0.913832i \(0.366891\pi\)
\(198\) 0 0
\(199\) −98.0000 −0.492462 −0.246231 0.969211i \(-0.579192\pi\)
−0.246231 + 0.969211i \(0.579192\pi\)
\(200\) 0 0
\(201\) 132.000 0.656716
\(202\) 0 0
\(203\) 346.410i 1.70645i
\(204\) 0 0
\(205\) − 138.564i − 0.675922i
\(206\) 0 0
\(207\) −60.0000 −0.289855
\(208\) 0 0
\(209\) 190.000 0.909091
\(210\) 0 0
\(211\) 173.205i 0.820877i 0.911888 + 0.410439i \(0.134624\pi\)
−0.911888 + 0.410439i \(0.865376\pi\)
\(212\) 0 0
\(213\) −180.000 −0.845070
\(214\) 0 0
\(215\) 40.0000 0.186047
\(216\) 0 0
\(217\) 173.205i 0.798180i
\(218\) 0 0
\(219\) − 17.3205i − 0.0790891i
\(220\) 0 0
\(221\) 242.487i 1.09723i
\(222\) 0 0
\(223\) − 79.6743i − 0.357284i −0.983914 0.178642i \(-0.942830\pi\)
0.983914 0.178642i \(-0.0571704\pi\)
\(224\) 0 0
\(225\) 27.0000 0.120000
\(226\) 0 0
\(227\) − 76.2102i − 0.335728i −0.985810 0.167864i \(-0.946313\pi\)
0.985810 0.167864i \(-0.0536869\pi\)
\(228\) 0 0
\(229\) 110.000 0.480349 0.240175 0.970730i \(-0.422795\pi\)
0.240175 + 0.970730i \(0.422795\pi\)
\(230\) 0 0
\(231\) − 173.205i − 0.749806i
\(232\) 0 0
\(233\) 190.000 0.815451 0.407725 0.913105i \(-0.366322\pi\)
0.407725 + 0.913105i \(0.366322\pi\)
\(234\) 0 0
\(235\) 320.000 1.36170
\(236\) 0 0
\(237\) 30.0000 0.126582
\(238\) 0 0
\(239\) 128.000 0.535565 0.267782 0.963479i \(-0.413709\pi\)
0.267782 + 0.963479i \(0.413709\pi\)
\(240\) 0 0
\(241\) − 138.564i − 0.574955i −0.957787 0.287477i \(-0.907183\pi\)
0.957787 0.287477i \(-0.0928166\pi\)
\(242\) 0 0
\(243\) 15.5885i 0.0641500i
\(244\) 0 0
\(245\) 204.000 0.832653
\(246\) 0 0
\(247\) − 460.726i − 1.86529i
\(248\) 0 0
\(249\) − 121.244i − 0.486922i
\(250\) 0 0
\(251\) 2.00000 0.00796813 0.00398406 0.999992i \(-0.498732\pi\)
0.00398406 + 0.999992i \(0.498732\pi\)
\(252\) 0 0
\(253\) −200.000 −0.790514
\(254\) 0 0
\(255\) 69.2820i 0.271694i
\(256\) 0 0
\(257\) − 491.902i − 1.91402i −0.290059 0.957009i \(-0.593675\pi\)
0.290059 0.957009i \(-0.406325\pi\)
\(258\) 0 0
\(259\) 103.923i 0.401247i
\(260\) 0 0
\(261\) − 103.923i − 0.398173i
\(262\) 0 0
\(263\) 200.000 0.760456 0.380228 0.924893i \(-0.375845\pi\)
0.380228 + 0.924893i \(0.375845\pi\)
\(264\) 0 0
\(265\) − 166.277i − 0.627460i
\(266\) 0 0
\(267\) −180.000 −0.674157
\(268\) 0 0
\(269\) 415.692i 1.54532i 0.634818 + 0.772662i \(0.281075\pi\)
−0.634818 + 0.772662i \(0.718925\pi\)
\(270\) 0 0
\(271\) −170.000 −0.627306 −0.313653 0.949538i \(-0.601553\pi\)
−0.313653 + 0.949538i \(0.601553\pi\)
\(272\) 0 0
\(273\) −420.000 −1.53846
\(274\) 0 0
\(275\) 90.0000 0.327273
\(276\) 0 0
\(277\) −10.0000 −0.0361011 −0.0180505 0.999837i \(-0.505746\pi\)
−0.0180505 + 0.999837i \(0.505746\pi\)
\(278\) 0 0
\(279\) − 51.9615i − 0.186242i
\(280\) 0 0
\(281\) − 381.051i − 1.35605i −0.735037 0.678027i \(-0.762835\pi\)
0.735037 0.678027i \(-0.237165\pi\)
\(282\) 0 0
\(283\) 70.0000 0.247350 0.123675 0.992323i \(-0.460532\pi\)
0.123675 + 0.992323i \(0.460532\pi\)
\(284\) 0 0
\(285\) − 131.636i − 0.461880i
\(286\) 0 0
\(287\) − 346.410i − 1.20700i
\(288\) 0 0
\(289\) −189.000 −0.653979
\(290\) 0 0
\(291\) 132.000 0.453608
\(292\) 0 0
\(293\) − 180.133i − 0.614789i −0.951582 0.307395i \(-0.900543\pi\)
0.951582 0.307395i \(-0.0994572\pi\)
\(294\) 0 0
\(295\) 138.564i 0.469709i
\(296\) 0 0
\(297\) 51.9615i 0.174955i
\(298\) 0 0
\(299\) 484.974i 1.62199i
\(300\) 0 0
\(301\) 100.000 0.332226
\(302\) 0 0
\(303\) 173.205i 0.571634i
\(304\) 0 0
\(305\) −40.0000 −0.131148
\(306\) 0 0
\(307\) − 145.492i − 0.473916i −0.971520 0.236958i \(-0.923850\pi\)
0.971520 0.236958i \(-0.0761504\pi\)
\(308\) 0 0
\(309\) −318.000 −1.02913
\(310\) 0 0
\(311\) −580.000 −1.86495 −0.932476 0.361232i \(-0.882356\pi\)
−0.932476 + 0.361232i \(0.882356\pi\)
\(312\) 0 0
\(313\) −370.000 −1.18211 −0.591054 0.806632i \(-0.701288\pi\)
−0.591054 + 0.806632i \(0.701288\pi\)
\(314\) 0 0
\(315\) −120.000 −0.380952
\(316\) 0 0
\(317\) − 27.7128i − 0.0874221i −0.999044 0.0437111i \(-0.986082\pi\)
0.999044 0.0437111i \(-0.0139181\pi\)
\(318\) 0 0
\(319\) − 346.410i − 1.08593i
\(320\) 0 0
\(321\) −108.000 −0.336449
\(322\) 0 0
\(323\) −190.000 −0.588235
\(324\) 0 0
\(325\) − 218.238i − 0.671503i
\(326\) 0 0
\(327\) −270.000 −0.825688
\(328\) 0 0
\(329\) 800.000 2.43161
\(330\) 0 0
\(331\) − 173.205i − 0.523278i −0.965166 0.261639i \(-0.915737\pi\)
0.965166 0.261639i \(-0.0842630\pi\)
\(332\) 0 0
\(333\) − 31.1769i − 0.0936244i
\(334\) 0 0
\(335\) − 304.841i − 0.909973i
\(336\) 0 0
\(337\) − 339.482i − 1.00736i −0.863889 0.503682i \(-0.831978\pi\)
0.863889 0.503682i \(-0.168022\pi\)
\(338\) 0 0
\(339\) −12.0000 −0.0353982
\(340\) 0 0
\(341\) − 173.205i − 0.507933i
\(342\) 0 0
\(343\) 20.0000 0.0583090
\(344\) 0 0
\(345\) 138.564i 0.401635i
\(346\) 0 0
\(347\) 590.000 1.70029 0.850144 0.526550i \(-0.176515\pi\)
0.850144 + 0.526550i \(0.176515\pi\)
\(348\) 0 0
\(349\) 98.0000 0.280802 0.140401 0.990095i \(-0.455161\pi\)
0.140401 + 0.990095i \(0.455161\pi\)
\(350\) 0 0
\(351\) 126.000 0.358974
\(352\) 0 0
\(353\) 190.000 0.538244 0.269122 0.963106i \(-0.413267\pi\)
0.269122 + 0.963106i \(0.413267\pi\)
\(354\) 0 0
\(355\) 415.692i 1.17096i
\(356\) 0 0
\(357\) 173.205i 0.485168i
\(358\) 0 0
\(359\) 200.000 0.557103 0.278552 0.960421i \(-0.410146\pi\)
0.278552 + 0.960421i \(0.410146\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) − 36.3731i − 0.100201i
\(364\) 0 0
\(365\) −40.0000 −0.109589
\(366\) 0 0
\(367\) −170.000 −0.463215 −0.231608 0.972809i \(-0.574399\pi\)
−0.231608 + 0.972809i \(0.574399\pi\)
\(368\) 0 0
\(369\) 103.923i 0.281634i
\(370\) 0 0
\(371\) − 415.692i − 1.12046i
\(372\) 0 0
\(373\) 356.802i 0.956575i 0.878203 + 0.478287i \(0.158742\pi\)
−0.878203 + 0.478287i \(0.841258\pi\)
\(374\) 0 0
\(375\) − 235.559i − 0.628157i
\(376\) 0 0
\(377\) −840.000 −2.22812
\(378\) 0 0
\(379\) 207.846i 0.548407i 0.961672 + 0.274203i \(0.0884141\pi\)
−0.961672 + 0.274203i \(0.911586\pi\)
\(380\) 0 0
\(381\) 198.000 0.519685
\(382\) 0 0
\(383\) − 630.466i − 1.64613i −0.567950 0.823063i \(-0.692263\pi\)
0.567950 0.823063i \(-0.307737\pi\)
\(384\) 0 0
\(385\) −400.000 −1.03896
\(386\) 0 0
\(387\) −30.0000 −0.0775194
\(388\) 0 0
\(389\) −128.000 −0.329049 −0.164524 0.986373i \(-0.552609\pi\)
−0.164524 + 0.986373i \(0.552609\pi\)
\(390\) 0 0
\(391\) 200.000 0.511509
\(392\) 0 0
\(393\) 65.8179i 0.167476i
\(394\) 0 0
\(395\) − 69.2820i − 0.175398i
\(396\) 0 0
\(397\) 650.000 1.63728 0.818640 0.574307i \(-0.194729\pi\)
0.818640 + 0.574307i \(0.194729\pi\)
\(398\) 0 0
\(399\) − 329.090i − 0.824786i
\(400\) 0 0
\(401\) 173.205i 0.431933i 0.976401 + 0.215966i \(0.0692902\pi\)
−0.976401 + 0.215966i \(0.930710\pi\)
\(402\) 0 0
\(403\) −420.000 −1.04218
\(404\) 0 0
\(405\) 36.0000 0.0888889
\(406\) 0 0
\(407\) − 103.923i − 0.255339i
\(408\) 0 0
\(409\) − 173.205i − 0.423484i −0.977326 0.211742i \(-0.932086\pi\)
0.977326 0.211742i \(-0.0679137\pi\)
\(410\) 0 0
\(411\) 329.090i 0.800705i
\(412\) 0 0
\(413\) 346.410i 0.838766i
\(414\) 0 0
\(415\) −280.000 −0.674699
\(416\) 0 0
\(417\) − 86.6025i − 0.207680i
\(418\) 0 0
\(419\) 38.0000 0.0906921 0.0453461 0.998971i \(-0.485561\pi\)
0.0453461 + 0.998971i \(0.485561\pi\)
\(420\) 0 0
\(421\) 17.3205i 0.0411413i 0.999788 + 0.0205707i \(0.00654831\pi\)
−0.999788 + 0.0205707i \(0.993452\pi\)
\(422\) 0 0
\(423\) −240.000 −0.567376
\(424\) 0 0
\(425\) −90.0000 −0.211765
\(426\) 0 0
\(427\) −100.000 −0.234192
\(428\) 0 0
\(429\) 420.000 0.979021
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 353.338i 0.816024i 0.912977 + 0.408012i \(0.133778\pi\)
−0.912977 + 0.408012i \(0.866222\pi\)
\(434\) 0 0
\(435\) −240.000 −0.551724
\(436\) 0 0
\(437\) −380.000 −0.869565
\(438\) 0 0
\(439\) − 121.244i − 0.276181i −0.990420 0.138091i \(-0.955903\pi\)
0.990420 0.138091i \(-0.0440965\pi\)
\(440\) 0 0
\(441\) −153.000 −0.346939
\(442\) 0 0
\(443\) 110.000 0.248307 0.124153 0.992263i \(-0.460378\pi\)
0.124153 + 0.992263i \(0.460378\pi\)
\(444\) 0 0
\(445\) 415.692i 0.934140i
\(446\) 0 0
\(447\) − 34.6410i − 0.0774967i
\(448\) 0 0
\(449\) − 311.769i − 0.694363i −0.937798 0.347182i \(-0.887139\pi\)
0.937798 0.347182i \(-0.112861\pi\)
\(450\) 0 0
\(451\) 346.410i 0.768093i
\(452\) 0 0
\(453\) 390.000 0.860927
\(454\) 0 0
\(455\) 969.948i 2.13175i
\(456\) 0 0
\(457\) 290.000 0.634573 0.317287 0.948330i \(-0.397228\pi\)
0.317287 + 0.948330i \(0.397228\pi\)
\(458\) 0 0
\(459\) − 51.9615i − 0.113206i
\(460\) 0 0
\(461\) −728.000 −1.57918 −0.789588 0.613638i \(-0.789706\pi\)
−0.789588 + 0.613638i \(0.789706\pi\)
\(462\) 0 0
\(463\) 790.000 1.70626 0.853132 0.521696i \(-0.174700\pi\)
0.853132 + 0.521696i \(0.174700\pi\)
\(464\) 0 0
\(465\) −120.000 −0.258065
\(466\) 0 0
\(467\) 530.000 1.13490 0.567452 0.823407i \(-0.307929\pi\)
0.567452 + 0.823407i \(0.307929\pi\)
\(468\) 0 0
\(469\) − 762.102i − 1.62495i
\(470\) 0 0
\(471\) 398.372i 0.845800i
\(472\) 0 0
\(473\) −100.000 −0.211416
\(474\) 0 0
\(475\) 171.000 0.360000
\(476\) 0 0
\(477\) 124.708i 0.261442i
\(478\) 0 0
\(479\) 80.0000 0.167015 0.0835073 0.996507i \(-0.473388\pi\)
0.0835073 + 0.996507i \(0.473388\pi\)
\(480\) 0 0
\(481\) −252.000 −0.523909
\(482\) 0 0
\(483\) 346.410i 0.717205i
\(484\) 0 0
\(485\) − 304.841i − 0.628538i
\(486\) 0 0
\(487\) − 509.223i − 1.04563i −0.852445 0.522816i \(-0.824881\pi\)
0.852445 0.522816i \(-0.175119\pi\)
\(488\) 0 0
\(489\) − 294.449i − 0.602144i
\(490\) 0 0
\(491\) −418.000 −0.851324 −0.425662 0.904882i \(-0.639959\pi\)
−0.425662 + 0.904882i \(0.639959\pi\)
\(492\) 0 0
\(493\) 346.410i 0.702658i
\(494\) 0 0
\(495\) 120.000 0.242424
\(496\) 0 0
\(497\) 1039.23i 2.09101i
\(498\) 0 0
\(499\) −470.000 −0.941884 −0.470942 0.882164i \(-0.656086\pi\)
−0.470942 + 0.882164i \(0.656086\pi\)
\(500\) 0 0
\(501\) 228.000 0.455090
\(502\) 0 0
\(503\) −100.000 −0.198807 −0.0994036 0.995047i \(-0.531693\pi\)
−0.0994036 + 0.995047i \(0.531693\pi\)
\(504\) 0 0
\(505\) 400.000 0.792079
\(506\) 0 0
\(507\) − 725.729i − 1.43142i
\(508\) 0 0
\(509\) − 450.333i − 0.884741i −0.896832 0.442371i \(-0.854138\pi\)
0.896832 0.442371i \(-0.145862\pi\)
\(510\) 0 0
\(511\) −100.000 −0.195695
\(512\) 0 0
\(513\) 98.7269i 0.192450i
\(514\) 0 0
\(515\) 734.390i 1.42600i
\(516\) 0 0
\(517\) −800.000 −1.54739
\(518\) 0 0
\(519\) 408.000 0.786127
\(520\) 0 0
\(521\) − 311.769i − 0.598405i −0.954190 0.299203i \(-0.903279\pi\)
0.954190 0.299203i \(-0.0967207\pi\)
\(522\) 0 0
\(523\) − 789.815i − 1.51016i −0.655631 0.755081i \(-0.727597\pi\)
0.655631 0.755081i \(-0.272403\pi\)
\(524\) 0 0
\(525\) − 155.885i − 0.296923i
\(526\) 0 0
\(527\) 173.205i 0.328662i
\(528\) 0 0
\(529\) −129.000 −0.243856
\(530\) 0 0
\(531\) − 103.923i − 0.195712i
\(532\) 0 0
\(533\) 840.000 1.57598
\(534\) 0 0
\(535\) 249.415i 0.466197i
\(536\) 0 0
\(537\) 180.000 0.335196
\(538\) 0 0
\(539\) −510.000 −0.946197
\(540\) 0 0
\(541\) 650.000 1.20148 0.600739 0.799445i \(-0.294873\pi\)
0.600739 + 0.799445i \(0.294873\pi\)
\(542\) 0 0
\(543\) 450.000 0.828729
\(544\) 0 0
\(545\) 623.538i 1.14411i
\(546\) 0 0
\(547\) 595.825i 1.08926i 0.838676 + 0.544630i \(0.183330\pi\)
−0.838676 + 0.544630i \(0.816670\pi\)
\(548\) 0 0
\(549\) 30.0000 0.0546448
\(550\) 0 0
\(551\) − 658.179i − 1.19452i
\(552\) 0 0
\(553\) − 173.205i − 0.313210i
\(554\) 0 0
\(555\) −72.0000 −0.129730
\(556\) 0 0
\(557\) −80.0000 −0.143627 −0.0718133 0.997418i \(-0.522879\pi\)
−0.0718133 + 0.997418i \(0.522879\pi\)
\(558\) 0 0
\(559\) 242.487i 0.433787i
\(560\) 0 0
\(561\) − 173.205i − 0.308743i
\(562\) 0 0
\(563\) − 339.482i − 0.602987i −0.953468 0.301494i \(-0.902515\pi\)
0.953468 0.301494i \(-0.0974852\pi\)
\(564\) 0 0
\(565\) 27.7128i 0.0490492i
\(566\) 0 0
\(567\) 90.0000 0.158730
\(568\) 0 0
\(569\) 658.179i 1.15673i 0.815778 + 0.578365i \(0.196309\pi\)
−0.815778 + 0.578365i \(0.803691\pi\)
\(570\) 0 0
\(571\) 610.000 1.06830 0.534151 0.845389i \(-0.320632\pi\)
0.534151 + 0.845389i \(0.320632\pi\)
\(572\) 0 0
\(573\) 575.041i 1.00356i
\(574\) 0 0
\(575\) −180.000 −0.313043
\(576\) 0 0
\(577\) 170.000 0.294627 0.147314 0.989090i \(-0.452937\pi\)
0.147314 + 0.989090i \(0.452937\pi\)
\(578\) 0 0
\(579\) −168.000 −0.290155
\(580\) 0 0
\(581\) −700.000 −1.20482
\(582\) 0 0
\(583\) 415.692i 0.713023i
\(584\) 0 0
\(585\) − 290.985i − 0.497409i
\(586\) 0 0
\(587\) 650.000 1.10733 0.553663 0.832741i \(-0.313230\pi\)
0.553663 + 0.832741i \(0.313230\pi\)
\(588\) 0 0
\(589\) − 329.090i − 0.558726i
\(590\) 0 0
\(591\) 277.128i 0.468914i
\(592\) 0 0
\(593\) 910.000 1.53457 0.767285 0.641306i \(-0.221607\pi\)
0.767285 + 0.641306i \(0.221607\pi\)
\(594\) 0 0
\(595\) 400.000 0.672269
\(596\) 0 0
\(597\) − 169.741i − 0.284323i
\(598\) 0 0
\(599\) 34.6410i 0.0578314i 0.999582 + 0.0289157i \(0.00920544\pi\)
−0.999582 + 0.0289157i \(0.990795\pi\)
\(600\) 0 0
\(601\) 173.205i 0.288195i 0.989564 + 0.144097i \(0.0460279\pi\)
−0.989564 + 0.144097i \(0.953972\pi\)
\(602\) 0 0
\(603\) 228.631i 0.379155i
\(604\) 0 0
\(605\) −84.0000 −0.138843
\(606\) 0 0
\(607\) 703.213i 1.15851i 0.815148 + 0.579253i \(0.196656\pi\)
−0.815148 + 0.579253i \(0.803344\pi\)
\(608\) 0 0
\(609\) −600.000 −0.985222
\(610\) 0 0
\(611\) 1939.90i 3.17495i
\(612\) 0 0
\(613\) 350.000 0.570962 0.285481 0.958384i \(-0.407847\pi\)
0.285481 + 0.958384i \(0.407847\pi\)
\(614\) 0 0
\(615\) 240.000 0.390244
\(616\) 0 0
\(617\) 610.000 0.988655 0.494327 0.869276i \(-0.335414\pi\)
0.494327 + 0.869276i \(0.335414\pi\)
\(618\) 0 0
\(619\) 10.0000 0.0161551 0.00807754 0.999967i \(-0.497429\pi\)
0.00807754 + 0.999967i \(0.497429\pi\)
\(620\) 0 0
\(621\) − 103.923i − 0.167348i
\(622\) 0 0
\(623\) 1039.23i 1.66811i
\(624\) 0 0
\(625\) −319.000 −0.510400
\(626\) 0 0
\(627\) 329.090i 0.524864i
\(628\) 0 0
\(629\) 103.923i 0.165219i
\(630\) 0 0
\(631\) −350.000 −0.554675 −0.277338 0.960773i \(-0.589452\pi\)
−0.277338 + 0.960773i \(0.589452\pi\)
\(632\) 0 0
\(633\) −300.000 −0.473934
\(634\) 0 0
\(635\) − 457.261i − 0.720097i
\(636\) 0 0
\(637\) 1236.68i 1.94142i
\(638\) 0 0
\(639\) − 311.769i − 0.487902i
\(640\) 0 0
\(641\) − 588.897i − 0.918716i −0.888251 0.459358i \(-0.848079\pi\)
0.888251 0.459358i \(-0.151921\pi\)
\(642\) 0 0
\(643\) −650.000 −1.01089 −0.505443 0.862860i \(-0.668671\pi\)
−0.505443 + 0.862860i \(0.668671\pi\)
\(644\) 0 0
\(645\) 69.2820i 0.107414i
\(646\) 0 0
\(647\) −820.000 −1.26739 −0.633694 0.773584i \(-0.718462\pi\)
−0.633694 + 0.773584i \(0.718462\pi\)
\(648\) 0 0
\(649\) − 346.410i − 0.533760i
\(650\) 0 0
\(651\) −300.000 −0.460829
\(652\) 0 0
\(653\) −560.000 −0.857580 −0.428790 0.903404i \(-0.641060\pi\)
−0.428790 + 0.903404i \(0.641060\pi\)
\(654\) 0 0
\(655\) 152.000 0.232061
\(656\) 0 0
\(657\) 30.0000 0.0456621
\(658\) 0 0
\(659\) − 450.333i − 0.683358i −0.939817 0.341679i \(-0.889004\pi\)
0.939817 0.341679i \(-0.110996\pi\)
\(660\) 0 0
\(661\) 398.372i 0.602680i 0.953517 + 0.301340i \(0.0974340\pi\)
−0.953517 + 0.301340i \(0.902566\pi\)
\(662\) 0 0
\(663\) −420.000 −0.633484
\(664\) 0 0
\(665\) −760.000 −1.14286
\(666\) 0 0
\(667\) 692.820i 1.03871i
\(668\) 0 0
\(669\) 138.000 0.206278
\(670\) 0 0
\(671\) 100.000 0.149031
\(672\) 0 0
\(673\) 630.466i 0.936800i 0.883516 + 0.468400i \(0.155169\pi\)
−0.883516 + 0.468400i \(0.844831\pi\)
\(674\) 0 0
\(675\) 46.7654i 0.0692820i
\(676\) 0 0
\(677\) − 526.543i − 0.777760i −0.921288 0.388880i \(-0.872862\pi\)
0.921288 0.388880i \(-0.127138\pi\)
\(678\) 0 0
\(679\) − 762.102i − 1.12239i
\(680\) 0 0
\(681\) 132.000 0.193833
\(682\) 0 0
\(683\) − 478.046i − 0.699921i −0.936764 0.349960i \(-0.886195\pi\)
0.936764 0.349960i \(-0.113805\pi\)
\(684\) 0 0
\(685\) 760.000 1.10949
\(686\) 0 0
\(687\) 190.526i 0.277330i
\(688\) 0 0
\(689\) 1008.00 1.46299
\(690\) 0 0
\(691\) −470.000 −0.680174 −0.340087 0.940394i \(-0.610456\pi\)
−0.340087 + 0.940394i \(0.610456\pi\)
\(692\) 0 0
\(693\) 300.000 0.432900
\(694\) 0 0
\(695\) −200.000 −0.287770
\(696\) 0 0
\(697\) − 346.410i − 0.497002i
\(698\) 0 0
\(699\) 329.090i 0.470801i
\(700\) 0 0
\(701\) −560.000 −0.798859 −0.399429 0.916764i \(-0.630792\pi\)
−0.399429 + 0.916764i \(0.630792\pi\)
\(702\) 0 0
\(703\) − 197.454i − 0.280873i
\(704\) 0 0
\(705\) 554.256i 0.786179i
\(706\) 0 0
\(707\) 1000.00 1.41443
\(708\) 0 0
\(709\) −982.000 −1.38505 −0.692525 0.721394i \(-0.743502\pi\)
−0.692525 + 0.721394i \(0.743502\pi\)
\(710\) 0 0
\(711\) 51.9615i 0.0730823i
\(712\) 0 0
\(713\) 346.410i 0.485849i
\(714\) 0 0
\(715\) − 969.948i − 1.35657i
\(716\) 0 0
\(717\) 221.703i 0.309209i
\(718\) 0 0
\(719\) −520.000 −0.723227 −0.361613 0.932328i \(-0.617774\pi\)
−0.361613 + 0.932328i \(0.617774\pi\)
\(720\) 0 0
\(721\) 1835.97i 2.54643i
\(722\) 0 0
\(723\) 240.000 0.331950
\(724\) 0 0
\(725\) − 311.769i − 0.430026i
\(726\) 0 0
\(727\) 790.000 1.08666 0.543329 0.839520i \(-0.317164\pi\)
0.543329 + 0.839520i \(0.317164\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) 100.000 0.136799
\(732\) 0 0
\(733\) −1150.00 −1.56889 −0.784447 0.620195i \(-0.787053\pi\)
−0.784447 + 0.620195i \(0.787053\pi\)
\(734\) 0 0
\(735\) 353.338i 0.480732i
\(736\) 0 0
\(737\) 762.102i 1.03406i
\(738\) 0 0
\(739\) −578.000 −0.782138 −0.391069 0.920361i \(-0.627895\pi\)
−0.391069 + 0.920361i \(0.627895\pi\)
\(740\) 0 0
\(741\) 798.000 1.07692
\(742\) 0 0
\(743\) − 235.559i − 0.317038i −0.987356 0.158519i \(-0.949328\pi\)
0.987356 0.158519i \(-0.0506718\pi\)
\(744\) 0 0
\(745\) −80.0000 −0.107383
\(746\) 0 0
\(747\) 210.000 0.281124
\(748\) 0 0
\(749\) 623.538i 0.832494i
\(750\) 0 0
\(751\) 952.628i 1.26848i 0.773137 + 0.634240i \(0.218687\pi\)
−0.773137 + 0.634240i \(0.781313\pi\)
\(752\) 0 0
\(753\) 3.46410i 0.00460040i
\(754\) 0 0
\(755\) − 900.666i − 1.19294i
\(756\) 0 0
\(757\) −250.000 −0.330251 −0.165125 0.986273i \(-0.552803\pi\)
−0.165125 + 0.986273i \(0.552803\pi\)
\(758\) 0 0
\(759\) − 346.410i − 0.456403i
\(760\) 0 0
\(761\) −770.000 −1.01183 −0.505913 0.862584i \(-0.668844\pi\)
−0.505913 + 0.862584i \(0.668844\pi\)
\(762\) 0 0
\(763\) 1558.85i 2.04305i
\(764\) 0 0
\(765\) −120.000 −0.156863
\(766\) 0 0
\(767\) −840.000 −1.09518
\(768\) 0 0
\(769\) 110.000 0.143043 0.0715215 0.997439i \(-0.477215\pi\)
0.0715215 + 0.997439i \(0.477215\pi\)
\(770\) 0 0
\(771\) 852.000 1.10506
\(772\) 0 0
\(773\) 145.492i 0.188218i 0.995562 + 0.0941088i \(0.0300002\pi\)
−0.995562 + 0.0941088i \(0.970000\pi\)
\(774\) 0 0
\(775\) − 155.885i − 0.201141i
\(776\) 0 0
\(777\) −180.000 −0.231660
\(778\) 0 0
\(779\) 658.179i 0.844903i
\(780\) 0 0
\(781\) − 1039.23i − 1.33064i
\(782\) 0 0
\(783\) 180.000 0.229885
\(784\) 0 0
\(785\) 920.000 1.17197
\(786\) 0 0
\(787\) − 96.9948i − 0.123246i −0.998099 0.0616232i \(-0.980372\pi\)
0.998099 0.0616232i \(-0.0196277\pi\)
\(788\) 0 0
\(789\) 346.410i 0.439050i
\(790\) 0 0
\(791\) 69.2820i 0.0875879i
\(792\) 0 0
\(793\) − 242.487i − 0.305785i
\(794\) 0 0
\(795\) 288.000 0.362264
\(796\) 0 0
\(797\) 339.482i 0.425950i 0.977058 + 0.212975i \(0.0683153\pi\)
−0.977058 + 0.212975i \(0.931685\pi\)
\(798\) 0 0
\(799\) 800.000 1.00125
\(800\) 0 0
\(801\) − 311.769i − 0.389225i
\(802\) 0 0
\(803\) 100.000 0.124533
\(804\) 0 0
\(805\) 800.000 0.993789
\(806\) 0 0
\(807\) −720.000 −0.892193
\(808\) 0 0
\(809\) −182.000 −0.224969 −0.112485 0.993653i \(-0.535881\pi\)
−0.112485 + 0.993653i \(0.535881\pi\)
\(810\) 0 0
\(811\) − 831.384i − 1.02513i −0.858647 0.512567i \(-0.828694\pi\)
0.858647 0.512567i \(-0.171306\pi\)
\(812\) 0 0
\(813\) − 294.449i − 0.362175i
\(814\) 0 0
\(815\) −680.000 −0.834356
\(816\) 0 0
\(817\) −190.000 −0.232558
\(818\) 0 0
\(819\) − 727.461i − 0.888231i
\(820\) 0 0
\(821\) −8.00000 −0.00974421 −0.00487211 0.999988i \(-0.501551\pi\)
−0.00487211 + 0.999988i \(0.501551\pi\)
\(822\) 0 0
\(823\) −950.000 −1.15431 −0.577157 0.816633i \(-0.695838\pi\)
−0.577157 + 0.816633i \(0.695838\pi\)
\(824\) 0 0
\(825\) 155.885i 0.188951i
\(826\) 0 0
\(827\) − 478.046i − 0.578048i −0.957322 0.289024i \(-0.906669\pi\)
0.957322 0.289024i \(-0.0933308\pi\)
\(828\) 0 0
\(829\) − 1195.12i − 1.44163i −0.693125 0.720817i \(-0.743767\pi\)
0.693125 0.720817i \(-0.256233\pi\)
\(830\) 0 0
\(831\) − 17.3205i − 0.0208430i
\(832\) 0 0
\(833\) 510.000 0.612245
\(834\) 0 0
\(835\) − 526.543i − 0.630591i
\(836\) 0 0
\(837\) 90.0000 0.107527
\(838\) 0 0
\(839\) − 1177.79i − 1.40381i −0.712272 0.701904i \(-0.752334\pi\)
0.712272 0.701904i \(-0.247666\pi\)
\(840\) 0 0
\(841\) −359.000 −0.426873
\(842\) 0 0
\(843\) 660.000 0.782918
\(844\) 0 0
\(845\) −1676.00 −1.98343
\(846\) 0 0
\(847\) −210.000 −0.247934
\(848\) 0 0
\(849\) 121.244i 0.142807i
\(850\) 0 0
\(851\) 207.846i 0.244237i
\(852\) 0 0
\(853\) 890.000 1.04338 0.521688 0.853136i \(-0.325302\pi\)
0.521688 + 0.853136i \(0.325302\pi\)
\(854\) 0 0
\(855\) 228.000 0.266667
\(856\) 0 0
\(857\) − 1254.00i − 1.46325i −0.681708 0.731625i \(-0.738762\pi\)
0.681708 0.731625i \(-0.261238\pi\)
\(858\) 0 0
\(859\) −182.000 −0.211874 −0.105937 0.994373i \(-0.533784\pi\)
−0.105937 + 0.994373i \(0.533784\pi\)
\(860\) 0 0
\(861\) 600.000 0.696864
\(862\) 0 0
\(863\) 1080.80i 1.25238i 0.779672 + 0.626188i \(0.215386\pi\)
−0.779672 + 0.626188i \(0.784614\pi\)
\(864\) 0 0
\(865\) − 942.236i − 1.08929i
\(866\) 0 0
\(867\) − 327.358i − 0.377575i
\(868\) 0 0
\(869\) 173.205i 0.199315i
\(870\) 0 0
\(871\) 1848.00 2.12170
\(872\) 0 0
\(873\) 228.631i 0.261891i
\(874\) 0 0
\(875\) −1360.00 −1.55429
\(876\) 0 0
\(877\) 1188.19i 1.35483i 0.735601 + 0.677416i \(0.236900\pi\)
−0.735601 + 0.677416i \(0.763100\pi\)
\(878\) 0 0
\(879\) 312.000 0.354949
\(880\) 0 0
\(881\) 550.000 0.624291 0.312145 0.950034i \(-0.398952\pi\)
0.312145 + 0.950034i \(0.398952\pi\)
\(882\) 0 0
\(883\) 1450.00 1.64213 0.821065 0.570835i \(-0.193381\pi\)
0.821065 + 0.570835i \(0.193381\pi\)
\(884\) 0 0
\(885\) −240.000 −0.271186
\(886\) 0 0
\(887\) − 1254.00i − 1.41376i −0.707334 0.706880i \(-0.750102\pi\)
0.707334 0.706880i \(-0.249898\pi\)
\(888\) 0 0
\(889\) − 1143.15i − 1.28589i
\(890\) 0 0
\(891\) −90.0000 −0.101010
\(892\) 0 0
\(893\) −1520.00 −1.70213
\(894\) 0 0
\(895\) − 415.692i − 0.464461i
\(896\) 0 0
\(897\) −840.000 −0.936455
\(898\) 0 0
\(899\) −600.000 −0.667408
\(900\) 0 0
\(901\) − 415.692i − 0.461368i
\(902\) 0 0
\(903\) 173.205i 0.191811i
\(904\) 0 0
\(905\) − 1039.23i − 1.14832i
\(906\) 0 0
\(907\) − 110.851i − 0.122217i −0.998131 0.0611087i \(-0.980536\pi\)
0.998131 0.0611087i \(-0.0194636\pi\)
\(908\) 0 0
\(909\) −300.000 −0.330033
\(910\) 0 0
\(911\) 796.743i 0.874581i 0.899320 + 0.437291i \(0.144062\pi\)
−0.899320 + 0.437291i \(0.855938\pi\)
\(912\) 0 0
\(913\) 700.000 0.766703
\(914\) 0 0
\(915\) − 69.2820i − 0.0757181i
\(916\) 0 0
\(917\) 380.000 0.414395
\(918\) 0 0
\(919\) −62.0000 −0.0674646 −0.0337323 0.999431i \(-0.510739\pi\)
−0.0337323 + 0.999431i \(0.510739\pi\)
\(920\) 0 0
\(921\) 252.000 0.273616
\(922\) 0 0
\(923\) −2520.00 −2.73023
\(924\) 0 0
\(925\) − 93.5307i − 0.101114i
\(926\) 0 0
\(927\) − 550.792i − 0.594166i
\(928\) 0 0
\(929\) −242.000 −0.260495 −0.130248 0.991482i \(-0.541577\pi\)
−0.130248 + 0.991482i \(0.541577\pi\)
\(930\) 0 0
\(931\) −969.000 −1.04082
\(932\) 0 0
\(933\) − 1004.59i − 1.07673i
\(934\) 0 0
\(935\) −400.000 −0.427807
\(936\) 0 0
\(937\) 110.000 0.117396 0.0586980 0.998276i \(-0.481305\pi\)
0.0586980 + 0.998276i \(0.481305\pi\)
\(938\) 0 0
\(939\) − 640.859i − 0.682491i
\(940\) 0 0
\(941\) 796.743i 0.846699i 0.905967 + 0.423349i \(0.139146\pi\)
−0.905967 + 0.423349i \(0.860854\pi\)
\(942\) 0 0
\(943\) − 692.820i − 0.734698i
\(944\) 0 0
\(945\) − 207.846i − 0.219943i
\(946\) 0 0
\(947\) −1450.00 −1.53115 −0.765576 0.643346i \(-0.777546\pi\)
−0.765576 + 0.643346i \(0.777546\pi\)
\(948\) 0 0
\(949\) − 242.487i − 0.255519i
\(950\) 0 0
\(951\) 48.0000 0.0504732
\(952\) 0 0
\(953\) − 353.338i − 0.370764i −0.982667 0.185382i \(-0.940648\pi\)
0.982667 0.185382i \(-0.0593523\pi\)
\(954\) 0 0
\(955\) 1328.00 1.39058
\(956\) 0 0
\(957\) 600.000 0.626959
\(958\) 0 0
\(959\) 1900.00 1.98123
\(960\) 0 0
\(961\) 661.000 0.687825
\(962\) 0 0
\(963\) − 187.061i − 0.194249i
\(964\) 0 0
\(965\) 387.979i 0.402051i
\(966\) 0 0
\(967\) −470.000 −0.486039 −0.243020 0.970021i \(-0.578138\pi\)
−0.243020 + 0.970021i \(0.578138\pi\)
\(968\) 0 0
\(969\) − 329.090i − 0.339618i
\(970\) 0 0
\(971\) − 519.615i − 0.535134i −0.963539 0.267567i \(-0.913780\pi\)
0.963539 0.267567i \(-0.0862197\pi\)
\(972\) 0 0
\(973\) −500.000 −0.513875
\(974\) 0 0
\(975\) 378.000 0.387692
\(976\) 0 0
\(977\) 1669.70i 1.70900i 0.519448 + 0.854502i \(0.326138\pi\)
−0.519448 + 0.854502i \(0.673862\pi\)
\(978\) 0 0
\(979\) − 1039.23i − 1.06152i
\(980\) 0 0
\(981\) − 467.654i − 0.476711i
\(982\) 0 0
\(983\) 1690.48i 1.71972i 0.510533 + 0.859858i \(0.329448\pi\)
−0.510533 + 0.859858i \(0.670552\pi\)
\(984\) 0 0
\(985\) 640.000 0.649746
\(986\) 0 0
\(987\) 1385.64i 1.40389i
\(988\) 0 0
\(989\) 200.000 0.202224
\(990\) 0 0
\(991\) 571.577i 0.576768i 0.957515 + 0.288384i \(0.0931179\pi\)
−0.957515 + 0.288384i \(0.906882\pi\)
\(992\) 0 0
\(993\) 300.000 0.302115
\(994\) 0 0
\(995\) −392.000 −0.393970
\(996\) 0 0
\(997\) 1550.00 1.55466 0.777332 0.629091i \(-0.216573\pi\)
0.777332 + 0.629091i \(0.216573\pi\)
\(998\) 0 0
\(999\) 54.0000 0.0540541
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 912.3.o.a.721.2 2
3.2 odd 2 2736.3.o.e.721.2 2
4.3 odd 2 57.3.c.a.37.1 2
12.11 even 2 171.3.c.c.37.2 2
19.18 odd 2 inner 912.3.o.a.721.1 2
57.56 even 2 2736.3.o.e.721.1 2
76.75 even 2 57.3.c.a.37.2 yes 2
228.227 odd 2 171.3.c.c.37.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.3.c.a.37.1 2 4.3 odd 2
57.3.c.a.37.2 yes 2 76.75 even 2
171.3.c.c.37.1 2 228.227 odd 2
171.3.c.c.37.2 2 12.11 even 2
912.3.o.a.721.1 2 19.18 odd 2 inner
912.3.o.a.721.2 2 1.1 even 1 trivial
2736.3.o.e.721.1 2 57.56 even 2
2736.3.o.e.721.2 2 3.2 odd 2