Properties

Label 1792.3.d.j.1023.15
Level $1792$
Weight $3$
Character 1792.1023
Analytic conductor $48.828$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1792,3,Mod(1023,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.1023");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1792.d (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: \(48.8284633734\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 5x^{14} + 48x^{12} + 180x^{10} + 1056x^{8} + 2880x^{6} + 12288x^{4} + 20480x^{2} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{38} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1023.15
Root \(-1.45617 - 1.37098i\) of defining polynomial
Character \(\chi\) \(=\) 1792.1023
Dual form 1792.3.d.j.1023.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.22363i q^{3} -6.26788 q^{5} +2.64575i q^{7} -18.2863 q^{9} +O(q^{10})\) \(q+5.22363i q^{3} -6.26788 q^{5} +2.64575i q^{7} -18.2863 q^{9} +9.80688i q^{11} +2.41653 q^{13} -32.7411i q^{15} +6.89452 q^{17} -2.77637i q^{19} -13.8204 q^{21} -42.8332i q^{23} +14.2863 q^{25} -48.5083i q^{27} -37.3505 q^{29} -7.16835i q^{31} -51.2275 q^{33} -16.5833i q^{35} +0.202653 q^{37} +12.6231i q^{39} -63.5494 q^{41} -35.3384i q^{43} +114.616 q^{45} -37.9129i q^{47} -7.00000 q^{49} +36.0144i q^{51} +54.6651 q^{53} -61.4684i q^{55} +14.5027 q^{57} +104.795i q^{59} +43.7668 q^{61} -48.3810i q^{63} -15.1465 q^{65} -31.1021i q^{67} +223.745 q^{69} -23.1294i q^{71} +69.2275 q^{73} +74.6264i q^{75} -25.9466 q^{77} +19.9328i q^{79} +88.8125 q^{81} +5.11617i q^{83} -43.2140 q^{85} -195.105i q^{87} +17.9889 q^{89} +6.39353i q^{91} +37.4448 q^{93} +17.4019i q^{95} +12.4864 q^{97} -179.332i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 96 q^{9} - 160 q^{17} + 32 q^{25} + 64 q^{33} - 256 q^{41} - 112 q^{49} - 112 q^{57} - 144 q^{65} + 224 q^{73} + 96 q^{81} + 1024 q^{89} + 128 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 5.22363i 1.74121i 0.491982 + 0.870605i \(0.336272\pi\)
−0.491982 + 0.870605i \(0.663728\pi\)
\(4\) 0 0
\(5\) −6.26788 −1.25358 −0.626788 0.779190i \(-0.715631\pi\)
−0.626788 + 0.779190i \(0.715631\pi\)
\(6\) 0 0
\(7\) 2.64575i 0.377964i
\(8\) 0 0
\(9\) −18.2863 −2.03181
\(10\) 0 0
\(11\) 9.80688i 0.891535i 0.895149 + 0.445767i \(0.147069\pi\)
−0.895149 + 0.445767i \(0.852931\pi\)
\(12\) 0 0
\(13\) 2.41653 0.185887 0.0929434 0.995671i \(-0.470372\pi\)
0.0929434 + 0.995671i \(0.470372\pi\)
\(14\) 0 0
\(15\) − 32.7411i − 2.18274i
\(16\) 0 0
\(17\) 6.89452 0.405560 0.202780 0.979224i \(-0.435002\pi\)
0.202780 + 0.979224i \(0.435002\pi\)
\(18\) 0 0
\(19\) − 2.77637i − 0.146125i −0.997327 0.0730624i \(-0.976723\pi\)
0.997327 0.0730624i \(-0.0232772\pi\)
\(20\) 0 0
\(21\) −13.8204 −0.658116
\(22\) 0 0
\(23\) − 42.8332i − 1.86231i −0.364617 0.931157i \(-0.618800\pi\)
0.364617 0.931157i \(-0.381200\pi\)
\(24\) 0 0
\(25\) 14.2863 0.571453
\(26\) 0 0
\(27\) − 48.5083i − 1.79660i
\(28\) 0 0
\(29\) −37.3505 −1.28795 −0.643974 0.765048i \(-0.722715\pi\)
−0.643974 + 0.765048i \(0.722715\pi\)
\(30\) 0 0
\(31\) − 7.16835i − 0.231237i −0.993294 0.115619i \(-0.963115\pi\)
0.993294 0.115619i \(-0.0368850\pi\)
\(32\) 0 0
\(33\) −51.2275 −1.55235
\(34\) 0 0
\(35\) − 16.5833i − 0.473807i
\(36\) 0 0
\(37\) 0.202653 0.00547709 0.00273855 0.999996i \(-0.499128\pi\)
0.00273855 + 0.999996i \(0.499128\pi\)
\(38\) 0 0
\(39\) 12.6231i 0.323668i
\(40\) 0 0
\(41\) −63.5494 −1.54999 −0.774993 0.631970i \(-0.782247\pi\)
−0.774993 + 0.631970i \(0.782247\pi\)
\(42\) 0 0
\(43\) − 35.3384i − 0.821823i −0.911675 0.410911i \(-0.865211\pi\)
0.911675 0.410911i \(-0.134789\pi\)
\(44\) 0 0
\(45\) 114.616 2.54703
\(46\) 0 0
\(47\) − 37.9129i − 0.806657i −0.915055 0.403329i \(-0.867853\pi\)
0.915055 0.403329i \(-0.132147\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) 36.0144i 0.706165i
\(52\) 0 0
\(53\) 54.6651 1.03142 0.515709 0.856764i \(-0.327529\pi\)
0.515709 + 0.856764i \(0.327529\pi\)
\(54\) 0 0
\(55\) − 61.4684i − 1.11761i
\(56\) 0 0
\(57\) 14.5027 0.254434
\(58\) 0 0
\(59\) 104.795i 1.77619i 0.459665 + 0.888093i \(0.347970\pi\)
−0.459665 + 0.888093i \(0.652030\pi\)
\(60\) 0 0
\(61\) 43.7668 0.717489 0.358745 0.933436i \(-0.383205\pi\)
0.358745 + 0.933436i \(0.383205\pi\)
\(62\) 0 0
\(63\) − 48.3810i − 0.767953i
\(64\) 0 0
\(65\) −15.1465 −0.233023
\(66\) 0 0
\(67\) − 31.1021i − 0.464210i −0.972691 0.232105i \(-0.925439\pi\)
0.972691 0.232105i \(-0.0745613\pi\)
\(68\) 0 0
\(69\) 223.745 3.24268
\(70\) 0 0
\(71\) − 23.1294i − 0.325766i −0.986645 0.162883i \(-0.947921\pi\)
0.986645 0.162883i \(-0.0520794\pi\)
\(72\) 0 0
\(73\) 69.2275 0.948322 0.474161 0.880438i \(-0.342751\pi\)
0.474161 + 0.880438i \(0.342751\pi\)
\(74\) 0 0
\(75\) 74.6264i 0.995019i
\(76\) 0 0
\(77\) −25.9466 −0.336968
\(78\) 0 0
\(79\) 19.9328i 0.252315i 0.992010 + 0.126157i \(0.0402644\pi\)
−0.992010 + 0.126157i \(0.959736\pi\)
\(80\) 0 0
\(81\) 88.8125 1.09645
\(82\) 0 0
\(83\) 5.11617i 0.0616406i 0.999525 + 0.0308203i \(0.00981197\pi\)
−0.999525 + 0.0308203i \(0.990188\pi\)
\(84\) 0 0
\(85\) −43.2140 −0.508400
\(86\) 0 0
\(87\) − 195.105i − 2.24259i
\(88\) 0 0
\(89\) 17.9889 0.202122 0.101061 0.994880i \(-0.467776\pi\)
0.101061 + 0.994880i \(0.467776\pi\)
\(90\) 0 0
\(91\) 6.39353i 0.0702586i
\(92\) 0 0
\(93\) 37.4448 0.402632
\(94\) 0 0
\(95\) 17.4019i 0.183178i
\(96\) 0 0
\(97\) 12.4864 0.128726 0.0643629 0.997927i \(-0.479498\pi\)
0.0643629 + 0.997927i \(0.479498\pi\)
\(98\) 0 0
\(99\) − 179.332i − 1.81143i
\(100\) 0 0
\(101\) 68.0753 0.674013 0.337006 0.941502i \(-0.390586\pi\)
0.337006 + 0.941502i \(0.390586\pi\)
\(102\) 0 0
\(103\) 58.2931i 0.565952i 0.959127 + 0.282976i \(0.0913217\pi\)
−0.959127 + 0.282976i \(0.908678\pi\)
\(104\) 0 0
\(105\) 86.6248 0.824998
\(106\) 0 0
\(107\) 135.868i 1.26979i 0.772597 + 0.634897i \(0.218957\pi\)
−0.772597 + 0.634897i \(0.781043\pi\)
\(108\) 0 0
\(109\) 44.4981 0.408239 0.204120 0.978946i \(-0.434567\pi\)
0.204120 + 0.978946i \(0.434567\pi\)
\(110\) 0 0
\(111\) 1.05858i 0.00953677i
\(112\) 0 0
\(113\) −133.391 −1.18045 −0.590224 0.807240i \(-0.700961\pi\)
−0.590224 + 0.807240i \(0.700961\pi\)
\(114\) 0 0
\(115\) 268.474i 2.33455i
\(116\) 0 0
\(117\) −44.1894 −0.377687
\(118\) 0 0
\(119\) 18.2412i 0.153287i
\(120\) 0 0
\(121\) 24.8251 0.205166
\(122\) 0 0
\(123\) − 331.959i − 2.69885i
\(124\) 0 0
\(125\) 67.1521 0.537217
\(126\) 0 0
\(127\) − 130.977i − 1.03131i −0.856795 0.515657i \(-0.827548\pi\)
0.856795 0.515657i \(-0.172452\pi\)
\(128\) 0 0
\(129\) 184.595 1.43097
\(130\) 0 0
\(131\) 53.3311i 0.407108i 0.979064 + 0.203554i \(0.0652492\pi\)
−0.979064 + 0.203554i \(0.934751\pi\)
\(132\) 0 0
\(133\) 7.34558 0.0552299
\(134\) 0 0
\(135\) 304.044i 2.25218i
\(136\) 0 0
\(137\) 57.7179 0.421299 0.210649 0.977562i \(-0.432442\pi\)
0.210649 + 0.977562i \(0.432442\pi\)
\(138\) 0 0
\(139\) − 172.422i − 1.24045i −0.784425 0.620224i \(-0.787042\pi\)
0.784425 0.620224i \(-0.212958\pi\)
\(140\) 0 0
\(141\) 198.043 1.40456
\(142\) 0 0
\(143\) 23.6986i 0.165725i
\(144\) 0 0
\(145\) 234.108 1.61454
\(146\) 0 0
\(147\) − 36.5654i − 0.248744i
\(148\) 0 0
\(149\) −219.529 −1.47335 −0.736673 0.676249i \(-0.763604\pi\)
−0.736673 + 0.676249i \(0.763604\pi\)
\(150\) 0 0
\(151\) 185.668i 1.22959i 0.788687 + 0.614795i \(0.210761\pi\)
−0.788687 + 0.614795i \(0.789239\pi\)
\(152\) 0 0
\(153\) −126.075 −0.824022
\(154\) 0 0
\(155\) 44.9304i 0.289873i
\(156\) 0 0
\(157\) −188.182 −1.19861 −0.599305 0.800521i \(-0.704556\pi\)
−0.599305 + 0.800521i \(0.704556\pi\)
\(158\) 0 0
\(159\) 285.550i 1.79591i
\(160\) 0 0
\(161\) 113.326 0.703889
\(162\) 0 0
\(163\) − 54.5154i − 0.334450i −0.985919 0.167225i \(-0.946519\pi\)
0.985919 0.167225i \(-0.0534806\pi\)
\(164\) 0 0
\(165\) 321.088 1.94599
\(166\) 0 0
\(167\) − 266.435i − 1.59542i −0.603042 0.797709i \(-0.706045\pi\)
0.603042 0.797709i \(-0.293955\pi\)
\(168\) 0 0
\(169\) −163.160 −0.965446
\(170\) 0 0
\(171\) 50.7696i 0.296898i
\(172\) 0 0
\(173\) 114.835 0.663786 0.331893 0.943317i \(-0.392313\pi\)
0.331893 + 0.943317i \(0.392313\pi\)
\(174\) 0 0
\(175\) 37.7980i 0.215989i
\(176\) 0 0
\(177\) −547.410 −3.09271
\(178\) 0 0
\(179\) − 112.849i − 0.630439i −0.949019 0.315220i \(-0.897922\pi\)
0.949019 0.315220i \(-0.102078\pi\)
\(180\) 0 0
\(181\) −60.9470 −0.336724 −0.168362 0.985725i \(-0.553848\pi\)
−0.168362 + 0.985725i \(0.553848\pi\)
\(182\) 0 0
\(183\) 228.622i 1.24930i
\(184\) 0 0
\(185\) −1.27020 −0.00686595
\(186\) 0 0
\(187\) 67.6138i 0.361571i
\(188\) 0 0
\(189\) 128.341 0.679052
\(190\) 0 0
\(191\) − 178.459i − 0.934342i −0.884167 0.467171i \(-0.845273\pi\)
0.884167 0.467171i \(-0.154727\pi\)
\(192\) 0 0
\(193\) 221.588 1.14812 0.574062 0.818812i \(-0.305367\pi\)
0.574062 + 0.818812i \(0.305367\pi\)
\(194\) 0 0
\(195\) − 79.1198i − 0.405742i
\(196\) 0 0
\(197\) 242.298 1.22994 0.614970 0.788550i \(-0.289168\pi\)
0.614970 + 0.788550i \(0.289168\pi\)
\(198\) 0 0
\(199\) 297.047i 1.49270i 0.665555 + 0.746349i \(0.268195\pi\)
−0.665555 + 0.746349i \(0.731805\pi\)
\(200\) 0 0
\(201\) 162.466 0.808287
\(202\) 0 0
\(203\) − 98.8201i − 0.486798i
\(204\) 0 0
\(205\) 398.320 1.94302
\(206\) 0 0
\(207\) 783.262i 3.78388i
\(208\) 0 0
\(209\) 27.2275 0.130275
\(210\) 0 0
\(211\) 141.020i 0.668341i 0.942513 + 0.334171i \(0.108456\pi\)
−0.942513 + 0.334171i \(0.891544\pi\)
\(212\) 0 0
\(213\) 120.820 0.567228
\(214\) 0 0
\(215\) 221.497i 1.03022i
\(216\) 0 0
\(217\) 18.9657 0.0873994
\(218\) 0 0
\(219\) 361.619i 1.65123i
\(220\) 0 0
\(221\) 16.6608 0.0753883
\(222\) 0 0
\(223\) − 40.8267i − 0.183079i −0.995801 0.0915396i \(-0.970821\pi\)
0.995801 0.0915396i \(-0.0291788\pi\)
\(224\) 0 0
\(225\) −261.244 −1.16108
\(226\) 0 0
\(227\) − 8.18598i − 0.0360616i −0.999837 0.0180308i \(-0.994260\pi\)
0.999837 0.0180308i \(-0.00573969\pi\)
\(228\) 0 0
\(229\) 332.252 1.45088 0.725440 0.688285i \(-0.241636\pi\)
0.725440 + 0.688285i \(0.241636\pi\)
\(230\) 0 0
\(231\) − 135.535i − 0.586733i
\(232\) 0 0
\(233\) −329.260 −1.41314 −0.706568 0.707646i \(-0.749757\pi\)
−0.706568 + 0.707646i \(0.749757\pi\)
\(234\) 0 0
\(235\) 237.633i 1.01121i
\(236\) 0 0
\(237\) −104.122 −0.439333
\(238\) 0 0
\(239\) 137.719i 0.576230i 0.957596 + 0.288115i \(0.0930286\pi\)
−0.957596 + 0.288115i \(0.906971\pi\)
\(240\) 0 0
\(241\) 201.854 0.837567 0.418783 0.908086i \(-0.362457\pi\)
0.418783 + 0.908086i \(0.362457\pi\)
\(242\) 0 0
\(243\) 27.3492i 0.112548i
\(244\) 0 0
\(245\) 43.8752 0.179082
\(246\) 0 0
\(247\) − 6.70917i − 0.0271627i
\(248\) 0 0
\(249\) −26.7250 −0.107329
\(250\) 0 0
\(251\) − 269.203i − 1.07252i −0.844052 0.536261i \(-0.819836\pi\)
0.844052 0.536261i \(-0.180164\pi\)
\(252\) 0 0
\(253\) 420.061 1.66032
\(254\) 0 0
\(255\) − 225.734i − 0.885232i
\(256\) 0 0
\(257\) −242.359 −0.943032 −0.471516 0.881858i \(-0.656293\pi\)
−0.471516 + 0.881858i \(0.656293\pi\)
\(258\) 0 0
\(259\) 0.536168i 0.00207015i
\(260\) 0 0
\(261\) 683.002 2.61687
\(262\) 0 0
\(263\) 33.8470i 0.128696i 0.997928 + 0.0643479i \(0.0204967\pi\)
−0.997928 + 0.0643479i \(0.979503\pi\)
\(264\) 0 0
\(265\) −342.634 −1.29296
\(266\) 0 0
\(267\) 93.9673i 0.351937i
\(268\) 0 0
\(269\) −165.598 −0.615606 −0.307803 0.951450i \(-0.599594\pi\)
−0.307803 + 0.951450i \(0.599594\pi\)
\(270\) 0 0
\(271\) − 148.308i − 0.547263i −0.961835 0.273632i \(-0.911775\pi\)
0.961835 0.273632i \(-0.0882249\pi\)
\(272\) 0 0
\(273\) −33.3975 −0.122335
\(274\) 0 0
\(275\) 140.104i 0.509470i
\(276\) 0 0
\(277\) −478.358 −1.72693 −0.863463 0.504413i \(-0.831709\pi\)
−0.863463 + 0.504413i \(0.831709\pi\)
\(278\) 0 0
\(279\) 131.083i 0.469830i
\(280\) 0 0
\(281\) 226.066 0.804506 0.402253 0.915528i \(-0.368227\pi\)
0.402253 + 0.915528i \(0.368227\pi\)
\(282\) 0 0
\(283\) − 254.628i − 0.899745i −0.893093 0.449873i \(-0.851469\pi\)
0.893093 0.449873i \(-0.148531\pi\)
\(284\) 0 0
\(285\) −90.9014 −0.318952
\(286\) 0 0
\(287\) − 168.136i − 0.585840i
\(288\) 0 0
\(289\) −241.466 −0.835521
\(290\) 0 0
\(291\) 65.2244i 0.224139i
\(292\) 0 0
\(293\) −149.558 −0.510437 −0.255218 0.966883i \(-0.582147\pi\)
−0.255218 + 0.966883i \(0.582147\pi\)
\(294\) 0 0
\(295\) − 656.842i − 2.22658i
\(296\) 0 0
\(297\) 475.715 1.60173
\(298\) 0 0
\(299\) − 103.508i − 0.346180i
\(300\) 0 0
\(301\) 93.4966 0.310620
\(302\) 0 0
\(303\) 355.600i 1.17360i
\(304\) 0 0
\(305\) −274.325 −0.899427
\(306\) 0 0
\(307\) − 271.779i − 0.885272i −0.896701 0.442636i \(-0.854043\pi\)
0.896701 0.442636i \(-0.145957\pi\)
\(308\) 0 0
\(309\) −304.502 −0.985442
\(310\) 0 0
\(311\) − 534.180i − 1.71762i −0.512293 0.858811i \(-0.671204\pi\)
0.512293 0.858811i \(-0.328796\pi\)
\(312\) 0 0
\(313\) 556.232 1.77710 0.888550 0.458780i \(-0.151713\pi\)
0.888550 + 0.458780i \(0.151713\pi\)
\(314\) 0 0
\(315\) 303.247i 0.962688i
\(316\) 0 0
\(317\) −387.459 −1.22227 −0.611134 0.791527i \(-0.709286\pi\)
−0.611134 + 0.791527i \(0.709286\pi\)
\(318\) 0 0
\(319\) − 366.292i − 1.14825i
\(320\) 0 0
\(321\) −709.724 −2.21098
\(322\) 0 0
\(323\) − 19.1417i − 0.0592624i
\(324\) 0 0
\(325\) 34.5233 0.106225
\(326\) 0 0
\(327\) 232.442i 0.710830i
\(328\) 0 0
\(329\) 100.308 0.304888
\(330\) 0 0
\(331\) − 383.205i − 1.15772i −0.815427 0.578860i \(-0.803498\pi\)
0.815427 0.578860i \(-0.196502\pi\)
\(332\) 0 0
\(333\) −3.70577 −0.0111284
\(334\) 0 0
\(335\) 194.944i 0.581923i
\(336\) 0 0
\(337\) 563.726 1.67278 0.836388 0.548138i \(-0.184663\pi\)
0.836388 + 0.548138i \(0.184663\pi\)
\(338\) 0 0
\(339\) − 696.783i − 2.05541i
\(340\) 0 0
\(341\) 70.2992 0.206156
\(342\) 0 0
\(343\) − 18.5203i − 0.0539949i
\(344\) 0 0
\(345\) −1402.41 −4.06495
\(346\) 0 0
\(347\) 51.5890i 0.148671i 0.997233 + 0.0743357i \(0.0236836\pi\)
−0.997233 + 0.0743357i \(0.976316\pi\)
\(348\) 0 0
\(349\) −586.383 −1.68018 −0.840090 0.542447i \(-0.817498\pi\)
−0.840090 + 0.542447i \(0.817498\pi\)
\(350\) 0 0
\(351\) − 117.222i − 0.333965i
\(352\) 0 0
\(353\) 303.844 0.860746 0.430373 0.902651i \(-0.358382\pi\)
0.430373 + 0.902651i \(0.358382\pi\)
\(354\) 0 0
\(355\) 144.972i 0.408373i
\(356\) 0 0
\(357\) −95.2852 −0.266905
\(358\) 0 0
\(359\) − 116.130i − 0.323481i −0.986833 0.161741i \(-0.948289\pi\)
0.986833 0.161741i \(-0.0517108\pi\)
\(360\) 0 0
\(361\) 353.292 0.978648
\(362\) 0 0
\(363\) 129.677i 0.357237i
\(364\) 0 0
\(365\) −433.910 −1.18879
\(366\) 0 0
\(367\) − 476.288i − 1.29779i −0.760879 0.648894i \(-0.775232\pi\)
0.760879 0.648894i \(-0.224768\pi\)
\(368\) 0 0
\(369\) 1162.08 3.14928
\(370\) 0 0
\(371\) 144.630i 0.389839i
\(372\) 0 0
\(373\) −49.2857 −0.132133 −0.0660666 0.997815i \(-0.521045\pi\)
−0.0660666 + 0.997815i \(0.521045\pi\)
\(374\) 0 0
\(375\) 350.778i 0.935407i
\(376\) 0 0
\(377\) −90.2585 −0.239412
\(378\) 0 0
\(379\) 167.511i 0.441983i 0.975276 + 0.220991i \(0.0709292\pi\)
−0.975276 + 0.220991i \(0.929071\pi\)
\(380\) 0 0
\(381\) 684.174 1.79573
\(382\) 0 0
\(383\) − 513.207i − 1.33997i −0.742376 0.669983i \(-0.766301\pi\)
0.742376 0.669983i \(-0.233699\pi\)
\(384\) 0 0
\(385\) 162.630 0.422416
\(386\) 0 0
\(387\) 646.209i 1.66979i
\(388\) 0 0
\(389\) −709.398 −1.82365 −0.911823 0.410584i \(-0.865325\pi\)
−0.911823 + 0.410584i \(0.865325\pi\)
\(390\) 0 0
\(391\) − 295.315i − 0.755281i
\(392\) 0 0
\(393\) −278.582 −0.708860
\(394\) 0 0
\(395\) − 124.937i − 0.316295i
\(396\) 0 0
\(397\) 309.404 0.779355 0.389677 0.920951i \(-0.372587\pi\)
0.389677 + 0.920951i \(0.372587\pi\)
\(398\) 0 0
\(399\) 38.3706i 0.0961669i
\(400\) 0 0
\(401\) −528.073 −1.31689 −0.658445 0.752629i \(-0.728785\pi\)
−0.658445 + 0.752629i \(0.728785\pi\)
\(402\) 0 0
\(403\) − 17.3225i − 0.0429839i
\(404\) 0 0
\(405\) −556.666 −1.37448
\(406\) 0 0
\(407\) 1.98739i 0.00488302i
\(408\) 0 0
\(409\) 612.830 1.49836 0.749181 0.662366i \(-0.230448\pi\)
0.749181 + 0.662366i \(0.230448\pi\)
\(410\) 0 0
\(411\) 301.497i 0.733569i
\(412\) 0 0
\(413\) −277.261 −0.671335
\(414\) 0 0
\(415\) − 32.0676i − 0.0772712i
\(416\) 0 0
\(417\) 900.670 2.15988
\(418\) 0 0
\(419\) 237.642i 0.567165i 0.958948 + 0.283583i \(0.0915230\pi\)
−0.958948 + 0.283583i \(0.908477\pi\)
\(420\) 0 0
\(421\) 394.516 0.937092 0.468546 0.883439i \(-0.344778\pi\)
0.468546 + 0.883439i \(0.344778\pi\)
\(422\) 0 0
\(423\) 693.287i 1.63898i
\(424\) 0 0
\(425\) 98.4973 0.231758
\(426\) 0 0
\(427\) 115.796i 0.271185i
\(428\) 0 0
\(429\) −123.793 −0.288561
\(430\) 0 0
\(431\) − 363.359i − 0.843059i −0.906815 0.421530i \(-0.861493\pi\)
0.906815 0.421530i \(-0.138507\pi\)
\(432\) 0 0
\(433\) 119.733 0.276520 0.138260 0.990396i \(-0.455849\pi\)
0.138260 + 0.990396i \(0.455849\pi\)
\(434\) 0 0
\(435\) 1222.89i 2.81125i
\(436\) 0 0
\(437\) −118.921 −0.272130
\(438\) 0 0
\(439\) − 871.477i − 1.98514i −0.121672 0.992570i \(-0.538826\pi\)
0.121672 0.992570i \(-0.461174\pi\)
\(440\) 0 0
\(441\) 128.004 0.290259
\(442\) 0 0
\(443\) − 343.956i − 0.776424i −0.921570 0.388212i \(-0.873093\pi\)
0.921570 0.388212i \(-0.126907\pi\)
\(444\) 0 0
\(445\) −112.752 −0.253376
\(446\) 0 0
\(447\) − 1146.74i − 2.56541i
\(448\) 0 0
\(449\) −242.849 −0.540866 −0.270433 0.962739i \(-0.587167\pi\)
−0.270433 + 0.962739i \(0.587167\pi\)
\(450\) 0 0
\(451\) − 623.222i − 1.38187i
\(452\) 0 0
\(453\) −969.861 −2.14097
\(454\) 0 0
\(455\) − 40.0739i − 0.0880745i
\(456\) 0 0
\(457\) 42.2571 0.0924662 0.0462331 0.998931i \(-0.485278\pi\)
0.0462331 + 0.998931i \(0.485278\pi\)
\(458\) 0 0
\(459\) − 334.441i − 0.728631i
\(460\) 0 0
\(461\) 816.370 1.77087 0.885434 0.464766i \(-0.153861\pi\)
0.885434 + 0.464766i \(0.153861\pi\)
\(462\) 0 0
\(463\) 115.161i 0.248727i 0.992237 + 0.124363i \(0.0396889\pi\)
−0.992237 + 0.124363i \(0.960311\pi\)
\(464\) 0 0
\(465\) −234.700 −0.504730
\(466\) 0 0
\(467\) − 603.424i − 1.29213i −0.763283 0.646064i \(-0.776414\pi\)
0.763283 0.646064i \(-0.223586\pi\)
\(468\) 0 0
\(469\) 82.2884 0.175455
\(470\) 0 0
\(471\) − 982.992i − 2.08703i
\(472\) 0 0
\(473\) 346.559 0.732684
\(474\) 0 0
\(475\) − 39.6641i − 0.0835034i
\(476\) 0 0
\(477\) −999.624 −2.09565
\(478\) 0 0
\(479\) 158.595i 0.331097i 0.986202 + 0.165548i \(0.0529394\pi\)
−0.986202 + 0.165548i \(0.947061\pi\)
\(480\) 0 0
\(481\) 0.489715 0.00101812
\(482\) 0 0
\(483\) 591.974i 1.22562i
\(484\) 0 0
\(485\) −78.2633 −0.161368
\(486\) 0 0
\(487\) 106.987i 0.219687i 0.993949 + 0.109843i \(0.0350349\pi\)
−0.993949 + 0.109843i \(0.964965\pi\)
\(488\) 0 0
\(489\) 284.768 0.582348
\(490\) 0 0
\(491\) 616.591i 1.25579i 0.778299 + 0.627893i \(0.216083\pi\)
−0.778299 + 0.627893i \(0.783917\pi\)
\(492\) 0 0
\(493\) −257.514 −0.522340
\(494\) 0 0
\(495\) 1124.03i 2.27077i
\(496\) 0 0
\(497\) 61.1947 0.123128
\(498\) 0 0
\(499\) − 554.090i − 1.11040i −0.831717 0.555200i \(-0.812642\pi\)
0.831717 0.555200i \(-0.187358\pi\)
\(500\) 0 0
\(501\) 1391.76 2.77796
\(502\) 0 0
\(503\) − 148.158i − 0.294548i −0.989096 0.147274i \(-0.952950\pi\)
0.989096 0.147274i \(-0.0470499\pi\)
\(504\) 0 0
\(505\) −426.688 −0.844926
\(506\) 0 0
\(507\) − 852.290i − 1.68104i
\(508\) 0 0
\(509\) −182.889 −0.359310 −0.179655 0.983730i \(-0.557498\pi\)
−0.179655 + 0.983730i \(0.557498\pi\)
\(510\) 0 0
\(511\) 183.159i 0.358432i
\(512\) 0 0
\(513\) −134.677 −0.262528
\(514\) 0 0
\(515\) − 365.374i − 0.709464i
\(516\) 0 0
\(517\) 371.807 0.719163
\(518\) 0 0
\(519\) 599.856i 1.15579i
\(520\) 0 0
\(521\) −98.2461 −0.188572 −0.0942861 0.995545i \(-0.530057\pi\)
−0.0942861 + 0.995545i \(0.530057\pi\)
\(522\) 0 0
\(523\) − 574.764i − 1.09898i −0.835502 0.549488i \(-0.814823\pi\)
0.835502 0.549488i \(-0.185177\pi\)
\(524\) 0 0
\(525\) −197.443 −0.376082
\(526\) 0 0
\(527\) − 49.4223i − 0.0937805i
\(528\) 0 0
\(529\) −1305.69 −2.46822
\(530\) 0 0
\(531\) − 1916.31i − 3.60888i
\(532\) 0 0
\(533\) −153.569 −0.288122
\(534\) 0 0
\(535\) − 851.604i − 1.59178i
\(536\) 0 0
\(537\) 589.480 1.09773
\(538\) 0 0
\(539\) − 68.6482i − 0.127362i
\(540\) 0 0
\(541\) −370.654 −0.685128 −0.342564 0.939495i \(-0.611295\pi\)
−0.342564 + 0.939495i \(0.611295\pi\)
\(542\) 0 0
\(543\) − 318.365i − 0.586307i
\(544\) 0 0
\(545\) −278.909 −0.511759
\(546\) 0 0
\(547\) 56.5966i 0.103467i 0.998661 + 0.0517336i \(0.0164747\pi\)
−0.998661 + 0.0517336i \(0.983525\pi\)
\(548\) 0 0
\(549\) −800.334 −1.45780
\(550\) 0 0
\(551\) 103.699i 0.188201i
\(552\) 0 0
\(553\) −52.7374 −0.0953659
\(554\) 0 0
\(555\) − 6.63506i − 0.0119551i
\(556\) 0 0
\(557\) −151.525 −0.272037 −0.136019 0.990706i \(-0.543431\pi\)
−0.136019 + 0.990706i \(0.543431\pi\)
\(558\) 0 0
\(559\) − 85.3962i − 0.152766i
\(560\) 0 0
\(561\) −353.189 −0.629571
\(562\) 0 0
\(563\) 318.048i 0.564917i 0.959280 + 0.282458i \(0.0911499\pi\)
−0.959280 + 0.282458i \(0.908850\pi\)
\(564\) 0 0
\(565\) 836.076 1.47978
\(566\) 0 0
\(567\) 234.976i 0.414419i
\(568\) 0 0
\(569\) −356.654 −0.626808 −0.313404 0.949620i \(-0.601469\pi\)
−0.313404 + 0.949620i \(0.601469\pi\)
\(570\) 0 0
\(571\) 831.014i 1.45537i 0.685914 + 0.727683i \(0.259403\pi\)
−0.685914 + 0.727683i \(0.740597\pi\)
\(572\) 0 0
\(573\) 932.205 1.62689
\(574\) 0 0
\(575\) − 611.929i − 1.06422i
\(576\) 0 0
\(577\) 771.483 1.33706 0.668530 0.743685i \(-0.266924\pi\)
0.668530 + 0.743685i \(0.266924\pi\)
\(578\) 0 0
\(579\) 1157.49i 1.99912i
\(580\) 0 0
\(581\) −13.5361 −0.0232980
\(582\) 0 0
\(583\) 536.094i 0.919544i
\(584\) 0 0
\(585\) 276.974 0.473460
\(586\) 0 0
\(587\) 144.376i 0.245956i 0.992409 + 0.122978i \(0.0392445\pi\)
−0.992409 + 0.122978i \(0.960756\pi\)
\(588\) 0 0
\(589\) −19.9020 −0.0337894
\(590\) 0 0
\(591\) 1265.68i 2.14158i
\(592\) 0 0
\(593\) 838.926 1.41471 0.707357 0.706856i \(-0.249887\pi\)
0.707357 + 0.706856i \(0.249887\pi\)
\(594\) 0 0
\(595\) − 114.334i − 0.192157i
\(596\) 0 0
\(597\) −1551.66 −2.59910
\(598\) 0 0
\(599\) − 711.341i − 1.18755i −0.804632 0.593774i \(-0.797637\pi\)
0.804632 0.593774i \(-0.202363\pi\)
\(600\) 0 0
\(601\) 356.394 0.593002 0.296501 0.955033i \(-0.404180\pi\)
0.296501 + 0.955033i \(0.404180\pi\)
\(602\) 0 0
\(603\) 568.742i 0.943188i
\(604\) 0 0
\(605\) −155.601 −0.257191
\(606\) 0 0
\(607\) 60.3719i 0.0994595i 0.998763 + 0.0497298i \(0.0158360\pi\)
−0.998763 + 0.0497298i \(0.984164\pi\)
\(608\) 0 0
\(609\) 516.199 0.847618
\(610\) 0 0
\(611\) − 91.6176i − 0.149947i
\(612\) 0 0
\(613\) −482.989 −0.787911 −0.393955 0.919130i \(-0.628894\pi\)
−0.393955 + 0.919130i \(0.628894\pi\)
\(614\) 0 0
\(615\) 2080.68i 3.38321i
\(616\) 0 0
\(617\) −712.490 −1.15476 −0.577382 0.816474i \(-0.695926\pi\)
−0.577382 + 0.816474i \(0.695926\pi\)
\(618\) 0 0
\(619\) − 93.0817i − 0.150374i −0.997169 0.0751872i \(-0.976045\pi\)
0.997169 0.0751872i \(-0.0239554\pi\)
\(620\) 0 0
\(621\) −2077.77 −3.34584
\(622\) 0 0
\(623\) 47.5941i 0.0763951i
\(624\) 0 0
\(625\) −778.059 −1.24489
\(626\) 0 0
\(627\) 142.227i 0.226837i
\(628\) 0 0
\(629\) 1.39719 0.00222129
\(630\) 0 0
\(631\) − 610.573i − 0.967628i −0.875171 0.483814i \(-0.839251\pi\)
0.875171 0.483814i \(-0.160749\pi\)
\(632\) 0 0
\(633\) −736.636 −1.16372
\(634\) 0 0
\(635\) 820.947i 1.29283i
\(636\) 0 0
\(637\) −16.9157 −0.0265553
\(638\) 0 0
\(639\) 422.952i 0.661897i
\(640\) 0 0
\(641\) 590.153 0.920676 0.460338 0.887744i \(-0.347728\pi\)
0.460338 + 0.887744i \(0.347728\pi\)
\(642\) 0 0
\(643\) 257.971i 0.401199i 0.979673 + 0.200600i \(0.0642890\pi\)
−0.979673 + 0.200600i \(0.935711\pi\)
\(644\) 0 0
\(645\) −1157.02 −1.79383
\(646\) 0 0
\(647\) − 379.964i − 0.587271i −0.955918 0.293635i \(-0.905135\pi\)
0.955918 0.293635i \(-0.0948651\pi\)
\(648\) 0 0
\(649\) −1027.71 −1.58353
\(650\) 0 0
\(651\) 99.0696i 0.152181i
\(652\) 0 0
\(653\) −952.773 −1.45907 −0.729535 0.683943i \(-0.760264\pi\)
−0.729535 + 0.683943i \(0.760264\pi\)
\(654\) 0 0
\(655\) − 334.273i − 0.510340i
\(656\) 0 0
\(657\) −1265.92 −1.92681
\(658\) 0 0
\(659\) 963.119i 1.46149i 0.682653 + 0.730743i \(0.260826\pi\)
−0.682653 + 0.730743i \(0.739174\pi\)
\(660\) 0 0
\(661\) −71.6817 −0.108444 −0.0542222 0.998529i \(-0.517268\pi\)
−0.0542222 + 0.998529i \(0.517268\pi\)
\(662\) 0 0
\(663\) 87.0299i 0.131267i
\(664\) 0 0
\(665\) −46.0412 −0.0692349
\(666\) 0 0
\(667\) 1599.84i 2.39856i
\(668\) 0 0
\(669\) 213.263 0.318779
\(670\) 0 0
\(671\) 429.216i 0.639667i
\(672\) 0 0
\(673\) −712.783 −1.05911 −0.529556 0.848275i \(-0.677642\pi\)
−0.529556 + 0.848275i \(0.677642\pi\)
\(674\) 0 0
\(675\) − 693.005i − 1.02667i
\(676\) 0 0
\(677\) −767.527 −1.13372 −0.566859 0.823815i \(-0.691841\pi\)
−0.566859 + 0.823815i \(0.691841\pi\)
\(678\) 0 0
\(679\) 33.0359i 0.0486538i
\(680\) 0 0
\(681\) 42.7605 0.0627908
\(682\) 0 0
\(683\) − 484.354i − 0.709156i −0.935026 0.354578i \(-0.884625\pi\)
0.935026 0.354578i \(-0.115375\pi\)
\(684\) 0 0
\(685\) −361.769 −0.528130
\(686\) 0 0
\(687\) 1735.56i 2.52629i
\(688\) 0 0
\(689\) 132.100 0.191727
\(690\) 0 0
\(691\) 574.851i 0.831912i 0.909385 + 0.415956i \(0.136553\pi\)
−0.909385 + 0.415956i \(0.863447\pi\)
\(692\) 0 0
\(693\) 474.467 0.684657
\(694\) 0 0
\(695\) 1080.72i 1.55500i
\(696\) 0 0
\(697\) −438.143 −0.628612
\(698\) 0 0
\(699\) − 1719.94i − 2.46057i
\(700\) 0 0
\(701\) −143.138 −0.204191 −0.102096 0.994775i \(-0.532555\pi\)
−0.102096 + 0.994775i \(0.532555\pi\)
\(702\) 0 0
\(703\) − 0.562638i 0 0.000800339i
\(704\) 0 0
\(705\) −1241.31 −1.76072
\(706\) 0 0
\(707\) 180.110i 0.254753i
\(708\) 0 0
\(709\) −255.311 −0.360101 −0.180050 0.983657i \(-0.557626\pi\)
−0.180050 + 0.983657i \(0.557626\pi\)
\(710\) 0 0
\(711\) − 364.498i − 0.512656i
\(712\) 0 0
\(713\) −307.044 −0.430636
\(714\) 0 0
\(715\) − 148.540i − 0.207748i
\(716\) 0 0
\(717\) −719.393 −1.00334
\(718\) 0 0
\(719\) − 415.630i − 0.578067i −0.957319 0.289034i \(-0.906666\pi\)
0.957319 0.289034i \(-0.0933340\pi\)
\(720\) 0 0
\(721\) −154.229 −0.213910
\(722\) 0 0
\(723\) 1054.41i 1.45838i
\(724\) 0 0
\(725\) −533.601 −0.736001
\(726\) 0 0
\(727\) 896.838i 1.23361i 0.787114 + 0.616807i \(0.211574\pi\)
−0.787114 + 0.616807i \(0.788426\pi\)
\(728\) 0 0
\(729\) 656.451 0.900481
\(730\) 0 0
\(731\) − 243.641i − 0.333299i
\(732\) 0 0
\(733\) 509.059 0.694487 0.347244 0.937775i \(-0.387118\pi\)
0.347244 + 0.937775i \(0.387118\pi\)
\(734\) 0 0
\(735\) 229.188i 0.311820i
\(736\) 0 0
\(737\) 305.014 0.413859
\(738\) 0 0
\(739\) − 741.427i − 1.00328i −0.865075 0.501642i \(-0.832729\pi\)
0.865075 0.501642i \(-0.167271\pi\)
\(740\) 0 0
\(741\) 35.0463 0.0472959
\(742\) 0 0
\(743\) 1344.98i 1.81021i 0.425191 + 0.905104i \(0.360207\pi\)
−0.425191 + 0.905104i \(0.639793\pi\)
\(744\) 0 0
\(745\) 1375.98 1.84695
\(746\) 0 0
\(747\) − 93.5560i − 0.125242i
\(748\) 0 0
\(749\) −359.473 −0.479937
\(750\) 0 0
\(751\) − 27.6931i − 0.0368749i −0.999830 0.0184375i \(-0.994131\pi\)
0.999830 0.0184375i \(-0.00586916\pi\)
\(752\) 0 0
\(753\) 1406.22 1.86749
\(754\) 0 0
\(755\) − 1163.74i − 1.54138i
\(756\) 0 0
\(757\) 1341.69 1.77238 0.886192 0.463318i \(-0.153341\pi\)
0.886192 + 0.463318i \(0.153341\pi\)
\(758\) 0 0
\(759\) 2194.24i 2.89096i
\(760\) 0 0
\(761\) 112.001 0.147176 0.0735881 0.997289i \(-0.476555\pi\)
0.0735881 + 0.997289i \(0.476555\pi\)
\(762\) 0 0
\(763\) 117.731i 0.154300i
\(764\) 0 0
\(765\) 790.226 1.03297
\(766\) 0 0
\(767\) 253.240i 0.330169i
\(768\) 0 0
\(769\) 140.749 0.183028 0.0915142 0.995804i \(-0.470829\pi\)
0.0915142 + 0.995804i \(0.470829\pi\)
\(770\) 0 0
\(771\) − 1265.99i − 1.64202i
\(772\) 0 0
\(773\) −1325.14 −1.71428 −0.857138 0.515087i \(-0.827760\pi\)
−0.857138 + 0.515087i \(0.827760\pi\)
\(774\) 0 0
\(775\) − 102.409i − 0.132141i
\(776\) 0 0
\(777\) −2.80074 −0.00360456
\(778\) 0 0
\(779\) 176.437i 0.226491i
\(780\) 0 0
\(781\) 226.827 0.290432
\(782\) 0 0
\(783\) 1811.81i 2.31393i
\(784\) 0 0
\(785\) 1179.50 1.50255
\(786\) 0 0
\(787\) 327.801i 0.416519i 0.978074 + 0.208260i \(0.0667799\pi\)
−0.978074 + 0.208260i \(0.933220\pi\)
\(788\) 0 0
\(789\) −176.804 −0.224086
\(790\) 0 0
\(791\) − 352.918i − 0.446167i
\(792\) 0 0
\(793\) 105.764 0.133372
\(794\) 0 0
\(795\) − 1789.80i − 2.25132i
\(796\) 0 0
\(797\) −393.650 −0.493915 −0.246958 0.969026i \(-0.579431\pi\)
−0.246958 + 0.969026i \(0.579431\pi\)
\(798\) 0 0
\(799\) − 261.391i − 0.327148i
\(800\) 0 0
\(801\) −328.950 −0.410675
\(802\) 0 0
\(803\) 678.906i 0.845462i
\(804\) 0 0
\(805\) −710.314 −0.882378
\(806\) 0 0
\(807\) − 865.023i − 1.07190i
\(808\) 0 0
\(809\) −416.641 −0.515008 −0.257504 0.966277i \(-0.582900\pi\)
−0.257504 + 0.966277i \(0.582900\pi\)
\(810\) 0 0
\(811\) − 748.707i − 0.923190i −0.887091 0.461595i \(-0.847277\pi\)
0.887091 0.461595i \(-0.152723\pi\)
\(812\) 0 0
\(813\) 774.708 0.952901
\(814\) 0 0
\(815\) 341.696i 0.419259i
\(816\) 0 0
\(817\) −98.1124 −0.120089
\(818\) 0 0
\(819\) − 116.914i − 0.142752i
\(820\) 0 0
\(821\) 554.169 0.674993 0.337496 0.941327i \(-0.390420\pi\)
0.337496 + 0.941327i \(0.390420\pi\)
\(822\) 0 0
\(823\) 121.452i 0.147572i 0.997274 + 0.0737861i \(0.0235082\pi\)
−0.997274 + 0.0737861i \(0.976492\pi\)
\(824\) 0 0
\(825\) −731.853 −0.887094
\(826\) 0 0
\(827\) − 1516.61i − 1.83386i −0.399043 0.916932i \(-0.630657\pi\)
0.399043 0.916932i \(-0.369343\pi\)
\(828\) 0 0
\(829\) 325.042 0.392089 0.196044 0.980595i \(-0.437190\pi\)
0.196044 + 0.980595i \(0.437190\pi\)
\(830\) 0 0
\(831\) − 2498.77i − 3.00694i
\(832\) 0 0
\(833\) −48.2617 −0.0579372
\(834\) 0 0
\(835\) 1669.98i 1.99998i
\(836\) 0 0
\(837\) −347.724 −0.415441
\(838\) 0 0
\(839\) − 1165.70i − 1.38939i −0.719303 0.694696i \(-0.755539\pi\)
0.719303 0.694696i \(-0.244461\pi\)
\(840\) 0 0
\(841\) 554.058 0.658808
\(842\) 0 0
\(843\) 1180.89i 1.40081i
\(844\) 0 0
\(845\) 1022.67 1.21026
\(846\) 0 0
\(847\) 65.6810i 0.0775454i
\(848\) 0 0
\(849\) 1330.08 1.56665
\(850\) 0 0
\(851\) − 8.68026i − 0.0102001i
\(852\) 0 0
\(853\) −151.949 −0.178134 −0.0890672 0.996026i \(-0.528389\pi\)
−0.0890672 + 0.996026i \(0.528389\pi\)
\(854\) 0 0
\(855\) − 318.218i − 0.372184i
\(856\) 0 0
\(857\) −412.018 −0.480768 −0.240384 0.970678i \(-0.577273\pi\)
−0.240384 + 0.970678i \(0.577273\pi\)
\(858\) 0 0
\(859\) 159.993i 0.186255i 0.995654 + 0.0931274i \(0.0296864\pi\)
−0.995654 + 0.0931274i \(0.970314\pi\)
\(860\) 0 0
\(861\) 878.280 1.02007
\(862\) 0 0
\(863\) − 992.910i − 1.15053i −0.817966 0.575266i \(-0.804898\pi\)
0.817966 0.575266i \(-0.195102\pi\)
\(864\) 0 0
\(865\) −719.772 −0.832107
\(866\) 0 0
\(867\) − 1261.33i − 1.45482i
\(868\) 0 0
\(869\) −195.479 −0.224947
\(870\) 0 0
\(871\) − 75.1590i − 0.0862905i
\(872\) 0 0
\(873\) −228.330 −0.261547
\(874\) 0 0
\(875\) 177.668i 0.203049i
\(876\) 0 0
\(877\) 825.096 0.940817 0.470408 0.882449i \(-0.344107\pi\)
0.470408 + 0.882449i \(0.344107\pi\)
\(878\) 0 0
\(879\) − 781.236i − 0.888778i
\(880\) 0 0
\(881\) 1352.83 1.53556 0.767780 0.640714i \(-0.221361\pi\)
0.767780 + 0.640714i \(0.221361\pi\)
\(882\) 0 0
\(883\) 1013.40i 1.14768i 0.818969 + 0.573838i \(0.194546\pi\)
−0.818969 + 0.573838i \(0.805454\pi\)
\(884\) 0 0
\(885\) 3431.10 3.87695
\(886\) 0 0
\(887\) − 233.760i − 0.263540i −0.991280 0.131770i \(-0.957934\pi\)
0.991280 0.131770i \(-0.0420661\pi\)
\(888\) 0 0
\(889\) 346.532 0.389800
\(890\) 0 0
\(891\) 870.974i 0.977524i
\(892\) 0 0
\(893\) −105.260 −0.117873
\(894\) 0 0
\(895\) 707.322i 0.790304i
\(896\) 0 0
\(897\) 540.686 0.602772
\(898\) 0 0
\(899\) 267.741i 0.297821i
\(900\) 0 0
\(901\) 376.890 0.418302
\(902\) 0 0
\(903\) 488.392i 0.540854i
\(904\) 0 0
\(905\) 382.008 0.422109
\(906\) 0 0
\(907\) − 203.590i − 0.224465i −0.993682 0.112233i \(-0.964200\pi\)
0.993682 0.112233i \(-0.0358002\pi\)
\(908\) 0 0
\(909\) −1244.85 −1.36947
\(910\) 0 0
\(911\) − 712.022i − 0.781583i −0.920479 0.390792i \(-0.872201\pi\)
0.920479 0.390792i \(-0.127799\pi\)
\(912\) 0 0
\(913\) −50.1737 −0.0549548
\(914\) 0 0
\(915\) − 1432.97i − 1.56609i
\(916\) 0 0
\(917\) −141.101 −0.153872
\(918\) 0 0
\(919\) − 786.783i − 0.856129i −0.903748 0.428065i \(-0.859196\pi\)
0.903748 0.428065i \(-0.140804\pi\)
\(920\) 0 0
\(921\) 1419.67 1.54145
\(922\) 0 0
\(923\) − 55.8929i − 0.0605557i
\(924\) 0 0
\(925\) 2.89516 0.00312990
\(926\) 0 0
\(927\) − 1065.97i − 1.14991i
\(928\) 0 0
\(929\) −1657.99 −1.78471 −0.892353 0.451338i \(-0.850947\pi\)
−0.892353 + 0.451338i \(0.850947\pi\)
\(930\) 0 0
\(931\) 19.4346i 0.0208750i
\(932\) 0 0
\(933\) 2790.36 2.99074
\(934\) 0 0
\(935\) − 423.795i − 0.453257i
\(936\) 0 0
\(937\) −333.736 −0.356175 −0.178088 0.984015i \(-0.556991\pi\)
−0.178088 + 0.984015i \(0.556991\pi\)
\(938\) 0 0
\(939\) 2905.55i 3.09430i
\(940\) 0 0
\(941\) −543.324 −0.577390 −0.288695 0.957421i \(-0.593221\pi\)
−0.288695 + 0.957421i \(0.593221\pi\)
\(942\) 0 0
\(943\) 2722.03i 2.88656i
\(944\) 0 0
\(945\) −804.425 −0.851243
\(946\) 0 0
\(947\) 359.728i 0.379861i 0.981798 + 0.189930i \(0.0608262\pi\)
−0.981798 + 0.189930i \(0.939174\pi\)
\(948\) 0 0
\(949\) 167.290 0.176281
\(950\) 0 0
\(951\) − 2023.94i − 2.12823i
\(952\) 0 0
\(953\) 904.225 0.948820 0.474410 0.880304i \(-0.342661\pi\)
0.474410 + 0.880304i \(0.342661\pi\)
\(954\) 0 0
\(955\) 1118.56i 1.17127i
\(956\) 0 0
\(957\) 1913.37 1.99934
\(958\) 0 0
\(959\) 152.707i 0.159236i
\(960\) 0 0
\(961\) 909.615 0.946529
\(962\) 0 0
\(963\) − 2484.52i − 2.57998i
\(964\) 0 0
\(965\) −1388.89 −1.43926
\(966\) 0 0
\(967\) − 920.961i − 0.952390i −0.879340 0.476195i \(-0.842016\pi\)
0.879340 0.476195i \(-0.157984\pi\)
\(968\) 0 0
\(969\) 99.9894 0.103188
\(970\) 0 0
\(971\) 1327.31i 1.36695i 0.729973 + 0.683476i \(0.239532\pi\)
−0.729973 + 0.683476i \(0.760468\pi\)
\(972\) 0 0
\(973\) 456.186 0.468845
\(974\) 0 0
\(975\) 180.337i 0.184961i
\(976\) 0 0
\(977\) −688.490 −0.704698 −0.352349 0.935869i \(-0.614617\pi\)
−0.352349 + 0.935869i \(0.614617\pi\)
\(978\) 0 0
\(979\) 176.415i 0.180199i
\(980\) 0 0
\(981\) −813.706 −0.829466
\(982\) 0 0
\(983\) 1687.08i 1.71625i 0.513438 + 0.858127i \(0.328372\pi\)
−0.513438 + 0.858127i \(0.671628\pi\)
\(984\) 0 0
\(985\) −1518.70 −1.54182
\(986\) 0 0
\(987\) 523.972i 0.530874i
\(988\) 0 0
\(989\) −1513.66 −1.53049
\(990\) 0 0
\(991\) − 1139.22i − 1.14957i −0.818304 0.574785i \(-0.805086\pi\)
0.818304 0.574785i \(-0.194914\pi\)
\(992\) 0 0
\(993\) 2001.72 2.01583
\(994\) 0 0
\(995\) − 1861.85i − 1.87121i
\(996\) 0 0
\(997\) −206.085 −0.206705 −0.103353 0.994645i \(-0.532957\pi\)
−0.103353 + 0.994645i \(0.532957\pi\)
\(998\) 0 0
\(999\) − 9.83033i − 0.00984017i
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 1792.3.d.j.1023.15 16
4.3 odd 2 inner 1792.3.d.j.1023.1 16
8.3 odd 2 inner 1792.3.d.j.1023.16 16
8.5 even 2 inner 1792.3.d.j.1023.2 16
16.3 odd 4 224.3.g.b.15.8 8
16.5 even 4 224.3.g.b.15.7 8
16.11 odd 4 56.3.g.b.43.5 8
16.13 even 4 56.3.g.b.43.6 yes 8
48.5 odd 4 2016.3.g.b.1135.8 8
48.11 even 4 504.3.g.b.379.4 8
48.29 odd 4 504.3.g.b.379.3 8
48.35 even 4 2016.3.g.b.1135.1 8
112.11 odd 12 392.3.k.o.275.6 16
112.13 odd 4 392.3.g.m.99.6 8
112.27 even 4 392.3.g.m.99.5 8
112.45 odd 12 392.3.k.n.275.1 16
112.59 even 12 392.3.k.n.275.6 16
112.61 odd 12 392.3.k.n.67.6 16
112.69 odd 4 1568.3.g.m.687.2 8
112.75 even 12 392.3.k.n.67.1 16
112.83 even 4 1568.3.g.m.687.1 8
112.93 even 12 392.3.k.o.67.6 16
112.107 odd 12 392.3.k.o.67.1 16
112.109 even 12 392.3.k.o.275.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.g.b.43.5 8 16.11 odd 4
56.3.g.b.43.6 yes 8 16.13 even 4
224.3.g.b.15.7 8 16.5 even 4
224.3.g.b.15.8 8 16.3 odd 4
392.3.g.m.99.5 8 112.27 even 4
392.3.g.m.99.6 8 112.13 odd 4
392.3.k.n.67.1 16 112.75 even 12
392.3.k.n.67.6 16 112.61 odd 12
392.3.k.n.275.1 16 112.45 odd 12
392.3.k.n.275.6 16 112.59 even 12
392.3.k.o.67.1 16 112.107 odd 12
392.3.k.o.67.6 16 112.93 even 12
392.3.k.o.275.1 16 112.109 even 12
392.3.k.o.275.6 16 112.11 odd 12
504.3.g.b.379.3 8 48.29 odd 4
504.3.g.b.379.4 8 48.11 even 4
1568.3.g.m.687.1 8 112.83 even 4
1568.3.g.m.687.2 8 112.69 odd 4
1792.3.d.j.1023.1 16 4.3 odd 2 inner
1792.3.d.j.1023.2 16 8.5 even 2 inner
1792.3.d.j.1023.15 16 1.1 even 1 trivial
1792.3.d.j.1023.16 16 8.3 odd 2 inner
2016.3.g.b.1135.1 8 48.35 even 4
2016.3.g.b.1135.8 8 48.5 odd 4