Properties

Label 1200.3.bg.i.193.2
Level $1200$
Weight $3$
Character 1200.193
Analytic conductor $32.698$
Analytic rank $0$
Dimension $4$
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] = [1200,3,Mod(193,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.193");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.bg (of order \(4\), degree \(2\), 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: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 193.2
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1200.193
Dual form 1200.3.bg.i.1057.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.22474 + 1.22474i) q^{3} +(3.67423 - 3.67423i) q^{7} +3.00000i q^{9} +O(q^{10})\) \(q+(1.22474 + 1.22474i) q^{3} +(3.67423 - 3.67423i) q^{7} +3.00000i q^{9} -6.00000 q^{11} +(-6.12372 - 6.12372i) q^{13} +(-22.0454 + 22.0454i) q^{17} -25.0000i q^{19} +9.00000 q^{21} +(-7.34847 - 7.34847i) q^{23} +(-3.67423 + 3.67423i) q^{27} +42.0000i q^{29} -49.0000 q^{31} +(-7.34847 - 7.34847i) q^{33} +(-4.89898 + 4.89898i) q^{37} -15.0000i q^{39} -60.0000 q^{41} +(-1.22474 - 1.22474i) q^{43} +(51.4393 - 51.4393i) q^{47} +22.0000i q^{49} -54.0000 q^{51} +(14.6969 + 14.6969i) q^{53} +(30.6186 - 30.6186i) q^{57} +78.0000i q^{59} -13.0000 q^{61} +(11.0227 + 11.0227i) q^{63} +(-52.6640 + 52.6640i) q^{67} -18.0000i q^{69} +60.0000 q^{71} +(-63.6867 - 63.6867i) q^{73} +(-22.0454 + 22.0454i) q^{77} +106.000i q^{79} -9.00000 q^{81} +(-80.8332 - 80.8332i) q^{83} +(-51.4393 + 51.4393i) q^{87} -60.0000i q^{89} -45.0000 q^{91} +(-60.0125 - 60.0125i) q^{93} +(121.250 - 121.250i) q^{97} -18.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 24 q^{11} + 36 q^{21} - 196 q^{31} - 240 q^{41} - 216 q^{51} - 52 q^{61} + 240 q^{71} - 36 q^{81} - 180 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\)

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
\(3\) 1.22474 + 1.22474i 0.408248 + 0.408248i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.67423 3.67423i 0.524891 0.524891i −0.394154 0.919044i \(-0.628962\pi\)
0.919044 + 0.394154i \(0.128962\pi\)
\(8\) 0 0
\(9\) 3.00000i 0.333333i
\(10\) 0 0
\(11\) −6.00000 −0.545455 −0.272727 0.962091i \(-0.587926\pi\)
−0.272727 + 0.962091i \(0.587926\pi\)
\(12\) 0 0
\(13\) −6.12372 6.12372i −0.471056 0.471056i 0.431200 0.902256i \(-0.358090\pi\)
−0.902256 + 0.431200i \(0.858090\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −22.0454 + 22.0454i −1.29679 + 1.29679i −0.366287 + 0.930502i \(0.619371\pi\)
−0.930502 + 0.366287i \(0.880629\pi\)
\(18\) 0 0
\(19\) 25.0000i 1.31579i −0.753110 0.657895i \(-0.771447\pi\)
0.753110 0.657895i \(-0.228553\pi\)
\(20\) 0 0
\(21\) 9.00000 0.428571
\(22\) 0 0
\(23\) −7.34847 7.34847i −0.319499 0.319499i 0.529076 0.848575i \(-0.322539\pi\)
−0.848575 + 0.529076i \(0.822539\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −3.67423 + 3.67423i −0.136083 + 0.136083i
\(28\) 0 0
\(29\) 42.0000i 1.44828i 0.689655 + 0.724138i \(0.257762\pi\)
−0.689655 + 0.724138i \(0.742238\pi\)
\(30\) 0 0
\(31\) −49.0000 −1.58065 −0.790323 0.612691i \(-0.790087\pi\)
−0.790323 + 0.612691i \(0.790087\pi\)
\(32\) 0 0
\(33\) −7.34847 7.34847i −0.222681 0.222681i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.89898 + 4.89898i −0.132405 + 0.132405i −0.770203 0.637798i \(-0.779845\pi\)
0.637798 + 0.770203i \(0.279845\pi\)
\(38\) 0 0
\(39\) 15.0000i 0.384615i
\(40\) 0 0
\(41\) −60.0000 −1.46341 −0.731707 0.681619i \(-0.761276\pi\)
−0.731707 + 0.681619i \(0.761276\pi\)
\(42\) 0 0
\(43\) −1.22474 1.22474i −0.0284824 0.0284824i 0.692722 0.721205i \(-0.256411\pi\)
−0.721205 + 0.692722i \(0.756411\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 51.4393 51.4393i 1.09445 1.09445i 0.0994059 0.995047i \(-0.468306\pi\)
0.995047 0.0994059i \(-0.0316942\pi\)
\(48\) 0 0
\(49\) 22.0000i 0.448980i
\(50\) 0 0
\(51\) −54.0000 −1.05882
\(52\) 0 0
\(53\) 14.6969 + 14.6969i 0.277301 + 0.277301i 0.832031 0.554730i \(-0.187178\pi\)
−0.554730 + 0.832031i \(0.687178\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 30.6186 30.6186i 0.537169 0.537169i
\(58\) 0 0
\(59\) 78.0000i 1.32203i 0.750371 + 0.661017i \(0.229875\pi\)
−0.750371 + 0.661017i \(0.770125\pi\)
\(60\) 0 0
\(61\) −13.0000 −0.213115 −0.106557 0.994307i \(-0.533983\pi\)
−0.106557 + 0.994307i \(0.533983\pi\)
\(62\) 0 0
\(63\) 11.0227 + 11.0227i 0.174964 + 0.174964i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −52.6640 + 52.6640i −0.786030 + 0.786030i −0.980841 0.194811i \(-0.937591\pi\)
0.194811 + 0.980841i \(0.437591\pi\)
\(68\) 0 0
\(69\) 18.0000i 0.260870i
\(70\) 0 0
\(71\) 60.0000 0.845070 0.422535 0.906347i \(-0.361140\pi\)
0.422535 + 0.906347i \(0.361140\pi\)
\(72\) 0 0
\(73\) −63.6867 63.6867i −0.872421 0.872421i 0.120315 0.992736i \(-0.461610\pi\)
−0.992736 + 0.120315i \(0.961610\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −22.0454 + 22.0454i −0.286304 + 0.286304i
\(78\) 0 0
\(79\) 106.000i 1.34177i 0.741560 + 0.670886i \(0.234086\pi\)
−0.741560 + 0.670886i \(0.765914\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) 0 0
\(83\) −80.8332 80.8332i −0.973894 0.973894i 0.0257743 0.999668i \(-0.491795\pi\)
−0.999668 + 0.0257743i \(0.991795\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −51.4393 + 51.4393i −0.591256 + 0.591256i
\(88\) 0 0
\(89\) 60.0000i 0.674157i −0.941476 0.337079i \(-0.890561\pi\)
0.941476 0.337079i \(-0.109439\pi\)
\(90\) 0 0
\(91\) −45.0000 −0.494505
\(92\) 0 0
\(93\) −60.0125 60.0125i −0.645296 0.645296i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 121.250 121.250i 1.25000 1.25000i 0.294277 0.955720i \(-0.404921\pi\)
0.955720 0.294277i \(-0.0950789\pi\)
\(98\) 0 0
\(99\) 18.0000i 0.181818i
\(100\) 0 0
\(101\) −192.000 −1.90099 −0.950495 0.310740i \(-0.899423\pi\)
−0.950495 + 0.310740i \(0.899423\pi\)
\(102\) 0 0
\(103\) −78.3837 78.3837i −0.761007 0.761007i 0.215498 0.976504i \(-0.430863\pi\)
−0.976504 + 0.215498i \(0.930863\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 29.3939 29.3939i 0.274709 0.274709i −0.556283 0.830993i \(-0.687773\pi\)
0.830993 + 0.556283i \(0.187773\pi\)
\(108\) 0 0
\(109\) 1.00000i 0.00917431i −0.999989 0.00458716i \(-0.998540\pi\)
0.999989 0.00458716i \(-0.00146014\pi\)
\(110\) 0 0
\(111\) −12.0000 −0.108108
\(112\) 0 0
\(113\) 58.7878 + 58.7878i 0.520246 + 0.520246i 0.917646 0.397400i \(-0.130087\pi\)
−0.397400 + 0.917646i \(0.630087\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 18.3712 18.3712i 0.157019 0.157019i
\(118\) 0 0
\(119\) 162.000i 1.36134i
\(120\) 0 0
\(121\) −85.0000 −0.702479
\(122\) 0 0
\(123\) −73.4847 73.4847i −0.597437 0.597437i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −14.6969 + 14.6969i −0.115724 + 0.115724i −0.762597 0.646873i \(-0.776076\pi\)
0.646873 + 0.762597i \(0.276076\pi\)
\(128\) 0 0
\(129\) 3.00000i 0.0232558i
\(130\) 0 0
\(131\) −12.0000 −0.0916031 −0.0458015 0.998951i \(-0.514584\pi\)
−0.0458015 + 0.998951i \(0.514584\pi\)
\(132\) 0 0
\(133\) −91.8559 91.8559i −0.690646 0.690646i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 80.8332 80.8332i 0.590023 0.590023i −0.347614 0.937638i \(-0.613008\pi\)
0.937638 + 0.347614i \(0.113008\pi\)
\(138\) 0 0
\(139\) 58.0000i 0.417266i −0.977994 0.208633i \(-0.933099\pi\)
0.977994 0.208633i \(-0.0669014\pi\)
\(140\) 0 0
\(141\) 126.000 0.893617
\(142\) 0 0
\(143\) 36.7423 + 36.7423i 0.256939 + 0.256939i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −26.9444 + 26.9444i −0.183295 + 0.183295i
\(148\) 0 0
\(149\) 234.000i 1.57047i 0.619198 + 0.785235i \(0.287458\pi\)
−0.619198 + 0.785235i \(0.712542\pi\)
\(150\) 0 0
\(151\) −85.0000 −0.562914 −0.281457 0.959574i \(-0.590818\pi\)
−0.281457 + 0.959574i \(0.590818\pi\)
\(152\) 0 0
\(153\) −66.1362 66.1362i −0.432263 0.432263i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −74.7094 + 74.7094i −0.475856 + 0.475856i −0.903804 0.427947i \(-0.859237\pi\)
0.427947 + 0.903804i \(0.359237\pi\)
\(158\) 0 0
\(159\) 36.0000i 0.226415i
\(160\) 0 0
\(161\) −54.0000 −0.335404
\(162\) 0 0
\(163\) −79.6084 79.6084i −0.488395 0.488395i 0.419404 0.907800i \(-0.362239\pi\)
−0.907800 + 0.419404i \(0.862239\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 176.363 176.363i 1.05607 1.05607i 0.0577355 0.998332i \(-0.481612\pi\)
0.998332 0.0577355i \(-0.0183880\pi\)
\(168\) 0 0
\(169\) 94.0000i 0.556213i
\(170\) 0 0
\(171\) 75.0000 0.438596
\(172\) 0 0
\(173\) −80.8332 80.8332i −0.467244 0.467244i 0.433777 0.901020i \(-0.357181\pi\)
−0.901020 + 0.433777i \(0.857181\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −95.5301 + 95.5301i −0.539718 + 0.539718i
\(178\) 0 0
\(179\) 42.0000i 0.234637i −0.993094 0.117318i \(-0.962570\pi\)
0.993094 0.117318i \(-0.0374298\pi\)
\(180\) 0 0
\(181\) 145.000 0.801105 0.400552 0.916274i \(-0.368818\pi\)
0.400552 + 0.916274i \(0.368818\pi\)
\(182\) 0 0
\(183\) −15.9217 15.9217i −0.0870037 0.0870037i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 132.272 132.272i 0.707339 0.707339i
\(188\) 0 0
\(189\) 27.0000i 0.142857i
\(190\) 0 0
\(191\) −294.000 −1.53927 −0.769634 0.638486i \(-0.779561\pi\)
−0.769634 + 0.638486i \(0.779561\pi\)
\(192\) 0 0
\(193\) −25.7196 25.7196i −0.133262 0.133262i 0.637329 0.770592i \(-0.280039\pi\)
−0.770592 + 0.637329i \(0.780039\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.0454 22.0454i 0.111906 0.111906i −0.648937 0.760842i \(-0.724786\pi\)
0.760842 + 0.648937i \(0.224786\pi\)
\(198\) 0 0
\(199\) 179.000i 0.899497i −0.893155 0.449749i \(-0.851514\pi\)
0.893155 0.449749i \(-0.148486\pi\)
\(200\) 0 0
\(201\) −129.000 −0.641791
\(202\) 0 0
\(203\) 154.318 + 154.318i 0.760186 + 0.760186i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 22.0454 22.0454i 0.106500 0.106500i
\(208\) 0 0
\(209\) 150.000i 0.717703i
\(210\) 0 0
\(211\) 85.0000 0.402844 0.201422 0.979505i \(-0.435444\pi\)
0.201422 + 0.979505i \(0.435444\pi\)
\(212\) 0 0
\(213\) 73.4847 + 73.4847i 0.344999 + 0.344999i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −180.037 + 180.037i −0.829666 + 0.829666i
\(218\) 0 0
\(219\) 156.000i 0.712329i
\(220\) 0 0
\(221\) 270.000 1.22172
\(222\) 0 0
\(223\) 104.103 + 104.103i 0.466831 + 0.466831i 0.900886 0.434055i \(-0.142918\pi\)
−0.434055 + 0.900886i \(0.642918\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(228\) 0 0
\(229\) 373.000i 1.62882i 0.580289 + 0.814410i \(0.302940\pi\)
−0.580289 + 0.814410i \(0.697060\pi\)
\(230\) 0 0
\(231\) −54.0000 −0.233766
\(232\) 0 0
\(233\) 220.454 + 220.454i 0.946155 + 0.946155i 0.998623 0.0524678i \(-0.0167087\pi\)
−0.0524678 + 0.998623i \(0.516709\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −129.823 + 129.823i −0.547776 + 0.547776i
\(238\) 0 0
\(239\) 36.0000i 0.150628i 0.997160 + 0.0753138i \(0.0239959\pi\)
−0.997160 + 0.0753138i \(0.976004\pi\)
\(240\) 0 0
\(241\) −97.0000 −0.402490 −0.201245 0.979541i \(-0.564499\pi\)
−0.201245 + 0.979541i \(0.564499\pi\)
\(242\) 0 0
\(243\) −11.0227 11.0227i −0.0453609 0.0453609i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −153.093 + 153.093i −0.619810 + 0.619810i
\(248\) 0 0
\(249\) 198.000i 0.795181i
\(250\) 0 0
\(251\) −288.000 −1.14741 −0.573705 0.819062i \(-0.694494\pi\)
−0.573705 + 0.819062i \(0.694494\pi\)
\(252\) 0 0
\(253\) 44.0908 + 44.0908i 0.174272 + 0.174272i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 338.030 338.030i 1.31529 1.31529i 0.397832 0.917458i \(-0.369763\pi\)
0.917458 0.397832i \(-0.130237\pi\)
\(258\) 0 0
\(259\) 36.0000i 0.138996i
\(260\) 0 0
\(261\) −126.000 −0.482759
\(262\) 0 0
\(263\) 249.848 + 249.848i 0.949992 + 0.949992i 0.998808 0.0488156i \(-0.0155447\pi\)
−0.0488156 + 0.998808i \(0.515545\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 73.4847 73.4847i 0.275224 0.275224i
\(268\) 0 0
\(269\) 318.000i 1.18216i 0.806614 + 0.591078i \(0.201297\pi\)
−0.806614 + 0.591078i \(0.798703\pi\)
\(270\) 0 0
\(271\) 226.000 0.833948 0.416974 0.908918i \(-0.363091\pi\)
0.416974 + 0.908918i \(0.363091\pi\)
\(272\) 0 0
\(273\) −55.1135 55.1135i −0.201881 0.201881i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −380.896 + 380.896i −1.37507 + 1.37507i −0.522333 + 0.852742i \(0.674938\pi\)
−0.852742 + 0.522333i \(0.825062\pi\)
\(278\) 0 0
\(279\) 147.000i 0.526882i
\(280\) 0 0
\(281\) 126.000 0.448399 0.224199 0.974543i \(-0.428023\pi\)
0.224199 + 0.974543i \(0.428023\pi\)
\(282\) 0 0
\(283\) −395.593 395.593i −1.39785 1.39785i −0.806171 0.591683i \(-0.798464\pi\)
−0.591683 0.806171i \(-0.701536\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −220.454 + 220.454i −0.768133 + 0.768133i
\(288\) 0 0
\(289\) 683.000i 2.36332i
\(290\) 0 0
\(291\) 297.000 1.02062
\(292\) 0 0
\(293\) 22.0454 + 22.0454i 0.0752403 + 0.0752403i 0.743725 0.668485i \(-0.233057\pi\)
−0.668485 + 0.743725i \(0.733057\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 22.0454 22.0454i 0.0742270 0.0742270i
\(298\) 0 0
\(299\) 90.0000i 0.301003i
\(300\) 0 0
\(301\) −9.00000 −0.0299003
\(302\) 0 0
\(303\) −235.151 235.151i −0.776076 0.776076i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −86.9569 + 86.9569i −0.283247 + 0.283247i −0.834403 0.551155i \(-0.814187\pi\)
0.551155 + 0.834403i \(0.314187\pi\)
\(308\) 0 0
\(309\) 192.000i 0.621359i
\(310\) 0 0
\(311\) 246.000 0.790997 0.395498 0.918467i \(-0.370572\pi\)
0.395498 + 0.918467i \(0.370572\pi\)
\(312\) 0 0
\(313\) −37.9671 37.9671i −0.121301 0.121301i 0.643851 0.765151i \(-0.277336\pi\)
−0.765151 + 0.643851i \(0.777336\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.6969 14.6969i 0.0463626 0.0463626i −0.683545 0.729908i \(-0.739563\pi\)
0.729908 + 0.683545i \(0.239563\pi\)
\(318\) 0 0
\(319\) 252.000i 0.789969i
\(320\) 0 0
\(321\) 72.0000 0.224299
\(322\) 0 0
\(323\) 551.135 + 551.135i 1.70630 + 1.70630i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.22474 1.22474i 0.00374540 0.00374540i
\(328\) 0 0
\(329\) 378.000i 1.14894i
\(330\) 0 0
\(331\) 178.000 0.537764 0.268882 0.963173i \(-0.413346\pi\)
0.268882 + 0.963173i \(0.413346\pi\)
\(332\) 0 0
\(333\) −14.6969 14.6969i −0.0441350 0.0441350i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 417.638 417.638i 1.23928 1.23928i 0.278987 0.960295i \(-0.410001\pi\)
0.960295 0.278987i \(-0.0899987\pi\)
\(338\) 0 0
\(339\) 144.000i 0.424779i
\(340\) 0 0
\(341\) 294.000 0.862170
\(342\) 0 0
\(343\) 260.871 + 260.871i 0.760556 + 0.760556i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −448.257 + 448.257i −1.29181 + 1.29181i −0.358137 + 0.933669i \(0.616588\pi\)
−0.933669 + 0.358137i \(0.883412\pi\)
\(348\) 0 0
\(349\) 542.000i 1.55301i −0.630112 0.776504i \(-0.716991\pi\)
0.630112 0.776504i \(-0.283009\pi\)
\(350\) 0 0
\(351\) 45.0000 0.128205
\(352\) 0 0
\(353\) 242.499 + 242.499i 0.686967 + 0.686967i 0.961561 0.274593i \(-0.0885432\pi\)
−0.274593 + 0.961561i \(0.588543\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −198.409 + 198.409i −0.555767 + 0.555767i
\(358\) 0 0
\(359\) 408.000i 1.13649i 0.822859 + 0.568245i \(0.192377\pi\)
−0.822859 + 0.568245i \(0.807623\pi\)
\(360\) 0 0
\(361\) −264.000 −0.731302
\(362\) 0 0
\(363\) −104.103 104.103i −0.286786 0.286786i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 199.633 199.633i 0.543960 0.543960i −0.380727 0.924687i \(-0.624326\pi\)
0.924687 + 0.380727i \(0.124326\pi\)
\(368\) 0 0
\(369\) 180.000i 0.487805i
\(370\) 0 0
\(371\) 108.000 0.291105
\(372\) 0 0
\(373\) 319.658 + 319.658i 0.856993 + 0.856993i 0.990983 0.133990i \(-0.0427789\pi\)
−0.133990 + 0.990983i \(0.542779\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 257.196 257.196i 0.682219 0.682219i
\(378\) 0 0
\(379\) 71.0000i 0.187335i 0.995604 + 0.0936675i \(0.0298591\pi\)
−0.995604 + 0.0936675i \(0.970141\pi\)
\(380\) 0 0
\(381\) −36.0000 −0.0944882
\(382\) 0 0
\(383\) −58.7878 58.7878i −0.153493 0.153493i 0.626183 0.779676i \(-0.284616\pi\)
−0.779676 + 0.626183i \(0.784616\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.67423 3.67423i 0.00949415 0.00949415i
\(388\) 0 0
\(389\) 288.000i 0.740360i 0.928960 + 0.370180i \(0.120704\pi\)
−0.928960 + 0.370180i \(0.879296\pi\)
\(390\) 0 0
\(391\) 324.000 0.828645
\(392\) 0 0
\(393\) −14.6969 14.6969i −0.0373968 0.0373968i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 74.7094 74.7094i 0.188185 0.188185i −0.606726 0.794911i \(-0.707517\pi\)
0.794911 + 0.606726i \(0.207517\pi\)
\(398\) 0 0
\(399\) 225.000i 0.563910i
\(400\) 0 0
\(401\) 36.0000 0.0897756 0.0448878 0.998992i \(-0.485707\pi\)
0.0448878 + 0.998992i \(0.485707\pi\)
\(402\) 0 0
\(403\) 300.062 + 300.062i 0.744572 + 0.744572i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 29.3939 29.3939i 0.0722208 0.0722208i
\(408\) 0 0
\(409\) 13.0000i 0.0317848i 0.999874 + 0.0158924i \(0.00505893\pi\)
−0.999874 + 0.0158924i \(0.994941\pi\)
\(410\) 0 0
\(411\) 198.000 0.481752
\(412\) 0 0
\(413\) 286.590 + 286.590i 0.693923 + 0.693923i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 71.0352 71.0352i 0.170348 0.170348i
\(418\) 0 0
\(419\) 144.000i 0.343675i 0.985125 + 0.171838i \(0.0549705\pi\)
−0.985125 + 0.171838i \(0.945030\pi\)
\(420\) 0 0
\(421\) 610.000 1.44893 0.724466 0.689311i \(-0.242087\pi\)
0.724466 + 0.689311i \(0.242087\pi\)
\(422\) 0 0
\(423\) 154.318 + 154.318i 0.364818 + 0.364818i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −47.7650 + 47.7650i −0.111862 + 0.111862i
\(428\) 0 0
\(429\) 90.0000i 0.209790i
\(430\) 0 0
\(431\) 402.000 0.932715 0.466357 0.884596i \(-0.345566\pi\)
0.466357 + 0.884596i \(0.345566\pi\)
\(432\) 0 0
\(433\) 292.714 + 292.714i 0.676014 + 0.676014i 0.959096 0.283082i \(-0.0913568\pi\)
−0.283082 + 0.959096i \(0.591357\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −183.712 + 183.712i −0.420393 + 0.420393i
\(438\) 0 0
\(439\) 119.000i 0.271071i −0.990773 0.135535i \(-0.956725\pi\)
0.990773 0.135535i \(-0.0432754\pi\)
\(440\) 0 0
\(441\) −66.0000 −0.149660
\(442\) 0 0
\(443\) 191.060 + 191.060i 0.431287 + 0.431287i 0.889066 0.457779i \(-0.151355\pi\)
−0.457779 + 0.889066i \(0.651355\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −286.590 + 286.590i −0.641142 + 0.641142i
\(448\) 0 0
\(449\) 84.0000i 0.187082i 0.995615 + 0.0935412i \(0.0298187\pi\)
−0.995615 + 0.0935412i \(0.970181\pi\)
\(450\) 0 0
\(451\) 360.000 0.798226
\(452\) 0 0
\(453\) −104.103 104.103i −0.229809 0.229809i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −146.969 + 146.969i −0.321596 + 0.321596i −0.849379 0.527783i \(-0.823023\pi\)
0.527783 + 0.849379i \(0.323023\pi\)
\(458\) 0 0
\(459\) 162.000i 0.352941i
\(460\) 0 0
\(461\) 516.000 1.11931 0.559653 0.828727i \(-0.310934\pi\)
0.559653 + 0.828727i \(0.310934\pi\)
\(462\) 0 0
\(463\) −524.191 524.191i −1.13216 1.13216i −0.989817 0.142344i \(-0.954536\pi\)
−0.142344 0.989817i \(-0.545464\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 507.044 507.044i 1.08575 1.08575i 0.0897872 0.995961i \(-0.471381\pi\)
0.995961 0.0897872i \(-0.0286187\pi\)
\(468\) 0 0
\(469\) 387.000i 0.825160i
\(470\) 0 0
\(471\) −183.000 −0.388535
\(472\) 0 0
\(473\) 7.34847 + 7.34847i 0.0155359 + 0.0155359i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −44.0908 + 44.0908i −0.0924336 + 0.0924336i
\(478\) 0 0
\(479\) 102.000i 0.212944i −0.994316 0.106472i \(-0.966045\pi\)
0.994316 0.106472i \(-0.0339554\pi\)
\(480\) 0 0
\(481\) 60.0000 0.124740
\(482\) 0 0
\(483\) −66.1362 66.1362i −0.136928 0.136928i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −243.724 + 243.724i −0.500460 + 0.500460i −0.911581 0.411121i \(-0.865138\pi\)
0.411121 + 0.911581i \(0.365138\pi\)
\(488\) 0 0
\(489\) 195.000i 0.398773i
\(490\) 0 0
\(491\) −444.000 −0.904277 −0.452138 0.891948i \(-0.649339\pi\)
−0.452138 + 0.891948i \(0.649339\pi\)
\(492\) 0 0
\(493\) −925.907 925.907i −1.87811 1.87811i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 220.454 220.454i 0.443570 0.443570i
\(498\) 0 0
\(499\) 707.000i 1.41683i −0.705794 0.708417i \(-0.749410\pi\)
0.705794 0.708417i \(-0.250590\pi\)
\(500\) 0 0
\(501\) 432.000 0.862275
\(502\) 0 0
\(503\) −176.363 176.363i −0.350623 0.350623i 0.509718 0.860341i \(-0.329750\pi\)
−0.860341 + 0.509718i \(0.829750\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 115.126 115.126i 0.227073 0.227073i
\(508\) 0 0
\(509\) 666.000i 1.30845i 0.756301 + 0.654224i \(0.227005\pi\)
−0.756301 + 0.654224i \(0.772995\pi\)
\(510\) 0 0
\(511\) −468.000 −0.915851
\(512\) 0 0
\(513\) 91.8559 + 91.8559i 0.179056 + 0.179056i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −308.636 + 308.636i −0.596974 + 0.596974i
\(518\) 0 0
\(519\) 198.000i 0.381503i
\(520\) 0 0
\(521\) −558.000 −1.07102 −0.535509 0.844530i \(-0.679880\pi\)
−0.535509 + 0.844530i \(0.679880\pi\)
\(522\) 0 0
\(523\) 547.461 + 547.461i 1.04677 + 1.04677i 0.998851 + 0.0479193i \(0.0152590\pi\)
0.0479193 + 0.998851i \(0.484741\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1080.22 1080.22i 2.04976 2.04976i
\(528\) 0 0
\(529\) 421.000i 0.795841i
\(530\) 0 0
\(531\) −234.000 −0.440678
\(532\) 0 0
\(533\) 367.423 + 367.423i 0.689350 + 0.689350i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 51.4393 51.4393i 0.0957901 0.0957901i
\(538\) 0 0
\(539\) 132.000i 0.244898i
\(540\) 0 0
\(541\) 803.000 1.48429 0.742144 0.670240i \(-0.233809\pi\)
0.742144 + 0.670240i \(0.233809\pi\)
\(542\) 0 0
\(543\) 177.588 + 177.588i 0.327050 + 0.327050i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 215.555 215.555i 0.394068 0.394068i −0.482067 0.876135i \(-0.660114\pi\)
0.876135 + 0.482067i \(0.160114\pi\)
\(548\) 0 0
\(549\) 39.0000i 0.0710383i
\(550\) 0 0
\(551\) 1050.00 1.90563
\(552\) 0 0
\(553\) 389.469 + 389.469i 0.704284 + 0.704284i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −448.257 + 448.257i −0.804770 + 0.804770i −0.983837 0.179067i \(-0.942692\pi\)
0.179067 + 0.983837i \(0.442692\pi\)
\(558\) 0 0
\(559\) 15.0000i 0.0268336i
\(560\) 0 0
\(561\) 324.000 0.577540
\(562\) 0 0
\(563\) 257.196 + 257.196i 0.456832 + 0.456832i 0.897614 0.440782i \(-0.145299\pi\)
−0.440782 + 0.897614i \(0.645299\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −33.0681 + 33.0681i −0.0583212 + 0.0583212i
\(568\) 0 0
\(569\) 582.000i 1.02285i −0.859329 0.511424i \(-0.829118\pi\)
0.859329 0.511424i \(-0.170882\pi\)
\(570\) 0 0
\(571\) −215.000 −0.376532 −0.188266 0.982118i \(-0.560287\pi\)
−0.188266 + 0.982118i \(0.560287\pi\)
\(572\) 0 0
\(573\) −360.075 360.075i −0.628403 0.628403i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −246.174 + 246.174i −0.426644 + 0.426644i −0.887484 0.460839i \(-0.847548\pi\)
0.460839 + 0.887484i \(0.347548\pi\)
\(578\) 0 0
\(579\) 63.0000i 0.108808i
\(580\) 0 0
\(581\) −594.000 −1.02238
\(582\) 0 0
\(583\) −88.1816 88.1816i −0.151255 0.151255i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −14.6969 + 14.6969i −0.0250374 + 0.0250374i −0.719515 0.694477i \(-0.755636\pi\)
0.694477 + 0.719515i \(0.255636\pi\)
\(588\) 0 0
\(589\) 1225.00i 2.07980i
\(590\) 0 0
\(591\) 54.0000 0.0913706
\(592\) 0 0
\(593\) −323.333 323.333i −0.545249 0.545249i 0.379814 0.925063i \(-0.375988\pi\)
−0.925063 + 0.379814i \(0.875988\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 219.229 219.229i 0.367218 0.367218i
\(598\) 0 0
\(599\) 528.000i 0.881469i −0.897637 0.440735i \(-0.854718\pi\)
0.897637 0.440735i \(-0.145282\pi\)
\(600\) 0 0
\(601\) 493.000 0.820300 0.410150 0.912018i \(-0.365476\pi\)
0.410150 + 0.912018i \(0.365476\pi\)
\(602\) 0 0
\(603\) −157.992 157.992i −0.262010 0.262010i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −729.948 + 729.948i −1.20255 + 1.20255i −0.229162 + 0.973388i \(0.573598\pi\)
−0.973388 + 0.229162i \(0.926402\pi\)
\(608\) 0 0
\(609\) 378.000i 0.620690i
\(610\) 0 0
\(611\) −630.000 −1.03110
\(612\) 0 0
\(613\) −112.677 112.677i −0.183812 0.183812i 0.609203 0.793014i \(-0.291490\pi\)
−0.793014 + 0.609203i \(0.791490\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −382.120 + 382.120i −0.619320 + 0.619320i −0.945357 0.326037i \(-0.894287\pi\)
0.326037 + 0.945357i \(0.394287\pi\)
\(618\) 0 0
\(619\) 733.000i 1.18417i 0.805876 + 0.592084i \(0.201695\pi\)
−0.805876 + 0.592084i \(0.798305\pi\)
\(620\) 0 0
\(621\) 54.0000 0.0869565
\(622\) 0 0
\(623\) −220.454 220.454i −0.353859 0.353859i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −183.712 + 183.712i −0.293001 + 0.293001i
\(628\) 0 0
\(629\) 216.000i 0.343402i
\(630\) 0 0
\(631\) −647.000 −1.02536 −0.512678 0.858581i \(-0.671347\pi\)
−0.512678 + 0.858581i \(0.671347\pi\)
\(632\) 0 0
\(633\) 104.103 + 104.103i 0.164460 + 0.164460i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 134.722 134.722i 0.211494 0.211494i
\(638\) 0 0
\(639\) 180.000i 0.281690i
\(640\) 0 0
\(641\) −636.000 −0.992200 −0.496100 0.868265i \(-0.665235\pi\)
−0.496100 + 0.868265i \(0.665235\pi\)
\(642\) 0 0
\(643\) −396.817 396.817i −0.617134 0.617134i 0.327661 0.944795i \(-0.393740\pi\)
−0.944795 + 0.327661i \(0.893740\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 29.3939 29.3939i 0.0454310 0.0454310i −0.684026 0.729457i \(-0.739773\pi\)
0.729457 + 0.684026i \(0.239773\pi\)
\(648\) 0 0
\(649\) 468.000i 0.721109i
\(650\) 0 0
\(651\) −441.000 −0.677419
\(652\) 0 0
\(653\) 521.741 + 521.741i 0.798991 + 0.798991i 0.982936 0.183945i \(-0.0588869\pi\)
−0.183945 + 0.982936i \(0.558887\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 191.060 191.060i 0.290807 0.290807i
\(658\) 0 0
\(659\) 36.0000i 0.0546282i 0.999627 + 0.0273141i \(0.00869543\pi\)
−0.999627 + 0.0273141i \(0.991305\pi\)
\(660\) 0 0
\(661\) −1022.00 −1.54614 −0.773071 0.634319i \(-0.781280\pi\)
−0.773071 + 0.634319i \(0.781280\pi\)
\(662\) 0 0
\(663\) 330.681 + 330.681i 0.498765 + 0.498765i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 308.636 308.636i 0.462722 0.462722i
\(668\) 0 0
\(669\) 255.000i 0.381166i
\(670\) 0 0
\(671\) 78.0000 0.116244
\(672\) 0 0
\(673\) −857.321 857.321i −1.27388 1.27388i −0.944035 0.329845i \(-0.893003\pi\)
−0.329845 0.944035i \(-0.606997\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −536.438 + 536.438i −0.792376 + 0.792376i −0.981880 0.189504i \(-0.939312\pi\)
0.189504 + 0.981880i \(0.439312\pi\)
\(678\) 0 0
\(679\) 891.000i 1.31222i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −764.241 764.241i −1.11895 1.11895i −0.991896 0.127051i \(-0.959449\pi\)
−0.127051 0.991896i \(-0.540551\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −456.830 + 456.830i −0.664963 + 0.664963i
\(688\) 0 0
\(689\) 180.000i 0.261248i
\(690\) 0 0
\(691\) 842.000 1.21852 0.609262 0.792969i \(-0.291466\pi\)
0.609262 + 0.792969i \(0.291466\pi\)
\(692\) 0 0
\(693\) −66.1362 66.1362i −0.0954347 0.0954347i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1322.72 1322.72i 1.89774 1.89774i
\(698\) 0 0
\(699\) 540.000i 0.772532i
\(700\) 0 0
\(701\) −534.000 −0.761769 −0.380884 0.924623i \(-0.624380\pi\)
−0.380884 + 0.924623i \(0.624380\pi\)
\(702\) 0 0
\(703\) 122.474 + 122.474i 0.174217 + 0.174217i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −705.453 + 705.453i −0.997812 + 0.997812i
\(708\) 0 0
\(709\) 743.000i 1.04795i −0.851732 0.523977i \(-0.824448\pi\)
0.851732 0.523977i \(-0.175552\pi\)
\(710\) 0 0
\(711\) −318.000 −0.447257
\(712\) 0 0
\(713\) 360.075 + 360.075i 0.505014 + 0.505014i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −44.0908 + 44.0908i −0.0614935 + 0.0614935i
\(718\) 0 0
\(719\) 906.000i 1.26008i 0.776561 + 0.630042i \(0.216962\pi\)
−0.776561 + 0.630042i \(0.783038\pi\)
\(720\) 0 0
\(721\) −576.000 −0.798890
\(722\) 0 0
\(723\) −118.800 118.800i −0.164316 0.164316i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 74.7094 74.7094i 0.102764 0.102764i −0.653855 0.756619i \(-0.726850\pi\)
0.756619 + 0.653855i \(0.226850\pi\)
\(728\) 0 0
\(729\) 27.0000i 0.0370370i
\(730\) 0 0
\(731\) 54.0000 0.0738714
\(732\) 0 0
\(733\) 631.968 + 631.968i 0.862167 + 0.862167i 0.991590 0.129423i \(-0.0413124\pi\)
−0.129423 + 0.991590i \(0.541312\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 315.984 315.984i 0.428744 0.428744i
\(738\) 0 0
\(739\) 470.000i 0.635995i 0.948092 + 0.317997i \(0.103010\pi\)
−0.948092 + 0.317997i \(0.896990\pi\)
\(740\) 0 0
\(741\) −375.000 −0.506073
\(742\) 0 0
\(743\) −249.848 249.848i −0.336269 0.336269i 0.518692 0.854961i \(-0.326419\pi\)
−0.854961 + 0.518692i \(0.826419\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 242.499 242.499i 0.324631 0.324631i
\(748\) 0 0
\(749\) 216.000i 0.288385i
\(750\) 0 0
\(751\) −1282.00 −1.70706 −0.853529 0.521046i \(-0.825542\pi\)
−0.853529 + 0.521046i \(0.825542\pi\)
\(752\) 0 0
\(753\) −352.727 352.727i −0.468428 0.468428i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −106.553 + 106.553i −0.140757 + 0.140757i −0.773974 0.633217i \(-0.781734\pi\)
0.633217 + 0.773974i \(0.281734\pi\)
\(758\) 0 0
\(759\) 108.000i 0.142292i
\(760\) 0 0
\(761\) −768.000 −1.00920 −0.504599 0.863354i \(-0.668360\pi\)
−0.504599 + 0.863354i \(0.668360\pi\)
\(762\) 0 0
\(763\) −3.67423 3.67423i −0.00481551 0.00481551i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 477.650 477.650i 0.622752 0.622752i
\(768\) 0 0
\(769\) 491.000i 0.638492i 0.947672 + 0.319246i \(0.103430\pi\)
−0.947672 + 0.319246i \(0.896570\pi\)
\(770\) 0 0
\(771\) 828.000 1.07393
\(772\) 0 0
\(773\) −793.635 793.635i −1.02669 1.02669i −0.999634 0.0270605i \(-0.991385\pi\)
−0.0270605 0.999634i \(-0.508615\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −44.0908 + 44.0908i −0.0567449 + 0.0567449i
\(778\) 0 0
\(779\) 1500.00i 1.92555i
\(780\) 0 0
\(781\) −360.000 −0.460948
\(782\) 0 0
\(783\) −154.318 154.318i −0.197085 0.197085i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −378.446 + 378.446i −0.480872 + 0.480872i −0.905410 0.424538i \(-0.860436\pi\)
0.424538 + 0.905410i \(0.360436\pi\)
\(788\) 0 0
\(789\) 612.000i 0.775665i
\(790\) 0 0
\(791\) 432.000 0.546144
\(792\) 0 0
\(793\) 79.6084 + 79.6084i 0.100389 + 0.100389i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 617.271 617.271i 0.774494 0.774494i −0.204395 0.978889i \(-0.565523\pi\)
0.978889 + 0.204395i \(0.0655227\pi\)
\(798\) 0 0
\(799\) 2268.00i 2.83855i
\(800\) 0 0
\(801\) 180.000 0.224719
\(802\) 0 0
\(803\) 382.120 + 382.120i 0.475866 + 0.475866i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −389.469 + 389.469i −0.482613 + 0.482613i
\(808\) 0 0
\(809\) 348.000i 0.430161i 0.976596 + 0.215080i \(0.0690013\pi\)
−0.976596 + 0.215080i \(0.930999\pi\)
\(810\) 0 0
\(811\) −1045.00 −1.28853 −0.644266 0.764801i \(-0.722837\pi\)
−0.644266 + 0.764801i \(0.722837\pi\)
\(812\) 0 0
\(813\) 276.792 + 276.792i 0.340458 + 0.340458i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −30.6186 + 30.6186i −0.0374769 + 0.0374769i
\(818\) 0 0
\(819\) 135.000i 0.164835i
\(820\) 0 0
\(821\) 606.000 0.738124 0.369062 0.929405i \(-0.379679\pi\)
0.369062 + 0.929405i \(0.379679\pi\)
\(822\) 0 0
\(823\) 302.512 + 302.512i 0.367572 + 0.367572i 0.866591 0.499019i \(-0.166306\pi\)
−0.499019 + 0.866591i \(0.666306\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −396.817 + 396.817i −0.479827 + 0.479827i −0.905076 0.425249i \(-0.860187\pi\)
0.425249 + 0.905076i \(0.360187\pi\)
\(828\) 0 0
\(829\) 406.000i 0.489747i 0.969555 + 0.244873i \(0.0787464\pi\)
−0.969555 + 0.244873i \(0.921254\pi\)
\(830\) 0 0
\(831\) −933.000 −1.12274
\(832\) 0 0
\(833\) −484.999 484.999i −0.582232 0.582232i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 180.037 180.037i 0.215099 0.215099i
\(838\) 0 0
\(839\) 462.000i 0.550656i −0.961350 0.275328i \(-0.911214\pi\)
0.961350 0.275328i \(-0.0887864\pi\)
\(840\) 0 0
\(841\) −923.000 −1.09750
\(842\) 0 0
\(843\) 154.318 + 154.318i 0.183058 + 0.183058i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −312.310 + 312.310i −0.368725 + 0.368725i
\(848\) 0 0
\(849\) 969.000i 1.14134i
\(850\) 0 0
\(851\) 72.0000 0.0846063
\(852\) 0 0
\(853\) −285.366 285.366i −0.334543 0.334543i 0.519766 0.854309i \(-0.326019\pi\)
−0.854309 + 0.519766i \(0.826019\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −191.060 + 191.060i −0.222941 + 0.222941i −0.809736 0.586795i \(-0.800389\pi\)
0.586795 + 0.809736i \(0.300389\pi\)
\(858\) 0 0
\(859\) 410.000i 0.477299i −0.971106 0.238650i \(-0.923295\pi\)
0.971106 0.238650i \(-0.0767047\pi\)
\(860\) 0 0
\(861\) −540.000 −0.627178
\(862\) 0 0
\(863\) −1116.97 1116.97i −1.29428 1.29428i −0.932111 0.362174i \(-0.882035\pi\)
−0.362174 0.932111i \(-0.617965\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 836.501 836.501i 0.964822 0.964822i
\(868\) 0 0
\(869\) 636.000i 0.731876i
\(870\) 0 0
\(871\) 645.000 0.740528
\(872\) 0 0
\(873\) 363.749 + 363.749i 0.416666 + 0.416666i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −180.037 + 180.037i −0.205288 + 0.205288i −0.802261 0.596973i \(-0.796370\pi\)
0.596973 + 0.802261i \(0.296370\pi\)
\(878\) 0 0
\(879\) 54.0000i 0.0614334i
\(880\) 0 0
\(881\) −120.000 −0.136209 −0.0681044 0.997678i \(-0.521695\pi\)
−0.0681044 + 0.997678i \(0.521695\pi\)
\(882\) 0 0
\(883\) −584.203 584.203i −0.661612 0.661612i 0.294148 0.955760i \(-0.404964\pi\)
−0.955760 + 0.294148i \(0.904964\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −198.409 + 198.409i −0.223685 + 0.223685i −0.810048 0.586363i \(-0.800559\pi\)
0.586363 + 0.810048i \(0.300559\pi\)
\(888\) 0 0
\(889\) 108.000i 0.121485i
\(890\) 0 0
\(891\) 54.0000 0.0606061
\(892\) 0 0
\(893\) −1285.98 1285.98i −1.44007 1.44007i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −110.227 + 110.227i −0.122884 + 0.122884i
\(898\) 0 0
\(899\) 2058.00i 2.28921i
\(900\) 0 0
\(901\) −648.000 −0.719201
\(902\) 0 0
\(903\) −11.0227 11.0227i −0.0122068 0.0122068i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1033.68 1033.68i 1.13967 1.13967i 0.151166 0.988508i \(-0.451697\pi\)
0.988508 0.151166i \(-0.0483028\pi\)
\(908\) 0 0
\(909\) 576.000i 0.633663i
\(910\) 0 0
\(911\) −1218.00 −1.33699 −0.668496 0.743716i \(-0.733062\pi\)
−0.668496 + 0.743716i \(0.733062\pi\)
\(912\) 0 0
\(913\) 484.999 + 484.999i 0.531215 + 0.531215i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −44.0908 + 44.0908i −0.0480816 + 0.0480816i
\(918\) 0 0
\(919\) 469.000i 0.510337i −0.966897 0.255169i \(-0.917869\pi\)
0.966897 0.255169i \(-0.0821310\pi\)
\(920\) 0 0
\(921\) −213.000 −0.231270
\(922\) 0 0
\(923\) −367.423 367.423i −0.398075 0.398075i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 235.151 235.151i 0.253669 0.253669i
\(928\) 0 0
\(929\) 474.000i 0.510226i 0.966911 + 0.255113i \(0.0821127\pi\)
−0.966911 + 0.255113i \(0.917887\pi\)
\(930\) 0 0
\(931\) 550.000 0.590763
\(932\) 0 0
\(933\) 301.287 + 301.287i 0.322923 + 0.322923i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −709.127 + 709.127i −0.756806 + 0.756806i −0.975740 0.218934i \(-0.929742\pi\)
0.218934 + 0.975740i \(0.429742\pi\)
\(938\) 0 0
\(939\) 93.0000i 0.0990415i
\(940\) 0 0
\(941\) 990.000 1.05207 0.526036 0.850462i \(-0.323678\pi\)
0.526036 + 0.850462i \(0.323678\pi\)
\(942\) 0 0
\(943\) 440.908 + 440.908i 0.467559 + 0.467559i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1036.13 + 1036.13i −1.09412 + 1.09412i −0.0990391 + 0.995084i \(0.531577\pi\)
−0.995084 + 0.0990391i \(0.968423\pi\)
\(948\) 0 0
\(949\) 780.000i 0.821918i
\(950\) 0 0
\(951\) 36.0000 0.0378549
\(952\) 0 0
\(953\) −631.968 631.968i −0.663136 0.663136i 0.292982 0.956118i \(-0.405352\pi\)
−0.956118 + 0.292982i \(0.905352\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 308.636 308.636i 0.322503 0.322503i
\(958\) 0 0
\(959\) 594.000i 0.619395i
\(960\) 0 0
\(961\) 1440.00 1.49844
\(962\) 0 0
\(963\) 88.1816 + 88.1816i 0.0915697 + 0.0915697i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 563.383 563.383i 0.582609 0.582609i −0.353011 0.935619i \(-0.614842\pi\)
0.935619 + 0.353011i \(0.114842\pi\)
\(968\) 0 0
\(969\) 1350.00i 1.39319i
\(970\) 0 0
\(971\) −1338.00 −1.37796 −0.688980 0.724780i \(-0.741941\pi\)
−0.688980 + 0.724780i \(0.741941\pi\)
\(972\) 0 0
\(973\) −213.106 213.106i −0.219019 0.219019i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −301.287 + 301.287i −0.308380 + 0.308380i −0.844281 0.535901i \(-0.819972\pi\)
0.535901 + 0.844281i \(0.319972\pi\)
\(978\) 0 0
\(979\) 360.000i 0.367722i
\(980\) 0 0
\(981\) 3.00000 0.00305810
\(982\) 0 0
\(983\) −426.211 426.211i −0.433582 0.433582i 0.456263 0.889845i \(-0.349188\pi\)
−0.889845 + 0.456263i \(0.849188\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 462.954 462.954i 0.469051 0.469051i
\(988\) 0 0
\(989\) 18.0000i 0.0182002i
\(990\) 0 0
\(991\) −83.0000 −0.0837538 −0.0418769 0.999123i \(-0.513334\pi\)
−0.0418769 + 0.999123i \(0.513334\pi\)
\(992\) 0 0
\(993\) 218.005 + 218.005i 0.219541 + 0.219541i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 573.181 573.181i 0.574905 0.574905i −0.358590 0.933495i \(-0.616742\pi\)
0.933495 + 0.358590i \(0.116742\pi\)
\(998\) 0 0
\(999\) 36.0000i 0.0360360i
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 1200.3.bg.i.193.2 4
4.3 odd 2 300.3.k.c.193.1 yes 4
5.2 odd 4 inner 1200.3.bg.i.1057.2 4
5.3 odd 4 inner 1200.3.bg.i.1057.1 4
5.4 even 2 inner 1200.3.bg.i.193.1 4
12.11 even 2 900.3.l.d.793.1 4
20.3 even 4 300.3.k.c.157.2 yes 4
20.7 even 4 300.3.k.c.157.1 4
20.19 odd 2 300.3.k.c.193.2 yes 4
60.23 odd 4 900.3.l.d.757.2 4
60.47 odd 4 900.3.l.d.757.1 4
60.59 even 2 900.3.l.d.793.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.3.k.c.157.1 4 20.7 even 4
300.3.k.c.157.2 yes 4 20.3 even 4
300.3.k.c.193.1 yes 4 4.3 odd 2
300.3.k.c.193.2 yes 4 20.19 odd 2
900.3.l.d.757.1 4 60.47 odd 4
900.3.l.d.757.2 4 60.23 odd 4
900.3.l.d.793.1 4 12.11 even 2
900.3.l.d.793.2 4 60.59 even 2
1200.3.bg.i.193.1 4 5.4 even 2 inner
1200.3.bg.i.193.2 4 1.1 even 1 trivial
1200.3.bg.i.1057.1 4 5.3 odd 4 inner
1200.3.bg.i.1057.2 4 5.2 odd 4 inner