Properties

Label 1200.3.bg.p.1057.1
Level $1200$
Weight $3$
Character 1200.1057
Analytic conductor $32.698$
Analytic rank $0$
Dimension $4$
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] = [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 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1057.1
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1200.1057
Dual form 1200.3.bg.p.193.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.22474 + 1.22474i) q^{3} +(7.22474 + 7.22474i) q^{7} -3.00000i q^{9} +O(q^{10})\) \(q+(-1.22474 + 1.22474i) q^{3} +(7.22474 + 7.22474i) q^{7} -3.00000i q^{9} +8.69694 q^{11} +(15.6742 - 15.6742i) q^{13} +(-13.3485 - 13.3485i) q^{17} +4.30306i q^{19} -17.6969 q^{21} +(28.0454 - 28.0454i) q^{23} +(3.67423 + 3.67423i) q^{27} -20.6969i q^{29} -39.0908 q^{31} +(-10.6515 + 10.6515i) q^{33} +(-12.4949 - 12.4949i) q^{37} +38.3939i q^{39} +62.6969 q^{41} +(-11.8763 + 11.8763i) q^{43} +(-58.0454 - 58.0454i) q^{47} +55.3939i q^{49} +32.6969 q^{51} +(-0.606123 + 0.606123i) q^{53} +(-5.27015 - 5.27015i) q^{57} +30.0000i q^{59} +69.7878 q^{61} +(21.6742 - 21.6742i) q^{63} +(-5.02270 - 5.02270i) q^{67} +68.6969i q^{69} +38.6969 q^{71} +(46.2929 - 46.2929i) q^{73} +(62.8332 + 62.8332i) q^{77} +31.3939i q^{79} -9.00000 q^{81} +(39.4393 - 39.4393i) q^{83} +(25.3485 + 25.3485i) q^{87} -41.3939i q^{89} +226.485 q^{91} +(47.8763 - 47.8763i) q^{93} +(-54.1237 - 54.1237i) q^{97} -26.0908i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 24 q^{7} - 24 q^{11} + 48 q^{13} - 24 q^{17} - 12 q^{21} + 24 q^{23} + 20 q^{31} - 72 q^{33} + 48 q^{37} + 192 q^{41} - 72 q^{43} - 144 q^{47} + 72 q^{51} - 120 q^{53} + 72 q^{57} + 44 q^{61} + 72 q^{63} + 24 q^{67} + 96 q^{71} + 48 q^{73} - 72 q^{77} - 36 q^{81} - 48 q^{83} + 72 q^{87} + 612 q^{91} + 216 q^{93} - 192 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(7\) 7.22474 + 7.22474i 1.03211 + 1.03211i 0.999467 + 0.0326392i \(0.0103912\pi\)
0.0326392 + 0.999467i \(0.489609\pi\)
\(8\) 0 0
\(9\) 3.00000i 0.333333i
\(10\) 0 0
\(11\) 8.69694 0.790631 0.395315 0.918545i \(-0.370635\pi\)
0.395315 + 0.918545i \(0.370635\pi\)
\(12\) 0 0
\(13\) 15.6742 15.6742i 1.20571 1.20571i 0.233307 0.972403i \(-0.425045\pi\)
0.972403 0.233307i \(-0.0749548\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −13.3485 13.3485i −0.785204 0.785204i 0.195500 0.980704i \(-0.437367\pi\)
−0.980704 + 0.195500i \(0.937367\pi\)
\(18\) 0 0
\(19\) 4.30306i 0.226477i 0.993568 + 0.113238i \(0.0361224\pi\)
−0.993568 + 0.113238i \(0.963878\pi\)
\(20\) 0 0
\(21\) −17.6969 −0.842711
\(22\) 0 0
\(23\) 28.0454 28.0454i 1.21937 1.21937i 0.251511 0.967854i \(-0.419072\pi\)
0.967854 0.251511i \(-0.0809275\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.67423 + 3.67423i 0.136083 + 0.136083i
\(28\) 0 0
\(29\) 20.6969i 0.713688i −0.934164 0.356844i \(-0.883853\pi\)
0.934164 0.356844i \(-0.116147\pi\)
\(30\) 0 0
\(31\) −39.0908 −1.26099 −0.630497 0.776192i \(-0.717149\pi\)
−0.630497 + 0.776192i \(0.717149\pi\)
\(32\) 0 0
\(33\) −10.6515 + 10.6515i −0.322774 + 0.322774i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −12.4949 12.4949i −0.337700 0.337700i 0.517801 0.855501i \(-0.326751\pi\)
−0.855501 + 0.517801i \(0.826751\pi\)
\(38\) 0 0
\(39\) 38.3939i 0.984458i
\(40\) 0 0
\(41\) 62.6969 1.52919 0.764597 0.644509i \(-0.222938\pi\)
0.764597 + 0.644509i \(0.222938\pi\)
\(42\) 0 0
\(43\) −11.8763 + 11.8763i −0.276192 + 0.276192i −0.831587 0.555395i \(-0.812567\pi\)
0.555395 + 0.831587i \(0.312567\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −58.0454 58.0454i −1.23501 1.23501i −0.962016 0.272993i \(-0.911987\pi\)
−0.272993 0.962016i \(-0.588013\pi\)
\(48\) 0 0
\(49\) 55.3939i 1.13049i
\(50\) 0 0
\(51\) 32.6969 0.641116
\(52\) 0 0
\(53\) −0.606123 + 0.606123i −0.0114363 + 0.0114363i −0.712802 0.701366i \(-0.752574\pi\)
0.701366 + 0.712802i \(0.252574\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.27015 5.27015i −0.0924588 0.0924588i
\(58\) 0 0
\(59\) 30.0000i 0.508475i 0.967142 + 0.254237i \(0.0818244\pi\)
−0.967142 + 0.254237i \(0.918176\pi\)
\(60\) 0 0
\(61\) 69.7878 1.14406 0.572031 0.820232i \(-0.306156\pi\)
0.572031 + 0.820232i \(0.306156\pi\)
\(62\) 0 0
\(63\) 21.6742 21.6742i 0.344035 0.344035i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.02270 5.02270i −0.0749657 0.0749657i 0.668630 0.743595i \(-0.266881\pi\)
−0.743595 + 0.668630i \(0.766881\pi\)
\(68\) 0 0
\(69\) 68.6969i 0.995608i
\(70\) 0 0
\(71\) 38.6969 0.545027 0.272514 0.962152i \(-0.412145\pi\)
0.272514 + 0.962152i \(0.412145\pi\)
\(72\) 0 0
\(73\) 46.2929 46.2929i 0.634149 0.634149i −0.314957 0.949106i \(-0.601990\pi\)
0.949106 + 0.314957i \(0.101990\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 62.8332 + 62.8332i 0.816015 + 0.816015i
\(78\) 0 0
\(79\) 31.3939i 0.397391i 0.980061 + 0.198695i \(0.0636705\pi\)
−0.980061 + 0.198695i \(0.936330\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) 0 0
\(83\) 39.4393 39.4393i 0.475172 0.475172i −0.428412 0.903584i \(-0.640927\pi\)
0.903584 + 0.428412i \(0.140927\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 25.3485 + 25.3485i 0.291362 + 0.291362i
\(88\) 0 0
\(89\) 41.3939i 0.465100i −0.972584 0.232550i \(-0.925293\pi\)
0.972584 0.232550i \(-0.0747069\pi\)
\(90\) 0 0
\(91\) 226.485 2.48884
\(92\) 0 0
\(93\) 47.8763 47.8763i 0.514799 0.514799i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −54.1237 54.1237i −0.557977 0.557977i 0.370754 0.928731i \(-0.379099\pi\)
−0.928731 + 0.370754i \(0.879099\pi\)
\(98\) 0 0
\(99\) 26.0908i 0.263544i
\(100\) 0 0
\(101\) −42.8786 −0.424540 −0.212270 0.977211i \(-0.568086\pi\)
−0.212270 + 0.977211i \(0.568086\pi\)
\(102\) 0 0
\(103\) 60.9898 60.9898i 0.592134 0.592134i −0.346073 0.938207i \(-0.612485\pi\)
0.938207 + 0.346073i \(0.112485\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 45.9092 + 45.9092i 0.429058 + 0.429058i 0.888307 0.459250i \(-0.151882\pi\)
−0.459250 + 0.888307i \(0.651882\pi\)
\(108\) 0 0
\(109\) 145.000i 1.33028i 0.746721 + 0.665138i \(0.231627\pi\)
−0.746721 + 0.665138i \(0.768373\pi\)
\(110\) 0 0
\(111\) 30.6061 0.275731
\(112\) 0 0
\(113\) 130.788 130.788i 1.15741 1.15741i 0.172384 0.985030i \(-0.444853\pi\)
0.985030 0.172384i \(-0.0551470\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −47.0227 47.0227i −0.401903 0.401903i
\(118\) 0 0
\(119\) 192.879i 1.62083i
\(120\) 0 0
\(121\) −45.3633 −0.374903
\(122\) 0 0
\(123\) −76.7878 + 76.7878i −0.624291 + 0.624291i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −21.3031 21.3031i −0.167741 0.167741i 0.618245 0.785986i \(-0.287844\pi\)
−0.785986 + 0.618245i \(0.787844\pi\)
\(128\) 0 0
\(129\) 29.0908i 0.225510i
\(130\) 0 0
\(131\) −108.272 −0.826507 −0.413254 0.910616i \(-0.635608\pi\)
−0.413254 + 0.910616i \(0.635608\pi\)
\(132\) 0 0
\(133\) −31.0885 + 31.0885i −0.233748 + 0.233748i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 76.6515 + 76.6515i 0.559500 + 0.559500i 0.929165 0.369665i \(-0.120528\pi\)
−0.369665 + 0.929165i \(0.620528\pi\)
\(138\) 0 0
\(139\) 152.788i 1.09919i 0.835430 + 0.549596i \(0.185218\pi\)
−0.835430 + 0.549596i \(0.814782\pi\)
\(140\) 0 0
\(141\) 142.182 1.00838
\(142\) 0 0
\(143\) 136.318 136.318i 0.953272 0.953272i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −67.8434 67.8434i −0.461520 0.461520i
\(148\) 0 0
\(149\) 112.788i 0.756965i 0.925608 + 0.378482i \(0.123554\pi\)
−0.925608 + 0.378482i \(0.876446\pi\)
\(150\) 0 0
\(151\) 167.879 1.11178 0.555889 0.831256i \(-0.312378\pi\)
0.555889 + 0.831256i \(0.312378\pi\)
\(152\) 0 0
\(153\) −40.0454 + 40.0454i −0.261735 + 0.261735i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 114.866 + 114.866i 0.731631 + 0.731631i 0.970943 0.239312i \(-0.0769218\pi\)
−0.239312 + 0.970943i \(0.576922\pi\)
\(158\) 0 0
\(159\) 1.48469i 0.00933769i
\(160\) 0 0
\(161\) 405.242 2.51703
\(162\) 0 0
\(163\) 10.1441 10.1441i 0.0622340 0.0622340i −0.675305 0.737539i \(-0.735988\pi\)
0.737539 + 0.675305i \(0.235988\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 155.666 + 155.666i 0.932134 + 0.932134i 0.997839 0.0657054i \(-0.0209298\pi\)
−0.0657054 + 0.997839i \(0.520930\pi\)
\(168\) 0 0
\(169\) 322.363i 1.90747i
\(170\) 0 0
\(171\) 12.9092 0.0754923
\(172\) 0 0
\(173\) 86.8332 86.8332i 0.501926 0.501926i −0.410110 0.912036i \(-0.634510\pi\)
0.912036 + 0.410110i \(0.134510\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −36.7423 36.7423i −0.207584 0.207584i
\(178\) 0 0
\(179\) 241.151i 1.34721i 0.739090 + 0.673606i \(0.235256\pi\)
−0.739090 + 0.673606i \(0.764744\pi\)
\(180\) 0 0
\(181\) 41.1816 0.227523 0.113761 0.993508i \(-0.463710\pi\)
0.113761 + 0.993508i \(0.463710\pi\)
\(182\) 0 0
\(183\) −85.4722 + 85.4722i −0.467061 + 0.467061i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −116.091 116.091i −0.620806 0.620806i
\(188\) 0 0
\(189\) 53.0908i 0.280904i
\(190\) 0 0
\(191\) 110.091 0.576392 0.288196 0.957571i \(-0.406945\pi\)
0.288196 + 0.957571i \(0.406945\pi\)
\(192\) 0 0
\(193\) −266.487 + 266.487i −1.38076 + 1.38076i −0.537494 + 0.843268i \(0.680629\pi\)
−0.843268 + 0.537494i \(0.819371\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 39.4393 + 39.4393i 0.200199 + 0.200199i 0.800085 0.599886i \(-0.204787\pi\)
−0.599886 + 0.800085i \(0.704787\pi\)
\(198\) 0 0
\(199\) 92.9092i 0.466880i 0.972371 + 0.233440i \(0.0749983\pi\)
−0.972371 + 0.233440i \(0.925002\pi\)
\(200\) 0 0
\(201\) 12.3031 0.0612093
\(202\) 0 0
\(203\) 149.530 149.530i 0.736601 0.736601i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −84.1362 84.1362i −0.406455 0.406455i
\(208\) 0 0
\(209\) 37.4235i 0.179060i
\(210\) 0 0
\(211\) −293.272 −1.38992 −0.694958 0.719050i \(-0.744577\pi\)
−0.694958 + 0.719050i \(0.744577\pi\)
\(212\) 0 0
\(213\) −47.3939 + 47.3939i −0.222506 + 0.222506i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −282.421 282.421i −1.30148 1.30148i
\(218\) 0 0
\(219\) 113.394i 0.517780i
\(220\) 0 0
\(221\) −418.454 −1.89346
\(222\) 0 0
\(223\) −74.8207 + 74.8207i −0.335519 + 0.335519i −0.854678 0.519159i \(-0.826245\pi\)
0.519159 + 0.854678i \(0.326245\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −47.1214 47.1214i −0.207583 0.207583i 0.595656 0.803240i \(-0.296892\pi\)
−0.803240 + 0.595656i \(0.796892\pi\)
\(228\) 0 0
\(229\) 275.000i 1.20087i 0.799672 + 0.600437i \(0.205007\pi\)
−0.799672 + 0.600437i \(0.794993\pi\)
\(230\) 0 0
\(231\) −153.909 −0.666274
\(232\) 0 0
\(233\) 187.757 187.757i 0.805825 0.805825i −0.178174 0.983999i \(-0.557019\pi\)
0.983999 + 0.178174i \(0.0570191\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −38.4495 38.4495i −0.162234 0.162234i
\(238\) 0 0
\(239\) 17.6663i 0.0739177i 0.999317 + 0.0369588i \(0.0117670\pi\)
−0.999317 + 0.0369588i \(0.988233\pi\)
\(240\) 0 0
\(241\) 167.000 0.692946 0.346473 0.938060i \(-0.387379\pi\)
0.346473 + 0.938060i \(0.387379\pi\)
\(242\) 0 0
\(243\) 11.0227 11.0227i 0.0453609 0.0453609i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 67.4472 + 67.4472i 0.273066 + 0.273066i
\(248\) 0 0
\(249\) 96.6061i 0.387976i
\(250\) 0 0
\(251\) −26.4245 −0.105277 −0.0526384 0.998614i \(-0.516763\pi\)
−0.0526384 + 0.998614i \(0.516763\pi\)
\(252\) 0 0
\(253\) 243.909 243.909i 0.964068 0.964068i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −241.485 241.485i −0.939629 0.939629i 0.0586495 0.998279i \(-0.481321\pi\)
−0.998279 + 0.0586495i \(0.981321\pi\)
\(258\) 0 0
\(259\) 180.545i 0.697085i
\(260\) 0 0
\(261\) −62.0908 −0.237896
\(262\) 0 0
\(263\) −142.182 + 142.182i −0.540615 + 0.540615i −0.923709 0.383095i \(-0.874858\pi\)
0.383095 + 0.923709i \(0.374858\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 50.6969 + 50.6969i 0.189876 + 0.189876i
\(268\) 0 0
\(269\) 521.605i 1.93905i −0.244988 0.969526i \(-0.578784\pi\)
0.244988 0.969526i \(-0.421216\pi\)
\(270\) 0 0
\(271\) −39.2122 −0.144695 −0.0723473 0.997379i \(-0.523049\pi\)
−0.0723473 + 0.997379i \(0.523049\pi\)
\(272\) 0 0
\(273\) −277.386 + 277.386i −1.01607 + 1.01607i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 69.5880 + 69.5880i 0.251220 + 0.251220i 0.821471 0.570251i \(-0.193154\pi\)
−0.570251 + 0.821471i \(0.693154\pi\)
\(278\) 0 0
\(279\) 117.272i 0.420331i
\(280\) 0 0
\(281\) −519.848 −1.84999 −0.924996 0.379976i \(-0.875932\pi\)
−0.924996 + 0.379976i \(0.875932\pi\)
\(282\) 0 0
\(283\) −153.427 + 153.427i −0.542144 + 0.542144i −0.924157 0.382013i \(-0.875231\pi\)
0.382013 + 0.924157i \(0.375231\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 452.969 + 452.969i 1.57829 + 1.57829i
\(288\) 0 0
\(289\) 67.3633i 0.233091i
\(290\) 0 0
\(291\) 132.576 0.455586
\(292\) 0 0
\(293\) 19.6209 19.6209i 0.0669656 0.0669656i −0.672831 0.739796i \(-0.734922\pi\)
0.739796 + 0.672831i \(0.234922\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 31.9546 + 31.9546i 0.107591 + 0.107591i
\(298\) 0 0
\(299\) 879.181i 2.94040i
\(300\) 0 0
\(301\) −171.606 −0.570120
\(302\) 0 0
\(303\) 52.5153 52.5153i 0.173318 0.173318i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −10.9115 10.9115i −0.0355423 0.0355423i 0.689112 0.724655i \(-0.258001\pi\)
−0.724655 + 0.689112i \(0.758001\pi\)
\(308\) 0 0
\(309\) 149.394i 0.483475i
\(310\) 0 0
\(311\) −64.7878 −0.208321 −0.104160 0.994561i \(-0.533216\pi\)
−0.104160 + 0.994561i \(0.533216\pi\)
\(312\) 0 0
\(313\) 25.4472 25.4472i 0.0813009 0.0813009i −0.665287 0.746588i \(-0.731691\pi\)
0.746588 + 0.665287i \(0.231691\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 431.060 + 431.060i 1.35981 + 1.35981i 0.874143 + 0.485668i \(0.161424\pi\)
0.485668 + 0.874143i \(0.338576\pi\)
\(318\) 0 0
\(319\) 180.000i 0.564263i
\(320\) 0 0
\(321\) −112.454 −0.350324
\(322\) 0 0
\(323\) 57.4393 57.4393i 0.177831 0.177831i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −177.588 177.588i −0.543083 0.543083i
\(328\) 0 0
\(329\) 838.727i 2.54932i
\(330\) 0 0
\(331\) −295.939 −0.894075 −0.447037 0.894515i \(-0.647521\pi\)
−0.447037 + 0.894515i \(0.647521\pi\)
\(332\) 0 0
\(333\) −37.4847 + 37.4847i −0.112567 + 0.112567i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 141.538 + 141.538i 0.419994 + 0.419994i 0.885202 0.465208i \(-0.154020\pi\)
−0.465208 + 0.885202i \(0.654020\pi\)
\(338\) 0 0
\(339\) 320.363i 0.945024i
\(340\) 0 0
\(341\) −339.970 −0.996981
\(342\) 0 0
\(343\) −46.1941 + 46.1941i −0.134677 + 0.134677i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −362.227 362.227i −1.04388 1.04388i −0.998992 0.0448900i \(-0.985706\pi\)
−0.0448900 0.998992i \(-0.514294\pi\)
\(348\) 0 0
\(349\) 403.939i 1.15742i 0.815534 + 0.578709i \(0.196443\pi\)
−0.815534 + 0.578709i \(0.803557\pi\)
\(350\) 0 0
\(351\) 115.182 0.328153
\(352\) 0 0
\(353\) −18.7423 + 18.7423i −0.0530945 + 0.0530945i −0.733156 0.680061i \(-0.761953\pi\)
0.680061 + 0.733156i \(0.261953\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 236.227 + 236.227i 0.661700 + 0.661700i
\(358\) 0 0
\(359\) 383.728i 1.06888i −0.845207 0.534439i \(-0.820523\pi\)
0.845207 0.534439i \(-0.179477\pi\)
\(360\) 0 0
\(361\) 342.484 0.948708
\(362\) 0 0
\(363\) 55.5584 55.5584i 0.153054 0.153054i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −387.341 387.341i −1.05542 1.05542i −0.998371 0.0570527i \(-0.981830\pi\)
−0.0570527 0.998371i \(-0.518170\pi\)
\(368\) 0 0
\(369\) 188.091i 0.509731i
\(370\) 0 0
\(371\) −8.75817 −0.0236069
\(372\) 0 0
\(373\) −419.815 + 419.815i −1.12551 + 1.12551i −0.134611 + 0.990899i \(0.542978\pi\)
−0.990899 + 0.134611i \(0.957022\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −324.409 324.409i −0.860500 0.860500i
\(378\) 0 0
\(379\) 472.181i 1.24586i 0.782278 + 0.622930i \(0.214058\pi\)
−0.782278 + 0.622930i \(0.785942\pi\)
\(380\) 0 0
\(381\) 52.1816 0.136960
\(382\) 0 0
\(383\) 67.8184 67.8184i 0.177071 0.177071i −0.613006 0.790078i \(-0.710040\pi\)
0.790078 + 0.613006i \(0.210040\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 35.6288 + 35.6288i 0.0920642 + 0.0920642i
\(388\) 0 0
\(389\) 603.242i 1.55075i 0.631501 + 0.775375i \(0.282439\pi\)
−0.631501 + 0.775375i \(0.717561\pi\)
\(390\) 0 0
\(391\) −748.727 −1.91490
\(392\) 0 0
\(393\) 132.606 132.606i 0.337420 0.337420i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 188.648 + 188.648i 0.475184 + 0.475184i 0.903588 0.428403i \(-0.140924\pi\)
−0.428403 + 0.903588i \(0.640924\pi\)
\(398\) 0 0
\(399\) 76.1510i 0.190855i
\(400\) 0 0
\(401\) 302.697 0.754855 0.377428 0.926039i \(-0.376809\pi\)
0.377428 + 0.926039i \(0.376809\pi\)
\(402\) 0 0
\(403\) −612.719 + 612.719i −1.52039 + 1.52039i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −108.667 108.667i −0.266996 0.266996i
\(408\) 0 0
\(409\) 20.8184i 0.0509007i −0.999676 0.0254503i \(-0.991898\pi\)
0.999676 0.0254503i \(-0.00810197\pi\)
\(410\) 0 0
\(411\) −187.757 −0.456830
\(412\) 0 0
\(413\) −216.742 + 216.742i −0.524800 + 0.524800i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −187.126 187.126i −0.448743 0.448743i
\(418\) 0 0
\(419\) 466.515i 1.11340i −0.830713 0.556701i \(-0.812067\pi\)
0.830713 0.556701i \(-0.187933\pi\)
\(420\) 0 0
\(421\) −636.120 −1.51097 −0.755487 0.655163i \(-0.772600\pi\)
−0.755487 + 0.655163i \(0.772600\pi\)
\(422\) 0 0
\(423\) −174.136 + 174.136i −0.411670 + 0.411670i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 504.199 + 504.199i 1.18079 + 1.18079i
\(428\) 0 0
\(429\) 333.909i 0.778343i
\(430\) 0 0
\(431\) −165.242 −0.383392 −0.191696 0.981454i \(-0.561399\pi\)
−0.191696 + 0.981454i \(0.561399\pi\)
\(432\) 0 0
\(433\) −282.619 + 282.619i −0.652699 + 0.652699i −0.953642 0.300943i \(-0.902699\pi\)
0.300943 + 0.953642i \(0.402699\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 120.681 + 120.681i 0.276158 + 0.276158i
\(438\) 0 0
\(439\) 272.909i 0.621661i 0.950465 + 0.310831i \(0.100607\pi\)
−0.950465 + 0.310831i \(0.899393\pi\)
\(440\) 0 0
\(441\) 166.182 0.376829
\(442\) 0 0
\(443\) 176.636 176.636i 0.398726 0.398726i −0.479057 0.877784i \(-0.659021\pi\)
0.877784 + 0.479057i \(0.159021\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −138.136 138.136i −0.309030 0.309030i
\(448\) 0 0
\(449\) 751.151i 1.67294i −0.548011 0.836471i \(-0.684615\pi\)
0.548011 0.836471i \(-0.315385\pi\)
\(450\) 0 0
\(451\) 545.271 1.20903
\(452\) 0 0
\(453\) −205.608 + 205.608i −0.453882 + 0.453882i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 26.4245 + 26.4245i 0.0578216 + 0.0578216i 0.735426 0.677605i \(-0.236982\pi\)
−0.677605 + 0.735426i \(0.736982\pi\)
\(458\) 0 0
\(459\) 98.0908i 0.213705i
\(460\) 0 0
\(461\) −38.6969 −0.0839413 −0.0419706 0.999119i \(-0.513364\pi\)
−0.0419706 + 0.999119i \(0.513364\pi\)
\(462\) 0 0
\(463\) −203.505 + 203.505i −0.439536 + 0.439536i −0.891856 0.452320i \(-0.850597\pi\)
0.452320 + 0.891856i \(0.350597\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 557.530 + 557.530i 1.19385 + 1.19385i 0.975976 + 0.217879i \(0.0699137\pi\)
0.217879 + 0.975976i \(0.430086\pi\)
\(468\) 0 0
\(469\) 72.5755i 0.154745i
\(470\) 0 0
\(471\) −281.363 −0.597374
\(472\) 0 0
\(473\) −103.287 + 103.287i −0.218366 + 0.218366i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.81837 + 1.81837i 0.00381209 + 0.00381209i
\(478\) 0 0
\(479\) 265.818i 0.554944i 0.960734 + 0.277472i \(0.0894966\pi\)
−0.960734 + 0.277472i \(0.910503\pi\)
\(480\) 0 0
\(481\) −391.696 −0.814337
\(482\) 0 0
\(483\) −496.318 + 496.318i −1.02757 + 1.02757i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −304.083 304.083i −0.624400 0.624400i 0.322253 0.946653i \(-0.395560\pi\)
−0.946653 + 0.322253i \(0.895560\pi\)
\(488\) 0 0
\(489\) 24.8480i 0.0508138i
\(490\) 0 0
\(491\) −169.212 −0.344628 −0.172314 0.985042i \(-0.555124\pi\)
−0.172314 + 0.985042i \(0.555124\pi\)
\(492\) 0 0
\(493\) −276.272 + 276.272i −0.560390 + 0.560390i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 279.576 + 279.576i 0.562526 + 0.562526i
\(498\) 0 0
\(499\) 699.636i 1.40208i −0.713124 0.701038i \(-0.752720\pi\)
0.713124 0.701038i \(-0.247280\pi\)
\(500\) 0 0
\(501\) −381.303 −0.761084
\(502\) 0 0
\(503\) −279.848 + 279.848i −0.556358 + 0.556358i −0.928268 0.371911i \(-0.878703\pi\)
0.371911 + 0.928268i \(0.378703\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 394.813 + 394.813i 0.778723 + 0.778723i
\(508\) 0 0
\(509\) 193.696i 0.380542i 0.981732 + 0.190271i \(0.0609367\pi\)
−0.981732 + 0.190271i \(0.939063\pi\)
\(510\) 0 0
\(511\) 668.908 1.30902
\(512\) 0 0
\(513\) −15.8105 + 15.8105i −0.0308196 + 0.0308196i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −504.817 504.817i −0.976436 0.976436i
\(518\) 0 0
\(519\) 212.697i 0.409821i
\(520\) 0 0
\(521\) 167.121 0.320771 0.160385 0.987054i \(-0.448726\pi\)
0.160385 + 0.987054i \(0.448726\pi\)
\(522\) 0 0
\(523\) 675.860 675.860i 1.29228 1.29228i 0.358900 0.933376i \(-0.383152\pi\)
0.933376 0.358900i \(-0.116848\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 521.803 + 521.803i 0.990138 + 0.990138i
\(528\) 0 0
\(529\) 1044.09i 1.97370i
\(530\) 0 0
\(531\) 90.0000 0.169492
\(532\) 0 0
\(533\) 982.727 982.727i 1.84376 1.84376i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −295.348 295.348i −0.549997 0.549997i
\(538\) 0 0
\(539\) 481.757i 0.893798i
\(540\) 0 0
\(541\) −131.120 −0.242367 −0.121183 0.992630i \(-0.538669\pi\)
−0.121183 + 0.992630i \(0.538669\pi\)
\(542\) 0 0
\(543\) −50.4370 + 50.4370i −0.0928858 + 0.0928858i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 423.959 + 423.959i 0.775062 + 0.775062i 0.978987 0.203924i \(-0.0653696\pi\)
−0.203924 + 0.978987i \(0.565370\pi\)
\(548\) 0 0
\(549\) 209.363i 0.381354i
\(550\) 0 0
\(551\) 89.0602 0.161634
\(552\) 0 0
\(553\) −226.813 + 226.813i −0.410150 + 0.410150i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 341.741 + 341.741i 0.613539 + 0.613539i 0.943866 0.330327i \(-0.107159\pi\)
−0.330327 + 0.943866i \(0.607159\pi\)
\(558\) 0 0
\(559\) 372.303i 0.666016i
\(560\) 0 0
\(561\) 284.363 0.506886
\(562\) 0 0
\(563\) 13.6209 13.6209i 0.0241935 0.0241935i −0.694907 0.719100i \(-0.744554\pi\)
0.719100 + 0.694907i \(0.244554\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −65.0227 65.0227i −0.114678 0.114678i
\(568\) 0 0
\(569\) 22.9990i 0.0404200i 0.999796 + 0.0202100i \(0.00643348\pi\)
−0.999796 + 0.0202100i \(0.993567\pi\)
\(570\) 0 0
\(571\) 660.666 1.15703 0.578517 0.815670i \(-0.303632\pi\)
0.578517 + 0.815670i \(0.303632\pi\)
\(572\) 0 0
\(573\) −134.833 + 134.833i −0.235311 + 0.235311i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −421.547 421.547i −0.730584 0.730584i 0.240151 0.970736i \(-0.422803\pi\)
−0.970736 + 0.240151i \(0.922803\pi\)
\(578\) 0 0
\(579\) 652.757i 1.12739i
\(580\) 0 0
\(581\) 569.878 0.980856
\(582\) 0 0
\(583\) −5.27142 + 5.27142i −0.00904188 + 0.00904188i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −441.514 441.514i −0.752154 0.752154i 0.222727 0.974881i \(-0.428504\pi\)
−0.974881 + 0.222727i \(0.928504\pi\)
\(588\) 0 0
\(589\) 168.210i 0.285586i
\(590\) 0 0
\(591\) −96.6061 −0.163462
\(592\) 0 0
\(593\) −171.303 + 171.303i −0.288875 + 0.288875i −0.836635 0.547760i \(-0.815481\pi\)
0.547760 + 0.836635i \(0.315481\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −113.790 113.790i −0.190603 0.190603i
\(598\) 0 0
\(599\) 64.1816i 0.107148i −0.998564 0.0535740i \(-0.982939\pi\)
0.998564 0.0535740i \(-0.0170613\pi\)
\(600\) 0 0
\(601\) 469.545 0.781273 0.390636 0.920545i \(-0.372255\pi\)
0.390636 + 0.920545i \(0.372255\pi\)
\(602\) 0 0
\(603\) −15.0681 + 15.0681i −0.0249886 + 0.0249886i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −121.930 121.930i −0.200872 0.200872i 0.599501 0.800374i \(-0.295366\pi\)
−0.800374 + 0.599501i \(0.795366\pi\)
\(608\) 0 0
\(609\) 366.272i 0.601433i
\(610\) 0 0
\(611\) −1819.63 −2.97813
\(612\) 0 0
\(613\) 542.252 542.252i 0.884587 0.884587i −0.109409 0.993997i \(-0.534896\pi\)
0.993997 + 0.109409i \(0.0348960\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 117.364 + 117.364i 0.190218 + 0.190218i 0.795790 0.605572i \(-0.207056\pi\)
−0.605572 + 0.795790i \(0.707056\pi\)
\(618\) 0 0
\(619\) 295.697i 0.477701i 0.971056 + 0.238851i \(0.0767706\pi\)
−0.971056 + 0.238851i \(0.923229\pi\)
\(620\) 0 0
\(621\) 206.091 0.331869
\(622\) 0 0
\(623\) 299.060 299.060i 0.480032 0.480032i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −45.8342 45.8342i −0.0731008 0.0731008i
\(628\) 0 0
\(629\) 333.576i 0.530327i
\(630\) 0 0
\(631\) −897.454 −1.42227 −0.711136 0.703054i \(-0.751819\pi\)
−0.711136 + 0.703054i \(0.751819\pi\)
\(632\) 0 0
\(633\) 359.184 359.184i 0.567431 0.567431i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 868.257 + 868.257i 1.36304 + 1.36304i
\(638\) 0 0
\(639\) 116.091i 0.181676i
\(640\) 0 0
\(641\) −226.120 −0.352762 −0.176381 0.984322i \(-0.556439\pi\)
−0.176381 + 0.984322i \(0.556439\pi\)
\(642\) 0 0
\(643\) 472.454 472.454i 0.734765 0.734765i −0.236794 0.971560i \(-0.576097\pi\)
0.971560 + 0.236794i \(0.0760968\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −583.151 583.151i −0.901315 0.901315i 0.0942347 0.995550i \(-0.469960\pi\)
−0.995550 + 0.0942347i \(0.969960\pi\)
\(648\) 0 0
\(649\) 260.908i 0.402016i
\(650\) 0 0
\(651\) 691.788 1.06265
\(652\) 0 0
\(653\) −36.1975 + 36.1975i −0.0554325 + 0.0554325i −0.734280 0.678847i \(-0.762480\pi\)
0.678847 + 0.734280i \(0.262480\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −138.879 138.879i −0.211383 0.211383i
\(658\) 0 0
\(659\) 615.787i 0.934426i 0.884145 + 0.467213i \(0.154742\pi\)
−0.884145 + 0.467213i \(0.845258\pi\)
\(660\) 0 0
\(661\) −426.849 −0.645762 −0.322881 0.946439i \(-0.604651\pi\)
−0.322881 + 0.946439i \(0.604651\pi\)
\(662\) 0 0
\(663\) 512.499 512.499i 0.773001 0.773001i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −580.454 580.454i −0.870246 0.870246i
\(668\) 0 0
\(669\) 183.272i 0.273950i
\(670\) 0 0
\(671\) 606.940 0.904530
\(672\) 0 0
\(673\) 350.474 350.474i 0.520764 0.520764i −0.397038 0.917802i \(-0.629962\pi\)
0.917802 + 0.397038i \(0.129962\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 208.985 + 208.985i 0.308693 + 0.308693i 0.844402 0.535709i \(-0.179956\pi\)
−0.535709 + 0.844402i \(0.679956\pi\)
\(678\) 0 0
\(679\) 782.060i 1.15178i
\(680\) 0 0
\(681\) 115.423 0.169491
\(682\) 0 0
\(683\) 701.271 701.271i 1.02675 1.02675i 0.0271195 0.999632i \(-0.491367\pi\)
0.999632 0.0271195i \(-0.00863347\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −336.805 336.805i −0.490254 0.490254i
\(688\) 0 0
\(689\) 19.0010i 0.0275777i
\(690\) 0 0
\(691\) 132.910 0.192345 0.0961724 0.995365i \(-0.469340\pi\)
0.0961724 + 0.995365i \(0.469340\pi\)
\(692\) 0 0
\(693\) 188.499 188.499i 0.272005 0.272005i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −836.908 836.908i −1.20073 1.20073i
\(698\) 0 0
\(699\) 459.909i 0.657953i
\(700\) 0 0
\(701\) 158.758 0.226474 0.113237 0.993568i \(-0.463878\pi\)
0.113237 + 0.993568i \(0.463878\pi\)
\(702\) 0 0
\(703\) 53.7663 53.7663i 0.0764812 0.0764812i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −309.787 309.787i −0.438171 0.438171i
\(708\) 0 0
\(709\) 674.514i 0.951360i 0.879618 + 0.475680i \(0.157798\pi\)
−0.879618 + 0.475680i \(0.842202\pi\)
\(710\) 0 0
\(711\) 94.1816 0.132464
\(712\) 0 0
\(713\) −1096.32 + 1096.32i −1.53761 + 1.53761i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −21.6367 21.6367i −0.0301768 0.0301768i
\(718\) 0 0
\(719\) 103.485i 0.143929i 0.997407 + 0.0719643i \(0.0229268\pi\)
−0.997407 + 0.0719643i \(0.977073\pi\)
\(720\) 0 0
\(721\) 881.271 1.22229
\(722\) 0 0
\(723\) −204.532 + 204.532i −0.282894 + 0.282894i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 658.392 + 658.392i 0.905628 + 0.905628i 0.995916 0.0902877i \(-0.0287787\pi\)
−0.0902877 + 0.995916i \(0.528779\pi\)
\(728\) 0 0
\(729\) 27.0000i 0.0370370i
\(730\) 0 0
\(731\) 317.060 0.433735
\(732\) 0 0
\(733\) −165.970 + 165.970i −0.226426 + 0.226426i −0.811198 0.584772i \(-0.801184\pi\)
0.584772 + 0.811198i \(0.301184\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −43.6821 43.6821i −0.0592702 0.0592702i
\(738\) 0 0
\(739\) 122.545i 0.165825i −0.996557 0.0829126i \(-0.973578\pi\)
0.996557 0.0829126i \(-0.0264222\pi\)
\(740\) 0 0
\(741\) −165.211 −0.222957
\(742\) 0 0
\(743\) −541.485 + 541.485i −0.728782 + 0.728782i −0.970377 0.241595i \(-0.922329\pi\)
0.241595 + 0.970377i \(0.422329\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −118.318 118.318i −0.158391 0.158391i
\(748\) 0 0
\(749\) 663.364i 0.885667i
\(750\) 0 0
\(751\) 495.212 0.659404 0.329702 0.944085i \(-0.393052\pi\)
0.329702 + 0.944085i \(0.393052\pi\)
\(752\) 0 0
\(753\) 32.3633 32.3633i 0.0429791 0.0429791i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 565.547 + 565.547i 0.747090 + 0.747090i 0.973932 0.226842i \(-0.0728400\pi\)
−0.226842 + 0.973932i \(0.572840\pi\)
\(758\) 0 0
\(759\) 597.453i 0.787158i
\(760\) 0 0
\(761\) −737.271 −0.968819 −0.484410 0.874841i \(-0.660966\pi\)
−0.484410 + 0.874841i \(0.660966\pi\)
\(762\) 0 0
\(763\) −1047.59 + 1047.59i −1.37299 + 1.37299i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 470.227 + 470.227i 0.613073 + 0.613073i
\(768\) 0 0
\(769\) 1014.27i 1.31895i 0.751727 + 0.659474i \(0.229221\pi\)
−0.751727 + 0.659474i \(0.770779\pi\)
\(770\) 0 0
\(771\) 591.514 0.767204
\(772\) 0 0
\(773\) 6.39491 6.39491i 0.00827284 0.00827284i −0.702958 0.711231i \(-0.748138\pi\)
0.711231 + 0.702958i \(0.248138\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 221.121 + 221.121i 0.284584 + 0.284584i
\(778\) 0 0
\(779\) 269.789i 0.346327i
\(780\) 0 0
\(781\) 336.545 0.430915
\(782\) 0 0
\(783\) 76.0454 76.0454i 0.0971206 0.0971206i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −565.806 565.806i −0.718940 0.718940i 0.249448 0.968388i \(-0.419751\pi\)
−0.968388 + 0.249448i \(0.919751\pi\)
\(788\) 0 0
\(789\) 348.272i 0.441410i
\(790\) 0 0
\(791\) 1889.82 2.38915
\(792\) 0 0
\(793\) 1093.87 1093.87i 1.37941 1.37941i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −352.182 352.182i −0.441884 0.441884i 0.450761 0.892645i \(-0.351153\pi\)
−0.892645 + 0.450761i \(0.851153\pi\)
\(798\) 0 0
\(799\) 1549.63i 1.93947i
\(800\) 0 0
\(801\) −124.182 −0.155033
\(802\) 0 0
\(803\) 402.606 402.606i 0.501377 0.501377i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 638.833 + 638.833i 0.791615 + 0.791615i
\(808\) 0 0
\(809\) 563.728i 0.696820i 0.937342 + 0.348410i \(0.113278\pi\)
−0.937342 + 0.348410i \(0.886722\pi\)
\(810\) 0 0
\(811\) 205.091 0.252886 0.126443 0.991974i \(-0.459644\pi\)
0.126443 + 0.991974i \(0.459644\pi\)
\(812\) 0 0
\(813\) 48.0250 48.0250i 0.0590713 0.0590713i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −51.1043 51.1043i −0.0625512 0.0625512i
\(818\) 0 0
\(819\) 679.454i 0.829614i
\(820\) 0 0
\(821\) 867.576 1.05673 0.528365 0.849017i \(-0.322805\pi\)
0.528365 + 0.849017i \(0.322805\pi\)
\(822\) 0 0
\(823\) 80.8094 80.8094i 0.0981889 0.0981889i −0.656306 0.754495i \(-0.727882\pi\)
0.754495 + 0.656306i \(0.227882\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −935.939 935.939i −1.13173 1.13173i −0.989890 0.141838i \(-0.954699\pi\)
−0.141838 0.989890i \(-0.545301\pi\)
\(828\) 0 0
\(829\) 1097.39i 1.32375i 0.749613 + 0.661877i \(0.230240\pi\)
−0.749613 + 0.661877i \(0.769760\pi\)
\(830\) 0 0
\(831\) −170.455 −0.205120
\(832\) 0 0
\(833\) 739.423 739.423i 0.887663 0.887663i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −143.629 143.629i −0.171600 0.171600i
\(838\) 0 0
\(839\) 1047.18i 1.24813i −0.781373 0.624065i \(-0.785480\pi\)
0.781373 0.624065i \(-0.214520\pi\)
\(840\) 0 0
\(841\) 412.637 0.490650
\(842\) 0 0
\(843\) 636.681 636.681i 0.755256 0.755256i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −327.738 327.738i −0.386940 0.386940i
\(848\) 0 0
\(849\) 375.817i 0.442659i
\(850\) 0 0
\(851\) −700.849 −0.823559
\(852\) 0 0
\(853\) 902.612 902.612i 1.05816 1.05816i 0.0599610 0.998201i \(-0.480902\pi\)
0.998201 0.0599610i \(-0.0190976\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −928.454 928.454i −1.08338 1.08338i −0.996192 0.0871848i \(-0.972213\pi\)
−0.0871848 0.996192i \(-0.527787\pi\)
\(858\) 0 0
\(859\) 581.151i 0.676544i −0.941048 0.338272i \(-0.890158\pi\)
0.941048 0.338272i \(-0.109842\pi\)
\(860\) 0 0
\(861\) −1109.54 −1.28867
\(862\) 0 0
\(863\) −317.271 + 317.271i −0.367638 + 0.367638i −0.866615 0.498977i \(-0.833709\pi\)
0.498977 + 0.866615i \(0.333709\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −82.5028 82.5028i −0.0951589 0.0951589i
\(868\) 0 0
\(869\) 273.031i 0.314189i
\(870\) 0 0
\(871\) −157.454 −0.180774
\(872\) 0 0
\(873\) −162.371 + 162.371i −0.185992 + 0.185992i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1109.75 1109.75i −1.26540 1.26540i −0.948439 0.316958i \(-0.897338\pi\)
−0.316958 0.948439i \(-0.602662\pi\)
\(878\) 0 0
\(879\) 48.0612i 0.0546772i
\(880\) 0 0
\(881\) 119.455 0.135590 0.0677952 0.997699i \(-0.478404\pi\)
0.0677952 + 0.997699i \(0.478404\pi\)
\(882\) 0 0
\(883\) 252.619 252.619i 0.286091 0.286091i −0.549441 0.835532i \(-0.685159\pi\)
0.835532 + 0.549441i \(0.185159\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 516.347 + 516.347i 0.582128 + 0.582128i 0.935488 0.353360i \(-0.114961\pi\)
−0.353360 + 0.935488i \(0.614961\pi\)
\(888\) 0 0
\(889\) 307.818i 0.346252i
\(890\) 0 0
\(891\) −78.2724 −0.0878479
\(892\) 0 0
\(893\) 249.773 249.773i 0.279701 0.279701i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 1076.77 + 1076.77i 1.20041 + 1.20041i
\(898\) 0 0
\(899\) 809.060i 0.899956i
\(900\) 0 0
\(901\) 16.1816 0.0179596
\(902\) 0 0
\(903\) 210.174 210.174i 0.232751 0.232751i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 983.160 + 983.160i 1.08397 + 1.08397i 0.996135 + 0.0878342i \(0.0279946\pi\)
0.0878342 + 0.996135i \(0.472005\pi\)
\(908\) 0 0
\(909\) 128.636i 0.141513i
\(910\) 0 0
\(911\) 1698.00 1.86389 0.931943 0.362605i \(-0.118113\pi\)
0.931943 + 0.362605i \(0.118113\pi\)
\(912\) 0 0
\(913\) 343.001 343.001i 0.375686 0.375686i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −782.241 782.241i −0.853043 0.853043i
\(918\) 0 0
\(919\) 581.272i 0.632505i −0.948675 0.316253i \(-0.897575\pi\)
0.948675 0.316253i \(-0.102425\pi\)
\(920\) 0 0
\(921\) 26.7276 0.0290201
\(922\) 0 0
\(923\) 606.545 606.545i 0.657145 0.657145i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −182.969 182.969i −0.197378 0.197378i
\(928\) 0 0
\(929\) 1328.57i 1.43011i −0.699067 0.715056i \(-0.746401\pi\)
0.699067 0.715056i \(-0.253599\pi\)
\(930\) 0 0
\(931\) −238.363 −0.256029
\(932\) 0 0
\(933\) 79.3485 79.3485i 0.0850466 0.0850466i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 626.487 + 626.487i 0.668609 + 0.668609i 0.957394 0.288785i \(-0.0932512\pi\)
−0.288785 + 0.957394i \(0.593251\pi\)
\(938\) 0 0
\(939\) 62.3326i 0.0663819i
\(940\) 0 0
\(941\) 762.422 0.810226 0.405113 0.914267i \(-0.367232\pi\)
0.405113 + 0.914267i \(0.367232\pi\)
\(942\) 0 0
\(943\) 1758.36 1758.36i 1.86465 1.86465i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −614.983 614.983i −0.649401 0.649401i 0.303447 0.952848i \(-0.401862\pi\)
−0.952848 + 0.303447i \(0.901862\pi\)
\(948\) 0 0
\(949\) 1451.21i 1.52920i
\(950\) 0 0
\(951\) −1055.88 −1.11028
\(952\) 0 0
\(953\) 328.454 328.454i 0.344653 0.344653i −0.513460 0.858113i \(-0.671637\pi\)
0.858113 + 0.513460i \(0.171637\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 220.454 + 220.454i 0.230360 + 0.230360i
\(958\) 0 0
\(959\) 1107.58i 1.15493i
\(960\) 0 0
\(961\) 567.092 0.590106
\(962\) 0 0
\(963\) 137.728 137.728i 0.143019 0.143019i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 172.009 + 172.009i 0.177879 + 0.177879i 0.790431 0.612551i \(-0.209857\pi\)
−0.612551 + 0.790431i \(0.709857\pi\)
\(968\) 0 0
\(969\) 140.697i 0.145198i
\(970\) 0 0
\(971\) 1273.82 1.31186 0.655931 0.754821i \(-0.272276\pi\)
0.655931 + 0.754821i \(0.272276\pi\)
\(972\) 0 0
\(973\) −1103.85 + 1103.85i −1.13448 + 1.13448i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1013.86 + 1013.86i 1.03773 + 1.03773i 0.999260 + 0.0384719i \(0.0122490\pi\)
0.0384719 + 0.999260i \(0.487751\pi\)
\(978\) 0 0
\(979\) 360.000i 0.367722i
\(980\) 0 0
\(981\) 435.000 0.443425
\(982\) 0 0
\(983\) −856.332 + 856.332i −0.871141 + 0.871141i −0.992597 0.121456i \(-0.961244\pi\)
0.121456 + 0.992597i \(0.461244\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 1027.23 + 1027.23i 1.04076 + 1.04076i
\(988\) 0 0
\(989\) 666.150i 0.673559i
\(990\) 0 0
\(991\) −688.605 −0.694859 −0.347429 0.937706i \(-0.612945\pi\)
−0.347429 + 0.937706i \(0.612945\pi\)
\(992\) 0 0
\(993\) 362.449 362.449i 0.365005 0.365005i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1260.82 + 1260.82i 1.26461 + 1.26461i 0.948833 + 0.315778i \(0.102266\pi\)
0.315778 + 0.948833i \(0.397734\pi\)
\(998\) 0 0
\(999\) 91.8184i 0.0919103i
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.p.1057.1 4
4.3 odd 2 150.3.f.a.7.2 4
5.2 odd 4 1200.3.bg.a.193.2 4
5.3 odd 4 inner 1200.3.bg.p.193.1 4
5.4 even 2 1200.3.bg.a.1057.2 4
12.11 even 2 450.3.g.h.307.1 4
20.3 even 4 150.3.f.a.43.2 yes 4
20.7 even 4 150.3.f.c.43.1 yes 4
20.19 odd 2 150.3.f.c.7.1 yes 4
60.23 odd 4 450.3.g.h.343.1 4
60.47 odd 4 450.3.g.g.343.2 4
60.59 even 2 450.3.g.g.307.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.3.f.a.7.2 4 4.3 odd 2
150.3.f.a.43.2 yes 4 20.3 even 4
150.3.f.c.7.1 yes 4 20.19 odd 2
150.3.f.c.43.1 yes 4 20.7 even 4
450.3.g.g.307.2 4 60.59 even 2
450.3.g.g.343.2 4 60.47 odd 4
450.3.g.h.307.1 4 12.11 even 2
450.3.g.h.343.1 4 60.23 odd 4
1200.3.bg.a.193.2 4 5.2 odd 4
1200.3.bg.a.1057.2 4 5.4 even 2
1200.3.bg.p.193.1 4 5.3 odd 4 inner
1200.3.bg.p.1057.1 4 1.1 even 1 trivial