Properties

Label 1200.3.bg.m.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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 600)
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.m.193.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.22474 + 1.22474i) q^{3} +(6.44949 + 6.44949i) q^{7} -3.00000i q^{9} +O(q^{10})\) \(q+(-1.22474 + 1.22474i) q^{3} +(6.44949 + 6.44949i) q^{7} -3.00000i q^{9} +11.7980 q^{11} +(2.44949 - 2.44949i) q^{13} +(0.898979 + 0.898979i) q^{17} -33.5959i q^{19} -15.7980 q^{21} +(21.7980 - 21.7980i) q^{23} +(3.67423 + 3.67423i) q^{27} +3.59592i q^{29} +25.1918 q^{31} +(-14.4495 + 14.4495i) q^{33} +(9.14643 + 9.14643i) q^{37} +6.00000i q^{39} -60.7878 q^{41} +(-22.2020 + 22.2020i) q^{43} +(4.49490 + 4.49490i) q^{47} +34.1918i q^{49} -2.20204 q^{51} +(52.0908 - 52.0908i) q^{53} +(41.1464 + 41.1464i) q^{57} -86.9898i q^{59} -108.384 q^{61} +(19.3485 - 19.3485i) q^{63} +(77.8888 + 77.8888i) q^{67} +53.3939i q^{69} +66.7878 q^{71} +(78.6969 - 78.6969i) q^{73} +(76.0908 + 76.0908i) q^{77} +66.0000i q^{79} -9.00000 q^{81} +(84.0908 - 84.0908i) q^{83} +(-4.40408 - 4.40408i) q^{87} +9.59592i q^{89} +31.5959 q^{91} +(-30.8536 + 30.8536i) q^{93} +(53.0806 + 53.0806i) q^{97} -35.3939i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{7} + 8 q^{11} - 16 q^{17} - 24 q^{21} + 48 q^{23} - 56 q^{31} - 48 q^{33} - 32 q^{37} - 8 q^{41} - 128 q^{43} - 80 q^{47} - 48 q^{51} + 32 q^{53} + 96 q^{57} - 120 q^{61} + 48 q^{63} + 96 q^{67} + 32 q^{71} + 256 q^{73} + 128 q^{77} - 36 q^{81} + 160 q^{83} - 96 q^{87} + 48 q^{91} - 192 q^{93} - 160 q^{97}+O(q^{100}) \) 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
\(7\) 6.44949 + 6.44949i 0.921356 + 0.921356i 0.997125 0.0757697i \(-0.0241414\pi\)
−0.0757697 + 0.997125i \(0.524141\pi\)
\(8\) 0 0
\(9\) 3.00000i 0.333333i
\(10\) 0 0
\(11\) 11.7980 1.07254 0.536271 0.844046i \(-0.319833\pi\)
0.536271 + 0.844046i \(0.319833\pi\)
\(12\) 0 0
\(13\) 2.44949 2.44949i 0.188422 0.188422i −0.606591 0.795014i \(-0.707464\pi\)
0.795014 + 0.606591i \(0.207464\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.898979 + 0.898979i 0.0528811 + 0.0528811i 0.733053 0.680172i \(-0.238095\pi\)
−0.680172 + 0.733053i \(0.738095\pi\)
\(18\) 0 0
\(19\) 33.5959i 1.76821i −0.467292 0.884103i \(-0.654770\pi\)
0.467292 0.884103i \(-0.345230\pi\)
\(20\) 0 0
\(21\) −15.7980 −0.752284
\(22\) 0 0
\(23\) 21.7980 21.7980i 0.947737 0.947737i −0.0509632 0.998701i \(-0.516229\pi\)
0.998701 + 0.0509632i \(0.0162291\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.67423 + 3.67423i 0.136083 + 0.136083i
\(28\) 0 0
\(29\) 3.59592i 0.123997i 0.998076 + 0.0619986i \(0.0197474\pi\)
−0.998076 + 0.0619986i \(0.980253\pi\)
\(30\) 0 0
\(31\) 25.1918 0.812640 0.406320 0.913731i \(-0.366812\pi\)
0.406320 + 0.913731i \(0.366812\pi\)
\(32\) 0 0
\(33\) −14.4495 + 14.4495i −0.437863 + 0.437863i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.14643 + 9.14643i 0.247201 + 0.247201i 0.819821 0.572620i \(-0.194073\pi\)
−0.572620 + 0.819821i \(0.694073\pi\)
\(38\) 0 0
\(39\) 6.00000i 0.153846i
\(40\) 0 0
\(41\) −60.7878 −1.48263 −0.741314 0.671158i \(-0.765797\pi\)
−0.741314 + 0.671158i \(0.765797\pi\)
\(42\) 0 0
\(43\) −22.2020 + 22.2020i −0.516327 + 0.516327i −0.916458 0.400131i \(-0.868965\pi\)
0.400131 + 0.916458i \(0.368965\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.49490 + 4.49490i 0.0956361 + 0.0956361i 0.753306 0.657670i \(-0.228458\pi\)
−0.657670 + 0.753306i \(0.728458\pi\)
\(48\) 0 0
\(49\) 34.1918i 0.697793i
\(50\) 0 0
\(51\) −2.20204 −0.0431773
\(52\) 0 0
\(53\) 52.0908 52.0908i 0.982846 0.982846i −0.0170098 0.999855i \(-0.505415\pi\)
0.999855 + 0.0170098i \(0.00541463\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 41.1464 + 41.1464i 0.721867 + 0.721867i
\(58\) 0 0
\(59\) 86.9898i 1.47440i −0.675673 0.737202i \(-0.736147\pi\)
0.675673 0.737202i \(-0.263853\pi\)
\(60\) 0 0
\(61\) −108.384 −1.77678 −0.888391 0.459088i \(-0.848176\pi\)
−0.888391 + 0.459088i \(0.848176\pi\)
\(62\) 0 0
\(63\) 19.3485 19.3485i 0.307119 0.307119i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 77.8888 + 77.8888i 1.16252 + 1.16252i 0.983922 + 0.178597i \(0.0571557\pi\)
0.178597 + 0.983922i \(0.442844\pi\)
\(68\) 0 0
\(69\) 53.3939i 0.773824i
\(70\) 0 0
\(71\) 66.7878 0.940673 0.470336 0.882487i \(-0.344133\pi\)
0.470336 + 0.882487i \(0.344133\pi\)
\(72\) 0 0
\(73\) 78.6969 78.6969i 1.07804 1.07804i 0.0813551 0.996685i \(-0.474075\pi\)
0.996685 0.0813551i \(-0.0259248\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 76.0908 + 76.0908i 0.988192 + 0.988192i
\(78\) 0 0
\(79\) 66.0000i 0.835443i 0.908575 + 0.417722i \(0.137171\pi\)
−0.908575 + 0.417722i \(0.862829\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) 0 0
\(83\) 84.0908 84.0908i 1.01314 1.01314i 0.0132299 0.999912i \(-0.495789\pi\)
0.999912 0.0132299i \(-0.00421132\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −4.40408 4.40408i −0.0506216 0.0506216i
\(88\) 0 0
\(89\) 9.59592i 0.107819i 0.998546 + 0.0539097i \(0.0171683\pi\)
−0.998546 + 0.0539097i \(0.982832\pi\)
\(90\) 0 0
\(91\) 31.5959 0.347208
\(92\) 0 0
\(93\) −30.8536 + 30.8536i −0.331759 + 0.331759i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 53.0806 + 53.0806i 0.547223 + 0.547223i 0.925637 0.378414i \(-0.123530\pi\)
−0.378414 + 0.925637i \(0.623530\pi\)
\(98\) 0 0
\(99\) 35.3939i 0.357514i
\(100\) 0 0
\(101\) 155.192 1.53655 0.768276 0.640118i \(-0.221115\pi\)
0.768276 + 0.640118i \(0.221115\pi\)
\(102\) 0 0
\(103\) −42.3587 + 42.3587i −0.411249 + 0.411249i −0.882174 0.470924i \(-0.843920\pi\)
0.470924 + 0.882174i \(0.343920\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 29.8888 + 29.8888i 0.279334 + 0.279334i 0.832843 0.553509i \(-0.186712\pi\)
−0.553509 + 0.832843i \(0.686712\pi\)
\(108\) 0 0
\(109\) 17.1918i 0.157723i −0.996886 0.0788616i \(-0.974871\pi\)
0.996886 0.0788616i \(-0.0251285\pi\)
\(110\) 0 0
\(111\) −22.4041 −0.201839
\(112\) 0 0
\(113\) −134.293 + 134.293i −1.18843 + 1.18843i −0.210932 + 0.977501i \(0.567650\pi\)
−0.977501 + 0.210932i \(0.932350\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −7.34847 7.34847i −0.0628074 0.0628074i
\(118\) 0 0
\(119\) 11.5959i 0.0974447i
\(120\) 0 0
\(121\) 18.1918 0.150346
\(122\) 0 0
\(123\) 74.4495 74.4495i 0.605280 0.605280i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 153.237 + 153.237i 1.20659 + 1.20659i 0.972124 + 0.234469i \(0.0753351\pi\)
0.234469 + 0.972124i \(0.424665\pi\)
\(128\) 0 0
\(129\) 54.3837i 0.421579i
\(130\) 0 0
\(131\) 196.606 1.50081 0.750405 0.660978i \(-0.229858\pi\)
0.750405 + 0.660978i \(0.229858\pi\)
\(132\) 0 0
\(133\) 216.677 216.677i 1.62915 1.62915i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 81.0806 + 81.0806i 0.591829 + 0.591829i 0.938125 0.346296i \(-0.112561\pi\)
−0.346296 + 0.938125i \(0.612561\pi\)
\(138\) 0 0
\(139\) 37.1918i 0.267567i 0.991011 + 0.133784i \(0.0427127\pi\)
−0.991011 + 0.133784i \(0.957287\pi\)
\(140\) 0 0
\(141\) −11.0102 −0.0780866
\(142\) 0 0
\(143\) 28.8990 28.8990i 0.202091 0.202091i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −41.8763 41.8763i −0.284873 0.284873i
\(148\) 0 0
\(149\) 86.7878i 0.582468i −0.956652 0.291234i \(-0.905934\pi\)
0.956652 0.291234i \(-0.0940658\pi\)
\(150\) 0 0
\(151\) −266.767 −1.76667 −0.883336 0.468741i \(-0.844708\pi\)
−0.883336 + 0.468741i \(0.844708\pi\)
\(152\) 0 0
\(153\) 2.69694 2.69694i 0.0176270 0.0176270i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −203.439 203.439i −1.29579 1.29579i −0.931147 0.364645i \(-0.881190\pi\)
−0.364645 0.931147i \(-0.618810\pi\)
\(158\) 0 0
\(159\) 127.596i 0.802490i
\(160\) 0 0
\(161\) 281.171 1.74641
\(162\) 0 0
\(163\) −29.2122 + 29.2122i −0.179216 + 0.179216i −0.791014 0.611798i \(-0.790447\pi\)
0.611798 + 0.791014i \(0.290447\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 116.182 + 116.182i 0.695698 + 0.695698i 0.963480 0.267781i \(-0.0862905\pi\)
−0.267781 + 0.963480i \(0.586290\pi\)
\(168\) 0 0
\(169\) 157.000i 0.928994i
\(170\) 0 0
\(171\) −100.788 −0.589402
\(172\) 0 0
\(173\) 60.2724 60.2724i 0.348396 0.348396i −0.511116 0.859512i \(-0.670768\pi\)
0.859512 + 0.511116i \(0.170768\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 106.540 + 106.540i 0.601923 + 0.601923i
\(178\) 0 0
\(179\) 176.161i 0.984141i 0.870555 + 0.492070i \(0.163760\pi\)
−0.870555 + 0.492070i \(0.836240\pi\)
\(180\) 0 0
\(181\) 161.959 0.894802 0.447401 0.894333i \(-0.352350\pi\)
0.447401 + 0.894333i \(0.352350\pi\)
\(182\) 0 0
\(183\) 132.742 132.742i 0.725368 0.725368i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 10.6061 + 10.6061i 0.0567172 + 0.0567172i
\(188\) 0 0
\(189\) 47.3939i 0.250761i
\(190\) 0 0
\(191\) −250.747 −1.31281 −0.656406 0.754408i \(-0.727924\pi\)
−0.656406 + 0.754408i \(0.727924\pi\)
\(192\) 0 0
\(193\) 8.62653 8.62653i 0.0446971 0.0446971i −0.684405 0.729102i \(-0.739938\pi\)
0.729102 + 0.684405i \(0.239938\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 182.202 + 182.202i 0.924883 + 0.924883i 0.997369 0.0724860i \(-0.0230933\pi\)
−0.0724860 + 0.997369i \(0.523093\pi\)
\(198\) 0 0
\(199\) 81.1918i 0.407999i 0.978971 + 0.204000i \(0.0653941\pi\)
−0.978971 + 0.204000i \(0.934606\pi\)
\(200\) 0 0
\(201\) −190.788 −0.949193
\(202\) 0 0
\(203\) −23.1918 + 23.1918i −0.114245 + 0.114245i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −65.3939 65.3939i −0.315912 0.315912i
\(208\) 0 0
\(209\) 396.363i 1.89647i
\(210\) 0 0
\(211\) −199.576 −0.945855 −0.472928 0.881101i \(-0.656803\pi\)
−0.472928 + 0.881101i \(0.656803\pi\)
\(212\) 0 0
\(213\) −81.7980 + 81.7980i −0.384028 + 0.384028i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 162.474 + 162.474i 0.748730 + 0.748730i
\(218\) 0 0
\(219\) 192.767i 0.880216i
\(220\) 0 0
\(221\) 4.40408 0.0199280
\(222\) 0 0
\(223\) −134.944 + 134.944i −0.605132 + 0.605132i −0.941670 0.336538i \(-0.890744\pi\)
0.336538 + 0.941670i \(0.390744\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −60.8990 60.8990i −0.268277 0.268277i 0.560128 0.828406i \(-0.310752\pi\)
−0.828406 + 0.560128i \(0.810752\pi\)
\(228\) 0 0
\(229\) 138.767i 0.605971i −0.952995 0.302985i \(-0.902017\pi\)
0.952995 0.302985i \(-0.0979834\pi\)
\(230\) 0 0
\(231\) −186.384 −0.806856
\(232\) 0 0
\(233\) −254.656 + 254.656i −1.09294 + 1.09294i −0.0977319 + 0.995213i \(0.531159\pi\)
−0.995213 + 0.0977319i \(0.968841\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −80.8332 80.8332i −0.341068 0.341068i
\(238\) 0 0
\(239\) 422.788i 1.76899i −0.466553 0.884493i \(-0.654504\pi\)
0.466553 0.884493i \(-0.345496\pi\)
\(240\) 0 0
\(241\) 10.7673 0.0446778 0.0223389 0.999750i \(-0.492889\pi\)
0.0223389 + 0.999750i \(0.492889\pi\)
\(242\) 0 0
\(243\) 11.0227 11.0227i 0.0453609 0.0453609i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −82.2929 82.2929i −0.333169 0.333169i
\(248\) 0 0
\(249\) 205.980i 0.827227i
\(250\) 0 0
\(251\) 177.818 0.708440 0.354220 0.935162i \(-0.384746\pi\)
0.354220 + 0.935162i \(0.384746\pi\)
\(252\) 0 0
\(253\) 257.171 257.171i 1.01649 1.01649i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −142.293 142.293i −0.553669 0.553669i 0.373829 0.927498i \(-0.378045\pi\)
−0.927498 + 0.373829i \(0.878045\pi\)
\(258\) 0 0
\(259\) 117.980i 0.455520i
\(260\) 0 0
\(261\) 10.7878 0.0413324
\(262\) 0 0
\(263\) 36.0000 36.0000i 0.136882 0.136882i −0.635346 0.772228i \(-0.719142\pi\)
0.772228 + 0.635346i \(0.219142\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −11.7526 11.7526i −0.0440170 0.0440170i
\(268\) 0 0
\(269\) 38.8286i 0.144344i −0.997392 0.0721721i \(-0.977007\pi\)
0.997392 0.0721721i \(-0.0229931\pi\)
\(270\) 0 0
\(271\) 11.5755 0.0427141 0.0213570 0.999772i \(-0.493201\pi\)
0.0213570 + 0.999772i \(0.493201\pi\)
\(272\) 0 0
\(273\) −38.6969 + 38.6969i −0.141747 + 0.141747i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −282.499 282.499i −1.01985 1.01985i −0.999799 0.0200549i \(-0.993616\pi\)
−0.0200549 0.999799i \(-0.506384\pi\)
\(278\) 0 0
\(279\) 75.5755i 0.270880i
\(280\) 0 0
\(281\) 432.384 1.53873 0.769366 0.638808i \(-0.220572\pi\)
0.769366 + 0.638808i \(0.220572\pi\)
\(282\) 0 0
\(283\) −223.868 + 223.868i −0.791054 + 0.791054i −0.981666 0.190611i \(-0.938953\pi\)
0.190611 + 0.981666i \(0.438953\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −392.050 392.050i −1.36603 1.36603i
\(288\) 0 0
\(289\) 287.384i 0.994407i
\(290\) 0 0
\(291\) −130.020 −0.446806
\(292\) 0 0
\(293\) 196.767 196.767i 0.671561 0.671561i −0.286515 0.958076i \(-0.592497\pi\)
0.958076 + 0.286515i \(0.0924968\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 43.3485 + 43.3485i 0.145954 + 0.145954i
\(298\) 0 0
\(299\) 106.788i 0.357150i
\(300\) 0 0
\(301\) −286.384 −0.951441
\(302\) 0 0
\(303\) −190.070 + 190.070i −0.627295 + 0.627295i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 141.707 + 141.707i 0.461587 + 0.461587i 0.899175 0.437589i \(-0.144167\pi\)
−0.437589 + 0.899175i \(0.644167\pi\)
\(308\) 0 0
\(309\) 103.757i 0.335784i
\(310\) 0 0
\(311\) 411.555 1.32333 0.661664 0.749800i \(-0.269850\pi\)
0.661664 + 0.749800i \(0.269850\pi\)
\(312\) 0 0
\(313\) −365.939 + 365.939i −1.16913 + 1.16913i −0.186720 + 0.982413i \(0.559786\pi\)
−0.982413 + 0.186720i \(0.940214\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −338.343 338.343i −1.06733 1.06733i −0.997564 0.0697641i \(-0.977775\pi\)
−0.0697641 0.997564i \(-0.522225\pi\)
\(318\) 0 0
\(319\) 42.4245i 0.132992i
\(320\) 0 0
\(321\) −73.2122 −0.228076
\(322\) 0 0
\(323\) 30.2020 30.2020i 0.0935048 0.0935048i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 21.0556 + 21.0556i 0.0643903 + 0.0643903i
\(328\) 0 0
\(329\) 57.9796i 0.176230i
\(330\) 0 0
\(331\) 97.5551 0.294728 0.147364 0.989082i \(-0.452921\pi\)
0.147364 + 0.989082i \(0.452921\pi\)
\(332\) 0 0
\(333\) 27.4393 27.4393i 0.0824003 0.0824003i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −72.4949 72.4949i −0.215118 0.215118i 0.591319 0.806438i \(-0.298607\pi\)
−0.806438 + 0.591319i \(0.798607\pi\)
\(338\) 0 0
\(339\) 328.949i 0.970351i
\(340\) 0 0
\(341\) 297.212 0.871590
\(342\) 0 0
\(343\) 95.5051 95.5051i 0.278441 0.278441i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −40.8582 40.8582i −0.117747 0.117747i 0.645778 0.763525i \(-0.276533\pi\)
−0.763525 + 0.645778i \(0.776533\pi\)
\(348\) 0 0
\(349\) 332.343i 0.952272i 0.879372 + 0.476136i \(0.157963\pi\)
−0.879372 + 0.476136i \(0.842037\pi\)
\(350\) 0 0
\(351\) 18.0000 0.0512821
\(352\) 0 0
\(353\) −260.232 + 260.232i −0.737200 + 0.737200i −0.972035 0.234835i \(-0.924545\pi\)
0.234835 + 0.972035i \(0.424545\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −14.2020 14.2020i −0.0397816 0.0397816i
\(358\) 0 0
\(359\) 332.000i 0.924791i −0.886674 0.462396i \(-0.846990\pi\)
0.886674 0.462396i \(-0.153010\pi\)
\(360\) 0 0
\(361\) −767.686 −2.12655
\(362\) 0 0
\(363\) −22.2804 + 22.2804i −0.0613784 + 0.0613784i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −210.409 210.409i −0.573321 0.573321i 0.359734 0.933055i \(-0.382867\pi\)
−0.933055 + 0.359734i \(0.882867\pi\)
\(368\) 0 0
\(369\) 182.363i 0.494209i
\(370\) 0 0
\(371\) 671.918 1.81110
\(372\) 0 0
\(373\) −222.672 + 222.672i −0.596976 + 0.596976i −0.939507 0.342531i \(-0.888716\pi\)
0.342531 + 0.939507i \(0.388716\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.80816 + 8.80816i 0.0233638 + 0.0233638i
\(378\) 0 0
\(379\) 222.767i 0.587777i −0.955840 0.293888i \(-0.905051\pi\)
0.955840 0.293888i \(-0.0949494\pi\)
\(380\) 0 0
\(381\) −375.353 −0.985179
\(382\) 0 0
\(383\) 120.717 120.717i 0.315189 0.315189i −0.531727 0.846916i \(-0.678457\pi\)
0.846916 + 0.531727i \(0.178457\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 66.6061 + 66.6061i 0.172109 + 0.172109i
\(388\) 0 0
\(389\) 80.8082i 0.207733i −0.994591 0.103867i \(-0.966879\pi\)
0.994591 0.103867i \(-0.0331215\pi\)
\(390\) 0 0
\(391\) 39.1918 0.100235
\(392\) 0 0
\(393\) −240.792 + 240.792i −0.612703 + 0.612703i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 21.9954 + 21.9954i 0.0554041 + 0.0554041i 0.734266 0.678862i \(-0.237527\pi\)
−0.678862 + 0.734266i \(0.737527\pi\)
\(398\) 0 0
\(399\) 530.747i 1.33019i
\(400\) 0 0
\(401\) −267.131 −0.666161 −0.333081 0.942898i \(-0.608088\pi\)
−0.333081 + 0.942898i \(0.608088\pi\)
\(402\) 0 0
\(403\) 61.7071 61.7071i 0.153119 0.153119i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 107.909 + 107.909i 0.265133 + 0.265133i
\(408\) 0 0
\(409\) 510.767i 1.24882i 0.781097 + 0.624410i \(0.214661\pi\)
−0.781097 + 0.624410i \(0.785339\pi\)
\(410\) 0 0
\(411\) −198.606 −0.483227
\(412\) 0 0
\(413\) 561.040 561.040i 1.35845 1.35845i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −45.5505 45.5505i −0.109234 0.109234i
\(418\) 0 0
\(419\) 40.1204i 0.0957527i −0.998853 0.0478764i \(-0.984755\pi\)
0.998853 0.0478764i \(-0.0152454\pi\)
\(420\) 0 0
\(421\) −132.343 −0.314354 −0.157177 0.987570i \(-0.550239\pi\)
−0.157177 + 0.987570i \(0.550239\pi\)
\(422\) 0 0
\(423\) 13.4847 13.4847i 0.0318787 0.0318787i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −699.019 699.019i −1.63705 1.63705i
\(428\) 0 0
\(429\) 70.7878i 0.165006i
\(430\) 0 0
\(431\) 510.343 1.18409 0.592045 0.805905i \(-0.298321\pi\)
0.592045 + 0.805905i \(0.298321\pi\)
\(432\) 0 0
\(433\) −443.828 + 443.828i −1.02501 + 1.02501i −0.0253266 + 0.999679i \(0.508063\pi\)
−0.999679 + 0.0253266i \(0.991937\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −732.322 732.322i −1.67580 1.67580i
\(438\) 0 0
\(439\) 306.808i 0.698880i 0.936959 + 0.349440i \(0.113628\pi\)
−0.936959 + 0.349440i \(0.886372\pi\)
\(440\) 0 0
\(441\) 102.576 0.232598
\(442\) 0 0
\(443\) −443.646 + 443.646i −1.00146 + 1.00146i −0.00145911 + 0.999999i \(0.500464\pi\)
−0.999999 + 0.00145911i \(0.999536\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 106.293 + 106.293i 0.237792 + 0.237792i
\(448\) 0 0
\(449\) 230.727i 0.513868i 0.966429 + 0.256934i \(0.0827122\pi\)
−0.966429 + 0.256934i \(0.917288\pi\)
\(450\) 0 0
\(451\) −717.171 −1.59018
\(452\) 0 0
\(453\) 326.722 326.722i 0.721240 0.721240i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −27.2827 27.2827i −0.0596995 0.0596995i 0.676627 0.736326i \(-0.263441\pi\)
−0.736326 + 0.676627i \(0.763441\pi\)
\(458\) 0 0
\(459\) 6.60612i 0.0143924i
\(460\) 0 0
\(461\) −256.322 −0.556014 −0.278007 0.960579i \(-0.589674\pi\)
−0.278007 + 0.960579i \(0.589674\pi\)
\(462\) 0 0
\(463\) −173.196 + 173.196i −0.374074 + 0.374074i −0.868959 0.494884i \(-0.835210\pi\)
0.494884 + 0.868959i \(0.335210\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −285.707 285.707i −0.611793 0.611793i 0.331620 0.943413i \(-0.392405\pi\)
−0.943413 + 0.331620i \(0.892405\pi\)
\(468\) 0 0
\(469\) 1004.69i 2.14219i
\(470\) 0 0
\(471\) 498.322 1.05801
\(472\) 0 0
\(473\) −261.939 + 261.939i −0.553782 + 0.553782i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −156.272 156.272i −0.327615 0.327615i
\(478\) 0 0
\(479\) 356.404i 0.744059i −0.928221 0.372029i \(-0.878662\pi\)
0.928221 0.372029i \(-0.121338\pi\)
\(480\) 0 0
\(481\) 44.8082 0.0931563
\(482\) 0 0
\(483\) −344.363 + 344.363i −0.712967 + 0.712967i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 55.6209 + 55.6209i 0.114211 + 0.114211i 0.761903 0.647691i \(-0.224265\pi\)
−0.647691 + 0.761903i \(0.724265\pi\)
\(488\) 0 0
\(489\) 71.5551i 0.146329i
\(490\) 0 0
\(491\) −106.141 −0.216173 −0.108086 0.994142i \(-0.534472\pi\)
−0.108086 + 0.994142i \(0.534472\pi\)
\(492\) 0 0
\(493\) −3.23266 + 3.23266i −0.00655711 + 0.00655711i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 430.747 + 430.747i 0.866694 + 0.866694i
\(498\) 0 0
\(499\) 298.363i 0.597922i −0.954265 0.298961i \(-0.903360\pi\)
0.954265 0.298961i \(-0.0966401\pi\)
\(500\) 0 0
\(501\) −284.586 −0.568035
\(502\) 0 0
\(503\) −655.101 + 655.101i −1.30239 + 1.30239i −0.375610 + 0.926778i \(0.622567\pi\)
−0.926778 + 0.375610i \(0.877433\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −192.285 192.285i −0.379260 0.379260i
\(508\) 0 0
\(509\) 683.110i 1.34206i −0.741429 0.671032i \(-0.765851\pi\)
0.741429 0.671032i \(-0.234149\pi\)
\(510\) 0 0
\(511\) 1015.11 1.98652
\(512\) 0 0
\(513\) 123.439 123.439i 0.240622 0.240622i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 53.0306 + 53.0306i 0.102574 + 0.102574i
\(518\) 0 0
\(519\) 147.637i 0.284464i
\(520\) 0 0
\(521\) −6.76734 −0.0129891 −0.00649457 0.999979i \(-0.502067\pi\)
−0.00649457 + 0.999979i \(0.502067\pi\)
\(522\) 0 0
\(523\) 600.990 600.990i 1.14912 1.14912i 0.162394 0.986726i \(-0.448078\pi\)
0.986726 0.162394i \(-0.0519216\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 22.6469 + 22.6469i 0.0429733 + 0.0429733i
\(528\) 0 0
\(529\) 421.302i 0.796412i
\(530\) 0 0
\(531\) −260.969 −0.491468
\(532\) 0 0
\(533\) −148.899 + 148.899i −0.279360 + 0.279360i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −215.753 215.753i −0.401774 0.401774i
\(538\) 0 0
\(539\) 403.394i 0.748412i
\(540\) 0 0
\(541\) 177.110 0.327376 0.163688 0.986512i \(-0.447661\pi\)
0.163688 + 0.986512i \(0.447661\pi\)
\(542\) 0 0
\(543\) −198.359 + 198.359i −0.365301 + 0.365301i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 35.3326 + 35.3326i 0.0645935 + 0.0645935i 0.738666 0.674072i \(-0.235456\pi\)
−0.674072 + 0.738666i \(0.735456\pi\)
\(548\) 0 0
\(549\) 325.151i 0.592261i
\(550\) 0 0
\(551\) 120.808 0.219253
\(552\) 0 0
\(553\) −425.666 + 425.666i −0.769740 + 0.769740i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 121.930 + 121.930i 0.218904 + 0.218904i 0.808037 0.589132i \(-0.200530\pi\)
−0.589132 + 0.808037i \(0.700530\pi\)
\(558\) 0 0
\(559\) 108.767i 0.194575i
\(560\) 0 0
\(561\) −25.9796 −0.0463094
\(562\) 0 0
\(563\) 436.636 436.636i 0.775552 0.775552i −0.203519 0.979071i \(-0.565238\pi\)
0.979071 + 0.203519i \(0.0652379\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −58.0454 58.0454i −0.102373 0.102373i
\(568\) 0 0
\(569\) 394.363i 0.693081i 0.938035 + 0.346541i \(0.112644\pi\)
−0.938035 + 0.346541i \(0.887356\pi\)
\(570\) 0 0
\(571\) −32.0612 −0.0561493 −0.0280746 0.999606i \(-0.508938\pi\)
−0.0280746 + 0.999606i \(0.508938\pi\)
\(572\) 0 0
\(573\) 307.101 307.101i 0.535953 0.535953i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 545.848 + 545.848i 0.946010 + 0.946010i 0.998615 0.0526051i \(-0.0167524\pi\)
−0.0526051 + 0.998615i \(0.516752\pi\)
\(578\) 0 0
\(579\) 21.1306i 0.0364950i
\(580\) 0 0
\(581\) 1084.69 1.86693
\(582\) 0 0
\(583\) 614.565 614.565i 1.05414 1.05414i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 307.728 + 307.728i 0.524238 + 0.524238i 0.918848 0.394611i \(-0.129121\pi\)
−0.394611 + 0.918848i \(0.629121\pi\)
\(588\) 0 0
\(589\) 846.343i 1.43691i
\(590\) 0 0
\(591\) −446.302 −0.755164
\(592\) 0 0
\(593\) 228.940 228.940i 0.386070 0.386070i −0.487213 0.873283i \(-0.661986\pi\)
0.873283 + 0.487213i \(0.161986\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −99.4393 99.4393i −0.166565 0.166565i
\(598\) 0 0
\(599\) 345.657i 0.577057i −0.957471 0.288529i \(-0.906834\pi\)
0.957471 0.288529i \(-0.0931660\pi\)
\(600\) 0 0
\(601\) 893.069 1.48597 0.742986 0.669307i \(-0.233409\pi\)
0.742986 + 0.669307i \(0.233409\pi\)
\(602\) 0 0
\(603\) 233.666 233.666i 0.387506 0.387506i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −560.974 560.974i −0.924175 0.924175i 0.0731466 0.997321i \(-0.476696\pi\)
−0.997321 + 0.0731466i \(0.976696\pi\)
\(608\) 0 0
\(609\) 56.8082i 0.0932811i
\(610\) 0 0
\(611\) 22.0204 0.0360400
\(612\) 0 0
\(613\) 439.448 439.448i 0.716882 0.716882i −0.251084 0.967965i \(-0.580787\pi\)
0.967965 + 0.251084i \(0.0807869\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 199.201 + 199.201i 0.322854 + 0.322854i 0.849861 0.527007i \(-0.176686\pi\)
−0.527007 + 0.849861i \(0.676686\pi\)
\(618\) 0 0
\(619\) 121.151i 0.195721i −0.995200 0.0978603i \(-0.968800\pi\)
0.995200 0.0978603i \(-0.0311998\pi\)
\(620\) 0 0
\(621\) 160.182 0.257941
\(622\) 0 0
\(623\) −61.8888 + 61.8888i −0.0993399 + 0.0993399i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 485.444 + 485.444i 0.774233 + 0.774233i
\(628\) 0 0
\(629\) 16.4449i 0.0261445i
\(630\) 0 0
\(631\) −247.494 −0.392225 −0.196112 0.980581i \(-0.562832\pi\)
−0.196112 + 0.980581i \(0.562832\pi\)
\(632\) 0 0
\(633\) 244.429 244.429i 0.386144 0.386144i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 83.7526 + 83.7526i 0.131480 + 0.131480i
\(638\) 0 0
\(639\) 200.363i 0.313558i
\(640\) 0 0
\(641\) 70.6857 0.110274 0.0551371 0.998479i \(-0.482440\pi\)
0.0551371 + 0.998479i \(0.482440\pi\)
\(642\) 0 0
\(643\) 231.373 231.373i 0.359834 0.359834i −0.503917 0.863752i \(-0.668108\pi\)
0.863752 + 0.503917i \(0.168108\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 500.958 + 500.958i 0.774278 + 0.774278i 0.978851 0.204573i \(-0.0655805\pi\)
−0.204573 + 0.978851i \(0.565581\pi\)
\(648\) 0 0
\(649\) 1026.30i 1.58136i
\(650\) 0 0
\(651\) −397.980 −0.611336
\(652\) 0 0
\(653\) −261.031 + 261.031i −0.399741 + 0.399741i −0.878142 0.478401i \(-0.841217\pi\)
0.478401 + 0.878142i \(0.341217\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −236.091 236.091i −0.359347 0.359347i
\(658\) 0 0
\(659\) 800.524i 1.21476i 0.794413 + 0.607378i \(0.207779\pi\)
−0.794413 + 0.607378i \(0.792221\pi\)
\(660\) 0 0
\(661\) −313.192 −0.473815 −0.236908 0.971532i \(-0.576134\pi\)
−0.236908 + 0.971532i \(0.576134\pi\)
\(662\) 0 0
\(663\) −5.39388 + 5.39388i −0.00813556 + 0.00813556i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 78.3837 + 78.3837i 0.117517 + 0.117517i
\(668\) 0 0
\(669\) 330.545i 0.494088i
\(670\) 0 0
\(671\) −1278.71 −1.90567
\(672\) 0 0
\(673\) −688.677 + 688.677i −1.02329 + 1.02329i −0.0235713 + 0.999722i \(0.507504\pi\)
−0.999722 + 0.0235713i \(0.992496\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −660.767 660.767i −0.976023 0.976023i 0.0236965 0.999719i \(-0.492456\pi\)
−0.999719 + 0.0236965i \(0.992456\pi\)
\(678\) 0 0
\(679\) 684.686i 1.00837i
\(680\) 0 0
\(681\) 149.171 0.219048
\(682\) 0 0
\(683\) −952.232 + 952.232i −1.39419 + 1.39419i −0.578525 + 0.815664i \(0.696372\pi\)
−0.815664 + 0.578525i \(0.803628\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 169.955 + 169.955i 0.247387 + 0.247387i
\(688\) 0 0
\(689\) 255.192i 0.370380i
\(690\) 0 0
\(691\) −568.665 −0.822960 −0.411480 0.911419i \(-0.634988\pi\)
−0.411480 + 0.911419i \(0.634988\pi\)
\(692\) 0 0
\(693\) 228.272 228.272i 0.329397 0.329397i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −54.6469 54.6469i −0.0784031 0.0784031i
\(698\) 0 0
\(699\) 623.778i 0.892386i
\(700\) 0 0
\(701\) −125.698 −0.179312 −0.0896562 0.995973i \(-0.528577\pi\)
−0.0896562 + 0.995973i \(0.528577\pi\)
\(702\) 0 0
\(703\) 307.283 307.283i 0.437102 0.437102i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1000.91 + 1000.91i 1.41571 + 1.41571i
\(708\) 0 0
\(709\) 201.069i 0.283596i −0.989896 0.141798i \(-0.954712\pi\)
0.989896 0.141798i \(-0.0452883\pi\)
\(710\) 0 0
\(711\) 198.000 0.278481
\(712\) 0 0
\(713\) 549.131 549.131i 0.770169 0.770169i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 517.807 + 517.807i 0.722186 + 0.722186i
\(718\) 0 0
\(719\) 785.576i 1.09259i −0.837591 0.546297i \(-0.816037\pi\)
0.837591 0.546297i \(-0.183963\pi\)
\(720\) 0 0
\(721\) −546.384 −0.757814
\(722\) 0 0
\(723\) −13.1872 + 13.1872i −0.0182396 + 0.0182396i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 282.459 + 282.459i 0.388526 + 0.388526i 0.874162 0.485635i \(-0.161412\pi\)
−0.485635 + 0.874162i \(0.661412\pi\)
\(728\) 0 0
\(729\) 27.0000i 0.0370370i
\(730\) 0 0
\(731\) −39.9184 −0.0546079
\(732\) 0 0
\(733\) 54.9036 54.9036i 0.0749025 0.0749025i −0.668663 0.743566i \(-0.733133\pi\)
0.743566 + 0.668663i \(0.233133\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 918.929 + 918.929i 1.24685 + 1.24685i
\(738\) 0 0
\(739\) 276.624i 0.374323i 0.982329 + 0.187161i \(0.0599287\pi\)
−0.982329 + 0.187161i \(0.940071\pi\)
\(740\) 0 0
\(741\) 201.576 0.272032
\(742\) 0 0
\(743\) 143.101 143.101i 0.192599 0.192599i −0.604219 0.796818i \(-0.706515\pi\)
0.796818 + 0.604219i \(0.206515\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −252.272 252.272i −0.337714 0.337714i
\(748\) 0 0
\(749\) 385.535i 0.514733i
\(750\) 0 0
\(751\) −871.453 −1.16039 −0.580195 0.814478i \(-0.697024\pi\)
−0.580195 + 0.814478i \(0.697024\pi\)
\(752\) 0 0
\(753\) −217.782 + 217.782i −0.289219 + 0.289219i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 304.702 + 304.702i 0.402512 + 0.402512i 0.879117 0.476605i \(-0.158133\pi\)
−0.476605 + 0.879117i \(0.658133\pi\)
\(758\) 0 0
\(759\) 629.939i 0.829959i
\(760\) 0 0
\(761\) 613.959 0.806779 0.403390 0.915028i \(-0.367832\pi\)
0.403390 + 0.915028i \(0.367832\pi\)
\(762\) 0 0
\(763\) 110.879 110.879i 0.145319 0.145319i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −213.081 213.081i −0.277810 0.277810i
\(768\) 0 0
\(769\) 887.412i 1.15398i 0.816750 + 0.576991i \(0.195773\pi\)
−0.816750 + 0.576991i \(0.804227\pi\)
\(770\) 0 0
\(771\) 348.545 0.452069
\(772\) 0 0
\(773\) 605.757 605.757i 0.783644 0.783644i −0.196799 0.980444i \(-0.563055\pi\)
0.980444 + 0.196799i \(0.0630548\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −144.495 144.495i −0.185965 0.185965i
\(778\) 0 0
\(779\) 2042.22i 2.62159i
\(780\) 0 0
\(781\) 787.959 1.00891
\(782\) 0 0
\(783\) −13.2122 + 13.2122i −0.0168739 + 0.0168739i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 616.463 + 616.463i 0.783308 + 0.783308i 0.980387 0.197080i \(-0.0631458\pi\)
−0.197080 + 0.980387i \(0.563146\pi\)
\(788\) 0 0
\(789\) 88.1816i 0.111764i
\(790\) 0 0
\(791\) −1732.24 −2.18994
\(792\) 0 0
\(793\) −265.485 + 265.485i −0.334785 + 0.334785i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −247.768 247.768i −0.310876 0.310876i 0.534373 0.845249i \(-0.320548\pi\)
−0.845249 + 0.534373i \(0.820548\pi\)
\(798\) 0 0
\(799\) 8.08164i 0.0101147i
\(800\) 0 0
\(801\) 28.7878 0.0359398
\(802\) 0 0
\(803\) 928.463 928.463i 1.15624 1.15624i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 47.5551 + 47.5551i 0.0589282 + 0.0589282i
\(808\) 0 0
\(809\) 1556.10i 1.92348i −0.273958 0.961742i \(-0.588333\pi\)
0.273958 0.961742i \(-0.411667\pi\)
\(810\) 0 0
\(811\) −277.192 −0.341790 −0.170895 0.985289i \(-0.554666\pi\)
−0.170895 + 0.985289i \(0.554666\pi\)
\(812\) 0 0
\(813\) −14.1770 + 14.1770i −0.0174379 + 0.0174379i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 745.898 + 745.898i 0.912972 + 0.912972i
\(818\) 0 0
\(819\) 94.7878i 0.115736i
\(820\) 0 0
\(821\) −1485.86 −1.80981 −0.904907 0.425610i \(-0.860060\pi\)
−0.904907 + 0.425610i \(0.860060\pi\)
\(822\) 0 0
\(823\) 809.519 809.519i 0.983620 0.983620i −0.0162485 0.999868i \(-0.505172\pi\)
0.999868 + 0.0162485i \(0.00517228\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1122.66 1122.66i −1.35750 1.35750i −0.876983 0.480521i \(-0.840447\pi\)
−0.480521 0.876983i \(-0.659553\pi\)
\(828\) 0 0
\(829\) 849.151i 1.02431i −0.858894 0.512154i \(-0.828848\pi\)
0.858894 0.512154i \(-0.171152\pi\)
\(830\) 0 0
\(831\) 691.980 0.832707
\(832\) 0 0
\(833\) −30.7378 + 30.7378i −0.0369001 + 0.0369001i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 92.5607 + 92.5607i 0.110586 + 0.110586i
\(838\) 0 0
\(839\) 1606.26i 1.91449i −0.289270 0.957247i \(-0.593413\pi\)
0.289270 0.957247i \(-0.406587\pi\)
\(840\) 0 0
\(841\) 828.069 0.984625
\(842\) 0 0
\(843\) −529.560 + 529.560i −0.628185 + 0.628185i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 117.328 + 117.328i 0.138522 + 0.138522i
\(848\) 0 0
\(849\) 548.363i 0.645893i
\(850\) 0 0
\(851\) 398.747 0.468563
\(852\) 0 0
\(853\) 632.702 632.702i 0.741737 0.741737i −0.231175 0.972912i \(-0.574257\pi\)
0.972912 + 0.231175i \(0.0742571\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 258.879 + 258.879i 0.302075 + 0.302075i 0.841825 0.539750i \(-0.181481\pi\)
−0.539750 + 0.841825i \(0.681481\pi\)
\(858\) 0 0
\(859\) 534.082i 0.621748i −0.950451 0.310874i \(-0.899378\pi\)
0.950451 0.310874i \(-0.100622\pi\)
\(860\) 0 0
\(861\) 960.322 1.11536
\(862\) 0 0
\(863\) 711.878 711.878i 0.824887 0.824887i −0.161917 0.986804i \(-0.551768\pi\)
0.986804 + 0.161917i \(0.0517678\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 351.972 + 351.972i 0.405965 + 0.405965i
\(868\) 0 0
\(869\) 778.665i 0.896048i
\(870\) 0 0
\(871\) 381.576 0.438089
\(872\) 0 0
\(873\) 159.242 159.242i 0.182408 0.182408i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 742.590 + 742.590i 0.846739 + 0.846739i 0.989725 0.142986i \(-0.0456702\pi\)
−0.142986 + 0.989725i \(0.545670\pi\)
\(878\) 0 0
\(879\) 481.980i 0.548327i
\(880\) 0 0
\(881\) −1000.75 −1.13592 −0.567961 0.823056i \(-0.692267\pi\)
−0.567961 + 0.823056i \(0.692267\pi\)
\(882\) 0 0
\(883\) −546.788 + 546.788i −0.619239 + 0.619239i −0.945336 0.326097i \(-0.894266\pi\)
0.326097 + 0.945336i \(0.394266\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1138.48 + 1138.48i 1.28352 + 1.28352i 0.938650 + 0.344872i \(0.112078\pi\)
0.344872 + 0.938650i \(0.387922\pi\)
\(888\) 0 0
\(889\) 1976.60i 2.22340i
\(890\) 0 0
\(891\) −106.182 −0.119171
\(892\) 0 0
\(893\) 151.010 151.010i 0.169104 0.169104i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 130.788 + 130.788i 0.145806 + 0.145806i
\(898\) 0 0
\(899\) 90.5878i 0.100765i
\(900\) 0 0
\(901\) 93.6571 0.103948
\(902\) 0 0
\(903\) 350.747 350.747i 0.388424 0.388424i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −162.706 162.706i −0.179389 0.179389i 0.611700 0.791090i \(-0.290486\pi\)
−0.791090 + 0.611700i \(0.790486\pi\)
\(908\) 0 0
\(909\) 465.576i 0.512184i
\(910\) 0 0
\(911\) 386.988 0.424794 0.212397 0.977183i \(-0.431873\pi\)
0.212397 + 0.977183i \(0.431873\pi\)
\(912\) 0 0
\(913\) 992.100 992.100i 1.08664 1.08664i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1268.01 + 1268.01i 1.38278 + 1.38278i
\(918\) 0 0
\(919\) 1101.80i 1.19891i −0.800409 0.599454i \(-0.795385\pi\)
0.800409 0.599454i \(-0.204615\pi\)
\(920\) 0 0
\(921\) −347.110 −0.376884
\(922\) 0 0
\(923\) 163.596 163.596i 0.177244 0.177244i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 127.076 + 127.076i 0.137083 + 0.137083i
\(928\) 0 0
\(929\) 261.273i 0.281242i 0.990064 + 0.140621i \(0.0449098\pi\)
−0.990064 + 0.140621i \(0.955090\pi\)
\(930\) 0 0
\(931\) 1148.71 1.23384
\(932\) 0 0
\(933\) −504.050 + 504.050i −0.540247 + 0.540247i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −57.7980 57.7980i −0.0616841 0.0616841i 0.675592 0.737276i \(-0.263888\pi\)
−0.737276 + 0.675592i \(0.763888\pi\)
\(938\) 0 0
\(939\) 896.363i 0.954593i
\(940\) 0 0
\(941\) 399.837 0.424906 0.212453 0.977171i \(-0.431855\pi\)
0.212453 + 0.977171i \(0.431855\pi\)
\(942\) 0 0
\(943\) −1325.05 + 1325.05i −1.40514 + 1.40514i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 464.958 + 464.958i 0.490980 + 0.490980i 0.908615 0.417635i \(-0.137141\pi\)
−0.417635 + 0.908615i \(0.637141\pi\)
\(948\) 0 0
\(949\) 385.535i 0.406254i
\(950\) 0 0
\(951\) 828.767 0.871469
\(952\) 0 0
\(953\) 627.242 627.242i 0.658176 0.658176i −0.296772 0.954948i \(-0.595910\pi\)
0.954948 + 0.296772i \(0.0959102\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −51.9592 51.9592i −0.0542938 0.0542938i
\(958\) 0 0
\(959\) 1045.86i 1.09057i
\(960\) 0 0
\(961\) −326.371 −0.339616
\(962\) 0 0
\(963\) 89.6663 89.6663i 0.0931114 0.0931114i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1334.02 + 1334.02i 1.37955 + 1.37955i 0.845375 + 0.534173i \(0.179377\pi\)
0.534173 + 0.845375i \(0.320623\pi\)
\(968\) 0 0
\(969\) 73.9796i 0.0763463i
\(970\) 0 0
\(971\) 1489.17 1.53365 0.766823 0.641859i \(-0.221837\pi\)
0.766823 + 0.641859i \(0.221837\pi\)
\(972\) 0 0
\(973\) −239.868 + 239.868i −0.246525 + 0.246525i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1134.66 1134.66i −1.16137 1.16137i −0.984176 0.177191i \(-0.943299\pi\)
−0.177191 0.984176i \(-0.556701\pi\)
\(978\) 0 0
\(979\) 113.212i 0.115641i
\(980\) 0 0
\(981\) −51.5755 −0.0525744
\(982\) 0 0
\(983\) 261.485 261.485i 0.266007 0.266007i −0.561482 0.827489i \(-0.689769\pi\)
0.827489 + 0.561482i \(0.189769\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −71.0102 71.0102i −0.0719455 0.0719455i
\(988\) 0 0
\(989\) 967.918i 0.978684i
\(990\) 0 0
\(991\) 950.645 0.959278 0.479639 0.877466i \(-0.340768\pi\)
0.479639 + 0.877466i \(0.340768\pi\)
\(992\) 0 0
\(993\) −119.480 + 119.480i −0.120322 + 0.120322i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −96.0566 96.0566i −0.0963457 0.0963457i 0.657291 0.753637i \(-0.271702\pi\)
−0.753637 + 0.657291i \(0.771702\pi\)
\(998\) 0 0
\(999\) 67.2122i 0.0672795i
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.m.1057.1 4
4.3 odd 2 600.3.u.a.457.2 yes 4
5.2 odd 4 1200.3.bg.b.193.2 4
5.3 odd 4 inner 1200.3.bg.m.193.1 4
5.4 even 2 1200.3.bg.b.1057.2 4
12.11 even 2 1800.3.v.j.1657.1 4
20.3 even 4 600.3.u.a.193.2 4
20.7 even 4 600.3.u.f.193.1 yes 4
20.19 odd 2 600.3.u.f.457.1 yes 4
60.23 odd 4 1800.3.v.j.793.1 4
60.47 odd 4 1800.3.v.q.793.2 4
60.59 even 2 1800.3.v.q.1657.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.3.u.a.193.2 4 20.3 even 4
600.3.u.a.457.2 yes 4 4.3 odd 2
600.3.u.f.193.1 yes 4 20.7 even 4
600.3.u.f.457.1 yes 4 20.19 odd 2
1200.3.bg.b.193.2 4 5.2 odd 4
1200.3.bg.b.1057.2 4 5.4 even 2
1200.3.bg.m.193.1 4 5.3 odd 4 inner
1200.3.bg.m.1057.1 4 1.1 even 1 trivial
1800.3.v.j.793.1 4 60.23 odd 4
1800.3.v.j.1657.1 4 12.11 even 2
1800.3.v.q.793.2 4 60.47 odd 4
1800.3.v.q.1657.2 4 60.59 even 2