Properties

Label 1200.3.bg.l.193.2
Level $1200$
Weight $3$
Character 1200.193
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 193.2
Root \(1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1200.193
Dual form 1200.3.bg.l.1057.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.22474 + 1.22474i) q^{3} +(0.775255 - 0.775255i) q^{7} +3.00000i q^{9} +O(q^{10})\) \(q+(1.22474 + 1.22474i) q^{3} +(0.775255 - 0.775255i) q^{7} +3.00000i q^{9} -2.89898 q^{11} +(-5.87628 - 5.87628i) q^{13} +(-4.44949 + 4.44949i) q^{17} +0.101021i q^{19} +1.89898 q^{21} +(25.3485 + 25.3485i) q^{23} +(-3.67423 + 3.67423i) q^{27} +32.2929i q^{29} +3.69694 q^{31} +(-3.55051 - 3.55051i) q^{33} +(-42.6969 + 42.6969i) q^{37} -14.3939i q^{39} -12.8990 q^{41} +(49.2702 + 49.2702i) q^{43} +(2.85357 - 2.85357i) q^{47} +47.7980i q^{49} -10.8990 q^{51} +(13.1918 + 13.1918i) q^{53} +(-0.123724 + 0.123724i) q^{57} -76.3837i q^{59} -103.788 q^{61} +(2.32577 + 2.32577i) q^{63} +(47.6288 - 47.6288i) q^{67} +62.0908i q^{69} -29.7071 q^{71} +(-3.50510 - 3.50510i) q^{73} +(-2.24745 + 2.24745i) q^{77} +87.7980i q^{79} -9.00000 q^{81} +(81.7321 + 81.7321i) q^{83} +(-39.5505 + 39.5505i) q^{87} +96.5857i q^{89} -9.11123 q^{91} +(4.52781 + 4.52781i) q^{93} +(-54.2804 + 54.2804i) q^{97} -8.69694i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{7} + 8 q^{11} - 48 q^{13} - 8 q^{17} - 12 q^{21} + 72 q^{23} - 44 q^{31} - 24 q^{33} - 112 q^{37} - 32 q^{41} + 104 q^{43} + 80 q^{47} - 24 q^{51} - 104 q^{53} + 24 q^{57} - 180 q^{61} + 24 q^{63} + 264 q^{67} - 256 q^{71} - 112 q^{73} + 40 q^{77} - 36 q^{81} - 16 q^{83} - 168 q^{87} - 252 q^{91} + 72 q^{93} - 320 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(7\) 0.775255 0.775255i 0.110751 0.110751i −0.649560 0.760311i \(-0.725047\pi\)
0.760311 + 0.649560i \(0.225047\pi\)
\(8\) 0 0
\(9\) 3.00000i 0.333333i
\(10\) 0 0
\(11\) −2.89898 −0.263544 −0.131772 0.991280i \(-0.542067\pi\)
−0.131772 + 0.991280i \(0.542067\pi\)
\(12\) 0 0
\(13\) −5.87628 5.87628i −0.452021 0.452021i 0.444004 0.896025i \(-0.353558\pi\)
−0.896025 + 0.444004i \(0.853558\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.44949 + 4.44949i −0.261735 + 0.261735i −0.825759 0.564024i \(-0.809253\pi\)
0.564024 + 0.825759i \(0.309253\pi\)
\(18\) 0 0
\(19\) 0.101021i 0.00531687i 0.999996 + 0.00265843i \(0.000846207\pi\)
−0.999996 + 0.00265843i \(0.999154\pi\)
\(20\) 0 0
\(21\) 1.89898 0.0904276
\(22\) 0 0
\(23\) 25.3485 + 25.3485i 1.10211 + 1.10211i 0.994156 + 0.107951i \(0.0344290\pi\)
0.107951 + 0.994156i \(0.465571\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −3.67423 + 3.67423i −0.136083 + 0.136083i
\(28\) 0 0
\(29\) 32.2929i 1.11355i 0.830664 + 0.556773i \(0.187961\pi\)
−0.830664 + 0.556773i \(0.812039\pi\)
\(30\) 0 0
\(31\) 3.69694 0.119256 0.0596280 0.998221i \(-0.481009\pi\)
0.0596280 + 0.998221i \(0.481009\pi\)
\(32\) 0 0
\(33\) −3.55051 3.55051i −0.107591 0.107591i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −42.6969 + 42.6969i −1.15397 + 1.15397i −0.168222 + 0.985749i \(0.553803\pi\)
−0.985749 + 0.168222i \(0.946197\pi\)
\(38\) 0 0
\(39\) 14.3939i 0.369074i
\(40\) 0 0
\(41\) −12.8990 −0.314609 −0.157305 0.987550i \(-0.550280\pi\)
−0.157305 + 0.987550i \(0.550280\pi\)
\(42\) 0 0
\(43\) 49.2702 + 49.2702i 1.14582 + 1.14582i 0.987367 + 0.158451i \(0.0506499\pi\)
0.158451 + 0.987367i \(0.449350\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.85357 2.85357i 0.0607143 0.0607143i −0.676098 0.736812i \(-0.736330\pi\)
0.736812 + 0.676098i \(0.236330\pi\)
\(48\) 0 0
\(49\) 47.7980i 0.975469i
\(50\) 0 0
\(51\) −10.8990 −0.213705
\(52\) 0 0
\(53\) 13.1918 + 13.1918i 0.248903 + 0.248903i 0.820520 0.571618i \(-0.193684\pi\)
−0.571618 + 0.820520i \(0.693684\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.123724 + 0.123724i −0.00217060 + 0.00217060i
\(58\) 0 0
\(59\) 76.3837i 1.29464i −0.762219 0.647319i \(-0.775890\pi\)
0.762219 0.647319i \(-0.224110\pi\)
\(60\) 0 0
\(61\) −103.788 −1.70144 −0.850719 0.525620i \(-0.823833\pi\)
−0.850719 + 0.525620i \(0.823833\pi\)
\(62\) 0 0
\(63\) 2.32577 + 2.32577i 0.0369169 + 0.0369169i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 47.6288 47.6288i 0.710878 0.710878i −0.255841 0.966719i \(-0.582352\pi\)
0.966719 + 0.255841i \(0.0823523\pi\)
\(68\) 0 0
\(69\) 62.0908i 0.899867i
\(70\) 0 0
\(71\) −29.7071 −0.418410 −0.209205 0.977872i \(-0.567088\pi\)
−0.209205 + 0.977872i \(0.567088\pi\)
\(72\) 0 0
\(73\) −3.50510 3.50510i −0.0480151 0.0480151i 0.682692 0.730707i \(-0.260809\pi\)
−0.730707 + 0.682692i \(0.760809\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.24745 + 2.24745i −0.0291876 + 0.0291876i
\(78\) 0 0
\(79\) 87.7980i 1.11137i 0.831394 + 0.555683i \(0.187543\pi\)
−0.831394 + 0.555683i \(0.812457\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) 0 0
\(83\) 81.7321 + 81.7321i 0.984725 + 0.984725i 0.999885 0.0151605i \(-0.00482592\pi\)
−0.0151605 + 0.999885i \(0.504826\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −39.5505 + 39.5505i −0.454604 + 0.454604i
\(88\) 0 0
\(89\) 96.5857i 1.08523i 0.839981 + 0.542616i \(0.182566\pi\)
−0.839981 + 0.542616i \(0.817434\pi\)
\(90\) 0 0
\(91\) −9.11123 −0.100123
\(92\) 0 0
\(93\) 4.52781 + 4.52781i 0.0486861 + 0.0486861i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −54.2804 + 54.2804i −0.559591 + 0.559591i −0.929191 0.369600i \(-0.879495\pi\)
0.369600 + 0.929191i \(0.379495\pi\)
\(98\) 0 0
\(99\) 8.69694i 0.0878479i
\(100\) 0 0
\(101\) −50.5153 −0.500152 −0.250076 0.968226i \(-0.580456\pi\)
−0.250076 + 0.968226i \(0.580456\pi\)
\(102\) 0 0
\(103\) 26.2020 + 26.2020i 0.254389 + 0.254389i 0.822767 0.568378i \(-0.192429\pi\)
−0.568378 + 0.822767i \(0.692429\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.48469 7.48469i 0.0699504 0.0699504i −0.671266 0.741216i \(-0.734249\pi\)
0.741216 + 0.671266i \(0.234249\pi\)
\(108\) 0 0
\(109\) 94.5755i 0.867665i −0.900993 0.433833i \(-0.857161\pi\)
0.900993 0.433833i \(-0.142839\pi\)
\(110\) 0 0
\(111\) −104.586 −0.942214
\(112\) 0 0
\(113\) 89.9796 + 89.9796i 0.796280 + 0.796280i 0.982507 0.186227i \(-0.0596260\pi\)
−0.186227 + 0.982507i \(0.559626\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 17.6288 17.6288i 0.150674 0.150674i
\(118\) 0 0
\(119\) 6.89898i 0.0579746i
\(120\) 0 0
\(121\) −112.596 −0.930545
\(122\) 0 0
\(123\) −15.7980 15.7980i −0.128439 0.128439i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 37.3031 37.3031i 0.293725 0.293725i −0.544825 0.838550i \(-0.683404\pi\)
0.838550 + 0.544825i \(0.183404\pi\)
\(128\) 0 0
\(129\) 120.687i 0.935556i
\(130\) 0 0
\(131\) −192.677 −1.47081 −0.735407 0.677626i \(-0.763009\pi\)
−0.735407 + 0.677626i \(0.763009\pi\)
\(132\) 0 0
\(133\) 0.0783167 + 0.0783167i 0.000588847 + 0.000588847i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 165.328 165.328i 1.20677 1.20677i 0.234708 0.972066i \(-0.424587\pi\)
0.972066 0.234708i \(-0.0754135\pi\)
\(138\) 0 0
\(139\) 256.747i 1.84710i 0.383478 + 0.923550i \(0.374726\pi\)
−0.383478 + 0.923550i \(0.625274\pi\)
\(140\) 0 0
\(141\) 6.98979 0.0495730
\(142\) 0 0
\(143\) 17.0352 + 17.0352i 0.119127 + 0.119127i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −58.5403 + 58.5403i −0.398233 + 0.398233i
\(148\) 0 0
\(149\) 192.788i 1.29388i −0.762542 0.646939i \(-0.776049\pi\)
0.762542 0.646939i \(-0.223951\pi\)
\(150\) 0 0
\(151\) −98.9092 −0.655028 −0.327514 0.944846i \(-0.606211\pi\)
−0.327514 + 0.944846i \(0.606211\pi\)
\(152\) 0 0
\(153\) −13.3485 13.3485i −0.0872449 0.0872449i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −103.452 + 103.452i −0.658929 + 0.658929i −0.955127 0.296198i \(-0.904281\pi\)
0.296198 + 0.955127i \(0.404281\pi\)
\(158\) 0 0
\(159\) 32.3133i 0.203228i
\(160\) 0 0
\(161\) 39.3031 0.244118
\(162\) 0 0
\(163\) −19.1339 19.1339i −0.117386 0.117386i 0.645974 0.763360i \(-0.276452\pi\)
−0.763360 + 0.645974i \(0.776452\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −96.6969 + 96.6969i −0.579024 + 0.579024i −0.934634 0.355611i \(-0.884273\pi\)
0.355611 + 0.934634i \(0.384273\pi\)
\(168\) 0 0
\(169\) 99.9388i 0.591354i
\(170\) 0 0
\(171\) −0.303062 −0.00177229
\(172\) 0 0
\(173\) 180.136 + 180.136i 1.04125 + 1.04125i 0.999112 + 0.0421380i \(0.0134169\pi\)
0.0421380 + 0.999112i \(0.486583\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 93.5505 93.5505i 0.528534 0.528534i
\(178\) 0 0
\(179\) 134.000i 0.748603i −0.927307 0.374302i \(-0.877882\pi\)
0.927307 0.374302i \(-0.122118\pi\)
\(180\) 0 0
\(181\) −171.586 −0.947987 −0.473994 0.880528i \(-0.657188\pi\)
−0.473994 + 0.880528i \(0.657188\pi\)
\(182\) 0 0
\(183\) −127.114 127.114i −0.694609 0.694609i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 12.8990 12.8990i 0.0689785 0.0689785i
\(188\) 0 0
\(189\) 5.69694i 0.0301425i
\(190\) 0 0
\(191\) −26.2724 −0.137552 −0.0687760 0.997632i \(-0.521909\pi\)
−0.0687760 + 0.997632i \(0.521909\pi\)
\(192\) 0 0
\(193\) 74.1237 + 74.1237i 0.384061 + 0.384061i 0.872563 0.488502i \(-0.162457\pi\)
−0.488502 + 0.872563i \(0.662457\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 47.7526 47.7526i 0.242399 0.242399i −0.575443 0.817842i \(-0.695170\pi\)
0.817842 + 0.575443i \(0.195170\pi\)
\(198\) 0 0
\(199\) 355.454i 1.78620i 0.449857 + 0.893101i \(0.351475\pi\)
−0.449857 + 0.893101i \(0.648525\pi\)
\(200\) 0 0
\(201\) 116.666 0.580429
\(202\) 0 0
\(203\) 25.0352 + 25.0352i 0.123326 + 0.123326i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −76.0454 + 76.0454i −0.367369 + 0.367369i
\(208\) 0 0
\(209\) 0.292856i 0.00140123i
\(210\) 0 0
\(211\) 145.474 0.689453 0.344726 0.938703i \(-0.387972\pi\)
0.344726 + 0.938703i \(0.387972\pi\)
\(212\) 0 0
\(213\) −36.3837 36.3837i −0.170815 0.170815i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 2.86607 2.86607i 0.0132077 0.0132077i
\(218\) 0 0
\(219\) 8.58571i 0.0392042i
\(220\) 0 0
\(221\) 52.2929 0.236619
\(222\) 0 0
\(223\) −230.351 230.351i −1.03296 1.03296i −0.999438 0.0335252i \(-0.989327\pi\)
−0.0335252 0.999438i \(-0.510673\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 40.4745 40.4745i 0.178302 0.178302i −0.612313 0.790615i \(-0.709761\pi\)
0.790615 + 0.612313i \(0.209761\pi\)
\(228\) 0 0
\(229\) 210.192i 0.917868i −0.888470 0.458934i \(-0.848231\pi\)
0.888470 0.458934i \(-0.151769\pi\)
\(230\) 0 0
\(231\) −5.50510 −0.0238316
\(232\) 0 0
\(233\) 296.384 + 296.384i 1.27203 + 1.27203i 0.945020 + 0.327013i \(0.106042\pi\)
0.327013 + 0.945020i \(0.393958\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −107.530 + 107.530i −0.453714 + 0.453714i
\(238\) 0 0
\(239\) 89.3031i 0.373653i −0.982393 0.186826i \(-0.940180\pi\)
0.982393 0.186826i \(-0.0598202\pi\)
\(240\) 0 0
\(241\) 120.616 0.500483 0.250241 0.968183i \(-0.419490\pi\)
0.250241 + 0.968183i \(0.419490\pi\)
\(242\) 0 0
\(243\) −11.0227 11.0227i −0.0453609 0.0453609i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.593624 0.593624i 0.00240334 0.00240334i
\(248\) 0 0
\(249\) 200.202i 0.804024i
\(250\) 0 0
\(251\) 197.576 0.787153 0.393577 0.919292i \(-0.371238\pi\)
0.393577 + 0.919292i \(0.371238\pi\)
\(252\) 0 0
\(253\) −73.4847 73.4847i −0.290453 0.290453i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 223.868 223.868i 0.871083 0.871083i −0.121507 0.992591i \(-0.538773\pi\)
0.992591 + 0.121507i \(0.0387728\pi\)
\(258\) 0 0
\(259\) 66.2020i 0.255606i
\(260\) 0 0
\(261\) −96.8786 −0.371182
\(262\) 0 0
\(263\) 90.7673 + 90.7673i 0.345123 + 0.345123i 0.858289 0.513166i \(-0.171528\pi\)
−0.513166 + 0.858289i \(0.671528\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −118.293 + 118.293i −0.443044 + 0.443044i
\(268\) 0 0
\(269\) 361.909i 1.34539i −0.739921 0.672694i \(-0.765137\pi\)
0.739921 0.672694i \(-0.234863\pi\)
\(270\) 0 0
\(271\) 216.788 0.799955 0.399977 0.916525i \(-0.369018\pi\)
0.399977 + 0.916525i \(0.369018\pi\)
\(272\) 0 0
\(273\) −11.1589 11.1589i −0.0408752 0.0408752i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −205.406 + 205.406i −0.741539 + 0.741539i −0.972874 0.231335i \(-0.925691\pi\)
0.231335 + 0.972874i \(0.425691\pi\)
\(278\) 0 0
\(279\) 11.0908i 0.0397520i
\(280\) 0 0
\(281\) 334.899 1.19181 0.595906 0.803054i \(-0.296793\pi\)
0.595906 + 0.803054i \(0.296793\pi\)
\(282\) 0 0
\(283\) −74.3508 74.3508i −0.262724 0.262724i 0.563436 0.826160i \(-0.309479\pi\)
−0.826160 + 0.563436i \(0.809479\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −10.0000 + 10.0000i −0.0348432 + 0.0348432i
\(288\) 0 0
\(289\) 249.404i 0.862990i
\(290\) 0 0
\(291\) −132.959 −0.456904
\(292\) 0 0
\(293\) −116.874 116.874i −0.398887 0.398887i 0.478953 0.877840i \(-0.341016\pi\)
−0.877840 + 0.478953i \(0.841016\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 10.6515 10.6515i 0.0358637 0.0358637i
\(298\) 0 0
\(299\) 297.909i 0.996352i
\(300\) 0 0
\(301\) 76.3939 0.253800
\(302\) 0 0
\(303\) −61.8684 61.8684i −0.204186 0.204186i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −135.250 + 135.250i −0.440553 + 0.440553i −0.892198 0.451645i \(-0.850837\pi\)
0.451645 + 0.892198i \(0.350837\pi\)
\(308\) 0 0
\(309\) 64.1816i 0.207708i
\(310\) 0 0
\(311\) 239.212 0.769171 0.384586 0.923089i \(-0.374344\pi\)
0.384586 + 0.923089i \(0.374344\pi\)
\(312\) 0 0
\(313\) −271.386 271.386i −0.867048 0.867048i 0.125097 0.992145i \(-0.460076\pi\)
−0.992145 + 0.125097i \(0.960076\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −175.060 + 175.060i −0.552240 + 0.552240i −0.927087 0.374847i \(-0.877695\pi\)
0.374847 + 0.927087i \(0.377695\pi\)
\(318\) 0 0
\(319\) 93.6163i 0.293468i
\(320\) 0 0
\(321\) 18.3337 0.0571143
\(322\) 0 0
\(323\) −0.449490 0.449490i −0.00139161 0.00139161i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 115.831 115.831i 0.354223 0.354223i
\(328\) 0 0
\(329\) 4.42449i 0.0134483i
\(330\) 0 0
\(331\) 134.445 0.406178 0.203089 0.979160i \(-0.434902\pi\)
0.203089 + 0.979160i \(0.434902\pi\)
\(332\) 0 0
\(333\) −128.091 128.091i −0.384657 0.384657i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 137.654 137.654i 0.408468 0.408468i −0.472736 0.881204i \(-0.656734\pi\)
0.881204 + 0.472736i \(0.156734\pi\)
\(338\) 0 0
\(339\) 220.404i 0.650160i
\(340\) 0 0
\(341\) −10.7173 −0.0314292
\(342\) 0 0
\(343\) 75.0431 + 75.0431i 0.218785 + 0.218785i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 365.823 365.823i 1.05424 1.05424i 0.0558030 0.998442i \(-0.482228\pi\)
0.998442 0.0558030i \(-0.0177719\pi\)
\(348\) 0 0
\(349\) 140.020i 0.401205i −0.979673 0.200602i \(-0.935710\pi\)
0.979673 0.200602i \(-0.0642899\pi\)
\(350\) 0 0
\(351\) 43.1816 0.123025
\(352\) 0 0
\(353\) −208.672 208.672i −0.591139 0.591139i 0.346800 0.937939i \(-0.387268\pi\)
−0.937939 + 0.346800i \(0.887268\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −8.44949 + 8.44949i −0.0236680 + 0.0236680i
\(358\) 0 0
\(359\) 583.019i 1.62401i −0.583651 0.812005i \(-0.698376\pi\)
0.583651 0.812005i \(-0.301624\pi\)
\(360\) 0 0
\(361\) 360.990 0.999972
\(362\) 0 0
\(363\) −137.901 137.901i −0.379893 0.379893i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 123.341 123.341i 0.336078 0.336078i −0.518811 0.854889i \(-0.673625\pi\)
0.854889 + 0.518811i \(0.173625\pi\)
\(368\) 0 0
\(369\) 38.6969i 0.104870i
\(370\) 0 0
\(371\) 20.4541 0.0551323
\(372\) 0 0
\(373\) −345.052 345.052i −0.925073 0.925073i 0.0723091 0.997382i \(-0.476963\pi\)
−0.997382 + 0.0723091i \(0.976963\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 189.762 189.762i 0.503347 0.503347i
\(378\) 0 0
\(379\) 421.393i 1.11185i −0.831231 0.555927i \(-0.812363\pi\)
0.831231 0.555927i \(-0.187637\pi\)
\(380\) 0 0
\(381\) 91.3735 0.239825
\(382\) 0 0
\(383\) 344.586 + 344.586i 0.899702 + 0.899702i 0.995409 0.0957079i \(-0.0305115\pi\)
−0.0957079 + 0.995409i \(0.530511\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −147.810 + 147.810i −0.381939 + 0.381939i
\(388\) 0 0
\(389\) 486.111i 1.24964i −0.780768 0.624822i \(-0.785172\pi\)
0.780768 0.624822i \(-0.214828\pi\)
\(390\) 0 0
\(391\) −225.576 −0.576919
\(392\) 0 0
\(393\) −235.980 235.980i −0.600457 0.600457i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −312.062 + 312.062i −0.786052 + 0.786052i −0.980844 0.194793i \(-0.937597\pi\)
0.194793 + 0.980844i \(0.437597\pi\)
\(398\) 0 0
\(399\) 0.191836i 0.000480792i
\(400\) 0 0
\(401\) 590.252 1.47195 0.735975 0.677009i \(-0.236724\pi\)
0.735975 + 0.677009i \(0.236724\pi\)
\(402\) 0 0
\(403\) −21.7242 21.7242i −0.0539063 0.0539063i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 123.778 123.778i 0.304122 0.304122i
\(408\) 0 0
\(409\) 736.696i 1.80121i 0.434636 + 0.900606i \(0.356877\pi\)
−0.434636 + 0.900606i \(0.643123\pi\)
\(410\) 0 0
\(411\) 404.969 0.985327
\(412\) 0 0
\(413\) −59.2168 59.2168i −0.143382 0.143382i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −314.449 + 314.449i −0.754076 + 0.754076i
\(418\) 0 0
\(419\) 38.0296i 0.0907627i 0.998970 + 0.0453814i \(0.0144503\pi\)
−0.998970 + 0.0453814i \(0.985550\pi\)
\(420\) 0 0
\(421\) −331.394 −0.787159 −0.393579 0.919291i \(-0.628763\pi\)
−0.393579 + 0.919291i \(0.628763\pi\)
\(422\) 0 0
\(423\) 8.56072 + 8.56072i 0.0202381 + 0.0202381i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −80.4620 + 80.4620i −0.188436 + 0.188436i
\(428\) 0 0
\(429\) 41.7276i 0.0972670i
\(430\) 0 0
\(431\) 843.040 1.95601 0.978004 0.208584i \(-0.0668856\pi\)
0.978004 + 0.208584i \(0.0668856\pi\)
\(432\) 0 0
\(433\) −149.381 149.381i −0.344992 0.344992i 0.513248 0.858240i \(-0.328442\pi\)
−0.858240 + 0.513248i \(0.828442\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.56072 + 2.56072i −0.00585976 + 0.00585976i
\(438\) 0 0
\(439\) 545.999i 1.24373i −0.783123 0.621867i \(-0.786375\pi\)
0.783123 0.621867i \(-0.213625\pi\)
\(440\) 0 0
\(441\) −143.394 −0.325156
\(442\) 0 0
\(443\) −45.0398 45.0398i −0.101670 0.101670i 0.654442 0.756112i \(-0.272904\pi\)
−0.756112 + 0.654442i \(0.772904\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 236.116 236.116i 0.528223 0.528223i
\(448\) 0 0
\(449\) 320.767i 0.714404i 0.934027 + 0.357202i \(0.116269\pi\)
−0.934027 + 0.357202i \(0.883731\pi\)
\(450\) 0 0
\(451\) 37.3939 0.0829133
\(452\) 0 0
\(453\) −121.139 121.139i −0.267414 0.267414i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 599.918 599.918i 1.31273 1.31273i 0.393337 0.919394i \(-0.371320\pi\)
0.919394 0.393337i \(-0.128680\pi\)
\(458\) 0 0
\(459\) 32.6969i 0.0712352i
\(460\) 0 0
\(461\) −376.595 −0.816909 −0.408454 0.912779i \(-0.633932\pi\)
−0.408454 + 0.912779i \(0.633932\pi\)
\(462\) 0 0
\(463\) −214.838 214.838i −0.464012 0.464012i 0.435956 0.899968i \(-0.356410\pi\)
−0.899968 + 0.435956i \(0.856410\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 463.419 463.419i 0.992332 0.992332i −0.00763918 0.999971i \(-0.502432\pi\)
0.999971 + 0.00763918i \(0.00243165\pi\)
\(468\) 0 0
\(469\) 73.8490i 0.157461i
\(470\) 0 0
\(471\) −253.404 −0.538013
\(472\) 0 0
\(473\) −142.833 142.833i −0.301973 0.301973i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −39.5755 + 39.5755i −0.0829675 + 0.0829675i
\(478\) 0 0
\(479\) 415.394i 0.867211i −0.901103 0.433605i \(-0.857241\pi\)
0.901103 0.433605i \(-0.142759\pi\)
\(480\) 0 0
\(481\) 501.798 1.04324
\(482\) 0 0
\(483\) 48.1362 + 48.1362i 0.0996609 + 0.0996609i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 38.1033 38.1033i 0.0782409 0.0782409i −0.666903 0.745144i \(-0.732381\pi\)
0.745144 + 0.666903i \(0.232381\pi\)
\(488\) 0 0
\(489\) 46.8684i 0.0958453i
\(490\) 0 0
\(491\) −383.514 −0.781088 −0.390544 0.920584i \(-0.627713\pi\)
−0.390544 + 0.920584i \(0.627713\pi\)
\(492\) 0 0
\(493\) −143.687 143.687i −0.291454 0.291454i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −23.0306 + 23.0306i −0.0463393 + 0.0463393i
\(498\) 0 0
\(499\) 81.4133i 0.163153i −0.996667 0.0815764i \(-0.974005\pi\)
0.996667 0.0815764i \(-0.0259955\pi\)
\(500\) 0 0
\(501\) −236.858 −0.472771
\(502\) 0 0
\(503\) 171.626 + 171.626i 0.341204 + 0.341204i 0.856820 0.515616i \(-0.172437\pi\)
−0.515616 + 0.856820i \(0.672437\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 122.399 122.399i 0.241419 0.241419i
\(508\) 0 0
\(509\) 452.202i 0.888413i −0.895925 0.444206i \(-0.853486\pi\)
0.895925 0.444206i \(-0.146514\pi\)
\(510\) 0 0
\(511\) −5.43470 −0.0106354
\(512\) 0 0
\(513\) −0.371173 0.371173i −0.000723534 0.000723534i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −8.27245 + 8.27245i −0.0160009 + 0.0160009i
\(518\) 0 0
\(519\) 441.242i 0.850177i
\(520\) 0 0
\(521\) −773.928 −1.48547 −0.742733 0.669588i \(-0.766471\pi\)
−0.742733 + 0.669588i \(0.766471\pi\)
\(522\) 0 0
\(523\) −474.507 474.507i −0.907280 0.907280i 0.0887721 0.996052i \(-0.471706\pi\)
−0.996052 + 0.0887721i \(0.971706\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −16.4495 + 16.4495i −0.0312135 + 0.0312135i
\(528\) 0 0
\(529\) 756.090i 1.42928i
\(530\) 0 0
\(531\) 229.151 0.431546
\(532\) 0 0
\(533\) 75.7980 + 75.7980i 0.142210 + 0.142210i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 164.116 164.116i 0.305616 0.305616i
\(538\) 0 0
\(539\) 138.565i 0.257078i
\(540\) 0 0
\(541\) 511.867 0.946150 0.473075 0.881022i \(-0.343144\pi\)
0.473075 + 0.881022i \(0.343144\pi\)
\(542\) 0 0
\(543\) −210.149 210.149i −0.387014 0.387014i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −373.980 + 373.980i −0.683692 + 0.683692i −0.960830 0.277138i \(-0.910614\pi\)
0.277138 + 0.960830i \(0.410614\pi\)
\(548\) 0 0
\(549\) 311.363i 0.567146i
\(550\) 0 0
\(551\) −3.26224 −0.00592058
\(552\) 0 0
\(553\) 68.0658 + 68.0658i 0.123085 + 0.123085i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −421.015 + 421.015i −0.755861 + 0.755861i −0.975566 0.219705i \(-0.929491\pi\)
0.219705 + 0.975566i \(0.429491\pi\)
\(558\) 0 0
\(559\) 579.050i 1.03587i
\(560\) 0 0
\(561\) 31.5959 0.0563207
\(562\) 0 0
\(563\) 7.77296 + 7.77296i 0.0138063 + 0.0138063i 0.713976 0.700170i \(-0.246892\pi\)
−0.700170 + 0.713976i \(0.746892\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −6.97730 + 6.97730i −0.0123056 + 0.0123056i
\(568\) 0 0
\(569\) 473.787i 0.832666i −0.909212 0.416333i \(-0.863315\pi\)
0.909212 0.416333i \(-0.136685\pi\)
\(570\) 0 0
\(571\) 120.344 0.210760 0.105380 0.994432i \(-0.466394\pi\)
0.105380 + 0.994432i \(0.466394\pi\)
\(572\) 0 0
\(573\) −32.1770 32.1770i −0.0561554 0.0561554i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 266.033 266.033i 0.461062 0.461062i −0.437941 0.899004i \(-0.644292\pi\)
0.899004 + 0.437941i \(0.144292\pi\)
\(578\) 0 0
\(579\) 181.565i 0.313584i
\(580\) 0 0
\(581\) 126.727 0.218118
\(582\) 0 0
\(583\) −38.2429 38.2429i −0.0655967 0.0655967i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 114.141 114.141i 0.194448 0.194448i −0.603167 0.797615i \(-0.706095\pi\)
0.797615 + 0.603167i \(0.206095\pi\)
\(588\) 0 0
\(589\) 0.373467i 0.000634069i
\(590\) 0 0
\(591\) 116.969 0.197918
\(592\) 0 0
\(593\) 566.636 + 566.636i 0.955541 + 0.955541i 0.999053 0.0435121i \(-0.0138547\pi\)
−0.0435121 + 0.999053i \(0.513855\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −435.341 + 435.341i −0.729214 + 0.729214i
\(598\) 0 0
\(599\) 1002.44i 1.67353i 0.547564 + 0.836764i \(0.315555\pi\)
−0.547564 + 0.836764i \(0.684445\pi\)
\(600\) 0 0
\(601\) −20.8796 −0.0347414 −0.0173707 0.999849i \(-0.505530\pi\)
−0.0173707 + 0.999849i \(0.505530\pi\)
\(602\) 0 0
\(603\) 142.886 + 142.886i 0.236959 + 0.236959i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −305.748 + 305.748i −0.503703 + 0.503703i −0.912587 0.408883i \(-0.865918\pi\)
0.408883 + 0.912587i \(0.365918\pi\)
\(608\) 0 0
\(609\) 61.3235i 0.100695i
\(610\) 0 0
\(611\) −33.5367 −0.0548883
\(612\) 0 0
\(613\) 153.303 + 153.303i 0.250087 + 0.250087i 0.821006 0.570919i \(-0.193413\pi\)
−0.570919 + 0.821006i \(0.693413\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 276.879 276.879i 0.448750 0.448750i −0.446189 0.894939i \(-0.647219\pi\)
0.894939 + 0.446189i \(0.147219\pi\)
\(618\) 0 0
\(619\) 389.352i 0.629002i 0.949257 + 0.314501i \(0.101837\pi\)
−0.949257 + 0.314501i \(0.898163\pi\)
\(620\) 0 0
\(621\) −186.272 −0.299956
\(622\) 0 0
\(623\) 74.8786 + 74.8786i 0.120190 + 0.120190i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0.358674 0.358674i 0.000572048 0.000572048i
\(628\) 0 0
\(629\) 379.959i 0.604069i
\(630\) 0 0
\(631\) −576.201 −0.913155 −0.456578 0.889684i \(-0.650925\pi\)
−0.456578 + 0.889684i \(0.650925\pi\)
\(632\) 0 0
\(633\) 178.169 + 178.169i 0.281468 + 0.281468i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 280.874 280.874i 0.440932 0.440932i
\(638\) 0 0
\(639\) 89.1214i 0.139470i
\(640\) 0 0
\(641\) 780.827 1.21814 0.609069 0.793117i \(-0.291543\pi\)
0.609069 + 0.793117i \(0.291543\pi\)
\(642\) 0 0
\(643\) 403.787 + 403.787i 0.627973 + 0.627973i 0.947558 0.319585i \(-0.103543\pi\)
−0.319585 + 0.947558i \(0.603543\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −381.898 + 381.898i −0.590260 + 0.590260i −0.937702 0.347442i \(-0.887050\pi\)
0.347442 + 0.937702i \(0.387050\pi\)
\(648\) 0 0
\(649\) 221.435i 0.341194i
\(650\) 0 0
\(651\) 7.02041 0.0107840
\(652\) 0 0
\(653\) −149.864 149.864i −0.229500 0.229500i 0.582984 0.812484i \(-0.301885\pi\)
−0.812484 + 0.582984i \(0.801885\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 10.5153 10.5153i 0.0160050 0.0160050i
\(658\) 0 0
\(659\) 552.495i 0.838384i 0.907898 + 0.419192i \(0.137687\pi\)
−0.907898 + 0.419192i \(0.862313\pi\)
\(660\) 0 0
\(661\) −274.767 −0.415684 −0.207842 0.978162i \(-0.566644\pi\)
−0.207842 + 0.978162i \(0.566644\pi\)
\(662\) 0 0
\(663\) 64.0454 + 64.0454i 0.0965994 + 0.0965994i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −818.574 + 818.574i −1.22725 + 1.22725i
\(668\) 0 0
\(669\) 564.242i 0.843411i
\(670\) 0 0
\(671\) 300.879 0.448403
\(672\) 0 0
\(673\) 624.272 + 624.272i 0.927597 + 0.927597i 0.997550 0.0699537i \(-0.0222852\pi\)
−0.0699537 + 0.997550i \(0.522285\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.04541 4.04541i 0.00597549 0.00597549i −0.704113 0.710088i \(-0.748655\pi\)
0.710088 + 0.704113i \(0.248655\pi\)
\(678\) 0 0
\(679\) 84.1623i 0.123950i
\(680\) 0 0
\(681\) 99.1418 0.145583
\(682\) 0 0
\(683\) −913.757 913.757i −1.33786 1.33786i −0.898132 0.439726i \(-0.855075\pi\)
−0.439726 0.898132i \(-0.644925\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 257.431 257.431i 0.374718 0.374718i
\(688\) 0 0
\(689\) 155.038i 0.225018i
\(690\) 0 0
\(691\) −1286.24 −1.86142 −0.930710 0.365759i \(-0.880810\pi\)
−0.930710 + 0.365759i \(0.880810\pi\)
\(692\) 0 0
\(693\) −6.74235 6.74235i −0.00972922 0.00972922i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 57.3939 57.3939i 0.0823442 0.0823442i
\(698\) 0 0
\(699\) 725.989i 1.03861i
\(700\) 0 0
\(701\) −527.181 −0.752041 −0.376020 0.926611i \(-0.622708\pi\)
−0.376020 + 0.926611i \(0.622708\pi\)
\(702\) 0 0
\(703\) −4.31327 4.31327i −0.00613551 0.00613551i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −39.1623 + 39.1623i −0.0553922 + 0.0553922i
\(708\) 0 0
\(709\) 204.514i 0.288455i 0.989545 + 0.144227i \(0.0460696\pi\)
−0.989545 + 0.144227i \(0.953930\pi\)
\(710\) 0 0
\(711\) −263.394 −0.370456
\(712\) 0 0
\(713\) 93.7117 + 93.7117i 0.131433 + 0.131433i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 109.373 109.373i 0.152543 0.152543i
\(718\) 0 0
\(719\) 511.989i 0.712085i 0.934470 + 0.356042i \(0.115874\pi\)
−0.934470 + 0.356042i \(0.884126\pi\)
\(720\) 0 0
\(721\) 40.6265 0.0563475
\(722\) 0 0
\(723\) 147.724 + 147.724i 0.204321 + 0.204321i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 211.992 211.992i 0.291598 0.291598i −0.546113 0.837712i \(-0.683893\pi\)
0.837712 + 0.546113i \(0.183893\pi\)
\(728\) 0 0
\(729\) 27.0000i 0.0370370i
\(730\) 0 0
\(731\) −438.454 −0.599800
\(732\) 0 0
\(733\) 702.979 + 702.979i 0.959043 + 0.959043i 0.999194 0.0401506i \(-0.0127838\pi\)
−0.0401506 + 0.999194i \(0.512784\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −138.075 + 138.075i −0.187347 + 0.187347i
\(738\) 0 0
\(739\) 838.586i 1.13476i 0.823457 + 0.567379i \(0.192042\pi\)
−0.823457 + 0.567379i \(0.807958\pi\)
\(740\) 0 0
\(741\) 1.45408 0.00196232
\(742\) 0 0
\(743\) 962.534 + 962.534i 1.29547 + 1.29547i 0.931354 + 0.364115i \(0.118629\pi\)
0.364115 + 0.931354i \(0.381371\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −245.196 + 245.196i −0.328242 + 0.328242i
\(748\) 0 0
\(749\) 11.6051i 0.0154941i
\(750\) 0 0
\(751\) −594.241 −0.791266 −0.395633 0.918409i \(-0.629475\pi\)
−0.395633 + 0.918409i \(0.629475\pi\)
\(752\) 0 0
\(753\) 241.980 + 241.980i 0.321354 + 0.321354i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 399.402 399.402i 0.527611 0.527611i −0.392248 0.919859i \(-0.628303\pi\)
0.919859 + 0.392248i \(0.128303\pi\)
\(758\) 0 0
\(759\) 180.000i 0.237154i
\(760\) 0 0
\(761\) 127.292 0.167269 0.0836346 0.996496i \(-0.473347\pi\)
0.0836346 + 0.996496i \(0.473347\pi\)
\(762\) 0 0
\(763\) −73.3201 73.3201i −0.0960946 0.0960946i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −448.852 + 448.852i −0.585204 + 0.585204i
\(768\) 0 0
\(769\) 699.847i 0.910074i −0.890473 0.455037i \(-0.849626\pi\)
0.890473 0.455037i \(-0.150374\pi\)
\(770\) 0 0
\(771\) 548.363 0.711236
\(772\) 0 0
\(773\) 627.485 + 627.485i 0.811753 + 0.811753i 0.984897 0.173144i \(-0.0553926\pi\)
−0.173144 + 0.984897i \(0.555393\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −81.0806 + 81.0806i −0.104351 + 0.104351i
\(778\) 0 0
\(779\) 1.30306i 0.00167274i
\(780\) 0 0
\(781\) 86.1204 0.110269
\(782\) 0 0
\(783\) −118.652 118.652i −0.151535 0.151535i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −363.992 + 363.992i −0.462506 + 0.462506i −0.899476 0.436970i \(-0.856052\pi\)
0.436970 + 0.899476i \(0.356052\pi\)
\(788\) 0 0
\(789\) 222.334i 0.281792i
\(790\) 0 0
\(791\) 139.514 0.176377
\(792\) 0 0
\(793\) 609.885 + 609.885i 0.769086 + 0.769086i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −506.080 + 506.080i −0.634981 + 0.634981i −0.949313 0.314332i \(-0.898219\pi\)
0.314332 + 0.949313i \(0.398219\pi\)
\(798\) 0 0
\(799\) 25.3939i 0.0317821i
\(800\) 0 0
\(801\) −289.757 −0.361744
\(802\) 0 0
\(803\) 10.1612 + 10.1612i 0.0126541 + 0.0126541i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 443.246 443.246i 0.549252 0.549252i
\(808\) 0 0
\(809\) 1165.28i 1.44040i 0.693768 + 0.720198i \(0.255949\pi\)
−0.693768 + 0.720198i \(0.744051\pi\)
\(810\) 0 0
\(811\) 1507.67 1.85903 0.929516 0.368783i \(-0.120225\pi\)
0.929516 + 0.368783i \(0.120225\pi\)
\(812\) 0 0
\(813\) 265.510 + 265.510i 0.326580 + 0.326580i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −4.97730 + 4.97730i −0.00609216 + 0.00609216i
\(818\) 0 0
\(819\) 27.3337i 0.0333745i
\(820\) 0 0
\(821\) −965.959 −1.17656 −0.588282 0.808656i \(-0.700196\pi\)
−0.588282 + 0.808656i \(0.700196\pi\)
\(822\) 0 0
\(823\) −88.9727 88.9727i −0.108108 0.108108i 0.650984 0.759092i \(-0.274357\pi\)
−0.759092 + 0.650984i \(0.774357\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 275.271 275.271i 0.332855 0.332855i −0.520814 0.853670i \(-0.674372\pi\)
0.853670 + 0.520814i \(0.174372\pi\)
\(828\) 0 0
\(829\) 351.980i 0.424583i 0.977206 + 0.212292i \(0.0680927\pi\)
−0.977206 + 0.212292i \(0.931907\pi\)
\(830\) 0 0
\(831\) −503.141 −0.605464
\(832\) 0 0
\(833\) −212.677 212.677i −0.255314 0.255314i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −13.5834 + 13.5834i −0.0162287 + 0.0162287i
\(838\) 0 0
\(839\) 1408.43i 1.67871i 0.543587 + 0.839353i \(0.317066\pi\)
−0.543587 + 0.839353i \(0.682934\pi\)
\(840\) 0 0
\(841\) −201.829 −0.239986
\(842\) 0 0
\(843\) 410.166 + 410.166i 0.486555 + 0.486555i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −87.2906 + 87.2906i −0.103059 + 0.103059i
\(848\) 0 0
\(849\) 182.121i 0.214513i
\(850\) 0 0
\(851\) −2164.60 −2.54360
\(852\) 0 0
\(853\) −50.5528 50.5528i −0.0592647 0.0592647i 0.676853 0.736118i \(-0.263343\pi\)
−0.736118 + 0.676853i \(0.763343\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 210.434 210.434i 0.245547 0.245547i −0.573593 0.819140i \(-0.694451\pi\)
0.819140 + 0.573593i \(0.194451\pi\)
\(858\) 0 0
\(859\) 1255.45i 1.46153i −0.682630 0.730764i \(-0.739164\pi\)
0.682630 0.730764i \(-0.260836\pi\)
\(860\) 0 0
\(861\) −24.4949 −0.0284494
\(862\) 0 0
\(863\) −1007.70 1007.70i −1.16767 1.16767i −0.982755 0.184911i \(-0.940800\pi\)
−0.184911 0.982755i \(-0.559200\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −305.456 + 305.456i −0.352314 + 0.352314i
\(868\) 0 0
\(869\) 254.524i 0.292894i
\(870\) 0 0
\(871\) −559.760 −0.642664
\(872\) 0 0
\(873\) −162.841 162.841i −0.186530 0.186530i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −282.346 + 282.346i −0.321945 + 0.321945i −0.849513 0.527568i \(-0.823104\pi\)
0.527568 + 0.849513i \(0.323104\pi\)
\(878\) 0 0
\(879\) 286.282i 0.325690i
\(880\) 0 0
\(881\) −1420.34 −1.61219 −0.806096 0.591785i \(-0.798423\pi\)
−0.806096 + 0.591785i \(0.798423\pi\)
\(882\) 0 0
\(883\) 1170.15 + 1170.15i 1.32519 + 1.32519i 0.909508 + 0.415686i \(0.136459\pi\)
0.415686 + 0.909508i \(0.363541\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −756.711 + 756.711i −0.853112 + 0.853112i −0.990515 0.137403i \(-0.956125\pi\)
0.137403 + 0.990515i \(0.456125\pi\)
\(888\) 0 0
\(889\) 57.8388i 0.0650605i
\(890\) 0 0
\(891\) 26.0908 0.0292826
\(892\) 0 0
\(893\) 0.288269 + 0.288269i 0.000322810 + 0.000322810i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 364.863 364.863i 0.406759 0.406759i
\(898\) 0 0
\(899\) 119.385i 0.132797i
\(900\) 0 0
\(901\) −117.394 −0.130293
\(902\) 0 0
\(903\) 93.5630 + 93.5630i 0.103614 + 0.103614i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 942.979 942.979i 1.03967 1.03967i 0.0404876 0.999180i \(-0.487109\pi\)
0.999180 0.0404876i \(-0.0128911\pi\)
\(908\) 0 0
\(909\) 151.546i 0.166717i
\(910\) 0 0
\(911\) 1181.15 1.29654 0.648272 0.761409i \(-0.275492\pi\)
0.648272 + 0.761409i \(0.275492\pi\)
\(912\) 0 0
\(913\) −236.940 236.940i −0.259518 0.259518i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −149.373 + 149.373i −0.162894 + 0.162894i
\(918\) 0 0
\(919\) 473.595i 0.515337i 0.966233 + 0.257669i \(0.0829543\pi\)
−0.966233 + 0.257669i \(0.917046\pi\)
\(920\) 0 0
\(921\) −331.293 −0.359710
\(922\) 0 0
\(923\) 174.567 + 174.567i 0.189130 + 0.189130i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −78.6061 + 78.6061i −0.0847962 + 0.0847962i
\(928\) 0 0
\(929\) 1094.43i 1.17807i 0.808106 + 0.589037i \(0.200493\pi\)
−0.808106 + 0.589037i \(0.799507\pi\)
\(930\) 0 0
\(931\) −4.82857 −0.00518644
\(932\) 0 0
\(933\) 292.974 + 292.974i 0.314013 + 0.314013i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −654.891 + 654.891i −0.698923 + 0.698923i −0.964178 0.265255i \(-0.914544\pi\)
0.265255 + 0.964178i \(0.414544\pi\)
\(938\) 0 0
\(939\) 664.757i 0.707942i
\(940\) 0 0
\(941\) 766.382 0.814433 0.407217 0.913332i \(-0.366499\pi\)
0.407217 + 0.913332i \(0.366499\pi\)
\(942\) 0 0
\(943\) −326.969 326.969i −0.346733 0.346733i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 325.348 325.348i 0.343557 0.343557i −0.514146 0.857703i \(-0.671891\pi\)
0.857703 + 0.514146i \(0.171891\pi\)
\(948\) 0 0
\(949\) 41.1939i 0.0434077i
\(950\) 0 0
\(951\) −428.808 −0.450902
\(952\) 0 0
\(953\) 31.1010 + 31.1010i 0.0326349 + 0.0326349i 0.723236 0.690601i \(-0.242654\pi\)
−0.690601 + 0.723236i \(0.742654\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 114.656 114.656i 0.119808 0.119808i
\(958\) 0 0
\(959\) 256.343i 0.267302i
\(960\) 0 0
\(961\) −947.333 −0.985778
\(962\) 0 0
\(963\) 22.4541 + 22.4541i 0.0233168 + 0.0233168i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 314.656 314.656i 0.325394 0.325394i −0.525438 0.850832i \(-0.676098\pi\)
0.850832 + 0.525438i \(0.176098\pi\)
\(968\) 0 0
\(969\) 1.10102i 0.00113624i
\(970\) 0 0
\(971\) 253.614 0.261189 0.130594 0.991436i \(-0.458311\pi\)
0.130594 + 0.991436i \(0.458311\pi\)
\(972\) 0 0
\(973\) 199.044 + 199.044i 0.204568 + 0.204568i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −580.974 + 580.974i −0.594651 + 0.594651i −0.938884 0.344233i \(-0.888139\pi\)
0.344233 + 0.938884i \(0.388139\pi\)
\(978\) 0 0
\(979\) 280.000i 0.286006i
\(980\) 0 0
\(981\) 283.727 0.289222
\(982\) 0 0
\(983\) 734.070 + 734.070i 0.746765 + 0.746765i 0.973870 0.227105i \(-0.0729260\pi\)
−0.227105 + 0.973870i \(0.572926\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 5.41887 5.41887i 0.00549025 0.00549025i
\(988\) 0 0
\(989\) 2497.85i 2.52563i
\(990\) 0 0
\(991\) −616.887 −0.622489 −0.311245 0.950330i \(-0.600746\pi\)
−0.311245 + 0.950330i \(0.600746\pi\)
\(992\) 0 0
\(993\) 164.661 + 164.661i 0.165821 + 0.165821i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 460.191 460.191i 0.461576 0.461576i −0.437596 0.899172i \(-0.644170\pi\)
0.899172 + 0.437596i \(0.144170\pi\)
\(998\) 0 0
\(999\) 313.757i 0.314071i
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.l.193.2 4
4.3 odd 2 600.3.u.c.193.1 4
5.2 odd 4 inner 1200.3.bg.l.1057.2 4
5.3 odd 4 1200.3.bg.f.1057.1 4
5.4 even 2 1200.3.bg.f.193.1 4
12.11 even 2 1800.3.v.l.793.2 4
20.3 even 4 600.3.u.d.457.2 yes 4
20.7 even 4 600.3.u.c.457.1 yes 4
20.19 odd 2 600.3.u.d.193.2 yes 4
60.23 odd 4 1800.3.v.m.1657.1 4
60.47 odd 4 1800.3.v.l.1657.2 4
60.59 even 2 1800.3.v.m.793.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.3.u.c.193.1 4 4.3 odd 2
600.3.u.c.457.1 yes 4 20.7 even 4
600.3.u.d.193.2 yes 4 20.19 odd 2
600.3.u.d.457.2 yes 4 20.3 even 4
1200.3.bg.f.193.1 4 5.4 even 2
1200.3.bg.f.1057.1 4 5.3 odd 4
1200.3.bg.l.193.2 4 1.1 even 1 trivial
1200.3.bg.l.1057.2 4 5.2 odd 4 inner
1800.3.v.l.793.2 4 12.11 even 2
1800.3.v.l.1657.2 4 60.47 odd 4
1800.3.v.m.793.1 4 60.59 even 2
1800.3.v.m.1657.1 4 60.23 odd 4