Properties

Label 1200.4.f.k.49.1
Level $1200$
Weight $4$
Character 1200.49
Analytic conductor $70.802$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,4,Mod(49,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1200.49
Dual form 1200.4.f.k.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} +13.0000i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} +13.0000i q^{7} -9.00000 q^{9} -6.00000 q^{11} -5.00000i q^{13} -78.0000i q^{17} +65.0000 q^{19} +39.0000 q^{21} +138.000i q^{23} +27.0000i q^{27} -66.0000 q^{29} -299.000 q^{31} +18.0000i q^{33} -214.000i q^{37} -15.0000 q^{39} +360.000 q^{41} +203.000i q^{43} -78.0000i q^{47} +174.000 q^{49} -234.000 q^{51} -636.000i q^{53} -195.000i q^{57} +786.000 q^{59} +467.000 q^{61} -117.000i q^{63} +217.000i q^{67} +414.000 q^{69} +360.000 q^{71} +286.000i q^{73} -78.0000i q^{77} +272.000 q^{79} +81.0000 q^{81} +498.000i q^{83} +198.000i q^{87} +65.0000 q^{91} +897.000i q^{93} -511.000i q^{97} +54.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} - 12 q^{11} + 130 q^{19} + 78 q^{21} - 132 q^{29} - 598 q^{31} - 30 q^{39} + 720 q^{41} + 348 q^{49} - 468 q^{51} + 1572 q^{59} + 934 q^{61} + 828 q^{69} + 720 q^{71} + 544 q^{79} + 162 q^{81} + 130 q^{91} + 108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 3.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 13.0000i 0.701934i 0.936388 + 0.350967i \(0.114147\pi\)
−0.936388 + 0.350967i \(0.885853\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −6.00000 −0.164461 −0.0822304 0.996613i \(-0.526204\pi\)
−0.0822304 + 0.996613i \(0.526204\pi\)
\(12\) 0 0
\(13\) − 5.00000i − 0.106673i −0.998577 0.0533366i \(-0.983014\pi\)
0.998577 0.0533366i \(-0.0169856\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 78.0000i − 1.11281i −0.830911 0.556405i \(-0.812180\pi\)
0.830911 0.556405i \(-0.187820\pi\)
\(18\) 0 0
\(19\) 65.0000 0.784843 0.392422 0.919785i \(-0.371637\pi\)
0.392422 + 0.919785i \(0.371637\pi\)
\(20\) 0 0
\(21\) 39.0000 0.405262
\(22\) 0 0
\(23\) 138.000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) −66.0000 −0.422617 −0.211308 0.977419i \(-0.567772\pi\)
−0.211308 + 0.977419i \(0.567772\pi\)
\(30\) 0 0
\(31\) −299.000 −1.73232 −0.866161 0.499765i \(-0.833420\pi\)
−0.866161 + 0.499765i \(0.833420\pi\)
\(32\) 0 0
\(33\) 18.0000i 0.0949514i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 214.000i − 0.950848i −0.879757 0.475424i \(-0.842295\pi\)
0.879757 0.475424i \(-0.157705\pi\)
\(38\) 0 0
\(39\) −15.0000 −0.0615878
\(40\) 0 0
\(41\) 360.000 1.37128 0.685641 0.727940i \(-0.259522\pi\)
0.685641 + 0.727940i \(0.259522\pi\)
\(42\) 0 0
\(43\) 203.000i 0.719935i 0.932965 + 0.359968i \(0.117212\pi\)
−0.932965 + 0.359968i \(0.882788\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 78.0000i − 0.242074i −0.992648 0.121037i \(-0.961378\pi\)
0.992648 0.121037i \(-0.0386219\pi\)
\(48\) 0 0
\(49\) 174.000 0.507289
\(50\) 0 0
\(51\) −234.000 −0.642481
\(52\) 0 0
\(53\) − 636.000i − 1.64833i −0.566352 0.824163i \(-0.691646\pi\)
0.566352 0.824163i \(-0.308354\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 195.000i − 0.453129i
\(58\) 0 0
\(59\) 786.000 1.73438 0.867191 0.497976i \(-0.165923\pi\)
0.867191 + 0.497976i \(0.165923\pi\)
\(60\) 0 0
\(61\) 467.000 0.980217 0.490108 0.871662i \(-0.336957\pi\)
0.490108 + 0.871662i \(0.336957\pi\)
\(62\) 0 0
\(63\) − 117.000i − 0.233978i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 217.000i 0.395683i 0.980234 + 0.197842i \(0.0633932\pi\)
−0.980234 + 0.197842i \(0.936607\pi\)
\(68\) 0 0
\(69\) 414.000 0.722315
\(70\) 0 0
\(71\) 360.000 0.601748 0.300874 0.953664i \(-0.402722\pi\)
0.300874 + 0.953664i \(0.402722\pi\)
\(72\) 0 0
\(73\) 286.000i 0.458545i 0.973362 + 0.229272i \(0.0736347\pi\)
−0.973362 + 0.229272i \(0.926365\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 78.0000i − 0.115441i
\(78\) 0 0
\(79\) 272.000 0.387372 0.193686 0.981064i \(-0.437956\pi\)
0.193686 + 0.981064i \(0.437956\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 498.000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 198.000i 0.243998i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 65.0000 0.0748775
\(92\) 0 0
\(93\) 897.000i 1.00016i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 511.000i − 0.534889i −0.963573 0.267444i \(-0.913821\pi\)
0.963573 0.267444i \(-0.0861791\pi\)
\(98\) 0 0
\(99\) 54.0000 0.0548202
\(100\) 0 0
\(101\) −1812.00 −1.78516 −0.892578 0.450893i \(-0.851106\pi\)
−0.892578 + 0.450893i \(0.851106\pi\)
\(102\) 0 0
\(103\) − 1708.00i − 1.63392i −0.576691 0.816962i \(-0.695656\pi\)
0.576691 0.816962i \(-0.304344\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1236.00i − 1.11672i −0.829600 0.558358i \(-0.811432\pi\)
0.829600 0.558358i \(-0.188568\pi\)
\(108\) 0 0
\(109\) 1543.00 1.35590 0.677948 0.735110i \(-0.262870\pi\)
0.677948 + 0.735110i \(0.262870\pi\)
\(110\) 0 0
\(111\) −642.000 −0.548972
\(112\) 0 0
\(113\) − 1884.00i − 1.56842i −0.620494 0.784212i \(-0.713068\pi\)
0.620494 0.784212i \(-0.286932\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 45.0000i 0.0355577i
\(118\) 0 0
\(119\) 1014.00 0.781120
\(120\) 0 0
\(121\) −1295.00 −0.972953
\(122\) 0 0
\(123\) − 1080.00i − 0.791710i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 2072.00i − 1.44772i −0.689948 0.723859i \(-0.742366\pi\)
0.689948 0.723859i \(-0.257634\pi\)
\(128\) 0 0
\(129\) 609.000 0.415655
\(130\) 0 0
\(131\) 2508.00 1.67271 0.836355 0.548188i \(-0.184682\pi\)
0.836355 + 0.548188i \(0.184682\pi\)
\(132\) 0 0
\(133\) 845.000i 0.550908i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1566.00i 0.976587i 0.872679 + 0.488293i \(0.162380\pi\)
−0.872679 + 0.488293i \(0.837620\pi\)
\(138\) 0 0
\(139\) −196.000 −0.119601 −0.0598004 0.998210i \(-0.519046\pi\)
−0.0598004 + 0.998210i \(0.519046\pi\)
\(140\) 0 0
\(141\) −234.000 −0.139761
\(142\) 0 0
\(143\) 30.0000i 0.0175435i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 522.000i − 0.292883i
\(148\) 0 0
\(149\) 1278.00 0.702670 0.351335 0.936250i \(-0.385728\pi\)
0.351335 + 0.936250i \(0.385728\pi\)
\(150\) 0 0
\(151\) −1385.00 −0.746422 −0.373211 0.927747i \(-0.621743\pi\)
−0.373211 + 0.927747i \(0.621743\pi\)
\(152\) 0 0
\(153\) 702.000i 0.370937i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 481.000i − 0.244509i −0.992499 0.122255i \(-0.960988\pi\)
0.992499 0.122255i \(-0.0390125\pi\)
\(158\) 0 0
\(159\) −1908.00 −0.951662
\(160\) 0 0
\(161\) −1794.00 −0.878180
\(162\) 0 0
\(163\) − 2815.00i − 1.35269i −0.736587 0.676343i \(-0.763564\pi\)
0.736587 0.676343i \(-0.236436\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 1956.00i − 0.906346i −0.891423 0.453173i \(-0.850292\pi\)
0.891423 0.453173i \(-0.149708\pi\)
\(168\) 0 0
\(169\) 2172.00 0.988621
\(170\) 0 0
\(171\) −585.000 −0.261614
\(172\) 0 0
\(173\) − 2382.00i − 1.04682i −0.852081 0.523411i \(-0.824659\pi\)
0.852081 0.523411i \(-0.175341\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 2358.00i − 1.00135i
\(178\) 0 0
\(179\) 4386.00 1.83142 0.915712 0.401834i \(-0.131627\pi\)
0.915712 + 0.401834i \(0.131627\pi\)
\(180\) 0 0
\(181\) −2275.00 −0.934251 −0.467125 0.884191i \(-0.654710\pi\)
−0.467125 + 0.884191i \(0.654710\pi\)
\(182\) 0 0
\(183\) − 1401.00i − 0.565928i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 468.000i 0.183014i
\(188\) 0 0
\(189\) −351.000 −0.135087
\(190\) 0 0
\(191\) −714.000 −0.270488 −0.135244 0.990812i \(-0.543182\pi\)
−0.135244 + 0.990812i \(0.543182\pi\)
\(192\) 0 0
\(193\) − 1547.00i − 0.576971i −0.957484 0.288486i \(-0.906848\pi\)
0.957484 0.288486i \(-0.0931518\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 438.000i 0.158407i 0.996858 + 0.0792036i \(0.0252377\pi\)
−0.996858 + 0.0792036i \(0.974762\pi\)
\(198\) 0 0
\(199\) 437.000 0.155669 0.0778344 0.996966i \(-0.475199\pi\)
0.0778344 + 0.996966i \(0.475199\pi\)
\(200\) 0 0
\(201\) 651.000 0.228448
\(202\) 0 0
\(203\) − 858.000i − 0.296649i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 1242.00i − 0.417029i
\(208\) 0 0
\(209\) −390.000 −0.129076
\(210\) 0 0
\(211\) −1625.00 −0.530188 −0.265094 0.964223i \(-0.585403\pi\)
−0.265094 + 0.964223i \(0.585403\pi\)
\(212\) 0 0
\(213\) − 1080.00i − 0.347420i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 3887.00i − 1.21598i
\(218\) 0 0
\(219\) 858.000 0.264741
\(220\) 0 0
\(221\) −390.000 −0.118707
\(222\) 0 0
\(223\) 875.000i 0.262755i 0.991332 + 0.131377i \(0.0419400\pi\)
−0.991332 + 0.131377i \(0.958060\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4680.00i 1.36838i 0.729303 + 0.684191i \(0.239844\pi\)
−0.729303 + 0.684191i \(0.760156\pi\)
\(228\) 0 0
\(229\) −1469.00 −0.423905 −0.211953 0.977280i \(-0.567982\pi\)
−0.211953 + 0.977280i \(0.567982\pi\)
\(230\) 0 0
\(231\) −234.000 −0.0666497
\(232\) 0 0
\(233\) − 3960.00i − 1.11343i −0.830705 0.556713i \(-0.812062\pi\)
0.830705 0.556713i \(-0.187938\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 816.000i − 0.223649i
\(238\) 0 0
\(239\) 2652.00 0.717756 0.358878 0.933385i \(-0.383159\pi\)
0.358878 + 0.933385i \(0.383159\pi\)
\(240\) 0 0
\(241\) 5753.00 1.53769 0.768845 0.639435i \(-0.220832\pi\)
0.768845 + 0.639435i \(0.220832\pi\)
\(242\) 0 0
\(243\) − 243.000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 325.000i − 0.0837217i
\(248\) 0 0
\(249\) 1494.00 0.380235
\(250\) 0 0
\(251\) −3588.00 −0.902281 −0.451141 0.892453i \(-0.648983\pi\)
−0.451141 + 0.892453i \(0.648983\pi\)
\(252\) 0 0
\(253\) − 828.000i − 0.205755i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 564.000i − 0.136892i −0.997655 0.0684462i \(-0.978196\pi\)
0.997655 0.0684462i \(-0.0218042\pi\)
\(258\) 0 0
\(259\) 2782.00 0.667433
\(260\) 0 0
\(261\) 594.000 0.140872
\(262\) 0 0
\(263\) 1248.00i 0.292604i 0.989240 + 0.146302i \(0.0467372\pi\)
−0.989240 + 0.146302i \(0.953263\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7266.00 1.64690 0.823450 0.567390i \(-0.192047\pi\)
0.823450 + 0.567390i \(0.192047\pi\)
\(270\) 0 0
\(271\) −3224.00 −0.722672 −0.361336 0.932436i \(-0.617679\pi\)
−0.361336 + 0.932436i \(0.617679\pi\)
\(272\) 0 0
\(273\) − 195.000i − 0.0432305i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 151.000i − 0.0327535i −0.999866 0.0163767i \(-0.994787\pi\)
0.999866 0.0163767i \(-0.00521311\pi\)
\(278\) 0 0
\(279\) 2691.00 0.577441
\(280\) 0 0
\(281\) 7566.00 1.60623 0.803113 0.595826i \(-0.203175\pi\)
0.803113 + 0.595826i \(0.203175\pi\)
\(282\) 0 0
\(283\) 1469.00i 0.308562i 0.988027 + 0.154281i \(0.0493061\pi\)
−0.988027 + 0.154281i \(0.950694\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4680.00i 0.962549i
\(288\) 0 0
\(289\) −1171.00 −0.238347
\(290\) 0 0
\(291\) −1533.00 −0.308818
\(292\) 0 0
\(293\) 306.000i 0.0610127i 0.999535 + 0.0305063i \(0.00971197\pi\)
−0.999535 + 0.0305063i \(0.990288\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 162.000i − 0.0316505i
\(298\) 0 0
\(299\) 690.000 0.133457
\(300\) 0 0
\(301\) −2639.00 −0.505347
\(302\) 0 0
\(303\) 5436.00i 1.03066i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 4661.00i − 0.866506i −0.901272 0.433253i \(-0.857366\pi\)
0.901272 0.433253i \(-0.142634\pi\)
\(308\) 0 0
\(309\) −5124.00 −0.943347
\(310\) 0 0
\(311\) −2514.00 −0.458379 −0.229189 0.973382i \(-0.573608\pi\)
−0.229189 + 0.973382i \(0.573608\pi\)
\(312\) 0 0
\(313\) − 6707.00i − 1.21119i −0.795774 0.605594i \(-0.792935\pi\)
0.795774 0.605594i \(-0.207065\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4632.00i 0.820691i 0.911930 + 0.410345i \(0.134592\pi\)
−0.911930 + 0.410345i \(0.865408\pi\)
\(318\) 0 0
\(319\) 396.000 0.0695039
\(320\) 0 0
\(321\) −3708.00 −0.644736
\(322\) 0 0
\(323\) − 5070.00i − 0.873382i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 4629.00i − 0.782827i
\(328\) 0 0
\(329\) 1014.00 0.169920
\(330\) 0 0
\(331\) 988.000 0.164065 0.0820323 0.996630i \(-0.473859\pi\)
0.0820323 + 0.996630i \(0.473859\pi\)
\(332\) 0 0
\(333\) 1926.00i 0.316949i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 9079.00i − 1.46755i −0.679392 0.733775i \(-0.737756\pi\)
0.679392 0.733775i \(-0.262244\pi\)
\(338\) 0 0
\(339\) −5652.00 −0.905530
\(340\) 0 0
\(341\) 1794.00 0.284899
\(342\) 0 0
\(343\) 6721.00i 1.05802i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5994.00i 0.927305i 0.886017 + 0.463652i \(0.153461\pi\)
−0.886017 + 0.463652i \(0.846539\pi\)
\(348\) 0 0
\(349\) 286.000 0.0438660 0.0219330 0.999759i \(-0.493018\pi\)
0.0219330 + 0.999759i \(0.493018\pi\)
\(350\) 0 0
\(351\) 135.000 0.0205293
\(352\) 0 0
\(353\) 11466.0i 1.72882i 0.502787 + 0.864410i \(0.332308\pi\)
−0.502787 + 0.864410i \(0.667692\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 3042.00i − 0.450980i
\(358\) 0 0
\(359\) −624.000 −0.0917367 −0.0458683 0.998947i \(-0.514605\pi\)
−0.0458683 + 0.998947i \(0.514605\pi\)
\(360\) 0 0
\(361\) −2634.00 −0.384021
\(362\) 0 0
\(363\) 3885.00i 0.561734i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 8477.00i − 1.20571i −0.797851 0.602855i \(-0.794030\pi\)
0.797851 0.602855i \(-0.205970\pi\)
\(368\) 0 0
\(369\) −3240.00 −0.457094
\(370\) 0 0
\(371\) 8268.00 1.15702
\(372\) 0 0
\(373\) 11257.0i 1.56264i 0.624130 + 0.781321i \(0.285454\pi\)
−0.624130 + 0.781321i \(0.714546\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 330.000i 0.0450819i
\(378\) 0 0
\(379\) −1213.00 −0.164400 −0.0822000 0.996616i \(-0.526195\pi\)
−0.0822000 + 0.996616i \(0.526195\pi\)
\(380\) 0 0
\(381\) −6216.00 −0.835841
\(382\) 0 0
\(383\) − 11076.0i − 1.47769i −0.673873 0.738847i \(-0.735370\pi\)
0.673873 0.738847i \(-0.264630\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 1827.00i − 0.239978i
\(388\) 0 0
\(389\) 8016.00 1.04480 0.522400 0.852700i \(-0.325037\pi\)
0.522400 + 0.852700i \(0.325037\pi\)
\(390\) 0 0
\(391\) 10764.0 1.39222
\(392\) 0 0
\(393\) − 7524.00i − 0.965739i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 6851.00i 0.866100i 0.901370 + 0.433050i \(0.142563\pi\)
−0.901370 + 0.433050i \(0.857437\pi\)
\(398\) 0 0
\(399\) 2535.00 0.318067
\(400\) 0 0
\(401\) 12636.0 1.57360 0.786798 0.617211i \(-0.211738\pi\)
0.786798 + 0.617211i \(0.211738\pi\)
\(402\) 0 0
\(403\) 1495.00i 0.184792i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1284.00i 0.156377i
\(408\) 0 0
\(409\) −10829.0 −1.30919 −0.654596 0.755979i \(-0.727161\pi\)
−0.654596 + 0.755979i \(0.727161\pi\)
\(410\) 0 0
\(411\) 4698.00 0.563833
\(412\) 0 0
\(413\) 10218.0i 1.21742i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 588.000i 0.0690515i
\(418\) 0 0
\(419\) −6252.00 −0.728950 −0.364475 0.931213i \(-0.618752\pi\)
−0.364475 + 0.931213i \(0.618752\pi\)
\(420\) 0 0
\(421\) −15730.0 −1.82098 −0.910491 0.413529i \(-0.864296\pi\)
−0.910491 + 0.413529i \(0.864296\pi\)
\(422\) 0 0
\(423\) 702.000i 0.0806913i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 6071.00i 0.688047i
\(428\) 0 0
\(429\) 90.0000 0.0101288
\(430\) 0 0
\(431\) 10062.0 1.12452 0.562262 0.826959i \(-0.309931\pi\)
0.562262 + 0.826959i \(0.309931\pi\)
\(432\) 0 0
\(433\) − 707.000i − 0.0784671i −0.999230 0.0392335i \(-0.987508\pi\)
0.999230 0.0392335i \(-0.0124916\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8970.00i 0.981907i
\(438\) 0 0
\(439\) −12493.0 −1.35822 −0.679110 0.734037i \(-0.737634\pi\)
−0.679110 + 0.734037i \(0.737634\pi\)
\(440\) 0 0
\(441\) −1566.00 −0.169096
\(442\) 0 0
\(443\) − 5928.00i − 0.635774i −0.948129 0.317887i \(-0.897027\pi\)
0.948129 0.317887i \(-0.102973\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 3834.00i − 0.405687i
\(448\) 0 0
\(449\) 2028.00 0.213156 0.106578 0.994304i \(-0.466011\pi\)
0.106578 + 0.994304i \(0.466011\pi\)
\(450\) 0 0
\(451\) −2160.00 −0.225522
\(452\) 0 0
\(453\) 4155.00i 0.430947i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 17030.0i 1.74317i 0.490242 + 0.871586i \(0.336908\pi\)
−0.490242 + 0.871586i \(0.663092\pi\)
\(458\) 0 0
\(459\) 2106.00 0.214160
\(460\) 0 0
\(461\) −1044.00 −0.105475 −0.0527374 0.998608i \(-0.516795\pi\)
−0.0527374 + 0.998608i \(0.516795\pi\)
\(462\) 0 0
\(463\) 9704.00i 0.974046i 0.873389 + 0.487023i \(0.161917\pi\)
−0.873389 + 0.487023i \(0.838083\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4194.00i 0.415579i 0.978174 + 0.207789i \(0.0666268\pi\)
−0.978174 + 0.207789i \(0.933373\pi\)
\(468\) 0 0
\(469\) −2821.00 −0.277743
\(470\) 0 0
\(471\) −1443.00 −0.141168
\(472\) 0 0
\(473\) − 1218.00i − 0.118401i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 5724.00i 0.549442i
\(478\) 0 0
\(479\) −12174.0 −1.16126 −0.580631 0.814167i \(-0.697194\pi\)
−0.580631 + 0.814167i \(0.697194\pi\)
\(480\) 0 0
\(481\) −1070.00 −0.101430
\(482\) 0 0
\(483\) 5382.00i 0.507018i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1651.00i 0.153622i 0.997046 + 0.0768110i \(0.0244738\pi\)
−0.997046 + 0.0768110i \(0.975526\pi\)
\(488\) 0 0
\(489\) −8445.00 −0.780974
\(490\) 0 0
\(491\) 6456.00 0.593391 0.296696 0.954972i \(-0.404115\pi\)
0.296696 + 0.954972i \(0.404115\pi\)
\(492\) 0 0
\(493\) 5148.00i 0.470293i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4680.00i 0.422388i
\(498\) 0 0
\(499\) −559.000 −0.0501489 −0.0250744 0.999686i \(-0.507982\pi\)
−0.0250744 + 0.999686i \(0.507982\pi\)
\(500\) 0 0
\(501\) −5868.00 −0.523279
\(502\) 0 0
\(503\) − 14208.0i − 1.25945i −0.776818 0.629725i \(-0.783168\pi\)
0.776818 0.629725i \(-0.216832\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 6516.00i − 0.570781i
\(508\) 0 0
\(509\) 8082.00 0.703789 0.351894 0.936040i \(-0.385538\pi\)
0.351894 + 0.936040i \(0.385538\pi\)
\(510\) 0 0
\(511\) −3718.00 −0.321868
\(512\) 0 0
\(513\) 1755.00i 0.151043i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 468.000i 0.0398116i
\(518\) 0 0
\(519\) −7146.00 −0.604383
\(520\) 0 0
\(521\) 20502.0 1.72401 0.862005 0.506900i \(-0.169209\pi\)
0.862005 + 0.506900i \(0.169209\pi\)
\(522\) 0 0
\(523\) 2069.00i 0.172985i 0.996253 + 0.0864924i \(0.0275658\pi\)
−0.996253 + 0.0864924i \(0.972434\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 23322.0i 1.92775i
\(528\) 0 0
\(529\) −6877.00 −0.565217
\(530\) 0 0
\(531\) −7074.00 −0.578127
\(532\) 0 0
\(533\) − 1800.00i − 0.146279i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 13158.0i − 1.05737i
\(538\) 0 0
\(539\) −1044.00 −0.0834291
\(540\) 0 0
\(541\) 1553.00 0.123417 0.0617086 0.998094i \(-0.480345\pi\)
0.0617086 + 0.998094i \(0.480345\pi\)
\(542\) 0 0
\(543\) 6825.00i 0.539390i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 12584.0i − 0.983643i −0.870696 0.491822i \(-0.836331\pi\)
0.870696 0.491822i \(-0.163669\pi\)
\(548\) 0 0
\(549\) −4203.00 −0.326739
\(550\) 0 0
\(551\) −4290.00 −0.331688
\(552\) 0 0
\(553\) 3536.00i 0.271910i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 20754.0i 1.57877i 0.613898 + 0.789385i \(0.289601\pi\)
−0.613898 + 0.789385i \(0.710399\pi\)
\(558\) 0 0
\(559\) 1015.00 0.0767977
\(560\) 0 0
\(561\) 1404.00 0.105663
\(562\) 0 0
\(563\) 14370.0i 1.07571i 0.843038 + 0.537854i \(0.180765\pi\)
−0.843038 + 0.537854i \(0.819235\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1053.00i 0.0779927i
\(568\) 0 0
\(569\) −15774.0 −1.16218 −0.581090 0.813839i \(-0.697374\pi\)
−0.581090 + 0.813839i \(0.697374\pi\)
\(570\) 0 0
\(571\) 3055.00 0.223902 0.111951 0.993714i \(-0.464290\pi\)
0.111951 + 0.993714i \(0.464290\pi\)
\(572\) 0 0
\(573\) 2142.00i 0.156166i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 25021.0i − 1.80526i −0.430412 0.902632i \(-0.641632\pi\)
0.430412 0.902632i \(-0.358368\pi\)
\(578\) 0 0
\(579\) −4641.00 −0.333115
\(580\) 0 0
\(581\) −6474.00 −0.462284
\(582\) 0 0
\(583\) 3816.00i 0.271085i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18828.0i 1.32388i 0.749559 + 0.661938i \(0.230266\pi\)
−0.749559 + 0.661938i \(0.769734\pi\)
\(588\) 0 0
\(589\) −19435.0 −1.35960
\(590\) 0 0
\(591\) 1314.00 0.0914564
\(592\) 0 0
\(593\) − 19968.0i − 1.38278i −0.722483 0.691389i \(-0.756999\pi\)
0.722483 0.691389i \(-0.243001\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 1311.00i − 0.0898755i
\(598\) 0 0
\(599\) 9684.00 0.660563 0.330282 0.943882i \(-0.392856\pi\)
0.330282 + 0.943882i \(0.392856\pi\)
\(600\) 0 0
\(601\) 23243.0 1.57754 0.788770 0.614688i \(-0.210718\pi\)
0.788770 + 0.614688i \(0.210718\pi\)
\(602\) 0 0
\(603\) − 1953.00i − 0.131894i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 22984.0i 1.53689i 0.639916 + 0.768445i \(0.278969\pi\)
−0.639916 + 0.768445i \(0.721031\pi\)
\(608\) 0 0
\(609\) −2574.00 −0.171271
\(610\) 0 0
\(611\) −390.000 −0.0258228
\(612\) 0 0
\(613\) − 13754.0i − 0.906230i −0.891452 0.453115i \(-0.850313\pi\)
0.891452 0.453115i \(-0.149687\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9468.00i 0.617775i 0.951099 + 0.308888i \(0.0999567\pi\)
−0.951099 + 0.308888i \(0.900043\pi\)
\(618\) 0 0
\(619\) −28099.0 −1.82455 −0.912273 0.409582i \(-0.865674\pi\)
−0.912273 + 0.409582i \(0.865674\pi\)
\(620\) 0 0
\(621\) −3726.00 −0.240772
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1170.00i 0.0745220i
\(628\) 0 0
\(629\) −16692.0 −1.05811
\(630\) 0 0
\(631\) −14807.0 −0.934164 −0.467082 0.884214i \(-0.654695\pi\)
−0.467082 + 0.884214i \(0.654695\pi\)
\(632\) 0 0
\(633\) 4875.00i 0.306104i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 870.000i − 0.0541141i
\(638\) 0 0
\(639\) −3240.00 −0.200583
\(640\) 0 0
\(641\) 20844.0 1.28438 0.642191 0.766545i \(-0.278026\pi\)
0.642191 + 0.766545i \(0.278026\pi\)
\(642\) 0 0
\(643\) 3692.00i 0.226436i 0.993570 + 0.113218i \(0.0361158\pi\)
−0.993570 + 0.113218i \(0.963884\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 20004.0i 1.21552i 0.794123 + 0.607758i \(0.207931\pi\)
−0.794123 + 0.607758i \(0.792069\pi\)
\(648\) 0 0
\(649\) −4716.00 −0.285238
\(650\) 0 0
\(651\) −11661.0 −0.702044
\(652\) 0 0
\(653\) 17862.0i 1.07044i 0.844714 + 0.535218i \(0.179770\pi\)
−0.844714 + 0.535218i \(0.820230\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 2574.00i − 0.152848i
\(658\) 0 0
\(659\) 18552.0 1.09664 0.548318 0.836270i \(-0.315268\pi\)
0.548318 + 0.836270i \(0.315268\pi\)
\(660\) 0 0
\(661\) −12382.0 −0.728599 −0.364300 0.931282i \(-0.618692\pi\)
−0.364300 + 0.931282i \(0.618692\pi\)
\(662\) 0 0
\(663\) 1170.00i 0.0685355i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 9108.00i − 0.528730i
\(668\) 0 0
\(669\) 2625.00 0.151702
\(670\) 0 0
\(671\) −2802.00 −0.161207
\(672\) 0 0
\(673\) 1690.00i 0.0967975i 0.998828 + 0.0483987i \(0.0154118\pi\)
−0.998828 + 0.0483987i \(0.984588\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5682.00i 0.322566i 0.986908 + 0.161283i \(0.0515631\pi\)
−0.986908 + 0.161283i \(0.948437\pi\)
\(678\) 0 0
\(679\) 6643.00 0.375456
\(680\) 0 0
\(681\) 14040.0 0.790035
\(682\) 0 0
\(683\) 31512.0i 1.76541i 0.469930 + 0.882704i \(0.344279\pi\)
−0.469930 + 0.882704i \(0.655721\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 4407.00i 0.244742i
\(688\) 0 0
\(689\) −3180.00 −0.175832
\(690\) 0 0
\(691\) −20648.0 −1.13674 −0.568370 0.822773i \(-0.692426\pi\)
−0.568370 + 0.822773i \(0.692426\pi\)
\(692\) 0 0
\(693\) 702.000i 0.0384802i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 28080.0i − 1.52598i
\(698\) 0 0
\(699\) −11880.0 −0.642837
\(700\) 0 0
\(701\) −11874.0 −0.639764 −0.319882 0.947457i \(-0.603643\pi\)
−0.319882 + 0.947457i \(0.603643\pi\)
\(702\) 0 0
\(703\) − 13910.0i − 0.746267i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 23556.0i − 1.25306i
\(708\) 0 0
\(709\) 10699.0 0.566727 0.283363 0.959013i \(-0.408550\pi\)
0.283363 + 0.959013i \(0.408550\pi\)
\(710\) 0 0
\(711\) −2448.00 −0.129124
\(712\) 0 0
\(713\) − 41262.0i − 2.16728i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 7956.00i − 0.414396i
\(718\) 0 0
\(719\) −5598.00 −0.290362 −0.145181 0.989405i \(-0.546376\pi\)
−0.145181 + 0.989405i \(0.546376\pi\)
\(720\) 0 0
\(721\) 22204.0 1.14691
\(722\) 0 0
\(723\) − 17259.0i − 0.887786i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 22859.0i − 1.16615i −0.812417 0.583077i \(-0.801849\pi\)
0.812417 0.583077i \(-0.198151\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 15834.0 0.801151
\(732\) 0 0
\(733\) 31642.0i 1.59444i 0.603689 + 0.797220i \(0.293697\pi\)
−0.603689 + 0.797220i \(0.706303\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 1302.00i − 0.0650743i
\(738\) 0 0
\(739\) −12220.0 −0.608281 −0.304141 0.952627i \(-0.598369\pi\)
−0.304141 + 0.952627i \(0.598369\pi\)
\(740\) 0 0
\(741\) −975.000 −0.0483367
\(742\) 0 0
\(743\) 17892.0i 0.883437i 0.897154 + 0.441719i \(0.145631\pi\)
−0.897154 + 0.441719i \(0.854369\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 4482.00i − 0.219529i
\(748\) 0 0
\(749\) 16068.0 0.783861
\(750\) 0 0
\(751\) 16648.0 0.808914 0.404457 0.914557i \(-0.367461\pi\)
0.404457 + 0.914557i \(0.367461\pi\)
\(752\) 0 0
\(753\) 10764.0i 0.520932i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 31733.0i 1.52359i 0.647820 + 0.761794i \(0.275681\pi\)
−0.647820 + 0.761794i \(0.724319\pi\)
\(758\) 0 0
\(759\) −2484.00 −0.118792
\(760\) 0 0
\(761\) −10068.0 −0.479586 −0.239793 0.970824i \(-0.577080\pi\)
−0.239793 + 0.970824i \(0.577080\pi\)
\(762\) 0 0
\(763\) 20059.0i 0.951749i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 3930.00i − 0.185012i
\(768\) 0 0
\(769\) −24323.0 −1.14058 −0.570292 0.821442i \(-0.693170\pi\)
−0.570292 + 0.821442i \(0.693170\pi\)
\(770\) 0 0
\(771\) −1692.00 −0.0790349
\(772\) 0 0
\(773\) 2184.00i 0.101621i 0.998708 + 0.0508105i \(0.0161804\pi\)
−0.998708 + 0.0508105i \(0.983820\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 8346.00i − 0.385342i
\(778\) 0 0
\(779\) 23400.0 1.07624
\(780\) 0 0
\(781\) −2160.00 −0.0989640
\(782\) 0 0
\(783\) − 1782.00i − 0.0813327i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 37531.0i 1.69992i 0.526849 + 0.849959i \(0.323373\pi\)
−0.526849 + 0.849959i \(0.676627\pi\)
\(788\) 0 0
\(789\) 3744.00 0.168935
\(790\) 0 0
\(791\) 24492.0 1.10093
\(792\) 0 0
\(793\) − 2335.00i − 0.104563i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 28644.0i 1.27305i 0.771255 + 0.636526i \(0.219629\pi\)
−0.771255 + 0.636526i \(0.780371\pi\)
\(798\) 0 0
\(799\) −6084.00 −0.269382
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 1716.00i − 0.0754126i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 21798.0i − 0.950838i
\(808\) 0 0
\(809\) 17316.0 0.752532 0.376266 0.926512i \(-0.377208\pi\)
0.376266 + 0.926512i \(0.377208\pi\)
\(810\) 0 0
\(811\) −425.000 −0.0184017 −0.00920084 0.999958i \(-0.502929\pi\)
−0.00920084 + 0.999958i \(0.502929\pi\)
\(812\) 0 0
\(813\) 9672.00i 0.417235i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 13195.0i 0.565036i
\(818\) 0 0
\(819\) −585.000 −0.0249592
\(820\) 0 0
\(821\) −25194.0 −1.07098 −0.535491 0.844541i \(-0.679874\pi\)
−0.535491 + 0.844541i \(0.679874\pi\)
\(822\) 0 0
\(823\) − 33961.0i − 1.43840i −0.694801 0.719202i \(-0.744508\pi\)
0.694801 0.719202i \(-0.255492\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1956.00i 0.0822452i 0.999154 + 0.0411226i \(0.0130934\pi\)
−0.999154 + 0.0411226i \(0.986907\pi\)
\(828\) 0 0
\(829\) 47302.0 1.98174 0.990872 0.134803i \(-0.0430403\pi\)
0.990872 + 0.134803i \(0.0430403\pi\)
\(830\) 0 0
\(831\) −453.000 −0.0189102
\(832\) 0 0
\(833\) − 13572.0i − 0.564516i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 8073.00i − 0.333386i
\(838\) 0 0
\(839\) 30726.0 1.26434 0.632169 0.774831i \(-0.282165\pi\)
0.632169 + 0.774831i \(0.282165\pi\)
\(840\) 0 0
\(841\) −20033.0 −0.821395
\(842\) 0 0
\(843\) − 22698.0i − 0.927355i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 16835.0i − 0.682949i
\(848\) 0 0
\(849\) 4407.00 0.178148
\(850\) 0 0
\(851\) 29532.0 1.18959
\(852\) 0 0
\(853\) 34489.0i 1.38439i 0.721713 + 0.692193i \(0.243355\pi\)
−0.721713 + 0.692193i \(0.756645\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 8484.00i 0.338166i 0.985602 + 0.169083i \(0.0540805\pi\)
−0.985602 + 0.169083i \(0.945919\pi\)
\(858\) 0 0
\(859\) −520.000 −0.0206544 −0.0103272 0.999947i \(-0.503287\pi\)
−0.0103272 + 0.999947i \(0.503287\pi\)
\(860\) 0 0
\(861\) 14040.0 0.555728
\(862\) 0 0
\(863\) − 6864.00i − 0.270745i −0.990795 0.135373i \(-0.956777\pi\)
0.990795 0.135373i \(-0.0432232\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 3513.00i 0.137610i
\(868\) 0 0
\(869\) −1632.00 −0.0637075
\(870\) 0 0
\(871\) 1085.00 0.0422088
\(872\) 0 0
\(873\) 4599.00i 0.178296i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 38857.0i − 1.49613i −0.663624 0.748066i \(-0.730983\pi\)
0.663624 0.748066i \(-0.269017\pi\)
\(878\) 0 0
\(879\) 918.000 0.0352257
\(880\) 0 0
\(881\) −10080.0 −0.385475 −0.192738 0.981250i \(-0.561737\pi\)
−0.192738 + 0.981250i \(0.561737\pi\)
\(882\) 0 0
\(883\) − 34549.0i − 1.31672i −0.752702 0.658362i \(-0.771250\pi\)
0.752702 0.658362i \(-0.228750\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 18798.0i 0.711584i 0.934565 + 0.355792i \(0.115789\pi\)
−0.934565 + 0.355792i \(0.884211\pi\)
\(888\) 0 0
\(889\) 26936.0 1.01620
\(890\) 0 0
\(891\) −486.000 −0.0182734
\(892\) 0 0
\(893\) − 5070.00i − 0.189990i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 2070.00i − 0.0770516i
\(898\) 0 0
\(899\) 19734.0 0.732109
\(900\) 0 0
\(901\) −49608.0 −1.83428
\(902\) 0 0
\(903\) 7917.00i 0.291762i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 50276.0i − 1.84056i −0.391261 0.920280i \(-0.627961\pi\)
0.391261 0.920280i \(-0.372039\pi\)
\(908\) 0 0
\(909\) 16308.0 0.595052
\(910\) 0 0
\(911\) 20502.0 0.745622 0.372811 0.927907i \(-0.378394\pi\)
0.372811 + 0.927907i \(0.378394\pi\)
\(912\) 0 0
\(913\) − 2988.00i − 0.108311i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 32604.0i 1.17413i
\(918\) 0 0
\(919\) 37817.0 1.35742 0.678709 0.734407i \(-0.262540\pi\)
0.678709 + 0.734407i \(0.262540\pi\)
\(920\) 0 0
\(921\) −13983.0 −0.500277
\(922\) 0 0
\(923\) − 1800.00i − 0.0641904i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 15372.0i 0.544642i
\(928\) 0 0
\(929\) 11778.0 0.415957 0.207978 0.978133i \(-0.433312\pi\)
0.207978 + 0.978133i \(0.433312\pi\)
\(930\) 0 0
\(931\) 11310.0 0.398142
\(932\) 0 0
\(933\) 7542.00i 0.264645i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 18409.0i − 0.641831i −0.947108 0.320916i \(-0.896009\pi\)
0.947108 0.320916i \(-0.103991\pi\)
\(938\) 0 0
\(939\) −20121.0 −0.699280
\(940\) 0 0
\(941\) −9330.00 −0.323219 −0.161610 0.986855i \(-0.551669\pi\)
−0.161610 + 0.986855i \(0.551669\pi\)
\(942\) 0 0
\(943\) 49680.0i 1.71559i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 49146.0i − 1.68641i −0.537592 0.843205i \(-0.680666\pi\)
0.537592 0.843205i \(-0.319334\pi\)
\(948\) 0 0
\(949\) 1430.00 0.0489144
\(950\) 0 0
\(951\) 13896.0 0.473826
\(952\) 0 0
\(953\) 5928.00i 0.201497i 0.994912 + 0.100749i \(0.0321238\pi\)
−0.994912 + 0.100749i \(0.967876\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 1188.00i − 0.0401281i
\(958\) 0 0
\(959\) −20358.0 −0.685500
\(960\) 0 0
\(961\) 59610.0 2.00094
\(962\) 0 0
\(963\) 11124.0i 0.372239i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 9520.00i 0.316590i 0.987392 + 0.158295i \(0.0505997\pi\)
−0.987392 + 0.158295i \(0.949400\pi\)
\(968\) 0 0
\(969\) −15210.0 −0.504247
\(970\) 0 0
\(971\) −12558.0 −0.415042 −0.207521 0.978231i \(-0.566539\pi\)
−0.207521 + 0.978231i \(0.566539\pi\)
\(972\) 0 0
\(973\) − 2548.00i − 0.0839518i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 46206.0i − 1.51306i −0.653958 0.756531i \(-0.726893\pi\)
0.653958 0.756531i \(-0.273107\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −13887.0 −0.451965
\(982\) 0 0
\(983\) − 32676.0i − 1.06023i −0.847927 0.530113i \(-0.822149\pi\)
0.847927 0.530113i \(-0.177851\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 3042.00i − 0.0981033i
\(988\) 0 0
\(989\) −28014.0 −0.900701
\(990\) 0 0
\(991\) 8137.00 0.260828 0.130414 0.991460i \(-0.458369\pi\)
0.130414 + 0.991460i \(0.458369\pi\)
\(992\) 0 0
\(993\) − 2964.00i − 0.0947228i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 18578.0i 0.590142i 0.955475 + 0.295071i \(0.0953432\pi\)
−0.955475 + 0.295071i \(0.904657\pi\)
\(998\) 0 0
\(999\) 5778.00 0.182991
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 1200.4.f.k.49.1 2
4.3 odd 2 300.4.d.c.49.2 2
5.2 odd 4 1200.4.a.d.1.1 1
5.3 odd 4 1200.4.a.bi.1.1 1
5.4 even 2 inner 1200.4.f.k.49.2 2
12.11 even 2 900.4.d.e.649.1 2
20.3 even 4 300.4.a.a.1.1 1
20.7 even 4 300.4.a.h.1.1 yes 1
20.19 odd 2 300.4.d.c.49.1 2
60.23 odd 4 900.4.a.f.1.1 1
60.47 odd 4 900.4.a.l.1.1 1
60.59 even 2 900.4.d.e.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.4.a.a.1.1 1 20.3 even 4
300.4.a.h.1.1 yes 1 20.7 even 4
300.4.d.c.49.1 2 20.19 odd 2
300.4.d.c.49.2 2 4.3 odd 2
900.4.a.f.1.1 1 60.23 odd 4
900.4.a.l.1.1 1 60.47 odd 4
900.4.d.e.649.1 2 12.11 even 2
900.4.d.e.649.2 2 60.59 even 2
1200.4.a.d.1.1 1 5.2 odd 4
1200.4.a.bi.1.1 1 5.3 odd 4
1200.4.f.k.49.1 2 1.1 even 1 trivial
1200.4.f.k.49.2 2 5.4 even 2 inner