Properties

Label 252.4.k.c.109.2
Level $252$
Weight $4$
Character 252.109
Analytic conductor $14.868$
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] = [252,4,Mod(37,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.37");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.k (of order \(3\), degree \(2\), 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: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{37})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 10x^{2} + 9x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 109.2
Root \(-1.27069 - 2.20090i\) of defining polynomial
Character \(\chi\) \(=\) 252.109
Dual form 252.4.k.c.37.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.58276 + 4.47348i) q^{5} +(18.1655 + 3.60745i) q^{7} +O(q^{10})\) \(q+(2.58276 + 4.47348i) q^{5} +(18.1655 + 3.60745i) q^{7} +(29.2897 - 50.7312i) q^{11} -38.3311 q^{13} +(-26.3345 + 45.6126i) q^{17} +(65.1241 + 112.798i) q^{19} +(38.2897 + 66.3197i) q^{23} +(49.1587 - 85.1453i) q^{25} +288.317 q^{29} +(70.2897 - 121.745i) q^{31} +(30.7794 + 90.5802i) q^{35} +(41.2380 + 71.4263i) q^{37} +282.993 q^{41} -172.000 q^{43} +(66.6207 + 115.390i) q^{47} +(316.973 + 131.062i) q^{49} +(18.2380 - 31.5892i) q^{53} +302.593 q^{55} +(-126.531 + 219.158i) q^{59} +(-249.562 - 432.254i) q^{61} +(-99.0000 - 171.473i) q^{65} +(-105.290 + 182.367i) q^{67} -107.365 q^{71} +(-180.672 + 312.934i) q^{73} +(715.072 - 815.898i) q^{77} +(417.504 + 723.138i) q^{79} -731.283 q^{83} -272.063 q^{85} +(-108.983 - 188.764i) q^{89} +(-696.304 - 138.277i) q^{91} +(-336.400 + 582.663i) q^{95} +252.290 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{5} + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 14 q^{5} + 24 q^{7} + 32 q^{11} - 56 q^{13} - 154 q^{17} + 224 q^{19} + 68 q^{23} - 144 q^{25} + 472 q^{29} + 196 q^{31} - 400 q^{35} - 346 q^{37} + 840 q^{41} - 688 q^{43} + 84 q^{47} + 100 q^{49} - 438 q^{53} + 1624 q^{55} - 56 q^{59} - 98 q^{61} - 396 q^{65} - 336 q^{67} - 1792 q^{71} - 966 q^{73} + 2398 q^{77} + 52 q^{79} - 784 q^{83} + 3340 q^{85} + 294 q^{89} - 1520 q^{91} + 1124 q^{95} - 840 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.58276 + 4.47348i 0.231009 + 0.400120i 0.958105 0.286416i \(-0.0924639\pi\)
−0.727096 + 0.686536i \(0.759131\pi\)
\(6\) 0 0
\(7\) 18.1655 + 3.60745i 0.980846 + 0.194784i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 29.2897 50.7312i 0.802833 1.39055i −0.114911 0.993376i \(-0.536658\pi\)
0.917744 0.397172i \(-0.130009\pi\)
\(12\) 0 0
\(13\) −38.3311 −0.817779 −0.408889 0.912584i \(-0.634084\pi\)
−0.408889 + 0.912584i \(0.634084\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −26.3345 + 45.6126i −0.375709 + 0.650747i −0.990433 0.137996i \(-0.955934\pi\)
0.614724 + 0.788742i \(0.289267\pi\)
\(18\) 0 0
\(19\) 65.1241 + 112.798i 0.786342 + 1.36198i 0.928194 + 0.372097i \(0.121361\pi\)
−0.141852 + 0.989888i \(0.545306\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 38.2897 + 66.3197i 0.347128 + 0.601244i 0.985738 0.168287i \(-0.0538235\pi\)
−0.638610 + 0.769531i \(0.720490\pi\)
\(24\) 0 0
\(25\) 49.1587 85.1453i 0.393269 0.681163i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 288.317 1.84618 0.923089 0.384585i \(-0.125656\pi\)
0.923089 + 0.384585i \(0.125656\pi\)
\(30\) 0 0
\(31\) 70.2897 121.745i 0.407239 0.705358i −0.587341 0.809340i \(-0.699825\pi\)
0.994579 + 0.103982i \(0.0331584\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 30.7794 + 90.5802i 0.148648 + 0.437453i
\(36\) 0 0
\(37\) 41.2380 + 71.4263i 0.183229 + 0.317363i 0.942978 0.332854i \(-0.108012\pi\)
−0.759749 + 0.650216i \(0.774678\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 282.993 1.07795 0.538977 0.842321i \(-0.318811\pi\)
0.538977 + 0.842321i \(0.318811\pi\)
\(42\) 0 0
\(43\) −172.000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 66.6207 + 115.390i 0.206758 + 0.358116i 0.950691 0.310138i \(-0.100375\pi\)
−0.743933 + 0.668254i \(0.767042\pi\)
\(48\) 0 0
\(49\) 316.973 + 131.062i 0.924118 + 0.382106i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 18.2380 31.5892i 0.0472676 0.0818699i −0.841424 0.540376i \(-0.818282\pi\)
0.888691 + 0.458506i \(0.151615\pi\)
\(54\) 0 0
\(55\) 302.593 0.741848
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −126.531 + 219.158i −0.279202 + 0.483593i −0.971187 0.238320i \(-0.923403\pi\)
0.691984 + 0.721913i \(0.256737\pi\)
\(60\) 0 0
\(61\) −249.562 432.254i −0.523822 0.907287i −0.999615 0.0277298i \(-0.991172\pi\)
0.475793 0.879557i \(-0.342161\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −99.0000 171.473i −0.188914 0.327209i
\(66\) 0 0
\(67\) −105.290 + 182.367i −0.191988 + 0.332533i −0.945909 0.324432i \(-0.894827\pi\)
0.753921 + 0.656965i \(0.228160\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −107.365 −0.179464 −0.0897318 0.995966i \(-0.528601\pi\)
−0.0897318 + 0.995966i \(0.528601\pi\)
\(72\) 0 0
\(73\) −180.672 + 312.934i −0.289673 + 0.501728i −0.973732 0.227699i \(-0.926880\pi\)
0.684059 + 0.729427i \(0.260213\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 715.072 815.898i 1.05831 1.20753i
\(78\) 0 0
\(79\) 417.504 + 723.138i 0.594593 + 1.02987i 0.993604 + 0.112919i \(0.0360201\pi\)
−0.399011 + 0.916946i \(0.630647\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −731.283 −0.967093 −0.483547 0.875319i \(-0.660652\pi\)
−0.483547 + 0.875319i \(0.660652\pi\)
\(84\) 0 0
\(85\) −272.063 −0.347169
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −108.983 188.764i −0.129800 0.224819i 0.793799 0.608180i \(-0.208100\pi\)
−0.923599 + 0.383360i \(0.874767\pi\)
\(90\) 0 0
\(91\) −696.304 138.277i −0.802115 0.159290i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −336.400 + 582.663i −0.363305 + 0.629262i
\(96\) 0 0
\(97\) 252.290 0.264084 0.132042 0.991244i \(-0.457847\pi\)
0.132042 + 0.991244i \(0.457847\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −577.886 + 1000.93i −0.569325 + 0.986100i 0.427308 + 0.904106i \(0.359462\pi\)
−0.996633 + 0.0819939i \(0.973871\pi\)
\(102\) 0 0
\(103\) −525.628 910.414i −0.502831 0.870929i −0.999995 0.00327239i \(-0.998958\pi\)
0.497163 0.867657i \(-0.334375\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1037.61 1797.19i −0.937470 1.62375i −0.770170 0.637839i \(-0.779828\pi\)
−0.167300 0.985906i \(-0.553505\pi\)
\(108\) 0 0
\(109\) 608.079 1053.22i 0.534343 0.925510i −0.464851 0.885389i \(-0.653892\pi\)
0.999195 0.0401213i \(-0.0127744\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1585.59 −1.32000 −0.659998 0.751268i \(-0.729443\pi\)
−0.659998 + 0.751268i \(0.729443\pi\)
\(114\) 0 0
\(115\) −197.786 + 342.576i −0.160380 + 0.277786i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −642.925 + 733.577i −0.495267 + 0.565100i
\(120\) 0 0
\(121\) −1050.27 1819.12i −0.789083 1.36673i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1153.55 0.825414
\(126\) 0 0
\(127\) −1107.37 −0.773723 −0.386861 0.922138i \(-0.626441\pi\)
−0.386861 + 0.922138i \(0.626441\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 442.642 + 766.678i 0.295220 + 0.511336i 0.975036 0.222047i \(-0.0712738\pi\)
−0.679816 + 0.733383i \(0.737940\pi\)
\(132\) 0 0
\(133\) 776.100 + 2283.97i 0.505988 + 1.48906i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1076.77 1865.02i 0.671494 1.16306i −0.305987 0.952036i \(-0.598986\pi\)
0.977481 0.211025i \(-0.0676802\pi\)
\(138\) 0 0
\(139\) −672.580 −0.410414 −0.205207 0.978719i \(-0.565787\pi\)
−0.205207 + 0.978719i \(0.565787\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1122.70 + 1944.58i −0.656540 + 1.13716i
\(144\) 0 0
\(145\) 744.655 + 1289.78i 0.426485 + 0.738693i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −901.666 1561.73i −0.495754 0.858671i 0.504234 0.863567i \(-0.331775\pi\)
−0.999988 + 0.00489580i \(0.998442\pi\)
\(150\) 0 0
\(151\) −557.504 + 965.625i −0.300457 + 0.520407i −0.976240 0.216694i \(-0.930473\pi\)
0.675783 + 0.737101i \(0.263806\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 726.166 0.376304
\(156\) 0 0
\(157\) 839.156 1453.46i 0.426573 0.738846i −0.569993 0.821649i \(-0.693054\pi\)
0.996566 + 0.0828038i \(0.0263875\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 456.307 + 1342.86i 0.223367 + 0.657343i
\(162\) 0 0
\(163\) −792.297 1372.30i −0.380721 0.659428i 0.610445 0.792059i \(-0.290991\pi\)
−0.991165 + 0.132631i \(0.957657\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −619.669 −0.287134 −0.143567 0.989641i \(-0.545857\pi\)
−0.143567 + 0.989641i \(0.545857\pi\)
\(168\) 0 0
\(169\) −727.731 −0.331238
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1790.38 3101.03i −0.786823 1.36282i −0.927904 0.372819i \(-0.878391\pi\)
0.141082 0.989998i \(-0.454942\pi\)
\(174\) 0 0
\(175\) 1200.15 1369.37i 0.518416 0.591513i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −287.773 + 498.438i −0.120163 + 0.208128i −0.919832 0.392313i \(-0.871675\pi\)
0.799669 + 0.600441i \(0.205008\pi\)
\(180\) 0 0
\(181\) 1556.51 0.639197 0.319598 0.947553i \(-0.396452\pi\)
0.319598 + 0.947553i \(0.396452\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −213.016 + 368.955i −0.0846554 + 0.146627i
\(186\) 0 0
\(187\) 1542.66 + 2671.96i 0.603263 + 1.04488i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 399.559 + 692.057i 0.151367 + 0.262175i 0.931730 0.363151i \(-0.118299\pi\)
−0.780363 + 0.625326i \(0.784966\pi\)
\(192\) 0 0
\(193\) 279.024 483.284i 0.104065 0.180246i −0.809291 0.587408i \(-0.800148\pi\)
0.913356 + 0.407162i \(0.133482\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1729.59 0.625523 0.312761 0.949832i \(-0.398746\pi\)
0.312761 + 0.949832i \(0.398746\pi\)
\(198\) 0 0
\(199\) −2142.77 + 3711.39i −0.763302 + 1.32208i 0.177837 + 0.984060i \(0.443090\pi\)
−0.941139 + 0.338018i \(0.890243\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 5237.44 + 1040.09i 1.81082 + 0.359606i
\(204\) 0 0
\(205\) 730.904 + 1265.96i 0.249017 + 0.431311i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 7629.86 2.52521
\(210\) 0 0
\(211\) 849.904 0.277298 0.138649 0.990342i \(-0.455724\pi\)
0.138649 + 0.990342i \(0.455724\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −444.235 769.438i −0.140914 0.244071i
\(216\) 0 0
\(217\) 1716.04 1958.00i 0.536831 0.612524i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1009.43 1748.38i 0.307247 0.532167i
\(222\) 0 0
\(223\) 4377.66 1.31457 0.657286 0.753641i \(-0.271704\pi\)
0.657286 + 0.753641i \(0.271704\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2559.83 + 4433.75i −0.748466 + 1.29638i 0.200092 + 0.979777i \(0.435876\pi\)
−0.948558 + 0.316604i \(0.897457\pi\)
\(228\) 0 0
\(229\) −2940.65 5093.36i −0.848576 1.46978i −0.882479 0.470351i \(-0.844127\pi\)
0.0339035 0.999425i \(-0.489206\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 106.039 + 183.665i 0.0298147 + 0.0516406i 0.880548 0.473957i \(-0.157175\pi\)
−0.850733 + 0.525598i \(0.823842\pi\)
\(234\) 0 0
\(235\) −344.131 + 596.052i −0.0955261 + 0.165456i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −5202.95 −1.40816 −0.704082 0.710119i \(-0.748641\pi\)
−0.704082 + 0.710119i \(0.748641\pi\)
\(240\) 0 0
\(241\) −1418.56 + 2457.02i −0.379160 + 0.656725i −0.990940 0.134303i \(-0.957120\pi\)
0.611780 + 0.791028i \(0.290454\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 232.360 + 1756.47i 0.0605916 + 0.458028i
\(246\) 0 0
\(247\) −2496.28 4323.68i −0.643054 1.11380i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4697.05 1.18118 0.590588 0.806973i \(-0.298896\pi\)
0.590588 + 0.806973i \(0.298896\pi\)
\(252\) 0 0
\(253\) 4485.97 1.11474
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3501.86 + 6065.40i 0.849961 + 1.47218i 0.881242 + 0.472665i \(0.156708\pi\)
−0.0312810 + 0.999511i \(0.509959\pi\)
\(258\) 0 0
\(259\) 491.443 + 1446.26i 0.117903 + 0.346974i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 614.153 1063.74i 0.143994 0.249404i −0.785003 0.619491i \(-0.787339\pi\)
0.928997 + 0.370087i \(0.120672\pi\)
\(264\) 0 0
\(265\) 188.418 0.0436770
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 70.9423 122.876i 0.0160797 0.0278508i −0.857874 0.513861i \(-0.828215\pi\)
0.873953 + 0.486010i \(0.161548\pi\)
\(270\) 0 0
\(271\) 182.387 + 315.904i 0.0408828 + 0.0708111i 0.885743 0.464176i \(-0.153650\pi\)
−0.844860 + 0.534988i \(0.820316\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2879.68 4987.76i −0.631460 1.09372i
\(276\) 0 0
\(277\) −994.238 + 1722.07i −0.215661 + 0.373535i −0.953477 0.301467i \(-0.902524\pi\)
0.737816 + 0.675002i \(0.235857\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4903.93 −1.04108 −0.520541 0.853837i \(-0.674270\pi\)
−0.520541 + 0.853837i \(0.674270\pi\)
\(282\) 0 0
\(283\) −1630.04 + 2823.31i −0.342387 + 0.593032i −0.984876 0.173263i \(-0.944569\pi\)
0.642488 + 0.766296i \(0.277902\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5140.72 + 1020.88i 1.05731 + 0.209968i
\(288\) 0 0
\(289\) 1069.49 + 1852.41i 0.217686 + 0.377043i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1613.67 −0.321746 −0.160873 0.986975i \(-0.551431\pi\)
−0.160873 + 0.986975i \(0.551431\pi\)
\(294\) 0 0
\(295\) −1307.20 −0.257993
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1467.68 2542.10i −0.283874 0.491684i
\(300\) 0 0
\(301\) −3124.47 620.481i −0.598311 0.118817i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1289.12 2232.82i 0.242016 0.419183i
\(306\) 0 0
\(307\) −2846.16 −0.529116 −0.264558 0.964370i \(-0.585226\pi\)
−0.264558 + 0.964370i \(0.585226\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1391.70 2410.49i 0.253749 0.439506i −0.710806 0.703388i \(-0.751670\pi\)
0.964555 + 0.263882i \(0.0850030\pi\)
\(312\) 0 0
\(313\) 1501.07 + 2599.92i 0.271071 + 0.469509i 0.969136 0.246525i \(-0.0792887\pi\)
−0.698065 + 0.716034i \(0.745955\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1886.91 3268.22i −0.334319 0.579058i 0.649035 0.760759i \(-0.275173\pi\)
−0.983354 + 0.181701i \(0.941840\pi\)
\(318\) 0 0
\(319\) 8444.72 14626.7i 1.48217 2.56720i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −6860.04 −1.18174
\(324\) 0 0
\(325\) −1884.30 + 3263.71i −0.321607 + 0.557040i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 793.935 + 2336.46i 0.133043 + 0.391529i
\(330\) 0 0
\(331\) 487.939 + 845.135i 0.0810259 + 0.140341i 0.903691 0.428186i \(-0.140847\pi\)
−0.822665 + 0.568526i \(0.807514\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1087.75 −0.177404
\(336\) 0 0
\(337\) −3563.93 −0.576083 −0.288041 0.957618i \(-0.593004\pi\)
−0.288041 + 0.957618i \(0.593004\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −4117.52 7131.76i −0.653890 1.13257i
\(342\) 0 0
\(343\) 5285.17 + 3524.28i 0.831990 + 0.554791i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −577.677 + 1000.57i −0.0893698 + 0.154793i −0.907245 0.420603i \(-0.861819\pi\)
0.817875 + 0.575396i \(0.195152\pi\)
\(348\) 0 0
\(349\) −4693.29 −0.719845 −0.359922 0.932982i \(-0.617197\pi\)
−0.359922 + 0.932982i \(0.617197\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6628.54 + 11481.0i −0.999439 + 1.73108i −0.470704 + 0.882291i \(0.656000\pi\)
−0.528734 + 0.848787i \(0.677333\pi\)
\(354\) 0 0
\(355\) −277.299 480.296i −0.0414578 0.0718070i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −893.312 1547.26i −0.131329 0.227469i 0.792860 0.609404i \(-0.208591\pi\)
−0.924189 + 0.381935i \(0.875258\pi\)
\(360\) 0 0
\(361\) −5052.81 + 8751.72i −0.736668 + 1.27595i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1866.54 −0.267668
\(366\) 0 0
\(367\) −3201.75 + 5545.60i −0.455395 + 0.788768i −0.998711 0.0507606i \(-0.983835\pi\)
0.543315 + 0.839529i \(0.317169\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 445.259 508.041i 0.0623092 0.0710948i
\(372\) 0 0
\(373\) −6085.11 10539.7i −0.844705 1.46307i −0.885877 0.463920i \(-0.846442\pi\)
0.0411720 0.999152i \(-0.486891\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −11051.5 −1.50977
\(378\) 0 0
\(379\) −2565.68 −0.347732 −0.173866 0.984769i \(-0.555626\pi\)
−0.173866 + 0.984769i \(0.555626\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3304.58 5723.70i −0.440878 0.763623i 0.556877 0.830595i \(-0.311999\pi\)
−0.997755 + 0.0669723i \(0.978666\pi\)
\(384\) 0 0
\(385\) 5496.76 + 1091.59i 0.727639 + 0.144500i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 4443.74 7696.79i 0.579195 1.00319i −0.416377 0.909192i \(-0.636700\pi\)
0.995572 0.0940027i \(-0.0299662\pi\)
\(390\) 0 0
\(391\) −4033.35 −0.521676
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2156.63 + 3735.39i −0.274713 + 0.475817i
\(396\) 0 0
\(397\) −4435.95 7683.29i −0.560791 0.971318i −0.997428 0.0716802i \(-0.977164\pi\)
0.436637 0.899638i \(-0.356169\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −641.120 1110.45i −0.0798404 0.138288i 0.823341 0.567548i \(-0.192108\pi\)
−0.903181 + 0.429260i \(0.858774\pi\)
\(402\) 0 0
\(403\) −2694.28 + 4666.62i −0.333031 + 0.576827i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4831.39 0.588411
\(408\) 0 0
\(409\) 3843.32 6656.82i 0.464645 0.804788i −0.534541 0.845143i \(-0.679515\pi\)
0.999185 + 0.0403544i \(0.0128487\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3089.11 + 3524.67i −0.368051 + 0.419946i
\(414\) 0 0
\(415\) −1888.73 3271.38i −0.223408 0.386953i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 13364.1 1.55818 0.779092 0.626909i \(-0.215680\pi\)
0.779092 + 0.626909i \(0.215680\pi\)
\(420\) 0 0
\(421\) 1781.20 0.206201 0.103100 0.994671i \(-0.467124\pi\)
0.103100 + 0.994671i \(0.467124\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2589.14 + 4484.51i 0.295509 + 0.511837i
\(426\) 0 0
\(427\) −2974.09 8752.41i −0.337064 0.991941i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −3791.18 + 6566.51i −0.423700 + 0.733869i −0.996298 0.0859668i \(-0.972602\pi\)
0.572598 + 0.819836i \(0.305935\pi\)
\(432\) 0 0
\(433\) 8642.25 0.959169 0.479585 0.877496i \(-0.340787\pi\)
0.479585 + 0.877496i \(0.340787\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4987.16 + 8638.02i −0.545923 + 0.945567i
\(438\) 0 0
\(439\) 7106.31 + 12308.5i 0.772586 + 1.33816i 0.936141 + 0.351625i \(0.114371\pi\)
−0.163555 + 0.986534i \(0.552296\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3884.78 6728.64i −0.416640 0.721642i 0.578959 0.815357i \(-0.303459\pi\)
−0.995599 + 0.0937150i \(0.970126\pi\)
\(444\) 0 0
\(445\) 562.954 975.065i 0.0599698 0.103871i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 7691.17 0.808393 0.404197 0.914672i \(-0.367551\pi\)
0.404197 + 0.914672i \(0.367551\pi\)
\(450\) 0 0
\(451\) 8288.78 14356.6i 0.865417 1.49895i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1179.81 3472.04i −0.121561 0.357740i
\(456\) 0 0
\(457\) 6692.10 + 11591.1i 0.684996 + 1.18645i 0.973438 + 0.228951i \(0.0735295\pi\)
−0.288442 + 0.957497i \(0.593137\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5451.80 0.550793 0.275396 0.961331i \(-0.411191\pi\)
0.275396 + 0.961331i \(0.411191\pi\)
\(462\) 0 0
\(463\) −13193.4 −1.32430 −0.662148 0.749373i \(-0.730355\pi\)
−0.662148 + 0.749373i \(0.730355\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1169.09 + 2024.93i 0.115844 + 0.200648i 0.918117 0.396310i \(-0.129709\pi\)
−0.802273 + 0.596958i \(0.796376\pi\)
\(468\) 0 0
\(469\) −2570.52 + 2932.97i −0.253082 + 0.288767i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5037.82 + 8725.77i −0.489724 + 0.848226i
\(474\) 0 0
\(475\) 12805.7 1.23698
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9262.11 16042.5i 0.883501 1.53027i 0.0360779 0.999349i \(-0.488514\pi\)
0.847423 0.530919i \(-0.178153\pi\)
\(480\) 0 0
\(481\) −1580.70 2737.85i −0.149841 0.259532i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 651.605 + 1128.61i 0.0610059 + 0.105665i
\(486\) 0 0
\(487\) −2878.54 + 4985.77i −0.267842 + 0.463916i −0.968304 0.249774i \(-0.919644\pi\)
0.700462 + 0.713689i \(0.252977\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 10593.6 0.973692 0.486846 0.873488i \(-0.338147\pi\)
0.486846 + 0.873488i \(0.338147\pi\)
\(492\) 0 0
\(493\) −7592.69 + 13150.9i −0.693626 + 1.20139i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1950.35 387.315i −0.176026 0.0349566i
\(498\) 0 0
\(499\) 1548.31 + 2681.75i 0.138901 + 0.240584i 0.927081 0.374861i \(-0.122310\pi\)
−0.788180 + 0.615445i \(0.788976\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 3149.14 0.279151 0.139576 0.990211i \(-0.455426\pi\)
0.139576 + 0.990211i \(0.455426\pi\)
\(504\) 0 0
\(505\) −5970.17 −0.526078
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −4566.92 7910.14i −0.397692 0.688823i 0.595749 0.803171i \(-0.296855\pi\)
−0.993441 + 0.114348i \(0.963522\pi\)
\(510\) 0 0
\(511\) −4410.90 + 5032.84i −0.381853 + 0.435694i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2715.14 4702.76i 0.232317 0.402386i
\(516\) 0 0
\(517\) 7805.20 0.663969
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −2558.89 + 4432.12i −0.215176 + 0.372696i −0.953327 0.301939i \(-0.902366\pi\)
0.738151 + 0.674636i \(0.235699\pi\)
\(522\) 0 0
\(523\) −2934.26 5082.28i −0.245327 0.424919i 0.716896 0.697180i \(-0.245562\pi\)
−0.962224 + 0.272261i \(0.912229\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3702.08 + 6412.20i 0.306006 + 0.530018i
\(528\) 0 0
\(529\) 3151.30 5458.22i 0.259004 0.448608i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −10847.4 −0.881527
\(534\) 0 0
\(535\) 5359.79 9283.42i 0.433128 0.750200i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 15933.0 12241.6i 1.27325 0.978263i
\(540\) 0 0
\(541\) 4140.16 + 7170.97i 0.329019 + 0.569878i 0.982318 0.187223i \(-0.0599486\pi\)
−0.653298 + 0.757101i \(0.726615\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6282.10 0.493753
\(546\) 0 0
\(547\) 16832.7 1.31575 0.657875 0.753127i \(-0.271455\pi\)
0.657875 + 0.753127i \(0.271455\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 18776.4 + 32521.7i 1.45173 + 2.51447i
\(552\) 0 0
\(553\) 4975.49 + 14642.3i 0.382603 + 1.12596i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5254.65 9101.32i 0.399725 0.692344i −0.593967 0.804489i \(-0.702439\pi\)
0.993692 + 0.112146i \(0.0357724\pi\)
\(558\) 0 0
\(559\) 6592.94 0.498840
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 166.502 288.390i 0.0124640 0.0215883i −0.859726 0.510755i \(-0.829366\pi\)
0.872190 + 0.489167i \(0.162699\pi\)
\(564\) 0 0
\(565\) −4095.19 7093.08i −0.304931 0.528156i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 7544.12 + 13066.8i 0.555828 + 0.962722i 0.997839 + 0.0657121i \(0.0209319\pi\)
−0.442011 + 0.897010i \(0.645735\pi\)
\(570\) 0 0
\(571\) 917.219 1588.67i 0.0672232 0.116434i −0.830455 0.557086i \(-0.811919\pi\)
0.897678 + 0.440652i \(0.145253\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 7529.08 0.546060
\(576\) 0 0
\(577\) −4258.15 + 7375.33i −0.307225 + 0.532130i −0.977754 0.209753i \(-0.932734\pi\)
0.670529 + 0.741883i \(0.266067\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −13284.1 2638.07i −0.948570 0.188374i
\(582\) 0 0
\(583\) −1068.37 1850.47i −0.0758960 0.131456i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −23957.0 −1.68452 −0.842258 0.539074i \(-0.818774\pi\)
−0.842258 + 0.539074i \(0.818774\pi\)
\(588\) 0 0
\(589\) 18310.2 1.28092
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5865.45 + 10159.3i 0.406181 + 0.703525i 0.994458 0.105134i \(-0.0335270\pi\)
−0.588277 + 0.808659i \(0.700194\pi\)
\(594\) 0 0
\(595\) −4942.16 981.453i −0.340519 0.0676229i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2315.67 4010.86i 0.157956 0.273588i −0.776175 0.630517i \(-0.782843\pi\)
0.934132 + 0.356929i \(0.116176\pi\)
\(600\) 0 0
\(601\) −25359.8 −1.72121 −0.860606 0.509272i \(-0.829915\pi\)
−0.860606 + 0.509272i \(0.829915\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5425.19 9396.71i 0.364571 0.631456i
\(606\) 0 0
\(607\) −62.3696 108.027i −0.00417052 0.00722355i 0.863933 0.503607i \(-0.167994\pi\)
−0.868103 + 0.496384i \(0.834661\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2553.64 4423.04i −0.169082 0.292859i
\(612\) 0 0
\(613\) 8592.54 14882.7i 0.566149 0.980599i −0.430793 0.902451i \(-0.641766\pi\)
0.996942 0.0781479i \(-0.0249006\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 25988.1 1.69569 0.847844 0.530246i \(-0.177900\pi\)
0.847844 + 0.530246i \(0.177900\pi\)
\(618\) 0 0
\(619\) 9442.62 16355.1i 0.613135 1.06198i −0.377573 0.925980i \(-0.623241\pi\)
0.990709 0.136002i \(-0.0434254\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1298.77 3822.15i −0.0835222 0.245796i
\(624\) 0 0
\(625\) −3165.49 5482.78i −0.202591 0.350898i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −4343.93 −0.275364
\(630\) 0 0
\(631\) −6552.95 −0.413421 −0.206710 0.978402i \(-0.566276\pi\)
−0.206710 + 0.978402i \(0.566276\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2860.06 4953.77i −0.178737 0.309582i
\(636\) 0 0
\(637\) −12149.9 5023.76i −0.755724 0.312478i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −13835.5 + 23963.8i −0.852525 + 1.47662i 0.0263968 + 0.999652i \(0.491597\pi\)
−0.878922 + 0.476965i \(0.841737\pi\)
\(642\) 0 0
\(643\) −27154.8 −1.66544 −0.832722 0.553691i \(-0.813219\pi\)
−0.832722 + 0.553691i \(0.813219\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13661.3 23662.1i 0.830110 1.43779i −0.0678409 0.997696i \(-0.521611\pi\)
0.897951 0.440096i \(-0.145056\pi\)
\(648\) 0 0
\(649\) 7412.11 + 12838.1i 0.448306 + 0.776489i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 716.430 + 1240.89i 0.0429342 + 0.0743643i 0.886694 0.462357i \(-0.152996\pi\)
−0.843760 + 0.536721i \(0.819663\pi\)
\(654\) 0 0
\(655\) −2286.48 + 3960.30i −0.136397 + 0.236247i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12914.0 −0.763367 −0.381684 0.924293i \(-0.624656\pi\)
−0.381684 + 0.924293i \(0.624656\pi\)
\(660\) 0 0
\(661\) 5034.42 8719.87i 0.296243 0.513107i −0.679031 0.734110i \(-0.737600\pi\)
0.975273 + 0.221003i \(0.0709330\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8212.82 + 9370.82i −0.478916 + 0.546444i
\(666\) 0 0
\(667\) 11039.6 + 19121.1i 0.640861 + 1.11000i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −29238.4 −1.68217
\(672\) 0 0
\(673\) −1686.86 −0.0966174 −0.0483087 0.998832i \(-0.515383\pi\)
−0.0483087 + 0.998832i \(0.515383\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −11267.5 19515.9i −0.639654 1.10791i −0.985509 0.169625i \(-0.945744\pi\)
0.345855 0.938288i \(-0.387589\pi\)
\(678\) 0 0
\(679\) 4582.98 + 910.124i 0.259026 + 0.0514394i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −14703.1 + 25466.4i −0.823714 + 1.42671i 0.0791844 + 0.996860i \(0.474768\pi\)
−0.902898 + 0.429854i \(0.858565\pi\)
\(684\) 0 0
\(685\) 11124.2 0.620485
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −699.082 + 1210.85i −0.0386544 + 0.0669515i
\(690\) 0 0
\(691\) −125.763 217.828i −0.00692365 0.0119921i 0.862543 0.505984i \(-0.168871\pi\)
−0.869466 + 0.493992i \(0.835537\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1737.11 3008.77i −0.0948093 0.164215i
\(696\) 0 0
\(697\) −7452.48 + 12908.1i −0.404997 + 0.701475i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 6188.42 0.333429 0.166714 0.986005i \(-0.446684\pi\)
0.166714 + 0.986005i \(0.446684\pi\)
\(702\) 0 0
\(703\) −5371.18 + 9303.16i −0.288162 + 0.499111i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −14108.4 + 16097.7i −0.750497 + 0.856317i
\(708\) 0 0
\(709\) −13719.7 23763.2i −0.726734 1.25874i −0.958256 0.285911i \(-0.907704\pi\)
0.231522 0.972830i \(-0.425629\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 10765.5 0.565456
\(714\) 0 0
\(715\) −11598.7 −0.606667
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −6139.25 10633.5i −0.318436 0.551547i 0.661726 0.749746i \(-0.269824\pi\)
−0.980162 + 0.198199i \(0.936491\pi\)
\(720\) 0 0
\(721\) −6264.03 18434.3i −0.323557 0.952191i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 14173.3 24548.9i 0.726046 1.25755i
\(726\) 0 0
\(727\) 8576.11 0.437510 0.218755 0.975780i \(-0.429800\pi\)
0.218755 + 0.975780i \(0.429800\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4529.53 7845.38i 0.229180 0.396952i
\(732\) 0 0
\(733\) 6437.42 + 11149.9i 0.324381 + 0.561845i 0.981387 0.192041i \(-0.0615106\pi\)
−0.657006 + 0.753886i \(0.728177\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6167.80 + 10682.9i 0.308268 + 0.533937i
\(738\) 0 0
\(739\) −9958.63 + 17248.9i −0.495716 + 0.858606i −0.999988 0.00493950i \(-0.998428\pi\)
0.504272 + 0.863545i \(0.331761\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −22672.6 −1.11948 −0.559741 0.828667i \(-0.689100\pi\)
−0.559741 + 0.828667i \(0.689100\pi\)
\(744\) 0 0
\(745\) 4657.58 8067.16i 0.229048 0.396722i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −12365.4 36390.0i −0.603234 1.77525i
\(750\) 0 0
\(751\) 10476.8 + 18146.3i 0.509059 + 0.881716i 0.999945 + 0.0104920i \(0.00333977\pi\)
−0.490886 + 0.871224i \(0.663327\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −5759.60 −0.277633
\(756\) 0 0
\(757\) −12684.5 −0.609018 −0.304509 0.952510i \(-0.598492\pi\)
−0.304509 + 0.952510i \(0.598492\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −16831.8 29153.6i −0.801779 1.38872i −0.918444 0.395550i \(-0.870554\pi\)
0.116666 0.993171i \(-0.462779\pi\)
\(762\) 0 0
\(763\) 14845.5 16938.8i 0.704383 0.803701i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4850.07 8400.57i 0.228326 0.395472i
\(768\) 0 0
\(769\) −20358.4 −0.954672 −0.477336 0.878721i \(-0.658398\pi\)
−0.477336 + 0.878721i \(0.658398\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −165.421 + 286.517i −0.00769698 + 0.0133316i −0.869848 0.493319i \(-0.835783\pi\)
0.862151 + 0.506651i \(0.169117\pi\)
\(774\) 0 0
\(775\) −6910.69 11969.7i −0.320309 0.554791i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 18429.7 + 31921.2i 0.847641 + 1.46816i
\(780\) 0 0
\(781\) −3144.69 + 5446.77i −0.144079 + 0.249553i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 8669.36 0.394169
\(786\) 0 0
\(787\) −3381.10 + 5856.24i −0.153143 + 0.265251i −0.932381 0.361477i \(-0.882273\pi\)
0.779239 + 0.626727i \(0.215606\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −28803.0 5719.93i −1.29471 0.257114i
\(792\) 0 0
\(793\) 9565.98 + 16568.8i 0.428371 + 0.741960i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −23848.2 −1.05991 −0.529954 0.848027i \(-0.677791\pi\)
−0.529954 + 0.848027i \(0.677791\pi\)
\(798\) 0 0
\(799\) −7017.69 −0.310723
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10583.7 + 18331.5i 0.465118 + 0.805608i
\(804\) 0 0
\(805\) −4828.72 + 5509.57i −0.211416 + 0.241226i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −463.570 + 802.926i −0.0201462 + 0.0348942i −0.875923 0.482451i \(-0.839746\pi\)
0.855777 + 0.517346i \(0.173080\pi\)
\(810\) 0 0
\(811\) −4288.65 −0.185690 −0.0928451 0.995681i \(-0.529596\pi\)
−0.0928451 + 0.995681i \(0.529596\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 4092.63 7088.64i 0.175900 0.304668i
\(816\) 0 0
\(817\) −11201.4 19401.3i −0.479664 0.830803i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 13876.0 + 24034.0i 0.589862 + 1.02167i 0.994250 + 0.107083i \(0.0341510\pi\)
−0.404389 + 0.914587i \(0.632516\pi\)
\(822\) 0 0
\(823\) −4345.08 + 7525.89i −0.184034 + 0.318756i −0.943251 0.332082i \(-0.892249\pi\)
0.759217 + 0.650838i \(0.225582\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −29503.0 −1.24053 −0.620267 0.784391i \(-0.712976\pi\)
−0.620267 + 0.784391i \(0.712976\pi\)
\(828\) 0 0
\(829\) 10138.2 17559.8i 0.424744 0.735678i −0.571653 0.820496i \(-0.693697\pi\)
0.996396 + 0.0848178i \(0.0270308\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −14325.4 + 11006.5i −0.595854 + 0.457806i
\(834\) 0 0
\(835\) −1600.46 2772.07i −0.0663307 0.114888i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −5229.63 −0.215193 −0.107596 0.994195i \(-0.534315\pi\)
−0.107596 + 0.994195i \(0.534315\pi\)
\(840\) 0 0
\(841\) 58737.9 2.40838
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1879.56 3255.49i −0.0765191 0.132535i
\(846\) 0 0
\(847\) −12516.3 36834.1i −0.507751 1.49425i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −3157.98 + 5469.78i −0.127208 + 0.220331i
\(852\) 0 0
\(853\) −16423.9 −0.659255 −0.329628 0.944111i \(-0.606923\pi\)
−0.329628 + 0.944111i \(0.606923\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 15495.9 26839.7i 0.617655 1.06981i −0.372258 0.928129i \(-0.621416\pi\)
0.989913 0.141680i \(-0.0452504\pi\)
\(858\) 0 0
\(859\) 16257.1 + 28158.2i 0.645735 + 1.11845i 0.984131 + 0.177442i \(0.0567823\pi\)
−0.338396 + 0.941004i \(0.609884\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 8682.78 + 15039.0i 0.342486 + 0.593203i 0.984894 0.173160i \(-0.0553978\pi\)
−0.642408 + 0.766363i \(0.722065\pi\)
\(864\) 0 0
\(865\) 9248.27 16018.5i 0.363527 0.629647i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 48914.2 1.90944
\(870\) 0 0
\(871\) 4035.86 6990.32i 0.157003 0.271938i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 20954.9 + 4161.38i 0.809604 + 0.160777i
\(876\) 0 0
\(877\) 10303.2 + 17845.6i 0.396709 + 0.687119i 0.993318 0.115413i \(-0.0368190\pi\)
−0.596609 + 0.802532i \(0.703486\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 4704.88 0.179922 0.0899610 0.995945i \(-0.471326\pi\)
0.0899610 + 0.995945i \(0.471326\pi\)
\(882\) 0 0
\(883\) −13075.1 −0.498313 −0.249157 0.968463i \(-0.580153\pi\)
−0.249157 + 0.968463i \(0.580153\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 22284.6 + 38598.1i 0.843568 + 1.46110i 0.886859 + 0.462039i \(0.152882\pi\)
−0.0432918 + 0.999062i \(0.513785\pi\)
\(888\) 0 0
\(889\) −20115.9 3994.77i −0.758903 0.150709i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −8677.23 + 15029.4i −0.325165 + 0.563203i
\(894\) 0 0
\(895\) −2973.00 −0.111035
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 20265.7 35101.3i 0.751835 1.30222i
\(900\) 0 0
\(901\) 960.577 + 1663.77i 0.0355177 + 0.0615185i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4020.10 + 6963.02i 0.147660 + 0.255755i
\(906\) 0 0
\(907\) 8001.65 13859.3i 0.292933 0.507375i −0.681569 0.731754i \(-0.738702\pi\)
0.974502 + 0.224379i \(0.0720353\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −10072.4 −0.366315 −0.183157 0.983084i \(-0.558632\pi\)
−0.183157 + 0.983084i \(0.558632\pi\)
\(912\) 0 0
\(913\) −21419.0 + 37098.9i −0.776415 + 1.34479i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5275.07 + 15523.9i 0.189965 + 0.559046i
\(918\) 0 0
\(919\) −22639.1 39212.0i −0.812617 1.40749i −0.911027 0.412347i \(-0.864709\pi\)
0.0984103 0.995146i \(-0.468624\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 4115.42 0.146761
\(924\) 0 0
\(925\) 8108.82 0.288234
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −4068.29 7046.48i −0.143677 0.248856i 0.785201 0.619240i \(-0.212559\pi\)
−0.928879 + 0.370384i \(0.879226\pi\)
\(930\) 0 0
\(931\) 5858.94 + 44289.3i 0.206250 + 1.55910i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −7968.63 + 13802.1i −0.278719 + 0.482755i
\(936\) 0 0
\(937\) 34741.9 1.21128 0.605640 0.795739i \(-0.292917\pi\)
0.605640 + 0.795739i \(0.292917\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 18728.8 32439.2i 0.648822 1.12379i −0.334583 0.942366i \(-0.608595\pi\)
0.983405 0.181426i \(-0.0580713\pi\)
\(942\) 0 0
\(943\) 10835.7 + 18768.0i 0.374188 + 0.648113i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −3776.15 6540.48i −0.129576 0.224432i 0.793937 0.608001i \(-0.208028\pi\)
−0.923512 + 0.383569i \(0.874695\pi\)
\(948\) 0 0
\(949\) 6925.36 11995.1i 0.236888 0.410302i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 25745.6 0.875111 0.437556 0.899191i \(-0.355844\pi\)
0.437556 + 0.899191i \(0.355844\pi\)
\(954\) 0 0
\(955\) −2063.93 + 3574.84i −0.0699343 + 0.121130i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 26288.0 29994.7i 0.885178 1.00999i
\(960\) 0 0
\(961\) 5014.22 + 8684.89i 0.168313 + 0.291527i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2882.61 0.0961601
\(966\) 0 0
\(967\) 31718.6 1.05481 0.527406 0.849614i \(-0.323165\pi\)
0.527406 + 0.849614i \(0.323165\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 19654.4 + 34042.4i 0.649578 + 1.12510i 0.983224 + 0.182404i \(0.0583877\pi\)
−0.333646 + 0.942699i \(0.608279\pi\)
\(972\) 0 0
\(973\) −12217.8 2426.30i −0.402553 0.0799420i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2619.30 + 4536.77i −0.0857717 + 0.148561i −0.905720 0.423877i \(-0.860669\pi\)
0.819948 + 0.572438i \(0.194002\pi\)
\(978\) 0 0
\(979\) −12768.3 −0.416830
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −12581.8 + 21792.2i −0.408236 + 0.707085i −0.994692 0.102896i \(-0.967189\pi\)
0.586456 + 0.809981i \(0.300523\pi\)
\(984\) 0 0
\(985\) 4467.11 + 7737.26i 0.144502 + 0.250284i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −6585.82 11407.0i −0.211746 0.366755i
\(990\) 0 0
\(991\) −21780.6 + 37725.1i −0.698166 + 1.20926i 0.270935 + 0.962598i \(0.412667\pi\)
−0.969102 + 0.246662i \(0.920666\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −22137.1 −0.705320
\(996\) 0 0
\(997\) 25420.2 44029.0i 0.807487 1.39861i −0.107112 0.994247i \(-0.534160\pi\)
0.914599 0.404362i \(-0.132506\pi\)
\(998\) 0 0
\(999\) 0 0
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 252.4.k.c.109.2 4
3.2 odd 2 28.4.e.a.25.2 yes 4
7.2 even 3 inner 252.4.k.c.37.2 4
7.3 odd 6 1764.4.a.n.1.2 2
7.4 even 3 1764.4.a.z.1.1 2
7.5 odd 6 1764.4.k.ba.1549.1 4
7.6 odd 2 1764.4.k.ba.361.1 4
12.11 even 2 112.4.i.d.81.1 4
15.2 even 4 700.4.r.d.249.2 8
15.8 even 4 700.4.r.d.249.3 8
15.14 odd 2 700.4.i.g.501.1 4
21.2 odd 6 28.4.e.a.9.2 4
21.5 even 6 196.4.e.g.177.1 4
21.11 odd 6 196.4.a.e.1.1 2
21.17 even 6 196.4.a.g.1.2 2
21.20 even 2 196.4.e.g.165.1 4
24.5 odd 2 448.4.i.h.193.1 4
24.11 even 2 448.4.i.g.193.2 4
84.11 even 6 784.4.a.u.1.2 2
84.23 even 6 112.4.i.d.65.1 4
84.59 odd 6 784.4.a.ba.1.1 2
105.2 even 12 700.4.r.d.149.3 8
105.23 even 12 700.4.r.d.149.2 8
105.44 odd 6 700.4.i.g.401.1 4
168.107 even 6 448.4.i.g.65.2 4
168.149 odd 6 448.4.i.h.65.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.4.e.a.9.2 4 21.2 odd 6
28.4.e.a.25.2 yes 4 3.2 odd 2
112.4.i.d.65.1 4 84.23 even 6
112.4.i.d.81.1 4 12.11 even 2
196.4.a.e.1.1 2 21.11 odd 6
196.4.a.g.1.2 2 21.17 even 6
196.4.e.g.165.1 4 21.20 even 2
196.4.e.g.177.1 4 21.5 even 6
252.4.k.c.37.2 4 7.2 even 3 inner
252.4.k.c.109.2 4 1.1 even 1 trivial
448.4.i.g.65.2 4 168.107 even 6
448.4.i.g.193.2 4 24.11 even 2
448.4.i.h.65.1 4 168.149 odd 6
448.4.i.h.193.1 4 24.5 odd 2
700.4.i.g.401.1 4 105.44 odd 6
700.4.i.g.501.1 4 15.14 odd 2
700.4.r.d.149.2 8 105.23 even 12
700.4.r.d.149.3 8 105.2 even 12
700.4.r.d.249.2 8 15.2 even 4
700.4.r.d.249.3 8 15.8 even 4
784.4.a.u.1.2 2 84.11 even 6
784.4.a.ba.1.1 2 84.59 odd 6
1764.4.a.n.1.2 2 7.3 odd 6
1764.4.a.z.1.1 2 7.4 even 3
1764.4.k.ba.361.1 4 7.6 odd 2
1764.4.k.ba.1549.1 4 7.5 odd 6