Properties

Label 784.3.s.g.129.1
Level $784$
Weight $3$
Character 784.129
Analytic conductor $21.362$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [784,3,Mod(129,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.129");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 784.s (of order \(6\), 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: \(21.3624527258\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 7^{4} \)
Twist minimal: no (minimal twist has level 196)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 129.1
Root \(-0.662827 - 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 784.129
Dual form 784.3.s.g.705.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.86324 - 2.23044i) q^{3} +(-7.33820 + 4.23671i) q^{5} +(5.44975 + 9.43924i) q^{9} +O(q^{10})\) \(q+(-3.86324 - 2.23044i) q^{3} +(-7.33820 + 4.23671i) q^{5} +(5.44975 + 9.43924i) q^{9} +(1.94975 - 3.37706i) q^{11} +19.1886i q^{13} +37.7990 q^{15} +(11.5897 + 6.69133i) q^{17} +(-5.80462 + 3.35130i) q^{19} +(13.0000 + 22.5167i) q^{23} +(23.3995 - 40.5291i) q^{25} -8.47343i q^{27} -11.7990 q^{29} +(-31.6825 - 18.2919i) q^{31} +(-15.0647 + 8.69760i) q^{33} +(16.0000 + 27.7128i) q^{37} +(42.7990 - 74.1300i) q^{39} -20.9594i q^{41} -79.2965 q^{43} +(-79.9827 - 46.1780i) q^{45} +(12.3858 - 7.15093i) q^{47} +(-29.8492 - 51.7004i) q^{51} +(6.89949 - 11.9503i) q^{53} +33.0421i q^{55} +29.8995 q^{57} +(7.31869 + 4.22545i) q^{59} +(27.4114 - 15.8260i) q^{61} +(-81.2965 - 140.810i) q^{65} +(-15.6985 + 27.1906i) q^{67} -115.983i q^{69} -95.5980 q^{71} +(-15.8607 - 9.15721i) q^{73} +(-180.796 + 104.382i) q^{75} +(39.8995 + 69.1080i) q^{79} +(30.1482 - 52.2183i) q^{81} -141.381i q^{83} -113.397 q^{85} +(45.5823 + 26.3170i) q^{87} +(-100.852 + 58.2269i) q^{89} +(81.5980 + 141.332i) q^{93} +(28.3970 - 49.1850i) q^{95} -137.346i q^{97} +42.5025 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} - 24 q^{11} + 144 q^{15} + 104 q^{23} + 108 q^{25} + 64 q^{29} + 128 q^{37} + 184 q^{39} - 80 q^{43} - 120 q^{51} - 24 q^{53} + 160 q^{57} - 96 q^{65} + 112 q^{67} - 448 q^{71} + 240 q^{79} - 36 q^{81} - 432 q^{85} + 336 q^{93} - 248 q^{95} + 736 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(3\) −3.86324 2.23044i −1.28775 0.743481i −0.309495 0.950901i \(-0.600160\pi\)
−0.978252 + 0.207420i \(0.933493\pi\)
\(4\) 0 0
\(5\) −7.33820 + 4.23671i −1.46764 + 0.847343i −0.999344 0.0362281i \(-0.988466\pi\)
−0.468297 + 0.883571i \(0.655132\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 5.44975 + 9.43924i 0.605527 + 1.04880i
\(10\) 0 0
\(11\) 1.94975 3.37706i 0.177250 0.307006i −0.763688 0.645586i \(-0.776613\pi\)
0.940938 + 0.338580i \(0.109947\pi\)
\(12\) 0 0
\(13\) 19.1886i 1.47604i 0.674777 + 0.738022i \(0.264240\pi\)
−0.674777 + 0.738022i \(0.735760\pi\)
\(14\) 0 0
\(15\) 37.7990 2.51993
\(16\) 0 0
\(17\) 11.5897 + 6.69133i 0.681748 + 0.393607i 0.800513 0.599315i \(-0.204560\pi\)
−0.118765 + 0.992922i \(0.537894\pi\)
\(18\) 0 0
\(19\) −5.80462 + 3.35130i −0.305506 + 0.176384i −0.644914 0.764255i \(-0.723107\pi\)
0.339408 + 0.940639i \(0.389773\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 13.0000 + 22.5167i 0.565217 + 0.978985i 0.997029 + 0.0770216i \(0.0245410\pi\)
−0.431812 + 0.901964i \(0.642126\pi\)
\(24\) 0 0
\(25\) 23.3995 40.5291i 0.935980 1.62116i
\(26\) 0 0
\(27\) 8.47343i 0.313831i
\(28\) 0 0
\(29\) −11.7990 −0.406862 −0.203431 0.979089i \(-0.565209\pi\)
−0.203431 + 0.979089i \(0.565209\pi\)
\(30\) 0 0
\(31\) −31.6825 18.2919i −1.02202 0.590061i −0.107328 0.994224i \(-0.534230\pi\)
−0.914687 + 0.404163i \(0.867563\pi\)
\(32\) 0 0
\(33\) −15.0647 + 8.69760i −0.456506 + 0.263564i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 16.0000 + 27.7128i 0.432432 + 0.748995i 0.997082 0.0763357i \(-0.0243221\pi\)
−0.564650 + 0.825331i \(0.690989\pi\)
\(38\) 0 0
\(39\) 42.7990 74.1300i 1.09741 1.90077i
\(40\) 0 0
\(41\) 20.9594i 0.511205i −0.966782 0.255602i \(-0.917726\pi\)
0.966782 0.255602i \(-0.0822738\pi\)
\(42\) 0 0
\(43\) −79.2965 −1.84410 −0.922052 0.387066i \(-0.873488\pi\)
−0.922052 + 0.387066i \(0.873488\pi\)
\(44\) 0 0
\(45\) −79.9827 46.1780i −1.77739 1.02618i
\(46\) 0 0
\(47\) 12.3858 7.15093i 0.263527 0.152148i −0.362415 0.932017i \(-0.618048\pi\)
0.625943 + 0.779869i \(0.284714\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −29.8492 51.7004i −0.585279 1.01373i
\(52\) 0 0
\(53\) 6.89949 11.9503i 0.130179 0.225477i −0.793566 0.608484i \(-0.791778\pi\)
0.923746 + 0.383007i \(0.125111\pi\)
\(54\) 0 0
\(55\) 33.0421i 0.600765i
\(56\) 0 0
\(57\) 29.8995 0.524553
\(58\) 0 0
\(59\) 7.31869 + 4.22545i 0.124046 + 0.0716178i 0.560739 0.827993i \(-0.310517\pi\)
−0.436693 + 0.899610i \(0.643850\pi\)
\(60\) 0 0
\(61\) 27.4114 15.8260i 0.449368 0.259443i −0.258195 0.966093i \(-0.583128\pi\)
0.707563 + 0.706650i \(0.249794\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −81.2965 140.810i −1.25071 2.16630i
\(66\) 0 0
\(67\) −15.6985 + 27.1906i −0.234306 + 0.405829i −0.959071 0.283167i \(-0.908615\pi\)
0.724765 + 0.688996i \(0.241948\pi\)
\(68\) 0 0
\(69\) 115.983i 1.68091i
\(70\) 0 0
\(71\) −95.5980 −1.34645 −0.673225 0.739438i \(-0.735092\pi\)
−0.673225 + 0.739438i \(0.735092\pi\)
\(72\) 0 0
\(73\) −15.8607 9.15721i −0.217270 0.125441i 0.387415 0.921905i \(-0.373368\pi\)
−0.604686 + 0.796464i \(0.706701\pi\)
\(74\) 0 0
\(75\) −180.796 + 104.382i −2.41061 + 1.39177i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 39.8995 + 69.1080i 0.505057 + 0.874784i 0.999983 + 0.00584908i \(0.00186183\pi\)
−0.494926 + 0.868935i \(0.664805\pi\)
\(80\) 0 0
\(81\) 30.1482 52.2183i 0.372200 0.644670i
\(82\) 0 0
\(83\) 141.381i 1.70338i −0.524044 0.851691i \(-0.675577\pi\)
0.524044 0.851691i \(-0.324423\pi\)
\(84\) 0 0
\(85\) −113.397 −1.33408
\(86\) 0 0
\(87\) 45.5823 + 26.3170i 0.523935 + 0.302494i
\(88\) 0 0
\(89\) −100.852 + 58.2269i −1.13317 + 0.654235i −0.944730 0.327850i \(-0.893676\pi\)
−0.188439 + 0.982085i \(0.560343\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 81.5980 + 141.332i 0.877398 + 1.51970i
\(94\) 0 0
\(95\) 28.3970 49.1850i 0.298915 0.517737i
\(96\) 0 0
\(97\) 137.346i 1.41593i −0.706245 0.707967i \(-0.749612\pi\)
0.706245 0.707967i \(-0.250388\pi\)
\(98\) 0 0
\(99\) 42.5025 0.429318
\(100\) 0 0
\(101\) −78.3906 45.2588i −0.776145 0.448107i 0.0589176 0.998263i \(-0.481235\pi\)
−0.835062 + 0.550156i \(0.814568\pi\)
\(102\) 0 0
\(103\) 12.3468 7.12840i 0.119871 0.0692078i −0.438865 0.898553i \(-0.644620\pi\)
0.558737 + 0.829345i \(0.311286\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −24.7990 42.9531i −0.231766 0.401431i 0.726562 0.687101i \(-0.241117\pi\)
−0.958328 + 0.285670i \(0.907784\pi\)
\(108\) 0 0
\(109\) −73.5980 + 127.475i −0.675211 + 1.16950i 0.301196 + 0.953562i \(0.402614\pi\)
−0.976407 + 0.215937i \(0.930719\pi\)
\(110\) 0 0
\(111\) 142.748i 1.28602i
\(112\) 0 0
\(113\) 74.3015 0.657536 0.328768 0.944411i \(-0.393367\pi\)
0.328768 + 0.944411i \(0.393367\pi\)
\(114\) 0 0
\(115\) −190.793 110.155i −1.65907 0.957866i
\(116\) 0 0
\(117\) −181.125 + 104.573i −1.54808 + 0.893785i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 52.8970 + 91.6202i 0.437165 + 0.757192i
\(122\) 0 0
\(123\) −46.7487 + 80.9712i −0.380071 + 0.658302i
\(124\) 0 0
\(125\) 184.712i 1.47770i
\(126\) 0 0
\(127\) 76.9949 0.606259 0.303130 0.952949i \(-0.401968\pi\)
0.303130 + 0.952949i \(0.401968\pi\)
\(128\) 0 0
\(129\) 306.341 + 176.866i 2.37474 + 1.37106i
\(130\) 0 0
\(131\) 134.398 77.5946i 1.02594 0.592325i 0.110119 0.993918i \(-0.464877\pi\)
0.915818 + 0.401593i \(0.131543\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 35.8995 + 62.1797i 0.265922 + 0.460591i
\(136\) 0 0
\(137\) 65.0503 112.670i 0.474819 0.822411i −0.524765 0.851247i \(-0.675847\pi\)
0.999584 + 0.0288360i \(0.00918006\pi\)
\(138\) 0 0
\(139\) 200.740i 1.44417i 0.691804 + 0.722086i \(0.256816\pi\)
−0.691804 + 0.722086i \(0.743184\pi\)
\(140\) 0 0
\(141\) −63.7990 −0.452475
\(142\) 0 0
\(143\) 64.8010 + 37.4129i 0.453154 + 0.261628i
\(144\) 0 0
\(145\) 86.5834 49.9889i 0.597127 0.344751i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −84.6985 146.702i −0.568446 0.984578i −0.996720 0.0809286i \(-0.974211\pi\)
0.428274 0.903649i \(-0.359122\pi\)
\(150\) 0 0
\(151\) −1.10051 + 1.90613i −0.00728811 + 0.0126234i −0.869646 0.493675i \(-0.835653\pi\)
0.862358 + 0.506298i \(0.168987\pi\)
\(152\) 0 0
\(153\) 145.864i 0.953361i
\(154\) 0 0
\(155\) 309.990 1.99993
\(156\) 0 0
\(157\) −139.504 80.5426i −0.888560 0.513010i −0.0150888 0.999886i \(-0.504803\pi\)
−0.873471 + 0.486876i \(0.838136\pi\)
\(158\) 0 0
\(159\) −53.3088 + 30.7779i −0.335276 + 0.193571i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −111.146 192.510i −0.681876 1.18104i −0.974408 0.224787i \(-0.927831\pi\)
0.292532 0.956256i \(-0.405502\pi\)
\(164\) 0 0
\(165\) 73.6985 127.650i 0.446657 0.773633i
\(166\) 0 0
\(167\) 207.084i 1.24003i −0.784592 0.620013i \(-0.787127\pi\)
0.784592 0.620013i \(-0.212873\pi\)
\(168\) 0 0
\(169\) −199.201 −1.17870
\(170\) 0 0
\(171\) −63.2674 36.5274i −0.369985 0.213611i
\(172\) 0 0
\(173\) 240.608 138.915i 1.39080 0.802976i 0.397392 0.917649i \(-0.369915\pi\)
0.993403 + 0.114673i \(0.0365819\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −18.8492 32.6478i −0.106493 0.184451i
\(178\) 0 0
\(179\) 170.196 294.788i 0.950815 1.64686i 0.207149 0.978309i \(-0.433581\pi\)
0.743666 0.668551i \(-0.233085\pi\)
\(180\) 0 0
\(181\) 184.174i 1.01753i 0.860904 + 0.508767i \(0.169899\pi\)
−0.860904 + 0.508767i \(0.830101\pi\)
\(182\) 0 0
\(183\) −141.196 −0.771563
\(184\) 0 0
\(185\) −234.823 135.575i −1.26931 0.732837i
\(186\) 0 0
\(187\) 45.1941 26.0928i 0.241679 0.139534i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 114.794 + 198.829i 0.601015 + 1.04099i 0.992668 + 0.120876i \(0.0385703\pi\)
−0.391652 + 0.920113i \(0.628096\pi\)
\(192\) 0 0
\(193\) 20.7487 35.9379i 0.107506 0.186207i −0.807253 0.590205i \(-0.799047\pi\)
0.914759 + 0.403999i \(0.132380\pi\)
\(194\) 0 0
\(195\) 725.308i 3.71953i
\(196\) 0 0
\(197\) 253.598 1.28730 0.643650 0.765320i \(-0.277419\pi\)
0.643650 + 0.765320i \(0.277419\pi\)
\(198\) 0 0
\(199\) 97.2990 + 56.1756i 0.488940 + 0.282289i 0.724135 0.689659i \(-0.242239\pi\)
−0.235195 + 0.971948i \(0.575573\pi\)
\(200\) 0 0
\(201\) 121.294 70.0291i 0.603453 0.348404i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 88.7990 + 153.804i 0.433166 + 0.750265i
\(206\) 0 0
\(207\) −141.693 + 245.420i −0.684509 + 1.18560i
\(208\) 0 0
\(209\) 26.1367i 0.125056i
\(210\) 0 0
\(211\) −353.789 −1.67672 −0.838362 0.545113i \(-0.816487\pi\)
−0.838362 + 0.545113i \(0.816487\pi\)
\(212\) 0 0
\(213\) 369.318 + 213.226i 1.73389 + 1.00106i
\(214\) 0 0
\(215\) 581.894 335.956i 2.70648 1.56259i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 40.8492 + 70.7530i 0.186526 + 0.323073i
\(220\) 0 0
\(221\) −128.397 + 222.390i −0.580982 + 1.00629i
\(222\) 0 0
\(223\) 126.653i 0.567951i −0.958832 0.283976i \(-0.908347\pi\)
0.958832 0.283976i \(-0.0916534\pi\)
\(224\) 0 0
\(225\) 510.085 2.26705
\(226\) 0 0
\(227\) 195.589 + 112.923i 0.861626 + 0.497460i 0.864557 0.502535i \(-0.167599\pi\)
−0.00293018 + 0.999996i \(0.500933\pi\)
\(228\) 0 0
\(229\) 223.680 129.141i 0.976767 0.563937i 0.0754744 0.997148i \(-0.475953\pi\)
0.901292 + 0.433211i \(0.142620\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −28.8492 49.9684i −0.123816 0.214456i 0.797453 0.603381i \(-0.206180\pi\)
−0.921270 + 0.388924i \(0.872847\pi\)
\(234\) 0 0
\(235\) −60.5929 + 104.950i −0.257842 + 0.446596i
\(236\) 0 0
\(237\) 355.974i 1.50200i
\(238\) 0 0
\(239\) −82.6030 −0.345619 −0.172810 0.984955i \(-0.555285\pi\)
−0.172810 + 0.984955i \(0.555285\pi\)
\(240\) 0 0
\(241\) −113.160 65.3328i −0.469543 0.271091i 0.246506 0.969141i \(-0.420718\pi\)
−0.716048 + 0.698051i \(0.754051\pi\)
\(242\) 0 0
\(243\) −298.984 + 172.618i −1.23038 + 0.710363i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −64.3066 111.382i −0.260350 0.450940i
\(248\) 0 0
\(249\) −315.342 + 546.188i −1.26643 + 2.19353i
\(250\) 0 0
\(251\) 149.854i 0.597029i 0.954405 + 0.298514i \(0.0964911\pi\)
−0.954405 + 0.298514i \(0.903509\pi\)
\(252\) 0 0
\(253\) 101.387 0.400739
\(254\) 0 0
\(255\) 438.080 + 252.925i 1.71796 + 0.991864i
\(256\) 0 0
\(257\) 4.52272 2.61119i 0.0175981 0.0101603i −0.491175 0.871061i \(-0.663432\pi\)
0.508773 + 0.860901i \(0.330099\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −64.3015 111.373i −0.246366 0.426718i
\(262\) 0 0
\(263\) −66.7035 + 115.534i −0.253626 + 0.439292i −0.964521 0.264005i \(-0.914956\pi\)
0.710896 + 0.703297i \(0.248290\pi\)
\(264\) 0 0
\(265\) 116.925i 0.441225i
\(266\) 0 0
\(267\) 519.487 1.94565
\(268\) 0 0
\(269\) 3.57252 + 2.06260i 0.0132808 + 0.00766765i 0.506626 0.862166i \(-0.330893\pi\)
−0.493345 + 0.869834i \(0.664226\pi\)
\(270\) 0 0
\(271\) −273.455 + 157.879i −1.00906 + 0.582580i −0.910916 0.412592i \(-0.864624\pi\)
−0.0981429 + 0.995172i \(0.531290\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −91.2462 158.043i −0.331804 0.574702i
\(276\) 0 0
\(277\) 130.095 225.332i 0.469659 0.813473i −0.529740 0.848160i \(-0.677710\pi\)
0.999398 + 0.0346877i \(0.0110436\pi\)
\(278\) 0 0
\(279\) 398.745i 1.42919i
\(280\) 0 0
\(281\) −372.894 −1.32703 −0.663513 0.748165i \(-0.730935\pi\)
−0.663513 + 0.748165i \(0.730935\pi\)
\(282\) 0 0
\(283\) 69.2476 + 39.9801i 0.244691 + 0.141273i 0.617331 0.786704i \(-0.288214\pi\)
−0.372640 + 0.927976i \(0.621547\pi\)
\(284\) 0 0
\(285\) −219.409 + 126.676i −0.769855 + 0.444476i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −54.9523 95.1801i −0.190146 0.329343i
\(290\) 0 0
\(291\) −306.342 + 530.599i −1.05272 + 1.82337i
\(292\) 0 0
\(293\) 112.038i 0.382382i −0.981553 0.191191i \(-0.938765\pi\)
0.981553 0.191191i \(-0.0612351\pi\)
\(294\) 0 0
\(295\) −71.6081 −0.242739
\(296\) 0 0
\(297\) −28.6153 16.5210i −0.0963478 0.0556264i
\(298\) 0 0
\(299\) −432.062 + 249.451i −1.44502 + 0.834285i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 201.894 + 349.691i 0.666318 + 1.15410i
\(304\) 0 0
\(305\) −134.101 + 232.269i −0.439674 + 0.761537i
\(306\) 0 0
\(307\) 146.177i 0.476148i 0.971247 + 0.238074i \(0.0765160\pi\)
−0.971247 + 0.238074i \(0.923484\pi\)
\(308\) 0 0
\(309\) −63.5980 −0.205819
\(310\) 0 0
\(311\) −366.290 211.477i −1.17778 0.679992i −0.222280 0.974983i \(-0.571350\pi\)
−0.955500 + 0.294991i \(0.904683\pi\)
\(312\) 0 0
\(313\) −75.9634 + 43.8575i −0.242695 + 0.140120i −0.616415 0.787422i \(-0.711415\pi\)
0.373720 + 0.927542i \(0.378082\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −215.693 373.592i −0.680421 1.17852i −0.974853 0.222851i \(-0.928464\pi\)
0.294432 0.955673i \(-0.404870\pi\)
\(318\) 0 0
\(319\) −23.0051 + 39.8459i −0.0721161 + 0.124909i
\(320\) 0 0
\(321\) 221.251i 0.689255i
\(322\) 0 0
\(323\) −89.6985 −0.277704
\(324\) 0 0
\(325\) 777.696 + 449.003i 2.39291 + 1.38155i
\(326\) 0 0
\(327\) 568.653 328.312i 1.73900 1.00401i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −170.744 295.737i −0.515842 0.893464i −0.999831 0.0183904i \(-0.994146\pi\)
0.483989 0.875074i \(-0.339187\pi\)
\(332\) 0 0
\(333\) −174.392 + 302.056i −0.523699 + 0.907074i
\(334\) 0 0
\(335\) 266.040i 0.794149i
\(336\) 0 0
\(337\) −391.377 −1.16136 −0.580678 0.814134i \(-0.697212\pi\)
−0.580678 + 0.814134i \(0.697212\pi\)
\(338\) 0 0
\(339\) −287.045 165.725i −0.846739 0.488865i
\(340\) 0 0
\(341\) −123.546 + 71.3291i −0.362304 + 0.209176i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 491.387 + 851.107i 1.42431 + 2.46698i
\(346\) 0 0
\(347\) −304.241 + 526.961i −0.876776 + 1.51862i −0.0219166 + 0.999760i \(0.506977\pi\)
−0.854859 + 0.518860i \(0.826357\pi\)
\(348\) 0 0
\(349\) 93.1627i 0.266942i 0.991053 + 0.133471i \(0.0426123\pi\)
−0.991053 + 0.133471i \(0.957388\pi\)
\(350\) 0 0
\(351\) 162.593 0.463228
\(352\) 0 0
\(353\) −80.0022 46.1893i −0.226635 0.130848i 0.382384 0.924004i \(-0.375103\pi\)
−0.609019 + 0.793156i \(0.708437\pi\)
\(354\) 0 0
\(355\) 701.518 405.021i 1.97611 1.14091i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −144.588 250.434i −0.402752 0.697586i 0.591305 0.806448i \(-0.298613\pi\)
−0.994057 + 0.108861i \(0.965280\pi\)
\(360\) 0 0
\(361\) −158.038 + 273.729i −0.437777 + 0.758253i
\(362\) 0 0
\(363\) 471.935i 1.30010i
\(364\) 0 0
\(365\) 155.186 0.425167
\(366\) 0 0
\(367\) 458.968 + 264.986i 1.25060 + 0.722032i 0.971228 0.238153i \(-0.0765419\pi\)
0.279368 + 0.960184i \(0.409875\pi\)
\(368\) 0 0
\(369\) 197.841 114.223i 0.536154 0.309549i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −232.899 403.394i −0.624395 1.08148i −0.988657 0.150188i \(-0.952012\pi\)
0.364262 0.931297i \(-0.381321\pi\)
\(374\) 0 0
\(375\) 411.990 713.587i 1.09864 1.90290i
\(376\) 0 0
\(377\) 226.406i 0.600546i
\(378\) 0 0
\(379\) −91.8793 −0.242426 −0.121213 0.992627i \(-0.538678\pi\)
−0.121213 + 0.992627i \(0.538678\pi\)
\(380\) 0 0
\(381\) −297.450 171.733i −0.780709 0.450742i
\(382\) 0 0
\(383\) 269.572 155.638i 0.703844 0.406364i −0.104934 0.994479i \(-0.533463\pi\)
0.808777 + 0.588115i \(0.200130\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −432.146 748.498i −1.11666 1.93410i
\(388\) 0 0
\(389\) 271.296 469.899i 0.697420 1.20797i −0.271938 0.962315i \(-0.587664\pi\)
0.969358 0.245652i \(-0.0790022\pi\)
\(390\) 0 0
\(391\) 347.949i 0.889895i
\(392\) 0 0
\(393\) −692.281 −1.76153
\(394\) 0 0
\(395\) −585.581 338.086i −1.48248 0.855913i
\(396\) 0 0
\(397\) −141.834 + 81.8877i −0.357263 + 0.206266i −0.667880 0.744269i \(-0.732798\pi\)
0.310616 + 0.950535i \(0.399465\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 71.7035 + 124.194i 0.178812 + 0.309711i 0.941474 0.337086i \(-0.109441\pi\)
−0.762662 + 0.646797i \(0.776108\pi\)
\(402\) 0 0
\(403\) 350.995 607.941i 0.870955 1.50854i
\(404\) 0 0
\(405\) 510.918i 1.26153i
\(406\) 0 0
\(407\) 124.784 0.306594
\(408\) 0 0
\(409\) 251.150 + 145.001i 0.614058 + 0.354526i 0.774552 0.632510i \(-0.217975\pi\)
−0.160494 + 0.987037i \(0.551309\pi\)
\(410\) 0 0
\(411\) −502.609 + 290.182i −1.22289 + 0.706038i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 598.990 + 1037.48i 1.44335 + 2.49995i
\(416\) 0 0
\(417\) 447.739 775.506i 1.07371 1.85973i
\(418\) 0 0
\(419\) 340.170i 0.811860i −0.913904 0.405930i \(-0.866948\pi\)
0.913904 0.405930i \(-0.133052\pi\)
\(420\) 0 0
\(421\) −6.18081 −0.0146813 −0.00734063 0.999973i \(-0.502337\pi\)
−0.00734063 + 0.999973i \(0.502337\pi\)
\(422\) 0 0
\(423\) 134.999 + 77.9416i 0.319146 + 0.184259i
\(424\) 0 0
\(425\) 542.387 313.147i 1.27621 0.736817i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −166.894 289.070i −0.389031 0.673822i
\(430\) 0 0
\(431\) −14.2965 + 24.7622i −0.0331705 + 0.0574529i −0.882134 0.470998i \(-0.843894\pi\)
0.848964 + 0.528451i \(0.177227\pi\)
\(432\) 0 0
\(433\) 243.736i 0.562900i −0.959576 0.281450i \(-0.909185\pi\)
0.959576 0.281450i \(-0.0908153\pi\)
\(434\) 0 0
\(435\) −445.990 −1.02526
\(436\) 0 0
\(437\) −150.920 87.1337i −0.345355 0.199391i
\(438\) 0 0
\(439\) −428.605 + 247.455i −0.976321 + 0.563679i −0.901157 0.433492i \(-0.857281\pi\)
−0.0751637 + 0.997171i \(0.523948\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 39.6985 + 68.7598i 0.0896128 + 0.155214i 0.907348 0.420381i \(-0.138104\pi\)
−0.817735 + 0.575595i \(0.804770\pi\)
\(444\) 0 0
\(445\) 493.382 854.562i 1.10872 1.92036i
\(446\) 0 0
\(447\) 755.660i 1.69052i
\(448\) 0 0
\(449\) 804.362 1.79145 0.895726 0.444607i \(-0.146657\pi\)
0.895726 + 0.444607i \(0.146657\pi\)
\(450\) 0 0
\(451\) −70.7812 40.8655i −0.156943 0.0906109i
\(452\) 0 0
\(453\) 8.50303 4.90923i 0.0187705 0.0108371i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −268.739 465.469i −0.588050 1.01853i −0.994488 0.104853i \(-0.966563\pi\)
0.406438 0.913678i \(-0.366771\pi\)
\(458\) 0 0
\(459\) 56.6985 98.2047i 0.123526 0.213953i
\(460\) 0 0
\(461\) 524.278i 1.13726i −0.822593 0.568631i \(-0.807473\pi\)
0.822593 0.568631i \(-0.192527\pi\)
\(462\) 0 0
\(463\) 506.995 1.09502 0.547511 0.836799i \(-0.315576\pi\)
0.547511 + 0.836799i \(0.315576\pi\)
\(464\) 0 0
\(465\) −1197.57 691.415i −2.57541 1.48691i
\(466\) 0 0
\(467\) −178.234 + 102.903i −0.381657 + 0.220350i −0.678539 0.734564i \(-0.737387\pi\)
0.296882 + 0.954914i \(0.404053\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 359.291 + 622.311i 0.762827 + 1.32125i
\(472\) 0 0
\(473\) −154.608 + 267.789i −0.326867 + 0.566150i
\(474\) 0 0
\(475\) 313.675i 0.660368i
\(476\) 0 0
\(477\) 150.402 0.315308
\(478\) 0 0
\(479\) −429.577 248.016i −0.896820 0.517779i −0.0206527 0.999787i \(-0.506574\pi\)
−0.876167 + 0.482008i \(0.839908\pi\)
\(480\) 0 0
\(481\) −531.769 + 307.017i −1.10555 + 0.638289i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 581.894 + 1007.87i 1.19978 + 2.07808i
\(486\) 0 0
\(487\) 236.688 409.956i 0.486013 0.841799i −0.513858 0.857875i \(-0.671784\pi\)
0.999871 + 0.0160761i \(0.00511739\pi\)
\(488\) 0 0
\(489\) 991.616i 2.02785i
\(490\) 0 0
\(491\) 143.417 0.292092 0.146046 0.989278i \(-0.453345\pi\)
0.146046 + 0.989278i \(0.453345\pi\)
\(492\) 0 0
\(493\) −136.747 78.9509i −0.277377 0.160144i
\(494\) 0 0
\(495\) −311.892 + 180.071i −0.630085 + 0.363780i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −223.497 387.109i −0.447891 0.775770i 0.550358 0.834929i \(-0.314491\pi\)
−0.998249 + 0.0591594i \(0.981158\pi\)
\(500\) 0 0
\(501\) −461.889 + 800.016i −0.921935 + 1.59684i
\(502\) 0 0
\(503\) 660.346i 1.31282i 0.754406 + 0.656408i \(0.227925\pi\)
−0.754406 + 0.656408i \(0.772075\pi\)
\(504\) 0 0
\(505\) 766.995 1.51880
\(506\) 0 0
\(507\) 769.561 + 444.306i 1.51787 + 0.876344i
\(508\) 0 0
\(509\) 421.228 243.196i 0.827559 0.477792i −0.0254569 0.999676i \(-0.508104\pi\)
0.853016 + 0.521884i \(0.174771\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 28.3970 + 49.1850i 0.0553547 + 0.0958772i
\(514\) 0 0
\(515\) −60.4020 + 104.619i −0.117285 + 0.203144i
\(516\) 0 0
\(517\) 55.7701i 0.107872i
\(518\) 0 0
\(519\) −1239.37 −2.38799
\(520\) 0 0
\(521\) −89.0106 51.3903i −0.170846 0.0986378i 0.412139 0.911121i \(-0.364782\pi\)
−0.582985 + 0.812483i \(0.698115\pi\)
\(522\) 0 0
\(523\) −447.960 + 258.630i −0.856520 + 0.494512i −0.862846 0.505468i \(-0.831320\pi\)
0.00632508 + 0.999980i \(0.497987\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −244.794 423.996i −0.464505 0.804546i
\(528\) 0 0
\(529\) −73.5000 + 127.306i −0.138941 + 0.240654i
\(530\) 0 0
\(531\) 92.1105i 0.173466i
\(532\) 0 0
\(533\) 402.181 0.754561
\(534\) 0 0
\(535\) 363.960 + 210.132i 0.680299 + 0.392771i
\(536\) 0 0
\(537\) −1315.02 + 759.225i −2.44882 + 1.41383i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 225.201 + 390.060i 0.416268 + 0.720997i 0.995561 0.0941223i \(-0.0300045\pi\)
−0.579293 + 0.815120i \(0.696671\pi\)
\(542\) 0 0
\(543\) 410.789 711.507i 0.756517 1.31033i
\(544\) 0 0
\(545\) 1247.25i 2.28854i
\(546\) 0 0
\(547\) −195.095 −0.356664 −0.178332 0.983970i \(-0.557070\pi\)
−0.178332 + 0.983970i \(0.557070\pi\)
\(548\) 0 0
\(549\) 298.771 + 172.495i 0.544209 + 0.314199i
\(550\) 0 0
\(551\) 68.4886 39.5419i 0.124299 0.0717639i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 604.784 + 1047.52i 1.08970 + 1.88742i
\(556\) 0 0
\(557\) 344.980 597.523i 0.619353 1.07275i −0.370251 0.928932i \(-0.620728\pi\)
0.989604 0.143820i \(-0.0459385\pi\)
\(558\) 0 0
\(559\) 1521.59i 2.72198i
\(560\) 0 0
\(561\) −232.794 −0.414962
\(562\) 0 0
\(563\) −724.755 418.438i −1.28731 0.743229i −0.309136 0.951018i \(-0.600040\pi\)
−0.978174 + 0.207789i \(0.933373\pi\)
\(564\) 0 0
\(565\) −545.240 + 314.794i −0.965026 + 0.557158i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3.90960 6.77162i −0.00687100 0.0119009i 0.862569 0.505939i \(-0.168854\pi\)
−0.869440 + 0.494038i \(0.835520\pi\)
\(570\) 0 0
\(571\) 0.854293 1.47968i 0.00149613 0.00259138i −0.865276 0.501295i \(-0.832857\pi\)
0.866773 + 0.498704i \(0.166190\pi\)
\(572\) 0 0
\(573\) 1024.17i 1.78737i
\(574\) 0 0
\(575\) 1216.77 2.11613
\(576\) 0 0
\(577\) 187.824 + 108.440i 0.325518 + 0.187938i 0.653849 0.756625i \(-0.273153\pi\)
−0.328332 + 0.944563i \(0.606486\pi\)
\(578\) 0 0
\(579\) −160.315 + 92.5577i −0.276882 + 0.159858i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −26.9045 46.6000i −0.0461484 0.0799315i
\(584\) 0 0
\(585\) 886.090 1534.75i 1.51468 2.62351i
\(586\) 0 0
\(587\) 50.9983i 0.0868795i −0.999056 0.0434398i \(-0.986168\pi\)
0.999056 0.0434398i \(-0.0138317\pi\)
\(588\) 0 0
\(589\) 245.206 0.416309
\(590\) 0 0
\(591\) −979.710 565.636i −1.65772 0.957082i
\(592\) 0 0
\(593\) 563.586 325.387i 0.950398 0.548713i 0.0571937 0.998363i \(-0.481785\pi\)
0.893205 + 0.449650i \(0.148451\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −250.593 434.040i −0.419754 0.727035i
\(598\) 0 0
\(599\) −16.8944 + 29.2620i −0.0282044 + 0.0488515i −0.879783 0.475375i \(-0.842312\pi\)
0.851579 + 0.524227i \(0.175646\pi\)
\(600\) 0 0
\(601\) 415.133i 0.690737i 0.938467 + 0.345368i \(0.112246\pi\)
−0.938467 + 0.345368i \(0.887754\pi\)
\(602\) 0 0
\(603\) −342.211 −0.567514
\(604\) 0 0
\(605\) −776.338 448.219i −1.28320 0.740857i
\(606\) 0 0
\(607\) −377.083 + 217.709i −0.621225 + 0.358664i −0.777346 0.629074i \(-0.783434\pi\)
0.156121 + 0.987738i \(0.450101\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 137.216 + 237.665i 0.224576 + 0.388978i
\(612\) 0 0
\(613\) −212.382 + 367.856i −0.346463 + 0.600092i −0.985618 0.168986i \(-0.945951\pi\)
0.639155 + 0.769078i \(0.279284\pi\)
\(614\) 0 0
\(615\) 792.244i 1.28820i
\(616\) 0 0
\(617\) −726.563 −1.17757 −0.588787 0.808289i \(-0.700394\pi\)
−0.588787 + 0.808289i \(0.700394\pi\)
\(618\) 0 0
\(619\) 613.943 + 354.460i 0.991830 + 0.572633i 0.905821 0.423661i \(-0.139255\pi\)
0.0860094 + 0.996294i \(0.472588\pi\)
\(620\) 0 0
\(621\) 190.793 110.155i 0.307236 0.177383i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −197.585 342.228i −0.316137 0.547565i
\(626\) 0 0
\(627\) 58.2965 100.972i 0.0929768 0.161041i
\(628\) 0 0
\(629\) 428.245i 0.680835i
\(630\) 0 0
\(631\) −207.176 −0.328329 −0.164165 0.986433i \(-0.552493\pi\)
−0.164165 + 0.986433i \(0.552493\pi\)
\(632\) 0 0
\(633\) 1366.77 + 789.106i 2.15920 + 1.24661i
\(634\) 0 0
\(635\) −565.005 + 326.206i −0.889771 + 0.513710i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −520.985 902.372i −0.815313 1.41216i
\(640\) 0 0
\(641\) −456.291 + 790.320i −0.711843 + 1.23295i 0.252321 + 0.967643i \(0.418806\pi\)
−0.964165 + 0.265305i \(0.914527\pi\)
\(642\) 0 0
\(643\) 656.917i 1.02164i −0.859686 0.510822i \(-0.829341\pi\)
0.859686 0.510822i \(-0.170659\pi\)
\(644\) 0 0
\(645\) −2997.33 −4.64702
\(646\) 0 0
\(647\) −33.0014 19.0534i −0.0510068 0.0294488i 0.474280 0.880374i \(-0.342709\pi\)
−0.525287 + 0.850925i \(0.676042\pi\)
\(648\) 0 0
\(649\) 28.5392 16.4771i 0.0439741 0.0253885i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 188.291 + 326.130i 0.288348 + 0.499434i 0.973416 0.229046i \(-0.0735606\pi\)
−0.685067 + 0.728480i \(0.740227\pi\)
\(654\) 0 0
\(655\) −657.492 + 1138.81i −1.00381 + 1.73864i
\(656\) 0 0
\(657\) 199.618i 0.303832i
\(658\) 0 0
\(659\) 954.291 1.44809 0.724045 0.689753i \(-0.242281\pi\)
0.724045 + 0.689753i \(0.242281\pi\)
\(660\) 0 0
\(661\) −1010.11 583.187i −1.52815 0.882281i −0.999439 0.0334815i \(-0.989341\pi\)
−0.528716 0.848799i \(-0.677326\pi\)
\(662\) 0 0
\(663\) 992.057 572.764i 1.49631 0.863898i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −153.387 265.674i −0.229965 0.398312i
\(668\) 0 0
\(669\) −282.492 + 489.291i −0.422261 + 0.731377i
\(670\) 0 0
\(671\) 123.427i 0.183945i
\(672\) 0 0
\(673\) −774.101 −1.15022 −0.575112 0.818075i \(-0.695041\pi\)
−0.575112 + 0.818075i \(0.695041\pi\)
\(674\) 0 0
\(675\) −343.421 198.274i −0.508771 0.293739i
\(676\) 0 0
\(677\) −486.263 + 280.744i −0.718261 + 0.414688i −0.814112 0.580707i \(-0.802776\pi\)
0.0958511 + 0.995396i \(0.469443\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −503.739 872.501i −0.739704 1.28121i
\(682\) 0 0
\(683\) 176.080 304.980i 0.257804 0.446530i −0.707849 0.706364i \(-0.750334\pi\)
0.965653 + 0.259834i \(0.0836677\pi\)
\(684\) 0 0
\(685\) 1102.40i 1.60934i
\(686\) 0 0
\(687\) −1152.17 −1.67710
\(688\) 0 0
\(689\) 229.309 + 132.391i 0.332814 + 0.192150i
\(690\) 0 0
\(691\) 907.163 523.751i 1.31283 0.757961i 0.330263 0.943889i \(-0.392863\pi\)
0.982563 + 0.185929i \(0.0595293\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −850.477 1473.07i −1.22371 2.11952i
\(696\) 0 0
\(697\) 140.246 242.914i 0.201214 0.348513i
\(698\) 0 0
\(699\) 257.386i 0.368221i
\(700\) 0 0
\(701\) 809.990 1.15548 0.577739 0.816222i \(-0.303935\pi\)
0.577739 + 0.816222i \(0.303935\pi\)
\(702\) 0 0
\(703\) −185.748 107.241i −0.264221 0.152548i
\(704\) 0 0
\(705\) 468.170 270.298i 0.664071 0.383402i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 229.980 + 398.337i 0.324372 + 0.561829i 0.981385 0.192050i \(-0.0615137\pi\)
−0.657013 + 0.753879i \(0.728180\pi\)
\(710\) 0 0
\(711\) −434.884 + 753.242i −0.611652 + 1.05941i
\(712\) 0 0
\(713\) 951.178i 1.33405i
\(714\) 0 0
\(715\) −634.030 −0.886756
\(716\) 0 0
\(717\) 319.115 + 184.241i 0.445070 + 0.256961i
\(718\) 0 0
\(719\) −74.5098 + 43.0183i −0.103630 + 0.0598307i −0.550919 0.834559i \(-0.685723\pi\)
0.447289 + 0.894389i \(0.352389\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 291.442 + 504.793i 0.403101 + 0.698192i
\(724\) 0 0
\(725\) −276.090 + 478.203i −0.380814 + 0.659590i
\(726\) 0 0
\(727\) 78.4124i 0.107858i 0.998545 + 0.0539288i \(0.0171744\pi\)
−0.998545 + 0.0539288i \(0.982826\pi\)
\(728\) 0 0
\(729\) 997.392 1.36816
\(730\) 0 0
\(731\) −919.024 530.599i −1.25721 0.725853i
\(732\) 0 0
\(733\) −449.960 + 259.785i −0.613861 + 0.354413i −0.774475 0.632604i \(-0.781986\pi\)
0.160614 + 0.987017i \(0.448653\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 61.2162 + 106.030i 0.0830613 + 0.143866i
\(738\) 0 0
\(739\) 581.739 1007.60i 0.787197 1.36347i −0.140481 0.990083i \(-0.544865\pi\)
0.927678 0.373382i \(-0.121802\pi\)
\(740\) 0 0
\(741\) 573.728i 0.774262i
\(742\) 0 0
\(743\) 66.8040 0.0899112 0.0449556 0.998989i \(-0.485685\pi\)
0.0449556 + 0.998989i \(0.485685\pi\)
\(744\) 0 0
\(745\) 1243.07 + 717.687i 1.66855 + 0.963338i
\(746\) 0 0
\(747\) 1334.53 770.489i 1.78652 1.03145i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1.69343 + 2.93311i 0.00225491 + 0.00390561i 0.867151 0.498046i \(-0.165949\pi\)
−0.864896 + 0.501952i \(0.832616\pi\)
\(752\) 0 0
\(753\) 334.241 578.923i 0.443879 0.768822i
\(754\) 0 0
\(755\) 18.6501i 0.0247021i
\(756\) 0 0
\(757\) −700.402 −0.925234 −0.462617 0.886558i \(-0.653089\pi\)
−0.462617 + 0.886558i \(0.653089\pi\)
\(758\) 0 0
\(759\) −391.682 226.138i −0.516050 0.297941i
\(760\) 0 0
\(761\) −529.962 + 305.974i −0.696403 + 0.402068i −0.806006 0.591907i \(-0.798375\pi\)
0.109604 + 0.993975i \(0.465042\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −617.985 1070.38i −0.807823 1.39919i
\(766\) 0 0
\(767\) −81.0803 + 140.435i −0.105711 + 0.183097i
\(768\) 0 0
\(769\) 638.310i 0.830052i −0.909810 0.415026i \(-0.863773\pi\)
0.909810 0.415026i \(-0.136227\pi\)
\(770\) 0 0
\(771\) −23.2965 −0.0302159
\(772\) 0 0
\(773\) −411.367 237.503i −0.532169 0.307248i 0.209730 0.977759i \(-0.432741\pi\)
−0.741899 + 0.670511i \(0.766075\pi\)
\(774\) 0 0
\(775\) −1482.71 + 856.042i −1.91317 + 1.10457i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 70.2412 + 121.661i 0.0901684 + 0.156176i
\(780\) 0 0
\(781\) −186.392 + 322.840i −0.238658 + 0.413368i
\(782\) 0 0
\(783\) 99.9779i 0.127686i
\(784\) 0 0
\(785\) 1364.94 1.73878
\(786\) 0 0
\(787\) 1277.76 + 737.717i 1.62359 + 0.937378i 0.985950 + 0.167041i \(0.0534212\pi\)
0.637637 + 0.770337i \(0.279912\pi\)
\(788\) 0 0
\(789\) 515.383 297.557i 0.653211 0.377132i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 303.678 + 525.986i 0.382949 + 0.663287i
\(794\) 0 0
\(795\) 260.794 451.708i 0.328043 0.568187i
\(796\) 0 0
\(797\) 330.869i 0.415143i −0.978220 0.207572i \(-0.933444\pi\)
0.978220 0.207572i \(-0.0665560\pi\)
\(798\) 0 0
\(799\) 191.397 0.239546
\(800\) 0 0
\(801\) −1099.24 634.644i −1.37233 0.792315i
\(802\) 0 0
\(803\) −61.8489 + 35.7085i −0.0770223 + 0.0444688i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −9.20101 15.9366i −0.0114015 0.0197480i
\(808\) 0 0
\(809\) 367.945 637.299i 0.454814 0.787761i −0.543863 0.839174i \(-0.683039\pi\)
0.998678 + 0.0514125i \(0.0163723\pi\)
\(810\) 0 0
\(811\) 1129.32i 1.39251i 0.717796 + 0.696253i \(0.245151\pi\)
−0.717796 + 0.696253i \(0.754849\pi\)
\(812\) 0 0
\(813\) 1408.56 1.73255
\(814\) 0 0
\(815\) 1631.22 + 941.785i 2.00150 + 1.15556i
\(816\) 0 0
\(817\) 460.285 265.746i 0.563385 0.325270i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 726.980 + 1259.17i 0.885481 + 1.53370i 0.845162 + 0.534511i \(0.179504\pi\)
0.0403194 + 0.999187i \(0.487162\pi\)
\(822\) 0 0
\(823\) −292.482 + 506.594i −0.355386 + 0.615546i −0.987184 0.159587i \(-0.948984\pi\)
0.631798 + 0.775133i \(0.282317\pi\)
\(824\) 0 0
\(825\) 814.078i 0.986761i
\(826\) 0 0
\(827\) 1453.16 1.75714 0.878570 0.477613i \(-0.158498\pi\)
0.878570 + 0.477613i \(0.158498\pi\)
\(828\) 0 0
\(829\) −349.516 201.793i −0.421611 0.243417i 0.274155 0.961685i \(-0.411602\pi\)
−0.695767 + 0.718268i \(0.744935\pi\)
\(830\) 0 0
\(831\) −1005.18 + 580.341i −1.20960 + 0.698364i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 877.357 + 1519.63i 1.05073 + 1.81991i
\(836\) 0 0
\(837\) −154.995 + 268.459i −0.185179 + 0.320740i
\(838\) 0 0
\(839\) 191.525i 0.228278i 0.993465 + 0.114139i \(0.0364109\pi\)
−0.993465 + 0.114139i \(0.963589\pi\)
\(840\) 0 0
\(841\) −701.784 −0.834464
\(842\) 0 0
\(843\) 1440.58 + 831.720i 1.70887 + 0.986619i
\(844\) 0 0
\(845\) 1461.78 843.958i 1.72991 0.998767i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −178.347 308.906i −0.210067 0.363846i
\(850\) 0 0
\(851\) −416.000 + 720.533i −0.488837 + 0.846690i
\(852\) 0 0
\(853\) 220.893i 0.258960i 0.991582 + 0.129480i \(0.0413308\pi\)
−0.991582 + 0.129480i \(0.958669\pi\)
\(854\) 0 0
\(855\) 619.025 0.724006
\(856\) 0 0
\(857\) 813.631 + 469.750i 0.949395 + 0.548133i 0.892893 0.450269i \(-0.148672\pi\)
0.0565020 + 0.998402i \(0.482005\pi\)
\(858\) 0 0
\(859\) 805.866 465.267i 0.938144 0.541638i 0.0487662 0.998810i \(-0.484471\pi\)
0.889378 + 0.457172i \(0.151138\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 586.080 + 1015.12i 0.679120 + 1.17627i 0.975246 + 0.221121i \(0.0709716\pi\)
−0.296127 + 0.955149i \(0.595695\pi\)
\(864\) 0 0
\(865\) −1177.09 + 2038.77i −1.36079 + 2.35696i
\(866\) 0 0
\(867\) 490.272i 0.565480i
\(868\) 0 0
\(869\) 311.176 0.358085
\(870\) 0 0
\(871\) −521.748 301.231i −0.599022 0.345845i
\(872\) 0 0
\(873\) 1296.44 748.499i 1.48504 0.857388i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 250.704 + 434.231i 0.285865 + 0.495133i 0.972819 0.231569i \(-0.0743858\pi\)
−0.686954 + 0.726701i \(0.741052\pi\)
\(878\) 0 0
\(879\) −249.894 + 432.830i −0.284294 + 0.492412i
\(880\) 0 0
\(881\) 1037.14i 1.17723i 0.808412 + 0.588616i \(0.200327\pi\)
−0.808412 + 0.588616i \(0.799673\pi\)
\(882\) 0 0
\(883\) −103.226 −0.116904 −0.0584520 0.998290i \(-0.518616\pi\)
−0.0584520 + 0.998290i \(0.518616\pi\)
\(884\) 0 0
\(885\) 276.639 + 159.718i 0.312587 + 0.180472i
\(886\) 0 0
\(887\) −467.667 + 270.007i −0.527245 + 0.304405i −0.739894 0.672723i \(-0.765124\pi\)
0.212649 + 0.977129i \(0.431791\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −117.563 203.625i −0.131945 0.228535i
\(892\) 0 0
\(893\) −47.9298 + 83.0168i −0.0536728 + 0.0929640i
\(894\) 0 0
\(895\) 2884.29i 3.22267i
\(896\) 0 0
\(897\) 2225.55 2.48110
\(898\) 0 0
\(899\) 373.821 + 215.826i 0.415819 + 0.240073i
\(900\) 0 0
\(901\) 159.926 92.3336i 0.177499 0.102479i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −780.291 1351.50i −0.862200 1.49337i
\(906\) 0 0
\(907\) −361.191 + 625.601i −0.398226 + 0.689748i −0.993507 0.113770i \(-0.963707\pi\)
0.595281 + 0.803517i \(0.297041\pi\)
\(908\) 0 0
\(909\) 986.597i 1.08537i
\(910\) 0 0
\(911\) −1292.59 −1.41887 −0.709436 0.704770i \(-0.751050\pi\)
−0.709436 + 0.704770i \(0.751050\pi\)
\(912\) 0 0
\(913\) −477.452 275.657i −0.522948 0.301924i
\(914\) 0 0
\(915\) 1036.12 598.207i 1.13238 0.653778i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 538.492 + 932.696i 0.585955 + 1.01490i 0.994756 + 0.102280i \(0.0326137\pi\)
−0.408801 + 0.912624i \(0.634053\pi\)
\(920\) 0 0
\(921\) 326.040 564.718i 0.354007 0.613158i
\(922\) 0 0
\(923\) 1834.39i 1.98742i
\(924\) 0 0
\(925\) 1497.57 1.61899
\(926\) 0 0
\(927\) 134.573 + 77.6960i 0.145171 + 0.0838145i
\(928\) 0 0
\(929\) −1084.29 + 626.015i −1.16716 + 0.673859i −0.953009 0.302942i \(-0.902031\pi\)
−0.214150 + 0.976801i \(0.568698\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 943.377 + 1633.98i 1.01112 + 1.75131i
\(934\) 0 0
\(935\) −221.095 + 382.949i −0.236466 + 0.409571i
\(936\) 0 0
\(937\) 857.272i 0.914911i 0.889232 + 0.457456i \(0.151239\pi\)
−0.889232 + 0.457456i \(0.848761\pi\)
\(938\) 0 0
\(939\) 391.286 0.416705
\(940\) 0 0
\(941\) 613.422 + 354.159i 0.651883 + 0.376365i 0.789177 0.614165i \(-0.210507\pi\)
−0.137294 + 0.990530i \(0.543841\pi\)
\(942\) 0 0
\(943\) 471.936 272.472i 0.500462 0.288942i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 241.548 + 418.373i 0.255066 + 0.441788i 0.964914 0.262568i \(-0.0845694\pi\)
−0.709847 + 0.704356i \(0.751236\pi\)
\(948\) 0 0
\(949\) 175.714 304.345i 0.185157 0.320701i
\(950\) 0 0
\(951\) 1924.37i 2.02352i
\(952\) 0 0
\(953\) 67.5778 0.0709106 0.0354553 0.999371i \(-0.488712\pi\)
0.0354553 + 0.999371i \(0.488712\pi\)
\(954\) 0 0
\(955\) −1684.76 972.698i −1.76415 1.01853i
\(956\) 0 0
\(957\) 177.748 102.623i 0.185735 0.107234i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 188.686 + 326.813i 0.196343 + 0.340076i
\(962\) 0 0
\(963\) 270.296 468.167i 0.280682 0.486155i
\(964\) 0 0
\(965\) 351.626i 0.364379i
\(966\) 0 0
\(967\) 806.382 0.833901 0.416950 0.908929i \(-0.363099\pi\)
0.416950 + 0.908929i \(0.363099\pi\)
\(968\) 0 0
\(969\) 346.527 + 200.067i 0.357613 + 0.206468i
\(970\) 0 0
\(971\) 598.531 345.562i 0.616407 0.355883i −0.159062 0.987269i \(-0.550847\pi\)
0.775469 + 0.631386i \(0.217514\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −2002.95 3469.21i −2.05431 3.55816i
\(976\) 0 0
\(977\) −179.824 + 311.464i −0.184057 + 0.318797i −0.943258 0.332059i \(-0.892257\pi\)
0.759201 + 0.650856i \(0.225590\pi\)
\(978\) 0 0
\(979\) 454.111i 0.463852i
\(980\) 0 0
\(981\) −1604.36 −1.63543
\(982\) 0 0
\(983\) 339.267 + 195.876i 0.345134 + 0.199263i 0.662540 0.749027i \(-0.269478\pi\)
−0.317406 + 0.948290i \(0.602812\pi\)
\(984\) 0 0
\(985\) −1860.95 + 1074.42i −1.88929 + 1.09078i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1030.85 1785.49i −1.04232 1.80535i
\(990\) 0 0
\(991\) 704.693 1220.56i 0.711093 1.23165i −0.253354 0.967374i \(-0.581534\pi\)
0.964447 0.264276i \(-0.0851330\pi\)
\(992\) 0 0
\(993\) 1523.34i 1.53407i
\(994\) 0 0
\(995\) −952.000 −0.956784
\(996\) 0 0
\(997\) 398.864 + 230.284i 0.400064 + 0.230977i 0.686512 0.727119i \(-0.259141\pi\)
−0.286448 + 0.958096i \(0.592474\pi\)
\(998\) 0 0
\(999\) 234.823 135.575i 0.235058 0.135711i
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 784.3.s.g.129.1 8
4.3 odd 2 196.3.h.c.129.4 8
7.2 even 3 inner 784.3.s.g.705.4 8
7.3 odd 6 784.3.c.d.97.1 4
7.4 even 3 784.3.c.d.97.4 4
7.5 odd 6 inner 784.3.s.g.705.1 8
7.6 odd 2 inner 784.3.s.g.129.4 8
12.11 even 2 1764.3.z.k.325.4 8
28.3 even 6 196.3.b.b.97.4 yes 4
28.11 odd 6 196.3.b.b.97.1 4
28.19 even 6 196.3.h.c.117.4 8
28.23 odd 6 196.3.h.c.117.1 8
28.27 even 2 196.3.h.c.129.1 8
84.11 even 6 1764.3.d.e.685.4 4
84.23 even 6 1764.3.z.k.901.1 8
84.47 odd 6 1764.3.z.k.901.4 8
84.59 odd 6 1764.3.d.e.685.1 4
84.83 odd 2 1764.3.z.k.325.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
196.3.b.b.97.1 4 28.11 odd 6
196.3.b.b.97.4 yes 4 28.3 even 6
196.3.h.c.117.1 8 28.23 odd 6
196.3.h.c.117.4 8 28.19 even 6
196.3.h.c.129.1 8 28.27 even 2
196.3.h.c.129.4 8 4.3 odd 2
784.3.c.d.97.1 4 7.3 odd 6
784.3.c.d.97.4 4 7.4 even 3
784.3.s.g.129.1 8 1.1 even 1 trivial
784.3.s.g.129.4 8 7.6 odd 2 inner
784.3.s.g.705.1 8 7.5 odd 6 inner
784.3.s.g.705.4 8 7.2 even 3 inner
1764.3.d.e.685.1 4 84.59 odd 6
1764.3.d.e.685.4 4 84.11 even 6
1764.3.z.k.325.1 8 84.83 odd 2
1764.3.z.k.325.4 8 12.11 even 2
1764.3.z.k.901.1 8 84.23 even 6
1764.3.z.k.901.4 8 84.47 odd 6