Properties

Label 1764.3.z.k.901.4
Level $1764$
Weight $3$
Character 1764.901
Analytic conductor $48.066$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,3,Mod(325,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, 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("1764.325");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1764.z (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: \(48.0655186332\)
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: \(\Z[a_1, \ldots, a_{25}]\)
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 901.4
Root \(-0.662827 + 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 1764.901
Dual form 1764.3.z.k.325.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(7.33820 + 4.23671i) q^{5} +O(q^{10})\) \(q+(7.33820 + 4.23671i) q^{5} +(1.94975 + 3.37706i) q^{11} -19.1886i q^{13} +(-11.5897 + 6.69133i) q^{17} +(5.80462 + 3.35130i) q^{19} +(13.0000 - 22.5167i) q^{23} +(23.3995 + 40.5291i) q^{25} +11.7990 q^{29} +(31.6825 - 18.2919i) q^{31} +(16.0000 - 27.7128i) q^{37} -20.9594i q^{41} +79.2965 q^{43} +(12.3858 + 7.15093i) q^{47} +(-6.89949 - 11.9503i) q^{53} +33.0421i q^{55} +(7.31869 - 4.22545i) q^{59} +(27.4114 + 15.8260i) q^{61} +(81.2965 - 140.810i) q^{65} +(15.6985 + 27.1906i) q^{67} -95.5980 q^{71} +(-15.8607 + 9.15721i) q^{73} +(-39.8995 + 69.1080i) q^{79} +141.381i q^{83} -113.397 q^{85} +(100.852 + 58.2269i) q^{89} +(28.3970 + 49.1850i) q^{95} +137.346i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{11} + 104 q^{23} + 108 q^{25} - 64 q^{29} + 128 q^{37} + 80 q^{43} + 24 q^{53} + 96 q^{65} - 112 q^{67} - 448 q^{71} - 240 q^{79} - 432 q^{85} - 248 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 7.33820 + 4.23671i 1.46764 + 0.847343i 0.999344 0.0362281i \(-0.0115343\pi\)
0.468297 + 0.883571i \(0.344868\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.94975 + 3.37706i 0.177250 + 0.307006i 0.940938 0.338580i \(-0.109947\pi\)
−0.763688 + 0.645586i \(0.776613\pi\)
\(12\) 0 0
\(13\) 19.1886i 1.47604i −0.674777 0.738022i \(-0.735760\pi\)
0.674777 0.738022i \(-0.264240\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −11.5897 + 6.69133i −0.681748 + 0.393607i −0.800513 0.599315i \(-0.795440\pi\)
0.118765 + 0.992922i \(0.462106\pi\)
\(18\) 0 0
\(19\) 5.80462 + 3.35130i 0.305506 + 0.176384i 0.644914 0.764255i \(-0.276893\pi\)
−0.339408 + 0.940639i \(0.610227\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 13.0000 22.5167i 0.565217 0.978985i −0.431812 0.901964i \(-0.642126\pi\)
0.997029 0.0770216i \(-0.0245410\pi\)
\(24\) 0 0
\(25\) 23.3995 + 40.5291i 0.935980 + 1.62116i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 11.7990 0.406862 0.203431 0.979089i \(-0.434791\pi\)
0.203431 + 0.979089i \(0.434791\pi\)
\(30\) 0 0
\(31\) 31.6825 18.2919i 1.02202 0.590061i 0.107328 0.994224i \(-0.465770\pi\)
0.914687 + 0.404163i \(0.132437\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 16.0000 27.7128i 0.432432 0.748995i −0.564650 0.825331i \(-0.690989\pi\)
0.997082 + 0.0763357i \(0.0243221\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 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 \(-0.126512\pi\)
0.922052 + 0.387066i \(0.126512\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.3858 + 7.15093i 0.263527 + 0.152148i 0.625943 0.779869i \(-0.284714\pi\)
−0.362415 + 0.932017i \(0.618048\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.89949 11.9503i −0.130179 0.225477i 0.793566 0.608484i \(-0.208222\pi\)
−0.923746 + 0.383007i \(0.874889\pi\)
\(54\) 0 0
\(55\) 33.0421i 0.600765i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.31869 4.22545i 0.124046 0.0716178i −0.436693 0.899610i \(-0.643850\pi\)
0.560739 + 0.827993i \(0.310517\pi\)
\(60\) 0 0
\(61\) 27.4114 + 15.8260i 0.449368 + 0.259443i 0.707563 0.706650i \(-0.249794\pi\)
−0.258195 + 0.966093i \(0.583128\pi\)
\(62\) 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 \(-0.0913850\pi\)
−0.724765 + 0.688996i \(0.758052\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 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 −0.604686 0.796464i \(-0.706701\pi\)
0.387415 + 0.921905i \(0.373368\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −39.8995 + 69.1080i −0.505057 + 0.874784i 0.494926 + 0.868935i \(0.335195\pi\)
−0.999983 + 0.00584908i \(0.998138\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 141.381i 1.70338i 0.524044 + 0.851691i \(0.324423\pi\)
−0.524044 + 0.851691i \(0.675577\pi\)
\(84\) 0 0
\(85\) −113.397 −1.33408
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 100.852 + 58.2269i 1.13317 + 0.654235i 0.944730 0.327850i \(-0.106324\pi\)
0.188439 + 0.982085i \(0.439657\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 28.3970 + 49.1850i 0.298915 + 0.517737i
\(96\) 0 0
\(97\) 137.346i 1.41593i 0.706245 + 0.707967i \(0.250388\pi\)
−0.706245 + 0.707967i \(0.749612\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 78.3906 45.2588i 0.776145 0.448107i −0.0589176 0.998263i \(-0.518765\pi\)
0.835062 + 0.550156i \(0.185432\pi\)
\(102\) 0 0
\(103\) −12.3468 7.12840i −0.119871 0.0692078i 0.438865 0.898553i \(-0.355380\pi\)
−0.558737 + 0.829345i \(0.688714\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −24.7990 + 42.9531i −0.231766 + 0.401431i −0.958328 0.285670i \(-0.907784\pi\)
0.726562 + 0.687101i \(0.241117\pi\)
\(108\) 0 0
\(109\) −73.5980 127.475i −0.675211 1.16950i −0.976407 0.215937i \(-0.930719\pi\)
0.301196 0.953562i \(-0.402614\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −74.3015 −0.657536 −0.328768 0.944411i \(-0.606633\pi\)
−0.328768 + 0.944411i \(0.606633\pi\)
\(114\) 0 0
\(115\) 190.793 110.155i 1.65907 0.957866i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 52.8970 91.6202i 0.437165 0.757192i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 184.712i 1.47770i
\(126\) 0 0
\(127\) −76.9949 −0.606259 −0.303130 0.952949i \(-0.598032\pi\)
−0.303130 + 0.952949i \(0.598032\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 134.398 + 77.5946i 1.02594 + 0.592325i 0.915818 0.401593i \(-0.131543\pi\)
0.110119 + 0.993918i \(0.464877\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −65.0503 112.670i −0.474819 0.822411i 0.524765 0.851247i \(-0.324153\pi\)
−0.999584 + 0.0288360i \(0.990820\pi\)
\(138\) 0 0
\(139\) 200.740i 1.44417i 0.691804 + 0.722086i \(0.256816\pi\)
−0.691804 + 0.722086i \(0.743184\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 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 −0.428274 0.903649i \(-0.640878\pi\)
0.996720 0.0809286i \(-0.0257886\pi\)
\(150\) 0 0
\(151\) 1.10051 + 1.90613i 0.00728811 + 0.0126234i 0.869646 0.493675i \(-0.164347\pi\)
−0.862358 + 0.506298i \(0.831013\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 309.990 1.99993
\(156\) 0 0
\(157\) −139.504 + 80.5426i −0.888560 + 0.513010i −0.873471 0.486876i \(-0.838136\pi\)
−0.0150888 + 0.999886i \(0.504803\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 111.146 192.510i 0.681876 1.18104i −0.292532 0.956256i \(-0.594498\pi\)
0.974408 0.224787i \(-0.0721688\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 207.084i 1.24003i 0.784592 + 0.620013i \(0.212873\pi\)
−0.784592 + 0.620013i \(0.787127\pi\)
\(168\) 0 0
\(169\) −199.201 −1.17870
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −240.608 138.915i −1.39080 0.802976i −0.397392 0.917649i \(-0.630085\pi\)
−0.993403 + 0.114673i \(0.963418\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 170.196 + 294.788i 0.950815 + 1.64686i 0.743666 + 0.668551i \(0.233085\pi\)
0.207149 + 0.978309i \(0.433581\pi\)
\(180\) 0 0
\(181\) 184.174i 1.01753i −0.860904 0.508767i \(-0.830101\pi\)
0.860904 0.508767i \(-0.169899\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 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 −0.391652 0.920113i \(-0.628096\pi\)
0.992668 0.120876i \(-0.0385703\pi\)
\(192\) 0 0
\(193\) 20.7487 + 35.9379i 0.107506 + 0.186207i 0.914759 0.403999i \(-0.132380\pi\)
−0.807253 + 0.590205i \(0.799047\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −253.598 −1.28730 −0.643650 0.765320i \(-0.722581\pi\)
−0.643650 + 0.765320i \(0.722581\pi\)
\(198\) 0 0
\(199\) −97.2990 + 56.1756i −0.488940 + 0.282289i −0.724135 0.689659i \(-0.757761\pi\)
0.235195 + 0.971948i \(0.424427\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 88.7990 153.804i 0.433166 0.750265i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 26.1367i 0.125056i
\(210\) 0 0
\(211\) 353.789 1.67672 0.838362 0.545113i \(-0.183513\pi\)
0.838362 + 0.545113i \(0.183513\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 581.894 + 335.956i 2.70648 + 1.56259i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 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
\(225\) 0 0
\(226\) 0 0
\(227\) 195.589 112.923i 0.861626 0.497460i −0.00293018 0.999996i \(-0.500933\pi\)
0.864557 + 0.502535i \(0.167599\pi\)
\(228\) 0 0
\(229\) 223.680 + 129.141i 0.976767 + 0.563937i 0.901292 0.433211i \(-0.142620\pi\)
0.0754744 + 0.997148i \(0.475953\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 28.8492 49.9684i 0.123816 0.214456i −0.797453 0.603381i \(-0.793820\pi\)
0.921270 + 0.388924i \(0.127153\pi\)
\(234\) 0 0
\(235\) 60.5929 + 104.950i 0.257842 + 0.446596i
\(236\) 0 0
\(237\) 0 0
\(238\) 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 −0.716048 0.698051i \(-0.754051\pi\)
0.246506 + 0.969141i \(0.420718\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 64.3066 111.382i 0.260350 0.450940i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 149.854i 0.597029i −0.954405 0.298514i \(-0.903509\pi\)
0.954405 0.298514i \(-0.0964911\pi\)
\(252\) 0 0
\(253\) 101.387 0.400739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4.52272 2.61119i −0.0175981 0.0101603i 0.491175 0.871061i \(-0.336568\pi\)
−0.508773 + 0.860901i \(0.669901\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −66.7035 115.534i −0.253626 0.439292i 0.710896 0.703297i \(-0.248290\pi\)
−0.964521 + 0.264005i \(0.914956\pi\)
\(264\) 0 0
\(265\) 116.925i 0.441225i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.57252 + 2.06260i −0.0132808 + 0.00766765i −0.506626 0.862166i \(-0.669107\pi\)
0.493345 + 0.869834i \(0.335774\pi\)
\(270\) 0 0
\(271\) 273.455 + 157.879i 1.00906 + 0.582580i 0.910916 0.412592i \(-0.135376\pi\)
0.0981429 + 0.995172i \(0.468710\pi\)
\(272\) 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 0.999398 0.0346877i \(-0.0110436\pi\)
−0.529740 + 0.848160i \(0.677710\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 372.894 1.32703 0.663513 0.748165i \(-0.269065\pi\)
0.663513 + 0.748165i \(0.269065\pi\)
\(282\) 0 0
\(283\) −69.2476 + 39.9801i −0.244691 + 0.141273i −0.617331 0.786704i \(-0.711786\pi\)
0.372640 + 0.927976i \(0.378453\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −54.9523 + 95.1801i −0.190146 + 0.329343i
\(290\) 0 0
\(291\) 0 0
\(292\) 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
\(297\) 0 0
\(298\) 0 0
\(299\) −432.062 249.451i −1.44502 0.834285i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 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
\(309\) 0 0
\(310\) 0 0
\(311\) −366.290 + 211.477i −1.17778 + 0.679992i −0.955500 0.294991i \(-0.904683\pi\)
−0.222280 + 0.974983i \(0.571350\pi\)
\(312\) 0 0
\(313\) −75.9634 43.8575i −0.242695 0.140120i 0.373720 0.927542i \(-0.378082\pi\)
−0.616415 + 0.787422i \(0.711415\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 215.693 373.592i 0.680421 1.17852i −0.294432 0.955673i \(-0.595130\pi\)
0.974853 0.222851i \(-0.0715364\pi\)
\(318\) 0 0
\(319\) 23.0051 + 39.8459i 0.0721161 + 0.124909i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −89.6985 −0.277704
\(324\) 0 0
\(325\) 777.696 449.003i 2.39291 1.38155i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 170.744 295.737i 0.515842 0.893464i −0.483989 0.875074i \(-0.660813\pi\)
0.999831 0.0183904i \(-0.00585416\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 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
\(339\) 0 0
\(340\) 0 0
\(341\) 123.546 + 71.3291i 0.362304 + 0.209176i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −304.241 526.961i −0.876776 1.51862i −0.854859 0.518860i \(-0.826357\pi\)
−0.0219166 0.999760i \(-0.506977\pi\)
\(348\) 0 0
\(349\) 93.1627i 0.266942i −0.991053 0.133471i \(-0.957388\pi\)
0.991053 0.133471i \(-0.0426123\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 80.0022 46.1893i 0.226635 0.130848i −0.382384 0.924004i \(-0.624897\pi\)
0.609019 + 0.793156i \(0.291563\pi\)
\(354\) 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 −0.994057 0.108861i \(-0.965280\pi\)
0.591305 + 0.806448i \(0.298613\pi\)
\(360\) 0 0
\(361\) −158.038 273.729i −0.437777 0.758253i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −155.186 −0.425167
\(366\) 0 0
\(367\) −458.968 + 264.986i −1.25060 + 0.722032i −0.971228 0.238153i \(-0.923458\pi\)
−0.279368 + 0.960184i \(0.590125\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −232.899 + 403.394i −0.624395 + 1.08148i 0.364262 + 0.931297i \(0.381321\pi\)
−0.988657 + 0.150188i \(0.952012\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 226.406i 0.600546i
\(378\) 0 0
\(379\) 91.8793 0.242426 0.121213 0.992627i \(-0.461322\pi\)
0.121213 + 0.992627i \(0.461322\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 269.572 + 155.638i 0.703844 + 0.406364i 0.808777 0.588115i \(-0.200130\pi\)
−0.104934 + 0.994479i \(0.533463\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −271.296 469.899i −0.697420 1.20797i −0.969358 0.245652i \(-0.920998\pi\)
0.271938 0.962315i \(-0.412336\pi\)
\(390\) 0 0
\(391\) 347.949i 0.889895i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −585.581 + 338.086i −1.48248 + 0.855913i
\(396\) 0 0
\(397\) −141.834 81.8877i −0.357263 0.206266i 0.310616 0.950535i \(-0.399465\pi\)
−0.667880 + 0.744269i \(0.732798\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −71.7035 + 124.194i −0.178812 + 0.309711i −0.941474 0.337086i \(-0.890559\pi\)
0.762662 + 0.646797i \(0.223892\pi\)
\(402\) 0 0
\(403\) −350.995 607.941i −0.870955 1.50854i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 124.784 0.306594
\(408\) 0 0
\(409\) 251.150 145.001i 0.614058 0.354526i −0.160494 0.987037i \(-0.551309\pi\)
0.774552 + 0.632510i \(0.217975\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −598.990 + 1037.48i −1.44335 + 2.49995i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 340.170i 0.811860i 0.913904 + 0.405930i \(0.133052\pi\)
−0.913904 + 0.405930i \(0.866948\pi\)
\(420\) 0 0
\(421\) −6.18081 −0.0146813 −0.00734063 0.999973i \(-0.502337\pi\)
−0.00734063 + 0.999973i \(0.502337\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −542.387 313.147i −1.27621 0.736817i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −14.2965 24.7622i −0.0331705 0.0574529i 0.848964 0.528451i \(-0.177227\pi\)
−0.882134 + 0.470998i \(0.843894\pi\)
\(432\) 0 0
\(433\) 243.736i 0.562900i 0.959576 + 0.281450i \(0.0908153\pi\)
−0.959576 + 0.281450i \(0.909185\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 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 \(-0.142719\pi\)
0.0751637 + 0.997171i \(0.476052\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 39.6985 68.7598i 0.0896128 0.155214i −0.817735 0.575595i \(-0.804770\pi\)
0.907348 + 0.420381i \(0.138104\pi\)
\(444\) 0 0
\(445\) 493.382 + 854.562i 1.10872 + 1.92036i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −804.362 −1.79145 −0.895726 0.444607i \(-0.853343\pi\)
−0.895726 + 0.444607i \(0.853343\pi\)
\(450\) 0 0
\(451\) 70.7812 40.8655i 0.156943 0.0906109i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −268.739 + 465.469i −0.588050 + 1.01853i 0.406438 + 0.913678i \(0.366771\pi\)
−0.994488 + 0.104853i \(0.966563\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 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 \(-0.684424\pi\)
−0.547511 + 0.836799i \(0.684424\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −178.234 102.903i −0.381657 0.220350i 0.296882 0.954914i \(-0.404053\pi\)
−0.678539 + 0.734564i \(0.737387\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 154.608 + 267.789i 0.326867 + 0.566150i
\(474\) 0 0
\(475\) 313.675i 0.660368i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −429.577 + 248.016i −0.896820 + 0.517779i −0.876167 0.482008i \(-0.839908\pi\)
−0.0206527 + 0.999787i \(0.506574\pi\)
\(480\) 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 \(-0.328216\pi\)
−0.999871 + 0.0160761i \(0.994883\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 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
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 223.497 387.109i 0.447891 0.775770i −0.550358 0.834929i \(-0.685509\pi\)
0.998249 + 0.0591594i \(0.0188420\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 660.346i 1.31282i −0.754406 0.656408i \(-0.772075\pi\)
0.754406 0.656408i \(-0.227925\pi\)
\(504\) 0 0
\(505\) 766.995 1.51880
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −421.228 243.196i −0.827559 0.477792i 0.0254569 0.999676i \(-0.491896\pi\)
−0.853016 + 0.521884i \(0.825229\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −60.4020 104.619i −0.117285 0.203144i
\(516\) 0 0
\(517\) 55.7701i 0.107872i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 89.0106 51.3903i 0.170846 0.0986378i −0.412139 0.911121i \(-0.635218\pi\)
0.582985 + 0.812483i \(0.301885\pi\)
\(522\) 0 0
\(523\) 447.960 + 258.630i 0.856520 + 0.494512i 0.862846 0.505468i \(-0.168680\pi\)
−0.00632508 + 0.999980i \(0.502013\pi\)
\(524\) 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
\(531\) 0 0
\(532\) 0 0
\(533\) −402.181 −0.754561
\(534\) 0 0
\(535\) −363.960 + 210.132i −0.680299 + 0.392771i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 225.201 390.060i 0.416268 0.720997i −0.579293 0.815120i \(-0.696671\pi\)
0.995561 + 0.0941223i \(0.0300045\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1247.25i 2.28854i
\(546\) 0 0
\(547\) 195.095 0.356664 0.178332 0.983970i \(-0.442930\pi\)
0.178332 + 0.983970i \(0.442930\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 68.4886 + 39.5419i 0.124299 + 0.0717639i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −344.980 597.523i −0.619353 1.07275i −0.989604 0.143820i \(-0.954062\pi\)
0.370251 0.928932i \(-0.379272\pi\)
\(558\) 0 0
\(559\) 1521.59i 2.72198i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −724.755 + 418.438i −1.28731 + 0.743229i −0.978174 0.207789i \(-0.933373\pi\)
−0.309136 + 0.951018i \(0.600040\pi\)
\(564\) 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 \(-0.831146\pi\)
0.869440 + 0.494038i \(0.164480\pi\)
\(570\) 0 0
\(571\) −0.854293 1.47968i −0.00149613 0.00259138i 0.865276 0.501295i \(-0.167143\pi\)
−0.866773 + 0.498704i \(0.833810\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1216.77 2.11613
\(576\) 0 0
\(577\) 187.824 108.440i 0.325518 0.187938i −0.328332 0.944563i \(-0.606486\pi\)
0.653849 + 0.756625i \(0.273153\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 26.9045 46.6000i 0.0461484 0.0799315i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 50.9983i 0.0868795i 0.999056 + 0.0434398i \(0.0138317\pi\)
−0.999056 + 0.0434398i \(0.986168\pi\)
\(588\) 0 0
\(589\) 245.206 0.416309
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −563.586 325.387i −0.950398 0.548713i −0.0571937 0.998363i \(-0.518215\pi\)
−0.893205 + 0.449650i \(0.851549\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −16.8944 29.2620i −0.0282044 0.0488515i 0.851579 0.524227i \(-0.175646\pi\)
−0.879783 + 0.475375i \(0.842312\pi\)
\(600\) 0 0
\(601\) 415.133i 0.690737i −0.938467 0.345368i \(-0.887754\pi\)
0.938467 0.345368i \(-0.112246\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 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 \(-0.216566\pi\)
−0.156121 + 0.987738i \(0.549899\pi\)
\(608\) 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 0.639155 0.769078i \(-0.279284\pi\)
−0.985618 + 0.168986i \(0.945951\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 726.563 1.17757 0.588787 0.808289i \(-0.299606\pi\)
0.588787 + 0.808289i \(0.299606\pi\)
\(618\) 0 0
\(619\) −613.943 + 354.460i −0.991830 + 0.572633i −0.905821 0.423661i \(-0.860745\pi\)
−0.0860094 + 0.996294i \(0.527412\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −197.585 + 342.228i −0.316137 + 0.547565i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 428.245i 0.680835i
\(630\) 0 0
\(631\) 207.176 0.328329 0.164165 0.986433i \(-0.447507\pi\)
0.164165 + 0.986433i \(0.447507\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −565.005 326.206i −0.889771 0.513710i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 456.291 + 790.320i 0.711843 + 1.23295i 0.964165 + 0.265305i \(0.0854726\pi\)
−0.252321 + 0.967643i \(0.581194\pi\)
\(642\) 0 0
\(643\) 656.917i 1.02164i −0.859686 0.510822i \(-0.829341\pi\)
0.859686 0.510822i \(-0.170659\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −33.0014 + 19.0534i −0.0510068 + 0.0294488i −0.525287 0.850925i \(-0.676042\pi\)
0.474280 + 0.880374i \(0.342709\pi\)
\(648\) 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 \(-0.926439\pi\)
0.685067 + 0.728480i \(0.259773\pi\)
\(654\) 0 0
\(655\) 657.492 + 1138.81i 1.00381 + 1.73864i
\(656\) 0 0
\(657\) 0 0
\(658\) 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 −0.528716 + 0.848799i \(0.677326\pi\)
−0.999439 + 0.0334815i \(0.989341\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 153.387 265.674i 0.229965 0.398312i
\(668\) 0 0
\(669\) 0 0
\(670\) 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
\(675\) 0 0
\(676\) 0 0
\(677\) 486.263 + 280.744i 0.718261 + 0.414688i 0.814112 0.580707i \(-0.197224\pi\)
−0.0958511 + 0.995396i \(0.530557\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 176.080 + 304.980i 0.257804 + 0.446530i 0.965653 0.259834i \(-0.0836677\pi\)
−0.707849 + 0.706364i \(0.750334\pi\)
\(684\) 0 0
\(685\) 1102.40i 1.60934i
\(686\) 0 0
\(687\) 0 0
\(688\) 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 \(-0.607137\pi\)
−0.982563 + 0.185929i \(0.940471\pi\)
\(692\) 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
\(699\) 0 0
\(700\) 0 0
\(701\) −809.990 −1.15548 −0.577739 0.816222i \(-0.696065\pi\)
−0.577739 + 0.816222i \(0.696065\pi\)
\(702\) 0 0
\(703\) 185.748 107.241i 0.264221 0.152548i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 229.980 398.337i 0.324372 0.561829i −0.657013 0.753879i \(-0.728180\pi\)
0.981385 + 0.192050i \(0.0615137\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 951.178i 1.33405i
\(714\) 0 0
\(715\) 634.030 0.886756
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −74.5098 43.0183i −0.103630 0.0598307i 0.447289 0.894389i \(-0.352389\pi\)
−0.550919 + 0.834559i \(0.685723\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 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
\(729\) 0 0
\(730\) 0 0
\(731\) −919.024 + 530.599i −1.25721 + 0.725853i
\(732\) 0 0
\(733\) −449.960 259.785i −0.613861 0.354413i 0.160614 0.987017i \(-0.448653\pi\)
−0.774475 + 0.632604i \(0.781986\pi\)
\(734\) 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 −0.927678 0.373382i \(-0.878198\pi\)
0.140481 0.990083i \(-0.455135\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 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
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.69343 + 2.93311i −0.00225491 + 0.00390561i −0.867151 0.498046i \(-0.834051\pi\)
0.864896 + 0.501952i \(0.167384\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 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
\(759\) 0 0
\(760\) 0 0
\(761\) 529.962 + 305.974i 0.696403 + 0.402068i 0.806006 0.591907i \(-0.201625\pi\)
−0.109604 + 0.993975i \(0.534958\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −81.0803 140.435i −0.105711 0.183097i
\(768\) 0 0
\(769\) 638.310i 0.830052i 0.909810 + 0.415026i \(0.136227\pi\)
−0.909810 + 0.415026i \(0.863773\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 411.367 237.503i 0.532169 0.307248i −0.209730 0.977759i \(-0.567259\pi\)
0.741899 + 0.670511i \(0.233925\pi\)
\(774\) 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
\(783\) 0 0
\(784\) 0 0
\(785\) −1364.94 −1.73878
\(786\) 0 0
\(787\) −1277.76 + 737.717i −1.62359 + 0.937378i −0.637637 + 0.770337i \(0.720088\pi\)
−0.985950 + 0.167041i \(0.946579\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 303.678 525.986i 0.382949 0.663287i
\(794\) 0 0
\(795\) 0 0
\(796\) 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
\(801\) 0 0
\(802\) 0 0
\(803\) −61.8489 35.7085i −0.0770223 0.0444688i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −367.945 637.299i −0.454814 0.787761i 0.543863 0.839174i \(-0.316961\pi\)
−0.998678 + 0.0514125i \(0.983628\pi\)
\(810\) 0 0
\(811\) 1129.32i 1.39251i 0.717796 + 0.696253i \(0.245151\pi\)
−0.717796 + 0.696253i \(0.754849\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 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 −0.0403194 + 0.999187i \(0.512838\pi\)
−0.845162 + 0.534511i \(0.820496\pi\)
\(822\) 0 0
\(823\) 292.482 + 506.594i 0.355386 + 0.615546i 0.987184 0.159587i \(-0.0510162\pi\)
−0.631798 + 0.775133i \(0.717683\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 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 −0.695767 0.718268i \(-0.744935\pi\)
0.274155 + 0.961685i \(0.411602\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −877.357 + 1519.63i −1.05073 + 1.81991i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 191.525i 0.228278i −0.993465 0.114139i \(-0.963589\pi\)
0.993465 0.114139i \(-0.0364109\pi\)
\(840\) 0 0
\(841\) −701.784 −0.834464
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1461.78 843.958i −1.72991 0.998767i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −416.000 720.533i −0.488837 0.846690i
\(852\) 0 0
\(853\) 220.893i 0.258960i −0.991582 0.129480i \(-0.958669\pi\)
0.991582 0.129480i \(-0.0413308\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −813.631 + 469.750i −0.949395 + 0.548133i −0.892893 0.450269i \(-0.851328\pi\)
−0.0565020 + 0.998402i \(0.517995\pi\)
\(858\) 0 0
\(859\) −805.866 465.267i −0.938144 0.541638i −0.0487662 0.998810i \(-0.515529\pi\)
−0.889378 + 0.457172i \(0.848862\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 586.080 1015.12i 0.679120 1.17627i −0.296127 0.955149i \(-0.595695\pi\)
0.975246 0.221121i \(-0.0709716\pi\)
\(864\) 0 0
\(865\) −1177.09 2038.77i −1.36079 2.35696i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −311.176 −0.358085
\(870\) 0 0
\(871\) 521.748 301.231i 0.599022 0.345845i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 250.704 434.231i 0.285865 0.495133i −0.686954 0.726701i \(-0.741052\pi\)
0.972819 + 0.231569i \(0.0743858\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 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 \(-0.481384\pi\)
0.0584520 + 0.998290i \(0.481384\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −467.667 270.007i −0.527245 0.304405i 0.212649 0.977129i \(-0.431791\pi\)
−0.739894 + 0.672723i \(0.765124\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 47.9298 + 83.0168i 0.0536728 + 0.0929640i
\(894\) 0 0
\(895\) 2884.29i 3.22267i
\(896\) 0 0
\(897\) 0 0
\(898\) 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 \(-0.0362927\pi\)
−0.595281 + 0.803517i \(0.702959\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 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
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −538.492 + 932.696i −0.585955 + 1.01490i 0.408801 + 0.912624i \(0.365947\pi\)
−0.994756 + 0.102280i \(0.967386\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1834.39i 1.98742i
\(924\) 0 0
\(925\) 1497.57 1.61899
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1084.29 + 626.015i 1.16716 + 0.673859i 0.953009 0.302942i \(-0.0979687\pi\)
0.214150 + 0.976801i \(0.431302\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −221.095 382.949i −0.236466 0.409571i
\(936\) 0 0
\(937\) 857.272i 0.914911i −0.889232 0.457456i \(-0.848761\pi\)
0.889232 0.457456i \(-0.151239\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −613.422 + 354.159i −0.651883 + 0.376365i −0.789177 0.614165i \(-0.789493\pi\)
0.137294 + 0.990530i \(0.456159\pi\)
\(942\) 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 −0.709847 0.704356i \(-0.751236\pi\)
0.964914 + 0.262568i \(0.0845694\pi\)
\(948\) 0 0
\(949\) 175.714 + 304.345i 0.185157 + 0.320701i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −67.5778 −0.0709106 −0.0354553 0.999371i \(-0.511288\pi\)
−0.0354553 + 0.999371i \(0.511288\pi\)
\(954\) 0 0
\(955\) 1684.76 972.698i 1.76415 1.01853i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 188.686 326.813i 0.196343 0.340076i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 351.626i 0.364379i
\(966\) 0 0
\(967\) −806.382 −0.833901 −0.416950 0.908929i \(-0.636901\pi\)
−0.416950 + 0.908929i \(0.636901\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 598.531 + 345.562i 0.616407 + 0.355883i 0.775469 0.631386i \(-0.217514\pi\)
−0.159062 + 0.987269i \(0.550847\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 179.824 + 311.464i 0.184057 + 0.318797i 0.943258 0.332059i \(-0.107743\pi\)
−0.759201 + 0.650856i \(0.774410\pi\)
\(978\) 0 0
\(979\) 454.111i 0.463852i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 339.267 195.876i 0.345134 0.199263i −0.317406 0.948290i \(-0.602812\pi\)
0.662540 + 0.749027i \(0.269478\pi\)
\(984\) 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 −0.964447 0.264276i \(-0.914867\pi\)
0.253354 0.967374i \(-0.418466\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −952.000 −0.956784
\(996\) 0 0
\(997\) 398.864 230.284i 0.400064 0.230977i −0.286448 0.958096i \(-0.592474\pi\)
0.686512 + 0.727119i \(0.259141\pi\)
\(998\) 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 1764.3.z.k.901.4 8
3.2 odd 2 196.3.h.c.117.4 8
7.2 even 3 1764.3.d.e.685.1 4
7.3 odd 6 inner 1764.3.z.k.325.4 8
7.4 even 3 inner 1764.3.z.k.325.1 8
7.5 odd 6 1764.3.d.e.685.4 4
7.6 odd 2 inner 1764.3.z.k.901.1 8
12.11 even 2 784.3.s.g.705.1 8
21.2 odd 6 196.3.b.b.97.4 yes 4
21.5 even 6 196.3.b.b.97.1 4
21.11 odd 6 196.3.h.c.129.1 8
21.17 even 6 196.3.h.c.129.4 8
21.20 even 2 196.3.h.c.117.1 8
84.11 even 6 784.3.s.g.129.4 8
84.23 even 6 784.3.c.d.97.1 4
84.47 odd 6 784.3.c.d.97.4 4
84.59 odd 6 784.3.s.g.129.1 8
84.83 odd 2 784.3.s.g.705.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
196.3.b.b.97.1 4 21.5 even 6
196.3.b.b.97.4 yes 4 21.2 odd 6
196.3.h.c.117.1 8 21.20 even 2
196.3.h.c.117.4 8 3.2 odd 2
196.3.h.c.129.1 8 21.11 odd 6
196.3.h.c.129.4 8 21.17 even 6
784.3.c.d.97.1 4 84.23 even 6
784.3.c.d.97.4 4 84.47 odd 6
784.3.s.g.129.1 8 84.59 odd 6
784.3.s.g.129.4 8 84.11 even 6
784.3.s.g.705.1 8 12.11 even 2
784.3.s.g.705.4 8 84.83 odd 2
1764.3.d.e.685.1 4 7.2 even 3
1764.3.d.e.685.4 4 7.5 odd 6
1764.3.z.k.325.1 8 7.4 even 3 inner
1764.3.z.k.325.4 8 7.3 odd 6 inner
1764.3.z.k.901.1 8 7.6 odd 2 inner
1764.3.z.k.901.4 8 1.1 even 1 trivial