Properties

Label 1764.3.z.n.325.3
Level $1764$
Weight $3$
Character 1764.325
Analytic conductor $48.066$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 12x^{14} + 114x^{12} - 336x^{10} + 755x^{8} - 336x^{6} + 114x^{4} - 12x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{6}\cdot 7^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 325.3
Root \(-2.55641 - 1.47595i\) of defining polynomial
Character \(\chi\) \(=\) 1764.325
Dual form 1764.3.z.n.901.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.68448 + 2.12723i) q^{5} +O(q^{10})\) \(q+(-3.68448 + 2.12723i) q^{5} +(-9.48935 + 16.4360i) q^{11} -4.46088i q^{13} +(-11.0534 - 6.38170i) q^{17} +(8.11475 - 4.68506i) q^{19} +(-8.53562 - 14.7841i) q^{23} +(-3.44975 + 5.97514i) q^{25} -20.8862 q^{29} +(13.5116 + 7.80092i) q^{31} +(-22.8492 - 39.5760i) q^{37} +63.3894i q^{41} +2.20101 q^{43} +(43.8434 - 25.3130i) q^{47} +(18.9787 - 32.8721i) q^{53} -80.7443i q^{55} +(58.5813 + 33.8220i) q^{59} +(54.8814 - 31.6858i) q^{61} +(9.48935 + 16.4360i) q^{65} +(-13.6985 + 23.7265i) q^{67} -15.1638 q^{71} +(72.5860 + 41.9075i) q^{73} +(-66.5980 - 115.351i) q^{79} -109.761i q^{83} +54.3015 q^{85} +(-69.2645 + 39.9899i) q^{89} +(-19.9324 + 34.5240i) q^{95} -119.547i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 24 q^{25} - 128 q^{37} + 352 q^{43} + 256 q^{67} - 432 q^{79} + 1344 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 0 0
\(4\) 0 0
\(5\) −3.68448 + 2.12723i −0.736896 + 0.425447i −0.820940 0.571015i \(-0.806550\pi\)
0.0840439 + 0.996462i \(0.473216\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −9.48935 + 16.4360i −0.862668 + 1.49418i 0.00667619 + 0.999978i \(0.497875\pi\)
−0.869344 + 0.494207i \(0.835458\pi\)
\(12\) 0 0
\(13\) 4.46088i 0.343145i −0.985171 0.171572i \(-0.945115\pi\)
0.985171 0.171572i \(-0.0548848\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −11.0534 6.38170i −0.650202 0.375394i 0.138332 0.990386i \(-0.455826\pi\)
−0.788534 + 0.614992i \(0.789159\pi\)
\(18\) 0 0
\(19\) 8.11475 4.68506i 0.427092 0.246582i −0.271015 0.962575i \(-0.587359\pi\)
0.698107 + 0.715993i \(0.254026\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.53562 14.7841i −0.371114 0.642788i 0.618623 0.785688i \(-0.287691\pi\)
−0.989737 + 0.142900i \(0.954357\pi\)
\(24\) 0 0
\(25\) −3.44975 + 5.97514i −0.137990 + 0.239006i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −20.8862 −0.720212 −0.360106 0.932911i \(-0.617259\pi\)
−0.360106 + 0.932911i \(0.617259\pi\)
\(30\) 0 0
\(31\) 13.5116 + 7.80092i 0.435858 + 0.251642i 0.701839 0.712336i \(-0.252363\pi\)
−0.265981 + 0.963978i \(0.585696\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −22.8492 39.5760i −0.617547 1.06962i −0.989932 0.141545i \(-0.954793\pi\)
0.372385 0.928078i \(-0.378540\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 63.3894i 1.54608i 0.634355 + 0.773042i \(0.281266\pi\)
−0.634355 + 0.773042i \(0.718734\pi\)
\(42\) 0 0
\(43\) 2.20101 0.0511863 0.0255931 0.999672i \(-0.491853\pi\)
0.0255931 + 0.999672i \(0.491853\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 43.8434 25.3130i 0.932839 0.538575i 0.0451306 0.998981i \(-0.485630\pi\)
0.887708 + 0.460406i \(0.152296\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 18.9787 32.8721i 0.358089 0.620228i −0.629553 0.776958i \(-0.716762\pi\)
0.987641 + 0.156730i \(0.0500952\pi\)
\(54\) 0 0
\(55\) 80.7443i 1.46808i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 58.5813 + 33.8220i 0.992904 + 0.573253i 0.906141 0.422976i \(-0.139014\pi\)
0.0867630 + 0.996229i \(0.472348\pi\)
\(60\) 0 0
\(61\) 54.8814 31.6858i 0.899695 0.519439i 0.0225940 0.999745i \(-0.492807\pi\)
0.877101 + 0.480305i \(0.159474\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.48935 + 16.4360i 0.145990 + 0.252862i
\(66\) 0 0
\(67\) −13.6985 + 23.7265i −0.204455 + 0.354126i −0.949959 0.312375i \(-0.898876\pi\)
0.745504 + 0.666501i \(0.232209\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −15.1638 −0.213574 −0.106787 0.994282i \(-0.534056\pi\)
−0.106787 + 0.994282i \(0.534056\pi\)
\(72\) 0 0
\(73\) 72.5860 + 41.9075i 0.994329 + 0.574076i 0.906565 0.422066i \(-0.138695\pi\)
0.0877632 + 0.996141i \(0.472028\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −66.5980 115.351i −0.843012 1.46014i −0.887336 0.461123i \(-0.847447\pi\)
0.0443238 0.999017i \(-0.485887\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 109.761i 1.32242i −0.750200 0.661211i \(-0.770043\pi\)
0.750200 0.661211i \(-0.229957\pi\)
\(84\) 0 0
\(85\) 54.3015 0.638841
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −69.2645 + 39.9899i −0.778252 + 0.449324i −0.835811 0.549018i \(-0.815002\pi\)
0.0575582 + 0.998342i \(0.481669\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −19.9324 + 34.5240i −0.209815 + 0.363410i
\(96\) 0 0
\(97\) 119.547i 1.23245i −0.787572 0.616223i \(-0.788662\pi\)
0.787572 0.616223i \(-0.211338\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −127.846 73.8118i −1.26580 0.730810i −0.291610 0.956537i \(-0.594191\pi\)
−0.974191 + 0.225727i \(0.927524\pi\)
\(102\) 0 0
\(103\) 159.577 92.1319i 1.54929 0.894485i 0.551097 0.834441i \(-0.314209\pi\)
0.998196 0.0600435i \(-0.0191240\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 23.6994 + 41.0486i 0.221490 + 0.383632i 0.955261 0.295766i \(-0.0955747\pi\)
−0.733771 + 0.679397i \(0.762241\pi\)
\(108\) 0 0
\(109\) −49.4472 + 85.6451i −0.453644 + 0.785735i −0.998609 0.0527240i \(-0.983210\pi\)
0.544965 + 0.838459i \(0.316543\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 187.880 1.66265 0.831325 0.555786i \(-0.187583\pi\)
0.831325 + 0.555786i \(0.187583\pi\)
\(114\) 0 0
\(115\) 62.8986 + 36.3145i 0.546944 + 0.315779i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −119.595 207.145i −0.988392 1.71195i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 135.715i 1.08572i
\(126\) 0 0
\(127\) −22.9949 −0.181063 −0.0905313 0.995894i \(-0.528857\pi\)
−0.0905313 + 0.995894i \(0.528857\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −124.532 + 71.8984i −0.950623 + 0.548843i −0.893274 0.449512i \(-0.851598\pi\)
−0.0573487 + 0.998354i \(0.518265\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 33.2367 57.5676i 0.242604 0.420202i −0.718852 0.695164i \(-0.755332\pi\)
0.961455 + 0.274962i \(0.0886653\pi\)
\(138\) 0 0
\(139\) 246.268i 1.77171i −0.463961 0.885856i \(-0.653572\pi\)
0.463961 0.885856i \(-0.346428\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 73.3193 + 42.3309i 0.512722 + 0.296020i
\(144\) 0 0
\(145\) 76.9546 44.4297i 0.530721 0.306412i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −39.8648 69.0479i −0.267549 0.463409i 0.700679 0.713477i \(-0.252880\pi\)
−0.968228 + 0.250068i \(0.919547\pi\)
\(150\) 0 0
\(151\) −16.2965 + 28.2263i −0.107924 + 0.186929i −0.914929 0.403615i \(-0.867754\pi\)
0.807005 + 0.590544i \(0.201087\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −66.3775 −0.428242
\(156\) 0 0
\(157\) 220.515 + 127.314i 1.40455 + 0.810919i 0.994856 0.101302i \(-0.0323008\pi\)
0.409698 + 0.912221i \(0.365634\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −121.497 210.440i −0.745383 1.29104i −0.950015 0.312203i \(-0.898933\pi\)
0.204632 0.978839i \(-0.434400\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 144.224i 0.863619i 0.901965 + 0.431809i \(0.142125\pi\)
−0.901965 + 0.431809i \(0.857875\pi\)
\(168\) 0 0
\(169\) 149.101 0.882252
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 289.593 167.196i 1.67395 0.966453i 0.708550 0.705660i \(-0.249349\pi\)
0.965395 0.260793i \(-0.0839839\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 58.7956 101.837i 0.328467 0.568922i −0.653741 0.756719i \(-0.726801\pi\)
0.982208 + 0.187797i \(0.0601347\pi\)
\(180\) 0 0
\(181\) 173.481i 0.958459i −0.877690 0.479230i \(-0.840916\pi\)
0.877690 0.479230i \(-0.159084\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 168.375 + 97.2114i 0.910135 + 0.525467i
\(186\) 0 0
\(187\) 209.780 121.116i 1.12182 0.647681i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 130.991 + 226.884i 0.685819 + 1.18787i 0.973179 + 0.230050i \(0.0738890\pi\)
−0.287360 + 0.957823i \(0.592778\pi\)
\(192\) 0 0
\(193\) −14.7990 + 25.6326i −0.0766787 + 0.132811i −0.901815 0.432122i \(-0.857765\pi\)
0.825136 + 0.564934i \(0.191098\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −237.282 −1.20448 −0.602238 0.798317i \(-0.705724\pi\)
−0.602238 + 0.798317i \(0.705724\pi\)
\(198\) 0 0
\(199\) −25.5871 14.7727i −0.128579 0.0742349i 0.434331 0.900753i \(-0.356985\pi\)
−0.562910 + 0.826518i \(0.690318\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −134.844 233.557i −0.657777 1.13930i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 177.832i 0.850873i
\(210\) 0 0
\(211\) −157.990 −0.748767 −0.374384 0.927274i \(-0.622146\pi\)
−0.374384 + 0.927274i \(0.622146\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.10957 + 4.68206i −0.0377189 + 0.0217770i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −28.4680 + 49.3081i −0.128815 + 0.223114i
\(222\) 0 0
\(223\) 225.779i 1.01246i −0.862397 0.506232i \(-0.831038\pi\)
0.862397 0.506232i \(-0.168962\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −124.902 72.1122i −0.550229 0.317675i 0.198985 0.980002i \(-0.436235\pi\)
−0.749214 + 0.662328i \(0.769569\pi\)
\(228\) 0 0
\(229\) 107.880 62.2845i 0.471092 0.271985i −0.245605 0.969370i \(-0.578987\pi\)
0.716697 + 0.697385i \(0.245653\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −213.486 369.769i −0.916251 1.58699i −0.805060 0.593193i \(-0.797867\pi\)
−0.111190 0.993799i \(-0.535466\pi\)
\(234\) 0 0
\(235\) −107.693 + 186.530i −0.458270 + 0.793747i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −28.5160 −0.119314 −0.0596568 0.998219i \(-0.519001\pi\)
−0.0596568 + 0.998219i \(0.519001\pi\)
\(240\) 0 0
\(241\) 203.548 + 117.518i 0.844597 + 0.487628i 0.858824 0.512271i \(-0.171195\pi\)
−0.0142272 + 0.999899i \(0.504529\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −20.8995 36.1990i −0.0846133 0.146555i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 193.568i 0.771186i 0.922669 + 0.385593i \(0.126003\pi\)
−0.922669 + 0.385593i \(0.873997\pi\)
\(252\) 0 0
\(253\) 323.990 1.28059
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −128.586 + 74.2394i −0.500336 + 0.288869i −0.728852 0.684671i \(-0.759946\pi\)
0.228516 + 0.973540i \(0.426613\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −223.882 + 387.774i −0.851261 + 1.47443i 0.0288108 + 0.999585i \(0.490828\pi\)
−0.880071 + 0.474842i \(0.842505\pi\)
\(264\) 0 0
\(265\) 161.489i 0.609391i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −158.803 91.6849i −0.590345 0.340836i 0.174889 0.984588i \(-0.444043\pi\)
−0.765234 + 0.643752i \(0.777377\pi\)
\(270\) 0 0
\(271\) 116.984 67.5407i 0.431675 0.249228i −0.268385 0.963312i \(-0.586490\pi\)
0.700060 + 0.714084i \(0.253157\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −65.4717 113.400i −0.238079 0.412365i
\(276\) 0 0
\(277\) −10.3015 + 17.8427i −0.0371896 + 0.0644143i −0.884021 0.467447i \(-0.845174\pi\)
0.846832 + 0.531861i \(0.178507\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 241.001 0.857654 0.428827 0.903387i \(-0.358927\pi\)
0.428827 + 0.903387i \(0.358927\pi\)
\(282\) 0 0
\(283\) −76.8765 44.3847i −0.271648 0.156836i 0.357988 0.933726i \(-0.383463\pi\)
−0.629637 + 0.776890i \(0.716796\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −63.0477 109.202i −0.218158 0.377861i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 275.258i 0.939446i −0.882814 0.469723i \(-0.844354\pi\)
0.882814 0.469723i \(-0.155646\pi\)
\(294\) 0 0
\(295\) −287.789 −0.975556
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −65.9503 + 38.0764i −0.220570 + 0.127346i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −134.806 + 233.491i −0.441988 + 0.765545i
\(306\) 0 0
\(307\) 366.196i 1.19282i −0.802680 0.596410i \(-0.796593\pi\)
0.802680 0.596410i \(-0.203407\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −256.062 147.837i −0.823350 0.475361i 0.0282202 0.999602i \(-0.491016\pi\)
−0.851570 + 0.524240i \(0.824349\pi\)
\(312\) 0 0
\(313\) −200.791 + 115.927i −0.641504 + 0.370373i −0.785194 0.619250i \(-0.787437\pi\)
0.143689 + 0.989623i \(0.454103\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 154.691 + 267.932i 0.487983 + 0.845212i 0.999904 0.0138204i \(-0.00439930\pi\)
−0.511921 + 0.859033i \(0.671066\pi\)
\(318\) 0 0
\(319\) 198.196 343.285i 0.621304 1.07613i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −119.595 −0.370262
\(324\) 0 0
\(325\) 26.6544 + 15.3889i 0.0820135 + 0.0473505i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 116.593 + 201.945i 0.352244 + 0.610105i 0.986642 0.162901i \(-0.0520852\pi\)
−0.634398 + 0.773007i \(0.718752\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 116.560i 0.347939i
\(336\) 0 0
\(337\) 172.894 0.513040 0.256520 0.966539i \(-0.417424\pi\)
0.256520 + 0.966539i \(0.417424\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −256.432 + 148.051i −0.752001 + 0.434168i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 232.465 402.641i 0.669928 1.16035i −0.307996 0.951388i \(-0.599658\pi\)
0.977924 0.208962i \(-0.0670084\pi\)
\(348\) 0 0
\(349\) 426.071i 1.22083i 0.792080 + 0.610417i \(0.208998\pi\)
−0.792080 + 0.610417i \(0.791002\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 349.655 + 201.873i 0.990524 + 0.571880i 0.905431 0.424493i \(-0.139548\pi\)
0.0850934 + 0.996373i \(0.472881\pi\)
\(354\) 0 0
\(355\) 55.8706 32.2569i 0.157382 0.0908646i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −37.0516 64.1752i −0.103208 0.178761i 0.809797 0.586710i \(-0.199577\pi\)
−0.913005 + 0.407949i \(0.866244\pi\)
\(360\) 0 0
\(361\) −136.601 + 236.599i −0.378395 + 0.655399i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −356.589 −0.976955
\(366\) 0 0
\(367\) −264.136 152.499i −0.719718 0.415529i 0.0949312 0.995484i \(-0.469737\pi\)
−0.814649 + 0.579955i \(0.803070\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 64.7035 + 112.070i 0.173468 + 0.300455i 0.939630 0.342192i \(-0.111169\pi\)
−0.766162 + 0.642647i \(0.777836\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 93.1707i 0.247137i
\(378\) 0 0
\(379\) −276.804 −0.730354 −0.365177 0.930938i \(-0.618991\pi\)
−0.365177 + 0.930938i \(0.618991\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 573.297 330.993i 1.49686 0.864212i 0.496866 0.867827i \(-0.334484\pi\)
0.999993 + 0.00361454i \(0.00115055\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 204.045 353.416i 0.524537 0.908525i −0.475055 0.879956i \(-0.657572\pi\)
0.999592 0.0285687i \(-0.00909494\pi\)
\(390\) 0 0
\(391\) 217.887i 0.557256i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 490.758 + 283.339i 1.24242 + 0.717314i
\(396\) 0 0
\(397\) −614.993 + 355.066i −1.54910 + 0.894373i −0.550889 + 0.834578i \(0.685711\pi\)
−0.998211 + 0.0597948i \(0.980955\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 336.800 + 583.355i 0.839900 + 1.45475i 0.889978 + 0.456004i \(0.150720\pi\)
−0.0500773 + 0.998745i \(0.515947\pi\)
\(402\) 0 0
\(403\) 34.7990 60.2736i 0.0863499 0.149562i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 867.298 2.13095
\(408\) 0 0
\(409\) 256.703 + 148.207i 0.627635 + 0.362365i 0.779835 0.625985i \(-0.215303\pi\)
−0.152201 + 0.988350i \(0.548636\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 233.487 + 404.412i 0.562620 + 0.974487i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 357.803i 0.853945i −0.904265 0.426973i \(-0.859580\pi\)
0.904265 0.426973i \(-0.140420\pi\)
\(420\) 0 0
\(421\) 185.186 0.439871 0.219936 0.975514i \(-0.429415\pi\)
0.219936 + 0.975514i \(0.429415\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 76.2631 44.0305i 0.179443 0.103601i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 360.643 624.652i 0.836759 1.44931i −0.0558311 0.998440i \(-0.517781\pi\)
0.892590 0.450869i \(-0.148886\pi\)
\(432\) 0 0
\(433\) 585.095i 1.35126i 0.737242 + 0.675629i \(0.236128\pi\)
−0.737242 + 0.675629i \(0.763872\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −138.529 79.9797i −0.317000 0.183020i
\(438\) 0 0
\(439\) 131.893 76.1482i 0.300439 0.173458i −0.342201 0.939627i \(-0.611172\pi\)
0.642640 + 0.766168i \(0.277839\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −358.736 621.348i −0.809787 1.40259i −0.913011 0.407934i \(-0.866249\pi\)
0.103224 0.994658i \(-0.467084\pi\)
\(444\) 0 0
\(445\) 170.136 294.684i 0.382327 0.662210i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 13.3522 0.0297376 0.0148688 0.999889i \(-0.495267\pi\)
0.0148688 + 0.999889i \(0.495267\pi\)
\(450\) 0 0
\(451\) −1041.87 601.524i −2.31013 1.33376i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 277.095 + 479.943i 0.606336 + 1.05020i 0.991839 + 0.127498i \(0.0406946\pi\)
−0.385503 + 0.922707i \(0.625972\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 336.103i 0.729074i 0.931189 + 0.364537i \(0.118773\pi\)
−0.931189 + 0.364537i \(0.881227\pi\)
\(462\) 0 0
\(463\) −481.186 −1.03928 −0.519639 0.854386i \(-0.673934\pi\)
−0.519639 + 0.854386i \(0.673934\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −334.899 + 193.354i −0.717128 + 0.414034i −0.813695 0.581293i \(-0.802547\pi\)
0.0965670 + 0.995326i \(0.469214\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −20.8862 + 36.1759i −0.0441568 + 0.0764818i
\(474\) 0 0
\(475\) 64.6490i 0.136103i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −232.474 134.219i −0.485332 0.280206i 0.237304 0.971435i \(-0.423736\pi\)
−0.722636 + 0.691229i \(0.757070\pi\)
\(480\) 0 0
\(481\) −176.544 + 101.928i −0.367036 + 0.211908i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 254.305 + 440.469i 0.524340 + 0.908184i
\(486\) 0 0
\(487\) 42.8995 74.3041i 0.0880893 0.152575i −0.818614 0.574344i \(-0.805257\pi\)
0.906703 + 0.421769i \(0.138591\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −465.122 −0.947295 −0.473647 0.880715i \(-0.657063\pi\)
−0.473647 + 0.880715i \(0.657063\pi\)
\(492\) 0 0
\(493\) 230.864 + 133.289i 0.468283 + 0.270364i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 427.784 + 740.943i 0.857282 + 1.48486i 0.874511 + 0.485005i \(0.161182\pi\)
−0.0172292 + 0.999852i \(0.505485\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 432.245i 0.859335i −0.902987 0.429667i \(-0.858631\pi\)
0.902987 0.429667i \(-0.141369\pi\)
\(504\) 0 0
\(505\) 628.060 1.24368
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −231.752 + 133.802i −0.455308 + 0.262872i −0.710069 0.704132i \(-0.751336\pi\)
0.254761 + 0.967004i \(0.418003\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −391.972 + 678.916i −0.761111 + 1.31828i
\(516\) 0 0
\(517\) 960.816i 1.85844i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −290.703 167.838i −0.557972 0.322145i 0.194359 0.980930i \(-0.437737\pi\)
−0.752331 + 0.658785i \(0.771071\pi\)
\(522\) 0 0
\(523\) −840.091 + 485.027i −1.60629 + 0.927393i −0.616102 + 0.787666i \(0.711289\pi\)
−0.990190 + 0.139727i \(0.955378\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −99.5663 172.454i −0.188930 0.327237i
\(528\) 0 0
\(529\) 118.786 205.744i 0.224549 0.388930i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 282.773 0.530531
\(534\) 0 0
\(535\) −174.640 100.828i −0.326430 0.188464i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 175.090 + 303.265i 0.323642 + 0.560565i 0.981237 0.192807i \(-0.0617593\pi\)
−0.657594 + 0.753372i \(0.728426\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 420.743i 0.772006i
\(546\) 0 0
\(547\) −648.794 −1.18609 −0.593047 0.805167i \(-0.702075\pi\)
−0.593047 + 0.805167i \(0.702075\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −169.486 + 97.8528i −0.307597 + 0.177591i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −439.371 + 761.013i −0.788817 + 1.36627i 0.137875 + 0.990450i \(0.455973\pi\)
−0.926692 + 0.375822i \(0.877360\pi\)
\(558\) 0 0
\(559\) 9.81845i 0.0175643i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −508.828 293.772i −0.903780 0.521798i −0.0253555 0.999678i \(-0.508072\pi\)
−0.878425 + 0.477881i \(0.841405\pi\)
\(564\) 0 0
\(565\) −692.238 + 399.664i −1.22520 + 0.707369i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −52.1195 90.2737i −0.0915985 0.158653i 0.816585 0.577225i \(-0.195864\pi\)
−0.908184 + 0.418571i \(0.862531\pi\)
\(570\) 0 0
\(571\) 235.693 408.233i 0.412773 0.714944i −0.582419 0.812889i \(-0.697894\pi\)
0.995192 + 0.0979450i \(0.0312269\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 117.783 0.204840
\(576\) 0 0
\(577\) −411.347 237.491i −0.712907 0.411597i 0.0992296 0.995065i \(-0.468362\pi\)
−0.812136 + 0.583468i \(0.801696\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 360.191 + 623.869i 0.617823 + 1.07010i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 539.013i 0.918251i 0.888371 + 0.459125i \(0.151837\pi\)
−0.888371 + 0.459125i \(0.848163\pi\)
\(588\) 0 0
\(589\) 146.191 0.248202
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 89.8901 51.8981i 0.151585 0.0875178i −0.422289 0.906461i \(-0.638773\pi\)
0.573874 + 0.818944i \(0.305440\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −275.095 + 476.479i −0.459257 + 0.795457i −0.998922 0.0464229i \(-0.985218\pi\)
0.539664 + 0.841880i \(0.318551\pi\)
\(600\) 0 0
\(601\) 848.487i 1.41179i 0.708315 + 0.705896i \(0.249456\pi\)
−0.708315 + 0.705896i \(0.750544\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 881.294 + 508.815i 1.45668 + 0.841017i
\(606\) 0 0
\(607\) 515.032 297.354i 0.848488 0.489875i −0.0116523 0.999932i \(-0.503709\pi\)
0.860140 + 0.510057i \(0.170376\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −112.918 195.580i −0.184809 0.320099i
\(612\) 0 0
\(613\) −258.754 + 448.175i −0.422111 + 0.731117i −0.996146 0.0877137i \(-0.972044\pi\)
0.574035 + 0.818831i \(0.305377\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −681.421 −1.10441 −0.552205 0.833708i \(-0.686214\pi\)
−0.552205 + 0.833708i \(0.686214\pi\)
\(618\) 0 0
\(619\) −774.084 446.918i −1.25054 0.721999i −0.279323 0.960197i \(-0.590110\pi\)
−0.971217 + 0.238198i \(0.923443\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 202.455 + 350.662i 0.323928 + 0.561059i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 583.268i 0.927295i
\(630\) 0 0
\(631\) −949.588 −1.50489 −0.752447 0.658653i \(-0.771127\pi\)
−0.752447 + 0.658653i \(0.771127\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 84.7244 48.9156i 0.133424 0.0770325i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 215.394 373.073i 0.336028 0.582017i −0.647654 0.761935i \(-0.724250\pi\)
0.983682 + 0.179917i \(0.0575831\pi\)
\(642\) 0 0
\(643\) 540.999i 0.841368i 0.907207 + 0.420684i \(0.138210\pi\)
−0.907207 + 0.420684i \(0.861790\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −85.8353 49.5570i −0.132667 0.0765951i 0.432198 0.901779i \(-0.357738\pi\)
−0.564864 + 0.825184i \(0.691072\pi\)
\(648\) 0 0
\(649\) −1111.80 + 641.897i −1.71309 + 0.989055i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −414.622 718.147i −0.634950 1.09977i −0.986526 0.163606i \(-0.947687\pi\)
0.351576 0.936159i \(-0.385646\pi\)
\(654\) 0 0
\(655\) 305.889 529.816i 0.467007 0.808879i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −584.716 −0.887278 −0.443639 0.896206i \(-0.646313\pi\)
−0.443639 + 0.896206i \(0.646313\pi\)
\(660\) 0 0
\(661\) 333.657 + 192.637i 0.504776 + 0.291433i 0.730684 0.682716i \(-0.239201\pi\)
−0.225908 + 0.974149i \(0.572535\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 178.276 + 308.784i 0.267281 + 0.462944i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1202.71i 1.79241i
\(672\) 0 0
\(673\) 797.266 1.18465 0.592323 0.805701i \(-0.298211\pi\)
0.592323 + 0.805701i \(0.298211\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 415.605 239.950i 0.613893 0.354431i −0.160595 0.987020i \(-0.551341\pi\)
0.774487 + 0.632589i \(0.218008\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 535.170 926.942i 0.783559 1.35716i −0.146298 0.989241i \(-0.546736\pi\)
0.929856 0.367923i \(-0.119931\pi\)
\(684\) 0 0
\(685\) 282.809i 0.412860i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −146.639 84.6618i −0.212828 0.122876i
\(690\) 0 0
\(691\) 25.9344 14.9732i 0.0375317 0.0216690i −0.481117 0.876657i \(-0.659769\pi\)
0.518648 + 0.854988i \(0.326435\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 523.870 + 907.369i 0.753769 + 1.30557i
\(696\) 0 0
\(697\) 404.533 700.671i 0.580391 1.00527i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 396.453 0.565554 0.282777 0.959186i \(-0.408744\pi\)
0.282777 + 0.959186i \(0.408744\pi\)
\(702\) 0 0
\(703\) −370.832 214.100i −0.527499 0.304552i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 173.749 + 300.942i 0.245062 + 0.424459i 0.962149 0.272524i \(-0.0878584\pi\)
−0.717087 + 0.696983i \(0.754525\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 266.343i 0.373552i
\(714\) 0 0
\(715\) −360.191 −0.503764
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 910.047 525.416i 1.26571 0.730760i 0.291539 0.956559i \(-0.405833\pi\)
0.974174 + 0.225799i \(0.0724994\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 72.0519 124.798i 0.0993820 0.172135i
\(726\) 0 0
\(727\) 47.3417i 0.0651193i 0.999470 + 0.0325596i \(0.0103659\pi\)
−0.999470 + 0.0325596i \(0.989634\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −24.3287 14.0462i −0.0332814 0.0192150i
\(732\) 0 0
\(733\) 53.9898 31.1710i 0.0736559 0.0425252i −0.462720 0.886505i \(-0.653126\pi\)
0.536376 + 0.843979i \(0.319793\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −259.979 450.297i −0.352754 0.610987i
\(738\) 0 0
\(739\) −669.176 + 1159.05i −0.905515 + 1.56840i −0.0852911 + 0.996356i \(0.527182\pi\)
−0.820224 + 0.572042i \(0.806151\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 242.908 0.326929 0.163464 0.986549i \(-0.447733\pi\)
0.163464 + 0.986549i \(0.447733\pi\)
\(744\) 0 0
\(745\) 293.762 + 169.604i 0.394312 + 0.227656i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 496.291 + 859.602i 0.660841 + 1.14461i 0.980395 + 0.197042i \(0.0631335\pi\)
−0.319554 + 0.947568i \(0.603533\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 138.666i 0.183663i
\(756\) 0 0
\(757\) 780.060 1.03046 0.515231 0.857051i \(-0.327706\pi\)
0.515231 + 0.857051i \(0.327706\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 402.349 232.296i 0.528711 0.305251i −0.211781 0.977317i \(-0.567926\pi\)
0.740491 + 0.672066i \(0.234593\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 150.876 261.325i 0.196709 0.340710i
\(768\) 0 0
\(769\) 611.409i 0.795071i 0.917587 + 0.397535i \(0.130134\pi\)
−0.917587 + 0.397535i \(0.869866\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 204.128 + 117.853i 0.264072 + 0.152462i 0.626191 0.779670i \(-0.284613\pi\)
−0.362119 + 0.932132i \(0.617947\pi\)
\(774\) 0 0
\(775\) −93.2231 + 53.8224i −0.120288 + 0.0694482i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 296.983 + 514.390i 0.381236 + 0.660321i
\(780\) 0 0
\(781\) 143.894 249.232i 0.184244 0.319120i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1083.31 −1.38001
\(786\) 0 0
\(787\) −515.266 297.489i −0.654722 0.378004i 0.135541 0.990772i \(-0.456723\pi\)
−0.790263 + 0.612768i \(0.790056\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −141.347 244.820i −0.178243 0.308726i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 547.544i 0.687006i −0.939152 0.343503i \(-0.888386\pi\)
0.939152 0.343503i \(-0.111614\pi\)
\(798\) 0 0
\(799\) −646.161 −0.808712
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1377.59 + 795.350i −1.71555 + 0.990474i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −256.069 + 443.524i −0.316525 + 0.548237i −0.979760 0.200174i \(-0.935849\pi\)
0.663236 + 0.748411i \(0.269183\pi\)
\(810\) 0 0
\(811\) 1615.33i 1.99178i −0.0905851 0.995889i \(-0.528874\pi\)
0.0905851 0.995889i \(-0.471126\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 895.310 + 516.907i 1.09854 + 0.634242i
\(816\) 0 0
\(817\) 17.8607 10.3119i 0.0218613 0.0126216i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −490.777 850.050i −0.597779 1.03538i −0.993148 0.116861i \(-0.962717\pi\)
0.395369 0.918522i \(-0.370617\pi\)
\(822\) 0 0
\(823\) 678.276 1174.81i 0.824151 1.42747i −0.0784154 0.996921i \(-0.524986\pi\)
0.902566 0.430551i \(-0.141681\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 601.883 0.727791 0.363896 0.931440i \(-0.381446\pi\)
0.363896 + 0.931440i \(0.381446\pi\)
\(828\) 0 0
\(829\) −154.783 89.3641i −0.186711 0.107798i 0.403731 0.914878i \(-0.367713\pi\)
−0.590442 + 0.807080i \(0.701046\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −306.799 531.391i −0.367424 0.636397i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 852.989i 1.01667i −0.861158 0.508337i \(-0.830261\pi\)
0.861158 0.508337i \(-0.169739\pi\)
\(840\) 0 0
\(841\) −404.769 −0.481295
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −549.358 + 317.172i −0.650127 + 0.375351i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −390.065 + 675.612i −0.458361 + 0.793904i
\(852\) 0 0
\(853\) 855.303i 1.00270i 0.865245 + 0.501350i \(0.167163\pi\)
−0.865245 + 0.501350i \(0.832837\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1251.22 722.394i −1.46000 0.842933i −0.460992 0.887404i \(-0.652506\pi\)
−0.999011 + 0.0444708i \(0.985840\pi\)
\(858\) 0 0
\(859\) 367.259 212.037i 0.427543 0.246842i −0.270756 0.962648i \(-0.587274\pi\)
0.698299 + 0.715806i \(0.253940\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 674.602 + 1168.44i 0.781694 + 1.35393i 0.930955 + 0.365135i \(0.118977\pi\)
−0.149261 + 0.988798i \(0.547689\pi\)
\(864\) 0 0
\(865\) −711.332 + 1232.06i −0.822349 + 1.42435i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2527.89 2.90896
\(870\) 0 0
\(871\) 105.841 + 61.1074i 0.121517 + 0.0701577i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −321.829 557.424i −0.366966 0.635603i 0.622124 0.782919i \(-0.286270\pi\)
−0.989090 + 0.147315i \(0.952937\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1206.54i 1.36951i −0.728774 0.684754i \(-0.759909\pi\)
0.728774 0.684754i \(-0.240091\pi\)
\(882\) 0 0
\(883\) 652.754 0.739245 0.369623 0.929182i \(-0.379487\pi\)
0.369623 + 0.929182i \(0.379487\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −910.418 + 525.630i −1.02640 + 0.592593i −0.915952 0.401289i \(-0.868562\pi\)
−0.110450 + 0.993882i \(0.535229\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 237.186 410.818i 0.265606 0.460042i
\(894\) 0 0
\(895\) 500.288i 0.558981i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −282.205 162.931i −0.313910 0.181236i
\(900\) 0 0
\(901\) −419.560 + 242.233i −0.465660 + 0.268849i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 369.035 + 639.187i 0.407773 + 0.706284i
\(906\) 0 0
\(907\) −143.080 + 247.822i −0.157751 + 0.273233i −0.934057 0.357123i \(-0.883758\pi\)
0.776306 + 0.630356i \(0.217091\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −713.465 −0.783167 −0.391583 0.920143i \(-0.628073\pi\)
−0.391583 + 0.920143i \(0.628073\pi\)
\(912\) 0 0
\(913\) 1804.04 + 1041.56i 1.97594 + 1.14081i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −170.698 295.658i −0.185744 0.321718i 0.758083 0.652158i \(-0.226136\pi\)
−0.943827 + 0.330440i \(0.892803\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 67.6439i 0.0732870i
\(924\) 0 0
\(925\) 315.296 0.340861
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −79.5587 + 45.9332i −0.0856390 + 0.0494437i −0.542208 0.840244i \(-0.682412\pi\)
0.456569 + 0.889688i \(0.349078\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −515.286 + 892.501i −0.551108 + 0.954547i
\(936\) 0 0
\(937\) 906.902i 0.967879i −0.875102 0.483939i \(-0.839206\pi\)
0.875102 0.483939i \(-0.160794\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 337.880 + 195.075i 0.359064 + 0.207306i 0.668670 0.743559i \(-0.266864\pi\)
−0.309606 + 0.950865i \(0.600197\pi\)
\(942\) 0 0
\(943\) 937.158 541.068i 0.993804 0.573773i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −304.757 527.854i −0.321813 0.557396i 0.659049 0.752100i \(-0.270959\pi\)
−0.980862 + 0.194704i \(0.937625\pi\)
\(948\) 0 0
\(949\) 186.945 323.798i 0.196991 0.341199i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −110.441 −0.115887 −0.0579437 0.998320i \(-0.518454\pi\)
−0.0579437 + 0.998320i \(0.518454\pi\)
\(954\) 0 0
\(955\) −965.269 557.299i −1.01075 0.583559i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −358.791 621.445i −0.373352 0.646665i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 125.924i 0.130491i
\(966\) 0 0
\(967\) −1125.53 −1.16394 −0.581969 0.813211i \(-0.697717\pi\)
−0.581969 + 0.813211i \(0.697717\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 977.849 564.562i 1.00705 0.581423i 0.0967262 0.995311i \(-0.469163\pi\)
0.910328 + 0.413888i \(0.135830\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 756.430 1310.18i 0.774238 1.34102i −0.160984 0.986957i \(-0.551467\pi\)
0.935222 0.354062i \(-0.115200\pi\)
\(978\) 0 0
\(979\) 1517.91i 1.55047i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −501.830 289.731i −0.510508 0.294742i 0.222534 0.974925i \(-0.428567\pi\)
−0.733043 + 0.680183i \(0.761900\pi\)
\(984\) 0 0
\(985\) 874.259 504.754i 0.887573 0.512440i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −18.7870 32.5400i −0.0189959 0.0329019i
\(990\) 0 0
\(991\) −86.3768 + 149.609i −0.0871612 + 0.150968i −0.906310 0.422613i \(-0.861113\pi\)
0.819149 + 0.573581i \(0.194446\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 125.700 0.126332
\(996\) 0 0
\(997\) 333.267 + 192.412i 0.334270 + 0.192991i 0.657735 0.753249i \(-0.271515\pi\)
−0.323465 + 0.946240i \(0.604848\pi\)
\(998\) 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.n.325.3 16
3.2 odd 2 inner 1764.3.z.n.325.6 16
7.2 even 3 inner 1764.3.z.n.901.5 16
7.3 odd 6 1764.3.d.g.685.6 yes 8
7.4 even 3 1764.3.d.g.685.4 yes 8
7.5 odd 6 inner 1764.3.z.n.901.3 16
7.6 odd 2 inner 1764.3.z.n.325.5 16
21.2 odd 6 inner 1764.3.z.n.901.4 16
21.5 even 6 inner 1764.3.z.n.901.6 16
21.11 odd 6 1764.3.d.g.685.5 yes 8
21.17 even 6 1764.3.d.g.685.3 8
21.20 even 2 inner 1764.3.z.n.325.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1764.3.d.g.685.3 8 21.17 even 6
1764.3.d.g.685.4 yes 8 7.4 even 3
1764.3.d.g.685.5 yes 8 21.11 odd 6
1764.3.d.g.685.6 yes 8 7.3 odd 6
1764.3.z.n.325.3 16 1.1 even 1 trivial
1764.3.z.n.325.4 16 21.20 even 2 inner
1764.3.z.n.325.5 16 7.6 odd 2 inner
1764.3.z.n.325.6 16 3.2 odd 2 inner
1764.3.z.n.901.3 16 7.5 odd 6 inner
1764.3.z.n.901.4 16 21.2 odd 6 inner
1764.3.z.n.901.5 16 7.2 even 3 inner
1764.3.z.n.901.6 16 21.5 even 6 inner