Properties

Label 1728.3.q.g.449.2
Level $1728$
Weight $3$
Character 1728.449
Analytic conductor $47.085$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,3,Mod(449,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.q (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: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 449.2
Root \(-1.18614 - 1.26217i\) of defining polynomial
Character \(\chi\) \(=\) 1728.449
Dual form 1728.3.q.g.1601.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(6.55842 + 3.78651i) q^{5} +(-4.55842 - 7.89542i) q^{7} +O(q^{10})\) \(q+(6.55842 + 3.78651i) q^{5} +(-4.55842 - 7.89542i) q^{7} +(-0.383156 + 0.221215i) q^{11} +(-5.55842 + 9.62747i) q^{13} +8.01544i q^{17} +8.11684 q^{19} +(-20.4416 - 11.8020i) q^{23} +(16.1753 + 28.0164i) q^{25} +(-45.9090 + 26.5055i) q^{29} +(-14.6753 + 25.4183i) q^{31} -69.0420i q^{35} -18.4674 q^{37} +(38.9674 + 22.4978i) q^{41} +(11.5000 + 19.9186i) q^{43} +(7.32473 - 4.22894i) q^{47} +(-17.0584 + 29.5461i) q^{49} +60.5841i q^{53} -3.35053 q^{55} +(65.9674 + 38.0863i) q^{59} +(2.67527 + 4.63370i) q^{61} +(-72.9090 + 42.0940i) q^{65} +(-54.8505 + 95.0039i) q^{67} -16.0309i q^{71} -4.35053 q^{73} +(3.49317 + 2.01678i) q^{77} +(-0.792110 - 1.37197i) q^{79} +(-7.32473 + 4.22894i) q^{83} +(-30.3505 + 52.5687i) q^{85} -64.1236i q^{89} +101.351 q^{91} +(53.2337 + 30.7345i) q^{95} +(-57.6168 - 99.7953i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 9 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 9 q^{5} - q^{7} - 36 q^{11} - 5 q^{13} - 2 q^{19} - 99 q^{23} + 13 q^{25} - 63 q^{29} - 7 q^{31} + 64 q^{37} + 18 q^{41} + 46 q^{43} + 81 q^{47} - 51 q^{49} + 90 q^{55} + 126 q^{59} - 41 q^{61} - 171 q^{65} - 116 q^{67} + 86 q^{73} - 279 q^{77} + 83 q^{79} - 81 q^{83} - 18 q^{85} + 302 q^{91} + 144 q^{95} - 196 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(5\) 6.55842 + 3.78651i 1.31168 + 0.757301i 0.982375 0.186921i \(-0.0598508\pi\)
0.329309 + 0.944222i \(0.393184\pi\)
\(6\) 0 0
\(7\) −4.55842 7.89542i −0.651203 1.12792i −0.982831 0.184507i \(-0.940931\pi\)
0.331628 0.943410i \(-0.392402\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.383156 + 0.221215i −0.0348324 + 0.0201105i −0.517315 0.855795i \(-0.673068\pi\)
0.482483 + 0.875906i \(0.339735\pi\)
\(12\) 0 0
\(13\) −5.55842 + 9.62747i −0.427571 + 0.740575i −0.996657 0.0817036i \(-0.973964\pi\)
0.569086 + 0.822278i \(0.307297\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.01544i 0.471497i 0.971814 + 0.235748i \(0.0757541\pi\)
−0.971814 + 0.235748i \(0.924246\pi\)
\(18\) 0 0
\(19\) 8.11684 0.427202 0.213601 0.976921i \(-0.431481\pi\)
0.213601 + 0.976921i \(0.431481\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −20.4416 11.8020i −0.888764 0.513128i −0.0152262 0.999884i \(-0.504847\pi\)
−0.873538 + 0.486756i \(0.838180\pi\)
\(24\) 0 0
\(25\) 16.1753 + 28.0164i 0.647011 + 1.12066i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −45.9090 + 26.5055i −1.58307 + 0.913984i −0.588659 + 0.808381i \(0.700344\pi\)
−0.994408 + 0.105603i \(0.966323\pi\)
\(30\) 0 0
\(31\) −14.6753 + 25.4183i −0.473396 + 0.819945i −0.999536 0.0304523i \(-0.990305\pi\)
0.526141 + 0.850398i \(0.323639\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 69.0420i 1.97263i
\(36\) 0 0
\(37\) −18.4674 −0.499118 −0.249559 0.968360i \(-0.580286\pi\)
−0.249559 + 0.968360i \(0.580286\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 38.9674 + 22.4978i 0.950424 + 0.548727i 0.893213 0.449635i \(-0.148446\pi\)
0.0572112 + 0.998362i \(0.481779\pi\)
\(42\) 0 0
\(43\) 11.5000 + 19.9186i 0.267442 + 0.463223i 0.968200 0.250176i \(-0.0804883\pi\)
−0.700759 + 0.713398i \(0.747155\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.32473 4.22894i 0.155845 0.0899774i −0.420049 0.907501i \(-0.637987\pi\)
0.575895 + 0.817524i \(0.304654\pi\)
\(48\) 0 0
\(49\) −17.0584 + 29.5461i −0.348131 + 0.602981i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 60.5841i 1.14310i 0.820569 + 0.571548i \(0.193657\pi\)
−0.820569 + 0.571548i \(0.806343\pi\)
\(54\) 0 0
\(55\) −3.35053 −0.0609188
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 65.9674 + 38.0863i 1.11809 + 0.645530i 0.940913 0.338649i \(-0.109970\pi\)
0.177178 + 0.984179i \(0.443303\pi\)
\(60\) 0 0
\(61\) 2.67527 + 4.63370i 0.0438568 + 0.0759622i 0.887121 0.461538i \(-0.152702\pi\)
−0.843264 + 0.537500i \(0.819369\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −72.9090 + 42.0940i −1.12168 + 0.647600i
\(66\) 0 0
\(67\) −54.8505 + 95.0039i −0.818665 + 1.41797i 0.0880017 + 0.996120i \(0.471952\pi\)
−0.906666 + 0.421848i \(0.861381\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 16.0309i 0.225787i −0.993607 0.112894i \(-0.963988\pi\)
0.993607 0.112894i \(-0.0360119\pi\)
\(72\) 0 0
\(73\) −4.35053 −0.0595963 −0.0297982 0.999556i \(-0.509486\pi\)
−0.0297982 + 0.999556i \(0.509486\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.49317 + 2.01678i 0.0453659 + 0.0261920i
\(78\) 0 0
\(79\) −0.792110 1.37197i −0.0100267 0.0173668i 0.860969 0.508658i \(-0.169858\pi\)
−0.870995 + 0.491291i \(0.836525\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.32473 + 4.22894i −0.0882498 + 0.0509511i −0.543475 0.839425i \(-0.682892\pi\)
0.455226 + 0.890376i \(0.349559\pi\)
\(84\) 0 0
\(85\) −30.3505 + 52.5687i −0.357065 + 0.618455i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 64.1236i 0.720489i −0.932858 0.360245i \(-0.882693\pi\)
0.932858 0.360245i \(-0.117307\pi\)
\(90\) 0 0
\(91\) 101.351 1.11374
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 53.2337 + 30.7345i 0.560355 + 0.323521i
\(96\) 0 0
\(97\) −57.6168 99.7953i −0.593988 1.02882i −0.993689 0.112172i \(-0.964219\pi\)
0.399701 0.916646i \(-0.369114\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 114.558 66.1403i 1.13424 0.654855i 0.189244 0.981930i \(-0.439396\pi\)
0.944998 + 0.327075i \(0.106063\pi\)
\(102\) 0 0
\(103\) −62.6753 + 108.557i −0.608498 + 1.05395i 0.382990 + 0.923752i \(0.374894\pi\)
−0.991488 + 0.130197i \(0.958439\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 36.5378i 0.341475i 0.985317 + 0.170737i \(0.0546149\pi\)
−0.985317 + 0.170737i \(0.945385\pi\)
\(108\) 0 0
\(109\) −134.701 −1.23579 −0.617895 0.786261i \(-0.712014\pi\)
−0.617895 + 0.786261i \(0.712014\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 164.727 + 95.1051i 1.45776 + 0.841638i 0.998901 0.0468711i \(-0.0149250\pi\)
0.458859 + 0.888509i \(0.348258\pi\)
\(114\) 0 0
\(115\) −89.3763 154.804i −0.777185 1.34612i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 63.2853 36.5378i 0.531809 0.307040i
\(120\) 0 0
\(121\) −60.4021 + 104.620i −0.499191 + 0.864624i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 55.6657i 0.445325i
\(126\) 0 0
\(127\) 184.103 1.44963 0.724816 0.688943i \(-0.241925\pi\)
0.724816 + 0.688943i \(0.241925\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −109.194 63.0433i −0.833544 0.481247i 0.0215207 0.999768i \(-0.493149\pi\)
−0.855064 + 0.518522i \(0.826483\pi\)
\(132\) 0 0
\(133\) −37.0000 64.0859i −0.278195 0.481849i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −107.617 + 62.1326i −0.785524 + 0.453523i −0.838385 0.545079i \(-0.816500\pi\)
0.0528602 + 0.998602i \(0.483166\pi\)
\(138\) 0 0
\(139\) 13.3832 23.1803i 0.0962817 0.166765i −0.813861 0.581059i \(-0.802638\pi\)
0.910143 + 0.414295i \(0.135972\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.91843i 0.0343946i
\(144\) 0 0
\(145\) −401.454 −2.76865
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 69.8437 + 40.3243i 0.468750 + 0.270633i 0.715716 0.698391i \(-0.246100\pi\)
−0.246966 + 0.969024i \(0.579434\pi\)
\(150\) 0 0
\(151\) −49.9742 86.5579i −0.330955 0.573231i 0.651744 0.758439i \(-0.274037\pi\)
−0.982699 + 0.185208i \(0.940704\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −192.493 + 111.136i −1.24189 + 0.717006i
\(156\) 0 0
\(157\) −34.7269 + 60.1487i −0.221190 + 0.383113i −0.955170 0.296059i \(-0.904327\pi\)
0.733979 + 0.679172i \(0.237661\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 215.193i 1.33660i
\(162\) 0 0
\(163\) −162.467 −0.996732 −0.498366 0.866967i \(-0.666066\pi\)
−0.498366 + 0.866967i \(0.666066\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −83.7269 48.3397i −0.501358 0.289459i 0.227916 0.973681i \(-0.426809\pi\)
−0.729274 + 0.684221i \(0.760142\pi\)
\(168\) 0 0
\(169\) 22.7079 + 39.3312i 0.134366 + 0.232729i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −24.2731 + 14.0141i −0.140307 + 0.0810064i −0.568510 0.822676i \(-0.692480\pi\)
0.428203 + 0.903682i \(0.359147\pi\)
\(174\) 0 0
\(175\) 147.467 255.421i 0.842671 1.45955i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 35.6012i 0.198889i −0.995043 0.0994447i \(-0.968293\pi\)
0.995043 0.0994447i \(-0.0317067\pi\)
\(180\) 0 0
\(181\) 19.6358 0.108485 0.0542426 0.998528i \(-0.482726\pi\)
0.0542426 + 0.998528i \(0.482726\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −121.117 69.9268i −0.654686 0.377983i
\(186\) 0 0
\(187\) −1.77314 3.07117i −0.00948202 0.0164233i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −188.662 + 108.924i −0.987757 + 0.570282i −0.904603 0.426255i \(-0.859833\pi\)
−0.0831540 + 0.996537i \(0.526499\pi\)
\(192\) 0 0
\(193\) 24.5000 42.4352i 0.126943 0.219872i −0.795548 0.605891i \(-0.792817\pi\)
0.922491 + 0.386019i \(0.126150\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 359.965i 1.82723i 0.406575 + 0.913617i \(0.366723\pi\)
−0.406575 + 0.913617i \(0.633277\pi\)
\(198\) 0 0
\(199\) −61.0652 −0.306861 −0.153430 0.988159i \(-0.549032\pi\)
−0.153430 + 0.988159i \(0.549032\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 418.545 + 241.647i 2.06180 + 1.19038i
\(204\) 0 0
\(205\) 170.376 + 295.100i 0.831104 + 1.43951i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3.11002 + 1.79557i −0.0148805 + 0.00859124i
\(210\) 0 0
\(211\) 193.493 335.140i 0.917029 1.58834i 0.113126 0.993581i \(-0.463913\pi\)
0.803903 0.594761i \(-0.202753\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 174.179i 0.810136i
\(216\) 0 0
\(217\) 267.584 1.23311
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −77.1684 44.5532i −0.349178 0.201598i
\(222\) 0 0
\(223\) −51.3763 88.9864i −0.230387 0.399042i 0.727535 0.686071i \(-0.240666\pi\)
−0.957922 + 0.287028i \(0.907333\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −293.552 + 169.482i −1.29318 + 0.746617i −0.979216 0.202818i \(-0.934990\pi\)
−0.313962 + 0.949435i \(0.601657\pi\)
\(228\) 0 0
\(229\) −148.376 + 256.995i −0.647932 + 1.12225i 0.335685 + 0.941974i \(0.391032\pi\)
−0.983616 + 0.180276i \(0.942301\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 346.537i 1.48728i 0.668578 + 0.743642i \(0.266903\pi\)
−0.668578 + 0.743642i \(0.733097\pi\)
\(234\) 0 0
\(235\) 64.0516 0.272560
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 140.026 + 80.8439i 0.585882 + 0.338259i 0.763468 0.645846i \(-0.223495\pi\)
−0.177586 + 0.984105i \(0.556829\pi\)
\(240\) 0 0
\(241\) 162.370 + 281.232i 0.673732 + 1.16694i 0.976838 + 0.213982i \(0.0686433\pi\)
−0.303105 + 0.952957i \(0.598023\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −223.753 + 129.184i −0.913276 + 0.527280i
\(246\) 0 0
\(247\) −45.1168 + 78.1447i −0.182659 + 0.316375i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 384.012i 1.52993i −0.644074 0.764963i \(-0.722757\pi\)
0.644074 0.764963i \(-0.277243\pi\)
\(252\) 0 0
\(253\) 10.4431 0.0412770
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 11.2011 + 6.46694i 0.0435839 + 0.0251632i 0.521634 0.853170i \(-0.325323\pi\)
−0.478050 + 0.878333i \(0.658656\pi\)
\(258\) 0 0
\(259\) 84.1821 + 145.808i 0.325027 + 0.562964i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 42.8437 24.7358i 0.162904 0.0940526i −0.416332 0.909213i \(-0.636684\pi\)
0.579236 + 0.815160i \(0.303351\pi\)
\(264\) 0 0
\(265\) −229.402 + 397.336i −0.865668 + 1.49938i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 21.4434i 0.0797154i 0.999205 + 0.0398577i \(0.0126905\pi\)
−0.999205 + 0.0398577i \(0.987310\pi\)
\(270\) 0 0
\(271\) −326.907 −1.20630 −0.603150 0.797628i \(-0.706088\pi\)
−0.603150 + 0.797628i \(0.706088\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −12.3953 7.15643i −0.0450738 0.0260234i
\(276\) 0 0
\(277\) −238.727 413.487i −0.861830 1.49273i −0.870161 0.492768i \(-0.835985\pi\)
0.00833105 0.999965i \(-0.497348\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 103.064 59.5039i 0.366775 0.211758i −0.305274 0.952265i \(-0.598748\pi\)
0.672049 + 0.740507i \(0.265415\pi\)
\(282\) 0 0
\(283\) −195.675 + 338.920i −0.691432 + 1.19760i 0.279937 + 0.960018i \(0.409687\pi\)
−0.971369 + 0.237577i \(0.923647\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 410.218i 1.42933i
\(288\) 0 0
\(289\) 224.753 0.777691
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 62.0910 + 35.8483i 0.211915 + 0.122349i 0.602201 0.798345i \(-0.294291\pi\)
−0.390286 + 0.920694i \(0.627624\pi\)
\(294\) 0 0
\(295\) 288.428 + 499.572i 0.977722 + 1.69346i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 227.246 131.200i 0.760020 0.438797i
\(300\) 0 0
\(301\) 104.844 181.595i 0.348318 0.603304i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 40.5196i 0.132851i
\(306\) 0 0
\(307\) 172.351 0.561402 0.280701 0.959795i \(-0.409433\pi\)
0.280701 + 0.959795i \(0.409433\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 524.246 + 302.673i 1.68568 + 0.973227i 0.957763 + 0.287559i \(0.0928438\pi\)
0.727915 + 0.685667i \(0.240489\pi\)
\(312\) 0 0
\(313\) −163.734 283.595i −0.523111 0.906055i −0.999638 0.0268949i \(-0.991438\pi\)
0.476527 0.879160i \(-0.341895\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −59.7921 + 34.5210i −0.188619 + 0.108899i −0.591336 0.806425i \(-0.701399\pi\)
0.402717 + 0.915325i \(0.368066\pi\)
\(318\) 0 0
\(319\) 11.7269 20.3115i 0.0367613 0.0636725i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 65.0601i 0.201424i
\(324\) 0 0
\(325\) −359.636 −1.10657
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −66.7785 38.5546i −0.202974 0.117187i
\(330\) 0 0
\(331\) 254.895 + 441.492i 0.770076 + 1.33381i 0.937521 + 0.347930i \(0.113115\pi\)
−0.167444 + 0.985882i \(0.553551\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −719.466 + 415.384i −2.14766 + 1.23995i
\(336\) 0 0
\(337\) 168.720 292.232i 0.500653 0.867156i −0.499347 0.866402i \(-0.666427\pi\)
1.00000 0.000754096i \(-0.000240036\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 12.9856i 0.0380808i
\(342\) 0 0
\(343\) −135.687 −0.395590
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 186.407 + 107.622i 0.537197 + 0.310151i 0.743942 0.668244i \(-0.232954\pi\)
−0.206745 + 0.978395i \(0.566287\pi\)
\(348\) 0 0
\(349\) 181.012 + 313.522i 0.518659 + 0.898345i 0.999765 + 0.0216818i \(0.00690207\pi\)
−0.481105 + 0.876663i \(0.659765\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 506.486 292.420i 1.43481 0.828385i 0.437323 0.899304i \(-0.355926\pi\)
0.997482 + 0.0709189i \(0.0225932\pi\)
\(354\) 0 0
\(355\) 60.7011 105.137i 0.170989 0.296161i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 393.693i 1.09664i −0.836269 0.548319i \(-0.815268\pi\)
0.836269 0.548319i \(-0.184732\pi\)
\(360\) 0 0
\(361\) −295.117 −0.817498
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −28.5326 16.4733i −0.0781716 0.0451324i
\(366\) 0 0
\(367\) −190.428 329.831i −0.518877 0.898722i −0.999759 0.0219364i \(-0.993017\pi\)
0.480882 0.876785i \(-0.340316\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 478.337 276.168i 1.28932 0.744388i
\(372\) 0 0
\(373\) 66.4416 115.080i 0.178128 0.308526i −0.763112 0.646267i \(-0.776329\pi\)
0.941239 + 0.337741i \(0.109663\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 589.316i 1.56317i
\(378\) 0 0
\(379\) −507.622 −1.33937 −0.669686 0.742644i \(-0.733571\pi\)
−0.669686 + 0.742644i \(0.733571\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −287.466 165.968i −0.750564 0.433338i 0.0753339 0.997158i \(-0.475998\pi\)
−0.825898 + 0.563820i \(0.809331\pi\)
\(384\) 0 0
\(385\) 15.2731 + 26.4539i 0.0396705 + 0.0687113i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 296.662 171.278i 0.762626 0.440302i −0.0676116 0.997712i \(-0.521538\pi\)
0.830238 + 0.557409i \(0.188205\pi\)
\(390\) 0 0
\(391\) 94.5979 163.848i 0.241938 0.419049i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 11.9973i 0.0303730i
\(396\) 0 0
\(397\) 24.8043 0.0624792 0.0312396 0.999512i \(-0.490055\pi\)
0.0312396 + 0.999512i \(0.490055\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −52.0842 30.0708i −0.129886 0.0749896i 0.433649 0.901082i \(-0.357226\pi\)
−0.563535 + 0.826092i \(0.690559\pi\)
\(402\) 0 0
\(403\) −163.143 282.571i −0.404820 0.701170i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.07589 4.08526i 0.0173855 0.0100375i
\(408\) 0 0
\(409\) 240.720 416.939i 0.588558 1.01941i −0.405864 0.913933i \(-0.633029\pi\)
0.994422 0.105478i \(-0.0336373\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 694.453i 1.68149i
\(414\) 0 0
\(415\) −64.0516 −0.154341
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −479.531 276.857i −1.14447 0.660758i −0.196933 0.980417i \(-0.563098\pi\)
−0.947533 + 0.319659i \(0.896432\pi\)
\(420\) 0 0
\(421\) −190.947 330.730i −0.453556 0.785581i 0.545048 0.838405i \(-0.316511\pi\)
−0.998604 + 0.0528233i \(0.983178\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −224.564 + 129.652i −0.528385 + 0.305063i
\(426\) 0 0
\(427\) 24.3900 42.2447i 0.0571194 0.0989337i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 821.321i 1.90562i −0.303570 0.952809i \(-0.598179\pi\)
0.303570 0.952809i \(-0.401821\pi\)
\(432\) 0 0
\(433\) −199.155 −0.459942 −0.229971 0.973198i \(-0.573863\pi\)
−0.229971 + 0.973198i \(0.573863\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −165.921 95.7946i −0.379682 0.219210i
\(438\) 0 0
\(439\) 240.830 + 417.130i 0.548588 + 0.950182i 0.998372 + 0.0570445i \(0.0181677\pi\)
−0.449784 + 0.893137i \(0.648499\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −467.902 + 270.143i −1.05621 + 0.609805i −0.924382 0.381468i \(-0.875419\pi\)
−0.131830 + 0.991272i \(0.542085\pi\)
\(444\) 0 0
\(445\) 242.804 420.549i 0.545628 0.945055i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 300.318i 0.668859i −0.942421 0.334429i \(-0.891456\pi\)
0.942421 0.334429i \(-0.108544\pi\)
\(450\) 0 0
\(451\) −19.9074 −0.0441407
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 664.700 + 383.764i 1.46088 + 0.843438i
\(456\) 0 0
\(457\) −77.8505 134.841i −0.170351 0.295057i 0.768191 0.640220i \(-0.221157\pi\)
−0.938543 + 0.345163i \(0.887824\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 261.143 150.771i 0.566470 0.327052i −0.189268 0.981925i \(-0.560612\pi\)
0.755738 + 0.654874i \(0.227278\pi\)
\(462\) 0 0
\(463\) −119.390 + 206.790i −0.257862 + 0.446630i −0.965669 0.259776i \(-0.916351\pi\)
0.707807 + 0.706406i \(0.249685\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 423.152i 0.906107i −0.891483 0.453054i \(-0.850335\pi\)
0.891483 0.453054i \(-0.149665\pi\)
\(468\) 0 0
\(469\) 1000.13 2.13247
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.81259 5.08795i −0.0186313 0.0107568i
\(474\) 0 0
\(475\) 131.292 + 227.405i 0.276404 + 0.478747i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 379.284 218.980i 0.791824 0.457160i −0.0487802 0.998810i \(-0.515533\pi\)
0.840604 + 0.541650i \(0.182200\pi\)
\(480\) 0 0
\(481\) 102.649 177.794i 0.213408 0.369634i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 872.666i 1.79931i
\(486\) 0 0
\(487\) 401.945 0.825350 0.412675 0.910878i \(-0.364595\pi\)
0.412675 + 0.910878i \(0.364595\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −241.084 139.190i −0.491007 0.283483i 0.233985 0.972240i \(-0.424823\pi\)
−0.724992 + 0.688757i \(0.758157\pi\)
\(492\) 0 0
\(493\) −212.454 367.981i −0.430941 0.746411i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −126.571 + 73.0756i −0.254669 + 0.147033i
\(498\) 0 0
\(499\) 272.655 472.252i 0.546402 0.946397i −0.452115 0.891960i \(-0.649330\pi\)
0.998517 0.0544369i \(-0.0173364\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 306.460i 0.609264i 0.952470 + 0.304632i \(0.0985335\pi\)
−0.952470 + 0.304632i \(0.901466\pi\)
\(504\) 0 0
\(505\) 1001.76 1.98369
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 480.208 + 277.248i 0.943434 + 0.544692i 0.891035 0.453934i \(-0.149980\pi\)
0.0523989 + 0.998626i \(0.483313\pi\)
\(510\) 0 0
\(511\) 19.8316 + 34.3493i 0.0388093 + 0.0672197i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −822.102 + 474.641i −1.59631 + 0.921632i
\(516\) 0 0
\(517\) −1.87101 + 3.24069i −0.00361898 + 0.00626825i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 154.167i 0.295905i −0.988994 0.147953i \(-0.952732\pi\)
0.988994 0.147953i \(-0.0472683\pi\)
\(522\) 0 0
\(523\) −480.598 −0.918925 −0.459463 0.888197i \(-0.651958\pi\)
−0.459463 + 0.888197i \(0.651958\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −203.739 117.629i −0.386602 0.223204i
\(528\) 0 0
\(529\) 14.0721 + 24.3735i 0.0266013 + 0.0460748i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −433.194 + 250.105i −0.812747 + 0.469240i
\(534\) 0 0
\(535\) −138.351 + 239.630i −0.258599 + 0.447907i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 15.0943i 0.0280043i
\(540\) 0 0
\(541\) 300.543 0.555533 0.277766 0.960649i \(-0.410406\pi\)
0.277766 + 0.960649i \(0.410406\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −883.426 510.046i −1.62097 0.935865i
\(546\) 0 0
\(547\) −50.6032 87.6473i −0.0925104 0.160233i 0.816056 0.577972i \(-0.196156\pi\)
−0.908567 + 0.417739i \(0.862822\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −372.636 + 215.141i −0.676290 + 0.390456i
\(552\) 0 0
\(553\) −7.22154 + 12.5081i −0.0130588 + 0.0226186i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 433.041i 0.777452i −0.921353 0.388726i \(-0.872915\pi\)
0.921353 0.388726i \(-0.127085\pi\)
\(558\) 0 0
\(559\) −255.687 −0.457401
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 902.201 + 520.886i 1.60249 + 0.925197i 0.990988 + 0.133954i \(0.0427673\pi\)
0.611501 + 0.791244i \(0.290566\pi\)
\(564\) 0 0
\(565\) 720.232 + 1247.48i 1.27475 + 2.20793i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −257.445 + 148.636i −0.452452 + 0.261223i −0.708865 0.705344i \(-0.750793\pi\)
0.256413 + 0.966567i \(0.417459\pi\)
\(570\) 0 0
\(571\) −339.524 + 588.073i −0.594613 + 1.02990i 0.398988 + 0.916956i \(0.369362\pi\)
−0.993601 + 0.112945i \(0.963972\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 763.599i 1.32800i
\(576\) 0 0
\(577\) −148.351 −0.257107 −0.128553 0.991703i \(-0.541033\pi\)
−0.128553 + 0.991703i \(0.541033\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 66.7785 + 38.5546i 0.114937 + 0.0663590i
\(582\) 0 0
\(583\) −13.4021 23.2132i −0.0229882 0.0398168i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 456.497 263.559i 0.777678 0.448993i −0.0579287 0.998321i \(-0.518450\pi\)
0.835607 + 0.549328i \(0.185116\pi\)
\(588\) 0 0
\(589\) −119.117 + 206.316i −0.202236 + 0.350283i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 473.848i 0.799069i 0.916718 + 0.399534i \(0.130828\pi\)
−0.916718 + 0.399534i \(0.869172\pi\)
\(594\) 0 0
\(595\) 553.402 0.930088
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 601.414 + 347.227i 1.00403 + 0.579677i 0.909438 0.415839i \(-0.136512\pi\)
0.0945922 + 0.995516i \(0.469845\pi\)
\(600\) 0 0
\(601\) −93.3559 161.697i −0.155334 0.269047i 0.777846 0.628454i \(-0.216312\pi\)
−0.933181 + 0.359408i \(0.882979\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −792.285 + 457.426i −1.30956 + 0.756076i
\(606\) 0 0
\(607\) 411.194 712.209i 0.677420 1.17333i −0.298335 0.954461i \(-0.596431\pi\)
0.975755 0.218865i \(-0.0702355\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 94.0249i 0.153887i
\(612\) 0 0
\(613\) 482.206 0.786634 0.393317 0.919403i \(-0.371328\pi\)
0.393317 + 0.919403i \(0.371328\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 595.916 + 344.052i 0.965828 + 0.557621i 0.897962 0.440073i \(-0.145048\pi\)
0.0678661 + 0.997694i \(0.478381\pi\)
\(618\) 0 0
\(619\) −380.253 658.617i −0.614302 1.06400i −0.990507 0.137465i \(-0.956105\pi\)
0.376205 0.926536i \(-0.377229\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −506.282 + 292.302i −0.812652 + 0.469185i
\(624\) 0 0
\(625\) 193.603 335.331i 0.309765 0.536529i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 148.024i 0.235333i
\(630\) 0 0
\(631\) 1008.08 1.59758 0.798792 0.601607i \(-0.205473\pi\)
0.798792 + 0.601607i \(0.205473\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1207.43 + 697.108i 1.90146 + 1.09781i
\(636\) 0 0
\(637\) −189.636 328.459i −0.297701 0.515634i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 488.095 281.802i 0.761458 0.439628i −0.0683607 0.997661i \(-0.521777\pi\)
0.829819 + 0.558032i \(0.188444\pi\)
\(642\) 0 0
\(643\) −288.500 + 499.697i −0.448678 + 0.777133i −0.998300 0.0582801i \(-0.981438\pi\)
0.549622 + 0.835413i \(0.314772\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1024.52i 1.58349i 0.610853 + 0.791744i \(0.290827\pi\)
−0.610853 + 0.791744i \(0.709173\pi\)
\(648\) 0 0
\(649\) −33.7011 −0.0519277
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 345.885 + 199.697i 0.529686 + 0.305814i 0.740888 0.671628i \(-0.234405\pi\)
−0.211203 + 0.977442i \(0.567738\pi\)
\(654\) 0 0
\(655\) −477.428 826.929i −0.728898 1.26249i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −646.308 + 373.146i −0.980741 + 0.566231i −0.902494 0.430703i \(-0.858266\pi\)
−0.0782470 + 0.996934i \(0.524932\pi\)
\(660\) 0 0
\(661\) 475.624 823.804i 0.719552 1.24630i −0.241626 0.970369i \(-0.577681\pi\)
0.961178 0.275931i \(-0.0889860\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 560.403i 0.842711i
\(666\) 0 0
\(667\) 1251.27 1.87596
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2.05009 1.18362i −0.00305527 0.00176396i
\(672\) 0 0
\(673\) 172.115 + 298.113i 0.255743 + 0.442961i 0.965097 0.261892i \(-0.0843464\pi\)
−0.709354 + 0.704853i \(0.751013\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 853.610 492.832i 1.26087 0.727965i 0.287628 0.957742i \(-0.407133\pi\)
0.973243 + 0.229778i \(0.0737999\pi\)
\(678\) 0 0
\(679\) −525.284 + 909.818i −0.773614 + 1.33994i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 166.658i 0.244009i 0.992530 + 0.122004i \(0.0389322\pi\)
−0.992530 + 0.122004i \(0.961068\pi\)
\(684\) 0 0
\(685\) −941.062 −1.37381
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −583.272 336.752i −0.846548 0.488755i
\(690\) 0 0
\(691\) −449.077 777.825i −0.649895 1.12565i −0.983148 0.182813i \(-0.941480\pi\)
0.333253 0.942838i \(-0.391854\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 175.545 101.351i 0.252582 0.145829i
\(696\) 0 0
\(697\) −180.330 + 312.341i −0.258723 + 0.448122i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 730.549i 1.04215i 0.853510 + 0.521076i \(0.174469\pi\)
−0.853510 + 0.521076i \(0.825531\pi\)
\(702\) 0 0
\(703\) −149.897 −0.213224
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1044.41 602.991i −1.47724 0.852887i
\(708\) 0 0
\(709\) −114.961 199.118i −0.162145 0.280843i 0.773493 0.633805i \(-0.218508\pi\)
−0.935638 + 0.352962i \(0.885174\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 599.971 346.394i 0.841474 0.485825i
\(714\) 0 0
\(715\) 18.6237 32.2571i 0.0260471 0.0451149i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 907.095i 1.26161i 0.775943 + 0.630803i \(0.217275\pi\)
−0.775943 + 0.630803i \(0.782725\pi\)
\(720\) 0 0
\(721\) 1142.80 1.58502
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1485.18 857.469i −2.04852 1.18272i
\(726\) 0 0
\(727\) −107.871 186.838i −0.148378 0.256999i 0.782250 0.622965i \(-0.214072\pi\)
−0.930628 + 0.365966i \(0.880739\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −159.656 + 92.1776i −0.218408 + 0.126098i
\(732\) 0 0
\(733\) 314.634 544.963i 0.429242 0.743469i −0.567564 0.823329i \(-0.692114\pi\)
0.996806 + 0.0798604i \(0.0254475\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 48.5351i 0.0658549i
\(738\) 0 0
\(739\) −547.649 −0.741068 −0.370534 0.928819i \(-0.620825\pi\)
−0.370534 + 0.928819i \(0.620825\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 462.583 + 267.072i 0.622588 + 0.359451i 0.777876 0.628418i \(-0.216297\pi\)
−0.155288 + 0.987869i \(0.549631\pi\)
\(744\) 0 0
\(745\) 305.376 + 528.927i 0.409901 + 0.709970i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 288.481 166.555i 0.385155 0.222369i
\(750\) 0 0
\(751\) 225.545 390.655i 0.300326 0.520180i −0.675884 0.737008i \(-0.736238\pi\)
0.976210 + 0.216828i \(0.0695712\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 756.911i 1.00253i
\(756\) 0 0
\(757\) 352.391 0.465511 0.232755 0.972535i \(-0.425226\pi\)
0.232755 + 0.972535i \(0.425226\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 929.923 + 536.891i 1.22197 + 0.705507i 0.965339 0.261001i \(-0.0840525\pi\)
0.256636 + 0.966508i \(0.417386\pi\)
\(762\) 0 0
\(763\) 614.024 + 1063.52i 0.804750 + 1.39387i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −733.349 + 423.399i −0.956126 + 0.552020i
\(768\) 0 0
\(769\) −177.988 + 308.284i −0.231454 + 0.400889i −0.958236 0.285978i \(-0.907681\pi\)
0.726782 + 0.686868i \(0.241015\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 370.790i 0.479677i 0.970813 + 0.239838i \(0.0770945\pi\)
−0.970813 + 0.239838i \(0.922906\pi\)
\(774\) 0 0
\(775\) −949.505 −1.22517
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 316.292 + 182.611i 0.406023 + 0.234418i
\(780\) 0 0
\(781\) 3.54628 + 6.14233i 0.00454069 + 0.00786470i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −455.507 + 262.987i −0.580263 + 0.335015i
\(786\) 0 0
\(787\) 576.531 998.581i 0.732568 1.26885i −0.223214 0.974769i \(-0.571655\pi\)
0.955782 0.294076i \(-0.0950118\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1734.12i 2.19231i
\(792\) 0 0
\(793\) −59.4810 −0.0750076
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −761.882 439.873i −0.955937 0.551910i −0.0610167 0.998137i \(-0.519434\pi\)
−0.894920 + 0.446226i \(0.852768\pi\)
\(798\) 0 0
\(799\) 33.8968 + 58.7110i 0.0424240 + 0.0734806i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.66693 0.962404i 0.00207588 0.00119851i
\(804\) 0 0
\(805\) −814.830 + 1411.33i −1.01221 + 1.75320i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 884.508i 1.09334i −0.837350 0.546668i \(-0.815896\pi\)
0.837350 0.546668i \(-0.184104\pi\)
\(810\) 0 0
\(811\) 961.464 1.18553 0.592765 0.805376i \(-0.298036\pi\)
0.592765 + 0.805376i \(0.298036\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1065.53 615.184i −1.30740 0.754827i
\(816\) 0 0
\(817\) 93.3437 + 161.676i 0.114252 + 0.197890i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 778.064 449.215i 0.947702 0.547156i 0.0553360 0.998468i \(-0.482377\pi\)
0.892366 + 0.451312i \(0.149044\pi\)
\(822\) 0 0
\(823\) −108.091 + 187.219i −0.131338 + 0.227484i −0.924193 0.381927i \(-0.875261\pi\)
0.792855 + 0.609411i \(0.208594\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 113.883i 0.137706i 0.997627 + 0.0688528i \(0.0219339\pi\)
−0.997627 + 0.0688528i \(0.978066\pi\)
\(828\) 0 0
\(829\) −101.326 −0.122227 −0.0611135 0.998131i \(-0.519465\pi\)
−0.0611135 + 0.998131i \(0.519465\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −236.825 136.731i −0.284303 0.164143i
\(834\) 0 0
\(835\) −366.077 634.065i −0.438416 0.759359i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −60.4689 + 34.9117i −0.0720726 + 0.0416111i −0.535603 0.844470i \(-0.679916\pi\)
0.463531 + 0.886081i \(0.346582\pi\)
\(840\) 0 0
\(841\) 984.588 1705.36i 1.17073 2.02777i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 343.934i 0.407023i
\(846\) 0 0
\(847\) 1101.35 1.30030
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 377.502 + 217.951i 0.443598 + 0.256112i
\(852\) 0 0
\(853\) −573.325 993.028i −0.672127 1.16416i −0.977300 0.211862i \(-0.932047\pi\)
0.305172 0.952297i \(-0.401286\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −217.871 + 125.788i −0.254225 + 0.146777i −0.621697 0.783258i \(-0.713557\pi\)
0.367472 + 0.930035i \(0.380223\pi\)
\(858\) 0 0
\(859\) −244.266 + 423.082i −0.284361 + 0.492528i −0.972454 0.233095i \(-0.925115\pi\)
0.688093 + 0.725623i \(0.258448\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 596.889i 0.691644i 0.938300 + 0.345822i \(0.112400\pi\)
−0.938300 + 0.345822i \(0.887600\pi\)
\(864\) 0 0
\(865\) −212.258 −0.245385
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0.607003 + 0.350454i 0.000698508 + 0.000403284i
\(870\) 0 0
\(871\) −609.765 1056.14i −0.700074 1.21256i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 439.504 253.748i 0.502290 0.289997i
\(876\) 0 0
\(877\) 358.208 620.434i 0.408447 0.707451i −0.586269 0.810116i \(-0.699404\pi\)
0.994716 + 0.102666i \(0.0327372\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1200.86i 1.36306i −0.731790 0.681531i \(-0.761315\pi\)
0.731790 0.681531i \(-0.238685\pi\)
\(882\) 0 0
\(883\) −22.8938 −0.0259273 −0.0129636 0.999916i \(-0.504127\pi\)
−0.0129636 + 0.999916i \(0.504127\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −521.857 301.294i −0.588340 0.339678i 0.176101 0.984372i \(-0.443651\pi\)
−0.764441 + 0.644694i \(0.776985\pi\)
\(888\) 0 0
\(889\) −839.220 1453.57i −0.944005 1.63506i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 59.4537 34.3256i 0.0665775 0.0384385i
\(894\) 0 0
\(895\) 134.804 233.488i 0.150619 0.260880i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1555.90i 1.73071i
\(900\) 0 0
\(901\) −485.609 −0.538966
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 128.780 + 74.3511i 0.142298 + 0.0821560i
\(906\) 0 0
\(907\) 211.473 + 366.281i 0.233156 + 0.403838i 0.958735 0.284300i \(-0.0917613\pi\)
−0.725579 + 0.688139i \(0.758428\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −125.376 + 72.3861i −0.137625 + 0.0794578i −0.567232 0.823558i \(-0.691986\pi\)
0.429607 + 0.903016i \(0.358652\pi\)
\(912\) 0 0
\(913\) 1.87101 3.24069i 0.00204930 0.00354949i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1149.51i 1.25356i
\(918\) 0 0
\(919\) −869.093 −0.945694 −0.472847 0.881145i \(-0.656774\pi\)
−0.472847 + 0.881145i \(0.656774\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 154.337 + 89.1064i 0.167212 + 0.0965400i
\(924\) 0 0
\(925\) −298.715 517.389i −0.322935 0.559340i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −82.2838 + 47.5066i −0.0885724 + 0.0511373i −0.543632 0.839324i \(-0.682951\pi\)
0.455060 + 0.890461i \(0.349618\pi\)
\(930\) 0 0
\(931\) −138.461 + 239.821i −0.148722 + 0.257595i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 26.8560i 0.0287230i
\(936\) 0 0
\(937\) −1555.55 −1.66014 −0.830071 0.557657i \(-0.811700\pi\)
−0.830071 + 0.557657i \(0.811700\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 335.246 + 193.554i 0.356265 + 0.205690i 0.667441 0.744662i \(-0.267389\pi\)
−0.311176 + 0.950352i \(0.600723\pi\)
\(942\) 0 0
\(943\) −531.036 919.782i −0.563135 0.975379i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −839.861 + 484.894i −0.886865 + 0.512032i −0.872916 0.487871i \(-0.837774\pi\)
−0.0139492 + 0.999903i \(0.504440\pi\)
\(948\) 0 0
\(949\) 24.1821 41.8846i 0.0254817 0.0441355i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1294.65i 1.35850i 0.733909 + 0.679248i \(0.237694\pi\)
−0.733909 + 0.679248i \(0.762306\pi\)
\(954\) 0 0
\(955\) −1649.76 −1.72750
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 981.126 + 566.453i 1.02307 + 0.590671i
\(960\) 0 0
\(961\) 49.7731 + 86.2096i 0.0517931 + 0.0897082i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 321.363 185.539i 0.333018 0.192268i
\(966\) 0 0
\(967\) −412.036 + 713.668i −0.426098 + 0.738023i −0.996522 0.0833272i \(-0.973445\pi\)
0.570425 + 0.821350i \(0.306779\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1518.35i 1.56370i 0.623469 + 0.781848i \(0.285723\pi\)
−0.623469 + 0.781848i \(0.714277\pi\)
\(972\) 0 0
\(973\) −244.024 −0.250796
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 321.497 + 185.616i 0.329065 + 0.189986i 0.655426 0.755259i \(-0.272489\pi\)
−0.326361 + 0.945245i \(0.605822\pi\)
\(978\) 0 0
\(979\) 14.1851 + 24.5693i 0.0144894 + 0.0250963i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 808.039 466.522i 0.822014 0.474590i −0.0290967 0.999577i \(-0.509263\pi\)
0.851110 + 0.524987i \(0.175930\pi\)
\(984\) 0 0
\(985\) −1363.01 + 2360.80i −1.38377 + 2.39676i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 542.890i 0.548928i
\(990\) 0 0
\(991\) 1615.53 1.63020 0.815099 0.579321i \(-0.196682\pi\)
0.815099 + 0.579321i \(0.196682\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −400.492 231.224i −0.402504 0.232386i
\(996\) 0 0
\(997\) 519.376 + 899.586i 0.520939 + 0.902293i 0.999704 + 0.0243496i \(0.00775149\pi\)
−0.478764 + 0.877943i \(0.658915\pi\)
\(998\) 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 1728.3.q.g.449.2 4
3.2 odd 2 576.3.q.d.257.2 4
4.3 odd 2 1728.3.q.h.449.2 4
8.3 odd 2 432.3.q.b.17.1 4
8.5 even 2 108.3.g.a.17.1 4
9.2 odd 6 inner 1728.3.q.g.1601.2 4
9.7 even 3 576.3.q.d.65.2 4
12.11 even 2 576.3.q.g.257.1 4
24.5 odd 2 36.3.g.a.5.1 4
24.11 even 2 144.3.q.b.113.2 4
36.7 odd 6 576.3.q.g.65.1 4
36.11 even 6 1728.3.q.h.1601.2 4
40.13 odd 4 2700.3.u.b.449.1 8
40.29 even 2 2700.3.p.b.2501.2 4
40.37 odd 4 2700.3.u.b.449.4 8
72.5 odd 6 324.3.c.b.161.1 4
72.11 even 6 432.3.q.b.305.1 4
72.13 even 6 324.3.c.b.161.4 4
72.29 odd 6 108.3.g.a.89.1 4
72.43 odd 6 144.3.q.b.65.2 4
72.59 even 6 1296.3.e.e.161.1 4
72.61 even 6 36.3.g.a.29.1 yes 4
72.67 odd 6 1296.3.e.e.161.4 4
120.29 odd 2 900.3.p.a.401.2 4
120.53 even 4 900.3.u.a.149.1 8
120.77 even 4 900.3.u.a.149.4 8
360.29 odd 6 2700.3.p.b.1601.2 4
360.133 odd 12 900.3.u.a.749.4 8
360.173 even 12 2700.3.u.b.2249.4 8
360.277 odd 12 900.3.u.a.749.1 8
360.317 even 12 2700.3.u.b.2249.1 8
360.349 even 6 900.3.p.a.101.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.3.g.a.5.1 4 24.5 odd 2
36.3.g.a.29.1 yes 4 72.61 even 6
108.3.g.a.17.1 4 8.5 even 2
108.3.g.a.89.1 4 72.29 odd 6
144.3.q.b.65.2 4 72.43 odd 6
144.3.q.b.113.2 4 24.11 even 2
324.3.c.b.161.1 4 72.5 odd 6
324.3.c.b.161.4 4 72.13 even 6
432.3.q.b.17.1 4 8.3 odd 2
432.3.q.b.305.1 4 72.11 even 6
576.3.q.d.65.2 4 9.7 even 3
576.3.q.d.257.2 4 3.2 odd 2
576.3.q.g.65.1 4 36.7 odd 6
576.3.q.g.257.1 4 12.11 even 2
900.3.p.a.101.2 4 360.349 even 6
900.3.p.a.401.2 4 120.29 odd 2
900.3.u.a.149.1 8 120.53 even 4
900.3.u.a.149.4 8 120.77 even 4
900.3.u.a.749.1 8 360.277 odd 12
900.3.u.a.749.4 8 360.133 odd 12
1296.3.e.e.161.1 4 72.59 even 6
1296.3.e.e.161.4 4 72.67 odd 6
1728.3.q.g.449.2 4 1.1 even 1 trivial
1728.3.q.g.1601.2 4 9.2 odd 6 inner
1728.3.q.h.449.2 4 4.3 odd 2
1728.3.q.h.1601.2 4 36.11 even 6
2700.3.p.b.1601.2 4 360.29 odd 6
2700.3.p.b.2501.2 4 40.29 even 2
2700.3.u.b.449.1 8 40.13 odd 4
2700.3.u.b.449.4 8 40.37 odd 4
2700.3.u.b.2249.1 8 360.317 even 12
2700.3.u.b.2249.4 8 360.173 even 12