Properties

Label 2700.3.u.b.449.4
Level $2700$
Weight $3$
Character 2700.449
Analytic conductor $73.570$
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] = [2700,3,Mod(449,2700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2700, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2700.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2700.u (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: \(73.5696713773\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 449.4
Root \(0.396143 + 1.68614i\) of defining polynomial
Character \(\chi\) \(=\) 2700.449
Dual form 2700.3.u.b.2249.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(7.89542 - 4.55842i) q^{7} +O(q^{10})\) \(q+(7.89542 - 4.55842i) q^{7} +(0.383156 - 0.221215i) q^{11} +(-9.62747 - 5.55842i) q^{13} -8.01544 q^{17} +8.11684 q^{19} +(-11.8020 + 20.4416i) q^{23} +(-45.9090 + 26.5055i) q^{29} +(-14.6753 + 25.4183i) q^{31} +18.4674i q^{37} +(38.9674 + 22.4978i) q^{41} +(-19.9186 + 11.5000i) q^{43} +(4.22894 + 7.32473i) q^{47} +(17.0584 - 29.5461i) q^{49} -60.5841 q^{53} +(65.9674 + 38.0863i) q^{59} +(-2.67527 - 4.63370i) q^{61} +(95.0039 + 54.8505i) q^{67} -16.0309i q^{71} +4.35053i q^{73} +(2.01678 - 3.49317i) q^{77} +(0.792110 + 1.37197i) q^{79} +(-4.22894 - 7.32473i) q^{83} +64.1236i q^{89} -101.351 q^{91} +(99.7953 - 57.6168i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 72 q^{11} - 4 q^{19} - 126 q^{29} - 14 q^{31} + 36 q^{41} + 102 q^{49} + 252 q^{59} + 82 q^{61} - 166 q^{79} - 604 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1351\) \(2377\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 7.89542 4.55842i 1.12792 0.651203i 0.184507 0.982831i \(-0.440931\pi\)
0.943410 + 0.331628i \(0.107598\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.383156 0.221215i 0.0348324 0.0201105i −0.482483 0.875906i \(-0.660265\pi\)
0.517315 + 0.855795i \(0.326932\pi\)
\(12\) 0 0
\(13\) −9.62747 5.55842i −0.740575 0.427571i 0.0817036 0.996657i \(-0.473964\pi\)
−0.822278 + 0.569086i \(0.807297\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −8.01544 −0.471497 −0.235748 0.971814i \(-0.575754\pi\)
−0.235748 + 0.971814i \(0.575754\pi\)
\(18\) 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
\(23\) −11.8020 + 20.4416i −0.513128 + 0.888764i 0.486756 + 0.873538i \(0.338180\pi\)
−0.999884 + 0.0152262i \(0.995153\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 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
\(35\) 0 0
\(36\) 0 0
\(37\) 18.4674i 0.499118i 0.968360 + 0.249559i \(0.0802857\pi\)
−0.968360 + 0.249559i \(0.919714\pi\)
\(38\) 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
\(43\) −19.9186 + 11.5000i −0.463223 + 0.267442i −0.713398 0.700759i \(-0.752845\pi\)
0.250176 + 0.968200i \(0.419512\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.22894 + 7.32473i 0.0899774 + 0.155845i 0.907501 0.420049i \(-0.137987\pi\)
−0.817524 + 0.575895i \(0.804654\pi\)
\(48\) 0 0
\(49\) 17.0584 29.5461i 0.348131 0.602981i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −60.5841 −1.14310 −0.571548 0.820569i \(-0.693657\pi\)
−0.571548 + 0.820569i \(0.693657\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 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 0.843264 0.537500i \(-0.180631\pi\)
−0.887121 + 0.461538i \(0.847298\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 95.0039 + 54.8505i 1.41797 + 0.818665i 0.996120 0.0880017i \(-0.0280481\pi\)
0.421848 + 0.906666i \(0.361381\pi\)
\(68\) 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
\(73\) 4.35053i 0.0595963i 0.999556 + 0.0297982i \(0.00948645\pi\)
−0.999556 + 0.0297982i \(0.990514\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.01678 3.49317i 0.0261920 0.0453659i
\(78\) 0 0
\(79\) 0.792110 + 1.37197i 0.0100267 + 0.0173668i 0.870995 0.491291i \(-0.163475\pi\)
−0.860969 + 0.508658i \(0.830142\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.22894 7.32473i −0.0509511 0.0882498i 0.839425 0.543475i \(-0.182892\pi\)
−0.890376 + 0.455226i \(0.849559\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 64.1236i 0.720489i 0.932858 + 0.360245i \(0.117307\pi\)
−0.932858 + 0.360245i \(0.882693\pi\)
\(90\) 0 0
\(91\) −101.351 −1.11374
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 99.7953 57.6168i 1.02882 0.593988i 0.112172 0.993689i \(-0.464219\pi\)
0.916646 + 0.399701i \(0.130886\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −114.558 + 66.1403i −1.13424 + 0.654855i −0.944998 0.327075i \(-0.893937\pi\)
−0.189244 + 0.981930i \(0.560604\pi\)
\(102\) 0 0
\(103\) 108.557 + 62.6753i 1.05395 + 0.608498i 0.923752 0.382990i \(-0.125106\pi\)
0.130197 + 0.991488i \(0.458439\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 36.5378 0.341475 0.170737 0.985317i \(-0.445385\pi\)
0.170737 + 0.985317i \(0.445385\pi\)
\(108\) 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
\(113\) 95.1051 164.727i 0.841638 1.45776i −0.0468711 0.998901i \(-0.514925\pi\)
0.888509 0.458859i \(-0.151742\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 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
\(125\) 0 0
\(126\) 0 0
\(127\) 184.103i 1.44963i 0.688943 + 0.724816i \(0.258075\pi\)
−0.688943 + 0.724816i \(0.741925\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 109.194 + 63.0433i 0.833544 + 0.481247i 0.855064 0.518522i \(-0.173517\pi\)
−0.0215207 + 0.999768i \(0.506851\pi\)
\(132\) 0 0
\(133\) 64.0859 37.0000i 0.481849 0.278195i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −62.1326 107.617i −0.453523 0.785524i 0.545079 0.838385i \(-0.316500\pi\)
−0.998602 + 0.0528602i \(0.983166\pi\)
\(138\) 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
\(143\) −4.91843 −0.0343946
\(144\) 0 0
\(145\) 0 0
\(146\) 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
\(155\) 0 0
\(156\) 0 0
\(157\) 60.1487 + 34.7269i 0.383113 + 0.221190i 0.679172 0.733979i \(-0.262339\pi\)
−0.296059 + 0.955170i \(0.595673\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 215.193i 1.33660i
\(162\) 0 0
\(163\) 162.467i 0.996732i −0.866967 0.498366i \(-0.833934\pi\)
0.866967 0.498366i \(-0.166066\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 48.3397 83.7269i 0.289459 0.501358i −0.684221 0.729274i \(-0.739858\pi\)
0.973681 + 0.227916i \(0.0731911\pi\)
\(168\) 0 0
\(169\) −22.7079 39.3312i −0.134366 0.232729i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −14.0141 24.2731i −0.0810064 0.140307i 0.822676 0.568510i \(-0.192480\pi\)
−0.903682 + 0.428203i \(0.859147\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 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 \(-0.517274\pi\)
−0.0542426 + 0.998528i \(0.517274\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3.07117 + 1.77314i −0.0164233 + 0.00948202i
\(188\) 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
\(193\) −42.4352 24.5000i −0.219872 0.126943i 0.386019 0.922491i \(-0.373850\pi\)
−0.605891 + 0.795548i \(0.707183\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 359.965 1.82723 0.913617 0.406575i \(-0.133277\pi\)
0.913617 + 0.406575i \(0.133277\pi\)
\(198\) 0 0
\(199\) 61.0652 0.306861 0.153430 0.988159i \(-0.450968\pi\)
0.153430 + 0.988159i \(0.450968\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −241.647 + 418.545i −1.19038 + 2.06180i
\(204\) 0 0
\(205\) 0 0
\(206\) 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 \(0.536087\pi\)
−0.803903 + 0.594761i \(0.797247\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 267.584i 1.23311i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 77.1684 + 44.5532i 0.349178 + 0.201598i
\(222\) 0 0
\(223\) −88.9864 + 51.3763i −0.399042 + 0.230387i −0.686071 0.727535i \(-0.740666\pi\)
0.287028 + 0.957922i \(0.407333\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 169.482 + 293.552i 0.746617 + 1.29318i 0.949435 + 0.313962i \(0.101657\pi\)
−0.202818 + 0.979216i \(0.565010\pi\)
\(228\) 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
\(233\) 346.537 1.48728 0.743642 0.668578i \(-0.233097\pi\)
0.743642 + 0.668578i \(0.233097\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −140.026 80.8439i −0.585882 0.338259i 0.177586 0.984105i \(-0.443171\pi\)
−0.763468 + 0.645846i \(0.776505\pi\)
\(240\) 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
\(245\) 0 0
\(246\) 0 0
\(247\) −78.1447 45.1168i −0.316375 0.182659i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 384.012i 1.52993i 0.644074 + 0.764963i \(0.277243\pi\)
−0.644074 + 0.764963i \(0.722757\pi\)
\(252\) 0 0
\(253\) 10.4431i 0.0412770i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.46694 + 11.2011i −0.0251632 + 0.0435839i −0.878333 0.478050i \(-0.841344\pi\)
0.853170 + 0.521634i \(0.174677\pi\)
\(258\) 0 0
\(259\) 84.1821 + 145.808i 0.325027 + 0.562964i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −24.7358 42.8437i −0.0940526 0.162904i 0.815160 0.579236i \(-0.196649\pi\)
−0.909213 + 0.416332i \(0.863316\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 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
\(275\) 0 0
\(276\) 0 0
\(277\) −413.487 + 238.727i −1.49273 + 0.861830i −0.999965 0.00833105i \(-0.997348\pi\)
−0.492768 + 0.870161i \(0.664015\pi\)
\(278\) 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
\(283\) −338.920 195.675i −1.19760 0.691432i −0.237577 0.971369i \(-0.576353\pi\)
−0.960018 + 0.279937i \(0.909687\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 410.218 1.42933
\(288\) 0 0
\(289\) −224.753 −0.777691
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −35.8483 + 62.0910i −0.122349 + 0.211915i −0.920694 0.390286i \(-0.872376\pi\)
0.798345 + 0.602201i \(0.205709\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) 172.351i 0.561402i −0.959795 0.280701i \(-0.909433\pi\)
0.959795 0.280701i \(-0.0905670\pi\)
\(308\) 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
\(313\) −283.595 + 163.734i −0.906055 + 0.523111i −0.879160 0.476527i \(-0.841895\pi\)
−0.0268949 + 0.999638i \(0.508562\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 34.5210 + 59.7921i 0.108899 + 0.188619i 0.915325 0.402717i \(-0.131934\pi\)
−0.806425 + 0.591336i \(0.798601\pi\)
\(318\) 0 0
\(319\) −11.7269 + 20.3115i −0.0367613 + 0.0636725i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −65.0601 −0.201424
\(324\) 0 0
\(325\) 0 0
\(326\) 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 \(-0.886885\pi\)
0.167444 0.985882i \(-0.446449\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 292.232 + 168.720i 0.867156 + 0.500653i 0.866402 0.499347i \(-0.166427\pi\)
0.000754096 1.00000i \(0.499760\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 12.9856i 0.0380808i
\(342\) 0 0
\(343\) 135.687i 0.395590i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 107.622 186.407i 0.310151 0.537197i −0.668244 0.743942i \(-0.732954\pi\)
0.978395 + 0.206745i \(0.0662871\pi\)
\(348\) 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
\(353\) −292.420 506.486i −0.828385 1.43481i −0.899304 0.437323i \(-0.855926\pi\)
0.0709189 0.997482i \(-0.477407\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 393.693i 1.09664i 0.836269 + 0.548319i \(0.184732\pi\)
−0.836269 + 0.548319i \(0.815268\pi\)
\(360\) 0 0
\(361\) −295.117 −0.817498
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 329.831 190.428i 0.898722 0.518877i 0.0219364 0.999759i \(-0.493017\pi\)
0.876785 + 0.480882i \(0.159684\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −478.337 + 276.168i −1.28932 + 0.744388i
\(372\) 0 0
\(373\) 115.080 + 66.4416i 0.308526 + 0.178128i 0.646267 0.763112i \(-0.276329\pi\)
−0.337741 + 0.941239i \(0.609663\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 589.316 1.56317
\(378\) 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
\(383\) −165.968 + 287.466i −0.433338 + 0.750564i −0.997158 0.0753339i \(-0.975998\pi\)
0.563820 + 0.825898i \(0.309331\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 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
\(395\) 0 0
\(396\) 0 0
\(397\) 24.8043i 0.0624792i −0.999512 0.0312396i \(-0.990055\pi\)
0.999512 0.0312396i \(-0.00994550\pi\)
\(398\) 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
\(403\) 282.571 163.143i 0.701170 0.404820i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.08526 + 7.07589i 0.0100375 + 0.0173855i
\(408\) 0 0
\(409\) −240.720 + 416.939i −0.588558 + 1.01941i 0.405864 + 0.913933i \(0.366971\pi\)
−0.994422 + 0.105478i \(0.966363\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 694.453 1.68149
\(414\) 0 0
\(415\) 0 0
\(416\) 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 0.998604 0.0528233i \(-0.0168220\pi\)
−0.545048 + 0.838405i \(0.683489\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −42.2447 24.3900i −0.0989337 0.0571194i
\(428\) 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
\(433\) 199.155i 0.459942i 0.973198 + 0.229971i \(0.0738631\pi\)
−0.973198 + 0.229971i \(0.926137\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −95.7946 + 165.921i −0.219210 + 0.379682i
\(438\) 0 0
\(439\) −240.830 417.130i −0.548588 0.950182i −0.998372 0.0570445i \(-0.981832\pi\)
0.449784 0.893137i \(-0.351501\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −270.143 467.902i −0.609805 1.05621i −0.991272 0.131830i \(-0.957915\pi\)
0.381468 0.924382i \(-0.375419\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 300.318i 0.668859i 0.942421 + 0.334429i \(0.108544\pi\)
−0.942421 + 0.334429i \(0.891456\pi\)
\(450\) 0 0
\(451\) 19.9074 0.0441407
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 134.841 77.8505i 0.295057 0.170351i −0.345163 0.938543i \(-0.612176\pi\)
0.640220 + 0.768191i \(0.278843\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −261.143 + 150.771i −0.566470 + 0.327052i −0.755738 0.654874i \(-0.772722\pi\)
0.189268 + 0.981925i \(0.439388\pi\)
\(462\) 0 0
\(463\) 206.790 + 119.390i 0.446630 + 0.257862i 0.706406 0.707807i \(-0.250315\pi\)
−0.259776 + 0.965669i \(0.583649\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −423.152 −0.906107 −0.453054 0.891483i \(-0.649665\pi\)
−0.453054 + 0.891483i \(0.649665\pi\)
\(468\) 0 0
\(469\) 1000.13 2.13247
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5.08795 + 8.81259i −0.0107568 + 0.0186313i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −379.284 + 218.980i −0.791824 + 0.457160i −0.840604 0.541650i \(-0.817800\pi\)
0.0487802 + 0.998810i \(0.484467\pi\)
\(480\) 0 0
\(481\) 102.649 177.794i 0.213408 0.369634i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 401.945i 0.825350i 0.910878 + 0.412675i \(0.135405\pi\)
−0.910878 + 0.412675i \(0.864595\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 241.084 + 139.190i 0.491007 + 0.283483i 0.724992 0.688757i \(-0.241843\pi\)
−0.233985 + 0.972240i \(0.575177\pi\)
\(492\) 0 0
\(493\) 367.981 212.454i 0.746411 0.430941i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −73.0756 126.571i −0.147033 0.254669i
\(498\) 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
\(503\) 306.460 0.609264 0.304632 0.952470i \(-0.401466\pi\)
0.304632 + 0.952470i \(0.401466\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 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
\(515\) 0 0
\(516\) 0 0
\(517\) 3.24069 + 1.87101i 0.00626825 + 0.00361898i
\(518\) 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
\(523\) 480.598i 0.918925i −0.888197 0.459463i \(-0.848042\pi\)
0.888197 0.459463i \(-0.151958\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 117.629 203.739i 0.223204 0.386602i
\(528\) 0 0
\(529\) −14.0721 24.3735i −0.0266013 0.0460748i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −250.105 433.194i −0.469240 0.812747i
\(534\) 0 0
\(535\) 0 0
\(536\) 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 \(-0.589594\pi\)
−0.277766 + 0.960649i \(0.589594\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −87.6473 + 50.6032i −0.160233 + 0.0925104i −0.577972 0.816056i \(-0.696156\pi\)
0.417739 + 0.908567i \(0.362822\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −372.636 + 215.141i −0.676290 + 0.390456i
\(552\) 0 0
\(553\) 12.5081 + 7.22154i 0.0226186 + 0.0130588i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −433.041 −0.777452 −0.388726 0.921353i \(-0.627085\pi\)
−0.388726 + 0.921353i \(0.627085\pi\)
\(558\) 0 0
\(559\) 255.687 0.457401
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −520.886 + 902.201i −0.925197 + 1.60249i −0.133954 + 0.990988i \(0.542767\pi\)
−0.791244 + 0.611501i \(0.790566\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 257.445 148.636i 0.452452 0.261223i −0.256413 0.966567i \(-0.582541\pi\)
0.708865 + 0.705344i \(0.249207\pi\)
\(570\) 0 0
\(571\) 339.524 588.073i 0.594613 1.02990i −0.398988 0.916956i \(-0.630638\pi\)
0.993601 0.112945i \(-0.0360282\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 148.351i 0.257107i −0.991703 0.128553i \(-0.958967\pi\)
0.991703 0.128553i \(-0.0410333\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −66.7785 38.5546i −0.114937 0.0663590i
\(582\) 0 0
\(583\) −23.2132 + 13.4021i −0.0398168 + 0.0229882i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −263.559 456.497i −0.448993 0.777678i 0.549328 0.835607i \(-0.314884\pi\)
−0.998321 + 0.0579287i \(0.981550\pi\)
\(588\) 0 0
\(589\) −119.117 + 206.316i −0.202236 + 0.350283i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 473.848 0.799069 0.399534 0.916718i \(-0.369172\pi\)
0.399534 + 0.916718i \(0.369172\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −601.414 347.227i −1.00403 0.579677i −0.0945922 0.995516i \(-0.530155\pi\)
−0.909438 + 0.415839i \(0.863488\pi\)
\(600\) 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
\(605\) 0 0
\(606\) 0 0
\(607\) 712.209 + 411.194i 1.17333 + 0.677420i 0.954461 0.298335i \(-0.0964311\pi\)
0.218865 + 0.975755i \(0.429764\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 94.0249i 0.153887i
\(612\) 0 0
\(613\) 482.206i 0.786634i 0.919403 + 0.393317i \(0.128672\pi\)
−0.919403 + 0.393317i \(0.871328\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −344.052 + 595.916i −0.557621 + 0.965828i 0.440073 + 0.897962i \(0.354952\pi\)
−0.997694 + 0.0678661i \(0.978381\pi\)
\(618\) 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
\(623\) 292.302 + 506.282i 0.469185 + 0.812652i
\(624\) 0 0
\(625\) 0 0
\(626\) 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
\(635\) 0 0
\(636\) 0 0
\(637\) −328.459 + 189.636i −0.515634 + 0.297701i
\(638\) 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
\(643\) −499.697 288.500i −0.777133 0.448678i 0.0582801 0.998300i \(-0.481438\pi\)
−0.835413 + 0.549622i \(0.814772\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1024.52 −1.58349 −0.791744 0.610853i \(-0.790827\pi\)
−0.791744 + 0.610853i \(0.790827\pi\)
\(648\) 0 0
\(649\) 33.7011 0.0519277
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −199.697 + 345.885i −0.305814 + 0.529686i −0.977442 0.211203i \(-0.932262\pi\)
0.671628 + 0.740888i \(0.265595\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 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 \(0.422319\pi\)
−0.961178 + 0.275931i \(0.911014\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1251.27i 1.87596i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2.05009 1.18362i −0.00305527 0.00176396i
\(672\) 0 0
\(673\) 298.113 172.115i 0.442961 0.255743i −0.261892 0.965097i \(-0.584346\pi\)
0.704853 + 0.709354i \(0.251013\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −492.832 853.610i −0.727965 1.26087i −0.957742 0.287628i \(-0.907133\pi\)
0.229778 0.973243i \(-0.426200\pi\)
\(678\) 0 0
\(679\) 525.284 909.818i 0.773614 1.33994i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −166.658 −0.244009 −0.122004 0.992530i \(-0.538932\pi\)
−0.122004 + 0.992530i \(0.538932\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 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 \(0.0585204\pi\)
−0.333253 + 0.942838i \(0.608146\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −312.341 180.330i −0.448122 0.258723i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 730.549i 1.04215i −0.853510 0.521076i \(-0.825531\pi\)
0.853510 0.521076i \(-0.174469\pi\)
\(702\) 0 0
\(703\) 149.897i 0.213224i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −602.991 + 1044.41i −0.852887 + 1.47724i
\(708\) 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
\(713\) −346.394 599.971i −0.485825 0.841474i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 907.095i 1.26161i −0.775943 0.630803i \(-0.782725\pi\)
0.775943 0.630803i \(-0.217275\pi\)
\(720\) 0 0
\(721\) 1142.80 1.58502
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 186.838 107.871i 0.256999 0.148378i −0.365966 0.930628i \(-0.619261\pi\)
0.622965 + 0.782250i \(0.285928\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 159.656 92.1776i 0.218408 0.126098i
\(732\) 0 0
\(733\) 544.963 + 314.634i 0.743469 + 0.429242i 0.823329 0.567564i \(-0.192114\pi\)
−0.0798604 + 0.996806i \(0.525447\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 48.5351 0.0658549
\(738\) 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
\(743\) 267.072 462.583i 0.359451 0.622588i −0.628418 0.777876i \(-0.716297\pi\)
0.987869 + 0.155288i \(0.0496306\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 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
\(755\) 0 0
\(756\) 0 0
\(757\) 352.391i 0.465511i −0.972535 0.232755i \(-0.925226\pi\)
0.972535 0.232755i \(-0.0747741\pi\)
\(758\) 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
\(763\) −1063.52 + 614.024i −1.39387 + 0.804750i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −423.399 733.349i −0.552020 0.956126i
\(768\) 0 0
\(769\) 177.988 308.284i 0.231454 0.400889i −0.726782 0.686868i \(-0.758985\pi\)
0.958236 + 0.285978i \(0.0923185\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −370.790 −0.479677 −0.239838 0.970813i \(-0.577094\pi\)
−0.239838 + 0.970813i \(0.577094\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 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
\(785\) 0 0
\(786\) 0 0
\(787\) −998.581 576.531i −1.26885 0.732568i −0.294076 0.955782i \(-0.595012\pi\)
−0.974769 + 0.223214i \(0.928345\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1734.12i 2.19231i
\(792\) 0 0
\(793\) 59.4810i 0.0750076i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −439.873 + 761.882i −0.551910 + 0.955937i 0.446226 + 0.894920i \(0.352768\pi\)
−0.998137 + 0.0610167i \(0.980566\pi\)
\(798\) 0 0
\(799\) −33.8968 58.7110i −0.0424240 0.0734806i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0.962404 + 1.66693i 0.00119851 + 0.00207588i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 884.508i 1.09334i 0.837350 + 0.546668i \(0.184104\pi\)
−0.837350 + 0.546668i \(0.815896\pi\)
\(810\) 0 0
\(811\) −961.464 −1.18553 −0.592765 0.805376i \(-0.701964\pi\)
−0.592765 + 0.805376i \(0.701964\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −161.676 + 93.3437i −0.197890 + 0.114252i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −778.064 + 449.215i −0.947702 + 0.547156i −0.892366 0.451312i \(-0.850956\pi\)
−0.0553360 + 0.998468i \(0.517623\pi\)
\(822\) 0 0
\(823\) 187.219 + 108.091i 0.227484 + 0.131338i 0.609411 0.792855i \(-0.291406\pi\)
−0.381927 + 0.924193i \(0.624739\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 113.883 0.137706 0.0688528 0.997627i \(-0.478066\pi\)
0.0688528 + 0.997627i \(0.478066\pi\)
\(828\) 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
\(833\) −136.731 + 236.825i −0.164143 + 0.284303i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 60.4689 34.9117i 0.0720726 0.0416111i −0.463531 0.886081i \(-0.653418\pi\)
0.535603 + 0.844470i \(0.320084\pi\)
\(840\) 0 0
\(841\) 984.588 1705.36i 1.17073 2.02777i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1101.35i 1.30030i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −377.502 217.951i −0.443598 0.256112i
\(852\) 0 0
\(853\) 993.028 573.325i 1.16416 0.672127i 0.211862 0.977300i \(-0.432047\pi\)
0.952297 + 0.305172i \(0.0987140\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −125.788 217.871i −0.146777 0.254225i 0.783258 0.621697i \(-0.213557\pi\)
−0.930035 + 0.367472i \(0.880223\pi\)
\(858\) 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
\(863\) 596.889 0.691644 0.345822 0.938300i \(-0.387600\pi\)
0.345822 + 0.938300i \(0.387600\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 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
\(875\) 0 0
\(876\) 0 0
\(877\) −620.434 358.208i −0.707451 0.408447i 0.102666 0.994716i \(-0.467263\pi\)
−0.810116 + 0.586269i \(0.800596\pi\)
\(878\) 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
\(883\) 22.8938i 0.0259273i −0.999916 0.0129636i \(-0.995873\pi\)
0.999916 0.0129636i \(-0.00412657\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 301.294 521.857i 0.339678 0.588340i −0.644694 0.764441i \(-0.723015\pi\)
0.984372 + 0.176101i \(0.0563485\pi\)
\(888\) 0 0
\(889\) 839.220 + 1453.57i 0.944005 + 1.63506i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 34.3256 + 59.4537i 0.0384385 + 0.0665775i
\(894\) 0 0
\(895\) 0 0
\(896\) 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
\(905\) 0 0
\(906\) 0 0
\(907\) 366.281 211.473i 0.403838 0.233156i −0.284300 0.958735i \(-0.591761\pi\)
0.688139 + 0.725579i \(0.258428\pi\)
\(908\) 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
\(913\) −3.24069 1.87101i −0.00354949 0.00204930i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1149.51 1.25356
\(918\) 0 0
\(919\) 869.093 0.945694 0.472847 0.881145i \(-0.343226\pi\)
0.472847 + 0.881145i \(0.343226\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −89.1064 + 154.337i −0.0965400 + 0.167212i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 82.2838 47.5066i 0.0885724 0.0511373i −0.455060 0.890461i \(-0.650382\pi\)
0.543632 + 0.839324i \(0.317049\pi\)
\(930\) 0 0
\(931\) 138.461 239.821i 0.148722 0.257595i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1555.55i 1.66014i −0.557657 0.830071i \(-0.688300\pi\)
0.557657 0.830071i \(-0.311700\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −335.246 193.554i −0.356265 0.205690i 0.311176 0.950352i \(-0.399277\pi\)
−0.667441 + 0.744662i \(0.732611\pi\)
\(942\) 0 0
\(943\) −919.782 + 531.036i −0.975379 + 0.563135i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 484.894 + 839.861i 0.512032 + 0.886865i 0.999903 + 0.0139492i \(0.00444033\pi\)
−0.487871 + 0.872916i \(0.662226\pi\)
\(948\) 0 0
\(949\) 24.1821 41.8846i 0.0254817 0.0441355i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1294.65 1.35850 0.679248 0.733909i \(-0.262306\pi\)
0.679248 + 0.733909i \(0.262306\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 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
\(965\) 0 0
\(966\) 0 0
\(967\) −713.668 412.036i −0.738023 0.426098i 0.0833272 0.996522i \(-0.473445\pi\)
−0.821350 + 0.570425i \(0.806779\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1518.35i 1.56370i −0.623469 0.781848i \(-0.714277\pi\)
0.623469 0.781848i \(-0.285723\pi\)
\(972\) 0 0
\(973\) 244.024i 0.250796i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −185.616 + 321.497i −0.189986 + 0.329065i −0.945245 0.326361i \(-0.894178\pi\)
0.755259 + 0.655426i \(0.227511\pi\)
\(978\) 0 0
\(979\) 14.1851 + 24.5693i 0.0144894 + 0.0250963i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −466.522 808.039i −0.474590 0.822014i 0.524987 0.851110i \(-0.324070\pi\)
−0.999577 + 0.0290967i \(0.990737\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 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
\(995\) 0 0
\(996\) 0 0
\(997\) 899.586 519.376i 0.902293 0.520939i 0.0243496 0.999704i \(-0.492249\pi\)
0.877943 + 0.478764i \(0.158915\pi\)
\(998\) 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 2700.3.u.b.449.4 8
3.2 odd 2 900.3.u.a.149.4 8
5.2 odd 4 2700.3.p.b.2501.2 4
5.3 odd 4 108.3.g.a.17.1 4
5.4 even 2 inner 2700.3.u.b.449.1 8
9.2 odd 6 inner 2700.3.u.b.2249.1 8
9.7 even 3 900.3.u.a.749.1 8
15.2 even 4 900.3.p.a.401.2 4
15.8 even 4 36.3.g.a.5.1 4
15.14 odd 2 900.3.u.a.149.1 8
20.3 even 4 432.3.q.b.17.1 4
40.3 even 4 1728.3.q.h.449.2 4
40.13 odd 4 1728.3.q.g.449.2 4
45.2 even 12 2700.3.p.b.1601.2 4
45.7 odd 12 900.3.p.a.101.2 4
45.13 odd 12 324.3.c.b.161.4 4
45.23 even 12 324.3.c.b.161.1 4
45.29 odd 6 inner 2700.3.u.b.2249.4 8
45.34 even 6 900.3.u.a.749.4 8
45.38 even 12 108.3.g.a.89.1 4
45.43 odd 12 36.3.g.a.29.1 yes 4
60.23 odd 4 144.3.q.b.113.2 4
120.53 even 4 576.3.q.d.257.2 4
120.83 odd 4 576.3.q.g.257.1 4
180.23 odd 12 1296.3.e.e.161.1 4
180.43 even 12 144.3.q.b.65.2 4
180.83 odd 12 432.3.q.b.305.1 4
180.103 even 12 1296.3.e.e.161.4 4
360.43 even 12 576.3.q.g.65.1 4
360.83 odd 12 1728.3.q.h.1601.2 4
360.133 odd 12 576.3.q.d.65.2 4
360.173 even 12 1728.3.q.g.1601.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.3.g.a.5.1 4 15.8 even 4
36.3.g.a.29.1 yes 4 45.43 odd 12
108.3.g.a.17.1 4 5.3 odd 4
108.3.g.a.89.1 4 45.38 even 12
144.3.q.b.65.2 4 180.43 even 12
144.3.q.b.113.2 4 60.23 odd 4
324.3.c.b.161.1 4 45.23 even 12
324.3.c.b.161.4 4 45.13 odd 12
432.3.q.b.17.1 4 20.3 even 4
432.3.q.b.305.1 4 180.83 odd 12
576.3.q.d.65.2 4 360.133 odd 12
576.3.q.d.257.2 4 120.53 even 4
576.3.q.g.65.1 4 360.43 even 12
576.3.q.g.257.1 4 120.83 odd 4
900.3.p.a.101.2 4 45.7 odd 12
900.3.p.a.401.2 4 15.2 even 4
900.3.u.a.149.1 8 15.14 odd 2
900.3.u.a.149.4 8 3.2 odd 2
900.3.u.a.749.1 8 9.7 even 3
900.3.u.a.749.4 8 45.34 even 6
1296.3.e.e.161.1 4 180.23 odd 12
1296.3.e.e.161.4 4 180.103 even 12
1728.3.q.g.449.2 4 40.13 odd 4
1728.3.q.g.1601.2 4 360.173 even 12
1728.3.q.h.449.2 4 40.3 even 4
1728.3.q.h.1601.2 4 360.83 odd 12
2700.3.p.b.1601.2 4 45.2 even 12
2700.3.p.b.2501.2 4 5.2 odd 4
2700.3.u.b.449.1 8 5.4 even 2 inner
2700.3.u.b.449.4 8 1.1 even 1 trivial
2700.3.u.b.2249.1 8 9.2 odd 6 inner
2700.3.u.b.2249.4 8 45.29 odd 6 inner