Properties

Label 900.3.u.c.149.5
Level $900$
Weight $3$
Character 900.149
Analytic conductor $24.523$
Analytic rank $0$
Dimension $24$
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] = [900,3,Mod(149,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, 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("900.149");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.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: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 149.5
Character \(\chi\) \(=\) 900.149
Dual form 900.3.u.c.749.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.783177 - 2.89597i) q^{3} +(4.08158 - 2.35650i) q^{7} +(-7.77327 + 4.53611i) q^{9} +O(q^{10})\) \(q+(-0.783177 - 2.89597i) q^{3} +(4.08158 - 2.35650i) q^{7} +(-7.77327 + 4.53611i) q^{9} +(16.6882 - 9.63493i) q^{11} +(-8.11434 - 4.68482i) q^{13} -16.2489 q^{17} +19.0505 q^{19} +(-10.0210 - 9.97458i) q^{21} +(8.30327 - 14.3817i) q^{23} +(19.2243 + 18.9586i) q^{27} +(3.87581 - 2.23770i) q^{29} +(-7.19383 + 12.4601i) q^{31} +(-40.9723 - 40.7826i) q^{33} -55.6092i q^{37} +(-7.21211 + 27.1679i) q^{39} +(26.7067 + 15.4191i) q^{41} +(37.3322 - 21.5538i) q^{43} +(-20.7446 - 35.9308i) q^{47} +(-13.3938 + 23.1987i) q^{49} +(12.7257 + 47.0562i) q^{51} +30.7037 q^{53} +(-14.9199 - 55.1696i) q^{57} +(-74.4236 - 42.9685i) q^{59} +(-51.7462 - 89.6270i) q^{61} +(-21.0379 + 36.8323i) q^{63} +(-107.020 - 61.7879i) q^{67} +(-48.1518 - 12.7826i) q^{69} +111.409i q^{71} +90.5781i q^{73} +(45.4095 - 78.6515i) q^{77} +(-41.6569 - 72.1519i) q^{79} +(39.8473 - 70.5208i) q^{81} +(-17.9251 - 31.0471i) q^{83} +(-9.51576 - 9.47171i) q^{87} +84.9144i q^{89} -44.1591 q^{91} +(41.7181 + 11.0747i) q^{93} +(-91.1508 + 52.6259i) q^{97} +(-86.0166 + 150.594i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 52 q^{9} + 96 q^{11} - 144 q^{19} - 256 q^{21} - 300 q^{29} - 24 q^{31} - 80 q^{39} + 180 q^{41} - 96 q^{49} - 288 q^{51} - 96 q^{59} - 156 q^{61} + 300 q^{69} - 240 q^{79} + 868 q^{81} + 240 q^{91} + 240 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\)

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
\(3\) −0.783177 2.89597i −0.261059 0.965323i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.08158 2.35650i 0.583083 0.336643i −0.179274 0.983799i \(-0.557375\pi\)
0.762358 + 0.647156i \(0.224042\pi\)
\(8\) 0 0
\(9\) −7.77327 + 4.53611i −0.863696 + 0.504013i
\(10\) 0 0
\(11\) 16.6882 9.63493i 1.51711 0.875902i 0.517310 0.855798i \(-0.326934\pi\)
0.999798 0.0201041i \(-0.00639976\pi\)
\(12\) 0 0
\(13\) −8.11434 4.68482i −0.624180 0.360370i 0.154315 0.988022i \(-0.450683\pi\)
−0.778495 + 0.627651i \(0.784016\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −16.2489 −0.955816 −0.477908 0.878410i \(-0.658605\pi\)
−0.477908 + 0.878410i \(0.658605\pi\)
\(18\) 0 0
\(19\) 19.0505 1.00266 0.501328 0.865257i \(-0.332845\pi\)
0.501328 + 0.865257i \(0.332845\pi\)
\(20\) 0 0
\(21\) −10.0210 9.97458i −0.477189 0.474980i
\(22\) 0 0
\(23\) 8.30327 14.3817i 0.361012 0.625290i −0.627116 0.778926i \(-0.715765\pi\)
0.988128 + 0.153635i \(0.0490981\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 19.2243 + 18.9586i 0.712011 + 0.702169i
\(28\) 0 0
\(29\) 3.87581 2.23770i 0.133649 0.0771621i −0.431685 0.902024i \(-0.642081\pi\)
0.565334 + 0.824862i \(0.308747\pi\)
\(30\) 0 0
\(31\) −7.19383 + 12.4601i −0.232059 + 0.401938i −0.958414 0.285382i \(-0.907880\pi\)
0.726355 + 0.687320i \(0.241213\pi\)
\(32\) 0 0
\(33\) −40.9723 40.7826i −1.24158 1.23584i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 55.6092i 1.50295i −0.659761 0.751476i \(-0.729342\pi\)
0.659761 0.751476i \(-0.270658\pi\)
\(38\) 0 0
\(39\) −7.21211 + 27.1679i −0.184926 + 0.696613i
\(40\) 0 0
\(41\) 26.7067 + 15.4191i 0.651384 + 0.376077i 0.788986 0.614411i \(-0.210606\pi\)
−0.137602 + 0.990488i \(0.543940\pi\)
\(42\) 0 0
\(43\) 37.3322 21.5538i 0.868191 0.501250i 0.00144406 0.999999i \(-0.499540\pi\)
0.866747 + 0.498749i \(0.166207\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −20.7446 35.9308i −0.441375 0.764484i 0.556417 0.830903i \(-0.312176\pi\)
−0.997792 + 0.0664192i \(0.978843\pi\)
\(48\) 0 0
\(49\) −13.3938 + 23.1987i −0.273343 + 0.473443i
\(50\) 0 0
\(51\) 12.7257 + 47.0562i 0.249524 + 0.922671i
\(52\) 0 0
\(53\) 30.7037 0.579316 0.289658 0.957130i \(-0.406458\pi\)
0.289658 + 0.957130i \(0.406458\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −14.9199 55.1696i −0.261753 0.967887i
\(58\) 0 0
\(59\) −74.4236 42.9685i −1.26142 0.728279i −0.288068 0.957610i \(-0.593013\pi\)
−0.973348 + 0.229331i \(0.926346\pi\)
\(60\) 0 0
\(61\) −51.7462 89.6270i −0.848298 1.46930i −0.882726 0.469889i \(-0.844294\pi\)
0.0344274 0.999407i \(-0.489039\pi\)
\(62\) 0 0
\(63\) −21.0379 + 36.8323i −0.333934 + 0.584639i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −107.020 61.7879i −1.59731 0.922207i −0.992003 0.126215i \(-0.959717\pi\)
−0.605307 0.795992i \(-0.706950\pi\)
\(68\) 0 0
\(69\) −48.1518 12.7826i −0.697853 0.185255i
\(70\) 0 0
\(71\) 111.409i 1.56914i 0.620043 + 0.784568i \(0.287115\pi\)
−0.620043 + 0.784568i \(0.712885\pi\)
\(72\) 0 0
\(73\) 90.5781i 1.24080i 0.784287 + 0.620398i \(0.213029\pi\)
−0.784287 + 0.620398i \(0.786971\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 45.4095 78.6515i 0.589733 1.02145i
\(78\) 0 0
\(79\) −41.6569 72.1519i −0.527302 0.913315i −0.999494 0.0318185i \(-0.989870\pi\)
0.472191 0.881496i \(-0.343463\pi\)
\(80\) 0 0
\(81\) 39.8473 70.5208i 0.491942 0.870628i
\(82\) 0 0
\(83\) −17.9251 31.0471i −0.215965 0.374062i 0.737606 0.675231i \(-0.235956\pi\)
−0.953571 + 0.301170i \(0.902623\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −9.51576 9.47171i −0.109377 0.108870i
\(88\) 0 0
\(89\) 84.9144i 0.954095i 0.878878 + 0.477047i \(0.158293\pi\)
−0.878878 + 0.477047i \(0.841707\pi\)
\(90\) 0 0
\(91\) −44.1591 −0.485265
\(92\) 0 0
\(93\) 41.7181 + 11.0747i 0.448581 + 0.119082i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −91.1508 + 52.6259i −0.939699 + 0.542535i −0.889866 0.456222i \(-0.849202\pi\)
−0.0498329 + 0.998758i \(0.515869\pi\)
\(98\) 0 0
\(99\) −86.0166 + 150.594i −0.868854 + 1.52116i
\(100\) 0 0
\(101\) 69.4641 40.1051i 0.687763 0.397080i −0.115010 0.993364i \(-0.536690\pi\)
0.802773 + 0.596284i \(0.203357\pi\)
\(102\) 0 0
\(103\) −112.947 65.2103i −1.09658 0.633109i −0.161258 0.986912i \(-0.551555\pi\)
−0.935320 + 0.353803i \(0.884888\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 49.8390 0.465785 0.232893 0.972502i \(-0.425181\pi\)
0.232893 + 0.972502i \(0.425181\pi\)
\(108\) 0 0
\(109\) −121.431 −1.11404 −0.557022 0.830498i \(-0.688056\pi\)
−0.557022 + 0.830498i \(0.688056\pi\)
\(110\) 0 0
\(111\) −161.043 + 43.5519i −1.45083 + 0.392359i
\(112\) 0 0
\(113\) −14.1371 + 24.4862i −0.125107 + 0.216692i −0.921775 0.387725i \(-0.873261\pi\)
0.796668 + 0.604418i \(0.206594\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 84.3258 0.391246i 0.720733 0.00334398i
\(118\) 0 0
\(119\) −66.3211 + 38.2905i −0.557320 + 0.321769i
\(120\) 0 0
\(121\) 125.164 216.790i 1.03441 1.79165i
\(122\) 0 0
\(123\) 23.7372 89.4178i 0.192986 0.726974i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 44.8753i 0.353349i −0.984269 0.176674i \(-0.943466\pi\)
0.984269 0.176674i \(-0.0565339\pi\)
\(128\) 0 0
\(129\) −91.6567 91.2324i −0.710517 0.707228i
\(130\) 0 0
\(131\) 63.7326 + 36.7960i 0.486508 + 0.280886i 0.723125 0.690717i \(-0.242705\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(132\) 0 0
\(133\) 77.7561 44.8925i 0.584632 0.337538i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −109.091 188.952i −0.796286 1.37921i −0.922019 0.387144i \(-0.873461\pi\)
0.125733 0.992064i \(-0.459872\pi\)
\(138\) 0 0
\(139\) −33.0004 + 57.1584i −0.237413 + 0.411212i −0.959971 0.280099i \(-0.909633\pi\)
0.722558 + 0.691310i \(0.242966\pi\)
\(140\) 0 0
\(141\) −87.8076 + 88.2160i −0.622749 + 0.625645i
\(142\) 0 0
\(143\) −180.551 −1.26260
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 77.6725 + 20.6193i 0.528384 + 0.140267i
\(148\) 0 0
\(149\) 169.401 + 97.8035i 1.13692 + 0.656399i 0.945665 0.325142i \(-0.105412\pi\)
0.191252 + 0.981541i \(0.438745\pi\)
\(150\) 0 0
\(151\) 79.5523 + 137.789i 0.526836 + 0.912507i 0.999511 + 0.0312701i \(0.00995522\pi\)
−0.472675 + 0.881237i \(0.656711\pi\)
\(152\) 0 0
\(153\) 126.307 73.7067i 0.825534 0.481743i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 111.185 + 64.1927i 0.708185 + 0.408871i 0.810389 0.585893i \(-0.199256\pi\)
−0.102204 + 0.994764i \(0.532589\pi\)
\(158\) 0 0
\(159\) −24.0465 88.9170i −0.151236 0.559227i
\(160\) 0 0
\(161\) 78.2667i 0.486129i
\(162\) 0 0
\(163\) 21.6963i 0.133106i −0.997783 0.0665531i \(-0.978800\pi\)
0.997783 0.0665531i \(-0.0212002\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 126.957 219.896i 0.760223 1.31674i −0.182513 0.983204i \(-0.558423\pi\)
0.942736 0.333541i \(-0.108244\pi\)
\(168\) 0 0
\(169\) −40.6050 70.3299i −0.240266 0.416153i
\(170\) 0 0
\(171\) −148.084 + 86.4151i −0.865991 + 0.505352i
\(172\) 0 0
\(173\) 29.6768 + 51.4018i 0.171542 + 0.297120i 0.938959 0.344028i \(-0.111792\pi\)
−0.767417 + 0.641148i \(0.778458\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −66.1485 + 249.180i −0.373720 + 1.40780i
\(178\) 0 0
\(179\) 5.43887i 0.0303848i −0.999885 0.0151924i \(-0.995164\pi\)
0.999885 0.0151924i \(-0.00483607\pi\)
\(180\) 0 0
\(181\) 45.4874 0.251312 0.125656 0.992074i \(-0.459896\pi\)
0.125656 + 0.992074i \(0.459896\pi\)
\(182\) 0 0
\(183\) −219.031 + 220.049i −1.19689 + 1.20245i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −271.164 + 156.557i −1.45007 + 0.837201i
\(188\) 0 0
\(189\) 123.141 + 32.0788i 0.651542 + 0.169729i
\(190\) 0 0
\(191\) 199.987 115.463i 1.04705 0.604516i 0.125230 0.992128i \(-0.460033\pi\)
0.921823 + 0.387612i \(0.126700\pi\)
\(192\) 0 0
\(193\) 63.1231 + 36.4441i 0.327063 + 0.188830i 0.654536 0.756031i \(-0.272864\pi\)
−0.327474 + 0.944860i \(0.606197\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 39.2251 0.199112 0.0995561 0.995032i \(-0.468258\pi\)
0.0995561 + 0.995032i \(0.468258\pi\)
\(198\) 0 0
\(199\) 268.068 1.34707 0.673537 0.739154i \(-0.264774\pi\)
0.673537 + 0.739154i \(0.264774\pi\)
\(200\) 0 0
\(201\) −95.1203 + 358.317i −0.473235 + 1.78267i
\(202\) 0 0
\(203\) 10.5463 18.2667i 0.0519522 0.0899838i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.693436 + 149.457i 0.00334993 + 0.722015i
\(208\) 0 0
\(209\) 317.918 183.550i 1.52114 0.878229i
\(210\) 0 0
\(211\) 173.445 300.416i 0.822015 1.42377i −0.0821652 0.996619i \(-0.526184\pi\)
0.904180 0.427152i \(-0.140483\pi\)
\(212\) 0 0
\(213\) 322.636 87.2527i 1.51472 0.409637i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 67.8092i 0.312485i
\(218\) 0 0
\(219\) 262.311 70.9387i 1.19777 0.323921i
\(220\) 0 0
\(221\) 131.849 + 76.1229i 0.596601 + 0.344448i
\(222\) 0 0
\(223\) 61.2866 35.3838i 0.274828 0.158672i −0.356252 0.934390i \(-0.615945\pi\)
0.631080 + 0.775718i \(0.282612\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 134.429 + 232.838i 0.592199 + 1.02572i 0.993936 + 0.109963i \(0.0350733\pi\)
−0.401737 + 0.915755i \(0.631593\pi\)
\(228\) 0 0
\(229\) 103.497 179.261i 0.451950 0.782801i −0.546557 0.837422i \(-0.684062\pi\)
0.998507 + 0.0546210i \(0.0173951\pi\)
\(230\) 0 0
\(231\) −263.336 69.9063i −1.13998 0.302625i
\(232\) 0 0
\(233\) 47.4893 0.203817 0.101908 0.994794i \(-0.467505\pi\)
0.101908 + 0.994794i \(0.467505\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −176.325 + 177.145i −0.743986 + 0.747446i
\(238\) 0 0
\(239\) 288.824 + 166.753i 1.20847 + 0.697710i 0.962425 0.271548i \(-0.0875356\pi\)
0.246045 + 0.969258i \(0.420869\pi\)
\(240\) 0 0
\(241\) −213.356 369.544i −0.885296 1.53338i −0.845374 0.534176i \(-0.820622\pi\)
−0.0399229 0.999203i \(-0.512711\pi\)
\(242\) 0 0
\(243\) −235.434 60.1663i −0.968863 0.247598i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −154.582 89.2480i −0.625838 0.361328i
\(248\) 0 0
\(249\) −75.8730 + 76.2258i −0.304711 + 0.306128i
\(250\) 0 0
\(251\) 394.340i 1.57107i 0.618814 + 0.785537i \(0.287613\pi\)
−0.618814 + 0.785537i \(0.712387\pi\)
\(252\) 0 0
\(253\) 320.005i 1.26484i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 181.057 313.600i 0.704503 1.22023i −0.262368 0.964968i \(-0.584503\pi\)
0.966871 0.255267i \(-0.0821634\pi\)
\(258\) 0 0
\(259\) −131.043 226.974i −0.505959 0.876346i
\(260\) 0 0
\(261\) −19.9772 + 34.9754i −0.0765412 + 0.134005i
\(262\) 0 0
\(263\) 169.779 + 294.065i 0.645546 + 1.11812i 0.984175 + 0.177198i \(0.0567033\pi\)
−0.338630 + 0.940920i \(0.609963\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 245.910 66.5031i 0.921010 0.249075i
\(268\) 0 0
\(269\) 23.2060i 0.0862678i −0.999069 0.0431339i \(-0.986266\pi\)
0.999069 0.0431339i \(-0.0137342\pi\)
\(270\) 0 0
\(271\) −433.939 −1.60125 −0.800626 0.599165i \(-0.795499\pi\)
−0.800626 + 0.599165i \(0.795499\pi\)
\(272\) 0 0
\(273\) 34.5844 + 127.883i 0.126683 + 0.468438i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −295.704 + 170.725i −1.06752 + 0.616336i −0.927504 0.373813i \(-0.878050\pi\)
−0.140021 + 0.990149i \(0.544717\pi\)
\(278\) 0 0
\(279\) −0.600783 129.488i −0.00215334 0.464113i
\(280\) 0 0
\(281\) 175.833 101.517i 0.625741 0.361272i −0.153360 0.988170i \(-0.549009\pi\)
0.779101 + 0.626899i \(0.215676\pi\)
\(282\) 0 0
\(283\) 135.815 + 78.4126i 0.479910 + 0.277076i 0.720379 0.693581i \(-0.243968\pi\)
−0.240469 + 0.970657i \(0.577301\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 145.341 0.506415
\(288\) 0 0
\(289\) −24.9744 −0.0864167
\(290\) 0 0
\(291\) 223.790 + 222.754i 0.769039 + 0.765479i
\(292\) 0 0
\(293\) 31.7203 54.9412i 0.108260 0.187513i −0.806805 0.590818i \(-0.798805\pi\)
0.915066 + 0.403305i \(0.132139\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 503.483 + 131.159i 1.69523 + 0.441613i
\(298\) 0 0
\(299\) −134.751 + 77.7986i −0.450672 + 0.260196i
\(300\) 0 0
\(301\) 101.583 175.947i 0.337485 0.584541i
\(302\) 0 0
\(303\) −170.546 169.756i −0.562857 0.560252i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 183.760i 0.598568i 0.954164 + 0.299284i \(0.0967478\pi\)
−0.954164 + 0.299284i \(0.903252\pi\)
\(308\) 0 0
\(309\) −100.389 + 378.164i −0.324883 + 1.22383i
\(310\) 0 0
\(311\) 328.599 + 189.717i 1.05659 + 0.610021i 0.924486 0.381215i \(-0.124494\pi\)
0.132101 + 0.991236i \(0.457828\pi\)
\(312\) 0 0
\(313\) −353.868 + 204.306i −1.13057 + 0.652734i −0.944078 0.329723i \(-0.893045\pi\)
−0.186491 + 0.982457i \(0.559711\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 259.421 + 449.330i 0.818362 + 1.41744i 0.906889 + 0.421370i \(0.138451\pi\)
−0.0885271 + 0.996074i \(0.528216\pi\)
\(318\) 0 0
\(319\) 43.1202 74.6863i 0.135173 0.234126i
\(320\) 0 0
\(321\) −39.0328 144.332i −0.121597 0.449633i
\(322\) 0 0
\(323\) −309.549 −0.958355
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 95.1019 + 351.660i 0.290831 + 1.07541i
\(328\) 0 0
\(329\) −169.342 97.7696i −0.514717 0.297172i
\(330\) 0 0
\(331\) −147.622 255.689i −0.445988 0.772474i 0.552133 0.833756i \(-0.313814\pi\)
−0.998120 + 0.0612827i \(0.980481\pi\)
\(332\) 0 0
\(333\) 252.250 + 432.265i 0.757507 + 1.29809i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 93.3245 + 53.8810i 0.276927 + 0.159884i 0.632032 0.774943i \(-0.282221\pi\)
−0.355104 + 0.934827i \(0.615555\pi\)
\(338\) 0 0
\(339\) 81.9833 + 21.7636i 0.241839 + 0.0641995i
\(340\) 0 0
\(341\) 277.248i 0.813045i
\(342\) 0 0
\(343\) 357.187i 1.04136i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −220.726 + 382.308i −0.636097 + 1.10175i 0.350185 + 0.936681i \(0.386119\pi\)
−0.986282 + 0.165071i \(0.947215\pi\)
\(348\) 0 0
\(349\) 94.7382 + 164.091i 0.271456 + 0.470176i 0.969235 0.246138i \(-0.0791615\pi\)
−0.697779 + 0.716313i \(0.745828\pi\)
\(350\) 0 0
\(351\) −67.1751 243.898i −0.191382 0.694867i
\(352\) 0 0
\(353\) 251.430 + 435.490i 0.712267 + 1.23368i 0.964004 + 0.265887i \(0.0856649\pi\)
−0.251737 + 0.967796i \(0.581002\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 162.829 + 162.076i 0.456104 + 0.453993i
\(358\) 0 0
\(359\) 2.93084i 0.00816391i 0.999992 + 0.00408196i \(0.00129933\pi\)
−0.999992 + 0.00408196i \(0.998701\pi\)
\(360\) 0 0
\(361\) 1.92045 0.00531981
\(362\) 0 0
\(363\) −725.842 192.685i −1.99956 0.530813i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 47.4055 27.3696i 0.129170 0.0745765i −0.434023 0.900902i \(-0.642906\pi\)
0.563193 + 0.826326i \(0.309573\pi\)
\(368\) 0 0
\(369\) −277.542 + 1.28771i −0.752145 + 0.00348973i
\(370\) 0 0
\(371\) 125.320 72.3534i 0.337789 0.195023i
\(372\) 0 0
\(373\) 586.707 + 338.735i 1.57294 + 0.908138i 0.995806 + 0.0914894i \(0.0291627\pi\)
0.577135 + 0.816649i \(0.304171\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −41.9329 −0.111228
\(378\) 0 0
\(379\) 431.714 1.13909 0.569543 0.821961i \(-0.307120\pi\)
0.569543 + 0.821961i \(0.307120\pi\)
\(380\) 0 0
\(381\) −129.957 + 35.1453i −0.341095 + 0.0922449i
\(382\) 0 0
\(383\) 292.269 506.226i 0.763106 1.32174i −0.178137 0.984006i \(-0.557007\pi\)
0.941242 0.337732i \(-0.109660\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −192.423 + 336.886i −0.497217 + 0.870507i
\(388\) 0 0
\(389\) 49.9565 28.8424i 0.128423 0.0741450i −0.434412 0.900714i \(-0.643044\pi\)
0.562835 + 0.826569i \(0.309711\pi\)
\(390\) 0 0
\(391\) −134.919 + 233.686i −0.345061 + 0.597662i
\(392\) 0 0
\(393\) 56.6462 213.385i 0.144138 0.542965i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 25.9853i 0.0654542i 0.999464 + 0.0327271i \(0.0104192\pi\)
−0.999464 + 0.0327271i \(0.989581\pi\)
\(398\) 0 0
\(399\) −190.904 190.020i −0.478456 0.476242i
\(400\) 0 0
\(401\) −229.069 132.253i −0.571244 0.329808i 0.186402 0.982474i \(-0.440317\pi\)
−0.757646 + 0.652666i \(0.773651\pi\)
\(402\) 0 0
\(403\) 116.746 67.4036i 0.289693 0.167255i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −535.791 928.017i −1.31644 2.28014i
\(408\) 0 0
\(409\) 154.678 267.911i 0.378187 0.655039i −0.612612 0.790384i \(-0.709881\pi\)
0.990798 + 0.135345i \(0.0432144\pi\)
\(410\) 0 0
\(411\) −461.760 + 463.907i −1.12350 + 1.12873i
\(412\) 0 0
\(413\) −405.021 −0.980681
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 191.374 + 50.8030i 0.458931 + 0.121830i
\(418\) 0 0
\(419\) 58.4329 + 33.7363i 0.139458 + 0.0805161i 0.568106 0.822956i \(-0.307676\pi\)
−0.428648 + 0.903472i \(0.641010\pi\)
\(420\) 0 0
\(421\) −101.577 175.936i −0.241275 0.417901i 0.719803 0.694179i \(-0.244232\pi\)
−0.961078 + 0.276278i \(0.910899\pi\)
\(422\) 0 0
\(423\) 324.240 + 185.199i 0.766524 + 0.437824i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −422.413 243.880i −0.989257 0.571148i
\(428\) 0 0
\(429\) 141.404 + 522.871i 0.329613 + 1.21881i
\(430\) 0 0
\(431\) 319.947i 0.742337i 0.928566 + 0.371168i \(0.121043\pi\)
−0.928566 + 0.371168i \(0.878957\pi\)
\(432\) 0 0
\(433\) 454.101i 1.04873i 0.851493 + 0.524366i \(0.175697\pi\)
−0.851493 + 0.524366i \(0.824303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 158.181 273.978i 0.361971 0.626951i
\(438\) 0 0
\(439\) 45.7154 + 79.1813i 0.104135 + 0.180367i 0.913385 0.407098i \(-0.133459\pi\)
−0.809249 + 0.587465i \(0.800126\pi\)
\(440\) 0 0
\(441\) −1.11856 241.086i −0.00253643 0.546679i
\(442\) 0 0
\(443\) 108.559 + 188.030i 0.245055 + 0.424448i 0.962147 0.272531i \(-0.0878607\pi\)
−0.717092 + 0.696978i \(0.754527\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 150.565 567.176i 0.336835 1.26885i
\(448\) 0 0
\(449\) 832.270i 1.85361i −0.375545 0.926804i \(-0.622544\pi\)
0.375545 0.926804i \(-0.377456\pi\)
\(450\) 0 0
\(451\) 594.249 1.31763
\(452\) 0 0
\(453\) 336.728 338.294i 0.743329 0.746785i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 693.510 400.398i 1.51753 0.876145i 0.517739 0.855538i \(-0.326774\pi\)
0.999788 0.0206062i \(-0.00655963\pi\)
\(458\) 0 0
\(459\) −312.373 308.055i −0.680551 0.671144i
\(460\) 0 0
\(461\) 504.549 291.301i 1.09447 0.631890i 0.159704 0.987165i \(-0.448946\pi\)
0.934762 + 0.355275i \(0.115613\pi\)
\(462\) 0 0
\(463\) −635.871 367.120i −1.37337 0.792916i −0.382021 0.924154i \(-0.624772\pi\)
−0.991351 + 0.131237i \(0.958105\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −670.573 −1.43592 −0.717958 0.696086i \(-0.754923\pi\)
−0.717958 + 0.696086i \(0.754923\pi\)
\(468\) 0 0
\(469\) −582.413 −1.24182
\(470\) 0 0
\(471\) 98.8225 372.263i 0.209814 0.790367i
\(472\) 0 0
\(473\) 415.338 719.386i 0.878092 1.52090i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −238.668 + 139.276i −0.500353 + 0.291982i
\(478\) 0 0
\(479\) −148.723 + 85.8655i −0.310487 + 0.179260i −0.647144 0.762367i \(-0.724037\pi\)
0.336657 + 0.941627i \(0.390704\pi\)
\(480\) 0 0
\(481\) −260.519 + 451.232i −0.541619 + 0.938112i
\(482\) 0 0
\(483\) −226.658 + 61.2967i −0.469271 + 0.126908i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 22.0906i 0.0453606i 0.999743 + 0.0226803i \(0.00721998\pi\)
−0.999743 + 0.0226803i \(0.992780\pi\)
\(488\) 0 0
\(489\) −62.8319 + 16.9921i −0.128491 + 0.0347486i
\(490\) 0 0
\(491\) −458.788 264.881i −0.934395 0.539473i −0.0461960 0.998932i \(-0.514710\pi\)
−0.888199 + 0.459459i \(0.848043\pi\)
\(492\) 0 0
\(493\) −62.9775 + 36.3601i −0.127743 + 0.0737527i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 262.535 + 454.724i 0.528239 + 0.914937i
\(498\) 0 0
\(499\) 84.6486 146.616i 0.169636 0.293819i −0.768656 0.639663i \(-0.779074\pi\)
0.938292 + 0.345844i \(0.112407\pi\)
\(500\) 0 0
\(501\) −736.243 195.446i −1.46955 0.390112i
\(502\) 0 0
\(503\) −22.5687 −0.0448682 −0.0224341 0.999748i \(-0.507142\pi\)
−0.0224341 + 0.999748i \(0.507142\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −171.872 + 172.672i −0.338999 + 0.340575i
\(508\) 0 0
\(509\) 654.896 + 378.104i 1.28663 + 0.742838i 0.978052 0.208361i \(-0.0668128\pi\)
0.308580 + 0.951198i \(0.400146\pi\)
\(510\) 0 0
\(511\) 213.448 + 369.702i 0.417706 + 0.723487i
\(512\) 0 0
\(513\) 366.232 + 361.169i 0.713902 + 0.704034i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −692.380 399.746i −1.33923 0.773203i
\(518\) 0 0
\(519\) 125.616 126.200i 0.242034 0.243160i
\(520\) 0 0
\(521\) 157.921i 0.303112i 0.988449 + 0.151556i \(0.0484283\pi\)
−0.988449 + 0.151556i \(0.951572\pi\)
\(522\) 0 0
\(523\) 535.341i 1.02360i −0.859106 0.511798i \(-0.828979\pi\)
0.859106 0.511798i \(-0.171021\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 116.892 202.462i 0.221806 0.384179i
\(528\) 0 0
\(529\) 126.612 + 219.298i 0.239341 + 0.414551i
\(530\) 0 0
\(531\) 773.424 3.58845i 1.45654 0.00675791i
\(532\) 0 0
\(533\) −144.472 250.232i −0.271054 0.469479i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −15.7508 + 4.25960i −0.0293311 + 0.00793222i
\(538\) 0 0
\(539\) 516.193i 0.957686i
\(540\) 0 0
\(541\) 446.519 0.825358 0.412679 0.910877i \(-0.364593\pi\)
0.412679 + 0.910877i \(0.364593\pi\)
\(542\) 0 0
\(543\) −35.6247 131.730i −0.0656072 0.242597i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −180.850 + 104.414i −0.330622 + 0.190885i −0.656117 0.754659i \(-0.727802\pi\)
0.325495 + 0.945544i \(0.394469\pi\)
\(548\) 0 0
\(549\) 808.795 + 461.968i 1.47322 + 0.841472i
\(550\) 0 0
\(551\) 73.8360 42.6292i 0.134004 0.0773671i
\(552\) 0 0
\(553\) −340.052 196.329i −0.614923 0.355026i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 428.915 0.770044 0.385022 0.922907i \(-0.374194\pi\)
0.385022 + 0.922907i \(0.374194\pi\)
\(558\) 0 0
\(559\) −403.901 −0.722543
\(560\) 0 0
\(561\) 665.753 + 662.671i 1.18672 + 1.18123i
\(562\) 0 0
\(563\) −394.727 + 683.687i −0.701114 + 1.21436i 0.266962 + 0.963707i \(0.413980\pi\)
−0.968076 + 0.250657i \(0.919353\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −3.54236 381.737i −0.00624756 0.673258i
\(568\) 0 0
\(569\) −153.834 + 88.8163i −0.270359 + 0.156092i −0.629051 0.777364i \(-0.716556\pi\)
0.358692 + 0.933456i \(0.383223\pi\)
\(570\) 0 0
\(571\) −9.18137 + 15.9026i −0.0160795 + 0.0278504i −0.873953 0.486010i \(-0.838452\pi\)
0.857874 + 0.513861i \(0.171785\pi\)
\(572\) 0 0
\(573\) −491.001 488.729i −0.856896 0.852929i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 240.588i 0.416963i 0.978026 + 0.208482i \(0.0668521\pi\)
−0.978026 + 0.208482i \(0.933148\pi\)
\(578\) 0 0
\(579\) 56.1045 211.345i 0.0968989 0.365017i
\(580\) 0 0
\(581\) −146.325 84.4809i −0.251851 0.145406i
\(582\) 0 0
\(583\) 512.389 295.828i 0.878884 0.507424i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 82.6260 + 143.112i 0.140760 + 0.243803i 0.927783 0.373120i \(-0.121712\pi\)
−0.787023 + 0.616923i \(0.788379\pi\)
\(588\) 0 0
\(589\) −137.046 + 237.370i −0.232676 + 0.403006i
\(590\) 0 0
\(591\) −30.7202 113.595i −0.0519801 0.192208i
\(592\) 0 0
\(593\) −901.661 −1.52051 −0.760254 0.649626i \(-0.774925\pi\)
−0.760254 + 0.649626i \(0.774925\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −209.945 776.316i −0.351666 1.30036i
\(598\) 0 0
\(599\) −893.571 515.904i −1.49177 0.861275i −0.491817 0.870699i \(-0.663667\pi\)
−0.999956 + 0.00942385i \(0.997000\pi\)
\(600\) 0 0
\(601\) −481.489 833.964i −0.801147 1.38763i −0.918862 0.394580i \(-0.870890\pi\)
0.117715 0.993047i \(-0.462443\pi\)
\(602\) 0 0
\(603\) 1112.17 5.16013i 1.84439 0.00855743i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 813.387 + 469.609i 1.34001 + 0.773656i 0.986809 0.161891i \(-0.0517594\pi\)
0.353202 + 0.935547i \(0.385093\pi\)
\(608\) 0 0
\(609\) −61.1595 16.2357i −0.100426 0.0266595i
\(610\) 0 0
\(611\) 388.739i 0.636234i
\(612\) 0 0
\(613\) 364.929i 0.595316i −0.954673 0.297658i \(-0.903794\pi\)
0.954673 0.297658i \(-0.0962055\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 133.518 231.260i 0.216398 0.374813i −0.737306 0.675559i \(-0.763902\pi\)
0.953704 + 0.300746i \(0.0972356\pi\)
\(618\) 0 0
\(619\) 453.122 + 784.830i 0.732022 + 1.26790i 0.956017 + 0.293310i \(0.0947567\pi\)
−0.223995 + 0.974590i \(0.571910\pi\)
\(620\) 0 0
\(621\) 432.280 119.060i 0.696103 0.191722i
\(622\) 0 0
\(623\) 200.101 + 346.585i 0.321190 + 0.556317i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −780.541 776.928i −1.24488 1.23912i
\(628\) 0 0
\(629\) 903.586i 1.43654i
\(630\) 0 0
\(631\) 597.193 0.946424 0.473212 0.880949i \(-0.343095\pi\)
0.473212 + 0.880949i \(0.343095\pi\)
\(632\) 0 0
\(633\) −1005.83 267.013i −1.58899 0.421821i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 217.363 125.495i 0.341230 0.197009i
\(638\) 0 0
\(639\) −505.362 866.009i −0.790864 1.35526i
\(640\) 0 0
\(641\) 78.1431 45.1160i 0.121908 0.0703837i −0.437806 0.899070i \(-0.644244\pi\)
0.559714 + 0.828686i \(0.310911\pi\)
\(642\) 0 0
\(643\) −336.770 194.434i −0.523748 0.302386i 0.214719 0.976676i \(-0.431117\pi\)
−0.738467 + 0.674290i \(0.764450\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 607.624 0.939141 0.469570 0.882895i \(-0.344409\pi\)
0.469570 + 0.882895i \(0.344409\pi\)
\(648\) 0 0
\(649\) −1655.99 −2.55161
\(650\) 0 0
\(651\) 196.373 53.1066i 0.301649 0.0815770i
\(652\) 0 0
\(653\) 566.508 981.221i 0.867547 1.50264i 0.00305096 0.999995i \(-0.499029\pi\)
0.864496 0.502640i \(-0.167638\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −410.873 704.088i −0.625377 1.07167i
\(658\) 0 0
\(659\) −772.309 + 445.893i −1.17194 + 0.676621i −0.954137 0.299372i \(-0.903223\pi\)
−0.217805 + 0.975992i \(0.569890\pi\)
\(660\) 0 0
\(661\) −219.766 + 380.645i −0.332474 + 0.575863i −0.982996 0.183625i \(-0.941217\pi\)
0.650522 + 0.759487i \(0.274550\pi\)
\(662\) 0 0
\(663\) 117.189 441.448i 0.176755 0.665834i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 74.3209i 0.111426i
\(668\) 0 0
\(669\) −150.469 149.772i −0.224916 0.223875i
\(670\) 0 0
\(671\) −1727.10 997.142i −2.57392 1.48605i
\(672\) 0 0
\(673\) −53.9165 + 31.1287i −0.0801137 + 0.0462537i −0.539522 0.841972i \(-0.681395\pi\)
0.459408 + 0.888225i \(0.348062\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −67.0416 116.119i −0.0990275 0.171521i 0.812255 0.583303i \(-0.198240\pi\)
−0.911282 + 0.411782i \(0.864907\pi\)
\(678\) 0 0
\(679\) −248.026 + 429.594i −0.365282 + 0.632687i
\(680\) 0 0
\(681\) 569.010 571.656i 0.835550 0.839436i
\(682\) 0 0
\(683\) −299.004 −0.437780 −0.218890 0.975750i \(-0.570244\pi\)
−0.218890 + 0.975750i \(0.570244\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −600.192 159.329i −0.873641 0.231921i
\(688\) 0 0
\(689\) −249.140 143.841i −0.361597 0.208768i
\(690\) 0 0
\(691\) 188.288 + 326.124i 0.272486 + 0.471960i 0.969498 0.245100i \(-0.0788208\pi\)
−0.697012 + 0.717060i \(0.745487\pi\)
\(692\) 0 0
\(693\) 3.79231 + 817.362i 0.00547231 + 1.17945i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −433.954 250.544i −0.622603 0.359460i
\(698\) 0 0
\(699\) −37.1925 137.527i −0.0532082 0.196749i
\(700\) 0 0
\(701\) 909.992i 1.29813i −0.760731 0.649067i \(-0.775159\pi\)
0.760731 0.649067i \(-0.224841\pi\)
\(702\) 0 0
\(703\) 1059.38i 1.50694i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 189.016 327.385i 0.267349 0.463062i
\(708\) 0 0
\(709\) −671.694 1163.41i −0.947382 1.64091i −0.750911 0.660404i \(-0.770385\pi\)
−0.196471 0.980510i \(-0.562948\pi\)
\(710\) 0 0
\(711\) 651.099 + 371.895i 0.915751 + 0.523059i
\(712\) 0 0
\(713\) 119.465 + 206.919i 0.167552 + 0.290209i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 256.710 967.023i 0.358034 1.34871i
\(718\) 0 0
\(719\) 288.276i 0.400940i −0.979700 0.200470i \(-0.935753\pi\)
0.979700 0.200470i \(-0.0642469\pi\)
\(720\) 0 0
\(721\) −614.673 −0.852528
\(722\) 0 0
\(723\) −903.092 + 907.292i −1.24909 + 1.25490i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 421.147 243.149i 0.579294 0.334456i −0.181559 0.983380i \(-0.558114\pi\)
0.760853 + 0.648925i \(0.224781\pi\)
\(728\) 0 0
\(729\) 10.1466 + 728.929i 0.0139185 + 0.999903i
\(730\) 0 0
\(731\) −606.606 + 350.224i −0.829830 + 0.479103i
\(732\) 0 0
\(733\) 1089.83 + 629.211i 1.48680 + 0.858406i 0.999887 0.0150435i \(-0.00478868\pi\)
0.486915 + 0.873449i \(0.338122\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2381.29 −3.23105
\(738\) 0 0
\(739\) 1192.17 1.61322 0.806609 0.591086i \(-0.201300\pi\)
0.806609 + 0.591086i \(0.201300\pi\)
\(740\) 0 0
\(741\) −137.394 + 517.562i −0.185417 + 0.698464i
\(742\) 0 0
\(743\) −98.7194 + 170.987i −0.132866 + 0.230131i −0.924780 0.380502i \(-0.875751\pi\)
0.791914 + 0.610632i \(0.209085\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 280.170 + 160.027i 0.375060 + 0.214227i
\(748\) 0 0
\(749\) 203.422 117.446i 0.271592 0.156803i
\(750\) 0 0
\(751\) −25.0633 + 43.4109i −0.0333733 + 0.0578042i −0.882230 0.470819i \(-0.843958\pi\)
0.848856 + 0.528624i \(0.177292\pi\)
\(752\) 0 0
\(753\) 1142.00 308.838i 1.51659 0.410143i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 282.361i 0.373000i −0.982455 0.186500i \(-0.940286\pi\)
0.982455 0.186500i \(-0.0597144\pi\)
\(758\) 0 0
\(759\) −926.726 + 250.621i −1.22098 + 0.330199i
\(760\) 0 0
\(761\) 89.9243 + 51.9178i 0.118166 + 0.0682231i 0.557918 0.829896i \(-0.311600\pi\)
−0.439752 + 0.898119i \(0.644934\pi\)
\(762\) 0 0
\(763\) −495.630 + 286.152i −0.649581 + 0.375036i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 402.599 + 697.321i 0.524900 + 0.909154i
\(768\) 0 0
\(769\) −190.567 + 330.073i −0.247812 + 0.429223i −0.962918 0.269793i \(-0.913045\pi\)
0.715106 + 0.699016i \(0.246378\pi\)
\(770\) 0 0
\(771\) −1049.98 278.731i −1.36184 0.361519i
\(772\) 0 0
\(773\) −276.633 −0.357870 −0.178935 0.983861i \(-0.557265\pi\)
−0.178935 + 0.983861i \(0.557265\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −554.678 + 557.258i −0.713872 + 0.717191i
\(778\) 0 0
\(779\) 508.776 + 293.742i 0.653114 + 0.377076i
\(780\) 0 0
\(781\) 1073.41 + 1859.21i 1.37441 + 2.38055i
\(782\) 0 0
\(783\) 116.933 + 30.4616i 0.149340 + 0.0389037i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1153.66 + 666.067i 1.46590 + 0.846336i 0.999273 0.0381190i \(-0.0121366\pi\)
0.466625 + 0.884455i \(0.345470\pi\)
\(788\) 0 0
\(789\) 718.636 721.978i 0.910819 0.915055i
\(790\) 0 0
\(791\) 133.257i 0.168466i
\(792\) 0 0
\(793\) 969.686i 1.22281i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −329.882 + 571.372i −0.413904 + 0.716903i −0.995313 0.0967084i \(-0.969169\pi\)
0.581408 + 0.813612i \(0.302502\pi\)
\(798\) 0 0
\(799\) 337.077 + 583.834i 0.421873 + 0.730706i
\(800\) 0 0
\(801\) −385.182 660.063i −0.480876 0.824048i
\(802\) 0 0
\(803\) 872.713 + 1511.58i 1.08682 + 1.88242i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −67.2039 + 18.1744i −0.0832762 + 0.0225210i
\(808\) 0 0
\(809\) 654.023i 0.808433i −0.914663 0.404217i \(-0.867544\pi\)
0.914663 0.404217i \(-0.132456\pi\)
\(810\) 0 0
\(811\) −408.003 −0.503086 −0.251543 0.967846i \(-0.580938\pi\)
−0.251543 + 0.967846i \(0.580938\pi\)
\(812\) 0 0
\(813\) 339.851 + 1256.67i 0.418021 + 1.54572i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 711.196 410.609i 0.870497 0.502582i
\(818\) 0 0
\(819\) 343.261 200.311i 0.419122 0.244580i
\(820\) 0 0
\(821\) 762.425 440.186i 0.928654 0.536158i 0.0422680 0.999106i \(-0.486542\pi\)
0.886385 + 0.462948i \(0.153208\pi\)
\(822\) 0 0
\(823\) 112.279 + 64.8244i 0.136427 + 0.0787660i 0.566660 0.823952i \(-0.308235\pi\)
−0.430233 + 0.902718i \(0.641569\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −646.300 −0.781499 −0.390750 0.920497i \(-0.627784\pi\)
−0.390750 + 0.920497i \(0.627784\pi\)
\(828\) 0 0
\(829\) 416.080 0.501906 0.250953 0.967999i \(-0.419256\pi\)
0.250953 + 0.967999i \(0.419256\pi\)
\(830\) 0 0
\(831\) 726.003 + 722.642i 0.873650 + 0.869606i
\(832\) 0 0
\(833\) 217.634 376.953i 0.261265 0.452524i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −374.522 + 103.152i −0.447457 + 0.123240i
\(838\) 0 0
\(839\) −240.544 + 138.878i −0.286703 + 0.165528i −0.636454 0.771315i \(-0.719600\pi\)
0.349751 + 0.936843i \(0.386266\pi\)
\(840\) 0 0
\(841\) −410.485 + 710.982i −0.488092 + 0.845400i
\(842\) 0 0
\(843\) −431.700 429.701i −0.512099 0.509729i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1179.79i 1.39291i
\(848\) 0 0
\(849\) 120.714 454.726i 0.142183 0.535602i
\(850\) 0 0
\(851\) −799.754 461.738i −0.939781 0.542583i
\(852\) 0 0
\(853\) −1139.76 + 658.042i −1.33618 + 0.771444i −0.986239 0.165327i \(-0.947132\pi\)
−0.349942 + 0.936771i \(0.613799\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 479.696 + 830.857i 0.559738 + 0.969495i 0.997518 + 0.0704129i \(0.0224317\pi\)
−0.437780 + 0.899082i \(0.644235\pi\)
\(858\) 0 0
\(859\) −439.753 + 761.675i −0.511936 + 0.886699i 0.487968 + 0.872861i \(0.337738\pi\)
−0.999904 + 0.0138379i \(0.995595\pi\)
\(860\) 0 0
\(861\) −113.828 420.903i −0.132204 0.488854i
\(862\) 0 0
\(863\) 326.481 0.378310 0.189155 0.981947i \(-0.439425\pi\)
0.189155 + 0.981947i \(0.439425\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 19.5594 + 72.3251i 0.0225599 + 0.0834200i
\(868\) 0 0
\(869\) −1390.36 802.722i −1.59995 0.923731i
\(870\) 0 0
\(871\) 578.930 + 1002.74i 0.664673 + 1.15125i
\(872\) 0 0
\(873\) 469.822 822.546i 0.538170 0.942206i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 99.0445 + 57.1834i 0.112936 + 0.0652034i 0.555404 0.831581i \(-0.312564\pi\)
−0.442468 + 0.896784i \(0.645897\pi\)
\(878\) 0 0
\(879\) −183.951 48.8323i −0.209273 0.0555544i
\(880\) 0 0
\(881\) 429.252i 0.487233i 0.969872 + 0.243617i \(0.0783338\pi\)
−0.969872 + 0.243617i \(0.921666\pi\)
\(882\) 0 0
\(883\) 294.993i 0.334080i −0.985950 0.167040i \(-0.946579\pi\)
0.985950 0.167040i \(-0.0534209\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −200.830 + 347.848i −0.226415 + 0.392162i −0.956743 0.290934i \(-0.906034\pi\)
0.730328 + 0.683097i \(0.239367\pi\)
\(888\) 0 0
\(889\) −105.749 183.162i −0.118952 0.206032i
\(890\) 0 0
\(891\) −14.4835 1560.79i −0.0162553 1.75173i
\(892\) 0 0
\(893\) −395.195 684.498i −0.442548 0.766515i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 330.836 + 329.305i 0.368825 + 0.367118i
\(898\) 0 0
\(899\) 64.3906i 0.0716247i
\(900\) 0 0
\(901\) −498.901 −0.553719
\(902\) 0 0
\(903\) −589.094 156.383i −0.652374 0.173182i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 367.089 211.939i 0.404729 0.233670i −0.283794 0.958885i \(-0.591593\pi\)
0.688522 + 0.725215i \(0.258260\pi\)
\(908\) 0 0
\(909\) −358.041 + 626.844i −0.393885 + 0.689598i
\(910\) 0 0
\(911\) −51.7248 + 29.8633i −0.0567780 + 0.0327808i −0.528120 0.849170i \(-0.677103\pi\)
0.471342 + 0.881950i \(0.343770\pi\)
\(912\) 0 0
\(913\) −598.273 345.413i −0.655283 0.378328i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 346.840 0.378233
\(918\) 0 0
\(919\) 522.062 0.568077 0.284038 0.958813i \(-0.408326\pi\)
0.284038 + 0.958813i \(0.408326\pi\)
\(920\) 0 0
\(921\) 532.164 143.917i 0.577811 0.156262i
\(922\) 0 0
\(923\) 521.929 904.008i 0.565470 0.979423i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1173.77 5.44595i 1.26621 0.00587481i
\(928\) 0 0
\(929\) 1309.83 756.229i 1.40993 0.814025i 0.414552 0.910026i \(-0.363938\pi\)
0.995381 + 0.0960006i \(0.0306051\pi\)
\(930\) 0 0
\(931\) −255.158 + 441.947i −0.274069 + 0.474701i
\(932\) 0 0
\(933\) 292.062 1100.19i 0.313036 1.17920i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 272.803i 0.291145i 0.989348 + 0.145573i \(0.0465025\pi\)
−0.989348 + 0.145573i \(0.953498\pi\)
\(938\) 0 0
\(939\) 868.805 + 864.783i 0.925245 + 0.920962i
\(940\) 0 0
\(941\) −1149.68 663.770i −1.22177 0.705388i −0.256473 0.966551i \(-0.582560\pi\)
−0.965295 + 0.261164i \(0.915894\pi\)
\(942\) 0 0
\(943\) 443.506 256.059i 0.470314 0.271536i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 11.3917 + 19.7310i 0.0120292 + 0.0208352i 0.871977 0.489546i \(-0.162838\pi\)
−0.859948 + 0.510381i \(0.829504\pi\)
\(948\) 0 0
\(949\) 424.342 734.981i 0.447146 0.774480i
\(950\) 0 0
\(951\) 1098.07 1103.18i 1.15465 1.16002i
\(952\) 0 0
\(953\) −563.843 −0.591650 −0.295825 0.955242i \(-0.595595\pi\)
−0.295825 + 0.955242i \(0.595595\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −250.060 66.3820i −0.261296 0.0693647i
\(958\) 0 0
\(959\) −890.530 514.148i −0.928603 0.536129i
\(960\) 0 0
\(961\) 376.998 + 652.979i 0.392297 + 0.679479i
\(962\) 0 0
\(963\) −387.412 + 226.075i −0.402297 + 0.234762i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1103.60 + 637.165i 1.14126 + 0.658909i 0.946743 0.321990i \(-0.104352\pi\)
0.194520 + 0.980899i \(0.437685\pi\)
\(968\) 0 0
\(969\) 242.431 + 896.443i 0.250187 + 0.925121i
\(970\) 0 0
\(971\) 119.681i 0.123255i 0.998099 + 0.0616276i \(0.0196291\pi\)
−0.998099 + 0.0616276i \(0.980371\pi\)
\(972\) 0 0
\(973\) 311.062i 0.319694i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −754.597 + 1307.00i −0.772361 + 1.33777i 0.163905 + 0.986476i \(0.447591\pi\)
−0.936266 + 0.351293i \(0.885742\pi\)
\(978\) 0 0
\(979\) 818.144 + 1417.07i 0.835694 + 1.44746i
\(980\) 0 0
\(981\) 943.914 550.824i 0.962196 0.561492i
\(982\) 0 0
\(983\) −258.736 448.144i −0.263211 0.455894i 0.703883 0.710316i \(-0.251448\pi\)
−0.967093 + 0.254422i \(0.918115\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −150.513 + 566.980i −0.152495 + 0.574448i
\(988\) 0 0
\(989\) 715.866i 0.723828i
\(990\) 0 0
\(991\) −201.958 −0.203793 −0.101896 0.994795i \(-0.532491\pi\)
−0.101896 + 0.994795i \(0.532491\pi\)
\(992\) 0 0
\(993\) −624.852 + 627.758i −0.629257 + 0.632184i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −181.297 + 104.672i −0.181842 + 0.104987i −0.588158 0.808746i \(-0.700147\pi\)
0.406316 + 0.913733i \(0.366813\pi\)
\(998\) 0 0
\(999\) 1054.27 1069.05i 1.05533 1.07012i
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 900.3.u.c.149.5 24
3.2 odd 2 2700.3.u.c.449.10 24
5.2 odd 4 900.3.p.c.401.1 12
5.3 odd 4 180.3.o.b.41.6 12
5.4 even 2 inner 900.3.u.c.149.8 24
9.2 odd 6 inner 900.3.u.c.749.8 24
9.7 even 3 2700.3.u.c.2249.3 24
15.2 even 4 2700.3.p.c.2501.5 12
15.8 even 4 540.3.o.b.341.4 12
15.14 odd 2 2700.3.u.c.449.3 24
20.3 even 4 720.3.bs.b.401.1 12
45.2 even 12 900.3.p.c.101.1 12
45.7 odd 12 2700.3.p.c.1601.5 12
45.13 odd 12 1620.3.g.b.161.11 12
45.23 even 12 1620.3.g.b.161.5 12
45.29 odd 6 inner 900.3.u.c.749.5 24
45.34 even 6 2700.3.u.c.2249.10 24
45.38 even 12 180.3.o.b.101.6 yes 12
45.43 odd 12 540.3.o.b.521.4 12
60.23 odd 4 2160.3.bs.b.881.6 12
180.43 even 12 2160.3.bs.b.1601.6 12
180.83 odd 12 720.3.bs.b.641.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.3.o.b.41.6 12 5.3 odd 4
180.3.o.b.101.6 yes 12 45.38 even 12
540.3.o.b.341.4 12 15.8 even 4
540.3.o.b.521.4 12 45.43 odd 12
720.3.bs.b.401.1 12 20.3 even 4
720.3.bs.b.641.1 12 180.83 odd 12
900.3.p.c.101.1 12 45.2 even 12
900.3.p.c.401.1 12 5.2 odd 4
900.3.u.c.149.5 24 1.1 even 1 trivial
900.3.u.c.149.8 24 5.4 even 2 inner
900.3.u.c.749.5 24 45.29 odd 6 inner
900.3.u.c.749.8 24 9.2 odd 6 inner
1620.3.g.b.161.5 12 45.23 even 12
1620.3.g.b.161.11 12 45.13 odd 12
2160.3.bs.b.881.6 12 60.23 odd 4
2160.3.bs.b.1601.6 12 180.43 even 12
2700.3.p.c.1601.5 12 45.7 odd 12
2700.3.p.c.2501.5 12 15.2 even 4
2700.3.u.c.449.3 24 15.14 odd 2
2700.3.u.c.449.10 24 3.2 odd 2
2700.3.u.c.2249.3 24 9.7 even 3
2700.3.u.c.2249.10 24 45.34 even 6