Properties

Label 882.2.g.l.667.2
Level $882$
Weight $2$
Character 882.667
Analytic conductor $7.043$
Analytic rank $0$
Dimension $4$
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] = [882,2,Mod(361,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.g (of order \(3\), 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: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 98)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 667.2
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 882.667
Dual form 882.2.g.l.361.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.41421 - 2.44949i) q^{5} -1.00000 q^{8} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.41421 - 2.44949i) q^{5} -1.00000 q^{8} +(-1.41421 - 2.44949i) q^{10} +(-1.00000 - 1.73205i) q^{11} +(-0.500000 + 0.866025i) q^{16} +(-0.707107 - 1.22474i) q^{17} +(3.53553 - 6.12372i) q^{19} -2.82843 q^{20} -2.00000 q^{22} +(-2.00000 + 3.46410i) q^{23} +(-1.50000 - 2.59808i) q^{25} -2.00000 q^{29} +(-4.24264 - 7.34847i) q^{31} +(0.500000 + 0.866025i) q^{32} -1.41421 q^{34} +(-5.00000 + 8.66025i) q^{37} +(-3.53553 - 6.12372i) q^{38} +(-1.41421 + 2.44949i) q^{40} +9.89949 q^{41} +2.00000 q^{43} +(-1.00000 + 1.73205i) q^{44} +(2.00000 + 3.46410i) q^{46} +(1.41421 - 2.44949i) q^{47} -3.00000 q^{50} +(-1.00000 - 1.73205i) q^{53} -5.65685 q^{55} +(-1.00000 + 1.73205i) q^{58} +(-0.707107 - 1.22474i) q^{59} +(-1.41421 + 2.44949i) q^{61} -8.48528 q^{62} +1.00000 q^{64} +(-6.00000 - 10.3923i) q^{67} +(-0.707107 + 1.22474i) q^{68} +12.0000 q^{71} +(0.707107 + 1.22474i) q^{73} +(5.00000 + 8.66025i) q^{74} -7.07107 q^{76} +(2.00000 - 3.46410i) q^{79} +(1.41421 + 2.44949i) q^{80} +(4.94975 - 8.57321i) q^{82} -9.89949 q^{83} -4.00000 q^{85} +(1.00000 - 1.73205i) q^{86} +(1.00000 + 1.73205i) q^{88} +(-3.53553 + 6.12372i) q^{89} +4.00000 q^{92} +(-1.41421 - 2.44949i) q^{94} +(-10.0000 - 17.3205i) q^{95} +9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 4 q^{11} - 2 q^{16} - 8 q^{22} - 8 q^{23} - 6 q^{25} - 8 q^{29} + 2 q^{32} - 20 q^{37} + 8 q^{43} - 4 q^{44} + 8 q^{46} - 12 q^{50} - 4 q^{53} - 4 q^{58} + 4 q^{64} - 24 q^{67} + 48 q^{71} + 20 q^{74} + 8 q^{79} - 16 q^{85} + 4 q^{86} + 4 q^{88} + 16 q^{92} - 40 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 0.866025i 0.353553 0.612372i
\(3\) 0 0
\(4\) −0.500000 0.866025i −0.250000 0.433013i
\(5\) 1.41421 2.44949i 0.632456 1.09545i −0.354593 0.935021i \(-0.615380\pi\)
0.987048 0.160424i \(-0.0512862\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.41421 2.44949i −0.447214 0.774597i
\(11\) −1.00000 1.73205i −0.301511 0.522233i 0.674967 0.737848i \(-0.264158\pi\)
−0.976478 + 0.215615i \(0.930824\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −0.707107 1.22474i −0.171499 0.297044i 0.767445 0.641114i \(-0.221528\pi\)
−0.938944 + 0.344070i \(0.888194\pi\)
\(18\) 0 0
\(19\) 3.53553 6.12372i 0.811107 1.40488i −0.100983 0.994888i \(-0.532199\pi\)
0.912090 0.409991i \(-0.134468\pi\)
\(20\) −2.82843 −0.632456
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) −2.00000 + 3.46410i −0.417029 + 0.722315i −0.995639 0.0932891i \(-0.970262\pi\)
0.578610 + 0.815604i \(0.303595\pi\)
\(24\) 0 0
\(25\) −1.50000 2.59808i −0.300000 0.519615i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −4.24264 7.34847i −0.762001 1.31982i −0.941818 0.336124i \(-0.890884\pi\)
0.179817 0.983700i \(-0.442449\pi\)
\(32\) 0.500000 + 0.866025i 0.0883883 + 0.153093i
\(33\) 0 0
\(34\) −1.41421 −0.242536
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00000 + 8.66025i −0.821995 + 1.42374i 0.0821995 + 0.996616i \(0.473806\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) −3.53553 6.12372i −0.573539 0.993399i
\(39\) 0 0
\(40\) −1.41421 + 2.44949i −0.223607 + 0.387298i
\(41\) 9.89949 1.54604 0.773021 0.634381i \(-0.218745\pi\)
0.773021 + 0.634381i \(0.218745\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −1.00000 + 1.73205i −0.150756 + 0.261116i
\(45\) 0 0
\(46\) 2.00000 + 3.46410i 0.294884 + 0.510754i
\(47\) 1.41421 2.44949i 0.206284 0.357295i −0.744257 0.667893i \(-0.767196\pi\)
0.950541 + 0.310599i \(0.100530\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −3.00000 −0.424264
\(51\) 0 0
\(52\) 0 0
\(53\) −1.00000 1.73205i −0.137361 0.237915i 0.789136 0.614218i \(-0.210529\pi\)
−0.926497 + 0.376303i \(0.877195\pi\)
\(54\) 0 0
\(55\) −5.65685 −0.762770
\(56\) 0 0
\(57\) 0 0
\(58\) −1.00000 + 1.73205i −0.131306 + 0.227429i
\(59\) −0.707107 1.22474i −0.0920575 0.159448i 0.816319 0.577601i \(-0.196011\pi\)
−0.908377 + 0.418153i \(0.862678\pi\)
\(60\) 0 0
\(61\) −1.41421 + 2.44949i −0.181071 + 0.313625i −0.942246 0.334922i \(-0.891290\pi\)
0.761174 + 0.648547i \(0.224623\pi\)
\(62\) −8.48528 −1.07763
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −6.00000 10.3923i −0.733017 1.26962i −0.955588 0.294706i \(-0.904778\pi\)
0.222571 0.974916i \(-0.428555\pi\)
\(68\) −0.707107 + 1.22474i −0.0857493 + 0.148522i
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 0.707107 + 1.22474i 0.0827606 + 0.143346i 0.904435 0.426612i \(-0.140293\pi\)
−0.821674 + 0.569958i \(0.806960\pi\)
\(74\) 5.00000 + 8.66025i 0.581238 + 1.00673i
\(75\) 0 0
\(76\) −7.07107 −0.811107
\(77\) 0 0
\(78\) 0 0
\(79\) 2.00000 3.46410i 0.225018 0.389742i −0.731307 0.682048i \(-0.761089\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) 1.41421 + 2.44949i 0.158114 + 0.273861i
\(81\) 0 0
\(82\) 4.94975 8.57321i 0.546608 0.946753i
\(83\) −9.89949 −1.08661 −0.543305 0.839535i \(-0.682827\pi\)
−0.543305 + 0.839535i \(0.682827\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 1.00000 1.73205i 0.107833 0.186772i
\(87\) 0 0
\(88\) 1.00000 + 1.73205i 0.106600 + 0.184637i
\(89\) −3.53553 + 6.12372i −0.374766 + 0.649113i −0.990292 0.139003i \(-0.955610\pi\)
0.615526 + 0.788116i \(0.288944\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) −1.41421 2.44949i −0.145865 0.252646i
\(95\) −10.0000 17.3205i −1.02598 1.77705i
\(96\) 0 0
\(97\) 9.89949 1.00514 0.502571 0.864536i \(-0.332388\pi\)
0.502571 + 0.864536i \(0.332388\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.50000 + 2.59808i −0.150000 + 0.259808i
\(101\) 4.24264 + 7.34847i 0.422159 + 0.731200i 0.996150 0.0876610i \(-0.0279392\pi\)
−0.573992 + 0.818861i \(0.694606\pi\)
\(102\) 0 0
\(103\) −1.41421 + 2.44949i −0.139347 + 0.241355i −0.927249 0.374444i \(-0.877834\pi\)
0.787903 + 0.615800i \(0.211167\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −2.00000 + 3.46410i −0.193347 + 0.334887i −0.946357 0.323122i \(-0.895268\pi\)
0.753010 + 0.658009i \(0.228601\pi\)
\(108\) 0 0
\(109\) 1.00000 + 1.73205i 0.0957826 + 0.165900i 0.909935 0.414751i \(-0.136131\pi\)
−0.814152 + 0.580651i \(0.802798\pi\)
\(110\) −2.82843 + 4.89898i −0.269680 + 0.467099i
\(111\) 0 0
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) 5.65685 + 9.79796i 0.527504 + 0.913664i
\(116\) 1.00000 + 1.73205i 0.0928477 + 0.160817i
\(117\) 0 0
\(118\) −1.41421 −0.130189
\(119\) 0 0
\(120\) 0 0
\(121\) 3.50000 6.06218i 0.318182 0.551107i
\(122\) 1.41421 + 2.44949i 0.128037 + 0.221766i
\(123\) 0 0
\(124\) −4.24264 + 7.34847i −0.381000 + 0.659912i
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0.500000 0.866025i 0.0441942 0.0765466i
\(129\) 0 0
\(130\) 0 0
\(131\) 6.36396 11.0227i 0.556022 0.963058i −0.441801 0.897113i \(-0.645660\pi\)
0.997823 0.0659452i \(-0.0210063\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 0.707107 + 1.22474i 0.0606339 + 0.105021i
\(137\) 6.00000 + 10.3923i 0.512615 + 0.887875i 0.999893 + 0.0146279i \(0.00465636\pi\)
−0.487278 + 0.873247i \(0.662010\pi\)
\(138\) 0 0
\(139\) 9.89949 0.839664 0.419832 0.907602i \(-0.362089\pi\)
0.419832 + 0.907602i \(0.362089\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.00000 10.3923i 0.503509 0.872103i
\(143\) 0 0
\(144\) 0 0
\(145\) −2.82843 + 4.89898i −0.234888 + 0.406838i
\(146\) 1.41421 0.117041
\(147\) 0 0
\(148\) 10.0000 0.821995
\(149\) 5.00000 8.66025i 0.409616 0.709476i −0.585231 0.810867i \(-0.698996\pi\)
0.994847 + 0.101391i \(0.0323294\pi\)
\(150\) 0 0
\(151\) 8.00000 + 13.8564i 0.651031 + 1.12762i 0.982873 + 0.184284i \(0.0589965\pi\)
−0.331842 + 0.943335i \(0.607670\pi\)
\(152\) −3.53553 + 6.12372i −0.286770 + 0.496700i
\(153\) 0 0
\(154\) 0 0
\(155\) −24.0000 −1.92773
\(156\) 0 0
\(157\) 5.65685 + 9.79796i 0.451466 + 0.781962i 0.998477 0.0551630i \(-0.0175678\pi\)
−0.547011 + 0.837125i \(0.684235\pi\)
\(158\) −2.00000 3.46410i −0.159111 0.275589i
\(159\) 0 0
\(160\) 2.82843 0.223607
\(161\) 0 0
\(162\) 0 0
\(163\) −5.00000 + 8.66025i −0.391630 + 0.678323i −0.992665 0.120900i \(-0.961422\pi\)
0.601035 + 0.799223i \(0.294755\pi\)
\(164\) −4.94975 8.57321i −0.386510 0.669456i
\(165\) 0 0
\(166\) −4.94975 + 8.57321i −0.384175 + 0.665410i
\(167\) 19.7990 1.53209 0.766046 0.642786i \(-0.222221\pi\)
0.766046 + 0.642786i \(0.222221\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) −2.00000 + 3.46410i −0.153393 + 0.265684i
\(171\) 0 0
\(172\) −1.00000 1.73205i −0.0762493 0.132068i
\(173\) −8.48528 + 14.6969i −0.645124 + 1.11739i 0.339149 + 0.940733i \(0.389861\pi\)
−0.984273 + 0.176655i \(0.943472\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) 3.53553 + 6.12372i 0.264999 + 0.458993i
\(179\) 6.00000 + 10.3923i 0.448461 + 0.776757i 0.998286 0.0585225i \(-0.0186389\pi\)
−0.549825 + 0.835280i \(0.685306\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2.00000 3.46410i 0.147442 0.255377i
\(185\) 14.1421 + 24.4949i 1.03975 + 1.80090i
\(186\) 0 0
\(187\) −1.41421 + 2.44949i −0.103418 + 0.179124i
\(188\) −2.82843 −0.206284
\(189\) 0 0
\(190\) −20.0000 −1.45095
\(191\) −2.00000 + 3.46410i −0.144715 + 0.250654i −0.929267 0.369410i \(-0.879560\pi\)
0.784552 + 0.620063i \(0.212893\pi\)
\(192\) 0 0
\(193\) 8.00000 + 13.8564i 0.575853 + 0.997406i 0.995948 + 0.0899262i \(0.0286631\pi\)
−0.420096 + 0.907480i \(0.638004\pi\)
\(194\) 4.94975 8.57321i 0.355371 0.615521i
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −4.24264 7.34847i −0.300753 0.520919i 0.675554 0.737311i \(-0.263905\pi\)
−0.976307 + 0.216391i \(0.930571\pi\)
\(200\) 1.50000 + 2.59808i 0.106066 + 0.183712i
\(201\) 0 0
\(202\) 8.48528 0.597022
\(203\) 0 0
\(204\) 0 0
\(205\) 14.0000 24.2487i 0.977802 1.69360i
\(206\) 1.41421 + 2.44949i 0.0985329 + 0.170664i
\(207\) 0 0
\(208\) 0 0
\(209\) −14.1421 −0.978232
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −1.00000 + 1.73205i −0.0686803 + 0.118958i
\(213\) 0 0
\(214\) 2.00000 + 3.46410i 0.136717 + 0.236801i
\(215\) 2.82843 4.89898i 0.192897 0.334108i
\(216\) 0 0
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) 2.82843 + 4.89898i 0.190693 + 0.330289i
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 6.00000 10.3923i 0.399114 0.691286i
\(227\) −10.6066 18.3712i −0.703985 1.21934i −0.967057 0.254561i \(-0.918069\pi\)
0.263072 0.964776i \(-0.415264\pi\)
\(228\) 0 0
\(229\) 8.48528 14.6969i 0.560723 0.971201i −0.436710 0.899602i \(-0.643857\pi\)
0.997434 0.0715988i \(-0.0228101\pi\)
\(230\) 11.3137 0.746004
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) 12.0000 20.7846i 0.786146 1.36165i −0.142166 0.989843i \(-0.545407\pi\)
0.928312 0.371802i \(-0.121260\pi\)
\(234\) 0 0
\(235\) −4.00000 6.92820i −0.260931 0.451946i
\(236\) −0.707107 + 1.22474i −0.0460287 + 0.0797241i
\(237\) 0 0
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 10.6066 + 18.3712i 0.683231 + 1.18339i 0.973989 + 0.226595i \(0.0727593\pi\)
−0.290758 + 0.956797i \(0.593907\pi\)
\(242\) −3.50000 6.06218i −0.224989 0.389692i
\(243\) 0 0
\(244\) 2.82843 0.181071
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 4.24264 + 7.34847i 0.269408 + 0.466628i
\(249\) 0 0
\(250\) 2.82843 4.89898i 0.178885 0.309839i
\(251\) −9.89949 −0.624851 −0.312425 0.949942i \(-0.601141\pi\)
−0.312425 + 0.949942i \(0.601141\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 8.00000 13.8564i 0.501965 0.869428i
\(255\) 0 0
\(256\) −0.500000 0.866025i −0.0312500 0.0541266i
\(257\) 6.36396 11.0227i 0.396973 0.687577i −0.596378 0.802704i \(-0.703394\pi\)
0.993351 + 0.115126i \(0.0367273\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −6.36396 11.0227i −0.393167 0.680985i
\(263\) 6.00000 + 10.3923i 0.369976 + 0.640817i 0.989561 0.144112i \(-0.0460326\pi\)
−0.619586 + 0.784929i \(0.712699\pi\)
\(264\) 0 0
\(265\) −5.65685 −0.347498
\(266\) 0 0
\(267\) 0 0
\(268\) −6.00000 + 10.3923i −0.366508 + 0.634811i
\(269\) −5.65685 9.79796i −0.344904 0.597392i 0.640432 0.768015i \(-0.278755\pi\)
−0.985336 + 0.170623i \(0.945422\pi\)
\(270\) 0 0
\(271\) −11.3137 + 19.5959i −0.687259 + 1.19037i 0.285462 + 0.958390i \(0.407853\pi\)
−0.972721 + 0.231977i \(0.925480\pi\)
\(272\) 1.41421 0.0857493
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) −3.00000 + 5.19615i −0.180907 + 0.313340i
\(276\) 0 0
\(277\) 1.00000 + 1.73205i 0.0600842 + 0.104069i 0.894503 0.447062i \(-0.147530\pi\)
−0.834419 + 0.551131i \(0.814196\pi\)
\(278\) 4.94975 8.57321i 0.296866 0.514187i
\(279\) 0 0
\(280\) 0 0
\(281\) −16.0000 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 0 0
\(283\) 0.707107 + 1.22474i 0.0420331 + 0.0728035i 0.886277 0.463156i \(-0.153283\pi\)
−0.844243 + 0.535960i \(0.819950\pi\)
\(284\) −6.00000 10.3923i −0.356034 0.616670i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 7.50000 12.9904i 0.441176 0.764140i
\(290\) 2.82843 + 4.89898i 0.166091 + 0.287678i
\(291\) 0 0
\(292\) 0.707107 1.22474i 0.0413803 0.0716728i
\(293\) 19.7990 1.15667 0.578335 0.815800i \(-0.303703\pi\)
0.578335 + 0.815800i \(0.303703\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 5.00000 8.66025i 0.290619 0.503367i
\(297\) 0 0
\(298\) −5.00000 8.66025i −0.289642 0.501675i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) 0 0
\(304\) 3.53553 + 6.12372i 0.202777 + 0.351220i
\(305\) 4.00000 + 6.92820i 0.229039 + 0.396708i
\(306\) 0 0
\(307\) −9.89949 −0.564994 −0.282497 0.959268i \(-0.591163\pi\)
−0.282497 + 0.959268i \(0.591163\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −12.0000 + 20.7846i −0.681554 + 1.18049i
\(311\) −5.65685 9.79796i −0.320771 0.555591i 0.659877 0.751374i \(-0.270609\pi\)
−0.980647 + 0.195783i \(0.937275\pi\)
\(312\) 0 0
\(313\) −6.36396 + 11.0227i −0.359712 + 0.623040i −0.987913 0.155012i \(-0.950459\pi\)
0.628200 + 0.778052i \(0.283792\pi\)
\(314\) 11.3137 0.638470
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 5.00000 8.66025i 0.280828 0.486408i −0.690761 0.723083i \(-0.742724\pi\)
0.971589 + 0.236675i \(0.0760576\pi\)
\(318\) 0 0
\(319\) 2.00000 + 3.46410i 0.111979 + 0.193952i
\(320\) 1.41421 2.44949i 0.0790569 0.136931i
\(321\) 0 0
\(322\) 0 0
\(323\) −10.0000 −0.556415
\(324\) 0 0
\(325\) 0 0
\(326\) 5.00000 + 8.66025i 0.276924 + 0.479647i
\(327\) 0 0
\(328\) −9.89949 −0.546608
\(329\) 0 0
\(330\) 0 0
\(331\) −5.00000 + 8.66025i −0.274825 + 0.476011i −0.970091 0.242742i \(-0.921953\pi\)
0.695266 + 0.718752i \(0.255287\pi\)
\(332\) 4.94975 + 8.57321i 0.271653 + 0.470516i
\(333\) 0 0
\(334\) 9.89949 17.1464i 0.541676 0.938211i
\(335\) −33.9411 −1.85440
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −6.50000 + 11.2583i −0.353553 + 0.612372i
\(339\) 0 0
\(340\) 2.00000 + 3.46410i 0.108465 + 0.187867i
\(341\) −8.48528 + 14.6969i −0.459504 + 0.795884i
\(342\) 0 0
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) 8.48528 + 14.6969i 0.456172 + 0.790112i
\(347\) −15.0000 25.9808i −0.805242 1.39472i −0.916127 0.400887i \(-0.868702\pi\)
0.110885 0.993833i \(-0.464631\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000 1.73205i 0.0533002 0.0923186i
\(353\) −0.707107 1.22474i −0.0376355 0.0651866i 0.846594 0.532239i \(-0.178649\pi\)
−0.884230 + 0.467052i \(0.845316\pi\)
\(354\) 0 0
\(355\) 16.9706 29.3939i 0.900704 1.56007i
\(356\) 7.07107 0.374766
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −16.0000 + 27.7128i −0.844448 + 1.46263i 0.0416523 + 0.999132i \(0.486738\pi\)
−0.886100 + 0.463494i \(0.846596\pi\)
\(360\) 0 0
\(361\) −15.5000 26.8468i −0.815789 1.41299i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) −14.1421 24.4949i −0.738213 1.27862i −0.953299 0.302028i \(-0.902336\pi\)
0.215086 0.976595i \(-0.430997\pi\)
\(368\) −2.00000 3.46410i −0.104257 0.180579i
\(369\) 0 0
\(370\) 28.2843 1.47043
\(371\) 0 0
\(372\) 0 0
\(373\) −5.00000 + 8.66025i −0.258890 + 0.448411i −0.965945 0.258748i \(-0.916690\pi\)
0.707055 + 0.707159i \(0.250023\pi\)
\(374\) 1.41421 + 2.44949i 0.0731272 + 0.126660i
\(375\) 0 0
\(376\) −1.41421 + 2.44949i −0.0729325 + 0.126323i
\(377\) 0 0
\(378\) 0 0
\(379\) −26.0000 −1.33553 −0.667765 0.744372i \(-0.732749\pi\)
−0.667765 + 0.744372i \(0.732749\pi\)
\(380\) −10.0000 + 17.3205i −0.512989 + 0.888523i
\(381\) 0 0
\(382\) 2.00000 + 3.46410i 0.102329 + 0.177239i
\(383\) −18.3848 + 31.8434i −0.939418 + 1.62712i −0.172859 + 0.984947i \(0.555300\pi\)
−0.766559 + 0.642173i \(0.778033\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 16.0000 0.814379
\(387\) 0 0
\(388\) −4.94975 8.57321i −0.251285 0.435239i
\(389\) 13.0000 + 22.5167i 0.659126 + 1.14164i 0.980842 + 0.194804i \(0.0624070\pi\)
−0.321716 + 0.946836i \(0.604260\pi\)
\(390\) 0 0
\(391\) 5.65685 0.286079
\(392\) 0 0
\(393\) 0 0
\(394\) −1.00000 + 1.73205i −0.0503793 + 0.0872595i
\(395\) −5.65685 9.79796i −0.284627 0.492989i
\(396\) 0 0
\(397\) −11.3137 + 19.5959i −0.567819 + 0.983491i 0.428963 + 0.903322i \(0.358879\pi\)
−0.996781 + 0.0801687i \(0.974454\pi\)
\(398\) −8.48528 −0.425329
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) −9.00000 + 15.5885i −0.449439 + 0.778450i −0.998350 0.0574304i \(-0.981709\pi\)
0.548911 + 0.835881i \(0.315043\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 4.24264 7.34847i 0.211079 0.365600i
\(405\) 0 0
\(406\) 0 0
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) −19.0919 33.0681i −0.944033 1.63511i −0.757676 0.652631i \(-0.773665\pi\)
−0.186357 0.982482i \(-0.559668\pi\)
\(410\) −14.0000 24.2487i −0.691411 1.19756i
\(411\) 0 0
\(412\) 2.82843 0.139347
\(413\) 0 0
\(414\) 0 0
\(415\) −14.0000 + 24.2487i −0.687233 + 1.19032i
\(416\) 0 0
\(417\) 0 0
\(418\) −7.07107 + 12.2474i −0.345857 + 0.599042i
\(419\) 9.89949 0.483622 0.241811 0.970323i \(-0.422259\pi\)
0.241811 + 0.970323i \(0.422259\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) −6.00000 + 10.3923i −0.292075 + 0.505889i
\(423\) 0 0
\(424\) 1.00000 + 1.73205i 0.0485643 + 0.0841158i
\(425\) −2.12132 + 3.67423i −0.102899 + 0.178227i
\(426\) 0 0
\(427\) 0 0
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) −2.82843 4.89898i −0.136399 0.236250i
\(431\) 6.00000 + 10.3923i 0.289010 + 0.500580i 0.973574 0.228373i \(-0.0733406\pi\)
−0.684564 + 0.728953i \(0.740007\pi\)
\(432\) 0 0
\(433\) −29.6985 −1.42722 −0.713609 0.700544i \(-0.752941\pi\)
−0.713609 + 0.700544i \(0.752941\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.00000 1.73205i 0.0478913 0.0829502i
\(437\) 14.1421 + 24.4949i 0.676510 + 1.17175i
\(438\) 0 0
\(439\) 8.48528 14.6969i 0.404980 0.701447i −0.589339 0.807886i \(-0.700612\pi\)
0.994319 + 0.106439i \(0.0339450\pi\)
\(440\) 5.65685 0.269680
\(441\) 0 0
\(442\) 0 0
\(443\) −2.00000 + 3.46410i −0.0950229 + 0.164584i −0.909618 0.415445i \(-0.863626\pi\)
0.814595 + 0.580030i \(0.196959\pi\)
\(444\) 0 0
\(445\) 10.0000 + 17.3205i 0.474045 + 0.821071i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) −9.89949 17.1464i −0.466149 0.807394i
\(452\) −6.00000 10.3923i −0.282216 0.488813i
\(453\) 0 0
\(454\) −21.2132 −0.995585
\(455\) 0 0
\(456\) 0 0
\(457\) −12.0000 + 20.7846i −0.561336 + 0.972263i 0.436044 + 0.899925i \(0.356379\pi\)
−0.997380 + 0.0723376i \(0.976954\pi\)
\(458\) −8.48528 14.6969i −0.396491 0.686743i
\(459\) 0 0
\(460\) 5.65685 9.79796i 0.263752 0.456832i
\(461\) −39.5980 −1.84426 −0.922131 0.386878i \(-0.873553\pi\)
−0.922131 + 0.386878i \(0.873553\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 1.00000 1.73205i 0.0464238 0.0804084i
\(465\) 0 0
\(466\) −12.0000 20.7846i −0.555889 0.962828i
\(467\) 16.2635 28.1691i 0.752583 1.30351i −0.193984 0.981005i \(-0.562141\pi\)
0.946567 0.322507i \(-0.104526\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) 0.707107 + 1.22474i 0.0325472 + 0.0563735i
\(473\) −2.00000 3.46410i −0.0919601 0.159280i
\(474\) 0 0
\(475\) −21.2132 −0.973329
\(476\) 0 0
\(477\) 0 0
\(478\) 6.00000 10.3923i 0.274434 0.475333i
\(479\) −15.5563 26.9444i −0.710788 1.23112i −0.964562 0.263857i \(-0.915005\pi\)
0.253774 0.967264i \(-0.418328\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 21.2132 0.966235
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 14.0000 24.2487i 0.635707 1.10108i
\(486\) 0 0
\(487\) −6.00000 10.3923i −0.271886 0.470920i 0.697459 0.716625i \(-0.254314\pi\)
−0.969345 + 0.245705i \(0.920981\pi\)
\(488\) 1.41421 2.44949i 0.0640184 0.110883i
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 1.41421 + 2.44949i 0.0636930 + 0.110319i
\(494\) 0 0
\(495\) 0 0
\(496\) 8.48528 0.381000
\(497\) 0 0
\(498\) 0 0
\(499\) 2.00000 3.46410i 0.0895323 0.155074i −0.817781 0.575529i \(-0.804796\pi\)
0.907314 + 0.420455i \(0.138129\pi\)
\(500\) −2.82843 4.89898i −0.126491 0.219089i
\(501\) 0 0
\(502\) −4.94975 + 8.57321i −0.220918 + 0.382641i
\(503\) −39.5980 −1.76559 −0.882793 0.469762i \(-0.844340\pi\)
−0.882793 + 0.469762i \(0.844340\pi\)
\(504\) 0 0
\(505\) 24.0000 1.06799
\(506\) 4.00000 6.92820i 0.177822 0.307996i
\(507\) 0 0
\(508\) −8.00000 13.8564i −0.354943 0.614779i
\(509\) 11.3137 19.5959i 0.501471 0.868574i −0.498527 0.866874i \(-0.666126\pi\)
0.999999 0.00169976i \(-0.000541051\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −6.36396 11.0227i −0.280702 0.486191i
\(515\) 4.00000 + 6.92820i 0.176261 + 0.305293i
\(516\) 0 0
\(517\) −5.65685 −0.248788
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −0.707107 1.22474i −0.0309789 0.0536570i 0.850120 0.526589i \(-0.176529\pi\)
−0.881099 + 0.472931i \(0.843196\pi\)
\(522\) 0 0
\(523\) −6.36396 + 11.0227i −0.278277 + 0.481989i −0.970957 0.239256i \(-0.923097\pi\)
0.692680 + 0.721245i \(0.256430\pi\)
\(524\) −12.7279 −0.556022
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) −6.00000 + 10.3923i −0.261364 + 0.452696i
\(528\) 0 0
\(529\) 3.50000 + 6.06218i 0.152174 + 0.263573i
\(530\) −2.82843 + 4.89898i −0.122859 + 0.212798i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 5.65685 + 9.79796i 0.244567 + 0.423603i
\(536\) 6.00000 + 10.3923i 0.259161 + 0.448879i
\(537\) 0 0
\(538\) −11.3137 −0.487769
\(539\) 0 0
\(540\) 0 0
\(541\) −5.00000 + 8.66025i −0.214967 + 0.372333i −0.953262 0.302144i \(-0.902298\pi\)
0.738296 + 0.674477i \(0.235631\pi\)
\(542\) 11.3137 + 19.5959i 0.485965 + 0.841717i
\(543\) 0 0
\(544\) 0.707107 1.22474i 0.0303170 0.0525105i
\(545\) 5.65685 0.242313
\(546\) 0 0
\(547\) −26.0000 −1.11168 −0.555840 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) 6.00000 10.3923i 0.256307 0.443937i
\(549\) 0 0
\(550\) 3.00000 + 5.19615i 0.127920 + 0.221565i
\(551\) −7.07107 + 12.2474i −0.301238 + 0.521759i
\(552\) 0 0
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −4.94975 8.57321i −0.209916 0.363585i
\(557\) −15.0000 25.9808i −0.635570 1.10084i −0.986394 0.164399i \(-0.947432\pi\)
0.350824 0.936442i \(-0.385902\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −8.00000 + 13.8564i −0.337460 + 0.584497i
\(563\) −0.707107 1.22474i −0.0298010 0.0516168i 0.850740 0.525586i \(-0.176154\pi\)
−0.880541 + 0.473970i \(0.842821\pi\)
\(564\) 0 0
\(565\) 16.9706 29.3939i 0.713957 1.23661i
\(566\) 1.41421 0.0594438
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) 5.00000 8.66025i 0.209611 0.363057i −0.741981 0.670421i \(-0.766114\pi\)
0.951592 + 0.307364i \(0.0994469\pi\)
\(570\) 0 0
\(571\) 1.00000 + 1.73205i 0.0418487 + 0.0724841i 0.886191 0.463320i \(-0.153342\pi\)
−0.844342 + 0.535804i \(0.820009\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) 10.6066 + 18.3712i 0.441559 + 0.764802i 0.997805 0.0662152i \(-0.0210924\pi\)
−0.556247 + 0.831017i \(0.687759\pi\)
\(578\) −7.50000 12.9904i −0.311959 0.540329i
\(579\) 0 0
\(580\) 5.65685 0.234888
\(581\) 0 0
\(582\) 0 0
\(583\) −2.00000 + 3.46410i −0.0828315 + 0.143468i
\(584\) −0.707107 1.22474i −0.0292603 0.0506803i
\(585\) 0 0
\(586\) 9.89949 17.1464i 0.408944 0.708312i
\(587\) 29.6985 1.22579 0.612894 0.790165i \(-0.290005\pi\)
0.612894 + 0.790165i \(0.290005\pi\)
\(588\) 0 0
\(589\) −60.0000 −2.47226
\(590\) −2.00000 + 3.46410i −0.0823387 + 0.142615i
\(591\) 0 0
\(592\) −5.00000 8.66025i −0.205499 0.355934i
\(593\) −3.53553 + 6.12372i −0.145187 + 0.251471i −0.929443 0.368967i \(-0.879712\pi\)
0.784256 + 0.620438i \(0.213045\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) 0 0
\(599\) −8.00000 13.8564i −0.326871 0.566157i 0.655018 0.755613i \(-0.272661\pi\)
−0.981889 + 0.189456i \(0.939328\pi\)
\(600\) 0 0
\(601\) 29.6985 1.21143 0.605713 0.795683i \(-0.292888\pi\)
0.605713 + 0.795683i \(0.292888\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 8.00000 13.8564i 0.325515 0.563809i
\(605\) −9.89949 17.1464i −0.402472 0.697101i
\(606\) 0 0
\(607\) 8.48528 14.6969i 0.344407 0.596530i −0.640839 0.767675i \(-0.721413\pi\)
0.985246 + 0.171145i \(0.0547467\pi\)
\(608\) 7.07107 0.286770
\(609\) 0 0
\(610\) 8.00000 0.323911
\(611\) 0 0
\(612\) 0 0
\(613\) 15.0000 + 25.9808i 0.605844 + 1.04935i 0.991917 + 0.126885i \(0.0404979\pi\)
−0.386073 + 0.922468i \(0.626169\pi\)
\(614\) −4.94975 + 8.57321i −0.199756 + 0.345987i
\(615\) 0 0
\(616\) 0 0
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) 0 0
\(619\) −9.19239 15.9217i −0.369473 0.639946i 0.620010 0.784594i \(-0.287129\pi\)
−0.989483 + 0.144647i \(0.953795\pi\)
\(620\) 12.0000 + 20.7846i 0.481932 + 0.834730i
\(621\) 0 0
\(622\) −11.3137 −0.453638
\(623\) 0 0
\(624\) 0 0
\(625\) 15.5000 26.8468i 0.620000 1.07387i
\(626\) 6.36396 + 11.0227i 0.254355 + 0.440556i
\(627\) 0 0
\(628\) 5.65685 9.79796i 0.225733 0.390981i
\(629\) 14.1421 0.563884
\(630\) 0 0
\(631\) 44.0000 1.75161 0.875806 0.482663i \(-0.160330\pi\)
0.875806 + 0.482663i \(0.160330\pi\)
\(632\) −2.00000 + 3.46410i −0.0795557 + 0.137795i
\(633\) 0 0
\(634\) −5.00000 8.66025i −0.198575 0.343943i
\(635\) 22.6274 39.1918i 0.897942 1.55528i
\(636\) 0 0
\(637\) 0 0
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) −1.41421 2.44949i −0.0559017 0.0968246i
\(641\) 13.0000 + 22.5167i 0.513469 + 0.889355i 0.999878 + 0.0156233i \(0.00497325\pi\)
−0.486409 + 0.873731i \(0.661693\pi\)
\(642\) 0 0
\(643\) 9.89949 0.390398 0.195199 0.980764i \(-0.437465\pi\)
0.195199 + 0.980764i \(0.437465\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −5.00000 + 8.66025i −0.196722 + 0.340733i
\(647\) 4.24264 + 7.34847i 0.166795 + 0.288898i 0.937291 0.348547i \(-0.113325\pi\)
−0.770496 + 0.637445i \(0.779991\pi\)
\(648\) 0 0
\(649\) −1.41421 + 2.44949i −0.0555127 + 0.0961509i
\(650\) 0 0
\(651\) 0 0
\(652\) 10.0000 0.391630
\(653\) −9.00000 + 15.5885i −0.352197 + 0.610023i −0.986634 0.162951i \(-0.947899\pi\)
0.634437 + 0.772975i \(0.281232\pi\)
\(654\) 0 0
\(655\) −18.0000 31.1769i −0.703318 1.21818i
\(656\) −4.94975 + 8.57321i −0.193255 + 0.334728i
\(657\) 0 0
\(658\) 0 0
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) −4.24264 7.34847i −0.165020 0.285822i 0.771643 0.636056i \(-0.219435\pi\)
−0.936662 + 0.350234i \(0.886102\pi\)
\(662\) 5.00000 + 8.66025i 0.194331 + 0.336590i
\(663\) 0 0
\(664\) 9.89949 0.384175
\(665\) 0 0
\(666\) 0 0
\(667\) 4.00000 6.92820i 0.154881 0.268261i
\(668\) −9.89949 17.1464i −0.383023 0.663415i
\(669\) 0 0
\(670\) −16.9706 + 29.3939i −0.655630 + 1.13558i
\(671\) 5.65685 0.218380
\(672\) 0 0
\(673\) −12.0000 −0.462566 −0.231283 0.972887i \(-0.574292\pi\)
−0.231283 + 0.972887i \(0.574292\pi\)
\(674\) 1.00000 1.73205i 0.0385186 0.0667161i
\(675\) 0 0
\(676\) 6.50000 + 11.2583i 0.250000 + 0.433013i
\(677\) −8.48528 + 14.6969i −0.326116 + 0.564849i −0.981738 0.190240i \(-0.939073\pi\)
0.655622 + 0.755090i \(0.272407\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 4.00000 0.153393
\(681\) 0 0
\(682\) 8.48528 + 14.6969i 0.324918 + 0.562775i
\(683\) 6.00000 + 10.3923i 0.229584 + 0.397650i 0.957685 0.287819i \(-0.0929302\pi\)
−0.728101 + 0.685470i \(0.759597\pi\)
\(684\) 0 0
\(685\) 33.9411 1.29682
\(686\) 0 0
\(687\) 0 0
\(688\) −1.00000 + 1.73205i −0.0381246 + 0.0660338i
\(689\) 0 0
\(690\) 0 0
\(691\) −6.36396 + 11.0227i −0.242096 + 0.419323i −0.961311 0.275464i \(-0.911168\pi\)
0.719215 + 0.694788i \(0.244502\pi\)
\(692\) 16.9706 0.645124
\(693\) 0 0
\(694\) −30.0000 −1.13878
\(695\) 14.0000 24.2487i 0.531050 0.919806i
\(696\) 0 0
\(697\) −7.00000 12.1244i −0.265144 0.459243i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 35.3553 + 61.2372i 1.33345 + 2.30961i
\(704\) −1.00000 1.73205i −0.0376889 0.0652791i
\(705\) 0 0
\(706\) −1.41421 −0.0532246
\(707\) 0 0
\(708\) 0 0
\(709\) −5.00000 + 8.66025i −0.187779 + 0.325243i −0.944509 0.328484i \(-0.893462\pi\)
0.756730 + 0.653727i \(0.226796\pi\)
\(710\) −16.9706 29.3939i −0.636894 1.10313i
\(711\) 0 0
\(712\) 3.53553 6.12372i 0.132500 0.229496i
\(713\) 33.9411 1.27111
\(714\) 0 0
\(715\) 0 0
\(716\) 6.00000 10.3923i 0.224231 0.388379i
\(717\) 0 0
\(718\) 16.0000 + 27.7128i 0.597115 + 1.03423i
\(719\) 1.41421 2.44949i 0.0527413 0.0913506i −0.838449 0.544979i \(-0.816537\pi\)
0.891191 + 0.453629i \(0.149871\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −31.0000 −1.15370
\(723\) 0 0
\(724\) 0 0
\(725\) 3.00000 + 5.19615i 0.111417 + 0.192980i
\(726\) 0 0
\(727\) −19.7990 −0.734304 −0.367152 0.930161i \(-0.619667\pi\)
−0.367152 + 0.930161i \(0.619667\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 2.00000 3.46410i 0.0740233 0.128212i
\(731\) −1.41421 2.44949i −0.0523066 0.0905977i
\(732\) 0 0
\(733\) −21.2132 + 36.7423i −0.783528 + 1.35711i 0.146347 + 0.989233i \(0.453248\pi\)
−0.929875 + 0.367876i \(0.880085\pi\)
\(734\) −28.2843 −1.04399
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −12.0000 + 20.7846i −0.442026 + 0.765611i
\(738\) 0 0
\(739\) 15.0000 + 25.9808i 0.551784 + 0.955718i 0.998146 + 0.0608653i \(0.0193860\pi\)
−0.446362 + 0.894852i \(0.647281\pi\)
\(740\) 14.1421 24.4949i 0.519875 0.900450i
\(741\) 0 0
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) −14.1421 24.4949i −0.518128 0.897424i
\(746\) 5.00000 + 8.66025i 0.183063 + 0.317074i
\(747\) 0 0
\(748\) 2.82843 0.103418
\(749\) 0 0
\(750\) 0 0
\(751\) 2.00000 3.46410i 0.0729810 0.126407i −0.827225 0.561870i \(-0.810082\pi\)
0.900207 + 0.435463i \(0.143415\pi\)
\(752\) 1.41421 + 2.44949i 0.0515711 + 0.0893237i
\(753\) 0 0
\(754\) 0 0
\(755\) 45.2548 1.64699
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −13.0000 + 22.5167i −0.472181 + 0.817842i
\(759\) 0 0
\(760\) 10.0000 + 17.3205i 0.362738 + 0.628281i
\(761\) −3.53553 + 6.12372i −0.128163 + 0.221985i −0.922965 0.384884i \(-0.874241\pi\)
0.794802 + 0.606869i \(0.207575\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 4.00000 0.144715
\(765\) 0 0
\(766\) 18.3848 + 31.8434i 0.664269 + 1.15055i
\(767\) 0 0
\(768\) 0 0
\(769\) 29.6985 1.07095 0.535477 0.844550i \(-0.320132\pi\)
0.535477 + 0.844550i \(0.320132\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.00000 13.8564i 0.287926 0.498703i
\(773\) 24.0416 + 41.6413i 0.864717 + 1.49773i 0.867328 + 0.497738i \(0.165836\pi\)
−0.00261021 + 0.999997i \(0.500831\pi\)
\(774\) 0 0
\(775\) −12.7279 + 22.0454i −0.457200 + 0.791894i
\(776\) −9.89949 −0.355371
\(777\) 0 0
\(778\) 26.0000 0.932145
\(779\) 35.0000 60.6218i 1.25401 2.17200i
\(780\) 0 0
\(781\) −12.0000 20.7846i −0.429394 0.743732i
\(782\) 2.82843 4.89898i 0.101144 0.175187i
\(783\) 0 0
\(784\) 0 0
\(785\) 32.0000 1.14213
\(786\) 0 0
\(787\) 0.707107 + 1.22474i 0.0252056 + 0.0436574i 0.878353 0.478012i \(-0.158643\pi\)
−0.853147 + 0.521670i \(0.825309\pi\)
\(788\) 1.00000 + 1.73205i 0.0356235 + 0.0617018i
\(789\) 0 0
\(790\) −11.3137 −0.402524
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 11.3137 + 19.5959i 0.401508 + 0.695433i
\(795\) 0 0
\(796\) −4.24264 + 7.34847i −0.150376 + 0.260460i
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) −4.00000 −0.141510
\(800\) 1.50000 2.59808i 0.0530330 0.0918559i
\(801\) 0 0
\(802\) 9.00000 + 15.5885i 0.317801 + 0.550448i
\(803\) 1.41421 2.44949i 0.0499065 0.0864406i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −4.24264 7.34847i −0.149256 0.258518i
\(809\) −8.00000 13.8564i −0.281265 0.487165i 0.690432 0.723398i \(-0.257421\pi\)
−0.971697 + 0.236232i \(0.924087\pi\)
\(810\) 0 0
\(811\) −29.6985 −1.04285 −0.521427 0.853296i \(-0.674600\pi\)
−0.521427 + 0.853296i \(0.674600\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 10.0000 17.3205i 0.350500 0.607083i
\(815\) 14.1421 + 24.4949i 0.495377 + 0.858019i
\(816\) 0 0
\(817\) 7.07107 12.2474i 0.247385 0.428484i
\(818\) −38.1838 −1.33506
\(819\) 0 0
\(820\) −28.0000 −0.977802
\(821\) −9.00000 + 15.5885i −0.314102 + 0.544041i −0.979246 0.202674i \(-0.935037\pi\)
0.665144 + 0.746715i \(0.268370\pi\)
\(822\) 0 0
\(823\) −20.0000 34.6410i −0.697156 1.20751i −0.969448 0.245295i \(-0.921115\pi\)
0.272292 0.962215i \(-0.412218\pi\)
\(824\) 1.41421 2.44949i 0.0492665 0.0853320i
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 15.5563 + 26.9444i 0.540294 + 0.935817i 0.998887 + 0.0471706i \(0.0150204\pi\)
−0.458593 + 0.888647i \(0.651646\pi\)
\(830\) 14.0000 + 24.2487i 0.485947 + 0.841685i
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 28.0000 48.4974i 0.968980 1.67832i
\(836\) 7.07107 + 12.2474i 0.244558 + 0.423587i
\(837\) 0 0
\(838\) 4.94975 8.57321i 0.170986 0.296157i
\(839\) −19.7990 −0.683537 −0.341769 0.939784i \(-0.611026\pi\)
−0.341769 + 0.939784i \(0.611026\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 15.0000 25.9808i 0.516934 0.895356i
\(843\) 0 0
\(844\) 6.00000 + 10.3923i 0.206529 + 0.357718i
\(845\) −18.3848 + 31.8434i −0.632456 + 1.09545i
\(846\) 0 0
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) 2.12132 + 3.67423i 0.0727607 + 0.126025i
\(851\) −20.0000 34.6410i −0.685591 1.18748i
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 2.00000 3.46410i 0.0683586 0.118401i
\(857\) 9.19239 + 15.9217i 0.314006 + 0.543874i 0.979226 0.202773i \(-0.0649955\pi\)
−0.665220 + 0.746648i \(0.731662\pi\)
\(858\) 0 0
\(859\) 13.4350 23.2702i 0.458397 0.793967i −0.540479 0.841357i \(-0.681757\pi\)
0.998876 + 0.0473900i \(0.0150904\pi\)
\(860\) −5.65685 −0.192897
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) −2.00000 + 3.46410i −0.0680808 + 0.117919i −0.898056 0.439880i \(-0.855021\pi\)
0.829976 + 0.557800i \(0.188354\pi\)
\(864\) 0 0
\(865\) 24.0000 + 41.5692i 0.816024 + 1.41340i
\(866\) −14.8492 + 25.7196i −0.504598 + 0.873989i
\(867\) 0 0
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) 0 0
\(872\) −1.00000 1.73205i −0.0338643 0.0586546i
\(873\) 0 0
\(874\) 28.2843 0.956730
\(875\) 0 0
\(876\) 0 0
\(877\) 23.0000 39.8372i 0.776655 1.34521i −0.157205 0.987566i \(-0.550248\pi\)
0.933860 0.357640i \(-0.116418\pi\)
\(878\) −8.48528 14.6969i −0.286364 0.495998i
\(879\) 0 0
\(880\) 2.82843 4.89898i 0.0953463 0.165145i
\(881\) −29.6985 −1.00057 −0.500284 0.865862i \(-0.666771\pi\)
−0.500284 + 0.865862i \(0.666771\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 2.00000 + 3.46410i 0.0671913 + 0.116379i
\(887\) −18.3848 + 31.8434i −0.617300 + 1.06920i 0.372676 + 0.927962i \(0.378440\pi\)
−0.989976 + 0.141234i \(0.954893\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 20.0000 0.670402
\(891\) 0 0
\(892\) 0 0
\(893\) −10.0000 17.3205i −0.334637 0.579609i
\(894\) 0 0
\(895\) 33.9411 1.13453
\(896\) 0 0
\(897\) 0 0
\(898\) −15.0000 + 25.9808i −0.500556 + 0.866989i
\(899\) 8.48528 + 14.6969i 0.283000 + 0.490170i
\(900\) 0 0
\(901\) −1.41421 + 2.44949i −0.0471143 + 0.0816043i
\(902\) −19.7990 −0.659234
\(903\) 0 0
\(904\) −12.0000 −0.399114
\(905\) 0 0
\(906\) 0 0
\(907\) 22.0000 + 38.1051i 0.730498 + 1.26526i 0.956671 + 0.291172i \(0.0940453\pi\)
−0.226173 + 0.974087i \(0.572621\pi\)
\(908\) −10.6066 + 18.3712i −0.351992 + 0.609669i
\(909\) 0 0
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) 9.89949 + 17.1464i 0.327625 + 0.567464i
\(914\) 12.0000 + 20.7846i 0.396925 + 0.687494i
\(915\) 0 0
\(916\) −16.9706 −0.560723
\(917\) 0 0
\(918\) 0 0
\(919\) 16.0000 27.7128i 0.527791 0.914161i −0.471684 0.881768i \(-0.656354\pi\)
0.999475 0.0323936i \(-0.0103130\pi\)
\(920\) −5.65685 9.79796i −0.186501 0.323029i
\(921\) 0 0
\(922\) −19.7990 + 34.2929i −0.652045 + 1.12938i
\(923\) 0 0
\(924\) 0 0
\(925\) 30.0000 0.986394
\(926\) 8.00000 13.8564i 0.262896 0.455350i
\(927\) 0 0
\(928\) −1.00000 1.73205i −0.0328266 0.0568574i
\(929\) 16.2635 28.1691i 0.533587 0.924199i −0.465644 0.884972i \(-0.654177\pi\)
0.999230 0.0392269i \(-0.0124895\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −24.0000 −0.786146
\(933\) 0 0
\(934\) −16.2635 28.1691i −0.532157 0.921722i
\(935\) 4.00000 + 6.92820i 0.130814 + 0.226576i
\(936\) 0 0
\(937\) 9.89949 0.323402 0.161701 0.986840i \(-0.448302\pi\)
0.161701 + 0.986840i \(0.448302\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −4.00000 + 6.92820i −0.130466 + 0.225973i
\(941\) −15.5563 26.9444i −0.507122 0.878362i −0.999966 0.00824396i \(-0.997376\pi\)
0.492844 0.870118i \(-0.335957\pi\)
\(942\) 0 0
\(943\) −19.7990 + 34.2929i −0.644744 + 1.11673i
\(944\) 1.41421 0.0460287
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −9.00000 + 15.5885i −0.292461 + 0.506557i −0.974391 0.224860i \(-0.927807\pi\)
0.681930 + 0.731417i \(0.261141\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −10.6066 + 18.3712i −0.344124 + 0.596040i
\(951\) 0 0
\(952\) 0 0
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 0 0
\(955\) 5.65685 + 9.79796i 0.183052 + 0.317055i
\(956\) −6.00000 10.3923i −0.194054 0.336111i
\(957\) 0 0
\(958\) −31.1127 −1.00521
\(959\) 0 0
\(960\) 0 0
\(961\) −20.5000 + 35.5070i −0.661290 + 1.14539i
\(962\) 0 0
\(963\) 0 0
\(964\) 10.6066 18.3712i 0.341616 0.591696i
\(965\) 45.2548 1.45680
\(966\) 0 0
\(967\) −12.0000 −0.385894 −0.192947 0.981209i \(-0.561805\pi\)
−0.192947 + 0.981209i \(0.561805\pi\)
\(968\) −3.50000 + 6.06218i −0.112494 + 0.194846i
\(969\) 0 0
\(970\) −14.0000 24.2487i −0.449513 0.778579i
\(971\) 16.2635 28.1691i 0.521919 0.903990i −0.477756 0.878493i \(-0.658550\pi\)
0.999675 0.0254978i \(-0.00811707\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −12.0000 −0.384505
\(975\) 0 0
\(976\) −1.41421 2.44949i −0.0452679 0.0784063i
\(977\) 6.00000 + 10.3923i 0.191957 + 0.332479i 0.945899 0.324462i \(-0.105183\pi\)
−0.753942 + 0.656941i \(0.771850\pi\)
\(978\) 0 0
\(979\) 14.1421 0.451985
\(980\) 0 0
\(981\) 0 0
\(982\) 6.00000 10.3923i 0.191468 0.331632i
\(983\) 24.0416 + 41.6413i 0.766809 + 1.32815i 0.939285 + 0.343138i \(0.111490\pi\)
−0.172476 + 0.985014i \(0.555177\pi\)
\(984\) 0 0
\(985\) −2.82843 + 4.89898i −0.0901212 + 0.156094i
\(986\) 2.82843 0.0900755
\(987\) 0 0
\(988\) 0 0
\(989\) −4.00000 + 6.92820i −0.127193 + 0.220304i
\(990\) 0 0
\(991\) 8.00000 + 13.8564i 0.254128 + 0.440163i 0.964658 0.263504i \(-0.0848781\pi\)
−0.710530 + 0.703667i \(0.751545\pi\)
\(992\) 4.24264 7.34847i 0.134704 0.233314i
\(993\) 0 0
\(994\) 0 0
\(995\) −24.0000 −0.760851
\(996\) 0 0
\(997\) 15.5563 + 26.9444i 0.492675 + 0.853337i 0.999964 0.00843818i \(-0.00268599\pi\)
−0.507290 + 0.861775i \(0.669353\pi\)
\(998\) −2.00000 3.46410i −0.0633089 0.109654i
\(999\) 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 882.2.g.l.667.2 4
3.2 odd 2 98.2.c.c.79.2 4
7.2 even 3 882.2.a.n.1.1 2
7.3 odd 6 inner 882.2.g.l.361.1 4
7.4 even 3 inner 882.2.g.l.361.2 4
7.5 odd 6 882.2.a.n.1.2 2
7.6 odd 2 inner 882.2.g.l.667.1 4
12.11 even 2 784.2.i.m.177.1 4
21.2 odd 6 98.2.a.b.1.1 2
21.5 even 6 98.2.a.b.1.2 yes 2
21.11 odd 6 98.2.c.c.67.2 4
21.17 even 6 98.2.c.c.67.1 4
21.20 even 2 98.2.c.c.79.1 4
28.19 even 6 7056.2.a.cl.1.2 2
28.23 odd 6 7056.2.a.cl.1.1 2
84.11 even 6 784.2.i.m.753.1 4
84.23 even 6 784.2.a.l.1.2 2
84.47 odd 6 784.2.a.l.1.1 2
84.59 odd 6 784.2.i.m.753.2 4
84.83 odd 2 784.2.i.m.177.2 4
105.2 even 12 2450.2.c.v.99.4 4
105.23 even 12 2450.2.c.v.99.1 4
105.44 odd 6 2450.2.a.bj.1.2 2
105.47 odd 12 2450.2.c.v.99.3 4
105.68 odd 12 2450.2.c.v.99.2 4
105.89 even 6 2450.2.a.bj.1.1 2
168.5 even 6 3136.2.a.bn.1.1 2
168.107 even 6 3136.2.a.bm.1.1 2
168.131 odd 6 3136.2.a.bm.1.2 2
168.149 odd 6 3136.2.a.bn.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.2.a.b.1.1 2 21.2 odd 6
98.2.a.b.1.2 yes 2 21.5 even 6
98.2.c.c.67.1 4 21.17 even 6
98.2.c.c.67.2 4 21.11 odd 6
98.2.c.c.79.1 4 21.20 even 2
98.2.c.c.79.2 4 3.2 odd 2
784.2.a.l.1.1 2 84.47 odd 6
784.2.a.l.1.2 2 84.23 even 6
784.2.i.m.177.1 4 12.11 even 2
784.2.i.m.177.2 4 84.83 odd 2
784.2.i.m.753.1 4 84.11 even 6
784.2.i.m.753.2 4 84.59 odd 6
882.2.a.n.1.1 2 7.2 even 3
882.2.a.n.1.2 2 7.5 odd 6
882.2.g.l.361.1 4 7.3 odd 6 inner
882.2.g.l.361.2 4 7.4 even 3 inner
882.2.g.l.667.1 4 7.6 odd 2 inner
882.2.g.l.667.2 4 1.1 even 1 trivial
2450.2.a.bj.1.1 2 105.89 even 6
2450.2.a.bj.1.2 2 105.44 odd 6
2450.2.c.v.99.1 4 105.23 even 12
2450.2.c.v.99.2 4 105.68 odd 12
2450.2.c.v.99.3 4 105.47 odd 12
2450.2.c.v.99.4 4 105.2 even 12
3136.2.a.bm.1.1 2 168.107 even 6
3136.2.a.bm.1.2 2 168.131 odd 6
3136.2.a.bn.1.1 2 168.5 even 6
3136.2.a.bn.1.2 2 168.149 odd 6
7056.2.a.cl.1.1 2 28.23 odd 6
7056.2.a.cl.1.2 2 28.19 even 6