Properties

Label 882.4.g.bg.667.1
Level $882$
Weight $4$
Character 882.667
Analytic conductor $52.040$
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] = [882,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.361");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(52.0396846251\)
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 667.1
Root \(-0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 882.667
Dual form 882.4.g.bg.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-3.53553 + 6.12372i) q^{5} -8.00000 q^{8} +O(q^{10})\) \(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-3.53553 + 6.12372i) q^{5} -8.00000 q^{8} +(7.07107 + 12.2474i) q^{10} +(-20.0000 - 34.6410i) q^{11} +63.6396 q^{13} +(-8.00000 + 13.8564i) q^{16} +(0.707107 + 1.22474i) q^{17} +(5.65685 - 9.79796i) q^{19} +28.2843 q^{20} -80.0000 q^{22} +(-34.0000 + 58.8897i) q^{23} +(37.5000 + 64.9519i) q^{25} +(63.6396 - 110.227i) q^{26} +110.000 q^{29} +(-59.3970 - 102.879i) q^{31} +(16.0000 + 27.7128i) q^{32} +2.82843 q^{34} +(10.0000 - 17.3205i) q^{37} +(-11.3137 - 19.5959i) q^{38} +(28.2843 - 48.9898i) q^{40} +49.4975 q^{41} -340.000 q^{43} +(-80.0000 + 138.564i) q^{44} +(68.0000 + 117.779i) q^{46} +(45.2548 - 78.3837i) q^{47} +150.000 q^{50} +(-127.279 - 220.454i) q^{52} +(-314.000 - 543.864i) q^{53} +282.843 q^{55} +(110.000 - 190.526i) q^{58} +(-438.406 - 759.342i) q^{59} +(458.912 - 794.859i) q^{61} -237.588 q^{62} +64.0000 q^{64} +(-225.000 + 389.711i) q^{65} +(-270.000 - 467.654i) q^{67} +(2.82843 - 4.89898i) q^{68} -420.000 q^{71} +(144.957 + 251.073i) q^{73} +(-20.0000 - 34.6410i) q^{74} -45.2548 q^{76} +(380.000 - 658.179i) q^{79} +(-56.5685 - 97.9796i) q^{80} +(49.4975 - 85.7321i) q^{82} -944.695 q^{83} -10.0000 q^{85} +(-340.000 + 588.897i) q^{86} +(160.000 + 277.128i) q^{88} +(576.292 - 998.167i) q^{89} +272.000 q^{92} +(-90.5097 - 156.767i) q^{94} +(40.0000 + 69.2820i) q^{95} +502.046 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 8 q^{4} - 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 8 q^{4} - 32 q^{8} - 80 q^{11} - 32 q^{16} - 320 q^{22} - 136 q^{23} + 150 q^{25} + 440 q^{29} + 64 q^{32} + 40 q^{37} - 1360 q^{43} - 320 q^{44} + 272 q^{46} + 600 q^{50} - 1256 q^{53} + 440 q^{58} + 256 q^{64} - 900 q^{65} - 1080 q^{67} - 1680 q^{71} - 80 q^{74} + 1520 q^{79} - 40 q^{85} - 1360 q^{86} + 640 q^{88} + 1088 q^{92} + 160 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 1.73205i 0.353553 0.612372i
\(3\) 0 0
\(4\) −2.00000 3.46410i −0.250000 0.433013i
\(5\) −3.53553 + 6.12372i −0.316228 + 0.547723i −0.979698 0.200480i \(-0.935750\pi\)
0.663470 + 0.748203i \(0.269083\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 7.07107 + 12.2474i 0.223607 + 0.387298i
\(11\) −20.0000 34.6410i −0.548202 0.949514i −0.998398 0.0565844i \(-0.981979\pi\)
0.450195 0.892930i \(-0.351354\pi\)
\(12\) 0 0
\(13\) 63.6396 1.35773 0.678864 0.734264i \(-0.262473\pi\)
0.678864 + 0.734264i \(0.262473\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −8.00000 + 13.8564i −0.125000 + 0.216506i
\(17\) 0.707107 + 1.22474i 0.0100882 + 0.0174732i 0.871025 0.491238i \(-0.163455\pi\)
−0.860937 + 0.508711i \(0.830122\pi\)
\(18\) 0 0
\(19\) 5.65685 9.79796i 0.0683038 0.118306i −0.829851 0.557985i \(-0.811575\pi\)
0.898155 + 0.439679i \(0.144908\pi\)
\(20\) 28.2843 0.316228
\(21\) 0 0
\(22\) −80.0000 −0.775275
\(23\) −34.0000 + 58.8897i −0.308239 + 0.533885i −0.977977 0.208712i \(-0.933073\pi\)
0.669738 + 0.742597i \(0.266406\pi\)
\(24\) 0 0
\(25\) 37.5000 + 64.9519i 0.300000 + 0.519615i
\(26\) 63.6396 110.227i 0.480029 0.831435i
\(27\) 0 0
\(28\) 0 0
\(29\) 110.000 0.704362 0.352181 0.935932i \(-0.385440\pi\)
0.352181 + 0.935932i \(0.385440\pi\)
\(30\) 0 0
\(31\) −59.3970 102.879i −0.344129 0.596050i 0.641066 0.767486i \(-0.278493\pi\)
−0.985195 + 0.171436i \(0.945159\pi\)
\(32\) 16.0000 + 27.7128i 0.0883883 + 0.153093i
\(33\) 0 0
\(34\) 2.82843 0.0142668
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000 17.3205i 0.0444322 0.0769588i −0.842954 0.537986i \(-0.819185\pi\)
0.887386 + 0.461027i \(0.152519\pi\)
\(38\) −11.3137 19.5959i −0.0482980 0.0836547i
\(39\) 0 0
\(40\) 28.2843 48.9898i 0.111803 0.193649i
\(41\) 49.4975 0.188542 0.0942708 0.995547i \(-0.469948\pi\)
0.0942708 + 0.995547i \(0.469948\pi\)
\(42\) 0 0
\(43\) −340.000 −1.20580 −0.602901 0.797816i \(-0.705989\pi\)
−0.602901 + 0.797816i \(0.705989\pi\)
\(44\) −80.0000 + 138.564i −0.274101 + 0.474757i
\(45\) 0 0
\(46\) 68.0000 + 117.779i 0.217958 + 0.377514i
\(47\) 45.2548 78.3837i 0.140449 0.243265i −0.787217 0.616676i \(-0.788479\pi\)
0.927666 + 0.373412i \(0.121812\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 150.000 0.424264
\(51\) 0 0
\(52\) −127.279 220.454i −0.339432 0.587913i
\(53\) −314.000 543.864i −0.813797 1.40954i −0.910189 0.414194i \(-0.864064\pi\)
0.0963923 0.995343i \(-0.469270\pi\)
\(54\) 0 0
\(55\) 282.843 0.693427
\(56\) 0 0
\(57\) 0 0
\(58\) 110.000 190.526i 0.249029 0.431332i
\(59\) −438.406 759.342i −0.967383 1.67556i −0.703070 0.711121i \(-0.748188\pi\)
−0.264313 0.964437i \(-0.585145\pi\)
\(60\) 0 0
\(61\) 458.912 794.859i 0.963241 1.66838i 0.248972 0.968511i \(-0.419907\pi\)
0.714269 0.699872i \(-0.246760\pi\)
\(62\) −237.588 −0.486672
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −225.000 + 389.711i −0.429351 + 0.743658i
\(66\) 0 0
\(67\) −270.000 467.654i −0.492325 0.852731i 0.507636 0.861571i \(-0.330519\pi\)
−0.999961 + 0.00884020i \(0.997186\pi\)
\(68\) 2.82843 4.89898i 0.00504408 0.00873660i
\(69\) 0 0
\(70\) 0 0
\(71\) −420.000 −0.702040 −0.351020 0.936368i \(-0.614165\pi\)
−0.351020 + 0.936368i \(0.614165\pi\)
\(72\) 0 0
\(73\) 144.957 + 251.073i 0.232410 + 0.402546i 0.958517 0.285036i \(-0.0920056\pi\)
−0.726107 + 0.687582i \(0.758672\pi\)
\(74\) −20.0000 34.6410i −0.0314183 0.0544181i
\(75\) 0 0
\(76\) −45.2548 −0.0683038
\(77\) 0 0
\(78\) 0 0
\(79\) 380.000 658.179i 0.541182 0.937354i −0.457655 0.889130i \(-0.651311\pi\)
0.998837 0.0482240i \(-0.0153561\pi\)
\(80\) −56.5685 97.9796i −0.0790569 0.136931i
\(81\) 0 0
\(82\) 49.4975 85.7321i 0.0666595 0.115458i
\(83\) −944.695 −1.24932 −0.624661 0.780896i \(-0.714763\pi\)
−0.624661 + 0.780896i \(0.714763\pi\)
\(84\) 0 0
\(85\) −10.0000 −0.0127606
\(86\) −340.000 + 588.897i −0.426316 + 0.738400i
\(87\) 0 0
\(88\) 160.000 + 277.128i 0.193819 + 0.335704i
\(89\) 576.292 998.167i 0.686369 1.18883i −0.286636 0.958040i \(-0.592537\pi\)
0.973005 0.230786i \(-0.0741298\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 272.000 0.308239
\(93\) 0 0
\(94\) −90.5097 156.767i −0.0993123 0.172014i
\(95\) 40.0000 + 69.2820i 0.0431991 + 0.0748230i
\(96\) 0 0
\(97\) 502.046 0.525516 0.262758 0.964862i \(-0.415368\pi\)
0.262758 + 0.964862i \(0.415368\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 150.000 259.808i 0.150000 0.259808i
\(101\) −880.348 1524.81i −0.867306 1.50222i −0.864739 0.502221i \(-0.832516\pi\)
−0.00256667 0.999997i \(-0.500817\pi\)
\(102\) 0 0
\(103\) 113.137 195.959i 0.108230 0.187461i −0.806823 0.590793i \(-0.798815\pi\)
0.915053 + 0.403333i \(0.132148\pi\)
\(104\) −509.117 −0.480029
\(105\) 0 0
\(106\) −1256.00 −1.15088
\(107\) −1012.00 + 1752.84i −0.914334 + 1.58367i −0.106460 + 0.994317i \(0.533952\pi\)
−0.807874 + 0.589356i \(0.799382\pi\)
\(108\) 0 0
\(109\) 202.000 + 349.874i 0.177505 + 0.307448i 0.941025 0.338336i \(-0.109864\pi\)
−0.763520 + 0.645784i \(0.776531\pi\)
\(110\) 282.843 489.898i 0.245164 0.424636i
\(111\) 0 0
\(112\) 0 0
\(113\) 1008.00 0.839156 0.419578 0.907719i \(-0.362178\pi\)
0.419578 + 0.907719i \(0.362178\pi\)
\(114\) 0 0
\(115\) −240.416 416.413i −0.194947 0.337659i
\(116\) −220.000 381.051i −0.176090 0.304998i
\(117\) 0 0
\(118\) −1753.62 −1.36809
\(119\) 0 0
\(120\) 0 0
\(121\) −134.500 + 232.961i −0.101052 + 0.175027i
\(122\) −917.825 1589.72i −0.681114 1.17972i
\(123\) 0 0
\(124\) −237.588 + 411.514i −0.172065 + 0.298025i
\(125\) −1414.21 −1.01193
\(126\) 0 0
\(127\) 1000.00 0.698706 0.349353 0.936991i \(-0.386401\pi\)
0.349353 + 0.936991i \(0.386401\pi\)
\(128\) 64.0000 110.851i 0.0441942 0.0765466i
\(129\) 0 0
\(130\) 450.000 + 779.423i 0.303597 + 0.525845i
\(131\) 42.4264 73.4847i 0.0282963 0.0490106i −0.851530 0.524305i \(-0.824325\pi\)
0.879827 + 0.475294i \(0.157658\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1080.00 −0.696252
\(135\) 0 0
\(136\) −5.65685 9.79796i −0.00356670 0.00617771i
\(137\) −1017.00 1761.50i −0.634220 1.09850i −0.986680 0.162675i \(-0.947988\pi\)
0.352460 0.935827i \(-0.385345\pi\)
\(138\) 0 0
\(139\) 1736.65 1.05972 0.529860 0.848085i \(-0.322244\pi\)
0.529860 + 0.848085i \(0.322244\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −420.000 + 727.461i −0.248209 + 0.429910i
\(143\) −1272.79 2204.54i −0.744309 1.28918i
\(144\) 0 0
\(145\) −388.909 + 673.610i −0.222739 + 0.385795i
\(146\) 579.828 0.328677
\(147\) 0 0
\(148\) −80.0000 −0.0444322
\(149\) −1070.00 + 1853.29i −0.588307 + 1.01898i 0.406147 + 0.913808i \(0.366872\pi\)
−0.994454 + 0.105171i \(0.966461\pi\)
\(150\) 0 0
\(151\) −1060.00 1835.97i −0.571269 0.989466i −0.996436 0.0843517i \(-0.973118\pi\)
0.425167 0.905115i \(-0.360215\pi\)
\(152\) −45.2548 + 78.3837i −0.0241490 + 0.0418273i
\(153\) 0 0
\(154\) 0 0
\(155\) 840.000 0.435293
\(156\) 0 0
\(157\) 873.277 + 1512.56i 0.443918 + 0.768888i 0.997976 0.0635898i \(-0.0202549\pi\)
−0.554058 + 0.832478i \(0.686922\pi\)
\(158\) −760.000 1316.36i −0.382673 0.662809i
\(159\) 0 0
\(160\) −226.274 −0.111803
\(161\) 0 0
\(162\) 0 0
\(163\) −1670.00 + 2892.52i −0.802482 + 1.38994i 0.115497 + 0.993308i \(0.463154\pi\)
−0.917978 + 0.396631i \(0.870179\pi\)
\(164\) −98.9949 171.464i −0.0471354 0.0816409i
\(165\) 0 0
\(166\) −944.695 + 1636.26i −0.441702 + 0.765050i
\(167\) −367.696 −0.170378 −0.0851890 0.996365i \(-0.527149\pi\)
−0.0851890 + 0.996365i \(0.527149\pi\)
\(168\) 0 0
\(169\) 1853.00 0.843423
\(170\) −10.0000 + 17.3205i −0.00451156 + 0.00781425i
\(171\) 0 0
\(172\) 680.000 + 1177.79i 0.301451 + 0.522128i
\(173\) 1694.93 2935.71i 0.744876 1.29016i −0.205377 0.978683i \(-0.565842\pi\)
0.950253 0.311480i \(-0.100825\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 640.000 0.274101
\(177\) 0 0
\(178\) −1152.58 1996.33i −0.485336 0.840627i
\(179\) −360.000 623.538i −0.150322 0.260366i 0.781024 0.624501i \(-0.214698\pi\)
−0.931346 + 0.364136i \(0.881364\pi\)
\(180\) 0 0
\(181\) 1854.03 0.761377 0.380689 0.924703i \(-0.375687\pi\)
0.380689 + 0.924703i \(0.375687\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 272.000 471.118i 0.108979 0.188757i
\(185\) 70.7107 + 122.474i 0.0281014 + 0.0486730i
\(186\) 0 0
\(187\) 28.2843 48.9898i 0.0110607 0.0191577i
\(188\) −362.039 −0.140449
\(189\) 0 0
\(190\) 160.000 0.0610927
\(191\) −1990.00 + 3446.78i −0.753881 + 1.30576i 0.192047 + 0.981386i \(0.438487\pi\)
−0.945929 + 0.324375i \(0.894846\pi\)
\(192\) 0 0
\(193\) −1855.00 3212.95i −0.691844 1.19831i −0.971233 0.238130i \(-0.923465\pi\)
0.279390 0.960178i \(-0.409868\pi\)
\(194\) 502.046 869.569i 0.185798 0.321811i
\(195\) 0 0
\(196\) 0 0
\(197\) 956.000 0.345747 0.172874 0.984944i \(-0.444695\pi\)
0.172874 + 0.984944i \(0.444695\pi\)
\(198\) 0 0
\(199\) 2044.95 + 3541.96i 0.728457 + 1.26172i 0.957535 + 0.288316i \(0.0930953\pi\)
−0.229079 + 0.973408i \(0.573571\pi\)
\(200\) −300.000 519.615i −0.106066 0.183712i
\(201\) 0 0
\(202\) −3521.39 −1.22656
\(203\) 0 0
\(204\) 0 0
\(205\) −175.000 + 303.109i −0.0596221 + 0.103269i
\(206\) −226.274 391.918i −0.0765304 0.132555i
\(207\) 0 0
\(208\) −509.117 + 881.816i −0.169716 + 0.293957i
\(209\) −452.548 −0.149777
\(210\) 0 0
\(211\) 2868.00 0.935741 0.467870 0.883797i \(-0.345021\pi\)
0.467870 + 0.883797i \(0.345021\pi\)
\(212\) −1256.00 + 2175.46i −0.406898 + 0.704768i
\(213\) 0 0
\(214\) 2024.00 + 3505.67i 0.646532 + 1.11983i
\(215\) 1202.08 2082.07i 0.381308 0.660445i
\(216\) 0 0
\(217\) 0 0
\(218\) 808.000 0.251031
\(219\) 0 0
\(220\) −565.685 979.796i −0.173357 0.300263i
\(221\) 45.0000 + 77.9423i 0.0136970 + 0.0237238i
\(222\) 0 0
\(223\) −2630.44 −0.789897 −0.394949 0.918703i \(-0.629238\pi\)
−0.394949 + 0.918703i \(0.629238\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1008.00 1745.91i 0.296687 0.513876i
\(227\) −84.8528 146.969i −0.0248100 0.0429722i 0.853354 0.521332i \(-0.174565\pi\)
−0.878164 + 0.478360i \(0.841231\pi\)
\(228\) 0 0
\(229\) −1571.90 + 2722.61i −0.453598 + 0.785655i −0.998606 0.0527757i \(-0.983193\pi\)
0.545008 + 0.838431i \(0.316527\pi\)
\(230\) −961.665 −0.275697
\(231\) 0 0
\(232\) −880.000 −0.249029
\(233\) 2241.00 3881.53i 0.630098 1.09136i −0.357433 0.933939i \(-0.616348\pi\)
0.987531 0.157423i \(-0.0503186\pi\)
\(234\) 0 0
\(235\) 320.000 + 554.256i 0.0888277 + 0.153854i
\(236\) −1753.62 + 3037.37i −0.483692 + 0.837779i
\(237\) 0 0
\(238\) 0 0
\(239\) −1740.00 −0.470926 −0.235463 0.971883i \(-0.575661\pi\)
−0.235463 + 0.971883i \(0.575661\pi\)
\(240\) 0 0
\(241\) 630.032 + 1091.25i 0.168398 + 0.291674i 0.937857 0.347023i \(-0.112807\pi\)
−0.769459 + 0.638697i \(0.779474\pi\)
\(242\) 269.000 + 465.922i 0.0714544 + 0.123763i
\(243\) 0 0
\(244\) −3671.30 −0.963241
\(245\) 0 0
\(246\) 0 0
\(247\) 360.000 623.538i 0.0927379 0.160627i
\(248\) 475.176 + 823.029i 0.121668 + 0.210735i
\(249\) 0 0
\(250\) −1414.21 + 2449.49i −0.357771 + 0.619677i
\(251\) 5826.56 1.46522 0.732608 0.680651i \(-0.238303\pi\)
0.732608 + 0.680651i \(0.238303\pi\)
\(252\) 0 0
\(253\) 2720.00 0.675909
\(254\) 1000.00 1732.05i 0.247030 0.427868i
\(255\) 0 0
\(256\) −128.000 221.703i −0.0312500 0.0541266i
\(257\) −2344.06 + 4060.03i −0.568943 + 0.985438i 0.427728 + 0.903908i \(0.359314\pi\)
−0.996671 + 0.0815308i \(0.974019\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1800.00 0.429351
\(261\) 0 0
\(262\) −84.8528 146.969i −0.0200085 0.0346557i
\(263\) −1086.00 1881.01i −0.254622 0.441019i 0.710171 0.704030i \(-0.248618\pi\)
−0.964793 + 0.263011i \(0.915284\pi\)
\(264\) 0 0
\(265\) 4440.63 1.02938
\(266\) 0 0
\(267\) 0 0
\(268\) −1080.00 + 1870.61i −0.246162 + 0.426366i
\(269\) −1354.11 2345.39i −0.306920 0.531601i 0.670767 0.741668i \(-0.265965\pi\)
−0.977687 + 0.210067i \(0.932632\pi\)
\(270\) 0 0
\(271\) 3094.30 5359.48i 0.693599 1.20135i −0.277052 0.960855i \(-0.589357\pi\)
0.970651 0.240494i \(-0.0773093\pi\)
\(272\) −22.6274 −0.00504408
\(273\) 0 0
\(274\) −4068.00 −0.896923
\(275\) 1500.00 2598.08i 0.328921 0.569709i
\(276\) 0 0
\(277\) −3065.00 5308.74i −0.664830 1.15152i −0.979331 0.202263i \(-0.935170\pi\)
0.314501 0.949257i \(-0.398163\pi\)
\(278\) 1736.65 3007.97i 0.374668 0.648943i
\(279\) 0 0
\(280\) 0 0
\(281\) −1970.00 −0.418222 −0.209111 0.977892i \(-0.567057\pi\)
−0.209111 + 0.977892i \(0.567057\pi\)
\(282\) 0 0
\(283\) −777.817 1347.22i −0.163380 0.282982i 0.772699 0.634773i \(-0.218906\pi\)
−0.936079 + 0.351791i \(0.885573\pi\)
\(284\) 840.000 + 1454.92i 0.175510 + 0.303992i
\(285\) 0 0
\(286\) −5091.17 −1.05261
\(287\) 0 0
\(288\) 0 0
\(289\) 2455.50 4253.05i 0.499796 0.865673i
\(290\) 777.817 + 1347.22i 0.157500 + 0.272798i
\(291\) 0 0
\(292\) 579.828 1004.29i 0.116205 0.201273i
\(293\) −7686.25 −1.53254 −0.766272 0.642516i \(-0.777891\pi\)
−0.766272 + 0.642516i \(0.777891\pi\)
\(294\) 0 0
\(295\) 6200.00 1.22365
\(296\) −80.0000 + 138.564i −0.0157091 + 0.0272090i
\(297\) 0 0
\(298\) 2140.00 + 3706.59i 0.415996 + 0.720527i
\(299\) −2163.75 + 3747.72i −0.418504 + 0.724870i
\(300\) 0 0
\(301\) 0 0
\(302\) −4240.00 −0.807896
\(303\) 0 0
\(304\) 90.5097 + 156.767i 0.0170759 + 0.0295764i
\(305\) 3245.00 + 5620.50i 0.609207 + 1.05518i
\(306\) 0 0
\(307\) 8598.42 1.59849 0.799247 0.601003i \(-0.205232\pi\)
0.799247 + 0.601003i \(0.205232\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 840.000 1454.92i 0.153899 0.266561i
\(311\) 3790.09 + 6564.63i 0.691050 + 1.19693i 0.971494 + 0.237063i \(0.0761847\pi\)
−0.280445 + 0.959870i \(0.590482\pi\)
\(312\) 0 0
\(313\) 1537.96 2663.82i 0.277733 0.481048i −0.693088 0.720853i \(-0.743750\pi\)
0.970821 + 0.239805i \(0.0770836\pi\)
\(314\) 3493.11 0.627794
\(315\) 0 0
\(316\) −3040.00 −0.541182
\(317\) 1162.00 2012.64i 0.205881 0.356597i −0.744532 0.667587i \(-0.767327\pi\)
0.950413 + 0.310990i \(0.100661\pi\)
\(318\) 0 0
\(319\) −2200.00 3810.51i −0.386133 0.668802i
\(320\) −226.274 + 391.918i −0.0395285 + 0.0684653i
\(321\) 0 0
\(322\) 0 0
\(323\) 16.0000 0.00275623
\(324\) 0 0
\(325\) 2386.49 + 4133.51i 0.407318 + 0.705496i
\(326\) 3340.00 + 5785.05i 0.567440 + 0.982835i
\(327\) 0 0
\(328\) −395.980 −0.0666595
\(329\) 0 0
\(330\) 0 0
\(331\) −4754.00 + 8234.17i −0.789436 + 1.36734i 0.136876 + 0.990588i \(0.456294\pi\)
−0.926313 + 0.376756i \(0.877040\pi\)
\(332\) 1889.39 + 3272.52i 0.312330 + 0.540972i
\(333\) 0 0
\(334\) −367.696 + 636.867i −0.0602377 + 0.104335i
\(335\) 3818.38 0.622747
\(336\) 0 0
\(337\) −4720.00 −0.762952 −0.381476 0.924379i \(-0.624584\pi\)
−0.381476 + 0.924379i \(0.624584\pi\)
\(338\) 1853.00 3209.49i 0.298195 0.516489i
\(339\) 0 0
\(340\) 20.0000 + 34.6410i 0.00319015 + 0.00552551i
\(341\) −2375.88 + 4115.14i −0.377305 + 0.653512i
\(342\) 0 0
\(343\) 0 0
\(344\) 2720.00 0.426316
\(345\) 0 0
\(346\) −3389.87 5871.43i −0.526707 0.912283i
\(347\) 3252.00 + 5632.63i 0.503102 + 0.871399i 0.999994 + 0.00358597i \(0.00114145\pi\)
−0.496891 + 0.867813i \(0.665525\pi\)
\(348\) 0 0
\(349\) 5256.63 0.806249 0.403125 0.915145i \(-0.367924\pi\)
0.403125 + 0.915145i \(0.367924\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 640.000 1108.51i 0.0969094 0.167852i
\(353\) 6105.87 + 10575.7i 0.920630 + 1.59458i 0.798442 + 0.602072i \(0.205658\pi\)
0.122188 + 0.992507i \(0.461009\pi\)
\(354\) 0 0
\(355\) 1484.92 2571.96i 0.222004 0.384523i
\(356\) −4610.34 −0.686369
\(357\) 0 0
\(358\) −1440.00 −0.212588
\(359\) 3670.00 6356.63i 0.539541 0.934512i −0.459388 0.888236i \(-0.651931\pi\)
0.998929 0.0462765i \(-0.0147355\pi\)
\(360\) 0 0
\(361\) 3365.50 + 5829.22i 0.490669 + 0.849864i
\(362\) 1854.03 3211.28i 0.269187 0.466246i
\(363\) 0 0
\(364\) 0 0
\(365\) −2050.00 −0.293978
\(366\) 0 0
\(367\) −3832.52 6638.12i −0.545111 0.944160i −0.998600 0.0528984i \(-0.983154\pi\)
0.453489 0.891262i \(-0.350179\pi\)
\(368\) −544.000 942.236i −0.0770597 0.133471i
\(369\) 0 0
\(370\) 282.843 0.0397413
\(371\) 0 0
\(372\) 0 0
\(373\) 1495.00 2589.42i 0.207529 0.359450i −0.743407 0.668840i \(-0.766791\pi\)
0.950935 + 0.309389i \(0.100125\pi\)
\(374\) −56.5685 97.9796i −0.00782110 0.0135465i
\(375\) 0 0
\(376\) −362.039 + 627.069i −0.0496562 + 0.0860070i
\(377\) 7000.36 0.956331
\(378\) 0 0
\(379\) −11900.0 −1.61283 −0.806414 0.591351i \(-0.798595\pi\)
−0.806414 + 0.591351i \(0.798595\pi\)
\(380\) 160.000 277.128i 0.0215995 0.0374115i
\(381\) 0 0
\(382\) 3980.00 + 6893.56i 0.533075 + 0.923312i
\(383\) −4856.41 + 8411.55i −0.647914 + 1.12222i 0.335707 + 0.941967i \(0.391025\pi\)
−0.983620 + 0.180253i \(0.942308\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −7420.00 −0.978415
\(387\) 0 0
\(388\) −1004.09 1739.14i −0.131379 0.227555i
\(389\) −1075.00 1861.95i −0.140115 0.242686i 0.787425 0.616411i \(-0.211414\pi\)
−0.927540 + 0.373725i \(0.878081\pi\)
\(390\) 0 0
\(391\) −96.1665 −0.0124382
\(392\) 0 0
\(393\) 0 0
\(394\) 956.000 1655.84i 0.122240 0.211726i
\(395\) 2687.01 + 4654.03i 0.342273 + 0.592835i
\(396\) 0 0
\(397\) 1700.59 2945.51i 0.214988 0.372370i −0.738281 0.674493i \(-0.764362\pi\)
0.953269 + 0.302123i \(0.0976954\pi\)
\(398\) 8179.81 1.03019
\(399\) 0 0
\(400\) −1200.00 −0.150000
\(401\) 6045.00 10470.2i 0.752800 1.30389i −0.193660 0.981069i \(-0.562036\pi\)
0.946461 0.322820i \(-0.104631\pi\)
\(402\) 0 0
\(403\) −3780.00 6547.15i −0.467234 0.809273i
\(404\) −3521.39 + 6099.23i −0.433653 + 0.751109i
\(405\) 0 0
\(406\) 0 0
\(407\) −800.000 −0.0974313
\(408\) 0 0
\(409\) 4096.27 + 7094.95i 0.495226 + 0.857757i 0.999985 0.00550362i \(-0.00175187\pi\)
−0.504759 + 0.863260i \(0.668419\pi\)
\(410\) 350.000 + 606.218i 0.0421592 + 0.0730219i
\(411\) 0 0
\(412\) −905.097 −0.108230
\(413\) 0 0
\(414\) 0 0
\(415\) 3340.00 5785.05i 0.395070 0.684282i
\(416\) 1018.23 + 1763.63i 0.120007 + 0.207859i
\(417\) 0 0
\(418\) −452.548 + 783.837i −0.0529542 + 0.0917194i
\(419\) 1046.52 0.122019 0.0610093 0.998137i \(-0.480568\pi\)
0.0610093 + 0.998137i \(0.480568\pi\)
\(420\) 0 0
\(421\) −3870.00 −0.448010 −0.224005 0.974588i \(-0.571913\pi\)
−0.224005 + 0.974588i \(0.571913\pi\)
\(422\) 2868.00 4967.52i 0.330834 0.573022i
\(423\) 0 0
\(424\) 2512.00 + 4350.91i 0.287721 + 0.498347i
\(425\) −53.0330 + 91.8559i −0.00605289 + 0.0104839i
\(426\) 0 0
\(427\) 0 0
\(428\) 8096.00 0.914334
\(429\) 0 0
\(430\) −2404.16 4164.13i −0.269626 0.467005i
\(431\) 1350.00 + 2338.27i 0.150875 + 0.261324i 0.931549 0.363615i \(-0.118458\pi\)
−0.780674 + 0.624938i \(0.785124\pi\)
\(432\) 0 0
\(433\) 5876.06 0.652160 0.326080 0.945342i \(-0.394272\pi\)
0.326080 + 0.945342i \(0.394272\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 808.000 1399.50i 0.0887527 0.153724i
\(437\) 384.666 + 666.261i 0.0421077 + 0.0729327i
\(438\) 0 0
\(439\) 3173.50 5496.65i 0.345017 0.597588i −0.640339 0.768092i \(-0.721206\pi\)
0.985357 + 0.170504i \(0.0545397\pi\)
\(440\) −2262.74 −0.245164
\(441\) 0 0
\(442\) 180.000 0.0193704
\(443\) 3464.00 5999.82i 0.371512 0.643477i −0.618287 0.785953i \(-0.712173\pi\)
0.989798 + 0.142476i \(0.0455063\pi\)
\(444\) 0 0
\(445\) 4075.00 + 7058.11i 0.434098 + 0.751879i
\(446\) −2630.44 + 4556.05i −0.279271 + 0.483711i
\(447\) 0 0
\(448\) 0 0
\(449\) −1320.00 −0.138741 −0.0693704 0.997591i \(-0.522099\pi\)
−0.0693704 + 0.997591i \(0.522099\pi\)
\(450\) 0 0
\(451\) −989.949 1714.64i −0.103359 0.179023i
\(452\) −2016.00 3491.81i −0.209789 0.363365i
\(453\) 0 0
\(454\) −339.411 −0.0350867
\(455\) 0 0
\(456\) 0 0
\(457\) −645.000 + 1117.17i −0.0660215 + 0.114353i −0.897147 0.441733i \(-0.854364\pi\)
0.831125 + 0.556085i \(0.187697\pi\)
\(458\) 3143.80 + 5445.22i 0.320742 + 0.555542i
\(459\) 0 0
\(460\) −961.665 + 1665.65i −0.0974736 + 0.168829i
\(461\) 17642.3 1.78240 0.891198 0.453615i \(-0.149866\pi\)
0.891198 + 0.453615i \(0.149866\pi\)
\(462\) 0 0
\(463\) 5680.00 0.570134 0.285067 0.958508i \(-0.407984\pi\)
0.285067 + 0.958508i \(0.407984\pi\)
\(464\) −880.000 + 1524.20i −0.0880452 + 0.152499i
\(465\) 0 0
\(466\) −4482.00 7763.05i −0.445546 0.771709i
\(467\) 3846.66 6662.61i 0.381161 0.660190i −0.610067 0.792350i \(-0.708858\pi\)
0.991229 + 0.132159i \(0.0421910\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1280.00 0.125621
\(471\) 0 0
\(472\) 3507.25 + 6074.73i 0.342022 + 0.592399i
\(473\) 6800.00 + 11777.9i 0.661024 + 1.14493i
\(474\) 0 0
\(475\) 848.528 0.0819645
\(476\) 0 0
\(477\) 0 0
\(478\) −1740.00 + 3013.77i −0.166497 + 0.288382i
\(479\) 8259.01 + 14305.0i 0.787816 + 1.36454i 0.927303 + 0.374313i \(0.122121\pi\)
−0.139487 + 0.990224i \(0.544545\pi\)
\(480\) 0 0
\(481\) 636.396 1102.27i 0.0603267 0.104489i
\(482\) 2520.13 0.238151
\(483\) 0 0
\(484\) 1076.00 0.101052
\(485\) −1775.00 + 3074.39i −0.166183 + 0.287837i
\(486\) 0 0
\(487\) −6840.00 11847.2i −0.636448 1.10236i −0.986206 0.165520i \(-0.947070\pi\)
0.349759 0.936840i \(-0.386264\pi\)
\(488\) −3671.30 + 6358.88i −0.340557 + 0.589862i
\(489\) 0 0
\(490\) 0 0
\(491\) 2280.00 0.209562 0.104781 0.994495i \(-0.466586\pi\)
0.104781 + 0.994495i \(0.466586\pi\)
\(492\) 0 0
\(493\) 77.7817 + 134.722i 0.00710571 + 0.0123074i
\(494\) −720.000 1247.08i −0.0655756 0.113580i
\(495\) 0 0
\(496\) 1900.70 0.172065
\(497\) 0 0
\(498\) 0 0
\(499\) −430.000 + 744.782i −0.0385760 + 0.0668157i −0.884669 0.466220i \(-0.845616\pi\)
0.846093 + 0.533036i \(0.178949\pi\)
\(500\) 2828.43 + 4898.98i 0.252982 + 0.438178i
\(501\) 0 0
\(502\) 5826.56 10091.9i 0.518032 0.897258i
\(503\) −5730.39 −0.507963 −0.253982 0.967209i \(-0.581740\pi\)
−0.253982 + 0.967209i \(0.581740\pi\)
\(504\) 0 0
\(505\) 12450.0 1.09706
\(506\) 2720.00 4711.18i 0.238970 0.413908i
\(507\) 0 0
\(508\) −2000.00 3464.10i −0.174676 0.302549i
\(509\) 2294.56 3974.30i 0.199813 0.346086i −0.748655 0.662960i \(-0.769300\pi\)
0.948468 + 0.316874i \(0.102633\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) 4688.12 + 8120.06i 0.402304 + 0.696810i
\(515\) 800.000 + 1385.64i 0.0684509 + 0.118560i
\(516\) 0 0
\(517\) −3620.39 −0.307978
\(518\) 0 0
\(519\) 0 0
\(520\) 1800.00 3117.69i 0.151799 0.262923i
\(521\) 7293.81 + 12633.2i 0.613335 + 1.06233i 0.990674 + 0.136252i \(0.0435057\pi\)
−0.377339 + 0.926075i \(0.623161\pi\)
\(522\) 0 0
\(523\) −3054.70 + 5290.90i −0.255397 + 0.442361i −0.965003 0.262238i \(-0.915540\pi\)
0.709606 + 0.704599i \(0.248873\pi\)
\(524\) −339.411 −0.0282963
\(525\) 0 0
\(526\) −4344.00 −0.360090
\(527\) 84.0000 145.492i 0.00694326 0.0120261i
\(528\) 0 0
\(529\) 3771.50 + 6532.43i 0.309978 + 0.536897i
\(530\) 4440.63 7691.40i 0.363941 0.630364i
\(531\) 0 0
\(532\) 0 0
\(533\) 3150.00 0.255988
\(534\) 0 0
\(535\) −7155.92 12394.4i −0.578276 1.00160i
\(536\) 2160.00 + 3741.23i 0.174063 + 0.301486i
\(537\) 0 0
\(538\) −5416.44 −0.434051
\(539\) 0 0
\(540\) 0 0
\(541\) 8605.00 14904.3i 0.683841 1.18445i −0.289959 0.957039i \(-0.593642\pi\)
0.973800 0.227408i \(-0.0730250\pi\)
\(542\) −6188.60 10719.0i −0.490448 0.849482i
\(543\) 0 0
\(544\) −22.6274 + 39.1918i −0.00178335 + 0.00308885i
\(545\) −2856.71 −0.224529
\(546\) 0 0
\(547\) 4060.00 0.317355 0.158677 0.987330i \(-0.449277\pi\)
0.158677 + 0.987330i \(0.449277\pi\)
\(548\) −4068.00 + 7045.98i −0.317110 + 0.549251i
\(549\) 0 0
\(550\) −3000.00 5196.15i −0.232583 0.402845i
\(551\) 622.254 1077.78i 0.0481105 0.0833299i
\(552\) 0 0
\(553\) 0 0
\(554\) −12260.0 −0.940212
\(555\) 0 0
\(556\) −3473.31 6015.95i −0.264930 0.458872i
\(557\) −5178.00 8968.56i −0.393894 0.682244i 0.599065 0.800700i \(-0.295539\pi\)
−0.992959 + 0.118456i \(0.962206\pi\)
\(558\) 0 0
\(559\) −21637.5 −1.63715
\(560\) 0 0
\(561\) 0 0
\(562\) −1970.00 + 3412.14i −0.147864 + 0.256108i
\(563\) −11605.0 20100.5i −0.868728 1.50468i −0.863297 0.504696i \(-0.831605\pi\)
−0.00543113 0.999985i \(-0.501729\pi\)
\(564\) 0 0
\(565\) −3563.82 + 6172.71i −0.265365 + 0.459625i
\(566\) −3111.27 −0.231054
\(567\) 0 0
\(568\) 3360.00 0.248209
\(569\) −2945.00 + 5100.89i −0.216979 + 0.375818i −0.953883 0.300179i \(-0.902954\pi\)
0.736904 + 0.675997i \(0.236287\pi\)
\(570\) 0 0
\(571\) 2806.00 + 4860.13i 0.205652 + 0.356200i 0.950340 0.311212i \(-0.100735\pi\)
−0.744688 + 0.667413i \(0.767402\pi\)
\(572\) −5091.17 + 8818.16i −0.372155 + 0.644591i
\(573\) 0 0
\(574\) 0 0
\(575\) −5100.00 −0.369886
\(576\) 0 0
\(577\) 8898.94 + 15413.4i 0.642058 + 1.11208i 0.984973 + 0.172711i \(0.0552527\pi\)
−0.342914 + 0.939367i \(0.611414\pi\)
\(578\) −4911.00 8506.10i −0.353409 0.612123i
\(579\) 0 0
\(580\) 3111.27 0.222739
\(581\) 0 0
\(582\) 0 0
\(583\) −12560.0 + 21754.6i −0.892251 + 1.54542i
\(584\) −1159.66 2008.58i −0.0821693 0.142321i
\(585\) 0 0
\(586\) −7686.25 + 13313.0i −0.541836 + 0.938488i
\(587\) −7942.22 −0.558451 −0.279225 0.960226i \(-0.590078\pi\)
−0.279225 + 0.960226i \(0.590078\pi\)
\(588\) 0 0
\(589\) −1344.00 −0.0940213
\(590\) 6200.00 10738.7i 0.432627 0.749332i
\(591\) 0 0
\(592\) 160.000 + 277.128i 0.0111080 + 0.0192397i
\(593\) −7039.25 + 12192.3i −0.487466 + 0.844316i −0.999896 0.0144132i \(-0.995412\pi\)
0.512430 + 0.858729i \(0.328745\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 8560.00 0.588307
\(597\) 0 0
\(598\) 4327.49 + 7495.44i 0.295927 + 0.512561i
\(599\) 3650.00 + 6321.99i 0.248973 + 0.431234i 0.963241 0.268638i \(-0.0865736\pi\)
−0.714268 + 0.699872i \(0.753240\pi\)
\(600\) 0 0
\(601\) −8727.11 −0.592323 −0.296162 0.955138i \(-0.595707\pi\)
−0.296162 + 0.955138i \(0.595707\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −4240.00 + 7343.90i −0.285634 + 0.494733i
\(605\) −951.059 1647.28i −0.0639108 0.110697i
\(606\) 0 0
\(607\) −5303.30 + 9185.59i −0.354620 + 0.614220i −0.987053 0.160395i \(-0.948723\pi\)
0.632433 + 0.774615i \(0.282056\pi\)
\(608\) 362.039 0.0241490
\(609\) 0 0
\(610\) 12980.0 0.861549
\(611\) 2880.00 4988.31i 0.190691 0.330287i
\(612\) 0 0
\(613\) 6990.00 + 12107.0i 0.460560 + 0.797714i 0.998989 0.0449573i \(-0.0143152\pi\)
−0.538429 + 0.842671i \(0.680982\pi\)
\(614\) 8598.42 14892.9i 0.565153 0.978874i
\(615\) 0 0
\(616\) 0 0
\(617\) −2654.00 −0.173170 −0.0865851 0.996244i \(-0.527595\pi\)
−0.0865851 + 0.996244i \(0.527595\pi\)
\(618\) 0 0
\(619\) −11941.6 20683.5i −0.775403 1.34304i −0.934568 0.355785i \(-0.884214\pi\)
0.159165 0.987252i \(-0.449120\pi\)
\(620\) −1680.00 2909.85i −0.108823 0.188487i
\(621\) 0 0
\(622\) 15160.4 0.977292
\(623\) 0 0
\(624\) 0 0
\(625\) 312.500 541.266i 0.0200000 0.0346410i
\(626\) −3075.91 5327.64i −0.196387 0.340152i
\(627\) 0 0
\(628\) 3493.11 6050.24i 0.221959 0.384444i
\(629\) 28.2843 0.00179295
\(630\) 0 0
\(631\) −6400.00 −0.403772 −0.201886 0.979409i \(-0.564707\pi\)
−0.201886 + 0.979409i \(0.564707\pi\)
\(632\) −3040.00 + 5265.43i −0.191337 + 0.331405i
\(633\) 0 0
\(634\) −2324.00 4025.29i −0.145580 0.252152i
\(635\) −3535.53 + 6123.72i −0.220950 + 0.382697i
\(636\) 0 0
\(637\) 0 0
\(638\) −8800.00 −0.546074
\(639\) 0 0
\(640\) 452.548 + 783.837i 0.0279508 + 0.0484123i
\(641\) −7675.00 13293.5i −0.472924 0.819128i 0.526596 0.850116i \(-0.323468\pi\)
−0.999520 + 0.0309874i \(0.990135\pi\)
\(642\) 0 0
\(643\) −17847.4 −1.09461 −0.547303 0.836934i \(-0.684345\pi\)
−0.547303 + 0.836934i \(0.684345\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 16.0000 27.7128i 0.000974476 0.00168784i
\(647\) −7000.36 12125.0i −0.425367 0.736757i 0.571088 0.820889i \(-0.306522\pi\)
−0.996455 + 0.0841319i \(0.973188\pi\)
\(648\) 0 0
\(649\) −17536.2 + 30373.7i −1.06064 + 1.83709i
\(650\) 9545.94 0.576035
\(651\) 0 0
\(652\) 13360.0 0.802482
\(653\) −13191.0 + 22847.5i −0.790511 + 1.36921i 0.135140 + 0.990827i \(0.456852\pi\)
−0.925651 + 0.378379i \(0.876482\pi\)
\(654\) 0 0
\(655\) 300.000 + 519.615i 0.0178961 + 0.0309970i
\(656\) −395.980 + 685.857i −0.0235677 + 0.0408205i
\(657\) 0 0
\(658\) 0 0
\(659\) 14400.0 0.851205 0.425603 0.904910i \(-0.360062\pi\)
0.425603 + 0.904910i \(0.360062\pi\)
\(660\) 0 0
\(661\) 14291.3 + 24753.3i 0.840951 + 1.45657i 0.889092 + 0.457729i \(0.151337\pi\)
−0.0481409 + 0.998841i \(0.515330\pi\)
\(662\) 9508.00 + 16468.3i 0.558216 + 0.966858i
\(663\) 0 0
\(664\) 7557.56 0.441702
\(665\) 0 0
\(666\) 0 0
\(667\) −3740.00 + 6477.87i −0.217112 + 0.376048i
\(668\) 735.391 + 1273.73i 0.0425945 + 0.0737759i
\(669\) 0 0
\(670\) 3818.38 6613.62i 0.220174 0.381353i
\(671\) −36713.0 −2.11220
\(672\) 0 0
\(673\) −18120.0 −1.03785 −0.518926 0.854819i \(-0.673668\pi\)
−0.518926 + 0.854819i \(0.673668\pi\)
\(674\) −4720.00 + 8175.28i −0.269744 + 0.467211i
\(675\) 0 0
\(676\) −3706.00 6418.98i −0.210856 0.365213i
\(677\) 8898.94 15413.4i 0.505191 0.875016i −0.494791 0.869012i \(-0.664756\pi\)
0.999982 0.00600394i \(-0.00191112\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 80.0000 0.00451156
\(681\) 0 0
\(682\) 4751.76 + 8230.29i 0.266795 + 0.462103i
\(683\) −864.000 1496.49i −0.0484042 0.0838385i 0.840808 0.541333i \(-0.182080\pi\)
−0.889212 + 0.457495i \(0.848747\pi\)
\(684\) 0 0
\(685\) 14382.6 0.802232
\(686\) 0 0
\(687\) 0 0
\(688\) 2720.00 4711.18i 0.150725 0.261064i
\(689\) −19982.8 34611.3i −1.10491 1.91377i
\(690\) 0 0
\(691\) 8544.68 14799.8i 0.470412 0.814778i −0.529015 0.848612i \(-0.677439\pi\)
0.999427 + 0.0338344i \(0.0107719\pi\)
\(692\) −13559.5 −0.744876
\(693\) 0 0
\(694\) 13008.0 0.711494
\(695\) −6140.00 + 10634.8i −0.335113 + 0.580433i
\(696\) 0 0
\(697\) 35.0000 + 60.6218i 0.00190204 + 0.00329442i
\(698\) 5256.63 9104.75i 0.285052 0.493725i
\(699\) 0 0
\(700\) 0 0
\(701\) 13410.0 0.722523 0.361262 0.932465i \(-0.382346\pi\)
0.361262 + 0.932465i \(0.382346\pi\)
\(702\) 0 0
\(703\) −113.137 195.959i −0.00606977 0.0105131i
\(704\) −1280.00 2217.03i −0.0685253 0.118689i
\(705\) 0 0
\(706\) 24423.5 1.30197
\(707\) 0 0
\(708\) 0 0
\(709\) 70.0000 121.244i 0.00370791 0.00642228i −0.864165 0.503208i \(-0.832153\pi\)
0.867873 + 0.496785i \(0.165486\pi\)
\(710\) −2969.85 5143.93i −0.156981 0.271899i
\(711\) 0 0
\(712\) −4610.34 + 7985.34i −0.242668 + 0.420313i
\(713\) 8077.99 0.424296
\(714\) 0 0
\(715\) 18000.0 0.941485
\(716\) −1440.00 + 2494.15i −0.0751611 + 0.130183i
\(717\) 0 0
\(718\) −7340.00 12713.3i −0.381513 0.660800i
\(719\) 12968.3 22461.8i 0.672653 1.16507i −0.304496 0.952514i \(-0.598488\pi\)
0.977149 0.212555i \(-0.0681786\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 13462.0 0.693911
\(723\) 0 0
\(724\) −3708.07 6422.56i −0.190344 0.329686i
\(725\) 4125.00 + 7144.71i 0.211308 + 0.365997i
\(726\) 0 0
\(727\) −9277.24 −0.473279 −0.236639 0.971598i \(-0.576046\pi\)
−0.236639 + 0.971598i \(0.576046\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2050.00 + 3550.70i −0.103937 + 0.180024i
\(731\) −240.416 416.413i −0.0121643 0.0210692i
\(732\) 0 0
\(733\) −12738.5 + 22063.8i −0.641894 + 1.11179i 0.343116 + 0.939293i \(0.388518\pi\)
−0.985010 + 0.172500i \(0.944816\pi\)
\(734\) −15330.1 −0.770904
\(735\) 0 0
\(736\) −2176.00 −0.108979
\(737\) −10800.0 + 18706.1i −0.539787 + 0.934939i
\(738\) 0 0
\(739\) 3462.00 + 5996.36i 0.172330 + 0.298484i 0.939234 0.343278i \(-0.111537\pi\)
−0.766904 + 0.641762i \(0.778204\pi\)
\(740\) 282.843 489.898i 0.0140507 0.0243365i
\(741\) 0 0
\(742\) 0 0
\(743\) −29108.0 −1.43724 −0.718620 0.695403i \(-0.755226\pi\)
−0.718620 + 0.695403i \(0.755226\pi\)
\(744\) 0 0
\(745\) −7566.04 13104.8i −0.372078 0.644459i
\(746\) −2990.00 5178.83i −0.146745 0.254170i
\(747\) 0 0
\(748\) −226.274 −0.0110607
\(749\) 0 0
\(750\) 0 0
\(751\) −15724.0 + 27234.8i −0.764017 + 1.32332i 0.176747 + 0.984256i \(0.443442\pi\)
−0.940765 + 0.339060i \(0.889891\pi\)
\(752\) 724.077 + 1254.14i 0.0351122 + 0.0608161i
\(753\) 0 0
\(754\) 7000.36 12125.0i 0.338114 0.585631i
\(755\) 14990.7 0.722604
\(756\) 0 0
\(757\) −13300.0 −0.638569 −0.319284 0.947659i \(-0.603443\pi\)
−0.319284 + 0.947659i \(0.603443\pi\)
\(758\) −11900.0 + 20611.4i −0.570221 + 0.987652i
\(759\) 0 0
\(760\) −320.000 554.256i −0.0152732 0.0264539i
\(761\) 2400.63 4158.01i 0.114353 0.198065i −0.803168 0.595753i \(-0.796854\pi\)
0.917521 + 0.397687i \(0.130187\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 15920.0 0.753881
\(765\) 0 0
\(766\) 9712.82 + 16823.1i 0.458144 + 0.793529i
\(767\) −27900.0 48324.2i −1.31344 2.27495i
\(768\) 0 0
\(769\) −16932.4 −0.794015 −0.397007 0.917815i \(-0.629951\pi\)
−0.397007 + 0.917815i \(0.629951\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7420.00 + 12851.8i −0.345922 + 0.599154i
\(773\) −9720.60 16836.6i −0.452297 0.783401i 0.546231 0.837634i \(-0.316062\pi\)
−0.998528 + 0.0542330i \(0.982729\pi\)
\(774\) 0 0
\(775\) 4454.77 7715.89i 0.206478 0.357630i
\(776\) −4016.37 −0.185798
\(777\) 0 0
\(778\) −4300.00 −0.198152
\(779\) 280.000 484.974i 0.0128781 0.0223055i
\(780\) 0 0
\(781\) 8400.00 + 14549.2i 0.384860 + 0.666597i
\(782\) −96.1665 + 166.565i −0.00439758 + 0.00761683i
\(783\) 0 0
\(784\) 0 0
\(785\) −12350.0 −0.561516
\(786\) 0 0
\(787\) −10366.2 17954.8i −0.469523 0.813238i 0.529870 0.848079i \(-0.322241\pi\)
−0.999393 + 0.0348413i \(0.988907\pi\)
\(788\) −1912.00 3311.68i −0.0864368 0.149713i
\(789\) 0 0
\(790\) 10748.0 0.484047
\(791\) 0 0
\(792\) 0 0
\(793\) 29205.0 50584.5i 1.30782 2.26521i
\(794\) −3401.18 5891.02i −0.152020 0.263306i
\(795\) 0 0
\(796\) 8179.81 14167.8i 0.364228 0.630862i
\(797\) −28582.7 −1.27033 −0.635163 0.772378i \(-0.719067\pi\)
−0.635163 + 0.772378i \(0.719067\pi\)
\(798\) 0 0
\(799\) 128.000 0.00566748
\(800\) −1200.00 + 2078.46i −0.0530330 + 0.0918559i
\(801\) 0 0
\(802\) −12090.0 20940.5i −0.532310 0.921988i
\(803\) 5798.28 10042.9i 0.254815 0.441353i
\(804\) 0 0
\(805\) 0 0
\(806\) −15120.0 −0.660768
\(807\) 0 0
\(808\) 7042.78 + 12198.5i 0.306639 + 0.531114i
\(809\) −11140.0 19295.0i −0.484130 0.838539i 0.515703 0.856767i \(-0.327531\pi\)
−0.999834 + 0.0182286i \(0.994197\pi\)
\(810\) 0 0
\(811\) −26100.7 −1.13011 −0.565056 0.825053i \(-0.691145\pi\)
−0.565056 + 0.825053i \(0.691145\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −800.000 + 1385.64i −0.0344472 + 0.0596642i
\(815\) −11808.7 20453.2i −0.507534 0.879075i
\(816\) 0 0
\(817\) −1923.33 + 3331.31i −0.0823609 + 0.142653i
\(818\) 16385.1 0.700356
\(819\) 0 0
\(820\) 1400.00 0.0596221
\(821\) 1770.00 3065.73i 0.0752417 0.130322i −0.825950 0.563744i \(-0.809361\pi\)
0.901191 + 0.433421i \(0.142694\pi\)
\(822\) 0 0
\(823\) 14200.0 + 24595.1i 0.601435 + 1.04172i 0.992604 + 0.121397i \(0.0387374\pi\)
−0.391169 + 0.920319i \(0.627929\pi\)
\(824\) −905.097 + 1567.67i −0.0382652 + 0.0662773i
\(825\) 0 0
\(826\) 0 0
\(827\) 38736.0 1.62876 0.814379 0.580334i \(-0.197078\pi\)
0.814379 + 0.580334i \(0.197078\pi\)
\(828\) 0 0
\(829\) −11704.7 20273.2i −0.490377 0.849358i 0.509562 0.860434i \(-0.329808\pi\)
−0.999939 + 0.0110765i \(0.996474\pi\)
\(830\) −6680.00 11570.1i −0.279357 0.483860i
\(831\) 0 0
\(832\) 4072.94 0.169716
\(833\) 0 0
\(834\) 0 0
\(835\) 1300.00 2251.67i 0.0538783 0.0933199i
\(836\) 905.097 + 1567.67i 0.0374443 + 0.0648554i
\(837\) 0 0
\(838\) 1046.52 1812.62i 0.0431401 0.0747208i
\(839\) 29387.4 1.20925 0.604627 0.796509i \(-0.293322\pi\)
0.604627 + 0.796509i \(0.293322\pi\)
\(840\) 0 0
\(841\) −12289.0 −0.503875
\(842\) −3870.00 + 6703.04i −0.158395 + 0.274349i
\(843\) 0 0
\(844\) −5736.00 9935.04i −0.233935 0.405188i
\(845\) −6551.34 + 11347.3i −0.266714 + 0.461962i
\(846\) 0 0
\(847\) 0 0
\(848\) 10048.0 0.406898
\(849\) 0 0
\(850\) 106.066 + 183.712i 0.00428004 + 0.00741325i
\(851\) 680.000 + 1177.79i 0.0273914 + 0.0474433i
\(852\) 0 0
\(853\) −22252.7 −0.893220 −0.446610 0.894729i \(-0.647369\pi\)
−0.446610 + 0.894729i \(0.647369\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 8096.00 14022.7i 0.323266 0.559913i
\(857\) −6916.21 11979.2i −0.275675 0.477483i 0.694630 0.719367i \(-0.255568\pi\)
−0.970305 + 0.241884i \(0.922235\pi\)
\(858\) 0 0
\(859\) 19832.9 34351.6i 0.787766 1.36445i −0.139567 0.990213i \(-0.544571\pi\)
0.927333 0.374238i \(-0.122096\pi\)
\(860\) −9616.65 −0.381308
\(861\) 0 0
\(862\) 5400.00 0.213370
\(863\) −15994.0 + 27702.4i −0.630871 + 1.09270i 0.356502 + 0.934294i \(0.383969\pi\)
−0.987374 + 0.158407i \(0.949364\pi\)
\(864\) 0 0
\(865\) 11985.0 + 20758.6i 0.471101 + 0.815971i
\(866\) 5876.06 10177.6i 0.230573 0.399365i
\(867\) 0 0
\(868\) 0 0
\(869\) −30400.0 −1.18671
\(870\) 0 0
\(871\) −17182.7 29761.3i −0.668442 1.15778i
\(872\) −1616.00 2798.99i −0.0627576 0.108699i
\(873\) 0 0
\(874\) 1538.66 0.0595493
\(875\) 0 0
\(876\) 0 0
\(877\) 16730.0 28977.2i 0.644164 1.11573i −0.340330 0.940306i \(-0.610539\pi\)
0.984494 0.175419i \(-0.0561280\pi\)
\(878\) −6346.99 10993.3i −0.243964 0.422558i
\(879\) 0 0
\(880\) −2262.74 + 3919.18i −0.0866784 + 0.150131i
\(881\) −12791.6 −0.489170 −0.244585 0.969628i \(-0.578652\pi\)
−0.244585 + 0.969628i \(0.578652\pi\)
\(882\) 0 0
\(883\) −35260.0 −1.34382 −0.671910 0.740633i \(-0.734526\pi\)
−0.671910 + 0.740633i \(0.734526\pi\)
\(884\) 180.000 311.769i 0.00684848 0.0118619i
\(885\) 0 0
\(886\) −6928.00 11999.6i −0.262698 0.455007i
\(887\) 7718.78 13369.3i 0.292188 0.506085i −0.682138 0.731223i \(-0.738950\pi\)
0.974327 + 0.225138i \(0.0722832\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 16300.0 0.613907
\(891\) 0 0
\(892\) 5260.87 + 9112.10i 0.197474 + 0.342036i
\(893\) −512.000 886.810i −0.0191864 0.0332318i
\(894\) 0 0
\(895\) 5091.17 0.190144
\(896\) 0 0
\(897\) 0 0
\(898\) −1320.00 + 2286.31i −0.0490523 + 0.0849611i
\(899\) −6533.67 11316.6i −0.242392 0.419834i
\(900\) 0 0
\(901\) 444.063 769.140i 0.0164194 0.0284392i
\(902\) −3959.80 −0.146172
\(903\) 0 0
\(904\) −8064.00 −0.296687
\(905\) −6555.00 + 11353.6i −0.240769 + 0.417023i
\(906\) 0 0
\(907\) −16550.0 28665.4i −0.605881 1.04942i −0.991912 0.126930i \(-0.959488\pi\)
0.386031 0.922486i \(-0.373846\pi\)
\(908\) −339.411 + 587.878i −0.0124050 + 0.0214861i
\(909\) 0 0
\(910\) 0 0
\(911\) 39620.0 1.44091 0.720455 0.693502i \(-0.243933\pi\)
0.720455 + 0.693502i \(0.243933\pi\)
\(912\) 0 0
\(913\) 18893.9 + 32725.2i 0.684881 + 1.18625i
\(914\) 1290.00 + 2234.35i 0.0466843 + 0.0808595i
\(915\) 0 0
\(916\) 12575.2 0.453598
\(917\) 0 0
\(918\) 0 0
\(919\) −10472.0 + 18138.0i −0.375886 + 0.651054i −0.990459 0.137806i \(-0.955995\pi\)
0.614573 + 0.788860i \(0.289328\pi\)
\(920\) 1923.33 + 3331.31i 0.0689243 + 0.119380i
\(921\) 0 0
\(922\) 17642.3 30557.4i 0.630172 1.09149i
\(923\) −26728.6 −0.953179
\(924\) 0 0
\(925\) 1500.00 0.0533186
\(926\) 5680.00 9838.05i 0.201573 0.349134i
\(927\) 0 0
\(928\) 1760.00 + 3048.41i 0.0622574 + 0.107833i
\(929\) 23118.9 40043.0i 0.816475 1.41418i −0.0917893 0.995778i \(-0.529259\pi\)
0.908264 0.418397i \(-0.137408\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −17928.0 −0.630098
\(933\) 0 0
\(934\) −7693.32 13325.2i −0.269522 0.466825i
\(935\) 200.000 + 346.410i 0.00699540 + 0.0121164i
\(936\) 0 0
\(937\) 50522.8 1.76148 0.880740 0.473600i \(-0.157046\pi\)
0.880740 + 0.473600i \(0.157046\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 1280.00 2217.03i 0.0444138 0.0769270i
\(941\) 4939.14 + 8554.84i 0.171107 + 0.296365i 0.938807 0.344444i \(-0.111932\pi\)
−0.767700 + 0.640809i \(0.778599\pi\)
\(942\) 0 0
\(943\) −1682.91 + 2914.89i −0.0581158 + 0.100660i
\(944\) 14029.0 0.483692
\(945\) 0 0
\(946\) 27200.0 0.934829
\(947\) −15108.0 + 26167.8i −0.518420 + 0.897930i 0.481351 + 0.876528i \(0.340146\pi\)
−0.999771 + 0.0214022i \(0.993187\pi\)
\(948\) 0 0
\(949\) 9225.00 + 15978.2i 0.315549 + 0.546547i
\(950\) 848.528 1469.69i 0.0289788 0.0501928i
\(951\) 0 0
\(952\) 0 0
\(953\) 35512.0 1.20708 0.603540 0.797333i \(-0.293757\pi\)
0.603540 + 0.797333i \(0.293757\pi\)
\(954\) 0 0
\(955\) −14071.4 24372.4i −0.476796 0.825836i
\(956\) 3480.00 + 6027.54i 0.117731 + 0.203917i
\(957\) 0 0
\(958\) 33036.0 1.11414
\(959\) 0 0
\(960\) 0 0
\(961\) 7839.50 13578.4i 0.263150 0.455789i
\(962\) −1272.79 2204.54i −0.0426575 0.0738849i
\(963\) 0 0
\(964\) 2520.13 4364.99i 0.0841990 0.145837i
\(965\) 26233.7 0.875121
\(966\) 0 0
\(967\) −720.000 −0.0239438 −0.0119719 0.999928i \(-0.503811\pi\)
−0.0119719 + 0.999928i \(0.503811\pi\)
\(968\) 1076.00 1863.69i 0.0357272 0.0618814i
\(969\) 0 0
\(970\) 3550.00 + 6148.78i 0.117509 + 0.203531i
\(971\) −7693.32 + 13325.2i −0.254264 + 0.440398i −0.964695 0.263368i \(-0.915167\pi\)
0.710431 + 0.703767i \(0.248500\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −27360.0 −0.900073
\(975\) 0 0
\(976\) 7342.60 + 12717.8i 0.240810 + 0.417096i
\(977\) 20787.0 + 36004.1i 0.680691 + 1.17899i 0.974770 + 0.223210i \(0.0716537\pi\)
−0.294079 + 0.955781i \(0.595013\pi\)
\(978\) 0 0
\(979\) −46103.4 −1.50508
\(980\) 0 0
\(981\) 0 0
\(982\) 2280.00 3949.08i 0.0740914 0.128330i
\(983\) 22118.3 + 38310.0i 0.717665 + 1.24303i 0.961923 + 0.273321i \(0.0881222\pi\)
−0.244258 + 0.969710i \(0.578544\pi\)
\(984\) 0 0
\(985\) −3379.97 + 5854.28i −0.109335 + 0.189373i
\(986\) 311.127 0.0100490
\(987\) 0 0
\(988\) −2880.00 −0.0927379
\(989\) 11560.0 20022.5i 0.371675 0.643760i
\(990\) 0 0
\(991\) −6136.00 10627.9i −0.196687 0.340671i 0.750765 0.660569i \(-0.229685\pi\)
−0.947452 + 0.319898i \(0.896351\pi\)
\(992\) 1900.70 3292.11i 0.0608341 0.105368i
\(993\) 0 0
\(994\) 0 0
\(995\) −28920.0 −0.921433
\(996\) 0 0
\(997\) 28846.4 + 49963.5i 0.916324 + 1.58712i 0.804951 + 0.593342i \(0.202192\pi\)
0.111374 + 0.993779i \(0.464475\pi\)
\(998\) 860.000 + 1489.56i 0.0272774 + 0.0472458i
\(999\) 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 882.4.g.bg.667.1 4
3.2 odd 2 882.4.g.bc.667.2 4
7.2 even 3 882.4.a.y.1.2 yes 2
7.3 odd 6 inner 882.4.g.bg.361.2 4
7.4 even 3 inner 882.4.g.bg.361.1 4
7.5 odd 6 882.4.a.y.1.1 2
7.6 odd 2 inner 882.4.g.bg.667.2 4
21.2 odd 6 882.4.a.be.1.1 yes 2
21.5 even 6 882.4.a.be.1.2 yes 2
21.11 odd 6 882.4.g.bc.361.2 4
21.17 even 6 882.4.g.bc.361.1 4
21.20 even 2 882.4.g.bc.667.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.4.a.y.1.1 2 7.5 odd 6
882.4.a.y.1.2 yes 2 7.2 even 3
882.4.a.be.1.1 yes 2 21.2 odd 6
882.4.a.be.1.2 yes 2 21.5 even 6
882.4.g.bc.361.1 4 21.17 even 6
882.4.g.bc.361.2 4 21.11 odd 6
882.4.g.bc.667.1 4 21.20 even 2
882.4.g.bc.667.2 4 3.2 odd 2
882.4.g.bg.361.1 4 7.4 even 3 inner
882.4.g.bg.361.2 4 7.3 odd 6 inner
882.4.g.bg.667.1 4 1.1 even 1 trivial
882.4.g.bg.667.2 4 7.6 odd 2 inner