Properties

Label 882.4.g.be.667.1
Level $882$
Weight $4$
Character 882.667
Analytic conductor $52.040$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: no (minimal twist has level 294)
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.be.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-3.70711 + 6.42090i) q^{5} -8.00000 q^{8} +O(q^{10})\) \(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-3.70711 + 6.42090i) q^{5} -8.00000 q^{8} +(7.41421 + 12.8418i) q^{10} +(5.24264 + 9.08052i) q^{11} -2.78680 q^{13} +(-8.00000 + 13.8564i) q^{16} +(-25.2218 - 43.6855i) q^{17} +(62.5269 - 108.300i) q^{19} +29.6569 q^{20} +20.9706 q^{22} +(-91.1249 + 157.833i) q^{23} +(35.0147 + 60.6473i) q^{25} +(-2.78680 + 4.82687i) q^{26} -156.132 q^{29} +(69.8162 + 120.925i) q^{31} +(16.0000 + 27.7128i) q^{32} -100.887 q^{34} +(197.279 - 341.698i) q^{37} +(-125.054 - 216.600i) q^{38} +(29.6569 - 51.3672i) q^{40} +197.605 q^{41} +343.294 q^{43} +(20.9706 - 36.3221i) q^{44} +(182.250 + 315.666i) q^{46} +(305.002 - 528.279i) q^{47} +140.059 q^{50} +(5.57359 + 9.65375i) q^{52} +(-68.7645 - 119.104i) q^{53} -77.7401 q^{55} +(-156.132 + 270.429i) q^{58} +(-294.718 - 510.466i) q^{59} +(123.609 - 214.097i) q^{61} +279.265 q^{62} +64.0000 q^{64} +(10.3310 - 17.8937i) q^{65} +(197.823 + 342.640i) q^{67} +(-100.887 + 174.742i) q^{68} -285.661 q^{71} +(-498.729 - 863.823i) q^{73} +(-394.558 - 683.395i) q^{74} -500.215 q^{76} +(424.132 - 734.618i) q^{79} +(-59.3137 - 102.734i) q^{80} +(197.605 - 342.262i) q^{82} -210.863 q^{83} +374.000 q^{85} +(343.294 - 594.602i) q^{86} +(-41.9411 - 72.6442i) q^{88} +(276.744 - 479.334i) q^{89} +728.999 q^{92} +(-610.004 - 1056.56i) q^{94} +(463.588 + 802.958i) q^{95} +903.910 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 8 q^{4} - 12 q^{5} - 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 8 q^{4} - 12 q^{5} - 32 q^{8} + 24 q^{10} + 4 q^{11} - 96 q^{13} - 32 q^{16} - 132 q^{17} + 120 q^{19} + 96 q^{20} + 16 q^{22} - 76 q^{23} + 174 q^{25} - 96 q^{26} + 224 q^{29} + 432 q^{31} + 64 q^{32} - 528 q^{34} + 280 q^{37} - 240 q^{38} + 96 q^{40} + 72 q^{41} - 256 q^{43} + 16 q^{44} + 152 q^{46} + 264 q^{47} + 696 q^{50} + 192 q^{52} + 268 q^{53} - 96 q^{55} + 224 q^{58} - 336 q^{59} - 504 q^{61} + 1728 q^{62} + 256 q^{64} + 228 q^{65} + 384 q^{67} - 528 q^{68} + 792 q^{71} - 312 q^{73} - 560 q^{74} - 960 q^{76} + 848 q^{79} - 192 q^{80} + 72 q^{82} - 1296 q^{83} + 1496 q^{85} - 256 q^{86} - 32 q^{88} + 612 q^{89} + 608 q^{92} - 528 q^{94} + 904 q^{95} + 4368 q^{97}+O(q^{100}) \) 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
\(5\) −3.70711 + 6.42090i −0.331574 + 0.574303i −0.982821 0.184563i \(-0.940913\pi\)
0.651247 + 0.758866i \(0.274246\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 7.41421 + 12.8418i 0.234458 + 0.406093i
\(11\) 5.24264 + 9.08052i 0.143701 + 0.248898i 0.928888 0.370361i \(-0.120766\pi\)
−0.785186 + 0.619260i \(0.787433\pi\)
\(12\) 0 0
\(13\) −2.78680 −0.0594553 −0.0297276 0.999558i \(-0.509464\pi\)
−0.0297276 + 0.999558i \(0.509464\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −8.00000 + 13.8564i −0.125000 + 0.216506i
\(17\) −25.2218 43.6855i −0.359835 0.623252i 0.628098 0.778134i \(-0.283834\pi\)
−0.987933 + 0.154882i \(0.950500\pi\)
\(18\) 0 0
\(19\) 62.5269 108.300i 0.754982 1.30767i −0.190401 0.981706i \(-0.560979\pi\)
0.945383 0.325961i \(-0.105688\pi\)
\(20\) 29.6569 0.331574
\(21\) 0 0
\(22\) 20.9706 0.203225
\(23\) −91.1249 + 157.833i −0.826124 + 1.43089i 0.0749331 + 0.997189i \(0.476126\pi\)
−0.901057 + 0.433700i \(0.857208\pi\)
\(24\) 0 0
\(25\) 35.0147 + 60.6473i 0.280118 + 0.485178i
\(26\) −2.78680 + 4.82687i −0.0210206 + 0.0364088i
\(27\) 0 0
\(28\) 0 0
\(29\) −156.132 −0.999758 −0.499879 0.866095i \(-0.666622\pi\)
−0.499879 + 0.866095i \(0.666622\pi\)
\(30\) 0 0
\(31\) 69.8162 + 120.925i 0.404496 + 0.700607i 0.994263 0.106966i \(-0.0341137\pi\)
−0.589767 + 0.807573i \(0.700780\pi\)
\(32\) 16.0000 + 27.7128i 0.0883883 + 0.153093i
\(33\) 0 0
\(34\) −100.887 −0.508883
\(35\) 0 0
\(36\) 0 0
\(37\) 197.279 341.698i 0.876554 1.51824i 0.0214563 0.999770i \(-0.493170\pi\)
0.855098 0.518467i \(-0.173497\pi\)
\(38\) −125.054 216.600i −0.533853 0.924660i
\(39\) 0 0
\(40\) 29.6569 51.3672i 0.117229 0.203047i
\(41\) 197.605 0.752701 0.376350 0.926477i \(-0.377179\pi\)
0.376350 + 0.926477i \(0.377179\pi\)
\(42\) 0 0
\(43\) 343.294 1.21748 0.608741 0.793369i \(-0.291675\pi\)
0.608741 + 0.793369i \(0.291675\pi\)
\(44\) 20.9706 36.3221i 0.0718507 0.124449i
\(45\) 0 0
\(46\) 182.250 + 315.666i 0.584158 + 1.01179i
\(47\) 305.002 528.279i 0.946577 1.63952i 0.194015 0.980999i \(-0.437849\pi\)
0.752562 0.658521i \(-0.228818\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 140.059 0.396146
\(51\) 0 0
\(52\) 5.57359 + 9.65375i 0.0148638 + 0.0257449i
\(53\) −68.7645 119.104i −0.178218 0.308682i 0.763053 0.646336i \(-0.223700\pi\)
−0.941270 + 0.337655i \(0.890366\pi\)
\(54\) 0 0
\(55\) −77.7401 −0.190590
\(56\) 0 0
\(57\) 0 0
\(58\) −156.132 + 270.429i −0.353468 + 0.612224i
\(59\) −294.718 510.466i −0.650322 1.12639i −0.983045 0.183366i \(-0.941301\pi\)
0.332723 0.943025i \(-0.392033\pi\)
\(60\) 0 0
\(61\) 123.609 214.097i 0.259450 0.449381i −0.706644 0.707569i \(-0.749792\pi\)
0.966095 + 0.258188i \(0.0831253\pi\)
\(62\) 279.265 0.572043
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 10.3310 17.8937i 0.0197138 0.0341453i
\(66\) 0 0
\(67\) 197.823 + 342.640i 0.360716 + 0.624778i 0.988079 0.153949i \(-0.0491990\pi\)
−0.627363 + 0.778727i \(0.715866\pi\)
\(68\) −100.887 + 174.742i −0.179917 + 0.311626i
\(69\) 0 0
\(70\) 0 0
\(71\) −285.661 −0.477489 −0.238745 0.971082i \(-0.576736\pi\)
−0.238745 + 0.971082i \(0.576736\pi\)
\(72\) 0 0
\(73\) −498.729 863.823i −0.799613 1.38497i −0.919868 0.392228i \(-0.871705\pi\)
0.120255 0.992743i \(-0.461629\pi\)
\(74\) −394.558 683.395i −0.619817 1.07356i
\(75\) 0 0
\(76\) −500.215 −0.754982
\(77\) 0 0
\(78\) 0 0
\(79\) 424.132 734.618i 0.604033 1.04622i −0.388171 0.921587i \(-0.626893\pi\)
0.992204 0.124628i \(-0.0397737\pi\)
\(80\) −59.3137 102.734i −0.0828934 0.143576i
\(81\) 0 0
\(82\) 197.605 342.262i 0.266120 0.460933i
\(83\) −210.863 −0.278858 −0.139429 0.990232i \(-0.544527\pi\)
−0.139429 + 0.990232i \(0.544527\pi\)
\(84\) 0 0
\(85\) 374.000 0.477247
\(86\) 343.294 594.602i 0.430445 0.745553i
\(87\) 0 0
\(88\) −41.9411 72.6442i −0.0508061 0.0879988i
\(89\) 276.744 479.334i 0.329604 0.570891i −0.652829 0.757505i \(-0.726418\pi\)
0.982433 + 0.186614i \(0.0597514\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 728.999 0.826124
\(93\) 0 0
\(94\) −610.004 1056.56i −0.669331 1.15932i
\(95\) 463.588 + 802.958i 0.500664 + 0.867176i
\(96\) 0 0
\(97\) 903.910 0.946166 0.473083 0.881018i \(-0.343141\pi\)
0.473083 + 0.881018i \(0.343141\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 140.059 242.589i 0.140059 0.242589i
\(101\) 156.806 + 271.595i 0.154482 + 0.267572i 0.932870 0.360212i \(-0.117296\pi\)
−0.778388 + 0.627784i \(0.783962\pi\)
\(102\) 0 0
\(103\) 115.487 200.030i 0.110479 0.191355i −0.805485 0.592617i \(-0.798095\pi\)
0.915963 + 0.401262i \(0.131428\pi\)
\(104\) 22.2944 0.0210206
\(105\) 0 0
\(106\) −275.058 −0.252038
\(107\) 62.7431 108.674i 0.0566879 0.0981863i −0.836289 0.548289i \(-0.815279\pi\)
0.892977 + 0.450103i \(0.148613\pi\)
\(108\) 0 0
\(109\) −372.764 645.646i −0.327562 0.567354i 0.654465 0.756092i \(-0.272894\pi\)
−0.982028 + 0.188738i \(0.939560\pi\)
\(110\) −77.7401 + 134.650i −0.0673839 + 0.116712i
\(111\) 0 0
\(112\) 0 0
\(113\) 1043.76 0.868929 0.434464 0.900689i \(-0.356938\pi\)
0.434464 + 0.900689i \(0.356938\pi\)
\(114\) 0 0
\(115\) −675.619 1170.21i −0.547842 0.948890i
\(116\) 312.264 + 540.857i 0.249940 + 0.432908i
\(117\) 0 0
\(118\) −1178.87 −0.919694
\(119\) 0 0
\(120\) 0 0
\(121\) 610.529 1057.47i 0.458700 0.794491i
\(122\) −247.217 428.193i −0.183459 0.317760i
\(123\) 0 0
\(124\) 279.265 483.701i 0.202248 0.350304i
\(125\) −1445.99 −1.03467
\(126\) 0 0
\(127\) −2080.17 −1.45343 −0.726715 0.686939i \(-0.758954\pi\)
−0.726715 + 0.686939i \(0.758954\pi\)
\(128\) 64.0000 110.851i 0.0441942 0.0765466i
\(129\) 0 0
\(130\) −20.6619 35.7875i −0.0139398 0.0241444i
\(131\) 634.764 1099.44i 0.423355 0.733273i −0.572910 0.819618i \(-0.694186\pi\)
0.996265 + 0.0863453i \(0.0275188\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 791.294 0.510129
\(135\) 0 0
\(136\) 201.775 + 349.484i 0.127221 + 0.220353i
\(137\) 1536.59 + 2661.46i 0.958249 + 1.65974i 0.726752 + 0.686900i \(0.241029\pi\)
0.231497 + 0.972836i \(0.425638\pi\)
\(138\) 0 0
\(139\) −1013.60 −0.618504 −0.309252 0.950980i \(-0.600079\pi\)
−0.309252 + 0.950980i \(0.600079\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −285.661 + 494.779i −0.168818 + 0.292401i
\(143\) −14.6102 25.3056i −0.00854380 0.0147983i
\(144\) 0 0
\(145\) 578.798 1002.51i 0.331494 0.574164i
\(146\) −1994.91 −1.13082
\(147\) 0 0
\(148\) −1578.23 −0.876554
\(149\) 615.735 1066.48i 0.338544 0.586375i −0.645615 0.763663i \(-0.723399\pi\)
0.984159 + 0.177288i \(0.0567324\pi\)
\(150\) 0 0
\(151\) 1122.37 + 1944.00i 0.604881 + 1.04768i 0.992070 + 0.125684i \(0.0401125\pi\)
−0.387190 + 0.922000i \(0.626554\pi\)
\(152\) −500.215 + 866.398i −0.266926 + 0.462330i
\(153\) 0 0
\(154\) 0 0
\(155\) −1035.26 −0.536481
\(156\) 0 0
\(157\) −1893.58 3279.77i −0.962573 1.66723i −0.715998 0.698102i \(-0.754028\pi\)
−0.246575 0.969124i \(-0.579305\pi\)
\(158\) −848.264 1469.24i −0.427116 0.739786i
\(159\) 0 0
\(160\) −237.255 −0.117229
\(161\) 0 0
\(162\) 0 0
\(163\) 1054.28 1826.07i 0.506611 0.877476i −0.493360 0.869825i \(-0.664231\pi\)
0.999971 0.00765060i \(-0.00243528\pi\)
\(164\) −395.210 684.524i −0.188175 0.325929i
\(165\) 0 0
\(166\) −210.863 + 365.225i −0.0985912 + 0.170765i
\(167\) 1502.41 0.696170 0.348085 0.937463i \(-0.386832\pi\)
0.348085 + 0.937463i \(0.386832\pi\)
\(168\) 0 0
\(169\) −2189.23 −0.996465
\(170\) 374.000 647.787i 0.168732 0.292253i
\(171\) 0 0
\(172\) −686.587 1189.20i −0.304371 0.527186i
\(173\) −235.628 + 408.120i −0.103552 + 0.179357i −0.913146 0.407634i \(-0.866354\pi\)
0.809594 + 0.586991i \(0.199687\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −167.765 −0.0718507
\(177\) 0 0
\(178\) −553.487 958.668i −0.233065 0.403681i
\(179\) 666.244 + 1153.97i 0.278198 + 0.481852i 0.970937 0.239336i \(-0.0769296\pi\)
−0.692739 + 0.721188i \(0.743596\pi\)
\(180\) 0 0
\(181\) 997.727 0.409726 0.204863 0.978791i \(-0.434325\pi\)
0.204863 + 0.978791i \(0.434325\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 728.999 1262.66i 0.292079 0.505896i
\(185\) 1462.67 + 2533.42i 0.581285 + 1.00681i
\(186\) 0 0
\(187\) 264.458 458.055i 0.103418 0.179124i
\(188\) −2440.02 −0.946577
\(189\) 0 0
\(190\) 1854.35 0.708046
\(191\) 613.360 1062.37i 0.232362 0.402463i −0.726141 0.687546i \(-0.758688\pi\)
0.958503 + 0.285083i \(0.0920212\pi\)
\(192\) 0 0
\(193\) 1739.65 + 3013.16i 0.648821 + 1.12379i 0.983405 + 0.181425i \(0.0580709\pi\)
−0.334584 + 0.942366i \(0.608596\pi\)
\(194\) 903.910 1565.62i 0.334520 0.579406i
\(195\) 0 0
\(196\) 0 0
\(197\) −3193.47 −1.15495 −0.577476 0.816408i \(-0.695962\pi\)
−0.577476 + 0.816408i \(0.695962\pi\)
\(198\) 0 0
\(199\) 532.574 + 922.446i 0.189715 + 0.328595i 0.945155 0.326622i \(-0.105910\pi\)
−0.755440 + 0.655217i \(0.772577\pi\)
\(200\) −280.118 485.178i −0.0990366 0.171536i
\(201\) 0 0
\(202\) 627.222 0.218471
\(203\) 0 0
\(204\) 0 0
\(205\) −732.543 + 1268.80i −0.249576 + 0.432278i
\(206\) −230.975 400.060i −0.0781203 0.135308i
\(207\) 0 0
\(208\) 22.2944 38.6150i 0.00743191 0.0128724i
\(209\) 1311.22 0.433968
\(210\) 0 0
\(211\) 2057.50 0.671298 0.335649 0.941987i \(-0.391044\pi\)
0.335649 + 0.941987i \(0.391044\pi\)
\(212\) −275.058 + 476.414i −0.0891088 + 0.154341i
\(213\) 0 0
\(214\) −125.486 217.348i −0.0400844 0.0694282i
\(215\) −1272.63 + 2204.25i −0.403685 + 0.699204i
\(216\) 0 0
\(217\) 0 0
\(218\) −1491.05 −0.463243
\(219\) 0 0
\(220\) 155.480 + 269.300i 0.0476476 + 0.0825281i
\(221\) 70.2881 + 121.743i 0.0213941 + 0.0370556i
\(222\) 0 0
\(223\) −2028.27 −0.609071 −0.304536 0.952501i \(-0.598501\pi\)
−0.304536 + 0.952501i \(0.598501\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1043.76 1807.85i 0.307213 0.532108i
\(227\) −1211.98 2099.21i −0.354369 0.613786i 0.632640 0.774446i \(-0.281971\pi\)
−0.987010 + 0.160660i \(0.948638\pi\)
\(228\) 0 0
\(229\) −983.584 + 1703.62i −0.283830 + 0.491608i −0.972325 0.233633i \(-0.924938\pi\)
0.688495 + 0.725241i \(0.258272\pi\)
\(230\) −2702.48 −0.774766
\(231\) 0 0
\(232\) 1249.06 0.353468
\(233\) 2239.17 3878.35i 0.629582 1.09047i −0.358053 0.933701i \(-0.616559\pi\)
0.987636 0.156767i \(-0.0501073\pi\)
\(234\) 0 0
\(235\) 2261.35 + 3916.77i 0.627720 + 1.08724i
\(236\) −1178.87 + 2041.86i −0.325161 + 0.563195i
\(237\) 0 0
\(238\) 0 0
\(239\) −6116.92 −1.65553 −0.827763 0.561078i \(-0.810387\pi\)
−0.827763 + 0.561078i \(0.810387\pi\)
\(240\) 0 0
\(241\) 3114.19 + 5393.94i 0.832376 + 1.44172i 0.896149 + 0.443754i \(0.146353\pi\)
−0.0637726 + 0.997964i \(0.520313\pi\)
\(242\) −1221.06 2114.94i −0.324350 0.561790i
\(243\) 0 0
\(244\) −988.870 −0.259450
\(245\) 0 0
\(246\) 0 0
\(247\) −174.250 + 301.809i −0.0448876 + 0.0777477i
\(248\) −558.530 967.402i −0.143011 0.247702i
\(249\) 0 0
\(250\) −1445.99 + 2504.53i −0.365810 + 0.633601i
\(251\) 5904.42 1.48479 0.742397 0.669960i \(-0.233689\pi\)
0.742397 + 0.669960i \(0.233689\pi\)
\(252\) 0 0
\(253\) −1910.94 −0.474861
\(254\) −2080.17 + 3602.97i −0.513865 + 0.890041i
\(255\) 0 0
\(256\) −128.000 221.703i −0.0312500 0.0541266i
\(257\) 204.112 353.532i 0.0495414 0.0858082i −0.840191 0.542290i \(-0.817557\pi\)
0.889733 + 0.456482i \(0.150891\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −82.6476 −0.0197138
\(261\) 0 0
\(262\) −1269.53 2198.89i −0.299357 0.518502i
\(263\) −2313.00 4006.24i −0.542304 0.939298i −0.998771 0.0495577i \(-0.984219\pi\)
0.456467 0.889740i \(-0.349115\pi\)
\(264\) 0 0
\(265\) 1019.67 0.236369
\(266\) 0 0
\(267\) 0 0
\(268\) 791.294 1370.56i 0.180358 0.312389i
\(269\) 435.646 + 754.562i 0.0987429 + 0.171028i 0.911165 0.412043i \(-0.135185\pi\)
−0.812422 + 0.583070i \(0.801851\pi\)
\(270\) 0 0
\(271\) 3236.17 5605.22i 0.725401 1.25643i −0.233408 0.972379i \(-0.574988\pi\)
0.958809 0.284052i \(-0.0916788\pi\)
\(272\) 807.098 0.179917
\(273\) 0 0
\(274\) 6146.38 1.35517
\(275\) −367.139 + 635.904i −0.0805066 + 0.139442i
\(276\) 0 0
\(277\) −2355.94 4080.61i −0.511028 0.885126i −0.999918 0.0127811i \(-0.995932\pi\)
0.488890 0.872345i \(-0.337402\pi\)
\(278\) −1013.60 + 1755.60i −0.218674 + 0.378755i
\(279\) 0 0
\(280\) 0 0
\(281\) 7165.66 1.52124 0.760618 0.649200i \(-0.224896\pi\)
0.760618 + 0.649200i \(0.224896\pi\)
\(282\) 0 0
\(283\) 3173.50 + 5496.67i 0.666590 + 1.15457i 0.978851 + 0.204572i \(0.0655803\pi\)
−0.312261 + 0.949996i \(0.601086\pi\)
\(284\) 571.322 + 989.559i 0.119372 + 0.206759i
\(285\) 0 0
\(286\) −58.4407 −0.0120828
\(287\) 0 0
\(288\) 0 0
\(289\) 1184.22 2051.13i 0.241038 0.417490i
\(290\) −1157.60 2005.02i −0.234401 0.405995i
\(291\) 0 0
\(292\) −1994.91 + 3455.29i −0.399807 + 0.692485i
\(293\) 9233.78 1.84110 0.920552 0.390621i \(-0.127740\pi\)
0.920552 + 0.390621i \(0.127740\pi\)
\(294\) 0 0
\(295\) 4370.20 0.862519
\(296\) −1578.23 + 2733.58i −0.309909 + 0.536778i
\(297\) 0 0
\(298\) −1231.47 2132.97i −0.239386 0.414629i
\(299\) 253.947 439.848i 0.0491174 0.0850739i
\(300\) 0 0
\(301\) 0 0
\(302\) 4489.47 0.855430
\(303\) 0 0
\(304\) 1000.43 + 1732.80i 0.188745 + 0.326917i
\(305\) 916.461 + 1587.36i 0.172054 + 0.298006i
\(306\) 0 0
\(307\) −6786.53 −1.26165 −0.630827 0.775923i \(-0.717284\pi\)
−0.630827 + 0.775923i \(0.717284\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1035.26 + 1793.13i −0.189675 + 0.328526i
\(311\) −2568.38 4448.57i −0.468295 0.811111i 0.531048 0.847341i \(-0.321798\pi\)
−0.999343 + 0.0362307i \(0.988465\pi\)
\(312\) 0 0
\(313\) −1881.78 + 3259.34i −0.339822 + 0.588590i −0.984399 0.175950i \(-0.943700\pi\)
0.644577 + 0.764540i \(0.277034\pi\)
\(314\) −7574.31 −1.36128
\(315\) 0 0
\(316\) −3393.06 −0.604033
\(317\) −517.473 + 896.289i −0.0916851 + 0.158803i −0.908220 0.418492i \(-0.862559\pi\)
0.816535 + 0.577296i \(0.195892\pi\)
\(318\) 0 0
\(319\) −818.544 1417.76i −0.143667 0.248838i
\(320\) −237.255 + 410.937i −0.0414467 + 0.0717878i
\(321\) 0 0
\(322\) 0 0
\(323\) −6308.17 −1.08668
\(324\) 0 0
\(325\) −97.5789 169.012i −0.0166545 0.0288464i
\(326\) −2108.56 3652.13i −0.358228 0.620469i
\(327\) 0 0
\(328\) −1580.84 −0.266120
\(329\) 0 0
\(330\) 0 0
\(331\) −4400.03 + 7621.08i −0.730657 + 1.26554i 0.225945 + 0.974140i \(0.427453\pi\)
−0.956603 + 0.291395i \(0.905880\pi\)
\(332\) 421.726 + 730.451i 0.0697145 + 0.120749i
\(333\) 0 0
\(334\) 1502.41 2602.26i 0.246133 0.426315i
\(335\) −2933.41 −0.478416
\(336\) 0 0
\(337\) −5859.78 −0.947189 −0.473595 0.880743i \(-0.657044\pi\)
−0.473595 + 0.880743i \(0.657044\pi\)
\(338\) −2189.23 + 3791.86i −0.352304 + 0.610208i
\(339\) 0 0
\(340\) −748.000 1295.57i −0.119312 0.206654i
\(341\) −732.043 + 1267.94i −0.116253 + 0.201356i
\(342\) 0 0
\(343\) 0 0
\(344\) −2746.35 −0.430445
\(345\) 0 0
\(346\) 471.256 + 816.239i 0.0732222 + 0.126825i
\(347\) 3969.27 + 6874.98i 0.614068 + 1.06360i 0.990547 + 0.137172i \(0.0438013\pi\)
−0.376479 + 0.926425i \(0.622865\pi\)
\(348\) 0 0
\(349\) 9927.75 1.52269 0.761347 0.648344i \(-0.224538\pi\)
0.761347 + 0.648344i \(0.224538\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −167.765 + 290.577i −0.0254031 + 0.0439994i
\(353\) 5051.52 + 8749.49i 0.761658 + 1.31923i 0.941996 + 0.335625i \(0.108948\pi\)
−0.180338 + 0.983605i \(0.557719\pi\)
\(354\) 0 0
\(355\) 1058.98 1834.20i 0.158323 0.274223i
\(356\) −2213.95 −0.329604
\(357\) 0 0
\(358\) 2664.97 0.393431
\(359\) −2412.64 + 4178.81i −0.354691 + 0.614343i −0.987065 0.160320i \(-0.948747\pi\)
0.632374 + 0.774663i \(0.282081\pi\)
\(360\) 0 0
\(361\) −4389.73 7603.23i −0.639996 1.10850i
\(362\) 997.727 1728.11i 0.144860 0.250905i
\(363\) 0 0
\(364\) 0 0
\(365\) 7395.36 1.06052
\(366\) 0 0
\(367\) 3467.65 + 6006.15i 0.493215 + 0.854274i 0.999969 0.00781688i \(-0.00248822\pi\)
−0.506754 + 0.862091i \(0.669155\pi\)
\(368\) −1458.00 2525.33i −0.206531 0.357722i
\(369\) 0 0
\(370\) 5850.68 0.822061
\(371\) 0 0
\(372\) 0 0
\(373\) −7046.55 + 12205.0i −0.978168 + 1.69424i −0.309110 + 0.951026i \(0.600031\pi\)
−0.669058 + 0.743210i \(0.733302\pi\)
\(374\) −528.916 916.109i −0.0731272 0.126660i
\(375\) 0 0
\(376\) −2440.02 + 4226.23i −0.334666 + 0.579658i
\(377\) 435.108 0.0594409
\(378\) 0 0
\(379\) 5354.17 0.725661 0.362830 0.931855i \(-0.381810\pi\)
0.362830 + 0.931855i \(0.381810\pi\)
\(380\) 1854.35 3211.83i 0.250332 0.433588i
\(381\) 0 0
\(382\) −1226.72 2124.74i −0.164305 0.284585i
\(383\) 4485.24 7768.66i 0.598394 1.03645i −0.394664 0.918825i \(-0.629139\pi\)
0.993058 0.117623i \(-0.0375276\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 6958.58 0.917571
\(387\) 0 0
\(388\) −1807.82 3131.23i −0.236542 0.409702i
\(389\) −1851.29 3206.53i −0.241296 0.417937i 0.719788 0.694194i \(-0.244239\pi\)
−0.961084 + 0.276257i \(0.910906\pi\)
\(390\) 0 0
\(391\) 9193.34 1.18907
\(392\) 0 0
\(393\) 0 0
\(394\) −3193.47 + 5531.26i −0.408337 + 0.707260i
\(395\) 3144.61 + 5446.62i 0.400563 + 0.693795i
\(396\) 0 0
\(397\) −1041.62 + 1804.13i −0.131681 + 0.228078i −0.924325 0.381607i \(-0.875371\pi\)
0.792644 + 0.609685i \(0.208704\pi\)
\(398\) 2130.30 0.268297
\(399\) 0 0
\(400\) −1120.47 −0.140059
\(401\) −5317.01 + 9209.33i −0.662141 + 1.14686i 0.317910 + 0.948121i \(0.397019\pi\)
−0.980052 + 0.198742i \(0.936314\pi\)
\(402\) 0 0
\(403\) −194.564 336.994i −0.0240494 0.0416548i
\(404\) 627.222 1086.38i 0.0772412 0.133786i
\(405\) 0 0
\(406\) 0 0
\(407\) 4137.06 0.503848
\(408\) 0 0
\(409\) −3258.18 5643.34i −0.393904 0.682262i 0.599056 0.800707i \(-0.295542\pi\)
−0.992961 + 0.118445i \(0.962209\pi\)
\(410\) 1465.09 + 2537.60i 0.176477 + 0.305667i
\(411\) 0 0
\(412\) −923.899 −0.110479
\(413\) 0 0
\(414\) 0 0
\(415\) 781.691 1353.93i 0.0924620 0.160149i
\(416\) −44.5887 77.2300i −0.00525515 0.00910219i
\(417\) 0 0
\(418\) 1311.22 2271.11i 0.153431 0.265750i
\(419\) 6079.92 0.708887 0.354443 0.935077i \(-0.384670\pi\)
0.354443 + 0.935077i \(0.384670\pi\)
\(420\) 0 0
\(421\) −5631.58 −0.651939 −0.325969 0.945380i \(-0.605691\pi\)
−0.325969 + 0.945380i \(0.605691\pi\)
\(422\) 2057.50 3563.69i 0.237340 0.411084i
\(423\) 0 0
\(424\) 550.116 + 952.829i 0.0630094 + 0.109136i
\(425\) 1766.27 3059.27i 0.201592 0.349168i
\(426\) 0 0
\(427\) 0 0
\(428\) −501.945 −0.0566879
\(429\) 0 0
\(430\) 2545.25 + 4408.50i 0.285449 + 0.494412i
\(431\) −1868.45 3236.25i −0.208817 0.361681i 0.742525 0.669818i \(-0.233628\pi\)
−0.951342 + 0.308137i \(0.900295\pi\)
\(432\) 0 0
\(433\) 5757.46 0.638998 0.319499 0.947587i \(-0.396485\pi\)
0.319499 + 0.947587i \(0.396485\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1491.05 + 2582.58i −0.163781 + 0.283677i
\(437\) 11395.5 + 19737.6i 1.24742 + 2.16059i
\(438\) 0 0
\(439\) −4906.38 + 8498.10i −0.533414 + 0.923900i 0.465824 + 0.884877i \(0.345758\pi\)
−0.999238 + 0.0390229i \(0.987575\pi\)
\(440\) 621.921 0.0673839
\(441\) 0 0
\(442\) 281.152 0.0302558
\(443\) 2915.15 5049.19i 0.312648 0.541522i −0.666287 0.745695i \(-0.732117\pi\)
0.978935 + 0.204174i \(0.0654507\pi\)
\(444\) 0 0
\(445\) 2051.84 + 3553.89i 0.218576 + 0.378585i
\(446\) −2028.27 + 3513.06i −0.215339 + 0.372978i
\(447\) 0 0
\(448\) 0 0
\(449\) 8674.94 0.911794 0.455897 0.890033i \(-0.349319\pi\)
0.455897 + 0.890033i \(0.349319\pi\)
\(450\) 0 0
\(451\) 1035.97 + 1794.36i 0.108164 + 0.187346i
\(452\) −2087.53 3615.70i −0.217232 0.376257i
\(453\) 0 0
\(454\) −4847.91 −0.501154
\(455\) 0 0
\(456\) 0 0
\(457\) −4553.41 + 7886.74i −0.466082 + 0.807278i −0.999250 0.0387315i \(-0.987668\pi\)
0.533167 + 0.846010i \(0.321002\pi\)
\(458\) 1967.17 + 3407.24i 0.200698 + 0.347619i
\(459\) 0 0
\(460\) −2702.48 + 4680.83i −0.273921 + 0.474445i
\(461\) −8729.69 −0.881957 −0.440979 0.897518i \(-0.645369\pi\)
−0.440979 + 0.897518i \(0.645369\pi\)
\(462\) 0 0
\(463\) −1795.62 −0.180237 −0.0901184 0.995931i \(-0.528725\pi\)
−0.0901184 + 0.995931i \(0.528725\pi\)
\(464\) 1249.06 2163.43i 0.124970 0.216454i
\(465\) 0 0
\(466\) −4478.33 7756.70i −0.445182 0.771078i
\(467\) −65.2797 + 113.068i −0.00646849 + 0.0112038i −0.869242 0.494388i \(-0.835392\pi\)
0.862773 + 0.505591i \(0.168726\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 9045.40 0.887730
\(471\) 0 0
\(472\) 2357.74 + 4083.73i 0.229924 + 0.398239i
\(473\) 1799.76 + 3117.28i 0.174954 + 0.303029i
\(474\) 0 0
\(475\) 8757.45 0.845935
\(476\) 0 0
\(477\) 0 0
\(478\) −6116.92 + 10594.8i −0.585317 + 1.01380i
\(479\) −5922.90 10258.8i −0.564978 0.978570i −0.997052 0.0767319i \(-0.975551\pi\)
0.432074 0.901838i \(-0.357782\pi\)
\(480\) 0 0
\(481\) −549.777 + 952.242i −0.0521158 + 0.0902671i
\(482\) 12456.8 1.17716
\(483\) 0 0
\(484\) −4884.24 −0.458700
\(485\) −3350.89 + 5803.91i −0.313724 + 0.543386i
\(486\) 0 0
\(487\) 2403.61 + 4163.17i 0.223650 + 0.387374i 0.955914 0.293648i \(-0.0948693\pi\)
−0.732263 + 0.681022i \(0.761536\pi\)
\(488\) −988.870 + 1712.77i −0.0917296 + 0.158880i
\(489\) 0 0
\(490\) 0 0
\(491\) −6068.04 −0.557733 −0.278866 0.960330i \(-0.589959\pi\)
−0.278866 + 0.960330i \(0.589959\pi\)
\(492\) 0 0
\(493\) 3937.93 + 6820.70i 0.359748 + 0.623101i
\(494\) 348.500 + 603.619i 0.0317404 + 0.0549759i
\(495\) 0 0
\(496\) −2234.12 −0.202248
\(497\) 0 0
\(498\) 0 0
\(499\) 7753.21 13429.0i 0.695554 1.20473i −0.274440 0.961604i \(-0.588492\pi\)
0.969994 0.243130i \(-0.0781742\pi\)
\(500\) 2891.98 + 5009.06i 0.258667 + 0.448024i
\(501\) 0 0
\(502\) 5904.42 10226.7i 0.524954 0.909247i
\(503\) −1496.79 −0.132681 −0.0663405 0.997797i \(-0.521132\pi\)
−0.0663405 + 0.997797i \(0.521132\pi\)
\(504\) 0 0
\(505\) −2325.18 −0.204889
\(506\) −1910.94 + 3309.85i −0.167889 + 0.290792i
\(507\) 0 0
\(508\) 4160.35 + 7205.94i 0.363358 + 0.629354i
\(509\) 2526.87 4376.66i 0.220042 0.381124i −0.734778 0.678307i \(-0.762714\pi\)
0.954821 + 0.297183i \(0.0960472\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) −408.223 707.063i −0.0350310 0.0606755i
\(515\) 856.248 + 1483.07i 0.0732637 + 0.126896i
\(516\) 0 0
\(517\) 6396.07 0.544098
\(518\) 0 0
\(519\) 0 0
\(520\) −82.6476 + 143.150i −0.00696988 + 0.0120722i
\(521\) −4868.36 8432.25i −0.409380 0.709067i 0.585441 0.810715i \(-0.300922\pi\)
−0.994820 + 0.101649i \(0.967588\pi\)
\(522\) 0 0
\(523\) −5898.34 + 10216.2i −0.493148 + 0.854157i −0.999969 0.00789446i \(-0.997487\pi\)
0.506821 + 0.862051i \(0.330820\pi\)
\(524\) −5078.11 −0.423355
\(525\) 0 0
\(526\) −9252.01 −0.766933
\(527\) 3521.79 6099.91i 0.291103 0.504206i
\(528\) 0 0
\(529\) −10524.0 18228.1i −0.864962 1.49816i
\(530\) 1019.67 1766.12i 0.0835691 0.144746i
\(531\) 0 0
\(532\) 0 0
\(533\) −550.685 −0.0447520
\(534\) 0 0
\(535\) 465.191 + 805.734i 0.0375924 + 0.0651120i
\(536\) −1582.59 2741.12i −0.127532 0.220893i
\(537\) 0 0
\(538\) 1742.59 0.139644
\(539\) 0 0
\(540\) 0 0
\(541\) 2155.41 3733.28i 0.171291 0.296684i −0.767581 0.640952i \(-0.778540\pi\)
0.938871 + 0.344268i \(0.111873\pi\)
\(542\) −6472.35 11210.4i −0.512936 0.888431i
\(543\) 0 0
\(544\) 807.098 1397.94i 0.0636104 0.110176i
\(545\) 5527.50 0.434444
\(546\) 0 0
\(547\) 17015.9 1.33007 0.665034 0.746813i \(-0.268417\pi\)
0.665034 + 0.746813i \(0.268417\pi\)
\(548\) 6146.38 10645.8i 0.479124 0.829868i
\(549\) 0 0
\(550\) 734.278 + 1271.81i 0.0569268 + 0.0986001i
\(551\) −9762.45 + 16909.1i −0.754799 + 1.30735i
\(552\) 0 0
\(553\) 0 0
\(554\) −9423.76 −0.722703
\(555\) 0 0
\(556\) 2027.19 + 3511.20i 0.154626 + 0.267820i
\(557\) 870.588 + 1507.90i 0.0662262 + 0.114707i 0.897237 0.441549i \(-0.145571\pi\)
−0.831011 + 0.556256i \(0.812237\pi\)
\(558\) 0 0
\(559\) −956.689 −0.0723858
\(560\) 0 0
\(561\) 0 0
\(562\) 7165.66 12411.3i 0.537838 0.931563i
\(563\) −5673.42 9826.66i −0.424700 0.735602i 0.571692 0.820468i \(-0.306287\pi\)
−0.996392 + 0.0848658i \(0.972954\pi\)
\(564\) 0 0
\(565\) −3869.34 + 6701.89i −0.288114 + 0.499028i
\(566\) 12694.0 0.942701
\(567\) 0 0
\(568\) 2285.29 0.168818
\(569\) −9208.72 + 15950.0i −0.678470 + 1.17514i 0.296972 + 0.954886i \(0.404023\pi\)
−0.975442 + 0.220258i \(0.929310\pi\)
\(570\) 0 0
\(571\) −4999.25 8658.95i −0.366396 0.634616i 0.622603 0.782538i \(-0.286075\pi\)
−0.988999 + 0.147922i \(0.952742\pi\)
\(572\) −58.4407 + 101.222i −0.00427190 + 0.00739915i
\(573\) 0 0
\(574\) 0 0
\(575\) −12762.8 −0.925648
\(576\) 0 0
\(577\) −700.855 1213.92i −0.0505667 0.0875840i 0.839634 0.543152i \(-0.182769\pi\)
−0.890201 + 0.455568i \(0.849436\pi\)
\(578\) −2368.44 4102.26i −0.170440 0.295210i
\(579\) 0 0
\(580\) −4630.38 −0.331494
\(581\) 0 0
\(582\) 0 0
\(583\) 721.015 1248.83i 0.0512202 0.0887160i
\(584\) 3989.83 + 6910.59i 0.282706 + 0.489661i
\(585\) 0 0
\(586\) 9233.78 15993.4i 0.650928 1.12744i
\(587\) −10851.3 −0.763001 −0.381500 0.924369i \(-0.624593\pi\)
−0.381500 + 0.924369i \(0.624593\pi\)
\(588\) 0 0
\(589\) 17461.6 1.22155
\(590\) 4370.20 7569.41i 0.304946 0.528183i
\(591\) 0 0
\(592\) 3156.47 + 5467.16i 0.219139 + 0.379559i
\(593\) −10231.0 + 17720.6i −0.708493 + 1.22715i 0.256923 + 0.966432i \(0.417291\pi\)
−0.965416 + 0.260715i \(0.916042\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4925.88 −0.338544
\(597\) 0 0
\(598\) −507.893 879.697i −0.0347313 0.0601563i
\(599\) 4995.21 + 8651.96i 0.340733 + 0.590166i 0.984569 0.174997i \(-0.0559916\pi\)
−0.643836 + 0.765163i \(0.722658\pi\)
\(600\) 0 0
\(601\) −17435.9 −1.18341 −0.591703 0.806156i \(-0.701544\pi\)
−0.591703 + 0.806156i \(0.701544\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 4489.47 7775.99i 0.302440 0.523842i
\(605\) 4526.60 + 7840.29i 0.304186 + 0.526865i
\(606\) 0 0
\(607\) 8350.16 14462.9i 0.558356 0.967102i −0.439277 0.898351i \(-0.644766\pi\)
0.997634 0.0687503i \(-0.0219012\pi\)
\(608\) 4001.72 0.266926
\(609\) 0 0
\(610\) 3665.85 0.243321
\(611\) −849.979 + 1472.21i −0.0562790 + 0.0974781i
\(612\) 0 0
\(613\) 13351.5 + 23125.4i 0.879707 + 1.52370i 0.851662 + 0.524091i \(0.175595\pi\)
0.0280452 + 0.999607i \(0.491072\pi\)
\(614\) −6786.53 + 11754.6i −0.446062 + 0.772602i
\(615\) 0 0
\(616\) 0 0
\(617\) −27790.4 −1.81329 −0.906645 0.421894i \(-0.861365\pi\)
−0.906645 + 0.421894i \(0.861365\pi\)
\(618\) 0 0
\(619\) 868.040 + 1503.49i 0.0563642 + 0.0976257i 0.892831 0.450392i \(-0.148716\pi\)
−0.836467 + 0.548018i \(0.815383\pi\)
\(620\) 2070.53 + 3586.26i 0.134120 + 0.232303i
\(621\) 0 0
\(622\) −10273.5 −0.662269
\(623\) 0 0
\(624\) 0 0
\(625\) 983.599 1703.64i 0.0629503 0.109033i
\(626\) 3763.56 + 6518.67i 0.240291 + 0.416196i
\(627\) 0 0
\(628\) −7574.31 + 13119.1i −0.481287 + 0.833613i
\(629\) −19903.0 −1.26166
\(630\) 0 0
\(631\) −8990.27 −0.567190 −0.283595 0.958944i \(-0.591527\pi\)
−0.283595 + 0.958944i \(0.591527\pi\)
\(632\) −3393.06 + 5876.95i −0.213558 + 0.369893i
\(633\) 0 0
\(634\) 1034.95 + 1792.58i 0.0648311 + 0.112291i
\(635\) 7711.43 13356.6i 0.481919 0.834709i
\(636\) 0 0
\(637\) 0 0
\(638\) −3274.18 −0.203175
\(639\) 0 0
\(640\) 474.510 + 821.875i 0.0293073 + 0.0507617i
\(641\) −6884.81 11924.8i −0.424233 0.734794i 0.572115 0.820173i \(-0.306123\pi\)
−0.996348 + 0.0853796i \(0.972790\pi\)
\(642\) 0 0
\(643\) −26969.9 −1.65411 −0.827053 0.562124i \(-0.809985\pi\)
−0.827053 + 0.562124i \(0.809985\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −6308.17 + 10926.1i −0.384198 + 0.665450i
\(647\) −12200.9 21132.5i −0.741368 1.28409i −0.951872 0.306495i \(-0.900844\pi\)
0.210504 0.977593i \(-0.432490\pi\)
\(648\) 0 0
\(649\) 3090.20 5352.38i 0.186904 0.323728i
\(650\) −390.316 −0.0235530
\(651\) 0 0
\(652\) −8434.24 −0.506611
\(653\) 7984.79 13830.1i 0.478513 0.828809i −0.521183 0.853445i \(-0.674509\pi\)
0.999696 + 0.0246357i \(0.00784257\pi\)
\(654\) 0 0
\(655\) 4706.27 + 8151.50i 0.280747 + 0.486268i
\(656\) −1580.84 + 2738.10i −0.0940876 + 0.162965i
\(657\) 0 0
\(658\) 0 0
\(659\) 11596.2 0.685467 0.342733 0.939433i \(-0.388647\pi\)
0.342733 + 0.939433i \(0.388647\pi\)
\(660\) 0 0
\(661\) 6301.11 + 10913.8i 0.370779 + 0.642208i 0.989686 0.143257i \(-0.0457575\pi\)
−0.618907 + 0.785464i \(0.712424\pi\)
\(662\) 8800.06 + 15242.2i 0.516653 + 0.894869i
\(663\) 0 0
\(664\) 1686.90 0.0985912
\(665\) 0 0
\(666\) 0 0
\(667\) 14227.5 24642.8i 0.825924 1.43054i
\(668\) −3004.83 5204.52i −0.174042 0.301450i
\(669\) 0 0
\(670\) −2933.41 + 5080.81i −0.169146 + 0.292969i
\(671\) 2592.14 0.149134
\(672\) 0 0
\(673\) 2126.29 0.121787 0.0608934 0.998144i \(-0.480605\pi\)
0.0608934 + 0.998144i \(0.480605\pi\)
\(674\) −5859.78 + 10149.4i −0.334882 + 0.580032i
\(675\) 0 0
\(676\) 4378.47 + 7583.73i 0.249116 + 0.431482i
\(677\) −1309.69 + 2268.45i −0.0743508 + 0.128779i −0.900804 0.434226i \(-0.857022\pi\)
0.826453 + 0.563006i \(0.190355\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2992.00 −0.168732
\(681\) 0 0
\(682\) 1464.09 + 2535.87i 0.0822034 + 0.142381i
\(683\) −14964.9 25919.9i −0.838383 1.45212i −0.891246 0.453520i \(-0.850168\pi\)
0.0528633 0.998602i \(-0.483165\pi\)
\(684\) 0 0
\(685\) −22785.3 −1.27092
\(686\) 0 0
\(687\) 0 0
\(688\) −2746.35 + 4756.81i −0.152185 + 0.263593i
\(689\) 191.633 + 331.918i 0.0105960 + 0.0183528i
\(690\) 0 0
\(691\) 3380.45 5855.11i 0.186105 0.322343i −0.757844 0.652436i \(-0.773747\pi\)
0.943948 + 0.330094i \(0.107080\pi\)
\(692\) 1885.02 0.103552
\(693\) 0 0
\(694\) 15877.1 0.868423
\(695\) 3757.51 6508.20i 0.205080 0.355209i
\(696\) 0 0
\(697\) −4983.96 8632.48i −0.270848 0.469122i
\(698\) 9927.75 17195.4i 0.538354 0.932456i
\(699\) 0 0
\(700\) 0 0
\(701\) −467.205 −0.0251727 −0.0125864 0.999921i \(-0.504006\pi\)
−0.0125864 + 0.999921i \(0.504006\pi\)
\(702\) 0 0
\(703\) −24670.5 42730.6i −1.32357 2.29248i
\(704\) 335.529 + 581.153i 0.0179627 + 0.0311123i
\(705\) 0 0
\(706\) 20206.1 1.07715
\(707\) 0 0
\(708\) 0 0
\(709\) −4412.32 + 7642.36i −0.233721 + 0.404816i −0.958900 0.283744i \(-0.908424\pi\)
0.725179 + 0.688560i \(0.241757\pi\)
\(710\) −2117.95 3668.40i −0.111951 0.193905i
\(711\) 0 0
\(712\) −2213.95 + 3834.67i −0.116533 + 0.201841i
\(713\) −25448.0 −1.33665
\(714\) 0 0
\(715\) 216.646 0.0113316
\(716\) 2664.97 4615.87i 0.139099 0.240926i
\(717\) 0 0
\(718\) 4825.27 + 8357.62i 0.250804 + 0.434406i
\(719\) 10549.4 18272.2i 0.547188 0.947757i −0.451278 0.892383i \(-0.649032\pi\)
0.998466 0.0553733i \(-0.0176349\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −17558.9 −0.905090
\(723\) 0 0
\(724\) −1995.45 3456.23i −0.102432 0.177417i
\(725\) −5466.92 9468.98i −0.280050 0.485061i
\(726\) 0 0
\(727\) 4616.48 0.235510 0.117755 0.993043i \(-0.462430\pi\)
0.117755 + 0.993043i \(0.462430\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 7395.36 12809.1i 0.374951 0.649435i
\(731\) −8658.49 14996.9i −0.438093 0.758799i
\(732\) 0 0
\(733\) −5344.03 + 9256.13i −0.269285 + 0.466416i −0.968678 0.248322i \(-0.920121\pi\)
0.699392 + 0.714738i \(0.253454\pi\)
\(734\) 13870.6 0.697511
\(735\) 0 0
\(736\) −5831.99 −0.292079
\(737\) −2074.23 + 3592.68i −0.103671 + 0.179563i
\(738\) 0 0
\(739\) 7683.78 + 13308.7i 0.382480 + 0.662474i 0.991416 0.130745i \(-0.0417369\pi\)
−0.608936 + 0.793219i \(0.708404\pi\)
\(740\) 5850.68 10133.7i 0.290642 0.503407i
\(741\) 0 0
\(742\) 0 0
\(743\) −6502.58 −0.321072 −0.160536 0.987030i \(-0.551322\pi\)
−0.160536 + 0.987030i \(0.551322\pi\)
\(744\) 0 0
\(745\) 4565.19 + 7907.14i 0.224504 + 0.388853i
\(746\) 14093.1 + 24410.0i 0.691669 + 1.19801i
\(747\) 0 0
\(748\) −2115.66 −0.103418
\(749\) 0 0
\(750\) 0 0
\(751\) 9937.07 17211.5i 0.482835 0.836295i −0.516971 0.856003i \(-0.672940\pi\)
0.999806 + 0.0197085i \(0.00627380\pi\)
\(752\) 4880.03 + 8452.47i 0.236644 + 0.409880i
\(753\) 0 0
\(754\) 435.108 753.630i 0.0210155 0.0364000i
\(755\) −16642.9 −0.802250
\(756\) 0 0
\(757\) −15157.8 −0.727765 −0.363883 0.931445i \(-0.618549\pi\)
−0.363883 + 0.931445i \(0.618549\pi\)
\(758\) 5354.17 9273.70i 0.256560 0.444375i
\(759\) 0 0
\(760\) −3708.70 6423.66i −0.177012 0.306593i
\(761\) −17609.9 + 30501.3i −0.838842 + 1.45292i 0.0520212 + 0.998646i \(0.483434\pi\)
−0.890863 + 0.454271i \(0.849900\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −4906.88 −0.232362
\(765\) 0 0
\(766\) −8970.47 15537.3i −0.423128 0.732880i
\(767\) 821.319 + 1422.57i 0.0386651 + 0.0669698i
\(768\) 0 0
\(769\) 2264.35 0.106183 0.0530915 0.998590i \(-0.483093\pi\)
0.0530915 + 0.998590i \(0.483093\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 6958.58 12052.6i 0.324410 0.561895i
\(773\) 4766.39 + 8255.64i 0.221779 + 0.384133i 0.955348 0.295482i \(-0.0954803\pi\)
−0.733569 + 0.679615i \(0.762147\pi\)
\(774\) 0 0
\(775\) −4889.19 + 8468.33i −0.226613 + 0.392505i
\(776\) −7231.28 −0.334520
\(777\) 0 0
\(778\) −7405.16 −0.341244
\(779\) 12355.6 21400.6i 0.568276 0.984282i
\(780\) 0 0
\(781\) −1497.62 2593.95i −0.0686159 0.118846i
\(782\) 9193.34 15923.3i 0.420401 0.728155i
\(783\) 0 0
\(784\) 0 0
\(785\) 28078.8 1.27666
\(786\) 0 0
\(787\) −16606.5 28763.3i −0.752170 1.30280i −0.946769 0.321913i \(-0.895674\pi\)
0.194599 0.980883i \(-0.437659\pi\)
\(788\) 6386.94 + 11062.5i 0.288738 + 0.500109i
\(789\) 0 0
\(790\) 12578.4 0.566481
\(791\) 0 0
\(792\) 0 0
\(793\) −344.472 + 596.643i −0.0154257 + 0.0267181i
\(794\) 2083.24 + 3608.27i 0.0931124 + 0.161275i
\(795\) 0 0
\(796\) 2130.30 3689.78i 0.0948573 0.164298i
\(797\) 42065.6 1.86956 0.934781 0.355223i \(-0.115595\pi\)
0.934781 + 0.355223i \(0.115595\pi\)
\(798\) 0 0
\(799\) −30770.8 −1.36245
\(800\) −1120.47 + 1940.71i −0.0495183 + 0.0857682i
\(801\) 0 0
\(802\) 10634.0 + 18418.7i 0.468205 + 0.810954i
\(803\) 5229.31 9057.43i 0.229811 0.398044i
\(804\) 0 0
\(805\) 0 0
\(806\) −778.255 −0.0340110
\(807\) 0 0
\(808\) −1254.44 2172.76i −0.0546178 0.0946008i
\(809\) −388.780 673.387i −0.0168959 0.0292646i 0.857454 0.514561i \(-0.172045\pi\)
−0.874350 + 0.485296i \(0.838712\pi\)
\(810\) 0 0
\(811\) −16559.4 −0.716991 −0.358496 0.933531i \(-0.616710\pi\)
−0.358496 + 0.933531i \(0.616710\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 4137.06 7165.59i 0.178137 0.308543i
\(815\) 7816.66 + 13538.8i 0.335958 + 0.581896i
\(816\) 0 0
\(817\) 21465.1 37178.6i 0.919178 1.59206i
\(818\) −13032.7 −0.557065
\(819\) 0 0
\(820\) 5860.35 0.249576
\(821\) 115.612 200.246i 0.00491460 0.00851235i −0.863558 0.504250i \(-0.831769\pi\)
0.868472 + 0.495738i \(0.165102\pi\)
\(822\) 0 0
\(823\) 2611.43 + 4523.13i 0.110606 + 0.191575i 0.916015 0.401145i \(-0.131388\pi\)
−0.805409 + 0.592720i \(0.798054\pi\)
\(824\) −923.899 + 1600.24i −0.0390601 + 0.0676541i
\(825\) 0 0
\(826\) 0 0
\(827\) 46225.5 1.94367 0.971836 0.235658i \(-0.0757246\pi\)
0.971836 + 0.235658i \(0.0757246\pi\)
\(828\) 0 0
\(829\) −19797.1 34289.6i −0.829410 1.43658i −0.898502 0.438970i \(-0.855343\pi\)
0.0690913 0.997610i \(-0.477990\pi\)
\(830\) −1563.38 2707.86i −0.0653805 0.113242i
\(831\) 0 0
\(832\) −178.355 −0.00743191
\(833\) 0 0
\(834\) 0 0
\(835\) −5569.61 + 9646.85i −0.230832 + 0.399812i
\(836\) −2622.45 4542.22i −0.108492 0.187914i
\(837\) 0 0
\(838\) 6079.92 10530.7i 0.250629 0.434103i
\(839\) −45737.6 −1.88205 −0.941023 0.338344i \(-0.890133\pi\)
−0.941023 + 0.338344i \(0.890133\pi\)
\(840\) 0 0
\(841\) −11.7878 −0.000483326
\(842\) −5631.58 + 9754.18i −0.230495 + 0.399229i
\(843\) 0 0
\(844\) −4114.99 7127.37i −0.167824 0.290680i
\(845\) 8115.72 14056.8i 0.330402 0.572272i
\(846\) 0 0
\(847\) 0 0
\(848\) 2200.46 0.0891088
\(849\) 0 0
\(850\) −3532.54 6118.54i −0.142547 0.246899i
\(851\) 35954.1 + 62274.3i 1.44828 + 2.50850i
\(852\) 0 0
\(853\) 8795.55 0.353053 0.176526 0.984296i \(-0.443514\pi\)
0.176526 + 0.984296i \(0.443514\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −501.945 + 869.394i −0.0200422 + 0.0347141i
\(857\) −15127.5 26201.6i −0.602969 1.04437i −0.992369 0.123305i \(-0.960651\pi\)
0.389400 0.921069i \(-0.372683\pi\)
\(858\) 0 0
\(859\) 22496.7 38965.5i 0.893573 1.54771i 0.0580118 0.998316i \(-0.481524\pi\)
0.835561 0.549398i \(-0.185143\pi\)
\(860\) 10181.0 0.403685
\(861\) 0 0
\(862\) −7473.79 −0.295311
\(863\) 22333.9 38683.5i 0.880946 1.52584i 0.0306555 0.999530i \(-0.490241\pi\)
0.850291 0.526313i \(-0.176426\pi\)
\(864\) 0 0
\(865\) −1747.00 3025.89i −0.0686701 0.118940i
\(866\) 5757.46 9972.22i 0.225920 0.391305i
\(867\) 0 0
\(868\) 0 0
\(869\) 8894.29 0.347201
\(870\) 0 0
\(871\) −551.294 954.868i −0.0214465 0.0371464i
\(872\) 2982.11 + 5165.16i 0.115811 + 0.200590i
\(873\) 0 0
\(874\) 45582.1 1.76411
\(875\) 0 0
\(876\) 0 0
\(877\) −2993.42 + 5184.76i −0.115257 + 0.199631i −0.917883 0.396852i \(-0.870103\pi\)
0.802625 + 0.596484i \(0.203436\pi\)
\(878\) 9812.76 + 16996.2i 0.377181 + 0.653296i
\(879\) 0 0
\(880\) 621.921 1077.20i 0.0238238 0.0412640i
\(881\) 37911.8 1.44981 0.724904 0.688850i \(-0.241884\pi\)
0.724904 + 0.688850i \(0.241884\pi\)
\(882\) 0 0
\(883\) −16293.6 −0.620978 −0.310489 0.950577i \(-0.600493\pi\)
−0.310489 + 0.950577i \(0.600493\pi\)
\(884\) 281.152 486.970i 0.0106970 0.0185278i
\(885\) 0 0
\(886\) −5830.30 10098.4i −0.221075 0.382914i
\(887\) −4426.71 + 7667.28i −0.167570 + 0.290239i −0.937565 0.347811i \(-0.886925\pi\)
0.769995 + 0.638050i \(0.220259\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 8207.35 0.309113
\(891\) 0 0
\(892\) 4056.54 + 7026.12i 0.152268 + 0.263735i
\(893\) −38141.7 66063.3i −1.42930 2.47562i
\(894\) 0 0
\(895\) −9879.34 −0.368972
\(896\) 0 0
\(897\) 0 0
\(898\) 8674.94 15025.4i 0.322368 0.558358i
\(899\) −10900.6 18880.3i −0.404398 0.700438i
\(900\) 0 0
\(901\) −3468.73 + 6008.02i −0.128258 + 0.222149i
\(902\) 4143.89 0.152967
\(903\) 0 0
\(904\) −8350.10 −0.307213
\(905\) −3698.68 + 6406.30i −0.135854 + 0.235307i
\(906\) 0 0
\(907\) −8331.23 14430.1i −0.304999 0.528274i 0.672262 0.740313i \(-0.265323\pi\)
−0.977261 + 0.212039i \(0.931989\pi\)
\(908\) −4847.91 + 8396.83i −0.177185 + 0.306893i
\(909\) 0 0
\(910\) 0 0
\(911\) 29071.2 1.05727 0.528635 0.848849i \(-0.322704\pi\)
0.528635 + 0.848849i \(0.322704\pi\)
\(912\) 0 0
\(913\) −1105.48 1914.74i −0.0400723 0.0694072i
\(914\) 9106.82 + 15773.5i 0.329570 + 0.570832i
\(915\) 0 0
\(916\) 7868.67 0.283830
\(917\) 0 0
\(918\) 0 0
\(919\) −3986.70 + 6905.16i −0.143100 + 0.247857i −0.928663 0.370926i \(-0.879040\pi\)
0.785562 + 0.618782i \(0.212374\pi\)
\(920\) 5404.96 + 9361.66i 0.193691 + 0.335483i
\(921\) 0 0
\(922\) −8729.69 + 15120.3i −0.311819 + 0.540086i
\(923\) 796.079 0.0283892
\(924\) 0 0
\(925\) 27630.7 0.982154
\(926\) −1795.62 + 3110.11i −0.0637233 + 0.110372i
\(927\) 0 0
\(928\) −2498.11 4326.86i −0.0883670 0.153056i
\(929\) −15070.7 + 26103.3i −0.532244 + 0.921874i 0.467047 + 0.884233i \(0.345318\pi\)
−0.999291 + 0.0376418i \(0.988015\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −17913.3 −0.629582
\(933\) 0 0
\(934\) 130.559 + 226.136i 0.00457391 + 0.00792225i
\(935\) 1960.75 + 3396.11i 0.0685811 + 0.118786i
\(936\) 0 0
\(937\) −1126.37 −0.0392709 −0.0196354 0.999807i \(-0.506251\pi\)
−0.0196354 + 0.999807i \(0.506251\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 9045.40 15667.1i 0.313860 0.543622i
\(941\) 5166.64 + 8948.89i 0.178988 + 0.310016i 0.941534 0.336917i \(-0.109384\pi\)
−0.762546 + 0.646934i \(0.776051\pi\)
\(942\) 0 0
\(943\) −18006.7 + 31188.6i −0.621824 + 1.07703i
\(944\) 9430.97 0.325161
\(945\) 0 0
\(946\) 7199.06 0.247422
\(947\) 12405.1 21486.3i 0.425673 0.737288i −0.570810 0.821082i \(-0.693371\pi\)
0.996483 + 0.0837944i \(0.0267039\pi\)
\(948\) 0 0
\(949\) 1389.85 + 2407.30i 0.0475412 + 0.0823438i
\(950\) 8757.45 15168.3i 0.299083 0.518028i
\(951\) 0 0
\(952\) 0 0
\(953\) 15048.6 0.511513 0.255757 0.966741i \(-0.417675\pi\)
0.255757 + 0.966741i \(0.417675\pi\)
\(954\) 0 0
\(955\) 4547.58 + 7876.65i 0.154090 + 0.266893i
\(956\) 12233.8 + 21189.6i 0.413881 + 0.716864i
\(957\) 0 0
\(958\) −23691.6 −0.798999
\(959\) 0 0
\(960\) 0 0
\(961\) 5146.89 8914.67i 0.172767 0.299240i
\(962\) 1099.55 + 1904.48i 0.0368514 + 0.0638285i
\(963\) 0 0
\(964\) 12456.8 21575.7i 0.416188 0.720859i
\(965\) −25796.2 −0.860528
\(966\) 0 0
\(967\) −15619.9 −0.519442 −0.259721 0.965684i \(-0.583631\pi\)
−0.259721 + 0.965684i \(0.583631\pi\)
\(968\) −4884.24 + 8459.74i −0.162175 + 0.280895i
\(969\) 0 0
\(970\) 6701.78 + 11607.8i 0.221836 + 0.384232i
\(971\) 12416.5 21506.0i 0.410365 0.710773i −0.584565 0.811347i \(-0.698735\pi\)
0.994930 + 0.100574i \(0.0320680\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 9614.42 0.316289
\(975\) 0 0
\(976\) 1977.74 + 3425.54i 0.0648626 + 0.112345i
\(977\) 19328.4 + 33477.8i 0.632928 + 1.09626i 0.986950 + 0.161026i \(0.0514804\pi\)
−0.354022 + 0.935237i \(0.615186\pi\)
\(978\) 0 0
\(979\) 5803.47 0.189458
\(980\) 0 0
\(981\) 0 0
\(982\) −6068.04 + 10510.2i −0.197188 + 0.341540i
\(983\) −13332.1 23091.8i −0.432581 0.749252i 0.564514 0.825424i \(-0.309064\pi\)
−0.997095 + 0.0761716i \(0.975730\pi\)
\(984\) 0 0
\(985\) 11838.5 20505.0i 0.382952 0.663292i
\(986\) 15751.7 0.508760
\(987\) 0 0
\(988\) 1394.00 0.0448876
\(989\) −31282.6 + 54183.0i −1.00579 + 1.74208i
\(990\) 0 0
\(991\) −13614.1 23580.3i −0.436393 0.755855i 0.561015 0.827806i \(-0.310411\pi\)
−0.997408 + 0.0719507i \(0.977078\pi\)
\(992\) −2234.12 + 3869.61i −0.0715054 + 0.123851i
\(993\) 0 0
\(994\) 0 0
\(995\) −7897.24 −0.251617
\(996\) 0 0
\(997\) 2581.46 + 4471.22i 0.0820016 + 0.142031i 0.904110 0.427301i \(-0.140535\pi\)
−0.822108 + 0.569332i \(0.807202\pi\)
\(998\) −15506.4 26857.9i −0.491831 0.851876i
\(999\) 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.be.667.1 4
3.2 odd 2 294.4.e.m.79.2 4
7.2 even 3 882.4.a.bb.1.2 2
7.3 odd 6 882.4.g.bk.361.2 4
7.4 even 3 inner 882.4.g.be.361.1 4
7.5 odd 6 882.4.a.t.1.1 2
7.6 odd 2 882.4.g.bk.667.2 4
21.2 odd 6 294.4.a.l.1.1 2
21.5 even 6 294.4.a.o.1.2 yes 2
21.11 odd 6 294.4.e.m.67.2 4
21.17 even 6 294.4.e.k.67.1 4
21.20 even 2 294.4.e.k.79.1 4
84.23 even 6 2352.4.a.bw.1.1 2
84.47 odd 6 2352.4.a.bu.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.4.a.l.1.1 2 21.2 odd 6
294.4.a.o.1.2 yes 2 21.5 even 6
294.4.e.k.67.1 4 21.17 even 6
294.4.e.k.79.1 4 21.20 even 2
294.4.e.m.67.2 4 21.11 odd 6
294.4.e.m.79.2 4 3.2 odd 2
882.4.a.t.1.1 2 7.5 odd 6
882.4.a.bb.1.2 2 7.2 even 3
882.4.g.be.361.1 4 7.4 even 3 inner
882.4.g.be.667.1 4 1.1 even 1 trivial
882.4.g.bk.361.2 4 7.3 odd 6
882.4.g.bk.667.2 4 7.6 odd 2
2352.4.a.bu.1.2 2 84.47 odd 6
2352.4.a.bw.1.1 2 84.23 even 6