Properties

Label 882.4.g.y.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: \( 7^{2} \)
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.y.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-7.94975 + 13.7694i) q^{5} +8.00000 q^{8} +O(q^{10})\) \(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-7.94975 + 13.7694i) q^{5} +8.00000 q^{8} +(-15.8995 - 27.5387i) q^{10} +(28.6985 + 49.7072i) q^{11} +5.69848 q^{13} +(-8.00000 + 13.8564i) q^{16} +(-25.9497 - 44.9463i) q^{17} +(-8.10051 + 14.0305i) q^{19} +63.5980 q^{20} -114.794 q^{22} +(-106.698 + 184.807i) q^{23} +(-63.8970 - 110.673i) q^{25} +(-5.69848 + 9.87007i) q^{26} +218.191 q^{29} +(125.698 + 217.716i) q^{31} +(-16.0000 - 27.7128i) q^{32} +103.799 q^{34} +(-193.397 + 334.973i) q^{37} +(-16.2010 - 28.0610i) q^{38} +(-63.5980 + 110.155i) q^{40} +328.503 q^{41} -37.5879 q^{43} +(114.794 - 198.829i) q^{44} +(-213.397 - 369.614i) q^{46} +(-127.497 + 220.832i) q^{47} +255.588 q^{50} +(-11.3970 - 19.7401i) q^{52} +(105.794 + 183.240i) q^{53} -912.583 q^{55} +(-218.191 + 377.918i) q^{58} +(-206.101 - 356.977i) q^{59} +(418.347 - 724.598i) q^{61} -502.794 q^{62} +64.0000 q^{64} +(-45.3015 + 78.4645i) q^{65} +(82.7939 + 143.403i) q^{67} +(-103.799 + 179.785i) q^{68} +465.015 q^{71} +(-224.829 - 389.415i) q^{73} +(-386.794 - 669.947i) q^{74} +64.8040 q^{76} +(171.779 - 297.530i) q^{79} +(-127.196 - 220.310i) q^{80} +(-328.503 + 568.983i) q^{82} -1502.33 q^{83} +825.176 q^{85} +(37.5879 - 65.1041i) q^{86} +(229.588 + 397.658i) q^{88} +(-170.543 + 295.389i) q^{89} +853.588 q^{92} +(-254.995 - 441.664i) q^{94} +(-128.794 - 223.078i) q^{95} -865.437 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 8 q^{4} - 12 q^{5} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 8 q^{4} - 12 q^{5} + 32 q^{8} - 24 q^{10} - 4 q^{11} - 96 q^{13} - 32 q^{16} - 84 q^{17} - 72 q^{19} + 96 q^{20} + 16 q^{22} - 308 q^{23} - 18 q^{25} + 96 q^{26} + 160 q^{29} + 384 q^{31} - 64 q^{32} + 336 q^{34} - 536 q^{37} - 144 q^{38} - 96 q^{40} + 1512 q^{41} + 800 q^{43} - 16 q^{44} - 616 q^{46} - 312 q^{47} + 72 q^{50} + 192 q^{52} - 52 q^{53} - 2304 q^{55} - 160 q^{58} - 864 q^{59} + 1416 q^{61} - 1536 q^{62} + 256 q^{64} - 300 q^{65} - 144 q^{67} - 336 q^{68} + 3048 q^{71} + 744 q^{73} - 1072 q^{74} + 576 q^{76} - 976 q^{79} - 192 q^{80} - 1512 q^{82} - 624 q^{83} + 1400 q^{85} - 800 q^{86} - 32 q^{88} - 108 q^{89} + 2464 q^{92} - 624 q^{94} - 40 q^{95} + 1488 q^{97}+O(q^{100}) \) 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
\(5\) −7.94975 + 13.7694i −0.711047 + 1.23157i 0.253417 + 0.967357i \(0.418445\pi\)
−0.964464 + 0.264213i \(0.914888\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) −15.8995 27.5387i −0.502786 0.870851i
\(11\) 28.6985 + 49.7072i 0.786629 + 1.36248i 0.928021 + 0.372528i \(0.121509\pi\)
−0.141392 + 0.989954i \(0.545158\pi\)
\(12\) 0 0
\(13\) 5.69848 0.121575 0.0607875 0.998151i \(-0.480639\pi\)
0.0607875 + 0.998151i \(0.480639\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −8.00000 + 13.8564i −0.125000 + 0.216506i
\(17\) −25.9497 44.9463i −0.370220 0.641240i 0.619379 0.785092i \(-0.287384\pi\)
−0.989599 + 0.143852i \(0.954051\pi\)
\(18\) 0 0
\(19\) −8.10051 + 14.0305i −0.0978096 + 0.169411i −0.910778 0.412897i \(-0.864517\pi\)
0.812968 + 0.582308i \(0.197850\pi\)
\(20\) 63.5980 0.711047
\(21\) 0 0
\(22\) −114.794 −1.11246
\(23\) −106.698 + 184.807i −0.967312 + 1.67543i −0.264041 + 0.964512i \(0.585055\pi\)
−0.703271 + 0.710922i \(0.748278\pi\)
\(24\) 0 0
\(25\) −63.8970 110.673i −0.511176 0.885382i
\(26\) −5.69848 + 9.87007i −0.0429833 + 0.0744492i
\(27\) 0 0
\(28\) 0 0
\(29\) 218.191 1.39714 0.698570 0.715542i \(-0.253820\pi\)
0.698570 + 0.715542i \(0.253820\pi\)
\(30\) 0 0
\(31\) 125.698 + 217.716i 0.728262 + 1.26139i 0.957617 + 0.288044i \(0.0930048\pi\)
−0.229356 + 0.973343i \(0.573662\pi\)
\(32\) −16.0000 27.7128i −0.0883883 0.153093i
\(33\) 0 0
\(34\) 103.799 0.523570
\(35\) 0 0
\(36\) 0 0
\(37\) −193.397 + 334.973i −0.859304 + 1.48836i 0.0132889 + 0.999912i \(0.495770\pi\)
−0.872593 + 0.488447i \(0.837563\pi\)
\(38\) −16.2010 28.0610i −0.0691619 0.119792i
\(39\) 0 0
\(40\) −63.5980 + 110.155i −0.251393 + 0.435426i
\(41\) 328.503 1.25130 0.625652 0.780102i \(-0.284833\pi\)
0.625652 + 0.780102i \(0.284833\pi\)
\(42\) 0 0
\(43\) −37.5879 −0.133305 −0.0666523 0.997776i \(-0.521232\pi\)
−0.0666523 + 0.997776i \(0.521232\pi\)
\(44\) 114.794 198.829i 0.393314 0.681241i
\(45\) 0 0
\(46\) −213.397 369.614i −0.683993 1.18471i
\(47\) −127.497 + 220.832i −0.395690 + 0.685355i −0.993189 0.116515i \(-0.962828\pi\)
0.597499 + 0.801869i \(0.296161\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 255.588 0.722912
\(51\) 0 0
\(52\) −11.3970 19.7401i −0.0303938 0.0526435i
\(53\) 105.794 + 183.240i 0.274187 + 0.474906i 0.969930 0.243385i \(-0.0782579\pi\)
−0.695743 + 0.718291i \(0.744925\pi\)
\(54\) 0 0
\(55\) −912.583 −2.23732
\(56\) 0 0
\(57\) 0 0
\(58\) −218.191 + 377.918i −0.493963 + 0.855569i
\(59\) −206.101 356.977i −0.454780 0.787701i 0.543896 0.839153i \(-0.316949\pi\)
−0.998676 + 0.0514512i \(0.983615\pi\)
\(60\) 0 0
\(61\) 418.347 724.598i 0.878095 1.52091i 0.0246666 0.999696i \(-0.492148\pi\)
0.853429 0.521210i \(-0.174519\pi\)
\(62\) −502.794 −1.02992
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −45.3015 + 78.4645i −0.0864456 + 0.149728i
\(66\) 0 0
\(67\) 82.7939 + 143.403i 0.150969 + 0.261485i 0.931584 0.363527i \(-0.118427\pi\)
−0.780615 + 0.625012i \(0.785094\pi\)
\(68\) −103.799 + 179.785i −0.185110 + 0.320620i
\(69\) 0 0
\(70\) 0 0
\(71\) 465.015 0.777284 0.388642 0.921389i \(-0.372944\pi\)
0.388642 + 0.921389i \(0.372944\pi\)
\(72\) 0 0
\(73\) −224.829 389.415i −0.360469 0.624351i 0.627569 0.778561i \(-0.284050\pi\)
−0.988038 + 0.154210i \(0.950717\pi\)
\(74\) −386.794 669.947i −0.607620 1.05243i
\(75\) 0 0
\(76\) 64.8040 0.0978096
\(77\) 0 0
\(78\) 0 0
\(79\) 171.779 297.530i 0.244641 0.423730i −0.717390 0.696672i \(-0.754663\pi\)
0.962031 + 0.272942i \(0.0879966\pi\)
\(80\) −127.196 220.310i −0.177762 0.307892i
\(81\) 0 0
\(82\) −328.503 + 568.983i −0.442403 + 0.766264i
\(83\) −1502.33 −1.98677 −0.993387 0.114812i \(-0.963373\pi\)
−0.993387 + 0.114812i \(0.963373\pi\)
\(84\) 0 0
\(85\) 825.176 1.05298
\(86\) 37.5879 65.1041i 0.0471303 0.0816321i
\(87\) 0 0
\(88\) 229.588 + 397.658i 0.278115 + 0.481710i
\(89\) −170.543 + 295.389i −0.203118 + 0.351810i −0.949531 0.313672i \(-0.898441\pi\)
0.746414 + 0.665482i \(0.231774\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 853.588 0.967312
\(93\) 0 0
\(94\) −254.995 441.664i −0.279795 0.484619i
\(95\) −128.794 223.078i −0.139095 0.240919i
\(96\) 0 0
\(97\) −865.437 −0.905895 −0.452947 0.891537i \(-0.649627\pi\)
−0.452947 + 0.891537i \(0.649627\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −255.588 + 442.691i −0.255588 + 0.442691i
\(101\) −121.628 210.666i −0.119826 0.207545i 0.799873 0.600170i \(-0.204900\pi\)
−0.919699 + 0.392625i \(0.871567\pi\)
\(102\) 0 0
\(103\) 476.673 825.622i 0.456000 0.789815i −0.542745 0.839898i \(-0.682615\pi\)
0.998745 + 0.0500822i \(0.0159483\pi\)
\(104\) 45.5879 0.0429833
\(105\) 0 0
\(106\) −423.176 −0.387759
\(107\) −672.477 + 1164.76i −0.607578 + 1.05236i 0.384061 + 0.923308i \(0.374525\pi\)
−0.991638 + 0.129048i \(0.958808\pi\)
\(108\) 0 0
\(109\) −867.176 1501.99i −0.762022 1.31986i −0.941807 0.336155i \(-0.890874\pi\)
0.179785 0.983706i \(-0.442460\pi\)
\(110\) 912.583 1580.64i 0.791012 1.37007i
\(111\) 0 0
\(112\) 0 0
\(113\) −1441.18 −1.19977 −0.599887 0.800085i \(-0.704788\pi\)
−0.599887 + 0.800085i \(0.704788\pi\)
\(114\) 0 0
\(115\) −1696.45 2938.34i −1.37561 2.38262i
\(116\) −436.382 755.835i −0.349285 0.604979i
\(117\) 0 0
\(118\) 824.402 0.643156
\(119\) 0 0
\(120\) 0 0
\(121\) −981.706 + 1700.36i −0.737570 + 1.27751i
\(122\) 836.693 + 1449.20i 0.620907 + 1.07544i
\(123\) 0 0
\(124\) 502.794 870.865i 0.364131 0.630693i
\(125\) 44.4222 0.0317860
\(126\) 0 0
\(127\) 1184.70 0.827759 0.413880 0.910332i \(-0.364173\pi\)
0.413880 + 0.910332i \(0.364173\pi\)
\(128\) −64.0000 + 110.851i −0.0441942 + 0.0765466i
\(129\) 0 0
\(130\) −90.6030 156.929i −0.0611262 0.105874i
\(131\) 148.794 257.719i 0.0992381 0.171885i −0.812131 0.583475i \(-0.801693\pi\)
0.911369 + 0.411589i \(0.135026\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −331.176 −0.213502
\(135\) 0 0
\(136\) −207.598 359.570i −0.130892 0.226712i
\(137\) 310.492 + 537.789i 0.193629 + 0.335375i 0.946450 0.322850i \(-0.104641\pi\)
−0.752821 + 0.658225i \(0.771308\pi\)
\(138\) 0 0
\(139\) −898.754 −0.548426 −0.274213 0.961669i \(-0.588417\pi\)
−0.274213 + 0.961669i \(0.588417\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −465.015 + 805.430i −0.274811 + 0.475987i
\(143\) 163.538 + 283.256i 0.0956344 + 0.165644i
\(144\) 0 0
\(145\) −1734.56 + 3004.35i −0.993432 + 1.72067i
\(146\) 899.316 0.509780
\(147\) 0 0
\(148\) 1547.18 0.859304
\(149\) −1527.35 + 2645.45i −0.839769 + 1.45452i 0.0503195 + 0.998733i \(0.483976\pi\)
−0.890088 + 0.455789i \(0.849357\pi\)
\(150\) 0 0
\(151\) 32.5727 + 56.4176i 0.0175545 + 0.0304053i 0.874669 0.484720i \(-0.161079\pi\)
−0.857115 + 0.515126i \(0.827745\pi\)
\(152\) −64.8040 + 112.244i −0.0345809 + 0.0598959i
\(153\) 0 0
\(154\) 0 0
\(155\) −3997.08 −2.07131
\(156\) 0 0
\(157\) −771.110 1335.60i −0.391983 0.678934i 0.600728 0.799453i \(-0.294877\pi\)
−0.992711 + 0.120519i \(0.961544\pi\)
\(158\) 343.558 + 595.059i 0.172987 + 0.299623i
\(159\) 0 0
\(160\) 508.784 0.251393
\(161\) 0 0
\(162\) 0 0
\(163\) 1257.37 2177.82i 0.604200 1.04650i −0.387978 0.921669i \(-0.626826\pi\)
0.992177 0.124836i \(-0.0398404\pi\)
\(164\) −657.005 1137.97i −0.312826 0.541831i
\(165\) 0 0
\(166\) 1502.33 2602.11i 0.702431 1.21665i
\(167\) −528.643 −0.244956 −0.122478 0.992471i \(-0.539084\pi\)
−0.122478 + 0.992471i \(0.539084\pi\)
\(168\) 0 0
\(169\) −2164.53 −0.985220
\(170\) −825.176 + 1429.25i −0.372283 + 0.644813i
\(171\) 0 0
\(172\) 75.1758 + 130.208i 0.0333261 + 0.0577226i
\(173\) 48.4220 83.8693i 0.0212801 0.0368582i −0.855189 0.518316i \(-0.826559\pi\)
0.876469 + 0.481458i \(0.159892\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −918.352 −0.393314
\(177\) 0 0
\(178\) −341.085 590.777i −0.143626 0.248768i
\(179\) −267.271 462.927i −0.111602 0.193301i 0.804814 0.593527i \(-0.202265\pi\)
−0.916416 + 0.400226i \(0.868932\pi\)
\(180\) 0 0
\(181\) 2087.00 0.857049 0.428524 0.903530i \(-0.359034\pi\)
0.428524 + 0.903530i \(0.359034\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −853.588 + 1478.46i −0.341996 + 0.592355i
\(185\) −3074.91 5325.91i −1.22201 2.11659i
\(186\) 0 0
\(187\) 1489.44 2579.78i 0.582451 1.00884i
\(188\) 1019.98 0.395690
\(189\) 0 0
\(190\) 515.176 0.196709
\(191\) 1693.84 2933.82i 0.641687 1.11143i −0.343369 0.939201i \(-0.611568\pi\)
0.985056 0.172234i \(-0.0550986\pi\)
\(192\) 0 0
\(193\) 954.176 + 1652.68i 0.355871 + 0.616386i 0.987267 0.159074i \(-0.0508510\pi\)
−0.631396 + 0.775461i \(0.717518\pi\)
\(194\) 865.437 1498.98i 0.320282 0.554745i
\(195\) 0 0
\(196\) 0 0
\(197\) 2061.88 0.745699 0.372850 0.927892i \(-0.378381\pi\)
0.372850 + 0.927892i \(0.378381\pi\)
\(198\) 0 0
\(199\) −1585.75 2746.60i −0.564878 0.978397i −0.997061 0.0766115i \(-0.975590\pi\)
0.432183 0.901786i \(-0.357743\pi\)
\(200\) −511.176 885.382i −0.180728 0.313030i
\(201\) 0 0
\(202\) 486.512 0.169460
\(203\) 0 0
\(204\) 0 0
\(205\) −2611.51 + 4523.27i −0.889736 + 1.54107i
\(206\) 953.346 + 1651.24i 0.322441 + 0.558484i
\(207\) 0 0
\(208\) −45.5879 + 78.9605i −0.0151969 + 0.0263218i
\(209\) −929.889 −0.307760
\(210\) 0 0
\(211\) 1349.97 0.440454 0.220227 0.975449i \(-0.429320\pi\)
0.220227 + 0.975449i \(0.429320\pi\)
\(212\) 423.176 732.962i 0.137094 0.237453i
\(213\) 0 0
\(214\) −1344.95 2329.53i −0.429622 0.744128i
\(215\) 298.814 517.561i 0.0947858 0.164174i
\(216\) 0 0
\(217\) 0 0
\(218\) 3468.70 1.07766
\(219\) 0 0
\(220\) 1825.17 + 3161.28i 0.559330 + 0.968788i
\(221\) −147.874 256.126i −0.0450095 0.0779587i
\(222\) 0 0
\(223\) −1361.85 −0.408951 −0.204476 0.978872i \(-0.565549\pi\)
−0.204476 + 0.978872i \(0.565549\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1441.18 2496.19i 0.424184 0.734708i
\(227\) 930.905 + 1612.37i 0.272186 + 0.471441i 0.969421 0.245402i \(-0.0789199\pi\)
−0.697235 + 0.716843i \(0.745587\pi\)
\(228\) 0 0
\(229\) 2679.39 4640.84i 0.773184 1.33919i −0.162625 0.986688i \(-0.551996\pi\)
0.935809 0.352506i \(-0.114671\pi\)
\(230\) 6785.81 1.94540
\(231\) 0 0
\(232\) 1745.53 0.493963
\(233\) 2720.56 4712.15i 0.764935 1.32491i −0.175345 0.984507i \(-0.556104\pi\)
0.940281 0.340400i \(-0.110562\pi\)
\(234\) 0 0
\(235\) −2027.15 3511.12i −0.562708 0.974639i
\(236\) −824.402 + 1427.91i −0.227390 + 0.393851i
\(237\) 0 0
\(238\) 0 0
\(239\) 1157.28 0.313213 0.156607 0.987661i \(-0.449945\pi\)
0.156607 + 0.987661i \(0.449945\pi\)
\(240\) 0 0
\(241\) 1984.69 + 3437.58i 0.530477 + 0.918814i 0.999368 + 0.0355573i \(0.0113206\pi\)
−0.468890 + 0.883256i \(0.655346\pi\)
\(242\) −1963.41 3400.73i −0.521541 0.903335i
\(243\) 0 0
\(244\) −3346.77 −0.878095
\(245\) 0 0
\(246\) 0 0
\(247\) −46.1606 + 79.9525i −0.0118912 + 0.0205962i
\(248\) 1005.59 + 1741.73i 0.257479 + 0.445967i
\(249\) 0 0
\(250\) −44.4222 + 76.9415i −0.0112380 + 0.0194648i
\(251\) −5978.75 −1.50349 −0.751744 0.659455i \(-0.770787\pi\)
−0.751744 + 0.659455i \(0.770787\pi\)
\(252\) 0 0
\(253\) −12248.3 −3.04366
\(254\) −1184.70 + 2051.97i −0.292657 + 0.506897i
\(255\) 0 0
\(256\) −128.000 221.703i −0.0312500 0.0541266i
\(257\) 2325.07 4027.15i 0.564335 0.977458i −0.432776 0.901502i \(-0.642466\pi\)
0.997111 0.0759560i \(-0.0242008\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 362.412 0.0864456
\(261\) 0 0
\(262\) 297.588 + 515.437i 0.0701719 + 0.121541i
\(263\) 1847.68 + 3200.28i 0.433205 + 0.750334i 0.997147 0.0754807i \(-0.0240491\pi\)
−0.563942 + 0.825815i \(0.690716\pi\)
\(264\) 0 0
\(265\) −3364.14 −0.779840
\(266\) 0 0
\(267\) 0 0
\(268\) 331.176 573.613i 0.0754843 0.130743i
\(269\) −3578.84 6198.74i −0.811175 1.40500i −0.912042 0.410096i \(-0.865495\pi\)
0.100868 0.994900i \(-0.467838\pi\)
\(270\) 0 0
\(271\) −2019.19 + 3497.33i −0.452608 + 0.783940i −0.998547 0.0538847i \(-0.982840\pi\)
0.545939 + 0.837825i \(0.316173\pi\)
\(272\) 830.392 0.185110
\(273\) 0 0
\(274\) −1241.97 −0.273833
\(275\) 3667.49 6352.28i 0.804211 1.39293i
\(276\) 0 0
\(277\) 1377.41 + 2385.75i 0.298775 + 0.517493i 0.975856 0.218415i \(-0.0700887\pi\)
−0.677081 + 0.735909i \(0.736755\pi\)
\(278\) 898.754 1556.69i 0.193898 0.335841i
\(279\) 0 0
\(280\) 0 0
\(281\) 772.742 0.164050 0.0820248 0.996630i \(-0.473861\pi\)
0.0820248 + 0.996630i \(0.473861\pi\)
\(282\) 0 0
\(283\) 3372.74 + 5841.76i 0.708441 + 1.22706i 0.965435 + 0.260643i \(0.0839344\pi\)
−0.256995 + 0.966413i \(0.582732\pi\)
\(284\) −930.030 1610.86i −0.194321 0.336574i
\(285\) 0 0
\(286\) −654.152 −0.135248
\(287\) 0 0
\(288\) 0 0
\(289\) 1109.72 1922.09i 0.225874 0.391226i
\(290\) −3469.13 6008.70i −0.702462 1.21670i
\(291\) 0 0
\(292\) −899.316 + 1557.66i −0.180235 + 0.312175i
\(293\) −1922.69 −0.383362 −0.191681 0.981457i \(-0.561394\pi\)
−0.191681 + 0.981457i \(0.561394\pi\)
\(294\) 0 0
\(295\) 6553.79 1.29348
\(296\) −1547.18 + 2679.79i −0.303810 + 0.526214i
\(297\) 0 0
\(298\) −3054.70 5290.90i −0.593806 1.02850i
\(299\) −608.020 + 1053.12i −0.117601 + 0.203691i
\(300\) 0 0
\(301\) 0 0
\(302\) −130.291 −0.0248258
\(303\) 0 0
\(304\) −129.608 224.488i −0.0244524 0.0423528i
\(305\) 6651.50 + 11520.7i 1.24873 + 2.16287i
\(306\) 0 0
\(307\) 2016.68 0.374913 0.187456 0.982273i \(-0.439976\pi\)
0.187456 + 0.982273i \(0.439976\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 3997.08 6923.15i 0.732320 1.26842i
\(311\) 3574.99 + 6192.07i 0.651831 + 1.12900i 0.982678 + 0.185320i \(0.0593320\pi\)
−0.330848 + 0.943684i \(0.607335\pi\)
\(312\) 0 0
\(313\) −4298.25 + 7444.78i −0.776202 + 1.34442i 0.157914 + 0.987453i \(0.449523\pi\)
−0.934116 + 0.356969i \(0.883810\pi\)
\(314\) 3084.44 0.554347
\(315\) 0 0
\(316\) −1374.23 −0.244641
\(317\) −1426.62 + 2470.98i −0.252766 + 0.437804i −0.964286 0.264862i \(-0.914674\pi\)
0.711520 + 0.702666i \(0.248007\pi\)
\(318\) 0 0
\(319\) 6261.75 + 10845.7i 1.09903 + 1.90358i
\(320\) −508.784 + 881.239i −0.0888809 + 0.153946i
\(321\) 0 0
\(322\) 0 0
\(323\) 840.824 0.144844
\(324\) 0 0
\(325\) −364.116 630.667i −0.0621462 0.107640i
\(326\) 2514.73 + 4355.65i 0.427234 + 0.739991i
\(327\) 0 0
\(328\) 2628.02 0.442403
\(329\) 0 0
\(330\) 0 0
\(331\) −809.558 + 1402.19i −0.134433 + 0.232845i −0.925381 0.379039i \(-0.876255\pi\)
0.790948 + 0.611884i \(0.209588\pi\)
\(332\) 3004.66 + 5204.23i 0.496694 + 0.860299i
\(333\) 0 0
\(334\) 528.643 915.637i 0.0866050 0.150004i
\(335\) −2632.76 −0.429383
\(336\) 0 0
\(337\) −3278.67 −0.529972 −0.264986 0.964252i \(-0.585367\pi\)
−0.264986 + 0.964252i \(0.585367\pi\)
\(338\) 2164.53 3749.07i 0.348328 0.603321i
\(339\) 0 0
\(340\) −1650.35 2858.49i −0.263244 0.455952i
\(341\) −7214.71 + 12496.2i −1.14574 + 1.98449i
\(342\) 0 0
\(343\) 0 0
\(344\) −300.703 −0.0471303
\(345\) 0 0
\(346\) 96.8439 + 167.739i 0.0150473 + 0.0260627i
\(347\) −1425.15 2468.43i −0.220479 0.381880i 0.734475 0.678636i \(-0.237429\pi\)
−0.954953 + 0.296756i \(0.904095\pi\)
\(348\) 0 0
\(349\) −4725.32 −0.724758 −0.362379 0.932031i \(-0.618035\pi\)
−0.362379 + 0.932031i \(0.618035\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 918.352 1590.63i 0.139058 0.240855i
\(353\) 3363.72 + 5826.13i 0.507175 + 0.878452i 0.999966 + 0.00830453i \(0.00264345\pi\)
−0.492791 + 0.870148i \(0.664023\pi\)
\(354\) 0 0
\(355\) −3696.75 + 6402.96i −0.552685 + 0.957279i
\(356\) 1364.34 0.203118
\(357\) 0 0
\(358\) 1069.08 0.157829
\(359\) −3665.94 + 6349.60i −0.538945 + 0.933479i 0.460017 + 0.887910i \(0.347843\pi\)
−0.998961 + 0.0455691i \(0.985490\pi\)
\(360\) 0 0
\(361\) 3298.26 + 5712.76i 0.480867 + 0.832885i
\(362\) −2087.00 + 3614.80i −0.303012 + 0.524833i
\(363\) 0 0
\(364\) 0 0
\(365\) 7149.34 1.02524
\(366\) 0 0
\(367\) −1387.22 2402.73i −0.197308 0.341748i 0.750347 0.661045i \(-0.229887\pi\)
−0.947655 + 0.319297i \(0.896553\pi\)
\(368\) −1707.18 2956.92i −0.241828 0.418858i
\(369\) 0 0
\(370\) 12299.7 1.72819
\(371\) 0 0
\(372\) 0 0
\(373\) −1263.67 + 2188.75i −0.175417 + 0.303831i −0.940305 0.340332i \(-0.889461\pi\)
0.764889 + 0.644163i \(0.222794\pi\)
\(374\) 2978.87 + 5159.56i 0.411855 + 0.713354i
\(375\) 0 0
\(376\) −1019.98 + 1766.66i −0.139897 + 0.242309i
\(377\) 1243.36 0.169857
\(378\) 0 0
\(379\) 3116.40 0.422371 0.211186 0.977446i \(-0.432268\pi\)
0.211186 + 0.977446i \(0.432268\pi\)
\(380\) −515.176 + 892.311i −0.0695473 + 0.120459i
\(381\) 0 0
\(382\) 3387.69 + 5867.65i 0.453741 + 0.785903i
\(383\) 759.035 1314.69i 0.101266 0.175398i −0.810940 0.585129i \(-0.801044\pi\)
0.912207 + 0.409731i \(0.134377\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3816.70 −0.503277
\(387\) 0 0
\(388\) 1730.87 + 2997.96i 0.226474 + 0.392264i
\(389\) −1623.39 2811.79i −0.211591 0.366487i 0.740622 0.671922i \(-0.234531\pi\)
−0.952213 + 0.305436i \(0.901198\pi\)
\(390\) 0 0
\(391\) 11075.2 1.43247
\(392\) 0 0
\(393\) 0 0
\(394\) −2061.88 + 3571.28i −0.263645 + 0.456646i
\(395\) 2731.20 + 4730.57i 0.347902 + 0.602584i
\(396\) 0 0
\(397\) 915.312 1585.37i 0.115713 0.200421i −0.802351 0.596852i \(-0.796418\pi\)
0.918065 + 0.396431i \(0.129751\pi\)
\(398\) 6342.99 0.798858
\(399\) 0 0
\(400\) 2044.70 0.255588
\(401\) −1692.90 + 2932.20i −0.210822 + 0.365154i −0.951972 0.306185i \(-0.900947\pi\)
0.741150 + 0.671339i \(0.234281\pi\)
\(402\) 0 0
\(403\) 716.291 + 1240.65i 0.0885384 + 0.153353i
\(404\) −486.512 + 842.664i −0.0599131 + 0.103772i
\(405\) 0 0
\(406\) 0 0
\(407\) −22200.8 −2.70382
\(408\) 0 0
\(409\) 4626.59 + 8013.48i 0.559340 + 0.968805i 0.997552 + 0.0699333i \(0.0222787\pi\)
−0.438212 + 0.898872i \(0.644388\pi\)
\(410\) −5223.02 9046.54i −0.629138 1.08970i
\(411\) 0 0
\(412\) −3813.39 −0.456000
\(413\) 0 0
\(414\) 0 0
\(415\) 11943.2 20686.2i 1.41269 2.44685i
\(416\) −91.1758 157.921i −0.0107458 0.0186123i
\(417\) 0 0
\(418\) 929.889 1610.61i 0.108809 0.188464i
\(419\) 3547.52 0.413622 0.206811 0.978381i \(-0.433692\pi\)
0.206811 + 0.978381i \(0.433692\pi\)
\(420\) 0 0
\(421\) 7848.87 0.908624 0.454312 0.890843i \(-0.349885\pi\)
0.454312 + 0.890843i \(0.349885\pi\)
\(422\) −1349.97 + 2338.22i −0.155724 + 0.269722i
\(423\) 0 0
\(424\) 846.352 + 1465.92i 0.0969398 + 0.167905i
\(425\) −3316.22 + 5743.86i −0.378495 + 0.655572i
\(426\) 0 0
\(427\) 0 0
\(428\) 5379.82 0.607578
\(429\) 0 0
\(430\) 597.628 + 1035.12i 0.0670237 + 0.116088i
\(431\) −2223.66 3851.50i −0.248515 0.430441i 0.714599 0.699534i \(-0.246609\pi\)
−0.963114 + 0.269094i \(0.913276\pi\)
\(432\) 0 0
\(433\) 6994.82 0.776327 0.388164 0.921590i \(-0.373110\pi\)
0.388164 + 0.921590i \(0.373110\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3468.70 + 6007.97i −0.381011 + 0.659930i
\(437\) −1728.62 2994.06i −0.189225 0.327747i
\(438\) 0 0
\(439\) 318.091 550.950i 0.0345823 0.0598984i −0.848216 0.529650i \(-0.822323\pi\)
0.882799 + 0.469752i \(0.155657\pi\)
\(440\) −7300.66 −0.791012
\(441\) 0 0
\(442\) 591.497 0.0636530
\(443\) −2237.12 + 3874.80i −0.239929 + 0.415570i −0.960694 0.277610i \(-0.910458\pi\)
0.720764 + 0.693180i \(0.243791\pi\)
\(444\) 0 0
\(445\) −2711.54 4696.53i −0.288853 0.500308i
\(446\) 1361.85 2358.79i 0.144586 0.250430i
\(447\) 0 0
\(448\) 0 0
\(449\) 2389.42 0.251144 0.125572 0.992085i \(-0.459923\pi\)
0.125572 + 0.992085i \(0.459923\pi\)
\(450\) 0 0
\(451\) 9427.52 + 16329.0i 0.984312 + 1.70488i
\(452\) 2882.35 + 4992.38i 0.299943 + 0.519517i
\(453\) 0 0
\(454\) −3723.62 −0.384930
\(455\) 0 0
\(456\) 0 0
\(457\) 2219.17 3843.71i 0.227152 0.393438i −0.729811 0.683649i \(-0.760392\pi\)
0.956963 + 0.290210i \(0.0937253\pi\)
\(458\) 5358.78 + 9281.68i 0.546724 + 0.946953i
\(459\) 0 0
\(460\) −6785.81 + 11753.4i −0.687804 + 1.19131i
\(461\) −14079.8 −1.42248 −0.711240 0.702949i \(-0.751866\pi\)
−0.711240 + 0.702949i \(0.751866\pi\)
\(462\) 0 0
\(463\) 4687.50 0.470511 0.235255 0.971934i \(-0.424407\pi\)
0.235255 + 0.971934i \(0.424407\pi\)
\(464\) −1745.53 + 3023.34i −0.174642 + 0.302489i
\(465\) 0 0
\(466\) 5441.12 + 9424.30i 0.540891 + 0.936851i
\(467\) −4223.63 + 7315.54i −0.418514 + 0.724888i −0.995790 0.0916611i \(-0.970782\pi\)
0.577276 + 0.816549i \(0.304116\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 8108.58 0.795789
\(471\) 0 0
\(472\) −1648.80 2855.81i −0.160789 0.278495i
\(473\) −1078.72 1868.39i −0.104861 0.181625i
\(474\) 0 0
\(475\) 2070.39 0.199992
\(476\) 0 0
\(477\) 0 0
\(478\) −1157.28 + 2004.46i −0.110738 + 0.191803i
\(479\) 2184.70 + 3784.02i 0.208396 + 0.360952i 0.951209 0.308546i \(-0.0998424\pi\)
−0.742814 + 0.669498i \(0.766509\pi\)
\(480\) 0 0
\(481\) −1102.07 + 1908.84i −0.104470 + 0.180947i
\(482\) −7938.75 −0.750208
\(483\) 0 0
\(484\) 7853.65 0.737570
\(485\) 6880.00 11916.5i 0.644134 1.11567i
\(486\) 0 0
\(487\) 7238.86 + 12538.1i 0.673561 + 1.16664i 0.976887 + 0.213755i \(0.0685695\pi\)
−0.303326 + 0.952887i \(0.598097\pi\)
\(488\) 3346.77 5796.78i 0.310454 0.537721i
\(489\) 0 0
\(490\) 0 0
\(491\) −9306.12 −0.855355 −0.427677 0.903931i \(-0.640668\pi\)
−0.427677 + 0.903931i \(0.640668\pi\)
\(492\) 0 0
\(493\) −5662.00 9806.87i −0.517249 0.895901i
\(494\) −92.3212 159.905i −0.00840836 0.0145637i
\(495\) 0 0
\(496\) −4022.35 −0.364131
\(497\) 0 0
\(498\) 0 0
\(499\) 6118.77 10598.0i 0.548926 0.950767i −0.449423 0.893319i \(-0.648370\pi\)
0.998349 0.0574479i \(-0.0182963\pi\)
\(500\) −88.8444 153.883i −0.00794649 0.0137637i
\(501\) 0 0
\(502\) 5978.75 10355.5i 0.531563 0.920695i
\(503\) 5524.30 0.489694 0.244847 0.969562i \(-0.421262\pi\)
0.244847 + 0.969562i \(0.421262\pi\)
\(504\) 0 0
\(505\) 3867.65 0.340808
\(506\) 12248.3 21214.7i 1.07610 1.86385i
\(507\) 0 0
\(508\) −2369.41 4103.93i −0.206940 0.358430i
\(509\) 5039.81 8729.20i 0.438871 0.760148i −0.558731 0.829349i \(-0.688712\pi\)
0.997603 + 0.0692012i \(0.0220450\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) 4650.15 + 8054.30i 0.399045 + 0.691167i
\(515\) 7578.86 + 13127.0i 0.648475 + 1.12319i
\(516\) 0 0
\(517\) −14635.9 −1.24504
\(518\) 0 0
\(519\) 0 0
\(520\) −362.412 + 627.716i −0.0305631 + 0.0529369i
\(521\) −2853.31 4942.07i −0.239934 0.415578i 0.720761 0.693184i \(-0.243792\pi\)
−0.960695 + 0.277606i \(0.910459\pi\)
\(522\) 0 0
\(523\) −5328.63 + 9229.46i −0.445516 + 0.771656i −0.998088 0.0618088i \(-0.980313\pi\)
0.552572 + 0.833465i \(0.313646\pi\)
\(524\) −1190.35 −0.0992381
\(525\) 0 0
\(526\) −7390.73 −0.612645
\(527\) 6523.69 11299.4i 0.539234 0.933981i
\(528\) 0 0
\(529\) −16685.6 28900.4i −1.37138 2.37531i
\(530\) 3364.14 5826.86i 0.275715 0.477552i
\(531\) 0 0
\(532\) 0 0
\(533\) 1871.97 0.152127
\(534\) 0 0
\(535\) −10692.0 18519.2i −0.864033 1.49655i
\(536\) 662.352 + 1147.23i 0.0533754 + 0.0924489i
\(537\) 0 0
\(538\) 14315.4 1.14717
\(539\) 0 0
\(540\) 0 0
\(541\) 2005.24 3473.18i 0.159357 0.276014i −0.775280 0.631618i \(-0.782391\pi\)
0.934637 + 0.355603i \(0.115725\pi\)
\(542\) −4038.37 6994.66i −0.320042 0.554329i
\(543\) 0 0
\(544\) −830.392 + 1438.28i −0.0654462 + 0.113356i
\(545\) 27575.3 2.16733
\(546\) 0 0
\(547\) −17619.8 −1.37728 −0.688638 0.725105i \(-0.741791\pi\)
−0.688638 + 0.725105i \(0.741791\pi\)
\(548\) 1241.97 2151.15i 0.0968144 0.167688i
\(549\) 0 0
\(550\) 7334.98 + 12704.6i 0.568663 + 0.984954i
\(551\) −1767.46 + 3061.32i −0.136654 + 0.236691i
\(552\) 0 0
\(553\) 0 0
\(554\) −5509.65 −0.422532
\(555\) 0 0
\(556\) 1797.51 + 3113.37i 0.137107 + 0.237476i
\(557\) −5168.85 8952.71i −0.393198 0.681038i 0.599672 0.800246i \(-0.295298\pi\)
−0.992869 + 0.119208i \(0.961965\pi\)
\(558\) 0 0
\(559\) −214.194 −0.0162065
\(560\) 0 0
\(561\) 0 0
\(562\) −772.742 + 1338.43i −0.0580003 + 0.100459i
\(563\) 12011.8 + 20805.1i 0.899180 + 1.55742i 0.828545 + 0.559923i \(0.189169\pi\)
0.0706347 + 0.997502i \(0.477498\pi\)
\(564\) 0 0
\(565\) 11457.0 19844.1i 0.853095 1.47760i
\(566\) −13491.0 −1.00189
\(567\) 0 0
\(568\) 3720.12 0.274811
\(569\) −6589.59 + 11413.5i −0.485501 + 0.840912i −0.999861 0.0166623i \(-0.994696\pi\)
0.514361 + 0.857574i \(0.328029\pi\)
\(570\) 0 0
\(571\) −3888.13 6734.44i −0.284962 0.493568i 0.687638 0.726054i \(-0.258648\pi\)
−0.972600 + 0.232485i \(0.925314\pi\)
\(572\) 654.152 1133.02i 0.0478172 0.0828219i
\(573\) 0 0
\(574\) 0 0
\(575\) 27270.8 1.97787
\(576\) 0 0
\(577\) 10083.5 + 17465.2i 0.727525 + 1.26011i 0.957926 + 0.287015i \(0.0926630\pi\)
−0.230401 + 0.973096i \(0.574004\pi\)
\(578\) 2219.44 + 3844.19i 0.159717 + 0.276639i
\(579\) 0 0
\(580\) 13876.5 0.993432
\(581\) 0 0
\(582\) 0 0
\(583\) −6072.25 + 10517.4i −0.431367 + 0.747150i
\(584\) −1798.63 3115.32i −0.127445 0.220741i
\(585\) 0 0
\(586\) 1922.69 3330.20i 0.135539 0.234760i
\(587\) −8365.08 −0.588184 −0.294092 0.955777i \(-0.595017\pi\)
−0.294092 + 0.955777i \(0.595017\pi\)
\(588\) 0 0
\(589\) −4072.88 −0.284924
\(590\) −6553.79 + 11351.5i −0.457314 + 0.792091i
\(591\) 0 0
\(592\) −3094.35 5359.57i −0.214826 0.372090i
\(593\) 13810.9 23921.2i 0.956403 1.65654i 0.225279 0.974294i \(-0.427671\pi\)
0.731124 0.682244i \(-0.238996\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12218.8 0.839769
\(597\) 0 0
\(598\) −1216.04 2106.24i −0.0831564 0.144031i
\(599\) 269.159 + 466.197i 0.0183598 + 0.0318002i 0.875059 0.484016i \(-0.160822\pi\)
−0.856700 + 0.515816i \(0.827489\pi\)
\(600\) 0 0
\(601\) −6958.64 −0.472294 −0.236147 0.971717i \(-0.575885\pi\)
−0.236147 + 0.971717i \(0.575885\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 130.291 225.670i 0.00877725 0.0152027i
\(605\) −15608.6 27034.9i −1.04889 1.81674i
\(606\) 0 0
\(607\) −8648.80 + 14980.2i −0.578326 + 1.00169i 0.417345 + 0.908748i \(0.362961\pi\)
−0.995671 + 0.0929425i \(0.970373\pi\)
\(608\) 518.432 0.0345809
\(609\) 0 0
\(610\) −26606.0 −1.76598
\(611\) −726.542 + 1258.41i −0.0481060 + 0.0833220i
\(612\) 0 0
\(613\) 419.579 + 726.732i 0.0276454 + 0.0478832i 0.879517 0.475867i \(-0.157866\pi\)
−0.851872 + 0.523751i \(0.824532\pi\)
\(614\) −2016.68 + 3493.00i −0.132552 + 0.229586i
\(615\) 0 0
\(616\) 0 0
\(617\) 16040.0 1.04659 0.523295 0.852152i \(-0.324703\pi\)
0.523295 + 0.852152i \(0.324703\pi\)
\(618\) 0 0
\(619\) −2714.64 4701.90i −0.176269 0.305307i 0.764331 0.644825i \(-0.223070\pi\)
−0.940600 + 0.339517i \(0.889736\pi\)
\(620\) 7994.17 + 13846.3i 0.517828 + 0.896905i
\(621\) 0 0
\(622\) −14300.0 −0.921828
\(623\) 0 0
\(624\) 0 0
\(625\) 7633.98 13222.4i 0.488574 0.846236i
\(626\) −8596.49 14889.6i −0.548858 0.950649i
\(627\) 0 0
\(628\) −3084.44 + 5342.41i −0.195991 + 0.339467i
\(629\) 20074.4 1.27253
\(630\) 0 0
\(631\) −1807.86 −0.114057 −0.0570284 0.998373i \(-0.518163\pi\)
−0.0570284 + 0.998373i \(0.518163\pi\)
\(632\) 1374.23 2380.24i 0.0864936 0.149811i
\(633\) 0 0
\(634\) −2853.24 4941.95i −0.178733 0.309574i
\(635\) −9418.09 + 16312.6i −0.588576 + 1.01944i
\(636\) 0 0
\(637\) 0 0
\(638\) −25047.0 −1.55426
\(639\) 0 0
\(640\) −1017.57 1762.48i −0.0628483 0.108856i
\(641\) −2952.28 5113.50i −0.181916 0.315087i 0.760617 0.649201i \(-0.224897\pi\)
−0.942533 + 0.334113i \(0.891563\pi\)
\(642\) 0 0
\(643\) −8092.42 −0.496320 −0.248160 0.968719i \(-0.579826\pi\)
−0.248160 + 0.968719i \(0.579826\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −840.824 + 1456.35i −0.0512102 + 0.0886987i
\(647\) −10096.3 17487.4i −0.613490 1.06260i −0.990647 0.136447i \(-0.956432\pi\)
0.377157 0.926149i \(-0.376902\pi\)
\(648\) 0 0
\(649\) 11829.5 20489.4i 0.715486 1.23926i
\(650\) 1456.46 0.0878880
\(651\) 0 0
\(652\) −10058.9 −0.604200
\(653\) −10590.4 + 18343.2i −0.634664 + 1.09927i 0.351923 + 0.936029i \(0.385528\pi\)
−0.986586 + 0.163240i \(0.947805\pi\)
\(654\) 0 0
\(655\) 2365.75 + 4097.60i 0.141126 + 0.244437i
\(656\) −2628.02 + 4551.86i −0.156413 + 0.270915i
\(657\) 0 0
\(658\) 0 0
\(659\) −28411.3 −1.67944 −0.839718 0.543023i \(-0.817280\pi\)
−0.839718 + 0.543023i \(0.817280\pi\)
\(660\) 0 0
\(661\) 8352.47 + 14466.9i 0.491488 + 0.851282i 0.999952 0.00980149i \(-0.00311996\pi\)
−0.508464 + 0.861083i \(0.669787\pi\)
\(662\) −1619.12 2804.39i −0.0950585 0.164646i
\(663\) 0 0
\(664\) −12018.7 −0.702431
\(665\) 0 0
\(666\) 0 0
\(667\) −23280.6 + 40323.3i −1.35147 + 2.34081i
\(668\) 1057.29 + 1831.27i 0.0612390 + 0.106069i
\(669\) 0 0
\(670\) 2632.76 4560.08i 0.151810 0.262942i
\(671\) 48023.7 2.76294
\(672\) 0 0
\(673\) −9047.09 −0.518187 −0.259093 0.965852i \(-0.583424\pi\)
−0.259093 + 0.965852i \(0.583424\pi\)
\(674\) 3278.67 5678.83i 0.187374 0.324540i
\(675\) 0 0
\(676\) 4329.05 + 7498.14i 0.246305 + 0.426613i
\(677\) 3922.13 6793.32i 0.222658 0.385655i −0.732956 0.680276i \(-0.761860\pi\)
0.955614 + 0.294621i \(0.0951933\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 6601.41 0.372283
\(681\) 0 0
\(682\) −14429.4 24992.5i −0.810163 1.40324i
\(683\) 12883.0 + 22314.1i 0.721751 + 1.25011i 0.960298 + 0.278978i \(0.0899956\pi\)
−0.238547 + 0.971131i \(0.576671\pi\)
\(684\) 0 0
\(685\) −9873.35 −0.550717
\(686\) 0 0
\(687\) 0 0
\(688\) 300.703 520.833i 0.0166631 0.0288613i
\(689\) 602.865 + 1044.19i 0.0333343 + 0.0577367i
\(690\) 0 0
\(691\) 12337.1 21368.4i 0.679195 1.17640i −0.296029 0.955179i \(-0.595662\pi\)
0.975224 0.221221i \(-0.0710042\pi\)
\(692\) −387.376 −0.0212801
\(693\) 0 0
\(694\) 5700.60 0.311804
\(695\) 7144.86 12375.3i 0.389957 0.675425i
\(696\) 0 0
\(697\) −8524.56 14765.0i −0.463258 0.802386i
\(698\) 4725.32 8184.49i 0.256241 0.443822i
\(699\) 0 0
\(700\) 0 0
\(701\) −29377.9 −1.58286 −0.791431 0.611258i \(-0.790664\pi\)
−0.791431 + 0.611258i \(0.790664\pi\)
\(702\) 0 0
\(703\) −3133.23 5426.91i −0.168097 0.291152i
\(704\) 1836.70 + 3181.26i 0.0983286 + 0.170310i
\(705\) 0 0
\(706\) −13454.9 −0.717253
\(707\) 0 0
\(708\) 0 0
\(709\) −15297.1 + 26495.4i −0.810291 + 1.40347i 0.102369 + 0.994746i \(0.467358\pi\)
−0.912660 + 0.408719i \(0.865976\pi\)
\(710\) −7393.51 12805.9i −0.390808 0.676898i
\(711\) 0 0
\(712\) −1364.34 + 2363.11i −0.0718130 + 0.124384i
\(713\) −53647.4 −2.81782
\(714\) 0 0
\(715\) −5200.34 −0.272002
\(716\) −1069.08 + 1851.71i −0.0558011 + 0.0966503i
\(717\) 0 0
\(718\) −7331.89 12699.2i −0.381091 0.660070i
\(719\) −973.469 + 1686.10i −0.0504927 + 0.0874559i −0.890167 0.455634i \(-0.849412\pi\)
0.839674 + 0.543090i \(0.182746\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −13193.1 −0.680048
\(723\) 0 0
\(724\) −4174.01 7229.60i −0.214262 0.371113i
\(725\) −13941.7 24147.8i −0.714184 1.23700i
\(726\) 0 0
\(727\) 15750.6 0.803518 0.401759 0.915745i \(-0.368399\pi\)
0.401759 + 0.915745i \(0.368399\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −7149.34 + 12383.0i −0.362478 + 0.627830i
\(731\) 975.396 + 1689.44i 0.0493520 + 0.0854802i
\(732\) 0 0
\(733\) −7174.78 + 12427.1i −0.361537 + 0.626200i −0.988214 0.153079i \(-0.951081\pi\)
0.626677 + 0.779279i \(0.284414\pi\)
\(734\) 5548.86 0.279036
\(735\) 0 0
\(736\) 6828.70 0.341996
\(737\) −4752.12 + 8230.92i −0.237512 + 0.411384i
\(738\) 0 0
\(739\) 8879.03 + 15378.9i 0.441976 + 0.765525i 0.997836 0.0657506i \(-0.0209442\pi\)
−0.555860 + 0.831276i \(0.687611\pi\)
\(740\) −12299.7 + 21303.6i −0.611006 + 1.05829i
\(741\) 0 0
\(742\) 0 0
\(743\) 29187.6 1.44117 0.720586 0.693366i \(-0.243873\pi\)
0.720586 + 0.693366i \(0.243873\pi\)
\(744\) 0 0
\(745\) −24284.1 42061.3i −1.19423 2.06847i
\(746\) −2527.35 4377.49i −0.124038 0.214841i
\(747\) 0 0
\(748\) −11915.5 −0.582451
\(749\) 0 0
\(750\) 0 0
\(751\) −6890.97 + 11935.5i −0.334827 + 0.579938i −0.983452 0.181171i \(-0.942011\pi\)
0.648624 + 0.761109i \(0.275345\pi\)
\(752\) −2039.96 3533.31i −0.0989224 0.171339i
\(753\) 0 0
\(754\) −1243.36 + 2153.56i −0.0600536 + 0.104016i
\(755\) −1035.78 −0.0499283
\(756\) 0 0
\(757\) 36952.7 1.77420 0.887099 0.461579i \(-0.152717\pi\)
0.887099 + 0.461579i \(0.152717\pi\)
\(758\) −3116.40 + 5397.76i −0.149331 + 0.258649i
\(759\) 0 0
\(760\) −1030.35 1784.62i −0.0491773 0.0851776i
\(761\) 14408.2 24955.8i 0.686332 1.18876i −0.286684 0.958025i \(-0.592553\pi\)
0.973016 0.230737i \(-0.0741136\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −13550.8 −0.641687
\(765\) 0 0
\(766\) 1518.07 + 2629.38i 0.0716059 + 0.124025i
\(767\) −1174.46 2034.23i −0.0552898 0.0957648i
\(768\) 0 0
\(769\) 25285.2 1.18571 0.592854 0.805310i \(-0.298001\pi\)
0.592854 + 0.805310i \(0.298001\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3816.70 6610.72i 0.177935 0.308193i
\(773\) −9209.10 15950.6i −0.428497 0.742179i 0.568243 0.822861i \(-0.307624\pi\)
−0.996740 + 0.0806820i \(0.974290\pi\)
\(774\) 0 0
\(775\) 16063.5 27822.8i 0.744540 1.28958i
\(776\) −6923.49 −0.320282
\(777\) 0 0
\(778\) 6493.55 0.299235
\(779\) −2661.04 + 4609.05i −0.122390 + 0.211985i
\(780\) 0 0
\(781\) 13345.2 + 23114.6i 0.611434 + 1.05903i
\(782\) −11075.2 + 19182.8i −0.506455 + 0.877207i
\(783\) 0 0
\(784\) 0 0
\(785\) 24520.5 1.11487
\(786\) 0 0
\(787\) 5537.90 + 9591.92i 0.250832 + 0.434454i 0.963755 0.266788i \(-0.0859625\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) −4123.76 7142.56i −0.186425 0.322897i
\(789\) 0 0
\(790\) −10924.8 −0.492008
\(791\) 0 0
\(792\) 0 0
\(793\) 2383.94 4129.11i 0.106754 0.184904i
\(794\) 1830.62 + 3170.74i 0.0818217 + 0.141719i
\(795\) 0 0
\(796\) −6342.99 + 10986.4i −0.282439 + 0.489199i
\(797\) 4838.83 0.215057 0.107528 0.994202i \(-0.465706\pi\)
0.107528 + 0.994202i \(0.465706\pi\)
\(798\) 0 0
\(799\) 13234.1 0.585969
\(800\) −2044.70 + 3541.53i −0.0903640 + 0.156515i
\(801\) 0 0
\(802\) −3385.81 5864.39i −0.149074 0.258203i
\(803\) 12904.5 22351.3i 0.567111 0.982265i
\(804\) 0 0
\(805\) 0 0
\(806\) −2865.16 −0.125212
\(807\) 0 0
\(808\) −973.024 1685.33i −0.0423649 0.0733782i
\(809\) −15754.9 27288.4i −0.684690 1.18592i −0.973534 0.228542i \(-0.926604\pi\)
0.288844 0.957376i \(-0.406729\pi\)
\(810\) 0 0
\(811\) 29463.3 1.27570 0.637851 0.770160i \(-0.279823\pi\)
0.637851 + 0.770160i \(0.279823\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 22200.8 38452.9i 0.955943 1.65574i
\(815\) 19991.5 + 34626.3i 0.859229 + 1.48823i
\(816\) 0 0
\(817\) 304.481 527.376i 0.0130385 0.0225833i
\(818\) −18506.3 −0.791026
\(819\) 0 0
\(820\) 20892.1 0.889736
\(821\) 1751.11 3033.01i 0.0744386 0.128932i −0.826403 0.563079i \(-0.809617\pi\)
0.900842 + 0.434147i \(0.142950\pi\)
\(822\) 0 0
\(823\) −19996.5 34635.0i −0.846943 1.46695i −0.883923 0.467632i \(-0.845107\pi\)
0.0369799 0.999316i \(-0.488226\pi\)
\(824\) 3813.39 6604.98i 0.161220 0.279242i
\(825\) 0 0
\(826\) 0 0
\(827\) 10733.6 0.451322 0.225661 0.974206i \(-0.427546\pi\)
0.225661 + 0.974206i \(0.427546\pi\)
\(828\) 0 0
\(829\) −7268.74 12589.8i −0.304528 0.527458i 0.672628 0.739981i \(-0.265165\pi\)
−0.977156 + 0.212523i \(0.931832\pi\)
\(830\) 23886.3 + 41372.3i 0.998923 + 1.73018i
\(831\) 0 0
\(832\) 364.703 0.0151969
\(833\) 0 0
\(834\) 0 0
\(835\) 4202.58 7279.09i 0.174175 0.301680i
\(836\) 1859.78 + 3221.23i 0.0769399 + 0.133264i
\(837\) 0 0
\(838\) −3547.52 + 6144.48i −0.146237 + 0.253291i
\(839\) 7353.57 0.302591 0.151295 0.988489i \(-0.451656\pi\)
0.151295 + 0.988489i \(0.451656\pi\)
\(840\) 0 0
\(841\) 23218.3 0.951998
\(842\) −7848.87 + 13594.6i −0.321247 + 0.556416i
\(843\) 0 0
\(844\) −2699.94 4676.43i −0.110113 0.190722i
\(845\) 17207.4 29804.2i 0.700537 1.21337i
\(846\) 0 0
\(847\) 0 0
\(848\) −3385.41 −0.137094
\(849\) 0 0
\(850\) −6632.44 11487.7i −0.267636 0.463560i
\(851\) −41270.3 71482.3i −1.66243 2.87941i
\(852\) 0 0
\(853\) −19293.6 −0.774442 −0.387221 0.921987i \(-0.626565\pi\)
−0.387221 + 0.921987i \(0.626565\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −5379.82 + 9318.12i −0.214811 + 0.372064i
\(857\) 7080.78 + 12264.3i 0.282234 + 0.488844i 0.971935 0.235251i \(-0.0755912\pi\)
−0.689701 + 0.724095i \(0.742258\pi\)
\(858\) 0 0
\(859\) 4109.57 7117.98i 0.163232 0.282727i −0.772794 0.634657i \(-0.781141\pi\)
0.936026 + 0.351930i \(0.114475\pi\)
\(860\) −2390.51 −0.0947858
\(861\) 0 0
\(862\) 8894.65 0.351454
\(863\) 1287.47 2229.97i 0.0507835 0.0879595i −0.839516 0.543335i \(-0.817161\pi\)
0.890300 + 0.455375i \(0.150495\pi\)
\(864\) 0 0
\(865\) 769.885 + 1333.48i 0.0302623 + 0.0524158i
\(866\) −6994.82 + 12115.4i −0.274473 + 0.475401i
\(867\) 0 0
\(868\) 0 0
\(869\) 19719.2 0.769766
\(870\) 0 0
\(871\) 471.800 + 817.182i 0.0183540 + 0.0317901i
\(872\) −6937.41 12015.9i −0.269415 0.466641i
\(873\) 0 0
\(874\) 6914.49 0.267604
\(875\) 0 0
\(876\) 0 0
\(877\) −15490.6 + 26830.5i −0.596442 + 1.03307i 0.396900 + 0.917862i \(0.370086\pi\)
−0.993342 + 0.115206i \(0.963247\pi\)
\(878\) 636.182 + 1101.90i 0.0244534 + 0.0423546i
\(879\) 0 0
\(880\) 7300.66 12645.1i 0.279665 0.484394i
\(881\) −41781.8 −1.59780 −0.798902 0.601461i \(-0.794585\pi\)
−0.798902 + 0.601461i \(0.794585\pi\)
\(882\) 0 0
\(883\) −39289.6 −1.49740 −0.748699 0.662911i \(-0.769321\pi\)
−0.748699 + 0.662911i \(0.769321\pi\)
\(884\) −591.497 + 1024.50i −0.0225047 + 0.0389794i
\(885\) 0 0
\(886\) −4474.24 7749.61i −0.169656 0.293852i
\(887\) −2916.22 + 5051.04i −0.110391 + 0.191203i −0.915928 0.401342i \(-0.868544\pi\)
0.805537 + 0.592546i \(0.201877\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 10846.2 0.408499
\(891\) 0 0
\(892\) 2723.70 + 4717.58i 0.102238 + 0.177081i
\(893\) −2065.59 3577.70i −0.0774045 0.134069i
\(894\) 0 0
\(895\) 8498.95 0.317418
\(896\) 0 0
\(897\) 0 0
\(898\) −2389.42 + 4138.59i −0.0887928 + 0.153794i
\(899\) 27426.3 + 47503.7i 1.01748 + 1.76233i
\(900\) 0 0
\(901\) 5490.65 9510.09i 0.203019 0.351639i
\(902\) −37710.1 −1.39203
\(903\) 0 0
\(904\) −11529.4 −0.424184
\(905\) −16591.2 + 28736.7i −0.609402 + 1.05552i
\(906\) 0 0
\(907\) −21194.0 36709.0i −0.775892 1.34388i −0.934292 0.356509i \(-0.883967\pi\)
0.158400 0.987375i \(-0.449367\pi\)
\(908\) 3723.62 6449.50i 0.136093 0.235720i
\(909\) 0 0
\(910\) 0 0
\(911\) −2275.12 −0.0827423 −0.0413711 0.999144i \(-0.513173\pi\)
−0.0413711 + 0.999144i \(0.513173\pi\)
\(912\) 0 0
\(913\) −43114.6 74676.7i −1.56285 2.70694i
\(914\) 4438.34 + 7687.43i 0.160621 + 0.278203i
\(915\) 0 0
\(916\) −21435.1 −0.773184
\(917\) 0 0
\(918\) 0 0
\(919\) 15642.1 27093.0i 0.561465 0.972486i −0.435904 0.899993i \(-0.643571\pi\)
0.997369 0.0724930i \(-0.0230955\pi\)
\(920\) −13571.6 23506.7i −0.486351 0.842385i
\(921\) 0 0
\(922\) 14079.8 24387.0i 0.502923 0.871088i
\(923\) 2649.88 0.0944983
\(924\) 0 0
\(925\) 49429.9 1.75702
\(926\) −4687.50 + 8118.98i −0.166351 + 0.288128i
\(927\) 0 0
\(928\) −3491.05 6046.68i −0.123491 0.213892i
\(929\) −16098.3 + 27883.1i −0.568535 + 0.984731i 0.428176 + 0.903695i \(0.359156\pi\)
−0.996711 + 0.0810358i \(0.974177\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −21764.5 −0.764935
\(933\) 0 0
\(934\) −8447.26 14631.1i −0.295934 0.512573i
\(935\) 23681.3 + 41017.2i 0.828301 + 1.43466i
\(936\) 0 0
\(937\) −22293.6 −0.777269 −0.388635 0.921392i \(-0.627053\pi\)
−0.388635 + 0.921392i \(0.627053\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −8108.58 + 14044.5i −0.281354 + 0.487319i
\(941\) 15904.8 + 27548.0i 0.550991 + 0.954345i 0.998203 + 0.0599168i \(0.0190835\pi\)
−0.447212 + 0.894428i \(0.647583\pi\)
\(942\) 0 0
\(943\) −35050.7 + 60709.6i −1.21040 + 2.09648i
\(944\) 6595.22 0.227390
\(945\) 0 0
\(946\) 4314.86 0.148296
\(947\) 9491.32 16439.4i 0.325688 0.564108i −0.655963 0.754793i \(-0.727737\pi\)
0.981651 + 0.190685i \(0.0610708\pi\)
\(948\) 0 0
\(949\) −1281.18 2219.08i −0.0438240 0.0759055i
\(950\) −2070.39 + 3586.02i −0.0707077 + 0.122469i
\(951\) 0 0
\(952\) 0 0
\(953\) −9254.58 −0.314570 −0.157285 0.987553i \(-0.550274\pi\)
−0.157285 + 0.987553i \(0.550274\pi\)
\(954\) 0 0
\(955\) 26931.3 + 46646.3i 0.912539 + 1.58056i
\(956\) −2314.55 4008.92i −0.0783033 0.135625i
\(957\) 0 0
\(958\) −8738.81 −0.294716
\(959\) 0 0
\(960\) 0 0
\(961\) −16704.7 + 28933.4i −0.560730 + 0.971213i
\(962\) −2204.14 3817.68i −0.0738714 0.127949i
\(963\) 0 0
\(964\) 7938.75 13750.3i 0.265239 0.459407i
\(965\) −30341.8 −1.01216
\(966\) 0 0
\(967\) 15317.5 0.509387 0.254694 0.967022i \(-0.418025\pi\)
0.254694 + 0.967022i \(0.418025\pi\)
\(968\) −7853.65 + 13602.9i −0.260770 + 0.451668i
\(969\) 0 0
\(970\) 13760.0 + 23833.0i 0.455471 + 0.788900i
\(971\) −11216.3 + 19427.2i −0.370699 + 0.642069i −0.989673 0.143342i \(-0.954215\pi\)
0.618974 + 0.785411i \(0.287548\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −28955.5 −0.952559
\(975\) 0 0
\(976\) 6693.55 + 11593.6i 0.219524 + 0.380226i
\(977\) −313.932 543.746i −0.0102800 0.0178055i 0.860840 0.508876i \(-0.169939\pi\)
−0.871120 + 0.491071i \(0.836606\pi\)
\(978\) 0 0
\(979\) −19577.3 −0.639114
\(980\) 0 0
\(981\) 0 0
\(982\) 9306.12 16118.7i 0.302414 0.523796i
\(983\) −22516.3 38999.4i −0.730579 1.26540i −0.956636 0.291286i \(-0.905917\pi\)
0.226057 0.974114i \(-0.427416\pi\)
\(984\) 0 0
\(985\) −16391.4 + 28390.8i −0.530227 + 0.918381i
\(986\) 22648.0 0.731500
\(987\) 0 0
\(988\) 369.285 0.0118912
\(989\) 4010.57 6946.51i 0.128947 0.223343i
\(990\) 0 0
\(991\) 15990.5 + 27696.4i 0.512569 + 0.887795i 0.999894 + 0.0145745i \(0.00463937\pi\)
−0.487325 + 0.873221i \(0.662027\pi\)
\(992\) 4022.35 6966.92i 0.128740 0.222984i
\(993\) 0 0
\(994\) 0 0
\(995\) 50425.2 1.60662
\(996\) 0 0
\(997\) 6189.41 + 10720.4i 0.196611 + 0.340539i 0.947427 0.319971i \(-0.103673\pi\)
−0.750817 + 0.660511i \(0.770340\pi\)
\(998\) 12237.5 + 21196.1i 0.388149 + 0.672294i
\(999\) 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.y.667.1 4
3.2 odd 2 294.4.e.n.79.2 4
7.2 even 3 882.4.a.bi.1.2 2
7.3 odd 6 882.4.g.bd.361.2 4
7.4 even 3 inner 882.4.g.y.361.1 4
7.5 odd 6 882.4.a.bc.1.1 2
7.6 odd 2 882.4.g.bd.667.2 4
21.2 odd 6 294.4.a.k.1.1 yes 2
21.5 even 6 294.4.a.j.1.2 2
21.11 odd 6 294.4.e.n.67.2 4
21.17 even 6 294.4.e.o.67.1 4
21.20 even 2 294.4.e.o.79.1 4
84.23 even 6 2352.4.a.bn.1.1 2
84.47 odd 6 2352.4.a.cd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.4.a.j.1.2 2 21.5 even 6
294.4.a.k.1.1 yes 2 21.2 odd 6
294.4.e.n.67.2 4 21.11 odd 6
294.4.e.n.79.2 4 3.2 odd 2
294.4.e.o.67.1 4 21.17 even 6
294.4.e.o.79.1 4 21.20 even 2
882.4.a.bc.1.1 2 7.5 odd 6
882.4.a.bi.1.2 2 7.2 even 3
882.4.g.y.361.1 4 7.4 even 3 inner
882.4.g.y.667.1 4 1.1 even 1 trivial
882.4.g.bd.361.2 4 7.3 odd 6
882.4.g.bd.667.2 4 7.6 odd 2
2352.4.a.bn.1.1 2 84.23 even 6
2352.4.a.cd.1.2 2 84.47 odd 6