Properties

Label 2205.4.a.u.1.2
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2205,4,Mod(1,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2205.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

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: yes
Analytic conductor: \(130.099211563\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.58579 q^{2} -1.31371 q^{4} -5.00000 q^{5} +24.0833 q^{8} +O(q^{10})\) \(q-2.58579 q^{2} -1.31371 q^{4} -5.00000 q^{5} +24.0833 q^{8} +12.9289 q^{10} -38.2548 q^{11} -19.3431 q^{13} -51.7645 q^{16} -87.2254 q^{17} +44.2254 q^{19} +6.56854 q^{20} +98.9188 q^{22} -218.167 q^{23} +25.0000 q^{25} +50.0172 q^{26} +46.9411 q^{29} -194.558 q^{31} -58.8141 q^{32} +225.546 q^{34} +366.853 q^{37} -114.357 q^{38} -120.416 q^{40} -339.362 q^{41} -226.167 q^{43} +50.2557 q^{44} +564.132 q^{46} +11.6762 q^{47} -64.6447 q^{50} +25.4113 q^{52} +209.019 q^{53} +191.274 q^{55} -121.380 q^{58} -616.000 q^{59} -320.735 q^{61} +503.087 q^{62} +566.197 q^{64} +96.7157 q^{65} +14.5097 q^{67} +114.589 q^{68} +952.000 q^{71} -824.489 q^{73} -948.603 q^{74} -58.0993 q^{76} +156.275 q^{79} +258.823 q^{80} +877.519 q^{82} -1036.53 q^{83} +436.127 q^{85} +584.818 q^{86} -921.301 q^{88} -170.225 q^{89} +286.607 q^{92} -30.1921 q^{94} -221.127 q^{95} -1059.87 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} + 20 q^{4} - 10 q^{5} - 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} + 20 q^{4} - 10 q^{5} - 48 q^{8} + 40 q^{10} + 14 q^{11} - 50 q^{13} + 168 q^{16} - 50 q^{17} - 36 q^{19} - 100 q^{20} - 184 q^{22} - 244 q^{23} + 50 q^{25} + 216 q^{26} + 26 q^{29} + 120 q^{31} - 672 q^{32} + 24 q^{34} + 564 q^{37} + 320 q^{38} + 240 q^{40} - 328 q^{41} - 260 q^{43} + 1164 q^{44} + 704 q^{46} - 350 q^{47} - 200 q^{50} - 628 q^{52} + 56 q^{53} - 70 q^{55} - 8 q^{58} - 1232 q^{59} - 336 q^{61} - 1200 q^{62} + 2128 q^{64} + 250 q^{65} - 152 q^{67} + 908 q^{68} + 1904 q^{71} - 676 q^{73} - 2016 q^{74} - 1768 q^{76} + 1014 q^{79} - 840 q^{80} + 816 q^{82} - 376 q^{83} + 250 q^{85} + 768 q^{86} - 4688 q^{88} - 216 q^{89} - 264 q^{92} + 1928 q^{94} + 180 q^{95} - 2742 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.58579 −0.914214 −0.457107 0.889412i \(-0.651114\pi\)
−0.457107 + 0.889412i \(0.651114\pi\)
\(3\) 0 0
\(4\) −1.31371 −0.164214
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 24.0833 1.06434
\(9\) 0 0
\(10\) 12.9289 0.408849
\(11\) −38.2548 −1.04857 −0.524285 0.851543i \(-0.675667\pi\)
−0.524285 + 0.851543i \(0.675667\pi\)
\(12\) 0 0
\(13\) −19.3431 −0.412679 −0.206339 0.978480i \(-0.566155\pi\)
−0.206339 + 0.978480i \(0.566155\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −51.7645 −0.808820
\(17\) −87.2254 −1.24443 −0.622214 0.782847i \(-0.713767\pi\)
−0.622214 + 0.782847i \(0.713767\pi\)
\(18\) 0 0
\(19\) 44.2254 0.534000 0.267000 0.963697i \(-0.413968\pi\)
0.267000 + 0.963697i \(0.413968\pi\)
\(20\) 6.56854 0.0734385
\(21\) 0 0
\(22\) 98.9188 0.958617
\(23\) −218.167 −1.97786 −0.988932 0.148371i \(-0.952597\pi\)
−0.988932 + 0.148371i \(0.952597\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 50.0172 0.377276
\(27\) 0 0
\(28\) 0 0
\(29\) 46.9411 0.300578 0.150289 0.988642i \(-0.451980\pi\)
0.150289 + 0.988642i \(0.451980\pi\)
\(30\) 0 0
\(31\) −194.558 −1.12722 −0.563609 0.826042i \(-0.690587\pi\)
−0.563609 + 0.826042i \(0.690587\pi\)
\(32\) −58.8141 −0.324905
\(33\) 0 0
\(34\) 225.546 1.13767
\(35\) 0 0
\(36\) 0 0
\(37\) 366.853 1.63001 0.815003 0.579457i \(-0.196735\pi\)
0.815003 + 0.579457i \(0.196735\pi\)
\(38\) −114.357 −0.488190
\(39\) 0 0
\(40\) −120.416 −0.475987
\(41\) −339.362 −1.29267 −0.646336 0.763053i \(-0.723699\pi\)
−0.646336 + 0.763053i \(0.723699\pi\)
\(42\) 0 0
\(43\) −226.167 −0.802095 −0.401047 0.916057i \(-0.631354\pi\)
−0.401047 + 0.916057i \(0.631354\pi\)
\(44\) 50.2557 0.172189
\(45\) 0 0
\(46\) 564.132 1.80819
\(47\) 11.6762 0.0362372 0.0181186 0.999836i \(-0.494232\pi\)
0.0181186 + 0.999836i \(0.494232\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −64.6447 −0.182843
\(51\) 0 0
\(52\) 25.4113 0.0677674
\(53\) 209.019 0.541717 0.270859 0.962619i \(-0.412692\pi\)
0.270859 + 0.962619i \(0.412692\pi\)
\(54\) 0 0
\(55\) 191.274 0.468935
\(56\) 0 0
\(57\) 0 0
\(58\) −121.380 −0.274792
\(59\) −616.000 −1.35926 −0.679630 0.733555i \(-0.737860\pi\)
−0.679630 + 0.733555i \(0.737860\pi\)
\(60\) 0 0
\(61\) −320.735 −0.673212 −0.336606 0.941646i \(-0.609279\pi\)
−0.336606 + 0.941646i \(0.609279\pi\)
\(62\) 503.087 1.03052
\(63\) 0 0
\(64\) 566.197 1.10585
\(65\) 96.7157 0.184556
\(66\) 0 0
\(67\) 14.5097 0.0264573 0.0132286 0.999912i \(-0.495789\pi\)
0.0132286 + 0.999912i \(0.495789\pi\)
\(68\) 114.589 0.204352
\(69\) 0 0
\(70\) 0 0
\(71\) 952.000 1.59129 0.795645 0.605763i \(-0.207132\pi\)
0.795645 + 0.605763i \(0.207132\pi\)
\(72\) 0 0
\(73\) −824.489 −1.32191 −0.660953 0.750427i \(-0.729848\pi\)
−0.660953 + 0.750427i \(0.729848\pi\)
\(74\) −948.603 −1.49017
\(75\) 0 0
\(76\) −58.0993 −0.0876901
\(77\) 0 0
\(78\) 0 0
\(79\) 156.275 0.222561 0.111280 0.993789i \(-0.464505\pi\)
0.111280 + 0.993789i \(0.464505\pi\)
\(80\) 258.823 0.361715
\(81\) 0 0
\(82\) 877.519 1.18178
\(83\) −1036.53 −1.37077 −0.685384 0.728182i \(-0.740366\pi\)
−0.685384 + 0.728182i \(0.740366\pi\)
\(84\) 0 0
\(85\) 436.127 0.556525
\(86\) 584.818 0.733286
\(87\) 0 0
\(88\) −921.301 −1.11603
\(89\) −170.225 −0.202740 −0.101370 0.994849i \(-0.532323\pi\)
−0.101370 + 0.994849i \(0.532323\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 286.607 0.324792
\(93\) 0 0
\(94\) −30.1921 −0.0331285
\(95\) −221.127 −0.238812
\(96\) 0 0
\(97\) −1059.87 −1.10942 −0.554710 0.832044i \(-0.687171\pi\)
−0.554710 + 0.832044i \(0.687171\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −32.8427 −0.0328427
\(101\) −241.833 −0.238251 −0.119125 0.992879i \(-0.538009\pi\)
−0.119125 + 0.992879i \(0.538009\pi\)
\(102\) 0 0
\(103\) 1679.58 1.60673 0.803367 0.595484i \(-0.203040\pi\)
0.803367 + 0.595484i \(0.203040\pi\)
\(104\) −465.846 −0.439230
\(105\) 0 0
\(106\) −540.479 −0.495245
\(107\) −1506.88 −1.36146 −0.680728 0.732537i \(-0.738336\pi\)
−0.680728 + 0.732537i \(0.738336\pi\)
\(108\) 0 0
\(109\) −1252.41 −1.10054 −0.550271 0.834986i \(-0.685476\pi\)
−0.550271 + 0.834986i \(0.685476\pi\)
\(110\) −494.594 −0.428706
\(111\) 0 0
\(112\) 0 0
\(113\) −1370.20 −1.14069 −0.570345 0.821405i \(-0.693190\pi\)
−0.570345 + 0.821405i \(0.693190\pi\)
\(114\) 0 0
\(115\) 1090.83 0.884528
\(116\) −61.6670 −0.0493589
\(117\) 0 0
\(118\) 1592.84 1.24265
\(119\) 0 0
\(120\) 0 0
\(121\) 132.432 0.0994984
\(122\) 829.352 0.615459
\(123\) 0 0
\(124\) 255.593 0.185104
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 1213.49 0.847873 0.423936 0.905692i \(-0.360648\pi\)
0.423936 + 0.905692i \(0.360648\pi\)
\(128\) −993.551 −0.686081
\(129\) 0 0
\(130\) −250.086 −0.168723
\(131\) −1982.42 −1.32217 −0.661087 0.750309i \(-0.729904\pi\)
−0.661087 + 0.750309i \(0.729904\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −37.5189 −0.0241876
\(135\) 0 0
\(136\) −2100.67 −1.32449
\(137\) −2210.95 −1.37879 −0.689394 0.724386i \(-0.742123\pi\)
−0.689394 + 0.724386i \(0.742123\pi\)
\(138\) 0 0
\(139\) −528.039 −0.322213 −0.161107 0.986937i \(-0.551506\pi\)
−0.161107 + 0.986937i \(0.551506\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −2461.67 −1.45478
\(143\) 739.969 0.432722
\(144\) 0 0
\(145\) −234.706 −0.134422
\(146\) 2131.95 1.20851
\(147\) 0 0
\(148\) −481.938 −0.267669
\(149\) 328.372 0.180545 0.0902727 0.995917i \(-0.471226\pi\)
0.0902727 + 0.995917i \(0.471226\pi\)
\(150\) 0 0
\(151\) 1029.43 0.554793 0.277396 0.960756i \(-0.410528\pi\)
0.277396 + 0.960756i \(0.410528\pi\)
\(152\) 1065.09 0.568358
\(153\) 0 0
\(154\) 0 0
\(155\) 972.792 0.504107
\(156\) 0 0
\(157\) −525.098 −0.266926 −0.133463 0.991054i \(-0.542610\pi\)
−0.133463 + 0.991054i \(0.542610\pi\)
\(158\) −404.094 −0.203468
\(159\) 0 0
\(160\) 294.071 0.145302
\(161\) 0 0
\(162\) 0 0
\(163\) 1002.63 0.481790 0.240895 0.970551i \(-0.422559\pi\)
0.240895 + 0.970551i \(0.422559\pi\)
\(164\) 445.823 0.212274
\(165\) 0 0
\(166\) 2680.24 1.25317
\(167\) −359.422 −0.166544 −0.0832722 0.996527i \(-0.526537\pi\)
−0.0832722 + 0.996527i \(0.526537\pi\)
\(168\) 0 0
\(169\) −1822.84 −0.829696
\(170\) −1127.73 −0.508783
\(171\) 0 0
\(172\) 297.117 0.131715
\(173\) 3293.65 1.44747 0.723733 0.690080i \(-0.242425\pi\)
0.723733 + 0.690080i \(0.242425\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1980.24 0.848104
\(177\) 0 0
\(178\) 440.167 0.185348
\(179\) −2978.82 −1.24384 −0.621921 0.783080i \(-0.713647\pi\)
−0.621921 + 0.783080i \(0.713647\pi\)
\(180\) 0 0
\(181\) −1462.31 −0.600514 −0.300257 0.953858i \(-0.597072\pi\)
−0.300257 + 0.953858i \(0.597072\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −5254.16 −2.10512
\(185\) −1834.26 −0.728961
\(186\) 0 0
\(187\) 3336.79 1.30487
\(188\) −15.3391 −0.00595064
\(189\) 0 0
\(190\) 571.787 0.218325
\(191\) 374.923 0.142034 0.0710169 0.997475i \(-0.477376\pi\)
0.0710169 + 0.997475i \(0.477376\pi\)
\(192\) 0 0
\(193\) 733.028 0.273391 0.136696 0.990613i \(-0.456352\pi\)
0.136696 + 0.990613i \(0.456352\pi\)
\(194\) 2740.61 1.01425
\(195\) 0 0
\(196\) 0 0
\(197\) 2093.24 0.757043 0.378521 0.925593i \(-0.376433\pi\)
0.378521 + 0.925593i \(0.376433\pi\)
\(198\) 0 0
\(199\) −2865.04 −1.02059 −0.510295 0.860000i \(-0.670464\pi\)
−0.510295 + 0.860000i \(0.670464\pi\)
\(200\) 602.082 0.212868
\(201\) 0 0
\(202\) 625.330 0.217812
\(203\) 0 0
\(204\) 0 0
\(205\) 1696.81 0.578100
\(206\) −4343.03 −1.46890
\(207\) 0 0
\(208\) 1001.29 0.333783
\(209\) −1691.84 −0.559936
\(210\) 0 0
\(211\) 5643.65 1.84135 0.920674 0.390331i \(-0.127640\pi\)
0.920674 + 0.390331i \(0.127640\pi\)
\(212\) −274.590 −0.0889573
\(213\) 0 0
\(214\) 3896.47 1.24466
\(215\) 1130.83 0.358708
\(216\) 0 0
\(217\) 0 0
\(218\) 3238.46 1.00613
\(219\) 0 0
\(220\) −251.279 −0.0770054
\(221\) 1687.21 0.513549
\(222\) 0 0
\(223\) 6369.16 1.91260 0.956302 0.292381i \(-0.0944477\pi\)
0.956302 + 0.292381i \(0.0944477\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 3543.05 1.04283
\(227\) −1015.67 −0.296972 −0.148486 0.988914i \(-0.547440\pi\)
−0.148486 + 0.988914i \(0.547440\pi\)
\(228\) 0 0
\(229\) −4108.35 −1.18554 −0.592768 0.805373i \(-0.701965\pi\)
−0.592768 + 0.805373i \(0.701965\pi\)
\(230\) −2820.66 −0.808647
\(231\) 0 0
\(232\) 1130.50 0.319917
\(233\) −608.431 −0.171071 −0.0855357 0.996335i \(-0.527260\pi\)
−0.0855357 + 0.996335i \(0.527260\pi\)
\(234\) 0 0
\(235\) −58.3810 −0.0162058
\(236\) 809.244 0.223209
\(237\) 0 0
\(238\) 0 0
\(239\) 5054.44 1.36797 0.683985 0.729496i \(-0.260245\pi\)
0.683985 + 0.729496i \(0.260245\pi\)
\(240\) 0 0
\(241\) −4.86782 −0.00130109 −0.000650547 1.00000i \(-0.500207\pi\)
−0.000650547 1.00000i \(0.500207\pi\)
\(242\) −342.442 −0.0909628
\(243\) 0 0
\(244\) 421.352 0.110551
\(245\) 0 0
\(246\) 0 0
\(247\) −855.458 −0.220370
\(248\) −4685.60 −1.19974
\(249\) 0 0
\(250\) 323.223 0.0817697
\(251\) −547.921 −0.137787 −0.0688934 0.997624i \(-0.521947\pi\)
−0.0688934 + 0.997624i \(0.521947\pi\)
\(252\) 0 0
\(253\) 8345.92 2.07393
\(254\) −3137.83 −0.775137
\(255\) 0 0
\(256\) −1960.46 −0.478629
\(257\) −1774.61 −0.430729 −0.215364 0.976534i \(-0.569094\pi\)
−0.215364 + 0.976534i \(0.569094\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −127.056 −0.0303065
\(261\) 0 0
\(262\) 5126.11 1.20875
\(263\) 1199.09 0.281138 0.140569 0.990071i \(-0.455107\pi\)
0.140569 + 0.990071i \(0.455107\pi\)
\(264\) 0 0
\(265\) −1045.10 −0.242263
\(266\) 0 0
\(267\) 0 0
\(268\) −19.0615 −0.00434464
\(269\) 3250.29 0.736706 0.368353 0.929686i \(-0.379922\pi\)
0.368353 + 0.929686i \(0.379922\pi\)
\(270\) 0 0
\(271\) 896.143 0.200874 0.100437 0.994943i \(-0.467976\pi\)
0.100437 + 0.994943i \(0.467976\pi\)
\(272\) 4515.18 1.00652
\(273\) 0 0
\(274\) 5717.04 1.26051
\(275\) −956.371 −0.209714
\(276\) 0 0
\(277\) −386.562 −0.0838492 −0.0419246 0.999121i \(-0.513349\pi\)
−0.0419246 + 0.999121i \(0.513349\pi\)
\(278\) 1365.40 0.294572
\(279\) 0 0
\(280\) 0 0
\(281\) 3335.10 0.708025 0.354013 0.935241i \(-0.384817\pi\)
0.354013 + 0.935241i \(0.384817\pi\)
\(282\) 0 0
\(283\) −5412.26 −1.13684 −0.568419 0.822739i \(-0.692445\pi\)
−0.568419 + 0.822739i \(0.692445\pi\)
\(284\) −1250.65 −0.261311
\(285\) 0 0
\(286\) −1913.40 −0.395601
\(287\) 0 0
\(288\) 0 0
\(289\) 2695.27 0.548600
\(290\) 606.899 0.122891
\(291\) 0 0
\(292\) 1083.14 0.217075
\(293\) 282.211 0.0562695 0.0281347 0.999604i \(-0.491043\pi\)
0.0281347 + 0.999604i \(0.491043\pi\)
\(294\) 0 0
\(295\) 3080.00 0.607880
\(296\) 8835.01 1.73488
\(297\) 0 0
\(298\) −849.099 −0.165057
\(299\) 4220.03 0.816222
\(300\) 0 0
\(301\) 0 0
\(302\) −2661.88 −0.507199
\(303\) 0 0
\(304\) −2289.31 −0.431910
\(305\) 1603.68 0.301069
\(306\) 0 0
\(307\) −1919.67 −0.356878 −0.178439 0.983951i \(-0.557105\pi\)
−0.178439 + 0.983951i \(0.557105\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2515.43 −0.460861
\(311\) 1213.31 0.221223 0.110612 0.993864i \(-0.464719\pi\)
0.110612 + 0.993864i \(0.464719\pi\)
\(312\) 0 0
\(313\) 1434.00 0.258960 0.129480 0.991582i \(-0.458669\pi\)
0.129480 + 0.991582i \(0.458669\pi\)
\(314\) 1357.79 0.244028
\(315\) 0 0
\(316\) −205.300 −0.0365475
\(317\) −6496.95 −1.15112 −0.575560 0.817760i \(-0.695216\pi\)
−0.575560 + 0.817760i \(0.695216\pi\)
\(318\) 0 0
\(319\) −1795.72 −0.315176
\(320\) −2830.98 −0.494553
\(321\) 0 0
\(322\) 0 0
\(323\) −3857.58 −0.664524
\(324\) 0 0
\(325\) −483.579 −0.0825357
\(326\) −2592.58 −0.440459
\(327\) 0 0
\(328\) −8172.96 −1.37584
\(329\) 0 0
\(330\) 0 0
\(331\) −9683.88 −1.60808 −0.804039 0.594576i \(-0.797320\pi\)
−0.804039 + 0.594576i \(0.797320\pi\)
\(332\) 1361.70 0.225099
\(333\) 0 0
\(334\) 929.389 0.152257
\(335\) −72.5483 −0.0118321
\(336\) 0 0
\(337\) 29.1319 0.00470895 0.00235447 0.999997i \(-0.499251\pi\)
0.00235447 + 0.999997i \(0.499251\pi\)
\(338\) 4713.48 0.758520
\(339\) 0 0
\(340\) −572.944 −0.0913889
\(341\) 7442.80 1.18197
\(342\) 0 0
\(343\) 0 0
\(344\) −5446.83 −0.853701
\(345\) 0 0
\(346\) −8516.68 −1.32329
\(347\) 7848.58 1.21422 0.607110 0.794618i \(-0.292329\pi\)
0.607110 + 0.794618i \(0.292329\pi\)
\(348\) 0 0
\(349\) 10269.6 1.57513 0.787567 0.616229i \(-0.211341\pi\)
0.787567 + 0.616229i \(0.211341\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2249.93 0.340686
\(353\) 2799.93 0.422168 0.211084 0.977468i \(-0.432301\pi\)
0.211084 + 0.977468i \(0.432301\pi\)
\(354\) 0 0
\(355\) −4760.00 −0.711647
\(356\) 223.627 0.0332927
\(357\) 0 0
\(358\) 7702.60 1.13714
\(359\) 3163.29 0.465048 0.232524 0.972591i \(-0.425302\pi\)
0.232524 + 0.972591i \(0.425302\pi\)
\(360\) 0 0
\(361\) −4903.11 −0.714844
\(362\) 3781.23 0.548998
\(363\) 0 0
\(364\) 0 0
\(365\) 4122.45 0.591175
\(366\) 0 0
\(367\) −3182.85 −0.452706 −0.226353 0.974045i \(-0.572680\pi\)
−0.226353 + 0.974045i \(0.572680\pi\)
\(368\) 11293.3 1.59974
\(369\) 0 0
\(370\) 4743.02 0.666426
\(371\) 0 0
\(372\) 0 0
\(373\) −2615.14 −0.363021 −0.181510 0.983389i \(-0.558099\pi\)
−0.181510 + 0.983389i \(0.558099\pi\)
\(374\) −8628.23 −1.19293
\(375\) 0 0
\(376\) 281.201 0.0385687
\(377\) −907.989 −0.124042
\(378\) 0 0
\(379\) −672.434 −0.0911362 −0.0455681 0.998961i \(-0.514510\pi\)
−0.0455681 + 0.998961i \(0.514510\pi\)
\(380\) 290.496 0.0392162
\(381\) 0 0
\(382\) −969.470 −0.129849
\(383\) 1169.86 0.156075 0.0780377 0.996950i \(-0.475135\pi\)
0.0780377 + 0.996950i \(0.475135\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1895.45 −0.249938
\(387\) 0 0
\(388\) 1392.36 0.182182
\(389\) 1122.22 0.146269 0.0731347 0.997322i \(-0.476700\pi\)
0.0731347 + 0.997322i \(0.476700\pi\)
\(390\) 0 0
\(391\) 19029.7 2.46131
\(392\) 0 0
\(393\) 0 0
\(394\) −5412.68 −0.692099
\(395\) −781.375 −0.0995323
\(396\) 0 0
\(397\) 1985.93 0.251060 0.125530 0.992090i \(-0.459937\pi\)
0.125530 + 0.992090i \(0.459937\pi\)
\(398\) 7408.38 0.933037
\(399\) 0 0
\(400\) −1294.11 −0.161764
\(401\) 4172.38 0.519597 0.259799 0.965663i \(-0.416344\pi\)
0.259799 + 0.965663i \(0.416344\pi\)
\(402\) 0 0
\(403\) 3763.37 0.465178
\(404\) 317.699 0.0391240
\(405\) 0 0
\(406\) 0 0
\(407\) −14033.9 −1.70918
\(408\) 0 0
\(409\) 11700.8 1.41459 0.707295 0.706919i \(-0.249915\pi\)
0.707295 + 0.706919i \(0.249915\pi\)
\(410\) −4387.59 −0.528507
\(411\) 0 0
\(412\) −2206.47 −0.263848
\(413\) 0 0
\(414\) 0 0
\(415\) 5182.64 0.613026
\(416\) 1137.65 0.134082
\(417\) 0 0
\(418\) 4374.72 0.511901
\(419\) 2733.20 0.318677 0.159339 0.987224i \(-0.449064\pi\)
0.159339 + 0.987224i \(0.449064\pi\)
\(420\) 0 0
\(421\) 13549.4 1.56854 0.784272 0.620417i \(-0.213037\pi\)
0.784272 + 0.620417i \(0.213037\pi\)
\(422\) −14593.3 −1.68339
\(423\) 0 0
\(424\) 5033.87 0.576571
\(425\) −2180.63 −0.248885
\(426\) 0 0
\(427\) 0 0
\(428\) 1979.60 0.223569
\(429\) 0 0
\(430\) −2924.09 −0.327935
\(431\) 6429.25 0.718530 0.359265 0.933236i \(-0.383027\pi\)
0.359265 + 0.933236i \(0.383027\pi\)
\(432\) 0 0
\(433\) −8022.03 −0.890333 −0.445166 0.895448i \(-0.646855\pi\)
−0.445166 + 0.895448i \(0.646855\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1645.30 0.180724
\(437\) −9648.50 −1.05618
\(438\) 0 0
\(439\) 5569.88 0.605549 0.302774 0.953062i \(-0.402087\pi\)
0.302774 + 0.953062i \(0.402087\pi\)
\(440\) 4606.51 0.499106
\(441\) 0 0
\(442\) −4362.77 −0.469493
\(443\) 5486.21 0.588392 0.294196 0.955745i \(-0.404948\pi\)
0.294196 + 0.955745i \(0.404948\pi\)
\(444\) 0 0
\(445\) 851.127 0.0906681
\(446\) −16469.3 −1.74853
\(447\) 0 0
\(448\) 0 0
\(449\) 7232.67 0.760203 0.380101 0.924945i \(-0.375889\pi\)
0.380101 + 0.924945i \(0.375889\pi\)
\(450\) 0 0
\(451\) 12982.3 1.35546
\(452\) 1800.05 0.187317
\(453\) 0 0
\(454\) 2626.32 0.271496
\(455\) 0 0
\(456\) 0 0
\(457\) −2900.51 −0.296893 −0.148446 0.988920i \(-0.547427\pi\)
−0.148446 + 0.988920i \(0.547427\pi\)
\(458\) 10623.3 1.08383
\(459\) 0 0
\(460\) −1433.04 −0.145251
\(461\) 6073.57 0.613611 0.306805 0.951772i \(-0.400740\pi\)
0.306805 + 0.951772i \(0.400740\pi\)
\(462\) 0 0
\(463\) −18922.8 −1.89939 −0.949693 0.313183i \(-0.898605\pi\)
−0.949693 + 0.313183i \(0.898605\pi\)
\(464\) −2429.88 −0.243113
\(465\) 0 0
\(466\) 1573.27 0.156396
\(467\) −6776.71 −0.671496 −0.335748 0.941952i \(-0.608989\pi\)
−0.335748 + 0.941952i \(0.608989\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 150.961 0.0148155
\(471\) 0 0
\(472\) −14835.3 −1.44672
\(473\) 8651.96 0.841052
\(474\) 0 0
\(475\) 1105.63 0.106800
\(476\) 0 0
\(477\) 0 0
\(478\) −13069.7 −1.25062
\(479\) 2397.32 0.228677 0.114338 0.993442i \(-0.463525\pi\)
0.114338 + 0.993442i \(0.463525\pi\)
\(480\) 0 0
\(481\) −7096.09 −0.672669
\(482\) 12.5871 0.00118948
\(483\) 0 0
\(484\) −173.977 −0.0163390
\(485\) 5299.37 0.496148
\(486\) 0 0
\(487\) 5586.17 0.519781 0.259890 0.965638i \(-0.416314\pi\)
0.259890 + 0.965638i \(0.416314\pi\)
\(488\) −7724.35 −0.716526
\(489\) 0 0
\(490\) 0 0
\(491\) −537.392 −0.0493934 −0.0246967 0.999695i \(-0.507862\pi\)
−0.0246967 + 0.999695i \(0.507862\pi\)
\(492\) 0 0
\(493\) −4094.46 −0.374047
\(494\) 2212.03 0.201466
\(495\) 0 0
\(496\) 10071.2 0.911716
\(497\) 0 0
\(498\) 0 0
\(499\) 598.965 0.0537342 0.0268671 0.999639i \(-0.491447\pi\)
0.0268671 + 0.999639i \(0.491447\pi\)
\(500\) 164.214 0.0146877
\(501\) 0 0
\(502\) 1416.81 0.125966
\(503\) 4426.76 0.392405 0.196202 0.980563i \(-0.437139\pi\)
0.196202 + 0.980563i \(0.437139\pi\)
\(504\) 0 0
\(505\) 1209.17 0.106549
\(506\) −21580.8 −1.89601
\(507\) 0 0
\(508\) −1594.17 −0.139232
\(509\) 17727.7 1.54374 0.771872 0.635779i \(-0.219321\pi\)
0.771872 + 0.635779i \(0.219321\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 13017.7 1.12365
\(513\) 0 0
\(514\) 4588.77 0.393778
\(515\) −8397.88 −0.718553
\(516\) 0 0
\(517\) −446.671 −0.0379972
\(518\) 0 0
\(519\) 0 0
\(520\) 2329.23 0.196430
\(521\) 8662.79 0.728453 0.364226 0.931310i \(-0.381333\pi\)
0.364226 + 0.931310i \(0.381333\pi\)
\(522\) 0 0
\(523\) 7770.40 0.649667 0.324833 0.945771i \(-0.394692\pi\)
0.324833 + 0.945771i \(0.394692\pi\)
\(524\) 2604.32 0.217119
\(525\) 0 0
\(526\) −3100.60 −0.257020
\(527\) 16970.4 1.40274
\(528\) 0 0
\(529\) 35429.6 2.91194
\(530\) 2702.40 0.221480
\(531\) 0 0
\(532\) 0 0
\(533\) 6564.34 0.533458
\(534\) 0 0
\(535\) 7534.41 0.608861
\(536\) 349.440 0.0281595
\(537\) 0 0
\(538\) −8404.56 −0.673506
\(539\) 0 0
\(540\) 0 0
\(541\) 21641.0 1.71981 0.859906 0.510453i \(-0.170522\pi\)
0.859906 + 0.510453i \(0.170522\pi\)
\(542\) −2317.23 −0.183642
\(543\) 0 0
\(544\) 5130.09 0.404321
\(545\) 6262.05 0.492177
\(546\) 0 0
\(547\) 7489.29 0.585409 0.292705 0.956203i \(-0.405445\pi\)
0.292705 + 0.956203i \(0.405445\pi\)
\(548\) 2904.54 0.226416
\(549\) 0 0
\(550\) 2472.97 0.191723
\(551\) 2075.99 0.160508
\(552\) 0 0
\(553\) 0 0
\(554\) 999.566 0.0766561
\(555\) 0 0
\(556\) 693.689 0.0529118
\(557\) −25297.9 −1.92443 −0.962214 0.272295i \(-0.912217\pi\)
−0.962214 + 0.272295i \(0.912217\pi\)
\(558\) 0 0
\(559\) 4374.77 0.331007
\(560\) 0 0
\(561\) 0 0
\(562\) −8623.85 −0.647286
\(563\) 15661.3 1.17237 0.586186 0.810177i \(-0.300629\pi\)
0.586186 + 0.810177i \(0.300629\pi\)
\(564\) 0 0
\(565\) 6851.02 0.510132
\(566\) 13994.9 1.03931
\(567\) 0 0
\(568\) 22927.3 1.69367
\(569\) 9982.75 0.735498 0.367749 0.929925i \(-0.380129\pi\)
0.367749 + 0.929925i \(0.380129\pi\)
\(570\) 0 0
\(571\) −11583.6 −0.848966 −0.424483 0.905436i \(-0.639544\pi\)
−0.424483 + 0.905436i \(0.639544\pi\)
\(572\) −972.103 −0.0710589
\(573\) 0 0
\(574\) 0 0
\(575\) −5454.16 −0.395573
\(576\) 0 0
\(577\) 595.378 0.0429565 0.0214783 0.999769i \(-0.493163\pi\)
0.0214783 + 0.999769i \(0.493163\pi\)
\(578\) −6969.39 −0.501537
\(579\) 0 0
\(580\) 308.335 0.0220740
\(581\) 0 0
\(582\) 0 0
\(583\) −7996.00 −0.568028
\(584\) −19856.4 −1.40696
\(585\) 0 0
\(586\) −729.738 −0.0514423
\(587\) −15750.3 −1.10747 −0.553736 0.832693i \(-0.686798\pi\)
−0.553736 + 0.832693i \(0.686798\pi\)
\(588\) 0 0
\(589\) −8604.42 −0.601934
\(590\) −7964.22 −0.555732
\(591\) 0 0
\(592\) −18990.0 −1.31838
\(593\) −417.878 −0.0289379 −0.0144690 0.999895i \(-0.504606\pi\)
−0.0144690 + 0.999895i \(0.504606\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −431.385 −0.0296480
\(597\) 0 0
\(598\) −10912.1 −0.746201
\(599\) 19997.3 1.36406 0.682028 0.731326i \(-0.261098\pi\)
0.682028 + 0.731326i \(0.261098\pi\)
\(600\) 0 0
\(601\) 15992.6 1.08545 0.542723 0.839912i \(-0.317393\pi\)
0.542723 + 0.839912i \(0.317393\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1352.37 −0.0911045
\(605\) −662.162 −0.0444970
\(606\) 0 0
\(607\) −14159.2 −0.946793 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(608\) −2601.08 −0.173499
\(609\) 0 0
\(610\) −4146.76 −0.275242
\(611\) −225.854 −0.0149543
\(612\) 0 0
\(613\) −4629.41 −0.305025 −0.152512 0.988302i \(-0.548736\pi\)
−0.152512 + 0.988302i \(0.548736\pi\)
\(614\) 4963.87 0.326263
\(615\) 0 0
\(616\) 0 0
\(617\) 23165.3 1.51151 0.755753 0.654857i \(-0.227271\pi\)
0.755753 + 0.654857i \(0.227271\pi\)
\(618\) 0 0
\(619\) −12370.6 −0.803258 −0.401629 0.915803i \(-0.631556\pi\)
−0.401629 + 0.915803i \(0.631556\pi\)
\(620\) −1277.97 −0.0827812
\(621\) 0 0
\(622\) −3137.36 −0.202245
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −3708.02 −0.236745
\(627\) 0 0
\(628\) 689.826 0.0438329
\(629\) −31998.9 −2.02842
\(630\) 0 0
\(631\) 13980.2 0.882002 0.441001 0.897507i \(-0.354623\pi\)
0.441001 + 0.897507i \(0.354623\pi\)
\(632\) 3763.61 0.236880
\(633\) 0 0
\(634\) 16799.7 1.05237
\(635\) −6067.45 −0.379180
\(636\) 0 0
\(637\) 0 0
\(638\) 4643.36 0.288139
\(639\) 0 0
\(640\) 4967.75 0.306825
\(641\) 16060.9 0.989655 0.494828 0.868991i \(-0.335231\pi\)
0.494828 + 0.868991i \(0.335231\pi\)
\(642\) 0 0
\(643\) 4502.17 0.276125 0.138063 0.990424i \(-0.455913\pi\)
0.138063 + 0.990424i \(0.455913\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 9974.87 0.607517
\(647\) 29414.8 1.78735 0.893675 0.448715i \(-0.148118\pi\)
0.893675 + 0.448715i \(0.148118\pi\)
\(648\) 0 0
\(649\) 23565.0 1.42528
\(650\) 1250.43 0.0754553
\(651\) 0 0
\(652\) −1317.16 −0.0791164
\(653\) −13013.6 −0.779882 −0.389941 0.920840i \(-0.627505\pi\)
−0.389941 + 0.920840i \(0.627505\pi\)
\(654\) 0 0
\(655\) 9912.09 0.591294
\(656\) 17566.9 1.04554
\(657\) 0 0
\(658\) 0 0
\(659\) −23474.2 −1.38759 −0.693797 0.720171i \(-0.744063\pi\)
−0.693797 + 0.720171i \(0.744063\pi\)
\(660\) 0 0
\(661\) 9266.36 0.545264 0.272632 0.962118i \(-0.412106\pi\)
0.272632 + 0.962118i \(0.412106\pi\)
\(662\) 25040.4 1.47013
\(663\) 0 0
\(664\) −24963.0 −1.45896
\(665\) 0 0
\(666\) 0 0
\(667\) −10241.0 −0.594501
\(668\) 472.176 0.0273489
\(669\) 0 0
\(670\) 187.595 0.0108170
\(671\) 12269.7 0.705909
\(672\) 0 0
\(673\) −25067.2 −1.43576 −0.717882 0.696164i \(-0.754888\pi\)
−0.717882 + 0.696164i \(0.754888\pi\)
\(674\) −75.3288 −0.00430498
\(675\) 0 0
\(676\) 2394.68 0.136247
\(677\) −22409.6 −1.27219 −0.636093 0.771613i \(-0.719450\pi\)
−0.636093 + 0.771613i \(0.719450\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 10503.4 0.592332
\(681\) 0 0
\(682\) −19245.5 −1.08057
\(683\) 8757.53 0.490626 0.245313 0.969444i \(-0.421109\pi\)
0.245313 + 0.969444i \(0.421109\pi\)
\(684\) 0 0
\(685\) 11054.7 0.616613
\(686\) 0 0
\(687\) 0 0
\(688\) 11707.4 0.648750
\(689\) −4043.09 −0.223555
\(690\) 0 0
\(691\) −8468.42 −0.466214 −0.233107 0.972451i \(-0.574889\pi\)
−0.233107 + 0.972451i \(0.574889\pi\)
\(692\) −4326.90 −0.237694
\(693\) 0 0
\(694\) −20294.8 −1.11006
\(695\) 2640.19 0.144098
\(696\) 0 0
\(697\) 29601.0 1.60864
\(698\) −26555.1 −1.44001
\(699\) 0 0
\(700\) 0 0
\(701\) −15996.9 −0.861906 −0.430953 0.902374i \(-0.641823\pi\)
−0.430953 + 0.902374i \(0.641823\pi\)
\(702\) 0 0
\(703\) 16224.2 0.870423
\(704\) −21659.8 −1.15956
\(705\) 0 0
\(706\) −7240.03 −0.385952
\(707\) 0 0
\(708\) 0 0
\(709\) 19903.0 1.05426 0.527131 0.849784i \(-0.323268\pi\)
0.527131 + 0.849784i \(0.323268\pi\)
\(710\) 12308.3 0.650597
\(711\) 0 0
\(712\) −4099.58 −0.215784
\(713\) 42446.1 2.22948
\(714\) 0 0
\(715\) −3699.84 −0.193519
\(716\) 3913.30 0.204256
\(717\) 0 0
\(718\) −8179.59 −0.425153
\(719\) −11073.1 −0.574347 −0.287174 0.957879i \(-0.592716\pi\)
−0.287174 + 0.957879i \(0.592716\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 12678.4 0.653520
\(723\) 0 0
\(724\) 1921.06 0.0986125
\(725\) 1173.53 0.0601155
\(726\) 0 0
\(727\) −31652.7 −1.61476 −0.807382 0.590029i \(-0.799116\pi\)
−0.807382 + 0.590029i \(0.799116\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −10659.8 −0.540460
\(731\) 19727.5 0.998149
\(732\) 0 0
\(733\) −16958.3 −0.854528 −0.427264 0.904127i \(-0.640523\pi\)
−0.427264 + 0.904127i \(0.640523\pi\)
\(734\) 8230.16 0.413870
\(735\) 0 0
\(736\) 12831.3 0.642618
\(737\) −555.065 −0.0277423
\(738\) 0 0
\(739\) −11616.6 −0.578245 −0.289123 0.957292i \(-0.593364\pi\)
−0.289123 + 0.957292i \(0.593364\pi\)
\(740\) 2409.69 0.119705
\(741\) 0 0
\(742\) 0 0
\(743\) −15928.0 −0.786464 −0.393232 0.919439i \(-0.628643\pi\)
−0.393232 + 0.919439i \(0.628643\pi\)
\(744\) 0 0
\(745\) −1641.86 −0.0807423
\(746\) 6762.19 0.331879
\(747\) 0 0
\(748\) −4383.57 −0.214277
\(749\) 0 0
\(750\) 0 0
\(751\) 25571.9 1.24252 0.621260 0.783604i \(-0.286621\pi\)
0.621260 + 0.783604i \(0.286621\pi\)
\(752\) −604.412 −0.0293094
\(753\) 0 0
\(754\) 2347.87 0.113401
\(755\) −5147.14 −0.248111
\(756\) 0 0
\(757\) 6202.41 0.297794 0.148897 0.988853i \(-0.452428\pi\)
0.148897 + 0.988853i \(0.452428\pi\)
\(758\) 1738.77 0.0833179
\(759\) 0 0
\(760\) −5325.46 −0.254177
\(761\) −29199.1 −1.39089 −0.695444 0.718580i \(-0.744792\pi\)
−0.695444 + 0.718580i \(0.744792\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −492.539 −0.0233239
\(765\) 0 0
\(766\) −3025.00 −0.142686
\(767\) 11915.4 0.560938
\(768\) 0 0
\(769\) −21838.2 −1.02407 −0.512033 0.858966i \(-0.671108\pi\)
−0.512033 + 0.858966i \(0.671108\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −962.985 −0.0448945
\(773\) −25544.8 −1.18859 −0.594296 0.804246i \(-0.702569\pi\)
−0.594296 + 0.804246i \(0.702569\pi\)
\(774\) 0 0
\(775\) −4863.96 −0.225443
\(776\) −25525.2 −1.18080
\(777\) 0 0
\(778\) −2901.82 −0.133721
\(779\) −15008.4 −0.690286
\(780\) 0 0
\(781\) −36418.6 −1.66858
\(782\) −49206.6 −2.25016
\(783\) 0 0
\(784\) 0 0
\(785\) 2625.49 0.119373
\(786\) 0 0
\(787\) 37223.2 1.68598 0.842989 0.537931i \(-0.180794\pi\)
0.842989 + 0.537931i \(0.180794\pi\)
\(788\) −2749.91 −0.124317
\(789\) 0 0
\(790\) 2020.47 0.0909938
\(791\) 0 0
\(792\) 0 0
\(793\) 6204.03 0.277820
\(794\) −5135.18 −0.229522
\(795\) 0 0
\(796\) 3763.83 0.167595
\(797\) −40384.6 −1.79485 −0.897425 0.441168i \(-0.854564\pi\)
−0.897425 + 0.441168i \(0.854564\pi\)
\(798\) 0 0
\(799\) −1018.46 −0.0450945
\(800\) −1470.35 −0.0649811
\(801\) 0 0
\(802\) −10788.9 −0.475023
\(803\) 31540.7 1.38611
\(804\) 0 0
\(805\) 0 0
\(806\) −9731.28 −0.425272
\(807\) 0 0
\(808\) −5824.14 −0.253580
\(809\) 1955.76 0.0849948 0.0424974 0.999097i \(-0.486469\pi\)
0.0424974 + 0.999097i \(0.486469\pi\)
\(810\) 0 0
\(811\) 34301.8 1.48520 0.742600 0.669735i \(-0.233592\pi\)
0.742600 + 0.669735i \(0.233592\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 36288.7 1.56255
\(815\) −5013.13 −0.215463
\(816\) 0 0
\(817\) −10002.3 −0.428319
\(818\) −30255.8 −1.29324
\(819\) 0 0
\(820\) −2229.12 −0.0949319
\(821\) 13665.6 0.580918 0.290459 0.956887i \(-0.406192\pi\)
0.290459 + 0.956887i \(0.406192\pi\)
\(822\) 0 0
\(823\) −21519.5 −0.911449 −0.455724 0.890121i \(-0.650620\pi\)
−0.455724 + 0.890121i \(0.650620\pi\)
\(824\) 40449.7 1.71011
\(825\) 0 0
\(826\) 0 0
\(827\) −35220.6 −1.48094 −0.740471 0.672088i \(-0.765398\pi\)
−0.740471 + 0.672088i \(0.765398\pi\)
\(828\) 0 0
\(829\) −31365.5 −1.31408 −0.657039 0.753857i \(-0.728191\pi\)
−0.657039 + 0.753857i \(0.728191\pi\)
\(830\) −13401.2 −0.560437
\(831\) 0 0
\(832\) −10952.0 −0.456362
\(833\) 0 0
\(834\) 0 0
\(835\) 1797.11 0.0744810
\(836\) 2222.58 0.0919491
\(837\) 0 0
\(838\) −7067.48 −0.291339
\(839\) −28287.1 −1.16398 −0.581990 0.813196i \(-0.697726\pi\)
−0.581990 + 0.813196i \(0.697726\pi\)
\(840\) 0 0
\(841\) −22185.5 −0.909653
\(842\) −35035.8 −1.43398
\(843\) 0 0
\(844\) −7414.11 −0.302374
\(845\) 9114.21 0.371051
\(846\) 0 0
\(847\) 0 0
\(848\) −10819.8 −0.438152
\(849\) 0 0
\(850\) 5638.66 0.227534
\(851\) −80035.0 −3.22393
\(852\) 0 0
\(853\) −9405.41 −0.377533 −0.188766 0.982022i \(-0.560449\pi\)
−0.188766 + 0.982022i \(0.560449\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −36290.6 −1.44905
\(857\) −27966.9 −1.11474 −0.557369 0.830265i \(-0.688189\pi\)
−0.557369 + 0.830265i \(0.688189\pi\)
\(858\) 0 0
\(859\) −6281.11 −0.249486 −0.124743 0.992189i \(-0.539811\pi\)
−0.124743 + 0.992189i \(0.539811\pi\)
\(860\) −1485.58 −0.0589047
\(861\) 0 0
\(862\) −16624.7 −0.656890
\(863\) 4757.13 0.187642 0.0938208 0.995589i \(-0.470092\pi\)
0.0938208 + 0.995589i \(0.470092\pi\)
\(864\) 0 0
\(865\) −16468.3 −0.647327
\(866\) 20743.3 0.813954
\(867\) 0 0
\(868\) 0 0
\(869\) −5978.28 −0.233371
\(870\) 0 0
\(871\) −280.663 −0.0109184
\(872\) −30162.1 −1.17135
\(873\) 0 0
\(874\) 24949.0 0.965574
\(875\) 0 0
\(876\) 0 0
\(877\) −30240.5 −1.16437 −0.582184 0.813057i \(-0.697802\pi\)
−0.582184 + 0.813057i \(0.697802\pi\)
\(878\) −14402.5 −0.553601
\(879\) 0 0
\(880\) −9901.21 −0.379284
\(881\) 44875.5 1.71611 0.858056 0.513556i \(-0.171672\pi\)
0.858056 + 0.513556i \(0.171672\pi\)
\(882\) 0 0
\(883\) 4892.13 0.186448 0.0932238 0.995645i \(-0.470283\pi\)
0.0932238 + 0.995645i \(0.470283\pi\)
\(884\) −2216.51 −0.0843317
\(885\) 0 0
\(886\) −14186.2 −0.537916
\(887\) 1761.40 0.0666765 0.0333382 0.999444i \(-0.489386\pi\)
0.0333382 + 0.999444i \(0.489386\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −2200.83 −0.0828900
\(891\) 0 0
\(892\) −8367.22 −0.314075
\(893\) 516.384 0.0193507
\(894\) 0 0
\(895\) 14894.1 0.556263
\(896\) 0 0
\(897\) 0 0
\(898\) −18702.2 −0.694988
\(899\) −9132.79 −0.338816
\(900\) 0 0
\(901\) −18231.8 −0.674128
\(902\) −33569.3 −1.23918
\(903\) 0 0
\(904\) −32999.0 −1.21408
\(905\) 7311.57 0.268558
\(906\) 0 0
\(907\) 23689.1 0.867238 0.433619 0.901096i \(-0.357236\pi\)
0.433619 + 0.901096i \(0.357236\pi\)
\(908\) 1334.30 0.0487669
\(909\) 0 0
\(910\) 0 0
\(911\) 13877.3 0.504692 0.252346 0.967637i \(-0.418798\pi\)
0.252346 + 0.967637i \(0.418798\pi\)
\(912\) 0 0
\(913\) 39652.2 1.43735
\(914\) 7500.09 0.271423
\(915\) 0 0
\(916\) 5397.18 0.194681
\(917\) 0 0
\(918\) 0 0
\(919\) 14331.6 0.514426 0.257213 0.966355i \(-0.417196\pi\)
0.257213 + 0.966355i \(0.417196\pi\)
\(920\) 26270.8 0.941438
\(921\) 0 0
\(922\) −15705.0 −0.560971
\(923\) −18414.7 −0.656692
\(924\) 0 0
\(925\) 9171.32 0.326001
\(926\) 48930.2 1.73644
\(927\) 0 0
\(928\) −2760.80 −0.0976592
\(929\) 16668.4 0.588668 0.294334 0.955703i \(-0.404902\pi\)
0.294334 + 0.955703i \(0.404902\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 799.300 0.0280922
\(933\) 0 0
\(934\) 17523.1 0.613891
\(935\) −16684.0 −0.583555
\(936\) 0 0
\(937\) −30384.9 −1.05937 −0.529685 0.848194i \(-0.677690\pi\)
−0.529685 + 0.848194i \(0.677690\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 76.6956 0.00266121
\(941\) −1196.35 −0.0414452 −0.0207226 0.999785i \(-0.506597\pi\)
−0.0207226 + 0.999785i \(0.506597\pi\)
\(942\) 0 0
\(943\) 74037.5 2.55673
\(944\) 31886.9 1.09940
\(945\) 0 0
\(946\) −22372.1 −0.768901
\(947\) −1788.41 −0.0613681 −0.0306840 0.999529i \(-0.509769\pi\)
−0.0306840 + 0.999529i \(0.509769\pi\)
\(948\) 0 0
\(949\) 15948.2 0.545523
\(950\) −2858.94 −0.0976380
\(951\) 0 0
\(952\) 0 0
\(953\) −8578.60 −0.291593 −0.145796 0.989315i \(-0.546574\pi\)
−0.145796 + 0.989315i \(0.546574\pi\)
\(954\) 0 0
\(955\) −1874.61 −0.0635194
\(956\) −6640.06 −0.224639
\(957\) 0 0
\(958\) −6198.95 −0.209059
\(959\) 0 0
\(960\) 0 0
\(961\) 8061.99 0.270618
\(962\) 18349.0 0.614963
\(963\) 0 0
\(964\) 6.39489 0.000213657 0
\(965\) −3665.14 −0.122264
\(966\) 0 0
\(967\) −55459.3 −1.84431 −0.922156 0.386818i \(-0.873574\pi\)
−0.922156 + 0.386818i \(0.873574\pi\)
\(968\) 3189.40 0.105900
\(969\) 0 0
\(970\) −13703.0 −0.453585
\(971\) 22047.3 0.728662 0.364331 0.931270i \(-0.381298\pi\)
0.364331 + 0.931270i \(0.381298\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −14444.6 −0.475191
\(975\) 0 0
\(976\) 16602.7 0.544507
\(977\) −14402.3 −0.471617 −0.235809 0.971800i \(-0.575774\pi\)
−0.235809 + 0.971800i \(0.575774\pi\)
\(978\) 0 0
\(979\) 6511.94 0.212587
\(980\) 0 0
\(981\) 0 0
\(982\) 1389.58 0.0451561
\(983\) 7817.11 0.253639 0.126819 0.991926i \(-0.459523\pi\)
0.126819 + 0.991926i \(0.459523\pi\)
\(984\) 0 0
\(985\) −10466.2 −0.338560
\(986\) 10587.4 0.341959
\(987\) 0 0
\(988\) 1123.82 0.0361878
\(989\) 49342.0 1.58643
\(990\) 0 0
\(991\) 24501.6 0.785386 0.392693 0.919670i \(-0.371543\pi\)
0.392693 + 0.919670i \(0.371543\pi\)
\(992\) 11442.8 0.366239
\(993\) 0 0
\(994\) 0 0
\(995\) 14325.2 0.456421
\(996\) 0 0
\(997\) 50696.0 1.61039 0.805195 0.593010i \(-0.202060\pi\)
0.805195 + 0.593010i \(0.202060\pi\)
\(998\) −1548.80 −0.0491245
\(999\) 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 2205.4.a.u.1.2 2
3.2 odd 2 245.4.a.k.1.1 2
7.6 odd 2 315.4.a.f.1.2 2
15.14 odd 2 1225.4.a.m.1.2 2
21.2 odd 6 245.4.e.i.116.2 4
21.5 even 6 245.4.e.h.116.2 4
21.11 odd 6 245.4.e.i.226.2 4
21.17 even 6 245.4.e.h.226.2 4
21.20 even 2 35.4.a.b.1.1 2
35.34 odd 2 1575.4.a.z.1.1 2
84.83 odd 2 560.4.a.r.1.1 2
105.62 odd 4 175.4.b.c.99.3 4
105.83 odd 4 175.4.b.c.99.2 4
105.104 even 2 175.4.a.c.1.2 2
168.83 odd 2 2240.4.a.bo.1.2 2
168.125 even 2 2240.4.a.bn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.1 2 21.20 even 2
175.4.a.c.1.2 2 105.104 even 2
175.4.b.c.99.2 4 105.83 odd 4
175.4.b.c.99.3 4 105.62 odd 4
245.4.a.k.1.1 2 3.2 odd 2
245.4.e.h.116.2 4 21.5 even 6
245.4.e.h.226.2 4 21.17 even 6
245.4.e.i.116.2 4 21.2 odd 6
245.4.e.i.226.2 4 21.11 odd 6
315.4.a.f.1.2 2 7.6 odd 2
560.4.a.r.1.1 2 84.83 odd 2
1225.4.a.m.1.2 2 15.14 odd 2
1575.4.a.z.1.1 2 35.34 odd 2
2205.4.a.u.1.2 2 1.1 even 1 trivial
2240.4.a.bn.1.1 2 168.125 even 2
2240.4.a.bo.1.2 2 168.83 odd 2