Properties

Label 72.4.d.c.37.4
Level $72$
Weight $4$
Character 72.37
Analytic conductor $4.248$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,4,Mod(37,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.37");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 72.d (of order \(2\), degree \(1\), 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: \(4.24813752041\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-10}, \sqrt{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 37.4
Root \(2.34521 + 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 72.37
Dual form 72.4.d.c.37.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.34521 + 1.58114i) q^{2} +(3.00000 + 7.41620i) q^{4} +6.32456i q^{5} +10.0000 q^{7} +(-4.69042 + 22.1359i) q^{8} +O(q^{10})\) \(q+(2.34521 + 1.58114i) q^{2} +(3.00000 + 7.41620i) q^{4} +6.32456i q^{5} +10.0000 q^{7} +(-4.69042 + 22.1359i) q^{8} +(-10.0000 + 14.8324i) q^{10} +37.9473i q^{11} -59.3296i q^{13} +(23.4521 + 15.8114i) q^{14} +(-46.0000 + 44.4972i) q^{16} +75.0467 q^{17} -118.659i q^{19} +(-46.9042 + 18.9737i) q^{20} +(-60.0000 + 88.9944i) q^{22} -150.093 q^{23} +85.0000 q^{25} +(93.8083 - 139.140i) q^{26} +(30.0000 + 74.1620i) q^{28} -246.658i q^{29} +62.0000 q^{31} +(-178.236 + 31.6228i) q^{32} +(176.000 + 118.659i) q^{34} +63.2456i q^{35} -59.3296i q^{37} +(187.617 - 278.280i) q^{38} +(-140.000 - 29.6648i) q^{40} +375.233 q^{41} +118.659i q^{43} +(-281.425 + 113.842i) q^{44} +(-352.000 - 237.318i) q^{46} -450.280 q^{47} -243.000 q^{49} +(199.343 + 134.397i) q^{50} +(440.000 - 177.989i) q^{52} -132.816i q^{53} -240.000 q^{55} +(-46.9042 + 221.359i) q^{56} +(390.000 - 578.463i) q^{58} +733.648i q^{59} +533.966i q^{61} +(145.403 + 98.0306i) q^{62} +(-468.000 - 207.654i) q^{64} +375.233 q^{65} +711.955i q^{67} +(225.140 + 556.561i) q^{68} +(-100.000 + 148.324i) q^{70} +30.0000 q^{73} +(93.8083 - 139.140i) q^{74} +(880.000 - 355.978i) q^{76} +379.473i q^{77} +94.0000 q^{79} +(-281.425 - 290.930i) q^{80} +(880.000 + 593.296i) q^{82} -670.403i q^{83} +474.637i q^{85} +(-187.617 + 278.280i) q^{86} +(-840.000 - 177.989i) q^{88} -750.467 q^{89} -593.296i q^{91} +(-450.280 - 1113.12i) q^{92} +(-1056.00 - 711.955i) q^{94} +750.467 q^{95} +130.000 q^{97} +(-569.886 - 384.217i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{4} + 40 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{4} + 40 q^{7} - 40 q^{10} - 184 q^{16} - 240 q^{22} + 340 q^{25} + 120 q^{28} + 248 q^{31} + 704 q^{34} - 560 q^{40} - 1408 q^{46} - 972 q^{49} + 1760 q^{52} - 960 q^{55} + 1560 q^{58} - 1872 q^{64} - 400 q^{70} + 120 q^{73} + 3520 q^{76} + 376 q^{79} + 3520 q^{82} - 3360 q^{88} - 4224 q^{94} + 520 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.34521 + 1.58114i 0.829156 + 0.559017i
\(3\) 0 0
\(4\) 3.00000 + 7.41620i 0.375000 + 0.927025i
\(5\) 6.32456i 0.565685i 0.959166 + 0.282843i \(0.0912774\pi\)
−0.959166 + 0.282843i \(0.908723\pi\)
\(6\) 0 0
\(7\) 10.0000 0.539949 0.269975 0.962867i \(-0.412985\pi\)
0.269975 + 0.962867i \(0.412985\pi\)
\(8\) −4.69042 + 22.1359i −0.207289 + 0.978280i
\(9\) 0 0
\(10\) −10.0000 + 14.8324i −0.316228 + 0.469042i
\(11\) 37.9473i 1.04014i 0.854123 + 0.520071i \(0.174094\pi\)
−0.854123 + 0.520071i \(0.825906\pi\)
\(12\) 0 0
\(13\) 59.3296i 1.26577i −0.774244 0.632887i \(-0.781870\pi\)
0.774244 0.632887i \(-0.218130\pi\)
\(14\) 23.4521 + 15.8114i 0.447702 + 0.301841i
\(15\) 0 0
\(16\) −46.0000 + 44.4972i −0.718750 + 0.695269i
\(17\) 75.0467 1.07068 0.535338 0.844638i \(-0.320184\pi\)
0.535338 + 0.844638i \(0.320184\pi\)
\(18\) 0 0
\(19\) 118.659i 1.43275i −0.697715 0.716376i \(-0.745800\pi\)
0.697715 0.716376i \(-0.254200\pi\)
\(20\) −46.9042 + 18.9737i −0.524404 + 0.212132i
\(21\) 0 0
\(22\) −60.0000 + 88.9944i −0.581456 + 0.862439i
\(23\) −150.093 −1.36072 −0.680361 0.732877i \(-0.738177\pi\)
−0.680361 + 0.732877i \(0.738177\pi\)
\(24\) 0 0
\(25\) 85.0000 0.680000
\(26\) 93.8083 139.140i 0.707589 1.04952i
\(27\) 0 0
\(28\) 30.0000 + 74.1620i 0.202481 + 0.500546i
\(29\) 246.658i 1.57942i −0.613480 0.789710i \(-0.710231\pi\)
0.613480 0.789710i \(-0.289769\pi\)
\(30\) 0 0
\(31\) 62.0000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −178.236 + 31.6228i −0.984623 + 0.174693i
\(33\) 0 0
\(34\) 176.000 + 118.659i 0.887757 + 0.598526i
\(35\) 63.2456i 0.305441i
\(36\) 0 0
\(37\) 59.3296i 0.263614i −0.991275 0.131807i \(-0.957922\pi\)
0.991275 0.131807i \(-0.0420779\pi\)
\(38\) 187.617 278.280i 0.800933 1.18797i
\(39\) 0 0
\(40\) −140.000 29.6648i −0.553399 0.117260i
\(41\) 375.233 1.42931 0.714654 0.699479i \(-0.246584\pi\)
0.714654 + 0.699479i \(0.246584\pi\)
\(42\) 0 0
\(43\) 118.659i 0.420822i 0.977613 + 0.210411i \(0.0674802\pi\)
−0.977613 + 0.210411i \(0.932520\pi\)
\(44\) −281.425 + 113.842i −0.964237 + 0.390053i
\(45\) 0 0
\(46\) −352.000 237.318i −1.12825 0.760667i
\(47\) −450.280 −1.39745 −0.698724 0.715391i \(-0.746249\pi\)
−0.698724 + 0.715391i \(0.746249\pi\)
\(48\) 0 0
\(49\) −243.000 −0.708455
\(50\) 199.343 + 134.397i 0.563826 + 0.380132i
\(51\) 0 0
\(52\) 440.000 177.989i 1.17340 0.474665i
\(53\) 132.816i 0.344220i −0.985078 0.172110i \(-0.944942\pi\)
0.985078 0.172110i \(-0.0550584\pi\)
\(54\) 0 0
\(55\) −240.000 −0.588393
\(56\) −46.9042 + 221.359i −0.111926 + 0.528221i
\(57\) 0 0
\(58\) 390.000 578.463i 0.882923 1.30959i
\(59\) 733.648i 1.61886i 0.587215 + 0.809431i \(0.300224\pi\)
−0.587215 + 0.809431i \(0.699776\pi\)
\(60\) 0 0
\(61\) 533.966i 1.12078i 0.828230 + 0.560388i \(0.189348\pi\)
−0.828230 + 0.560388i \(0.810652\pi\)
\(62\) 145.403 + 98.0306i 0.297842 + 0.200805i
\(63\) 0 0
\(64\) −468.000 207.654i −0.914062 0.405573i
\(65\) 375.233 0.716030
\(66\) 0 0
\(67\) 711.955i 1.29820i 0.760705 + 0.649098i \(0.224854\pi\)
−0.760705 + 0.649098i \(0.775146\pi\)
\(68\) 225.140 + 556.561i 0.401503 + 0.992543i
\(69\) 0 0
\(70\) −100.000 + 148.324i −0.170747 + 0.253259i
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 30.0000 0.0480991 0.0240496 0.999711i \(-0.492344\pi\)
0.0240496 + 0.999711i \(0.492344\pi\)
\(74\) 93.8083 139.140i 0.147365 0.218577i
\(75\) 0 0
\(76\) 880.000 355.978i 1.32820 0.537282i
\(77\) 379.473i 0.561623i
\(78\) 0 0
\(79\) 94.0000 0.133871 0.0669356 0.997757i \(-0.478678\pi\)
0.0669356 + 0.997757i \(0.478678\pi\)
\(80\) −281.425 290.930i −0.393303 0.406586i
\(81\) 0 0
\(82\) 880.000 + 593.296i 1.18512 + 0.799007i
\(83\) 670.403i 0.886582i −0.896378 0.443291i \(-0.853811\pi\)
0.896378 0.443291i \(-0.146189\pi\)
\(84\) 0 0
\(85\) 474.637i 0.605666i
\(86\) −187.617 + 278.280i −0.235247 + 0.348927i
\(87\) 0 0
\(88\) −840.000 177.989i −1.01755 0.215610i
\(89\) −750.467 −0.893812 −0.446906 0.894581i \(-0.647474\pi\)
−0.446906 + 0.894581i \(0.647474\pi\)
\(90\) 0 0
\(91\) 593.296i 0.683454i
\(92\) −450.280 1113.12i −0.510271 1.26142i
\(93\) 0 0
\(94\) −1056.00 711.955i −1.15870 0.781197i
\(95\) 750.467 0.810487
\(96\) 0 0
\(97\) 130.000 0.136077 0.0680387 0.997683i \(-0.478326\pi\)
0.0680387 + 0.997683i \(0.478326\pi\)
\(98\) −569.886 384.217i −0.587420 0.396038i
\(99\) 0 0
\(100\) 255.000 + 630.377i 0.255000 + 0.630377i
\(101\) 562.885i 0.554546i −0.960791 0.277273i \(-0.910569\pi\)
0.960791 0.277273i \(-0.0894307\pi\)
\(102\) 0 0
\(103\) −570.000 −0.545279 −0.272640 0.962116i \(-0.587897\pi\)
−0.272640 + 0.962116i \(0.587897\pi\)
\(104\) 1313.32 + 278.280i 1.23828 + 0.262381i
\(105\) 0 0
\(106\) 210.000 311.480i 0.192425 0.285412i
\(107\) 1290.21i 1.16569i −0.812582 0.582847i \(-0.801939\pi\)
0.812582 0.582847i \(-0.198061\pi\)
\(108\) 0 0
\(109\) 1720.56i 1.51192i −0.654616 0.755961i \(-0.727170\pi\)
0.654616 0.755961i \(-0.272830\pi\)
\(110\) −562.850 379.473i −0.487869 0.328921i
\(111\) 0 0
\(112\) −460.000 + 444.972i −0.388089 + 0.375410i
\(113\) −1350.84 −1.12457 −0.562285 0.826944i \(-0.690077\pi\)
−0.562285 + 0.826944i \(0.690077\pi\)
\(114\) 0 0
\(115\) 949.273i 0.769741i
\(116\) 1829.26 739.973i 1.46416 0.592282i
\(117\) 0 0
\(118\) −1160.00 + 1720.56i −0.904972 + 1.34229i
\(119\) 750.467 0.578111
\(120\) 0 0
\(121\) −109.000 −0.0818933
\(122\) −844.275 + 1252.26i −0.626533 + 0.929299i
\(123\) 0 0
\(124\) 186.000 + 459.804i 0.134704 + 0.332997i
\(125\) 1328.16i 0.950352i
\(126\) 0 0
\(127\) 2530.00 1.76773 0.883863 0.467746i \(-0.154934\pi\)
0.883863 + 0.467746i \(0.154934\pi\)
\(128\) −769.228 1226.96i −0.531178 0.847260i
\(129\) 0 0
\(130\) 880.000 + 593.296i 0.593701 + 0.400273i
\(131\) 252.982i 0.168726i −0.996435 0.0843632i \(-0.973114\pi\)
0.996435 0.0843632i \(-0.0268856\pi\)
\(132\) 0 0
\(133\) 1186.59i 0.773613i
\(134\) −1125.70 + 1669.68i −0.725714 + 1.07641i
\(135\) 0 0
\(136\) −352.000 + 1661.23i −0.221939 + 1.04742i
\(137\) −675.420 −0.421204 −0.210602 0.977572i \(-0.567542\pi\)
−0.210602 + 0.977572i \(0.567542\pi\)
\(138\) 0 0
\(139\) 237.318i 0.144814i 0.997375 + 0.0724068i \(0.0230680\pi\)
−0.997375 + 0.0724068i \(0.976932\pi\)
\(140\) −469.042 + 189.737i −0.283152 + 0.114541i
\(141\) 0 0
\(142\) 0 0
\(143\) 2251.40 1.31658
\(144\) 0 0
\(145\) 1560.00 0.893455
\(146\) 70.3562 + 47.4342i 0.0398817 + 0.0268882i
\(147\) 0 0
\(148\) 440.000 177.989i 0.244377 0.0988553i
\(149\) 2940.92i 1.61698i 0.588513 + 0.808488i \(0.299714\pi\)
−0.588513 + 0.808488i \(0.700286\pi\)
\(150\) 0 0
\(151\) −2262.00 −1.21907 −0.609533 0.792761i \(-0.708643\pi\)
−0.609533 + 0.792761i \(0.708643\pi\)
\(152\) 2626.63 + 556.561i 1.40163 + 0.296994i
\(153\) 0 0
\(154\) −600.000 + 889.944i −0.313957 + 0.465673i
\(155\) 392.122i 0.203200i
\(156\) 0 0
\(157\) 1720.56i 0.874621i 0.899311 + 0.437310i \(0.144069\pi\)
−0.899311 + 0.437310i \(0.855931\pi\)
\(158\) 220.450 + 148.627i 0.111000 + 0.0748363i
\(159\) 0 0
\(160\) −200.000 1127.26i −0.0988212 0.556987i
\(161\) −1500.93 −0.734721
\(162\) 0 0
\(163\) 2017.21i 0.969324i −0.874702 0.484662i \(-0.838943\pi\)
0.874702 0.484662i \(-0.161057\pi\)
\(164\) 1125.70 + 2782.80i 0.535990 + 1.32500i
\(165\) 0 0
\(166\) 1060.00 1572.23i 0.495614 0.735115i
\(167\) 1050.65 0.486838 0.243419 0.969921i \(-0.421731\pi\)
0.243419 + 0.969921i \(0.421731\pi\)
\(168\) 0 0
\(169\) −1323.00 −0.602185
\(170\) −750.467 + 1113.12i −0.338577 + 0.502191i
\(171\) 0 0
\(172\) −880.000 + 355.978i −0.390113 + 0.157808i
\(173\) 1423.02i 0.625379i −0.949855 0.312690i \(-0.898770\pi\)
0.949855 0.312690i \(-0.101230\pi\)
\(174\) 0 0
\(175\) 850.000 0.367165
\(176\) −1688.55 1745.58i −0.723177 0.747601i
\(177\) 0 0
\(178\) −1760.00 1186.59i −0.741110 0.499656i
\(179\) 126.491i 0.0528178i 0.999651 + 0.0264089i \(0.00840719\pi\)
−0.999651 + 0.0264089i \(0.991593\pi\)
\(180\) 0 0
\(181\) 59.3296i 0.0243643i 0.999926 + 0.0121821i \(0.00387779\pi\)
−0.999926 + 0.0121821i \(0.996122\pi\)
\(182\) 938.083 1391.40i 0.382062 0.566690i
\(183\) 0 0
\(184\) 704.000 3322.46i 0.282063 1.33117i
\(185\) 375.233 0.149123
\(186\) 0 0
\(187\) 2847.82i 1.11365i
\(188\) −1350.84 3339.37i −0.524043 1.29547i
\(189\) 0 0
\(190\) 1760.00 + 1186.59i 0.672020 + 0.453076i
\(191\) −3752.33 −1.42151 −0.710757 0.703437i \(-0.751648\pi\)
−0.710757 + 0.703437i \(0.751648\pi\)
\(192\) 0 0
\(193\) −1350.00 −0.503498 −0.251749 0.967793i \(-0.581006\pi\)
−0.251749 + 0.967793i \(0.581006\pi\)
\(194\) 304.877 + 205.548i 0.112829 + 0.0760695i
\(195\) 0 0
\(196\) −729.000 1802.14i −0.265671 0.656755i
\(197\) 1448.32i 0.523801i −0.965095 0.261900i \(-0.915651\pi\)
0.965095 0.261900i \(-0.0843492\pi\)
\(198\) 0 0
\(199\) −3194.00 −1.13777 −0.568886 0.822416i \(-0.692625\pi\)
−0.568886 + 0.822416i \(0.692625\pi\)
\(200\) −398.685 + 1881.56i −0.140957 + 0.665230i
\(201\) 0 0
\(202\) 890.000 1320.08i 0.310001 0.459806i
\(203\) 2466.58i 0.852807i
\(204\) 0 0
\(205\) 2373.18i 0.808538i
\(206\) −1336.77 901.249i −0.452122 0.304820i
\(207\) 0 0
\(208\) 2640.00 + 2729.16i 0.880053 + 0.909775i
\(209\) 4502.80 1.49026
\(210\) 0 0
\(211\) 5458.32i 1.78088i 0.455097 + 0.890442i \(0.349604\pi\)
−0.455097 + 0.890442i \(0.650396\pi\)
\(212\) 984.987 398.447i 0.319100 0.129082i
\(213\) 0 0
\(214\) 2040.00 3025.81i 0.651643 0.966542i
\(215\) −750.467 −0.238053
\(216\) 0 0
\(217\) 620.000 0.193955
\(218\) 2720.44 4035.07i 0.845190 1.25362i
\(219\) 0 0
\(220\) −720.000 1779.89i −0.220647 0.545455i
\(221\) 4452.49i 1.35523i
\(222\) 0 0
\(223\) 5330.00 1.60055 0.800276 0.599632i \(-0.204686\pi\)
0.800276 + 0.599632i \(0.204686\pi\)
\(224\) −1782.36 + 316.228i −0.531646 + 0.0943253i
\(225\) 0 0
\(226\) −3168.00 2135.87i −0.932443 0.628653i
\(227\) 4768.71i 1.39432i 0.716915 + 0.697160i \(0.245553\pi\)
−0.716915 + 0.697160i \(0.754447\pi\)
\(228\) 0 0
\(229\) 3619.10i 1.04435i 0.852837 + 0.522177i \(0.174880\pi\)
−0.852837 + 0.522177i \(0.825120\pi\)
\(230\) 1500.93 2226.24i 0.430298 0.638235i
\(231\) 0 0
\(232\) 5460.00 + 1156.93i 1.54511 + 0.327396i
\(233\) 150.093 0.0422015 0.0211007 0.999777i \(-0.493283\pi\)
0.0211007 + 0.999777i \(0.493283\pi\)
\(234\) 0 0
\(235\) 2847.82i 0.790516i
\(236\) −5440.88 + 2200.95i −1.50073 + 0.607073i
\(237\) 0 0
\(238\) 1760.00 + 1186.59i 0.479344 + 0.323174i
\(239\) −2251.40 −0.609334 −0.304667 0.952459i \(-0.598545\pi\)
−0.304667 + 0.952459i \(0.598545\pi\)
\(240\) 0 0
\(241\) −1162.00 −0.310585 −0.155293 0.987869i \(-0.549632\pi\)
−0.155293 + 0.987869i \(0.549632\pi\)
\(242\) −255.628 172.344i −0.0679023 0.0457798i
\(243\) 0 0
\(244\) −3960.00 + 1601.90i −1.03899 + 0.420291i
\(245\) 1536.87i 0.400763i
\(246\) 0 0
\(247\) −7040.00 −1.81354
\(248\) −290.806 + 1372.43i −0.0744604 + 0.351408i
\(249\) 0 0
\(250\) −2100.00 + 3114.80i −0.531263 + 0.787990i
\(251\) 645.105i 0.162226i 0.996705 + 0.0811128i \(0.0258474\pi\)
−0.996705 + 0.0811128i \(0.974153\pi\)
\(252\) 0 0
\(253\) 5695.64i 1.41534i
\(254\) 5933.38 + 4000.28i 1.46572 + 0.988189i
\(255\) 0 0
\(256\) 136.000 4093.74i 0.0332031 0.999449i
\(257\) 1801.12 0.437162 0.218581 0.975819i \(-0.429857\pi\)
0.218581 + 0.975819i \(0.429857\pi\)
\(258\) 0 0
\(259\) 593.296i 0.142338i
\(260\) 1125.70 + 2782.80i 0.268511 + 0.663778i
\(261\) 0 0
\(262\) 400.000 593.296i 0.0943209 0.139901i
\(263\) −6604.11 −1.54839 −0.774195 0.632947i \(-0.781845\pi\)
−0.774195 + 0.632947i \(0.781845\pi\)
\(264\) 0 0
\(265\) 840.000 0.194720
\(266\) 1876.17 2782.80i 0.432463 0.641446i
\(267\) 0 0
\(268\) −5280.00 + 2135.87i −1.20346 + 0.486824i
\(269\) 6052.60i 1.37187i 0.727662 + 0.685936i \(0.240607\pi\)
−0.727662 + 0.685936i \(0.759393\pi\)
\(270\) 0 0
\(271\) 6402.00 1.43503 0.717516 0.696542i \(-0.245279\pi\)
0.717516 + 0.696542i \(0.245279\pi\)
\(272\) −3452.15 + 3339.37i −0.769548 + 0.744407i
\(273\) 0 0
\(274\) −1584.00 1067.93i −0.349244 0.235460i
\(275\) 3225.52i 0.707296i
\(276\) 0 0
\(277\) 2076.54i 0.450422i −0.974310 0.225211i \(-0.927693\pi\)
0.974310 0.225211i \(-0.0723072\pi\)
\(278\) −375.233 + 556.561i −0.0809532 + 0.120073i
\(279\) 0 0
\(280\) −1400.00 296.648i −0.298807 0.0633147i
\(281\) 9005.60 1.91185 0.955923 0.293616i \(-0.0948587\pi\)
0.955923 + 0.293616i \(0.0948587\pi\)
\(282\) 0 0
\(283\) 5221.00i 1.09667i −0.836260 0.548333i \(-0.815263\pi\)
0.836260 0.548333i \(-0.184737\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 5280.00 + 3559.78i 1.09165 + 0.735993i
\(287\) 3752.33 0.771753
\(288\) 0 0
\(289\) 719.000 0.146346
\(290\) 3658.52 + 2466.58i 0.740814 + 0.499456i
\(291\) 0 0
\(292\) 90.0000 + 222.486i 0.0180372 + 0.0445891i
\(293\) 5293.65i 1.05549i −0.849403 0.527745i \(-0.823038\pi\)
0.849403 0.527745i \(-0.176962\pi\)
\(294\) 0 0
\(295\) −4640.00 −0.915767
\(296\) 1313.32 + 278.280i 0.257888 + 0.0546443i
\(297\) 0 0
\(298\) −4650.00 + 6897.06i −0.903917 + 1.34073i
\(299\) 8904.97i 1.72237i
\(300\) 0 0
\(301\) 1186.59i 0.227223i
\(302\) −5304.86 3576.54i −1.01080 0.681479i
\(303\) 0 0
\(304\) 5280.00 + 5458.32i 0.996147 + 1.02979i
\(305\) −3377.10 −0.634007
\(306\) 0 0
\(307\) 4983.69i 0.926495i −0.886229 0.463247i \(-0.846684\pi\)
0.886229 0.463247i \(-0.153316\pi\)
\(308\) −2814.25 + 1138.42i −0.520639 + 0.210609i
\(309\) 0 0
\(310\) −620.000 + 919.609i −0.113592 + 0.168485i
\(311\) −1500.93 −0.273666 −0.136833 0.990594i \(-0.543692\pi\)
−0.136833 + 0.990594i \(0.543692\pi\)
\(312\) 0 0
\(313\) 6350.00 1.14672 0.573360 0.819304i \(-0.305640\pi\)
0.573360 + 0.819304i \(0.305640\pi\)
\(314\) −2720.44 + 4035.07i −0.488928 + 0.725197i
\(315\) 0 0
\(316\) 282.000 + 697.123i 0.0502017 + 0.124102i
\(317\) 1739.25i 0.308158i −0.988059 0.154079i \(-0.950759\pi\)
0.988059 0.154079i \(-0.0492411\pi\)
\(318\) 0 0
\(319\) 9360.00 1.64282
\(320\) 1313.32 2959.89i 0.229427 0.517072i
\(321\) 0 0
\(322\) −3520.00 2373.18i −0.609199 0.410722i
\(323\) 8904.97i 1.53401i
\(324\) 0 0
\(325\) 5043.01i 0.860727i
\(326\) 3189.48 4730.77i 0.541868 0.803721i
\(327\) 0 0
\(328\) −1760.00 + 8306.14i −0.296280 + 1.39826i
\(329\) −4502.80 −0.754551
\(330\) 0 0
\(331\) 3322.46i 0.551718i 0.961198 + 0.275859i \(0.0889623\pi\)
−0.961198 + 0.275859i \(0.911038\pi\)
\(332\) 4971.84 2011.21i 0.821883 0.332468i
\(333\) 0 0
\(334\) 2464.00 + 1661.23i 0.403665 + 0.272151i
\(335\) −4502.80 −0.734371
\(336\) 0 0
\(337\) −1430.00 −0.231149 −0.115574 0.993299i \(-0.536871\pi\)
−0.115574 + 0.993299i \(0.536871\pi\)
\(338\) −3102.71 2091.85i −0.499305 0.336632i
\(339\) 0 0
\(340\) −3520.00 + 1423.91i −0.561467 + 0.227125i
\(341\) 2352.73i 0.373630i
\(342\) 0 0
\(343\) −5860.00 −0.922479
\(344\) −2626.63 556.561i −0.411682 0.0872318i
\(345\) 0 0
\(346\) 2250.00 3337.29i 0.349598 0.518537i
\(347\) 10106.6i 1.56355i −0.623559 0.781776i \(-0.714314\pi\)
0.623559 0.781776i \(-0.285686\pi\)
\(348\) 0 0
\(349\) 1127.26i 0.172897i −0.996256 0.0864484i \(-0.972448\pi\)
0.996256 0.0864484i \(-0.0275518\pi\)
\(350\) 1993.43 + 1343.97i 0.304438 + 0.205252i
\(351\) 0 0
\(352\) −1200.00 6763.57i −0.181705 1.02415i
\(353\) 1350.84 0.203677 0.101838 0.994801i \(-0.467528\pi\)
0.101838 + 0.994801i \(0.467528\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2251.40 5565.61i −0.335180 0.828586i
\(357\) 0 0
\(358\) −200.000 + 296.648i −0.0295261 + 0.0437942i
\(359\) 12007.5 1.76526 0.882632 0.470065i \(-0.155769\pi\)
0.882632 + 0.470065i \(0.155769\pi\)
\(360\) 0 0
\(361\) −7221.00 −1.05278
\(362\) −93.8083 + 139.140i −0.0136200 + 0.0202018i
\(363\) 0 0
\(364\) 4400.00 1779.89i 0.633579 0.256295i
\(365\) 189.737i 0.0272090i
\(366\) 0 0
\(367\) 5070.00 0.721122 0.360561 0.932736i \(-0.382585\pi\)
0.360561 + 0.932736i \(0.382585\pi\)
\(368\) 6904.29 6678.73i 0.978019 0.946068i
\(369\) 0 0
\(370\) 880.000 + 593.296i 0.123646 + 0.0833621i
\(371\) 1328.16i 0.185861i
\(372\) 0 0
\(373\) 2432.51i 0.337670i −0.985644 0.168835i \(-0.946000\pi\)
0.985644 0.168835i \(-0.0540004\pi\)
\(374\) −4502.80 + 6678.73i −0.622551 + 0.923393i
\(375\) 0 0
\(376\) 2112.00 9967.37i 0.289676 1.36710i
\(377\) −14634.1 −1.99919
\(378\) 0 0
\(379\) 7712.85i 1.04534i 0.852536 + 0.522668i \(0.175063\pi\)
−0.852536 + 0.522668i \(0.824937\pi\)
\(380\) 2251.40 + 5565.61i 0.303933 + 0.751341i
\(381\) 0 0
\(382\) −8800.00 5932.96i −1.17866 0.794651i
\(383\) 5853.64 0.780958 0.390479 0.920612i \(-0.372309\pi\)
0.390479 + 0.920612i \(0.372309\pi\)
\(384\) 0 0
\(385\) −2400.00 −0.317702
\(386\) −3166.03 2134.54i −0.417479 0.281464i
\(387\) 0 0
\(388\) 390.000 + 964.106i 0.0510290 + 0.126147i
\(389\) 2156.67i 0.281099i 0.990074 + 0.140550i \(0.0448870\pi\)
−0.990074 + 0.140550i \(0.955113\pi\)
\(390\) 0 0
\(391\) −11264.0 −1.45689
\(392\) 1139.77 5379.03i 0.146855 0.693067i
\(393\) 0 0
\(394\) 2290.00 3396.62i 0.292814 0.434313i
\(395\) 594.508i 0.0757290i
\(396\) 0 0
\(397\) 12162.6i 1.53759i 0.639498 + 0.768793i \(0.279142\pi\)
−0.639498 + 0.768793i \(0.720858\pi\)
\(398\) −7490.59 5050.16i −0.943391 0.636034i
\(399\) 0 0
\(400\) −3910.00 + 3782.26i −0.488750 + 0.472783i
\(401\) −3377.10 −0.420559 −0.210280 0.977641i \(-0.567437\pi\)
−0.210280 + 0.977641i \(0.567437\pi\)
\(402\) 0 0
\(403\) 3678.43i 0.454680i
\(404\) 4174.47 1688.66i 0.514078 0.207955i
\(405\) 0 0
\(406\) 3900.00 5784.63i 0.476733 0.707110i
\(407\) 2251.40 0.274196
\(408\) 0 0
\(409\) 3526.00 0.426282 0.213141 0.977021i \(-0.431631\pi\)
0.213141 + 0.977021i \(0.431631\pi\)
\(410\) −3752.33 + 5565.61i −0.451987 + 0.670404i
\(411\) 0 0
\(412\) −1710.00 4227.23i −0.204480 0.505487i
\(413\) 7336.48i 0.874104i
\(414\) 0 0
\(415\) 4240.00 0.501526
\(416\) 1876.17 + 10574.7i 0.221122 + 1.24631i
\(417\) 0 0
\(418\) 10560.0 + 7119.55i 1.23566 + 0.833083i
\(419\) 2466.58i 0.287590i −0.989608 0.143795i \(-0.954069\pi\)
0.989608 0.143795i \(-0.0459306\pi\)
\(420\) 0 0
\(421\) 13586.5i 1.57284i 0.617694 + 0.786418i \(0.288067\pi\)
−0.617694 + 0.786418i \(0.711933\pi\)
\(422\) −8630.36 + 12800.9i −0.995544 + 1.47663i
\(423\) 0 0
\(424\) 2940.00 + 622.961i 0.336743 + 0.0713529i
\(425\) 6378.97 0.728059
\(426\) 0 0
\(427\) 5339.66i 0.605163i
\(428\) 9568.45 3870.63i 1.08063 0.437135i
\(429\) 0 0
\(430\) −1760.00 1186.59i −0.197383 0.133076i
\(431\) 6754.20 0.754845 0.377423 0.926041i \(-0.376810\pi\)
0.377423 + 0.926041i \(0.376810\pi\)
\(432\) 0 0
\(433\) 7790.00 0.864581 0.432290 0.901734i \(-0.357706\pi\)
0.432290 + 0.901734i \(0.357706\pi\)
\(434\) 1454.03 + 980.306i 0.160819 + 0.108424i
\(435\) 0 0
\(436\) 12760.0 5161.67i 1.40159 0.566971i
\(437\) 17809.9i 1.94958i
\(438\) 0 0
\(439\) −9354.00 −1.01695 −0.508476 0.861076i \(-0.669791\pi\)
−0.508476 + 0.861076i \(0.669791\pi\)
\(440\) 1125.70 5312.63i 0.121967 0.575613i
\(441\) 0 0
\(442\) 7040.00 10442.0i 0.757599 1.12370i
\(443\) 6488.99i 0.695940i 0.937506 + 0.347970i \(0.113129\pi\)
−0.937506 + 0.347970i \(0.886871\pi\)
\(444\) 0 0
\(445\) 4746.37i 0.505617i
\(446\) 12500.0 + 8427.47i 1.32711 + 0.894736i
\(447\) 0 0
\(448\) −4680.00 2076.54i −0.493547 0.218989i
\(449\) −10131.3 −1.06487 −0.532434 0.846472i \(-0.678722\pi\)
−0.532434 + 0.846472i \(0.678722\pi\)
\(450\) 0 0
\(451\) 14239.1i 1.48668i
\(452\) −4052.52 10018.1i −0.421713 1.04250i
\(453\) 0 0
\(454\) −7540.00 + 11183.6i −0.779449 + 1.15611i
\(455\) 3752.33 0.386620
\(456\) 0 0
\(457\) −12010.0 −1.22933 −0.614665 0.788788i \(-0.710709\pi\)
−0.614665 + 0.788788i \(0.710709\pi\)
\(458\) −5722.31 + 8487.55i −0.583812 + 0.865933i
\(459\) 0 0
\(460\) 7040.00 2847.82i 0.713569 0.288653i
\(461\) 9316.07i 0.941199i −0.882347 0.470599i \(-0.844038\pi\)
0.882347 0.470599i \(-0.155962\pi\)
\(462\) 0 0
\(463\) 14770.0 1.48255 0.741274 0.671202i \(-0.234222\pi\)
0.741274 + 0.671202i \(0.234222\pi\)
\(464\) 10975.6 + 11346.3i 1.09812 + 1.13521i
\(465\) 0 0
\(466\) 352.000 + 237.318i 0.0349916 + 0.0235913i
\(467\) 8740.54i 0.866089i −0.901372 0.433045i \(-0.857439\pi\)
0.901372 0.433045i \(-0.142561\pi\)
\(468\) 0 0
\(469\) 7119.55i 0.700960i
\(470\) 4502.80 6678.73i 0.441912 0.655461i
\(471\) 0 0
\(472\) −16240.0 3441.12i −1.58370 0.335572i
\(473\) −4502.80 −0.437714
\(474\) 0 0
\(475\) 10086.0i 0.974271i
\(476\) 2251.40 + 5565.61i 0.216791 + 0.535923i
\(477\) 0 0
\(478\) −5280.00 3559.78i −0.505233 0.340628i
\(479\) −3001.87 −0.286344 −0.143172 0.989698i \(-0.545730\pi\)
−0.143172 + 0.989698i \(0.545730\pi\)
\(480\) 0 0
\(481\) −3520.00 −0.333676
\(482\) −2725.13 1837.28i −0.257524 0.173622i
\(483\) 0 0
\(484\) −327.000 808.366i −0.0307100 0.0759171i
\(485\) 822.192i 0.0769770i
\(486\) 0 0
\(487\) −9910.00 −0.922105 −0.461052 0.887373i \(-0.652528\pi\)
−0.461052 + 0.887373i \(0.652528\pi\)
\(488\) −11819.8 2504.52i −1.09643 0.232325i
\(489\) 0 0
\(490\) 2430.00 3604.27i 0.224033 0.332295i
\(491\) 3389.96i 0.311582i 0.987790 + 0.155791i \(0.0497927\pi\)
−0.987790 + 0.155791i \(0.950207\pi\)
\(492\) 0 0
\(493\) 18510.8i 1.69105i
\(494\) −16510.3 11131.2i −1.50371 1.01380i
\(495\) 0 0
\(496\) −2852.00 + 2758.83i −0.258183 + 0.249748i
\(497\) 0 0
\(498\) 0 0
\(499\) 4271.73i 0.383224i −0.981471 0.191612i \(-0.938628\pi\)
0.981471 0.191612i \(-0.0613716\pi\)
\(500\) −9849.87 + 3984.47i −0.880999 + 0.356382i
\(501\) 0 0
\(502\) −1020.00 + 1512.90i −0.0906869 + 0.134510i
\(503\) 900.560 0.0798290 0.0399145 0.999203i \(-0.487291\pi\)
0.0399145 + 0.999203i \(0.487291\pi\)
\(504\) 0 0
\(505\) 3560.00 0.313699
\(506\) 9005.60 13357.5i 0.791201 1.17354i
\(507\) 0 0
\(508\) 7590.00 + 18763.0i 0.662897 + 1.63873i
\(509\) 853.815i 0.0743510i −0.999309 0.0371755i \(-0.988164\pi\)
0.999309 0.0371755i \(-0.0118361\pi\)
\(510\) 0 0
\(511\) 300.000 0.0259711
\(512\) 6791.72 9385.64i 0.586239 0.810138i
\(513\) 0 0
\(514\) 4224.00 + 2847.82i 0.362476 + 0.244381i
\(515\) 3605.00i 0.308457i
\(516\) 0 0
\(517\) 17086.9i 1.45354i
\(518\) 938.083 1391.40i 0.0795695 0.118021i
\(519\) 0 0
\(520\) −1760.00 + 8306.14i −0.148425 + 0.700478i
\(521\) 7879.90 0.662619 0.331310 0.943522i \(-0.392510\pi\)
0.331310 + 0.943522i \(0.392510\pi\)
\(522\) 0 0
\(523\) 10323.3i 0.863114i −0.902086 0.431557i \(-0.857964\pi\)
0.902086 0.431557i \(-0.142036\pi\)
\(524\) 1876.17 758.947i 0.156414 0.0632724i
\(525\) 0 0
\(526\) −15488.0 10442.0i −1.28386 0.865576i
\(527\) 4652.89 0.384598
\(528\) 0 0
\(529\) 10361.0 0.851566
\(530\) 1969.97 + 1328.16i 0.161453 + 0.108852i
\(531\) 0 0
\(532\) 8800.00 3559.78i 0.717159 0.290105i
\(533\) 22262.4i 1.80918i
\(534\) 0 0
\(535\) 8160.00 0.659416
\(536\) −15759.8 3339.37i −1.27000 0.269102i
\(537\) 0 0
\(538\) −9570.00 + 14194.6i −0.766900 + 1.13750i
\(539\) 9221.20i 0.736893i
\(540\) 0 0
\(541\) 15603.7i 1.24003i 0.784591 + 0.620014i \(0.212873\pi\)
−0.784591 + 0.620014i \(0.787127\pi\)
\(542\) 15014.0 + 10122.5i 1.18987 + 0.802208i
\(543\) 0 0
\(544\) −13376.0 + 2373.18i −1.05421 + 0.187039i
\(545\) 10881.8 0.855273
\(546\) 0 0
\(547\) 2254.52i 0.176228i −0.996110 0.0881138i \(-0.971916\pi\)
0.996110 0.0881138i \(-0.0280839\pi\)
\(548\) −2026.26 5009.05i −0.157952 0.390467i
\(549\) 0 0
\(550\) −5100.00 + 7564.52i −0.395390 + 0.586459i
\(551\) −29268.2 −2.26292
\(552\) 0 0
\(553\) 940.000 0.0722837
\(554\) 3283.29 4869.91i 0.251794 0.373470i
\(555\) 0 0
\(556\) −1760.00 + 711.955i −0.134246 + 0.0543051i
\(557\) 284.605i 0.0216501i −0.999941 0.0108250i \(-0.996554\pi\)
0.999941 0.0108250i \(-0.00344579\pi\)
\(558\) 0 0
\(559\) 7040.00 0.532666
\(560\) −2814.25 2909.30i −0.212364 0.219536i
\(561\) 0 0
\(562\) 21120.0 + 14239.1i 1.58522 + 1.06875i
\(563\) 18328.6i 1.37204i 0.727584 + 0.686018i \(0.240643\pi\)
−0.727584 + 0.686018i \(0.759357\pi\)
\(564\) 0 0
\(565\) 8543.46i 0.636152i
\(566\) 8255.13 12244.3i 0.613055 0.909307i
\(567\) 0 0
\(568\) 0 0
\(569\) 7879.90 0.580567 0.290283 0.956941i \(-0.406250\pi\)
0.290283 + 0.956941i \(0.406250\pi\)
\(570\) 0 0
\(571\) 8306.14i 0.608759i −0.952551 0.304379i \(-0.901551\pi\)
0.952551 0.304379i \(-0.0984491\pi\)
\(572\) 6754.20 + 16696.8i 0.493719 + 1.22051i
\(573\) 0 0
\(574\) 8800.00 + 5932.96i 0.639904 + 0.431423i
\(575\) −12757.9 −0.925291
\(576\) 0 0
\(577\) −6330.00 −0.456709 −0.228355 0.973578i \(-0.573335\pi\)
−0.228355 + 0.973578i \(0.573335\pi\)
\(578\) 1686.20 + 1136.84i 0.121344 + 0.0818101i
\(579\) 0 0
\(580\) 4680.00 + 11569.3i 0.335046 + 0.828255i
\(581\) 6704.03i 0.478709i
\(582\) 0 0
\(583\) 5040.00 0.358037
\(584\) −140.712 + 664.078i −0.00997042 + 0.0470544i
\(585\) 0 0
\(586\) 8370.00 12414.7i 0.590037 0.875166i
\(587\) 1922.66i 0.135191i −0.997713 0.0675953i \(-0.978467\pi\)
0.997713 0.0675953i \(-0.0215327\pi\)
\(588\) 0 0
\(589\) 7356.87i 0.514660i
\(590\) −10881.8 7336.48i −0.759314 0.511929i
\(591\) 0 0
\(592\) 2640.00 + 2729.16i 0.183283 + 0.189473i
\(593\) 10356.4 0.717180 0.358590 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(594\) 0 0
\(595\) 4746.37i 0.327029i
\(596\) −21810.4 + 8822.75i −1.49898 + 0.606366i
\(597\) 0 0
\(598\) −14080.0 + 20884.0i −0.962833 + 1.42811i
\(599\) 750.467 0.0511907 0.0255954 0.999672i \(-0.491852\pi\)
0.0255954 + 0.999672i \(0.491852\pi\)
\(600\) 0 0
\(601\) 18578.0 1.26092 0.630460 0.776222i \(-0.282866\pi\)
0.630460 + 0.776222i \(0.282866\pi\)
\(602\) −1876.17 + 2782.80i −0.127021 + 0.188403i
\(603\) 0 0
\(604\) −6786.00 16775.4i −0.457150 1.13010i
\(605\) 689.377i 0.0463259i
\(606\) 0 0
\(607\) −8030.00 −0.536948 −0.268474 0.963287i \(-0.586519\pi\)
−0.268474 + 0.963287i \(0.586519\pi\)
\(608\) 3752.33 + 21149.3i 0.250291 + 1.41072i
\(609\) 0 0
\(610\) −7920.00 5339.66i −0.525691 0.354421i
\(611\) 26714.9i 1.76885i
\(612\) 0 0
\(613\) 3856.42i 0.254094i −0.991897 0.127047i \(-0.959450\pi\)
0.991897 0.127047i \(-0.0405499\pi\)
\(614\) 7879.90 11687.8i 0.517926 0.768209i
\(615\) 0 0
\(616\) −8400.00 1779.89i −0.549425 0.116418i
\(617\) 300.187 0.0195868 0.00979340 0.999952i \(-0.496883\pi\)
0.00979340 + 0.999952i \(0.496883\pi\)
\(618\) 0 0
\(619\) 1423.91i 0.0924584i 0.998931 + 0.0462292i \(0.0147205\pi\)
−0.998931 + 0.0462292i \(0.985280\pi\)
\(620\) −2908.06 + 1176.37i −0.188372 + 0.0762001i
\(621\) 0 0
\(622\) −3520.00 2373.18i −0.226912 0.152984i
\(623\) −7504.67 −0.482613
\(624\) 0 0
\(625\) 2225.00 0.142400
\(626\) 14892.1 + 10040.2i 0.950810 + 0.641036i
\(627\) 0 0
\(628\) −12760.0 + 5161.67i −0.810795 + 0.327983i
\(629\) 4452.49i 0.282245i
\(630\) 0 0
\(631\) −12902.0 −0.813979 −0.406989 0.913433i \(-0.633421\pi\)
−0.406989 + 0.913433i \(0.633421\pi\)
\(632\) −440.899 + 2080.78i −0.0277500 + 0.130964i
\(633\) 0 0
\(634\) 2750.00 4078.91i 0.172266 0.255511i
\(635\) 16001.1i 0.999977i
\(636\) 0 0
\(637\) 14417.1i 0.896744i
\(638\) 21951.1 + 14799.5i 1.36215 + 0.918364i
\(639\) 0 0
\(640\) 7760.00 4865.03i 0.479283 0.300480i
\(641\) 19887.4 1.22543 0.612717 0.790302i \(-0.290076\pi\)
0.612717 + 0.790302i \(0.290076\pi\)
\(642\) 0 0
\(643\) 29783.5i 1.82666i −0.407216 0.913332i \(-0.633500\pi\)
0.407216 0.913332i \(-0.366500\pi\)
\(644\) −4502.80 11131.2i −0.275520 0.681105i
\(645\) 0 0
\(646\) 14080.0 20884.0i 0.857539 1.27194i
\(647\) 13958.7 0.848180 0.424090 0.905620i \(-0.360594\pi\)
0.424090 + 0.905620i \(0.360594\pi\)
\(648\) 0 0
\(649\) −27840.0 −1.68385
\(650\) 7973.71 11826.9i 0.481161 0.713677i
\(651\) 0 0
\(652\) 14960.0 6051.62i 0.898587 0.363496i
\(653\) 18461.4i 1.10635i 0.833064 + 0.553177i \(0.186585\pi\)
−0.833064 + 0.553177i \(0.813415\pi\)
\(654\) 0 0
\(655\) 1600.00 0.0954461
\(656\) −17260.7 + 16696.8i −1.02731 + 0.993752i
\(657\) 0 0
\(658\) −10560.0 7119.55i −0.625641 0.421807i
\(659\) 5160.84i 0.305065i −0.988298 0.152532i \(-0.951257\pi\)
0.988298 0.152532i \(-0.0487428\pi\)
\(660\) 0 0
\(661\) 13467.8i 0.792492i 0.918144 + 0.396246i \(0.129687\pi\)
−0.918144 + 0.396246i \(0.870313\pi\)
\(662\) −5253.27 + 7791.85i −0.308420 + 0.457461i
\(663\) 0 0
\(664\) 14840.0 + 3144.47i 0.867325 + 0.183779i
\(665\) 7504.67 0.437622
\(666\) 0 0
\(667\) 37021.7i 2.14915i
\(668\) 3151.96 + 7791.85i 0.182564 + 0.451311i
\(669\) 0 0
\(670\) −10560.0 7119.55i −0.608908 0.410526i
\(671\) −20262.6 −1.16577
\(672\) 0 0
\(673\) −15010.0 −0.859722 −0.429861 0.902895i \(-0.641437\pi\)
−0.429861 + 0.902895i \(0.641437\pi\)
\(674\) −3353.65 2261.03i −0.191658 0.129216i
\(675\) 0 0
\(676\) −3969.00 9811.63i −0.225819 0.558240i
\(677\) 6280.28i 0.356530i 0.983983 + 0.178265i \(0.0570485\pi\)
−0.983983 + 0.178265i \(0.942952\pi\)
\(678\) 0 0
\(679\) 1300.00 0.0734748
\(680\) −10506.5 2226.24i −0.592510 0.125548i
\(681\) 0 0
\(682\) −3720.00 + 5517.65i −0.208865 + 0.309797i
\(683\) 12510.0i 0.700850i −0.936591 0.350425i \(-0.886037\pi\)
0.936591 0.350425i \(-0.113963\pi\)
\(684\) 0 0
\(685\) 4271.73i 0.238269i
\(686\) −13742.9 9265.47i −0.764879 0.515681i
\(687\) 0 0
\(688\) −5280.00 5458.32i −0.292584 0.302466i
\(689\) −7879.90 −0.435704
\(690\) 0 0
\(691\) 28359.5i 1.56128i 0.624978 + 0.780642i \(0.285108\pi\)
−0.624978 + 0.780642i \(0.714892\pi\)
\(692\) 10553.4 4269.07i 0.579742 0.234517i
\(693\) 0 0
\(694\) 15980.0 23702.2i 0.874053 1.29643i
\(695\) −1500.93 −0.0819189
\(696\) 0 0
\(697\) 28160.0 1.53032
\(698\) 1782.36 2643.66i 0.0966522 0.143358i
\(699\) 0 0
\(700\) 2550.00 + 6303.77i 0.137687 + 0.340372i
\(701\) 19802.2i 1.06693i −0.845822 0.533465i \(-0.820890\pi\)
0.845822 0.533465i \(-0.179110\pi\)
\(702\) 0 0
\(703\) −7040.00 −0.377694
\(704\) 7879.90 17759.4i 0.421853 0.950754i
\(705\) 0 0
\(706\) 3168.00 + 2135.87i 0.168880 + 0.113859i
\(707\) 5628.85i 0.299427i
\(708\) 0 0
\(709\) 12874.5i 0.681964i 0.940070 + 0.340982i \(0.110760\pi\)
−0.940070 + 0.340982i \(0.889240\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 3520.00 16612.3i 0.185277 0.874398i
\(713\) −9305.78 −0.488786
\(714\) 0 0
\(715\) 14239.1i 0.744772i
\(716\) −938.083 + 379.473i −0.0489634 + 0.0198067i
\(717\) 0 0
\(718\) 28160.0 + 18985.5i 1.46368 + 0.986813i
\(719\) −1500.93 −0.0778517 −0.0389258 0.999242i \(-0.512394\pi\)
−0.0389258 + 0.999242i \(0.512394\pi\)
\(720\) 0 0
\(721\) −5700.00 −0.294423
\(722\) −16934.7 11417.4i −0.872917 0.588520i
\(723\) 0 0
\(724\) −440.000 + 177.989i −0.0225863 + 0.00913660i
\(725\) 20965.9i 1.07401i
\(726\) 0 0
\(727\) 23850.0 1.21671 0.608355 0.793665i \(-0.291830\pi\)
0.608355 + 0.793665i \(0.291830\pi\)
\(728\) 13133.2 + 2782.80i 0.668609 + 0.141673i
\(729\) 0 0
\(730\) −300.000 + 444.972i −0.0152103 + 0.0225605i
\(731\) 8904.97i 0.450564i
\(732\) 0 0
\(733\) 2195.19i 0.110616i −0.998469 0.0553079i \(-0.982386\pi\)
0.998469 0.0553079i \(-0.0176140\pi\)
\(734\) 11890.2 + 8016.37i 0.597923 + 0.403120i
\(735\) 0 0
\(736\) 26752.0 4746.37i 1.33980 0.237708i
\(737\) −27016.8 −1.35031
\(738\) 0 0
\(739\) 2610.50i 0.129944i 0.997887 + 0.0649722i \(0.0206959\pi\)
−0.997887 + 0.0649722i \(0.979304\pi\)
\(740\) 1125.70 + 2782.80i 0.0559210 + 0.138240i
\(741\) 0 0
\(742\) 2100.00 3114.80i 0.103899 0.154108i
\(743\) 2851.77 0.140809 0.0704047 0.997519i \(-0.477571\pi\)
0.0704047 + 0.997519i \(0.477571\pi\)
\(744\) 0 0
\(745\) −18600.0 −0.914700
\(746\) 3846.14 5704.75i 0.188763 0.279981i
\(747\) 0 0
\(748\) −21120.0 + 8543.46i −1.03238 + 0.417620i
\(749\) 12902.1i 0.629416i
\(750\) 0 0
\(751\) −20578.0 −0.999869 −0.499935 0.866063i \(-0.666643\pi\)
−0.499935 + 0.866063i \(0.666643\pi\)
\(752\) 20712.9 20036.2i 1.00442 0.971602i
\(753\) 0 0
\(754\) −34320.0 23138.5i −1.65764 1.11758i
\(755\) 14306.1i 0.689608i
\(756\) 0 0
\(757\) 20468.7i 0.982758i 0.870946 + 0.491379i \(0.163507\pi\)
−0.870946 + 0.491379i \(0.836493\pi\)
\(758\) −12195.1 + 18088.2i −0.584361 + 0.866747i
\(759\) 0 0
\(760\) −3520.00 + 16612.3i −0.168005 + 0.792883i
\(761\) 22138.8 1.05457 0.527286 0.849688i \(-0.323210\pi\)
0.527286 + 0.849688i \(0.323210\pi\)
\(762\) 0 0
\(763\) 17205.6i 0.816362i
\(764\) −11257.0 27828.0i −0.533068 1.31778i
\(765\) 0 0
\(766\) 13728.0 + 9255.42i 0.647536 + 0.436569i
\(767\) 43527.1 2.04911
\(768\) 0 0
\(769\) −6854.00 −0.321406 −0.160703 0.987003i \(-0.551376\pi\)
−0.160703 + 0.987003i \(0.551376\pi\)
\(770\) −5628.50 3794.73i −0.263425 0.177601i
\(771\) 0 0
\(772\) −4050.00 10011.9i −0.188812 0.466755i
\(773\) 38902.3i 1.81012i 0.425288 + 0.905058i \(0.360173\pi\)
−0.425288 + 0.905058i \(0.639827\pi\)
\(774\) 0 0
\(775\) 5270.00 0.244263
\(776\) −609.754 + 2877.67i −0.0282073 + 0.133122i
\(777\) 0 0
\(778\) −3410.00 + 5057.85i −0.157139 + 0.233075i
\(779\) 44524.9i 2.04784i
\(780\) 0 0
\(781\) 0 0
\(782\) −26416.4 17809.9i −1.20799 0.814428i
\(783\) 0 0
\(784\) 11178.0 10812.8i 0.509202 0.492566i
\(785\) −10881.8 −0.494760
\(786\) 0 0
\(787\) 13171.2i 0.596571i −0.954477 0.298286i \(-0.903585\pi\)
0.954477 0.298286i \(-0.0964148\pi\)
\(788\) 10741.1 4344.97i 0.485576 0.196425i
\(789\) 0 0
\(790\) −940.000 + 1394.25i −0.0423338 + 0.0627912i
\(791\) −13508.4 −0.607210
\(792\) 0 0
\(793\) 31680.0 1.41865
\(794\) −19230.7 + 28523.7i −0.859537 + 1.27490i
\(795\) 0 0
\(796\) −9582.00 23687.3i −0.426665 1.05474i
\(797\) 13730.6i 0.610242i 0.952314 + 0.305121i \(0.0986970\pi\)
−0.952314 + 0.305121i \(0.901303\pi\)
\(798\) 0 0
\(799\) −33792.0 −1.49621
\(800\) −15150.0 + 2687.94i −0.669544 + 0.118791i
\(801\) 0 0
\(802\) −7920.00 5339.66i −0.348709 0.235100i
\(803\) 1138.42i 0.0500298i
\(804\) 0 0
\(805\) 9492.73i 0.415621i
\(806\) 5816.12 8626.69i 0.254174 0.377000i
\(807\) 0 0
\(808\) 12460.0 + 2640.17i 0.542502 + 0.114951i
\(809\) −30393.9 −1.32088 −0.660440 0.750879i \(-0.729630\pi\)
−0.660440 + 0.750879i \(0.729630\pi\)
\(810\) 0 0
\(811\) 27647.6i 1.19709i −0.801090 0.598544i \(-0.795746\pi\)
0.801090 0.598544i \(-0.204254\pi\)
\(812\) 18292.6 7399.73i 0.790573 0.319802i
\(813\) 0 0
\(814\) 5280.00 + 3559.78i 0.227351 + 0.153280i
\(815\) 12757.9 0.548332
\(816\) 0 0
\(817\) 14080.0 0.602934
\(818\) 8269.20 + 5575.10i 0.353455 + 0.238299i
\(819\) 0 0
\(820\) −17600.0 + 7119.55i −0.749535 + 0.303202i
\(821\) 26303.8i 1.11816i 0.829114 + 0.559080i \(0.188846\pi\)
−0.829114 + 0.559080i \(0.811154\pi\)
\(822\) 0 0
\(823\) 22970.0 0.972884 0.486442 0.873713i \(-0.338294\pi\)
0.486442 + 0.873713i \(0.338294\pi\)
\(824\) 2673.54 12617.5i 0.113030 0.533436i
\(825\) 0 0
\(826\) −11600.0 + 17205.6i −0.488639 + 0.724768i
\(827\) 7665.36i 0.322310i 0.986929 + 0.161155i \(0.0515220\pi\)
−0.986929 + 0.161155i \(0.948478\pi\)
\(828\) 0 0
\(829\) 17383.6i 0.728295i −0.931341 0.364147i \(-0.881360\pi\)
0.931341 0.364147i \(-0.118640\pi\)
\(830\) 9943.68 + 6704.03i 0.415844 + 0.280362i
\(831\) 0 0
\(832\) −12320.0 + 27766.2i −0.513364 + 1.15700i
\(833\) −18236.3 −0.758525
\(834\) 0 0
\(835\) 6644.91i 0.275397i
\(836\) 13508.4 + 33393.7i 0.558849 + 1.38151i
\(837\) 0 0
\(838\) 3900.00 5784.63i 0.160768 0.238457i
\(839\) 33020.5 1.35875 0.679377 0.733789i \(-0.262250\pi\)
0.679377 + 0.733789i \(0.262250\pi\)
\(840\) 0 0
\(841\) −36451.0 −1.49457
\(842\) −21482.1 + 31863.1i −0.879243 + 1.30413i
\(843\) 0 0
\(844\) −40480.0 + 16375.0i −1.65092 + 0.667831i
\(845\) 8367.39i 0.340647i
\(846\) 0 0
\(847\) −1090.00 −0.0442182
\(848\) 5909.92 + 6109.52i 0.239325 + 0.247408i
\(849\) 0 0
\(850\) 14960.0 + 10086.0i 0.603675 + 0.406998i
\(851\) 8904.97i 0.358706i
\(852\) 0 0
\(853\) 46099.1i 1.85041i −0.379463 0.925207i \(-0.623891\pi\)
0.379463 0.925207i \(-0.376109\pi\)
\(854\) −8442.75 + 12522.6i −0.338296 + 0.501774i
\(855\) 0 0
\(856\) 28560.0 + 6051.62i 1.14037 + 0.241636i
\(857\) −44352.6 −1.76786 −0.883929 0.467620i \(-0.845111\pi\)
−0.883929 + 0.467620i \(0.845111\pi\)
\(858\) 0 0
\(859\) 7712.85i 0.306355i −0.988199 0.153177i \(-0.951049\pi\)
0.988199 0.153177i \(-0.0489506\pi\)
\(860\) −2251.40 5565.61i −0.0892699 0.220681i
\(861\) 0 0
\(862\) 15840.0 + 10679.3i 0.625885 + 0.421971i
\(863\) 21163.2 0.834765 0.417383 0.908731i \(-0.362948\pi\)
0.417383 + 0.908731i \(0.362948\pi\)
\(864\) 0 0
\(865\) 9000.00 0.353768
\(866\) 18269.2 + 12317.1i 0.716873 + 0.483315i
\(867\) 0 0
\(868\) 1860.00 + 4598.04i 0.0727333 + 0.179802i
\(869\) 3567.05i 0.139245i
\(870\) 0 0
\(871\) 42240.0 1.64322
\(872\) 38086.2 + 8070.13i 1.47908 + 0.313405i
\(873\) 0 0
\(874\) −28160.0 + 41768.0i −1.08985 + 1.61650i
\(875\) 13281.6i 0.513142i
\(876\) 0 0
\(877\) 17146.3i 0.660191i 0.943947 + 0.330096i \(0.107081\pi\)
−0.943947 + 0.330096i \(0.892919\pi\)
\(878\) −21937.1 14790.0i −0.843212 0.568494i
\(879\) 0 0
\(880\) 11040.0 10679.3i 0.422907 0.409091i
\(881\) 42026.1 1.60715 0.803573 0.595206i \(-0.202929\pi\)
0.803573 + 0.595206i \(0.202929\pi\)
\(882\) 0 0
\(883\) 4865.03i 0.185415i −0.995693 0.0927073i \(-0.970448\pi\)
0.995693 0.0927073i \(-0.0295521\pi\)
\(884\) 33020.5 13357.5i 1.25634 0.508213i
\(885\) 0 0
\(886\) −10260.0 + 15218.0i −0.389042 + 0.577043i
\(887\) 11707.3 0.443170 0.221585 0.975141i \(-0.428877\pi\)
0.221585 + 0.975141i \(0.428877\pi\)
\(888\) 0 0
\(889\) 25300.0 0.954482
\(890\) 7504.67 11131.2i 0.282648 0.419235i
\(891\) 0 0
\(892\) 15990.0 + 39528.3i 0.600207 + 1.48375i
\(893\) 53429.8i 2.00220i
\(894\) 0 0
\(895\) −800.000 −0.0298783
\(896\) −7692.28 12269.6i −0.286809 0.457477i
\(897\) 0 0
\(898\) −23760.0 16019.0i −0.882942 0.595279i
\(899\) 15292.8i 0.567344i
\(900\) 0 0
\(901\) 9967.37i 0.368547i
\(902\) −22514.0 + 33393.7i −0.831080 + 1.23269i
\(903\) 0 0
\(904\) 6336.00 29902.1i 0.233111 1.10014i
\(905\) −375.233 −0.0137825
\(906\) 0 0
\(907\) 12933.9i 0.473497i −0.971571 0.236748i \(-0.923918\pi\)
0.971571 0.236748i \(-0.0760817\pi\)
\(908\) −35365.7 + 14306.1i −1.29257 + 0.522870i
\(909\) 0 0
\(910\) 8800.00 + 5932.96i 0.320568 + 0.216127i
\(911\) −20262.6 −0.736915 −0.368458 0.929645i \(-0.620114\pi\)
−0.368458 + 0.929645i \(0.620114\pi\)
\(912\) 0 0
\(913\) 25440.0 0.922170
\(914\) −28165.9 18989.5i −1.01931 0.687217i
\(915\) 0 0
\(916\) −26840.0 + 10857.3i −0.968143 + 0.391633i
\(917\) 2529.82i 0.0911037i
\(918\) 0 0
\(919\) 22746.0 0.816454 0.408227 0.912880i \(-0.366147\pi\)
0.408227 + 0.912880i \(0.366147\pi\)
\(920\) 21013.1 + 4452.49i 0.753022 + 0.159559i
\(921\) 0 0
\(922\) 14730.0 21848.1i 0.526146 0.780401i
\(923\) 0 0
\(924\) 0 0
\(925\) 5043.01i 0.179258i
\(926\) 34638.7 + 23353.4i 1.22926 + 0.828770i
\(927\) 0 0
\(928\) 7800.00 + 43963.2i 0.275913 + 1.55513i
\(929\) −13883.6 −0.490320 −0.245160 0.969483i \(-0.578840\pi\)
−0.245160 + 0.969483i \(0.578840\pi\)
\(930\) 0 0
\(931\) 28834.2i 1.01504i
\(932\) 450.280 + 1113.12i 0.0158255 + 0.0391218i
\(933\) 0 0
\(934\) 13820.0 20498.4i 0.484159 0.718123i
\(935\) −18011.2 −0.629978
\(936\) 0 0
\(937\) −27850.0 −0.970992 −0.485496 0.874239i \(-0.661361\pi\)
−0.485496 + 0.874239i \(0.661361\pi\)
\(938\) −11257.0 + 16696.8i −0.391849 + 0.581205i
\(939\) 0 0
\(940\) 21120.0 8543.46i 0.732828 0.296444i
\(941\) 36789.9i 1.27451i −0.770651 0.637257i \(-0.780069\pi\)
0.770651 0.637257i \(-0.219931\pi\)
\(942\) 0 0
\(943\) −56320.0 −1.94489
\(944\) −32645.3 33747.8i −1.12554 1.16356i
\(945\) 0 0
\(946\) −10560.0 7119.55i −0.362934 0.244690i
\(947\) 42728.7i 1.46620i −0.680118 0.733102i \(-0.738072\pi\)
0.680118 0.733102i \(-0.261928\pi\)
\(948\) 0 0
\(949\) 1779.89i 0.0608826i
\(950\) 15947.4 23653.8i 0.544634 0.807823i
\(951\) 0 0
\(952\) −3520.00 + 16612.3i −0.119836 + 0.565554i
\(953\) 24990.5 0.849447 0.424723 0.905323i \(-0.360371\pi\)
0.424723 + 0.905323i \(0.360371\pi\)
\(954\) 0 0
\(955\) 23731.8i 0.804130i
\(956\) −6754.20 16696.8i −0.228500 0.564868i
\(957\) 0 0
\(958\) −7040.00 4746.37i −0.237424 0.160071i
\(959\) −6754.20 −0.227429
\(960\) 0 0
\(961\) −25947.0 −0.870968
\(962\) −8255.13 5565.61i −0.276670 0.186531i
\(963\) 0 0
\(964\) −3486.00 8617.62i −0.116469 0.287920i
\(965\) 8538.15i 0.284822i
\(966\) 0 0
\(967\) 10550.0 0.350843 0.175421 0.984493i \(-0.443871\pi\)
0.175421 + 0.984493i \(0.443871\pi\)
\(968\) 511.255 2412.82i 0.0169756 0.0801146i
\(969\) 0 0
\(970\) −1300.00 + 1928.21i −0.0430314 + 0.0638259i
\(971\) 53012.4i 1.75206i 0.482257 + 0.876030i \(0.339817\pi\)
−0.482257 + 0.876030i \(0.660183\pi\)
\(972\) 0 0
\(973\) 2373.18i 0.0781920i
\(974\) −23241.0 15669.1i −0.764569 0.515472i
\(975\) 0 0
\(976\) −23760.0 24562.4i −0.779241 0.805558i
\(977\) −34596.5 −1.13290 −0.566448 0.824097i \(-0.691683\pi\)
−0.566448 + 0.824097i \(0.691683\pi\)
\(978\) 0 0
\(979\) 28478.2i 0.929691i
\(980\) 11397.7 4610.60i 0.371517 0.150286i
\(981\) 0 0
\(982\) −5360.00 + 7950.16i −0.174180 + 0.258350i
\(983\) 33921.1 1.10063 0.550313 0.834959i \(-0.314509\pi\)
0.550313 + 0.834959i \(0.314509\pi\)
\(984\) 0 0
\(985\) 9160.00 0.296306
\(986\) 29268.2 43411.7i 0.945324 1.40214i
\(987\) 0 0
\(988\) −21120.0 52210.0i −0.680078 1.68120i
\(989\) 17809.9i 0.572622i
\(990\) 0 0
\(991\) −8818.00 −0.282657 −0.141328 0.989963i \(-0.545137\pi\)
−0.141328 + 0.989963i \(0.545137\pi\)
\(992\) −11050.6 + 1960.61i −0.353687 + 0.0627515i
\(993\) 0 0
\(994\) 0 0
\(995\) 20200.6i 0.643621i
\(996\) 0 0
\(997\) 25096.4i 0.797203i 0.917124 + 0.398602i \(0.130504\pi\)
−0.917124 + 0.398602i \(0.869496\pi\)
\(998\) 6754.20 10018.1i 0.214229 0.317753i
\(999\) 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 72.4.d.c.37.4 yes 4
3.2 odd 2 inner 72.4.d.c.37.1 4
4.3 odd 2 288.4.d.c.145.3 4
8.3 odd 2 288.4.d.c.145.2 4
8.5 even 2 inner 72.4.d.c.37.3 yes 4
12.11 even 2 288.4.d.c.145.1 4
16.3 odd 4 2304.4.a.cc.1.4 4
16.5 even 4 2304.4.a.bx.1.1 4
16.11 odd 4 2304.4.a.cc.1.1 4
16.13 even 4 2304.4.a.bx.1.4 4
24.5 odd 2 inner 72.4.d.c.37.2 yes 4
24.11 even 2 288.4.d.c.145.4 4
48.5 odd 4 2304.4.a.bx.1.3 4
48.11 even 4 2304.4.a.cc.1.3 4
48.29 odd 4 2304.4.a.bx.1.2 4
48.35 even 4 2304.4.a.cc.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.4.d.c.37.1 4 3.2 odd 2 inner
72.4.d.c.37.2 yes 4 24.5 odd 2 inner
72.4.d.c.37.3 yes 4 8.5 even 2 inner
72.4.d.c.37.4 yes 4 1.1 even 1 trivial
288.4.d.c.145.1 4 12.11 even 2
288.4.d.c.145.2 4 8.3 odd 2
288.4.d.c.145.3 4 4.3 odd 2
288.4.d.c.145.4 4 24.11 even 2
2304.4.a.bx.1.1 4 16.5 even 4
2304.4.a.bx.1.2 4 48.29 odd 4
2304.4.a.bx.1.3 4 48.5 odd 4
2304.4.a.bx.1.4 4 16.13 even 4
2304.4.a.cc.1.1 4 16.11 odd 4
2304.4.a.cc.1.2 4 48.35 even 4
2304.4.a.cc.1.3 4 48.11 even 4
2304.4.a.cc.1.4 4 16.3 odd 4