Properties

Label 208.6.f.a.129.1
Level $208$
Weight $6$
Character 208.129
Analytic conductor $33.360$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [208,6,Mod(129,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.129");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 208.f (of order \(2\), degree \(1\), 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: \(33.3598345211\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 208.129
Dual form 208.6.f.a.129.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{3} -68.0000i q^{5} +82.0000i q^{7} -227.000 q^{9} +O(q^{10})\) \(q-4.00000 q^{3} -68.0000i q^{5} +82.0000i q^{7} -227.000 q^{9} +390.000i q^{11} +(507.000 + 338.000i) q^{13} +272.000i q^{15} +1738.00 q^{17} -1074.00i q^{19} -328.000i q^{21} -2104.00 q^{23} -1499.00 q^{25} +1880.00 q^{27} -1690.00 q^{29} -1430.00i q^{31} -1560.00i q^{33} +5576.00 q^{35} -8852.00i q^{37} +(-2028.00 - 1352.00i) q^{39} -6760.00i q^{41} +16916.0 q^{43} +15436.0i q^{45} -25158.0i q^{47} +10083.0 q^{49} -6952.00 q^{51} +38214.0 q^{53} +26520.0 q^{55} +4296.00i q^{57} +21286.0i q^{59} -5458.00 q^{61} -18614.0i q^{63} +(22984.0 - 34476.0i) q^{65} +44542.0i q^{67} +8416.00 q^{69} -17790.0i q^{71} -31064.0i q^{73} +5996.00 q^{75} -31980.0 q^{77} +45360.0 q^{79} +47641.0 q^{81} -124546. i q^{83} -118184. i q^{85} +6760.00 q^{87} +18744.0i q^{89} +(-27716.0 + 41574.0i) q^{91} +5720.00i q^{93} -73032.0 q^{95} +121488. i q^{97} -88530.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{3} - 454 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{3} - 454 q^{9} + 1014 q^{13} + 3476 q^{17} - 4208 q^{23} - 2998 q^{25} + 3760 q^{27} - 3380 q^{29} + 11152 q^{35} - 4056 q^{39} + 33832 q^{43} + 20166 q^{49} - 13904 q^{51} + 76428 q^{53} + 53040 q^{55} - 10916 q^{61} + 45968 q^{65} + 16832 q^{69} + 11992 q^{75} - 63960 q^{77} + 90720 q^{79} + 95282 q^{81} + 13520 q^{87} - 55432 q^{91} - 146064 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −4.00000 −0.256600 −0.128300 0.991735i \(-0.540952\pi\)
−0.128300 + 0.991735i \(0.540952\pi\)
\(4\) 0 0
\(5\) 68.0000i 1.21642i −0.793776 0.608210i \(-0.791888\pi\)
0.793776 0.608210i \(-0.208112\pi\)
\(6\) 0 0
\(7\) 82.0000i 0.632512i 0.948674 + 0.316256i \(0.102426\pi\)
−0.948674 + 0.316256i \(0.897574\pi\)
\(8\) 0 0
\(9\) −227.000 −0.934156
\(10\) 0 0
\(11\) 390.000i 0.971813i 0.874011 + 0.485907i \(0.161511\pi\)
−0.874011 + 0.485907i \(0.838489\pi\)
\(12\) 0 0
\(13\) 507.000 + 338.000i 0.832050 + 0.554700i
\(14\) 0 0
\(15\) 272.000i 0.312134i
\(16\) 0 0
\(17\) 1738.00 1.45857 0.729285 0.684210i \(-0.239853\pi\)
0.729285 + 0.684210i \(0.239853\pi\)
\(18\) 0 0
\(19\) 1074.00i 0.682528i −0.939968 0.341264i \(-0.889145\pi\)
0.939968 0.341264i \(-0.110855\pi\)
\(20\) 0 0
\(21\) 328.000i 0.162303i
\(22\) 0 0
\(23\) −2104.00 −0.829328 −0.414664 0.909975i \(-0.636101\pi\)
−0.414664 + 0.909975i \(0.636101\pi\)
\(24\) 0 0
\(25\) −1499.00 −0.479680
\(26\) 0 0
\(27\) 1880.00 0.496305
\(28\) 0 0
\(29\) −1690.00 −0.373157 −0.186579 0.982440i \(-0.559740\pi\)
−0.186579 + 0.982440i \(0.559740\pi\)
\(30\) 0 0
\(31\) 1430.00i 0.267259i −0.991031 0.133629i \(-0.957337\pi\)
0.991031 0.133629i \(-0.0426632\pi\)
\(32\) 0 0
\(33\) 1560.00i 0.249367i
\(34\) 0 0
\(35\) 5576.00 0.769401
\(36\) 0 0
\(37\) 8852.00i 1.06301i −0.847055 0.531505i \(-0.821627\pi\)
0.847055 0.531505i \(-0.178373\pi\)
\(38\) 0 0
\(39\) −2028.00 1352.00i −0.213504 0.142336i
\(40\) 0 0
\(41\) 6760.00i 0.628040i −0.949416 0.314020i \(-0.898324\pi\)
0.949416 0.314020i \(-0.101676\pi\)
\(42\) 0 0
\(43\) 16916.0 1.39517 0.697584 0.716503i \(-0.254258\pi\)
0.697584 + 0.716503i \(0.254258\pi\)
\(44\) 0 0
\(45\) 15436.0i 1.13633i
\(46\) 0 0
\(47\) 25158.0i 1.66124i −0.556842 0.830618i \(-0.687987\pi\)
0.556842 0.830618i \(-0.312013\pi\)
\(48\) 0 0
\(49\) 10083.0 0.599929
\(50\) 0 0
\(51\) −6952.00 −0.374269
\(52\) 0 0
\(53\) 38214.0 1.86867 0.934335 0.356395i \(-0.115994\pi\)
0.934335 + 0.356395i \(0.115994\pi\)
\(54\) 0 0
\(55\) 26520.0 1.18213
\(56\) 0 0
\(57\) 4296.00i 0.175137i
\(58\) 0 0
\(59\) 21286.0i 0.796093i 0.917365 + 0.398047i \(0.130312\pi\)
−0.917365 + 0.398047i \(0.869688\pi\)
\(60\) 0 0
\(61\) −5458.00 −0.187806 −0.0939029 0.995581i \(-0.529934\pi\)
−0.0939029 + 0.995581i \(0.529934\pi\)
\(62\) 0 0
\(63\) 18614.0i 0.590865i
\(64\) 0 0
\(65\) 22984.0 34476.0i 0.674749 1.01212i
\(66\) 0 0
\(67\) 44542.0i 1.21222i 0.795379 + 0.606112i \(0.207272\pi\)
−0.795379 + 0.606112i \(0.792728\pi\)
\(68\) 0 0
\(69\) 8416.00 0.212806
\(70\) 0 0
\(71\) 17790.0i 0.418823i −0.977828 0.209411i \(-0.932845\pi\)
0.977828 0.209411i \(-0.0671547\pi\)
\(72\) 0 0
\(73\) 31064.0i 0.682260i −0.940016 0.341130i \(-0.889190\pi\)
0.940016 0.341130i \(-0.110810\pi\)
\(74\) 0 0
\(75\) 5996.00 0.123086
\(76\) 0 0
\(77\) −31980.0 −0.614684
\(78\) 0 0
\(79\) 45360.0 0.817721 0.408861 0.912597i \(-0.365926\pi\)
0.408861 + 0.912597i \(0.365926\pi\)
\(80\) 0 0
\(81\) 47641.0 0.806805
\(82\) 0 0
\(83\) 124546.i 1.98442i −0.124559 0.992212i \(-0.539752\pi\)
0.124559 0.992212i \(-0.460248\pi\)
\(84\) 0 0
\(85\) 118184.i 1.77424i
\(86\) 0 0
\(87\) 6760.00 0.0957522
\(88\) 0 0
\(89\) 18744.0i 0.250834i 0.992104 + 0.125417i \(0.0400270\pi\)
−0.992104 + 0.125417i \(0.959973\pi\)
\(90\) 0 0
\(91\) −27716.0 + 41574.0i −0.350855 + 0.526282i
\(92\) 0 0
\(93\) 5720.00i 0.0685786i
\(94\) 0 0
\(95\) −73032.0 −0.830241
\(96\) 0 0
\(97\) 121488.i 1.31100i 0.755193 + 0.655502i \(0.227543\pi\)
−0.755193 + 0.655502i \(0.772457\pi\)
\(98\) 0 0
\(99\) 88530.0i 0.907826i
\(100\) 0 0
\(101\) −14218.0 −0.138687 −0.0693434 0.997593i \(-0.522090\pi\)
−0.0693434 + 0.997593i \(0.522090\pi\)
\(102\) 0 0
\(103\) 62776.0 0.583043 0.291521 0.956564i \(-0.405839\pi\)
0.291521 + 0.956564i \(0.405839\pi\)
\(104\) 0 0
\(105\) −22304.0 −0.197428
\(106\) 0 0
\(107\) 79252.0 0.669192 0.334596 0.942362i \(-0.391400\pi\)
0.334596 + 0.942362i \(0.391400\pi\)
\(108\) 0 0
\(109\) 218084.i 1.75816i 0.476677 + 0.879078i \(0.341841\pi\)
−0.476677 + 0.879078i \(0.658159\pi\)
\(110\) 0 0
\(111\) 35408.0i 0.272768i
\(112\) 0 0
\(113\) 44234.0 0.325882 0.162941 0.986636i \(-0.447902\pi\)
0.162941 + 0.986636i \(0.447902\pi\)
\(114\) 0 0
\(115\) 143072.i 1.00881i
\(116\) 0 0
\(117\) −115089. 76726.0i −0.777265 0.518177i
\(118\) 0 0
\(119\) 142516.i 0.922563i
\(120\) 0 0
\(121\) 8951.00 0.0555787
\(122\) 0 0
\(123\) 27040.0i 0.161155i
\(124\) 0 0
\(125\) 110568.i 0.632928i
\(126\) 0 0
\(127\) 310432. 1.70788 0.853940 0.520372i \(-0.174207\pi\)
0.853940 + 0.520372i \(0.174207\pi\)
\(128\) 0 0
\(129\) −67664.0 −0.358000
\(130\) 0 0
\(131\) −310372. −1.58017 −0.790086 0.612996i \(-0.789964\pi\)
−0.790086 + 0.612996i \(0.789964\pi\)
\(132\) 0 0
\(133\) 88068.0 0.431707
\(134\) 0 0
\(135\) 127840.i 0.603716i
\(136\) 0 0
\(137\) 281032.i 1.27925i −0.768688 0.639623i \(-0.779090\pi\)
0.768688 0.639623i \(-0.220910\pi\)
\(138\) 0 0
\(139\) −363820. −1.59716 −0.798582 0.601886i \(-0.794416\pi\)
−0.798582 + 0.601886i \(0.794416\pi\)
\(140\) 0 0
\(141\) 100632.i 0.426273i
\(142\) 0 0
\(143\) −131820. + 197730.i −0.539065 + 0.808598i
\(144\) 0 0
\(145\) 114920.i 0.453916i
\(146\) 0 0
\(147\) −40332.0 −0.153942
\(148\) 0 0
\(149\) 274204.i 1.01183i 0.862583 + 0.505916i \(0.168845\pi\)
−0.862583 + 0.505916i \(0.831155\pi\)
\(150\) 0 0
\(151\) 344030.i 1.22787i −0.789355 0.613937i \(-0.789585\pi\)
0.789355 0.613937i \(-0.210415\pi\)
\(152\) 0 0
\(153\) −394526. −1.36253
\(154\) 0 0
\(155\) −97240.0 −0.325099
\(156\) 0 0
\(157\) 20518.0 0.0664333 0.0332167 0.999448i \(-0.489425\pi\)
0.0332167 + 0.999448i \(0.489425\pi\)
\(158\) 0 0
\(159\) −152856. −0.479501
\(160\) 0 0
\(161\) 172528.i 0.524560i
\(162\) 0 0
\(163\) 36626.0i 0.107974i −0.998542 0.0539872i \(-0.982807\pi\)
0.998542 0.0539872i \(-0.0171930\pi\)
\(164\) 0 0
\(165\) −106080. −0.303336
\(166\) 0 0
\(167\) 269442.i 0.747608i 0.927508 + 0.373804i \(0.121947\pi\)
−0.927508 + 0.373804i \(0.878053\pi\)
\(168\) 0 0
\(169\) 142805. + 342732.i 0.384615 + 0.923077i
\(170\) 0 0
\(171\) 243798.i 0.637588i
\(172\) 0 0
\(173\) 282654. 0.718026 0.359013 0.933333i \(-0.383113\pi\)
0.359013 + 0.933333i \(0.383113\pi\)
\(174\) 0 0
\(175\) 122918.i 0.303403i
\(176\) 0 0
\(177\) 85144.0i 0.204278i
\(178\) 0 0
\(179\) −333780. −0.778624 −0.389312 0.921106i \(-0.627287\pi\)
−0.389312 + 0.921106i \(0.627287\pi\)
\(180\) 0 0
\(181\) −459938. −1.04352 −0.521762 0.853091i \(-0.674725\pi\)
−0.521762 + 0.853091i \(0.674725\pi\)
\(182\) 0 0
\(183\) 21832.0 0.0481910
\(184\) 0 0
\(185\) −601936. −1.29307
\(186\) 0 0
\(187\) 677820.i 1.41746i
\(188\) 0 0
\(189\) 154160.i 0.313919i
\(190\) 0 0
\(191\) 917088. 1.81898 0.909489 0.415727i \(-0.136473\pi\)
0.909489 + 0.415727i \(0.136473\pi\)
\(192\) 0 0
\(193\) 639056.i 1.23494i 0.786595 + 0.617470i \(0.211842\pi\)
−0.786595 + 0.617470i \(0.788158\pi\)
\(194\) 0 0
\(195\) −91936.0 + 137904.i −0.173141 + 0.259711i
\(196\) 0 0
\(197\) 358292.i 0.657766i −0.944371 0.328883i \(-0.893328\pi\)
0.944371 0.328883i \(-0.106672\pi\)
\(198\) 0 0
\(199\) −370440. −0.663109 −0.331555 0.943436i \(-0.607573\pi\)
−0.331555 + 0.943436i \(0.607573\pi\)
\(200\) 0 0
\(201\) 178168.i 0.311057i
\(202\) 0 0
\(203\) 138580.i 0.236026i
\(204\) 0 0
\(205\) −459680. −0.763961
\(206\) 0 0
\(207\) 477608. 0.774722
\(208\) 0 0
\(209\) 418860. 0.663290
\(210\) 0 0
\(211\) 177228. 0.274048 0.137024 0.990568i \(-0.456246\pi\)
0.137024 + 0.990568i \(0.456246\pi\)
\(212\) 0 0
\(213\) 71160.0i 0.107470i
\(214\) 0 0
\(215\) 1.15029e6i 1.69711i
\(216\) 0 0
\(217\) 117260. 0.169044
\(218\) 0 0
\(219\) 124256.i 0.175068i
\(220\) 0 0
\(221\) 881166. + 587444.i 1.21360 + 0.809069i
\(222\) 0 0
\(223\) 1.11297e6i 1.49872i −0.662164 0.749359i \(-0.730362\pi\)
0.662164 0.749359i \(-0.269638\pi\)
\(224\) 0 0
\(225\) 340273. 0.448096
\(226\) 0 0
\(227\) 1.39158e6i 1.79244i 0.443612 + 0.896219i \(0.353697\pi\)
−0.443612 + 0.896219i \(0.646303\pi\)
\(228\) 0 0
\(229\) 909796.i 1.14645i −0.819398 0.573225i \(-0.805692\pi\)
0.819398 0.573225i \(-0.194308\pi\)
\(230\) 0 0
\(231\) 127920. 0.157728
\(232\) 0 0
\(233\) 266154. 0.321176 0.160588 0.987022i \(-0.448661\pi\)
0.160588 + 0.987022i \(0.448661\pi\)
\(234\) 0 0
\(235\) −1.71074e6 −2.02076
\(236\) 0 0
\(237\) −181440. −0.209827
\(238\) 0 0
\(239\) 254614.i 0.288328i −0.989554 0.144164i \(-0.953951\pi\)
0.989554 0.144164i \(-0.0460494\pi\)
\(240\) 0 0
\(241\) 313600.i 0.347803i 0.984763 + 0.173902i \(0.0556375\pi\)
−0.984763 + 0.173902i \(0.944363\pi\)
\(242\) 0 0
\(243\) −647404. −0.703331
\(244\) 0 0
\(245\) 685644.i 0.729766i
\(246\) 0 0
\(247\) 363012. 544518.i 0.378598 0.567897i
\(248\) 0 0
\(249\) 498184.i 0.509204i
\(250\) 0 0
\(251\) 1.07127e6 1.07328 0.536641 0.843811i \(-0.319693\pi\)
0.536641 + 0.843811i \(0.319693\pi\)
\(252\) 0 0
\(253\) 820560.i 0.805952i
\(254\) 0 0
\(255\) 472736.i 0.455269i
\(256\) 0 0
\(257\) −188382. −0.177913 −0.0889563 0.996036i \(-0.528353\pi\)
−0.0889563 + 0.996036i \(0.528353\pi\)
\(258\) 0 0
\(259\) 725864. 0.672366
\(260\) 0 0
\(261\) 383630. 0.348587
\(262\) 0 0
\(263\) 1.48678e6 1.32543 0.662714 0.748873i \(-0.269404\pi\)
0.662714 + 0.748873i \(0.269404\pi\)
\(264\) 0 0
\(265\) 2.59855e6i 2.27309i
\(266\) 0 0
\(267\) 74976.0i 0.0643642i
\(268\) 0 0
\(269\) 743990. 0.626883 0.313441 0.949608i \(-0.398518\pi\)
0.313441 + 0.949608i \(0.398518\pi\)
\(270\) 0 0
\(271\) 455590.i 0.376835i −0.982089 0.188417i \(-0.939664\pi\)
0.982089 0.188417i \(-0.0603358\pi\)
\(272\) 0 0
\(273\) 110864. 166296.i 0.0900293 0.135044i
\(274\) 0 0
\(275\) 584610.i 0.466159i
\(276\) 0 0
\(277\) 460198. 0.360367 0.180184 0.983633i \(-0.442331\pi\)
0.180184 + 0.983633i \(0.442331\pi\)
\(278\) 0 0
\(279\) 324610.i 0.249661i
\(280\) 0 0
\(281\) 49240.0i 0.0372008i 0.999827 + 0.0186004i \(0.00592103\pi\)
−0.999827 + 0.0186004i \(0.994079\pi\)
\(282\) 0 0
\(283\) 544196. 0.403914 0.201957 0.979394i \(-0.435270\pi\)
0.201957 + 0.979394i \(0.435270\pi\)
\(284\) 0 0
\(285\) 292128. 0.213040
\(286\) 0 0
\(287\) 554320. 0.397243
\(288\) 0 0
\(289\) 1.60079e6 1.12743
\(290\) 0 0
\(291\) 485952.i 0.336404i
\(292\) 0 0
\(293\) 1.02504e6i 0.697542i 0.937208 + 0.348771i \(0.113401\pi\)
−0.937208 + 0.348771i \(0.886599\pi\)
\(294\) 0 0
\(295\) 1.44745e6 0.968385
\(296\) 0 0
\(297\) 733200.i 0.482316i
\(298\) 0 0
\(299\) −1.06673e6 711152.i −0.690042 0.460028i
\(300\) 0 0
\(301\) 1.38711e6i 0.882461i
\(302\) 0 0
\(303\) 56872.0 0.0355870
\(304\) 0 0
\(305\) 371144.i 0.228451i
\(306\) 0 0
\(307\) 1.57766e6i 0.955362i 0.878533 + 0.477681i \(0.158523\pi\)
−0.878533 + 0.477681i \(0.841477\pi\)
\(308\) 0 0
\(309\) −251104. −0.149609
\(310\) 0 0
\(311\) 330088. 0.193521 0.0967606 0.995308i \(-0.469152\pi\)
0.0967606 + 0.995308i \(0.469152\pi\)
\(312\) 0 0
\(313\) −1.78677e6 −1.03088 −0.515438 0.856927i \(-0.672371\pi\)
−0.515438 + 0.856927i \(0.672371\pi\)
\(314\) 0 0
\(315\) −1.26575e6 −0.718741
\(316\) 0 0
\(317\) 182148.i 0.101807i 0.998704 + 0.0509033i \(0.0162100\pi\)
−0.998704 + 0.0509033i \(0.983790\pi\)
\(318\) 0 0
\(319\) 659100.i 0.362639i
\(320\) 0 0
\(321\) −317008. −0.171715
\(322\) 0 0
\(323\) 1.86661e6i 0.995515i
\(324\) 0 0
\(325\) −759993. 506662.i −0.399118 0.266079i
\(326\) 0 0
\(327\) 872336.i 0.451143i
\(328\) 0 0
\(329\) 2.06296e6 1.05075
\(330\) 0 0
\(331\) 216230.i 0.108479i 0.998528 + 0.0542395i \(0.0172735\pi\)
−0.998528 + 0.0542395i \(0.982727\pi\)
\(332\) 0 0
\(333\) 2.00940e6i 0.993017i
\(334\) 0 0
\(335\) 3.02886e6 1.47457
\(336\) 0 0
\(337\) −2.05314e6 −0.984791 −0.492396 0.870371i \(-0.663879\pi\)
−0.492396 + 0.870371i \(0.663879\pi\)
\(338\) 0 0
\(339\) −176936. −0.0836213
\(340\) 0 0
\(341\) 557700. 0.259726
\(342\) 0 0
\(343\) 2.20498e6i 1.01197i
\(344\) 0 0
\(345\) 572288.i 0.258861i
\(346\) 0 0
\(347\) −4.28819e6 −1.91183 −0.955917 0.293637i \(-0.905134\pi\)
−0.955917 + 0.293637i \(0.905134\pi\)
\(348\) 0 0
\(349\) 3.55152e6i 1.56081i −0.625274 0.780405i \(-0.715013\pi\)
0.625274 0.780405i \(-0.284987\pi\)
\(350\) 0 0
\(351\) 953160. + 635440.i 0.412951 + 0.275300i
\(352\) 0 0
\(353\) 2.08678e6i 0.891335i −0.895199 0.445667i \(-0.852966\pi\)
0.895199 0.445667i \(-0.147034\pi\)
\(354\) 0 0
\(355\) −1.20972e6 −0.509465
\(356\) 0 0
\(357\) 570064.i 0.236730i
\(358\) 0 0
\(359\) 500654.i 0.205023i −0.994732 0.102511i \(-0.967312\pi\)
0.994732 0.102511i \(-0.0326878\pi\)
\(360\) 0 0
\(361\) 1.32262e6 0.534156
\(362\) 0 0
\(363\) −35804.0 −0.0142615
\(364\) 0 0
\(365\) −2.11235e6 −0.829916
\(366\) 0 0
\(367\) 1.28027e6 0.496178 0.248089 0.968737i \(-0.420198\pi\)
0.248089 + 0.968737i \(0.420198\pi\)
\(368\) 0 0
\(369\) 1.53452e6i 0.586687i
\(370\) 0 0
\(371\) 3.13355e6i 1.18196i
\(372\) 0 0
\(373\) −405666. −0.150972 −0.0754860 0.997147i \(-0.524051\pi\)
−0.0754860 + 0.997147i \(0.524051\pi\)
\(374\) 0 0
\(375\) 442272.i 0.162409i
\(376\) 0 0
\(377\) −856830. 571220.i −0.310485 0.206990i
\(378\) 0 0
\(379\) 4.66217e6i 1.66721i 0.552363 + 0.833604i \(0.313726\pi\)
−0.552363 + 0.833604i \(0.686274\pi\)
\(380\) 0 0
\(381\) −1.24173e6 −0.438242
\(382\) 0 0
\(383\) 4.35473e6i 1.51692i −0.651717 0.758462i \(-0.725951\pi\)
0.651717 0.758462i \(-0.274049\pi\)
\(384\) 0 0
\(385\) 2.17464e6i 0.747714i
\(386\) 0 0
\(387\) −3.83993e6 −1.30331
\(388\) 0 0
\(389\) 786990. 0.263691 0.131845 0.991270i \(-0.457910\pi\)
0.131845 + 0.991270i \(0.457910\pi\)
\(390\) 0 0
\(391\) −3.65675e6 −1.20963
\(392\) 0 0
\(393\) 1.24149e6 0.405472
\(394\) 0 0
\(395\) 3.08448e6i 0.994693i
\(396\) 0 0
\(397\) 3.97023e6i 1.26427i 0.774859 + 0.632134i \(0.217821\pi\)
−0.774859 + 0.632134i \(0.782179\pi\)
\(398\) 0 0
\(399\) −352272. −0.110776
\(400\) 0 0
\(401\) 344640.i 0.107030i 0.998567 + 0.0535149i \(0.0170425\pi\)
−0.998567 + 0.0535149i \(0.982958\pi\)
\(402\) 0 0
\(403\) 483340. 725010.i 0.148248 0.222373i
\(404\) 0 0
\(405\) 3.23959e6i 0.981414i
\(406\) 0 0
\(407\) 3.45228e6 1.03305
\(408\) 0 0
\(409\) 2.55466e6i 0.755137i 0.925982 + 0.377568i \(0.123240\pi\)
−0.925982 + 0.377568i \(0.876760\pi\)
\(410\) 0 0
\(411\) 1.12413e6i 0.328255i
\(412\) 0 0
\(413\) −1.74545e6 −0.503539
\(414\) 0 0
\(415\) −8.46913e6 −2.41390
\(416\) 0 0
\(417\) 1.45528e6 0.409833
\(418\) 0 0
\(419\) 2.51894e6 0.700943 0.350472 0.936573i \(-0.386021\pi\)
0.350472 + 0.936573i \(0.386021\pi\)
\(420\) 0 0
\(421\) 4.83670e6i 1.32998i 0.746854 + 0.664988i \(0.231563\pi\)
−0.746854 + 0.664988i \(0.768437\pi\)
\(422\) 0 0
\(423\) 5.71087e6i 1.55185i
\(424\) 0 0
\(425\) −2.60526e6 −0.699647
\(426\) 0 0
\(427\) 447556.i 0.118789i
\(428\) 0 0
\(429\) 527280. 790920.i 0.138324 0.207486i
\(430\) 0 0
\(431\) 219110.i 0.0568158i −0.999596 0.0284079i \(-0.990956\pi\)
0.999596 0.0284079i \(-0.00904373\pi\)
\(432\) 0 0
\(433\) −3.03477e6 −0.777867 −0.388934 0.921266i \(-0.627156\pi\)
−0.388934 + 0.921266i \(0.627156\pi\)
\(434\) 0 0
\(435\) 459680.i 0.116475i
\(436\) 0 0
\(437\) 2.25970e6i 0.566039i
\(438\) 0 0
\(439\) −4.16940e6 −1.03255 −0.516276 0.856422i \(-0.672682\pi\)
−0.516276 + 0.856422i \(0.672682\pi\)
\(440\) 0 0
\(441\) −2.28884e6 −0.560427
\(442\) 0 0
\(443\) 6.30548e6 1.52654 0.763271 0.646079i \(-0.223592\pi\)
0.763271 + 0.646079i \(0.223592\pi\)
\(444\) 0 0
\(445\) 1.27459e6 0.305120
\(446\) 0 0
\(447\) 1.09682e6i 0.259636i
\(448\) 0 0
\(449\) 7.41586e6i 1.73598i −0.496579 0.867991i \(-0.665411\pi\)
0.496579 0.867991i \(-0.334589\pi\)
\(450\) 0 0
\(451\) 2.63640e6 0.610337
\(452\) 0 0
\(453\) 1.37612e6i 0.315073i
\(454\) 0 0
\(455\) 2.82703e6 + 1.88469e6i 0.640180 + 0.426787i
\(456\) 0 0
\(457\) 4.71529e6i 1.05613i 0.849204 + 0.528065i \(0.177082\pi\)
−0.849204 + 0.528065i \(0.822918\pi\)
\(458\) 0 0
\(459\) 3.26744e6 0.723896
\(460\) 0 0
\(461\) 3.34566e6i 0.733212i −0.930376 0.366606i \(-0.880520\pi\)
0.930376 0.366606i \(-0.119480\pi\)
\(462\) 0 0
\(463\) 1.65791e6i 0.359426i 0.983719 + 0.179713i \(0.0575169\pi\)
−0.983719 + 0.179713i \(0.942483\pi\)
\(464\) 0 0
\(465\) 388960. 0.0834205
\(466\) 0 0
\(467\) −823668. −0.174767 −0.0873836 0.996175i \(-0.527851\pi\)
−0.0873836 + 0.996175i \(0.527851\pi\)
\(468\) 0 0
\(469\) −3.65244e6 −0.766746
\(470\) 0 0
\(471\) −82072.0 −0.0170468
\(472\) 0 0
\(473\) 6.59724e6i 1.35584i
\(474\) 0 0
\(475\) 1.60993e6i 0.327395i
\(476\) 0 0
\(477\) −8.67458e6 −1.74563
\(478\) 0 0
\(479\) 3.59011e6i 0.714938i 0.933925 + 0.357469i \(0.116360\pi\)
−0.933925 + 0.357469i \(0.883640\pi\)
\(480\) 0 0
\(481\) 2.99198e6 4.48796e6i 0.589652 0.884477i
\(482\) 0 0
\(483\) 690112.i 0.134602i
\(484\) 0 0
\(485\) 8.26118e6 1.59473
\(486\) 0 0
\(487\) 9.67688e6i 1.84890i −0.381306 0.924449i \(-0.624526\pi\)
0.381306 0.924449i \(-0.375474\pi\)
\(488\) 0 0
\(489\) 146504.i 0.0277062i
\(490\) 0 0
\(491\) −3.45633e6 −0.647011 −0.323506 0.946226i \(-0.604861\pi\)
−0.323506 + 0.946226i \(0.604861\pi\)
\(492\) 0 0
\(493\) −2.93722e6 −0.544276
\(494\) 0 0
\(495\) −6.02004e6 −1.10430
\(496\) 0 0
\(497\) 1.45878e6 0.264910
\(498\) 0 0
\(499\) 2.09109e6i 0.375942i 0.982175 + 0.187971i \(0.0601911\pi\)
−0.982175 + 0.187971i \(0.939809\pi\)
\(500\) 0 0
\(501\) 1.07777e6i 0.191836i
\(502\) 0 0
\(503\) −5.58626e6 −0.984468 −0.492234 0.870463i \(-0.663820\pi\)
−0.492234 + 0.870463i \(0.663820\pi\)
\(504\) 0 0
\(505\) 966824.i 0.168702i
\(506\) 0 0
\(507\) −571220. 1.37093e6i −0.0986924 0.236862i
\(508\) 0 0
\(509\) 4.15504e6i 0.710854i −0.934704 0.355427i \(-0.884335\pi\)
0.934704 0.355427i \(-0.115665\pi\)
\(510\) 0 0
\(511\) 2.54725e6 0.431538
\(512\) 0 0
\(513\) 2.01912e6i 0.338742i
\(514\) 0 0
\(515\) 4.26877e6i 0.709226i
\(516\) 0 0
\(517\) 9.81162e6 1.61441
\(518\) 0 0
\(519\) −1.13062e6 −0.184245
\(520\) 0 0
\(521\) −9.84416e6 −1.58886 −0.794428 0.607359i \(-0.792229\pi\)
−0.794428 + 0.607359i \(0.792229\pi\)
\(522\) 0 0
\(523\) −481324. −0.0769455 −0.0384728 0.999260i \(-0.512249\pi\)
−0.0384728 + 0.999260i \(0.512249\pi\)
\(524\) 0 0
\(525\) 491672.i 0.0778533i
\(526\) 0 0
\(527\) 2.48534e6i 0.389816i
\(528\) 0 0
\(529\) −2.00953e6 −0.312216
\(530\) 0 0
\(531\) 4.83192e6i 0.743676i
\(532\) 0 0
\(533\) 2.28488e6 3.42732e6i 0.348374 0.522561i
\(534\) 0 0
\(535\) 5.38914e6i 0.814019i
\(536\) 0 0
\(537\) 1.33512e6 0.199795
\(538\) 0 0
\(539\) 3.93237e6i 0.583019i
\(540\) 0 0
\(541\) 263980.i 0.0387773i −0.999812 0.0193887i \(-0.993828\pi\)
0.999812 0.0193887i \(-0.00617199\pi\)
\(542\) 0 0
\(543\) 1.83975e6 0.267769
\(544\) 0 0
\(545\) 1.48297e7 2.13866
\(546\) 0 0
\(547\) −2.80023e6 −0.400152 −0.200076 0.979780i \(-0.564119\pi\)
−0.200076 + 0.979780i \(0.564119\pi\)
\(548\) 0 0
\(549\) 1.23897e6 0.175440
\(550\) 0 0
\(551\) 1.81506e6i 0.254690i
\(552\) 0 0
\(553\) 3.71952e6i 0.517219i
\(554\) 0 0
\(555\) 2.40774e6 0.331801
\(556\) 0 0
\(557\) 2.70983e6i 0.370087i 0.982730 + 0.185043i \(0.0592426\pi\)
−0.982730 + 0.185043i \(0.940757\pi\)
\(558\) 0 0
\(559\) 8.57641e6 + 5.71761e6i 1.16085 + 0.773900i
\(560\) 0 0
\(561\) 2.71128e6i 0.363720i
\(562\) 0 0
\(563\) 1.14870e7 1.52733 0.763667 0.645610i \(-0.223397\pi\)
0.763667 + 0.645610i \(0.223397\pi\)
\(564\) 0 0
\(565\) 3.00791e6i 0.396409i
\(566\) 0 0
\(567\) 3.90656e6i 0.510314i
\(568\) 0 0
\(569\) 7.85065e6 1.01654 0.508271 0.861197i \(-0.330285\pi\)
0.508271 + 0.861197i \(0.330285\pi\)
\(570\) 0 0
\(571\) 6.34071e6 0.813856 0.406928 0.913460i \(-0.366600\pi\)
0.406928 + 0.913460i \(0.366600\pi\)
\(572\) 0 0
\(573\) −3.66835e6 −0.466750
\(574\) 0 0
\(575\) 3.15390e6 0.397812
\(576\) 0 0
\(577\) 7.20867e6i 0.901396i −0.892676 0.450698i \(-0.851175\pi\)
0.892676 0.450698i \(-0.148825\pi\)
\(578\) 0 0
\(579\) 2.55622e6i 0.316886i
\(580\) 0 0
\(581\) 1.02128e7 1.25517
\(582\) 0 0
\(583\) 1.49035e7i 1.81600i
\(584\) 0 0
\(585\) −5.21737e6 + 7.82605e6i −0.630321 + 0.945482i
\(586\) 0 0
\(587\) 2.48138e6i 0.297234i 0.988895 + 0.148617i \(0.0474821\pi\)
−0.988895 + 0.148617i \(0.952518\pi\)
\(588\) 0 0
\(589\) −1.53582e6 −0.182411
\(590\) 0 0
\(591\) 1.43317e6i 0.168783i
\(592\) 0 0
\(593\) 1.38811e7i 1.62102i −0.585728 0.810508i \(-0.699191\pi\)
0.585728 0.810508i \(-0.300809\pi\)
\(594\) 0 0
\(595\) 9.69109e6 1.12223
\(596\) 0 0
\(597\) 1.48176e6 0.170154
\(598\) 0 0
\(599\) −3.85356e6 −0.438829 −0.219414 0.975632i \(-0.570415\pi\)
−0.219414 + 0.975632i \(0.570415\pi\)
\(600\) 0 0
\(601\) 1.32728e6 0.149892 0.0749458 0.997188i \(-0.476122\pi\)
0.0749458 + 0.997188i \(0.476122\pi\)
\(602\) 0 0
\(603\) 1.01110e7i 1.13241i
\(604\) 0 0
\(605\) 608668.i 0.0676071i
\(606\) 0 0
\(607\) −9.73197e6 −1.07208 −0.536042 0.844191i \(-0.680081\pi\)
−0.536042 + 0.844191i \(0.680081\pi\)
\(608\) 0 0
\(609\) 554320.i 0.0605644i
\(610\) 0 0
\(611\) 8.50340e6 1.27551e7i 0.921488 1.38223i
\(612\) 0 0
\(613\) 1.40465e7i 1.50979i 0.655846 + 0.754894i \(0.272312\pi\)
−0.655846 + 0.754894i \(0.727688\pi\)
\(614\) 0 0
\(615\) 1.83872e6 0.196032
\(616\) 0 0
\(617\) 3.72561e6i 0.393989i 0.980405 + 0.196995i \(0.0631181\pi\)
−0.980405 + 0.196995i \(0.936882\pi\)
\(618\) 0 0
\(619\) 8.96911e6i 0.940855i −0.882439 0.470428i \(-0.844100\pi\)
0.882439 0.470428i \(-0.155900\pi\)
\(620\) 0 0
\(621\) −3.95552e6 −0.411599
\(622\) 0 0
\(623\) −1.53701e6 −0.158656
\(624\) 0 0
\(625\) −1.22030e7 −1.24959
\(626\) 0 0
\(627\) −1.67544e6 −0.170200
\(628\) 0 0
\(629\) 1.53848e7i 1.55047i
\(630\) 0 0
\(631\) 1.72189e7i 1.72160i −0.508943 0.860800i \(-0.669964\pi\)
0.508943 0.860800i \(-0.330036\pi\)
\(632\) 0 0
\(633\) −708912. −0.0703207
\(634\) 0 0
\(635\) 2.11094e7i 2.07750i
\(636\) 0 0
\(637\) 5.11208e6 + 3.40805e6i 0.499171 + 0.332781i
\(638\) 0 0
\(639\) 4.03833e6i 0.391246i
\(640\) 0 0
\(641\) 8.51692e6 0.818724 0.409362 0.912372i \(-0.365751\pi\)
0.409362 + 0.912372i \(0.365751\pi\)
\(642\) 0 0
\(643\) 8.14145e6i 0.776559i 0.921542 + 0.388280i \(0.126931\pi\)
−0.921542 + 0.388280i \(0.873069\pi\)
\(644\) 0 0
\(645\) 4.60115e6i 0.435479i
\(646\) 0 0
\(647\) 2.39391e6 0.224826 0.112413 0.993662i \(-0.464142\pi\)
0.112413 + 0.993662i \(0.464142\pi\)
\(648\) 0 0
\(649\) −8.30154e6 −0.773654
\(650\) 0 0
\(651\) −469040. −0.0433768
\(652\) 0 0
\(653\) 1.17900e7 1.08201 0.541003 0.841020i \(-0.318045\pi\)
0.541003 + 0.841020i \(0.318045\pi\)
\(654\) 0 0
\(655\) 2.11053e7i 1.92215i
\(656\) 0 0
\(657\) 7.05153e6i 0.637338i
\(658\) 0 0
\(659\) −4.84562e6 −0.434646 −0.217323 0.976100i \(-0.569733\pi\)
−0.217323 + 0.976100i \(0.569733\pi\)
\(660\) 0 0
\(661\) 1.14461e7i 1.01895i −0.860485 0.509476i \(-0.829839\pi\)
0.860485 0.509476i \(-0.170161\pi\)
\(662\) 0 0
\(663\) −3.52466e6 2.34978e6i −0.311411 0.207607i
\(664\) 0 0
\(665\) 5.98862e6i 0.525137i
\(666\) 0 0
\(667\) 3.55576e6 0.309470
\(668\) 0 0
\(669\) 4.45186e6i 0.384571i
\(670\) 0 0
\(671\) 2.12862e6i 0.182512i
\(672\) 0 0
\(673\) −5.34001e6 −0.454469 −0.227234 0.973840i \(-0.572968\pi\)
−0.227234 + 0.973840i \(0.572968\pi\)
\(674\) 0 0
\(675\) −2.81812e6 −0.238067
\(676\) 0 0
\(677\) −7.06132e6 −0.592126 −0.296063 0.955168i \(-0.595674\pi\)
−0.296063 + 0.955168i \(0.595674\pi\)
\(678\) 0 0
\(679\) −9.96202e6 −0.829226
\(680\) 0 0
\(681\) 5.56633e6i 0.459940i
\(682\) 0 0
\(683\) 3.50035e6i 0.287117i −0.989642 0.143559i \(-0.954145\pi\)
0.989642 0.143559i \(-0.0458546\pi\)
\(684\) 0 0
\(685\) −1.91102e7 −1.55610
\(686\) 0 0
\(687\) 3.63918e6i 0.294179i
\(688\) 0 0
\(689\) 1.93745e7 + 1.29163e7i 1.55483 + 1.03655i
\(690\) 0 0
\(691\) 302510.i 0.0241015i 0.999927 + 0.0120508i \(0.00383597\pi\)
−0.999927 + 0.0120508i \(0.996164\pi\)
\(692\) 0 0
\(693\) 7.25946e6 0.574211
\(694\) 0 0
\(695\) 2.47398e7i 1.94282i
\(696\) 0 0
\(697\) 1.17489e7i 0.916040i
\(698\) 0 0
\(699\) −1.06462e6 −0.0824138
\(700\) 0 0
\(701\) −1.03212e7 −0.793294 −0.396647 0.917971i \(-0.629826\pi\)
−0.396647 + 0.917971i \(0.629826\pi\)
\(702\) 0 0
\(703\) −9.50705e6 −0.725533
\(704\) 0 0
\(705\) 6.84298e6 0.518528
\(706\) 0 0
\(707\) 1.16588e6i 0.0877211i
\(708\) 0 0
\(709\) 5.27524e6i 0.394118i −0.980392 0.197059i \(-0.936861\pi\)
0.980392 0.197059i \(-0.0631391\pi\)
\(710\) 0 0
\(711\) −1.02967e7 −0.763880
\(712\) 0 0
\(713\) 3.00872e6i 0.221645i
\(714\) 0 0
\(715\) 1.34456e7 + 8.96376e6i 0.983595 + 0.655730i
\(716\) 0 0
\(717\) 1.01846e6i 0.0739851i
\(718\) 0 0
\(719\) 5.02216e6 0.362300 0.181150 0.983455i \(-0.442018\pi\)
0.181150 + 0.983455i \(0.442018\pi\)
\(720\) 0 0
\(721\) 5.14763e6i 0.368782i
\(722\) 0 0
\(723\) 1.25440e6i 0.0892463i
\(724\) 0 0
\(725\) 2.53331e6 0.178996
\(726\) 0 0
\(727\) −8.80441e6 −0.617823 −0.308912 0.951091i \(-0.599965\pi\)
−0.308912 + 0.951091i \(0.599965\pi\)
\(728\) 0 0
\(729\) −8.98715e6 −0.626330
\(730\) 0 0
\(731\) 2.94000e7 2.03495
\(732\) 0 0
\(733\) 3.05052e6i 0.209708i −0.994488 0.104854i \(-0.966563\pi\)
0.994488 0.104854i \(-0.0334375\pi\)
\(734\) 0 0
\(735\) 2.74258e6i 0.187258i
\(736\) 0 0
\(737\) −1.73714e7 −1.17806
\(738\) 0 0
\(739\) 7.62605e6i 0.513675i 0.966455 + 0.256837i \(0.0826805\pi\)
−0.966455 + 0.256837i \(0.917320\pi\)
\(740\) 0 0
\(741\) −1.45205e6 + 2.17807e6i −0.0971484 + 0.145723i
\(742\) 0 0
\(743\) 2.18236e7i 1.45029i −0.688595 0.725146i \(-0.741772\pi\)
0.688595 0.725146i \(-0.258228\pi\)
\(744\) 0 0
\(745\) 1.86459e7 1.23081
\(746\) 0 0
\(747\) 2.82719e7i 1.85376i
\(748\) 0 0
\(749\) 6.49866e6i 0.423272i
\(750\) 0 0
\(751\) −1.69030e7 −1.09361 −0.546807 0.837259i \(-0.684157\pi\)
−0.546807 + 0.837259i \(0.684157\pi\)
\(752\) 0 0
\(753\) −4.28507e6 −0.275404
\(754\) 0 0
\(755\) −2.33940e7 −1.49361
\(756\) 0 0
\(757\) −8.90252e6 −0.564642 −0.282321 0.959320i \(-0.591104\pi\)
−0.282321 + 0.959320i \(0.591104\pi\)
\(758\) 0 0
\(759\) 3.28224e6i 0.206807i
\(760\) 0 0
\(761\) 6.98052e6i 0.436944i 0.975843 + 0.218472i \(0.0701073\pi\)
−0.975843 + 0.218472i \(0.929893\pi\)
\(762\) 0 0
\(763\) −1.78829e7 −1.11206
\(764\) 0 0
\(765\) 2.68278e7i 1.65741i
\(766\) 0 0
\(767\) −7.19467e6 + 1.07920e7i −0.441593 + 0.662390i
\(768\) 0 0
\(769\) 2.67789e7i 1.63296i 0.577372 + 0.816481i \(0.304078\pi\)
−0.577372 + 0.816481i \(0.695922\pi\)
\(770\) 0 0
\(771\) 753528. 0.0456524
\(772\) 0 0
\(773\) 710244.i 0.0427522i −0.999772 0.0213761i \(-0.993195\pi\)
0.999772 0.0213761i \(-0.00680475\pi\)
\(774\) 0 0
\(775\) 2.14357e6i 0.128199i
\(776\) 0 0
\(777\) −2.90346e6 −0.172529
\(778\) 0 0
\(779\) −7.26024e6 −0.428654
\(780\) 0 0
\(781\) 6.93810e6 0.407017
\(782\) 0 0
\(783\) −3.17720e6 −0.185200
\(784\) 0 0
\(785\) 1.39522e6i 0.0808109i
\(786\) 0 0
\(787\) 5.18538e6i 0.298431i −0.988805 0.149215i \(-0.952325\pi\)
0.988805 0.149215i \(-0.0476748\pi\)
\(788\) 0 0