Properties

Label 208.6.f.b.129.2
Level $208$
Weight $6$
Character 208.129
Analytic conductor $33.360$
Analytic rank $1$
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: \(1\)
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: \( 3 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+13.0000 q^{3} +51.0000i q^{5} -105.000i q^{7} -74.0000 q^{9} +O(q^{10})\) \(q+13.0000 q^{3} +51.0000i q^{5} -105.000i q^{7} -74.0000 q^{9} -120.000i q^{11} +(-598.000 + 117.000i) q^{13} +663.000i q^{15} -1101.00 q^{17} +1170.00i q^{19} -1365.00i q^{21} -1050.00 q^{23} +524.000 q^{25} -4121.00 q^{27} -4104.00 q^{29} -9624.00i q^{31} -1560.00i q^{33} +5355.00 q^{35} +8709.00i q^{37} +(-7774.00 + 1521.00i) q^{39} -9480.00i q^{41} -9995.00 q^{43} -3774.00i q^{45} +2943.00i q^{47} +5782.00 q^{49} -14313.0 q^{51} -750.000 q^{53} +6120.00 q^{55} +15210.0i q^{57} +40938.0i q^{59} -57920.0 q^{61} +7770.00i q^{63} +(-5967.00 - 30498.0i) q^{65} -22812.0i q^{67} -13650.0 q^{69} -63741.0i q^{71} +58866.0i q^{73} +6812.00 q^{75} -12600.0 q^{77} -63202.0 q^{79} -35591.0 q^{81} -55458.0i q^{83} -56151.0i q^{85} -53352.0 q^{87} -104778. i q^{89} +(12285.0 + 62790.0i) q^{91} -125112. i q^{93} -59670.0 q^{95} +160452. i q^{97} +8880.00i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 26 q^{3} - 148 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 26 q^{3} - 148 q^{9} - 1196 q^{13} - 2202 q^{17} - 2100 q^{23} + 1048 q^{25} - 8242 q^{27} - 8208 q^{29} + 10710 q^{35} - 15548 q^{39} - 19990 q^{43} + 11564 q^{49} - 28626 q^{51} - 1500 q^{53} + 12240 q^{55} - 115840 q^{61} - 11934 q^{65} - 27300 q^{69} + 13624 q^{75} - 25200 q^{77} - 126404 q^{79} - 71182 q^{81} - 106704 q^{87} + 24570 q^{91} - 119340 q^{95}+O(q^{100}) \) 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
\(3\) 13.0000 0.833950 0.416975 0.908918i \(-0.363090\pi\)
0.416975 + 0.908918i \(0.363090\pi\)
\(4\) 0 0
\(5\) 51.0000i 0.912316i 0.889899 + 0.456158i \(0.150775\pi\)
−0.889899 + 0.456158i \(0.849225\pi\)
\(6\) 0 0
\(7\) 105.000i 0.809924i −0.914334 0.404962i \(-0.867285\pi\)
0.914334 0.404962i \(-0.132715\pi\)
\(8\) 0 0
\(9\) −74.0000 −0.304527
\(10\) 0 0
\(11\) 120.000i 0.299020i −0.988760 0.149510i \(-0.952230\pi\)
0.988760 0.149510i \(-0.0477695\pi\)
\(12\) 0 0
\(13\) −598.000 + 117.000i −0.981393 + 0.192012i
\(14\) 0 0
\(15\) 663.000i 0.760826i
\(16\) 0 0
\(17\) −1101.00 −0.923985 −0.461993 0.886884i \(-0.652865\pi\)
−0.461993 + 0.886884i \(0.652865\pi\)
\(18\) 0 0
\(19\) 1170.00i 0.743536i 0.928326 + 0.371768i \(0.121248\pi\)
−0.928326 + 0.371768i \(0.878752\pi\)
\(20\) 0 0
\(21\) 1365.00i 0.675436i
\(22\) 0 0
\(23\) −1050.00 −0.413875 −0.206938 0.978354i \(-0.566350\pi\)
−0.206938 + 0.978354i \(0.566350\pi\)
\(24\) 0 0
\(25\) 524.000 0.167680
\(26\) 0 0
\(27\) −4121.00 −1.08791
\(28\) 0 0
\(29\) −4104.00 −0.906176 −0.453088 0.891466i \(-0.649678\pi\)
−0.453088 + 0.891466i \(0.649678\pi\)
\(30\) 0 0
\(31\) 9624.00i 1.79867i −0.437261 0.899335i \(-0.644051\pi\)
0.437261 0.899335i \(-0.355949\pi\)
\(32\) 0 0
\(33\) 1560.00i 0.249367i
\(34\) 0 0
\(35\) 5355.00 0.738906
\(36\) 0 0
\(37\) 8709.00i 1.04584i 0.852383 + 0.522918i \(0.175157\pi\)
−0.852383 + 0.522918i \(0.824843\pi\)
\(38\) 0 0
\(39\) −7774.00 + 1521.00i −0.818433 + 0.160128i
\(40\) 0 0
\(41\) 9480.00i 0.880742i −0.897816 0.440371i \(-0.854847\pi\)
0.897816 0.440371i \(-0.145153\pi\)
\(42\) 0 0
\(43\) −9995.00 −0.824350 −0.412175 0.911105i \(-0.635231\pi\)
−0.412175 + 0.911105i \(0.635231\pi\)
\(44\) 0 0
\(45\) 3774.00i 0.277825i
\(46\) 0 0
\(47\) 2943.00i 0.194333i 0.995268 + 0.0971663i \(0.0309779\pi\)
−0.995268 + 0.0971663i \(0.969022\pi\)
\(48\) 0 0
\(49\) 5782.00 0.344023
\(50\) 0 0
\(51\) −14313.0 −0.770558
\(52\) 0 0
\(53\) −750.000 −0.0366751 −0.0183376 0.999832i \(-0.505837\pi\)
−0.0183376 + 0.999832i \(0.505837\pi\)
\(54\) 0 0
\(55\) 6120.00 0.272800
\(56\) 0 0
\(57\) 15210.0i 0.620072i
\(58\) 0 0
\(59\) 40938.0i 1.53108i 0.643391 + 0.765538i \(0.277527\pi\)
−0.643391 + 0.765538i \(0.722473\pi\)
\(60\) 0 0
\(61\) −57920.0 −1.99298 −0.996492 0.0836839i \(-0.973331\pi\)
−0.996492 + 0.0836839i \(0.973331\pi\)
\(62\) 0 0
\(63\) 7770.00i 0.246643i
\(64\) 0 0
\(65\) −5967.00 30498.0i −0.175175 0.895340i
\(66\) 0 0
\(67\) 22812.0i 0.620835i −0.950600 0.310418i \(-0.899531\pi\)
0.950600 0.310418i \(-0.100469\pi\)
\(68\) 0 0
\(69\) −13650.0 −0.345152
\(70\) 0 0
\(71\) 63741.0i 1.50063i −0.661082 0.750314i \(-0.729902\pi\)
0.661082 0.750314i \(-0.270098\pi\)
\(72\) 0 0
\(73\) 58866.0i 1.29288i 0.762966 + 0.646439i \(0.223742\pi\)
−0.762966 + 0.646439i \(0.776258\pi\)
\(74\) 0 0
\(75\) 6812.00 0.139837
\(76\) 0 0
\(77\) −12600.0 −0.242183
\(78\) 0 0
\(79\) −63202.0 −1.13937 −0.569683 0.821865i \(-0.692934\pi\)
−0.569683 + 0.821865i \(0.692934\pi\)
\(80\) 0 0
\(81\) −35591.0 −0.602737
\(82\) 0 0
\(83\) 55458.0i 0.883627i −0.897107 0.441813i \(-0.854335\pi\)
0.897107 0.441813i \(-0.145665\pi\)
\(84\) 0 0
\(85\) 56151.0i 0.842966i
\(86\) 0 0
\(87\) −53352.0 −0.755705
\(88\) 0 0
\(89\) 104778.i 1.40215i −0.713087 0.701076i \(-0.752703\pi\)
0.713087 0.701076i \(-0.247297\pi\)
\(90\) 0 0
\(91\) 12285.0 + 62790.0i 0.155515 + 0.794853i
\(92\) 0 0
\(93\) 125112.i 1.50000i
\(94\) 0 0
\(95\) −59670.0 −0.678339
\(96\) 0 0
\(97\) 160452.i 1.73147i 0.500500 + 0.865737i \(0.333150\pi\)
−0.500500 + 0.865737i \(0.666850\pi\)
\(98\) 0 0
\(99\) 8880.00i 0.0910594i
\(100\) 0 0
\(101\) −113124. −1.10345 −0.551723 0.834027i \(-0.686030\pi\)
−0.551723 + 0.834027i \(0.686030\pi\)
\(102\) 0 0
\(103\) −25046.0 −0.232619 −0.116310 0.993213i \(-0.537106\pi\)
−0.116310 + 0.993213i \(0.537106\pi\)
\(104\) 0 0
\(105\) 69615.0 0.616211
\(106\) 0 0
\(107\) −24924.0 −0.210455 −0.105227 0.994448i \(-0.533557\pi\)
−0.105227 + 0.994448i \(0.533557\pi\)
\(108\) 0 0
\(109\) 144831.i 1.16760i 0.811896 + 0.583802i \(0.198435\pi\)
−0.811896 + 0.583802i \(0.801565\pi\)
\(110\) 0 0
\(111\) 113217.i 0.872176i
\(112\) 0 0
\(113\) 100266. 0.738682 0.369341 0.929294i \(-0.379583\pi\)
0.369341 + 0.929294i \(0.379583\pi\)
\(114\) 0 0
\(115\) 53550.0i 0.377585i
\(116\) 0 0
\(117\) 44252.0 8658.00i 0.298860 0.0584727i
\(118\) 0 0
\(119\) 115605.i 0.748358i
\(120\) 0 0
\(121\) 146651. 0.910587
\(122\) 0 0
\(123\) 123240.i 0.734495i
\(124\) 0 0
\(125\) 186099.i 1.06529i
\(126\) 0 0
\(127\) 202754. 1.11548 0.557738 0.830017i \(-0.311669\pi\)
0.557738 + 0.830017i \(0.311669\pi\)
\(128\) 0 0
\(129\) −129935. −0.687467
\(130\) 0 0
\(131\) 303855. 1.54699 0.773496 0.633801i \(-0.218506\pi\)
0.773496 + 0.633801i \(0.218506\pi\)
\(132\) 0 0
\(133\) 122850. 0.602207
\(134\) 0 0
\(135\) 210171.i 0.992518i
\(136\) 0 0
\(137\) 63738.0i 0.290133i −0.989422 0.145066i \(-0.953660\pi\)
0.989422 0.145066i \(-0.0463396\pi\)
\(138\) 0 0
\(139\) −13841.0 −0.0607618 −0.0303809 0.999538i \(-0.509672\pi\)
−0.0303809 + 0.999538i \(0.509672\pi\)
\(140\) 0 0
\(141\) 38259.0i 0.162064i
\(142\) 0 0
\(143\) 14040.0 + 71760.0i 0.0574152 + 0.293456i
\(144\) 0 0
\(145\) 209304.i 0.826718i
\(146\) 0 0
\(147\) 75166.0 0.286898
\(148\) 0 0
\(149\) 276426.i 1.02003i −0.860165 0.510015i \(-0.829640\pi\)
0.860165 0.510015i \(-0.170360\pi\)
\(150\) 0 0
\(151\) 321333.i 1.14687i 0.819252 + 0.573433i \(0.194389\pi\)
−0.819252 + 0.573433i \(0.805611\pi\)
\(152\) 0 0
\(153\) 81474.0 0.281378
\(154\) 0 0
\(155\) 490824. 1.64095
\(156\) 0 0
\(157\) 339506. 1.09925 0.549627 0.835410i \(-0.314770\pi\)
0.549627 + 0.835410i \(0.314770\pi\)
\(158\) 0 0
\(159\) −9750.00 −0.0305852
\(160\) 0 0
\(161\) 110250.i 0.335208i
\(162\) 0 0
\(163\) 395718.i 1.16659i 0.812262 + 0.583293i \(0.198236\pi\)
−0.812262 + 0.583293i \(0.801764\pi\)
\(164\) 0 0
\(165\) 79560.0 0.227502
\(166\) 0 0
\(167\) 426708.i 1.18397i −0.805950 0.591984i \(-0.798345\pi\)
0.805950 0.591984i \(-0.201655\pi\)
\(168\) 0 0
\(169\) 343915. 139932.i 0.926263 0.376878i
\(170\) 0 0
\(171\) 86580.0i 0.226427i
\(172\) 0 0
\(173\) 16026.0 0.0407108 0.0203554 0.999793i \(-0.493520\pi\)
0.0203554 + 0.999793i \(0.493520\pi\)
\(174\) 0 0
\(175\) 55020.0i 0.135808i
\(176\) 0 0
\(177\) 532194.i 1.27684i
\(178\) 0 0
\(179\) 690045. 1.60970 0.804850 0.593479i \(-0.202246\pi\)
0.804850 + 0.593479i \(0.202246\pi\)
\(180\) 0 0
\(181\) −96478.0 −0.218893 −0.109446 0.993993i \(-0.534908\pi\)
−0.109446 + 0.993993i \(0.534908\pi\)
\(182\) 0 0
\(183\) −752960. −1.66205
\(184\) 0 0
\(185\) −444159. −0.954134
\(186\) 0 0
\(187\) 132120.i 0.276290i
\(188\) 0 0
\(189\) 432705.i 0.881125i
\(190\) 0 0
\(191\) −708180. −1.40462 −0.702312 0.711869i \(-0.747849\pi\)
−0.702312 + 0.711869i \(0.747849\pi\)
\(192\) 0 0
\(193\) 347862.i 0.672224i −0.941822 0.336112i \(-0.890888\pi\)
0.941822 0.336112i \(-0.109112\pi\)
\(194\) 0 0
\(195\) −77571.0 396474.i −0.146087 0.746669i
\(196\) 0 0
\(197\) 899589.i 1.65150i 0.564036 + 0.825750i \(0.309248\pi\)
−0.564036 + 0.825750i \(0.690752\pi\)
\(198\) 0 0
\(199\) −143116. −0.256186 −0.128093 0.991762i \(-0.540886\pi\)
−0.128093 + 0.991762i \(0.540886\pi\)
\(200\) 0 0
\(201\) 296556.i 0.517746i
\(202\) 0 0
\(203\) 430920.i 0.733933i
\(204\) 0 0
\(205\) 483480. 0.803515
\(206\) 0 0
\(207\) 77700.0 0.126036
\(208\) 0 0
\(209\) 140400. 0.222332
\(210\) 0 0
\(211\) 339731. 0.525326 0.262663 0.964888i \(-0.415399\pi\)
0.262663 + 0.964888i \(0.415399\pi\)
\(212\) 0 0
\(213\) 828633.i 1.25145i
\(214\) 0 0
\(215\) 509745.i 0.752068i
\(216\) 0 0
\(217\) −1.01052e6 −1.45679
\(218\) 0 0
\(219\) 765258.i 1.07820i
\(220\) 0 0
\(221\) 658398. 128817.i 0.906792 0.177416i
\(222\) 0 0
\(223\) 623757.i 0.839950i −0.907536 0.419975i \(-0.862039\pi\)
0.907536 0.419975i \(-0.137961\pi\)
\(224\) 0 0
\(225\) −38776.0 −0.0510630
\(226\) 0 0
\(227\) 177612.i 0.228775i 0.993436 + 0.114387i \(0.0364905\pi\)
−0.993436 + 0.114387i \(0.963510\pi\)
\(228\) 0 0
\(229\) 1.18705e6i 1.49582i −0.663799 0.747911i \(-0.731057\pi\)
0.663799 0.747911i \(-0.268943\pi\)
\(230\) 0 0
\(231\) −163800. −0.201969
\(232\) 0 0
\(233\) −112317. −0.135536 −0.0677682 0.997701i \(-0.521588\pi\)
−0.0677682 + 0.997701i \(0.521588\pi\)
\(234\) 0 0
\(235\) −150093. −0.177293
\(236\) 0 0
\(237\) −821626. −0.950174
\(238\) 0 0
\(239\) 1.19805e6i 1.35669i 0.734743 + 0.678346i \(0.237303\pi\)
−0.734743 + 0.678346i \(0.762697\pi\)
\(240\) 0 0
\(241\) 1.16629e6i 1.29349i 0.762707 + 0.646744i \(0.223870\pi\)
−0.762707 + 0.646744i \(0.776130\pi\)
\(242\) 0 0
\(243\) 538720. 0.585258
\(244\) 0 0
\(245\) 294882.i 0.313858i
\(246\) 0 0
\(247\) −136890. 699660.i −0.142767 0.729701i
\(248\) 0 0
\(249\) 720954.i 0.736901i
\(250\) 0 0
\(251\) −648996. −0.650216 −0.325108 0.945677i \(-0.605401\pi\)
−0.325108 + 0.945677i \(0.605401\pi\)
\(252\) 0 0
\(253\) 126000.i 0.123757i
\(254\) 0 0
\(255\) 729963.i 0.702992i
\(256\) 0 0
\(257\) −945885. −0.893317 −0.446658 0.894705i \(-0.647386\pi\)
−0.446658 + 0.894705i \(0.647386\pi\)
\(258\) 0 0
\(259\) 914445. 0.847048
\(260\) 0 0
\(261\) 303696. 0.275955
\(262\) 0 0
\(263\) −1.01222e6 −0.902375 −0.451188 0.892429i \(-0.649000\pi\)
−0.451188 + 0.892429i \(0.649000\pi\)
\(264\) 0 0
\(265\) 38250.0i 0.0334593i
\(266\) 0 0
\(267\) 1.36211e6i 1.16933i
\(268\) 0 0
\(269\) −1.01772e6 −0.857527 −0.428763 0.903417i \(-0.641051\pi\)
−0.428763 + 0.903417i \(0.641051\pi\)
\(270\) 0 0
\(271\) 463461.i 0.383345i −0.981459 0.191673i \(-0.938609\pi\)
0.981459 0.191673i \(-0.0613912\pi\)
\(272\) 0 0
\(273\) 159705. + 816270.i 0.129692 + 0.662868i
\(274\) 0 0
\(275\) 62880.0i 0.0501396i
\(276\) 0 0
\(277\) 332528. 0.260393 0.130196 0.991488i \(-0.458439\pi\)
0.130196 + 0.991488i \(0.458439\pi\)
\(278\) 0 0
\(279\) 712176.i 0.547743i
\(280\) 0 0
\(281\) 49122.0i 0.0371116i −0.999828 0.0185558i \(-0.994093\pi\)
0.999828 0.0185558i \(-0.00590684\pi\)
\(282\) 0 0
\(283\) 1.55848e6 1.15674 0.578371 0.815774i \(-0.303689\pi\)
0.578371 + 0.815774i \(0.303689\pi\)
\(284\) 0 0
\(285\) −775710. −0.565701
\(286\) 0 0
\(287\) −995400. −0.713334
\(288\) 0 0
\(289\) −207656. −0.146251
\(290\) 0 0
\(291\) 2.08588e6i 1.44396i
\(292\) 0 0
\(293\) 218463.i 0.148665i −0.997234 0.0743325i \(-0.976317\pi\)
0.997234 0.0743325i \(-0.0236826\pi\)
\(294\) 0 0
\(295\) −2.08784e6 −1.39682
\(296\) 0 0
\(297\) 494520.i 0.325306i
\(298\) 0 0
\(299\) 627900. 122850.i 0.406174 0.0794689i
\(300\) 0 0
\(301\) 1.04948e6i 0.667661i
\(302\) 0 0
\(303\) −1.47061e6 −0.920220
\(304\) 0 0
\(305\) 2.95392e6i 1.81823i
\(306\) 0 0
\(307\) 321102.i 0.194445i −0.995263 0.0972226i \(-0.969004\pi\)
0.995263 0.0972226i \(-0.0309959\pi\)
\(308\) 0 0
\(309\) −325598. −0.193993
\(310\) 0 0
\(311\) −3.33725e6 −1.95654 −0.978269 0.207340i \(-0.933519\pi\)
−0.978269 + 0.207340i \(0.933519\pi\)
\(312\) 0 0
\(313\) 1.16568e6 0.672538 0.336269 0.941766i \(-0.390835\pi\)
0.336269 + 0.941766i \(0.390835\pi\)
\(314\) 0 0
\(315\) −396270. −0.225017
\(316\) 0 0
\(317\) 73518.0i 0.0410909i 0.999789 + 0.0205454i \(0.00654028\pi\)
−0.999789 + 0.0205454i \(0.993460\pi\)
\(318\) 0 0
\(319\) 492480.i 0.270964i
\(320\) 0 0
\(321\) −324012. −0.175509
\(322\) 0 0
\(323\) 1.28817e6i 0.687016i
\(324\) 0 0
\(325\) −313352. + 61308.0i −0.164560 + 0.0321965i
\(326\) 0 0
\(327\) 1.88280e6i 0.973723i
\(328\) 0 0
\(329\) 309015. 0.157395
\(330\) 0 0
\(331\) 632682.i 0.317406i −0.987326 0.158703i \(-0.949269\pi\)
0.987326 0.158703i \(-0.0507313\pi\)
\(332\) 0 0
\(333\) 644466.i 0.318485i
\(334\) 0 0
\(335\) 1.16341e6 0.566398
\(336\) 0 0
\(337\) −326843. −0.156771 −0.0783853 0.996923i \(-0.524976\pi\)
−0.0783853 + 0.996923i \(0.524976\pi\)
\(338\) 0 0
\(339\) 1.30346e6 0.616024
\(340\) 0 0
\(341\) −1.15488e6 −0.537837
\(342\) 0 0
\(343\) 2.37184e6i 1.08856i
\(344\) 0 0
\(345\) 696150.i 0.314887i
\(346\) 0 0
\(347\) −2.96275e6 −1.32090 −0.660452 0.750868i \(-0.729635\pi\)
−0.660452 + 0.750868i \(0.729635\pi\)
\(348\) 0 0
\(349\) 866325.i 0.380730i 0.981713 + 0.190365i \(0.0609672\pi\)
−0.981713 + 0.190365i \(0.939033\pi\)
\(350\) 0 0
\(351\) 2.46436e6 482157.i 1.06767 0.208891i
\(352\) 0 0
\(353\) 1.66291e6i 0.710282i 0.934813 + 0.355141i \(0.115567\pi\)
−0.934813 + 0.355141i \(0.884433\pi\)
\(354\) 0 0
\(355\) 3.25079e6 1.36905
\(356\) 0 0
\(357\) 1.50286e6i 0.624093i
\(358\) 0 0
\(359\) 625536.i 0.256163i −0.991764 0.128081i \(-0.959118\pi\)
0.991764 0.128081i \(-0.0408819\pi\)
\(360\) 0 0
\(361\) 1.10720e6 0.447155
\(362\) 0 0
\(363\) 1.90646e6 0.759385
\(364\) 0 0
\(365\) −3.00217e6 −1.17951
\(366\) 0 0
\(367\) −1.08327e6 −0.419829 −0.209914 0.977720i \(-0.567319\pi\)
−0.209914 + 0.977720i \(0.567319\pi\)
\(368\) 0 0
\(369\) 701520.i 0.268209i
\(370\) 0 0
\(371\) 78750.0i 0.0297041i
\(372\) 0 0
\(373\) −1.78896e6 −0.665775 −0.332888 0.942967i \(-0.608023\pi\)
−0.332888 + 0.942967i \(0.608023\pi\)
\(374\) 0 0
\(375\) 2.41929e6i 0.888401i
\(376\) 0 0
\(377\) 2.45419e6 480168.i 0.889314 0.173996i
\(378\) 0 0
\(379\) 868614.i 0.310620i −0.987866 0.155310i \(-0.950362\pi\)
0.987866 0.155310i \(-0.0496376\pi\)
\(380\) 0 0
\(381\) 2.63580e6 0.930251
\(382\) 0 0
\(383\) 1.07972e6i 0.376108i 0.982159 + 0.188054i \(0.0602179\pi\)
−0.982159 + 0.188054i \(0.939782\pi\)
\(384\) 0 0
\(385\) 642600.i 0.220947i
\(386\) 0 0
\(387\) 739630. 0.251037
\(388\) 0 0
\(389\) 1.28822e6 0.431634 0.215817 0.976434i \(-0.430759\pi\)
0.215817 + 0.976434i \(0.430759\pi\)
\(390\) 0 0
\(391\) 1.15605e6 0.382415
\(392\) 0 0
\(393\) 3.95012e6 1.29011
\(394\) 0 0
\(395\) 3.22330e6i 1.03946i
\(396\) 0 0
\(397\) 5.46909e6i 1.74156i −0.491672 0.870781i \(-0.663614\pi\)
0.491672 0.870781i \(-0.336386\pi\)
\(398\) 0 0
\(399\) 1.59705e6 0.502211
\(400\) 0 0
\(401\) 1.58612e6i 0.492577i −0.969196 0.246289i \(-0.920789\pi\)
0.969196 0.246289i \(-0.0792112\pi\)
\(402\) 0 0
\(403\) 1.12601e6 + 5.75515e6i 0.345365 + 1.76520i
\(404\) 0 0
\(405\) 1.81514e6i 0.549886i
\(406\) 0 0
\(407\) 1.04508e6 0.312726
\(408\) 0 0
\(409\) 6.44192e6i 1.90418i 0.305825 + 0.952088i \(0.401068\pi\)
−0.305825 + 0.952088i \(0.598932\pi\)
\(410\) 0 0
\(411\) 828594.i 0.241956i
\(412\) 0 0
\(413\) 4.29849e6 1.24005
\(414\) 0 0
\(415\) 2.82836e6 0.806147
\(416\) 0 0
\(417\) −179933. −0.0506723
\(418\) 0 0
\(419\) 4.30545e6 1.19807 0.599037 0.800721i \(-0.295550\pi\)
0.599037 + 0.800721i \(0.295550\pi\)
\(420\) 0 0
\(421\) 1.51346e6i 0.416164i 0.978111 + 0.208082i \(0.0667221\pi\)
−0.978111 + 0.208082i \(0.933278\pi\)
\(422\) 0 0
\(423\) 217782.i 0.0591795i
\(424\) 0 0
\(425\) −576924. −0.154934
\(426\) 0 0
\(427\) 6.08160e6i 1.61417i
\(428\) 0 0
\(429\) 182520. + 932880.i 0.0478814 + 0.244727i
\(430\) 0 0
\(431\) 1.43116e6i 0.371105i 0.982634 + 0.185552i \(0.0594074\pi\)
−0.982634 + 0.185552i \(0.940593\pi\)
\(432\) 0 0
\(433\) 429613. 0.110118 0.0550589 0.998483i \(-0.482465\pi\)
0.0550589 + 0.998483i \(0.482465\pi\)
\(434\) 0 0
\(435\) 2.72095e6i 0.689442i
\(436\) 0 0
\(437\) 1.22850e6i 0.307731i
\(438\) 0 0
\(439\) −552038. −0.136712 −0.0683562 0.997661i \(-0.521775\pi\)
−0.0683562 + 0.997661i \(0.521775\pi\)
\(440\) 0 0
\(441\) −427868. −0.104764
\(442\) 0 0
\(443\) −2.15255e6 −0.521128 −0.260564 0.965457i \(-0.583908\pi\)
−0.260564 + 0.965457i \(0.583908\pi\)
\(444\) 0 0
\(445\) 5.34368e6 1.27921
\(446\) 0 0
\(447\) 3.59354e6i 0.850655i
\(448\) 0 0
\(449\) 1.40429e6i 0.328731i 0.986400 + 0.164365i \(0.0525576\pi\)
−0.986400 + 0.164365i \(0.947442\pi\)
\(450\) 0 0
\(451\) −1.13760e6 −0.263359
\(452\) 0 0
\(453\) 4.17733e6i 0.956430i
\(454\) 0 0
\(455\) −3.20229e6 + 626535.i −0.725157 + 0.141879i
\(456\) 0 0
\(457\) 1.32818e6i 0.297485i −0.988876 0.148743i \(-0.952477\pi\)
0.988876 0.148743i \(-0.0475226\pi\)
\(458\) 0 0
\(459\) 4.53722e6 1.00521
\(460\) 0 0
\(461\) 5.89070e6i 1.29096i 0.763775 + 0.645482i \(0.223344\pi\)
−0.763775 + 0.645482i \(0.776656\pi\)
\(462\) 0 0
\(463\) 2.37139e6i 0.514104i −0.966398 0.257052i \(-0.917249\pi\)
0.966398 0.257052i \(-0.0827511\pi\)
\(464\) 0 0
\(465\) 6.38071e6 1.36847
\(466\) 0 0
\(467\) −7.17827e6 −1.52310 −0.761548 0.648108i \(-0.775560\pi\)
−0.761548 + 0.648108i \(0.775560\pi\)
\(468\) 0 0
\(469\) −2.39526e6 −0.502829
\(470\) 0 0
\(471\) 4.41358e6 0.916724
\(472\) 0 0
\(473\) 1.19940e6i 0.246497i
\(474\) 0 0
\(475\) 613080.i 0.124676i
\(476\) 0 0
\(477\) 55500.0 0.0111686
\(478\) 0 0
\(479\) 7.25193e6i 1.44416i −0.691810 0.722079i \(-0.743186\pi\)
0.691810 0.722079i \(-0.256814\pi\)
\(480\) 0 0
\(481\) −1.01895e6 5.20798e6i −0.200813 1.02638i
\(482\) 0 0
\(483\) 1.43325e6i 0.279547i
\(484\) 0 0
\(485\) −8.18305e6 −1.57965
\(486\) 0 0
\(487\) 2.53364e6i 0.484087i 0.970265 + 0.242043i \(0.0778176\pi\)
−0.970265 + 0.242043i \(0.922182\pi\)
\(488\) 0 0
\(489\) 5.14433e6i 0.972875i
\(490\) 0 0
\(491\) −8.46186e6 −1.58403 −0.792013 0.610504i \(-0.790967\pi\)
−0.792013 + 0.610504i \(0.790967\pi\)
\(492\) 0 0
\(493\) 4.51850e6 0.837293
\(494\) 0 0
\(495\) −452880. −0.0830750
\(496\) 0 0
\(497\) −6.69280e6 −1.21539
\(498\) 0 0
\(499\) 1.95383e6i 0.351265i 0.984456 + 0.175633i \(0.0561971\pi\)
−0.984456 + 0.175633i \(0.943803\pi\)
\(500\) 0 0
\(501\) 5.54720e6i 0.987370i
\(502\) 0 0
\(503\) −119778. −0.0211085 −0.0105542 0.999944i \(-0.503360\pi\)
−0.0105542 + 0.999944i \(0.503360\pi\)
\(504\) 0 0
\(505\) 5.76932e6i 1.00669i
\(506\) 0 0
\(507\) 4.47090e6 1.81912e6i 0.772457 0.314297i
\(508\) 0 0
\(509\) 1.03653e7i 1.77332i −0.462420 0.886661i \(-0.653019\pi\)
0.462420 0.886661i \(-0.346981\pi\)
\(510\) 0 0
\(511\) 6.18093e6 1.04713
\(512\) 0 0
\(513\) 4.82157e6i 0.808900i
\(514\) 0 0
\(515\) 1.27735e6i 0.212222i
\(516\) 0 0
\(517\) 353160. 0.0581092
\(518\) 0 0
\(519\) 208338. 0.0339508
\(520\) 0 0
\(521\) −1.04899e7 −1.69307 −0.846537 0.532330i \(-0.821316\pi\)
−0.846537 + 0.532330i \(0.821316\pi\)
\(522\) 0 0
\(523\) −4.42662e6 −0.707649 −0.353824 0.935312i \(-0.615119\pi\)
−0.353824 + 0.935312i \(0.615119\pi\)
\(524\) 0 0
\(525\) 715260.i 0.113257i
\(526\) 0 0
\(527\) 1.05960e7i 1.66194i
\(528\) 0 0
\(529\) −5.33384e6 −0.828707
\(530\) 0 0
\(531\) 3.02941e6i 0.466253i
\(532\) 0 0
\(533\) 1.10916e6 + 5.66904e6i 0.169113 + 0.864354i
\(534\) 0 0
\(535\) 1.27112e6i 0.192001i
\(536\) 0 0
\(537\) 8.97058e6 1.34241
\(538\) 0 0
\(539\) 693840.i 0.102870i
\(540\) 0 0
\(541\) 2.26377e6i 0.332536i 0.986081 + 0.166268i \(0.0531717\pi\)
−0.986081 + 0.166268i \(0.946828\pi\)
\(542\) 0 0
\(543\) −1.25421e6 −0.182546
\(544\) 0 0
\(545\) −7.38638e6 −1.06522
\(546\) 0 0
\(547\) −7.21090e6 −1.03044 −0.515218 0.857059i \(-0.672289\pi\)
−0.515218 + 0.857059i \(0.672289\pi\)
\(548\) 0 0
\(549\) 4.28608e6 0.606917
\(550\) 0 0
\(551\) 4.80168e6i 0.673774i
\(552\) 0 0
\(553\) 6.63621e6i 0.922799i
\(554\) 0 0
\(555\) −5.77407e6 −0.795700
\(556\) 0 0
\(557\) 273507.i 0.0373534i 0.999826 + 0.0186767i \(0.00594533\pi\)
−0.999826 + 0.0186767i \(0.994055\pi\)
\(558\) 0 0
\(559\) 5.97701e6 1.16942e6i 0.809011 0.158285i
\(560\) 0 0
\(561\) 1.71756e6i 0.230412i
\(562\) 0 0
\(563\) 959349. 0.127557 0.0637787 0.997964i \(-0.479685\pi\)
0.0637787 + 0.997964i \(0.479685\pi\)
\(564\) 0 0
\(565\) 5.11357e6i 0.673911i
\(566\) 0 0
\(567\) 3.73706e6i 0.488171i
\(568\) 0 0
\(569\) −1.19403e7 −1.54609 −0.773044 0.634352i \(-0.781267\pi\)
−0.773044 + 0.634352i \(0.781267\pi\)
\(570\) 0 0
\(571\) −7.20205e6 −0.924413 −0.462206 0.886772i \(-0.652942\pi\)
−0.462206 + 0.886772i \(0.652942\pi\)
\(572\) 0 0
\(573\) −9.20634e6 −1.17139
\(574\) 0 0
\(575\) −550200. −0.0693986
\(576\) 0 0
\(577\) 1.66990e6i 0.208810i −0.994535 0.104405i \(-0.966706\pi\)
0.994535 0.104405i \(-0.0332938\pi\)
\(578\) 0 0
\(579\) 4.52221e6i 0.560601i
\(580\) 0 0
\(581\) −5.82309e6 −0.715671
\(582\) 0 0
\(583\) 90000.0i 0.0109666i
\(584\) 0 0
\(585\) 441558. + 2.25685e6i 0.0533455 + 0.272655i
\(586\) 0 0
\(587\) 8.29913e6i 0.994117i −0.867717 0.497059i \(-0.834413\pi\)
0.867717 0.497059i \(-0.165587\pi\)
\(588\) 0 0
\(589\) 1.12601e7 1.33738
\(590\) 0 0
\(591\) 1.16947e7i 1.37727i
\(592\) 0 0
\(593\) 4.48969e6i 0.524300i −0.965027 0.262150i \(-0.915568\pi\)
0.965027 0.262150i \(-0.0844315\pi\)
\(594\) 0 0
\(595\) −5.89586e6 −0.682738
\(596\) 0 0
\(597\) −1.86051e6 −0.213646
\(598\) 0 0
\(599\) −1.38261e6 −0.157446 −0.0787232 0.996897i \(-0.525084\pi\)
−0.0787232 + 0.996897i \(0.525084\pi\)
\(600\) 0 0
\(601\) 1.04021e7 1.17472 0.587359 0.809327i \(-0.300168\pi\)
0.587359 + 0.809327i \(0.300168\pi\)
\(602\) 0 0
\(603\) 1.68809e6i 0.189061i
\(604\) 0 0
\(605\) 7.47920e6i 0.830743i
\(606\) 0 0
\(607\) 4.78668e6 0.527306 0.263653 0.964618i \(-0.415073\pi\)
0.263653 + 0.964618i \(0.415073\pi\)
\(608\) 0 0
\(609\) 5.60196e6i 0.612064i
\(610\) 0 0
\(611\) −344331. 1.75991e6i −0.0373141 0.190717i
\(612\) 0 0
\(613\) 1.04783e7i 1.12627i 0.826366 + 0.563134i \(0.190404\pi\)
−0.826366 + 0.563134i \(0.809596\pi\)
\(614\) 0 0
\(615\) 6.28524e6 0.670091
\(616\) 0 0
\(617\) 1.79106e7i 1.89407i −0.321128 0.947036i \(-0.604062\pi\)
0.321128 0.947036i \(-0.395938\pi\)
\(618\) 0 0
\(619\) 4.43222e6i 0.464938i 0.972604 + 0.232469i \(0.0746804\pi\)
−0.972604 + 0.232469i \(0.925320\pi\)
\(620\) 0 0
\(621\) 4.32705e6 0.450260
\(622\) 0 0
\(623\) −1.10017e7 −1.13564
\(624\) 0 0
\(625\) −7.85355e6 −0.804203
\(626\) 0 0
\(627\) 1.82520e6 0.185414
\(628\) 0 0
\(629\) 9.58861e6i 0.966338i
\(630\) 0 0
\(631\) 1.43291e7i 1.43267i −0.697756 0.716335i \(-0.745818\pi\)
0.697756 0.716335i \(-0.254182\pi\)
\(632\) 0 0
\(633\) 4.41650e6 0.438096
\(634\) 0 0
\(635\) 1.03405e7i 1.01767i
\(636\) 0 0
\(637\) −3.45764e6 + 676494.i −0.337622 + 0.0660565i
\(638\) 0 0
\(639\) 4.71683e6i 0.456981i
\(640\) 0 0
\(641\) 6.65869e6 0.640094 0.320047 0.947402i \(-0.396301\pi\)
0.320047 + 0.947402i \(0.396301\pi\)
\(642\) 0 0
\(643\) 1.55224e7i 1.48058i −0.672286 0.740291i \(-0.734688\pi\)
0.672286 0.740291i \(-0.265312\pi\)
\(644\) 0 0
\(645\) 6.62668e6i 0.627187i
\(646\) 0 0
\(647\) 2.44454e6 0.229581 0.114791 0.993390i \(-0.463380\pi\)
0.114791 + 0.993390i \(0.463380\pi\)
\(648\) 0 0
\(649\) 4.91256e6 0.457821
\(650\) 0 0
\(651\) −1.31368e7 −1.21489
\(652\) 0 0
\(653\) 1.16500e7 1.06916 0.534580 0.845118i \(-0.320470\pi\)
0.534580 + 0.845118i \(0.320470\pi\)
\(654\) 0 0
\(655\) 1.54966e7i 1.41135i
\(656\) 0 0
\(657\) 4.35608e6i 0.393716i
\(658\) 0 0
\(659\) −1.33185e7 −1.19465 −0.597326 0.801999i \(-0.703770\pi\)
−0.597326 + 0.801999i \(0.703770\pi\)
\(660\) 0 0
\(661\) 1.35722e7i 1.20822i −0.796900 0.604112i \(-0.793528\pi\)
0.796900 0.604112i \(-0.206472\pi\)
\(662\) 0 0
\(663\) 8.55917e6 1.67462e6i 0.756220 0.147956i
\(664\) 0 0
\(665\) 6.26535e6i 0.549403i
\(666\) 0 0
\(667\) 4.30920e6 0.375044
\(668\) 0 0
\(669\) 8.10884e6i 0.700476i
\(670\) 0 0
\(671\) 6.95040e6i 0.595941i
\(672\) 0 0
\(673\) −1.58674e7 −1.35042 −0.675209 0.737626i \(-0.735947\pi\)
−0.675209 + 0.737626i \(0.735947\pi\)
\(674\) 0 0
\(675\) −2.15940e6 −0.182421
\(676\) 0 0
\(677\) −2.24264e7 −1.88056 −0.940281 0.340398i \(-0.889438\pi\)
−0.940281 + 0.340398i \(0.889438\pi\)
\(678\) 0 0
\(679\) 1.68475e7 1.40236
\(680\) 0 0
\(681\) 2.30896e6i 0.190787i
\(682\) 0 0
\(683\) 8.11034e6i 0.665254i −0.943059 0.332627i \(-0.892065\pi\)
0.943059 0.332627i \(-0.107935\pi\)
\(684\) 0 0
\(685\) 3.25064e6 0.264693
\(686\) 0 0
\(687\) 1.54316e7i 1.24744i
\(688\) 0 0
\(689\) 448500. 87750.0i 0.0359927 0.00704205i
\(690\) 0 0
\(691\) 2.00020e7i 1.59359i −0.604246 0.796797i \(-0.706526\pi\)
0.604246 0.796797i \(-0.293474\pi\)
\(692\) 0 0
\(693\) 932400. 0.0737512
\(694\) 0 0
\(695\) 705891.i 0.0554339i
\(696\) 0 0
\(697\) 1.04375e7i 0.813793i
\(698\) 0 0
\(699\) −1.46012e6 −0.113031
\(700\) 0 0
\(701\) −2.22272e6 −0.170840 −0.0854200 0.996345i \(-0.527223\pi\)
−0.0854200 + 0.996345i \(0.527223\pi\)
\(702\) 0 0
\(703\) −1.01895e7 −0.777617
\(704\) 0 0
\(705\) −1.95121e6 −0.147853
\(706\) 0 0
\(707\) 1.18780e7i 0.893708i
\(708\) 0 0
\(709\) 2.03634e7i 1.52137i 0.649122 + 0.760684i \(0.275136\pi\)
−0.649122 + 0.760684i \(0.724864\pi\)
\(710\) 0 0
\(711\) 4.67695e6 0.346967
\(712\) 0 0
\(713\) 1.01052e7i 0.744425i
\(714\) 0 0
\(715\) −3.65976e6 + 716040.i −0.267724 + 0.0523808i
\(716\) 0 0
\(717\) 1.55747e7i 1.13141i
\(718\) 0 0
\(719\) 1.98255e7 1.43022 0.715108 0.699014i \(-0.246377\pi\)
0.715108 + 0.699014i \(0.246377\pi\)
\(720\) 0 0
\(721\) 2.62983e6i 0.188404i
\(722\) 0 0
\(723\) 1.51617e7i 1.07870i
\(724\) 0 0
\(725\) −2.15050e6 −0.151948
\(726\) 0 0
\(727\) 9.24667e6 0.648857 0.324429 0.945910i \(-0.394828\pi\)
0.324429 + 0.945910i \(0.394828\pi\)
\(728\) 0 0
\(729\) 1.56520e7 1.09081
\(730\) 0 0
\(731\) 1.10045e7 0.761687
\(732\) 0 0
\(733\) 1.48114e7i 1.01821i −0.860704 0.509105i \(-0.829976\pi\)
0.860704 0.509105i \(-0.170024\pi\)
\(734\) 0 0
\(735\) 3.83347e6i 0.261742i
\(736\) 0 0
\(737\) −2.73744e6 −0.185642
\(738\) 0 0
\(739\) 5.67210e6i 0.382061i 0.981584 + 0.191031i \(0.0611830\pi\)
−0.981584 + 0.191031i \(0.938817\pi\)
\(740\) 0 0
\(741\) −1.77957e6 9.09558e6i −0.119061 0.608534i
\(742\) 0 0
\(743\) 2.75704e7i 1.83219i 0.400960 + 0.916095i \(0.368677\pi\)
−0.400960 + 0.916095i \(0.631323\pi\)
\(744\) 0 0
\(745\) 1.40977e7 0.930590
\(746\) 0 0
\(747\) 4.10389e6i 0.269088i
\(748\) 0 0
\(749\) 2.61702e6i 0.170452i
\(750\) 0 0
\(751\) 4.09636e6 0.265032 0.132516 0.991181i \(-0.457694\pi\)
0.132516 + 0.991181i \(0.457694\pi\)
\(752\) 0 0
\(753\) −8.43695e6 −0.542248
\(754\) 0 0
\(755\) −1.63880e7 −1.04630
\(756\) 0 0
\(757\) 1.09396e7 0.693844 0.346922 0.937894i \(-0.387227\pi\)
0.346922 + 0.937894i \(0.387227\pi\)
\(758\) 0 0
\(759\) 1.63800e6i 0.103207i
\(760\) 0 0
\(761\) 1.36940e6i 0.0857172i 0.999081 + 0.0428586i \(0.0136465\pi\)
−0.999081 + 0.0428586i \(0.986354\pi\)
\(762\) 0 0
\(763\) 1.52073e7 0.945670
\(764\) 0 0
\(765\) 4.15517e6i 0.256706i
\(766\) 0 0
\(767\) −4.78975e6 2.44809e7i −0.293984 1.50259i
\(768\) 0 0
\(769\) 1.08375e7i 0.660867i −0.943829 0.330433i \(-0.892805\pi\)
0.943829 0.330433i \(-0.107195\pi\)
\(770\) 0 0
\(771\) −1.22965e7 −0.744982
\(772\) 0 0
\(773\) 2.05445e7i 1.23665i −0.785922 0.618325i \(-0.787812\pi\)
0.785922 0.618325i \(-0.212188\pi\)
\(774\) 0 0
\(775\) 5.04298e6i 0.301601i
\(776\) 0 0
\(777\) 1.18878e7 0.706396
\(778\) 0 0
\(779\) 1.10916e7 0.654863
\(780\) 0 0
\(781\) −7.64892e6 −0.448717
\(782\) 0 0
\(783\) 1.69126e7 0.985838
\(784\) 0 0
\(785\) 1.73148e7i 1.00287i
\(786\) 0 0
\(787\) 1.34637e7i 0.774869i 0.921897 + 0.387435i \(0.126639\pi\)
−0.921897 + 0.387435i \(0.873361\pi\)
\(788\) 0 0
\(789\) −1.31589e7 −0.752536