Properties

Label 384.6.f.f.191.5
Level $384$
Weight $6$
Character 384.191
Analytic conductor $61.587$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,6,Mod(191,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.191");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 384.f (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: \(61.5873868082\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 5192x^{16} + 8441320x^{12} + 4098006217x^{8} + 8949568544x^{4} + 8386816 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{87}\cdot 3^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 191.5
Root \(-0.860472 - 0.860472i\) of defining polynomial
Character \(\chi\) \(=\) 384.191
Dual form 384.6.f.f.191.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-10.7340 - 11.3040i) q^{3} -4.25229 q^{5} +187.352i q^{7} +(-12.5625 + 242.675i) q^{9} +O(q^{10})\) \(q+(-10.7340 - 11.3040i) q^{3} -4.25229 q^{5} +187.352i q^{7} +(-12.5625 + 242.675i) q^{9} +712.864i q^{11} +804.242i q^{13} +(45.6440 + 48.0680i) q^{15} -309.664i q^{17} +2467.73 q^{19} +(2117.83 - 2011.03i) q^{21} -3069.31 q^{23} -3106.92 q^{25} +(2878.05 - 2462.87i) q^{27} -6325.12 q^{29} -9005.91i q^{31} +(8058.25 - 7651.89i) q^{33} -796.673i q^{35} +7350.14i q^{37} +(9091.18 - 8632.73i) q^{39} +10389.8i q^{41} +3834.27 q^{43} +(53.4195 - 1031.92i) q^{45} +11195.2 q^{47} -18293.7 q^{49} +(-3500.45 + 3323.93i) q^{51} -7390.12 q^{53} -3031.30i q^{55} +(-26488.6 - 27895.3i) q^{57} -9680.57i q^{59} +15417.8i q^{61} +(-45465.6 - 2353.62i) q^{63} -3419.87i q^{65} +34082.9 q^{67} +(32946.0 + 34695.6i) q^{69} -17459.6 q^{71} +23208.7 q^{73} +(33349.7 + 35120.7i) q^{75} -133556. q^{77} +41087.3i q^{79} +(-58733.4 - 6097.23i) q^{81} -2074.09i q^{83} +1316.78i q^{85} +(67893.8 + 71499.4i) q^{87} -136300. i q^{89} -150676. q^{91} +(-101803. + 96669.4i) q^{93} -10493.5 q^{95} -94090.8 q^{97} +(-172994. - 8955.39i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 44 q^{9} + 3568 q^{15} - 6112 q^{23} + 15228 q^{25} + 7592 q^{33} + 2800 q^{39} - 26112 q^{47} - 81044 q^{49} - 89296 q^{57} + 14816 q^{63} + 72224 q^{71} - 61256 q^{73} + 89588 q^{81} + 145648 q^{87} - 385504 q^{95} + 92808 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) −10.7340 11.3040i −0.688586 0.725154i
\(4\) 0 0
\(5\) −4.25229 −0.0760672 −0.0380336 0.999276i \(-0.512109\pi\)
−0.0380336 + 0.999276i \(0.512109\pi\)
\(6\) 0 0
\(7\) 187.352i 1.44515i 0.691293 + 0.722575i \(0.257041\pi\)
−0.691293 + 0.722575i \(0.742959\pi\)
\(8\) 0 0
\(9\) −12.5625 + 242.675i −0.0516977 + 0.998663i
\(10\) 0 0
\(11\) 712.864i 1.77634i 0.459518 + 0.888168i \(0.348022\pi\)
−0.459518 + 0.888168i \(0.651978\pi\)
\(12\) 0 0
\(13\) 804.242i 1.31986i 0.751327 + 0.659930i \(0.229414\pi\)
−0.751327 + 0.659930i \(0.770586\pi\)
\(14\) 0 0
\(15\) 45.6440 + 48.0680i 0.0523788 + 0.0551605i
\(16\) 0 0
\(17\) 309.664i 0.259877i −0.991522 0.129939i \(-0.958522\pi\)
0.991522 0.129939i \(-0.0414780\pi\)
\(18\) 0 0
\(19\) 2467.73 1.56824 0.784121 0.620608i \(-0.213114\pi\)
0.784121 + 0.620608i \(0.213114\pi\)
\(20\) 0 0
\(21\) 2117.83 2011.03i 1.04796 0.995110i
\(22\) 0 0
\(23\) −3069.31 −1.20982 −0.604911 0.796293i \(-0.706791\pi\)
−0.604911 + 0.796293i \(0.706791\pi\)
\(24\) 0 0
\(25\) −3106.92 −0.994214
\(26\) 0 0
\(27\) 2878.05 2462.87i 0.759783 0.650177i
\(28\) 0 0
\(29\) −6325.12 −1.39661 −0.698303 0.715803i \(-0.746061\pi\)
−0.698303 + 0.715803i \(0.746061\pi\)
\(30\) 0 0
\(31\) 9005.91i 1.68315i −0.540138 0.841576i \(-0.681628\pi\)
0.540138 0.841576i \(-0.318372\pi\)
\(32\) 0 0
\(33\) 8058.25 7651.89i 1.28812 1.22316i
\(34\) 0 0
\(35\) 796.673i 0.109928i
\(36\) 0 0
\(37\) 7350.14i 0.882656i 0.897346 + 0.441328i \(0.145492\pi\)
−0.897346 + 0.441328i \(0.854508\pi\)
\(38\) 0 0
\(39\) 9091.18 8632.73i 0.957103 0.908838i
\(40\) 0 0
\(41\) 10389.8i 0.965265i 0.875823 + 0.482632i \(0.160319\pi\)
−0.875823 + 0.482632i \(0.839681\pi\)
\(42\) 0 0
\(43\) 3834.27 0.316236 0.158118 0.987420i \(-0.449457\pi\)
0.158118 + 0.987420i \(0.449457\pi\)
\(44\) 0 0
\(45\) 53.4195 1031.92i 0.00393250 0.0759655i
\(46\) 0 0
\(47\) 11195.2 0.739244 0.369622 0.929182i \(-0.379487\pi\)
0.369622 + 0.929182i \(0.379487\pi\)
\(48\) 0 0
\(49\) −18293.7 −1.08846
\(50\) 0 0
\(51\) −3500.45 + 3323.93i −0.188451 + 0.178948i
\(52\) 0 0
\(53\) −7390.12 −0.361378 −0.180689 0.983540i \(-0.557833\pi\)
−0.180689 + 0.983540i \(0.557833\pi\)
\(54\) 0 0
\(55\) 3031.30i 0.135121i
\(56\) 0 0
\(57\) −26488.6 27895.3i −1.07987 1.13722i
\(58\) 0 0
\(59\) 9680.57i 0.362052i −0.983478 0.181026i \(-0.942058\pi\)
0.983478 0.181026i \(-0.0579418\pi\)
\(60\) 0 0
\(61\) 15417.8i 0.530514i 0.964178 + 0.265257i \(0.0854568\pi\)
−0.964178 + 0.265257i \(0.914543\pi\)
\(62\) 0 0
\(63\) −45465.6 2353.62i −1.44322 0.0747110i
\(64\) 0 0
\(65\) 3419.87i 0.100398i
\(66\) 0 0
\(67\) 34082.9 0.927577 0.463788 0.885946i \(-0.346490\pi\)
0.463788 + 0.885946i \(0.346490\pi\)
\(68\) 0 0
\(69\) 32946.0 + 34695.6i 0.833067 + 0.877308i
\(70\) 0 0
\(71\) −17459.6 −0.411044 −0.205522 0.978652i \(-0.565889\pi\)
−0.205522 + 0.978652i \(0.565889\pi\)
\(72\) 0 0
\(73\) 23208.7 0.509735 0.254867 0.966976i \(-0.417968\pi\)
0.254867 + 0.966976i \(0.417968\pi\)
\(74\) 0 0
\(75\) 33349.7 + 35120.7i 0.684602 + 0.720958i
\(76\) 0 0
\(77\) −133556. −2.56707
\(78\) 0 0
\(79\) 41087.3i 0.740696i 0.928893 + 0.370348i \(0.120762\pi\)
−0.928893 + 0.370348i \(0.879238\pi\)
\(80\) 0 0
\(81\) −58733.4 6097.23i −0.994655 0.103257i
\(82\) 0 0
\(83\) 2074.09i 0.0330470i −0.999863 0.0165235i \(-0.994740\pi\)
0.999863 0.0165235i \(-0.00525983\pi\)
\(84\) 0 0
\(85\) 1316.78i 0.0197681i
\(86\) 0 0
\(87\) 67893.8 + 71499.4i 0.961683 + 1.01275i
\(88\) 0 0
\(89\) 136300.i 1.82398i −0.410214 0.911989i \(-0.634546\pi\)
0.410214 0.911989i \(-0.365454\pi\)
\(90\) 0 0
\(91\) −150676. −1.90740
\(92\) 0 0
\(93\) −101803. + 96669.4i −1.22055 + 1.15900i
\(94\) 0 0
\(95\) −10493.5 −0.119292
\(96\) 0 0
\(97\) −94090.8 −1.01536 −0.507678 0.861547i \(-0.669496\pi\)
−0.507678 + 0.861547i \(0.669496\pi\)
\(98\) 0 0
\(99\) −172994. 8955.39i −1.77396 0.0918326i
\(100\) 0 0
\(101\) 83946.4 0.818840 0.409420 0.912346i \(-0.365731\pi\)
0.409420 + 0.912346i \(0.365731\pi\)
\(102\) 0 0
\(103\) 101368.i 0.941472i 0.882274 + 0.470736i \(0.156012\pi\)
−0.882274 + 0.470736i \(0.843988\pi\)
\(104\) 0 0
\(105\) −9005.63 + 8551.49i −0.0797151 + 0.0756952i
\(106\) 0 0
\(107\) 10903.6i 0.0920680i −0.998940 0.0460340i \(-0.985342\pi\)
0.998940 0.0460340i \(-0.0146583\pi\)
\(108\) 0 0
\(109\) 128645.i 1.03711i 0.855044 + 0.518556i \(0.173530\pi\)
−0.855044 + 0.518556i \(0.826470\pi\)
\(110\) 0 0
\(111\) 83086.3 78896.4i 0.640062 0.607785i
\(112\) 0 0
\(113\) 163476.i 1.20436i −0.798360 0.602181i \(-0.794299\pi\)
0.798360 0.602181i \(-0.205701\pi\)
\(114\) 0 0
\(115\) 13051.6 0.0920278
\(116\) 0 0
\(117\) −195169. 10103.3i −1.31810 0.0682338i
\(118\) 0 0
\(119\) 58016.0 0.375561
\(120\) 0 0
\(121\) −347125. −2.15537
\(122\) 0 0
\(123\) 117446. 111524.i 0.699966 0.664668i
\(124\) 0 0
\(125\) 26499.9 0.151694
\(126\) 0 0
\(127\) 199225.i 1.09606i −0.836458 0.548031i \(-0.815378\pi\)
0.836458 0.548031i \(-0.184622\pi\)
\(128\) 0 0
\(129\) −41157.0 43342.7i −0.217756 0.229320i
\(130\) 0 0
\(131\) 50397.5i 0.256585i −0.991736 0.128292i \(-0.959050\pi\)
0.991736 0.128292i \(-0.0409496\pi\)
\(132\) 0 0
\(133\) 462333.i 2.26634i
\(134\) 0 0
\(135\) −12238.3 + 10472.8i −0.0577946 + 0.0494571i
\(136\) 0 0
\(137\) 43239.6i 0.196825i −0.995146 0.0984125i \(-0.968624\pi\)
0.995146 0.0984125i \(-0.0313764\pi\)
\(138\) 0 0
\(139\) 82492.8 0.362142 0.181071 0.983470i \(-0.442044\pi\)
0.181071 + 0.983470i \(0.442044\pi\)
\(140\) 0 0
\(141\) −120169. 126551.i −0.509033 0.536066i
\(142\) 0 0
\(143\) −573315. −2.34452
\(144\) 0 0
\(145\) 26896.2 0.106236
\(146\) 0 0
\(147\) 196365. + 206793.i 0.749497 + 0.789299i
\(148\) 0 0
\(149\) −444609. −1.64064 −0.820319 0.571906i \(-0.806204\pi\)
−0.820319 + 0.571906i \(0.806204\pi\)
\(150\) 0 0
\(151\) 198371.i 0.708004i 0.935245 + 0.354002i \(0.115179\pi\)
−0.935245 + 0.354002i \(0.884821\pi\)
\(152\) 0 0
\(153\) 75147.6 + 3890.16i 0.259530 + 0.0134351i
\(154\) 0 0
\(155\) 38295.7i 0.128033i
\(156\) 0 0
\(157\) 82535.0i 0.267232i 0.991033 + 0.133616i \(0.0426590\pi\)
−0.991033 + 0.133616i \(0.957341\pi\)
\(158\) 0 0
\(159\) 79325.6 + 83538.2i 0.248840 + 0.262055i
\(160\) 0 0
\(161\) 575042.i 1.74837i
\(162\) 0 0
\(163\) 328393. 0.968111 0.484056 0.875037i \(-0.339163\pi\)
0.484056 + 0.875037i \(0.339163\pi\)
\(164\) 0 0
\(165\) −34266.0 + 32538.0i −0.0979835 + 0.0930424i
\(166\) 0 0
\(167\) 650470. 1.80483 0.902414 0.430870i \(-0.141793\pi\)
0.902414 + 0.430870i \(0.141793\pi\)
\(168\) 0 0
\(169\) −275512. −0.742033
\(170\) 0 0
\(171\) −31000.9 + 598856.i −0.0810745 + 1.56614i
\(172\) 0 0
\(173\) 86336.1 0.219320 0.109660 0.993969i \(-0.465024\pi\)
0.109660 + 0.993969i \(0.465024\pi\)
\(174\) 0 0
\(175\) 582087.i 1.43679i
\(176\) 0 0
\(177\) −109430. + 103911.i −0.262544 + 0.249304i
\(178\) 0 0
\(179\) 417917.i 0.974894i −0.873153 0.487447i \(-0.837928\pi\)
0.873153 0.487447i \(-0.162072\pi\)
\(180\) 0 0
\(181\) 478809.i 1.08634i −0.839622 0.543170i \(-0.817224\pi\)
0.839622 0.543170i \(-0.182776\pi\)
\(182\) 0 0
\(183\) 174283. 165494.i 0.384705 0.365305i
\(184\) 0 0
\(185\) 31254.9i 0.0671412i
\(186\) 0 0
\(187\) 220748. 0.461629
\(188\) 0 0
\(189\) 461422. + 539209.i 0.939603 + 1.09800i
\(190\) 0 0
\(191\) −714309. −1.41678 −0.708391 0.705820i \(-0.750579\pi\)
−0.708391 + 0.705820i \(0.750579\pi\)
\(192\) 0 0
\(193\) −154226. −0.298033 −0.149017 0.988835i \(-0.547611\pi\)
−0.149017 + 0.988835i \(0.547611\pi\)
\(194\) 0 0
\(195\) −38658.3 + 36708.8i −0.0728041 + 0.0691328i
\(196\) 0 0
\(197\) −321250. −0.589764 −0.294882 0.955534i \(-0.595280\pi\)
−0.294882 + 0.955534i \(0.595280\pi\)
\(198\) 0 0
\(199\) 100493.i 0.179889i 0.995947 + 0.0899443i \(0.0286689\pi\)
−0.995947 + 0.0899443i \(0.971331\pi\)
\(200\) 0 0
\(201\) −365846. 385275.i −0.638717 0.672636i
\(202\) 0 0
\(203\) 1.18502e6i 2.01830i
\(204\) 0 0
\(205\) 44180.3i 0.0734250i
\(206\) 0 0
\(207\) 38558.4 744846.i 0.0625451 1.20821i
\(208\) 0 0
\(209\) 1.75915e6i 2.78572i
\(210\) 0 0
\(211\) −648944. −1.00346 −0.501731 0.865024i \(-0.667303\pi\)
−0.501731 + 0.865024i \(0.667303\pi\)
\(212\) 0 0
\(213\) 187411. + 197364.i 0.283039 + 0.298070i
\(214\) 0 0
\(215\) −16304.4 −0.0240552
\(216\) 0 0
\(217\) 1.68727e6 2.43241
\(218\) 0 0
\(219\) −249122. 262352.i −0.350996 0.369636i
\(220\) 0 0
\(221\) 249044. 0.343002
\(222\) 0 0
\(223\) 574520.i 0.773648i −0.922154 0.386824i \(-0.873572\pi\)
0.922154 0.386824i \(-0.126428\pi\)
\(224\) 0 0
\(225\) 39030.8 753972.i 0.0513986 0.992884i
\(226\) 0 0
\(227\) 412664.i 0.531536i 0.964037 + 0.265768i \(0.0856255\pi\)
−0.964037 + 0.265768i \(0.914375\pi\)
\(228\) 0 0
\(229\) 32427.2i 0.0408621i 0.999791 + 0.0204311i \(0.00650386\pi\)
−0.999791 + 0.0204311i \(0.993496\pi\)
\(230\) 0 0
\(231\) 1.43359e6 + 1.50973e6i 1.76765 + 1.86152i
\(232\) 0 0
\(233\) 455816.i 0.550047i 0.961438 + 0.275023i \(0.0886856\pi\)
−0.961438 + 0.275023i \(0.911314\pi\)
\(234\) 0 0
\(235\) −47605.2 −0.0562322
\(236\) 0 0
\(237\) 464453. 441031.i 0.537119 0.510033i
\(238\) 0 0
\(239\) −784252. −0.888098 −0.444049 0.896002i \(-0.646458\pi\)
−0.444049 + 0.896002i \(0.646458\pi\)
\(240\) 0 0
\(241\) −74831.8 −0.0829934 −0.0414967 0.999139i \(-0.513213\pi\)
−0.0414967 + 0.999139i \(0.513213\pi\)
\(242\) 0 0
\(243\) 561520. + 729372.i 0.610028 + 0.792380i
\(244\) 0 0
\(245\) 77790.0 0.0827959
\(246\) 0 0
\(247\) 1.98465e6i 2.06986i
\(248\) 0 0
\(249\) −23445.6 + 22263.3i −0.0239642 + 0.0227557i
\(250\) 0 0
\(251\) 1.22569e6i 1.22799i 0.789308 + 0.613997i \(0.210439\pi\)
−0.789308 + 0.613997i \(0.789561\pi\)
\(252\) 0 0
\(253\) 2.18801e6i 2.14905i
\(254\) 0 0
\(255\) 14884.9 14134.3i 0.0143349 0.0136121i
\(256\) 0 0
\(257\) 525291.i 0.496098i −0.968748 0.248049i \(-0.920211\pi\)
0.968748 0.248049i \(-0.0797894\pi\)
\(258\) 0 0
\(259\) −1.37706e6 −1.27557
\(260\) 0 0
\(261\) 79459.6 1.53495e6i 0.0722013 1.39474i
\(262\) 0 0
\(263\) −1.47722e6 −1.31691 −0.658457 0.752619i \(-0.728790\pi\)
−0.658457 + 0.752619i \(0.728790\pi\)
\(264\) 0 0
\(265\) 31424.9 0.0274890
\(266\) 0 0
\(267\) −1.54074e6 + 1.46304e6i −1.32267 + 1.25597i
\(268\) 0 0
\(269\) 1.20638e6 1.01649 0.508245 0.861212i \(-0.330294\pi\)
0.508245 + 0.861212i \(0.330294\pi\)
\(270\) 0 0
\(271\) 1.18886e6i 0.983347i 0.870780 + 0.491673i \(0.163615\pi\)
−0.870780 + 0.491673i \(0.836385\pi\)
\(272\) 0 0
\(273\) 1.61736e6 + 1.70325e6i 1.31341 + 1.38316i
\(274\) 0 0
\(275\) 2.21481e6i 1.76606i
\(276\) 0 0
\(277\) 920507.i 0.720821i 0.932794 + 0.360411i \(0.117363\pi\)
−0.932794 + 0.360411i \(0.882637\pi\)
\(278\) 0 0
\(279\) 2.18551e6 + 113137.i 1.68090 + 0.0870152i
\(280\) 0 0
\(281\) 703851.i 0.531759i 0.964006 + 0.265879i \(0.0856623\pi\)
−0.964006 + 0.265879i \(0.914338\pi\)
\(282\) 0 0
\(283\) 1.10317e6 0.818799 0.409400 0.912355i \(-0.365738\pi\)
0.409400 + 0.912355i \(0.365738\pi\)
\(284\) 0 0
\(285\) 112637. + 118619.i 0.0821427 + 0.0865049i
\(286\) 0 0
\(287\) −1.94654e6 −1.39495
\(288\) 0 0
\(289\) 1.32397e6 0.932464
\(290\) 0 0
\(291\) 1.00997e6 + 1.06361e6i 0.699160 + 0.736289i
\(292\) 0 0
\(293\) 1.59372e6 1.08453 0.542266 0.840207i \(-0.317567\pi\)
0.542266 + 0.840207i \(0.317567\pi\)
\(294\) 0 0
\(295\) 41164.5i 0.0275403i
\(296\) 0 0
\(297\) 1.75569e6 + 2.05166e6i 1.15493 + 1.34963i
\(298\) 0 0
\(299\) 2.46847e6i 1.59680i
\(300\) 0 0
\(301\) 718357.i 0.457008i
\(302\) 0 0
\(303\) −901081. 948934.i −0.563842 0.593785i
\(304\) 0 0
\(305\) 65560.8i 0.0403547i
\(306\) 0 0
\(307\) 900531. 0.545322 0.272661 0.962110i \(-0.412096\pi\)
0.272661 + 0.962110i \(0.412096\pi\)
\(308\) 0 0
\(309\) 1.14587e6 1.08808e6i 0.682713 0.648285i
\(310\) 0 0
\(311\) 1.15505e6 0.677173 0.338587 0.940935i \(-0.390051\pi\)
0.338587 + 0.940935i \(0.390051\pi\)
\(312\) 0 0
\(313\) −2.17256e6 −1.25346 −0.626732 0.779235i \(-0.715608\pi\)
−0.626732 + 0.779235i \(0.715608\pi\)
\(314\) 0 0
\(315\) 193333. + 10008.2i 0.109781 + 0.00568305i
\(316\) 0 0
\(317\) −1.30916e6 −0.731721 −0.365860 0.930670i \(-0.619225\pi\)
−0.365860 + 0.930670i \(0.619225\pi\)
\(318\) 0 0
\(319\) 4.50895e6i 2.48084i
\(320\) 0 0
\(321\) −123254. + 117039.i −0.0667635 + 0.0633967i
\(322\) 0 0
\(323\) 764165.i 0.407550i
\(324\) 0 0
\(325\) 2.49871e6i 1.31222i
\(326\) 0 0
\(327\) 1.45420e6 1.38087e6i 0.752067 0.714141i
\(328\) 0 0
\(329\) 2.09744e6i 1.06832i
\(330\) 0 0
\(331\) −685986. −0.344148 −0.172074 0.985084i \(-0.555047\pi\)
−0.172074 + 0.985084i \(0.555047\pi\)
\(332\) 0 0
\(333\) −1.78370e6 92336.5i −0.881476 0.0456313i
\(334\) 0 0
\(335\) −144930. −0.0705582
\(336\) 0 0
\(337\) 1.99207e6 0.955500 0.477750 0.878496i \(-0.341453\pi\)
0.477750 + 0.878496i \(0.341453\pi\)
\(338\) 0 0
\(339\) −1.84794e6 + 1.75475e6i −0.873348 + 0.829307i
\(340\) 0 0
\(341\) 6.41999e6 2.98985
\(342\) 0 0
\(343\) 278536.i 0.127834i
\(344\) 0 0
\(345\) −140096. 147536.i −0.0633691 0.0667344i
\(346\) 0 0
\(347\) 2.29595e6i 1.02362i −0.859099 0.511810i \(-0.828975\pi\)
0.859099 0.511810i \(-0.171025\pi\)
\(348\) 0 0
\(349\) 1.29935e6i 0.571036i −0.958373 0.285518i \(-0.907834\pi\)
0.958373 0.285518i \(-0.0921656\pi\)
\(350\) 0 0
\(351\) 1.98074e6 + 2.31465e6i 0.858143 + 1.00281i
\(352\) 0 0
\(353\) 876536.i 0.374398i −0.982322 0.187199i \(-0.940059\pi\)
0.982322 0.187199i \(-0.0599409\pi\)
\(354\) 0 0
\(355\) 74243.2 0.0312670
\(356\) 0 0
\(357\) −622744. 655816.i −0.258606 0.272340i
\(358\) 0 0
\(359\) −62753.1 −0.0256980 −0.0128490 0.999917i \(-0.504090\pi\)
−0.0128490 + 0.999917i \(0.504090\pi\)
\(360\) 0 0
\(361\) 3.61357e6 1.45938
\(362\) 0 0
\(363\) 3.72604e6 + 3.92391e6i 1.48416 + 1.56298i
\(364\) 0 0
\(365\) −98690.2 −0.0387741
\(366\) 0 0
\(367\) 928044.i 0.359669i 0.983697 + 0.179835i \(0.0575563\pi\)
−0.983697 + 0.179835i \(0.942444\pi\)
\(368\) 0 0
\(369\) −2.52134e6 130522.i −0.963974 0.0499020i
\(370\) 0 0
\(371\) 1.38455e6i 0.522246i
\(372\) 0 0
\(373\) 1.81032e6i 0.673726i 0.941554 + 0.336863i \(0.109366\pi\)
−0.941554 + 0.336863i \(0.890634\pi\)
\(374\) 0 0
\(375\) −284450. 299556.i −0.104455 0.110002i
\(376\) 0 0
\(377\) 5.08692e6i 1.84332i
\(378\) 0 0
\(379\) 776668. 0.277739 0.138870 0.990311i \(-0.455653\pi\)
0.138870 + 0.990311i \(0.455653\pi\)
\(380\) 0 0
\(381\) −2.25205e6 + 2.13848e6i −0.794814 + 0.754733i
\(382\) 0 0
\(383\) −5.03072e6 −1.75240 −0.876199 0.481949i \(-0.839929\pi\)
−0.876199 + 0.481949i \(0.839929\pi\)
\(384\) 0 0
\(385\) 567920. 0.195270
\(386\) 0 0
\(387\) −48168.2 + 930481.i −0.0163487 + 0.315813i
\(388\) 0 0
\(389\) −4.51637e6 −1.51327 −0.756633 0.653839i \(-0.773157\pi\)
−0.756633 + 0.653839i \(0.773157\pi\)
\(390\) 0 0
\(391\) 950455.i 0.314405i
\(392\) 0 0
\(393\) −569695. + 540967.i −0.186063 + 0.176681i
\(394\) 0 0
\(395\) 174715.i 0.0563427i
\(396\) 0 0
\(397\) 4.80508e6i 1.53012i −0.643961 0.765058i \(-0.722710\pi\)
0.643961 0.765058i \(-0.277290\pi\)
\(398\) 0 0
\(399\) 5.22623e6 4.96268e6i 1.64345 1.56057i
\(400\) 0 0
\(401\) 5.82811e6i 1.80995i 0.425463 + 0.904976i \(0.360111\pi\)
−0.425463 + 0.904976i \(0.639889\pi\)
\(402\) 0 0
\(403\) 7.24293e6 2.22153
\(404\) 0 0
\(405\) 249751. + 25927.2i 0.0756606 + 0.00785449i
\(406\) 0 0
\(407\) −5.23965e6 −1.56789
\(408\) 0 0
\(409\) 21312.8 0.00629987 0.00314994 0.999995i \(-0.498997\pi\)
0.00314994 + 0.999995i \(0.498997\pi\)
\(410\) 0 0
\(411\) −488782. + 464134.i −0.142728 + 0.135531i
\(412\) 0 0
\(413\) 1.81367e6 0.523219
\(414\) 0 0
\(415\) 8819.61i 0.00251379i
\(416\) 0 0
\(417\) −885478. 932502.i −0.249366 0.262609i
\(418\) 0 0
\(419\) 653173.i 0.181758i 0.995862 + 0.0908789i \(0.0289676\pi\)
−0.995862 + 0.0908789i \(0.971032\pi\)
\(420\) 0 0
\(421\) 4.56795e6i 1.25608i 0.778183 + 0.628038i \(0.216142\pi\)
−0.778183 + 0.628038i \(0.783858\pi\)
\(422\) 0 0
\(423\) −140640. + 2.71680e6i −0.0382172 + 0.738255i
\(424\) 0 0
\(425\) 962100.i 0.258373i
\(426\) 0 0
\(427\) −2.88855e6 −0.766672
\(428\) 0 0
\(429\) 6.15397e6 + 6.48078e6i 1.61440 + 1.70014i
\(430\) 0 0
\(431\) −4.33039e6 −1.12288 −0.561441 0.827517i \(-0.689753\pi\)
−0.561441 + 0.827517i \(0.689753\pi\)
\(432\) 0 0
\(433\) −4.95867e6 −1.27100 −0.635500 0.772101i \(-0.719206\pi\)
−0.635500 + 0.772101i \(0.719206\pi\)
\(434\) 0 0
\(435\) −288704. 304036.i −0.0731526 0.0770374i
\(436\) 0 0
\(437\) −7.57423e6 −1.89729
\(438\) 0 0
\(439\) 1.57813e6i 0.390825i −0.980721 0.195412i \(-0.937395\pi\)
0.980721 0.195412i \(-0.0626045\pi\)
\(440\) 0 0
\(441\) 229815. 4.43942e6i 0.0562708 1.08700i
\(442\) 0 0
\(443\) 2.24569e6i 0.543676i −0.962343 0.271838i \(-0.912368\pi\)
0.962343 0.271838i \(-0.0876315\pi\)
\(444\) 0 0
\(445\) 579585.i 0.138745i
\(446\) 0 0
\(447\) 4.77243e6 + 5.02588e6i 1.12972 + 1.18972i
\(448\) 0 0
\(449\) 6.24299e6i 1.46143i −0.682685 0.730713i \(-0.739188\pi\)
0.682685 0.730713i \(-0.260812\pi\)
\(450\) 0 0
\(451\) −7.40650e6 −1.71464
\(452\) 0 0
\(453\) 2.24239e6 2.12931e6i 0.513413 0.487522i
\(454\) 0 0
\(455\) 640718. 0.145090
\(456\) 0 0
\(457\) −5.12647e6 −1.14823 −0.574113 0.818776i \(-0.694653\pi\)
−0.574113 + 0.818776i \(0.694653\pi\)
\(458\) 0 0
\(459\) −762660. 891229.i −0.168966 0.197450i
\(460\) 0 0
\(461\) −6.78319e6 −1.48656 −0.743278 0.668982i \(-0.766730\pi\)
−0.743278 + 0.668982i \(0.766730\pi\)
\(462\) 0 0
\(463\) 7.59144e6i 1.64578i 0.568201 + 0.822890i \(0.307639\pi\)
−0.568201 + 0.822890i \(0.692361\pi\)
\(464\) 0 0
\(465\) 432896. 411066.i 0.0928435 0.0881616i
\(466\) 0 0
\(467\) 6.08139e6i 1.29036i −0.764031 0.645179i \(-0.776783\pi\)
0.764031 0.645179i \(-0.223217\pi\)
\(468\) 0 0
\(469\) 6.38550e6i 1.34049i
\(470\) 0 0
\(471\) 932979. 885931.i 0.193785 0.184013i
\(472\) 0 0
\(473\) 2.73331e6i 0.561741i
\(474\) 0 0
\(475\) −7.66702e6 −1.55917
\(476\) 0 0
\(477\) 92838.8 1.79340e6i 0.0186824 0.360895i
\(478\) 0 0
\(479\) 9.75608e6 1.94284 0.971419 0.237373i \(-0.0762863\pi\)
0.971419 + 0.237373i \(0.0762863\pi\)
\(480\) 0 0
\(481\) −5.91129e6 −1.16498
\(482\) 0 0
\(483\) −6.50029e6 + 6.17250e6i −1.26784 + 1.20391i
\(484\) 0 0
\(485\) 400101. 0.0772352
\(486\) 0 0
\(487\) 5.95397e6i 1.13759i −0.822481 0.568793i \(-0.807411\pi\)
0.822481 0.568793i \(-0.192589\pi\)
\(488\) 0 0
\(489\) −3.52497e6 3.71217e6i −0.666628 0.702030i
\(490\) 0 0
\(491\) 3.28879e6i 0.615649i 0.951443 + 0.307824i \(0.0996009\pi\)
−0.951443 + 0.307824i \(0.900399\pi\)
\(492\) 0 0
\(493\) 1.95866e6i 0.362946i
\(494\) 0 0
\(495\) 735622. + 38080.9i 0.134940 + 0.00698545i
\(496\) 0 0
\(497\) 3.27109e6i 0.594020i
\(498\) 0 0
\(499\) −845037. −0.151923 −0.0759616 0.997111i \(-0.524203\pi\)
−0.0759616 + 0.997111i \(0.524203\pi\)
\(500\) 0 0
\(501\) −6.98214e6 7.35293e6i −1.24278 1.30878i
\(502\) 0 0
\(503\) −1.09603e6 −0.193153 −0.0965765 0.995326i \(-0.530789\pi\)
−0.0965765 + 0.995326i \(0.530789\pi\)
\(504\) 0 0
\(505\) −356964. −0.0622868
\(506\) 0 0
\(507\) 2.95734e6 + 3.11439e6i 0.510954 + 0.538088i
\(508\) 0 0
\(509\) −1.06884e6 −0.182861 −0.0914303 0.995811i \(-0.529144\pi\)
−0.0914303 + 0.995811i \(0.529144\pi\)
\(510\) 0 0
\(511\) 4.34820e6i 0.736643i
\(512\) 0 0
\(513\) 7.10225e6 6.07768e6i 1.19152 1.01963i
\(514\) 0 0
\(515\) 431045.i 0.0716152i
\(516\) 0 0
\(517\) 7.98067e6i 1.31315i
\(518\) 0 0
\(519\) −926732. 975947.i −0.151020 0.159041i
\(520\) 0 0
\(521\) 4.98135e6i 0.803995i −0.915641 0.401997i \(-0.868316\pi\)
0.915641 0.401997i \(-0.131684\pi\)
\(522\) 0 0
\(523\) 3.41829e6 0.546455 0.273228 0.961949i \(-0.411909\pi\)
0.273228 + 0.961949i \(0.411909\pi\)
\(524\) 0 0
\(525\) −6.57993e6 + 6.24812e6i −1.04189 + 0.989352i
\(526\) 0 0
\(527\) −2.78880e6 −0.437413
\(528\) 0 0
\(529\) 2.98435e6 0.463671
\(530\) 0 0
\(531\) 2.34923e6 + 121613.i 0.361568 + 0.0187173i
\(532\) 0 0
\(533\) −8.35589e6 −1.27402
\(534\) 0 0
\(535\) 46365.0i 0.00700335i
\(536\) 0 0
\(537\) −4.72415e6 + 4.48592e6i −0.706949 + 0.671299i
\(538\) 0 0
\(539\) 1.30409e7i 1.93347i
\(540\) 0 0
\(541\) 7.98082e6i 1.17234i 0.810187 + 0.586171i \(0.199365\pi\)
−0.810187 + 0.586171i \(0.800635\pi\)
\(542\) 0 0
\(543\) −5.41248e6 + 5.13954e6i −0.787765 + 0.748039i
\(544\) 0 0
\(545\) 547034.i 0.0788902i
\(546\) 0 0
\(547\) 7.94344e6 1.13512 0.567558 0.823333i \(-0.307888\pi\)
0.567558 + 0.823333i \(0.307888\pi\)
\(548\) 0 0
\(549\) −3.74151e6 193686.i −0.529805 0.0274264i
\(550\) 0 0
\(551\) −1.56087e7 −2.19021
\(552\) 0 0
\(553\) −7.69779e6 −1.07042
\(554\) 0 0
\(555\) −353307. + 335490.i −0.0486877 + 0.0462325i
\(556\) 0 0
\(557\) 9.79798e6 1.33813 0.669065 0.743203i \(-0.266695\pi\)
0.669065 + 0.743203i \(0.266695\pi\)
\(558\) 0 0
\(559\) 3.08368e6i 0.417387i
\(560\) 0 0
\(561\) −2.36951e6 2.49535e6i −0.317871 0.334752i
\(562\) 0 0
\(563\) 671056.i 0.0892252i 0.999004 + 0.0446126i \(0.0142054\pi\)
−0.999004 + 0.0446126i \(0.985795\pi\)
\(564\) 0 0
\(565\) 695145.i 0.0916124i
\(566\) 0 0
\(567\) 1.14233e6 1.10038e7i 0.149222 1.43742i
\(568\) 0 0
\(569\) 3.33810e6i 0.432233i 0.976368 + 0.216117i \(0.0693392\pi\)
−0.976368 + 0.216117i \(0.930661\pi\)
\(570\) 0 0
\(571\) 1.06507e7 1.36706 0.683528 0.729924i \(-0.260445\pi\)
0.683528 + 0.729924i \(0.260445\pi\)
\(572\) 0 0
\(573\) 7.66740e6 + 8.07458e6i 0.975576 + 1.02739i
\(574\) 0 0
\(575\) 9.53611e6 1.20282
\(576\) 0 0
\(577\) −8.00078e6 −1.00044 −0.500222 0.865897i \(-0.666748\pi\)
−0.500222 + 0.865897i \(0.666748\pi\)
\(578\) 0 0
\(579\) 1.65546e6 + 1.74338e6i 0.205221 + 0.216120i
\(580\) 0 0
\(581\) 388584. 0.0477578
\(582\) 0 0
\(583\) 5.26816e6i 0.641929i
\(584\) 0 0
\(585\) 829916. + 42962.2i 0.100264 + 0.00519036i
\(586\) 0 0
\(587\) 1.20246e7i 1.44037i 0.693780 + 0.720187i \(0.255944\pi\)
−0.693780 + 0.720187i \(0.744056\pi\)
\(588\) 0 0
\(589\) 2.22241e7i 2.63959i
\(590\) 0 0
\(591\) 3.44830e6 + 3.63143e6i 0.406103 + 0.427670i
\(592\) 0 0
\(593\) 9.80287e6i 1.14477i −0.819987 0.572383i \(-0.806019\pi\)
0.819987 0.572383i \(-0.193981\pi\)
\(594\) 0 0
\(595\) −246701. −0.0285679
\(596\) 0 0
\(597\) 1.13598e6 1.07869e6i 0.130447 0.123869i
\(598\) 0 0
\(599\) −9.41689e6 −1.07236 −0.536180 0.844104i \(-0.680133\pi\)
−0.536180 + 0.844104i \(0.680133\pi\)
\(600\) 0 0
\(601\) 4.79726e6 0.541760 0.270880 0.962613i \(-0.412685\pi\)
0.270880 + 0.962613i \(0.412685\pi\)
\(602\) 0 0
\(603\) −428168. + 8.27108e6i −0.0479536 + 0.926337i
\(604\) 0 0
\(605\) 1.47607e6 0.163953
\(606\) 0 0
\(607\) 7.86218e6i 0.866106i −0.901368 0.433053i \(-0.857436\pi\)
0.901368 0.433053i \(-0.142564\pi\)
\(608\) 0 0
\(609\) −1.33955e7 + 1.27200e7i −1.46358 + 1.38978i
\(610\) 0 0
\(611\) 9.00365e6i 0.975699i
\(612\) 0 0
\(613\) 1.35943e7i 1.46119i 0.682811 + 0.730595i \(0.260757\pi\)
−0.682811 + 0.730595i \(0.739243\pi\)
\(614\) 0 0
\(615\) −499416. + 474231.i −0.0532445 + 0.0505594i
\(616\) 0 0
\(617\) 1.62327e7i 1.71664i −0.513115 0.858320i \(-0.671509\pi\)
0.513115 0.858320i \(-0.328491\pi\)
\(618\) 0 0
\(619\) −4.88310e6 −0.512235 −0.256117 0.966646i \(-0.582443\pi\)
−0.256117 + 0.966646i \(0.582443\pi\)
\(620\) 0 0
\(621\) −8.83365e6 + 7.55931e6i −0.919203 + 0.786599i
\(622\) 0 0
\(623\) 2.55360e7 2.63592
\(624\) 0 0
\(625\) 9.59643e6 0.982675
\(626\) 0 0
\(627\) 1.98855e7 1.88828e7i 2.02008 1.91821i
\(628\) 0 0
\(629\) 2.27607e6 0.229382
\(630\) 0 0
\(631\) 2.15393e6i 0.215356i −0.994186 0.107678i \(-0.965658\pi\)
0.994186 0.107678i \(-0.0343416\pi\)
\(632\) 0 0
\(633\) 6.96577e6 + 7.33569e6i 0.690971 + 0.727665i
\(634\) 0 0
\(635\) 847163.i 0.0833744i
\(636\) 0 0
\(637\) 1.47126e7i 1.43661i
\(638\) 0 0
\(639\) 219337. 4.23701e6i 0.0212500 0.410494i
\(640\) 0 0
\(641\) 1.32161e7i 1.27045i 0.772327 + 0.635226i \(0.219093\pi\)
−0.772327 + 0.635226i \(0.780907\pi\)
\(642\) 0 0
\(643\) −1.78560e7 −1.70316 −0.851581 0.524223i \(-0.824356\pi\)
−0.851581 + 0.524223i \(0.824356\pi\)
\(644\) 0 0
\(645\) 175011. + 184305.i 0.0165641 + 0.0174437i
\(646\) 0 0
\(647\) 640030. 0.0601090 0.0300545 0.999548i \(-0.490432\pi\)
0.0300545 + 0.999548i \(0.490432\pi\)
\(648\) 0 0
\(649\) 6.90093e6 0.643126
\(650\) 0 0
\(651\) −1.81112e7 1.90730e7i −1.67492 1.76387i
\(652\) 0 0
\(653\) −6.04740e6 −0.554991 −0.277496 0.960727i \(-0.589504\pi\)
−0.277496 + 0.960727i \(0.589504\pi\)
\(654\) 0 0
\(655\) 214304.i 0.0195177i
\(656\) 0 0
\(657\) −291561. + 5.63218e6i −0.0263521 + 0.509053i
\(658\) 0 0
\(659\) 9.15627e6i 0.821306i −0.911792 0.410653i \(-0.865301\pi\)
0.911792 0.410653i \(-0.134699\pi\)
\(660\) 0 0
\(661\) 267971.i 0.0238553i −0.999929 0.0119276i \(-0.996203\pi\)
0.999929 0.0119276i \(-0.00379677\pi\)
\(662\) 0 0
\(663\) −2.67324e6 2.81521e6i −0.236186 0.248729i
\(664\) 0 0
\(665\) 1.96597e6i 0.172394i
\(666\) 0 0
\(667\) 1.94138e7 1.68964
\(668\) 0 0
\(669\) −6.49440e6 + 6.16690e6i −0.561014 + 0.532723i
\(670\) 0 0
\(671\) −1.09908e7 −0.942372
\(672\) 0 0
\(673\) 1.72475e7 1.46787 0.733935 0.679220i \(-0.237682\pi\)
0.733935 + 0.679220i \(0.237682\pi\)
\(674\) 0 0
\(675\) −8.94188e6 + 7.65192e6i −0.755387 + 0.646415i
\(676\) 0 0
\(677\) 2.03336e7 1.70507 0.852535 0.522670i \(-0.175064\pi\)
0.852535 + 0.522670i \(0.175064\pi\)
\(678\) 0 0
\(679\) 1.76281e7i 1.46734i
\(680\) 0 0
\(681\) 4.66477e6 4.42954e6i 0.385445 0.366008i
\(682\) 0 0
\(683\) 2.64704e6i 0.217125i 0.994090 + 0.108562i \(0.0346247\pi\)
−0.994090 + 0.108562i \(0.965375\pi\)
\(684\) 0 0
\(685\) 183867.i 0.0149719i
\(686\) 0 0
\(687\) 366558. 348074.i 0.0296313 0.0281371i
\(688\) 0 0
\(689\) 5.94344e6i 0.476969i
\(690\) 0 0
\(691\) 6.33199e6 0.504481 0.252240 0.967665i \(-0.418833\pi\)
0.252240 + 0.967665i \(0.418833\pi\)
\(692\) 0 0
\(693\) 1.67781e6 3.24108e7i 0.132712 2.56364i
\(694\) 0 0
\(695\) −350783. −0.0275471
\(696\) 0 0
\(697\) 3.21734e6 0.250850
\(698\) 0 0
\(699\) 5.15256e6 4.89273e6i 0.398869 0.378755i
\(700\) 0 0
\(701\) −6.12150e6 −0.470503 −0.235252 0.971934i \(-0.575591\pi\)
−0.235252 + 0.971934i \(0.575591\pi\)
\(702\) 0 0
\(703\) 1.81381e7i 1.38422i
\(704\) 0 0
\(705\) 510994. + 538131.i 0.0387207 + 0.0407770i
\(706\) 0 0
\(707\) 1.57275e7i 1.18335i
\(708\) 0 0
\(709\) 1.03996e7i 0.776965i −0.921456 0.388483i \(-0.872999\pi\)
0.921456 0.388483i \(-0.127001\pi\)
\(710\) 0 0
\(711\) −9.97087e6 516162.i −0.739706 0.0382923i
\(712\) 0 0
\(713\) 2.76420e7i 2.03632i
\(714\) 0 0
\(715\) 2.43790e6 0.178341
\(716\) 0 0
\(717\) 8.41816e6 + 8.86522e6i 0.611532 + 0.644008i
\(718\) 0 0
\(719\) −1.48619e6 −0.107214 −0.0536071 0.998562i \(-0.517072\pi\)
−0.0536071 + 0.998562i \(0.517072\pi\)
\(720\) 0 0
\(721\) −1.89915e7 −1.36057
\(722\) 0 0
\(723\) 803244. + 845901.i 0.0571481 + 0.0601830i
\(724\) 0 0
\(725\) 1.96516e7 1.38852
\(726\) 0 0
\(727\) 47337.8i 0.00332179i 0.999999 + 0.00166089i \(0.000528679\pi\)
−0.999999 + 0.00166089i \(0.999471\pi\)
\(728\) 0 0
\(729\) 2.21749e6 1.41765e7i 0.154541 0.987986i
\(730\) 0 0
\(731\) 1.18733e6i 0.0821824i
\(732\) 0 0
\(733\) 9.51842e6i 0.654342i −0.944965 0.327171i \(-0.893905\pi\)
0.944965 0.327171i \(-0.106095\pi\)
\(734\) 0 0
\(735\) −834998. 879341.i −0.0570121 0.0600398i
\(736\) 0 0
\(737\) 2.42965e7i 1.64769i
\(738\) 0 0
\(739\) −1.02926e7 −0.693292 −0.346646 0.937996i \(-0.612679\pi\)
−0.346646 + 0.937996i \(0.612679\pi\)
\(740\) 0 0
\(741\) 2.24345e7 2.13032e7i 1.50097 1.42528i
\(742\) 0 0
\(743\) 2.27872e7 1.51433 0.757163 0.653226i \(-0.226585\pi\)
0.757163 + 0.653226i \(0.226585\pi\)
\(744\) 0 0
\(745\) 1.89061e6 0.124799
\(746\) 0 0
\(747\) 503329. + 26055.8i 0.0330028 + 0.00170845i
\(748\) 0 0
\(749\) 2.04280e6 0.133052
\(750\) 0 0
\(751\) 2.50137e7i 1.61837i 0.587554 + 0.809185i \(0.300091\pi\)
−0.587554 + 0.809185i \(0.699909\pi\)
\(752\) 0 0
\(753\) 1.38553e7 1.31566e7i 0.890486 0.845580i
\(754\) 0 0
\(755\) 843530.i 0.0538559i
\(756\) 0 0
\(757\) 2.21276e7i 1.40344i 0.712452 + 0.701721i \(0.247585\pi\)
−0.712452 + 0.701721i \(0.752415\pi\)
\(758\) 0 0
\(759\) −2.47333e7 + 2.34860e7i −1.55839 + 1.47981i
\(760\) 0 0
\(761\) 1.81968e7i 1.13902i 0.821983 + 0.569512i \(0.192868\pi\)
−0.821983 + 0.569512i \(0.807132\pi\)
\(762\) 0 0
\(763\) −2.41018e7 −1.49878
\(764\) 0 0
\(765\) −319549. 16542.1i −0.0197417 0.00102197i
\(766\) 0 0
\(767\) 7.78552e6 0.477858
\(768\) 0 0
\(769\) 1.64592e7 1.00367 0.501837 0.864962i \(-0.332658\pi\)
0.501837 + 0.864962i \(0.332658\pi\)
\(770\) 0 0
\(771\) −5.93791e6 + 5.63847e6i −0.359747 + 0.341606i
\(772\) 0 0
\(773\) −8.17589e6 −0.492137 −0.246069 0.969252i \(-0.579139\pi\)
−0.246069 + 0.969252i \(0.579139\pi\)
\(774\) 0 0
\(775\) 2.79806e7i 1.67341i
\(776\) 0 0
\(777\) 1.47814e7 + 1.55664e7i 0.878340 + 0.924985i
\(778\) 0 0
\(779\) 2.56391e7i 1.51377i
\(780\) 0 0
\(781\) 1.24463e7i 0.730152i
\(782\) 0 0
\(783\) −1.82040e7 + 1.55779e7i −1.06112 + 0.908040i
\(784\) 0 0
\(785\) 350963.i 0.0203276i
\(786\) 0 0
\(787\) 5.85675e6 0.337070 0.168535 0.985696i \(-0.446096\pi\)
0.168535 + 0.985696i \(0.446096\pi\)
\(788\) 0 0
\(789\) 1.58565e7 + 1.66986e7i 0.906809 + 0.954965i
\(790\) 0 0
\(791\) 3.06275e7 1.74048
\(792\) 0 0
\(793\) −1.23996e7 −0.700205
\(794\) 0 0
\(795\) −337315. 355228.i −0.0189286 0.0199338i
\(796\) 0 0
\(797\) −1.92534e6 −0.107365 −0.0536825 0.998558i \(-0.517096\pi\)
−0.0536825 + 0.998558i \(0.517096\pi\)
\(798\) 0 0
\(799\) 3.46675e6i 0.192112i
\(800\) 0 0
\(801\) 3.30765e7 + 1.71227e6i 1.82154 + 0.0942955i
\(802\) 0 0
\(803\) 1.65447e7i 0.905460i
\(804\) 0 0
\(805\) 2.44524e6i 0.132994i
\(806\) 0 0
\(807\) −1.29493e7 1.36370e7i −0.699941 0.737112i
\(808\) 0 0
\(809\) 5.47170e6i 0.293935i 0.989141 + 0.146967i \(0.0469512\pi\)
−0.989141 + 0.146967i \(0.953049\pi\)
\(810\) 0 0
\(811\) −2.39885e7 −1.28071 −0.640354 0.768080i \(-0.721212\pi\)
−0.640354 + 0.768080i \(0.721212\pi\)
\(812\) 0 0
\(813\) 1.34389e7 1.27612e7i 0.713078 0.677119i
\(814\) 0 0
\(815\) −1.39642e6 −0.0736415
\(816\) 0 0
\(817\) 9.46192e6 0.495934
\(818\) 0 0
\(819\) 1.89288e6 3.65653e7i 0.0986081 1.90485i
\(820\) 0 0
\(821\) 2.38393e6 0.123434 0.0617170 0.998094i \(-0.480342\pi\)
0.0617170 + 0.998094i \(0.480342\pi\)
\(822\) 0 0
\(823\) 1.45230e7i 0.747407i −0.927548 0.373703i \(-0.878088\pi\)
0.927548 0.373703i \(-0.121912\pi\)
\(824\) 0 0
\(825\) −2.50363e7 + 2.37738e7i −1.28066 + 1.21608i
\(826\) 0 0
\(827\) 124756.i 0.00634303i 0.999995 + 0.00317152i \(0.00100953\pi\)
−0.999995 + 0.00317152i \(0.998990\pi\)
\(828\) 0 0
\(829\) 3.63133e7i 1.83518i −0.397527 0.917590i \(-0.630132\pi\)
0.397527 0.917590i \(-0.369868\pi\)
\(830\) 0 0
\(831\) 1.04054e7 9.88072e6i 0.522707 0.496348i
\(832\) 0 0
\(833\) 5.66489e6i 0.282865i
\(834\) 0 0
\(835\) −2.76598e6 −0.137288
\(836\) 0 0
\(837\) −2.21804e7 2.59195e7i −1.09435 1.27883i
\(838\) 0 0
\(839\) 8.33073e6 0.408581 0.204291 0.978910i \(-0.434511\pi\)
0.204291 + 0.978910i \(0.434511\pi\)
\(840\) 0 0
\(841\) 1.94960e7 0.950506
\(842\) 0 0
\(843\) 7.95635e6 7.55513e6i 0.385607 0.366162i
\(844\) 0 0
\(845\) 1.17155e6 0.0564444
\(846\) 0 0
\(847\) 6.50344e7i 3.11483i
\(848\) 0 0
\(849\) −1.18415e7 1.24703e7i −0.563814 0.593756i
\(850\) 0 0
\(851\) 2.25599e7i 1.06786i
\(852\) 0 0
\(853\) 750959.i 0.0353381i 0.999844 + 0.0176691i \(0.00562453\pi\)
−0.999844 + 0.0176691i \(0.994375\pi\)
\(854\) 0 0
\(855\) 131825. 2.54651e6i 0.00616711 0.119132i
\(856\) 0 0
\(857\) 2.08548e7i 0.969960i −0.874525 0.484980i \(-0.838827\pi\)
0.874525 0.484980i \(-0.161173\pi\)
\(858\) 0 0
\(859\) 2.96062e7 1.36899 0.684494 0.729019i \(-0.260023\pi\)
0.684494 + 0.729019i \(0.260023\pi\)
\(860\) 0 0
\(861\) 2.08942e7 + 2.20038e7i 0.960545 + 1.01156i
\(862\) 0 0
\(863\) 205022. 0.00937074 0.00468537 0.999989i \(-0.498509\pi\)
0.00468537 + 0.999989i \(0.498509\pi\)
\(864\) 0 0
\(865\) −367126. −0.0166830
\(866\) 0 0
\(867\) −1.42114e7 1.49662e7i −0.642082 0.676180i
\(868\) 0 0
\(869\) −2.92897e7 −1.31573
\(870\) 0 0
\(871\) 2.74109e7i 1.22427i
\(872\) 0 0
\(873\) 1.18202e6 2.28335e7i 0.0524916 1.01400i
\(874\) 0 0
\(875\) 4.96480e6i 0.219221i
\(876\) 0 0
\(877\) 2.06342e6i 0.0905918i 0.998974 + 0.0452959i \(0.0144231\pi\)
−0.998974 + 0.0452959i \(0.985577\pi\)
\(878\) 0 0
\(879\) −1.71070e7 1.80154e7i −0.746794 0.786453i
\(880\) 0 0
\(881\) 1.20461e7i 0.522887i −0.965219 0.261444i \(-0.915801\pi\)
0.965219 0.261444i \(-0.0841985\pi\)
\(882\) 0 0
\(883\) 3.90309e7 1.68464 0.842319 0.538979i \(-0.181190\pi\)
0.842319 + 0.538979i \(0.181190\pi\)
\(884\) 0 0
\(885\) 465326. 441860.i 0.0199709 0.0189639i
\(886\) 0 0
\(887\) 1.19857e7 0.511509 0.255754 0.966742i \(-0.417676\pi\)
0.255754 + 0.966742i \(0.417676\pi\)
\(888\) 0 0
\(889\) 3.73252e7 1.58397
\(890\) 0 0
\(891\) 4.34650e6 4.18689e7i 0.183420 1.76684i
\(892\) 0 0
\(893\) 2.76267e7 1.15931
\(894\) 0 0
\(895\) 1.77710e6i 0.0741575i
\(896\) 0 0
\(897\) −2.79037e7 + 2.64966e7i −1.15792 + 1.09953i
\(898\) 0 0
\(899\) 5.69635e7i 2.35070i
\(900\) 0 0
\(901\) 2.28845e6i 0.0939139i
\(902\) 0 0
\(903\) 8.12033e6 7.71084e6i 0.331401 0.314690i
\(904\) 0 0
\(905\) 2.03603e6i 0.0826349i
\(906\) 0 0
\(907\) −5.52158e6 −0.222867 −0.111433 0.993772i \(-0.535544\pi\)
−0.111433 + 0.993772i \(0.535544\pi\)
\(908\) 0 0
\(909\) −1.05458e6 + 2.03717e7i −0.0423321 + 0.817745i
\(910\) 0 0
\(911\) −521804. −0.0208310 −0.0104155 0.999946i \(-0.503315\pi\)
−0.0104155 + 0.999946i \(0.503315\pi\)
\(912\) 0 0
\(913\) 1.47854e6 0.0587026
\(914\) 0 0
\(915\) −741101. + 703729.i −0.0292634 + 0.0277877i
\(916\) 0 0
\(917\) 9.44206e6 0.370803
\(918\) 0 0
\(919\) 6.42906e6i 0.251107i 0.992087 + 0.125554i \(0.0400707\pi\)
−0.992087 + 0.125554i \(0.959929\pi\)
\(920\) 0 0
\(921\) −9.66630e6 1.01796e7i −0.375501 0.395442i
\(922\) 0 0
\(923\) 1.40417e7i 0.542521i
\(924\) 0 0
\(925\) 2.28363e7i 0.877549i
\(926\) 0 0
\(927\) −2.45995e7 1.27344e6i −0.940213 0.0486720i
\(928\) 0 0
\(929\) 1.52604e7i 0.580132i 0.957007 + 0.290066i \(0.0936772\pi\)
−0.957007 + 0.290066i \(0.906323\pi\)
\(930\) 0 0
\(931\) −4.51438e7 −1.70696
\(932\) 0 0
\(933\) −1.23983e7 1.30567e7i −0.466292 0.491055i
\(934\) 0 0
\(935\) −938684. −0.0351148
\(936\) 0 0
\(937\) −1.99702e7 −0.743078 −0.371539 0.928417i \(-0.621170\pi\)
−0.371539 + 0.928417i \(0.621170\pi\)
\(938\) 0 0
\(939\) 2.33203e7 + 2.45587e7i 0.863118 + 0.908954i
\(940\) 0 0
\(941\) 1.96538e7 0.723557 0.361778 0.932264i \(-0.382170\pi\)
0.361778 + 0.932264i \(0.382170\pi\)
\(942\) 0 0
\(943\) 3.18895e7i 1.16780i
\(944\) 0 0
\(945\) −1.96210e6 2.29287e6i −0.0714729 0.0835218i
\(946\) 0 0
\(947\) 1.78564e7i 0.647024i 0.946224 + 0.323512i \(0.104864\pi\)
−0.946224 + 0.323512i \(0.895136\pi\)
\(948\) 0 0
\(949\) 1.86654e7i 0.672779i
\(950\) 0 0
\(951\) 1.40525e7 + 1.47988e7i 0.503853 + 0.530610i
\(952\) 0 0
\(953\) 2.73344e7i 0.974939i 0.873140 + 0.487469i \(0.162080\pi\)
−0.873140 + 0.487469i \(0.837920\pi\)
\(954\) 0 0
\(955\) 3.03745e6 0.107771
\(956\) 0 0
\(957\) −5.09694e7 + 4.83991e7i −1.79899 + 1.70827i
\(958\) 0 0
\(959\) 8.10101e6 0.284441
\(960\) 0 0
\(961\) −5.24773e7 −1.83300
\(962\) 0 0
\(963\) 2.64602e6 + 136976.i 0.0919448 + 0.00475970i
\(964\) 0 0
\(965\) 655813. 0.0226705
\(966\) 0 0
\(967\) 4.89954e6i 0.168496i 0.996445 + 0.0842479i \(0.0268488\pi\)
−0.996445 + 0.0842479i \(0.973151\pi\)
\(968\) 0 0
\(969\) −8.63815e6 + 8.20255e6i −0.295537 + 0.280633i
\(970\) 0 0
\(971\) 4.88073e7i 1.66126i 0.556828 + 0.830628i \(0.312018\pi\)
−0.556828 + 0.830628i \(0.687982\pi\)
\(972\) 0 0
\(973\) 1.54552e7i 0.523350i
\(974\) 0 0
\(975\) −2.82455e7 + 2.68212e7i −0.951565 + 0.903579i
\(976\) 0 0
\(977\) 1.11727e7i 0.374474i −0.982315 0.187237i \(-0.940047\pi\)
0.982315 0.187237i \(-0.0599533\pi\)
\(978\) 0 0
\(979\) 9.71632e7 3.24000
\(980\) 0 0
\(981\) −3.12189e7 1.61611e6i −1.03573 0.0536164i
\(982\) 0 0
\(983\) 2.07531e7 0.685014 0.342507 0.939515i \(-0.388724\pi\)
0.342507 + 0.939515i \(0.388724\pi\)
\(984\) 0 0
\(985\) 1.36605e6 0.0448617
\(986\) 0 0
\(987\) 2.37096e7 2.25139e7i 0.774695 0.735629i
\(988\) 0 0
\(989\) −1.17686e7 −0.382589
\(990\) 0 0
\(991\) 1.17133e7i 0.378873i 0.981893 + 0.189436i \(0.0606661\pi\)
−0.981893 + 0.189436i \(0.939334\pi\)
\(992\) 0 0
\(993\) 7.36337e6 + 7.75441e6i 0.236976 + 0.249560i
\(994\) 0 0
\(995\) 427325.i 0.0136836i
\(996\) 0 0
\(997\) 3.69725e7i 1.17799i 0.808137 + 0.588995i \(0.200476\pi\)
−0.808137 + 0.588995i \(0.799524\pi\)
\(998\) 0 0
\(999\) 1.81024e7 + 2.11541e7i 0.573882 + 0.670627i
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 384.6.f.f.191.5 yes 20
3.2 odd 2 384.6.f.e.191.6 yes 20
4.3 odd 2 384.6.f.e.191.16 yes 20
8.3 odd 2 384.6.f.e.191.5 20
8.5 even 2 inner 384.6.f.f.191.16 yes 20
12.11 even 2 inner 384.6.f.f.191.15 yes 20
24.5 odd 2 384.6.f.e.191.15 yes 20
24.11 even 2 inner 384.6.f.f.191.6 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.6.f.e.191.5 20 8.3 odd 2
384.6.f.e.191.6 yes 20 3.2 odd 2
384.6.f.e.191.15 yes 20 24.5 odd 2
384.6.f.e.191.16 yes 20 4.3 odd 2
384.6.f.f.191.5 yes 20 1.1 even 1 trivial
384.6.f.f.191.6 yes 20 24.11 even 2 inner
384.6.f.f.191.15 yes 20 12.11 even 2 inner
384.6.f.f.191.16 yes 20 8.5 even 2 inner