Properties

Label 384.6.c.c.383.18
Level $384$
Weight $6$
Character 384.383
Analytic conductor $61.587$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,6,Mod(383,384)]
 
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, 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("384.383");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 384.c (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: \( x^{20} + 306 x^{18} + 37827 x^{16} + 2442168 x^{14} + 88368509 x^{12} + 1774000974 x^{10} + 18093172325 x^{8} + 74958811500 x^{6} + 79355888475 x^{4} + \cdots + 2870280625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{88}\cdot 3^{14}\cdot 41^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 383.18
Root \(0.671758i\) of defining polynomial
Character \(\chi\) \(=\) 384.383
Dual form 384.6.c.c.383.17

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(15.1145 + 3.81453i) q^{3} -40.5648i q^{5} +81.4905i q^{7} +(213.899 + 115.310i) q^{9} +O(q^{10})\) \(q+(15.1145 + 3.81453i) q^{3} -40.5648i q^{5} +81.4905i q^{7} +(213.899 + 115.310i) q^{9} -499.309 q^{11} +262.278 q^{13} +(154.735 - 613.118i) q^{15} -565.287i q^{17} -2137.81i q^{19} +(-310.848 + 1231.69i) q^{21} +3650.06 q^{23} +1479.50 q^{25} +(2793.13 + 2558.77i) q^{27} +7310.25i q^{29} -2216.91i q^{31} +(-7546.83 - 1904.63i) q^{33} +3305.64 q^{35} +10097.4 q^{37} +(3964.21 + 1000.47i) q^{39} -2865.65i q^{41} -8282.13i q^{43} +(4677.51 - 8676.75i) q^{45} +3920.49 q^{47} +10166.3 q^{49} +(2156.30 - 8544.05i) q^{51} -1497.09i q^{53} +20254.4i q^{55} +(8154.71 - 32311.9i) q^{57} +6360.57 q^{59} -5095.51 q^{61} +(-9396.64 + 17430.7i) q^{63} -10639.2i q^{65} -58778.9i q^{67} +(55169.0 + 13923.2i) q^{69} +24686.6 q^{71} +34838.3 q^{73} +(22362.0 + 5643.59i) q^{75} -40689.0i q^{77} -37197.4i q^{79} +(32456.4 + 49329.2i) q^{81} -77701.0 q^{83} -22930.7 q^{85} +(-27885.1 + 110491. i) q^{87} +38427.8i q^{89} +21373.2i q^{91} +(8456.48 - 33507.6i) q^{93} -86719.6 q^{95} +148735. q^{97} +(-106802. - 57575.2i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{3} + 948 q^{11} - 852 q^{15} + 1640 q^{21} - 328 q^{23} - 12500 q^{25} + 2030 q^{27} + 2836 q^{33} - 7184 q^{35} + 15056 q^{37} + 12980 q^{39} + 11800 q^{45} - 36640 q^{47} - 33388 q^{49} + 1936 q^{51} + 15404 q^{57} + 62908 q^{59} + 73264 q^{61} - 23608 q^{63} - 84024 q^{69} - 34888 q^{71} + 52568 q^{73} + 115698 q^{75} + 55444 q^{81} - 225172 q^{83} - 30112 q^{85} + 225700 q^{87} - 148016 q^{93} - 418616 q^{95} + 7600 q^{97} + 378260 q^{99}+O(q^{100}) \) 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
\(3\) 15.1145 + 3.81453i 0.969598 + 0.244702i
\(4\) 0 0
\(5\) 40.5648i 0.725645i −0.931858 0.362822i \(-0.881813\pi\)
0.931858 0.362822i \(-0.118187\pi\)
\(6\) 0 0
\(7\) 81.4905i 0.628582i 0.949327 + 0.314291i \(0.101767\pi\)
−0.949327 + 0.314291i \(0.898233\pi\)
\(8\) 0 0
\(9\) 213.899 + 115.310i 0.880242 + 0.474525i
\(10\) 0 0
\(11\) −499.309 −1.24419 −0.622097 0.782940i \(-0.713719\pi\)
−0.622097 + 0.782940i \(0.713719\pi\)
\(12\) 0 0
\(13\) 262.278 0.430431 0.215216 0.976567i \(-0.430955\pi\)
0.215216 + 0.976567i \(0.430955\pi\)
\(14\) 0 0
\(15\) 154.735 613.118i 0.177567 0.703584i
\(16\) 0 0
\(17\) 565.287i 0.474402i −0.971461 0.237201i \(-0.923770\pi\)
0.971461 0.237201i \(-0.0762300\pi\)
\(18\) 0 0
\(19\) 2137.81i 1.35858i −0.733872 0.679288i \(-0.762289\pi\)
0.733872 0.679288i \(-0.237711\pi\)
\(20\) 0 0
\(21\) −310.848 + 1231.69i −0.153815 + 0.609472i
\(22\) 0 0
\(23\) 3650.06 1.43873 0.719367 0.694630i \(-0.244432\pi\)
0.719367 + 0.694630i \(0.244432\pi\)
\(24\) 0 0
\(25\) 1479.50 0.473440
\(26\) 0 0
\(27\) 2793.13 + 2558.77i 0.737364 + 0.675496i
\(28\) 0 0
\(29\) 7310.25i 1.61413i 0.590466 + 0.807063i \(0.298944\pi\)
−0.590466 + 0.807063i \(0.701056\pi\)
\(30\) 0 0
\(31\) 2216.91i 0.414328i −0.978306 0.207164i \(-0.933577\pi\)
0.978306 0.207164i \(-0.0664234\pi\)
\(32\) 0 0
\(33\) −7546.83 1904.63i −1.20637 0.304457i
\(34\) 0 0
\(35\) 3305.64 0.456127
\(36\) 0 0
\(37\) 10097.4 1.21256 0.606280 0.795251i \(-0.292661\pi\)
0.606280 + 0.795251i \(0.292661\pi\)
\(38\) 0 0
\(39\) 3964.21 + 1000.47i 0.417345 + 0.105327i
\(40\) 0 0
\(41\) 2865.65i 0.266234i −0.991100 0.133117i \(-0.957501\pi\)
0.991100 0.133117i \(-0.0424987\pi\)
\(42\) 0 0
\(43\) 8282.13i 0.683079i −0.939867 0.341539i \(-0.889052\pi\)
0.939867 0.341539i \(-0.110948\pi\)
\(44\) 0 0
\(45\) 4677.51 8676.75i 0.344337 0.638743i
\(46\) 0 0
\(47\) 3920.49 0.258878 0.129439 0.991587i \(-0.458682\pi\)
0.129439 + 0.991587i \(0.458682\pi\)
\(48\) 0 0
\(49\) 10166.3 0.604884
\(50\) 0 0
\(51\) 2156.30 8544.05i 0.116087 0.459979i
\(52\) 0 0
\(53\) 1497.09i 0.0732080i −0.999330 0.0366040i \(-0.988346\pi\)
0.999330 0.0366040i \(-0.0116540\pi\)
\(54\) 0 0
\(55\) 20254.4i 0.902843i
\(56\) 0 0
\(57\) 8154.71 32311.9i 0.332446 1.31727i
\(58\) 0 0
\(59\) 6360.57 0.237885 0.118942 0.992901i \(-0.462050\pi\)
0.118942 + 0.992901i \(0.462050\pi\)
\(60\) 0 0
\(61\) −5095.51 −0.175333 −0.0876664 0.996150i \(-0.527941\pi\)
−0.0876664 + 0.996150i \(0.527941\pi\)
\(62\) 0 0
\(63\) −9396.64 + 17430.7i −0.298278 + 0.553304i
\(64\) 0 0
\(65\) 10639.2i 0.312340i
\(66\) 0 0
\(67\) 58778.9i 1.59969i −0.600209 0.799843i \(-0.704916\pi\)
0.600209 0.799843i \(-0.295084\pi\)
\(68\) 0 0
\(69\) 55169.0 + 13923.2i 1.39499 + 0.352061i
\(70\) 0 0
\(71\) 24686.6 0.581185 0.290593 0.956847i \(-0.406148\pi\)
0.290593 + 0.956847i \(0.406148\pi\)
\(72\) 0 0
\(73\) 34838.3 0.765157 0.382578 0.923923i \(-0.375036\pi\)
0.382578 + 0.923923i \(0.375036\pi\)
\(74\) 0 0
\(75\) 22362.0 + 5643.59i 0.459047 + 0.115852i
\(76\) 0 0
\(77\) 40689.0i 0.782078i
\(78\) 0 0
\(79\) 37197.4i 0.670572i −0.942116 0.335286i \(-0.891167\pi\)
0.942116 0.335286i \(-0.108833\pi\)
\(80\) 0 0
\(81\) 32456.4 + 49329.2i 0.549652 + 0.835394i
\(82\) 0 0
\(83\) −77701.0 −1.23803 −0.619015 0.785379i \(-0.712468\pi\)
−0.619015 + 0.785379i \(0.712468\pi\)
\(84\) 0 0
\(85\) −22930.7 −0.344247
\(86\) 0 0
\(87\) −27885.1 + 110491.i −0.394980 + 1.56505i
\(88\) 0 0
\(89\) 38427.8i 0.514245i 0.966379 + 0.257123i \(0.0827745\pi\)
−0.966379 + 0.257123i \(0.917226\pi\)
\(90\) 0 0
\(91\) 21373.2i 0.270561i
\(92\) 0 0
\(93\) 8456.48 33507.6i 0.101387 0.401732i
\(94\) 0 0
\(95\) −86719.6 −0.985844
\(96\) 0 0
\(97\) 148735. 1.60503 0.802515 0.596632i \(-0.203495\pi\)
0.802515 + 0.596632i \(0.203495\pi\)
\(98\) 0 0
\(99\) −106802. 57575.2i −1.09519 0.590401i
\(100\) 0 0
\(101\) 43258.7i 0.421959i −0.977490 0.210980i \(-0.932335\pi\)
0.977490 0.210980i \(-0.0676654\pi\)
\(102\) 0 0
\(103\) 177132.i 1.64514i 0.568662 + 0.822571i \(0.307461\pi\)
−0.568662 + 0.822571i \(0.692539\pi\)
\(104\) 0 0
\(105\) 49963.3 + 12609.5i 0.442260 + 0.111615i
\(106\) 0 0
\(107\) −108837. −0.919006 −0.459503 0.888176i \(-0.651973\pi\)
−0.459503 + 0.888176i \(0.651973\pi\)
\(108\) 0 0
\(109\) 242786. 1.95730 0.978651 0.205530i \(-0.0658919\pi\)
0.978651 + 0.205530i \(0.0658919\pi\)
\(110\) 0 0
\(111\) 152617. + 38516.6i 1.17570 + 0.296716i
\(112\) 0 0
\(113\) 111435.i 0.820964i 0.911869 + 0.410482i \(0.134640\pi\)
−0.911869 + 0.410482i \(0.865360\pi\)
\(114\) 0 0
\(115\) 148064.i 1.04401i
\(116\) 0 0
\(117\) 56101.0 + 30243.2i 0.378883 + 0.204250i
\(118\) 0 0
\(119\) 46065.5 0.298201
\(120\) 0 0
\(121\) 88258.9 0.548018
\(122\) 0 0
\(123\) 10931.1 43313.0i 0.0651480 0.258140i
\(124\) 0 0
\(125\) 186780.i 1.06919i
\(126\) 0 0
\(127\) 310454.i 1.70800i 0.520272 + 0.854001i \(0.325831\pi\)
−0.520272 + 0.854001i \(0.674169\pi\)
\(128\) 0 0
\(129\) 31592.4 125181.i 0.167151 0.662312i
\(130\) 0 0
\(131\) 126488. 0.643980 0.321990 0.946743i \(-0.395648\pi\)
0.321990 + 0.946743i \(0.395648\pi\)
\(132\) 0 0
\(133\) 174211. 0.853977
\(134\) 0 0
\(135\) 103796. 113303.i 0.490170 0.535064i
\(136\) 0 0
\(137\) 407875.i 1.85663i −0.371795 0.928315i \(-0.621258\pi\)
0.371795 0.928315i \(-0.378742\pi\)
\(138\) 0 0
\(139\) 345151.i 1.51521i −0.652714 0.757604i \(-0.726370\pi\)
0.652714 0.757604i \(-0.273630\pi\)
\(140\) 0 0
\(141\) 59256.4 + 14954.8i 0.251008 + 0.0633481i
\(142\) 0 0
\(143\) −130958. −0.535540
\(144\) 0 0
\(145\) 296539. 1.17128
\(146\) 0 0
\(147\) 153659. + 38779.6i 0.586495 + 0.148016i
\(148\) 0 0
\(149\) 255290.i 0.942036i 0.882124 + 0.471018i \(0.156113\pi\)
−0.882124 + 0.471018i \(0.843887\pi\)
\(150\) 0 0
\(151\) 39203.8i 0.139922i −0.997550 0.0699610i \(-0.977713\pi\)
0.997550 0.0699610i \(-0.0222875\pi\)
\(152\) 0 0
\(153\) 65183.0 120914.i 0.225116 0.417589i
\(154\) 0 0
\(155\) −89928.6 −0.300655
\(156\) 0 0
\(157\) −90738.1 −0.293792 −0.146896 0.989152i \(-0.546928\pi\)
−0.146896 + 0.989152i \(0.546928\pi\)
\(158\) 0 0
\(159\) 5710.69 22627.8i 0.0179141 0.0709824i
\(160\) 0 0
\(161\) 297445.i 0.904362i
\(162\) 0 0
\(163\) 82660.0i 0.243684i −0.992550 0.121842i \(-0.961120\pi\)
0.992550 0.121842i \(-0.0388801\pi\)
\(164\) 0 0
\(165\) −77260.8 + 306136.i −0.220927 + 0.875395i
\(166\) 0 0
\(167\) 1354.48 0.00375821 0.00187910 0.999998i \(-0.499402\pi\)
0.00187910 + 0.999998i \(0.499402\pi\)
\(168\) 0 0
\(169\) −302503. −0.814729
\(170\) 0 0
\(171\) 246510. 457274.i 0.644679 1.19588i
\(172\) 0 0
\(173\) 752693.i 1.91207i −0.293260 0.956033i \(-0.594740\pi\)
0.293260 0.956033i \(-0.405260\pi\)
\(174\) 0 0
\(175\) 120565.i 0.297596i
\(176\) 0 0
\(177\) 96137.2 + 24262.6i 0.230652 + 0.0582108i
\(178\) 0 0
\(179\) 145447. 0.339292 0.169646 0.985505i \(-0.445738\pi\)
0.169646 + 0.985505i \(0.445738\pi\)
\(180\) 0 0
\(181\) 816899. 1.85341 0.926706 0.375787i \(-0.122628\pi\)
0.926706 + 0.375787i \(0.122628\pi\)
\(182\) 0 0
\(183\) −77016.4 19437.0i −0.170002 0.0429043i
\(184\) 0 0
\(185\) 409597.i 0.879888i
\(186\) 0 0
\(187\) 282253.i 0.590248i
\(188\) 0 0
\(189\) −208516. + 227614.i −0.424605 + 0.463494i
\(190\) 0 0
\(191\) 570025. 1.13060 0.565302 0.824884i \(-0.308760\pi\)
0.565302 + 0.824884i \(0.308760\pi\)
\(192\) 0 0
\(193\) −402670. −0.778138 −0.389069 0.921209i \(-0.627203\pi\)
−0.389069 + 0.921209i \(0.627203\pi\)
\(194\) 0 0
\(195\) 40583.7 160807.i 0.0764302 0.302844i
\(196\) 0 0
\(197\) 697392.i 1.28030i 0.768250 + 0.640150i \(0.221128\pi\)
−0.768250 + 0.640150i \(0.778872\pi\)
\(198\) 0 0
\(199\) 17654.7i 0.0316029i −0.999875 0.0158015i \(-0.994970\pi\)
0.999875 0.0158015i \(-0.00502997\pi\)
\(200\) 0 0
\(201\) 224214. 888417.i 0.391446 1.55105i
\(202\) 0 0
\(203\) −595716. −1.01461
\(204\) 0 0
\(205\) −116245. −0.193191
\(206\) 0 0
\(207\) 780743. + 420887.i 1.26643 + 0.682715i
\(208\) 0 0
\(209\) 1.06743e6i 1.69033i
\(210\) 0 0
\(211\) 705719.i 1.09125i 0.838028 + 0.545627i \(0.183708\pi\)
−0.838028 + 0.545627i \(0.816292\pi\)
\(212\) 0 0
\(213\) 373126. + 94167.5i 0.563516 + 0.142217i
\(214\) 0 0
\(215\) −335963. −0.495673
\(216\) 0 0
\(217\) 180658. 0.260439
\(218\) 0 0
\(219\) 526566. + 132892.i 0.741895 + 0.187235i
\(220\) 0 0
\(221\) 148262.i 0.204197i
\(222\) 0 0
\(223\) 514390.i 0.692676i −0.938110 0.346338i \(-0.887425\pi\)
0.938110 0.346338i \(-0.112575\pi\)
\(224\) 0 0
\(225\) 316463. + 170601.i 0.416742 + 0.224659i
\(226\) 0 0
\(227\) −689433. −0.888030 −0.444015 0.896019i \(-0.646446\pi\)
−0.444015 + 0.896019i \(0.646446\pi\)
\(228\) 0 0
\(229\) −1.16383e6 −1.46656 −0.733280 0.679927i \(-0.762011\pi\)
−0.733280 + 0.679927i \(0.762011\pi\)
\(230\) 0 0
\(231\) 155209. 614996.i 0.191376 0.758302i
\(232\) 0 0
\(233\) 918501.i 1.10838i 0.832389 + 0.554192i \(0.186973\pi\)
−0.832389 + 0.554192i \(0.813027\pi\)
\(234\) 0 0
\(235\) 159034.i 0.187854i
\(236\) 0 0
\(237\) 141891. 562222.i 0.164090 0.650185i
\(238\) 0 0
\(239\) −1.74289e6 −1.97367 −0.986836 0.161722i \(-0.948295\pi\)
−0.986836 + 0.161722i \(0.948295\pi\)
\(240\) 0 0
\(241\) −819192. −0.908538 −0.454269 0.890865i \(-0.650100\pi\)
−0.454269 + 0.890865i \(0.650100\pi\)
\(242\) 0 0
\(243\) 302396. + 869394.i 0.328519 + 0.944497i
\(244\) 0 0
\(245\) 412393.i 0.438931i
\(246\) 0 0
\(247\) 560699.i 0.584774i
\(248\) 0 0
\(249\) −1.17442e6 296393.i −1.20039 0.302949i
\(250\) 0 0
\(251\) −594929. −0.596047 −0.298024 0.954559i \(-0.596327\pi\)
−0.298024 + 0.954559i \(0.596327\pi\)
\(252\) 0 0
\(253\) −1.82251e6 −1.79006
\(254\) 0 0
\(255\) −346587. 87469.9i −0.333782 0.0842380i
\(256\) 0 0
\(257\) 1.60886e6i 1.51944i 0.650248 + 0.759722i \(0.274665\pi\)
−0.650248 + 0.759722i \(0.725335\pi\)
\(258\) 0 0
\(259\) 822839.i 0.762194i
\(260\) 0 0
\(261\) −842942. + 1.56365e6i −0.765943 + 1.42082i
\(262\) 0 0
\(263\) −1.44015e6 −1.28386 −0.641929 0.766764i \(-0.721866\pi\)
−0.641929 + 0.766764i \(0.721866\pi\)
\(264\) 0 0
\(265\) −60729.1 −0.0531230
\(266\) 0 0
\(267\) −146584. + 580819.i −0.125837 + 0.498611i
\(268\) 0 0
\(269\) 1.85670e6i 1.56445i −0.622996 0.782225i \(-0.714085\pi\)
0.622996 0.782225i \(-0.285915\pi\)
\(270\) 0 0
\(271\) 210714.i 0.174289i −0.996196 0.0871447i \(-0.972226\pi\)
0.996196 0.0871447i \(-0.0277742\pi\)
\(272\) 0 0
\(273\) −81528.6 + 323046.i −0.0662069 + 0.262336i
\(274\) 0 0
\(275\) −738728. −0.589051
\(276\) 0 0
\(277\) −366298. −0.286837 −0.143419 0.989662i \(-0.545810\pi\)
−0.143419 + 0.989662i \(0.545810\pi\)
\(278\) 0 0
\(279\) 255632. 474195.i 0.196609 0.364709i
\(280\) 0 0
\(281\) 962614.i 0.727254i −0.931545 0.363627i \(-0.881538\pi\)
0.931545 0.363627i \(-0.118462\pi\)
\(282\) 0 0
\(283\) 1.59949e6i 1.18717i 0.804770 + 0.593587i \(0.202289\pi\)
−0.804770 + 0.593587i \(0.797711\pi\)
\(284\) 0 0
\(285\) −1.31073e6 330794.i −0.955873 0.241238i
\(286\) 0 0
\(287\) 233524. 0.167350
\(288\) 0 0
\(289\) 1.10031e6 0.774943
\(290\) 0 0
\(291\) 2.24806e6 + 567352.i 1.55623 + 0.392754i
\(292\) 0 0
\(293\) 451436.i 0.307204i −0.988133 0.153602i \(-0.950913\pi\)
0.988133 0.153602i \(-0.0490874\pi\)
\(294\) 0 0
\(295\) 258015.i 0.172620i
\(296\) 0 0
\(297\) −1.39464e6 1.27762e6i −0.917424 0.840448i
\(298\) 0 0
\(299\) 957331. 0.619276
\(300\) 0 0
\(301\) 674915. 0.429371
\(302\) 0 0
\(303\) 165012. 653836.i 0.103254 0.409131i
\(304\) 0 0
\(305\) 206698.i 0.127229i
\(306\) 0 0
\(307\) 2.49010e6i 1.50790i −0.656934 0.753948i \(-0.728147\pi\)
0.656934 0.753948i \(-0.271853\pi\)
\(308\) 0 0
\(309\) −675674. + 2.67727e6i −0.402569 + 1.59513i
\(310\) 0 0
\(311\) −752051. −0.440906 −0.220453 0.975398i \(-0.570754\pi\)
−0.220453 + 0.975398i \(0.570754\pi\)
\(312\) 0 0
\(313\) −2.30499e6 −1.32987 −0.664934 0.746902i \(-0.731540\pi\)
−0.664934 + 0.746902i \(0.731540\pi\)
\(314\) 0 0
\(315\) 707073. + 381173.i 0.401502 + 0.216444i
\(316\) 0 0
\(317\) 1.11295e6i 0.622054i −0.950401 0.311027i \(-0.899327\pi\)
0.950401 0.311027i \(-0.100673\pi\)
\(318\) 0 0
\(319\) 3.65008e6i 2.00828i
\(320\) 0 0
\(321\) −1.64503e6 415163.i −0.891067 0.224883i
\(322\) 0 0
\(323\) −1.20847e6 −0.644511
\(324\) 0 0
\(325\) 388040. 0.203783
\(326\) 0 0
\(327\) 3.66960e6 + 926114.i 1.89780 + 0.478955i
\(328\) 0 0
\(329\) 319483.i 0.162726i
\(330\) 0 0
\(331\) 2.51218e6i 1.26032i 0.776465 + 0.630160i \(0.217011\pi\)
−0.776465 + 0.630160i \(0.782989\pi\)
\(332\) 0 0
\(333\) 2.15981e6 + 1.16432e6i 1.06735 + 0.575391i
\(334\) 0 0
\(335\) −2.38435e6 −1.16080
\(336\) 0 0
\(337\) −39468.6 −0.0189311 −0.00946556 0.999955i \(-0.503013\pi\)
−0.00946556 + 0.999955i \(0.503013\pi\)
\(338\) 0 0
\(339\) −425070. + 1.68428e6i −0.200892 + 0.796005i
\(340\) 0 0
\(341\) 1.10693e6i 0.515505i
\(342\) 0 0
\(343\) 2.19807e6i 1.00880i
\(344\) 0 0
\(345\) 564793. 2.23792e6i 0.255471 1.01227i
\(346\) 0 0
\(347\) 602701. 0.268707 0.134353 0.990933i \(-0.457104\pi\)
0.134353 + 0.990933i \(0.457104\pi\)
\(348\) 0 0
\(349\) −3.47666e6 −1.52791 −0.763956 0.645268i \(-0.776746\pi\)
−0.763956 + 0.645268i \(0.776746\pi\)
\(350\) 0 0
\(351\) 732577. + 671111.i 0.317384 + 0.290754i
\(352\) 0 0
\(353\) 3.70548e6i 1.58273i 0.611341 + 0.791367i \(0.290630\pi\)
−0.611341 + 0.791367i \(0.709370\pi\)
\(354\) 0 0
\(355\) 1.00140e6i 0.421734i
\(356\) 0 0
\(357\) 696259. + 175718.i 0.289135 + 0.0729703i
\(358\) 0 0
\(359\) 1.00364e6 0.411001 0.205501 0.978657i \(-0.434118\pi\)
0.205501 + 0.978657i \(0.434118\pi\)
\(360\) 0 0
\(361\) −2.09411e6 −0.845730
\(362\) 0 0
\(363\) 1.33399e6 + 336666.i 0.531358 + 0.134101i
\(364\) 0 0
\(365\) 1.41321e6i 0.555232i
\(366\) 0 0
\(367\) 2.82819e6i 1.09608i −0.836451 0.548042i \(-0.815373\pi\)
0.836451 0.548042i \(-0.184627\pi\)
\(368\) 0 0
\(369\) 330437. 612960.i 0.126335 0.234351i
\(370\) 0 0
\(371\) 121999. 0.0460172
\(372\) 0 0
\(373\) −1.58345e6 −0.589296 −0.294648 0.955606i \(-0.595202\pi\)
−0.294648 + 0.955606i \(0.595202\pi\)
\(374\) 0 0
\(375\) 712479. 2.82310e6i 0.261634 1.03669i
\(376\) 0 0
\(377\) 1.91732e6i 0.694770i
\(378\) 0 0
\(379\) 1.10073e6i 0.393624i 0.980441 + 0.196812i \(0.0630588\pi\)
−0.980441 + 0.196812i \(0.936941\pi\)
\(380\) 0 0
\(381\) −1.18424e6 + 4.69237e6i −0.417951 + 1.65608i
\(382\) 0 0
\(383\) 729536. 0.254126 0.127063 0.991895i \(-0.459445\pi\)
0.127063 + 0.991895i \(0.459445\pi\)
\(384\) 0 0
\(385\) −1.65054e6 −0.567511
\(386\) 0 0
\(387\) 955009. 1.77154e6i 0.324138 0.601275i
\(388\) 0 0
\(389\) 1.70809e6i 0.572319i 0.958182 + 0.286159i \(0.0923787\pi\)
−0.958182 + 0.286159i \(0.907621\pi\)
\(390\) 0 0
\(391\) 2.06333e6i 0.682538i
\(392\) 0 0
\(393\) 1.91181e6 + 482493.i 0.624402 + 0.157583i
\(394\) 0 0
\(395\) −1.50890e6 −0.486597
\(396\) 0 0
\(397\) 4.33954e6 1.38187 0.690935 0.722917i \(-0.257199\pi\)
0.690935 + 0.722917i \(0.257199\pi\)
\(398\) 0 0
\(399\) 2.63312e6 + 664532.i 0.828015 + 0.208970i
\(400\) 0 0
\(401\) 3.02307e6i 0.938832i −0.882977 0.469416i \(-0.844464\pi\)
0.882977 0.469416i \(-0.155536\pi\)
\(402\) 0 0
\(403\) 581448.i 0.178340i
\(404\) 0 0
\(405\) 2.00103e6 1.31659e6i 0.606199 0.398852i
\(406\) 0 0
\(407\) −5.04171e6 −1.50866
\(408\) 0 0
\(409\) 3.62103e6 1.07035 0.535173 0.844743i \(-0.320246\pi\)
0.535173 + 0.844743i \(0.320246\pi\)
\(410\) 0 0
\(411\) 1.55585e6 6.16484e6i 0.454321 1.80019i
\(412\) 0 0
\(413\) 518327.i 0.149530i
\(414\) 0 0
\(415\) 3.15192e6i 0.898371i
\(416\) 0 0
\(417\) 1.31659e6 5.21680e6i 0.370775 1.46914i
\(418\) 0 0
\(419\) −3.48851e6 −0.970744 −0.485372 0.874308i \(-0.661316\pi\)
−0.485372 + 0.874308i \(0.661316\pi\)
\(420\) 0 0
\(421\) 870260. 0.239301 0.119650 0.992816i \(-0.461823\pi\)
0.119650 + 0.992816i \(0.461823\pi\)
\(422\) 0 0
\(423\) 838588. + 452070.i 0.227876 + 0.122844i
\(424\) 0 0
\(425\) 836342.i 0.224601i
\(426\) 0 0
\(427\) 415236.i 0.110211i
\(428\) 0 0
\(429\) −1.97937e6 499542.i −0.519258 0.131048i
\(430\) 0 0
\(431\) 2.04631e6 0.530614 0.265307 0.964164i \(-0.414527\pi\)
0.265307 + 0.964164i \(0.414527\pi\)
\(432\) 0 0
\(433\) 3382.30 0.000866946 0.000433473 1.00000i \(-0.499862\pi\)
0.000433473 1.00000i \(0.499862\pi\)
\(434\) 0 0
\(435\) 4.48204e6 + 1.13115e6i 1.13567 + 0.286615i
\(436\) 0 0
\(437\) 7.80312e6i 1.95463i
\(438\) 0 0
\(439\) 6.09542e6i 1.50953i 0.655993 + 0.754767i \(0.272250\pi\)
−0.655993 + 0.754767i \(0.727750\pi\)
\(440\) 0 0
\(441\) 2.17456e6 + 1.17227e6i 0.532445 + 0.287033i
\(442\) 0 0
\(443\) −3.51063e6 −0.849915 −0.424957 0.905213i \(-0.639711\pi\)
−0.424957 + 0.905213i \(0.639711\pi\)
\(444\) 0 0
\(445\) 1.55881e6 0.373159
\(446\) 0 0
\(447\) −973809. + 3.85858e6i −0.230518 + 0.913396i
\(448\) 0 0
\(449\) 3.42542e6i 0.801859i 0.916109 + 0.400929i \(0.131313\pi\)
−0.916109 + 0.400929i \(0.868687\pi\)
\(450\) 0 0
\(451\) 1.43085e6i 0.331247i
\(452\) 0 0
\(453\) 149544. 592548.i 0.0342392 0.135668i
\(454\) 0 0
\(455\) 866998. 0.196331
\(456\) 0 0
\(457\) 5.02677e6 1.12590 0.562948 0.826492i \(-0.309667\pi\)
0.562948 + 0.826492i \(0.309667\pi\)
\(458\) 0 0
\(459\) 1.44644e6 1.57892e6i 0.320457 0.349807i
\(460\) 0 0
\(461\) 4.17924e6i 0.915894i −0.888979 0.457947i \(-0.848585\pi\)
0.888979 0.457947i \(-0.151415\pi\)
\(462\) 0 0
\(463\) 3.01723e6i 0.654118i 0.945004 + 0.327059i \(0.106058\pi\)
−0.945004 + 0.327059i \(0.893942\pi\)
\(464\) 0 0
\(465\) −1.35923e6 343035.i −0.291515 0.0735709i
\(466\) 0 0
\(467\) −6.42968e6 −1.36426 −0.682130 0.731231i \(-0.738946\pi\)
−0.682130 + 0.731231i \(0.738946\pi\)
\(468\) 0 0
\(469\) 4.78993e6 1.00553
\(470\) 0 0
\(471\) −1.37146e6 346123.i −0.284860 0.0718915i
\(472\) 0 0
\(473\) 4.13535e6i 0.849883i
\(474\) 0 0
\(475\) 3.16288e6i 0.643204i
\(476\) 0 0
\(477\) 172629. 320226.i 0.0347390 0.0644407i
\(478\) 0 0
\(479\) −5.05852e6 −1.00736 −0.503680 0.863890i \(-0.668021\pi\)
−0.503680 + 0.863890i \(0.668021\pi\)
\(480\) 0 0
\(481\) 2.64832e6 0.521924
\(482\) 0 0
\(483\) −1.13461e6 + 4.49575e6i −0.221299 + 0.876868i
\(484\) 0 0
\(485\) 6.03339e6i 1.16468i
\(486\) 0 0
\(487\) 6.19956e6i 1.18451i 0.805751 + 0.592255i \(0.201762\pi\)
−0.805751 + 0.592255i \(0.798238\pi\)
\(488\) 0 0
\(489\) 315309. 1.24937e6i 0.0596299 0.236275i
\(490\) 0 0
\(491\) −2.31690e6 −0.433714 −0.216857 0.976203i \(-0.569581\pi\)
−0.216857 + 0.976203i \(0.569581\pi\)
\(492\) 0 0
\(493\) 4.13239e6 0.765744
\(494\) 0 0
\(495\) −2.33552e6 + 4.33239e6i −0.428422 + 0.794720i
\(496\) 0 0
\(497\) 2.01172e6i 0.365323i
\(498\) 0 0
\(499\) 9.84932e6i 1.77074i −0.464887 0.885370i \(-0.653905\pi\)
0.464887 0.885370i \(-0.346095\pi\)
\(500\) 0 0
\(501\) 20472.3 + 5166.69i 0.00364395 + 0.000919641i
\(502\) 0 0
\(503\) −6.70258e6 −1.18120 −0.590598 0.806966i \(-0.701108\pi\)
−0.590598 + 0.806966i \(0.701108\pi\)
\(504\) 0 0
\(505\) −1.75478e6 −0.306192
\(506\) 0 0
\(507\) −4.57220e6 1.15391e6i −0.789960 0.199366i
\(508\) 0 0
\(509\) 3.85622e6i 0.659732i 0.944028 + 0.329866i \(0.107004\pi\)
−0.944028 + 0.329866i \(0.892996\pi\)
\(510\) 0 0
\(511\) 2.83900e6i 0.480964i
\(512\) 0 0
\(513\) 5.47016e6 5.97117e6i 0.917713 1.00177i
\(514\) 0 0
\(515\) 7.18531e6 1.19379
\(516\) 0 0
\(517\) −1.95754e6 −0.322095
\(518\) 0 0
\(519\) 2.87117e6 1.13766e7i 0.467886 1.85394i
\(520\) 0 0
\(521\) 8.16460e6i 1.31777i 0.752243 + 0.658886i \(0.228972\pi\)
−0.752243 + 0.658886i \(0.771028\pi\)
\(522\) 0 0
\(523\) 2.95007e6i 0.471605i 0.971801 + 0.235802i \(0.0757718\pi\)
−0.971801 + 0.235802i \(0.924228\pi\)
\(524\) 0 0
\(525\) −459899. + 1.82229e6i −0.0728223 + 0.288548i
\(526\) 0 0
\(527\) −1.25319e6 −0.196558
\(528\) 0 0
\(529\) 6.88659e6 1.06995
\(530\) 0 0
\(531\) 1.36052e6 + 733435.i 0.209396 + 0.112882i
\(532\) 0 0
\(533\) 751598.i 0.114596i
\(534\) 0 0
\(535\) 4.41496e6i 0.666872i
\(536\) 0 0
\(537\) 2.19837e6 + 554813.i 0.328977 + 0.0830254i
\(538\) 0 0
\(539\) −5.07613e6 −0.752594
\(540\) 0 0
\(541\) 7.20843e6 1.05888 0.529441 0.848347i \(-0.322402\pi\)
0.529441 + 0.848347i \(0.322402\pi\)
\(542\) 0 0
\(543\) 1.23471e7 + 3.11608e6i 1.79707 + 0.453534i
\(544\) 0 0
\(545\) 9.84857e6i 1.42031i
\(546\) 0 0
\(547\) 1.12166e6i 0.160285i −0.996783 0.0801427i \(-0.974462\pi\)
0.996783 0.0801427i \(-0.0255376\pi\)
\(548\) 0 0
\(549\) −1.08992e6 587562.i −0.154335 0.0831999i
\(550\) 0 0
\(551\) 1.56279e7 2.19291
\(552\) 0 0
\(553\) 3.03124e6 0.421509
\(554\) 0 0
\(555\) 1.56242e6 6.19087e6i 0.215310 0.853138i
\(556\) 0 0
\(557\) 1.22251e7i 1.66961i 0.550549 + 0.834803i \(0.314419\pi\)
−0.550549 + 0.834803i \(0.685581\pi\)
\(558\) 0 0
\(559\) 2.17222e6i 0.294018i
\(560\) 0 0
\(561\) −1.07666e6 + 4.26613e6i −0.144435 + 0.572304i
\(562\) 0 0
\(563\) 3.55071e6 0.472111 0.236056 0.971740i \(-0.424145\pi\)
0.236056 + 0.971740i \(0.424145\pi\)
\(564\) 0 0
\(565\) 4.52032e6 0.595728
\(566\) 0 0
\(567\) −4.01986e6 + 2.64489e6i −0.525114 + 0.345501i
\(568\) 0 0
\(569\) 692500.i 0.0896683i 0.998994 + 0.0448341i \(0.0142759\pi\)
−0.998994 + 0.0448341i \(0.985724\pi\)
\(570\) 0 0
\(571\) 6.28266e6i 0.806406i −0.915111 0.403203i \(-0.867897\pi\)
0.915111 0.403203i \(-0.132103\pi\)
\(572\) 0 0
\(573\) 8.61566e6 + 2.17437e6i 1.09623 + 0.276661i
\(574\) 0 0
\(575\) 5.40026e6 0.681154
\(576\) 0 0
\(577\) 57039.3 0.00713238 0.00356619 0.999994i \(-0.498865\pi\)
0.00356619 + 0.999994i \(0.498865\pi\)
\(578\) 0 0
\(579\) −6.08618e6 1.53600e6i −0.754481 0.190412i
\(580\) 0 0
\(581\) 6.33190e6i 0.778204i
\(582\) 0 0
\(583\) 747512.i 0.0910849i
\(584\) 0 0
\(585\) 1.22681e6 2.27572e6i 0.148213 0.274935i
\(586\) 0 0
\(587\) −4.50165e6 −0.539233 −0.269616 0.962968i \(-0.586897\pi\)
−0.269616 + 0.962968i \(0.586897\pi\)
\(588\) 0 0
\(589\) −4.73933e6 −0.562897
\(590\) 0 0
\(591\) −2.66022e6 + 1.05408e7i −0.313292 + 1.24138i
\(592\) 0 0
\(593\) 1.63168e7i 1.90545i −0.303832 0.952725i \(-0.598266\pi\)
0.303832 0.952725i \(-0.401734\pi\)
\(594\) 0 0
\(595\) 1.86864e6i 0.216388i
\(596\) 0 0
\(597\) 67344.2 266842.i 0.00773330 0.0306421i
\(598\) 0 0
\(599\) −6.53911e6 −0.744650 −0.372325 0.928103i \(-0.621439\pi\)
−0.372325 + 0.928103i \(0.621439\pi\)
\(600\) 0 0
\(601\) 3.41684e6 0.385868 0.192934 0.981212i \(-0.438200\pi\)
0.192934 + 0.981212i \(0.438200\pi\)
\(602\) 0 0
\(603\) 6.77778e6 1.25727e7i 0.759091 1.40811i
\(604\) 0 0
\(605\) 3.58020e6i 0.397667i
\(606\) 0 0
\(607\) 1.20758e7i 1.33028i −0.746720 0.665139i \(-0.768372\pi\)
0.746720 0.665139i \(-0.231628\pi\)
\(608\) 0 0
\(609\) −9.00398e6 2.27237e6i −0.983764 0.248277i
\(610\) 0 0
\(611\) 1.02826e6 0.111429
\(612\) 0 0
\(613\) −4.10117e6 −0.440815 −0.220407 0.975408i \(-0.570739\pi\)
−0.220407 + 0.975408i \(0.570739\pi\)
\(614\) 0 0
\(615\) −1.75698e6 443418.i −0.187318 0.0472743i
\(616\) 0 0
\(617\) 1.22186e7i 1.29214i 0.763278 + 0.646070i \(0.223588\pi\)
−0.763278 + 0.646070i \(0.776412\pi\)
\(618\) 0 0
\(619\) 4.74806e6i 0.498069i 0.968495 + 0.249035i \(0.0801133\pi\)
−0.968495 + 0.249035i \(0.919887\pi\)
\(620\) 0 0
\(621\) 1.01951e7 + 9.33968e6i 1.06087 + 0.971858i
\(622\) 0 0
\(623\) −3.13150e6 −0.323246
\(624\) 0 0
\(625\) −2.95327e6 −0.302415
\(626\) 0 0
\(627\) −4.07173e6 + 1.61337e7i −0.413628 + 1.63894i
\(628\) 0 0
\(629\) 5.70790e6i 0.575241i
\(630\) 0 0
\(631\) 8.28679e6i 0.828539i −0.910154 0.414269i \(-0.864037\pi\)
0.910154 0.414269i \(-0.135963\pi\)
\(632\) 0 0
\(633\) −2.69198e6 + 1.06666e7i −0.267032 + 1.05808i
\(634\) 0 0
\(635\) 1.25935e7 1.23940
\(636\) 0 0
\(637\) 2.66640e6 0.260361
\(638\) 0 0
\(639\) 5.28043e6 + 2.84660e6i 0.511584 + 0.275787i
\(640\) 0 0
\(641\) 1.29886e7i 1.24858i −0.781193 0.624289i \(-0.785389\pi\)
0.781193 0.624289i \(-0.214611\pi\)
\(642\) 0 0
\(643\) 1.08387e7i 1.03383i −0.856038 0.516914i \(-0.827081\pi\)
0.856038 0.516914i \(-0.172919\pi\)
\(644\) 0 0
\(645\) −5.07792e6 1.28154e6i −0.480603 0.121292i
\(646\) 0 0
\(647\) 7.26979e6 0.682749 0.341375 0.939927i \(-0.389107\pi\)
0.341375 + 0.939927i \(0.389107\pi\)
\(648\) 0 0
\(649\) −3.17589e6 −0.295975
\(650\) 0 0
\(651\) 2.73056e6 + 689123.i 0.252522 + 0.0637300i
\(652\) 0 0
\(653\) 1.41342e7i 1.29714i 0.761154 + 0.648571i \(0.224633\pi\)
−0.761154 + 0.648571i \(0.775367\pi\)
\(654\) 0 0
\(655\) 5.13097e6i 0.467301i
\(656\) 0 0
\(657\) 7.45188e6 + 4.01720e6i 0.673523 + 0.363086i
\(658\) 0 0
\(659\) 3.84668e6 0.345042 0.172521 0.985006i \(-0.444809\pi\)
0.172521 + 0.985006i \(0.444809\pi\)
\(660\) 0 0
\(661\) −1.41825e7 −1.26255 −0.631274 0.775560i \(-0.717468\pi\)
−0.631274 + 0.775560i \(0.717468\pi\)
\(662\) 0 0
\(663\) 565551. 2.24092e6i 0.0499675 0.197989i
\(664\) 0 0
\(665\) 7.06682e6i 0.619684i
\(666\) 0 0
\(667\) 2.66828e7i 2.32230i
\(668\) 0 0
\(669\) 1.96215e6 7.77476e6i 0.169499 0.671617i
\(670\) 0 0
\(671\) 2.54424e6 0.218148
\(672\) 0 0
\(673\) −1.88309e7 −1.60263 −0.801315 0.598243i \(-0.795866\pi\)
−0.801315 + 0.598243i \(0.795866\pi\)
\(674\) 0 0
\(675\) 4.13244e6 + 3.78571e6i 0.349097 + 0.319807i
\(676\) 0 0
\(677\) 2.13428e6i 0.178970i −0.995988 0.0894850i \(-0.971478\pi\)
0.995988 0.0894850i \(-0.0285221\pi\)
\(678\) 0 0
\(679\) 1.21205e7i 1.00889i
\(680\) 0 0
\(681\) −1.04205e7 2.62986e6i −0.861032 0.217303i
\(682\) 0 0
\(683\) 1.35008e7 1.10741 0.553704 0.832714i \(-0.313214\pi\)
0.553704 + 0.832714i \(0.313214\pi\)
\(684\) 0 0
\(685\) −1.65453e7 −1.34725
\(686\) 0 0
\(687\) −1.75907e7 4.43945e6i −1.42197 0.358870i
\(688\) 0 0
\(689\) 392654.i 0.0315110i
\(690\) 0 0
\(691\) 8.61029e6i 0.685998i −0.939336 0.342999i \(-0.888557\pi\)
0.939336 0.342999i \(-0.111443\pi\)
\(692\) 0 0
\(693\) 4.69183e6 8.70333e6i 0.371116 0.688418i
\(694\) 0 0
\(695\) −1.40010e7 −1.09950
\(696\) 0 0
\(697\) −1.61992e6 −0.126302
\(698\) 0 0
\(699\) −3.50365e6 + 1.38827e7i −0.271224 + 1.07469i
\(700\) 0 0
\(701\) 1.35843e7i 1.04410i 0.852915 + 0.522050i \(0.174832\pi\)
−0.852915 + 0.522050i \(0.825168\pi\)
\(702\) 0 0
\(703\) 2.15862e7i 1.64736i
\(704\) 0 0
\(705\) 606639. 2.40372e6i 0.0459682 0.182143i
\(706\) 0 0
\(707\) 3.52518e6 0.265236
\(708\) 0 0
\(709\) 1.58912e7 1.18725 0.593623 0.804743i \(-0.297697\pi\)
0.593623 + 0.804743i \(0.297697\pi\)
\(710\) 0 0
\(711\) 4.28922e6 7.95648e6i 0.318203 0.590265i
\(712\) 0 0
\(713\) 8.09187e6i 0.596108i
\(714\) 0 0
\(715\) 5.31228e6i 0.388612i
\(716\) 0 0
\(717\) −2.63430e7 6.64830e6i −1.91367 0.482962i
\(718\) 0 0
\(719\) −6.32529e6 −0.456308 −0.228154 0.973625i \(-0.573269\pi\)
−0.228154 + 0.973625i \(0.573269\pi\)
\(720\) 0 0
\(721\) −1.44346e7 −1.03411
\(722\) 0 0
\(723\) −1.23817e7 3.12483e6i −0.880917 0.222321i
\(724\) 0 0
\(725\) 1.08155e7i 0.764191i
\(726\) 0 0
\(727\) 1.91829e7i 1.34610i 0.739596 + 0.673051i \(0.235017\pi\)
−0.739596 + 0.673051i \(0.764983\pi\)
\(728\) 0 0
\(729\) 1.25425e6 + 1.42940e7i 0.0874109 + 0.996172i
\(730\) 0 0
\(731\) −4.68178e6 −0.324054
\(732\) 0 0
\(733\) −2.15672e6 −0.148264 −0.0741318 0.997248i \(-0.523619\pi\)
−0.0741318 + 0.997248i \(0.523619\pi\)
\(734\) 0 0
\(735\) 1.57308e6 6.23314e6i 0.107407 0.425587i
\(736\) 0 0
\(737\) 2.93489e7i 1.99032i
\(738\) 0 0
\(739\) 8.62944e6i 0.581262i −0.956835 0.290631i \(-0.906135\pi\)
0.956835 0.290631i \(-0.0938651\pi\)
\(740\) 0 0
\(741\) 2.13880e6 8.47472e6i 0.143095 0.566996i
\(742\) 0 0
\(743\) 2.75125e7 1.82834 0.914171 0.405328i \(-0.132843\pi\)
0.914171 + 0.405328i \(0.132843\pi\)
\(744\) 0 0
\(745\) 1.03558e7 0.683583
\(746\) 0 0
\(747\) −1.66202e7 8.95968e6i −1.08977 0.587477i
\(748\) 0 0
\(749\) 8.86922e6i 0.577671i
\(750\) 0 0
\(751\) 2.11517e7i 1.36850i −0.729247 0.684250i \(-0.760129\pi\)
0.729247 0.684250i \(-0.239871\pi\)
\(752\) 0 0
\(753\) −8.99207e6 2.26937e6i −0.577926 0.145854i
\(754\) 0 0
\(755\) −1.59029e6 −0.101534
\(756\) 0 0
\(757\) −2.55766e7 −1.62219 −0.811096 0.584913i \(-0.801129\pi\)
−0.811096 + 0.584913i \(0.801129\pi\)
\(758\) 0 0
\(759\) −2.75464e7 6.95201e6i −1.73564 0.438032i
\(760\) 0 0
\(761\) 3.87679e6i 0.242667i 0.992612 + 0.121333i \(0.0387170\pi\)
−0.992612 + 0.121333i \(0.961283\pi\)
\(762\) 0 0
\(763\) 1.97848e7i 1.23032i
\(764\) 0 0
\(765\) −4.90485e6 2.64413e6i −0.303021 0.163354i
\(766\) 0 0
\(767\) 1.66824e6 0.102393
\(768\) 0 0
\(769\) 2.92438e7 1.78327 0.891637 0.452750i \(-0.149557\pi\)
0.891637 + 0.452750i \(0.149557\pi\)
\(770\) 0 0
\(771\) −6.13703e6 + 2.43171e7i −0.371811 + 1.47325i
\(772\) 0 0
\(773\) 3.45316e6i 0.207859i 0.994585 + 0.103929i \(0.0331416\pi\)
−0.994585 + 0.103929i \(0.966858\pi\)
\(774\) 0 0
\(775\) 3.27992e6i 0.196160i
\(776\) 0 0
\(777\) −3.13874e6 + 1.24368e7i −0.186510 + 0.739022i
\(778\) 0 0
\(779\) −6.12621e6 −0.361700
\(780\) 0 0
\(781\) −1.23262e7 −0.723107
\(782\) 0 0
\(783\) −1.87053e7 + 2.04185e7i −1.09033 + 1.19020i
\(784\) 0 0
\(785\) 3.68077e6i 0.213189i
\(786\) 0 0
\(787\) 1.36417e7i 0.785110i 0.919728 + 0.392555i \(0.128409\pi\)
−0.919728 + 0.392555i \(0.871591\pi\)
\(788\) 0 0
\(789\) −2.17671e7 5.49347e6i −1.24483 0.314163i
\(790\) 0 0
\(791\) −9.08087e6 −0.516043
\(792\) 0 0
\(793\) −1.33644e6 −0.0754687
\(794\) 0 0
\(795\) −917893. 231653.i −0.0515080 0.0129993i
\(796\) 0 0
\(797\) 7.88629e6i 0.439771i −0.975526 0.219886i \(-0.929432\pi\)
0.975526 0.219886i \(-0.0705684\pi\)
\(798\) 0 0
\(799\) 2.21620e6i 0.122812i
\(800\) 0 0
\(801\) −4.43109e6 + 8.21966e6i −0.244022 + 0.452660i
\(802\) 0 0
\(803\) −1.73951e7 −0.952003
\(804\) 0 0
\(805\) 1.20658e7 0.656246
\(806\) 0 0
\(807\) 7.08244e6 2.80632e7i 0.382824 1.51689i
\(808\) 0 0
\(809\) 7.30535e6i 0.392437i −0.980560 0.196218i \(-0.937134\pi\)
0.980560 0.196218i \(-0.0628661\pi\)
\(810\) 0 0
\(811\) 1.67815e7i 0.895941i 0.894048 + 0.447971i \(0.147853\pi\)
−0.894048 + 0.447971i \(0.852147\pi\)
\(812\) 0 0
\(813\) 803775. 3.18485e6i 0.0426489 0.168991i
\(814\) 0 0
\(815\) −3.35308e6 −0.176828
\(816\) 0 0
\(817\) −1.77056e7 −0.928015
\(818\) 0 0
\(819\) −2.46453e6 + 4.57170e6i −0.128388 + 0.238159i
\(820\) 0 0
\(821\) 3.03619e7i 1.57207i 0.618185 + 0.786033i \(0.287868\pi\)
−0.618185 + 0.786033i \(0.712132\pi\)
\(822\) 0 0
\(823\) 1.48318e6i 0.0763296i 0.999271 + 0.0381648i \(0.0121512\pi\)
−0.999271 + 0.0381648i \(0.987849\pi\)
\(824\) 0 0
\(825\) −1.11655e7 2.81790e6i −0.571143 0.144142i
\(826\) 0 0
\(827\) 1.47869e7 0.751819 0.375910 0.926656i \(-0.377330\pi\)
0.375910 + 0.926656i \(0.377330\pi\)
\(828\) 0 0
\(829\) 1.09627e7 0.554025 0.277013 0.960866i \(-0.410656\pi\)
0.277013 + 0.960866i \(0.410656\pi\)
\(830\) 0 0
\(831\) −5.53643e6 1.39725e6i −0.278117 0.0701896i
\(832\) 0 0
\(833\) 5.74687e6i 0.286958i
\(834\) 0 0
\(835\) 54944.1i 0.00272712i
\(836\) 0 0
\(837\) 5.67258e6 6.19213e6i 0.279877 0.305511i
\(838\) 0 0
\(839\) 3.22320e7 1.58082 0.790409 0.612580i \(-0.209868\pi\)
0.790409 + 0.612580i \(0.209868\pi\)
\(840\) 0 0
\(841\) −3.29286e7 −1.60540
\(842\) 0 0
\(843\) 3.67191e6 1.45495e7i 0.177960 0.705144i
\(844\) 0 0
\(845\) 1.22710e7i 0.591204i
\(846\) 0 0
\(847\) 7.19227e6i 0.344475i
\(848\) 0 0
\(849\) −6.10128e6 + 2.41755e7i −0.290504 + 1.15108i
\(850\) 0 0
\(851\) 3.68560e7 1.74455
\(852\) 0 0
\(853\) −7.55584e6 −0.355558 −0.177779 0.984070i \(-0.556891\pi\)
−0.177779 + 0.984070i \(0.556891\pi\)
\(854\) 0 0
\(855\) −1.85492e7 9.99960e6i −0.867781 0.467808i
\(856\) 0 0
\(857\) 9.11651e6i 0.424010i −0.977269 0.212005i \(-0.932001\pi\)
0.977269 0.212005i \(-0.0679994\pi\)
\(858\) 0 0
\(859\) 1.22074e7i 0.564471i 0.959345 + 0.282235i \(0.0910759\pi\)
−0.959345 + 0.282235i \(0.908924\pi\)
\(860\) 0 0
\(861\) 3.52960e6 + 890782.i 0.162262 + 0.0409509i
\(862\) 0 0
\(863\) 3.22980e7 1.47621 0.738106 0.674685i \(-0.235721\pi\)
0.738106 + 0.674685i \(0.235721\pi\)
\(864\) 0 0
\(865\) −3.05328e7 −1.38748
\(866\) 0 0
\(867\) 1.66306e7 + 4.19715e6i 0.751383 + 0.189630i
\(868\) 0 0
\(869\) 1.85730e7i 0.834321i
\(870\) 0 0
\(871\) 1.54164e7i 0.688555i
\(872\) 0 0
\(873\) 3.18142e7 + 1.71505e7i 1.41281 + 0.761627i
\(874\) 0 0
\(875\) 1.52208e7 0.672076
\(876\) 0 0
\(877\) −1.11768e7 −0.490703 −0.245352 0.969434i \(-0.578903\pi\)
−0.245352 + 0.969434i \(0.578903\pi\)
\(878\) 0 0
\(879\) 1.72202e6 6.82325e6i 0.0751735 0.297865i
\(880\) 0 0
\(881\) 3.55914e7i 1.54492i −0.635065 0.772459i \(-0.719027\pi\)
0.635065 0.772459i \(-0.280973\pi\)
\(882\) 0 0
\(883\) 1.56424e7i 0.675152i 0.941298 + 0.337576i \(0.109607\pi\)
−0.941298 + 0.337576i \(0.890393\pi\)
\(884\) 0 0
\(885\) 984206. 3.89978e6i 0.0422404 0.167372i
\(886\) 0 0
\(887\) −3.40474e7 −1.45303 −0.726515 0.687150i \(-0.758861\pi\)
−0.726515 + 0.687150i \(0.758861\pi\)
\(888\) 0 0
\(889\) −2.52991e7 −1.07362
\(890\) 0 0
\(891\) −1.62058e7 2.46305e7i −0.683873 1.03939i
\(892\) 0 0
\(893\) 8.38125e6i 0.351706i
\(894\) 0 0
\(895\) 5.90004e6i 0.246205i
\(896\) 0 0
\(897\) 1.44696e7 + 3.65176e6i 0.600449 + 0.151538i
\(898\) 0 0
\(899\) 1.62062e7 0.668778
\(900\) 0 0
\(901\) −846286. −0.0347300
\(902\) 0 0
\(903\) 1.02010e7 + 2.57448e6i 0.416318 + 0.105068i
\(904\) 0 0
\(905\) 3.31373e7i 1.34492i
\(906\) 0 0
\(907\) 1.54115e7i 0.622052i −0.950401 0.311026i \(-0.899327\pi\)
0.950401 0.311026i \(-0.100673\pi\)
\(908\) 0 0
\(909\) 4.98815e6 9.25299e6i 0.200230 0.371426i
\(910\) 0 0
\(911\) 1.35161e7 0.539580 0.269790 0.962919i \(-0.413046\pi\)
0.269790 + 0.962919i \(0.413046\pi\)
\(912\) 0 0
\(913\) 3.87969e7 1.54035
\(914\) 0 0
\(915\) −788456. + 3.12415e6i −0.0311333 + 0.123361i
\(916\) 0 0
\(917\) 1.03076e7i 0.404794i
\(918\) 0 0
\(919\) 4.00733e7i 1.56519i 0.622533 + 0.782594i \(0.286104\pi\)
−0.622533 + 0.782594i \(0.713896\pi\)
\(920\) 0 0
\(921\) 9.49856e6 3.76368e7i 0.368985 1.46205i
\(922\) 0 0
\(923\) 6.47474e6 0.250160
\(924\) 0 0
\(925\) 1.49390e7 0.574075
\(926\) 0 0
\(927\) −2.04250e7 + 3.78883e7i −0.780661 + 1.44812i
\(928\) 0 0
\(929\) 1.38484e7i 0.526454i 0.964734 + 0.263227i \(0.0847868\pi\)
−0.964734 + 0.263227i \(0.915213\pi\)
\(930\) 0 0
\(931\) 2.17336e7i 0.821782i
\(932\) 0 0
\(933\) −1.13669e7 2.86872e6i −0.427502 0.107891i
\(934\) 0 0
\(935\) 1.14495e7 0.428310
\(936\) 0 0
\(937\) −1.64331e7 −0.611464 −0.305732 0.952118i \(-0.598901\pi\)
−0.305732 + 0.952118i \(0.598901\pi\)
\(938\) 0 0
\(939\) −3.48389e7 8.79245e6i −1.28944 0.325421i
\(940\) 0 0
\(941\) 1.22708e7i 0.451752i 0.974156 + 0.225876i \(0.0725245\pi\)
−0.974156 + 0.225876i \(0.927476\pi\)
\(942\) 0 0
\(943\) 1.04598e7i 0.383040i
\(944\) 0 0
\(945\) 9.23310e6 + 8.45840e6i 0.336332 + 0.308112i
\(946\) 0 0
\(947\) −4.94103e6 −0.179037 −0.0895185 0.995985i \(-0.528533\pi\)
−0.0895185 + 0.995985i \(0.528533\pi\)
\(948\) 0 0
\(949\) 9.13733e6 0.329347
\(950\) 0 0
\(951\) 4.24538e6 1.68217e7i 0.152218 0.603142i
\(952\) 0 0
\(953\) 3.15735e7i 1.12614i −0.826410 0.563068i \(-0.809621\pi\)
0.826410 0.563068i \(-0.190379\pi\)
\(954\) 0 0
\(955\) 2.31229e7i 0.820416i
\(956\) 0 0
\(957\) 1.39233e7 5.51692e7i 0.491431 1.94723i
\(958\) 0 0
\(959\) 3.32379e7 1.16704
\(960\) 0 0
\(961\) 2.37144e7 0.828332
\(962\) 0 0
\(963\) −2.32802e7 1.25500e7i −0.808948 0.436092i
\(964\) 0 0
\(965\) 1.63342e7i 0.564651i
\(966\) 0 0
\(967\) 3.69660e7i 1.27127i −0.771991 0.635634i \(-0.780739\pi\)
0.771991 0.635634i \(-0.219261\pi\)
\(968\) 0 0
\(969\) −1.82655e7 4.60975e6i −0.624917 0.157713i
\(970\) 0 0
\(971\) 4.40891e7 1.50066 0.750331 0.661062i \(-0.229894\pi\)
0.750331 + 0.661062i \(0.229894\pi\)
\(972\) 0 0
\(973\) 2.81266e7 0.952433
\(974\) 0 0
\(975\) 5.86505e6 + 1.48019e6i 0.197588 + 0.0498662i
\(976\) 0 0
\(977\) 1.62078e6i 0.0543236i 0.999631 + 0.0271618i \(0.00864694\pi\)
−0.999631 + 0.0271618i \(0.991353\pi\)
\(978\) 0 0
\(979\) 1.91874e7i 0.639821i
\(980\) 0 0
\(981\) 5.19317e7 + 2.79956e7i 1.72290 + 0.928789i
\(982\) 0 0
\(983\) −4.23292e7 −1.39719 −0.698596 0.715516i \(-0.746191\pi\)
−0.698596 + 0.715516i \(0.746191\pi\)
\(984\) 0 0
\(985\) 2.82895e7 0.929042
\(986\) 0 0
\(987\) −1.21868e6 + 4.82884e6i −0.0398195 + 0.157779i
\(988\) 0 0
\(989\) 3.02303e7i 0.982769i
\(990\) 0 0
\(991\) 1.76998e6i 0.0572513i 0.999590 + 0.0286256i \(0.00911307\pi\)
−0.999590 + 0.0286256i \(0.990887\pi\)
\(992\) 0 0
\(993\) −9.58278e6 + 3.79705e7i −0.308403 + 1.22200i
\(994\) 0 0
\(995\) −716158. −0.0229325
\(996\) 0 0
\(997\) 872512. 0.0277993 0.0138996 0.999903i \(-0.495575\pi\)
0.0138996 + 0.999903i \(0.495575\pi\)
\(998\) 0 0
\(999\) 2.82032e7 + 2.58369e7i 0.894099 + 0.819080i
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.c.c.383.18 yes 20
3.2 odd 2 384.6.c.b.383.4 yes 20
4.3 odd 2 384.6.c.b.383.3 yes 20
8.3 odd 2 384.6.c.d.383.18 yes 20
8.5 even 2 384.6.c.a.383.3 20
12.11 even 2 inner 384.6.c.c.383.17 yes 20
24.5 odd 2 384.6.c.d.383.17 yes 20
24.11 even 2 384.6.c.a.383.4 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.6.c.a.383.3 20 8.5 even 2
384.6.c.a.383.4 yes 20 24.11 even 2
384.6.c.b.383.3 yes 20 4.3 odd 2
384.6.c.b.383.4 yes 20 3.2 odd 2
384.6.c.c.383.17 yes 20 12.11 even 2 inner
384.6.c.c.383.18 yes 20 1.1 even 1 trivial
384.6.c.d.383.17 yes 20 24.5 odd 2
384.6.c.d.383.18 yes 20 8.3 odd 2