Properties

Label 384.6.c.a.383.10
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 383.10
Root \(-5.94193i\) of defining polynomial
Character \(\chi\) \(=\) 384.383
Dual form 384.6.c.a.383.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.25525 + 15.4245i) q^{3} +86.6842i q^{5} -110.468i q^{7} +(-232.828 - 69.5720i) q^{9} +O(q^{10})\) \(q+(-2.25525 + 15.4245i) q^{3} +86.6842i q^{5} -110.468i q^{7} +(-232.828 - 69.5720i) q^{9} +212.115 q^{11} +539.316 q^{13} +(-1337.06 - 195.494i) q^{15} +24.1619i q^{17} -1930.54i q^{19} +(1703.91 + 249.133i) q^{21} -275.110 q^{23} -4389.15 q^{25} +(1598.19 - 3434.34i) q^{27} -7337.17i q^{29} -7478.16i q^{31} +(-478.373 + 3271.76i) q^{33} +9575.83 q^{35} -7681.01 q^{37} +(-1216.29 + 8318.65i) q^{39} -19394.9i q^{41} +4494.69i q^{43} +(6030.79 - 20182.5i) q^{45} +3629.25 q^{47} +4603.83 q^{49} +(-372.684 - 54.4911i) q^{51} +3676.39i q^{53} +18387.1i q^{55} +(29777.5 + 4353.85i) q^{57} +23949.3 q^{59} -43340.3 q^{61} +(-7685.47 + 25720.0i) q^{63} +46750.1i q^{65} +65528.4i q^{67} +(620.442 - 4243.43i) q^{69} +30068.2 q^{71} -43260.2 q^{73} +(9898.63 - 67700.3i) q^{75} -23432.0i q^{77} -90539.6i q^{79} +(49368.5 + 32396.6i) q^{81} +109867. q^{83} -2094.46 q^{85} +(113172. + 16547.1i) q^{87} +138798. i q^{89} -59577.1i q^{91} +(115347. + 16865.1i) q^{93} +167347. q^{95} -4237.49 q^{97} +(-49386.3 - 14757.3i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{3}+O(q^{10}) \) 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
\(3\) −2.25525 + 15.4245i −0.144674 + 0.989479i
\(4\) 0 0
\(5\) 86.6842i 1.55065i 0.631560 + 0.775327i \(0.282415\pi\)
−0.631560 + 0.775327i \(0.717585\pi\)
\(6\) 0 0
\(7\) 110.468i 0.852101i −0.904699 0.426051i \(-0.859905\pi\)
0.904699 0.426051i \(-0.140095\pi\)
\(8\) 0 0
\(9\) −232.828 69.5720i −0.958139 0.286304i
\(10\) 0 0
\(11\) 212.115 0.528555 0.264278 0.964447i \(-0.414866\pi\)
0.264278 + 0.964447i \(0.414866\pi\)
\(12\) 0 0
\(13\) 539.316 0.885084 0.442542 0.896748i \(-0.354077\pi\)
0.442542 + 0.896748i \(0.354077\pi\)
\(14\) 0 0
\(15\) −1337.06 195.494i −1.53434 0.224340i
\(16\) 0 0
\(17\) 24.1619i 0.0202773i 0.999949 + 0.0101386i \(0.00322728\pi\)
−0.999949 + 0.0101386i \(0.996773\pi\)
\(18\) 0 0
\(19\) 1930.54i 1.22686i −0.789749 0.613430i \(-0.789789\pi\)
0.789749 0.613430i \(-0.210211\pi\)
\(20\) 0 0
\(21\) 1703.91 + 249.133i 0.843137 + 0.123277i
\(22\) 0 0
\(23\) −275.110 −0.108439 −0.0542197 0.998529i \(-0.517267\pi\)
−0.0542197 + 0.998529i \(0.517267\pi\)
\(24\) 0 0
\(25\) −4389.15 −1.40453
\(26\) 0 0
\(27\) 1598.19 3434.34i 0.421910 0.906638i
\(28\) 0 0
\(29\) 7337.17i 1.62007i −0.586382 0.810034i \(-0.699448\pi\)
0.586382 0.810034i \(-0.300552\pi\)
\(30\) 0 0
\(31\) 7478.16i 1.39762i −0.715305 0.698812i \(-0.753712\pi\)
0.715305 0.698812i \(-0.246288\pi\)
\(32\) 0 0
\(33\) −478.373 + 3271.76i −0.0764683 + 0.522995i
\(34\) 0 0
\(35\) 9575.83 1.32131
\(36\) 0 0
\(37\) −7681.01 −0.922389 −0.461195 0.887299i \(-0.652579\pi\)
−0.461195 + 0.887299i \(0.652579\pi\)
\(38\) 0 0
\(39\) −1216.29 + 8318.65i −0.128049 + 0.875773i
\(40\) 0 0
\(41\) 19394.9i 1.80188i −0.433939 0.900942i \(-0.642877\pi\)
0.433939 0.900942i \(-0.357123\pi\)
\(42\) 0 0
\(43\) 4494.69i 0.370705i 0.982672 + 0.185352i \(0.0593427\pi\)
−0.982672 + 0.185352i \(0.940657\pi\)
\(44\) 0 0
\(45\) 6030.79 20182.5i 0.443959 1.48574i
\(46\) 0 0
\(47\) 3629.25 0.239647 0.119824 0.992795i \(-0.461767\pi\)
0.119824 + 0.992795i \(0.461767\pi\)
\(48\) 0 0
\(49\) 4603.83 0.273923
\(50\) 0 0
\(51\) −372.684 54.4911i −0.0200639 0.00293360i
\(52\) 0 0
\(53\) 3676.39i 0.179776i 0.995952 + 0.0898880i \(0.0286509\pi\)
−0.995952 + 0.0898880i \(0.971349\pi\)
\(54\) 0 0
\(55\) 18387.1i 0.819606i
\(56\) 0 0
\(57\) 29777.5 + 4353.85i 1.21395 + 0.177495i
\(58\) 0 0
\(59\) 23949.3 0.895702 0.447851 0.894108i \(-0.352189\pi\)
0.447851 + 0.894108i \(0.352189\pi\)
\(60\) 0 0
\(61\) −43340.3 −1.49131 −0.745653 0.666334i \(-0.767863\pi\)
−0.745653 + 0.666334i \(0.767863\pi\)
\(62\) 0 0
\(63\) −7685.47 + 25720.0i −0.243960 + 0.816431i
\(64\) 0 0
\(65\) 46750.1i 1.37246i
\(66\) 0 0
\(67\) 65528.4i 1.78337i 0.452652 + 0.891687i \(0.350478\pi\)
−0.452652 + 0.891687i \(0.649522\pi\)
\(68\) 0 0
\(69\) 620.442 4243.43i 0.0156884 0.107299i
\(70\) 0 0
\(71\) 30068.2 0.707882 0.353941 0.935268i \(-0.384841\pi\)
0.353941 + 0.935268i \(0.384841\pi\)
\(72\) 0 0
\(73\) −43260.2 −0.950127 −0.475063 0.879952i \(-0.657575\pi\)
−0.475063 + 0.879952i \(0.657575\pi\)
\(74\) 0 0
\(75\) 9898.63 67700.3i 0.203199 1.38975i
\(76\) 0 0
\(77\) 23432.0i 0.450383i
\(78\) 0 0
\(79\) 90539.6i 1.63219i −0.577917 0.816096i \(-0.696134\pi\)
0.577917 0.816096i \(-0.303866\pi\)
\(80\) 0 0
\(81\) 49368.5 + 32396.6i 0.836060 + 0.548639i
\(82\) 0 0
\(83\) 109867. 1.75053 0.875266 0.483641i \(-0.160686\pi\)
0.875266 + 0.483641i \(0.160686\pi\)
\(84\) 0 0
\(85\) −2094.46 −0.0314430
\(86\) 0 0
\(87\) 113172. + 16547.1i 1.60302 + 0.234382i
\(88\) 0 0
\(89\) 138798.i 1.85741i 0.370822 + 0.928704i \(0.379076\pi\)
−0.370822 + 0.928704i \(0.620924\pi\)
\(90\) 0 0
\(91\) 59577.1i 0.754181i
\(92\) 0 0
\(93\) 115347. + 16865.1i 1.38292 + 0.202200i
\(94\) 0 0
\(95\) 167347. 1.90243
\(96\) 0 0
\(97\) −4237.49 −0.0457277 −0.0228638 0.999739i \(-0.507278\pi\)
−0.0228638 + 0.999739i \(0.507278\pi\)
\(98\) 0 0
\(99\) −49386.3 14757.3i −0.506429 0.151328i
\(100\) 0 0
\(101\) 162305.i 1.58317i −0.611057 0.791587i \(-0.709255\pi\)
0.611057 0.791587i \(-0.290745\pi\)
\(102\) 0 0
\(103\) 91064.8i 0.845780i −0.906181 0.422890i \(-0.861016\pi\)
0.906181 0.422890i \(-0.138984\pi\)
\(104\) 0 0
\(105\) −21595.9 + 147702.i −0.191160 + 1.30741i
\(106\) 0 0
\(107\) −156580. −1.32214 −0.661068 0.750326i \(-0.729897\pi\)
−0.661068 + 0.750326i \(0.729897\pi\)
\(108\) 0 0
\(109\) 113832. 0.917692 0.458846 0.888516i \(-0.348263\pi\)
0.458846 + 0.888516i \(0.348263\pi\)
\(110\) 0 0
\(111\) 17322.6 118475.i 0.133446 0.912685i
\(112\) 0 0
\(113\) 9212.03i 0.0678671i −0.999424 0.0339336i \(-0.989197\pi\)
0.999424 0.0339336i \(-0.0108035\pi\)
\(114\) 0 0
\(115\) 23847.7i 0.168152i
\(116\) 0 0
\(117\) −125568. 37521.2i −0.848033 0.253403i
\(118\) 0 0
\(119\) 2669.12 0.0172783
\(120\) 0 0
\(121\) −116058. −0.720629
\(122\) 0 0
\(123\) 299155. + 43740.2i 1.78293 + 0.260686i
\(124\) 0 0
\(125\) 109582.i 0.627284i
\(126\) 0 0
\(127\) 82080.7i 0.451577i 0.974176 + 0.225788i \(0.0724958\pi\)
−0.974176 + 0.225788i \(0.927504\pi\)
\(128\) 0 0
\(129\) −69328.1 10136.6i −0.366805 0.0536315i
\(130\) 0 0
\(131\) 192564. 0.980387 0.490193 0.871614i \(-0.336926\pi\)
0.490193 + 0.871614i \(0.336926\pi\)
\(132\) 0 0
\(133\) −213263. −1.04541
\(134\) 0 0
\(135\) 297703. + 138538.i 1.40588 + 0.654237i
\(136\) 0 0
\(137\) 262306.i 1.19401i 0.802238 + 0.597004i \(0.203642\pi\)
−0.802238 + 0.597004i \(0.796358\pi\)
\(138\) 0 0
\(139\) 33737.6i 0.148108i 0.997254 + 0.0740539i \(0.0235937\pi\)
−0.997254 + 0.0740539i \(0.976406\pi\)
\(140\) 0 0
\(141\) −8184.86 + 55979.2i −0.0346708 + 0.237126i
\(142\) 0 0
\(143\) 114397. 0.467816
\(144\) 0 0
\(145\) 636017. 2.51217
\(146\) 0 0
\(147\) −10382.8 + 71011.5i −0.0396296 + 0.271041i
\(148\) 0 0
\(149\) 322123.i 1.18866i −0.804223 0.594328i \(-0.797418\pi\)
0.804223 0.594328i \(-0.202582\pi\)
\(150\) 0 0
\(151\) 190296.i 0.679183i 0.940573 + 0.339592i \(0.110289\pi\)
−0.940573 + 0.339592i \(0.889711\pi\)
\(152\) 0 0
\(153\) 1680.99 5625.56i 0.00580547 0.0194284i
\(154\) 0 0
\(155\) 648238. 2.16723
\(156\) 0 0
\(157\) 6464.74 0.0209316 0.0104658 0.999945i \(-0.496669\pi\)
0.0104658 + 0.999945i \(0.496669\pi\)
\(158\) 0 0
\(159\) −56706.3 8291.16i −0.177885 0.0260089i
\(160\) 0 0
\(161\) 30390.9i 0.0924014i
\(162\) 0 0
\(163\) 333250.i 0.982428i −0.871039 0.491214i \(-0.836553\pi\)
0.871039 0.491214i \(-0.163447\pi\)
\(164\) 0 0
\(165\) −283610. 41467.4i −0.810984 0.118576i
\(166\) 0 0
\(167\) −430857. −1.19548 −0.597740 0.801690i \(-0.703934\pi\)
−0.597740 + 0.801690i \(0.703934\pi\)
\(168\) 0 0
\(169\) −80431.7 −0.216626
\(170\) 0 0
\(171\) −134311. + 449483.i −0.351255 + 1.17550i
\(172\) 0 0
\(173\) 720397.i 1.83002i −0.403428 0.915011i \(-0.632181\pi\)
0.403428 0.915011i \(-0.367819\pi\)
\(174\) 0 0
\(175\) 484861.i 1.19680i
\(176\) 0 0
\(177\) −54011.7 + 369406.i −0.129585 + 0.886279i
\(178\) 0 0
\(179\) 822982. 1.91981 0.959904 0.280329i \(-0.0904437\pi\)
0.959904 + 0.280329i \(0.0904437\pi\)
\(180\) 0 0
\(181\) −86567.9 −0.196409 −0.0982043 0.995166i \(-0.531310\pi\)
−0.0982043 + 0.995166i \(0.531310\pi\)
\(182\) 0 0
\(183\) 97743.1 668500.i 0.215754 1.47562i
\(184\) 0 0
\(185\) 665822.i 1.43031i
\(186\) 0 0
\(187\) 5125.11i 0.0107176i
\(188\) 0 0
\(189\) −379384. 176549.i −0.772547 0.359510i
\(190\) 0 0
\(191\) 93397.9 0.185248 0.0926240 0.995701i \(-0.470475\pi\)
0.0926240 + 0.995701i \(0.470475\pi\)
\(192\) 0 0
\(193\) 63345.4 0.122411 0.0612057 0.998125i \(-0.480505\pi\)
0.0612057 + 0.998125i \(0.480505\pi\)
\(194\) 0 0
\(195\) −721096. 105433.i −1.35802 0.198560i
\(196\) 0 0
\(197\) 603734.i 1.10836i 0.832397 + 0.554179i \(0.186968\pi\)
−0.832397 + 0.554179i \(0.813032\pi\)
\(198\) 0 0
\(199\) 324247.i 0.580420i −0.956963 0.290210i \(-0.906275\pi\)
0.956963 0.290210i \(-0.0937252\pi\)
\(200\) 0 0
\(201\) −1.01074e6 147783.i −1.76461 0.258008i
\(202\) 0 0
\(203\) −810522. −1.38046
\(204\) 0 0
\(205\) 1.68123e6 2.79410
\(206\) 0 0
\(207\) 64053.3 + 19140.0i 0.103900 + 0.0310467i
\(208\) 0 0
\(209\) 409497.i 0.648463i
\(210\) 0 0
\(211\) 660510.i 1.02135i −0.859775 0.510673i \(-0.829396\pi\)
0.859775 0.510673i \(-0.170604\pi\)
\(212\) 0 0
\(213\) −67811.2 + 463785.i −0.102412 + 0.700435i
\(214\) 0 0
\(215\) −389618. −0.574835
\(216\) 0 0
\(217\) −826097. −1.19092
\(218\) 0 0
\(219\) 97562.5 667265.i 0.137459 0.940131i
\(220\) 0 0
\(221\) 13030.9i 0.0179471i
\(222\) 0 0
\(223\) 162536.i 0.218870i 0.993994 + 0.109435i \(0.0349042\pi\)
−0.993994 + 0.109435i \(0.965096\pi\)
\(224\) 0 0
\(225\) 1.02192e6 + 305362.i 1.34573 + 0.402123i
\(226\) 0 0
\(227\) −667969. −0.860383 −0.430192 0.902738i \(-0.641554\pi\)
−0.430192 + 0.902738i \(0.641554\pi\)
\(228\) 0 0
\(229\) −945355. −1.19126 −0.595630 0.803259i \(-0.703097\pi\)
−0.595630 + 0.803259i \(0.703097\pi\)
\(230\) 0 0
\(231\) 361425. + 52844.9i 0.445644 + 0.0651588i
\(232\) 0 0
\(233\) 666944.i 0.804821i −0.915459 0.402411i \(-0.868172\pi\)
0.915459 0.402411i \(-0.131828\pi\)
\(234\) 0 0
\(235\) 314599.i 0.371610i
\(236\) 0 0
\(237\) 1.39652e6 + 204189.i 1.61502 + 0.236136i
\(238\) 0 0
\(239\) −409633. −0.463874 −0.231937 0.972731i \(-0.574506\pi\)
−0.231937 + 0.972731i \(0.574506\pi\)
\(240\) 0 0
\(241\) 427196. 0.473789 0.236894 0.971535i \(-0.423870\pi\)
0.236894 + 0.971535i \(0.423870\pi\)
\(242\) 0 0
\(243\) −611038. + 688420.i −0.663823 + 0.747890i
\(244\) 0 0
\(245\) 399079.i 0.424760i
\(246\) 0 0
\(247\) 1.04117e6i 1.08587i
\(248\) 0 0
\(249\) −247776. + 1.69463e6i −0.253257 + 1.73212i
\(250\) 0 0
\(251\) 881998. 0.883656 0.441828 0.897100i \(-0.354330\pi\)
0.441828 + 0.897100i \(0.354330\pi\)
\(252\) 0 0
\(253\) −58355.1 −0.0573162
\(254\) 0 0
\(255\) 4723.52 32305.9i 0.00454899 0.0311122i
\(256\) 0 0
\(257\) 599689.i 0.566361i −0.959067 0.283181i \(-0.908610\pi\)
0.959067 0.283181i \(-0.0913896\pi\)
\(258\) 0 0
\(259\) 848506.i 0.785969i
\(260\) 0 0
\(261\) −510461. + 1.70830e6i −0.463833 + 1.55225i
\(262\) 0 0
\(263\) −456966. −0.407375 −0.203687 0.979036i \(-0.565293\pi\)
−0.203687 + 0.979036i \(0.565293\pi\)
\(264\) 0 0
\(265\) −318685. −0.278770
\(266\) 0 0
\(267\) −2.14088e6 313023.i −1.83787 0.268719i
\(268\) 0 0
\(269\) 1.10405e6i 0.930267i 0.885241 + 0.465134i \(0.153994\pi\)
−0.885241 + 0.465134i \(0.846006\pi\)
\(270\) 0 0
\(271\) 33195.9i 0.0274575i −0.999906 0.0137287i \(-0.995630\pi\)
0.999906 0.0137287i \(-0.00437013\pi\)
\(272\) 0 0
\(273\) 918944. + 134361.i 0.746247 + 0.109111i
\(274\) 0 0
\(275\) −931006. −0.742371
\(276\) 0 0
\(277\) −1.31847e6 −1.03246 −0.516229 0.856451i \(-0.672664\pi\)
−0.516229 + 0.856451i \(0.672664\pi\)
\(278\) 0 0
\(279\) −520270. + 1.74112e6i −0.400146 + 1.33912i
\(280\) 0 0
\(281\) 401388.i 0.303248i −0.988438 0.151624i \(-0.951550\pi\)
0.988438 0.151624i \(-0.0484504\pi\)
\(282\) 0 0
\(283\) 2.13872e6i 1.58741i 0.608305 + 0.793703i \(0.291850\pi\)
−0.608305 + 0.793703i \(0.708150\pi\)
\(284\) 0 0
\(285\) −377410. + 2.58124e6i −0.275233 + 1.88242i
\(286\) 0 0
\(287\) −2.14251e6 −1.53539
\(288\) 0 0
\(289\) 1.41927e6 0.999589
\(290\) 0 0
\(291\) 9556.58 65360.9i 0.00661561 0.0452466i
\(292\) 0 0
\(293\) 1.22189e6i 0.831499i −0.909479 0.415750i \(-0.863519\pi\)
0.909479 0.415750i \(-0.136481\pi\)
\(294\) 0 0
\(295\) 2.07603e6i 1.38892i
\(296\) 0 0
\(297\) 339001. 728476.i 0.223003 0.479208i
\(298\) 0 0
\(299\) −148371. −0.0959781
\(300\) 0 0
\(301\) 496519. 0.315878
\(302\) 0 0
\(303\) 2.50347e6 + 366038.i 1.56652 + 0.229044i
\(304\) 0 0
\(305\) 3.75692e6i 2.31250i
\(306\) 0 0
\(307\) 58956.8i 0.0357016i −0.999841 0.0178508i \(-0.994318\pi\)
0.999841 0.0178508i \(-0.00568240\pi\)
\(308\) 0 0
\(309\) 1.40463e6 + 205374.i 0.836882 + 0.122363i
\(310\) 0 0
\(311\) −1.02518e6 −0.601034 −0.300517 0.953776i \(-0.597159\pi\)
−0.300517 + 0.953776i \(0.597159\pi\)
\(312\) 0 0
\(313\) −1.63335e6 −0.942365 −0.471183 0.882036i \(-0.656173\pi\)
−0.471183 + 0.882036i \(0.656173\pi\)
\(314\) 0 0
\(315\) −2.22952e6 666209.i −1.26600 0.378298i
\(316\) 0 0
\(317\) 596082.i 0.333164i 0.986028 + 0.166582i \(0.0532731\pi\)
−0.986028 + 0.166582i \(0.946727\pi\)
\(318\) 0 0
\(319\) 1.55633e6i 0.856296i
\(320\) 0 0
\(321\) 353126. 2.41516e6i 0.191279 1.30823i
\(322\) 0 0
\(323\) 46645.5 0.0248773
\(324\) 0 0
\(325\) −2.36714e6 −1.24313
\(326\) 0 0
\(327\) −256719. + 1.75579e6i −0.132766 + 0.908038i
\(328\) 0 0
\(329\) 400916.i 0.204204i
\(330\) 0 0
\(331\) 1.40214e6i 0.703428i −0.936107 0.351714i \(-0.885599\pi\)
0.936107 0.351714i \(-0.114401\pi\)
\(332\) 0 0
\(333\) 1.78835e6 + 534383.i 0.883777 + 0.264084i
\(334\) 0 0
\(335\) −5.68028e6 −2.76540
\(336\) 0 0
\(337\) −2.84572e6 −1.36495 −0.682477 0.730907i \(-0.739097\pi\)
−0.682477 + 0.730907i \(0.739097\pi\)
\(338\) 0 0
\(339\) 142091. + 20775.4i 0.0671531 + 0.00981862i
\(340\) 0 0
\(341\) 1.58623e6i 0.738722i
\(342\) 0 0
\(343\) 2.36521e6i 1.08551i
\(344\) 0 0
\(345\) 367838. + 53782.5i 0.166383 + 0.0243273i
\(346\) 0 0
\(347\) 121815. 0.0543098 0.0271549 0.999631i \(-0.491355\pi\)
0.0271549 + 0.999631i \(0.491355\pi\)
\(348\) 0 0
\(349\) 324699. 0.142698 0.0713488 0.997451i \(-0.477270\pi\)
0.0713488 + 0.997451i \(0.477270\pi\)
\(350\) 0 0
\(351\) 861931. 1.85219e6i 0.373426 0.802451i
\(352\) 0 0
\(353\) 244816.i 0.104569i −0.998632 0.0522846i \(-0.983350\pi\)
0.998632 0.0522846i \(-0.0166503\pi\)
\(354\) 0 0
\(355\) 2.60643e6i 1.09768i
\(356\) 0 0
\(357\) −6019.52 + 41169.7i −0.00249972 + 0.0170965i
\(358\) 0 0
\(359\) −793556. −0.324969 −0.162484 0.986711i \(-0.551951\pi\)
−0.162484 + 0.986711i \(0.551951\pi\)
\(360\) 0 0
\(361\) −1.25088e6 −0.505183
\(362\) 0 0
\(363\) 261740. 1.79013e6i 0.104257 0.713048i
\(364\) 0 0
\(365\) 3.74998e6i 1.47332i
\(366\) 0 0
\(367\) 785525.i 0.304435i −0.988347 0.152218i \(-0.951359\pi\)
0.988347 0.152218i \(-0.0486415\pi\)
\(368\) 0 0
\(369\) −1.34934e6 + 4.51566e6i −0.515887 + 1.72646i
\(370\) 0 0
\(371\) 406123. 0.153187
\(372\) 0 0
\(373\) 1.97945e6 0.736671 0.368335 0.929693i \(-0.379928\pi\)
0.368335 + 0.929693i \(0.379928\pi\)
\(374\) 0 0
\(375\) 1.69024e6 + 247135.i 0.620684 + 0.0907518i
\(376\) 0 0
\(377\) 3.95705e6i 1.43390i
\(378\) 0 0
\(379\) 3.50429e6i 1.25315i −0.779362 0.626574i \(-0.784456\pi\)
0.779362 0.626574i \(-0.215544\pi\)
\(380\) 0 0
\(381\) −1.26605e6 185112.i −0.446826 0.0653315i
\(382\) 0 0
\(383\) −1.16489e6 −0.405779 −0.202889 0.979202i \(-0.565033\pi\)
−0.202889 + 0.979202i \(0.565033\pi\)
\(384\) 0 0
\(385\) 2.03118e6 0.698388
\(386\) 0 0
\(387\) 312704. 1.04649e6i 0.106134 0.355187i
\(388\) 0 0
\(389\) 65857.4i 0.0220664i −0.999939 0.0110332i \(-0.996488\pi\)
0.999939 0.0110332i \(-0.00351204\pi\)
\(390\) 0 0
\(391\) 6647.19i 0.00219885i
\(392\) 0 0
\(393\) −434280. + 2.97020e6i −0.141837 + 0.970073i
\(394\) 0 0
\(395\) 7.84836e6 2.53096
\(396\) 0 0
\(397\) 4.01996e6 1.28011 0.640053 0.768331i \(-0.278913\pi\)
0.640053 + 0.768331i \(0.278913\pi\)
\(398\) 0 0
\(399\) 480960. 3.28946e6i 0.151244 1.03441i
\(400\) 0 0
\(401\) 1.82280e6i 0.566081i −0.959108 0.283040i \(-0.908657\pi\)
0.959108 0.283040i \(-0.0913430\pi\)
\(402\) 0 0
\(403\) 4.03309e6i 1.23702i
\(404\) 0 0
\(405\) −2.80827e6 + 4.27947e6i −0.850749 + 1.29644i
\(406\) 0 0
\(407\) −1.62926e6 −0.487534
\(408\) 0 0
\(409\) −1.23169e6 −0.364077 −0.182039 0.983291i \(-0.558270\pi\)
−0.182039 + 0.983291i \(0.558270\pi\)
\(410\) 0 0
\(411\) −4.04593e6 591566.i −1.18145 0.172742i
\(412\) 0 0
\(413\) 2.64564e6i 0.763229i
\(414\) 0 0
\(415\) 9.52369e6i 2.71447i
\(416\) 0 0
\(417\) −520385. 76086.8i −0.146550 0.0214274i
\(418\) 0 0
\(419\) 3.26757e6 0.909264 0.454632 0.890679i \(-0.349771\pi\)
0.454632 + 0.890679i \(0.349771\pi\)
\(420\) 0 0
\(421\) 2.58190e6 0.709961 0.354981 0.934874i \(-0.384487\pi\)
0.354981 + 0.934874i \(0.384487\pi\)
\(422\) 0 0
\(423\) −844990. 252494.i −0.229615 0.0686120i
\(424\) 0 0
\(425\) 106050.i 0.0284800i
\(426\) 0 0
\(427\) 4.78771e6i 1.27074i
\(428\) 0 0
\(429\) −257994. + 1.76451e6i −0.0676809 + 0.462894i
\(430\) 0 0
\(431\) 3.97814e6 1.03154 0.515771 0.856727i \(-0.327506\pi\)
0.515771 + 0.856727i \(0.327506\pi\)
\(432\) 0 0
\(433\) 5.30560e6 1.35992 0.679962 0.733247i \(-0.261996\pi\)
0.679962 + 0.733247i \(0.261996\pi\)
\(434\) 0 0
\(435\) −1.43438e6 + 9.81021e6i −0.363446 + 2.48574i
\(436\) 0 0
\(437\) 531111.i 0.133040i
\(438\) 0 0
\(439\) 2.79565e6i 0.692344i −0.938171 0.346172i \(-0.887481\pi\)
0.938171 0.346172i \(-0.112519\pi\)
\(440\) 0 0
\(441\) −1.07190e6 320297.i −0.262456 0.0784254i
\(442\) 0 0
\(443\) −3.79176e6 −0.917976 −0.458988 0.888442i \(-0.651788\pi\)
−0.458988 + 0.888442i \(0.651788\pi\)
\(444\) 0 0
\(445\) −1.20316e7 −2.88020
\(446\) 0 0
\(447\) 4.96858e6 + 726468.i 1.17615 + 0.171968i
\(448\) 0 0
\(449\) 6.90181e6i 1.61565i 0.589422 + 0.807825i \(0.299355\pi\)
−0.589422 + 0.807825i \(0.700645\pi\)
\(450\) 0 0
\(451\) 4.11395e6i 0.952396i
\(452\) 0 0
\(453\) −2.93521e6 429164.i −0.672038 0.0982603i
\(454\) 0 0
\(455\) 5.16439e6 1.16947
\(456\) 0 0
\(457\) 5.98001e6 1.33940 0.669702 0.742630i \(-0.266422\pi\)
0.669702 + 0.742630i \(0.266422\pi\)
\(458\) 0 0
\(459\) 82980.2 + 38615.4i 0.0183841 + 0.00855518i
\(460\) 0 0
\(461\) 5.66582e6i 1.24168i −0.783936 0.620841i \(-0.786791\pi\)
0.783936 0.620841i \(-0.213209\pi\)
\(462\) 0 0
\(463\) 1.78722e6i 0.387458i −0.981055 0.193729i \(-0.937942\pi\)
0.981055 0.193729i \(-0.0620583\pi\)
\(464\) 0 0
\(465\) −1.46194e6 + 9.99872e6i −0.313543 + 2.14443i
\(466\) 0 0
\(467\) −654822. −0.138941 −0.0694706 0.997584i \(-0.522131\pi\)
−0.0694706 + 0.997584i \(0.522131\pi\)
\(468\) 0 0
\(469\) 7.23879e6 1.51962
\(470\) 0 0
\(471\) −14579.6 + 99715.0i −0.00302826 + 0.0207114i
\(472\) 0 0
\(473\) 953392.i 0.195938i
\(474\) 0 0
\(475\) 8.47343e6i 1.72316i
\(476\) 0 0
\(477\) 255773. 855965.i 0.0514706 0.172250i
\(478\) 0 0
\(479\) 4.61770e6 0.919575 0.459788 0.888029i \(-0.347926\pi\)
0.459788 + 0.888029i \(0.347926\pi\)
\(480\) 0 0
\(481\) −4.14249e6 −0.816392
\(482\) 0 0
\(483\) −468763. 68539.0i −0.0914293 0.0133681i
\(484\) 0 0
\(485\) 367323.i 0.0709078i
\(486\) 0 0
\(487\) 2.36141e6i 0.451178i 0.974223 + 0.225589i \(0.0724307\pi\)
−0.974223 + 0.225589i \(0.927569\pi\)
\(488\) 0 0
\(489\) 5.14020e6 + 751561.i 0.972093 + 0.142132i
\(490\) 0 0
\(491\) 8.12401e6 1.52078 0.760390 0.649466i \(-0.225008\pi\)
0.760390 + 0.649466i \(0.225008\pi\)
\(492\) 0 0
\(493\) 177280. 0.0328505
\(494\) 0 0
\(495\) 1.27922e6 4.28102e6i 0.234657 0.785297i
\(496\) 0 0
\(497\) 3.32157e6i 0.603187i
\(498\) 0 0
\(499\) 6.53711e6i 1.17526i −0.809129 0.587631i \(-0.800061\pi\)
0.809129 0.587631i \(-0.199939\pi\)
\(500\) 0 0
\(501\) 971689. 6.64573e6i 0.172955 1.18290i
\(502\) 0 0
\(503\) 7.65689e6 1.34937 0.674687 0.738104i \(-0.264279\pi\)
0.674687 + 0.738104i \(0.264279\pi\)
\(504\) 0 0
\(505\) 1.40693e7 2.45495
\(506\) 0 0
\(507\) 181393. 1.24061e6i 0.0313402 0.214347i
\(508\) 0 0
\(509\) 2.52236e6i 0.431532i 0.976445 + 0.215766i \(0.0692248\pi\)
−0.976445 + 0.215766i \(0.930775\pi\)
\(510\) 0 0
\(511\) 4.77887e6i 0.809604i
\(512\) 0 0
\(513\) −6.63013e6 3.08538e6i −1.11232 0.517624i
\(514\) 0 0
\(515\) 7.89388e6 1.31151
\(516\) 0 0
\(517\) 769820. 0.126667
\(518\) 0 0
\(519\) 1.11117e7 + 1.62467e6i 1.81077 + 0.264757i
\(520\) 0 0
\(521\) 4.05284e6i 0.654131i 0.945002 + 0.327065i \(0.106060\pi\)
−0.945002 + 0.327065i \(0.893940\pi\)
\(522\) 0 0
\(523\) 4.38894e6i 0.701625i −0.936446 0.350813i \(-0.885905\pi\)
0.936446 0.350813i \(-0.114095\pi\)
\(524\) 0 0
\(525\) −7.47871e6 1.09348e6i −1.18421 0.173146i
\(526\) 0 0
\(527\) 180687. 0.0283400
\(528\) 0 0
\(529\) −6.36066e6 −0.988241
\(530\) 0 0
\(531\) −5.57607e6 1.66620e6i −0.858207 0.256443i
\(532\) 0 0
\(533\) 1.04600e7i 1.59482i
\(534\) 0 0
\(535\) 1.35730e7i 2.05018i
\(536\) 0 0
\(537\) −1.85603e6 + 1.26940e7i −0.277747 + 1.89961i
\(538\) 0 0
\(539\) 976543. 0.144784
\(540\) 0 0
\(541\) 5.53968e6 0.813751 0.406875 0.913484i \(-0.366618\pi\)
0.406875 + 0.913484i \(0.366618\pi\)
\(542\) 0 0
\(543\) 195232. 1.33526e6i 0.0284153 0.194342i
\(544\) 0 0
\(545\) 9.86741e6i 1.42302i
\(546\) 0 0
\(547\) 1.15951e7i 1.65694i −0.560034 0.828470i \(-0.689212\pi\)
0.560034 0.828470i \(-0.310788\pi\)
\(548\) 0 0
\(549\) 1.00908e7 + 3.01527e6i 1.42888 + 0.426968i
\(550\) 0 0
\(551\) −1.41647e7 −1.98760
\(552\) 0 0
\(553\) −1.00017e7 −1.39079
\(554\) 0 0
\(555\) 1.02699e7 + 1.50159e6i 1.41526 + 0.206929i
\(556\) 0 0
\(557\) 7.96541e6i 1.08785i 0.839133 + 0.543927i \(0.183063\pi\)
−0.839133 + 0.543927i \(0.816937\pi\)
\(558\) 0 0
\(559\) 2.42405e6i 0.328105i
\(560\) 0 0
\(561\) −79052.1 11558.4i −0.0106049 0.00155057i
\(562\) 0 0
\(563\) 5.14209e6 0.683705 0.341853 0.939754i \(-0.388946\pi\)
0.341853 + 0.939754i \(0.388946\pi\)
\(564\) 0 0
\(565\) 798538. 0.105238
\(566\) 0 0
\(567\) 3.57878e6 5.45364e6i 0.467496 0.712408i
\(568\) 0 0
\(569\) 3.04215e6i 0.393913i 0.980412 + 0.196956i \(0.0631057\pi\)
−0.980412 + 0.196956i \(0.936894\pi\)
\(570\) 0 0
\(571\) 9.13371e6i 1.17235i 0.810185 + 0.586175i \(0.199367\pi\)
−0.810185 + 0.586175i \(0.800633\pi\)
\(572\) 0 0
\(573\) −210635. + 1.44061e6i −0.0268006 + 0.183299i
\(574\) 0 0
\(575\) 1.20750e6 0.152306
\(576\) 0 0
\(577\) −9.31021e6 −1.16418 −0.582090 0.813125i \(-0.697765\pi\)
−0.582090 + 0.813125i \(0.697765\pi\)
\(578\) 0 0
\(579\) −142860. + 977069.i −0.0177098 + 0.121124i
\(580\) 0 0
\(581\) 1.21367e7i 1.49163i
\(582\) 0 0
\(583\) 779818.i 0.0950215i
\(584\) 0 0
\(585\) 3.25250e6 1.08847e7i 0.392941 1.31501i
\(586\) 0 0
\(587\) −5.65177e6 −0.677001 −0.338500 0.940966i \(-0.609920\pi\)
−0.338500 + 0.940966i \(0.609920\pi\)
\(588\) 0 0
\(589\) −1.44369e7 −1.71469
\(590\) 0 0
\(591\) −9.31227e6 1.36157e6i −1.09670 0.160351i
\(592\) 0 0
\(593\) 1.25646e7i 1.46728i −0.679541 0.733638i \(-0.737821\pi\)
0.679541 0.733638i \(-0.262179\pi\)
\(594\) 0 0
\(595\) 231370.i 0.0267926i
\(596\) 0 0
\(597\) 5.00133e6 + 731256.i 0.574314 + 0.0839719i
\(598\) 0 0
\(599\) −1.66264e7 −1.89335 −0.946676 0.322186i \(-0.895582\pi\)
−0.946676 + 0.322186i \(0.895582\pi\)
\(600\) 0 0
\(601\) 4.65512e6 0.525708 0.262854 0.964836i \(-0.415336\pi\)
0.262854 + 0.964836i \(0.415336\pi\)
\(602\) 0 0
\(603\) 4.55894e6 1.52568e7i 0.510588 1.70872i
\(604\) 0 0
\(605\) 1.00604e7i 1.11745i
\(606\) 0 0
\(607\) 4.60458e6i 0.507245i −0.967303 0.253623i \(-0.918378\pi\)
0.967303 0.253623i \(-0.0816222\pi\)
\(608\) 0 0
\(609\) 1.82793e6 1.25019e7i 0.199717 1.36594i
\(610\) 0 0
\(611\) 1.95731e6 0.212108
\(612\) 0 0
\(613\) 5.06680e6 0.544606 0.272303 0.962212i \(-0.412215\pi\)
0.272303 + 0.962212i \(0.412215\pi\)
\(614\) 0 0
\(615\) −3.79159e6 + 2.59320e7i −0.404234 + 2.76470i
\(616\) 0 0
\(617\) 5.69843e6i 0.602618i 0.953527 + 0.301309i \(0.0974235\pi\)
−0.953527 + 0.301309i \(0.902576\pi\)
\(618\) 0 0
\(619\) 4.94045e6i 0.518251i 0.965844 + 0.259125i \(0.0834343\pi\)
−0.965844 + 0.259125i \(0.916566\pi\)
\(620\) 0 0
\(621\) −439680. + 944822.i −0.0457517 + 0.0983153i
\(622\) 0 0
\(623\) 1.53327e7 1.58270
\(624\) 0 0
\(625\) −4.21707e6 −0.431828
\(626\) 0 0
\(627\) 6.31627e6 + 923518.i 0.641641 + 0.0938159i
\(628\) 0 0
\(629\) 185588.i 0.0187035i
\(630\) 0 0
\(631\) 1.04200e7i 1.04182i −0.853612 0.520910i \(-0.825593\pi\)
0.853612 0.520910i \(-0.174407\pi\)
\(632\) 0 0
\(633\) 1.01880e7 + 1.48961e6i 1.01060 + 0.147762i
\(634\) 0 0
\(635\) −7.11510e6 −0.700240
\(636\) 0 0
\(637\) 2.48292e6 0.242445
\(638\) 0 0
\(639\) −7.00070e6 2.09190e6i −0.678249 0.202670i
\(640\) 0 0
\(641\) 2.75177e6i 0.264525i 0.991215 + 0.132263i \(0.0422242\pi\)
−0.991215 + 0.132263i \(0.957776\pi\)
\(642\) 0 0
\(643\) 1.96874e7i 1.87785i 0.344123 + 0.938925i \(0.388176\pi\)
−0.344123 + 0.938925i \(0.611824\pi\)
\(644\) 0 0
\(645\) 878686. 6.00965e6i 0.0831638 0.568787i
\(646\) 0 0
\(647\) 2.00089e7 1.87915 0.939575 0.342343i \(-0.111220\pi\)
0.939575 + 0.342343i \(0.111220\pi\)
\(648\) 0 0
\(649\) 5.08002e6 0.473428
\(650\) 0 0
\(651\) 1.86305e6 1.27421e7i 0.172295 1.17839i
\(652\) 0 0
\(653\) 3.44288e6i 0.315965i 0.987442 + 0.157982i \(0.0504989\pi\)
−0.987442 + 0.157982i \(0.949501\pi\)
\(654\) 0 0
\(655\) 1.66923e7i 1.52024i
\(656\) 0 0
\(657\) 1.00722e7 + 3.00970e6i 0.910353 + 0.272025i
\(658\) 0 0
\(659\) 2.86205e6 0.256722 0.128361 0.991727i \(-0.459028\pi\)
0.128361 + 0.991727i \(0.459028\pi\)
\(660\) 0 0
\(661\) −1.51348e7 −1.34733 −0.673663 0.739039i \(-0.735280\pi\)
−0.673663 + 0.739039i \(0.735280\pi\)
\(662\) 0 0
\(663\) −200995. 29387.9i −0.0177583 0.00259648i
\(664\) 0 0
\(665\) 1.84865e7i 1.62107i
\(666\) 0 0
\(667\) 2.01853e6i 0.175679i
\(668\) 0 0
\(669\) −2.50703e6 366559.i −0.216568 0.0316649i
\(670\) 0 0
\(671\) −9.19314e6 −0.788238
\(672\) 0 0
\(673\) −6.06636e6 −0.516286 −0.258143 0.966107i \(-0.583111\pi\)
−0.258143 + 0.966107i \(0.583111\pi\)
\(674\) 0 0
\(675\) −7.01472e6 + 1.50738e7i −0.592585 + 1.27340i
\(676\) 0 0
\(677\) 1.23983e7i 1.03966i 0.854270 + 0.519829i \(0.174005\pi\)
−0.854270 + 0.519829i \(0.825995\pi\)
\(678\) 0 0
\(679\) 468106.i 0.0389646i
\(680\) 0 0
\(681\) 1.50644e6 1.03031e7i 0.124475 0.851331i
\(682\) 0 0
\(683\) −1.41080e7 −1.15722 −0.578608 0.815605i \(-0.696404\pi\)
−0.578608 + 0.815605i \(0.696404\pi\)
\(684\) 0 0
\(685\) −2.27378e7 −1.85149
\(686\) 0 0
\(687\) 2.13201e6 1.45816e7i 0.172345 1.17873i
\(688\) 0 0
\(689\) 1.98273e6i 0.159117i
\(690\) 0 0
\(691\) 5.34373e6i 0.425745i −0.977080 0.212872i \(-0.931718\pi\)
0.977080 0.212872i \(-0.0682818\pi\)
\(692\) 0 0
\(693\) −1.63021e6 + 5.45561e6i −0.128947 + 0.431529i
\(694\) 0 0
\(695\) −2.92452e6 −0.229664
\(696\) 0 0
\(697\) 468617. 0.0365373
\(698\) 0 0
\(699\) 1.02872e7 + 1.50412e6i 0.796354 + 0.116437i
\(700\) 0 0
\(701\) 4.64930e6i 0.357349i 0.983908 + 0.178674i \(0.0571809\pi\)
−0.983908 + 0.178674i \(0.942819\pi\)
\(702\) 0 0
\(703\) 1.48285e7i 1.13164i
\(704\) 0 0
\(705\) −4.85251e6 709498.i −0.367700 0.0537624i
\(706\) 0 0
\(707\) −1.79295e7 −1.34902
\(708\) 0 0
\(709\) 7.37828e6 0.551239 0.275619 0.961267i \(-0.411117\pi\)
0.275619 + 0.961267i \(0.411117\pi\)
\(710\) 0 0
\(711\) −6.29902e6 + 2.10801e7i −0.467303 + 1.56387i
\(712\) 0 0
\(713\) 2.05732e6i 0.151558i
\(714\) 0 0
\(715\) 9.91642e6i 0.725421i
\(716\) 0 0
\(717\) 923823. 6.31836e6i 0.0671106 0.458994i
\(718\) 0 0
\(719\) −2.11646e7 −1.52682 −0.763410 0.645914i \(-0.776477\pi\)
−0.763410 + 0.645914i \(0.776477\pi\)
\(720\) 0 0
\(721\) −1.00597e7 −0.720691
\(722\) 0 0
\(723\) −963434. + 6.58927e6i −0.0685451 + 0.468804i
\(724\) 0 0
\(725\) 3.22039e7i 2.27543i
\(726\) 0 0
\(727\) 1.97307e7i 1.38454i −0.721636 0.692272i \(-0.756610\pi\)
0.721636 0.692272i \(-0.243390\pi\)
\(728\) 0 0
\(729\) −9.24046e6 1.09775e7i −0.643984 0.765039i
\(730\) 0 0
\(731\) −108600. −0.00751688
\(732\) 0 0
\(733\) 1.30762e7 0.898924 0.449462 0.893299i \(-0.351616\pi\)
0.449462 + 0.893299i \(0.351616\pi\)
\(734\) 0 0
\(735\) −6.15558e6 900023.i −0.420291 0.0614519i
\(736\) 0 0
\(737\) 1.38996e7i 0.942612i
\(738\) 0 0
\(739\) 9.42776e6i 0.635035i −0.948252 0.317517i \(-0.897151\pi\)
0.948252 0.317517i \(-0.102849\pi\)
\(740\) 0 0
\(741\) 1.60595e7 + 2.34810e6i 1.07445 + 0.157098i
\(742\) 0 0
\(743\) −5.00117e6 −0.332353 −0.166177 0.986096i \(-0.553142\pi\)
−0.166177 + 0.986096i \(0.553142\pi\)
\(744\) 0 0
\(745\) 2.79230e7 1.84320
\(746\) 0 0
\(747\) −2.55800e7 7.64363e6i −1.67725 0.501185i
\(748\) 0 0
\(749\) 1.72971e7i 1.12659i
\(750\) 0 0
\(751\) 2.94559e7i 1.90578i −0.303314 0.952891i \(-0.598093\pi\)
0.303314 0.952891i \(-0.401907\pi\)
\(752\) 0 0
\(753\) −1.98912e6 + 1.36043e7i −0.127842 + 0.874360i
\(754\) 0 0
\(755\) −1.64956e7 −1.05318
\(756\) 0 0
\(757\) 2.64703e7 1.67888 0.839440 0.543452i \(-0.182883\pi\)
0.839440 + 0.543452i \(0.182883\pi\)
\(758\) 0 0
\(759\) 131605. 900096.i 0.00829218 0.0567132i
\(760\) 0 0
\(761\) 1.16223e7i 0.727494i −0.931498 0.363747i \(-0.881497\pi\)
0.931498 0.363747i \(-0.118503\pi\)
\(762\) 0 0
\(763\) 1.25748e7i 0.781967i
\(764\) 0 0
\(765\) 487648. + 145715.i 0.0301268 + 0.00900227i
\(766\) 0 0
\(767\) 1.29163e7 0.792772
\(768\) 0 0
\(769\) −4.74960e6 −0.289628 −0.144814 0.989459i \(-0.546258\pi\)
−0.144814 + 0.989459i \(0.546258\pi\)
\(770\) 0 0
\(771\) 9.24988e6 + 1.35245e6i 0.560403 + 0.0819379i
\(772\) 0 0
\(773\) 2.39153e7i 1.43955i 0.694208 + 0.719774i \(0.255755\pi\)
−0.694208 + 0.719774i \(0.744245\pi\)
\(774\) 0 0
\(775\) 3.28228e7i 1.96300i
\(776\) 0 0
\(777\) −1.30877e7 1.91359e6i −0.777700 0.113709i
\(778\) 0 0
\(779\) −3.74425e7 −2.21066
\(780\) 0 0
\(781\) 6.37792e6 0.374155
\(782\) 0 0
\(783\) −2.51983e7 1.17262e7i −1.46882 0.683524i
\(784\) 0 0
\(785\) 560391.i 0.0324576i
\(786\) 0 0
\(787\) 773120.i 0.0444949i −0.999752 0.0222475i \(-0.992918\pi\)
0.999752 0.0222475i \(-0.00708217\pi\)
\(788\) 0 0
\(789\) 1.03057e6 7.04844e6i 0.0589366 0.403089i
\(790\) 0 0
\(791\) −1.01763e6 −0.0578297
\(792\) 0 0
\(793\) −2.33741e7 −1.31993
\(794\) 0 0
\(795\) 718713. 4.91554e6i 0.0403309 0.275837i
\(796\) 0 0
\(797\) 2.84342e7i 1.58560i 0.609480 + 0.792802i \(0.291378\pi\)
−0.609480 + 0.792802i \(0.708622\pi\)
\(798\) 0 0
\(799\) 87689.7i 0.00485939i
\(800\) 0 0
\(801\) 9.65643e6 3.23160e7i 0.531784 1.77965i
\(802\) 0 0
\(803\) −9.17616e6 −0.502194
\(804\) 0 0
\(805\) −2.63441e6 −0.143283
\(806\) 0 0
\(807\) −1.70294e7 2.48990e6i −0.920480 0.134586i
\(808\) 0 0
\(809\) 1.04767e7i 0.562800i 0.959591 + 0.281400i \(0.0907988\pi\)
−0.959591 + 0.281400i \(0.909201\pi\)
\(810\) 0 0
\(811\) 1.11593e7i 0.595776i 0.954601 + 0.297888i \(0.0962822\pi\)
−0.954601 + 0.297888i \(0.903718\pi\)
\(812\) 0 0
\(813\) 512028. + 74864.9i 0.0271686 + 0.00397239i
\(814\) 0 0
\(815\) 2.88875e7 1.52341
\(816\) 0 0
\(817\) 8.67717e6 0.454803
\(818\) 0 0
\(819\) −4.14490e6 + 1.38712e7i −0.215925 + 0.722611i
\(820\) 0 0
\(821\) 1.20649e7i 0.624690i 0.949969 + 0.312345i \(0.101114\pi\)
−0.949969 + 0.312345i \(0.898886\pi\)
\(822\) 0 0
\(823\) 2.97522e7i 1.53115i −0.643344 0.765577i \(-0.722454\pi\)
0.643344 0.765577i \(-0.277546\pi\)
\(824\) 0 0
\(825\) 2.09965e6 1.43603e7i 0.107402 0.734561i
\(826\) 0 0
\(827\) 2.54258e7 1.29274 0.646369 0.763025i \(-0.276287\pi\)
0.646369 + 0.763025i \(0.276287\pi\)
\(828\) 0 0
\(829\) 2.74678e7 1.38815 0.694077 0.719901i \(-0.255813\pi\)
0.694077 + 0.719901i \(0.255813\pi\)
\(830\) 0 0
\(831\) 2.97349e6 2.03367e7i 0.149370 1.02159i
\(832\) 0 0
\(833\) 111237.i 0.00555441i
\(834\) 0 0
\(835\) 3.73485e7i 1.85377i
\(836\) 0 0
\(837\) −2.56825e7 1.19515e7i −1.26714 0.589672i
\(838\) 0 0
\(839\) 2.10051e7 1.03019 0.515097 0.857132i \(-0.327756\pi\)
0.515097 + 0.857132i \(0.327756\pi\)
\(840\) 0 0
\(841\) −3.33229e7 −1.62462
\(842\) 0 0
\(843\) 6.19119e6 + 905230.i 0.300058 + 0.0438722i
\(844\) 0 0
\(845\) 6.97216e6i 0.335912i
\(846\) 0 0
\(847\) 1.28207e7i 0.614049i
\(848\) 0 0
\(849\) −3.29886e7 4.82335e6i −1.57071 0.229657i
\(850\) 0 0
\(851\) 2.11313e6 0.100023
\(852\) 0 0
\(853\) −2.25309e7 −1.06024 −0.530121 0.847922i \(-0.677854\pi\)
−0.530121 + 0.847922i \(0.677854\pi\)
\(854\) 0 0
\(855\) −3.89631e7 1.16427e7i −1.82280 0.544675i
\(856\) 0 0
\(857\) 2.37134e7i 1.10292i 0.834203 + 0.551458i \(0.185928\pi\)
−0.834203 + 0.551458i \(0.814072\pi\)
\(858\) 0 0
\(859\) 1.08107e7i 0.499887i 0.968260 + 0.249944i \(0.0804121\pi\)
−0.968260 + 0.249944i \(0.919588\pi\)
\(860\) 0 0
\(861\) 4.83189e6 3.30471e7i 0.222131 1.51924i
\(862\) 0 0
\(863\) −1.86848e7 −0.854006 −0.427003 0.904250i \(-0.640431\pi\)
−0.427003 + 0.904250i \(0.640431\pi\)
\(864\) 0 0
\(865\) 6.24470e7 2.83773
\(866\) 0 0
\(867\) −3.20081e6 + 2.18915e7i −0.144615 + 0.989072i
\(868\) 0 0
\(869\) 1.92048e7i 0.862703i
\(870\) 0 0
\(871\) 3.53405e7i 1.57844i
\(872\) 0 0
\(873\) 986604. + 294810.i 0.0438134 + 0.0130920i
\(874\) 0 0
\(875\) −1.21053e7 −0.534509
\(876\) 0 0
\(877\) −1.16971e7 −0.513544 −0.256772 0.966472i \(-0.582659\pi\)
−0.256772 + 0.966472i \(0.582659\pi\)
\(878\) 0 0
\(879\) 1.88469e7 + 2.75566e6i 0.822751 + 0.120296i
\(880\) 0 0
\(881\) 2.81207e7i 1.22064i −0.792156 0.610318i \(-0.791042\pi\)
0.792156 0.610318i \(-0.208958\pi\)
\(882\) 0 0
\(883\) 1.71174e7i 0.738816i −0.929267 0.369408i \(-0.879561\pi\)
0.929267 0.369408i \(-0.120439\pi\)
\(884\) 0 0
\(885\) −3.20216e7 4.68196e6i −1.37431 0.200942i
\(886\) 0 0
\(887\) 2.58331e7 1.10247 0.551237 0.834349i \(-0.314156\pi\)
0.551237 + 0.834349i \(0.314156\pi\)
\(888\) 0 0
\(889\) 9.06729e6 0.384789
\(890\) 0 0
\(891\) 1.04718e7 + 6.87181e6i 0.441904 + 0.289986i
\(892\) 0 0
\(893\) 7.00641e6i 0.294013i
\(894\) 0 0
\(895\) 7.13395e7i 2.97696i
\(896\) 0 0
\(897\) 334614. 2.28855e6i 0.0138856 0.0949683i
\(898\) 0 0
\(899\) −5.48685e7 −2.26425
\(900\) 0 0
\(901\) −88828.6 −0.00364536
\(902\) 0 0
\(903\) −1.11977e6 + 7.65854e6i −0.0456994 + 0.312555i
\(904\) 0 0
\(905\) 7.50407e6i 0.304562i
\(906\) 0 0
\(907\) 2.84483e7i 1.14825i 0.818766 + 0.574127i \(0.194659\pi\)
−0.818766 + 0.574127i \(0.805341\pi\)
\(908\) 0 0
\(909\) −1.12919e7 + 3.77891e7i −0.453269 + 1.51690i
\(910\) 0 0
\(911\) −4.51021e7 −1.80053 −0.900266 0.435340i \(-0.856628\pi\)
−0.900266 + 0.435340i \(0.856628\pi\)
\(912\) 0 0
\(913\) 2.33044e7 0.925253
\(914\) 0 0
\(915\) 5.79484e7 + 8.47278e6i 2.28817 + 0.334559i
\(916\) 0 0
\(917\) 2.12722e7i 0.835389i
\(918\) 0 0
\(919\) 4.39345e7i 1.71600i −0.513651 0.857999i \(-0.671707\pi\)
0.513651 0.857999i \(-0.328293\pi\)
\(920\) 0 0
\(921\) 909377. + 132962.i 0.0353260 + 0.00516511i
\(922\) 0 0
\(923\) 1.62162e7 0.626535
\(924\) 0 0
\(925\) 3.37131e7 1.29552
\(926\) 0 0
\(927\) −6.33556e6 + 2.12024e7i −0.242151 + 0.810375i
\(928\) 0 0
\(929\) 2.25804e7i 0.858405i −0.903208 0.429203i \(-0.858795\pi\)
0.903208 0.429203i \(-0.141205\pi\)
\(930\) 0 0
\(931\) 8.88787e6i 0.336065i
\(932\) 0 0
\(933\) 2.31204e6 1.58128e7i 0.0869542 0.594711i
\(934\) 0 0
\(935\) −444266. −0.0166194
\(936\) 0 0
\(937\) 8.13233e6 0.302598 0.151299 0.988488i \(-0.451654\pi\)
0.151299 + 0.988488i \(0.451654\pi\)
\(938\) 0 0
\(939\) 3.68362e6 2.51936e7i 0.136336 0.932451i
\(940\) 0 0
\(941\) 1.83529e7i 0.675665i 0.941206 + 0.337833i \(0.109694\pi\)
−0.941206 + 0.337833i \(0.890306\pi\)
\(942\) 0 0
\(943\) 5.33573e6i 0.195395i
\(944\) 0 0
\(945\) 1.53040e7 3.28866e7i 0.557476 1.19795i
\(946\) 0 0
\(947\) 3.42017e6 0.123929 0.0619644 0.998078i \(-0.480263\pi\)
0.0619644 + 0.998078i \(0.480263\pi\)
\(948\) 0 0
\(949\) −2.33309e7 −0.840942
\(950\) 0 0
\(951\) −9.19425e6 1.34431e6i −0.329659 0.0482003i
\(952\) 0 0
\(953\) 4.57241e7i 1.63084i 0.578867 + 0.815422i \(0.303495\pi\)
−0.578867 + 0.815422i \(0.696505\pi\)
\(954\) 0 0
\(955\) 8.09612e6i 0.287256i
\(956\) 0 0
\(957\) 2.40055e7 + 3.50990e6i 0.847287 + 0.123884i
\(958\) 0 0
\(959\) 2.89765e7 1.01742
\(960\) 0 0
\(961\) −2.72937e7 −0.953354
\(962\) 0 0
\(963\) 3.64561e7 + 1.08936e7i 1.26679 + 0.378533i
\(964\) 0 0
\(965\) 5.49105e6i 0.189818i
\(966\) 0 0
\(967\) 6.01434e6i 0.206834i −0.994638 0.103417i \(-0.967022\pi\)
0.994638 0.103417i \(-0.0329776\pi\)
\(968\) 0 0
\(969\) −105197. + 719482.i −0.00359911 + 0.0246156i
\(970\) 0 0
\(971\) 4.11774e7 1.40156 0.700779 0.713378i \(-0.252836\pi\)
0.700779 + 0.713378i \(0.252836\pi\)
\(972\) 0 0
\(973\) 3.72693e6 0.126203
\(974\) 0 0
\(975\) 5.33848e6 3.65118e7i 0.179848 1.23005i
\(976\) 0 0
\(977\) 2.82069e7i 0.945409i 0.881221 + 0.472704i \(0.156722\pi\)
−0.881221 + 0.472704i \(0.843278\pi\)
\(978\) 0 0
\(979\) 2.94411e7i 0.981743i
\(980\) 0 0
\(981\) −2.65032e7 7.91950e6i −0.879277 0.262739i
\(982\) 0 0
\(983\) −6.06646e6 −0.200240 −0.100120 0.994975i \(-0.531923\pi\)
−0.100120 + 0.994975i \(0.531923\pi\)
\(984\) 0 0
\(985\) −5.23342e7 −1.71868
\(986\) 0 0
\(987\) 6.18391e6 + 904165.i 0.202055 + 0.0295430i
\(988\) 0 0
\(989\) 1.23653e6i 0.0401990i
\(990\) 0 0
\(991\) 2.99468e7i 0.968649i 0.874888 + 0.484324i \(0.160935\pi\)
−0.874888 + 0.484324i \(0.839065\pi\)
\(992\) 0 0
\(993\) 2.16272e7 + 3.16216e6i 0.696028 + 0.101768i
\(994\) 0 0
\(995\) 2.81071e7 0.900031
\(996\) 0 0
\(997\) −3.82255e7 −1.21791 −0.608954 0.793205i \(-0.708411\pi\)
−0.608954 + 0.793205i \(0.708411\pi\)
\(998\) 0 0
\(999\) −1.22757e7 + 2.63792e7i −0.389165 + 0.836273i
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.a.383.10 yes 20
3.2 odd 2 384.6.c.d.383.12 yes 20
4.3 odd 2 384.6.c.d.383.11 yes 20
8.3 odd 2 384.6.c.b.383.10 yes 20
8.5 even 2 384.6.c.c.383.11 yes 20
12.11 even 2 inner 384.6.c.a.383.9 20
24.5 odd 2 384.6.c.b.383.9 yes 20
24.11 even 2 384.6.c.c.383.12 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.6.c.a.383.9 20 12.11 even 2 inner
384.6.c.a.383.10 yes 20 1.1 even 1 trivial
384.6.c.b.383.9 yes 20 24.5 odd 2
384.6.c.b.383.10 yes 20 8.3 odd 2
384.6.c.c.383.11 yes 20 8.5 even 2
384.6.c.c.383.12 yes 20 24.11 even 2
384.6.c.d.383.11 yes 20 4.3 odd 2
384.6.c.d.383.12 yes 20 3.2 odd 2