Properties

Label 688.6.a.h.1.1
Level $688$
Weight $6$
Character 688.1
Self dual yes
Analytic conductor $110.344$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [688,6,Mod(1,688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(688, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("688.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 688.a (trivial)

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: yes
Analytic conductor: \(110.344068031\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + 8437093 x^{2} - 5752252 x - 22734604 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(11.5305\) of defining polynomial
Character \(\chi\) \(=\) 688.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-27.4953 q^{3} +86.8464 q^{5} +19.8137 q^{7} +512.989 q^{9} +O(q^{10})\) \(q-27.4953 q^{3} +86.8464 q^{5} +19.8137 q^{7} +512.989 q^{9} +85.3712 q^{11} -229.081 q^{13} -2387.86 q^{15} +1356.51 q^{17} +2795.35 q^{19} -544.783 q^{21} +1856.11 q^{23} +4417.29 q^{25} -7423.41 q^{27} +7312.96 q^{29} +2937.36 q^{31} -2347.30 q^{33} +1720.75 q^{35} +2577.36 q^{37} +6298.65 q^{39} -3532.54 q^{41} -1849.00 q^{43} +44551.2 q^{45} +7065.73 q^{47} -16414.4 q^{49} -37297.6 q^{51} -3852.63 q^{53} +7414.18 q^{55} -76858.8 q^{57} -27996.1 q^{59} -39244.4 q^{61} +10164.2 q^{63} -19894.9 q^{65} +14809.1 q^{67} -51034.2 q^{69} -8956.13 q^{71} +35168.6 q^{73} -121455. q^{75} +1691.52 q^{77} +13263.6 q^{79} +79452.3 q^{81} +9812.47 q^{83} +117808. q^{85} -201072. q^{87} -87124.9 q^{89} -4538.95 q^{91} -80763.5 q^{93} +242766. q^{95} +83982.4 q^{97} +43794.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9} - 745 q^{11} + 1917 q^{13} - 1688 q^{15} + 4017 q^{17} + 2404 q^{19} - 228 q^{21} - 1733 q^{23} + 7120 q^{25} + 2324 q^{27} + 6996 q^{29} + 4899 q^{31} - 15734 q^{33} - 7084 q^{35} + 1466 q^{37} + 26542 q^{39} + 10297 q^{41} - 18490 q^{43} + 73822 q^{45} - 48592 q^{47} + 29458 q^{49} - 92972 q^{51} + 127165 q^{53} - 106672 q^{55} + 34060 q^{57} - 99372 q^{59} + 17408 q^{61} - 2244 q^{63} + 54484 q^{65} + 2021 q^{67} + 1654 q^{69} - 11286 q^{71} + 49892 q^{73} + 44662 q^{75} + 98144 q^{77} + 91524 q^{79} - 26450 q^{81} + 105203 q^{83} - 87212 q^{85} - 181200 q^{87} - 62682 q^{89} + 295304 q^{91} - 238430 q^{93} + 305340 q^{95} + 108383 q^{97} + 270499 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −27.4953 −1.76382 −0.881911 0.471417i \(-0.843743\pi\)
−0.881911 + 0.471417i \(0.843743\pi\)
\(4\) 0 0
\(5\) 86.8464 1.55356 0.776778 0.629775i \(-0.216853\pi\)
0.776778 + 0.629775i \(0.216853\pi\)
\(6\) 0 0
\(7\) 19.8137 0.152834 0.0764171 0.997076i \(-0.475652\pi\)
0.0764171 + 0.997076i \(0.475652\pi\)
\(8\) 0 0
\(9\) 512.989 2.11107
\(10\) 0 0
\(11\) 85.3712 0.212730 0.106365 0.994327i \(-0.466079\pi\)
0.106365 + 0.994327i \(0.466079\pi\)
\(12\) 0 0
\(13\) −229.081 −0.375951 −0.187975 0.982174i \(-0.560193\pi\)
−0.187975 + 0.982174i \(0.560193\pi\)
\(14\) 0 0
\(15\) −2387.86 −2.74019
\(16\) 0 0
\(17\) 1356.51 1.13841 0.569207 0.822194i \(-0.307250\pi\)
0.569207 + 0.822194i \(0.307250\pi\)
\(18\) 0 0
\(19\) 2795.35 1.77645 0.888223 0.459413i \(-0.151940\pi\)
0.888223 + 0.459413i \(0.151940\pi\)
\(20\) 0 0
\(21\) −544.783 −0.269572
\(22\) 0 0
\(23\) 1856.11 0.731618 0.365809 0.930690i \(-0.380792\pi\)
0.365809 + 0.930690i \(0.380792\pi\)
\(24\) 0 0
\(25\) 4417.29 1.41353
\(26\) 0 0
\(27\) −7423.41 −1.95972
\(28\) 0 0
\(29\) 7312.96 1.61472 0.807362 0.590057i \(-0.200895\pi\)
0.807362 + 0.590057i \(0.200895\pi\)
\(30\) 0 0
\(31\) 2937.36 0.548975 0.274488 0.961591i \(-0.411492\pi\)
0.274488 + 0.961591i \(0.411492\pi\)
\(32\) 0 0
\(33\) −2347.30 −0.375218
\(34\) 0 0
\(35\) 1720.75 0.237436
\(36\) 0 0
\(37\) 2577.36 0.309507 0.154754 0.987953i \(-0.450542\pi\)
0.154754 + 0.987953i \(0.450542\pi\)
\(38\) 0 0
\(39\) 6298.65 0.663110
\(40\) 0 0
\(41\) −3532.54 −0.328192 −0.164096 0.986444i \(-0.552471\pi\)
−0.164096 + 0.986444i \(0.552471\pi\)
\(42\) 0 0
\(43\) −1849.00 −0.152499
\(44\) 0 0
\(45\) 44551.2 3.27966
\(46\) 0 0
\(47\) 7065.73 0.466566 0.233283 0.972409i \(-0.425053\pi\)
0.233283 + 0.972409i \(0.425053\pi\)
\(48\) 0 0
\(49\) −16414.4 −0.976642
\(50\) 0 0
\(51\) −37297.6 −2.00796
\(52\) 0 0
\(53\) −3852.63 −0.188394 −0.0941972 0.995554i \(-0.530028\pi\)
−0.0941972 + 0.995554i \(0.530028\pi\)
\(54\) 0 0
\(55\) 7414.18 0.330488
\(56\) 0 0
\(57\) −76858.8 −3.13333
\(58\) 0 0
\(59\) −27996.1 −1.04705 −0.523524 0.852011i \(-0.675383\pi\)
−0.523524 + 0.852011i \(0.675383\pi\)
\(60\) 0 0
\(61\) −39244.4 −1.35037 −0.675186 0.737648i \(-0.735937\pi\)
−0.675186 + 0.737648i \(0.735937\pi\)
\(62\) 0 0
\(63\) 10164.2 0.322643
\(64\) 0 0
\(65\) −19894.9 −0.584060
\(66\) 0 0
\(67\) 14809.1 0.403034 0.201517 0.979485i \(-0.435413\pi\)
0.201517 + 0.979485i \(0.435413\pi\)
\(68\) 0 0
\(69\) −51034.2 −1.29044
\(70\) 0 0
\(71\) −8956.13 −0.210850 −0.105425 0.994427i \(-0.533620\pi\)
−0.105425 + 0.994427i \(0.533620\pi\)
\(72\) 0 0
\(73\) 35168.6 0.772411 0.386205 0.922413i \(-0.373786\pi\)
0.386205 + 0.922413i \(0.373786\pi\)
\(74\) 0 0
\(75\) −121455. −2.49322
\(76\) 0 0
\(77\) 1691.52 0.0325125
\(78\) 0 0
\(79\) 13263.6 0.239108 0.119554 0.992828i \(-0.461854\pi\)
0.119554 + 0.992828i \(0.461854\pi\)
\(80\) 0 0
\(81\) 79452.3 1.34553
\(82\) 0 0
\(83\) 9812.47 0.156345 0.0781724 0.996940i \(-0.475092\pi\)
0.0781724 + 0.996940i \(0.475092\pi\)
\(84\) 0 0
\(85\) 117808. 1.76859
\(86\) 0 0
\(87\) −201072. −2.84808
\(88\) 0 0
\(89\) −87124.9 −1.16592 −0.582958 0.812502i \(-0.698105\pi\)
−0.582958 + 0.812502i \(0.698105\pi\)
\(90\) 0 0
\(91\) −4538.95 −0.0574581
\(92\) 0 0
\(93\) −80763.5 −0.968295
\(94\) 0 0
\(95\) 242766. 2.75981
\(96\) 0 0
\(97\) 83982.4 0.906272 0.453136 0.891441i \(-0.350305\pi\)
0.453136 + 0.891441i \(0.350305\pi\)
\(98\) 0 0
\(99\) 43794.5 0.449088
\(100\) 0 0
\(101\) 74775.4 0.729382 0.364691 0.931129i \(-0.381175\pi\)
0.364691 + 0.931129i \(0.381175\pi\)
\(102\) 0 0
\(103\) −29400.7 −0.273064 −0.136532 0.990636i \(-0.543596\pi\)
−0.136532 + 0.990636i \(0.543596\pi\)
\(104\) 0 0
\(105\) −47312.4 −0.418795
\(106\) 0 0
\(107\) 199822. 1.68727 0.843635 0.536917i \(-0.180411\pi\)
0.843635 + 0.536917i \(0.180411\pi\)
\(108\) 0 0
\(109\) −39465.7 −0.318166 −0.159083 0.987265i \(-0.550854\pi\)
−0.159083 + 0.987265i \(0.550854\pi\)
\(110\) 0 0
\(111\) −70865.2 −0.545915
\(112\) 0 0
\(113\) 41487.8 0.305650 0.152825 0.988253i \(-0.451163\pi\)
0.152825 + 0.988253i \(0.451163\pi\)
\(114\) 0 0
\(115\) 161196. 1.13661
\(116\) 0 0
\(117\) −117516. −0.793657
\(118\) 0 0
\(119\) 26877.5 0.173989
\(120\) 0 0
\(121\) −153763. −0.954746
\(122\) 0 0
\(123\) 97128.1 0.578871
\(124\) 0 0
\(125\) 112231. 0.642448
\(126\) 0 0
\(127\) −357769. −1.96831 −0.984156 0.177305i \(-0.943262\pi\)
−0.984156 + 0.177305i \(0.943262\pi\)
\(128\) 0 0
\(129\) 50838.7 0.268980
\(130\) 0 0
\(131\) 61207.9 0.311623 0.155811 0.987787i \(-0.450201\pi\)
0.155811 + 0.987787i \(0.450201\pi\)
\(132\) 0 0
\(133\) 55386.2 0.271502
\(134\) 0 0
\(135\) −644697. −3.04453
\(136\) 0 0
\(137\) −309777. −1.41009 −0.705047 0.709161i \(-0.749074\pi\)
−0.705047 + 0.709161i \(0.749074\pi\)
\(138\) 0 0
\(139\) 193740. 0.850517 0.425259 0.905072i \(-0.360183\pi\)
0.425259 + 0.905072i \(0.360183\pi\)
\(140\) 0 0
\(141\) −194274. −0.822938
\(142\) 0 0
\(143\) −19556.9 −0.0799762
\(144\) 0 0
\(145\) 635104. 2.50856
\(146\) 0 0
\(147\) 451319. 1.72262
\(148\) 0 0
\(149\) −455017. −1.67904 −0.839521 0.543327i \(-0.817164\pi\)
−0.839521 + 0.543327i \(0.817164\pi\)
\(150\) 0 0
\(151\) 505868. 1.80549 0.902744 0.430179i \(-0.141549\pi\)
0.902744 + 0.430179i \(0.141549\pi\)
\(152\) 0 0
\(153\) 695874. 2.40327
\(154\) 0 0
\(155\) 255099. 0.852864
\(156\) 0 0
\(157\) 348495. 1.12836 0.564179 0.825652i \(-0.309193\pi\)
0.564179 + 0.825652i \(0.309193\pi\)
\(158\) 0 0
\(159\) 105929. 0.332294
\(160\) 0 0
\(161\) 36776.4 0.111816
\(162\) 0 0
\(163\) −104915. −0.309291 −0.154645 0.987970i \(-0.549424\pi\)
−0.154645 + 0.987970i \(0.549424\pi\)
\(164\) 0 0
\(165\) −203855. −0.582923
\(166\) 0 0
\(167\) −186982. −0.518809 −0.259405 0.965769i \(-0.583526\pi\)
−0.259405 + 0.965769i \(0.583526\pi\)
\(168\) 0 0
\(169\) −318815. −0.858661
\(170\) 0 0
\(171\) 1.43398e6 3.75019
\(172\) 0 0
\(173\) 491718. 1.24911 0.624555 0.780981i \(-0.285280\pi\)
0.624555 + 0.780981i \(0.285280\pi\)
\(174\) 0 0
\(175\) 87522.9 0.216036
\(176\) 0 0
\(177\) 769758. 1.84681
\(178\) 0 0
\(179\) −374509. −0.873634 −0.436817 0.899550i \(-0.643894\pi\)
−0.436817 + 0.899550i \(0.643894\pi\)
\(180\) 0 0
\(181\) 460794. 1.04547 0.522733 0.852496i \(-0.324912\pi\)
0.522733 + 0.852496i \(0.324912\pi\)
\(182\) 0 0
\(183\) 1.07904e6 2.38181
\(184\) 0 0
\(185\) 223834. 0.480837
\(186\) 0 0
\(187\) 115807. 0.242175
\(188\) 0 0
\(189\) −147085. −0.299512
\(190\) 0 0
\(191\) 22464.4 0.0445565 0.0222783 0.999752i \(-0.492908\pi\)
0.0222783 + 0.999752i \(0.492908\pi\)
\(192\) 0 0
\(193\) 830124. 1.60417 0.802083 0.597212i \(-0.203725\pi\)
0.802083 + 0.597212i \(0.203725\pi\)
\(194\) 0 0
\(195\) 547015. 1.03018
\(196\) 0 0
\(197\) 802058. 1.47245 0.736224 0.676738i \(-0.236607\pi\)
0.736224 + 0.676738i \(0.236607\pi\)
\(198\) 0 0
\(199\) 187945. 0.336432 0.168216 0.985750i \(-0.446199\pi\)
0.168216 + 0.985750i \(0.446199\pi\)
\(200\) 0 0
\(201\) −407180. −0.710879
\(202\) 0 0
\(203\) 144897. 0.246785
\(204\) 0 0
\(205\) −306788. −0.509864
\(206\) 0 0
\(207\) 952164. 1.54449
\(208\) 0 0
\(209\) 238642. 0.377904
\(210\) 0 0
\(211\) −327613. −0.506589 −0.253294 0.967389i \(-0.581514\pi\)
−0.253294 + 0.967389i \(0.581514\pi\)
\(212\) 0 0
\(213\) 246251. 0.371902
\(214\) 0 0
\(215\) −160579. −0.236915
\(216\) 0 0
\(217\) 58200.0 0.0839022
\(218\) 0 0
\(219\) −966970. −1.36239
\(220\) 0 0
\(221\) −310751. −0.427988
\(222\) 0 0
\(223\) 166692. 0.224467 0.112234 0.993682i \(-0.464199\pi\)
0.112234 + 0.993682i \(0.464199\pi\)
\(224\) 0 0
\(225\) 2.26602e6 2.98406
\(226\) 0 0
\(227\) 155259. 0.199982 0.0999912 0.994988i \(-0.468119\pi\)
0.0999912 + 0.994988i \(0.468119\pi\)
\(228\) 0 0
\(229\) −1.01590e6 −1.28016 −0.640079 0.768309i \(-0.721098\pi\)
−0.640079 + 0.768309i \(0.721098\pi\)
\(230\) 0 0
\(231\) −46508.7 −0.0573462
\(232\) 0 0
\(233\) 23299.8 0.0281166 0.0140583 0.999901i \(-0.495525\pi\)
0.0140583 + 0.999901i \(0.495525\pi\)
\(234\) 0 0
\(235\) 613633. 0.724835
\(236\) 0 0
\(237\) −364686. −0.421743
\(238\) 0 0
\(239\) −4871.64 −0.00551671 −0.00275836 0.999996i \(-0.500878\pi\)
−0.00275836 + 0.999996i \(0.500878\pi\)
\(240\) 0 0
\(241\) 918873. 1.01909 0.509545 0.860444i \(-0.329814\pi\)
0.509545 + 0.860444i \(0.329814\pi\)
\(242\) 0 0
\(243\) −380672. −0.413557
\(244\) 0 0
\(245\) −1.42553e6 −1.51727
\(246\) 0 0
\(247\) −640362. −0.667856
\(248\) 0 0
\(249\) −269796. −0.275764
\(250\) 0 0
\(251\) −1.47118e6 −1.47395 −0.736973 0.675922i \(-0.763746\pi\)
−0.736973 + 0.675922i \(0.763746\pi\)
\(252\) 0 0
\(253\) 158458. 0.155637
\(254\) 0 0
\(255\) −3.23916e6 −3.11948
\(256\) 0 0
\(257\) −813506. −0.768295 −0.384147 0.923272i \(-0.625505\pi\)
−0.384147 + 0.923272i \(0.625505\pi\)
\(258\) 0 0
\(259\) 51067.0 0.0473033
\(260\) 0 0
\(261\) 3.75147e6 3.40879
\(262\) 0 0
\(263\) −691865. −0.616782 −0.308391 0.951260i \(-0.599791\pi\)
−0.308391 + 0.951260i \(0.599791\pi\)
\(264\) 0 0
\(265\) −334587. −0.292681
\(266\) 0 0
\(267\) 2.39552e6 2.05647
\(268\) 0 0
\(269\) −759828. −0.640228 −0.320114 0.947379i \(-0.603721\pi\)
−0.320114 + 0.947379i \(0.603721\pi\)
\(270\) 0 0
\(271\) −954020. −0.789104 −0.394552 0.918874i \(-0.629100\pi\)
−0.394552 + 0.918874i \(0.629100\pi\)
\(272\) 0 0
\(273\) 124799. 0.101346
\(274\) 0 0
\(275\) 377110. 0.300702
\(276\) 0 0
\(277\) 1.03398e6 0.809680 0.404840 0.914387i \(-0.367327\pi\)
0.404840 + 0.914387i \(0.367327\pi\)
\(278\) 0 0
\(279\) 1.50683e6 1.15892
\(280\) 0 0
\(281\) 2.05117e6 1.54966 0.774829 0.632171i \(-0.217836\pi\)
0.774829 + 0.632171i \(0.217836\pi\)
\(282\) 0 0
\(283\) 2.37044e6 1.75940 0.879698 0.475533i \(-0.157745\pi\)
0.879698 + 0.475533i \(0.157745\pi\)
\(284\) 0 0
\(285\) −6.67491e6 −4.86780
\(286\) 0 0
\(287\) −69992.7 −0.0501589
\(288\) 0 0
\(289\) 420259. 0.295987
\(290\) 0 0
\(291\) −2.30912e6 −1.59850
\(292\) 0 0
\(293\) 1.48623e6 1.01138 0.505692 0.862714i \(-0.331237\pi\)
0.505692 + 0.862714i \(0.331237\pi\)
\(294\) 0 0
\(295\) −2.43136e6 −1.62665
\(296\) 0 0
\(297\) −633746. −0.416892
\(298\) 0 0
\(299\) −425200. −0.275052
\(300\) 0 0
\(301\) −36635.5 −0.0233070
\(302\) 0 0
\(303\) −2.05597e6 −1.28650
\(304\) 0 0
\(305\) −3.40824e6 −2.09788
\(306\) 0 0
\(307\) 1.62571e6 0.984456 0.492228 0.870466i \(-0.336183\pi\)
0.492228 + 0.870466i \(0.336183\pi\)
\(308\) 0 0
\(309\) 808378. 0.481636
\(310\) 0 0
\(311\) −2.07269e6 −1.21516 −0.607581 0.794258i \(-0.707860\pi\)
−0.607581 + 0.794258i \(0.707860\pi\)
\(312\) 0 0
\(313\) −345918. −0.199578 −0.0997889 0.995009i \(-0.531817\pi\)
−0.0997889 + 0.995009i \(0.531817\pi\)
\(314\) 0 0
\(315\) 882725. 0.501244
\(316\) 0 0
\(317\) 574435. 0.321065 0.160532 0.987031i \(-0.448679\pi\)
0.160532 + 0.987031i \(0.448679\pi\)
\(318\) 0 0
\(319\) 624316. 0.343501
\(320\) 0 0
\(321\) −5.49417e6 −2.97604
\(322\) 0 0
\(323\) 3.79191e6 2.02233
\(324\) 0 0
\(325\) −1.01192e6 −0.531419
\(326\) 0 0
\(327\) 1.08512e6 0.561187
\(328\) 0 0
\(329\) 139998. 0.0713072
\(330\) 0 0
\(331\) 1.12898e6 0.566393 0.283196 0.959062i \(-0.408605\pi\)
0.283196 + 0.959062i \(0.408605\pi\)
\(332\) 0 0
\(333\) 1.32216e6 0.653390
\(334\) 0 0
\(335\) 1.28612e6 0.626135
\(336\) 0 0
\(337\) −370207. −0.177570 −0.0887850 0.996051i \(-0.528298\pi\)
−0.0887850 + 0.996051i \(0.528298\pi\)
\(338\) 0 0
\(339\) −1.14072e6 −0.539112
\(340\) 0 0
\(341\) 250766. 0.116784
\(342\) 0 0
\(343\) −658239. −0.302098
\(344\) 0 0
\(345\) −4.43214e6 −2.00477
\(346\) 0 0
\(347\) −372676. −0.166153 −0.0830764 0.996543i \(-0.526475\pi\)
−0.0830764 + 0.996543i \(0.526475\pi\)
\(348\) 0 0
\(349\) −2.14587e6 −0.943060 −0.471530 0.881850i \(-0.656298\pi\)
−0.471530 + 0.881850i \(0.656298\pi\)
\(350\) 0 0
\(351\) 1.70056e6 0.736759
\(352\) 0 0
\(353\) −2.87740e6 −1.22903 −0.614517 0.788904i \(-0.710649\pi\)
−0.614517 + 0.788904i \(0.710649\pi\)
\(354\) 0 0
\(355\) −777807. −0.327568
\(356\) 0 0
\(357\) −739003. −0.306885
\(358\) 0 0
\(359\) −647359. −0.265100 −0.132550 0.991176i \(-0.542316\pi\)
−0.132550 + 0.991176i \(0.542316\pi\)
\(360\) 0 0
\(361\) 5.33787e6 2.15576
\(362\) 0 0
\(363\) 4.22775e6 1.68400
\(364\) 0 0
\(365\) 3.05427e6 1.19998
\(366\) 0 0
\(367\) −3.06021e6 −1.18600 −0.593001 0.805202i \(-0.702057\pi\)
−0.593001 + 0.805202i \(0.702057\pi\)
\(368\) 0 0
\(369\) −1.81215e6 −0.692834
\(370\) 0 0
\(371\) −76335.0 −0.0287931
\(372\) 0 0
\(373\) 3.54156e6 1.31802 0.659010 0.752134i \(-0.270975\pi\)
0.659010 + 0.752134i \(0.270975\pi\)
\(374\) 0 0
\(375\) −3.08582e6 −1.13316
\(376\) 0 0
\(377\) −1.67526e6 −0.607057
\(378\) 0 0
\(379\) 3.92757e6 1.40451 0.702257 0.711923i \(-0.252176\pi\)
0.702257 + 0.711923i \(0.252176\pi\)
\(380\) 0 0
\(381\) 9.83696e6 3.47175
\(382\) 0 0
\(383\) 1.29688e6 0.451754 0.225877 0.974156i \(-0.427475\pi\)
0.225877 + 0.974156i \(0.427475\pi\)
\(384\) 0 0
\(385\) 146902. 0.0505099
\(386\) 0 0
\(387\) −948517. −0.321934
\(388\) 0 0
\(389\) −100541. −0.0336874 −0.0168437 0.999858i \(-0.505362\pi\)
−0.0168437 + 0.999858i \(0.505362\pi\)
\(390\) 0 0
\(391\) 2.51783e6 0.832884
\(392\) 0 0
\(393\) −1.68293e6 −0.549647
\(394\) 0 0
\(395\) 1.15190e6 0.371467
\(396\) 0 0
\(397\) 3.36979e6 1.07307 0.536533 0.843879i \(-0.319734\pi\)
0.536533 + 0.843879i \(0.319734\pi\)
\(398\) 0 0
\(399\) −1.52286e6 −0.478880
\(400\) 0 0
\(401\) 4.73407e6 1.47019 0.735095 0.677964i \(-0.237138\pi\)
0.735095 + 0.677964i \(0.237138\pi\)
\(402\) 0 0
\(403\) −672894. −0.206388
\(404\) 0 0
\(405\) 6.90015e6 2.09036
\(406\) 0 0
\(407\) 220032. 0.0658416
\(408\) 0 0
\(409\) −84169.8 −0.0248799 −0.0124399 0.999923i \(-0.503960\pi\)
−0.0124399 + 0.999923i \(0.503960\pi\)
\(410\) 0 0
\(411\) 8.51740e6 2.48715
\(412\) 0 0
\(413\) −554705. −0.160025
\(414\) 0 0
\(415\) 852178. 0.242890
\(416\) 0 0
\(417\) −5.32694e6 −1.50016
\(418\) 0 0
\(419\) 1.45191e6 0.404022 0.202011 0.979383i \(-0.435252\pi\)
0.202011 + 0.979383i \(0.435252\pi\)
\(420\) 0 0
\(421\) 4.69910e6 1.29214 0.646070 0.763278i \(-0.276411\pi\)
0.646070 + 0.763278i \(0.276411\pi\)
\(422\) 0 0
\(423\) 3.62464e6 0.984950
\(424\) 0 0
\(425\) 5.99210e6 1.60919
\(426\) 0 0
\(427\) −777577. −0.206383
\(428\) 0 0
\(429\) 537723. 0.141064
\(430\) 0 0
\(431\) −3.34282e6 −0.866803 −0.433401 0.901201i \(-0.642687\pi\)
−0.433401 + 0.901201i \(0.642687\pi\)
\(432\) 0 0
\(433\) 629642. 0.161389 0.0806945 0.996739i \(-0.474286\pi\)
0.0806945 + 0.996739i \(0.474286\pi\)
\(434\) 0 0
\(435\) −1.74623e7 −4.42465
\(436\) 0 0
\(437\) 5.18847e6 1.29968
\(438\) 0 0
\(439\) −6.24963e6 −1.54772 −0.773861 0.633355i \(-0.781677\pi\)
−0.773861 + 0.633355i \(0.781677\pi\)
\(440\) 0 0
\(441\) −8.42041e6 −2.06175
\(442\) 0 0
\(443\) −6.58305e6 −1.59374 −0.796871 0.604149i \(-0.793513\pi\)
−0.796871 + 0.604149i \(0.793513\pi\)
\(444\) 0 0
\(445\) −7.56648e6 −1.81131
\(446\) 0 0
\(447\) 1.25108e7 2.96153
\(448\) 0 0
\(449\) 1.62106e6 0.379476 0.189738 0.981835i \(-0.439236\pi\)
0.189738 + 0.981835i \(0.439236\pi\)
\(450\) 0 0
\(451\) −301577. −0.0698163
\(452\) 0 0
\(453\) −1.39090e7 −3.18456
\(454\) 0 0
\(455\) −394191. −0.0892644
\(456\) 0 0
\(457\) −6.01194e6 −1.34656 −0.673278 0.739390i \(-0.735114\pi\)
−0.673278 + 0.739390i \(0.735114\pi\)
\(458\) 0 0
\(459\) −1.00699e7 −2.23097
\(460\) 0 0
\(461\) −1.97072e6 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(462\) 0 0
\(463\) 925538. 0.200651 0.100326 0.994955i \(-0.468012\pi\)
0.100326 + 0.994955i \(0.468012\pi\)
\(464\) 0 0
\(465\) −7.01401e6 −1.50430
\(466\) 0 0
\(467\) −1.87994e6 −0.398890 −0.199445 0.979909i \(-0.563914\pi\)
−0.199445 + 0.979909i \(0.563914\pi\)
\(468\) 0 0
\(469\) 293423. 0.0615973
\(470\) 0 0
\(471\) −9.58195e6 −1.99022
\(472\) 0 0
\(473\) −157851. −0.0324411
\(474\) 0 0
\(475\) 1.23479e7 2.51107
\(476\) 0 0
\(477\) −1.97636e6 −0.397713
\(478\) 0 0
\(479\) −8.18229e6 −1.62943 −0.814716 0.579860i \(-0.803107\pi\)
−0.814716 + 0.579860i \(0.803107\pi\)
\(480\) 0 0
\(481\) −590425. −0.116360
\(482\) 0 0
\(483\) −1.01118e6 −0.197224
\(484\) 0 0
\(485\) 7.29356e6 1.40794
\(486\) 0 0
\(487\) 1.84424e6 0.352367 0.176184 0.984357i \(-0.443625\pi\)
0.176184 + 0.984357i \(0.443625\pi\)
\(488\) 0 0
\(489\) 2.88465e6 0.545534
\(490\) 0 0
\(491\) 4.29014e6 0.803096 0.401548 0.915838i \(-0.368472\pi\)
0.401548 + 0.915838i \(0.368472\pi\)
\(492\) 0 0
\(493\) 9.92009e6 1.83822
\(494\) 0 0
\(495\) 3.80339e6 0.697683
\(496\) 0 0
\(497\) −177454. −0.0322252
\(498\) 0 0
\(499\) 7.80338e6 1.40291 0.701457 0.712711i \(-0.252533\pi\)
0.701457 + 0.712711i \(0.252533\pi\)
\(500\) 0 0
\(501\) 5.14111e6 0.915087
\(502\) 0 0
\(503\) −673490. −0.118689 −0.0593446 0.998238i \(-0.518901\pi\)
−0.0593446 + 0.998238i \(0.518901\pi\)
\(504\) 0 0
\(505\) 6.49397e6 1.13314
\(506\) 0 0
\(507\) 8.76589e6 1.51452
\(508\) 0 0
\(509\) 9.44081e6 1.61516 0.807579 0.589760i \(-0.200778\pi\)
0.807579 + 0.589760i \(0.200778\pi\)
\(510\) 0 0
\(511\) 696821. 0.118051
\(512\) 0 0
\(513\) −2.07510e7 −3.48134
\(514\) 0 0
\(515\) −2.55334e6 −0.424220
\(516\) 0 0
\(517\) 603210. 0.0992527
\(518\) 0 0
\(519\) −1.35199e7 −2.20321
\(520\) 0 0
\(521\) −1.01037e7 −1.63074 −0.815369 0.578941i \(-0.803466\pi\)
−0.815369 + 0.578941i \(0.803466\pi\)
\(522\) 0 0
\(523\) −1.22441e6 −0.195737 −0.0978687 0.995199i \(-0.531203\pi\)
−0.0978687 + 0.995199i \(0.531203\pi\)
\(524\) 0 0
\(525\) −2.40647e6 −0.381049
\(526\) 0 0
\(527\) 3.98455e6 0.624961
\(528\) 0 0
\(529\) −2.99120e6 −0.464735
\(530\) 0 0
\(531\) −1.43617e7 −2.21039
\(532\) 0 0
\(533\) 809239. 0.123384
\(534\) 0 0
\(535\) 1.73539e7 2.62127
\(536\) 0 0
\(537\) 1.02972e7 1.54093
\(538\) 0 0
\(539\) −1.40132e6 −0.207761
\(540\) 0 0
\(541\) −2.66077e6 −0.390853 −0.195426 0.980718i \(-0.562609\pi\)
−0.195426 + 0.980718i \(0.562609\pi\)
\(542\) 0 0
\(543\) −1.26696e7 −1.84402
\(544\) 0 0
\(545\) −3.42745e6 −0.494288
\(546\) 0 0
\(547\) −7.44368e6 −1.06370 −0.531850 0.846839i \(-0.678503\pi\)
−0.531850 + 0.846839i \(0.678503\pi\)
\(548\) 0 0
\(549\) −2.01320e7 −2.85072
\(550\) 0 0
\(551\) 2.04423e7 2.86847
\(552\) 0 0
\(553\) 262801. 0.0365438
\(554\) 0 0
\(555\) −6.15438e6 −0.848110
\(556\) 0 0
\(557\) 5.55246e6 0.758311 0.379155 0.925333i \(-0.376215\pi\)
0.379155 + 0.925333i \(0.376215\pi\)
\(558\) 0 0
\(559\) 423571. 0.0573320
\(560\) 0 0
\(561\) −3.18414e6 −0.427154
\(562\) 0 0
\(563\) 2.05400e6 0.273105 0.136552 0.990633i \(-0.456398\pi\)
0.136552 + 0.990633i \(0.456398\pi\)
\(564\) 0 0
\(565\) 3.60307e6 0.474845
\(566\) 0 0
\(567\) 1.57424e6 0.205643
\(568\) 0 0
\(569\) 5.13151e6 0.664453 0.332227 0.943200i \(-0.392200\pi\)
0.332227 + 0.943200i \(0.392200\pi\)
\(570\) 0 0
\(571\) 1.13394e7 1.45546 0.727730 0.685863i \(-0.240575\pi\)
0.727730 + 0.685863i \(0.240575\pi\)
\(572\) 0 0
\(573\) −617664. −0.0785898
\(574\) 0 0
\(575\) 8.19899e6 1.03417
\(576\) 0 0
\(577\) −9.19195e6 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(578\) 0 0
\(579\) −2.28245e7 −2.82946
\(580\) 0 0
\(581\) 194421. 0.0238948
\(582\) 0 0
\(583\) −328904. −0.0400772
\(584\) 0 0
\(585\) −1.02058e7 −1.23299
\(586\) 0 0
\(587\) −7.32138e6 −0.876997 −0.438498 0.898732i \(-0.644490\pi\)
−0.438498 + 0.898732i \(0.644490\pi\)
\(588\) 0 0
\(589\) 8.21094e6 0.975225
\(590\) 0 0
\(591\) −2.20528e7 −2.59714
\(592\) 0 0
\(593\) 8.89330e6 1.03855 0.519274 0.854608i \(-0.326203\pi\)
0.519274 + 0.854608i \(0.326203\pi\)
\(594\) 0 0
\(595\) 2.33421e6 0.270301
\(596\) 0 0
\(597\) −5.16758e6 −0.593406
\(598\) 0 0
\(599\) −7.52622e6 −0.857057 −0.428529 0.903528i \(-0.640968\pi\)
−0.428529 + 0.903528i \(0.640968\pi\)
\(600\) 0 0
\(601\) 1.53849e7 1.73743 0.868716 0.495311i \(-0.164946\pi\)
0.868716 + 0.495311i \(0.164946\pi\)
\(602\) 0 0
\(603\) 7.59690e6 0.850830
\(604\) 0 0
\(605\) −1.33537e7 −1.48325
\(606\) 0 0
\(607\) 1.10155e7 1.21348 0.606740 0.794900i \(-0.292477\pi\)
0.606740 + 0.794900i \(0.292477\pi\)
\(608\) 0 0
\(609\) −3.98397e6 −0.435285
\(610\) 0 0
\(611\) −1.61863e6 −0.175406
\(612\) 0 0
\(613\) −8.89814e6 −0.956419 −0.478209 0.878246i \(-0.658714\pi\)
−0.478209 + 0.878246i \(0.658714\pi\)
\(614\) 0 0
\(615\) 8.43522e6 0.899309
\(616\) 0 0
\(617\) 7.40581e6 0.783177 0.391588 0.920140i \(-0.371926\pi\)
0.391588 + 0.920140i \(0.371926\pi\)
\(618\) 0 0
\(619\) −1.41475e7 −1.48406 −0.742032 0.670364i \(-0.766138\pi\)
−0.742032 + 0.670364i \(0.766138\pi\)
\(620\) 0 0
\(621\) −1.37787e7 −1.43377
\(622\) 0 0
\(623\) −1.72627e6 −0.178192
\(624\) 0 0
\(625\) −4.05719e6 −0.415456
\(626\) 0 0
\(627\) −6.56153e6 −0.666555
\(628\) 0 0
\(629\) 3.49621e6 0.352347
\(630\) 0 0
\(631\) −9.57148e6 −0.956986 −0.478493 0.878091i \(-0.658817\pi\)
−0.478493 + 0.878091i \(0.658817\pi\)
\(632\) 0 0
\(633\) 9.00781e6 0.893532
\(634\) 0 0
\(635\) −3.10710e7 −3.05788
\(636\) 0 0
\(637\) 3.76023e6 0.367169
\(638\) 0 0
\(639\) −4.59440e6 −0.445119
\(640\) 0 0
\(641\) −637566. −0.0612887 −0.0306443 0.999530i \(-0.509756\pi\)
−0.0306443 + 0.999530i \(0.509756\pi\)
\(642\) 0 0
\(643\) 1.89702e7 1.80944 0.904718 0.426010i \(-0.140081\pi\)
0.904718 + 0.426010i \(0.140081\pi\)
\(644\) 0 0
\(645\) 4.41516e6 0.417876
\(646\) 0 0
\(647\) −2.02807e7 −1.90468 −0.952338 0.305044i \(-0.901329\pi\)
−0.952338 + 0.305044i \(0.901329\pi\)
\(648\) 0 0
\(649\) −2.39006e6 −0.222739
\(650\) 0 0
\(651\) −1.60022e6 −0.147989
\(652\) 0 0
\(653\) −3.76628e6 −0.345645 −0.172822 0.984953i \(-0.555289\pi\)
−0.172822 + 0.984953i \(0.555289\pi\)
\(654\) 0 0
\(655\) 5.31568e6 0.484123
\(656\) 0 0
\(657\) 1.80411e7 1.63061
\(658\) 0 0
\(659\) 1.70994e7 1.53379 0.766897 0.641770i \(-0.221800\pi\)
0.766897 + 0.641770i \(0.221800\pi\)
\(660\) 0 0
\(661\) 875240. 0.0779154 0.0389577 0.999241i \(-0.487596\pi\)
0.0389577 + 0.999241i \(0.487596\pi\)
\(662\) 0 0
\(663\) 8.54417e6 0.754894
\(664\) 0 0
\(665\) 4.81009e6 0.421793
\(666\) 0 0
\(667\) 1.35737e7 1.18136
\(668\) 0 0
\(669\) −4.58324e6 −0.395920
\(670\) 0 0
\(671\) −3.35034e6 −0.287265
\(672\) 0 0
\(673\) −1.59835e7 −1.36030 −0.680149 0.733074i \(-0.738085\pi\)
−0.680149 + 0.733074i \(0.738085\pi\)
\(674\) 0 0
\(675\) −3.27914e7 −2.77013
\(676\) 0 0
\(677\) −2.27744e7 −1.90974 −0.954871 0.297020i \(-0.904007\pi\)
−0.954871 + 0.297020i \(0.904007\pi\)
\(678\) 0 0
\(679\) 1.66400e6 0.138509
\(680\) 0 0
\(681\) −4.26888e6 −0.352733
\(682\) 0 0
\(683\) 1.16349e7 0.954354 0.477177 0.878807i \(-0.341660\pi\)
0.477177 + 0.878807i \(0.341660\pi\)
\(684\) 0 0
\(685\) −2.69030e7 −2.19066
\(686\) 0 0
\(687\) 2.79325e7 2.25797
\(688\) 0 0
\(689\) 882566. 0.0708271
\(690\) 0 0
\(691\) 5.50626e6 0.438694 0.219347 0.975647i \(-0.429607\pi\)
0.219347 + 0.975647i \(0.429607\pi\)
\(692\) 0 0
\(693\) 867731. 0.0686360
\(694\) 0 0
\(695\) 1.68257e7 1.32133
\(696\) 0 0
\(697\) −4.79192e6 −0.373618
\(698\) 0 0
\(699\) −640633. −0.0495926
\(700\) 0 0
\(701\) −1.26842e7 −0.974916 −0.487458 0.873146i \(-0.662076\pi\)
−0.487458 + 0.873146i \(0.662076\pi\)
\(702\) 0 0
\(703\) 7.20462e6 0.549823
\(704\) 0 0
\(705\) −1.68720e7 −1.27848
\(706\) 0 0
\(707\) 1.48158e6 0.111475
\(708\) 0 0
\(709\) 919145. 0.0686702 0.0343351 0.999410i \(-0.489069\pi\)
0.0343351 + 0.999410i \(0.489069\pi\)
\(710\) 0 0
\(711\) 6.80408e6 0.504772
\(712\) 0 0
\(713\) 5.45207e6 0.401640
\(714\) 0 0
\(715\) −1.69845e6 −0.124247
\(716\) 0 0
\(717\) 133947. 0.00973049
\(718\) 0 0
\(719\) −7.21135e6 −0.520228 −0.260114 0.965578i \(-0.583760\pi\)
−0.260114 + 0.965578i \(0.583760\pi\)
\(720\) 0 0
\(721\) −582536. −0.0417335
\(722\) 0 0
\(723\) −2.52646e7 −1.79749
\(724\) 0 0
\(725\) 3.23035e7 2.28247
\(726\) 0 0
\(727\) 4.15671e6 0.291685 0.145842 0.989308i \(-0.453411\pi\)
0.145842 + 0.989308i \(0.453411\pi\)
\(728\) 0 0
\(729\) −8.84023e6 −0.616091
\(730\) 0 0
\(731\) −2.50818e6 −0.173607
\(732\) 0 0
\(733\) 4.05351e6 0.278658 0.139329 0.990246i \(-0.455505\pi\)
0.139329 + 0.990246i \(0.455505\pi\)
\(734\) 0 0
\(735\) 3.91954e7 2.67619
\(736\) 0 0
\(737\) 1.26427e6 0.0857375
\(738\) 0 0
\(739\) −1.27176e7 −0.856629 −0.428315 0.903630i \(-0.640892\pi\)
−0.428315 + 0.903630i \(0.640892\pi\)
\(740\) 0 0
\(741\) 1.76069e7 1.17798
\(742\) 0 0
\(743\) 1.86660e6 0.124045 0.0620224 0.998075i \(-0.480245\pi\)
0.0620224 + 0.998075i \(0.480245\pi\)
\(744\) 0 0
\(745\) −3.95165e7 −2.60848
\(746\) 0 0
\(747\) 5.03369e6 0.330054
\(748\) 0 0
\(749\) 3.95922e6 0.257873
\(750\) 0 0
\(751\) −2.61334e7 −1.69081 −0.845407 0.534122i \(-0.820642\pi\)
−0.845407 + 0.534122i \(0.820642\pi\)
\(752\) 0 0
\(753\) 4.04505e7 2.59978
\(754\) 0 0
\(755\) 4.39328e7 2.80492
\(756\) 0 0
\(757\) 1.20591e6 0.0764846 0.0382423 0.999268i \(-0.487824\pi\)
0.0382423 + 0.999268i \(0.487824\pi\)
\(758\) 0 0
\(759\) −4.35685e6 −0.274516
\(760\) 0 0
\(761\) −1.63050e6 −0.102061 −0.0510303 0.998697i \(-0.516251\pi\)
−0.0510303 + 0.998697i \(0.516251\pi\)
\(762\) 0 0
\(763\) −781961. −0.0486266
\(764\) 0 0
\(765\) 6.04341e7 3.73361
\(766\) 0 0
\(767\) 6.41337e6 0.393639
\(768\) 0 0
\(769\) −3.03077e6 −0.184815 −0.0924074 0.995721i \(-0.529456\pi\)
−0.0924074 + 0.995721i \(0.529456\pi\)
\(770\) 0 0
\(771\) 2.23676e7 1.35513
\(772\) 0 0
\(773\) 1.56440e7 0.941669 0.470834 0.882222i \(-0.343953\pi\)
0.470834 + 0.882222i \(0.343953\pi\)
\(774\) 0 0
\(775\) 1.29752e7 0.775995
\(776\) 0 0
\(777\) −1.40410e6 −0.0834346
\(778\) 0 0
\(779\) −9.87468e6 −0.583015
\(780\) 0 0
\(781\) −764595. −0.0448543
\(782\) 0 0
\(783\) −5.42871e7 −3.16441
\(784\) 0 0
\(785\) 3.02655e7 1.75297
\(786\) 0 0
\(787\) −2.77750e7 −1.59852 −0.799260 0.600986i \(-0.794775\pi\)
−0.799260 + 0.600986i \(0.794775\pi\)
\(788\) 0 0
\(789\) 1.90230e7 1.08789
\(790\) 0 0
\(791\) 822028. 0.0467138
\(792\) 0 0
\(793\) 8.99016e6 0.507673
\(794\) 0 0
\(795\) 9.19957e6 0.516237
\(796\) 0 0
\(797\) −646020. −0.0360247 −0.0180123 0.999838i \(-0.505734\pi\)
−0.0180123 + 0.999838i \(0.505734\pi\)
\(798\) 0 0
\(799\) 9.58473e6 0.531145
\(800\) 0 0
\(801\) −4.46941e7 −2.46132
\(802\) 0 0
\(803\) 3.00239e6 0.164315
\(804\) 0 0
\(805\) 3.19390e6 0.173713
\(806\) 0 0
\(807\) 2.08917e7 1.12925
\(808\) 0 0
\(809\) −1.67691e6 −0.0900823 −0.0450412 0.998985i \(-0.514342\pi\)
−0.0450412 + 0.998985i \(0.514342\pi\)
\(810\) 0 0
\(811\) −1.09231e7 −0.583169 −0.291585 0.956545i \(-0.594183\pi\)
−0.291585 + 0.956545i \(0.594183\pi\)
\(812\) 0 0
\(813\) 2.62310e7 1.39184
\(814\) 0 0
\(815\) −9.11145e6 −0.480500
\(816\) 0 0
\(817\) −5.16860e6 −0.270905
\(818\) 0 0
\(819\) −2.32843e6 −0.121298
\(820\) 0 0
\(821\) 1.94364e7 1.00637 0.503186 0.864178i \(-0.332161\pi\)
0.503186 + 0.864178i \(0.332161\pi\)
\(822\) 0 0
\(823\) 3.08462e7 1.58746 0.793730 0.608271i \(-0.208136\pi\)
0.793730 + 0.608271i \(0.208136\pi\)
\(824\) 0 0
\(825\) −1.03687e7 −0.530384
\(826\) 0 0
\(827\) 1.97925e7 1.00632 0.503161 0.864193i \(-0.332170\pi\)
0.503161 + 0.864193i \(0.332170\pi\)
\(828\) 0 0
\(829\) −1.75755e7 −0.888223 −0.444111 0.895972i \(-0.646481\pi\)
−0.444111 + 0.895972i \(0.646481\pi\)
\(830\) 0 0
\(831\) −2.84296e7 −1.42813
\(832\) 0 0
\(833\) −2.22663e7 −1.11182
\(834\) 0 0
\(835\) −1.62387e7 −0.805999
\(836\) 0 0
\(837\) −2.18052e7 −1.07584
\(838\) 0 0
\(839\) 1.50238e7 0.736842 0.368421 0.929659i \(-0.379899\pi\)
0.368421 + 0.929659i \(0.379899\pi\)
\(840\) 0 0
\(841\) 3.29682e7 1.60733
\(842\) 0 0
\(843\) −5.63974e7 −2.73332
\(844\) 0 0
\(845\) −2.76879e7 −1.33398
\(846\) 0 0
\(847\) −3.04661e6 −0.145918
\(848\) 0 0
\(849\) −6.51759e7 −3.10326
\(850\) 0 0
\(851\) 4.78387e6 0.226441
\(852\) 0 0
\(853\) 3.88390e7 1.82766 0.913829 0.406099i \(-0.133111\pi\)
0.913829 + 0.406099i \(0.133111\pi\)
\(854\) 0 0
\(855\) 1.24536e8 5.82613
\(856\) 0 0
\(857\) −3.57330e6 −0.166195 −0.0830975 0.996541i \(-0.526481\pi\)
−0.0830975 + 0.996541i \(0.526481\pi\)
\(858\) 0 0
\(859\) −3.27850e6 −0.151598 −0.0757988 0.997123i \(-0.524151\pi\)
−0.0757988 + 0.997123i \(0.524151\pi\)
\(860\) 0 0
\(861\) 1.92447e6 0.0884713
\(862\) 0 0
\(863\) −1.93441e7 −0.884139 −0.442070 0.896981i \(-0.645756\pi\)
−0.442070 + 0.896981i \(0.645756\pi\)
\(864\) 0 0
\(865\) 4.27039e7 1.94056
\(866\) 0 0
\(867\) −1.15551e7 −0.522068
\(868\) 0 0
\(869\) 1.13233e6 0.0508655
\(870\) 0 0
\(871\) −3.39248e6 −0.151521
\(872\) 0 0
\(873\) 4.30820e7 1.91320
\(874\) 0 0
\(875\) 2.22371e6 0.0981880
\(876\) 0 0
\(877\) −2.57531e7 −1.13066 −0.565329 0.824866i \(-0.691251\pi\)
−0.565329 + 0.824866i \(0.691251\pi\)
\(878\) 0 0
\(879\) −4.08642e7 −1.78390
\(880\) 0 0
\(881\) 4.39384e6 0.190724 0.0953618 0.995443i \(-0.469599\pi\)
0.0953618 + 0.995443i \(0.469599\pi\)
\(882\) 0 0
\(883\) −2.21529e7 −0.956156 −0.478078 0.878317i \(-0.658666\pi\)
−0.478078 + 0.878317i \(0.658666\pi\)
\(884\) 0 0
\(885\) 6.68507e7 2.86912
\(886\) 0 0
\(887\) 4.36187e7 1.86150 0.930750 0.365655i \(-0.119155\pi\)
0.930750 + 0.365655i \(0.119155\pi\)
\(888\) 0 0
\(889\) −7.08874e6 −0.300825
\(890\) 0 0
\(891\) 6.78294e6 0.286236
\(892\) 0 0
\(893\) 1.97512e7 0.828828
\(894\) 0 0
\(895\) −3.25247e7 −1.35724
\(896\) 0 0
\(897\) 1.16910e7 0.485143
\(898\) 0 0
\(899\) 2.14808e7 0.886443
\(900\) 0 0
\(901\) −5.22613e6 −0.214471
\(902\) 0 0
\(903\) 1.00730e6 0.0411094
\(904\) 0 0
\(905\) 4.00183e7 1.62419
\(906\) 0 0
\(907\) −1.38621e7 −0.559515 −0.279758 0.960071i \(-0.590254\pi\)
−0.279758 + 0.960071i \(0.590254\pi\)
\(908\) 0 0
\(909\) 3.83589e7 1.53977
\(910\) 0 0
\(911\) 2.31037e7 0.922329 0.461164 0.887315i \(-0.347432\pi\)
0.461164 + 0.887315i \(0.347432\pi\)
\(912\) 0 0
\(913\) 837702. 0.0332593
\(914\) 0 0
\(915\) 9.37103e7 3.70028
\(916\) 0 0
\(917\) 1.21275e6 0.0476266
\(918\) 0 0
\(919\) 2.27313e7 0.887842 0.443921 0.896066i \(-0.353587\pi\)
0.443921 + 0.896066i \(0.353587\pi\)
\(920\) 0 0
\(921\) −4.46992e7 −1.73640
\(922\) 0 0
\(923\) 2.05168e6 0.0792694
\(924\) 0 0
\(925\) 1.13850e7 0.437499
\(926\) 0 0
\(927\) −1.50822e7 −0.576455
\(928\) 0 0
\(929\) −3.32467e7 −1.26389 −0.631945 0.775013i \(-0.717743\pi\)
−0.631945 + 0.775013i \(0.717743\pi\)
\(930\) 0 0
\(931\) −4.58840e7 −1.73495
\(932\) 0 0
\(933\) 5.69892e7 2.14333
\(934\) 0 0
\(935\) 1.00574e7 0.376233
\(936\) 0 0
\(937\) 1.72533e7 0.641981 0.320991 0.947082i \(-0.395984\pi\)
0.320991 + 0.947082i \(0.395984\pi\)
\(938\) 0 0
\(939\) 9.51110e6 0.352019
\(940\) 0 0
\(941\) 1.12167e6 0.0412945 0.0206473 0.999787i \(-0.493427\pi\)
0.0206473 + 0.999787i \(0.493427\pi\)
\(942\) 0 0
\(943\) −6.55679e6 −0.240111
\(944\) 0 0
\(945\) −1.27738e7 −0.465309
\(946\) 0 0
\(947\) −3.89113e7 −1.40994 −0.704970 0.709237i \(-0.749040\pi\)
−0.704970 + 0.709237i \(0.749040\pi\)
\(948\) 0 0
\(949\) −8.05647e6 −0.290388
\(950\) 0 0
\(951\) −1.57942e7 −0.566301
\(952\) 0 0
\(953\) −1.79943e7 −0.641803 −0.320902 0.947113i \(-0.603986\pi\)
−0.320902 + 0.947113i \(0.603986\pi\)
\(954\) 0 0
\(955\) 1.95095e6 0.0692210
\(956\) 0 0
\(957\) −1.71657e7 −0.605874
\(958\) 0 0
\(959\) −6.13783e6 −0.215510
\(960\) 0 0
\(961\) −2.00011e7 −0.698626
\(962\) 0 0
\(963\) 1.02507e8 3.56194
\(964\) 0 0
\(965\) 7.20932e7 2.49216
\(966\) 0 0
\(967\) −2.31110e7 −0.794791 −0.397395 0.917648i \(-0.630086\pi\)
−0.397395 + 0.917648i \(0.630086\pi\)
\(968\) 0 0
\(969\) −1.04260e8 −3.56703
\(970\) 0 0
\(971\) 3.50927e7 1.19445 0.597226 0.802073i \(-0.296270\pi\)
0.597226 + 0.802073i \(0.296270\pi\)
\(972\) 0 0
\(973\) 3.83871e6 0.129988
\(974\) 0 0
\(975\) 2.78230e7 0.937329
\(976\) 0 0
\(977\) 7.20029e6 0.241331 0.120666 0.992693i \(-0.461497\pi\)
0.120666 + 0.992693i \(0.461497\pi\)
\(978\) 0 0
\(979\) −7.43795e6 −0.248026
\(980\) 0 0
\(981\) −2.02455e7 −0.671669
\(982\) 0 0
\(983\) 3.28324e7 1.08373 0.541863 0.840467i \(-0.317719\pi\)
0.541863 + 0.840467i \(0.317719\pi\)
\(984\) 0 0
\(985\) 6.96558e7 2.28753
\(986\) 0 0
\(987\) −3.84929e6 −0.125773
\(988\) 0 0
\(989\) −3.43195e6 −0.111571
\(990\) 0 0
\(991\) −5.19670e7 −1.68090 −0.840452 0.541885i \(-0.817711\pi\)
−0.840452 + 0.541885i \(0.817711\pi\)
\(992\) 0 0
\(993\) −3.10417e7 −0.999015
\(994\) 0 0
\(995\) 1.63223e7 0.522665
\(996\) 0 0
\(997\) −5.30582e6 −0.169050 −0.0845249 0.996421i \(-0.526937\pi\)
−0.0845249 + 0.996421i \(0.526937\pi\)
\(998\) 0 0
\(999\) −1.91328e7 −0.606548
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 688.6.a.h.1.1 10
4.3 odd 2 43.6.a.b.1.1 10
12.11 even 2 387.6.a.e.1.10 10
20.19 odd 2 1075.6.a.b.1.10 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.b.1.1 10 4.3 odd 2
387.6.a.e.1.10 10 12.11 even 2
688.6.a.h.1.1 10 1.1 even 1 trivial
1075.6.a.b.1.10 10 20.19 odd 2