Properties

Label 688.6.a.h.1.2
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 1.2
Root \(-4.38824\) 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)
 
\(f(q)\) \(=\) \(q-25.0462 q^{3} -0.456695 q^{5} -166.517 q^{7} +384.312 q^{9} +O(q^{10})\) \(q-25.0462 q^{3} -0.456695 q^{5} -166.517 q^{7} +384.312 q^{9} +65.6764 q^{11} -689.371 q^{13} +11.4385 q^{15} +737.385 q^{17} +609.887 q^{19} +4170.63 q^{21} -1312.45 q^{23} -3124.79 q^{25} -3539.34 q^{27} -8969.74 q^{29} -5858.93 q^{31} -1644.94 q^{33} +76.0477 q^{35} -55.9069 q^{37} +17266.1 q^{39} -10450.3 q^{41} -1849.00 q^{43} -175.514 q^{45} -1942.82 q^{47} +10921.1 q^{49} -18468.7 q^{51} +30129.2 q^{53} -29.9941 q^{55} -15275.3 q^{57} -52167.4 q^{59} -14040.1 q^{61} -63994.7 q^{63} +314.832 q^{65} -51437.7 q^{67} +32872.0 q^{69} -11268.6 q^{71} -61124.5 q^{73} +78264.2 q^{75} -10936.3 q^{77} -96018.3 q^{79} -4740.91 q^{81} -30378.0 q^{83} -336.760 q^{85} +224658. q^{87} -65040.4 q^{89} +114792. q^{91} +146744. q^{93} -278.532 q^{95} +12067.6 q^{97} +25240.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9}+O(q^{10}) \) 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
\(3\) −25.0462 −1.60671 −0.803357 0.595497i \(-0.796955\pi\)
−0.803357 + 0.595497i \(0.796955\pi\)
\(4\) 0 0
\(5\) −0.456695 −0.00816961 −0.00408481 0.999992i \(-0.501300\pi\)
−0.00408481 + 0.999992i \(0.501300\pi\)
\(6\) 0 0
\(7\) −166.517 −1.28444 −0.642221 0.766519i \(-0.721987\pi\)
−0.642221 + 0.766519i \(0.721987\pi\)
\(8\) 0 0
\(9\) 384.312 1.58153
\(10\) 0 0
\(11\) 65.6764 0.163654 0.0818272 0.996647i \(-0.473924\pi\)
0.0818272 + 0.996647i \(0.473924\pi\)
\(12\) 0 0
\(13\) −689.371 −1.13134 −0.565672 0.824630i \(-0.691383\pi\)
−0.565672 + 0.824630i \(0.691383\pi\)
\(14\) 0 0
\(15\) 11.4385 0.0131262
\(16\) 0 0
\(17\) 737.385 0.618831 0.309415 0.950927i \(-0.399867\pi\)
0.309415 + 0.950927i \(0.399867\pi\)
\(18\) 0 0
\(19\) 609.887 0.387583 0.193792 0.981043i \(-0.437921\pi\)
0.193792 + 0.981043i \(0.437921\pi\)
\(20\) 0 0
\(21\) 4170.63 2.06373
\(22\) 0 0
\(23\) −1312.45 −0.517326 −0.258663 0.965968i \(-0.583282\pi\)
−0.258663 + 0.965968i \(0.583282\pi\)
\(24\) 0 0
\(25\) −3124.79 −0.999933
\(26\) 0 0
\(27\) −3539.34 −0.934357
\(28\) 0 0
\(29\) −8969.74 −1.98055 −0.990273 0.139137i \(-0.955567\pi\)
−0.990273 + 0.139137i \(0.955567\pi\)
\(30\) 0 0
\(31\) −5858.93 −1.09500 −0.547500 0.836806i \(-0.684420\pi\)
−0.547500 + 0.836806i \(0.684420\pi\)
\(32\) 0 0
\(33\) −1644.94 −0.262946
\(34\) 0 0
\(35\) 76.0477 0.0104934
\(36\) 0 0
\(37\) −55.9069 −0.00671369 −0.00335685 0.999994i \(-0.501069\pi\)
−0.00335685 + 0.999994i \(0.501069\pi\)
\(38\) 0 0
\(39\) 17266.1 1.81775
\(40\) 0 0
\(41\) −10450.3 −0.970889 −0.485445 0.874267i \(-0.661342\pi\)
−0.485445 + 0.874267i \(0.661342\pi\)
\(42\) 0 0
\(43\) −1849.00 −0.152499
\(44\) 0 0
\(45\) −175.514 −0.0129205
\(46\) 0 0
\(47\) −1942.82 −0.128288 −0.0641442 0.997941i \(-0.520432\pi\)
−0.0641442 + 0.997941i \(0.520432\pi\)
\(48\) 0 0
\(49\) 10921.1 0.649792
\(50\) 0 0
\(51\) −18468.7 −0.994284
\(52\) 0 0
\(53\) 30129.2 1.47332 0.736661 0.676262i \(-0.236401\pi\)
0.736661 + 0.676262i \(0.236401\pi\)
\(54\) 0 0
\(55\) −29.9941 −0.00133699
\(56\) 0 0
\(57\) −15275.3 −0.622736
\(58\) 0 0
\(59\) −52167.4 −1.95106 −0.975528 0.219877i \(-0.929434\pi\)
−0.975528 + 0.219877i \(0.929434\pi\)
\(60\) 0 0
\(61\) −14040.1 −0.483108 −0.241554 0.970387i \(-0.577657\pi\)
−0.241554 + 0.970387i \(0.577657\pi\)
\(62\) 0 0
\(63\) −63994.7 −2.03139
\(64\) 0 0
\(65\) 314.832 0.00924264
\(66\) 0 0
\(67\) −51437.7 −1.39989 −0.699946 0.714196i \(-0.746792\pi\)
−0.699946 + 0.714196i \(0.746792\pi\)
\(68\) 0 0
\(69\) 32872.0 0.831196
\(70\) 0 0
\(71\) −11268.6 −0.265292 −0.132646 0.991163i \(-0.542347\pi\)
−0.132646 + 0.991163i \(0.542347\pi\)
\(72\) 0 0
\(73\) −61124.5 −1.34248 −0.671241 0.741239i \(-0.734238\pi\)
−0.671241 + 0.741239i \(0.734238\pi\)
\(74\) 0 0
\(75\) 78264.2 1.60661
\(76\) 0 0
\(77\) −10936.3 −0.210205
\(78\) 0 0
\(79\) −96018.3 −1.73096 −0.865478 0.500947i \(-0.832985\pi\)
−0.865478 + 0.500947i \(0.832985\pi\)
\(80\) 0 0
\(81\) −4740.91 −0.0802878
\(82\) 0 0
\(83\) −30378.0 −0.484021 −0.242010 0.970274i \(-0.577807\pi\)
−0.242010 + 0.970274i \(0.577807\pi\)
\(84\) 0 0
\(85\) −336.760 −0.00505561
\(86\) 0 0
\(87\) 224658. 3.18217
\(88\) 0 0
\(89\) −65040.4 −0.870378 −0.435189 0.900339i \(-0.643319\pi\)
−0.435189 + 0.900339i \(0.643319\pi\)
\(90\) 0 0
\(91\) 114792. 1.45315
\(92\) 0 0
\(93\) 146744. 1.75935
\(94\) 0 0
\(95\) −278.532 −0.00316641
\(96\) 0 0
\(97\) 12067.6 0.130224 0.0651119 0.997878i \(-0.479260\pi\)
0.0651119 + 0.997878i \(0.479260\pi\)
\(98\) 0 0
\(99\) 25240.2 0.258825
\(100\) 0 0
\(101\) 107046. 1.04416 0.522079 0.852897i \(-0.325157\pi\)
0.522079 + 0.852897i \(0.325157\pi\)
\(102\) 0 0
\(103\) 86209.7 0.800688 0.400344 0.916365i \(-0.368891\pi\)
0.400344 + 0.916365i \(0.368891\pi\)
\(104\) 0 0
\(105\) −1904.71 −0.0168599
\(106\) 0 0
\(107\) −197153. −1.66473 −0.832365 0.554227i \(-0.813014\pi\)
−0.832365 + 0.554227i \(0.813014\pi\)
\(108\) 0 0
\(109\) 165311. 1.33271 0.666355 0.745634i \(-0.267853\pi\)
0.666355 + 0.745634i \(0.267853\pi\)
\(110\) 0 0
\(111\) 1400.26 0.0107870
\(112\) 0 0
\(113\) −179790. −1.32455 −0.662275 0.749260i \(-0.730409\pi\)
−0.662275 + 0.749260i \(0.730409\pi\)
\(114\) 0 0
\(115\) 599.391 0.00422635
\(116\) 0 0
\(117\) −264934. −1.78926
\(118\) 0 0
\(119\) −122787. −0.794852
\(120\) 0 0
\(121\) −156738. −0.973217
\(122\) 0 0
\(123\) 261741. 1.55994
\(124\) 0 0
\(125\) 2854.25 0.0163387
\(126\) 0 0
\(127\) 286585. 1.57668 0.788341 0.615238i \(-0.210940\pi\)
0.788341 + 0.615238i \(0.210940\pi\)
\(128\) 0 0
\(129\) 46310.4 0.245022
\(130\) 0 0
\(131\) −67619.1 −0.344263 −0.172132 0.985074i \(-0.555065\pi\)
−0.172132 + 0.985074i \(0.555065\pi\)
\(132\) 0 0
\(133\) −101557. −0.497829
\(134\) 0 0
\(135\) 1616.40 0.00763333
\(136\) 0 0
\(137\) 133580. 0.608050 0.304025 0.952664i \(-0.401669\pi\)
0.304025 + 0.952664i \(0.401669\pi\)
\(138\) 0 0
\(139\) 262195. 1.15103 0.575516 0.817790i \(-0.304801\pi\)
0.575516 + 0.817790i \(0.304801\pi\)
\(140\) 0 0
\(141\) 48660.2 0.206123
\(142\) 0 0
\(143\) −45275.4 −0.185149
\(144\) 0 0
\(145\) 4096.44 0.0161803
\(146\) 0 0
\(147\) −273531. −1.04403
\(148\) 0 0
\(149\) 187414. 0.691569 0.345785 0.938314i \(-0.387613\pi\)
0.345785 + 0.938314i \(0.387613\pi\)
\(150\) 0 0
\(151\) 131632. 0.469807 0.234904 0.972019i \(-0.424523\pi\)
0.234904 + 0.972019i \(0.424523\pi\)
\(152\) 0 0
\(153\) 283386. 0.978701
\(154\) 0 0
\(155\) 2675.75 0.00894573
\(156\) 0 0
\(157\) −252270. −0.816800 −0.408400 0.912803i \(-0.633913\pi\)
−0.408400 + 0.912803i \(0.633913\pi\)
\(158\) 0 0
\(159\) −754622. −2.36721
\(160\) 0 0
\(161\) 218547. 0.664476
\(162\) 0 0
\(163\) 518121. 1.52743 0.763717 0.645552i \(-0.223373\pi\)
0.763717 + 0.645552i \(0.223373\pi\)
\(164\) 0 0
\(165\) 751.238 0.00214817
\(166\) 0 0
\(167\) 681328. 1.89045 0.945224 0.326422i \(-0.105843\pi\)
0.945224 + 0.326422i \(0.105843\pi\)
\(168\) 0 0
\(169\) 103939. 0.279939
\(170\) 0 0
\(171\) 234387. 0.612976
\(172\) 0 0
\(173\) −148596. −0.377478 −0.188739 0.982027i \(-0.560440\pi\)
−0.188739 + 0.982027i \(0.560440\pi\)
\(174\) 0 0
\(175\) 520332. 1.28436
\(176\) 0 0
\(177\) 1.30660e6 3.13479
\(178\) 0 0
\(179\) −103068. −0.240432 −0.120216 0.992748i \(-0.538359\pi\)
−0.120216 + 0.992748i \(0.538359\pi\)
\(180\) 0 0
\(181\) 309316. 0.701789 0.350894 0.936415i \(-0.385878\pi\)
0.350894 + 0.936415i \(0.385878\pi\)
\(182\) 0 0
\(183\) 351650. 0.776217
\(184\) 0 0
\(185\) 25.5324 5.48483e−5 0
\(186\) 0 0
\(187\) 48428.8 0.101274
\(188\) 0 0
\(189\) 589362. 1.20013
\(190\) 0 0
\(191\) −373258. −0.740330 −0.370165 0.928966i \(-0.620699\pi\)
−0.370165 + 0.928966i \(0.620699\pi\)
\(192\) 0 0
\(193\) 85782.8 0.165770 0.0828852 0.996559i \(-0.473587\pi\)
0.0828852 + 0.996559i \(0.473587\pi\)
\(194\) 0 0
\(195\) −7885.36 −0.0148503
\(196\) 0 0
\(197\) −178146. −0.327047 −0.163523 0.986539i \(-0.552286\pi\)
−0.163523 + 0.986539i \(0.552286\pi\)
\(198\) 0 0
\(199\) −154725. −0.276967 −0.138484 0.990365i \(-0.544223\pi\)
−0.138484 + 0.990365i \(0.544223\pi\)
\(200\) 0 0
\(201\) 1.28832e6 2.24923
\(202\) 0 0
\(203\) 1.49362e6 2.54390
\(204\) 0 0
\(205\) 4772.61 0.00793179
\(206\) 0 0
\(207\) −504392. −0.818168
\(208\) 0 0
\(209\) 40055.2 0.0634297
\(210\) 0 0
\(211\) 682180. 1.05485 0.527427 0.849600i \(-0.323157\pi\)
0.527427 + 0.849600i \(0.323157\pi\)
\(212\) 0 0
\(213\) 282236. 0.426249
\(214\) 0 0
\(215\) 844.429 0.00124585
\(216\) 0 0
\(217\) 975615. 1.40647
\(218\) 0 0
\(219\) 1.53094e6 2.15698
\(220\) 0 0
\(221\) −508332. −0.700110
\(222\) 0 0
\(223\) 464004. 0.624827 0.312413 0.949946i \(-0.398863\pi\)
0.312413 + 0.949946i \(0.398863\pi\)
\(224\) 0 0
\(225\) −1.20090e6 −1.58143
\(226\) 0 0
\(227\) −1.13876e6 −1.46679 −0.733396 0.679801i \(-0.762066\pi\)
−0.733396 + 0.679801i \(0.762066\pi\)
\(228\) 0 0
\(229\) 648718. 0.817462 0.408731 0.912655i \(-0.365971\pi\)
0.408731 + 0.912655i \(0.365971\pi\)
\(230\) 0 0
\(231\) 273912. 0.337739
\(232\) 0 0
\(233\) −588684. −0.710383 −0.355191 0.934794i \(-0.615584\pi\)
−0.355191 + 0.934794i \(0.615584\pi\)
\(234\) 0 0
\(235\) 887.275 0.00104807
\(236\) 0 0
\(237\) 2.40489e6 2.78115
\(238\) 0 0
\(239\) −739129. −0.837000 −0.418500 0.908217i \(-0.637444\pi\)
−0.418500 + 0.908217i \(0.637444\pi\)
\(240\) 0 0
\(241\) −1.57360e6 −1.74523 −0.872615 0.488409i \(-0.837577\pi\)
−0.872615 + 0.488409i \(0.837577\pi\)
\(242\) 0 0
\(243\) 978801. 1.06336
\(244\) 0 0
\(245\) −4987.60 −0.00530855
\(246\) 0 0
\(247\) −420438. −0.438490
\(248\) 0 0
\(249\) 760854. 0.777684
\(250\) 0 0
\(251\) −456789. −0.457648 −0.228824 0.973468i \(-0.573488\pi\)
−0.228824 + 0.973468i \(0.573488\pi\)
\(252\) 0 0
\(253\) −86197.3 −0.0846627
\(254\) 0 0
\(255\) 8434.56 0.00812292
\(256\) 0 0
\(257\) 1.12132e6 1.05900 0.529499 0.848310i \(-0.322380\pi\)
0.529499 + 0.848310i \(0.322380\pi\)
\(258\) 0 0
\(259\) 9309.48 0.00862335
\(260\) 0 0
\(261\) −3.44718e6 −3.13230
\(262\) 0 0
\(263\) −1.75620e6 −1.56562 −0.782808 0.622263i \(-0.786214\pi\)
−0.782808 + 0.622263i \(0.786214\pi\)
\(264\) 0 0
\(265\) −13759.9 −0.0120365
\(266\) 0 0
\(267\) 1.62901e6 1.39845
\(268\) 0 0
\(269\) 1.22933e6 1.03583 0.517915 0.855432i \(-0.326708\pi\)
0.517915 + 0.855432i \(0.326708\pi\)
\(270\) 0 0
\(271\) −1.32305e6 −1.09435 −0.547173 0.837020i \(-0.684296\pi\)
−0.547173 + 0.837020i \(0.684296\pi\)
\(272\) 0 0
\(273\) −2.87511e6 −2.33479
\(274\) 0 0
\(275\) −205225. −0.163643
\(276\) 0 0
\(277\) 1.96854e6 1.54151 0.770754 0.637133i \(-0.219880\pi\)
0.770754 + 0.637133i \(0.219880\pi\)
\(278\) 0 0
\(279\) −2.25166e6 −1.73178
\(280\) 0 0
\(281\) −1.11820e6 −0.844801 −0.422400 0.906409i \(-0.638812\pi\)
−0.422400 + 0.906409i \(0.638812\pi\)
\(282\) 0 0
\(283\) −1.25727e6 −0.933174 −0.466587 0.884475i \(-0.654517\pi\)
−0.466587 + 0.884475i \(0.654517\pi\)
\(284\) 0 0
\(285\) 6976.18 0.00508751
\(286\) 0 0
\(287\) 1.74016e6 1.24705
\(288\) 0 0
\(289\) −876121. −0.617049
\(290\) 0 0
\(291\) −302247. −0.209233
\(292\) 0 0
\(293\) −260406. −0.177207 −0.0886037 0.996067i \(-0.528240\pi\)
−0.0886037 + 0.996067i \(0.528240\pi\)
\(294\) 0 0
\(295\) 23824.6 0.0159394
\(296\) 0 0
\(297\) −232451. −0.152912
\(298\) 0 0
\(299\) 904768. 0.585274
\(300\) 0 0
\(301\) 307891. 0.195876
\(302\) 0 0
\(303\) −2.68109e6 −1.67766
\(304\) 0 0
\(305\) 6412.03 0.00394681
\(306\) 0 0
\(307\) 1.56144e6 0.945536 0.472768 0.881187i \(-0.343255\pi\)
0.472768 + 0.881187i \(0.343255\pi\)
\(308\) 0 0
\(309\) −2.15923e6 −1.28648
\(310\) 0 0
\(311\) −655137. −0.384088 −0.192044 0.981386i \(-0.561512\pi\)
−0.192044 + 0.981386i \(0.561512\pi\)
\(312\) 0 0
\(313\) −212890. −0.122827 −0.0614136 0.998112i \(-0.519561\pi\)
−0.0614136 + 0.998112i \(0.519561\pi\)
\(314\) 0 0
\(315\) 29226.1 0.0165956
\(316\) 0 0
\(317\) −385858. −0.215665 −0.107832 0.994169i \(-0.534391\pi\)
−0.107832 + 0.994169i \(0.534391\pi\)
\(318\) 0 0
\(319\) −589100. −0.324125
\(320\) 0 0
\(321\) 4.93794e6 2.67475
\(322\) 0 0
\(323\) 449721. 0.239849
\(324\) 0 0
\(325\) 2.15414e6 1.13127
\(326\) 0 0
\(327\) −4.14042e6 −2.14129
\(328\) 0 0
\(329\) 323513. 0.164779
\(330\) 0 0
\(331\) 1.62905e6 0.817269 0.408634 0.912698i \(-0.366005\pi\)
0.408634 + 0.912698i \(0.366005\pi\)
\(332\) 0 0
\(333\) −21485.7 −0.0106179
\(334\) 0 0
\(335\) 23491.3 0.0114366
\(336\) 0 0
\(337\) −2.32344e6 −1.11444 −0.557220 0.830365i \(-0.688132\pi\)
−0.557220 + 0.830365i \(0.688132\pi\)
\(338\) 0 0
\(339\) 4.50305e6 2.12818
\(340\) 0 0
\(341\) −384794. −0.179202
\(342\) 0 0
\(343\) 980112. 0.449822
\(344\) 0 0
\(345\) −15012.5 −0.00679055
\(346\) 0 0
\(347\) 2.45789e6 1.09582 0.547910 0.836537i \(-0.315424\pi\)
0.547910 + 0.836537i \(0.315424\pi\)
\(348\) 0 0
\(349\) −4.36489e6 −1.91827 −0.959135 0.282950i \(-0.908687\pi\)
−0.959135 + 0.282950i \(0.908687\pi\)
\(350\) 0 0
\(351\) 2.43992e6 1.05708
\(352\) 0 0
\(353\) 898354. 0.383717 0.191858 0.981423i \(-0.438549\pi\)
0.191858 + 0.981423i \(0.438549\pi\)
\(354\) 0 0
\(355\) 5146.32 0.00216734
\(356\) 0 0
\(357\) 3.07536e6 1.27710
\(358\) 0 0
\(359\) −1.81866e6 −0.744759 −0.372379 0.928081i \(-0.621458\pi\)
−0.372379 + 0.928081i \(0.621458\pi\)
\(360\) 0 0
\(361\) −2.10414e6 −0.849779
\(362\) 0 0
\(363\) 3.92568e6 1.56368
\(364\) 0 0
\(365\) 27915.3 0.0109675
\(366\) 0 0
\(367\) 2.34762e6 0.909836 0.454918 0.890533i \(-0.349669\pi\)
0.454918 + 0.890533i \(0.349669\pi\)
\(368\) 0 0
\(369\) −4.01619e6 −1.53549
\(370\) 0 0
\(371\) −5.01703e6 −1.89240
\(372\) 0 0
\(373\) 2.97881e6 1.10859 0.554295 0.832321i \(-0.312988\pi\)
0.554295 + 0.832321i \(0.312988\pi\)
\(374\) 0 0
\(375\) −71488.1 −0.0262516
\(376\) 0 0
\(377\) 6.18348e6 2.24068
\(378\) 0 0
\(379\) −1.77718e6 −0.635527 −0.317764 0.948170i \(-0.602932\pi\)
−0.317764 + 0.948170i \(0.602932\pi\)
\(380\) 0 0
\(381\) −7.17787e6 −2.53328
\(382\) 0 0
\(383\) 2.08193e6 0.725220 0.362610 0.931941i \(-0.381886\pi\)
0.362610 + 0.931941i \(0.381886\pi\)
\(384\) 0 0
\(385\) 4994.54 0.00171729
\(386\) 0 0
\(387\) −710594. −0.241181
\(388\) 0 0
\(389\) −1.81080e6 −0.606730 −0.303365 0.952874i \(-0.598110\pi\)
−0.303365 + 0.952874i \(0.598110\pi\)
\(390\) 0 0
\(391\) −967784. −0.320137
\(392\) 0 0
\(393\) 1.69360e6 0.553133
\(394\) 0 0
\(395\) 43851.1 0.0141412
\(396\) 0 0
\(397\) 185900. 0.0591975 0.0295988 0.999562i \(-0.490577\pi\)
0.0295988 + 0.999562i \(0.490577\pi\)
\(398\) 0 0
\(399\) 2.54361e6 0.799869
\(400\) 0 0
\(401\) −3.41892e6 −1.06176 −0.530882 0.847446i \(-0.678139\pi\)
−0.530882 + 0.847446i \(0.678139\pi\)
\(402\) 0 0
\(403\) 4.03898e6 1.23882
\(404\) 0 0
\(405\) 2165.15 0.000655920 0
\(406\) 0 0
\(407\) −3671.77 −0.00109873
\(408\) 0 0
\(409\) −428179. −0.126566 −0.0632830 0.997996i \(-0.520157\pi\)
−0.0632830 + 0.997996i \(0.520157\pi\)
\(410\) 0 0
\(411\) −3.34567e6 −0.976963
\(412\) 0 0
\(413\) 8.68679e6 2.50602
\(414\) 0 0
\(415\) 13873.5 0.00395426
\(416\) 0 0
\(417\) −6.56699e6 −1.84938
\(418\) 0 0
\(419\) −3.84011e6 −1.06858 −0.534292 0.845300i \(-0.679422\pi\)
−0.534292 + 0.845300i \(0.679422\pi\)
\(420\) 0 0
\(421\) 7.05884e6 1.94101 0.970506 0.241078i \(-0.0775010\pi\)
0.970506 + 0.241078i \(0.0775010\pi\)
\(422\) 0 0
\(423\) −746649. −0.202892
\(424\) 0 0
\(425\) −2.30417e6 −0.618789
\(426\) 0 0
\(427\) 2.33792e6 0.620525
\(428\) 0 0
\(429\) 1.13398e6 0.297482
\(430\) 0 0
\(431\) 2.11813e6 0.549237 0.274619 0.961553i \(-0.411448\pi\)
0.274619 + 0.961553i \(0.411448\pi\)
\(432\) 0 0
\(433\) −1.58858e6 −0.407182 −0.203591 0.979056i \(-0.565261\pi\)
−0.203591 + 0.979056i \(0.565261\pi\)
\(434\) 0 0
\(435\) −102600. −0.0259971
\(436\) 0 0
\(437\) −800448. −0.200507
\(438\) 0 0
\(439\) −7.11302e6 −1.76154 −0.880771 0.473543i \(-0.842975\pi\)
−0.880771 + 0.473543i \(0.842975\pi\)
\(440\) 0 0
\(441\) 4.19710e6 1.02767
\(442\) 0 0
\(443\) −2.52423e6 −0.611110 −0.305555 0.952174i \(-0.598842\pi\)
−0.305555 + 0.952174i \(0.598842\pi\)
\(444\) 0 0
\(445\) 29703.6 0.00711065
\(446\) 0 0
\(447\) −4.69400e6 −1.11115
\(448\) 0 0
\(449\) −471868. −0.110460 −0.0552300 0.998474i \(-0.517589\pi\)
−0.0552300 + 0.998474i \(0.517589\pi\)
\(450\) 0 0
\(451\) −686339. −0.158890
\(452\) 0 0
\(453\) −3.29689e6 −0.754846
\(454\) 0 0
\(455\) −52425.1 −0.0118716
\(456\) 0 0
\(457\) −6.11039e6 −1.36861 −0.684303 0.729198i \(-0.739893\pi\)
−0.684303 + 0.729198i \(0.739893\pi\)
\(458\) 0 0
\(459\) −2.60985e6 −0.578209
\(460\) 0 0
\(461\) 7.47383e6 1.63791 0.818957 0.573855i \(-0.194553\pi\)
0.818957 + 0.573855i \(0.194553\pi\)
\(462\) 0 0
\(463\) 4.14981e6 0.899654 0.449827 0.893116i \(-0.351486\pi\)
0.449827 + 0.893116i \(0.351486\pi\)
\(464\) 0 0
\(465\) −67017.3 −0.0143732
\(466\) 0 0
\(467\) 1.18789e6 0.252048 0.126024 0.992027i \(-0.459778\pi\)
0.126024 + 0.992027i \(0.459778\pi\)
\(468\) 0 0
\(469\) 8.56527e6 1.79808
\(470\) 0 0
\(471\) 6.31840e6 1.31237
\(472\) 0 0
\(473\) −121436. −0.0249571
\(474\) 0 0
\(475\) −1.90577e6 −0.387558
\(476\) 0 0
\(477\) 1.15790e7 2.33011
\(478\) 0 0
\(479\) −3.74102e6 −0.744992 −0.372496 0.928034i \(-0.621498\pi\)
−0.372496 + 0.928034i \(0.621498\pi\)
\(480\) 0 0
\(481\) 38540.6 0.00759549
\(482\) 0 0
\(483\) −5.47376e6 −1.06762
\(484\) 0 0
\(485\) −5511.20 −0.00106388
\(486\) 0 0
\(487\) −7.18109e6 −1.37204 −0.686021 0.727581i \(-0.740644\pi\)
−0.686021 + 0.727581i \(0.740644\pi\)
\(488\) 0 0
\(489\) −1.29770e7 −2.45415
\(490\) 0 0
\(491\) 7.37443e6 1.38046 0.690231 0.723589i \(-0.257509\pi\)
0.690231 + 0.723589i \(0.257509\pi\)
\(492\) 0 0
\(493\) −6.61415e6 −1.22562
\(494\) 0 0
\(495\) −11527.1 −0.00211450
\(496\) 0 0
\(497\) 1.87642e6 0.340753
\(498\) 0 0
\(499\) −2.61086e6 −0.469388 −0.234694 0.972069i \(-0.575409\pi\)
−0.234694 + 0.972069i \(0.575409\pi\)
\(500\) 0 0
\(501\) −1.70647e7 −3.03741
\(502\) 0 0
\(503\) 398438. 0.0702167 0.0351084 0.999384i \(-0.488822\pi\)
0.0351084 + 0.999384i \(0.488822\pi\)
\(504\) 0 0
\(505\) −48887.3 −0.00853036
\(506\) 0 0
\(507\) −2.60329e6 −0.449782
\(508\) 0 0
\(509\) 556201. 0.0951563 0.0475781 0.998868i \(-0.484850\pi\)
0.0475781 + 0.998868i \(0.484850\pi\)
\(510\) 0 0
\(511\) 1.01783e7 1.72434
\(512\) 0 0
\(513\) −2.15860e6 −0.362141
\(514\) 0 0
\(515\) −39371.6 −0.00654131
\(516\) 0 0
\(517\) −127597. −0.0209950
\(518\) 0 0
\(519\) 3.72176e6 0.606499
\(520\) 0 0
\(521\) 3.39465e6 0.547898 0.273949 0.961744i \(-0.411670\pi\)
0.273949 + 0.961744i \(0.411670\pi\)
\(522\) 0 0
\(523\) 9.40326e6 1.50323 0.751613 0.659605i \(-0.229276\pi\)
0.751613 + 0.659605i \(0.229276\pi\)
\(524\) 0 0
\(525\) −1.30323e7 −2.06359
\(526\) 0 0
\(527\) −4.32029e6 −0.677620
\(528\) 0 0
\(529\) −4.71381e6 −0.732373
\(530\) 0 0
\(531\) −2.00486e7 −3.08566
\(532\) 0 0
\(533\) 7.20414e6 1.09841
\(534\) 0 0
\(535\) 90038.8 0.0136002
\(536\) 0 0
\(537\) 2.58147e6 0.386306
\(538\) 0 0
\(539\) 717256. 0.106341
\(540\) 0 0
\(541\) 4.76159e6 0.699453 0.349727 0.936852i \(-0.386274\pi\)
0.349727 + 0.936852i \(0.386274\pi\)
\(542\) 0 0
\(543\) −7.74720e6 −1.12757
\(544\) 0 0
\(545\) −75496.8 −0.0108877
\(546\) 0 0
\(547\) −4.68380e6 −0.669314 −0.334657 0.942340i \(-0.608621\pi\)
−0.334657 + 0.942340i \(0.608621\pi\)
\(548\) 0 0
\(549\) −5.39577e6 −0.764051
\(550\) 0 0
\(551\) −5.47053e6 −0.767627
\(552\) 0 0
\(553\) 1.59887e7 2.22331
\(554\) 0 0
\(555\) −639.490 −8.81255e−5 0
\(556\) 0 0
\(557\) 4.37082e6 0.596931 0.298466 0.954420i \(-0.403525\pi\)
0.298466 + 0.954420i \(0.403525\pi\)
\(558\) 0 0
\(559\) 1.27465e6 0.172528
\(560\) 0 0
\(561\) −1.21296e6 −0.162719
\(562\) 0 0
\(563\) 9.98439e6 1.32755 0.663774 0.747933i \(-0.268954\pi\)
0.663774 + 0.747933i \(0.268954\pi\)
\(564\) 0 0
\(565\) 82109.1 0.0108211
\(566\) 0 0
\(567\) 789445. 0.103125
\(568\) 0 0
\(569\) −3.03616e6 −0.393137 −0.196568 0.980490i \(-0.562980\pi\)
−0.196568 + 0.980490i \(0.562980\pi\)
\(570\) 0 0
\(571\) 4.66426e6 0.598677 0.299339 0.954147i \(-0.403234\pi\)
0.299339 + 0.954147i \(0.403234\pi\)
\(572\) 0 0
\(573\) 9.34869e6 1.18950
\(574\) 0 0
\(575\) 4.10115e6 0.517292
\(576\) 0 0
\(577\) 2.53940e6 0.317536 0.158768 0.987316i \(-0.449248\pi\)
0.158768 + 0.987316i \(0.449248\pi\)
\(578\) 0 0
\(579\) −2.14853e6 −0.266346
\(580\) 0 0
\(581\) 5.05847e6 0.621697
\(582\) 0 0
\(583\) 1.97878e6 0.241116
\(584\) 0 0
\(585\) 120994. 0.0146175
\(586\) 0 0
\(587\) −1.18921e7 −1.42450 −0.712248 0.701927i \(-0.752323\pi\)
−0.712248 + 0.701927i \(0.752323\pi\)
\(588\) 0 0
\(589\) −3.57329e6 −0.424404
\(590\) 0 0
\(591\) 4.46187e6 0.525471
\(592\) 0 0
\(593\) 1.49053e6 0.174062 0.0870311 0.996206i \(-0.472262\pi\)
0.0870311 + 0.996206i \(0.472262\pi\)
\(594\) 0 0
\(595\) 56076.4 0.00649363
\(596\) 0 0
\(597\) 3.87528e6 0.445008
\(598\) 0 0
\(599\) 1.13339e7 1.29066 0.645328 0.763905i \(-0.276721\pi\)
0.645328 + 0.763905i \(0.276721\pi\)
\(600\) 0 0
\(601\) 1.07157e7 1.21014 0.605069 0.796173i \(-0.293145\pi\)
0.605069 + 0.796173i \(0.293145\pi\)
\(602\) 0 0
\(603\) −1.97681e7 −2.21397
\(604\) 0 0
\(605\) 71581.3 0.00795081
\(606\) 0 0
\(607\) 871321. 0.0959857 0.0479929 0.998848i \(-0.484718\pi\)
0.0479929 + 0.998848i \(0.484718\pi\)
\(608\) 0 0
\(609\) −3.74095e7 −4.08732
\(610\) 0 0
\(611\) 1.33932e6 0.145138
\(612\) 0 0
\(613\) −1.62264e7 −1.74410 −0.872051 0.489415i \(-0.837210\pi\)
−0.872051 + 0.489415i \(0.837210\pi\)
\(614\) 0 0
\(615\) −119536. −0.0127441
\(616\) 0 0
\(617\) 1.18192e6 0.124990 0.0624950 0.998045i \(-0.480094\pi\)
0.0624950 + 0.998045i \(0.480094\pi\)
\(618\) 0 0
\(619\) −1.78346e7 −1.87084 −0.935418 0.353544i \(-0.884977\pi\)
−0.935418 + 0.353544i \(0.884977\pi\)
\(620\) 0 0
\(621\) 4.64522e6 0.483367
\(622\) 0 0
\(623\) 1.08304e7 1.11795
\(624\) 0 0
\(625\) 9.76367e6 0.999800
\(626\) 0 0
\(627\) −1.00323e6 −0.101913
\(628\) 0 0
\(629\) −41224.9 −0.00415464
\(630\) 0 0
\(631\) 1.56569e7 1.56543 0.782715 0.622381i \(-0.213834\pi\)
0.782715 + 0.622381i \(0.213834\pi\)
\(632\) 0 0
\(633\) −1.70860e7 −1.69485
\(634\) 0 0
\(635\) −130882. −0.0128809
\(636\) 0 0
\(637\) −7.52866e6 −0.735139
\(638\) 0 0
\(639\) −4.33067e6 −0.419568
\(640\) 0 0
\(641\) −9.56677e6 −0.919645 −0.459822 0.888011i \(-0.652087\pi\)
−0.459822 + 0.888011i \(0.652087\pi\)
\(642\) 0 0
\(643\) −6.58449e6 −0.628050 −0.314025 0.949415i \(-0.601678\pi\)
−0.314025 + 0.949415i \(0.601678\pi\)
\(644\) 0 0
\(645\) −21149.7 −0.00200173
\(646\) 0 0
\(647\) 5.77942e6 0.542779 0.271390 0.962470i \(-0.412517\pi\)
0.271390 + 0.962470i \(0.412517\pi\)
\(648\) 0 0
\(649\) −3.42617e6 −0.319299
\(650\) 0 0
\(651\) −2.44354e7 −2.25979
\(652\) 0 0
\(653\) −1.76847e7 −1.62298 −0.811492 0.584364i \(-0.801344\pi\)
−0.811492 + 0.584364i \(0.801344\pi\)
\(654\) 0 0
\(655\) 30881.3 0.00281250
\(656\) 0 0
\(657\) −2.34909e7 −2.12318
\(658\) 0 0
\(659\) −1.07891e7 −0.967769 −0.483884 0.875132i \(-0.660774\pi\)
−0.483884 + 0.875132i \(0.660774\pi\)
\(660\) 0 0
\(661\) −1.27678e7 −1.13662 −0.568308 0.822816i \(-0.692402\pi\)
−0.568308 + 0.822816i \(0.692402\pi\)
\(662\) 0 0
\(663\) 1.27318e7 1.12488
\(664\) 0 0
\(665\) 46380.5 0.00406707
\(666\) 0 0
\(667\) 1.17724e7 1.02459
\(668\) 0 0
\(669\) −1.16215e7 −1.00392
\(670\) 0 0
\(671\) −922101. −0.0790628
\(672\) 0 0
\(673\) 5.05831e6 0.430495 0.215247 0.976560i \(-0.430944\pi\)
0.215247 + 0.976560i \(0.430944\pi\)
\(674\) 0 0
\(675\) 1.10597e7 0.934294
\(676\) 0 0
\(677\) 1.48127e7 1.24211 0.621057 0.783766i \(-0.286704\pi\)
0.621057 + 0.783766i \(0.286704\pi\)
\(678\) 0 0
\(679\) −2.00946e6 −0.167265
\(680\) 0 0
\(681\) 2.85217e7 2.35672
\(682\) 0 0
\(683\) 9.52658e6 0.781421 0.390711 0.920514i \(-0.372229\pi\)
0.390711 + 0.920514i \(0.372229\pi\)
\(684\) 0 0
\(685\) −61005.2 −0.00496753
\(686\) 0 0
\(687\) −1.62479e7 −1.31343
\(688\) 0 0
\(689\) −2.07702e7 −1.66683
\(690\) 0 0
\(691\) 4.04497e6 0.322270 0.161135 0.986932i \(-0.448484\pi\)
0.161135 + 0.986932i \(0.448484\pi\)
\(692\) 0 0
\(693\) −4.20294e6 −0.332445
\(694\) 0 0
\(695\) −119743. −0.00940349
\(696\) 0 0
\(697\) −7.70590e6 −0.600816
\(698\) 0 0
\(699\) 1.47443e7 1.14138
\(700\) 0 0
\(701\) 1.78401e6 0.137120 0.0685602 0.997647i \(-0.478159\pi\)
0.0685602 + 0.997647i \(0.478159\pi\)
\(702\) 0 0
\(703\) −34096.9 −0.00260212
\(704\) 0 0
\(705\) −22222.9 −0.00168394
\(706\) 0 0
\(707\) −1.78250e7 −1.34116
\(708\) 0 0
\(709\) −1.22322e7 −0.913878 −0.456939 0.889498i \(-0.651054\pi\)
−0.456939 + 0.889498i \(0.651054\pi\)
\(710\) 0 0
\(711\) −3.69010e7 −2.73756
\(712\) 0 0
\(713\) 7.68958e6 0.566473
\(714\) 0 0
\(715\) 20677.1 0.00151260
\(716\) 0 0
\(717\) 1.85124e7 1.34482
\(718\) 0 0
\(719\) −8.42924e6 −0.608088 −0.304044 0.952658i \(-0.598337\pi\)
−0.304044 + 0.952658i \(0.598337\pi\)
\(720\) 0 0
\(721\) −1.43554e7 −1.02844
\(722\) 0 0
\(723\) 3.94128e7 2.80409
\(724\) 0 0
\(725\) 2.80286e7 1.98041
\(726\) 0 0
\(727\) −1.10329e7 −0.774201 −0.387101 0.922037i \(-0.626523\pi\)
−0.387101 + 0.922037i \(0.626523\pi\)
\(728\) 0 0
\(729\) −2.33632e7 −1.62822
\(730\) 0 0
\(731\) −1.36342e6 −0.0943708
\(732\) 0 0
\(733\) −5.02000e6 −0.345099 −0.172550 0.985001i \(-0.555201\pi\)
−0.172550 + 0.985001i \(0.555201\pi\)
\(734\) 0 0
\(735\) 124920. 0.00852933
\(736\) 0 0
\(737\) −3.37824e6 −0.229098
\(738\) 0 0
\(739\) −2.49061e6 −0.167762 −0.0838810 0.996476i \(-0.526732\pi\)
−0.0838810 + 0.996476i \(0.526732\pi\)
\(740\) 0 0
\(741\) 1.05304e7 0.704529
\(742\) 0 0
\(743\) 1.21111e7 0.804847 0.402423 0.915454i \(-0.368168\pi\)
0.402423 + 0.915454i \(0.368168\pi\)
\(744\) 0 0
\(745\) −85590.9 −0.00564985
\(746\) 0 0
\(747\) −1.16746e7 −0.765495
\(748\) 0 0
\(749\) 3.28294e7 2.13825
\(750\) 0 0
\(751\) −2.21669e7 −1.43418 −0.717092 0.696978i \(-0.754528\pi\)
−0.717092 + 0.696978i \(0.754528\pi\)
\(752\) 0 0
\(753\) 1.14408e7 0.735310
\(754\) 0 0
\(755\) −60115.8 −0.00383814
\(756\) 0 0
\(757\) 1.03112e7 0.653986 0.326993 0.945027i \(-0.393965\pi\)
0.326993 + 0.945027i \(0.393965\pi\)
\(758\) 0 0
\(759\) 2.15891e6 0.136029
\(760\) 0 0
\(761\) 5.23003e6 0.327373 0.163686 0.986512i \(-0.447661\pi\)
0.163686 + 0.986512i \(0.447661\pi\)
\(762\) 0 0
\(763\) −2.75272e7 −1.71179
\(764\) 0 0
\(765\) −129421. −0.00799560
\(766\) 0 0
\(767\) 3.59627e7 2.20731
\(768\) 0 0
\(769\) 1.16812e7 0.712315 0.356157 0.934426i \(-0.384087\pi\)
0.356157 + 0.934426i \(0.384087\pi\)
\(770\) 0 0
\(771\) −2.80847e7 −1.70151
\(772\) 0 0
\(773\) 1.35127e7 0.813379 0.406690 0.913566i \(-0.366683\pi\)
0.406690 + 0.913566i \(0.366683\pi\)
\(774\) 0 0
\(775\) 1.83079e7 1.09493
\(776\) 0 0
\(777\) −233167. −0.0138553
\(778\) 0 0
\(779\) −6.37351e6 −0.376301
\(780\) 0 0
\(781\) −740082. −0.0434162
\(782\) 0 0
\(783\) 3.17470e7 1.85054
\(784\) 0 0
\(785\) 115210. 0.00667294
\(786\) 0 0
\(787\) 2.08780e7 1.20158 0.600789 0.799407i \(-0.294853\pi\)
0.600789 + 0.799407i \(0.294853\pi\)
\(788\) 0 0
\(789\) 4.39862e7 2.51550
\(790\) 0 0
\(791\) 2.99381e7 1.70131
\(792\) 0 0
\(793\) 9.67881e6 0.546562
\(794\) 0 0
\(795\) 344632. 0.0193392
\(796\) 0 0
\(797\) −1.17024e7 −0.652573 −0.326286 0.945271i \(-0.605797\pi\)
−0.326286 + 0.945271i \(0.605797\pi\)
\(798\) 0 0
\(799\) −1.43260e6 −0.0793888
\(800\) 0 0
\(801\) −2.49958e7 −1.37653
\(802\) 0 0
\(803\) −4.01444e6 −0.219703
\(804\) 0 0
\(805\) −99809.1 −0.00542851
\(806\) 0 0
\(807\) −3.07901e7 −1.66428
\(808\) 0 0
\(809\) −2.76467e7 −1.48516 −0.742578 0.669760i \(-0.766397\pi\)
−0.742578 + 0.669760i \(0.766397\pi\)
\(810\) 0 0
\(811\) 8.32113e6 0.444253 0.222126 0.975018i \(-0.428700\pi\)
0.222126 + 0.975018i \(0.428700\pi\)
\(812\) 0 0
\(813\) 3.31375e7 1.75830
\(814\) 0 0
\(815\) −236623. −0.0124785
\(816\) 0 0
\(817\) −1.12768e6 −0.0591059
\(818\) 0 0
\(819\) 4.41161e7 2.29820
\(820\) 0 0
\(821\) 8.01626e6 0.415063 0.207531 0.978228i \(-0.433457\pi\)
0.207531 + 0.978228i \(0.433457\pi\)
\(822\) 0 0
\(823\) −4.63830e6 −0.238704 −0.119352 0.992852i \(-0.538082\pi\)
−0.119352 + 0.992852i \(0.538082\pi\)
\(824\) 0 0
\(825\) 5.14011e6 0.262928
\(826\) 0 0
\(827\) −5.55667e6 −0.282521 −0.141261 0.989972i \(-0.545116\pi\)
−0.141261 + 0.989972i \(0.545116\pi\)
\(828\) 0 0
\(829\) 2.47536e7 1.25098 0.625491 0.780231i \(-0.284899\pi\)
0.625491 + 0.780231i \(0.284899\pi\)
\(830\) 0 0
\(831\) −4.93046e7 −2.47676
\(832\) 0 0
\(833\) 8.05302e6 0.402111
\(834\) 0 0
\(835\) −311159. −0.0154442
\(836\) 0 0
\(837\) 2.07367e7 1.02312
\(838\) 0 0
\(839\) 3.24512e7 1.59157 0.795785 0.605579i \(-0.207058\pi\)
0.795785 + 0.605579i \(0.207058\pi\)
\(840\) 0 0
\(841\) 5.99451e7 2.92256
\(842\) 0 0
\(843\) 2.80067e7 1.35735
\(844\) 0 0
\(845\) −47468.6 −0.00228699
\(846\) 0 0
\(847\) 2.60995e7 1.25004
\(848\) 0 0
\(849\) 3.14899e7 1.49935
\(850\) 0 0
\(851\) 73375.3 0.00347317
\(852\) 0 0
\(853\) 2.47564e7 1.16497 0.582485 0.812841i \(-0.302080\pi\)
0.582485 + 0.812841i \(0.302080\pi\)
\(854\) 0 0
\(855\) −107043. −0.00500777
\(856\) 0 0
\(857\) 1.28652e7 0.598363 0.299182 0.954196i \(-0.403286\pi\)
0.299182 + 0.954196i \(0.403286\pi\)
\(858\) 0 0
\(859\) −1.98227e6 −0.0916601 −0.0458300 0.998949i \(-0.514593\pi\)
−0.0458300 + 0.998949i \(0.514593\pi\)
\(860\) 0 0
\(861\) −4.35844e7 −2.00366
\(862\) 0 0
\(863\) 3.11840e6 0.142529 0.0712647 0.997457i \(-0.477296\pi\)
0.0712647 + 0.997457i \(0.477296\pi\)
\(864\) 0 0
\(865\) 67863.0 0.00308384
\(866\) 0 0
\(867\) 2.19435e7 0.991421
\(868\) 0 0
\(869\) −6.30613e6 −0.283279
\(870\) 0 0
\(871\) 3.54596e7 1.58376
\(872\) 0 0
\(873\) 4.63772e6 0.205953
\(874\) 0 0
\(875\) −475282. −0.0209861
\(876\) 0 0
\(877\) −2.15790e7 −0.947398 −0.473699 0.880687i \(-0.657082\pi\)
−0.473699 + 0.880687i \(0.657082\pi\)
\(878\) 0 0
\(879\) 6.52218e6 0.284722
\(880\) 0 0
\(881\) −1.27720e7 −0.554393 −0.277196 0.960813i \(-0.589405\pi\)
−0.277196 + 0.960813i \(0.589405\pi\)
\(882\) 0 0
\(883\) −8.80113e6 −0.379871 −0.189936 0.981797i \(-0.560828\pi\)
−0.189936 + 0.981797i \(0.560828\pi\)
\(884\) 0 0
\(885\) −596716. −0.0256100
\(886\) 0 0
\(887\) 3.47818e7 1.48437 0.742187 0.670192i \(-0.233788\pi\)
0.742187 + 0.670192i \(0.233788\pi\)
\(888\) 0 0
\(889\) −4.77214e7 −2.02516
\(890\) 0 0
\(891\) −311366. −0.0131394
\(892\) 0 0
\(893\) −1.18490e6 −0.0497225
\(894\) 0 0
\(895\) 47070.8 0.00196424
\(896\) 0 0
\(897\) −2.26610e7 −0.940368
\(898\) 0 0
\(899\) 5.25531e7 2.16870
\(900\) 0 0
\(901\) 2.22168e7 0.911737
\(902\) 0 0
\(903\) −7.71150e6 −0.314716
\(904\) 0 0
\(905\) −141263. −0.00573334
\(906\) 0 0
\(907\) −1.91865e7 −0.774420 −0.387210 0.921992i \(-0.626561\pi\)
−0.387210 + 0.921992i \(0.626561\pi\)
\(908\) 0 0
\(909\) 4.11390e7 1.65137
\(910\) 0 0
\(911\) −834307. −0.0333066 −0.0166533 0.999861i \(-0.505301\pi\)
−0.0166533 + 0.999861i \(0.505301\pi\)
\(912\) 0 0
\(913\) −1.99512e6 −0.0792121
\(914\) 0 0
\(915\) −160597. −0.00634139
\(916\) 0 0
\(917\) 1.12598e7 0.442187
\(918\) 0 0
\(919\) −1.11037e7 −0.433690 −0.216845 0.976206i \(-0.569577\pi\)
−0.216845 + 0.976206i \(0.569577\pi\)
\(920\) 0 0
\(921\) −3.91080e7 −1.51921
\(922\) 0 0
\(923\) 7.76826e6 0.300137
\(924\) 0 0
\(925\) 174698. 0.00671324
\(926\) 0 0
\(927\) 3.31315e7 1.26631
\(928\) 0 0
\(929\) −1.03857e7 −0.394816 −0.197408 0.980321i \(-0.563252\pi\)
−0.197408 + 0.980321i \(0.563252\pi\)
\(930\) 0 0
\(931\) 6.66061e6 0.251849
\(932\) 0 0
\(933\) 1.64087e7 0.617120
\(934\) 0 0
\(935\) −22117.2 −0.000827372 0
\(936\) 0 0
\(937\) −2.10627e7 −0.783729 −0.391865 0.920023i \(-0.628170\pi\)
−0.391865 + 0.920023i \(0.628170\pi\)
\(938\) 0 0
\(939\) 5.33209e6 0.197348
\(940\) 0 0
\(941\) −3.67199e7 −1.35185 −0.675924 0.736971i \(-0.736255\pi\)
−0.675924 + 0.736971i \(0.736255\pi\)
\(942\) 0 0
\(943\) 1.37156e7 0.502267
\(944\) 0 0
\(945\) −269159. −0.00980457
\(946\) 0 0
\(947\) −1.07332e7 −0.388914 −0.194457 0.980911i \(-0.562294\pi\)
−0.194457 + 0.980911i \(0.562294\pi\)
\(948\) 0 0
\(949\) 4.21375e7 1.51881
\(950\) 0 0
\(951\) 9.66428e6 0.346512
\(952\) 0 0
\(953\) 4.10240e7 1.46321 0.731603 0.681731i \(-0.238772\pi\)
0.731603 + 0.681731i \(0.238772\pi\)
\(954\) 0 0
\(955\) 170465. 0.00604821
\(956\) 0 0
\(957\) 1.47547e7 0.520776
\(958\) 0 0
\(959\) −2.22434e7 −0.781005
\(960\) 0 0
\(961\) 5.69795e6 0.199026
\(962\) 0 0
\(963\) −7.57683e7 −2.63283
\(964\) 0 0
\(965\) −39176.6 −0.00135428
\(966\) 0 0
\(967\) −2.93253e7 −1.00850 −0.504251 0.863557i \(-0.668231\pi\)
−0.504251 + 0.863557i \(0.668231\pi\)
\(968\) 0 0
\(969\) −1.12638e7 −0.385368
\(970\) 0 0
\(971\) −3.44554e6 −0.117276 −0.0586381 0.998279i \(-0.518676\pi\)
−0.0586381 + 0.998279i \(0.518676\pi\)
\(972\) 0 0
\(973\) −4.36601e7 −1.47843
\(974\) 0 0
\(975\) −5.39530e7 −1.81763
\(976\) 0 0
\(977\) 3.77647e7 1.26576 0.632878 0.774252i \(-0.281874\pi\)
0.632878 + 0.774252i \(0.281874\pi\)
\(978\) 0 0
\(979\) −4.27162e6 −0.142441
\(980\) 0 0
\(981\) 6.35311e7 2.10772
\(982\) 0 0
\(983\) −1.69058e7 −0.558024 −0.279012 0.960288i \(-0.590007\pi\)
−0.279012 + 0.960288i \(0.590007\pi\)
\(984\) 0 0
\(985\) 81358.2 0.00267184
\(986\) 0 0
\(987\) −8.10277e6 −0.264753
\(988\) 0 0
\(989\) 2.42673e6 0.0788915
\(990\) 0 0
\(991\) −1.83818e7 −0.594570 −0.297285 0.954789i \(-0.596081\pi\)
−0.297285 + 0.954789i \(0.596081\pi\)
\(992\) 0 0
\(993\) −4.08016e7 −1.31312
\(994\) 0 0
\(995\) 70662.3 0.00226272
\(996\) 0 0
\(997\) −2.05937e7 −0.656139 −0.328069 0.944654i \(-0.606398\pi\)
−0.328069 + 0.944654i \(0.606398\pi\)
\(998\) 0 0
\(999\) 197874. 0.00627298
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.1.2 10
4.3 odd 2 43.6.a.b.1.7 10
12.11 even 2 387.6.a.e.1.4 10
20.19 odd 2 1075.6.a.b.1.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.b.1.7 10 4.3 odd 2
387.6.a.e.1.4 10 12.11 even 2
688.6.a.h.1.2 10 1.1 even 1 trivial
1075.6.a.b.1.4 10 20.19 odd 2