Properties

Label 400.4.a.c.1.1
Level $400$
Weight $4$
Character 400.1
Self dual yes
Analytic conductor $23.601$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,4,Mod(1,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.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: \(23.6007640023\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-7.00000 q^{3} -6.00000 q^{7} +22.0000 q^{9} +O(q^{10})\) \(q-7.00000 q^{3} -6.00000 q^{7} +22.0000 q^{9} +43.0000 q^{11} -28.0000 q^{13} +91.0000 q^{17} +35.0000 q^{19} +42.0000 q^{21} -162.000 q^{23} +35.0000 q^{27} +160.000 q^{29} -42.0000 q^{31} -301.000 q^{33} -314.000 q^{37} +196.000 q^{39} -203.000 q^{41} -92.0000 q^{43} -196.000 q^{47} -307.000 q^{49} -637.000 q^{51} +82.0000 q^{53} -245.000 q^{57} +280.000 q^{59} -518.000 q^{61} -132.000 q^{63} -141.000 q^{67} +1134.00 q^{69} -412.000 q^{71} -763.000 q^{73} -258.000 q^{77} -510.000 q^{79} -839.000 q^{81} -777.000 q^{83} -1120.00 q^{87} -945.000 q^{89} +168.000 q^{91} +294.000 q^{93} +1246.00 q^{97} +946.000 q^{99} +O(q^{100})\)

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
\(3\) −7.00000 −1.34715 −0.673575 0.739119i \(-0.735242\pi\)
−0.673575 + 0.739119i \(0.735242\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −6.00000 −0.323970 −0.161985 0.986793i \(-0.551790\pi\)
−0.161985 + 0.986793i \(0.551790\pi\)
\(8\) 0 0
\(9\) 22.0000 0.814815
\(10\) 0 0
\(11\) 43.0000 1.17864 0.589318 0.807901i \(-0.299397\pi\)
0.589318 + 0.807901i \(0.299397\pi\)
\(12\) 0 0
\(13\) −28.0000 −0.597369 −0.298685 0.954352i \(-0.596548\pi\)
−0.298685 + 0.954352i \(0.596548\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 91.0000 1.29828 0.649139 0.760669i \(-0.275129\pi\)
0.649139 + 0.760669i \(0.275129\pi\)
\(18\) 0 0
\(19\) 35.0000 0.422608 0.211304 0.977420i \(-0.432229\pi\)
0.211304 + 0.977420i \(0.432229\pi\)
\(20\) 0 0
\(21\) 42.0000 0.436436
\(22\) 0 0
\(23\) −162.000 −1.46867 −0.734333 0.678789i \(-0.762505\pi\)
−0.734333 + 0.678789i \(0.762505\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 35.0000 0.249472
\(28\) 0 0
\(29\) 160.000 1.02453 0.512263 0.858829i \(-0.328807\pi\)
0.512263 + 0.858829i \(0.328807\pi\)
\(30\) 0 0
\(31\) −42.0000 −0.243336 −0.121668 0.992571i \(-0.538824\pi\)
−0.121668 + 0.992571i \(0.538824\pi\)
\(32\) 0 0
\(33\) −301.000 −1.58780
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −314.000 −1.39517 −0.697585 0.716502i \(-0.745742\pi\)
−0.697585 + 0.716502i \(0.745742\pi\)
\(38\) 0 0
\(39\) 196.000 0.804747
\(40\) 0 0
\(41\) −203.000 −0.773251 −0.386625 0.922237i \(-0.626359\pi\)
−0.386625 + 0.922237i \(0.626359\pi\)
\(42\) 0 0
\(43\) −92.0000 −0.326276 −0.163138 0.986603i \(-0.552162\pi\)
−0.163138 + 0.986603i \(0.552162\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −196.000 −0.608288 −0.304144 0.952626i \(-0.598370\pi\)
−0.304144 + 0.952626i \(0.598370\pi\)
\(48\) 0 0
\(49\) −307.000 −0.895044
\(50\) 0 0
\(51\) −637.000 −1.74898
\(52\) 0 0
\(53\) 82.0000 0.212520 0.106260 0.994338i \(-0.466112\pi\)
0.106260 + 0.994338i \(0.466112\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −245.000 −0.569317
\(58\) 0 0
\(59\) 280.000 0.617846 0.308923 0.951087i \(-0.400032\pi\)
0.308923 + 0.951087i \(0.400032\pi\)
\(60\) 0 0
\(61\) −518.000 −1.08726 −0.543632 0.839324i \(-0.682951\pi\)
−0.543632 + 0.839324i \(0.682951\pi\)
\(62\) 0 0
\(63\) −132.000 −0.263975
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −141.000 −0.257103 −0.128551 0.991703i \(-0.541033\pi\)
−0.128551 + 0.991703i \(0.541033\pi\)
\(68\) 0 0
\(69\) 1134.00 1.97852
\(70\) 0 0
\(71\) −412.000 −0.688668 −0.344334 0.938847i \(-0.611895\pi\)
−0.344334 + 0.938847i \(0.611895\pi\)
\(72\) 0 0
\(73\) −763.000 −1.22332 −0.611660 0.791121i \(-0.709498\pi\)
−0.611660 + 0.791121i \(0.709498\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −258.000 −0.381842
\(78\) 0 0
\(79\) −510.000 −0.726323 −0.363161 0.931726i \(-0.618303\pi\)
−0.363161 + 0.931726i \(0.618303\pi\)
\(80\) 0 0
\(81\) −839.000 −1.15089
\(82\) 0 0
\(83\) −777.000 −1.02755 −0.513776 0.857924i \(-0.671754\pi\)
−0.513776 + 0.857924i \(0.671754\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1120.00 −1.38019
\(88\) 0 0
\(89\) −945.000 −1.12550 −0.562752 0.826626i \(-0.690257\pi\)
−0.562752 + 0.826626i \(0.690257\pi\)
\(90\) 0 0
\(91\) 168.000 0.193530
\(92\) 0 0
\(93\) 294.000 0.327811
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1246.00 1.30425 0.652124 0.758112i \(-0.273878\pi\)
0.652124 + 0.758112i \(0.273878\pi\)
\(98\) 0 0
\(99\) 946.000 0.960369
\(100\) 0 0
\(101\) 1302.00 1.28271 0.641356 0.767244i \(-0.278372\pi\)
0.641356 + 0.767244i \(0.278372\pi\)
\(102\) 0 0
\(103\) −532.000 −0.508927 −0.254464 0.967082i \(-0.581899\pi\)
−0.254464 + 0.967082i \(0.581899\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1269.00 1.14653 0.573266 0.819370i \(-0.305676\pi\)
0.573266 + 0.819370i \(0.305676\pi\)
\(108\) 0 0
\(109\) 1070.00 0.940251 0.470126 0.882599i \(-0.344209\pi\)
0.470126 + 0.882599i \(0.344209\pi\)
\(110\) 0 0
\(111\) 2198.00 1.87950
\(112\) 0 0
\(113\) −503.000 −0.418746 −0.209373 0.977836i \(-0.567142\pi\)
−0.209373 + 0.977836i \(0.567142\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −616.000 −0.486745
\(118\) 0 0
\(119\) −546.000 −0.420603
\(120\) 0 0
\(121\) 518.000 0.389181
\(122\) 0 0
\(123\) 1421.00 1.04169
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 874.000 0.610669 0.305334 0.952245i \(-0.401232\pi\)
0.305334 + 0.952245i \(0.401232\pi\)
\(128\) 0 0
\(129\) 644.000 0.439543
\(130\) 0 0
\(131\) −1092.00 −0.728309 −0.364155 0.931339i \(-0.618642\pi\)
−0.364155 + 0.931339i \(0.618642\pi\)
\(132\) 0 0
\(133\) −210.000 −0.136912
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 411.000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) 595.000 0.363074 0.181537 0.983384i \(-0.441893\pi\)
0.181537 + 0.983384i \(0.441893\pi\)
\(140\) 0 0
\(141\) 1372.00 0.819456
\(142\) 0 0
\(143\) −1204.00 −0.704081
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2149.00 1.20576
\(148\) 0 0
\(149\) −3200.00 −1.75942 −0.879712 0.475507i \(-0.842265\pi\)
−0.879712 + 0.475507i \(0.842265\pi\)
\(150\) 0 0
\(151\) −202.000 −0.108864 −0.0544322 0.998517i \(-0.517335\pi\)
−0.0544322 + 0.998517i \(0.517335\pi\)
\(152\) 0 0
\(153\) 2002.00 1.05786
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 406.000 0.206384 0.103192 0.994661i \(-0.467094\pi\)
0.103192 + 0.994661i \(0.467094\pi\)
\(158\) 0 0
\(159\) −574.000 −0.286297
\(160\) 0 0
\(161\) 972.000 0.475803
\(162\) 0 0
\(163\) 3803.00 1.82745 0.913724 0.406336i \(-0.133194\pi\)
0.913724 + 0.406336i \(0.133194\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4116.00 −1.90722 −0.953610 0.301046i \(-0.902664\pi\)
−0.953610 + 0.301046i \(0.902664\pi\)
\(168\) 0 0
\(169\) −1413.00 −0.643150
\(170\) 0 0
\(171\) 770.000 0.344347
\(172\) 0 0
\(173\) 1512.00 0.664481 0.332241 0.943195i \(-0.392195\pi\)
0.332241 + 0.943195i \(0.392195\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1960.00 −0.832331
\(178\) 0 0
\(179\) −2585.00 −1.07940 −0.539698 0.841859i \(-0.681462\pi\)
−0.539698 + 0.841859i \(0.681462\pi\)
\(180\) 0 0
\(181\) −2758.00 −1.13260 −0.566300 0.824199i \(-0.691626\pi\)
−0.566300 + 0.824199i \(0.691626\pi\)
\(182\) 0 0
\(183\) 3626.00 1.46471
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3913.00 1.53020
\(188\) 0 0
\(189\) −210.000 −0.0808214
\(190\) 0 0
\(191\) 2378.00 0.900869 0.450435 0.892809i \(-0.351269\pi\)
0.450435 + 0.892809i \(0.351269\pi\)
\(192\) 0 0
\(193\) 3067.00 1.14387 0.571937 0.820298i \(-0.306192\pi\)
0.571937 + 0.820298i \(0.306192\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2346.00 0.848455 0.424227 0.905556i \(-0.360546\pi\)
0.424227 + 0.905556i \(0.360546\pi\)
\(198\) 0 0
\(199\) −4900.00 −1.74549 −0.872743 0.488180i \(-0.837661\pi\)
−0.872743 + 0.488180i \(0.837661\pi\)
\(200\) 0 0
\(201\) 987.000 0.346356
\(202\) 0 0
\(203\) −960.000 −0.331915
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −3564.00 −1.19669
\(208\) 0 0
\(209\) 1505.00 0.498101
\(210\) 0 0
\(211\) −4307.00 −1.40524 −0.702621 0.711564i \(-0.747987\pi\)
−0.702621 + 0.711564i \(0.747987\pi\)
\(212\) 0 0
\(213\) 2884.00 0.927739
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 252.000 0.0788335
\(218\) 0 0
\(219\) 5341.00 1.64800
\(220\) 0 0
\(221\) −2548.00 −0.775552
\(222\) 0 0
\(223\) −2212.00 −0.664244 −0.332122 0.943236i \(-0.607765\pi\)
−0.332122 + 0.943236i \(0.607765\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −476.000 −0.139177 −0.0695886 0.997576i \(-0.522169\pi\)
−0.0695886 + 0.997576i \(0.522169\pi\)
\(228\) 0 0
\(229\) −2940.00 −0.848387 −0.424194 0.905572i \(-0.639442\pi\)
−0.424194 + 0.905572i \(0.639442\pi\)
\(230\) 0 0
\(231\) 1806.00 0.514399
\(232\) 0 0
\(233\) 1002.00 0.281730 0.140865 0.990029i \(-0.455012\pi\)
0.140865 + 0.990029i \(0.455012\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 3570.00 0.978466
\(238\) 0 0
\(239\) −2480.00 −0.671204 −0.335602 0.942004i \(-0.608940\pi\)
−0.335602 + 0.942004i \(0.608940\pi\)
\(240\) 0 0
\(241\) 1897.00 0.507039 0.253520 0.967330i \(-0.418412\pi\)
0.253520 + 0.967330i \(0.418412\pi\)
\(242\) 0 0
\(243\) 4928.00 1.30095
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −980.000 −0.252453
\(248\) 0 0
\(249\) 5439.00 1.38427
\(250\) 0 0
\(251\) 2373.00 0.596743 0.298371 0.954450i \(-0.403557\pi\)
0.298371 + 0.954450i \(0.403557\pi\)
\(252\) 0 0
\(253\) −6966.00 −1.73102
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4494.00 −1.09077 −0.545385 0.838185i \(-0.683617\pi\)
−0.545385 + 0.838185i \(0.683617\pi\)
\(258\) 0 0
\(259\) 1884.00 0.451993
\(260\) 0 0
\(261\) 3520.00 0.834799
\(262\) 0 0
\(263\) −722.000 −0.169279 −0.0846396 0.996412i \(-0.526974\pi\)
−0.0846396 + 0.996412i \(0.526974\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6615.00 1.51622
\(268\) 0 0
\(269\) −6160.00 −1.39621 −0.698107 0.715993i \(-0.745974\pi\)
−0.698107 + 0.715993i \(0.745974\pi\)
\(270\) 0 0
\(271\) 7238.00 1.62243 0.811213 0.584751i \(-0.198808\pi\)
0.811213 + 0.584751i \(0.198808\pi\)
\(272\) 0 0
\(273\) −1176.00 −0.260713
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1776.00 0.385233 0.192616 0.981274i \(-0.438303\pi\)
0.192616 + 0.981274i \(0.438303\pi\)
\(278\) 0 0
\(279\) −924.000 −0.198274
\(280\) 0 0
\(281\) 4542.00 0.964246 0.482123 0.876104i \(-0.339866\pi\)
0.482123 + 0.876104i \(0.339866\pi\)
\(282\) 0 0
\(283\) −7077.00 −1.48652 −0.743258 0.669005i \(-0.766720\pi\)
−0.743258 + 0.669005i \(0.766720\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1218.00 0.250510
\(288\) 0 0
\(289\) 3368.00 0.685528
\(290\) 0 0
\(291\) −8722.00 −1.75702
\(292\) 0 0
\(293\) −4158.00 −0.829054 −0.414527 0.910037i \(-0.636053\pi\)
−0.414527 + 0.910037i \(0.636053\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1505.00 0.294037
\(298\) 0 0
\(299\) 4536.00 0.877337
\(300\) 0 0
\(301\) 552.000 0.105703
\(302\) 0 0
\(303\) −9114.00 −1.72801
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2569.00 0.477591 0.238796 0.971070i \(-0.423247\pi\)
0.238796 + 0.971070i \(0.423247\pi\)
\(308\) 0 0
\(309\) 3724.00 0.685602
\(310\) 0 0
\(311\) −2982.00 −0.543710 −0.271855 0.962338i \(-0.587637\pi\)
−0.271855 + 0.962338i \(0.587637\pi\)
\(312\) 0 0
\(313\) 2422.00 0.437379 0.218689 0.975795i \(-0.429822\pi\)
0.218689 + 0.975795i \(0.429822\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9484.00 −1.68036 −0.840181 0.542307i \(-0.817551\pi\)
−0.840181 + 0.542307i \(0.817551\pi\)
\(318\) 0 0
\(319\) 6880.00 1.20754
\(320\) 0 0
\(321\) −8883.00 −1.54455
\(322\) 0 0
\(323\) 3185.00 0.548663
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −7490.00 −1.26666
\(328\) 0 0
\(329\) 1176.00 0.197067
\(330\) 0 0
\(331\) 183.000 0.0303885 0.0151942 0.999885i \(-0.495163\pi\)
0.0151942 + 0.999885i \(0.495163\pi\)
\(332\) 0 0
\(333\) −6908.00 −1.13681
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2861.00 0.462459 0.231229 0.972899i \(-0.425725\pi\)
0.231229 + 0.972899i \(0.425725\pi\)
\(338\) 0 0
\(339\) 3521.00 0.564113
\(340\) 0 0
\(341\) −1806.00 −0.286805
\(342\) 0 0
\(343\) 3900.00 0.613936
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 629.000 0.0973098 0.0486549 0.998816i \(-0.484507\pi\)
0.0486549 + 0.998816i \(0.484507\pi\)
\(348\) 0 0
\(349\) 5950.00 0.912597 0.456298 0.889827i \(-0.349175\pi\)
0.456298 + 0.889827i \(0.349175\pi\)
\(350\) 0 0
\(351\) −980.000 −0.149027
\(352\) 0 0
\(353\) −11718.0 −1.76682 −0.883408 0.468604i \(-0.844757\pi\)
−0.883408 + 0.468604i \(0.844757\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 3822.00 0.566615
\(358\) 0 0
\(359\) −8070.00 −1.18640 −0.593201 0.805054i \(-0.702136\pi\)
−0.593201 + 0.805054i \(0.702136\pi\)
\(360\) 0 0
\(361\) −5634.00 −0.821403
\(362\) 0 0
\(363\) −3626.00 −0.524286
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −8316.00 −1.18281 −0.591406 0.806374i \(-0.701427\pi\)
−0.591406 + 0.806374i \(0.701427\pi\)
\(368\) 0 0
\(369\) −4466.00 −0.630056
\(370\) 0 0
\(371\) −492.000 −0.0688500
\(372\) 0 0
\(373\) 12062.0 1.67439 0.837194 0.546906i \(-0.184195\pi\)
0.837194 + 0.546906i \(0.184195\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4480.00 −0.612021
\(378\) 0 0
\(379\) −1735.00 −0.235148 −0.117574 0.993064i \(-0.537512\pi\)
−0.117574 + 0.993064i \(0.537512\pi\)
\(380\) 0 0
\(381\) −6118.00 −0.822663
\(382\) 0 0
\(383\) −7602.00 −1.01421 −0.507107 0.861883i \(-0.669285\pi\)
−0.507107 + 0.861883i \(0.669285\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2024.00 −0.265855
\(388\) 0 0
\(389\) 3030.00 0.394928 0.197464 0.980310i \(-0.436729\pi\)
0.197464 + 0.980310i \(0.436729\pi\)
\(390\) 0 0
\(391\) −14742.0 −1.90674
\(392\) 0 0
\(393\) 7644.00 0.981142
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1204.00 −0.152209 −0.0761046 0.997100i \(-0.524248\pi\)
−0.0761046 + 0.997100i \(0.524248\pi\)
\(398\) 0 0
\(399\) 1470.00 0.184441
\(400\) 0 0
\(401\) 1077.00 0.134122 0.0670609 0.997749i \(-0.478638\pi\)
0.0670609 + 0.997749i \(0.478638\pi\)
\(402\) 0 0
\(403\) 1176.00 0.145362
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −13502.0 −1.64440
\(408\) 0 0
\(409\) −3955.00 −0.478147 −0.239074 0.971001i \(-0.576844\pi\)
−0.239074 + 0.971001i \(0.576844\pi\)
\(410\) 0 0
\(411\) −2877.00 −0.345285
\(412\) 0 0
\(413\) −1680.00 −0.200163
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −4165.00 −0.489115
\(418\) 0 0
\(419\) −6265.00 −0.730466 −0.365233 0.930916i \(-0.619011\pi\)
−0.365233 + 0.930916i \(0.619011\pi\)
\(420\) 0 0
\(421\) −3788.00 −0.438517 −0.219259 0.975667i \(-0.570364\pi\)
−0.219259 + 0.975667i \(0.570364\pi\)
\(422\) 0 0
\(423\) −4312.00 −0.495642
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 3108.00 0.352240
\(428\) 0 0
\(429\) 8428.00 0.948503
\(430\) 0 0
\(431\) 15258.0 1.70523 0.852613 0.522544i \(-0.175017\pi\)
0.852613 + 0.522544i \(0.175017\pi\)
\(432\) 0 0
\(433\) −13573.0 −1.50641 −0.753206 0.657784i \(-0.771494\pi\)
−0.753206 + 0.657784i \(0.771494\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5670.00 −0.620670
\(438\) 0 0
\(439\) 8120.00 0.882794 0.441397 0.897312i \(-0.354483\pi\)
0.441397 + 0.897312i \(0.354483\pi\)
\(440\) 0 0
\(441\) −6754.00 −0.729295
\(442\) 0 0
\(443\) 6183.00 0.663122 0.331561 0.943434i \(-0.392425\pi\)
0.331561 + 0.943434i \(0.392425\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 22400.0 2.37021
\(448\) 0 0
\(449\) −1975.00 −0.207586 −0.103793 0.994599i \(-0.533098\pi\)
−0.103793 + 0.994599i \(0.533098\pi\)
\(450\) 0 0
\(451\) −8729.00 −0.911380
\(452\) 0 0
\(453\) 1414.00 0.146657
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11831.0 1.21101 0.605504 0.795842i \(-0.292971\pi\)
0.605504 + 0.795842i \(0.292971\pi\)
\(458\) 0 0
\(459\) 3185.00 0.323885
\(460\) 0 0
\(461\) 1932.00 0.195189 0.0975946 0.995226i \(-0.468885\pi\)
0.0975946 + 0.995226i \(0.468885\pi\)
\(462\) 0 0
\(463\) 9228.00 0.926267 0.463133 0.886289i \(-0.346725\pi\)
0.463133 + 0.886289i \(0.346725\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −13916.0 −1.37892 −0.689460 0.724324i \(-0.742152\pi\)
−0.689460 + 0.724324i \(0.742152\pi\)
\(468\) 0 0
\(469\) 846.000 0.0832935
\(470\) 0 0
\(471\) −2842.00 −0.278031
\(472\) 0 0
\(473\) −3956.00 −0.384560
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1804.00 0.173165
\(478\) 0 0
\(479\) −2310.00 −0.220348 −0.110174 0.993912i \(-0.535141\pi\)
−0.110174 + 0.993912i \(0.535141\pi\)
\(480\) 0 0
\(481\) 8792.00 0.833432
\(482\) 0 0
\(483\) −6804.00 −0.640979
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 17114.0 1.59242 0.796211 0.605019i \(-0.206835\pi\)
0.796211 + 0.605019i \(0.206835\pi\)
\(488\) 0 0
\(489\) −26621.0 −2.46185
\(490\) 0 0
\(491\) 17228.0 1.58348 0.791740 0.610858i \(-0.209175\pi\)
0.791740 + 0.610858i \(0.209175\pi\)
\(492\) 0 0
\(493\) 14560.0 1.33012
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2472.00 0.223107
\(498\) 0 0
\(499\) 12500.0 1.12140 0.560698 0.828020i \(-0.310533\pi\)
0.560698 + 0.828020i \(0.310533\pi\)
\(500\) 0 0
\(501\) 28812.0 2.56931
\(502\) 0 0
\(503\) 868.000 0.0769428 0.0384714 0.999260i \(-0.487751\pi\)
0.0384714 + 0.999260i \(0.487751\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9891.00 0.866420
\(508\) 0 0
\(509\) 13370.0 1.16427 0.582136 0.813091i \(-0.302217\pi\)
0.582136 + 0.813091i \(0.302217\pi\)
\(510\) 0 0
\(511\) 4578.00 0.396319
\(512\) 0 0
\(513\) 1225.00 0.105429
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −8428.00 −0.716950
\(518\) 0 0
\(519\) −10584.0 −0.895156
\(520\) 0 0
\(521\) 21637.0 1.81945 0.909726 0.415210i \(-0.136292\pi\)
0.909726 + 0.415210i \(0.136292\pi\)
\(522\) 0 0
\(523\) −287.000 −0.0239955 −0.0119977 0.999928i \(-0.503819\pi\)
−0.0119977 + 0.999928i \(0.503819\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3822.00 −0.315918
\(528\) 0 0
\(529\) 14077.0 1.15698
\(530\) 0 0
\(531\) 6160.00 0.503430
\(532\) 0 0
\(533\) 5684.00 0.461916
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 18095.0 1.45411
\(538\) 0 0
\(539\) −13201.0 −1.05493
\(540\) 0 0
\(541\) −5328.00 −0.423417 −0.211709 0.977333i \(-0.567903\pi\)
−0.211709 + 0.977333i \(0.567903\pi\)
\(542\) 0 0
\(543\) 19306.0 1.52578
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −71.0000 −0.00554980 −0.00277490 0.999996i \(-0.500883\pi\)
−0.00277490 + 0.999996i \(0.500883\pi\)
\(548\) 0 0
\(549\) −11396.0 −0.885919
\(550\) 0 0
\(551\) 5600.00 0.432973
\(552\) 0 0
\(553\) 3060.00 0.235306
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18444.0 −1.40305 −0.701524 0.712646i \(-0.747497\pi\)
−0.701524 + 0.712646i \(0.747497\pi\)
\(558\) 0 0
\(559\) 2576.00 0.194907
\(560\) 0 0
\(561\) −27391.0 −2.06141
\(562\) 0 0
\(563\) −672.000 −0.0503045 −0.0251522 0.999684i \(-0.508007\pi\)
−0.0251522 + 0.999684i \(0.508007\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 5034.00 0.372854
\(568\) 0 0
\(569\) −10935.0 −0.805657 −0.402829 0.915275i \(-0.631973\pi\)
−0.402829 + 0.915275i \(0.631973\pi\)
\(570\) 0 0
\(571\) 13588.0 0.995867 0.497934 0.867215i \(-0.334092\pi\)
0.497934 + 0.867215i \(0.334092\pi\)
\(572\) 0 0
\(573\) −16646.0 −1.21361
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 8701.00 0.627777 0.313889 0.949460i \(-0.398368\pi\)
0.313889 + 0.949460i \(0.398368\pi\)
\(578\) 0 0
\(579\) −21469.0 −1.54097
\(580\) 0 0
\(581\) 4662.00 0.332896
\(582\) 0 0
\(583\) 3526.00 0.250484
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −11361.0 −0.798839 −0.399420 0.916768i \(-0.630788\pi\)
−0.399420 + 0.916768i \(0.630788\pi\)
\(588\) 0 0
\(589\) −1470.00 −0.102836
\(590\) 0 0
\(591\) −16422.0 −1.14300
\(592\) 0 0
\(593\) 11417.0 0.790624 0.395312 0.918547i \(-0.370636\pi\)
0.395312 + 0.918547i \(0.370636\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 34300.0 2.35143
\(598\) 0 0
\(599\) 21050.0 1.43586 0.717930 0.696116i \(-0.245090\pi\)
0.717930 + 0.696116i \(0.245090\pi\)
\(600\) 0 0
\(601\) 7427.00 0.504083 0.252041 0.967716i \(-0.418898\pi\)
0.252041 + 0.967716i \(0.418898\pi\)
\(602\) 0 0
\(603\) −3102.00 −0.209491
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4144.00 0.277100 0.138550 0.990355i \(-0.455756\pi\)
0.138550 + 0.990355i \(0.455756\pi\)
\(608\) 0 0
\(609\) 6720.00 0.447140
\(610\) 0 0
\(611\) 5488.00 0.363373
\(612\) 0 0
\(613\) 30122.0 1.98469 0.992346 0.123489i \(-0.0394084\pi\)
0.992346 + 0.123489i \(0.0394084\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −11934.0 −0.778679 −0.389339 0.921094i \(-0.627297\pi\)
−0.389339 + 0.921094i \(0.627297\pi\)
\(618\) 0 0
\(619\) −8540.00 −0.554526 −0.277263 0.960794i \(-0.589427\pi\)
−0.277263 + 0.960794i \(0.589427\pi\)
\(620\) 0 0
\(621\) −5670.00 −0.366392
\(622\) 0 0
\(623\) 5670.00 0.364629
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −10535.0 −0.671017
\(628\) 0 0
\(629\) −28574.0 −1.81132
\(630\) 0 0
\(631\) 3158.00 0.199236 0.0996181 0.995026i \(-0.468238\pi\)
0.0996181 + 0.995026i \(0.468238\pi\)
\(632\) 0 0
\(633\) 30149.0 1.89307
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 8596.00 0.534672
\(638\) 0 0
\(639\) −9064.00 −0.561137
\(640\) 0 0
\(641\) −4278.00 −0.263605 −0.131803 0.991276i \(-0.542076\pi\)
−0.131803 + 0.991276i \(0.542076\pi\)
\(642\) 0 0
\(643\) 11508.0 0.705803 0.352901 0.935661i \(-0.385195\pi\)
0.352901 + 0.935661i \(0.385195\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8204.00 0.498505 0.249252 0.968439i \(-0.419815\pi\)
0.249252 + 0.968439i \(0.419815\pi\)
\(648\) 0 0
\(649\) 12040.0 0.728215
\(650\) 0 0
\(651\) −1764.00 −0.106201
\(652\) 0 0
\(653\) −5518.00 −0.330683 −0.165342 0.986236i \(-0.552873\pi\)
−0.165342 + 0.986236i \(0.552873\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −16786.0 −0.996780
\(658\) 0 0
\(659\) −13295.0 −0.785887 −0.392944 0.919563i \(-0.628543\pi\)
−0.392944 + 0.919563i \(0.628543\pi\)
\(660\) 0 0
\(661\) −9968.00 −0.586551 −0.293276 0.956028i \(-0.594745\pi\)
−0.293276 + 0.956028i \(0.594745\pi\)
\(662\) 0 0
\(663\) 17836.0 1.04479
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −25920.0 −1.50469
\(668\) 0 0
\(669\) 15484.0 0.894837
\(670\) 0 0
\(671\) −22274.0 −1.28149
\(672\) 0 0
\(673\) −15738.0 −0.901419 −0.450710 0.892671i \(-0.648829\pi\)
−0.450710 + 0.892671i \(0.648829\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −19824.0 −1.12540 −0.562702 0.826660i \(-0.690238\pi\)
−0.562702 + 0.826660i \(0.690238\pi\)
\(678\) 0 0
\(679\) −7476.00 −0.422537
\(680\) 0 0
\(681\) 3332.00 0.187493
\(682\) 0 0
\(683\) 11073.0 0.620346 0.310173 0.950680i \(-0.399613\pi\)
0.310173 + 0.950680i \(0.399613\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 20580.0 1.14291
\(688\) 0 0
\(689\) −2296.00 −0.126953
\(690\) 0 0
\(691\) 6503.00 0.358011 0.179006 0.983848i \(-0.442712\pi\)
0.179006 + 0.983848i \(0.442712\pi\)
\(692\) 0 0
\(693\) −5676.00 −0.311130
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −18473.0 −1.00389
\(698\) 0 0
\(699\) −7014.00 −0.379533
\(700\) 0 0
\(701\) −10148.0 −0.546768 −0.273384 0.961905i \(-0.588143\pi\)
−0.273384 + 0.961905i \(0.588143\pi\)
\(702\) 0 0
\(703\) −10990.0 −0.589610
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −7812.00 −0.415559
\(708\) 0 0
\(709\) −9980.00 −0.528641 −0.264321 0.964435i \(-0.585148\pi\)
−0.264321 + 0.964435i \(0.585148\pi\)
\(710\) 0 0
\(711\) −11220.0 −0.591818
\(712\) 0 0
\(713\) 6804.00 0.357380
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 17360.0 0.904214
\(718\) 0 0
\(719\) 27510.0 1.42691 0.713456 0.700700i \(-0.247129\pi\)
0.713456 + 0.700700i \(0.247129\pi\)
\(720\) 0 0
\(721\) 3192.00 0.164877
\(722\) 0 0
\(723\) −13279.0 −0.683059
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 17024.0 0.868480 0.434240 0.900797i \(-0.357017\pi\)
0.434240 + 0.900797i \(0.357017\pi\)
\(728\) 0 0
\(729\) −11843.0 −0.601687
\(730\) 0 0
\(731\) −8372.00 −0.423597
\(732\) 0 0
\(733\) −34748.0 −1.75095 −0.875475 0.483263i \(-0.839451\pi\)
−0.875475 + 0.483263i \(0.839451\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6063.00 −0.303030
\(738\) 0 0
\(739\) 12020.0 0.598326 0.299163 0.954202i \(-0.403293\pi\)
0.299163 + 0.954202i \(0.403293\pi\)
\(740\) 0 0
\(741\) 6860.00 0.340092
\(742\) 0 0
\(743\) −28642.0 −1.41423 −0.707115 0.707098i \(-0.750004\pi\)
−0.707115 + 0.707098i \(0.750004\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −17094.0 −0.837265
\(748\) 0 0
\(749\) −7614.00 −0.371441
\(750\) 0 0
\(751\) −8752.00 −0.425253 −0.212627 0.977134i \(-0.568202\pi\)
−0.212627 + 0.977134i \(0.568202\pi\)
\(752\) 0 0
\(753\) −16611.0 −0.803902
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 10256.0 0.492418 0.246209 0.969217i \(-0.420815\pi\)
0.246209 + 0.969217i \(0.420815\pi\)
\(758\) 0 0
\(759\) 48762.0 2.33195
\(760\) 0 0
\(761\) 33957.0 1.61753 0.808765 0.588132i \(-0.200136\pi\)
0.808765 + 0.588132i \(0.200136\pi\)
\(762\) 0 0
\(763\) −6420.00 −0.304613
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −7840.00 −0.369082
\(768\) 0 0
\(769\) 27965.0 1.31137 0.655685 0.755034i \(-0.272380\pi\)
0.655685 + 0.755034i \(0.272380\pi\)
\(770\) 0 0
\(771\) 31458.0 1.46943
\(772\) 0 0
\(773\) 9912.00 0.461203 0.230601 0.973048i \(-0.425931\pi\)
0.230601 + 0.973048i \(0.425931\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −13188.0 −0.608902
\(778\) 0 0
\(779\) −7105.00 −0.326782
\(780\) 0 0
\(781\) −17716.0 −0.811688
\(782\) 0 0
\(783\) 5600.00 0.255591
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 25564.0 1.15789 0.578944 0.815367i \(-0.303465\pi\)
0.578944 + 0.815367i \(0.303465\pi\)
\(788\) 0 0
\(789\) 5054.00 0.228045
\(790\) 0 0
\(791\) 3018.00 0.135661
\(792\) 0 0
\(793\) 14504.0 0.649498
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12446.0 0.553149 0.276575 0.960992i \(-0.410801\pi\)
0.276575 + 0.960992i \(0.410801\pi\)
\(798\) 0 0
\(799\) −17836.0 −0.789728
\(800\) 0 0
\(801\) −20790.0 −0.917077
\(802\) 0 0
\(803\) −32809.0 −1.44185
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 43120.0 1.88091
\(808\) 0 0
\(809\) 33970.0 1.47629 0.738147 0.674640i \(-0.235701\pi\)
0.738147 + 0.674640i \(0.235701\pi\)
\(810\) 0 0
\(811\) −18732.0 −0.811060 −0.405530 0.914082i \(-0.632913\pi\)
−0.405530 + 0.914082i \(0.632913\pi\)
\(812\) 0 0
\(813\) −50666.0 −2.18565
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −3220.00 −0.137887
\(818\) 0 0
\(819\) 3696.00 0.157691
\(820\) 0 0
\(821\) 6162.00 0.261943 0.130972 0.991386i \(-0.458190\pi\)
0.130972 + 0.991386i \(0.458190\pi\)
\(822\) 0 0
\(823\) 25388.0 1.07530 0.537649 0.843169i \(-0.319313\pi\)
0.537649 + 0.843169i \(0.319313\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −25201.0 −1.05964 −0.529821 0.848109i \(-0.677741\pi\)
−0.529821 + 0.848109i \(0.677741\pi\)
\(828\) 0 0
\(829\) −19740.0 −0.827019 −0.413509 0.910500i \(-0.635697\pi\)
−0.413509 + 0.910500i \(0.635697\pi\)
\(830\) 0 0
\(831\) −12432.0 −0.518967
\(832\) 0 0
\(833\) −27937.0 −1.16202
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1470.00 −0.0607057
\(838\) 0 0
\(839\) −29680.0 −1.22130 −0.610648 0.791902i \(-0.709091\pi\)
−0.610648 + 0.791902i \(0.709091\pi\)
\(840\) 0 0
\(841\) 1211.00 0.0496535
\(842\) 0 0
\(843\) −31794.0 −1.29898
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −3108.00 −0.126083
\(848\) 0 0
\(849\) 49539.0 2.00256
\(850\) 0 0
\(851\) 50868.0 2.04904
\(852\) 0 0
\(853\) −1218.00 −0.0488904 −0.0244452 0.999701i \(-0.507782\pi\)
−0.0244452 + 0.999701i \(0.507782\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 38731.0 1.54379 0.771894 0.635752i \(-0.219310\pi\)
0.771894 + 0.635752i \(0.219310\pi\)
\(858\) 0 0
\(859\) 23555.0 0.935607 0.467803 0.883833i \(-0.345046\pi\)
0.467803 + 0.883833i \(0.345046\pi\)
\(860\) 0 0
\(861\) −8526.00 −0.337474
\(862\) 0 0
\(863\) −24872.0 −0.981058 −0.490529 0.871425i \(-0.663196\pi\)
−0.490529 + 0.871425i \(0.663196\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −23576.0 −0.923510
\(868\) 0 0
\(869\) −21930.0 −0.856069
\(870\) 0 0
\(871\) 3948.00 0.153585
\(872\) 0 0
\(873\) 27412.0 1.06272
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −17124.0 −0.659335 −0.329667 0.944097i \(-0.606937\pi\)
−0.329667 + 0.944097i \(0.606937\pi\)
\(878\) 0 0
\(879\) 29106.0 1.11686
\(880\) 0 0
\(881\) −658.000 −0.0251630 −0.0125815 0.999921i \(-0.504005\pi\)
−0.0125815 + 0.999921i \(0.504005\pi\)
\(882\) 0 0
\(883\) −33727.0 −1.28540 −0.642698 0.766120i \(-0.722185\pi\)
−0.642698 + 0.766120i \(0.722185\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −36036.0 −1.36412 −0.682058 0.731298i \(-0.738915\pi\)
−0.682058 + 0.731298i \(0.738915\pi\)
\(888\) 0 0
\(889\) −5244.00 −0.197838
\(890\) 0 0
\(891\) −36077.0 −1.35648
\(892\) 0 0
\(893\) −6860.00 −0.257067
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −31752.0 −1.18190
\(898\) 0 0
\(899\) −6720.00 −0.249304
\(900\) 0 0
\(901\) 7462.00 0.275910
\(902\) 0 0
\(903\) −3864.00 −0.142399
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −39156.0 −1.43347 −0.716733 0.697348i \(-0.754363\pi\)
−0.716733 + 0.697348i \(0.754363\pi\)
\(908\) 0 0
\(909\) 28644.0 1.04517
\(910\) 0 0
\(911\) −43532.0 −1.58318 −0.791591 0.611051i \(-0.790747\pi\)
−0.791591 + 0.611051i \(0.790747\pi\)
\(912\) 0 0
\(913\) −33411.0 −1.21111
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6552.00 0.235950
\(918\) 0 0
\(919\) 28610.0 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(920\) 0 0
\(921\) −17983.0 −0.643388
\(922\) 0 0
\(923\) 11536.0 0.411389
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −11704.0 −0.414682
\(928\) 0 0
\(929\) −24290.0 −0.857835 −0.428918 0.903344i \(-0.641105\pi\)
−0.428918 + 0.903344i \(0.641105\pi\)
\(930\) 0 0
\(931\) −10745.0 −0.378253
\(932\) 0 0
\(933\) 20874.0 0.732459
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 34461.0 1.20149 0.600743 0.799442i \(-0.294872\pi\)
0.600743 + 0.799442i \(0.294872\pi\)
\(938\) 0 0
\(939\) −16954.0 −0.589215
\(940\) 0 0
\(941\) −40628.0 −1.40748 −0.703738 0.710460i \(-0.748487\pi\)
−0.703738 + 0.710460i \(0.748487\pi\)
\(942\) 0 0
\(943\) 32886.0 1.13565
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 20904.0 0.717306 0.358653 0.933471i \(-0.383236\pi\)
0.358653 + 0.933471i \(0.383236\pi\)
\(948\) 0 0
\(949\) 21364.0 0.730774
\(950\) 0 0
\(951\) 66388.0 2.26370
\(952\) 0 0
\(953\) 1807.00 0.0614213 0.0307106 0.999528i \(-0.490223\pi\)
0.0307106 + 0.999528i \(0.490223\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −48160.0 −1.62674
\(958\) 0 0
\(959\) −2466.00 −0.0830358
\(960\) 0 0
\(961\) −28027.0 −0.940787
\(962\) 0 0
\(963\) 27918.0 0.934211
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 57584.0 1.91497 0.957485 0.288482i \(-0.0931505\pi\)
0.957485 + 0.288482i \(0.0931505\pi\)
\(968\) 0 0
\(969\) −22295.0 −0.739132
\(970\) 0 0
\(971\) −27237.0 −0.900182 −0.450091 0.892983i \(-0.648608\pi\)
−0.450091 + 0.892983i \(0.648608\pi\)
\(972\) 0 0
\(973\) −3570.00 −0.117625
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −13649.0 −0.446950 −0.223475 0.974710i \(-0.571740\pi\)
−0.223475 + 0.974710i \(0.571740\pi\)
\(978\) 0 0
\(979\) −40635.0 −1.32656
\(980\) 0 0
\(981\) 23540.0 0.766131
\(982\) 0 0
\(983\) −16002.0 −0.519211 −0.259606 0.965715i \(-0.583593\pi\)
−0.259606 + 0.965715i \(0.583593\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −8232.00 −0.265479
\(988\) 0 0
\(989\) 14904.0 0.479191
\(990\) 0 0
\(991\) −37022.0 −1.18672 −0.593362 0.804936i \(-0.702200\pi\)
−0.593362 + 0.804936i \(0.702200\pi\)
\(992\) 0 0
\(993\) −1281.00 −0.0409379
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 18396.0 0.584360 0.292180 0.956363i \(-0.405619\pi\)
0.292180 + 0.956363i \(0.405619\pi\)
\(998\) 0 0
\(999\) −10990.0 −0.348056
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 400.4.a.c.1.1 1
4.3 odd 2 25.4.a.b.1.1 yes 1
5.2 odd 4 400.4.c.e.49.2 2
5.3 odd 4 400.4.c.e.49.1 2
5.4 even 2 400.4.a.s.1.1 1
8.3 odd 2 1600.4.a.i.1.1 1
8.5 even 2 1600.4.a.bs.1.1 1
12.11 even 2 225.4.a.c.1.1 1
20.3 even 4 25.4.b.b.24.1 2
20.7 even 4 25.4.b.b.24.2 2
20.19 odd 2 25.4.a.a.1.1 1
28.27 even 2 1225.4.a.i.1.1 1
40.19 odd 2 1600.4.a.bt.1.1 1
40.29 even 2 1600.4.a.h.1.1 1
60.23 odd 4 225.4.b.f.199.2 2
60.47 odd 4 225.4.b.f.199.1 2
60.59 even 2 225.4.a.e.1.1 1
140.139 even 2 1225.4.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.4.a.a.1.1 1 20.19 odd 2
25.4.a.b.1.1 yes 1 4.3 odd 2
25.4.b.b.24.1 2 20.3 even 4
25.4.b.b.24.2 2 20.7 even 4
225.4.a.c.1.1 1 12.11 even 2
225.4.a.e.1.1 1 60.59 even 2
225.4.b.f.199.1 2 60.47 odd 4
225.4.b.f.199.2 2 60.23 odd 4
400.4.a.c.1.1 1 1.1 even 1 trivial
400.4.a.s.1.1 1 5.4 even 2
400.4.c.e.49.1 2 5.3 odd 4
400.4.c.e.49.2 2 5.2 odd 4
1225.4.a.h.1.1 1 140.139 even 2
1225.4.a.i.1.1 1 28.27 even 2
1600.4.a.h.1.1 1 40.29 even 2
1600.4.a.i.1.1 1 8.3 odd 2
1600.4.a.bs.1.1 1 8.5 even 2
1600.4.a.bt.1.1 1 40.19 odd 2