Properties

Label 400.4.c.e.49.1
Level $400$
Weight $4$
Character 400.49
Analytic conductor $23.601$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,4,Mod(49,400)]
 
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, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.6007640023\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 400.49
Dual form 400.4.c.e.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-7.00000i q^{3} +6.00000i q^{7} -22.0000 q^{9} +O(q^{10})\) \(q-7.00000i q^{3} +6.00000i q^{7} -22.0000 q^{9} +43.0000 q^{11} -28.0000i q^{13} -91.0000i q^{17} -35.0000 q^{19} +42.0000 q^{21} -162.000i q^{23} -35.0000i q^{27} -160.000 q^{29} -42.0000 q^{31} -301.000i q^{33} +314.000i q^{37} -196.000 q^{39} -203.000 q^{41} -92.0000i q^{43} +196.000i q^{47} +307.000 q^{49} -637.000 q^{51} +82.0000i q^{53} +245.000i q^{57} -280.000 q^{59} -518.000 q^{61} -132.000i q^{63} +141.000i q^{67} -1134.00 q^{69} -412.000 q^{71} -763.000i q^{73} +258.000i q^{77} +510.000 q^{79} -839.000 q^{81} -777.000i q^{83} +1120.00i q^{87} +945.000 q^{89} +168.000 q^{91} +294.000i q^{93} -1246.00i q^{97} -946.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 44 q^{9} + 86 q^{11} - 70 q^{19} + 84 q^{21} - 320 q^{29} - 84 q^{31} - 392 q^{39} - 406 q^{41} + 614 q^{49} - 1274 q^{51} - 560 q^{59} - 1036 q^{61} - 2268 q^{69} - 824 q^{71} + 1020 q^{79} - 1678 q^{81} + 1890 q^{89} + 336 q^{91} - 1892 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 7.00000i − 1.34715i −0.739119 0.673575i \(-0.764758\pi\)
0.739119 0.673575i \(-0.235242\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 6.00000i 0.323970i 0.986793 + 0.161985i \(0.0517895\pi\)
−0.986793 + 0.161985i \(0.948210\pi\)
\(8\) 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
\(13\) − 28.0000i − 0.597369i −0.954352 0.298685i \(-0.903452\pi\)
0.954352 0.298685i \(-0.0965479\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 91.0000i − 1.29828i −0.760669 0.649139i \(-0.775129\pi\)
0.760669 0.649139i \(-0.224871\pi\)
\(18\) 0 0
\(19\) −35.0000 −0.422608 −0.211304 0.977420i \(-0.567771\pi\)
−0.211304 + 0.977420i \(0.567771\pi\)
\(20\) 0 0
\(21\) 42.0000 0.436436
\(22\) 0 0
\(23\) − 162.000i − 1.46867i −0.678789 0.734333i \(-0.737495\pi\)
0.678789 0.734333i \(-0.262505\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 35.0000i − 0.249472i
\(28\) 0 0
\(29\) −160.000 −1.02453 −0.512263 0.858829i \(-0.671193\pi\)
−0.512263 + 0.858829i \(0.671193\pi\)
\(30\) 0 0
\(31\) −42.0000 −0.243336 −0.121668 0.992571i \(-0.538824\pi\)
−0.121668 + 0.992571i \(0.538824\pi\)
\(32\) 0 0
\(33\) − 301.000i − 1.58780i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 314.000i 1.39517i 0.716502 + 0.697585i \(0.245742\pi\)
−0.716502 + 0.697585i \(0.754258\pi\)
\(38\) 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
\(43\) − 92.0000i − 0.326276i −0.986603 0.163138i \(-0.947838\pi\)
0.986603 0.163138i \(-0.0521616\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 196.000i 0.608288i 0.952626 + 0.304144i \(0.0983704\pi\)
−0.952626 + 0.304144i \(0.901630\pi\)
\(48\) 0 0
\(49\) 307.000 0.895044
\(50\) 0 0
\(51\) −637.000 −1.74898
\(52\) 0 0
\(53\) 82.0000i 0.212520i 0.994338 + 0.106260i \(0.0338876\pi\)
−0.994338 + 0.106260i \(0.966112\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 245.000i 0.569317i
\(58\) 0 0
\(59\) −280.000 −0.617846 −0.308923 0.951087i \(-0.599968\pi\)
−0.308923 + 0.951087i \(0.599968\pi\)
\(60\) 0 0
\(61\) −518.000 −1.08726 −0.543632 0.839324i \(-0.682951\pi\)
−0.543632 + 0.839324i \(0.682951\pi\)
\(62\) 0 0
\(63\) − 132.000i − 0.263975i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 141.000i 0.257103i 0.991703 + 0.128551i \(0.0410327\pi\)
−0.991703 + 0.128551i \(0.958967\pi\)
\(68\) 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
\(73\) − 763.000i − 1.22332i −0.791121 0.611660i \(-0.790502\pi\)
0.791121 0.611660i \(-0.209498\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 258.000i 0.381842i
\(78\) 0 0
\(79\) 510.000 0.726323 0.363161 0.931726i \(-0.381697\pi\)
0.363161 + 0.931726i \(0.381697\pi\)
\(80\) 0 0
\(81\) −839.000 −1.15089
\(82\) 0 0
\(83\) − 777.000i − 1.02755i −0.857924 0.513776i \(-0.828246\pi\)
0.857924 0.513776i \(-0.171754\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1120.00i 1.38019i
\(88\) 0 0
\(89\) 945.000 1.12550 0.562752 0.826626i \(-0.309743\pi\)
0.562752 + 0.826626i \(0.309743\pi\)
\(90\) 0 0
\(91\) 168.000 0.193530
\(92\) 0 0
\(93\) 294.000i 0.327811i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 1246.00i − 1.30425i −0.758112 0.652124i \(-0.773878\pi\)
0.758112 0.652124i \(-0.226122\pi\)
\(98\) 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
\(103\) − 532.000i − 0.508927i −0.967082 0.254464i \(-0.918101\pi\)
0.967082 0.254464i \(-0.0818989\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1269.00i − 1.14653i −0.819370 0.573266i \(-0.805676\pi\)
0.819370 0.573266i \(-0.194324\pi\)
\(108\) 0 0
\(109\) −1070.00 −0.940251 −0.470126 0.882599i \(-0.655791\pi\)
−0.470126 + 0.882599i \(0.655791\pi\)
\(110\) 0 0
\(111\) 2198.00 1.87950
\(112\) 0 0
\(113\) − 503.000i − 0.418746i −0.977836 0.209373i \(-0.932858\pi\)
0.977836 0.209373i \(-0.0671422\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 616.000i 0.486745i
\(118\) 0 0
\(119\) 546.000 0.420603
\(120\) 0 0
\(121\) 518.000 0.389181
\(122\) 0 0
\(123\) 1421.00i 1.04169i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 874.000i − 0.610669i −0.952245 0.305334i \(-0.901232\pi\)
0.952245 0.305334i \(-0.0987683\pi\)
\(128\) 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
\(133\) − 210.000i − 0.136912i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 411.000i − 0.256307i −0.991754 0.128154i \(-0.959095\pi\)
0.991754 0.128154i \(-0.0409051\pi\)
\(138\) 0 0
\(139\) −595.000 −0.363074 −0.181537 0.983384i \(-0.558107\pi\)
−0.181537 + 0.983384i \(0.558107\pi\)
\(140\) 0 0
\(141\) 1372.00 0.819456
\(142\) 0 0
\(143\) − 1204.00i − 0.704081i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 2149.00i − 1.20576i
\(148\) 0 0
\(149\) 3200.00 1.75942 0.879712 0.475507i \(-0.157735\pi\)
0.879712 + 0.475507i \(0.157735\pi\)
\(150\) 0 0
\(151\) −202.000 −0.108864 −0.0544322 0.998517i \(-0.517335\pi\)
−0.0544322 + 0.998517i \(0.517335\pi\)
\(152\) 0 0
\(153\) 2002.00i 1.05786i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 406.000i − 0.206384i −0.994661 0.103192i \(-0.967094\pi\)
0.994661 0.103192i \(-0.0329057\pi\)
\(158\) 0 0
\(159\) 574.000 0.286297
\(160\) 0 0
\(161\) 972.000 0.475803
\(162\) 0 0
\(163\) 3803.00i 1.82745i 0.406336 + 0.913724i \(0.366806\pi\)
−0.406336 + 0.913724i \(0.633194\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4116.00i 1.90722i 0.301046 + 0.953610i \(0.402664\pi\)
−0.301046 + 0.953610i \(0.597336\pi\)
\(168\) 0 0
\(169\) 1413.00 0.643150
\(170\) 0 0
\(171\) 770.000 0.344347
\(172\) 0 0
\(173\) 1512.00i 0.664481i 0.943195 + 0.332241i \(0.107805\pi\)
−0.943195 + 0.332241i \(0.892195\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1960.00i 0.832331i
\(178\) 0 0
\(179\) 2585.00 1.07940 0.539698 0.841859i \(-0.318538\pi\)
0.539698 + 0.841859i \(0.318538\pi\)
\(180\) 0 0
\(181\) −2758.00 −1.13260 −0.566300 0.824199i \(-0.691626\pi\)
−0.566300 + 0.824199i \(0.691626\pi\)
\(182\) 0 0
\(183\) 3626.00i 1.46471i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 3913.00i − 1.53020i
\(188\) 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
\(193\) 3067.00i 1.14387i 0.820298 + 0.571937i \(0.193808\pi\)
−0.820298 + 0.571937i \(0.806192\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 2346.00i − 0.848455i −0.905556 0.424227i \(-0.860546\pi\)
0.905556 0.424227i \(-0.139454\pi\)
\(198\) 0 0
\(199\) 4900.00 1.74549 0.872743 0.488180i \(-0.162339\pi\)
0.872743 + 0.488180i \(0.162339\pi\)
\(200\) 0 0
\(201\) 987.000 0.346356
\(202\) 0 0
\(203\) − 960.000i − 0.331915i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 3564.00i 1.19669i
\(208\) 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
\(213\) 2884.00i 0.927739i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 252.000i − 0.0788335i
\(218\) 0 0
\(219\) −5341.00 −1.64800
\(220\) 0 0
\(221\) −2548.00 −0.775552
\(222\) 0 0
\(223\) − 2212.00i − 0.664244i −0.943236 0.332122i \(-0.892235\pi\)
0.943236 0.332122i \(-0.107765\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 476.000i 0.139177i 0.997576 + 0.0695886i \(0.0221687\pi\)
−0.997576 + 0.0695886i \(0.977831\pi\)
\(228\) 0 0
\(229\) 2940.00 0.848387 0.424194 0.905572i \(-0.360558\pi\)
0.424194 + 0.905572i \(0.360558\pi\)
\(230\) 0 0
\(231\) 1806.00 0.514399
\(232\) 0 0
\(233\) 1002.00i 0.281730i 0.990029 + 0.140865i \(0.0449884\pi\)
−0.990029 + 0.140865i \(0.955012\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 3570.00i − 0.978466i
\(238\) 0 0
\(239\) 2480.00 0.671204 0.335602 0.942004i \(-0.391060\pi\)
0.335602 + 0.942004i \(0.391060\pi\)
\(240\) 0 0
\(241\) 1897.00 0.507039 0.253520 0.967330i \(-0.418412\pi\)
0.253520 + 0.967330i \(0.418412\pi\)
\(242\) 0 0
\(243\) 4928.00i 1.30095i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 980.000i 0.252453i
\(248\) 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
\(253\) − 6966.00i − 1.73102i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4494.00i 1.09077i 0.838185 + 0.545385i \(0.183617\pi\)
−0.838185 + 0.545385i \(0.816383\pi\)
\(258\) 0 0
\(259\) −1884.00 −0.451993
\(260\) 0 0
\(261\) 3520.00 0.834799
\(262\) 0 0
\(263\) − 722.000i − 0.169279i −0.996412 0.0846396i \(-0.973026\pi\)
0.996412 0.0846396i \(-0.0269739\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 6615.00i − 1.51622i
\(268\) 0 0
\(269\) 6160.00 1.39621 0.698107 0.715993i \(-0.254026\pi\)
0.698107 + 0.715993i \(0.254026\pi\)
\(270\) 0 0
\(271\) 7238.00 1.62243 0.811213 0.584751i \(-0.198808\pi\)
0.811213 + 0.584751i \(0.198808\pi\)
\(272\) 0 0
\(273\) − 1176.00i − 0.260713i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 1776.00i − 0.385233i −0.981274 0.192616i \(-0.938303\pi\)
0.981274 0.192616i \(-0.0616973\pi\)
\(278\) 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
\(283\) − 7077.00i − 1.48652i −0.669005 0.743258i \(-0.733280\pi\)
0.669005 0.743258i \(-0.266720\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 1218.00i − 0.250510i
\(288\) 0 0
\(289\) −3368.00 −0.685528
\(290\) 0 0
\(291\) −8722.00 −1.75702
\(292\) 0 0
\(293\) − 4158.00i − 0.829054i −0.910037 0.414527i \(-0.863947\pi\)
0.910037 0.414527i \(-0.136053\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 1505.00i − 0.294037i
\(298\) 0 0
\(299\) −4536.00 −0.877337
\(300\) 0 0
\(301\) 552.000 0.105703
\(302\) 0 0
\(303\) − 9114.00i − 1.72801i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 2569.00i − 0.477591i −0.971070 0.238796i \(-0.923247\pi\)
0.971070 0.238796i \(-0.0767526\pi\)
\(308\) 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
\(313\) 2422.00i 0.437379i 0.975795 + 0.218689i \(0.0701781\pi\)
−0.975795 + 0.218689i \(0.929822\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9484.00i 1.68036i 0.542307 + 0.840181i \(0.317551\pi\)
−0.542307 + 0.840181i \(0.682449\pi\)
\(318\) 0 0
\(319\) −6880.00 −1.20754
\(320\) 0 0
\(321\) −8883.00 −1.54455
\(322\) 0 0
\(323\) 3185.00i 0.548663i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 7490.00i 1.26666i
\(328\) 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
\(333\) − 6908.00i − 1.13681i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 2861.00i − 0.462459i −0.972899 0.231229i \(-0.925725\pi\)
0.972899 0.231229i \(-0.0742748\pi\)
\(338\) 0 0
\(339\) −3521.00 −0.564113
\(340\) 0 0
\(341\) −1806.00 −0.286805
\(342\) 0 0
\(343\) 3900.00i 0.613936i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 629.000i − 0.0973098i −0.998816 0.0486549i \(-0.984507\pi\)
0.998816 0.0486549i \(-0.0154934\pi\)
\(348\) 0 0
\(349\) −5950.00 −0.912597 −0.456298 0.889827i \(-0.650825\pi\)
−0.456298 + 0.889827i \(0.650825\pi\)
\(350\) 0 0
\(351\) −980.000 −0.149027
\(352\) 0 0
\(353\) − 11718.0i − 1.76682i −0.468604 0.883408i \(-0.655243\pi\)
0.468604 0.883408i \(-0.344757\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 3822.00i − 0.566615i
\(358\) 0 0
\(359\) 8070.00 1.18640 0.593201 0.805054i \(-0.297864\pi\)
0.593201 + 0.805054i \(0.297864\pi\)
\(360\) 0 0
\(361\) −5634.00 −0.821403
\(362\) 0 0
\(363\) − 3626.00i − 0.524286i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8316.00i 1.18281i 0.806374 + 0.591406i \(0.201427\pi\)
−0.806374 + 0.591406i \(0.798573\pi\)
\(368\) 0 0
\(369\) 4466.00 0.630056
\(370\) 0 0
\(371\) −492.000 −0.0688500
\(372\) 0 0
\(373\) 12062.0i 1.67439i 0.546906 + 0.837194i \(0.315805\pi\)
−0.546906 + 0.837194i \(0.684195\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4480.00i 0.612021i
\(378\) 0 0
\(379\) 1735.00 0.235148 0.117574 0.993064i \(-0.462488\pi\)
0.117574 + 0.993064i \(0.462488\pi\)
\(380\) 0 0
\(381\) −6118.00 −0.822663
\(382\) 0 0
\(383\) − 7602.00i − 1.01421i −0.861883 0.507107i \(-0.830715\pi\)
0.861883 0.507107i \(-0.169285\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2024.00i 0.265855i
\(388\) 0 0
\(389\) −3030.00 −0.394928 −0.197464 0.980310i \(-0.563271\pi\)
−0.197464 + 0.980310i \(0.563271\pi\)
\(390\) 0 0
\(391\) −14742.0 −1.90674
\(392\) 0 0
\(393\) 7644.00i 0.981142i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1204.00i 0.152209i 0.997100 + 0.0761046i \(0.0242483\pi\)
−0.997100 + 0.0761046i \(0.975752\pi\)
\(398\) 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
\(403\) 1176.00i 0.145362i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 13502.0i 1.64440i
\(408\) 0 0
\(409\) 3955.00 0.478147 0.239074 0.971001i \(-0.423156\pi\)
0.239074 + 0.971001i \(0.423156\pi\)
\(410\) 0 0
\(411\) −2877.00 −0.345285
\(412\) 0 0
\(413\) − 1680.00i − 0.200163i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4165.00i 0.489115i
\(418\) 0 0
\(419\) 6265.00 0.730466 0.365233 0.930916i \(-0.380989\pi\)
0.365233 + 0.930916i \(0.380989\pi\)
\(420\) 0 0
\(421\) −3788.00 −0.438517 −0.219259 0.975667i \(-0.570364\pi\)
−0.219259 + 0.975667i \(0.570364\pi\)
\(422\) 0 0
\(423\) − 4312.00i − 0.495642i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 3108.00i − 0.352240i
\(428\) 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
\(433\) − 13573.0i − 1.50641i −0.657784 0.753206i \(-0.728506\pi\)
0.657784 0.753206i \(-0.271494\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5670.00i 0.620670i
\(438\) 0 0
\(439\) −8120.00 −0.882794 −0.441397 0.897312i \(-0.645517\pi\)
−0.441397 + 0.897312i \(0.645517\pi\)
\(440\) 0 0
\(441\) −6754.00 −0.729295
\(442\) 0 0
\(443\) 6183.00i 0.663122i 0.943434 + 0.331561i \(0.107575\pi\)
−0.943434 + 0.331561i \(0.892425\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 22400.0i − 2.37021i
\(448\) 0 0
\(449\) 1975.00 0.207586 0.103793 0.994599i \(-0.466902\pi\)
0.103793 + 0.994599i \(0.466902\pi\)
\(450\) 0 0
\(451\) −8729.00 −0.911380
\(452\) 0 0
\(453\) 1414.00i 0.146657i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 11831.0i − 1.21101i −0.795842 0.605504i \(-0.792971\pi\)
0.795842 0.605504i \(-0.207029\pi\)
\(458\) 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
\(463\) 9228.00i 0.926267i 0.886289 + 0.463133i \(0.153275\pi\)
−0.886289 + 0.463133i \(0.846725\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 13916.0i 1.37892i 0.724324 + 0.689460i \(0.242152\pi\)
−0.724324 + 0.689460i \(0.757848\pi\)
\(468\) 0 0
\(469\) −846.000 −0.0832935
\(470\) 0 0
\(471\) −2842.00 −0.278031
\(472\) 0 0
\(473\) − 3956.00i − 0.384560i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 1804.00i − 0.173165i
\(478\) 0 0
\(479\) 2310.00 0.220348 0.110174 0.993912i \(-0.464859\pi\)
0.110174 + 0.993912i \(0.464859\pi\)
\(480\) 0 0
\(481\) 8792.00 0.833432
\(482\) 0 0
\(483\) − 6804.00i − 0.640979i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 17114.0i − 1.59242i −0.605019 0.796211i \(-0.706835\pi\)
0.605019 0.796211i \(-0.293165\pi\)
\(488\) 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
\(493\) 14560.0i 1.33012i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 2472.00i − 0.223107i
\(498\) 0 0
\(499\) −12500.0 −1.12140 −0.560698 0.828020i \(-0.689467\pi\)
−0.560698 + 0.828020i \(0.689467\pi\)
\(500\) 0 0
\(501\) 28812.0 2.56931
\(502\) 0 0
\(503\) 868.000i 0.0769428i 0.999260 + 0.0384714i \(0.0122488\pi\)
−0.999260 + 0.0384714i \(0.987751\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 9891.00i − 0.866420i
\(508\) 0 0
\(509\) −13370.0 −1.16427 −0.582136 0.813091i \(-0.697783\pi\)
−0.582136 + 0.813091i \(0.697783\pi\)
\(510\) 0 0
\(511\) 4578.00 0.396319
\(512\) 0 0
\(513\) 1225.00i 0.105429i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 8428.00i 0.716950i
\(518\) 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
\(523\) − 287.000i − 0.0239955i −0.999928 0.0119977i \(-0.996181\pi\)
0.999928 0.0119977i \(-0.00381909\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3822.00i 0.315918i
\(528\) 0 0
\(529\) −14077.0 −1.15698
\(530\) 0 0
\(531\) 6160.00 0.503430
\(532\) 0 0
\(533\) 5684.00i 0.461916i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 18095.0i − 1.45411i
\(538\) 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
\(543\) 19306.0i 1.52578i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 71.0000i 0.00554980i 0.999996 + 0.00277490i \(0.000883279\pi\)
−0.999996 + 0.00277490i \(0.999117\pi\)
\(548\) 0 0
\(549\) 11396.0 0.885919
\(550\) 0 0
\(551\) 5600.00 0.432973
\(552\) 0 0
\(553\) 3060.00i 0.235306i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18444.0i 1.40305i 0.712646 + 0.701524i \(0.247497\pi\)
−0.712646 + 0.701524i \(0.752503\pi\)
\(558\) 0 0
\(559\) −2576.00 −0.194907
\(560\) 0 0
\(561\) −27391.0 −2.06141
\(562\) 0 0
\(563\) − 672.000i − 0.0503045i −0.999684 0.0251522i \(-0.991993\pi\)
0.999684 0.0251522i \(-0.00800705\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 5034.00i − 0.372854i
\(568\) 0 0
\(569\) 10935.0 0.805657 0.402829 0.915275i \(-0.368027\pi\)
0.402829 + 0.915275i \(0.368027\pi\)
\(570\) 0 0
\(571\) 13588.0 0.995867 0.497934 0.867215i \(-0.334092\pi\)
0.497934 + 0.867215i \(0.334092\pi\)
\(572\) 0 0
\(573\) − 16646.0i − 1.21361i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 8701.00i − 0.627777i −0.949460 0.313889i \(-0.898368\pi\)
0.949460 0.313889i \(-0.101632\pi\)
\(578\) 0 0
\(579\) 21469.0 1.54097
\(580\) 0 0
\(581\) 4662.00 0.332896
\(582\) 0 0
\(583\) 3526.00i 0.250484i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11361.0i 0.798839i 0.916768 + 0.399420i \(0.130788\pi\)
−0.916768 + 0.399420i \(0.869212\pi\)
\(588\) 0 0
\(589\) 1470.00 0.102836
\(590\) 0 0
\(591\) −16422.0 −1.14300
\(592\) 0 0
\(593\) 11417.0i 0.790624i 0.918547 + 0.395312i \(0.129364\pi\)
−0.918547 + 0.395312i \(0.870636\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 34300.0i − 2.35143i
\(598\) 0 0
\(599\) −21050.0 −1.43586 −0.717930 0.696116i \(-0.754910\pi\)
−0.717930 + 0.696116i \(0.754910\pi\)
\(600\) 0 0
\(601\) 7427.00 0.504083 0.252041 0.967716i \(-0.418898\pi\)
0.252041 + 0.967716i \(0.418898\pi\)
\(602\) 0 0
\(603\) − 3102.00i − 0.209491i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 4144.00i − 0.277100i −0.990355 0.138550i \(-0.955756\pi\)
0.990355 0.138550i \(-0.0442442\pi\)
\(608\) 0 0
\(609\) −6720.00 −0.447140
\(610\) 0 0
\(611\) 5488.00 0.363373
\(612\) 0 0
\(613\) 30122.0i 1.98469i 0.123489 + 0.992346i \(0.460592\pi\)
−0.123489 + 0.992346i \(0.539408\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 11934.0i 0.778679i 0.921094 + 0.389339i \(0.127297\pi\)
−0.921094 + 0.389339i \(0.872703\pi\)
\(618\) 0 0
\(619\) 8540.00 0.554526 0.277263 0.960794i \(-0.410573\pi\)
0.277263 + 0.960794i \(0.410573\pi\)
\(620\) 0 0
\(621\) −5670.00 −0.366392
\(622\) 0 0
\(623\) 5670.00i 0.364629i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 10535.0i 0.671017i
\(628\) 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
\(633\) 30149.0i 1.89307i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 8596.00i − 0.534672i
\(638\) 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
\(643\) 11508.0i 0.705803i 0.935661 + 0.352901i \(0.114805\pi\)
−0.935661 + 0.352901i \(0.885195\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 8204.00i − 0.498505i −0.968439 0.249252i \(-0.919815\pi\)
0.968439 0.249252i \(-0.0801849\pi\)
\(648\) 0 0
\(649\) −12040.0 −0.728215
\(650\) 0 0
\(651\) −1764.00 −0.106201
\(652\) 0 0
\(653\) − 5518.00i − 0.330683i −0.986236 0.165342i \(-0.947127\pi\)
0.986236 0.165342i \(-0.0528726\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 16786.0i 0.996780i
\(658\) 0 0
\(659\) 13295.0 0.785887 0.392944 0.919563i \(-0.371457\pi\)
0.392944 + 0.919563i \(0.371457\pi\)
\(660\) 0 0
\(661\) −9968.00 −0.586551 −0.293276 0.956028i \(-0.594745\pi\)
−0.293276 + 0.956028i \(0.594745\pi\)
\(662\) 0 0
\(663\) 17836.0i 1.04479i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 25920.0i 1.50469i
\(668\) 0 0
\(669\) −15484.0 −0.894837
\(670\) 0 0
\(671\) −22274.0 −1.28149
\(672\) 0 0
\(673\) − 15738.0i − 0.901419i −0.892671 0.450710i \(-0.851171\pi\)
0.892671 0.450710i \(-0.148829\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19824.0i 1.12540i 0.826660 + 0.562702i \(0.190238\pi\)
−0.826660 + 0.562702i \(0.809762\pi\)
\(678\) 0 0
\(679\) 7476.00 0.422537
\(680\) 0 0
\(681\) 3332.00 0.187493
\(682\) 0 0
\(683\) 11073.0i 0.620346i 0.950680 + 0.310173i \(0.100387\pi\)
−0.950680 + 0.310173i \(0.899613\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 20580.0i − 1.14291i
\(688\) 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
\(693\) − 5676.00i − 0.311130i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 18473.0i 1.00389i
\(698\) 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
\(703\) − 10990.0i − 0.589610i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7812.00i 0.415559i
\(708\) 0 0
\(709\) 9980.00 0.528641 0.264321 0.964435i \(-0.414852\pi\)
0.264321 + 0.964435i \(0.414852\pi\)
\(710\) 0 0
\(711\) −11220.0 −0.591818
\(712\) 0 0
\(713\) 6804.00i 0.357380i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 17360.0i − 0.904214i
\(718\) 0 0
\(719\) −27510.0 −1.42691 −0.713456 0.700700i \(-0.752871\pi\)
−0.713456 + 0.700700i \(0.752871\pi\)
\(720\) 0 0
\(721\) 3192.00 0.164877
\(722\) 0 0
\(723\) − 13279.0i − 0.683059i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 17024.0i − 0.868480i −0.900797 0.434240i \(-0.857017\pi\)
0.900797 0.434240i \(-0.142983\pi\)
\(728\) 0 0
\(729\) 11843.0 0.601687
\(730\) 0 0
\(731\) −8372.00 −0.423597
\(732\) 0 0
\(733\) − 34748.0i − 1.75095i −0.483263 0.875475i \(-0.660549\pi\)
0.483263 0.875475i \(-0.339451\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6063.00i 0.303030i
\(738\) 0 0
\(739\) −12020.0 −0.598326 −0.299163 0.954202i \(-0.596707\pi\)
−0.299163 + 0.954202i \(0.596707\pi\)
\(740\) 0 0
\(741\) 6860.00 0.340092
\(742\) 0 0
\(743\) − 28642.0i − 1.41423i −0.707098 0.707115i \(-0.749996\pi\)
0.707098 0.707115i \(-0.250004\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 17094.0i 0.837265i
\(748\) 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
\(753\) − 16611.0i − 0.803902i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 10256.0i − 0.492418i −0.969217 0.246209i \(-0.920815\pi\)
0.969217 0.246209i \(-0.0791850\pi\)
\(758\) 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
\(763\) − 6420.00i − 0.304613i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7840.00i 0.369082i
\(768\) 0 0
\(769\) −27965.0 −1.31137 −0.655685 0.755034i \(-0.727620\pi\)
−0.655685 + 0.755034i \(0.727620\pi\)
\(770\) 0 0
\(771\) 31458.0 1.46943
\(772\) 0 0
\(773\) 9912.00i 0.461203i 0.973048 + 0.230601i \(0.0740694\pi\)
−0.973048 + 0.230601i \(0.925931\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 13188.0i 0.608902i
\(778\) 0 0
\(779\) 7105.00 0.326782
\(780\) 0 0
\(781\) −17716.0 −0.811688
\(782\) 0 0
\(783\) 5600.00i 0.255591i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 25564.0i − 1.15789i −0.815367 0.578944i \(-0.803465\pi\)
0.815367 0.578944i \(-0.196535\pi\)
\(788\) 0 0
\(789\) −5054.00 −0.228045
\(790\) 0 0
\(791\) 3018.00 0.135661
\(792\) 0 0
\(793\) 14504.0i 0.649498i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 12446.0i − 0.553149i −0.960992 0.276575i \(-0.910801\pi\)
0.960992 0.276575i \(-0.0891993\pi\)
\(798\) 0 0
\(799\) 17836.0 0.789728
\(800\) 0 0
\(801\) −20790.0 −0.917077
\(802\) 0 0
\(803\) − 32809.0i − 1.44185i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 43120.0i − 1.88091i
\(808\) 0 0
\(809\) −33970.0 −1.47629 −0.738147 0.674640i \(-0.764299\pi\)
−0.738147 + 0.674640i \(0.764299\pi\)
\(810\) 0 0
\(811\) −18732.0 −0.811060 −0.405530 0.914082i \(-0.632913\pi\)
−0.405530 + 0.914082i \(0.632913\pi\)
\(812\) 0 0
\(813\) − 50666.0i − 2.18565i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 3220.00i 0.137887i
\(818\) 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
\(823\) 25388.0i 1.07530i 0.843169 + 0.537649i \(0.180687\pi\)
−0.843169 + 0.537649i \(0.819313\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 25201.0i 1.05964i 0.848109 + 0.529821i \(0.177741\pi\)
−0.848109 + 0.529821i \(0.822259\pi\)
\(828\) 0 0
\(829\) 19740.0 0.827019 0.413509 0.910500i \(-0.364303\pi\)
0.413509 + 0.910500i \(0.364303\pi\)
\(830\) 0 0
\(831\) −12432.0 −0.518967
\(832\) 0 0
\(833\) − 27937.0i − 1.16202i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1470.00i 0.0607057i
\(838\) 0 0
\(839\) 29680.0 1.22130 0.610648 0.791902i \(-0.290909\pi\)
0.610648 + 0.791902i \(0.290909\pi\)
\(840\) 0 0
\(841\) 1211.00 0.0496535
\(842\) 0 0
\(843\) − 31794.0i − 1.29898i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3108.00i 0.126083i
\(848\) 0 0
\(849\) −49539.0 −2.00256
\(850\) 0 0
\(851\) 50868.0 2.04904
\(852\) 0 0
\(853\) − 1218.00i − 0.0488904i −0.999701 0.0244452i \(-0.992218\pi\)
0.999701 0.0244452i \(-0.00778193\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 38731.0i − 1.54379i −0.635752 0.771894i \(-0.719310\pi\)
0.635752 0.771894i \(-0.280690\pi\)
\(858\) 0 0
\(859\) −23555.0 −0.935607 −0.467803 0.883833i \(-0.654954\pi\)
−0.467803 + 0.883833i \(0.654954\pi\)
\(860\) 0 0
\(861\) −8526.00 −0.337474
\(862\) 0 0
\(863\) − 24872.0i − 0.981058i −0.871425 0.490529i \(-0.836804\pi\)
0.871425 0.490529i \(-0.163196\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 23576.0i 0.923510i
\(868\) 0 0
\(869\) 21930.0 0.856069
\(870\) 0 0
\(871\) 3948.00 0.153585
\(872\) 0 0
\(873\) 27412.0i 1.06272i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 17124.0i 0.659335i 0.944097 + 0.329667i \(0.106937\pi\)
−0.944097 + 0.329667i \(0.893063\pi\)
\(878\) 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
\(883\) − 33727.0i − 1.28540i −0.766120 0.642698i \(-0.777815\pi\)
0.766120 0.642698i \(-0.222185\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 36036.0i 1.36412i 0.731298 + 0.682058i \(0.238915\pi\)
−0.731298 + 0.682058i \(0.761085\pi\)
\(888\) 0 0
\(889\) 5244.00 0.197838
\(890\) 0 0
\(891\) −36077.0 −1.35648
\(892\) 0 0
\(893\) − 6860.00i − 0.257067i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 31752.0i 1.18190i
\(898\) 0 0
\(899\) 6720.00 0.249304
\(900\) 0 0
\(901\) 7462.00 0.275910
\(902\) 0 0
\(903\) − 3864.00i − 0.142399i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 39156.0i 1.43347i 0.697348 + 0.716733i \(0.254363\pi\)
−0.697348 + 0.716733i \(0.745637\pi\)
\(908\) 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
\(913\) − 33411.0i − 1.21111i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 6552.00i − 0.235950i
\(918\) 0 0
\(919\) −28610.0 −1.02694 −0.513469 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(920\) 0 0
\(921\) −17983.0 −0.643388
\(922\) 0 0
\(923\) 11536.0i 0.411389i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 11704.0i 0.414682i
\(928\) 0 0
\(929\) 24290.0 0.857835 0.428918 0.903344i \(-0.358895\pi\)
0.428918 + 0.903344i \(0.358895\pi\)
\(930\) 0 0
\(931\) −10745.0 −0.378253
\(932\) 0 0
\(933\) 20874.0i 0.732459i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 34461.0i − 1.20149i −0.799442 0.600743i \(-0.794872\pi\)
0.799442 0.600743i \(-0.205128\pi\)
\(938\) 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
\(943\) 32886.0i 1.13565i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 20904.0i − 0.717306i −0.933471 0.358653i \(-0.883236\pi\)
0.933471 0.358653i \(-0.116764\pi\)
\(948\) 0 0
\(949\) −21364.0 −0.730774
\(950\) 0 0
\(951\) 66388.0 2.26370
\(952\) 0 0
\(953\) 1807.00i 0.0614213i 0.999528 + 0.0307106i \(0.00977704\pi\)
−0.999528 + 0.0307106i \(0.990223\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 48160.0i 1.62674i
\(958\) 0 0
\(959\) 2466.00 0.0830358
\(960\) 0 0
\(961\) −28027.0 −0.940787
\(962\) 0 0
\(963\) 27918.0i 0.934211i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 57584.0i − 1.91497i −0.288482 0.957485i \(-0.593151\pi\)
0.288482 0.957485i \(-0.406849\pi\)
\(968\) 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
\(973\) − 3570.00i − 0.117625i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 13649.0i 0.446950i 0.974710 + 0.223475i \(0.0717401\pi\)
−0.974710 + 0.223475i \(0.928260\pi\)
\(978\) 0 0
\(979\) 40635.0 1.32656
\(980\) 0 0
\(981\) 23540.0 0.766131
\(982\) 0 0
\(983\) − 16002.0i − 0.519211i −0.965715 0.259606i \(-0.916407\pi\)
0.965715 0.259606i \(-0.0835925\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 8232.00i 0.265479i
\(988\) 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
\(993\) − 1281.00i − 0.0409379i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 18396.0i − 0.584360i −0.956363 0.292180i \(-0.905619\pi\)
0.956363 0.292180i \(-0.0943807\pi\)
\(998\) 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.c.e.49.1 2
4.3 odd 2 25.4.b.b.24.1 2
5.2 odd 4 400.4.a.c.1.1 1
5.3 odd 4 400.4.a.s.1.1 1
5.4 even 2 inner 400.4.c.e.49.2 2
12.11 even 2 225.4.b.f.199.2 2
20.3 even 4 25.4.a.a.1.1 1
20.7 even 4 25.4.a.b.1.1 yes 1
20.19 odd 2 25.4.b.b.24.2 2
40.3 even 4 1600.4.a.bt.1.1 1
40.13 odd 4 1600.4.a.h.1.1 1
40.27 even 4 1600.4.a.i.1.1 1
40.37 odd 4 1600.4.a.bs.1.1 1
60.23 odd 4 225.4.a.e.1.1 1
60.47 odd 4 225.4.a.c.1.1 1
60.59 even 2 225.4.b.f.199.1 2
140.27 odd 4 1225.4.a.i.1.1 1
140.83 odd 4 1225.4.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.4.a.a.1.1 1 20.3 even 4
25.4.a.b.1.1 yes 1 20.7 even 4
25.4.b.b.24.1 2 4.3 odd 2
25.4.b.b.24.2 2 20.19 odd 2
225.4.a.c.1.1 1 60.47 odd 4
225.4.a.e.1.1 1 60.23 odd 4
225.4.b.f.199.1 2 60.59 even 2
225.4.b.f.199.2 2 12.11 even 2
400.4.a.c.1.1 1 5.2 odd 4
400.4.a.s.1.1 1 5.3 odd 4
400.4.c.e.49.1 2 1.1 even 1 trivial
400.4.c.e.49.2 2 5.4 even 2 inner
1225.4.a.h.1.1 1 140.83 odd 4
1225.4.a.i.1.1 1 140.27 odd 4
1600.4.a.h.1.1 1 40.13 odd 4
1600.4.a.i.1.1 1 40.27 even 4
1600.4.a.bs.1.1 1 40.37 odd 4
1600.4.a.bt.1.1 1 40.3 even 4