Properties

Label 2500.2.c.a.1249.4
Level $2500$
Weight $2$
Character 2500.1249
Analytic conductor $19.963$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2500,2,Mod(1249,2500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2500, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2500.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2500 = 2^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2500.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: \(19.9626005053\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.4
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 2500.1249
Dual form 2500.2.c.a.1249.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.61803i q^{3} +1.61803i q^{7} +0.381966 q^{9} +O(q^{10})\) \(q+1.61803i q^{3} +1.61803i q^{7} +0.381966 q^{9} +3.61803 q^{11} -3.00000i q^{13} +5.47214i q^{17} +2.61803 q^{19} -2.61803 q^{21} -0.381966i q^{23} +5.47214i q^{27} +5.70820 q^{29} -8.47214 q^{31} +5.85410i q^{33} -1.47214i q^{37} +4.85410 q^{39} +10.4721 q^{41} -10.0000i q^{43} -4.32624i q^{47} +4.38197 q^{49} -8.85410 q^{51} +8.47214i q^{53} +4.23607i q^{57} +5.76393 q^{59} +12.3262 q^{61} +0.618034i q^{63} +9.18034i q^{67} +0.618034 q^{69} -5.09017 q^{71} -4.38197i q^{73} +5.85410i q^{77} -13.0000 q^{79} -7.70820 q^{81} +4.94427i q^{83} +9.23607i q^{87} -5.38197 q^{89} +4.85410 q^{91} -13.7082i q^{93} +18.8541i q^{97} +1.38197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{9} + 10 q^{11} + 6 q^{19} - 6 q^{21} - 4 q^{29} - 16 q^{31} + 6 q^{39} + 24 q^{41} + 22 q^{49} - 22 q^{51} + 32 q^{59} + 18 q^{61} - 2 q^{69} + 2 q^{71} - 52 q^{79} - 4 q^{81} - 26 q^{89} + 6 q^{91} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2500\mathbb{Z}\right)^\times\).

\(n\) \(1251\) \(1877\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) 1.61803i 0.934172i 0.884212 + 0.467086i \(0.154696\pi\)
−0.884212 + 0.467086i \(0.845304\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.61803i 0.611559i 0.952102 + 0.305780i \(0.0989171\pi\)
−0.952102 + 0.305780i \(0.901083\pi\)
\(8\) 0 0
\(9\) 0.381966 0.127322
\(10\) 0 0
\(11\) 3.61803 1.09088 0.545439 0.838150i \(-0.316363\pi\)
0.545439 + 0.838150i \(0.316363\pi\)
\(12\) 0 0
\(13\) − 3.00000i − 0.832050i −0.909353 0.416025i \(-0.863423\pi\)
0.909353 0.416025i \(-0.136577\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.47214i 1.32719i 0.748093 + 0.663594i \(0.230970\pi\)
−0.748093 + 0.663594i \(0.769030\pi\)
\(18\) 0 0
\(19\) 2.61803 0.600618 0.300309 0.953842i \(-0.402910\pi\)
0.300309 + 0.953842i \(0.402910\pi\)
\(20\) 0 0
\(21\) −2.61803 −0.571302
\(22\) 0 0
\(23\) − 0.381966i − 0.0796454i −0.999207 0.0398227i \(-0.987321\pi\)
0.999207 0.0398227i \(-0.0126793\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.47214i 1.05311i
\(28\) 0 0
\(29\) 5.70820 1.05999 0.529993 0.848002i \(-0.322194\pi\)
0.529993 + 0.848002i \(0.322194\pi\)
\(30\) 0 0
\(31\) −8.47214 −1.52164 −0.760820 0.648963i \(-0.775203\pi\)
−0.760820 + 0.648963i \(0.775203\pi\)
\(32\) 0 0
\(33\) 5.85410i 1.01907i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 1.47214i − 0.242018i −0.992651 0.121009i \(-0.961387\pi\)
0.992651 0.121009i \(-0.0386129\pi\)
\(38\) 0 0
\(39\) 4.85410 0.777278
\(40\) 0 0
\(41\) 10.4721 1.63547 0.817736 0.575593i \(-0.195229\pi\)
0.817736 + 0.575593i \(0.195229\pi\)
\(42\) 0 0
\(43\) − 10.0000i − 1.52499i −0.646997 0.762493i \(-0.723975\pi\)
0.646997 0.762493i \(-0.276025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 4.32624i − 0.631047i −0.948918 0.315523i \(-0.897820\pi\)
0.948918 0.315523i \(-0.102180\pi\)
\(48\) 0 0
\(49\) 4.38197 0.625995
\(50\) 0 0
\(51\) −8.85410 −1.23982
\(52\) 0 0
\(53\) 8.47214i 1.16374i 0.813283 + 0.581869i \(0.197678\pi\)
−0.813283 + 0.581869i \(0.802322\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.23607i 0.561081i
\(58\) 0 0
\(59\) 5.76393 0.750400 0.375200 0.926944i \(-0.377574\pi\)
0.375200 + 0.926944i \(0.377574\pi\)
\(60\) 0 0
\(61\) 12.3262 1.57821 0.789107 0.614256i \(-0.210544\pi\)
0.789107 + 0.614256i \(0.210544\pi\)
\(62\) 0 0
\(63\) 0.618034i 0.0778650i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.18034i 1.12156i 0.827966 + 0.560779i \(0.189498\pi\)
−0.827966 + 0.560779i \(0.810502\pi\)
\(68\) 0 0
\(69\) 0.618034 0.0744025
\(70\) 0 0
\(71\) −5.09017 −0.604092 −0.302046 0.953293i \(-0.597670\pi\)
−0.302046 + 0.953293i \(0.597670\pi\)
\(72\) 0 0
\(73\) − 4.38197i − 0.512870i −0.966561 0.256435i \(-0.917452\pi\)
0.966561 0.256435i \(-0.0825480\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.85410i 0.667137i
\(78\) 0 0
\(79\) −13.0000 −1.46261 −0.731307 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) −7.70820 −0.856467
\(82\) 0 0
\(83\) 4.94427i 0.542704i 0.962480 + 0.271352i \(0.0874708\pi\)
−0.962480 + 0.271352i \(0.912529\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.23607i 0.990210i
\(88\) 0 0
\(89\) −5.38197 −0.570487 −0.285244 0.958455i \(-0.592075\pi\)
−0.285244 + 0.958455i \(0.592075\pi\)
\(90\) 0 0
\(91\) 4.85410 0.508848
\(92\) 0 0
\(93\) − 13.7082i − 1.42147i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 18.8541i 1.91434i 0.289520 + 0.957172i \(0.406504\pi\)
−0.289520 + 0.957172i \(0.593496\pi\)
\(98\) 0 0
\(99\) 1.38197 0.138893
\(100\) 0 0
\(101\) −10.7082 −1.06551 −0.532753 0.846271i \(-0.678843\pi\)
−0.532753 + 0.846271i \(0.678843\pi\)
\(102\) 0 0
\(103\) 16.0344i 1.57992i 0.613158 + 0.789960i \(0.289899\pi\)
−0.613158 + 0.789960i \(0.710101\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.5623i 1.31112i 0.755144 + 0.655559i \(0.227567\pi\)
−0.755144 + 0.655559i \(0.772433\pi\)
\(108\) 0 0
\(109\) −4.56231 −0.436990 −0.218495 0.975838i \(-0.570115\pi\)
−0.218495 + 0.975838i \(0.570115\pi\)
\(110\) 0 0
\(111\) 2.38197 0.226086
\(112\) 0 0
\(113\) − 18.2361i − 1.71550i −0.514063 0.857752i \(-0.671860\pi\)
0.514063 0.857752i \(-0.328140\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 1.14590i − 0.105938i
\(118\) 0 0
\(119\) −8.85410 −0.811654
\(120\) 0 0
\(121\) 2.09017 0.190015
\(122\) 0 0
\(123\) 16.9443i 1.52781i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1.09017i 0.0967369i 0.998830 + 0.0483685i \(0.0154022\pi\)
−0.998830 + 0.0483685i \(0.984598\pi\)
\(128\) 0 0
\(129\) 16.1803 1.42460
\(130\) 0 0
\(131\) 6.18034 0.539979 0.269989 0.962863i \(-0.412980\pi\)
0.269989 + 0.962863i \(0.412980\pi\)
\(132\) 0 0
\(133\) 4.23607i 0.367314i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.76393i 0.834189i 0.908863 + 0.417095i \(0.136952\pi\)
−0.908863 + 0.417095i \(0.863048\pi\)
\(138\) 0 0
\(139\) 0.708204 0.0600691 0.0300345 0.999549i \(-0.490438\pi\)
0.0300345 + 0.999549i \(0.490438\pi\)
\(140\) 0 0
\(141\) 7.00000 0.589506
\(142\) 0 0
\(143\) − 10.8541i − 0.907666i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 7.09017i 0.584787i
\(148\) 0 0
\(149\) −16.0344 −1.31359 −0.656796 0.754068i \(-0.728089\pi\)
−0.656796 + 0.754068i \(0.728089\pi\)
\(150\) 0 0
\(151\) −21.1803 −1.72363 −0.861816 0.507221i \(-0.830673\pi\)
−0.861816 + 0.507221i \(0.830673\pi\)
\(152\) 0 0
\(153\) 2.09017i 0.168980i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 7.47214i − 0.596341i −0.954513 0.298171i \(-0.903624\pi\)
0.954513 0.298171i \(-0.0963765\pi\)
\(158\) 0 0
\(159\) −13.7082 −1.08713
\(160\) 0 0
\(161\) 0.618034 0.0487079
\(162\) 0 0
\(163\) 3.61803i 0.283386i 0.989911 + 0.141693i \(0.0452546\pi\)
−0.989911 + 0.141693i \(0.954745\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 1.29180i − 0.0999622i −0.998750 0.0499811i \(-0.984084\pi\)
0.998750 0.0499811i \(-0.0159161\pi\)
\(168\) 0 0
\(169\) 4.00000 0.307692
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) − 23.2361i − 1.76661i −0.468803 0.883303i \(-0.655315\pi\)
0.468803 0.883303i \(-0.344685\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 9.32624i 0.701003i
\(178\) 0 0
\(179\) −17.4721 −1.30593 −0.652964 0.757389i \(-0.726475\pi\)
−0.652964 + 0.757389i \(0.726475\pi\)
\(180\) 0 0
\(181\) 15.8541 1.17843 0.589213 0.807978i \(-0.299438\pi\)
0.589213 + 0.807978i \(0.299438\pi\)
\(182\) 0 0
\(183\) 19.9443i 1.47432i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 19.7984i 1.44780i
\(188\) 0 0
\(189\) −8.85410 −0.644041
\(190\) 0 0
\(191\) 18.1246 1.31145 0.655725 0.754999i \(-0.272363\pi\)
0.655725 + 0.754999i \(0.272363\pi\)
\(192\) 0 0
\(193\) 6.00000i 0.431889i 0.976406 + 0.215945i \(0.0692831\pi\)
−0.976406 + 0.215945i \(0.930717\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.52786i 0.536338i 0.963372 + 0.268169i \(0.0864186\pi\)
−0.963372 + 0.268169i \(0.913581\pi\)
\(198\) 0 0
\(199\) 6.14590 0.435671 0.217836 0.975985i \(-0.430100\pi\)
0.217836 + 0.975985i \(0.430100\pi\)
\(200\) 0 0
\(201\) −14.8541 −1.04773
\(202\) 0 0
\(203\) 9.23607i 0.648245i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 0.145898i − 0.0101406i
\(208\) 0 0
\(209\) 9.47214 0.655201
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) 0 0
\(213\) − 8.23607i − 0.564326i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 13.7082i − 0.930574i
\(218\) 0 0
\(219\) 7.09017 0.479109
\(220\) 0 0
\(221\) 16.4164 1.10429
\(222\) 0 0
\(223\) − 2.56231i − 0.171585i −0.996313 0.0857923i \(-0.972658\pi\)
0.996313 0.0857923i \(-0.0273422\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 24.6525i − 1.63624i −0.575046 0.818121i \(-0.695016\pi\)
0.575046 0.818121i \(-0.304984\pi\)
\(228\) 0 0
\(229\) 3.76393 0.248728 0.124364 0.992237i \(-0.460311\pi\)
0.124364 + 0.992237i \(0.460311\pi\)
\(230\) 0 0
\(231\) −9.47214 −0.623221
\(232\) 0 0
\(233\) − 14.6180i − 0.957659i −0.877908 0.478830i \(-0.841061\pi\)
0.877908 0.478830i \(-0.158939\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 21.0344i − 1.36633i
\(238\) 0 0
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) 10.3820 0.668761 0.334381 0.942438i \(-0.391473\pi\)
0.334381 + 0.942438i \(0.391473\pi\)
\(242\) 0 0
\(243\) 3.94427i 0.253025i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 7.85410i − 0.499745i
\(248\) 0 0
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) 5.00000 0.315597 0.157799 0.987471i \(-0.449560\pi\)
0.157799 + 0.987471i \(0.449560\pi\)
\(252\) 0 0
\(253\) − 1.38197i − 0.0868835i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 0.180340i − 0.0112493i −0.999984 0.00562465i \(-0.998210\pi\)
0.999984 0.00562465i \(-0.00179039\pi\)
\(258\) 0 0
\(259\) 2.38197 0.148008
\(260\) 0 0
\(261\) 2.18034 0.134960
\(262\) 0 0
\(263\) 16.1459i 0.995599i 0.867292 + 0.497799i \(0.165858\pi\)
−0.867292 + 0.497799i \(0.834142\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 8.70820i − 0.532933i
\(268\) 0 0
\(269\) −28.6525 −1.74697 −0.873486 0.486849i \(-0.838146\pi\)
−0.873486 + 0.486849i \(0.838146\pi\)
\(270\) 0 0
\(271\) −1.18034 −0.0717005 −0.0358503 0.999357i \(-0.511414\pi\)
−0.0358503 + 0.999357i \(0.511414\pi\)
\(272\) 0 0
\(273\) 7.85410i 0.475352i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 20.7082i 1.24424i 0.782924 + 0.622118i \(0.213727\pi\)
−0.782924 + 0.622118i \(0.786273\pi\)
\(278\) 0 0
\(279\) −3.23607 −0.193738
\(280\) 0 0
\(281\) −3.47214 −0.207130 −0.103565 0.994623i \(-0.533025\pi\)
−0.103565 + 0.994623i \(0.533025\pi\)
\(282\) 0 0
\(283\) − 3.76393i − 0.223743i −0.993723 0.111871i \(-0.964316\pi\)
0.993723 0.111871i \(-0.0356844\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 16.9443i 1.00019i
\(288\) 0 0
\(289\) −12.9443 −0.761428
\(290\) 0 0
\(291\) −30.5066 −1.78833
\(292\) 0 0
\(293\) − 14.2705i − 0.833692i −0.908977 0.416846i \(-0.863135\pi\)
0.908977 0.416846i \(-0.136865\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 19.7984i 1.14882i
\(298\) 0 0
\(299\) −1.14590 −0.0662690
\(300\) 0 0
\(301\) 16.1803 0.932619
\(302\) 0 0
\(303\) − 17.3262i − 0.995366i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 10.0344i 0.572696i 0.958126 + 0.286348i \(0.0924413\pi\)
−0.958126 + 0.286348i \(0.907559\pi\)
\(308\) 0 0
\(309\) −25.9443 −1.47592
\(310\) 0 0
\(311\) −4.50658 −0.255545 −0.127772 0.991804i \(-0.540783\pi\)
−0.127772 + 0.991804i \(0.540783\pi\)
\(312\) 0 0
\(313\) − 6.14590i − 0.347387i −0.984800 0.173693i \(-0.944430\pi\)
0.984800 0.173693i \(-0.0555702\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 29.6180i 1.66351i 0.555139 + 0.831757i \(0.312665\pi\)
−0.555139 + 0.831757i \(0.687335\pi\)
\(318\) 0 0
\(319\) 20.6525 1.15632
\(320\) 0 0
\(321\) −21.9443 −1.22481
\(322\) 0 0
\(323\) 14.3262i 0.797133i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 7.38197i − 0.408224i
\(328\) 0 0
\(329\) 7.00000 0.385922
\(330\) 0 0
\(331\) 3.00000 0.164895 0.0824475 0.996595i \(-0.473726\pi\)
0.0824475 + 0.996595i \(0.473726\pi\)
\(332\) 0 0
\(333\) − 0.562306i − 0.0308142i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 30.9443i − 1.68564i −0.538194 0.842821i \(-0.680893\pi\)
0.538194 0.842821i \(-0.319107\pi\)
\(338\) 0 0
\(339\) 29.5066 1.60258
\(340\) 0 0
\(341\) −30.6525 −1.65992
\(342\) 0 0
\(343\) 18.4164i 0.994393i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 29.4721i − 1.58215i −0.611721 0.791074i \(-0.709522\pi\)
0.611721 0.791074i \(-0.290478\pi\)
\(348\) 0 0
\(349\) 24.5967 1.31663 0.658317 0.752741i \(-0.271269\pi\)
0.658317 + 0.752741i \(0.271269\pi\)
\(350\) 0 0
\(351\) 16.4164 0.876243
\(352\) 0 0
\(353\) − 21.5967i − 1.14948i −0.818336 0.574739i \(-0.805103\pi\)
0.818336 0.574739i \(-0.194897\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 14.3262i − 0.758225i
\(358\) 0 0
\(359\) 33.3607 1.76071 0.880355 0.474316i \(-0.157305\pi\)
0.880355 + 0.474316i \(0.157305\pi\)
\(360\) 0 0
\(361\) −12.1459 −0.639258
\(362\) 0 0
\(363\) 3.38197i 0.177507i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 12.4721i − 0.651040i −0.945535 0.325520i \(-0.894461\pi\)
0.945535 0.325520i \(-0.105539\pi\)
\(368\) 0 0
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) −13.7082 −0.711694
\(372\) 0 0
\(373\) 9.09017i 0.470671i 0.971914 + 0.235336i \(0.0756189\pi\)
−0.971914 + 0.235336i \(0.924381\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 17.1246i − 0.881962i
\(378\) 0 0
\(379\) −8.18034 −0.420196 −0.210098 0.977680i \(-0.567378\pi\)
−0.210098 + 0.977680i \(0.567378\pi\)
\(380\) 0 0
\(381\) −1.76393 −0.0903690
\(382\) 0 0
\(383\) − 21.0344i − 1.07481i −0.843324 0.537405i \(-0.819405\pi\)
0.843324 0.537405i \(-0.180595\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 3.81966i − 0.194164i
\(388\) 0 0
\(389\) −11.0344 −0.559468 −0.279734 0.960077i \(-0.590246\pi\)
−0.279734 + 0.960077i \(0.590246\pi\)
\(390\) 0 0
\(391\) 2.09017 0.105704
\(392\) 0 0
\(393\) 10.0000i 0.504433i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 26.4721i − 1.32860i −0.747467 0.664299i \(-0.768730\pi\)
0.747467 0.664299i \(-0.231270\pi\)
\(398\) 0 0
\(399\) −6.85410 −0.343134
\(400\) 0 0
\(401\) 31.3820 1.56714 0.783570 0.621303i \(-0.213396\pi\)
0.783570 + 0.621303i \(0.213396\pi\)
\(402\) 0 0
\(403\) 25.4164i 1.26608i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 5.32624i − 0.264012i
\(408\) 0 0
\(409\) 23.5967 1.16678 0.583392 0.812191i \(-0.301725\pi\)
0.583392 + 0.812191i \(0.301725\pi\)
\(410\) 0 0
\(411\) −15.7984 −0.779276
\(412\) 0 0
\(413\) 9.32624i 0.458914i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.14590i 0.0561149i
\(418\) 0 0
\(419\) 10.0557 0.491254 0.245627 0.969364i \(-0.421006\pi\)
0.245627 + 0.969364i \(0.421006\pi\)
\(420\) 0 0
\(421\) 21.6525 1.05528 0.527639 0.849469i \(-0.323078\pi\)
0.527639 + 0.849469i \(0.323078\pi\)
\(422\) 0 0
\(423\) − 1.65248i − 0.0803461i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 19.9443i 0.965171i
\(428\) 0 0
\(429\) 17.5623 0.847916
\(430\) 0 0
\(431\) −0.583592 −0.0281106 −0.0140553 0.999901i \(-0.504474\pi\)
−0.0140553 + 0.999901i \(0.504474\pi\)
\(432\) 0 0
\(433\) − 11.2148i − 0.538948i −0.963008 0.269474i \(-0.913150\pi\)
0.963008 0.269474i \(-0.0868498\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 1.00000i − 0.0478365i
\(438\) 0 0
\(439\) 12.2016 0.582352 0.291176 0.956670i \(-0.405954\pi\)
0.291176 + 0.956670i \(0.405954\pi\)
\(440\) 0 0
\(441\) 1.67376 0.0797030
\(442\) 0 0
\(443\) 26.6525i 1.26630i 0.774030 + 0.633149i \(0.218238\pi\)
−0.774030 + 0.633149i \(0.781762\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 25.9443i − 1.22712i
\(448\) 0 0
\(449\) 29.6525 1.39939 0.699693 0.714443i \(-0.253320\pi\)
0.699693 + 0.714443i \(0.253320\pi\)
\(450\) 0 0
\(451\) 37.8885 1.78410
\(452\) 0 0
\(453\) − 34.2705i − 1.61017i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 19.9443i − 0.932954i −0.884533 0.466477i \(-0.845523\pi\)
0.884533 0.466477i \(-0.154477\pi\)
\(458\) 0 0
\(459\) −29.9443 −1.39768
\(460\) 0 0
\(461\) 10.8541 0.505526 0.252763 0.967528i \(-0.418661\pi\)
0.252763 + 0.967528i \(0.418661\pi\)
\(462\) 0 0
\(463\) 36.5623i 1.69919i 0.527432 + 0.849597i \(0.323155\pi\)
−0.527432 + 0.849597i \(0.676845\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.4377i 0.575548i 0.957698 + 0.287774i \(0.0929152\pi\)
−0.957698 + 0.287774i \(0.907085\pi\)
\(468\) 0 0
\(469\) −14.8541 −0.685899
\(470\) 0 0
\(471\) 12.0902 0.557086
\(472\) 0 0
\(473\) − 36.1803i − 1.66357i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.23607i 0.148169i
\(478\) 0 0
\(479\) −1.12461 −0.0513848 −0.0256924 0.999670i \(-0.508179\pi\)
−0.0256924 + 0.999670i \(0.508179\pi\)
\(480\) 0 0
\(481\) −4.41641 −0.201371
\(482\) 0 0
\(483\) 1.00000i 0.0455016i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 42.7082i − 1.93529i −0.252309 0.967647i \(-0.581190\pi\)
0.252309 0.967647i \(-0.418810\pi\)
\(488\) 0 0
\(489\) −5.85410 −0.264732
\(490\) 0 0
\(491\) −6.58359 −0.297113 −0.148557 0.988904i \(-0.547463\pi\)
−0.148557 + 0.988904i \(0.547463\pi\)
\(492\) 0 0
\(493\) 31.2361i 1.40680i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 8.23607i − 0.369438i
\(498\) 0 0
\(499\) −23.7639 −1.06382 −0.531910 0.846801i \(-0.678525\pi\)
−0.531910 + 0.846801i \(0.678525\pi\)
\(500\) 0 0
\(501\) 2.09017 0.0933819
\(502\) 0 0
\(503\) − 40.4853i − 1.80515i −0.430533 0.902575i \(-0.641674\pi\)
0.430533 0.902575i \(-0.358326\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6.47214i 0.287438i
\(508\) 0 0
\(509\) 28.2705 1.25307 0.626534 0.779394i \(-0.284473\pi\)
0.626534 + 0.779394i \(0.284473\pi\)
\(510\) 0 0
\(511\) 7.09017 0.313651
\(512\) 0 0
\(513\) 14.3262i 0.632519i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 15.6525i − 0.688395i
\(518\) 0 0
\(519\) 37.5967 1.65031
\(520\) 0 0
\(521\) −9.94427 −0.435666 −0.217833 0.975986i \(-0.569899\pi\)
−0.217833 + 0.975986i \(0.569899\pi\)
\(522\) 0 0
\(523\) − 2.18034i − 0.0953396i −0.998863 0.0476698i \(-0.984820\pi\)
0.998863 0.0476698i \(-0.0151795\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 46.3607i − 2.01950i
\(528\) 0 0
\(529\) 22.8541 0.993657
\(530\) 0 0
\(531\) 2.20163 0.0955424
\(532\) 0 0
\(533\) − 31.4164i − 1.36080i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 28.2705i − 1.21996i
\(538\) 0 0
\(539\) 15.8541 0.682885
\(540\) 0 0
\(541\) −30.0689 −1.29276 −0.646381 0.763015i \(-0.723718\pi\)
−0.646381 + 0.763015i \(0.723718\pi\)
\(542\) 0 0
\(543\) 25.6525i 1.10085i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.36068i 0.0581785i 0.999577 + 0.0290892i \(0.00926070\pi\)
−0.999577 + 0.0290892i \(0.990739\pi\)
\(548\) 0 0
\(549\) 4.70820 0.200941
\(550\) 0 0
\(551\) 14.9443 0.636647
\(552\) 0 0
\(553\) − 21.0344i − 0.894475i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 35.4508i − 1.50210i −0.660245 0.751050i \(-0.729548\pi\)
0.660245 0.751050i \(-0.270452\pi\)
\(558\) 0 0
\(559\) −30.0000 −1.26886
\(560\) 0 0
\(561\) −32.0344 −1.35250
\(562\) 0 0
\(563\) − 46.0132i − 1.93922i −0.244649 0.969612i \(-0.578673\pi\)
0.244649 0.969612i \(-0.421327\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 12.4721i − 0.523780i
\(568\) 0 0
\(569\) −17.7639 −0.744703 −0.372351 0.928092i \(-0.621448\pi\)
−0.372351 + 0.928092i \(0.621448\pi\)
\(570\) 0 0
\(571\) −5.94427 −0.248760 −0.124380 0.992235i \(-0.539694\pi\)
−0.124380 + 0.992235i \(0.539694\pi\)
\(572\) 0 0
\(573\) 29.3262i 1.22512i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 18.1246i − 0.754537i −0.926104 0.377269i \(-0.876863\pi\)
0.926104 0.377269i \(-0.123137\pi\)
\(578\) 0 0
\(579\) −9.70820 −0.403459
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) 30.6525i 1.26950i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 30.7984i 1.27118i 0.772025 + 0.635592i \(0.219244\pi\)
−0.772025 + 0.635592i \(0.780756\pi\)
\(588\) 0 0
\(589\) −22.1803 −0.913925
\(590\) 0 0
\(591\) −12.1803 −0.501032
\(592\) 0 0
\(593\) − 23.8328i − 0.978696i −0.872088 0.489348i \(-0.837235\pi\)
0.872088 0.489348i \(-0.162765\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 9.94427i 0.406992i
\(598\) 0 0
\(599\) −46.1803 −1.88688 −0.943439 0.331547i \(-0.892430\pi\)
−0.943439 + 0.331547i \(0.892430\pi\)
\(600\) 0 0
\(601\) 29.5967 1.20728 0.603638 0.797258i \(-0.293717\pi\)
0.603638 + 0.797258i \(0.293717\pi\)
\(602\) 0 0
\(603\) 3.50658i 0.142799i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 26.4721i 1.07447i 0.843432 + 0.537235i \(0.180531\pi\)
−0.843432 + 0.537235i \(0.819469\pi\)
\(608\) 0 0
\(609\) −14.9443 −0.605572
\(610\) 0 0
\(611\) −12.9787 −0.525063
\(612\) 0 0
\(613\) 19.2361i 0.776937i 0.921462 + 0.388469i \(0.126996\pi\)
−0.921462 + 0.388469i \(0.873004\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 30.6312i − 1.23317i −0.787290 0.616583i \(-0.788517\pi\)
0.787290 0.616583i \(-0.211483\pi\)
\(618\) 0 0
\(619\) −10.1246 −0.406943 −0.203471 0.979081i \(-0.565222\pi\)
−0.203471 + 0.979081i \(0.565222\pi\)
\(620\) 0 0
\(621\) 2.09017 0.0838756
\(622\) 0 0
\(623\) − 8.70820i − 0.348887i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 15.3262i 0.612071i
\(628\) 0 0
\(629\) 8.05573 0.321203
\(630\) 0 0
\(631\) 23.1803 0.922795 0.461397 0.887194i \(-0.347348\pi\)
0.461397 + 0.887194i \(0.347348\pi\)
\(632\) 0 0
\(633\) − 1.61803i − 0.0643111i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 13.1459i − 0.520859i
\(638\) 0 0
\(639\) −1.94427 −0.0769142
\(640\) 0 0
\(641\) −28.6525 −1.13170 −0.565852 0.824507i \(-0.691453\pi\)
−0.565852 + 0.824507i \(0.691453\pi\)
\(642\) 0 0
\(643\) 40.3050i 1.58947i 0.606955 + 0.794736i \(0.292391\pi\)
−0.606955 + 0.794736i \(0.707609\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 43.8328i − 1.72325i −0.507549 0.861623i \(-0.669448\pi\)
0.507549 0.861623i \(-0.330552\pi\)
\(648\) 0 0
\(649\) 20.8541 0.818595
\(650\) 0 0
\(651\) 22.1803 0.869316
\(652\) 0 0
\(653\) − 0.0344419i − 0.00134781i −1.00000 0.000673907i \(-0.999785\pi\)
1.00000 0.000673907i \(-0.000214511\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 1.67376i − 0.0652997i
\(658\) 0 0
\(659\) 23.2705 0.906490 0.453245 0.891386i \(-0.350266\pi\)
0.453245 + 0.891386i \(0.350266\pi\)
\(660\) 0 0
\(661\) −15.8197 −0.615313 −0.307657 0.951497i \(-0.599545\pi\)
−0.307657 + 0.951497i \(0.599545\pi\)
\(662\) 0 0
\(663\) 26.5623i 1.03159i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 2.18034i − 0.0844231i
\(668\) 0 0
\(669\) 4.14590 0.160290
\(670\) 0 0
\(671\) 44.5967 1.72164
\(672\) 0 0
\(673\) 14.5279i 0.560008i 0.959999 + 0.280004i \(0.0903358\pi\)
−0.959999 + 0.280004i \(0.909664\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 7.72949i 0.297068i 0.988907 + 0.148534i \(0.0474555\pi\)
−0.988907 + 0.148534i \(0.952544\pi\)
\(678\) 0 0
\(679\) −30.5066 −1.17074
\(680\) 0 0
\(681\) 39.8885 1.52853
\(682\) 0 0
\(683\) − 46.7984i − 1.79069i −0.445373 0.895345i \(-0.646929\pi\)
0.445373 0.895345i \(-0.353071\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 6.09017i 0.232354i
\(688\) 0 0
\(689\) 25.4164 0.968288
\(690\) 0 0
\(691\) −18.0557 −0.686872 −0.343436 0.939176i \(-0.611591\pi\)
−0.343436 + 0.939176i \(0.611591\pi\)
\(692\) 0 0
\(693\) 2.23607i 0.0849412i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 57.3050i 2.17058i
\(698\) 0 0
\(699\) 23.6525 0.894619
\(700\) 0 0
\(701\) −27.3607 −1.03340 −0.516699 0.856167i \(-0.672839\pi\)
−0.516699 + 0.856167i \(0.672839\pi\)
\(702\) 0 0
\(703\) − 3.85410i − 0.145360i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 17.3262i − 0.651620i
\(708\) 0 0
\(709\) −13.7639 −0.516915 −0.258458 0.966023i \(-0.583214\pi\)
−0.258458 + 0.966023i \(0.583214\pi\)
\(710\) 0 0
\(711\) −4.96556 −0.186223
\(712\) 0 0
\(713\) 3.23607i 0.121192i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.47214i 0.241706i
\(718\) 0 0
\(719\) −24.6180 −0.918098 −0.459049 0.888411i \(-0.651810\pi\)
−0.459049 + 0.888411i \(0.651810\pi\)
\(720\) 0 0
\(721\) −25.9443 −0.966215
\(722\) 0 0
\(723\) 16.7984i 0.624738i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 18.6738i − 0.692571i −0.938129 0.346286i \(-0.887443\pi\)
0.938129 0.346286i \(-0.112557\pi\)
\(728\) 0 0
\(729\) −29.5066 −1.09284
\(730\) 0 0
\(731\) 54.7214 2.02394
\(732\) 0 0
\(733\) − 47.2492i − 1.74519i −0.488445 0.872595i \(-0.662436\pi\)
0.488445 0.872595i \(-0.337564\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 33.2148i 1.22348i
\(738\) 0 0
\(739\) −27.7639 −1.02131 −0.510656 0.859785i \(-0.670598\pi\)
−0.510656 + 0.859785i \(0.670598\pi\)
\(740\) 0 0
\(741\) 12.7082 0.466848
\(742\) 0 0
\(743\) 15.6738i 0.575015i 0.957778 + 0.287507i \(0.0928266\pi\)
−0.957778 + 0.287507i \(0.907173\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.88854i 0.0690982i
\(748\) 0 0
\(749\) −21.9443 −0.801826
\(750\) 0 0
\(751\) −31.0344 −1.13246 −0.566231 0.824246i \(-0.691599\pi\)
−0.566231 + 0.824246i \(0.691599\pi\)
\(752\) 0 0
\(753\) 8.09017i 0.294822i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.47214i 0.0535057i 0.999642 + 0.0267528i \(0.00851671\pi\)
−0.999642 + 0.0267528i \(0.991483\pi\)
\(758\) 0 0
\(759\) 2.23607 0.0811641
\(760\) 0 0
\(761\) −23.9787 −0.869228 −0.434614 0.900617i \(-0.643115\pi\)
−0.434614 + 0.900617i \(0.643115\pi\)
\(762\) 0 0
\(763\) − 7.38197i − 0.267245i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 17.2918i − 0.624371i
\(768\) 0 0
\(769\) 6.81966 0.245923 0.122962 0.992411i \(-0.460761\pi\)
0.122962 + 0.992411i \(0.460761\pi\)
\(770\) 0 0
\(771\) 0.291796 0.0105088
\(772\) 0 0
\(773\) − 21.5836i − 0.776308i −0.921595 0.388154i \(-0.873113\pi\)
0.921595 0.388154i \(-0.126887\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 3.85410i 0.138265i
\(778\) 0 0
\(779\) 27.4164 0.982295
\(780\) 0 0
\(781\) −18.4164 −0.658991
\(782\) 0 0
\(783\) 31.2361i 1.11629i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 24.1246i 0.859950i 0.902841 + 0.429975i \(0.141478\pi\)
−0.902841 + 0.429975i \(0.858522\pi\)
\(788\) 0 0
\(789\) −26.1246 −0.930061
\(790\) 0 0
\(791\) 29.5066 1.04913
\(792\) 0 0
\(793\) − 36.9787i − 1.31315i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 5.43769i 0.192613i 0.995352 + 0.0963065i \(0.0307029\pi\)
−0.995352 + 0.0963065i \(0.969297\pi\)
\(798\) 0 0
\(799\) 23.6738 0.837517
\(800\) 0 0
\(801\) −2.05573 −0.0726356
\(802\) 0 0
\(803\) − 15.8541i − 0.559479i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 46.3607i − 1.63197i
\(808\) 0 0
\(809\) 38.1591 1.34160 0.670800 0.741638i \(-0.265951\pi\)
0.670800 + 0.741638i \(0.265951\pi\)
\(810\) 0 0
\(811\) −18.8885 −0.663266 −0.331633 0.943408i \(-0.607600\pi\)
−0.331633 + 0.943408i \(0.607600\pi\)
\(812\) 0 0
\(813\) − 1.90983i − 0.0669807i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 26.1803i − 0.915934i
\(818\) 0 0
\(819\) 1.85410 0.0647876
\(820\) 0 0
\(821\) 10.4721 0.365480 0.182740 0.983161i \(-0.441503\pi\)
0.182740 + 0.983161i \(0.441503\pi\)
\(822\) 0 0
\(823\) − 10.4164i − 0.363093i −0.983382 0.181547i \(-0.941890\pi\)
0.983382 0.181547i \(-0.0581103\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 24.0689i − 0.836957i −0.908227 0.418479i \(-0.862564\pi\)
0.908227 0.418479i \(-0.137436\pi\)
\(828\) 0 0
\(829\) −47.1246 −1.63671 −0.818353 0.574716i \(-0.805112\pi\)
−0.818353 + 0.574716i \(0.805112\pi\)
\(830\) 0 0
\(831\) −33.5066 −1.16233
\(832\) 0 0
\(833\) 23.9787i 0.830813i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 46.3607i − 1.60246i
\(838\) 0 0
\(839\) 6.09017 0.210256 0.105128 0.994459i \(-0.466475\pi\)
0.105128 + 0.994459i \(0.466475\pi\)
\(840\) 0 0
\(841\) 3.58359 0.123572
\(842\) 0 0
\(843\) − 5.61803i − 0.193495i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3.38197i 0.116206i
\(848\) 0 0
\(849\) 6.09017 0.209014
\(850\) 0 0
\(851\) −0.562306 −0.0192756
\(852\) 0 0
\(853\) − 16.9098i − 0.578982i −0.957181 0.289491i \(-0.906514\pi\)
0.957181 0.289491i \(-0.0934860\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 23.7082i 0.809857i 0.914348 + 0.404928i \(0.132704\pi\)
−0.914348 + 0.404928i \(0.867296\pi\)
\(858\) 0 0
\(859\) −55.7214 −1.90119 −0.950594 0.310436i \(-0.899525\pi\)
−0.950594 + 0.310436i \(0.899525\pi\)
\(860\) 0 0
\(861\) −27.4164 −0.934349
\(862\) 0 0
\(863\) 9.29180i 0.316296i 0.987415 + 0.158148i \(0.0505524\pi\)
−0.987415 + 0.158148i \(0.949448\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 20.9443i − 0.711305i
\(868\) 0 0
\(869\) −47.0344 −1.59553
\(870\) 0 0
\(871\) 27.5410 0.933192
\(872\) 0 0
\(873\) 7.20163i 0.243738i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7.74265i 0.261451i 0.991419 + 0.130725i \(0.0417306\pi\)
−0.991419 + 0.130725i \(0.958269\pi\)
\(878\) 0 0
\(879\) 23.0902 0.778812
\(880\) 0 0
\(881\) −4.25735 −0.143434 −0.0717170 0.997425i \(-0.522848\pi\)
−0.0717170 + 0.997425i \(0.522848\pi\)
\(882\) 0 0
\(883\) 39.6525i 1.33441i 0.744873 + 0.667206i \(0.232510\pi\)
−0.744873 + 0.667206i \(0.767490\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 24.7639i 0.831491i 0.909481 + 0.415746i \(0.136479\pi\)
−0.909481 + 0.415746i \(0.863521\pi\)
\(888\) 0 0
\(889\) −1.76393 −0.0591604
\(890\) 0 0
\(891\) −27.8885 −0.934301
\(892\) 0 0
\(893\) − 11.3262i − 0.379018i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 1.85410i − 0.0619067i
\(898\) 0 0
\(899\) −48.3607 −1.61292
\(900\) 0 0
\(901\) −46.3607 −1.54450
\(902\) 0 0
\(903\) 26.1803i 0.871227i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 3.56231i 0.118284i 0.998250 + 0.0591422i \(0.0188365\pi\)
−0.998250 + 0.0591422i \(0.981163\pi\)
\(908\) 0 0
\(909\) −4.09017 −0.135662
\(910\) 0 0
\(911\) 43.1803 1.43063 0.715314 0.698803i \(-0.246284\pi\)
0.715314 + 0.698803i \(0.246284\pi\)
\(912\) 0 0
\(913\) 17.8885i 0.592024i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 10.0000i 0.330229i
\(918\) 0 0
\(919\) 34.0132 1.12199 0.560995 0.827819i \(-0.310419\pi\)
0.560995 + 0.827819i \(0.310419\pi\)
\(920\) 0 0
\(921\) −16.2361 −0.534997
\(922\) 0 0
\(923\) 15.2705i 0.502635i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 6.12461i 0.201159i
\(928\) 0 0
\(929\) 40.0132 1.31279 0.656395 0.754418i \(-0.272081\pi\)
0.656395 + 0.754418i \(0.272081\pi\)
\(930\) 0 0
\(931\) 11.4721 0.375984
\(932\) 0 0
\(933\) − 7.29180i − 0.238723i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 12.5967i − 0.411518i −0.978603 0.205759i \(-0.934034\pi\)
0.978603 0.205759i \(-0.0659663\pi\)
\(938\) 0 0
\(939\) 9.94427 0.324519
\(940\) 0 0
\(941\) 7.65248 0.249464 0.124732 0.992190i \(-0.460193\pi\)
0.124732 + 0.992190i \(0.460193\pi\)
\(942\) 0 0
\(943\) − 4.00000i − 0.130258i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 18.7082i 0.607935i 0.952682 + 0.303967i \(0.0983114\pi\)
−0.952682 + 0.303967i \(0.901689\pi\)
\(948\) 0 0
\(949\) −13.1459 −0.426734
\(950\) 0 0
\(951\) −47.9230 −1.55401
\(952\) 0 0
\(953\) − 8.56231i − 0.277360i −0.990337 0.138680i \(-0.955714\pi\)
0.990337 0.138680i \(-0.0442860\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 33.4164i 1.08020i
\(958\) 0 0
\(959\) −15.7984 −0.510156
\(960\) 0 0
\(961\) 40.7771 1.31539
\(962\) 0 0
\(963\) 5.18034i 0.166934i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 19.3607i 0.622597i 0.950312 + 0.311299i \(0.100764\pi\)
−0.950312 + 0.311299i \(0.899236\pi\)
\(968\) 0 0
\(969\) −23.1803 −0.744660
\(970\) 0 0
\(971\) −14.4508 −0.463750 −0.231875 0.972746i \(-0.574486\pi\)
−0.231875 + 0.972746i \(0.574486\pi\)
\(972\) 0 0
\(973\) 1.14590i 0.0367358i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 39.1803i − 1.25349i −0.779224 0.626745i \(-0.784387\pi\)
0.779224 0.626745i \(-0.215613\pi\)
\(978\) 0 0
\(979\) −19.4721 −0.622332
\(980\) 0 0
\(981\) −1.74265 −0.0556384
\(982\) 0 0
\(983\) 9.88854i 0.315396i 0.987487 + 0.157698i \(0.0504072\pi\)
−0.987487 + 0.157698i \(0.949593\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 11.3262i 0.360518i
\(988\) 0 0
\(989\) −3.81966 −0.121458
\(990\) 0 0
\(991\) 51.1246 1.62403 0.812013 0.583639i \(-0.198372\pi\)
0.812013 + 0.583639i \(0.198372\pi\)
\(992\) 0 0
\(993\) 4.85410i 0.154040i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 8.03444i − 0.254453i −0.991874 0.127227i \(-0.959392\pi\)
0.991874 0.127227i \(-0.0406076\pi\)
\(998\) 0 0
\(999\) 8.05573 0.254872
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 2500.2.c.a.1249.4 4
5.2 odd 4 2500.2.a.b.1.2 yes 2
5.3 odd 4 2500.2.a.a.1.1 2
5.4 even 2 inner 2500.2.c.a.1249.1 4
20.3 even 4 10000.2.a.j.1.2 2
20.7 even 4 10000.2.a.e.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2500.2.a.a.1.1 2 5.3 odd 4
2500.2.a.b.1.2 yes 2 5.2 odd 4
2500.2.c.a.1249.1 4 5.4 even 2 inner
2500.2.c.a.1249.4 4 1.1 even 1 trivial
10000.2.a.e.1.1 2 20.7 even 4
10000.2.a.j.1.2 2 20.3 even 4