Properties

Label 2320.2.a.n.1.3
Level $2320$
Weight $2$
Character 2320.1
Self dual yes
Analytic conductor $18.525$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2320,2,Mod(1,2320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2320.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2320.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: \(18.5252932689\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 2320.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.70928 q^{3} +1.00000 q^{5} -3.70928 q^{7} -0.0783777 q^{9} +O(q^{10})\) \(q+1.70928 q^{3} +1.00000 q^{5} -3.70928 q^{7} -0.0783777 q^{9} +0.630898 q^{11} -4.34017 q^{13} +1.70928 q^{15} -1.55252 q^{17} +5.70928 q^{19} -6.34017 q^{21} -6.63090 q^{23} +1.00000 q^{25} -5.26180 q^{27} -1.00000 q^{29} +2.29072 q^{31} +1.07838 q^{33} -3.70928 q^{35} -2.44748 q^{37} -7.41855 q^{39} +5.60197 q^{41} -12.5464 q^{43} -0.0783777 q^{45} -2.29072 q^{47} +6.75872 q^{49} -2.65368 q^{51} +0.921622 q^{53} +0.630898 q^{55} +9.75872 q^{57} +3.60197 q^{59} -13.0205 q^{61} +0.290725 q^{63} -4.34017 q^{65} -10.6309 q^{67} -11.3340 q^{69} -15.6020 q^{71} -10.9444 q^{73} +1.70928 q^{75} -2.34017 q^{77} +10.2062 q^{79} -8.75872 q^{81} +3.12783 q^{83} -1.55252 q^{85} -1.70928 q^{87} +1.41855 q^{89} +16.0989 q^{91} +3.91548 q^{93} +5.70928 q^{95} +13.4680 q^{97} -0.0494483 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 2 q^{11} - 2 q^{13} - 2 q^{15} - 4 q^{17} + 10 q^{19} - 8 q^{21} - 16 q^{23} + 3 q^{25} - 8 q^{27} - 3 q^{29} + 14 q^{31} - 4 q^{35} - 8 q^{37} - 8 q^{39} - 2 q^{41} - 2 q^{43} + 3 q^{45} - 14 q^{47} - 5 q^{49} + 16 q^{51} + 6 q^{53} - 2 q^{55} + 4 q^{57} - 8 q^{59} - 6 q^{61} + 8 q^{63} - 2 q^{65} - 28 q^{67} + 12 q^{69} - 28 q^{71} - 16 q^{73} - 2 q^{75} + 4 q^{77} + 6 q^{79} - q^{81} - 12 q^{83} - 4 q^{85} + 2 q^{87} - 10 q^{89} + 12 q^{91} - 20 q^{93} + 10 q^{95} + 8 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.70928 0.986851 0.493425 0.869788i \(-0.335745\pi\)
0.493425 + 0.869788i \(0.335745\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.70928 −1.40197 −0.700987 0.713174i \(-0.747257\pi\)
−0.700987 + 0.713174i \(0.747257\pi\)
\(8\) 0 0
\(9\) −0.0783777 −0.0261259
\(10\) 0 0
\(11\) 0.630898 0.190223 0.0951114 0.995467i \(-0.469679\pi\)
0.0951114 + 0.995467i \(0.469679\pi\)
\(12\) 0 0
\(13\) −4.34017 −1.20375 −0.601874 0.798591i \(-0.705579\pi\)
−0.601874 + 0.798591i \(0.705579\pi\)
\(14\) 0 0
\(15\) 1.70928 0.441333
\(16\) 0 0
\(17\) −1.55252 −0.376541 −0.188271 0.982117i \(-0.560288\pi\)
−0.188271 + 0.982117i \(0.560288\pi\)
\(18\) 0 0
\(19\) 5.70928 1.30980 0.654899 0.755717i \(-0.272711\pi\)
0.654899 + 0.755717i \(0.272711\pi\)
\(20\) 0 0
\(21\) −6.34017 −1.38354
\(22\) 0 0
\(23\) −6.63090 −1.38264 −0.691319 0.722550i \(-0.742970\pi\)
−0.691319 + 0.722550i \(0.742970\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.26180 −1.01263
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 2.29072 0.411426 0.205713 0.978612i \(-0.434049\pi\)
0.205713 + 0.978612i \(0.434049\pi\)
\(32\) 0 0
\(33\) 1.07838 0.187721
\(34\) 0 0
\(35\) −3.70928 −0.626982
\(36\) 0 0
\(37\) −2.44748 −0.402363 −0.201182 0.979554i \(-0.564478\pi\)
−0.201182 + 0.979554i \(0.564478\pi\)
\(38\) 0 0
\(39\) −7.41855 −1.18792
\(40\) 0 0
\(41\) 5.60197 0.874880 0.437440 0.899247i \(-0.355885\pi\)
0.437440 + 0.899247i \(0.355885\pi\)
\(42\) 0 0
\(43\) −12.5464 −1.91330 −0.956652 0.291233i \(-0.905935\pi\)
−0.956652 + 0.291233i \(0.905935\pi\)
\(44\) 0 0
\(45\) −0.0783777 −0.0116839
\(46\) 0 0
\(47\) −2.29072 −0.334137 −0.167068 0.985945i \(-0.553430\pi\)
−0.167068 + 0.985945i \(0.553430\pi\)
\(48\) 0 0
\(49\) 6.75872 0.965532
\(50\) 0 0
\(51\) −2.65368 −0.371590
\(52\) 0 0
\(53\) 0.921622 0.126595 0.0632973 0.997995i \(-0.479838\pi\)
0.0632973 + 0.997995i \(0.479838\pi\)
\(54\) 0 0
\(55\) 0.630898 0.0850702
\(56\) 0 0
\(57\) 9.75872 1.29257
\(58\) 0 0
\(59\) 3.60197 0.468936 0.234468 0.972124i \(-0.424665\pi\)
0.234468 + 0.972124i \(0.424665\pi\)
\(60\) 0 0
\(61\) −13.0205 −1.66711 −0.833553 0.552439i \(-0.813697\pi\)
−0.833553 + 0.552439i \(0.813697\pi\)
\(62\) 0 0
\(63\) 0.290725 0.0366279
\(64\) 0 0
\(65\) −4.34017 −0.538332
\(66\) 0 0
\(67\) −10.6309 −1.29877 −0.649385 0.760459i \(-0.724974\pi\)
−0.649385 + 0.760459i \(0.724974\pi\)
\(68\) 0 0
\(69\) −11.3340 −1.36446
\(70\) 0 0
\(71\) −15.6020 −1.85161 −0.925806 0.377998i \(-0.876613\pi\)
−0.925806 + 0.377998i \(0.876613\pi\)
\(72\) 0 0
\(73\) −10.9444 −1.28095 −0.640473 0.767981i \(-0.721262\pi\)
−0.640473 + 0.767981i \(0.721262\pi\)
\(74\) 0 0
\(75\) 1.70928 0.197370
\(76\) 0 0
\(77\) −2.34017 −0.266687
\(78\) 0 0
\(79\) 10.2062 1.14829 0.574144 0.818754i \(-0.305335\pi\)
0.574144 + 0.818754i \(0.305335\pi\)
\(80\) 0 0
\(81\) −8.75872 −0.973192
\(82\) 0 0
\(83\) 3.12783 0.343324 0.171662 0.985156i \(-0.445086\pi\)
0.171662 + 0.985156i \(0.445086\pi\)
\(84\) 0 0
\(85\) −1.55252 −0.168394
\(86\) 0 0
\(87\) −1.70928 −0.183254
\(88\) 0 0
\(89\) 1.41855 0.150366 0.0751830 0.997170i \(-0.476046\pi\)
0.0751830 + 0.997170i \(0.476046\pi\)
\(90\) 0 0
\(91\) 16.0989 1.68762
\(92\) 0 0
\(93\) 3.91548 0.406016
\(94\) 0 0
\(95\) 5.70928 0.585759
\(96\) 0 0
\(97\) 13.4680 1.36747 0.683734 0.729731i \(-0.260355\pi\)
0.683734 + 0.729731i \(0.260355\pi\)
\(98\) 0 0
\(99\) −0.0494483 −0.00496974
\(100\) 0 0
\(101\) 1.10504 0.109956 0.0549778 0.998488i \(-0.482491\pi\)
0.0549778 + 0.998488i \(0.482491\pi\)
\(102\) 0 0
\(103\) −15.6248 −1.53955 −0.769776 0.638314i \(-0.779632\pi\)
−0.769776 + 0.638314i \(0.779632\pi\)
\(104\) 0 0
\(105\) −6.34017 −0.618738
\(106\) 0 0
\(107\) −2.81432 −0.272070 −0.136035 0.990704i \(-0.543436\pi\)
−0.136035 + 0.990704i \(0.543436\pi\)
\(108\) 0 0
\(109\) 5.91548 0.566600 0.283300 0.959031i \(-0.408571\pi\)
0.283300 + 0.959031i \(0.408571\pi\)
\(110\) 0 0
\(111\) −4.18342 −0.397072
\(112\) 0 0
\(113\) −1.95055 −0.183492 −0.0917462 0.995782i \(-0.529245\pi\)
−0.0917462 + 0.995782i \(0.529245\pi\)
\(114\) 0 0
\(115\) −6.63090 −0.618334
\(116\) 0 0
\(117\) 0.340173 0.0314490
\(118\) 0 0
\(119\) 5.75872 0.527901
\(120\) 0 0
\(121\) −10.6020 −0.963815
\(122\) 0 0
\(123\) 9.57531 0.863376
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 22.4885 1.99553 0.997767 0.0667962i \(-0.0212777\pi\)
0.997767 + 0.0667962i \(0.0212777\pi\)
\(128\) 0 0
\(129\) −21.4452 −1.88815
\(130\) 0 0
\(131\) −3.86603 −0.337777 −0.168888 0.985635i \(-0.554018\pi\)
−0.168888 + 0.985635i \(0.554018\pi\)
\(132\) 0 0
\(133\) −21.1773 −1.83630
\(134\) 0 0
\(135\) −5.26180 −0.452863
\(136\) 0 0
\(137\) 21.2846 1.81846 0.909232 0.416289i \(-0.136670\pi\)
0.909232 + 0.416289i \(0.136670\pi\)
\(138\) 0 0
\(139\) −8.09890 −0.686939 −0.343470 0.939164i \(-0.611602\pi\)
−0.343470 + 0.939164i \(0.611602\pi\)
\(140\) 0 0
\(141\) −3.91548 −0.329743
\(142\) 0 0
\(143\) −2.73820 −0.228980
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 0 0
\(147\) 11.5525 0.952836
\(148\) 0 0
\(149\) 12.8371 1.05166 0.525828 0.850591i \(-0.323755\pi\)
0.525828 + 0.850591i \(0.323755\pi\)
\(150\) 0 0
\(151\) 20.5958 1.67606 0.838032 0.545621i \(-0.183706\pi\)
0.838032 + 0.545621i \(0.183706\pi\)
\(152\) 0 0
\(153\) 0.121683 0.00983749
\(154\) 0 0
\(155\) 2.29072 0.183995
\(156\) 0 0
\(157\) −6.04945 −0.482799 −0.241399 0.970426i \(-0.577606\pi\)
−0.241399 + 0.970426i \(0.577606\pi\)
\(158\) 0 0
\(159\) 1.57531 0.124930
\(160\) 0 0
\(161\) 24.5958 1.93842
\(162\) 0 0
\(163\) 15.9649 1.25047 0.625235 0.780437i \(-0.285003\pi\)
0.625235 + 0.780437i \(0.285003\pi\)
\(164\) 0 0
\(165\) 1.07838 0.0839516
\(166\) 0 0
\(167\) 11.3112 0.875290 0.437645 0.899148i \(-0.355813\pi\)
0.437645 + 0.899148i \(0.355813\pi\)
\(168\) 0 0
\(169\) 5.83710 0.449008
\(170\) 0 0
\(171\) −0.447480 −0.0342197
\(172\) 0 0
\(173\) −10.4969 −0.798067 −0.399033 0.916936i \(-0.630654\pi\)
−0.399033 + 0.916936i \(0.630654\pi\)
\(174\) 0 0
\(175\) −3.70928 −0.280395
\(176\) 0 0
\(177\) 6.15676 0.462770
\(178\) 0 0
\(179\) 12.3135 0.920355 0.460178 0.887827i \(-0.347786\pi\)
0.460178 + 0.887827i \(0.347786\pi\)
\(180\) 0 0
\(181\) −1.60197 −0.119073 −0.0595367 0.998226i \(-0.518962\pi\)
−0.0595367 + 0.998226i \(0.518962\pi\)
\(182\) 0 0
\(183\) −22.2557 −1.64519
\(184\) 0 0
\(185\) −2.44748 −0.179942
\(186\) 0 0
\(187\) −0.979481 −0.0716268
\(188\) 0 0
\(189\) 19.5174 1.41969
\(190\) 0 0
\(191\) −13.6248 −0.985853 −0.492926 0.870071i \(-0.664073\pi\)
−0.492926 + 0.870071i \(0.664073\pi\)
\(192\) 0 0
\(193\) −16.9711 −1.22160 −0.610802 0.791783i \(-0.709153\pi\)
−0.610802 + 0.791783i \(0.709153\pi\)
\(194\) 0 0
\(195\) −7.41855 −0.531253
\(196\) 0 0
\(197\) −0.0578588 −0.00412227 −0.00206114 0.999998i \(-0.500656\pi\)
−0.00206114 + 0.999998i \(0.500656\pi\)
\(198\) 0 0
\(199\) −5.39189 −0.382221 −0.191110 0.981569i \(-0.561209\pi\)
−0.191110 + 0.981569i \(0.561209\pi\)
\(200\) 0 0
\(201\) −18.1711 −1.28169
\(202\) 0 0
\(203\) 3.70928 0.260340
\(204\) 0 0
\(205\) 5.60197 0.391258
\(206\) 0 0
\(207\) 0.519715 0.0361227
\(208\) 0 0
\(209\) 3.60197 0.249153
\(210\) 0 0
\(211\) −4.14834 −0.285584 −0.142792 0.989753i \(-0.545608\pi\)
−0.142792 + 0.989753i \(0.545608\pi\)
\(212\) 0 0
\(213\) −26.6681 −1.82727
\(214\) 0 0
\(215\) −12.5464 −0.855656
\(216\) 0 0
\(217\) −8.49693 −0.576809
\(218\) 0 0
\(219\) −18.7070 −1.26410
\(220\) 0 0
\(221\) 6.73820 0.453261
\(222\) 0 0
\(223\) 6.72979 0.450660 0.225330 0.974282i \(-0.427654\pi\)
0.225330 + 0.974282i \(0.427654\pi\)
\(224\) 0 0
\(225\) −0.0783777 −0.00522518
\(226\) 0 0
\(227\) 22.2472 1.47660 0.738301 0.674472i \(-0.235629\pi\)
0.738301 + 0.674472i \(0.235629\pi\)
\(228\) 0 0
\(229\) −7.16290 −0.473338 −0.236669 0.971590i \(-0.576056\pi\)
−0.236669 + 0.971590i \(0.576056\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 30.1978 1.97832 0.989162 0.146831i \(-0.0469073\pi\)
0.989162 + 0.146831i \(0.0469073\pi\)
\(234\) 0 0
\(235\) −2.29072 −0.149430
\(236\) 0 0
\(237\) 17.4452 1.13319
\(238\) 0 0
\(239\) 6.43907 0.416509 0.208254 0.978075i \(-0.433222\pi\)
0.208254 + 0.978075i \(0.433222\pi\)
\(240\) 0 0
\(241\) 10.9939 0.708177 0.354088 0.935212i \(-0.384791\pi\)
0.354088 + 0.935212i \(0.384791\pi\)
\(242\) 0 0
\(243\) 0.814315 0.0522383
\(244\) 0 0
\(245\) 6.75872 0.431799
\(246\) 0 0
\(247\) −24.7792 −1.57667
\(248\) 0 0
\(249\) 5.34632 0.338809
\(250\) 0 0
\(251\) −9.41014 −0.593963 −0.296981 0.954883i \(-0.595980\pi\)
−0.296981 + 0.954883i \(0.595980\pi\)
\(252\) 0 0
\(253\) −4.18342 −0.263009
\(254\) 0 0
\(255\) −2.65368 −0.166180
\(256\) 0 0
\(257\) 5.81658 0.362828 0.181414 0.983407i \(-0.441933\pi\)
0.181414 + 0.983407i \(0.441933\pi\)
\(258\) 0 0
\(259\) 9.07838 0.564103
\(260\) 0 0
\(261\) 0.0783777 0.00485146
\(262\) 0 0
\(263\) 8.91321 0.549612 0.274806 0.961500i \(-0.411386\pi\)
0.274806 + 0.961500i \(0.411386\pi\)
\(264\) 0 0
\(265\) 0.921622 0.0566148
\(266\) 0 0
\(267\) 2.42469 0.148389
\(268\) 0 0
\(269\) −16.4391 −1.00231 −0.501154 0.865358i \(-0.667091\pi\)
−0.501154 + 0.865358i \(0.667091\pi\)
\(270\) 0 0
\(271\) −29.4101 −1.78654 −0.893269 0.449522i \(-0.851594\pi\)
−0.893269 + 0.449522i \(0.851594\pi\)
\(272\) 0 0
\(273\) 27.5174 1.66543
\(274\) 0 0
\(275\) 0.630898 0.0380446
\(276\) 0 0
\(277\) 11.0784 0.665635 0.332818 0.942991i \(-0.392001\pi\)
0.332818 + 0.942991i \(0.392001\pi\)
\(278\) 0 0
\(279\) −0.179542 −0.0107489
\(280\) 0 0
\(281\) −21.1194 −1.25988 −0.629939 0.776644i \(-0.716920\pi\)
−0.629939 + 0.776644i \(0.716920\pi\)
\(282\) 0 0
\(283\) 13.7815 0.819226 0.409613 0.912259i \(-0.365664\pi\)
0.409613 + 0.912259i \(0.365664\pi\)
\(284\) 0 0
\(285\) 9.75872 0.578057
\(286\) 0 0
\(287\) −20.7792 −1.22656
\(288\) 0 0
\(289\) −14.5897 −0.858217
\(290\) 0 0
\(291\) 23.0205 1.34949
\(292\) 0 0
\(293\) 6.14834 0.359190 0.179595 0.983741i \(-0.442521\pi\)
0.179595 + 0.983741i \(0.442521\pi\)
\(294\) 0 0
\(295\) 3.60197 0.209715
\(296\) 0 0
\(297\) −3.31965 −0.192626
\(298\) 0 0
\(299\) 28.7792 1.66435
\(300\) 0 0
\(301\) 46.5380 2.68240
\(302\) 0 0
\(303\) 1.88882 0.108510
\(304\) 0 0
\(305\) −13.0205 −0.745553
\(306\) 0 0
\(307\) −10.3896 −0.592967 −0.296484 0.955038i \(-0.595814\pi\)
−0.296484 + 0.955038i \(0.595814\pi\)
\(308\) 0 0
\(309\) −26.7070 −1.51931
\(310\) 0 0
\(311\) −18.7565 −1.06358 −0.531791 0.846876i \(-0.678481\pi\)
−0.531791 + 0.846876i \(0.678481\pi\)
\(312\) 0 0
\(313\) 12.3402 0.697508 0.348754 0.937214i \(-0.386605\pi\)
0.348754 + 0.937214i \(0.386605\pi\)
\(314\) 0 0
\(315\) 0.290725 0.0163805
\(316\) 0 0
\(317\) −30.6986 −1.72421 −0.862103 0.506734i \(-0.830853\pi\)
−0.862103 + 0.506734i \(0.830853\pi\)
\(318\) 0 0
\(319\) −0.630898 −0.0353235
\(320\) 0 0
\(321\) −4.81044 −0.268493
\(322\) 0 0
\(323\) −8.86376 −0.493193
\(324\) 0 0
\(325\) −4.34017 −0.240749
\(326\) 0 0
\(327\) 10.1112 0.559150
\(328\) 0 0
\(329\) 8.49693 0.468451
\(330\) 0 0
\(331\) 4.08065 0.224293 0.112146 0.993692i \(-0.464227\pi\)
0.112146 + 0.993692i \(0.464227\pi\)
\(332\) 0 0
\(333\) 0.191828 0.0105121
\(334\) 0 0
\(335\) −10.6309 −0.580828
\(336\) 0 0
\(337\) −18.3630 −1.00029 −0.500147 0.865940i \(-0.666721\pi\)
−0.500147 + 0.865940i \(0.666721\pi\)
\(338\) 0 0
\(339\) −3.33403 −0.181080
\(340\) 0 0
\(341\) 1.44521 0.0782627
\(342\) 0 0
\(343\) 0.894960 0.0483233
\(344\) 0 0
\(345\) −11.3340 −0.610204
\(346\) 0 0
\(347\) −8.97107 −0.481592 −0.240796 0.970576i \(-0.577409\pi\)
−0.240796 + 0.970576i \(0.577409\pi\)
\(348\) 0 0
\(349\) 26.1978 1.40234 0.701168 0.712996i \(-0.252662\pi\)
0.701168 + 0.712996i \(0.252662\pi\)
\(350\) 0 0
\(351\) 22.8371 1.21895
\(352\) 0 0
\(353\) −26.2823 −1.39887 −0.699433 0.714698i \(-0.746564\pi\)
−0.699433 + 0.714698i \(0.746564\pi\)
\(354\) 0 0
\(355\) −15.6020 −0.828066
\(356\) 0 0
\(357\) 9.84324 0.520960
\(358\) 0 0
\(359\) −22.8722 −1.20715 −0.603574 0.797307i \(-0.706257\pi\)
−0.603574 + 0.797307i \(0.706257\pi\)
\(360\) 0 0
\(361\) 13.5958 0.715570
\(362\) 0 0
\(363\) −18.1217 −0.951142
\(364\) 0 0
\(365\) −10.9444 −0.572857
\(366\) 0 0
\(367\) 11.0700 0.577848 0.288924 0.957352i \(-0.406703\pi\)
0.288924 + 0.957352i \(0.406703\pi\)
\(368\) 0 0
\(369\) −0.439070 −0.0228571
\(370\) 0 0
\(371\) −3.41855 −0.177482
\(372\) 0 0
\(373\) 11.5753 0.599347 0.299673 0.954042i \(-0.403122\pi\)
0.299673 + 0.954042i \(0.403122\pi\)
\(374\) 0 0
\(375\) 1.70928 0.0882666
\(376\) 0 0
\(377\) 4.34017 0.223530
\(378\) 0 0
\(379\) 9.31124 0.478286 0.239143 0.970984i \(-0.423133\pi\)
0.239143 + 0.970984i \(0.423133\pi\)
\(380\) 0 0
\(381\) 38.4391 1.96929
\(382\) 0 0
\(383\) −33.9649 −1.73553 −0.867763 0.496978i \(-0.834443\pi\)
−0.867763 + 0.496978i \(0.834443\pi\)
\(384\) 0 0
\(385\) −2.34017 −0.119266
\(386\) 0 0
\(387\) 0.983357 0.0499868
\(388\) 0 0
\(389\) −4.12556 −0.209174 −0.104587 0.994516i \(-0.533352\pi\)
−0.104587 + 0.994516i \(0.533352\pi\)
\(390\) 0 0
\(391\) 10.2946 0.520620
\(392\) 0 0
\(393\) −6.60811 −0.333335
\(394\) 0 0
\(395\) 10.2062 0.513530
\(396\) 0 0
\(397\) 17.1050 0.858477 0.429239 0.903191i \(-0.358782\pi\)
0.429239 + 0.903191i \(0.358782\pi\)
\(398\) 0 0
\(399\) −36.1978 −1.81216
\(400\) 0 0
\(401\) −0.554787 −0.0277048 −0.0138524 0.999904i \(-0.504409\pi\)
−0.0138524 + 0.999904i \(0.504409\pi\)
\(402\) 0 0
\(403\) −9.94214 −0.495253
\(404\) 0 0
\(405\) −8.75872 −0.435224
\(406\) 0 0
\(407\) −1.54411 −0.0765387
\(408\) 0 0
\(409\) −20.6537 −1.02126 −0.510629 0.859801i \(-0.670588\pi\)
−0.510629 + 0.859801i \(0.670588\pi\)
\(410\) 0 0
\(411\) 36.3812 1.79455
\(412\) 0 0
\(413\) −13.3607 −0.657437
\(414\) 0 0
\(415\) 3.12783 0.153539
\(416\) 0 0
\(417\) −13.8432 −0.677907
\(418\) 0 0
\(419\) 6.02666 0.294422 0.147211 0.989105i \(-0.452970\pi\)
0.147211 + 0.989105i \(0.452970\pi\)
\(420\) 0 0
\(421\) −12.5380 −0.611063 −0.305532 0.952182i \(-0.598834\pi\)
−0.305532 + 0.952182i \(0.598834\pi\)
\(422\) 0 0
\(423\) 0.179542 0.00872962
\(424\) 0 0
\(425\) −1.55252 −0.0753083
\(426\) 0 0
\(427\) 48.2967 2.33724
\(428\) 0 0
\(429\) −4.68035 −0.225969
\(430\) 0 0
\(431\) 18.0410 0.869006 0.434503 0.900670i \(-0.356924\pi\)
0.434503 + 0.900670i \(0.356924\pi\)
\(432\) 0 0
\(433\) 18.8143 0.904158 0.452079 0.891978i \(-0.350682\pi\)
0.452079 + 0.891978i \(0.350682\pi\)
\(434\) 0 0
\(435\) −1.70928 −0.0819535
\(436\) 0 0
\(437\) −37.8576 −1.81098
\(438\) 0 0
\(439\) −5.54411 −0.264606 −0.132303 0.991209i \(-0.542237\pi\)
−0.132303 + 0.991209i \(0.542237\pi\)
\(440\) 0 0
\(441\) −0.529734 −0.0252254
\(442\) 0 0
\(443\) −17.8082 −0.846092 −0.423046 0.906108i \(-0.639039\pi\)
−0.423046 + 0.906108i \(0.639039\pi\)
\(444\) 0 0
\(445\) 1.41855 0.0672458
\(446\) 0 0
\(447\) 21.9421 1.03783
\(448\) 0 0
\(449\) −10.6947 −0.504715 −0.252358 0.967634i \(-0.581206\pi\)
−0.252358 + 0.967634i \(0.581206\pi\)
\(450\) 0 0
\(451\) 3.53427 0.166422
\(452\) 0 0
\(453\) 35.2039 1.65403
\(454\) 0 0
\(455\) 16.0989 0.754728
\(456\) 0 0
\(457\) −21.7998 −1.01975 −0.509875 0.860249i \(-0.670308\pi\)
−0.509875 + 0.860249i \(0.670308\pi\)
\(458\) 0 0
\(459\) 8.16904 0.381298
\(460\) 0 0
\(461\) 22.4124 1.04385 0.521925 0.852991i \(-0.325214\pi\)
0.521925 + 0.852991i \(0.325214\pi\)
\(462\) 0 0
\(463\) 2.10277 0.0977241 0.0488621 0.998806i \(-0.484441\pi\)
0.0488621 + 0.998806i \(0.484441\pi\)
\(464\) 0 0
\(465\) 3.91548 0.181576
\(466\) 0 0
\(467\) 18.6042 0.860901 0.430451 0.902614i \(-0.358355\pi\)
0.430451 + 0.902614i \(0.358355\pi\)
\(468\) 0 0
\(469\) 39.4329 1.82084
\(470\) 0 0
\(471\) −10.3402 −0.476450
\(472\) 0 0
\(473\) −7.91548 −0.363954
\(474\) 0 0
\(475\) 5.70928 0.261960
\(476\) 0 0
\(477\) −0.0722347 −0.00330740
\(478\) 0 0
\(479\) −8.89884 −0.406598 −0.203299 0.979117i \(-0.565166\pi\)
−0.203299 + 0.979117i \(0.565166\pi\)
\(480\) 0 0
\(481\) 10.6225 0.484344
\(482\) 0 0
\(483\) 42.0410 1.91293
\(484\) 0 0
\(485\) 13.4680 0.611550
\(486\) 0 0
\(487\) 12.9711 0.587775 0.293888 0.955840i \(-0.405051\pi\)
0.293888 + 0.955840i \(0.405051\pi\)
\(488\) 0 0
\(489\) 27.2885 1.23403
\(490\) 0 0
\(491\) 13.8615 0.625561 0.312780 0.949826i \(-0.398740\pi\)
0.312780 + 0.949826i \(0.398740\pi\)
\(492\) 0 0
\(493\) 1.55252 0.0699220
\(494\) 0 0
\(495\) −0.0494483 −0.00222254
\(496\) 0 0
\(497\) 57.8720 2.59591
\(498\) 0 0
\(499\) 22.3545 1.00073 0.500364 0.865815i \(-0.333200\pi\)
0.500364 + 0.865815i \(0.333200\pi\)
\(500\) 0 0
\(501\) 19.3340 0.863781
\(502\) 0 0
\(503\) −34.4885 −1.53777 −0.768884 0.639389i \(-0.779187\pi\)
−0.768884 + 0.639389i \(0.779187\pi\)
\(504\) 0 0
\(505\) 1.10504 0.0491736
\(506\) 0 0
\(507\) 9.97721 0.443104
\(508\) 0 0
\(509\) −28.7526 −1.27444 −0.637218 0.770684i \(-0.719915\pi\)
−0.637218 + 0.770684i \(0.719915\pi\)
\(510\) 0 0
\(511\) 40.5958 1.79585
\(512\) 0 0
\(513\) −30.0410 −1.32634
\(514\) 0 0
\(515\) −15.6248 −0.688509
\(516\) 0 0
\(517\) −1.44521 −0.0635604
\(518\) 0 0
\(519\) −17.9421 −0.787573
\(520\) 0 0
\(521\) −21.6020 −0.946399 −0.473200 0.880955i \(-0.656901\pi\)
−0.473200 + 0.880955i \(0.656901\pi\)
\(522\) 0 0
\(523\) −4.20620 −0.183924 −0.0919622 0.995762i \(-0.529314\pi\)
−0.0919622 + 0.995762i \(0.529314\pi\)
\(524\) 0 0
\(525\) −6.34017 −0.276708
\(526\) 0 0
\(527\) −3.55640 −0.154919
\(528\) 0 0
\(529\) 20.9688 0.911687
\(530\) 0 0
\(531\) −0.282314 −0.0122514
\(532\) 0 0
\(533\) −24.3135 −1.05314
\(534\) 0 0
\(535\) −2.81432 −0.121673
\(536\) 0 0
\(537\) 21.0472 0.908253
\(538\) 0 0
\(539\) 4.26406 0.183666
\(540\) 0 0
\(541\) 26.7792 1.15133 0.575665 0.817686i \(-0.304743\pi\)
0.575665 + 0.817686i \(0.304743\pi\)
\(542\) 0 0
\(543\) −2.73820 −0.117508
\(544\) 0 0
\(545\) 5.91548 0.253391
\(546\) 0 0
\(547\) −2.33176 −0.0996990 −0.0498495 0.998757i \(-0.515874\pi\)
−0.0498495 + 0.998757i \(0.515874\pi\)
\(548\) 0 0
\(549\) 1.02052 0.0435547
\(550\) 0 0
\(551\) −5.70928 −0.243223
\(552\) 0 0
\(553\) −37.8576 −1.60987
\(554\) 0 0
\(555\) −4.18342 −0.177576
\(556\) 0 0
\(557\) 30.8781 1.30835 0.654174 0.756344i \(-0.273016\pi\)
0.654174 + 0.756344i \(0.273016\pi\)
\(558\) 0 0
\(559\) 54.4534 2.30314
\(560\) 0 0
\(561\) −1.67420 −0.0706849
\(562\) 0 0
\(563\) 34.1750 1.44030 0.720152 0.693816i \(-0.244072\pi\)
0.720152 + 0.693816i \(0.244072\pi\)
\(564\) 0 0
\(565\) −1.95055 −0.0820603
\(566\) 0 0
\(567\) 32.4885 1.36439
\(568\) 0 0
\(569\) −30.6947 −1.28679 −0.643395 0.765535i \(-0.722475\pi\)
−0.643395 + 0.765535i \(0.722475\pi\)
\(570\) 0 0
\(571\) 25.7275 1.07666 0.538332 0.842733i \(-0.319055\pi\)
0.538332 + 0.842733i \(0.319055\pi\)
\(572\) 0 0
\(573\) −23.2885 −0.972889
\(574\) 0 0
\(575\) −6.63090 −0.276528
\(576\) 0 0
\(577\) 16.0228 0.667037 0.333519 0.942743i \(-0.391764\pi\)
0.333519 + 0.942743i \(0.391764\pi\)
\(578\) 0 0
\(579\) −29.0082 −1.20554
\(580\) 0 0
\(581\) −11.6020 −0.481331
\(582\) 0 0
\(583\) 0.581449 0.0240812
\(584\) 0 0
\(585\) 0.340173 0.0140644
\(586\) 0 0
\(587\) −19.6248 −0.810000 −0.405000 0.914317i \(-0.632729\pi\)
−0.405000 + 0.914317i \(0.632729\pi\)
\(588\) 0 0
\(589\) 13.0784 0.538885
\(590\) 0 0
\(591\) −0.0988967 −0.00406807
\(592\) 0 0
\(593\) 30.9627 1.27148 0.635742 0.771902i \(-0.280694\pi\)
0.635742 + 0.771902i \(0.280694\pi\)
\(594\) 0 0
\(595\) 5.75872 0.236085
\(596\) 0 0
\(597\) −9.21622 −0.377195
\(598\) 0 0
\(599\) −24.4619 −0.999484 −0.499742 0.866174i \(-0.666572\pi\)
−0.499742 + 0.866174i \(0.666572\pi\)
\(600\) 0 0
\(601\) 16.3857 0.668389 0.334194 0.942504i \(-0.391536\pi\)
0.334194 + 0.942504i \(0.391536\pi\)
\(602\) 0 0
\(603\) 0.833226 0.0339316
\(604\) 0 0
\(605\) −10.6020 −0.431031
\(606\) 0 0
\(607\) −16.6986 −0.677775 −0.338888 0.940827i \(-0.610051\pi\)
−0.338888 + 0.940827i \(0.610051\pi\)
\(608\) 0 0
\(609\) 6.34017 0.256917
\(610\) 0 0
\(611\) 9.94214 0.402216
\(612\) 0 0
\(613\) 5.83096 0.235510 0.117755 0.993043i \(-0.462430\pi\)
0.117755 + 0.993043i \(0.462430\pi\)
\(614\) 0 0
\(615\) 9.57531 0.386114
\(616\) 0 0
\(617\) −11.7237 −0.471976 −0.235988 0.971756i \(-0.575833\pi\)
−0.235988 + 0.971756i \(0.575833\pi\)
\(618\) 0 0
\(619\) 8.41628 0.338279 0.169139 0.985592i \(-0.445901\pi\)
0.169139 + 0.985592i \(0.445901\pi\)
\(620\) 0 0
\(621\) 34.8904 1.40010
\(622\) 0 0
\(623\) −5.26180 −0.210809
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 6.15676 0.245877
\(628\) 0 0
\(629\) 3.79976 0.151506
\(630\) 0 0
\(631\) 12.7792 0.508734 0.254367 0.967108i \(-0.418133\pi\)
0.254367 + 0.967108i \(0.418133\pi\)
\(632\) 0 0
\(633\) −7.09066 −0.281829
\(634\) 0 0
\(635\) 22.4885 0.892430
\(636\) 0 0
\(637\) −29.3340 −1.16226
\(638\) 0 0
\(639\) 1.22285 0.0483751
\(640\) 0 0
\(641\) −0.0722347 −0.00285310 −0.00142655 0.999999i \(-0.500454\pi\)
−0.00142655 + 0.999999i \(0.500454\pi\)
\(642\) 0 0
\(643\) −32.7175 −1.29025 −0.645126 0.764076i \(-0.723195\pi\)
−0.645126 + 0.764076i \(0.723195\pi\)
\(644\) 0 0
\(645\) −21.4452 −0.844404
\(646\) 0 0
\(647\) 15.8082 0.621483 0.310742 0.950494i \(-0.399423\pi\)
0.310742 + 0.950494i \(0.399423\pi\)
\(648\) 0 0
\(649\) 2.27247 0.0892024
\(650\) 0 0
\(651\) −14.5236 −0.569224
\(652\) 0 0
\(653\) −15.3112 −0.599175 −0.299588 0.954069i \(-0.596849\pi\)
−0.299588 + 0.954069i \(0.596849\pi\)
\(654\) 0 0
\(655\) −3.86603 −0.151058
\(656\) 0 0
\(657\) 0.857798 0.0334659
\(658\) 0 0
\(659\) −17.1278 −0.667205 −0.333603 0.942714i \(-0.608264\pi\)
−0.333603 + 0.942714i \(0.608264\pi\)
\(660\) 0 0
\(661\) −26.2290 −1.02019 −0.510095 0.860118i \(-0.670390\pi\)
−0.510095 + 0.860118i \(0.670390\pi\)
\(662\) 0 0
\(663\) 11.5174 0.447301
\(664\) 0 0
\(665\) −21.1773 −0.821219
\(666\) 0 0
\(667\) 6.63090 0.256749
\(668\) 0 0
\(669\) 11.5031 0.444734
\(670\) 0 0
\(671\) −8.21461 −0.317122
\(672\) 0 0
\(673\) 46.4657 1.79112 0.895561 0.444938i \(-0.146774\pi\)
0.895561 + 0.444938i \(0.146774\pi\)
\(674\) 0 0
\(675\) −5.26180 −0.202527
\(676\) 0 0
\(677\) −27.8394 −1.06995 −0.534977 0.844867i \(-0.679680\pi\)
−0.534977 + 0.844867i \(0.679680\pi\)
\(678\) 0 0
\(679\) −49.9565 −1.91716
\(680\) 0 0
\(681\) 38.0267 1.45718
\(682\) 0 0
\(683\) −39.0966 −1.49599 −0.747995 0.663704i \(-0.768984\pi\)
−0.747995 + 0.663704i \(0.768984\pi\)
\(684\) 0 0
\(685\) 21.2846 0.813242
\(686\) 0 0
\(687\) −12.2434 −0.467114
\(688\) 0 0
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) −24.7480 −0.941460 −0.470730 0.882277i \(-0.656009\pi\)
−0.470730 + 0.882277i \(0.656009\pi\)
\(692\) 0 0
\(693\) 0.183417 0.00696745
\(694\) 0 0
\(695\) −8.09890 −0.307209
\(696\) 0 0
\(697\) −8.69717 −0.329429
\(698\) 0 0
\(699\) 51.6163 1.95231
\(700\) 0 0
\(701\) 0.187952 0.00709886 0.00354943 0.999994i \(-0.498870\pi\)
0.00354943 + 0.999994i \(0.498870\pi\)
\(702\) 0 0
\(703\) −13.9733 −0.527014
\(704\) 0 0
\(705\) −3.91548 −0.147465
\(706\) 0 0
\(707\) −4.09890 −0.154155
\(708\) 0 0
\(709\) 13.6020 0.510833 0.255416 0.966831i \(-0.417787\pi\)
0.255416 + 0.966831i \(0.417787\pi\)
\(710\) 0 0
\(711\) −0.799939 −0.0300001
\(712\) 0 0
\(713\) −15.1896 −0.568854
\(714\) 0 0
\(715\) −2.73820 −0.102403
\(716\) 0 0
\(717\) 11.0061 0.411032
\(718\) 0 0
\(719\) 9.27617 0.345943 0.172971 0.984927i \(-0.444663\pi\)
0.172971 + 0.984927i \(0.444663\pi\)
\(720\) 0 0
\(721\) 57.9565 2.15841
\(722\) 0 0
\(723\) 18.7915 0.698864
\(724\) 0 0
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) −29.0121 −1.07600 −0.538000 0.842945i \(-0.680820\pi\)
−0.538000 + 0.842945i \(0.680820\pi\)
\(728\) 0 0
\(729\) 27.6681 1.02474
\(730\) 0 0
\(731\) 19.4785 0.720438
\(732\) 0 0
\(733\) −34.0638 −1.25818 −0.629088 0.777334i \(-0.716572\pi\)
−0.629088 + 0.777334i \(0.716572\pi\)
\(734\) 0 0
\(735\) 11.5525 0.426121
\(736\) 0 0
\(737\) −6.70701 −0.247056
\(738\) 0 0
\(739\) −1.49466 −0.0549820 −0.0274910 0.999622i \(-0.508752\pi\)
−0.0274910 + 0.999622i \(0.508752\pi\)
\(740\) 0 0
\(741\) −42.3545 −1.55593
\(742\) 0 0
\(743\) −17.8082 −0.653318 −0.326659 0.945142i \(-0.605923\pi\)
−0.326659 + 0.945142i \(0.605923\pi\)
\(744\) 0 0
\(745\) 12.8371 0.470315
\(746\) 0 0
\(747\) −0.245152 −0.00896964
\(748\) 0 0
\(749\) 10.4391 0.381435
\(750\) 0 0
\(751\) 23.7503 0.866661 0.433331 0.901235i \(-0.357338\pi\)
0.433331 + 0.901235i \(0.357338\pi\)
\(752\) 0 0
\(753\) −16.0845 −0.586153
\(754\) 0 0
\(755\) 20.5958 0.749559
\(756\) 0 0
\(757\) 26.1939 0.952034 0.476017 0.879436i \(-0.342080\pi\)
0.476017 + 0.879436i \(0.342080\pi\)
\(758\) 0 0
\(759\) −7.15061 −0.259551
\(760\) 0 0
\(761\) −44.7214 −1.62115 −0.810574 0.585636i \(-0.800845\pi\)
−0.810574 + 0.585636i \(0.800845\pi\)
\(762\) 0 0
\(763\) −21.9421 −0.794359
\(764\) 0 0
\(765\) 0.121683 0.00439946
\(766\) 0 0
\(767\) −15.6332 −0.564481
\(768\) 0 0
\(769\) 10.8950 0.392882 0.196441 0.980516i \(-0.437062\pi\)
0.196441 + 0.980516i \(0.437062\pi\)
\(770\) 0 0
\(771\) 9.94214 0.358057
\(772\) 0 0
\(773\) 34.1171 1.22711 0.613554 0.789653i \(-0.289739\pi\)
0.613554 + 0.789653i \(0.289739\pi\)
\(774\) 0 0
\(775\) 2.29072 0.0822853
\(776\) 0 0
\(777\) 15.5174 0.556685
\(778\) 0 0
\(779\) 31.9832 1.14592
\(780\) 0 0
\(781\) −9.84324 −0.352219
\(782\) 0 0
\(783\) 5.26180 0.188041
\(784\) 0 0
\(785\) −6.04945 −0.215914
\(786\) 0 0
\(787\) −39.7548 −1.41711 −0.708554 0.705657i \(-0.750652\pi\)
−0.708554 + 0.705657i \(0.750652\pi\)
\(788\) 0 0
\(789\) 15.2351 0.542385
\(790\) 0 0
\(791\) 7.23513 0.257252
\(792\) 0 0
\(793\) 56.5113 2.00678
\(794\) 0 0
\(795\) 1.57531 0.0558704
\(796\) 0 0
\(797\) 18.7298 0.663443 0.331722 0.943377i \(-0.392371\pi\)
0.331722 + 0.943377i \(0.392371\pi\)
\(798\) 0 0
\(799\) 3.55640 0.125816
\(800\) 0 0
\(801\) −0.111183 −0.00392845
\(802\) 0 0
\(803\) −6.90480 −0.243665
\(804\) 0 0
\(805\) 24.5958 0.866889
\(806\) 0 0
\(807\) −28.0989 −0.989128
\(808\) 0 0
\(809\) −31.9421 −1.12303 −0.561513 0.827468i \(-0.689781\pi\)
−0.561513 + 0.827468i \(0.689781\pi\)
\(810\) 0 0
\(811\) 17.8888 0.628161 0.314081 0.949396i \(-0.398304\pi\)
0.314081 + 0.949396i \(0.398304\pi\)
\(812\) 0 0
\(813\) −50.2700 −1.76305
\(814\) 0 0
\(815\) 15.9649 0.559227
\(816\) 0 0
\(817\) −71.6307 −2.50604
\(818\) 0 0
\(819\) −1.26180 −0.0440907
\(820\) 0 0
\(821\) 30.9939 1.08169 0.540847 0.841121i \(-0.318104\pi\)
0.540847 + 0.841121i \(0.318104\pi\)
\(822\) 0 0
\(823\) −16.5008 −0.575182 −0.287591 0.957753i \(-0.592854\pi\)
−0.287591 + 0.957753i \(0.592854\pi\)
\(824\) 0 0
\(825\) 1.07838 0.0375443
\(826\) 0 0
\(827\) −43.1155 −1.49927 −0.749637 0.661849i \(-0.769772\pi\)
−0.749637 + 0.661849i \(0.769772\pi\)
\(828\) 0 0
\(829\) 22.5958 0.784785 0.392393 0.919798i \(-0.371647\pi\)
0.392393 + 0.919798i \(0.371647\pi\)
\(830\) 0 0
\(831\) 18.9360 0.656882
\(832\) 0 0
\(833\) −10.4931 −0.363563
\(834\) 0 0
\(835\) 11.3112 0.391442
\(836\) 0 0
\(837\) −12.0533 −0.416624
\(838\) 0 0
\(839\) −1.21235 −0.0418549 −0.0209274 0.999781i \(-0.506662\pi\)
−0.0209274 + 0.999781i \(0.506662\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −36.0989 −1.24331
\(844\) 0 0
\(845\) 5.83710 0.200802
\(846\) 0 0
\(847\) 39.3256 1.35124
\(848\) 0 0
\(849\) 23.5564 0.808453
\(850\) 0 0
\(851\) 16.2290 0.556323
\(852\) 0 0
\(853\) 5.93618 0.203251 0.101625 0.994823i \(-0.467596\pi\)
0.101625 + 0.994823i \(0.467596\pi\)
\(854\) 0 0
\(855\) −0.447480 −0.0153035
\(856\) 0 0
\(857\) 14.5503 0.497027 0.248514 0.968628i \(-0.420058\pi\)
0.248514 + 0.968628i \(0.420058\pi\)
\(858\) 0 0
\(859\) 6.32192 0.215701 0.107851 0.994167i \(-0.465603\pi\)
0.107851 + 0.994167i \(0.465603\pi\)
\(860\) 0 0
\(861\) −35.5174 −1.21043
\(862\) 0 0
\(863\) 3.28005 0.111654 0.0558270 0.998440i \(-0.482220\pi\)
0.0558270 + 0.998440i \(0.482220\pi\)
\(864\) 0 0
\(865\) −10.4969 −0.356906
\(866\) 0 0
\(867\) −24.9378 −0.846932
\(868\) 0 0
\(869\) 6.43907 0.218430
\(870\) 0 0
\(871\) 46.1399 1.56339
\(872\) 0 0
\(873\) −1.05559 −0.0357264
\(874\) 0 0
\(875\) −3.70928 −0.125396
\(876\) 0 0
\(877\) 18.2823 0.617350 0.308675 0.951168i \(-0.400114\pi\)
0.308675 + 0.951168i \(0.400114\pi\)
\(878\) 0 0
\(879\) 10.5092 0.354467
\(880\) 0 0
\(881\) 29.7464 1.00218 0.501091 0.865394i \(-0.332932\pi\)
0.501091 + 0.865394i \(0.332932\pi\)
\(882\) 0 0
\(883\) −13.8127 −0.464835 −0.232417 0.972616i \(-0.574664\pi\)
−0.232417 + 0.972616i \(0.574664\pi\)
\(884\) 0 0
\(885\) 6.15676 0.206957
\(886\) 0 0
\(887\) 56.3318 1.89144 0.945718 0.324989i \(-0.105361\pi\)
0.945718 + 0.324989i \(0.105361\pi\)
\(888\) 0 0
\(889\) −83.4161 −2.79769
\(890\) 0 0
\(891\) −5.52586 −0.185123
\(892\) 0 0
\(893\) −13.0784 −0.437651
\(894\) 0 0
\(895\) 12.3135 0.411595
\(896\) 0 0
\(897\) 49.1917 1.64246
\(898\) 0 0
\(899\) −2.29072 −0.0763999
\(900\) 0 0
\(901\) −1.43084 −0.0476681
\(902\) 0 0
\(903\) 79.5462 2.64713
\(904\) 0 0
\(905\) −1.60197 −0.0532512
\(906\) 0 0
\(907\) −21.8082 −0.724128 −0.362064 0.932153i \(-0.617928\pi\)
−0.362064 + 0.932153i \(0.617928\pi\)
\(908\) 0 0
\(909\) −0.0866105 −0.00287269
\(910\) 0 0
\(911\) −4.76099 −0.157739 −0.0788693 0.996885i \(-0.525131\pi\)
−0.0788693 + 0.996885i \(0.525131\pi\)
\(912\) 0 0
\(913\) 1.97334 0.0653080
\(914\) 0 0
\(915\) −22.2557 −0.735749
\(916\) 0 0
\(917\) 14.3402 0.473554
\(918\) 0 0
\(919\) −34.1256 −1.12570 −0.562849 0.826560i \(-0.690295\pi\)
−0.562849 + 0.826560i \(0.690295\pi\)
\(920\) 0 0
\(921\) −17.7587 −0.585170
\(922\) 0 0
\(923\) 67.7152 2.22887
\(924\) 0 0
\(925\) −2.44748 −0.0804727
\(926\) 0 0
\(927\) 1.22463 0.0402222
\(928\) 0 0
\(929\) −12.5769 −0.412635 −0.206318 0.978485i \(-0.566148\pi\)
−0.206318 + 0.978485i \(0.566148\pi\)
\(930\) 0 0
\(931\) 38.5874 1.26465
\(932\) 0 0
\(933\) −32.0599 −1.04960
\(934\) 0 0
\(935\) −0.979481 −0.0320325
\(936\) 0 0
\(937\) −29.7464 −0.971774 −0.485887 0.874022i \(-0.661503\pi\)
−0.485887 + 0.874022i \(0.661503\pi\)
\(938\) 0 0
\(939\) 21.0928 0.688336
\(940\) 0 0
\(941\) −7.47641 −0.243724 −0.121862 0.992547i \(-0.538887\pi\)
−0.121862 + 0.992547i \(0.538887\pi\)
\(942\) 0 0
\(943\) −37.1461 −1.20964
\(944\) 0 0
\(945\) 19.5174 0.634903
\(946\) 0 0
\(947\) 15.2846 0.496682 0.248341 0.968673i \(-0.420115\pi\)
0.248341 + 0.968673i \(0.420115\pi\)
\(948\) 0 0
\(949\) 47.5006 1.54194
\(950\) 0 0
\(951\) −52.4724 −1.70153
\(952\) 0 0
\(953\) −54.0288 −1.75016 −0.875081 0.483976i \(-0.839192\pi\)
−0.875081 + 0.483976i \(0.839192\pi\)
\(954\) 0 0
\(955\) −13.6248 −0.440887
\(956\) 0 0
\(957\) −1.07838 −0.0348590
\(958\) 0 0
\(959\) −78.9504 −2.54944
\(960\) 0 0
\(961\) −25.7526 −0.830728
\(962\) 0 0
\(963\) 0.220580 0.00710808
\(964\) 0 0
\(965\) −16.9711 −0.546318
\(966\) 0 0
\(967\) 15.7671 0.507037 0.253518 0.967331i \(-0.418412\pi\)
0.253518 + 0.967331i \(0.418412\pi\)
\(968\) 0 0
\(969\) −15.1506 −0.486708
\(970\) 0 0
\(971\) −48.1627 −1.54562 −0.772808 0.634640i \(-0.781148\pi\)
−0.772808 + 0.634640i \(0.781148\pi\)
\(972\) 0 0
\(973\) 30.0410 0.963071
\(974\) 0 0
\(975\) −7.41855 −0.237584
\(976\) 0 0
\(977\) −8.28685 −0.265120 −0.132560 0.991175i \(-0.542320\pi\)
−0.132560 + 0.991175i \(0.542320\pi\)
\(978\) 0 0
\(979\) 0.894960 0.0286031
\(980\) 0 0
\(981\) −0.463642 −0.0148029
\(982\) 0 0
\(983\) −22.9177 −0.730963 −0.365481 0.930819i \(-0.619096\pi\)
−0.365481 + 0.930819i \(0.619096\pi\)
\(984\) 0 0
\(985\) −0.0578588 −0.00184354
\(986\) 0 0
\(987\) 14.5236 0.462291
\(988\) 0 0
\(989\) 83.1937 2.64541
\(990\) 0 0
\(991\) −22.3234 −0.709125 −0.354562 0.935032i \(-0.615370\pi\)
−0.354562 + 0.935032i \(0.615370\pi\)
\(992\) 0 0
\(993\) 6.97495 0.221343
\(994\) 0 0
\(995\) −5.39189 −0.170934
\(996\) 0 0
\(997\) 14.3630 0.454879 0.227440 0.973792i \(-0.426965\pi\)
0.227440 + 0.973792i \(0.426965\pi\)
\(998\) 0 0
\(999\) 12.8781 0.407446
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 2320.2.a.n.1.3 3
4.3 odd 2 145.2.a.c.1.3 3
8.3 odd 2 9280.2.a.bj.1.3 3
8.5 even 2 9280.2.a.br.1.1 3
12.11 even 2 1305.2.a.p.1.1 3
20.3 even 4 725.2.b.e.349.1 6
20.7 even 4 725.2.b.e.349.6 6
20.19 odd 2 725.2.a.e.1.1 3
28.27 even 2 7105.2.a.o.1.3 3
60.59 even 2 6525.2.a.be.1.3 3
116.115 odd 2 4205.2.a.f.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.c.1.3 3 4.3 odd 2
725.2.a.e.1.1 3 20.19 odd 2
725.2.b.e.349.1 6 20.3 even 4
725.2.b.e.349.6 6 20.7 even 4
1305.2.a.p.1.1 3 12.11 even 2
2320.2.a.n.1.3 3 1.1 even 1 trivial
4205.2.a.f.1.1 3 116.115 odd 2
6525.2.a.be.1.3 3 60.59 even 2
7105.2.a.o.1.3 3 28.27 even 2
9280.2.a.bj.1.3 3 8.3 odd 2
9280.2.a.br.1.1 3 8.5 even 2