Properties

Label 3312.2.a.bh.1.2
Level $3312$
Weight $2$
Character 3312.1
Self dual yes
Analytic conductor $26.446$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3312,2,Mod(1,3312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3312, 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("3312.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3312.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: \(26.4464531494\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.44688.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 5x^{2} + 6x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1656)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.27733\) of defining polynomial
Character \(\chi\) \(=\) 3312.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.64575 q^{5} +1.01109 q^{7} +5.21151 q^{11} +2.55467 q^{13} +3.56576 q^{17} -3.09108 q^{19} +1.00000 q^{23} -2.29150 q^{25} +1.44533 q^{29} -1.44533 q^{31} -1.66401 q^{35} +6.10217 q^{37} +4.73683 q^{41} -11.3319 q^{43} -0.0221841 q^{47} -5.97769 q^{49} +7.66794 q^{53} -8.57685 q^{55} +3.28535 q^{59} +7.21151 q^{61} -4.20435 q^{65} -11.4919 q^{67} -0.736834 q^{71} +11.6862 q^{73} +5.26932 q^{77} +2.12043 q^{79} +9.21151 q^{83} -5.86836 q^{85} -7.96660 q^{89} +2.58301 q^{91} +5.08715 q^{95} -7.31369 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} - 2 q^{7} - 2 q^{11} + 4 q^{13} + 2 q^{17} - 8 q^{19} + 4 q^{23} + 12 q^{25} + 12 q^{29} - 12 q^{31} + 12 q^{35} + 14 q^{37} + 4 q^{41} - 4 q^{43} + 12 q^{47} + 28 q^{49} + 8 q^{53} - 16 q^{55}+ \cdots + 4 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.64575 −0.736002 −0.368001 0.929825i \(-0.619958\pi\)
−0.368001 + 0.929825i \(0.619958\pi\)
\(6\) 0 0
\(7\) 1.01109 0.382157 0.191078 0.981575i \(-0.438802\pi\)
0.191078 + 0.981575i \(0.438802\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.21151 1.57133 0.785665 0.618652i \(-0.212321\pi\)
0.785665 + 0.618652i \(0.212321\pi\)
\(12\) 0 0
\(13\) 2.55467 0.708538 0.354269 0.935144i \(-0.384730\pi\)
0.354269 + 0.935144i \(0.384730\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.56576 0.864824 0.432412 0.901676i \(-0.357663\pi\)
0.432412 + 0.901676i \(0.357663\pi\)
\(18\) 0 0
\(19\) −3.09108 −0.709143 −0.354571 0.935029i \(-0.615373\pi\)
−0.354571 + 0.935029i \(0.615373\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −2.29150 −0.458301
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.44533 0.268391 0.134196 0.990955i \(-0.457155\pi\)
0.134196 + 0.990955i \(0.457155\pi\)
\(30\) 0 0
\(31\) −1.44533 −0.259589 −0.129795 0.991541i \(-0.541432\pi\)
−0.129795 + 0.991541i \(0.541432\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.66401 −0.281268
\(36\) 0 0
\(37\) 6.10217 1.00319 0.501596 0.865102i \(-0.332747\pi\)
0.501596 + 0.865102i \(0.332747\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.73683 0.739769 0.369885 0.929078i \(-0.379397\pi\)
0.369885 + 0.929078i \(0.379397\pi\)
\(42\) 0 0
\(43\) −11.3319 −1.72810 −0.864052 0.503402i \(-0.832082\pi\)
−0.864052 + 0.503402i \(0.832082\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.0221841 −0.00323589 −0.00161794 0.999999i \(-0.500515\pi\)
−0.00161794 + 0.999999i \(0.500515\pi\)
\(48\) 0 0
\(49\) −5.97769 −0.853956
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.66794 1.05327 0.526636 0.850091i \(-0.323453\pi\)
0.526636 + 0.850091i \(0.323453\pi\)
\(54\) 0 0
\(55\) −8.57685 −1.15650
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.28535 0.427716 0.213858 0.976865i \(-0.431397\pi\)
0.213858 + 0.976865i \(0.431397\pi\)
\(60\) 0 0
\(61\) 7.21151 0.923340 0.461670 0.887052i \(-0.347251\pi\)
0.461670 + 0.887052i \(0.347251\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.20435 −0.521485
\(66\) 0 0
\(67\) −11.4919 −1.40396 −0.701981 0.712196i \(-0.747701\pi\)
−0.701981 + 0.712196i \(0.747701\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.736834 −0.0874461 −0.0437230 0.999044i \(-0.513922\pi\)
−0.0437230 + 0.999044i \(0.513922\pi\)
\(72\) 0 0
\(73\) 11.6862 1.36777 0.683883 0.729592i \(-0.260290\pi\)
0.683883 + 0.729592i \(0.260290\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.26932 0.600495
\(78\) 0 0
\(79\) 2.12043 0.238567 0.119283 0.992860i \(-0.461940\pi\)
0.119283 + 0.992860i \(0.461940\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.21151 1.01109 0.505547 0.862799i \(-0.331291\pi\)
0.505547 + 0.862799i \(0.331291\pi\)
\(84\) 0 0
\(85\) −5.86836 −0.636513
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.96660 −0.844458 −0.422229 0.906489i \(-0.638752\pi\)
−0.422229 + 0.906489i \(0.638752\pi\)
\(90\) 0 0
\(91\) 2.58301 0.270773
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.08715 0.521931
\(96\) 0 0
\(97\) −7.31369 −0.742592 −0.371296 0.928514i \(-0.621087\pi\)
−0.371296 + 0.928514i \(0.621087\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.42315 0.340616 0.170308 0.985391i \(-0.445524\pi\)
0.170308 + 0.985391i \(0.445524\pi\)
\(102\) 0 0
\(103\) 7.43412 0.732505 0.366253 0.930515i \(-0.380641\pi\)
0.366253 + 0.930515i \(0.380641\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.5252 −1.01751 −0.508755 0.860912i \(-0.669894\pi\)
−0.508755 + 0.860912i \(0.669894\pi\)
\(108\) 0 0
\(109\) 17.7945 1.70441 0.852203 0.523212i \(-0.175266\pi\)
0.852203 + 0.523212i \(0.175266\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.835079 0.0785576 0.0392788 0.999228i \(-0.487494\pi\)
0.0392788 + 0.999228i \(0.487494\pi\)
\(114\) 0 0
\(115\) −1.64575 −0.153467
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.60531 0.330498
\(120\) 0 0
\(121\) 16.1599 1.46908
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 8.60519 0.763587 0.381794 0.924248i \(-0.375307\pi\)
0.381794 + 0.924248i \(0.375307\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 13.3137 1.16322 0.581611 0.813467i \(-0.302423\pi\)
0.581611 + 0.813467i \(0.302423\pi\)
\(132\) 0 0
\(133\) −3.12537 −0.271004
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 15.8066 1.35045 0.675225 0.737612i \(-0.264046\pi\)
0.675225 + 0.737612i \(0.264046\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 13.3137 1.11335
\(144\) 0 0
\(145\) −2.37866 −0.197537
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 19.9088 1.63099 0.815496 0.578763i \(-0.196464\pi\)
0.815496 + 0.578763i \(0.196464\pi\)
\(150\) 0 0
\(151\) −10.0505 −0.817900 −0.408950 0.912557i \(-0.634105\pi\)
−0.408950 + 0.912557i \(0.634105\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.37866 0.191058
\(156\) 0 0
\(157\) 4.48083 0.357609 0.178805 0.983885i \(-0.442777\pi\)
0.178805 + 0.983885i \(0.442777\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.01109 0.0796852
\(162\) 0 0
\(163\) 20.1376 1.57729 0.788647 0.614846i \(-0.210782\pi\)
0.788647 + 0.614846i \(0.210782\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −14.9999 −1.16073 −0.580363 0.814358i \(-0.697089\pi\)
−0.580363 + 0.814358i \(0.697089\pi\)
\(168\) 0 0
\(169\) −6.47367 −0.497974
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.29138 −0.478325 −0.239162 0.970980i \(-0.576873\pi\)
−0.239162 + 0.970980i \(0.576873\pi\)
\(174\) 0 0
\(175\) −2.31692 −0.175143
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 16.2914 1.21767 0.608837 0.793295i \(-0.291636\pi\)
0.608837 + 0.793295i \(0.291636\pi\)
\(180\) 0 0
\(181\) 12.5252 0.930991 0.465495 0.885050i \(-0.345876\pi\)
0.465495 + 0.885050i \(0.345876\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −10.0427 −0.738351
\(186\) 0 0
\(187\) 18.5830 1.35892
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.2693 0.960134 0.480067 0.877232i \(-0.340612\pi\)
0.480067 + 0.877232i \(0.340612\pi\)
\(192\) 0 0
\(193\) 7.50402 0.540152 0.270076 0.962839i \(-0.412951\pi\)
0.270076 + 0.962839i \(0.412951\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.75902 −0.196572 −0.0982859 0.995158i \(-0.531336\pi\)
−0.0982859 + 0.995158i \(0.531336\pi\)
\(198\) 0 0
\(199\) −15.2741 −1.08275 −0.541377 0.840780i \(-0.682097\pi\)
−0.541377 + 0.840780i \(0.682097\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.46136 0.102568
\(204\) 0 0
\(205\) −7.79565 −0.544472
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −16.1092 −1.11430
\(210\) 0 0
\(211\) −7.33587 −0.505022 −0.252511 0.967594i \(-0.581256\pi\)
−0.252511 + 0.967594i \(0.581256\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 18.6496 1.27189
\(216\) 0 0
\(217\) −1.46136 −0.0992038
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 9.10934 0.612760
\(222\) 0 0
\(223\) 11.1155 0.744348 0.372174 0.928163i \(-0.378612\pi\)
0.372174 + 0.928163i \(0.378612\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −17.1082 −1.13551 −0.567756 0.823197i \(-0.692188\pi\)
−0.567756 + 0.823197i \(0.692188\pi\)
\(228\) 0 0
\(229\) −13.7945 −0.911567 −0.455784 0.890091i \(-0.650641\pi\)
−0.455784 + 0.890091i \(0.650641\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3.95563 −0.259142 −0.129571 0.991570i \(-0.541360\pi\)
−0.129571 + 0.991570i \(0.541360\pi\)
\(234\) 0 0
\(235\) 0.0365096 0.00238162
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.04437 0.261608 0.130804 0.991408i \(-0.458244\pi\)
0.130804 + 0.991408i \(0.458244\pi\)
\(240\) 0 0
\(241\) −12.3787 −0.797379 −0.398690 0.917086i \(-0.630535\pi\)
−0.398690 + 0.917086i \(0.630535\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 9.83780 0.628514
\(246\) 0 0
\(247\) −7.89669 −0.502454
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.6852 1.17940 0.589699 0.807623i \(-0.299246\pi\)
0.589699 + 0.807623i \(0.299246\pi\)
\(252\) 0 0
\(253\) 5.21151 0.327645
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −20.4169 −1.27357 −0.636785 0.771042i \(-0.719736\pi\)
−0.636785 + 0.771042i \(0.719736\pi\)
\(258\) 0 0
\(259\) 6.16986 0.383376
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −9.10934 −0.561706 −0.280853 0.959751i \(-0.590617\pi\)
−0.280853 + 0.959751i \(0.590617\pi\)
\(264\) 0 0
\(265\) −12.6195 −0.775211
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 19.5607 1.19264 0.596318 0.802748i \(-0.296630\pi\)
0.596318 + 0.802748i \(0.296630\pi\)
\(270\) 0 0
\(271\) −2.02218 −0.122839 −0.0614195 0.998112i \(-0.519563\pi\)
−0.0614195 + 0.998112i \(0.519563\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −11.9422 −0.720141
\(276\) 0 0
\(277\) −24.4230 −1.46744 −0.733719 0.679453i \(-0.762217\pi\)
−0.733719 + 0.679453i \(0.762217\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 13.4403 0.801779 0.400890 0.916126i \(-0.368701\pi\)
0.400890 + 0.916126i \(0.368701\pi\)
\(282\) 0 0
\(283\) 7.13545 0.424159 0.212079 0.977252i \(-0.431976\pi\)
0.212079 + 0.977252i \(0.431976\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.78938 0.282708
\(288\) 0 0
\(289\) −4.28535 −0.252079
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 28.2731 1.65173 0.825867 0.563865i \(-0.190686\pi\)
0.825867 + 0.563865i \(0.190686\pi\)
\(294\) 0 0
\(295\) −5.40687 −0.314800
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.55467 0.147740
\(300\) 0 0
\(301\) −11.4576 −0.660407
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −11.8684 −0.679580
\(306\) 0 0
\(307\) 5.07283 0.289522 0.144761 0.989467i \(-0.453759\pi\)
0.144761 + 0.989467i \(0.453759\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −29.9250 −1.69689 −0.848446 0.529281i \(-0.822462\pi\)
−0.848446 + 0.529281i \(0.822462\pi\)
\(312\) 0 0
\(313\) 7.35806 0.415902 0.207951 0.978139i \(-0.433321\pi\)
0.207951 + 0.978139i \(0.433321\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 31.3563 1.76115 0.880574 0.473909i \(-0.157157\pi\)
0.880574 + 0.473909i \(0.157157\pi\)
\(318\) 0 0
\(319\) 7.53236 0.421731
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −11.0221 −0.613284
\(324\) 0 0
\(325\) −5.85403 −0.324723
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −0.0224302 −0.00123662
\(330\) 0 0
\(331\) 25.8745 1.42219 0.711096 0.703095i \(-0.248199\pi\)
0.711096 + 0.703095i \(0.248199\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 18.9128 1.03332
\(336\) 0 0
\(337\) −29.7811 −1.62228 −0.811139 0.584853i \(-0.801152\pi\)
−0.811139 + 0.584853i \(0.801152\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −7.53236 −0.407900
\(342\) 0 0
\(343\) −13.1216 −0.708502
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.2347 −0.656793 −0.328397 0.944540i \(-0.606508\pi\)
−0.328397 + 0.944540i \(0.606508\pi\)
\(348\) 0 0
\(349\) −21.0221 −1.12529 −0.562643 0.826700i \(-0.690215\pi\)
−0.562643 + 0.826700i \(0.690215\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4.91285 −0.261484 −0.130742 0.991416i \(-0.541736\pi\)
−0.130742 + 0.991416i \(0.541736\pi\)
\(354\) 0 0
\(355\) 1.21265 0.0643605
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8.15998 −0.430667 −0.215334 0.976541i \(-0.569084\pi\)
−0.215334 + 0.976541i \(0.569084\pi\)
\(360\) 0 0
\(361\) −9.44521 −0.497116
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −19.2326 −1.00668
\(366\) 0 0
\(367\) 24.7034 1.28951 0.644754 0.764390i \(-0.276960\pi\)
0.644754 + 0.764390i \(0.276960\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 7.75299 0.402515
\(372\) 0 0
\(373\) 4.16087 0.215442 0.107721 0.994181i \(-0.465645\pi\)
0.107721 + 0.994181i \(0.465645\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.69234 0.190165
\(378\) 0 0
\(379\) −11.1276 −0.571586 −0.285793 0.958291i \(-0.592257\pi\)
−0.285793 + 0.958291i \(0.592257\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2.89066 −0.147706 −0.0738530 0.997269i \(-0.523530\pi\)
−0.0738530 + 0.997269i \(0.523530\pi\)
\(384\) 0 0
\(385\) −8.67199 −0.441965
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −17.7266 −0.898776 −0.449388 0.893337i \(-0.648358\pi\)
−0.449388 + 0.893337i \(0.648358\pi\)
\(390\) 0 0
\(391\) 3.56576 0.180328
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.48970 −0.175586
\(396\) 0 0
\(397\) 7.47367 0.375093 0.187546 0.982256i \(-0.439947\pi\)
0.187546 + 0.982256i \(0.439947\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.56576 0.178066 0.0890328 0.996029i \(-0.471622\pi\)
0.0890328 + 0.996029i \(0.471622\pi\)
\(402\) 0 0
\(403\) −3.69234 −0.183929
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 31.8016 1.57634
\(408\) 0 0
\(409\) 24.7144 1.22205 0.611024 0.791612i \(-0.290758\pi\)
0.611024 + 0.791612i \(0.290758\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 3.32179 0.163455
\(414\) 0 0
\(415\) −15.1599 −0.744168
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −3.15282 −0.154025 −0.0770126 0.997030i \(-0.524538\pi\)
−0.0770126 + 0.997030i \(0.524538\pi\)
\(420\) 0 0
\(421\) −10.0578 −0.490187 −0.245094 0.969499i \(-0.578819\pi\)
−0.245094 + 0.969499i \(0.578819\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −8.17095 −0.396349
\(426\) 0 0
\(427\) 7.29150 0.352861
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.9494 −0.623749 −0.311874 0.950123i \(-0.600957\pi\)
−0.311874 + 0.950123i \(0.600957\pi\)
\(432\) 0 0
\(433\) 10.5243 0.505766 0.252883 0.967497i \(-0.418621\pi\)
0.252883 + 0.967497i \(0.418621\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.09108 −0.147867
\(438\) 0 0
\(439\) 18.6274 0.889036 0.444518 0.895770i \(-0.353375\pi\)
0.444518 + 0.895770i \(0.353375\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.69234 0.175428 0.0877142 0.996146i \(-0.472044\pi\)
0.0877142 + 0.996146i \(0.472044\pi\)
\(444\) 0 0
\(445\) 13.1110 0.621523
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −17.1012 −0.807054 −0.403527 0.914968i \(-0.632216\pi\)
−0.403527 + 0.914968i \(0.632216\pi\)
\(450\) 0 0
\(451\) 24.6861 1.16242
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4.25098 −0.199289
\(456\) 0 0
\(457\) −35.3137 −1.65190 −0.825952 0.563740i \(-0.809362\pi\)
−0.825952 + 0.563740i \(0.809362\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 29.7794 1.38696 0.693482 0.720474i \(-0.256076\pi\)
0.693482 + 0.720474i \(0.256076\pi\)
\(462\) 0 0
\(463\) 6.17431 0.286944 0.143472 0.989654i \(-0.454173\pi\)
0.143472 + 0.989654i \(0.454173\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 38.1732 1.76644 0.883222 0.468955i \(-0.155369\pi\)
0.883222 + 0.468955i \(0.155369\pi\)
\(468\) 0 0
\(469\) −11.6194 −0.536534
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −59.0566 −2.71542
\(474\) 0 0
\(475\) 7.08322 0.325001
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4.80168 −0.219394 −0.109697 0.993965i \(-0.534988\pi\)
−0.109697 + 0.993965i \(0.534988\pi\)
\(480\) 0 0
\(481\) 15.5890 0.710799
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.0365 0.546550
\(486\) 0 0
\(487\) 25.2104 1.14239 0.571196 0.820814i \(-0.306480\pi\)
0.571196 + 0.820814i \(0.306480\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −11.4453 −0.516521 −0.258260 0.966075i \(-0.583149\pi\)
−0.258260 + 0.966075i \(0.583149\pi\)
\(492\) 0 0
\(493\) 5.15371 0.232111
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −0.745007 −0.0334181
\(498\) 0 0
\(499\) −30.7083 −1.37469 −0.687345 0.726331i \(-0.741224\pi\)
−0.687345 + 0.726331i \(0.741224\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −31.2247 −1.39224 −0.696120 0.717925i \(-0.745092\pi\)
−0.696120 + 0.717925i \(0.745092\pi\)
\(504\) 0 0
\(505\) −5.63365 −0.250694
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −35.7207 −1.58329 −0.791646 0.610981i \(-0.790775\pi\)
−0.791646 + 0.610981i \(0.790775\pi\)
\(510\) 0 0
\(511\) 11.8158 0.522701
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −12.2347 −0.539126
\(516\) 0 0
\(517\) −0.115613 −0.00508465
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −25.6446 −1.12351 −0.561756 0.827303i \(-0.689874\pi\)
−0.561756 + 0.827303i \(0.689874\pi\)
\(522\) 0 0
\(523\) −6.53026 −0.285548 −0.142774 0.989755i \(-0.545602\pi\)
−0.142774 + 0.989755i \(0.545602\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5.15371 −0.224499
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.1010 0.524154
\(534\) 0 0
\(535\) 17.3219 0.748889
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −31.1528 −1.34185
\(540\) 0 0
\(541\) 4.61336 0.198344 0.0991720 0.995070i \(-0.468381\pi\)
0.0991720 + 0.995070i \(0.468381\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −29.2854 −1.25445
\(546\) 0 0
\(547\) −17.6112 −0.753001 −0.376501 0.926416i \(-0.622873\pi\)
−0.376501 + 0.926416i \(0.622873\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −4.46764 −0.190328
\(552\) 0 0
\(553\) 2.14395 0.0911700
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.44140 0.0610742 0.0305371 0.999534i \(-0.490278\pi\)
0.0305371 + 0.999534i \(0.490278\pi\)
\(558\) 0 0
\(559\) −28.9494 −1.22443
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8.84718 −0.372864 −0.186432 0.982468i \(-0.559692\pi\)
−0.186432 + 0.982468i \(0.559692\pi\)
\(564\) 0 0
\(565\) −1.37433 −0.0578186
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.47861 0.187753 0.0938765 0.995584i \(-0.470074\pi\)
0.0938765 + 0.995584i \(0.470074\pi\)
\(570\) 0 0
\(571\) 7.49192 0.313527 0.156764 0.987636i \(-0.449894\pi\)
0.156764 + 0.987636i \(0.449894\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.29150 −0.0955623
\(576\) 0 0
\(577\) 14.2265 0.592258 0.296129 0.955148i \(-0.404304\pi\)
0.296129 + 0.955148i \(0.404304\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 9.31369 0.386397
\(582\) 0 0
\(583\) 39.9615 1.65504
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.94131 −0.0801263 −0.0400631 0.999197i \(-0.512756\pi\)
−0.0400631 + 0.999197i \(0.512756\pi\)
\(588\) 0 0
\(589\) 4.46764 0.184086
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 40.1659 1.64942 0.824708 0.565559i \(-0.191340\pi\)
0.824708 + 0.565559i \(0.191340\pi\)
\(594\) 0 0
\(595\) −5.93345 −0.243248
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −17.4170 −0.711639 −0.355820 0.934555i \(-0.615798\pi\)
−0.355820 + 0.934555i \(0.615798\pi\)
\(600\) 0 0
\(601\) 43.3925 1.77002 0.885009 0.465573i \(-0.154152\pi\)
0.885009 + 0.465573i \(0.154152\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −26.5951 −1.08124
\(606\) 0 0
\(607\) 20.2692 0.822701 0.411351 0.911477i \(-0.365057\pi\)
0.411351 + 0.911477i \(0.365057\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −0.0566731 −0.00229275
\(612\) 0 0
\(613\) 11.6792 0.471716 0.235858 0.971787i \(-0.424210\pi\)
0.235858 + 0.971787i \(0.424210\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −43.6589 −1.75764 −0.878821 0.477151i \(-0.841670\pi\)
−0.878821 + 0.477151i \(0.841670\pi\)
\(618\) 0 0
\(619\) 10.1193 0.406729 0.203364 0.979103i \(-0.434812\pi\)
0.203364 + 0.979103i \(0.434812\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −8.05497 −0.322715
\(624\) 0 0
\(625\) −8.29150 −0.331660
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 21.7589 0.867584
\(630\) 0 0
\(631\) −42.8188 −1.70459 −0.852295 0.523062i \(-0.824790\pi\)
−0.852295 + 0.523062i \(0.824790\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −14.1620 −0.562002
\(636\) 0 0
\(637\) −15.2710 −0.605060
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.13646 0.242376 0.121188 0.992630i \(-0.461330\pi\)
0.121188 + 0.992630i \(0.461330\pi\)
\(642\) 0 0
\(643\) −35.3319 −1.39336 −0.696678 0.717384i \(-0.745339\pi\)
−0.696678 + 0.717384i \(0.745339\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 14.5403 0.571640 0.285820 0.958283i \(-0.407734\pi\)
0.285820 + 0.958283i \(0.407734\pi\)
\(648\) 0 0
\(649\) 17.1216 0.672083
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −20.3136 −0.794931 −0.397466 0.917617i \(-0.630110\pi\)
−0.397466 + 0.917617i \(0.630110\pi\)
\(654\) 0 0
\(655\) −21.9110 −0.856134
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −28.9605 −1.12814 −0.564071 0.825726i \(-0.690766\pi\)
−0.564071 + 0.825726i \(0.690766\pi\)
\(660\) 0 0
\(661\) −17.1669 −0.667715 −0.333857 0.942624i \(-0.608350\pi\)
−0.333857 + 0.942624i \(0.608350\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5.14358 0.199459
\(666\) 0 0
\(667\) 1.44533 0.0559635
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 37.5829 1.45087
\(672\) 0 0
\(673\) 15.3359 0.591154 0.295577 0.955319i \(-0.404488\pi\)
0.295577 + 0.955319i \(0.404488\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8.79970 0.338200 0.169100 0.985599i \(-0.445914\pi\)
0.169100 + 0.985599i \(0.445914\pi\)
\(678\) 0 0
\(679\) −7.39481 −0.283787
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −16.6861 −0.638475 −0.319237 0.947675i \(-0.603427\pi\)
−0.319237 + 0.947675i \(0.603427\pi\)
\(684\) 0 0
\(685\) −26.0138 −0.993935
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 19.5890 0.746283
\(690\) 0 0
\(691\) −2.97154 −0.113043 −0.0565214 0.998401i \(-0.518001\pi\)
−0.0565214 + 0.998401i \(0.518001\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.58301 0.249708
\(696\) 0 0
\(697\) 16.8904 0.639770
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 28.8216 1.08858 0.544289 0.838898i \(-0.316799\pi\)
0.544289 + 0.838898i \(0.316799\pi\)
\(702\) 0 0
\(703\) −18.8623 −0.711406
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.46112 0.130169
\(708\) 0 0
\(709\) 29.6345 1.11295 0.556474 0.830865i \(-0.312154\pi\)
0.556474 + 0.830865i \(0.312154\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.44533 −0.0541281
\(714\) 0 0
\(715\) −21.9110 −0.819426
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −20.0870 −0.749120 −0.374560 0.927203i \(-0.622206\pi\)
−0.374560 + 0.927203i \(0.622206\pi\)
\(720\) 0 0
\(721\) 7.51658 0.279932
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.31198 −0.123004
\(726\) 0 0
\(727\) 19.8227 0.735181 0.367591 0.929988i \(-0.380183\pi\)
0.367591 + 0.929988i \(0.380183\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −40.4070 −1.49451
\(732\) 0 0
\(733\) −43.6343 −1.61167 −0.805835 0.592141i \(-0.798283\pi\)
−0.805835 + 0.592141i \(0.798283\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −59.8903 −2.20609
\(738\) 0 0
\(739\) −35.4611 −1.30446 −0.652229 0.758022i \(-0.726166\pi\)
−0.652229 + 0.758022i \(0.726166\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 21.9431 0.805014 0.402507 0.915417i \(-0.368139\pi\)
0.402507 + 0.915417i \(0.368139\pi\)
\(744\) 0 0
\(745\) −32.7649 −1.20041
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −10.6419 −0.388848
\(750\) 0 0
\(751\) 44.6445 1.62910 0.814550 0.580093i \(-0.196984\pi\)
0.814550 + 0.580093i \(0.196984\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 16.5407 0.601976
\(756\) 0 0
\(757\) −1.85346 −0.0673650 −0.0336825 0.999433i \(-0.510724\pi\)
−0.0336825 + 0.999433i \(0.510724\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.93345 −0.0700874 −0.0350437 0.999386i \(-0.511157\pi\)
−0.0350437 + 0.999386i \(0.511157\pi\)
\(762\) 0 0
\(763\) 17.9919 0.651350
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.39298 0.303053
\(768\) 0 0
\(769\) 37.8523 1.36499 0.682495 0.730890i \(-0.260895\pi\)
0.682495 + 0.730890i \(0.260895\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 27.5568 0.991148 0.495574 0.868566i \(-0.334958\pi\)
0.495574 + 0.868566i \(0.334958\pi\)
\(774\) 0 0
\(775\) 3.31198 0.118970
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −14.6419 −0.524602
\(780\) 0 0
\(781\) −3.84002 −0.137407
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −7.37433 −0.263201
\(786\) 0 0
\(787\) −28.3380 −1.01014 −0.505070 0.863079i \(-0.668533\pi\)
−0.505070 + 0.863079i \(0.668533\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0.844342 0.0300213
\(792\) 0 0
\(793\) 18.4230 0.654221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.73918 0.132449 0.0662243 0.997805i \(-0.478905\pi\)
0.0662243 + 0.997805i \(0.478905\pi\)
\(798\) 0 0
\(799\) −0.0791033 −0.00279847
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 60.9027 2.14921
\(804\) 0 0
\(805\) −1.66401 −0.0586485
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −42.0222 −1.47742 −0.738711 0.674023i \(-0.764565\pi\)
−0.738711 + 0.674023i \(0.764565\pi\)
\(810\) 0 0
\(811\) −28.8382 −1.01265 −0.506323 0.862344i \(-0.668996\pi\)
−0.506323 + 0.862344i \(0.668996\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −33.1414 −1.16089
\(816\) 0 0
\(817\) 35.0280 1.22547
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0.269195 0.00939499 0.00469749 0.999989i \(-0.498505\pi\)
0.00469749 + 0.999989i \(0.498505\pi\)
\(822\) 0 0
\(823\) −43.4007 −1.51285 −0.756427 0.654078i \(-0.773057\pi\)
−0.756427 + 0.654078i \(0.773057\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −48.7439 −1.69499 −0.847495 0.530803i \(-0.821890\pi\)
−0.847495 + 0.530803i \(0.821890\pi\)
\(828\) 0 0
\(829\) 46.1758 1.60375 0.801875 0.597491i \(-0.203836\pi\)
0.801875 + 0.597491i \(0.203836\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −21.3150 −0.738522
\(834\) 0 0
\(835\) 24.6861 0.854297
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −54.7571 −1.89042 −0.945212 0.326457i \(-0.894145\pi\)
−0.945212 + 0.326457i \(0.894145\pi\)
\(840\) 0 0
\(841\) −26.9110 −0.927966
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 10.6540 0.366510
\(846\) 0 0
\(847\) 16.3391 0.561418
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 6.10217 0.209180
\(852\) 0 0
\(853\) 29.7811 1.01968 0.509842 0.860268i \(-0.329704\pi\)
0.509842 + 0.860268i \(0.329704\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 5.42930 0.185461 0.0927307 0.995691i \(-0.470440\pi\)
0.0927307 + 0.995691i \(0.470440\pi\)
\(858\) 0 0
\(859\) −7.32801 −0.250029 −0.125014 0.992155i \(-0.539898\pi\)
−0.125014 + 0.992155i \(0.539898\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 17.5484 0.597354 0.298677 0.954354i \(-0.403455\pi\)
0.298677 + 0.954354i \(0.403455\pi\)
\(864\) 0 0
\(865\) 10.3540 0.352048
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 11.0506 0.374867
\(870\) 0 0
\(871\) −29.3581 −0.994760
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 12.1331 0.410174
\(876\) 0 0
\(877\) 2.86062 0.0965963 0.0482981 0.998833i \(-0.484620\pi\)
0.0482981 + 0.998833i \(0.484620\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −55.1425 −1.85780 −0.928899 0.370334i \(-0.879243\pi\)
−0.928899 + 0.370334i \(0.879243\pi\)
\(882\) 0 0
\(883\) 13.8622 0.466500 0.233250 0.972417i \(-0.425064\pi\)
0.233250 + 0.972417i \(0.425064\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −24.6049 −0.826153 −0.413077 0.910696i \(-0.635546\pi\)
−0.413077 + 0.910696i \(0.635546\pi\)
\(888\) 0 0
\(889\) 8.70064 0.291810
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0.0685730 0.00229471
\(894\) 0 0
\(895\) −26.8116 −0.896212
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.08898 −0.0696715
\(900\) 0 0
\(901\) 27.3420 0.910895
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −20.6134 −0.685211
\(906\) 0 0
\(907\) 32.2571 1.07108 0.535540 0.844510i \(-0.320108\pi\)
0.535540 + 0.844510i \(0.320108\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −26.2041 −0.868181 −0.434090 0.900869i \(-0.642930\pi\)
−0.434090 + 0.900869i \(0.642930\pi\)
\(912\) 0 0
\(913\) 48.0059 1.58876
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 13.4614 0.444533
\(918\) 0 0
\(919\) −27.9584 −0.922263 −0.461132 0.887332i \(-0.652556\pi\)
−0.461132 + 0.887332i \(0.652556\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.88237 −0.0619589
\(924\) 0 0
\(925\) −13.9831 −0.459763
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 46.7810 1.53483 0.767417 0.641149i \(-0.221542\pi\)
0.767417 + 0.641149i \(0.221542\pi\)
\(930\) 0 0
\(931\) 18.4775 0.605577
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −30.5830 −1.00017
\(936\) 0 0
\(937\) −14.8017 −0.483550 −0.241775 0.970332i \(-0.577730\pi\)
−0.241775 + 0.970332i \(0.577730\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −26.5951 −0.866976 −0.433488 0.901159i \(-0.642717\pi\)
−0.433488 + 0.901159i \(0.642717\pi\)
\(942\) 0 0
\(943\) 4.73683 0.154253
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 57.0200 1.85290 0.926451 0.376415i \(-0.122843\pi\)
0.926451 + 0.376415i \(0.122843\pi\)
\(948\) 0 0
\(949\) 29.8543 0.969113
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 36.1266 1.17025 0.585127 0.810941i \(-0.301045\pi\)
0.585127 + 0.810941i \(0.301045\pi\)
\(954\) 0 0
\(955\) −21.8380 −0.706661
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 15.9819 0.516084
\(960\) 0 0
\(961\) −28.9110 −0.932613
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −12.3498 −0.397553
\(966\) 0 0
\(967\) 10.2470 0.329522 0.164761 0.986334i \(-0.447315\pi\)
0.164761 + 0.986334i \(0.447315\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −8.37954 −0.268912 −0.134456 0.990920i \(-0.542929\pi\)
−0.134456 + 0.990920i \(0.542929\pi\)
\(972\) 0 0
\(973\) −4.04437 −0.129656
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −50.9280 −1.62933 −0.814666 0.579931i \(-0.803080\pi\)
−0.814666 + 0.579931i \(0.803080\pi\)
\(978\) 0 0
\(979\) −41.5180 −1.32692
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −17.9857 −0.573654 −0.286827 0.957982i \(-0.592600\pi\)
−0.286827 + 0.957982i \(0.592600\pi\)
\(984\) 0 0
\(985\) 4.54066 0.144677
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −11.3319 −0.360335
\(990\) 0 0
\(991\) −47.0931 −1.49596 −0.747980 0.663721i \(-0.768976\pi\)
−0.747980 + 0.663721i \(0.768976\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 25.1374 0.796910
\(996\) 0 0
\(997\) 14.4391 0.457289 0.228645 0.973510i \(-0.426571\pi\)
0.228645 + 0.973510i \(0.426571\pi\)
\(998\) 0 0
\(999\) 0 0
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 3312.2.a.bh.1.2 4
3.2 odd 2 3312.2.a.bg.1.4 4
4.3 odd 2 1656.2.a.p.1.1 yes 4
12.11 even 2 1656.2.a.o.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1656.2.a.o.1.3 4 12.11 even 2
1656.2.a.p.1.1 yes 4 4.3 odd 2
3312.2.a.bg.1.4 4 3.2 odd 2
3312.2.a.bh.1.2 4 1.1 even 1 trivial