Properties

Label 7744.2.a.cb.1.2
Level $7744$
Weight $2$
Character 7744.1
Self dual yes
Analytic conductor $61.836$
Analytic rank $0$
Dimension $2$
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] = [7744,2,Mod(1,7744)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7744, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7744.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7744 = 2^{6} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7744.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: \(61.8361513253\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 7744.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.618034 q^{3} +1.61803 q^{5} +0.618034 q^{7} -2.61803 q^{9} +O(q^{10})\) \(q+0.618034 q^{3} +1.61803 q^{5} +0.618034 q^{7} -2.61803 q^{9} -0.854102 q^{13} +1.00000 q^{15} +4.85410 q^{17} -1.85410 q^{19} +0.381966 q^{21} +4.00000 q^{23} -2.38197 q^{25} -3.47214 q^{27} +8.32624 q^{29} +10.0902 q^{31} +1.00000 q^{35} +7.38197 q^{37} -0.527864 q^{39} -7.38197 q^{41} +10.4721 q^{43} -4.23607 q^{45} -9.56231 q^{47} -6.61803 q^{49} +3.00000 q^{51} +1.61803 q^{53} -1.14590 q^{57} -4.61803 q^{59} -4.85410 q^{61} -1.61803 q^{63} -1.38197 q^{65} -5.52786 q^{67} +2.47214 q^{69} -6.09017 q^{71} -9.85410 q^{73} -1.47214 q^{75} -2.85410 q^{79} +5.70820 q^{81} +11.6180 q^{83} +7.85410 q^{85} +5.14590 q^{87} +4.47214 q^{89} -0.527864 q^{91} +6.23607 q^{93} -3.00000 q^{95} +3.32624 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + q^{5} - q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + q^{5} - q^{7} - 3 q^{9} + 5 q^{13} + 2 q^{15} + 3 q^{17} + 3 q^{19} + 3 q^{21} + 8 q^{23} - 7 q^{25} + 2 q^{27} + q^{29} + 9 q^{31} + 2 q^{35} + 17 q^{37} - 10 q^{39} - 17 q^{41} + 12 q^{43} - 4 q^{45} + q^{47} - 11 q^{49} + 6 q^{51} + q^{53} - 9 q^{57} - 7 q^{59} - 3 q^{61} - q^{63} - 5 q^{65} - 20 q^{67} - 4 q^{69} - q^{71} - 13 q^{73} + 6 q^{75} + q^{79} - 2 q^{81} + 21 q^{83} + 9 q^{85} + 17 q^{87} - 10 q^{91} + 8 q^{93} - 6 q^{95} - 9 q^{97}+O(q^{100}) \) 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
\(3\) 0.618034 0.356822 0.178411 0.983956i \(-0.442904\pi\)
0.178411 + 0.983956i \(0.442904\pi\)
\(4\) 0 0
\(5\) 1.61803 0.723607 0.361803 0.932254i \(-0.382161\pi\)
0.361803 + 0.932254i \(0.382161\pi\)
\(6\) 0 0
\(7\) 0.618034 0.233595 0.116797 0.993156i \(-0.462737\pi\)
0.116797 + 0.993156i \(0.462737\pi\)
\(8\) 0 0
\(9\) −2.61803 −0.872678
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −0.854102 −0.236885 −0.118443 0.992961i \(-0.537790\pi\)
−0.118443 + 0.992961i \(0.537790\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 4.85410 1.17729 0.588646 0.808391i \(-0.299661\pi\)
0.588646 + 0.808391i \(0.299661\pi\)
\(18\) 0 0
\(19\) −1.85410 −0.425360 −0.212680 0.977122i \(-0.568219\pi\)
−0.212680 + 0.977122i \(0.568219\pi\)
\(20\) 0 0
\(21\) 0.381966 0.0833518
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −2.38197 −0.476393
\(26\) 0 0
\(27\) −3.47214 −0.668213
\(28\) 0 0
\(29\) 8.32624 1.54614 0.773072 0.634319i \(-0.218719\pi\)
0.773072 + 0.634319i \(0.218719\pi\)
\(30\) 0 0
\(31\) 10.0902 1.81225 0.906124 0.423012i \(-0.139027\pi\)
0.906124 + 0.423012i \(0.139027\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 7.38197 1.21359 0.606794 0.794859i \(-0.292455\pi\)
0.606794 + 0.794859i \(0.292455\pi\)
\(38\) 0 0
\(39\) −0.527864 −0.0845259
\(40\) 0 0
\(41\) −7.38197 −1.15287 −0.576435 0.817143i \(-0.695556\pi\)
−0.576435 + 0.817143i \(0.695556\pi\)
\(42\) 0 0
\(43\) 10.4721 1.59699 0.798493 0.602004i \(-0.205631\pi\)
0.798493 + 0.602004i \(0.205631\pi\)
\(44\) 0 0
\(45\) −4.23607 −0.631476
\(46\) 0 0
\(47\) −9.56231 −1.39481 −0.697403 0.716679i \(-0.745661\pi\)
−0.697403 + 0.716679i \(0.745661\pi\)
\(48\) 0 0
\(49\) −6.61803 −0.945433
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) 1.61803 0.222254 0.111127 0.993806i \(-0.464554\pi\)
0.111127 + 0.993806i \(0.464554\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.14590 −0.151778
\(58\) 0 0
\(59\) −4.61803 −0.601217 −0.300608 0.953748i \(-0.597190\pi\)
−0.300608 + 0.953748i \(0.597190\pi\)
\(60\) 0 0
\(61\) −4.85410 −0.621504 −0.310752 0.950491i \(-0.600581\pi\)
−0.310752 + 0.950491i \(0.600581\pi\)
\(62\) 0 0
\(63\) −1.61803 −0.203853
\(64\) 0 0
\(65\) −1.38197 −0.171412
\(66\) 0 0
\(67\) −5.52786 −0.675336 −0.337668 0.941265i \(-0.609638\pi\)
−0.337668 + 0.941265i \(0.609638\pi\)
\(68\) 0 0
\(69\) 2.47214 0.297610
\(70\) 0 0
\(71\) −6.09017 −0.722770 −0.361385 0.932417i \(-0.617696\pi\)
−0.361385 + 0.932417i \(0.617696\pi\)
\(72\) 0 0
\(73\) −9.85410 −1.15334 −0.576668 0.816979i \(-0.695647\pi\)
−0.576668 + 0.816979i \(0.695647\pi\)
\(74\) 0 0
\(75\) −1.47214 −0.169988
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.85410 −0.321112 −0.160556 0.987027i \(-0.551329\pi\)
−0.160556 + 0.987027i \(0.551329\pi\)
\(80\) 0 0
\(81\) 5.70820 0.634245
\(82\) 0 0
\(83\) 11.6180 1.27524 0.637622 0.770349i \(-0.279918\pi\)
0.637622 + 0.770349i \(0.279918\pi\)
\(84\) 0 0
\(85\) 7.85410 0.851897
\(86\) 0 0
\(87\) 5.14590 0.551698
\(88\) 0 0
\(89\) 4.47214 0.474045 0.237023 0.971504i \(-0.423828\pi\)
0.237023 + 0.971504i \(0.423828\pi\)
\(90\) 0 0
\(91\) −0.527864 −0.0553352
\(92\) 0 0
\(93\) 6.23607 0.646650
\(94\) 0 0
\(95\) −3.00000 −0.307794
\(96\) 0 0
\(97\) 3.32624 0.337728 0.168864 0.985639i \(-0.445990\pi\)
0.168864 + 0.985639i \(0.445990\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.09017 0.805002 0.402501 0.915420i \(-0.368141\pi\)
0.402501 + 0.915420i \(0.368141\pi\)
\(102\) 0 0
\(103\) −0.909830 −0.0896482 −0.0448241 0.998995i \(-0.514273\pi\)
−0.0448241 + 0.998995i \(0.514273\pi\)
\(104\) 0 0
\(105\) 0.618034 0.0603139
\(106\) 0 0
\(107\) 15.0902 1.45882 0.729411 0.684076i \(-0.239794\pi\)
0.729411 + 0.684076i \(0.239794\pi\)
\(108\) 0 0
\(109\) 2.94427 0.282010 0.141005 0.990009i \(-0.454967\pi\)
0.141005 + 0.990009i \(0.454967\pi\)
\(110\) 0 0
\(111\) 4.56231 0.433035
\(112\) 0 0
\(113\) 2.14590 0.201869 0.100935 0.994893i \(-0.467817\pi\)
0.100935 + 0.994893i \(0.467817\pi\)
\(114\) 0 0
\(115\) 6.47214 0.603530
\(116\) 0 0
\(117\) 2.23607 0.206725
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −4.56231 −0.411369
\(124\) 0 0
\(125\) −11.9443 −1.06833
\(126\) 0 0
\(127\) 21.3262 1.89240 0.946199 0.323586i \(-0.104888\pi\)
0.946199 + 0.323586i \(0.104888\pi\)
\(128\) 0 0
\(129\) 6.47214 0.569840
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) −1.14590 −0.0993620
\(134\) 0 0
\(135\) −5.61803 −0.483523
\(136\) 0 0
\(137\) −2.56231 −0.218913 −0.109456 0.993992i \(-0.534911\pi\)
−0.109456 + 0.993992i \(0.534911\pi\)
\(138\) 0 0
\(139\) 17.2705 1.46487 0.732433 0.680839i \(-0.238385\pi\)
0.732433 + 0.680839i \(0.238385\pi\)
\(140\) 0 0
\(141\) −5.90983 −0.497697
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 13.4721 1.11880
\(146\) 0 0
\(147\) −4.09017 −0.337352
\(148\) 0 0
\(149\) 11.1459 0.913108 0.456554 0.889696i \(-0.349084\pi\)
0.456554 + 0.889696i \(0.349084\pi\)
\(150\) 0 0
\(151\) 8.32624 0.677580 0.338790 0.940862i \(-0.389982\pi\)
0.338790 + 0.940862i \(0.389982\pi\)
\(152\) 0 0
\(153\) −12.7082 −1.02740
\(154\) 0 0
\(155\) 16.3262 1.31135
\(156\) 0 0
\(157\) 16.3262 1.30298 0.651488 0.758659i \(-0.274145\pi\)
0.651488 + 0.758659i \(0.274145\pi\)
\(158\) 0 0
\(159\) 1.00000 0.0793052
\(160\) 0 0
\(161\) 2.47214 0.194832
\(162\) 0 0
\(163\) 15.7984 1.23742 0.618712 0.785618i \(-0.287655\pi\)
0.618712 + 0.785618i \(0.287655\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.32624 −0.102627 −0.0513137 0.998683i \(-0.516341\pi\)
−0.0513137 + 0.998683i \(0.516341\pi\)
\(168\) 0 0
\(169\) −12.2705 −0.943885
\(170\) 0 0
\(171\) 4.85410 0.371202
\(172\) 0 0
\(173\) −2.14590 −0.163150 −0.0815748 0.996667i \(-0.525995\pi\)
−0.0815748 + 0.996667i \(0.525995\pi\)
\(174\) 0 0
\(175\) −1.47214 −0.111283
\(176\) 0 0
\(177\) −2.85410 −0.214527
\(178\) 0 0
\(179\) −9.27051 −0.692910 −0.346455 0.938067i \(-0.612615\pi\)
−0.346455 + 0.938067i \(0.612615\pi\)
\(180\) 0 0
\(181\) −24.2705 −1.80401 −0.902006 0.431723i \(-0.857906\pi\)
−0.902006 + 0.431723i \(0.857906\pi\)
\(182\) 0 0
\(183\) −3.00000 −0.221766
\(184\) 0 0
\(185\) 11.9443 0.878160
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −2.14590 −0.156091
\(190\) 0 0
\(191\) −21.8541 −1.58131 −0.790654 0.612264i \(-0.790259\pi\)
−0.790654 + 0.612264i \(0.790259\pi\)
\(192\) 0 0
\(193\) −9.61803 −0.692321 −0.346161 0.938175i \(-0.612515\pi\)
−0.346161 + 0.938175i \(0.612515\pi\)
\(194\) 0 0
\(195\) −0.854102 −0.0611635
\(196\) 0 0
\(197\) 8.47214 0.603615 0.301807 0.953369i \(-0.402410\pi\)
0.301807 + 0.953369i \(0.402410\pi\)
\(198\) 0 0
\(199\) 24.3607 1.72688 0.863441 0.504449i \(-0.168304\pi\)
0.863441 + 0.504449i \(0.168304\pi\)
\(200\) 0 0
\(201\) −3.41641 −0.240975
\(202\) 0 0
\(203\) 5.14590 0.361171
\(204\) 0 0
\(205\) −11.9443 −0.834224
\(206\) 0 0
\(207\) −10.4721 −0.727864
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 1.90983 0.131478 0.0657391 0.997837i \(-0.479060\pi\)
0.0657391 + 0.997837i \(0.479060\pi\)
\(212\) 0 0
\(213\) −3.76393 −0.257900
\(214\) 0 0
\(215\) 16.9443 1.15559
\(216\) 0 0
\(217\) 6.23607 0.423332
\(218\) 0 0
\(219\) −6.09017 −0.411536
\(220\) 0 0
\(221\) −4.14590 −0.278883
\(222\) 0 0
\(223\) 7.38197 0.494333 0.247167 0.968973i \(-0.420501\pi\)
0.247167 + 0.968973i \(0.420501\pi\)
\(224\) 0 0
\(225\) 6.23607 0.415738
\(226\) 0 0
\(227\) 13.8541 0.919529 0.459765 0.888041i \(-0.347934\pi\)
0.459765 + 0.888041i \(0.347934\pi\)
\(228\) 0 0
\(229\) 0.673762 0.0445235 0.0222617 0.999752i \(-0.492913\pi\)
0.0222617 + 0.999752i \(0.492913\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −13.0344 −0.853915 −0.426957 0.904272i \(-0.640415\pi\)
−0.426957 + 0.904272i \(0.640415\pi\)
\(234\) 0 0
\(235\) −15.4721 −1.00929
\(236\) 0 0
\(237\) −1.76393 −0.114580
\(238\) 0 0
\(239\) −14.5066 −0.938353 −0.469176 0.883105i \(-0.655449\pi\)
−0.469176 + 0.883105i \(0.655449\pi\)
\(240\) 0 0
\(241\) 18.9443 1.22031 0.610154 0.792283i \(-0.291108\pi\)
0.610154 + 0.792283i \(0.291108\pi\)
\(242\) 0 0
\(243\) 13.9443 0.894525
\(244\) 0 0
\(245\) −10.7082 −0.684122
\(246\) 0 0
\(247\) 1.58359 0.100762
\(248\) 0 0
\(249\) 7.18034 0.455036
\(250\) 0 0
\(251\) 8.38197 0.529065 0.264533 0.964377i \(-0.414782\pi\)
0.264533 + 0.964377i \(0.414782\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 4.85410 0.303976
\(256\) 0 0
\(257\) 17.5623 1.09551 0.547753 0.836640i \(-0.315483\pi\)
0.547753 + 0.836640i \(0.315483\pi\)
\(258\) 0 0
\(259\) 4.56231 0.283488
\(260\) 0 0
\(261\) −21.7984 −1.34929
\(262\) 0 0
\(263\) 19.4164 1.19727 0.598633 0.801023i \(-0.295711\pi\)
0.598633 + 0.801023i \(0.295711\pi\)
\(264\) 0 0
\(265\) 2.61803 0.160825
\(266\) 0 0
\(267\) 2.76393 0.169150
\(268\) 0 0
\(269\) 12.0902 0.737151 0.368575 0.929598i \(-0.379846\pi\)
0.368575 + 0.929598i \(0.379846\pi\)
\(270\) 0 0
\(271\) 28.6180 1.73842 0.869211 0.494442i \(-0.164627\pi\)
0.869211 + 0.494442i \(0.164627\pi\)
\(272\) 0 0
\(273\) −0.326238 −0.0197448
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.61803 0.337555 0.168777 0.985654i \(-0.446018\pi\)
0.168777 + 0.985654i \(0.446018\pi\)
\(278\) 0 0
\(279\) −26.4164 −1.58151
\(280\) 0 0
\(281\) −6.56231 −0.391474 −0.195737 0.980656i \(-0.562710\pi\)
−0.195737 + 0.980656i \(0.562710\pi\)
\(282\) 0 0
\(283\) −10.7984 −0.641897 −0.320948 0.947097i \(-0.604002\pi\)
−0.320948 + 0.947097i \(0.604002\pi\)
\(284\) 0 0
\(285\) −1.85410 −0.109828
\(286\) 0 0
\(287\) −4.56231 −0.269304
\(288\) 0 0
\(289\) 6.56231 0.386018
\(290\) 0 0
\(291\) 2.05573 0.120509
\(292\) 0 0
\(293\) −9.56231 −0.558636 −0.279318 0.960199i \(-0.590108\pi\)
−0.279318 + 0.960199i \(0.590108\pi\)
\(294\) 0 0
\(295\) −7.47214 −0.435045
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.41641 −0.197576
\(300\) 0 0
\(301\) 6.47214 0.373048
\(302\) 0 0
\(303\) 5.00000 0.287242
\(304\) 0 0
\(305\) −7.85410 −0.449725
\(306\) 0 0
\(307\) −1.88854 −0.107785 −0.0538924 0.998547i \(-0.517163\pi\)
−0.0538924 + 0.998547i \(0.517163\pi\)
\(308\) 0 0
\(309\) −0.562306 −0.0319885
\(310\) 0 0
\(311\) 1.27051 0.0720440 0.0360220 0.999351i \(-0.488531\pi\)
0.0360220 + 0.999351i \(0.488531\pi\)
\(312\) 0 0
\(313\) 14.7426 0.833304 0.416652 0.909066i \(-0.363203\pi\)
0.416652 + 0.909066i \(0.363203\pi\)
\(314\) 0 0
\(315\) −2.61803 −0.147510
\(316\) 0 0
\(317\) −15.3262 −0.860807 −0.430404 0.902637i \(-0.641629\pi\)
−0.430404 + 0.902637i \(0.641629\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 9.32624 0.520540
\(322\) 0 0
\(323\) −9.00000 −0.500773
\(324\) 0 0
\(325\) 2.03444 0.112851
\(326\) 0 0
\(327\) 1.81966 0.100627
\(328\) 0 0
\(329\) −5.90983 −0.325819
\(330\) 0 0
\(331\) 8.94427 0.491622 0.245811 0.969318i \(-0.420946\pi\)
0.245811 + 0.969318i \(0.420946\pi\)
\(332\) 0 0
\(333\) −19.3262 −1.05907
\(334\) 0 0
\(335\) −8.94427 −0.488678
\(336\) 0 0
\(337\) 28.6180 1.55892 0.779462 0.626450i \(-0.215493\pi\)
0.779462 + 0.626450i \(0.215493\pi\)
\(338\) 0 0
\(339\) 1.32624 0.0720314
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −8.41641 −0.454443
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) 29.5066 1.58400 0.791998 0.610524i \(-0.209041\pi\)
0.791998 + 0.610524i \(0.209041\pi\)
\(348\) 0 0
\(349\) 13.2705 0.710354 0.355177 0.934799i \(-0.384421\pi\)
0.355177 + 0.934799i \(0.384421\pi\)
\(350\) 0 0
\(351\) 2.96556 0.158290
\(352\) 0 0
\(353\) −25.4164 −1.35278 −0.676389 0.736544i \(-0.736456\pi\)
−0.676389 + 0.736544i \(0.736456\pi\)
\(354\) 0 0
\(355\) −9.85410 −0.523001
\(356\) 0 0
\(357\) 1.85410 0.0981295
\(358\) 0 0
\(359\) 22.5066 1.18785 0.593926 0.804520i \(-0.297577\pi\)
0.593926 + 0.804520i \(0.297577\pi\)
\(360\) 0 0
\(361\) −15.5623 −0.819069
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −15.9443 −0.834561
\(366\) 0 0
\(367\) 11.6738 0.609365 0.304683 0.952454i \(-0.401450\pi\)
0.304683 + 0.952454i \(0.401450\pi\)
\(368\) 0 0
\(369\) 19.3262 1.00608
\(370\) 0 0
\(371\) 1.00000 0.0519174
\(372\) 0 0
\(373\) −21.4164 −1.10890 −0.554450 0.832217i \(-0.687071\pi\)
−0.554450 + 0.832217i \(0.687071\pi\)
\(374\) 0 0
\(375\) −7.38197 −0.381203
\(376\) 0 0
\(377\) −7.11146 −0.366259
\(378\) 0 0
\(379\) −14.2705 −0.733027 −0.366513 0.930413i \(-0.619449\pi\)
−0.366513 + 0.930413i \(0.619449\pi\)
\(380\) 0 0
\(381\) 13.1803 0.675249
\(382\) 0 0
\(383\) 31.0344 1.58579 0.792893 0.609361i \(-0.208574\pi\)
0.792893 + 0.609361i \(0.208574\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −27.4164 −1.39365
\(388\) 0 0
\(389\) 15.7426 0.798184 0.399092 0.916911i \(-0.369325\pi\)
0.399092 + 0.916911i \(0.369325\pi\)
\(390\) 0 0
\(391\) 19.4164 0.981930
\(392\) 0 0
\(393\) 4.94427 0.249406
\(394\) 0 0
\(395\) −4.61803 −0.232359
\(396\) 0 0
\(397\) 29.4164 1.47637 0.738184 0.674600i \(-0.235684\pi\)
0.738184 + 0.674600i \(0.235684\pi\)
\(398\) 0 0
\(399\) −0.708204 −0.0354545
\(400\) 0 0
\(401\) −4.09017 −0.204253 −0.102127 0.994771i \(-0.532565\pi\)
−0.102127 + 0.994771i \(0.532565\pi\)
\(402\) 0 0
\(403\) −8.61803 −0.429295
\(404\) 0 0
\(405\) 9.23607 0.458944
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −18.5623 −0.917847 −0.458923 0.888476i \(-0.651765\pi\)
−0.458923 + 0.888476i \(0.651765\pi\)
\(410\) 0 0
\(411\) −1.58359 −0.0781128
\(412\) 0 0
\(413\) −2.85410 −0.140441
\(414\) 0 0
\(415\) 18.7984 0.922776
\(416\) 0 0
\(417\) 10.6738 0.522696
\(418\) 0 0
\(419\) 20.9443 1.02319 0.511597 0.859225i \(-0.329054\pi\)
0.511597 + 0.859225i \(0.329054\pi\)
\(420\) 0 0
\(421\) −3.09017 −0.150606 −0.0753028 0.997161i \(-0.523992\pi\)
−0.0753028 + 0.997161i \(0.523992\pi\)
\(422\) 0 0
\(423\) 25.0344 1.21722
\(424\) 0 0
\(425\) −11.5623 −0.560854
\(426\) 0 0
\(427\) −3.00000 −0.145180
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −36.1591 −1.74172 −0.870860 0.491531i \(-0.836438\pi\)
−0.870860 + 0.491531i \(0.836438\pi\)
\(432\) 0 0
\(433\) −26.7984 −1.28785 −0.643924 0.765090i \(-0.722695\pi\)
−0.643924 + 0.765090i \(0.722695\pi\)
\(434\) 0 0
\(435\) 8.32624 0.399213
\(436\) 0 0
\(437\) −7.41641 −0.354775
\(438\) 0 0
\(439\) −16.9443 −0.808706 −0.404353 0.914603i \(-0.632503\pi\)
−0.404353 + 0.914603i \(0.632503\pi\)
\(440\) 0 0
\(441\) 17.3262 0.825059
\(442\) 0 0
\(443\) −18.7984 −0.893138 −0.446569 0.894749i \(-0.647354\pi\)
−0.446569 + 0.894749i \(0.647354\pi\)
\(444\) 0 0
\(445\) 7.23607 0.343023
\(446\) 0 0
\(447\) 6.88854 0.325817
\(448\) 0 0
\(449\) −16.4508 −0.776364 −0.388182 0.921583i \(-0.626897\pi\)
−0.388182 + 0.921583i \(0.626897\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 5.14590 0.241775
\(454\) 0 0
\(455\) −0.854102 −0.0400409
\(456\) 0 0
\(457\) −10.5623 −0.494084 −0.247042 0.969005i \(-0.579459\pi\)
−0.247042 + 0.969005i \(0.579459\pi\)
\(458\) 0 0
\(459\) −16.8541 −0.786682
\(460\) 0 0
\(461\) −14.3607 −0.668844 −0.334422 0.942424i \(-0.608541\pi\)
−0.334422 + 0.942424i \(0.608541\pi\)
\(462\) 0 0
\(463\) 11.4164 0.530565 0.265283 0.964171i \(-0.414535\pi\)
0.265283 + 0.964171i \(0.414535\pi\)
\(464\) 0 0
\(465\) 10.0902 0.467920
\(466\) 0 0
\(467\) −27.7984 −1.28636 −0.643178 0.765717i \(-0.722384\pi\)
−0.643178 + 0.765717i \(0.722384\pi\)
\(468\) 0 0
\(469\) −3.41641 −0.157755
\(470\) 0 0
\(471\) 10.0902 0.464930
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.41641 0.202639
\(476\) 0 0
\(477\) −4.23607 −0.193956
\(478\) 0 0
\(479\) −23.9787 −1.09562 −0.547808 0.836604i \(-0.684537\pi\)
−0.547808 + 0.836604i \(0.684537\pi\)
\(480\) 0 0
\(481\) −6.30495 −0.287481
\(482\) 0 0
\(483\) 1.52786 0.0695202
\(484\) 0 0
\(485\) 5.38197 0.244382
\(486\) 0 0
\(487\) 3.96556 0.179697 0.0898483 0.995955i \(-0.471362\pi\)
0.0898483 + 0.995955i \(0.471362\pi\)
\(488\) 0 0
\(489\) 9.76393 0.441540
\(490\) 0 0
\(491\) −18.5066 −0.835190 −0.417595 0.908633i \(-0.637127\pi\)
−0.417595 + 0.908633i \(0.637127\pi\)
\(492\) 0 0
\(493\) 40.4164 1.82026
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3.76393 −0.168835
\(498\) 0 0
\(499\) −23.0902 −1.03366 −0.516829 0.856089i \(-0.672888\pi\)
−0.516829 + 0.856089i \(0.672888\pi\)
\(500\) 0 0
\(501\) −0.819660 −0.0366197
\(502\) 0 0
\(503\) −3.09017 −0.137784 −0.0688919 0.997624i \(-0.521946\pi\)
−0.0688919 + 0.997624i \(0.521946\pi\)
\(504\) 0 0
\(505\) 13.0902 0.582505
\(506\) 0 0
\(507\) −7.58359 −0.336799
\(508\) 0 0
\(509\) −18.5066 −0.820290 −0.410145 0.912020i \(-0.634522\pi\)
−0.410145 + 0.912020i \(0.634522\pi\)
\(510\) 0 0
\(511\) −6.09017 −0.269413
\(512\) 0 0
\(513\) 6.43769 0.284231
\(514\) 0 0
\(515\) −1.47214 −0.0648701
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −1.32624 −0.0582154
\(520\) 0 0
\(521\) 4.03444 0.176752 0.0883761 0.996087i \(-0.471832\pi\)
0.0883761 + 0.996087i \(0.471832\pi\)
\(522\) 0 0
\(523\) 25.5066 1.11532 0.557662 0.830068i \(-0.311698\pi\)
0.557662 + 0.830068i \(0.311698\pi\)
\(524\) 0 0
\(525\) −0.909830 −0.0397082
\(526\) 0 0
\(527\) 48.9787 2.13355
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 12.0902 0.524669
\(532\) 0 0
\(533\) 6.30495 0.273098
\(534\) 0 0
\(535\) 24.4164 1.05561
\(536\) 0 0
\(537\) −5.72949 −0.247246
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −36.2705 −1.55939 −0.779696 0.626159i \(-0.784626\pi\)
−0.779696 + 0.626159i \(0.784626\pi\)
\(542\) 0 0
\(543\) −15.0000 −0.643712
\(544\) 0 0
\(545\) 4.76393 0.204064
\(546\) 0 0
\(547\) −34.7984 −1.48787 −0.743936 0.668251i \(-0.767043\pi\)
−0.743936 + 0.668251i \(0.767043\pi\)
\(548\) 0 0
\(549\) 12.7082 0.542373
\(550\) 0 0
\(551\) −15.4377 −0.657668
\(552\) 0 0
\(553\) −1.76393 −0.0750100
\(554\) 0 0
\(555\) 7.38197 0.313347
\(556\) 0 0
\(557\) −26.1459 −1.10784 −0.553919 0.832571i \(-0.686868\pi\)
−0.553919 + 0.832571i \(0.686868\pi\)
\(558\) 0 0
\(559\) −8.94427 −0.378302
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4.56231 −0.192278 −0.0961391 0.995368i \(-0.530649\pi\)
−0.0961391 + 0.995368i \(0.530649\pi\)
\(564\) 0 0
\(565\) 3.47214 0.146074
\(566\) 0 0
\(567\) 3.52786 0.148156
\(568\) 0 0
\(569\) 1.20163 0.0503748 0.0251874 0.999683i \(-0.491982\pi\)
0.0251874 + 0.999683i \(0.491982\pi\)
\(570\) 0 0
\(571\) 22.8328 0.955524 0.477762 0.878489i \(-0.341448\pi\)
0.477762 + 0.878489i \(0.341448\pi\)
\(572\) 0 0
\(573\) −13.5066 −0.564245
\(574\) 0 0
\(575\) −9.52786 −0.397339
\(576\) 0 0
\(577\) −21.6180 −0.899971 −0.449985 0.893036i \(-0.648571\pi\)
−0.449985 + 0.893036i \(0.648571\pi\)
\(578\) 0 0
\(579\) −5.94427 −0.247036
\(580\) 0 0
\(581\) 7.18034 0.297891
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 3.61803 0.149587
\(586\) 0 0
\(587\) −11.0902 −0.457740 −0.228870 0.973457i \(-0.573503\pi\)
−0.228870 + 0.973457i \(0.573503\pi\)
\(588\) 0 0
\(589\) −18.7082 −0.770858
\(590\) 0 0
\(591\) 5.23607 0.215383
\(592\) 0 0
\(593\) −8.47214 −0.347909 −0.173954 0.984754i \(-0.555655\pi\)
−0.173954 + 0.984754i \(0.555655\pi\)
\(594\) 0 0
\(595\) 4.85410 0.198999
\(596\) 0 0
\(597\) 15.0557 0.616190
\(598\) 0 0
\(599\) 13.9098 0.568340 0.284170 0.958774i \(-0.408282\pi\)
0.284170 + 0.958774i \(0.408282\pi\)
\(600\) 0 0
\(601\) −37.8541 −1.54410 −0.772051 0.635561i \(-0.780769\pi\)
−0.772051 + 0.635561i \(0.780769\pi\)
\(602\) 0 0
\(603\) 14.4721 0.589351
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 31.6180 1.28334 0.641668 0.766982i \(-0.278243\pi\)
0.641668 + 0.766982i \(0.278243\pi\)
\(608\) 0 0
\(609\) 3.18034 0.128874
\(610\) 0 0
\(611\) 8.16718 0.330409
\(612\) 0 0
\(613\) −4.97871 −0.201088 −0.100544 0.994933i \(-0.532058\pi\)
−0.100544 + 0.994933i \(0.532058\pi\)
\(614\) 0 0
\(615\) −7.38197 −0.297670
\(616\) 0 0
\(617\) 1.41641 0.0570224 0.0285112 0.999593i \(-0.490923\pi\)
0.0285112 + 0.999593i \(0.490923\pi\)
\(618\) 0 0
\(619\) −13.8541 −0.556843 −0.278422 0.960459i \(-0.589811\pi\)
−0.278422 + 0.960459i \(0.589811\pi\)
\(620\) 0 0
\(621\) −13.8885 −0.557328
\(622\) 0 0
\(623\) 2.76393 0.110735
\(624\) 0 0
\(625\) −7.41641 −0.296656
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 35.8328 1.42875
\(630\) 0 0
\(631\) −0.909830 −0.0362198 −0.0181099 0.999836i \(-0.505765\pi\)
−0.0181099 + 0.999836i \(0.505765\pi\)
\(632\) 0 0
\(633\) 1.18034 0.0469143
\(634\) 0 0
\(635\) 34.5066 1.36935
\(636\) 0 0
\(637\) 5.65248 0.223959
\(638\) 0 0
\(639\) 15.9443 0.630746
\(640\) 0 0
\(641\) 33.5623 1.32563 0.662816 0.748783i \(-0.269361\pi\)
0.662816 + 0.748783i \(0.269361\pi\)
\(642\) 0 0
\(643\) −24.3820 −0.961531 −0.480765 0.876849i \(-0.659641\pi\)
−0.480765 + 0.876849i \(0.659641\pi\)
\(644\) 0 0
\(645\) 10.4721 0.412340
\(646\) 0 0
\(647\) 32.5623 1.28016 0.640078 0.768310i \(-0.278902\pi\)
0.640078 + 0.768310i \(0.278902\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 3.85410 0.151054
\(652\) 0 0
\(653\) −25.9230 −1.01444 −0.507222 0.861815i \(-0.669328\pi\)
−0.507222 + 0.861815i \(0.669328\pi\)
\(654\) 0 0
\(655\) 12.9443 0.505775
\(656\) 0 0
\(657\) 25.7984 1.00649
\(658\) 0 0
\(659\) 32.0000 1.24654 0.623272 0.782006i \(-0.285803\pi\)
0.623272 + 0.782006i \(0.285803\pi\)
\(660\) 0 0
\(661\) 10.9443 0.425683 0.212841 0.977087i \(-0.431728\pi\)
0.212841 + 0.977087i \(0.431728\pi\)
\(662\) 0 0
\(663\) −2.56231 −0.0995117
\(664\) 0 0
\(665\) −1.85410 −0.0718990
\(666\) 0 0
\(667\) 33.3050 1.28957
\(668\) 0 0
\(669\) 4.56231 0.176389
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 34.7426 1.33923 0.669615 0.742708i \(-0.266459\pi\)
0.669615 + 0.742708i \(0.266459\pi\)
\(674\) 0 0
\(675\) 8.27051 0.318332
\(676\) 0 0
\(677\) 4.67376 0.179627 0.0898136 0.995959i \(-0.471373\pi\)
0.0898136 + 0.995959i \(0.471373\pi\)
\(678\) 0 0
\(679\) 2.05573 0.0788916
\(680\) 0 0
\(681\) 8.56231 0.328108
\(682\) 0 0
\(683\) 16.9443 0.648355 0.324177 0.945996i \(-0.394913\pi\)
0.324177 + 0.945996i \(0.394913\pi\)
\(684\) 0 0
\(685\) −4.14590 −0.158407
\(686\) 0 0
\(687\) 0.416408 0.0158870
\(688\) 0 0
\(689\) −1.38197 −0.0526487
\(690\) 0 0
\(691\) 3.43769 0.130776 0.0653880 0.997860i \(-0.479171\pi\)
0.0653880 + 0.997860i \(0.479171\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 27.9443 1.05999
\(696\) 0 0
\(697\) −35.8328 −1.35726
\(698\) 0 0
\(699\) −8.05573 −0.304696
\(700\) 0 0
\(701\) 49.2705 1.86092 0.930461 0.366392i \(-0.119407\pi\)
0.930461 + 0.366392i \(0.119407\pi\)
\(702\) 0 0
\(703\) −13.6869 −0.516212
\(704\) 0 0
\(705\) −9.56231 −0.360137
\(706\) 0 0
\(707\) 5.00000 0.188044
\(708\) 0 0
\(709\) −22.7426 −0.854118 −0.427059 0.904224i \(-0.640450\pi\)
−0.427059 + 0.904224i \(0.640450\pi\)
\(710\) 0 0
\(711\) 7.47214 0.280227
\(712\) 0 0
\(713\) 40.3607 1.51152
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −8.96556 −0.334825
\(718\) 0 0
\(719\) 11.0902 0.413594 0.206797 0.978384i \(-0.433696\pi\)
0.206797 + 0.978384i \(0.433696\pi\)
\(720\) 0 0
\(721\) −0.562306 −0.0209414
\(722\) 0 0
\(723\) 11.7082 0.435433
\(724\) 0 0
\(725\) −19.8328 −0.736572
\(726\) 0 0
\(727\) 11.0557 0.410034 0.205017 0.978758i \(-0.434275\pi\)
0.205017 + 0.978758i \(0.434275\pi\)
\(728\) 0 0
\(729\) −8.50658 −0.315058
\(730\) 0 0
\(731\) 50.8328 1.88012
\(732\) 0 0
\(733\) −16.0344 −0.592246 −0.296123 0.955150i \(-0.595694\pi\)
−0.296123 + 0.955150i \(0.595694\pi\)
\(734\) 0 0
\(735\) −6.61803 −0.244110
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 14.8541 0.546417 0.273208 0.961955i \(-0.411915\pi\)
0.273208 + 0.961955i \(0.411915\pi\)
\(740\) 0 0
\(741\) 0.978714 0.0359539
\(742\) 0 0
\(743\) 13.6869 0.502124 0.251062 0.967971i \(-0.419220\pi\)
0.251062 + 0.967971i \(0.419220\pi\)
\(744\) 0 0
\(745\) 18.0344 0.660731
\(746\) 0 0
\(747\) −30.4164 −1.11288
\(748\) 0 0
\(749\) 9.32624 0.340773
\(750\) 0 0
\(751\) 12.3262 0.449791 0.224895 0.974383i \(-0.427796\pi\)
0.224895 + 0.974383i \(0.427796\pi\)
\(752\) 0 0
\(753\) 5.18034 0.188782
\(754\) 0 0
\(755\) 13.4721 0.490301
\(756\) 0 0
\(757\) −42.1591 −1.53230 −0.766148 0.642664i \(-0.777829\pi\)
−0.766148 + 0.642664i \(0.777829\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −31.5066 −1.14211 −0.571056 0.820911i \(-0.693466\pi\)
−0.571056 + 0.820911i \(0.693466\pi\)
\(762\) 0 0
\(763\) 1.81966 0.0658761
\(764\) 0 0
\(765\) −20.5623 −0.743432
\(766\) 0 0
\(767\) 3.94427 0.142419
\(768\) 0 0
\(769\) −44.2492 −1.59567 −0.797834 0.602877i \(-0.794021\pi\)
−0.797834 + 0.602877i \(0.794021\pi\)
\(770\) 0 0
\(771\) 10.8541 0.390901
\(772\) 0 0
\(773\) 37.6312 1.35350 0.676750 0.736213i \(-0.263388\pi\)
0.676750 + 0.736213i \(0.263388\pi\)
\(774\) 0 0
\(775\) −24.0344 −0.863343
\(776\) 0 0
\(777\) 2.81966 0.101155
\(778\) 0 0
\(779\) 13.6869 0.490385
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −28.9098 −1.03315
\(784\) 0 0
\(785\) 26.4164 0.942842
\(786\) 0 0
\(787\) −3.43769 −0.122541 −0.0612703 0.998121i \(-0.519515\pi\)
−0.0612703 + 0.998121i \(0.519515\pi\)
\(788\) 0 0
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) 1.32624 0.0471556
\(792\) 0 0
\(793\) 4.14590 0.147225
\(794\) 0 0
\(795\) 1.61803 0.0573858
\(796\) 0 0
\(797\) 44.4508 1.57453 0.787265 0.616615i \(-0.211496\pi\)
0.787265 + 0.616615i \(0.211496\pi\)
\(798\) 0 0
\(799\) −46.4164 −1.64209
\(800\) 0 0
\(801\) −11.7082 −0.413689
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) 0 0
\(807\) 7.47214 0.263032
\(808\) 0 0
\(809\) −39.5066 −1.38898 −0.694489 0.719504i \(-0.744369\pi\)
−0.694489 + 0.719504i \(0.744369\pi\)
\(810\) 0 0
\(811\) 0.257354 0.00903693 0.00451846 0.999990i \(-0.498562\pi\)
0.00451846 + 0.999990i \(0.498562\pi\)
\(812\) 0 0
\(813\) 17.6869 0.620307
\(814\) 0 0
\(815\) 25.5623 0.895409
\(816\) 0 0
\(817\) −19.4164 −0.679294
\(818\) 0 0
\(819\) 1.38197 0.0482898
\(820\) 0 0
\(821\) 26.4377 0.922682 0.461341 0.887223i \(-0.347368\pi\)
0.461341 + 0.887223i \(0.347368\pi\)
\(822\) 0 0
\(823\) −29.3262 −1.02225 −0.511124 0.859507i \(-0.670771\pi\)
−0.511124 + 0.859507i \(0.670771\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 29.3262 1.01977 0.509887 0.860242i \(-0.329687\pi\)
0.509887 + 0.860242i \(0.329687\pi\)
\(828\) 0 0
\(829\) −12.0344 −0.417973 −0.208987 0.977918i \(-0.567017\pi\)
−0.208987 + 0.977918i \(0.567017\pi\)
\(830\) 0 0
\(831\) 3.47214 0.120447
\(832\) 0 0
\(833\) −32.1246 −1.11305
\(834\) 0 0
\(835\) −2.14590 −0.0742619
\(836\) 0 0
\(837\) −35.0344 −1.21097
\(838\) 0 0
\(839\) −48.0344 −1.65833 −0.829167 0.559002i \(-0.811185\pi\)
−0.829167 + 0.559002i \(0.811185\pi\)
\(840\) 0 0
\(841\) 40.3262 1.39056
\(842\) 0 0
\(843\) −4.05573 −0.139687
\(844\) 0 0
\(845\) −19.8541 −0.683002
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −6.67376 −0.229043
\(850\) 0 0
\(851\) 29.5279 1.01220
\(852\) 0 0
\(853\) 44.0902 1.50962 0.754809 0.655944i \(-0.227729\pi\)
0.754809 + 0.655944i \(0.227729\pi\)
\(854\) 0 0
\(855\) 7.85410 0.268605
\(856\) 0 0
\(857\) 18.3607 0.627189 0.313594 0.949557i \(-0.398467\pi\)
0.313594 + 0.949557i \(0.398467\pi\)
\(858\) 0 0
\(859\) −21.8885 −0.746827 −0.373414 0.927665i \(-0.621813\pi\)
−0.373414 + 0.927665i \(0.621813\pi\)
\(860\) 0 0
\(861\) −2.81966 −0.0960938
\(862\) 0 0
\(863\) −25.5066 −0.868254 −0.434127 0.900852i \(-0.642943\pi\)
−0.434127 + 0.900852i \(0.642943\pi\)
\(864\) 0 0
\(865\) −3.47214 −0.118056
\(866\) 0 0
\(867\) 4.05573 0.137740
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 4.72136 0.159977
\(872\) 0 0
\(873\) −8.70820 −0.294728
\(874\) 0 0
\(875\) −7.38197 −0.249556
\(876\) 0 0
\(877\) −48.0344 −1.62201 −0.811004 0.585041i \(-0.801079\pi\)
−0.811004 + 0.585041i \(0.801079\pi\)
\(878\) 0 0
\(879\) −5.90983 −0.199334
\(880\) 0 0
\(881\) 40.4721 1.36354 0.681770 0.731566i \(-0.261210\pi\)
0.681770 + 0.731566i \(0.261210\pi\)
\(882\) 0 0
\(883\) 6.14590 0.206826 0.103413 0.994639i \(-0.467024\pi\)
0.103413 + 0.994639i \(0.467024\pi\)
\(884\) 0 0
\(885\) −4.61803 −0.155234
\(886\) 0 0
\(887\) −46.5755 −1.56385 −0.781925 0.623372i \(-0.785762\pi\)
−0.781925 + 0.623372i \(0.785762\pi\)
\(888\) 0 0
\(889\) 13.1803 0.442054
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 17.7295 0.593295
\(894\) 0 0
\(895\) −15.0000 −0.501395
\(896\) 0 0
\(897\) −2.11146 −0.0704995
\(898\) 0 0
\(899\) 84.0132 2.80200
\(900\) 0 0
\(901\) 7.85410 0.261658
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 0 0
\(905\) −39.2705 −1.30540
\(906\) 0 0
\(907\) −6.27051 −0.208209 −0.104104 0.994566i \(-0.533198\pi\)
−0.104104 + 0.994566i \(0.533198\pi\)
\(908\) 0 0
\(909\) −21.1803 −0.702508
\(910\) 0 0
\(911\) −0.0212862 −0.000705244 0 −0.000352622 1.00000i \(-0.500112\pi\)
−0.000352622 1.00000i \(0.500112\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −4.85410 −0.160472
\(916\) 0 0
\(917\) 4.94427 0.163274
\(918\) 0 0
\(919\) 16.3820 0.540391 0.270196 0.962805i \(-0.412912\pi\)
0.270196 + 0.962805i \(0.412912\pi\)
\(920\) 0 0
\(921\) −1.16718 −0.0384600
\(922\) 0 0
\(923\) 5.20163 0.171214
\(924\) 0 0
\(925\) −17.5836 −0.578145
\(926\) 0 0
\(927\) 2.38197 0.0782340
\(928\) 0 0
\(929\) −34.5623 −1.13395 −0.566976 0.823734i \(-0.691887\pi\)
−0.566976 + 0.823734i \(0.691887\pi\)
\(930\) 0 0
\(931\) 12.2705 0.402150
\(932\) 0 0
\(933\) 0.785218 0.0257069
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −18.5623 −0.606404 −0.303202 0.952926i \(-0.598056\pi\)
−0.303202 + 0.952926i \(0.598056\pi\)
\(938\) 0 0
\(939\) 9.11146 0.297341
\(940\) 0 0
\(941\) −4.27051 −0.139215 −0.0696073 0.997574i \(-0.522175\pi\)
−0.0696073 + 0.997574i \(0.522175\pi\)
\(942\) 0 0
\(943\) −29.5279 −0.961560
\(944\) 0 0
\(945\) −3.47214 −0.112949
\(946\) 0 0
\(947\) −34.8328 −1.13191 −0.565957 0.824435i \(-0.691493\pi\)
−0.565957 + 0.824435i \(0.691493\pi\)
\(948\) 0 0
\(949\) 8.41641 0.273208
\(950\) 0 0
\(951\) −9.47214 −0.307155
\(952\) 0 0
\(953\) 16.9787 0.549994 0.274997 0.961445i \(-0.411323\pi\)
0.274997 + 0.961445i \(0.411323\pi\)
\(954\) 0 0
\(955\) −35.3607 −1.14424
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.58359 −0.0511369
\(960\) 0 0
\(961\) 70.8115 2.28424
\(962\) 0 0
\(963\) −39.5066 −1.27308
\(964\) 0 0
\(965\) −15.5623 −0.500968
\(966\) 0 0
\(967\) 29.3050 0.942384 0.471192 0.882031i \(-0.343824\pi\)
0.471192 + 0.882031i \(0.343824\pi\)
\(968\) 0 0
\(969\) −5.56231 −0.178687
\(970\) 0 0
\(971\) −17.8541 −0.572965 −0.286483 0.958085i \(-0.592486\pi\)
−0.286483 + 0.958085i \(0.592486\pi\)
\(972\) 0 0
\(973\) 10.6738 0.342185
\(974\) 0 0
\(975\) 1.25735 0.0402676
\(976\) 0 0
\(977\) 51.6869 1.65361 0.826805 0.562488i \(-0.190156\pi\)
0.826805 + 0.562488i \(0.190156\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −7.70820 −0.246104
\(982\) 0 0
\(983\) 3.09017 0.0985611 0.0492806 0.998785i \(-0.484307\pi\)
0.0492806 + 0.998785i \(0.484307\pi\)
\(984\) 0 0
\(985\) 13.7082 0.436780
\(986\) 0 0
\(987\) −3.65248 −0.116260
\(988\) 0 0
\(989\) 41.8885 1.33198
\(990\) 0 0
\(991\) −44.9443 −1.42770 −0.713851 0.700298i \(-0.753051\pi\)
−0.713851 + 0.700298i \(0.753051\pi\)
\(992\) 0 0
\(993\) 5.52786 0.175421
\(994\) 0 0
\(995\) 39.4164 1.24958
\(996\) 0 0
\(997\) 49.6312 1.57184 0.785918 0.618331i \(-0.212191\pi\)
0.785918 + 0.618331i \(0.212191\pi\)
\(998\) 0 0
\(999\) −25.6312 −0.810935
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 7744.2.a.cb.1.2 2
4.3 odd 2 7744.2.a.cr.1.1 2
8.3 odd 2 968.2.a.i.1.2 2
8.5 even 2 1936.2.a.t.1.1 2
11.5 even 5 704.2.m.g.641.1 4
11.9 even 5 704.2.m.g.257.1 4
11.10 odd 2 7744.2.a.cc.1.2 2
24.11 even 2 8712.2.a.bp.1.2 2
44.27 odd 10 704.2.m.b.641.1 4
44.31 odd 10 704.2.m.b.257.1 4
44.43 even 2 7744.2.a.cq.1.1 2
88.3 odd 10 968.2.i.d.9.1 4
88.5 even 10 176.2.m.a.113.1 4
88.19 even 10 968.2.i.c.9.1 4
88.21 odd 2 1936.2.a.u.1.1 2
88.27 odd 10 88.2.i.a.25.1 4
88.35 even 10 968.2.i.k.81.1 4
88.43 even 2 968.2.a.h.1.2 2
88.51 even 10 968.2.i.c.753.1 4
88.53 even 10 176.2.m.a.81.1 4
88.59 odd 10 968.2.i.d.753.1 4
88.75 odd 10 88.2.i.a.81.1 yes 4
88.83 even 10 968.2.i.k.729.1 4
264.131 odd 2 8712.2.a.bm.1.2 2
264.203 even 10 792.2.r.b.289.1 4
264.251 even 10 792.2.r.b.433.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
88.2.i.a.25.1 4 88.27 odd 10
88.2.i.a.81.1 yes 4 88.75 odd 10
176.2.m.a.81.1 4 88.53 even 10
176.2.m.a.113.1 4 88.5 even 10
704.2.m.b.257.1 4 44.31 odd 10
704.2.m.b.641.1 4 44.27 odd 10
704.2.m.g.257.1 4 11.9 even 5
704.2.m.g.641.1 4 11.5 even 5
792.2.r.b.289.1 4 264.203 even 10
792.2.r.b.433.1 4 264.251 even 10
968.2.a.h.1.2 2 88.43 even 2
968.2.a.i.1.2 2 8.3 odd 2
968.2.i.c.9.1 4 88.19 even 10
968.2.i.c.753.1 4 88.51 even 10
968.2.i.d.9.1 4 88.3 odd 10
968.2.i.d.753.1 4 88.59 odd 10
968.2.i.k.81.1 4 88.35 even 10
968.2.i.k.729.1 4 88.83 even 10
1936.2.a.t.1.1 2 8.5 even 2
1936.2.a.u.1.1 2 88.21 odd 2
7744.2.a.cb.1.2 2 1.1 even 1 trivial
7744.2.a.cc.1.2 2 11.10 odd 2
7744.2.a.cq.1.1 2 44.43 even 2
7744.2.a.cr.1.1 2 4.3 odd 2
8712.2.a.bm.1.2 2 264.131 odd 2
8712.2.a.bp.1.2 2 24.11 even 2