Properties

Label 9680.2.a.dc.1.4
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $1$
Dimension $6$
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] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9680, 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("9680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.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: \(77.2951891566\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.25903625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5 \) 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 440)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.03795\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.03795 q^{3} -1.00000 q^{5} +2.60210 q^{7} -1.92266 q^{9} +O(q^{10})\) \(q+1.03795 q^{3} -1.00000 q^{5} +2.60210 q^{7} -1.92266 q^{9} -2.87756 q^{13} -1.03795 q^{15} +0.810293 q^{17} -5.15069 q^{19} +2.70085 q^{21} -4.98262 q^{23} +1.00000 q^{25} -5.10948 q^{27} +6.72933 q^{29} +4.14084 q^{31} -2.60210 q^{35} +4.05133 q^{37} -2.98677 q^{39} +9.76082 q^{41} +5.92912 q^{43} +1.92266 q^{45} +0.967904 q^{47} -0.229090 q^{49} +0.841046 q^{51} +4.47445 q^{53} -5.34617 q^{57} -6.80756 q^{59} -2.61307 q^{61} -5.00294 q^{63} +2.87756 q^{65} +15.8408 q^{67} -5.17172 q^{69} +1.41346 q^{71} -13.1429 q^{73} +1.03795 q^{75} -5.69123 q^{79} +0.464568 q^{81} -7.96967 q^{83} -0.810293 q^{85} +6.98472 q^{87} -14.5196 q^{89} -7.48768 q^{91} +4.29799 q^{93} +5.15069 q^{95} -2.99325 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9} - q^{13} - 3 q^{15} - 6 q^{17} - 7 q^{19} - 14 q^{21} + 9 q^{23} + 6 q^{25} + 21 q^{27} - 10 q^{29} - q^{31} + 7 q^{35} + 3 q^{37} - q^{39} + 6 q^{41} + 18 q^{43} - 5 q^{45} + 3 q^{47} + 17 q^{49} + 15 q^{51} - 23 q^{53} - 9 q^{57} + 2 q^{59} - 6 q^{61} - 49 q^{63} + q^{65} + 22 q^{67} + 2 q^{69} + 13 q^{71} - 10 q^{73} + 3 q^{75} - 22 q^{79} + 10 q^{81} + 10 q^{83} + 6 q^{85} - 3 q^{87} - 25 q^{89} - 12 q^{91} + 19 q^{93} + 7 q^{95} - 33 q^{97}+O(q^{100}) \) 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
\(3\) 1.03795 0.599262 0.299631 0.954055i \(-0.403136\pi\)
0.299631 + 0.954055i \(0.403136\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.60210 0.983500 0.491750 0.870736i \(-0.336357\pi\)
0.491750 + 0.870736i \(0.336357\pi\)
\(8\) 0 0
\(9\) −1.92266 −0.640885
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −2.87756 −0.798091 −0.399045 0.916931i \(-0.630658\pi\)
−0.399045 + 0.916931i \(0.630658\pi\)
\(14\) 0 0
\(15\) −1.03795 −0.267998
\(16\) 0 0
\(17\) 0.810293 0.196525 0.0982625 0.995161i \(-0.468672\pi\)
0.0982625 + 0.995161i \(0.468672\pi\)
\(18\) 0 0
\(19\) −5.15069 −1.18165 −0.590824 0.806800i \(-0.701197\pi\)
−0.590824 + 0.806800i \(0.701197\pi\)
\(20\) 0 0
\(21\) 2.70085 0.589374
\(22\) 0 0
\(23\) −4.98262 −1.03895 −0.519474 0.854486i \(-0.673872\pi\)
−0.519474 + 0.854486i \(0.673872\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.10948 −0.983320
\(28\) 0 0
\(29\) 6.72933 1.24961 0.624803 0.780783i \(-0.285179\pi\)
0.624803 + 0.780783i \(0.285179\pi\)
\(30\) 0 0
\(31\) 4.14084 0.743717 0.371858 0.928290i \(-0.378721\pi\)
0.371858 + 0.928290i \(0.378721\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.60210 −0.439835
\(36\) 0 0
\(37\) 4.05133 0.666034 0.333017 0.942921i \(-0.391933\pi\)
0.333017 + 0.942921i \(0.391933\pi\)
\(38\) 0 0
\(39\) −2.98677 −0.478266
\(40\) 0 0
\(41\) 9.76082 1.52438 0.762192 0.647351i \(-0.224123\pi\)
0.762192 + 0.647351i \(0.224123\pi\)
\(42\) 0 0
\(43\) 5.92912 0.904182 0.452091 0.891972i \(-0.350678\pi\)
0.452091 + 0.891972i \(0.350678\pi\)
\(44\) 0 0
\(45\) 1.92266 0.286613
\(46\) 0 0
\(47\) 0.967904 0.141183 0.0705917 0.997505i \(-0.477511\pi\)
0.0705917 + 0.997505i \(0.477511\pi\)
\(48\) 0 0
\(49\) −0.229090 −0.0327272
\(50\) 0 0
\(51\) 0.841046 0.117770
\(52\) 0 0
\(53\) 4.47445 0.614613 0.307307 0.951611i \(-0.400572\pi\)
0.307307 + 0.951611i \(0.400572\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.34617 −0.708117
\(58\) 0 0
\(59\) −6.80756 −0.886269 −0.443135 0.896455i \(-0.646134\pi\)
−0.443135 + 0.896455i \(0.646134\pi\)
\(60\) 0 0
\(61\) −2.61307 −0.334569 −0.167285 0.985909i \(-0.553500\pi\)
−0.167285 + 0.985909i \(0.553500\pi\)
\(62\) 0 0
\(63\) −5.00294 −0.630311
\(64\) 0 0
\(65\) 2.87756 0.356917
\(66\) 0 0
\(67\) 15.8408 1.93526 0.967629 0.252378i \(-0.0812125\pi\)
0.967629 + 0.252378i \(0.0812125\pi\)
\(68\) 0 0
\(69\) −5.17172 −0.622602
\(70\) 0 0
\(71\) 1.41346 0.167746 0.0838732 0.996476i \(-0.473271\pi\)
0.0838732 + 0.996476i \(0.473271\pi\)
\(72\) 0 0
\(73\) −13.1429 −1.53826 −0.769131 0.639091i \(-0.779311\pi\)
−0.769131 + 0.639091i \(0.779311\pi\)
\(74\) 0 0
\(75\) 1.03795 0.119852
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −5.69123 −0.640314 −0.320157 0.947365i \(-0.603736\pi\)
−0.320157 + 0.947365i \(0.603736\pi\)
\(80\) 0 0
\(81\) 0.464568 0.0516187
\(82\) 0 0
\(83\) −7.96967 −0.874785 −0.437392 0.899271i \(-0.644098\pi\)
−0.437392 + 0.899271i \(0.644098\pi\)
\(84\) 0 0
\(85\) −0.810293 −0.0878887
\(86\) 0 0
\(87\) 6.98472 0.748841
\(88\) 0 0
\(89\) −14.5196 −1.53907 −0.769535 0.638605i \(-0.779512\pi\)
−0.769535 + 0.638605i \(0.779512\pi\)
\(90\) 0 0
\(91\) −7.48768 −0.784923
\(92\) 0 0
\(93\) 4.29799 0.445681
\(94\) 0 0
\(95\) 5.15069 0.528449
\(96\) 0 0
\(97\) −2.99325 −0.303919 −0.151959 0.988387i \(-0.548558\pi\)
−0.151959 + 0.988387i \(0.548558\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −15.3700 −1.52938 −0.764688 0.644401i \(-0.777107\pi\)
−0.764688 + 0.644401i \(0.777107\pi\)
\(102\) 0 0
\(103\) −5.49309 −0.541251 −0.270625 0.962685i \(-0.587230\pi\)
−0.270625 + 0.962685i \(0.587230\pi\)
\(104\) 0 0
\(105\) −2.70085 −0.263576
\(106\) 0 0
\(107\) 6.23672 0.602927 0.301463 0.953478i \(-0.402525\pi\)
0.301463 + 0.953478i \(0.402525\pi\)
\(108\) 0 0
\(109\) −19.1460 −1.83385 −0.916927 0.399056i \(-0.869338\pi\)
−0.916927 + 0.399056i \(0.869338\pi\)
\(110\) 0 0
\(111\) 4.20508 0.399129
\(112\) 0 0
\(113\) −10.8681 −1.02238 −0.511192 0.859467i \(-0.670796\pi\)
−0.511192 + 0.859467i \(0.670796\pi\)
\(114\) 0 0
\(115\) 4.98262 0.464632
\(116\) 0 0
\(117\) 5.53255 0.511485
\(118\) 0 0
\(119\) 2.10846 0.193282
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 10.1313 0.913505
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −18.9400 −1.68065 −0.840325 0.542083i \(-0.817636\pi\)
−0.840325 + 0.542083i \(0.817636\pi\)
\(128\) 0 0
\(129\) 6.15414 0.541842
\(130\) 0 0
\(131\) −14.5409 −1.27045 −0.635224 0.772328i \(-0.719092\pi\)
−0.635224 + 0.772328i \(0.719092\pi\)
\(132\) 0 0
\(133\) −13.4026 −1.16215
\(134\) 0 0
\(135\) 5.10948 0.439754
\(136\) 0 0
\(137\) −19.9840 −1.70735 −0.853675 0.520806i \(-0.825632\pi\)
−0.853675 + 0.520806i \(0.825632\pi\)
\(138\) 0 0
\(139\) 2.07071 0.175635 0.0878176 0.996137i \(-0.472011\pi\)
0.0878176 + 0.996137i \(0.472011\pi\)
\(140\) 0 0
\(141\) 1.00464 0.0846058
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −6.72933 −0.558840
\(146\) 0 0
\(147\) −0.237785 −0.0196122
\(148\) 0 0
\(149\) 0.504694 0.0413461 0.0206731 0.999786i \(-0.493419\pi\)
0.0206731 + 0.999786i \(0.493419\pi\)
\(150\) 0 0
\(151\) −12.1188 −0.986216 −0.493108 0.869968i \(-0.664139\pi\)
−0.493108 + 0.869968i \(0.664139\pi\)
\(152\) 0 0
\(153\) −1.55791 −0.125950
\(154\) 0 0
\(155\) −4.14084 −0.332600
\(156\) 0 0
\(157\) −11.8853 −0.948548 −0.474274 0.880377i \(-0.657289\pi\)
−0.474274 + 0.880377i \(0.657289\pi\)
\(158\) 0 0
\(159\) 4.64427 0.368314
\(160\) 0 0
\(161\) −12.9653 −1.02181
\(162\) 0 0
\(163\) −0.0631196 −0.00494391 −0.00247195 0.999997i \(-0.500787\pi\)
−0.00247195 + 0.999997i \(0.500787\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.54352 0.583735 0.291868 0.956459i \(-0.405723\pi\)
0.291868 + 0.956459i \(0.405723\pi\)
\(168\) 0 0
\(169\) −4.71966 −0.363051
\(170\) 0 0
\(171\) 9.90300 0.757301
\(172\) 0 0
\(173\) −1.41262 −0.107400 −0.0536998 0.998557i \(-0.517101\pi\)
−0.0536998 + 0.998557i \(0.517101\pi\)
\(174\) 0 0
\(175\) 2.60210 0.196700
\(176\) 0 0
\(177\) −7.06593 −0.531108
\(178\) 0 0
\(179\) 15.8217 1.18257 0.591285 0.806463i \(-0.298621\pi\)
0.591285 + 0.806463i \(0.298621\pi\)
\(180\) 0 0
\(181\) 18.9773 1.41057 0.705287 0.708922i \(-0.250818\pi\)
0.705287 + 0.708922i \(0.250818\pi\)
\(182\) 0 0
\(183\) −2.71224 −0.200495
\(184\) 0 0
\(185\) −4.05133 −0.297860
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −13.2954 −0.967096
\(190\) 0 0
\(191\) −10.3666 −0.750098 −0.375049 0.927005i \(-0.622374\pi\)
−0.375049 + 0.927005i \(0.622374\pi\)
\(192\) 0 0
\(193\) −0.608652 −0.0438117 −0.0219059 0.999760i \(-0.506973\pi\)
−0.0219059 + 0.999760i \(0.506973\pi\)
\(194\) 0 0
\(195\) 2.98677 0.213887
\(196\) 0 0
\(197\) 11.1521 0.794553 0.397277 0.917699i \(-0.369955\pi\)
0.397277 + 0.917699i \(0.369955\pi\)
\(198\) 0 0
\(199\) −8.91481 −0.631954 −0.315977 0.948767i \(-0.602332\pi\)
−0.315977 + 0.948767i \(0.602332\pi\)
\(200\) 0 0
\(201\) 16.4420 1.15973
\(202\) 0 0
\(203\) 17.5104 1.22899
\(204\) 0 0
\(205\) −9.76082 −0.681725
\(206\) 0 0
\(207\) 9.57987 0.665847
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −17.4046 −1.19818 −0.599092 0.800680i \(-0.704472\pi\)
−0.599092 + 0.800680i \(0.704472\pi\)
\(212\) 0 0
\(213\) 1.46710 0.100524
\(214\) 0 0
\(215\) −5.92912 −0.404362
\(216\) 0 0
\(217\) 10.7749 0.731445
\(218\) 0 0
\(219\) −13.6417 −0.921822
\(220\) 0 0
\(221\) −2.33167 −0.156845
\(222\) 0 0
\(223\) 1.53670 0.102905 0.0514524 0.998675i \(-0.483615\pi\)
0.0514524 + 0.998675i \(0.483615\pi\)
\(224\) 0 0
\(225\) −1.92266 −0.128177
\(226\) 0 0
\(227\) −10.0981 −0.670231 −0.335116 0.942177i \(-0.608775\pi\)
−0.335116 + 0.942177i \(0.608775\pi\)
\(228\) 0 0
\(229\) 2.39276 0.158118 0.0790590 0.996870i \(-0.474808\pi\)
0.0790590 + 0.996870i \(0.474808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −8.28726 −0.542916 −0.271458 0.962450i \(-0.587506\pi\)
−0.271458 + 0.962450i \(0.587506\pi\)
\(234\) 0 0
\(235\) −0.967904 −0.0631391
\(236\) 0 0
\(237\) −5.90723 −0.383716
\(238\) 0 0
\(239\) −4.56206 −0.295095 −0.147548 0.989055i \(-0.547138\pi\)
−0.147548 + 0.989055i \(0.547138\pi\)
\(240\) 0 0
\(241\) 23.4148 1.50828 0.754139 0.656715i \(-0.228054\pi\)
0.754139 + 0.656715i \(0.228054\pi\)
\(242\) 0 0
\(243\) 15.8106 1.01425
\(244\) 0 0
\(245\) 0.229090 0.0146361
\(246\) 0 0
\(247\) 14.8214 0.943063
\(248\) 0 0
\(249\) −8.27214 −0.524225
\(250\) 0 0
\(251\) 27.7446 1.75122 0.875611 0.483017i \(-0.160459\pi\)
0.875611 + 0.483017i \(0.160459\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −0.841046 −0.0526683
\(256\) 0 0
\(257\) −25.4843 −1.58966 −0.794832 0.606829i \(-0.792441\pi\)
−0.794832 + 0.606829i \(0.792441\pi\)
\(258\) 0 0
\(259\) 10.5419 0.655045
\(260\) 0 0
\(261\) −12.9382 −0.800853
\(262\) 0 0
\(263\) −16.1271 −0.994442 −0.497221 0.867624i \(-0.665646\pi\)
−0.497221 + 0.867624i \(0.665646\pi\)
\(264\) 0 0
\(265\) −4.47445 −0.274863
\(266\) 0 0
\(267\) −15.0706 −0.922306
\(268\) 0 0
\(269\) 17.5236 1.06843 0.534216 0.845348i \(-0.320607\pi\)
0.534216 + 0.845348i \(0.320607\pi\)
\(270\) 0 0
\(271\) −7.33746 −0.445719 −0.222860 0.974851i \(-0.571539\pi\)
−0.222860 + 0.974851i \(0.571539\pi\)
\(272\) 0 0
\(273\) −7.77186 −0.470374
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −17.4068 −1.04588 −0.522938 0.852371i \(-0.675164\pi\)
−0.522938 + 0.852371i \(0.675164\pi\)
\(278\) 0 0
\(279\) −7.96140 −0.476637
\(280\) 0 0
\(281\) 13.7827 0.822206 0.411103 0.911589i \(-0.365144\pi\)
0.411103 + 0.911589i \(0.365144\pi\)
\(282\) 0 0
\(283\) −11.5904 −0.688980 −0.344490 0.938790i \(-0.611948\pi\)
−0.344490 + 0.938790i \(0.611948\pi\)
\(284\) 0 0
\(285\) 5.34617 0.316680
\(286\) 0 0
\(287\) 25.3986 1.49923
\(288\) 0 0
\(289\) −16.3434 −0.961378
\(290\) 0 0
\(291\) −3.10685 −0.182127
\(292\) 0 0
\(293\) −16.1449 −0.943197 −0.471598 0.881813i \(-0.656323\pi\)
−0.471598 + 0.881813i \(0.656323\pi\)
\(294\) 0 0
\(295\) 6.80756 0.396352
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 14.3378 0.829176
\(300\) 0 0
\(301\) 15.4281 0.889263
\(302\) 0 0
\(303\) −15.9534 −0.916497
\(304\) 0 0
\(305\) 2.61307 0.149624
\(306\) 0 0
\(307\) 14.3107 0.816753 0.408377 0.912814i \(-0.366095\pi\)
0.408377 + 0.912814i \(0.366095\pi\)
\(308\) 0 0
\(309\) −5.70157 −0.324351
\(310\) 0 0
\(311\) 22.3744 1.26873 0.634367 0.773032i \(-0.281261\pi\)
0.634367 + 0.773032i \(0.281261\pi\)
\(312\) 0 0
\(313\) 11.5368 0.652100 0.326050 0.945352i \(-0.394282\pi\)
0.326050 + 0.945352i \(0.394282\pi\)
\(314\) 0 0
\(315\) 5.00294 0.281883
\(316\) 0 0
\(317\) −25.3314 −1.42275 −0.711376 0.702812i \(-0.751928\pi\)
−0.711376 + 0.702812i \(0.751928\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 6.47342 0.361311
\(322\) 0 0
\(323\) −4.17357 −0.232224
\(324\) 0 0
\(325\) −2.87756 −0.159618
\(326\) 0 0
\(327\) −19.8726 −1.09896
\(328\) 0 0
\(329\) 2.51858 0.138854
\(330\) 0 0
\(331\) 20.9669 1.15244 0.576221 0.817294i \(-0.304527\pi\)
0.576221 + 0.817294i \(0.304527\pi\)
\(332\) 0 0
\(333\) −7.78931 −0.426851
\(334\) 0 0
\(335\) −15.8408 −0.865474
\(336\) 0 0
\(337\) −18.1430 −0.988311 −0.494156 0.869373i \(-0.664523\pi\)
−0.494156 + 0.869373i \(0.664523\pi\)
\(338\) 0 0
\(339\) −11.2806 −0.612675
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −18.8108 −1.01569
\(344\) 0 0
\(345\) 5.17172 0.278436
\(346\) 0 0
\(347\) −18.9900 −1.01944 −0.509718 0.860342i \(-0.670250\pi\)
−0.509718 + 0.860342i \(0.670250\pi\)
\(348\) 0 0
\(349\) 16.1172 0.862732 0.431366 0.902177i \(-0.358032\pi\)
0.431366 + 0.902177i \(0.358032\pi\)
\(350\) 0 0
\(351\) 14.7028 0.784779
\(352\) 0 0
\(353\) 28.0252 1.49163 0.745816 0.666152i \(-0.232060\pi\)
0.745816 + 0.666152i \(0.232060\pi\)
\(354\) 0 0
\(355\) −1.41346 −0.0750185
\(356\) 0 0
\(357\) 2.18848 0.115827
\(358\) 0 0
\(359\) −5.23896 −0.276502 −0.138251 0.990397i \(-0.544148\pi\)
−0.138251 + 0.990397i \(0.544148\pi\)
\(360\) 0 0
\(361\) 7.52958 0.396294
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 13.1429 0.687932
\(366\) 0 0
\(367\) 26.3079 1.37326 0.686632 0.727005i \(-0.259089\pi\)
0.686632 + 0.727005i \(0.259089\pi\)
\(368\) 0 0
\(369\) −18.7667 −0.976955
\(370\) 0 0
\(371\) 11.6430 0.604472
\(372\) 0 0
\(373\) −36.0081 −1.86443 −0.932215 0.361905i \(-0.882127\pi\)
−0.932215 + 0.361905i \(0.882127\pi\)
\(374\) 0 0
\(375\) −1.03795 −0.0535996
\(376\) 0 0
\(377\) −19.3640 −0.997298
\(378\) 0 0
\(379\) 14.0945 0.723986 0.361993 0.932181i \(-0.382096\pi\)
0.361993 + 0.932181i \(0.382096\pi\)
\(380\) 0 0
\(381\) −19.6588 −1.00715
\(382\) 0 0
\(383\) −2.14120 −0.109410 −0.0547052 0.998503i \(-0.517422\pi\)
−0.0547052 + 0.998503i \(0.517422\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −11.3996 −0.579477
\(388\) 0 0
\(389\) −29.2825 −1.48468 −0.742340 0.670023i \(-0.766284\pi\)
−0.742340 + 0.670023i \(0.766284\pi\)
\(390\) 0 0
\(391\) −4.03739 −0.204179
\(392\) 0 0
\(393\) −15.0928 −0.761331
\(394\) 0 0
\(395\) 5.69123 0.286357
\(396\) 0 0
\(397\) 9.09620 0.456525 0.228263 0.973600i \(-0.426696\pi\)
0.228263 + 0.973600i \(0.426696\pi\)
\(398\) 0 0
\(399\) −13.9112 −0.696434
\(400\) 0 0
\(401\) 32.3614 1.61605 0.808027 0.589146i \(-0.200536\pi\)
0.808027 + 0.589146i \(0.200536\pi\)
\(402\) 0 0
\(403\) −11.9155 −0.593553
\(404\) 0 0
\(405\) −0.464568 −0.0230846
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −20.8620 −1.03156 −0.515779 0.856722i \(-0.672498\pi\)
−0.515779 + 0.856722i \(0.672498\pi\)
\(410\) 0 0
\(411\) −20.7425 −1.02315
\(412\) 0 0
\(413\) −17.7139 −0.871646
\(414\) 0 0
\(415\) 7.96967 0.391216
\(416\) 0 0
\(417\) 2.14930 0.105251
\(418\) 0 0
\(419\) 36.4444 1.78043 0.890213 0.455545i \(-0.150556\pi\)
0.890213 + 0.455545i \(0.150556\pi\)
\(420\) 0 0
\(421\) −32.4285 −1.58047 −0.790235 0.612804i \(-0.790041\pi\)
−0.790235 + 0.612804i \(0.790041\pi\)
\(422\) 0 0
\(423\) −1.86095 −0.0904823
\(424\) 0 0
\(425\) 0.810293 0.0393050
\(426\) 0 0
\(427\) −6.79946 −0.329049
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −28.8732 −1.39077 −0.695387 0.718636i \(-0.744767\pi\)
−0.695387 + 0.718636i \(0.744767\pi\)
\(432\) 0 0
\(433\) 7.55542 0.363090 0.181545 0.983383i \(-0.441890\pi\)
0.181545 + 0.983383i \(0.441890\pi\)
\(434\) 0 0
\(435\) −6.98472 −0.334892
\(436\) 0 0
\(437\) 25.6639 1.22767
\(438\) 0 0
\(439\) 1.96072 0.0935801 0.0467900 0.998905i \(-0.485101\pi\)
0.0467900 + 0.998905i \(0.485101\pi\)
\(440\) 0 0
\(441\) 0.440462 0.0209744
\(442\) 0 0
\(443\) −6.03304 −0.286638 −0.143319 0.989677i \(-0.545778\pi\)
−0.143319 + 0.989677i \(0.545778\pi\)
\(444\) 0 0
\(445\) 14.5196 0.688293
\(446\) 0 0
\(447\) 0.523848 0.0247772
\(448\) 0 0
\(449\) 9.82399 0.463623 0.231811 0.972761i \(-0.425535\pi\)
0.231811 + 0.972761i \(0.425535\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −12.5788 −0.591002
\(454\) 0 0
\(455\) 7.48768 0.351028
\(456\) 0 0
\(457\) −7.82880 −0.366216 −0.183108 0.983093i \(-0.558616\pi\)
−0.183108 + 0.983093i \(0.558616\pi\)
\(458\) 0 0
\(459\) −4.14018 −0.193247
\(460\) 0 0
\(461\) −3.19972 −0.149026 −0.0745130 0.997220i \(-0.523740\pi\)
−0.0745130 + 0.997220i \(0.523740\pi\)
\(462\) 0 0
\(463\) 21.2096 0.985696 0.492848 0.870115i \(-0.335956\pi\)
0.492848 + 0.870115i \(0.335956\pi\)
\(464\) 0 0
\(465\) −4.29799 −0.199315
\(466\) 0 0
\(467\) 31.2967 1.44824 0.724118 0.689676i \(-0.242247\pi\)
0.724118 + 0.689676i \(0.242247\pi\)
\(468\) 0 0
\(469\) 41.2192 1.90333
\(470\) 0 0
\(471\) −12.3363 −0.568429
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −5.15069 −0.236330
\(476\) 0 0
\(477\) −8.60283 −0.393896
\(478\) 0 0
\(479\) −18.0022 −0.822542 −0.411271 0.911513i \(-0.634915\pi\)
−0.411271 + 0.911513i \(0.634915\pi\)
\(480\) 0 0
\(481\) −11.6579 −0.531556
\(482\) 0 0
\(483\) −13.4573 −0.612330
\(484\) 0 0
\(485\) 2.99325 0.135917
\(486\) 0 0
\(487\) −21.0873 −0.955558 −0.477779 0.878480i \(-0.658558\pi\)
−0.477779 + 0.878480i \(0.658558\pi\)
\(488\) 0 0
\(489\) −0.0655151 −0.00296270
\(490\) 0 0
\(491\) 22.0057 0.993104 0.496552 0.868007i \(-0.334599\pi\)
0.496552 + 0.868007i \(0.334599\pi\)
\(492\) 0 0
\(493\) 5.45273 0.245579
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.67795 0.164979
\(498\) 0 0
\(499\) −16.1255 −0.721877 −0.360939 0.932590i \(-0.617544\pi\)
−0.360939 + 0.932590i \(0.617544\pi\)
\(500\) 0 0
\(501\) 7.82982 0.349810
\(502\) 0 0
\(503\) −3.16694 −0.141207 −0.0706035 0.997504i \(-0.522493\pi\)
−0.0706035 + 0.997504i \(0.522493\pi\)
\(504\) 0 0
\(505\) 15.3700 0.683957
\(506\) 0 0
\(507\) −4.89878 −0.217563
\(508\) 0 0
\(509\) 28.9843 1.28470 0.642352 0.766409i \(-0.277959\pi\)
0.642352 + 0.766409i \(0.277959\pi\)
\(510\) 0 0
\(511\) −34.1992 −1.51288
\(512\) 0 0
\(513\) 26.3173 1.16194
\(514\) 0 0
\(515\) 5.49309 0.242055
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −1.46623 −0.0643605
\(520\) 0 0
\(521\) −15.8497 −0.694388 −0.347194 0.937793i \(-0.612865\pi\)
−0.347194 + 0.937793i \(0.612865\pi\)
\(522\) 0 0
\(523\) 24.0929 1.05351 0.526754 0.850018i \(-0.323409\pi\)
0.526754 + 0.850018i \(0.323409\pi\)
\(524\) 0 0
\(525\) 2.70085 0.117875
\(526\) 0 0
\(527\) 3.35529 0.146159
\(528\) 0 0
\(529\) 1.82653 0.0794143
\(530\) 0 0
\(531\) 13.0886 0.567997
\(532\) 0 0
\(533\) −28.0873 −1.21660
\(534\) 0 0
\(535\) −6.23672 −0.269637
\(536\) 0 0
\(537\) 16.4222 0.708669
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −21.3207 −0.916648 −0.458324 0.888785i \(-0.651550\pi\)
−0.458324 + 0.888785i \(0.651550\pi\)
\(542\) 0 0
\(543\) 19.6976 0.845303
\(544\) 0 0
\(545\) 19.1460 0.820124
\(546\) 0 0
\(547\) 29.3180 1.25355 0.626775 0.779201i \(-0.284375\pi\)
0.626775 + 0.779201i \(0.284375\pi\)
\(548\) 0 0
\(549\) 5.02403 0.214420
\(550\) 0 0
\(551\) −34.6607 −1.47659
\(552\) 0 0
\(553\) −14.8091 −0.629749
\(554\) 0 0
\(555\) −4.20508 −0.178496
\(556\) 0 0
\(557\) −1.68279 −0.0713019 −0.0356510 0.999364i \(-0.511350\pi\)
−0.0356510 + 0.999364i \(0.511350\pi\)
\(558\) 0 0
\(559\) −17.0614 −0.721619
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.14833 −0.0483963 −0.0241981 0.999707i \(-0.507703\pi\)
−0.0241981 + 0.999707i \(0.507703\pi\)
\(564\) 0 0
\(565\) 10.8681 0.457224
\(566\) 0 0
\(567\) 1.20885 0.0507670
\(568\) 0 0
\(569\) −6.68468 −0.280237 −0.140118 0.990135i \(-0.544748\pi\)
−0.140118 + 0.990135i \(0.544748\pi\)
\(570\) 0 0
\(571\) −17.7964 −0.744756 −0.372378 0.928081i \(-0.621458\pi\)
−0.372378 + 0.928081i \(0.621458\pi\)
\(572\) 0 0
\(573\) −10.7600 −0.449505
\(574\) 0 0
\(575\) −4.98262 −0.207790
\(576\) 0 0
\(577\) −45.9807 −1.91420 −0.957102 0.289752i \(-0.906427\pi\)
−0.957102 + 0.289752i \(0.906427\pi\)
\(578\) 0 0
\(579\) −0.631752 −0.0262547
\(580\) 0 0
\(581\) −20.7379 −0.860351
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −5.53255 −0.228743
\(586\) 0 0
\(587\) −4.72240 −0.194914 −0.0974571 0.995240i \(-0.531071\pi\)
−0.0974571 + 0.995240i \(0.531071\pi\)
\(588\) 0 0
\(589\) −21.3282 −0.878812
\(590\) 0 0
\(591\) 11.5753 0.476146
\(592\) 0 0
\(593\) 3.67545 0.150932 0.0754662 0.997148i \(-0.475955\pi\)
0.0754662 + 0.997148i \(0.475955\pi\)
\(594\) 0 0
\(595\) −2.10846 −0.0864385
\(596\) 0 0
\(597\) −9.25315 −0.378706
\(598\) 0 0
\(599\) 39.8714 1.62910 0.814550 0.580093i \(-0.196984\pi\)
0.814550 + 0.580093i \(0.196984\pi\)
\(600\) 0 0
\(601\) −12.3476 −0.503670 −0.251835 0.967770i \(-0.581034\pi\)
−0.251835 + 0.967770i \(0.581034\pi\)
\(602\) 0 0
\(603\) −30.4563 −1.24028
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 19.7067 0.799869 0.399935 0.916544i \(-0.369033\pi\)
0.399935 + 0.916544i \(0.369033\pi\)
\(608\) 0 0
\(609\) 18.1749 0.736485
\(610\) 0 0
\(611\) −2.78520 −0.112677
\(612\) 0 0
\(613\) 37.7646 1.52530 0.762650 0.646812i \(-0.223898\pi\)
0.762650 + 0.646812i \(0.223898\pi\)
\(614\) 0 0
\(615\) −10.1313 −0.408532
\(616\) 0 0
\(617\) −3.49353 −0.140644 −0.0703222 0.997524i \(-0.522403\pi\)
−0.0703222 + 0.997524i \(0.522403\pi\)
\(618\) 0 0
\(619\) −6.48840 −0.260791 −0.130395 0.991462i \(-0.541625\pi\)
−0.130395 + 0.991462i \(0.541625\pi\)
\(620\) 0 0
\(621\) 25.4586 1.02162
\(622\) 0 0
\(623\) −37.7813 −1.51368
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3.28276 0.130892
\(630\) 0 0
\(631\) −32.7333 −1.30309 −0.651547 0.758609i \(-0.725880\pi\)
−0.651547 + 0.758609i \(0.725880\pi\)
\(632\) 0 0
\(633\) −18.0652 −0.718026
\(634\) 0 0
\(635\) 18.9400 0.751609
\(636\) 0 0
\(637\) 0.659221 0.0261193
\(638\) 0 0
\(639\) −2.71759 −0.107506
\(640\) 0 0
\(641\) −7.19375 −0.284136 −0.142068 0.989857i \(-0.545375\pi\)
−0.142068 + 0.989857i \(0.545375\pi\)
\(642\) 0 0
\(643\) −8.21093 −0.323808 −0.161904 0.986807i \(-0.551763\pi\)
−0.161904 + 0.986807i \(0.551763\pi\)
\(644\) 0 0
\(645\) −6.15414 −0.242319
\(646\) 0 0
\(647\) 49.0008 1.92642 0.963210 0.268750i \(-0.0866106\pi\)
0.963210 + 0.268750i \(0.0866106\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 11.1838 0.438327
\(652\) 0 0
\(653\) −43.8455 −1.71581 −0.857903 0.513811i \(-0.828233\pi\)
−0.857903 + 0.513811i \(0.828233\pi\)
\(654\) 0 0
\(655\) 14.5409 0.568161
\(656\) 0 0
\(657\) 25.2693 0.985850
\(658\) 0 0
\(659\) 37.2905 1.45263 0.726315 0.687362i \(-0.241231\pi\)
0.726315 + 0.687362i \(0.241231\pi\)
\(660\) 0 0
\(661\) 35.5579 1.38304 0.691520 0.722357i \(-0.256941\pi\)
0.691520 + 0.722357i \(0.256941\pi\)
\(662\) 0 0
\(663\) −2.42016 −0.0939912
\(664\) 0 0
\(665\) 13.4026 0.519730
\(666\) 0 0
\(667\) −33.5297 −1.29828
\(668\) 0 0
\(669\) 1.59502 0.0616669
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −13.6219 −0.525084 −0.262542 0.964921i \(-0.584561\pi\)
−0.262542 + 0.964921i \(0.584561\pi\)
\(674\) 0 0
\(675\) −5.10948 −0.196664
\(676\) 0 0
\(677\) 6.89247 0.264899 0.132449 0.991190i \(-0.457716\pi\)
0.132449 + 0.991190i \(0.457716\pi\)
\(678\) 0 0
\(679\) −7.78874 −0.298904
\(680\) 0 0
\(681\) −10.4813 −0.401644
\(682\) 0 0
\(683\) −8.29642 −0.317454 −0.158727 0.987323i \(-0.550739\pi\)
−0.158727 + 0.987323i \(0.550739\pi\)
\(684\) 0 0
\(685\) 19.9840 0.763550
\(686\) 0 0
\(687\) 2.48357 0.0947541
\(688\) 0 0
\(689\) −12.8755 −0.490517
\(690\) 0 0
\(691\) −18.3562 −0.698303 −0.349151 0.937066i \(-0.613530\pi\)
−0.349151 + 0.937066i \(0.613530\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2.07071 −0.0785464
\(696\) 0 0
\(697\) 7.90913 0.299580
\(698\) 0 0
\(699\) −8.60178 −0.325349
\(700\) 0 0
\(701\) −28.0624 −1.05990 −0.529951 0.848028i \(-0.677790\pi\)
−0.529951 + 0.848028i \(0.677790\pi\)
\(702\) 0 0
\(703\) −20.8671 −0.787018
\(704\) 0 0
\(705\) −1.00464 −0.0378369
\(706\) 0 0
\(707\) −39.9943 −1.50414
\(708\) 0 0
\(709\) −16.2588 −0.610612 −0.305306 0.952254i \(-0.598759\pi\)
−0.305306 + 0.952254i \(0.598759\pi\)
\(710\) 0 0
\(711\) 10.9423 0.410368
\(712\) 0 0
\(713\) −20.6322 −0.772683
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −4.73520 −0.176839
\(718\) 0 0
\(719\) 3.75469 0.140026 0.0700132 0.997546i \(-0.477696\pi\)
0.0700132 + 0.997546i \(0.477696\pi\)
\(720\) 0 0
\(721\) −14.2936 −0.532320
\(722\) 0 0
\(723\) 24.3034 0.903854
\(724\) 0 0
\(725\) 6.72933 0.249921
\(726\) 0 0
\(727\) −35.1985 −1.30544 −0.652720 0.757599i \(-0.726372\pi\)
−0.652720 + 0.757599i \(0.726372\pi\)
\(728\) 0 0
\(729\) 15.0170 0.556185
\(730\) 0 0
\(731\) 4.80432 0.177694
\(732\) 0 0
\(733\) −20.2869 −0.749313 −0.374656 0.927164i \(-0.622239\pi\)
−0.374656 + 0.927164i \(0.622239\pi\)
\(734\) 0 0
\(735\) 0.237785 0.00877083
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −15.6336 −0.575093 −0.287546 0.957767i \(-0.592840\pi\)
−0.287546 + 0.957767i \(0.592840\pi\)
\(740\) 0 0
\(741\) 15.3839 0.565142
\(742\) 0 0
\(743\) 19.9836 0.733129 0.366564 0.930393i \(-0.380534\pi\)
0.366564 + 0.930393i \(0.380534\pi\)
\(744\) 0 0
\(745\) −0.504694 −0.0184905
\(746\) 0 0
\(747\) 15.3229 0.560637
\(748\) 0 0
\(749\) 16.2286 0.592979
\(750\) 0 0
\(751\) −31.4457 −1.14747 −0.573734 0.819041i \(-0.694506\pi\)
−0.573734 + 0.819041i \(0.694506\pi\)
\(752\) 0 0
\(753\) 28.7975 1.04944
\(754\) 0 0
\(755\) 12.1188 0.441049
\(756\) 0 0
\(757\) 45.3850 1.64955 0.824773 0.565463i \(-0.191303\pi\)
0.824773 + 0.565463i \(0.191303\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 35.6102 1.29087 0.645435 0.763816i \(-0.276676\pi\)
0.645435 + 0.763816i \(0.276676\pi\)
\(762\) 0 0
\(763\) −49.8197 −1.80360
\(764\) 0 0
\(765\) 1.55791 0.0563265
\(766\) 0 0
\(767\) 19.5892 0.707324
\(768\) 0 0
\(769\) −18.9907 −0.684821 −0.342411 0.939550i \(-0.611243\pi\)
−0.342411 + 0.939550i \(0.611243\pi\)
\(770\) 0 0
\(771\) −26.4514 −0.952625
\(772\) 0 0
\(773\) 43.6808 1.57109 0.785544 0.618805i \(-0.212383\pi\)
0.785544 + 0.618805i \(0.212383\pi\)
\(774\) 0 0
\(775\) 4.14084 0.148743
\(776\) 0 0
\(777\) 10.9420 0.392543
\(778\) 0 0
\(779\) −50.2749 −1.80129
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −34.3834 −1.22876
\(784\) 0 0
\(785\) 11.8853 0.424204
\(786\) 0 0
\(787\) 50.8854 1.81387 0.906934 0.421272i \(-0.138416\pi\)
0.906934 + 0.421272i \(0.138416\pi\)
\(788\) 0 0
\(789\) −16.7392 −0.595931
\(790\) 0 0
\(791\) −28.2798 −1.00551
\(792\) 0 0
\(793\) 7.51925 0.267017
\(794\) 0 0
\(795\) −4.64427 −0.164715
\(796\) 0 0
\(797\) −18.9156 −0.670025 −0.335013 0.942214i \(-0.608741\pi\)
−0.335013 + 0.942214i \(0.608741\pi\)
\(798\) 0 0
\(799\) 0.784286 0.0277461
\(800\) 0 0
\(801\) 27.9161 0.986367
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 12.9653 0.456966
\(806\) 0 0
\(807\) 18.1887 0.640271
\(808\) 0 0
\(809\) 20.8204 0.732005 0.366002 0.930614i \(-0.380726\pi\)
0.366002 + 0.930614i \(0.380726\pi\)
\(810\) 0 0
\(811\) 36.0642 1.26639 0.633193 0.773994i \(-0.281744\pi\)
0.633193 + 0.773994i \(0.281744\pi\)
\(812\) 0 0
\(813\) −7.61594 −0.267103
\(814\) 0 0
\(815\) 0.0631196 0.00221098
\(816\) 0 0
\(817\) −30.5390 −1.06843
\(818\) 0 0
\(819\) 14.3962 0.503045
\(820\) 0 0
\(821\) −37.9278 −1.32369 −0.661845 0.749641i \(-0.730226\pi\)
−0.661845 + 0.749641i \(0.730226\pi\)
\(822\) 0 0
\(823\) 47.3524 1.65060 0.825300 0.564695i \(-0.191006\pi\)
0.825300 + 0.564695i \(0.191006\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 32.7566 1.13906 0.569530 0.821970i \(-0.307125\pi\)
0.569530 + 0.821970i \(0.307125\pi\)
\(828\) 0 0
\(829\) −24.3065 −0.844199 −0.422099 0.906550i \(-0.638707\pi\)
−0.422099 + 0.906550i \(0.638707\pi\)
\(830\) 0 0
\(831\) −18.0675 −0.626753
\(832\) 0 0
\(833\) −0.185630 −0.00643171
\(834\) 0 0
\(835\) −7.54352 −0.261054
\(836\) 0 0
\(837\) −21.1575 −0.731311
\(838\) 0 0
\(839\) 38.6436 1.33413 0.667063 0.745001i \(-0.267551\pi\)
0.667063 + 0.745001i \(0.267551\pi\)
\(840\) 0 0
\(841\) 16.2839 0.561513
\(842\) 0 0
\(843\) 14.3058 0.492717
\(844\) 0 0
\(845\) 4.71966 0.162361
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −12.0303 −0.412879
\(850\) 0 0
\(851\) −20.1862 −0.691975
\(852\) 0 0
\(853\) 28.7982 0.986033 0.493017 0.870020i \(-0.335894\pi\)
0.493017 + 0.870020i \(0.335894\pi\)
\(854\) 0 0
\(855\) −9.90300 −0.338675
\(856\) 0 0
\(857\) −44.6116 −1.52390 −0.761952 0.647633i \(-0.775759\pi\)
−0.761952 + 0.647633i \(0.775759\pi\)
\(858\) 0 0
\(859\) −46.4941 −1.58636 −0.793180 0.608988i \(-0.791576\pi\)
−0.793180 + 0.608988i \(0.791576\pi\)
\(860\) 0 0
\(861\) 26.3625 0.898433
\(862\) 0 0
\(863\) −15.6647 −0.533231 −0.266616 0.963803i \(-0.585905\pi\)
−0.266616 + 0.963803i \(0.585905\pi\)
\(864\) 0 0
\(865\) 1.41262 0.0480306
\(866\) 0 0
\(867\) −16.9637 −0.576117
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −45.5827 −1.54451
\(872\) 0 0
\(873\) 5.75499 0.194777
\(874\) 0 0
\(875\) −2.60210 −0.0879669
\(876\) 0 0
\(877\) −33.6285 −1.13555 −0.567777 0.823182i \(-0.692196\pi\)
−0.567777 + 0.823182i \(0.692196\pi\)
\(878\) 0 0
\(879\) −16.7577 −0.565222
\(880\) 0 0
\(881\) 42.4530 1.43028 0.715139 0.698982i \(-0.246363\pi\)
0.715139 + 0.698982i \(0.246363\pi\)
\(882\) 0 0
\(883\) 8.63358 0.290543 0.145272 0.989392i \(-0.453594\pi\)
0.145272 + 0.989392i \(0.453594\pi\)
\(884\) 0 0
\(885\) 7.06593 0.237519
\(886\) 0 0
\(887\) 4.32578 0.145246 0.0726228 0.997359i \(-0.476863\pi\)
0.0726228 + 0.997359i \(0.476863\pi\)
\(888\) 0 0
\(889\) −49.2836 −1.65292
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −4.98537 −0.166829
\(894\) 0 0
\(895\) −15.8217 −0.528862
\(896\) 0 0
\(897\) 14.8819 0.496893
\(898\) 0 0
\(899\) 27.8651 0.929352
\(900\) 0 0
\(901\) 3.62562 0.120787
\(902\) 0 0
\(903\) 16.0137 0.532902
\(904\) 0 0
\(905\) −18.9773 −0.630828
\(906\) 0 0
\(907\) 47.4842 1.57669 0.788344 0.615234i \(-0.210939\pi\)
0.788344 + 0.615234i \(0.210939\pi\)
\(908\) 0 0
\(909\) 29.5513 0.980154
\(910\) 0 0
\(911\) 22.7476 0.753662 0.376831 0.926282i \(-0.377014\pi\)
0.376831 + 0.926282i \(0.377014\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 2.71224 0.0896639
\(916\) 0 0
\(917\) −37.8369 −1.24949
\(918\) 0 0
\(919\) −37.0964 −1.22370 −0.611849 0.790975i \(-0.709574\pi\)
−0.611849 + 0.790975i \(0.709574\pi\)
\(920\) 0 0
\(921\) 14.8538 0.489449
\(922\) 0 0
\(923\) −4.06730 −0.133877
\(924\) 0 0
\(925\) 4.05133 0.133207
\(926\) 0 0
\(927\) 10.5613 0.346879
\(928\) 0 0
\(929\) 43.0548 1.41258 0.706290 0.707922i \(-0.250367\pi\)
0.706290 + 0.707922i \(0.250367\pi\)
\(930\) 0 0
\(931\) 1.17997 0.0386721
\(932\) 0 0
\(933\) 23.2235 0.760304
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 23.5075 0.767956 0.383978 0.923342i \(-0.374554\pi\)
0.383978 + 0.923342i \(0.374554\pi\)
\(938\) 0 0
\(939\) 11.9747 0.390779
\(940\) 0 0
\(941\) −37.1410 −1.21076 −0.605382 0.795935i \(-0.706979\pi\)
−0.605382 + 0.795935i \(0.706979\pi\)
\(942\) 0 0
\(943\) −48.6345 −1.58376
\(944\) 0 0
\(945\) 13.2954 0.432498
\(946\) 0 0
\(947\) 16.3128 0.530094 0.265047 0.964235i \(-0.414612\pi\)
0.265047 + 0.964235i \(0.414612\pi\)
\(948\) 0 0
\(949\) 37.8195 1.22767
\(950\) 0 0
\(951\) −26.2928 −0.852601
\(952\) 0 0
\(953\) 58.7967 1.90461 0.952306 0.305146i \(-0.0987052\pi\)
0.952306 + 0.305146i \(0.0987052\pi\)
\(954\) 0 0
\(955\) 10.3666 0.335454
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −52.0004 −1.67918
\(960\) 0 0
\(961\) −13.8535 −0.446886
\(962\) 0 0
\(963\) −11.9911 −0.386407
\(964\) 0 0
\(965\) 0.608652 0.0195932
\(966\) 0 0
\(967\) 60.4542 1.94408 0.972039 0.234821i \(-0.0754502\pi\)
0.972039 + 0.234821i \(0.0754502\pi\)
\(968\) 0 0
\(969\) −4.33196 −0.139163
\(970\) 0 0
\(971\) −41.6267 −1.33586 −0.667932 0.744222i \(-0.732820\pi\)
−0.667932 + 0.744222i \(0.732820\pi\)
\(972\) 0 0
\(973\) 5.38818 0.172737
\(974\) 0 0
\(975\) −2.98677 −0.0956531
\(976\) 0 0
\(977\) 47.7290 1.52699 0.763493 0.645816i \(-0.223483\pi\)
0.763493 + 0.645816i \(0.223483\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 36.8111 1.17529
\(982\) 0 0
\(983\) 5.92744 0.189056 0.0945280 0.995522i \(-0.469866\pi\)
0.0945280 + 0.995522i \(0.469866\pi\)
\(984\) 0 0
\(985\) −11.1521 −0.355335
\(986\) 0 0
\(987\) 2.61417 0.0832098
\(988\) 0 0
\(989\) −29.5425 −0.939398
\(990\) 0 0
\(991\) −34.5440 −1.09733 −0.548663 0.836044i \(-0.684863\pi\)
−0.548663 + 0.836044i \(0.684863\pi\)
\(992\) 0 0
\(993\) 21.7626 0.690615
\(994\) 0 0
\(995\) 8.91481 0.282618
\(996\) 0 0
\(997\) 34.4542 1.09117 0.545587 0.838054i \(-0.316307\pi\)
0.545587 + 0.838054i \(0.316307\pi\)
\(998\) 0 0
\(999\) −20.7002 −0.654925
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 9680.2.a.dc.1.4 6
4.3 odd 2 4840.2.a.bb.1.3 6
11.5 even 5 880.2.bo.i.641.2 12
11.9 even 5 880.2.bo.i.81.2 12
11.10 odd 2 9680.2.a.dd.1.4 6
44.27 odd 10 440.2.y.c.201.2 yes 12
44.31 odd 10 440.2.y.c.81.2 12
44.43 even 2 4840.2.a.ba.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.c.81.2 12 44.31 odd 10
440.2.y.c.201.2 yes 12 44.27 odd 10
880.2.bo.i.81.2 12 11.9 even 5
880.2.bo.i.641.2 12 11.5 even 5
4840.2.a.ba.1.3 6 44.43 even 2
4840.2.a.bb.1.3 6 4.3 odd 2
9680.2.a.dc.1.4 6 1.1 even 1 trivial
9680.2.a.dd.1.4 6 11.10 odd 2