Properties

Label 4600.2.e.q.4049.6
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4600,2,Mod(4049,4600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4600.4049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.3356224.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 8x^{4} + 16x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.6
Root \(-2.11491i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.q.4049.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.47283i q^{3} -0.527166i q^{7} -3.11491 q^{9} +O(q^{10})\) \(q+2.47283i q^{3} -0.527166i q^{7} -3.11491 q^{9} +3.11491 q^{11} +4.11491i q^{13} +4.39905i q^{17} +3.70265 q^{19} +1.30359 q^{21} -1.00000i q^{23} -0.284147i q^{27} +9.10170 q^{29} +4.83076 q^{31} +7.70265i q^{33} -9.74378i q^{37} -10.1755 q^{39} +6.93246 q^{41} -4.45963i q^{43} -0.642074i q^{47} +6.72210 q^{49} -10.8781 q^{51} -3.89134i q^{53} +9.15604i q^{57} -8.79811 q^{59} -3.45339 q^{61} +1.64207i q^{63} +8.60719i q^{67} +2.47283 q^{69} +12.3642 q^{71} -5.81756i q^{73} -1.64207i q^{77} +3.17548 q^{79} -8.64207 q^{81} +4.71585i q^{83} +22.5070i q^{87} +5.43171 q^{89} +2.16924 q^{91} +11.9457i q^{93} +4.06058i q^{97} -9.70265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{9} + 6 q^{11} - 14 q^{19} - 12 q^{21} + 2 q^{29} + 20 q^{31} - 14 q^{39} - 20 q^{41} - 12 q^{49} + 18 q^{51} - 20 q^{59} - 26 q^{61} + 4 q^{69} + 20 q^{71} - 28 q^{79} - 50 q^{81} + 40 q^{89} + 22 q^{91} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.47283i 1.42769i 0.700303 + 0.713846i \(0.253048\pi\)
−0.700303 + 0.713846i \(0.746952\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 0.527166i − 0.199250i −0.995025 0.0996250i \(-0.968236\pi\)
0.995025 0.0996250i \(-0.0317643\pi\)
\(8\) 0 0
\(9\) −3.11491 −1.03830
\(10\) 0 0
\(11\) 3.11491 0.939180 0.469590 0.882885i \(-0.344402\pi\)
0.469590 + 0.882885i \(0.344402\pi\)
\(12\) 0 0
\(13\) 4.11491i 1.14127i 0.821204 + 0.570635i \(0.193303\pi\)
−0.821204 + 0.570635i \(0.806697\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.39905i 1.06693i 0.845823 + 0.533464i \(0.179110\pi\)
−0.845823 + 0.533464i \(0.820890\pi\)
\(18\) 0 0
\(19\) 3.70265 0.849446 0.424723 0.905323i \(-0.360372\pi\)
0.424723 + 0.905323i \(0.360372\pi\)
\(20\) 0 0
\(21\) 1.30359 0.284468
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 0.284147i − 0.0546842i
\(28\) 0 0
\(29\) 9.10170 1.69014 0.845072 0.534653i \(-0.179558\pi\)
0.845072 + 0.534653i \(0.179558\pi\)
\(30\) 0 0
\(31\) 4.83076 0.867630 0.433815 0.901002i \(-0.357167\pi\)
0.433815 + 0.901002i \(0.357167\pi\)
\(32\) 0 0
\(33\) 7.70265i 1.34086i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 9.74378i − 1.60187i −0.598753 0.800934i \(-0.704337\pi\)
0.598753 0.800934i \(-0.295663\pi\)
\(38\) 0 0
\(39\) −10.1755 −1.62938
\(40\) 0 0
\(41\) 6.93246 1.08267 0.541334 0.840807i \(-0.317919\pi\)
0.541334 + 0.840807i \(0.317919\pi\)
\(42\) 0 0
\(43\) − 4.45963i − 0.680087i −0.940410 0.340044i \(-0.889558\pi\)
0.940410 0.340044i \(-0.110442\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 0.642074i − 0.0936561i −0.998903 0.0468280i \(-0.985089\pi\)
0.998903 0.0468280i \(-0.0149113\pi\)
\(48\) 0 0
\(49\) 6.72210 0.960299
\(50\) 0 0
\(51\) −10.8781 −1.52324
\(52\) 0 0
\(53\) − 3.89134i − 0.534516i −0.963625 0.267258i \(-0.913882\pi\)
0.963625 0.267258i \(-0.0861176\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 9.15604i 1.21275i
\(58\) 0 0
\(59\) −8.79811 −1.14542 −0.572708 0.819759i \(-0.694107\pi\)
−0.572708 + 0.819759i \(0.694107\pi\)
\(60\) 0 0
\(61\) −3.45339 −0.442161 −0.221080 0.975256i \(-0.570958\pi\)
−0.221080 + 0.975256i \(0.570958\pi\)
\(62\) 0 0
\(63\) 1.64207i 0.206882i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.60719i 1.05154i 0.850628 + 0.525768i \(0.176222\pi\)
−0.850628 + 0.525768i \(0.823778\pi\)
\(68\) 0 0
\(69\) 2.47283 0.297694
\(70\) 0 0
\(71\) 12.3642 1.46736 0.733678 0.679497i \(-0.237802\pi\)
0.733678 + 0.679497i \(0.237802\pi\)
\(72\) 0 0
\(73\) − 5.81756i − 0.680893i −0.940264 0.340447i \(-0.889422\pi\)
0.940264 0.340447i \(-0.110578\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.64207i − 0.187132i
\(78\) 0 0
\(79\) 3.17548 0.357270 0.178635 0.983915i \(-0.442832\pi\)
0.178635 + 0.983915i \(0.442832\pi\)
\(80\) 0 0
\(81\) −8.64207 −0.960230
\(82\) 0 0
\(83\) 4.71585i 0.517632i 0.965927 + 0.258816i \(0.0833323\pi\)
−0.965927 + 0.258816i \(0.916668\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 22.5070i 2.41300i
\(88\) 0 0
\(89\) 5.43171 0.575760 0.287880 0.957667i \(-0.407050\pi\)
0.287880 + 0.957667i \(0.407050\pi\)
\(90\) 0 0
\(91\) 2.16924 0.227398
\(92\) 0 0
\(93\) 11.9457i 1.23871i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.06058i 0.412289i 0.978522 + 0.206144i \(0.0660917\pi\)
−0.978522 + 0.206144i \(0.933908\pi\)
\(98\) 0 0
\(99\) −9.70265 −0.975153
\(100\) 0 0
\(101\) −15.5529 −1.54757 −0.773784 0.633450i \(-0.781638\pi\)
−0.773784 + 0.633450i \(0.781638\pi\)
\(102\) 0 0
\(103\) 17.9519i 1.76885i 0.466678 + 0.884427i \(0.345451\pi\)
−0.466678 + 0.884427i \(0.654549\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.2298i 1.18230i 0.806561 + 0.591150i \(0.201326\pi\)
−0.806561 + 0.591150i \(0.798674\pi\)
\(108\) 0 0
\(109\) −9.10795 −0.872383 −0.436192 0.899854i \(-0.643673\pi\)
−0.436192 + 0.899854i \(0.643673\pi\)
\(110\) 0 0
\(111\) 24.0947 2.28697
\(112\) 0 0
\(113\) − 13.1755i − 1.23945i −0.784821 0.619723i \(-0.787245\pi\)
0.784821 0.619723i \(-0.212755\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 12.8176i − 1.18498i
\(118\) 0 0
\(119\) 2.31903 0.212585
\(120\) 0 0
\(121\) −1.29735 −0.117941
\(122\) 0 0
\(123\) 17.1428i 1.54572i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 19.5613i − 1.73579i −0.496750 0.867894i \(-0.665473\pi\)
0.496750 0.867894i \(-0.334527\pi\)
\(128\) 0 0
\(129\) 11.0279 0.970955
\(130\) 0 0
\(131\) −4.15604 −0.363115 −0.181557 0.983380i \(-0.558114\pi\)
−0.181557 + 0.983380i \(0.558114\pi\)
\(132\) 0 0
\(133\) − 1.95191i − 0.169252i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.1344i 0.951272i 0.879642 + 0.475636i \(0.157782\pi\)
−0.879642 + 0.475636i \(0.842218\pi\)
\(138\) 0 0
\(139\) 6.49452 0.550858 0.275429 0.961321i \(-0.411180\pi\)
0.275429 + 0.961321i \(0.411180\pi\)
\(140\) 0 0
\(141\) 1.58774 0.133712
\(142\) 0 0
\(143\) 12.8176i 1.07186i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 16.6226i 1.37101i
\(148\) 0 0
\(149\) 5.47283 0.448352 0.224176 0.974549i \(-0.428031\pi\)
0.224176 + 0.974549i \(0.428031\pi\)
\(150\) 0 0
\(151\) −0.0954606 −0.00776848 −0.00388424 0.999992i \(-0.501236\pi\)
−0.00388424 + 0.999992i \(0.501236\pi\)
\(152\) 0 0
\(153\) − 13.7026i − 1.10779i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3.40530i 0.271772i 0.990724 + 0.135886i \(0.0433881\pi\)
−0.990724 + 0.135886i \(0.956612\pi\)
\(158\) 0 0
\(159\) 9.62263 0.763124
\(160\) 0 0
\(161\) −0.527166 −0.0415465
\(162\) 0 0
\(163\) 18.5745i 1.45487i 0.686177 + 0.727435i \(0.259288\pi\)
−0.686177 + 0.727435i \(0.740712\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.4596i 0.964155i 0.876129 + 0.482078i \(0.160118\pi\)
−0.876129 + 0.482078i \(0.839882\pi\)
\(168\) 0 0
\(169\) −3.93246 −0.302497
\(170\) 0 0
\(171\) −11.5334 −0.881982
\(172\) 0 0
\(173\) 3.49228i 0.265513i 0.991149 + 0.132757i \(0.0423829\pi\)
−0.991149 + 0.132757i \(0.957617\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 21.7563i − 1.63530i
\(178\) 0 0
\(179\) −6.38585 −0.477301 −0.238650 0.971106i \(-0.576705\pi\)
−0.238650 + 0.971106i \(0.576705\pi\)
\(180\) 0 0
\(181\) −20.8781 −1.55186 −0.775930 0.630819i \(-0.782719\pi\)
−0.775930 + 0.630819i \(0.782719\pi\)
\(182\) 0 0
\(183\) − 8.53965i − 0.631269i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 13.7026i 1.00204i
\(188\) 0 0
\(189\) −0.149793 −0.0108958
\(190\) 0 0
\(191\) −6.83700 −0.494708 −0.247354 0.968925i \(-0.579561\pi\)
−0.247354 + 0.968925i \(0.579561\pi\)
\(192\) 0 0
\(193\) 7.10170i 0.511192i 0.966784 + 0.255596i \(0.0822717\pi\)
−0.966784 + 0.255596i \(0.917728\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 0.850207i − 0.0605748i −0.999541 0.0302874i \(-0.990358\pi\)
0.999541 0.0302874i \(-0.00964225\pi\)
\(198\) 0 0
\(199\) −8.56829 −0.607390 −0.303695 0.952769i \(-0.598220\pi\)
−0.303695 + 0.952769i \(0.598220\pi\)
\(200\) 0 0
\(201\) −21.2841 −1.50127
\(202\) 0 0
\(203\) − 4.79811i − 0.336761i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 3.11491i 0.216501i
\(208\) 0 0
\(209\) 11.5334 0.797783
\(210\) 0 0
\(211\) 9.63511 0.663309 0.331654 0.943401i \(-0.392393\pi\)
0.331654 + 0.943401i \(0.392393\pi\)
\(212\) 0 0
\(213\) 30.5745i 2.09493i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 2.54661i − 0.172875i
\(218\) 0 0
\(219\) 14.3859 0.972106
\(220\) 0 0
\(221\) −18.1017 −1.21765
\(222\) 0 0
\(223\) 9.43171i 0.631594i 0.948827 + 0.315797i \(0.102272\pi\)
−0.948827 + 0.315797i \(0.897728\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.91774i 0.260030i 0.991512 + 0.130015i \(0.0415025\pi\)
−0.991512 + 0.130015i \(0.958497\pi\)
\(228\) 0 0
\(229\) 22.0947 1.46006 0.730031 0.683414i \(-0.239506\pi\)
0.730031 + 0.683414i \(0.239506\pi\)
\(230\) 0 0
\(231\) 4.06058 0.267166
\(232\) 0 0
\(233\) − 12.5334i − 0.821091i −0.911840 0.410545i \(-0.865338\pi\)
0.911840 0.410545i \(-0.134662\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 7.85244i 0.510071i
\(238\) 0 0
\(239\) 12.4123 0.802882 0.401441 0.915885i \(-0.368509\pi\)
0.401441 + 0.915885i \(0.368509\pi\)
\(240\) 0 0
\(241\) 5.74378 0.369989 0.184995 0.982740i \(-0.440773\pi\)
0.184995 + 0.982740i \(0.440773\pi\)
\(242\) 0 0
\(243\) − 22.2229i − 1.42560i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 15.2361i 0.969447i
\(248\) 0 0
\(249\) −11.6615 −0.739019
\(250\) 0 0
\(251\) −6.33624 −0.399940 −0.199970 0.979802i \(-0.564085\pi\)
−0.199970 + 0.979802i \(0.564085\pi\)
\(252\) 0 0
\(253\) − 3.11491i − 0.195833i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 22.2772i − 1.38961i −0.719197 0.694806i \(-0.755490\pi\)
0.719197 0.694806i \(-0.244510\pi\)
\(258\) 0 0
\(259\) −5.13659 −0.319172
\(260\) 0 0
\(261\) −28.3510 −1.75488
\(262\) 0 0
\(263\) − 4.66776i − 0.287827i −0.989590 0.143913i \(-0.954031\pi\)
0.989590 0.143913i \(-0.0459687\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 13.4317i 0.822007i
\(268\) 0 0
\(269\) −9.70889 −0.591962 −0.295981 0.955194i \(-0.595646\pi\)
−0.295981 + 0.955194i \(0.595646\pi\)
\(270\) 0 0
\(271\) −6.75698 −0.410457 −0.205229 0.978714i \(-0.565794\pi\)
−0.205229 + 0.978714i \(0.565794\pi\)
\(272\) 0 0
\(273\) 5.36417i 0.324654i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 19.0194i − 1.14277i −0.820683 0.571384i \(-0.806407\pi\)
0.820683 0.571384i \(-0.193593\pi\)
\(278\) 0 0
\(279\) −15.0474 −0.900863
\(280\) 0 0
\(281\) −32.8370 −1.95889 −0.979446 0.201708i \(-0.935351\pi\)
−0.979446 + 0.201708i \(0.935351\pi\)
\(282\) 0 0
\(283\) 13.7827i 0.819295i 0.912244 + 0.409647i \(0.134348\pi\)
−0.912244 + 0.409647i \(0.865652\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 3.65456i − 0.215722i
\(288\) 0 0
\(289\) −2.35168 −0.138334
\(290\) 0 0
\(291\) −10.0411 −0.588621
\(292\) 0 0
\(293\) 15.0668i 0.880213i 0.897946 + 0.440106i \(0.145059\pi\)
−0.897946 + 0.440106i \(0.854941\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 0.885092i − 0.0513583i
\(298\) 0 0
\(299\) 4.11491 0.237971
\(300\) 0 0
\(301\) −2.35097 −0.135507
\(302\) 0 0
\(303\) − 38.4596i − 2.20945i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5.91302i 0.337474i 0.985661 + 0.168737i \(0.0539688\pi\)
−0.985661 + 0.168737i \(0.946031\pi\)
\(308\) 0 0
\(309\) −44.3921 −2.52538
\(310\) 0 0
\(311\) −25.7911 −1.46248 −0.731241 0.682119i \(-0.761058\pi\)
−0.731241 + 0.682119i \(0.761058\pi\)
\(312\) 0 0
\(313\) 16.7306i 0.945668i 0.881152 + 0.472834i \(0.156769\pi\)
−0.881152 + 0.472834i \(0.843231\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 28.1623i − 1.58175i −0.611977 0.790876i \(-0.709625\pi\)
0.611977 0.790876i \(-0.290375\pi\)
\(318\) 0 0
\(319\) 28.3510 1.58735
\(320\) 0 0
\(321\) −30.2423 −1.68796
\(322\) 0 0
\(323\) 16.2882i 0.906297i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 22.5224i − 1.24549i
\(328\) 0 0
\(329\) −0.338479 −0.0186610
\(330\) 0 0
\(331\) −25.8176 −1.41906 −0.709531 0.704675i \(-0.751093\pi\)
−0.709531 + 0.704675i \(0.751093\pi\)
\(332\) 0 0
\(333\) 30.3510i 1.66322i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 33.5676i − 1.82854i −0.405103 0.914271i \(-0.632764\pi\)
0.405103 0.914271i \(-0.367236\pi\)
\(338\) 0 0
\(339\) 32.5808 1.76955
\(340\) 0 0
\(341\) 15.0474 0.814861
\(342\) 0 0
\(343\) − 7.23382i − 0.390590i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 7.63984i − 0.410128i −0.978749 0.205064i \(-0.934260\pi\)
0.978749 0.205064i \(-0.0657403\pi\)
\(348\) 0 0
\(349\) −3.01945 −0.161627 −0.0808136 0.996729i \(-0.525752\pi\)
−0.0808136 + 0.996729i \(0.525752\pi\)
\(350\) 0 0
\(351\) 1.16924 0.0624094
\(352\) 0 0
\(353\) − 25.1840i − 1.34041i −0.742177 0.670203i \(-0.766207\pi\)
0.742177 0.670203i \(-0.233793\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 5.73458i 0.303506i
\(358\) 0 0
\(359\) 5.32304 0.280939 0.140470 0.990085i \(-0.455139\pi\)
0.140470 + 0.990085i \(0.455139\pi\)
\(360\) 0 0
\(361\) −5.29039 −0.278442
\(362\) 0 0
\(363\) − 3.20813i − 0.168383i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 8.24230i − 0.430245i −0.976587 0.215122i \(-0.930985\pi\)
0.976587 0.215122i \(-0.0690150\pi\)
\(368\) 0 0
\(369\) −21.5940 −1.12414
\(370\) 0 0
\(371\) −2.05138 −0.106502
\(372\) 0 0
\(373\) 1.91774i 0.0992970i 0.998767 + 0.0496485i \(0.0158101\pi\)
−0.998767 + 0.0496485i \(0.984190\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 37.4527i 1.92891i
\(378\) 0 0
\(379\) −30.1692 −1.54969 −0.774845 0.632151i \(-0.782172\pi\)
−0.774845 + 0.632151i \(0.782172\pi\)
\(380\) 0 0
\(381\) 48.3719 2.47817
\(382\) 0 0
\(383\) − 1.64903i − 0.0842617i −0.999112 0.0421309i \(-0.986585\pi\)
0.999112 0.0421309i \(-0.0134146\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 13.8913i 0.706136i
\(388\) 0 0
\(389\) 16.6414 0.843750 0.421875 0.906654i \(-0.361372\pi\)
0.421875 + 0.906654i \(0.361372\pi\)
\(390\) 0 0
\(391\) 4.39905 0.222470
\(392\) 0 0
\(393\) − 10.2772i − 0.518415i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 36.0257i − 1.80808i −0.427450 0.904039i \(-0.640588\pi\)
0.427450 0.904039i \(-0.359412\pi\)
\(398\) 0 0
\(399\) 4.82675 0.241640
\(400\) 0 0
\(401\) −5.62263 −0.280781 −0.140390 0.990096i \(-0.544836\pi\)
−0.140390 + 0.990096i \(0.544836\pi\)
\(402\) 0 0
\(403\) 19.8781i 0.990200i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 30.3510i − 1.50444i
\(408\) 0 0
\(409\) 2.76546 0.136743 0.0683716 0.997660i \(-0.478220\pi\)
0.0683716 + 0.997660i \(0.478220\pi\)
\(410\) 0 0
\(411\) −27.5334 −1.35812
\(412\) 0 0
\(413\) 4.63807i 0.228224i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 16.0599i 0.786455i
\(418\) 0 0
\(419\) 17.5962 0.859632 0.429816 0.902917i \(-0.358579\pi\)
0.429816 + 0.902917i \(0.358579\pi\)
\(420\) 0 0
\(421\) 25.5676 1.24609 0.623044 0.782187i \(-0.285896\pi\)
0.623044 + 0.782187i \(0.285896\pi\)
\(422\) 0 0
\(423\) 2.00000i 0.0972433i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.82051i 0.0881006i
\(428\) 0 0
\(429\) −31.6957 −1.53028
\(430\) 0 0
\(431\) 17.9736 0.865757 0.432879 0.901452i \(-0.357498\pi\)
0.432879 + 0.901452i \(0.357498\pi\)
\(432\) 0 0
\(433\) 37.0382i 1.77994i 0.456018 + 0.889971i \(0.349275\pi\)
−0.456018 + 0.889971i \(0.650725\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 3.70265i − 0.177122i
\(438\) 0 0
\(439\) −38.8168 −1.85263 −0.926313 0.376754i \(-0.877040\pi\)
−0.926313 + 0.376754i \(0.877040\pi\)
\(440\) 0 0
\(441\) −20.9387 −0.997081
\(442\) 0 0
\(443\) 25.3726i 1.20549i 0.797934 + 0.602745i \(0.205927\pi\)
−0.797934 + 0.602745i \(0.794073\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 13.5334i 0.640108i
\(448\) 0 0
\(449\) −33.4589 −1.57902 −0.789512 0.613735i \(-0.789666\pi\)
−0.789512 + 0.613735i \(0.789666\pi\)
\(450\) 0 0
\(451\) 21.5940 1.01682
\(452\) 0 0
\(453\) − 0.236058i − 0.0110910i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 27.5264i 1.28763i 0.765180 + 0.643816i \(0.222650\pi\)
−0.765180 + 0.643816i \(0.777350\pi\)
\(458\) 0 0
\(459\) 1.24998 0.0583440
\(460\) 0 0
\(461\) 36.3161 1.69141 0.845704 0.533652i \(-0.179181\pi\)
0.845704 + 0.533652i \(0.179181\pi\)
\(462\) 0 0
\(463\) − 22.5544i − 1.04819i −0.851660 0.524095i \(-0.824404\pi\)
0.851660 0.524095i \(-0.175596\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 22.8370i − 1.05677i −0.849005 0.528385i \(-0.822798\pi\)
0.849005 0.528385i \(-0.177202\pi\)
\(468\) 0 0
\(469\) 4.53742 0.209518
\(470\) 0 0
\(471\) −8.42074 −0.388007
\(472\) 0 0
\(473\) − 13.8913i − 0.638724i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 12.1212i 0.554989i
\(478\) 0 0
\(479\) 35.5264 1.62324 0.811622 0.584182i \(-0.198585\pi\)
0.811622 + 0.584182i \(0.198585\pi\)
\(480\) 0 0
\(481\) 40.0947 1.82816
\(482\) 0 0
\(483\) − 1.30359i − 0.0593156i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 10.4945i 0.475552i 0.971320 + 0.237776i \(0.0764184\pi\)
−0.971320 + 0.237776i \(0.923582\pi\)
\(488\) 0 0
\(489\) −45.9317 −2.07711
\(490\) 0 0
\(491\) 42.1685 1.90304 0.951519 0.307589i \(-0.0995221\pi\)
0.951519 + 0.307589i \(0.0995221\pi\)
\(492\) 0 0
\(493\) 40.0389i 1.80326i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 6.51797i − 0.292371i
\(498\) 0 0
\(499\) −30.1560 −1.34997 −0.674985 0.737832i \(-0.735850\pi\)
−0.674985 + 0.737832i \(0.735850\pi\)
\(500\) 0 0
\(501\) −30.8106 −1.37652
\(502\) 0 0
\(503\) 22.9729i 1.02431i 0.858893 + 0.512155i \(0.171153\pi\)
−0.858893 + 0.512155i \(0.828847\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 9.72433i − 0.431873i
\(508\) 0 0
\(509\) −11.8176 −0.523804 −0.261902 0.965094i \(-0.584350\pi\)
−0.261902 + 0.965094i \(0.584350\pi\)
\(510\) 0 0
\(511\) −3.06682 −0.135668
\(512\) 0 0
\(513\) − 1.05210i − 0.0464512i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 2.00000i − 0.0879599i
\(518\) 0 0
\(519\) −8.63583 −0.379071
\(520\) 0 0
\(521\) −35.7438 −1.56596 −0.782982 0.622045i \(-0.786302\pi\)
−0.782982 + 0.622045i \(0.786302\pi\)
\(522\) 0 0
\(523\) 14.3899i 0.629225i 0.949220 + 0.314612i \(0.101875\pi\)
−0.949220 + 0.314612i \(0.898125\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 21.2508i 0.925698i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 27.4053 1.18929
\(532\) 0 0
\(533\) 28.5264i 1.23562i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 15.7911i − 0.681438i
\(538\) 0 0
\(539\) 20.9387 0.901894
\(540\) 0 0
\(541\) 14.5209 0.624303 0.312152 0.950032i \(-0.398950\pi\)
0.312152 + 0.950032i \(0.398950\pi\)
\(542\) 0 0
\(543\) − 51.6282i − 2.21558i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 19.6289i − 0.839270i −0.907693 0.419635i \(-0.862158\pi\)
0.907693 0.419635i \(-0.137842\pi\)
\(548\) 0 0
\(549\) 10.7570 0.459097
\(550\) 0 0
\(551\) 33.7004 1.43569
\(552\) 0 0
\(553\) − 1.67401i − 0.0711860i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 1.13659i − 0.0481588i −0.999710 0.0240794i \(-0.992335\pi\)
0.999710 0.0240794i \(-0.00766546\pi\)
\(558\) 0 0
\(559\) 18.3510 0.776163
\(560\) 0 0
\(561\) −33.8844 −1.43060
\(562\) 0 0
\(563\) 25.8913i 1.09119i 0.838049 + 0.545595i \(0.183696\pi\)
−0.838049 + 0.545595i \(0.816304\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.55581i 0.191326i
\(568\) 0 0
\(569\) 44.0558 1.84692 0.923459 0.383698i \(-0.125350\pi\)
0.923459 + 0.383698i \(0.125350\pi\)
\(570\) 0 0
\(571\) −41.8991 −1.75342 −0.876711 0.481017i \(-0.840268\pi\)
−0.876711 + 0.481017i \(0.840268\pi\)
\(572\) 0 0
\(573\) − 16.9068i − 0.706291i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 1.34544i − 0.0560114i −0.999608 0.0280057i \(-0.991084\pi\)
0.999608 0.0280057i \(-0.00891566\pi\)
\(578\) 0 0
\(579\) −17.5613 −0.729824
\(580\) 0 0
\(581\) 2.48604 0.103138
\(582\) 0 0
\(583\) − 12.1212i − 0.502007i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 25.0411i − 1.03356i −0.856119 0.516779i \(-0.827131\pi\)
0.856119 0.516779i \(-0.172869\pi\)
\(588\) 0 0
\(589\) 17.8866 0.737005
\(590\) 0 0
\(591\) 2.10242 0.0864821
\(592\) 0 0
\(593\) − 22.5947i − 0.927853i −0.885874 0.463927i \(-0.846440\pi\)
0.885874 0.463927i \(-0.153560\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 21.1880i − 0.867166i
\(598\) 0 0
\(599\) −14.7111 −0.601080 −0.300540 0.953769i \(-0.597167\pi\)
−0.300540 + 0.953769i \(0.597167\pi\)
\(600\) 0 0
\(601\) 19.2904 0.786871 0.393436 0.919352i \(-0.371286\pi\)
0.393436 + 0.919352i \(0.371286\pi\)
\(602\) 0 0
\(603\) − 26.8106i − 1.09181i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 20.7547i 0.842409i 0.906966 + 0.421205i \(0.138393\pi\)
−0.906966 + 0.421205i \(0.861607\pi\)
\(608\) 0 0
\(609\) 11.8649 0.480791
\(610\) 0 0
\(611\) 2.64207 0.106887
\(612\) 0 0
\(613\) − 43.3091i − 1.74924i −0.484810 0.874619i \(-0.661111\pi\)
0.484810 0.874619i \(-0.338889\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 13.5621i − 0.545988i −0.962016 0.272994i \(-0.911986\pi\)
0.962016 0.272994i \(-0.0880139\pi\)
\(618\) 0 0
\(619\) 12.9045 0.518677 0.259339 0.965786i \(-0.416495\pi\)
0.259339 + 0.965786i \(0.416495\pi\)
\(620\) 0 0
\(621\) −0.284147 −0.0114024
\(622\) 0 0
\(623\) − 2.86341i − 0.114720i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 28.5202i 1.13899i
\(628\) 0 0
\(629\) 42.8634 1.70908
\(630\) 0 0
\(631\) 45.0404 1.79303 0.896515 0.443013i \(-0.146090\pi\)
0.896515 + 0.443013i \(0.146090\pi\)
\(632\) 0 0
\(633\) 23.8260i 0.947000i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 27.6608i 1.09596i
\(638\) 0 0
\(639\) −38.5132 −1.52356
\(640\) 0 0
\(641\) −3.08074 −0.121682 −0.0608410 0.998147i \(-0.519378\pi\)
−0.0608410 + 0.998147i \(0.519378\pi\)
\(642\) 0 0
\(643\) 4.52493i 0.178446i 0.996012 + 0.0892229i \(0.0284384\pi\)
−0.996012 + 0.0892229i \(0.971562\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 34.8719i − 1.37096i −0.728094 0.685478i \(-0.759593\pi\)
0.728094 0.685478i \(-0.240407\pi\)
\(648\) 0 0
\(649\) −27.4053 −1.07575
\(650\) 0 0
\(651\) 6.29735 0.246813
\(652\) 0 0
\(653\) − 35.3726i − 1.38424i −0.721783 0.692119i \(-0.756677\pi\)
0.721783 0.692119i \(-0.243323\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 18.1212i 0.706973i
\(658\) 0 0
\(659\) 2.10866 0.0821419 0.0410710 0.999156i \(-0.486923\pi\)
0.0410710 + 0.999156i \(0.486923\pi\)
\(660\) 0 0
\(661\) 20.0342 0.779239 0.389619 0.920976i \(-0.372607\pi\)
0.389619 + 0.920976i \(0.372607\pi\)
\(662\) 0 0
\(663\) − 44.7625i − 1.73843i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 9.10170i − 0.352419i
\(668\) 0 0
\(669\) −23.3230 −0.901721
\(670\) 0 0
\(671\) −10.7570 −0.415269
\(672\) 0 0
\(673\) − 5.07530i − 0.195638i −0.995204 0.0978191i \(-0.968813\pi\)
0.995204 0.0978191i \(-0.0311867\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13.4876i 0.518369i 0.965828 + 0.259184i \(0.0834537\pi\)
−0.965828 + 0.259184i \(0.916546\pi\)
\(678\) 0 0
\(679\) 2.14060 0.0821486
\(680\) 0 0
\(681\) −9.68793 −0.371242
\(682\) 0 0
\(683\) 25.4379i 0.973356i 0.873581 + 0.486678i \(0.161792\pi\)
−0.873581 + 0.486678i \(0.838208\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 54.6366i 2.08452i
\(688\) 0 0
\(689\) 16.0125 0.610027
\(690\) 0 0
\(691\) 22.6506 0.861668 0.430834 0.902431i \(-0.358220\pi\)
0.430834 + 0.902431i \(0.358220\pi\)
\(692\) 0 0
\(693\) 5.11491i 0.194299i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 30.4963i 1.15513i
\(698\) 0 0
\(699\) 30.9930 1.17226
\(700\) 0 0
\(701\) −41.8099 −1.57914 −0.789569 0.613662i \(-0.789696\pi\)
−0.789569 + 0.613662i \(0.789696\pi\)
\(702\) 0 0
\(703\) − 36.0778i − 1.36070i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.19894i 0.308353i
\(708\) 0 0
\(709\) −2.87117 −0.107829 −0.0539146 0.998546i \(-0.517170\pi\)
−0.0539146 + 0.998546i \(0.517170\pi\)
\(710\) 0 0
\(711\) −9.89134 −0.370954
\(712\) 0 0
\(713\) − 4.83076i − 0.180913i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 30.6935i 1.14627i
\(718\) 0 0
\(719\) 50.1623 1.87074 0.935369 0.353674i \(-0.115068\pi\)
0.935369 + 0.353674i \(0.115068\pi\)
\(720\) 0 0
\(721\) 9.46364 0.352444
\(722\) 0 0
\(723\) 14.2034i 0.528230i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 26.8198i 0.994691i 0.867553 + 0.497345i \(0.165692\pi\)
−0.867553 + 0.497345i \(0.834308\pi\)
\(728\) 0 0
\(729\) 29.0272 1.07508
\(730\) 0 0
\(731\) 19.6182 0.725604
\(732\) 0 0
\(733\) 35.9861i 1.32918i 0.747210 + 0.664588i \(0.231393\pi\)
−0.747210 + 0.664588i \(0.768607\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 26.8106i 0.987581i
\(738\) 0 0
\(739\) −5.84396 −0.214974 −0.107487 0.994207i \(-0.534280\pi\)
−0.107487 + 0.994207i \(0.534280\pi\)
\(740\) 0 0
\(741\) −37.6762 −1.38407
\(742\) 0 0
\(743\) 14.2012i 0.520991i 0.965475 + 0.260495i \(0.0838858\pi\)
−0.965475 + 0.260495i \(0.916114\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 14.6894i − 0.537459i
\(748\) 0 0
\(749\) 6.44714 0.235574
\(750\) 0 0
\(751\) 33.0404 1.20566 0.602831 0.797869i \(-0.294039\pi\)
0.602831 + 0.797869i \(0.294039\pi\)
\(752\) 0 0
\(753\) − 15.6685i − 0.570991i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 18.2034i − 0.661614i −0.943698 0.330807i \(-0.892679\pi\)
0.943698 0.330807i \(-0.107321\pi\)
\(758\) 0 0
\(759\) 7.70265 0.279588
\(760\) 0 0
\(761\) 34.1010 1.23616 0.618080 0.786115i \(-0.287911\pi\)
0.618080 + 0.786115i \(0.287911\pi\)
\(762\) 0 0
\(763\) 4.80140i 0.173822i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 36.2034i − 1.30723i
\(768\) 0 0
\(769\) −35.1366 −1.26706 −0.633529 0.773719i \(-0.718394\pi\)
−0.633529 + 0.773719i \(0.718394\pi\)
\(770\) 0 0
\(771\) 55.0878 1.98394
\(772\) 0 0
\(773\) 1.13659i 0.0408803i 0.999791 + 0.0204401i \(0.00650675\pi\)
−0.999791 + 0.0204401i \(0.993493\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 12.7019i − 0.455679i
\(778\) 0 0
\(779\) 25.6685 0.919669
\(780\) 0 0
\(781\) 38.5132 1.37811
\(782\) 0 0
\(783\) − 2.58622i − 0.0924241i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 8.33848i 0.297235i 0.988895 + 0.148617i \(0.0474823\pi\)
−0.988895 + 0.148617i \(0.952518\pi\)
\(788\) 0 0
\(789\) 11.5426 0.410928
\(790\) 0 0
\(791\) −6.94567 −0.246960
\(792\) 0 0
\(793\) − 14.2104i − 0.504625i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0.190921i 0.00676278i 0.999994 + 0.00338139i \(0.00107633\pi\)
−0.999994 + 0.00338139i \(0.998924\pi\)
\(798\) 0 0
\(799\) 2.82452 0.0999242
\(800\) 0 0
\(801\) −16.9193 −0.597813
\(802\) 0 0
\(803\) − 18.1212i − 0.639482i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 24.0085i − 0.845138i
\(808\) 0 0
\(809\) 14.5202 0.510503 0.255252 0.966875i \(-0.417842\pi\)
0.255252 + 0.966875i \(0.417842\pi\)
\(810\) 0 0
\(811\) −51.5474 −1.81007 −0.905037 0.425332i \(-0.860157\pi\)
−0.905037 + 0.425332i \(0.860157\pi\)
\(812\) 0 0
\(813\) − 16.7089i − 0.586006i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 16.5124i − 0.577697i
\(818\) 0 0
\(819\) −6.75698 −0.236108
\(820\) 0 0
\(821\) 13.7827 0.481019 0.240509 0.970647i \(-0.422686\pi\)
0.240509 + 0.970647i \(0.422686\pi\)
\(822\) 0 0
\(823\) − 37.1531i − 1.29508i −0.762034 0.647538i \(-0.775799\pi\)
0.762034 0.647538i \(-0.224201\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 39.8510i 1.38576i 0.721055 + 0.692878i \(0.243657\pi\)
−0.721055 + 0.692878i \(0.756343\pi\)
\(828\) 0 0
\(829\) −20.5933 −0.715234 −0.357617 0.933868i \(-0.616411\pi\)
−0.357617 + 0.933868i \(0.616411\pi\)
\(830\) 0 0
\(831\) 47.0319 1.63152
\(832\) 0 0
\(833\) 29.5709i 1.02457i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 1.37265i − 0.0474456i
\(838\) 0 0
\(839\) 36.5669 1.26243 0.631214 0.775609i \(-0.282557\pi\)
0.631214 + 0.775609i \(0.282557\pi\)
\(840\) 0 0
\(841\) 53.8410 1.85659
\(842\) 0 0
\(843\) − 81.2005i − 2.79669i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0.683919i 0.0234998i
\(848\) 0 0
\(849\) −34.0823 −1.16970
\(850\) 0 0
\(851\) −9.74378 −0.334012
\(852\) 0 0
\(853\) − 19.3208i − 0.661532i −0.943713 0.330766i \(-0.892693\pi\)
0.943713 0.330766i \(-0.107307\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 54.6017i − 1.86516i −0.360963 0.932580i \(-0.617552\pi\)
0.360963 0.932580i \(-0.382448\pi\)
\(858\) 0 0
\(859\) 32.9541 1.12438 0.562190 0.827008i \(-0.309959\pi\)
0.562190 + 0.827008i \(0.309959\pi\)
\(860\) 0 0
\(861\) 9.03712 0.307984
\(862\) 0 0
\(863\) − 54.6840i − 1.86147i −0.365701 0.930733i \(-0.619171\pi\)
0.365701 0.930733i \(-0.380829\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 5.81532i − 0.197499i
\(868\) 0 0
\(869\) 9.89134 0.335541
\(870\) 0 0
\(871\) −35.4178 −1.20009
\(872\) 0 0
\(873\) − 12.6483i − 0.428081i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 16.7500i − 0.565608i −0.959178 0.282804i \(-0.908735\pi\)
0.959178 0.282804i \(-0.0912646\pi\)
\(878\) 0 0
\(879\) −37.2577 −1.25667
\(880\) 0 0
\(881\) −31.7049 −1.06816 −0.534082 0.845432i \(-0.679343\pi\)
−0.534082 + 0.845432i \(0.679343\pi\)
\(882\) 0 0
\(883\) 51.9185i 1.74720i 0.486646 + 0.873599i \(0.338220\pi\)
−0.486646 + 0.873599i \(0.661780\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 36.2897i − 1.21849i −0.792983 0.609244i \(-0.791473\pi\)
0.792983 0.609244i \(-0.208527\pi\)
\(888\) 0 0
\(889\) −10.3121 −0.345856
\(890\) 0 0
\(891\) −26.9193 −0.901829
\(892\) 0 0
\(893\) − 2.37737i − 0.0795558i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 10.1755i 0.339749i
\(898\) 0 0
\(899\) 43.9681 1.46642
\(900\) 0 0
\(901\) 17.1182 0.570290
\(902\) 0 0
\(903\) − 5.81355i − 0.193463i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 25.4442i 0.844861i 0.906395 + 0.422430i \(0.138823\pi\)
−0.906395 + 0.422430i \(0.861177\pi\)
\(908\) 0 0
\(909\) 48.4457 1.60684
\(910\) 0 0
\(911\) 47.9597 1.58897 0.794487 0.607281i \(-0.207740\pi\)
0.794487 + 0.607281i \(0.207740\pi\)
\(912\) 0 0
\(913\) 14.6894i 0.486150i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.19092i 0.0723506i
\(918\) 0 0
\(919\) −2.78562 −0.0918892 −0.0459446 0.998944i \(-0.514630\pi\)
−0.0459446 + 0.998944i \(0.514630\pi\)
\(920\) 0 0
\(921\) −14.6219 −0.481808
\(922\) 0 0
\(923\) 50.8774i 1.67465i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 55.9185i − 1.83661i
\(928\) 0 0
\(929\) −40.2508 −1.32059 −0.660293 0.751008i \(-0.729568\pi\)
−0.660293 + 0.751008i \(0.729568\pi\)
\(930\) 0 0
\(931\) 24.8896 0.815722
\(932\) 0 0
\(933\) − 63.7772i − 2.08797i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 38.9868i 1.27364i 0.771011 + 0.636822i \(0.219751\pi\)
−0.771011 + 0.636822i \(0.780249\pi\)
\(938\) 0 0
\(939\) −41.3719 −1.35012
\(940\) 0 0
\(941\) 38.3682 1.25077 0.625383 0.780318i \(-0.284943\pi\)
0.625383 + 0.780318i \(0.284943\pi\)
\(942\) 0 0
\(943\) − 6.93246i − 0.225752i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 15.0217i − 0.488139i −0.969758 0.244070i \(-0.921517\pi\)
0.969758 0.244070i \(-0.0784825\pi\)
\(948\) 0 0
\(949\) 23.9387 0.777083
\(950\) 0 0
\(951\) 69.6406 2.25825
\(952\) 0 0
\(953\) − 12.1303i − 0.392940i −0.980510 0.196470i \(-0.937052\pi\)
0.980510 0.196470i \(-0.0629479\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 70.1072i 2.26624i
\(958\) 0 0
\(959\) 5.86965 0.189541
\(960\) 0 0
\(961\) −7.66376 −0.247218
\(962\) 0 0
\(963\) − 38.0947i − 1.22759i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 21.2882i − 0.684581i −0.939594 0.342290i \(-0.888797\pi\)
0.939594 0.342290i \(-0.111203\pi\)
\(968\) 0 0
\(969\) −40.2779 −1.29391
\(970\) 0 0
\(971\) 11.4006 0.365862 0.182931 0.983126i \(-0.441442\pi\)
0.182931 + 0.983126i \(0.441442\pi\)
\(972\) 0 0
\(973\) − 3.42369i − 0.109758i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 41.2849i − 1.32082i −0.750904 0.660411i \(-0.770382\pi\)
0.750904 0.660411i \(-0.229618\pi\)
\(978\) 0 0
\(979\) 16.9193 0.540742
\(980\) 0 0
\(981\) 28.3704 0.905798
\(982\) 0 0
\(983\) − 42.8448i − 1.36654i −0.730168 0.683268i \(-0.760558\pi\)
0.730168 0.683268i \(-0.239442\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 0.837003i − 0.0266421i
\(988\) 0 0
\(989\) −4.45963 −0.141808
\(990\) 0 0
\(991\) 8.73307 0.277415 0.138707 0.990333i \(-0.455705\pi\)
0.138707 + 0.990333i \(0.455705\pi\)
\(992\) 0 0
\(993\) − 63.8425i − 2.02598i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 44.0808i 1.39605i 0.716072 + 0.698027i \(0.245938\pi\)
−0.716072 + 0.698027i \(0.754062\pi\)
\(998\) 0 0
\(999\) −2.76867 −0.0875968
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 4600.2.e.q.4049.6 6
5.2 odd 4 920.2.a.i.1.3 3
5.3 odd 4 4600.2.a.v.1.1 3
5.4 even 2 inner 4600.2.e.q.4049.1 6
15.2 even 4 8280.2.a.bl.1.1 3
20.3 even 4 9200.2.a.ci.1.3 3
20.7 even 4 1840.2.a.q.1.1 3
40.27 even 4 7360.2.a.cf.1.3 3
40.37 odd 4 7360.2.a.bw.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.i.1.3 3 5.2 odd 4
1840.2.a.q.1.1 3 20.7 even 4
4600.2.a.v.1.1 3 5.3 odd 4
4600.2.e.q.4049.1 6 5.4 even 2 inner
4600.2.e.q.4049.6 6 1.1 even 1 trivial
7360.2.a.bw.1.1 3 40.37 odd 4
7360.2.a.cf.1.3 3 40.27 even 4
8280.2.a.bl.1.1 3 15.2 even 4
9200.2.a.ci.1.3 3 20.3 even 4