Properties

Label 4020.2.a.b.1.4
Level $4020$
Weight $2$
Character 4020.1
Self dual yes
Analytic conductor $32.100$
Analytic rank $1$
Dimension $4$
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] = [4020,2,Mod(1,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, 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("4020.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.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: \(32.0998616126\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.9301.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.60312\) of defining polynomial
Character \(\chi\) \(=\) 4020.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +3.77625 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +3.77625 q^{7} +1.00000 q^{9} +3.60312 q^{11} -5.22690 q^{13} +1.00000 q^{15} +1.62378 q^{17} -3.15246 q^{19} -3.77625 q^{21} -6.55249 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.87493 q^{29} -9.40003 q^{31} -3.60312 q^{33} -3.77625 q^{35} +1.90132 q^{37} +5.22690 q^{39} -10.8036 q^{41} +0.674415 q^{43} -1.00000 q^{45} -1.78198 q^{47} +7.26003 q^{49} -1.62378 q^{51} +6.01246 q^{53} -3.60312 q^{55} +3.15246 q^{57} -0.839338 q^{59} -2.26261 q^{61} +3.77625 q^{63} +5.22690 q^{65} +1.00000 q^{67} +6.55249 q^{69} -14.9433 q^{71} +7.34378 q^{73} -1.00000 q^{75} +13.6063 q^{77} +7.98822 q^{79} +1.00000 q^{81} +4.72820 q^{83} -1.62378 q^{85} -1.87493 q^{87} -2.96688 q^{89} -19.7381 q^{91} +9.40003 q^{93} +3.15246 q^{95} -16.1556 q^{97} +3.60312 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{5} - q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 4 q^{5} - q^{7} + 4 q^{9} + 5 q^{11} - q^{13} + 4 q^{15} - 4 q^{17} - 7 q^{19} + q^{21} + 6 q^{23} + 4 q^{25} - 4 q^{27} - q^{29} - 11 q^{31} - 5 q^{33} + q^{35} + q^{39} - 13 q^{41} + 15 q^{43} - 4 q^{45} + 7 q^{47} - 3 q^{49} + 4 q^{51} + 13 q^{53} - 5 q^{55} + 7 q^{57} + q^{59} - 15 q^{61} - q^{63} + q^{65} + 4 q^{67} - 6 q^{69} - 6 q^{71} + 8 q^{73} - 4 q^{75} + 9 q^{77} - q^{79} + 4 q^{81} + 18 q^{83} + 4 q^{85} + q^{87} - 24 q^{89} - 29 q^{91} + 11 q^{93} + 7 q^{95} - 23 q^{97} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.77625 1.42729 0.713643 0.700509i \(-0.247044\pi\)
0.713643 + 0.700509i \(0.247044\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.60312 1.08638 0.543191 0.839609i \(-0.317216\pi\)
0.543191 + 0.839609i \(0.317216\pi\)
\(12\) 0 0
\(13\) −5.22690 −1.44968 −0.724841 0.688916i \(-0.758087\pi\)
−0.724841 + 0.688916i \(0.758087\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 1.62378 0.393825 0.196913 0.980421i \(-0.436908\pi\)
0.196913 + 0.980421i \(0.436908\pi\)
\(18\) 0 0
\(19\) −3.15246 −0.723224 −0.361612 0.932329i \(-0.617774\pi\)
−0.361612 + 0.932329i \(0.617774\pi\)
\(20\) 0 0
\(21\) −3.77625 −0.824044
\(22\) 0 0
\(23\) −6.55249 −1.36629 −0.683144 0.730283i \(-0.739388\pi\)
−0.683144 + 0.730283i \(0.739388\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.87493 0.348165 0.174082 0.984731i \(-0.444304\pi\)
0.174082 + 0.984731i \(0.444304\pi\)
\(30\) 0 0
\(31\) −9.40003 −1.68829 −0.844147 0.536111i \(-0.819893\pi\)
−0.844147 + 0.536111i \(0.819893\pi\)
\(32\) 0 0
\(33\) −3.60312 −0.627223
\(34\) 0 0
\(35\) −3.77625 −0.638302
\(36\) 0 0
\(37\) 1.90132 0.312575 0.156288 0.987712i \(-0.450047\pi\)
0.156288 + 0.987712i \(0.450047\pi\)
\(38\) 0 0
\(39\) 5.22690 0.836975
\(40\) 0 0
\(41\) −10.8036 −1.68724 −0.843622 0.536938i \(-0.819581\pi\)
−0.843622 + 0.536938i \(0.819581\pi\)
\(42\) 0 0
\(43\) 0.674415 0.102847 0.0514237 0.998677i \(-0.483624\pi\)
0.0514237 + 0.998677i \(0.483624\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −1.78198 −0.259928 −0.129964 0.991519i \(-0.541486\pi\)
−0.129964 + 0.991519i \(0.541486\pi\)
\(48\) 0 0
\(49\) 7.26003 1.03715
\(50\) 0 0
\(51\) −1.62378 −0.227375
\(52\) 0 0
\(53\) 6.01246 0.825875 0.412938 0.910759i \(-0.364503\pi\)
0.412938 + 0.910759i \(0.364503\pi\)
\(54\) 0 0
\(55\) −3.60312 −0.485845
\(56\) 0 0
\(57\) 3.15246 0.417554
\(58\) 0 0
\(59\) −0.839338 −0.109273 −0.0546363 0.998506i \(-0.517400\pi\)
−0.0546363 + 0.998506i \(0.517400\pi\)
\(60\) 0 0
\(61\) −2.26261 −0.289698 −0.144849 0.989454i \(-0.546270\pi\)
−0.144849 + 0.989454i \(0.546270\pi\)
\(62\) 0 0
\(63\) 3.77625 0.475762
\(64\) 0 0
\(65\) 5.22690 0.648318
\(66\) 0 0
\(67\) 1.00000 0.122169
\(68\) 0 0
\(69\) 6.55249 0.788827
\(70\) 0 0
\(71\) −14.9433 −1.77345 −0.886723 0.462301i \(-0.847024\pi\)
−0.886723 + 0.462301i \(0.847024\pi\)
\(72\) 0 0
\(73\) 7.34378 0.859524 0.429762 0.902942i \(-0.358597\pi\)
0.429762 + 0.902942i \(0.358597\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 13.6063 1.55058
\(78\) 0 0
\(79\) 7.98822 0.898745 0.449373 0.893344i \(-0.351648\pi\)
0.449373 + 0.893344i \(0.351648\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.72820 0.518987 0.259494 0.965745i \(-0.416444\pi\)
0.259494 + 0.965745i \(0.416444\pi\)
\(84\) 0 0
\(85\) −1.62378 −0.176124
\(86\) 0 0
\(87\) −1.87493 −0.201013
\(88\) 0 0
\(89\) −2.96688 −0.314488 −0.157244 0.987560i \(-0.550261\pi\)
−0.157244 + 0.987560i \(0.550261\pi\)
\(90\) 0 0
\(91\) −19.7381 −2.06911
\(92\) 0 0
\(93\) 9.40003 0.974737
\(94\) 0 0
\(95\) 3.15246 0.323436
\(96\) 0 0
\(97\) −16.1556 −1.64035 −0.820177 0.572110i \(-0.806125\pi\)
−0.820177 + 0.572110i \(0.806125\pi\)
\(98\) 0 0
\(99\) 3.60312 0.362127
\(100\) 0 0
\(101\) −2.33736 −0.232576 −0.116288 0.993216i \(-0.537100\pi\)
−0.116288 + 0.993216i \(0.537100\pi\)
\(102\) 0 0
\(103\) −0.674415 −0.0664521 −0.0332260 0.999448i \(-0.510578\pi\)
−0.0332260 + 0.999448i \(0.510578\pi\)
\(104\) 0 0
\(105\) 3.77625 0.368524
\(106\) 0 0
\(107\) 5.72573 0.553527 0.276764 0.960938i \(-0.410738\pi\)
0.276764 + 0.960938i \(0.410738\pi\)
\(108\) 0 0
\(109\) −9.89312 −0.947589 −0.473795 0.880635i \(-0.657116\pi\)
−0.473795 + 0.880635i \(0.657116\pi\)
\(110\) 0 0
\(111\) −1.90132 −0.180465
\(112\) 0 0
\(113\) 8.70742 0.819125 0.409562 0.912282i \(-0.365681\pi\)
0.409562 + 0.912282i \(0.365681\pi\)
\(114\) 0 0
\(115\) 6.55249 0.611023
\(116\) 0 0
\(117\) −5.22690 −0.483228
\(118\) 0 0
\(119\) 6.13180 0.562101
\(120\) 0 0
\(121\) 1.98249 0.180226
\(122\) 0 0
\(123\) 10.8036 0.974131
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −14.5582 −1.29183 −0.645917 0.763408i \(-0.723525\pi\)
−0.645917 + 0.763408i \(0.723525\pi\)
\(128\) 0 0
\(129\) −0.674415 −0.0593789
\(130\) 0 0
\(131\) 21.3501 1.86537 0.932683 0.360698i \(-0.117462\pi\)
0.932683 + 0.360698i \(0.117462\pi\)
\(132\) 0 0
\(133\) −11.9045 −1.03225
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −4.53195 −0.387190 −0.193595 0.981081i \(-0.562015\pi\)
−0.193595 + 0.981081i \(0.562015\pi\)
\(138\) 0 0
\(139\) 4.08117 0.346160 0.173080 0.984908i \(-0.444628\pi\)
0.173080 + 0.984908i \(0.444628\pi\)
\(140\) 0 0
\(141\) 1.78198 0.150070
\(142\) 0 0
\(143\) −18.8332 −1.57491
\(144\) 0 0
\(145\) −1.87493 −0.155704
\(146\) 0 0
\(147\) −7.26003 −0.598797
\(148\) 0 0
\(149\) 13.0099 1.06581 0.532905 0.846175i \(-0.321100\pi\)
0.532905 + 0.846175i \(0.321100\pi\)
\(150\) 0 0
\(151\) −8.90952 −0.725046 −0.362523 0.931975i \(-0.618085\pi\)
−0.362523 + 0.931975i \(0.618085\pi\)
\(152\) 0 0
\(153\) 1.62378 0.131275
\(154\) 0 0
\(155\) 9.40003 0.755028
\(156\) 0 0
\(157\) −12.1798 −0.972058 −0.486029 0.873943i \(-0.661555\pi\)
−0.486029 + 0.873943i \(0.661555\pi\)
\(158\) 0 0
\(159\) −6.01246 −0.476819
\(160\) 0 0
\(161\) −24.7438 −1.95009
\(162\) 0 0
\(163\) −21.4730 −1.68189 −0.840947 0.541117i \(-0.818002\pi\)
−0.840947 + 0.541117i \(0.818002\pi\)
\(164\) 0 0
\(165\) 3.60312 0.280503
\(166\) 0 0
\(167\) 8.11330 0.627826 0.313913 0.949452i \(-0.398360\pi\)
0.313913 + 0.949452i \(0.398360\pi\)
\(168\) 0 0
\(169\) 14.3205 1.10158
\(170\) 0 0
\(171\) −3.15246 −0.241075
\(172\) 0 0
\(173\) −5.16393 −0.392606 −0.196303 0.980543i \(-0.562894\pi\)
−0.196303 + 0.980543i \(0.562894\pi\)
\(174\) 0 0
\(175\) 3.77625 0.285457
\(176\) 0 0
\(177\) 0.839338 0.0630885
\(178\) 0 0
\(179\) −11.8905 −0.888740 −0.444370 0.895843i \(-0.646573\pi\)
−0.444370 + 0.895843i \(0.646573\pi\)
\(180\) 0 0
\(181\) −20.0123 −1.48751 −0.743753 0.668455i \(-0.766956\pi\)
−0.743753 + 0.668455i \(0.766956\pi\)
\(182\) 0 0
\(183\) 2.26261 0.167257
\(184\) 0 0
\(185\) −1.90132 −0.139788
\(186\) 0 0
\(187\) 5.85069 0.427845
\(188\) 0 0
\(189\) −3.77625 −0.274681
\(190\) 0 0
\(191\) 1.63525 0.118323 0.0591613 0.998248i \(-0.481157\pi\)
0.0591613 + 0.998248i \(0.481157\pi\)
\(192\) 0 0
\(193\) 4.90805 0.353289 0.176644 0.984275i \(-0.443476\pi\)
0.176644 + 0.984275i \(0.443476\pi\)
\(194\) 0 0
\(195\) −5.22690 −0.374306
\(196\) 0 0
\(197\) 23.5681 1.67916 0.839579 0.543238i \(-0.182802\pi\)
0.839579 + 0.543238i \(0.182802\pi\)
\(198\) 0 0
\(199\) −26.1887 −1.85647 −0.928235 0.371994i \(-0.878674\pi\)
−0.928235 + 0.371994i \(0.878674\pi\)
\(200\) 0 0
\(201\) −1.00000 −0.0705346
\(202\) 0 0
\(203\) 7.08018 0.496931
\(204\) 0 0
\(205\) 10.8036 0.754558
\(206\) 0 0
\(207\) −6.55249 −0.455430
\(208\) 0 0
\(209\) −11.3587 −0.785698
\(210\) 0 0
\(211\) 4.31054 0.296750 0.148375 0.988931i \(-0.452596\pi\)
0.148375 + 0.988931i \(0.452596\pi\)
\(212\) 0 0
\(213\) 14.9433 1.02390
\(214\) 0 0
\(215\) −0.674415 −0.0459947
\(216\) 0 0
\(217\) −35.4968 −2.40968
\(218\) 0 0
\(219\) −7.34378 −0.496247
\(220\) 0 0
\(221\) −8.48736 −0.570921
\(222\) 0 0
\(223\) −12.6662 −0.848193 −0.424096 0.905617i \(-0.639408\pi\)
−0.424096 + 0.905617i \(0.639408\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 10.0812 0.669111 0.334555 0.942376i \(-0.391414\pi\)
0.334555 + 0.942376i \(0.391414\pi\)
\(228\) 0 0
\(229\) −5.91063 −0.390585 −0.195293 0.980745i \(-0.562566\pi\)
−0.195293 + 0.980745i \(0.562566\pi\)
\(230\) 0 0
\(231\) −13.6063 −0.895227
\(232\) 0 0
\(233\) 12.1073 0.793173 0.396586 0.917997i \(-0.370195\pi\)
0.396586 + 0.917997i \(0.370195\pi\)
\(234\) 0 0
\(235\) 1.78198 0.116243
\(236\) 0 0
\(237\) −7.98822 −0.518891
\(238\) 0 0
\(239\) −9.08105 −0.587404 −0.293702 0.955897i \(-0.594887\pi\)
−0.293702 + 0.955897i \(0.594887\pi\)
\(240\) 0 0
\(241\) −26.0627 −1.67884 −0.839422 0.543480i \(-0.817106\pi\)
−0.839422 + 0.543480i \(0.817106\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −7.26003 −0.463826
\(246\) 0 0
\(247\) 16.4776 1.04845
\(248\) 0 0
\(249\) −4.72820 −0.299637
\(250\) 0 0
\(251\) −0.877201 −0.0553684 −0.0276842 0.999617i \(-0.508813\pi\)
−0.0276842 + 0.999617i \(0.508813\pi\)
\(252\) 0 0
\(253\) −23.6094 −1.48431
\(254\) 0 0
\(255\) 1.62378 0.101685
\(256\) 0 0
\(257\) −11.0413 −0.688739 −0.344369 0.938834i \(-0.611907\pi\)
−0.344369 + 0.938834i \(0.611907\pi\)
\(258\) 0 0
\(259\) 7.17985 0.446134
\(260\) 0 0
\(261\) 1.87493 0.116055
\(262\) 0 0
\(263\) −3.65018 −0.225080 −0.112540 0.993647i \(-0.535899\pi\)
−0.112540 + 0.993647i \(0.535899\pi\)
\(264\) 0 0
\(265\) −6.01246 −0.369343
\(266\) 0 0
\(267\) 2.96688 0.181570
\(268\) 0 0
\(269\) −18.9112 −1.15304 −0.576518 0.817084i \(-0.695589\pi\)
−0.576518 + 0.817084i \(0.695589\pi\)
\(270\) 0 0
\(271\) 16.1827 0.983028 0.491514 0.870870i \(-0.336444\pi\)
0.491514 + 0.870870i \(0.336444\pi\)
\(272\) 0 0
\(273\) 19.7381 1.19460
\(274\) 0 0
\(275\) 3.60312 0.217276
\(276\) 0 0
\(277\) −29.3557 −1.76381 −0.881906 0.471425i \(-0.843740\pi\)
−0.881906 + 0.471425i \(0.843740\pi\)
\(278\) 0 0
\(279\) −9.40003 −0.562765
\(280\) 0 0
\(281\) 3.14957 0.187888 0.0939438 0.995578i \(-0.470053\pi\)
0.0939438 + 0.995578i \(0.470053\pi\)
\(282\) 0 0
\(283\) 8.73955 0.519512 0.259756 0.965674i \(-0.416358\pi\)
0.259756 + 0.965674i \(0.416358\pi\)
\(284\) 0 0
\(285\) −3.15246 −0.186736
\(286\) 0 0
\(287\) −40.7972 −2.40818
\(288\) 0 0
\(289\) −14.3633 −0.844902
\(290\) 0 0
\(291\) 16.1556 0.947059
\(292\) 0 0
\(293\) −32.2250 −1.88260 −0.941302 0.337564i \(-0.890397\pi\)
−0.941302 + 0.337564i \(0.890397\pi\)
\(294\) 0 0
\(295\) 0.839338 0.0488681
\(296\) 0 0
\(297\) −3.60312 −0.209074
\(298\) 0 0
\(299\) 34.2492 1.98068
\(300\) 0 0
\(301\) 2.54676 0.146793
\(302\) 0 0
\(303\) 2.33736 0.134278
\(304\) 0 0
\(305\) 2.26261 0.129557
\(306\) 0 0
\(307\) 17.2079 0.982108 0.491054 0.871129i \(-0.336612\pi\)
0.491054 + 0.871129i \(0.336612\pi\)
\(308\) 0 0
\(309\) 0.674415 0.0383661
\(310\) 0 0
\(311\) 18.2154 1.03290 0.516451 0.856317i \(-0.327253\pi\)
0.516451 + 0.856317i \(0.327253\pi\)
\(312\) 0 0
\(313\) −8.66363 −0.489697 −0.244849 0.969561i \(-0.578738\pi\)
−0.244849 + 0.969561i \(0.578738\pi\)
\(314\) 0 0
\(315\) −3.77625 −0.212767
\(316\) 0 0
\(317\) 33.8591 1.90172 0.950858 0.309627i \(-0.100204\pi\)
0.950858 + 0.309627i \(0.100204\pi\)
\(318\) 0 0
\(319\) 6.75558 0.378240
\(320\) 0 0
\(321\) −5.72573 −0.319579
\(322\) 0 0
\(323\) −5.11891 −0.284824
\(324\) 0 0
\(325\) −5.22690 −0.289937
\(326\) 0 0
\(327\) 9.89312 0.547091
\(328\) 0 0
\(329\) −6.72919 −0.370992
\(330\) 0 0
\(331\) −5.34625 −0.293856 −0.146928 0.989147i \(-0.546939\pi\)
−0.146928 + 0.989147i \(0.546939\pi\)
\(332\) 0 0
\(333\) 1.90132 0.104192
\(334\) 0 0
\(335\) −1.00000 −0.0546358
\(336\) 0 0
\(337\) −11.1638 −0.608132 −0.304066 0.952651i \(-0.598344\pi\)
−0.304066 + 0.952651i \(0.598344\pi\)
\(338\) 0 0
\(339\) −8.70742 −0.472922
\(340\) 0 0
\(341\) −33.8694 −1.83413
\(342\) 0 0
\(343\) 0.981924 0.0530189
\(344\) 0 0
\(345\) −6.55249 −0.352774
\(346\) 0 0
\(347\) 21.8414 1.17251 0.586253 0.810128i \(-0.300602\pi\)
0.586253 + 0.810128i \(0.300602\pi\)
\(348\) 0 0
\(349\) 12.3056 0.658704 0.329352 0.944207i \(-0.393170\pi\)
0.329352 + 0.944207i \(0.393170\pi\)
\(350\) 0 0
\(351\) 5.22690 0.278992
\(352\) 0 0
\(353\) −16.2258 −0.863612 −0.431806 0.901967i \(-0.642123\pi\)
−0.431806 + 0.901967i \(0.642123\pi\)
\(354\) 0 0
\(355\) 14.9433 0.793109
\(356\) 0 0
\(357\) −6.13180 −0.324529
\(358\) 0 0
\(359\) −10.4341 −0.550693 −0.275346 0.961345i \(-0.588793\pi\)
−0.275346 + 0.961345i \(0.588793\pi\)
\(360\) 0 0
\(361\) −9.06198 −0.476946
\(362\) 0 0
\(363\) −1.98249 −0.104054
\(364\) 0 0
\(365\) −7.34378 −0.384391
\(366\) 0 0
\(367\) −5.90459 −0.308217 −0.154108 0.988054i \(-0.549251\pi\)
−0.154108 + 0.988054i \(0.549251\pi\)
\(368\) 0 0
\(369\) −10.8036 −0.562415
\(370\) 0 0
\(371\) 22.7045 1.17876
\(372\) 0 0
\(373\) −12.5743 −0.651071 −0.325535 0.945530i \(-0.605544\pi\)
−0.325535 + 0.945530i \(0.605544\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −9.80006 −0.504729
\(378\) 0 0
\(379\) 22.4124 1.15125 0.575623 0.817715i \(-0.304760\pi\)
0.575623 + 0.817715i \(0.304760\pi\)
\(380\) 0 0
\(381\) 14.5582 0.745840
\(382\) 0 0
\(383\) 7.87739 0.402516 0.201258 0.979538i \(-0.435497\pi\)
0.201258 + 0.979538i \(0.435497\pi\)
\(384\) 0 0
\(385\) −13.6063 −0.693440
\(386\) 0 0
\(387\) 0.674415 0.0342824
\(388\) 0 0
\(389\) −12.0405 −0.610479 −0.305240 0.952276i \(-0.598737\pi\)
−0.305240 + 0.952276i \(0.598737\pi\)
\(390\) 0 0
\(391\) −10.6398 −0.538079
\(392\) 0 0
\(393\) −21.3501 −1.07697
\(394\) 0 0
\(395\) −7.98822 −0.401931
\(396\) 0 0
\(397\) 29.6133 1.48625 0.743123 0.669154i \(-0.233344\pi\)
0.743123 + 0.669154i \(0.233344\pi\)
\(398\) 0 0
\(399\) 11.9045 0.595969
\(400\) 0 0
\(401\) −13.2243 −0.660391 −0.330196 0.943913i \(-0.607115\pi\)
−0.330196 + 0.943913i \(0.607115\pi\)
\(402\) 0 0
\(403\) 49.1331 2.44749
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 6.85069 0.339576
\(408\) 0 0
\(409\) −33.8798 −1.67525 −0.837623 0.546249i \(-0.816055\pi\)
−0.837623 + 0.546249i \(0.816055\pi\)
\(410\) 0 0
\(411\) 4.53195 0.223545
\(412\) 0 0
\(413\) −3.16955 −0.155963
\(414\) 0 0
\(415\) −4.72820 −0.232098
\(416\) 0 0
\(417\) −4.08117 −0.199856
\(418\) 0 0
\(419\) 1.71931 0.0839939 0.0419970 0.999118i \(-0.486628\pi\)
0.0419970 + 0.999118i \(0.486628\pi\)
\(420\) 0 0
\(421\) 4.37233 0.213094 0.106547 0.994308i \(-0.466020\pi\)
0.106547 + 0.994308i \(0.466020\pi\)
\(422\) 0 0
\(423\) −1.78198 −0.0866428
\(424\) 0 0
\(425\) 1.62378 0.0787650
\(426\) 0 0
\(427\) −8.54417 −0.413481
\(428\) 0 0
\(429\) 18.8332 0.909274
\(430\) 0 0
\(431\) −9.67887 −0.466215 −0.233107 0.972451i \(-0.574889\pi\)
−0.233107 + 0.972451i \(0.574889\pi\)
\(432\) 0 0
\(433\) 23.4480 1.12684 0.563418 0.826172i \(-0.309486\pi\)
0.563418 + 0.826172i \(0.309486\pi\)
\(434\) 0 0
\(435\) 1.87493 0.0898958
\(436\) 0 0
\(437\) 20.6565 0.988133
\(438\) 0 0
\(439\) 2.45727 0.117279 0.0586395 0.998279i \(-0.481324\pi\)
0.0586395 + 0.998279i \(0.481324\pi\)
\(440\) 0 0
\(441\) 7.26003 0.345716
\(442\) 0 0
\(443\) 2.88682 0.137157 0.0685785 0.997646i \(-0.478154\pi\)
0.0685785 + 0.997646i \(0.478154\pi\)
\(444\) 0 0
\(445\) 2.96688 0.140644
\(446\) 0 0
\(447\) −13.0099 −0.615346
\(448\) 0 0
\(449\) 14.6595 0.691824 0.345912 0.938267i \(-0.387570\pi\)
0.345912 + 0.938267i \(0.387570\pi\)
\(450\) 0 0
\(451\) −38.9268 −1.83299
\(452\) 0 0
\(453\) 8.90952 0.418606
\(454\) 0 0
\(455\) 19.7381 0.925335
\(456\) 0 0
\(457\) 30.6905 1.43564 0.717820 0.696229i \(-0.245140\pi\)
0.717820 + 0.696229i \(0.245140\pi\)
\(458\) 0 0
\(459\) −1.62378 −0.0757917
\(460\) 0 0
\(461\) −1.85820 −0.0865451 −0.0432726 0.999063i \(-0.513778\pi\)
−0.0432726 + 0.999063i \(0.513778\pi\)
\(462\) 0 0
\(463\) 24.9450 1.15929 0.579646 0.814868i \(-0.303191\pi\)
0.579646 + 0.814868i \(0.303191\pi\)
\(464\) 0 0
\(465\) −9.40003 −0.435916
\(466\) 0 0
\(467\) −14.4090 −0.666770 −0.333385 0.942791i \(-0.608191\pi\)
−0.333385 + 0.942791i \(0.608191\pi\)
\(468\) 0 0
\(469\) 3.77625 0.174371
\(470\) 0 0
\(471\) 12.1798 0.561218
\(472\) 0 0
\(473\) 2.43000 0.111731
\(474\) 0 0
\(475\) −3.15246 −0.144645
\(476\) 0 0
\(477\) 6.01246 0.275292
\(478\) 0 0
\(479\) 29.4255 1.34449 0.672243 0.740331i \(-0.265331\pi\)
0.672243 + 0.740331i \(0.265331\pi\)
\(480\) 0 0
\(481\) −9.93802 −0.453135
\(482\) 0 0
\(483\) 24.7438 1.12588
\(484\) 0 0
\(485\) 16.1556 0.733589
\(486\) 0 0
\(487\) −4.17571 −0.189219 −0.0946097 0.995514i \(-0.530160\pi\)
−0.0946097 + 0.995514i \(0.530160\pi\)
\(488\) 0 0
\(489\) 21.4730 0.971043
\(490\) 0 0
\(491\) 38.9816 1.75921 0.879607 0.475701i \(-0.157806\pi\)
0.879607 + 0.475701i \(0.157806\pi\)
\(492\) 0 0
\(493\) 3.04447 0.137116
\(494\) 0 0
\(495\) −3.60312 −0.161948
\(496\) 0 0
\(497\) −56.4297 −2.53122
\(498\) 0 0
\(499\) −35.9639 −1.60996 −0.804982 0.593299i \(-0.797825\pi\)
−0.804982 + 0.593299i \(0.797825\pi\)
\(500\) 0 0
\(501\) −8.11330 −0.362475
\(502\) 0 0
\(503\) 5.50018 0.245241 0.122620 0.992454i \(-0.460870\pi\)
0.122620 + 0.992454i \(0.460870\pi\)
\(504\) 0 0
\(505\) 2.33736 0.104011
\(506\) 0 0
\(507\) −14.3205 −0.635997
\(508\) 0 0
\(509\) −23.0234 −1.02049 −0.510247 0.860028i \(-0.670446\pi\)
−0.510247 + 0.860028i \(0.670446\pi\)
\(510\) 0 0
\(511\) 27.7319 1.22679
\(512\) 0 0
\(513\) 3.15246 0.139185
\(514\) 0 0
\(515\) 0.674415 0.0297183
\(516\) 0 0
\(517\) −6.42069 −0.282382
\(518\) 0 0
\(519\) 5.16393 0.226671
\(520\) 0 0
\(521\) −29.2379 −1.28094 −0.640469 0.767984i \(-0.721260\pi\)
−0.640469 + 0.767984i \(0.721260\pi\)
\(522\) 0 0
\(523\) 45.3475 1.98291 0.991454 0.130458i \(-0.0416449\pi\)
0.991454 + 0.130458i \(0.0416449\pi\)
\(524\) 0 0
\(525\) −3.77625 −0.164809
\(526\) 0 0
\(527\) −15.2636 −0.664893
\(528\) 0 0
\(529\) 19.9351 0.866745
\(530\) 0 0
\(531\) −0.839338 −0.0364242
\(532\) 0 0
\(533\) 56.4696 2.44597
\(534\) 0 0
\(535\) −5.72573 −0.247545
\(536\) 0 0
\(537\) 11.8905 0.513115
\(538\) 0 0
\(539\) 26.1588 1.12674
\(540\) 0 0
\(541\) −4.97915 −0.214070 −0.107035 0.994255i \(-0.534136\pi\)
−0.107035 + 0.994255i \(0.534136\pi\)
\(542\) 0 0
\(543\) 20.0123 0.858812
\(544\) 0 0
\(545\) 9.89312 0.423775
\(546\) 0 0
\(547\) −26.7029 −1.14174 −0.570868 0.821042i \(-0.693393\pi\)
−0.570868 + 0.821042i \(0.693393\pi\)
\(548\) 0 0
\(549\) −2.26261 −0.0965659
\(550\) 0 0
\(551\) −5.91063 −0.251801
\(552\) 0 0
\(553\) 30.1655 1.28277
\(554\) 0 0
\(555\) 1.90132 0.0807065
\(556\) 0 0
\(557\) 25.3824 1.07549 0.537743 0.843109i \(-0.319277\pi\)
0.537743 + 0.843109i \(0.319277\pi\)
\(558\) 0 0
\(559\) −3.52510 −0.149096
\(560\) 0 0
\(561\) −5.85069 −0.247016
\(562\) 0 0
\(563\) −32.6969 −1.37801 −0.689005 0.724757i \(-0.741952\pi\)
−0.689005 + 0.724757i \(0.741952\pi\)
\(564\) 0 0
\(565\) −8.70742 −0.366324
\(566\) 0 0
\(567\) 3.77625 0.158587
\(568\) 0 0
\(569\) −41.4469 −1.73754 −0.868771 0.495213i \(-0.835090\pi\)
−0.868771 + 0.495213i \(0.835090\pi\)
\(570\) 0 0
\(571\) 19.0228 0.796080 0.398040 0.917368i \(-0.369690\pi\)
0.398040 + 0.917368i \(0.369690\pi\)
\(572\) 0 0
\(573\) −1.63525 −0.0683136
\(574\) 0 0
\(575\) −6.55249 −0.273258
\(576\) 0 0
\(577\) 4.22937 0.176071 0.0880355 0.996117i \(-0.471941\pi\)
0.0880355 + 0.996117i \(0.471941\pi\)
\(578\) 0 0
\(579\) −4.90805 −0.203971
\(580\) 0 0
\(581\) 17.8548 0.740743
\(582\) 0 0
\(583\) 21.6636 0.897216
\(584\) 0 0
\(585\) 5.22690 0.216106
\(586\) 0 0
\(587\) 8.85069 0.365307 0.182653 0.983177i \(-0.441531\pi\)
0.182653 + 0.983177i \(0.441531\pi\)
\(588\) 0 0
\(589\) 29.6332 1.22102
\(590\) 0 0
\(591\) −23.5681 −0.969462
\(592\) 0 0
\(593\) −12.4167 −0.509895 −0.254947 0.966955i \(-0.582058\pi\)
−0.254947 + 0.966955i \(0.582058\pi\)
\(594\) 0 0
\(595\) −6.13180 −0.251379
\(596\) 0 0
\(597\) 26.1887 1.07183
\(598\) 0 0
\(599\) 12.6469 0.516739 0.258369 0.966046i \(-0.416815\pi\)
0.258369 + 0.966046i \(0.416815\pi\)
\(600\) 0 0
\(601\) 8.06809 0.329104 0.164552 0.986368i \(-0.447382\pi\)
0.164552 + 0.986368i \(0.447382\pi\)
\(602\) 0 0
\(603\) 1.00000 0.0407231
\(604\) 0 0
\(605\) −1.98249 −0.0805997
\(606\) 0 0
\(607\) 17.8958 0.726370 0.363185 0.931717i \(-0.381689\pi\)
0.363185 + 0.931717i \(0.381689\pi\)
\(608\) 0 0
\(609\) −7.08018 −0.286903
\(610\) 0 0
\(611\) 9.31424 0.376814
\(612\) 0 0
\(613\) 22.3202 0.901505 0.450753 0.892649i \(-0.351156\pi\)
0.450753 + 0.892649i \(0.351156\pi\)
\(614\) 0 0
\(615\) −10.8036 −0.435645
\(616\) 0 0
\(617\) 31.0658 1.25066 0.625331 0.780359i \(-0.284964\pi\)
0.625331 + 0.780359i \(0.284964\pi\)
\(618\) 0 0
\(619\) 32.4628 1.30479 0.652395 0.757879i \(-0.273764\pi\)
0.652395 + 0.757879i \(0.273764\pi\)
\(620\) 0 0
\(621\) 6.55249 0.262942
\(622\) 0 0
\(623\) −11.2037 −0.448865
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 11.3587 0.453623
\(628\) 0 0
\(629\) 3.08733 0.123100
\(630\) 0 0
\(631\) −2.89102 −0.115090 −0.0575448 0.998343i \(-0.518327\pi\)
−0.0575448 + 0.998343i \(0.518327\pi\)
\(632\) 0 0
\(633\) −4.31054 −0.171329
\(634\) 0 0
\(635\) 14.5582 0.577726
\(636\) 0 0
\(637\) −37.9475 −1.50353
\(638\) 0 0
\(639\) −14.9433 −0.591149
\(640\) 0 0
\(641\) −34.8389 −1.37605 −0.688027 0.725685i \(-0.741523\pi\)
−0.688027 + 0.725685i \(0.741523\pi\)
\(642\) 0 0
\(643\) −6.86068 −0.270559 −0.135279 0.990807i \(-0.543193\pi\)
−0.135279 + 0.990807i \(0.543193\pi\)
\(644\) 0 0
\(645\) 0.674415 0.0265551
\(646\) 0 0
\(647\) 8.55014 0.336141 0.168070 0.985775i \(-0.446246\pi\)
0.168070 + 0.985775i \(0.446246\pi\)
\(648\) 0 0
\(649\) −3.02424 −0.118712
\(650\) 0 0
\(651\) 35.4968 1.39123
\(652\) 0 0
\(653\) −38.1123 −1.49145 −0.745724 0.666255i \(-0.767896\pi\)
−0.745724 + 0.666255i \(0.767896\pi\)
\(654\) 0 0
\(655\) −21.3501 −0.834217
\(656\) 0 0
\(657\) 7.34378 0.286508
\(658\) 0 0
\(659\) 18.8356 0.733732 0.366866 0.930274i \(-0.380431\pi\)
0.366866 + 0.930274i \(0.380431\pi\)
\(660\) 0 0
\(661\) 32.0398 1.24621 0.623103 0.782140i \(-0.285872\pi\)
0.623103 + 0.782140i \(0.285872\pi\)
\(662\) 0 0
\(663\) 8.48736 0.329622
\(664\) 0 0
\(665\) 11.9045 0.461636
\(666\) 0 0
\(667\) −12.2854 −0.475694
\(668\) 0 0
\(669\) 12.6662 0.489704
\(670\) 0 0
\(671\) −8.15246 −0.314722
\(672\) 0 0
\(673\) 37.9297 1.46208 0.731041 0.682333i \(-0.239035\pi\)
0.731041 + 0.682333i \(0.239035\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 6.06323 0.233029 0.116514 0.993189i \(-0.462828\pi\)
0.116514 + 0.993189i \(0.462828\pi\)
\(678\) 0 0
\(679\) −61.0076 −2.34125
\(680\) 0 0
\(681\) −10.0812 −0.386311
\(682\) 0 0
\(683\) 24.7163 0.945743 0.472872 0.881131i \(-0.343217\pi\)
0.472872 + 0.881131i \(0.343217\pi\)
\(684\) 0 0
\(685\) 4.53195 0.173157
\(686\) 0 0
\(687\) 5.91063 0.225505
\(688\) 0 0
\(689\) −31.4266 −1.19726
\(690\) 0 0
\(691\) −45.7806 −1.74158 −0.870789 0.491657i \(-0.836391\pi\)
−0.870789 + 0.491657i \(0.836391\pi\)
\(692\) 0 0
\(693\) 13.6063 0.516860
\(694\) 0 0
\(695\) −4.08117 −0.154808
\(696\) 0 0
\(697\) −17.5428 −0.664479
\(698\) 0 0
\(699\) −12.1073 −0.457938
\(700\) 0 0
\(701\) −43.2338 −1.63292 −0.816458 0.577404i \(-0.804066\pi\)
−0.816458 + 0.577404i \(0.804066\pi\)
\(702\) 0 0
\(703\) −5.99384 −0.226062
\(704\) 0 0
\(705\) −1.78198 −0.0671132
\(706\) 0 0
\(707\) −8.82645 −0.331953
\(708\) 0 0
\(709\) 4.40219 0.165328 0.0826638 0.996577i \(-0.473657\pi\)
0.0826638 + 0.996577i \(0.473657\pi\)
\(710\) 0 0
\(711\) 7.98822 0.299582
\(712\) 0 0
\(713\) 61.5936 2.30670
\(714\) 0 0
\(715\) 18.8332 0.704321
\(716\) 0 0
\(717\) 9.08105 0.339138
\(718\) 0 0
\(719\) −35.8469 −1.33686 −0.668432 0.743774i \(-0.733034\pi\)
−0.668432 + 0.743774i \(0.733034\pi\)
\(720\) 0 0
\(721\) −2.54676 −0.0948461
\(722\) 0 0
\(723\) 26.0627 0.969281
\(724\) 0 0
\(725\) 1.87493 0.0696330
\(726\) 0 0
\(727\) 11.6395 0.431686 0.215843 0.976428i \(-0.430750\pi\)
0.215843 + 0.976428i \(0.430750\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 1.09510 0.0405039
\(732\) 0 0
\(733\) 27.6633 1.02177 0.510884 0.859650i \(-0.329318\pi\)
0.510884 + 0.859650i \(0.329318\pi\)
\(734\) 0 0
\(735\) 7.26003 0.267790
\(736\) 0 0
\(737\) 3.60312 0.132723
\(738\) 0 0
\(739\) −24.8592 −0.914461 −0.457230 0.889348i \(-0.651159\pi\)
−0.457230 + 0.889348i \(0.651159\pi\)
\(740\) 0 0
\(741\) −16.4776 −0.605321
\(742\) 0 0
\(743\) 34.7609 1.27525 0.637627 0.770345i \(-0.279916\pi\)
0.637627 + 0.770345i \(0.279916\pi\)
\(744\) 0 0
\(745\) −13.0099 −0.476645
\(746\) 0 0
\(747\) 4.72820 0.172996
\(748\) 0 0
\(749\) 21.6218 0.790042
\(750\) 0 0
\(751\) −46.1218 −1.68301 −0.841503 0.540252i \(-0.818329\pi\)
−0.841503 + 0.540252i \(0.818329\pi\)
\(752\) 0 0
\(753\) 0.877201 0.0319670
\(754\) 0 0
\(755\) 8.90952 0.324251
\(756\) 0 0
\(757\) −49.7737 −1.80906 −0.904528 0.426413i \(-0.859777\pi\)
−0.904528 + 0.426413i \(0.859777\pi\)
\(758\) 0 0
\(759\) 23.6094 0.856968
\(760\) 0 0
\(761\) 17.4073 0.631015 0.315507 0.948923i \(-0.397825\pi\)
0.315507 + 0.948923i \(0.397825\pi\)
\(762\) 0 0
\(763\) −37.3588 −1.35248
\(764\) 0 0
\(765\) −1.62378 −0.0587080
\(766\) 0 0
\(767\) 4.38714 0.158410
\(768\) 0 0
\(769\) −33.2180 −1.19787 −0.598935 0.800797i \(-0.704409\pi\)
−0.598935 + 0.800797i \(0.704409\pi\)
\(770\) 0 0
\(771\) 11.0413 0.397643
\(772\) 0 0
\(773\) −25.0868 −0.902309 −0.451154 0.892446i \(-0.648988\pi\)
−0.451154 + 0.892446i \(0.648988\pi\)
\(774\) 0 0
\(775\) −9.40003 −0.337659
\(776\) 0 0
\(777\) −7.17985 −0.257576
\(778\) 0 0
\(779\) 34.0580 1.22026
\(780\) 0 0
\(781\) −53.8426 −1.92664
\(782\) 0 0
\(783\) −1.87493 −0.0670044
\(784\) 0 0
\(785\) 12.1798 0.434717
\(786\) 0 0
\(787\) −19.6568 −0.700689 −0.350344 0.936621i \(-0.613935\pi\)
−0.350344 + 0.936621i \(0.613935\pi\)
\(788\) 0 0
\(789\) 3.65018 0.129950
\(790\) 0 0
\(791\) 32.8813 1.16913
\(792\) 0 0
\(793\) 11.8265 0.419970
\(794\) 0 0
\(795\) 6.01246 0.213240
\(796\) 0 0
\(797\) −21.3425 −0.755990 −0.377995 0.925808i \(-0.623386\pi\)
−0.377995 + 0.925808i \(0.623386\pi\)
\(798\) 0 0
\(799\) −2.89355 −0.102366
\(800\) 0 0
\(801\) −2.96688 −0.104829
\(802\) 0 0
\(803\) 26.4605 0.933772
\(804\) 0 0
\(805\) 24.7438 0.872105
\(806\) 0 0
\(807\) 18.9112 0.665706
\(808\) 0 0
\(809\) 20.8446 0.732859 0.366429 0.930446i \(-0.380580\pi\)
0.366429 + 0.930446i \(0.380580\pi\)
\(810\) 0 0
\(811\) −12.5594 −0.441020 −0.220510 0.975385i \(-0.570772\pi\)
−0.220510 + 0.975385i \(0.570772\pi\)
\(812\) 0 0
\(813\) −16.1827 −0.567552
\(814\) 0 0
\(815\) 21.4730 0.752166
\(816\) 0 0
\(817\) −2.12607 −0.0743817
\(818\) 0 0
\(819\) −19.7381 −0.689704
\(820\) 0 0
\(821\) 8.51352 0.297124 0.148562 0.988903i \(-0.452536\pi\)
0.148562 + 0.988903i \(0.452536\pi\)
\(822\) 0 0
\(823\) 7.89589 0.275234 0.137617 0.990486i \(-0.456056\pi\)
0.137617 + 0.990486i \(0.456056\pi\)
\(824\) 0 0
\(825\) −3.60312 −0.125445
\(826\) 0 0
\(827\) 32.1443 1.11777 0.558883 0.829247i \(-0.311230\pi\)
0.558883 + 0.829247i \(0.311230\pi\)
\(828\) 0 0
\(829\) 17.6986 0.614698 0.307349 0.951597i \(-0.400558\pi\)
0.307349 + 0.951597i \(0.400558\pi\)
\(830\) 0 0
\(831\) 29.3557 1.01834
\(832\) 0 0
\(833\) 11.7887 0.408454
\(834\) 0 0
\(835\) −8.11330 −0.280772
\(836\) 0 0
\(837\) 9.40003 0.324912
\(838\) 0 0
\(839\) 48.8281 1.68573 0.842867 0.538122i \(-0.180866\pi\)
0.842867 + 0.538122i \(0.180866\pi\)
\(840\) 0 0
\(841\) −25.4847 −0.878781
\(842\) 0 0
\(843\) −3.14957 −0.108477
\(844\) 0 0
\(845\) −14.3205 −0.492641
\(846\) 0 0
\(847\) 7.48637 0.257235
\(848\) 0 0
\(849\) −8.73955 −0.299940
\(850\) 0 0
\(851\) −12.4584 −0.427068
\(852\) 0 0
\(853\) −21.5404 −0.737529 −0.368765 0.929523i \(-0.620219\pi\)
−0.368765 + 0.929523i \(0.620219\pi\)
\(854\) 0 0
\(855\) 3.15246 0.107812
\(856\) 0 0
\(857\) 25.5713 0.873497 0.436749 0.899584i \(-0.356130\pi\)
0.436749 + 0.899584i \(0.356130\pi\)
\(858\) 0 0
\(859\) −24.4722 −0.834981 −0.417490 0.908681i \(-0.637090\pi\)
−0.417490 + 0.908681i \(0.637090\pi\)
\(860\) 0 0
\(861\) 40.7972 1.39036
\(862\) 0 0
\(863\) 3.16054 0.107586 0.0537931 0.998552i \(-0.482869\pi\)
0.0537931 + 0.998552i \(0.482869\pi\)
\(864\) 0 0
\(865\) 5.16393 0.175579
\(866\) 0 0
\(867\) 14.3633 0.487804
\(868\) 0 0
\(869\) 28.7825 0.976381
\(870\) 0 0
\(871\) −5.22690 −0.177107
\(872\) 0 0
\(873\) −16.1556 −0.546785
\(874\) 0 0
\(875\) −3.77625 −0.127660
\(876\) 0 0
\(877\) 2.25292 0.0760758 0.0380379 0.999276i \(-0.487889\pi\)
0.0380379 + 0.999276i \(0.487889\pi\)
\(878\) 0 0
\(879\) 32.2250 1.08692
\(880\) 0 0
\(881\) −16.7432 −0.564094 −0.282047 0.959401i \(-0.591013\pi\)
−0.282047 + 0.959401i \(0.591013\pi\)
\(882\) 0 0
\(883\) −47.7769 −1.60782 −0.803910 0.594751i \(-0.797251\pi\)
−0.803910 + 0.594751i \(0.797251\pi\)
\(884\) 0 0
\(885\) −0.839338 −0.0282140
\(886\) 0 0
\(887\) −23.9579 −0.804429 −0.402215 0.915545i \(-0.631759\pi\)
−0.402215 + 0.915545i \(0.631759\pi\)
\(888\) 0 0
\(889\) −54.9754 −1.84382
\(890\) 0 0
\(891\) 3.60312 0.120709
\(892\) 0 0
\(893\) 5.61762 0.187987
\(894\) 0 0
\(895\) 11.8905 0.397457
\(896\) 0 0
\(897\) −34.2492 −1.14355
\(898\) 0 0
\(899\) −17.6243 −0.587805
\(900\) 0 0
\(901\) 9.76293 0.325250
\(902\) 0 0
\(903\) −2.54676 −0.0847507
\(904\) 0 0
\(905\) 20.0123 0.665233
\(906\) 0 0
\(907\) 21.3367 0.708475 0.354237 0.935155i \(-0.384740\pi\)
0.354237 + 0.935155i \(0.384740\pi\)
\(908\) 0 0
\(909\) −2.33736 −0.0775254
\(910\) 0 0
\(911\) −9.53867 −0.316030 −0.158015 0.987437i \(-0.550509\pi\)
−0.158015 + 0.987437i \(0.550509\pi\)
\(912\) 0 0
\(913\) 17.0363 0.563818
\(914\) 0 0
\(915\) −2.26261 −0.0747996
\(916\) 0 0
\(917\) 80.6231 2.66241
\(918\) 0 0
\(919\) 32.6636 1.07747 0.538736 0.842475i \(-0.318902\pi\)
0.538736 + 0.842475i \(0.318902\pi\)
\(920\) 0 0
\(921\) −17.2079 −0.567020
\(922\) 0 0
\(923\) 78.1073 2.57093
\(924\) 0 0
\(925\) 1.90132 0.0625150
\(926\) 0 0
\(927\) −0.674415 −0.0221507
\(928\) 0 0
\(929\) 51.0801 1.67588 0.837941 0.545760i \(-0.183759\pi\)
0.837941 + 0.545760i \(0.183759\pi\)
\(930\) 0 0
\(931\) −22.8870 −0.750090
\(932\) 0 0
\(933\) −18.2154 −0.596347
\(934\) 0 0
\(935\) −5.85069 −0.191338
\(936\) 0 0
\(937\) 1.88651 0.0616297 0.0308148 0.999525i \(-0.490190\pi\)
0.0308148 + 0.999525i \(0.490190\pi\)
\(938\) 0 0
\(939\) 8.66363 0.282727
\(940\) 0 0
\(941\) 9.91748 0.323300 0.161650 0.986848i \(-0.448318\pi\)
0.161650 + 0.986848i \(0.448318\pi\)
\(942\) 0 0
\(943\) 70.7907 2.30526
\(944\) 0 0
\(945\) 3.77625 0.122841
\(946\) 0 0
\(947\) −33.5684 −1.09083 −0.545413 0.838168i \(-0.683627\pi\)
−0.545413 + 0.838168i \(0.683627\pi\)
\(948\) 0 0
\(949\) −38.3852 −1.24604
\(950\) 0 0
\(951\) −33.8591 −1.09796
\(952\) 0 0
\(953\) 36.9301 1.19628 0.598142 0.801390i \(-0.295906\pi\)
0.598142 + 0.801390i \(0.295906\pi\)
\(954\) 0 0
\(955\) −1.63525 −0.0529155
\(956\) 0 0
\(957\) −6.75558 −0.218377
\(958\) 0 0
\(959\) −17.1137 −0.552632
\(960\) 0 0
\(961\) 57.3605 1.85034
\(962\) 0 0
\(963\) 5.72573 0.184509
\(964\) 0 0
\(965\) −4.90805 −0.157996
\(966\) 0 0
\(967\) 2.84706 0.0915552 0.0457776 0.998952i \(-0.485423\pi\)
0.0457776 + 0.998952i \(0.485423\pi\)
\(968\) 0 0
\(969\) 5.11891 0.164443
\(970\) 0 0
\(971\) 48.8727 1.56840 0.784200 0.620508i \(-0.213074\pi\)
0.784200 + 0.620508i \(0.213074\pi\)
\(972\) 0 0
\(973\) 15.4115 0.494070
\(974\) 0 0
\(975\) 5.22690 0.167395
\(976\) 0 0
\(977\) 30.2666 0.968315 0.484158 0.874981i \(-0.339126\pi\)
0.484158 + 0.874981i \(0.339126\pi\)
\(978\) 0 0
\(979\) −10.6900 −0.341655
\(980\) 0 0
\(981\) −9.89312 −0.315863
\(982\) 0 0
\(983\) 34.7693 1.10897 0.554484 0.832194i \(-0.312916\pi\)
0.554484 + 0.832194i \(0.312916\pi\)
\(984\) 0 0
\(985\) −23.5681 −0.750942
\(986\) 0 0
\(987\) 6.72919 0.214192
\(988\) 0 0
\(989\) −4.41910 −0.140519
\(990\) 0 0
\(991\) 51.3396 1.63085 0.815427 0.578859i \(-0.196502\pi\)
0.815427 + 0.578859i \(0.196502\pi\)
\(992\) 0 0
\(993\) 5.34625 0.169658
\(994\) 0 0
\(995\) 26.1887 0.830239
\(996\) 0 0
\(997\) 58.3360 1.84752 0.923760 0.382971i \(-0.125099\pi\)
0.923760 + 0.382971i \(0.125099\pi\)
\(998\) 0 0
\(999\) −1.90132 −0.0601551
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 4020.2.a.b.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4020.2.a.b.1.4 4 1.1 even 1 trivial