Properties

Label 4008.2.a.h.1.1
Level $4008$
Weight $2$
Character 4008.1
Self dual yes
Analytic conductor $32.004$
Analytic rank $1$
Dimension $8$
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] = [4008,2,Mod(1,4008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4008, 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("4008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4008.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: \(32.0040411301\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 23x^{6} - 3x^{5} + 163x^{4} + 13x^{3} - 418x^{2} + 4x + 269 \) 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 1.1
Root \(-3.24252\) of defining polynomial
Character \(\chi\) \(=\) 4008.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.24252 q^{5} -3.21024 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.24252 q^{5} -3.21024 q^{7} +1.00000 q^{9} +2.82128 q^{11} -5.85040 q^{13} +3.24252 q^{15} +1.86063 q^{17} +5.37894 q^{19} +3.21024 q^{21} +4.21024 q^{23} +5.51393 q^{25} -1.00000 q^{27} +3.33063 q^{29} +1.27146 q^{31} -2.82128 q^{33} +10.4092 q^{35} -2.06358 q^{37} +5.85040 q^{39} +2.10689 q^{41} +6.96695 q^{43} -3.24252 q^{45} -6.68686 q^{47} +3.30561 q^{49} -1.86063 q^{51} +9.34827 q^{53} -9.14806 q^{55} -5.37894 q^{57} -11.2378 q^{59} +7.78735 q^{61} -3.21024 q^{63} +18.9700 q^{65} +3.51420 q^{67} -4.21024 q^{69} -11.6222 q^{71} -5.43704 q^{73} -5.51393 q^{75} -9.05698 q^{77} -4.16325 q^{79} +1.00000 q^{81} -1.27361 q^{83} -6.03313 q^{85} -3.33063 q^{87} -5.91690 q^{89} +18.7811 q^{91} -1.27146 q^{93} -17.4413 q^{95} +4.01613 q^{97} +2.82128 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} - q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} - q^{7} + 8 q^{9} - 3 q^{11} - 8 q^{13} - 7 q^{17} + q^{21} + 9 q^{23} + 6 q^{25} - 8 q^{27} + 17 q^{29} - 23 q^{31} + 3 q^{33} - 15 q^{35} + 8 q^{37} + 8 q^{39} - 8 q^{41} - 2 q^{43} - 34 q^{47} + 5 q^{49} + 7 q^{51} + 12 q^{53} - 7 q^{55} - 16 q^{59} - 2 q^{61} - q^{63} - 14 q^{65} + 21 q^{67} - 9 q^{69} - 29 q^{71} - 38 q^{73} - 6 q^{75} + 20 q^{77} - 12 q^{79} + 8 q^{81} - 32 q^{83} + 23 q^{85} - 17 q^{87} + 11 q^{89} - 5 q^{91} + 23 q^{93} - 67 q^{95} + 8 q^{97} - 3 q^{99}+O(q^{100}) \) 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
\(5\) −3.24252 −1.45010 −0.725049 0.688697i \(-0.758183\pi\)
−0.725049 + 0.688697i \(0.758183\pi\)
\(6\) 0 0
\(7\) −3.21024 −1.21335 −0.606677 0.794948i \(-0.707498\pi\)
−0.606677 + 0.794948i \(0.707498\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.82128 0.850649 0.425324 0.905041i \(-0.360160\pi\)
0.425324 + 0.905041i \(0.360160\pi\)
\(12\) 0 0
\(13\) −5.85040 −1.62261 −0.811304 0.584625i \(-0.801242\pi\)
−0.811304 + 0.584625i \(0.801242\pi\)
\(14\) 0 0
\(15\) 3.24252 0.837215
\(16\) 0 0
\(17\) 1.86063 0.451269 0.225635 0.974212i \(-0.427554\pi\)
0.225635 + 0.974212i \(0.427554\pi\)
\(18\) 0 0
\(19\) 5.37894 1.23401 0.617007 0.786958i \(-0.288345\pi\)
0.617007 + 0.786958i \(0.288345\pi\)
\(20\) 0 0
\(21\) 3.21024 0.700531
\(22\) 0 0
\(23\) 4.21024 0.877895 0.438947 0.898513i \(-0.355351\pi\)
0.438947 + 0.898513i \(0.355351\pi\)
\(24\) 0 0
\(25\) 5.51393 1.10279
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.33063 0.618483 0.309241 0.950984i \(-0.399925\pi\)
0.309241 + 0.950984i \(0.399925\pi\)
\(30\) 0 0
\(31\) 1.27146 0.228361 0.114180 0.993460i \(-0.463576\pi\)
0.114180 + 0.993460i \(0.463576\pi\)
\(32\) 0 0
\(33\) −2.82128 −0.491122
\(34\) 0 0
\(35\) 10.4092 1.75948
\(36\) 0 0
\(37\) −2.06358 −0.339251 −0.169625 0.985509i \(-0.554256\pi\)
−0.169625 + 0.985509i \(0.554256\pi\)
\(38\) 0 0
\(39\) 5.85040 0.936813
\(40\) 0 0
\(41\) 2.10689 0.329041 0.164521 0.986374i \(-0.447392\pi\)
0.164521 + 0.986374i \(0.447392\pi\)
\(42\) 0 0
\(43\) 6.96695 1.06245 0.531225 0.847231i \(-0.321732\pi\)
0.531225 + 0.847231i \(0.321732\pi\)
\(44\) 0 0
\(45\) −3.24252 −0.483366
\(46\) 0 0
\(47\) −6.68686 −0.975380 −0.487690 0.873017i \(-0.662160\pi\)
−0.487690 + 0.873017i \(0.662160\pi\)
\(48\) 0 0
\(49\) 3.30561 0.472230
\(50\) 0 0
\(51\) −1.86063 −0.260540
\(52\) 0 0
\(53\) 9.34827 1.28408 0.642042 0.766670i \(-0.278088\pi\)
0.642042 + 0.766670i \(0.278088\pi\)
\(54\) 0 0
\(55\) −9.14806 −1.23352
\(56\) 0 0
\(57\) −5.37894 −0.712458
\(58\) 0 0
\(59\) −11.2378 −1.46304 −0.731518 0.681822i \(-0.761188\pi\)
−0.731518 + 0.681822i \(0.761188\pi\)
\(60\) 0 0
\(61\) 7.78735 0.997068 0.498534 0.866870i \(-0.333872\pi\)
0.498534 + 0.866870i \(0.333872\pi\)
\(62\) 0 0
\(63\) −3.21024 −0.404452
\(64\) 0 0
\(65\) 18.9700 2.35294
\(66\) 0 0
\(67\) 3.51420 0.429327 0.214664 0.976688i \(-0.431134\pi\)
0.214664 + 0.976688i \(0.431134\pi\)
\(68\) 0 0
\(69\) −4.21024 −0.506853
\(70\) 0 0
\(71\) −11.6222 −1.37930 −0.689651 0.724142i \(-0.742236\pi\)
−0.689651 + 0.724142i \(0.742236\pi\)
\(72\) 0 0
\(73\) −5.43704 −0.636357 −0.318179 0.948031i \(-0.603071\pi\)
−0.318179 + 0.948031i \(0.603071\pi\)
\(74\) 0 0
\(75\) −5.51393 −0.636694
\(76\) 0 0
\(77\) −9.05698 −1.03214
\(78\) 0 0
\(79\) −4.16325 −0.468402 −0.234201 0.972188i \(-0.575247\pi\)
−0.234201 + 0.972188i \(0.575247\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.27361 −0.139797 −0.0698987 0.997554i \(-0.522268\pi\)
−0.0698987 + 0.997554i \(0.522268\pi\)
\(84\) 0 0
\(85\) −6.03313 −0.654385
\(86\) 0 0
\(87\) −3.33063 −0.357081
\(88\) 0 0
\(89\) −5.91690 −0.627190 −0.313595 0.949557i \(-0.601533\pi\)
−0.313595 + 0.949557i \(0.601533\pi\)
\(90\) 0 0
\(91\) 18.7811 1.96880
\(92\) 0 0
\(93\) −1.27146 −0.131844
\(94\) 0 0
\(95\) −17.4413 −1.78944
\(96\) 0 0
\(97\) 4.01613 0.407776 0.203888 0.978994i \(-0.434642\pi\)
0.203888 + 0.978994i \(0.434642\pi\)
\(98\) 0 0
\(99\) 2.82128 0.283550
\(100\) 0 0
\(101\) 9.09153 0.904641 0.452321 0.891855i \(-0.350596\pi\)
0.452321 + 0.891855i \(0.350596\pi\)
\(102\) 0 0
\(103\) −15.5592 −1.53309 −0.766545 0.642190i \(-0.778026\pi\)
−0.766545 + 0.642190i \(0.778026\pi\)
\(104\) 0 0
\(105\) −10.4092 −1.01584
\(106\) 0 0
\(107\) 4.95284 0.478809 0.239404 0.970920i \(-0.423048\pi\)
0.239404 + 0.970920i \(0.423048\pi\)
\(108\) 0 0
\(109\) 9.59455 0.918991 0.459495 0.888180i \(-0.348030\pi\)
0.459495 + 0.888180i \(0.348030\pi\)
\(110\) 0 0
\(111\) 2.06358 0.195866
\(112\) 0 0
\(113\) −14.1378 −1.32998 −0.664988 0.746854i \(-0.731563\pi\)
−0.664988 + 0.746854i \(0.731563\pi\)
\(114\) 0 0
\(115\) −13.6518 −1.27303
\(116\) 0 0
\(117\) −5.85040 −0.540869
\(118\) 0 0
\(119\) −5.97306 −0.547550
\(120\) 0 0
\(121\) −3.04036 −0.276397
\(122\) 0 0
\(123\) −2.10689 −0.189972
\(124\) 0 0
\(125\) −1.66643 −0.149050
\(126\) 0 0
\(127\) 2.06152 0.182930 0.0914651 0.995808i \(-0.470845\pi\)
0.0914651 + 0.995808i \(0.470845\pi\)
\(128\) 0 0
\(129\) −6.96695 −0.613406
\(130\) 0 0
\(131\) −14.0106 −1.22411 −0.612055 0.790815i \(-0.709657\pi\)
−0.612055 + 0.790815i \(0.709657\pi\)
\(132\) 0 0
\(133\) −17.2677 −1.49730
\(134\) 0 0
\(135\) 3.24252 0.279072
\(136\) 0 0
\(137\) 9.83399 0.840174 0.420087 0.907484i \(-0.362000\pi\)
0.420087 + 0.907484i \(0.362000\pi\)
\(138\) 0 0
\(139\) 4.61893 0.391773 0.195886 0.980627i \(-0.437242\pi\)
0.195886 + 0.980627i \(0.437242\pi\)
\(140\) 0 0
\(141\) 6.68686 0.563136
\(142\) 0 0
\(143\) −16.5056 −1.38027
\(144\) 0 0
\(145\) −10.7996 −0.896861
\(146\) 0 0
\(147\) −3.30561 −0.272642
\(148\) 0 0
\(149\) 11.6341 0.953105 0.476553 0.879146i \(-0.341886\pi\)
0.476553 + 0.879146i \(0.341886\pi\)
\(150\) 0 0
\(151\) 13.1949 1.07379 0.536893 0.843651i \(-0.319598\pi\)
0.536893 + 0.843651i \(0.319598\pi\)
\(152\) 0 0
\(153\) 1.86063 0.150423
\(154\) 0 0
\(155\) −4.12273 −0.331146
\(156\) 0 0
\(157\) −5.22817 −0.417254 −0.208627 0.977995i \(-0.566899\pi\)
−0.208627 + 0.977995i \(0.566899\pi\)
\(158\) 0 0
\(159\) −9.34827 −0.741366
\(160\) 0 0
\(161\) −13.5158 −1.06520
\(162\) 0 0
\(163\) 0.261930 0.0205160 0.0102580 0.999947i \(-0.496735\pi\)
0.0102580 + 0.999947i \(0.496735\pi\)
\(164\) 0 0
\(165\) 9.14806 0.712176
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) 21.2271 1.63286
\(170\) 0 0
\(171\) 5.37894 0.411338
\(172\) 0 0
\(173\) −19.5287 −1.48474 −0.742370 0.669990i \(-0.766298\pi\)
−0.742370 + 0.669990i \(0.766298\pi\)
\(174\) 0 0
\(175\) −17.7010 −1.33807
\(176\) 0 0
\(177\) 11.2378 0.844685
\(178\) 0 0
\(179\) −19.0968 −1.42737 −0.713683 0.700469i \(-0.752974\pi\)
−0.713683 + 0.700469i \(0.752974\pi\)
\(180\) 0 0
\(181\) −8.26626 −0.614427 −0.307213 0.951641i \(-0.599396\pi\)
−0.307213 + 0.951641i \(0.599396\pi\)
\(182\) 0 0
\(183\) −7.78735 −0.575658
\(184\) 0 0
\(185\) 6.69120 0.491947
\(186\) 0 0
\(187\) 5.24936 0.383872
\(188\) 0 0
\(189\) 3.21024 0.233510
\(190\) 0 0
\(191\) −21.0760 −1.52501 −0.762504 0.646984i \(-0.776030\pi\)
−0.762504 + 0.646984i \(0.776030\pi\)
\(192\) 0 0
\(193\) 22.3043 1.60550 0.802749 0.596318i \(-0.203370\pi\)
0.802749 + 0.596318i \(0.203370\pi\)
\(194\) 0 0
\(195\) −18.9700 −1.35847
\(196\) 0 0
\(197\) −10.7503 −0.765925 −0.382962 0.923764i \(-0.625096\pi\)
−0.382962 + 0.923764i \(0.625096\pi\)
\(198\) 0 0
\(199\) −12.7649 −0.904877 −0.452439 0.891796i \(-0.649446\pi\)
−0.452439 + 0.891796i \(0.649446\pi\)
\(200\) 0 0
\(201\) −3.51420 −0.247872
\(202\) 0 0
\(203\) −10.6921 −0.750439
\(204\) 0 0
\(205\) −6.83164 −0.477143
\(206\) 0 0
\(207\) 4.21024 0.292632
\(208\) 0 0
\(209\) 15.1755 1.04971
\(210\) 0 0
\(211\) 16.9680 1.16812 0.584061 0.811710i \(-0.301463\pi\)
0.584061 + 0.811710i \(0.301463\pi\)
\(212\) 0 0
\(213\) 11.6222 0.796340
\(214\) 0 0
\(215\) −22.5905 −1.54066
\(216\) 0 0
\(217\) −4.08168 −0.277083
\(218\) 0 0
\(219\) 5.43704 0.367401
\(220\) 0 0
\(221\) −10.8854 −0.732233
\(222\) 0 0
\(223\) −14.7467 −0.987514 −0.493757 0.869600i \(-0.664377\pi\)
−0.493757 + 0.869600i \(0.664377\pi\)
\(224\) 0 0
\(225\) 5.51393 0.367595
\(226\) 0 0
\(227\) 0.120587 0.00800367 0.00400183 0.999992i \(-0.498726\pi\)
0.00400183 + 0.999992i \(0.498726\pi\)
\(228\) 0 0
\(229\) 1.33643 0.0883139 0.0441570 0.999025i \(-0.485940\pi\)
0.0441570 + 0.999025i \(0.485940\pi\)
\(230\) 0 0
\(231\) 9.05698 0.595906
\(232\) 0 0
\(233\) −8.77423 −0.574819 −0.287409 0.957808i \(-0.592794\pi\)
−0.287409 + 0.957808i \(0.592794\pi\)
\(234\) 0 0
\(235\) 21.6823 1.41440
\(236\) 0 0
\(237\) 4.16325 0.270432
\(238\) 0 0
\(239\) −5.99261 −0.387630 −0.193815 0.981038i \(-0.562086\pi\)
−0.193815 + 0.981038i \(0.562086\pi\)
\(240\) 0 0
\(241\) 20.6690 1.33141 0.665704 0.746216i \(-0.268132\pi\)
0.665704 + 0.746216i \(0.268132\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −10.7185 −0.684780
\(246\) 0 0
\(247\) −31.4689 −2.00232
\(248\) 0 0
\(249\) 1.27361 0.0807121
\(250\) 0 0
\(251\) 9.72679 0.613950 0.306975 0.951718i \(-0.400683\pi\)
0.306975 + 0.951718i \(0.400683\pi\)
\(252\) 0 0
\(253\) 11.8783 0.746780
\(254\) 0 0
\(255\) 6.03313 0.377809
\(256\) 0 0
\(257\) −6.72200 −0.419307 −0.209654 0.977776i \(-0.567234\pi\)
−0.209654 + 0.977776i \(0.567234\pi\)
\(258\) 0 0
\(259\) 6.62458 0.411631
\(260\) 0 0
\(261\) 3.33063 0.206161
\(262\) 0 0
\(263\) −20.9835 −1.29390 −0.646948 0.762534i \(-0.723955\pi\)
−0.646948 + 0.762534i \(0.723955\pi\)
\(264\) 0 0
\(265\) −30.3119 −1.86205
\(266\) 0 0
\(267\) 5.91690 0.362109
\(268\) 0 0
\(269\) −2.08333 −0.127023 −0.0635115 0.997981i \(-0.520230\pi\)
−0.0635115 + 0.997981i \(0.520230\pi\)
\(270\) 0 0
\(271\) 10.9907 0.667639 0.333819 0.942637i \(-0.391662\pi\)
0.333819 + 0.942637i \(0.391662\pi\)
\(272\) 0 0
\(273\) −18.7811 −1.13669
\(274\) 0 0
\(275\) 15.5564 0.938084
\(276\) 0 0
\(277\) −25.5833 −1.53715 −0.768577 0.639758i \(-0.779035\pi\)
−0.768577 + 0.639758i \(0.779035\pi\)
\(278\) 0 0
\(279\) 1.27146 0.0761203
\(280\) 0 0
\(281\) −11.2733 −0.672507 −0.336253 0.941772i \(-0.609160\pi\)
−0.336253 + 0.941772i \(0.609160\pi\)
\(282\) 0 0
\(283\) 23.5911 1.40235 0.701174 0.712991i \(-0.252660\pi\)
0.701174 + 0.712991i \(0.252660\pi\)
\(284\) 0 0
\(285\) 17.4413 1.03313
\(286\) 0 0
\(287\) −6.76362 −0.399244
\(288\) 0 0
\(289\) −13.5381 −0.796356
\(290\) 0 0
\(291\) −4.01613 −0.235430
\(292\) 0 0
\(293\) −9.21613 −0.538412 −0.269206 0.963083i \(-0.586761\pi\)
−0.269206 + 0.963083i \(0.586761\pi\)
\(294\) 0 0
\(295\) 36.4388 2.12155
\(296\) 0 0
\(297\) −2.82128 −0.163707
\(298\) 0 0
\(299\) −24.6315 −1.42448
\(300\) 0 0
\(301\) −22.3656 −1.28913
\(302\) 0 0
\(303\) −9.09153 −0.522295
\(304\) 0 0
\(305\) −25.2506 −1.44585
\(306\) 0 0
\(307\) −13.3545 −0.762183 −0.381091 0.924537i \(-0.624452\pi\)
−0.381091 + 0.924537i \(0.624452\pi\)
\(308\) 0 0
\(309\) 15.5592 0.885130
\(310\) 0 0
\(311\) −18.9770 −1.07609 −0.538043 0.842918i \(-0.680836\pi\)
−0.538043 + 0.842918i \(0.680836\pi\)
\(312\) 0 0
\(313\) 0.865518 0.0489220 0.0244610 0.999701i \(-0.492213\pi\)
0.0244610 + 0.999701i \(0.492213\pi\)
\(314\) 0 0
\(315\) 10.4092 0.586495
\(316\) 0 0
\(317\) 6.57367 0.369214 0.184607 0.982812i \(-0.440899\pi\)
0.184607 + 0.982812i \(0.440899\pi\)
\(318\) 0 0
\(319\) 9.39666 0.526112
\(320\) 0 0
\(321\) −4.95284 −0.276440
\(322\) 0 0
\(323\) 10.0082 0.556872
\(324\) 0 0
\(325\) −32.2587 −1.78939
\(326\) 0 0
\(327\) −9.59455 −0.530580
\(328\) 0 0
\(329\) 21.4664 1.18348
\(330\) 0 0
\(331\) 5.69256 0.312892 0.156446 0.987687i \(-0.449996\pi\)
0.156446 + 0.987687i \(0.449996\pi\)
\(332\) 0 0
\(333\) −2.06358 −0.113084
\(334\) 0 0
\(335\) −11.3948 −0.622567
\(336\) 0 0
\(337\) −10.1210 −0.551323 −0.275662 0.961255i \(-0.588897\pi\)
−0.275662 + 0.961255i \(0.588897\pi\)
\(338\) 0 0
\(339\) 14.1378 0.767862
\(340\) 0 0
\(341\) 3.58715 0.194255
\(342\) 0 0
\(343\) 11.8599 0.640372
\(344\) 0 0
\(345\) 13.6518 0.734987
\(346\) 0 0
\(347\) −35.4498 −1.90304 −0.951521 0.307583i \(-0.900480\pi\)
−0.951521 + 0.307583i \(0.900480\pi\)
\(348\) 0 0
\(349\) 26.6545 1.42678 0.713392 0.700765i \(-0.247158\pi\)
0.713392 + 0.700765i \(0.247158\pi\)
\(350\) 0 0
\(351\) 5.85040 0.312271
\(352\) 0 0
\(353\) 7.94068 0.422640 0.211320 0.977417i \(-0.432224\pi\)
0.211320 + 0.977417i \(0.432224\pi\)
\(354\) 0 0
\(355\) 37.6852 2.00012
\(356\) 0 0
\(357\) 5.97306 0.316128
\(358\) 0 0
\(359\) 15.7685 0.832228 0.416114 0.909312i \(-0.363392\pi\)
0.416114 + 0.909312i \(0.363392\pi\)
\(360\) 0 0
\(361\) 9.93298 0.522789
\(362\) 0 0
\(363\) 3.04036 0.159578
\(364\) 0 0
\(365\) 17.6297 0.922781
\(366\) 0 0
\(367\) 2.32595 0.121414 0.0607068 0.998156i \(-0.480665\pi\)
0.0607068 + 0.998156i \(0.480665\pi\)
\(368\) 0 0
\(369\) 2.10689 0.109680
\(370\) 0 0
\(371\) −30.0101 −1.55805
\(372\) 0 0
\(373\) 4.77773 0.247382 0.123691 0.992321i \(-0.460527\pi\)
0.123691 + 0.992321i \(0.460527\pi\)
\(374\) 0 0
\(375\) 1.66643 0.0860539
\(376\) 0 0
\(377\) −19.4855 −1.00356
\(378\) 0 0
\(379\) −14.7452 −0.757410 −0.378705 0.925517i \(-0.623631\pi\)
−0.378705 + 0.925517i \(0.623631\pi\)
\(380\) 0 0
\(381\) −2.06152 −0.105615
\(382\) 0 0
\(383\) 7.62816 0.389781 0.194890 0.980825i \(-0.437565\pi\)
0.194890 + 0.980825i \(0.437565\pi\)
\(384\) 0 0
\(385\) 29.3674 1.49670
\(386\) 0 0
\(387\) 6.96695 0.354150
\(388\) 0 0
\(389\) 29.0716 1.47399 0.736993 0.675900i \(-0.236245\pi\)
0.736993 + 0.675900i \(0.236245\pi\)
\(390\) 0 0
\(391\) 7.83369 0.396167
\(392\) 0 0
\(393\) 14.0106 0.706740
\(394\) 0 0
\(395\) 13.4994 0.679229
\(396\) 0 0
\(397\) −11.2476 −0.564499 −0.282249 0.959341i \(-0.591081\pi\)
−0.282249 + 0.959341i \(0.591081\pi\)
\(398\) 0 0
\(399\) 17.2677 0.864464
\(400\) 0 0
\(401\) 8.80206 0.439554 0.219777 0.975550i \(-0.429467\pi\)
0.219777 + 0.975550i \(0.429467\pi\)
\(402\) 0 0
\(403\) −7.43854 −0.370540
\(404\) 0 0
\(405\) −3.24252 −0.161122
\(406\) 0 0
\(407\) −5.82195 −0.288583
\(408\) 0 0
\(409\) 35.7694 1.76868 0.884340 0.466843i \(-0.154609\pi\)
0.884340 + 0.466843i \(0.154609\pi\)
\(410\) 0 0
\(411\) −9.83399 −0.485075
\(412\) 0 0
\(413\) 36.0760 1.77518
\(414\) 0 0
\(415\) 4.12972 0.202720
\(416\) 0 0
\(417\) −4.61893 −0.226190
\(418\) 0 0
\(419\) −24.5130 −1.19754 −0.598770 0.800921i \(-0.704344\pi\)
−0.598770 + 0.800921i \(0.704344\pi\)
\(420\) 0 0
\(421\) 4.85223 0.236483 0.118242 0.992985i \(-0.462274\pi\)
0.118242 + 0.992985i \(0.462274\pi\)
\(422\) 0 0
\(423\) −6.68686 −0.325127
\(424\) 0 0
\(425\) 10.2594 0.497653
\(426\) 0 0
\(427\) −24.9992 −1.20980
\(428\) 0 0
\(429\) 16.5056 0.796899
\(430\) 0 0
\(431\) 16.5070 0.795116 0.397558 0.917577i \(-0.369858\pi\)
0.397558 + 0.917577i \(0.369858\pi\)
\(432\) 0 0
\(433\) −20.5559 −0.987851 −0.493926 0.869504i \(-0.664439\pi\)
−0.493926 + 0.869504i \(0.664439\pi\)
\(434\) 0 0
\(435\) 10.7996 0.517803
\(436\) 0 0
\(437\) 22.6466 1.08333
\(438\) 0 0
\(439\) 3.71337 0.177229 0.0886147 0.996066i \(-0.471756\pi\)
0.0886147 + 0.996066i \(0.471756\pi\)
\(440\) 0 0
\(441\) 3.30561 0.157410
\(442\) 0 0
\(443\) −4.80205 −0.228153 −0.114076 0.993472i \(-0.536391\pi\)
−0.114076 + 0.993472i \(0.536391\pi\)
\(444\) 0 0
\(445\) 19.1857 0.909488
\(446\) 0 0
\(447\) −11.6341 −0.550276
\(448\) 0 0
\(449\) 11.6862 0.551507 0.275753 0.961228i \(-0.411073\pi\)
0.275753 + 0.961228i \(0.411073\pi\)
\(450\) 0 0
\(451\) 5.94414 0.279899
\(452\) 0 0
\(453\) −13.1949 −0.619950
\(454\) 0 0
\(455\) −60.8982 −2.85495
\(456\) 0 0
\(457\) 6.66756 0.311895 0.155948 0.987765i \(-0.450157\pi\)
0.155948 + 0.987765i \(0.450157\pi\)
\(458\) 0 0
\(459\) −1.86063 −0.0868468
\(460\) 0 0
\(461\) −5.11682 −0.238314 −0.119157 0.992875i \(-0.538019\pi\)
−0.119157 + 0.992875i \(0.538019\pi\)
\(462\) 0 0
\(463\) −25.7320 −1.19587 −0.597934 0.801546i \(-0.704011\pi\)
−0.597934 + 0.801546i \(0.704011\pi\)
\(464\) 0 0
\(465\) 4.12273 0.191187
\(466\) 0 0
\(467\) −10.1888 −0.471482 −0.235741 0.971816i \(-0.575752\pi\)
−0.235741 + 0.971816i \(0.575752\pi\)
\(468\) 0 0
\(469\) −11.2814 −0.520927
\(470\) 0 0
\(471\) 5.22817 0.240901
\(472\) 0 0
\(473\) 19.6557 0.903772
\(474\) 0 0
\(475\) 29.6591 1.36085
\(476\) 0 0
\(477\) 9.34827 0.428028
\(478\) 0 0
\(479\) 22.1075 1.01012 0.505059 0.863085i \(-0.331471\pi\)
0.505059 + 0.863085i \(0.331471\pi\)
\(480\) 0 0
\(481\) 12.0728 0.550471
\(482\) 0 0
\(483\) 13.5158 0.614992
\(484\) 0 0
\(485\) −13.0224 −0.591316
\(486\) 0 0
\(487\) −31.9470 −1.44766 −0.723829 0.689980i \(-0.757619\pi\)
−0.723829 + 0.689980i \(0.757619\pi\)
\(488\) 0 0
\(489\) −0.261930 −0.0118449
\(490\) 0 0
\(491\) 34.9820 1.57871 0.789357 0.613934i \(-0.210414\pi\)
0.789357 + 0.613934i \(0.210414\pi\)
\(492\) 0 0
\(493\) 6.19708 0.279102
\(494\) 0 0
\(495\) −9.14806 −0.411175
\(496\) 0 0
\(497\) 37.3100 1.67358
\(498\) 0 0
\(499\) −26.2314 −1.17428 −0.587140 0.809486i \(-0.699746\pi\)
−0.587140 + 0.809486i \(0.699746\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) −24.3106 −1.08396 −0.541979 0.840392i \(-0.682325\pi\)
−0.541979 + 0.840392i \(0.682325\pi\)
\(504\) 0 0
\(505\) −29.4795 −1.31182
\(506\) 0 0
\(507\) −21.2271 −0.942730
\(508\) 0 0
\(509\) −32.5371 −1.44218 −0.721090 0.692841i \(-0.756359\pi\)
−0.721090 + 0.692841i \(0.756359\pi\)
\(510\) 0 0
\(511\) 17.4542 0.772127
\(512\) 0 0
\(513\) −5.37894 −0.237486
\(514\) 0 0
\(515\) 50.4509 2.22313
\(516\) 0 0
\(517\) −18.8655 −0.829705
\(518\) 0 0
\(519\) 19.5287 0.857215
\(520\) 0 0
\(521\) 32.9927 1.44543 0.722717 0.691144i \(-0.242893\pi\)
0.722717 + 0.691144i \(0.242893\pi\)
\(522\) 0 0
\(523\) −19.4021 −0.848393 −0.424197 0.905570i \(-0.639443\pi\)
−0.424197 + 0.905570i \(0.639443\pi\)
\(524\) 0 0
\(525\) 17.7010 0.772536
\(526\) 0 0
\(527\) 2.36572 0.103052
\(528\) 0 0
\(529\) −5.27392 −0.229301
\(530\) 0 0
\(531\) −11.2378 −0.487679
\(532\) 0 0
\(533\) −12.3262 −0.533905
\(534\) 0 0
\(535\) −16.0597 −0.694320
\(536\) 0 0
\(537\) 19.0968 0.824090
\(538\) 0 0
\(539\) 9.32606 0.401702
\(540\) 0 0
\(541\) 20.4863 0.880773 0.440386 0.897808i \(-0.354841\pi\)
0.440386 + 0.897808i \(0.354841\pi\)
\(542\) 0 0
\(543\) 8.26626 0.354739
\(544\) 0 0
\(545\) −31.1105 −1.33263
\(546\) 0 0
\(547\) −2.16456 −0.0925498 −0.0462749 0.998929i \(-0.514735\pi\)
−0.0462749 + 0.998929i \(0.514735\pi\)
\(548\) 0 0
\(549\) 7.78735 0.332356
\(550\) 0 0
\(551\) 17.9153 0.763216
\(552\) 0 0
\(553\) 13.3650 0.568338
\(554\) 0 0
\(555\) −6.69120 −0.284026
\(556\) 0 0
\(557\) −1.20589 −0.0510952 −0.0255476 0.999674i \(-0.508133\pi\)
−0.0255476 + 0.999674i \(0.508133\pi\)
\(558\) 0 0
\(559\) −40.7594 −1.72394
\(560\) 0 0
\(561\) −5.24936 −0.221628
\(562\) 0 0
\(563\) −1.05092 −0.0442909 −0.0221455 0.999755i \(-0.507050\pi\)
−0.0221455 + 0.999755i \(0.507050\pi\)
\(564\) 0 0
\(565\) 45.8422 1.92860
\(566\) 0 0
\(567\) −3.21024 −0.134817
\(568\) 0 0
\(569\) 4.65108 0.194983 0.0974917 0.995236i \(-0.468918\pi\)
0.0974917 + 0.995236i \(0.468918\pi\)
\(570\) 0 0
\(571\) −23.8655 −0.998739 −0.499370 0.866389i \(-0.666435\pi\)
−0.499370 + 0.866389i \(0.666435\pi\)
\(572\) 0 0
\(573\) 21.0760 0.880463
\(574\) 0 0
\(575\) 23.2149 0.968130
\(576\) 0 0
\(577\) −44.4676 −1.85121 −0.925605 0.378490i \(-0.876443\pi\)
−0.925605 + 0.378490i \(0.876443\pi\)
\(578\) 0 0
\(579\) −22.3043 −0.926934
\(580\) 0 0
\(581\) 4.08860 0.169624
\(582\) 0 0
\(583\) 26.3741 1.09230
\(584\) 0 0
\(585\) 18.9700 0.784314
\(586\) 0 0
\(587\) −48.2366 −1.99094 −0.995469 0.0950882i \(-0.969687\pi\)
−0.995469 + 0.0950882i \(0.969687\pi\)
\(588\) 0 0
\(589\) 6.83910 0.281800
\(590\) 0 0
\(591\) 10.7503 0.442207
\(592\) 0 0
\(593\) 4.81080 0.197556 0.0987779 0.995110i \(-0.468507\pi\)
0.0987779 + 0.995110i \(0.468507\pi\)
\(594\) 0 0
\(595\) 19.3678 0.794001
\(596\) 0 0
\(597\) 12.7649 0.522431
\(598\) 0 0
\(599\) −23.7293 −0.969552 −0.484776 0.874638i \(-0.661099\pi\)
−0.484776 + 0.874638i \(0.661099\pi\)
\(600\) 0 0
\(601\) −15.1161 −0.616597 −0.308299 0.951290i \(-0.599760\pi\)
−0.308299 + 0.951290i \(0.599760\pi\)
\(602\) 0 0
\(603\) 3.51420 0.143109
\(604\) 0 0
\(605\) 9.85843 0.400802
\(606\) 0 0
\(607\) −37.6031 −1.52626 −0.763132 0.646242i \(-0.776340\pi\)
−0.763132 + 0.646242i \(0.776340\pi\)
\(608\) 0 0
\(609\) 10.6921 0.433266
\(610\) 0 0
\(611\) 39.1208 1.58266
\(612\) 0 0
\(613\) −13.0653 −0.527701 −0.263850 0.964564i \(-0.584992\pi\)
−0.263850 + 0.964564i \(0.584992\pi\)
\(614\) 0 0
\(615\) 6.83164 0.275478
\(616\) 0 0
\(617\) −30.4263 −1.22492 −0.612459 0.790502i \(-0.709820\pi\)
−0.612459 + 0.790502i \(0.709820\pi\)
\(618\) 0 0
\(619\) 22.9971 0.924332 0.462166 0.886793i \(-0.347072\pi\)
0.462166 + 0.886793i \(0.347072\pi\)
\(620\) 0 0
\(621\) −4.21024 −0.168951
\(622\) 0 0
\(623\) 18.9947 0.761005
\(624\) 0 0
\(625\) −22.1662 −0.886649
\(626\) 0 0
\(627\) −15.1755 −0.606051
\(628\) 0 0
\(629\) −3.83956 −0.153093
\(630\) 0 0
\(631\) −45.3341 −1.80472 −0.902360 0.430982i \(-0.858167\pi\)
−0.902360 + 0.430982i \(0.858167\pi\)
\(632\) 0 0
\(633\) −16.9680 −0.674416
\(634\) 0 0
\(635\) −6.68452 −0.265267
\(636\) 0 0
\(637\) −19.3391 −0.766244
\(638\) 0 0
\(639\) −11.6222 −0.459767
\(640\) 0 0
\(641\) −4.60629 −0.181937 −0.0909687 0.995854i \(-0.528996\pi\)
−0.0909687 + 0.995854i \(0.528996\pi\)
\(642\) 0 0
\(643\) 10.7066 0.422227 0.211114 0.977462i \(-0.432291\pi\)
0.211114 + 0.977462i \(0.432291\pi\)
\(644\) 0 0
\(645\) 22.5905 0.889499
\(646\) 0 0
\(647\) −25.1350 −0.988158 −0.494079 0.869417i \(-0.664495\pi\)
−0.494079 + 0.869417i \(0.664495\pi\)
\(648\) 0 0
\(649\) −31.7050 −1.24453
\(650\) 0 0
\(651\) 4.08168 0.159974
\(652\) 0 0
\(653\) −29.5769 −1.15743 −0.578717 0.815528i \(-0.696447\pi\)
−0.578717 + 0.815528i \(0.696447\pi\)
\(654\) 0 0
\(655\) 45.4296 1.77508
\(656\) 0 0
\(657\) −5.43704 −0.212119
\(658\) 0 0
\(659\) −26.7918 −1.04366 −0.521831 0.853049i \(-0.674751\pi\)
−0.521831 + 0.853049i \(0.674751\pi\)
\(660\) 0 0
\(661\) 5.89353 0.229232 0.114616 0.993410i \(-0.463436\pi\)
0.114616 + 0.993410i \(0.463436\pi\)
\(662\) 0 0
\(663\) 10.8854 0.422755
\(664\) 0 0
\(665\) 55.9907 2.17123
\(666\) 0 0
\(667\) 14.0227 0.542963
\(668\) 0 0
\(669\) 14.7467 0.570141
\(670\) 0 0
\(671\) 21.9703 0.848155
\(672\) 0 0
\(673\) −3.26057 −0.125686 −0.0628428 0.998023i \(-0.520017\pi\)
−0.0628428 + 0.998023i \(0.520017\pi\)
\(674\) 0 0
\(675\) −5.51393 −0.212231
\(676\) 0 0
\(677\) 4.15513 0.159695 0.0798474 0.996807i \(-0.474557\pi\)
0.0798474 + 0.996807i \(0.474557\pi\)
\(678\) 0 0
\(679\) −12.8927 −0.494777
\(680\) 0 0
\(681\) −0.120587 −0.00462092
\(682\) 0 0
\(683\) 51.4838 1.96997 0.984987 0.172629i \(-0.0552261\pi\)
0.984987 + 0.172629i \(0.0552261\pi\)
\(684\) 0 0
\(685\) −31.8869 −1.21834
\(686\) 0 0
\(687\) −1.33643 −0.0509881
\(688\) 0 0
\(689\) −54.6911 −2.08356
\(690\) 0 0
\(691\) 36.5334 1.38980 0.694899 0.719108i \(-0.255449\pi\)
0.694899 + 0.719108i \(0.255449\pi\)
\(692\) 0 0
\(693\) −9.05698 −0.344046
\(694\) 0 0
\(695\) −14.9770 −0.568109
\(696\) 0 0
\(697\) 3.92015 0.148486
\(698\) 0 0
\(699\) 8.77423 0.331872
\(700\) 0 0
\(701\) −20.6445 −0.779731 −0.389866 0.920872i \(-0.627479\pi\)
−0.389866 + 0.920872i \(0.627479\pi\)
\(702\) 0 0
\(703\) −11.0999 −0.418640
\(704\) 0 0
\(705\) −21.6823 −0.816602
\(706\) 0 0
\(707\) −29.1860 −1.09765
\(708\) 0 0
\(709\) 33.5779 1.26105 0.630523 0.776171i \(-0.282840\pi\)
0.630523 + 0.776171i \(0.282840\pi\)
\(710\) 0 0
\(711\) −4.16325 −0.156134
\(712\) 0 0
\(713\) 5.35314 0.200477
\(714\) 0 0
\(715\) 53.5198 2.00153
\(716\) 0 0
\(717\) 5.99261 0.223798
\(718\) 0 0
\(719\) 22.2353 0.829236 0.414618 0.909996i \(-0.363915\pi\)
0.414618 + 0.909996i \(0.363915\pi\)
\(720\) 0 0
\(721\) 49.9486 1.86018
\(722\) 0 0
\(723\) −20.6690 −0.768688
\(724\) 0 0
\(725\) 18.3649 0.682054
\(726\) 0 0
\(727\) 6.90158 0.255966 0.127983 0.991776i \(-0.459150\pi\)
0.127983 + 0.991776i \(0.459150\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 12.9629 0.479451
\(732\) 0 0
\(733\) 47.0601 1.73821 0.869103 0.494632i \(-0.164697\pi\)
0.869103 + 0.494632i \(0.164697\pi\)
\(734\) 0 0
\(735\) 10.7185 0.395358
\(736\) 0 0
\(737\) 9.91454 0.365207
\(738\) 0 0
\(739\) 3.09124 0.113713 0.0568565 0.998382i \(-0.481892\pi\)
0.0568565 + 0.998382i \(0.481892\pi\)
\(740\) 0 0
\(741\) 31.4689 1.15604
\(742\) 0 0
\(743\) −26.7232 −0.980380 −0.490190 0.871616i \(-0.663073\pi\)
−0.490190 + 0.871616i \(0.663073\pi\)
\(744\) 0 0
\(745\) −37.7239 −1.38210
\(746\) 0 0
\(747\) −1.27361 −0.0465991
\(748\) 0 0
\(749\) −15.8998 −0.580965
\(750\) 0 0
\(751\) 13.6355 0.497566 0.248783 0.968559i \(-0.419969\pi\)
0.248783 + 0.968559i \(0.419969\pi\)
\(752\) 0 0
\(753\) −9.72679 −0.354464
\(754\) 0 0
\(755\) −42.7847 −1.55709
\(756\) 0 0
\(757\) 45.5231 1.65456 0.827282 0.561787i \(-0.189886\pi\)
0.827282 + 0.561787i \(0.189886\pi\)
\(758\) 0 0
\(759\) −11.8783 −0.431154
\(760\) 0 0
\(761\) −54.7160 −1.98345 −0.991727 0.128367i \(-0.959026\pi\)
−0.991727 + 0.128367i \(0.959026\pi\)
\(762\) 0 0
\(763\) −30.8007 −1.11506
\(764\) 0 0
\(765\) −6.03313 −0.218128
\(766\) 0 0
\(767\) 65.7456 2.37393
\(768\) 0 0
\(769\) 10.8454 0.391097 0.195548 0.980694i \(-0.437351\pi\)
0.195548 + 0.980694i \(0.437351\pi\)
\(770\) 0 0
\(771\) 6.72200 0.242087
\(772\) 0 0
\(773\) −8.57334 −0.308362 −0.154181 0.988043i \(-0.549274\pi\)
−0.154181 + 0.988043i \(0.549274\pi\)
\(774\) 0 0
\(775\) 7.01074 0.251833
\(776\) 0 0
\(777\) −6.62458 −0.237655
\(778\) 0 0
\(779\) 11.3329 0.406042
\(780\) 0 0
\(781\) −32.7895 −1.17330
\(782\) 0 0
\(783\) −3.33063 −0.119027
\(784\) 0 0
\(785\) 16.9524 0.605059
\(786\) 0 0
\(787\) 54.0373 1.92622 0.963111 0.269104i \(-0.0867277\pi\)
0.963111 + 0.269104i \(0.0867277\pi\)
\(788\) 0 0
\(789\) 20.9835 0.747031
\(790\) 0 0
\(791\) 45.3858 1.61373
\(792\) 0 0
\(793\) −45.5591 −1.61785
\(794\) 0 0
\(795\) 30.3119 1.07505
\(796\) 0 0
\(797\) 22.5826 0.799917 0.399958 0.916533i \(-0.369025\pi\)
0.399958 + 0.916533i \(0.369025\pi\)
\(798\) 0 0
\(799\) −12.4418 −0.440159
\(800\) 0 0
\(801\) −5.91690 −0.209063
\(802\) 0 0
\(803\) −15.3394 −0.541316
\(804\) 0 0
\(805\) 43.8254 1.54464
\(806\) 0 0
\(807\) 2.08333 0.0733368
\(808\) 0 0
\(809\) −40.2340 −1.41455 −0.707277 0.706937i \(-0.750076\pi\)
−0.707277 + 0.706937i \(0.750076\pi\)
\(810\) 0 0
\(811\) −49.9998 −1.75573 −0.877865 0.478909i \(-0.841032\pi\)
−0.877865 + 0.478909i \(0.841032\pi\)
\(812\) 0 0
\(813\) −10.9907 −0.385462
\(814\) 0 0
\(815\) −0.849314 −0.0297502
\(816\) 0 0
\(817\) 37.4748 1.31108
\(818\) 0 0
\(819\) 18.7811 0.656266
\(820\) 0 0
\(821\) 39.4031 1.37518 0.687589 0.726100i \(-0.258669\pi\)
0.687589 + 0.726100i \(0.258669\pi\)
\(822\) 0 0
\(823\) −43.3232 −1.51015 −0.755077 0.655636i \(-0.772400\pi\)
−0.755077 + 0.655636i \(0.772400\pi\)
\(824\) 0 0
\(825\) −15.5564 −0.541603
\(826\) 0 0
\(827\) 42.1930 1.46719 0.733597 0.679585i \(-0.237840\pi\)
0.733597 + 0.679585i \(0.237840\pi\)
\(828\) 0 0
\(829\) 27.1711 0.943693 0.471846 0.881681i \(-0.343588\pi\)
0.471846 + 0.881681i \(0.343588\pi\)
\(830\) 0 0
\(831\) 25.5833 0.887476
\(832\) 0 0
\(833\) 6.15052 0.213103
\(834\) 0 0
\(835\) −3.24252 −0.112212
\(836\) 0 0
\(837\) −1.27146 −0.0439481
\(838\) 0 0
\(839\) −35.5522 −1.22740 −0.613698 0.789541i \(-0.710319\pi\)
−0.613698 + 0.789541i \(0.710319\pi\)
\(840\) 0 0
\(841\) −17.9069 −0.617479
\(842\) 0 0
\(843\) 11.2733 0.388272
\(844\) 0 0
\(845\) −68.8294 −2.36780
\(846\) 0 0
\(847\) 9.76028 0.335367
\(848\) 0 0
\(849\) −23.5911 −0.809646
\(850\) 0 0
\(851\) −8.68816 −0.297826
\(852\) 0 0
\(853\) 56.9119 1.94863 0.974313 0.225200i \(-0.0723034\pi\)
0.974313 + 0.225200i \(0.0723034\pi\)
\(854\) 0 0
\(855\) −17.4413 −0.596480
\(856\) 0 0
\(857\) −34.1803 −1.16758 −0.583788 0.811906i \(-0.698430\pi\)
−0.583788 + 0.811906i \(0.698430\pi\)
\(858\) 0 0
\(859\) −16.9625 −0.578754 −0.289377 0.957215i \(-0.593448\pi\)
−0.289377 + 0.957215i \(0.593448\pi\)
\(860\) 0 0
\(861\) 6.76362 0.230504
\(862\) 0 0
\(863\) 44.6654 1.52043 0.760214 0.649672i \(-0.225094\pi\)
0.760214 + 0.649672i \(0.225094\pi\)
\(864\) 0 0
\(865\) 63.3222 2.15302
\(866\) 0 0
\(867\) 13.5381 0.459776
\(868\) 0 0
\(869\) −11.7457 −0.398446
\(870\) 0 0
\(871\) −20.5594 −0.696630
\(872\) 0 0
\(873\) 4.01613 0.135925
\(874\) 0 0
\(875\) 5.34962 0.180850
\(876\) 0 0
\(877\) 1.65783 0.0559808 0.0279904 0.999608i \(-0.491089\pi\)
0.0279904 + 0.999608i \(0.491089\pi\)
\(878\) 0 0
\(879\) 9.21613 0.310852
\(880\) 0 0
\(881\) −11.0757 −0.373150 −0.186575 0.982441i \(-0.559739\pi\)
−0.186575 + 0.982441i \(0.559739\pi\)
\(882\) 0 0
\(883\) 6.29928 0.211988 0.105994 0.994367i \(-0.466198\pi\)
0.105994 + 0.994367i \(0.466198\pi\)
\(884\) 0 0
\(885\) −36.4388 −1.22488
\(886\) 0 0
\(887\) −13.7138 −0.460463 −0.230231 0.973136i \(-0.573948\pi\)
−0.230231 + 0.973136i \(0.573948\pi\)
\(888\) 0 0
\(889\) −6.61796 −0.221959
\(890\) 0 0
\(891\) 2.82128 0.0945165
\(892\) 0 0
\(893\) −35.9682 −1.20363
\(894\) 0 0
\(895\) 61.9219 2.06982
\(896\) 0 0
\(897\) 24.6315 0.822423
\(898\) 0 0
\(899\) 4.23476 0.141237
\(900\) 0 0
\(901\) 17.3937 0.579467
\(902\) 0 0
\(903\) 22.3656 0.744279
\(904\) 0 0
\(905\) 26.8035 0.890979
\(906\) 0 0
\(907\) −35.5148 −1.17925 −0.589625 0.807677i \(-0.700725\pi\)
−0.589625 + 0.807677i \(0.700725\pi\)
\(908\) 0 0
\(909\) 9.09153 0.301547
\(910\) 0 0
\(911\) 38.4589 1.27420 0.637100 0.770781i \(-0.280134\pi\)
0.637100 + 0.770781i \(0.280134\pi\)
\(912\) 0 0
\(913\) −3.59323 −0.118918
\(914\) 0 0
\(915\) 25.2506 0.834760
\(916\) 0 0
\(917\) 44.9773 1.48528
\(918\) 0 0
\(919\) 40.7347 1.34371 0.671856 0.740682i \(-0.265497\pi\)
0.671856 + 0.740682i \(0.265497\pi\)
\(920\) 0 0
\(921\) 13.3545 0.440046
\(922\) 0 0
\(923\) 67.9945 2.23807
\(924\) 0 0
\(925\) −11.3784 −0.374121
\(926\) 0 0
\(927\) −15.5592 −0.511030
\(928\) 0 0
\(929\) 52.9417 1.73696 0.868481 0.495723i \(-0.165097\pi\)
0.868481 + 0.495723i \(0.165097\pi\)
\(930\) 0 0
\(931\) 17.7807 0.582738
\(932\) 0 0
\(933\) 18.9770 0.621278
\(934\) 0 0
\(935\) −17.0212 −0.556652
\(936\) 0 0
\(937\) 22.6766 0.740811 0.370406 0.928870i \(-0.379219\pi\)
0.370406 + 0.928870i \(0.379219\pi\)
\(938\) 0 0
\(939\) −0.865518 −0.0282451
\(940\) 0 0
\(941\) −49.9912 −1.62967 −0.814834 0.579694i \(-0.803172\pi\)
−0.814834 + 0.579694i \(0.803172\pi\)
\(942\) 0 0
\(943\) 8.87052 0.288864
\(944\) 0 0
\(945\) −10.4092 −0.338613
\(946\) 0 0
\(947\) −21.4522 −0.697103 −0.348552 0.937290i \(-0.613326\pi\)
−0.348552 + 0.937290i \(0.613326\pi\)
\(948\) 0 0
\(949\) 31.8088 1.03256
\(950\) 0 0
\(951\) −6.57367 −0.213166
\(952\) 0 0
\(953\) −35.3902 −1.14640 −0.573201 0.819415i \(-0.694298\pi\)
−0.573201 + 0.819415i \(0.694298\pi\)
\(954\) 0 0
\(955\) 68.3394 2.21141
\(956\) 0 0
\(957\) −9.39666 −0.303751
\(958\) 0 0
\(959\) −31.5694 −1.01943
\(960\) 0 0
\(961\) −29.3834 −0.947851
\(962\) 0 0
\(963\) 4.95284 0.159603
\(964\) 0 0
\(965\) −72.3221 −2.32813
\(966\) 0 0
\(967\) 41.2924 1.32787 0.663937 0.747789i \(-0.268884\pi\)
0.663937 + 0.747789i \(0.268884\pi\)
\(968\) 0 0
\(969\) −10.0082 −0.321510
\(970\) 0 0
\(971\) 16.6855 0.535464 0.267732 0.963493i \(-0.413726\pi\)
0.267732 + 0.963493i \(0.413726\pi\)
\(972\) 0 0
\(973\) −14.8279 −0.475359
\(974\) 0 0
\(975\) 32.2587 1.03310
\(976\) 0 0
\(977\) 17.3143 0.553932 0.276966 0.960880i \(-0.410671\pi\)
0.276966 + 0.960880i \(0.410671\pi\)
\(978\) 0 0
\(979\) −16.6933 −0.533519
\(980\) 0 0
\(981\) 9.59455 0.306330
\(982\) 0 0
\(983\) 32.8464 1.04764 0.523819 0.851830i \(-0.324507\pi\)
0.523819 + 0.851830i \(0.324507\pi\)
\(984\) 0 0
\(985\) 34.8579 1.11067
\(986\) 0 0
\(987\) −21.4664 −0.683283
\(988\) 0 0
\(989\) 29.3325 0.932719
\(990\) 0 0
\(991\) 25.3128 0.804087 0.402044 0.915621i \(-0.368300\pi\)
0.402044 + 0.915621i \(0.368300\pi\)
\(992\) 0 0
\(993\) −5.69256 −0.180648
\(994\) 0 0
\(995\) 41.3903 1.31216
\(996\) 0 0
\(997\) −56.1380 −1.77791 −0.888954 0.457996i \(-0.848567\pi\)
−0.888954 + 0.457996i \(0.848567\pi\)
\(998\) 0 0
\(999\) 2.06358 0.0652888
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 4008.2.a.h.1.1 8
4.3 odd 2 8016.2.a.y.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.h.1.1 8 1.1 even 1 trivial
8016.2.a.y.1.1 8 4.3 odd 2