Properties

Label 6021.2.a.k.1.4
Level $6021$
Weight $2$
Character 6021.1
Self dual yes
Analytic conductor $48.078$
Analytic rank $1$
Dimension $4$
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] = [6021,2,Mod(1,6021)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6021, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6021.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6021 = 3^{3} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6021.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: \(48.0779270570\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.874032\) of defining polynomial
Character \(\chi\) \(=\) 6021.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} +2.28825 q^{5} +1.00000 q^{7} -2.82843 q^{8} +O(q^{10})\) \(q+1.41421 q^{2} +2.28825 q^{5} +1.00000 q^{7} -2.82843 q^{8} +3.23607 q^{10} +4.57649 q^{11} +1.00000 q^{13} +1.41421 q^{14} -4.00000 q^{16} -5.99070 q^{17} -6.70820 q^{19} +6.47214 q^{22} -8.27895 q^{23} +0.236068 q^{25} +1.41421 q^{26} -6.86474 q^{29} -7.23607 q^{31} -8.47214 q^{34} +2.28825 q^{35} -9.00000 q^{37} -9.48683 q^{38} -6.47214 q^{40} +1.95440 q^{41} +8.47214 q^{43} -11.7082 q^{46} +13.0618 q^{47} -6.00000 q^{49} +0.333851 q^{50} -7.61125 q^{53} +10.4721 q^{55} -2.82843 q^{56} -9.70820 q^{58} -13.7295 q^{59} +2.70820 q^{61} -10.2333 q^{62} +8.00000 q^{64} +2.28825 q^{65} -9.00000 q^{67} +3.23607 q^{70} -12.3153 q^{71} +13.1803 q^{73} -12.7279 q^{74} +4.57649 q^{77} +5.76393 q^{79} -9.15298 q^{80} +2.76393 q^{82} +11.4412 q^{83} -13.7082 q^{85} +11.9814 q^{86} -12.9443 q^{88} -4.57649 q^{89} +1.00000 q^{91} +18.4721 q^{94} -15.3500 q^{95} +2.70820 q^{97} -8.48528 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{7} + 4 q^{10} + 4 q^{13} - 16 q^{16} + 8 q^{22} - 8 q^{25} - 20 q^{31} - 16 q^{34} - 36 q^{37} - 8 q^{40} + 16 q^{43} - 20 q^{46} - 24 q^{49} + 24 q^{55} - 12 q^{58} - 16 q^{61} + 32 q^{64} - 36 q^{67} + 4 q^{70} + 8 q^{73} + 32 q^{79} + 20 q^{82} - 28 q^{85} - 16 q^{88} + 4 q^{91} + 56 q^{94} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(3\) 0 0
\(4\) 0 0
\(5\) 2.28825 1.02333 0.511667 0.859184i \(-0.329028\pi\)
0.511667 + 0.859184i \(0.329028\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −2.82843 −1.00000
\(9\) 0 0
\(10\) 3.23607 1.02333
\(11\) 4.57649 1.37986 0.689932 0.723874i \(-0.257640\pi\)
0.689932 + 0.723874i \(0.257640\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 1.41421 0.377964
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −5.99070 −1.45296 −0.726480 0.687188i \(-0.758845\pi\)
−0.726480 + 0.687188i \(0.758845\pi\)
\(18\) 0 0
\(19\) −6.70820 −1.53897 −0.769484 0.638666i \(-0.779486\pi\)
−0.769484 + 0.638666i \(0.779486\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.47214 1.37986
\(23\) −8.27895 −1.72628 −0.863140 0.504964i \(-0.831506\pi\)
−0.863140 + 0.504964i \(0.831506\pi\)
\(24\) 0 0
\(25\) 0.236068 0.0472136
\(26\) 1.41421 0.277350
\(27\) 0 0
\(28\) 0 0
\(29\) −6.86474 −1.27475 −0.637375 0.770554i \(-0.719980\pi\)
−0.637375 + 0.770554i \(0.719980\pi\)
\(30\) 0 0
\(31\) −7.23607 −1.29964 −0.649818 0.760090i \(-0.725155\pi\)
−0.649818 + 0.760090i \(0.725155\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) −8.47214 −1.45296
\(35\) 2.28825 0.386784
\(36\) 0 0
\(37\) −9.00000 −1.47959 −0.739795 0.672832i \(-0.765078\pi\)
−0.739795 + 0.672832i \(0.765078\pi\)
\(38\) −9.48683 −1.53897
\(39\) 0 0
\(40\) −6.47214 −1.02333
\(41\) 1.95440 0.305225 0.152613 0.988286i \(-0.451231\pi\)
0.152613 + 0.988286i \(0.451231\pi\)
\(42\) 0 0
\(43\) 8.47214 1.29199 0.645994 0.763342i \(-0.276443\pi\)
0.645994 + 0.763342i \(0.276443\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −11.7082 −1.72628
\(47\) 13.0618 1.90526 0.952628 0.304139i \(-0.0983687\pi\)
0.952628 + 0.304139i \(0.0983687\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0.333851 0.0472136
\(51\) 0 0
\(52\) 0 0
\(53\) −7.61125 −1.04549 −0.522743 0.852490i \(-0.675091\pi\)
−0.522743 + 0.852490i \(0.675091\pi\)
\(54\) 0 0
\(55\) 10.4721 1.41206
\(56\) −2.82843 −0.377964
\(57\) 0 0
\(58\) −9.70820 −1.27475
\(59\) −13.7295 −1.78743 −0.893713 0.448640i \(-0.851909\pi\)
−0.893713 + 0.448640i \(0.851909\pi\)
\(60\) 0 0
\(61\) 2.70820 0.346750 0.173375 0.984856i \(-0.444533\pi\)
0.173375 + 0.984856i \(0.444533\pi\)
\(62\) −10.2333 −1.29964
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 2.28825 0.283822
\(66\) 0 0
\(67\) −9.00000 −1.09952 −0.549762 0.835321i \(-0.685282\pi\)
−0.549762 + 0.835321i \(0.685282\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 3.23607 0.386784
\(71\) −12.3153 −1.46155 −0.730776 0.682617i \(-0.760842\pi\)
−0.730776 + 0.682617i \(0.760842\pi\)
\(72\) 0 0
\(73\) 13.1803 1.54264 0.771321 0.636446i \(-0.219596\pi\)
0.771321 + 0.636446i \(0.219596\pi\)
\(74\) −12.7279 −1.47959
\(75\) 0 0
\(76\) 0 0
\(77\) 4.57649 0.521540
\(78\) 0 0
\(79\) 5.76393 0.648493 0.324247 0.945973i \(-0.394889\pi\)
0.324247 + 0.945973i \(0.394889\pi\)
\(80\) −9.15298 −1.02333
\(81\) 0 0
\(82\) 2.76393 0.305225
\(83\) 11.4412 1.25584 0.627919 0.778279i \(-0.283907\pi\)
0.627919 + 0.778279i \(0.283907\pi\)
\(84\) 0 0
\(85\) −13.7082 −1.48686
\(86\) 11.9814 1.29199
\(87\) 0 0
\(88\) −12.9443 −1.37986
\(89\) −4.57649 −0.485107 −0.242554 0.970138i \(-0.577985\pi\)
−0.242554 + 0.970138i \(0.577985\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 18.4721 1.90526
\(95\) −15.3500 −1.57488
\(96\) 0 0
\(97\) 2.70820 0.274976 0.137488 0.990503i \(-0.456097\pi\)
0.137488 + 0.990503i \(0.456097\pi\)
\(98\) −8.48528 −0.857143
\(99\) 0 0
\(100\) 0 0
\(101\) 13.5231 1.34560 0.672801 0.739823i \(-0.265091\pi\)
0.672801 + 0.739823i \(0.265091\pi\)
\(102\) 0 0
\(103\) 5.47214 0.539186 0.269593 0.962974i \(-0.413111\pi\)
0.269593 + 0.962974i \(0.413111\pi\)
\(104\) −2.82843 −0.277350
\(105\) 0 0
\(106\) −10.7639 −1.04549
\(107\) 2.28825 0.221213 0.110607 0.993864i \(-0.464721\pi\)
0.110607 + 0.993864i \(0.464721\pi\)
\(108\) 0 0
\(109\) 13.4164 1.28506 0.642529 0.766261i \(-0.277885\pi\)
0.642529 + 0.766261i \(0.277885\pi\)
\(110\) 14.8098 1.41206
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) −5.65685 −0.532152 −0.266076 0.963952i \(-0.585727\pi\)
−0.266076 + 0.963952i \(0.585727\pi\)
\(114\) 0 0
\(115\) −18.9443 −1.76656
\(116\) 0 0
\(117\) 0 0
\(118\) −19.4164 −1.78743
\(119\) −5.99070 −0.549167
\(120\) 0 0
\(121\) 9.94427 0.904025
\(122\) 3.82998 0.346750
\(123\) 0 0
\(124\) 0 0
\(125\) −10.9010 −0.975019
\(126\) 0 0
\(127\) −4.76393 −0.422731 −0.211365 0.977407i \(-0.567791\pi\)
−0.211365 + 0.977407i \(0.567791\pi\)
\(128\) 11.3137 1.00000
\(129\) 0 0
\(130\) 3.23607 0.283822
\(131\) 15.6839 1.37031 0.685153 0.728399i \(-0.259735\pi\)
0.685153 + 0.728399i \(0.259735\pi\)
\(132\) 0 0
\(133\) −6.70820 −0.581675
\(134\) −12.7279 −1.09952
\(135\) 0 0
\(136\) 16.9443 1.45296
\(137\) 5.99070 0.511820 0.255910 0.966701i \(-0.417625\pi\)
0.255910 + 0.966701i \(0.417625\pi\)
\(138\) 0 0
\(139\) 3.00000 0.254457 0.127228 0.991873i \(-0.459392\pi\)
0.127228 + 0.991873i \(0.459392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −17.4164 −1.46155
\(143\) 4.57649 0.382705
\(144\) 0 0
\(145\) −15.7082 −1.30450
\(146\) 18.6398 1.54264
\(147\) 0 0
\(148\) 0 0
\(149\) −0.874032 −0.0716035 −0.0358017 0.999359i \(-0.511398\pi\)
−0.0358017 + 0.999359i \(0.511398\pi\)
\(150\) 0 0
\(151\) 1.94427 0.158223 0.0791113 0.996866i \(-0.474792\pi\)
0.0791113 + 0.996866i \(0.474792\pi\)
\(152\) 18.9737 1.53897
\(153\) 0 0
\(154\) 6.47214 0.521540
\(155\) −16.5579 −1.32996
\(156\) 0 0
\(157\) −13.4164 −1.07075 −0.535373 0.844616i \(-0.679829\pi\)
−0.535373 + 0.844616i \(0.679829\pi\)
\(158\) 8.15143 0.648493
\(159\) 0 0
\(160\) 0 0
\(161\) −8.27895 −0.652473
\(162\) 0 0
\(163\) −0.708204 −0.0554708 −0.0277354 0.999615i \(-0.508830\pi\)
−0.0277354 + 0.999615i \(0.508830\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 16.1803 1.25584
\(167\) 6.86474 0.531209 0.265605 0.964082i \(-0.414428\pi\)
0.265605 + 0.964082i \(0.414428\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) −19.3863 −1.48686
\(171\) 0 0
\(172\) 0 0
\(173\) 15.5563 1.18273 0.591364 0.806405i \(-0.298590\pi\)
0.591364 + 0.806405i \(0.298590\pi\)
\(174\) 0 0
\(175\) 0.236068 0.0178451
\(176\) −18.3060 −1.37986
\(177\) 0 0
\(178\) −6.47214 −0.485107
\(179\) −0.952843 −0.0712189 −0.0356094 0.999366i \(-0.511337\pi\)
−0.0356094 + 0.999366i \(0.511337\pi\)
\(180\) 0 0
\(181\) 11.6525 0.866122 0.433061 0.901365i \(-0.357434\pi\)
0.433061 + 0.901365i \(0.357434\pi\)
\(182\) 1.41421 0.104828
\(183\) 0 0
\(184\) 23.4164 1.72628
\(185\) −20.5942 −1.51412
\(186\) 0 0
\(187\) −27.4164 −2.00489
\(188\) 0 0
\(189\) 0 0
\(190\) −21.7082 −1.57488
\(191\) −2.49458 −0.180501 −0.0902506 0.995919i \(-0.528767\pi\)
−0.0902506 + 0.995919i \(0.528767\pi\)
\(192\) 0 0
\(193\) 9.00000 0.647834 0.323917 0.946085i \(-0.395000\pi\)
0.323917 + 0.946085i \(0.395000\pi\)
\(194\) 3.82998 0.274976
\(195\) 0 0
\(196\) 0 0
\(197\) −19.7990 −1.41062 −0.705310 0.708899i \(-0.749192\pi\)
−0.705310 + 0.708899i \(0.749192\pi\)
\(198\) 0 0
\(199\) −1.76393 −0.125042 −0.0625209 0.998044i \(-0.519914\pi\)
−0.0625209 + 0.998044i \(0.519914\pi\)
\(200\) −0.667701 −0.0472136
\(201\) 0 0
\(202\) 19.1246 1.34560
\(203\) −6.86474 −0.481810
\(204\) 0 0
\(205\) 4.47214 0.312348
\(206\) 7.73877 0.539186
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) −30.7000 −2.12357
\(210\) 0 0
\(211\) −6.23607 −0.429309 −0.214654 0.976690i \(-0.568862\pi\)
−0.214654 + 0.976690i \(0.568862\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 3.23607 0.221213
\(215\) 19.3863 1.32214
\(216\) 0 0
\(217\) −7.23607 −0.491216
\(218\) 18.9737 1.28506
\(219\) 0 0
\(220\) 0 0
\(221\) −5.99070 −0.402978
\(222\) 0 0
\(223\) 1.00000 0.0669650
\(224\) 0 0
\(225\) 0 0
\(226\) −8.00000 −0.532152
\(227\) −28.1266 −1.86683 −0.933416 0.358797i \(-0.883187\pi\)
−0.933416 + 0.358797i \(0.883187\pi\)
\(228\) 0 0
\(229\) −6.47214 −0.427691 −0.213845 0.976868i \(-0.568599\pi\)
−0.213845 + 0.976868i \(0.568599\pi\)
\(230\) −26.7912 −1.76656
\(231\) 0 0
\(232\) 19.4164 1.27475
\(233\) −29.4922 −1.93210 −0.966048 0.258364i \(-0.916817\pi\)
−0.966048 + 0.258364i \(0.916817\pi\)
\(234\) 0 0
\(235\) 29.8885 1.94971
\(236\) 0 0
\(237\) 0 0
\(238\) −8.47214 −0.549167
\(239\) 18.7186 1.21081 0.605404 0.795919i \(-0.293012\pi\)
0.605404 + 0.795919i \(0.293012\pi\)
\(240\) 0 0
\(241\) 28.4164 1.83046 0.915231 0.402930i \(-0.132008\pi\)
0.915231 + 0.402930i \(0.132008\pi\)
\(242\) 14.0633 0.904025
\(243\) 0 0
\(244\) 0 0
\(245\) −13.7295 −0.877144
\(246\) 0 0
\(247\) −6.70820 −0.426833
\(248\) 20.4667 1.29964
\(249\) 0 0
\(250\) −15.4164 −0.975019
\(251\) 27.0463 1.70715 0.853573 0.520973i \(-0.174431\pi\)
0.853573 + 0.520973i \(0.174431\pi\)
\(252\) 0 0
\(253\) −37.8885 −2.38203
\(254\) −6.73722 −0.422731
\(255\) 0 0
\(256\) 0 0
\(257\) 10.1058 0.630384 0.315192 0.949028i \(-0.397931\pi\)
0.315192 + 0.949028i \(0.397931\pi\)
\(258\) 0 0
\(259\) −9.00000 −0.559233
\(260\) 0 0
\(261\) 0 0
\(262\) 22.1803 1.37031
\(263\) 7.65996 0.472333 0.236167 0.971713i \(-0.424109\pi\)
0.236167 + 0.971713i \(0.424109\pi\)
\(264\) 0 0
\(265\) −17.4164 −1.06988
\(266\) −9.48683 −0.581675
\(267\) 0 0
\(268\) 0 0
\(269\) −4.44897 −0.271259 −0.135629 0.990760i \(-0.543306\pi\)
−0.135629 + 0.990760i \(0.543306\pi\)
\(270\) 0 0
\(271\) −12.8885 −0.782923 −0.391462 0.920194i \(-0.628030\pi\)
−0.391462 + 0.920194i \(0.628030\pi\)
\(272\) 23.9628 1.45296
\(273\) 0 0
\(274\) 8.47214 0.511820
\(275\) 1.08036 0.0651483
\(276\) 0 0
\(277\) −14.3607 −0.862850 −0.431425 0.902149i \(-0.641989\pi\)
−0.431425 + 0.902149i \(0.641989\pi\)
\(278\) 4.24264 0.254457
\(279\) 0 0
\(280\) −6.47214 −0.386784
\(281\) −12.5216 −0.746975 −0.373488 0.927635i \(-0.621838\pi\)
−0.373488 + 0.927635i \(0.621838\pi\)
\(282\) 0 0
\(283\) −11.7082 −0.695980 −0.347990 0.937498i \(-0.613136\pi\)
−0.347990 + 0.937498i \(0.613136\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 6.47214 0.382705
\(287\) 1.95440 0.115364
\(288\) 0 0
\(289\) 18.8885 1.11109
\(290\) −22.2148 −1.30450
\(291\) 0 0
\(292\) 0 0
\(293\) −1.66925 −0.0975188 −0.0487594 0.998811i \(-0.515527\pi\)
−0.0487594 + 0.998811i \(0.515527\pi\)
\(294\) 0 0
\(295\) −31.4164 −1.82913
\(296\) 25.4558 1.47959
\(297\) 0 0
\(298\) −1.23607 −0.0716035
\(299\) −8.27895 −0.478784
\(300\) 0 0
\(301\) 8.47214 0.488326
\(302\) 2.74962 0.158223
\(303\) 0 0
\(304\) 26.8328 1.53897
\(305\) 6.19704 0.354841
\(306\) 0 0
\(307\) −19.4164 −1.10815 −0.554076 0.832466i \(-0.686928\pi\)
−0.554076 + 0.832466i \(0.686928\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −23.4164 −1.32996
\(311\) −13.3168 −0.755127 −0.377564 0.925984i \(-0.623238\pi\)
−0.377564 + 0.925984i \(0.623238\pi\)
\(312\) 0 0
\(313\) −26.7082 −1.50964 −0.754818 0.655934i \(-0.772275\pi\)
−0.754818 + 0.655934i \(0.772275\pi\)
\(314\) −18.9737 −1.07075
\(315\) 0 0
\(316\) 0 0
\(317\) 22.9613 1.28963 0.644817 0.764337i \(-0.276934\pi\)
0.644817 + 0.764337i \(0.276934\pi\)
\(318\) 0 0
\(319\) −31.4164 −1.75898
\(320\) 18.3060 1.02333
\(321\) 0 0
\(322\) −11.7082 −0.652473
\(323\) 40.1869 2.23606
\(324\) 0 0
\(325\) 0.236068 0.0130947
\(326\) −1.00155 −0.0554708
\(327\) 0 0
\(328\) −5.52786 −0.305225
\(329\) 13.0618 0.720119
\(330\) 0 0
\(331\) 3.94427 0.216797 0.108398 0.994108i \(-0.465428\pi\)
0.108398 + 0.994108i \(0.465428\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 9.70820 0.531209
\(335\) −20.5942 −1.12518
\(336\) 0 0
\(337\) −7.76393 −0.422928 −0.211464 0.977386i \(-0.567823\pi\)
−0.211464 + 0.977386i \(0.567823\pi\)
\(338\) −16.9706 −0.923077
\(339\) 0 0
\(340\) 0 0
\(341\) −33.1158 −1.79332
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) −23.9628 −1.29199
\(345\) 0 0
\(346\) 22.0000 1.18273
\(347\) −21.4195 −1.14986 −0.574930 0.818202i \(-0.694971\pi\)
−0.574930 + 0.818202i \(0.694971\pi\)
\(348\) 0 0
\(349\) −10.4164 −0.557578 −0.278789 0.960352i \(-0.589933\pi\)
−0.278789 + 0.960352i \(0.589933\pi\)
\(350\) 0.333851 0.0178451
\(351\) 0 0
\(352\) 0 0
\(353\) 24.2480 1.29059 0.645294 0.763934i \(-0.276735\pi\)
0.645294 + 0.763934i \(0.276735\pi\)
\(354\) 0 0
\(355\) −28.1803 −1.49566
\(356\) 0 0
\(357\) 0 0
\(358\) −1.34752 −0.0712189
\(359\) 10.7735 0.568605 0.284303 0.958735i \(-0.408238\pi\)
0.284303 + 0.958735i \(0.408238\pi\)
\(360\) 0 0
\(361\) 26.0000 1.36842
\(362\) 16.4791 0.866122
\(363\) 0 0
\(364\) 0 0
\(365\) 30.1599 1.57864
\(366\) 0 0
\(367\) −32.7082 −1.70735 −0.853677 0.520803i \(-0.825633\pi\)
−0.853677 + 0.520803i \(0.825633\pi\)
\(368\) 33.1158 1.72628
\(369\) 0 0
\(370\) −29.1246 −1.51412
\(371\) −7.61125 −0.395156
\(372\) 0 0
\(373\) 4.70820 0.243782 0.121891 0.992544i \(-0.461104\pi\)
0.121891 + 0.992544i \(0.461104\pi\)
\(374\) −38.7727 −2.00489
\(375\) 0 0
\(376\) −36.9443 −1.90526
\(377\) −6.86474 −0.353552
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −3.52786 −0.180501
\(383\) −9.61435 −0.491270 −0.245635 0.969362i \(-0.578997\pi\)
−0.245635 + 0.969362i \(0.578997\pi\)
\(384\) 0 0
\(385\) 10.4721 0.533709
\(386\) 12.7279 0.647834
\(387\) 0 0
\(388\) 0 0
\(389\) 23.0100 1.16665 0.583326 0.812238i \(-0.301751\pi\)
0.583326 + 0.812238i \(0.301751\pi\)
\(390\) 0 0
\(391\) 49.5967 2.50822
\(392\) 16.9706 0.857143
\(393\) 0 0
\(394\) −28.0000 −1.41062
\(395\) 13.1893 0.663625
\(396\) 0 0
\(397\) −29.4164 −1.47637 −0.738184 0.674600i \(-0.764316\pi\)
−0.738184 + 0.674600i \(0.764316\pi\)
\(398\) −2.49458 −0.125042
\(399\) 0 0
\(400\) −0.944272 −0.0472136
\(401\) 14.1421 0.706225 0.353112 0.935581i \(-0.385123\pi\)
0.353112 + 0.935581i \(0.385123\pi\)
\(402\) 0 0
\(403\) −7.23607 −0.360454
\(404\) 0 0
\(405\) 0 0
\(406\) −9.70820 −0.481810
\(407\) −41.1884 −2.04163
\(408\) 0 0
\(409\) 7.58359 0.374984 0.187492 0.982266i \(-0.439964\pi\)
0.187492 + 0.982266i \(0.439964\pi\)
\(410\) 6.32456 0.312348
\(411\) 0 0
\(412\) 0 0
\(413\) −13.7295 −0.675583
\(414\) 0 0
\(415\) 26.1803 1.28514
\(416\) 0 0
\(417\) 0 0
\(418\) −43.4164 −2.12357
\(419\) −5.91189 −0.288815 −0.144407 0.989518i \(-0.546128\pi\)
−0.144407 + 0.989518i \(0.546128\pi\)
\(420\) 0 0
\(421\) 6.12461 0.298495 0.149248 0.988800i \(-0.452315\pi\)
0.149248 + 0.988800i \(0.452315\pi\)
\(422\) −8.81913 −0.429309
\(423\) 0 0
\(424\) 21.5279 1.04549
\(425\) −1.41421 −0.0685994
\(426\) 0 0
\(427\) 2.70820 0.131059
\(428\) 0 0
\(429\) 0 0
\(430\) 27.4164 1.32214
\(431\) 2.28825 0.110221 0.0551105 0.998480i \(-0.482449\pi\)
0.0551105 + 0.998480i \(0.482449\pi\)
\(432\) 0 0
\(433\) −1.41641 −0.0680682 −0.0340341 0.999421i \(-0.510835\pi\)
−0.0340341 + 0.999421i \(0.510835\pi\)
\(434\) −10.2333 −0.491216
\(435\) 0 0
\(436\) 0 0
\(437\) 55.5369 2.65669
\(438\) 0 0
\(439\) −0.180340 −0.00860715 −0.00430358 0.999991i \(-0.501370\pi\)
−0.00430358 + 0.999991i \(0.501370\pi\)
\(440\) −29.6197 −1.41206
\(441\) 0 0
\(442\) −8.47214 −0.402978
\(443\) 31.9079 1.51599 0.757995 0.652260i \(-0.226179\pi\)
0.757995 + 0.652260i \(0.226179\pi\)
\(444\) 0 0
\(445\) −10.4721 −0.496427
\(446\) 1.41421 0.0669650
\(447\) 0 0
\(448\) 8.00000 0.377964
\(449\) 1.82688 0.0862156 0.0431078 0.999070i \(-0.486274\pi\)
0.0431078 + 0.999070i \(0.486274\pi\)
\(450\) 0 0
\(451\) 8.94427 0.421169
\(452\) 0 0
\(453\) 0 0
\(454\) −39.7771 −1.86683
\(455\) 2.28825 0.107275
\(456\) 0 0
\(457\) 7.12461 0.333275 0.166638 0.986018i \(-0.446709\pi\)
0.166638 + 0.986018i \(0.446709\pi\)
\(458\) −9.15298 −0.427691
\(459\) 0 0
\(460\) 0 0
\(461\) 20.1815 0.939948 0.469974 0.882680i \(-0.344263\pi\)
0.469974 + 0.882680i \(0.344263\pi\)
\(462\) 0 0
\(463\) −28.2361 −1.31224 −0.656121 0.754656i \(-0.727804\pi\)
−0.656121 + 0.754656i \(0.727804\pi\)
\(464\) 27.4589 1.27475
\(465\) 0 0
\(466\) −41.7082 −1.93210
\(467\) 24.6305 1.13976 0.569882 0.821726i \(-0.306989\pi\)
0.569882 + 0.821726i \(0.306989\pi\)
\(468\) 0 0
\(469\) −9.00000 −0.415581
\(470\) 42.2688 1.94971
\(471\) 0 0
\(472\) 38.8328 1.78743
\(473\) 38.7727 1.78277
\(474\) 0 0
\(475\) −1.58359 −0.0726602
\(476\) 0 0
\(477\) 0 0
\(478\) 26.4721 1.21081
\(479\) 5.86319 0.267896 0.133948 0.990988i \(-0.457235\pi\)
0.133948 + 0.990988i \(0.457235\pi\)
\(480\) 0 0
\(481\) −9.00000 −0.410365
\(482\) 40.1869 1.83046
\(483\) 0 0
\(484\) 0 0
\(485\) 6.19704 0.281393
\(486\) 0 0
\(487\) 5.29180 0.239794 0.119897 0.992786i \(-0.461744\pi\)
0.119897 + 0.992786i \(0.461744\pi\)
\(488\) −7.65996 −0.346750
\(489\) 0 0
\(490\) −19.4164 −0.877144
\(491\) 3.28980 0.148466 0.0742332 0.997241i \(-0.476349\pi\)
0.0742332 + 0.997241i \(0.476349\pi\)
\(492\) 0 0
\(493\) 41.1246 1.85216
\(494\) −9.48683 −0.426833
\(495\) 0 0
\(496\) 28.9443 1.29964
\(497\) −12.3153 −0.552415
\(498\) 0 0
\(499\) −33.7082 −1.50899 −0.754493 0.656308i \(-0.772117\pi\)
−0.754493 + 0.656308i \(0.772117\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 38.2492 1.70715
\(503\) 30.4149 1.35613 0.678067 0.735001i \(-0.262818\pi\)
0.678067 + 0.735001i \(0.262818\pi\)
\(504\) 0 0
\(505\) 30.9443 1.37700
\(506\) −53.5825 −2.38203
\(507\) 0 0
\(508\) 0 0
\(509\) 17.6383 0.781802 0.390901 0.920433i \(-0.372163\pi\)
0.390901 + 0.920433i \(0.372163\pi\)
\(510\) 0 0
\(511\) 13.1803 0.583064
\(512\) −22.6274 −1.00000
\(513\) 0 0
\(514\) 14.2918 0.630384
\(515\) 12.5216 0.551767
\(516\) 0 0
\(517\) 59.7771 2.62899
\(518\) −12.7279 −0.559233
\(519\) 0 0
\(520\) −6.47214 −0.283822
\(521\) −31.3677 −1.37425 −0.687123 0.726541i \(-0.741127\pi\)
−0.687123 + 0.726541i \(0.741127\pi\)
\(522\) 0 0
\(523\) −16.8885 −0.738484 −0.369242 0.929333i \(-0.620383\pi\)
−0.369242 + 0.929333i \(0.620383\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 10.8328 0.472333
\(527\) 43.3491 1.88832
\(528\) 0 0
\(529\) 45.5410 1.98004
\(530\) −24.6305 −1.06988
\(531\) 0 0
\(532\) 0 0
\(533\) 1.95440 0.0846542
\(534\) 0 0
\(535\) 5.23607 0.226375
\(536\) 25.4558 1.09952
\(537\) 0 0
\(538\) −6.29180 −0.271259
\(539\) −27.4589 −1.18274
\(540\) 0 0
\(541\) 17.4721 0.751186 0.375593 0.926785i \(-0.377439\pi\)
0.375593 + 0.926785i \(0.377439\pi\)
\(542\) −18.2272 −0.782923
\(543\) 0 0
\(544\) 0 0
\(545\) 30.7000 1.31505
\(546\) 0 0
\(547\) −0.236068 −0.0100935 −0.00504677 0.999987i \(-0.501606\pi\)
−0.00504677 + 0.999987i \(0.501606\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 1.52786 0.0651483
\(551\) 46.0501 1.96180
\(552\) 0 0
\(553\) 5.76393 0.245107
\(554\) −20.3091 −0.862850
\(555\) 0 0
\(556\) 0 0
\(557\) 21.3894 0.906299 0.453150 0.891434i \(-0.350300\pi\)
0.453150 + 0.891434i \(0.350300\pi\)
\(558\) 0 0
\(559\) 8.47214 0.358333
\(560\) −9.15298 −0.386784
\(561\) 0 0
\(562\) −17.7082 −0.746975
\(563\) −18.1784 −0.766130 −0.383065 0.923721i \(-0.625131\pi\)
−0.383065 + 0.923721i \(0.625131\pi\)
\(564\) 0 0
\(565\) −12.9443 −0.544570
\(566\) −16.5579 −0.695980
\(567\) 0 0
\(568\) 34.8328 1.46155
\(569\) 7.19859 0.301780 0.150890 0.988551i \(-0.451786\pi\)
0.150890 + 0.988551i \(0.451786\pi\)
\(570\) 0 0
\(571\) 24.7082 1.03401 0.517003 0.855984i \(-0.327048\pi\)
0.517003 + 0.855984i \(0.327048\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 2.76393 0.115364
\(575\) −1.95440 −0.0815039
\(576\) 0 0
\(577\) −34.2361 −1.42527 −0.712633 0.701537i \(-0.752498\pi\)
−0.712633 + 0.701537i \(0.752498\pi\)
\(578\) 26.7124 1.11109
\(579\) 0 0
\(580\) 0 0
\(581\) 11.4412 0.474662
\(582\) 0 0
\(583\) −34.8328 −1.44263
\(584\) −37.2796 −1.54264
\(585\) 0 0
\(586\) −2.36068 −0.0975188
\(587\) −11.4899 −0.474240 −0.237120 0.971480i \(-0.576204\pi\)
−0.237120 + 0.971480i \(0.576204\pi\)
\(588\) 0 0
\(589\) 48.5410 2.00010
\(590\) −44.4295 −1.82913
\(591\) 0 0
\(592\) 36.0000 1.47959
\(593\) −18.9737 −0.779155 −0.389578 0.920994i \(-0.627379\pi\)
−0.389578 + 0.920994i \(0.627379\pi\)
\(594\) 0 0
\(595\) −13.7082 −0.561982
\(596\) 0 0
\(597\) 0 0
\(598\) −11.7082 −0.478784
\(599\) 33.5285 1.36994 0.684968 0.728573i \(-0.259816\pi\)
0.684968 + 0.728573i \(0.259816\pi\)
\(600\) 0 0
\(601\) −35.4164 −1.44467 −0.722333 0.691546i \(-0.756930\pi\)
−0.722333 + 0.691546i \(0.756930\pi\)
\(602\) 11.9814 0.488326
\(603\) 0 0
\(604\) 0 0
\(605\) 22.7549 0.925120
\(606\) 0 0
\(607\) 32.1246 1.30390 0.651949 0.758263i \(-0.273952\pi\)
0.651949 + 0.758263i \(0.273952\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 8.76393 0.354841
\(611\) 13.0618 0.528423
\(612\) 0 0
\(613\) 5.29180 0.213734 0.106867 0.994273i \(-0.465918\pi\)
0.106867 + 0.994273i \(0.465918\pi\)
\(614\) −27.4589 −1.10815
\(615\) 0 0
\(616\) −12.9443 −0.521540
\(617\) −22.5486 −0.907773 −0.453886 0.891060i \(-0.649963\pi\)
−0.453886 + 0.891060i \(0.649963\pi\)
\(618\) 0 0
\(619\) −21.5836 −0.867518 −0.433759 0.901029i \(-0.642813\pi\)
−0.433759 + 0.901029i \(0.642813\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −18.8328 −0.755127
\(623\) −4.57649 −0.183353
\(624\) 0 0
\(625\) −26.1246 −1.04498
\(626\) −37.7711 −1.50964
\(627\) 0 0
\(628\) 0 0
\(629\) 53.9163 2.14979
\(630\) 0 0
\(631\) −37.6525 −1.49892 −0.749461 0.662049i \(-0.769687\pi\)
−0.749461 + 0.662049i \(0.769687\pi\)
\(632\) −16.3029 −0.648493
\(633\) 0 0
\(634\) 32.4721 1.28963
\(635\) −10.9010 −0.432595
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) −44.4295 −1.75898
\(639\) 0 0
\(640\) 25.8885 1.02333
\(641\) −5.45052 −0.215283 −0.107641 0.994190i \(-0.534330\pi\)
−0.107641 + 0.994190i \(0.534330\pi\)
\(642\) 0 0
\(643\) −33.8885 −1.33643 −0.668217 0.743967i \(-0.732942\pi\)
−0.668217 + 0.743967i \(0.732942\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 56.8328 2.23606
\(647\) 27.0764 1.06448 0.532241 0.846593i \(-0.321350\pi\)
0.532241 + 0.846593i \(0.321350\pi\)
\(648\) 0 0
\(649\) −62.8328 −2.46640
\(650\) 0.333851 0.0130947
\(651\) 0 0
\(652\) 0 0
\(653\) −30.4937 −1.19331 −0.596655 0.802498i \(-0.703504\pi\)
−0.596655 + 0.802498i \(0.703504\pi\)
\(654\) 0 0
\(655\) 35.8885 1.40228
\(656\) −7.81758 −0.305225
\(657\) 0 0
\(658\) 18.4721 0.720119
\(659\) −20.4667 −0.797269 −0.398635 0.917110i \(-0.630516\pi\)
−0.398635 + 0.917110i \(0.630516\pi\)
\(660\) 0 0
\(661\) −45.8328 −1.78269 −0.891345 0.453326i \(-0.850237\pi\)
−0.891345 + 0.453326i \(0.850237\pi\)
\(662\) 5.57804 0.216797
\(663\) 0 0
\(664\) −32.3607 −1.25584
\(665\) −15.3500 −0.595248
\(666\) 0 0
\(667\) 56.8328 2.20058
\(668\) 0 0
\(669\) 0 0
\(670\) −29.1246 −1.12518
\(671\) 12.3941 0.478468
\(672\) 0 0
\(673\) 6.52786 0.251631 0.125815 0.992054i \(-0.459845\pi\)
0.125815 + 0.992054i \(0.459845\pi\)
\(674\) −10.9799 −0.422928
\(675\) 0 0
\(676\) 0 0
\(677\) 17.3531 0.666935 0.333467 0.942762i \(-0.391781\pi\)
0.333467 + 0.942762i \(0.391781\pi\)
\(678\) 0 0
\(679\) 2.70820 0.103931
\(680\) 38.7727 1.48686
\(681\) 0 0
\(682\) −46.8328 −1.79332
\(683\) −15.0649 −0.576441 −0.288221 0.957564i \(-0.593064\pi\)
−0.288221 + 0.957564i \(0.593064\pi\)
\(684\) 0 0
\(685\) 13.7082 0.523764
\(686\) −18.3848 −0.701934
\(687\) 0 0
\(688\) −33.8885 −1.29199
\(689\) −7.61125 −0.289966
\(690\) 0 0
\(691\) −14.3607 −0.546306 −0.273153 0.961971i \(-0.588067\pi\)
−0.273153 + 0.961971i \(0.588067\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −30.2918 −1.14986
\(695\) 6.86474 0.260394
\(696\) 0 0
\(697\) −11.7082 −0.443480
\(698\) −14.7310 −0.557578
\(699\) 0 0
\(700\) 0 0
\(701\) 30.9064 1.16732 0.583659 0.811999i \(-0.301621\pi\)
0.583659 + 0.811999i \(0.301621\pi\)
\(702\) 0 0
\(703\) 60.3738 2.27704
\(704\) 36.6119 1.37986
\(705\) 0 0
\(706\) 34.2918 1.29059
\(707\) 13.5231 0.508590
\(708\) 0 0
\(709\) −46.1246 −1.73225 −0.866123 0.499831i \(-0.833396\pi\)
−0.866123 + 0.499831i \(0.833396\pi\)
\(710\) −39.8530 −1.49566
\(711\) 0 0
\(712\) 12.9443 0.485107
\(713\) 59.9070 2.24354
\(714\) 0 0
\(715\) 10.4721 0.391636
\(716\) 0 0
\(717\) 0 0
\(718\) 15.2361 0.568605
\(719\) −30.1111 −1.12296 −0.561478 0.827492i \(-0.689767\pi\)
−0.561478 + 0.827492i \(0.689767\pi\)
\(720\) 0 0
\(721\) 5.47214 0.203793
\(722\) 36.7696 1.36842
\(723\) 0 0
\(724\) 0 0
\(725\) −1.62054 −0.0601855
\(726\) 0 0
\(727\) −15.4164 −0.571763 −0.285881 0.958265i \(-0.592286\pi\)
−0.285881 + 0.958265i \(0.592286\pi\)
\(728\) −2.82843 −0.104828
\(729\) 0 0
\(730\) 42.6525 1.57864
\(731\) −50.7541 −1.87721
\(732\) 0 0
\(733\) 47.1246 1.74059 0.870294 0.492533i \(-0.163929\pi\)
0.870294 + 0.492533i \(0.163929\pi\)
\(734\) −46.2564 −1.70735
\(735\) 0 0
\(736\) 0 0
\(737\) −41.1884 −1.51719
\(738\) 0 0
\(739\) −43.4164 −1.59710 −0.798549 0.601930i \(-0.794399\pi\)
−0.798549 + 0.601930i \(0.794399\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −10.7639 −0.395156
\(743\) −39.1552 −1.43647 −0.718233 0.695803i \(-0.755049\pi\)
−0.718233 + 0.695803i \(0.755049\pi\)
\(744\) 0 0
\(745\) −2.00000 −0.0732743
\(746\) 6.65841 0.243782
\(747\) 0 0
\(748\) 0 0
\(749\) 2.28825 0.0836107
\(750\) 0 0
\(751\) −9.94427 −0.362872 −0.181436 0.983403i \(-0.558074\pi\)
−0.181436 + 0.983403i \(0.558074\pi\)
\(752\) −52.2471 −1.90526
\(753\) 0 0
\(754\) −9.70820 −0.353552
\(755\) 4.44897 0.161915
\(756\) 0 0
\(757\) 6.52786 0.237259 0.118630 0.992939i \(-0.462150\pi\)
0.118630 + 0.992939i \(0.462150\pi\)
\(758\) −7.07107 −0.256833
\(759\) 0 0
\(760\) 43.4164 1.57488
\(761\) 46.8453 1.69814 0.849070 0.528280i \(-0.177163\pi\)
0.849070 + 0.528280i \(0.177163\pi\)
\(762\) 0 0
\(763\) 13.4164 0.485707
\(764\) 0 0
\(765\) 0 0
\(766\) −13.5967 −0.491270
\(767\) −13.7295 −0.495743
\(768\) 0 0
\(769\) 5.76393 0.207853 0.103926 0.994585i \(-0.466859\pi\)
0.103926 + 0.994585i \(0.466859\pi\)
\(770\) 14.8098 0.533709
\(771\) 0 0
\(772\) 0 0
\(773\) 10.8222 0.389249 0.194624 0.980878i \(-0.437651\pi\)
0.194624 + 0.980878i \(0.437651\pi\)
\(774\) 0 0
\(775\) −1.70820 −0.0613605
\(776\) −7.65996 −0.274976
\(777\) 0 0
\(778\) 32.5410 1.16665
\(779\) −13.1105 −0.469732
\(780\) 0 0
\(781\) −56.3607 −2.01674
\(782\) 70.1404 2.50822
\(783\) 0 0
\(784\) 24.0000 0.857143
\(785\) −30.7000 −1.09573
\(786\) 0 0
\(787\) −25.1803 −0.897582 −0.448791 0.893637i \(-0.648145\pi\)
−0.448791 + 0.893637i \(0.648145\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 18.6525 0.663625
\(791\) −5.65685 −0.201135
\(792\) 0 0
\(793\) 2.70820 0.0961711
\(794\) −41.6011 −1.47637
\(795\) 0 0
\(796\) 0 0
\(797\) 14.5548 0.515557 0.257779 0.966204i \(-0.417009\pi\)
0.257779 + 0.966204i \(0.417009\pi\)
\(798\) 0 0
\(799\) −78.2492 −2.76826
\(800\) 0 0
\(801\) 0 0
\(802\) 20.0000 0.706225
\(803\) 60.3197 2.12864
\(804\) 0 0
\(805\) −18.9443 −0.667698
\(806\) −10.2333 −0.360454
\(807\) 0 0
\(808\) −38.2492 −1.34560
\(809\) 26.2511 0.922938 0.461469 0.887156i \(-0.347323\pi\)
0.461469 + 0.887156i \(0.347323\pi\)
\(810\) 0 0
\(811\) −15.4164 −0.541343 −0.270672 0.962672i \(-0.587246\pi\)
−0.270672 + 0.962672i \(0.587246\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −58.2492 −2.04163
\(815\) −1.62054 −0.0567652
\(816\) 0 0
\(817\) −56.8328 −1.98833
\(818\) 10.7248 0.374984
\(819\) 0 0
\(820\) 0 0
\(821\) −43.0640 −1.50294 −0.751472 0.659765i \(-0.770656\pi\)
−0.751472 + 0.659765i \(0.770656\pi\)
\(822\) 0 0
\(823\) 44.9574 1.56712 0.783559 0.621318i \(-0.213402\pi\)
0.783559 + 0.621318i \(0.213402\pi\)
\(824\) −15.4775 −0.539186
\(825\) 0 0
\(826\) −19.4164 −0.675583
\(827\) 12.7279 0.442593 0.221297 0.975207i \(-0.428971\pi\)
0.221297 + 0.975207i \(0.428971\pi\)
\(828\) 0 0
\(829\) 5.00000 0.173657 0.0868286 0.996223i \(-0.472327\pi\)
0.0868286 + 0.996223i \(0.472327\pi\)
\(830\) 37.0246 1.28514
\(831\) 0 0
\(832\) 8.00000 0.277350
\(833\) 35.9442 1.24539
\(834\) 0 0
\(835\) 15.7082 0.543605
\(836\) 0 0
\(837\) 0 0
\(838\) −8.36068 −0.288815
\(839\) −40.1382 −1.38572 −0.692862 0.721071i \(-0.743650\pi\)
−0.692862 + 0.721071i \(0.743650\pi\)
\(840\) 0 0
\(841\) 18.1246 0.624987
\(842\) 8.66151 0.298495
\(843\) 0 0
\(844\) 0 0
\(845\) −27.4589 −0.944617
\(846\) 0 0
\(847\) 9.94427 0.341689
\(848\) 30.4450 1.04549
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) 74.5106 2.55419
\(852\) 0 0
\(853\) −2.70820 −0.0927271 −0.0463636 0.998925i \(-0.514763\pi\)
−0.0463636 + 0.998925i \(0.514763\pi\)
\(854\) 3.82998 0.131059
\(855\) 0 0
\(856\) −6.47214 −0.221213
\(857\) −12.1390 −0.414661 −0.207331 0.978271i \(-0.566478\pi\)
−0.207331 + 0.978271i \(0.566478\pi\)
\(858\) 0 0
\(859\) −4.05573 −0.138380 −0.0691898 0.997604i \(-0.522041\pi\)
−0.0691898 + 0.997604i \(0.522041\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 3.23607 0.110221
\(863\) −13.8570 −0.471698 −0.235849 0.971790i \(-0.575787\pi\)
−0.235849 + 0.971790i \(0.575787\pi\)
\(864\) 0 0
\(865\) 35.5967 1.21033
\(866\) −2.00310 −0.0680682
\(867\) 0 0
\(868\) 0 0
\(869\) 26.3786 0.894832
\(870\) 0 0
\(871\) −9.00000 −0.304953
\(872\) −37.9473 −1.28506
\(873\) 0 0
\(874\) 78.5410 2.65669
\(875\) −10.9010 −0.368523
\(876\) 0 0
\(877\) −40.0132 −1.35115 −0.675574 0.737292i \(-0.736104\pi\)
−0.675574 + 0.737292i \(0.736104\pi\)
\(878\) −0.255039 −0.00860715
\(879\) 0 0
\(880\) −41.8885 −1.41206
\(881\) 50.5778 1.70401 0.852005 0.523533i \(-0.175386\pi\)
0.852005 + 0.523533i \(0.175386\pi\)
\(882\) 0 0
\(883\) 29.0000 0.975928 0.487964 0.872864i \(-0.337740\pi\)
0.487964 + 0.872864i \(0.337740\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 45.1246 1.51599
\(887\) 2.82843 0.0949693 0.0474846 0.998872i \(-0.484879\pi\)
0.0474846 + 0.998872i \(0.484879\pi\)
\(888\) 0 0
\(889\) −4.76393 −0.159777
\(890\) −14.8098 −0.496427
\(891\) 0 0
\(892\) 0 0
\(893\) −87.6210 −2.93213
\(894\) 0 0
\(895\) −2.18034 −0.0728807
\(896\) 11.3137 0.377964
\(897\) 0 0
\(898\) 2.58359 0.0862156
\(899\) 49.6737 1.65671
\(900\) 0 0
\(901\) 45.5967 1.51905
\(902\) 12.6491 0.421169
\(903\) 0 0
\(904\) 16.0000 0.532152
\(905\) 26.6637 0.886332
\(906\) 0 0
\(907\) −15.5836 −0.517445 −0.258722 0.965952i \(-0.583301\pi\)
−0.258722 + 0.965952i \(0.583301\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 3.23607 0.107275
\(911\) −49.0547 −1.62526 −0.812628 0.582783i \(-0.801964\pi\)
−0.812628 + 0.582783i \(0.801964\pi\)
\(912\) 0 0
\(913\) 52.3607 1.73289
\(914\) 10.0757 0.333275
\(915\) 0 0
\(916\) 0 0
\(917\) 15.6839 0.517927
\(918\) 0 0
\(919\) 16.9443 0.558940 0.279470 0.960154i \(-0.409841\pi\)
0.279470 + 0.960154i \(0.409841\pi\)
\(920\) 53.5825 1.76656
\(921\) 0 0
\(922\) 28.5410 0.939948
\(923\) −12.3153 −0.405362
\(924\) 0 0
\(925\) −2.12461 −0.0698568
\(926\) −39.9318 −1.31224
\(927\) 0 0
\(928\) 0 0
\(929\) −3.00465 −0.0985795 −0.0492898 0.998785i \(-0.515696\pi\)
−0.0492898 + 0.998785i \(0.515696\pi\)
\(930\) 0 0
\(931\) 40.2492 1.31912
\(932\) 0 0
\(933\) 0 0
\(934\) 34.8328 1.13976
\(935\) −62.7355 −2.05167
\(936\) 0 0
\(937\) −25.5410 −0.834389 −0.417194 0.908817i \(-0.636987\pi\)
−0.417194 + 0.908817i \(0.636987\pi\)
\(938\) −12.7279 −0.415581
\(939\) 0 0
\(940\) 0 0
\(941\) −26.4574 −0.862486 −0.431243 0.902236i \(-0.641925\pi\)
−0.431243 + 0.902236i \(0.641925\pi\)
\(942\) 0 0
\(943\) −16.1803 −0.526904
\(944\) 54.9179 1.78743
\(945\) 0 0
\(946\) 54.8328 1.78277
\(947\) −20.2117 −0.656790 −0.328395 0.944540i \(-0.606508\pi\)
−0.328395 + 0.944540i \(0.606508\pi\)
\(948\) 0 0
\(949\) 13.1803 0.427852
\(950\) −2.23954 −0.0726602
\(951\) 0 0
\(952\) 16.9443 0.549167
\(953\) −4.65530 −0.150800 −0.0754000 0.997153i \(-0.524023\pi\)
−0.0754000 + 0.997153i \(0.524023\pi\)
\(954\) 0 0
\(955\) −5.70820 −0.184713
\(956\) 0 0
\(957\) 0 0
\(958\) 8.29180 0.267896
\(959\) 5.99070 0.193450
\(960\) 0 0
\(961\) 21.3607 0.689054
\(962\) −12.7279 −0.410365
\(963\) 0 0
\(964\) 0 0
\(965\) 20.5942 0.662951
\(966\) 0 0
\(967\) −16.4164 −0.527916 −0.263958 0.964534i \(-0.585028\pi\)
−0.263958 + 0.964534i \(0.585028\pi\)
\(968\) −28.1266 −0.904025
\(969\) 0 0
\(970\) 8.76393 0.281393
\(971\) −43.3979 −1.39270 −0.696352 0.717701i \(-0.745195\pi\)
−0.696352 + 0.717701i \(0.745195\pi\)
\(972\) 0 0
\(973\) 3.00000 0.0961756
\(974\) 7.48373 0.239794
\(975\) 0 0
\(976\) −10.8328 −0.346750
\(977\) −7.19859 −0.230303 −0.115152 0.993348i \(-0.536735\pi\)
−0.115152 + 0.993348i \(0.536735\pi\)
\(978\) 0 0
\(979\) −20.9443 −0.669382
\(980\) 0 0
\(981\) 0 0
\(982\) 4.65248 0.148466
\(983\) −18.1784 −0.579802 −0.289901 0.957057i \(-0.593622\pi\)
−0.289901 + 0.957057i \(0.593622\pi\)
\(984\) 0 0
\(985\) −45.3050 −1.44354
\(986\) 58.1590 1.85216
\(987\) 0 0
\(988\) 0 0
\(989\) −70.1404 −2.23033
\(990\) 0 0
\(991\) 10.7082 0.340157 0.170079 0.985430i \(-0.445598\pi\)
0.170079 + 0.985430i \(0.445598\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −17.4164 −0.552415
\(995\) −4.03631 −0.127960
\(996\) 0 0
\(997\) 34.9443 1.10670 0.553348 0.832950i \(-0.313350\pi\)
0.553348 + 0.832950i \(0.313350\pi\)
\(998\) −47.6706 −1.50899
\(999\) 0 0
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 6021.2.a.k.1.4 yes 4
3.2 odd 2 inner 6021.2.a.k.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6021.2.a.k.1.1 4 3.2 odd 2 inner
6021.2.a.k.1.4 yes 4 1.1 even 1 trivial