Properties

Label 3344.2.a.q.1.2
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
Analytic rank $0$
Dimension $3$
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] = [3344,2,Mod(1,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, 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("3344.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.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: \(26.7019744359\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.523976\) of defining polynomial
Character \(\chi\) \(=\) 3344.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.523976 q^{3} +2.72545 q^{5} +4.67750 q^{7} -2.72545 q^{9} +O(q^{10})\) \(q+0.523976 q^{3} +2.72545 q^{5} +4.67750 q^{7} -2.72545 q^{9} +1.00000 q^{11} -2.67750 q^{13} +1.42807 q^{15} +0.201472 q^{17} +1.00000 q^{19} +2.45090 q^{21} +1.79853 q^{23} +2.42807 q^{25} -3.00000 q^{27} -4.92692 q^{29} +2.57193 q^{31} +0.523976 q^{33} +12.7483 q^{35} +4.40294 q^{37} -1.40294 q^{39} +4.97487 q^{41} +11.7734 q^{43} -7.42807 q^{45} +12.4989 q^{47} +14.8790 q^{49} +0.105567 q^{51} -4.10557 q^{53} +2.72545 q^{55} +0.523976 q^{57} +5.24943 q^{59} +1.59706 q^{61} -12.7483 q^{63} -7.29738 q^{65} -9.08044 q^{67} +0.942386 q^{69} -12.8214 q^{71} +2.20147 q^{73} +1.27225 q^{75} +4.67750 q^{77} -11.8538 q^{79} +6.60442 q^{81} +13.6775 q^{83} +0.549103 q^{85} -2.58159 q^{87} -5.45090 q^{89} -12.5240 q^{91} +1.34763 q^{93} +2.72545 q^{95} +16.8059 q^{97} -2.72545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} + 6 q^{7} + 3 q^{9} + 3 q^{11} + 9 q^{15} - 9 q^{17} + 3 q^{19} - 15 q^{21} + 15 q^{23} + 12 q^{25} - 9 q^{27} + 6 q^{29} + 3 q^{31} - 6 q^{37} + 15 q^{39} - 9 q^{41} + 21 q^{43} - 27 q^{45} + 12 q^{47} + 27 q^{49} - 3 q^{51} - 9 q^{53} - 3 q^{55} + 3 q^{59} + 24 q^{61} - 6 q^{65} + 3 q^{69} - 21 q^{71} - 3 q^{73} - 36 q^{75} + 6 q^{77} + 6 q^{79} - 9 q^{81} + 33 q^{83} + 24 q^{85} - 6 q^{87} + 6 q^{89} - 36 q^{91} + 36 q^{93} - 3 q^{95} + 12 q^{97} + 3 q^{99}+O(q^{100}) \) 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
\(3\) 0.523976 0.302518 0.151259 0.988494i \(-0.451667\pi\)
0.151259 + 0.988494i \(0.451667\pi\)
\(4\) 0 0
\(5\) 2.72545 1.21886 0.609429 0.792841i \(-0.291399\pi\)
0.609429 + 0.792841i \(0.291399\pi\)
\(6\) 0 0
\(7\) 4.67750 1.76793 0.883964 0.467556i \(-0.154865\pi\)
0.883964 + 0.467556i \(0.154865\pi\)
\(8\) 0 0
\(9\) −2.72545 −0.908483
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −2.67750 −0.742604 −0.371302 0.928512i \(-0.621089\pi\)
−0.371302 + 0.928512i \(0.621089\pi\)
\(14\) 0 0
\(15\) 1.42807 0.368726
\(16\) 0 0
\(17\) 0.201472 0.0488642 0.0244321 0.999701i \(-0.492222\pi\)
0.0244321 + 0.999701i \(0.492222\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 2.45090 0.534830
\(22\) 0 0
\(23\) 1.79853 0.375019 0.187509 0.982263i \(-0.439958\pi\)
0.187509 + 0.982263i \(0.439958\pi\)
\(24\) 0 0
\(25\) 2.42807 0.485614
\(26\) 0 0
\(27\) −3.00000 −0.577350
\(28\) 0 0
\(29\) −4.92692 −0.914906 −0.457453 0.889234i \(-0.651238\pi\)
−0.457453 + 0.889234i \(0.651238\pi\)
\(30\) 0 0
\(31\) 2.57193 0.461932 0.230966 0.972962i \(-0.425811\pi\)
0.230966 + 0.972962i \(0.425811\pi\)
\(32\) 0 0
\(33\) 0.523976 0.0912126
\(34\) 0 0
\(35\) 12.7483 2.15485
\(36\) 0 0
\(37\) 4.40294 0.723840 0.361920 0.932209i \(-0.382121\pi\)
0.361920 + 0.932209i \(0.382121\pi\)
\(38\) 0 0
\(39\) −1.40294 −0.224651
\(40\) 0 0
\(41\) 4.97487 0.776945 0.388472 0.921460i \(-0.373003\pi\)
0.388472 + 0.921460i \(0.373003\pi\)
\(42\) 0 0
\(43\) 11.7734 1.79543 0.897713 0.440580i \(-0.145227\pi\)
0.897713 + 0.440580i \(0.145227\pi\)
\(44\) 0 0
\(45\) −7.42807 −1.10731
\(46\) 0 0
\(47\) 12.4989 1.82314 0.911572 0.411140i \(-0.134869\pi\)
0.911572 + 0.411140i \(0.134869\pi\)
\(48\) 0 0
\(49\) 14.8790 2.12557
\(50\) 0 0
\(51\) 0.105567 0.0147823
\(52\) 0 0
\(53\) −4.10557 −0.563943 −0.281971 0.959423i \(-0.590988\pi\)
−0.281971 + 0.959423i \(0.590988\pi\)
\(54\) 0 0
\(55\) 2.72545 0.367499
\(56\) 0 0
\(57\) 0.523976 0.0694024
\(58\) 0 0
\(59\) 5.24943 0.683417 0.341708 0.939806i \(-0.388994\pi\)
0.341708 + 0.939806i \(0.388994\pi\)
\(60\) 0 0
\(61\) 1.59706 0.204482 0.102241 0.994760i \(-0.467399\pi\)
0.102241 + 0.994760i \(0.467399\pi\)
\(62\) 0 0
\(63\) −12.7483 −1.60613
\(64\) 0 0
\(65\) −7.29738 −0.905128
\(66\) 0 0
\(67\) −9.08044 −1.10935 −0.554676 0.832066i \(-0.687158\pi\)
−0.554676 + 0.832066i \(0.687158\pi\)
\(68\) 0 0
\(69\) 0.942386 0.113450
\(70\) 0 0
\(71\) −12.8214 −1.52161 −0.760807 0.648978i \(-0.775197\pi\)
−0.760807 + 0.648978i \(0.775197\pi\)
\(72\) 0 0
\(73\) 2.20147 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(74\) 0 0
\(75\) 1.27225 0.146907
\(76\) 0 0
\(77\) 4.67750 0.533050
\(78\) 0 0
\(79\) −11.8538 −1.33366 −0.666831 0.745209i \(-0.732350\pi\)
−0.666831 + 0.745209i \(0.732350\pi\)
\(80\) 0 0
\(81\) 6.60442 0.733824
\(82\) 0 0
\(83\) 13.6775 1.50130 0.750650 0.660700i \(-0.229740\pi\)
0.750650 + 0.660700i \(0.229740\pi\)
\(84\) 0 0
\(85\) 0.549103 0.0595585
\(86\) 0 0
\(87\) −2.58159 −0.276776
\(88\) 0 0
\(89\) −5.45090 −0.577794 −0.288897 0.957360i \(-0.593289\pi\)
−0.288897 + 0.957360i \(0.593289\pi\)
\(90\) 0 0
\(91\) −12.5240 −1.31287
\(92\) 0 0
\(93\) 1.34763 0.139743
\(94\) 0 0
\(95\) 2.72545 0.279625
\(96\) 0 0
\(97\) 16.8059 1.70638 0.853190 0.521601i \(-0.174665\pi\)
0.853190 + 0.521601i \(0.174665\pi\)
\(98\) 0 0
\(99\) −2.72545 −0.273918
\(100\) 0 0
\(101\) 7.45090 0.741392 0.370696 0.928754i \(-0.379119\pi\)
0.370696 + 0.928754i \(0.379119\pi\)
\(102\) 0 0
\(103\) 14.4834 1.42709 0.713545 0.700609i \(-0.247088\pi\)
0.713545 + 0.700609i \(0.247088\pi\)
\(104\) 0 0
\(105\) 6.67980 0.651881
\(106\) 0 0
\(107\) −1.15352 −0.111515 −0.0557575 0.998444i \(-0.517757\pi\)
−0.0557575 + 0.998444i \(0.517757\pi\)
\(108\) 0 0
\(109\) −9.55646 −0.915343 −0.457672 0.889121i \(-0.651316\pi\)
−0.457672 + 0.889121i \(0.651316\pi\)
\(110\) 0 0
\(111\) 2.30704 0.218974
\(112\) 0 0
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) 0 0
\(115\) 4.90179 0.457095
\(116\) 0 0
\(117\) 7.29738 0.674643
\(118\) 0 0
\(119\) 0.942386 0.0863884
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 2.60672 0.235040
\(124\) 0 0
\(125\) −7.00966 −0.626963
\(126\) 0 0
\(127\) −5.75794 −0.510934 −0.255467 0.966818i \(-0.582229\pi\)
−0.255467 + 0.966818i \(0.582229\pi\)
\(128\) 0 0
\(129\) 6.16898 0.543149
\(130\) 0 0
\(131\) −1.42807 −0.124771 −0.0623856 0.998052i \(-0.519871\pi\)
−0.0623856 + 0.998052i \(0.519871\pi\)
\(132\) 0 0
\(133\) 4.67750 0.405590
\(134\) 0 0
\(135\) −8.17635 −0.703708
\(136\) 0 0
\(137\) −14.5313 −1.24150 −0.620748 0.784010i \(-0.713171\pi\)
−0.620748 + 0.784010i \(0.713171\pi\)
\(138\) 0 0
\(139\) −16.1763 −1.37206 −0.686030 0.727573i \(-0.740648\pi\)
−0.686030 + 0.727573i \(0.740648\pi\)
\(140\) 0 0
\(141\) 6.54910 0.551534
\(142\) 0 0
\(143\) −2.67750 −0.223903
\(144\) 0 0
\(145\) −13.4281 −1.11514
\(146\) 0 0
\(147\) 7.79623 0.643022
\(148\) 0 0
\(149\) 6.30704 0.516693 0.258346 0.966052i \(-0.416822\pi\)
0.258346 + 0.966052i \(0.416822\pi\)
\(150\) 0 0
\(151\) −0.498850 −0.0405959 −0.0202979 0.999794i \(-0.506461\pi\)
−0.0202979 + 0.999794i \(0.506461\pi\)
\(152\) 0 0
\(153\) −0.549103 −0.0443923
\(154\) 0 0
\(155\) 7.00966 0.563030
\(156\) 0 0
\(157\) −13.6272 −1.08757 −0.543786 0.839224i \(-0.683010\pi\)
−0.543786 + 0.839224i \(0.683010\pi\)
\(158\) 0 0
\(159\) −2.15122 −0.170603
\(160\) 0 0
\(161\) 8.41261 0.663006
\(162\) 0 0
\(163\) −15.9497 −1.24928 −0.624640 0.780913i \(-0.714754\pi\)
−0.624640 + 0.780913i \(0.714754\pi\)
\(164\) 0 0
\(165\) 1.42807 0.111175
\(166\) 0 0
\(167\) −18.8059 −1.45524 −0.727622 0.685979i \(-0.759374\pi\)
−0.727622 + 0.685979i \(0.759374\pi\)
\(168\) 0 0
\(169\) −5.83102 −0.448540
\(170\) 0 0
\(171\) −2.72545 −0.208420
\(172\) 0 0
\(173\) −3.22430 −0.245139 −0.122569 0.992460i \(-0.539113\pi\)
−0.122569 + 0.992460i \(0.539113\pi\)
\(174\) 0 0
\(175\) 11.3573 0.858531
\(176\) 0 0
\(177\) 2.75057 0.206746
\(178\) 0 0
\(179\) −13.3778 −0.999905 −0.499953 0.866053i \(-0.666649\pi\)
−0.499953 + 0.866053i \(0.666649\pi\)
\(180\) 0 0
\(181\) 15.9497 1.18554 0.592768 0.805373i \(-0.298035\pi\)
0.592768 + 0.805373i \(0.298035\pi\)
\(182\) 0 0
\(183\) 0.836819 0.0618595
\(184\) 0 0
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) 0.201472 0.0147331
\(188\) 0 0
\(189\) −14.0325 −1.02071
\(190\) 0 0
\(191\) 24.0553 1.74058 0.870291 0.492538i \(-0.163931\pi\)
0.870291 + 0.492538i \(0.163931\pi\)
\(192\) 0 0
\(193\) 13.0325 0.938099 0.469050 0.883172i \(-0.344597\pi\)
0.469050 + 0.883172i \(0.344597\pi\)
\(194\) 0 0
\(195\) −3.82365 −0.273818
\(196\) 0 0
\(197\) −5.59706 −0.398774 −0.199387 0.979921i \(-0.563895\pi\)
−0.199387 + 0.979921i \(0.563895\pi\)
\(198\) 0 0
\(199\) 7.39558 0.524259 0.262129 0.965033i \(-0.415575\pi\)
0.262129 + 0.965033i \(0.415575\pi\)
\(200\) 0 0
\(201\) −4.75794 −0.335599
\(202\) 0 0
\(203\) −23.0457 −1.61749
\(204\) 0 0
\(205\) 13.5588 0.946985
\(206\) 0 0
\(207\) −4.90179 −0.340698
\(208\) 0 0
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 14.2974 0.984272 0.492136 0.870518i \(-0.336216\pi\)
0.492136 + 0.870518i \(0.336216\pi\)
\(212\) 0 0
\(213\) −6.71809 −0.460316
\(214\) 0 0
\(215\) 32.0878 2.18837
\(216\) 0 0
\(217\) 12.0302 0.816662
\(218\) 0 0
\(219\) 1.15352 0.0779476
\(220\) 0 0
\(221\) −0.539441 −0.0362868
\(222\) 0 0
\(223\) 0.402945 0.0269832 0.0134916 0.999909i \(-0.495705\pi\)
0.0134916 + 0.999909i \(0.495705\pi\)
\(224\) 0 0
\(225\) −6.61758 −0.441172
\(226\) 0 0
\(227\) 21.0074 1.39431 0.697154 0.716922i \(-0.254449\pi\)
0.697154 + 0.716922i \(0.254449\pi\)
\(228\) 0 0
\(229\) −14.3299 −0.946944 −0.473472 0.880809i \(-0.657000\pi\)
−0.473472 + 0.880809i \(0.657000\pi\)
\(230\) 0 0
\(231\) 2.45090 0.161257
\(232\) 0 0
\(233\) −24.4029 −1.59869 −0.799345 0.600872i \(-0.794820\pi\)
−0.799345 + 0.600872i \(0.794820\pi\)
\(234\) 0 0
\(235\) 34.0650 2.22215
\(236\) 0 0
\(237\) −6.21113 −0.403456
\(238\) 0 0
\(239\) 12.6849 0.820515 0.410258 0.911970i \(-0.365439\pi\)
0.410258 + 0.911970i \(0.365439\pi\)
\(240\) 0 0
\(241\) −23.9343 −1.54174 −0.770871 0.636991i \(-0.780179\pi\)
−0.770871 + 0.636991i \(0.780179\pi\)
\(242\) 0 0
\(243\) 12.4606 0.799345
\(244\) 0 0
\(245\) 40.5519 2.59076
\(246\) 0 0
\(247\) −2.67750 −0.170365
\(248\) 0 0
\(249\) 7.16669 0.454170
\(250\) 0 0
\(251\) −18.4989 −1.16764 −0.583819 0.811884i \(-0.698442\pi\)
−0.583819 + 0.811884i \(0.698442\pi\)
\(252\) 0 0
\(253\) 1.79853 0.113072
\(254\) 0 0
\(255\) 0.287717 0.0180175
\(256\) 0 0
\(257\) 27.7579 1.73149 0.865746 0.500483i \(-0.166844\pi\)
0.865746 + 0.500483i \(0.166844\pi\)
\(258\) 0 0
\(259\) 20.5948 1.27970
\(260\) 0 0
\(261\) 13.4281 0.831177
\(262\) 0 0
\(263\) −20.2723 −1.25004 −0.625020 0.780608i \(-0.714909\pi\)
−0.625020 + 0.780608i \(0.714909\pi\)
\(264\) 0 0
\(265\) −11.1895 −0.687366
\(266\) 0 0
\(267\) −2.85614 −0.174793
\(268\) 0 0
\(269\) −15.6427 −0.953753 −0.476876 0.878970i \(-0.658231\pi\)
−0.476876 + 0.878970i \(0.658231\pi\)
\(270\) 0 0
\(271\) 12.3395 0.749573 0.374786 0.927111i \(-0.377716\pi\)
0.374786 + 0.927111i \(0.377716\pi\)
\(272\) 0 0
\(273\) −6.56227 −0.397167
\(274\) 0 0
\(275\) 2.42807 0.146418
\(276\) 0 0
\(277\) 8.49885 0.510646 0.255323 0.966856i \(-0.417818\pi\)
0.255323 + 0.966856i \(0.417818\pi\)
\(278\) 0 0
\(279\) −7.00966 −0.419657
\(280\) 0 0
\(281\) −7.12839 −0.425244 −0.212622 0.977134i \(-0.568200\pi\)
−0.212622 + 0.977134i \(0.568200\pi\)
\(282\) 0 0
\(283\) −11.5240 −0.685029 −0.342515 0.939512i \(-0.611279\pi\)
−0.342515 + 0.939512i \(0.611279\pi\)
\(284\) 0 0
\(285\) 1.42807 0.0845916
\(286\) 0 0
\(287\) 23.2700 1.37358
\(288\) 0 0
\(289\) −16.9594 −0.997612
\(290\) 0 0
\(291\) 8.80589 0.516210
\(292\) 0 0
\(293\) 10.6775 0.623786 0.311893 0.950117i \(-0.399037\pi\)
0.311893 + 0.950117i \(0.399037\pi\)
\(294\) 0 0
\(295\) 14.3070 0.832988
\(296\) 0 0
\(297\) −3.00000 −0.174078
\(298\) 0 0
\(299\) −4.81555 −0.278490
\(300\) 0 0
\(301\) 55.0700 3.17418
\(302\) 0 0
\(303\) 3.90409 0.224284
\(304\) 0 0
\(305\) 4.35269 0.249234
\(306\) 0 0
\(307\) 30.2568 1.72685 0.863423 0.504481i \(-0.168316\pi\)
0.863423 + 0.504481i \(0.168316\pi\)
\(308\) 0 0
\(309\) 7.58895 0.431720
\(310\) 0 0
\(311\) −8.79623 −0.498788 −0.249394 0.968402i \(-0.580231\pi\)
−0.249394 + 0.968402i \(0.580231\pi\)
\(312\) 0 0
\(313\) 10.1860 0.575747 0.287874 0.957668i \(-0.407052\pi\)
0.287874 + 0.957668i \(0.407052\pi\)
\(314\) 0 0
\(315\) −34.7448 −1.95765
\(316\) 0 0
\(317\) −12.0097 −0.674530 −0.337265 0.941410i \(-0.609502\pi\)
−0.337265 + 0.941410i \(0.609502\pi\)
\(318\) 0 0
\(319\) −4.92692 −0.275855
\(320\) 0 0
\(321\) −0.604417 −0.0337353
\(322\) 0 0
\(323\) 0.201472 0.0112102
\(324\) 0 0
\(325\) −6.50115 −0.360619
\(326\) 0 0
\(327\) −5.00736 −0.276908
\(328\) 0 0
\(329\) 58.4633 3.22319
\(330\) 0 0
\(331\) −26.1667 −1.43825 −0.719126 0.694880i \(-0.755457\pi\)
−0.719126 + 0.694880i \(0.755457\pi\)
\(332\) 0 0
\(333\) −12.0000 −0.657596
\(334\) 0 0
\(335\) −24.7483 −1.35214
\(336\) 0 0
\(337\) 17.3778 0.946630 0.473315 0.880893i \(-0.343057\pi\)
0.473315 + 0.880893i \(0.343057\pi\)
\(338\) 0 0
\(339\) −4.19181 −0.227668
\(340\) 0 0
\(341\) 2.57193 0.139278
\(342\) 0 0
\(343\) 36.8538 1.98992
\(344\) 0 0
\(345\) 2.56842 0.138279
\(346\) 0 0
\(347\) 12.4029 0.665825 0.332912 0.942958i \(-0.391969\pi\)
0.332912 + 0.942958i \(0.391969\pi\)
\(348\) 0 0
\(349\) −23.8036 −1.27418 −0.637088 0.770791i \(-0.719861\pi\)
−0.637088 + 0.770791i \(0.719861\pi\)
\(350\) 0 0
\(351\) 8.03249 0.428742
\(352\) 0 0
\(353\) −24.2664 −1.29157 −0.645786 0.763518i \(-0.723470\pi\)
−0.645786 + 0.763518i \(0.723470\pi\)
\(354\) 0 0
\(355\) −34.9439 −1.85463
\(356\) 0 0
\(357\) 0.493788 0.0261340
\(358\) 0 0
\(359\) −20.3395 −1.07348 −0.536740 0.843748i \(-0.680344\pi\)
−0.536740 + 0.843748i \(0.680344\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0.523976 0.0275016
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 23.0170 1.20148 0.600739 0.799445i \(-0.294873\pi\)
0.600739 + 0.799445i \(0.294873\pi\)
\(368\) 0 0
\(369\) −13.5588 −0.705841
\(370\) 0 0
\(371\) −19.2038 −0.997010
\(372\) 0 0
\(373\) −18.5890 −0.962499 −0.481250 0.876584i \(-0.659817\pi\)
−0.481250 + 0.876584i \(0.659817\pi\)
\(374\) 0 0
\(375\) −3.67290 −0.189668
\(376\) 0 0
\(377\) 13.1918 0.679413
\(378\) 0 0
\(379\) 10.5816 0.543540 0.271770 0.962362i \(-0.412391\pi\)
0.271770 + 0.962362i \(0.412391\pi\)
\(380\) 0 0
\(381\) −3.01702 −0.154567
\(382\) 0 0
\(383\) 26.0878 1.33302 0.666512 0.745494i \(-0.267786\pi\)
0.666512 + 0.745494i \(0.267786\pi\)
\(384\) 0 0
\(385\) 12.7483 0.649712
\(386\) 0 0
\(387\) −32.0878 −1.63111
\(388\) 0 0
\(389\) −9.46636 −0.479964 −0.239982 0.970777i \(-0.577141\pi\)
−0.239982 + 0.970777i \(0.577141\pi\)
\(390\) 0 0
\(391\) 0.362354 0.0183250
\(392\) 0 0
\(393\) −0.748275 −0.0377455
\(394\) 0 0
\(395\) −32.3070 −1.62554
\(396\) 0 0
\(397\) −25.1667 −1.26308 −0.631540 0.775343i \(-0.717577\pi\)
−0.631540 + 0.775343i \(0.717577\pi\)
\(398\) 0 0
\(399\) 2.45090 0.122698
\(400\) 0 0
\(401\) −16.0650 −0.802247 −0.401123 0.916024i \(-0.631380\pi\)
−0.401123 + 0.916024i \(0.631380\pi\)
\(402\) 0 0
\(403\) −6.88633 −0.343033
\(404\) 0 0
\(405\) 18.0000 0.894427
\(406\) 0 0
\(407\) 4.40294 0.218246
\(408\) 0 0
\(409\) −19.5313 −0.965763 −0.482881 0.875686i \(-0.660410\pi\)
−0.482881 + 0.875686i \(0.660410\pi\)
\(410\) 0 0
\(411\) −7.61408 −0.375575
\(412\) 0 0
\(413\) 24.5542 1.20823
\(414\) 0 0
\(415\) 37.2773 1.82987
\(416\) 0 0
\(417\) −8.47602 −0.415073
\(418\) 0 0
\(419\) −24.9018 −1.21653 −0.608266 0.793733i \(-0.708135\pi\)
−0.608266 + 0.793733i \(0.708135\pi\)
\(420\) 0 0
\(421\) 26.6044 1.29662 0.648310 0.761377i \(-0.275476\pi\)
0.648310 + 0.761377i \(0.275476\pi\)
\(422\) 0 0
\(423\) −34.0650 −1.65630
\(424\) 0 0
\(425\) 0.489189 0.0237292
\(426\) 0 0
\(427\) 7.47022 0.361509
\(428\) 0 0
\(429\) −1.40294 −0.0677348
\(430\) 0 0
\(431\) 32.5948 1.57003 0.785017 0.619474i \(-0.212654\pi\)
0.785017 + 0.619474i \(0.212654\pi\)
\(432\) 0 0
\(433\) 19.7386 0.948577 0.474289 0.880369i \(-0.342705\pi\)
0.474289 + 0.880369i \(0.342705\pi\)
\(434\) 0 0
\(435\) −7.03599 −0.337350
\(436\) 0 0
\(437\) 1.79853 0.0860352
\(438\) 0 0
\(439\) 24.4029 1.16469 0.582345 0.812942i \(-0.302135\pi\)
0.582345 + 0.812942i \(0.302135\pi\)
\(440\) 0 0
\(441\) −40.5519 −1.93104
\(442\) 0 0
\(443\) 16.5948 0.788441 0.394220 0.919016i \(-0.371015\pi\)
0.394220 + 0.919016i \(0.371015\pi\)
\(444\) 0 0
\(445\) −14.8561 −0.704249
\(446\) 0 0
\(447\) 3.30474 0.156309
\(448\) 0 0
\(449\) −20.9520 −0.988788 −0.494394 0.869238i \(-0.664610\pi\)
−0.494394 + 0.869238i \(0.664610\pi\)
\(450\) 0 0
\(451\) 4.97487 0.234258
\(452\) 0 0
\(453\) −0.261386 −0.0122810
\(454\) 0 0
\(455\) −34.1335 −1.60020
\(456\) 0 0
\(457\) 23.3144 1.09060 0.545301 0.838240i \(-0.316415\pi\)
0.545301 + 0.838240i \(0.316415\pi\)
\(458\) 0 0
\(459\) −0.604417 −0.0282118
\(460\) 0 0
\(461\) 20.8059 0.969027 0.484513 0.874784i \(-0.338997\pi\)
0.484513 + 0.874784i \(0.338997\pi\)
\(462\) 0 0
\(463\) 9.29002 0.431744 0.215872 0.976422i \(-0.430741\pi\)
0.215872 + 0.976422i \(0.430741\pi\)
\(464\) 0 0
\(465\) 3.67290 0.170327
\(466\) 0 0
\(467\) −12.0457 −0.557406 −0.278703 0.960377i \(-0.589905\pi\)
−0.278703 + 0.960377i \(0.589905\pi\)
\(468\) 0 0
\(469\) −42.4737 −1.96125
\(470\) 0 0
\(471\) −7.14035 −0.329010
\(472\) 0 0
\(473\) 11.7734 0.541342
\(474\) 0 0
\(475\) 2.42807 0.111408
\(476\) 0 0
\(477\) 11.1895 0.512333
\(478\) 0 0
\(479\) −0.188307 −0.00860396 −0.00430198 0.999991i \(-0.501369\pi\)
−0.00430198 + 0.999991i \(0.501369\pi\)
\(480\) 0 0
\(481\) −11.7889 −0.537526
\(482\) 0 0
\(483\) 4.40801 0.200571
\(484\) 0 0
\(485\) 45.8036 2.07983
\(486\) 0 0
\(487\) 33.3852 1.51283 0.756413 0.654094i \(-0.226950\pi\)
0.756413 + 0.654094i \(0.226950\pi\)
\(488\) 0 0
\(489\) −8.35729 −0.377930
\(490\) 0 0
\(491\) −12.5217 −0.565095 −0.282548 0.959253i \(-0.591180\pi\)
−0.282548 + 0.959253i \(0.591180\pi\)
\(492\) 0 0
\(493\) −0.992638 −0.0447062
\(494\) 0 0
\(495\) −7.42807 −0.333867
\(496\) 0 0
\(497\) −59.9718 −2.69010
\(498\) 0 0
\(499\) −16.3070 −0.730003 −0.365002 0.931007i \(-0.618932\pi\)
−0.365002 + 0.931007i \(0.618932\pi\)
\(500\) 0 0
\(501\) −9.85384 −0.440237
\(502\) 0 0
\(503\) 18.9726 0.845945 0.422973 0.906142i \(-0.360987\pi\)
0.422973 + 0.906142i \(0.360987\pi\)
\(504\) 0 0
\(505\) 20.3070 0.903651
\(506\) 0 0
\(507\) −3.05531 −0.135691
\(508\) 0 0
\(509\) −30.6907 −1.36034 −0.680170 0.733055i \(-0.738094\pi\)
−0.680170 + 0.733055i \(0.738094\pi\)
\(510\) 0 0
\(511\) 10.2974 0.455529
\(512\) 0 0
\(513\) −3.00000 −0.132453
\(514\) 0 0
\(515\) 39.4737 1.73942
\(516\) 0 0
\(517\) 12.4989 0.549699
\(518\) 0 0
\(519\) −1.68946 −0.0741589
\(520\) 0 0
\(521\) −42.1106 −1.84490 −0.922450 0.386116i \(-0.873816\pi\)
−0.922450 + 0.386116i \(0.873816\pi\)
\(522\) 0 0
\(523\) −37.8133 −1.65346 −0.826729 0.562600i \(-0.809801\pi\)
−0.826729 + 0.562600i \(0.809801\pi\)
\(524\) 0 0
\(525\) 5.95095 0.259721
\(526\) 0 0
\(527\) 0.518173 0.0225720
\(528\) 0 0
\(529\) −19.7653 −0.859361
\(530\) 0 0
\(531\) −14.3070 −0.620873
\(532\) 0 0
\(533\) −13.3202 −0.576962
\(534\) 0 0
\(535\) −3.14386 −0.135921
\(536\) 0 0
\(537\) −7.00966 −0.302489
\(538\) 0 0
\(539\) 14.8790 0.640883
\(540\) 0 0
\(541\) 13.5661 0.583253 0.291627 0.956532i \(-0.405804\pi\)
0.291627 + 0.956532i \(0.405804\pi\)
\(542\) 0 0
\(543\) 8.35729 0.358646
\(544\) 0 0
\(545\) −26.0457 −1.11567
\(546\) 0 0
\(547\) −46.4177 −1.98468 −0.992338 0.123552i \(-0.960571\pi\)
−0.992338 + 0.123552i \(0.960571\pi\)
\(548\) 0 0
\(549\) −4.35269 −0.185768
\(550\) 0 0
\(551\) −4.92692 −0.209894
\(552\) 0 0
\(553\) −55.4463 −2.35782
\(554\) 0 0
\(555\) 6.28772 0.266899
\(556\) 0 0
\(557\) −38.2065 −1.61886 −0.809431 0.587214i \(-0.800225\pi\)
−0.809431 + 0.587214i \(0.800225\pi\)
\(558\) 0 0
\(559\) −31.5232 −1.33329
\(560\) 0 0
\(561\) 0.105567 0.00445703
\(562\) 0 0
\(563\) 7.01702 0.295732 0.147866 0.989007i \(-0.452760\pi\)
0.147866 + 0.989007i \(0.452760\pi\)
\(564\) 0 0
\(565\) −21.8036 −0.917284
\(566\) 0 0
\(567\) 30.8921 1.29735
\(568\) 0 0
\(569\) −41.4235 −1.73656 −0.868281 0.496072i \(-0.834775\pi\)
−0.868281 + 0.496072i \(0.834775\pi\)
\(570\) 0 0
\(571\) −12.1381 −0.507962 −0.253981 0.967209i \(-0.581740\pi\)
−0.253981 + 0.967209i \(0.581740\pi\)
\(572\) 0 0
\(573\) 12.6044 0.526557
\(574\) 0 0
\(575\) 4.36695 0.182115
\(576\) 0 0
\(577\) 12.8384 0.534469 0.267234 0.963632i \(-0.413890\pi\)
0.267234 + 0.963632i \(0.413890\pi\)
\(578\) 0 0
\(579\) 6.82872 0.283792
\(580\) 0 0
\(581\) 63.9764 2.65419
\(582\) 0 0
\(583\) −4.10557 −0.170035
\(584\) 0 0
\(585\) 19.8886 0.822294
\(586\) 0 0
\(587\) −4.00000 −0.165098 −0.0825488 0.996587i \(-0.526306\pi\)
−0.0825488 + 0.996587i \(0.526306\pi\)
\(588\) 0 0
\(589\) 2.57193 0.105974
\(590\) 0 0
\(591\) −2.93272 −0.120636
\(592\) 0 0
\(593\) 40.4177 1.65975 0.829877 0.557946i \(-0.188410\pi\)
0.829877 + 0.557946i \(0.188410\pi\)
\(594\) 0 0
\(595\) 2.56842 0.105295
\(596\) 0 0
\(597\) 3.87511 0.158598
\(598\) 0 0
\(599\) −0.679795 −0.0277757 −0.0138878 0.999904i \(-0.504421\pi\)
−0.0138878 + 0.999904i \(0.504421\pi\)
\(600\) 0 0
\(601\) 21.5240 0.877981 0.438991 0.898492i \(-0.355336\pi\)
0.438991 + 0.898492i \(0.355336\pi\)
\(602\) 0 0
\(603\) 24.7483 1.00783
\(604\) 0 0
\(605\) 2.72545 0.110805
\(606\) 0 0
\(607\) −13.3550 −0.542062 −0.271031 0.962571i \(-0.587365\pi\)
−0.271031 + 0.962571i \(0.587365\pi\)
\(608\) 0 0
\(609\) −12.0754 −0.489319
\(610\) 0 0
\(611\) −33.4656 −1.35387
\(612\) 0 0
\(613\) 9.11293 0.368068 0.184034 0.982920i \(-0.441084\pi\)
0.184034 + 0.982920i \(0.441084\pi\)
\(614\) 0 0
\(615\) 7.10447 0.286480
\(616\) 0 0
\(617\) 0.329866 0.0132799 0.00663995 0.999978i \(-0.497886\pi\)
0.00663995 + 0.999978i \(0.497886\pi\)
\(618\) 0 0
\(619\) −8.92112 −0.358570 −0.179285 0.983797i \(-0.557378\pi\)
−0.179285 + 0.983797i \(0.557378\pi\)
\(620\) 0 0
\(621\) −5.39558 −0.216517
\(622\) 0 0
\(623\) −25.4966 −1.02150
\(624\) 0 0
\(625\) −31.2448 −1.24979
\(626\) 0 0
\(627\) 0.523976 0.0209256
\(628\) 0 0
\(629\) 0.887072 0.0353699
\(630\) 0 0
\(631\) −44.6597 −1.77788 −0.888938 0.458028i \(-0.848556\pi\)
−0.888938 + 0.458028i \(0.848556\pi\)
\(632\) 0 0
\(633\) 7.49149 0.297760
\(634\) 0 0
\(635\) −15.6930 −0.622756
\(636\) 0 0
\(637\) −39.8384 −1.57845
\(638\) 0 0
\(639\) 34.9439 1.38236
\(640\) 0 0
\(641\) −6.93272 −0.273826 −0.136913 0.990583i \(-0.543718\pi\)
−0.136913 + 0.990583i \(0.543718\pi\)
\(642\) 0 0
\(643\) −23.4966 −0.926614 −0.463307 0.886198i \(-0.653337\pi\)
−0.463307 + 0.886198i \(0.653337\pi\)
\(644\) 0 0
\(645\) 16.8133 0.662021
\(646\) 0 0
\(647\) −20.9880 −0.825125 −0.412562 0.910929i \(-0.635366\pi\)
−0.412562 + 0.910929i \(0.635366\pi\)
\(648\) 0 0
\(649\) 5.24943 0.206058
\(650\) 0 0
\(651\) 6.30353 0.247055
\(652\) 0 0
\(653\) −44.1335 −1.72708 −0.863538 0.504284i \(-0.831756\pi\)
−0.863538 + 0.504284i \(0.831756\pi\)
\(654\) 0 0
\(655\) −3.89213 −0.152078
\(656\) 0 0
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) −24.0862 −0.938267 −0.469133 0.883127i \(-0.655434\pi\)
−0.469133 + 0.883127i \(0.655434\pi\)
\(660\) 0 0
\(661\) −9.65237 −0.375434 −0.187717 0.982223i \(-0.560109\pi\)
−0.187717 + 0.982223i \(0.560109\pi\)
\(662\) 0 0
\(663\) −0.282655 −0.0109774
\(664\) 0 0
\(665\) 12.7483 0.494357
\(666\) 0 0
\(667\) −8.86120 −0.343107
\(668\) 0 0
\(669\) 0.211133 0.00816289
\(670\) 0 0
\(671\) 1.59706 0.0616536
\(672\) 0 0
\(673\) 24.6176 0.948938 0.474469 0.880272i \(-0.342640\pi\)
0.474469 + 0.880272i \(0.342640\pi\)
\(674\) 0 0
\(675\) −7.28421 −0.280369
\(676\) 0 0
\(677\) 21.5719 0.829077 0.414538 0.910032i \(-0.363943\pi\)
0.414538 + 0.910032i \(0.363943\pi\)
\(678\) 0 0
\(679\) 78.6095 3.01675
\(680\) 0 0
\(681\) 11.0074 0.421803
\(682\) 0 0
\(683\) 0.997701 0.0381759 0.0190880 0.999818i \(-0.493924\pi\)
0.0190880 + 0.999818i \(0.493924\pi\)
\(684\) 0 0
\(685\) −39.6044 −1.51321
\(686\) 0 0
\(687\) −7.50851 −0.286468
\(688\) 0 0
\(689\) 10.9926 0.418786
\(690\) 0 0
\(691\) 9.00230 0.342464 0.171232 0.985231i \(-0.445225\pi\)
0.171232 + 0.985231i \(0.445225\pi\)
\(692\) 0 0
\(693\) −12.7483 −0.484267
\(694\) 0 0
\(695\) −44.0878 −1.67235
\(696\) 0 0
\(697\) 1.00230 0.0379648
\(698\) 0 0
\(699\) −12.7866 −0.483632
\(700\) 0 0
\(701\) −15.0936 −0.570078 −0.285039 0.958516i \(-0.592006\pi\)
−0.285039 + 0.958516i \(0.592006\pi\)
\(702\) 0 0
\(703\) 4.40294 0.166060
\(704\) 0 0
\(705\) 17.8492 0.672241
\(706\) 0 0
\(707\) 34.8515 1.31073
\(708\) 0 0
\(709\) −22.9246 −0.860952 −0.430476 0.902602i \(-0.641654\pi\)
−0.430476 + 0.902602i \(0.641654\pi\)
\(710\) 0 0
\(711\) 32.3070 1.21161
\(712\) 0 0
\(713\) 4.62569 0.173233
\(714\) 0 0
\(715\) −7.29738 −0.272906
\(716\) 0 0
\(717\) 6.64657 0.248221
\(718\) 0 0
\(719\) −6.70032 −0.249880 −0.124940 0.992164i \(-0.539874\pi\)
−0.124940 + 0.992164i \(0.539874\pi\)
\(720\) 0 0
\(721\) 67.7460 2.52299
\(722\) 0 0
\(723\) −12.5410 −0.466405
\(724\) 0 0
\(725\) −11.9629 −0.444291
\(726\) 0 0
\(727\) 18.4583 0.684579 0.342289 0.939595i \(-0.388798\pi\)
0.342289 + 0.939595i \(0.388798\pi\)
\(728\) 0 0
\(729\) −13.2842 −0.492008
\(730\) 0 0
\(731\) 2.37201 0.0877321
\(732\) 0 0
\(733\) 41.8995 1.54759 0.773797 0.633434i \(-0.218355\pi\)
0.773797 + 0.633434i \(0.218355\pi\)
\(734\) 0 0
\(735\) 21.2482 0.783752
\(736\) 0 0
\(737\) −9.08044 −0.334482
\(738\) 0 0
\(739\) 34.8361 1.28147 0.640733 0.767764i \(-0.278631\pi\)
0.640733 + 0.767764i \(0.278631\pi\)
\(740\) 0 0
\(741\) −1.40294 −0.0515385
\(742\) 0 0
\(743\) 25.7889 0.946102 0.473051 0.881035i \(-0.343153\pi\)
0.473051 + 0.881035i \(0.343153\pi\)
\(744\) 0 0
\(745\) 17.1895 0.629775
\(746\) 0 0
\(747\) −37.2773 −1.36391
\(748\) 0 0
\(749\) −5.39558 −0.197150
\(750\) 0 0
\(751\) 33.5011 1.22247 0.611237 0.791447i \(-0.290672\pi\)
0.611237 + 0.791447i \(0.290672\pi\)
\(752\) 0 0
\(753\) −9.69296 −0.353231
\(754\) 0 0
\(755\) −1.35959 −0.0494806
\(756\) 0 0
\(757\) 7.53134 0.273731 0.136866 0.990590i \(-0.456297\pi\)
0.136866 + 0.990590i \(0.456297\pi\)
\(758\) 0 0
\(759\) 0.942386 0.0342064
\(760\) 0 0
\(761\) 38.1969 1.38464 0.692318 0.721593i \(-0.256590\pi\)
0.692318 + 0.721593i \(0.256590\pi\)
\(762\) 0 0
\(763\) −44.7003 −1.61826
\(764\) 0 0
\(765\) −1.49655 −0.0541079
\(766\) 0 0
\(767\) −14.0553 −0.507508
\(768\) 0 0
\(769\) 16.6501 0.600417 0.300208 0.953874i \(-0.402944\pi\)
0.300208 + 0.953874i \(0.402944\pi\)
\(770\) 0 0
\(771\) 14.5445 0.523808
\(772\) 0 0
\(773\) −23.6330 −0.850022 −0.425011 0.905188i \(-0.639730\pi\)
−0.425011 + 0.905188i \(0.639730\pi\)
\(774\) 0 0
\(775\) 6.24483 0.224321
\(776\) 0 0
\(777\) 10.7912 0.387131
\(778\) 0 0
\(779\) 4.97487 0.178243
\(780\) 0 0
\(781\) −12.8214 −0.458784
\(782\) 0 0
\(783\) 14.7808 0.528221
\(784\) 0 0
\(785\) −37.1404 −1.32560
\(786\) 0 0
\(787\) 20.3933 0.726942 0.363471 0.931606i \(-0.381592\pi\)
0.363471 + 0.931606i \(0.381592\pi\)
\(788\) 0 0
\(789\) −10.6222 −0.378160
\(790\) 0 0
\(791\) −37.4200 −1.33050
\(792\) 0 0
\(793\) −4.27611 −0.151849
\(794\) 0 0
\(795\) −5.86304 −0.207941
\(796\) 0 0
\(797\) −9.70262 −0.343685 −0.171842 0.985124i \(-0.554972\pi\)
−0.171842 + 0.985124i \(0.554972\pi\)
\(798\) 0 0
\(799\) 2.51817 0.0890865
\(800\) 0 0
\(801\) 14.8561 0.524916
\(802\) 0 0
\(803\) 2.20147 0.0776883
\(804\) 0 0
\(805\) 22.9281 0.808110
\(806\) 0 0
\(807\) −8.19641 −0.288527
\(808\) 0 0
\(809\) −27.0723 −0.951813 −0.475906 0.879496i \(-0.657880\pi\)
−0.475906 + 0.879496i \(0.657880\pi\)
\(810\) 0 0
\(811\) −51.9548 −1.82438 −0.912190 0.409767i \(-0.865610\pi\)
−0.912190 + 0.409767i \(0.865610\pi\)
\(812\) 0 0
\(813\) 6.46562 0.226759
\(814\) 0 0
\(815\) −43.4702 −1.52270
\(816\) 0 0
\(817\) 11.7734 0.411899
\(818\) 0 0
\(819\) 34.1335 1.19272
\(820\) 0 0
\(821\) −25.6574 −0.895451 −0.447725 0.894171i \(-0.647766\pi\)
−0.447725 + 0.894171i \(0.647766\pi\)
\(822\) 0 0
\(823\) 41.8635 1.45927 0.729635 0.683837i \(-0.239690\pi\)
0.729635 + 0.683837i \(0.239690\pi\)
\(824\) 0 0
\(825\) 1.27225 0.0442941
\(826\) 0 0
\(827\) −6.15582 −0.214059 −0.107029 0.994256i \(-0.534134\pi\)
−0.107029 + 0.994256i \(0.534134\pi\)
\(828\) 0 0
\(829\) −0.247126 −0.00858303 −0.00429151 0.999991i \(-0.501366\pi\)
−0.00429151 + 0.999991i \(0.501366\pi\)
\(830\) 0 0
\(831\) 4.45320 0.154480
\(832\) 0 0
\(833\) 2.99770 0.103864
\(834\) 0 0
\(835\) −51.2545 −1.77373
\(836\) 0 0
\(837\) −7.71579 −0.266697
\(838\) 0 0
\(839\) −16.9320 −0.584557 −0.292278 0.956333i \(-0.594413\pi\)
−0.292278 + 0.956333i \(0.594413\pi\)
\(840\) 0 0
\(841\) −4.72545 −0.162947
\(842\) 0 0
\(843\) −3.73511 −0.128644
\(844\) 0 0
\(845\) −15.8921 −0.546706
\(846\) 0 0
\(847\) 4.67750 0.160721
\(848\) 0 0
\(849\) −6.03829 −0.207234
\(850\) 0 0
\(851\) 7.91882 0.271454
\(852\) 0 0
\(853\) 21.6930 0.742753 0.371376 0.928482i \(-0.378886\pi\)
0.371376 + 0.928482i \(0.378886\pi\)
\(854\) 0 0
\(855\) −7.42807 −0.254035
\(856\) 0 0
\(857\) 34.7328 1.18645 0.593225 0.805037i \(-0.297854\pi\)
0.593225 + 0.805037i \(0.297854\pi\)
\(858\) 0 0
\(859\) 15.3047 0.522191 0.261095 0.965313i \(-0.415916\pi\)
0.261095 + 0.965313i \(0.415916\pi\)
\(860\) 0 0
\(861\) 12.1929 0.415533
\(862\) 0 0
\(863\) −35.8457 −1.22020 −0.610102 0.792323i \(-0.708871\pi\)
−0.610102 + 0.792323i \(0.708871\pi\)
\(864\) 0 0
\(865\) −8.78766 −0.298789
\(866\) 0 0
\(867\) −8.88633 −0.301796
\(868\) 0 0
\(869\) −11.8538 −0.402114
\(870\) 0 0
\(871\) 24.3128 0.823809
\(872\) 0 0
\(873\) −45.8036 −1.55022
\(874\) 0 0
\(875\) −32.7877 −1.10843
\(876\) 0 0
\(877\) −45.4907 −1.53611 −0.768057 0.640382i \(-0.778776\pi\)
−0.768057 + 0.640382i \(0.778776\pi\)
\(878\) 0 0
\(879\) 5.59476 0.188706
\(880\) 0 0
\(881\) 31.3322 1.05561 0.527804 0.849366i \(-0.323016\pi\)
0.527804 + 0.849366i \(0.323016\pi\)
\(882\) 0 0
\(883\) −35.3550 −1.18979 −0.594895 0.803803i \(-0.702806\pi\)
−0.594895 + 0.803803i \(0.702806\pi\)
\(884\) 0 0
\(885\) 7.49655 0.251994
\(886\) 0 0
\(887\) 16.6141 0.557846 0.278923 0.960313i \(-0.410023\pi\)
0.278923 + 0.960313i \(0.410023\pi\)
\(888\) 0 0
\(889\) −26.9327 −0.903295
\(890\) 0 0
\(891\) 6.60442 0.221256
\(892\) 0 0
\(893\) 12.4989 0.418258
\(894\) 0 0
\(895\) −36.4606 −1.21874
\(896\) 0 0
\(897\) −2.52323 −0.0842484
\(898\) 0 0
\(899\) −12.6717 −0.422625
\(900\) 0 0
\(901\) −0.827158 −0.0275566
\(902\) 0 0
\(903\) 28.8554 0.960248
\(904\) 0 0
\(905\) 43.4702 1.44500
\(906\) 0 0
\(907\) −54.8612 −1.82164 −0.910818 0.412808i \(-0.864548\pi\)
−0.910818 + 0.412808i \(0.864548\pi\)
\(908\) 0 0
\(909\) −20.3070 −0.673542
\(910\) 0 0
\(911\) −2.98298 −0.0988304 −0.0494152 0.998778i \(-0.515736\pi\)
−0.0494152 + 0.998778i \(0.515736\pi\)
\(912\) 0 0
\(913\) 13.6775 0.452659
\(914\) 0 0
\(915\) 2.28071 0.0753979
\(916\) 0 0
\(917\) −6.67980 −0.220586
\(918\) 0 0
\(919\) 26.6701 0.879767 0.439883 0.898055i \(-0.355020\pi\)
0.439883 + 0.898055i \(0.355020\pi\)
\(920\) 0 0
\(921\) 15.8538 0.522402
\(922\) 0 0
\(923\) 34.3291 1.12996
\(924\) 0 0
\(925\) 10.6907 0.351507
\(926\) 0 0
\(927\) −39.4737 −1.29649
\(928\) 0 0
\(929\) 43.7595 1.43570 0.717851 0.696197i \(-0.245126\pi\)
0.717851 + 0.696197i \(0.245126\pi\)
\(930\) 0 0
\(931\) 14.8790 0.487638
\(932\) 0 0
\(933\) −4.60902 −0.150892
\(934\) 0 0
\(935\) 0.549103 0.0179576
\(936\) 0 0
\(937\) 47.7823 1.56098 0.780490 0.625168i \(-0.214970\pi\)
0.780490 + 0.625168i \(0.214970\pi\)
\(938\) 0 0
\(939\) 5.33723 0.174174
\(940\) 0 0
\(941\) 36.2664 1.18225 0.591126 0.806579i \(-0.298684\pi\)
0.591126 + 0.806579i \(0.298684\pi\)
\(942\) 0 0
\(943\) 8.94745 0.291369
\(944\) 0 0
\(945\) −38.2448 −1.24410
\(946\) 0 0
\(947\) −35.5468 −1.15512 −0.577558 0.816350i \(-0.695994\pi\)
−0.577558 + 0.816350i \(0.695994\pi\)
\(948\) 0 0
\(949\) −5.89443 −0.191341
\(950\) 0 0
\(951\) −6.29278 −0.204057
\(952\) 0 0
\(953\) 23.2088 0.751808 0.375904 0.926659i \(-0.377332\pi\)
0.375904 + 0.926659i \(0.377332\pi\)
\(954\) 0 0
\(955\) 65.5615 2.12152
\(956\) 0 0
\(957\) −2.58159 −0.0834510
\(958\) 0 0
\(959\) −67.9703 −2.19487
\(960\) 0 0
\(961\) −24.3852 −0.786619
\(962\) 0 0
\(963\) 3.14386 0.101309
\(964\) 0 0
\(965\) 35.5194 1.14341
\(966\) 0 0
\(967\) 6.09591 0.196031 0.0980156 0.995185i \(-0.468751\pi\)
0.0980156 + 0.995185i \(0.468751\pi\)
\(968\) 0 0
\(969\) 0.105567 0.00339129
\(970\) 0 0
\(971\) −34.9822 −1.12263 −0.561317 0.827601i \(-0.689705\pi\)
−0.561317 + 0.827601i \(0.689705\pi\)
\(972\) 0 0
\(973\) −75.6648 −2.42570
\(974\) 0 0
\(975\) −3.40645 −0.109094
\(976\) 0 0
\(977\) −10.9018 −0.348779 −0.174390 0.984677i \(-0.555795\pi\)
−0.174390 + 0.984677i \(0.555795\pi\)
\(978\) 0 0
\(979\) −5.45090 −0.174211
\(980\) 0 0
\(981\) 26.0457 0.831574
\(982\) 0 0
\(983\) 25.4281 0.811030 0.405515 0.914089i \(-0.367092\pi\)
0.405515 + 0.914089i \(0.367092\pi\)
\(984\) 0 0
\(985\) −15.2545 −0.486048
\(986\) 0 0
\(987\) 30.6334 0.975072
\(988\) 0 0
\(989\) 21.1748 0.673319
\(990\) 0 0
\(991\) −49.3925 −1.56901 −0.784503 0.620125i \(-0.787082\pi\)
−0.784503 + 0.620125i \(0.787082\pi\)
\(992\) 0 0
\(993\) −13.7107 −0.435097
\(994\) 0 0
\(995\) 20.1563 0.638997
\(996\) 0 0
\(997\) 20.7409 0.656871 0.328436 0.944526i \(-0.393479\pi\)
0.328436 + 0.944526i \(0.393479\pi\)
\(998\) 0 0
\(999\) −13.2088 −0.417909
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 3344.2.a.q.1.2 3
4.3 odd 2 418.2.a.g.1.2 3
12.11 even 2 3762.2.a.bg.1.1 3
44.43 even 2 4598.2.a.bo.1.2 3
76.75 even 2 7942.2.a.bi.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.g.1.2 3 4.3 odd 2
3344.2.a.q.1.2 3 1.1 even 1 trivial
3762.2.a.bg.1.1 3 12.11 even 2
4598.2.a.bo.1.2 3 44.43 even 2
7942.2.a.bi.1.2 3 76.75 even 2