Properties

Label 1452.4.a.t.1.1
Level $1452$
Weight $4$
Character 1452.1
Self dual yes
Analytic conductor $85.671$
Analytic rank $1$
Dimension $6$
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] = [1452,4,Mod(1,1452)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1452, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1452.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1452.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: \(85.6707733283\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 174x^{4} + 63x^{3} + 7614x^{2} + 1579x - 12611 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 11 \)
Twist minimal: no (minimal twist has level 132)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(8.85455\) of defining polynomial
Character \(\chi\) \(=\) 1452.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -17.0090 q^{5} +13.6696 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -17.0090 q^{5} +13.6696 q^{7} +9.00000 q^{9} -11.1904 q^{13} -51.0271 q^{15} +88.8916 q^{17} -78.1973 q^{19} +41.0089 q^{21} -106.563 q^{23} +164.307 q^{25} +27.0000 q^{27} +233.050 q^{29} -243.874 q^{31} -232.507 q^{35} +368.957 q^{37} -33.5712 q^{39} +35.2574 q^{41} -187.520 q^{43} -153.081 q^{45} -286.165 q^{47} -156.141 q^{49} +266.675 q^{51} -183.810 q^{53} -234.592 q^{57} +257.195 q^{59} +472.784 q^{61} +123.027 q^{63} +190.338 q^{65} -138.217 q^{67} -319.688 q^{69} +838.988 q^{71} -1146.43 q^{73} +492.920 q^{75} +203.650 q^{79} +81.0000 q^{81} -624.221 q^{83} -1511.96 q^{85} +699.149 q^{87} +1082.95 q^{89} -152.969 q^{91} -731.622 q^{93} +1330.06 q^{95} -519.492 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{3} + 13 q^{5} - 23 q^{7} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 18 q^{3} + 13 q^{5} - 23 q^{7} + 54 q^{9} - 66 q^{13} + 39 q^{15} - 44 q^{17} - 270 q^{19} - 69 q^{21} - 124 q^{23} + 131 q^{25} + 162 q^{27} - 141 q^{29} - 253 q^{31} - 884 q^{35} + 288 q^{37} - 198 q^{39} - 428 q^{41} - 1006 q^{43} + 117 q^{45} - 674 q^{47} + 181 q^{49} - 132 q^{51} - 773 q^{53} - 810 q^{57} - 17 q^{59} - 1016 q^{61} - 207 q^{63} - 1220 q^{65} + 1836 q^{67} - 372 q^{69} - 208 q^{71} - 1521 q^{73} + 393 q^{75} - 1425 q^{79} + 486 q^{81} - 3065 q^{83} - 1304 q^{85} - 423 q^{87} + 1444 q^{89} + 1328 q^{91} - 759 q^{93} - 1760 q^{95} - 3887 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −17.0090 −1.52133 −0.760666 0.649143i \(-0.775128\pi\)
−0.760666 + 0.649143i \(0.775128\pi\)
\(6\) 0 0
\(7\) 13.6696 0.738091 0.369045 0.929411i \(-0.379685\pi\)
0.369045 + 0.929411i \(0.379685\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −11.1904 −0.238743 −0.119371 0.992850i \(-0.538088\pi\)
−0.119371 + 0.992850i \(0.538088\pi\)
\(14\) 0 0
\(15\) −51.0271 −0.878342
\(16\) 0 0
\(17\) 88.8916 1.26820 0.634100 0.773251i \(-0.281371\pi\)
0.634100 + 0.773251i \(0.281371\pi\)
\(18\) 0 0
\(19\) −78.1973 −0.944194 −0.472097 0.881547i \(-0.656503\pi\)
−0.472097 + 0.881547i \(0.656503\pi\)
\(20\) 0 0
\(21\) 41.0089 0.426137
\(22\) 0 0
\(23\) −106.563 −0.966081 −0.483041 0.875598i \(-0.660468\pi\)
−0.483041 + 0.875598i \(0.660468\pi\)
\(24\) 0 0
\(25\) 164.307 1.31445
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 233.050 1.49228 0.746142 0.665787i \(-0.231904\pi\)
0.746142 + 0.665787i \(0.231904\pi\)
\(30\) 0 0
\(31\) −243.874 −1.41294 −0.706469 0.707744i \(-0.749713\pi\)
−0.706469 + 0.707744i \(0.749713\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −232.507 −1.12288
\(36\) 0 0
\(37\) 368.957 1.63936 0.819679 0.572823i \(-0.194152\pi\)
0.819679 + 0.572823i \(0.194152\pi\)
\(38\) 0 0
\(39\) −33.5712 −0.137838
\(40\) 0 0
\(41\) 35.2574 0.134300 0.0671498 0.997743i \(-0.478609\pi\)
0.0671498 + 0.997743i \(0.478609\pi\)
\(42\) 0 0
\(43\) −187.520 −0.665035 −0.332518 0.943097i \(-0.607898\pi\)
−0.332518 + 0.943097i \(0.607898\pi\)
\(44\) 0 0
\(45\) −153.081 −0.507111
\(46\) 0 0
\(47\) −286.165 −0.888117 −0.444059 0.895998i \(-0.646462\pi\)
−0.444059 + 0.895998i \(0.646462\pi\)
\(48\) 0 0
\(49\) −156.141 −0.455222
\(50\) 0 0
\(51\) 266.675 0.732195
\(52\) 0 0
\(53\) −183.810 −0.476383 −0.238191 0.971218i \(-0.576555\pi\)
−0.238191 + 0.971218i \(0.576555\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −234.592 −0.545131
\(58\) 0 0
\(59\) 257.195 0.567525 0.283763 0.958895i \(-0.408417\pi\)
0.283763 + 0.958895i \(0.408417\pi\)
\(60\) 0 0
\(61\) 472.784 0.992356 0.496178 0.868221i \(-0.334736\pi\)
0.496178 + 0.868221i \(0.334736\pi\)
\(62\) 0 0
\(63\) 123.027 0.246030
\(64\) 0 0
\(65\) 190.338 0.363207
\(66\) 0 0
\(67\) −138.217 −0.252029 −0.126014 0.992028i \(-0.540219\pi\)
−0.126014 + 0.992028i \(0.540219\pi\)
\(68\) 0 0
\(69\) −319.688 −0.557767
\(70\) 0 0
\(71\) 838.988 1.40239 0.701194 0.712971i \(-0.252651\pi\)
0.701194 + 0.712971i \(0.252651\pi\)
\(72\) 0 0
\(73\) −1146.43 −1.83807 −0.919036 0.394173i \(-0.871031\pi\)
−0.919036 + 0.394173i \(0.871031\pi\)
\(74\) 0 0
\(75\) 492.920 0.758900
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 203.650 0.290030 0.145015 0.989429i \(-0.453677\pi\)
0.145015 + 0.989429i \(0.453677\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −624.221 −0.825509 −0.412754 0.910842i \(-0.635433\pi\)
−0.412754 + 0.910842i \(0.635433\pi\)
\(84\) 0 0
\(85\) −1511.96 −1.92935
\(86\) 0 0
\(87\) 699.149 0.861571
\(88\) 0 0
\(89\) 1082.95 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(90\) 0 0
\(91\) −152.969 −0.176214
\(92\) 0 0
\(93\) −731.622 −0.815760
\(94\) 0 0
\(95\) 1330.06 1.43643
\(96\) 0 0
\(97\) −519.492 −0.543777 −0.271889 0.962329i \(-0.587648\pi\)
−0.271889 + 0.962329i \(0.587648\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −876.418 −0.863434 −0.431717 0.902009i \(-0.642092\pi\)
−0.431717 + 0.902009i \(0.642092\pi\)
\(102\) 0 0
\(103\) −1480.19 −1.41599 −0.707996 0.706217i \(-0.750401\pi\)
−0.707996 + 0.706217i \(0.750401\pi\)
\(104\) 0 0
\(105\) −697.521 −0.648296
\(106\) 0 0
\(107\) −1565.47 −1.41439 −0.707194 0.707020i \(-0.750039\pi\)
−0.707194 + 0.707020i \(0.750039\pi\)
\(108\) 0 0
\(109\) −1753.56 −1.54092 −0.770460 0.637489i \(-0.779973\pi\)
−0.770460 + 0.637489i \(0.779973\pi\)
\(110\) 0 0
\(111\) 1106.87 0.946484
\(112\) 0 0
\(113\) −346.101 −0.288128 −0.144064 0.989568i \(-0.546017\pi\)
−0.144064 + 0.989568i \(0.546017\pi\)
\(114\) 0 0
\(115\) 1812.53 1.46973
\(116\) 0 0
\(117\) −100.714 −0.0795810
\(118\) 0 0
\(119\) 1215.12 0.936046
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 105.772 0.0775379
\(124\) 0 0
\(125\) −668.567 −0.478388
\(126\) 0 0
\(127\) −1436.30 −1.00355 −0.501775 0.864998i \(-0.667319\pi\)
−0.501775 + 0.864998i \(0.667319\pi\)
\(128\) 0 0
\(129\) −562.560 −0.383958
\(130\) 0 0
\(131\) −679.818 −0.453405 −0.226702 0.973964i \(-0.572794\pi\)
−0.226702 + 0.973964i \(0.572794\pi\)
\(132\) 0 0
\(133\) −1068.93 −0.696901
\(134\) 0 0
\(135\) −459.243 −0.292781
\(136\) 0 0
\(137\) 278.540 0.173703 0.0868513 0.996221i \(-0.472320\pi\)
0.0868513 + 0.996221i \(0.472320\pi\)
\(138\) 0 0
\(139\) −2588.80 −1.57971 −0.789853 0.613296i \(-0.789843\pi\)
−0.789853 + 0.613296i \(0.789843\pi\)
\(140\) 0 0
\(141\) −858.496 −0.512755
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −3963.95 −2.27026
\(146\) 0 0
\(147\) −468.424 −0.262823
\(148\) 0 0
\(149\) −1970.11 −1.08321 −0.541603 0.840634i \(-0.682182\pi\)
−0.541603 + 0.840634i \(0.682182\pi\)
\(150\) 0 0
\(151\) −2052.67 −1.10625 −0.553126 0.833098i \(-0.686565\pi\)
−0.553126 + 0.833098i \(0.686565\pi\)
\(152\) 0 0
\(153\) 800.025 0.422733
\(154\) 0 0
\(155\) 4148.06 2.14955
\(156\) 0 0
\(157\) 2424.74 1.23258 0.616291 0.787518i \(-0.288634\pi\)
0.616291 + 0.787518i \(0.288634\pi\)
\(158\) 0 0
\(159\) −551.431 −0.275040
\(160\) 0 0
\(161\) −1456.67 −0.713055
\(162\) 0 0
\(163\) −2168.97 −1.04225 −0.521125 0.853481i \(-0.674487\pi\)
−0.521125 + 0.853481i \(0.674487\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2742.10 1.27060 0.635300 0.772266i \(-0.280877\pi\)
0.635300 + 0.772266i \(0.280877\pi\)
\(168\) 0 0
\(169\) −2071.77 −0.943002
\(170\) 0 0
\(171\) −703.775 −0.314731
\(172\) 0 0
\(173\) 736.423 0.323637 0.161819 0.986821i \(-0.448264\pi\)
0.161819 + 0.986821i \(0.448264\pi\)
\(174\) 0 0
\(175\) 2246.01 0.970185
\(176\) 0 0
\(177\) 771.586 0.327661
\(178\) 0 0
\(179\) −2861.85 −1.19500 −0.597499 0.801870i \(-0.703839\pi\)
−0.597499 + 0.801870i \(0.703839\pi\)
\(180\) 0 0
\(181\) 2696.48 1.10733 0.553667 0.832738i \(-0.313228\pi\)
0.553667 + 0.832738i \(0.313228\pi\)
\(182\) 0 0
\(183\) 1418.35 0.572937
\(184\) 0 0
\(185\) −6275.60 −2.49401
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 369.080 0.142046
\(190\) 0 0
\(191\) −3801.42 −1.44011 −0.720055 0.693917i \(-0.755884\pi\)
−0.720055 + 0.693917i \(0.755884\pi\)
\(192\) 0 0
\(193\) −4745.36 −1.76984 −0.884918 0.465746i \(-0.845786\pi\)
−0.884918 + 0.465746i \(0.845786\pi\)
\(194\) 0 0
\(195\) 571.013 0.209698
\(196\) 0 0
\(197\) 3421.35 1.23737 0.618683 0.785641i \(-0.287667\pi\)
0.618683 + 0.785641i \(0.287667\pi\)
\(198\) 0 0
\(199\) −1065.15 −0.379429 −0.189715 0.981839i \(-0.560756\pi\)
−0.189715 + 0.981839i \(0.560756\pi\)
\(200\) 0 0
\(201\) −414.652 −0.145509
\(202\) 0 0
\(203\) 3185.70 1.10144
\(204\) 0 0
\(205\) −599.694 −0.204314
\(206\) 0 0
\(207\) −959.065 −0.322027
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −410.962 −0.134084 −0.0670422 0.997750i \(-0.521356\pi\)
−0.0670422 + 0.997750i \(0.521356\pi\)
\(212\) 0 0
\(213\) 2516.96 0.809669
\(214\) 0 0
\(215\) 3189.53 1.01174
\(216\) 0 0
\(217\) −3333.67 −1.04288
\(218\) 0 0
\(219\) −3439.29 −1.06121
\(220\) 0 0
\(221\) −994.733 −0.302774
\(222\) 0 0
\(223\) 1013.85 0.304450 0.152225 0.988346i \(-0.451356\pi\)
0.152225 + 0.988346i \(0.451356\pi\)
\(224\) 0 0
\(225\) 1478.76 0.438151
\(226\) 0 0
\(227\) 4161.85 1.21688 0.608440 0.793600i \(-0.291796\pi\)
0.608440 + 0.793600i \(0.291796\pi\)
\(228\) 0 0
\(229\) −2113.51 −0.609889 −0.304944 0.952370i \(-0.598638\pi\)
−0.304944 + 0.952370i \(0.598638\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3236.09 −0.909885 −0.454943 0.890521i \(-0.650340\pi\)
−0.454943 + 0.890521i \(0.650340\pi\)
\(234\) 0 0
\(235\) 4867.39 1.35112
\(236\) 0 0
\(237\) 610.949 0.167449
\(238\) 0 0
\(239\) −6039.49 −1.63457 −0.817285 0.576233i \(-0.804522\pi\)
−0.817285 + 0.576233i \(0.804522\pi\)
\(240\) 0 0
\(241\) 4937.58 1.31974 0.659871 0.751379i \(-0.270611\pi\)
0.659871 + 0.751379i \(0.270611\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 2655.81 0.692545
\(246\) 0 0
\(247\) 875.059 0.225420
\(248\) 0 0
\(249\) −1872.66 −0.476608
\(250\) 0 0
\(251\) −2922.10 −0.734825 −0.367413 0.930058i \(-0.619756\pi\)
−0.367413 + 0.930058i \(0.619756\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −4535.88 −1.11391
\(256\) 0 0
\(257\) −285.451 −0.0692837 −0.0346419 0.999400i \(-0.511029\pi\)
−0.0346419 + 0.999400i \(0.511029\pi\)
\(258\) 0 0
\(259\) 5043.51 1.20999
\(260\) 0 0
\(261\) 2097.45 0.497428
\(262\) 0 0
\(263\) 5421.45 1.27111 0.635553 0.772057i \(-0.280772\pi\)
0.635553 + 0.772057i \(0.280772\pi\)
\(264\) 0 0
\(265\) 3126.43 0.724737
\(266\) 0 0
\(267\) 3248.84 0.744667
\(268\) 0 0
\(269\) 4300.61 0.974769 0.487385 0.873187i \(-0.337951\pi\)
0.487385 + 0.873187i \(0.337951\pi\)
\(270\) 0 0
\(271\) 6661.56 1.49321 0.746607 0.665266i \(-0.231682\pi\)
0.746607 + 0.665266i \(0.231682\pi\)
\(272\) 0 0
\(273\) −458.906 −0.101737
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3054.34 −0.662518 −0.331259 0.943540i \(-0.607473\pi\)
−0.331259 + 0.943540i \(0.607473\pi\)
\(278\) 0 0
\(279\) −2194.87 −0.470979
\(280\) 0 0
\(281\) 2984.12 0.633514 0.316757 0.948507i \(-0.397406\pi\)
0.316757 + 0.948507i \(0.397406\pi\)
\(282\) 0 0
\(283\) −70.2804 −0.0147623 −0.00738116 0.999973i \(-0.502350\pi\)
−0.00738116 + 0.999973i \(0.502350\pi\)
\(284\) 0 0
\(285\) 3990.18 0.829325
\(286\) 0 0
\(287\) 481.956 0.0991253
\(288\) 0 0
\(289\) 2988.72 0.608330
\(290\) 0 0
\(291\) −1558.48 −0.313950
\(292\) 0 0
\(293\) −2750.45 −0.548406 −0.274203 0.961672i \(-0.588414\pi\)
−0.274203 + 0.961672i \(0.588414\pi\)
\(294\) 0 0
\(295\) −4374.64 −0.863395
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1192.48 0.230645
\(300\) 0 0
\(301\) −2563.33 −0.490856
\(302\) 0 0
\(303\) −2629.25 −0.498504
\(304\) 0 0
\(305\) −8041.58 −1.50970
\(306\) 0 0
\(307\) −1627.60 −0.302581 −0.151290 0.988489i \(-0.548343\pi\)
−0.151290 + 0.988489i \(0.548343\pi\)
\(308\) 0 0
\(309\) −4440.56 −0.817523
\(310\) 0 0
\(311\) −2282.57 −0.416183 −0.208091 0.978109i \(-0.566725\pi\)
−0.208091 + 0.978109i \(0.566725\pi\)
\(312\) 0 0
\(313\) 766.776 0.138469 0.0692344 0.997600i \(-0.477944\pi\)
0.0692344 + 0.997600i \(0.477944\pi\)
\(314\) 0 0
\(315\) −2092.56 −0.374294
\(316\) 0 0
\(317\) 5445.98 0.964911 0.482455 0.875921i \(-0.339745\pi\)
0.482455 + 0.875921i \(0.339745\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −4696.40 −0.816597
\(322\) 0 0
\(323\) −6951.08 −1.19743
\(324\) 0 0
\(325\) −1838.66 −0.313816
\(326\) 0 0
\(327\) −5260.67 −0.889650
\(328\) 0 0
\(329\) −3911.77 −0.655511
\(330\) 0 0
\(331\) 7101.94 1.17933 0.589665 0.807648i \(-0.299260\pi\)
0.589665 + 0.807648i \(0.299260\pi\)
\(332\) 0 0
\(333\) 3320.62 0.546453
\(334\) 0 0
\(335\) 2350.94 0.383420
\(336\) 0 0
\(337\) −4390.41 −0.709676 −0.354838 0.934928i \(-0.615464\pi\)
−0.354838 + 0.934928i \(0.615464\pi\)
\(338\) 0 0
\(339\) −1038.30 −0.166351
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −6823.08 −1.07409
\(344\) 0 0
\(345\) 5437.58 0.848550
\(346\) 0 0
\(347\) −5223.56 −0.808114 −0.404057 0.914734i \(-0.632400\pi\)
−0.404057 + 0.914734i \(0.632400\pi\)
\(348\) 0 0
\(349\) 2847.06 0.436675 0.218337 0.975873i \(-0.429937\pi\)
0.218337 + 0.975873i \(0.429937\pi\)
\(350\) 0 0
\(351\) −302.141 −0.0459461
\(352\) 0 0
\(353\) −7358.14 −1.10944 −0.554722 0.832035i \(-0.687176\pi\)
−0.554722 + 0.832035i \(0.687176\pi\)
\(354\) 0 0
\(355\) −14270.4 −2.13350
\(356\) 0 0
\(357\) 3645.35 0.540426
\(358\) 0 0
\(359\) −5011.39 −0.736743 −0.368372 0.929679i \(-0.620085\pi\)
−0.368372 + 0.929679i \(0.620085\pi\)
\(360\) 0 0
\(361\) −744.188 −0.108498
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 19499.6 2.79632
\(366\) 0 0
\(367\) 9522.11 1.35436 0.677180 0.735817i \(-0.263202\pi\)
0.677180 + 0.735817i \(0.263202\pi\)
\(368\) 0 0
\(369\) 317.317 0.0447665
\(370\) 0 0
\(371\) −2512.62 −0.351614
\(372\) 0 0
\(373\) −6551.60 −0.909462 −0.454731 0.890629i \(-0.650265\pi\)
−0.454731 + 0.890629i \(0.650265\pi\)
\(374\) 0 0
\(375\) −2005.70 −0.276197
\(376\) 0 0
\(377\) −2607.92 −0.356272
\(378\) 0 0
\(379\) −8699.52 −1.17906 −0.589531 0.807746i \(-0.700687\pi\)
−0.589531 + 0.807746i \(0.700687\pi\)
\(380\) 0 0
\(381\) −4308.89 −0.579400
\(382\) 0 0
\(383\) 6276.58 0.837385 0.418692 0.908128i \(-0.362489\pi\)
0.418692 + 0.908128i \(0.362489\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1687.68 −0.221678
\(388\) 0 0
\(389\) −7985.49 −1.04082 −0.520412 0.853915i \(-0.674222\pi\)
−0.520412 + 0.853915i \(0.674222\pi\)
\(390\) 0 0
\(391\) −9472.54 −1.22518
\(392\) 0 0
\(393\) −2039.45 −0.261773
\(394\) 0 0
\(395\) −3463.88 −0.441232
\(396\) 0 0
\(397\) 12404.4 1.56816 0.784082 0.620657i \(-0.213134\pi\)
0.784082 + 0.620657i \(0.213134\pi\)
\(398\) 0 0
\(399\) −3206.78 −0.402356
\(400\) 0 0
\(401\) 11474.4 1.42894 0.714469 0.699667i \(-0.246668\pi\)
0.714469 + 0.699667i \(0.246668\pi\)
\(402\) 0 0
\(403\) 2729.05 0.337329
\(404\) 0 0
\(405\) −1377.73 −0.169037
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 6930.77 0.837908 0.418954 0.908007i \(-0.362397\pi\)
0.418954 + 0.908007i \(0.362397\pi\)
\(410\) 0 0
\(411\) 835.619 0.100287
\(412\) 0 0
\(413\) 3515.77 0.418885
\(414\) 0 0
\(415\) 10617.4 1.25587
\(416\) 0 0
\(417\) −7766.40 −0.912044
\(418\) 0 0
\(419\) 13548.6 1.57970 0.789848 0.613303i \(-0.210160\pi\)
0.789848 + 0.613303i \(0.210160\pi\)
\(420\) 0 0
\(421\) −2802.75 −0.324460 −0.162230 0.986753i \(-0.551869\pi\)
−0.162230 + 0.986753i \(0.551869\pi\)
\(422\) 0 0
\(423\) −2575.49 −0.296039
\(424\) 0 0
\(425\) 14605.5 1.66699
\(426\) 0 0
\(427\) 6462.78 0.732449
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7975.96 −0.891388 −0.445694 0.895185i \(-0.647043\pi\)
−0.445694 + 0.895185i \(0.647043\pi\)
\(432\) 0 0
\(433\) −11119.7 −1.23413 −0.617066 0.786912i \(-0.711679\pi\)
−0.617066 + 0.786912i \(0.711679\pi\)
\(434\) 0 0
\(435\) −11891.8 −1.31074
\(436\) 0 0
\(437\) 8332.92 0.912168
\(438\) 0 0
\(439\) −3314.64 −0.360363 −0.180181 0.983633i \(-0.557668\pi\)
−0.180181 + 0.983633i \(0.557668\pi\)
\(440\) 0 0
\(441\) −1405.27 −0.151741
\(442\) 0 0
\(443\) −6135.28 −0.658004 −0.329002 0.944329i \(-0.606712\pi\)
−0.329002 + 0.944329i \(0.606712\pi\)
\(444\) 0 0
\(445\) −18419.9 −1.96222
\(446\) 0 0
\(447\) −5910.33 −0.625389
\(448\) 0 0
\(449\) −16110.8 −1.69335 −0.846676 0.532109i \(-0.821400\pi\)
−0.846676 + 0.532109i \(0.821400\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −6158.01 −0.638695
\(454\) 0 0
\(455\) 2601.85 0.268080
\(456\) 0 0
\(457\) −16431.5 −1.68191 −0.840955 0.541105i \(-0.818006\pi\)
−0.840955 + 0.541105i \(0.818006\pi\)
\(458\) 0 0
\(459\) 2400.07 0.244065
\(460\) 0 0
\(461\) −2394.67 −0.241933 −0.120966 0.992657i \(-0.538599\pi\)
−0.120966 + 0.992657i \(0.538599\pi\)
\(462\) 0 0
\(463\) −16785.6 −1.68487 −0.842433 0.538801i \(-0.818878\pi\)
−0.842433 + 0.538801i \(0.818878\pi\)
\(464\) 0 0
\(465\) 12444.2 1.24104
\(466\) 0 0
\(467\) 2812.41 0.278678 0.139339 0.990245i \(-0.455502\pi\)
0.139339 + 0.990245i \(0.455502\pi\)
\(468\) 0 0
\(469\) −1889.38 −0.186020
\(470\) 0 0
\(471\) 7274.22 0.711632
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −12848.3 −1.24110
\(476\) 0 0
\(477\) −1654.29 −0.158794
\(478\) 0 0
\(479\) 10540.6 1.00545 0.502727 0.864445i \(-0.332330\pi\)
0.502727 + 0.864445i \(0.332330\pi\)
\(480\) 0 0
\(481\) −4128.78 −0.391385
\(482\) 0 0
\(483\) −4370.02 −0.411683
\(484\) 0 0
\(485\) 8836.05 0.827266
\(486\) 0 0
\(487\) 7150.11 0.665302 0.332651 0.943050i \(-0.392057\pi\)
0.332651 + 0.943050i \(0.392057\pi\)
\(488\) 0 0
\(489\) −6506.90 −0.601743
\(490\) 0 0
\(491\) −11064.7 −1.01699 −0.508496 0.861064i \(-0.669798\pi\)
−0.508496 + 0.861064i \(0.669798\pi\)
\(492\) 0 0
\(493\) 20716.2 1.89251
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 11468.7 1.03509
\(498\) 0 0
\(499\) 14846.7 1.33193 0.665963 0.745985i \(-0.268021\pi\)
0.665963 + 0.745985i \(0.268021\pi\)
\(500\) 0 0
\(501\) 8226.30 0.733581
\(502\) 0 0
\(503\) −14684.7 −1.30170 −0.650851 0.759205i \(-0.725588\pi\)
−0.650851 + 0.759205i \(0.725588\pi\)
\(504\) 0 0
\(505\) 14907.0 1.31357
\(506\) 0 0
\(507\) −6215.32 −0.544442
\(508\) 0 0
\(509\) −1352.83 −0.117806 −0.0589028 0.998264i \(-0.518760\pi\)
−0.0589028 + 0.998264i \(0.518760\pi\)
\(510\) 0 0
\(511\) −15671.3 −1.35666
\(512\) 0 0
\(513\) −2111.33 −0.181710
\(514\) 0 0
\(515\) 25176.5 2.15419
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 2209.27 0.186852
\(520\) 0 0
\(521\) −14915.9 −1.25427 −0.627137 0.778909i \(-0.715774\pi\)
−0.627137 + 0.778909i \(0.715774\pi\)
\(522\) 0 0
\(523\) 7593.40 0.634868 0.317434 0.948280i \(-0.397179\pi\)
0.317434 + 0.948280i \(0.397179\pi\)
\(524\) 0 0
\(525\) 6738.03 0.560137
\(526\) 0 0
\(527\) −21678.4 −1.79189
\(528\) 0 0
\(529\) −811.381 −0.0666870
\(530\) 0 0
\(531\) 2314.76 0.189175
\(532\) 0 0
\(533\) −394.545 −0.0320631
\(534\) 0 0
\(535\) 26627.1 2.15175
\(536\) 0 0
\(537\) −8585.55 −0.689932
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 15670.6 1.24534 0.622672 0.782483i \(-0.286047\pi\)
0.622672 + 0.782483i \(0.286047\pi\)
\(542\) 0 0
\(543\) 8089.43 0.639320
\(544\) 0 0
\(545\) 29826.3 2.34425
\(546\) 0 0
\(547\) 7933.16 0.620105 0.310053 0.950719i \(-0.399653\pi\)
0.310053 + 0.950719i \(0.399653\pi\)
\(548\) 0 0
\(549\) 4255.05 0.330785
\(550\) 0 0
\(551\) −18223.8 −1.40901
\(552\) 0 0
\(553\) 2783.82 0.214069
\(554\) 0 0
\(555\) −18826.8 −1.43992
\(556\) 0 0
\(557\) 23335.9 1.77518 0.887588 0.460638i \(-0.152380\pi\)
0.887588 + 0.460638i \(0.152380\pi\)
\(558\) 0 0
\(559\) 2098.42 0.158772
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 15063.1 1.12759 0.563794 0.825915i \(-0.309341\pi\)
0.563794 + 0.825915i \(0.309341\pi\)
\(564\) 0 0
\(565\) 5886.84 0.438338
\(566\) 0 0
\(567\) 1107.24 0.0820101
\(568\) 0 0
\(569\) 15560.3 1.14643 0.573217 0.819403i \(-0.305695\pi\)
0.573217 + 0.819403i \(0.305695\pi\)
\(570\) 0 0
\(571\) 9652.00 0.707397 0.353698 0.935360i \(-0.384924\pi\)
0.353698 + 0.935360i \(0.384924\pi\)
\(572\) 0 0
\(573\) −11404.3 −0.831448
\(574\) 0 0
\(575\) −17509.0 −1.26987
\(576\) 0 0
\(577\) 23301.8 1.68122 0.840612 0.541638i \(-0.182196\pi\)
0.840612 + 0.541638i \(0.182196\pi\)
\(578\) 0 0
\(579\) −14236.1 −1.02182
\(580\) 0 0
\(581\) −8532.88 −0.609300
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 1713.04 0.121069
\(586\) 0 0
\(587\) 8578.17 0.603167 0.301583 0.953440i \(-0.402485\pi\)
0.301583 + 0.953440i \(0.402485\pi\)
\(588\) 0 0
\(589\) 19070.3 1.33409
\(590\) 0 0
\(591\) 10264.0 0.714393
\(592\) 0 0
\(593\) 20171.7 1.39689 0.698443 0.715666i \(-0.253876\pi\)
0.698443 + 0.715666i \(0.253876\pi\)
\(594\) 0 0
\(595\) −20667.9 −1.42404
\(596\) 0 0
\(597\) −3195.45 −0.219064
\(598\) 0 0
\(599\) 9659.62 0.658901 0.329450 0.944173i \(-0.393137\pi\)
0.329450 + 0.944173i \(0.393137\pi\)
\(600\) 0 0
\(601\) −25891.2 −1.75728 −0.878639 0.477486i \(-0.841548\pi\)
−0.878639 + 0.477486i \(0.841548\pi\)
\(602\) 0 0
\(603\) −1243.96 −0.0840096
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −20344.4 −1.36038 −0.680191 0.733035i \(-0.738103\pi\)
−0.680191 + 0.733035i \(0.738103\pi\)
\(608\) 0 0
\(609\) 9557.11 0.635917
\(610\) 0 0
\(611\) 3202.30 0.212032
\(612\) 0 0
\(613\) 23223.2 1.53014 0.765070 0.643948i \(-0.222705\pi\)
0.765070 + 0.643948i \(0.222705\pi\)
\(614\) 0 0
\(615\) −1799.08 −0.117961
\(616\) 0 0
\(617\) 10667.2 0.696023 0.348012 0.937490i \(-0.386857\pi\)
0.348012 + 0.937490i \(0.386857\pi\)
\(618\) 0 0
\(619\) 26297.5 1.70757 0.853784 0.520627i \(-0.174302\pi\)
0.853784 + 0.520627i \(0.174302\pi\)
\(620\) 0 0
\(621\) −2877.19 −0.185922
\(622\) 0 0
\(623\) 14803.5 0.951989
\(624\) 0 0
\(625\) −9166.66 −0.586666
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 32797.2 2.07903
\(630\) 0 0
\(631\) 21004.2 1.32514 0.662569 0.749001i \(-0.269466\pi\)
0.662569 + 0.749001i \(0.269466\pi\)
\(632\) 0 0
\(633\) −1232.89 −0.0774136
\(634\) 0 0
\(635\) 24430.0 1.52673
\(636\) 0 0
\(637\) 1747.28 0.108681
\(638\) 0 0
\(639\) 7550.89 0.467463
\(640\) 0 0
\(641\) 17238.1 1.06219 0.531095 0.847313i \(-0.321781\pi\)
0.531095 + 0.847313i \(0.321781\pi\)
\(642\) 0 0
\(643\) −6609.52 −0.405372 −0.202686 0.979244i \(-0.564967\pi\)
−0.202686 + 0.979244i \(0.564967\pi\)
\(644\) 0 0
\(645\) 9568.59 0.584128
\(646\) 0 0
\(647\) 508.333 0.0308881 0.0154441 0.999881i \(-0.495084\pi\)
0.0154441 + 0.999881i \(0.495084\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −10001.0 −0.602105
\(652\) 0 0
\(653\) 17477.2 1.04737 0.523687 0.851911i \(-0.324556\pi\)
0.523687 + 0.851911i \(0.324556\pi\)
\(654\) 0 0
\(655\) 11563.0 0.689779
\(656\) 0 0
\(657\) −10317.9 −0.612691
\(658\) 0 0
\(659\) −10391.7 −0.614269 −0.307134 0.951666i \(-0.599370\pi\)
−0.307134 + 0.951666i \(0.599370\pi\)
\(660\) 0 0
\(661\) −25556.6 −1.50384 −0.751918 0.659256i \(-0.770871\pi\)
−0.751918 + 0.659256i \(0.770871\pi\)
\(662\) 0 0
\(663\) −2984.20 −0.174806
\(664\) 0 0
\(665\) 18181.4 1.06022
\(666\) 0 0
\(667\) −24834.4 −1.44167
\(668\) 0 0
\(669\) 3041.54 0.175774
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −7311.20 −0.418761 −0.209381 0.977834i \(-0.567145\pi\)
−0.209381 + 0.977834i \(0.567145\pi\)
\(674\) 0 0
\(675\) 4436.28 0.252967
\(676\) 0 0
\(677\) 34051.9 1.93312 0.966558 0.256449i \(-0.0825524\pi\)
0.966558 + 0.256449i \(0.0825524\pi\)
\(678\) 0 0
\(679\) −7101.26 −0.401357
\(680\) 0 0
\(681\) 12485.5 0.702566
\(682\) 0 0
\(683\) 7640.50 0.428046 0.214023 0.976829i \(-0.431343\pi\)
0.214023 + 0.976829i \(0.431343\pi\)
\(684\) 0 0
\(685\) −4737.69 −0.264259
\(686\) 0 0
\(687\) −6340.52 −0.352119
\(688\) 0 0
\(689\) 2056.91 0.113733
\(690\) 0 0
\(691\) 13217.1 0.727645 0.363823 0.931468i \(-0.381471\pi\)
0.363823 + 0.931468i \(0.381471\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 44033.0 2.40326
\(696\) 0 0
\(697\) 3134.09 0.170319
\(698\) 0 0
\(699\) −9708.27 −0.525323
\(700\) 0 0
\(701\) 19328.3 1.04140 0.520699 0.853740i \(-0.325671\pi\)
0.520699 + 0.853740i \(0.325671\pi\)
\(702\) 0 0
\(703\) −28851.5 −1.54787
\(704\) 0 0
\(705\) 14602.2 0.780071
\(706\) 0 0
\(707\) −11980.3 −0.637292
\(708\) 0 0
\(709\) −23199.8 −1.22889 −0.614447 0.788958i \(-0.710621\pi\)
−0.614447 + 0.788958i \(0.710621\pi\)
\(710\) 0 0
\(711\) 1832.85 0.0966767
\(712\) 0 0
\(713\) 25987.9 1.36501
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −18118.5 −0.943720
\(718\) 0 0
\(719\) −11586.0 −0.600952 −0.300476 0.953789i \(-0.597145\pi\)
−0.300476 + 0.953789i \(0.597145\pi\)
\(720\) 0 0
\(721\) −20233.6 −1.04513
\(722\) 0 0
\(723\) 14812.7 0.761953
\(724\) 0 0
\(725\) 38291.6 1.96154
\(726\) 0 0
\(727\) −16861.4 −0.860183 −0.430092 0.902785i \(-0.641519\pi\)
−0.430092 + 0.902785i \(0.641519\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −16669.0 −0.843397
\(732\) 0 0
\(733\) −15077.6 −0.759762 −0.379881 0.925035i \(-0.624035\pi\)
−0.379881 + 0.925035i \(0.624035\pi\)
\(734\) 0 0
\(735\) 7967.43 0.399841
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 2779.99 0.138381 0.0691905 0.997603i \(-0.477958\pi\)
0.0691905 + 0.997603i \(0.477958\pi\)
\(740\) 0 0
\(741\) 2625.18 0.130146
\(742\) 0 0
\(743\) 612.132 0.0302247 0.0151124 0.999886i \(-0.495189\pi\)
0.0151124 + 0.999886i \(0.495189\pi\)
\(744\) 0 0
\(745\) 33509.6 1.64792
\(746\) 0 0
\(747\) −5617.99 −0.275170
\(748\) 0 0
\(749\) −21399.4 −1.04395
\(750\) 0 0
\(751\) 21672.2 1.05304 0.526518 0.850164i \(-0.323497\pi\)
0.526518 + 0.850164i \(0.323497\pi\)
\(752\) 0 0
\(753\) −8766.29 −0.424251
\(754\) 0 0
\(755\) 34913.9 1.68298
\(756\) 0 0
\(757\) −18363.8 −0.881697 −0.440849 0.897582i \(-0.645322\pi\)
−0.440849 + 0.897582i \(0.645322\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9641.68 0.459278 0.229639 0.973276i \(-0.426245\pi\)
0.229639 + 0.973276i \(0.426245\pi\)
\(762\) 0 0
\(763\) −23970.5 −1.13734
\(764\) 0 0
\(765\) −13607.6 −0.643118
\(766\) 0 0
\(767\) −2878.12 −0.135493
\(768\) 0 0
\(769\) −11597.5 −0.543844 −0.271922 0.962319i \(-0.587659\pi\)
−0.271922 + 0.962319i \(0.587659\pi\)
\(770\) 0 0
\(771\) −856.352 −0.0400010
\(772\) 0 0
\(773\) 450.086 0.0209424 0.0104712 0.999945i \(-0.496667\pi\)
0.0104712 + 0.999945i \(0.496667\pi\)
\(774\) 0 0
\(775\) −40070.1 −1.85724
\(776\) 0 0
\(777\) 15130.5 0.698591
\(778\) 0 0
\(779\) −2757.03 −0.126805
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6292.34 0.287190
\(784\) 0 0
\(785\) −41242.5 −1.87517
\(786\) 0 0
\(787\) 13793.5 0.624761 0.312380 0.949957i \(-0.398874\pi\)
0.312380 + 0.949957i \(0.398874\pi\)
\(788\) 0 0
\(789\) 16264.4 0.733874
\(790\) 0 0
\(791\) −4731.07 −0.212664
\(792\) 0 0
\(793\) −5290.64 −0.236918
\(794\) 0 0
\(795\) 9379.30 0.418427
\(796\) 0 0
\(797\) −15831.3 −0.703606 −0.351803 0.936074i \(-0.614431\pi\)
−0.351803 + 0.936074i \(0.614431\pi\)
\(798\) 0 0
\(799\) −25437.7 −1.12631
\(800\) 0 0
\(801\) 9746.53 0.429933
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 24776.6 1.08479
\(806\) 0 0
\(807\) 12901.8 0.562783
\(808\) 0 0
\(809\) −24782.0 −1.07700 −0.538498 0.842627i \(-0.681008\pi\)
−0.538498 + 0.842627i \(0.681008\pi\)
\(810\) 0 0
\(811\) −33326.0 −1.44295 −0.721476 0.692439i \(-0.756536\pi\)
−0.721476 + 0.692439i \(0.756536\pi\)
\(812\) 0 0
\(813\) 19984.7 0.862107
\(814\) 0 0
\(815\) 36892.0 1.58561
\(816\) 0 0
\(817\) 14663.5 0.627922
\(818\) 0 0
\(819\) −1376.72 −0.0587380
\(820\) 0 0
\(821\) 7011.15 0.298040 0.149020 0.988834i \(-0.452388\pi\)
0.149020 + 0.988834i \(0.452388\pi\)
\(822\) 0 0
\(823\) −9745.23 −0.412755 −0.206377 0.978472i \(-0.566167\pi\)
−0.206377 + 0.978472i \(0.566167\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12260.2 −0.515514 −0.257757 0.966210i \(-0.582983\pi\)
−0.257757 + 0.966210i \(0.582983\pi\)
\(828\) 0 0
\(829\) −11845.2 −0.496262 −0.248131 0.968726i \(-0.579816\pi\)
−0.248131 + 0.968726i \(0.579816\pi\)
\(830\) 0 0
\(831\) −9163.01 −0.382505
\(832\) 0 0
\(833\) −13879.7 −0.577313
\(834\) 0 0
\(835\) −46640.4 −1.93300
\(836\) 0 0
\(837\) −6584.60 −0.271920
\(838\) 0 0
\(839\) −35335.1 −1.45400 −0.726998 0.686639i \(-0.759085\pi\)
−0.726998 + 0.686639i \(0.759085\pi\)
\(840\) 0 0
\(841\) 29923.2 1.22691
\(842\) 0 0
\(843\) 8952.35 0.365760
\(844\) 0 0
\(845\) 35238.9 1.43462
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −210.841 −0.00852302
\(850\) 0 0
\(851\) −39317.1 −1.58375
\(852\) 0 0
\(853\) −7408.14 −0.297362 −0.148681 0.988885i \(-0.547503\pi\)
−0.148681 + 0.988885i \(0.547503\pi\)
\(854\) 0 0
\(855\) 11970.5 0.478811
\(856\) 0 0
\(857\) −3199.41 −0.127526 −0.0637630 0.997965i \(-0.520310\pi\)
−0.0637630 + 0.997965i \(0.520310\pi\)
\(858\) 0 0
\(859\) 7809.90 0.310210 0.155105 0.987898i \(-0.450428\pi\)
0.155105 + 0.987898i \(0.450428\pi\)
\(860\) 0 0
\(861\) 1445.87 0.0572300
\(862\) 0 0
\(863\) 26408.7 1.04167 0.520836 0.853657i \(-0.325620\pi\)
0.520836 + 0.853657i \(0.325620\pi\)
\(864\) 0 0
\(865\) −12525.8 −0.492360
\(866\) 0 0
\(867\) 8966.17 0.351219
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 1546.71 0.0601701
\(872\) 0 0
\(873\) −4675.43 −0.181259
\(874\) 0 0
\(875\) −9139.07 −0.353094
\(876\) 0 0
\(877\) 31918.9 1.22899 0.614495 0.788921i \(-0.289360\pi\)
0.614495 + 0.788921i \(0.289360\pi\)
\(878\) 0 0
\(879\) −8251.35 −0.316622
\(880\) 0 0
\(881\) 28928.7 1.10628 0.553140 0.833089i \(-0.313430\pi\)
0.553140 + 0.833089i \(0.313430\pi\)
\(882\) 0 0
\(883\) 6087.67 0.232012 0.116006 0.993249i \(-0.462991\pi\)
0.116006 + 0.993249i \(0.462991\pi\)
\(884\) 0 0
\(885\) −13123.9 −0.498481
\(886\) 0 0
\(887\) −255.738 −0.00968076 −0.00484038 0.999988i \(-0.501541\pi\)
−0.00484038 + 0.999988i \(0.501541\pi\)
\(888\) 0 0
\(889\) −19633.7 −0.740711
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 22377.3 0.838555
\(894\) 0 0
\(895\) 48677.2 1.81799
\(896\) 0 0
\(897\) 3577.44 0.133163
\(898\) 0 0
\(899\) −56834.8 −2.10851
\(900\) 0 0
\(901\) −16339.2 −0.604148
\(902\) 0 0
\(903\) −7689.98 −0.283396
\(904\) 0 0
\(905\) −45864.4 −1.68462
\(906\) 0 0
\(907\) 44242.1 1.61966 0.809832 0.586662i \(-0.199558\pi\)
0.809832 + 0.586662i \(0.199558\pi\)
\(908\) 0 0
\(909\) −7887.76 −0.287811
\(910\) 0 0
\(911\) 11877.9 0.431979 0.215989 0.976396i \(-0.430702\pi\)
0.215989 + 0.976396i \(0.430702\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −24124.8 −0.871628
\(916\) 0 0
\(917\) −9292.86 −0.334654
\(918\) 0 0
\(919\) 20659.7 0.741567 0.370783 0.928719i \(-0.379089\pi\)
0.370783 + 0.928719i \(0.379089\pi\)
\(920\) 0 0
\(921\) −4882.81 −0.174695
\(922\) 0 0
\(923\) −9388.61 −0.334810
\(924\) 0 0
\(925\) 60622.2 2.15486
\(926\) 0 0
\(927\) −13321.7 −0.471997
\(928\) 0 0
\(929\) −32231.3 −1.13829 −0.569146 0.822236i \(-0.692726\pi\)
−0.569146 + 0.822236i \(0.692726\pi\)
\(930\) 0 0
\(931\) 12209.8 0.429818
\(932\) 0 0
\(933\) −6847.72 −0.240283
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −32719.1 −1.14075 −0.570376 0.821384i \(-0.693203\pi\)
−0.570376 + 0.821384i \(0.693203\pi\)
\(938\) 0 0
\(939\) 2300.33 0.0799450
\(940\) 0 0
\(941\) 5457.02 0.189048 0.0945239 0.995523i \(-0.469867\pi\)
0.0945239 + 0.995523i \(0.469867\pi\)
\(942\) 0 0
\(943\) −3757.13 −0.129744
\(944\) 0 0
\(945\) −6277.69 −0.216099
\(946\) 0 0
\(947\) −24764.2 −0.849765 −0.424882 0.905249i \(-0.639685\pi\)
−0.424882 + 0.905249i \(0.639685\pi\)
\(948\) 0 0
\(949\) 12829.0 0.438827
\(950\) 0 0
\(951\) 16337.9 0.557091
\(952\) 0 0
\(953\) −48547.3 −1.65016 −0.825080 0.565017i \(-0.808870\pi\)
−0.825080 + 0.565017i \(0.808870\pi\)
\(954\) 0 0
\(955\) 64658.4 2.19089
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3807.53 0.128208
\(960\) 0 0
\(961\) 29683.6 0.996394
\(962\) 0 0
\(963\) −14089.2 −0.471463
\(964\) 0 0
\(965\) 80713.9 2.69251
\(966\) 0 0
\(967\) −38.9052 −0.00129380 −0.000646902 1.00000i \(-0.500206\pi\)
−0.000646902 1.00000i \(0.500206\pi\)
\(968\) 0 0
\(969\) −20853.2 −0.691334
\(970\) 0 0
\(971\) 1675.99 0.0553915 0.0276958 0.999616i \(-0.491183\pi\)
0.0276958 + 0.999616i \(0.491183\pi\)
\(972\) 0 0
\(973\) −35387.9 −1.16597
\(974\) 0 0
\(975\) −5515.97 −0.181182
\(976\) 0 0
\(977\) −32919.6 −1.07798 −0.538992 0.842311i \(-0.681195\pi\)
−0.538992 + 0.842311i \(0.681195\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −15782.0 −0.513640
\(982\) 0 0
\(983\) 31547.9 1.02362 0.511812 0.859097i \(-0.328974\pi\)
0.511812 + 0.859097i \(0.328974\pi\)
\(984\) 0 0
\(985\) −58193.8 −1.88244
\(986\) 0 0
\(987\) −11735.3 −0.378459
\(988\) 0 0
\(989\) 19982.6 0.642478
\(990\) 0 0
\(991\) −28778.3 −0.922476 −0.461238 0.887276i \(-0.652595\pi\)
−0.461238 + 0.887276i \(0.652595\pi\)
\(992\) 0 0
\(993\) 21305.8 0.680886
\(994\) 0 0
\(995\) 18117.1 0.577238
\(996\) 0 0
\(997\) 4901.93 0.155713 0.0778563 0.996965i \(-0.475192\pi\)
0.0778563 + 0.996965i \(0.475192\pi\)
\(998\) 0 0
\(999\) 9961.85 0.315495
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 1452.4.a.t.1.1 6
11.2 odd 10 132.4.i.c.37.3 yes 12
11.6 odd 10 132.4.i.c.25.3 12
11.10 odd 2 1452.4.a.u.1.1 6
33.2 even 10 396.4.j.c.37.1 12
33.17 even 10 396.4.j.c.289.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
132.4.i.c.25.3 12 11.6 odd 10
132.4.i.c.37.3 yes 12 11.2 odd 10
396.4.j.c.37.1 12 33.2 even 10
396.4.j.c.289.1 12 33.17 even 10
1452.4.a.t.1.1 6 1.1 even 1 trivial
1452.4.a.u.1.1 6 11.10 odd 2