Properties

Label 1232.4.a.o.1.1
Level $1232$
Weight $4$
Character 1232.1
Self dual yes
Analytic conductor $72.690$
Analytic rank $1$
Dimension $2$
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] = [1232,4,Mod(1,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1232.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1232.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: \(72.6903531271\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) 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 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.27492\) of defining polynomial
Character \(\chi\) \(=\) 1232.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.27492 q^{3} -19.8248 q^{5} +7.00000 q^{7} -25.3746 q^{9} +O(q^{10})\) \(q-1.27492 q^{3} -19.8248 q^{5} +7.00000 q^{7} -25.3746 q^{9} +11.0000 q^{11} -35.4502 q^{13} +25.2749 q^{15} +65.1478 q^{17} +83.0997 q^{19} -8.92442 q^{21} +70.7733 q^{23} +268.021 q^{25} +66.7733 q^{27} +243.196 q^{29} +69.4743 q^{31} -14.0241 q^{33} -138.773 q^{35} -131.674 q^{37} +45.1960 q^{39} -482.646 q^{41} -498.296 q^{43} +503.045 q^{45} +504.839 q^{47} +49.0000 q^{49} -83.0581 q^{51} -573.286 q^{53} -218.072 q^{55} -105.945 q^{57} -262.615 q^{59} +884.585 q^{61} -177.622 q^{63} +702.791 q^{65} -396.478 q^{67} -90.2300 q^{69} +414.629 q^{71} -205.547 q^{73} -341.704 q^{75} +77.0000 q^{77} -232.254 q^{79} +599.983 q^{81} +584.453 q^{83} -1291.54 q^{85} -310.055 q^{87} -629.715 q^{89} -248.151 q^{91} -88.5739 q^{93} -1647.43 q^{95} +320.120 q^{97} -279.120 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{3} - 17 q^{5} + 14 q^{7} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{3} - 17 q^{5} + 14 q^{7} - 13 q^{9} + 22 q^{11} - 86 q^{13} + 43 q^{15} - 66 q^{17} + 136 q^{19} + 35 q^{21} - 17 q^{23} + 151 q^{25} - 25 q^{27} + 124 q^{29} + 71 q^{31} + 55 q^{33} - 119 q^{35} - 135 q^{37} - 272 q^{39} - 618 q^{41} - 604 q^{43} + 538 q^{45} - 2 q^{47} + 98 q^{49} - 906 q^{51} + 152 q^{53} - 187 q^{55} + 226 q^{57} + 343 q^{59} + 380 q^{61} - 91 q^{63} + 560 q^{65} - 1027 q^{67} - 641 q^{69} + 1169 q^{71} - 94 q^{73} - 1076 q^{75} + 154 q^{77} - 842 q^{79} - 310 q^{81} + 1486 q^{83} - 1662 q^{85} - 1058 q^{87} - 361 q^{89} - 602 q^{91} - 79 q^{93} - 1498 q^{95} + 225 q^{97} - 143 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.27492 −0.245358 −0.122679 0.992446i \(-0.539149\pi\)
−0.122679 + 0.992446i \(0.539149\pi\)
\(4\) 0 0
\(5\) −19.8248 −1.77318 −0.886590 0.462556i \(-0.846932\pi\)
−0.886590 + 0.462556i \(0.846932\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) −25.3746 −0.939799
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) −35.4502 −0.756316 −0.378158 0.925741i \(-0.623442\pi\)
−0.378158 + 0.925741i \(0.623442\pi\)
\(14\) 0 0
\(15\) 25.2749 0.435064
\(16\) 0 0
\(17\) 65.1478 0.929451 0.464726 0.885455i \(-0.346153\pi\)
0.464726 + 0.885455i \(0.346153\pi\)
\(18\) 0 0
\(19\) 83.0997 1.00339 0.501694 0.865045i \(-0.332710\pi\)
0.501694 + 0.865045i \(0.332710\pi\)
\(20\) 0 0
\(21\) −8.92442 −0.0927366
\(22\) 0 0
\(23\) 70.7733 0.641619 0.320810 0.947144i \(-0.396045\pi\)
0.320810 + 0.947144i \(0.396045\pi\)
\(24\) 0 0
\(25\) 268.021 2.14417
\(26\) 0 0
\(27\) 66.7733 0.475945
\(28\) 0 0
\(29\) 243.196 1.55725 0.778627 0.627487i \(-0.215916\pi\)
0.778627 + 0.627487i \(0.215916\pi\)
\(30\) 0 0
\(31\) 69.4743 0.402514 0.201257 0.979538i \(-0.435497\pi\)
0.201257 + 0.979538i \(0.435497\pi\)
\(32\) 0 0
\(33\) −14.0241 −0.0739782
\(34\) 0 0
\(35\) −138.773 −0.670199
\(36\) 0 0
\(37\) −131.674 −0.585054 −0.292527 0.956257i \(-0.594496\pi\)
−0.292527 + 0.956257i \(0.594496\pi\)
\(38\) 0 0
\(39\) 45.1960 0.185568
\(40\) 0 0
\(41\) −482.646 −1.83846 −0.919228 0.393726i \(-0.871186\pi\)
−0.919228 + 0.393726i \(0.871186\pi\)
\(42\) 0 0
\(43\) −498.296 −1.76719 −0.883597 0.468247i \(-0.844886\pi\)
−0.883597 + 0.468247i \(0.844886\pi\)
\(44\) 0 0
\(45\) 503.045 1.66643
\(46\) 0 0
\(47\) 504.839 1.56677 0.783386 0.621535i \(-0.213491\pi\)
0.783386 + 0.621535i \(0.213491\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −83.0581 −0.228048
\(52\) 0 0
\(53\) −573.286 −1.48579 −0.742895 0.669408i \(-0.766548\pi\)
−0.742895 + 0.669408i \(0.766548\pi\)
\(54\) 0 0
\(55\) −218.072 −0.534634
\(56\) 0 0
\(57\) −105.945 −0.246189
\(58\) 0 0
\(59\) −262.615 −0.579485 −0.289743 0.957105i \(-0.593570\pi\)
−0.289743 + 0.957105i \(0.593570\pi\)
\(60\) 0 0
\(61\) 884.585 1.85671 0.928356 0.371691i \(-0.121222\pi\)
0.928356 + 0.371691i \(0.121222\pi\)
\(62\) 0 0
\(63\) −177.622 −0.355211
\(64\) 0 0
\(65\) 702.791 1.34108
\(66\) 0 0
\(67\) −396.478 −0.722947 −0.361473 0.932382i \(-0.617726\pi\)
−0.361473 + 0.932382i \(0.617726\pi\)
\(68\) 0 0
\(69\) −90.2300 −0.157426
\(70\) 0 0
\(71\) 414.629 0.693062 0.346531 0.938039i \(-0.387360\pi\)
0.346531 + 0.938039i \(0.387360\pi\)
\(72\) 0 0
\(73\) −205.547 −0.329553 −0.164777 0.986331i \(-0.552690\pi\)
−0.164777 + 0.986331i \(0.552690\pi\)
\(74\) 0 0
\(75\) −341.704 −0.526088
\(76\) 0 0
\(77\) 77.0000 0.113961
\(78\) 0 0
\(79\) −232.254 −0.330767 −0.165384 0.986229i \(-0.552886\pi\)
−0.165384 + 0.986229i \(0.552886\pi\)
\(80\) 0 0
\(81\) 599.983 0.823023
\(82\) 0 0
\(83\) 584.453 0.772917 0.386458 0.922307i \(-0.373698\pi\)
0.386458 + 0.922307i \(0.373698\pi\)
\(84\) 0 0
\(85\) −1291.54 −1.64808
\(86\) 0 0
\(87\) −310.055 −0.382085
\(88\) 0 0
\(89\) −629.715 −0.749996 −0.374998 0.927026i \(-0.622357\pi\)
−0.374998 + 0.927026i \(0.622357\pi\)
\(90\) 0 0
\(91\) −248.151 −0.285861
\(92\) 0 0
\(93\) −88.5739 −0.0987601
\(94\) 0 0
\(95\) −1647.43 −1.77919
\(96\) 0 0
\(97\) 320.120 0.335086 0.167543 0.985865i \(-0.446417\pi\)
0.167543 + 0.985865i \(0.446417\pi\)
\(98\) 0 0
\(99\) −279.120 −0.283360
\(100\) 0 0
\(101\) −311.836 −0.307216 −0.153608 0.988132i \(-0.549089\pi\)
−0.153608 + 0.988132i \(0.549089\pi\)
\(102\) 0 0
\(103\) −110.434 −0.105644 −0.0528221 0.998604i \(-0.516822\pi\)
−0.0528221 + 0.998604i \(0.516822\pi\)
\(104\) 0 0
\(105\) 176.924 0.164439
\(106\) 0 0
\(107\) −408.762 −0.369314 −0.184657 0.982803i \(-0.559117\pi\)
−0.184657 + 0.982803i \(0.559117\pi\)
\(108\) 0 0
\(109\) −1400.23 −1.23044 −0.615221 0.788355i \(-0.710933\pi\)
−0.615221 + 0.788355i \(0.710933\pi\)
\(110\) 0 0
\(111\) 167.873 0.143548
\(112\) 0 0
\(113\) −1113.23 −0.926763 −0.463381 0.886159i \(-0.653364\pi\)
−0.463381 + 0.886159i \(0.653364\pi\)
\(114\) 0 0
\(115\) −1403.06 −1.13771
\(116\) 0 0
\(117\) 899.533 0.710785
\(118\) 0 0
\(119\) 456.035 0.351300
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) 615.334 0.451080
\(124\) 0 0
\(125\) −2835.35 −2.02881
\(126\) 0 0
\(127\) 1574.44 1.10007 0.550035 0.835141i \(-0.314614\pi\)
0.550035 + 0.835141i \(0.314614\pi\)
\(128\) 0 0
\(129\) 635.286 0.433595
\(130\) 0 0
\(131\) 2128.09 1.41933 0.709664 0.704540i \(-0.248847\pi\)
0.709664 + 0.704540i \(0.248847\pi\)
\(132\) 0 0
\(133\) 581.698 0.379245
\(134\) 0 0
\(135\) −1323.76 −0.843936
\(136\) 0 0
\(137\) −406.841 −0.253714 −0.126857 0.991921i \(-0.540489\pi\)
−0.126857 + 0.991921i \(0.540489\pi\)
\(138\) 0 0
\(139\) −159.814 −0.0975197 −0.0487599 0.998811i \(-0.515527\pi\)
−0.0487599 + 0.998811i \(0.515527\pi\)
\(140\) 0 0
\(141\) −643.628 −0.384420
\(142\) 0 0
\(143\) −389.952 −0.228038
\(144\) 0 0
\(145\) −4821.30 −2.76129
\(146\) 0 0
\(147\) −62.4709 −0.0350511
\(148\) 0 0
\(149\) −1106.69 −0.608479 −0.304240 0.952596i \(-0.598402\pi\)
−0.304240 + 0.952596i \(0.598402\pi\)
\(150\) 0 0
\(151\) 2950.54 1.59014 0.795070 0.606517i \(-0.207434\pi\)
0.795070 + 0.606517i \(0.207434\pi\)
\(152\) 0 0
\(153\) −1653.10 −0.873498
\(154\) 0 0
\(155\) −1377.31 −0.713730
\(156\) 0 0
\(157\) −3439.07 −1.74820 −0.874101 0.485744i \(-0.838549\pi\)
−0.874101 + 0.485744i \(0.838549\pi\)
\(158\) 0 0
\(159\) 730.892 0.364550
\(160\) 0 0
\(161\) 495.413 0.242509
\(162\) 0 0
\(163\) −3347.56 −1.60860 −0.804298 0.594226i \(-0.797459\pi\)
−0.804298 + 0.594226i \(0.797459\pi\)
\(164\) 0 0
\(165\) 278.024 0.131177
\(166\) 0 0
\(167\) −424.145 −0.196535 −0.0982673 0.995160i \(-0.531330\pi\)
−0.0982673 + 0.995160i \(0.531330\pi\)
\(168\) 0 0
\(169\) −940.286 −0.427986
\(170\) 0 0
\(171\) −2108.62 −0.942984
\(172\) 0 0
\(173\) −326.762 −0.143603 −0.0718014 0.997419i \(-0.522875\pi\)
−0.0718014 + 0.997419i \(0.522875\pi\)
\(174\) 0 0
\(175\) 1876.15 0.810419
\(176\) 0 0
\(177\) 334.813 0.142181
\(178\) 0 0
\(179\) −3982.89 −1.66310 −0.831550 0.555450i \(-0.812546\pi\)
−0.831550 + 0.555450i \(0.812546\pi\)
\(180\) 0 0
\(181\) 704.346 0.289247 0.144623 0.989487i \(-0.453803\pi\)
0.144623 + 0.989487i \(0.453803\pi\)
\(182\) 0 0
\(183\) −1127.77 −0.455559
\(184\) 0 0
\(185\) 2610.40 1.03741
\(186\) 0 0
\(187\) 716.626 0.280240
\(188\) 0 0
\(189\) 467.413 0.179890
\(190\) 0 0
\(191\) 3296.18 1.24871 0.624355 0.781141i \(-0.285362\pi\)
0.624355 + 0.781141i \(0.285362\pi\)
\(192\) 0 0
\(193\) 1754.99 0.654544 0.327272 0.944930i \(-0.393871\pi\)
0.327272 + 0.944930i \(0.393871\pi\)
\(194\) 0 0
\(195\) −896.000 −0.329046
\(196\) 0 0
\(197\) −917.890 −0.331964 −0.165982 0.986129i \(-0.553079\pi\)
−0.165982 + 0.986129i \(0.553079\pi\)
\(198\) 0 0
\(199\) −5300.55 −1.88817 −0.944085 0.329701i \(-0.893052\pi\)
−0.944085 + 0.329701i \(0.893052\pi\)
\(200\) 0 0
\(201\) 505.476 0.177381
\(202\) 0 0
\(203\) 1702.37 0.588587
\(204\) 0 0
\(205\) 9568.34 3.25991
\(206\) 0 0
\(207\) −1795.84 −0.602994
\(208\) 0 0
\(209\) 914.096 0.302533
\(210\) 0 0
\(211\) −3649.87 −1.19084 −0.595420 0.803415i \(-0.703014\pi\)
−0.595420 + 0.803415i \(0.703014\pi\)
\(212\) 0 0
\(213\) −528.617 −0.170048
\(214\) 0 0
\(215\) 9878.59 3.13355
\(216\) 0 0
\(217\) 486.320 0.152136
\(218\) 0 0
\(219\) 262.055 0.0808585
\(220\) 0 0
\(221\) −2309.50 −0.702959
\(222\) 0 0
\(223\) 1529.82 0.459392 0.229696 0.973262i \(-0.426227\pi\)
0.229696 + 0.973262i \(0.426227\pi\)
\(224\) 0 0
\(225\) −6800.92 −2.01509
\(226\) 0 0
\(227\) 256.195 0.0749085 0.0374542 0.999298i \(-0.488075\pi\)
0.0374542 + 0.999298i \(0.488075\pi\)
\(228\) 0 0
\(229\) −591.724 −0.170752 −0.0853760 0.996349i \(-0.527209\pi\)
−0.0853760 + 0.996349i \(0.527209\pi\)
\(230\) 0 0
\(231\) −98.1686 −0.0279611
\(232\) 0 0
\(233\) 1648.22 0.463426 0.231713 0.972784i \(-0.425567\pi\)
0.231713 + 0.972784i \(0.425567\pi\)
\(234\) 0 0
\(235\) −10008.3 −2.77817
\(236\) 0 0
\(237\) 296.105 0.0811564
\(238\) 0 0
\(239\) −1427.79 −0.386428 −0.193214 0.981157i \(-0.561891\pi\)
−0.193214 + 0.981157i \(0.561891\pi\)
\(240\) 0 0
\(241\) 1971.21 0.526874 0.263437 0.964677i \(-0.415144\pi\)
0.263437 + 0.964677i \(0.415144\pi\)
\(242\) 0 0
\(243\) −2567.81 −0.677880
\(244\) 0 0
\(245\) −971.413 −0.253311
\(246\) 0 0
\(247\) −2945.90 −0.758878
\(248\) 0 0
\(249\) −745.130 −0.189641
\(250\) 0 0
\(251\) 4442.18 1.11708 0.558542 0.829476i \(-0.311361\pi\)
0.558542 + 0.829476i \(0.311361\pi\)
\(252\) 0 0
\(253\) 778.506 0.193456
\(254\) 0 0
\(255\) 1646.61 0.404371
\(256\) 0 0
\(257\) 7113.32 1.72652 0.863262 0.504757i \(-0.168418\pi\)
0.863262 + 0.504757i \(0.168418\pi\)
\(258\) 0 0
\(259\) −921.715 −0.221130
\(260\) 0 0
\(261\) −6171.00 −1.46351
\(262\) 0 0
\(263\) −1000.42 −0.234557 −0.117279 0.993099i \(-0.537417\pi\)
−0.117279 + 0.993099i \(0.537417\pi\)
\(264\) 0 0
\(265\) 11365.2 2.63457
\(266\) 0 0
\(267\) 802.835 0.184018
\(268\) 0 0
\(269\) 187.128 0.0424141 0.0212071 0.999775i \(-0.493249\pi\)
0.0212071 + 0.999775i \(0.493249\pi\)
\(270\) 0 0
\(271\) 605.814 0.135795 0.0678977 0.997692i \(-0.478371\pi\)
0.0678977 + 0.997692i \(0.478371\pi\)
\(272\) 0 0
\(273\) 316.372 0.0701382
\(274\) 0 0
\(275\) 2948.23 0.646490
\(276\) 0 0
\(277\) −1454.30 −0.315452 −0.157726 0.987483i \(-0.550416\pi\)
−0.157726 + 0.987483i \(0.550416\pi\)
\(278\) 0 0
\(279\) −1762.88 −0.378283
\(280\) 0 0
\(281\) −5940.16 −1.26107 −0.630535 0.776161i \(-0.717164\pi\)
−0.630535 + 0.776161i \(0.717164\pi\)
\(282\) 0 0
\(283\) −4483.67 −0.941790 −0.470895 0.882189i \(-0.656069\pi\)
−0.470895 + 0.882189i \(0.656069\pi\)
\(284\) 0 0
\(285\) 2100.34 0.436538
\(286\) 0 0
\(287\) −3378.52 −0.694871
\(288\) 0 0
\(289\) −668.758 −0.136120
\(290\) 0 0
\(291\) −408.127 −0.0822159
\(292\) 0 0
\(293\) 3239.72 0.645961 0.322980 0.946406i \(-0.395315\pi\)
0.322980 + 0.946406i \(0.395315\pi\)
\(294\) 0 0
\(295\) 5206.29 1.02753
\(296\) 0 0
\(297\) 734.506 0.143503
\(298\) 0 0
\(299\) −2508.92 −0.485267
\(300\) 0 0
\(301\) −3488.07 −0.667937
\(302\) 0 0
\(303\) 397.565 0.0753778
\(304\) 0 0
\(305\) −17536.7 −3.29229
\(306\) 0 0
\(307\) −1296.15 −0.240961 −0.120481 0.992716i \(-0.538444\pi\)
−0.120481 + 0.992716i \(0.538444\pi\)
\(308\) 0 0
\(309\) 140.794 0.0259206
\(310\) 0 0
\(311\) −4513.53 −0.822954 −0.411477 0.911420i \(-0.634987\pi\)
−0.411477 + 0.911420i \(0.634987\pi\)
\(312\) 0 0
\(313\) −6918.74 −1.24943 −0.624713 0.780855i \(-0.714784\pi\)
−0.624713 + 0.780855i \(0.714784\pi\)
\(314\) 0 0
\(315\) 3521.31 0.629853
\(316\) 0 0
\(317\) 3844.49 0.681161 0.340580 0.940215i \(-0.389376\pi\)
0.340580 + 0.940215i \(0.389376\pi\)
\(318\) 0 0
\(319\) 2675.16 0.469530
\(320\) 0 0
\(321\) 521.138 0.0906140
\(322\) 0 0
\(323\) 5413.76 0.932600
\(324\) 0 0
\(325\) −9501.38 −1.62167
\(326\) 0 0
\(327\) 1785.18 0.301899
\(328\) 0 0
\(329\) 3533.87 0.592184
\(330\) 0 0
\(331\) −7497.78 −1.24506 −0.622530 0.782596i \(-0.713895\pi\)
−0.622530 + 0.782596i \(0.713895\pi\)
\(332\) 0 0
\(333\) 3341.16 0.549834
\(334\) 0 0
\(335\) 7860.07 1.28191
\(336\) 0 0
\(337\) −9372.82 −1.51504 −0.757522 0.652809i \(-0.773590\pi\)
−0.757522 + 0.652809i \(0.773590\pi\)
\(338\) 0 0
\(339\) 1419.28 0.227389
\(340\) 0 0
\(341\) 764.217 0.121363
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) 1788.79 0.279145
\(346\) 0 0
\(347\) 8942.10 1.38339 0.691696 0.722189i \(-0.256864\pi\)
0.691696 + 0.722189i \(0.256864\pi\)
\(348\) 0 0
\(349\) 6516.32 0.999457 0.499728 0.866182i \(-0.333433\pi\)
0.499728 + 0.866182i \(0.333433\pi\)
\(350\) 0 0
\(351\) −2367.12 −0.359965
\(352\) 0 0
\(353\) 11607.3 1.75012 0.875060 0.484014i \(-0.160822\pi\)
0.875060 + 0.484014i \(0.160822\pi\)
\(354\) 0 0
\(355\) −8219.91 −1.22892
\(356\) 0 0
\(357\) −581.407 −0.0861941
\(358\) 0 0
\(359\) 7431.08 1.09247 0.546236 0.837631i \(-0.316060\pi\)
0.546236 + 0.837631i \(0.316060\pi\)
\(360\) 0 0
\(361\) 46.5550 0.00678743
\(362\) 0 0
\(363\) −154.265 −0.0223053
\(364\) 0 0
\(365\) 4074.91 0.584357
\(366\) 0 0
\(367\) −4233.75 −0.602179 −0.301090 0.953596i \(-0.597350\pi\)
−0.301090 + 0.953596i \(0.597350\pi\)
\(368\) 0 0
\(369\) 12246.9 1.72778
\(370\) 0 0
\(371\) −4013.00 −0.561576
\(372\) 0 0
\(373\) −5807.15 −0.806120 −0.403060 0.915173i \(-0.632053\pi\)
−0.403060 + 0.915173i \(0.632053\pi\)
\(374\) 0 0
\(375\) 3614.84 0.497785
\(376\) 0 0
\(377\) −8621.34 −1.17778
\(378\) 0 0
\(379\) −11312.8 −1.53324 −0.766622 0.642098i \(-0.778064\pi\)
−0.766622 + 0.642098i \(0.778064\pi\)
\(380\) 0 0
\(381\) −2007.28 −0.269911
\(382\) 0 0
\(383\) −13099.0 −1.74759 −0.873794 0.486296i \(-0.838348\pi\)
−0.873794 + 0.486296i \(0.838348\pi\)
\(384\) 0 0
\(385\) −1526.51 −0.202073
\(386\) 0 0
\(387\) 12644.0 1.66081
\(388\) 0 0
\(389\) −62.8377 −0.00819022 −0.00409511 0.999992i \(-0.501304\pi\)
−0.00409511 + 0.999992i \(0.501304\pi\)
\(390\) 0 0
\(391\) 4610.73 0.596354
\(392\) 0 0
\(393\) −2713.14 −0.348244
\(394\) 0 0
\(395\) 4604.38 0.586510
\(396\) 0 0
\(397\) −276.394 −0.0349416 −0.0174708 0.999847i \(-0.505561\pi\)
−0.0174708 + 0.999847i \(0.505561\pi\)
\(398\) 0 0
\(399\) −741.616 −0.0930508
\(400\) 0 0
\(401\) −7970.49 −0.992587 −0.496293 0.868155i \(-0.665306\pi\)
−0.496293 + 0.868155i \(0.665306\pi\)
\(402\) 0 0
\(403\) −2462.87 −0.304428
\(404\) 0 0
\(405\) −11894.5 −1.45937
\(406\) 0 0
\(407\) −1448.41 −0.176400
\(408\) 0 0
\(409\) 908.394 0.109822 0.0549110 0.998491i \(-0.482512\pi\)
0.0549110 + 0.998491i \(0.482512\pi\)
\(410\) 0 0
\(411\) 518.689 0.0622507
\(412\) 0 0
\(413\) −1838.31 −0.219025
\(414\) 0 0
\(415\) −11586.6 −1.37052
\(416\) 0 0
\(417\) 203.750 0.0239272
\(418\) 0 0
\(419\) 36.0397 0.00420204 0.00210102 0.999998i \(-0.499331\pi\)
0.00210102 + 0.999998i \(0.499331\pi\)
\(420\) 0 0
\(421\) −8332.89 −0.964656 −0.482328 0.875991i \(-0.660209\pi\)
−0.482328 + 0.875991i \(0.660209\pi\)
\(422\) 0 0
\(423\) −12810.1 −1.47245
\(424\) 0 0
\(425\) 17461.0 1.99290
\(426\) 0 0
\(427\) 6192.09 0.701771
\(428\) 0 0
\(429\) 497.156 0.0559509
\(430\) 0 0
\(431\) 11169.9 1.24834 0.624171 0.781288i \(-0.285437\pi\)
0.624171 + 0.781288i \(0.285437\pi\)
\(432\) 0 0
\(433\) −10710.9 −1.18875 −0.594377 0.804186i \(-0.702601\pi\)
−0.594377 + 0.804186i \(0.702601\pi\)
\(434\) 0 0
\(435\) 6146.76 0.677505
\(436\) 0 0
\(437\) 5881.23 0.643793
\(438\) 0 0
\(439\) −3900.57 −0.424064 −0.212032 0.977263i \(-0.568008\pi\)
−0.212032 + 0.977263i \(0.568008\pi\)
\(440\) 0 0
\(441\) −1243.35 −0.134257
\(442\) 0 0
\(443\) −3396.29 −0.364250 −0.182125 0.983275i \(-0.558298\pi\)
−0.182125 + 0.983275i \(0.558298\pi\)
\(444\) 0 0
\(445\) 12483.9 1.32988
\(446\) 0 0
\(447\) 1410.94 0.149295
\(448\) 0 0
\(449\) 17354.1 1.82403 0.912014 0.410160i \(-0.134527\pi\)
0.912014 + 0.410160i \(0.134527\pi\)
\(450\) 0 0
\(451\) −5309.11 −0.554315
\(452\) 0 0
\(453\) −3761.69 −0.390154
\(454\) 0 0
\(455\) 4919.54 0.506882
\(456\) 0 0
\(457\) −3040.01 −0.311172 −0.155586 0.987822i \(-0.549727\pi\)
−0.155586 + 0.987822i \(0.549727\pi\)
\(458\) 0 0
\(459\) 4350.13 0.442368
\(460\) 0 0
\(461\) 8808.45 0.889914 0.444957 0.895552i \(-0.353219\pi\)
0.444957 + 0.895552i \(0.353219\pi\)
\(462\) 0 0
\(463\) −9797.22 −0.983402 −0.491701 0.870764i \(-0.663625\pi\)
−0.491701 + 0.870764i \(0.663625\pi\)
\(464\) 0 0
\(465\) 1755.96 0.175119
\(466\) 0 0
\(467\) 5800.53 0.574768 0.287384 0.957815i \(-0.407214\pi\)
0.287384 + 0.957815i \(0.407214\pi\)
\(468\) 0 0
\(469\) −2775.34 −0.273248
\(470\) 0 0
\(471\) 4384.53 0.428935
\(472\) 0 0
\(473\) −5481.25 −0.532829
\(474\) 0 0
\(475\) 22272.4 2.15143
\(476\) 0 0
\(477\) 14546.9 1.39634
\(478\) 0 0
\(479\) 19752.6 1.88418 0.942089 0.335363i \(-0.108859\pi\)
0.942089 + 0.335363i \(0.108859\pi\)
\(480\) 0 0
\(481\) 4667.85 0.442486
\(482\) 0 0
\(483\) −631.610 −0.0595016
\(484\) 0 0
\(485\) −6346.31 −0.594167
\(486\) 0 0
\(487\) 1315.06 0.122363 0.0611817 0.998127i \(-0.480513\pi\)
0.0611817 + 0.998127i \(0.480513\pi\)
\(488\) 0 0
\(489\) 4267.86 0.394682
\(490\) 0 0
\(491\) 751.005 0.0690273 0.0345136 0.999404i \(-0.489012\pi\)
0.0345136 + 0.999404i \(0.489012\pi\)
\(492\) 0 0
\(493\) 15843.7 1.44739
\(494\) 0 0
\(495\) 5533.49 0.502449
\(496\) 0 0
\(497\) 2902.40 0.261953
\(498\) 0 0
\(499\) −7619.81 −0.683586 −0.341793 0.939775i \(-0.611034\pi\)
−0.341793 + 0.939775i \(0.611034\pi\)
\(500\) 0 0
\(501\) 540.749 0.0482213
\(502\) 0 0
\(503\) −9779.34 −0.866877 −0.433439 0.901183i \(-0.642700\pi\)
−0.433439 + 0.901183i \(0.642700\pi\)
\(504\) 0 0
\(505\) 6182.06 0.544749
\(506\) 0 0
\(507\) 1198.79 0.105010
\(508\) 0 0
\(509\) 184.110 0.0160325 0.00801625 0.999968i \(-0.497448\pi\)
0.00801625 + 0.999968i \(0.497448\pi\)
\(510\) 0 0
\(511\) −1438.83 −0.124559
\(512\) 0 0
\(513\) 5548.84 0.477558
\(514\) 0 0
\(515\) 2189.32 0.187326
\(516\) 0 0
\(517\) 5553.23 0.472400
\(518\) 0 0
\(519\) 416.595 0.0352341
\(520\) 0 0
\(521\) −20360.9 −1.71215 −0.856073 0.516855i \(-0.827102\pi\)
−0.856073 + 0.516855i \(0.827102\pi\)
\(522\) 0 0
\(523\) 12075.6 1.00961 0.504806 0.863233i \(-0.331564\pi\)
0.504806 + 0.863233i \(0.331564\pi\)
\(524\) 0 0
\(525\) −2391.93 −0.198843
\(526\) 0 0
\(527\) 4526.10 0.374118
\(528\) 0 0
\(529\) −7158.15 −0.588325
\(530\) 0 0
\(531\) 6663.76 0.544600
\(532\) 0 0
\(533\) 17109.9 1.39045
\(534\) 0 0
\(535\) 8103.61 0.654859
\(536\) 0 0
\(537\) 5077.85 0.408055
\(538\) 0 0
\(539\) 539.000 0.0430730
\(540\) 0 0
\(541\) −2616.08 −0.207900 −0.103950 0.994583i \(-0.533148\pi\)
−0.103950 + 0.994583i \(0.533148\pi\)
\(542\) 0 0
\(543\) −897.983 −0.0709690
\(544\) 0 0
\(545\) 27759.3 2.18179
\(546\) 0 0
\(547\) 13719.3 1.07239 0.536195 0.844094i \(-0.319861\pi\)
0.536195 + 0.844094i \(0.319861\pi\)
\(548\) 0 0
\(549\) −22446.0 −1.74494
\(550\) 0 0
\(551\) 20209.5 1.56253
\(552\) 0 0
\(553\) −1625.78 −0.125018
\(554\) 0 0
\(555\) −3328.04 −0.254536
\(556\) 0 0
\(557\) 10510.1 0.799510 0.399755 0.916622i \(-0.369095\pi\)
0.399755 + 0.916622i \(0.369095\pi\)
\(558\) 0 0
\(559\) 17664.7 1.33656
\(560\) 0 0
\(561\) −913.639 −0.0687591
\(562\) 0 0
\(563\) −7040.40 −0.527029 −0.263515 0.964655i \(-0.584882\pi\)
−0.263515 + 0.964655i \(0.584882\pi\)
\(564\) 0 0
\(565\) 22069.6 1.64332
\(566\) 0 0
\(567\) 4199.88 0.311073
\(568\) 0 0
\(569\) −11806.9 −0.869900 −0.434950 0.900455i \(-0.643234\pi\)
−0.434950 + 0.900455i \(0.643234\pi\)
\(570\) 0 0
\(571\) 16432.1 1.20431 0.602155 0.798379i \(-0.294309\pi\)
0.602155 + 0.798379i \(0.294309\pi\)
\(572\) 0 0
\(573\) −4202.36 −0.306381
\(574\) 0 0
\(575\) 18968.7 1.37574
\(576\) 0 0
\(577\) 9781.38 0.705727 0.352863 0.935675i \(-0.385208\pi\)
0.352863 + 0.935675i \(0.385208\pi\)
\(578\) 0 0
\(579\) −2237.47 −0.160598
\(580\) 0 0
\(581\) 4091.17 0.292135
\(582\) 0 0
\(583\) −6306.14 −0.447982
\(584\) 0 0
\(585\) −17833.0 −1.26035
\(586\) 0 0
\(587\) −8510.75 −0.598426 −0.299213 0.954186i \(-0.596724\pi\)
−0.299213 + 0.954186i \(0.596724\pi\)
\(588\) 0 0
\(589\) 5773.29 0.403878
\(590\) 0 0
\(591\) 1170.23 0.0814501
\(592\) 0 0
\(593\) 12400.7 0.858746 0.429373 0.903127i \(-0.358735\pi\)
0.429373 + 0.903127i \(0.358735\pi\)
\(594\) 0 0
\(595\) −9040.78 −0.622917
\(596\) 0 0
\(597\) 6757.76 0.463278
\(598\) 0 0
\(599\) −1568.75 −0.107007 −0.0535037 0.998568i \(-0.517039\pi\)
−0.0535037 + 0.998568i \(0.517039\pi\)
\(600\) 0 0
\(601\) −3991.41 −0.270903 −0.135452 0.990784i \(-0.543249\pi\)
−0.135452 + 0.990784i \(0.543249\pi\)
\(602\) 0 0
\(603\) 10060.5 0.679425
\(604\) 0 0
\(605\) −2398.79 −0.161198
\(606\) 0 0
\(607\) −9689.34 −0.647905 −0.323952 0.946073i \(-0.605012\pi\)
−0.323952 + 0.946073i \(0.605012\pi\)
\(608\) 0 0
\(609\) −2170.38 −0.144414
\(610\) 0 0
\(611\) −17896.6 −1.18498
\(612\) 0 0
\(613\) −5475.09 −0.360745 −0.180373 0.983598i \(-0.557730\pi\)
−0.180373 + 0.983598i \(0.557730\pi\)
\(614\) 0 0
\(615\) −12198.8 −0.799845
\(616\) 0 0
\(617\) 12908.8 0.842282 0.421141 0.906995i \(-0.361630\pi\)
0.421141 + 0.906995i \(0.361630\pi\)
\(618\) 0 0
\(619\) 13016.6 0.845206 0.422603 0.906315i \(-0.361116\pi\)
0.422603 + 0.906315i \(0.361116\pi\)
\(620\) 0 0
\(621\) 4725.76 0.305376
\(622\) 0 0
\(623\) −4408.01 −0.283472
\(624\) 0 0
\(625\) 22707.5 1.45328
\(626\) 0 0
\(627\) −1165.40 −0.0742288
\(628\) 0 0
\(629\) −8578.25 −0.543779
\(630\) 0 0
\(631\) 16555.5 1.04447 0.522236 0.852801i \(-0.325098\pi\)
0.522236 + 0.852801i \(0.325098\pi\)
\(632\) 0 0
\(633\) 4653.28 0.292182
\(634\) 0 0
\(635\) −31212.9 −1.95062
\(636\) 0 0
\(637\) −1737.06 −0.108045
\(638\) 0 0
\(639\) −10521.0 −0.651339
\(640\) 0 0
\(641\) −27993.2 −1.72491 −0.862454 0.506136i \(-0.831074\pi\)
−0.862454 + 0.506136i \(0.831074\pi\)
\(642\) 0 0
\(643\) −25489.3 −1.56330 −0.781648 0.623720i \(-0.785621\pi\)
−0.781648 + 0.623720i \(0.785621\pi\)
\(644\) 0 0
\(645\) −12594.4 −0.768842
\(646\) 0 0
\(647\) 21121.0 1.28339 0.641693 0.766962i \(-0.278232\pi\)
0.641693 + 0.766962i \(0.278232\pi\)
\(648\) 0 0
\(649\) −2888.77 −0.174721
\(650\) 0 0
\(651\) −620.017 −0.0373278
\(652\) 0 0
\(653\) −11406.7 −0.683583 −0.341792 0.939776i \(-0.611034\pi\)
−0.341792 + 0.939776i \(0.611034\pi\)
\(654\) 0 0
\(655\) −42188.9 −2.51672
\(656\) 0 0
\(657\) 5215.66 0.309714
\(658\) 0 0
\(659\) −27601.4 −1.63156 −0.815779 0.578364i \(-0.803691\pi\)
−0.815779 + 0.578364i \(0.803691\pi\)
\(660\) 0 0
\(661\) −26538.2 −1.56160 −0.780799 0.624782i \(-0.785188\pi\)
−0.780799 + 0.624782i \(0.785188\pi\)
\(662\) 0 0
\(663\) 2944.42 0.172477
\(664\) 0 0
\(665\) −11532.0 −0.672470
\(666\) 0 0
\(667\) 17211.8 0.999164
\(668\) 0 0
\(669\) −1950.39 −0.112715
\(670\) 0 0
\(671\) 9730.43 0.559820
\(672\) 0 0
\(673\) 16791.6 0.961767 0.480883 0.876785i \(-0.340316\pi\)
0.480883 + 0.876785i \(0.340316\pi\)
\(674\) 0 0
\(675\) 17896.6 1.02051
\(676\) 0 0
\(677\) 6405.21 0.363622 0.181811 0.983333i \(-0.441804\pi\)
0.181811 + 0.983333i \(0.441804\pi\)
\(678\) 0 0
\(679\) 2240.84 0.126650
\(680\) 0 0
\(681\) −326.627 −0.0183794
\(682\) 0 0
\(683\) −6951.93 −0.389470 −0.194735 0.980856i \(-0.562385\pi\)
−0.194735 + 0.980856i \(0.562385\pi\)
\(684\) 0 0
\(685\) 8065.53 0.449880
\(686\) 0 0
\(687\) 754.399 0.0418953
\(688\) 0 0
\(689\) 20323.1 1.12373
\(690\) 0 0
\(691\) 14914.5 0.821090 0.410545 0.911840i \(-0.365338\pi\)
0.410545 + 0.911840i \(0.365338\pi\)
\(692\) 0 0
\(693\) −1953.84 −0.107100
\(694\) 0 0
\(695\) 3168.27 0.172920
\(696\) 0 0
\(697\) −31443.4 −1.70875
\(698\) 0 0
\(699\) −2101.34 −0.113705
\(700\) 0 0
\(701\) −10125.3 −0.545544 −0.272772 0.962079i \(-0.587941\pi\)
−0.272772 + 0.962079i \(0.587941\pi\)
\(702\) 0 0
\(703\) −10942.0 −0.587036
\(704\) 0 0
\(705\) 12759.8 0.681646
\(706\) 0 0
\(707\) −2182.85 −0.116117
\(708\) 0 0
\(709\) −29376.9 −1.55609 −0.778047 0.628206i \(-0.783790\pi\)
−0.778047 + 0.628206i \(0.783790\pi\)
\(710\) 0 0
\(711\) 5893.35 0.310855
\(712\) 0 0
\(713\) 4916.92 0.258261
\(714\) 0 0
\(715\) 7730.70 0.404352
\(716\) 0 0
\(717\) 1820.32 0.0948132
\(718\) 0 0
\(719\) 34057.4 1.76652 0.883260 0.468883i \(-0.155343\pi\)
0.883260 + 0.468883i \(0.155343\pi\)
\(720\) 0 0
\(721\) −773.035 −0.0399297
\(722\) 0 0
\(723\) −2513.12 −0.129273
\(724\) 0 0
\(725\) 65181.6 3.33901
\(726\) 0 0
\(727\) −13915.1 −0.709880 −0.354940 0.934889i \(-0.615499\pi\)
−0.354940 + 0.934889i \(0.615499\pi\)
\(728\) 0 0
\(729\) −12925.8 −0.656699
\(730\) 0 0
\(731\) −32462.9 −1.64252
\(732\) 0 0
\(733\) −11153.8 −0.562039 −0.281020 0.959702i \(-0.590673\pi\)
−0.281020 + 0.959702i \(0.590673\pi\)
\(734\) 0 0
\(735\) 1238.47 0.0621520
\(736\) 0 0
\(737\) −4361.25 −0.217977
\(738\) 0 0
\(739\) −18406.4 −0.916226 −0.458113 0.888894i \(-0.651474\pi\)
−0.458113 + 0.888894i \(0.651474\pi\)
\(740\) 0 0
\(741\) 3755.77 0.186197
\(742\) 0 0
\(743\) 7902.81 0.390210 0.195105 0.980782i \(-0.437495\pi\)
0.195105 + 0.980782i \(0.437495\pi\)
\(744\) 0 0
\(745\) 21939.8 1.07894
\(746\) 0 0
\(747\) −14830.3 −0.726387
\(748\) 0 0
\(749\) −2861.34 −0.139587
\(750\) 0 0
\(751\) 23322.6 1.13323 0.566615 0.823983i \(-0.308253\pi\)
0.566615 + 0.823983i \(0.308253\pi\)
\(752\) 0 0
\(753\) −5663.41 −0.274085
\(754\) 0 0
\(755\) −58493.7 −2.81961
\(756\) 0 0
\(757\) −9085.76 −0.436232 −0.218116 0.975923i \(-0.569991\pi\)
−0.218116 + 0.975923i \(0.569991\pi\)
\(758\) 0 0
\(759\) −992.531 −0.0474658
\(760\) 0 0
\(761\) −38562.4 −1.83691 −0.918453 0.395529i \(-0.870561\pi\)
−0.918453 + 0.395529i \(0.870561\pi\)
\(762\) 0 0
\(763\) −9801.64 −0.465063
\(764\) 0 0
\(765\) 32772.3 1.54887
\(766\) 0 0
\(767\) 9309.76 0.438274
\(768\) 0 0
\(769\) 8816.49 0.413434 0.206717 0.978401i \(-0.433722\pi\)
0.206717 + 0.978401i \(0.433722\pi\)
\(770\) 0 0
\(771\) −9068.89 −0.423616
\(772\) 0 0
\(773\) 10849.6 0.504829 0.252414 0.967619i \(-0.418775\pi\)
0.252414 + 0.967619i \(0.418775\pi\)
\(774\) 0 0
\(775\) 18620.5 0.863058
\(776\) 0 0
\(777\) 1175.11 0.0542559
\(778\) 0 0
\(779\) −40107.7 −1.84468
\(780\) 0 0
\(781\) 4560.92 0.208966
\(782\) 0 0
\(783\) 16239.0 0.741168
\(784\) 0 0
\(785\) 68178.7 3.09988
\(786\) 0 0
\(787\) −33817.6 −1.53173 −0.765863 0.643004i \(-0.777688\pi\)
−0.765863 + 0.643004i \(0.777688\pi\)
\(788\) 0 0
\(789\) 1275.45 0.0575505
\(790\) 0 0
\(791\) −7792.63 −0.350283
\(792\) 0 0
\(793\) −31358.7 −1.40426
\(794\) 0 0
\(795\) −14489.8 −0.646413
\(796\) 0 0
\(797\) −35828.3 −1.59235 −0.796175 0.605066i \(-0.793147\pi\)
−0.796175 + 0.605066i \(0.793147\pi\)
\(798\) 0 0
\(799\) 32889.2 1.45624
\(800\) 0 0
\(801\) 15978.8 0.704846
\(802\) 0 0
\(803\) −2261.01 −0.0993641
\(804\) 0 0
\(805\) −9821.44 −0.430013
\(806\) 0 0
\(807\) −238.573 −0.0104066
\(808\) 0 0
\(809\) −8838.88 −0.384127 −0.192063 0.981383i \(-0.561518\pi\)
−0.192063 + 0.981383i \(0.561518\pi\)
\(810\) 0 0
\(811\) 16690.4 0.722662 0.361331 0.932438i \(-0.382323\pi\)
0.361331 + 0.932438i \(0.382323\pi\)
\(812\) 0 0
\(813\) −772.363 −0.0333185
\(814\) 0 0
\(815\) 66364.6 2.85233
\(816\) 0 0
\(817\) −41408.2 −1.77318
\(818\) 0 0
\(819\) 6296.73 0.268652
\(820\) 0 0
\(821\) −46102.2 −1.95978 −0.979889 0.199544i \(-0.936054\pi\)
−0.979889 + 0.199544i \(0.936054\pi\)
\(822\) 0 0
\(823\) 5544.15 0.234820 0.117410 0.993084i \(-0.462541\pi\)
0.117410 + 0.993084i \(0.462541\pi\)
\(824\) 0 0
\(825\) −3758.75 −0.158622
\(826\) 0 0
\(827\) −39002.0 −1.63994 −0.819972 0.572404i \(-0.806011\pi\)
−0.819972 + 0.572404i \(0.806011\pi\)
\(828\) 0 0
\(829\) 30509.4 1.27821 0.639106 0.769119i \(-0.279305\pi\)
0.639106 + 0.769119i \(0.279305\pi\)
\(830\) 0 0
\(831\) 1854.11 0.0773987
\(832\) 0 0
\(833\) 3192.24 0.132779
\(834\) 0 0
\(835\) 8408.56 0.348491
\(836\) 0 0
\(837\) 4639.02 0.191575
\(838\) 0 0
\(839\) −17283.7 −0.711204 −0.355602 0.934638i \(-0.615724\pi\)
−0.355602 + 0.934638i \(0.615724\pi\)
\(840\) 0 0
\(841\) 34755.3 1.42504
\(842\) 0 0
\(843\) 7573.22 0.309413
\(844\) 0 0
\(845\) 18640.9 0.758897
\(846\) 0 0
\(847\) 847.000 0.0343604
\(848\) 0 0
\(849\) 5716.31 0.231076
\(850\) 0 0
\(851\) −9318.97 −0.375382
\(852\) 0 0
\(853\) −9478.08 −0.380450 −0.190225 0.981741i \(-0.560922\pi\)
−0.190225 + 0.981741i \(0.560922\pi\)
\(854\) 0 0
\(855\) 41802.9 1.67208
\(856\) 0 0
\(857\) 4723.97 0.188294 0.0941469 0.995558i \(-0.469988\pi\)
0.0941469 + 0.995558i \(0.469988\pi\)
\(858\) 0 0
\(859\) 6148.17 0.244206 0.122103 0.992517i \(-0.461036\pi\)
0.122103 + 0.992517i \(0.461036\pi\)
\(860\) 0 0
\(861\) 4307.34 0.170492
\(862\) 0 0
\(863\) −8098.18 −0.319427 −0.159713 0.987163i \(-0.551057\pi\)
−0.159713 + 0.987163i \(0.551057\pi\)
\(864\) 0 0
\(865\) 6477.98 0.254634
\(866\) 0 0
\(867\) 852.611 0.0333981
\(868\) 0 0
\(869\) −2554.80 −0.0997302
\(870\) 0 0
\(871\) 14055.2 0.546776
\(872\) 0 0
\(873\) −8122.92 −0.314913
\(874\) 0 0
\(875\) −19847.5 −0.766819
\(876\) 0 0
\(877\) 44807.6 1.72525 0.862626 0.505843i \(-0.168818\pi\)
0.862626 + 0.505843i \(0.168818\pi\)
\(878\) 0 0
\(879\) −4130.38 −0.158492
\(880\) 0 0
\(881\) −20014.9 −0.765404 −0.382702 0.923872i \(-0.625006\pi\)
−0.382702 + 0.923872i \(0.625006\pi\)
\(882\) 0 0
\(883\) 38695.5 1.47475 0.737377 0.675481i \(-0.236064\pi\)
0.737377 + 0.675481i \(0.236064\pi\)
\(884\) 0 0
\(885\) −6637.58 −0.252113
\(886\) 0 0
\(887\) 39223.3 1.48477 0.742384 0.669974i \(-0.233695\pi\)
0.742384 + 0.669974i \(0.233695\pi\)
\(888\) 0 0
\(889\) 11021.1 0.415788
\(890\) 0 0
\(891\) 6599.82 0.248151
\(892\) 0 0
\(893\) 41951.9 1.57208
\(894\) 0 0
\(895\) 78959.7 2.94898
\(896\) 0 0
\(897\) 3198.67 0.119064
\(898\) 0 0
\(899\) 16895.9 0.626817
\(900\) 0 0
\(901\) −37348.3 −1.38097
\(902\) 0 0
\(903\) 4447.00 0.163884
\(904\) 0 0
\(905\) −13963.5 −0.512886
\(906\) 0 0
\(907\) −45745.8 −1.67471 −0.837357 0.546657i \(-0.815900\pi\)
−0.837357 + 0.546657i \(0.815900\pi\)
\(908\) 0 0
\(909\) 7912.70 0.288721
\(910\) 0 0
\(911\) −17363.7 −0.631487 −0.315744 0.948845i \(-0.602254\pi\)
−0.315744 + 0.948845i \(0.602254\pi\)
\(912\) 0 0
\(913\) 6428.99 0.233043
\(914\) 0 0
\(915\) 22357.8 0.807788
\(916\) 0 0
\(917\) 14896.6 0.536456
\(918\) 0 0
\(919\) 42719.1 1.53338 0.766688 0.642020i \(-0.221904\pi\)
0.766688 + 0.642020i \(0.221904\pi\)
\(920\) 0 0
\(921\) 1652.48 0.0591217
\(922\) 0 0
\(923\) −14698.7 −0.524174
\(924\) 0 0
\(925\) −35291.3 −1.25445
\(926\) 0 0
\(927\) 2802.21 0.0992843
\(928\) 0 0
\(929\) −8459.95 −0.298775 −0.149388 0.988779i \(-0.547730\pi\)
−0.149388 + 0.988779i \(0.547730\pi\)
\(930\) 0 0
\(931\) 4071.88 0.143341
\(932\) 0 0
\(933\) 5754.38 0.201918
\(934\) 0 0
\(935\) −14206.9 −0.496916
\(936\) 0 0
\(937\) −28780.9 −1.00345 −0.501724 0.865028i \(-0.667301\pi\)
−0.501724 + 0.865028i \(0.667301\pi\)
\(938\) 0 0
\(939\) 8820.81 0.306556
\(940\) 0 0
\(941\) −48285.4 −1.67275 −0.836376 0.548156i \(-0.815330\pi\)
−0.836376 + 0.548156i \(0.815330\pi\)
\(942\) 0 0
\(943\) −34158.4 −1.17959
\(944\) 0 0
\(945\) −9266.34 −0.318978
\(946\) 0 0
\(947\) −10090.3 −0.346240 −0.173120 0.984901i \(-0.555385\pi\)
−0.173120 + 0.984901i \(0.555385\pi\)
\(948\) 0 0
\(949\) 7286.66 0.249247
\(950\) 0 0
\(951\) −4901.40 −0.167128
\(952\) 0 0
\(953\) 2555.53 0.0868644 0.0434322 0.999056i \(-0.486171\pi\)
0.0434322 + 0.999056i \(0.486171\pi\)
\(954\) 0 0
\(955\) −65346.0 −2.21419
\(956\) 0 0
\(957\) −3410.60 −0.115203
\(958\) 0 0
\(959\) −2847.89 −0.0958948
\(960\) 0 0
\(961\) −24964.3 −0.837982
\(962\) 0 0
\(963\) 10372.2 0.347081
\(964\) 0 0
\(965\) −34792.2 −1.16062
\(966\) 0 0
\(967\) 41632.4 1.38450 0.692248 0.721660i \(-0.256620\pi\)
0.692248 + 0.721660i \(0.256620\pi\)
\(968\) 0 0
\(969\) −6902.10 −0.228821
\(970\) 0 0
\(971\) 26258.7 0.867849 0.433924 0.900949i \(-0.357129\pi\)
0.433924 + 0.900949i \(0.357129\pi\)
\(972\) 0 0
\(973\) −1118.70 −0.0368590
\(974\) 0 0
\(975\) 12113.5 0.397889
\(976\) 0 0
\(977\) −56691.6 −1.85642 −0.928211 0.372053i \(-0.878654\pi\)
−0.928211 + 0.372053i \(0.878654\pi\)
\(978\) 0 0
\(979\) −6926.87 −0.226132
\(980\) 0 0
\(981\) 35530.4 1.15637
\(982\) 0 0
\(983\) −38106.9 −1.23644 −0.618220 0.786005i \(-0.712146\pi\)
−0.618220 + 0.786005i \(0.712146\pi\)
\(984\) 0 0
\(985\) 18196.9 0.588632
\(986\) 0 0
\(987\) −4505.39 −0.145297
\(988\) 0 0
\(989\) −35266.0 −1.13387
\(990\) 0 0
\(991\) −37420.5 −1.19950 −0.599748 0.800189i \(-0.704732\pi\)
−0.599748 + 0.800189i \(0.704732\pi\)
\(992\) 0 0
\(993\) 9559.05 0.305486
\(994\) 0 0
\(995\) 105082. 3.34807
\(996\) 0 0
\(997\) −47293.6 −1.50231 −0.751156 0.660125i \(-0.770503\pi\)
−0.751156 + 0.660125i \(0.770503\pi\)
\(998\) 0 0
\(999\) −8792.28 −0.278454
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 1232.4.a.o.1.1 2
4.3 odd 2 154.4.a.h.1.2 2
12.11 even 2 1386.4.a.s.1.2 2
28.27 even 2 1078.4.a.o.1.1 2
44.43 even 2 1694.4.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.4.a.h.1.2 2 4.3 odd 2
1078.4.a.o.1.1 2 28.27 even 2
1232.4.a.o.1.1 2 1.1 even 1 trivial
1386.4.a.s.1.2 2 12.11 even 2
1694.4.a.h.1.2 2 44.43 even 2