Properties

Label 1232.4.a.bc.1.4
Level $1232$
Weight $4$
Character 1232.1
Self dual yes
Analytic conductor $72.690$
Analytic rank $0$
Dimension $7$
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: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 144x^{5} + 354x^{4} + 5172x^{3} - 6504x^{2} - 34432x + 18816 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 616)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.517294\) 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)
 
\(f(q)\) \(=\) \(q-0.517294 q^{3} -7.39668 q^{5} -7.00000 q^{7} -26.7324 q^{9} +O(q^{10})\) \(q-0.517294 q^{3} -7.39668 q^{5} -7.00000 q^{7} -26.7324 q^{9} -11.0000 q^{11} -78.8766 q^{13} +3.82626 q^{15} +57.7886 q^{17} -106.951 q^{19} +3.62106 q^{21} +11.8823 q^{23} -70.2891 q^{25} +27.7955 q^{27} +139.466 q^{29} -297.397 q^{31} +5.69024 q^{33} +51.7768 q^{35} -136.832 q^{37} +40.8024 q^{39} +126.644 q^{41} -265.571 q^{43} +197.731 q^{45} +587.057 q^{47} +49.0000 q^{49} -29.8937 q^{51} +154.322 q^{53} +81.3635 q^{55} +55.3253 q^{57} -199.082 q^{59} +385.652 q^{61} +187.127 q^{63} +583.425 q^{65} -772.165 q^{67} -6.14663 q^{69} -163.931 q^{71} +426.710 q^{73} +36.3601 q^{75} +77.0000 q^{77} -886.257 q^{79} +707.397 q^{81} +89.9744 q^{83} -427.444 q^{85} -72.1450 q^{87} +1391.62 q^{89} +552.136 q^{91} +153.842 q^{93} +791.085 q^{95} -1418.66 q^{97} +294.056 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 3 q^{3} + 11 q^{5} - 49 q^{7} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 3 q^{3} + 11 q^{5} - 49 q^{7} + 108 q^{9} - 77 q^{11} + 26 q^{13} - 67 q^{15} + 44 q^{17} - 34 q^{19} + 21 q^{21} + 29 q^{23} + 182 q^{25} - 99 q^{27} + 94 q^{29} + 173 q^{31} + 33 q^{33} - 77 q^{35} + 255 q^{37} + 60 q^{39} + 508 q^{41} - 656 q^{43} + 466 q^{45} + 18 q^{47} + 343 q^{49} - 850 q^{51} + 1806 q^{53} - 121 q^{55} + 1154 q^{57} - 665 q^{59} + 608 q^{61} - 756 q^{63} + 588 q^{65} - 669 q^{67} + 2067 q^{69} - 1169 q^{71} + 380 q^{73} - 1254 q^{75} + 539 q^{77} + 110 q^{79} + 2847 q^{81} + 496 q^{83} + 3446 q^{85} + 582 q^{87} + 1321 q^{89} - 182 q^{91} + 1493 q^{93} + 50 q^{95} + 3927 q^{97} - 1188 q^{99}+O(q^{100}) \) 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
\(3\) −0.517294 −0.0995533 −0.0497767 0.998760i \(-0.515851\pi\)
−0.0497767 + 0.998760i \(0.515851\pi\)
\(4\) 0 0
\(5\) −7.39668 −0.661580 −0.330790 0.943704i \(-0.607315\pi\)
−0.330790 + 0.943704i \(0.607315\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 0 0
\(9\) −26.7324 −0.990089
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) −78.8766 −1.68280 −0.841401 0.540411i \(-0.818269\pi\)
−0.841401 + 0.540411i \(0.818269\pi\)
\(14\) 0 0
\(15\) 3.82626 0.0658624
\(16\) 0 0
\(17\) 57.7886 0.824459 0.412229 0.911080i \(-0.364750\pi\)
0.412229 + 0.911080i \(0.364750\pi\)
\(18\) 0 0
\(19\) −106.951 −1.29138 −0.645692 0.763598i \(-0.723431\pi\)
−0.645692 + 0.763598i \(0.723431\pi\)
\(20\) 0 0
\(21\) 3.62106 0.0376276
\(22\) 0 0
\(23\) 11.8823 0.107723 0.0538614 0.998548i \(-0.482847\pi\)
0.0538614 + 0.998548i \(0.482847\pi\)
\(24\) 0 0
\(25\) −70.2891 −0.562313
\(26\) 0 0
\(27\) 27.7955 0.198120
\(28\) 0 0
\(29\) 139.466 0.893041 0.446521 0.894773i \(-0.352663\pi\)
0.446521 + 0.894773i \(0.352663\pi\)
\(30\) 0 0
\(31\) −297.397 −1.72303 −0.861517 0.507728i \(-0.830485\pi\)
−0.861517 + 0.507728i \(0.830485\pi\)
\(32\) 0 0
\(33\) 5.69024 0.0300165
\(34\) 0 0
\(35\) 51.7768 0.250054
\(36\) 0 0
\(37\) −136.832 −0.607972 −0.303986 0.952676i \(-0.598318\pi\)
−0.303986 + 0.952676i \(0.598318\pi\)
\(38\) 0 0
\(39\) 40.8024 0.167529
\(40\) 0 0
\(41\) 126.644 0.482401 0.241200 0.970475i \(-0.422459\pi\)
0.241200 + 0.970475i \(0.422459\pi\)
\(42\) 0 0
\(43\) −265.571 −0.941841 −0.470920 0.882176i \(-0.656078\pi\)
−0.470920 + 0.882176i \(0.656078\pi\)
\(44\) 0 0
\(45\) 197.731 0.655023
\(46\) 0 0
\(47\) 587.057 1.82194 0.910969 0.412474i \(-0.135335\pi\)
0.910969 + 0.412474i \(0.135335\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −29.8937 −0.0820776
\(52\) 0 0
\(53\) 154.322 0.399959 0.199979 0.979800i \(-0.435913\pi\)
0.199979 + 0.979800i \(0.435913\pi\)
\(54\) 0 0
\(55\) 81.3635 0.199474
\(56\) 0 0
\(57\) 55.3253 0.128562
\(58\) 0 0
\(59\) −199.082 −0.439292 −0.219646 0.975580i \(-0.570490\pi\)
−0.219646 + 0.975580i \(0.570490\pi\)
\(60\) 0 0
\(61\) 385.652 0.809471 0.404736 0.914434i \(-0.367364\pi\)
0.404736 + 0.914434i \(0.367364\pi\)
\(62\) 0 0
\(63\) 187.127 0.374219
\(64\) 0 0
\(65\) 583.425 1.11331
\(66\) 0 0
\(67\) −772.165 −1.40799 −0.703993 0.710207i \(-0.748601\pi\)
−0.703993 + 0.710207i \(0.748601\pi\)
\(68\) 0 0
\(69\) −6.14663 −0.0107242
\(70\) 0 0
\(71\) −163.931 −0.274015 −0.137008 0.990570i \(-0.543748\pi\)
−0.137008 + 0.990570i \(0.543748\pi\)
\(72\) 0 0
\(73\) 426.710 0.684146 0.342073 0.939673i \(-0.388871\pi\)
0.342073 + 0.939673i \(0.388871\pi\)
\(74\) 0 0
\(75\) 36.3601 0.0559801
\(76\) 0 0
\(77\) 77.0000 0.113961
\(78\) 0 0
\(79\) −886.257 −1.26217 −0.631087 0.775712i \(-0.717391\pi\)
−0.631087 + 0.775712i \(0.717391\pi\)
\(80\) 0 0
\(81\) 707.397 0.970366
\(82\) 0 0
\(83\) 89.9744 0.118988 0.0594938 0.998229i \(-0.481051\pi\)
0.0594938 + 0.998229i \(0.481051\pi\)
\(84\) 0 0
\(85\) −427.444 −0.545445
\(86\) 0 0
\(87\) −72.1450 −0.0889052
\(88\) 0 0
\(89\) 1391.62 1.65743 0.828717 0.559667i \(-0.189071\pi\)
0.828717 + 0.559667i \(0.189071\pi\)
\(90\) 0 0
\(91\) 552.136 0.636040
\(92\) 0 0
\(93\) 153.842 0.171534
\(94\) 0 0
\(95\) 791.085 0.854354
\(96\) 0 0
\(97\) −1418.66 −1.48498 −0.742489 0.669858i \(-0.766355\pi\)
−0.742489 + 0.669858i \(0.766355\pi\)
\(98\) 0 0
\(99\) 294.056 0.298523
\(100\) 0 0
\(101\) −614.172 −0.605073 −0.302537 0.953138i \(-0.597833\pi\)
−0.302537 + 0.953138i \(0.597833\pi\)
\(102\) 0 0
\(103\) 650.855 0.622627 0.311314 0.950307i \(-0.399231\pi\)
0.311314 + 0.950307i \(0.399231\pi\)
\(104\) 0 0
\(105\) −26.7838 −0.0248937
\(106\) 0 0
\(107\) −417.533 −0.377237 −0.188619 0.982050i \(-0.560401\pi\)
−0.188619 + 0.982050i \(0.560401\pi\)
\(108\) 0 0
\(109\) 1260.90 1.10801 0.554003 0.832515i \(-0.313100\pi\)
0.554003 + 0.832515i \(0.313100\pi\)
\(110\) 0 0
\(111\) 70.7821 0.0605256
\(112\) 0 0
\(113\) −484.972 −0.403737 −0.201869 0.979413i \(-0.564701\pi\)
−0.201869 + 0.979413i \(0.564701\pi\)
\(114\) 0 0
\(115\) −87.8895 −0.0712673
\(116\) 0 0
\(117\) 2108.56 1.66612
\(118\) 0 0
\(119\) −404.520 −0.311616
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) −65.5121 −0.0480246
\(124\) 0 0
\(125\) 1444.49 1.03359
\(126\) 0 0
\(127\) −469.263 −0.327877 −0.163939 0.986471i \(-0.552420\pi\)
−0.163939 + 0.986471i \(0.552420\pi\)
\(128\) 0 0
\(129\) 137.378 0.0937634
\(130\) 0 0
\(131\) −1005.42 −0.670563 −0.335282 0.942118i \(-0.608831\pi\)
−0.335282 + 0.942118i \(0.608831\pi\)
\(132\) 0 0
\(133\) 748.659 0.488098
\(134\) 0 0
\(135\) −205.594 −0.131072
\(136\) 0 0
\(137\) 486.764 0.303555 0.151778 0.988415i \(-0.451500\pi\)
0.151778 + 0.988415i \(0.451500\pi\)
\(138\) 0 0
\(139\) 540.533 0.329837 0.164919 0.986307i \(-0.447264\pi\)
0.164919 + 0.986307i \(0.447264\pi\)
\(140\) 0 0
\(141\) −303.681 −0.181380
\(142\) 0 0
\(143\) 867.643 0.507384
\(144\) 0 0
\(145\) −1031.59 −0.590818
\(146\) 0 0
\(147\) −25.3474 −0.0142219
\(148\) 0 0
\(149\) 2083.81 1.14572 0.572860 0.819654i \(-0.305834\pi\)
0.572860 + 0.819654i \(0.305834\pi\)
\(150\) 0 0
\(151\) 1831.37 0.986983 0.493492 0.869750i \(-0.335720\pi\)
0.493492 + 0.869750i \(0.335720\pi\)
\(152\) 0 0
\(153\) −1544.83 −0.816288
\(154\) 0 0
\(155\) 2199.75 1.13992
\(156\) 0 0
\(157\) −228.649 −0.116231 −0.0581153 0.998310i \(-0.518509\pi\)
−0.0581153 + 0.998310i \(0.518509\pi\)
\(158\) 0 0
\(159\) −79.8301 −0.0398172
\(160\) 0 0
\(161\) −83.1760 −0.0407154
\(162\) 0 0
\(163\) 2382.99 1.14509 0.572546 0.819872i \(-0.305956\pi\)
0.572546 + 0.819872i \(0.305956\pi\)
\(164\) 0 0
\(165\) −42.0889 −0.0198583
\(166\) 0 0
\(167\) −2700.28 −1.25122 −0.625610 0.780136i \(-0.715150\pi\)
−0.625610 + 0.780136i \(0.715150\pi\)
\(168\) 0 0
\(169\) 4024.52 1.83183
\(170\) 0 0
\(171\) 2859.07 1.27859
\(172\) 0 0
\(173\) 2567.05 1.12815 0.564073 0.825725i \(-0.309234\pi\)
0.564073 + 0.825725i \(0.309234\pi\)
\(174\) 0 0
\(175\) 492.023 0.212534
\(176\) 0 0
\(177\) 102.984 0.0437330
\(178\) 0 0
\(179\) −1810.02 −0.755793 −0.377897 0.925848i \(-0.623353\pi\)
−0.377897 + 0.925848i \(0.623353\pi\)
\(180\) 0 0
\(181\) −1046.75 −0.429859 −0.214929 0.976630i \(-0.568952\pi\)
−0.214929 + 0.976630i \(0.568952\pi\)
\(182\) 0 0
\(183\) −199.496 −0.0805855
\(184\) 0 0
\(185\) 1012.10 0.402222
\(186\) 0 0
\(187\) −635.675 −0.248584
\(188\) 0 0
\(189\) −194.568 −0.0748823
\(190\) 0 0
\(191\) −1387.86 −0.525771 −0.262886 0.964827i \(-0.584674\pi\)
−0.262886 + 0.964827i \(0.584674\pi\)
\(192\) 0 0
\(193\) −518.006 −0.193196 −0.0965981 0.995323i \(-0.530796\pi\)
−0.0965981 + 0.995323i \(0.530796\pi\)
\(194\) 0 0
\(195\) −301.803 −0.110833
\(196\) 0 0
\(197\) −1633.42 −0.590742 −0.295371 0.955383i \(-0.595443\pi\)
−0.295371 + 0.955383i \(0.595443\pi\)
\(198\) 0 0
\(199\) 2959.23 1.05414 0.527071 0.849821i \(-0.323290\pi\)
0.527071 + 0.849821i \(0.323290\pi\)
\(200\) 0 0
\(201\) 399.437 0.140170
\(202\) 0 0
\(203\) −976.263 −0.337538
\(204\) 0 0
\(205\) −936.744 −0.319147
\(206\) 0 0
\(207\) −317.642 −0.106655
\(208\) 0 0
\(209\) 1176.46 0.389367
\(210\) 0 0
\(211\) 622.680 0.203161 0.101581 0.994827i \(-0.467610\pi\)
0.101581 + 0.994827i \(0.467610\pi\)
\(212\) 0 0
\(213\) 84.8007 0.0272791
\(214\) 0 0
\(215\) 1964.34 0.623103
\(216\) 0 0
\(217\) 2081.78 0.651246
\(218\) 0 0
\(219\) −220.735 −0.0681090
\(220\) 0 0
\(221\) −4558.17 −1.38740
\(222\) 0 0
\(223\) 3635.58 1.09173 0.545866 0.837872i \(-0.316201\pi\)
0.545866 + 0.837872i \(0.316201\pi\)
\(224\) 0 0
\(225\) 1879.00 0.556740
\(226\) 0 0
\(227\) −2986.85 −0.873322 −0.436661 0.899626i \(-0.643839\pi\)
−0.436661 + 0.899626i \(0.643839\pi\)
\(228\) 0 0
\(229\) −268.330 −0.0774312 −0.0387156 0.999250i \(-0.512327\pi\)
−0.0387156 + 0.999250i \(0.512327\pi\)
\(230\) 0 0
\(231\) −39.8316 −0.0113452
\(232\) 0 0
\(233\) 3608.12 1.01449 0.507244 0.861803i \(-0.330664\pi\)
0.507244 + 0.861803i \(0.330664\pi\)
\(234\) 0 0
\(235\) −4342.28 −1.20536
\(236\) 0 0
\(237\) 458.455 0.125654
\(238\) 0 0
\(239\) 89.9305 0.0243394 0.0121697 0.999926i \(-0.496126\pi\)
0.0121697 + 0.999926i \(0.496126\pi\)
\(240\) 0 0
\(241\) −5090.89 −1.36072 −0.680359 0.732879i \(-0.738176\pi\)
−0.680359 + 0.732879i \(0.738176\pi\)
\(242\) 0 0
\(243\) −1116.41 −0.294723
\(244\) 0 0
\(245\) −362.438 −0.0945114
\(246\) 0 0
\(247\) 8435.96 2.17315
\(248\) 0 0
\(249\) −46.5432 −0.0118456
\(250\) 0 0
\(251\) 5539.36 1.39299 0.696496 0.717561i \(-0.254741\pi\)
0.696496 + 0.717561i \(0.254741\pi\)
\(252\) 0 0
\(253\) −130.705 −0.0324797
\(254\) 0 0
\(255\) 221.114 0.0543009
\(256\) 0 0
\(257\) −4669.77 −1.13343 −0.566716 0.823913i \(-0.691786\pi\)
−0.566716 + 0.823913i \(0.691786\pi\)
\(258\) 0 0
\(259\) 957.821 0.229792
\(260\) 0 0
\(261\) −3728.26 −0.884191
\(262\) 0 0
\(263\) 3691.28 0.865453 0.432726 0.901525i \(-0.357552\pi\)
0.432726 + 0.901525i \(0.357552\pi\)
\(264\) 0 0
\(265\) −1141.47 −0.264604
\(266\) 0 0
\(267\) −719.878 −0.165003
\(268\) 0 0
\(269\) −850.050 −0.192671 −0.0963354 0.995349i \(-0.530712\pi\)
−0.0963354 + 0.995349i \(0.530712\pi\)
\(270\) 0 0
\(271\) 7458.57 1.67187 0.835933 0.548831i \(-0.184927\pi\)
0.835933 + 0.548831i \(0.184927\pi\)
\(272\) 0 0
\(273\) −285.617 −0.0633199
\(274\) 0 0
\(275\) 773.180 0.169544
\(276\) 0 0
\(277\) −5368.97 −1.16458 −0.582292 0.812980i \(-0.697844\pi\)
−0.582292 + 0.812980i \(0.697844\pi\)
\(278\) 0 0
\(279\) 7950.14 1.70596
\(280\) 0 0
\(281\) 6478.43 1.37534 0.687670 0.726023i \(-0.258633\pi\)
0.687670 + 0.726023i \(0.258633\pi\)
\(282\) 0 0
\(283\) −3841.89 −0.806985 −0.403493 0.914983i \(-0.632204\pi\)
−0.403493 + 0.914983i \(0.632204\pi\)
\(284\) 0 0
\(285\) −409.224 −0.0850537
\(286\) 0 0
\(287\) −886.507 −0.182330
\(288\) 0 0
\(289\) −1573.47 −0.320267
\(290\) 0 0
\(291\) 733.863 0.147834
\(292\) 0 0
\(293\) 2179.76 0.434618 0.217309 0.976103i \(-0.430272\pi\)
0.217309 + 0.976103i \(0.430272\pi\)
\(294\) 0 0
\(295\) 1472.54 0.290627
\(296\) 0 0
\(297\) −305.750 −0.0597354
\(298\) 0 0
\(299\) −937.234 −0.181276
\(300\) 0 0
\(301\) 1859.00 0.355982
\(302\) 0 0
\(303\) 317.708 0.0602370
\(304\) 0 0
\(305\) −2852.55 −0.535529
\(306\) 0 0
\(307\) 7685.42 1.42876 0.714382 0.699756i \(-0.246708\pi\)
0.714382 + 0.699756i \(0.246708\pi\)
\(308\) 0 0
\(309\) −336.683 −0.0619846
\(310\) 0 0
\(311\) 2699.99 0.492291 0.246146 0.969233i \(-0.420836\pi\)
0.246146 + 0.969233i \(0.420836\pi\)
\(312\) 0 0
\(313\) 3484.31 0.629217 0.314609 0.949222i \(-0.398127\pi\)
0.314609 + 0.949222i \(0.398127\pi\)
\(314\) 0 0
\(315\) −1384.12 −0.247575
\(316\) 0 0
\(317\) −7.35952 −0.00130395 −0.000651975 1.00000i \(-0.500208\pi\)
−0.000651975 1.00000i \(0.500208\pi\)
\(318\) 0 0
\(319\) −1534.13 −0.269262
\(320\) 0 0
\(321\) 215.987 0.0375552
\(322\) 0 0
\(323\) −6180.57 −1.06469
\(324\) 0 0
\(325\) 5544.16 0.946261
\(326\) 0 0
\(327\) −652.258 −0.110306
\(328\) 0 0
\(329\) −4109.40 −0.688628
\(330\) 0 0
\(331\) −8152.84 −1.35384 −0.676919 0.736058i \(-0.736685\pi\)
−0.676919 + 0.736058i \(0.736685\pi\)
\(332\) 0 0
\(333\) 3657.84 0.601947
\(334\) 0 0
\(335\) 5711.46 0.931494
\(336\) 0 0
\(337\) −669.429 −0.108208 −0.0541041 0.998535i \(-0.517230\pi\)
−0.0541041 + 0.998535i \(0.517230\pi\)
\(338\) 0 0
\(339\) 250.873 0.0401934
\(340\) 0 0
\(341\) 3271.37 0.519515
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) 0 0
\(345\) 45.4647 0.00709489
\(346\) 0 0
\(347\) −5364.50 −0.829917 −0.414959 0.909840i \(-0.636204\pi\)
−0.414959 + 0.909840i \(0.636204\pi\)
\(348\) 0 0
\(349\) −1520.78 −0.233254 −0.116627 0.993176i \(-0.537208\pi\)
−0.116627 + 0.993176i \(0.537208\pi\)
\(350\) 0 0
\(351\) −2192.41 −0.333397
\(352\) 0 0
\(353\) −9736.69 −1.46808 −0.734040 0.679107i \(-0.762367\pi\)
−0.734040 + 0.679107i \(0.762367\pi\)
\(354\) 0 0
\(355\) 1212.55 0.181283
\(356\) 0 0
\(357\) 209.256 0.0310224
\(358\) 0 0
\(359\) −5802.44 −0.853040 −0.426520 0.904478i \(-0.640261\pi\)
−0.426520 + 0.904478i \(0.640261\pi\)
\(360\) 0 0
\(361\) 4579.58 0.667675
\(362\) 0 0
\(363\) −62.5926 −0.00905030
\(364\) 0 0
\(365\) −3156.24 −0.452617
\(366\) 0 0
\(367\) −2947.86 −0.419283 −0.209641 0.977778i \(-0.567230\pi\)
−0.209641 + 0.977778i \(0.567230\pi\)
\(368\) 0 0
\(369\) −3385.49 −0.477620
\(370\) 0 0
\(371\) −1080.26 −0.151170
\(372\) 0 0
\(373\) 11014.4 1.52896 0.764480 0.644647i \(-0.222996\pi\)
0.764480 + 0.644647i \(0.222996\pi\)
\(374\) 0 0
\(375\) −747.227 −0.102898
\(376\) 0 0
\(377\) −11000.6 −1.50281
\(378\) 0 0
\(379\) −9879.25 −1.33895 −0.669476 0.742833i \(-0.733481\pi\)
−0.669476 + 0.742833i \(0.733481\pi\)
\(380\) 0 0
\(381\) 242.747 0.0326412
\(382\) 0 0
\(383\) −4246.20 −0.566503 −0.283251 0.959046i \(-0.591413\pi\)
−0.283251 + 0.959046i \(0.591413\pi\)
\(384\) 0 0
\(385\) −569.545 −0.0753940
\(386\) 0 0
\(387\) 7099.34 0.932506
\(388\) 0 0
\(389\) −13848.9 −1.80505 −0.902525 0.430637i \(-0.858289\pi\)
−0.902525 + 0.430637i \(0.858289\pi\)
\(390\) 0 0
\(391\) 686.661 0.0888131
\(392\) 0 0
\(393\) 520.097 0.0667568
\(394\) 0 0
\(395\) 6555.36 0.835028
\(396\) 0 0
\(397\) 7536.27 0.952731 0.476366 0.879247i \(-0.341954\pi\)
0.476366 + 0.879247i \(0.341954\pi\)
\(398\) 0 0
\(399\) −387.277 −0.0485917
\(400\) 0 0
\(401\) 15424.3 1.92083 0.960413 0.278582i \(-0.0898643\pi\)
0.960413 + 0.278582i \(0.0898643\pi\)
\(402\) 0 0
\(403\) 23457.7 2.89953
\(404\) 0 0
\(405\) −5232.39 −0.641974
\(406\) 0 0
\(407\) 1505.15 0.183310
\(408\) 0 0
\(409\) 259.112 0.0313259 0.0156629 0.999877i \(-0.495014\pi\)
0.0156629 + 0.999877i \(0.495014\pi\)
\(410\) 0 0
\(411\) −251.800 −0.0302199
\(412\) 0 0
\(413\) 1393.57 0.166037
\(414\) 0 0
\(415\) −665.512 −0.0787198
\(416\) 0 0
\(417\) −279.614 −0.0328364
\(418\) 0 0
\(419\) −10193.3 −1.18848 −0.594241 0.804287i \(-0.702548\pi\)
−0.594241 + 0.804287i \(0.702548\pi\)
\(420\) 0 0
\(421\) −14180.9 −1.64166 −0.820828 0.571176i \(-0.806488\pi\)
−0.820828 + 0.571176i \(0.806488\pi\)
\(422\) 0 0
\(423\) −15693.5 −1.80388
\(424\) 0 0
\(425\) −4061.91 −0.463604
\(426\) 0 0
\(427\) −2699.57 −0.305951
\(428\) 0 0
\(429\) −448.827 −0.0505118
\(430\) 0 0
\(431\) 15237.2 1.70290 0.851448 0.524439i \(-0.175725\pi\)
0.851448 + 0.524439i \(0.175725\pi\)
\(432\) 0 0
\(433\) 17620.3 1.95561 0.977805 0.209518i \(-0.0671893\pi\)
0.977805 + 0.209518i \(0.0671893\pi\)
\(434\) 0 0
\(435\) 533.634 0.0588179
\(436\) 0 0
\(437\) −1270.83 −0.139112
\(438\) 0 0
\(439\) 7168.92 0.779394 0.389697 0.920943i \(-0.372580\pi\)
0.389697 + 0.920943i \(0.372580\pi\)
\(440\) 0 0
\(441\) −1309.89 −0.141441
\(442\) 0 0
\(443\) −10441.2 −1.11981 −0.559904 0.828558i \(-0.689162\pi\)
−0.559904 + 0.828558i \(0.689162\pi\)
\(444\) 0 0
\(445\) −10293.4 −1.09652
\(446\) 0 0
\(447\) −1077.94 −0.114060
\(448\) 0 0
\(449\) 5927.94 0.623066 0.311533 0.950235i \(-0.399158\pi\)
0.311533 + 0.950235i \(0.399158\pi\)
\(450\) 0 0
\(451\) −1393.08 −0.145449
\(452\) 0 0
\(453\) −947.355 −0.0982575
\(454\) 0 0
\(455\) −4083.98 −0.420791
\(456\) 0 0
\(457\) −15408.6 −1.57721 −0.788604 0.614901i \(-0.789196\pi\)
−0.788604 + 0.614901i \(0.789196\pi\)
\(458\) 0 0
\(459\) 1606.26 0.163342
\(460\) 0 0
\(461\) −3831.23 −0.387068 −0.193534 0.981094i \(-0.561995\pi\)
−0.193534 + 0.981094i \(0.561995\pi\)
\(462\) 0 0
\(463\) −3890.07 −0.390468 −0.195234 0.980757i \(-0.562547\pi\)
−0.195234 + 0.980757i \(0.562547\pi\)
\(464\) 0 0
\(465\) −1137.92 −0.113483
\(466\) 0 0
\(467\) −2931.13 −0.290443 −0.145221 0.989399i \(-0.546389\pi\)
−0.145221 + 0.989399i \(0.546389\pi\)
\(468\) 0 0
\(469\) 5405.16 0.532168
\(470\) 0 0
\(471\) 118.279 0.0115711
\(472\) 0 0
\(473\) 2921.28 0.283976
\(474\) 0 0
\(475\) 7517.51 0.726162
\(476\) 0 0
\(477\) −4125.41 −0.395995
\(478\) 0 0
\(479\) −3133.99 −0.298947 −0.149473 0.988766i \(-0.547758\pi\)
−0.149473 + 0.988766i \(0.547758\pi\)
\(480\) 0 0
\(481\) 10792.8 1.02310
\(482\) 0 0
\(483\) 43.0264 0.00405335
\(484\) 0 0
\(485\) 10493.4 0.982431
\(486\) 0 0
\(487\) −12714.8 −1.18309 −0.591544 0.806272i \(-0.701482\pi\)
−0.591544 + 0.806272i \(0.701482\pi\)
\(488\) 0 0
\(489\) −1232.71 −0.113998
\(490\) 0 0
\(491\) −1714.36 −0.157572 −0.0787862 0.996892i \(-0.525104\pi\)
−0.0787862 + 0.996892i \(0.525104\pi\)
\(492\) 0 0
\(493\) 8059.55 0.736276
\(494\) 0 0
\(495\) −2175.04 −0.197497
\(496\) 0 0
\(497\) 1147.52 0.103568
\(498\) 0 0
\(499\) −20481.3 −1.83741 −0.918706 0.394942i \(-0.870765\pi\)
−0.918706 + 0.394942i \(0.870765\pi\)
\(500\) 0 0
\(501\) 1396.84 0.124563
\(502\) 0 0
\(503\) −2345.46 −0.207911 −0.103955 0.994582i \(-0.533150\pi\)
−0.103955 + 0.994582i \(0.533150\pi\)
\(504\) 0 0
\(505\) 4542.84 0.400304
\(506\) 0 0
\(507\) −2081.86 −0.182364
\(508\) 0 0
\(509\) 18843.5 1.64091 0.820457 0.571708i \(-0.193719\pi\)
0.820457 + 0.571708i \(0.193719\pi\)
\(510\) 0 0
\(511\) −2986.97 −0.258583
\(512\) 0 0
\(513\) −2972.76 −0.255849
\(514\) 0 0
\(515\) −4814.17 −0.411917
\(516\) 0 0
\(517\) −6457.63 −0.549335
\(518\) 0 0
\(519\) −1327.92 −0.112311
\(520\) 0 0
\(521\) 9379.26 0.788700 0.394350 0.918960i \(-0.370970\pi\)
0.394350 + 0.918960i \(0.370970\pi\)
\(522\) 0 0
\(523\) −5745.10 −0.480336 −0.240168 0.970731i \(-0.577203\pi\)
−0.240168 + 0.970731i \(0.577203\pi\)
\(524\) 0 0
\(525\) −254.521 −0.0211585
\(526\) 0 0
\(527\) −17186.2 −1.42057
\(528\) 0 0
\(529\) −12025.8 −0.988396
\(530\) 0 0
\(531\) 5321.93 0.434938
\(532\) 0 0
\(533\) −9989.23 −0.811786
\(534\) 0 0
\(535\) 3088.36 0.249573
\(536\) 0 0
\(537\) 936.311 0.0752417
\(538\) 0 0
\(539\) −539.000 −0.0430730
\(540\) 0 0
\(541\) −13581.0 −1.07929 −0.539644 0.841893i \(-0.681441\pi\)
−0.539644 + 0.841893i \(0.681441\pi\)
\(542\) 0 0
\(543\) 541.478 0.0427938
\(544\) 0 0
\(545\) −9326.50 −0.733034
\(546\) 0 0
\(547\) −21441.9 −1.67603 −0.838017 0.545645i \(-0.816285\pi\)
−0.838017 + 0.545645i \(0.816285\pi\)
\(548\) 0 0
\(549\) −10309.4 −0.801449
\(550\) 0 0
\(551\) −14916.1 −1.15326
\(552\) 0 0
\(553\) 6203.80 0.477057
\(554\) 0 0
\(555\) −523.553 −0.0400425
\(556\) 0 0
\(557\) 7104.61 0.540452 0.270226 0.962797i \(-0.412902\pi\)
0.270226 + 0.962797i \(0.412902\pi\)
\(558\) 0 0
\(559\) 20947.3 1.58493
\(560\) 0 0
\(561\) 328.831 0.0247473
\(562\) 0 0
\(563\) 2015.82 0.150900 0.0754501 0.997150i \(-0.475961\pi\)
0.0754501 + 0.997150i \(0.475961\pi\)
\(564\) 0 0
\(565\) 3587.18 0.267104
\(566\) 0 0
\(567\) −4951.78 −0.366764
\(568\) 0 0
\(569\) −24979.6 −1.84042 −0.920212 0.391421i \(-0.871984\pi\)
−0.920212 + 0.391421i \(0.871984\pi\)
\(570\) 0 0
\(571\) −17670.1 −1.29505 −0.647523 0.762046i \(-0.724195\pi\)
−0.647523 + 0.762046i \(0.724195\pi\)
\(572\) 0 0
\(573\) 717.934 0.0523423
\(574\) 0 0
\(575\) −835.194 −0.0605739
\(576\) 0 0
\(577\) −7744.82 −0.558789 −0.279394 0.960176i \(-0.590134\pi\)
−0.279394 + 0.960176i \(0.590134\pi\)
\(578\) 0 0
\(579\) 267.961 0.0192333
\(580\) 0 0
\(581\) −629.821 −0.0449731
\(582\) 0 0
\(583\) −1697.55 −0.120592
\(584\) 0 0
\(585\) −15596.4 −1.10227
\(586\) 0 0
\(587\) −6418.51 −0.451312 −0.225656 0.974207i \(-0.572453\pi\)
−0.225656 + 0.974207i \(0.572453\pi\)
\(588\) 0 0
\(589\) 31807.0 2.22510
\(590\) 0 0
\(591\) 844.957 0.0588103
\(592\) 0 0
\(593\) 1577.72 0.109257 0.0546285 0.998507i \(-0.482603\pi\)
0.0546285 + 0.998507i \(0.482603\pi\)
\(594\) 0 0
\(595\) 2992.11 0.206159
\(596\) 0 0
\(597\) −1530.79 −0.104943
\(598\) 0 0
\(599\) −18754.0 −1.27925 −0.639623 0.768688i \(-0.720910\pi\)
−0.639623 + 0.768688i \(0.720910\pi\)
\(600\) 0 0
\(601\) −18851.7 −1.27949 −0.639747 0.768586i \(-0.720961\pi\)
−0.639747 + 0.768586i \(0.720961\pi\)
\(602\) 0 0
\(603\) 20641.8 1.39403
\(604\) 0 0
\(605\) −894.999 −0.0601436
\(606\) 0 0
\(607\) 1628.13 0.108869 0.0544347 0.998517i \(-0.482664\pi\)
0.0544347 + 0.998517i \(0.482664\pi\)
\(608\) 0 0
\(609\) 505.015 0.0336030
\(610\) 0 0
\(611\) −46305.1 −3.06596
\(612\) 0 0
\(613\) −29472.2 −1.94188 −0.970939 0.239327i \(-0.923073\pi\)
−0.970939 + 0.239327i \(0.923073\pi\)
\(614\) 0 0
\(615\) 484.572 0.0317721
\(616\) 0 0
\(617\) 26335.9 1.71839 0.859194 0.511650i \(-0.170966\pi\)
0.859194 + 0.511650i \(0.170966\pi\)
\(618\) 0 0
\(619\) 6196.40 0.402349 0.201175 0.979555i \(-0.435524\pi\)
0.201175 + 0.979555i \(0.435524\pi\)
\(620\) 0 0
\(621\) 330.273 0.0213421
\(622\) 0 0
\(623\) −9741.36 −0.626451
\(624\) 0 0
\(625\) −1898.31 −0.121492
\(626\) 0 0
\(627\) −608.578 −0.0387628
\(628\) 0 0
\(629\) −7907.31 −0.501248
\(630\) 0 0
\(631\) 2060.70 0.130008 0.0650042 0.997885i \(-0.479294\pi\)
0.0650042 + 0.997885i \(0.479294\pi\)
\(632\) 0 0
\(633\) −322.109 −0.0202254
\(634\) 0 0
\(635\) 3470.99 0.216917
\(636\) 0 0
\(637\) −3864.95 −0.240400
\(638\) 0 0
\(639\) 4382.28 0.271299
\(640\) 0 0
\(641\) 30191.4 1.86036 0.930178 0.367108i \(-0.119652\pi\)
0.930178 + 0.367108i \(0.119652\pi\)
\(642\) 0 0
\(643\) 4434.88 0.271998 0.135999 0.990709i \(-0.456576\pi\)
0.135999 + 0.990709i \(0.456576\pi\)
\(644\) 0 0
\(645\) −1016.14 −0.0620319
\(646\) 0 0
\(647\) 28348.5 1.72256 0.861279 0.508133i \(-0.169664\pi\)
0.861279 + 0.508133i \(0.169664\pi\)
\(648\) 0 0
\(649\) 2189.90 0.132452
\(650\) 0 0
\(651\) −1076.89 −0.0648337
\(652\) 0 0
\(653\) 27303.9 1.63627 0.818134 0.575028i \(-0.195009\pi\)
0.818134 + 0.575028i \(0.195009\pi\)
\(654\) 0 0
\(655\) 7436.76 0.443631
\(656\) 0 0
\(657\) −11407.0 −0.677366
\(658\) 0 0
\(659\) −10101.6 −0.597123 −0.298562 0.954390i \(-0.596507\pi\)
−0.298562 + 0.954390i \(0.596507\pi\)
\(660\) 0 0
\(661\) 6035.78 0.355166 0.177583 0.984106i \(-0.443172\pi\)
0.177583 + 0.984106i \(0.443172\pi\)
\(662\) 0 0
\(663\) 2357.92 0.138120
\(664\) 0 0
\(665\) −5537.60 −0.322915
\(666\) 0 0
\(667\) 1657.17 0.0962010
\(668\) 0 0
\(669\) −1880.66 −0.108686
\(670\) 0 0
\(671\) −4242.18 −0.244065
\(672\) 0 0
\(673\) 29125.1 1.66819 0.834094 0.551622i \(-0.185991\pi\)
0.834094 + 0.551622i \(0.185991\pi\)
\(674\) 0 0
\(675\) −1953.72 −0.111405
\(676\) 0 0
\(677\) 23472.4 1.33252 0.666260 0.745720i \(-0.267894\pi\)
0.666260 + 0.745720i \(0.267894\pi\)
\(678\) 0 0
\(679\) 9930.60 0.561269
\(680\) 0 0
\(681\) 1545.08 0.0869421
\(682\) 0 0
\(683\) −13568.7 −0.760162 −0.380081 0.924953i \(-0.624104\pi\)
−0.380081 + 0.924953i \(0.624104\pi\)
\(684\) 0 0
\(685\) −3600.44 −0.200826
\(686\) 0 0
\(687\) 138.806 0.00770853
\(688\) 0 0
\(689\) −12172.4 −0.673052
\(690\) 0 0
\(691\) 3113.95 0.171433 0.0857166 0.996320i \(-0.472682\pi\)
0.0857166 + 0.996320i \(0.472682\pi\)
\(692\) 0 0
\(693\) −2058.40 −0.112831
\(694\) 0 0
\(695\) −3998.15 −0.218214
\(696\) 0 0
\(697\) 7318.57 0.397720
\(698\) 0 0
\(699\) −1866.46 −0.100996
\(700\) 0 0
\(701\) 29130.5 1.56953 0.784766 0.619792i \(-0.212783\pi\)
0.784766 + 0.619792i \(0.212783\pi\)
\(702\) 0 0
\(703\) 14634.3 0.785126
\(704\) 0 0
\(705\) 2246.24 0.119997
\(706\) 0 0
\(707\) 4299.20 0.228696
\(708\) 0 0
\(709\) −17391.2 −0.921213 −0.460606 0.887605i \(-0.652368\pi\)
−0.460606 + 0.887605i \(0.652368\pi\)
\(710\) 0 0
\(711\) 23691.8 1.24966
\(712\) 0 0
\(713\) −3533.75 −0.185610
\(714\) 0 0
\(715\) −6417.68 −0.335675
\(716\) 0 0
\(717\) −46.5205 −0.00242307
\(718\) 0 0
\(719\) 13940.8 0.723095 0.361548 0.932354i \(-0.382249\pi\)
0.361548 + 0.932354i \(0.382249\pi\)
\(720\) 0 0
\(721\) −4555.98 −0.235331
\(722\) 0 0
\(723\) 2633.49 0.135464
\(724\) 0 0
\(725\) −9802.94 −0.502168
\(726\) 0 0
\(727\) 34506.6 1.76035 0.880177 0.474645i \(-0.157424\pi\)
0.880177 + 0.474645i \(0.157424\pi\)
\(728\) 0 0
\(729\) −18522.2 −0.941025
\(730\) 0 0
\(731\) −15347.0 −0.776509
\(732\) 0 0
\(733\) −20926.4 −1.05448 −0.527240 0.849717i \(-0.676773\pi\)
−0.527240 + 0.849717i \(0.676773\pi\)
\(734\) 0 0
\(735\) 187.487 0.00940892
\(736\) 0 0
\(737\) 8493.82 0.424524
\(738\) 0 0
\(739\) −16313.8 −0.812061 −0.406030 0.913860i \(-0.633087\pi\)
−0.406030 + 0.913860i \(0.633087\pi\)
\(740\) 0 0
\(741\) −4363.87 −0.216344
\(742\) 0 0
\(743\) 11806.0 0.582933 0.291466 0.956581i \(-0.405857\pi\)
0.291466 + 0.956581i \(0.405857\pi\)
\(744\) 0 0
\(745\) −15413.3 −0.757984
\(746\) 0 0
\(747\) −2405.23 −0.117808
\(748\) 0 0
\(749\) 2922.73 0.142582
\(750\) 0 0
\(751\) −21437.1 −1.04161 −0.520806 0.853675i \(-0.674368\pi\)
−0.520806 + 0.853675i \(0.674368\pi\)
\(752\) 0 0
\(753\) −2865.48 −0.138677
\(754\) 0 0
\(755\) −13546.0 −0.652968
\(756\) 0 0
\(757\) 16573.8 0.795753 0.397876 0.917439i \(-0.369747\pi\)
0.397876 + 0.917439i \(0.369747\pi\)
\(758\) 0 0
\(759\) 67.6130 0.00323346
\(760\) 0 0
\(761\) −19353.4 −0.921892 −0.460946 0.887428i \(-0.652490\pi\)
−0.460946 + 0.887428i \(0.652490\pi\)
\(762\) 0 0
\(763\) −8826.32 −0.418787
\(764\) 0 0
\(765\) 11426.6 0.540039
\(766\) 0 0
\(767\) 15702.9 0.739242
\(768\) 0 0
\(769\) −17968.4 −0.842595 −0.421298 0.906922i \(-0.638425\pi\)
−0.421298 + 0.906922i \(0.638425\pi\)
\(770\) 0 0
\(771\) 2415.64 0.112837
\(772\) 0 0
\(773\) −23031.6 −1.07166 −0.535828 0.844327i \(-0.680000\pi\)
−0.535828 + 0.844327i \(0.680000\pi\)
\(774\) 0 0
\(775\) 20903.8 0.968884
\(776\) 0 0
\(777\) −495.475 −0.0228765
\(778\) 0 0
\(779\) −13544.7 −0.622965
\(780\) 0 0
\(781\) 1803.25 0.0826187
\(782\) 0 0
\(783\) 3876.52 0.176929
\(784\) 0 0
\(785\) 1691.25 0.0768957
\(786\) 0 0
\(787\) −10354.5 −0.468992 −0.234496 0.972117i \(-0.575344\pi\)
−0.234496 + 0.972117i \(0.575344\pi\)
\(788\) 0 0
\(789\) −1909.48 −0.0861587
\(790\) 0 0
\(791\) 3394.80 0.152598
\(792\) 0 0
\(793\) −30419.0 −1.36218
\(794\) 0 0
\(795\) 590.478 0.0263422
\(796\) 0 0
\(797\) −27274.5 −1.21219 −0.606093 0.795394i \(-0.707264\pi\)
−0.606093 + 0.795394i \(0.707264\pi\)
\(798\) 0 0
\(799\) 33925.2 1.50211
\(800\) 0 0
\(801\) −37201.4 −1.64101
\(802\) 0 0
\(803\) −4693.81 −0.206278
\(804\) 0 0
\(805\) 615.226 0.0269365
\(806\) 0 0
\(807\) 439.726 0.0191810
\(808\) 0 0
\(809\) 30138.1 1.30976 0.654881 0.755732i \(-0.272719\pi\)
0.654881 + 0.755732i \(0.272719\pi\)
\(810\) 0 0
\(811\) 1563.55 0.0676989 0.0338494 0.999427i \(-0.489223\pi\)
0.0338494 + 0.999427i \(0.489223\pi\)
\(812\) 0 0
\(813\) −3858.27 −0.166440
\(814\) 0 0
\(815\) −17626.2 −0.757570
\(816\) 0 0
\(817\) 28403.1 1.21628
\(818\) 0 0
\(819\) −14759.9 −0.629736
\(820\) 0 0
\(821\) 3178.02 0.135096 0.0675478 0.997716i \(-0.478482\pi\)
0.0675478 + 0.997716i \(0.478482\pi\)
\(822\) 0 0
\(823\) −18229.6 −0.772105 −0.386053 0.922477i \(-0.626162\pi\)
−0.386053 + 0.922477i \(0.626162\pi\)
\(824\) 0 0
\(825\) −399.961 −0.0168786
\(826\) 0 0
\(827\) −3016.88 −0.126853 −0.0634263 0.997987i \(-0.520203\pi\)
−0.0634263 + 0.997987i \(0.520203\pi\)
\(828\) 0 0
\(829\) 22135.9 0.927397 0.463699 0.885993i \(-0.346522\pi\)
0.463699 + 0.885993i \(0.346522\pi\)
\(830\) 0 0
\(831\) 2777.33 0.115938
\(832\) 0 0
\(833\) 2831.64 0.117780
\(834\) 0 0
\(835\) 19973.1 0.827781
\(836\) 0 0
\(837\) −8266.29 −0.341368
\(838\) 0 0
\(839\) 42000.8 1.72828 0.864140 0.503251i \(-0.167863\pi\)
0.864140 + 0.503251i \(0.167863\pi\)
\(840\) 0 0
\(841\) −4938.21 −0.202477
\(842\) 0 0
\(843\) −3351.25 −0.136920
\(844\) 0 0
\(845\) −29768.1 −1.21190
\(846\) 0 0
\(847\) −847.000 −0.0343604
\(848\) 0 0
\(849\) 1987.39 0.0803381
\(850\) 0 0
\(851\) −1625.87 −0.0654925
\(852\) 0 0
\(853\) −6205.24 −0.249078 −0.124539 0.992215i \(-0.539745\pi\)
−0.124539 + 0.992215i \(0.539745\pi\)
\(854\) 0 0
\(855\) −21147.6 −0.845886
\(856\) 0 0
\(857\) −8749.81 −0.348761 −0.174380 0.984678i \(-0.555792\pi\)
−0.174380 + 0.984678i \(0.555792\pi\)
\(858\) 0 0
\(859\) −37194.1 −1.47735 −0.738677 0.674059i \(-0.764549\pi\)
−0.738677 + 0.674059i \(0.764549\pi\)
\(860\) 0 0
\(861\) 458.585 0.0181516
\(862\) 0 0
\(863\) 17442.3 0.687997 0.343999 0.938970i \(-0.388218\pi\)
0.343999 + 0.938970i \(0.388218\pi\)
\(864\) 0 0
\(865\) −18987.7 −0.746358
\(866\) 0 0
\(867\) 813.949 0.0318837
\(868\) 0 0
\(869\) 9748.82 0.380560
\(870\) 0 0
\(871\) 60905.8 2.36936
\(872\) 0 0
\(873\) 37924.1 1.47026
\(874\) 0 0
\(875\) −10111.4 −0.390662
\(876\) 0 0
\(877\) 35990.5 1.38576 0.692882 0.721051i \(-0.256341\pi\)
0.692882 + 0.721051i \(0.256341\pi\)
\(878\) 0 0
\(879\) −1127.58 −0.0432677
\(880\) 0 0
\(881\) −5116.00 −0.195644 −0.0978220 0.995204i \(-0.531188\pi\)
−0.0978220 + 0.995204i \(0.531188\pi\)
\(882\) 0 0
\(883\) 3155.66 0.120268 0.0601340 0.998190i \(-0.480847\pi\)
0.0601340 + 0.998190i \(0.480847\pi\)
\(884\) 0 0
\(885\) −761.739 −0.0289328
\(886\) 0 0
\(887\) 30997.1 1.17337 0.586687 0.809814i \(-0.300432\pi\)
0.586687 + 0.809814i \(0.300432\pi\)
\(888\) 0 0
\(889\) 3284.84 0.123926
\(890\) 0 0
\(891\) −7781.36 −0.292576
\(892\) 0 0
\(893\) −62786.6 −2.35282
\(894\) 0 0
\(895\) 13388.1 0.500017
\(896\) 0 0
\(897\) 484.826 0.0180467
\(898\) 0 0
\(899\) −41476.8 −1.53874
\(900\) 0 0
\(901\) 8918.08 0.329749
\(902\) 0 0
\(903\) −961.647 −0.0354392
\(904\) 0 0
\(905\) 7742.49 0.284386
\(906\) 0 0
\(907\) 26954.0 0.986760 0.493380 0.869814i \(-0.335761\pi\)
0.493380 + 0.869814i \(0.335761\pi\)
\(908\) 0 0
\(909\) 16418.3 0.599076
\(910\) 0 0
\(911\) 35664.9 1.29707 0.648536 0.761184i \(-0.275382\pi\)
0.648536 + 0.761184i \(0.275382\pi\)
\(912\) 0 0
\(913\) −989.718 −0.0358761
\(914\) 0 0
\(915\) 1475.61 0.0533137
\(916\) 0 0
\(917\) 7037.93 0.253449
\(918\) 0 0
\(919\) 17489.2 0.627764 0.313882 0.949462i \(-0.398370\pi\)
0.313882 + 0.949462i \(0.398370\pi\)
\(920\) 0 0
\(921\) −3975.62 −0.142238
\(922\) 0 0
\(923\) 12930.4 0.461113
\(924\) 0 0
\(925\) 9617.76 0.341870
\(926\) 0 0
\(927\) −17398.9 −0.616456
\(928\) 0 0
\(929\) 50623.1 1.78783 0.893913 0.448240i \(-0.147949\pi\)
0.893913 + 0.448240i \(0.147949\pi\)
\(930\) 0 0
\(931\) −5240.61 −0.184484
\(932\) 0 0
\(933\) −1396.69 −0.0490092
\(934\) 0 0
\(935\) 4701.89 0.164458
\(936\) 0 0
\(937\) −33135.9 −1.15529 −0.577643 0.816290i \(-0.696027\pi\)
−0.577643 + 0.816290i \(0.696027\pi\)
\(938\) 0 0
\(939\) −1802.41 −0.0626406
\(940\) 0 0
\(941\) 18090.3 0.626701 0.313351 0.949638i \(-0.398548\pi\)
0.313351 + 0.949638i \(0.398548\pi\)
\(942\) 0 0
\(943\) 1504.82 0.0519656
\(944\) 0 0
\(945\) 1439.16 0.0495406
\(946\) 0 0
\(947\) 37534.9 1.28798 0.643991 0.765033i \(-0.277277\pi\)
0.643991 + 0.765033i \(0.277277\pi\)
\(948\) 0 0
\(949\) −33657.5 −1.15128
\(950\) 0 0
\(951\) 3.80704 0.000129812 0
\(952\) 0 0
\(953\) 5610.39 0.190701 0.0953507 0.995444i \(-0.469603\pi\)
0.0953507 + 0.995444i \(0.469603\pi\)
\(954\) 0 0
\(955\) 10265.6 0.347840
\(956\) 0 0
\(957\) 793.595 0.0268059
\(958\) 0 0
\(959\) −3407.35 −0.114733
\(960\) 0 0
\(961\) 58654.0 1.96885
\(962\) 0 0
\(963\) 11161.7 0.373499
\(964\) 0 0
\(965\) 3831.52 0.127815
\(966\) 0 0
\(967\) 43421.0 1.44398 0.721989 0.691905i \(-0.243228\pi\)
0.721989 + 0.691905i \(0.243228\pi\)
\(968\) 0 0
\(969\) 3197.17 0.105994
\(970\) 0 0
\(971\) 34695.7 1.14669 0.573346 0.819313i \(-0.305645\pi\)
0.573346 + 0.819313i \(0.305645\pi\)
\(972\) 0 0
\(973\) −3783.73 −0.124667
\(974\) 0 0
\(975\) −2867.96 −0.0942034
\(976\) 0 0
\(977\) −41221.1 −1.34983 −0.674914 0.737897i \(-0.735819\pi\)
−0.674914 + 0.737897i \(0.735819\pi\)
\(978\) 0 0
\(979\) −15307.8 −0.499735
\(980\) 0 0
\(981\) −33707.0 −1.09702
\(982\) 0 0
\(983\) −29994.9 −0.973232 −0.486616 0.873616i \(-0.661769\pi\)
−0.486616 + 0.873616i \(0.661769\pi\)
\(984\) 0 0
\(985\) 12081.9 0.390823
\(986\) 0 0
\(987\) 2125.77 0.0685552
\(988\) 0 0
\(989\) −3155.59 −0.101458
\(990\) 0 0
\(991\) −32670.1 −1.04722 −0.523612 0.851957i \(-0.675416\pi\)
−0.523612 + 0.851957i \(0.675416\pi\)
\(992\) 0 0
\(993\) 4217.41 0.134779
\(994\) 0 0
\(995\) −21888.5 −0.697399
\(996\) 0 0
\(997\) −10699.6 −0.339879 −0.169939 0.985454i \(-0.554357\pi\)
−0.169939 + 0.985454i \(0.554357\pi\)
\(998\) 0 0
\(999\) −3803.30 −0.120451
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.bc.1.4 7
4.3 odd 2 616.4.a.k.1.4 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
616.4.a.k.1.4 7 4.3 odd 2
1232.4.a.bc.1.4 7 1.1 even 1 trivial