Properties

Label 8001.2.a.n.1.3
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $12$
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] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.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: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} - 3 x^{10} + 41 x^{9} - 11 x^{8} - 123 x^{7} + 44 x^{6} + 159 x^{5} - 39 x^{4} - 71 x^{3} + 16 x^{2} + 7 x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 889)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.94650\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.946499 q^{2} -1.10414 q^{4} +2.90523 q^{5} +1.00000 q^{7} +2.93806 q^{8} +O(q^{10})\) \(q-0.946499 q^{2} -1.10414 q^{4} +2.90523 q^{5} +1.00000 q^{7} +2.93806 q^{8} -2.74979 q^{10} +4.94698 q^{11} +2.17643 q^{13} -0.946499 q^{14} -0.572592 q^{16} +3.30137 q^{17} -3.52914 q^{19} -3.20778 q^{20} -4.68231 q^{22} -0.742903 q^{23} +3.44035 q^{25} -2.05998 q^{26} -1.10414 q^{28} -0.319007 q^{29} -7.09600 q^{31} -5.33417 q^{32} -3.12475 q^{34} +2.90523 q^{35} +5.44695 q^{37} +3.34032 q^{38} +8.53575 q^{40} -4.63255 q^{41} +2.59387 q^{43} -5.46216 q^{44} +0.703156 q^{46} +2.05606 q^{47} +1.00000 q^{49} -3.25629 q^{50} -2.40308 q^{52} +8.76118 q^{53} +14.3721 q^{55} +2.93806 q^{56} +0.301940 q^{58} +12.4619 q^{59} +10.4232 q^{61} +6.71636 q^{62} +6.19397 q^{64} +6.32301 q^{65} +3.00987 q^{67} -3.64518 q^{68} -2.74979 q^{70} +15.6703 q^{71} +6.60269 q^{73} -5.15553 q^{74} +3.89666 q^{76} +4.94698 q^{77} +2.76302 q^{79} -1.66351 q^{80} +4.38470 q^{82} -11.9081 q^{83} +9.59124 q^{85} -2.45510 q^{86} +14.5346 q^{88} -10.9952 q^{89} +2.17643 q^{91} +0.820269 q^{92} -1.94606 q^{94} -10.2529 q^{95} -15.7659 q^{97} -0.946499 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31} + 27 q^{32} - 5 q^{34} + 7 q^{35} - 4 q^{37} - 2 q^{38} + 16 q^{40} + 21 q^{41} + 11 q^{43} + 11 q^{44} + 31 q^{47} + 12 q^{49} + 21 q^{50} + 3 q^{52} + 38 q^{53} - 11 q^{55} + 15 q^{56} + 20 q^{58} + 15 q^{59} - 3 q^{61} + 4 q^{62} + 29 q^{64} + 32 q^{65} - q^{67} - 17 q^{68} - 2 q^{70} + 57 q^{71} - 7 q^{73} + 42 q^{74} - 44 q^{76} + 22 q^{77} - 18 q^{79} - q^{80} + 56 q^{82} + 21 q^{83} - 5 q^{85} + 32 q^{86} - 10 q^{88} - 6 q^{89} + 15 q^{92} + 35 q^{94} + 57 q^{95} + 4 q^{97} + 7 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.946499 −0.669276 −0.334638 0.942347i \(-0.608614\pi\)
−0.334638 + 0.942347i \(0.608614\pi\)
\(3\) 0 0
\(4\) −1.10414 −0.552070
\(5\) 2.90523 1.29926 0.649629 0.760252i \(-0.274924\pi\)
0.649629 + 0.760252i \(0.274924\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 2.93806 1.03876
\(9\) 0 0
\(10\) −2.74979 −0.869561
\(11\) 4.94698 1.49157 0.745786 0.666186i \(-0.232074\pi\)
0.745786 + 0.666186i \(0.232074\pi\)
\(12\) 0 0
\(13\) 2.17643 0.603632 0.301816 0.953366i \(-0.402407\pi\)
0.301816 + 0.953366i \(0.402407\pi\)
\(14\) −0.946499 −0.252962
\(15\) 0 0
\(16\) −0.572592 −0.143148
\(17\) 3.30137 0.800701 0.400350 0.916362i \(-0.368888\pi\)
0.400350 + 0.916362i \(0.368888\pi\)
\(18\) 0 0
\(19\) −3.52914 −0.809640 −0.404820 0.914397i \(-0.632666\pi\)
−0.404820 + 0.914397i \(0.632666\pi\)
\(20\) −3.20778 −0.717281
\(21\) 0 0
\(22\) −4.68231 −0.998272
\(23\) −0.742903 −0.154906 −0.0774530 0.996996i \(-0.524679\pi\)
−0.0774530 + 0.996996i \(0.524679\pi\)
\(24\) 0 0
\(25\) 3.44035 0.688070
\(26\) −2.05998 −0.403996
\(27\) 0 0
\(28\) −1.10414 −0.208663
\(29\) −0.319007 −0.0592381 −0.0296191 0.999561i \(-0.509429\pi\)
−0.0296191 + 0.999561i \(0.509429\pi\)
\(30\) 0 0
\(31\) −7.09600 −1.27448 −0.637240 0.770665i \(-0.719924\pi\)
−0.637240 + 0.770665i \(0.719924\pi\)
\(32\) −5.33417 −0.942957
\(33\) 0 0
\(34\) −3.12475 −0.535889
\(35\) 2.90523 0.491073
\(36\) 0 0
\(37\) 5.44695 0.895474 0.447737 0.894165i \(-0.352230\pi\)
0.447737 + 0.894165i \(0.352230\pi\)
\(38\) 3.34032 0.541872
\(39\) 0 0
\(40\) 8.53575 1.34962
\(41\) −4.63255 −0.723482 −0.361741 0.932279i \(-0.617818\pi\)
−0.361741 + 0.932279i \(0.617818\pi\)
\(42\) 0 0
\(43\) 2.59387 0.395562 0.197781 0.980246i \(-0.436627\pi\)
0.197781 + 0.980246i \(0.436627\pi\)
\(44\) −5.46216 −0.823452
\(45\) 0 0
\(46\) 0.703156 0.103675
\(47\) 2.05606 0.299907 0.149954 0.988693i \(-0.452088\pi\)
0.149954 + 0.988693i \(0.452088\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −3.25629 −0.460508
\(51\) 0 0
\(52\) −2.40308 −0.333247
\(53\) 8.76118 1.20344 0.601720 0.798707i \(-0.294482\pi\)
0.601720 + 0.798707i \(0.294482\pi\)
\(54\) 0 0
\(55\) 14.3721 1.93794
\(56\) 2.93806 0.392615
\(57\) 0 0
\(58\) 0.301940 0.0396466
\(59\) 12.4619 1.62240 0.811199 0.584770i \(-0.198815\pi\)
0.811199 + 0.584770i \(0.198815\pi\)
\(60\) 0 0
\(61\) 10.4232 1.33455 0.667277 0.744809i \(-0.267460\pi\)
0.667277 + 0.744809i \(0.267460\pi\)
\(62\) 6.71636 0.852978
\(63\) 0 0
\(64\) 6.19397 0.774246
\(65\) 6.32301 0.784273
\(66\) 0 0
\(67\) 3.00987 0.367714 0.183857 0.982953i \(-0.441142\pi\)
0.183857 + 0.982953i \(0.441142\pi\)
\(68\) −3.64518 −0.442043
\(69\) 0 0
\(70\) −2.74979 −0.328663
\(71\) 15.6703 1.85972 0.929859 0.367916i \(-0.119928\pi\)
0.929859 + 0.367916i \(0.119928\pi\)
\(72\) 0 0
\(73\) 6.60269 0.772787 0.386393 0.922334i \(-0.373721\pi\)
0.386393 + 0.922334i \(0.373721\pi\)
\(74\) −5.15553 −0.599319
\(75\) 0 0
\(76\) 3.89666 0.446978
\(77\) 4.94698 0.563761
\(78\) 0 0
\(79\) 2.76302 0.310865 0.155432 0.987847i \(-0.450323\pi\)
0.155432 + 0.987847i \(0.450323\pi\)
\(80\) −1.66351 −0.185986
\(81\) 0 0
\(82\) 4.38470 0.484209
\(83\) −11.9081 −1.30709 −0.653544 0.756888i \(-0.726719\pi\)
−0.653544 + 0.756888i \(0.726719\pi\)
\(84\) 0 0
\(85\) 9.59124 1.04032
\(86\) −2.45510 −0.264740
\(87\) 0 0
\(88\) 14.5346 1.54939
\(89\) −10.9952 −1.16549 −0.582746 0.812654i \(-0.698022\pi\)
−0.582746 + 0.812654i \(0.698022\pi\)
\(90\) 0 0
\(91\) 2.17643 0.228151
\(92\) 0.820269 0.0855190
\(93\) 0 0
\(94\) −1.94606 −0.200721
\(95\) −10.2529 −1.05193
\(96\) 0 0
\(97\) −15.7659 −1.60078 −0.800391 0.599478i \(-0.795375\pi\)
−0.800391 + 0.599478i \(0.795375\pi\)
\(98\) −0.946499 −0.0956108
\(99\) 0 0
\(100\) −3.79863 −0.379863
\(101\) −11.5305 −1.14733 −0.573665 0.819090i \(-0.694479\pi\)
−0.573665 + 0.819090i \(0.694479\pi\)
\(102\) 0 0
\(103\) 6.90350 0.680222 0.340111 0.940385i \(-0.389535\pi\)
0.340111 + 0.940385i \(0.389535\pi\)
\(104\) 6.39448 0.627030
\(105\) 0 0
\(106\) −8.29244 −0.805433
\(107\) −0.262006 −0.0253290 −0.0126645 0.999920i \(-0.504031\pi\)
−0.0126645 + 0.999920i \(0.504031\pi\)
\(108\) 0 0
\(109\) −10.3229 −0.988753 −0.494377 0.869248i \(-0.664604\pi\)
−0.494377 + 0.869248i \(0.664604\pi\)
\(110\) −13.6032 −1.29701
\(111\) 0 0
\(112\) −0.572592 −0.0541049
\(113\) 9.52058 0.895621 0.447811 0.894128i \(-0.352204\pi\)
0.447811 + 0.894128i \(0.352204\pi\)
\(114\) 0 0
\(115\) −2.15830 −0.201263
\(116\) 0.352229 0.0327036
\(117\) 0 0
\(118\) −11.7951 −1.08583
\(119\) 3.30137 0.302636
\(120\) 0 0
\(121\) 13.4726 1.22479
\(122\) −9.86555 −0.893185
\(123\) 0 0
\(124\) 7.83499 0.703603
\(125\) −4.53114 −0.405277
\(126\) 0 0
\(127\) 1.00000 0.0887357
\(128\) 4.80576 0.424773
\(129\) 0 0
\(130\) −5.98472 −0.524895
\(131\) 2.11923 0.185158 0.0925791 0.995705i \(-0.470489\pi\)
0.0925791 + 0.995705i \(0.470489\pi\)
\(132\) 0 0
\(133\) −3.52914 −0.306015
\(134\) −2.84883 −0.246102
\(135\) 0 0
\(136\) 9.69965 0.831738
\(137\) −1.54986 −0.132414 −0.0662069 0.997806i \(-0.521090\pi\)
−0.0662069 + 0.997806i \(0.521090\pi\)
\(138\) 0 0
\(139\) 9.95651 0.844500 0.422250 0.906479i \(-0.361240\pi\)
0.422250 + 0.906479i \(0.361240\pi\)
\(140\) −3.20778 −0.271107
\(141\) 0 0
\(142\) −14.8319 −1.24466
\(143\) 10.7667 0.900360
\(144\) 0 0
\(145\) −0.926788 −0.0769656
\(146\) −6.24944 −0.517207
\(147\) 0 0
\(148\) −6.01420 −0.494365
\(149\) −7.29141 −0.597335 −0.298668 0.954357i \(-0.596542\pi\)
−0.298668 + 0.954357i \(0.596542\pi\)
\(150\) 0 0
\(151\) −14.8054 −1.20485 −0.602424 0.798176i \(-0.705798\pi\)
−0.602424 + 0.798176i \(0.705798\pi\)
\(152\) −10.3688 −0.841023
\(153\) 0 0
\(154\) −4.68231 −0.377311
\(155\) −20.6155 −1.65588
\(156\) 0 0
\(157\) −21.6316 −1.72639 −0.863196 0.504868i \(-0.831541\pi\)
−0.863196 + 0.504868i \(0.831541\pi\)
\(158\) −2.61520 −0.208054
\(159\) 0 0
\(160\) −15.4970 −1.22514
\(161\) −0.742903 −0.0585489
\(162\) 0 0
\(163\) −3.35133 −0.262496 −0.131248 0.991350i \(-0.541898\pi\)
−0.131248 + 0.991350i \(0.541898\pi\)
\(164\) 5.11498 0.399413
\(165\) 0 0
\(166\) 11.2710 0.874802
\(167\) 9.37694 0.725609 0.362805 0.931865i \(-0.381819\pi\)
0.362805 + 0.931865i \(0.381819\pi\)
\(168\) 0 0
\(169\) −8.26317 −0.635629
\(170\) −9.07810 −0.696258
\(171\) 0 0
\(172\) −2.86400 −0.218378
\(173\) 7.49955 0.570180 0.285090 0.958501i \(-0.407977\pi\)
0.285090 + 0.958501i \(0.407977\pi\)
\(174\) 0 0
\(175\) 3.44035 0.260066
\(176\) −2.83261 −0.213516
\(177\) 0 0
\(178\) 10.4070 0.780035
\(179\) 8.83499 0.660358 0.330179 0.943918i \(-0.392891\pi\)
0.330179 + 0.943918i \(0.392891\pi\)
\(180\) 0 0
\(181\) −3.61913 −0.269008 −0.134504 0.990913i \(-0.542944\pi\)
−0.134504 + 0.990913i \(0.542944\pi\)
\(182\) −2.05998 −0.152696
\(183\) 0 0
\(184\) −2.18270 −0.160911
\(185\) 15.8246 1.16345
\(186\) 0 0
\(187\) 16.3318 1.19430
\(188\) −2.27018 −0.165570
\(189\) 0 0
\(190\) 9.70440 0.704031
\(191\) 7.37068 0.533324 0.266662 0.963790i \(-0.414079\pi\)
0.266662 + 0.963790i \(0.414079\pi\)
\(192\) 0 0
\(193\) 6.10872 0.439715 0.219858 0.975532i \(-0.429441\pi\)
0.219858 + 0.975532i \(0.429441\pi\)
\(194\) 14.9224 1.07136
\(195\) 0 0
\(196\) −1.10414 −0.0788672
\(197\) 23.9507 1.70642 0.853210 0.521568i \(-0.174653\pi\)
0.853210 + 0.521568i \(0.174653\pi\)
\(198\) 0 0
\(199\) 8.99542 0.637669 0.318834 0.947810i \(-0.396709\pi\)
0.318834 + 0.947810i \(0.396709\pi\)
\(200\) 10.1080 0.714741
\(201\) 0 0
\(202\) 10.9136 0.767880
\(203\) −0.319007 −0.0223899
\(204\) 0 0
\(205\) −13.4586 −0.939990
\(206\) −6.53415 −0.455256
\(207\) 0 0
\(208\) −1.24620 −0.0864087
\(209\) −17.4586 −1.20764
\(210\) 0 0
\(211\) 8.24920 0.567898 0.283949 0.958839i \(-0.408355\pi\)
0.283949 + 0.958839i \(0.408355\pi\)
\(212\) −9.67357 −0.664383
\(213\) 0 0
\(214\) 0.247988 0.0169521
\(215\) 7.53579 0.513936
\(216\) 0 0
\(217\) −7.09600 −0.481708
\(218\) 9.77060 0.661748
\(219\) 0 0
\(220\) −15.8688 −1.06988
\(221\) 7.18519 0.483328
\(222\) 0 0
\(223\) 5.19009 0.347554 0.173777 0.984785i \(-0.444403\pi\)
0.173777 + 0.984785i \(0.444403\pi\)
\(224\) −5.33417 −0.356404
\(225\) 0 0
\(226\) −9.01122 −0.599417
\(227\) 3.74134 0.248321 0.124161 0.992262i \(-0.460376\pi\)
0.124161 + 0.992262i \(0.460376\pi\)
\(228\) 0 0
\(229\) −9.06533 −0.599054 −0.299527 0.954088i \(-0.596829\pi\)
−0.299527 + 0.954088i \(0.596829\pi\)
\(230\) 2.04283 0.134700
\(231\) 0 0
\(232\) −0.937263 −0.0615343
\(233\) 5.38719 0.352927 0.176463 0.984307i \(-0.443534\pi\)
0.176463 + 0.984307i \(0.443534\pi\)
\(234\) 0 0
\(235\) 5.97333 0.389657
\(236\) −13.7597 −0.895678
\(237\) 0 0
\(238\) −3.12475 −0.202547
\(239\) −20.2277 −1.30842 −0.654212 0.756311i \(-0.727000\pi\)
−0.654212 + 0.756311i \(0.727000\pi\)
\(240\) 0 0
\(241\) −10.7927 −0.695221 −0.347610 0.937639i \(-0.613007\pi\)
−0.347610 + 0.937639i \(0.613007\pi\)
\(242\) −12.7518 −0.819719
\(243\) 0 0
\(244\) −11.5087 −0.736768
\(245\) 2.90523 0.185608
\(246\) 0 0
\(247\) −7.68090 −0.488724
\(248\) −20.8485 −1.32388
\(249\) 0 0
\(250\) 4.28872 0.271242
\(251\) 4.47720 0.282599 0.141299 0.989967i \(-0.454872\pi\)
0.141299 + 0.989967i \(0.454872\pi\)
\(252\) 0 0
\(253\) −3.67513 −0.231053
\(254\) −0.946499 −0.0593886
\(255\) 0 0
\(256\) −16.9366 −1.05854
\(257\) −14.0925 −0.879064 −0.439532 0.898227i \(-0.644856\pi\)
−0.439532 + 0.898227i \(0.644856\pi\)
\(258\) 0 0
\(259\) 5.44695 0.338457
\(260\) −6.98149 −0.432974
\(261\) 0 0
\(262\) −2.00585 −0.123922
\(263\) 26.2405 1.61806 0.809028 0.587770i \(-0.199994\pi\)
0.809028 + 0.587770i \(0.199994\pi\)
\(264\) 0 0
\(265\) 25.4532 1.56358
\(266\) 3.34032 0.204808
\(267\) 0 0
\(268\) −3.32332 −0.203004
\(269\) −25.1525 −1.53357 −0.766787 0.641901i \(-0.778146\pi\)
−0.766787 + 0.641901i \(0.778146\pi\)
\(270\) 0 0
\(271\) −28.8886 −1.75486 −0.877428 0.479709i \(-0.840742\pi\)
−0.877428 + 0.479709i \(0.840742\pi\)
\(272\) −1.89034 −0.114619
\(273\) 0 0
\(274\) 1.46694 0.0886213
\(275\) 17.0194 1.02631
\(276\) 0 0
\(277\) 13.0555 0.784431 0.392216 0.919873i \(-0.371709\pi\)
0.392216 + 0.919873i \(0.371709\pi\)
\(278\) −9.42382 −0.565203
\(279\) 0 0
\(280\) 8.53575 0.510108
\(281\) 18.9153 1.12839 0.564197 0.825640i \(-0.309186\pi\)
0.564197 + 0.825640i \(0.309186\pi\)
\(282\) 0 0
\(283\) 19.0469 1.13222 0.566112 0.824328i \(-0.308447\pi\)
0.566112 + 0.824328i \(0.308447\pi\)
\(284\) −17.3022 −1.02670
\(285\) 0 0
\(286\) −10.1907 −0.602589
\(287\) −4.63255 −0.273451
\(288\) 0 0
\(289\) −6.10093 −0.358878
\(290\) 0.877204 0.0515112
\(291\) 0 0
\(292\) −7.29030 −0.426633
\(293\) −19.7793 −1.15552 −0.577759 0.816207i \(-0.696073\pi\)
−0.577759 + 0.816207i \(0.696073\pi\)
\(294\) 0 0
\(295\) 36.2046 2.10791
\(296\) 16.0035 0.930185
\(297\) 0 0
\(298\) 6.90131 0.399782
\(299\) −1.61687 −0.0935061
\(300\) 0 0
\(301\) 2.59387 0.149508
\(302\) 14.0133 0.806375
\(303\) 0 0
\(304\) 2.02076 0.115898
\(305\) 30.2818 1.73393
\(306\) 0 0
\(307\) 7.97628 0.455230 0.227615 0.973751i \(-0.426907\pi\)
0.227615 + 0.973751i \(0.426907\pi\)
\(308\) −5.46216 −0.311236
\(309\) 0 0
\(310\) 19.5126 1.10824
\(311\) −8.94253 −0.507084 −0.253542 0.967324i \(-0.581596\pi\)
−0.253542 + 0.967324i \(0.581596\pi\)
\(312\) 0 0
\(313\) 3.35550 0.189664 0.0948320 0.995493i \(-0.469769\pi\)
0.0948320 + 0.995493i \(0.469769\pi\)
\(314\) 20.4743 1.15543
\(315\) 0 0
\(316\) −3.05077 −0.171619
\(317\) 32.1693 1.80681 0.903405 0.428788i \(-0.141059\pi\)
0.903405 + 0.428788i \(0.141059\pi\)
\(318\) 0 0
\(319\) −1.57812 −0.0883579
\(320\) 17.9949 1.00595
\(321\) 0 0
\(322\) 0.703156 0.0391854
\(323\) −11.6510 −0.648279
\(324\) 0 0
\(325\) 7.48766 0.415341
\(326\) 3.17202 0.175682
\(327\) 0 0
\(328\) −13.6107 −0.751526
\(329\) 2.05606 0.113354
\(330\) 0 0
\(331\) −8.65680 −0.475821 −0.237910 0.971287i \(-0.576462\pi\)
−0.237910 + 0.971287i \(0.576462\pi\)
\(332\) 13.1483 0.721604
\(333\) 0 0
\(334\) −8.87526 −0.485632
\(335\) 8.74435 0.477755
\(336\) 0 0
\(337\) 8.57382 0.467046 0.233523 0.972351i \(-0.424975\pi\)
0.233523 + 0.972351i \(0.424975\pi\)
\(338\) 7.82108 0.425411
\(339\) 0 0
\(340\) −10.5901 −0.574328
\(341\) −35.1038 −1.90098
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 7.62096 0.410895
\(345\) 0 0
\(346\) −7.09831 −0.381607
\(347\) 7.60641 0.408333 0.204167 0.978936i \(-0.434552\pi\)
0.204167 + 0.978936i \(0.434552\pi\)
\(348\) 0 0
\(349\) 31.7137 1.69760 0.848798 0.528718i \(-0.177327\pi\)
0.848798 + 0.528718i \(0.177327\pi\)
\(350\) −3.25629 −0.174056
\(351\) 0 0
\(352\) −26.3881 −1.40649
\(353\) −9.13856 −0.486397 −0.243198 0.969977i \(-0.578197\pi\)
−0.243198 + 0.969977i \(0.578197\pi\)
\(354\) 0 0
\(355\) 45.5257 2.41625
\(356\) 12.1403 0.643433
\(357\) 0 0
\(358\) −8.36230 −0.441962
\(359\) −20.8619 −1.10105 −0.550525 0.834819i \(-0.685572\pi\)
−0.550525 + 0.834819i \(0.685572\pi\)
\(360\) 0 0
\(361\) −6.54519 −0.344484
\(362\) 3.42550 0.180041
\(363\) 0 0
\(364\) −2.40308 −0.125956
\(365\) 19.1823 1.00405
\(366\) 0 0
\(367\) −33.9477 −1.77206 −0.886028 0.463631i \(-0.846546\pi\)
−0.886028 + 0.463631i \(0.846546\pi\)
\(368\) 0.425381 0.0221745
\(369\) 0 0
\(370\) −14.9780 −0.778669
\(371\) 8.76118 0.454858
\(372\) 0 0
\(373\) −25.1597 −1.30272 −0.651360 0.758769i \(-0.725801\pi\)
−0.651360 + 0.758769i \(0.725801\pi\)
\(374\) −15.4581 −0.799317
\(375\) 0 0
\(376\) 6.04084 0.311533
\(377\) −0.694295 −0.0357580
\(378\) 0 0
\(379\) 30.6120 1.57243 0.786216 0.617951i \(-0.212037\pi\)
0.786216 + 0.617951i \(0.212037\pi\)
\(380\) 11.3207 0.580739
\(381\) 0 0
\(382\) −6.97634 −0.356941
\(383\) 38.1453 1.94913 0.974567 0.224098i \(-0.0719435\pi\)
0.974567 + 0.224098i \(0.0719435\pi\)
\(384\) 0 0
\(385\) 14.3721 0.732471
\(386\) −5.78189 −0.294291
\(387\) 0 0
\(388\) 17.4077 0.883744
\(389\) 12.6250 0.640115 0.320057 0.947398i \(-0.396298\pi\)
0.320057 + 0.947398i \(0.396298\pi\)
\(390\) 0 0
\(391\) −2.45260 −0.124033
\(392\) 2.93806 0.148395
\(393\) 0 0
\(394\) −22.6693 −1.14206
\(395\) 8.02722 0.403893
\(396\) 0 0
\(397\) −28.5214 −1.43145 −0.715723 0.698384i \(-0.753903\pi\)
−0.715723 + 0.698384i \(0.753903\pi\)
\(398\) −8.51415 −0.426776
\(399\) 0 0
\(400\) −1.96992 −0.0984959
\(401\) −37.6239 −1.87885 −0.939424 0.342758i \(-0.888639\pi\)
−0.939424 + 0.342758i \(0.888639\pi\)
\(402\) 0 0
\(403\) −15.4439 −0.769317
\(404\) 12.7313 0.633407
\(405\) 0 0
\(406\) 0.301940 0.0149850
\(407\) 26.9460 1.33566
\(408\) 0 0
\(409\) −10.0228 −0.495597 −0.247799 0.968812i \(-0.579707\pi\)
−0.247799 + 0.968812i \(0.579707\pi\)
\(410\) 12.7386 0.629112
\(411\) 0 0
\(412\) −7.62243 −0.375530
\(413\) 12.4619 0.613209
\(414\) 0 0
\(415\) −34.5959 −1.69824
\(416\) −11.6094 −0.569199
\(417\) 0 0
\(418\) 16.5245 0.808241
\(419\) −15.8944 −0.776492 −0.388246 0.921556i \(-0.626919\pi\)
−0.388246 + 0.921556i \(0.626919\pi\)
\(420\) 0 0
\(421\) 24.9881 1.21784 0.608922 0.793230i \(-0.291602\pi\)
0.608922 + 0.793230i \(0.291602\pi\)
\(422\) −7.80785 −0.380080
\(423\) 0 0
\(424\) 25.7409 1.25009
\(425\) 11.3579 0.550938
\(426\) 0 0
\(427\) 10.4232 0.504414
\(428\) 0.289291 0.0139834
\(429\) 0 0
\(430\) −7.13261 −0.343965
\(431\) 21.3467 1.02823 0.514117 0.857720i \(-0.328120\pi\)
0.514117 + 0.857720i \(0.328120\pi\)
\(432\) 0 0
\(433\) 13.8838 0.667213 0.333607 0.942712i \(-0.391734\pi\)
0.333607 + 0.942712i \(0.391734\pi\)
\(434\) 6.71636 0.322395
\(435\) 0 0
\(436\) 11.3979 0.545861
\(437\) 2.62181 0.125418
\(438\) 0 0
\(439\) 5.27790 0.251900 0.125950 0.992037i \(-0.459802\pi\)
0.125950 + 0.992037i \(0.459802\pi\)
\(440\) 42.2262 2.01306
\(441\) 0 0
\(442\) −6.80077 −0.323480
\(443\) −1.78515 −0.0848152 −0.0424076 0.999100i \(-0.513503\pi\)
−0.0424076 + 0.999100i \(0.513503\pi\)
\(444\) 0 0
\(445\) −31.9436 −1.51427
\(446\) −4.91241 −0.232609
\(447\) 0 0
\(448\) 6.19397 0.292638
\(449\) 40.5984 1.91596 0.957979 0.286837i \(-0.0926038\pi\)
0.957979 + 0.286837i \(0.0926038\pi\)
\(450\) 0 0
\(451\) −22.9171 −1.07913
\(452\) −10.5121 −0.494446
\(453\) 0 0
\(454\) −3.54117 −0.166195
\(455\) 6.32301 0.296427
\(456\) 0 0
\(457\) −7.95828 −0.372273 −0.186136 0.982524i \(-0.559597\pi\)
−0.186136 + 0.982524i \(0.559597\pi\)
\(458\) 8.58032 0.400932
\(459\) 0 0
\(460\) 2.38307 0.111111
\(461\) 23.4497 1.09216 0.546080 0.837733i \(-0.316120\pi\)
0.546080 + 0.837733i \(0.316120\pi\)
\(462\) 0 0
\(463\) −10.8328 −0.503441 −0.251721 0.967800i \(-0.580996\pi\)
−0.251721 + 0.967800i \(0.580996\pi\)
\(464\) 0.182661 0.00847983
\(465\) 0 0
\(466\) −5.09897 −0.236205
\(467\) 34.7902 1.60990 0.804948 0.593345i \(-0.202193\pi\)
0.804948 + 0.593345i \(0.202193\pi\)
\(468\) 0 0
\(469\) 3.00987 0.138983
\(470\) −5.65374 −0.260788
\(471\) 0 0
\(472\) 36.6138 1.68529
\(473\) 12.8318 0.590009
\(474\) 0 0
\(475\) −12.1415 −0.557089
\(476\) −3.64518 −0.167077
\(477\) 0 0
\(478\) 19.1455 0.875696
\(479\) −10.0138 −0.457543 −0.228772 0.973480i \(-0.573471\pi\)
−0.228772 + 0.973480i \(0.573471\pi\)
\(480\) 0 0
\(481\) 11.8549 0.540536
\(482\) 10.2153 0.465294
\(483\) 0 0
\(484\) −14.8757 −0.676168
\(485\) −45.8035 −2.07983
\(486\) 0 0
\(487\) −32.0140 −1.45069 −0.725347 0.688384i \(-0.758320\pi\)
−0.725347 + 0.688384i \(0.758320\pi\)
\(488\) 30.6240 1.38629
\(489\) 0 0
\(490\) −2.74979 −0.124223
\(491\) −12.2449 −0.552604 −0.276302 0.961071i \(-0.589109\pi\)
−0.276302 + 0.961071i \(0.589109\pi\)
\(492\) 0 0
\(493\) −1.05316 −0.0474320
\(494\) 7.26996 0.327091
\(495\) 0 0
\(496\) 4.06312 0.182439
\(497\) 15.6703 0.702907
\(498\) 0 0
\(499\) −17.5460 −0.785465 −0.392732 0.919653i \(-0.628470\pi\)
−0.392732 + 0.919653i \(0.628470\pi\)
\(500\) 5.00302 0.223742
\(501\) 0 0
\(502\) −4.23767 −0.189136
\(503\) −1.16673 −0.0520218 −0.0260109 0.999662i \(-0.508280\pi\)
−0.0260109 + 0.999662i \(0.508280\pi\)
\(504\) 0 0
\(505\) −33.4988 −1.49068
\(506\) 3.47850 0.154638
\(507\) 0 0
\(508\) −1.10414 −0.0489883
\(509\) −28.2068 −1.25025 −0.625123 0.780526i \(-0.714951\pi\)
−0.625123 + 0.780526i \(0.714951\pi\)
\(510\) 0 0
\(511\) 6.60269 0.292086
\(512\) 6.41893 0.283679
\(513\) 0 0
\(514\) 13.3385 0.588336
\(515\) 20.0562 0.883783
\(516\) 0 0
\(517\) 10.1713 0.447333
\(518\) −5.15553 −0.226521
\(519\) 0 0
\(520\) 18.5774 0.814674
\(521\) −24.2587 −1.06279 −0.531397 0.847123i \(-0.678333\pi\)
−0.531397 + 0.847123i \(0.678333\pi\)
\(522\) 0 0
\(523\) 38.2576 1.67289 0.836443 0.548053i \(-0.184631\pi\)
0.836443 + 0.548053i \(0.184631\pi\)
\(524\) −2.33993 −0.102220
\(525\) 0 0
\(526\) −24.8366 −1.08293
\(527\) −23.4266 −1.02048
\(528\) 0 0
\(529\) −22.4481 −0.976004
\(530\) −24.0914 −1.04646
\(531\) 0 0
\(532\) 3.89666 0.168942
\(533\) −10.0824 −0.436717
\(534\) 0 0
\(535\) −0.761186 −0.0329089
\(536\) 8.84318 0.381967
\(537\) 0 0
\(538\) 23.8068 1.02638
\(539\) 4.94698 0.213082
\(540\) 0 0
\(541\) 8.38204 0.360372 0.180186 0.983633i \(-0.442330\pi\)
0.180186 + 0.983633i \(0.442330\pi\)
\(542\) 27.3430 1.17448
\(543\) 0 0
\(544\) −17.6101 −0.755026
\(545\) −29.9903 −1.28465
\(546\) 0 0
\(547\) −22.8688 −0.977798 −0.488899 0.872340i \(-0.662601\pi\)
−0.488899 + 0.872340i \(0.662601\pi\)
\(548\) 1.71127 0.0731017
\(549\) 0 0
\(550\) −16.1088 −0.686881
\(551\) 1.12582 0.0479615
\(552\) 0 0
\(553\) 2.76302 0.117496
\(554\) −12.3570 −0.525000
\(555\) 0 0
\(556\) −10.9934 −0.466223
\(557\) −21.8641 −0.926413 −0.463206 0.886250i \(-0.653301\pi\)
−0.463206 + 0.886250i \(0.653301\pi\)
\(558\) 0 0
\(559\) 5.64537 0.238774
\(560\) −1.66351 −0.0702962
\(561\) 0 0
\(562\) −17.9033 −0.755206
\(563\) 24.1831 1.01920 0.509599 0.860412i \(-0.329794\pi\)
0.509599 + 0.860412i \(0.329794\pi\)
\(564\) 0 0
\(565\) 27.6595 1.16364
\(566\) −18.0279 −0.757770
\(567\) 0 0
\(568\) 46.0402 1.93181
\(569\) −31.8186 −1.33391 −0.666953 0.745099i \(-0.732402\pi\)
−0.666953 + 0.745099i \(0.732402\pi\)
\(570\) 0 0
\(571\) 42.5806 1.78194 0.890972 0.454058i \(-0.150024\pi\)
0.890972 + 0.454058i \(0.150024\pi\)
\(572\) −11.8880 −0.497062
\(573\) 0 0
\(574\) 4.38470 0.183014
\(575\) −2.55585 −0.106586
\(576\) 0 0
\(577\) 18.6971 0.778372 0.389186 0.921159i \(-0.372756\pi\)
0.389186 + 0.921159i \(0.372756\pi\)
\(578\) 5.77452 0.240188
\(579\) 0 0
\(580\) 1.02330 0.0424904
\(581\) −11.9081 −0.494033
\(582\) 0 0
\(583\) 43.3414 1.79502
\(584\) 19.3991 0.802742
\(585\) 0 0
\(586\) 18.7211 0.773360
\(587\) 21.6873 0.895131 0.447565 0.894251i \(-0.352291\pi\)
0.447565 + 0.894251i \(0.352291\pi\)
\(588\) 0 0
\(589\) 25.0428 1.03187
\(590\) −34.2676 −1.41077
\(591\) 0 0
\(592\) −3.11889 −0.128185
\(593\) −37.1088 −1.52388 −0.761939 0.647649i \(-0.775752\pi\)
−0.761939 + 0.647649i \(0.775752\pi\)
\(594\) 0 0
\(595\) 9.59124 0.393203
\(596\) 8.05074 0.329771
\(597\) 0 0
\(598\) 1.53037 0.0625814
\(599\) 26.1898 1.07009 0.535043 0.844825i \(-0.320295\pi\)
0.535043 + 0.844825i \(0.320295\pi\)
\(600\) 0 0
\(601\) 12.5840 0.513311 0.256655 0.966503i \(-0.417379\pi\)
0.256655 + 0.966503i \(0.417379\pi\)
\(602\) −2.45510 −0.100062
\(603\) 0 0
\(604\) 16.3473 0.665161
\(605\) 39.1411 1.59131
\(606\) 0 0
\(607\) −44.5936 −1.81000 −0.904999 0.425413i \(-0.860129\pi\)
−0.904999 + 0.425413i \(0.860129\pi\)
\(608\) 18.8250 0.763455
\(609\) 0 0
\(610\) −28.6617 −1.16048
\(611\) 4.47486 0.181034
\(612\) 0 0
\(613\) 15.8923 0.641885 0.320942 0.947099i \(-0.396000\pi\)
0.320942 + 0.947099i \(0.396000\pi\)
\(614\) −7.54954 −0.304675
\(615\) 0 0
\(616\) 14.5346 0.585614
\(617\) 25.7330 1.03597 0.517985 0.855390i \(-0.326682\pi\)
0.517985 + 0.855390i \(0.326682\pi\)
\(618\) 0 0
\(619\) −33.1354 −1.33182 −0.665912 0.746030i \(-0.731957\pi\)
−0.665912 + 0.746030i \(0.731957\pi\)
\(620\) 22.7624 0.914161
\(621\) 0 0
\(622\) 8.46409 0.339379
\(623\) −10.9952 −0.440515
\(624\) 0 0
\(625\) −30.3657 −1.21463
\(626\) −3.17598 −0.126938
\(627\) 0 0
\(628\) 23.8844 0.953090
\(629\) 17.9824 0.717007
\(630\) 0 0
\(631\) 19.5658 0.778902 0.389451 0.921047i \(-0.372665\pi\)
0.389451 + 0.921047i \(0.372665\pi\)
\(632\) 8.11795 0.322915
\(633\) 0 0
\(634\) −30.4482 −1.20925
\(635\) 2.90523 0.115290
\(636\) 0 0
\(637\) 2.17643 0.0862331
\(638\) 1.49369 0.0591358
\(639\) 0 0
\(640\) 13.9618 0.551890
\(641\) 11.2433 0.444085 0.222043 0.975037i \(-0.428728\pi\)
0.222043 + 0.975037i \(0.428728\pi\)
\(642\) 0 0
\(643\) 6.66730 0.262933 0.131466 0.991321i \(-0.458031\pi\)
0.131466 + 0.991321i \(0.458031\pi\)
\(644\) 0.820269 0.0323231
\(645\) 0 0
\(646\) 11.0277 0.433877
\(647\) 3.29425 0.129510 0.0647551 0.997901i \(-0.479373\pi\)
0.0647551 + 0.997901i \(0.479373\pi\)
\(648\) 0 0
\(649\) 61.6487 2.41992
\(650\) −7.08706 −0.277977
\(651\) 0 0
\(652\) 3.70033 0.144916
\(653\) −44.1351 −1.72714 −0.863570 0.504230i \(-0.831777\pi\)
−0.863570 + 0.504230i \(0.831777\pi\)
\(654\) 0 0
\(655\) 6.15685 0.240568
\(656\) 2.65256 0.103565
\(657\) 0 0
\(658\) −1.94606 −0.0758653
\(659\) −26.9020 −1.04795 −0.523976 0.851733i \(-0.675552\pi\)
−0.523976 + 0.851733i \(0.675552\pi\)
\(660\) 0 0
\(661\) 19.1836 0.746157 0.373079 0.927800i \(-0.378302\pi\)
0.373079 + 0.927800i \(0.378302\pi\)
\(662\) 8.19364 0.318455
\(663\) 0 0
\(664\) −34.9869 −1.35775
\(665\) −10.2529 −0.397592
\(666\) 0 0
\(667\) 0.236991 0.00917634
\(668\) −10.3535 −0.400587
\(669\) 0 0
\(670\) −8.27651 −0.319750
\(671\) 51.5634 1.99058
\(672\) 0 0
\(673\) 1.61017 0.0620673 0.0310337 0.999518i \(-0.490120\pi\)
0.0310337 + 0.999518i \(0.490120\pi\)
\(674\) −8.11511 −0.312582
\(675\) 0 0
\(676\) 9.12370 0.350912
\(677\) 19.9709 0.767545 0.383773 0.923428i \(-0.374625\pi\)
0.383773 + 0.923428i \(0.374625\pi\)
\(678\) 0 0
\(679\) −15.7659 −0.605039
\(680\) 28.1797 1.08064
\(681\) 0 0
\(682\) 33.2257 1.27228
\(683\) −2.10058 −0.0803765 −0.0401882 0.999192i \(-0.512796\pi\)
−0.0401882 + 0.999192i \(0.512796\pi\)
\(684\) 0 0
\(685\) −4.50271 −0.172040
\(686\) −0.946499 −0.0361375
\(687\) 0 0
\(688\) −1.48523 −0.0566239
\(689\) 19.0680 0.726435
\(690\) 0 0
\(691\) 16.0483 0.610505 0.305252 0.952271i \(-0.401259\pi\)
0.305252 + 0.952271i \(0.401259\pi\)
\(692\) −8.28055 −0.314779
\(693\) 0 0
\(694\) −7.19945 −0.273287
\(695\) 28.9259 1.09722
\(696\) 0 0
\(697\) −15.2938 −0.579293
\(698\) −30.0170 −1.13616
\(699\) 0 0
\(700\) −3.79863 −0.143575
\(701\) 29.4967 1.11408 0.557038 0.830487i \(-0.311938\pi\)
0.557038 + 0.830487i \(0.311938\pi\)
\(702\) 0 0
\(703\) −19.2230 −0.725011
\(704\) 30.6415 1.15484
\(705\) 0 0
\(706\) 8.64964 0.325533
\(707\) −11.5305 −0.433650
\(708\) 0 0
\(709\) −29.1469 −1.09464 −0.547318 0.836925i \(-0.684351\pi\)
−0.547318 + 0.836925i \(0.684351\pi\)
\(710\) −43.0900 −1.61714
\(711\) 0 0
\(712\) −32.3047 −1.21067
\(713\) 5.27164 0.197425
\(714\) 0 0
\(715\) 31.2798 1.16980
\(716\) −9.75507 −0.364564
\(717\) 0 0
\(718\) 19.7458 0.736905
\(719\) 22.4686 0.837937 0.418968 0.908001i \(-0.362392\pi\)
0.418968 + 0.908001i \(0.362392\pi\)
\(720\) 0 0
\(721\) 6.90350 0.257100
\(722\) 6.19502 0.230555
\(723\) 0 0
\(724\) 3.99603 0.148511
\(725\) −1.09750 −0.0407600
\(726\) 0 0
\(727\) −1.54240 −0.0572045 −0.0286023 0.999591i \(-0.509106\pi\)
−0.0286023 + 0.999591i \(0.509106\pi\)
\(728\) 6.39448 0.236995
\(729\) 0 0
\(730\) −18.1560 −0.671985
\(731\) 8.56334 0.316727
\(732\) 0 0
\(733\) 29.5101 1.08998 0.544990 0.838442i \(-0.316533\pi\)
0.544990 + 0.838442i \(0.316533\pi\)
\(734\) 32.1315 1.18599
\(735\) 0 0
\(736\) 3.96277 0.146070
\(737\) 14.8898 0.548471
\(738\) 0 0
\(739\) 0.659012 0.0242421 0.0121211 0.999927i \(-0.496142\pi\)
0.0121211 + 0.999927i \(0.496142\pi\)
\(740\) −17.4726 −0.642307
\(741\) 0 0
\(742\) −8.29244 −0.304425
\(743\) 9.62625 0.353153 0.176576 0.984287i \(-0.443498\pi\)
0.176576 + 0.984287i \(0.443498\pi\)
\(744\) 0 0
\(745\) −21.1832 −0.776093
\(746\) 23.8136 0.871879
\(747\) 0 0
\(748\) −18.0326 −0.659339
\(749\) −0.262006 −0.00957347
\(750\) 0 0
\(751\) −52.8305 −1.92781 −0.963906 0.266242i \(-0.914218\pi\)
−0.963906 + 0.266242i \(0.914218\pi\)
\(752\) −1.17729 −0.0429312
\(753\) 0 0
\(754\) 0.657149 0.0239320
\(755\) −43.0131 −1.56541
\(756\) 0 0
\(757\) 26.2632 0.954551 0.477276 0.878754i \(-0.341624\pi\)
0.477276 + 0.878754i \(0.341624\pi\)
\(758\) −28.9742 −1.05239
\(759\) 0 0
\(760\) −30.1238 −1.09271
\(761\) 23.8727 0.865386 0.432693 0.901541i \(-0.357563\pi\)
0.432693 + 0.901541i \(0.357563\pi\)
\(762\) 0 0
\(763\) −10.3229 −0.373714
\(764\) −8.13827 −0.294432
\(765\) 0 0
\(766\) −36.1045 −1.30451
\(767\) 27.1223 0.979331
\(768\) 0 0
\(769\) 42.6177 1.53683 0.768416 0.639950i \(-0.221045\pi\)
0.768416 + 0.639950i \(0.221045\pi\)
\(770\) −13.6032 −0.490225
\(771\) 0 0
\(772\) −6.74488 −0.242754
\(773\) 52.5536 1.89022 0.945110 0.326751i \(-0.105954\pi\)
0.945110 + 0.326751i \(0.105954\pi\)
\(774\) 0 0
\(775\) −24.4127 −0.876931
\(776\) −46.3211 −1.66283
\(777\) 0 0
\(778\) −11.9496 −0.428413
\(779\) 16.3489 0.585760
\(780\) 0 0
\(781\) 77.5205 2.77390
\(782\) 2.32138 0.0830125
\(783\) 0 0
\(784\) −0.572592 −0.0204497
\(785\) −62.8448 −2.24303
\(786\) 0 0
\(787\) −1.10111 −0.0392504 −0.0196252 0.999807i \(-0.506247\pi\)
−0.0196252 + 0.999807i \(0.506247\pi\)
\(788\) −26.4450 −0.942063
\(789\) 0 0
\(790\) −7.59775 −0.270316
\(791\) 9.52058 0.338513
\(792\) 0 0
\(793\) 22.6853 0.805579
\(794\) 26.9954 0.958032
\(795\) 0 0
\(796\) −9.93221 −0.352038
\(797\) −46.4625 −1.64579 −0.822893 0.568197i \(-0.807641\pi\)
−0.822893 + 0.568197i \(0.807641\pi\)
\(798\) 0 0
\(799\) 6.78783 0.240136
\(800\) −18.3514 −0.648820
\(801\) 0 0
\(802\) 35.6110 1.25747
\(803\) 32.6634 1.15267
\(804\) 0 0
\(805\) −2.15830 −0.0760702
\(806\) 14.6177 0.514885
\(807\) 0 0
\(808\) −33.8774 −1.19180
\(809\) −9.63649 −0.338801 −0.169401 0.985547i \(-0.554183\pi\)
−0.169401 + 0.985547i \(0.554183\pi\)
\(810\) 0 0
\(811\) 32.3426 1.13570 0.567850 0.823132i \(-0.307775\pi\)
0.567850 + 0.823132i \(0.307775\pi\)
\(812\) 0.352229 0.0123608
\(813\) 0 0
\(814\) −25.5043 −0.893927
\(815\) −9.73636 −0.341050
\(816\) 0 0
\(817\) −9.15413 −0.320262
\(818\) 9.48660 0.331691
\(819\) 0 0
\(820\) 14.8602 0.518940
\(821\) −9.19978 −0.321075 −0.160537 0.987030i \(-0.551323\pi\)
−0.160537 + 0.987030i \(0.551323\pi\)
\(822\) 0 0
\(823\) 27.0062 0.941379 0.470689 0.882299i \(-0.344005\pi\)
0.470689 + 0.882299i \(0.344005\pi\)
\(824\) 20.2829 0.706589
\(825\) 0 0
\(826\) −11.7951 −0.410406
\(827\) −18.8167 −0.654320 −0.327160 0.944969i \(-0.606092\pi\)
−0.327160 + 0.944969i \(0.606092\pi\)
\(828\) 0 0
\(829\) 39.6289 1.37637 0.688184 0.725536i \(-0.258408\pi\)
0.688184 + 0.725536i \(0.258408\pi\)
\(830\) 32.7449 1.13659
\(831\) 0 0
\(832\) 13.4807 0.467360
\(833\) 3.30137 0.114386
\(834\) 0 0
\(835\) 27.2421 0.942753
\(836\) 19.2767 0.666700
\(837\) 0 0
\(838\) 15.0440 0.519687
\(839\) −15.2610 −0.526870 −0.263435 0.964677i \(-0.584855\pi\)
−0.263435 + 0.964677i \(0.584855\pi\)
\(840\) 0 0
\(841\) −28.8982 −0.996491
\(842\) −23.6512 −0.815074
\(843\) 0 0
\(844\) −9.10827 −0.313520
\(845\) −24.0064 −0.825845
\(846\) 0 0
\(847\) 13.4726 0.462926
\(848\) −5.01658 −0.172270
\(849\) 0 0
\(850\) −10.7502 −0.368729
\(851\) −4.04656 −0.138714
\(852\) 0 0
\(853\) 5.84519 0.200136 0.100068 0.994981i \(-0.468094\pi\)
0.100068 + 0.994981i \(0.468094\pi\)
\(854\) −9.86555 −0.337592
\(855\) 0 0
\(856\) −0.769789 −0.0263109
\(857\) −28.1450 −0.961416 −0.480708 0.876881i \(-0.659620\pi\)
−0.480708 + 0.876881i \(0.659620\pi\)
\(858\) 0 0
\(859\) 23.0135 0.785211 0.392606 0.919707i \(-0.371574\pi\)
0.392606 + 0.919707i \(0.371574\pi\)
\(860\) −8.32057 −0.283729
\(861\) 0 0
\(862\) −20.2046 −0.688172
\(863\) 15.7017 0.534492 0.267246 0.963628i \(-0.413886\pi\)
0.267246 + 0.963628i \(0.413886\pi\)
\(864\) 0 0
\(865\) 21.7879 0.740810
\(866\) −13.1410 −0.446549
\(867\) 0 0
\(868\) 7.83499 0.265937
\(869\) 13.6686 0.463677
\(870\) 0 0
\(871\) 6.55075 0.221964
\(872\) −30.3293 −1.02708
\(873\) 0 0
\(874\) −2.48154 −0.0839392
\(875\) −4.53114 −0.153180
\(876\) 0 0
\(877\) −45.1043 −1.52306 −0.761532 0.648127i \(-0.775553\pi\)
−0.761532 + 0.648127i \(0.775553\pi\)
\(878\) −4.99552 −0.168591
\(879\) 0 0
\(880\) −8.22936 −0.277412
\(881\) −15.6070 −0.525812 −0.262906 0.964821i \(-0.584681\pi\)
−0.262906 + 0.964821i \(0.584681\pi\)
\(882\) 0 0
\(883\) −18.1214 −0.609835 −0.304917 0.952379i \(-0.598629\pi\)
−0.304917 + 0.952379i \(0.598629\pi\)
\(884\) −7.93346 −0.266831
\(885\) 0 0
\(886\) 1.68964 0.0567647
\(887\) −32.4870 −1.09081 −0.545404 0.838173i \(-0.683624\pi\)
−0.545404 + 0.838173i \(0.683624\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) 30.2346 1.01347
\(891\) 0 0
\(892\) −5.73058 −0.191874
\(893\) −7.25612 −0.242817
\(894\) 0 0
\(895\) 25.6677 0.857975
\(896\) 4.80576 0.160549
\(897\) 0 0
\(898\) −38.4264 −1.28230
\(899\) 2.26368 0.0754978
\(900\) 0 0
\(901\) 28.9239 0.963595
\(902\) 21.6910 0.722232
\(903\) 0 0
\(904\) 27.9721 0.930338
\(905\) −10.5144 −0.349511
\(906\) 0 0
\(907\) 33.2443 1.10386 0.551930 0.833890i \(-0.313892\pi\)
0.551930 + 0.833890i \(0.313892\pi\)
\(908\) −4.13096 −0.137091
\(909\) 0 0
\(910\) −5.98472 −0.198392
\(911\) 6.23541 0.206588 0.103294 0.994651i \(-0.467062\pi\)
0.103294 + 0.994651i \(0.467062\pi\)
\(912\) 0 0
\(913\) −58.9094 −1.94962
\(914\) 7.53250 0.249153
\(915\) 0 0
\(916\) 10.0094 0.330720
\(917\) 2.11923 0.0699832
\(918\) 0 0
\(919\) 3.51224 0.115858 0.0579290 0.998321i \(-0.481550\pi\)
0.0579290 + 0.998321i \(0.481550\pi\)
\(920\) −6.34123 −0.209064
\(921\) 0 0
\(922\) −22.1951 −0.730956
\(923\) 34.1052 1.12259
\(924\) 0 0
\(925\) 18.7394 0.616149
\(926\) 10.2532 0.336941
\(927\) 0 0
\(928\) 1.70164 0.0558590
\(929\) −9.88716 −0.324387 −0.162194 0.986759i \(-0.551857\pi\)
−0.162194 + 0.986759i \(0.551857\pi\)
\(930\) 0 0
\(931\) −3.52914 −0.115663
\(932\) −5.94822 −0.194840
\(933\) 0 0
\(934\) −32.9288 −1.07746
\(935\) 47.4477 1.55171
\(936\) 0 0
\(937\) 12.5089 0.408648 0.204324 0.978903i \(-0.434500\pi\)
0.204324 + 0.978903i \(0.434500\pi\)
\(938\) −2.84883 −0.0930177
\(939\) 0 0
\(940\) −6.59539 −0.215118
\(941\) 21.9212 0.714612 0.357306 0.933987i \(-0.383695\pi\)
0.357306 + 0.933987i \(0.383695\pi\)
\(942\) 0 0
\(943\) 3.44153 0.112072
\(944\) −7.13557 −0.232243
\(945\) 0 0
\(946\) −12.1453 −0.394878
\(947\) 37.2266 1.20970 0.604851 0.796339i \(-0.293233\pi\)
0.604851 + 0.796339i \(0.293233\pi\)
\(948\) 0 0
\(949\) 14.3703 0.466479
\(950\) 11.4919 0.372846
\(951\) 0 0
\(952\) 9.69965 0.314367
\(953\) −52.1922 −1.69067 −0.845336 0.534235i \(-0.820600\pi\)
−0.845336 + 0.534235i \(0.820600\pi\)
\(954\) 0 0
\(955\) 21.4135 0.692925
\(956\) 22.3343 0.722342
\(957\) 0 0
\(958\) 9.47807 0.306222
\(959\) −1.54986 −0.0500477
\(960\) 0 0
\(961\) 19.3533 0.624299
\(962\) −11.2206 −0.361768
\(963\) 0 0
\(964\) 11.9167 0.383811
\(965\) 17.7472 0.571303
\(966\) 0 0
\(967\) −35.6045 −1.14496 −0.572482 0.819917i \(-0.694019\pi\)
−0.572482 + 0.819917i \(0.694019\pi\)
\(968\) 39.5835 1.27226
\(969\) 0 0
\(970\) 43.3529 1.39198
\(971\) −42.4387 −1.36192 −0.680962 0.732319i \(-0.738438\pi\)
−0.680962 + 0.732319i \(0.738438\pi\)
\(972\) 0 0
\(973\) 9.95651 0.319191
\(974\) 30.3012 0.970913
\(975\) 0 0
\(976\) −5.96825 −0.191039
\(977\) −4.99648 −0.159852 −0.0799258 0.996801i \(-0.525468\pi\)
−0.0799258 + 0.996801i \(0.525468\pi\)
\(978\) 0 0
\(979\) −54.3932 −1.73841
\(980\) −3.20778 −0.102469
\(981\) 0 0
\(982\) 11.5898 0.369844
\(983\) −7.70508 −0.245754 −0.122877 0.992422i \(-0.539212\pi\)
−0.122877 + 0.992422i \(0.539212\pi\)
\(984\) 0 0
\(985\) 69.5823 2.21708
\(986\) 0.996816 0.0317451
\(987\) 0 0
\(988\) 8.48080 0.269810
\(989\) −1.92699 −0.0612749
\(990\) 0 0
\(991\) −37.7733 −1.19991 −0.599954 0.800034i \(-0.704815\pi\)
−0.599954 + 0.800034i \(0.704815\pi\)
\(992\) 37.8513 1.20178
\(993\) 0 0
\(994\) −14.8319 −0.470439
\(995\) 26.1338 0.828496
\(996\) 0 0
\(997\) −8.61397 −0.272807 −0.136404 0.990653i \(-0.543554\pi\)
−0.136404 + 0.990653i \(0.543554\pi\)
\(998\) 16.6072 0.525692
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8001.2.a.n.1.3 12
3.2 odd 2 889.2.a.a.1.10 12
21.20 even 2 6223.2.a.i.1.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
889.2.a.a.1.10 12 3.2 odd 2
6223.2.a.i.1.10 12 21.20 even 2
8001.2.a.n.1.3 12 1.1 even 1 trivial