Properties

Label 289.2.a.f.1.1
Level $289$
Weight $2$
Character 289.1
Self dual yes
Analytic conductor $2.308$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [289,2,Mod(1,289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("289.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 289.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: \(2.30767661842\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.765367\) of defining polynomial
Character \(\chi\) \(=\) 289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.414214 q^{2} -2.61313 q^{3} -1.82843 q^{4} -1.84776 q^{5} +1.08239 q^{6} -1.08239 q^{7} +1.58579 q^{8} +3.82843 q^{9} +O(q^{10})\) \(q-0.414214 q^{2} -2.61313 q^{3} -1.82843 q^{4} -1.84776 q^{5} +1.08239 q^{6} -1.08239 q^{7} +1.58579 q^{8} +3.82843 q^{9} +0.765367 q^{10} +1.08239 q^{11} +4.77791 q^{12} -1.41421 q^{13} +0.448342 q^{14} +4.82843 q^{15} +3.00000 q^{16} -1.58579 q^{18} +4.82843 q^{19} +3.37849 q^{20} +2.82843 q^{21} -0.448342 q^{22} +4.14386 q^{23} -4.14386 q^{24} -1.58579 q^{25} +0.585786 q^{26} -2.16478 q^{27} +1.97908 q^{28} -4.46088 q^{29} -2.00000 q^{30} +3.24718 q^{31} -4.41421 q^{32} -2.82843 q^{33} +2.00000 q^{35} -7.00000 q^{36} -3.82683 q^{37} -2.00000 q^{38} +3.69552 q^{39} -2.93015 q^{40} +8.15640 q^{41} -1.17157 q^{42} -4.82843 q^{43} -1.97908 q^{44} -7.07401 q^{45} -1.71644 q^{46} +10.8284 q^{47} -7.83938 q^{48} -5.82843 q^{49} +0.656854 q^{50} +2.58579 q^{52} -1.41421 q^{53} +0.896683 q^{54} -2.00000 q^{55} -1.71644 q^{56} -12.6173 q^{57} +1.84776 q^{58} +6.00000 q^{59} -8.82843 q^{60} +9.23880 q^{61} -1.34502 q^{62} -4.14386 q^{63} -4.17157 q^{64} +2.61313 q^{65} +1.17157 q^{66} -6.82843 q^{67} -10.8284 q^{69} -0.828427 q^{70} +13.0656 q^{71} +6.07107 q^{72} +5.35757 q^{73} +1.58513 q^{74} +4.14386 q^{75} -8.82843 q^{76} -1.17157 q^{77} -1.53073 q^{78} +4.14386 q^{79} -5.54328 q^{80} -5.82843 q^{81} -3.37849 q^{82} +0.343146 q^{83} -5.17157 q^{84} +2.00000 q^{86} +11.6569 q^{87} +1.71644 q^{88} +9.41421 q^{89} +2.93015 q^{90} +1.53073 q^{91} -7.57675 q^{92} -8.48528 q^{93} -4.48528 q^{94} -8.92177 q^{95} +11.5349 q^{96} +6.43996 q^{97} +2.41421 q^{98} +4.14386 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} + 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{4} + 12 q^{8} + 4 q^{9} + 8 q^{15} + 12 q^{16} - 12 q^{18} + 8 q^{19} - 12 q^{25} + 8 q^{26} - 8 q^{30} - 12 q^{32} + 8 q^{35} - 28 q^{36} - 8 q^{38} - 16 q^{42} - 8 q^{43} + 32 q^{47} - 12 q^{49} - 20 q^{50} + 16 q^{52} - 8 q^{55} + 24 q^{59} - 24 q^{60} - 28 q^{64} + 16 q^{66} - 16 q^{67} - 32 q^{69} + 8 q^{70} - 4 q^{72} - 24 q^{76} - 16 q^{77} - 12 q^{81} + 24 q^{83} - 32 q^{84} + 8 q^{86} + 24 q^{87} + 32 q^{89} + 16 q^{94} + 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.414214 −0.292893 −0.146447 0.989219i \(-0.546784\pi\)
−0.146447 + 0.989219i \(0.546784\pi\)
\(3\) −2.61313 −1.50869 −0.754344 0.656479i \(-0.772045\pi\)
−0.754344 + 0.656479i \(0.772045\pi\)
\(4\) −1.82843 −0.914214
\(5\) −1.84776 −0.826343 −0.413171 0.910653i \(-0.635579\pi\)
−0.413171 + 0.910653i \(0.635579\pi\)
\(6\) 1.08239 0.441885
\(7\) −1.08239 −0.409106 −0.204553 0.978856i \(-0.565574\pi\)
−0.204553 + 0.978856i \(0.565574\pi\)
\(8\) 1.58579 0.560660
\(9\) 3.82843 1.27614
\(10\) 0.765367 0.242030
\(11\) 1.08239 0.326354 0.163177 0.986597i \(-0.447826\pi\)
0.163177 + 0.986597i \(0.447826\pi\)
\(12\) 4.77791 1.37926
\(13\) −1.41421 −0.392232 −0.196116 0.980581i \(-0.562833\pi\)
−0.196116 + 0.980581i \(0.562833\pi\)
\(14\) 0.448342 0.119824
\(15\) 4.82843 1.24669
\(16\) 3.00000 0.750000
\(17\) 0 0
\(18\) −1.58579 −0.373773
\(19\) 4.82843 1.10772 0.553859 0.832611i \(-0.313155\pi\)
0.553859 + 0.832611i \(0.313155\pi\)
\(20\) 3.37849 0.755454
\(21\) 2.82843 0.617213
\(22\) −0.448342 −0.0955867
\(23\) 4.14386 0.864054 0.432027 0.901861i \(-0.357798\pi\)
0.432027 + 0.901861i \(0.357798\pi\)
\(24\) −4.14386 −0.845862
\(25\) −1.58579 −0.317157
\(26\) 0.585786 0.114882
\(27\) −2.16478 −0.416613
\(28\) 1.97908 0.374010
\(29\) −4.46088 −0.828366 −0.414183 0.910194i \(-0.635933\pi\)
−0.414183 + 0.910194i \(0.635933\pi\)
\(30\) −2.00000 −0.365148
\(31\) 3.24718 0.583210 0.291605 0.956539i \(-0.405811\pi\)
0.291605 + 0.956539i \(0.405811\pi\)
\(32\) −4.41421 −0.780330
\(33\) −2.82843 −0.492366
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) −7.00000 −1.16667
\(37\) −3.82683 −0.629128 −0.314564 0.949236i \(-0.601858\pi\)
−0.314564 + 0.949236i \(0.601858\pi\)
\(38\) −2.00000 −0.324443
\(39\) 3.69552 0.591756
\(40\) −2.93015 −0.463298
\(41\) 8.15640 1.27382 0.636908 0.770940i \(-0.280213\pi\)
0.636908 + 0.770940i \(0.280213\pi\)
\(42\) −1.17157 −0.180778
\(43\) −4.82843 −0.736328 −0.368164 0.929761i \(-0.620014\pi\)
−0.368164 + 0.929761i \(0.620014\pi\)
\(44\) −1.97908 −0.298357
\(45\) −7.07401 −1.05453
\(46\) −1.71644 −0.253076
\(47\) 10.8284 1.57949 0.789744 0.613436i \(-0.210213\pi\)
0.789744 + 0.613436i \(0.210213\pi\)
\(48\) −7.83938 −1.13152
\(49\) −5.82843 −0.832632
\(50\) 0.656854 0.0928932
\(51\) 0 0
\(52\) 2.58579 0.358584
\(53\) −1.41421 −0.194257 −0.0971286 0.995272i \(-0.530966\pi\)
−0.0971286 + 0.995272i \(0.530966\pi\)
\(54\) 0.896683 0.122023
\(55\) −2.00000 −0.269680
\(56\) −1.71644 −0.229369
\(57\) −12.6173 −1.67120
\(58\) 1.84776 0.242623
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) −8.82843 −1.13975
\(61\) 9.23880 1.18291 0.591453 0.806339i \(-0.298554\pi\)
0.591453 + 0.806339i \(0.298554\pi\)
\(62\) −1.34502 −0.170818
\(63\) −4.14386 −0.522077
\(64\) −4.17157 −0.521447
\(65\) 2.61313 0.324118
\(66\) 1.17157 0.144211
\(67\) −6.82843 −0.834225 −0.417113 0.908855i \(-0.636958\pi\)
−0.417113 + 0.908855i \(0.636958\pi\)
\(68\) 0 0
\(69\) −10.8284 −1.30359
\(70\) −0.828427 −0.0990160
\(71\) 13.0656 1.55060 0.775302 0.631590i \(-0.217597\pi\)
0.775302 + 0.631590i \(0.217597\pi\)
\(72\) 6.07107 0.715482
\(73\) 5.35757 0.627056 0.313528 0.949579i \(-0.398489\pi\)
0.313528 + 0.949579i \(0.398489\pi\)
\(74\) 1.58513 0.184267
\(75\) 4.14386 0.478492
\(76\) −8.82843 −1.01269
\(77\) −1.17157 −0.133513
\(78\) −1.53073 −0.173321
\(79\) 4.14386 0.466221 0.233110 0.972450i \(-0.425110\pi\)
0.233110 + 0.972450i \(0.425110\pi\)
\(80\) −5.54328 −0.619757
\(81\) −5.82843 −0.647603
\(82\) −3.37849 −0.373092
\(83\) 0.343146 0.0376651 0.0188326 0.999823i \(-0.494005\pi\)
0.0188326 + 0.999823i \(0.494005\pi\)
\(84\) −5.17157 −0.564265
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) 11.6569 1.24975
\(88\) 1.71644 0.182973
\(89\) 9.41421 0.997905 0.498952 0.866629i \(-0.333718\pi\)
0.498952 + 0.866629i \(0.333718\pi\)
\(90\) 2.93015 0.308865
\(91\) 1.53073 0.160464
\(92\) −7.57675 −0.789930
\(93\) −8.48528 −0.879883
\(94\) −4.48528 −0.462621
\(95\) −8.92177 −0.915354
\(96\) 11.5349 1.17728
\(97\) 6.43996 0.653879 0.326939 0.945045i \(-0.393983\pi\)
0.326939 + 0.945045i \(0.393983\pi\)
\(98\) 2.41421 0.243872
\(99\) 4.14386 0.416474
\(100\) 2.89949 0.289949
\(101\) 13.4142 1.33476 0.667382 0.744715i \(-0.267415\pi\)
0.667382 + 0.744715i \(0.267415\pi\)
\(102\) 0 0
\(103\) −4.48528 −0.441948 −0.220974 0.975280i \(-0.570924\pi\)
−0.220974 + 0.975280i \(0.570924\pi\)
\(104\) −2.24264 −0.219909
\(105\) −5.22625 −0.510030
\(106\) 0.585786 0.0568966
\(107\) 6.30864 0.609880 0.304940 0.952372i \(-0.401364\pi\)
0.304940 + 0.952372i \(0.401364\pi\)
\(108\) 3.95815 0.380873
\(109\) −4.27518 −0.409488 −0.204744 0.978816i \(-0.565636\pi\)
−0.204744 + 0.978816i \(0.565636\pi\)
\(110\) 0.828427 0.0789874
\(111\) 10.0000 0.949158
\(112\) −3.24718 −0.306829
\(113\) −16.1815 −1.52223 −0.761113 0.648619i \(-0.775347\pi\)
−0.761113 + 0.648619i \(0.775347\pi\)
\(114\) 5.22625 0.489483
\(115\) −7.65685 −0.714005
\(116\) 8.15640 0.757303
\(117\) −5.41421 −0.500544
\(118\) −2.48528 −0.228789
\(119\) 0 0
\(120\) 7.65685 0.698972
\(121\) −9.82843 −0.893493
\(122\) −3.82683 −0.346465
\(123\) −21.3137 −1.92179
\(124\) −5.93723 −0.533179
\(125\) 12.1689 1.08842
\(126\) 1.71644 0.152913
\(127\) 17.3137 1.53634 0.768172 0.640244i \(-0.221167\pi\)
0.768172 + 0.640244i \(0.221167\pi\)
\(128\) 10.5563 0.933058
\(129\) 12.6173 1.11089
\(130\) −1.08239 −0.0949321
\(131\) 0.185709 0.0162255 0.00811274 0.999967i \(-0.497418\pi\)
0.00811274 + 0.999967i \(0.497418\pi\)
\(132\) 5.17157 0.450128
\(133\) −5.22625 −0.453174
\(134\) 2.82843 0.244339
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) 8.72792 0.745677 0.372838 0.927896i \(-0.378385\pi\)
0.372838 + 0.927896i \(0.378385\pi\)
\(138\) 4.48528 0.381813
\(139\) −14.9678 −1.26955 −0.634775 0.772697i \(-0.718907\pi\)
−0.634775 + 0.772697i \(0.718907\pi\)
\(140\) −3.65685 −0.309061
\(141\) −28.2960 −2.38296
\(142\) −5.41196 −0.454162
\(143\) −1.53073 −0.128006
\(144\) 11.4853 0.957107
\(145\) 8.24264 0.684514
\(146\) −2.21918 −0.183660
\(147\) 15.2304 1.25618
\(148\) 6.99709 0.575157
\(149\) 16.9706 1.39028 0.695141 0.718873i \(-0.255342\pi\)
0.695141 + 0.718873i \(0.255342\pi\)
\(150\) −1.71644 −0.140147
\(151\) 12.8284 1.04396 0.521981 0.852957i \(-0.325193\pi\)
0.521981 + 0.852957i \(0.325193\pi\)
\(152\) 7.65685 0.621053
\(153\) 0 0
\(154\) 0.485281 0.0391051
\(155\) −6.00000 −0.481932
\(156\) −6.75699 −0.540992
\(157\) 1.65685 0.132231 0.0661157 0.997812i \(-0.478939\pi\)
0.0661157 + 0.997812i \(0.478939\pi\)
\(158\) −1.71644 −0.136553
\(159\) 3.69552 0.293074
\(160\) 8.15640 0.644820
\(161\) −4.48528 −0.353490
\(162\) 2.41421 0.189679
\(163\) −5.67459 −0.444468 −0.222234 0.974993i \(-0.571335\pi\)
−0.222234 + 0.974993i \(0.571335\pi\)
\(164\) −14.9134 −1.16454
\(165\) 5.22625 0.406863
\(166\) −0.142136 −0.0110319
\(167\) −10.0042 −0.774145 −0.387073 0.922049i \(-0.626514\pi\)
−0.387073 + 0.922049i \(0.626514\pi\)
\(168\) 4.48528 0.346047
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) 18.4853 1.41360
\(172\) 8.82843 0.673161
\(173\) −3.37849 −0.256862 −0.128431 0.991718i \(-0.540994\pi\)
−0.128431 + 0.991718i \(0.540994\pi\)
\(174\) −4.82843 −0.366042
\(175\) 1.71644 0.129751
\(176\) 3.24718 0.244765
\(177\) −15.6788 −1.17849
\(178\) −3.89949 −0.292280
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 12.9343 0.964067
\(181\) 12.4860 0.928075 0.464037 0.885816i \(-0.346400\pi\)
0.464037 + 0.885816i \(0.346400\pi\)
\(182\) −0.634051 −0.0469990
\(183\) −24.1421 −1.78464
\(184\) 6.57128 0.484441
\(185\) 7.07107 0.519875
\(186\) 3.51472 0.257712
\(187\) 0 0
\(188\) −19.7990 −1.44399
\(189\) 2.34315 0.170439
\(190\) 3.69552 0.268101
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 10.9008 0.786701
\(193\) 5.54328 0.399014 0.199507 0.979896i \(-0.436066\pi\)
0.199507 + 0.979896i \(0.436066\pi\)
\(194\) −2.66752 −0.191517
\(195\) −6.82843 −0.488994
\(196\) 10.6569 0.761204
\(197\) −14.9134 −1.06253 −0.531267 0.847204i \(-0.678284\pi\)
−0.531267 + 0.847204i \(0.678284\pi\)
\(198\) −1.71644 −0.121982
\(199\) −1.71644 −0.121675 −0.0608377 0.998148i \(-0.519377\pi\)
−0.0608377 + 0.998148i \(0.519377\pi\)
\(200\) −2.51472 −0.177817
\(201\) 17.8435 1.25859
\(202\) −5.55635 −0.390943
\(203\) 4.82843 0.338889
\(204\) 0 0
\(205\) −15.0711 −1.05261
\(206\) 1.85786 0.129444
\(207\) 15.8645 1.10266
\(208\) −4.24264 −0.294174
\(209\) 5.22625 0.361507
\(210\) 2.16478 0.149384
\(211\) −14.9678 −1.03042 −0.515212 0.857063i \(-0.672287\pi\)
−0.515212 + 0.857063i \(0.672287\pi\)
\(212\) 2.58579 0.177593
\(213\) −34.1421 −2.33938
\(214\) −2.61313 −0.178630
\(215\) 8.92177 0.608460
\(216\) −3.43289 −0.233578
\(217\) −3.51472 −0.238595
\(218\) 1.77084 0.119936
\(219\) −14.0000 −0.946032
\(220\) 3.65685 0.246545
\(221\) 0 0
\(222\) −4.14214 −0.278002
\(223\) 0.828427 0.0554756 0.0277378 0.999615i \(-0.491170\pi\)
0.0277378 + 0.999615i \(0.491170\pi\)
\(224\) 4.77791 0.319238
\(225\) −6.07107 −0.404738
\(226\) 6.70259 0.445850
\(227\) −5.04054 −0.334553 −0.167276 0.985910i \(-0.553497\pi\)
−0.167276 + 0.985910i \(0.553497\pi\)
\(228\) 23.0698 1.52783
\(229\) −22.8284 −1.50854 −0.754272 0.656562i \(-0.772010\pi\)
−0.754272 + 0.656562i \(0.772010\pi\)
\(230\) 3.17157 0.209127
\(231\) 3.06147 0.201430
\(232\) −7.07401 −0.464432
\(233\) 10.1355 0.663997 0.331999 0.943280i \(-0.392277\pi\)
0.331999 + 0.943280i \(0.392277\pi\)
\(234\) 2.24264 0.146606
\(235\) −20.0083 −1.30520
\(236\) −10.9706 −0.714123
\(237\) −10.8284 −0.703382
\(238\) 0 0
\(239\) 9.17157 0.593260 0.296630 0.954993i \(-0.404137\pi\)
0.296630 + 0.954993i \(0.404137\pi\)
\(240\) 14.4853 0.935021
\(241\) 12.3003 0.792330 0.396165 0.918179i \(-0.370341\pi\)
0.396165 + 0.918179i \(0.370341\pi\)
\(242\) 4.07107 0.261698
\(243\) 21.7248 1.39364
\(244\) −16.8925 −1.08143
\(245\) 10.7695 0.688040
\(246\) 8.82843 0.562880
\(247\) −6.82843 −0.434482
\(248\) 5.14933 0.326983
\(249\) −0.896683 −0.0568250
\(250\) −5.04054 −0.318792
\(251\) 3.51472 0.221847 0.110924 0.993829i \(-0.464619\pi\)
0.110924 + 0.993829i \(0.464619\pi\)
\(252\) 7.57675 0.477290
\(253\) 4.48528 0.281987
\(254\) −7.17157 −0.449985
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) −22.1421 −1.38119 −0.690594 0.723242i \(-0.742651\pi\)
−0.690594 + 0.723242i \(0.742651\pi\)
\(258\) −5.22625 −0.325372
\(259\) 4.14214 0.257380
\(260\) −4.77791 −0.296313
\(261\) −17.0782 −1.05711
\(262\) −0.0769232 −0.00475233
\(263\) 6.48528 0.399900 0.199950 0.979806i \(-0.435922\pi\)
0.199950 + 0.979806i \(0.435922\pi\)
\(264\) −4.48528 −0.276050
\(265\) 2.61313 0.160523
\(266\) 2.16478 0.132731
\(267\) −24.6005 −1.50553
\(268\) 12.4853 0.762660
\(269\) −6.36304 −0.387961 −0.193981 0.981005i \(-0.562140\pi\)
−0.193981 + 0.981005i \(0.562140\pi\)
\(270\) −1.65685 −0.100833
\(271\) 6.14214 0.373108 0.186554 0.982445i \(-0.440268\pi\)
0.186554 + 0.982445i \(0.440268\pi\)
\(272\) 0 0
\(273\) −4.00000 −0.242091
\(274\) −3.61522 −0.218404
\(275\) −1.71644 −0.103505
\(276\) 19.7990 1.19176
\(277\) −22.0418 −1.32436 −0.662181 0.749344i \(-0.730369\pi\)
−0.662181 + 0.749344i \(0.730369\pi\)
\(278\) 6.19986 0.371843
\(279\) 12.4316 0.744259
\(280\) 3.17157 0.189538
\(281\) −17.8995 −1.06779 −0.533897 0.845549i \(-0.679273\pi\)
−0.533897 + 0.845549i \(0.679273\pi\)
\(282\) 11.7206 0.697952
\(283\) −22.8841 −1.36032 −0.680159 0.733065i \(-0.738089\pi\)
−0.680159 + 0.733065i \(0.738089\pi\)
\(284\) −23.8896 −1.41758
\(285\) 23.3137 1.38098
\(286\) 0.634051 0.0374922
\(287\) −8.82843 −0.521126
\(288\) −16.8995 −0.995812
\(289\) 0 0
\(290\) −3.41421 −0.200490
\(291\) −16.8284 −0.986500
\(292\) −9.79592 −0.573263
\(293\) −23.6569 −1.38205 −0.691024 0.722832i \(-0.742840\pi\)
−0.691024 + 0.722832i \(0.742840\pi\)
\(294\) −6.30864 −0.367928
\(295\) −11.0866 −0.645484
\(296\) −6.06854 −0.352727
\(297\) −2.34315 −0.135963
\(298\) −7.02944 −0.407204
\(299\) −5.86030 −0.338910
\(300\) −7.57675 −0.437444
\(301\) 5.22625 0.301236
\(302\) −5.31371 −0.305770
\(303\) −35.0530 −2.01374
\(304\) 14.4853 0.830788
\(305\) −17.0711 −0.977486
\(306\) 0 0
\(307\) 2.14214 0.122258 0.0611291 0.998130i \(-0.480530\pi\)
0.0611291 + 0.998130i \(0.480530\pi\)
\(308\) 2.14214 0.122060
\(309\) 11.7206 0.666762
\(310\) 2.48528 0.141154
\(311\) 4.51528 0.256038 0.128019 0.991772i \(-0.459138\pi\)
0.128019 + 0.991772i \(0.459138\pi\)
\(312\) 5.86030 0.331774
\(313\) 12.7486 0.720594 0.360297 0.932838i \(-0.382675\pi\)
0.360297 + 0.932838i \(0.382675\pi\)
\(314\) −0.686292 −0.0387297
\(315\) 7.65685 0.431415
\(316\) −7.57675 −0.426225
\(317\) 5.80591 0.326092 0.163046 0.986618i \(-0.447868\pi\)
0.163046 + 0.986618i \(0.447868\pi\)
\(318\) −1.53073 −0.0858393
\(319\) −4.82843 −0.270340
\(320\) 7.70806 0.430894
\(321\) −16.4853 −0.920119
\(322\) 1.85786 0.103535
\(323\) 0 0
\(324\) 10.6569 0.592047
\(325\) 2.24264 0.124399
\(326\) 2.35049 0.130182
\(327\) 11.1716 0.617789
\(328\) 12.9343 0.714178
\(329\) −11.7206 −0.646178
\(330\) −2.16478 −0.119167
\(331\) 17.7990 0.978321 0.489160 0.872194i \(-0.337303\pi\)
0.489160 + 0.872194i \(0.337303\pi\)
\(332\) −0.627417 −0.0344340
\(333\) −14.6508 −0.802857
\(334\) 4.14386 0.226742
\(335\) 12.6173 0.689356
\(336\) 8.48528 0.462910
\(337\) 34.4734 1.87788 0.938942 0.344075i \(-0.111808\pi\)
0.938942 + 0.344075i \(0.111808\pi\)
\(338\) 4.55635 0.247833
\(339\) 42.2843 2.29657
\(340\) 0 0
\(341\) 3.51472 0.190333
\(342\) −7.65685 −0.414035
\(343\) 13.8854 0.749741
\(344\) −7.65685 −0.412830
\(345\) 20.0083 1.07721
\(346\) 1.39942 0.0752332
\(347\) 3.88123 0.208355 0.104178 0.994559i \(-0.466779\pi\)
0.104178 + 0.994559i \(0.466779\pi\)
\(348\) −21.3137 −1.14253
\(349\) 4.24264 0.227103 0.113552 0.993532i \(-0.463777\pi\)
0.113552 + 0.993532i \(0.463777\pi\)
\(350\) −0.710974 −0.0380032
\(351\) 3.06147 0.163409
\(352\) −4.77791 −0.254663
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 6.49435 0.345171
\(355\) −24.1421 −1.28133
\(356\) −17.2132 −0.912298
\(357\) 0 0
\(358\) −2.48528 −0.131351
\(359\) 23.1716 1.22295 0.611474 0.791264i \(-0.290577\pi\)
0.611474 + 0.791264i \(0.290577\pi\)
\(360\) −11.2179 −0.591234
\(361\) 4.31371 0.227037
\(362\) −5.17186 −0.271827
\(363\) 25.6829 1.34800
\(364\) −2.79884 −0.146699
\(365\) −9.89949 −0.518163
\(366\) 10.0000 0.522708
\(367\) 26.3170 1.37373 0.686867 0.726783i \(-0.258985\pi\)
0.686867 + 0.726783i \(0.258985\pi\)
\(368\) 12.4316 0.648041
\(369\) 31.2262 1.62557
\(370\) −2.92893 −0.152268
\(371\) 1.53073 0.0794717
\(372\) 15.5147 0.804401
\(373\) 19.5563 1.01259 0.506295 0.862361i \(-0.331015\pi\)
0.506295 + 0.862361i \(0.331015\pi\)
\(374\) 0 0
\(375\) −31.7990 −1.64209
\(376\) 17.1716 0.885556
\(377\) 6.30864 0.324912
\(378\) −0.970563 −0.0499204
\(379\) −1.08239 −0.0555988 −0.0277994 0.999614i \(-0.508850\pi\)
−0.0277994 + 0.999614i \(0.508850\pi\)
\(380\) 16.3128 0.836829
\(381\) −45.2429 −2.31786
\(382\) −8.28427 −0.423860
\(383\) 5.51472 0.281789 0.140894 0.990025i \(-0.455002\pi\)
0.140894 + 0.990025i \(0.455002\pi\)
\(384\) −27.5851 −1.40769
\(385\) 2.16478 0.110328
\(386\) −2.29610 −0.116868
\(387\) −18.4853 −0.939660
\(388\) −11.7750 −0.597785
\(389\) 16.1421 0.818439 0.409219 0.912436i \(-0.365801\pi\)
0.409219 + 0.912436i \(0.365801\pi\)
\(390\) 2.82843 0.143223
\(391\) 0 0
\(392\) −9.24264 −0.466824
\(393\) −0.485281 −0.0244792
\(394\) 6.17733 0.311209
\(395\) −7.65685 −0.385258
\(396\) −7.57675 −0.380746
\(397\) 27.2680 1.36854 0.684272 0.729227i \(-0.260120\pi\)
0.684272 + 0.729227i \(0.260120\pi\)
\(398\) 0.710974 0.0356379
\(399\) 13.6569 0.683698
\(400\) −4.75736 −0.237868
\(401\) 17.0782 0.852843 0.426422 0.904525i \(-0.359774\pi\)
0.426422 + 0.904525i \(0.359774\pi\)
\(402\) −7.39104 −0.368631
\(403\) −4.59220 −0.228754
\(404\) −24.5269 −1.22026
\(405\) 10.7695 0.535142
\(406\) −2.00000 −0.0992583
\(407\) −4.14214 −0.205318
\(408\) 0 0
\(409\) 19.3137 0.955001 0.477501 0.878631i \(-0.341543\pi\)
0.477501 + 0.878631i \(0.341543\pi\)
\(410\) 6.24264 0.308302
\(411\) −22.8072 −1.12499
\(412\) 8.20101 0.404035
\(413\) −6.49435 −0.319566
\(414\) −6.57128 −0.322961
\(415\) −0.634051 −0.0311243
\(416\) 6.24264 0.306071
\(417\) 39.1127 1.91536
\(418\) −2.16478 −0.105883
\(419\) 26.9510 1.31664 0.658322 0.752737i \(-0.271267\pi\)
0.658322 + 0.752737i \(0.271267\pi\)
\(420\) 9.55582 0.466276
\(421\) 17.4142 0.848717 0.424358 0.905494i \(-0.360500\pi\)
0.424358 + 0.905494i \(0.360500\pi\)
\(422\) 6.19986 0.301804
\(423\) 41.4558 2.01565
\(424\) −2.24264 −0.108912
\(425\) 0 0
\(426\) 14.1421 0.685189
\(427\) −10.0000 −0.483934
\(428\) −11.5349 −0.557560
\(429\) 4.00000 0.193122
\(430\) −3.69552 −0.178214
\(431\) −39.8309 −1.91859 −0.959294 0.282408i \(-0.908867\pi\)
−0.959294 + 0.282408i \(0.908867\pi\)
\(432\) −6.49435 −0.312460
\(433\) 15.1716 0.729099 0.364550 0.931184i \(-0.381223\pi\)
0.364550 + 0.931184i \(0.381223\pi\)
\(434\) 1.45584 0.0698828
\(435\) −21.5391 −1.03272
\(436\) 7.81685 0.374359
\(437\) 20.0083 0.957128
\(438\) 5.79899 0.277086
\(439\) −10.9008 −0.520269 −0.260134 0.965572i \(-0.583767\pi\)
−0.260134 + 0.965572i \(0.583767\pi\)
\(440\) −3.17157 −0.151199
\(441\) −22.3137 −1.06256
\(442\) 0 0
\(443\) 15.7990 0.750633 0.375316 0.926897i \(-0.377534\pi\)
0.375316 + 0.926897i \(0.377534\pi\)
\(444\) −18.2843 −0.867733
\(445\) −17.3952 −0.824611
\(446\) −0.343146 −0.0162484
\(447\) −44.3462 −2.09750
\(448\) 4.51528 0.213327
\(449\) 18.7946 0.886973 0.443486 0.896281i \(-0.353741\pi\)
0.443486 + 0.896281i \(0.353741\pi\)
\(450\) 2.51472 0.118545
\(451\) 8.82843 0.415714
\(452\) 29.5867 1.39164
\(453\) −33.5223 −1.57501
\(454\) 2.08786 0.0979882
\(455\) −2.82843 −0.132599
\(456\) −20.0083 −0.936976
\(457\) 18.8284 0.880757 0.440378 0.897812i \(-0.354844\pi\)
0.440378 + 0.897812i \(0.354844\pi\)
\(458\) 9.45584 0.441843
\(459\) 0 0
\(460\) 14.0000 0.652753
\(461\) 24.0416 1.11973 0.559865 0.828584i \(-0.310853\pi\)
0.559865 + 0.828584i \(0.310853\pi\)
\(462\) −1.26810 −0.0589974
\(463\) −30.6274 −1.42338 −0.711688 0.702495i \(-0.752069\pi\)
−0.711688 + 0.702495i \(0.752069\pi\)
\(464\) −13.3827 −0.621274
\(465\) 15.6788 0.727085
\(466\) −4.19825 −0.194480
\(467\) −12.6274 −0.584327 −0.292164 0.956368i \(-0.594375\pi\)
−0.292164 + 0.956368i \(0.594375\pi\)
\(468\) 9.89949 0.457604
\(469\) 7.39104 0.341286
\(470\) 8.28772 0.382284
\(471\) −4.32957 −0.199496
\(472\) 9.51472 0.437950
\(473\) −5.22625 −0.240303
\(474\) 4.48528 0.206016
\(475\) −7.65685 −0.351321
\(476\) 0 0
\(477\) −5.41421 −0.247900
\(478\) −3.79899 −0.173762
\(479\) −34.6047 −1.58113 −0.790564 0.612379i \(-0.790213\pi\)
−0.790564 + 0.612379i \(0.790213\pi\)
\(480\) −21.3137 −0.972833
\(481\) 5.41196 0.246764
\(482\) −5.09494 −0.232068
\(483\) 11.7206 0.533306
\(484\) 17.9706 0.816844
\(485\) −11.8995 −0.540328
\(486\) −8.99869 −0.408189
\(487\) 4.40649 0.199677 0.0998386 0.995004i \(-0.468167\pi\)
0.0998386 + 0.995004i \(0.468167\pi\)
\(488\) 14.6508 0.663209
\(489\) 14.8284 0.670565
\(490\) −4.46088 −0.201522
\(491\) −25.1127 −1.13332 −0.566660 0.823952i \(-0.691765\pi\)
−0.566660 + 0.823952i \(0.691765\pi\)
\(492\) 38.9706 1.75693
\(493\) 0 0
\(494\) 2.82843 0.127257
\(495\) −7.65685 −0.344150
\(496\) 9.74153 0.437408
\(497\) −14.1421 −0.634361
\(498\) 0.371418 0.0166437
\(499\) 37.0321 1.65778 0.828892 0.559408i \(-0.188972\pi\)
0.828892 + 0.559408i \(0.188972\pi\)
\(500\) −22.2500 −0.995052
\(501\) 26.1421 1.16794
\(502\) −1.45584 −0.0649775
\(503\) 14.9678 0.667380 0.333690 0.942683i \(-0.391706\pi\)
0.333690 + 0.942683i \(0.391706\pi\)
\(504\) −6.57128 −0.292708
\(505\) −24.7862 −1.10297
\(506\) −1.85786 −0.0825921
\(507\) 28.7444 1.27658
\(508\) −31.6569 −1.40455
\(509\) 3.02944 0.134277 0.0671387 0.997744i \(-0.478613\pi\)
0.0671387 + 0.997744i \(0.478613\pi\)
\(510\) 0 0
\(511\) −5.79899 −0.256532
\(512\) −22.7574 −1.00574
\(513\) −10.4525 −0.461489
\(514\) 9.17157 0.404541
\(515\) 8.28772 0.365201
\(516\) −23.0698 −1.01559
\(517\) 11.7206 0.515472
\(518\) −1.71573 −0.0753848
\(519\) 8.82843 0.387525
\(520\) 4.14386 0.181720
\(521\) −3.11586 −0.136508 −0.0682542 0.997668i \(-0.521743\pi\)
−0.0682542 + 0.997668i \(0.521743\pi\)
\(522\) 7.07401 0.309621
\(523\) 6.82843 0.298586 0.149293 0.988793i \(-0.452300\pi\)
0.149293 + 0.988793i \(0.452300\pi\)
\(524\) −0.339556 −0.0148336
\(525\) −4.48528 −0.195754
\(526\) −2.68629 −0.117128
\(527\) 0 0
\(528\) −8.48528 −0.369274
\(529\) −5.82843 −0.253410
\(530\) −1.08239 −0.0470161
\(531\) 22.9706 0.996838
\(532\) 9.55582 0.414297
\(533\) −11.5349 −0.499632
\(534\) 10.1899 0.440959
\(535\) −11.6569 −0.503970
\(536\) −10.8284 −0.467717
\(537\) −15.6788 −0.676588
\(538\) 2.63566 0.113631
\(539\) −6.30864 −0.271733
\(540\) −7.31371 −0.314732
\(541\) 18.3463 0.788768 0.394384 0.918946i \(-0.370958\pi\)
0.394384 + 0.918946i \(0.370958\pi\)
\(542\) −2.54416 −0.109281
\(543\) −32.6274 −1.40018
\(544\) 0 0
\(545\) 7.89949 0.338377
\(546\) 1.65685 0.0709068
\(547\) 24.7862 1.05978 0.529891 0.848065i \(-0.322233\pi\)
0.529891 + 0.848065i \(0.322233\pi\)
\(548\) −15.9584 −0.681708
\(549\) 35.3701 1.50956
\(550\) 0.710974 0.0303160
\(551\) −21.5391 −0.917595
\(552\) −17.1716 −0.730871
\(553\) −4.48528 −0.190734
\(554\) 9.13001 0.387897
\(555\) −18.4776 −0.784330
\(556\) 27.3675 1.16064
\(557\) −28.2426 −1.19668 −0.598340 0.801243i \(-0.704173\pi\)
−0.598340 + 0.801243i \(0.704173\pi\)
\(558\) −5.14933 −0.217988
\(559\) 6.82843 0.288812
\(560\) 6.00000 0.253546
\(561\) 0 0
\(562\) 7.41421 0.312750
\(563\) 38.7696 1.63394 0.816971 0.576679i \(-0.195652\pi\)
0.816971 + 0.576679i \(0.195652\pi\)
\(564\) 51.7373 2.17853
\(565\) 29.8995 1.25788
\(566\) 9.47890 0.398428
\(567\) 6.30864 0.264938
\(568\) 20.7193 0.869362
\(569\) −36.0416 −1.51094 −0.755472 0.655181i \(-0.772592\pi\)
−0.755472 + 0.655181i \(0.772592\pi\)
\(570\) −9.65685 −0.404481
\(571\) −47.2220 −1.97618 −0.988089 0.153883i \(-0.950822\pi\)
−0.988089 + 0.153883i \(0.950822\pi\)
\(572\) 2.79884 0.117025
\(573\) −52.2625 −2.18330
\(574\) 3.65685 0.152634
\(575\) −6.57128 −0.274041
\(576\) −15.9706 −0.665440
\(577\) 12.9289 0.538238 0.269119 0.963107i \(-0.413267\pi\)
0.269119 + 0.963107i \(0.413267\pi\)
\(578\) 0 0
\(579\) −14.4853 −0.601988
\(580\) −15.0711 −0.625792
\(581\) −0.371418 −0.0154090
\(582\) 6.97056 0.288939
\(583\) −1.53073 −0.0633965
\(584\) 8.49596 0.351565
\(585\) 10.0042 0.413621
\(586\) 9.79899 0.404793
\(587\) −22.6863 −0.936363 −0.468182 0.883632i \(-0.655091\pi\)
−0.468182 + 0.883632i \(0.655091\pi\)
\(588\) −27.8477 −1.14842
\(589\) 15.6788 0.646032
\(590\) 4.59220 0.189058
\(591\) 38.9706 1.60303
\(592\) −11.4805 −0.471846
\(593\) 27.0711 1.11168 0.555838 0.831291i \(-0.312398\pi\)
0.555838 + 0.831291i \(0.312398\pi\)
\(594\) 0.970563 0.0398227
\(595\) 0 0
\(596\) −31.0294 −1.27102
\(597\) 4.48528 0.183570
\(598\) 2.42742 0.0992645
\(599\) 34.6274 1.41484 0.707419 0.706794i \(-0.249859\pi\)
0.707419 + 0.706794i \(0.249859\pi\)
\(600\) 6.57128 0.268271
\(601\) −20.3253 −0.829088 −0.414544 0.910029i \(-0.636059\pi\)
−0.414544 + 0.910029i \(0.636059\pi\)
\(602\) −2.16478 −0.0882300
\(603\) −26.1421 −1.06459
\(604\) −23.4558 −0.954405
\(605\) 18.1606 0.738332
\(606\) 14.5194 0.589812
\(607\) −34.3421 −1.39390 −0.696951 0.717119i \(-0.745460\pi\)
−0.696951 + 0.717119i \(0.745460\pi\)
\(608\) −21.3137 −0.864385
\(609\) −12.6173 −0.511278
\(610\) 7.07107 0.286299
\(611\) −15.3137 −0.619526
\(612\) 0 0
\(613\) −17.3137 −0.699294 −0.349647 0.936881i \(-0.613698\pi\)
−0.349647 + 0.936881i \(0.613698\pi\)
\(614\) −0.887302 −0.0358086
\(615\) 39.3826 1.58806
\(616\) −1.85786 −0.0748555
\(617\) 3.37849 0.136013 0.0680065 0.997685i \(-0.478336\pi\)
0.0680065 + 0.997685i \(0.478336\pi\)
\(618\) −4.85483 −0.195290
\(619\) −9.63274 −0.387173 −0.193586 0.981083i \(-0.562012\pi\)
−0.193586 + 0.981083i \(0.562012\pi\)
\(620\) 10.9706 0.440588
\(621\) −8.97056 −0.359976
\(622\) −1.87029 −0.0749918
\(623\) −10.1899 −0.408249
\(624\) 11.0866 0.443817
\(625\) −14.5563 −0.582254
\(626\) −5.28064 −0.211057
\(627\) −13.6569 −0.545402
\(628\) −3.02944 −0.120888
\(629\) 0 0
\(630\) −3.17157 −0.126358
\(631\) 6.68629 0.266177 0.133089 0.991104i \(-0.457511\pi\)
0.133089 + 0.991104i \(0.457511\pi\)
\(632\) 6.57128 0.261391
\(633\) 39.1127 1.55459
\(634\) −2.40489 −0.0955102
\(635\) −31.9916 −1.26955
\(636\) −6.75699 −0.267932
\(637\) 8.24264 0.326585
\(638\) 2.00000 0.0791808
\(639\) 50.0208 1.97879
\(640\) −19.5056 −0.771026
\(641\) −14.9903 −0.592082 −0.296041 0.955175i \(-0.595666\pi\)
−0.296041 + 0.955175i \(0.595666\pi\)
\(642\) 6.82843 0.269497
\(643\) 40.0936 1.58114 0.790568 0.612374i \(-0.209785\pi\)
0.790568 + 0.612374i \(0.209785\pi\)
\(644\) 8.20101 0.323165
\(645\) −23.3137 −0.917976
\(646\) 0 0
\(647\) 2.82843 0.111197 0.0555985 0.998453i \(-0.482293\pi\)
0.0555985 + 0.998453i \(0.482293\pi\)
\(648\) −9.24264 −0.363085
\(649\) 6.49435 0.254926
\(650\) −0.928932 −0.0364357
\(651\) 9.18440 0.359965
\(652\) 10.3756 0.406339
\(653\) 17.7122 0.693133 0.346566 0.938025i \(-0.387348\pi\)
0.346566 + 0.938025i \(0.387348\pi\)
\(654\) −4.62742 −0.180946
\(655\) −0.343146 −0.0134078
\(656\) 24.4692 0.955362
\(657\) 20.5111 0.800213
\(658\) 4.85483 0.189261
\(659\) 8.48528 0.330540 0.165270 0.986248i \(-0.447151\pi\)
0.165270 + 0.986248i \(0.447151\pi\)
\(660\) −9.55582 −0.371960
\(661\) −41.2132 −1.60301 −0.801504 0.597990i \(-0.795966\pi\)
−0.801504 + 0.597990i \(0.795966\pi\)
\(662\) −7.37258 −0.286544
\(663\) 0 0
\(664\) 0.544156 0.0211173
\(665\) 9.65685 0.374477
\(666\) 6.06854 0.235151
\(667\) −18.4853 −0.715753
\(668\) 18.2919 0.707734
\(669\) −2.16478 −0.0836954
\(670\) −5.22625 −0.201908
\(671\) 10.0000 0.386046
\(672\) −12.4853 −0.481630
\(673\) −0.317025 −0.0122204 −0.00611021 0.999981i \(-0.501945\pi\)
−0.00611021 + 0.999981i \(0.501945\pi\)
\(674\) −14.2793 −0.550020
\(675\) 3.43289 0.132132
\(676\) 20.1127 0.773565
\(677\) −43.9204 −1.68800 −0.843999 0.536344i \(-0.819805\pi\)
−0.843999 + 0.536344i \(0.819805\pi\)
\(678\) −17.5147 −0.672649
\(679\) −6.97056 −0.267506
\(680\) 0 0
\(681\) 13.1716 0.504736
\(682\) −1.45584 −0.0557472
\(683\) −31.2806 −1.19692 −0.598459 0.801153i \(-0.704220\pi\)
−0.598459 + 0.801153i \(0.704220\pi\)
\(684\) −33.7990 −1.29234
\(685\) −16.1271 −0.616185
\(686\) −5.75152 −0.219594
\(687\) 59.6536 2.27593
\(688\) −14.4853 −0.552246
\(689\) 2.00000 0.0761939
\(690\) −8.28772 −0.315508
\(691\) −40.7276 −1.54935 −0.774676 0.632358i \(-0.782087\pi\)
−0.774676 + 0.632358i \(0.782087\pi\)
\(692\) 6.17733 0.234827
\(693\) −4.48528 −0.170382
\(694\) −1.60766 −0.0610258
\(695\) 27.6569 1.04908
\(696\) 18.4853 0.700683
\(697\) 0 0
\(698\) −1.75736 −0.0665170
\(699\) −26.4853 −1.00177
\(700\) −3.13839 −0.118620
\(701\) −21.6985 −0.819540 −0.409770 0.912189i \(-0.634391\pi\)
−0.409770 + 0.912189i \(0.634391\pi\)
\(702\) −1.26810 −0.0478614
\(703\) −18.4776 −0.696896
\(704\) −4.51528 −0.170176
\(705\) 52.2843 1.96914
\(706\) 5.79899 0.218248
\(707\) −14.5194 −0.546060
\(708\) 28.6675 1.07739
\(709\) −11.5893 −0.435245 −0.217622 0.976033i \(-0.569830\pi\)
−0.217622 + 0.976033i \(0.569830\pi\)
\(710\) 10.0000 0.375293
\(711\) 15.8645 0.594964
\(712\) 14.9289 0.559485
\(713\) 13.4558 0.503925
\(714\) 0 0
\(715\) 2.82843 0.105777
\(716\) −10.9706 −0.409989
\(717\) −23.9665 −0.895044
\(718\) −9.59798 −0.358193
\(719\) −14.0711 −0.524763 −0.262382 0.964964i \(-0.584508\pi\)
−0.262382 + 0.964964i \(0.584508\pi\)
\(720\) −21.2220 −0.790898
\(721\) 4.85483 0.180803
\(722\) −1.78680 −0.0664977
\(723\) −32.1421 −1.19538
\(724\) −22.8297 −0.848459
\(725\) 7.07401 0.262722
\(726\) −10.6382 −0.394821
\(727\) 19.1127 0.708851 0.354425 0.935084i \(-0.384676\pi\)
0.354425 + 0.935084i \(0.384676\pi\)
\(728\) 2.42742 0.0899661
\(729\) −39.2843 −1.45497
\(730\) 4.10051 0.151767
\(731\) 0 0
\(732\) 44.1421 1.63154
\(733\) −12.0416 −0.444768 −0.222384 0.974959i \(-0.571384\pi\)
−0.222384 + 0.974959i \(0.571384\pi\)
\(734\) −10.9008 −0.402358
\(735\) −28.1421 −1.03804
\(736\) −18.2919 −0.674248
\(737\) −7.39104 −0.272252
\(738\) −12.9343 −0.476119
\(739\) −34.2843 −1.26117 −0.630584 0.776121i \(-0.717184\pi\)
−0.630584 + 0.776121i \(0.717184\pi\)
\(740\) −12.9289 −0.475277
\(741\) 17.8435 0.655499
\(742\) −0.634051 −0.0232767
\(743\) 49.2780 1.80783 0.903917 0.427708i \(-0.140679\pi\)
0.903917 + 0.427708i \(0.140679\pi\)
\(744\) −13.4558 −0.493315
\(745\) −31.3575 −1.14885
\(746\) −8.10051 −0.296581
\(747\) 1.31371 0.0480661
\(748\) 0 0
\(749\) −6.82843 −0.249505
\(750\) 13.1716 0.480958
\(751\) −26.2082 −0.956350 −0.478175 0.878265i \(-0.658702\pi\)
−0.478175 + 0.878265i \(0.658702\pi\)
\(752\) 32.4853 1.18462
\(753\) −9.18440 −0.334698
\(754\) −2.61313 −0.0951644
\(755\) −23.7038 −0.862671
\(756\) −4.28427 −0.155817
\(757\) 53.4558 1.94289 0.971443 0.237274i \(-0.0762538\pi\)
0.971443 + 0.237274i \(0.0762538\pi\)
\(758\) 0.448342 0.0162845
\(759\) −11.7206 −0.425431
\(760\) −14.1480 −0.513203
\(761\) −21.6985 −0.786569 −0.393285 0.919417i \(-0.628661\pi\)
−0.393285 + 0.919417i \(0.628661\pi\)
\(762\) 18.7402 0.678887
\(763\) 4.62742 0.167524
\(764\) −36.5685 −1.32300
\(765\) 0 0
\(766\) −2.28427 −0.0825341
\(767\) −8.48528 −0.306386
\(768\) −10.3756 −0.374397
\(769\) −12.7279 −0.458981 −0.229490 0.973311i \(-0.573706\pi\)
−0.229490 + 0.973311i \(0.573706\pi\)
\(770\) −0.896683 −0.0323142
\(771\) 57.8602 2.08378
\(772\) −10.1355 −0.364784
\(773\) 4.82843 0.173666 0.0868332 0.996223i \(-0.472325\pi\)
0.0868332 + 0.996223i \(0.472325\pi\)
\(774\) 7.65685 0.275220
\(775\) −5.14933 −0.184969
\(776\) 10.2124 0.366604
\(777\) −10.8239 −0.388306
\(778\) −6.68629 −0.239715
\(779\) 39.3826 1.41103
\(780\) 12.4853 0.447045
\(781\) 14.1421 0.506045
\(782\) 0 0
\(783\) 9.65685 0.345108
\(784\) −17.4853 −0.624474
\(785\) −3.06147 −0.109268
\(786\) 0.201010 0.00716979
\(787\) 49.6494 1.76981 0.884905 0.465772i \(-0.154223\pi\)
0.884905 + 0.465772i \(0.154223\pi\)
\(788\) 27.2680 0.971384
\(789\) −16.9469 −0.603324
\(790\) 3.17157 0.112839
\(791\) 17.5147 0.622752
\(792\) 6.57128 0.233500
\(793\) −13.0656 −0.463974
\(794\) −11.2948 −0.400837
\(795\) −6.82843 −0.242179
\(796\) 3.13839 0.111237
\(797\) 17.2132 0.609723 0.304861 0.952397i \(-0.401390\pi\)
0.304861 + 0.952397i \(0.401390\pi\)
\(798\) −5.65685 −0.200250
\(799\) 0 0
\(800\) 7.00000 0.247487
\(801\) 36.0416 1.27347
\(802\) −7.07401 −0.249792
\(803\) 5.79899 0.204642
\(804\) −32.6256 −1.15062
\(805\) 8.28772 0.292104
\(806\) 1.90215 0.0670004
\(807\) 16.6274 0.585313
\(808\) 21.2721 0.748349
\(809\) −29.7724 −1.04674 −0.523371 0.852105i \(-0.675326\pi\)
−0.523371 + 0.852105i \(0.675326\pi\)
\(810\) −4.46088 −0.156740
\(811\) 44.2693 1.55451 0.777253 0.629189i \(-0.216613\pi\)
0.777253 + 0.629189i \(0.216613\pi\)
\(812\) −8.82843 −0.309817
\(813\) −16.0502 −0.562904
\(814\) 1.71573 0.0601363
\(815\) 10.4853 0.367283
\(816\) 0 0
\(817\) −23.3137 −0.815643
\(818\) −8.00000 −0.279713
\(819\) 5.86030 0.204776
\(820\) 27.5563 0.962309
\(821\) 14.3881 0.502149 0.251074 0.967968i \(-0.419216\pi\)
0.251074 + 0.967968i \(0.419216\pi\)
\(822\) 9.44703 0.329503
\(823\) −23.5181 −0.819791 −0.409895 0.912133i \(-0.634435\pi\)
−0.409895 + 0.912133i \(0.634435\pi\)
\(824\) −7.11270 −0.247783
\(825\) 4.48528 0.156157
\(826\) 2.69005 0.0935988
\(827\) 34.7135 1.20711 0.603553 0.797323i \(-0.293751\pi\)
0.603553 + 0.797323i \(0.293751\pi\)
\(828\) −29.0070 −1.00806
\(829\) −13.9411 −0.484195 −0.242098 0.970252i \(-0.577835\pi\)
−0.242098 + 0.970252i \(0.577835\pi\)
\(830\) 0.262632 0.00911610
\(831\) 57.5980 1.99805
\(832\) 5.89949 0.204528
\(833\) 0 0
\(834\) −16.2010 −0.560995
\(835\) 18.4853 0.639710
\(836\) −9.55582 −0.330495
\(837\) −7.02944 −0.242973
\(838\) −11.1635 −0.385636
\(839\) −3.88123 −0.133995 −0.0669974 0.997753i \(-0.521342\pi\)
−0.0669974 + 0.997753i \(0.521342\pi\)
\(840\) −8.28772 −0.285953
\(841\) −9.10051 −0.313811
\(842\) −7.21320 −0.248583
\(843\) 46.7736 1.61097
\(844\) 27.3675 0.942028
\(845\) 20.3253 0.699213
\(846\) −17.1716 −0.590371
\(847\) 10.6382 0.365533
\(848\) −4.24264 −0.145693
\(849\) 59.7990 2.05230
\(850\) 0 0
\(851\) −15.8579 −0.543601
\(852\) 62.4264 2.13869
\(853\) 42.5754 1.45775 0.728877 0.684645i \(-0.240043\pi\)
0.728877 + 0.684645i \(0.240043\pi\)
\(854\) 4.14214 0.141741
\(855\) −34.1563 −1.16812
\(856\) 10.0042 0.341935
\(857\) 3.82683 0.130722 0.0653611 0.997862i \(-0.479180\pi\)
0.0653611 + 0.997862i \(0.479180\pi\)
\(858\) −1.65685 −0.0565641
\(859\) −1.02944 −0.0351239 −0.0175620 0.999846i \(-0.505590\pi\)
−0.0175620 + 0.999846i \(0.505590\pi\)
\(860\) −16.3128 −0.556262
\(861\) 23.0698 0.786216
\(862\) 16.4985 0.561942
\(863\) 34.6274 1.17873 0.589365 0.807867i \(-0.299378\pi\)
0.589365 + 0.807867i \(0.299378\pi\)
\(864\) 9.55582 0.325096
\(865\) 6.24264 0.212256
\(866\) −6.28427 −0.213548
\(867\) 0 0
\(868\) 6.42641 0.218126
\(869\) 4.48528 0.152153
\(870\) 8.92177 0.302476
\(871\) 9.65685 0.327210
\(872\) −6.77952 −0.229583
\(873\) 24.6549 0.834443
\(874\) −8.28772 −0.280336
\(875\) −13.1716 −0.445280
\(876\) 25.5980 0.864876
\(877\) −37.6436 −1.27113 −0.635567 0.772045i \(-0.719234\pi\)
−0.635567 + 0.772045i \(0.719234\pi\)
\(878\) 4.51528 0.152383
\(879\) 61.8183 2.08508
\(880\) −6.00000 −0.202260
\(881\) −18.5320 −0.624358 −0.312179 0.950023i \(-0.601059\pi\)
−0.312179 + 0.950023i \(0.601059\pi\)
\(882\) 9.24264 0.311216
\(883\) −8.00000 −0.269221 −0.134611 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) 0 0
\(885\) 28.9706 0.973835
\(886\) −6.54416 −0.219855
\(887\) 48.8615 1.64061 0.820304 0.571927i \(-0.193804\pi\)
0.820304 + 0.571927i \(0.193804\pi\)
\(888\) 15.8579 0.532155
\(889\) −18.7402 −0.628527
\(890\) 7.20533 0.241523
\(891\) −6.30864 −0.211348
\(892\) −1.51472 −0.0507165
\(893\) 52.2843 1.74963
\(894\) 18.3688 0.614345
\(895\) −11.0866 −0.370583
\(896\) −11.4261 −0.381720
\(897\) 15.3137 0.511310
\(898\) −7.78498 −0.259788
\(899\) −14.4853 −0.483111
\(900\) 11.1005 0.370017
\(901\) 0 0
\(902\) −3.65685 −0.121760
\(903\) −13.6569 −0.454472
\(904\) −25.6604 −0.853452
\(905\) −23.0711 −0.766908
\(906\) 13.8854 0.461311
\(907\) −25.0489 −0.831734 −0.415867 0.909425i \(-0.636522\pi\)
−0.415867 + 0.909425i \(0.636522\pi\)
\(908\) 9.21627 0.305852
\(909\) 51.3553 1.70335
\(910\) 1.17157 0.0388373
\(911\) −8.47343 −0.280737 −0.140369 0.990099i \(-0.544829\pi\)
−0.140369 + 0.990099i \(0.544829\pi\)
\(912\) −37.8519 −1.25340
\(913\) 0.371418 0.0122922
\(914\) −7.79899 −0.257968
\(915\) 44.6088 1.47472
\(916\) 41.7401 1.37913
\(917\) −0.201010 −0.00663794
\(918\) 0 0
\(919\) −3.31371 −0.109309 −0.0546546 0.998505i \(-0.517406\pi\)
−0.0546546 + 0.998505i \(0.517406\pi\)
\(920\) −12.1421 −0.400314
\(921\) −5.59767 −0.184450
\(922\) −9.95837 −0.327961
\(923\) −18.4776 −0.608197
\(924\) −5.59767 −0.184150
\(925\) 6.06854 0.199532
\(926\) 12.6863 0.416897
\(927\) −17.1716 −0.563988
\(928\) 19.6913 0.646399
\(929\) −20.9594 −0.687656 −0.343828 0.939033i \(-0.611724\pi\)
−0.343828 + 0.939033i \(0.611724\pi\)
\(930\) −6.49435 −0.212958
\(931\) −28.1421 −0.922321
\(932\) −18.5320 −0.607035
\(933\) −11.7990 −0.386282
\(934\) 5.23045 0.171145
\(935\) 0 0
\(936\) −8.58579 −0.280635
\(937\) 3.55635 0.116181 0.0580904 0.998311i \(-0.481499\pi\)
0.0580904 + 0.998311i \(0.481499\pi\)
\(938\) −3.06147 −0.0999605
\(939\) −33.3137 −1.08715
\(940\) 36.5838 1.19323
\(941\) 18.6858 0.609141 0.304570 0.952490i \(-0.401487\pi\)
0.304570 + 0.952490i \(0.401487\pi\)
\(942\) 1.79337 0.0584310
\(943\) 33.7990 1.10065
\(944\) 18.0000 0.585850
\(945\) −4.32957 −0.140841
\(946\) 2.16478 0.0703832
\(947\) 14.8590 0.482852 0.241426 0.970419i \(-0.422385\pi\)
0.241426 + 0.970419i \(0.422385\pi\)
\(948\) 19.7990 0.643041
\(949\) −7.57675 −0.245952
\(950\) 3.17157 0.102899
\(951\) −15.1716 −0.491972
\(952\) 0 0
\(953\) −9.69848 −0.314165 −0.157082 0.987586i \(-0.550209\pi\)
−0.157082 + 0.987586i \(0.550209\pi\)
\(954\) 2.24264 0.0726082
\(955\) −36.9552 −1.19584
\(956\) −16.7696 −0.542366
\(957\) 12.6173 0.407859
\(958\) 14.3337 0.463102
\(959\) −9.44703 −0.305061
\(960\) −20.1421 −0.650085
\(961\) −20.4558 −0.659866
\(962\) −2.24171 −0.0722756
\(963\) 24.1522 0.778293
\(964\) −22.4901 −0.724358
\(965\) −10.2426 −0.329722
\(966\) −4.85483 −0.156202
\(967\) −32.3431 −1.04009 −0.520043 0.854140i \(-0.674084\pi\)
−0.520043 + 0.854140i \(0.674084\pi\)
\(968\) −15.5858 −0.500946
\(969\) 0 0
\(970\) 4.92893 0.158258
\(971\) −55.7401 −1.78879 −0.894393 0.447283i \(-0.852392\pi\)
−0.894393 + 0.447283i \(0.852392\pi\)
\(972\) −39.7222 −1.27409
\(973\) 16.2010 0.519381
\(974\) −1.82523 −0.0584841
\(975\) −5.86030 −0.187680
\(976\) 27.7164 0.887180
\(977\) −1.61522 −0.0516756 −0.0258378 0.999666i \(-0.508225\pi\)
−0.0258378 + 0.999666i \(0.508225\pi\)
\(978\) −6.14214 −0.196404
\(979\) 10.1899 0.325670
\(980\) −19.6913 −0.629015
\(981\) −16.3672 −0.522564
\(982\) 10.4020 0.331942
\(983\) 23.5181 0.750112 0.375056 0.927002i \(-0.377623\pi\)
0.375056 + 0.927002i \(0.377623\pi\)
\(984\) −33.7990 −1.07747
\(985\) 27.5563 0.878018
\(986\) 0 0
\(987\) 30.6274 0.974881
\(988\) 12.4853 0.397210
\(989\) −20.0083 −0.636228
\(990\) 3.17157 0.100799
\(991\) −32.0685 −1.01869 −0.509345 0.860563i \(-0.670112\pi\)
−0.509345 + 0.860563i \(0.670112\pi\)
\(992\) −14.3337 −0.455096
\(993\) −46.5110 −1.47598
\(994\) 5.85786 0.185800
\(995\) 3.17157 0.100546
\(996\) 1.63952 0.0519502
\(997\) 7.70806 0.244117 0.122058 0.992523i \(-0.461050\pi\)
0.122058 + 0.992523i \(0.461050\pi\)
\(998\) −15.3392 −0.485554
\(999\) 8.28427 0.262103
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 289.2.a.f.1.1 4
3.2 odd 2 2601.2.a.bb.1.4 4
4.3 odd 2 4624.2.a.bp.1.4 4
5.4 even 2 7225.2.a.u.1.4 4
17.2 even 8 289.2.c.c.38.1 8
17.3 odd 16 17.2.d.a.9.1 yes 4
17.4 even 4 289.2.b.b.288.4 4
17.5 odd 16 289.2.d.c.110.1 4
17.6 odd 16 17.2.d.a.2.1 4
17.7 odd 16 289.2.d.c.134.1 4
17.8 even 8 289.2.c.c.251.4 8
17.9 even 8 289.2.c.c.251.3 8
17.10 odd 16 289.2.d.b.134.1 4
17.11 odd 16 289.2.d.a.155.1 4
17.12 odd 16 289.2.d.b.110.1 4
17.13 even 4 289.2.b.b.288.3 4
17.14 odd 16 289.2.d.a.179.1 4
17.15 even 8 289.2.c.c.38.2 8
17.16 even 2 inner 289.2.a.f.1.2 4
51.20 even 16 153.2.l.c.145.1 4
51.23 even 16 153.2.l.c.19.1 4
51.50 odd 2 2601.2.a.bb.1.3 4
68.3 even 16 272.2.v.d.145.1 4
68.23 even 16 272.2.v.d.257.1 4
68.67 odd 2 4624.2.a.bp.1.1 4
85.3 even 16 425.2.n.b.349.1 4
85.23 even 16 425.2.n.a.274.1 4
85.37 even 16 425.2.n.a.349.1 4
85.54 odd 16 425.2.m.a.26.1 4
85.57 even 16 425.2.n.b.274.1 4
85.74 odd 16 425.2.m.a.376.1 4
85.84 even 2 7225.2.a.u.1.3 4
119.3 even 48 833.2.v.a.128.1 8
119.6 even 16 833.2.l.a.393.1 4
119.20 even 16 833.2.l.a.638.1 4
119.23 odd 48 833.2.v.b.410.1 8
119.37 odd 48 833.2.v.b.655.1 8
119.40 even 48 833.2.v.a.410.1 8
119.54 even 48 833.2.v.a.655.1 8
119.74 odd 48 833.2.v.b.716.1 8
119.88 odd 48 833.2.v.b.128.1 8
119.108 even 48 833.2.v.a.716.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.2.d.a.2.1 4 17.6 odd 16
17.2.d.a.9.1 yes 4 17.3 odd 16
153.2.l.c.19.1 4 51.23 even 16
153.2.l.c.145.1 4 51.20 even 16
272.2.v.d.145.1 4 68.3 even 16
272.2.v.d.257.1 4 68.23 even 16
289.2.a.f.1.1 4 1.1 even 1 trivial
289.2.a.f.1.2 4 17.16 even 2 inner
289.2.b.b.288.3 4 17.13 even 4
289.2.b.b.288.4 4 17.4 even 4
289.2.c.c.38.1 8 17.2 even 8
289.2.c.c.38.2 8 17.15 even 8
289.2.c.c.251.3 8 17.9 even 8
289.2.c.c.251.4 8 17.8 even 8
289.2.d.a.155.1 4 17.11 odd 16
289.2.d.a.179.1 4 17.14 odd 16
289.2.d.b.110.1 4 17.12 odd 16
289.2.d.b.134.1 4 17.10 odd 16
289.2.d.c.110.1 4 17.5 odd 16
289.2.d.c.134.1 4 17.7 odd 16
425.2.m.a.26.1 4 85.54 odd 16
425.2.m.a.376.1 4 85.74 odd 16
425.2.n.a.274.1 4 85.23 even 16
425.2.n.a.349.1 4 85.37 even 16
425.2.n.b.274.1 4 85.57 even 16
425.2.n.b.349.1 4 85.3 even 16
833.2.l.a.393.1 4 119.6 even 16
833.2.l.a.638.1 4 119.20 even 16
833.2.v.a.128.1 8 119.3 even 48
833.2.v.a.410.1 8 119.40 even 48
833.2.v.a.655.1 8 119.54 even 48
833.2.v.a.716.1 8 119.108 even 48
833.2.v.b.128.1 8 119.88 odd 48
833.2.v.b.410.1 8 119.23 odd 48
833.2.v.b.655.1 8 119.37 odd 48
833.2.v.b.716.1 8 119.74 odd 48
2601.2.a.bb.1.3 4 51.50 odd 2
2601.2.a.bb.1.4 4 3.2 odd 2
4624.2.a.bp.1.1 4 68.67 odd 2
4624.2.a.bp.1.4 4 4.3 odd 2
7225.2.a.u.1.3 4 85.84 even 2
7225.2.a.u.1.4 4 5.4 even 2