Properties

Label 6422.2.a.t.1.3
Level $6422$
Weight $2$
Character 6422.1
Self dual yes
Analytic conductor $51.280$
Analytic rank $1$
Dimension $3$
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] = [6422,2,Mod(1,6422)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6422, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6422.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.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: \(51.2799281781\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 494)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 6422.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.04892 q^{3} +1.00000 q^{4} +0.198062 q^{5} +2.04892 q^{6} -0.890084 q^{7} +1.00000 q^{8} +1.19806 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.04892 q^{3} +1.00000 q^{4} +0.198062 q^{5} +2.04892 q^{6} -0.890084 q^{7} +1.00000 q^{8} +1.19806 q^{9} +0.198062 q^{10} -4.13706 q^{11} +2.04892 q^{12} -0.890084 q^{14} +0.405813 q^{15} +1.00000 q^{16} -2.66487 q^{17} +1.19806 q^{18} -1.00000 q^{19} +0.198062 q^{20} -1.82371 q^{21} -4.13706 q^{22} -8.45473 q^{23} +2.04892 q^{24} -4.96077 q^{25} -3.69202 q^{27} -0.890084 q^{28} -4.13706 q^{29} +0.405813 q^{30} +1.08815 q^{31} +1.00000 q^{32} -8.47650 q^{33} -2.66487 q^{34} -0.176292 q^{35} +1.19806 q^{36} +10.7681 q^{37} -1.00000 q^{38} +0.198062 q^{40} -6.66487 q^{41} -1.82371 q^{42} +6.19567 q^{43} -4.13706 q^{44} +0.237291 q^{45} -8.45473 q^{46} +7.38404 q^{47} +2.04892 q^{48} -6.20775 q^{49} -4.96077 q^{50} -5.46011 q^{51} -3.40581 q^{53} -3.69202 q^{54} -0.819396 q^{55} -0.890084 q^{56} -2.04892 q^{57} -4.13706 q^{58} +7.20775 q^{59} +0.405813 q^{60} -11.9215 q^{61} +1.08815 q^{62} -1.06638 q^{63} +1.00000 q^{64} -8.47650 q^{66} +3.87800 q^{67} -2.66487 q^{68} -17.3230 q^{69} -0.176292 q^{70} +6.07606 q^{71} +1.19806 q^{72} -5.16421 q^{73} +10.7681 q^{74} -10.1642 q^{75} -1.00000 q^{76} +3.68233 q^{77} -15.6582 q^{79} +0.198062 q^{80} -11.1588 q^{81} -6.66487 q^{82} -13.1588 q^{83} -1.82371 q^{84} -0.527811 q^{85} +6.19567 q^{86} -8.47650 q^{87} -4.13706 q^{88} -4.53319 q^{89} +0.237291 q^{90} -8.45473 q^{92} +2.22952 q^{93} +7.38404 q^{94} -0.198062 q^{95} +2.04892 q^{96} +8.18060 q^{97} -6.20775 q^{98} -4.95646 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 5 q^{5} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 5 q^{5} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 8 q^{9} + 5 q^{10} - 7 q^{11} - 3 q^{12} - 2 q^{14} - 12 q^{15} + 3 q^{16} - 9 q^{17} + 8 q^{18} - 3 q^{19} + 5 q^{20} + 2 q^{21} - 7 q^{22} - 3 q^{23} - 3 q^{24} - 2 q^{25} - 6 q^{27} - 2 q^{28} - 7 q^{29} - 12 q^{30} + 7 q^{31} + 3 q^{32} - 9 q^{34} - 8 q^{35} + 8 q^{36} + 12 q^{37} - 3 q^{38} + 5 q^{40} - 21 q^{41} + 2 q^{42} - 18 q^{43} - 7 q^{44} + 18 q^{45} - 3 q^{46} + 12 q^{47} - 3 q^{48} - q^{49} - 2 q^{50} + 9 q^{51} + 3 q^{53} - 6 q^{54} - 14 q^{55} - 2 q^{56} + 3 q^{57} - 7 q^{58} + 4 q^{59} - 12 q^{60} - 10 q^{61} + 7 q^{62} - 10 q^{63} + 3 q^{64} - 8 q^{67} - 9 q^{68} - 32 q^{69} - 8 q^{70} + 3 q^{71} + 8 q^{72} - 4 q^{73} + 12 q^{74} - 19 q^{75} - 3 q^{76} + 28 q^{77} - 26 q^{79} + 5 q^{80} - 25 q^{81} - 21 q^{82} - 31 q^{83} + 2 q^{84} - 8 q^{85} - 18 q^{86} - 7 q^{88} - 17 q^{89} + 18 q^{90} - 3 q^{92} - 14 q^{93} + 12 q^{94} - 5 q^{95} - 3 q^{96} + 13 q^{97} - q^{98} - 21 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.04892 1.18294 0.591471 0.806326i \(-0.298547\pi\)
0.591471 + 0.806326i \(0.298547\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.198062 0.0885761 0.0442881 0.999019i \(-0.485898\pi\)
0.0442881 + 0.999019i \(0.485898\pi\)
\(6\) 2.04892 0.836467
\(7\) −0.890084 −0.336420 −0.168210 0.985751i \(-0.553799\pi\)
−0.168210 + 0.985751i \(0.553799\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.19806 0.399354
\(10\) 0.198062 0.0626328
\(11\) −4.13706 −1.24737 −0.623686 0.781675i \(-0.714366\pi\)
−0.623686 + 0.781675i \(0.714366\pi\)
\(12\) 2.04892 0.591471
\(13\) 0 0
\(14\) −0.890084 −0.237885
\(15\) 0.405813 0.104781
\(16\) 1.00000 0.250000
\(17\) −2.66487 −0.646327 −0.323163 0.946343i \(-0.604746\pi\)
−0.323163 + 0.946343i \(0.604746\pi\)
\(18\) 1.19806 0.282386
\(19\) −1.00000 −0.229416
\(20\) 0.198062 0.0442881
\(21\) −1.82371 −0.397966
\(22\) −4.13706 −0.882025
\(23\) −8.45473 −1.76293 −0.881467 0.472246i \(-0.843443\pi\)
−0.881467 + 0.472246i \(0.843443\pi\)
\(24\) 2.04892 0.418234
\(25\) −4.96077 −0.992154
\(26\) 0 0
\(27\) −3.69202 −0.710530
\(28\) −0.890084 −0.168210
\(29\) −4.13706 −0.768233 −0.384117 0.923285i \(-0.625494\pi\)
−0.384117 + 0.923285i \(0.625494\pi\)
\(30\) 0.405813 0.0740910
\(31\) 1.08815 0.195437 0.0977184 0.995214i \(-0.468846\pi\)
0.0977184 + 0.995214i \(0.468846\pi\)
\(32\) 1.00000 0.176777
\(33\) −8.47650 −1.47557
\(34\) −2.66487 −0.457022
\(35\) −0.176292 −0.0297988
\(36\) 1.19806 0.199677
\(37\) 10.7681 1.77026 0.885131 0.465342i \(-0.154068\pi\)
0.885131 + 0.465342i \(0.154068\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0.198062 0.0313164
\(41\) −6.66487 −1.04088 −0.520439 0.853899i \(-0.674232\pi\)
−0.520439 + 0.853899i \(0.674232\pi\)
\(42\) −1.82371 −0.281404
\(43\) 6.19567 0.944831 0.472415 0.881376i \(-0.343382\pi\)
0.472415 + 0.881376i \(0.343382\pi\)
\(44\) −4.13706 −0.623686
\(45\) 0.237291 0.0353732
\(46\) −8.45473 −1.24658
\(47\) 7.38404 1.07707 0.538537 0.842602i \(-0.318977\pi\)
0.538537 + 0.842602i \(0.318977\pi\)
\(48\) 2.04892 0.295736
\(49\) −6.20775 −0.886822
\(50\) −4.96077 −0.701559
\(51\) −5.46011 −0.764568
\(52\) 0 0
\(53\) −3.40581 −0.467824 −0.233912 0.972258i \(-0.575153\pi\)
−0.233912 + 0.972258i \(0.575153\pi\)
\(54\) −3.69202 −0.502420
\(55\) −0.819396 −0.110487
\(56\) −0.890084 −0.118942
\(57\) −2.04892 −0.271386
\(58\) −4.13706 −0.543223
\(59\) 7.20775 0.938369 0.469185 0.883100i \(-0.344548\pi\)
0.469185 + 0.883100i \(0.344548\pi\)
\(60\) 0.405813 0.0523903
\(61\) −11.9215 −1.52640 −0.763199 0.646164i \(-0.776372\pi\)
−0.763199 + 0.646164i \(0.776372\pi\)
\(62\) 1.08815 0.138195
\(63\) −1.06638 −0.134351
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −8.47650 −1.04339
\(67\) 3.87800 0.473773 0.236887 0.971537i \(-0.423873\pi\)
0.236887 + 0.971537i \(0.423873\pi\)
\(68\) −2.66487 −0.323163
\(69\) −17.3230 −2.08545
\(70\) −0.176292 −0.0210709
\(71\) 6.07606 0.721096 0.360548 0.932741i \(-0.382590\pi\)
0.360548 + 0.932741i \(0.382590\pi\)
\(72\) 1.19806 0.141193
\(73\) −5.16421 −0.604425 −0.302213 0.953241i \(-0.597725\pi\)
−0.302213 + 0.953241i \(0.597725\pi\)
\(74\) 10.7681 1.25176
\(75\) −10.1642 −1.17366
\(76\) −1.00000 −0.114708
\(77\) 3.68233 0.419641
\(78\) 0 0
\(79\) −15.6582 −1.76168 −0.880841 0.473412i \(-0.843022\pi\)
−0.880841 + 0.473412i \(0.843022\pi\)
\(80\) 0.198062 0.0221440
\(81\) −11.1588 −1.23987
\(82\) −6.66487 −0.736012
\(83\) −13.1588 −1.44437 −0.722185 0.691700i \(-0.756862\pi\)
−0.722185 + 0.691700i \(0.756862\pi\)
\(84\) −1.82371 −0.198983
\(85\) −0.527811 −0.0572491
\(86\) 6.19567 0.668096
\(87\) −8.47650 −0.908776
\(88\) −4.13706 −0.441012
\(89\) −4.53319 −0.480517 −0.240258 0.970709i \(-0.577232\pi\)
−0.240258 + 0.970709i \(0.577232\pi\)
\(90\) 0.237291 0.0250127
\(91\) 0 0
\(92\) −8.45473 −0.881467
\(93\) 2.22952 0.231191
\(94\) 7.38404 0.761606
\(95\) −0.198062 −0.0203208
\(96\) 2.04892 0.209117
\(97\) 8.18060 0.830614 0.415307 0.909681i \(-0.363674\pi\)
0.415307 + 0.909681i \(0.363674\pi\)
\(98\) −6.20775 −0.627078
\(99\) −4.95646 −0.498143
\(100\) −4.96077 −0.496077
\(101\) 1.32975 0.132315 0.0661575 0.997809i \(-0.478926\pi\)
0.0661575 + 0.997809i \(0.478926\pi\)
\(102\) −5.46011 −0.540631
\(103\) −0.670251 −0.0660418 −0.0330209 0.999455i \(-0.510513\pi\)
−0.0330209 + 0.999455i \(0.510513\pi\)
\(104\) 0 0
\(105\) −0.361208 −0.0352503
\(106\) −3.40581 −0.330802
\(107\) 13.1075 1.26715 0.633576 0.773680i \(-0.281586\pi\)
0.633576 + 0.773680i \(0.281586\pi\)
\(108\) −3.69202 −0.355265
\(109\) −15.0858 −1.44495 −0.722477 0.691395i \(-0.756996\pi\)
−0.722477 + 0.691395i \(0.756996\pi\)
\(110\) −0.819396 −0.0781264
\(111\) 22.0629 2.09412
\(112\) −0.890084 −0.0841050
\(113\) 19.5254 1.83680 0.918398 0.395657i \(-0.129483\pi\)
0.918398 + 0.395657i \(0.129483\pi\)
\(114\) −2.04892 −0.191899
\(115\) −1.67456 −0.156154
\(116\) −4.13706 −0.384117
\(117\) 0 0
\(118\) 7.20775 0.663527
\(119\) 2.37196 0.217437
\(120\) 0.405813 0.0370455
\(121\) 6.11529 0.555936
\(122\) −11.9215 −1.07933
\(123\) −13.6558 −1.23130
\(124\) 1.08815 0.0977184
\(125\) −1.97285 −0.176457
\(126\) −1.06638 −0.0950003
\(127\) −13.2078 −1.17200 −0.585999 0.810312i \(-0.699298\pi\)
−0.585999 + 0.810312i \(0.699298\pi\)
\(128\) 1.00000 0.0883883
\(129\) 12.6944 1.11768
\(130\) 0 0
\(131\) 14.9879 1.30950 0.654750 0.755845i \(-0.272774\pi\)
0.654750 + 0.755845i \(0.272774\pi\)
\(132\) −8.47650 −0.737785
\(133\) 0.890084 0.0771800
\(134\) 3.87800 0.335008
\(135\) −0.731250 −0.0629360
\(136\) −2.66487 −0.228511
\(137\) 17.3056 1.47852 0.739258 0.673422i \(-0.235176\pi\)
0.739258 + 0.673422i \(0.235176\pi\)
\(138\) −17.3230 −1.47464
\(139\) 14.4155 1.22271 0.611353 0.791358i \(-0.290625\pi\)
0.611353 + 0.791358i \(0.290625\pi\)
\(140\) −0.176292 −0.0148994
\(141\) 15.1293 1.27412
\(142\) 6.07606 0.509892
\(143\) 0 0
\(144\) 1.19806 0.0998385
\(145\) −0.819396 −0.0680471
\(146\) −5.16421 −0.427393
\(147\) −12.7192 −1.04906
\(148\) 10.7681 0.885131
\(149\) 5.38942 0.441518 0.220759 0.975328i \(-0.429146\pi\)
0.220759 + 0.975328i \(0.429146\pi\)
\(150\) −10.1642 −0.829904
\(151\) −15.9366 −1.29690 −0.648451 0.761256i \(-0.724583\pi\)
−0.648451 + 0.761256i \(0.724583\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −3.19269 −0.258113
\(154\) 3.68233 0.296731
\(155\) 0.215521 0.0173110
\(156\) 0 0
\(157\) −10.5483 −0.841842 −0.420921 0.907097i \(-0.638293\pi\)
−0.420921 + 0.907097i \(0.638293\pi\)
\(158\) −15.6582 −1.24570
\(159\) −6.97823 −0.553410
\(160\) 0.198062 0.0156582
\(161\) 7.52542 0.593086
\(162\) −11.1588 −0.876721
\(163\) −5.59850 −0.438508 −0.219254 0.975668i \(-0.570362\pi\)
−0.219254 + 0.975668i \(0.570362\pi\)
\(164\) −6.66487 −0.520439
\(165\) −1.67887 −0.130700
\(166\) −13.1588 −1.02132
\(167\) 13.8485 1.07163 0.535813 0.844337i \(-0.320005\pi\)
0.535813 + 0.844337i \(0.320005\pi\)
\(168\) −1.82371 −0.140702
\(169\) 0 0
\(170\) −0.527811 −0.0404813
\(171\) −1.19806 −0.0916181
\(172\) 6.19567 0.472415
\(173\) 15.4523 1.17482 0.587410 0.809290i \(-0.300148\pi\)
0.587410 + 0.809290i \(0.300148\pi\)
\(174\) −8.47650 −0.642602
\(175\) 4.41550 0.333781
\(176\) −4.13706 −0.311843
\(177\) 14.7681 1.11004
\(178\) −4.53319 −0.339777
\(179\) −12.7114 −0.950095 −0.475047 0.879960i \(-0.657569\pi\)
−0.475047 + 0.879960i \(0.657569\pi\)
\(180\) 0.237291 0.0176866
\(181\) 13.4765 1.00170 0.500850 0.865534i \(-0.333021\pi\)
0.500850 + 0.865534i \(0.333021\pi\)
\(182\) 0 0
\(183\) −24.4263 −1.80564
\(184\) −8.45473 −0.623291
\(185\) 2.13275 0.156803
\(186\) 2.22952 0.163476
\(187\) 11.0248 0.806210
\(188\) 7.38404 0.538537
\(189\) 3.28621 0.239036
\(190\) −0.198062 −0.0143689
\(191\) 18.5157 1.33975 0.669876 0.742473i \(-0.266347\pi\)
0.669876 + 0.742473i \(0.266347\pi\)
\(192\) 2.04892 0.147868
\(193\) −3.55496 −0.255891 −0.127946 0.991781i \(-0.540838\pi\)
−0.127946 + 0.991781i \(0.540838\pi\)
\(194\) 8.18060 0.587333
\(195\) 0 0
\(196\) −6.20775 −0.443411
\(197\) −2.19567 −0.156435 −0.0782175 0.996936i \(-0.524923\pi\)
−0.0782175 + 0.996936i \(0.524923\pi\)
\(198\) −4.95646 −0.352240
\(199\) −15.2228 −1.07912 −0.539558 0.841948i \(-0.681409\pi\)
−0.539558 + 0.841948i \(0.681409\pi\)
\(200\) −4.96077 −0.350780
\(201\) 7.94571 0.560447
\(202\) 1.32975 0.0935608
\(203\) 3.68233 0.258449
\(204\) −5.46011 −0.382284
\(205\) −1.32006 −0.0921970
\(206\) −0.670251 −0.0466986
\(207\) −10.1293 −0.704035
\(208\) 0 0
\(209\) 4.13706 0.286167
\(210\) −0.361208 −0.0249257
\(211\) −17.2905 −1.19033 −0.595164 0.803604i \(-0.702913\pi\)
−0.595164 + 0.803604i \(0.702913\pi\)
\(212\) −3.40581 −0.233912
\(213\) 12.4494 0.853016
\(214\) 13.1075 0.896012
\(215\) 1.22713 0.0836895
\(216\) −3.69202 −0.251210
\(217\) −0.968541 −0.0657489
\(218\) −15.0858 −1.02174
\(219\) −10.5810 −0.715000
\(220\) −0.819396 −0.0552437
\(221\) 0 0
\(222\) 22.0629 1.48077
\(223\) 19.8726 1.33077 0.665385 0.746501i \(-0.268268\pi\)
0.665385 + 0.746501i \(0.268268\pi\)
\(224\) −0.890084 −0.0594712
\(225\) −5.94331 −0.396221
\(226\) 19.5254 1.29881
\(227\) −4.76809 −0.316469 −0.158234 0.987402i \(-0.550580\pi\)
−0.158234 + 0.987402i \(0.550580\pi\)
\(228\) −2.04892 −0.135693
\(229\) −23.8931 −1.57890 −0.789449 0.613816i \(-0.789634\pi\)
−0.789449 + 0.613816i \(0.789634\pi\)
\(230\) −1.67456 −0.110417
\(231\) 7.54480 0.496411
\(232\) −4.13706 −0.271612
\(233\) 11.7778 0.771588 0.385794 0.922585i \(-0.373928\pi\)
0.385794 + 0.922585i \(0.373928\pi\)
\(234\) 0 0
\(235\) 1.46250 0.0954030
\(236\) 7.20775 0.469185
\(237\) −32.0823 −2.08397
\(238\) 2.37196 0.153751
\(239\) −12.2983 −0.795510 −0.397755 0.917492i \(-0.630211\pi\)
−0.397755 + 0.917492i \(0.630211\pi\)
\(240\) 0.405813 0.0261951
\(241\) 0.613564 0.0395231 0.0197616 0.999805i \(-0.493709\pi\)
0.0197616 + 0.999805i \(0.493709\pi\)
\(242\) 6.11529 0.393106
\(243\) −11.7875 −0.756166
\(244\) −11.9215 −0.763199
\(245\) −1.22952 −0.0785512
\(246\) −13.6558 −0.870661
\(247\) 0 0
\(248\) 1.08815 0.0690973
\(249\) −26.9614 −1.70861
\(250\) −1.97285 −0.124774
\(251\) −21.1836 −1.33710 −0.668548 0.743669i \(-0.733084\pi\)
−0.668548 + 0.743669i \(0.733084\pi\)
\(252\) −1.06638 −0.0671754
\(253\) 34.9778 2.19903
\(254\) −13.2078 −0.828728
\(255\) −1.08144 −0.0677225
\(256\) 1.00000 0.0625000
\(257\) 15.1535 0.945247 0.472623 0.881265i \(-0.343307\pi\)
0.472623 + 0.881265i \(0.343307\pi\)
\(258\) 12.6944 0.790320
\(259\) −9.58450 −0.595552
\(260\) 0 0
\(261\) −4.95646 −0.306797
\(262\) 14.9879 0.925957
\(263\) −9.45712 −0.583151 −0.291576 0.956548i \(-0.594179\pi\)
−0.291576 + 0.956548i \(0.594179\pi\)
\(264\) −8.47650 −0.521693
\(265\) −0.674563 −0.0414381
\(266\) 0.890084 0.0545745
\(267\) −9.28813 −0.568424
\(268\) 3.87800 0.236887
\(269\) −13.4034 −0.817221 −0.408610 0.912709i \(-0.633987\pi\)
−0.408610 + 0.912709i \(0.633987\pi\)
\(270\) −0.731250 −0.0445025
\(271\) −24.7138 −1.50126 −0.750628 0.660725i \(-0.770249\pi\)
−0.750628 + 0.660725i \(0.770249\pi\)
\(272\) −2.66487 −0.161582
\(273\) 0 0
\(274\) 17.3056 1.04547
\(275\) 20.5230 1.23758
\(276\) −17.3230 −1.04272
\(277\) 8.78746 0.527987 0.263994 0.964524i \(-0.414960\pi\)
0.263994 + 0.964524i \(0.414960\pi\)
\(278\) 14.4155 0.864584
\(279\) 1.30367 0.0780485
\(280\) −0.176292 −0.0105355
\(281\) −17.7168 −1.05689 −0.528447 0.848966i \(-0.677226\pi\)
−0.528447 + 0.848966i \(0.677226\pi\)
\(282\) 15.1293 0.900936
\(283\) −9.96508 −0.592363 −0.296181 0.955132i \(-0.595713\pi\)
−0.296181 + 0.955132i \(0.595713\pi\)
\(284\) 6.07606 0.360548
\(285\) −0.405813 −0.0240383
\(286\) 0 0
\(287\) 5.93230 0.350172
\(288\) 1.19806 0.0705965
\(289\) −9.89844 −0.582261
\(290\) −0.819396 −0.0481166
\(291\) 16.7614 0.982570
\(292\) −5.16421 −0.302213
\(293\) 11.1099 0.649048 0.324524 0.945877i \(-0.394796\pi\)
0.324524 + 0.945877i \(0.394796\pi\)
\(294\) −12.7192 −0.741797
\(295\) 1.42758 0.0831171
\(296\) 10.7681 0.625882
\(297\) 15.2741 0.886295
\(298\) 5.38942 0.312201
\(299\) 0 0
\(300\) −10.1642 −0.586831
\(301\) −5.51466 −0.317860
\(302\) −15.9366 −0.917049
\(303\) 2.72455 0.156521
\(304\) −1.00000 −0.0573539
\(305\) −2.36121 −0.135202
\(306\) −3.19269 −0.182514
\(307\) 9.83446 0.561282 0.280641 0.959813i \(-0.409453\pi\)
0.280641 + 0.959813i \(0.409453\pi\)
\(308\) 3.68233 0.209820
\(309\) −1.37329 −0.0781237
\(310\) 0.215521 0.0122408
\(311\) 11.8726 0.673235 0.336617 0.941642i \(-0.390717\pi\)
0.336617 + 0.941642i \(0.390717\pi\)
\(312\) 0 0
\(313\) −8.81402 −0.498198 −0.249099 0.968478i \(-0.580134\pi\)
−0.249099 + 0.968478i \(0.580134\pi\)
\(314\) −10.5483 −0.595272
\(315\) −0.211209 −0.0119003
\(316\) −15.6582 −0.880841
\(317\) −12.7138 −0.714078 −0.357039 0.934090i \(-0.616214\pi\)
−0.357039 + 0.934090i \(0.616214\pi\)
\(318\) −6.97823 −0.391320
\(319\) 17.1153 0.958272
\(320\) 0.198062 0.0110720
\(321\) 26.8562 1.49897
\(322\) 7.52542 0.419375
\(323\) 2.66487 0.148278
\(324\) −11.1588 −0.619935
\(325\) 0 0
\(326\) −5.59850 −0.310072
\(327\) −30.9095 −1.70930
\(328\) −6.66487 −0.368006
\(329\) −6.57242 −0.362349
\(330\) −1.67887 −0.0924190
\(331\) −18.1715 −0.998796 −0.499398 0.866373i \(-0.666446\pi\)
−0.499398 + 0.866373i \(0.666446\pi\)
\(332\) −13.1588 −0.722185
\(333\) 12.9008 0.706962
\(334\) 13.8485 0.757754
\(335\) 0.768086 0.0419650
\(336\) −1.82371 −0.0994914
\(337\) 13.9323 0.758941 0.379470 0.925204i \(-0.376106\pi\)
0.379470 + 0.925204i \(0.376106\pi\)
\(338\) 0 0
\(339\) 40.0060 2.17283
\(340\) −0.527811 −0.0286246
\(341\) −4.50173 −0.243782
\(342\) −1.19806 −0.0647838
\(343\) 11.7560 0.634765
\(344\) 6.19567 0.334048
\(345\) −3.43104 −0.184721
\(346\) 15.4523 0.830723
\(347\) −8.02416 −0.430760 −0.215380 0.976530i \(-0.569099\pi\)
−0.215380 + 0.976530i \(0.569099\pi\)
\(348\) −8.47650 −0.454388
\(349\) 10.7192 0.573784 0.286892 0.957963i \(-0.407378\pi\)
0.286892 + 0.957963i \(0.407378\pi\)
\(350\) 4.41550 0.236019
\(351\) 0 0
\(352\) −4.13706 −0.220506
\(353\) 36.1172 1.92233 0.961163 0.275983i \(-0.0890032\pi\)
0.961163 + 0.275983i \(0.0890032\pi\)
\(354\) 14.7681 0.784915
\(355\) 1.20344 0.0638719
\(356\) −4.53319 −0.240258
\(357\) 4.85995 0.257216
\(358\) −12.7114 −0.671818
\(359\) 32.3129 1.70541 0.852704 0.522394i \(-0.174961\pi\)
0.852704 + 0.522394i \(0.174961\pi\)
\(360\) 0.237291 0.0125063
\(361\) 1.00000 0.0526316
\(362\) 13.4765 0.708309
\(363\) 12.5297 0.657640
\(364\) 0 0
\(365\) −1.02284 −0.0535376
\(366\) −24.4263 −1.27678
\(367\) 5.24027 0.273540 0.136770 0.990603i \(-0.456328\pi\)
0.136770 + 0.990603i \(0.456328\pi\)
\(368\) −8.45473 −0.440733
\(369\) −7.98493 −0.415679
\(370\) 2.13275 0.110876
\(371\) 3.03146 0.157386
\(372\) 2.22952 0.115595
\(373\) −14.6213 −0.757064 −0.378532 0.925588i \(-0.623571\pi\)
−0.378532 + 0.925588i \(0.623571\pi\)
\(374\) 11.0248 0.570076
\(375\) −4.04221 −0.208739
\(376\) 7.38404 0.380803
\(377\) 0 0
\(378\) 3.28621 0.169024
\(379\) −14.8116 −0.760822 −0.380411 0.924818i \(-0.624217\pi\)
−0.380411 + 0.924818i \(0.624217\pi\)
\(380\) −0.198062 −0.0101604
\(381\) −27.0616 −1.38641
\(382\) 18.5157 0.947347
\(383\) 12.2524 0.626066 0.313033 0.949742i \(-0.398655\pi\)
0.313033 + 0.949742i \(0.398655\pi\)
\(384\) 2.04892 0.104558
\(385\) 0.729331 0.0371702
\(386\) −3.55496 −0.180943
\(387\) 7.42280 0.377322
\(388\) 8.18060 0.415307
\(389\) 33.6969 1.70850 0.854251 0.519861i \(-0.174016\pi\)
0.854251 + 0.519861i \(0.174016\pi\)
\(390\) 0 0
\(391\) 22.5308 1.13943
\(392\) −6.20775 −0.313539
\(393\) 30.7090 1.54906
\(394\) −2.19567 −0.110616
\(395\) −3.10129 −0.156043
\(396\) −4.95646 −0.249071
\(397\) −0.198062 −0.00994046 −0.00497023 0.999988i \(-0.501582\pi\)
−0.00497023 + 0.999988i \(0.501582\pi\)
\(398\) −15.2228 −0.763051
\(399\) 1.82371 0.0912996
\(400\) −4.96077 −0.248039
\(401\) 7.22952 0.361025 0.180513 0.983573i \(-0.442224\pi\)
0.180513 + 0.983573i \(0.442224\pi\)
\(402\) 7.94571 0.396296
\(403\) 0 0
\(404\) 1.32975 0.0661575
\(405\) −2.21014 −0.109823
\(406\) 3.68233 0.182751
\(407\) −44.5483 −2.20817
\(408\) −5.46011 −0.270316
\(409\) 10.5483 0.521578 0.260789 0.965396i \(-0.416017\pi\)
0.260789 + 0.965396i \(0.416017\pi\)
\(410\) −1.32006 −0.0651931
\(411\) 35.4577 1.74900
\(412\) −0.670251 −0.0330209
\(413\) −6.41550 −0.315686
\(414\) −10.1293 −0.497828
\(415\) −2.60627 −0.127937
\(416\) 0 0
\(417\) 29.5362 1.44639
\(418\) 4.13706 0.202350
\(419\) −9.92154 −0.484699 −0.242350 0.970189i \(-0.577918\pi\)
−0.242350 + 0.970189i \(0.577918\pi\)
\(420\) −0.361208 −0.0176251
\(421\) −25.4577 −1.24073 −0.620367 0.784312i \(-0.713016\pi\)
−0.620367 + 0.784312i \(0.713016\pi\)
\(422\) −17.2905 −0.841689
\(423\) 8.84654 0.430134
\(424\) −3.40581 −0.165401
\(425\) 13.2198 0.641256
\(426\) 12.4494 0.603173
\(427\) 10.6112 0.513511
\(428\) 13.1075 0.633576
\(429\) 0 0
\(430\) 1.22713 0.0591774
\(431\) −1.26145 −0.0607621 −0.0303811 0.999538i \(-0.509672\pi\)
−0.0303811 + 0.999538i \(0.509672\pi\)
\(432\) −3.69202 −0.177632
\(433\) −5.78017 −0.277777 −0.138889 0.990308i \(-0.544353\pi\)
−0.138889 + 0.990308i \(0.544353\pi\)
\(434\) −0.968541 −0.0464915
\(435\) −1.67887 −0.0804959
\(436\) −15.0858 −0.722477
\(437\) 8.45473 0.404445
\(438\) −10.5810 −0.505582
\(439\) 15.6668 0.747735 0.373868 0.927482i \(-0.378031\pi\)
0.373868 + 0.927482i \(0.378031\pi\)
\(440\) −0.819396 −0.0390632
\(441\) −7.43727 −0.354156
\(442\) 0 0
\(443\) 4.89008 0.232335 0.116167 0.993230i \(-0.462939\pi\)
0.116167 + 0.993230i \(0.462939\pi\)
\(444\) 22.0629 1.04706
\(445\) −0.897853 −0.0425623
\(446\) 19.8726 0.940996
\(447\) 11.0425 0.522291
\(448\) −0.890084 −0.0420525
\(449\) 0.347207 0.0163857 0.00819286 0.999966i \(-0.497392\pi\)
0.00819286 + 0.999966i \(0.497392\pi\)
\(450\) −5.94331 −0.280170
\(451\) 27.5730 1.29836
\(452\) 19.5254 0.918398
\(453\) −32.6528 −1.53416
\(454\) −4.76809 −0.223777
\(455\) 0 0
\(456\) −2.04892 −0.0959493
\(457\) 31.7754 1.48639 0.743195 0.669075i \(-0.233310\pi\)
0.743195 + 0.669075i \(0.233310\pi\)
\(458\) −23.8931 −1.11645
\(459\) 9.83877 0.459235
\(460\) −1.67456 −0.0780769
\(461\) −28.2543 −1.31593 −0.657966 0.753047i \(-0.728583\pi\)
−0.657966 + 0.753047i \(0.728583\pi\)
\(462\) 7.54480 0.351016
\(463\) −2.42029 −0.112480 −0.0562402 0.998417i \(-0.517911\pi\)
−0.0562402 + 0.998417i \(0.517911\pi\)
\(464\) −4.13706 −0.192058
\(465\) 0.441584 0.0204780
\(466\) 11.7778 0.545595
\(467\) −10.5724 −0.489233 −0.244617 0.969620i \(-0.578662\pi\)
−0.244617 + 0.969620i \(0.578662\pi\)
\(468\) 0 0
\(469\) −3.45175 −0.159387
\(470\) 1.46250 0.0674601
\(471\) −21.6125 −0.995851
\(472\) 7.20775 0.331764
\(473\) −25.6319 −1.17855
\(474\) −32.0823 −1.47359
\(475\) 4.96077 0.227616
\(476\) 2.37196 0.108719
\(477\) −4.08038 −0.186828
\(478\) −12.2983 −0.562511
\(479\) −29.0616 −1.32786 −0.663929 0.747796i \(-0.731112\pi\)
−0.663929 + 0.747796i \(0.731112\pi\)
\(480\) 0.405813 0.0185228
\(481\) 0 0
\(482\) 0.613564 0.0279471
\(483\) 15.4190 0.701587
\(484\) 6.11529 0.277968
\(485\) 1.62027 0.0735726
\(486\) −11.7875 −0.534690
\(487\) 11.9366 0.540899 0.270450 0.962734i \(-0.412828\pi\)
0.270450 + 0.962734i \(0.412828\pi\)
\(488\) −11.9215 −0.539663
\(489\) −11.4709 −0.518730
\(490\) −1.22952 −0.0555441
\(491\) −29.3163 −1.32303 −0.661514 0.749933i \(-0.730086\pi\)
−0.661514 + 0.749933i \(0.730086\pi\)
\(492\) −13.6558 −0.615650
\(493\) 11.0248 0.496530
\(494\) 0 0
\(495\) −0.981688 −0.0441236
\(496\) 1.08815 0.0488592
\(497\) −5.40821 −0.242591
\(498\) −26.9614 −1.20817
\(499\) 1.30499 0.0584196 0.0292098 0.999573i \(-0.490701\pi\)
0.0292098 + 0.999573i \(0.490701\pi\)
\(500\) −1.97285 −0.0882287
\(501\) 28.3744 1.26767
\(502\) −21.1836 −0.945470
\(503\) −6.13275 −0.273446 −0.136723 0.990609i \(-0.543657\pi\)
−0.136723 + 0.990609i \(0.543657\pi\)
\(504\) −1.06638 −0.0475002
\(505\) 0.263373 0.0117199
\(506\) 34.9778 1.55495
\(507\) 0 0
\(508\) −13.2078 −0.585999
\(509\) −22.6219 −1.00270 −0.501350 0.865245i \(-0.667163\pi\)
−0.501350 + 0.865245i \(0.667163\pi\)
\(510\) −1.08144 −0.0478870
\(511\) 4.59658 0.203341
\(512\) 1.00000 0.0441942
\(513\) 3.69202 0.163007
\(514\) 15.1535 0.668390
\(515\) −0.132751 −0.00584973
\(516\) 12.6944 0.558840
\(517\) −30.5483 −1.34351
\(518\) −9.58450 −0.421119
\(519\) 31.6606 1.38974
\(520\) 0 0
\(521\) −11.0858 −0.485676 −0.242838 0.970067i \(-0.578078\pi\)
−0.242838 + 0.970067i \(0.578078\pi\)
\(522\) −4.95646 −0.216938
\(523\) −35.0834 −1.53409 −0.767044 0.641594i \(-0.778273\pi\)
−0.767044 + 0.641594i \(0.778273\pi\)
\(524\) 14.9879 0.654750
\(525\) 9.04700 0.394843
\(526\) −9.45712 −0.412350
\(527\) −2.89977 −0.126316
\(528\) −8.47650 −0.368892
\(529\) 48.4825 2.10793
\(530\) −0.674563 −0.0293011
\(531\) 8.63533 0.374742
\(532\) 0.890084 0.0385900
\(533\) 0 0
\(534\) −9.28813 −0.401937
\(535\) 2.59611 0.112239
\(536\) 3.87800 0.167504
\(537\) −26.0446 −1.12391
\(538\) −13.4034 −0.577862
\(539\) 25.6819 1.10620
\(540\) −0.731250 −0.0314680
\(541\) 17.8261 0.766404 0.383202 0.923665i \(-0.374821\pi\)
0.383202 + 0.923665i \(0.374821\pi\)
\(542\) −24.7138 −1.06155
\(543\) 27.6122 1.18495
\(544\) −2.66487 −0.114256
\(545\) −2.98792 −0.127988
\(546\) 0 0
\(547\) −13.1075 −0.560437 −0.280219 0.959936i \(-0.590407\pi\)
−0.280219 + 0.959936i \(0.590407\pi\)
\(548\) 17.3056 0.739258
\(549\) −14.2828 −0.609573
\(550\) 20.5230 0.875105
\(551\) 4.13706 0.176245
\(552\) −17.3230 −0.737318
\(553\) 13.9371 0.592665
\(554\) 8.78746 0.373344
\(555\) 4.36983 0.185489
\(556\) 14.4155 0.611353
\(557\) −35.4534 −1.50221 −0.751104 0.660183i \(-0.770479\pi\)
−0.751104 + 0.660183i \(0.770479\pi\)
\(558\) 1.30367 0.0551886
\(559\) 0 0
\(560\) −0.176292 −0.00744970
\(561\) 22.5888 0.953700
\(562\) −17.7168 −0.747337
\(563\) 3.14914 0.132721 0.0663603 0.997796i \(-0.478861\pi\)
0.0663603 + 0.997796i \(0.478861\pi\)
\(564\) 15.1293 0.637058
\(565\) 3.86725 0.162696
\(566\) −9.96508 −0.418864
\(567\) 9.93230 0.417117
\(568\) 6.07606 0.254946
\(569\) 20.3827 0.854488 0.427244 0.904136i \(-0.359485\pi\)
0.427244 + 0.904136i \(0.359485\pi\)
\(570\) −0.405813 −0.0169976
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 37.9372 1.58485
\(574\) 5.93230 0.247609
\(575\) 41.9420 1.74910
\(576\) 1.19806 0.0499193
\(577\) −30.6219 −1.27481 −0.637404 0.770530i \(-0.719992\pi\)
−0.637404 + 0.770530i \(0.719992\pi\)
\(578\) −9.89844 −0.411721
\(579\) −7.28382 −0.302705
\(580\) −0.819396 −0.0340236
\(581\) 11.7125 0.485915
\(582\) 16.7614 0.694782
\(583\) 14.0901 0.583551
\(584\) −5.16421 −0.213697
\(585\) 0 0
\(586\) 11.1099 0.458946
\(587\) 3.40342 0.140474 0.0702371 0.997530i \(-0.477624\pi\)
0.0702371 + 0.997530i \(0.477624\pi\)
\(588\) −12.7192 −0.524530
\(589\) −1.08815 −0.0448363
\(590\) 1.42758 0.0587727
\(591\) −4.49875 −0.185054
\(592\) 10.7681 0.442566
\(593\) −4.31767 −0.177305 −0.0886527 0.996063i \(-0.528256\pi\)
−0.0886527 + 0.996063i \(0.528256\pi\)
\(594\) 15.2741 0.626705
\(595\) 0.469796 0.0192598
\(596\) 5.38942 0.220759
\(597\) −31.1903 −1.27653
\(598\) 0 0
\(599\) −13.0616 −0.533682 −0.266841 0.963741i \(-0.585980\pi\)
−0.266841 + 0.963741i \(0.585980\pi\)
\(600\) −10.1642 −0.414952
\(601\) −42.7284 −1.74293 −0.871464 0.490460i \(-0.836829\pi\)
−0.871464 + 0.490460i \(0.836829\pi\)
\(602\) −5.51466 −0.224761
\(603\) 4.64609 0.189203
\(604\) −15.9366 −0.648451
\(605\) 1.21121 0.0492426
\(606\) 2.72455 0.110677
\(607\) −43.9168 −1.78253 −0.891263 0.453487i \(-0.850180\pi\)
−0.891263 + 0.453487i \(0.850180\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 7.54480 0.305731
\(610\) −2.36121 −0.0956025
\(611\) 0 0
\(612\) −3.19269 −0.129057
\(613\) −1.42519 −0.0575629 −0.0287815 0.999586i \(-0.509163\pi\)
−0.0287815 + 0.999586i \(0.509163\pi\)
\(614\) 9.83446 0.396887
\(615\) −2.70469 −0.109064
\(616\) 3.68233 0.148365
\(617\) 12.4832 0.502555 0.251277 0.967915i \(-0.419149\pi\)
0.251277 + 0.967915i \(0.419149\pi\)
\(618\) −1.37329 −0.0552418
\(619\) 36.1148 1.45158 0.725789 0.687918i \(-0.241475\pi\)
0.725789 + 0.687918i \(0.241475\pi\)
\(620\) 0.215521 0.00865552
\(621\) 31.2150 1.25262
\(622\) 11.8726 0.476049
\(623\) 4.03492 0.161656
\(624\) 0 0
\(625\) 24.4131 0.976524
\(626\) −8.81402 −0.352279
\(627\) 8.47650 0.338519
\(628\) −10.5483 −0.420921
\(629\) −28.6956 −1.14417
\(630\) −0.211209 −0.00841476
\(631\) −2.85517 −0.113662 −0.0568312 0.998384i \(-0.518100\pi\)
−0.0568312 + 0.998384i \(0.518100\pi\)
\(632\) −15.6582 −0.622849
\(633\) −35.4268 −1.40809
\(634\) −12.7138 −0.504929
\(635\) −2.61596 −0.103811
\(636\) −6.97823 −0.276705
\(637\) 0 0
\(638\) 17.1153 0.677601
\(639\) 7.27950 0.287973
\(640\) 0.198062 0.00782910
\(641\) −19.0508 −0.752463 −0.376231 0.926526i \(-0.622780\pi\)
−0.376231 + 0.926526i \(0.622780\pi\)
\(642\) 26.8562 1.05993
\(643\) −50.0877 −1.97526 −0.987632 0.156787i \(-0.949886\pi\)
−0.987632 + 0.156787i \(0.949886\pi\)
\(644\) 7.52542 0.296543
\(645\) 2.51428 0.0989999
\(646\) 2.66487 0.104848
\(647\) 4.00836 0.157585 0.0787925 0.996891i \(-0.474894\pi\)
0.0787925 + 0.996891i \(0.474894\pi\)
\(648\) −11.1588 −0.438360
\(649\) −29.8189 −1.17050
\(650\) 0 0
\(651\) −1.98446 −0.0777771
\(652\) −5.59850 −0.219254
\(653\) 14.8116 0.579624 0.289812 0.957084i \(-0.406407\pi\)
0.289812 + 0.957084i \(0.406407\pi\)
\(654\) −30.9095 −1.20866
\(655\) 2.96854 0.115990
\(656\) −6.66487 −0.260220
\(657\) −6.18705 −0.241380
\(658\) −6.57242 −0.256219
\(659\) 13.0019 0.506483 0.253241 0.967403i \(-0.418503\pi\)
0.253241 + 0.967403i \(0.418503\pi\)
\(660\) −1.67887 −0.0653501
\(661\) 2.18705 0.0850662 0.0425331 0.999095i \(-0.486457\pi\)
0.0425331 + 0.999095i \(0.486457\pi\)
\(662\) −18.1715 −0.706256
\(663\) 0 0
\(664\) −13.1588 −0.510662
\(665\) 0.176292 0.00683631
\(666\) 12.9008 0.499897
\(667\) 34.9778 1.35434
\(668\) 13.8485 0.535813
\(669\) 40.7174 1.57422
\(670\) 0.768086 0.0296737
\(671\) 49.3202 1.90398
\(672\) −1.82371 −0.0703511
\(673\) −40.6762 −1.56795 −0.783977 0.620790i \(-0.786812\pi\)
−0.783977 + 0.620790i \(0.786812\pi\)
\(674\) 13.9323 0.536652
\(675\) 18.3153 0.704955
\(676\) 0 0
\(677\) −1.47757 −0.0567875 −0.0283937 0.999597i \(-0.509039\pi\)
−0.0283937 + 0.999597i \(0.509039\pi\)
\(678\) 40.0060 1.53642
\(679\) −7.28142 −0.279435
\(680\) −0.527811 −0.0202406
\(681\) −9.76941 −0.374365
\(682\) −4.50173 −0.172380
\(683\) −48.2210 −1.84513 −0.922563 0.385847i \(-0.873909\pi\)
−0.922563 + 0.385847i \(0.873909\pi\)
\(684\) −1.19806 −0.0458091
\(685\) 3.42758 0.130961
\(686\) 11.7560 0.448846
\(687\) −48.9549 −1.86775
\(688\) 6.19567 0.236208
\(689\) 0 0
\(690\) −3.43104 −0.130618
\(691\) 9.73019 0.370154 0.185077 0.982724i \(-0.440747\pi\)
0.185077 + 0.982724i \(0.440747\pi\)
\(692\) 15.4523 0.587410
\(693\) 4.41166 0.167585
\(694\) −8.02416 −0.304593
\(695\) 2.85517 0.108303
\(696\) −8.47650 −0.321301
\(697\) 17.7611 0.672748
\(698\) 10.7192 0.405727
\(699\) 24.1317 0.912744
\(700\) 4.41550 0.166890
\(701\) 33.2030 1.25406 0.627029 0.778996i \(-0.284271\pi\)
0.627029 + 0.778996i \(0.284271\pi\)
\(702\) 0 0
\(703\) −10.7681 −0.406126
\(704\) −4.13706 −0.155921
\(705\) 2.99654 0.112856
\(706\) 36.1172 1.35929
\(707\) −1.18359 −0.0445134
\(708\) 14.7681 0.555019
\(709\) 12.1715 0.457111 0.228555 0.973531i \(-0.426600\pi\)
0.228555 + 0.973531i \(0.426600\pi\)
\(710\) 1.20344 0.0451643
\(711\) −18.7595 −0.703535
\(712\) −4.53319 −0.169888
\(713\) −9.19998 −0.344542
\(714\) 4.85995 0.181879
\(715\) 0 0
\(716\) −12.7114 −0.475047
\(717\) −25.1982 −0.941043
\(718\) 32.3129 1.20591
\(719\) −8.43967 −0.314746 −0.157373 0.987539i \(-0.550303\pi\)
−0.157373 + 0.987539i \(0.550303\pi\)
\(720\) 0.237291 0.00884331
\(721\) 0.596580 0.0222178
\(722\) 1.00000 0.0372161
\(723\) 1.25714 0.0467536
\(724\) 13.4765 0.500850
\(725\) 20.5230 0.762206
\(726\) 12.5297 0.465022
\(727\) 16.8498 0.624924 0.312462 0.949930i \(-0.398846\pi\)
0.312462 + 0.949930i \(0.398846\pi\)
\(728\) 0 0
\(729\) 9.32496 0.345369
\(730\) −1.02284 −0.0378568
\(731\) −16.5107 −0.610670
\(732\) −24.4263 −0.902820
\(733\) 50.3086 1.85819 0.929095 0.369842i \(-0.120588\pi\)
0.929095 + 0.369842i \(0.120588\pi\)
\(734\) 5.24027 0.193422
\(735\) −2.51919 −0.0929216
\(736\) −8.45473 −0.311646
\(737\) −16.0435 −0.590971
\(738\) −7.98493 −0.293930
\(739\) −9.67324 −0.355836 −0.177918 0.984045i \(-0.556936\pi\)
−0.177918 + 0.984045i \(0.556936\pi\)
\(740\) 2.13275 0.0784015
\(741\) 0 0
\(742\) 3.03146 0.111288
\(743\) 41.6364 1.52749 0.763746 0.645517i \(-0.223358\pi\)
0.763746 + 0.645517i \(0.223358\pi\)
\(744\) 2.22952 0.0817382
\(745\) 1.06744 0.0391080
\(746\) −14.6213 −0.535325
\(747\) −15.7651 −0.576815
\(748\) 11.0248 0.403105
\(749\) −11.6668 −0.426295
\(750\) −4.04221 −0.147601
\(751\) −43.1680 −1.57522 −0.787612 0.616171i \(-0.788683\pi\)
−0.787612 + 0.616171i \(0.788683\pi\)
\(752\) 7.38404 0.269268
\(753\) −43.4034 −1.58171
\(754\) 0 0
\(755\) −3.15644 −0.114875
\(756\) 3.28621 0.119518
\(757\) 23.1535 0.841527 0.420763 0.907170i \(-0.361762\pi\)
0.420763 + 0.907170i \(0.361762\pi\)
\(758\) −14.8116 −0.537982
\(759\) 71.6665 2.60133
\(760\) −0.198062 −0.00718447
\(761\) −22.2258 −0.805685 −0.402842 0.915269i \(-0.631978\pi\)
−0.402842 + 0.915269i \(0.631978\pi\)
\(762\) −27.0616 −0.980338
\(763\) 13.4276 0.486111
\(764\) 18.5157 0.669876
\(765\) −0.632351 −0.0228627
\(766\) 12.2524 0.442696
\(767\) 0 0
\(768\) 2.04892 0.0739339
\(769\) −14.8552 −0.535691 −0.267846 0.963462i \(-0.586312\pi\)
−0.267846 + 0.963462i \(0.586312\pi\)
\(770\) 0.729331 0.0262833
\(771\) 31.0482 1.11817
\(772\) −3.55496 −0.127946
\(773\) 0.537500 0.0193325 0.00966626 0.999953i \(-0.496923\pi\)
0.00966626 + 0.999953i \(0.496923\pi\)
\(774\) 7.42280 0.266807
\(775\) −5.39804 −0.193903
\(776\) 8.18060 0.293667
\(777\) −19.6378 −0.704504
\(778\) 33.6969 1.20809
\(779\) 6.66487 0.238794
\(780\) 0 0
\(781\) −25.1371 −0.899475
\(782\) 22.5308 0.805700
\(783\) 15.2741 0.545853
\(784\) −6.20775 −0.221705
\(785\) −2.08921 −0.0745671
\(786\) 30.7090 1.09535
\(787\) −36.1124 −1.28727 −0.643634 0.765333i \(-0.722574\pi\)
−0.643634 + 0.765333i \(0.722574\pi\)
\(788\) −2.19567 −0.0782175
\(789\) −19.3769 −0.689835
\(790\) −3.10129 −0.110339
\(791\) −17.3793 −0.617935
\(792\) −4.95646 −0.176120
\(793\) 0 0
\(794\) −0.198062 −0.00702897
\(795\) −1.38212 −0.0490189
\(796\) −15.2228 −0.539558
\(797\) 20.4413 0.724069 0.362034 0.932165i \(-0.382082\pi\)
0.362034 + 0.932165i \(0.382082\pi\)
\(798\) 1.82371 0.0645586
\(799\) −19.6775 −0.696142
\(800\) −4.96077 −0.175390
\(801\) −5.43104 −0.191896
\(802\) 7.22952 0.255283
\(803\) 21.3647 0.753943
\(804\) 7.94571 0.280223
\(805\) 1.49050 0.0525333
\(806\) 0 0
\(807\) −27.4625 −0.966726
\(808\) 1.32975 0.0467804
\(809\) 35.5271 1.24907 0.624533 0.780999i \(-0.285289\pi\)
0.624533 + 0.780999i \(0.285289\pi\)
\(810\) −2.21014 −0.0776565
\(811\) 12.9250 0.453858 0.226929 0.973911i \(-0.427131\pi\)
0.226929 + 0.973911i \(0.427131\pi\)
\(812\) 3.68233 0.129225
\(813\) −50.6365 −1.77590
\(814\) −44.5483 −1.56142
\(815\) −1.10885 −0.0388414
\(816\) −5.46011 −0.191142
\(817\) −6.19567 −0.216759
\(818\) 10.5483 0.368811
\(819\) 0 0
\(820\) −1.32006 −0.0460985
\(821\) 41.4905 1.44803 0.724014 0.689785i \(-0.242295\pi\)
0.724014 + 0.689785i \(0.242295\pi\)
\(822\) 35.4577 1.23673
\(823\) 2.29185 0.0798888 0.0399444 0.999202i \(-0.487282\pi\)
0.0399444 + 0.999202i \(0.487282\pi\)
\(824\) −0.670251 −0.0233493
\(825\) 42.0500 1.46399
\(826\) −6.41550 −0.223224
\(827\) −28.0194 −0.974329 −0.487165 0.873310i \(-0.661969\pi\)
−0.487165 + 0.873310i \(0.661969\pi\)
\(828\) −10.1293 −0.352017
\(829\) −20.1497 −0.699829 −0.349915 0.936782i \(-0.613789\pi\)
−0.349915 + 0.936782i \(0.613789\pi\)
\(830\) −2.60627 −0.0904649
\(831\) 18.0048 0.624579
\(832\) 0 0
\(833\) 16.5429 0.573177
\(834\) 29.5362 1.02275
\(835\) 2.74286 0.0949205
\(836\) 4.13706 0.143083
\(837\) −4.01746 −0.138864
\(838\) −9.92154 −0.342734
\(839\) 46.7348 1.61347 0.806733 0.590917i \(-0.201234\pi\)
0.806733 + 0.590917i \(0.201234\pi\)
\(840\) −0.361208 −0.0124629
\(841\) −11.8847 −0.409817
\(842\) −25.4577 −0.877331
\(843\) −36.3002 −1.25025
\(844\) −17.2905 −0.595164
\(845\) 0 0
\(846\) 8.84654 0.304150
\(847\) −5.44312 −0.187028
\(848\) −3.40581 −0.116956
\(849\) −20.4176 −0.700731
\(850\) 13.2198 0.453437
\(851\) −91.0413 −3.12085
\(852\) 12.4494 0.426508
\(853\) 9.02608 0.309047 0.154524 0.987989i \(-0.450616\pi\)
0.154524 + 0.987989i \(0.450616\pi\)
\(854\) 10.6112 0.363107
\(855\) −0.237291 −0.00811518
\(856\) 13.1075 0.448006
\(857\) −14.0194 −0.478893 −0.239446 0.970910i \(-0.576966\pi\)
−0.239446 + 0.970910i \(0.576966\pi\)
\(858\) 0 0
\(859\) −32.1500 −1.09694 −0.548472 0.836169i \(-0.684790\pi\)
−0.548472 + 0.836169i \(0.684790\pi\)
\(860\) 1.22713 0.0418447
\(861\) 12.1548 0.414234
\(862\) −1.26145 −0.0429653
\(863\) 31.5357 1.07349 0.536744 0.843745i \(-0.319654\pi\)
0.536744 + 0.843745i \(0.319654\pi\)
\(864\) −3.69202 −0.125605
\(865\) 3.06052 0.104061
\(866\) −5.78017 −0.196418
\(867\) −20.2811 −0.688782
\(868\) −0.968541 −0.0328744
\(869\) 64.7788 2.19747
\(870\) −1.67887 −0.0569192
\(871\) 0 0
\(872\) −15.0858 −0.510868
\(873\) 9.80087 0.331709
\(874\) 8.45473 0.285986
\(875\) 1.75600 0.0593638
\(876\) −10.5810 −0.357500
\(877\) 39.3551 1.32893 0.664464 0.747321i \(-0.268660\pi\)
0.664464 + 0.747321i \(0.268660\pi\)
\(878\) 15.6668 0.528729
\(879\) 22.7633 0.767787
\(880\) −0.819396 −0.0276218
\(881\) −39.9734 −1.34674 −0.673370 0.739306i \(-0.735154\pi\)
−0.673370 + 0.739306i \(0.735154\pi\)
\(882\) −7.43727 −0.250426
\(883\) 30.1366 1.01418 0.507088 0.861894i \(-0.330722\pi\)
0.507088 + 0.861894i \(0.330722\pi\)
\(884\) 0 0
\(885\) 2.92500 0.0983228
\(886\) 4.89008 0.164286
\(887\) −34.0823 −1.14437 −0.572186 0.820124i \(-0.693904\pi\)
−0.572186 + 0.820124i \(0.693904\pi\)
\(888\) 22.0629 0.740383
\(889\) 11.7560 0.394284
\(890\) −0.897853 −0.0300961
\(891\) 46.1648 1.54658
\(892\) 19.8726 0.665385
\(893\) −7.38404 −0.247098
\(894\) 11.0425 0.369316
\(895\) −2.51765 −0.0841557
\(896\) −0.890084 −0.0297356
\(897\) 0 0
\(898\) 0.347207 0.0115865
\(899\) −4.50173 −0.150141
\(900\) −5.94331 −0.198110
\(901\) 9.07606 0.302368
\(902\) 27.5730 0.918081
\(903\) −11.2991 −0.376010
\(904\) 19.5254 0.649406
\(905\) 2.66919 0.0887268
\(906\) −32.6528 −1.08482
\(907\) 35.2403 1.17013 0.585067 0.810985i \(-0.301068\pi\)
0.585067 + 0.810985i \(0.301068\pi\)
\(908\) −4.76809 −0.158234
\(909\) 1.59312 0.0528405
\(910\) 0 0
\(911\) 23.8237 0.789315 0.394657 0.918828i \(-0.370863\pi\)
0.394657 + 0.918828i \(0.370863\pi\)
\(912\) −2.04892 −0.0678464
\(913\) 54.4389 1.80167
\(914\) 31.7754 1.05104
\(915\) −4.83792 −0.159937
\(916\) −23.8931 −0.789449
\(917\) −13.3405 −0.440542
\(918\) 9.83877 0.324728
\(919\) −19.1675 −0.632276 −0.316138 0.948713i \(-0.602386\pi\)
−0.316138 + 0.948713i \(0.602386\pi\)
\(920\) −1.67456 −0.0552087
\(921\) 20.1500 0.663965
\(922\) −28.2543 −0.930505
\(923\) 0 0
\(924\) 7.54480 0.248206
\(925\) −53.4180 −1.75637
\(926\) −2.42029 −0.0795356
\(927\) −0.803003 −0.0263741
\(928\) −4.13706 −0.135806
\(929\) 59.9120 1.96565 0.982824 0.184545i \(-0.0590812\pi\)
0.982824 + 0.184545i \(0.0590812\pi\)
\(930\) 0.441584 0.0144801
\(931\) 6.20775 0.203451
\(932\) 11.7778 0.385794
\(933\) 24.3260 0.796398
\(934\) −10.5724 −0.345940
\(935\) 2.18359 0.0714110
\(936\) 0 0
\(937\) −45.9057 −1.49968 −0.749838 0.661622i \(-0.769868\pi\)
−0.749838 + 0.661622i \(0.769868\pi\)
\(938\) −3.45175 −0.112704
\(939\) −18.0592 −0.589340
\(940\) 1.46250 0.0477015
\(941\) −16.8009 −0.547693 −0.273846 0.961773i \(-0.588296\pi\)
−0.273846 + 0.961773i \(0.588296\pi\)
\(942\) −21.6125 −0.704173
\(943\) 56.3497 1.83500
\(944\) 7.20775 0.234592
\(945\) 0.650874 0.0211729
\(946\) −25.6319 −0.833364
\(947\) 8.02774 0.260866 0.130433 0.991457i \(-0.458363\pi\)
0.130433 + 0.991457i \(0.458363\pi\)
\(948\) −32.0823 −1.04198
\(949\) 0 0
\(950\) 4.96077 0.160949
\(951\) −26.0495 −0.844713
\(952\) 2.37196 0.0768757
\(953\) 23.1353 0.749425 0.374712 0.927141i \(-0.377741\pi\)
0.374712 + 0.927141i \(0.377741\pi\)
\(954\) −4.08038 −0.132107
\(955\) 3.66727 0.118670
\(956\) −12.2983 −0.397755
\(957\) 35.0678 1.13358
\(958\) −29.0616 −0.938937
\(959\) −15.4034 −0.497402
\(960\) 0.405813 0.0130976
\(961\) −29.8159 −0.961804
\(962\) 0 0
\(963\) 15.7036 0.506042
\(964\) 0.613564 0.0197616
\(965\) −0.704103 −0.0226659
\(966\) 15.4190 0.496097
\(967\) 3.93495 0.126540 0.0632698 0.997996i \(-0.479847\pi\)
0.0632698 + 0.997996i \(0.479847\pi\)
\(968\) 6.11529 0.196553
\(969\) 5.46011 0.175404
\(970\) 1.62027 0.0520237
\(971\) −9.46011 −0.303589 −0.151795 0.988412i \(-0.548505\pi\)
−0.151795 + 0.988412i \(0.548505\pi\)
\(972\) −11.7875 −0.378083
\(973\) −12.8310 −0.411343
\(974\) 11.9366 0.382474
\(975\) 0 0
\(976\) −11.9215 −0.381599
\(977\) −33.8340 −1.08245 −0.541223 0.840879i \(-0.682038\pi\)
−0.541223 + 0.840879i \(0.682038\pi\)
\(978\) −11.4709 −0.366798
\(979\) 18.7541 0.599383
\(980\) −1.22952 −0.0392756
\(981\) −18.0737 −0.577048
\(982\) −29.3163 −0.935522
\(983\) 13.7614 0.438920 0.219460 0.975622i \(-0.429570\pi\)
0.219460 + 0.975622i \(0.429570\pi\)
\(984\) −13.6558 −0.435330
\(985\) −0.434879 −0.0138564
\(986\) 11.0248 0.351100
\(987\) −13.4663 −0.428638
\(988\) 0 0
\(989\) −52.3827 −1.66567
\(990\) −0.981688 −0.0312001
\(991\) 24.7525 0.786291 0.393145 0.919476i \(-0.371387\pi\)
0.393145 + 0.919476i \(0.371387\pi\)
\(992\) 1.08815 0.0345487
\(993\) −37.2319 −1.18152
\(994\) −5.40821 −0.171538
\(995\) −3.01507 −0.0955840
\(996\) −26.9614 −0.854303
\(997\) 58.4951 1.85256 0.926280 0.376836i \(-0.122988\pi\)
0.926280 + 0.376836i \(0.122988\pi\)
\(998\) 1.30499 0.0413089
\(999\) −39.7560 −1.25782
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 6422.2.a.t.1.3 3
13.5 odd 4 494.2.d.b.77.3 6
13.8 odd 4 494.2.d.b.77.6 yes 6
13.12 even 2 6422.2.a.k.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
494.2.d.b.77.3 6 13.5 odd 4
494.2.d.b.77.6 yes 6 13.8 odd 4
6422.2.a.k.1.3 3 13.12 even 2
6422.2.a.t.1.3 3 1.1 even 1 trivial