Properties

Label 2432.2.c.j.1217.7
Level $2432$
Weight $2$
Character 2432.1217
Analytic conductor $19.420$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2432,2,Mod(1217,2432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2432, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2432.1217");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2432 = 2^{7} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2432.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(19.4196177716\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 170x^{16} + 6593x^{12} + 64168x^{8} + 95760x^{4} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{19} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1217.7
Root \(-0.814454 - 0.814454i\) of defining polynomial
Character \(\chi\) \(=\) 2432.1217
Dual form 2432.2.c.j.1217.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.15181i q^{3} -1.07587i q^{5} +0.238565 q^{7} +1.67333 q^{9} +O(q^{10})\) \(q-1.15181i q^{3} -1.07587i q^{5} +0.238565 q^{7} +1.67333 q^{9} +2.84249i q^{11} -1.31444i q^{13} -1.23921 q^{15} -4.47279 q^{17} +1.00000i q^{19} -0.274782i q^{21} +1.94338 q^{23} +3.84249 q^{25} -5.38280i q^{27} -8.83944i q^{29} +6.64877 q^{31} +3.27402 q^{33} -0.256666i q^{35} -4.06763i q^{37} -1.51399 q^{39} +2.78211 q^{41} +1.21363i q^{43} -1.80029i q^{45} +10.2695 q^{47} -6.94309 q^{49} +5.15181i q^{51} -8.36231i q^{53} +3.05817 q^{55} +1.15181 q^{57} -9.25916i q^{59} +4.66035i q^{61} +0.399197 q^{63} -1.41417 q^{65} +4.31849i q^{67} -2.23841i q^{69} +2.78069 q^{71} +0.134460 q^{73} -4.42583i q^{75} +0.678119i q^{77} -12.7841 q^{79} -1.17998 q^{81} +13.4028i q^{83} +4.81216i q^{85} -10.1814 q^{87} -9.20650 q^{89} -0.313579i q^{91} -7.65814i q^{93} +1.07587 q^{95} -0.247499 q^{97} +4.75643i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 28 q^{9} + 8 q^{17} - 20 q^{25} - 16 q^{33} + 24 q^{41} + 52 q^{49} - 8 q^{57} - 48 q^{65} - 24 q^{73} + 68 q^{81} + 40 q^{89} - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2432\mathbb{Z}\right)^\times\).

\(n\) \(1407\) \(1921\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.15181i − 0.664999i −0.943103 0.332500i \(-0.892108\pi\)
0.943103 0.332500i \(-0.107892\pi\)
\(4\) 0 0
\(5\) − 1.07587i − 0.481146i −0.970631 0.240573i \(-0.922665\pi\)
0.970631 0.240573i \(-0.0773353\pi\)
\(6\) 0 0
\(7\) 0.238565 0.0901690 0.0450845 0.998983i \(-0.485644\pi\)
0.0450845 + 0.998983i \(0.485644\pi\)
\(8\) 0 0
\(9\) 1.67333 0.557776
\(10\) 0 0
\(11\) 2.84249i 0.857044i 0.903531 + 0.428522i \(0.140966\pi\)
−0.903531 + 0.428522i \(0.859034\pi\)
\(12\) 0 0
\(13\) − 1.31444i − 0.364560i −0.983247 0.182280i \(-0.941652\pi\)
0.983247 0.182280i \(-0.0583477\pi\)
\(14\) 0 0
\(15\) −1.23921 −0.319962
\(16\) 0 0
\(17\) −4.47279 −1.08481 −0.542405 0.840117i \(-0.682486\pi\)
−0.542405 + 0.840117i \(0.682486\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i
\(20\) 0 0
\(21\) − 0.274782i − 0.0599623i
\(22\) 0 0
\(23\) 1.94338 0.405222 0.202611 0.979259i \(-0.435057\pi\)
0.202611 + 0.979259i \(0.435057\pi\)
\(24\) 0 0
\(25\) 3.84249 0.768499
\(26\) 0 0
\(27\) − 5.38280i − 1.03592i
\(28\) 0 0
\(29\) − 8.83944i − 1.64144i −0.571329 0.820721i \(-0.693572\pi\)
0.571329 0.820721i \(-0.306428\pi\)
\(30\) 0 0
\(31\) 6.64877 1.19415 0.597077 0.802184i \(-0.296329\pi\)
0.597077 + 0.802184i \(0.296329\pi\)
\(32\) 0 0
\(33\) 3.27402 0.569933
\(34\) 0 0
\(35\) − 0.256666i − 0.0433845i
\(36\) 0 0
\(37\) − 4.06763i − 0.668715i −0.942446 0.334357i \(-0.891481\pi\)
0.942446 0.334357i \(-0.108519\pi\)
\(38\) 0 0
\(39\) −1.51399 −0.242432
\(40\) 0 0
\(41\) 2.78211 0.434492 0.217246 0.976117i \(-0.430293\pi\)
0.217246 + 0.976117i \(0.430293\pi\)
\(42\) 0 0
\(43\) 1.21363i 0.185077i 0.995709 + 0.0925386i \(0.0294982\pi\)
−0.995709 + 0.0925386i \(0.970502\pi\)
\(44\) 0 0
\(45\) − 1.80029i − 0.268372i
\(46\) 0 0
\(47\) 10.2695 1.49796 0.748978 0.662595i \(-0.230545\pi\)
0.748978 + 0.662595i \(0.230545\pi\)
\(48\) 0 0
\(49\) −6.94309 −0.991870
\(50\) 0 0
\(51\) 5.15181i 0.721398i
\(52\) 0 0
\(53\) − 8.36231i − 1.14865i −0.818627 0.574326i \(-0.805264\pi\)
0.818627 0.574326i \(-0.194736\pi\)
\(54\) 0 0
\(55\) 3.05817 0.412363
\(56\) 0 0
\(57\) 1.15181 0.152561
\(58\) 0 0
\(59\) − 9.25916i − 1.20544i −0.797953 0.602720i \(-0.794084\pi\)
0.797953 0.602720i \(-0.205916\pi\)
\(60\) 0 0
\(61\) 4.66035i 0.596697i 0.954457 + 0.298349i \(0.0964358\pi\)
−0.954457 + 0.298349i \(0.903564\pi\)
\(62\) 0 0
\(63\) 0.399197 0.0502941
\(64\) 0 0
\(65\) −1.41417 −0.175407
\(66\) 0 0
\(67\) 4.31849i 0.527587i 0.964579 + 0.263794i \(0.0849738\pi\)
−0.964579 + 0.263794i \(0.915026\pi\)
\(68\) 0 0
\(69\) − 2.23841i − 0.269472i
\(70\) 0 0
\(71\) 2.78069 0.330007 0.165003 0.986293i \(-0.447236\pi\)
0.165003 + 0.986293i \(0.447236\pi\)
\(72\) 0 0
\(73\) 0.134460 0.0157373 0.00786866 0.999969i \(-0.497495\pi\)
0.00786866 + 0.999969i \(0.497495\pi\)
\(74\) 0 0
\(75\) − 4.42583i − 0.511051i
\(76\) 0 0
\(77\) 0.678119i 0.0772788i
\(78\) 0 0
\(79\) −12.7841 −1.43832 −0.719162 0.694843i \(-0.755474\pi\)
−0.719162 + 0.694843i \(0.755474\pi\)
\(80\) 0 0
\(81\) −1.17998 −0.131109
\(82\) 0 0
\(83\) 13.4028i 1.47115i 0.677445 + 0.735573i \(0.263087\pi\)
−0.677445 + 0.735573i \(0.736913\pi\)
\(84\) 0 0
\(85\) 4.81216i 0.521952i
\(86\) 0 0
\(87\) −10.1814 −1.09156
\(88\) 0 0
\(89\) −9.20650 −0.975887 −0.487944 0.872875i \(-0.662253\pi\)
−0.487944 + 0.872875i \(0.662253\pi\)
\(90\) 0 0
\(91\) − 0.313579i − 0.0328720i
\(92\) 0 0
\(93\) − 7.65814i − 0.794112i
\(94\) 0 0
\(95\) 1.07587 0.110382
\(96\) 0 0
\(97\) −0.247499 −0.0251297 −0.0125648 0.999921i \(-0.504000\pi\)
−0.0125648 + 0.999921i \(0.504000\pi\)
\(98\) 0 0
\(99\) 4.75643i 0.478039i
\(100\) 0 0
\(101\) − 16.8026i − 1.67193i −0.548786 0.835963i \(-0.684910\pi\)
0.548786 0.835963i \(-0.315090\pi\)
\(102\) 0 0
\(103\) 5.40957 0.533020 0.266510 0.963832i \(-0.414129\pi\)
0.266510 + 0.963832i \(0.414129\pi\)
\(104\) 0 0
\(105\) −0.295631 −0.0288506
\(106\) 0 0
\(107\) − 6.39113i − 0.617853i −0.951086 0.308927i \(-0.900030\pi\)
0.951086 0.308927i \(-0.0999698\pi\)
\(108\) 0 0
\(109\) − 14.8779i − 1.42505i −0.701648 0.712524i \(-0.747552\pi\)
0.701648 0.712524i \(-0.252448\pi\)
\(110\) 0 0
\(111\) −4.68515 −0.444695
\(112\) 0 0
\(113\) 8.59073 0.808148 0.404074 0.914726i \(-0.367594\pi\)
0.404074 + 0.914726i \(0.367594\pi\)
\(114\) 0 0
\(115\) − 2.09083i − 0.194971i
\(116\) 0 0
\(117\) − 2.19949i − 0.203343i
\(118\) 0 0
\(119\) −1.06705 −0.0978163
\(120\) 0 0
\(121\) 2.92023 0.265476
\(122\) 0 0
\(123\) − 3.20446i − 0.288937i
\(124\) 0 0
\(125\) − 9.51342i − 0.850906i
\(126\) 0 0
\(127\) 1.15983 0.102918 0.0514592 0.998675i \(-0.483613\pi\)
0.0514592 + 0.998675i \(0.483613\pi\)
\(128\) 0 0
\(129\) 1.39788 0.123076
\(130\) 0 0
\(131\) − 3.85559i − 0.336864i −0.985713 0.168432i \(-0.946130\pi\)
0.985713 0.168432i \(-0.0538704\pi\)
\(132\) 0 0
\(133\) 0.238565i 0.0206862i
\(134\) 0 0
\(135\) −5.79122 −0.498428
\(136\) 0 0
\(137\) 1.42976 0.122152 0.0610761 0.998133i \(-0.480547\pi\)
0.0610761 + 0.998133i \(0.480547\pi\)
\(138\) 0 0
\(139\) − 3.58190i − 0.303813i −0.988395 0.151907i \(-0.951459\pi\)
0.988395 0.151907i \(-0.0485413\pi\)
\(140\) 0 0
\(141\) − 11.8285i − 0.996139i
\(142\) 0 0
\(143\) 3.73629 0.312444
\(144\) 0 0
\(145\) −9.51013 −0.789773
\(146\) 0 0
\(147\) 7.99713i 0.659592i
\(148\) 0 0
\(149\) − 16.2763i − 1.33341i −0.745322 0.666705i \(-0.767704\pi\)
0.745322 0.666705i \(-0.232296\pi\)
\(150\) 0 0
\(151\) −14.3242 −1.16569 −0.582845 0.812583i \(-0.698061\pi\)
−0.582845 + 0.812583i \(0.698061\pi\)
\(152\) 0 0
\(153\) −7.48445 −0.605082
\(154\) 0 0
\(155\) − 7.15325i − 0.574563i
\(156\) 0 0
\(157\) 3.73495i 0.298081i 0.988831 + 0.149041i \(0.0476185\pi\)
−0.988831 + 0.149041i \(0.952381\pi\)
\(158\) 0 0
\(159\) −9.63181 −0.763852
\(160\) 0 0
\(161\) 0.463621 0.0365385
\(162\) 0 0
\(163\) − 3.98861i − 0.312412i −0.987724 0.156206i \(-0.950074\pi\)
0.987724 0.156206i \(-0.0499264\pi\)
\(164\) 0 0
\(165\) − 3.52243i − 0.274221i
\(166\) 0 0
\(167\) 8.17158 0.632336 0.316168 0.948703i \(-0.397604\pi\)
0.316168 + 0.948703i \(0.397604\pi\)
\(168\) 0 0
\(169\) 11.2722 0.867096
\(170\) 0 0
\(171\) 1.67333i 0.127963i
\(172\) 0 0
\(173\) 0.351356i 0.0267131i 0.999911 + 0.0133565i \(0.00425164\pi\)
−0.999911 + 0.0133565i \(0.995748\pi\)
\(174\) 0 0
\(175\) 0.916684 0.0692948
\(176\) 0 0
\(177\) −10.6648 −0.801616
\(178\) 0 0
\(179\) 15.2740i 1.14163i 0.821077 + 0.570817i \(0.193373\pi\)
−0.821077 + 0.570817i \(0.806627\pi\)
\(180\) 0 0
\(181\) − 8.94630i − 0.664974i −0.943108 0.332487i \(-0.892112\pi\)
0.943108 0.332487i \(-0.107888\pi\)
\(182\) 0 0
\(183\) 5.36785 0.396803
\(184\) 0 0
\(185\) −4.37626 −0.321749
\(186\) 0 0
\(187\) − 12.7139i − 0.929730i
\(188\) 0 0
\(189\) − 1.28415i − 0.0934079i
\(190\) 0 0
\(191\) −18.3946 −1.33098 −0.665492 0.746405i \(-0.731778\pi\)
−0.665492 + 0.746405i \(0.731778\pi\)
\(192\) 0 0
\(193\) −5.09902 −0.367035 −0.183518 0.983016i \(-0.558748\pi\)
−0.183518 + 0.983016i \(0.558748\pi\)
\(194\) 0 0
\(195\) 1.62886i 0.116645i
\(196\) 0 0
\(197\) 20.2916i 1.44572i 0.690995 + 0.722860i \(0.257173\pi\)
−0.690995 + 0.722860i \(0.742827\pi\)
\(198\) 0 0
\(199\) 0.310999 0.0220461 0.0110231 0.999939i \(-0.496491\pi\)
0.0110231 + 0.999939i \(0.496491\pi\)
\(200\) 0 0
\(201\) 4.97408 0.350845
\(202\) 0 0
\(203\) − 2.10878i − 0.148007i
\(204\) 0 0
\(205\) − 2.99320i − 0.209054i
\(206\) 0 0
\(207\) 3.25191 0.226023
\(208\) 0 0
\(209\) −2.84249 −0.196619
\(210\) 0 0
\(211\) 14.3284i 0.986406i 0.869914 + 0.493203i \(0.164174\pi\)
−0.869914 + 0.493203i \(0.835826\pi\)
\(212\) 0 0
\(213\) − 3.20283i − 0.219454i
\(214\) 0 0
\(215\) 1.30572 0.0890491
\(216\) 0 0
\(217\) 1.58616 0.107676
\(218\) 0 0
\(219\) − 0.154872i − 0.0104653i
\(220\) 0 0
\(221\) 5.87921i 0.395479i
\(222\) 0 0
\(223\) 12.8621 0.861312 0.430656 0.902516i \(-0.358282\pi\)
0.430656 + 0.902516i \(0.358282\pi\)
\(224\) 0 0
\(225\) 6.42976 0.428650
\(226\) 0 0
\(227\) 18.4491i 1.22451i 0.790659 + 0.612256i \(0.209738\pi\)
−0.790659 + 0.612256i \(0.790262\pi\)
\(228\) 0 0
\(229\) 4.73279i 0.312751i 0.987698 + 0.156376i \(0.0499811\pi\)
−0.987698 + 0.156376i \(0.950019\pi\)
\(230\) 0 0
\(231\) 0.781066 0.0513903
\(232\) 0 0
\(233\) 4.27809 0.280267 0.140133 0.990133i \(-0.455247\pi\)
0.140133 + 0.990133i \(0.455247\pi\)
\(234\) 0 0
\(235\) − 11.0487i − 0.720735i
\(236\) 0 0
\(237\) 14.7249i 0.956483i
\(238\) 0 0
\(239\) −20.3905 −1.31895 −0.659474 0.751727i \(-0.729221\pi\)
−0.659474 + 0.751727i \(0.729221\pi\)
\(240\) 0 0
\(241\) 25.5877 1.64825 0.824123 0.566410i \(-0.191668\pi\)
0.824123 + 0.566410i \(0.191668\pi\)
\(242\) 0 0
\(243\) − 14.7893i − 0.948732i
\(244\) 0 0
\(245\) 7.46989i 0.477234i
\(246\) 0 0
\(247\) 1.31444 0.0836358
\(248\) 0 0
\(249\) 15.4375 0.978311
\(250\) 0 0
\(251\) − 28.9847i − 1.82950i −0.404024 0.914749i \(-0.632389\pi\)
0.404024 0.914749i \(-0.367611\pi\)
\(252\) 0 0
\(253\) 5.52404i 0.347293i
\(254\) 0 0
\(255\) 5.54270 0.347098
\(256\) 0 0
\(257\) 5.10203 0.318256 0.159128 0.987258i \(-0.449132\pi\)
0.159128 + 0.987258i \(0.449132\pi\)
\(258\) 0 0
\(259\) − 0.970394i − 0.0602974i
\(260\) 0 0
\(261\) − 14.7913i − 0.915558i
\(262\) 0 0
\(263\) 11.6953 0.721160 0.360580 0.932728i \(-0.382579\pi\)
0.360580 + 0.932728i \(0.382579\pi\)
\(264\) 0 0
\(265\) −8.99680 −0.552669
\(266\) 0 0
\(267\) 10.6042i 0.648964i
\(268\) 0 0
\(269\) 0.163433i 0.00996466i 0.999988 + 0.00498233i \(0.00158593\pi\)
−0.999988 + 0.00498233i \(0.998414\pi\)
\(270\) 0 0
\(271\) −1.01216 −0.0614846 −0.0307423 0.999527i \(-0.509787\pi\)
−0.0307423 + 0.999527i \(0.509787\pi\)
\(272\) 0 0
\(273\) −0.361184 −0.0218599
\(274\) 0 0
\(275\) 10.9223i 0.658637i
\(276\) 0 0
\(277\) 3.14825i 0.189160i 0.995517 + 0.0945800i \(0.0301508\pi\)
−0.995517 + 0.0945800i \(0.969849\pi\)
\(278\) 0 0
\(279\) 11.1256 0.666071
\(280\) 0 0
\(281\) −24.6734 −1.47189 −0.735946 0.677040i \(-0.763262\pi\)
−0.735946 + 0.677040i \(0.763262\pi\)
\(282\) 0 0
\(283\) 13.8095i 0.820889i 0.911886 + 0.410445i \(0.134626\pi\)
−0.911886 + 0.410445i \(0.865374\pi\)
\(284\) 0 0
\(285\) − 1.23921i − 0.0734042i
\(286\) 0 0
\(287\) 0.663713 0.0391777
\(288\) 0 0
\(289\) 3.00584 0.176814
\(290\) 0 0
\(291\) 0.285072i 0.0167112i
\(292\) 0 0
\(293\) 8.24682i 0.481784i 0.970552 + 0.240892i \(0.0774400\pi\)
−0.970552 + 0.240892i \(0.922560\pi\)
\(294\) 0 0
\(295\) −9.96169 −0.579992
\(296\) 0 0
\(297\) 15.3006 0.887829
\(298\) 0 0
\(299\) − 2.55445i − 0.147728i
\(300\) 0 0
\(301\) 0.289530i 0.0166882i
\(302\) 0 0
\(303\) −19.3535 −1.11183
\(304\) 0 0
\(305\) 5.01396 0.287098
\(306\) 0 0
\(307\) 17.4540i 0.996153i 0.867133 + 0.498076i \(0.165960\pi\)
−0.867133 + 0.498076i \(0.834040\pi\)
\(308\) 0 0
\(309\) − 6.23080i − 0.354458i
\(310\) 0 0
\(311\) 10.5256 0.596851 0.298425 0.954433i \(-0.403539\pi\)
0.298425 + 0.954433i \(0.403539\pi\)
\(312\) 0 0
\(313\) −14.3498 −0.811098 −0.405549 0.914073i \(-0.632920\pi\)
−0.405549 + 0.914073i \(0.632920\pi\)
\(314\) 0 0
\(315\) − 0.429486i − 0.0241988i
\(316\) 0 0
\(317\) − 16.3470i − 0.918138i −0.888401 0.459069i \(-0.848183\pi\)
0.888401 0.459069i \(-0.151817\pi\)
\(318\) 0 0
\(319\) 25.1260 1.40679
\(320\) 0 0
\(321\) −7.36138 −0.410872
\(322\) 0 0
\(323\) − 4.47279i − 0.248873i
\(324\) 0 0
\(325\) − 5.05073i − 0.280164i
\(326\) 0 0
\(327\) −17.1366 −0.947656
\(328\) 0 0
\(329\) 2.44993 0.135069
\(330\) 0 0
\(331\) 15.0118i 0.825123i 0.910930 + 0.412562i \(0.135366\pi\)
−0.910930 + 0.412562i \(0.864634\pi\)
\(332\) 0 0
\(333\) − 6.80649i − 0.372993i
\(334\) 0 0
\(335\) 4.64615 0.253846
\(336\) 0 0
\(337\) 0.462187 0.0251769 0.0125885 0.999921i \(-0.495993\pi\)
0.0125885 + 0.999921i \(0.495993\pi\)
\(338\) 0 0
\(339\) − 9.89491i − 0.537418i
\(340\) 0 0
\(341\) 18.8991i 1.02344i
\(342\) 0 0
\(343\) −3.32633 −0.179605
\(344\) 0 0
\(345\) −2.40824 −0.129655
\(346\) 0 0
\(347\) − 17.2669i − 0.926935i −0.886114 0.463468i \(-0.846605\pi\)
0.886114 0.463468i \(-0.153395\pi\)
\(348\) 0 0
\(349\) 26.1466i 1.39960i 0.714341 + 0.699798i \(0.246727\pi\)
−0.714341 + 0.699798i \(0.753273\pi\)
\(350\) 0 0
\(351\) −7.07536 −0.377655
\(352\) 0 0
\(353\) 12.3930 0.659614 0.329807 0.944048i \(-0.393016\pi\)
0.329807 + 0.944048i \(0.393016\pi\)
\(354\) 0 0
\(355\) − 2.99167i − 0.158781i
\(356\) 0 0
\(357\) 1.22904i 0.0650478i
\(358\) 0 0
\(359\) −14.1664 −0.747673 −0.373837 0.927495i \(-0.621958\pi\)
−0.373837 + 0.927495i \(0.621958\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) − 3.36356i − 0.176541i
\(364\) 0 0
\(365\) − 0.144662i − 0.00757195i
\(366\) 0 0
\(367\) −3.00328 −0.156770 −0.0783850 0.996923i \(-0.524976\pi\)
−0.0783850 + 0.996923i \(0.524976\pi\)
\(368\) 0 0
\(369\) 4.65538 0.242349
\(370\) 0 0
\(371\) − 1.99495i − 0.103573i
\(372\) 0 0
\(373\) 38.0432i 1.96980i 0.173119 + 0.984901i \(0.444615\pi\)
−0.173119 + 0.984901i \(0.555385\pi\)
\(374\) 0 0
\(375\) −10.9577 −0.565852
\(376\) 0 0
\(377\) −11.6189 −0.598404
\(378\) 0 0
\(379\) − 28.6291i − 1.47058i −0.677753 0.735290i \(-0.737046\pi\)
0.677753 0.735290i \(-0.262954\pi\)
\(380\) 0 0
\(381\) − 1.33591i − 0.0684407i
\(382\) 0 0
\(383\) 24.1407 1.23353 0.616767 0.787146i \(-0.288442\pi\)
0.616767 + 0.787146i \(0.288442\pi\)
\(384\) 0 0
\(385\) 0.729571 0.0371824
\(386\) 0 0
\(387\) 2.03081i 0.103232i
\(388\) 0 0
\(389\) − 16.2953i − 0.826206i −0.910684 0.413103i \(-0.864445\pi\)
0.910684 0.413103i \(-0.135555\pi\)
\(390\) 0 0
\(391\) −8.69231 −0.439589
\(392\) 0 0
\(393\) −4.44091 −0.224014
\(394\) 0 0
\(395\) 13.7541i 0.692043i
\(396\) 0 0
\(397\) 36.8139i 1.84764i 0.382829 + 0.923819i \(0.374950\pi\)
−0.382829 + 0.923819i \(0.625050\pi\)
\(398\) 0 0
\(399\) 0.274782 0.0137563
\(400\) 0 0
\(401\) −23.6899 −1.18302 −0.591508 0.806299i \(-0.701467\pi\)
−0.591508 + 0.806299i \(0.701467\pi\)
\(402\) 0 0
\(403\) − 8.73941i − 0.435341i
\(404\) 0 0
\(405\) 1.26951i 0.0630827i
\(406\) 0 0
\(407\) 11.5622 0.573118
\(408\) 0 0
\(409\) 0.102363 0.00506154 0.00253077 0.999997i \(-0.499194\pi\)
0.00253077 + 0.999997i \(0.499194\pi\)
\(410\) 0 0
\(411\) − 1.64681i − 0.0812311i
\(412\) 0 0
\(413\) − 2.20891i − 0.108693i
\(414\) 0 0
\(415\) 14.4197 0.707836
\(416\) 0 0
\(417\) −4.12568 −0.202035
\(418\) 0 0
\(419\) 25.7326i 1.25712i 0.777761 + 0.628560i \(0.216355\pi\)
−0.777761 + 0.628560i \(0.783645\pi\)
\(420\) 0 0
\(421\) 18.0014i 0.877334i 0.898650 + 0.438667i \(0.144549\pi\)
−0.898650 + 0.438667i \(0.855451\pi\)
\(422\) 0 0
\(423\) 17.1842 0.835524
\(424\) 0 0
\(425\) −17.1867 −0.833675
\(426\) 0 0
\(427\) 1.11180i 0.0538036i
\(428\) 0 0
\(429\) − 4.30350i − 0.207775i
\(430\) 0 0
\(431\) 27.6392 1.33133 0.665667 0.746249i \(-0.268147\pi\)
0.665667 + 0.746249i \(0.268147\pi\)
\(432\) 0 0
\(433\) 38.5219 1.85124 0.925621 0.378451i \(-0.123543\pi\)
0.925621 + 0.378451i \(0.123543\pi\)
\(434\) 0 0
\(435\) 10.9539i 0.525198i
\(436\) 0 0
\(437\) 1.94338i 0.0929643i
\(438\) 0 0
\(439\) 23.1302 1.10394 0.551971 0.833863i \(-0.313876\pi\)
0.551971 + 0.833863i \(0.313876\pi\)
\(440\) 0 0
\(441\) −11.6181 −0.553241
\(442\) 0 0
\(443\) − 19.2436i − 0.914290i −0.889392 0.457145i \(-0.848872\pi\)
0.889392 0.457145i \(-0.151128\pi\)
\(444\) 0 0
\(445\) 9.90505i 0.469544i
\(446\) 0 0
\(447\) −18.7473 −0.886716
\(448\) 0 0
\(449\) 11.0578 0.521851 0.260926 0.965359i \(-0.415972\pi\)
0.260926 + 0.965359i \(0.415972\pi\)
\(450\) 0 0
\(451\) 7.90812i 0.372379i
\(452\) 0 0
\(453\) 16.4988i 0.775183i
\(454\) 0 0
\(455\) −0.337372 −0.0158162
\(456\) 0 0
\(457\) 4.08310 0.190999 0.0954996 0.995429i \(-0.469555\pi\)
0.0954996 + 0.995429i \(0.469555\pi\)
\(458\) 0 0
\(459\) 24.0761i 1.12378i
\(460\) 0 0
\(461\) 7.59017i 0.353509i 0.984255 + 0.176755i \(0.0565599\pi\)
−0.984255 + 0.176755i \(0.943440\pi\)
\(462\) 0 0
\(463\) 37.2637 1.73179 0.865894 0.500227i \(-0.166750\pi\)
0.865894 + 0.500227i \(0.166750\pi\)
\(464\) 0 0
\(465\) −8.23920 −0.382084
\(466\) 0 0
\(467\) − 8.92480i − 0.412991i −0.978448 0.206495i \(-0.933794\pi\)
0.978448 0.206495i \(-0.0662058\pi\)
\(468\) 0 0
\(469\) 1.03024i 0.0475720i
\(470\) 0 0
\(471\) 4.30196 0.198224
\(472\) 0 0
\(473\) −3.44974 −0.158619
\(474\) 0 0
\(475\) 3.84249i 0.176306i
\(476\) 0 0
\(477\) − 13.9929i − 0.640690i
\(478\) 0 0
\(479\) −19.6241 −0.896650 −0.448325 0.893871i \(-0.647979\pi\)
−0.448325 + 0.893871i \(0.647979\pi\)
\(480\) 0 0
\(481\) −5.34666 −0.243787
\(482\) 0 0
\(483\) − 0.534005i − 0.0242981i
\(484\) 0 0
\(485\) 0.266277i 0.0120910i
\(486\) 0 0
\(487\) 13.5448 0.613775 0.306887 0.951746i \(-0.400712\pi\)
0.306887 + 0.951746i \(0.400712\pi\)
\(488\) 0 0
\(489\) −4.59413 −0.207754
\(490\) 0 0
\(491\) 9.58255i 0.432454i 0.976343 + 0.216227i \(0.0693752\pi\)
−0.976343 + 0.216227i \(0.930625\pi\)
\(492\) 0 0
\(493\) 39.5369i 1.78065i
\(494\) 0 0
\(495\) 5.11732 0.230006
\(496\) 0 0
\(497\) 0.663374 0.0297564
\(498\) 0 0
\(499\) 7.24834i 0.324480i 0.986751 + 0.162240i \(0.0518719\pi\)
−0.986751 + 0.162240i \(0.948128\pi\)
\(500\) 0 0
\(501\) − 9.41213i − 0.420503i
\(502\) 0 0
\(503\) −2.19200 −0.0977367 −0.0488683 0.998805i \(-0.515561\pi\)
−0.0488683 + 0.998805i \(0.515561\pi\)
\(504\) 0 0
\(505\) −18.0775 −0.804440
\(506\) 0 0
\(507\) − 12.9835i − 0.576618i
\(508\) 0 0
\(509\) 13.5765i 0.601766i 0.953661 + 0.300883i \(0.0972814\pi\)
−0.953661 + 0.300883i \(0.902719\pi\)
\(510\) 0 0
\(511\) 0.0320774 0.00141902
\(512\) 0 0
\(513\) 5.38280 0.237656
\(514\) 0 0
\(515\) − 5.82002i − 0.256461i
\(516\) 0 0
\(517\) 29.1909i 1.28381i
\(518\) 0 0
\(519\) 0.404696 0.0177642
\(520\) 0 0
\(521\) 6.57594 0.288097 0.144049 0.989571i \(-0.453988\pi\)
0.144049 + 0.989571i \(0.453988\pi\)
\(522\) 0 0
\(523\) 22.2161i 0.971443i 0.874114 + 0.485722i \(0.161443\pi\)
−0.874114 + 0.485722i \(0.838557\pi\)
\(524\) 0 0
\(525\) − 1.05585i − 0.0460810i
\(526\) 0 0
\(527\) −29.7386 −1.29543
\(528\) 0 0
\(529\) −19.2233 −0.835795
\(530\) 0 0
\(531\) − 15.4936i − 0.672366i
\(532\) 0 0
\(533\) − 3.65691i − 0.158398i
\(534\) 0 0
\(535\) −6.87605 −0.297278
\(536\) 0 0
\(537\) 17.5928 0.759185
\(538\) 0 0
\(539\) − 19.7357i − 0.850076i
\(540\) 0 0
\(541\) 22.5004i 0.967368i 0.875243 + 0.483684i \(0.160702\pi\)
−0.875243 + 0.483684i \(0.839298\pi\)
\(542\) 0 0
\(543\) −10.3045 −0.442207
\(544\) 0 0
\(545\) −16.0068 −0.685656
\(546\) 0 0
\(547\) − 24.1551i − 1.03280i −0.856348 0.516399i \(-0.827272\pi\)
0.856348 0.516399i \(-0.172728\pi\)
\(548\) 0 0
\(549\) 7.79830i 0.332823i
\(550\) 0 0
\(551\) 8.83944 0.376573
\(552\) 0 0
\(553\) −3.04984 −0.129692
\(554\) 0 0
\(555\) 5.04063i 0.213963i
\(556\) 0 0
\(557\) − 21.4469i − 0.908734i −0.890815 0.454367i \(-0.849865\pi\)
0.890815 0.454367i \(-0.150135\pi\)
\(558\) 0 0
\(559\) 1.59525 0.0674717
\(560\) 0 0
\(561\) −14.6440 −0.618270
\(562\) 0 0
\(563\) 15.3594i 0.647322i 0.946173 + 0.323661i \(0.104914\pi\)
−0.946173 + 0.323661i \(0.895086\pi\)
\(564\) 0 0
\(565\) − 9.24256i − 0.388837i
\(566\) 0 0
\(567\) −0.281503 −0.0118220
\(568\) 0 0
\(569\) −2.05613 −0.0861973 −0.0430986 0.999071i \(-0.513723\pi\)
−0.0430986 + 0.999071i \(0.513723\pi\)
\(570\) 0 0
\(571\) − 35.4718i − 1.48445i −0.670151 0.742225i \(-0.733771\pi\)
0.670151 0.742225i \(-0.266229\pi\)
\(572\) 0 0
\(573\) 21.1871i 0.885103i
\(574\) 0 0
\(575\) 7.46741 0.311413
\(576\) 0 0
\(577\) 14.6910 0.611595 0.305798 0.952097i \(-0.401077\pi\)
0.305798 + 0.952097i \(0.401077\pi\)
\(578\) 0 0
\(579\) 5.87311i 0.244078i
\(580\) 0 0
\(581\) 3.19743i 0.132652i
\(582\) 0 0
\(583\) 23.7698 0.984445
\(584\) 0 0
\(585\) −2.36638 −0.0978376
\(586\) 0 0
\(587\) − 1.71339i − 0.0707193i −0.999375 0.0353596i \(-0.988742\pi\)
0.999375 0.0353596i \(-0.0112577\pi\)
\(588\) 0 0
\(589\) 6.64877i 0.273958i
\(590\) 0 0
\(591\) 23.3722 0.961402
\(592\) 0 0
\(593\) 20.3947 0.837509 0.418754 0.908100i \(-0.362467\pi\)
0.418754 + 0.908100i \(0.362467\pi\)
\(594\) 0 0
\(595\) 1.14801i 0.0470639i
\(596\) 0 0
\(597\) − 0.358212i − 0.0146607i
\(598\) 0 0
\(599\) −21.9447 −0.896637 −0.448318 0.893874i \(-0.647977\pi\)
−0.448318 + 0.893874i \(0.647977\pi\)
\(600\) 0 0
\(601\) 3.41894 0.139461 0.0697306 0.997566i \(-0.477786\pi\)
0.0697306 + 0.997566i \(0.477786\pi\)
\(602\) 0 0
\(603\) 7.22625i 0.294276i
\(604\) 0 0
\(605\) − 3.14180i − 0.127732i
\(606\) 0 0
\(607\) −23.1235 −0.938555 −0.469278 0.883051i \(-0.655486\pi\)
−0.469278 + 0.883051i \(0.655486\pi\)
\(608\) 0 0
\(609\) −2.42892 −0.0984247
\(610\) 0 0
\(611\) − 13.4986i − 0.546095i
\(612\) 0 0
\(613\) 46.3763i 1.87312i 0.350504 + 0.936561i \(0.386010\pi\)
−0.350504 + 0.936561i \(0.613990\pi\)
\(614\) 0 0
\(615\) −3.44760 −0.139021
\(616\) 0 0
\(617\) 25.2123 1.01501 0.507504 0.861649i \(-0.330568\pi\)
0.507504 + 0.861649i \(0.330568\pi\)
\(618\) 0 0
\(619\) 37.2031i 1.49532i 0.664082 + 0.747660i \(0.268822\pi\)
−0.664082 + 0.747660i \(0.731178\pi\)
\(620\) 0 0
\(621\) − 10.4608i − 0.419778i
\(622\) 0 0
\(623\) −2.19635 −0.0879948
\(624\) 0 0
\(625\) 8.97722 0.359089
\(626\) 0 0
\(627\) 3.27402i 0.130752i
\(628\) 0 0
\(629\) 18.1937i 0.725429i
\(630\) 0 0
\(631\) −40.8412 −1.62586 −0.812931 0.582360i \(-0.802129\pi\)
−0.812931 + 0.582360i \(0.802129\pi\)
\(632\) 0 0
\(633\) 16.5036 0.655959
\(634\) 0 0
\(635\) − 1.24783i − 0.0495188i
\(636\) 0 0
\(637\) 9.12627i 0.361596i
\(638\) 0 0
\(639\) 4.65300 0.184070
\(640\) 0 0
\(641\) 5.87446 0.232027 0.116014 0.993248i \(-0.462988\pi\)
0.116014 + 0.993248i \(0.462988\pi\)
\(642\) 0 0
\(643\) − 21.3344i − 0.841347i −0.907212 0.420673i \(-0.861794\pi\)
0.907212 0.420673i \(-0.138206\pi\)
\(644\) 0 0
\(645\) − 1.50394i − 0.0592176i
\(646\) 0 0
\(647\) −48.1625 −1.89346 −0.946732 0.322024i \(-0.895637\pi\)
−0.946732 + 0.322024i \(0.895637\pi\)
\(648\) 0 0
\(649\) 26.3191 1.03311
\(650\) 0 0
\(651\) − 1.82696i − 0.0716043i
\(652\) 0 0
\(653\) 11.3441i 0.443929i 0.975055 + 0.221965i \(0.0712469\pi\)
−0.975055 + 0.221965i \(0.928753\pi\)
\(654\) 0 0
\(655\) −4.14813 −0.162081
\(656\) 0 0
\(657\) 0.224995 0.00877791
\(658\) 0 0
\(659\) 24.6320i 0.959526i 0.877398 + 0.479763i \(0.159277\pi\)
−0.877398 + 0.479763i \(0.840723\pi\)
\(660\) 0 0
\(661\) 47.3409i 1.84135i 0.390335 + 0.920673i \(0.372359\pi\)
−0.390335 + 0.920673i \(0.627641\pi\)
\(662\) 0 0
\(663\) 6.77175 0.262993
\(664\) 0 0
\(665\) 0.256666 0.00995308
\(666\) 0 0
\(667\) − 17.1784i − 0.665149i
\(668\) 0 0
\(669\) − 14.8148i − 0.572772i
\(670\) 0 0
\(671\) −13.2470 −0.511396
\(672\) 0 0
\(673\) −35.1767 −1.35596 −0.677981 0.735079i \(-0.737145\pi\)
−0.677981 + 0.735079i \(0.737145\pi\)
\(674\) 0 0
\(675\) − 20.6834i − 0.796103i
\(676\) 0 0
\(677\) − 2.71085i − 0.104186i −0.998642 0.0520932i \(-0.983411\pi\)
0.998642 0.0520932i \(-0.0165893\pi\)
\(678\) 0 0
\(679\) −0.0590444 −0.00226592
\(680\) 0 0
\(681\) 21.2499 0.814300
\(682\) 0 0
\(683\) 48.4820i 1.85511i 0.373682 + 0.927557i \(0.378095\pi\)
−0.373682 + 0.927557i \(0.621905\pi\)
\(684\) 0 0
\(685\) − 1.53824i − 0.0587730i
\(686\) 0 0
\(687\) 5.45128 0.207979
\(688\) 0 0
\(689\) −10.9917 −0.418752
\(690\) 0 0
\(691\) 41.3295i 1.57225i 0.618068 + 0.786124i \(0.287915\pi\)
−0.618068 + 0.786124i \(0.712085\pi\)
\(692\) 0 0
\(693\) 1.13472i 0.0431043i
\(694\) 0 0
\(695\) −3.85368 −0.146178
\(696\) 0 0
\(697\) −12.4438 −0.471342
\(698\) 0 0
\(699\) − 4.92755i − 0.186377i
\(700\) 0 0
\(701\) 15.9692i 0.603147i 0.953443 + 0.301573i \(0.0975118\pi\)
−0.953443 + 0.301573i \(0.902488\pi\)
\(702\) 0 0
\(703\) 4.06763 0.153414
\(704\) 0 0
\(705\) −12.7260 −0.479288
\(706\) 0 0
\(707\) − 4.00852i − 0.150756i
\(708\) 0 0
\(709\) 1.27532i 0.0478957i 0.999713 + 0.0239478i \(0.00762356\pi\)
−0.999713 + 0.0239478i \(0.992376\pi\)
\(710\) 0 0
\(711\) −21.3920 −0.802263
\(712\) 0 0
\(713\) 12.9211 0.483898
\(714\) 0 0
\(715\) − 4.01978i − 0.150331i
\(716\) 0 0
\(717\) 23.4860i 0.877100i
\(718\) 0 0
\(719\) 38.6245 1.44045 0.720225 0.693740i \(-0.244038\pi\)
0.720225 + 0.693740i \(0.244038\pi\)
\(720\) 0 0
\(721\) 1.29053 0.0480619
\(722\) 0 0
\(723\) − 29.4722i − 1.09608i
\(724\) 0 0
\(725\) − 33.9655i − 1.26145i
\(726\) 0 0
\(727\) −3.96945 −0.147219 −0.0736095 0.997287i \(-0.523452\pi\)
−0.0736095 + 0.997287i \(0.523452\pi\)
\(728\) 0 0
\(729\) −20.5744 −0.762015
\(730\) 0 0
\(731\) − 5.42832i − 0.200774i
\(732\) 0 0
\(733\) 23.9821i 0.885800i 0.896571 + 0.442900i \(0.146050\pi\)
−0.896571 + 0.442900i \(0.853950\pi\)
\(734\) 0 0
\(735\) 8.60391 0.317360
\(736\) 0 0
\(737\) −12.2753 −0.452165
\(738\) 0 0
\(739\) − 34.2553i − 1.26010i −0.776554 0.630051i \(-0.783034\pi\)
0.776554 0.630051i \(-0.216966\pi\)
\(740\) 0 0
\(741\) − 1.51399i − 0.0556177i
\(742\) 0 0
\(743\) 31.0058 1.13749 0.568746 0.822513i \(-0.307429\pi\)
0.568746 + 0.822513i \(0.307429\pi\)
\(744\) 0 0
\(745\) −17.5113 −0.641564
\(746\) 0 0
\(747\) 22.4273i 0.820571i
\(748\) 0 0
\(749\) − 1.52470i − 0.0557112i
\(750\) 0 0
\(751\) −36.0417 −1.31518 −0.657590 0.753376i \(-0.728424\pi\)
−0.657590 + 0.753376i \(0.728424\pi\)
\(752\) 0 0
\(753\) −33.3849 −1.21661
\(754\) 0 0
\(755\) 15.4111i 0.560867i
\(756\) 0 0
\(757\) − 26.6253i − 0.967712i −0.875148 0.483856i \(-0.839236\pi\)
0.875148 0.483856i \(-0.160764\pi\)
\(758\) 0 0
\(759\) 6.36265 0.230950
\(760\) 0 0
\(761\) 40.0064 1.45023 0.725115 0.688628i \(-0.241787\pi\)
0.725115 + 0.688628i \(0.241787\pi\)
\(762\) 0 0
\(763\) − 3.54935i − 0.128495i
\(764\) 0 0
\(765\) 8.05233i 0.291133i
\(766\) 0 0
\(767\) −12.1706 −0.439455
\(768\) 0 0
\(769\) 19.1147 0.689295 0.344648 0.938732i \(-0.387998\pi\)
0.344648 + 0.938732i \(0.387998\pi\)
\(770\) 0 0
\(771\) − 5.87658i − 0.211640i
\(772\) 0 0
\(773\) − 42.2244i − 1.51871i −0.650679 0.759353i \(-0.725516\pi\)
0.650679 0.759353i \(-0.274484\pi\)
\(774\) 0 0
\(775\) 25.5479 0.917706
\(776\) 0 0
\(777\) −1.11771 −0.0400977
\(778\) 0 0
\(779\) 2.78211i 0.0996793i
\(780\) 0 0
\(781\) 7.90408i 0.282830i
\(782\) 0 0
\(783\) −47.5809 −1.70040
\(784\) 0 0
\(785\) 4.01833 0.143421
\(786\) 0 0
\(787\) − 43.6009i − 1.55420i −0.629375 0.777102i \(-0.716689\pi\)
0.629375 0.777102i \(-0.283311\pi\)
\(788\) 0 0
\(789\) − 13.4707i − 0.479571i
\(790\) 0 0
\(791\) 2.04945 0.0728699
\(792\) 0 0
\(793\) 6.12575 0.217532
\(794\) 0 0
\(795\) 10.3626i 0.367524i
\(796\) 0 0
\(797\) − 14.6901i − 0.520351i −0.965561 0.260176i \(-0.916220\pi\)
0.965561 0.260176i \(-0.0837805\pi\)
\(798\) 0 0
\(799\) −45.9332 −1.62500
\(800\) 0 0
\(801\) −15.4055 −0.544327
\(802\) 0 0
\(803\) 0.382201i 0.0134876i
\(804\) 0 0
\(805\) − 0.498799i − 0.0175803i
\(806\) 0 0
\(807\) 0.188244 0.00662649
\(808\) 0 0
\(809\) 4.76004 0.167354 0.0836771 0.996493i \(-0.473334\pi\)
0.0836771 + 0.996493i \(0.473334\pi\)
\(810\) 0 0
\(811\) 14.6172i 0.513279i 0.966507 + 0.256640i \(0.0826154\pi\)
−0.966507 + 0.256640i \(0.917385\pi\)
\(812\) 0 0
\(813\) 1.16582i 0.0408872i
\(814\) 0 0
\(815\) −4.29125 −0.150316
\(816\) 0 0
\(817\) −1.21363 −0.0424596
\(818\) 0 0
\(819\) − 0.524721i − 0.0183352i
\(820\) 0 0
\(821\) 32.3079i 1.12755i 0.825928 + 0.563776i \(0.190652\pi\)
−0.825928 + 0.563776i \(0.809348\pi\)
\(822\) 0 0
\(823\) −9.79982 −0.341600 −0.170800 0.985306i \(-0.554635\pi\)
−0.170800 + 0.985306i \(0.554635\pi\)
\(824\) 0 0
\(825\) 12.5804 0.437993
\(826\) 0 0
\(827\) 11.2066i 0.389691i 0.980834 + 0.194845i \(0.0624205\pi\)
−0.980834 + 0.194845i \(0.937579\pi\)
\(828\) 0 0
\(829\) − 45.4323i − 1.57793i −0.614438 0.788965i \(-0.710617\pi\)
0.614438 0.788965i \(-0.289383\pi\)
\(830\) 0 0
\(831\) 3.62619 0.125791
\(832\) 0 0
\(833\) 31.0550 1.07599
\(834\) 0 0
\(835\) − 8.79160i − 0.304246i
\(836\) 0 0
\(837\) − 35.7890i − 1.23705i
\(838\) 0 0
\(839\) −16.8997 −0.583441 −0.291721 0.956504i \(-0.594228\pi\)
−0.291721 + 0.956504i \(0.594228\pi\)
\(840\) 0 0
\(841\) −49.1356 −1.69433
\(842\) 0 0
\(843\) 28.4191i 0.978807i
\(844\) 0 0
\(845\) − 12.1275i − 0.417200i
\(846\) 0 0
\(847\) 0.696664 0.0239377
\(848\) 0 0
\(849\) 15.9059 0.545890
\(850\) 0 0
\(851\) − 7.90494i − 0.270978i
\(852\) 0 0
\(853\) 37.7943i 1.29405i 0.762467 + 0.647027i \(0.223988\pi\)
−0.762467 + 0.647027i \(0.776012\pi\)
\(854\) 0 0
\(855\) 1.80029 0.0615687
\(856\) 0 0
\(857\) 38.8619 1.32750 0.663749 0.747956i \(-0.268964\pi\)
0.663749 + 0.747956i \(0.268964\pi\)
\(858\) 0 0
\(859\) 41.1795i 1.40503i 0.711671 + 0.702513i \(0.247939\pi\)
−0.711671 + 0.702513i \(0.752061\pi\)
\(860\) 0 0
\(861\) − 0.764473i − 0.0260532i
\(862\) 0 0
\(863\) 25.5909 0.871125 0.435562 0.900159i \(-0.356550\pi\)
0.435562 + 0.900159i \(0.356550\pi\)
\(864\) 0 0
\(865\) 0.378015 0.0128529
\(866\) 0 0
\(867\) − 3.46216i − 0.117581i
\(868\) 0 0
\(869\) − 36.3387i − 1.23271i
\(870\) 0 0
\(871\) 5.67639 0.192337
\(872\) 0 0
\(873\) −0.414146 −0.0140167
\(874\) 0 0
\(875\) − 2.26957i − 0.0767253i
\(876\) 0 0
\(877\) 14.4790i 0.488920i 0.969659 + 0.244460i \(0.0786107\pi\)
−0.969659 + 0.244460i \(0.921389\pi\)
\(878\) 0 0
\(879\) 9.49879 0.320386
\(880\) 0 0
\(881\) 48.3407 1.62864 0.814320 0.580416i \(-0.197110\pi\)
0.814320 + 0.580416i \(0.197110\pi\)
\(882\) 0 0
\(883\) − 17.2669i − 0.581077i −0.956863 0.290539i \(-0.906166\pi\)
0.956863 0.290539i \(-0.0938344\pi\)
\(884\) 0 0
\(885\) 11.4740i 0.385694i
\(886\) 0 0
\(887\) 51.0650 1.71460 0.857298 0.514820i \(-0.172141\pi\)
0.857298 + 0.514820i \(0.172141\pi\)
\(888\) 0 0
\(889\) 0.276695 0.00928006
\(890\) 0 0
\(891\) − 3.35410i − 0.112366i
\(892\) 0 0
\(893\) 10.2695i 0.343655i
\(894\) 0 0
\(895\) 16.4329 0.549292
\(896\) 0 0
\(897\) −2.94225 −0.0982388
\(898\) 0 0
\(899\) − 58.7714i − 1.96014i
\(900\) 0 0
\(901\) 37.4028i 1.24607i
\(902\) 0 0
\(903\) 0.333484 0.0110977
\(904\) 0 0
\(905\) −9.62510 −0.319949
\(906\) 0 0
\(907\) 30.0974i 0.999367i 0.866208 + 0.499684i \(0.166550\pi\)
−0.866208 + 0.499684i \(0.833450\pi\)
\(908\) 0 0
\(909\) − 28.1164i − 0.932561i
\(910\) 0 0
\(911\) 9.15413 0.303290 0.151645 0.988435i \(-0.451543\pi\)
0.151645 + 0.988435i \(0.451543\pi\)
\(912\) 0 0
\(913\) −38.0973 −1.26084
\(914\) 0 0
\(915\) − 5.77514i − 0.190920i
\(916\) 0 0
\(917\) − 0.919807i − 0.0303747i
\(918\) 0 0
\(919\) −7.49270 −0.247161 −0.123581 0.992335i \(-0.539438\pi\)
−0.123581 + 0.992335i \(0.539438\pi\)
\(920\) 0 0
\(921\) 20.1037 0.662440
\(922\) 0 0
\(923\) − 3.65505i − 0.120307i
\(924\) 0 0
\(925\) − 15.6299i − 0.513906i
\(926\) 0 0
\(927\) 9.05198 0.297306
\(928\) 0 0
\(929\) −12.0792 −0.396306 −0.198153 0.980171i \(-0.563494\pi\)
−0.198153 + 0.980171i \(0.563494\pi\)
\(930\) 0 0
\(931\) − 6.94309i − 0.227550i
\(932\) 0 0
\(933\) − 12.1235i − 0.396905i
\(934\) 0 0
\(935\) −13.6785 −0.447336
\(936\) 0 0
\(937\) −16.6732 −0.544690 −0.272345 0.962200i \(-0.587799\pi\)
−0.272345 + 0.962200i \(0.587799\pi\)
\(938\) 0 0
\(939\) 16.5283i 0.539380i
\(940\) 0 0
\(941\) − 29.5263i − 0.962529i −0.876575 0.481265i \(-0.840178\pi\)
0.876575 0.481265i \(-0.159822\pi\)
\(942\) 0 0
\(943\) 5.40668 0.176066
\(944\) 0 0
\(945\) −1.38158 −0.0449428
\(946\) 0 0
\(947\) 24.2099i 0.786717i 0.919385 + 0.393358i \(0.128687\pi\)
−0.919385 + 0.393358i \(0.871313\pi\)
\(948\) 0 0
\(949\) − 0.176739i − 0.00573720i
\(950\) 0 0
\(951\) −18.8287 −0.610561
\(952\) 0 0
\(953\) −18.5304 −0.600258 −0.300129 0.953899i \(-0.597030\pi\)
−0.300129 + 0.953899i \(0.597030\pi\)
\(954\) 0 0
\(955\) 19.7903i 0.640398i
\(956\) 0 0
\(957\) − 28.9405i − 0.935513i
\(958\) 0 0
\(959\) 0.341089 0.0110143
\(960\) 0 0
\(961\) 13.2062 0.426005
\(962\) 0 0
\(963\) − 10.6945i − 0.344624i
\(964\) 0 0
\(965\) 5.48590i 0.176597i
\(966\) 0 0
\(967\) 54.2821 1.74559 0.872797 0.488084i \(-0.162304\pi\)
0.872797 + 0.488084i \(0.162304\pi\)
\(968\) 0 0
\(969\) −5.15181 −0.165500
\(970\) 0 0
\(971\) − 7.69284i − 0.246875i −0.992352 0.123437i \(-0.960608\pi\)
0.992352 0.123437i \(-0.0393918\pi\)
\(972\) 0 0
\(973\) − 0.854516i − 0.0273945i
\(974\) 0 0
\(975\) −5.81749 −0.186309
\(976\) 0 0
\(977\) −32.8587 −1.05124 −0.525621 0.850719i \(-0.676167\pi\)
−0.525621 + 0.850719i \(0.676167\pi\)
\(978\) 0 0
\(979\) − 26.1694i − 0.836378i
\(980\) 0 0
\(981\) − 24.8957i − 0.794858i
\(982\) 0 0
\(983\) −10.4144 −0.332169 −0.166084 0.986112i \(-0.553112\pi\)
−0.166084 + 0.986112i \(0.553112\pi\)
\(984\) 0 0
\(985\) 21.8313 0.695602
\(986\) 0 0
\(987\) − 2.82186i − 0.0898209i
\(988\) 0 0
\(989\) 2.35855i 0.0749974i
\(990\) 0 0
\(991\) −62.1164 −1.97319 −0.986596 0.163183i \(-0.947824\pi\)
−0.986596 + 0.163183i \(0.947824\pi\)
\(992\) 0 0
\(993\) 17.2908 0.548706
\(994\) 0 0
\(995\) − 0.334596i − 0.0106074i
\(996\) 0 0
\(997\) − 5.60362i − 0.177468i −0.996055 0.0887341i \(-0.971718\pi\)
0.996055 0.0887341i \(-0.0282822\pi\)
\(998\) 0 0
\(999\) −21.8952 −0.692735
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 2432.2.c.j.1217.7 20
4.3 odd 2 inner 2432.2.c.j.1217.13 yes 20
8.3 odd 2 inner 2432.2.c.j.1217.8 yes 20
8.5 even 2 inner 2432.2.c.j.1217.14 yes 20
16.3 odd 4 4864.2.a.bt.1.3 10
16.5 even 4 4864.2.a.bt.1.4 10
16.11 odd 4 4864.2.a.bs.1.8 10
16.13 even 4 4864.2.a.bs.1.7 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.j.1217.7 20 1.1 even 1 trivial
2432.2.c.j.1217.8 yes 20 8.3 odd 2 inner
2432.2.c.j.1217.13 yes 20 4.3 odd 2 inner
2432.2.c.j.1217.14 yes 20 8.5 even 2 inner
4864.2.a.bs.1.7 10 16.13 even 4
4864.2.a.bs.1.8 10 16.11 odd 4
4864.2.a.bt.1.3 10 16.3 odd 4
4864.2.a.bt.1.4 10 16.5 even 4