Properties

Label 2432.2.c.j.1217.8
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 1217.8
Root \(0.814454 + 0.814454i\) of defining polynomial
Character \(\chi\) \(=\) 2432.1217
Dual form 2432.2.c.j.1217.13

$q$-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 \(0.0773353\pi\)
−0.970631 + 0.240573i \(0.922665\pi\)
\(6\) 0 0
\(7\) −0.238565 −0.0901690 −0.0450845 0.998983i \(-0.514356\pi\)
−0.0450845 + 0.998983i \(0.514356\pi\)
\(8\) 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 \(0.0583477\pi\)
−0.983247 + 0.182280i \(0.941652\pi\)
\(14\) 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 \(-0.564943\pi\)
−0.202611 + 0.979259i \(0.564943\pi\)
\(24\) 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 \(0.306428\pi\)
−0.571329 + 0.820721i \(0.693572\pi\)
\(30\) 0 0
\(31\) −6.64877 −1.19415 −0.597077 0.802184i \(-0.703671\pi\)
−0.597077 + 0.802184i \(0.703671\pi\)
\(32\) 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 \(0.108519\pi\)
−0.942446 + 0.334357i \(0.891481\pi\)
\(38\) 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 \(-0.769455\pi\)
−0.748978 + 0.662595i \(0.769455\pi\)
\(48\) 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 \(0.194736\pi\)
−0.818627 + 0.574326i \(0.805264\pi\)
\(54\) 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 \(-0.903564\pi\)
0.954457 0.298349i \(-0.0964358\pi\)
\(62\) 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 \(-0.552764\pi\)
−0.165003 + 0.986293i \(0.552764\pi\)
\(72\) 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 \(-0.244526\pi\)
0.719162 + 0.694843i \(0.244526\pi\)
\(80\) 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 \(0.315090\pi\)
−0.548786 + 0.835963i \(0.684910\pi\)
\(102\) 0 0
\(103\) −5.40957 −0.533020 −0.266510 0.963832i \(-0.585871\pi\)
−0.266510 + 0.963832i \(0.585871\pi\)
\(104\) 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 \(0.252448\pi\)
−0.701648 + 0.712524i \(0.747552\pi\)
\(110\) 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 \(-0.516387\pi\)
−0.0514592 + 0.998675i \(0.516387\pi\)
\(128\) 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 \(0.232296\pi\)
−0.745322 + 0.666705i \(0.767704\pi\)
\(150\) 0 0
\(151\) 14.3242 1.16569 0.582845 0.812583i \(-0.301939\pi\)
0.582845 + 0.812583i \(0.301939\pi\)
\(152\) 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 \(-0.952381\pi\)
0.988831 0.149041i \(-0.0476185\pi\)
\(158\) 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 \(-0.602396\pi\)
−0.316168 + 0.948703i \(0.602396\pi\)
\(168\) 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 \(-0.995748\pi\)
0.999911 0.0133565i \(-0.00425164\pi\)
\(174\) 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 \(0.107888\pi\)
−0.943108 + 0.332487i \(0.892112\pi\)
\(182\) 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 \(-0.268222\pi\)
0.665492 + 0.746405i \(0.268222\pi\)
\(192\) 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 \(-0.742827\pi\)
0.690995 0.722860i \(-0.257173\pi\)
\(198\) 0 0
\(199\) −0.310999 −0.0220461 −0.0110231 0.999939i \(-0.503509\pi\)
−0.0110231 + 0.999939i \(0.503509\pi\)
\(200\) 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 \(-0.641718\pi\)
−0.430656 + 0.902516i \(0.641718\pi\)
\(224\) 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 \(-0.950019\pi\)
0.987698 0.156376i \(-0.0499811\pi\)
\(230\) 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 \(-0.270779\pi\)
0.659474 + 0.751727i \(0.270779\pi\)
\(240\) 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 \(-0.617421\pi\)
−0.360580 + 0.932728i \(0.617421\pi\)
\(264\) 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 \(-0.998414\pi\)
0.999988 0.00498233i \(-0.00158593\pi\)
\(270\) 0 0
\(271\) 1.01216 0.0614846 0.0307423 0.999527i \(-0.490213\pi\)
0.0307423 + 0.999527i \(0.490213\pi\)
\(272\) 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 \(-0.969849\pi\)
0.995517 0.0945800i \(-0.0301508\pi\)
\(278\) 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 \(-0.922560\pi\)
0.970552 0.240892i \(-0.0774400\pi\)
\(294\) 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 \(-0.596461\pi\)
−0.298425 + 0.954433i \(0.596461\pi\)
\(312\) 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 \(0.151817\pi\)
−0.888401 + 0.459069i \(0.848183\pi\)
\(318\) 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 \(-0.753273\pi\)
0.714341 0.699798i \(-0.246727\pi\)
\(350\) 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 \(-0.378042\pi\)
0.373837 + 0.927495i \(0.378042\pi\)
\(360\) 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 \(-0.475024\pi\)
0.0783850 + 0.996923i \(0.475024\pi\)
\(368\) 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 \(-0.555385\pi\)
0.173119 0.984901i \(-0.444615\pi\)
\(374\) 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 \(-0.711558\pi\)
−0.616767 + 0.787146i \(0.711558\pi\)
\(384\) 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 \(0.135555\pi\)
−0.910684 + 0.413103i \(0.864445\pi\)
\(390\) 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 \(-0.625050\pi\)
0.382829 0.923819i \(-0.374950\pi\)
\(398\) 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 \(-0.855451\pi\)
0.898650 0.438667i \(-0.144549\pi\)
\(422\) 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 \(-0.731853\pi\)
−0.665667 + 0.746249i \(0.731853\pi\)
\(432\) 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 \(-0.686124\pi\)
−0.551971 + 0.833863i \(0.686124\pi\)
\(440\) 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 \(-0.943440\pi\)
0.984255 0.176755i \(-0.0565599\pi\)
\(462\) 0 0
\(463\) −37.2637 −1.73179 −0.865894 0.500227i \(-0.833250\pi\)
−0.865894 + 0.500227i \(0.833250\pi\)
\(464\) 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 \(-0.352021\pi\)
0.448325 + 0.893871i \(0.352021\pi\)
\(480\) 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 \(-0.599288\pi\)
−0.306887 + 0.951746i \(0.599288\pi\)
\(488\) 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 \(-0.484439\pi\)
0.0488683 + 0.998805i \(0.484439\pi\)
\(504\) 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 \(-0.902719\pi\)
0.953661 0.300883i \(-0.0972814\pi\)
\(510\) 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 \(-0.839298\pi\)
0.875243 0.483684i \(-0.160702\pi\)
\(542\) 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 \(0.150135\pi\)
−0.890815 + 0.454367i \(0.849865\pi\)
\(558\) 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 \(-0.352023\pi\)
0.448318 + 0.893874i \(0.352023\pi\)
\(600\) 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 \(-0.344514\pi\)
0.469278 + 0.883051i \(0.344514\pi\)
\(608\) 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 \(-0.613990\pi\)
0.350504 0.936561i \(-0.386010\pi\)
\(614\) 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 \(-0.197871\pi\)
0.812931 + 0.582360i \(0.197871\pi\)
\(632\) 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 \(-0.104363\pi\)
0.946732 + 0.322024i \(0.104363\pi\)
\(648\) 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 \(-0.928753\pi\)
0.975055 0.221965i \(-0.0712469\pi\)
\(654\) 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 \(-0.627641\pi\)
0.390335 0.920673i \(-0.372359\pi\)
\(662\) 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 \(0.0165893\pi\)
−0.998642 + 0.0520932i \(0.983411\pi\)
\(678\) 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 \(-0.902488\pi\)
0.953443 0.301573i \(-0.0975118\pi\)
\(702\) 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 \(-0.992376\pi\)
0.999713 0.0239478i \(-0.00762356\pi\)
\(710\) 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 \(-0.755962\pi\)
−0.720225 + 0.693740i \(0.755962\pi\)
\(720\) 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 \(-0.476548\pi\)
0.0736095 + 0.997287i \(0.476548\pi\)
\(728\) 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 \(-0.853950\pi\)
0.896571 0.442900i \(-0.146050\pi\)
\(734\) 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 \(-0.692571\pi\)
−0.568746 + 0.822513i \(0.692571\pi\)
\(744\) 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 \(-0.271576\pi\)
0.657590 + 0.753376i \(0.271576\pi\)
\(752\) 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 \(0.160764\pi\)
−0.875148 + 0.483856i \(0.839236\pi\)
\(758\) 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 \(0.274484\pi\)
−0.650679 + 0.759353i \(0.725516\pi\)
\(774\) 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 \(0.0837805\pi\)
−0.965561 + 0.260176i \(0.916220\pi\)
\(798\) 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 \(-0.809348\pi\)
0.825928 0.563776i \(-0.190652\pi\)
\(822\) 0 0
\(823\) 9.79982 0.341600 0.170800 0.985306i \(-0.445365\pi\)
0.170800 + 0.985306i \(0.445365\pi\)
\(824\) 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 \(0.289383\pi\)
−0.614438 + 0.788965i \(0.710617\pi\)
\(830\) 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 \(-0.405772\pi\)
0.291721 + 0.956504i \(0.405772\pi\)
\(840\) 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 \(-0.776012\pi\)
0.762467 0.647027i \(-0.223988\pi\)
\(854\) 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 \(-0.643450\pi\)
−0.435562 + 0.900159i \(0.643450\pi\)
\(864\) 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 \(-0.921389\pi\)
0.969659 0.244460i \(-0.0786107\pi\)
\(878\) 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 \(-0.827859\pi\)
−0.857298 + 0.514820i \(0.827859\pi\)
\(888\) 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 \(-0.548457\pi\)
−0.151645 + 0.988435i \(0.548457\pi\)
\(912\) 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 \(-0.460562\pi\)
0.123581 + 0.992335i \(0.460562\pi\)
\(920\) 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 \(0.159822\pi\)
−0.876575 + 0.481265i \(0.840178\pi\)
\(942\) 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 \(-0.837696\pi\)
−0.872797 + 0.488084i \(0.837696\pi\)
\(968\) 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 \(-0.446888\pi\)
0.166084 + 0.986112i \(0.446888\pi\)
\(984\) 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 \(-0.0521761\pi\)
0.986596 + 0.163183i \(0.0521761\pi\)
\(992\) 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 \(0.0282822\pi\)
−0.996055 + 0.0887341i \(0.971718\pi\)
\(998\) 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.1217.8 yes 20
4.3 odd 2 inner 2432.2.c.j.1217.14 yes 20
8.3 odd 2 inner 2432.2.c.j.1217.7 20
8.5 even 2 inner 2432.2.c.j.1217.13 yes 20
16.3 odd 4 4864.2.a.bt.1.4 10
16.5 even 4 4864.2.a.bt.1.3 10
16.11 odd 4 4864.2.a.bs.1.7 10
16.13 even 4 4864.2.a.bs.1.8 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.j.1217.7 20 8.3 odd 2 inner
2432.2.c.j.1217.8 yes 20 1.1 even 1 trivial
2432.2.c.j.1217.13 yes 20 8.5 even 2 inner
2432.2.c.j.1217.14 yes 20 4.3 odd 2 inner
4864.2.a.bs.1.7 10 16.11 odd 4
4864.2.a.bs.1.8 10 16.13 even 4
4864.2.a.bt.1.3 10 16.5 even 4
4864.2.a.bt.1.4 10 16.3 odd 4