Properties

Label 2432.2.c.g.1217.1
Level $2432$
Weight $2$
Character 2432.1217
Analytic conductor $19.420$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(6\)
Coefficient field: 6.0.3182656.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{3} + 25x^{2} - 10x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1217.1
Root \(0.203364 - 0.203364i\) of defining polynomial
Character \(\chi\) \(=\) 2432.1217
Dual form 2432.2.c.g.1217.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.91729i q^{3} -3.91729i q^{5} +0.593272 q^{7} -5.51056 q^{9} +O(q^{10})\) \(q-2.91729i q^{3} -3.91729i q^{5} +0.593272 q^{7} -5.51056 q^{9} -3.51056i q^{11} -2.51056i q^{13} -11.4278 q^{15} +3.00000 q^{17} -1.00000i q^{19} -1.73074i q^{21} +7.32401 q^{23} -10.3451 q^{25} +7.32401i q^{27} -5.73074i q^{29} -4.40673 q^{31} -10.2413 q^{33} -2.32401i q^{35} -3.42784i q^{37} -7.32401 q^{39} +10.2413 q^{41} +4.69710i q^{43} +21.5864i q^{45} -2.08271 q^{47} -6.64803 q^{49} -8.75186i q^{51} +1.08271i q^{53} -13.7519 q^{55} -2.91729 q^{57} +8.51056i q^{59} -7.10383i q^{61} -3.26926 q^{63} -9.83457 q^{65} +12.9173i q^{67} -21.3662i q^{69} +12.8557 q^{71} +16.2288 q^{73} +30.1797i q^{75} -2.08271i q^{77} -4.40673 q^{79} +4.83457 q^{81} +13.4615i q^{83} -11.7519i q^{85} -16.7182 q^{87} -0.241300 q^{89} -1.48944i q^{91} +12.8557i q^{93} -3.91729 q^{95} +15.4615 q^{97} +19.3451i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{7} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{7} - 16 q^{9} - 32 q^{15} + 18 q^{17} + 22 q^{23} - 6 q^{25} - 24 q^{31} - 20 q^{33} - 22 q^{39} + 20 q^{41} - 32 q^{47} + 4 q^{49} - 24 q^{55} + 2 q^{57} - 44 q^{63} - 20 q^{65} + 4 q^{71} + 34 q^{73} - 24 q^{79} - 10 q^{81} - 54 q^{87} + 40 q^{89} - 4 q^{95} + 44 q^{97}+O(q^{100}) \) 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
\(3\) − 2.91729i − 1.68430i −0.539247 0.842148i \(-0.681291\pi\)
0.539247 0.842148i \(-0.318709\pi\)
\(4\) 0 0
\(5\) − 3.91729i − 1.75186i −0.482435 0.875932i \(-0.660248\pi\)
0.482435 0.875932i \(-0.339752\pi\)
\(6\) 0 0
\(7\) 0.593272 0.224236 0.112118 0.993695i \(-0.464237\pi\)
0.112118 + 0.993695i \(0.464237\pi\)
\(8\) 0 0
\(9\) −5.51056 −1.83685
\(10\) 0 0
\(11\) − 3.51056i − 1.05847i −0.848474 0.529236i \(-0.822478\pi\)
0.848474 0.529236i \(-0.177522\pi\)
\(12\) 0 0
\(13\) − 2.51056i − 0.696303i −0.937438 0.348152i \(-0.886809\pi\)
0.937438 0.348152i \(-0.113191\pi\)
\(14\) 0 0
\(15\) −11.4278 −2.95066
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) − 1.00000i − 0.229416i
\(20\) 0 0
\(21\) − 1.73074i − 0.377679i
\(22\) 0 0
\(23\) 7.32401 1.52716 0.763581 0.645712i \(-0.223439\pi\)
0.763581 + 0.645712i \(0.223439\pi\)
\(24\) 0 0
\(25\) −10.3451 −2.06903
\(26\) 0 0
\(27\) 7.32401i 1.40951i
\(28\) 0 0
\(29\) − 5.73074i − 1.06417i −0.846690 0.532086i \(-0.821408\pi\)
0.846690 0.532086i \(-0.178592\pi\)
\(30\) 0 0
\(31\) −4.40673 −0.791472 −0.395736 0.918364i \(-0.629510\pi\)
−0.395736 + 0.918364i \(0.629510\pi\)
\(32\) 0 0
\(33\) −10.2413 −1.78278
\(34\) 0 0
\(35\) − 2.32401i − 0.392830i
\(36\) 0 0
\(37\) − 3.42784i − 0.563534i −0.959483 0.281767i \(-0.909079\pi\)
0.959483 0.281767i \(-0.0909205\pi\)
\(38\) 0 0
\(39\) −7.32401 −1.17278
\(40\) 0 0
\(41\) 10.2413 1.59942 0.799711 0.600385i \(-0.204986\pi\)
0.799711 + 0.600385i \(0.204986\pi\)
\(42\) 0 0
\(43\) 4.69710i 0.716301i 0.933664 + 0.358151i \(0.116593\pi\)
−0.933664 + 0.358151i \(0.883407\pi\)
\(44\) 0 0
\(45\) 21.5864i 3.21791i
\(46\) 0 0
\(47\) −2.08271 −0.303795 −0.151898 0.988396i \(-0.548538\pi\)
−0.151898 + 0.988396i \(0.548538\pi\)
\(48\) 0 0
\(49\) −6.64803 −0.949718
\(50\) 0 0
\(51\) − 8.75186i − 1.22551i
\(52\) 0 0
\(53\) 1.08271i 0.148722i 0.997231 + 0.0743611i \(0.0236917\pi\)
−0.997231 + 0.0743611i \(0.976308\pi\)
\(54\) 0 0
\(55\) −13.7519 −1.85430
\(56\) 0 0
\(57\) −2.91729 −0.386404
\(58\) 0 0
\(59\) 8.51056i 1.10798i 0.832523 + 0.553990i \(0.186895\pi\)
−0.832523 + 0.553990i \(0.813105\pi\)
\(60\) 0 0
\(61\) − 7.10383i − 0.909552i −0.890606 0.454776i \(-0.849719\pi\)
0.890606 0.454776i \(-0.150281\pi\)
\(62\) 0 0
\(63\) −3.26926 −0.411888
\(64\) 0 0
\(65\) −9.83457 −1.21983
\(66\) 0 0
\(67\) 12.9173i 1.57810i 0.614330 + 0.789049i \(0.289426\pi\)
−0.614330 + 0.789049i \(0.710574\pi\)
\(68\) 0 0
\(69\) − 21.3662i − 2.57219i
\(70\) 0 0
\(71\) 12.8557 1.52569 0.762845 0.646582i \(-0.223802\pi\)
0.762845 + 0.646582i \(0.223802\pi\)
\(72\) 0 0
\(73\) 16.2288 1.89943 0.949717 0.313109i \(-0.101371\pi\)
0.949717 + 0.313109i \(0.101371\pi\)
\(74\) 0 0
\(75\) 30.1797i 3.48485i
\(76\) 0 0
\(77\) − 2.08271i − 0.237347i
\(78\) 0 0
\(79\) −4.40673 −0.495796 −0.247898 0.968786i \(-0.579740\pi\)
−0.247898 + 0.968786i \(0.579740\pi\)
\(80\) 0 0
\(81\) 4.83457 0.537175
\(82\) 0 0
\(83\) 13.4615i 1.47759i 0.673930 + 0.738795i \(0.264605\pi\)
−0.673930 + 0.738795i \(0.735395\pi\)
\(84\) 0 0
\(85\) − 11.7519i − 1.27467i
\(86\) 0 0
\(87\) −16.7182 −1.79238
\(88\) 0 0
\(89\) −0.241300 −0.0255778 −0.0127889 0.999918i \(-0.504071\pi\)
−0.0127889 + 0.999918i \(0.504071\pi\)
\(90\) 0 0
\(91\) − 1.48944i − 0.156136i
\(92\) 0 0
\(93\) 12.8557i 1.33307i
\(94\) 0 0
\(95\) −3.91729 −0.401905
\(96\) 0 0
\(97\) 15.4615 1.56988 0.784938 0.619574i \(-0.212695\pi\)
0.784938 + 0.619574i \(0.212695\pi\)
\(98\) 0 0
\(99\) 19.3451i 1.94426i
\(100\) 0 0
\(101\) − 15.8346i − 1.57560i −0.615932 0.787799i \(-0.711220\pi\)
0.615932 0.787799i \(-0.288780\pi\)
\(102\) 0 0
\(103\) 13.4615 1.32640 0.663200 0.748442i \(-0.269198\pi\)
0.663200 + 0.748442i \(0.269198\pi\)
\(104\) 0 0
\(105\) −6.77981 −0.661642
\(106\) 0 0
\(107\) 0.302899i 0.0292824i 0.999893 + 0.0146412i \(0.00466060\pi\)
−0.999893 + 0.0146412i \(0.995339\pi\)
\(108\) 0 0
\(109\) − 5.29037i − 0.506726i −0.967371 0.253363i \(-0.918463\pi\)
0.967371 0.253363i \(-0.0815367\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) 2.77981 0.261503 0.130751 0.991415i \(-0.458261\pi\)
0.130751 + 0.991415i \(0.458261\pi\)
\(114\) 0 0
\(115\) − 28.6903i − 2.67538i
\(116\) 0 0
\(117\) 13.8346i 1.27901i
\(118\) 0 0
\(119\) 1.77981 0.163155
\(120\) 0 0
\(121\) −1.32401 −0.120365
\(122\) 0 0
\(123\) − 29.8768i − 2.69390i
\(124\) 0 0
\(125\) 20.9384i 1.87279i
\(126\) 0 0
\(127\) 14.2835 1.26746 0.633729 0.773555i \(-0.281523\pi\)
0.633729 + 0.773555i \(0.281523\pi\)
\(128\) 0 0
\(129\) 13.7028 1.20646
\(130\) 0 0
\(131\) 16.3662i 1.42993i 0.699163 + 0.714963i \(0.253556\pi\)
−0.699163 + 0.714963i \(0.746444\pi\)
\(132\) 0 0
\(133\) − 0.593272i − 0.0514432i
\(134\) 0 0
\(135\) 28.6903 2.46926
\(136\) 0 0
\(137\) 4.66914 0.398912 0.199456 0.979907i \(-0.436083\pi\)
0.199456 + 0.979907i \(0.436083\pi\)
\(138\) 0 0
\(139\) 6.48944i 0.550427i 0.961383 + 0.275214i \(0.0887486\pi\)
−0.961383 + 0.275214i \(0.911251\pi\)
\(140\) 0 0
\(141\) 6.07587i 0.511681i
\(142\) 0 0
\(143\) −8.81346 −0.737018
\(144\) 0 0
\(145\) −22.4490 −1.86428
\(146\) 0 0
\(147\) 19.3942i 1.59961i
\(148\) 0 0
\(149\) 6.73074i 0.551404i 0.961243 + 0.275702i \(0.0889103\pi\)
−0.961243 + 0.275702i \(0.911090\pi\)
\(150\) 0 0
\(151\) 18.2077 1.48172 0.740859 0.671660i \(-0.234419\pi\)
0.740859 + 0.671660i \(0.234419\pi\)
\(152\) 0 0
\(153\) −16.5317 −1.33651
\(154\) 0 0
\(155\) 17.2624i 1.38655i
\(156\) 0 0
\(157\) 16.0422i 1.28031i 0.768246 + 0.640155i \(0.221130\pi\)
−0.768246 + 0.640155i \(0.778870\pi\)
\(158\) 0 0
\(159\) 3.15859 0.250492
\(160\) 0 0
\(161\) 4.34513 0.342444
\(162\) 0 0
\(163\) − 24.0422i − 1.88313i −0.336827 0.941566i \(-0.609354\pi\)
0.336827 0.941566i \(-0.390646\pi\)
\(164\) 0 0
\(165\) 40.1181i 3.12319i
\(166\) 0 0
\(167\) −19.6355 −1.51944 −0.759720 0.650250i \(-0.774664\pi\)
−0.759720 + 0.650250i \(0.774664\pi\)
\(168\) 0 0
\(169\) 6.69710 0.515162
\(170\) 0 0
\(171\) 5.51056i 0.421403i
\(172\) 0 0
\(173\) 8.37309i 0.636594i 0.947991 + 0.318297i \(0.103111\pi\)
−0.947991 + 0.318297i \(0.896889\pi\)
\(174\) 0 0
\(175\) −6.13747 −0.463949
\(176\) 0 0
\(177\) 24.8277 1.86617
\(178\) 0 0
\(179\) 10.2413i 0.765471i 0.923858 + 0.382735i \(0.125018\pi\)
−0.923858 + 0.382735i \(0.874982\pi\)
\(180\) 0 0
\(181\) 24.5248i 1.82292i 0.411393 + 0.911458i \(0.365042\pi\)
−0.411393 + 0.911458i \(0.634958\pi\)
\(182\) 0 0
\(183\) −20.7239 −1.53195
\(184\) 0 0
\(185\) −13.4278 −0.987235
\(186\) 0 0
\(187\) − 10.5317i − 0.770152i
\(188\) 0 0
\(189\) 4.34513i 0.316062i
\(190\) 0 0
\(191\) 9.28353 0.671733 0.335866 0.941910i \(-0.390971\pi\)
0.335866 + 0.941910i \(0.390971\pi\)
\(192\) 0 0
\(193\) −8.03364 −0.578274 −0.289137 0.957288i \(-0.593368\pi\)
−0.289137 + 0.957288i \(0.593368\pi\)
\(194\) 0 0
\(195\) 28.6903i 2.05455i
\(196\) 0 0
\(197\) − 5.62691i − 0.400901i −0.979704 0.200451i \(-0.935759\pi\)
0.979704 0.200451i \(-0.0642406\pi\)
\(198\) 0 0
\(199\) −18.0970 −1.28286 −0.641431 0.767181i \(-0.721659\pi\)
−0.641431 + 0.767181i \(0.721659\pi\)
\(200\) 0 0
\(201\) 37.6834 2.65798
\(202\) 0 0
\(203\) − 3.39989i − 0.238625i
\(204\) 0 0
\(205\) − 40.1181i − 2.80197i
\(206\) 0 0
\(207\) −40.3594 −2.80517
\(208\) 0 0
\(209\) −3.51056 −0.242830
\(210\) 0 0
\(211\) 0.302899i 0.0208524i 0.999946 + 0.0104262i \(0.00331883\pi\)
−0.999946 + 0.0104262i \(0.996681\pi\)
\(212\) 0 0
\(213\) − 37.5037i − 2.56971i
\(214\) 0 0
\(215\) 18.3999 1.25486
\(216\) 0 0
\(217\) −2.61439 −0.177476
\(218\) 0 0
\(219\) − 47.3440i − 3.19921i
\(220\) 0 0
\(221\) − 7.53167i − 0.506635i
\(222\) 0 0
\(223\) 0.847099 0.0567259 0.0283630 0.999598i \(-0.490971\pi\)
0.0283630 + 0.999598i \(0.490971\pi\)
\(224\) 0 0
\(225\) 57.0074 3.80050
\(226\) 0 0
\(227\) 12.5528i 0.833158i 0.909100 + 0.416579i \(0.136771\pi\)
−0.909100 + 0.416579i \(0.863229\pi\)
\(228\) 0 0
\(229\) 14.7730i 0.976226i 0.872781 + 0.488113i \(0.162315\pi\)
−0.872781 + 0.488113i \(0.837685\pi\)
\(230\) 0 0
\(231\) −6.07587 −0.399763
\(232\) 0 0
\(233\) −7.71822 −0.505637 −0.252819 0.967514i \(-0.581358\pi\)
−0.252819 + 0.967514i \(0.581358\pi\)
\(234\) 0 0
\(235\) 8.15859i 0.532207i
\(236\) 0 0
\(237\) 12.8557i 0.835067i
\(238\) 0 0
\(239\) −27.6144 −1.78623 −0.893113 0.449832i \(-0.851484\pi\)
−0.893113 + 0.449832i \(0.851484\pi\)
\(240\) 0 0
\(241\) −1.42784 −0.0919755 −0.0459877 0.998942i \(-0.514644\pi\)
−0.0459877 + 0.998942i \(0.514644\pi\)
\(242\) 0 0
\(243\) 7.86821i 0.504746i
\(244\) 0 0
\(245\) 26.0422i 1.66378i
\(246\) 0 0
\(247\) −2.51056 −0.159743
\(248\) 0 0
\(249\) 39.2710 2.48870
\(250\) 0 0
\(251\) − 20.5317i − 1.29595i −0.761663 0.647974i \(-0.775617\pi\)
0.761663 0.647974i \(-0.224383\pi\)
\(252\) 0 0
\(253\) − 25.7114i − 1.61646i
\(254\) 0 0
\(255\) −34.2835 −2.14692
\(256\) 0 0
\(257\) −25.5037 −1.59088 −0.795439 0.606034i \(-0.792760\pi\)
−0.795439 + 0.606034i \(0.792760\pi\)
\(258\) 0 0
\(259\) − 2.03364i − 0.126364i
\(260\) 0 0
\(261\) 31.5796i 1.95473i
\(262\) 0 0
\(263\) 13.7519 0.847976 0.423988 0.905668i \(-0.360630\pi\)
0.423988 + 0.905668i \(0.360630\pi\)
\(264\) 0 0
\(265\) 4.24130 0.260541
\(266\) 0 0
\(267\) 0.703942i 0.0430806i
\(268\) 0 0
\(269\) − 20.8220i − 1.26954i −0.772700 0.634771i \(-0.781094\pi\)
0.772700 0.634771i \(-0.218906\pi\)
\(270\) 0 0
\(271\) −14.8277 −0.900720 −0.450360 0.892847i \(-0.648704\pi\)
−0.450360 + 0.892847i \(0.648704\pi\)
\(272\) 0 0
\(273\) −4.34513 −0.262979
\(274\) 0 0
\(275\) 36.3172i 2.19001i
\(276\) 0 0
\(277\) − 15.9173i − 0.956377i −0.878257 0.478189i \(-0.841294\pi\)
0.878257 0.478189i \(-0.158706\pi\)
\(278\) 0 0
\(279\) 24.2835 1.45382
\(280\) 0 0
\(281\) −4.64803 −0.277278 −0.138639 0.990343i \(-0.544273\pi\)
−0.138639 + 0.990343i \(0.544273\pi\)
\(282\) 0 0
\(283\) 3.99316i 0.237369i 0.992932 + 0.118684i \(0.0378676\pi\)
−0.992932 + 0.118684i \(0.962132\pi\)
\(284\) 0 0
\(285\) 11.4278i 0.676927i
\(286\) 0 0
\(287\) 6.07587 0.358647
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) − 45.1056i − 2.64414i
\(292\) 0 0
\(293\) 3.11636i 0.182059i 0.995848 + 0.0910297i \(0.0290158\pi\)
−0.995848 + 0.0910297i \(0.970984\pi\)
\(294\) 0 0
\(295\) 33.3383 1.94103
\(296\) 0 0
\(297\) 25.7114 1.49193
\(298\) 0 0
\(299\) − 18.3874i − 1.06337i
\(300\) 0 0
\(301\) 2.78666i 0.160620i
\(302\) 0 0
\(303\) −46.1940 −2.65377
\(304\) 0 0
\(305\) −27.8277 −1.59341
\(306\) 0 0
\(307\) − 23.7028i − 1.35279i −0.736539 0.676395i \(-0.763541\pi\)
0.736539 0.676395i \(-0.236459\pi\)
\(308\) 0 0
\(309\) − 39.2710i − 2.23405i
\(310\) 0 0
\(311\) 1.90301 0.107910 0.0539550 0.998543i \(-0.482817\pi\)
0.0539550 + 0.998543i \(0.482817\pi\)
\(312\) 0 0
\(313\) −31.6834 −1.79085 −0.895426 0.445210i \(-0.853129\pi\)
−0.895426 + 0.445210i \(0.853129\pi\)
\(314\) 0 0
\(315\) 12.8066i 0.721571i
\(316\) 0 0
\(317\) − 12.1797i − 0.684080i −0.939685 0.342040i \(-0.888882\pi\)
0.939685 0.342040i \(-0.111118\pi\)
\(318\) 0 0
\(319\) −20.1181 −1.12640
\(320\) 0 0
\(321\) 0.883644 0.0493202
\(322\) 0 0
\(323\) − 3.00000i − 0.166924i
\(324\) 0 0
\(325\) 25.9720i 1.44067i
\(326\) 0 0
\(327\) −15.4335 −0.853476
\(328\) 0 0
\(329\) −1.23562 −0.0681217
\(330\) 0 0
\(331\) − 7.08271i − 0.389301i −0.980873 0.194651i \(-0.937643\pi\)
0.980873 0.194651i \(-0.0623573\pi\)
\(332\) 0 0
\(333\) 18.8893i 1.03513i
\(334\) 0 0
\(335\) 50.6007 2.76461
\(336\) 0 0
\(337\) −9.02112 −0.491411 −0.245706 0.969344i \(-0.579020\pi\)
−0.245706 + 0.969344i \(0.579020\pi\)
\(338\) 0 0
\(339\) − 8.10951i − 0.440448i
\(340\) 0 0
\(341\) 15.4701i 0.837751i
\(342\) 0 0
\(343\) −8.09699 −0.437196
\(344\) 0 0
\(345\) −83.6977 −4.50613
\(346\) 0 0
\(347\) 28.0354i 1.50502i 0.658582 + 0.752509i \(0.271157\pi\)
−0.658582 + 0.752509i \(0.728843\pi\)
\(348\) 0 0
\(349\) − 4.39989i − 0.235521i −0.993042 0.117760i \(-0.962429\pi\)
0.993042 0.117760i \(-0.0375714\pi\)
\(350\) 0 0
\(351\) 18.3874 0.981445
\(352\) 0 0
\(353\) −0.950928 −0.0506128 −0.0253064 0.999680i \(-0.508056\pi\)
−0.0253064 + 0.999680i \(0.508056\pi\)
\(354\) 0 0
\(355\) − 50.3594i − 2.67280i
\(356\) 0 0
\(357\) − 5.19223i − 0.274802i
\(358\) 0 0
\(359\) 12.3856 0.653688 0.326844 0.945078i \(-0.394015\pi\)
0.326844 + 0.945078i \(0.394015\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 3.86253i 0.202730i
\(364\) 0 0
\(365\) − 63.5727i − 3.32755i
\(366\) 0 0
\(367\) 20.4826 1.06918 0.534592 0.845111i \(-0.320465\pi\)
0.534592 + 0.845111i \(0.320465\pi\)
\(368\) 0 0
\(369\) −56.4353 −2.93790
\(370\) 0 0
\(371\) 0.642343i 0.0333488i
\(372\) 0 0
\(373\) − 4.88364i − 0.252865i −0.991975 0.126433i \(-0.959647\pi\)
0.991975 0.126433i \(-0.0403528\pi\)
\(374\) 0 0
\(375\) 61.0833 3.15433
\(376\) 0 0
\(377\) −14.3874 −0.740987
\(378\) 0 0
\(379\) 0.179702i 0.00923065i 0.999989 + 0.00461532i \(0.00146911\pi\)
−0.999989 + 0.00461532i \(0.998531\pi\)
\(380\) 0 0
\(381\) − 41.6691i − 2.13477i
\(382\) 0 0
\(383\) −7.86821 −0.402047 −0.201023 0.979586i \(-0.564427\pi\)
−0.201023 + 0.979586i \(0.564427\pi\)
\(384\) 0 0
\(385\) −8.15859 −0.415800
\(386\) 0 0
\(387\) − 25.8836i − 1.31574i
\(388\) 0 0
\(389\) 31.8614i 1.61544i 0.589569 + 0.807718i \(0.299298\pi\)
−0.589569 + 0.807718i \(0.700702\pi\)
\(390\) 0 0
\(391\) 21.9720 1.11117
\(392\) 0 0
\(393\) 47.7450 2.40842
\(394\) 0 0
\(395\) 17.2624i 0.868566i
\(396\) 0 0
\(397\) − 2.93840i − 0.147474i −0.997278 0.0737371i \(-0.976507\pi\)
0.997278 0.0737371i \(-0.0234926\pi\)
\(398\) 0 0
\(399\) −1.73074 −0.0866455
\(400\) 0 0
\(401\) 4.77981 0.238693 0.119346 0.992853i \(-0.461920\pi\)
0.119346 + 0.992853i \(0.461920\pi\)
\(402\) 0 0
\(403\) 11.0633i 0.551104i
\(404\) 0 0
\(405\) − 18.9384i − 0.941057i
\(406\) 0 0
\(407\) −12.0336 −0.596485
\(408\) 0 0
\(409\) 17.8009 0.880199 0.440100 0.897949i \(-0.354943\pi\)
0.440100 + 0.897949i \(0.354943\pi\)
\(410\) 0 0
\(411\) − 13.6212i − 0.671886i
\(412\) 0 0
\(413\) 5.04907i 0.248449i
\(414\) 0 0
\(415\) 52.7325 2.58854
\(416\) 0 0
\(417\) 18.9316 0.927082
\(418\) 0 0
\(419\) 16.9230i 0.826741i 0.910563 + 0.413371i \(0.135649\pi\)
−0.910563 + 0.413371i \(0.864351\pi\)
\(420\) 0 0
\(421\) 11.9384i 0.581842i 0.956747 + 0.290921i \(0.0939617\pi\)
−0.956747 + 0.290921i \(0.906038\pi\)
\(422\) 0 0
\(423\) 11.4769 0.558027
\(424\) 0 0
\(425\) −31.0354 −1.50544
\(426\) 0 0
\(427\) − 4.21450i − 0.203954i
\(428\) 0 0
\(429\) 25.7114i 1.24136i
\(430\) 0 0
\(431\) 29.1729 1.40521 0.702604 0.711581i \(-0.252021\pi\)
0.702604 + 0.711581i \(0.252021\pi\)
\(432\) 0 0
\(433\) −5.91044 −0.284038 −0.142019 0.989864i \(-0.545359\pi\)
−0.142019 + 0.989864i \(0.545359\pi\)
\(434\) 0 0
\(435\) 65.4900i 3.14001i
\(436\) 0 0
\(437\) − 7.32401i − 0.350355i
\(438\) 0 0
\(439\) −37.7450 −1.80147 −0.900736 0.434368i \(-0.856972\pi\)
−0.900736 + 0.434368i \(0.856972\pi\)
\(440\) 0 0
\(441\) 36.6343 1.74449
\(442\) 0 0
\(443\) − 9.95093i − 0.472783i −0.971658 0.236391i \(-0.924035\pi\)
0.971658 0.236391i \(-0.0759648\pi\)
\(444\) 0 0
\(445\) 0.945243i 0.0448088i
\(446\) 0 0
\(447\) 19.6355 0.928727
\(448\) 0 0
\(449\) 21.6269 1.02064 0.510319 0.859985i \(-0.329527\pi\)
0.510319 + 0.859985i \(0.329527\pi\)
\(450\) 0 0
\(451\) − 35.9527i − 1.69295i
\(452\) 0 0
\(453\) − 53.1169i − 2.49565i
\(454\) 0 0
\(455\) −5.83457 −0.273529
\(456\) 0 0
\(457\) −5.49628 −0.257105 −0.128553 0.991703i \(-0.541033\pi\)
−0.128553 + 0.991703i \(0.541033\pi\)
\(458\) 0 0
\(459\) 21.9720i 1.02557i
\(460\) 0 0
\(461\) − 5.42100i − 0.252481i −0.992000 0.126241i \(-0.959709\pi\)
0.992000 0.126241i \(-0.0402911\pi\)
\(462\) 0 0
\(463\) −7.31149 −0.339794 −0.169897 0.985462i \(-0.554343\pi\)
−0.169897 + 0.985462i \(0.554343\pi\)
\(464\) 0 0
\(465\) 50.3594 2.33536
\(466\) 0 0
\(467\) − 6.97204i − 0.322628i −0.986903 0.161314i \(-0.948427\pi\)
0.986903 0.161314i \(-0.0515731\pi\)
\(468\) 0 0
\(469\) 7.66346i 0.353866i
\(470\) 0 0
\(471\) 46.7998 2.15642
\(472\) 0 0
\(473\) 16.4894 0.758185
\(474\) 0 0
\(475\) 10.3451i 0.474667i
\(476\) 0 0
\(477\) − 5.96636i − 0.273181i
\(478\) 0 0
\(479\) −25.2288 −1.15273 −0.576366 0.817192i \(-0.695530\pi\)
−0.576366 + 0.817192i \(0.695530\pi\)
\(480\) 0 0
\(481\) −8.60580 −0.392391
\(482\) 0 0
\(483\) − 12.6760i − 0.576777i
\(484\) 0 0
\(485\) − 60.5671i − 2.75021i
\(486\) 0 0
\(487\) −1.06335 −0.0481848 −0.0240924 0.999710i \(-0.507670\pi\)
−0.0240924 + 0.999710i \(0.507670\pi\)
\(488\) 0 0
\(489\) −70.1381 −3.17175
\(490\) 0 0
\(491\) − 18.6903i − 0.843480i −0.906717 0.421740i \(-0.861420\pi\)
0.906717 0.421740i \(-0.138580\pi\)
\(492\) 0 0
\(493\) − 17.1922i − 0.774299i
\(494\) 0 0
\(495\) 75.7804 3.40608
\(496\) 0 0
\(497\) 7.62691 0.342114
\(498\) 0 0
\(499\) − 15.8836i − 0.711050i −0.934667 0.355525i \(-0.884302\pi\)
0.934667 0.355525i \(-0.115698\pi\)
\(500\) 0 0
\(501\) 57.2824i 2.55919i
\(502\) 0 0
\(503\) −21.2430 −0.947181 −0.473590 0.880745i \(-0.657042\pi\)
−0.473590 + 0.880745i \(0.657042\pi\)
\(504\) 0 0
\(505\) −62.0285 −2.76023
\(506\) 0 0
\(507\) − 19.5374i − 0.867685i
\(508\) 0 0
\(509\) − 20.6230i − 0.914097i −0.889442 0.457049i \(-0.848907\pi\)
0.889442 0.457049i \(-0.151093\pi\)
\(510\) 0 0
\(511\) 9.62807 0.425921
\(512\) 0 0
\(513\) 7.32401 0.323363
\(514\) 0 0
\(515\) − 52.7325i − 2.32367i
\(516\) 0 0
\(517\) 7.31149i 0.321559i
\(518\) 0 0
\(519\) 24.4267 1.07221
\(520\) 0 0
\(521\) 5.26242 0.230551 0.115275 0.993334i \(-0.463225\pi\)
0.115275 + 0.993334i \(0.463225\pi\)
\(522\) 0 0
\(523\) 2.67599i 0.117013i 0.998287 + 0.0585063i \(0.0186338\pi\)
−0.998287 + 0.0585063i \(0.981366\pi\)
\(524\) 0 0
\(525\) 17.9048i 0.781428i
\(526\) 0 0
\(527\) −13.2202 −0.575880
\(528\) 0 0
\(529\) 30.6412 1.33223
\(530\) 0 0
\(531\) − 46.8979i − 2.03520i
\(532\) 0 0
\(533\) − 25.7114i − 1.11368i
\(534\) 0 0
\(535\) 1.18654 0.0512987
\(536\) 0 0
\(537\) 29.8768 1.28928
\(538\) 0 0
\(539\) 23.3383i 1.00525i
\(540\) 0 0
\(541\) 22.5653i 0.970159i 0.874470 + 0.485079i \(0.161209\pi\)
−0.874470 + 0.485079i \(0.838791\pi\)
\(542\) 0 0
\(543\) 71.5459 3.07033
\(544\) 0 0
\(545\) −20.7239 −0.887714
\(546\) 0 0
\(547\) 8.08446i 0.345667i 0.984951 + 0.172833i \(0.0552922\pi\)
−0.984951 + 0.172833i \(0.944708\pi\)
\(548\) 0 0
\(549\) 39.1461i 1.67071i
\(550\) 0 0
\(551\) −5.73074 −0.244138
\(552\) 0 0
\(553\) −2.61439 −0.111175
\(554\) 0 0
\(555\) 39.1729i 1.66280i
\(556\) 0 0
\(557\) − 1.12888i − 0.0478323i −0.999714 0.0239162i \(-0.992387\pi\)
0.999714 0.0239162i \(-0.00761348\pi\)
\(558\) 0 0
\(559\) 11.7923 0.498763
\(560\) 0 0
\(561\) −30.7239 −1.29716
\(562\) 0 0
\(563\) 35.4701i 1.49489i 0.664326 + 0.747443i \(0.268719\pi\)
−0.664326 + 0.747443i \(0.731281\pi\)
\(564\) 0 0
\(565\) − 10.8893i − 0.458118i
\(566\) 0 0
\(567\) 2.86821 0.120454
\(568\) 0 0
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 12.2499i 0.512642i 0.966592 + 0.256321i \(0.0825104\pi\)
−0.966592 + 0.256321i \(0.917490\pi\)
\(572\) 0 0
\(573\) − 27.0827i − 1.13140i
\(574\) 0 0
\(575\) −75.7679 −3.15974
\(576\) 0 0
\(577\) −22.8768 −0.952374 −0.476187 0.879344i \(-0.657981\pi\)
−0.476187 + 0.879344i \(0.657981\pi\)
\(578\) 0 0
\(579\) 23.4364i 0.973985i
\(580\) 0 0
\(581\) 7.98632i 0.331328i
\(582\) 0 0
\(583\) 3.80093 0.157418
\(584\) 0 0
\(585\) 54.1940 2.24065
\(586\) 0 0
\(587\) 29.2219i 1.20612i 0.797697 + 0.603059i \(0.206052\pi\)
−0.797697 + 0.603059i \(0.793948\pi\)
\(588\) 0 0
\(589\) 4.40673i 0.181576i
\(590\) 0 0
\(591\) −16.4153 −0.675236
\(592\) 0 0
\(593\) −1.35197 −0.0555188 −0.0277594 0.999615i \(-0.508837\pi\)
−0.0277594 + 0.999615i \(0.508837\pi\)
\(594\) 0 0
\(595\) − 6.97204i − 0.285826i
\(596\) 0 0
\(597\) 52.7941i 2.16072i
\(598\) 0 0
\(599\) 41.1814 1.68263 0.841314 0.540546i \(-0.181782\pi\)
0.841314 + 0.540546i \(0.181782\pi\)
\(600\) 0 0
\(601\) 5.25383 0.214308 0.107154 0.994242i \(-0.465826\pi\)
0.107154 + 0.994242i \(0.465826\pi\)
\(602\) 0 0
\(603\) − 71.1814i − 2.89873i
\(604\) 0 0
\(605\) 5.18654i 0.210863i
\(606\) 0 0
\(607\) −9.17286 −0.372315 −0.186157 0.982520i \(-0.559603\pi\)
−0.186157 + 0.982520i \(0.559603\pi\)
\(608\) 0 0
\(609\) −9.91844 −0.401916
\(610\) 0 0
\(611\) 5.22877i 0.211534i
\(612\) 0 0
\(613\) − 28.3577i − 1.14535i −0.819781 0.572677i \(-0.805905\pi\)
0.819781 0.572677i \(-0.194095\pi\)
\(614\) 0 0
\(615\) −117.036 −4.71935
\(616\) 0 0
\(617\) −19.4546 −0.783214 −0.391607 0.920132i \(-0.628081\pi\)
−0.391607 + 0.920132i \(0.628081\pi\)
\(618\) 0 0
\(619\) 25.2538i 1.01504i 0.861641 + 0.507519i \(0.169437\pi\)
−0.861641 + 0.507519i \(0.830563\pi\)
\(620\) 0 0
\(621\) 53.6412i 2.15255i
\(622\) 0 0
\(623\) −0.143157 −0.00573545
\(624\) 0 0
\(625\) 30.2961 1.21184
\(626\) 0 0
\(627\) 10.2413i 0.408998i
\(628\) 0 0
\(629\) − 10.2835i − 0.410031i
\(630\) 0 0
\(631\) −23.6287 −0.940642 −0.470321 0.882495i \(-0.655862\pi\)
−0.470321 + 0.882495i \(0.655862\pi\)
\(632\) 0 0
\(633\) 0.883644 0.0351217
\(634\) 0 0
\(635\) − 55.9527i − 2.22041i
\(636\) 0 0
\(637\) 16.6903i 0.661292i
\(638\) 0 0
\(639\) −70.8420 −2.80247
\(640\) 0 0
\(641\) 26.9230 1.06339 0.531697 0.846935i \(-0.321555\pi\)
0.531697 + 0.846935i \(0.321555\pi\)
\(642\) 0 0
\(643\) − 13.4124i − 0.528934i −0.964395 0.264467i \(-0.914804\pi\)
0.964395 0.264467i \(-0.0851960\pi\)
\(644\) 0 0
\(645\) − 53.6777i − 2.11356i
\(646\) 0 0
\(647\) 6.55104 0.257548 0.128774 0.991674i \(-0.458896\pi\)
0.128774 + 0.991674i \(0.458896\pi\)
\(648\) 0 0
\(649\) 29.8768 1.17277
\(650\) 0 0
\(651\) 7.62691i 0.298922i
\(652\) 0 0
\(653\) 2.52308i 0.0987359i 0.998781 + 0.0493680i \(0.0157207\pi\)
−0.998781 + 0.0493680i \(0.984279\pi\)
\(654\) 0 0
\(655\) 64.1113 2.50503
\(656\) 0 0
\(657\) −89.4296 −3.48898
\(658\) 0 0
\(659\) 5.17227i 0.201483i 0.994913 + 0.100742i \(0.0321215\pi\)
−0.994913 + 0.100742i \(0.967878\pi\)
\(660\) 0 0
\(661\) 41.6412i 1.61965i 0.586668 + 0.809827i \(0.300439\pi\)
−0.586668 + 0.809827i \(0.699561\pi\)
\(662\) 0 0
\(663\) −21.9720 −0.853323
\(664\) 0 0
\(665\) −2.32401 −0.0901214
\(666\) 0 0
\(667\) − 41.9720i − 1.62516i
\(668\) 0 0
\(669\) − 2.47123i − 0.0955433i
\(670\) 0 0
\(671\) −24.9384 −0.962736
\(672\) 0 0
\(673\) −16.7325 −0.644990 −0.322495 0.946571i \(-0.604522\pi\)
−0.322495 + 0.946571i \(0.604522\pi\)
\(674\) 0 0
\(675\) − 75.7679i − 2.91631i
\(676\) 0 0
\(677\) 1.45580i 0.0559510i 0.999609 + 0.0279755i \(0.00890603\pi\)
−0.999609 + 0.0279755i \(0.991094\pi\)
\(678\) 0 0
\(679\) 9.17286 0.352022
\(680\) 0 0
\(681\) 36.6201 1.40328
\(682\) 0 0
\(683\) − 8.81346i − 0.337238i −0.985681 0.168619i \(-0.946069\pi\)
0.985681 0.168619i \(-0.0539307\pi\)
\(684\) 0 0
\(685\) − 18.2904i − 0.698839i
\(686\) 0 0
\(687\) 43.0970 1.64425
\(688\) 0 0
\(689\) 2.71822 0.103556
\(690\) 0 0
\(691\) 12.6834i 0.482500i 0.970463 + 0.241250i \(0.0775574\pi\)
−0.970463 + 0.241250i \(0.922443\pi\)
\(692\) 0 0
\(693\) 11.4769i 0.435972i
\(694\) 0 0
\(695\) 25.4210 0.964274
\(696\) 0 0
\(697\) 30.7239 1.16375
\(698\) 0 0
\(699\) 22.5162i 0.851643i
\(700\) 0 0
\(701\) 1.46149i 0.0551996i 0.999619 + 0.0275998i \(0.00878640\pi\)
−0.999619 + 0.0275998i \(0.991214\pi\)
\(702\) 0 0
\(703\) −3.42784 −0.129284
\(704\) 0 0
\(705\) 23.8009 0.896395
\(706\) 0 0
\(707\) − 9.39420i − 0.353305i
\(708\) 0 0
\(709\) − 19.7114i − 0.740276i −0.928977 0.370138i \(-0.879310\pi\)
0.928977 0.370138i \(-0.120690\pi\)
\(710\) 0 0
\(711\) 24.2835 0.910704
\(712\) 0 0
\(713\) −32.2749 −1.20871
\(714\) 0 0
\(715\) 34.5248i 1.29116i
\(716\) 0 0
\(717\) 80.5591i 3.00853i
\(718\) 0 0
\(719\) −20.5933 −0.767999 −0.384000 0.923333i \(-0.625454\pi\)
−0.384000 + 0.923333i \(0.625454\pi\)
\(720\) 0 0
\(721\) 7.98632 0.297426
\(722\) 0 0
\(723\) 4.16543i 0.154914i
\(724\) 0 0
\(725\) 59.2853i 2.20180i
\(726\) 0 0
\(727\) 4.66056 0.172850 0.0864252 0.996258i \(-0.472456\pi\)
0.0864252 + 0.996258i \(0.472456\pi\)
\(728\) 0 0
\(729\) 37.4575 1.38732
\(730\) 0 0
\(731\) 14.0913i 0.521186i
\(732\) 0 0
\(733\) − 34.4826i − 1.27364i −0.771011 0.636822i \(-0.780249\pi\)
0.771011 0.636822i \(-0.219751\pi\)
\(734\) 0 0
\(735\) 75.9726 2.80229
\(736\) 0 0
\(737\) 45.3469 1.67037
\(738\) 0 0
\(739\) − 12.4222i − 0.456956i −0.973549 0.228478i \(-0.926625\pi\)
0.973549 0.228478i \(-0.0733750\pi\)
\(740\) 0 0
\(741\) 7.32401i 0.269054i
\(742\) 0 0
\(743\) 6.77981 0.248727 0.124364 0.992237i \(-0.460311\pi\)
0.124364 + 0.992237i \(0.460311\pi\)
\(744\) 0 0
\(745\) 26.3662 0.965984
\(746\) 0 0
\(747\) − 74.1803i − 2.71411i
\(748\) 0 0
\(749\) 0.179702i 0.00656615i
\(750\) 0 0
\(751\) 40.4826 1.47723 0.738616 0.674127i \(-0.235480\pi\)
0.738616 + 0.674127i \(0.235480\pi\)
\(752\) 0 0
\(753\) −59.8968 −2.18276
\(754\) 0 0
\(755\) − 71.3246i − 2.59577i
\(756\) 0 0
\(757\) − 23.5191i − 0.854818i −0.904058 0.427409i \(-0.859426\pi\)
0.904058 0.427409i \(-0.140574\pi\)
\(758\) 0 0
\(759\) −75.0074 −2.72260
\(760\) 0 0
\(761\) 2.32692 0.0843507 0.0421753 0.999110i \(-0.486571\pi\)
0.0421753 + 0.999110i \(0.486571\pi\)
\(762\) 0 0
\(763\) − 3.13863i − 0.113626i
\(764\) 0 0
\(765\) 64.7593i 2.34138i
\(766\) 0 0
\(767\) 21.3662 0.771490
\(768\) 0 0
\(769\) 34.8768 1.25769 0.628845 0.777531i \(-0.283528\pi\)
0.628845 + 0.777531i \(0.283528\pi\)
\(770\) 0 0
\(771\) 74.4016i 2.67951i
\(772\) 0 0
\(773\) − 5.94699i − 0.213898i −0.994264 0.106949i \(-0.965892\pi\)
0.994264 0.106949i \(-0.0341082\pi\)
\(774\) 0 0
\(775\) 45.5882 1.63758
\(776\) 0 0
\(777\) −5.93272 −0.212835
\(778\) 0 0
\(779\) − 10.2413i − 0.366933i
\(780\) 0 0
\(781\) − 45.1306i − 1.61490i
\(782\) 0 0
\(783\) 41.9720 1.49996
\(784\) 0 0
\(785\) 62.8420 2.24293
\(786\) 0 0
\(787\) − 0.277845i − 0.00990411i −0.999988 0.00495206i \(-0.998424\pi\)
0.999988 0.00495206i \(-0.00157630\pi\)
\(788\) 0 0
\(789\) − 40.1181i − 1.42824i
\(790\) 0 0
\(791\) 1.64919 0.0586383
\(792\) 0 0
\(793\) −17.8346 −0.633324
\(794\) 0 0
\(795\) − 12.3731i − 0.438828i
\(796\) 0 0
\(797\) − 5.01543i − 0.177656i −0.996047 0.0888278i \(-0.971688\pi\)
0.996047 0.0888278i \(-0.0283121\pi\)
\(798\) 0 0
\(799\) −6.24814 −0.221043
\(800\) 0 0
\(801\) 1.32970 0.0469826
\(802\) 0 0
\(803\) − 56.9720i − 2.01050i
\(804\) 0 0
\(805\) − 17.0211i − 0.599915i
\(806\) 0 0
\(807\) −60.7439 −2.13829
\(808\) 0 0
\(809\) 18.9190 0.665158 0.332579 0.943075i \(-0.392081\pi\)
0.332579 + 0.943075i \(0.392081\pi\)
\(810\) 0 0
\(811\) − 45.3103i − 1.59106i −0.605914 0.795530i \(-0.707192\pi\)
0.605914 0.795530i \(-0.292808\pi\)
\(812\) 0 0
\(813\) 43.2567i 1.51708i
\(814\) 0 0
\(815\) −94.1803 −3.29899
\(816\) 0 0
\(817\) 4.69710 0.164331
\(818\) 0 0
\(819\) 8.20766i 0.286799i
\(820\) 0 0
\(821\) 12.6498i 0.441480i 0.975333 + 0.220740i \(0.0708473\pi\)
−0.975333 + 0.220740i \(0.929153\pi\)
\(822\) 0 0
\(823\) 43.9065 1.53048 0.765242 0.643742i \(-0.222619\pi\)
0.765242 + 0.643742i \(0.222619\pi\)
\(824\) 0 0
\(825\) 105.948 3.68862
\(826\) 0 0
\(827\) 13.0405i 0.453462i 0.973957 + 0.226731i \(0.0728038\pi\)
−0.973957 + 0.226731i \(0.927196\pi\)
\(828\) 0 0
\(829\) 49.4507i 1.71749i 0.512400 + 0.858747i \(0.328757\pi\)
−0.512400 + 0.858747i \(0.671243\pi\)
\(830\) 0 0
\(831\) −46.4353 −1.61082
\(832\) 0 0
\(833\) −19.9441 −0.691022
\(834\) 0 0
\(835\) 76.9179i 2.66185i
\(836\) 0 0
\(837\) − 32.2749i − 1.11559i
\(838\) 0 0
\(839\) −15.2288 −0.525756 −0.262878 0.964829i \(-0.584672\pi\)
−0.262878 + 0.964829i \(0.584672\pi\)
\(840\) 0 0
\(841\) −3.84141 −0.132463
\(842\) 0 0
\(843\) 13.5596i 0.467018i
\(844\) 0 0
\(845\) − 26.2345i − 0.902493i
\(846\) 0 0
\(847\) −0.785500 −0.0269901
\(848\) 0 0
\(849\) 11.6492 0.399799
\(850\) 0 0
\(851\) − 25.1056i − 0.860608i
\(852\) 0 0
\(853\) 15.4193i 0.527945i 0.964530 + 0.263973i \(0.0850329\pi\)
−0.964530 + 0.263973i \(0.914967\pi\)
\(854\) 0 0
\(855\) 21.5864 0.738240
\(856\) 0 0
\(857\) 14.3422 0.489921 0.244961 0.969533i \(-0.421225\pi\)
0.244961 + 0.969533i \(0.421225\pi\)
\(858\) 0 0
\(859\) − 46.2681i − 1.57865i −0.613977 0.789324i \(-0.710431\pi\)
0.613977 0.789324i \(-0.289569\pi\)
\(860\) 0 0
\(861\) − 17.7251i − 0.604068i
\(862\) 0 0
\(863\) −25.9527 −0.883439 −0.441720 0.897153i \(-0.645631\pi\)
−0.441720 + 0.897153i \(0.645631\pi\)
\(864\) 0 0
\(865\) 32.7998 1.11523
\(866\) 0 0
\(867\) 23.3383i 0.792610i
\(868\) 0 0
\(869\) 15.4701i 0.524786i
\(870\) 0 0
\(871\) 32.4296 1.09883
\(872\) 0 0
\(873\) −85.2014 −2.88363
\(874\) 0 0
\(875\) 12.4222i 0.419946i
\(876\) 0 0
\(877\) 21.3526i 0.721025i 0.932754 + 0.360512i \(0.117398\pi\)
−0.932754 + 0.360512i \(0.882602\pi\)
\(878\) 0 0
\(879\) 9.09130 0.306642
\(880\) 0 0
\(881\) 53.2642 1.79452 0.897258 0.441507i \(-0.145556\pi\)
0.897258 + 0.441507i \(0.145556\pi\)
\(882\) 0 0
\(883\) − 38.1586i − 1.28414i −0.766647 0.642069i \(-0.778076\pi\)
0.766647 0.642069i \(-0.221924\pi\)
\(884\) 0 0
\(885\) − 97.2573i − 3.26927i
\(886\) 0 0
\(887\) −11.1615 −0.374766 −0.187383 0.982287i \(-0.560001\pi\)
−0.187383 + 0.982287i \(0.560001\pi\)
\(888\) 0 0
\(889\) 8.47401 0.284209
\(890\) 0 0
\(891\) − 16.9720i − 0.568585i
\(892\) 0 0
\(893\) 2.08271i 0.0696954i
\(894\) 0 0
\(895\) 40.1181 1.34100
\(896\) 0 0
\(897\) −53.6412 −1.79103
\(898\) 0 0
\(899\) 25.2538i 0.842262i
\(900\) 0 0
\(901\) 3.24814i 0.108211i
\(902\) 0 0
\(903\) 8.12947 0.270532
\(904\) 0 0
\(905\) 96.0708 3.19350
\(906\) 0 0
\(907\) − 32.7941i − 1.08891i −0.838790 0.544455i \(-0.816737\pi\)
0.838790 0.544455i \(-0.183263\pi\)
\(908\) 0 0
\(909\) 87.2573i 2.89414i
\(910\) 0 0
\(911\) −46.8979 −1.55380 −0.776899 0.629626i \(-0.783208\pi\)
−0.776899 + 0.629626i \(0.783208\pi\)
\(912\) 0 0
\(913\) 47.2573 1.56399
\(914\) 0 0
\(915\) 81.1814i 2.68378i
\(916\) 0 0
\(917\) 9.70963i 0.320640i
\(918\) 0 0
\(919\) −45.1871 −1.49059 −0.745293 0.666737i \(-0.767690\pi\)
−0.745293 + 0.666737i \(0.767690\pi\)
\(920\) 0 0
\(921\) −69.1478 −2.27850
\(922\) 0 0
\(923\) − 32.2749i − 1.06234i
\(924\) 0 0
\(925\) 35.4615i 1.16597i
\(926\) 0 0
\(927\) −74.1803 −2.43640
\(928\) 0 0
\(929\) −51.0776 −1.67580 −0.837901 0.545822i \(-0.816217\pi\)
−0.837901 + 0.545822i \(0.816217\pi\)
\(930\) 0 0
\(931\) 6.64803i 0.217880i
\(932\) 0 0
\(933\) − 5.55163i − 0.181752i
\(934\) 0 0
\(935\) −41.2556 −1.34920
\(936\) 0 0
\(937\) −24.5037 −0.800502 −0.400251 0.916406i \(-0.631077\pi\)
−0.400251 + 0.916406i \(0.631077\pi\)
\(938\) 0 0
\(939\) 92.4296i 3.01633i
\(940\) 0 0
\(941\) 55.1871i 1.79905i 0.436870 + 0.899525i \(0.356087\pi\)
−0.436870 + 0.899525i \(0.643913\pi\)
\(942\) 0 0
\(943\) 75.0074 2.44258
\(944\) 0 0
\(945\) 17.0211 0.553697
\(946\) 0 0
\(947\) 29.9019i 0.971680i 0.874048 + 0.485840i \(0.161486\pi\)
−0.874048 + 0.485840i \(0.838514\pi\)
\(948\) 0 0
\(949\) − 40.7433i − 1.32258i
\(950\) 0 0
\(951\) −35.5317 −1.15219
\(952\) 0 0
\(953\) −26.0086 −0.842501 −0.421250 0.906944i \(-0.638409\pi\)
−0.421250 + 0.906944i \(0.638409\pi\)
\(954\) 0 0
\(955\) − 36.3662i − 1.17678i
\(956\) 0 0
\(957\) 58.6903i 1.89719i
\(958\) 0 0
\(959\) 2.77007 0.0894502
\(960\) 0 0
\(961\) −11.5807 −0.373572
\(962\) 0 0
\(963\) − 1.66914i − 0.0537874i
\(964\) 0 0
\(965\) 31.4701i 1.01306i
\(966\) 0 0
\(967\) 50.8420 1.63497 0.817484 0.575951i \(-0.195368\pi\)
0.817484 + 0.575951i \(0.195368\pi\)
\(968\) 0 0
\(969\) −8.75186 −0.281150
\(970\) 0 0
\(971\) − 22.1267i − 0.710079i −0.934851 0.355040i \(-0.884467\pi\)
0.934851 0.355040i \(-0.115533\pi\)
\(972\) 0 0
\(973\) 3.85000i 0.123425i
\(974\) 0 0
\(975\) 75.7679 2.42651
\(976\) 0 0
\(977\) 43.1814 1.38150 0.690748 0.723095i \(-0.257281\pi\)
0.690748 + 0.723095i \(0.257281\pi\)
\(978\) 0 0
\(979\) 0.847099i 0.0270734i
\(980\) 0 0
\(981\) 29.1529i 0.930780i
\(982\) 0 0
\(983\) 34.8842 1.11263 0.556317 0.830970i \(-0.312214\pi\)
0.556317 + 0.830970i \(0.312214\pi\)
\(984\) 0 0
\(985\) −22.0422 −0.702324
\(986\) 0 0
\(987\) 3.60464i 0.114737i
\(988\) 0 0
\(989\) 34.4016i 1.09391i
\(990\) 0 0
\(991\) −7.36056 −0.233816 −0.116908 0.993143i \(-0.537298\pi\)
−0.116908 + 0.993143i \(0.537298\pi\)
\(992\) 0 0
\(993\) −20.6623 −0.655698
\(994\) 0 0
\(995\) 70.8911i 2.24740i
\(996\) 0 0
\(997\) − 37.2978i − 1.18123i −0.806952 0.590617i \(-0.798885\pi\)
0.806952 0.590617i \(-0.201115\pi\)
\(998\) 0 0
\(999\) 25.1056 0.794305
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.g.1217.1 yes 6
4.3 odd 2 2432.2.c.f.1217.6 yes 6
8.3 odd 2 2432.2.c.f.1217.1 6
8.5 even 2 inner 2432.2.c.g.1217.6 yes 6
16.3 odd 4 4864.2.a.be.1.1 3
16.5 even 4 4864.2.a.bf.1.1 3
16.11 odd 4 4864.2.a.bd.1.3 3
16.13 even 4 4864.2.a.bc.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.f.1217.1 6 8.3 odd 2
2432.2.c.f.1217.6 yes 6 4.3 odd 2
2432.2.c.g.1217.1 yes 6 1.1 even 1 trivial
2432.2.c.g.1217.6 yes 6 8.5 even 2 inner
4864.2.a.bc.1.3 3 16.13 even 4
4864.2.a.bd.1.3 3 16.11 odd 4
4864.2.a.be.1.1 3 16.3 odd 4
4864.2.a.bf.1.1 3 16.5 even 4