Properties

Label 3822.2.c.h.883.1
Level $3822$
Weight $2$
Character 3822.883
Analytic conductor $30.519$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3822,2,Mod(883,3822)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3822, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3822.883");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.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: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 883.1
Root \(-1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 3822.883
Dual form 3822.2.c.h.883.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -0.561553i q^{5} -1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -0.561553i q^{5} -1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} -0.561553 q^{10} -1.43845i q^{11} -1.00000 q^{12} +(0.561553 + 3.56155i) q^{13} -0.561553i q^{15} +1.00000 q^{16} -5.68466 q^{17} -1.00000i q^{18} +2.56155i q^{19} +0.561553i q^{20} -1.43845 q^{22} +5.68466 q^{23} +1.00000i q^{24} +4.68466 q^{25} +(3.56155 - 0.561553i) q^{26} +1.00000 q^{27} -2.56155 q^{29} -0.561553 q^{30} +10.2462i q^{31} -1.00000i q^{32} -1.43845i q^{33} +5.68466i q^{34} -1.00000 q^{36} +1.68466i q^{37} +2.56155 q^{38} +(0.561553 + 3.56155i) q^{39} +0.561553 q^{40} +4.00000i q^{41} -10.5616 q^{43} +1.43845i q^{44} -0.561553i q^{45} -5.68466i q^{46} +6.24621i q^{47} +1.00000 q^{48} -4.68466i q^{50} -5.68466 q^{51} +(-0.561553 - 3.56155i) q^{52} +13.1231 q^{53} -1.00000i q^{54} -0.807764 q^{55} +2.56155i q^{57} +2.56155i q^{58} -12.2462i q^{59} +0.561553i q^{60} -2.56155 q^{61} +10.2462 q^{62} -1.00000 q^{64} +(2.00000 - 0.315342i) q^{65} -1.43845 q^{66} -7.12311i q^{67} +5.68466 q^{68} +5.68466 q^{69} +15.3693i q^{71} +1.00000i q^{72} +7.43845i q^{73} +1.68466 q^{74} +4.68466 q^{75} -2.56155i q^{76} +(3.56155 - 0.561553i) q^{78} +16.0000 q^{79} -0.561553i q^{80} +1.00000 q^{81} +4.00000 q^{82} -2.00000i q^{83} +3.19224i q^{85} +10.5616i q^{86} -2.56155 q^{87} +1.43845 q^{88} -8.00000i q^{89} -0.561553 q^{90} -5.68466 q^{92} +10.2462i q^{93} +6.24621 q^{94} +1.43845 q^{95} -1.00000i q^{96} +10.0000i q^{97} -1.43845i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} + 6 q^{10} - 4 q^{12} - 6 q^{13} + 4 q^{16} + 2 q^{17} - 14 q^{22} - 2 q^{23} - 6 q^{25} + 6 q^{26} + 4 q^{27} - 2 q^{29} + 6 q^{30} - 4 q^{36} + 2 q^{38} - 6 q^{39} - 6 q^{40} - 34 q^{43} + 4 q^{48} + 2 q^{51} + 6 q^{52} + 36 q^{53} + 38 q^{55} - 2 q^{61} + 8 q^{62} - 4 q^{64} + 8 q^{65} - 14 q^{66} - 2 q^{68} - 2 q^{69} - 18 q^{74} - 6 q^{75} + 6 q^{78} + 64 q^{79} + 4 q^{81} + 16 q^{82} - 2 q^{87} + 14 q^{88} + 6 q^{90} + 2 q^{92} - 8 q^{94} + 14 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3822\mathbb{Z}\right)^\times\).

\(n\) \(1471\) \(2549\) \(3433\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 0.561553i 0.251134i −0.992085 0.125567i \(-0.959925\pi\)
0.992085 0.125567i \(-0.0400750\pi\)
\(6\) 1.00000i 0.408248i
\(7\) 0 0
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) −0.561553 −0.177579
\(11\) 1.43845i 0.433708i −0.976204 0.216854i \(-0.930420\pi\)
0.976204 0.216854i \(-0.0695796\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0.561553 + 3.56155i 0.155747 + 0.987797i
\(14\) 0 0
\(15\) 0.561553i 0.144992i
\(16\) 1.00000 0.250000
\(17\) −5.68466 −1.37873 −0.689366 0.724413i \(-0.742111\pi\)
−0.689366 + 0.724413i \(0.742111\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 2.56155i 0.587661i 0.955858 + 0.293830i \(0.0949300\pi\)
−0.955858 + 0.293830i \(0.905070\pi\)
\(20\) 0.561553i 0.125567i
\(21\) 0 0
\(22\) −1.43845 −0.306678
\(23\) 5.68466 1.18533 0.592667 0.805448i \(-0.298075\pi\)
0.592667 + 0.805448i \(0.298075\pi\)
\(24\) 1.00000i 0.204124i
\(25\) 4.68466 0.936932
\(26\) 3.56155 0.561553i 0.698478 0.110130i
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.56155 −0.475668 −0.237834 0.971306i \(-0.576437\pi\)
−0.237834 + 0.971306i \(0.576437\pi\)
\(30\) −0.561553 −0.102525
\(31\) 10.2462i 1.84027i 0.391597 + 0.920137i \(0.371923\pi\)
−0.391597 + 0.920137i \(0.628077\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 1.43845i 0.250402i
\(34\) 5.68466i 0.974911i
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 1.68466i 0.276956i 0.990366 + 0.138478i \(0.0442210\pi\)
−0.990366 + 0.138478i \(0.955779\pi\)
\(38\) 2.56155 0.415539
\(39\) 0.561553 + 3.56155i 0.0899204 + 0.570305i
\(40\) 0.561553 0.0887893
\(41\) 4.00000i 0.624695i 0.949968 + 0.312348i \(0.101115\pi\)
−0.949968 + 0.312348i \(0.898885\pi\)
\(42\) 0 0
\(43\) −10.5616 −1.61062 −0.805311 0.592853i \(-0.798002\pi\)
−0.805311 + 0.592853i \(0.798002\pi\)
\(44\) 1.43845i 0.216854i
\(45\) 0.561553i 0.0837114i
\(46\) 5.68466i 0.838157i
\(47\) 6.24621i 0.911104i 0.890209 + 0.455552i \(0.150558\pi\)
−0.890209 + 0.455552i \(0.849442\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 4.68466i 0.662511i
\(51\) −5.68466 −0.796011
\(52\) −0.561553 3.56155i −0.0778734 0.493899i
\(53\) 13.1231 1.80260 0.901299 0.433198i \(-0.142615\pi\)
0.901299 + 0.433198i \(0.142615\pi\)
\(54\) 1.00000i 0.136083i
\(55\) −0.807764 −0.108919
\(56\) 0 0
\(57\) 2.56155i 0.339286i
\(58\) 2.56155i 0.336348i
\(59\) 12.2462i 1.59432i −0.603768 0.797160i \(-0.706334\pi\)
0.603768 0.797160i \(-0.293666\pi\)
\(60\) 0.561553i 0.0724962i
\(61\) −2.56155 −0.327973 −0.163987 0.986463i \(-0.552435\pi\)
−0.163987 + 0.986463i \(0.552435\pi\)
\(62\) 10.2462 1.30127
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 2.00000 0.315342i 0.248069 0.0391133i
\(66\) −1.43845 −0.177061
\(67\) 7.12311i 0.870226i −0.900376 0.435113i \(-0.856708\pi\)
0.900376 0.435113i \(-0.143292\pi\)
\(68\) 5.68466 0.689366
\(69\) 5.68466 0.684352
\(70\) 0 0
\(71\) 15.3693i 1.82400i 0.410188 + 0.912001i \(0.365463\pi\)
−0.410188 + 0.912001i \(0.634537\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 7.43845i 0.870604i 0.900284 + 0.435302i \(0.143358\pi\)
−0.900284 + 0.435302i \(0.856642\pi\)
\(74\) 1.68466 0.195838
\(75\) 4.68466 0.540938
\(76\) 2.56155i 0.293830i
\(77\) 0 0
\(78\) 3.56155 0.561553i 0.403266 0.0635833i
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0.561553i 0.0627835i
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) 2.00000i 0.219529i −0.993958 0.109764i \(-0.964990\pi\)
0.993958 0.109764i \(-0.0350096\pi\)
\(84\) 0 0
\(85\) 3.19224i 0.346247i
\(86\) 10.5616i 1.13888i
\(87\) −2.56155 −0.274627
\(88\) 1.43845 0.153339
\(89\) 8.00000i 0.847998i −0.905663 0.423999i \(-0.860626\pi\)
0.905663 0.423999i \(-0.139374\pi\)
\(90\) −0.561553 −0.0591929
\(91\) 0 0
\(92\) −5.68466 −0.592667
\(93\) 10.2462i 1.06248i
\(94\) 6.24621 0.644247
\(95\) 1.43845 0.147582
\(96\) 1.00000i 0.102062i
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) 1.43845i 0.144569i
\(100\) −4.68466 −0.468466
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 5.68466i 0.562865i
\(103\) 18.8078 1.85318 0.926592 0.376068i \(-0.122724\pi\)
0.926592 + 0.376068i \(0.122724\pi\)
\(104\) −3.56155 + 0.561553i −0.349239 + 0.0550648i
\(105\) 0 0
\(106\) 13.1231i 1.27463i
\(107\) −13.1231 −1.26866 −0.634329 0.773063i \(-0.718724\pi\)
−0.634329 + 0.773063i \(0.718724\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 15.9309i 1.52590i 0.646457 + 0.762950i \(0.276250\pi\)
−0.646457 + 0.762950i \(0.723750\pi\)
\(110\) 0.807764i 0.0770173i
\(111\) 1.68466i 0.159901i
\(112\) 0 0
\(113\) 3.75379 0.353127 0.176563 0.984289i \(-0.443502\pi\)
0.176563 + 0.984289i \(0.443502\pi\)
\(114\) 2.56155 0.239911
\(115\) 3.19224i 0.297678i
\(116\) 2.56155 0.237834
\(117\) 0.561553 + 3.56155i 0.0519156 + 0.329266i
\(118\) −12.2462 −1.12736
\(119\) 0 0
\(120\) 0.561553 0.0512625
\(121\) 8.93087 0.811897
\(122\) 2.56155i 0.231912i
\(123\) 4.00000i 0.360668i
\(124\) 10.2462i 0.920137i
\(125\) 5.43845i 0.486430i
\(126\) 0 0
\(127\) −6.24621 −0.554262 −0.277131 0.960832i \(-0.589384\pi\)
−0.277131 + 0.960832i \(0.589384\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −10.5616 −0.929893
\(130\) −0.315342 2.00000i −0.0276573 0.175412i
\(131\) −2.56155 −0.223804 −0.111902 0.993719i \(-0.535694\pi\)
−0.111902 + 0.993719i \(0.535694\pi\)
\(132\) 1.43845i 0.125201i
\(133\) 0 0
\(134\) −7.12311 −0.615343
\(135\) 0.561553i 0.0483308i
\(136\) 5.68466i 0.487455i
\(137\) 5.68466i 0.485673i 0.970067 + 0.242837i \(0.0780779\pi\)
−0.970067 + 0.242837i \(0.921922\pi\)
\(138\) 5.68466i 0.483910i
\(139\) 6.24621 0.529797 0.264898 0.964276i \(-0.414662\pi\)
0.264898 + 0.964276i \(0.414662\pi\)
\(140\) 0 0
\(141\) 6.24621i 0.526026i
\(142\) 15.3693 1.28976
\(143\) 5.12311 0.807764i 0.428416 0.0675486i
\(144\) 1.00000 0.0833333
\(145\) 1.43845i 0.119457i
\(146\) 7.43845 0.615610
\(147\) 0 0
\(148\) 1.68466i 0.138478i
\(149\) 10.0000i 0.819232i −0.912258 0.409616i \(-0.865663\pi\)
0.912258 0.409616i \(-0.134337\pi\)
\(150\) 4.68466i 0.382501i
\(151\) 4.31534i 0.351178i −0.984464 0.175589i \(-0.943817\pi\)
0.984464 0.175589i \(-0.0561829\pi\)
\(152\) −2.56155 −0.207769
\(153\) −5.68466 −0.459577
\(154\) 0 0
\(155\) 5.75379 0.462155
\(156\) −0.561553 3.56155i −0.0449602 0.285152i
\(157\) −8.80776 −0.702936 −0.351468 0.936200i \(-0.614317\pi\)
−0.351468 + 0.936200i \(0.614317\pi\)
\(158\) 16.0000i 1.27289i
\(159\) 13.1231 1.04073
\(160\) −0.561553 −0.0443946
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) 0.876894i 0.0686837i −0.999410 0.0343418i \(-0.989067\pi\)
0.999410 0.0343418i \(-0.0109335\pi\)
\(164\) 4.00000i 0.312348i
\(165\) −0.807764 −0.0628843
\(166\) −2.00000 −0.155230
\(167\) 8.80776i 0.681565i 0.940142 + 0.340783i \(0.110692\pi\)
−0.940142 + 0.340783i \(0.889308\pi\)
\(168\) 0 0
\(169\) −12.3693 + 4.00000i −0.951486 + 0.307692i
\(170\) 3.19224 0.244833
\(171\) 2.56155i 0.195887i
\(172\) 10.5616 0.805311
\(173\) 20.2462 1.53929 0.769645 0.638471i \(-0.220433\pi\)
0.769645 + 0.638471i \(0.220433\pi\)
\(174\) 2.56155i 0.194191i
\(175\) 0 0
\(176\) 1.43845i 0.108427i
\(177\) 12.2462i 0.920482i
\(178\) −8.00000 −0.599625
\(179\) 2.87689 0.215029 0.107515 0.994204i \(-0.465711\pi\)
0.107515 + 0.994204i \(0.465711\pi\)
\(180\) 0.561553i 0.0418557i
\(181\) −11.3693 −0.845075 −0.422537 0.906346i \(-0.638860\pi\)
−0.422537 + 0.906346i \(0.638860\pi\)
\(182\) 0 0
\(183\) −2.56155 −0.189355
\(184\) 5.68466i 0.419079i
\(185\) 0.946025 0.0695531
\(186\) 10.2462 0.751289
\(187\) 8.17708i 0.597967i
\(188\) 6.24621i 0.455552i
\(189\) 0 0
\(190\) 1.43845i 0.104356i
\(191\) 13.0540 0.944553 0.472276 0.881451i \(-0.343432\pi\)
0.472276 + 0.881451i \(0.343432\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 12.0000i 0.863779i 0.901927 + 0.431889i \(0.142153\pi\)
−0.901927 + 0.431889i \(0.857847\pi\)
\(194\) 10.0000 0.717958
\(195\) 2.00000 0.315342i 0.143223 0.0225821i
\(196\) 0 0
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) −1.43845 −0.102226
\(199\) 23.9309 1.69641 0.848207 0.529665i \(-0.177682\pi\)
0.848207 + 0.529665i \(0.177682\pi\)
\(200\) 4.68466i 0.331255i
\(201\) 7.12311i 0.502425i
\(202\) 6.00000i 0.422159i
\(203\) 0 0
\(204\) 5.68466 0.398006
\(205\) 2.24621 0.156882
\(206\) 18.8078i 1.31040i
\(207\) 5.68466 0.395111
\(208\) 0.561553 + 3.56155i 0.0389367 + 0.246949i
\(209\) 3.68466 0.254873
\(210\) 0 0
\(211\) −12.8078 −0.881723 −0.440861 0.897575i \(-0.645327\pi\)
−0.440861 + 0.897575i \(0.645327\pi\)
\(212\) −13.1231 −0.901299
\(213\) 15.3693i 1.05309i
\(214\) 13.1231i 0.897077i
\(215\) 5.93087i 0.404482i
\(216\) 1.00000i 0.0680414i
\(217\) 0 0
\(218\) 15.9309 1.07897
\(219\) 7.43845i 0.502644i
\(220\) 0.807764 0.0544594
\(221\) −3.19224 20.2462i −0.214733 1.36191i
\(222\) 1.68466 0.113067
\(223\) 18.2462i 1.22186i −0.791686 0.610928i \(-0.790796\pi\)
0.791686 0.610928i \(-0.209204\pi\)
\(224\) 0 0
\(225\) 4.68466 0.312311
\(226\) 3.75379i 0.249698i
\(227\) 23.6155i 1.56742i 0.621128 + 0.783709i \(0.286675\pi\)
−0.621128 + 0.783709i \(0.713325\pi\)
\(228\) 2.56155i 0.169643i
\(229\) 2.00000i 0.132164i 0.997814 + 0.0660819i \(0.0210498\pi\)
−0.997814 + 0.0660819i \(0.978950\pi\)
\(230\) −3.19224 −0.210490
\(231\) 0 0
\(232\) 2.56155i 0.168174i
\(233\) 2.00000 0.131024 0.0655122 0.997852i \(-0.479132\pi\)
0.0655122 + 0.997852i \(0.479132\pi\)
\(234\) 3.56155 0.561553i 0.232826 0.0367099i
\(235\) 3.50758 0.228809
\(236\) 12.2462i 0.797160i
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) 2.24621i 0.145295i 0.997358 + 0.0726477i \(0.0231449\pi\)
−0.997358 + 0.0726477i \(0.976855\pi\)
\(240\) 0.561553i 0.0362481i
\(241\) 6.00000i 0.386494i 0.981150 + 0.193247i \(0.0619019\pi\)
−0.981150 + 0.193247i \(0.938098\pi\)
\(242\) 8.93087i 0.574098i
\(243\) 1.00000 0.0641500
\(244\) 2.56155 0.163987
\(245\) 0 0
\(246\) 4.00000 0.255031
\(247\) −9.12311 + 1.43845i −0.580489 + 0.0915262i
\(248\) −10.2462 −0.650635
\(249\) 2.00000i 0.126745i
\(250\) −5.43845 −0.343958
\(251\) 15.0540 0.950198 0.475099 0.879932i \(-0.342412\pi\)
0.475099 + 0.879932i \(0.342412\pi\)
\(252\) 0 0
\(253\) 8.17708i 0.514089i
\(254\) 6.24621i 0.391922i
\(255\) 3.19224i 0.199906i
\(256\) 1.00000 0.0625000
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) 10.5616i 0.657534i
\(259\) 0 0
\(260\) −2.00000 + 0.315342i −0.124035 + 0.0195567i
\(261\) −2.56155 −0.158556
\(262\) 2.56155i 0.158253i
\(263\) 1.36932 0.0844357 0.0422178 0.999108i \(-0.486558\pi\)
0.0422178 + 0.999108i \(0.486558\pi\)
\(264\) 1.43845 0.0885303
\(265\) 7.36932i 0.452694i
\(266\) 0 0
\(267\) 8.00000i 0.489592i
\(268\) 7.12311i 0.435113i
\(269\) −6.63068 −0.404280 −0.202140 0.979357i \(-0.564790\pi\)
−0.202140 + 0.979357i \(0.564790\pi\)
\(270\) −0.561553 −0.0341750
\(271\) 8.00000i 0.485965i −0.970031 0.242983i \(-0.921874\pi\)
0.970031 0.242983i \(-0.0781258\pi\)
\(272\) −5.68466 −0.344683
\(273\) 0 0
\(274\) 5.68466 0.343423
\(275\) 6.73863i 0.406355i
\(276\) −5.68466 −0.342176
\(277\) −19.1231 −1.14900 −0.574498 0.818506i \(-0.694803\pi\)
−0.574498 + 0.818506i \(0.694803\pi\)
\(278\) 6.24621i 0.374623i
\(279\) 10.2462i 0.613425i
\(280\) 0 0
\(281\) 6.00000i 0.357930i −0.983855 0.178965i \(-0.942725\pi\)
0.983855 0.178965i \(-0.0572749\pi\)
\(282\) 6.24621 0.371956
\(283\) 13.1231 0.780088 0.390044 0.920796i \(-0.372460\pi\)
0.390044 + 0.920796i \(0.372460\pi\)
\(284\) 15.3693i 0.912001i
\(285\) 1.43845 0.0852063
\(286\) −0.807764 5.12311i −0.0477641 0.302936i
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) 15.3153 0.900902
\(290\) 1.43845 0.0844685
\(291\) 10.0000i 0.586210i
\(292\) 7.43845i 0.435302i
\(293\) 24.2462i 1.41648i −0.705972 0.708239i \(-0.749490\pi\)
0.705972 0.708239i \(-0.250510\pi\)
\(294\) 0 0
\(295\) −6.87689 −0.400388
\(296\) −1.68466 −0.0979188
\(297\) 1.43845i 0.0834672i
\(298\) −10.0000 −0.579284
\(299\) 3.19224 + 20.2462i 0.184612 + 1.17087i
\(300\) −4.68466 −0.270469
\(301\) 0 0
\(302\) −4.31534 −0.248320
\(303\) −6.00000 −0.344691
\(304\) 2.56155i 0.146915i
\(305\) 1.43845i 0.0823652i
\(306\) 5.68466i 0.324970i
\(307\) 9.75379i 0.556678i 0.960483 + 0.278339i \(0.0897839\pi\)
−0.960483 + 0.278339i \(0.910216\pi\)
\(308\) 0 0
\(309\) 18.8078 1.06994
\(310\) 5.75379i 0.326793i
\(311\) −32.4924 −1.84248 −0.921238 0.388999i \(-0.872821\pi\)
−0.921238 + 0.388999i \(0.872821\pi\)
\(312\) −3.56155 + 0.561553i −0.201633 + 0.0317917i
\(313\) −15.7538 −0.890457 −0.445228 0.895417i \(-0.646878\pi\)
−0.445228 + 0.895417i \(0.646878\pi\)
\(314\) 8.80776i 0.497051i
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) 21.3693i 1.20022i −0.799917 0.600110i \(-0.795123\pi\)
0.799917 0.600110i \(-0.204877\pi\)
\(318\) 13.1231i 0.735907i
\(319\) 3.68466i 0.206301i
\(320\) 0.561553i 0.0313918i
\(321\) −13.1231 −0.732460
\(322\) 0 0
\(323\) 14.5616i 0.810226i
\(324\) −1.00000 −0.0555556
\(325\) 2.63068 + 16.6847i 0.145924 + 0.925498i
\(326\) −0.876894 −0.0485667
\(327\) 15.9309i 0.880979i
\(328\) −4.00000 −0.220863
\(329\) 0 0
\(330\) 0.807764i 0.0444659i
\(331\) 7.12311i 0.391521i −0.980652 0.195761i \(-0.937282\pi\)
0.980652 0.195761i \(-0.0627176\pi\)
\(332\) 2.00000i 0.109764i
\(333\) 1.68466i 0.0923187i
\(334\) 8.80776 0.481939
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −17.0540 −0.928989 −0.464495 0.885576i \(-0.653764\pi\)
−0.464495 + 0.885576i \(0.653764\pi\)
\(338\) 4.00000 + 12.3693i 0.217571 + 0.672802i
\(339\) 3.75379 0.203878
\(340\) 3.19224i 0.173123i
\(341\) 14.7386 0.798142
\(342\) 2.56155 0.138513
\(343\) 0 0
\(344\) 10.5616i 0.569441i
\(345\) 3.19224i 0.171864i
\(346\) 20.2462i 1.08844i
\(347\) 8.49242 0.455897 0.227949 0.973673i \(-0.426798\pi\)
0.227949 + 0.973673i \(0.426798\pi\)
\(348\) 2.56155 0.137314
\(349\) 21.3693i 1.14387i −0.820298 0.571937i \(-0.806192\pi\)
0.820298 0.571937i \(-0.193808\pi\)
\(350\) 0 0
\(351\) 0.561553 + 3.56155i 0.0299735 + 0.190102i
\(352\) −1.43845 −0.0766695
\(353\) 1.75379i 0.0933448i −0.998910 0.0466724i \(-0.985138\pi\)
0.998910 0.0466724i \(-0.0148617\pi\)
\(354\) −12.2462 −0.650879
\(355\) 8.63068 0.458069
\(356\) 8.00000i 0.423999i
\(357\) 0 0
\(358\) 2.87689i 0.152049i
\(359\) 17.6155i 0.929712i 0.885386 + 0.464856i \(0.153894\pi\)
−0.885386 + 0.464856i \(0.846106\pi\)
\(360\) 0.561553 0.0295964
\(361\) 12.4384 0.654655
\(362\) 11.3693i 0.597558i
\(363\) 8.93087 0.468749
\(364\) 0 0
\(365\) 4.17708 0.218638
\(366\) 2.56155i 0.133895i
\(367\) −25.3693 −1.32427 −0.662134 0.749386i \(-0.730349\pi\)
−0.662134 + 0.749386i \(0.730349\pi\)
\(368\) 5.68466 0.296333
\(369\) 4.00000i 0.208232i
\(370\) 0.946025i 0.0491815i
\(371\) 0 0
\(372\) 10.2462i 0.531241i
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 8.17708 0.422827
\(375\) 5.43845i 0.280840i
\(376\) −6.24621 −0.322124
\(377\) −1.43845 9.12311i −0.0740838 0.469864i
\(378\) 0 0
\(379\) 27.1231i 1.39322i 0.717450 + 0.696610i \(0.245309\pi\)
−0.717450 + 0.696610i \(0.754691\pi\)
\(380\) −1.43845 −0.0737908
\(381\) −6.24621 −0.320003
\(382\) 13.0540i 0.667899i
\(383\) 13.4384i 0.686673i 0.939213 + 0.343336i \(0.111557\pi\)
−0.939213 + 0.343336i \(0.888443\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 0 0
\(386\) 12.0000 0.610784
\(387\) −10.5616 −0.536874
\(388\) 10.0000i 0.507673i
\(389\) 18.8769 0.957097 0.478548 0.878061i \(-0.341163\pi\)
0.478548 + 0.878061i \(0.341163\pi\)
\(390\) −0.315342 2.00000i −0.0159679 0.101274i
\(391\) −32.3153 −1.63426
\(392\) 0 0
\(393\) −2.56155 −0.129213
\(394\) 6.00000 0.302276
\(395\) 8.98485i 0.452077i
\(396\) 1.43845i 0.0722847i
\(397\) 9.36932i 0.470233i −0.971967 0.235116i \(-0.924453\pi\)
0.971967 0.235116i \(-0.0755471\pi\)
\(398\) 23.9309i 1.19955i
\(399\) 0 0
\(400\) 4.68466 0.234233
\(401\) 34.9848i 1.74706i −0.486771 0.873530i \(-0.661825\pi\)
0.486771 0.873530i \(-0.338175\pi\)
\(402\) −7.12311 −0.355268
\(403\) −36.4924 + 5.75379i −1.81782 + 0.286617i
\(404\) 6.00000 0.298511
\(405\) 0.561553i 0.0279038i
\(406\) 0 0
\(407\) 2.42329 0.120118
\(408\) 5.68466i 0.281433i
\(409\) 6.80776i 0.336622i −0.985734 0.168311i \(-0.946169\pi\)
0.985734 0.168311i \(-0.0538313\pi\)
\(410\) 2.24621i 0.110932i
\(411\) 5.68466i 0.280404i
\(412\) −18.8078 −0.926592
\(413\) 0 0
\(414\) 5.68466i 0.279386i
\(415\) −1.12311 −0.0551311
\(416\) 3.56155 0.561553i 0.174619 0.0275324i
\(417\) 6.24621 0.305878
\(418\) 3.68466i 0.180223i
\(419\) 8.31534 0.406231 0.203116 0.979155i \(-0.434893\pi\)
0.203116 + 0.979155i \(0.434893\pi\)
\(420\) 0 0
\(421\) 10.0000i 0.487370i 0.969854 + 0.243685i \(0.0783563\pi\)
−0.969854 + 0.243685i \(0.921644\pi\)
\(422\) 12.8078i 0.623472i
\(423\) 6.24621i 0.303701i
\(424\) 13.1231i 0.637314i
\(425\) −26.6307 −1.29178
\(426\) 15.3693 0.744646
\(427\) 0 0
\(428\) 13.1231 0.634329
\(429\) 5.12311 0.807764i 0.247346 0.0389992i
\(430\) 5.93087 0.286012
\(431\) 27.8617i 1.34205i 0.741433 + 0.671026i \(0.234146\pi\)
−0.741433 + 0.671026i \(0.765854\pi\)
\(432\) 1.00000 0.0481125
\(433\) 1.36932 0.0658052 0.0329026 0.999459i \(-0.489525\pi\)
0.0329026 + 0.999459i \(0.489525\pi\)
\(434\) 0 0
\(435\) 1.43845i 0.0689683i
\(436\) 15.9309i 0.762950i
\(437\) 14.5616i 0.696574i
\(438\) 7.43845 0.355423
\(439\) −19.9309 −0.951249 −0.475624 0.879649i \(-0.657778\pi\)
−0.475624 + 0.879649i \(0.657778\pi\)
\(440\) 0.807764i 0.0385086i
\(441\) 0 0
\(442\) −20.2462 + 3.19224i −0.963014 + 0.151839i
\(443\) 31.3693 1.49040 0.745201 0.666840i \(-0.232354\pi\)
0.745201 + 0.666840i \(0.232354\pi\)
\(444\) 1.68466i 0.0799504i
\(445\) −4.49242 −0.212961
\(446\) −18.2462 −0.863983
\(447\) 10.0000i 0.472984i
\(448\) 0 0
\(449\) 21.6847i 1.02336i 0.859175 + 0.511681i \(0.170977\pi\)
−0.859175 + 0.511681i \(0.829023\pi\)
\(450\) 4.68466i 0.220837i
\(451\) 5.75379 0.270935
\(452\) −3.75379 −0.176563
\(453\) 4.31534i 0.202752i
\(454\) 23.6155 1.10833
\(455\) 0 0
\(456\) −2.56155 −0.119956
\(457\) 16.0000i 0.748448i 0.927338 + 0.374224i \(0.122091\pi\)
−0.927338 + 0.374224i \(0.877909\pi\)
\(458\) 2.00000 0.0934539
\(459\) −5.68466 −0.265337
\(460\) 3.19224i 0.148839i
\(461\) 8.06913i 0.375817i 0.982187 + 0.187908i \(0.0601708\pi\)
−0.982187 + 0.187908i \(0.939829\pi\)
\(462\) 0 0
\(463\) 39.5464i 1.83788i −0.394401 0.918938i \(-0.629048\pi\)
0.394401 0.918938i \(-0.370952\pi\)
\(464\) −2.56155 −0.118917
\(465\) 5.75379 0.266826
\(466\) 2.00000i 0.0926482i
\(467\) 2.56155 0.118535 0.0592673 0.998242i \(-0.481124\pi\)
0.0592673 + 0.998242i \(0.481124\pi\)
\(468\) −0.561553 3.56155i −0.0259578 0.164633i
\(469\) 0 0
\(470\) 3.50758i 0.161792i
\(471\) −8.80776 −0.405840
\(472\) 12.2462 0.563678
\(473\) 15.1922i 0.698540i
\(474\) 16.0000i 0.734904i
\(475\) 12.0000i 0.550598i
\(476\) 0 0
\(477\) 13.1231 0.600866
\(478\) 2.24621 0.102739
\(479\) 41.3002i 1.88705i −0.331296 0.943527i \(-0.607486\pi\)
0.331296 0.943527i \(-0.392514\pi\)
\(480\) −0.561553 −0.0256313
\(481\) −6.00000 + 0.946025i −0.273576 + 0.0431350i
\(482\) 6.00000 0.273293
\(483\) 0 0
\(484\) −8.93087 −0.405949
\(485\) 5.61553 0.254988
\(486\) 1.00000i 0.0453609i
\(487\) 30.2462i 1.37059i 0.728267 + 0.685293i \(0.240326\pi\)
−0.728267 + 0.685293i \(0.759674\pi\)
\(488\) 2.56155i 0.115956i
\(489\) 0.876894i 0.0396545i
\(490\) 0 0
\(491\) 6.24621 0.281888 0.140944 0.990018i \(-0.454986\pi\)
0.140944 + 0.990018i \(0.454986\pi\)
\(492\) 4.00000i 0.180334i
\(493\) 14.5616 0.655819
\(494\) 1.43845 + 9.12311i 0.0647188 + 0.410468i
\(495\) −0.807764 −0.0363063
\(496\) 10.2462i 0.460068i
\(497\) 0 0
\(498\) −2.00000 −0.0896221
\(499\) 34.4924i 1.54409i 0.635566 + 0.772046i \(0.280767\pi\)
−0.635566 + 0.772046i \(0.719233\pi\)
\(500\) 5.43845i 0.243215i
\(501\) 8.80776i 0.393502i
\(502\) 15.0540i 0.671892i
\(503\) 5.61553 0.250384 0.125192 0.992133i \(-0.460045\pi\)
0.125192 + 0.992133i \(0.460045\pi\)
\(504\) 0 0
\(505\) 3.36932i 0.149933i
\(506\) −8.17708 −0.363516
\(507\) −12.3693 + 4.00000i −0.549341 + 0.177646i
\(508\) 6.24621 0.277131
\(509\) 9.68466i 0.429265i −0.976695 0.214632i \(-0.931145\pi\)
0.976695 0.214632i \(-0.0688554\pi\)
\(510\) 3.19224 0.141355
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) 2.56155i 0.113095i
\(514\) 26.0000i 1.14681i
\(515\) 10.5616i 0.465398i
\(516\) 10.5616 0.464946
\(517\) 8.98485 0.395153
\(518\) 0 0
\(519\) 20.2462 0.888710
\(520\) 0.315342 + 2.00000i 0.0138286 + 0.0877058i
\(521\) 29.0540 1.27288 0.636439 0.771327i \(-0.280407\pi\)
0.636439 + 0.771327i \(0.280407\pi\)
\(522\) 2.56155i 0.112116i
\(523\) −24.4924 −1.07098 −0.535489 0.844542i \(-0.679873\pi\)
−0.535489 + 0.844542i \(0.679873\pi\)
\(524\) 2.56155 0.111902
\(525\) 0 0
\(526\) 1.36932i 0.0597051i
\(527\) 58.2462i 2.53724i
\(528\) 1.43845i 0.0626004i
\(529\) 9.31534 0.405015
\(530\) −7.36932 −0.320103
\(531\) 12.2462i 0.531440i
\(532\) 0 0
\(533\) −14.2462 + 2.24621i −0.617072 + 0.0972942i
\(534\) −8.00000 −0.346194
\(535\) 7.36932i 0.318603i
\(536\) 7.12311 0.307671
\(537\) 2.87689 0.124147
\(538\) 6.63068i 0.285869i
\(539\) 0 0
\(540\) 0.561553i 0.0241654i
\(541\) 26.8078i 1.15256i 0.817254 + 0.576278i \(0.195495\pi\)
−0.817254 + 0.576278i \(0.804505\pi\)
\(542\) −8.00000 −0.343629
\(543\) −11.3693 −0.487904
\(544\) 5.68466i 0.243728i
\(545\) 8.94602 0.383206
\(546\) 0 0
\(547\) −36.9848 −1.58136 −0.790679 0.612231i \(-0.790272\pi\)
−0.790679 + 0.612231i \(0.790272\pi\)
\(548\) 5.68466i 0.242837i
\(549\) −2.56155 −0.109324
\(550\) −6.73863 −0.287336
\(551\) 6.56155i 0.279532i
\(552\) 5.68466i 0.241955i
\(553\) 0 0
\(554\) 19.1231i 0.812463i
\(555\) 0.946025 0.0401565
\(556\) −6.24621 −0.264898
\(557\) 0.246211i 0.0104323i 0.999986 + 0.00521615i \(0.00166036\pi\)
−0.999986 + 0.00521615i \(0.998340\pi\)
\(558\) 10.2462 0.433757
\(559\) −5.93087 37.6155i −0.250849 1.59097i
\(560\) 0 0
\(561\) 8.17708i 0.345237i
\(562\) −6.00000 −0.253095
\(563\) −0.946025 −0.0398702 −0.0199351 0.999801i \(-0.506346\pi\)
−0.0199351 + 0.999801i \(0.506346\pi\)
\(564\) 6.24621i 0.263013i
\(565\) 2.10795i 0.0886821i
\(566\) 13.1231i 0.551605i
\(567\) 0 0
\(568\) −15.3693 −0.644882
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 1.43845i 0.0602499i
\(571\) −1.75379 −0.0733938 −0.0366969 0.999326i \(-0.511684\pi\)
−0.0366969 + 0.999326i \(0.511684\pi\)
\(572\) −5.12311 + 0.807764i −0.214208 + 0.0337743i
\(573\) 13.0540 0.545338
\(574\) 0 0
\(575\) 26.6307 1.11058
\(576\) −1.00000 −0.0416667
\(577\) 24.2462i 1.00938i 0.863300 + 0.504691i \(0.168394\pi\)
−0.863300 + 0.504691i \(0.831606\pi\)
\(578\) 15.3153i 0.637034i
\(579\) 12.0000i 0.498703i
\(580\) 1.43845i 0.0597283i
\(581\) 0 0
\(582\) 10.0000 0.414513
\(583\) 18.8769i 0.781801i
\(584\) −7.43845 −0.307805
\(585\) 2.00000 0.315342i 0.0826898 0.0130378i
\(586\) −24.2462 −1.00160
\(587\) 28.8769i 1.19188i −0.803030 0.595938i \(-0.796780\pi\)
0.803030 0.595938i \(-0.203220\pi\)
\(588\) 0 0
\(589\) −26.2462 −1.08146
\(590\) 6.87689i 0.283117i
\(591\) 6.00000i 0.246807i
\(592\) 1.68466i 0.0692390i
\(593\) 28.0000i 1.14982i −0.818216 0.574911i \(-0.805037\pi\)
0.818216 0.574911i \(-0.194963\pi\)
\(594\) −1.43845 −0.0590202
\(595\) 0 0
\(596\) 10.0000i 0.409616i
\(597\) 23.9309 0.979425
\(598\) 20.2462 3.19224i 0.827929 0.130540i
\(599\) −23.3002 −0.952020 −0.476010 0.879440i \(-0.657917\pi\)
−0.476010 + 0.879440i \(0.657917\pi\)
\(600\) 4.68466i 0.191250i
\(601\) −25.8617 −1.05492 −0.527461 0.849579i \(-0.676856\pi\)
−0.527461 + 0.849579i \(0.676856\pi\)
\(602\) 0 0
\(603\) 7.12311i 0.290075i
\(604\) 4.31534i 0.175589i
\(605\) 5.01515i 0.203895i
\(606\) 6.00000i 0.243733i
\(607\) −41.5464 −1.68632 −0.843158 0.537666i \(-0.819306\pi\)
−0.843158 + 0.537666i \(0.819306\pi\)
\(608\) 2.56155 0.103885
\(609\) 0 0
\(610\) 1.43845 0.0582410
\(611\) −22.2462 + 3.50758i −0.899985 + 0.141901i
\(612\) 5.68466 0.229789
\(613\) 18.8078i 0.759638i −0.925061 0.379819i \(-0.875986\pi\)
0.925061 0.379819i \(-0.124014\pi\)
\(614\) 9.75379 0.393631
\(615\) 2.24621 0.0905760
\(616\) 0 0
\(617\) 26.8078i 1.07924i −0.841909 0.539620i \(-0.818568\pi\)
0.841909 0.539620i \(-0.181432\pi\)
\(618\) 18.8078i 0.756559i
\(619\) 2.06913i 0.0831654i −0.999135 0.0415827i \(-0.986760\pi\)
0.999135 0.0415827i \(-0.0132400\pi\)
\(620\) −5.75379 −0.231078
\(621\) 5.68466 0.228117
\(622\) 32.4924i 1.30283i
\(623\) 0 0
\(624\) 0.561553 + 3.56155i 0.0224801 + 0.142576i
\(625\) 20.3693 0.814773
\(626\) 15.7538i 0.629648i
\(627\) 3.68466 0.147151
\(628\) 8.80776 0.351468
\(629\) 9.57671i 0.381848i
\(630\) 0 0
\(631\) 0.315342i 0.0125535i −0.999980 0.00627677i \(-0.998002\pi\)
0.999980 0.00627677i \(-0.00199797\pi\)
\(632\) 16.0000i 0.636446i
\(633\) −12.8078 −0.509063
\(634\) −21.3693 −0.848684
\(635\) 3.50758i 0.139194i
\(636\) −13.1231 −0.520365
\(637\) 0 0
\(638\) 3.68466 0.145877
\(639\) 15.3693i 0.608001i
\(640\) 0.561553 0.0221973
\(641\) −44.1080 −1.74216 −0.871080 0.491142i \(-0.836580\pi\)
−0.871080 + 0.491142i \(0.836580\pi\)
\(642\) 13.1231i 0.517928i
\(643\) 20.8078i 0.820578i −0.911956 0.410289i \(-0.865428\pi\)
0.911956 0.410289i \(-0.134572\pi\)
\(644\) 0 0
\(645\) 5.93087i 0.233528i
\(646\) −14.5616 −0.572917
\(647\) −35.3693 −1.39051 −0.695256 0.718763i \(-0.744709\pi\)
−0.695256 + 0.718763i \(0.744709\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −17.6155 −0.691470
\(650\) 16.6847 2.63068i 0.654426 0.103184i
\(651\) 0 0
\(652\) 0.876894i 0.0343418i
\(653\) −25.4384 −0.995483 −0.497742 0.867325i \(-0.665837\pi\)
−0.497742 + 0.867325i \(0.665837\pi\)
\(654\) 15.9309 0.622946
\(655\) 1.43845i 0.0562048i
\(656\) 4.00000i 0.156174i
\(657\) 7.43845i 0.290201i
\(658\) 0 0
\(659\) 21.1231 0.822839 0.411420 0.911446i \(-0.365033\pi\)
0.411420 + 0.911446i \(0.365033\pi\)
\(660\) 0.807764 0.0314422
\(661\) 4.73863i 0.184311i 0.995745 + 0.0921557i \(0.0293758\pi\)
−0.995745 + 0.0921557i \(0.970624\pi\)
\(662\) −7.12311 −0.276847
\(663\) −3.19224 20.2462i −0.123976 0.786298i
\(664\) 2.00000 0.0776151
\(665\) 0 0
\(666\) 1.68466 0.0652792
\(667\) −14.5616 −0.563826
\(668\) 8.80776i 0.340783i
\(669\) 18.2462i 0.705439i
\(670\) 4.00000i 0.154533i
\(671\) 3.68466i 0.142245i
\(672\) 0 0
\(673\) 17.1922 0.662712 0.331356 0.943506i \(-0.392494\pi\)
0.331356 + 0.943506i \(0.392494\pi\)
\(674\) 17.0540i 0.656895i
\(675\) 4.68466 0.180313
\(676\) 12.3693 4.00000i 0.475743 0.153846i
\(677\) −6.63068 −0.254838 −0.127419 0.991849i \(-0.540669\pi\)
−0.127419 + 0.991849i \(0.540669\pi\)
\(678\) 3.75379i 0.144163i
\(679\) 0 0
\(680\) −3.19224 −0.122417
\(681\) 23.6155i 0.904949i
\(682\) 14.7386i 0.564371i
\(683\) 11.6847i 0.447101i 0.974692 + 0.223551i \(0.0717648\pi\)
−0.974692 + 0.223551i \(0.928235\pi\)
\(684\) 2.56155i 0.0979434i
\(685\) 3.19224 0.121969
\(686\) 0 0
\(687\) 2.00000i 0.0763048i
\(688\) −10.5616 −0.402655
\(689\) 7.36932 + 46.7386i 0.280749 + 1.78060i
\(690\) −3.19224 −0.121526
\(691\) 3.50758i 0.133435i 0.997772 + 0.0667173i \(0.0212526\pi\)
−0.997772 + 0.0667173i \(0.978747\pi\)
\(692\) −20.2462 −0.769645
\(693\) 0 0
\(694\) 8.49242i 0.322368i
\(695\) 3.50758i 0.133050i
\(696\) 2.56155i 0.0970954i
\(697\) 22.7386i 0.861287i
\(698\) −21.3693 −0.808841
\(699\) 2.00000 0.0756469
\(700\) 0 0
\(701\) 37.6155 1.42072 0.710359 0.703839i \(-0.248532\pi\)
0.710359 + 0.703839i \(0.248532\pi\)
\(702\) 3.56155 0.561553i 0.134422 0.0211944i
\(703\) −4.31534 −0.162756
\(704\) 1.43845i 0.0542135i
\(705\) 3.50758 0.132103
\(706\) −1.75379 −0.0660047
\(707\) 0 0
\(708\) 12.2462i 0.460241i
\(709\) 48.2462i 1.81192i −0.423358 0.905962i \(-0.639149\pi\)
0.423358 0.905962i \(-0.360851\pi\)
\(710\) 8.63068i 0.323904i
\(711\) 16.0000 0.600047
\(712\) 8.00000 0.299813
\(713\) 58.2462i 2.18134i
\(714\) 0 0
\(715\) −0.453602 2.87689i −0.0169638 0.107590i
\(716\) −2.87689 −0.107515
\(717\) 2.24621i 0.0838863i
\(718\) 17.6155 0.657406
\(719\) −30.2462 −1.12799 −0.563997 0.825777i \(-0.690737\pi\)
−0.563997 + 0.825777i \(0.690737\pi\)
\(720\) 0.561553i 0.0209278i
\(721\) 0 0
\(722\) 12.4384i 0.462911i
\(723\) 6.00000i 0.223142i
\(724\) 11.3693 0.422537
\(725\) −12.0000 −0.445669
\(726\) 8.93087i 0.331456i
\(727\) 20.0691 0.744323 0.372161 0.928168i \(-0.378617\pi\)
0.372161 + 0.928168i \(0.378617\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 4.17708i 0.154601i
\(731\) 60.0388 2.22062
\(732\) 2.56155 0.0946777
\(733\) 31.1231i 1.14956i −0.818309 0.574779i \(-0.805088\pi\)
0.818309 0.574779i \(-0.194912\pi\)
\(734\) 25.3693i 0.936399i
\(735\) 0 0
\(736\) 5.68466i 0.209539i
\(737\) −10.2462 −0.377424
\(738\) 4.00000 0.147242
\(739\) 23.6155i 0.868711i −0.900741 0.434356i \(-0.856976\pi\)
0.900741 0.434356i \(-0.143024\pi\)
\(740\) −0.946025 −0.0347766
\(741\) −9.12311 + 1.43845i −0.335146 + 0.0528427i
\(742\) 0 0
\(743\) 16.0000i 0.586983i 0.955962 + 0.293492i \(0.0948173\pi\)
−0.955962 + 0.293492i \(0.905183\pi\)
\(744\) −10.2462 −0.375644
\(745\) −5.61553 −0.205737
\(746\) 6.00000i 0.219676i
\(747\) 2.00000i 0.0731762i
\(748\) 8.17708i 0.298984i
\(749\) 0 0
\(750\) −5.43845 −0.198584
\(751\) −14.2462 −0.519852 −0.259926 0.965629i \(-0.583698\pi\)
−0.259926 + 0.965629i \(0.583698\pi\)
\(752\) 6.24621i 0.227776i
\(753\) 15.0540 0.548597
\(754\) −9.12311 + 1.43845i −0.332244 + 0.0523852i
\(755\) −2.42329 −0.0881926
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 27.1231 0.985156
\(759\) 8.17708i 0.296809i
\(760\) 1.43845i 0.0521780i
\(761\) 30.8769i 1.11929i −0.828734 0.559643i \(-0.810938\pi\)
0.828734 0.559643i \(-0.189062\pi\)
\(762\) 6.24621i 0.226276i
\(763\) 0 0
\(764\) −13.0540 −0.472276
\(765\) 3.19224i 0.115416i
\(766\) 13.4384 0.485551
\(767\) 43.6155 6.87689i 1.57487 0.248310i
\(768\) 1.00000 0.0360844
\(769\) 12.5616i 0.452981i −0.974013 0.226491i \(-0.927275\pi\)
0.974013 0.226491i \(-0.0727253\pi\)
\(770\) 0 0
\(771\) 26.0000 0.936367
\(772\) 12.0000i 0.431889i
\(773\) 15.9309i 0.572994i −0.958081 0.286497i \(-0.907509\pi\)
0.958081 0.286497i \(-0.0924908\pi\)
\(774\) 10.5616i 0.379627i
\(775\) 48.0000i 1.72421i
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) 18.8769i 0.676769i
\(779\) −10.2462 −0.367109
\(780\) −2.00000 + 0.315342i −0.0716115 + 0.0112910i
\(781\) 22.1080 0.791085
\(782\) 32.3153i 1.15559i
\(783\) −2.56155 −0.0915424
\(784\) 0 0
\(785\) 4.94602i 0.176531i
\(786\) 2.56155i 0.0913676i
\(787\) 45.9309i 1.63726i 0.574322 + 0.818629i \(0.305266\pi\)
−0.574322 + 0.818629i \(0.694734\pi\)
\(788\) 6.00000i 0.213741i
\(789\) 1.36932 0.0487490
\(790\) −8.98485 −0.319666
\(791\) 0 0
\(792\) 1.43845 0.0511130
\(793\) −1.43845 9.12311i −0.0510808 0.323971i
\(794\) −9.36932 −0.332505
\(795\) 7.36932i 0.261363i
\(796\) −23.9309 −0.848207
\(797\) −20.2462 −0.717158 −0.358579 0.933499i \(-0.616739\pi\)
−0.358579 + 0.933499i \(0.616739\pi\)
\(798\) 0 0
\(799\) 35.5076i 1.25617i
\(800\) 4.68466i 0.165628i
\(801\) 8.00000i 0.282666i
\(802\) −34.9848 −1.23536
\(803\) 10.6998 0.377588
\(804\) 7.12311i 0.251213i
\(805\) 0 0
\(806\) 5.75379 + 36.4924i 0.202669 + 1.28539i
\(807\) −6.63068 −0.233411
\(808\) 6.00000i 0.211079i
\(809\) −30.6307 −1.07692 −0.538459 0.842652i \(-0.680993\pi\)
−0.538459 + 0.842652i \(0.680993\pi\)
\(810\) −0.561553 −0.0197310
\(811\) 35.0540i 1.23091i 0.788171 + 0.615456i \(0.211028\pi\)
−0.788171 + 0.615456i \(0.788972\pi\)
\(812\) 0 0
\(813\) 8.00000i 0.280572i
\(814\) 2.42329i 0.0849363i
\(815\) −0.492423 −0.0172488
\(816\) −5.68466 −0.199003
\(817\) 27.0540i 0.946499i
\(818\) −6.80776 −0.238028
\(819\) 0 0
\(820\) −2.24621 −0.0784411
\(821\) 39.1231i 1.36541i 0.730696 + 0.682703i \(0.239196\pi\)
−0.730696 + 0.682703i \(0.760804\pi\)
\(822\) 5.68466 0.198275
\(823\) 18.7386 0.653188 0.326594 0.945165i \(-0.394099\pi\)
0.326594 + 0.945165i \(0.394099\pi\)
\(824\) 18.8078i 0.655200i
\(825\) 6.73863i 0.234609i
\(826\) 0 0
\(827\) 15.0540i 0.523478i 0.965139 + 0.261739i \(0.0842960\pi\)
−0.965139 + 0.261739i \(0.915704\pi\)
\(828\) −5.68466 −0.197556
\(829\) −4.94602 −0.171783 −0.0858913 0.996305i \(-0.527374\pi\)
−0.0858913 + 0.996305i \(0.527374\pi\)
\(830\) 1.12311i 0.0389836i
\(831\) −19.1231 −0.663373
\(832\) −0.561553 3.56155i −0.0194683 0.123475i
\(833\) 0 0
\(834\) 6.24621i 0.216289i
\(835\) 4.94602 0.171164
\(836\) −3.68466 −0.127437
\(837\) 10.2462i 0.354161i
\(838\) 8.31534i 0.287249i
\(839\) 42.2462i 1.45850i 0.684247 + 0.729251i \(0.260131\pi\)
−0.684247 + 0.729251i \(0.739869\pi\)
\(840\) 0 0
\(841\) −22.4384 −0.773740
\(842\) 10.0000 0.344623
\(843\) 6.00000i 0.206651i
\(844\) 12.8078 0.440861
\(845\) 2.24621 + 6.94602i 0.0772720 + 0.238951i
\(846\) 6.24621 0.214749
\(847\) 0 0
\(848\) 13.1231 0.450649
\(849\) 13.1231 0.450384
\(850\) 26.6307i 0.913425i
\(851\) 9.57671i 0.328285i
\(852\) 15.3693i 0.526544i
\(853\) 19.1231i 0.654763i 0.944892 + 0.327381i \(0.106166\pi\)
−0.944892 + 0.327381i \(0.893834\pi\)
\(854\) 0 0
\(855\) 1.43845 0.0491939
\(856\) 13.1231i 0.448539i
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) −0.807764 5.12311i −0.0275766 0.174900i
\(859\) −41.6155 −1.41990 −0.709952 0.704250i \(-0.751283\pi\)
−0.709952 + 0.704250i \(0.751283\pi\)
\(860\) 5.93087i 0.202241i
\(861\) 0 0
\(862\) 27.8617 0.948975
\(863\) 6.73863i 0.229386i −0.993401 0.114693i \(-0.963412\pi\)
0.993401 0.114693i \(-0.0365884\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) 11.3693i 0.386568i
\(866\) 1.36932i 0.0465313i
\(867\) 15.3153 0.520136
\(868\) 0 0
\(869\) 23.0152i 0.780736i
\(870\) 1.43845 0.0487679
\(871\) 25.3693 4.00000i 0.859607 0.135535i
\(872\) −15.9309 −0.539487
\(873\) 10.0000i 0.338449i
\(874\) 14.5616 0.492552
\(875\) 0 0
\(876\) 7.43845i 0.251322i
\(877\) 42.9848i 1.45150i −0.687961 0.725748i \(-0.741494\pi\)
0.687961 0.725748i \(-0.258506\pi\)
\(878\) 19.9309i 0.672634i
\(879\) 24.2462i 0.817804i
\(880\) −0.807764 −0.0272297
\(881\) 39.3002 1.32406 0.662028 0.749479i \(-0.269696\pi\)
0.662028 + 0.749479i \(0.269696\pi\)
\(882\) 0 0
\(883\) −18.5616 −0.624646 −0.312323 0.949976i \(-0.601107\pi\)
−0.312323 + 0.949976i \(0.601107\pi\)
\(884\) 3.19224 + 20.2462i 0.107367 + 0.680954i
\(885\) −6.87689 −0.231164
\(886\) 31.3693i 1.05387i
\(887\) −19.8617 −0.666892 −0.333446 0.942769i \(-0.608211\pi\)
−0.333446 + 0.942769i \(0.608211\pi\)
\(888\) −1.68466 −0.0565334
\(889\) 0 0
\(890\) 4.49242i 0.150586i
\(891\) 1.43845i 0.0481898i
\(892\) 18.2462i 0.610928i
\(893\) −16.0000 −0.535420
\(894\) −10.0000 −0.334450
\(895\) 1.61553i 0.0540011i
\(896\) 0 0
\(897\) 3.19224 + 20.2462i 0.106586 + 0.676001i
\(898\) 21.6847 0.723626
\(899\) 26.2462i 0.875360i
\(900\) −4.68466 −0.156155
\(901\) −74.6004 −2.48530
\(902\) 5.75379i 0.191580i
\(903\) 0 0
\(904\) 3.75379i 0.124849i
\(905\) 6.38447i 0.212227i
\(906\) −4.31534 −0.143368
\(907\) 8.49242 0.281986 0.140993 0.990011i \(-0.454970\pi\)
0.140993 + 0.990011i \(0.454970\pi\)
\(908\) 23.6155i 0.783709i
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −8.56155 −0.283657 −0.141828 0.989891i \(-0.545298\pi\)
−0.141828 + 0.989891i \(0.545298\pi\)
\(912\) 2.56155i 0.0848215i
\(913\) −2.87689 −0.0952113
\(914\) 16.0000 0.529233
\(915\) 1.43845i 0.0475536i
\(916\) 2.00000i 0.0660819i
\(917\) 0 0
\(918\) 5.68466i 0.187622i
\(919\) 39.8617 1.31492 0.657459 0.753491i \(-0.271631\pi\)
0.657459 + 0.753491i \(0.271631\pi\)
\(920\) 3.19224 0.105245
\(921\) 9.75379i 0.321398i
\(922\) 8.06913 0.265743
\(923\) −54.7386 + 8.63068i −1.80174 + 0.284082i
\(924\) 0 0
\(925\) 7.89205i 0.259489i
\(926\) −39.5464 −1.29958
\(927\) 18.8078 0.617728
\(928\) 2.56155i 0.0840871i
\(929\) 49.1231i 1.61168i −0.592136 0.805838i \(-0.701715\pi\)
0.592136 0.805838i \(-0.298285\pi\)
\(930\) 5.75379i 0.188674i
\(931\) 0 0
\(932\) −2.00000 −0.0655122
\(933\) −32.4924 −1.06375
\(934\) 2.56155i 0.0838166i
\(935\) 4.59186 0.150170
\(936\) −3.56155 + 0.561553i −0.116413 + 0.0183549i
\(937\) −37.8617 −1.23689 −0.618445 0.785828i \(-0.712237\pi\)
−0.618445 + 0.785828i \(0.712237\pi\)
\(938\) 0 0
\(939\) −15.7538 −0.514105
\(940\) −3.50758 −0.114405
\(941\) 6.49242i 0.211647i 0.994385 + 0.105823i \(0.0337478\pi\)
−0.994385 + 0.105823i \(0.966252\pi\)
\(942\) 8.80776i 0.286972i
\(943\) 22.7386i 0.740472i
\(944\) 12.2462i 0.398580i
\(945\) 0 0
\(946\) 15.1922 0.493942
\(947\) 7.19224i 0.233716i −0.993149 0.116858i \(-0.962718\pi\)
0.993149 0.116858i \(-0.0372823\pi\)
\(948\) −16.0000 −0.519656
\(949\) −26.4924 + 4.17708i −0.859980 + 0.135594i
\(950\) 12.0000 0.389331
\(951\) 21.3693i 0.692948i
\(952\) 0 0
\(953\) −48.7386 −1.57880 −0.789400 0.613880i \(-0.789608\pi\)
−0.789400 + 0.613880i \(0.789608\pi\)
\(954\) 13.1231i 0.424876i
\(955\) 7.33050i 0.237209i
\(956\) 2.24621i 0.0726477i
\(957\) 3.68466i 0.119108i
\(958\) −41.3002 −1.33435
\(959\) 0 0
\(960\) 0.561553i 0.0181240i
\(961\) −73.9848 −2.38661
\(962\) 0.946025 + 6.00000i 0.0305011 + 0.193448i
\(963\) −13.1231 −0.422886
\(964\) 6.00000i 0.193247i
\(965\) 6.73863 0.216924
\(966\) 0 0
\(967\) 25.9309i 0.833881i 0.908934 + 0.416940i \(0.136898\pi\)
−0.908934 + 0.416940i \(0.863102\pi\)
\(968\) 8.93087i 0.287049i
\(969\) 14.5616i 0.467784i
\(970\) 5.61553i 0.180304i
\(971\) −4.00000 −0.128366 −0.0641831 0.997938i \(-0.520444\pi\)
−0.0641831 + 0.997938i \(0.520444\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 30.2462 0.969151
\(975\) 2.63068 + 16.6847i 0.0842493 + 0.534337i
\(976\) −2.56155 −0.0819933
\(977\) 39.7926i 1.27308i −0.771244 0.636539i \(-0.780365\pi\)
0.771244 0.636539i \(-0.219635\pi\)
\(978\) −0.876894 −0.0280400
\(979\) −11.5076 −0.367784
\(980\) 0 0
\(981\) 15.9309i 0.508634i
\(982\) 6.24621i 0.199325i
\(983\) 6.56155i 0.209281i −0.994510 0.104641i \(-0.966631\pi\)
0.994510 0.104641i \(-0.0333692\pi\)
\(984\) −4.00000 −0.127515
\(985\) 3.36932 0.107355
\(986\) 14.5616i 0.463734i
\(987\) 0 0
\(988\) 9.12311 1.43845i 0.290245 0.0457631i
\(989\) −60.0388 −1.90912
\(990\) 0.807764i 0.0256724i
\(991\) −13.7538 −0.436903 −0.218452 0.975848i \(-0.570101\pi\)
−0.218452 + 0.975848i \(0.570101\pi\)
\(992\) 10.2462 0.325318
\(993\) 7.12311i 0.226045i
\(994\) 0 0
\(995\) 13.4384i 0.426027i
\(996\) 2.00000i 0.0633724i
\(997\) 59.3693 1.88025 0.940123 0.340837i \(-0.110710\pi\)
0.940123 + 0.340837i \(0.110710\pi\)
\(998\) 34.4924 1.09184
\(999\) 1.68466i 0.0533002i
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 3822.2.c.h.883.1 4
7.6 odd 2 546.2.c.e.337.2 4
13.12 even 2 inner 3822.2.c.h.883.4 4
21.20 even 2 1638.2.c.h.883.3 4
28.27 even 2 4368.2.h.n.337.3 4
91.34 even 4 7098.2.a.bv.1.1 2
91.83 even 4 7098.2.a.bg.1.2 2
91.90 odd 2 546.2.c.e.337.3 yes 4
273.272 even 2 1638.2.c.h.883.2 4
364.363 even 2 4368.2.h.n.337.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.c.e.337.2 4 7.6 odd 2
546.2.c.e.337.3 yes 4 91.90 odd 2
1638.2.c.h.883.2 4 273.272 even 2
1638.2.c.h.883.3 4 21.20 even 2
3822.2.c.h.883.1 4 1.1 even 1 trivial
3822.2.c.h.883.4 4 13.12 even 2 inner
4368.2.h.n.337.2 4 364.363 even 2
4368.2.h.n.337.3 4 28.27 even 2
7098.2.a.bg.1.2 2 91.83 even 4
7098.2.a.bv.1.1 2 91.34 even 4