Properties

Label 3380.2.f.c.3041.2
Level $3380$
Weight $2$
Character 3380.3041
Analytic conductor $26.989$
Analytic rank $1$
Dimension $2$
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] = [3380,2,Mod(3041,3380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3380, 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("3380.3041");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.f (of order \(2\), degree \(1\), not 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: \(26.9894358832\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3041.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3380.3041
Dual form 3380.2.f.c.3041.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000i q^{5} +5.00000i q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000i q^{5} +5.00000i q^{7} -2.00000 q^{9} -5.00000i q^{11} -1.00000i q^{15} +1.00000 q^{17} +3.00000i q^{19} -5.00000i q^{21} -3.00000 q^{23} -1.00000 q^{25} +5.00000 q^{27} -1.00000 q^{29} +5.00000i q^{33} -5.00000 q^{35} +7.00000i q^{37} +5.00000i q^{41} -5.00000 q^{43} -2.00000i q^{45} +12.0000i q^{47} -18.0000 q^{49} -1.00000 q^{51} +2.00000 q^{53} +5.00000 q^{55} -3.00000i q^{57} -11.0000i q^{59} -13.0000 q^{61} -10.0000i q^{63} -3.00000i q^{67} +3.00000 q^{69} -13.0000i q^{71} -2.00000i q^{73} +1.00000 q^{75} +25.0000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} +1.00000i q^{85} +1.00000 q^{87} +7.00000i q^{89} -3.00000 q^{95} -11.0000i q^{97} +10.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 4 q^{9} + 2 q^{17} - 6 q^{23} - 2 q^{25} + 10 q^{27} - 2 q^{29} - 10 q^{35} - 10 q^{43} - 36 q^{49} - 2 q^{51} + 4 q^{53} + 10 q^{55} - 26 q^{61} + 6 q^{69} + 2 q^{75} + 50 q^{77} - 8 q^{79} + 2 q^{81} + 2 q^{87} - 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1691\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 5.00000i 1.88982i 0.327327 + 0.944911i \(0.393852\pi\)
−0.327327 + 0.944911i \(0.606148\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) − 5.00000i − 1.50756i −0.657129 0.753778i \(-0.728229\pi\)
0.657129 0.753778i \(-0.271771\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) − 1.00000i − 0.258199i
\(16\) 0 0
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) 3.00000i 0.688247i 0.938924 + 0.344124i \(0.111824\pi\)
−0.938924 + 0.344124i \(0.888176\pi\)
\(20\) 0 0
\(21\) − 5.00000i − 1.09109i
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 5.00000i 0.870388i
\(34\) 0 0
\(35\) −5.00000 −0.845154
\(36\) 0 0
\(37\) 7.00000i 1.15079i 0.817875 + 0.575396i \(0.195152\pi\)
−0.817875 + 0.575396i \(0.804848\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.00000i 0.780869i 0.920631 + 0.390434i \(0.127675\pi\)
−0.920631 + 0.390434i \(0.872325\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) − 2.00000i − 0.298142i
\(46\) 0 0
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 0 0
\(49\) −18.0000 −2.57143
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) − 3.00000i − 0.397360i
\(58\) 0 0
\(59\) − 11.0000i − 1.43208i −0.698060 0.716039i \(-0.745953\pi\)
0.698060 0.716039i \(-0.254047\pi\)
\(60\) 0 0
\(61\) −13.0000 −1.66448 −0.832240 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) − 10.0000i − 1.25988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.00000i − 0.366508i −0.983066 0.183254i \(-0.941337\pi\)
0.983066 0.183254i \(-0.0586631\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) − 13.0000i − 1.54282i −0.636341 0.771408i \(-0.719553\pi\)
0.636341 0.771408i \(-0.280447\pi\)
\(72\) 0 0
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 25.0000 2.84901
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) 1.00000i 0.108465i
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) 7.00000i 0.741999i 0.928633 + 0.370999i \(0.120985\pi\)
−0.928633 + 0.370999i \(0.879015\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.00000 −0.307794
\(96\) 0 0
\(97\) − 11.0000i − 1.11688i −0.829545 0.558440i \(-0.811400\pi\)
0.829545 0.558440i \(-0.188600\pi\)
\(98\) 0 0
\(99\) 10.0000i 1.00504i
\(100\) 0 0
\(101\) 13.0000 1.29355 0.646774 0.762682i \(-0.276118\pi\)
0.646774 + 0.762682i \(0.276118\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 5.00000 0.487950
\(106\) 0 0
\(107\) −9.00000 −0.870063 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) 0 0
\(109\) − 18.0000i − 1.72409i −0.506834 0.862044i \(-0.669184\pi\)
0.506834 0.862044i \(-0.330816\pi\)
\(110\) 0 0
\(111\) − 7.00000i − 0.664411i
\(112\) 0 0
\(113\) −1.00000 −0.0940721 −0.0470360 0.998893i \(-0.514978\pi\)
−0.0470360 + 0.998893i \(0.514978\pi\)
\(114\) 0 0
\(115\) − 3.00000i − 0.279751i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.00000i 0.458349i
\(120\) 0 0
\(121\) −14.0000 −1.27273
\(122\) 0 0
\(123\) − 5.00000i − 0.450835i
\(124\) 0 0
\(125\) − 1.00000i − 0.0894427i
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 0 0
\(129\) 5.00000 0.440225
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) −15.0000 −1.30066
\(134\) 0 0
\(135\) 5.00000i 0.430331i
\(136\) 0 0
\(137\) 3.00000i 0.256307i 0.991754 + 0.128154i \(0.0409051\pi\)
−0.991754 + 0.128154i \(0.959095\pi\)
\(138\) 0 0
\(139\) 13.0000 1.10265 0.551323 0.834292i \(-0.314123\pi\)
0.551323 + 0.834292i \(0.314123\pi\)
\(140\) 0 0
\(141\) − 12.0000i − 1.01058i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) − 1.00000i − 0.0830455i
\(146\) 0 0
\(147\) 18.0000 1.48461
\(148\) 0 0
\(149\) − 11.0000i − 0.901155i −0.892737 0.450578i \(-0.851218\pi\)
0.892737 0.450578i \(-0.148782\pi\)
\(150\) 0 0
\(151\) − 24.0000i − 1.95309i −0.215308 0.976546i \(-0.569076\pi\)
0.215308 0.976546i \(-0.430924\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) − 15.0000i − 1.18217i
\(162\) 0 0
\(163\) 5.00000i 0.391630i 0.980641 + 0.195815i \(0.0627352\pi\)
−0.980641 + 0.195815i \(0.937265\pi\)
\(164\) 0 0
\(165\) −5.00000 −0.389249
\(166\) 0 0
\(167\) − 13.0000i − 1.00597i −0.864295 0.502985i \(-0.832235\pi\)
0.864295 0.502985i \(-0.167765\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) − 6.00000i − 0.458831i
\(172\) 0 0
\(173\) 17.0000 1.29249 0.646243 0.763132i \(-0.276339\pi\)
0.646243 + 0.763132i \(0.276339\pi\)
\(174\) 0 0
\(175\) − 5.00000i − 0.377964i
\(176\) 0 0
\(177\) 11.0000i 0.826811i
\(178\) 0 0
\(179\) −11.0000 −0.822179 −0.411089 0.911595i \(-0.634852\pi\)
−0.411089 + 0.911595i \(0.634852\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 13.0000 0.960988
\(184\) 0 0
\(185\) −7.00000 −0.514650
\(186\) 0 0
\(187\) − 5.00000i − 0.365636i
\(188\) 0 0
\(189\) 25.0000i 1.81848i
\(190\) 0 0
\(191\) −15.0000 −1.08536 −0.542681 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) 0 0
\(193\) 23.0000i 1.65558i 0.561041 + 0.827788i \(0.310401\pi\)
−0.561041 + 0.827788i \(0.689599\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 27.0000i − 1.92367i −0.273629 0.961835i \(-0.588224\pi\)
0.273629 0.961835i \(-0.411776\pi\)
\(198\) 0 0
\(199\) −21.0000 −1.48865 −0.744325 0.667817i \(-0.767229\pi\)
−0.744325 + 0.667817i \(0.767229\pi\)
\(200\) 0 0
\(201\) 3.00000i 0.211604i
\(202\) 0 0
\(203\) − 5.00000i − 0.350931i
\(204\) 0 0
\(205\) −5.00000 −0.349215
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 15.0000 1.03757
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 0 0
\(213\) 13.0000i 0.890745i
\(214\) 0 0
\(215\) − 5.00000i − 0.340997i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2.00000i 0.135147i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 19.0000i − 1.27233i −0.771551 0.636167i \(-0.780519\pi\)
0.771551 0.636167i \(-0.219481\pi\)
\(224\) 0 0
\(225\) 2.00000 0.133333
\(226\) 0 0
\(227\) − 17.0000i − 1.12833i −0.825662 0.564165i \(-0.809198\pi\)
0.825662 0.564165i \(-0.190802\pi\)
\(228\) 0 0
\(229\) 10.0000i 0.660819i 0.943838 + 0.330409i \(0.107187\pi\)
−0.943838 + 0.330409i \(0.892813\pi\)
\(230\) 0 0
\(231\) −25.0000 −1.64488
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) 8.00000i 0.517477i 0.965947 + 0.258738i \(0.0833068\pi\)
−0.965947 + 0.258738i \(0.916693\pi\)
\(240\) 0 0
\(241\) 11.0000i 0.708572i 0.935137 + 0.354286i \(0.115276\pi\)
−0.935137 + 0.354286i \(0.884724\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) − 18.0000i − 1.14998i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 12.0000i 0.760469i
\(250\) 0 0
\(251\) 15.0000 0.946792 0.473396 0.880850i \(-0.343028\pi\)
0.473396 + 0.880850i \(0.343028\pi\)
\(252\) 0 0
\(253\) 15.0000i 0.943042i
\(254\) 0 0
\(255\) − 1.00000i − 0.0626224i
\(256\) 0 0
\(257\) −15.0000 −0.935674 −0.467837 0.883815i \(-0.654967\pi\)
−0.467837 + 0.883815i \(0.654967\pi\)
\(258\) 0 0
\(259\) −35.0000 −2.17479
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) 11.0000 0.678289 0.339145 0.940734i \(-0.389862\pi\)
0.339145 + 0.940734i \(0.389862\pi\)
\(264\) 0 0
\(265\) 2.00000i 0.122859i
\(266\) 0 0
\(267\) − 7.00000i − 0.428393i
\(268\) 0 0
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 0 0
\(271\) 7.00000i 0.425220i 0.977137 + 0.212610i \(0.0681963\pi\)
−0.977137 + 0.212610i \(0.931804\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.00000i 0.301511i
\(276\) 0 0
\(277\) 13.0000 0.781094 0.390547 0.920583i \(-0.372286\pi\)
0.390547 + 0.920583i \(0.372286\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 30.0000i 1.78965i 0.446417 + 0.894825i \(0.352700\pi\)
−0.446417 + 0.894825i \(0.647300\pi\)
\(282\) 0 0
\(283\) −23.0000 −1.36721 −0.683604 0.729853i \(-0.739588\pi\)
−0.683604 + 0.729853i \(0.739588\pi\)
\(284\) 0 0
\(285\) 3.00000 0.177705
\(286\) 0 0
\(287\) −25.0000 −1.47570
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 11.0000i 0.644831i
\(292\) 0 0
\(293\) 7.00000i 0.408944i 0.978872 + 0.204472i \(0.0655478\pi\)
−0.978872 + 0.204472i \(0.934452\pi\)
\(294\) 0 0
\(295\) 11.0000 0.640445
\(296\) 0 0
\(297\) − 25.0000i − 1.45065i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 25.0000i − 1.44098i
\(302\) 0 0
\(303\) −13.0000 −0.746830
\(304\) 0 0
\(305\) − 13.0000i − 0.744378i
\(306\) 0 0
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0 0
\(315\) 10.0000 0.563436
\(316\) 0 0
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) 5.00000i 0.279946i
\(320\) 0 0
\(321\) 9.00000 0.502331
\(322\) 0 0
\(323\) 3.00000i 0.166924i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 18.0000i 0.995402i
\(328\) 0 0
\(329\) −60.0000 −3.30791
\(330\) 0 0
\(331\) − 1.00000i − 0.0549650i −0.999622 0.0274825i \(-0.991251\pi\)
0.999622 0.0274825i \(-0.00874905\pi\)
\(332\) 0 0
\(333\) − 14.0000i − 0.767195i
\(334\) 0 0
\(335\) 3.00000 0.163908
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 1.00000 0.0543125
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 55.0000i − 2.96972i
\(344\) 0 0
\(345\) 3.00000i 0.161515i
\(346\) 0 0
\(347\) −27.0000 −1.44944 −0.724718 0.689046i \(-0.758030\pi\)
−0.724718 + 0.689046i \(0.758030\pi\)
\(348\) 0 0
\(349\) 35.0000i 1.87351i 0.349990 + 0.936754i \(0.386185\pi\)
−0.349990 + 0.936754i \(0.613815\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5.00000i 0.266123i 0.991108 + 0.133062i \(0.0424808\pi\)
−0.991108 + 0.133062i \(0.957519\pi\)
\(354\) 0 0
\(355\) 13.0000 0.689968
\(356\) 0 0
\(357\) − 5.00000i − 0.264628i
\(358\) 0 0
\(359\) − 24.0000i − 1.26667i −0.773877 0.633336i \(-0.781685\pi\)
0.773877 0.633336i \(-0.218315\pi\)
\(360\) 0 0
\(361\) 10.0000 0.526316
\(362\) 0 0
\(363\) 14.0000 0.734809
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) 3.00000 0.156599 0.0782994 0.996930i \(-0.475051\pi\)
0.0782994 + 0.996930i \(0.475051\pi\)
\(368\) 0 0
\(369\) − 10.0000i − 0.520579i
\(370\) 0 0
\(371\) 10.0000i 0.519174i
\(372\) 0 0
\(373\) 19.0000 0.983783 0.491891 0.870657i \(-0.336306\pi\)
0.491891 + 0.870657i \(0.336306\pi\)
\(374\) 0 0
\(375\) 1.00000i 0.0516398i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 21.0000i 1.07870i 0.842082 + 0.539349i \(0.181330\pi\)
−0.842082 + 0.539349i \(0.818670\pi\)
\(380\) 0 0
\(381\) 7.00000 0.358621
\(382\) 0 0
\(383\) 3.00000i 0.153293i 0.997058 + 0.0766464i \(0.0244213\pi\)
−0.997058 + 0.0766464i \(0.975579\pi\)
\(384\) 0 0
\(385\) 25.0000i 1.27412i
\(386\) 0 0
\(387\) 10.0000 0.508329
\(388\) 0 0
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 0 0
\(395\) − 4.00000i − 0.201262i
\(396\) 0 0
\(397\) − 13.0000i − 0.652451i −0.945292 0.326226i \(-0.894223\pi\)
0.945292 0.326226i \(-0.105777\pi\)
\(398\) 0 0
\(399\) 15.0000 0.750939
\(400\) 0 0
\(401\) 27.0000i 1.34832i 0.738587 + 0.674158i \(0.235493\pi\)
−0.738587 + 0.674158i \(0.764507\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 35.0000 1.73489
\(408\) 0 0
\(409\) − 19.0000i − 0.939490i −0.882802 0.469745i \(-0.844346\pi\)
0.882802 0.469745i \(-0.155654\pi\)
\(410\) 0 0
\(411\) − 3.00000i − 0.147979i
\(412\) 0 0
\(413\) 55.0000 2.70637
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) −13.0000 −0.636613
\(418\) 0 0
\(419\) −17.0000 −0.830504 −0.415252 0.909706i \(-0.636307\pi\)
−0.415252 + 0.909706i \(0.636307\pi\)
\(420\) 0 0
\(421\) − 10.0000i − 0.487370i −0.969854 0.243685i \(-0.921644\pi\)
0.969854 0.243685i \(-0.0783563\pi\)
\(422\) 0 0
\(423\) − 24.0000i − 1.16692i
\(424\) 0 0
\(425\) −1.00000 −0.0485071
\(426\) 0 0
\(427\) − 65.0000i − 3.14557i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 21.0000i 1.01153i 0.862670 + 0.505767i \(0.168791\pi\)
−0.862670 + 0.505767i \(0.831209\pi\)
\(432\) 0 0
\(433\) −7.00000 −0.336399 −0.168199 0.985753i \(-0.553795\pi\)
−0.168199 + 0.985753i \(0.553795\pi\)
\(434\) 0 0
\(435\) 1.00000i 0.0479463i
\(436\) 0 0
\(437\) − 9.00000i − 0.430528i
\(438\) 0 0
\(439\) 29.0000 1.38409 0.692047 0.721852i \(-0.256709\pi\)
0.692047 + 0.721852i \(0.256709\pi\)
\(440\) 0 0
\(441\) 36.0000 1.71429
\(442\) 0 0
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 0 0
\(445\) −7.00000 −0.331832
\(446\) 0 0
\(447\) 11.0000i 0.520282i
\(448\) 0 0
\(449\) − 21.0000i − 0.991051i −0.868593 0.495526i \(-0.834975\pi\)
0.868593 0.495526i \(-0.165025\pi\)
\(450\) 0 0
\(451\) 25.0000 1.17720
\(452\) 0 0
\(453\) 24.0000i 1.12762i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 11.0000i − 0.514558i −0.966337 0.257279i \(-0.917174\pi\)
0.966337 0.257279i \(-0.0828260\pi\)
\(458\) 0 0
\(459\) 5.00000 0.233380
\(460\) 0 0
\(461\) 33.0000i 1.53696i 0.639872 + 0.768482i \(0.278987\pi\)
−0.639872 + 0.768482i \(0.721013\pi\)
\(462\) 0 0
\(463\) − 8.00000i − 0.371792i −0.982569 0.185896i \(-0.940481\pi\)
0.982569 0.185896i \(-0.0595187\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 15.0000 0.692636
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 0 0
\(473\) 25.0000i 1.14950i
\(474\) 0 0
\(475\) − 3.00000i − 0.137649i
\(476\) 0 0
\(477\) −4.00000 −0.183147
\(478\) 0 0
\(479\) 11.0000i 0.502603i 0.967909 + 0.251301i \(0.0808585\pi\)
−0.967909 + 0.251301i \(0.919141\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 15.0000i 0.682524i
\(484\) 0 0
\(485\) 11.0000 0.499484
\(486\) 0 0
\(487\) − 17.0000i − 0.770344i −0.922845 0.385172i \(-0.874142\pi\)
0.922845 0.385172i \(-0.125858\pi\)
\(488\) 0 0
\(489\) − 5.00000i − 0.226108i
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) 0 0
\(493\) −1.00000 −0.0450377
\(494\) 0 0
\(495\) −10.0000 −0.449467
\(496\) 0 0
\(497\) 65.0000 2.91565
\(498\) 0 0
\(499\) 4.00000i 0.179065i 0.995984 + 0.0895323i \(0.0285372\pi\)
−0.995984 + 0.0895323i \(0.971463\pi\)
\(500\) 0 0
\(501\) 13.0000i 0.580797i
\(502\) 0 0
\(503\) −11.0000 −0.490466 −0.245233 0.969464i \(-0.578864\pi\)
−0.245233 + 0.969464i \(0.578864\pi\)
\(504\) 0 0
\(505\) 13.0000i 0.578492i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 3.00000i − 0.132973i −0.997787 0.0664863i \(-0.978821\pi\)
0.997787 0.0664863i \(-0.0211789\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) 0 0
\(513\) 15.0000i 0.662266i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 60.0000 2.63880
\(518\) 0 0
\(519\) −17.0000 −0.746217
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) −21.0000 −0.918266 −0.459133 0.888368i \(-0.651840\pi\)
−0.459133 + 0.888368i \(0.651840\pi\)
\(524\) 0 0
\(525\) 5.00000i 0.218218i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 22.0000i 0.954719i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 9.00000i − 0.389104i
\(536\) 0 0
\(537\) 11.0000 0.474685
\(538\) 0 0
\(539\) 90.0000i 3.87657i
\(540\) 0 0
\(541\) − 2.00000i − 0.0859867i −0.999075 0.0429934i \(-0.986311\pi\)
0.999075 0.0429934i \(-0.0136894\pi\)
\(542\) 0 0
\(543\) 10.0000 0.429141
\(544\) 0 0
\(545\) 18.0000 0.771035
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 26.0000 1.10965
\(550\) 0 0
\(551\) − 3.00000i − 0.127804i
\(552\) 0 0
\(553\) − 20.0000i − 0.850487i
\(554\) 0 0
\(555\) 7.00000 0.297133
\(556\) 0 0
\(557\) − 13.0000i − 0.550828i −0.961326 0.275414i \(-0.911185\pi\)
0.961326 0.275414i \(-0.0888149\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 5.00000i 0.211100i
\(562\) 0 0
\(563\) −9.00000 −0.379305 −0.189652 0.981851i \(-0.560736\pi\)
−0.189652 + 0.981851i \(0.560736\pi\)
\(564\) 0 0
\(565\) − 1.00000i − 0.0420703i
\(566\) 0 0
\(567\) 5.00000i 0.209980i
\(568\) 0 0
\(569\) −39.0000 −1.63497 −0.817483 0.575953i \(-0.804631\pi\)
−0.817483 + 0.575953i \(0.804631\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 15.0000 0.626634
\(574\) 0 0
\(575\) 3.00000 0.125109
\(576\) 0 0
\(577\) 42.0000i 1.74848i 0.485491 + 0.874241i \(0.338641\pi\)
−0.485491 + 0.874241i \(0.661359\pi\)
\(578\) 0 0
\(579\) − 23.0000i − 0.955847i
\(580\) 0 0
\(581\) 60.0000 2.48922
\(582\) 0 0
\(583\) − 10.0000i − 0.414158i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 3.00000i − 0.123823i −0.998082 0.0619116i \(-0.980280\pi\)
0.998082 0.0619116i \(-0.0197197\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 27.0000i 1.11063i
\(592\) 0 0
\(593\) − 2.00000i − 0.0821302i −0.999156 0.0410651i \(-0.986925\pi\)
0.999156 0.0410651i \(-0.0130751\pi\)
\(594\) 0 0
\(595\) −5.00000 −0.204980
\(596\) 0 0
\(597\) 21.0000 0.859473
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) −5.00000 −0.203954 −0.101977 0.994787i \(-0.532517\pi\)
−0.101977 + 0.994787i \(0.532517\pi\)
\(602\) 0 0
\(603\) 6.00000i 0.244339i
\(604\) 0 0
\(605\) − 14.0000i − 0.569181i
\(606\) 0 0
\(607\) −31.0000 −1.25825 −0.629126 0.777304i \(-0.716587\pi\)
−0.629126 + 0.777304i \(0.716587\pi\)
\(608\) 0 0
\(609\) 5.00000i 0.202610i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 25.0000i 1.00974i 0.863195 + 0.504870i \(0.168460\pi\)
−0.863195 + 0.504870i \(0.831540\pi\)
\(614\) 0 0
\(615\) 5.00000 0.201619
\(616\) 0 0
\(617\) − 27.0000i − 1.08698i −0.839416 0.543490i \(-0.817103\pi\)
0.839416 0.543490i \(-0.182897\pi\)
\(618\) 0 0
\(619\) − 4.00000i − 0.160774i −0.996764 0.0803868i \(-0.974384\pi\)
0.996764 0.0803868i \(-0.0256155\pi\)
\(620\) 0 0
\(621\) −15.0000 −0.601929
\(622\) 0 0
\(623\) −35.0000 −1.40225
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −15.0000 −0.599042
\(628\) 0 0
\(629\) 7.00000i 0.279108i
\(630\) 0 0
\(631\) − 27.0000i − 1.07485i −0.843311 0.537427i \(-0.819397\pi\)
0.843311 0.537427i \(-0.180603\pi\)
\(632\) 0 0
\(633\) 5.00000 0.198732
\(634\) 0 0
\(635\) − 7.00000i − 0.277787i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 26.0000i 1.02854i
\(640\) 0 0
\(641\) −27.0000 −1.06644 −0.533218 0.845978i \(-0.679017\pi\)
−0.533218 + 0.845978i \(0.679017\pi\)
\(642\) 0 0
\(643\) − 5.00000i − 0.197181i −0.995128 0.0985904i \(-0.968567\pi\)
0.995128 0.0985904i \(-0.0314334\pi\)
\(644\) 0 0
\(645\) 5.00000i 0.196875i
\(646\) 0 0
\(647\) 9.00000 0.353827 0.176913 0.984226i \(-0.443389\pi\)
0.176913 + 0.984226i \(0.443389\pi\)
\(648\) 0 0
\(649\) −55.0000 −2.15894
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 39.0000 1.52619 0.763094 0.646288i \(-0.223679\pi\)
0.763094 + 0.646288i \(0.223679\pi\)
\(654\) 0 0
\(655\) 4.00000i 0.156293i
\(656\) 0 0
\(657\) 4.00000i 0.156055i
\(658\) 0 0
\(659\) 17.0000 0.662226 0.331113 0.943591i \(-0.392576\pi\)
0.331113 + 0.943591i \(0.392576\pi\)
\(660\) 0 0
\(661\) 3.00000i 0.116686i 0.998297 + 0.0583432i \(0.0185818\pi\)
−0.998297 + 0.0583432i \(0.981418\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 15.0000i − 0.581675i
\(666\) 0 0
\(667\) 3.00000 0.116160
\(668\) 0 0
\(669\) 19.0000i 0.734582i
\(670\) 0 0
\(671\) 65.0000i 2.50930i
\(672\) 0 0
\(673\) −11.0000 −0.424019 −0.212009 0.977268i \(-0.568001\pi\)
−0.212009 + 0.977268i \(0.568001\pi\)
\(674\) 0 0
\(675\) −5.00000 −0.192450
\(676\) 0 0
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 0 0
\(679\) 55.0000 2.11071
\(680\) 0 0
\(681\) 17.0000i 0.651441i
\(682\) 0 0
\(683\) 49.0000i 1.87493i 0.348076 + 0.937466i \(0.386835\pi\)
−0.348076 + 0.937466i \(0.613165\pi\)
\(684\) 0 0
\(685\) −3.00000 −0.114624
\(686\) 0 0
\(687\) − 10.0000i − 0.381524i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 5.00000i 0.190209i 0.995467 + 0.0951045i \(0.0303185\pi\)
−0.995467 + 0.0951045i \(0.969681\pi\)
\(692\) 0 0
\(693\) −50.0000 −1.89934
\(694\) 0 0
\(695\) 13.0000i 0.493118i
\(696\) 0 0
\(697\) 5.00000i 0.189389i
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −26.0000 −0.982006 −0.491003 0.871158i \(-0.663370\pi\)
−0.491003 + 0.871158i \(0.663370\pi\)
\(702\) 0 0
\(703\) −21.0000 −0.792030
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) 0 0
\(707\) 65.0000i 2.44458i
\(708\) 0 0
\(709\) 23.0000i 0.863783i 0.901926 + 0.431892i \(0.142154\pi\)
−0.901926 + 0.431892i \(0.857846\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 8.00000i − 0.298765i
\(718\) 0 0
\(719\) −33.0000 −1.23069 −0.615346 0.788257i \(-0.710984\pi\)
−0.615346 + 0.788257i \(0.710984\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 11.0000i − 0.409094i
\(724\) 0 0
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −5.00000 −0.184932
\(732\) 0 0
\(733\) 34.0000i 1.25582i 0.778287 + 0.627909i \(0.216089\pi\)
−0.778287 + 0.627909i \(0.783911\pi\)
\(734\) 0 0
\(735\) 18.0000i 0.663940i
\(736\) 0 0
\(737\) −15.0000 −0.552532
\(738\) 0 0
\(739\) 15.0000i 0.551784i 0.961189 + 0.275892i \(0.0889732\pi\)
−0.961189 + 0.275892i \(0.911027\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.00000i 0.330178i 0.986279 + 0.165089i \(0.0527911\pi\)
−0.986279 + 0.165089i \(0.947209\pi\)
\(744\) 0 0
\(745\) 11.0000 0.403009
\(746\) 0 0
\(747\) 24.0000i 0.878114i
\(748\) 0 0
\(749\) − 45.0000i − 1.64426i
\(750\) 0 0
\(751\) 25.0000 0.912263 0.456131 0.889912i \(-0.349235\pi\)
0.456131 + 0.889912i \(0.349235\pi\)
\(752\) 0 0
\(753\) −15.0000 −0.546630
\(754\) 0 0
\(755\) 24.0000 0.873449
\(756\) 0 0
\(757\) −9.00000 −0.327111 −0.163555 0.986534i \(-0.552296\pi\)
−0.163555 + 0.986534i \(0.552296\pi\)
\(758\) 0 0
\(759\) − 15.0000i − 0.544466i
\(760\) 0 0
\(761\) 19.0000i 0.688749i 0.938832 + 0.344375i \(0.111909\pi\)
−0.938832 + 0.344375i \(0.888091\pi\)
\(762\) 0 0
\(763\) 90.0000 3.25822
\(764\) 0 0
\(765\) − 2.00000i − 0.0723102i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) − 19.0000i − 0.685158i −0.939489 0.342579i \(-0.888700\pi\)
0.939489 0.342579i \(-0.111300\pi\)
\(770\) 0 0
\(771\) 15.0000 0.540212
\(772\) 0 0
\(773\) 37.0000i 1.33080i 0.746488 + 0.665399i \(0.231738\pi\)
−0.746488 + 0.665399i \(0.768262\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 35.0000 1.25562
\(778\) 0 0
\(779\) −15.0000 −0.537431
\(780\) 0 0
\(781\) −65.0000 −2.32588
\(782\) 0 0
\(783\) −5.00000 −0.178685
\(784\) 0 0
\(785\) 10.0000i 0.356915i
\(786\) 0 0
\(787\) − 11.0000i − 0.392108i −0.980593 0.196054i \(-0.937187\pi\)
0.980593 0.196054i \(-0.0628127\pi\)
\(788\) 0 0
\(789\) −11.0000 −0.391610
\(790\) 0 0
\(791\) − 5.00000i − 0.177780i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) − 2.00000i − 0.0709327i
\(796\) 0 0
\(797\) −23.0000 −0.814702 −0.407351 0.913272i \(-0.633547\pi\)
−0.407351 + 0.913272i \(0.633547\pi\)
\(798\) 0 0
\(799\) 12.0000i 0.424529i
\(800\) 0 0
\(801\) − 14.0000i − 0.494666i
\(802\) 0 0
\(803\) −10.0000 −0.352892
\(804\) 0 0
\(805\) 15.0000 0.528681
\(806\) 0 0
\(807\) 9.00000 0.316815
\(808\) 0 0
\(809\) −37.0000 −1.30085 −0.650425 0.759570i \(-0.725409\pi\)
−0.650425 + 0.759570i \(0.725409\pi\)
\(810\) 0 0
\(811\) 8.00000i 0.280918i 0.990086 + 0.140459i \(0.0448578\pi\)
−0.990086 + 0.140459i \(0.955142\pi\)
\(812\) 0 0
\(813\) − 7.00000i − 0.245501i
\(814\) 0 0
\(815\) −5.00000 −0.175142
\(816\) 0 0
\(817\) − 15.0000i − 0.524784i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 25.0000i 0.872506i 0.899824 + 0.436253i \(0.143695\pi\)
−0.899824 + 0.436253i \(0.856305\pi\)
\(822\) 0 0
\(823\) 31.0000 1.08059 0.540296 0.841475i \(-0.318312\pi\)
0.540296 + 0.841475i \(0.318312\pi\)
\(824\) 0 0
\(825\) − 5.00000i − 0.174078i
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 0 0
\(829\) 25.0000 0.868286 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) 0 0
\(831\) −13.0000 −0.450965
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) 13.0000 0.449884
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 43.0000i − 1.48452i −0.670109 0.742262i \(-0.733753\pi\)
0.670109 0.742262i \(-0.266247\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) − 30.0000i − 1.03325i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 70.0000i − 2.40523i
\(848\) 0 0
\(849\) 23.0000 0.789358
\(850\) 0 0
\(851\) − 21.0000i − 0.719871i
\(852\) 0 0
\(853\) 34.0000i 1.16414i 0.813139 + 0.582069i \(0.197757\pi\)
−0.813139 + 0.582069i \(0.802243\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) 0 0
\(857\) 26.0000 0.888143 0.444072 0.895991i \(-0.353534\pi\)
0.444072 + 0.895991i \(0.353534\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 25.0000 0.851998
\(862\) 0 0
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) 0 0
\(865\) 17.0000i 0.578017i
\(866\) 0 0
\(867\) 16.0000 0.543388
\(868\) 0 0
\(869\) 20.0000i 0.678454i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 22.0000i 0.744587i
\(874\) 0 0
\(875\) 5.00000 0.169031
\(876\) 0 0
\(877\) − 31.0000i − 1.04680i −0.852088 0.523398i \(-0.824664\pi\)
0.852088 0.523398i \(-0.175336\pi\)
\(878\) 0 0
\(879\) − 7.00000i − 0.236104i
\(880\) 0 0
\(881\) 45.0000 1.51609 0.758044 0.652203i \(-0.226155\pi\)
0.758044 + 0.652203i \(0.226155\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) −11.0000 −0.369761
\(886\) 0 0
\(887\) 31.0000 1.04088 0.520439 0.853899i \(-0.325768\pi\)
0.520439 + 0.853899i \(0.325768\pi\)
\(888\) 0 0
\(889\) − 35.0000i − 1.17386i
\(890\) 0 0
\(891\) − 5.00000i − 0.167506i
\(892\) 0 0
\(893\) −36.0000 −1.20469
\(894\) 0 0
\(895\) − 11.0000i − 0.367689i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 2.00000 0.0666297
\(902\) 0 0
\(903\) 25.0000i 0.831948i
\(904\) 0 0
\(905\) − 10.0000i − 0.332411i
\(906\) 0 0
\(907\) 17.0000 0.564476 0.282238 0.959344i \(-0.408923\pi\)
0.282238 + 0.959344i \(0.408923\pi\)
\(908\) 0 0
\(909\) −26.0000 −0.862366
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 0 0
\(913\) −60.0000 −1.98571
\(914\) 0 0
\(915\) 13.0000i 0.429767i
\(916\) 0 0
\(917\) 20.0000i 0.660458i
\(918\) 0 0
\(919\) 49.0000 1.61636 0.808180 0.588935i \(-0.200453\pi\)
0.808180 + 0.588935i \(0.200453\pi\)
\(920\) 0 0
\(921\) 12.0000i 0.395413i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) − 7.00000i − 0.230159i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 45.0000i 1.47640i 0.674581 + 0.738201i \(0.264324\pi\)
−0.674581 + 0.738201i \(0.735676\pi\)
\(930\) 0 0
\(931\) − 54.0000i − 1.76978i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5.00000 0.163517
\(936\) 0 0
\(937\) −18.0000 −0.588034 −0.294017 0.955800i \(-0.594992\pi\)
−0.294017 + 0.955800i \(0.594992\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) − 14.0000i − 0.456387i −0.973616 0.228193i \(-0.926718\pi\)
0.973616 0.228193i \(-0.0732819\pi\)
\(942\) 0 0
\(943\) − 15.0000i − 0.488467i
\(944\) 0 0
\(945\) −25.0000 −0.813250
\(946\) 0 0
\(947\) 31.0000i 1.00736i 0.863889 + 0.503682i \(0.168022\pi\)
−0.863889 + 0.503682i \(0.831978\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) − 18.0000i − 0.583690i
\(952\) 0 0
\(953\) −3.00000 −0.0971795 −0.0485898 0.998819i \(-0.515473\pi\)
−0.0485898 + 0.998819i \(0.515473\pi\)
\(954\) 0 0
\(955\) − 15.0000i − 0.485389i
\(956\) 0 0
\(957\) − 5.00000i − 0.161627i
\(958\) 0 0
\(959\) −15.0000 −0.484375
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 0 0
\(963\) 18.0000 0.580042
\(964\) 0 0
\(965\) −23.0000 −0.740396
\(966\) 0 0
\(967\) − 16.0000i − 0.514525i −0.966342 0.257263i \(-0.917179\pi\)
0.966342 0.257263i \(-0.0828206\pi\)
\(968\) 0 0
\(969\) − 3.00000i − 0.0963739i
\(970\) 0 0
\(971\) −7.00000 −0.224641 −0.112320 0.993672i \(-0.535828\pi\)
−0.112320 + 0.993672i \(0.535828\pi\)
\(972\) 0 0
\(973\) 65.0000i 2.08380i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 45.0000i 1.43968i 0.694141 + 0.719839i \(0.255784\pi\)
−0.694141 + 0.719839i \(0.744216\pi\)
\(978\) 0 0
\(979\) 35.0000 1.11860
\(980\) 0 0
\(981\) 36.0000i 1.14939i
\(982\) 0 0
\(983\) − 16.0000i − 0.510321i −0.966899 0.255160i \(-0.917872\pi\)
0.966899 0.255160i \(-0.0821283\pi\)
\(984\) 0 0
\(985\) 27.0000 0.860292
\(986\) 0 0
\(987\) 60.0000 1.90982
\(988\) 0 0
\(989\) 15.0000 0.476972
\(990\) 0 0
\(991\) 3.00000 0.0952981 0.0476491 0.998864i \(-0.484827\pi\)
0.0476491 + 0.998864i \(0.484827\pi\)
\(992\) 0 0
\(993\) 1.00000i 0.0317340i
\(994\) 0 0
\(995\) − 21.0000i − 0.665745i
\(996\) 0 0
\(997\) 7.00000 0.221692 0.110846 0.993838i \(-0.464644\pi\)
0.110846 + 0.993838i \(0.464644\pi\)
\(998\) 0 0
\(999\) 35.0000i 1.10735i
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 3380.2.f.c.3041.2 2
13.5 odd 4 3380.2.a.e.1.1 1
13.7 odd 12 260.2.i.c.81.1 yes 2
13.8 odd 4 3380.2.a.d.1.1 1
13.11 odd 12 260.2.i.c.61.1 2
13.12 even 2 inner 3380.2.f.c.3041.1 2
39.11 even 12 2340.2.q.c.1621.1 2
39.20 even 12 2340.2.q.c.2161.1 2
52.7 even 12 1040.2.q.f.81.1 2
52.11 even 12 1040.2.q.f.321.1 2
65.7 even 12 1300.2.bb.c.549.2 4
65.24 odd 12 1300.2.i.c.1101.1 2
65.33 even 12 1300.2.bb.c.549.1 4
65.37 even 12 1300.2.bb.c.1049.1 4
65.59 odd 12 1300.2.i.c.601.1 2
65.63 even 12 1300.2.bb.c.1049.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.i.c.61.1 2 13.11 odd 12
260.2.i.c.81.1 yes 2 13.7 odd 12
1040.2.q.f.81.1 2 52.7 even 12
1040.2.q.f.321.1 2 52.11 even 12
1300.2.i.c.601.1 2 65.59 odd 12
1300.2.i.c.1101.1 2 65.24 odd 12
1300.2.bb.c.549.1 4 65.33 even 12
1300.2.bb.c.549.2 4 65.7 even 12
1300.2.bb.c.1049.1 4 65.37 even 12
1300.2.bb.c.1049.2 4 65.63 even 12
2340.2.q.c.1621.1 2 39.11 even 12
2340.2.q.c.2161.1 2 39.20 even 12
3380.2.a.d.1.1 1 13.8 odd 4
3380.2.a.e.1.1 1 13.5 odd 4
3380.2.f.c.3041.1 2 13.12 even 2 inner
3380.2.f.c.3041.2 2 1.1 even 1 trivial