Properties

Label 3330.2.a.bg.1.1
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3330,2,Mod(1,3330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3330, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3330.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

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: yes
Analytic conductor: \(26.5901838731\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 8x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.59774\) of defining polynomial
Character \(\chi\) \(=\) 3330.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.59774 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.59774 q^{7} -1.00000 q^{8} -1.00000 q^{10} -4.74823 q^{11} +6.69193 q^{13} +2.59774 q^{14} +1.00000 q^{16} +0.748228 q^{17} -3.34596 q^{19} +1.00000 q^{20} +4.74823 q^{22} -1.49646 q^{23} +1.00000 q^{25} -6.69193 q^{26} -2.59774 q^{28} -3.94370 q^{29} +7.79321 q^{31} -1.00000 q^{32} -0.748228 q^{34} -2.59774 q^{35} -1.00000 q^{37} +3.34596 q^{38} -1.00000 q^{40} +6.44724 q^{41} +1.94370 q^{43} -4.74823 q^{44} +1.49646 q^{46} -1.84951 q^{47} -0.251772 q^{49} -1.00000 q^{50} +6.69193 q^{52} -10.4472 q^{53} -4.74823 q^{55} +2.59774 q^{56} +3.94370 q^{58} -5.84951 q^{59} +7.94370 q^{61} -7.79321 q^{62} +1.00000 q^{64} +6.69193 q^{65} +1.84951 q^{67} +0.748228 q^{68} +2.59774 q^{70} +3.88740 q^{71} -7.49646 q^{73} +1.00000 q^{74} -3.34596 q^{76} +12.3346 q^{77} -16.5414 q^{79} +1.00000 q^{80} -6.44724 q^{82} -15.2334 q^{83} +0.748228 q^{85} -1.94370 q^{86} +4.74823 q^{88} -6.00000 q^{89} -17.3839 q^{91} -1.49646 q^{92} +1.84951 q^{94} -3.34596 q^{95} -10.4472 q^{97} +0.251772 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8} - 3 q^{10} - 11 q^{11} + q^{14} + 3 q^{16} - q^{17} + 3 q^{20} + 11 q^{22} + 2 q^{23} + 3 q^{25} - q^{28} + 5 q^{29} + 3 q^{31} - 3 q^{32} + q^{34} - q^{35} - 3 q^{37} - 3 q^{40} + 9 q^{41} - 11 q^{43} - 11 q^{44} - 2 q^{46} - 2 q^{47} - 4 q^{49} - 3 q^{50} - 21 q^{53} - 11 q^{55} + q^{56} - 5 q^{58} - 14 q^{59} + 7 q^{61} - 3 q^{62} + 3 q^{64} + 2 q^{67} - q^{68} + q^{70} - 22 q^{71} - 16 q^{73} + 3 q^{74} - 7 q^{77} - 26 q^{79} + 3 q^{80} - 9 q^{82} - 2 q^{83} - q^{85} + 11 q^{86} + 11 q^{88} - 18 q^{89} - 12 q^{91} + 2 q^{92} + 2 q^{94} - 21 q^{97} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.59774 −0.981852 −0.490926 0.871201i \(-0.663341\pi\)
−0.490926 + 0.871201i \(0.663341\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −4.74823 −1.43164 −0.715822 0.698282i \(-0.753948\pi\)
−0.715822 + 0.698282i \(0.753948\pi\)
\(12\) 0 0
\(13\) 6.69193 1.85601 0.928003 0.372572i \(-0.121524\pi\)
0.928003 + 0.372572i \(0.121524\pi\)
\(14\) 2.59774 0.694274
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.748228 0.181472 0.0907360 0.995875i \(-0.471078\pi\)
0.0907360 + 0.995875i \(0.471078\pi\)
\(18\) 0 0
\(19\) −3.34596 −0.767617 −0.383808 0.923413i \(-0.625388\pi\)
−0.383808 + 0.923413i \(0.625388\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 4.74823 1.01233
\(23\) −1.49646 −0.312033 −0.156016 0.987754i \(-0.549865\pi\)
−0.156016 + 0.987754i \(0.549865\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −6.69193 −1.31239
\(27\) 0 0
\(28\) −2.59774 −0.490926
\(29\) −3.94370 −0.732326 −0.366163 0.930551i \(-0.619329\pi\)
−0.366163 + 0.930551i \(0.619329\pi\)
\(30\) 0 0
\(31\) 7.79321 1.39970 0.699851 0.714289i \(-0.253250\pi\)
0.699851 + 0.714289i \(0.253250\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −0.748228 −0.128320
\(35\) −2.59774 −0.439097
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 3.34596 0.542787
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 6.44724 1.00689 0.503445 0.864027i \(-0.332066\pi\)
0.503445 + 0.864027i \(0.332066\pi\)
\(42\) 0 0
\(43\) 1.94370 0.296411 0.148206 0.988957i \(-0.452650\pi\)
0.148206 + 0.988957i \(0.452650\pi\)
\(44\) −4.74823 −0.715822
\(45\) 0 0
\(46\) 1.49646 0.220640
\(47\) −1.84951 −0.269778 −0.134889 0.990861i \(-0.543068\pi\)
−0.134889 + 0.990861i \(0.543068\pi\)
\(48\) 0 0
\(49\) −0.251772 −0.0359674
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 6.69193 0.928003
\(53\) −10.4472 −1.43504 −0.717520 0.696538i \(-0.754723\pi\)
−0.717520 + 0.696538i \(0.754723\pi\)
\(54\) 0 0
\(55\) −4.74823 −0.640251
\(56\) 2.59774 0.347137
\(57\) 0 0
\(58\) 3.94370 0.517833
\(59\) −5.84951 −0.761541 −0.380770 0.924670i \(-0.624341\pi\)
−0.380770 + 0.924670i \(0.624341\pi\)
\(60\) 0 0
\(61\) 7.94370 1.01709 0.508543 0.861036i \(-0.330184\pi\)
0.508543 + 0.861036i \(0.330184\pi\)
\(62\) −7.79321 −0.989738
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.69193 0.830031
\(66\) 0 0
\(67\) 1.84951 0.225953 0.112977 0.993598i \(-0.463961\pi\)
0.112977 + 0.993598i \(0.463961\pi\)
\(68\) 0.748228 0.0907360
\(69\) 0 0
\(70\) 2.59774 0.310489
\(71\) 3.88740 0.461349 0.230675 0.973031i \(-0.425907\pi\)
0.230675 + 0.973031i \(0.425907\pi\)
\(72\) 0 0
\(73\) −7.49646 −0.877394 −0.438697 0.898635i \(-0.644560\pi\)
−0.438697 + 0.898635i \(0.644560\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −3.34596 −0.383808
\(77\) 12.3346 1.40566
\(78\) 0 0
\(79\) −16.5414 −1.86106 −0.930528 0.366220i \(-0.880652\pi\)
−0.930528 + 0.366220i \(0.880652\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −6.44724 −0.711979
\(83\) −15.2334 −1.67208 −0.836039 0.548670i \(-0.815134\pi\)
−0.836039 + 0.548670i \(0.815134\pi\)
\(84\) 0 0
\(85\) 0.748228 0.0811567
\(86\) −1.94370 −0.209594
\(87\) 0 0
\(88\) 4.74823 0.506163
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −17.3839 −1.82232
\(92\) −1.49646 −0.156016
\(93\) 0 0
\(94\) 1.84951 0.190762
\(95\) −3.34596 −0.343289
\(96\) 0 0
\(97\) −10.4472 −1.06076 −0.530378 0.847761i \(-0.677950\pi\)
−0.530378 + 0.847761i \(0.677950\pi\)
\(98\) 0.251772 0.0254328
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 12.1884 1.21279 0.606395 0.795164i \(-0.292615\pi\)
0.606395 + 0.795164i \(0.292615\pi\)
\(102\) 0 0
\(103\) −1.30807 −0.128888 −0.0644442 0.997921i \(-0.520527\pi\)
−0.0644442 + 0.997921i \(0.520527\pi\)
\(104\) −6.69193 −0.656197
\(105\) 0 0
\(106\) 10.4472 1.01473
\(107\) 3.04498 0.294369 0.147185 0.989109i \(-0.452979\pi\)
0.147185 + 0.989109i \(0.452979\pi\)
\(108\) 0 0
\(109\) −1.44015 −0.137942 −0.0689709 0.997619i \(-0.521972\pi\)
−0.0689709 + 0.997619i \(0.521972\pi\)
\(110\) 4.74823 0.452726
\(111\) 0 0
\(112\) −2.59774 −0.245463
\(113\) −11.1392 −1.04788 −0.523942 0.851754i \(-0.675539\pi\)
−0.523942 + 0.851754i \(0.675539\pi\)
\(114\) 0 0
\(115\) −1.49646 −0.139545
\(116\) −3.94370 −0.366163
\(117\) 0 0
\(118\) 5.84951 0.538491
\(119\) −1.94370 −0.178179
\(120\) 0 0
\(121\) 11.5457 1.04961
\(122\) −7.94370 −0.719189
\(123\) 0 0
\(124\) 7.79321 0.699851
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −8.84242 −0.784638 −0.392319 0.919829i \(-0.628327\pi\)
−0.392319 + 0.919829i \(0.628327\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −6.69193 −0.586921
\(131\) −6.15049 −0.537371 −0.268686 0.963228i \(-0.586589\pi\)
−0.268686 + 0.963228i \(0.586589\pi\)
\(132\) 0 0
\(133\) 8.69193 0.753686
\(134\) −1.84951 −0.159773
\(135\) 0 0
\(136\) −0.748228 −0.0641600
\(137\) −10.8945 −0.930779 −0.465389 0.885106i \(-0.654086\pi\)
−0.465389 + 0.885106i \(0.654086\pi\)
\(138\) 0 0
\(139\) 1.04921 0.0889932 0.0444966 0.999010i \(-0.485832\pi\)
0.0444966 + 0.999010i \(0.485832\pi\)
\(140\) −2.59774 −0.219549
\(141\) 0 0
\(142\) −3.88740 −0.326223
\(143\) −31.7748 −2.65714
\(144\) 0 0
\(145\) −3.94370 −0.327506
\(146\) 7.49646 0.620411
\(147\) 0 0
\(148\) −1.00000 −0.0821995
\(149\) 4.18838 0.343126 0.171563 0.985173i \(-0.445118\pi\)
0.171563 + 0.985173i \(0.445118\pi\)
\(150\) 0 0
\(151\) −6.69193 −0.544581 −0.272291 0.962215i \(-0.587781\pi\)
−0.272291 + 0.962215i \(0.587781\pi\)
\(152\) 3.34596 0.271393
\(153\) 0 0
\(154\) −12.3346 −0.993954
\(155\) 7.79321 0.625965
\(156\) 0 0
\(157\) −3.94370 −0.314741 −0.157371 0.987540i \(-0.550302\pi\)
−0.157371 + 0.987540i \(0.550302\pi\)
\(158\) 16.5414 1.31597
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 3.88740 0.306370
\(162\) 0 0
\(163\) −12.1463 −0.951368 −0.475684 0.879616i \(-0.657799\pi\)
−0.475684 + 0.879616i \(0.657799\pi\)
\(164\) 6.44724 0.503445
\(165\) 0 0
\(166\) 15.2334 1.18234
\(167\) −17.0829 −1.32191 −0.660956 0.750425i \(-0.729849\pi\)
−0.660956 + 0.750425i \(0.729849\pi\)
\(168\) 0 0
\(169\) 31.7819 2.44476
\(170\) −0.748228 −0.0573865
\(171\) 0 0
\(172\) 1.94370 0.148206
\(173\) 15.3417 1.16641 0.583205 0.812325i \(-0.301798\pi\)
0.583205 + 0.812325i \(0.301798\pi\)
\(174\) 0 0
\(175\) −2.59774 −0.196370
\(176\) −4.74823 −0.357911
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −2.45148 −0.183232 −0.0916161 0.995794i \(-0.529203\pi\)
−0.0916161 + 0.995794i \(0.529203\pi\)
\(180\) 0 0
\(181\) 13.1955 0.980812 0.490406 0.871494i \(-0.336849\pi\)
0.490406 + 0.871494i \(0.336849\pi\)
\(182\) 17.3839 1.28858
\(183\) 0 0
\(184\) 1.49646 0.110320
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) −3.55276 −0.259803
\(188\) −1.84951 −0.134889
\(189\) 0 0
\(190\) 3.34596 0.242742
\(191\) 21.7790 1.57588 0.787938 0.615755i \(-0.211149\pi\)
0.787938 + 0.615755i \(0.211149\pi\)
\(192\) 0 0
\(193\) −12.9929 −0.935250 −0.467625 0.883927i \(-0.654890\pi\)
−0.467625 + 0.883927i \(0.654890\pi\)
\(194\) 10.4472 0.750068
\(195\) 0 0
\(196\) −0.251772 −0.0179837
\(197\) −0.616147 −0.0438986 −0.0219493 0.999759i \(-0.506987\pi\)
−0.0219493 + 0.999759i \(0.506987\pi\)
\(198\) 0 0
\(199\) −1.54852 −0.109772 −0.0548859 0.998493i \(-0.517480\pi\)
−0.0548859 + 0.998493i \(0.517480\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −12.1884 −0.857572
\(203\) 10.2447 0.719036
\(204\) 0 0
\(205\) 6.44724 0.450295
\(206\) 1.30807 0.0911378
\(207\) 0 0
\(208\) 6.69193 0.464002
\(209\) 15.8874 1.09895
\(210\) 0 0
\(211\) 20.9366 1.44134 0.720668 0.693280i \(-0.243835\pi\)
0.720668 + 0.693280i \(0.243835\pi\)
\(212\) −10.4472 −0.717520
\(213\) 0 0
\(214\) −3.04498 −0.208150
\(215\) 1.94370 0.132559
\(216\) 0 0
\(217\) −20.2447 −1.37430
\(218\) 1.44015 0.0975396
\(219\) 0 0
\(220\) −4.74823 −0.320125
\(221\) 5.00709 0.336813
\(222\) 0 0
\(223\) −2.71034 −0.181498 −0.0907488 0.995874i \(-0.528926\pi\)
−0.0907488 + 0.995874i \(0.528926\pi\)
\(224\) 2.59774 0.173568
\(225\) 0 0
\(226\) 11.1392 0.740966
\(227\) −26.1321 −1.73445 −0.867224 0.497919i \(-0.834098\pi\)
−0.867224 + 0.497919i \(0.834098\pi\)
\(228\) 0 0
\(229\) −24.7677 −1.63670 −0.818348 0.574723i \(-0.805110\pi\)
−0.818348 + 0.574723i \(0.805110\pi\)
\(230\) 1.49646 0.0986734
\(231\) 0 0
\(232\) 3.94370 0.258916
\(233\) 12.2783 0.804381 0.402190 0.915556i \(-0.368249\pi\)
0.402190 + 0.915556i \(0.368249\pi\)
\(234\) 0 0
\(235\) −1.84951 −0.120649
\(236\) −5.84951 −0.380770
\(237\) 0 0
\(238\) 1.94370 0.125991
\(239\) 17.1771 1.11109 0.555546 0.831486i \(-0.312509\pi\)
0.555546 + 0.831486i \(0.312509\pi\)
\(240\) 0 0
\(241\) −15.2713 −0.983708 −0.491854 0.870678i \(-0.663681\pi\)
−0.491854 + 0.870678i \(0.663681\pi\)
\(242\) −11.5457 −0.742184
\(243\) 0 0
\(244\) 7.94370 0.508543
\(245\) −0.251772 −0.0160851
\(246\) 0 0
\(247\) −22.3909 −1.42470
\(248\) −7.79321 −0.494869
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −10.0379 −0.633586 −0.316793 0.948495i \(-0.602606\pi\)
−0.316793 + 0.948495i \(0.602606\pi\)
\(252\) 0 0
\(253\) 7.10552 0.446720
\(254\) 8.84242 0.554823
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 20.9703 1.30809 0.654045 0.756456i \(-0.273071\pi\)
0.654045 + 0.756456i \(0.273071\pi\)
\(258\) 0 0
\(259\) 2.59774 0.161415
\(260\) 6.69193 0.415016
\(261\) 0 0
\(262\) 6.15049 0.379979
\(263\) −22.3725 −1.37955 −0.689775 0.724024i \(-0.742290\pi\)
−0.689775 + 0.724024i \(0.742290\pi\)
\(264\) 0 0
\(265\) −10.4472 −0.641769
\(266\) −8.69193 −0.532936
\(267\) 0 0
\(268\) 1.84951 0.112977
\(269\) 4.18838 0.255370 0.127685 0.991815i \(-0.459245\pi\)
0.127685 + 0.991815i \(0.459245\pi\)
\(270\) 0 0
\(271\) −12.1126 −0.735788 −0.367894 0.929868i \(-0.619921\pi\)
−0.367894 + 0.929868i \(0.619921\pi\)
\(272\) 0.748228 0.0453680
\(273\) 0 0
\(274\) 10.8945 0.658160
\(275\) −4.74823 −0.286329
\(276\) 0 0
\(277\) −9.30807 −0.559268 −0.279634 0.960107i \(-0.590213\pi\)
−0.279634 + 0.960107i \(0.590213\pi\)
\(278\) −1.04921 −0.0629277
\(279\) 0 0
\(280\) 2.59774 0.155244
\(281\) −19.2713 −1.14963 −0.574813 0.818285i \(-0.694925\pi\)
−0.574813 + 0.818285i \(0.694925\pi\)
\(282\) 0 0
\(283\) −13.6848 −0.813479 −0.406740 0.913544i \(-0.633334\pi\)
−0.406740 + 0.913544i \(0.633334\pi\)
\(284\) 3.88740 0.230675
\(285\) 0 0
\(286\) 31.7748 1.87888
\(287\) −16.7482 −0.988617
\(288\) 0 0
\(289\) −16.4402 −0.967068
\(290\) 3.94370 0.231582
\(291\) 0 0
\(292\) −7.49646 −0.438697
\(293\) 10.4472 0.610334 0.305167 0.952299i \(-0.401288\pi\)
0.305167 + 0.952299i \(0.401288\pi\)
\(294\) 0 0
\(295\) −5.84951 −0.340571
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) −4.18838 −0.242627
\(299\) −10.0142 −0.579135
\(300\) 0 0
\(301\) −5.04921 −0.291032
\(302\) 6.69193 0.385077
\(303\) 0 0
\(304\) −3.34596 −0.191904
\(305\) 7.94370 0.454855
\(306\) 0 0
\(307\) −4.95502 −0.282798 −0.141399 0.989953i \(-0.545160\pi\)
−0.141399 + 0.989953i \(0.545160\pi\)
\(308\) 12.3346 0.702831
\(309\) 0 0
\(310\) −7.79321 −0.442624
\(311\) −3.68060 −0.208708 −0.104354 0.994540i \(-0.533277\pi\)
−0.104354 + 0.994540i \(0.533277\pi\)
\(312\) 0 0
\(313\) −0.992912 −0.0561227 −0.0280614 0.999606i \(-0.508933\pi\)
−0.0280614 + 0.999606i \(0.508933\pi\)
\(314\) 3.94370 0.222556
\(315\) 0 0
\(316\) −16.5414 −0.930528
\(317\) 6.55985 0.368438 0.184219 0.982885i \(-0.441024\pi\)
0.184219 + 0.982885i \(0.441024\pi\)
\(318\) 0 0
\(319\) 18.7256 1.04843
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −3.88740 −0.216636
\(323\) −2.50354 −0.139301
\(324\) 0 0
\(325\) 6.69193 0.371201
\(326\) 12.1463 0.672719
\(327\) 0 0
\(328\) −6.44724 −0.355989
\(329\) 4.80453 0.264882
\(330\) 0 0
\(331\) −18.0379 −0.991452 −0.495726 0.868479i \(-0.665098\pi\)
−0.495726 + 0.868479i \(0.665098\pi\)
\(332\) −15.2334 −0.836039
\(333\) 0 0
\(334\) 17.0829 0.934733
\(335\) 1.84951 0.101049
\(336\) 0 0
\(337\) −14.8945 −0.811354 −0.405677 0.914016i \(-0.632964\pi\)
−0.405677 + 0.914016i \(0.632964\pi\)
\(338\) −31.7819 −1.72871
\(339\) 0 0
\(340\) 0.748228 0.0405784
\(341\) −37.0039 −2.00387
\(342\) 0 0
\(343\) 18.8382 1.01717
\(344\) −1.94370 −0.104797
\(345\) 0 0
\(346\) −15.3417 −0.824776
\(347\) 9.38385 0.503752 0.251876 0.967760i \(-0.418953\pi\)
0.251876 + 0.967760i \(0.418953\pi\)
\(348\) 0 0
\(349\) −32.9703 −1.76486 −0.882429 0.470446i \(-0.844093\pi\)
−0.882429 + 0.470446i \(0.844093\pi\)
\(350\) 2.59774 0.138855
\(351\) 0 0
\(352\) 4.74823 0.253081
\(353\) −27.4260 −1.45974 −0.729869 0.683587i \(-0.760419\pi\)
−0.729869 + 0.683587i \(0.760419\pi\)
\(354\) 0 0
\(355\) 3.88740 0.206322
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 2.45148 0.129565
\(359\) 25.0829 1.32382 0.661912 0.749582i \(-0.269745\pi\)
0.661912 + 0.749582i \(0.269745\pi\)
\(360\) 0 0
\(361\) −7.80453 −0.410765
\(362\) −13.1955 −0.693539
\(363\) 0 0
\(364\) −17.3839 −0.911161
\(365\) −7.49646 −0.392382
\(366\) 0 0
\(367\) 2.48513 0.129723 0.0648614 0.997894i \(-0.479339\pi\)
0.0648614 + 0.997894i \(0.479339\pi\)
\(368\) −1.49646 −0.0780082
\(369\) 0 0
\(370\) 1.00000 0.0519875
\(371\) 27.1392 1.40900
\(372\) 0 0
\(373\) −35.1586 −1.82045 −0.910223 0.414119i \(-0.864090\pi\)
−0.910223 + 0.414119i \(0.864090\pi\)
\(374\) 3.55276 0.183709
\(375\) 0 0
\(376\) 1.84951 0.0953810
\(377\) −26.3909 −1.35920
\(378\) 0 0
\(379\) 24.6778 1.26761 0.633805 0.773492i \(-0.281492\pi\)
0.633805 + 0.773492i \(0.281492\pi\)
\(380\) −3.34596 −0.171644
\(381\) 0 0
\(382\) −21.7790 −1.11431
\(383\) 13.3839 0.683883 0.341941 0.939721i \(-0.388916\pi\)
0.341941 + 0.939721i \(0.388916\pi\)
\(384\) 0 0
\(385\) 12.3346 0.628631
\(386\) 12.9929 0.661322
\(387\) 0 0
\(388\) −10.4472 −0.530378
\(389\) −18.2220 −0.923894 −0.461947 0.886908i \(-0.652849\pi\)
−0.461947 + 0.886908i \(0.652849\pi\)
\(390\) 0 0
\(391\) −1.11969 −0.0566252
\(392\) 0.251772 0.0127164
\(393\) 0 0
\(394\) 0.616147 0.0310410
\(395\) −16.5414 −0.832290
\(396\) 0 0
\(397\) −2.22521 −0.111680 −0.0558399 0.998440i \(-0.517784\pi\)
−0.0558399 + 0.998440i \(0.517784\pi\)
\(398\) 1.54852 0.0776204
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −12.3909 −0.618774 −0.309387 0.950936i \(-0.600124\pi\)
−0.309387 + 0.950936i \(0.600124\pi\)
\(402\) 0 0
\(403\) 52.1516 2.59785
\(404\) 12.1884 0.606395
\(405\) 0 0
\(406\) −10.2447 −0.508435
\(407\) 4.74823 0.235361
\(408\) 0 0
\(409\) 7.27125 0.359540 0.179770 0.983709i \(-0.442465\pi\)
0.179770 + 0.983709i \(0.442465\pi\)
\(410\) −6.44724 −0.318407
\(411\) 0 0
\(412\) −1.30807 −0.0644442
\(413\) 15.1955 0.747720
\(414\) 0 0
\(415\) −15.2334 −0.747776
\(416\) −6.69193 −0.328099
\(417\) 0 0
\(418\) −15.8874 −0.777078
\(419\) −28.4809 −1.39138 −0.695691 0.718341i \(-0.744902\pi\)
−0.695691 + 0.718341i \(0.744902\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −20.9366 −1.01918
\(423\) 0 0
\(424\) 10.4472 0.507363
\(425\) 0.748228 0.0362944
\(426\) 0 0
\(427\) −20.6356 −0.998628
\(428\) 3.04498 0.147185
\(429\) 0 0
\(430\) −1.94370 −0.0937335
\(431\) −1.51487 −0.0729686 −0.0364843 0.999334i \(-0.511616\pi\)
−0.0364843 + 0.999334i \(0.511616\pi\)
\(432\) 0 0
\(433\) 24.9929 1.20108 0.600541 0.799594i \(-0.294952\pi\)
0.600541 + 0.799594i \(0.294952\pi\)
\(434\) 20.2447 0.971776
\(435\) 0 0
\(436\) −1.44015 −0.0689709
\(437\) 5.00709 0.239521
\(438\) 0 0
\(439\) −5.10128 −0.243471 −0.121735 0.992563i \(-0.538846\pi\)
−0.121735 + 0.992563i \(0.538846\pi\)
\(440\) 4.74823 0.226363
\(441\) 0 0
\(442\) −5.00709 −0.238163
\(443\) −25.7369 −1.22280 −0.611399 0.791323i \(-0.709393\pi\)
−0.611399 + 0.791323i \(0.709393\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 2.71034 0.128338
\(447\) 0 0
\(448\) −2.59774 −0.122731
\(449\) 17.7748 0.838844 0.419422 0.907791i \(-0.362233\pi\)
0.419422 + 0.907791i \(0.362233\pi\)
\(450\) 0 0
\(451\) −30.6130 −1.44151
\(452\) −11.1392 −0.523942
\(453\) 0 0
\(454\) 26.1321 1.22644
\(455\) −17.3839 −0.814968
\(456\) 0 0
\(457\) 34.3346 1.60611 0.803053 0.595907i \(-0.203207\pi\)
0.803053 + 0.595907i \(0.203207\pi\)
\(458\) 24.7677 1.15732
\(459\) 0 0
\(460\) −1.49646 −0.0697726
\(461\) 16.9508 0.789477 0.394738 0.918794i \(-0.370835\pi\)
0.394738 + 0.918794i \(0.370835\pi\)
\(462\) 0 0
\(463\) 10.9929 0.510884 0.255442 0.966824i \(-0.417779\pi\)
0.255442 + 0.966824i \(0.417779\pi\)
\(464\) −3.94370 −0.183082
\(465\) 0 0
\(466\) −12.2783 −0.568783
\(467\) −37.4175 −1.73148 −0.865738 0.500498i \(-0.833150\pi\)
−0.865738 + 0.500498i \(0.833150\pi\)
\(468\) 0 0
\(469\) −4.80453 −0.221853
\(470\) 1.84951 0.0853114
\(471\) 0 0
\(472\) 5.84951 0.269245
\(473\) −9.22912 −0.424356
\(474\) 0 0
\(475\) −3.34596 −0.153523
\(476\) −1.94370 −0.0890893
\(477\) 0 0
\(478\) −17.1771 −0.785660
\(479\) 0.465654 0.0212763 0.0106381 0.999943i \(-0.496614\pi\)
0.0106381 + 0.999943i \(0.496614\pi\)
\(480\) 0 0
\(481\) −6.69193 −0.305126
\(482\) 15.2713 0.695586
\(483\) 0 0
\(484\) 11.5457 0.524803
\(485\) −10.4472 −0.474385
\(486\) 0 0
\(487\) 34.9561 1.58401 0.792006 0.610514i \(-0.209037\pi\)
0.792006 + 0.610514i \(0.209037\pi\)
\(488\) −7.94370 −0.359594
\(489\) 0 0
\(490\) 0.251772 0.0113739
\(491\) 24.7819 1.11839 0.559195 0.829036i \(-0.311110\pi\)
0.559195 + 0.829036i \(0.311110\pi\)
\(492\) 0 0
\(493\) −2.95079 −0.132897
\(494\) 22.3909 1.00742
\(495\) 0 0
\(496\) 7.79321 0.349925
\(497\) −10.0984 −0.452976
\(498\) 0 0
\(499\) −2.33888 −0.104702 −0.0523512 0.998629i \(-0.516672\pi\)
−0.0523512 + 0.998629i \(0.516672\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 10.0379 0.448013
\(503\) 18.6161 0.830053 0.415026 0.909809i \(-0.363772\pi\)
0.415026 + 0.909809i \(0.363772\pi\)
\(504\) 0 0
\(505\) 12.1884 0.542376
\(506\) −7.10552 −0.315879
\(507\) 0 0
\(508\) −8.84242 −0.392319
\(509\) −6.89448 −0.305593 −0.152796 0.988258i \(-0.548828\pi\)
−0.152796 + 0.988258i \(0.548828\pi\)
\(510\) 0 0
\(511\) 19.4738 0.861471
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −20.9703 −0.924959
\(515\) −1.30807 −0.0576406
\(516\) 0 0
\(517\) 8.78188 0.386227
\(518\) −2.59774 −0.114138
\(519\) 0 0
\(520\) −6.69193 −0.293460
\(521\) −4.86083 −0.212957 −0.106478 0.994315i \(-0.533958\pi\)
−0.106478 + 0.994315i \(0.533958\pi\)
\(522\) 0 0
\(523\) 22.7677 0.995562 0.497781 0.867303i \(-0.334148\pi\)
0.497781 + 0.867303i \(0.334148\pi\)
\(524\) −6.15049 −0.268686
\(525\) 0 0
\(526\) 22.3725 0.975489
\(527\) 5.83110 0.254007
\(528\) 0 0
\(529\) −20.7606 −0.902636
\(530\) 10.4472 0.453799
\(531\) 0 0
\(532\) 8.69193 0.376843
\(533\) 43.1445 1.86879
\(534\) 0 0
\(535\) 3.04498 0.131646
\(536\) −1.84951 −0.0798865
\(537\) 0 0
\(538\) −4.18838 −0.180574
\(539\) 1.19547 0.0514926
\(540\) 0 0
\(541\) 8.99291 0.386636 0.193318 0.981136i \(-0.438075\pi\)
0.193318 + 0.981136i \(0.438075\pi\)
\(542\) 12.1126 0.520281
\(543\) 0 0
\(544\) −0.748228 −0.0320800
\(545\) −1.44015 −0.0616894
\(546\) 0 0
\(547\) −18.8382 −0.805463 −0.402731 0.915318i \(-0.631939\pi\)
−0.402731 + 0.915318i \(0.631939\pi\)
\(548\) −10.8945 −0.465389
\(549\) 0 0
\(550\) 4.74823 0.202465
\(551\) 13.1955 0.562146
\(552\) 0 0
\(553\) 42.9703 1.82728
\(554\) 9.30807 0.395462
\(555\) 0 0
\(556\) 1.04921 0.0444966
\(557\) −3.69901 −0.156732 −0.0783661 0.996925i \(-0.524970\pi\)
−0.0783661 + 0.996925i \(0.524970\pi\)
\(558\) 0 0
\(559\) 13.0071 0.550141
\(560\) −2.59774 −0.109774
\(561\) 0 0
\(562\) 19.2713 0.812909
\(563\) 14.3488 0.604730 0.302365 0.953192i \(-0.402224\pi\)
0.302365 + 0.953192i \(0.402224\pi\)
\(564\) 0 0
\(565\) −11.1392 −0.468628
\(566\) 13.6848 0.575217
\(567\) 0 0
\(568\) −3.88740 −0.163112
\(569\) −15.9858 −0.670161 −0.335080 0.942190i \(-0.608763\pi\)
−0.335080 + 0.942190i \(0.608763\pi\)
\(570\) 0 0
\(571\) −30.2079 −1.26416 −0.632080 0.774903i \(-0.717799\pi\)
−0.632080 + 0.774903i \(0.717799\pi\)
\(572\) −31.7748 −1.32857
\(573\) 0 0
\(574\) 16.7482 0.699058
\(575\) −1.49646 −0.0624065
\(576\) 0 0
\(577\) −19.1813 −0.798528 −0.399264 0.916836i \(-0.630734\pi\)
−0.399264 + 0.916836i \(0.630734\pi\)
\(578\) 16.4402 0.683820
\(579\) 0 0
\(580\) −3.94370 −0.163753
\(581\) 39.5722 1.64173
\(582\) 0 0
\(583\) 49.6059 2.05447
\(584\) 7.49646 0.310206
\(585\) 0 0
\(586\) −10.4472 −0.431572
\(587\) 26.6130 1.09844 0.549218 0.835679i \(-0.314926\pi\)
0.549218 + 0.835679i \(0.314926\pi\)
\(588\) 0 0
\(589\) −26.0758 −1.07443
\(590\) 5.84951 0.240820
\(591\) 0 0
\(592\) −1.00000 −0.0410997
\(593\) −26.2642 −1.07854 −0.539270 0.842133i \(-0.681300\pi\)
−0.539270 + 0.842133i \(0.681300\pi\)
\(594\) 0 0
\(595\) −1.94370 −0.0796839
\(596\) 4.18838 0.171563
\(597\) 0 0
\(598\) 10.0142 0.409510
\(599\) 38.2415 1.56251 0.781253 0.624215i \(-0.214581\pi\)
0.781253 + 0.624215i \(0.214581\pi\)
\(600\) 0 0
\(601\) 30.2447 1.23371 0.616853 0.787078i \(-0.288407\pi\)
0.616853 + 0.787078i \(0.288407\pi\)
\(602\) 5.04921 0.205791
\(603\) 0 0
\(604\) −6.69193 −0.272291
\(605\) 11.5457 0.469398
\(606\) 0 0
\(607\) −5.90157 −0.239537 −0.119769 0.992802i \(-0.538215\pi\)
−0.119769 + 0.992802i \(0.538215\pi\)
\(608\) 3.34596 0.135697
\(609\) 0 0
\(610\) −7.94370 −0.321631
\(611\) −12.3768 −0.500710
\(612\) 0 0
\(613\) 6.15473 0.248587 0.124294 0.992245i \(-0.460334\pi\)
0.124294 + 0.992245i \(0.460334\pi\)
\(614\) 4.95502 0.199968
\(615\) 0 0
\(616\) −12.3346 −0.496977
\(617\) −16.5035 −0.664408 −0.332204 0.943208i \(-0.607792\pi\)
−0.332204 + 0.943208i \(0.607792\pi\)
\(618\) 0 0
\(619\) 9.04921 0.363719 0.181859 0.983325i \(-0.441788\pi\)
0.181859 + 0.983325i \(0.441788\pi\)
\(620\) 7.79321 0.312983
\(621\) 0 0
\(622\) 3.68060 0.147579
\(623\) 15.5864 0.624456
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0.992912 0.0396848
\(627\) 0 0
\(628\) −3.94370 −0.157371
\(629\) −0.748228 −0.0298338
\(630\) 0 0
\(631\) −0.507780 −0.0202144 −0.0101072 0.999949i \(-0.503217\pi\)
−0.0101072 + 0.999949i \(0.503217\pi\)
\(632\) 16.5414 0.657983
\(633\) 0 0
\(634\) −6.55985 −0.260525
\(635\) −8.84242 −0.350901
\(636\) 0 0
\(637\) −1.68484 −0.0667558
\(638\) −18.7256 −0.741353
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 40.6356 1.60501 0.802505 0.596645i \(-0.203500\pi\)
0.802505 + 0.596645i \(0.203500\pi\)
\(642\) 0 0
\(643\) −9.53011 −0.375831 −0.187915 0.982185i \(-0.560173\pi\)
−0.187915 + 0.982185i \(0.560173\pi\)
\(644\) 3.88740 0.153185
\(645\) 0 0
\(646\) 2.50354 0.0985006
\(647\) 21.0071 0.825874 0.412937 0.910760i \(-0.364503\pi\)
0.412937 + 0.910760i \(0.364503\pi\)
\(648\) 0 0
\(649\) 27.7748 1.09026
\(650\) −6.69193 −0.262479
\(651\) 0 0
\(652\) −12.1463 −0.475684
\(653\) 45.5496 1.78249 0.891247 0.453519i \(-0.149832\pi\)
0.891247 + 0.453519i \(0.149832\pi\)
\(654\) 0 0
\(655\) −6.15049 −0.240320
\(656\) 6.44724 0.251723
\(657\) 0 0
\(658\) −4.80453 −0.187300
\(659\) −9.38385 −0.365543 −0.182772 0.983155i \(-0.558507\pi\)
−0.182772 + 0.983155i \(0.558507\pi\)
\(660\) 0 0
\(661\) 4.05630 0.157772 0.0788859 0.996884i \(-0.474864\pi\)
0.0788859 + 0.996884i \(0.474864\pi\)
\(662\) 18.0379 0.701062
\(663\) 0 0
\(664\) 15.2334 0.591169
\(665\) 8.69193 0.337058
\(666\) 0 0
\(667\) 5.90157 0.228510
\(668\) −17.0829 −0.660956
\(669\) 0 0
\(670\) −1.84951 −0.0714527
\(671\) −37.7185 −1.45611
\(672\) 0 0
\(673\) 7.60906 0.293308 0.146654 0.989188i \(-0.453150\pi\)
0.146654 + 0.989188i \(0.453150\pi\)
\(674\) 14.8945 0.573714
\(675\) 0 0
\(676\) 31.7819 1.22238
\(677\) −28.7677 −1.10563 −0.552816 0.833303i \(-0.686447\pi\)
−0.552816 + 0.833303i \(0.686447\pi\)
\(678\) 0 0
\(679\) 27.1392 1.04151
\(680\) −0.748228 −0.0286932
\(681\) 0 0
\(682\) 37.0039 1.41695
\(683\) −0.0704767 −0.00269671 −0.00134836 0.999999i \(-0.500429\pi\)
−0.00134836 + 0.999999i \(0.500429\pi\)
\(684\) 0 0
\(685\) −10.8945 −0.416257
\(686\) −18.8382 −0.719245
\(687\) 0 0
\(688\) 1.94370 0.0741028
\(689\) −69.9122 −2.66344
\(690\) 0 0
\(691\) 10.6498 0.405137 0.202569 0.979268i \(-0.435071\pi\)
0.202569 + 0.979268i \(0.435071\pi\)
\(692\) 15.3417 0.583205
\(693\) 0 0
\(694\) −9.38385 −0.356206
\(695\) 1.04921 0.0397990
\(696\) 0 0
\(697\) 4.82401 0.182722
\(698\) 32.9703 1.24794
\(699\) 0 0
\(700\) −2.59774 −0.0981852
\(701\) 7.98582 0.301620 0.150810 0.988563i \(-0.451812\pi\)
0.150810 + 0.988563i \(0.451812\pi\)
\(702\) 0 0
\(703\) 3.34596 0.126195
\(704\) −4.74823 −0.178956
\(705\) 0 0
\(706\) 27.4260 1.03219
\(707\) −31.6622 −1.19078
\(708\) 0 0
\(709\) −37.2149 −1.39764 −0.698818 0.715299i \(-0.746290\pi\)
−0.698818 + 0.715299i \(0.746290\pi\)
\(710\) −3.88740 −0.145891
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) −11.6622 −0.436752
\(714\) 0 0
\(715\) −31.7748 −1.18831
\(716\) −2.45148 −0.0916161
\(717\) 0 0
\(718\) −25.0829 −0.936084
\(719\) −30.1742 −1.12531 −0.562654 0.826692i \(-0.690220\pi\)
−0.562654 + 0.826692i \(0.690220\pi\)
\(720\) 0 0
\(721\) 3.39803 0.126549
\(722\) 7.80453 0.290455
\(723\) 0 0
\(724\) 13.1955 0.490406
\(725\) −3.94370 −0.146465
\(726\) 0 0
\(727\) −18.7677 −0.696056 −0.348028 0.937484i \(-0.613149\pi\)
−0.348028 + 0.937484i \(0.613149\pi\)
\(728\) 17.3839 0.644288
\(729\) 0 0
\(730\) 7.49646 0.277456
\(731\) 1.45433 0.0537903
\(732\) 0 0
\(733\) 26.1094 0.964374 0.482187 0.876068i \(-0.339843\pi\)
0.482187 + 0.876068i \(0.339843\pi\)
\(734\) −2.48513 −0.0917279
\(735\) 0 0
\(736\) 1.49646 0.0551601
\(737\) −8.78188 −0.323485
\(738\) 0 0
\(739\) 42.6130 1.56754 0.783772 0.621049i \(-0.213293\pi\)
0.783772 + 0.621049i \(0.213293\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 0 0
\(742\) −27.1392 −0.996310
\(743\) 35.4922 1.30208 0.651042 0.759042i \(-0.274332\pi\)
0.651042 + 0.759042i \(0.274332\pi\)
\(744\) 0 0
\(745\) 4.18838 0.153450
\(746\) 35.1586 1.28725
\(747\) 0 0
\(748\) −3.55276 −0.129902
\(749\) −7.91005 −0.289027
\(750\) 0 0
\(751\) 21.5722 0.787182 0.393591 0.919286i \(-0.371233\pi\)
0.393591 + 0.919286i \(0.371233\pi\)
\(752\) −1.84951 −0.0674446
\(753\) 0 0
\(754\) 26.3909 0.961101
\(755\) −6.69193 −0.243544
\(756\) 0 0
\(757\) −23.7890 −0.864625 −0.432312 0.901724i \(-0.642302\pi\)
−0.432312 + 0.901724i \(0.642302\pi\)
\(758\) −24.6778 −0.896336
\(759\) 0 0
\(760\) 3.34596 0.121371
\(761\) 18.6724 0.676876 0.338438 0.940989i \(-0.390102\pi\)
0.338438 + 0.940989i \(0.390102\pi\)
\(762\) 0 0
\(763\) 3.74114 0.135438
\(764\) 21.7790 0.787938
\(765\) 0 0
\(766\) −13.3839 −0.483578
\(767\) −39.1445 −1.41342
\(768\) 0 0
\(769\) −2.48937 −0.0897689 −0.0448845 0.998992i \(-0.514292\pi\)
−0.0448845 + 0.998992i \(0.514292\pi\)
\(770\) −12.3346 −0.444510
\(771\) 0 0
\(772\) −12.9929 −0.467625
\(773\) −0.950786 −0.0341974 −0.0170987 0.999854i \(-0.505443\pi\)
−0.0170987 + 0.999854i \(0.505443\pi\)
\(774\) 0 0
\(775\) 7.79321 0.279940
\(776\) 10.4472 0.375034
\(777\) 0 0
\(778\) 18.2220 0.653292
\(779\) −21.5722 −0.772906
\(780\) 0 0
\(781\) −18.4582 −0.660488
\(782\) 1.11969 0.0400401
\(783\) 0 0
\(784\) −0.251772 −0.00899185
\(785\) −3.94370 −0.140757
\(786\) 0 0
\(787\) −29.4359 −1.04928 −0.524639 0.851325i \(-0.675800\pi\)
−0.524639 + 0.851325i \(0.675800\pi\)
\(788\) −0.616147 −0.0219493
\(789\) 0 0
\(790\) 16.5414 0.588518
\(791\) 28.9366 1.02887
\(792\) 0 0
\(793\) 53.1586 1.88772
\(794\) 2.22521 0.0789696
\(795\) 0 0
\(796\) −1.54852 −0.0548859
\(797\) 14.1742 0.502076 0.251038 0.967977i \(-0.419228\pi\)
0.251038 + 0.967977i \(0.419228\pi\)
\(798\) 0 0
\(799\) −1.38385 −0.0489572
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 12.3909 0.437539
\(803\) 35.5949 1.25612
\(804\) 0 0
\(805\) 3.88740 0.137013
\(806\) −52.1516 −1.83696
\(807\) 0 0
\(808\) −12.1884 −0.428786
\(809\) 32.9929 1.15997 0.579985 0.814627i \(-0.303059\pi\)
0.579985 + 0.814627i \(0.303059\pi\)
\(810\) 0 0
\(811\) 40.5567 1.42414 0.712069 0.702110i \(-0.247758\pi\)
0.712069 + 0.702110i \(0.247758\pi\)
\(812\) 10.2447 0.359518
\(813\) 0 0
\(814\) −4.74823 −0.166425
\(815\) −12.1463 −0.425465
\(816\) 0 0
\(817\) −6.50354 −0.227530
\(818\) −7.27125 −0.254233
\(819\) 0 0
\(820\) 6.44724 0.225147
\(821\) 38.8661 1.35644 0.678219 0.734860i \(-0.262752\pi\)
0.678219 + 0.734860i \(0.262752\pi\)
\(822\) 0 0
\(823\) −49.6243 −1.72979 −0.864897 0.501949i \(-0.832616\pi\)
−0.864897 + 0.501949i \(0.832616\pi\)
\(824\) 1.30807 0.0455689
\(825\) 0 0
\(826\) −15.1955 −0.528718
\(827\) 30.0195 1.04388 0.521940 0.852982i \(-0.325209\pi\)
0.521940 + 0.852982i \(0.325209\pi\)
\(828\) 0 0
\(829\) 10.5598 0.366759 0.183379 0.983042i \(-0.441296\pi\)
0.183379 + 0.983042i \(0.441296\pi\)
\(830\) 15.2334 0.528758
\(831\) 0 0
\(832\) 6.69193 0.232001
\(833\) −0.188383 −0.00652708
\(834\) 0 0
\(835\) −17.0829 −0.591177
\(836\) 15.8874 0.549477
\(837\) 0 0
\(838\) 28.4809 0.983856
\(839\) −29.3839 −1.01444 −0.507222 0.861816i \(-0.669327\pi\)
−0.507222 + 0.861816i \(0.669327\pi\)
\(840\) 0 0
\(841\) −13.4472 −0.463698
\(842\) −22.0000 −0.758170
\(843\) 0 0
\(844\) 20.9366 0.720668
\(845\) 31.7819 1.09333
\(846\) 0 0
\(847\) −29.9926 −1.03056
\(848\) −10.4472 −0.358760
\(849\) 0 0
\(850\) −0.748228 −0.0256640
\(851\) 1.49646 0.0512979
\(852\) 0 0
\(853\) 6.22521 0.213147 0.106573 0.994305i \(-0.466012\pi\)
0.106573 + 0.994305i \(0.466012\pi\)
\(854\) 20.6356 0.706137
\(855\) 0 0
\(856\) −3.04498 −0.104075
\(857\) 3.65119 0.124722 0.0623611 0.998054i \(-0.480137\pi\)
0.0623611 + 0.998054i \(0.480137\pi\)
\(858\) 0 0
\(859\) −36.3162 −1.23909 −0.619547 0.784960i \(-0.712684\pi\)
−0.619547 + 0.784960i \(0.712684\pi\)
\(860\) 1.94370 0.0662796
\(861\) 0 0
\(862\) 1.51487 0.0515966
\(863\) −24.5751 −0.836546 −0.418273 0.908321i \(-0.637364\pi\)
−0.418273 + 0.908321i \(0.637364\pi\)
\(864\) 0 0
\(865\) 15.3417 0.521634
\(866\) −24.9929 −0.849294
\(867\) 0 0
\(868\) −20.2447 −0.687149
\(869\) 78.5425 2.66437
\(870\) 0 0
\(871\) 12.3768 0.419371
\(872\) 1.44015 0.0487698
\(873\) 0 0
\(874\) −5.00709 −0.169367
\(875\) −2.59774 −0.0878195
\(876\) 0 0
\(877\) 20.0563 0.677253 0.338627 0.940921i \(-0.390038\pi\)
0.338627 + 0.940921i \(0.390038\pi\)
\(878\) 5.10128 0.172160
\(879\) 0 0
\(880\) −4.74823 −0.160063
\(881\) 15.0266 0.506258 0.253129 0.967433i \(-0.418540\pi\)
0.253129 + 0.967433i \(0.418540\pi\)
\(882\) 0 0
\(883\) −6.62145 −0.222830 −0.111415 0.993774i \(-0.535538\pi\)
−0.111415 + 0.993774i \(0.535538\pi\)
\(884\) 5.00709 0.168407
\(885\) 0 0
\(886\) 25.7369 0.864648
\(887\) 37.5538 1.26093 0.630467 0.776216i \(-0.282863\pi\)
0.630467 + 0.776216i \(0.282863\pi\)
\(888\) 0 0
\(889\) 22.9703 0.770398
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −2.71034 −0.0907488
\(893\) 6.18838 0.207086
\(894\) 0 0
\(895\) −2.45148 −0.0819439
\(896\) 2.59774 0.0867842
\(897\) 0 0
\(898\) −17.7748 −0.593153
\(899\) −30.7341 −1.02504
\(900\) 0 0
\(901\) −7.81692 −0.260419
\(902\) 30.6130 1.01930
\(903\) 0 0
\(904\) 11.1392 0.370483
\(905\) 13.1955 0.438632
\(906\) 0 0
\(907\) 50.4667 1.67572 0.837860 0.545885i \(-0.183807\pi\)
0.837860 + 0.545885i \(0.183807\pi\)
\(908\) −26.1321 −0.867224
\(909\) 0 0
\(910\) 17.3839 0.576269
\(911\) −15.9395 −0.528098 −0.264049 0.964509i \(-0.585058\pi\)
−0.264049 + 0.964509i \(0.585058\pi\)
\(912\) 0 0
\(913\) 72.3315 2.39382
\(914\) −34.3346 −1.13569
\(915\) 0 0
\(916\) −24.7677 −0.818348
\(917\) 15.9774 0.527619
\(918\) 0 0
\(919\) 32.5414 1.07344 0.536721 0.843760i \(-0.319663\pi\)
0.536721 + 0.843760i \(0.319663\pi\)
\(920\) 1.49646 0.0493367
\(921\) 0 0
\(922\) −16.9508 −0.558244
\(923\) 26.0142 0.856267
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) −10.9929 −0.361250
\(927\) 0 0
\(928\) 3.94370 0.129458
\(929\) 34.1094 1.11909 0.559547 0.828799i \(-0.310975\pi\)
0.559547 + 0.828799i \(0.310975\pi\)
\(930\) 0 0
\(931\) 0.842420 0.0276092
\(932\) 12.2783 0.402190
\(933\) 0 0
\(934\) 37.4175 1.22434
\(935\) −3.55276 −0.116188
\(936\) 0 0
\(937\) 0.0141751 0.000463082 0 0.000231541 1.00000i \(-0.499926\pi\)
0.000231541 1.00000i \(0.499926\pi\)
\(938\) 4.80453 0.156873
\(939\) 0 0
\(940\) −1.84951 −0.0603243
\(941\) −11.1813 −0.364500 −0.182250 0.983252i \(-0.558338\pi\)
−0.182250 + 0.983252i \(0.558338\pi\)
\(942\) 0 0
\(943\) −9.64802 −0.314183
\(944\) −5.84951 −0.190385
\(945\) 0 0
\(946\) 9.22912 0.300065
\(947\) 53.6427 1.74315 0.871577 0.490259i \(-0.163098\pi\)
0.871577 + 0.490259i \(0.163098\pi\)
\(948\) 0 0
\(949\) −50.1657 −1.62845
\(950\) 3.34596 0.108557
\(951\) 0 0
\(952\) 1.94370 0.0629956
\(953\) 9.88740 0.320284 0.160142 0.987094i \(-0.448805\pi\)
0.160142 + 0.987094i \(0.448805\pi\)
\(954\) 0 0
\(955\) 21.7790 0.704753
\(956\) 17.1771 0.555546
\(957\) 0 0
\(958\) −0.465654 −0.0150446
\(959\) 28.3010 0.913886
\(960\) 0 0
\(961\) 29.7341 0.959163
\(962\) 6.69193 0.215756
\(963\) 0 0
\(964\) −15.2713 −0.491854
\(965\) −12.9929 −0.418257
\(966\) 0 0
\(967\) 51.6254 1.66016 0.830080 0.557644i \(-0.188295\pi\)
0.830080 + 0.557644i \(0.188295\pi\)
\(968\) −11.5457 −0.371092
\(969\) 0 0
\(970\) 10.4472 0.335441
\(971\) −20.5598 −0.659797 −0.329898 0.944016i \(-0.607014\pi\)
−0.329898 + 0.944016i \(0.607014\pi\)
\(972\) 0 0
\(973\) −2.72558 −0.0873781
\(974\) −34.9561 −1.12007
\(975\) 0 0
\(976\) 7.94370 0.254272
\(977\) −38.4472 −1.23004 −0.615018 0.788513i \(-0.710851\pi\)
−0.615018 + 0.788513i \(0.710851\pi\)
\(978\) 0 0
\(979\) 28.4894 0.910524
\(980\) −0.251772 −0.00804256
\(981\) 0 0
\(982\) −24.7819 −0.790822
\(983\) −11.8973 −0.379466 −0.189733 0.981836i \(-0.560762\pi\)
−0.189733 + 0.981836i \(0.560762\pi\)
\(984\) 0 0
\(985\) −0.616147 −0.0196321
\(986\) 2.95079 0.0939722
\(987\) 0 0
\(988\) −22.3909 −0.712351
\(989\) −2.90866 −0.0924900
\(990\) 0 0
\(991\) −19.7932 −0.628752 −0.314376 0.949299i \(-0.601795\pi\)
−0.314376 + 0.949299i \(0.601795\pi\)
\(992\) −7.79321 −0.247435
\(993\) 0 0
\(994\) 10.0984 0.320303
\(995\) −1.54852 −0.0490914
\(996\) 0 0
\(997\) 5.88740 0.186456 0.0932279 0.995645i \(-0.470281\pi\)
0.0932279 + 0.995645i \(0.470281\pi\)
\(998\) 2.33888 0.0740358
\(999\) 0 0
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 3330.2.a.bg.1.1 3
3.2 odd 2 370.2.a.g.1.1 3
12.11 even 2 2960.2.a.u.1.3 3
15.2 even 4 1850.2.b.o.149.6 6
15.8 even 4 1850.2.b.o.149.1 6
15.14 odd 2 1850.2.a.z.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.g.1.1 3 3.2 odd 2
1850.2.a.z.1.3 3 15.14 odd 2
1850.2.b.o.149.1 6 15.8 even 4
1850.2.b.o.149.6 6 15.2 even 4
2960.2.a.u.1.3 3 12.11 even 2
3330.2.a.bg.1.1 3 1.1 even 1 trivial