↑

Properties

Label 3344.2.a.e.1.1
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
Analytic rank $1$
Dimension $1$
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] = [3344,2,Mod(1,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3344.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.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: \(26.7019744359\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1672)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3344.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.00000 q^{5} +3.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.00000 q^{5} +3.00000 q^{7} -2.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.00000 q^{15} -7.00000 q^{17} +1.00000 q^{19} -3.00000 q^{21} -3.00000 q^{23} -1.00000 q^{25} +5.00000 q^{27} -1.00000 q^{29} -2.00000 q^{31} -1.00000 q^{33} +6.00000 q^{35} -10.0000 q^{37} +1.00000 q^{39} -6.00000 q^{41} -8.00000 q^{43} -4.00000 q^{45} +2.00000 q^{49} +7.00000 q^{51} +1.00000 q^{53} +2.00000 q^{55} -1.00000 q^{57} -5.00000 q^{59} -8.00000 q^{61} -6.00000 q^{63} -2.00000 q^{65} +3.00000 q^{67} +3.00000 q^{69} -6.00000 q^{71} +3.00000 q^{73} +1.00000 q^{75} +3.00000 q^{77} +4.00000 q^{79} +1.00000 q^{81} +8.00000 q^{83} -14.0000 q^{85} +1.00000 q^{87} +12.0000 q^{89} -3.00000 q^{91} +2.00000 q^{93} +2.00000 q^{95} +4.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 − 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) −7.00000 −1.69775 −0.848875 − 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 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
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 6.00000 1.01419
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) −4.00000 −0.596285
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 7.00000 0.980196
\(52\) 0 0
\(53\) 1.00000 0.137361 0.0686803 − 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −5.00000 −0.650945 −0.325472 − 0.945552i \(-0.605523\pi\)
−0.325472 + 0.945552i \(0.605523\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) −6.00000 −0.755929
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 3.00000 0.366508 0.183254 − 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 3.00000 0.351123 0.175562 − 0.984468i \(-0.443826\pi\)
0.175562 + 0.984468i \(0.443826\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.00000 0.878114 0.439057 − 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) −14.0000 −1.51851
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 − 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 0 0
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) 4.00000 0.406138 0.203069 − 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) −6.00000 −0.585540
\(106\) 0 0
\(107\) 11.0000 1.06341 0.531705 − 0.846930i \(-0.321551\pi\)
0.531705 + 0.846930i \(0.321551\pi\)
\(108\) 0 0
\(109\) −3.00000 −0.287348 −0.143674 − 0.989625i \(-0.545892\pi\)
−0.143674 + 0.989625i \(0.545892\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) 0 0
\(113\) 8.00000 0.752577 0.376288 − 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) −21.0000 −1.92507
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 − 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 − 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 3.00000 0.260133
\(134\) 0 0
\(135\) 10.0000 0.860663
\(136\) 0 0
\(137\) 15.0000 1.28154 0.640768 − 0.767734i \(-0.278616\pi\)
0.640768 + 0.767734i \(0.278616\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 − 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 − 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 − 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) 14.0000 1.13183
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 − 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 0 0
\(159\) −1.00000 −0.0793052
\(160\) 0 0
\(161\) −9.00000 −0.709299
\(162\) 0 0
\(163\) −10.0000 −0.783260 −0.391630 − 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 0 0
\(165\) −2.00000 −0.155700
\(166\) 0 0
\(167\) 2.00000 0.154765 0.0773823 − 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) 0 0
\(177\) 5.00000 0.375823
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 − 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) −20.0000 −1.47043
\(186\) 0 0
\(187\) −7.00000 −0.511891
\(188\) 0 0
\(189\) 15.0000 1.09109
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 − 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 0 0
\(193\) −12.0000 −0.863779 −0.431889 − 0.901927i \(-0.642153\pi\)
−0.431889 + 0.901927i \(0.642153\pi\)
\(194\) 0 0
\(195\) 2.00000 0.143223
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 − 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) −15.0000 −1.06332 −0.531661 − 0.846957i \(-0.678432\pi\)
−0.531661 + 0.846957i \(0.678432\pi\)
\(200\) 0 0
\(201\) −3.00000 −0.211604
\(202\) 0 0
\(203\) −3.00000 −0.210559
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −15.0000 −1.03264 −0.516321 − 0.856395i \(-0.672699\pi\)
−0.516321 + 0.856395i \(0.672699\pi\)
\(212\) 0 0
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) −16.0000 −1.09119
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) 0 0
\(219\) −3.00000 −0.202721
\(220\) 0 0
\(221\) 7.00000 0.470871
\(222\) 0 0
\(223\) 26.0000 1.74109 0.870544 − 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) 0 0
\(225\) 2.00000 0.133333
\(226\) 0 0
\(227\) −7.00000 −0.464606 −0.232303 − 0.972643i \(-0.574626\pi\)
−0.232303 + 0.972643i \(0.574626\pi\)
\(228\) 0 0
\(229\) 16.0000 1.05731 0.528655 − 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) −10.0000 −0.655122 −0.327561 − 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) 15.0000 0.970269 0.485135 − 0.874439i \(-0.338771\pi\)
0.485135 + 0.874439i \(0.338771\pi\)
\(240\) 0 0
\(241\) 12.0000 0.772988 0.386494 − 0.922292i \(-0.373686\pi\)
0.386494 + 0.922292i \(0.373686\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) 4.00000 0.255551
\(246\) 0 0
\(247\) −1.00000 −0.0636285
\(248\) 0 0
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) 10.0000 0.631194 0.315597 − 0.948893i \(-0.397795\pi\)
0.315597 + 0.948893i \(0.397795\pi\)
\(252\) 0 0
\(253\) −3.00000 −0.188608
\(254\) 0 0
\(255\) 14.0000 0.876714
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 − 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) −30.0000 −1.86411
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) −4.00000 −0.246651 −0.123325 − 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) 0 0
\(269\) −2.00000 −0.121942 −0.0609711 − 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) −1.00000 −0.0607457 −0.0303728 − 0.999539i \(-0.509669\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(272\) 0 0
\(273\) 3.00000 0.181568
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 12.0000 0.721010 0.360505 − 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 − 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 − 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 0 0
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) −18.0000 −1.06251
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) −4.00000 −0.234484
\(292\) 0 0
\(293\) 17.0000 0.993151 0.496575 − 0.867994i \(-0.334591\pi\)
0.496575 + 0.867994i \(0.334591\pi\)
\(294\) 0 0
\(295\) −10.0000 −0.582223
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) −16.0000 −0.916157
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 − 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −19.0000 −1.07739 −0.538696 − 0.842500i \(-0.681083\pi\)
−0.538696 + 0.842500i \(0.681083\pi\)
\(312\) 0 0
\(313\) −19.0000 −1.07394 −0.536972 − 0.843600i \(-0.680432\pi\)
−0.536972 + 0.843600i \(0.680432\pi\)
\(314\) 0 0
\(315\) −12.0000 −0.676123
\(316\) 0 0
\(317\) 3.00000 0.168497 0.0842484 − 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) 0 0
\(319\) −1.00000 −0.0559893
\(320\) 0 0
\(321\) −11.0000 −0.613960
\(322\) 0 0
\(323\) −7.00000 −0.389490
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) 3.00000 0.165900
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −19.0000 −1.04433 −0.522167 − 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) 0 0
\(333\) 20.0000 1.09599
\(334\) 0 0
\(335\) 6.00000 0.327815
\(336\) 0 0
\(337\) −24.0000 −1.30736 −0.653682 − 0.756770i \(-0.726776\pi\)
−0.653682 + 0.756770i \(0.726776\pi\)
\(338\) 0 0
\(339\) −8.00000 −0.434500
\(340\) 0 0
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 6.00000 0.323029
\(346\) 0 0
\(347\) −18.0000 −0.966291 −0.483145 − 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 − 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) 0 0
\(353\) −7.00000 −0.372572 −0.186286 − 0.982496i \(-0.559645\pi\)
−0.186286 + 0.982496i \(0.559645\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 0 0
\(357\) 21.0000 1.11144
\(358\) 0 0
\(359\) 17.0000 0.897226 0.448613 − 0.893726i \(-0.351918\pi\)
0.448613 + 0.893726i \(0.351918\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 − 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) 3.00000 0.155752
\(372\) 0 0
\(373\) 13.0000 0.673114 0.336557 − 0.941663i \(-0.390737\pi\)
0.336557 + 0.941663i \(0.390737\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) 1.00000 0.0515026
\(378\) 0 0
\(379\) 27.0000 1.38690 0.693448 − 0.720506i \(-0.256091\pi\)
0.693448 + 0.720506i \(0.256091\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 0 0
\(383\) 2.00000 0.102195 0.0510976 − 0.998694i \(-0.483728\pi\)
0.0510976 + 0.998694i \(0.483728\pi\)
\(384\) 0 0
\(385\) 6.00000 0.305788
\(386\) 0 0
\(387\) 16.0000 0.813326
\(388\) 0 0
\(389\) −14.0000 −0.709828 −0.354914 − 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 21.0000 1.06202
\(392\) 0 0
\(393\) 6.00000 0.302660
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −26.0000 −1.30490 −0.652451 − 0.757831i \(-0.726259\pi\)
−0.652451 + 0.757831i \(0.726259\pi\)
\(398\) 0 0
\(399\) −3.00000 −0.150188
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 − 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 2.00000 0.0996271
\(404\) 0 0
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) −10.0000 −0.495682
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 − 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −15.0000 −0.739895
\(412\) 0 0
\(413\) −15.0000 −0.738102
\(414\) 0 0
\(415\) 16.0000 0.785409
\(416\) 0 0
\(417\) 14.0000 0.685583
\(418\) 0 0
\(419\) 18.0000 0.879358 0.439679 − 0.898155i \(-0.355092\pi\)
0.439679 + 0.898155i \(0.355092\pi\)
\(420\) 0 0
\(421\) 41.0000 1.99822 0.999109 − 0.0422075i \(-0.0134391\pi\)
0.999109 + 0.0422075i \(0.0134391\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 7.00000 0.339550
\(426\) 0 0
\(427\) −24.0000 −1.16144
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 − 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 0 0
\(433\) 40.0000 1.92228 0.961139 − 0.276066i \(-0.0890309\pi\)
0.961139 + 0.276066i \(0.0890309\pi\)
\(434\) 0 0
\(435\) 2.00000 0.0958927
\(436\) 0 0
\(437\) −3.00000 −0.143509
\(438\) 0 0
\(439\) −22.0000 −1.05000 −0.525001 − 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 0 0
\(443\) −34.0000 −1.61539 −0.807694 − 0.589601i \(-0.799285\pi\)
−0.807694 + 0.589601i \(0.799285\pi\)
\(444\) 0 0
\(445\) 24.0000 1.13771
\(446\) 0 0
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) −24.0000 −1.13263 −0.566315 − 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 0 0
\(453\) −4.00000 −0.187936
\(454\) 0 0
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) 19.0000 0.888783 0.444391 − 0.895833i \(-0.353420\pi\)
0.444391 + 0.895833i \(0.353420\pi\)
\(458\) 0 0
\(459\) −35.0000 −1.63366
\(460\) 0 0
\(461\) −24.0000 −1.11779 −0.558896 − 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 − 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) 0 0
\(467\) 16.0000 0.740392 0.370196 − 0.928954i \(-0.379291\pi\)
0.370196 + 0.928954i \(0.379291\pi\)
\(468\) 0 0
\(469\) 9.00000 0.415581
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) 28.0000 1.27935 0.639676 − 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) 9.00000 0.409514
\(484\) 0 0
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) −4.00000 −0.181257 −0.0906287 − 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) 0 0
\(489\) 10.0000 0.452216
\(490\) 0 0
\(491\) −34.0000 −1.53440 −0.767199 − 0.641409i \(-0.778350\pi\)
−0.767199 + 0.641409i \(0.778350\pi\)
\(492\) 0 0
\(493\) 7.00000 0.315264
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) 0 0
\(497\) −18.0000 −0.807410
\(498\) 0 0
\(499\) 8.00000 0.358129 0.179065 − 0.983837i \(-0.442693\pi\)
0.179065 + 0.983837i \(0.442693\pi\)
\(500\) 0 0
\(501\) −2.00000 −0.0893534
\(502\) 0 0
\(503\) 27.0000 1.20387 0.601935 − 0.798545i \(-0.294397\pi\)
0.601935 + 0.798545i \(0.294397\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 0 0
\(509\) 34.0000 1.50702 0.753512 − 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) 0 0
\(511\) 9.00000 0.398137
\(512\) 0 0
\(513\) 5.00000 0.220755
\(514\) 0 0
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −20.0000 −0.876216 −0.438108 − 0.898922i \(-0.644351\pi\)
−0.438108 + 0.898922i \(0.644351\pi\)
\(522\) 0 0
\(523\) −5.00000 −0.218635 −0.109317 − 0.994007i \(-0.534866\pi\)
−0.109317 + 0.994007i \(0.534866\pi\)
\(524\) 0 0
\(525\) 3.00000 0.130931
\(526\) 0 0
\(527\) 14.0000 0.609850
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 10.0000 0.433963
\(532\) 0 0
\(533\) 6.00000 0.259889
\(534\) 0 0
\(535\) 22.0000 0.951143
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 − 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) 0 0
\(543\) 22.0000 0.944110
\(544\) 0 0
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 − 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) 16.0000 0.682863
\(550\) 0 0
\(551\) −1.00000 −0.0426014
\(552\) 0 0
\(553\) 12.0000 0.510292
\(554\) 0 0
\(555\) 20.0000 0.848953
\(556\) 0 0
\(557\) −34.0000 −1.44063 −0.720313 − 0.693649i \(-0.756002\pi\)
−0.720313 + 0.693649i \(0.756002\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 7.00000 0.295540
\(562\) 0 0
\(563\) 40.0000 1.68580 0.842900 − 0.538071i \(-0.180847\pi\)
0.842900 + 0.538071i \(0.180847\pi\)
\(564\) 0 0
\(565\) 16.0000 0.673125
\(566\) 0 0
\(567\) 3.00000 0.125988
\(568\) 0 0
\(569\) 20.0000 0.838444 0.419222 − 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 0 0
\(571\) −2.00000 −0.0836974 −0.0418487 − 0.999124i \(-0.513325\pi\)
−0.0418487 + 0.999124i \(0.513325\pi\)
\(572\) 0 0
\(573\) 3.00000 0.125327
\(574\) 0 0
\(575\) 3.00000 0.125109
\(576\) 0 0
\(577\) 47.0000 1.95664 0.978318 − 0.207109i \(-0.0664056\pi\)
0.978318 + 0.207109i \(0.0664056\pi\)
\(578\) 0 0
\(579\) 12.0000 0.498703
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) 1.00000 0.0414158
\(584\) 0 0
\(585\) 4.00000 0.165380
\(586\) 0 0
\(587\) 44.0000 1.81607 0.908037 − 0.418890i \(-0.137581\pi\)
0.908037 + 0.418890i \(0.137581\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) 0 0
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) −42.0000 −1.72183
\(596\) 0 0
\(597\) 15.0000 0.613909
\(598\) 0 0
\(599\) 4.00000 0.163436 0.0817178 − 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 − 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 0 0
\(603\) −6.00000 −0.244339
\(604\) 0 0
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −18.0000 −0.730597 −0.365299 − 0.930890i \(-0.619033\pi\)
−0.365299 + 0.930890i \(0.619033\pi\)
\(608\) 0 0
\(609\) 3.00000 0.121566
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 44.0000 1.77714 0.888572 − 0.458738i \(-0.151698\pi\)
0.888572 + 0.458738i \(0.151698\pi\)
\(614\) 0 0
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) −40.0000 −1.60774 −0.803868 − 0.594808i \(-0.797228\pi\)
−0.803868 + 0.594808i \(0.797228\pi\)
\(620\) 0 0
\(621\) −15.0000 −0.601929
\(622\) 0 0
\(623\) 36.0000 1.44231
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) −1.00000 −0.0399362
\(628\) 0 0
\(629\) 70.0000 2.79108
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 15.0000 0.596196
\(634\) 0 0
\(635\) 24.0000 0.952411
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 24.0000 0.947943 0.473972 − 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 0 0
\(643\) −2.00000 −0.0788723 −0.0394362 − 0.999222i \(-0.512556\pi\)
−0.0394362 + 0.999222i \(0.512556\pi\)
\(644\) 0 0
\(645\) 16.0000 0.629999
\(646\) 0 0
\(647\) 29.0000 1.14011 0.570054 − 0.821607i \(-0.306922\pi\)
0.570054 + 0.821607i \(0.306922\pi\)
\(648\) 0 0
\(649\) −5.00000 −0.196267
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) 0 0
\(653\) −24.0000 −0.939193 −0.469596 − 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) 0 0
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) −45.0000 −1.75295 −0.876476 − 0.481446i \(-0.840112\pi\)
−0.876476 + 0.481446i \(0.840112\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 − 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 0 0
\(663\) −7.00000 −0.271857
\(664\) 0 0
\(665\) 6.00000 0.232670
\(666\) 0 0
\(667\) 3.00000 0.116160
\(668\) 0 0
\(669\) −26.0000 −1.00522
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) −24.0000 −0.925132 −0.462566 − 0.886585i \(-0.653071\pi\)
−0.462566 + 0.886585i \(0.653071\pi\)
\(674\) 0 0
\(675\) −5.00000 −0.192450
\(676\) 0 0
\(677\) 13.0000 0.499631 0.249815 − 0.968294i \(-0.419630\pi\)
0.249815 + 0.968294i \(0.419630\pi\)
\(678\) 0 0
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) 7.00000 0.268241
\(682\) 0 0
\(683\) −8.00000 −0.306111 −0.153056 − 0.988218i \(-0.548911\pi\)
−0.153056 + 0.988218i \(0.548911\pi\)
\(684\) 0 0
\(685\) 30.0000 1.14624
\(686\) 0 0
\(687\) −16.0000 −0.610438
\(688\) 0 0
\(689\) −1.00000 −0.0380970
\(690\) 0 0
\(691\) −30.0000 −1.14125 −0.570627 − 0.821209i \(-0.693300\pi\)
−0.570627 + 0.821209i \(0.693300\pi\)
\(692\) 0 0
\(693\) −6.00000 −0.227921
\(694\) 0 0
\(695\) −28.0000 −1.06210
\(696\) 0 0
\(697\) 42.0000 1.59086
\(698\) 0 0
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −10.0000 −0.377157
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18.0000 0.676960
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 − 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) −2.00000 −0.0747958
\(716\) 0 0
\(717\) −15.0000 −0.560185
\(718\) 0 0
\(719\) 49.0000 1.82739 0.913696 − 0.406399i \(-0.133216\pi\)
0.913696 + 0.406399i \(0.133216\pi\)
\(720\) 0 0
\(721\) −12.0000 −0.446903
\(722\) 0 0
\(723\) −12.0000 −0.446285
\(724\) 0 0
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 17.0000 0.630495 0.315248 − 0.949009i \(-0.397912\pi\)
0.315248 + 0.949009i \(0.397912\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 56.0000 2.07123
\(732\) 0 0
\(733\) −50.0000 −1.84679 −0.923396 − 0.383849i \(-0.874598\pi\)
−0.923396 + 0.383849i \(0.874598\pi\)
\(734\) 0 0
\(735\) −4.00000 −0.147542
\(736\) 0 0
\(737\) 3.00000 0.110506
\(738\) 0 0
\(739\) −28.0000 −1.03000 −0.514998 − 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 0 0
\(741\) 1.00000 0.0367359
\(742\) 0 0
\(743\) 36.0000 1.32071 0.660356 − 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) −20.0000 −0.732743
\(746\) 0 0
\(747\) −16.0000 −0.585409
\(748\) 0 0
\(749\) 33.0000 1.20579
\(750\) 0 0
\(751\) 46.0000 1.67856 0.839282 − 0.543696i \(-0.182976\pi\)
0.839282 + 0.543696i \(0.182976\pi\)
\(752\) 0 0
\(753\) −10.0000 −0.364420
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 − 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 3.00000 0.108893
\(760\) 0 0
\(761\) 5.00000 0.181250 0.0906249 − 0.995885i \(-0.471114\pi\)
0.0906249 + 0.995885i \(0.471114\pi\)
\(762\) 0 0
\(763\) −9.00000 −0.325822
\(764\) 0 0
\(765\) 28.0000 1.01234
\(766\) 0 0
\(767\) 5.00000 0.180540
\(768\) 0 0
\(769\) −25.0000 −0.901523 −0.450762 − 0.892644i \(-0.648848\pi\)
−0.450762 + 0.892644i \(0.648848\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 0 0
\(773\) 31.0000 1.11499 0.557496 − 0.830179i \(-0.311762\pi\)
0.557496 + 0.830179i \(0.311762\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 0 0
\(777\) 30.0000 1.07624
\(778\) 0 0
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) 0 0
\(783\) −5.00000 −0.178685
\(784\) 0 0
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) −23.0000 −0.819861 −0.409931 − 0.912117i \(-0.634447\pi\)
−0.409931 + 0.912117i \(0.634447\pi\)
\(788\) 0 0
\(789\) 4.00000 0.142404
\(790\) 0 0
\(791\) 24.0000 0.853342
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 0 0
\(795\) −2.00000 −0.0709327
\(796\) 0 0
\(797\) −27.0000 −0.956389 −0.478195 − 0.878254i \(-0.658709\pi\)
−0.478195 + 0.878254i \(0.658709\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −24.0000 −0.847998
\(802\) 0 0
\(803\) 3.00000 0.105868
\(804\) 0 0
\(805\) −18.0000 −0.634417
\(806\) 0 0
\(807\) 2.00000 0.0704033
\(808\) 0 0
\(809\) 27.0000 0.949269 0.474635 − 0.880183i \(-0.342580\pi\)
0.474635 + 0.880183i \(0.342580\pi\)
\(810\) 0 0
\(811\) −33.0000 −1.15879 −0.579393 − 0.815048i \(-0.696710\pi\)
−0.579393 + 0.815048i \(0.696710\pi\)
\(812\) 0 0
\(813\) 1.00000 0.0350715
\(814\) 0 0
\(815\) −20.0000 −0.700569
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) 0 0
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) −44.0000 −1.53561 −0.767805 − 0.640683i \(-0.778651\pi\)
−0.767805 + 0.640683i \(0.778651\pi\)
\(822\) 0 0
\(823\) 35.0000 1.22002 0.610012 − 0.792392i \(-0.291165\pi\)
0.610012 + 0.792392i \(0.291165\pi\)
\(824\) 0 0
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) 37.0000 1.28662 0.643308 − 0.765607i \(-0.277561\pi\)
0.643308 + 0.765607i \(0.277561\pi\)
\(828\) 0 0
\(829\) 39.0000 1.35453 0.677263 − 0.735741i \(-0.263166\pi\)
0.677263 + 0.735741i \(0.263166\pi\)
\(830\) 0 0
\(831\) −12.0000 −0.416275
\(832\) 0 0
\(833\) −14.0000 −0.485071
\(834\) 0 0
\(835\) 4.00000 0.138426
\(836\) 0 0
\(837\) −10.0000 −0.345651
\(838\) 0 0
\(839\) −12.0000 −0.414286 −0.207143 − 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) −8.00000 −0.275535
\(844\) 0 0
\(845\) −24.0000 −0.825625
\(846\) 0 0
\(847\) 3.00000 0.103081
\(848\) 0 0
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) 30.0000 1.02839
\(852\) 0 0
\(853\) −42.0000 −1.43805 −0.719026 − 0.694983i \(-0.755412\pi\)
−0.719026 + 0.694983i \(0.755412\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) 22.0000 0.751506 0.375753 − 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 0 0
\(859\) 46.0000 1.56950 0.784750 − 0.619813i \(-0.212791\pi\)
0.784750 + 0.619813i \(0.212791\pi\)
\(860\) 0 0
\(861\) 18.0000 0.613438
\(862\) 0 0
\(863\) 30.0000 1.02121 0.510606 − 0.859815i \(-0.329421\pi\)
0.510606 + 0.859815i \(0.329421\pi\)
\(864\) 0 0
\(865\) −28.0000 −0.952029
\(866\) 0 0
\(867\) −32.0000 −1.08678
\(868\) 0 0
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) −3.00000 −0.101651
\(872\) 0 0
\(873\) −8.00000 −0.270759
\(874\) 0 0
\(875\) −36.0000 −1.21702
\(876\) 0 0
\(877\) −35.0000 −1.18187 −0.590933 − 0.806721i \(-0.701240\pi\)
−0.590933 + 0.806721i \(0.701240\pi\)
\(878\) 0 0
\(879\) −17.0000 −0.573396
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 − 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 0 0
\(883\) 16.0000 0.538443 0.269221 − 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 0 0
\(885\) 10.0000 0.336146
\(886\) 0 0
\(887\) −8.00000 −0.268614 −0.134307 − 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) 36.0000 1.20740
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) −3.00000 −0.100167
\(898\) 0 0
\(899\) 2.00000 0.0667037
\(900\) 0 0
\(901\) −7.00000 −0.233204
\(902\) 0 0
\(903\) 24.0000 0.798670
\(904\) 0 0
\(905\) −44.0000 −1.46261
\(906\) 0 0
\(907\) 5.00000 0.166022 0.0830111 − 0.996549i \(-0.473546\pi\)
0.0830111 + 0.996549i \(0.473546\pi\)
\(908\) 0 0
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −34.0000 −1.12647 −0.563235 − 0.826297i \(-0.690443\pi\)
−0.563235 + 0.826297i \(0.690443\pi\)
\(912\) 0 0
\(913\) 8.00000 0.264761
\(914\) 0 0
\(915\) 16.0000 0.528944
\(916\) 0 0
\(917\) −18.0000 −0.594412
\(918\) 0 0
\(919\) 5.00000 0.164935 0.0824674 − 0.996594i \(-0.473720\pi\)
0.0824674 + 0.996594i \(0.473720\pi\)
\(920\) 0 0
\(921\) 16.0000 0.527218
\(922\) 0 0
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) −3.00000 −0.0984268 −0.0492134 − 0.998788i \(-0.515671\pi\)
−0.0492134 + 0.998788i \(0.515671\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 0 0
\(933\) 19.0000 0.622032
\(934\) 0 0
\(935\) −14.0000 −0.457849
\(936\) 0 0
\(937\) −61.0000 −1.99278 −0.996392 − 0.0848755i \(-0.972951\pi\)
−0.996392 + 0.0848755i \(0.972951\pi\)
\(938\) 0 0
\(939\) 19.0000 0.620042
\(940\) 0 0
\(941\) −49.0000 −1.59735 −0.798677 − 0.601760i \(-0.794466\pi\)
−0.798677 + 0.601760i \(0.794466\pi\)
\(942\) 0 0
\(943\) 18.0000 0.586161
\(944\) 0 0
\(945\) 30.0000 0.975900
\(946\) 0 0
\(947\) 52.0000 1.68977 0.844886 − 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 0 0
\(949\) −3.00000 −0.0973841
\(950\) 0 0
\(951\) −3.00000 −0.0972817
\(952\) 0 0
\(953\) 52.0000 1.68445 0.842223 − 0.539130i \(-0.181247\pi\)
0.842223 + 0.539130i \(0.181247\pi\)
\(954\) 0 0
\(955\) −6.00000 −0.194155
\(956\) 0 0
\(957\) 1.00000 0.0323254
\(958\) 0 0
\(959\) 45.0000 1.45313
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −22.0000 −0.708940
\(964\) 0 0
\(965\) −24.0000 −0.772587
\(966\) 0 0
\(967\) −24.0000 −0.771788 −0.385894 − 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) 0 0
\(969\) 7.00000 0.224872
\(970\) 0 0
\(971\) −20.0000 −0.641831 −0.320915 − 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) −42.0000 −1.34646
\(974\) 0 0
\(975\) −1.00000 −0.0320256
\(976\) 0 0
\(977\) −32.0000 −1.02377 −0.511885 − 0.859054i \(-0.671053\pi\)
−0.511885 + 0.859054i \(0.671053\pi\)
\(978\) 0 0
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 0 0
\(983\) 46.0000 1.46717 0.733586 − 0.679597i \(-0.237845\pi\)
0.733586 + 0.679597i \(0.237845\pi\)
\(984\) 0 0
\(985\) −24.0000 −0.764704
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 − 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 19.0000 0.602947
\(994\) 0 0
\(995\) −30.0000 −0.951064
\(996\) 0 0
\(997\) −14.0000 −0.443384 −0.221692 − 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) 0 0
\(999\) −50.0000 −1.58193
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 3344.2.a.e.1.1 1
4.3 odd 2 1672.2.a.c.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1672.2.a.c.1.1 ✓ 1 4.3 odd 2
3344.2.a.e.1.1 1 1.1 even 1 trivial