Properties

Label 1088.2.a.o.1.1
Level $1088$
Weight $2$
Character 1088.1
Self dual yes
Analytic conductor $8.688$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1088,2,Mod(1,1088)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1088, 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("1088.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1088 = 2^{6} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1088.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: \(8.68772373992\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 136)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1088.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.23607 q^{3} -2.00000 q^{5} -3.23607 q^{7} +7.47214 q^{9} +O(q^{10})\) \(q-3.23607 q^{3} -2.00000 q^{5} -3.23607 q^{7} +7.47214 q^{9} +3.23607 q^{11} +4.47214 q^{13} +6.47214 q^{15} +1.00000 q^{17} +2.47214 q^{19} +10.4721 q^{21} -3.23607 q^{23} -1.00000 q^{25} -14.4721 q^{27} -2.00000 q^{29} +3.23607 q^{31} -10.4721 q^{33} +6.47214 q^{35} -6.94427 q^{37} -14.4721 q^{39} +2.00000 q^{41} -10.4721 q^{43} -14.9443 q^{45} +4.94427 q^{47} +3.47214 q^{49} -3.23607 q^{51} +2.00000 q^{53} -6.47214 q^{55} -8.00000 q^{57} +5.52786 q^{59} +10.9443 q^{61} -24.1803 q^{63} -8.94427 q^{65} -12.0000 q^{67} +10.4721 q^{69} -4.76393 q^{71} -2.94427 q^{73} +3.23607 q^{75} -10.4721 q^{77} +1.70820 q^{79} +24.4164 q^{81} +10.4721 q^{83} -2.00000 q^{85} +6.47214 q^{87} -16.4721 q^{89} -14.4721 q^{91} -10.4721 q^{93} -4.94427 q^{95} +2.00000 q^{97} +24.1803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 4 q^{5} - 2 q^{7} + 6 q^{9} + 2 q^{11} + 4 q^{15} + 2 q^{17} - 4 q^{19} + 12 q^{21} - 2 q^{23} - 2 q^{25} - 20 q^{27} - 4 q^{29} + 2 q^{31} - 12 q^{33} + 4 q^{35} + 4 q^{37} - 20 q^{39} + 4 q^{41} - 12 q^{43} - 12 q^{45} - 8 q^{47} - 2 q^{49} - 2 q^{51} + 4 q^{53} - 4 q^{55} - 16 q^{57} + 20 q^{59} + 4 q^{61} - 26 q^{63} - 24 q^{67} + 12 q^{69} - 14 q^{71} + 12 q^{73} + 2 q^{75} - 12 q^{77} - 10 q^{79} + 22 q^{81} + 12 q^{83} - 4 q^{85} + 4 q^{87} - 24 q^{89} - 20 q^{91} - 12 q^{93} + 8 q^{95} + 4 q^{97} + 26 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.23607 −1.86834 −0.934172 0.356822i \(-0.883860\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) 0 0
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −3.23607 −1.22312 −0.611559 0.791199i \(-0.709457\pi\)
−0.611559 + 0.791199i \(0.709457\pi\)
\(8\) 0 0
\(9\) 7.47214 2.49071
\(10\) 0 0
\(11\) 3.23607 0.975711 0.487856 0.872924i \(-0.337779\pi\)
0.487856 + 0.872924i \(0.337779\pi\)
\(12\) 0 0
\(13\) 4.47214 1.24035 0.620174 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 0 0
\(15\) 6.47214 1.67110
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) 2.47214 0.567147 0.283573 0.958951i \(-0.408480\pi\)
0.283573 + 0.958951i \(0.408480\pi\)
\(20\) 0 0
\(21\) 10.4721 2.28521
\(22\) 0 0
\(23\) −3.23607 −0.674767 −0.337383 0.941367i \(-0.609542\pi\)
−0.337383 + 0.941367i \(0.609542\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −14.4721 −2.78516
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 3.23607 0.581215 0.290607 0.956842i \(-0.406143\pi\)
0.290607 + 0.956842i \(0.406143\pi\)
\(32\) 0 0
\(33\) −10.4721 −1.82296
\(34\) 0 0
\(35\) 6.47214 1.09399
\(36\) 0 0
\(37\) −6.94427 −1.14163 −0.570816 0.821078i \(-0.693373\pi\)
−0.570816 + 0.821078i \(0.693373\pi\)
\(38\) 0 0
\(39\) −14.4721 −2.31740
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −10.4721 −1.59699 −0.798493 0.602004i \(-0.794369\pi\)
−0.798493 + 0.602004i \(0.794369\pi\)
\(44\) 0 0
\(45\) −14.9443 −2.22776
\(46\) 0 0
\(47\) 4.94427 0.721196 0.360598 0.932721i \(-0.382573\pi\)
0.360598 + 0.932721i \(0.382573\pi\)
\(48\) 0 0
\(49\) 3.47214 0.496019
\(50\) 0 0
\(51\) −3.23607 −0.453140
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −6.47214 −0.872703
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) 5.52786 0.719667 0.359833 0.933017i \(-0.382834\pi\)
0.359833 + 0.933017i \(0.382834\pi\)
\(60\) 0 0
\(61\) 10.9443 1.40127 0.700635 0.713520i \(-0.252900\pi\)
0.700635 + 0.713520i \(0.252900\pi\)
\(62\) 0 0
\(63\) −24.1803 −3.04644
\(64\) 0 0
\(65\) −8.94427 −1.10940
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 10.4721 1.26070
\(70\) 0 0
\(71\) −4.76393 −0.565375 −0.282687 0.959212i \(-0.591226\pi\)
−0.282687 + 0.959212i \(0.591226\pi\)
\(72\) 0 0
\(73\) −2.94427 −0.344601 −0.172300 0.985044i \(-0.555120\pi\)
−0.172300 + 0.985044i \(0.555120\pi\)
\(74\) 0 0
\(75\) 3.23607 0.373669
\(76\) 0 0
\(77\) −10.4721 −1.19341
\(78\) 0 0
\(79\) 1.70820 0.192188 0.0960940 0.995372i \(-0.469365\pi\)
0.0960940 + 0.995372i \(0.469365\pi\)
\(80\) 0 0
\(81\) 24.4164 2.71293
\(82\) 0 0
\(83\) 10.4721 1.14947 0.574733 0.818341i \(-0.305106\pi\)
0.574733 + 0.818341i \(0.305106\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 6.47214 0.693886
\(88\) 0 0
\(89\) −16.4721 −1.74604 −0.873021 0.487682i \(-0.837843\pi\)
−0.873021 + 0.487682i \(0.837843\pi\)
\(90\) 0 0
\(91\) −14.4721 −1.51709
\(92\) 0 0
\(93\) −10.4721 −1.08591
\(94\) 0 0
\(95\) −4.94427 −0.507272
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 24.1803 2.43022
\(100\) 0 0
\(101\) −13.4164 −1.33498 −0.667491 0.744618i \(-0.732632\pi\)
−0.667491 + 0.744618i \(0.732632\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) −20.9443 −2.04395
\(106\) 0 0
\(107\) 9.70820 0.938527 0.469264 0.883058i \(-0.344519\pi\)
0.469264 + 0.883058i \(0.344519\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 22.4721 2.13296
\(112\) 0 0
\(113\) −2.94427 −0.276974 −0.138487 0.990364i \(-0.544224\pi\)
−0.138487 + 0.990364i \(0.544224\pi\)
\(114\) 0 0
\(115\) 6.47214 0.603530
\(116\) 0 0
\(117\) 33.4164 3.08935
\(118\) 0 0
\(119\) −3.23607 −0.296650
\(120\) 0 0
\(121\) −0.527864 −0.0479876
\(122\) 0 0
\(123\) −6.47214 −0.583573
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 6.47214 0.574309 0.287155 0.957884i \(-0.407291\pi\)
0.287155 + 0.957884i \(0.407291\pi\)
\(128\) 0 0
\(129\) 33.8885 2.98372
\(130\) 0 0
\(131\) −21.1246 −1.84567 −0.922833 0.385200i \(-0.874132\pi\)
−0.922833 + 0.385200i \(0.874132\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 0 0
\(135\) 28.9443 2.49113
\(136\) 0 0
\(137\) −16.4721 −1.40731 −0.703655 0.710542i \(-0.748450\pi\)
−0.703655 + 0.710542i \(0.748450\pi\)
\(138\) 0 0
\(139\) −11.2361 −0.953031 −0.476515 0.879166i \(-0.658100\pi\)
−0.476515 + 0.879166i \(0.658100\pi\)
\(140\) 0 0
\(141\) −16.0000 −1.34744
\(142\) 0 0
\(143\) 14.4721 1.21022
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 0 0
\(147\) −11.2361 −0.926735
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −14.4721 −1.17773 −0.588863 0.808233i \(-0.700424\pi\)
−0.588863 + 0.808233i \(0.700424\pi\)
\(152\) 0 0
\(153\) 7.47214 0.604086
\(154\) 0 0
\(155\) −6.47214 −0.519854
\(156\) 0 0
\(157\) −23.8885 −1.90651 −0.953257 0.302162i \(-0.902292\pi\)
−0.953257 + 0.302162i \(0.902292\pi\)
\(158\) 0 0
\(159\) −6.47214 −0.513274
\(160\) 0 0
\(161\) 10.4721 0.825320
\(162\) 0 0
\(163\) 0.180340 0.0141253 0.00706266 0.999975i \(-0.497752\pi\)
0.00706266 + 0.999975i \(0.497752\pi\)
\(164\) 0 0
\(165\) 20.9443 1.63051
\(166\) 0 0
\(167\) 6.29180 0.486874 0.243437 0.969917i \(-0.421725\pi\)
0.243437 + 0.969917i \(0.421725\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 18.4721 1.41260
\(172\) 0 0
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) 3.23607 0.244624
\(176\) 0 0
\(177\) −17.8885 −1.34459
\(178\) 0 0
\(179\) 5.52786 0.413172 0.206586 0.978428i \(-0.433765\pi\)
0.206586 + 0.978428i \(0.433765\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) −35.4164 −2.61806
\(184\) 0 0
\(185\) 13.8885 1.02111
\(186\) 0 0
\(187\) 3.23607 0.236645
\(188\) 0 0
\(189\) 46.8328 3.40659
\(190\) 0 0
\(191\) −4.94427 −0.357755 −0.178877 0.983871i \(-0.557247\pi\)
−0.178877 + 0.983871i \(0.557247\pi\)
\(192\) 0 0
\(193\) −2.94427 −0.211933 −0.105967 0.994370i \(-0.533794\pi\)
−0.105967 + 0.994370i \(0.533794\pi\)
\(194\) 0 0
\(195\) 28.9443 2.07274
\(196\) 0 0
\(197\) 10.9443 0.779747 0.389874 0.920868i \(-0.372519\pi\)
0.389874 + 0.920868i \(0.372519\pi\)
\(198\) 0 0
\(199\) 0.180340 0.0127840 0.00639198 0.999980i \(-0.497965\pi\)
0.00639198 + 0.999980i \(0.497965\pi\)
\(200\) 0 0
\(201\) 38.8328 2.73906
\(202\) 0 0
\(203\) 6.47214 0.454255
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 0 0
\(207\) −24.1803 −1.68065
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 3.23607 0.222780 0.111390 0.993777i \(-0.464470\pi\)
0.111390 + 0.993777i \(0.464470\pi\)
\(212\) 0 0
\(213\) 15.4164 1.05631
\(214\) 0 0
\(215\) 20.9443 1.42839
\(216\) 0 0
\(217\) −10.4721 −0.710895
\(218\) 0 0
\(219\) 9.52786 0.643833
\(220\) 0 0
\(221\) 4.47214 0.300828
\(222\) 0 0
\(223\) −14.4721 −0.969126 −0.484563 0.874756i \(-0.661021\pi\)
−0.484563 + 0.874756i \(0.661021\pi\)
\(224\) 0 0
\(225\) −7.47214 −0.498142
\(226\) 0 0
\(227\) 3.23607 0.214785 0.107393 0.994217i \(-0.465750\pi\)
0.107393 + 0.994217i \(0.465750\pi\)
\(228\) 0 0
\(229\) 15.5279 1.02611 0.513055 0.858356i \(-0.328514\pi\)
0.513055 + 0.858356i \(0.328514\pi\)
\(230\) 0 0
\(231\) 33.8885 2.22970
\(232\) 0 0
\(233\) 11.8885 0.778844 0.389422 0.921059i \(-0.372675\pi\)
0.389422 + 0.921059i \(0.372675\pi\)
\(234\) 0 0
\(235\) −9.88854 −0.645057
\(236\) 0 0
\(237\) −5.52786 −0.359073
\(238\) 0 0
\(239\) 17.8885 1.15711 0.578557 0.815642i \(-0.303616\pi\)
0.578557 + 0.815642i \(0.303616\pi\)
\(240\) 0 0
\(241\) −18.9443 −1.22031 −0.610154 0.792283i \(-0.708892\pi\)
−0.610154 + 0.792283i \(0.708892\pi\)
\(242\) 0 0
\(243\) −35.5967 −2.28353
\(244\) 0 0
\(245\) −6.94427 −0.443653
\(246\) 0 0
\(247\) 11.0557 0.703459
\(248\) 0 0
\(249\) −33.8885 −2.14760
\(250\) 0 0
\(251\) 8.94427 0.564557 0.282279 0.959332i \(-0.408910\pi\)
0.282279 + 0.959332i \(0.408910\pi\)
\(252\) 0 0
\(253\) −10.4721 −0.658378
\(254\) 0 0
\(255\) 6.47214 0.405301
\(256\) 0 0
\(257\) 22.3607 1.39482 0.697410 0.716672i \(-0.254335\pi\)
0.697410 + 0.716672i \(0.254335\pi\)
\(258\) 0 0
\(259\) 22.4721 1.39635
\(260\) 0 0
\(261\) −14.9443 −0.925027
\(262\) 0 0
\(263\) −11.4164 −0.703966 −0.351983 0.936006i \(-0.614492\pi\)
−0.351983 + 0.936006i \(0.614492\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) 53.3050 3.26221
\(268\) 0 0
\(269\) 18.9443 1.15505 0.577526 0.816372i \(-0.304018\pi\)
0.577526 + 0.816372i \(0.304018\pi\)
\(270\) 0 0
\(271\) −17.8885 −1.08665 −0.543326 0.839522i \(-0.682835\pi\)
−0.543326 + 0.839522i \(0.682835\pi\)
\(272\) 0 0
\(273\) 46.8328 2.83445
\(274\) 0 0
\(275\) −3.23607 −0.195142
\(276\) 0 0
\(277\) −32.8328 −1.97273 −0.986366 0.164564i \(-0.947378\pi\)
−0.986366 + 0.164564i \(0.947378\pi\)
\(278\) 0 0
\(279\) 24.1803 1.44764
\(280\) 0 0
\(281\) −15.8885 −0.947831 −0.473916 0.880570i \(-0.657160\pi\)
−0.473916 + 0.880570i \(0.657160\pi\)
\(282\) 0 0
\(283\) 3.23607 0.192364 0.0961821 0.995364i \(-0.469337\pi\)
0.0961821 + 0.995364i \(0.469337\pi\)
\(284\) 0 0
\(285\) 16.0000 0.947758
\(286\) 0 0
\(287\) −6.47214 −0.382038
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −6.47214 −0.379403
\(292\) 0 0
\(293\) 10.0000 0.584206 0.292103 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 0 0
\(295\) −11.0557 −0.643689
\(296\) 0 0
\(297\) −46.8328 −2.71752
\(298\) 0 0
\(299\) −14.4721 −0.836945
\(300\) 0 0
\(301\) 33.8885 1.95330
\(302\) 0 0
\(303\) 43.4164 2.49421
\(304\) 0 0
\(305\) −21.8885 −1.25333
\(306\) 0 0
\(307\) −29.8885 −1.70583 −0.852915 0.522050i \(-0.825167\pi\)
−0.852915 + 0.522050i \(0.825167\pi\)
\(308\) 0 0
\(309\) −25.8885 −1.47275
\(310\) 0 0
\(311\) 19.2361 1.09078 0.545389 0.838183i \(-0.316382\pi\)
0.545389 + 0.838183i \(0.316382\pi\)
\(312\) 0 0
\(313\) 27.8885 1.57635 0.788177 0.615449i \(-0.211025\pi\)
0.788177 + 0.615449i \(0.211025\pi\)
\(314\) 0 0
\(315\) 48.3607 2.72482
\(316\) 0 0
\(317\) −19.8885 −1.11705 −0.558526 0.829487i \(-0.688633\pi\)
−0.558526 + 0.829487i \(0.688633\pi\)
\(318\) 0 0
\(319\) −6.47214 −0.362370
\(320\) 0 0
\(321\) −31.4164 −1.75349
\(322\) 0 0
\(323\) 2.47214 0.137553
\(324\) 0 0
\(325\) −4.47214 −0.248069
\(326\) 0 0
\(327\) 32.3607 1.78955
\(328\) 0 0
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) −10.4721 −0.575601 −0.287800 0.957690i \(-0.592924\pi\)
−0.287800 + 0.957690i \(0.592924\pi\)
\(332\) 0 0
\(333\) −51.8885 −2.84347
\(334\) 0 0
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) −23.8885 −1.30129 −0.650646 0.759381i \(-0.725502\pi\)
−0.650646 + 0.759381i \(0.725502\pi\)
\(338\) 0 0
\(339\) 9.52786 0.517483
\(340\) 0 0
\(341\) 10.4721 0.567098
\(342\) 0 0
\(343\) 11.4164 0.616428
\(344\) 0 0
\(345\) −20.9443 −1.12760
\(346\) 0 0
\(347\) −4.76393 −0.255741 −0.127871 0.991791i \(-0.540814\pi\)
−0.127871 + 0.991791i \(0.540814\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) −64.7214 −3.45457
\(352\) 0 0
\(353\) −23.8885 −1.27146 −0.635729 0.771912i \(-0.719301\pi\)
−0.635729 + 0.771912i \(0.719301\pi\)
\(354\) 0 0
\(355\) 9.52786 0.505687
\(356\) 0 0
\(357\) 10.4721 0.554244
\(358\) 0 0
\(359\) −22.4721 −1.18603 −0.593017 0.805190i \(-0.702063\pi\)
−0.593017 + 0.805190i \(0.702063\pi\)
\(360\) 0 0
\(361\) −12.8885 −0.678344
\(362\) 0 0
\(363\) 1.70820 0.0896575
\(364\) 0 0
\(365\) 5.88854 0.308220
\(366\) 0 0
\(367\) −19.2361 −1.00411 −0.502057 0.864834i \(-0.667423\pi\)
−0.502057 + 0.864834i \(0.667423\pi\)
\(368\) 0 0
\(369\) 14.9443 0.777968
\(370\) 0 0
\(371\) −6.47214 −0.336017
\(372\) 0 0
\(373\) −3.52786 −0.182666 −0.0913329 0.995820i \(-0.529113\pi\)
−0.0913329 + 0.995820i \(0.529113\pi\)
\(374\) 0 0
\(375\) −38.8328 −2.00532
\(376\) 0 0
\(377\) −8.94427 −0.460653
\(378\) 0 0
\(379\) −1.34752 −0.0692177 −0.0346088 0.999401i \(-0.511019\pi\)
−0.0346088 + 0.999401i \(0.511019\pi\)
\(380\) 0 0
\(381\) −20.9443 −1.07301
\(382\) 0 0
\(383\) −3.41641 −0.174570 −0.0872851 0.996183i \(-0.527819\pi\)
−0.0872851 + 0.996183i \(0.527819\pi\)
\(384\) 0 0
\(385\) 20.9443 1.06742
\(386\) 0 0
\(387\) −78.2492 −3.97763
\(388\) 0 0
\(389\) 25.4164 1.28866 0.644332 0.764746i \(-0.277136\pi\)
0.644332 + 0.764746i \(0.277136\pi\)
\(390\) 0 0
\(391\) −3.23607 −0.163655
\(392\) 0 0
\(393\) 68.3607 3.44834
\(394\) 0 0
\(395\) −3.41641 −0.171898
\(396\) 0 0
\(397\) −30.9443 −1.55305 −0.776524 0.630087i \(-0.783019\pi\)
−0.776524 + 0.630087i \(0.783019\pi\)
\(398\) 0 0
\(399\) 25.8885 1.29605
\(400\) 0 0
\(401\) 13.0557 0.651972 0.325986 0.945375i \(-0.394304\pi\)
0.325986 + 0.945375i \(0.394304\pi\)
\(402\) 0 0
\(403\) 14.4721 0.720908
\(404\) 0 0
\(405\) −48.8328 −2.42652
\(406\) 0 0
\(407\) −22.4721 −1.11390
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 53.3050 2.62934
\(412\) 0 0
\(413\) −17.8885 −0.880238
\(414\) 0 0
\(415\) −20.9443 −1.02811
\(416\) 0 0
\(417\) 36.3607 1.78059
\(418\) 0 0
\(419\) −20.7639 −1.01438 −0.507192 0.861833i \(-0.669317\pi\)
−0.507192 + 0.861833i \(0.669317\pi\)
\(420\) 0 0
\(421\) 38.3607 1.86959 0.934793 0.355194i \(-0.115585\pi\)
0.934793 + 0.355194i \(0.115585\pi\)
\(422\) 0 0
\(423\) 36.9443 1.79629
\(424\) 0 0
\(425\) −1.00000 −0.0485071
\(426\) 0 0
\(427\) −35.4164 −1.71392
\(428\) 0 0
\(429\) −46.8328 −2.26111
\(430\) 0 0
\(431\) 11.5967 0.558596 0.279298 0.960205i \(-0.409898\pi\)
0.279298 + 0.960205i \(0.409898\pi\)
\(432\) 0 0
\(433\) −19.5279 −0.938449 −0.469225 0.883079i \(-0.655467\pi\)
−0.469225 + 0.883079i \(0.655467\pi\)
\(434\) 0 0
\(435\) −12.9443 −0.620630
\(436\) 0 0
\(437\) −8.00000 −0.382692
\(438\) 0 0
\(439\) −30.6525 −1.46296 −0.731481 0.681861i \(-0.761171\pi\)
−0.731481 + 0.681861i \(0.761171\pi\)
\(440\) 0 0
\(441\) 25.9443 1.23544
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) 32.9443 1.56171
\(446\) 0 0
\(447\) 19.4164 0.918365
\(448\) 0 0
\(449\) 27.8885 1.31614 0.658071 0.752956i \(-0.271373\pi\)
0.658071 + 0.752956i \(0.271373\pi\)
\(450\) 0 0
\(451\) 6.47214 0.304761
\(452\) 0 0
\(453\) 46.8328 2.20040
\(454\) 0 0
\(455\) 28.9443 1.35693
\(456\) 0 0
\(457\) 10.5836 0.495080 0.247540 0.968878i \(-0.420378\pi\)
0.247540 + 0.968878i \(0.420378\pi\)
\(458\) 0 0
\(459\) −14.4721 −0.675501
\(460\) 0 0
\(461\) 3.88854 0.181108 0.0905538 0.995892i \(-0.471136\pi\)
0.0905538 + 0.995892i \(0.471136\pi\)
\(462\) 0 0
\(463\) 22.8328 1.06113 0.530565 0.847644i \(-0.321980\pi\)
0.530565 + 0.847644i \(0.321980\pi\)
\(464\) 0 0
\(465\) 20.9443 0.971267
\(466\) 0 0
\(467\) 41.3050 1.91137 0.955683 0.294399i \(-0.0951194\pi\)
0.955683 + 0.294399i \(0.0951194\pi\)
\(468\) 0 0
\(469\) 38.8328 1.79313
\(470\) 0 0
\(471\) 77.3050 3.56202
\(472\) 0 0
\(473\) −33.8885 −1.55820
\(474\) 0 0
\(475\) −2.47214 −0.113429
\(476\) 0 0
\(477\) 14.9443 0.684251
\(478\) 0 0
\(479\) −33.7082 −1.54017 −0.770084 0.637943i \(-0.779786\pi\)
−0.770084 + 0.637943i \(0.779786\pi\)
\(480\) 0 0
\(481\) −31.0557 −1.41602
\(482\) 0 0
\(483\) −33.8885 −1.54198
\(484\) 0 0
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) −43.2361 −1.95921 −0.979607 0.200925i \(-0.935605\pi\)
−0.979607 + 0.200925i \(0.935605\pi\)
\(488\) 0 0
\(489\) −0.583592 −0.0263909
\(490\) 0 0
\(491\) 10.4721 0.472601 0.236300 0.971680i \(-0.424065\pi\)
0.236300 + 0.971680i \(0.424065\pi\)
\(492\) 0 0
\(493\) −2.00000 −0.0900755
\(494\) 0 0
\(495\) −48.3607 −2.17365
\(496\) 0 0
\(497\) 15.4164 0.691520
\(498\) 0 0
\(499\) −15.8197 −0.708185 −0.354093 0.935210i \(-0.615210\pi\)
−0.354093 + 0.935210i \(0.615210\pi\)
\(500\) 0 0
\(501\) −20.3607 −0.909648
\(502\) 0 0
\(503\) −14.2918 −0.637240 −0.318620 0.947883i \(-0.603219\pi\)
−0.318620 + 0.947883i \(0.603219\pi\)
\(504\) 0 0
\(505\) 26.8328 1.19404
\(506\) 0 0
\(507\) −22.6525 −1.00603
\(508\) 0 0
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) 9.52786 0.421488
\(512\) 0 0
\(513\) −35.7771 −1.57960
\(514\) 0 0
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) 6.47214 0.284095
\(520\) 0 0
\(521\) 11.8885 0.520847 0.260423 0.965495i \(-0.416138\pi\)
0.260423 + 0.965495i \(0.416138\pi\)
\(522\) 0 0
\(523\) −7.05573 −0.308525 −0.154263 0.988030i \(-0.549300\pi\)
−0.154263 + 0.988030i \(0.549300\pi\)
\(524\) 0 0
\(525\) −10.4721 −0.457041
\(526\) 0 0
\(527\) 3.23607 0.140965
\(528\) 0 0
\(529\) −12.5279 −0.544690
\(530\) 0 0
\(531\) 41.3050 1.79248
\(532\) 0 0
\(533\) 8.94427 0.387419
\(534\) 0 0
\(535\) −19.4164 −0.839445
\(536\) 0 0
\(537\) −17.8885 −0.771948
\(538\) 0 0
\(539\) 11.2361 0.483972
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 0 0
\(543\) 32.3607 1.38873
\(544\) 0 0
\(545\) 20.0000 0.856706
\(546\) 0 0
\(547\) 1.70820 0.0730375 0.0365188 0.999333i \(-0.488373\pi\)
0.0365188 + 0.999333i \(0.488373\pi\)
\(548\) 0 0
\(549\) 81.7771 3.49016
\(550\) 0 0
\(551\) −4.94427 −0.210633
\(552\) 0 0
\(553\) −5.52786 −0.235069
\(554\) 0 0
\(555\) −44.9443 −1.90778
\(556\) 0 0
\(557\) −21.4164 −0.907442 −0.453721 0.891144i \(-0.649904\pi\)
−0.453721 + 0.891144i \(0.649904\pi\)
\(558\) 0 0
\(559\) −46.8328 −1.98082
\(560\) 0 0
\(561\) −10.4721 −0.442134
\(562\) 0 0
\(563\) 20.3607 0.858100 0.429050 0.903281i \(-0.358848\pi\)
0.429050 + 0.903281i \(0.358848\pi\)
\(564\) 0 0
\(565\) 5.88854 0.247733
\(566\) 0 0
\(567\) −79.0132 −3.31824
\(568\) 0 0
\(569\) 35.8885 1.50453 0.752263 0.658863i \(-0.228962\pi\)
0.752263 + 0.658863i \(0.228962\pi\)
\(570\) 0 0
\(571\) 3.23607 0.135425 0.0677126 0.997705i \(-0.478430\pi\)
0.0677126 + 0.997705i \(0.478430\pi\)
\(572\) 0 0
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) 3.23607 0.134953
\(576\) 0 0
\(577\) −8.47214 −0.352700 −0.176350 0.984328i \(-0.556429\pi\)
−0.176350 + 0.984328i \(0.556429\pi\)
\(578\) 0 0
\(579\) 9.52786 0.395965
\(580\) 0 0
\(581\) −33.8885 −1.40593
\(582\) 0 0
\(583\) 6.47214 0.268048
\(584\) 0 0
\(585\) −66.8328 −2.76320
\(586\) 0 0
\(587\) −12.3607 −0.510180 −0.255090 0.966917i \(-0.582105\pi\)
−0.255090 + 0.966917i \(0.582105\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −35.4164 −1.45684
\(592\) 0 0
\(593\) 11.8885 0.488204 0.244102 0.969750i \(-0.421507\pi\)
0.244102 + 0.969750i \(0.421507\pi\)
\(594\) 0 0
\(595\) 6.47214 0.265332
\(596\) 0 0
\(597\) −0.583592 −0.0238848
\(598\) 0 0
\(599\) −46.8328 −1.91354 −0.956768 0.290851i \(-0.906062\pi\)
−0.956768 + 0.290851i \(0.906062\pi\)
\(600\) 0 0
\(601\) 38.9443 1.58857 0.794285 0.607545i \(-0.207846\pi\)
0.794285 + 0.607545i \(0.207846\pi\)
\(602\) 0 0
\(603\) −89.6656 −3.65147
\(604\) 0 0
\(605\) 1.05573 0.0429215
\(606\) 0 0
\(607\) 6.29180 0.255376 0.127688 0.991814i \(-0.459244\pi\)
0.127688 + 0.991814i \(0.459244\pi\)
\(608\) 0 0
\(609\) −20.9443 −0.848705
\(610\) 0 0
\(611\) 22.1115 0.894534
\(612\) 0 0
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 0 0
\(615\) 12.9443 0.521963
\(616\) 0 0
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 0 0
\(619\) 1.34752 0.0541616 0.0270808 0.999633i \(-0.491379\pi\)
0.0270808 + 0.999633i \(0.491379\pi\)
\(620\) 0 0
\(621\) 46.8328 1.87934
\(622\) 0 0
\(623\) 53.3050 2.13562
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) −25.8885 −1.03389
\(628\) 0 0
\(629\) −6.94427 −0.276886
\(630\) 0 0
\(631\) 29.3050 1.16661 0.583306 0.812253i \(-0.301759\pi\)
0.583306 + 0.812253i \(0.301759\pi\)
\(632\) 0 0
\(633\) −10.4721 −0.416230
\(634\) 0 0
\(635\) −12.9443 −0.513678
\(636\) 0 0
\(637\) 15.5279 0.615236
\(638\) 0 0
\(639\) −35.5967 −1.40819
\(640\) 0 0
\(641\) −7.88854 −0.311579 −0.155789 0.987790i \(-0.549792\pi\)
−0.155789 + 0.987790i \(0.549792\pi\)
\(642\) 0 0
\(643\) 48.5410 1.91427 0.957135 0.289641i \(-0.0935358\pi\)
0.957135 + 0.289641i \(0.0935358\pi\)
\(644\) 0 0
\(645\) −67.7771 −2.66872
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) 17.8885 0.702187
\(650\) 0 0
\(651\) 33.8885 1.32820
\(652\) 0 0
\(653\) 20.8328 0.815251 0.407626 0.913149i \(-0.366357\pi\)
0.407626 + 0.913149i \(0.366357\pi\)
\(654\) 0 0
\(655\) 42.2492 1.65081
\(656\) 0 0
\(657\) −22.0000 −0.858302
\(658\) 0 0
\(659\) 18.8328 0.733622 0.366811 0.930295i \(-0.380450\pi\)
0.366811 + 0.930295i \(0.380450\pi\)
\(660\) 0 0
\(661\) −39.8885 −1.55148 −0.775742 0.631050i \(-0.782624\pi\)
−0.775742 + 0.631050i \(0.782624\pi\)
\(662\) 0 0
\(663\) −14.4721 −0.562051
\(664\) 0 0
\(665\) 16.0000 0.620453
\(666\) 0 0
\(667\) 6.47214 0.250602
\(668\) 0 0
\(669\) 46.8328 1.81066
\(670\) 0 0
\(671\) 35.4164 1.36724
\(672\) 0 0
\(673\) 37.7771 1.45620 0.728100 0.685471i \(-0.240404\pi\)
0.728100 + 0.685471i \(0.240404\pi\)
\(674\) 0 0
\(675\) 14.4721 0.557033
\(676\) 0 0
\(677\) −43.8885 −1.68677 −0.843387 0.537307i \(-0.819442\pi\)
−0.843387 + 0.537307i \(0.819442\pi\)
\(678\) 0 0
\(679\) −6.47214 −0.248378
\(680\) 0 0
\(681\) −10.4721 −0.401293
\(682\) 0 0
\(683\) −50.0689 −1.91583 −0.957916 0.287048i \(-0.907326\pi\)
−0.957916 + 0.287048i \(0.907326\pi\)
\(684\) 0 0
\(685\) 32.9443 1.25874
\(686\) 0 0
\(687\) −50.2492 −1.91713
\(688\) 0 0
\(689\) 8.94427 0.340750
\(690\) 0 0
\(691\) −11.5967 −0.441161 −0.220581 0.975369i \(-0.570795\pi\)
−0.220581 + 0.975369i \(0.570795\pi\)
\(692\) 0 0
\(693\) −78.2492 −2.97244
\(694\) 0 0
\(695\) 22.4721 0.852417
\(696\) 0 0
\(697\) 2.00000 0.0757554
\(698\) 0 0
\(699\) −38.4721 −1.45515
\(700\) 0 0
\(701\) 30.3607 1.14671 0.573354 0.819308i \(-0.305642\pi\)
0.573354 + 0.819308i \(0.305642\pi\)
\(702\) 0 0
\(703\) −17.1672 −0.647473
\(704\) 0 0
\(705\) 32.0000 1.20519
\(706\) 0 0
\(707\) 43.4164 1.63284
\(708\) 0 0
\(709\) 2.94427 0.110574 0.0552872 0.998470i \(-0.482393\pi\)
0.0552872 + 0.998470i \(0.482393\pi\)
\(710\) 0 0
\(711\) 12.7639 0.478685
\(712\) 0 0
\(713\) −10.4721 −0.392185
\(714\) 0 0
\(715\) −28.9443 −1.08245
\(716\) 0 0
\(717\) −57.8885 −2.16189
\(718\) 0 0
\(719\) −22.6525 −0.844795 −0.422397 0.906411i \(-0.638811\pi\)
−0.422397 + 0.906411i \(0.638811\pi\)
\(720\) 0 0
\(721\) −25.8885 −0.964140
\(722\) 0 0
\(723\) 61.3050 2.27996
\(724\) 0 0
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) −3.05573 −0.113331 −0.0566653 0.998393i \(-0.518047\pi\)
−0.0566653 + 0.998393i \(0.518047\pi\)
\(728\) 0 0
\(729\) 41.9443 1.55349
\(730\) 0 0
\(731\) −10.4721 −0.387326
\(732\) 0 0
\(733\) 0.111456 0.00411673 0.00205836 0.999998i \(-0.499345\pi\)
0.00205836 + 0.999998i \(0.499345\pi\)
\(734\) 0 0
\(735\) 22.4721 0.828897
\(736\) 0 0
\(737\) −38.8328 −1.43043
\(738\) 0 0
\(739\) −16.5836 −0.610037 −0.305019 0.952346i \(-0.598663\pi\)
−0.305019 + 0.952346i \(0.598663\pi\)
\(740\) 0 0
\(741\) −35.7771 −1.31430
\(742\) 0 0
\(743\) 17.3475 0.636419 0.318209 0.948020i \(-0.396918\pi\)
0.318209 + 0.948020i \(0.396918\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) 0 0
\(747\) 78.2492 2.86299
\(748\) 0 0
\(749\) −31.4164 −1.14793
\(750\) 0 0
\(751\) 22.2918 0.813439 0.406720 0.913553i \(-0.366673\pi\)
0.406720 + 0.913553i \(0.366673\pi\)
\(752\) 0 0
\(753\) −28.9443 −1.05479
\(754\) 0 0
\(755\) 28.9443 1.05339
\(756\) 0 0
\(757\) −16.4721 −0.598690 −0.299345 0.954145i \(-0.596768\pi\)
−0.299345 + 0.954145i \(0.596768\pi\)
\(758\) 0 0
\(759\) 33.8885 1.23008
\(760\) 0 0
\(761\) −10.3607 −0.375574 −0.187787 0.982210i \(-0.560132\pi\)
−0.187787 + 0.982210i \(0.560132\pi\)
\(762\) 0 0
\(763\) 32.3607 1.17154
\(764\) 0 0
\(765\) −14.9443 −0.540311
\(766\) 0 0
\(767\) 24.7214 0.892637
\(768\) 0 0
\(769\) 7.52786 0.271462 0.135731 0.990746i \(-0.456662\pi\)
0.135731 + 0.990746i \(0.456662\pi\)
\(770\) 0 0
\(771\) −72.3607 −2.60601
\(772\) 0 0
\(773\) 35.3050 1.26983 0.634915 0.772582i \(-0.281035\pi\)
0.634915 + 0.772582i \(0.281035\pi\)
\(774\) 0 0
\(775\) −3.23607 −0.116243
\(776\) 0 0
\(777\) −72.7214 −2.60886
\(778\) 0 0
\(779\) 4.94427 0.177147
\(780\) 0 0
\(781\) −15.4164 −0.551642
\(782\) 0 0
\(783\) 28.9443 1.03438
\(784\) 0 0
\(785\) 47.7771 1.70524
\(786\) 0 0
\(787\) 6.65248 0.237135 0.118568 0.992946i \(-0.462170\pi\)
0.118568 + 0.992946i \(0.462170\pi\)
\(788\) 0 0
\(789\) 36.9443 1.31525
\(790\) 0 0
\(791\) 9.52786 0.338772
\(792\) 0 0
\(793\) 48.9443 1.73806
\(794\) 0 0
\(795\) 12.9443 0.459086
\(796\) 0 0
\(797\) 19.8885 0.704488 0.352244 0.935908i \(-0.385419\pi\)
0.352244 + 0.935908i \(0.385419\pi\)
\(798\) 0 0
\(799\) 4.94427 0.174916
\(800\) 0 0
\(801\) −123.082 −4.34889
\(802\) 0 0
\(803\) −9.52786 −0.336231
\(804\) 0 0
\(805\) −20.9443 −0.738189
\(806\) 0 0
\(807\) −61.3050 −2.15804
\(808\) 0 0
\(809\) −25.0557 −0.880912 −0.440456 0.897774i \(-0.645183\pi\)
−0.440456 + 0.897774i \(0.645183\pi\)
\(810\) 0 0
\(811\) −13.1246 −0.460867 −0.230434 0.973088i \(-0.574014\pi\)
−0.230434 + 0.973088i \(0.574014\pi\)
\(812\) 0 0
\(813\) 57.8885 2.03024
\(814\) 0 0
\(815\) −0.360680 −0.0126341
\(816\) 0 0
\(817\) −25.8885 −0.905725
\(818\) 0 0
\(819\) −108.138 −3.77864
\(820\) 0 0
\(821\) −50.0000 −1.74501 −0.872506 0.488603i \(-0.837507\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(822\) 0 0
\(823\) 51.5967 1.79855 0.899275 0.437384i \(-0.144095\pi\)
0.899275 + 0.437384i \(0.144095\pi\)
\(824\) 0 0
\(825\) 10.4721 0.364593
\(826\) 0 0
\(827\) −20.7639 −0.722033 −0.361016 0.932559i \(-0.617570\pi\)
−0.361016 + 0.932559i \(0.617570\pi\)
\(828\) 0 0
\(829\) 8.11146 0.281723 0.140861 0.990029i \(-0.455013\pi\)
0.140861 + 0.990029i \(0.455013\pi\)
\(830\) 0 0
\(831\) 106.249 3.68574
\(832\) 0 0
\(833\) 3.47214 0.120302
\(834\) 0 0
\(835\) −12.5836 −0.435473
\(836\) 0 0
\(837\) −46.8328 −1.61878
\(838\) 0 0
\(839\) −26.0689 −0.899998 −0.449999 0.893029i \(-0.648576\pi\)
−0.449999 + 0.893029i \(0.648576\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 51.4164 1.77088
\(844\) 0 0
\(845\) −14.0000 −0.481615
\(846\) 0 0
\(847\) 1.70820 0.0586946
\(848\) 0 0
\(849\) −10.4721 −0.359403
\(850\) 0 0
\(851\) 22.4721 0.770335
\(852\) 0 0
\(853\) 33.7771 1.15651 0.578253 0.815858i \(-0.303735\pi\)
0.578253 + 0.815858i \(0.303735\pi\)
\(854\) 0 0
\(855\) −36.9443 −1.26347
\(856\) 0 0
\(857\) 27.8885 0.952655 0.476327 0.879268i \(-0.341968\pi\)
0.476327 + 0.879268i \(0.341968\pi\)
\(858\) 0 0
\(859\) −30.2492 −1.03209 −0.516045 0.856561i \(-0.672596\pi\)
−0.516045 + 0.856561i \(0.672596\pi\)
\(860\) 0 0
\(861\) 20.9443 0.713779
\(862\) 0 0
\(863\) −22.8328 −0.777238 −0.388619 0.921399i \(-0.627048\pi\)
−0.388619 + 0.921399i \(0.627048\pi\)
\(864\) 0 0
\(865\) 4.00000 0.136004
\(866\) 0 0
\(867\) −3.23607 −0.109903
\(868\) 0 0
\(869\) 5.52786 0.187520
\(870\) 0 0
\(871\) −53.6656 −1.81839
\(872\) 0 0
\(873\) 14.9443 0.505787
\(874\) 0 0
\(875\) −38.8328 −1.31279
\(876\) 0 0
\(877\) 31.8885 1.07680 0.538400 0.842690i \(-0.319029\pi\)
0.538400 + 0.842690i \(0.319029\pi\)
\(878\) 0 0
\(879\) −32.3607 −1.09150
\(880\) 0 0
\(881\) 27.8885 0.939589 0.469794 0.882776i \(-0.344328\pi\)
0.469794 + 0.882776i \(0.344328\pi\)
\(882\) 0 0
\(883\) −37.8885 −1.27505 −0.637526 0.770429i \(-0.720042\pi\)
−0.637526 + 0.770429i \(0.720042\pi\)
\(884\) 0 0
\(885\) 35.7771 1.20263
\(886\) 0 0
\(887\) 29.1246 0.977909 0.488954 0.872309i \(-0.337378\pi\)
0.488954 + 0.872309i \(0.337378\pi\)
\(888\) 0 0
\(889\) −20.9443 −0.702448
\(890\) 0 0
\(891\) 79.0132 2.64704
\(892\) 0 0
\(893\) 12.2229 0.409024
\(894\) 0 0
\(895\) −11.0557 −0.369552
\(896\) 0 0
\(897\) 46.8328 1.56370
\(898\) 0 0
\(899\) −6.47214 −0.215858
\(900\) 0 0
\(901\) 2.00000 0.0666297
\(902\) 0 0
\(903\) −109.666 −3.64944
\(904\) 0 0
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) −18.8754 −0.626747 −0.313373 0.949630i \(-0.601459\pi\)
−0.313373 + 0.949630i \(0.601459\pi\)
\(908\) 0 0
\(909\) −100.249 −3.32506
\(910\) 0 0
\(911\) −14.2918 −0.473508 −0.236754 0.971570i \(-0.576084\pi\)
−0.236754 + 0.971570i \(0.576084\pi\)
\(912\) 0 0
\(913\) 33.8885 1.12155
\(914\) 0 0
\(915\) 70.8328 2.34166
\(916\) 0 0
\(917\) 68.3607 2.25747
\(918\) 0 0
\(919\) 6.11146 0.201598 0.100799 0.994907i \(-0.467860\pi\)
0.100799 + 0.994907i \(0.467860\pi\)
\(920\) 0 0
\(921\) 96.7214 3.18708
\(922\) 0 0
\(923\) −21.3050 −0.701261
\(924\) 0 0
\(925\) 6.94427 0.228326
\(926\) 0 0
\(927\) 59.7771 1.96334
\(928\) 0 0
\(929\) −9.05573 −0.297109 −0.148554 0.988904i \(-0.547462\pi\)
−0.148554 + 0.988904i \(0.547462\pi\)
\(930\) 0 0
\(931\) 8.58359 0.281316
\(932\) 0 0
\(933\) −62.2492 −2.03795
\(934\) 0 0
\(935\) −6.47214 −0.211661
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 0 0
\(939\) −90.2492 −2.94517
\(940\) 0 0
\(941\) −45.7771 −1.49229 −0.746145 0.665783i \(-0.768098\pi\)
−0.746145 + 0.665783i \(0.768098\pi\)
\(942\) 0 0
\(943\) −6.47214 −0.210762
\(944\) 0 0
\(945\) −93.6656 −3.04694
\(946\) 0 0
\(947\) −39.0132 −1.26776 −0.633879 0.773433i \(-0.718538\pi\)
−0.633879 + 0.773433i \(0.718538\pi\)
\(948\) 0 0
\(949\) −13.1672 −0.427425
\(950\) 0 0
\(951\) 64.3607 2.08704
\(952\) 0 0
\(953\) −32.4721 −1.05188 −0.525938 0.850523i \(-0.676286\pi\)
−0.525938 + 0.850523i \(0.676286\pi\)
\(954\) 0 0
\(955\) 9.88854 0.319986
\(956\) 0 0
\(957\) 20.9443 0.677032
\(958\) 0 0
\(959\) 53.3050 1.72131
\(960\) 0 0
\(961\) −20.5279 −0.662189
\(962\) 0 0
\(963\) 72.5410 2.33760
\(964\) 0 0
\(965\) 5.88854 0.189559
\(966\) 0 0
\(967\) −6.47214 −0.208130 −0.104065 0.994571i \(-0.533185\pi\)
−0.104065 + 0.994571i \(0.533185\pi\)
\(968\) 0 0
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) 18.4721 0.592799 0.296400 0.955064i \(-0.404214\pi\)
0.296400 + 0.955064i \(0.404214\pi\)
\(972\) 0 0
\(973\) 36.3607 1.16567
\(974\) 0 0
\(975\) 14.4721 0.463479
\(976\) 0 0
\(977\) 34.0000 1.08776 0.543878 0.839164i \(-0.316955\pi\)
0.543878 + 0.839164i \(0.316955\pi\)
\(978\) 0 0
\(979\) −53.3050 −1.70363
\(980\) 0 0
\(981\) −74.7214 −2.38567
\(982\) 0 0
\(983\) 11.2361 0.358375 0.179187 0.983815i \(-0.442653\pi\)
0.179187 + 0.983815i \(0.442653\pi\)
\(984\) 0 0
\(985\) −21.8885 −0.697427
\(986\) 0 0
\(987\) 51.7771 1.64808
\(988\) 0 0
\(989\) 33.8885 1.07759
\(990\) 0 0
\(991\) −12.7639 −0.405460 −0.202730 0.979235i \(-0.564981\pi\)
−0.202730 + 0.979235i \(0.564981\pi\)
\(992\) 0 0
\(993\) 33.8885 1.07542
\(994\) 0 0
\(995\) −0.360680 −0.0114343
\(996\) 0 0
\(997\) −46.9443 −1.48674 −0.743370 0.668880i \(-0.766774\pi\)
−0.743370 + 0.668880i \(0.766774\pi\)
\(998\) 0 0
\(999\) 100.498 3.17963
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 1088.2.a.o.1.1 2
3.2 odd 2 9792.2.a.da.1.1 2
4.3 odd 2 1088.2.a.s.1.2 2
8.3 odd 2 136.2.a.c.1.1 2
8.5 even 2 272.2.a.f.1.2 2
12.11 even 2 9792.2.a.db.1.2 2
24.5 odd 2 2448.2.a.u.1.1 2
24.11 even 2 1224.2.a.i.1.2 2
40.3 even 4 3400.2.e.f.2449.1 4
40.19 odd 2 3400.2.a.i.1.2 2
40.27 even 4 3400.2.e.f.2449.4 4
40.29 even 2 6800.2.a.bd.1.1 2
56.27 even 2 6664.2.a.i.1.2 2
136.67 odd 2 2312.2.a.m.1.2 2
136.101 even 2 4624.2.a.h.1.1 2
136.115 odd 4 2312.2.b.g.577.1 4
136.123 odd 4 2312.2.b.g.577.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.2.a.c.1.1 2 8.3 odd 2
272.2.a.f.1.2 2 8.5 even 2
1088.2.a.o.1.1 2 1.1 even 1 trivial
1088.2.a.s.1.2 2 4.3 odd 2
1224.2.a.i.1.2 2 24.11 even 2
2312.2.a.m.1.2 2 136.67 odd 2
2312.2.b.g.577.1 4 136.115 odd 4
2312.2.b.g.577.4 4 136.123 odd 4
2448.2.a.u.1.1 2 24.5 odd 2
3400.2.a.i.1.2 2 40.19 odd 2
3400.2.e.f.2449.1 4 40.3 even 4
3400.2.e.f.2449.4 4 40.27 even 4
4624.2.a.h.1.1 2 136.101 even 2
6664.2.a.i.1.2 2 56.27 even 2
6800.2.a.bd.1.1 2 40.29 even 2
9792.2.a.da.1.1 2 3.2 odd 2
9792.2.a.db.1.2 2 12.11 even 2