Properties

Label 136.2.a.c.1.1
Level $136$
Weight $2$
Character 136.1
Self dual yes
Analytic conductor $1.086$
Analytic rank $0$
Dimension $2$
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] = [136,2,Mod(1,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, 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("136.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 136.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: \(1.08596546749\)
Analytic rank: \(0\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 136.1

$q$-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 \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 3.23607 1.22312 0.611559 0.791199i \(-0.290543\pi\)
0.611559 + 0.791199i \(0.290543\pi\)
\(8\) 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 \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 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 \(-0.390458\pi\)
0.337383 + 0.941367i \(0.390458\pi\)
\(24\) 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 \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −3.23607 −0.581215 −0.290607 0.956842i \(-0.593857\pi\)
−0.290607 + 0.956842i \(0.593857\pi\)
\(32\) 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 \(-0.306627\pi\)
0.570816 + 0.821078i \(0.306627\pi\)
\(38\) 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 \(-0.617427\pi\)
−0.360598 + 0.932721i \(0.617427\pi\)
\(48\) 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 \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 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 \(-0.747100\pi\)
−0.700635 + 0.713520i \(0.747100\pi\)
\(62\) 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 \(-0.408774\pi\)
0.282687 + 0.959212i \(0.408774\pi\)
\(72\) 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 \(-0.530635\pi\)
−0.0960940 + 0.995372i \(0.530635\pi\)
\(80\) 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 \(-0.267368\pi\)
0.667491 + 0.744618i \(0.267368\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 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 \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 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 \(-0.592709\pi\)
−0.287155 + 0.957884i \(0.592709\pi\)
\(128\) 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 \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 14.4721 1.17773 0.588863 0.808233i \(-0.299576\pi\)
0.588863 + 0.808233i \(0.299576\pi\)
\(152\) 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 \(-0.0977083\pi\)
0.953257 + 0.302162i \(0.0977083\pi\)
\(158\) 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 \(-0.578275\pi\)
−0.243437 + 0.969917i \(0.578275\pi\)
\(168\) 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 \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 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 \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 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 \(-0.442753\pi\)
0.178877 + 0.983871i \(0.442753\pi\)
\(192\) 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 \(-0.627481\pi\)
−0.389874 + 0.920868i \(0.627481\pi\)
\(198\) 0 0
\(199\) −0.180340 −0.0127840 −0.00639198 0.999980i \(-0.502035\pi\)
−0.00639198 + 0.999980i \(0.502035\pi\)
\(200\) 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 \(-0.338979\pi\)
0.484563 + 0.874756i \(0.338979\pi\)
\(224\) 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 \(-0.671486\pi\)
−0.513055 + 0.858356i \(0.671486\pi\)
\(230\) 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 \(-0.696384\pi\)
−0.578557 + 0.815642i \(0.696384\pi\)
\(240\) 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 \(-0.385508\pi\)
0.351983 + 0.936006i \(0.385508\pi\)
\(264\) 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 \(-0.695982\pi\)
−0.577526 + 0.816372i \(0.695982\pi\)
\(270\) 0 0
\(271\) 17.8885 1.08665 0.543326 0.839522i \(-0.317165\pi\)
0.543326 + 0.839522i \(0.317165\pi\)
\(272\) 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 \(-0.0526218\pi\)
0.986366 + 0.164564i \(0.0526218\pi\)
\(278\) 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 \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) 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 \(-0.683618\pi\)
−0.545389 + 0.838183i \(0.683618\pi\)
\(312\) 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 \(-0.311367\pi\)
0.558526 + 0.829487i \(0.311367\pi\)
\(318\) 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 \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 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 \(-0.297937\pi\)
0.593017 + 0.805190i \(0.297937\pi\)
\(360\) 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 \(-0.332577\pi\)
0.502057 + 0.864834i \(0.332577\pi\)
\(368\) 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 \(-0.470887\pi\)
0.0913329 + 0.995820i \(0.470887\pi\)
\(374\) 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 \(-0.472181\pi\)
0.0872851 + 0.996183i \(0.472181\pi\)
\(384\) 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 \(-0.722864\pi\)
−0.644332 + 0.764746i \(0.722864\pi\)
\(390\) 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 \(-0.216981\pi\)
0.776524 + 0.630087i \(0.216981\pi\)
\(398\) 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 \(-0.884415\pi\)
−0.934793 + 0.355194i \(0.884415\pi\)
\(422\) 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 \(-0.590102\pi\)
−0.279298 + 0.960205i \(0.590102\pi\)
\(432\) 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 \(-0.238829\pi\)
0.731481 + 0.681861i \(0.238829\pi\)
\(440\) 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 \(-0.528864\pi\)
−0.0905538 + 0.995892i \(0.528864\pi\)
\(462\) 0 0
\(463\) −22.8328 −1.06113 −0.530565 0.847644i \(-0.678020\pi\)
−0.530565 + 0.847644i \(0.678020\pi\)
\(464\) 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 \(-0.220214\pi\)
0.770084 + 0.637943i \(0.220214\pi\)
\(480\) 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 \(-0.0643948\pi\)
0.979607 + 0.200925i \(0.0643948\pi\)
\(488\) 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 \(-0.396781\pi\)
0.318620 + 0.947883i \(0.396781\pi\)
\(504\) 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 \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 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 \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 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 \(-0.350096\pi\)
0.453721 + 0.891144i \(0.350096\pi\)
\(558\) 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 \(-0.0939383\pi\)
0.956768 + 0.290851i \(0.0939383\pi\)
\(600\) 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 \(-0.540756\pi\)
−0.127688 + 0.991814i \(0.540756\pi\)
\(608\) 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 \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 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 \(-0.698241\pi\)
−0.583306 + 0.812253i \(0.698241\pi\)
\(632\) 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 \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 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 \(-0.633643\pi\)
−0.407626 + 0.913149i \(0.633643\pi\)
\(654\) 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 \(-0.217376\pi\)
0.775742 + 0.631050i \(0.217376\pi\)
\(662\) 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 \(-0.180558\pi\)
0.843387 + 0.537307i \(0.180558\pi\)
\(678\) 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 \(-0.694358\pi\)
−0.573354 + 0.819308i \(0.694358\pi\)
\(702\) 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 \(-0.517607\pi\)
−0.0552872 + 0.998470i \(0.517607\pi\)
\(710\) 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 \(-0.361189\pi\)
0.422397 + 0.906411i \(0.361189\pi\)
\(720\) 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 \(-0.481953\pi\)
0.0566653 + 0.998393i \(0.481953\pi\)
\(728\) 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 \(-0.500655\pi\)
−0.00205836 + 0.999998i \(0.500655\pi\)
\(734\) 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 \(-0.603082\pi\)
−0.318209 + 0.948020i \(0.603082\pi\)
\(744\) 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 \(-0.633327\pi\)
−0.406720 + 0.913553i \(0.633327\pi\)
\(752\) 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 \(-0.403232\pi\)
0.299345 + 0.954145i \(0.403232\pi\)
\(758\) 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 \(-0.718965\pi\)
−0.634915 + 0.772582i \(0.718965\pi\)
\(774\) 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 \(-0.614581\pi\)
−0.352244 + 0.935908i \(0.614581\pi\)
\(798\) 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 \(-0.162493\pi\)
0.872506 + 0.488603i \(0.162493\pi\)
\(822\) 0 0
\(823\) −51.5967 −1.79855 −0.899275 0.437384i \(-0.855905\pi\)
−0.899275 + 0.437384i \(0.855905\pi\)
\(824\) 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 \(-0.544987\pi\)
−0.140861 + 0.990029i \(0.544987\pi\)
\(830\) 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 \(-0.351424\pi\)
0.449999 + 0.893029i \(0.351424\pi\)
\(840\) 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 \(-0.696265\pi\)
−0.578253 + 0.815858i \(0.696265\pi\)
\(854\) 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 \(-0.372952\pi\)
0.388619 + 0.921399i \(0.372952\pi\)
\(864\) 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 \(-0.680971\pi\)
−0.538400 + 0.842690i \(0.680971\pi\)
\(878\) 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 \(-0.662622\pi\)
−0.488954 + 0.872309i \(0.662622\pi\)
\(888\) 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 \(-0.423916\pi\)
0.236754 + 0.971570i \(0.423916\pi\)
\(912\) 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 \(-0.532140\pi\)
−0.100799 + 0.994907i \(0.532140\pi\)
\(920\) 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 \(-0.231902\pi\)
0.746145 + 0.665783i \(0.231902\pi\)
\(942\) 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 \(-0.466815\pi\)
0.104065 + 0.994571i \(0.466815\pi\)
\(968\) 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 \(-0.557347\pi\)
−0.179187 + 0.983815i \(0.557347\pi\)
\(984\) 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 \(-0.435019\pi\)
0.202730 + 0.979235i \(0.435019\pi\)
\(992\) 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 \(-0.233226\pi\)
0.743370 + 0.668880i \(0.233226\pi\)
\(998\) 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 136.2.a.c.1.1 2
3.2 odd 2 1224.2.a.i.1.2 2
4.3 odd 2 272.2.a.f.1.2 2
5.2 odd 4 3400.2.e.f.2449.4 4
5.3 odd 4 3400.2.e.f.2449.1 4
5.4 even 2 3400.2.a.i.1.2 2
7.6 odd 2 6664.2.a.i.1.2 2
8.3 odd 2 1088.2.a.o.1.1 2
8.5 even 2 1088.2.a.s.1.2 2
12.11 even 2 2448.2.a.u.1.1 2
17.4 even 4 2312.2.b.g.577.4 4
17.13 even 4 2312.2.b.g.577.1 4
17.16 even 2 2312.2.a.m.1.2 2
20.19 odd 2 6800.2.a.bd.1.1 2
24.5 odd 2 9792.2.a.db.1.2 2
24.11 even 2 9792.2.a.da.1.1 2
68.67 odd 2 4624.2.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.2.a.c.1.1 2 1.1 even 1 trivial
272.2.a.f.1.2 2 4.3 odd 2
1088.2.a.o.1.1 2 8.3 odd 2
1088.2.a.s.1.2 2 8.5 even 2
1224.2.a.i.1.2 2 3.2 odd 2
2312.2.a.m.1.2 2 17.16 even 2
2312.2.b.g.577.1 4 17.13 even 4
2312.2.b.g.577.4 4 17.4 even 4
2448.2.a.u.1.1 2 12.11 even 2
3400.2.a.i.1.2 2 5.4 even 2
3400.2.e.f.2449.1 4 5.3 odd 4
3400.2.e.f.2449.4 4 5.2 odd 4
4624.2.a.h.1.1 2 68.67 odd 2
6664.2.a.i.1.2 2 7.6 odd 2
6800.2.a.bd.1.1 2 20.19 odd 2
9792.2.a.da.1.1 2 24.11 even 2
9792.2.a.db.1.2 2 24.5 odd 2