Properties

Label 9702.2.a.cx.1.1
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $0$
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] = [9702,2,Mod(1,9702)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9702, 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("9702.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.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: \(77.4708600410\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +0.585786 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +0.585786 q^{5} -1.00000 q^{8} -0.585786 q^{10} -1.00000 q^{11} -3.82843 q^{13} +1.00000 q^{16} -3.65685 q^{17} -0.585786 q^{19} +0.585786 q^{20} +1.00000 q^{22} +6.24264 q^{23} -4.65685 q^{25} +3.82843 q^{26} -2.65685 q^{29} -4.00000 q^{31} -1.00000 q^{32} +3.65685 q^{34} -9.41421 q^{37} +0.585786 q^{38} -0.585786 q^{40} +5.41421 q^{41} -5.65685 q^{43} -1.00000 q^{44} -6.24264 q^{46} +10.4853 q^{47} +4.65685 q^{50} -3.82843 q^{52} -7.89949 q^{53} -0.585786 q^{55} +2.65685 q^{58} +5.58579 q^{59} +11.8284 q^{61} +4.00000 q^{62} +1.00000 q^{64} -2.24264 q^{65} +2.75736 q^{67} -3.65685 q^{68} +11.0711 q^{71} -9.41421 q^{73} +9.41421 q^{74} -0.585786 q^{76} -13.2426 q^{79} +0.585786 q^{80} -5.41421 q^{82} +12.1421 q^{83} -2.14214 q^{85} +5.65685 q^{86} +1.00000 q^{88} -12.4853 q^{89} +6.24264 q^{92} -10.4853 q^{94} -0.343146 q^{95} -3.82843 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8} - 4 q^{10} - 2 q^{11} - 2 q^{13} + 2 q^{16} + 4 q^{17} - 4 q^{19} + 4 q^{20} + 2 q^{22} + 4 q^{23} + 2 q^{25} + 2 q^{26} + 6 q^{29} - 8 q^{31} - 2 q^{32} - 4 q^{34} - 16 q^{37} + 4 q^{38} - 4 q^{40} + 8 q^{41} - 2 q^{44} - 4 q^{46} + 4 q^{47} - 2 q^{50} - 2 q^{52} + 4 q^{53} - 4 q^{55} - 6 q^{58} + 14 q^{59} + 18 q^{61} + 8 q^{62} + 2 q^{64} + 4 q^{65} + 14 q^{67} + 4 q^{68} + 8 q^{71} - 16 q^{73} + 16 q^{74} - 4 q^{76} - 18 q^{79} + 4 q^{80} - 8 q^{82} - 4 q^{83} + 24 q^{85} + 2 q^{88} - 8 q^{89} + 4 q^{92} - 4 q^{94} - 12 q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0.585786 0.261972 0.130986 0.991384i \(-0.458186\pi\)
0.130986 + 0.991384i \(0.458186\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −0.585786 −0.185242
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −3.82843 −1.06181 −0.530907 0.847430i \(-0.678149\pi\)
−0.530907 + 0.847430i \(0.678149\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.65685 −0.886917 −0.443459 0.896295i \(-0.646249\pi\)
−0.443459 + 0.896295i \(0.646249\pi\)
\(18\) 0 0
\(19\) −0.585786 −0.134389 −0.0671943 0.997740i \(-0.521405\pi\)
−0.0671943 + 0.997740i \(0.521405\pi\)
\(20\) 0.585786 0.130986
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 6.24264 1.30168 0.650840 0.759215i \(-0.274417\pi\)
0.650840 + 0.759215i \(0.274417\pi\)
\(24\) 0 0
\(25\) −4.65685 −0.931371
\(26\) 3.82843 0.750816
\(27\) 0 0
\(28\) 0 0
\(29\) −2.65685 −0.493365 −0.246683 0.969096i \(-0.579341\pi\)
−0.246683 + 0.969096i \(0.579341\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.65685 0.627145
\(35\) 0 0
\(36\) 0 0
\(37\) −9.41421 −1.54769 −0.773844 0.633377i \(-0.781668\pi\)
−0.773844 + 0.633377i \(0.781668\pi\)
\(38\) 0.585786 0.0950271
\(39\) 0 0
\(40\) −0.585786 −0.0926210
\(41\) 5.41421 0.845558 0.422779 0.906233i \(-0.361055\pi\)
0.422779 + 0.906233i \(0.361055\pi\)
\(42\) 0 0
\(43\) −5.65685 −0.862662 −0.431331 0.902194i \(-0.641956\pi\)
−0.431331 + 0.902194i \(0.641956\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −6.24264 −0.920427
\(47\) 10.4853 1.52944 0.764718 0.644365i \(-0.222878\pi\)
0.764718 + 0.644365i \(0.222878\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 4.65685 0.658579
\(51\) 0 0
\(52\) −3.82843 −0.530907
\(53\) −7.89949 −1.08508 −0.542540 0.840030i \(-0.682537\pi\)
−0.542540 + 0.840030i \(0.682537\pi\)
\(54\) 0 0
\(55\) −0.585786 −0.0789874
\(56\) 0 0
\(57\) 0 0
\(58\) 2.65685 0.348862
\(59\) 5.58579 0.727207 0.363604 0.931554i \(-0.381546\pi\)
0.363604 + 0.931554i \(0.381546\pi\)
\(60\) 0 0
\(61\) 11.8284 1.51447 0.757237 0.653140i \(-0.226549\pi\)
0.757237 + 0.653140i \(0.226549\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.24264 −0.278165
\(66\) 0 0
\(67\) 2.75736 0.336865 0.168433 0.985713i \(-0.446129\pi\)
0.168433 + 0.985713i \(0.446129\pi\)
\(68\) −3.65685 −0.443459
\(69\) 0 0
\(70\) 0 0
\(71\) 11.0711 1.31389 0.656947 0.753937i \(-0.271848\pi\)
0.656947 + 0.753937i \(0.271848\pi\)
\(72\) 0 0
\(73\) −9.41421 −1.10185 −0.550925 0.834555i \(-0.685725\pi\)
−0.550925 + 0.834555i \(0.685725\pi\)
\(74\) 9.41421 1.09438
\(75\) 0 0
\(76\) −0.585786 −0.0671943
\(77\) 0 0
\(78\) 0 0
\(79\) −13.2426 −1.48991 −0.744957 0.667113i \(-0.767530\pi\)
−0.744957 + 0.667113i \(0.767530\pi\)
\(80\) 0.585786 0.0654929
\(81\) 0 0
\(82\) −5.41421 −0.597900
\(83\) 12.1421 1.33277 0.666386 0.745607i \(-0.267840\pi\)
0.666386 + 0.745607i \(0.267840\pi\)
\(84\) 0 0
\(85\) −2.14214 −0.232347
\(86\) 5.65685 0.609994
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −12.4853 −1.32344 −0.661719 0.749752i \(-0.730173\pi\)
−0.661719 + 0.749752i \(0.730173\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.24264 0.650840
\(93\) 0 0
\(94\) −10.4853 −1.08147
\(95\) −0.343146 −0.0352060
\(96\) 0 0
\(97\) −3.82843 −0.388718 −0.194359 0.980930i \(-0.562263\pi\)
−0.194359 + 0.980930i \(0.562263\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −4.65685 −0.465685
\(101\) −6.17157 −0.614094 −0.307047 0.951694i \(-0.599341\pi\)
−0.307047 + 0.951694i \(0.599341\pi\)
\(102\) 0 0
\(103\) 13.4142 1.32174 0.660871 0.750500i \(-0.270187\pi\)
0.660871 + 0.750500i \(0.270187\pi\)
\(104\) 3.82843 0.375408
\(105\) 0 0
\(106\) 7.89949 0.767267
\(107\) 3.07107 0.296891 0.148446 0.988921i \(-0.452573\pi\)
0.148446 + 0.988921i \(0.452573\pi\)
\(108\) 0 0
\(109\) 16.4853 1.57900 0.789502 0.613748i \(-0.210339\pi\)
0.789502 + 0.613748i \(0.210339\pi\)
\(110\) 0.585786 0.0558525
\(111\) 0 0
\(112\) 0 0
\(113\) 8.17157 0.768717 0.384358 0.923184i \(-0.374423\pi\)
0.384358 + 0.923184i \(0.374423\pi\)
\(114\) 0 0
\(115\) 3.65685 0.341003
\(116\) −2.65685 −0.246683
\(117\) 0 0
\(118\) −5.58579 −0.514213
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −11.8284 −1.07090
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) 15.7279 1.39563 0.697814 0.716279i \(-0.254156\pi\)
0.697814 + 0.716279i \(0.254156\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 2.24264 0.196693
\(131\) 0.585786 0.0511804 0.0255902 0.999673i \(-0.491853\pi\)
0.0255902 + 0.999673i \(0.491853\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −2.75736 −0.238200
\(135\) 0 0
\(136\) 3.65685 0.313573
\(137\) 16.6569 1.42309 0.711546 0.702640i \(-0.247996\pi\)
0.711546 + 0.702640i \(0.247996\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −11.0711 −0.929063
\(143\) 3.82843 0.320149
\(144\) 0 0
\(145\) −1.55635 −0.129248
\(146\) 9.41421 0.779126
\(147\) 0 0
\(148\) −9.41421 −0.773844
\(149\) −17.6569 −1.44651 −0.723253 0.690583i \(-0.757354\pi\)
−0.723253 + 0.690583i \(0.757354\pi\)
\(150\) 0 0
\(151\) −15.7279 −1.27992 −0.639960 0.768408i \(-0.721049\pi\)
−0.639960 + 0.768408i \(0.721049\pi\)
\(152\) 0.585786 0.0475136
\(153\) 0 0
\(154\) 0 0
\(155\) −2.34315 −0.188206
\(156\) 0 0
\(157\) 17.6569 1.40917 0.704585 0.709619i \(-0.251133\pi\)
0.704585 + 0.709619i \(0.251133\pi\)
\(158\) 13.2426 1.05353
\(159\) 0 0
\(160\) −0.585786 −0.0463105
\(161\) 0 0
\(162\) 0 0
\(163\) 9.72792 0.761950 0.380975 0.924585i \(-0.375588\pi\)
0.380975 + 0.924585i \(0.375588\pi\)
\(164\) 5.41421 0.422779
\(165\) 0 0
\(166\) −12.1421 −0.942412
\(167\) −13.7279 −1.06230 −0.531149 0.847278i \(-0.678240\pi\)
−0.531149 + 0.847278i \(0.678240\pi\)
\(168\) 0 0
\(169\) 1.65685 0.127450
\(170\) 2.14214 0.164294
\(171\) 0 0
\(172\) −5.65685 −0.431331
\(173\) 9.82843 0.747241 0.373621 0.927582i \(-0.378116\pi\)
0.373621 + 0.927582i \(0.378116\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 12.4853 0.935811
\(179\) −0.899495 −0.0672314 −0.0336157 0.999435i \(-0.510702\pi\)
−0.0336157 + 0.999435i \(0.510702\pi\)
\(180\) 0 0
\(181\) −7.65685 −0.569129 −0.284565 0.958657i \(-0.591849\pi\)
−0.284565 + 0.958657i \(0.591849\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −6.24264 −0.460214
\(185\) −5.51472 −0.405450
\(186\) 0 0
\(187\) 3.65685 0.267416
\(188\) 10.4853 0.764718
\(189\) 0 0
\(190\) 0.343146 0.0248944
\(191\) 7.17157 0.518917 0.259458 0.965754i \(-0.416456\pi\)
0.259458 + 0.965754i \(0.416456\pi\)
\(192\) 0 0
\(193\) 21.8995 1.57636 0.788180 0.615445i \(-0.211024\pi\)
0.788180 + 0.615445i \(0.211024\pi\)
\(194\) 3.82843 0.274865
\(195\) 0 0
\(196\) 0 0
\(197\) 0.514719 0.0366722 0.0183361 0.999832i \(-0.494163\pi\)
0.0183361 + 0.999832i \(0.494163\pi\)
\(198\) 0 0
\(199\) 0.100505 0.00712462 0.00356231 0.999994i \(-0.498866\pi\)
0.00356231 + 0.999994i \(0.498866\pi\)
\(200\) 4.65685 0.329289
\(201\) 0 0
\(202\) 6.17157 0.434230
\(203\) 0 0
\(204\) 0 0
\(205\) 3.17157 0.221512
\(206\) −13.4142 −0.934613
\(207\) 0 0
\(208\) −3.82843 −0.265454
\(209\) 0.585786 0.0405197
\(210\) 0 0
\(211\) 7.41421 0.510416 0.255208 0.966886i \(-0.417856\pi\)
0.255208 + 0.966886i \(0.417856\pi\)
\(212\) −7.89949 −0.542540
\(213\) 0 0
\(214\) −3.07107 −0.209934
\(215\) −3.31371 −0.225993
\(216\) 0 0
\(217\) 0 0
\(218\) −16.4853 −1.11652
\(219\) 0 0
\(220\) −0.585786 −0.0394937
\(221\) 14.0000 0.941742
\(222\) 0 0
\(223\) 8.58579 0.574947 0.287473 0.957789i \(-0.407185\pi\)
0.287473 + 0.957789i \(0.407185\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −8.17157 −0.543565
\(227\) 28.8284 1.91341 0.956705 0.291059i \(-0.0940078\pi\)
0.956705 + 0.291059i \(0.0940078\pi\)
\(228\) 0 0
\(229\) 23.3137 1.54061 0.770307 0.637674i \(-0.220103\pi\)
0.770307 + 0.637674i \(0.220103\pi\)
\(230\) −3.65685 −0.241126
\(231\) 0 0
\(232\) 2.65685 0.174431
\(233\) −1.41421 −0.0926482 −0.0463241 0.998926i \(-0.514751\pi\)
−0.0463241 + 0.998926i \(0.514751\pi\)
\(234\) 0 0
\(235\) 6.14214 0.400669
\(236\) 5.58579 0.363604
\(237\) 0 0
\(238\) 0 0
\(239\) −20.2132 −1.30748 −0.653742 0.756718i \(-0.726802\pi\)
−0.653742 + 0.756718i \(0.726802\pi\)
\(240\) 0 0
\(241\) 12.2426 0.788618 0.394309 0.918978i \(-0.370984\pi\)
0.394309 + 0.918978i \(0.370984\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 11.8284 0.757237
\(245\) 0 0
\(246\) 0 0
\(247\) 2.24264 0.142696
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) 5.65685 0.357771
\(251\) 26.1421 1.65008 0.825038 0.565077i \(-0.191153\pi\)
0.825038 + 0.565077i \(0.191153\pi\)
\(252\) 0 0
\(253\) −6.24264 −0.392471
\(254\) −15.7279 −0.986858
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 25.1421 1.56832 0.784162 0.620557i \(-0.213093\pi\)
0.784162 + 0.620557i \(0.213093\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2.24264 −0.139083
\(261\) 0 0
\(262\) −0.585786 −0.0361900
\(263\) 17.0416 1.05083 0.525416 0.850845i \(-0.323910\pi\)
0.525416 + 0.850845i \(0.323910\pi\)
\(264\) 0 0
\(265\) −4.62742 −0.284260
\(266\) 0 0
\(267\) 0 0
\(268\) 2.75736 0.168433
\(269\) −2.34315 −0.142864 −0.0714321 0.997445i \(-0.522757\pi\)
−0.0714321 + 0.997445i \(0.522757\pi\)
\(270\) 0 0
\(271\) 4.55635 0.276779 0.138389 0.990378i \(-0.455808\pi\)
0.138389 + 0.990378i \(0.455808\pi\)
\(272\) −3.65685 −0.221729
\(273\) 0 0
\(274\) −16.6569 −1.00628
\(275\) 4.65685 0.280819
\(276\) 0 0
\(277\) 1.82843 0.109860 0.0549298 0.998490i \(-0.482507\pi\)
0.0549298 + 0.998490i \(0.482507\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.72792 −0.520664 −0.260332 0.965519i \(-0.583832\pi\)
−0.260332 + 0.965519i \(0.583832\pi\)
\(282\) 0 0
\(283\) −23.4142 −1.39183 −0.695915 0.718124i \(-0.745001\pi\)
−0.695915 + 0.718124i \(0.745001\pi\)
\(284\) 11.0711 0.656947
\(285\) 0 0
\(286\) −3.82843 −0.226380
\(287\) 0 0
\(288\) 0 0
\(289\) −3.62742 −0.213377
\(290\) 1.55635 0.0913920
\(291\) 0 0
\(292\) −9.41421 −0.550925
\(293\) 10.8284 0.632603 0.316302 0.948659i \(-0.397559\pi\)
0.316302 + 0.948659i \(0.397559\pi\)
\(294\) 0 0
\(295\) 3.27208 0.190508
\(296\) 9.41421 0.547190
\(297\) 0 0
\(298\) 17.6569 1.02283
\(299\) −23.8995 −1.38214
\(300\) 0 0
\(301\) 0 0
\(302\) 15.7279 0.905040
\(303\) 0 0
\(304\) −0.585786 −0.0335972
\(305\) 6.92893 0.396750
\(306\) 0 0
\(307\) 9.89949 0.564994 0.282497 0.959268i \(-0.408837\pi\)
0.282497 + 0.959268i \(0.408837\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 2.34315 0.133082
\(311\) −16.7279 −0.948553 −0.474277 0.880376i \(-0.657290\pi\)
−0.474277 + 0.880376i \(0.657290\pi\)
\(312\) 0 0
\(313\) −20.6569 −1.16759 −0.583797 0.811900i \(-0.698434\pi\)
−0.583797 + 0.811900i \(0.698434\pi\)
\(314\) −17.6569 −0.996434
\(315\) 0 0
\(316\) −13.2426 −0.744957
\(317\) −8.68629 −0.487871 −0.243935 0.969791i \(-0.578438\pi\)
−0.243935 + 0.969791i \(0.578438\pi\)
\(318\) 0 0
\(319\) 2.65685 0.148755
\(320\) 0.585786 0.0327465
\(321\) 0 0
\(322\) 0 0
\(323\) 2.14214 0.119192
\(324\) 0 0
\(325\) 17.8284 0.988943
\(326\) −9.72792 −0.538780
\(327\) 0 0
\(328\) −5.41421 −0.298950
\(329\) 0 0
\(330\) 0 0
\(331\) −24.0711 −1.32307 −0.661533 0.749916i \(-0.730094\pi\)
−0.661533 + 0.749916i \(0.730094\pi\)
\(332\) 12.1421 0.666386
\(333\) 0 0
\(334\) 13.7279 0.751158
\(335\) 1.61522 0.0882491
\(336\) 0 0
\(337\) −28.2426 −1.53847 −0.769237 0.638963i \(-0.779364\pi\)
−0.769237 + 0.638963i \(0.779364\pi\)
\(338\) −1.65685 −0.0901210
\(339\) 0 0
\(340\) −2.14214 −0.116174
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 0 0
\(344\) 5.65685 0.304997
\(345\) 0 0
\(346\) −9.82843 −0.528380
\(347\) −17.4142 −0.934844 −0.467422 0.884034i \(-0.654817\pi\)
−0.467422 + 0.884034i \(0.654817\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 2.68629 0.142977 0.0714884 0.997441i \(-0.477225\pi\)
0.0714884 + 0.997441i \(0.477225\pi\)
\(354\) 0 0
\(355\) 6.48528 0.344203
\(356\) −12.4853 −0.661719
\(357\) 0 0
\(358\) 0.899495 0.0475398
\(359\) −19.2426 −1.01559 −0.507794 0.861479i \(-0.669539\pi\)
−0.507794 + 0.861479i \(0.669539\pi\)
\(360\) 0 0
\(361\) −18.6569 −0.981940
\(362\) 7.65685 0.402435
\(363\) 0 0
\(364\) 0 0
\(365\) −5.51472 −0.288654
\(366\) 0 0
\(367\) −10.7279 −0.559993 −0.279996 0.960001i \(-0.590333\pi\)
−0.279996 + 0.960001i \(0.590333\pi\)
\(368\) 6.24264 0.325420
\(369\) 0 0
\(370\) 5.51472 0.286697
\(371\) 0 0
\(372\) 0 0
\(373\) −19.9706 −1.03404 −0.517018 0.855974i \(-0.672958\pi\)
−0.517018 + 0.855974i \(0.672958\pi\)
\(374\) −3.65685 −0.189091
\(375\) 0 0
\(376\) −10.4853 −0.540737
\(377\) 10.1716 0.523863
\(378\) 0 0
\(379\) 27.8701 1.43159 0.715794 0.698311i \(-0.246065\pi\)
0.715794 + 0.698311i \(0.246065\pi\)
\(380\) −0.343146 −0.0176030
\(381\) 0 0
\(382\) −7.17157 −0.366930
\(383\) 6.38478 0.326247 0.163123 0.986606i \(-0.447843\pi\)
0.163123 + 0.986606i \(0.447843\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −21.8995 −1.11465
\(387\) 0 0
\(388\) −3.82843 −0.194359
\(389\) 22.7279 1.15235 0.576176 0.817326i \(-0.304544\pi\)
0.576176 + 0.817326i \(0.304544\pi\)
\(390\) 0 0
\(391\) −22.8284 −1.15448
\(392\) 0 0
\(393\) 0 0
\(394\) −0.514719 −0.0259311
\(395\) −7.75736 −0.390315
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) −0.100505 −0.00503786
\(399\) 0 0
\(400\) −4.65685 −0.232843
\(401\) −18.3137 −0.914543 −0.457271 0.889327i \(-0.651173\pi\)
−0.457271 + 0.889327i \(0.651173\pi\)
\(402\) 0 0
\(403\) 15.3137 0.762830
\(404\) −6.17157 −0.307047
\(405\) 0 0
\(406\) 0 0
\(407\) 9.41421 0.466645
\(408\) 0 0
\(409\) 2.72792 0.134887 0.0674435 0.997723i \(-0.478516\pi\)
0.0674435 + 0.997723i \(0.478516\pi\)
\(410\) −3.17157 −0.156633
\(411\) 0 0
\(412\) 13.4142 0.660871
\(413\) 0 0
\(414\) 0 0
\(415\) 7.11270 0.349149
\(416\) 3.82843 0.187704
\(417\) 0 0
\(418\) −0.585786 −0.0286518
\(419\) 26.1421 1.27713 0.638563 0.769569i \(-0.279529\pi\)
0.638563 + 0.769569i \(0.279529\pi\)
\(420\) 0 0
\(421\) 0.686292 0.0334478 0.0167239 0.999860i \(-0.494676\pi\)
0.0167239 + 0.999860i \(0.494676\pi\)
\(422\) −7.41421 −0.360918
\(423\) 0 0
\(424\) 7.89949 0.383633
\(425\) 17.0294 0.826049
\(426\) 0 0
\(427\) 0 0
\(428\) 3.07107 0.148446
\(429\) 0 0
\(430\) 3.31371 0.159801
\(431\) 17.5858 0.847078 0.423539 0.905878i \(-0.360788\pi\)
0.423539 + 0.905878i \(0.360788\pi\)
\(432\) 0 0
\(433\) 26.1421 1.25631 0.628155 0.778088i \(-0.283810\pi\)
0.628155 + 0.778088i \(0.283810\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 16.4853 0.789502
\(437\) −3.65685 −0.174931
\(438\) 0 0
\(439\) −27.3848 −1.30700 −0.653502 0.756925i \(-0.726701\pi\)
−0.653502 + 0.756925i \(0.726701\pi\)
\(440\) 0.585786 0.0279263
\(441\) 0 0
\(442\) −14.0000 −0.665912
\(443\) −12.6274 −0.599947 −0.299973 0.953948i \(-0.596978\pi\)
−0.299973 + 0.953948i \(0.596978\pi\)
\(444\) 0 0
\(445\) −7.31371 −0.346703
\(446\) −8.58579 −0.406549
\(447\) 0 0
\(448\) 0 0
\(449\) −22.3431 −1.05444 −0.527219 0.849729i \(-0.676765\pi\)
−0.527219 + 0.849729i \(0.676765\pi\)
\(450\) 0 0
\(451\) −5.41421 −0.254945
\(452\) 8.17157 0.384358
\(453\) 0 0
\(454\) −28.8284 −1.35299
\(455\) 0 0
\(456\) 0 0
\(457\) −11.6569 −0.545285 −0.272642 0.962115i \(-0.587898\pi\)
−0.272642 + 0.962115i \(0.587898\pi\)
\(458\) −23.3137 −1.08938
\(459\) 0 0
\(460\) 3.65685 0.170502
\(461\) 8.31371 0.387208 0.193604 0.981080i \(-0.437982\pi\)
0.193604 + 0.981080i \(0.437982\pi\)
\(462\) 0 0
\(463\) 12.8284 0.596188 0.298094 0.954537i \(-0.403649\pi\)
0.298094 + 0.954537i \(0.403649\pi\)
\(464\) −2.65685 −0.123341
\(465\) 0 0
\(466\) 1.41421 0.0655122
\(467\) 34.0000 1.57333 0.786666 0.617379i \(-0.211805\pi\)
0.786666 + 0.617379i \(0.211805\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −6.14214 −0.283316
\(471\) 0 0
\(472\) −5.58579 −0.257107
\(473\) 5.65685 0.260102
\(474\) 0 0
\(475\) 2.72792 0.125166
\(476\) 0 0
\(477\) 0 0
\(478\) 20.2132 0.924530
\(479\) 11.9289 0.545047 0.272523 0.962149i \(-0.412142\pi\)
0.272523 + 0.962149i \(0.412142\pi\)
\(480\) 0 0
\(481\) 36.0416 1.64336
\(482\) −12.2426 −0.557637
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −2.24264 −0.101833
\(486\) 0 0
\(487\) −9.65685 −0.437594 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(488\) −11.8284 −0.535448
\(489\) 0 0
\(490\) 0 0
\(491\) −19.1716 −0.865201 −0.432600 0.901586i \(-0.642404\pi\)
−0.432600 + 0.901586i \(0.642404\pi\)
\(492\) 0 0
\(493\) 9.71573 0.437574
\(494\) −2.24264 −0.100901
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) 22.1421 0.991218 0.495609 0.868546i \(-0.334945\pi\)
0.495609 + 0.868546i \(0.334945\pi\)
\(500\) −5.65685 −0.252982
\(501\) 0 0
\(502\) −26.1421 −1.16678
\(503\) −4.21320 −0.187857 −0.0939287 0.995579i \(-0.529943\pi\)
−0.0939287 + 0.995579i \(0.529943\pi\)
\(504\) 0 0
\(505\) −3.61522 −0.160875
\(506\) 6.24264 0.277519
\(507\) 0 0
\(508\) 15.7279 0.697814
\(509\) −9.31371 −0.412823 −0.206411 0.978465i \(-0.566179\pi\)
−0.206411 + 0.978465i \(0.566179\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −25.1421 −1.10897
\(515\) 7.85786 0.346259
\(516\) 0 0
\(517\) −10.4853 −0.461142
\(518\) 0 0
\(519\) 0 0
\(520\) 2.24264 0.0983463
\(521\) 28.2843 1.23916 0.619578 0.784935i \(-0.287304\pi\)
0.619578 + 0.784935i \(0.287304\pi\)
\(522\) 0 0
\(523\) −12.7279 −0.556553 −0.278277 0.960501i \(-0.589763\pi\)
−0.278277 + 0.960501i \(0.589763\pi\)
\(524\) 0.585786 0.0255902
\(525\) 0 0
\(526\) −17.0416 −0.743050
\(527\) 14.6274 0.637180
\(528\) 0 0
\(529\) 15.9706 0.694372
\(530\) 4.62742 0.201002
\(531\) 0 0
\(532\) 0 0
\(533\) −20.7279 −0.897826
\(534\) 0 0
\(535\) 1.79899 0.0777771
\(536\) −2.75736 −0.119100
\(537\) 0 0
\(538\) 2.34315 0.101020
\(539\) 0 0
\(540\) 0 0
\(541\) −12.8579 −0.552803 −0.276401 0.961042i \(-0.589142\pi\)
−0.276401 + 0.961042i \(0.589142\pi\)
\(542\) −4.55635 −0.195712
\(543\) 0 0
\(544\) 3.65685 0.156786
\(545\) 9.65685 0.413654
\(546\) 0 0
\(547\) 34.8701 1.49094 0.745468 0.666541i \(-0.232226\pi\)
0.745468 + 0.666541i \(0.232226\pi\)
\(548\) 16.6569 0.711546
\(549\) 0 0
\(550\) −4.65685 −0.198569
\(551\) 1.55635 0.0663027
\(552\) 0 0
\(553\) 0 0
\(554\) −1.82843 −0.0776824
\(555\) 0 0
\(556\) 0 0
\(557\) −7.51472 −0.318409 −0.159204 0.987246i \(-0.550893\pi\)
−0.159204 + 0.987246i \(0.550893\pi\)
\(558\) 0 0
\(559\) 21.6569 0.915987
\(560\) 0 0
\(561\) 0 0
\(562\) 8.72792 0.368165
\(563\) −9.07107 −0.382300 −0.191150 0.981561i \(-0.561222\pi\)
−0.191150 + 0.981561i \(0.561222\pi\)
\(564\) 0 0
\(565\) 4.78680 0.201382
\(566\) 23.4142 0.984173
\(567\) 0 0
\(568\) −11.0711 −0.464532
\(569\) 4.00000 0.167689 0.0838444 0.996479i \(-0.473280\pi\)
0.0838444 + 0.996479i \(0.473280\pi\)
\(570\) 0 0
\(571\) 26.3848 1.10417 0.552084 0.833788i \(-0.313833\pi\)
0.552084 + 0.833788i \(0.313833\pi\)
\(572\) 3.82843 0.160075
\(573\) 0 0
\(574\) 0 0
\(575\) −29.0711 −1.21235
\(576\) 0 0
\(577\) −9.68629 −0.403246 −0.201623 0.979463i \(-0.564622\pi\)
−0.201623 + 0.979463i \(0.564622\pi\)
\(578\) 3.62742 0.150881
\(579\) 0 0
\(580\) −1.55635 −0.0646239
\(581\) 0 0
\(582\) 0 0
\(583\) 7.89949 0.327164
\(584\) 9.41421 0.389563
\(585\) 0 0
\(586\) −10.8284 −0.447318
\(587\) 25.1005 1.03601 0.518004 0.855378i \(-0.326675\pi\)
0.518004 + 0.855378i \(0.326675\pi\)
\(588\) 0 0
\(589\) 2.34315 0.0965476
\(590\) −3.27208 −0.134709
\(591\) 0 0
\(592\) −9.41421 −0.386922
\(593\) −23.6985 −0.973180 −0.486590 0.873630i \(-0.661759\pi\)
−0.486590 + 0.873630i \(0.661759\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −17.6569 −0.723253
\(597\) 0 0
\(598\) 23.8995 0.977323
\(599\) −42.6274 −1.74171 −0.870855 0.491541i \(-0.836434\pi\)
−0.870855 + 0.491541i \(0.836434\pi\)
\(600\) 0 0
\(601\) 31.9411 1.30291 0.651453 0.758689i \(-0.274160\pi\)
0.651453 + 0.758689i \(0.274160\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −15.7279 −0.639960
\(605\) 0.585786 0.0238156
\(606\) 0 0
\(607\) −42.9706 −1.74412 −0.872061 0.489398i \(-0.837217\pi\)
−0.872061 + 0.489398i \(0.837217\pi\)
\(608\) 0.585786 0.0237568
\(609\) 0 0
\(610\) −6.92893 −0.280544
\(611\) −40.1421 −1.62398
\(612\) 0 0
\(613\) 28.6274 1.15625 0.578125 0.815948i \(-0.303785\pi\)
0.578125 + 0.815948i \(0.303785\pi\)
\(614\) −9.89949 −0.399511
\(615\) 0 0
\(616\) 0 0
\(617\) 41.9706 1.68967 0.844836 0.535026i \(-0.179698\pi\)
0.844836 + 0.535026i \(0.179698\pi\)
\(618\) 0 0
\(619\) −21.9411 −0.881888 −0.440944 0.897535i \(-0.645356\pi\)
−0.440944 + 0.897535i \(0.645356\pi\)
\(620\) −2.34315 −0.0941030
\(621\) 0 0
\(622\) 16.7279 0.670729
\(623\) 0 0
\(624\) 0 0
\(625\) 19.9706 0.798823
\(626\) 20.6569 0.825614
\(627\) 0 0
\(628\) 17.6569 0.704585
\(629\) 34.4264 1.37267
\(630\) 0 0
\(631\) 23.2721 0.926447 0.463223 0.886242i \(-0.346693\pi\)
0.463223 + 0.886242i \(0.346693\pi\)
\(632\) 13.2426 0.526764
\(633\) 0 0
\(634\) 8.68629 0.344977
\(635\) 9.21320 0.365615
\(636\) 0 0
\(637\) 0 0
\(638\) −2.65685 −0.105186
\(639\) 0 0
\(640\) −0.585786 −0.0231552
\(641\) −15.2843 −0.603692 −0.301846 0.953357i \(-0.597603\pi\)
−0.301846 + 0.953357i \(0.597603\pi\)
\(642\) 0 0
\(643\) 1.58579 0.0625373 0.0312687 0.999511i \(-0.490045\pi\)
0.0312687 + 0.999511i \(0.490045\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −2.14214 −0.0842812
\(647\) −30.1838 −1.18665 −0.593323 0.804964i \(-0.702184\pi\)
−0.593323 + 0.804964i \(0.702184\pi\)
\(648\) 0 0
\(649\) −5.58579 −0.219261
\(650\) −17.8284 −0.699288
\(651\) 0 0
\(652\) 9.72792 0.380975
\(653\) 18.3848 0.719452 0.359726 0.933058i \(-0.382870\pi\)
0.359726 + 0.933058i \(0.382870\pi\)
\(654\) 0 0
\(655\) 0.343146 0.0134078
\(656\) 5.41421 0.211390
\(657\) 0 0
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) 18.9706 0.737869 0.368935 0.929455i \(-0.379723\pi\)
0.368935 + 0.929455i \(0.379723\pi\)
\(662\) 24.0711 0.935549
\(663\) 0 0
\(664\) −12.1421 −0.471206
\(665\) 0 0
\(666\) 0 0
\(667\) −16.5858 −0.642204
\(668\) −13.7279 −0.531149
\(669\) 0 0
\(670\) −1.61522 −0.0624015
\(671\) −11.8284 −0.456631
\(672\) 0 0
\(673\) 5.55635 0.214182 0.107091 0.994249i \(-0.465846\pi\)
0.107091 + 0.994249i \(0.465846\pi\)
\(674\) 28.2426 1.08787
\(675\) 0 0
\(676\) 1.65685 0.0637252
\(677\) 35.3137 1.35722 0.678608 0.734501i \(-0.262584\pi\)
0.678608 + 0.734501i \(0.262584\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 2.14214 0.0821472
\(681\) 0 0
\(682\) −4.00000 −0.153168
\(683\) −13.5858 −0.519846 −0.259923 0.965629i \(-0.583697\pi\)
−0.259923 + 0.965629i \(0.583697\pi\)
\(684\) 0 0
\(685\) 9.75736 0.372810
\(686\) 0 0
\(687\) 0 0
\(688\) −5.65685 −0.215666
\(689\) 30.2426 1.15215
\(690\) 0 0
\(691\) 28.0711 1.06787 0.533937 0.845524i \(-0.320712\pi\)
0.533937 + 0.845524i \(0.320712\pi\)
\(692\) 9.82843 0.373621
\(693\) 0 0
\(694\) 17.4142 0.661035
\(695\) 0 0
\(696\) 0 0
\(697\) −19.7990 −0.749940
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 26.1127 0.986263 0.493132 0.869955i \(-0.335852\pi\)
0.493132 + 0.869955i \(0.335852\pi\)
\(702\) 0 0
\(703\) 5.51472 0.207992
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −2.68629 −0.101100
\(707\) 0 0
\(708\) 0 0
\(709\) −12.0416 −0.452233 −0.226116 0.974100i \(-0.572603\pi\)
−0.226116 + 0.974100i \(0.572603\pi\)
\(710\) −6.48528 −0.243388
\(711\) 0 0
\(712\) 12.4853 0.467906
\(713\) −24.9706 −0.935155
\(714\) 0 0
\(715\) 2.24264 0.0838700
\(716\) −0.899495 −0.0336157
\(717\) 0 0
\(718\) 19.2426 0.718129
\(719\) 1.51472 0.0564895 0.0282447 0.999601i \(-0.491008\pi\)
0.0282447 + 0.999601i \(0.491008\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 18.6569 0.694336
\(723\) 0 0
\(724\) −7.65685 −0.284565
\(725\) 12.3726 0.459506
\(726\) 0 0
\(727\) 36.4264 1.35098 0.675490 0.737369i \(-0.263932\pi\)
0.675490 + 0.737369i \(0.263932\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 5.51472 0.204109
\(731\) 20.6863 0.765110
\(732\) 0 0
\(733\) −13.0000 −0.480166 −0.240083 0.970752i \(-0.577175\pi\)
−0.240083 + 0.970752i \(0.577175\pi\)
\(734\) 10.7279 0.395975
\(735\) 0 0
\(736\) −6.24264 −0.230107
\(737\) −2.75736 −0.101569
\(738\) 0 0
\(739\) −52.4264 −1.92854 −0.964268 0.264928i \(-0.914652\pi\)
−0.964268 + 0.264928i \(0.914652\pi\)
\(740\) −5.51472 −0.202725
\(741\) 0 0
\(742\) 0 0
\(743\) 13.3137 0.488433 0.244216 0.969721i \(-0.421469\pi\)
0.244216 + 0.969721i \(0.421469\pi\)
\(744\) 0 0
\(745\) −10.3431 −0.378944
\(746\) 19.9706 0.731174
\(747\) 0 0
\(748\) 3.65685 0.133708
\(749\) 0 0
\(750\) 0 0
\(751\) 45.6569 1.66604 0.833021 0.553241i \(-0.186609\pi\)
0.833021 + 0.553241i \(0.186609\pi\)
\(752\) 10.4853 0.382359
\(753\) 0 0
\(754\) −10.1716 −0.370427
\(755\) −9.21320 −0.335303
\(756\) 0 0
\(757\) 8.34315 0.303237 0.151618 0.988439i \(-0.451552\pi\)
0.151618 + 0.988439i \(0.451552\pi\)
\(758\) −27.8701 −1.01229
\(759\) 0 0
\(760\) 0.343146 0.0124472
\(761\) −30.9706 −1.12268 −0.561341 0.827585i \(-0.689714\pi\)
−0.561341 + 0.827585i \(0.689714\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 7.17157 0.259458
\(765\) 0 0
\(766\) −6.38478 −0.230691
\(767\) −21.3848 −0.772160
\(768\) 0 0
\(769\) 22.9706 0.828340 0.414170 0.910200i \(-0.364072\pi\)
0.414170 + 0.910200i \(0.364072\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 21.8995 0.788180
\(773\) 21.9411 0.789167 0.394584 0.918860i \(-0.370889\pi\)
0.394584 + 0.918860i \(0.370889\pi\)
\(774\) 0 0
\(775\) 18.6274 0.669117
\(776\) 3.82843 0.137433
\(777\) 0 0
\(778\) −22.7279 −0.814835
\(779\) −3.17157 −0.113633
\(780\) 0 0
\(781\) −11.0711 −0.396154
\(782\) 22.8284 0.816343
\(783\) 0 0
\(784\) 0 0
\(785\) 10.3431 0.369163
\(786\) 0 0
\(787\) 13.5563 0.483232 0.241616 0.970372i \(-0.422323\pi\)
0.241616 + 0.970372i \(0.422323\pi\)
\(788\) 0.514719 0.0183361
\(789\) 0 0
\(790\) 7.75736 0.275994
\(791\) 0 0
\(792\) 0 0
\(793\) −45.2843 −1.60809
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) 0.100505 0.00356231
\(797\) 7.65685 0.271220 0.135610 0.990762i \(-0.456701\pi\)
0.135610 + 0.990762i \(0.456701\pi\)
\(798\) 0 0
\(799\) −38.3431 −1.35648
\(800\) 4.65685 0.164645
\(801\) 0 0
\(802\) 18.3137 0.646680
\(803\) 9.41421 0.332220
\(804\) 0 0
\(805\) 0 0
\(806\) −15.3137 −0.539402
\(807\) 0 0
\(808\) 6.17157 0.217115
\(809\) 46.6274 1.63933 0.819666 0.572841i \(-0.194159\pi\)
0.819666 + 0.572841i \(0.194159\pi\)
\(810\) 0 0
\(811\) −24.2843 −0.852736 −0.426368 0.904550i \(-0.640207\pi\)
−0.426368 + 0.904550i \(0.640207\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −9.41421 −0.329968
\(815\) 5.69848 0.199609
\(816\) 0 0
\(817\) 3.31371 0.115932
\(818\) −2.72792 −0.0953796
\(819\) 0 0
\(820\) 3.17157 0.110756
\(821\) 29.4853 1.02904 0.514522 0.857477i \(-0.327969\pi\)
0.514522 + 0.857477i \(0.327969\pi\)
\(822\) 0 0
\(823\) −43.4558 −1.51478 −0.757388 0.652965i \(-0.773525\pi\)
−0.757388 + 0.652965i \(0.773525\pi\)
\(824\) −13.4142 −0.467306
\(825\) 0 0
\(826\) 0 0
\(827\) 21.3553 0.742598 0.371299 0.928513i \(-0.378913\pi\)
0.371299 + 0.928513i \(0.378913\pi\)
\(828\) 0 0
\(829\) 21.7574 0.755664 0.377832 0.925874i \(-0.376670\pi\)
0.377832 + 0.925874i \(0.376670\pi\)
\(830\) −7.11270 −0.246885
\(831\) 0 0
\(832\) −3.82843 −0.132727
\(833\) 0 0
\(834\) 0 0
\(835\) −8.04163 −0.278292
\(836\) 0.585786 0.0202598
\(837\) 0 0
\(838\) −26.1421 −0.903065
\(839\) 18.4853 0.638183 0.319091 0.947724i \(-0.396622\pi\)
0.319091 + 0.947724i \(0.396622\pi\)
\(840\) 0 0
\(841\) −21.9411 −0.756591
\(842\) −0.686292 −0.0236512
\(843\) 0 0
\(844\) 7.41421 0.255208
\(845\) 0.970563 0.0333884
\(846\) 0 0
\(847\) 0 0
\(848\) −7.89949 −0.271270
\(849\) 0 0
\(850\) −17.0294 −0.584105
\(851\) −58.7696 −2.01459
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −3.07107 −0.104967
\(857\) 34.6274 1.18285 0.591425 0.806360i \(-0.298566\pi\)
0.591425 + 0.806360i \(0.298566\pi\)
\(858\) 0 0
\(859\) −5.72792 −0.195434 −0.0977171 0.995214i \(-0.531154\pi\)
−0.0977171 + 0.995214i \(0.531154\pi\)
\(860\) −3.31371 −0.112997
\(861\) 0 0
\(862\) −17.5858 −0.598974
\(863\) −35.2132 −1.19867 −0.599336 0.800498i \(-0.704569\pi\)
−0.599336 + 0.800498i \(0.704569\pi\)
\(864\) 0 0
\(865\) 5.75736 0.195756
\(866\) −26.1421 −0.888346
\(867\) 0 0
\(868\) 0 0
\(869\) 13.2426 0.449226
\(870\) 0 0
\(871\) −10.5563 −0.357688
\(872\) −16.4853 −0.558262
\(873\) 0 0
\(874\) 3.65685 0.123695
\(875\) 0 0
\(876\) 0 0
\(877\) 19.6274 0.662771 0.331385 0.943495i \(-0.392484\pi\)
0.331385 + 0.943495i \(0.392484\pi\)
\(878\) 27.3848 0.924191
\(879\) 0 0
\(880\) −0.585786 −0.0197469
\(881\) −6.45584 −0.217503 −0.108751 0.994069i \(-0.534685\pi\)
−0.108751 + 0.994069i \(0.534685\pi\)
\(882\) 0 0
\(883\) 15.7279 0.529287 0.264643 0.964346i \(-0.414746\pi\)
0.264643 + 0.964346i \(0.414746\pi\)
\(884\) 14.0000 0.470871
\(885\) 0 0
\(886\) 12.6274 0.424226
\(887\) 3.10051 0.104105 0.0520524 0.998644i \(-0.483424\pi\)
0.0520524 + 0.998644i \(0.483424\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 7.31371 0.245156
\(891\) 0 0
\(892\) 8.58579 0.287473
\(893\) −6.14214 −0.205539
\(894\) 0 0
\(895\) −0.526912 −0.0176127
\(896\) 0 0
\(897\) 0 0
\(898\) 22.3431 0.745600
\(899\) 10.6274 0.354444
\(900\) 0 0
\(901\) 28.8873 0.962376
\(902\) 5.41421 0.180274
\(903\) 0 0
\(904\) −8.17157 −0.271782
\(905\) −4.48528 −0.149096
\(906\) 0 0
\(907\) 10.6863 0.354832 0.177416 0.984136i \(-0.443226\pi\)
0.177416 + 0.984136i \(0.443226\pi\)
\(908\) 28.8284 0.956705
\(909\) 0 0
\(910\) 0 0
\(911\) 30.4853 1.01002 0.505011 0.863113i \(-0.331488\pi\)
0.505011 + 0.863113i \(0.331488\pi\)
\(912\) 0 0
\(913\) −12.1421 −0.401846
\(914\) 11.6569 0.385574
\(915\) 0 0
\(916\) 23.3137 0.770307
\(917\) 0 0
\(918\) 0 0
\(919\) 22.1421 0.730402 0.365201 0.930929i \(-0.381000\pi\)
0.365201 + 0.930929i \(0.381000\pi\)
\(920\) −3.65685 −0.120563
\(921\) 0 0
\(922\) −8.31371 −0.273797
\(923\) −42.3848 −1.39511
\(924\) 0 0
\(925\) 43.8406 1.44147
\(926\) −12.8284 −0.421568
\(927\) 0 0
\(928\) 2.65685 0.0872155
\(929\) 40.4558 1.32731 0.663657 0.748037i \(-0.269004\pi\)
0.663657 + 0.748037i \(0.269004\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.41421 −0.0463241
\(933\) 0 0
\(934\) −34.0000 −1.11251
\(935\) 2.14214 0.0700553
\(936\) 0 0
\(937\) −19.4142 −0.634235 −0.317117 0.948386i \(-0.602715\pi\)
−0.317117 + 0.948386i \(0.602715\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 6.14214 0.200334
\(941\) −24.6569 −0.803790 −0.401895 0.915686i \(-0.631648\pi\)
−0.401895 + 0.915686i \(0.631648\pi\)
\(942\) 0 0
\(943\) 33.7990 1.10065
\(944\) 5.58579 0.181802
\(945\) 0 0
\(946\) −5.65685 −0.183920
\(947\) −34.8284 −1.13177 −0.565886 0.824484i \(-0.691466\pi\)
−0.565886 + 0.824484i \(0.691466\pi\)
\(948\) 0 0
\(949\) 36.0416 1.16996
\(950\) −2.72792 −0.0885055
\(951\) 0 0
\(952\) 0 0
\(953\) −5.55635 −0.179988 −0.0899939 0.995942i \(-0.528685\pi\)
−0.0899939 + 0.995942i \(0.528685\pi\)
\(954\) 0 0
\(955\) 4.20101 0.135941
\(956\) −20.2132 −0.653742
\(957\) 0 0
\(958\) −11.9289 −0.385406
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −36.0416 −1.16203
\(963\) 0 0
\(964\) 12.2426 0.394309
\(965\) 12.8284 0.412962
\(966\) 0 0
\(967\) −36.2843 −1.16682 −0.583412 0.812177i \(-0.698283\pi\)
−0.583412 + 0.812177i \(0.698283\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 2.24264 0.0720069
\(971\) −22.2721 −0.714745 −0.357372 0.933962i \(-0.616327\pi\)
−0.357372 + 0.933962i \(0.616327\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 9.65685 0.309426
\(975\) 0 0
\(976\) 11.8284 0.378619
\(977\) −40.0000 −1.27971 −0.639857 0.768494i \(-0.721006\pi\)
−0.639857 + 0.768494i \(0.721006\pi\)
\(978\) 0 0
\(979\) 12.4853 0.399031
\(980\) 0 0
\(981\) 0 0
\(982\) 19.1716 0.611789
\(983\) −32.4264 −1.03424 −0.517121 0.855912i \(-0.672996\pi\)
−0.517121 + 0.855912i \(0.672996\pi\)
\(984\) 0 0
\(985\) 0.301515 0.00960707
\(986\) −9.71573 −0.309412
\(987\) 0 0
\(988\) 2.24264 0.0713479
\(989\) −35.3137 −1.12291
\(990\) 0 0
\(991\) −1.61522 −0.0513093 −0.0256546 0.999671i \(-0.508167\pi\)
−0.0256546 + 0.999671i \(0.508167\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) 0 0
\(995\) 0.0588745 0.00186645
\(996\) 0 0
\(997\) 13.1716 0.417148 0.208574 0.978007i \(-0.433118\pi\)
0.208574 + 0.978007i \(0.433118\pi\)
\(998\) −22.1421 −0.700897
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9702.2.a.cx.1.1 2
3.2 odd 2 1078.2.a.t.1.1 2
7.2 even 3 1386.2.k.t.991.2 4
7.4 even 3 1386.2.k.t.793.2 4
7.6 odd 2 9702.2.a.ch.1.2 2
12.11 even 2 8624.2.a.cc.1.2 2
21.2 odd 6 154.2.e.e.67.2 yes 4
21.5 even 6 1078.2.e.m.67.1 4
21.11 odd 6 154.2.e.e.23.2 4
21.17 even 6 1078.2.e.m.177.1 4
21.20 even 2 1078.2.a.x.1.2 2
84.11 even 6 1232.2.q.f.177.1 4
84.23 even 6 1232.2.q.f.529.1 4
84.83 odd 2 8624.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.e.23.2 4 21.11 odd 6
154.2.e.e.67.2 yes 4 21.2 odd 6
1078.2.a.t.1.1 2 3.2 odd 2
1078.2.a.x.1.2 2 21.20 even 2
1078.2.e.m.67.1 4 21.5 even 6
1078.2.e.m.177.1 4 21.17 even 6
1232.2.q.f.177.1 4 84.11 even 6
1232.2.q.f.529.1 4 84.23 even 6
1386.2.k.t.793.2 4 7.4 even 3
1386.2.k.t.991.2 4 7.2 even 3
8624.2.a.bh.1.1 2 84.83 odd 2
8624.2.a.cc.1.2 2 12.11 even 2
9702.2.a.ch.1.2 2 7.6 odd 2
9702.2.a.cx.1.1 2 1.1 even 1 trivial