Properties

Label 9702.2.a.cl.1.1
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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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: \(1\)
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: yes
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} -1.00000 q^{8} +1.41421 q^{10} -1.00000 q^{11} +1.41421 q^{13} +1.00000 q^{16} +1.41421 q^{17} -2.82843 q^{19} -1.41421 q^{20} +1.00000 q^{22} +4.00000 q^{23} -3.00000 q^{25} -1.41421 q^{26} +6.00000 q^{29} -2.82843 q^{31} -1.00000 q^{32} -1.41421 q^{34} -8.00000 q^{37} +2.82843 q^{38} +1.41421 q^{40} +7.07107 q^{41} -8.00000 q^{43} -1.00000 q^{44} -4.00000 q^{46} +8.48528 q^{47} +3.00000 q^{50} +1.41421 q^{52} +1.41421 q^{55} -6.00000 q^{58} -7.07107 q^{61} +2.82843 q^{62} +1.00000 q^{64} -2.00000 q^{65} -8.00000 q^{67} +1.41421 q^{68} +8.00000 q^{71} +1.41421 q^{73} +8.00000 q^{74} -2.82843 q^{76} +4.00000 q^{79} -1.41421 q^{80} -7.07107 q^{82} +2.82843 q^{83} -2.00000 q^{85} +8.00000 q^{86} +1.00000 q^{88} -9.89949 q^{89} +4.00000 q^{92} -8.48528 q^{94} +4.00000 q^{95} +15.5563 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{11} + 2 q^{16} + 2 q^{22} + 8 q^{23} - 6 q^{25} + 12 q^{29} - 2 q^{32} - 16 q^{37} - 16 q^{43} - 2 q^{44} - 8 q^{46} + 6 q^{50} - 12 q^{58} + 2 q^{64} - 4 q^{65} - 16 q^{67} + 16 q^{71} + 16 q^{74} + 8 q^{79} - 4 q^{85} + 16 q^{86} + 2 q^{88} + 8 q^{92} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.41421 −0.632456 −0.316228 0.948683i \(-0.602416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.41421 0.447214
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.41421 0.392232 0.196116 0.980581i \(-0.437167\pi\)
0.196116 + 0.980581i \(0.437167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.41421 0.342997 0.171499 0.985184i \(-0.445139\pi\)
0.171499 + 0.985184i \(0.445139\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) −1.41421 −0.316228
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) −1.41421 −0.277350
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −2.82843 −0.508001 −0.254000 0.967204i \(-0.581746\pi\)
−0.254000 + 0.967204i \(0.581746\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −1.41421 −0.242536
\(35\) 0 0
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 2.82843 0.458831
\(39\) 0 0
\(40\) 1.41421 0.223607
\(41\) 7.07107 1.10432 0.552158 0.833740i \(-0.313805\pi\)
0.552158 + 0.833740i \(0.313805\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 8.48528 1.23771 0.618853 0.785507i \(-0.287598\pi\)
0.618853 + 0.785507i \(0.287598\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 3.00000 0.424264
\(51\) 0 0
\(52\) 1.41421 0.196116
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 1.41421 0.190693
\(56\) 0 0
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −7.07107 −0.905357 −0.452679 0.891674i \(-0.649532\pi\)
−0.452679 + 0.891674i \(0.649532\pi\)
\(62\) 2.82843 0.359211
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 1.41421 0.171499
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 1.41421 0.165521 0.0827606 0.996569i \(-0.473626\pi\)
0.0827606 + 0.996569i \(0.473626\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −2.82843 −0.324443
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −1.41421 −0.158114
\(81\) 0 0
\(82\) −7.07107 −0.780869
\(83\) 2.82843 0.310460 0.155230 0.987878i \(-0.450388\pi\)
0.155230 + 0.987878i \(0.450388\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −9.89949 −1.04934 −0.524672 0.851304i \(-0.675812\pi\)
−0.524672 + 0.851304i \(0.675812\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) −8.48528 −0.875190
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 15.5563 1.57951 0.789754 0.613424i \(-0.210208\pi\)
0.789754 + 0.613424i \(0.210208\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −3.00000 −0.300000
\(101\) −15.5563 −1.54791 −0.773957 0.633238i \(-0.781726\pi\)
−0.773957 + 0.633238i \(0.781726\pi\)
\(102\) 0 0
\(103\) 8.48528 0.836080 0.418040 0.908429i \(-0.362717\pi\)
0.418040 + 0.908429i \(0.362717\pi\)
\(104\) −1.41421 −0.138675
\(105\) 0 0
\(106\) 0 0
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) −1.41421 −0.134840
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) −5.65685 −0.527504
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 7.07107 0.640184
\(123\) 0 0
\(124\) −2.82843 −0.254000
\(125\) 11.3137 1.01193
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) −2.82843 −0.247121 −0.123560 0.992337i \(-0.539431\pi\)
−0.123560 + 0.992337i \(0.539431\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −1.41421 −0.121268
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) 2.82843 0.239904 0.119952 0.992780i \(-0.461726\pi\)
0.119952 + 0.992780i \(0.461726\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) −1.41421 −0.118262
\(144\) 0 0
\(145\) −8.48528 −0.704664
\(146\) −1.41421 −0.117041
\(147\) 0 0
\(148\) −8.00000 −0.657596
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 2.82843 0.229416
\(153\) 0 0
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −1.41421 −0.112867 −0.0564333 0.998406i \(-0.517973\pi\)
−0.0564333 + 0.998406i \(0.517973\pi\)
\(158\) −4.00000 −0.318223
\(159\) 0 0
\(160\) 1.41421 0.111803
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 7.07107 0.552158
\(165\) 0 0
\(166\) −2.82843 −0.219529
\(167\) 11.3137 0.875481 0.437741 0.899101i \(-0.355779\pi\)
0.437741 + 0.899101i \(0.355779\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) −1.41421 −0.107521 −0.0537603 0.998554i \(-0.517121\pi\)
−0.0537603 + 0.998554i \(0.517121\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 9.89949 0.741999
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) 12.7279 0.946059 0.473029 0.881047i \(-0.343160\pi\)
0.473029 + 0.881047i \(0.343160\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) 11.3137 0.831800
\(186\) 0 0
\(187\) −1.41421 −0.103418
\(188\) 8.48528 0.618853
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) −15.5563 −1.11688
\(195\) 0 0
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) 25.4558 1.80452 0.902258 0.431196i \(-0.141908\pi\)
0.902258 + 0.431196i \(0.141908\pi\)
\(200\) 3.00000 0.212132
\(201\) 0 0
\(202\) 15.5563 1.09454
\(203\) 0 0
\(204\) 0 0
\(205\) −10.0000 −0.698430
\(206\) −8.48528 −0.591198
\(207\) 0 0
\(208\) 1.41421 0.0980581
\(209\) 2.82843 0.195646
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) 11.3137 0.771589
\(216\) 0 0
\(217\) 0 0
\(218\) −4.00000 −0.270914
\(219\) 0 0
\(220\) 1.41421 0.0953463
\(221\) 2.00000 0.134535
\(222\) 0 0
\(223\) 2.82843 0.189405 0.0947027 0.995506i \(-0.469810\pi\)
0.0947027 + 0.995506i \(0.469810\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −14.1421 −0.938647 −0.469323 0.883026i \(-0.655502\pi\)
−0.469323 + 0.883026i \(0.655502\pi\)
\(228\) 0 0
\(229\) −21.2132 −1.40181 −0.700904 0.713256i \(-0.747220\pi\)
−0.700904 + 0.713256i \(0.747220\pi\)
\(230\) 5.65685 0.373002
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 9.89949 0.637683 0.318841 0.947808i \(-0.396706\pi\)
0.318841 + 0.947808i \(0.396706\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) −7.07107 −0.452679
\(245\) 0 0
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 2.82843 0.179605
\(249\) 0 0
\(250\) −11.3137 −0.715542
\(251\) 11.3137 0.714115 0.357057 0.934082i \(-0.383780\pi\)
0.357057 + 0.934082i \(0.383780\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −7.07107 −0.441081 −0.220541 0.975378i \(-0.570782\pi\)
−0.220541 + 0.975378i \(0.570782\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) 2.82843 0.174741
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) −9.89949 −0.603583 −0.301791 0.953374i \(-0.597585\pi\)
−0.301791 + 0.953374i \(0.597585\pi\)
\(270\) 0 0
\(271\) −16.9706 −1.03089 −0.515444 0.856923i \(-0.672373\pi\)
−0.515444 + 0.856923i \(0.672373\pi\)
\(272\) 1.41421 0.0857493
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) 3.00000 0.180907
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −2.82843 −0.169638
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 8.48528 0.504398 0.252199 0.967675i \(-0.418846\pi\)
0.252199 + 0.967675i \(0.418846\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 1.41421 0.0836242
\(287\) 0 0
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) 8.48528 0.498273
\(291\) 0 0
\(292\) 1.41421 0.0827606
\(293\) −21.2132 −1.23929 −0.619644 0.784883i \(-0.712723\pi\)
−0.619644 + 0.784883i \(0.712723\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) 4.00000 0.231714
\(299\) 5.65685 0.327144
\(300\) 0 0
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) −2.82843 −0.162221
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) 2.82843 0.161427 0.0807134 0.996737i \(-0.474280\pi\)
0.0807134 + 0.996737i \(0.474280\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) −19.7990 −1.12270 −0.561349 0.827579i \(-0.689717\pi\)
−0.561349 + 0.827579i \(0.689717\pi\)
\(312\) 0 0
\(313\) 7.07107 0.399680 0.199840 0.979829i \(-0.435958\pi\)
0.199840 + 0.979829i \(0.435958\pi\)
\(314\) 1.41421 0.0798087
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) −8.00000 −0.449325 −0.224662 0.974437i \(-0.572128\pi\)
−0.224662 + 0.974437i \(0.572128\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) −1.41421 −0.0790569
\(321\) 0 0
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) −4.24264 −0.235339
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −7.07107 −0.390434
\(329\) 0 0
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 2.82843 0.155230
\(333\) 0 0
\(334\) −11.3137 −0.619059
\(335\) 11.3137 0.618134
\(336\) 0 0
\(337\) −20.0000 −1.08947 −0.544735 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) 11.0000 0.598321
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) 2.82843 0.153168
\(342\) 0 0
\(343\) 0 0
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) 1.41421 0.0760286
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 18.3848 0.984115 0.492057 0.870563i \(-0.336245\pi\)
0.492057 + 0.870563i \(0.336245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −26.8701 −1.43015 −0.715074 0.699048i \(-0.753607\pi\)
−0.715074 + 0.699048i \(0.753607\pi\)
\(354\) 0 0
\(355\) −11.3137 −0.600469
\(356\) −9.89949 −0.524672
\(357\) 0 0
\(358\) 16.0000 0.845626
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) −12.7279 −0.668965
\(363\) 0 0
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) −8.48528 −0.442928 −0.221464 0.975169i \(-0.571084\pi\)
−0.221464 + 0.975169i \(0.571084\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) −11.3137 −0.588172
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 1.41421 0.0731272
\(375\) 0 0
\(376\) −8.48528 −0.437595
\(377\) 8.48528 0.437014
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 4.00000 0.205196
\(381\) 0 0
\(382\) 12.0000 0.613973
\(383\) −14.1421 −0.722629 −0.361315 0.932444i \(-0.617672\pi\)
−0.361315 + 0.932444i \(0.617672\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) 0 0
\(388\) 15.5563 0.789754
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) 5.65685 0.286079
\(392\) 0 0
\(393\) 0 0
\(394\) 12.0000 0.604551
\(395\) −5.65685 −0.284627
\(396\) 0 0
\(397\) −9.89949 −0.496841 −0.248421 0.968652i \(-0.579912\pi\)
−0.248421 + 0.968652i \(0.579912\pi\)
\(398\) −25.4558 −1.27599
\(399\) 0 0
\(400\) −3.00000 −0.150000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) −15.5563 −0.773957
\(405\) 0 0
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 24.0416 1.18878 0.594391 0.804176i \(-0.297393\pi\)
0.594391 + 0.804176i \(0.297393\pi\)
\(410\) 10.0000 0.493865
\(411\) 0 0
\(412\) 8.48528 0.418040
\(413\) 0 0
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) −1.41421 −0.0693375
\(417\) 0 0
\(418\) −2.82843 −0.138343
\(419\) 16.9706 0.829066 0.414533 0.910034i \(-0.363945\pi\)
0.414533 + 0.910034i \(0.363945\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) 0 0
\(425\) −4.24264 −0.205798
\(426\) 0 0
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) −11.3137 −0.545595
\(431\) 28.0000 1.34871 0.674356 0.738406i \(-0.264421\pi\)
0.674356 + 0.738406i \(0.264421\pi\)
\(432\) 0 0
\(433\) 4.24264 0.203888 0.101944 0.994790i \(-0.467494\pi\)
0.101944 + 0.994790i \(0.467494\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) −11.3137 −0.541208
\(438\) 0 0
\(439\) −33.9411 −1.61992 −0.809961 0.586484i \(-0.800512\pi\)
−0.809961 + 0.586484i \(0.800512\pi\)
\(440\) −1.41421 −0.0674200
\(441\) 0 0
\(442\) −2.00000 −0.0951303
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 14.0000 0.663664
\(446\) −2.82843 −0.133930
\(447\) 0 0
\(448\) 0 0
\(449\) −24.0000 −1.13263 −0.566315 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) 0 0
\(451\) −7.07107 −0.332964
\(452\) 0 0
\(453\) 0 0
\(454\) 14.1421 0.663723
\(455\) 0 0
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 21.2132 0.991228
\(459\) 0 0
\(460\) −5.65685 −0.263752
\(461\) 26.8701 1.25146 0.625732 0.780038i \(-0.284800\pi\)
0.625732 + 0.780038i \(0.284800\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 16.9706 0.785304 0.392652 0.919687i \(-0.371558\pi\)
0.392652 + 0.919687i \(0.371558\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 12.0000 0.553519
\(471\) 0 0
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) 8.48528 0.389331
\(476\) 0 0
\(477\) 0 0
\(478\) −8.00000 −0.365911
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −11.3137 −0.515861
\(482\) −9.89949 −0.450910
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −22.0000 −0.998969
\(486\) 0 0
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) 7.07107 0.320092
\(489\) 0 0
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 0 0
\(493\) 8.48528 0.382158
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −2.82843 −0.127000
\(497\) 0 0
\(498\) 0 0
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 11.3137 0.505964
\(501\) 0 0
\(502\) −11.3137 −0.504956
\(503\) 5.65685 0.252227 0.126113 0.992016i \(-0.459750\pi\)
0.126113 + 0.992016i \(0.459750\pi\)
\(504\) 0 0
\(505\) 22.0000 0.978987
\(506\) 4.00000 0.177822
\(507\) 0 0
\(508\) −12.0000 −0.532414
\(509\) 12.7279 0.564155 0.282078 0.959392i \(-0.408976\pi\)
0.282078 + 0.959392i \(0.408976\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 7.07107 0.311891
\(515\) −12.0000 −0.528783
\(516\) 0 0
\(517\) −8.48528 −0.373182
\(518\) 0 0
\(519\) 0 0
\(520\) 2.00000 0.0877058
\(521\) 7.07107 0.309789 0.154895 0.987931i \(-0.450496\pi\)
0.154895 + 0.987931i \(0.450496\pi\)
\(522\) 0 0
\(523\) −42.4264 −1.85518 −0.927589 0.373603i \(-0.878122\pi\)
−0.927589 + 0.373603i \(0.878122\pi\)
\(524\) −2.82843 −0.123560
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 10.0000 0.433148
\(534\) 0 0
\(535\) 11.3137 0.489134
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) 9.89949 0.426798
\(539\) 0 0
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 16.9706 0.728948
\(543\) 0 0
\(544\) −1.41421 −0.0606339
\(545\) −5.65685 −0.242313
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 10.0000 0.427179
\(549\) 0 0
\(550\) −3.00000 −0.127920
\(551\) −16.9706 −0.722970
\(552\) 0 0
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 2.82843 0.119952
\(557\) −36.0000 −1.52537 −0.762684 0.646771i \(-0.776119\pi\)
−0.762684 + 0.646771i \(0.776119\pi\)
\(558\) 0 0
\(559\) −11.3137 −0.478519
\(560\) 0 0
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −2.82843 −0.119204 −0.0596020 0.998222i \(-0.518983\pi\)
−0.0596020 + 0.998222i \(0.518983\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −8.48528 −0.356663
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −1.41421 −0.0591312
\(573\) 0 0
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) 38.1838 1.58961 0.794805 0.606864i \(-0.207573\pi\)
0.794805 + 0.606864i \(0.207573\pi\)
\(578\) 15.0000 0.623918
\(579\) 0 0
\(580\) −8.48528 −0.352332
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −1.41421 −0.0585206
\(585\) 0 0
\(586\) 21.2132 0.876309
\(587\) −45.2548 −1.86787 −0.933933 0.357447i \(-0.883647\pi\)
−0.933933 + 0.357447i \(0.883647\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 0 0
\(592\) −8.00000 −0.328798
\(593\) 18.3848 0.754972 0.377486 0.926015i \(-0.376789\pi\)
0.377486 + 0.926015i \(0.376789\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4.00000 −0.163846
\(597\) 0 0
\(598\) −5.65685 −0.231326
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −7.07107 −0.288435 −0.144217 0.989546i \(-0.546066\pi\)
−0.144217 + 0.989546i \(0.546066\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) −1.41421 −0.0574960
\(606\) 0 0
\(607\) −16.9706 −0.688814 −0.344407 0.938820i \(-0.611920\pi\)
−0.344407 + 0.938820i \(0.611920\pi\)
\(608\) 2.82843 0.114708
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) −28.0000 −1.13091 −0.565455 0.824779i \(-0.691299\pi\)
−0.565455 + 0.824779i \(0.691299\pi\)
\(614\) −2.82843 −0.114146
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) 16.9706 0.682105 0.341052 0.940044i \(-0.389217\pi\)
0.341052 + 0.940044i \(0.389217\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) 19.7990 0.793867
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) −7.07107 −0.282617
\(627\) 0 0
\(628\) −1.41421 −0.0564333
\(629\) −11.3137 −0.451107
\(630\) 0 0
\(631\) −12.0000 −0.477712 −0.238856 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) −4.00000 −0.159111
\(633\) 0 0
\(634\) 8.00000 0.317721
\(635\) 16.9706 0.673456
\(636\) 0 0
\(637\) 0 0
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) 1.41421 0.0559017
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) −36.7696 −1.44556 −0.722780 0.691078i \(-0.757136\pi\)
−0.722780 + 0.691078i \(0.757136\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 4.24264 0.166410
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) 4.00000 0.156293
\(656\) 7.07107 0.276079
\(657\) 0 0
\(658\) 0 0
\(659\) 8.00000 0.311636 0.155818 0.987786i \(-0.450199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(660\) 0 0
\(661\) −29.6985 −1.15514 −0.577569 0.816342i \(-0.695998\pi\)
−0.577569 + 0.816342i \(0.695998\pi\)
\(662\) 8.00000 0.310929
\(663\) 0 0
\(664\) −2.82843 −0.109764
\(665\) 0 0
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 11.3137 0.437741
\(669\) 0 0
\(670\) −11.3137 −0.437087
\(671\) 7.07107 0.272976
\(672\) 0 0
\(673\) −4.00000 −0.154189 −0.0770943 0.997024i \(-0.524564\pi\)
−0.0770943 + 0.997024i \(0.524564\pi\)
\(674\) 20.0000 0.770371
\(675\) 0 0
\(676\) −11.0000 −0.423077
\(677\) 15.5563 0.597879 0.298940 0.954272i \(-0.403367\pi\)
0.298940 + 0.954272i \(0.403367\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 2.00000 0.0766965
\(681\) 0 0
\(682\) −2.82843 −0.108306
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −14.1421 −0.540343
\(686\) 0 0
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) 0 0
\(690\) 0 0
\(691\) −22.6274 −0.860788 −0.430394 0.902641i \(-0.641625\pi\)
−0.430394 + 0.902641i \(0.641625\pi\)
\(692\) −1.41421 −0.0537603
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 10.0000 0.378777
\(698\) −18.3848 −0.695874
\(699\) 0 0
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) 22.6274 0.853409
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 26.8701 1.01127
\(707\) 0 0
\(708\) 0 0
\(709\) −8.00000 −0.300446 −0.150223 0.988652i \(-0.547999\pi\)
−0.150223 + 0.988652i \(0.547999\pi\)
\(710\) 11.3137 0.424596
\(711\) 0 0
\(712\) 9.89949 0.370999
\(713\) −11.3137 −0.423702
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) −16.0000 −0.597948
\(717\) 0 0
\(718\) 4.00000 0.149279
\(719\) 36.7696 1.37127 0.685636 0.727944i \(-0.259524\pi\)
0.685636 + 0.727944i \(0.259524\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 11.0000 0.409378
\(723\) 0 0
\(724\) 12.7279 0.473029
\(725\) −18.0000 −0.668503
\(726\) 0 0
\(727\) 2.82843 0.104901 0.0524503 0.998624i \(-0.483297\pi\)
0.0524503 + 0.998624i \(0.483297\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 2.00000 0.0740233
\(731\) −11.3137 −0.418453
\(732\) 0 0
\(733\) −29.6985 −1.09694 −0.548469 0.836171i \(-0.684789\pi\)
−0.548469 + 0.836171i \(0.684789\pi\)
\(734\) 8.48528 0.313197
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 11.3137 0.415900
\(741\) 0 0
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) 5.65685 0.207251
\(746\) −10.0000 −0.366126
\(747\) 0 0
\(748\) −1.41421 −0.0517088
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 8.48528 0.309426
\(753\) 0 0
\(754\) −8.48528 −0.309016
\(755\) −22.6274 −0.823496
\(756\) 0 0
\(757\) 16.0000 0.581530 0.290765 0.956795i \(-0.406090\pi\)
0.290765 + 0.956795i \(0.406090\pi\)
\(758\) 12.0000 0.435860
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) −12.7279 −0.461387 −0.230693 0.973026i \(-0.574099\pi\)
−0.230693 + 0.973026i \(0.574099\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 14.1421 0.510976
\(767\) 0 0
\(768\) 0 0
\(769\) 29.6985 1.07095 0.535477 0.844550i \(-0.320132\pi\)
0.535477 + 0.844550i \(0.320132\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −18.0000 −0.647834
\(773\) 18.3848 0.661254 0.330627 0.943761i \(-0.392740\pi\)
0.330627 + 0.943761i \(0.392740\pi\)
\(774\) 0 0
\(775\) 8.48528 0.304800
\(776\) −15.5563 −0.558440
\(777\) 0 0
\(778\) 10.0000 0.358517
\(779\) −20.0000 −0.716574
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) −5.65685 −0.202289
\(783\) 0 0
\(784\) 0 0
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) −42.4264 −1.51234 −0.756169 0.654376i \(-0.772931\pi\)
−0.756169 + 0.654376i \(0.772931\pi\)
\(788\) −12.0000 −0.427482
\(789\) 0 0
\(790\) 5.65685 0.201262
\(791\) 0 0
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) 9.89949 0.351320
\(795\) 0 0
\(796\) 25.4558 0.902258
\(797\) 26.8701 0.951786 0.475893 0.879503i \(-0.342125\pi\)
0.475893 + 0.879503i \(0.342125\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 3.00000 0.106066
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) −1.41421 −0.0499065
\(804\) 0 0
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 0 0
\(808\) 15.5563 0.547270
\(809\) −36.0000 −1.26569 −0.632846 0.774277i \(-0.718114\pi\)
−0.632846 + 0.774277i \(0.718114\pi\)
\(810\) 0 0
\(811\) 8.48528 0.297959 0.148979 0.988840i \(-0.452401\pi\)
0.148979 + 0.988840i \(0.452401\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) 5.65685 0.198151
\(816\) 0 0
\(817\) 22.6274 0.791633
\(818\) −24.0416 −0.840596
\(819\) 0 0
\(820\) −10.0000 −0.349215
\(821\) −20.0000 −0.698005 −0.349002 0.937122i \(-0.613479\pi\)
−0.349002 + 0.937122i \(0.613479\pi\)
\(822\) 0 0
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) −8.48528 −0.295599
\(825\) 0 0
\(826\) 0 0
\(827\) 52.0000 1.80822 0.904109 0.427303i \(-0.140536\pi\)
0.904109 + 0.427303i \(0.140536\pi\)
\(828\) 0 0
\(829\) 9.89949 0.343824 0.171912 0.985112i \(-0.445006\pi\)
0.171912 + 0.985112i \(0.445006\pi\)
\(830\) 4.00000 0.138842
\(831\) 0 0
\(832\) 1.41421 0.0490290
\(833\) 0 0
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) 2.82843 0.0978232
\(837\) 0 0
\(838\) −16.9706 −0.586238
\(839\) −53.7401 −1.85531 −0.927657 0.373432i \(-0.878181\pi\)
−0.927657 + 0.373432i \(0.878181\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 22.0000 0.758170
\(843\) 0 0
\(844\) −8.00000 −0.275371
\(845\) 15.5563 0.535155
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 4.24264 0.145521
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) 41.0122 1.40423 0.702115 0.712063i \(-0.252239\pi\)
0.702115 + 0.712063i \(0.252239\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) −12.7279 −0.434778 −0.217389 0.976085i \(-0.569754\pi\)
−0.217389 + 0.976085i \(0.569754\pi\)
\(858\) 0 0
\(859\) 16.9706 0.579028 0.289514 0.957174i \(-0.406506\pi\)
0.289514 + 0.957174i \(0.406506\pi\)
\(860\) 11.3137 0.385794
\(861\) 0 0
\(862\) −28.0000 −0.953684
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 0 0
\(865\) 2.00000 0.0680020
\(866\) −4.24264 −0.144171
\(867\) 0 0
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) 0 0
\(871\) −11.3137 −0.383350
\(872\) −4.00000 −0.135457
\(873\) 0 0
\(874\) 11.3137 0.382692
\(875\) 0 0
\(876\) 0 0
\(877\) −52.0000 −1.75592 −0.877958 0.478738i \(-0.841094\pi\)
−0.877958 + 0.478738i \(0.841094\pi\)
\(878\) 33.9411 1.14546
\(879\) 0 0
\(880\) 1.41421 0.0476731
\(881\) 55.1543 1.85820 0.929098 0.369833i \(-0.120585\pi\)
0.929098 + 0.369833i \(0.120585\pi\)
\(882\) 0 0
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) 2.00000 0.0672673
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 45.2548 1.51951 0.759754 0.650210i \(-0.225319\pi\)
0.759754 + 0.650210i \(0.225319\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −14.0000 −0.469281
\(891\) 0 0
\(892\) 2.82843 0.0947027
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) 22.6274 0.756351
\(896\) 0 0
\(897\) 0 0
\(898\) 24.0000 0.800890
\(899\) −16.9706 −0.566000
\(900\) 0 0
\(901\) 0 0
\(902\) 7.07107 0.235441
\(903\) 0 0
\(904\) 0 0
\(905\) −18.0000 −0.598340
\(906\) 0 0
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) −14.1421 −0.469323
\(909\) 0 0
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) −2.82843 −0.0936073
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) −21.2132 −0.700904
\(917\) 0 0
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 5.65685 0.186501
\(921\) 0 0
\(922\) −26.8701 −0.884918
\(923\) 11.3137 0.372395
\(924\) 0 0
\(925\) 24.0000 0.789115
\(926\) 32.0000 1.05159
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) −43.8406 −1.43836 −0.719182 0.694822i \(-0.755483\pi\)
−0.719182 + 0.694822i \(0.755483\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) −16.9706 −0.555294
\(935\) 2.00000 0.0654070
\(936\) 0 0
\(937\) −46.6690 −1.52461 −0.762306 0.647217i \(-0.775933\pi\)
−0.762306 + 0.647217i \(0.775933\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −12.0000 −0.391397
\(941\) 15.5563 0.507122 0.253561 0.967319i \(-0.418398\pi\)
0.253561 + 0.967319i \(0.418398\pi\)
\(942\) 0 0
\(943\) 28.2843 0.921063
\(944\) 0 0
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 2.00000 0.0649227
\(950\) −8.48528 −0.275299
\(951\) 0 0
\(952\) 0 0
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 0 0
\(955\) 16.9706 0.549155
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 11.3137 0.364769
\(963\) 0 0
\(964\) 9.89949 0.318841
\(965\) 25.4558 0.819453
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 22.0000 0.706377
\(971\) 28.2843 0.907685 0.453843 0.891082i \(-0.350053\pi\)
0.453843 + 0.891082i \(0.350053\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −28.0000 −0.897178
\(975\) 0 0
\(976\) −7.07107 −0.226339
\(977\) −38.0000 −1.21573 −0.607864 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) 0 0
\(979\) 9.89949 0.316389
\(980\) 0 0
\(981\) 0 0
\(982\) 20.0000 0.638226
\(983\) −2.82843 −0.0902128 −0.0451064 0.998982i \(-0.514363\pi\)
−0.0451064 + 0.998982i \(0.514363\pi\)
\(984\) 0 0
\(985\) 16.9706 0.540727
\(986\) −8.48528 −0.270226
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) −32.0000 −1.01754
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 2.82843 0.0898027
\(993\) 0 0
\(994\) 0 0
\(995\) −36.0000 −1.14128
\(996\) 0 0
\(997\) 29.6985 0.940560 0.470280 0.882517i \(-0.344153\pi\)
0.470280 + 0.882517i \(0.344153\pi\)
\(998\) 24.0000 0.759707
\(999\) 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.cl.1.1 2
3.2 odd 2 9702.2.a.dp.1.2 yes 2
7.6 odd 2 inner 9702.2.a.cl.1.2 yes 2
21.20 even 2 9702.2.a.dp.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9702.2.a.cl.1.1 2 1.1 even 1 trivial
9702.2.a.cl.1.2 yes 2 7.6 odd 2 inner
9702.2.a.dp.1.1 yes 2 21.20 even 2
9702.2.a.dp.1.2 yes 2 3.2 odd 2