Properties

Label 1305.2.a.j.1.2
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
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] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.23607 q^{2} +3.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +2.23607 q^{8} +O(q^{10})\) \(q+2.23607 q^{2} +3.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +2.23607 q^{8} +2.23607 q^{10} +2.00000 q^{11} +2.00000 q^{13} +4.47214 q^{14} -1.00000 q^{16} -4.47214 q^{17} +2.00000 q^{19} +3.00000 q^{20} +4.47214 q^{22} +2.00000 q^{23} +1.00000 q^{25} +4.47214 q^{26} +6.00000 q^{28} +1.00000 q^{29} -2.00000 q^{31} -6.70820 q^{32} -10.0000 q^{34} +2.00000 q^{35} +8.47214 q^{37} +4.47214 q^{38} +2.23607 q^{40} -2.00000 q^{41} +4.00000 q^{43} +6.00000 q^{44} +4.47214 q^{46} -12.9443 q^{47} -3.00000 q^{49} +2.23607 q^{50} +6.00000 q^{52} -2.00000 q^{53} +2.00000 q^{55} +4.47214 q^{56} +2.23607 q^{58} -8.00000 q^{59} +6.94427 q^{61} -4.47214 q^{62} -13.0000 q^{64} +2.00000 q^{65} +2.94427 q^{67} -13.4164 q^{68} +4.47214 q^{70} -4.00000 q^{71} +3.52786 q^{73} +18.9443 q^{74} +6.00000 q^{76} +4.00000 q^{77} -2.94427 q^{79} -1.00000 q^{80} -4.47214 q^{82} +14.9443 q^{83} -4.47214 q^{85} +8.94427 q^{86} +4.47214 q^{88} +6.00000 q^{89} +4.00000 q^{91} +6.00000 q^{92} -28.9443 q^{94} +2.00000 q^{95} -17.4164 q^{97} -6.70820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{4} + 2 q^{5} + 4 q^{7} + 4 q^{11} + 4 q^{13} - 2 q^{16} + 4 q^{19} + 6 q^{20} + 4 q^{23} + 2 q^{25} + 12 q^{28} + 2 q^{29} - 4 q^{31} - 20 q^{34} + 4 q^{35} + 8 q^{37} - 4 q^{41} + 8 q^{43} + 12 q^{44} - 8 q^{47} - 6 q^{49} + 12 q^{52} - 4 q^{53} + 4 q^{55} - 16 q^{59} - 4 q^{61} - 26 q^{64} + 4 q^{65} - 12 q^{67} - 8 q^{71} + 16 q^{73} + 20 q^{74} + 12 q^{76} + 8 q^{77} + 12 q^{79} - 2 q^{80} + 12 q^{83} + 12 q^{89} + 8 q^{91} + 12 q^{92} - 40 q^{94} + 4 q^{95} - 8 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23607 1.58114 0.790569 0.612372i \(-0.209785\pi\)
0.790569 + 0.612372i \(0.209785\pi\)
\(3\) 0 0
\(4\) 3.00000 1.50000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 2.23607 0.790569
\(9\) 0 0
\(10\) 2.23607 0.707107
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 4.47214 1.19523
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 3.00000 0.670820
\(21\) 0 0
\(22\) 4.47214 0.953463
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.47214 0.877058
\(27\) 0 0
\(28\) 6.00000 1.13389
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −6.70820 −1.18585
\(33\) 0 0
\(34\) −10.0000 −1.71499
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 8.47214 1.39281 0.696405 0.717649i \(-0.254782\pi\)
0.696405 + 0.717649i \(0.254782\pi\)
\(38\) 4.47214 0.725476
\(39\) 0 0
\(40\) 2.23607 0.353553
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) 4.47214 0.659380
\(47\) −12.9443 −1.88812 −0.944058 0.329779i \(-0.893026\pi\)
−0.944058 + 0.329779i \(0.893026\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 2.23607 0.316228
\(51\) 0 0
\(52\) 6.00000 0.832050
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 4.47214 0.597614
\(57\) 0 0
\(58\) 2.23607 0.293610
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 6.94427 0.889123 0.444561 0.895748i \(-0.353360\pi\)
0.444561 + 0.895748i \(0.353360\pi\)
\(62\) −4.47214 −0.567962
\(63\) 0 0
\(64\) −13.0000 −1.62500
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 2.94427 0.359700 0.179850 0.983694i \(-0.442439\pi\)
0.179850 + 0.983694i \(0.442439\pi\)
\(68\) −13.4164 −1.62698
\(69\) 0 0
\(70\) 4.47214 0.534522
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 3.52786 0.412905 0.206453 0.978457i \(-0.433808\pi\)
0.206453 + 0.978457i \(0.433808\pi\)
\(74\) 18.9443 2.20223
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −2.94427 −0.331256 −0.165628 0.986188i \(-0.552965\pi\)
−0.165628 + 0.986188i \(0.552965\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −4.47214 −0.493865
\(83\) 14.9443 1.64035 0.820173 0.572115i \(-0.193877\pi\)
0.820173 + 0.572115i \(0.193877\pi\)
\(84\) 0 0
\(85\) −4.47214 −0.485071
\(86\) 8.94427 0.964486
\(87\) 0 0
\(88\) 4.47214 0.476731
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) −28.9443 −2.98537
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −17.4164 −1.76837 −0.884184 0.467139i \(-0.845285\pi\)
−0.884184 + 0.467139i \(0.845285\pi\)
\(98\) −6.70820 −0.677631
\(99\) 0 0
\(100\) 3.00000 0.300000
\(101\) 2.94427 0.292966 0.146483 0.989213i \(-0.453205\pi\)
0.146483 + 0.989213i \(0.453205\pi\)
\(102\) 0 0
\(103\) −14.9443 −1.47250 −0.736251 0.676708i \(-0.763406\pi\)
−0.736251 + 0.676708i \(0.763406\pi\)
\(104\) 4.47214 0.438529
\(105\) 0 0
\(106\) −4.47214 −0.434372
\(107\) −6.94427 −0.671328 −0.335664 0.941982i \(-0.608961\pi\)
−0.335664 + 0.941982i \(0.608961\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 4.47214 0.426401
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) 4.47214 0.420703 0.210352 0.977626i \(-0.432539\pi\)
0.210352 + 0.977626i \(0.432539\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 3.00000 0.278543
\(117\) 0 0
\(118\) −17.8885 −1.64677
\(119\) −8.94427 −0.819920
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 15.5279 1.40583
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −8.94427 −0.793676 −0.396838 0.917889i \(-0.629892\pi\)
−0.396838 + 0.917889i \(0.629892\pi\)
\(128\) −15.6525 −1.38350
\(129\) 0 0
\(130\) 4.47214 0.392232
\(131\) 10.9443 0.956205 0.478103 0.878304i \(-0.341325\pi\)
0.478103 + 0.878304i \(0.341325\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 6.58359 0.568736
\(135\) 0 0
\(136\) −10.0000 −0.857493
\(137\) −12.4721 −1.06557 −0.532783 0.846252i \(-0.678854\pi\)
−0.532783 + 0.846252i \(0.678854\pi\)
\(138\) 0 0
\(139\) −17.8885 −1.51729 −0.758643 0.651506i \(-0.774137\pi\)
−0.758643 + 0.651506i \(0.774137\pi\)
\(140\) 6.00000 0.507093
\(141\) 0 0
\(142\) −8.94427 −0.750587
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 7.88854 0.652861
\(147\) 0 0
\(148\) 25.4164 2.08922
\(149\) −15.8885 −1.30164 −0.650820 0.759232i \(-0.725575\pi\)
−0.650820 + 0.759232i \(0.725575\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 4.47214 0.362738
\(153\) 0 0
\(154\) 8.94427 0.720750
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) 7.52786 0.600789 0.300394 0.953815i \(-0.402882\pi\)
0.300394 + 0.953815i \(0.402882\pi\)
\(158\) −6.58359 −0.523762
\(159\) 0 0
\(160\) −6.70820 −0.530330
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 33.4164 2.59362
\(167\) 10.0000 0.773823 0.386912 0.922117i \(-0.373542\pi\)
0.386912 + 0.922117i \(0.373542\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −10.0000 −0.766965
\(171\) 0 0
\(172\) 12.0000 0.914991
\(173\) 2.94427 0.223849 0.111924 0.993717i \(-0.464299\pi\)
0.111924 + 0.993717i \(0.464299\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 13.4164 1.00560
\(179\) −13.8885 −1.03808 −0.519039 0.854750i \(-0.673710\pi\)
−0.519039 + 0.854750i \(0.673710\pi\)
\(180\) 0 0
\(181\) −19.8885 −1.47830 −0.739152 0.673539i \(-0.764773\pi\)
−0.739152 + 0.673539i \(0.764773\pi\)
\(182\) 8.94427 0.662994
\(183\) 0 0
\(184\) 4.47214 0.329690
\(185\) 8.47214 0.622884
\(186\) 0 0
\(187\) −8.94427 −0.654070
\(188\) −38.8328 −2.83217
\(189\) 0 0
\(190\) 4.47214 0.324443
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) 0 0
\(193\) −0.472136 −0.0339851 −0.0169925 0.999856i \(-0.505409\pi\)
−0.0169925 + 0.999856i \(0.505409\pi\)
\(194\) −38.9443 −2.79604
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) −14.9443 −1.06474 −0.532368 0.846513i \(-0.678698\pi\)
−0.532368 + 0.846513i \(0.678698\pi\)
\(198\) 0 0
\(199\) 21.8885 1.55164 0.775819 0.630956i \(-0.217337\pi\)
0.775819 + 0.630956i \(0.217337\pi\)
\(200\) 2.23607 0.158114
\(201\) 0 0
\(202\) 6.58359 0.463220
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) −33.4164 −2.32823
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −1.05573 −0.0726793 −0.0363397 0.999339i \(-0.511570\pi\)
−0.0363397 + 0.999339i \(0.511570\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −15.5279 −1.06146
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 4.47214 0.302891
\(219\) 0 0
\(220\) 6.00000 0.404520
\(221\) −8.94427 −0.601657
\(222\) 0 0
\(223\) −22.9443 −1.53646 −0.768231 0.640173i \(-0.778863\pi\)
−0.768231 + 0.640173i \(0.778863\pi\)
\(224\) −13.4164 −0.896421
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) 23.8885 1.58554 0.792769 0.609522i \(-0.208639\pi\)
0.792769 + 0.609522i \(0.208639\pi\)
\(228\) 0 0
\(229\) 5.05573 0.334092 0.167046 0.985949i \(-0.446577\pi\)
0.167046 + 0.985949i \(0.446577\pi\)
\(230\) 4.47214 0.294884
\(231\) 0 0
\(232\) 2.23607 0.146805
\(233\) 18.9443 1.24108 0.620540 0.784175i \(-0.286913\pi\)
0.620540 + 0.784175i \(0.286913\pi\)
\(234\) 0 0
\(235\) −12.9443 −0.844391
\(236\) −24.0000 −1.56227
\(237\) 0 0
\(238\) −20.0000 −1.29641
\(239\) 21.8885 1.41585 0.707926 0.706287i \(-0.249631\pi\)
0.707926 + 0.706287i \(0.249631\pi\)
\(240\) 0 0
\(241\) 15.8885 1.02347 0.511736 0.859143i \(-0.329003\pi\)
0.511736 + 0.859143i \(0.329003\pi\)
\(242\) −15.6525 −1.00618
\(243\) 0 0
\(244\) 20.8328 1.33368
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) −4.47214 −0.283981
\(249\) 0 0
\(250\) 2.23607 0.141421
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) −20.0000 −1.25491
\(255\) 0 0
\(256\) −9.00000 −0.562500
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 16.9443 1.05287
\(260\) 6.00000 0.372104
\(261\) 0 0
\(262\) 24.4721 1.51189
\(263\) 30.8328 1.90123 0.950616 0.310368i \(-0.100452\pi\)
0.950616 + 0.310368i \(0.100452\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) 8.94427 0.548408
\(267\) 0 0
\(268\) 8.83282 0.539550
\(269\) −30.9443 −1.88671 −0.943353 0.331791i \(-0.892347\pi\)
−0.943353 + 0.331791i \(0.892347\pi\)
\(270\) 0 0
\(271\) −9.05573 −0.550096 −0.275048 0.961430i \(-0.588694\pi\)
−0.275048 + 0.961430i \(0.588694\pi\)
\(272\) 4.47214 0.271163
\(273\) 0 0
\(274\) −27.8885 −1.68481
\(275\) 2.00000 0.120605
\(276\) 0 0
\(277\) −10.9443 −0.657578 −0.328789 0.944403i \(-0.606640\pi\)
−0.328789 + 0.944403i \(0.606640\pi\)
\(278\) −40.0000 −2.39904
\(279\) 0 0
\(280\) 4.47214 0.267261
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −1.05573 −0.0627565 −0.0313783 0.999508i \(-0.509990\pi\)
−0.0313783 + 0.999508i \(0.509990\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 8.94427 0.528886
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 2.23607 0.131306
\(291\) 0 0
\(292\) 10.5836 0.619358
\(293\) −15.5279 −0.907148 −0.453574 0.891219i \(-0.649851\pi\)
−0.453574 + 0.891219i \(0.649851\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 18.9443 1.10111
\(297\) 0 0
\(298\) −35.5279 −2.05807
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) −8.94427 −0.514685
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) 6.94427 0.397628
\(306\) 0 0
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 12.0000 0.683763
\(309\) 0 0
\(310\) −4.47214 −0.254000
\(311\) −20.8328 −1.18132 −0.590660 0.806920i \(-0.701133\pi\)
−0.590660 + 0.806920i \(0.701133\pi\)
\(312\) 0 0
\(313\) −2.94427 −0.166420 −0.0832100 0.996532i \(-0.526517\pi\)
−0.0832100 + 0.996532i \(0.526517\pi\)
\(314\) 16.8328 0.949931
\(315\) 0 0
\(316\) −8.83282 −0.496885
\(317\) 9.41641 0.528878 0.264439 0.964402i \(-0.414813\pi\)
0.264439 + 0.964402i \(0.414813\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) −13.0000 −0.726722
\(321\) 0 0
\(322\) 8.94427 0.498445
\(323\) −8.94427 −0.497673
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 35.7771 1.98151
\(327\) 0 0
\(328\) −4.47214 −0.246932
\(329\) −25.8885 −1.42728
\(330\) 0 0
\(331\) 19.8885 1.09317 0.546587 0.837403i \(-0.315927\pi\)
0.546587 + 0.837403i \(0.315927\pi\)
\(332\) 44.8328 2.46052
\(333\) 0 0
\(334\) 22.3607 1.22352
\(335\) 2.94427 0.160863
\(336\) 0 0
\(337\) 34.3607 1.87175 0.935873 0.352338i \(-0.114613\pi\)
0.935873 + 0.352338i \(0.114613\pi\)
\(338\) −20.1246 −1.09463
\(339\) 0 0
\(340\) −13.4164 −0.727607
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 8.94427 0.482243
\(345\) 0 0
\(346\) 6.58359 0.353936
\(347\) 30.9443 1.66118 0.830588 0.556888i \(-0.188005\pi\)
0.830588 + 0.556888i \(0.188005\pi\)
\(348\) 0 0
\(349\) −7.88854 −0.422264 −0.211132 0.977458i \(-0.567715\pi\)
−0.211132 + 0.977458i \(0.567715\pi\)
\(350\) 4.47214 0.239046
\(351\) 0 0
\(352\) −13.4164 −0.715097
\(353\) −30.9443 −1.64700 −0.823499 0.567318i \(-0.807981\pi\)
−0.823499 + 0.567318i \(0.807981\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) −31.0557 −1.64135
\(359\) −6.94427 −0.366505 −0.183252 0.983066i \(-0.558663\pi\)
−0.183252 + 0.983066i \(0.558663\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −44.4721 −2.33740
\(363\) 0 0
\(364\) 12.0000 0.628971
\(365\) 3.52786 0.184657
\(366\) 0 0
\(367\) 7.05573 0.368306 0.184153 0.982898i \(-0.441046\pi\)
0.184153 + 0.982898i \(0.441046\pi\)
\(368\) −2.00000 −0.104257
\(369\) 0 0
\(370\) 18.9443 0.984866
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) 19.8885 1.02979 0.514895 0.857253i \(-0.327831\pi\)
0.514895 + 0.857253i \(0.327831\pi\)
\(374\) −20.0000 −1.03418
\(375\) 0 0
\(376\) −28.9443 −1.49269
\(377\) 2.00000 0.103005
\(378\) 0 0
\(379\) −1.05573 −0.0542291 −0.0271146 0.999632i \(-0.508632\pi\)
−0.0271146 + 0.999632i \(0.508632\pi\)
\(380\) 6.00000 0.307794
\(381\) 0 0
\(382\) 22.3607 1.14407
\(383\) 17.0557 0.871507 0.435753 0.900066i \(-0.356482\pi\)
0.435753 + 0.900066i \(0.356482\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) −1.05573 −0.0537351
\(387\) 0 0
\(388\) −52.2492 −2.65255
\(389\) 28.8328 1.46188 0.730941 0.682441i \(-0.239081\pi\)
0.730941 + 0.682441i \(0.239081\pi\)
\(390\) 0 0
\(391\) −8.94427 −0.452331
\(392\) −6.70820 −0.338815
\(393\) 0 0
\(394\) −33.4164 −1.68349
\(395\) −2.94427 −0.148142
\(396\) 0 0
\(397\) 19.8885 0.998177 0.499089 0.866551i \(-0.333668\pi\)
0.499089 + 0.866551i \(0.333668\pi\)
\(398\) 48.9443 2.45335
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 23.8885 1.19294 0.596468 0.802637i \(-0.296570\pi\)
0.596468 + 0.802637i \(0.296570\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 8.83282 0.439449
\(405\) 0 0
\(406\) 4.47214 0.221948
\(407\) 16.9443 0.839896
\(408\) 0 0
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) −4.47214 −0.220863
\(411\) 0 0
\(412\) −44.8328 −2.20875
\(413\) −16.0000 −0.787309
\(414\) 0 0
\(415\) 14.9443 0.733585
\(416\) −13.4164 −0.657794
\(417\) 0 0
\(418\) 8.94427 0.437479
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) 24.8328 1.21028 0.605139 0.796120i \(-0.293118\pi\)
0.605139 + 0.796120i \(0.293118\pi\)
\(422\) −2.36068 −0.114916
\(423\) 0 0
\(424\) −4.47214 −0.217186
\(425\) −4.47214 −0.216930
\(426\) 0 0
\(427\) 13.8885 0.672114
\(428\) −20.8328 −1.00699
\(429\) 0 0
\(430\) 8.94427 0.431331
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 0 0
\(433\) 4.47214 0.214917 0.107459 0.994210i \(-0.465729\pi\)
0.107459 + 0.994210i \(0.465729\pi\)
\(434\) −8.94427 −0.429339
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 4.47214 0.213201
\(441\) 0 0
\(442\) −20.0000 −0.951303
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) −51.3050 −2.42936
\(447\) 0 0
\(448\) −26.0000 −1.22838
\(449\) 9.05573 0.427366 0.213683 0.976903i \(-0.431454\pi\)
0.213683 + 0.976903i \(0.431454\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) 13.4164 0.631055
\(453\) 0 0
\(454\) 53.4164 2.50696
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) 11.8885 0.556123 0.278061 0.960563i \(-0.410308\pi\)
0.278061 + 0.960563i \(0.410308\pi\)
\(458\) 11.3050 0.528246
\(459\) 0 0
\(460\) 6.00000 0.279751
\(461\) 10.9443 0.509726 0.254863 0.966977i \(-0.417970\pi\)
0.254863 + 0.966977i \(0.417970\pi\)
\(462\) 0 0
\(463\) 32.8328 1.52587 0.762935 0.646475i \(-0.223758\pi\)
0.762935 + 0.646475i \(0.223758\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) 42.3607 1.96232
\(467\) 6.11146 0.282804 0.141402 0.989952i \(-0.454839\pi\)
0.141402 + 0.989952i \(0.454839\pi\)
\(468\) 0 0
\(469\) 5.88854 0.271908
\(470\) −28.9443 −1.33510
\(471\) 0 0
\(472\) −17.8885 −0.823387
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) −26.8328 −1.22988
\(477\) 0 0
\(478\) 48.9443 2.23866
\(479\) 26.0000 1.18797 0.593985 0.804476i \(-0.297554\pi\)
0.593985 + 0.804476i \(0.297554\pi\)
\(480\) 0 0
\(481\) 16.9443 0.772592
\(482\) 35.5279 1.61825
\(483\) 0 0
\(484\) −21.0000 −0.954545
\(485\) −17.4164 −0.790838
\(486\) 0 0
\(487\) 27.8885 1.26375 0.631875 0.775070i \(-0.282285\pi\)
0.631875 + 0.775070i \(0.282285\pi\)
\(488\) 15.5279 0.702913
\(489\) 0 0
\(490\) −6.70820 −0.303046
\(491\) 10.9443 0.493908 0.246954 0.969027i \(-0.420570\pi\)
0.246954 + 0.969027i \(0.420570\pi\)
\(492\) 0 0
\(493\) −4.47214 −0.201415
\(494\) 8.94427 0.402422
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 3.00000 0.134164
\(501\) 0 0
\(502\) 40.2492 1.79641
\(503\) −18.8328 −0.839714 −0.419857 0.907590i \(-0.637920\pi\)
−0.419857 + 0.907590i \(0.637920\pi\)
\(504\) 0 0
\(505\) 2.94427 0.131018
\(506\) 8.94427 0.397621
\(507\) 0 0
\(508\) −26.8328 −1.19051
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) 7.05573 0.312127
\(512\) 11.1803 0.494106
\(513\) 0 0
\(514\) −40.2492 −1.77532
\(515\) −14.9443 −0.658523
\(516\) 0 0
\(517\) −25.8885 −1.13858
\(518\) 37.8885 1.66473
\(519\) 0 0
\(520\) 4.47214 0.196116
\(521\) −7.88854 −0.345603 −0.172802 0.984957i \(-0.555282\pi\)
−0.172802 + 0.984957i \(0.555282\pi\)
\(522\) 0 0
\(523\) −38.9443 −1.70291 −0.851457 0.524424i \(-0.824281\pi\)
−0.851457 + 0.524424i \(0.824281\pi\)
\(524\) 32.8328 1.43431
\(525\) 0 0
\(526\) 68.9443 3.00611
\(527\) 8.94427 0.389619
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) −4.47214 −0.194257
\(531\) 0 0
\(532\) 12.0000 0.520266
\(533\) −4.00000 −0.173259
\(534\) 0 0
\(535\) −6.94427 −0.300227
\(536\) 6.58359 0.284368
\(537\) 0 0
\(538\) −69.1935 −2.98314
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 0.111456 0.00479188 0.00239594 0.999997i \(-0.499237\pi\)
0.00239594 + 0.999997i \(0.499237\pi\)
\(542\) −20.2492 −0.869779
\(543\) 0 0
\(544\) 30.0000 1.28624
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −6.00000 −0.256541 −0.128271 0.991739i \(-0.540943\pi\)
−0.128271 + 0.991739i \(0.540943\pi\)
\(548\) −37.4164 −1.59835
\(549\) 0 0
\(550\) 4.47214 0.190693
\(551\) 2.00000 0.0852029
\(552\) 0 0
\(553\) −5.88854 −0.250406
\(554\) −24.4721 −1.03972
\(555\) 0 0
\(556\) −53.6656 −2.27593
\(557\) −35.8885 −1.52065 −0.760323 0.649545i \(-0.774959\pi\)
−0.760323 + 0.649545i \(0.774959\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) 22.3607 0.943228
\(563\) 35.7771 1.50782 0.753912 0.656975i \(-0.228164\pi\)
0.753912 + 0.656975i \(0.228164\pi\)
\(564\) 0 0
\(565\) 4.47214 0.188144
\(566\) −2.36068 −0.0992268
\(567\) 0 0
\(568\) −8.94427 −0.375293
\(569\) 34.9443 1.46494 0.732470 0.680799i \(-0.238367\pi\)
0.732470 + 0.680799i \(0.238367\pi\)
\(570\) 0 0
\(571\) −45.8885 −1.92038 −0.960188 0.279355i \(-0.909879\pi\)
−0.960188 + 0.279355i \(0.909879\pi\)
\(572\) 12.0000 0.501745
\(573\) 0 0
\(574\) −8.94427 −0.373327
\(575\) 2.00000 0.0834058
\(576\) 0 0
\(577\) −16.4721 −0.685744 −0.342872 0.939382i \(-0.611400\pi\)
−0.342872 + 0.939382i \(0.611400\pi\)
\(578\) 6.70820 0.279024
\(579\) 0 0
\(580\) 3.00000 0.124568
\(581\) 29.8885 1.23999
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) 7.88854 0.326430
\(585\) 0 0
\(586\) −34.7214 −1.43433
\(587\) −0.111456 −0.00460029 −0.00230014 0.999997i \(-0.500732\pi\)
−0.00230014 + 0.999997i \(0.500732\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) −17.8885 −0.736460
\(591\) 0 0
\(592\) −8.47214 −0.348203
\(593\) 26.9443 1.10647 0.553234 0.833026i \(-0.313393\pi\)
0.553234 + 0.833026i \(0.313393\pi\)
\(594\) 0 0
\(595\) −8.94427 −0.366679
\(596\) −47.6656 −1.95246
\(597\) 0 0
\(598\) 8.94427 0.365758
\(599\) −0.111456 −0.00455398 −0.00227699 0.999997i \(-0.500725\pi\)
−0.00227699 + 0.999997i \(0.500725\pi\)
\(600\) 0 0
\(601\) −18.9443 −0.772753 −0.386376 0.922341i \(-0.626273\pi\)
−0.386376 + 0.922341i \(0.626273\pi\)
\(602\) 17.8885 0.729083
\(603\) 0 0
\(604\) −12.0000 −0.488273
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) 36.9443 1.49952 0.749761 0.661709i \(-0.230169\pi\)
0.749761 + 0.661709i \(0.230169\pi\)
\(608\) −13.4164 −0.544107
\(609\) 0 0
\(610\) 15.5279 0.628705
\(611\) −25.8885 −1.04734
\(612\) 0 0
\(613\) 30.9443 1.24983 0.624914 0.780694i \(-0.285134\pi\)
0.624914 + 0.780694i \(0.285134\pi\)
\(614\) 17.8885 0.721923
\(615\) 0 0
\(616\) 8.94427 0.360375
\(617\) 15.5279 0.625128 0.312564 0.949897i \(-0.398812\pi\)
0.312564 + 0.949897i \(0.398812\pi\)
\(618\) 0 0
\(619\) 1.05573 0.0424333 0.0212166 0.999775i \(-0.493246\pi\)
0.0212166 + 0.999775i \(0.493246\pi\)
\(620\) −6.00000 −0.240966
\(621\) 0 0
\(622\) −46.5836 −1.86783
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −6.58359 −0.263133
\(627\) 0 0
\(628\) 22.5836 0.901183
\(629\) −37.8885 −1.51072
\(630\) 0 0
\(631\) −47.7771 −1.90198 −0.950988 0.309228i \(-0.899929\pi\)
−0.950988 + 0.309228i \(0.899929\pi\)
\(632\) −6.58359 −0.261881
\(633\) 0 0
\(634\) 21.0557 0.836230
\(635\) −8.94427 −0.354943
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) 4.47214 0.177054
\(639\) 0 0
\(640\) −15.6525 −0.618718
\(641\) −35.8885 −1.41751 −0.708756 0.705454i \(-0.750743\pi\)
−0.708756 + 0.705454i \(0.750743\pi\)
\(642\) 0 0
\(643\) 30.9443 1.22032 0.610161 0.792277i \(-0.291105\pi\)
0.610161 + 0.792277i \(0.291105\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) −20.0000 −0.786889
\(647\) −9.05573 −0.356017 −0.178009 0.984029i \(-0.556966\pi\)
−0.178009 + 0.984029i \(0.556966\pi\)
\(648\) 0 0
\(649\) −16.0000 −0.628055
\(650\) 4.47214 0.175412
\(651\) 0 0
\(652\) 48.0000 1.87983
\(653\) 23.3050 0.911993 0.455997 0.889982i \(-0.349283\pi\)
0.455997 + 0.889982i \(0.349283\pi\)
\(654\) 0 0
\(655\) 10.9443 0.427628
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) −57.8885 −2.25673
\(659\) −8.83282 −0.344078 −0.172039 0.985090i \(-0.555035\pi\)
−0.172039 + 0.985090i \(0.555035\pi\)
\(660\) 0 0
\(661\) 15.8885 0.617993 0.308996 0.951063i \(-0.400007\pi\)
0.308996 + 0.951063i \(0.400007\pi\)
\(662\) 44.4721 1.72846
\(663\) 0 0
\(664\) 33.4164 1.29681
\(665\) 4.00000 0.155113
\(666\) 0 0
\(667\) 2.00000 0.0774403
\(668\) 30.0000 1.16073
\(669\) 0 0
\(670\) 6.58359 0.254346
\(671\) 13.8885 0.536161
\(672\) 0 0
\(673\) 46.9443 1.80957 0.904784 0.425870i \(-0.140032\pi\)
0.904784 + 0.425870i \(0.140032\pi\)
\(674\) 76.8328 2.95949
\(675\) 0 0
\(676\) −27.0000 −1.03846
\(677\) −31.5279 −1.21171 −0.605857 0.795573i \(-0.707170\pi\)
−0.605857 + 0.795573i \(0.707170\pi\)
\(678\) 0 0
\(679\) −34.8328 −1.33676
\(680\) −10.0000 −0.383482
\(681\) 0 0
\(682\) −8.94427 −0.342494
\(683\) 45.7771 1.75161 0.875806 0.482664i \(-0.160331\pi\)
0.875806 + 0.482664i \(0.160331\pi\)
\(684\) 0 0
\(685\) −12.4721 −0.476536
\(686\) −44.7214 −1.70747
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) −2.11146 −0.0803236 −0.0401618 0.999193i \(-0.512787\pi\)
−0.0401618 + 0.999193i \(0.512787\pi\)
\(692\) 8.83282 0.335773
\(693\) 0 0
\(694\) 69.1935 2.62655
\(695\) −17.8885 −0.678551
\(696\) 0 0
\(697\) 8.94427 0.338788
\(698\) −17.6393 −0.667658
\(699\) 0 0
\(700\) 6.00000 0.226779
\(701\) 3.88854 0.146868 0.0734341 0.997300i \(-0.476604\pi\)
0.0734341 + 0.997300i \(0.476604\pi\)
\(702\) 0 0
\(703\) 16.9443 0.639065
\(704\) −26.0000 −0.979912
\(705\) 0 0
\(706\) −69.1935 −2.60413
\(707\) 5.88854 0.221461
\(708\) 0 0
\(709\) −41.7771 −1.56897 −0.784486 0.620147i \(-0.787073\pi\)
−0.784486 + 0.620147i \(0.787073\pi\)
\(710\) −8.94427 −0.335673
\(711\) 0 0
\(712\) 13.4164 0.502801
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) −41.6656 −1.55712
\(717\) 0 0
\(718\) −15.5279 −0.579495
\(719\) −39.7771 −1.48344 −0.741718 0.670712i \(-0.765988\pi\)
−0.741718 + 0.670712i \(0.765988\pi\)
\(720\) 0 0
\(721\) −29.8885 −1.11311
\(722\) −33.5410 −1.24827
\(723\) 0 0
\(724\) −59.6656 −2.21746
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 50.8328 1.88528 0.942642 0.333804i \(-0.108332\pi\)
0.942642 + 0.333804i \(0.108332\pi\)
\(728\) 8.94427 0.331497
\(729\) 0 0
\(730\) 7.88854 0.291968
\(731\) −17.8885 −0.661632
\(732\) 0 0
\(733\) 49.1935 1.81700 0.908502 0.417881i \(-0.137227\pi\)
0.908502 + 0.417881i \(0.137227\pi\)
\(734\) 15.7771 0.582343
\(735\) 0 0
\(736\) −13.4164 −0.494535
\(737\) 5.88854 0.216907
\(738\) 0 0
\(739\) −23.8885 −0.878754 −0.439377 0.898303i \(-0.644801\pi\)
−0.439377 + 0.898303i \(0.644801\pi\)
\(740\) 25.4164 0.934326
\(741\) 0 0
\(742\) −8.94427 −0.328355
\(743\) 0.944272 0.0346420 0.0173210 0.999850i \(-0.494486\pi\)
0.0173210 + 0.999850i \(0.494486\pi\)
\(744\) 0 0
\(745\) −15.8885 −0.582111
\(746\) 44.4721 1.62824
\(747\) 0 0
\(748\) −26.8328 −0.981105
\(749\) −13.8885 −0.507476
\(750\) 0 0
\(751\) −6.00000 −0.218943 −0.109472 0.993990i \(-0.534916\pi\)
−0.109472 + 0.993990i \(0.534916\pi\)
\(752\) 12.9443 0.472029
\(753\) 0 0
\(754\) 4.47214 0.162866
\(755\) −4.00000 −0.145575
\(756\) 0 0
\(757\) −21.4164 −0.778393 −0.389196 0.921155i \(-0.627247\pi\)
−0.389196 + 0.921155i \(0.627247\pi\)
\(758\) −2.36068 −0.0857438
\(759\) 0 0
\(760\) 4.47214 0.162221
\(761\) 29.7771 1.07942 0.539709 0.841851i \(-0.318534\pi\)
0.539709 + 0.841851i \(0.318534\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) 30.0000 1.08536
\(765\) 0 0
\(766\) 38.1378 1.37797
\(767\) −16.0000 −0.577727
\(768\) 0 0
\(769\) 5.05573 0.182314 0.0911571 0.995837i \(-0.470943\pi\)
0.0911571 + 0.995837i \(0.470943\pi\)
\(770\) 8.94427 0.322329
\(771\) 0 0
\(772\) −1.41641 −0.0509776
\(773\) −50.3607 −1.81135 −0.905674 0.423975i \(-0.860634\pi\)
−0.905674 + 0.423975i \(0.860634\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) −38.9443 −1.39802
\(777\) 0 0
\(778\) 64.4721 2.31144
\(779\) −4.00000 −0.143315
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) −20.0000 −0.715199
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 7.52786 0.268681
\(786\) 0 0
\(787\) −11.8885 −0.423781 −0.211890 0.977293i \(-0.567962\pi\)
−0.211890 + 0.977293i \(0.567962\pi\)
\(788\) −44.8328 −1.59710
\(789\) 0 0
\(790\) −6.58359 −0.234234
\(791\) 8.94427 0.318022
\(792\) 0 0
\(793\) 13.8885 0.493197
\(794\) 44.4721 1.57826
\(795\) 0 0
\(796\) 65.6656 2.32746
\(797\) 13.4164 0.475234 0.237617 0.971359i \(-0.423634\pi\)
0.237617 + 0.971359i \(0.423634\pi\)
\(798\) 0 0
\(799\) 57.8885 2.04795
\(800\) −6.70820 −0.237171
\(801\) 0 0
\(802\) 53.4164 1.88620
\(803\) 7.05573 0.248991
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) −8.94427 −0.315049
\(807\) 0 0
\(808\) 6.58359 0.231610
\(809\) −0.111456 −0.00391859 −0.00195930 0.999998i \(-0.500624\pi\)
−0.00195930 + 0.999998i \(0.500624\pi\)
\(810\) 0 0
\(811\) −41.8885 −1.47091 −0.735453 0.677576i \(-0.763031\pi\)
−0.735453 + 0.677576i \(0.763031\pi\)
\(812\) 6.00000 0.210559
\(813\) 0 0
\(814\) 37.8885 1.32799
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) 8.00000 0.279885
\(818\) −67.0820 −2.34547
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) −14.0000 −0.488603 −0.244302 0.969699i \(-0.578559\pi\)
−0.244302 + 0.969699i \(0.578559\pi\)
\(822\) 0 0
\(823\) 28.9443 1.00893 0.504467 0.863431i \(-0.331689\pi\)
0.504467 + 0.863431i \(0.331689\pi\)
\(824\) −33.4164 −1.16412
\(825\) 0 0
\(826\) −35.7771 −1.24484
\(827\) −29.8885 −1.03933 −0.519663 0.854371i \(-0.673943\pi\)
−0.519663 + 0.854371i \(0.673943\pi\)
\(828\) 0 0
\(829\) 29.7771 1.03420 0.517101 0.855925i \(-0.327011\pi\)
0.517101 + 0.855925i \(0.327011\pi\)
\(830\) 33.4164 1.15990
\(831\) 0 0
\(832\) −26.0000 −0.901388
\(833\) 13.4164 0.464851
\(834\) 0 0
\(835\) 10.0000 0.346064
\(836\) 12.0000 0.415029
\(837\) 0 0
\(838\) 8.94427 0.308975
\(839\) −12.1115 −0.418134 −0.209067 0.977901i \(-0.567043\pi\)
−0.209067 + 0.977901i \(0.567043\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 55.5279 1.91362
\(843\) 0 0
\(844\) −3.16718 −0.109019
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) −10.0000 −0.342997
\(851\) 16.9443 0.580842
\(852\) 0 0
\(853\) 27.5279 0.942536 0.471268 0.881990i \(-0.343796\pi\)
0.471268 + 0.881990i \(0.343796\pi\)
\(854\) 31.0557 1.06271
\(855\) 0 0
\(856\) −15.5279 −0.530731
\(857\) −26.0000 −0.888143 −0.444072 0.895991i \(-0.646466\pi\)
−0.444072 + 0.895991i \(0.646466\pi\)
\(858\) 0 0
\(859\) −38.9443 −1.32876 −0.664381 0.747394i \(-0.731305\pi\)
−0.664381 + 0.747394i \(0.731305\pi\)
\(860\) 12.0000 0.409197
\(861\) 0 0
\(862\) 17.8885 0.609286
\(863\) 34.9443 1.18952 0.594758 0.803904i \(-0.297248\pi\)
0.594758 + 0.803904i \(0.297248\pi\)
\(864\) 0 0
\(865\) 2.94427 0.100108
\(866\) 10.0000 0.339814
\(867\) 0 0
\(868\) −12.0000 −0.407307
\(869\) −5.88854 −0.199755
\(870\) 0 0
\(871\) 5.88854 0.199526
\(872\) 4.47214 0.151446
\(873\) 0 0
\(874\) 8.94427 0.302545
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) −9.05573 −0.305790 −0.152895 0.988242i \(-0.548860\pi\)
−0.152895 + 0.988242i \(0.548860\pi\)
\(878\) 17.8885 0.603709
\(879\) 0 0
\(880\) −2.00000 −0.0674200
\(881\) −35.8885 −1.20912 −0.604558 0.796561i \(-0.706650\pi\)
−0.604558 + 0.796561i \(0.706650\pi\)
\(882\) 0 0
\(883\) 43.8885 1.47697 0.738484 0.674271i \(-0.235542\pi\)
0.738484 + 0.674271i \(0.235542\pi\)
\(884\) −26.8328 −0.902485
\(885\) 0 0
\(886\) 53.6656 1.80293
\(887\) 16.9443 0.568933 0.284466 0.958686i \(-0.408184\pi\)
0.284466 + 0.958686i \(0.408184\pi\)
\(888\) 0 0
\(889\) −17.8885 −0.599963
\(890\) 13.4164 0.449719
\(891\) 0 0
\(892\) −68.8328 −2.30469
\(893\) −25.8885 −0.866327
\(894\) 0 0
\(895\) −13.8885 −0.464243
\(896\) −31.3050 −1.04583
\(897\) 0 0
\(898\) 20.2492 0.675725
\(899\) −2.00000 −0.0667037
\(900\) 0 0
\(901\) 8.94427 0.297977
\(902\) −8.94427 −0.297812
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) −19.8885 −0.661118
\(906\) 0 0
\(907\) 31.7771 1.05514 0.527570 0.849511i \(-0.323103\pi\)
0.527570 + 0.849511i \(0.323103\pi\)
\(908\) 71.6656 2.37831
\(909\) 0 0
\(910\) 8.94427 0.296500
\(911\) 28.8328 0.955274 0.477637 0.878557i \(-0.341493\pi\)
0.477637 + 0.878557i \(0.341493\pi\)
\(912\) 0 0
\(913\) 29.8885 0.989166
\(914\) 26.5836 0.879307
\(915\) 0 0
\(916\) 15.1672 0.501138
\(917\) 21.8885 0.722823
\(918\) 0 0
\(919\) 5.88854 0.194245 0.0971226 0.995272i \(-0.469036\pi\)
0.0971226 + 0.995272i \(0.469036\pi\)
\(920\) 4.47214 0.147442
\(921\) 0 0
\(922\) 24.4721 0.805947
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) 8.47214 0.278562
\(926\) 73.4164 2.41261
\(927\) 0 0
\(928\) −6.70820 −0.220208
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 56.8328 1.86162
\(933\) 0 0
\(934\) 13.6656 0.447153
\(935\) −8.94427 −0.292509
\(936\) 0 0
\(937\) −57.7771 −1.88750 −0.943748 0.330667i \(-0.892726\pi\)
−0.943748 + 0.330667i \(0.892726\pi\)
\(938\) 13.1672 0.429924
\(939\) 0 0
\(940\) −38.8328 −1.26659
\(941\) −31.8885 −1.03954 −0.519768 0.854307i \(-0.673982\pi\)
−0.519768 + 0.854307i \(0.673982\pi\)
\(942\) 0 0
\(943\) −4.00000 −0.130258
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) 17.8885 0.581607
\(947\) 40.0000 1.29983 0.649913 0.760009i \(-0.274805\pi\)
0.649913 + 0.760009i \(0.274805\pi\)
\(948\) 0 0
\(949\) 7.05573 0.229039
\(950\) 4.47214 0.145095
\(951\) 0 0
\(952\) −20.0000 −0.648204
\(953\) −14.9443 −0.484092 −0.242046 0.970265i \(-0.577819\pi\)
−0.242046 + 0.970265i \(0.577819\pi\)
\(954\) 0 0
\(955\) 10.0000 0.323592
\(956\) 65.6656 2.12378
\(957\) 0 0
\(958\) 58.1378 1.87835
\(959\) −24.9443 −0.805493
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 37.8885 1.22158
\(963\) 0 0
\(964\) 47.6656 1.53521
\(965\) −0.472136 −0.0151986
\(966\) 0 0
\(967\) −48.7214 −1.56677 −0.783387 0.621535i \(-0.786509\pi\)
−0.783387 + 0.621535i \(0.786509\pi\)
\(968\) −15.6525 −0.503090
\(969\) 0 0
\(970\) −38.9443 −1.25043
\(971\) 48.8328 1.56712 0.783560 0.621316i \(-0.213402\pi\)
0.783560 + 0.621316i \(0.213402\pi\)
\(972\) 0 0
\(973\) −35.7771 −1.14696
\(974\) 62.3607 1.99817
\(975\) 0 0
\(976\) −6.94427 −0.222281
\(977\) 15.8885 0.508320 0.254160 0.967162i \(-0.418201\pi\)
0.254160 + 0.967162i \(0.418201\pi\)
\(978\) 0 0
\(979\) 12.0000 0.383522
\(980\) −9.00000 −0.287494
\(981\) 0 0
\(982\) 24.4721 0.780937
\(983\) 31.0557 0.990524 0.495262 0.868744i \(-0.335072\pi\)
0.495262 + 0.868744i \(0.335072\pi\)
\(984\) 0 0
\(985\) −14.9443 −0.476164
\(986\) −10.0000 −0.318465
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 37.8885 1.20357 0.601785 0.798658i \(-0.294457\pi\)
0.601785 + 0.798658i \(0.294457\pi\)
\(992\) 13.4164 0.425971
\(993\) 0 0
\(994\) −17.8885 −0.567390
\(995\) 21.8885 0.693913
\(996\) 0 0
\(997\) −27.5279 −0.871816 −0.435908 0.899991i \(-0.643573\pi\)
−0.435908 + 0.899991i \(0.643573\pi\)
\(998\) 44.7214 1.41563
\(999\) 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 1305.2.a.j.1.2 2
3.2 odd 2 435.2.a.g.1.1 2
5.4 even 2 6525.2.a.y.1.1 2
12.11 even 2 6960.2.a.bp.1.2 2
15.2 even 4 2175.2.c.g.349.1 4
15.8 even 4 2175.2.c.g.349.4 4
15.14 odd 2 2175.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.g.1.1 2 3.2 odd 2
1305.2.a.j.1.2 2 1.1 even 1 trivial
2175.2.a.o.1.2 2 15.14 odd 2
2175.2.c.g.349.1 4 15.2 even 4
2175.2.c.g.349.4 4 15.8 even 4
6525.2.a.y.1.1 2 5.4 even 2
6960.2.a.bp.1.2 2 12.11 even 2