Properties

Label 7605.2.a.cb.1.3
Level $7605$
Weight $2$
Character 7605.1
Self dual yes
Analytic conductor $60.726$
Analytic rank $0$
Dimension $3$
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] = [7605,2,Mod(1,7605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7605, 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("7605.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7605.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: \(60.7262307372\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2535)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 7605.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.80194 q^{2} +1.24698 q^{4} +1.00000 q^{5} +2.80194 q^{7} -1.35690 q^{8} +O(q^{10})\) \(q+1.80194 q^{2} +1.24698 q^{4} +1.00000 q^{5} +2.80194 q^{7} -1.35690 q^{8} +1.80194 q^{10} -3.49396 q^{11} +5.04892 q^{14} -4.93900 q^{16} +7.60388 q^{17} +1.75302 q^{19} +1.24698 q^{20} -6.29590 q^{22} +6.44504 q^{23} +1.00000 q^{25} +3.49396 q^{28} +9.74094 q^{29} -9.59179 q^{31} -6.18598 q^{32} +13.7017 q^{34} +2.80194 q^{35} -6.85086 q^{37} +3.15883 q^{38} -1.35690 q^{40} +1.19806 q^{41} +4.00969 q^{43} -4.35690 q^{44} +11.6136 q^{46} +2.97823 q^{47} +0.850855 q^{49} +1.80194 q^{50} -4.51573 q^{53} -3.49396 q^{55} -3.80194 q^{56} +17.5526 q^{58} +7.18598 q^{59} +4.43296 q^{61} -17.2838 q^{62} -1.26875 q^{64} -2.45712 q^{67} +9.48188 q^{68} +5.04892 q^{70} -13.5090 q^{71} +1.75302 q^{73} -12.3448 q^{74} +2.18598 q^{76} -9.78986 q^{77} +4.85623 q^{79} -4.93900 q^{80} +2.15883 q^{82} +12.0586 q^{83} +7.60388 q^{85} +7.22521 q^{86} +4.74094 q^{88} +4.12498 q^{89} +8.03684 q^{92} +5.36658 q^{94} +1.75302 q^{95} +2.32975 q^{97} +1.53319 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} - q^{4} + 3 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} - q^{4} + 3 q^{5} + 4 q^{7} + q^{10} - q^{11} + 6 q^{14} - 5 q^{16} + 14 q^{17} + 10 q^{19} - q^{20} - 5 q^{22} + 19 q^{23} + 3 q^{25} + q^{28} + 15 q^{29} - q^{31} - 4 q^{32} + 14 q^{34} + 4 q^{35} - 7 q^{37} + q^{38} + 8 q^{41} - 10 q^{43} - 9 q^{44} + 4 q^{46} + 12 q^{47} - 11 q^{49} + q^{50} - q^{53} - q^{55} - 7 q^{56} + 12 q^{58} + 7 q^{59} - 6 q^{61} - 19 q^{62} + 4 q^{64} - 26 q^{67} + 6 q^{70} - 6 q^{71} + 10 q^{73} - 14 q^{74} - 8 q^{76} - 6 q^{77} - 2 q^{79} - 5 q^{80} - 2 q^{82} + 5 q^{83} + 14 q^{85} + 20 q^{86} - 12 q^{89} - 4 q^{92} - 10 q^{94} + 10 q^{95} + 9 q^{97} + 8 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.80194 1.27416 0.637081 0.770797i \(-0.280142\pi\)
0.637081 + 0.770797i \(0.280142\pi\)
\(3\) 0 0
\(4\) 1.24698 0.623490
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.80194 1.05903 0.529516 0.848300i \(-0.322373\pi\)
0.529516 + 0.848300i \(0.322373\pi\)
\(8\) −1.35690 −0.479735
\(9\) 0 0
\(10\) 1.80194 0.569823
\(11\) −3.49396 −1.05347 −0.526734 0.850030i \(-0.676584\pi\)
−0.526734 + 0.850030i \(0.676584\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 5.04892 1.34938
\(15\) 0 0
\(16\) −4.93900 −1.23475
\(17\) 7.60388 1.84421 0.922105 0.386939i \(-0.126468\pi\)
0.922105 + 0.386939i \(0.126468\pi\)
\(18\) 0 0
\(19\) 1.75302 0.402170 0.201085 0.979574i \(-0.435553\pi\)
0.201085 + 0.979574i \(0.435553\pi\)
\(20\) 1.24698 0.278833
\(21\) 0 0
\(22\) −6.29590 −1.34229
\(23\) 6.44504 1.34388 0.671942 0.740604i \(-0.265460\pi\)
0.671942 + 0.740604i \(0.265460\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 3.49396 0.660296
\(29\) 9.74094 1.80885 0.904423 0.426636i \(-0.140301\pi\)
0.904423 + 0.426636i \(0.140301\pi\)
\(30\) 0 0
\(31\) −9.59179 −1.72274 −0.861369 0.507981i \(-0.830392\pi\)
−0.861369 + 0.507981i \(0.830392\pi\)
\(32\) −6.18598 −1.09354
\(33\) 0 0
\(34\) 13.7017 2.34982
\(35\) 2.80194 0.473614
\(36\) 0 0
\(37\) −6.85086 −1.12627 −0.563137 0.826364i \(-0.690406\pi\)
−0.563137 + 0.826364i \(0.690406\pi\)
\(38\) 3.15883 0.512430
\(39\) 0 0
\(40\) −1.35690 −0.214544
\(41\) 1.19806 0.187106 0.0935529 0.995614i \(-0.470178\pi\)
0.0935529 + 0.995614i \(0.470178\pi\)
\(42\) 0 0
\(43\) 4.00969 0.611472 0.305736 0.952116i \(-0.401098\pi\)
0.305736 + 0.952116i \(0.401098\pi\)
\(44\) −4.35690 −0.656827
\(45\) 0 0
\(46\) 11.6136 1.71233
\(47\) 2.97823 0.434419 0.217210 0.976125i \(-0.430304\pi\)
0.217210 + 0.976125i \(0.430304\pi\)
\(48\) 0 0
\(49\) 0.850855 0.121551
\(50\) 1.80194 0.254832
\(51\) 0 0
\(52\) 0 0
\(53\) −4.51573 −0.620283 −0.310142 0.950690i \(-0.600376\pi\)
−0.310142 + 0.950690i \(0.600376\pi\)
\(54\) 0 0
\(55\) −3.49396 −0.471125
\(56\) −3.80194 −0.508055
\(57\) 0 0
\(58\) 17.5526 2.30476
\(59\) 7.18598 0.935535 0.467767 0.883852i \(-0.345058\pi\)
0.467767 + 0.883852i \(0.345058\pi\)
\(60\) 0 0
\(61\) 4.43296 0.567582 0.283791 0.958886i \(-0.408408\pi\)
0.283791 + 0.958886i \(0.408408\pi\)
\(62\) −17.2838 −2.19505
\(63\) 0 0
\(64\) −1.26875 −0.158594
\(65\) 0 0
\(66\) 0 0
\(67\) −2.45712 −0.300185 −0.150093 0.988672i \(-0.547957\pi\)
−0.150093 + 0.988672i \(0.547957\pi\)
\(68\) 9.48188 1.14985
\(69\) 0 0
\(70\) 5.04892 0.603461
\(71\) −13.5090 −1.60323 −0.801613 0.597843i \(-0.796025\pi\)
−0.801613 + 0.597843i \(0.796025\pi\)
\(72\) 0 0
\(73\) 1.75302 0.205176 0.102588 0.994724i \(-0.467288\pi\)
0.102588 + 0.994724i \(0.467288\pi\)
\(74\) −12.3448 −1.43506
\(75\) 0 0
\(76\) 2.18598 0.250749
\(77\) −9.78986 −1.11566
\(78\) 0 0
\(79\) 4.85623 0.546369 0.273184 0.961962i \(-0.411923\pi\)
0.273184 + 0.961962i \(0.411923\pi\)
\(80\) −4.93900 −0.552197
\(81\) 0 0
\(82\) 2.15883 0.238403
\(83\) 12.0586 1.32360 0.661802 0.749679i \(-0.269792\pi\)
0.661802 + 0.749679i \(0.269792\pi\)
\(84\) 0 0
\(85\) 7.60388 0.824756
\(86\) 7.22521 0.779114
\(87\) 0 0
\(88\) 4.74094 0.505386
\(89\) 4.12498 0.437247 0.218624 0.975809i \(-0.429843\pi\)
0.218624 + 0.975809i \(0.429843\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8.03684 0.837898
\(93\) 0 0
\(94\) 5.36658 0.553521
\(95\) 1.75302 0.179856
\(96\) 0 0
\(97\) 2.32975 0.236550 0.118275 0.992981i \(-0.462264\pi\)
0.118275 + 0.992981i \(0.462264\pi\)
\(98\) 1.53319 0.154875
\(99\) 0 0
\(100\) 1.24698 0.124698
\(101\) 11.4058 1.13492 0.567460 0.823401i \(-0.307926\pi\)
0.567460 + 0.823401i \(0.307926\pi\)
\(102\) 0 0
\(103\) −5.46011 −0.538000 −0.269000 0.963140i \(-0.586693\pi\)
−0.269000 + 0.963140i \(0.586693\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −8.13706 −0.790341
\(107\) 13.9879 1.35226 0.676132 0.736781i \(-0.263655\pi\)
0.676132 + 0.736781i \(0.263655\pi\)
\(108\) 0 0
\(109\) 8.21983 0.787317 0.393659 0.919257i \(-0.371209\pi\)
0.393659 + 0.919257i \(0.371209\pi\)
\(110\) −6.29590 −0.600290
\(111\) 0 0
\(112\) −13.8388 −1.30764
\(113\) 1.85325 0.174339 0.0871695 0.996193i \(-0.472218\pi\)
0.0871695 + 0.996193i \(0.472218\pi\)
\(114\) 0 0
\(115\) 6.44504 0.601003
\(116\) 12.1468 1.12780
\(117\) 0 0
\(118\) 12.9487 1.19202
\(119\) 21.3056 1.95308
\(120\) 0 0
\(121\) 1.20775 0.109796
\(122\) 7.98792 0.723192
\(123\) 0 0
\(124\) −11.9608 −1.07411
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 9.24027 0.819942 0.409971 0.912099i \(-0.365539\pi\)
0.409971 + 0.912099i \(0.365539\pi\)
\(128\) 10.0858 0.891463
\(129\) 0 0
\(130\) 0 0
\(131\) 12.4886 1.09113 0.545566 0.838068i \(-0.316315\pi\)
0.545566 + 0.838068i \(0.316315\pi\)
\(132\) 0 0
\(133\) 4.91185 0.425912
\(134\) −4.42758 −0.382485
\(135\) 0 0
\(136\) −10.3177 −0.884733
\(137\) 1.10454 0.0943672 0.0471836 0.998886i \(-0.484975\pi\)
0.0471836 + 0.998886i \(0.484975\pi\)
\(138\) 0 0
\(139\) −10.1903 −0.864329 −0.432165 0.901795i \(-0.642250\pi\)
−0.432165 + 0.901795i \(0.642250\pi\)
\(140\) 3.49396 0.295293
\(141\) 0 0
\(142\) −24.3424 −2.04277
\(143\) 0 0
\(144\) 0 0
\(145\) 9.74094 0.808941
\(146\) 3.15883 0.261427
\(147\) 0 0
\(148\) −8.54288 −0.702220
\(149\) −19.2403 −1.57622 −0.788112 0.615531i \(-0.788941\pi\)
−0.788112 + 0.615531i \(0.788941\pi\)
\(150\) 0 0
\(151\) 20.8267 1.69485 0.847426 0.530913i \(-0.178151\pi\)
0.847426 + 0.530913i \(0.178151\pi\)
\(152\) −2.37867 −0.192935
\(153\) 0 0
\(154\) −17.6407 −1.42153
\(155\) −9.59179 −0.770431
\(156\) 0 0
\(157\) −23.6722 −1.88924 −0.944622 0.328159i \(-0.893572\pi\)
−0.944622 + 0.328159i \(0.893572\pi\)
\(158\) 8.75063 0.696163
\(159\) 0 0
\(160\) −6.18598 −0.489045
\(161\) 18.0586 1.42322
\(162\) 0 0
\(163\) 6.79954 0.532581 0.266291 0.963893i \(-0.414202\pi\)
0.266291 + 0.963893i \(0.414202\pi\)
\(164\) 1.49396 0.116659
\(165\) 0 0
\(166\) 21.7289 1.68649
\(167\) 13.0194 1.00747 0.503735 0.863858i \(-0.331959\pi\)
0.503735 + 0.863858i \(0.331959\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 13.7017 1.05087
\(171\) 0 0
\(172\) 5.00000 0.381246
\(173\) −21.5459 −1.63810 −0.819051 0.573721i \(-0.805499\pi\)
−0.819051 + 0.573721i \(0.805499\pi\)
\(174\) 0 0
\(175\) 2.80194 0.211807
\(176\) 17.2567 1.30077
\(177\) 0 0
\(178\) 7.43296 0.557124
\(179\) 18.9051 1.41304 0.706519 0.707694i \(-0.250265\pi\)
0.706519 + 0.707694i \(0.250265\pi\)
\(180\) 0 0
\(181\) 12.9487 0.962469 0.481234 0.876592i \(-0.340189\pi\)
0.481234 + 0.876592i \(0.340189\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −8.74525 −0.644708
\(185\) −6.85086 −0.503685
\(186\) 0 0
\(187\) −26.5676 −1.94282
\(188\) 3.71379 0.270856
\(189\) 0 0
\(190\) 3.15883 0.229166
\(191\) 0.660563 0.0477966 0.0238983 0.999714i \(-0.492392\pi\)
0.0238983 + 0.999714i \(0.492392\pi\)
\(192\) 0 0
\(193\) 0.0325239 0.00234112 0.00117056 0.999999i \(-0.499627\pi\)
0.00117056 + 0.999999i \(0.499627\pi\)
\(194\) 4.19806 0.301403
\(195\) 0 0
\(196\) 1.06100 0.0757856
\(197\) −14.4426 −1.02900 −0.514498 0.857492i \(-0.672022\pi\)
−0.514498 + 0.857492i \(0.672022\pi\)
\(198\) 0 0
\(199\) −2.06398 −0.146312 −0.0731559 0.997321i \(-0.523307\pi\)
−0.0731559 + 0.997321i \(0.523307\pi\)
\(200\) −1.35690 −0.0959470
\(201\) 0 0
\(202\) 20.5526 1.44607
\(203\) 27.2935 1.91563
\(204\) 0 0
\(205\) 1.19806 0.0836763
\(206\) −9.83877 −0.685500
\(207\) 0 0
\(208\) 0 0
\(209\) −6.12498 −0.423674
\(210\) 0 0
\(211\) 23.5080 1.61835 0.809177 0.587564i \(-0.199913\pi\)
0.809177 + 0.587564i \(0.199913\pi\)
\(212\) −5.63102 −0.386740
\(213\) 0 0
\(214\) 25.2054 1.72300
\(215\) 4.00969 0.273458
\(216\) 0 0
\(217\) −26.8756 −1.82444
\(218\) 14.8116 1.00317
\(219\) 0 0
\(220\) −4.35690 −0.293742
\(221\) 0 0
\(222\) 0 0
\(223\) −12.6213 −0.845187 −0.422594 0.906319i \(-0.638880\pi\)
−0.422594 + 0.906319i \(0.638880\pi\)
\(224\) −17.3327 −1.15809
\(225\) 0 0
\(226\) 3.33944 0.222136
\(227\) 23.4088 1.55370 0.776848 0.629688i \(-0.216817\pi\)
0.776848 + 0.629688i \(0.216817\pi\)
\(228\) 0 0
\(229\) 23.0640 1.52411 0.762055 0.647512i \(-0.224190\pi\)
0.762055 + 0.647512i \(0.224190\pi\)
\(230\) 11.6136 0.765776
\(231\) 0 0
\(232\) −13.2174 −0.867767
\(233\) −0.945706 −0.0619553 −0.0309776 0.999520i \(-0.509862\pi\)
−0.0309776 + 0.999520i \(0.509862\pi\)
\(234\) 0 0
\(235\) 2.97823 0.194278
\(236\) 8.96077 0.583297
\(237\) 0 0
\(238\) 38.3913 2.48854
\(239\) −20.2567 −1.31029 −0.655147 0.755501i \(-0.727394\pi\)
−0.655147 + 0.755501i \(0.727394\pi\)
\(240\) 0 0
\(241\) −7.75840 −0.499762 −0.249881 0.968277i \(-0.580392\pi\)
−0.249881 + 0.968277i \(0.580392\pi\)
\(242\) 2.17629 0.139897
\(243\) 0 0
\(244\) 5.52781 0.353882
\(245\) 0.850855 0.0543591
\(246\) 0 0
\(247\) 0 0
\(248\) 13.0151 0.826457
\(249\) 0 0
\(250\) 1.80194 0.113965
\(251\) 10.1021 0.637642 0.318821 0.947815i \(-0.396713\pi\)
0.318821 + 0.947815i \(0.396713\pi\)
\(252\) 0 0
\(253\) −22.5187 −1.41574
\(254\) 16.6504 1.04474
\(255\) 0 0
\(256\) 20.7114 1.29446
\(257\) 23.9801 1.49584 0.747920 0.663789i \(-0.231053\pi\)
0.747920 + 0.663789i \(0.231053\pi\)
\(258\) 0 0
\(259\) −19.1957 −1.19276
\(260\) 0 0
\(261\) 0 0
\(262\) 22.5036 1.39028
\(263\) −31.9028 −1.96721 −0.983604 0.180341i \(-0.942280\pi\)
−0.983604 + 0.180341i \(0.942280\pi\)
\(264\) 0 0
\(265\) −4.51573 −0.277399
\(266\) 8.85086 0.542681
\(267\) 0 0
\(268\) −3.06398 −0.187163
\(269\) −14.4004 −0.878010 −0.439005 0.898485i \(-0.644669\pi\)
−0.439005 + 0.898485i \(0.644669\pi\)
\(270\) 0 0
\(271\) −2.22414 −0.135107 −0.0675536 0.997716i \(-0.521519\pi\)
−0.0675536 + 0.997716i \(0.521519\pi\)
\(272\) −37.5555 −2.27714
\(273\) 0 0
\(274\) 1.99031 0.120239
\(275\) −3.49396 −0.210694
\(276\) 0 0
\(277\) −24.8853 −1.49521 −0.747606 0.664142i \(-0.768797\pi\)
−0.747606 + 0.664142i \(0.768797\pi\)
\(278\) −18.3623 −1.10130
\(279\) 0 0
\(280\) −3.80194 −0.227209
\(281\) −13.8834 −0.828213 −0.414106 0.910228i \(-0.635906\pi\)
−0.414106 + 0.910228i \(0.635906\pi\)
\(282\) 0 0
\(283\) 1.44504 0.0858988 0.0429494 0.999077i \(-0.486325\pi\)
0.0429494 + 0.999077i \(0.486325\pi\)
\(284\) −16.8455 −0.999595
\(285\) 0 0
\(286\) 0 0
\(287\) 3.35690 0.198151
\(288\) 0 0
\(289\) 40.8189 2.40111
\(290\) 17.5526 1.03072
\(291\) 0 0
\(292\) 2.18598 0.127925
\(293\) 11.2741 0.658642 0.329321 0.944218i \(-0.393180\pi\)
0.329321 + 0.944218i \(0.393180\pi\)
\(294\) 0 0
\(295\) 7.18598 0.418384
\(296\) 9.29590 0.540313
\(297\) 0 0
\(298\) −34.6698 −2.00837
\(299\) 0 0
\(300\) 0 0
\(301\) 11.2349 0.647569
\(302\) 37.5284 2.15952
\(303\) 0 0
\(304\) −8.65817 −0.496580
\(305\) 4.43296 0.253831
\(306\) 0 0
\(307\) 3.90754 0.223015 0.111508 0.993764i \(-0.464432\pi\)
0.111508 + 0.993764i \(0.464432\pi\)
\(308\) −12.2078 −0.695601
\(309\) 0 0
\(310\) −17.2838 −0.981655
\(311\) 14.1347 0.801504 0.400752 0.916187i \(-0.368749\pi\)
0.400752 + 0.916187i \(0.368749\pi\)
\(312\) 0 0
\(313\) 14.4470 0.816591 0.408295 0.912850i \(-0.366123\pi\)
0.408295 + 0.912850i \(0.366123\pi\)
\(314\) −42.6558 −2.40720
\(315\) 0 0
\(316\) 6.05562 0.340655
\(317\) −4.28919 −0.240905 −0.120453 0.992719i \(-0.538435\pi\)
−0.120453 + 0.992719i \(0.538435\pi\)
\(318\) 0 0
\(319\) −34.0344 −1.90556
\(320\) −1.26875 −0.0709253
\(321\) 0 0
\(322\) 32.5405 1.81341
\(323\) 13.3297 0.741687
\(324\) 0 0
\(325\) 0 0
\(326\) 12.2524 0.678595
\(327\) 0 0
\(328\) −1.62565 −0.0897613
\(329\) 8.34481 0.460065
\(330\) 0 0
\(331\) −24.9420 −1.37094 −0.685468 0.728103i \(-0.740402\pi\)
−0.685468 + 0.728103i \(0.740402\pi\)
\(332\) 15.0368 0.825254
\(333\) 0 0
\(334\) 23.4601 1.28368
\(335\) −2.45712 −0.134247
\(336\) 0 0
\(337\) −30.1226 −1.64088 −0.820441 0.571731i \(-0.806272\pi\)
−0.820441 + 0.571731i \(0.806272\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 9.48188 0.514227
\(341\) 33.5133 1.81485
\(342\) 0 0
\(343\) −17.2295 −0.930307
\(344\) −5.44073 −0.293345
\(345\) 0 0
\(346\) −38.8243 −2.08721
\(347\) 21.3230 1.14468 0.572340 0.820016i \(-0.306036\pi\)
0.572340 + 0.820016i \(0.306036\pi\)
\(348\) 0 0
\(349\) 27.2218 1.45715 0.728573 0.684968i \(-0.240184\pi\)
0.728573 + 0.684968i \(0.240184\pi\)
\(350\) 5.04892 0.269876
\(351\) 0 0
\(352\) 21.6136 1.15201
\(353\) −12.0871 −0.643330 −0.321665 0.946853i \(-0.604243\pi\)
−0.321665 + 0.946853i \(0.604243\pi\)
\(354\) 0 0
\(355\) −13.5090 −0.716985
\(356\) 5.14377 0.272619
\(357\) 0 0
\(358\) 34.0659 1.80044
\(359\) −4.21014 −0.222203 −0.111101 0.993809i \(-0.535438\pi\)
−0.111101 + 0.993809i \(0.535438\pi\)
\(360\) 0 0
\(361\) −15.9269 −0.838259
\(362\) 23.3327 1.22634
\(363\) 0 0
\(364\) 0 0
\(365\) 1.75302 0.0917573
\(366\) 0 0
\(367\) −26.9071 −1.40454 −0.702269 0.711912i \(-0.747830\pi\)
−0.702269 + 0.711912i \(0.747830\pi\)
\(368\) −31.8321 −1.65936
\(369\) 0 0
\(370\) −12.3448 −0.641776
\(371\) −12.6528 −0.656900
\(372\) 0 0
\(373\) −28.8461 −1.49359 −0.746796 0.665053i \(-0.768409\pi\)
−0.746796 + 0.665053i \(0.768409\pi\)
\(374\) −47.8732 −2.47547
\(375\) 0 0
\(376\) −4.04115 −0.208406
\(377\) 0 0
\(378\) 0 0
\(379\) −8.57135 −0.440281 −0.220140 0.975468i \(-0.570652\pi\)
−0.220140 + 0.975468i \(0.570652\pi\)
\(380\) 2.18598 0.112138
\(381\) 0 0
\(382\) 1.19029 0.0609007
\(383\) −7.28382 −0.372186 −0.186093 0.982532i \(-0.559583\pi\)
−0.186093 + 0.982532i \(0.559583\pi\)
\(384\) 0 0
\(385\) −9.78986 −0.498937
\(386\) 0.0586060 0.00298297
\(387\) 0 0
\(388\) 2.90515 0.147487
\(389\) −15.5840 −0.790141 −0.395071 0.918651i \(-0.629280\pi\)
−0.395071 + 0.918651i \(0.629280\pi\)
\(390\) 0 0
\(391\) 49.0073 2.47841
\(392\) −1.15452 −0.0583122
\(393\) 0 0
\(394\) −26.0248 −1.31111
\(395\) 4.85623 0.244344
\(396\) 0 0
\(397\) −32.0538 −1.60874 −0.804368 0.594132i \(-0.797496\pi\)
−0.804368 + 0.594132i \(0.797496\pi\)
\(398\) −3.71917 −0.186425
\(399\) 0 0
\(400\) −4.93900 −0.246950
\(401\) −14.0073 −0.699491 −0.349745 0.936845i \(-0.613732\pi\)
−0.349745 + 0.936845i \(0.613732\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 14.2228 0.707612
\(405\) 0 0
\(406\) 49.1812 2.44082
\(407\) 23.9366 1.18649
\(408\) 0 0
\(409\) 20.0707 0.992432 0.496216 0.868199i \(-0.334722\pi\)
0.496216 + 0.868199i \(0.334722\pi\)
\(410\) 2.15883 0.106617
\(411\) 0 0
\(412\) −6.80864 −0.335438
\(413\) 20.1347 0.990762
\(414\) 0 0
\(415\) 12.0586 0.591934
\(416\) 0 0
\(417\) 0 0
\(418\) −11.0368 −0.539829
\(419\) −36.1299 −1.76506 −0.882530 0.470256i \(-0.844162\pi\)
−0.882530 + 0.470256i \(0.844162\pi\)
\(420\) 0 0
\(421\) −0.708415 −0.0345260 −0.0172630 0.999851i \(-0.505495\pi\)
−0.0172630 + 0.999851i \(0.505495\pi\)
\(422\) 42.3599 2.06205
\(423\) 0 0
\(424\) 6.12737 0.297572
\(425\) 7.60388 0.368842
\(426\) 0 0
\(427\) 12.4209 0.601088
\(428\) 17.4426 0.843122
\(429\) 0 0
\(430\) 7.22521 0.348431
\(431\) 23.6286 1.13815 0.569076 0.822285i \(-0.307301\pi\)
0.569076 + 0.822285i \(0.307301\pi\)
\(432\) 0 0
\(433\) −10.6450 −0.511567 −0.255784 0.966734i \(-0.582333\pi\)
−0.255784 + 0.966734i \(0.582333\pi\)
\(434\) −48.4282 −2.32463
\(435\) 0 0
\(436\) 10.2500 0.490884
\(437\) 11.2983 0.540470
\(438\) 0 0
\(439\) 11.4832 0.548064 0.274032 0.961721i \(-0.411643\pi\)
0.274032 + 0.961721i \(0.411643\pi\)
\(440\) 4.74094 0.226015
\(441\) 0 0
\(442\) 0 0
\(443\) 3.73423 0.177419 0.0887094 0.996058i \(-0.471726\pi\)
0.0887094 + 0.996058i \(0.471726\pi\)
\(444\) 0 0
\(445\) 4.12498 0.195543
\(446\) −22.7429 −1.07691
\(447\) 0 0
\(448\) −3.55496 −0.167956
\(449\) −4.52052 −0.213336 −0.106668 0.994295i \(-0.534018\pi\)
−0.106668 + 0.994295i \(0.534018\pi\)
\(450\) 0 0
\(451\) −4.18598 −0.197110
\(452\) 2.31096 0.108699
\(453\) 0 0
\(454\) 42.1812 1.97966
\(455\) 0 0
\(456\) 0 0
\(457\) 16.3860 0.766503 0.383251 0.923644i \(-0.374804\pi\)
0.383251 + 0.923644i \(0.374804\pi\)
\(458\) 41.5599 1.94196
\(459\) 0 0
\(460\) 8.03684 0.374719
\(461\) −13.8769 −0.646313 −0.323157 0.946345i \(-0.604744\pi\)
−0.323157 + 0.946345i \(0.604744\pi\)
\(462\) 0 0
\(463\) −22.9530 −1.06672 −0.533358 0.845889i \(-0.679070\pi\)
−0.533358 + 0.845889i \(0.679070\pi\)
\(464\) −48.1105 −2.23347
\(465\) 0 0
\(466\) −1.70410 −0.0789410
\(467\) 5.15346 0.238474 0.119237 0.992866i \(-0.461955\pi\)
0.119237 + 0.992866i \(0.461955\pi\)
\(468\) 0 0
\(469\) −6.88471 −0.317906
\(470\) 5.36658 0.247542
\(471\) 0 0
\(472\) −9.75063 −0.448809
\(473\) −14.0097 −0.644166
\(474\) 0 0
\(475\) 1.75302 0.0804341
\(476\) 26.5676 1.21773
\(477\) 0 0
\(478\) −36.5013 −1.66953
\(479\) 16.2252 0.741349 0.370674 0.928763i \(-0.379127\pi\)
0.370674 + 0.928763i \(0.379127\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −13.9801 −0.636778
\(483\) 0 0
\(484\) 1.50604 0.0684564
\(485\) 2.32975 0.105788
\(486\) 0 0
\(487\) 32.2543 1.46158 0.730790 0.682602i \(-0.239152\pi\)
0.730790 + 0.682602i \(0.239152\pi\)
\(488\) −6.01507 −0.272289
\(489\) 0 0
\(490\) 1.53319 0.0692624
\(491\) −12.1497 −0.548310 −0.274155 0.961686i \(-0.588398\pi\)
−0.274155 + 0.961686i \(0.588398\pi\)
\(492\) 0 0
\(493\) 74.0689 3.33589
\(494\) 0 0
\(495\) 0 0
\(496\) 47.3739 2.12715
\(497\) −37.8514 −1.69787
\(498\) 0 0
\(499\) −30.9202 −1.38418 −0.692089 0.721812i \(-0.743310\pi\)
−0.692089 + 0.721812i \(0.743310\pi\)
\(500\) 1.24698 0.0557666
\(501\) 0 0
\(502\) 18.2034 0.812459
\(503\) −22.0737 −0.984216 −0.492108 0.870534i \(-0.663774\pi\)
−0.492108 + 0.870534i \(0.663774\pi\)
\(504\) 0 0
\(505\) 11.4058 0.507552
\(506\) −40.5773 −1.80388
\(507\) 0 0
\(508\) 11.5224 0.511225
\(509\) 1.64071 0.0727232 0.0363616 0.999339i \(-0.488423\pi\)
0.0363616 + 0.999339i \(0.488423\pi\)
\(510\) 0 0
\(511\) 4.91185 0.217288
\(512\) 17.1491 0.757892
\(513\) 0 0
\(514\) 43.2107 1.90594
\(515\) −5.46011 −0.240601
\(516\) 0 0
\(517\) −10.4058 −0.457647
\(518\) −34.5894 −1.51977
\(519\) 0 0
\(520\) 0 0
\(521\) −34.1390 −1.49566 −0.747828 0.663893i \(-0.768903\pi\)
−0.747828 + 0.663893i \(0.768903\pi\)
\(522\) 0 0
\(523\) 12.5104 0.547040 0.273520 0.961866i \(-0.411812\pi\)
0.273520 + 0.961866i \(0.411812\pi\)
\(524\) 15.5730 0.680310
\(525\) 0 0
\(526\) −57.4868 −2.50654
\(527\) −72.9348 −3.17709
\(528\) 0 0
\(529\) 18.5386 0.806025
\(530\) −8.13706 −0.353451
\(531\) 0 0
\(532\) 6.12498 0.265552
\(533\) 0 0
\(534\) 0 0
\(535\) 13.9879 0.604750
\(536\) 3.33406 0.144009
\(537\) 0 0
\(538\) −25.9487 −1.11873
\(539\) −2.97285 −0.128050
\(540\) 0 0
\(541\) −29.5254 −1.26940 −0.634698 0.772760i \(-0.718876\pi\)
−0.634698 + 0.772760i \(0.718876\pi\)
\(542\) −4.00777 −0.172148
\(543\) 0 0
\(544\) −47.0374 −2.01671
\(545\) 8.21983 0.352099
\(546\) 0 0
\(547\) −6.07367 −0.259691 −0.129846 0.991534i \(-0.541448\pi\)
−0.129846 + 0.991534i \(0.541448\pi\)
\(548\) 1.37734 0.0588370
\(549\) 0 0
\(550\) −6.29590 −0.268458
\(551\) 17.0761 0.727465
\(552\) 0 0
\(553\) 13.6069 0.578623
\(554\) −44.8418 −1.90514
\(555\) 0 0
\(556\) −12.7071 −0.538901
\(557\) −20.0291 −0.848659 −0.424329 0.905508i \(-0.639490\pi\)
−0.424329 + 0.905508i \(0.639490\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −13.8388 −0.584795
\(561\) 0 0
\(562\) −25.0170 −1.05528
\(563\) 7.47219 0.314915 0.157458 0.987526i \(-0.449670\pi\)
0.157458 + 0.987526i \(0.449670\pi\)
\(564\) 0 0
\(565\) 1.85325 0.0779667
\(566\) 2.60388 0.109449
\(567\) 0 0
\(568\) 18.3303 0.769124
\(569\) −0.428911 −0.0179809 −0.00899045 0.999960i \(-0.502862\pi\)
−0.00899045 + 0.999960i \(0.502862\pi\)
\(570\) 0 0
\(571\) −0.00298391 −0.000124873 0 −6.24363e−5 1.00000i \(-0.500020\pi\)
−6.24363e−5 1.00000i \(0.500020\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 6.04892 0.252477
\(575\) 6.44504 0.268777
\(576\) 0 0
\(577\) −14.5743 −0.606738 −0.303369 0.952873i \(-0.598111\pi\)
−0.303369 + 0.952873i \(0.598111\pi\)
\(578\) 73.5532 3.05941
\(579\) 0 0
\(580\) 12.1468 0.504366
\(581\) 33.7875 1.40174
\(582\) 0 0
\(583\) 15.7778 0.653449
\(584\) −2.37867 −0.0984299
\(585\) 0 0
\(586\) 20.3153 0.839216
\(587\) −20.6329 −0.851613 −0.425806 0.904814i \(-0.640010\pi\)
−0.425806 + 0.904814i \(0.640010\pi\)
\(588\) 0 0
\(589\) −16.8146 −0.692834
\(590\) 12.9487 0.533089
\(591\) 0 0
\(592\) 33.8364 1.39067
\(593\) 30.0901 1.23565 0.617825 0.786315i \(-0.288014\pi\)
0.617825 + 0.786315i \(0.288014\pi\)
\(594\) 0 0
\(595\) 21.3056 0.873444
\(596\) −23.9922 −0.982760
\(597\) 0 0
\(598\) 0 0
\(599\) 2.72481 0.111333 0.0556663 0.998449i \(-0.482272\pi\)
0.0556663 + 0.998449i \(0.482272\pi\)
\(600\) 0 0
\(601\) −29.9584 −1.22203 −0.611014 0.791620i \(-0.709238\pi\)
−0.611014 + 0.791620i \(0.709238\pi\)
\(602\) 20.2446 0.825108
\(603\) 0 0
\(604\) 25.9705 1.05672
\(605\) 1.20775 0.0491021
\(606\) 0 0
\(607\) 35.1116 1.42513 0.712567 0.701604i \(-0.247532\pi\)
0.712567 + 0.701604i \(0.247532\pi\)
\(608\) −10.8442 −0.439788
\(609\) 0 0
\(610\) 7.98792 0.323421
\(611\) 0 0
\(612\) 0 0
\(613\) −41.4142 −1.67270 −0.836351 0.548194i \(-0.815316\pi\)
−0.836351 + 0.548194i \(0.815316\pi\)
\(614\) 7.04115 0.284158
\(615\) 0 0
\(616\) 13.2838 0.535220
\(617\) 21.1575 0.851769 0.425885 0.904778i \(-0.359963\pi\)
0.425885 + 0.904778i \(0.359963\pi\)
\(618\) 0 0
\(619\) 7.39134 0.297083 0.148541 0.988906i \(-0.452542\pi\)
0.148541 + 0.988906i \(0.452542\pi\)
\(620\) −11.9608 −0.480356
\(621\) 0 0
\(622\) 25.4698 1.02125
\(623\) 11.5579 0.463059
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 26.0325 1.04047
\(627\) 0 0
\(628\) −29.5187 −1.17792
\(629\) −52.0930 −2.07709
\(630\) 0 0
\(631\) −3.16660 −0.126060 −0.0630302 0.998012i \(-0.520076\pi\)
−0.0630302 + 0.998012i \(0.520076\pi\)
\(632\) −6.58940 −0.262112
\(633\) 0 0
\(634\) −7.72886 −0.306952
\(635\) 9.24027 0.366689
\(636\) 0 0
\(637\) 0 0
\(638\) −61.3279 −2.42800
\(639\) 0 0
\(640\) 10.0858 0.398674
\(641\) 9.94762 0.392908 0.196454 0.980513i \(-0.437057\pi\)
0.196454 + 0.980513i \(0.437057\pi\)
\(642\) 0 0
\(643\) 28.7211 1.13265 0.566325 0.824182i \(-0.308365\pi\)
0.566325 + 0.824182i \(0.308365\pi\)
\(644\) 22.5187 0.887362
\(645\) 0 0
\(646\) 24.0194 0.945030
\(647\) 9.03013 0.355011 0.177506 0.984120i \(-0.443197\pi\)
0.177506 + 0.984120i \(0.443197\pi\)
\(648\) 0 0
\(649\) −25.1075 −0.985557
\(650\) 0 0
\(651\) 0 0
\(652\) 8.47889 0.332059
\(653\) −8.55496 −0.334781 −0.167391 0.985891i \(-0.553534\pi\)
−0.167391 + 0.985891i \(0.553534\pi\)
\(654\) 0 0
\(655\) 12.4886 0.487969
\(656\) −5.91723 −0.231029
\(657\) 0 0
\(658\) 15.0368 0.586197
\(659\) 24.0847 0.938206 0.469103 0.883143i \(-0.344577\pi\)
0.469103 + 0.883143i \(0.344577\pi\)
\(660\) 0 0
\(661\) 25.5646 0.994350 0.497175 0.867650i \(-0.334371\pi\)
0.497175 + 0.867650i \(0.334371\pi\)
\(662\) −44.9439 −1.74679
\(663\) 0 0
\(664\) −16.3623 −0.634979
\(665\) 4.91185 0.190474
\(666\) 0 0
\(667\) 62.7808 2.43088
\(668\) 16.2349 0.628147
\(669\) 0 0
\(670\) −4.42758 −0.171052
\(671\) −15.4886 −0.597930
\(672\) 0 0
\(673\) 29.9681 1.15518 0.577592 0.816326i \(-0.303992\pi\)
0.577592 + 0.816326i \(0.303992\pi\)
\(674\) −54.2790 −2.09075
\(675\) 0 0
\(676\) 0 0
\(677\) −0.0736715 −0.00283143 −0.00141571 0.999999i \(-0.500451\pi\)
−0.00141571 + 0.999999i \(0.500451\pi\)
\(678\) 0 0
\(679\) 6.52781 0.250514
\(680\) −10.3177 −0.395664
\(681\) 0 0
\(682\) 60.3889 2.31241
\(683\) 3.58078 0.137015 0.0685073 0.997651i \(-0.478176\pi\)
0.0685073 + 0.997651i \(0.478176\pi\)
\(684\) 0 0
\(685\) 1.10454 0.0422023
\(686\) −31.0465 −1.18536
\(687\) 0 0
\(688\) −19.8039 −0.755015
\(689\) 0 0
\(690\) 0 0
\(691\) −19.1454 −0.728326 −0.364163 0.931335i \(-0.618645\pi\)
−0.364163 + 0.931335i \(0.618645\pi\)
\(692\) −26.8672 −1.02134
\(693\) 0 0
\(694\) 38.4228 1.45851
\(695\) −10.1903 −0.386540
\(696\) 0 0
\(697\) 9.10992 0.345063
\(698\) 49.0519 1.85664
\(699\) 0 0
\(700\) 3.49396 0.132059
\(701\) 18.3230 0.692052 0.346026 0.938225i \(-0.387531\pi\)
0.346026 + 0.938225i \(0.387531\pi\)
\(702\) 0 0
\(703\) −12.0097 −0.452954
\(704\) 4.43296 0.167073
\(705\) 0 0
\(706\) −21.7802 −0.819707
\(707\) 31.9584 1.20192
\(708\) 0 0
\(709\) 18.5084 0.695099 0.347549 0.937662i \(-0.387014\pi\)
0.347549 + 0.937662i \(0.387014\pi\)
\(710\) −24.3424 −0.913555
\(711\) 0 0
\(712\) −5.59717 −0.209763
\(713\) −61.8195 −2.31516
\(714\) 0 0
\(715\) 0 0
\(716\) 23.5743 0.881014
\(717\) 0 0
\(718\) −7.58642 −0.283123
\(719\) 14.8006 0.551970 0.275985 0.961162i \(-0.410996\pi\)
0.275985 + 0.961162i \(0.410996\pi\)
\(720\) 0 0
\(721\) −15.2989 −0.569760
\(722\) −28.6993 −1.06808
\(723\) 0 0
\(724\) 16.1468 0.600089
\(725\) 9.74094 0.361769
\(726\) 0 0
\(727\) 4.76749 0.176817 0.0884083 0.996084i \(-0.471822\pi\)
0.0884083 + 0.996084i \(0.471822\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 3.15883 0.116914
\(731\) 30.4892 1.12768
\(732\) 0 0
\(733\) −18.2416 −0.673769 −0.336884 0.941546i \(-0.609373\pi\)
−0.336884 + 0.941546i \(0.609373\pi\)
\(734\) −48.4849 −1.78961
\(735\) 0 0
\(736\) −39.8689 −1.46959
\(737\) 8.58509 0.316236
\(738\) 0 0
\(739\) −16.2218 −0.596727 −0.298363 0.954452i \(-0.596441\pi\)
−0.298363 + 0.954452i \(0.596441\pi\)
\(740\) −8.54288 −0.314042
\(741\) 0 0
\(742\) −22.7995 −0.836998
\(743\) 4.19375 0.153854 0.0769269 0.997037i \(-0.475489\pi\)
0.0769269 + 0.997037i \(0.475489\pi\)
\(744\) 0 0
\(745\) −19.2403 −0.704909
\(746\) −51.9788 −1.90308
\(747\) 0 0
\(748\) −33.1293 −1.21133
\(749\) 39.1933 1.43209
\(750\) 0 0
\(751\) −9.43727 −0.344371 −0.172185 0.985065i \(-0.555083\pi\)
−0.172185 + 0.985065i \(0.555083\pi\)
\(752\) −14.7095 −0.536400
\(753\) 0 0
\(754\) 0 0
\(755\) 20.8267 0.757961
\(756\) 0 0
\(757\) 8.80061 0.319864 0.159932 0.987128i \(-0.448873\pi\)
0.159932 + 0.987128i \(0.448873\pi\)
\(758\) −15.4450 −0.560989
\(759\) 0 0
\(760\) −2.37867 −0.0862833
\(761\) −44.4728 −1.61214 −0.806069 0.591822i \(-0.798409\pi\)
−0.806069 + 0.591822i \(0.798409\pi\)
\(762\) 0 0
\(763\) 23.0315 0.833795
\(764\) 0.823708 0.0298007
\(765\) 0 0
\(766\) −13.1250 −0.474225
\(767\) 0 0
\(768\) 0 0
\(769\) 0.521106 0.0187916 0.00939579 0.999956i \(-0.497009\pi\)
0.00939579 + 0.999956i \(0.497009\pi\)
\(770\) −17.6407 −0.635727
\(771\) 0 0
\(772\) 0.0405566 0.00145966
\(773\) 10.3690 0.372946 0.186473 0.982460i \(-0.440294\pi\)
0.186473 + 0.982460i \(0.440294\pi\)
\(774\) 0 0
\(775\) −9.59179 −0.344547
\(776\) −3.16123 −0.113481
\(777\) 0 0
\(778\) −28.0814 −1.00677
\(779\) 2.10023 0.0752485
\(780\) 0 0
\(781\) 47.2000 1.68895
\(782\) 88.3081 3.15789
\(783\) 0 0
\(784\) −4.20237 −0.150085
\(785\) −23.6722 −0.844896
\(786\) 0 0
\(787\) −23.1444 −0.825007 −0.412504 0.910956i \(-0.635346\pi\)
−0.412504 + 0.910956i \(0.635346\pi\)
\(788\) −18.0097 −0.641569
\(789\) 0 0
\(790\) 8.75063 0.311333
\(791\) 5.19269 0.184631
\(792\) 0 0
\(793\) 0 0
\(794\) −57.7590 −2.04979
\(795\) 0 0
\(796\) −2.57374 −0.0912240
\(797\) −34.6292 −1.22663 −0.613315 0.789838i \(-0.710164\pi\)
−0.613315 + 0.789838i \(0.710164\pi\)
\(798\) 0 0
\(799\) 22.6461 0.801161
\(800\) −6.18598 −0.218707
\(801\) 0 0
\(802\) −25.2403 −0.891265
\(803\) −6.12498 −0.216146
\(804\) 0 0
\(805\) 18.0586 0.636482
\(806\) 0 0
\(807\) 0 0
\(808\) −15.4765 −0.544461
\(809\) 42.8082 1.50506 0.752528 0.658561i \(-0.228834\pi\)
0.752528 + 0.658561i \(0.228834\pi\)
\(810\) 0 0
\(811\) 12.5410 0.440373 0.220186 0.975458i \(-0.429333\pi\)
0.220186 + 0.975458i \(0.429333\pi\)
\(812\) 34.0344 1.19437
\(813\) 0 0
\(814\) 43.1323 1.51179
\(815\) 6.79954 0.238178
\(816\) 0 0
\(817\) 7.02907 0.245916
\(818\) 36.1661 1.26452
\(819\) 0 0
\(820\) 1.49396 0.0521713
\(821\) −11.4324 −0.398992 −0.199496 0.979899i \(-0.563931\pi\)
−0.199496 + 0.979899i \(0.563931\pi\)
\(822\) 0 0
\(823\) 20.2349 0.705344 0.352672 0.935747i \(-0.385273\pi\)
0.352672 + 0.935747i \(0.385273\pi\)
\(824\) 7.40880 0.258098
\(825\) 0 0
\(826\) 36.2814 1.26239
\(827\) −47.0267 −1.63528 −0.817639 0.575731i \(-0.804717\pi\)
−0.817639 + 0.575731i \(0.804717\pi\)
\(828\) 0 0
\(829\) 17.8291 0.619230 0.309615 0.950862i \(-0.399800\pi\)
0.309615 + 0.950862i \(0.399800\pi\)
\(830\) 21.7289 0.754220
\(831\) 0 0
\(832\) 0 0
\(833\) 6.46980 0.224165
\(834\) 0 0
\(835\) 13.0194 0.450554
\(836\) −7.63773 −0.264156
\(837\) 0 0
\(838\) −65.1038 −2.24897
\(839\) 11.8931 0.410594 0.205297 0.978700i \(-0.434184\pi\)
0.205297 + 0.978700i \(0.434184\pi\)
\(840\) 0 0
\(841\) 65.8859 2.27193
\(842\) −1.27652 −0.0439918
\(843\) 0 0
\(844\) 29.3139 1.00903
\(845\) 0 0
\(846\) 0 0
\(847\) 3.38404 0.116277
\(848\) 22.3032 0.765895
\(849\) 0 0
\(850\) 13.7017 0.469965
\(851\) −44.1540 −1.51358
\(852\) 0 0
\(853\) 15.5749 0.533275 0.266638 0.963797i \(-0.414087\pi\)
0.266638 + 0.963797i \(0.414087\pi\)
\(854\) 22.3817 0.765884
\(855\) 0 0
\(856\) −18.9801 −0.648728
\(857\) −20.3327 −0.694553 −0.347276 0.937763i \(-0.612893\pi\)
−0.347276 + 0.937763i \(0.612893\pi\)
\(858\) 0 0
\(859\) 26.2965 0.897225 0.448612 0.893726i \(-0.351918\pi\)
0.448612 + 0.893726i \(0.351918\pi\)
\(860\) 5.00000 0.170499
\(861\) 0 0
\(862\) 42.5773 1.45019
\(863\) −28.6698 −0.975931 −0.487965 0.872863i \(-0.662261\pi\)
−0.487965 + 0.872863i \(0.662261\pi\)
\(864\) 0 0
\(865\) −21.5459 −0.732581
\(866\) −19.1817 −0.651820
\(867\) 0 0
\(868\) −33.5133 −1.13752
\(869\) −16.9675 −0.575582
\(870\) 0 0
\(871\) 0 0
\(872\) −11.1535 −0.377704
\(873\) 0 0
\(874\) 20.3588 0.688647
\(875\) 2.80194 0.0947228
\(876\) 0 0
\(877\) −10.8522 −0.366452 −0.183226 0.983071i \(-0.558654\pi\)
−0.183226 + 0.983071i \(0.558654\pi\)
\(878\) 20.6920 0.698322
\(879\) 0 0
\(880\) 17.2567 0.581722
\(881\) 47.6118 1.60408 0.802041 0.597270i \(-0.203748\pi\)
0.802041 + 0.597270i \(0.203748\pi\)
\(882\) 0 0
\(883\) −29.7469 −1.00106 −0.500532 0.865718i \(-0.666862\pi\)
−0.500532 + 0.865718i \(0.666862\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 6.72886 0.226060
\(887\) −18.1621 −0.609823 −0.304911 0.952381i \(-0.598627\pi\)
−0.304911 + 0.952381i \(0.598627\pi\)
\(888\) 0 0
\(889\) 25.8907 0.868345
\(890\) 7.43296 0.249153
\(891\) 0 0
\(892\) −15.7385 −0.526965
\(893\) 5.22090 0.174711
\(894\) 0 0
\(895\) 18.9051 0.631929
\(896\) 28.2597 0.944089
\(897\) 0 0
\(898\) −8.14569 −0.271825
\(899\) −93.4331 −3.11617
\(900\) 0 0
\(901\) −34.3370 −1.14393
\(902\) −7.54288 −0.251150
\(903\) 0 0
\(904\) −2.51466 −0.0836365
\(905\) 12.9487 0.430429
\(906\) 0 0
\(907\) −1.81641 −0.0603130 −0.0301565 0.999545i \(-0.509601\pi\)
−0.0301565 + 0.999545i \(0.509601\pi\)
\(908\) 29.1903 0.968714
\(909\) 0 0
\(910\) 0 0
\(911\) −52.5502 −1.74106 −0.870532 0.492111i \(-0.836225\pi\)
−0.870532 + 0.492111i \(0.836225\pi\)
\(912\) 0 0
\(913\) −42.1323 −1.39437
\(914\) 29.5265 0.976649
\(915\) 0 0
\(916\) 28.7603 0.950268
\(917\) 34.9922 1.15555
\(918\) 0 0
\(919\) −59.3135 −1.95657 −0.978285 0.207262i \(-0.933545\pi\)
−0.978285 + 0.207262i \(0.933545\pi\)
\(920\) −8.74525 −0.288322
\(921\) 0 0
\(922\) −25.0054 −0.823508
\(923\) 0 0
\(924\) 0 0
\(925\) −6.85086 −0.225255
\(926\) −41.3599 −1.35917
\(927\) 0 0
\(928\) −60.2573 −1.97804
\(929\) 55.0629 1.80656 0.903278 0.429056i \(-0.141154\pi\)
0.903278 + 0.429056i \(0.141154\pi\)
\(930\) 0 0
\(931\) 1.49157 0.0488841
\(932\) −1.17928 −0.0386285
\(933\) 0 0
\(934\) 9.28621 0.303854
\(935\) −26.5676 −0.868854
\(936\) 0 0
\(937\) −24.6673 −0.805844 −0.402922 0.915234i \(-0.632005\pi\)
−0.402922 + 0.915234i \(0.632005\pi\)
\(938\) −12.4058 −0.405064
\(939\) 0 0
\(940\) 3.71379 0.121131
\(941\) −48.8823 −1.59352 −0.796759 0.604297i \(-0.793454\pi\)
−0.796759 + 0.604297i \(0.793454\pi\)
\(942\) 0 0
\(943\) 7.72156 0.251449
\(944\) −35.4916 −1.15515
\(945\) 0 0
\(946\) −25.2446 −0.820772
\(947\) 10.2832 0.334160 0.167080 0.985943i \(-0.446566\pi\)
0.167080 + 0.985943i \(0.446566\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 3.15883 0.102486
\(951\) 0 0
\(952\) −28.9095 −0.936961
\(953\) −33.7458 −1.09314 −0.546568 0.837415i \(-0.684066\pi\)
−0.546568 + 0.837415i \(0.684066\pi\)
\(954\) 0 0
\(955\) 0.660563 0.0213753
\(956\) −25.2597 −0.816956
\(957\) 0 0
\(958\) 29.2368 0.944599
\(959\) 3.09485 0.0999379
\(960\) 0 0
\(961\) 61.0025 1.96782
\(962\) 0 0
\(963\) 0 0
\(964\) −9.67456 −0.311597
\(965\) 0.0325239 0.00104698
\(966\) 0 0
\(967\) −14.6517 −0.471168 −0.235584 0.971854i \(-0.575700\pi\)
−0.235584 + 0.971854i \(0.575700\pi\)
\(968\) −1.63879 −0.0526728
\(969\) 0 0
\(970\) 4.19806 0.134792
\(971\) 21.3884 0.686385 0.343192 0.939265i \(-0.388492\pi\)
0.343192 + 0.939265i \(0.388492\pi\)
\(972\) 0 0
\(973\) −28.5526 −0.915353
\(974\) 58.1202 1.86229
\(975\) 0 0
\(976\) −21.8944 −0.700823
\(977\) 58.2049 1.86214 0.931070 0.364842i \(-0.118877\pi\)
0.931070 + 0.364842i \(0.118877\pi\)
\(978\) 0 0
\(979\) −14.4125 −0.460626
\(980\) 1.06100 0.0338924
\(981\) 0 0
\(982\) −21.8931 −0.698636
\(983\) 33.1855 1.05845 0.529227 0.848481i \(-0.322482\pi\)
0.529227 + 0.848481i \(0.322482\pi\)
\(984\) 0 0
\(985\) −14.4426 −0.460181
\(986\) 133.468 4.25047
\(987\) 0 0
\(988\) 0 0
\(989\) 25.8426 0.821747
\(990\) 0 0
\(991\) −21.4166 −0.680320 −0.340160 0.940368i \(-0.610481\pi\)
−0.340160 + 0.940368i \(0.610481\pi\)
\(992\) 59.3347 1.88388
\(993\) 0 0
\(994\) −68.2059 −2.16336
\(995\) −2.06398 −0.0654327
\(996\) 0 0
\(997\) −9.76330 −0.309207 −0.154603 0.987977i \(-0.549410\pi\)
−0.154603 + 0.987977i \(0.549410\pi\)
\(998\) −55.7163 −1.76367
\(999\) 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 7605.2.a.cb.1.3 3
3.2 odd 2 2535.2.a.w.1.1 3
13.12 even 2 7605.2.a.bo.1.1 3
39.38 odd 2 2535.2.a.bf.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2535.2.a.w.1.1 3 3.2 odd 2
2535.2.a.bf.1.3 yes 3 39.38 odd 2
7605.2.a.bo.1.1 3 13.12 even 2
7605.2.a.cb.1.3 3 1.1 even 1 trivial