Properties

Label 1305.2.a.s.1.4
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $0$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.431560\) 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)
 
\(f(q)\) \(=\) \(q-0.431560 q^{2} -1.81376 q^{4} -1.00000 q^{5} -2.42505 q^{7} +1.64586 q^{8} +O(q^{10})\) \(q-0.431560 q^{2} -1.81376 q^{4} -1.00000 q^{5} -2.42505 q^{7} +1.64586 q^{8} +0.431560 q^{10} +0.00755077 q^{11} -2.73098 q^{13} +1.04655 q^{14} +2.91722 q^{16} -4.73098 q^{17} +2.74388 q^{19} +1.81376 q^{20} -0.00325861 q^{22} -3.68872 q^{23} +1.00000 q^{25} +1.17858 q^{26} +4.39845 q^{28} +1.00000 q^{29} +1.25612 q^{31} -4.55068 q^{32} +2.04170 q^{34} +2.42505 q^{35} +6.60420 q^{37} -1.18415 q^{38} -1.64586 q^{40} -0.160198 q^{41} +11.0103 q^{43} -0.0136953 q^{44} +1.59190 q^{46} -10.3736 q^{47} -1.11912 q^{49} -0.431560 q^{50} +4.95333 q^{52} +11.1099 q^{53} -0.00755077 q^{55} -3.99130 q^{56} -0.431560 q^{58} +8.85010 q^{59} +13.5237 q^{61} -0.542089 q^{62} -3.87056 q^{64} +2.73098 q^{65} +4.69881 q^{67} +8.58085 q^{68} -1.04655 q^{70} -1.48777 q^{71} +2.75695 q^{73} -2.85010 q^{74} -4.97674 q^{76} -0.0183110 q^{77} +2.72878 q^{79} -2.91722 q^{80} +0.0691351 q^{82} +7.81988 q^{83} +4.73098 q^{85} -4.75160 q^{86} +0.0124275 q^{88} -7.59619 q^{89} +6.62277 q^{91} +6.69044 q^{92} +4.47682 q^{94} -2.74388 q^{95} -7.02540 q^{97} +0.482968 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - q^{2} + 13 q^{4} - 7 q^{5} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - q^{2} + 13 q^{4} - 7 q^{5} + 10 q^{7} + q^{10} + 3 q^{11} + 6 q^{13} + 9 q^{14} + 21 q^{16} - 8 q^{17} + 10 q^{19} - 13 q^{20} + 9 q^{22} - 11 q^{23} + 7 q^{25} - 3 q^{26} + 25 q^{28} + 7 q^{29} + 18 q^{31} - q^{32} - q^{34} - 10 q^{35} + 13 q^{37} - 12 q^{38} + 13 q^{41} + 9 q^{43} + 37 q^{44} - 8 q^{46} - 2 q^{47} + 21 q^{49} - q^{50} - q^{52} - 5 q^{53} - 3 q^{55} + 30 q^{56} - q^{58} + 8 q^{59} + 14 q^{61} + 8 q^{62} + 8 q^{64} - 6 q^{65} + 14 q^{67} - 27 q^{68} - 9 q^{70} + 8 q^{71} + 3 q^{73} + 34 q^{74} + 4 q^{76} + 28 q^{77} + 4 q^{79} - 21 q^{80} - 20 q^{82} - 17 q^{83} + 8 q^{85} - 4 q^{86} + 26 q^{88} + 20 q^{89} + 12 q^{91} - 60 q^{92} - 21 q^{94} - 10 q^{95} + 13 q^{97} + 20 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.431560 −0.305159 −0.152579 0.988291i \(-0.548758\pi\)
−0.152579 + 0.988291i \(0.548758\pi\)
\(3\) 0 0
\(4\) −1.81376 −0.906878
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.42505 −0.916583 −0.458292 0.888802i \(-0.651539\pi\)
−0.458292 + 0.888802i \(0.651539\pi\)
\(8\) 1.64586 0.581901
\(9\) 0 0
\(10\) 0.431560 0.136471
\(11\) 0.00755077 0.00227664 0.00113832 0.999999i \(-0.499638\pi\)
0.00113832 + 0.999999i \(0.499638\pi\)
\(12\) 0 0
\(13\) −2.73098 −0.757438 −0.378719 0.925512i \(-0.623635\pi\)
−0.378719 + 0.925512i \(0.623635\pi\)
\(14\) 1.04655 0.279703
\(15\) 0 0
\(16\) 2.91722 0.729306
\(17\) −4.73098 −1.14743 −0.573716 0.819055i \(-0.694499\pi\)
−0.573716 + 0.819055i \(0.694499\pi\)
\(18\) 0 0
\(19\) 2.74388 0.629490 0.314745 0.949176i \(-0.398081\pi\)
0.314745 + 0.949176i \(0.398081\pi\)
\(20\) 1.81376 0.405568
\(21\) 0 0
\(22\) −0.00325861 −0.000694738 0
\(23\) −3.68872 −0.769151 −0.384575 0.923094i \(-0.625652\pi\)
−0.384575 + 0.923094i \(0.625652\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.17858 0.231139
\(27\) 0 0
\(28\) 4.39845 0.831230
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 1.25612 0.225605 0.112803 0.993617i \(-0.464017\pi\)
0.112803 + 0.993617i \(0.464017\pi\)
\(32\) −4.55068 −0.804455
\(33\) 0 0
\(34\) 2.04170 0.350149
\(35\) 2.42505 0.409909
\(36\) 0 0
\(37\) 6.60420 1.08572 0.542861 0.839822i \(-0.317341\pi\)
0.542861 + 0.839822i \(0.317341\pi\)
\(38\) −1.18415 −0.192094
\(39\) 0 0
\(40\) −1.64586 −0.260234
\(41\) −0.160198 −0.0250188 −0.0125094 0.999922i \(-0.503982\pi\)
−0.0125094 + 0.999922i \(0.503982\pi\)
\(42\) 0 0
\(43\) 11.0103 1.67906 0.839528 0.543317i \(-0.182832\pi\)
0.839528 + 0.543317i \(0.182832\pi\)
\(44\) −0.0136953 −0.00206464
\(45\) 0 0
\(46\) 1.59190 0.234713
\(47\) −10.3736 −1.51314 −0.756572 0.653910i \(-0.773127\pi\)
−0.756572 + 0.653910i \(0.773127\pi\)
\(48\) 0 0
\(49\) −1.11912 −0.159875
\(50\) −0.431560 −0.0610317
\(51\) 0 0
\(52\) 4.95333 0.686904
\(53\) 11.1099 1.52607 0.763033 0.646360i \(-0.223709\pi\)
0.763033 + 0.646360i \(0.223709\pi\)
\(54\) 0 0
\(55\) −0.00755077 −0.00101815
\(56\) −3.99130 −0.533360
\(57\) 0 0
\(58\) −0.431560 −0.0566666
\(59\) 8.85010 1.15219 0.576093 0.817384i \(-0.304577\pi\)
0.576093 + 0.817384i \(0.304577\pi\)
\(60\) 0 0
\(61\) 13.5237 1.73153 0.865766 0.500449i \(-0.166832\pi\)
0.865766 + 0.500449i \(0.166832\pi\)
\(62\) −0.542089 −0.0688454
\(63\) 0 0
\(64\) −3.87056 −0.483820
\(65\) 2.73098 0.338736
\(66\) 0 0
\(67\) 4.69881 0.574051 0.287026 0.957923i \(-0.407333\pi\)
0.287026 + 0.957923i \(0.407333\pi\)
\(68\) 8.58085 1.04058
\(69\) 0 0
\(70\) −1.04655 −0.125087
\(71\) −1.48777 −0.176566 −0.0882828 0.996095i \(-0.528138\pi\)
−0.0882828 + 0.996095i \(0.528138\pi\)
\(72\) 0 0
\(73\) 2.75695 0.322677 0.161339 0.986899i \(-0.448419\pi\)
0.161339 + 0.986899i \(0.448419\pi\)
\(74\) −2.85010 −0.331318
\(75\) 0 0
\(76\) −4.97674 −0.570871
\(77\) −0.0183110 −0.00208673
\(78\) 0 0
\(79\) 2.72878 0.307012 0.153506 0.988148i \(-0.450944\pi\)
0.153506 + 0.988148i \(0.450944\pi\)
\(80\) −2.91722 −0.326156
\(81\) 0 0
\(82\) 0.0691351 0.00763470
\(83\) 7.81988 0.858343 0.429172 0.903223i \(-0.358806\pi\)
0.429172 + 0.903223i \(0.358806\pi\)
\(84\) 0 0
\(85\) 4.73098 0.513147
\(86\) −4.75160 −0.512378
\(87\) 0 0
\(88\) 0.0124275 0.00132478
\(89\) −7.59619 −0.805194 −0.402597 0.915377i \(-0.631892\pi\)
−0.402597 + 0.915377i \(0.631892\pi\)
\(90\) 0 0
\(91\) 6.62277 0.694255
\(92\) 6.69044 0.697526
\(93\) 0 0
\(94\) 4.47682 0.461749
\(95\) −2.74388 −0.281517
\(96\) 0 0
\(97\) −7.02540 −0.713322 −0.356661 0.934234i \(-0.616085\pi\)
−0.356661 + 0.934234i \(0.616085\pi\)
\(98\) 0.482968 0.0487872
\(99\) 0 0
\(100\) −1.81376 −0.181376
\(101\) 15.4826 1.54058 0.770289 0.637695i \(-0.220112\pi\)
0.770289 + 0.637695i \(0.220112\pi\)
\(102\) 0 0
\(103\) 7.00119 0.689847 0.344924 0.938631i \(-0.387905\pi\)
0.344924 + 0.938631i \(0.387905\pi\)
\(104\) −4.49482 −0.440753
\(105\) 0 0
\(106\) −4.79460 −0.465692
\(107\) −11.3367 −1.09596 −0.547979 0.836492i \(-0.684603\pi\)
−0.547979 + 0.836492i \(0.684603\pi\)
\(108\) 0 0
\(109\) 15.5028 1.48490 0.742450 0.669901i \(-0.233664\pi\)
0.742450 + 0.669901i \(0.233664\pi\)
\(110\) 0.00325861 0.000310696 0
\(111\) 0 0
\(112\) −7.07442 −0.668470
\(113\) 0.883740 0.0831353 0.0415676 0.999136i \(-0.486765\pi\)
0.0415676 + 0.999136i \(0.486765\pi\)
\(114\) 0 0
\(115\) 3.68872 0.343975
\(116\) −1.81376 −0.168403
\(117\) 0 0
\(118\) −3.81935 −0.351599
\(119\) 11.4729 1.05172
\(120\) 0 0
\(121\) −10.9999 −0.999995
\(122\) −5.83628 −0.528392
\(123\) 0 0
\(124\) −2.27829 −0.204596
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 11.0254 0.978346 0.489173 0.872187i \(-0.337299\pi\)
0.489173 + 0.872187i \(0.337299\pi\)
\(128\) 10.7717 0.952096
\(129\) 0 0
\(130\) −1.17858 −0.103368
\(131\) −19.6037 −1.71278 −0.856391 0.516328i \(-0.827298\pi\)
−0.856391 + 0.516328i \(0.827298\pi\)
\(132\) 0 0
\(133\) −6.65406 −0.576980
\(134\) −2.02782 −0.175177
\(135\) 0 0
\(136\) −7.78655 −0.667691
\(137\) −7.12387 −0.608633 −0.304316 0.952571i \(-0.598428\pi\)
−0.304316 + 0.952571i \(0.598428\pi\)
\(138\) 0 0
\(139\) −9.68606 −0.821561 −0.410781 0.911734i \(-0.634744\pi\)
−0.410781 + 0.911734i \(0.634744\pi\)
\(140\) −4.39845 −0.371737
\(141\) 0 0
\(142\) 0.642061 0.0538805
\(143\) −0.0206210 −0.00172442
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) −1.18979 −0.0984677
\(147\) 0 0
\(148\) −11.9784 −0.984618
\(149\) 17.7002 1.45006 0.725029 0.688719i \(-0.241826\pi\)
0.725029 + 0.688719i \(0.241826\pi\)
\(150\) 0 0
\(151\) 20.7409 1.68787 0.843937 0.536443i \(-0.180232\pi\)
0.843937 + 0.536443i \(0.180232\pi\)
\(152\) 4.51606 0.366301
\(153\) 0 0
\(154\) 0.00790230 0.000636785 0
\(155\) −1.25612 −0.100894
\(156\) 0 0
\(157\) −6.65849 −0.531406 −0.265703 0.964055i \(-0.585604\pi\)
−0.265703 + 0.964055i \(0.585604\pi\)
\(158\) −1.17763 −0.0936874
\(159\) 0 0
\(160\) 4.55068 0.359763
\(161\) 8.94533 0.704991
\(162\) 0 0
\(163\) −5.89592 −0.461804 −0.230902 0.972977i \(-0.574168\pi\)
−0.230902 + 0.972977i \(0.574168\pi\)
\(164\) 0.290561 0.0226890
\(165\) 0 0
\(166\) −3.37474 −0.261931
\(167\) −10.1921 −0.788688 −0.394344 0.918963i \(-0.629028\pi\)
−0.394344 + 0.918963i \(0.629028\pi\)
\(168\) 0 0
\(169\) −5.54174 −0.426288
\(170\) −2.04170 −0.156591
\(171\) 0 0
\(172\) −19.9700 −1.52270
\(173\) −10.8564 −0.825394 −0.412697 0.910868i \(-0.635413\pi\)
−0.412697 + 0.910868i \(0.635413\pi\)
\(174\) 0 0
\(175\) −2.42505 −0.183317
\(176\) 0.0220273 0.00166037
\(177\) 0 0
\(178\) 3.27821 0.245712
\(179\) 20.7170 1.54846 0.774230 0.632904i \(-0.218137\pi\)
0.774230 + 0.632904i \(0.218137\pi\)
\(180\) 0 0
\(181\) −0.0561768 −0.00417559 −0.00208779 0.999998i \(-0.500665\pi\)
−0.00208779 + 0.999998i \(0.500665\pi\)
\(182\) −2.85812 −0.211858
\(183\) 0 0
\(184\) −6.07113 −0.447569
\(185\) −6.60420 −0.485550
\(186\) 0 0
\(187\) −0.0357226 −0.00261229
\(188\) 18.8152 1.37224
\(189\) 0 0
\(190\) 1.18415 0.0859072
\(191\) 3.24218 0.234596 0.117298 0.993097i \(-0.462577\pi\)
0.117298 + 0.993097i \(0.462577\pi\)
\(192\) 0 0
\(193\) −10.2522 −0.737967 −0.368983 0.929436i \(-0.620294\pi\)
−0.368983 + 0.929436i \(0.620294\pi\)
\(194\) 3.03188 0.217676
\(195\) 0 0
\(196\) 2.02982 0.144987
\(197\) −10.7593 −0.766570 −0.383285 0.923630i \(-0.625207\pi\)
−0.383285 + 0.923630i \(0.625207\pi\)
\(198\) 0 0
\(199\) 8.13396 0.576601 0.288301 0.957540i \(-0.406910\pi\)
0.288301 + 0.957540i \(0.406910\pi\)
\(200\) 1.64586 0.116380
\(201\) 0 0
\(202\) −6.68167 −0.470121
\(203\) −2.42505 −0.170205
\(204\) 0 0
\(205\) 0.160198 0.0111887
\(206\) −3.02143 −0.210513
\(207\) 0 0
\(208\) −7.96688 −0.552404
\(209\) 0.0207184 0.00143313
\(210\) 0 0
\(211\) 2.42349 0.166840 0.0834199 0.996514i \(-0.473416\pi\)
0.0834199 + 0.996514i \(0.473416\pi\)
\(212\) −20.1507 −1.38396
\(213\) 0 0
\(214\) 4.89246 0.334441
\(215\) −11.0103 −0.750896
\(216\) 0 0
\(217\) −3.04615 −0.206786
\(218\) −6.69039 −0.453130
\(219\) 0 0
\(220\) 0.0136953 0.000923334 0
\(221\) 12.9202 0.869108
\(222\) 0 0
\(223\) 16.1406 1.08085 0.540427 0.841391i \(-0.318263\pi\)
0.540427 + 0.841391i \(0.318263\pi\)
\(224\) 11.0356 0.737350
\(225\) 0 0
\(226\) −0.381387 −0.0253695
\(227\) 19.7831 1.31305 0.656527 0.754303i \(-0.272025\pi\)
0.656527 + 0.754303i \(0.272025\pi\)
\(228\) 0 0
\(229\) −1.99560 −0.131873 −0.0659366 0.997824i \(-0.521004\pi\)
−0.0659366 + 0.997824i \(0.521004\pi\)
\(230\) −1.59190 −0.104967
\(231\) 0 0
\(232\) 1.64586 0.108056
\(233\) −3.98606 −0.261136 −0.130568 0.991439i \(-0.541680\pi\)
−0.130568 + 0.991439i \(0.541680\pi\)
\(234\) 0 0
\(235\) 10.3736 0.676699
\(236\) −16.0519 −1.04489
\(237\) 0 0
\(238\) −4.95123 −0.320941
\(239\) 29.5973 1.91449 0.957245 0.289279i \(-0.0934155\pi\)
0.957245 + 0.289279i \(0.0934155\pi\)
\(240\) 0 0
\(241\) −13.1622 −0.847854 −0.423927 0.905696i \(-0.639349\pi\)
−0.423927 + 0.905696i \(0.639349\pi\)
\(242\) 4.74713 0.305157
\(243\) 0 0
\(244\) −24.5287 −1.57029
\(245\) 1.11912 0.0714982
\(246\) 0 0
\(247\) −7.49350 −0.476800
\(248\) 2.06739 0.131280
\(249\) 0 0
\(250\) 0.431560 0.0272942
\(251\) −6.84372 −0.431972 −0.215986 0.976397i \(-0.569296\pi\)
−0.215986 + 0.976397i \(0.569296\pi\)
\(252\) 0 0
\(253\) −0.0278527 −0.00175108
\(254\) −4.75812 −0.298551
\(255\) 0 0
\(256\) 3.09247 0.193279
\(257\) 8.42123 0.525302 0.262651 0.964891i \(-0.415403\pi\)
0.262651 + 0.964891i \(0.415403\pi\)
\(258\) 0 0
\(259\) −16.0155 −0.995156
\(260\) −4.95333 −0.307193
\(261\) 0 0
\(262\) 8.46016 0.522670
\(263\) 2.28447 0.140866 0.0704331 0.997517i \(-0.477562\pi\)
0.0704331 + 0.997517i \(0.477562\pi\)
\(264\) 0 0
\(265\) −11.1099 −0.682477
\(266\) 2.87162 0.176071
\(267\) 0 0
\(268\) −8.52250 −0.520595
\(269\) 1.62731 0.0992186 0.0496093 0.998769i \(-0.484202\pi\)
0.0496093 + 0.998769i \(0.484202\pi\)
\(270\) 0 0
\(271\) 6.18126 0.375485 0.187742 0.982218i \(-0.439883\pi\)
0.187742 + 0.982218i \(0.439883\pi\)
\(272\) −13.8013 −0.836829
\(273\) 0 0
\(274\) 3.07437 0.185730
\(275\) 0.00755077 0.000455329 0
\(276\) 0 0
\(277\) −12.5435 −0.753668 −0.376834 0.926281i \(-0.622987\pi\)
−0.376834 + 0.926281i \(0.622987\pi\)
\(278\) 4.18011 0.250707
\(279\) 0 0
\(280\) 3.99130 0.238526
\(281\) 3.78236 0.225637 0.112818 0.993616i \(-0.464012\pi\)
0.112818 + 0.993616i \(0.464012\pi\)
\(282\) 0 0
\(283\) −20.7316 −1.23237 −0.616183 0.787603i \(-0.711322\pi\)
−0.616183 + 0.787603i \(0.711322\pi\)
\(284\) 2.69845 0.160124
\(285\) 0 0
\(286\) 0.00889920 0.000526221 0
\(287\) 0.388489 0.0229318
\(288\) 0 0
\(289\) 5.38218 0.316599
\(290\) 0.431560 0.0253421
\(291\) 0 0
\(292\) −5.00044 −0.292629
\(293\) −15.3438 −0.896394 −0.448197 0.893935i \(-0.647934\pi\)
−0.448197 + 0.893935i \(0.647934\pi\)
\(294\) 0 0
\(295\) −8.85010 −0.515273
\(296\) 10.8696 0.631783
\(297\) 0 0
\(298\) −7.63869 −0.442498
\(299\) 10.0738 0.582584
\(300\) 0 0
\(301\) −26.7006 −1.53899
\(302\) −8.95095 −0.515069
\(303\) 0 0
\(304\) 8.00453 0.459091
\(305\) −13.5237 −0.774365
\(306\) 0 0
\(307\) −6.67480 −0.380951 −0.190476 0.981692i \(-0.561003\pi\)
−0.190476 + 0.981692i \(0.561003\pi\)
\(308\) 0.0332117 0.00189241
\(309\) 0 0
\(310\) 0.542089 0.0307886
\(311\) 4.07460 0.231049 0.115525 0.993305i \(-0.463145\pi\)
0.115525 + 0.993305i \(0.463145\pi\)
\(312\) 0 0
\(313\) 10.8923 0.615672 0.307836 0.951439i \(-0.400395\pi\)
0.307836 + 0.951439i \(0.400395\pi\)
\(314\) 2.87354 0.162163
\(315\) 0 0
\(316\) −4.94935 −0.278423
\(317\) −21.5993 −1.21314 −0.606569 0.795031i \(-0.707455\pi\)
−0.606569 + 0.795031i \(0.707455\pi\)
\(318\) 0 0
\(319\) 0.00755077 0.000422762 0
\(320\) 3.87056 0.216371
\(321\) 0 0
\(322\) −3.86044 −0.215134
\(323\) −12.9813 −0.722297
\(324\) 0 0
\(325\) −2.73098 −0.151488
\(326\) 2.54444 0.140924
\(327\) 0 0
\(328\) −0.263665 −0.0145584
\(329\) 25.1565 1.38692
\(330\) 0 0
\(331\) 15.1834 0.834558 0.417279 0.908778i \(-0.362984\pi\)
0.417279 + 0.908778i \(0.362984\pi\)
\(332\) −14.1834 −0.778413
\(333\) 0 0
\(334\) 4.39850 0.240675
\(335\) −4.69881 −0.256724
\(336\) 0 0
\(337\) 11.0960 0.604437 0.302219 0.953239i \(-0.402273\pi\)
0.302219 + 0.953239i \(0.402273\pi\)
\(338\) 2.39159 0.130086
\(339\) 0 0
\(340\) −8.58085 −0.465362
\(341\) 0.00948465 0.000513622 0
\(342\) 0 0
\(343\) 19.6893 1.06312
\(344\) 18.1215 0.977043
\(345\) 0 0
\(346\) 4.68517 0.251876
\(347\) −13.9866 −0.750838 −0.375419 0.926855i \(-0.622501\pi\)
−0.375419 + 0.926855i \(0.622501\pi\)
\(348\) 0 0
\(349\) −23.8346 −1.27584 −0.637918 0.770104i \(-0.720204\pi\)
−0.637918 + 0.770104i \(0.720204\pi\)
\(350\) 1.04655 0.0559407
\(351\) 0 0
\(352\) −0.0343612 −0.00183146
\(353\) 20.0478 1.06704 0.533518 0.845789i \(-0.320870\pi\)
0.533518 + 0.845789i \(0.320870\pi\)
\(354\) 0 0
\(355\) 1.48777 0.0789626
\(356\) 13.7776 0.730213
\(357\) 0 0
\(358\) −8.94061 −0.472526
\(359\) −22.5167 −1.18839 −0.594193 0.804322i \(-0.702528\pi\)
−0.594193 + 0.804322i \(0.702528\pi\)
\(360\) 0 0
\(361\) −11.4711 −0.603742
\(362\) 0.0242436 0.00127422
\(363\) 0 0
\(364\) −12.0121 −0.629605
\(365\) −2.75695 −0.144306
\(366\) 0 0
\(367\) 35.5683 1.85665 0.928324 0.371772i \(-0.121250\pi\)
0.928324 + 0.371772i \(0.121250\pi\)
\(368\) −10.7608 −0.560947
\(369\) 0 0
\(370\) 2.85010 0.148170
\(371\) −26.9422 −1.39877
\(372\) 0 0
\(373\) 6.61603 0.342565 0.171283 0.985222i \(-0.445209\pi\)
0.171283 + 0.985222i \(0.445209\pi\)
\(374\) 0.0154164 0.000797164 0
\(375\) 0 0
\(376\) −17.0735 −0.880499
\(377\) −2.73098 −0.140653
\(378\) 0 0
\(379\) 16.6561 0.855567 0.427784 0.903881i \(-0.359295\pi\)
0.427784 + 0.903881i \(0.359295\pi\)
\(380\) 4.97674 0.255301
\(381\) 0 0
\(382\) −1.39919 −0.0715890
\(383\) −16.6700 −0.851796 −0.425898 0.904771i \(-0.640042\pi\)
−0.425898 + 0.904771i \(0.640042\pi\)
\(384\) 0 0
\(385\) 0.0183110 0.000933216 0
\(386\) 4.42442 0.225197
\(387\) 0 0
\(388\) 12.7424 0.646896
\(389\) −37.3519 −1.89382 −0.946908 0.321505i \(-0.895811\pi\)
−0.946908 + 0.321505i \(0.895811\pi\)
\(390\) 0 0
\(391\) 17.4513 0.882548
\(392\) −1.84192 −0.0930312
\(393\) 0 0
\(394\) 4.64329 0.233926
\(395\) −2.72878 −0.137300
\(396\) 0 0
\(397\) 20.0742 1.00750 0.503748 0.863850i \(-0.331954\pi\)
0.503748 + 0.863850i \(0.331954\pi\)
\(398\) −3.51029 −0.175955
\(399\) 0 0
\(400\) 2.91722 0.145861
\(401\) 12.4073 0.619591 0.309795 0.950803i \(-0.399739\pi\)
0.309795 + 0.950803i \(0.399739\pi\)
\(402\) 0 0
\(403\) −3.43043 −0.170882
\(404\) −28.0817 −1.39712
\(405\) 0 0
\(406\) 1.04655 0.0519396
\(407\) 0.0498668 0.00247180
\(408\) 0 0
\(409\) −16.9907 −0.840138 −0.420069 0.907492i \(-0.637994\pi\)
−0.420069 + 0.907492i \(0.637994\pi\)
\(410\) −0.0691351 −0.00341434
\(411\) 0 0
\(412\) −12.6984 −0.625608
\(413\) −21.4620 −1.05607
\(414\) 0 0
\(415\) −7.81988 −0.383863
\(416\) 12.4278 0.609324
\(417\) 0 0
\(418\) −0.00894125 −0.000437331 0
\(419\) 8.07891 0.394680 0.197340 0.980335i \(-0.436770\pi\)
0.197340 + 0.980335i \(0.436770\pi\)
\(420\) 0 0
\(421\) 9.72062 0.473754 0.236877 0.971540i \(-0.423876\pi\)
0.236877 + 0.971540i \(0.423876\pi\)
\(422\) −1.04588 −0.0509126
\(423\) 0 0
\(424\) 18.2854 0.888019
\(425\) −4.73098 −0.229486
\(426\) 0 0
\(427\) −32.7957 −1.58709
\(428\) 20.5620 0.993901
\(429\) 0 0
\(430\) 4.75160 0.229143
\(431\) 38.4501 1.85208 0.926038 0.377429i \(-0.123192\pi\)
0.926038 + 0.377429i \(0.123192\pi\)
\(432\) 0 0
\(433\) 1.54078 0.0740452 0.0370226 0.999314i \(-0.488213\pi\)
0.0370226 + 0.999314i \(0.488213\pi\)
\(434\) 1.31459 0.0631025
\(435\) 0 0
\(436\) −28.1183 −1.34662
\(437\) −10.1214 −0.484173
\(438\) 0 0
\(439\) 13.0277 0.621778 0.310889 0.950446i \(-0.399373\pi\)
0.310889 + 0.950446i \(0.399373\pi\)
\(440\) −0.0124275 −0.000592460 0
\(441\) 0 0
\(442\) −5.57584 −0.265216
\(443\) 4.37921 0.208063 0.104031 0.994574i \(-0.466826\pi\)
0.104031 + 0.994574i \(0.466826\pi\)
\(444\) 0 0
\(445\) 7.59619 0.360094
\(446\) −6.96562 −0.329832
\(447\) 0 0
\(448\) 9.38630 0.443461
\(449\) 19.0425 0.898673 0.449336 0.893363i \(-0.351661\pi\)
0.449336 + 0.893363i \(0.351661\pi\)
\(450\) 0 0
\(451\) −0.00120962 −5.69588e−5 0
\(452\) −1.60289 −0.0753936
\(453\) 0 0
\(454\) −8.53761 −0.400690
\(455\) −6.62277 −0.310480
\(456\) 0 0
\(457\) −12.5512 −0.587122 −0.293561 0.955940i \(-0.594840\pi\)
−0.293561 + 0.955940i \(0.594840\pi\)
\(458\) 0.861222 0.0402423
\(459\) 0 0
\(460\) −6.69044 −0.311943
\(461\) 29.0335 1.35223 0.676113 0.736798i \(-0.263663\pi\)
0.676113 + 0.736798i \(0.263663\pi\)
\(462\) 0 0
\(463\) 2.74254 0.127457 0.0637284 0.997967i \(-0.479701\pi\)
0.0637284 + 0.997967i \(0.479701\pi\)
\(464\) 2.91722 0.135429
\(465\) 0 0
\(466\) 1.72022 0.0796878
\(467\) 7.53116 0.348501 0.174250 0.984701i \(-0.444250\pi\)
0.174250 + 0.984701i \(0.444250\pi\)
\(468\) 0 0
\(469\) −11.3949 −0.526166
\(470\) −4.47682 −0.206501
\(471\) 0 0
\(472\) 14.5661 0.670457
\(473\) 0.0831363 0.00382261
\(474\) 0 0
\(475\) 2.74388 0.125898
\(476\) −20.8090 −0.953779
\(477\) 0 0
\(478\) −12.7730 −0.584223
\(479\) −25.4783 −1.16414 −0.582068 0.813140i \(-0.697756\pi\)
−0.582068 + 0.813140i \(0.697756\pi\)
\(480\) 0 0
\(481\) −18.0359 −0.822368
\(482\) 5.68029 0.258730
\(483\) 0 0
\(484\) 19.9512 0.906873
\(485\) 7.02540 0.319007
\(486\) 0 0
\(487\) 2.38516 0.108082 0.0540409 0.998539i \(-0.482790\pi\)
0.0540409 + 0.998539i \(0.482790\pi\)
\(488\) 22.2582 1.00758
\(489\) 0 0
\(490\) −0.482968 −0.0218183
\(491\) 32.4925 1.46637 0.733184 0.680031i \(-0.238034\pi\)
0.733184 + 0.680031i \(0.238034\pi\)
\(492\) 0 0
\(493\) −4.73098 −0.213073
\(494\) 3.23389 0.145500
\(495\) 0 0
\(496\) 3.66437 0.164535
\(497\) 3.60792 0.161837
\(498\) 0 0
\(499\) −8.56521 −0.383431 −0.191716 0.981451i \(-0.561405\pi\)
−0.191716 + 0.981451i \(0.561405\pi\)
\(500\) 1.81376 0.0811136
\(501\) 0 0
\(502\) 2.95347 0.131820
\(503\) 3.39783 0.151502 0.0757510 0.997127i \(-0.475865\pi\)
0.0757510 + 0.997127i \(0.475865\pi\)
\(504\) 0 0
\(505\) −15.4826 −0.688968
\(506\) 0.0120201 0.000534358 0
\(507\) 0 0
\(508\) −19.9974 −0.887241
\(509\) 29.0172 1.28616 0.643082 0.765797i \(-0.277655\pi\)
0.643082 + 0.765797i \(0.277655\pi\)
\(510\) 0 0
\(511\) −6.68576 −0.295761
\(512\) −22.8781 −1.01108
\(513\) 0 0
\(514\) −3.63426 −0.160301
\(515\) −7.00119 −0.308509
\(516\) 0 0
\(517\) −0.0783287 −0.00344489
\(518\) 6.91165 0.303680
\(519\) 0 0
\(520\) 4.49482 0.197111
\(521\) −10.8098 −0.473588 −0.236794 0.971560i \(-0.576097\pi\)
−0.236794 + 0.971560i \(0.576097\pi\)
\(522\) 0 0
\(523\) 24.7294 1.08134 0.540671 0.841234i \(-0.318170\pi\)
0.540671 + 0.841234i \(0.318170\pi\)
\(524\) 35.5563 1.55328
\(525\) 0 0
\(526\) −0.985884 −0.0429866
\(527\) −5.94266 −0.258866
\(528\) 0 0
\(529\) −9.39336 −0.408407
\(530\) 4.79460 0.208264
\(531\) 0 0
\(532\) 12.0688 0.523251
\(533\) 0.437499 0.0189502
\(534\) 0 0
\(535\) 11.3367 0.490128
\(536\) 7.73360 0.334041
\(537\) 0 0
\(538\) −0.702279 −0.0302774
\(539\) −0.00845025 −0.000363978 0
\(540\) 0 0
\(541\) −10.2403 −0.440264 −0.220132 0.975470i \(-0.570649\pi\)
−0.220132 + 0.975470i \(0.570649\pi\)
\(542\) −2.66758 −0.114582
\(543\) 0 0
\(544\) 21.5292 0.923057
\(545\) −15.5028 −0.664068
\(546\) 0 0
\(547\) −1.68008 −0.0718349 −0.0359175 0.999355i \(-0.511435\pi\)
−0.0359175 + 0.999355i \(0.511435\pi\)
\(548\) 12.9210 0.551956
\(549\) 0 0
\(550\) −0.00325861 −0.000138948 0
\(551\) 2.74388 0.116893
\(552\) 0 0
\(553\) −6.61744 −0.281402
\(554\) 5.41329 0.229988
\(555\) 0 0
\(556\) 17.5682 0.745056
\(557\) −17.6075 −0.746053 −0.373026 0.927821i \(-0.621680\pi\)
−0.373026 + 0.927821i \(0.621680\pi\)
\(558\) 0 0
\(559\) −30.0689 −1.27178
\(560\) 7.07442 0.298949
\(561\) 0 0
\(562\) −1.63231 −0.0688550
\(563\) −38.3553 −1.61648 −0.808241 0.588852i \(-0.799580\pi\)
−0.808241 + 0.588852i \(0.799580\pi\)
\(564\) 0 0
\(565\) −0.883740 −0.0371792
\(566\) 8.94692 0.376067
\(567\) 0 0
\(568\) −2.44866 −0.102744
\(569\) −5.63212 −0.236111 −0.118055 0.993007i \(-0.537666\pi\)
−0.118055 + 0.993007i \(0.537666\pi\)
\(570\) 0 0
\(571\) 13.6537 0.571389 0.285695 0.958321i \(-0.407776\pi\)
0.285695 + 0.958321i \(0.407776\pi\)
\(572\) 0.0374015 0.00156384
\(573\) 0 0
\(574\) −0.167656 −0.00699784
\(575\) −3.68872 −0.153830
\(576\) 0 0
\(577\) 12.1911 0.507521 0.253760 0.967267i \(-0.418333\pi\)
0.253760 + 0.967267i \(0.418333\pi\)
\(578\) −2.32273 −0.0966129
\(579\) 0 0
\(580\) 1.81376 0.0753121
\(581\) −18.9636 −0.786743
\(582\) 0 0
\(583\) 0.0838886 0.00347431
\(584\) 4.53757 0.187766
\(585\) 0 0
\(586\) 6.62176 0.273543
\(587\) −16.0556 −0.662687 −0.331343 0.943510i \(-0.607502\pi\)
−0.331343 + 0.943510i \(0.607502\pi\)
\(588\) 0 0
\(589\) 3.44664 0.142016
\(590\) 3.81935 0.157240
\(591\) 0 0
\(592\) 19.2659 0.791824
\(593\) −18.6072 −0.764108 −0.382054 0.924140i \(-0.624783\pi\)
−0.382054 + 0.924140i \(0.624783\pi\)
\(594\) 0 0
\(595\) −11.4729 −0.470342
\(596\) −32.1039 −1.31503
\(597\) 0 0
\(598\) −4.34745 −0.177781
\(599\) 28.3057 1.15654 0.578269 0.815846i \(-0.303728\pi\)
0.578269 + 0.815846i \(0.303728\pi\)
\(600\) 0 0
\(601\) 30.1736 1.23081 0.615404 0.788212i \(-0.288993\pi\)
0.615404 + 0.788212i \(0.288993\pi\)
\(602\) 11.5229 0.469638
\(603\) 0 0
\(604\) −37.6190 −1.53070
\(605\) 10.9999 0.447211
\(606\) 0 0
\(607\) −3.99729 −0.162245 −0.0811224 0.996704i \(-0.525850\pi\)
−0.0811224 + 0.996704i \(0.525850\pi\)
\(608\) −12.4865 −0.506396
\(609\) 0 0
\(610\) 5.83628 0.236304
\(611\) 28.3301 1.14611
\(612\) 0 0
\(613\) 38.3571 1.54923 0.774614 0.632434i \(-0.217944\pi\)
0.774614 + 0.632434i \(0.217944\pi\)
\(614\) 2.88058 0.116251
\(615\) 0 0
\(616\) −0.0301374 −0.00121427
\(617\) −16.7381 −0.673852 −0.336926 0.941531i \(-0.609387\pi\)
−0.336926 + 0.941531i \(0.609387\pi\)
\(618\) 0 0
\(619\) 47.4248 1.90616 0.953081 0.302714i \(-0.0978929\pi\)
0.953081 + 0.302714i \(0.0978929\pi\)
\(620\) 2.27829 0.0914982
\(621\) 0 0
\(622\) −1.75843 −0.0705067
\(623\) 18.4211 0.738028
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −4.70070 −0.187878
\(627\) 0 0
\(628\) 12.0769 0.481920
\(629\) −31.2443 −1.24579
\(630\) 0 0
\(631\) 21.1985 0.843898 0.421949 0.906620i \(-0.361346\pi\)
0.421949 + 0.906620i \(0.361346\pi\)
\(632\) 4.49120 0.178650
\(633\) 0 0
\(634\) 9.32139 0.370200
\(635\) −11.0254 −0.437530
\(636\) 0 0
\(637\) 3.05630 0.121095
\(638\) −0.00325861 −0.000129010 0
\(639\) 0 0
\(640\) −10.7717 −0.425790
\(641\) 9.76078 0.385528 0.192764 0.981245i \(-0.438255\pi\)
0.192764 + 0.981245i \(0.438255\pi\)
\(642\) 0 0
\(643\) −3.87214 −0.152702 −0.0763511 0.997081i \(-0.524327\pi\)
−0.0763511 + 0.997081i \(0.524327\pi\)
\(644\) −16.2247 −0.639341
\(645\) 0 0
\(646\) 5.60219 0.220415
\(647\) 11.7270 0.461037 0.230518 0.973068i \(-0.425958\pi\)
0.230518 + 0.973068i \(0.425958\pi\)
\(648\) 0 0
\(649\) 0.0668251 0.00262312
\(650\) 1.17858 0.0462278
\(651\) 0 0
\(652\) 10.6938 0.418800
\(653\) 27.0409 1.05819 0.529097 0.848561i \(-0.322531\pi\)
0.529097 + 0.848561i \(0.322531\pi\)
\(654\) 0 0
\(655\) 19.6037 0.765979
\(656\) −0.467334 −0.0182463
\(657\) 0 0
\(658\) −10.8565 −0.423232
\(659\) 27.3773 1.06647 0.533235 0.845967i \(-0.320976\pi\)
0.533235 + 0.845967i \(0.320976\pi\)
\(660\) 0 0
\(661\) 42.6188 1.65768 0.828839 0.559487i \(-0.189002\pi\)
0.828839 + 0.559487i \(0.189002\pi\)
\(662\) −6.55256 −0.254673
\(663\) 0 0
\(664\) 12.8704 0.499470
\(665\) 6.65406 0.258033
\(666\) 0 0
\(667\) −3.68872 −0.142828
\(668\) 18.4860 0.715244
\(669\) 0 0
\(670\) 2.02782 0.0783414
\(671\) 0.102114 0.00394208
\(672\) 0 0
\(673\) 30.4014 1.17189 0.585943 0.810352i \(-0.300724\pi\)
0.585943 + 0.810352i \(0.300724\pi\)
\(674\) −4.78858 −0.184449
\(675\) 0 0
\(676\) 10.0514 0.386591
\(677\) 39.1078 1.50303 0.751517 0.659714i \(-0.229323\pi\)
0.751517 + 0.659714i \(0.229323\pi\)
\(678\) 0 0
\(679\) 17.0370 0.653819
\(680\) 7.78655 0.298600
\(681\) 0 0
\(682\) −0.00409319 −0.000156736 0
\(683\) 22.2515 0.851431 0.425716 0.904857i \(-0.360022\pi\)
0.425716 + 0.904857i \(0.360022\pi\)
\(684\) 0 0
\(685\) 7.12387 0.272189
\(686\) −8.49710 −0.324421
\(687\) 0 0
\(688\) 32.1195 1.22455
\(689\) −30.3410 −1.15590
\(690\) 0 0
\(691\) −19.8899 −0.756647 −0.378323 0.925674i \(-0.623499\pi\)
−0.378323 + 0.925674i \(0.623499\pi\)
\(692\) 19.6908 0.748531
\(693\) 0 0
\(694\) 6.03603 0.229125
\(695\) 9.68606 0.367413
\(696\) 0 0
\(697\) 0.757895 0.0287073
\(698\) 10.2861 0.389333
\(699\) 0 0
\(700\) 4.39845 0.166246
\(701\) 35.7640 1.35079 0.675393 0.737458i \(-0.263974\pi\)
0.675393 + 0.737458i \(0.263974\pi\)
\(702\) 0 0
\(703\) 18.1211 0.683452
\(704\) −0.0292257 −0.00110149
\(705\) 0 0
\(706\) −8.65182 −0.325615
\(707\) −37.5462 −1.41207
\(708\) 0 0
\(709\) 36.8609 1.38434 0.692171 0.721734i \(-0.256654\pi\)
0.692171 + 0.721734i \(0.256654\pi\)
\(710\) −0.642061 −0.0240961
\(711\) 0 0
\(712\) −12.5023 −0.468543
\(713\) −4.63346 −0.173524
\(714\) 0 0
\(715\) 0.0206210 0.000771182 0
\(716\) −37.5756 −1.40426
\(717\) 0 0
\(718\) 9.71730 0.362646
\(719\) 5.42783 0.202424 0.101212 0.994865i \(-0.467728\pi\)
0.101212 + 0.994865i \(0.467728\pi\)
\(720\) 0 0
\(721\) −16.9782 −0.632303
\(722\) 4.95046 0.184237
\(723\) 0 0
\(724\) 0.101891 0.00378675
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 24.6244 0.913267 0.456633 0.889655i \(-0.349055\pi\)
0.456633 + 0.889655i \(0.349055\pi\)
\(728\) 10.9002 0.403987
\(729\) 0 0
\(730\) 1.18979 0.0440361
\(731\) −52.0895 −1.92660
\(732\) 0 0
\(733\) −41.7803 −1.54319 −0.771594 0.636115i \(-0.780540\pi\)
−0.771594 + 0.636115i \(0.780540\pi\)
\(734\) −15.3498 −0.566572
\(735\) 0 0
\(736\) 16.7862 0.618747
\(737\) 0.0354797 0.00130691
\(738\) 0 0
\(739\) −1.46856 −0.0540217 −0.0270108 0.999635i \(-0.508599\pi\)
−0.0270108 + 0.999635i \(0.508599\pi\)
\(740\) 11.9784 0.440335
\(741\) 0 0
\(742\) 11.6271 0.426846
\(743\) 42.7834 1.56957 0.784785 0.619768i \(-0.212773\pi\)
0.784785 + 0.619768i \(0.212773\pi\)
\(744\) 0 0
\(745\) −17.7002 −0.648485
\(746\) −2.85521 −0.104537
\(747\) 0 0
\(748\) 0.0647920 0.00236903
\(749\) 27.4921 1.00454
\(750\) 0 0
\(751\) −32.9769 −1.20335 −0.601673 0.798743i \(-0.705499\pi\)
−0.601673 + 0.798743i \(0.705499\pi\)
\(752\) −30.2621 −1.10355
\(753\) 0 0
\(754\) 1.17858 0.0429214
\(755\) −20.7409 −0.754840
\(756\) 0 0
\(757\) 43.4442 1.57900 0.789502 0.613748i \(-0.210339\pi\)
0.789502 + 0.613748i \(0.210339\pi\)
\(758\) −7.18811 −0.261084
\(759\) 0 0
\(760\) −4.51606 −0.163815
\(761\) −12.9239 −0.468492 −0.234246 0.972177i \(-0.575262\pi\)
−0.234246 + 0.972177i \(0.575262\pi\)
\(762\) 0 0
\(763\) −37.5951 −1.36103
\(764\) −5.88052 −0.212750
\(765\) 0 0
\(766\) 7.19409 0.259933
\(767\) −24.1695 −0.872709
\(768\) 0 0
\(769\) 33.6451 1.21327 0.606636 0.794979i \(-0.292518\pi\)
0.606636 + 0.794979i \(0.292518\pi\)
\(770\) −0.00790230 −0.000284779 0
\(771\) 0 0
\(772\) 18.5949 0.669246
\(773\) −47.5209 −1.70921 −0.854603 0.519281i \(-0.826200\pi\)
−0.854603 + 0.519281i \(0.826200\pi\)
\(774\) 0 0
\(775\) 1.25612 0.0451210
\(776\) −11.5629 −0.415082
\(777\) 0 0
\(778\) 16.1196 0.577914
\(779\) −0.439566 −0.0157491
\(780\) 0 0
\(781\) −0.0112338 −0.000401977 0
\(782\) −7.53126 −0.269317
\(783\) 0 0
\(784\) −3.26473 −0.116598
\(785\) 6.65849 0.237652
\(786\) 0 0
\(787\) −2.20720 −0.0786783 −0.0393391 0.999226i \(-0.512525\pi\)
−0.0393391 + 0.999226i \(0.512525\pi\)
\(788\) 19.5148 0.695186
\(789\) 0 0
\(790\) 1.17763 0.0418983
\(791\) −2.14312 −0.0762004
\(792\) 0 0
\(793\) −36.9330 −1.31153
\(794\) −8.66323 −0.307446
\(795\) 0 0
\(796\) −14.7530 −0.522907
\(797\) 35.5149 1.25800 0.629001 0.777405i \(-0.283464\pi\)
0.629001 + 0.777405i \(0.283464\pi\)
\(798\) 0 0
\(799\) 49.0773 1.73623
\(800\) −4.55068 −0.160891
\(801\) 0 0
\(802\) −5.35449 −0.189074
\(803\) 0.0208171 0.000734621 0
\(804\) 0 0
\(805\) −8.94533 −0.315282
\(806\) 1.48043 0.0521461
\(807\) 0 0
\(808\) 25.4823 0.896463
\(809\) −16.1284 −0.567044 −0.283522 0.958966i \(-0.591503\pi\)
−0.283522 + 0.958966i \(0.591503\pi\)
\(810\) 0 0
\(811\) −21.5983 −0.758417 −0.379209 0.925311i \(-0.623804\pi\)
−0.379209 + 0.925311i \(0.623804\pi\)
\(812\) 4.39845 0.154355
\(813\) 0 0
\(814\) −0.0215205 −0.000754293 0
\(815\) 5.89592 0.206525
\(816\) 0 0
\(817\) 30.2110 1.05695
\(818\) 7.33252 0.256376
\(819\) 0 0
\(820\) −0.290561 −0.0101468
\(821\) −35.7338 −1.24712 −0.623559 0.781777i \(-0.714314\pi\)
−0.623559 + 0.781777i \(0.714314\pi\)
\(822\) 0 0
\(823\) 19.2184 0.669911 0.334956 0.942234i \(-0.391279\pi\)
0.334956 + 0.942234i \(0.391279\pi\)
\(824\) 11.5230 0.401423
\(825\) 0 0
\(826\) 9.26212 0.322270
\(827\) 15.1283 0.526061 0.263031 0.964787i \(-0.415278\pi\)
0.263031 + 0.964787i \(0.415278\pi\)
\(828\) 0 0
\(829\) 20.8410 0.723837 0.361918 0.932210i \(-0.382122\pi\)
0.361918 + 0.932210i \(0.382122\pi\)
\(830\) 3.37474 0.117139
\(831\) 0 0
\(832\) 10.5704 0.366463
\(833\) 5.29455 0.183445
\(834\) 0 0
\(835\) 10.1921 0.352712
\(836\) −0.0375782 −0.00129967
\(837\) 0 0
\(838\) −3.48653 −0.120440
\(839\) −17.3275 −0.598213 −0.299107 0.954220i \(-0.596689\pi\)
−0.299107 + 0.954220i \(0.596689\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −4.19503 −0.144570
\(843\) 0 0
\(844\) −4.39562 −0.151303
\(845\) 5.54174 0.190642
\(846\) 0 0
\(847\) 26.6754 0.916579
\(848\) 32.4102 1.11297
\(849\) 0 0
\(850\) 2.04170 0.0700297
\(851\) −24.3610 −0.835085
\(852\) 0 0
\(853\) −33.5458 −1.14859 −0.574293 0.818650i \(-0.694723\pi\)
−0.574293 + 0.818650i \(0.694723\pi\)
\(854\) 14.1533 0.484316
\(855\) 0 0
\(856\) −18.6586 −0.637739
\(857\) 55.0914 1.88189 0.940944 0.338563i \(-0.109941\pi\)
0.940944 + 0.338563i \(0.109941\pi\)
\(858\) 0 0
\(859\) 4.34649 0.148300 0.0741501 0.997247i \(-0.476376\pi\)
0.0741501 + 0.997247i \(0.476376\pi\)
\(860\) 19.9700 0.680972
\(861\) 0 0
\(862\) −16.5935 −0.565177
\(863\) −39.8846 −1.35769 −0.678844 0.734282i \(-0.737519\pi\)
−0.678844 + 0.734282i \(0.737519\pi\)
\(864\) 0 0
\(865\) 10.8564 0.369127
\(866\) −0.664939 −0.0225955
\(867\) 0 0
\(868\) 5.52497 0.187530
\(869\) 0.0206044 0.000698957 0
\(870\) 0 0
\(871\) −12.8324 −0.434808
\(872\) 25.5155 0.864064
\(873\) 0 0
\(874\) 4.36799 0.147750
\(875\) 2.42505 0.0819817
\(876\) 0 0
\(877\) −41.1414 −1.38925 −0.694623 0.719374i \(-0.744429\pi\)
−0.694623 + 0.719374i \(0.744429\pi\)
\(878\) −5.62223 −0.189741
\(879\) 0 0
\(880\) −0.0220273 −0.000742540 0
\(881\) 19.1383 0.644784 0.322392 0.946606i \(-0.395513\pi\)
0.322392 + 0.946606i \(0.395513\pi\)
\(882\) 0 0
\(883\) 10.3573 0.348550 0.174275 0.984697i \(-0.444242\pi\)
0.174275 + 0.984697i \(0.444242\pi\)
\(884\) −23.4341 −0.788175
\(885\) 0 0
\(886\) −1.88989 −0.0634921
\(887\) 11.5233 0.386915 0.193457 0.981109i \(-0.438030\pi\)
0.193457 + 0.981109i \(0.438030\pi\)
\(888\) 0 0
\(889\) −26.7372 −0.896736
\(890\) −3.27821 −0.109886
\(891\) 0 0
\(892\) −29.2751 −0.980202
\(893\) −28.4639 −0.952510
\(894\) 0 0
\(895\) −20.7170 −0.692492
\(896\) −26.1220 −0.872676
\(897\) 0 0
\(898\) −8.21799 −0.274238
\(899\) 1.25612 0.0418938
\(900\) 0 0
\(901\) −52.5609 −1.75106
\(902\) 0.000522024 0 1.73815e−5 0
\(903\) 0 0
\(904\) 1.45452 0.0483765
\(905\) 0.0561768 0.00186738
\(906\) 0 0
\(907\) 49.6527 1.64869 0.824345 0.566088i \(-0.191544\pi\)
0.824345 + 0.566088i \(0.191544\pi\)
\(908\) −35.8818 −1.19078
\(909\) 0 0
\(910\) 2.85812 0.0947458
\(911\) −46.1515 −1.52907 −0.764534 0.644583i \(-0.777031\pi\)
−0.764534 + 0.644583i \(0.777031\pi\)
\(912\) 0 0
\(913\) 0.0590461 0.00195414
\(914\) 5.41660 0.179165
\(915\) 0 0
\(916\) 3.61954 0.119593
\(917\) 47.5399 1.56991
\(918\) 0 0
\(919\) −49.4890 −1.63249 −0.816245 0.577706i \(-0.803948\pi\)
−0.816245 + 0.577706i \(0.803948\pi\)
\(920\) 6.07113 0.200159
\(921\) 0 0
\(922\) −12.5297 −0.412643
\(923\) 4.06307 0.133737
\(924\) 0 0
\(925\) 6.60420 0.217145
\(926\) −1.18357 −0.0388946
\(927\) 0 0
\(928\) −4.55068 −0.149383
\(929\) −26.7056 −0.876182 −0.438091 0.898931i \(-0.644345\pi\)
−0.438091 + 0.898931i \(0.644345\pi\)
\(930\) 0 0
\(931\) −3.07074 −0.100640
\(932\) 7.22975 0.236818
\(933\) 0 0
\(934\) −3.25014 −0.106348
\(935\) 0.0357226 0.00116825
\(936\) 0 0
\(937\) −35.2391 −1.15121 −0.575605 0.817728i \(-0.695233\pi\)
−0.575605 + 0.817728i \(0.695233\pi\)
\(938\) 4.91756 0.160564
\(939\) 0 0
\(940\) −18.8152 −0.613683
\(941\) 9.14156 0.298006 0.149003 0.988837i \(-0.452394\pi\)
0.149003 + 0.988837i \(0.452394\pi\)
\(942\) 0 0
\(943\) 0.590927 0.0192432
\(944\) 25.8177 0.840296
\(945\) 0 0
\(946\) −0.0358783 −0.00116650
\(947\) 19.2419 0.625277 0.312638 0.949872i \(-0.398787\pi\)
0.312638 + 0.949872i \(0.398787\pi\)
\(948\) 0 0
\(949\) −7.52919 −0.244408
\(950\) −1.18415 −0.0384189
\(951\) 0 0
\(952\) 18.8828 0.611994
\(953\) 22.3736 0.724751 0.362376 0.932032i \(-0.381966\pi\)
0.362376 + 0.932032i \(0.381966\pi\)
\(954\) 0 0
\(955\) −3.24218 −0.104914
\(956\) −53.6823 −1.73621
\(957\) 0 0
\(958\) 10.9954 0.355246
\(959\) 17.2757 0.557863
\(960\) 0 0
\(961\) −29.4222 −0.949102
\(962\) 7.78358 0.250953
\(963\) 0 0
\(964\) 23.8731 0.768900
\(965\) 10.2522 0.330029
\(966\) 0 0
\(967\) 14.7362 0.473885 0.236943 0.971524i \(-0.423855\pi\)
0.236943 + 0.971524i \(0.423855\pi\)
\(968\) −18.1044 −0.581898
\(969\) 0 0
\(970\) −3.03188 −0.0973478
\(971\) 21.7758 0.698819 0.349410 0.936970i \(-0.386382\pi\)
0.349410 + 0.936970i \(0.386382\pi\)
\(972\) 0 0
\(973\) 23.4892 0.753029
\(974\) −1.02934 −0.0329821
\(975\) 0 0
\(976\) 39.4517 1.26282
\(977\) −2.59126 −0.0829018 −0.0414509 0.999141i \(-0.513198\pi\)
−0.0414509 + 0.999141i \(0.513198\pi\)
\(978\) 0 0
\(979\) −0.0573571 −0.00183314
\(980\) −2.02982 −0.0648401
\(981\) 0 0
\(982\) −14.0225 −0.447475
\(983\) 41.5671 1.32578 0.662892 0.748715i \(-0.269329\pi\)
0.662892 + 0.748715i \(0.269329\pi\)
\(984\) 0 0
\(985\) 10.7593 0.342821
\(986\) 2.04170 0.0650210
\(987\) 0 0
\(988\) 13.5914 0.432399
\(989\) −40.6139 −1.29145
\(990\) 0 0
\(991\) 12.6958 0.403297 0.201648 0.979458i \(-0.435370\pi\)
0.201648 + 0.979458i \(0.435370\pi\)
\(992\) −5.71618 −0.181489
\(993\) 0 0
\(994\) −1.55703 −0.0493860
\(995\) −8.13396 −0.257864
\(996\) 0 0
\(997\) −54.4560 −1.72464 −0.862320 0.506364i \(-0.830989\pi\)
−0.862320 + 0.506364i \(0.830989\pi\)
\(998\) 3.69640 0.117007
\(999\) 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.s.1.4 7
3.2 odd 2 1305.2.a.t.1.4 yes 7
5.4 even 2 6525.2.a.bw.1.4 7
15.14 odd 2 6525.2.a.bv.1.4 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.a.s.1.4 7 1.1 even 1 trivial
1305.2.a.t.1.4 yes 7 3.2 odd 2
6525.2.a.bv.1.4 7 15.14 odd 2
6525.2.a.bw.1.4 7 5.4 even 2