Properties

Label 9075.2.a.da.1.3
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9075,2,Mod(1,9075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9075, 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("9075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.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: \(72.4642398343\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.18890\) of defining polynomial
Character \(\chi\) \(=\) 9075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.456850 q^{2} +1.00000 q^{3} -1.79129 q^{4} +0.456850 q^{6} -2.18890 q^{7} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.456850 q^{2} +1.00000 q^{3} -1.79129 q^{4} +0.456850 q^{6} -2.18890 q^{7} -1.73205 q^{8} +1.00000 q^{9} -1.79129 q^{12} -0.456850 q^{13} -1.00000 q^{14} +2.79129 q^{16} +1.73205 q^{17} +0.456850 q^{18} +1.73205 q^{19} -2.18890 q^{21} -1.20871 q^{23} -1.73205 q^{24} -0.208712 q^{26} +1.00000 q^{27} +3.92095 q^{28} +1.73205 q^{29} -3.20871 q^{31} +4.73930 q^{32} +0.791288 q^{34} -1.79129 q^{36} +1.58258 q^{37} +0.791288 q^{38} -0.456850 q^{39} -3.10260 q^{41} -1.00000 q^{42} +3.46410 q^{43} -0.552200 q^{46} -5.58258 q^{47} +2.79129 q^{48} -2.20871 q^{49} +1.73205 q^{51} +0.818350 q^{52} +6.79129 q^{53} +0.456850 q^{54} +3.79129 q^{56} +1.73205 q^{57} +0.791288 q^{58} -12.1652 q^{59} +7.11890 q^{61} -1.46590 q^{62} -2.18890 q^{63} -3.41742 q^{64} -9.16515 q^{67} -3.10260 q^{68} -1.20871 q^{69} +15.7477 q^{71} -1.73205 q^{72} +9.21245 q^{73} +0.723000 q^{74} -3.10260 q^{76} -0.208712 q^{78} -14.7701 q^{79} +1.00000 q^{81} -1.41742 q^{82} +14.5040 q^{83} +3.92095 q^{84} +1.58258 q^{86} +1.73205 q^{87} -2.62614 q^{89} +1.00000 q^{91} +2.16515 q^{92} -3.20871 q^{93} -2.55040 q^{94} +4.73930 q^{96} -13.5826 q^{97} -1.00905 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 2 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 2 q^{4} + 4 q^{9} + 2 q^{12} - 4 q^{14} + 2 q^{16} - 14 q^{23} - 10 q^{26} + 4 q^{27} - 22 q^{31} - 6 q^{34} + 2 q^{36} - 12 q^{37} - 6 q^{38} - 4 q^{42} - 4 q^{47} + 2 q^{48} - 18 q^{49} + 18 q^{53} + 6 q^{56} - 6 q^{58} - 12 q^{59} - 32 q^{64} - 14 q^{69} + 8 q^{71} - 10 q^{78} + 4 q^{81} - 24 q^{82} - 12 q^{86} - 38 q^{89} + 4 q^{91} - 28 q^{92} - 22 q^{93} - 36 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.456850 0.323042 0.161521 0.986869i \(-0.448360\pi\)
0.161521 + 0.986869i \(0.448360\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.79129 −0.895644
\(5\) 0 0
\(6\) 0.456850 0.186508
\(7\) −2.18890 −0.827327 −0.413663 0.910430i \(-0.635751\pi\)
−0.413663 + 0.910430i \(0.635751\pi\)
\(8\) −1.73205 −0.612372
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) −1.79129 −0.517100
\(13\) −0.456850 −0.126707 −0.0633537 0.997991i \(-0.520180\pi\)
−0.0633537 + 0.997991i \(0.520180\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 2.79129 0.697822
\(17\) 1.73205 0.420084 0.210042 0.977692i \(-0.432640\pi\)
0.210042 + 0.977692i \(0.432640\pi\)
\(18\) 0.456850 0.107681
\(19\) 1.73205 0.397360 0.198680 0.980064i \(-0.436335\pi\)
0.198680 + 0.980064i \(0.436335\pi\)
\(20\) 0 0
\(21\) −2.18890 −0.477657
\(22\) 0 0
\(23\) −1.20871 −0.252034 −0.126017 0.992028i \(-0.540219\pi\)
−0.126017 + 0.992028i \(0.540219\pi\)
\(24\) −1.73205 −0.353553
\(25\) 0 0
\(26\) −0.208712 −0.0409318
\(27\) 1.00000 0.192450
\(28\) 3.92095 0.740990
\(29\) 1.73205 0.321634 0.160817 0.986984i \(-0.448587\pi\)
0.160817 + 0.986984i \(0.448587\pi\)
\(30\) 0 0
\(31\) −3.20871 −0.576302 −0.288151 0.957585i \(-0.593040\pi\)
−0.288151 + 0.957585i \(0.593040\pi\)
\(32\) 4.73930 0.837798
\(33\) 0 0
\(34\) 0.791288 0.135705
\(35\) 0 0
\(36\) −1.79129 −0.298548
\(37\) 1.58258 0.260174 0.130087 0.991503i \(-0.458474\pi\)
0.130087 + 0.991503i \(0.458474\pi\)
\(38\) 0.791288 0.128364
\(39\) −0.456850 −0.0731546
\(40\) 0 0
\(41\) −3.10260 −0.484545 −0.242272 0.970208i \(-0.577893\pi\)
−0.242272 + 0.970208i \(0.577893\pi\)
\(42\) −1.00000 −0.154303
\(43\) 3.46410 0.528271 0.264135 0.964486i \(-0.414913\pi\)
0.264135 + 0.964486i \(0.414913\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −0.552200 −0.0814175
\(47\) −5.58258 −0.814302 −0.407151 0.913361i \(-0.633478\pi\)
−0.407151 + 0.913361i \(0.633478\pi\)
\(48\) 2.79129 0.402888
\(49\) −2.20871 −0.315530
\(50\) 0 0
\(51\) 1.73205 0.242536
\(52\) 0.818350 0.113485
\(53\) 6.79129 0.932855 0.466428 0.884559i \(-0.345541\pi\)
0.466428 + 0.884559i \(0.345541\pi\)
\(54\) 0.456850 0.0621694
\(55\) 0 0
\(56\) 3.79129 0.506632
\(57\) 1.73205 0.229416
\(58\) 0.791288 0.103901
\(59\) −12.1652 −1.58377 −0.791884 0.610672i \(-0.790900\pi\)
−0.791884 + 0.610672i \(0.790900\pi\)
\(60\) 0 0
\(61\) 7.11890 0.911482 0.455741 0.890112i \(-0.349374\pi\)
0.455741 + 0.890112i \(0.349374\pi\)
\(62\) −1.46590 −0.186170
\(63\) −2.18890 −0.275776
\(64\) −3.41742 −0.427178
\(65\) 0 0
\(66\) 0 0
\(67\) −9.16515 −1.11970 −0.559851 0.828593i \(-0.689142\pi\)
−0.559851 + 0.828593i \(0.689142\pi\)
\(68\) −3.10260 −0.376246
\(69\) −1.20871 −0.145512
\(70\) 0 0
\(71\) 15.7477 1.86891 0.934456 0.356079i \(-0.115887\pi\)
0.934456 + 0.356079i \(0.115887\pi\)
\(72\) −1.73205 −0.204124
\(73\) 9.21245 1.07824 0.539118 0.842230i \(-0.318758\pi\)
0.539118 + 0.842230i \(0.318758\pi\)
\(74\) 0.723000 0.0840471
\(75\) 0 0
\(76\) −3.10260 −0.355893
\(77\) 0 0
\(78\) −0.208712 −0.0236320
\(79\) −14.7701 −1.66177 −0.830883 0.556447i \(-0.812164\pi\)
−0.830883 + 0.556447i \(0.812164\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −1.41742 −0.156528
\(83\) 14.5040 1.59202 0.796008 0.605286i \(-0.206941\pi\)
0.796008 + 0.605286i \(0.206941\pi\)
\(84\) 3.92095 0.427811
\(85\) 0 0
\(86\) 1.58258 0.170654
\(87\) 1.73205 0.185695
\(88\) 0 0
\(89\) −2.62614 −0.278370 −0.139185 0.990266i \(-0.544448\pi\)
−0.139185 + 0.990266i \(0.544448\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 2.16515 0.225733
\(93\) −3.20871 −0.332728
\(94\) −2.55040 −0.263054
\(95\) 0 0
\(96\) 4.73930 0.483703
\(97\) −13.5826 −1.37910 −0.689551 0.724237i \(-0.742192\pi\)
−0.689551 + 0.724237i \(0.742192\pi\)
\(98\) −1.00905 −0.101930
\(99\) 0 0
\(100\) 0 0
\(101\) −15.4177 −1.53411 −0.767057 0.641579i \(-0.778280\pi\)
−0.767057 + 0.641579i \(0.778280\pi\)
\(102\) 0.791288 0.0783492
\(103\) 7.16515 0.706003 0.353002 0.935623i \(-0.385161\pi\)
0.353002 + 0.935623i \(0.385161\pi\)
\(104\) 0.791288 0.0775922
\(105\) 0 0
\(106\) 3.10260 0.301351
\(107\) 11.8582 1.14638 0.573188 0.819424i \(-0.305706\pi\)
0.573188 + 0.819424i \(0.305706\pi\)
\(108\) −1.79129 −0.172367
\(109\) 11.4014 1.09205 0.546026 0.837768i \(-0.316140\pi\)
0.546026 + 0.837768i \(0.316140\pi\)
\(110\) 0 0
\(111\) 1.58258 0.150211
\(112\) −6.10985 −0.577327
\(113\) −10.7477 −1.01106 −0.505531 0.862809i \(-0.668703\pi\)
−0.505531 + 0.862809i \(0.668703\pi\)
\(114\) 0.791288 0.0741109
\(115\) 0 0
\(116\) −3.10260 −0.288069
\(117\) −0.456850 −0.0422358
\(118\) −5.55765 −0.511623
\(119\) −3.79129 −0.347547
\(120\) 0 0
\(121\) 0 0
\(122\) 3.25227 0.294447
\(123\) −3.10260 −0.279752
\(124\) 5.74773 0.516161
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) −13.2288 −1.17386 −0.586931 0.809637i \(-0.699664\pi\)
−0.586931 + 0.809637i \(0.699664\pi\)
\(128\) −11.0399 −0.975795
\(129\) 3.46410 0.304997
\(130\) 0 0
\(131\) 1.27520 0.111415 0.0557074 0.998447i \(-0.482259\pi\)
0.0557074 + 0.998447i \(0.482259\pi\)
\(132\) 0 0
\(133\) −3.79129 −0.328746
\(134\) −4.18710 −0.361710
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 0.417424 0.0356630 0.0178315 0.999841i \(-0.494324\pi\)
0.0178315 + 0.999841i \(0.494324\pi\)
\(138\) −0.552200 −0.0470064
\(139\) 6.66205 0.565068 0.282534 0.959257i \(-0.408825\pi\)
0.282534 + 0.959257i \(0.408825\pi\)
\(140\) 0 0
\(141\) −5.58258 −0.470138
\(142\) 7.19435 0.603737
\(143\) 0 0
\(144\) 2.79129 0.232607
\(145\) 0 0
\(146\) 4.20871 0.348315
\(147\) −2.20871 −0.182172
\(148\) −2.83485 −0.233023
\(149\) −9.02175 −0.739091 −0.369545 0.929213i \(-0.620487\pi\)
−0.369545 + 0.929213i \(0.620487\pi\)
\(150\) 0 0
\(151\) 11.8582 0.965007 0.482504 0.875894i \(-0.339728\pi\)
0.482504 + 0.875894i \(0.339728\pi\)
\(152\) −3.00000 −0.243332
\(153\) 1.73205 0.140028
\(154\) 0 0
\(155\) 0 0
\(156\) 0.818350 0.0655205
\(157\) −7.74773 −0.618336 −0.309168 0.951007i \(-0.600051\pi\)
−0.309168 + 0.951007i \(0.600051\pi\)
\(158\) −6.74773 −0.536820
\(159\) 6.79129 0.538584
\(160\) 0 0
\(161\) 2.64575 0.208514
\(162\) 0.456850 0.0358935
\(163\) 16.3739 1.28250 0.641250 0.767332i \(-0.278416\pi\)
0.641250 + 0.767332i \(0.278416\pi\)
\(164\) 5.55765 0.433980
\(165\) 0 0
\(166\) 6.62614 0.514288
\(167\) −17.9681 −1.39041 −0.695205 0.718811i \(-0.744686\pi\)
−0.695205 + 0.718811i \(0.744686\pi\)
\(168\) 3.79129 0.292504
\(169\) −12.7913 −0.983945
\(170\) 0 0
\(171\) 1.73205 0.132453
\(172\) −6.20520 −0.473142
\(173\) −22.5167 −1.71191 −0.855955 0.517050i \(-0.827030\pi\)
−0.855955 + 0.517050i \(0.827030\pi\)
\(174\) 0.791288 0.0599874
\(175\) 0 0
\(176\) 0 0
\(177\) −12.1652 −0.914389
\(178\) −1.19975 −0.0899251
\(179\) −24.1652 −1.80619 −0.903094 0.429443i \(-0.858710\pi\)
−0.903094 + 0.429443i \(0.858710\pi\)
\(180\) 0 0
\(181\) −6.41742 −0.477003 −0.238502 0.971142i \(-0.576656\pi\)
−0.238502 + 0.971142i \(0.576656\pi\)
\(182\) 0.456850 0.0338640
\(183\) 7.11890 0.526244
\(184\) 2.09355 0.154339
\(185\) 0 0
\(186\) −1.46590 −0.107485
\(187\) 0 0
\(188\) 10.0000 0.729325
\(189\) −2.18890 −0.159219
\(190\) 0 0
\(191\) −9.62614 −0.696523 −0.348261 0.937397i \(-0.613228\pi\)
−0.348261 + 0.937397i \(0.613228\pi\)
\(192\) −3.41742 −0.246631
\(193\) −1.63670 −0.117812 −0.0589061 0.998264i \(-0.518761\pi\)
−0.0589061 + 0.998264i \(0.518761\pi\)
\(194\) −6.20520 −0.445508
\(195\) 0 0
\(196\) 3.95644 0.282603
\(197\) −16.2360 −1.15677 −0.578384 0.815765i \(-0.696316\pi\)
−0.578384 + 0.815765i \(0.696316\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) −9.16515 −0.646460
\(202\) −7.04356 −0.495583
\(203\) −3.79129 −0.266096
\(204\) −3.10260 −0.217226
\(205\) 0 0
\(206\) 3.27340 0.228069
\(207\) −1.20871 −0.0840113
\(208\) −1.27520 −0.0884192
\(209\) 0 0
\(210\) 0 0
\(211\) −15.7792 −1.08628 −0.543141 0.839641i \(-0.682765\pi\)
−0.543141 + 0.839641i \(0.682765\pi\)
\(212\) −12.1652 −0.835506
\(213\) 15.7477 1.07902
\(214\) 5.41742 0.370328
\(215\) 0 0
\(216\) −1.73205 −0.117851
\(217\) 7.02355 0.476790
\(218\) 5.20871 0.352779
\(219\) 9.21245 0.622520
\(220\) 0 0
\(221\) −0.791288 −0.0532278
\(222\) 0.723000 0.0485246
\(223\) −3.83485 −0.256800 −0.128400 0.991722i \(-0.540984\pi\)
−0.128400 + 0.991722i \(0.540984\pi\)
\(224\) −10.3739 −0.693133
\(225\) 0 0
\(226\) −4.91010 −0.326615
\(227\) 16.5022 1.09529 0.547643 0.836712i \(-0.315525\pi\)
0.547643 + 0.836712i \(0.315525\pi\)
\(228\) −3.10260 −0.205475
\(229\) −21.3739 −1.41242 −0.706212 0.708000i \(-0.749598\pi\)
−0.706212 + 0.708000i \(0.749598\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) 8.29875 0.543669 0.271835 0.962344i \(-0.412370\pi\)
0.271835 + 0.962344i \(0.412370\pi\)
\(234\) −0.208712 −0.0136439
\(235\) 0 0
\(236\) 21.7913 1.41849
\(237\) −14.7701 −0.959422
\(238\) −1.73205 −0.112272
\(239\) 16.5975 1.07360 0.536802 0.843708i \(-0.319632\pi\)
0.536802 + 0.843708i \(0.319632\pi\)
\(240\) 0 0
\(241\) −2.64575 −0.170428 −0.0852139 0.996363i \(-0.527157\pi\)
−0.0852139 + 0.996363i \(0.527157\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −12.7520 −0.816364
\(245\) 0 0
\(246\) −1.41742 −0.0903717
\(247\) −0.791288 −0.0503484
\(248\) 5.55765 0.352911
\(249\) 14.5040 0.919151
\(250\) 0 0
\(251\) 3.41742 0.215706 0.107853 0.994167i \(-0.465602\pi\)
0.107853 + 0.994167i \(0.465602\pi\)
\(252\) 3.92095 0.246997
\(253\) 0 0
\(254\) −6.04356 −0.379207
\(255\) 0 0
\(256\) 1.79129 0.111955
\(257\) −21.7913 −1.35930 −0.679652 0.733535i \(-0.737869\pi\)
−0.679652 + 0.733535i \(0.737869\pi\)
\(258\) 1.58258 0.0985269
\(259\) −3.46410 −0.215249
\(260\) 0 0
\(261\) 1.73205 0.107211
\(262\) 0.582576 0.0359916
\(263\) −14.1425 −0.872061 −0.436031 0.899932i \(-0.643616\pi\)
−0.436031 + 0.899932i \(0.643616\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1.73205 −0.106199
\(267\) −2.62614 −0.160717
\(268\) 16.4174 1.00285
\(269\) −9.95644 −0.607055 −0.303527 0.952823i \(-0.598164\pi\)
−0.303527 + 0.952823i \(0.598164\pi\)
\(270\) 0 0
\(271\) −8.48945 −0.515698 −0.257849 0.966185i \(-0.583014\pi\)
−0.257849 + 0.966185i \(0.583014\pi\)
\(272\) 4.83465 0.293144
\(273\) 1.00000 0.0605228
\(274\) 0.190700 0.0115206
\(275\) 0 0
\(276\) 2.16515 0.130327
\(277\) −9.76465 −0.586701 −0.293351 0.956005i \(-0.594770\pi\)
−0.293351 + 0.956005i \(0.594770\pi\)
\(278\) 3.04356 0.182541
\(279\) −3.20871 −0.192101
\(280\) 0 0
\(281\) −14.6947 −0.876610 −0.438305 0.898826i \(-0.644421\pi\)
−0.438305 + 0.898826i \(0.644421\pi\)
\(282\) −2.55040 −0.151874
\(283\) −11.5722 −0.687893 −0.343947 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(284\) −28.2087 −1.67388
\(285\) 0 0
\(286\) 0 0
\(287\) 6.79129 0.400877
\(288\) 4.73930 0.279266
\(289\) −14.0000 −0.823529
\(290\) 0 0
\(291\) −13.5826 −0.796225
\(292\) −16.5022 −0.965716
\(293\) −8.94630 −0.522649 −0.261324 0.965251i \(-0.584159\pi\)
−0.261324 + 0.965251i \(0.584159\pi\)
\(294\) −1.00905 −0.0588490
\(295\) 0 0
\(296\) −2.74110 −0.159323
\(297\) 0 0
\(298\) −4.12159 −0.238757
\(299\) 0.552200 0.0319346
\(300\) 0 0
\(301\) −7.58258 −0.437052
\(302\) 5.41742 0.311738
\(303\) −15.4177 −0.885721
\(304\) 4.83465 0.277286
\(305\) 0 0
\(306\) 0.791288 0.0452349
\(307\) −13.3241 −0.760447 −0.380223 0.924895i \(-0.624153\pi\)
−0.380223 + 0.924895i \(0.624153\pi\)
\(308\) 0 0
\(309\) 7.16515 0.407611
\(310\) 0 0
\(311\) −0.208712 −0.0118350 −0.00591749 0.999982i \(-0.501884\pi\)
−0.00591749 + 0.999982i \(0.501884\pi\)
\(312\) 0.791288 0.0447979
\(313\) −8.16515 −0.461522 −0.230761 0.973011i \(-0.574121\pi\)
−0.230761 + 0.973011i \(0.574121\pi\)
\(314\) −3.53955 −0.199748
\(315\) 0 0
\(316\) 26.4575 1.48835
\(317\) 0.165151 0.00927583 0.00463791 0.999989i \(-0.498524\pi\)
0.00463791 + 0.999989i \(0.498524\pi\)
\(318\) 3.10260 0.173985
\(319\) 0 0
\(320\) 0 0
\(321\) 11.8582 0.661861
\(322\) 1.20871 0.0673589
\(323\) 3.00000 0.166924
\(324\) −1.79129 −0.0995160
\(325\) 0 0
\(326\) 7.48040 0.414301
\(327\) 11.4014 0.630496
\(328\) 5.37386 0.296722
\(329\) 12.2197 0.673694
\(330\) 0 0
\(331\) −28.4955 −1.56625 −0.783126 0.621863i \(-0.786376\pi\)
−0.783126 + 0.621863i \(0.786376\pi\)
\(332\) −25.9808 −1.42588
\(333\) 1.58258 0.0867246
\(334\) −8.20871 −0.449161
\(335\) 0 0
\(336\) −6.10985 −0.333320
\(337\) 2.64575 0.144123 0.0720616 0.997400i \(-0.477042\pi\)
0.0720616 + 0.997400i \(0.477042\pi\)
\(338\) −5.84370 −0.317856
\(339\) −10.7477 −0.583736
\(340\) 0 0
\(341\) 0 0
\(342\) 0.791288 0.0427879
\(343\) 20.1570 1.08837
\(344\) −6.00000 −0.323498
\(345\) 0 0
\(346\) −10.2867 −0.553019
\(347\) 23.3549 1.25376 0.626879 0.779117i \(-0.284332\pi\)
0.626879 + 0.779117i \(0.284332\pi\)
\(348\) −3.10260 −0.166317
\(349\) 35.5746 1.90427 0.952133 0.305685i \(-0.0988853\pi\)
0.952133 + 0.305685i \(0.0988853\pi\)
\(350\) 0 0
\(351\) −0.456850 −0.0243849
\(352\) 0 0
\(353\) 4.58258 0.243906 0.121953 0.992536i \(-0.461084\pi\)
0.121953 + 0.992536i \(0.461084\pi\)
\(354\) −5.55765 −0.295386
\(355\) 0 0
\(356\) 4.70417 0.249320
\(357\) −3.79129 −0.200656
\(358\) −11.0399 −0.583474
\(359\) −19.9663 −1.05378 −0.526889 0.849934i \(-0.676642\pi\)
−0.526889 + 0.849934i \(0.676642\pi\)
\(360\) 0 0
\(361\) −16.0000 −0.842105
\(362\) −2.93180 −0.154092
\(363\) 0 0
\(364\) −1.79129 −0.0938890
\(365\) 0 0
\(366\) 3.25227 0.169999
\(367\) −4.25227 −0.221967 −0.110983 0.993822i \(-0.535400\pi\)
−0.110983 + 0.993822i \(0.535400\pi\)
\(368\) −3.37386 −0.175875
\(369\) −3.10260 −0.161515
\(370\) 0 0
\(371\) −14.8655 −0.771776
\(372\) 5.74773 0.298006
\(373\) 0.647551 0.0335289 0.0167645 0.999859i \(-0.494663\pi\)
0.0167645 + 0.999859i \(0.494663\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 9.66930 0.498656
\(377\) −0.791288 −0.0407534
\(378\) −1.00000 −0.0514344
\(379\) 10.6261 0.545828 0.272914 0.962038i \(-0.412013\pi\)
0.272914 + 0.962038i \(0.412013\pi\)
\(380\) 0 0
\(381\) −13.2288 −0.677730
\(382\) −4.39770 −0.225006
\(383\) 11.2087 0.572738 0.286369 0.958119i \(-0.407552\pi\)
0.286369 + 0.958119i \(0.407552\pi\)
\(384\) −11.0399 −0.563375
\(385\) 0 0
\(386\) −0.747727 −0.0380583
\(387\) 3.46410 0.176090
\(388\) 24.3303 1.23518
\(389\) −28.7477 −1.45757 −0.728784 0.684744i \(-0.759914\pi\)
−0.728784 + 0.684744i \(0.759914\pi\)
\(390\) 0 0
\(391\) −2.09355 −0.105875
\(392\) 3.82560 0.193222
\(393\) 1.27520 0.0643254
\(394\) −7.41742 −0.373684
\(395\) 0 0
\(396\) 0 0
\(397\) 11.3303 0.568652 0.284326 0.958728i \(-0.408230\pi\)
0.284326 + 0.958728i \(0.408230\pi\)
\(398\) 4.56850 0.228998
\(399\) −3.79129 −0.189802
\(400\) 0 0
\(401\) −11.4174 −0.570159 −0.285079 0.958504i \(-0.592020\pi\)
−0.285079 + 0.958504i \(0.592020\pi\)
\(402\) −4.18710 −0.208834
\(403\) 1.46590 0.0730217
\(404\) 27.6175 1.37402
\(405\) 0 0
\(406\) −1.73205 −0.0859602
\(407\) 0 0
\(408\) −3.00000 −0.148522
\(409\) −21.3368 −1.05504 −0.527519 0.849543i \(-0.676878\pi\)
−0.527519 + 0.849543i \(0.676878\pi\)
\(410\) 0 0
\(411\) 0.417424 0.0205900
\(412\) −12.8348 −0.632328
\(413\) 26.6283 1.31029
\(414\) −0.552200 −0.0271392
\(415\) 0 0
\(416\) −2.16515 −0.106155
\(417\) 6.66205 0.326242
\(418\) 0 0
\(419\) 2.79129 0.136363 0.0681817 0.997673i \(-0.478280\pi\)
0.0681817 + 0.997673i \(0.478280\pi\)
\(420\) 0 0
\(421\) −11.8348 −0.576795 −0.288398 0.957511i \(-0.593122\pi\)
−0.288398 + 0.957511i \(0.593122\pi\)
\(422\) −7.20871 −0.350915
\(423\) −5.58258 −0.271434
\(424\) −11.7629 −0.571255
\(425\) 0 0
\(426\) 7.19435 0.348568
\(427\) −15.5826 −0.754094
\(428\) −21.2415 −1.02674
\(429\) 0 0
\(430\) 0 0
\(431\) −24.9717 −1.20285 −0.601423 0.798931i \(-0.705399\pi\)
−0.601423 + 0.798931i \(0.705399\pi\)
\(432\) 2.79129 0.134296
\(433\) 23.5390 1.13121 0.565606 0.824675i \(-0.308642\pi\)
0.565606 + 0.824675i \(0.308642\pi\)
\(434\) 3.20871 0.154023
\(435\) 0 0
\(436\) −20.4231 −0.978090
\(437\) −2.09355 −0.100148
\(438\) 4.20871 0.201100
\(439\) 21.6983 1.03560 0.517802 0.855501i \(-0.326750\pi\)
0.517802 + 0.855501i \(0.326750\pi\)
\(440\) 0 0
\(441\) −2.20871 −0.105177
\(442\) −0.361500 −0.0171948
\(443\) −9.91288 −0.470975 −0.235488 0.971877i \(-0.575669\pi\)
−0.235488 + 0.971877i \(0.575669\pi\)
\(444\) −2.83485 −0.134536
\(445\) 0 0
\(446\) −1.75195 −0.0829573
\(447\) −9.02175 −0.426714
\(448\) 7.48040 0.353416
\(449\) −21.3303 −1.00664 −0.503320 0.864100i \(-0.667888\pi\)
−0.503320 + 0.864100i \(0.667888\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 19.2523 0.905551
\(453\) 11.8582 0.557147
\(454\) 7.53901 0.353824
\(455\) 0 0
\(456\) −3.00000 −0.140488
\(457\) −19.2433 −0.900162 −0.450081 0.892988i \(-0.648605\pi\)
−0.450081 + 0.892988i \(0.648605\pi\)
\(458\) −9.76465 −0.456272
\(459\) 1.73205 0.0808452
\(460\) 0 0
\(461\) 10.4877 0.488459 0.244229 0.969717i \(-0.421465\pi\)
0.244229 + 0.969717i \(0.421465\pi\)
\(462\) 0 0
\(463\) −27.7477 −1.28955 −0.644773 0.764374i \(-0.723048\pi\)
−0.644773 + 0.764374i \(0.723048\pi\)
\(464\) 4.83465 0.224443
\(465\) 0 0
\(466\) 3.79129 0.175628
\(467\) 34.1652 1.58097 0.790487 0.612478i \(-0.209827\pi\)
0.790487 + 0.612478i \(0.209827\pi\)
\(468\) 0.818350 0.0378283
\(469\) 20.0616 0.926359
\(470\) 0 0
\(471\) −7.74773 −0.356996
\(472\) 21.0707 0.969856
\(473\) 0 0
\(474\) −6.74773 −0.309933
\(475\) 0 0
\(476\) 6.79129 0.311278
\(477\) 6.79129 0.310952
\(478\) 7.58258 0.346819
\(479\) −38.4865 −1.75849 −0.879247 0.476366i \(-0.841954\pi\)
−0.879247 + 0.476366i \(0.841954\pi\)
\(480\) 0 0
\(481\) −0.723000 −0.0329660
\(482\) −1.20871 −0.0550553
\(483\) 2.64575 0.120386
\(484\) 0 0
\(485\) 0 0
\(486\) 0.456850 0.0207231
\(487\) −2.58258 −0.117028 −0.0585138 0.998287i \(-0.518636\pi\)
−0.0585138 + 0.998287i \(0.518636\pi\)
\(488\) −12.3303 −0.558167
\(489\) 16.3739 0.740452
\(490\) 0 0
\(491\) 33.2705 1.50148 0.750738 0.660601i \(-0.229698\pi\)
0.750738 + 0.660601i \(0.229698\pi\)
\(492\) 5.55765 0.250558
\(493\) 3.00000 0.135113
\(494\) −0.361500 −0.0162647
\(495\) 0 0
\(496\) −8.95644 −0.402156
\(497\) −34.4702 −1.54620
\(498\) 6.62614 0.296924
\(499\) −6.83485 −0.305970 −0.152985 0.988229i \(-0.548889\pi\)
−0.152985 + 0.988229i \(0.548889\pi\)
\(500\) 0 0
\(501\) −17.9681 −0.802754
\(502\) 1.56125 0.0696820
\(503\) −8.56490 −0.381890 −0.190945 0.981601i \(-0.561155\pi\)
−0.190945 + 0.981601i \(0.561155\pi\)
\(504\) 3.79129 0.168877
\(505\) 0 0
\(506\) 0 0
\(507\) −12.7913 −0.568081
\(508\) 23.6965 1.05136
\(509\) 6.25227 0.277127 0.138564 0.990354i \(-0.455751\pi\)
0.138564 + 0.990354i \(0.455751\pi\)
\(510\) 0 0
\(511\) −20.1652 −0.892054
\(512\) 22.8981 1.01196
\(513\) 1.73205 0.0764719
\(514\) −9.95536 −0.439112
\(515\) 0 0
\(516\) −6.20520 −0.273169
\(517\) 0 0
\(518\) −1.58258 −0.0695344
\(519\) −22.5167 −0.988372
\(520\) 0 0
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) 0.791288 0.0346337
\(523\) 21.3368 0.932994 0.466497 0.884523i \(-0.345516\pi\)
0.466497 + 0.884523i \(0.345516\pi\)
\(524\) −2.28425 −0.0997880
\(525\) 0 0
\(526\) −6.46099 −0.281712
\(527\) −5.55765 −0.242095
\(528\) 0 0
\(529\) −21.5390 −0.936479
\(530\) 0 0
\(531\) −12.1652 −0.527923
\(532\) 6.79129 0.294440
\(533\) 1.41742 0.0613955
\(534\) −1.19975 −0.0519183
\(535\) 0 0
\(536\) 15.8745 0.685674
\(537\) −24.1652 −1.04280
\(538\) −4.54860 −0.196104
\(539\) 0 0
\(540\) 0 0
\(541\) 18.0634 0.776607 0.388303 0.921532i \(-0.373061\pi\)
0.388303 + 0.921532i \(0.373061\pi\)
\(542\) −3.87841 −0.166592
\(543\) −6.41742 −0.275398
\(544\) 8.20871 0.351946
\(545\) 0 0
\(546\) 0.456850 0.0195514
\(547\) 40.7509 1.74238 0.871191 0.490945i \(-0.163348\pi\)
0.871191 + 0.490945i \(0.163348\pi\)
\(548\) −0.747727 −0.0319413
\(549\) 7.11890 0.303827
\(550\) 0 0
\(551\) 3.00000 0.127804
\(552\) 2.09355 0.0891074
\(553\) 32.3303 1.37482
\(554\) −4.46099 −0.189529
\(555\) 0 0
\(556\) −11.9337 −0.506100
\(557\) 29.2741 1.24038 0.620191 0.784451i \(-0.287055\pi\)
0.620191 + 0.784451i \(0.287055\pi\)
\(558\) −1.46590 −0.0620565
\(559\) −1.58258 −0.0669358
\(560\) 0 0
\(561\) 0 0
\(562\) −6.71326 −0.283182
\(563\) 23.1642 0.976255 0.488128 0.872772i \(-0.337680\pi\)
0.488128 + 0.872772i \(0.337680\pi\)
\(564\) 10.0000 0.421076
\(565\) 0 0
\(566\) −5.28674 −0.222218
\(567\) −2.18890 −0.0919252
\(568\) −27.2759 −1.14447
\(569\) 30.7400 1.28869 0.644343 0.764736i \(-0.277131\pi\)
0.644343 + 0.764736i \(0.277131\pi\)
\(570\) 0 0
\(571\) 40.3139 1.68709 0.843543 0.537062i \(-0.180466\pi\)
0.843543 + 0.537062i \(0.180466\pi\)
\(572\) 0 0
\(573\) −9.62614 −0.402138
\(574\) 3.10260 0.129500
\(575\) 0 0
\(576\) −3.41742 −0.142393
\(577\) 27.1216 1.12909 0.564543 0.825403i \(-0.309052\pi\)
0.564543 + 0.825403i \(0.309052\pi\)
\(578\) −6.39590 −0.266035
\(579\) −1.63670 −0.0680190
\(580\) 0 0
\(581\) −31.7477 −1.31712
\(582\) −6.20520 −0.257214
\(583\) 0 0
\(584\) −15.9564 −0.660282
\(585\) 0 0
\(586\) −4.08712 −0.168837
\(587\) −29.8693 −1.23284 −0.616419 0.787418i \(-0.711417\pi\)
−0.616419 + 0.787418i \(0.711417\pi\)
\(588\) 3.95644 0.163161
\(589\) −5.55765 −0.228999
\(590\) 0 0
\(591\) −16.2360 −0.667860
\(592\) 4.41742 0.181555
\(593\) −3.38865 −0.139155 −0.0695776 0.997577i \(-0.522165\pi\)
−0.0695776 + 0.997577i \(0.522165\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 16.1606 0.661962
\(597\) 10.0000 0.409273
\(598\) 0.252273 0.0103162
\(599\) −41.1216 −1.68018 −0.840091 0.542445i \(-0.817499\pi\)
−0.840091 + 0.542445i \(0.817499\pi\)
\(600\) 0 0
\(601\) 8.01270 0.326845 0.163422 0.986556i \(-0.447747\pi\)
0.163422 + 0.986556i \(0.447747\pi\)
\(602\) −3.46410 −0.141186
\(603\) −9.16515 −0.373234
\(604\) −21.2415 −0.864303
\(605\) 0 0
\(606\) −7.04356 −0.286125
\(607\) 19.7756 0.802665 0.401333 0.915932i \(-0.368547\pi\)
0.401333 + 0.915932i \(0.368547\pi\)
\(608\) 8.20871 0.332907
\(609\) −3.79129 −0.153631
\(610\) 0 0
\(611\) 2.55040 0.103178
\(612\) −3.10260 −0.125415
\(613\) 7.30960 0.295232 0.147616 0.989045i \(-0.452840\pi\)
0.147616 + 0.989045i \(0.452840\pi\)
\(614\) −6.08712 −0.245656
\(615\) 0 0
\(616\) 0 0
\(617\) 34.1216 1.37368 0.686842 0.726807i \(-0.258997\pi\)
0.686842 + 0.726807i \(0.258997\pi\)
\(618\) 3.27340 0.131676
\(619\) −5.00000 −0.200967 −0.100483 0.994939i \(-0.532039\pi\)
−0.100483 + 0.994939i \(0.532039\pi\)
\(620\) 0 0
\(621\) −1.20871 −0.0485039
\(622\) −0.0953502 −0.00382319
\(623\) 5.74835 0.230303
\(624\) −1.27520 −0.0510489
\(625\) 0 0
\(626\) −3.73025 −0.149091
\(627\) 0 0
\(628\) 13.8784 0.553809
\(629\) 2.74110 0.109295
\(630\) 0 0
\(631\) −13.5390 −0.538980 −0.269490 0.963003i \(-0.586855\pi\)
−0.269490 + 0.963003i \(0.586855\pi\)
\(632\) 25.5826 1.01762
\(633\) −15.7792 −0.627165
\(634\) 0.0754495 0.00299648
\(635\) 0 0
\(636\) −12.1652 −0.482380
\(637\) 1.00905 0.0399800
\(638\) 0 0
\(639\) 15.7477 0.622970
\(640\) 0 0
\(641\) −28.3739 −1.12070 −0.560350 0.828256i \(-0.689333\pi\)
−0.560350 + 0.828256i \(0.689333\pi\)
\(642\) 5.41742 0.213809
\(643\) −27.4955 −1.08431 −0.542157 0.840277i \(-0.682392\pi\)
−0.542157 + 0.840277i \(0.682392\pi\)
\(644\) −4.73930 −0.186755
\(645\) 0 0
\(646\) 1.37055 0.0539236
\(647\) 7.00000 0.275198 0.137599 0.990488i \(-0.456061\pi\)
0.137599 + 0.990488i \(0.456061\pi\)
\(648\) −1.73205 −0.0680414
\(649\) 0 0
\(650\) 0 0
\(651\) 7.02355 0.275275
\(652\) −29.3303 −1.14866
\(653\) 38.4955 1.50644 0.753222 0.657767i \(-0.228499\pi\)
0.753222 + 0.657767i \(0.228499\pi\)
\(654\) 5.20871 0.203677
\(655\) 0 0
\(656\) −8.66025 −0.338126
\(657\) 9.21245 0.359412
\(658\) 5.58258 0.217631
\(659\) −6.94810 −0.270660 −0.135330 0.990801i \(-0.543209\pi\)
−0.135330 + 0.990801i \(0.543209\pi\)
\(660\) 0 0
\(661\) −37.8693 −1.47295 −0.736473 0.676467i \(-0.763510\pi\)
−0.736473 + 0.676467i \(0.763510\pi\)
\(662\) −13.0182 −0.505965
\(663\) −0.791288 −0.0307311
\(664\) −25.1216 −0.974907
\(665\) 0 0
\(666\) 0.723000 0.0280157
\(667\) −2.09355 −0.0810626
\(668\) 32.1860 1.24531
\(669\) −3.83485 −0.148264
\(670\) 0 0
\(671\) 0 0
\(672\) −10.3739 −0.400180
\(673\) −0.170800 −0.00658384 −0.00329192 0.999995i \(-0.501048\pi\)
−0.00329192 + 0.999995i \(0.501048\pi\)
\(674\) 1.20871 0.0465579
\(675\) 0 0
\(676\) 22.9129 0.881265
\(677\) 22.9735 0.882944 0.441472 0.897275i \(-0.354456\pi\)
0.441472 + 0.897275i \(0.354456\pi\)
\(678\) −4.91010 −0.188571
\(679\) 29.7309 1.14097
\(680\) 0 0
\(681\) 16.5022 0.632364
\(682\) 0 0
\(683\) 13.3303 0.510070 0.255035 0.966932i \(-0.417913\pi\)
0.255035 + 0.966932i \(0.417913\pi\)
\(684\) −3.10260 −0.118631
\(685\) 0 0
\(686\) 9.20871 0.351590
\(687\) −21.3739 −0.815464
\(688\) 9.66930 0.368639
\(689\) −3.10260 −0.118200
\(690\) 0 0
\(691\) 7.79129 0.296395 0.148197 0.988958i \(-0.452653\pi\)
0.148197 + 0.988958i \(0.452653\pi\)
\(692\) 40.3338 1.53326
\(693\) 0 0
\(694\) 10.6697 0.405016
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) −5.37386 −0.203550
\(698\) 16.2523 0.615158
\(699\) 8.29875 0.313888
\(700\) 0 0
\(701\) −16.6929 −0.630481 −0.315240 0.949012i \(-0.602085\pi\)
−0.315240 + 0.949012i \(0.602085\pi\)
\(702\) −0.208712 −0.00787733
\(703\) 2.74110 0.103383
\(704\) 0 0
\(705\) 0 0
\(706\) 2.09355 0.0787918
\(707\) 33.7477 1.26921
\(708\) 21.7913 0.818967
\(709\) 22.1216 0.830794 0.415397 0.909640i \(-0.363643\pi\)
0.415397 + 0.909640i \(0.363643\pi\)
\(710\) 0 0
\(711\) −14.7701 −0.553922
\(712\) 4.54860 0.170466
\(713\) 3.87841 0.145248
\(714\) −1.73205 −0.0648204
\(715\) 0 0
\(716\) 43.2867 1.61770
\(717\) 16.5975 0.619845
\(718\) −9.12159 −0.340415
\(719\) 11.1652 0.416390 0.208195 0.978087i \(-0.433241\pi\)
0.208195 + 0.978087i \(0.433241\pi\)
\(720\) 0 0
\(721\) −15.6838 −0.584096
\(722\) −7.30960 −0.272035
\(723\) −2.64575 −0.0983965
\(724\) 11.4955 0.427225
\(725\) 0 0
\(726\) 0 0
\(727\) −20.7913 −0.771106 −0.385553 0.922686i \(-0.625989\pi\)
−0.385553 + 0.922686i \(0.625989\pi\)
\(728\) −1.73205 −0.0641941
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.00000 0.221918
\(732\) −12.7520 −0.471328
\(733\) −0.532300 −0.0196609 −0.00983047 0.999952i \(-0.503129\pi\)
−0.00983047 + 0.999952i \(0.503129\pi\)
\(734\) −1.94265 −0.0717046
\(735\) 0 0
\(736\) −5.72845 −0.211154
\(737\) 0 0
\(738\) −1.41742 −0.0521761
\(739\) −17.0743 −0.628087 −0.314043 0.949409i \(-0.601684\pi\)
−0.314043 + 0.949409i \(0.601684\pi\)
\(740\) 0 0
\(741\) −0.791288 −0.0290687
\(742\) −6.79129 −0.249316
\(743\) 37.9343 1.39168 0.695838 0.718199i \(-0.255033\pi\)
0.695838 + 0.718199i \(0.255033\pi\)
\(744\) 5.55765 0.203753
\(745\) 0 0
\(746\) 0.295834 0.0108312
\(747\) 14.5040 0.530672
\(748\) 0 0
\(749\) −25.9564 −0.948428
\(750\) 0 0
\(751\) −8.04356 −0.293514 −0.146757 0.989173i \(-0.546883\pi\)
−0.146757 + 0.989173i \(0.546883\pi\)
\(752\) −15.5826 −0.568238
\(753\) 3.41742 0.124538
\(754\) −0.361500 −0.0131651
\(755\) 0 0
\(756\) 3.92095 0.142604
\(757\) −54.2867 −1.97309 −0.986543 0.163504i \(-0.947720\pi\)
−0.986543 + 0.163504i \(0.947720\pi\)
\(758\) 4.85455 0.176325
\(759\) 0 0
\(760\) 0 0
\(761\) −45.1287 −1.63591 −0.817957 0.575280i \(-0.804893\pi\)
−0.817957 + 0.575280i \(0.804893\pi\)
\(762\) −6.04356 −0.218935
\(763\) −24.9564 −0.903484
\(764\) 17.2432 0.623836
\(765\) 0 0
\(766\) 5.12070 0.185019
\(767\) 5.55765 0.200675
\(768\) 1.79129 0.0646375
\(769\) −15.2469 −0.549816 −0.274908 0.961471i \(-0.588647\pi\)
−0.274908 + 0.961471i \(0.588647\pi\)
\(770\) 0 0
\(771\) −21.7913 −0.784794
\(772\) 2.93180 0.105518
\(773\) 5.37386 0.193284 0.0966422 0.995319i \(-0.469190\pi\)
0.0966422 + 0.995319i \(0.469190\pi\)
\(774\) 1.58258 0.0568845
\(775\) 0 0
\(776\) 23.5257 0.844524
\(777\) −3.46410 −0.124274
\(778\) −13.1334 −0.470855
\(779\) −5.37386 −0.192539
\(780\) 0 0
\(781\) 0 0
\(782\) −0.956439 −0.0342022
\(783\) 1.73205 0.0618984
\(784\) −6.16515 −0.220184
\(785\) 0 0
\(786\) 0.582576 0.0207798
\(787\) −12.0290 −0.428788 −0.214394 0.976747i \(-0.568778\pi\)
−0.214394 + 0.976747i \(0.568778\pi\)
\(788\) 29.0834 1.03605
\(789\) −14.1425 −0.503485
\(790\) 0 0
\(791\) 23.5257 0.836478
\(792\) 0 0
\(793\) −3.25227 −0.115492
\(794\) 5.17625 0.183698
\(795\) 0 0
\(796\) −17.9129 −0.634905
\(797\) 53.1652 1.88321 0.941603 0.336725i \(-0.109319\pi\)
0.941603 + 0.336725i \(0.109319\pi\)
\(798\) −1.73205 −0.0613139
\(799\) −9.66930 −0.342075
\(800\) 0 0
\(801\) −2.62614 −0.0927900
\(802\) −5.21605 −0.184185
\(803\) 0 0
\(804\) 16.4174 0.578998
\(805\) 0 0
\(806\) 0.669697 0.0235891
\(807\) −9.95644 −0.350483
\(808\) 26.7042 0.939449
\(809\) −20.5185 −0.721391 −0.360695 0.932684i \(-0.617461\pi\)
−0.360695 + 0.932684i \(0.617461\pi\)
\(810\) 0 0
\(811\) −6.18530 −0.217195 −0.108598 0.994086i \(-0.534636\pi\)
−0.108598 + 0.994086i \(0.534636\pi\)
\(812\) 6.79129 0.238327
\(813\) −8.48945 −0.297738
\(814\) 0 0
\(815\) 0 0
\(816\) 4.83465 0.169247
\(817\) 6.00000 0.209913
\(818\) −9.74773 −0.340821
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) 10.0308 0.350078 0.175039 0.984562i \(-0.443995\pi\)
0.175039 + 0.984562i \(0.443995\pi\)
\(822\) 0.190700 0.00665144
\(823\) −23.4174 −0.816280 −0.408140 0.912919i \(-0.633822\pi\)
−0.408140 + 0.912919i \(0.633822\pi\)
\(824\) −12.4104 −0.432337
\(825\) 0 0
\(826\) 12.1652 0.423280
\(827\) −22.0598 −0.767095 −0.383547 0.923521i \(-0.625298\pi\)
−0.383547 + 0.923521i \(0.625298\pi\)
\(828\) 2.16515 0.0752442
\(829\) −3.87841 −0.134703 −0.0673514 0.997729i \(-0.521455\pi\)
−0.0673514 + 0.997729i \(0.521455\pi\)
\(830\) 0 0
\(831\) −9.76465 −0.338732
\(832\) 1.56125 0.0541266
\(833\) −3.82560 −0.132549
\(834\) 3.04356 0.105390
\(835\) 0 0
\(836\) 0 0
\(837\) −3.20871 −0.110909
\(838\) 1.27520 0.0440511
\(839\) −12.6261 −0.435903 −0.217951 0.975960i \(-0.569937\pi\)
−0.217951 + 0.975960i \(0.569937\pi\)
\(840\) 0 0
\(841\) −26.0000 −0.896552
\(842\) −5.40675 −0.186329
\(843\) −14.6947 −0.506111
\(844\) 28.2650 0.972922
\(845\) 0 0
\(846\) −2.55040 −0.0876846
\(847\) 0 0
\(848\) 18.9564 0.650967
\(849\) −11.5722 −0.397155
\(850\) 0 0
\(851\) −1.91288 −0.0655726
\(852\) −28.2087 −0.966415
\(853\) −33.6320 −1.15154 −0.575768 0.817613i \(-0.695297\pi\)
−0.575768 + 0.817613i \(0.695297\pi\)
\(854\) −7.11890 −0.243604
\(855\) 0 0
\(856\) −20.5390 −0.702009
\(857\) −18.3097 −0.625446 −0.312723 0.949844i \(-0.601241\pi\)
−0.312723 + 0.949844i \(0.601241\pi\)
\(858\) 0 0
\(859\) 21.7913 0.743509 0.371755 0.928331i \(-0.378756\pi\)
0.371755 + 0.928331i \(0.378756\pi\)
\(860\) 0 0
\(861\) 6.79129 0.231446
\(862\) −11.4083 −0.388569
\(863\) 7.25227 0.246870 0.123435 0.992353i \(-0.460609\pi\)
0.123435 + 0.992353i \(0.460609\pi\)
\(864\) 4.73930 0.161234
\(865\) 0 0
\(866\) 10.7538 0.365429
\(867\) −14.0000 −0.475465
\(868\) −12.5812 −0.427034
\(869\) 0 0
\(870\) 0 0
\(871\) 4.18710 0.141875
\(872\) −19.7477 −0.668742
\(873\) −13.5826 −0.459701
\(874\) −0.956439 −0.0323520
\(875\) 0 0
\(876\) −16.5022 −0.557556
\(877\) −19.7955 −0.668445 −0.334223 0.942494i \(-0.608474\pi\)
−0.334223 + 0.942494i \(0.608474\pi\)
\(878\) 9.91288 0.334543
\(879\) −8.94630 −0.301751
\(880\) 0 0
\(881\) 36.2432 1.22106 0.610532 0.791992i \(-0.290956\pi\)
0.610532 + 0.791992i \(0.290956\pi\)
\(882\) −1.00905 −0.0339765
\(883\) 22.8693 0.769614 0.384807 0.922997i \(-0.374268\pi\)
0.384807 + 0.922997i \(0.374268\pi\)
\(884\) 1.41742 0.0476731
\(885\) 0 0
\(886\) −4.52870 −0.152145
\(887\) −19.7756 −0.663998 −0.331999 0.943280i \(-0.607723\pi\)
−0.331999 + 0.943280i \(0.607723\pi\)
\(888\) −2.74110 −0.0919853
\(889\) 28.9564 0.971168
\(890\) 0 0
\(891\) 0 0
\(892\) 6.86932 0.230002
\(893\) −9.66930 −0.323571
\(894\) −4.12159 −0.137847
\(895\) 0 0
\(896\) 24.1652 0.807301
\(897\) 0.552200 0.0184374
\(898\) −9.74475 −0.325187
\(899\) −5.55765 −0.185358
\(900\) 0 0
\(901\) 11.7629 0.391878
\(902\) 0 0
\(903\) −7.58258 −0.252332
\(904\) 18.6156 0.619146
\(905\) 0 0
\(906\) 5.41742 0.179982
\(907\) −29.2867 −0.972450 −0.486225 0.873834i \(-0.661627\pi\)
−0.486225 + 0.873834i \(0.661627\pi\)
\(908\) −29.5601 −0.980987
\(909\) −15.4177 −0.511371
\(910\) 0 0
\(911\) 31.8693 1.05588 0.527939 0.849282i \(-0.322965\pi\)
0.527939 + 0.849282i \(0.322965\pi\)
\(912\) 4.83465 0.160091
\(913\) 0 0
\(914\) −8.79129 −0.290790
\(915\) 0 0
\(916\) 38.2867 1.26503
\(917\) −2.79129 −0.0921764
\(918\) 0.791288 0.0261164
\(919\) 6.39590 0.210981 0.105491 0.994420i \(-0.466359\pi\)
0.105491 + 0.994420i \(0.466359\pi\)
\(920\) 0 0
\(921\) −13.3241 −0.439044
\(922\) 4.79129 0.157793
\(923\) −7.19435 −0.236805
\(924\) 0 0
\(925\) 0 0
\(926\) −12.6766 −0.416577
\(927\) 7.16515 0.235334
\(928\) 8.20871 0.269464
\(929\) −32.2867 −1.05929 −0.529647 0.848218i \(-0.677676\pi\)
−0.529647 + 0.848218i \(0.677676\pi\)
\(930\) 0 0
\(931\) −3.82560 −0.125379
\(932\) −14.8655 −0.486934
\(933\) −0.208712 −0.00683293
\(934\) 15.6084 0.510721
\(935\) 0 0
\(936\) 0.791288 0.0258641
\(937\) 58.5680 1.91333 0.956667 0.291184i \(-0.0940492\pi\)
0.956667 + 0.291184i \(0.0940492\pi\)
\(938\) 9.16515 0.299253
\(939\) −8.16515 −0.266460
\(940\) 0 0
\(941\) 53.0659 1.72990 0.864950 0.501858i \(-0.167350\pi\)
0.864950 + 0.501858i \(0.167350\pi\)
\(942\) −3.53955 −0.115325
\(943\) 3.75015 0.122122
\(944\) −33.9564 −1.10519
\(945\) 0 0
\(946\) 0 0
\(947\) 51.7042 1.68016 0.840080 0.542463i \(-0.182508\pi\)
0.840080 + 0.542463i \(0.182508\pi\)
\(948\) 26.4575 0.859300
\(949\) −4.20871 −0.136621
\(950\) 0 0
\(951\) 0.165151 0.00535540
\(952\) 6.56670 0.212828
\(953\) 30.6645 0.993321 0.496661 0.867945i \(-0.334559\pi\)
0.496661 + 0.867945i \(0.334559\pi\)
\(954\) 3.10260 0.100450
\(955\) 0 0
\(956\) −29.7309 −0.961566
\(957\) 0 0
\(958\) −17.5826 −0.568067
\(959\) −0.913701 −0.0295049
\(960\) 0 0
\(961\) −20.7042 −0.667876
\(962\) −0.330303 −0.0106494
\(963\) 11.8582 0.382125
\(964\) 4.73930 0.152643
\(965\) 0 0
\(966\) 1.20871 0.0388897
\(967\) 7.04345 0.226502 0.113251 0.993566i \(-0.463874\pi\)
0.113251 + 0.993566i \(0.463874\pi\)
\(968\) 0 0
\(969\) 3.00000 0.0963739
\(970\) 0 0
\(971\) −18.1652 −0.582947 −0.291474 0.956579i \(-0.594146\pi\)
−0.291474 + 0.956579i \(0.594146\pi\)
\(972\) −1.79129 −0.0574556
\(973\) −14.5826 −0.467496
\(974\) −1.17985 −0.0378048
\(975\) 0 0
\(976\) 19.8709 0.636052
\(977\) 21.7913 0.697165 0.348583 0.937278i \(-0.386663\pi\)
0.348583 + 0.937278i \(0.386663\pi\)
\(978\) 7.48040 0.239197
\(979\) 0 0
\(980\) 0 0
\(981\) 11.4014 0.364017
\(982\) 15.1996 0.485039
\(983\) 20.6606 0.658971 0.329485 0.944161i \(-0.393125\pi\)
0.329485 + 0.944161i \(0.393125\pi\)
\(984\) 5.37386 0.171313
\(985\) 0 0
\(986\) 1.37055 0.0436472
\(987\) 12.2197 0.388958
\(988\) 1.41742 0.0450943
\(989\) −4.18710 −0.133142
\(990\) 0 0
\(991\) −5.62614 −0.178720 −0.0893601 0.995999i \(-0.528482\pi\)
−0.0893601 + 0.995999i \(0.528482\pi\)
\(992\) −15.2071 −0.482825
\(993\) −28.4955 −0.904276
\(994\) −15.7477 −0.499488
\(995\) 0 0
\(996\) −25.9808 −0.823232
\(997\) −4.75920 −0.150725 −0.0753627 0.997156i \(-0.524011\pi\)
−0.0753627 + 0.997156i \(0.524011\pi\)
\(998\) −3.12250 −0.0988411
\(999\) 1.58258 0.0500705
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 9075.2.a.da.1.3 yes 4
5.4 even 2 9075.2.a.ct.1.2 4
11.10 odd 2 inner 9075.2.a.da.1.2 yes 4
55.54 odd 2 9075.2.a.ct.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9075.2.a.ct.1.2 4 5.4 even 2
9075.2.a.ct.1.3 yes 4 55.54 odd 2
9075.2.a.da.1.2 yes 4 11.10 odd 2 inner
9075.2.a.da.1.3 yes 4 1.1 even 1 trivial