Properties

Label 1305.2.a.q.1.2
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $1$
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] = [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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.469.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.772866\) 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)
 
\(f(q)\) \(=\) \(q-0.772866 q^{2} -1.40268 q^{4} -1.00000 q^{5} +2.17554 q^{7} +2.62981 q^{8} +O(q^{10})\) \(q-0.772866 q^{2} -1.40268 q^{4} -1.00000 q^{5} +2.17554 q^{7} +2.62981 q^{8} +0.772866 q^{10} -3.00000 q^{11} -0.629813 q^{13} -1.68140 q^{14} +0.772866 q^{16} -4.17554 q^{17} +4.80536 q^{19} +1.40268 q^{20} +2.31860 q^{22} -2.08408 q^{23} +1.00000 q^{25} +0.486761 q^{26} -3.05159 q^{28} +1.00000 q^{29} -5.85695 q^{32} +3.22713 q^{34} -2.17554 q^{35} +6.08408 q^{37} -3.71390 q^{38} -2.62981 q^{40} -0.824456 q^{41} -8.72128 q^{43} +4.20804 q^{44} +1.61072 q^{46} +8.98090 q^{47} -2.26701 q^{49} -0.772866 q^{50} +0.883426 q^{52} -6.88944 q^{53} +3.00000 q^{55} +5.72128 q^{56} -0.772866 q^{58} -6.45427 q^{59} -2.80536 q^{61} +2.98090 q^{64} +0.629813 q^{65} -11.0841 q^{67} +5.85695 q^{68} +1.68140 q^{70} -2.63719 q^{71} -14.7863 q^{73} -4.70218 q^{74} -6.74037 q^{76} -6.52663 q^{77} -12.0650 q^{79} -0.772866 q^{80} +0.637193 q^{82} +7.62981 q^{83} +4.17554 q^{85} +6.74037 q^{86} -7.88944 q^{88} -17.8894 q^{89} -1.37019 q^{91} +2.92330 q^{92} -6.94103 q^{94} -4.80536 q^{95} +0.538351 q^{97} +1.75209 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 5 q^{4} - 3 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 5 q^{4} - 3 q^{5} - 4 q^{7} + q^{10} - 9 q^{11} + 6 q^{13} - 9 q^{14} + q^{16} - 2 q^{17} - 4 q^{19} - 5 q^{20} + 3 q^{22} - q^{23} + 3 q^{25} - 13 q^{26} - 21 q^{28} + 3 q^{29} - 11 q^{32} + 11 q^{34} + 4 q^{35} + 13 q^{37} + 2 q^{38} - 13 q^{41} - 13 q^{43} - 15 q^{44} - 32 q^{46} - 2 q^{47} + 9 q^{49} - q^{50} + 25 q^{52} + 3 q^{53} + 9 q^{55} + 4 q^{56} - q^{58} - 22 q^{59} + 10 q^{61} - 20 q^{64} - 6 q^{65} - 28 q^{67} + 11 q^{68} + 9 q^{70} + 3 q^{73} + 28 q^{74} - 36 q^{76} + 12 q^{77} - 2 q^{79} - q^{80} - 6 q^{82} + 15 q^{83} + 2 q^{85} + 36 q^{86} - 30 q^{89} - 12 q^{91} + 14 q^{92} - 9 q^{94} + 4 q^{95} - q^{97} + 50 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.772866 −0.546498 −0.273249 0.961943i \(-0.588098\pi\)
−0.273249 + 0.961943i \(0.588098\pi\)
\(3\) 0 0
\(4\) −1.40268 −0.701339
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.17554 0.822278 0.411139 0.911573i \(-0.365131\pi\)
0.411139 + 0.911573i \(0.365131\pi\)
\(8\) 2.62981 0.929779
\(9\) 0 0
\(10\) 0.772866 0.244402
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) −0.629813 −0.174679 −0.0873394 0.996179i \(-0.527836\pi\)
−0.0873394 + 0.996179i \(0.527836\pi\)
\(14\) −1.68140 −0.449374
\(15\) 0 0
\(16\) 0.772866 0.193216
\(17\) −4.17554 −1.01272 −0.506359 0.862323i \(-0.669009\pi\)
−0.506359 + 0.862323i \(0.669009\pi\)
\(18\) 0 0
\(19\) 4.80536 1.10242 0.551212 0.834365i \(-0.314165\pi\)
0.551212 + 0.834365i \(0.314165\pi\)
\(20\) 1.40268 0.313649
\(21\) 0 0
\(22\) 2.31860 0.494326
\(23\) −2.08408 −0.434561 −0.217281 0.976109i \(-0.569719\pi\)
−0.217281 + 0.976109i \(0.569719\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0.486761 0.0954617
\(27\) 0 0
\(28\) −3.05159 −0.576696
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −5.85695 −1.03537
\(33\) 0 0
\(34\) 3.22713 0.553449
\(35\) −2.17554 −0.367734
\(36\) 0 0
\(37\) 6.08408 1.00022 0.500108 0.865963i \(-0.333293\pi\)
0.500108 + 0.865963i \(0.333293\pi\)
\(38\) −3.71390 −0.602473
\(39\) 0 0
\(40\) −2.62981 −0.415810
\(41\) −0.824456 −0.128758 −0.0643792 0.997926i \(-0.520507\pi\)
−0.0643792 + 0.997926i \(0.520507\pi\)
\(42\) 0 0
\(43\) −8.72128 −1.32998 −0.664991 0.746851i \(-0.731565\pi\)
−0.664991 + 0.746851i \(0.731565\pi\)
\(44\) 4.20804 0.634385
\(45\) 0 0
\(46\) 1.61072 0.237487
\(47\) 8.98090 1.31000 0.655000 0.755629i \(-0.272669\pi\)
0.655000 + 0.755629i \(0.272669\pi\)
\(48\) 0 0
\(49\) −2.26701 −0.323858
\(50\) −0.772866 −0.109300
\(51\) 0 0
\(52\) 0.883426 0.122509
\(53\) −6.88944 −0.946337 −0.473169 0.880972i \(-0.656890\pi\)
−0.473169 + 0.880972i \(0.656890\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 5.72128 0.764538
\(57\) 0 0
\(58\) −0.772866 −0.101482
\(59\) −6.45427 −0.840274 −0.420137 0.907461i \(-0.638018\pi\)
−0.420137 + 0.907461i \(0.638018\pi\)
\(60\) 0 0
\(61\) −2.80536 −0.359189 −0.179595 0.983741i \(-0.557479\pi\)
−0.179595 + 0.983741i \(0.557479\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 2.98090 0.372613
\(65\) 0.629813 0.0781187
\(66\) 0 0
\(67\) −11.0841 −1.35414 −0.677068 0.735920i \(-0.736750\pi\)
−0.677068 + 0.735920i \(0.736750\pi\)
\(68\) 5.85695 0.710259
\(69\) 0 0
\(70\) 1.68140 0.200966
\(71\) −2.63719 −0.312977 −0.156489 0.987680i \(-0.550017\pi\)
−0.156489 + 0.987680i \(0.550017\pi\)
\(72\) 0 0
\(73\) −14.7863 −1.73060 −0.865300 0.501254i \(-0.832872\pi\)
−0.865300 + 0.501254i \(0.832872\pi\)
\(74\) −4.70218 −0.546617
\(75\) 0 0
\(76\) −6.74037 −0.773174
\(77\) −6.52663 −0.743779
\(78\) 0 0
\(79\) −12.0650 −1.35742 −0.678708 0.734408i \(-0.737460\pi\)
−0.678708 + 0.734408i \(0.737460\pi\)
\(80\) −0.772866 −0.0864090
\(81\) 0 0
\(82\) 0.637193 0.0703662
\(83\) 7.62981 0.837481 0.418740 0.908106i \(-0.362472\pi\)
0.418740 + 0.908106i \(0.362472\pi\)
\(84\) 0 0
\(85\) 4.17554 0.452901
\(86\) 6.74037 0.726833
\(87\) 0 0
\(88\) −7.88944 −0.841017
\(89\) −17.8894 −1.89628 −0.948138 0.317858i \(-0.897037\pi\)
−0.948138 + 0.317858i \(0.897037\pi\)
\(90\) 0 0
\(91\) −1.37019 −0.143635
\(92\) 2.92330 0.304775
\(93\) 0 0
\(94\) −6.94103 −0.715913
\(95\) −4.80536 −0.493019
\(96\) 0 0
\(97\) 0.538351 0.0546613 0.0273306 0.999626i \(-0.491299\pi\)
0.0273306 + 0.999626i \(0.491299\pi\)
\(98\) 1.75209 0.176988
\(99\) 0 0
\(100\) −1.40268 −0.140268
\(101\) −13.8054 −1.37368 −0.686842 0.726807i \(-0.741004\pi\)
−0.686842 + 0.726807i \(0.741004\pi\)
\(102\) 0 0
\(103\) −13.4426 −1.32453 −0.662267 0.749268i \(-0.730406\pi\)
−0.662267 + 0.749268i \(0.730406\pi\)
\(104\) −1.65629 −0.162413
\(105\) 0 0
\(106\) 5.32461 0.517172
\(107\) 9.25963 0.895162 0.447581 0.894243i \(-0.352286\pi\)
0.447581 + 0.894243i \(0.352286\pi\)
\(108\) 0 0
\(109\) 8.61072 0.824757 0.412378 0.911013i \(-0.364698\pi\)
0.412378 + 0.911013i \(0.364698\pi\)
\(110\) −2.31860 −0.221070
\(111\) 0 0
\(112\) 1.68140 0.158878
\(113\) −11.2670 −1.05991 −0.529955 0.848026i \(-0.677791\pi\)
−0.529955 + 0.848026i \(0.677791\pi\)
\(114\) 0 0
\(115\) 2.08408 0.194342
\(116\) −1.40268 −0.130235
\(117\) 0 0
\(118\) 4.98828 0.459209
\(119\) −9.08408 −0.832736
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 2.16816 0.196296
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 16.7863 1.48954 0.744770 0.667321i \(-0.232559\pi\)
0.744770 + 0.667321i \(0.232559\pi\)
\(128\) 9.41006 0.831740
\(129\) 0 0
\(130\) −0.486761 −0.0426918
\(131\) −11.2670 −0.984403 −0.492201 0.870481i \(-0.663808\pi\)
−0.492201 + 0.870481i \(0.663808\pi\)
\(132\) 0 0
\(133\) 10.4543 0.906500
\(134\) 8.56651 0.740033
\(135\) 0 0
\(136\) −10.9809 −0.941605
\(137\) 18.8703 1.61220 0.806101 0.591778i \(-0.201574\pi\)
0.806101 + 0.591778i \(0.201574\pi\)
\(138\) 0 0
\(139\) 3.80536 0.322766 0.161383 0.986892i \(-0.448405\pi\)
0.161383 + 0.986892i \(0.448405\pi\)
\(140\) 3.05159 0.257906
\(141\) 0 0
\(142\) 2.03820 0.171042
\(143\) 1.88944 0.158003
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 11.4278 0.945771
\(147\) 0 0
\(148\) −8.53401 −0.701492
\(149\) −4.45427 −0.364908 −0.182454 0.983214i \(-0.558404\pi\)
−0.182454 + 0.983214i \(0.558404\pi\)
\(150\) 0 0
\(151\) 1.17554 0.0956644 0.0478322 0.998855i \(-0.484769\pi\)
0.0478322 + 0.998855i \(0.484769\pi\)
\(152\) 12.6372 1.02501
\(153\) 0 0
\(154\) 5.04421 0.406474
\(155\) 0 0
\(156\) 0 0
\(157\) −0.805358 −0.0642745 −0.0321373 0.999483i \(-0.510231\pi\)
−0.0321373 + 0.999483i \(0.510231\pi\)
\(158\) 9.32461 0.741826
\(159\) 0 0
\(160\) 5.85695 0.463032
\(161\) −4.53401 −0.357330
\(162\) 0 0
\(163\) −1.11056 −0.0869858 −0.0434929 0.999054i \(-0.513849\pi\)
−0.0434929 + 0.999054i \(0.513849\pi\)
\(164\) 1.15645 0.0903033
\(165\) 0 0
\(166\) −5.89682 −0.457682
\(167\) 8.70218 0.673395 0.336697 0.941613i \(-0.390690\pi\)
0.336697 + 0.941613i \(0.390690\pi\)
\(168\) 0 0
\(169\) −12.6033 −0.969487
\(170\) −3.22713 −0.247510
\(171\) 0 0
\(172\) 12.2331 0.932769
\(173\) 7.69480 0.585025 0.292512 0.956262i \(-0.405509\pi\)
0.292512 + 0.956262i \(0.405509\pi\)
\(174\) 0 0
\(175\) 2.17554 0.164456
\(176\) −2.31860 −0.174771
\(177\) 0 0
\(178\) 13.8261 1.03631
\(179\) 6.06498 0.453318 0.226659 0.973974i \(-0.427220\pi\)
0.226659 + 0.973974i \(0.427220\pi\)
\(180\) 0 0
\(181\) 4.09146 0.304116 0.152058 0.988372i \(-0.451410\pi\)
0.152058 + 0.988372i \(0.451410\pi\)
\(182\) 1.05897 0.0784961
\(183\) 0 0
\(184\) −5.48075 −0.404046
\(185\) −6.08408 −0.447311
\(186\) 0 0
\(187\) 12.5266 0.916038
\(188\) −12.5973 −0.918754
\(189\) 0 0
\(190\) 3.71390 0.269434
\(191\) 0.267007 0.0193199 0.00965996 0.999953i \(-0.496925\pi\)
0.00965996 + 0.999953i \(0.496925\pi\)
\(192\) 0 0
\(193\) 10.7022 0.770360 0.385180 0.922842i \(-0.374139\pi\)
0.385180 + 0.922842i \(0.374139\pi\)
\(194\) −0.416073 −0.0298723
\(195\) 0 0
\(196\) 3.17988 0.227134
\(197\) −8.08408 −0.575967 −0.287984 0.957635i \(-0.592985\pi\)
−0.287984 + 0.957635i \(0.592985\pi\)
\(198\) 0 0
\(199\) −15.8054 −1.12041 −0.560206 0.828353i \(-0.689278\pi\)
−0.560206 + 0.828353i \(0.689278\pi\)
\(200\) 2.62981 0.185956
\(201\) 0 0
\(202\) 10.6697 0.750716
\(203\) 2.17554 0.152693
\(204\) 0 0
\(205\) 0.824456 0.0575825
\(206\) 10.3893 0.723856
\(207\) 0 0
\(208\) −0.486761 −0.0337508
\(209\) −14.4161 −0.997181
\(210\) 0 0
\(211\) −25.2214 −1.73631 −0.868157 0.496289i \(-0.834696\pi\)
−0.868157 + 0.496289i \(0.834696\pi\)
\(212\) 9.66367 0.663704
\(213\) 0 0
\(214\) −7.15645 −0.489205
\(215\) 8.72128 0.594786
\(216\) 0 0
\(217\) 0 0
\(218\) −6.65493 −0.450728
\(219\) 0 0
\(220\) −4.20804 −0.283706
\(221\) 2.62981 0.176900
\(222\) 0 0
\(223\) −12.5266 −0.838845 −0.419423 0.907791i \(-0.637767\pi\)
−0.419423 + 0.907791i \(0.637767\pi\)
\(224\) −12.7420 −0.851364
\(225\) 0 0
\(226\) 8.70788 0.579240
\(227\) 7.46165 0.495247 0.247624 0.968856i \(-0.420350\pi\)
0.247624 + 0.968856i \(0.420350\pi\)
\(228\) 0 0
\(229\) 21.7936 1.44016 0.720082 0.693889i \(-0.244104\pi\)
0.720082 + 0.693889i \(0.244104\pi\)
\(230\) −1.61072 −0.106207
\(231\) 0 0
\(232\) 2.62981 0.172656
\(233\) 9.69480 0.635127 0.317564 0.948237i \(-0.397135\pi\)
0.317564 + 0.948237i \(0.397135\pi\)
\(234\) 0 0
\(235\) −8.98090 −0.585849
\(236\) 9.05327 0.589317
\(237\) 0 0
\(238\) 7.02077 0.455089
\(239\) 10.7022 0.692266 0.346133 0.938185i \(-0.387495\pi\)
0.346133 + 0.938185i \(0.387495\pi\)
\(240\) 0 0
\(241\) 11.1682 0.719405 0.359702 0.933067i \(-0.382878\pi\)
0.359702 + 0.933067i \(0.382878\pi\)
\(242\) 1.54573 0.0993634
\(243\) 0 0
\(244\) 3.93502 0.251914
\(245\) 2.26701 0.144834
\(246\) 0 0
\(247\) −3.02648 −0.192570
\(248\) 0 0
\(249\) 0 0
\(250\) 0.772866 0.0488803
\(251\) −14.3585 −0.906299 −0.453149 0.891435i \(-0.649700\pi\)
−0.453149 + 0.891435i \(0.649700\pi\)
\(252\) 0 0
\(253\) 6.25225 0.393075
\(254\) −12.9735 −0.814031
\(255\) 0 0
\(256\) −13.2345 −0.827157
\(257\) 7.11056 0.443545 0.221772 0.975098i \(-0.428816\pi\)
0.221772 + 0.975098i \(0.428816\pi\)
\(258\) 0 0
\(259\) 13.2362 0.822457
\(260\) −0.883426 −0.0547877
\(261\) 0 0
\(262\) 8.70788 0.537975
\(263\) −19.2214 −1.18524 −0.592622 0.805481i \(-0.701907\pi\)
−0.592622 + 0.805481i \(0.701907\pi\)
\(264\) 0 0
\(265\) 6.88944 0.423215
\(266\) −8.07974 −0.495401
\(267\) 0 0
\(268\) 15.5474 0.949709
\(269\) 9.88944 0.602970 0.301485 0.953471i \(-0.402518\pi\)
0.301485 + 0.953471i \(0.402518\pi\)
\(270\) 0 0
\(271\) −13.7936 −0.837904 −0.418952 0.908008i \(-0.637602\pi\)
−0.418952 + 0.908008i \(0.637602\pi\)
\(272\) −3.22713 −0.195674
\(273\) 0 0
\(274\) −14.5842 −0.881066
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) −14.9661 −0.899228 −0.449614 0.893223i \(-0.648439\pi\)
−0.449614 + 0.893223i \(0.648439\pi\)
\(278\) −2.94103 −0.176391
\(279\) 0 0
\(280\) −5.72128 −0.341912
\(281\) −11.7287 −0.699673 −0.349836 0.936811i \(-0.613763\pi\)
−0.349836 + 0.936811i \(0.613763\pi\)
\(282\) 0 0
\(283\) −3.96180 −0.235505 −0.117752 0.993043i \(-0.537569\pi\)
−0.117752 + 0.993043i \(0.537569\pi\)
\(284\) 3.69914 0.219503
\(285\) 0 0
\(286\) −1.46028 −0.0863483
\(287\) −1.79364 −0.105875
\(288\) 0 0
\(289\) 0.435171 0.0255983
\(290\) 0.772866 0.0453842
\(291\) 0 0
\(292\) 20.7404 1.21374
\(293\) 3.26701 0.190861 0.0954303 0.995436i \(-0.469577\pi\)
0.0954303 + 0.995436i \(0.469577\pi\)
\(294\) 0 0
\(295\) 6.45427 0.375782
\(296\) 16.0000 0.929981
\(297\) 0 0
\(298\) 3.44255 0.199422
\(299\) 1.31258 0.0759086
\(300\) 0 0
\(301\) −18.9735 −1.09362
\(302\) −0.908538 −0.0522805
\(303\) 0 0
\(304\) 3.71390 0.213007
\(305\) 2.80536 0.160634
\(306\) 0 0
\(307\) 20.4352 1.16630 0.583148 0.812366i \(-0.301821\pi\)
0.583148 + 0.812366i \(0.301821\pi\)
\(308\) 9.15477 0.521641
\(309\) 0 0
\(310\) 0 0
\(311\) −11.0000 −0.623753 −0.311876 0.950123i \(-0.600957\pi\)
−0.311876 + 0.950123i \(0.600957\pi\)
\(312\) 0 0
\(313\) 27.6683 1.56391 0.781953 0.623337i \(-0.214224\pi\)
0.781953 + 0.623337i \(0.214224\pi\)
\(314\) 0.622433 0.0351259
\(315\) 0 0
\(316\) 16.9233 0.952010
\(317\) 12.8777 0.723285 0.361642 0.932317i \(-0.382216\pi\)
0.361642 + 0.932317i \(0.382216\pi\)
\(318\) 0 0
\(319\) −3.00000 −0.167968
\(320\) −2.98090 −0.166637
\(321\) 0 0
\(322\) 3.50418 0.195280
\(323\) −20.0650 −1.11645
\(324\) 0 0
\(325\) −0.629813 −0.0349358
\(326\) 0.858314 0.0475376
\(327\) 0 0
\(328\) −2.16816 −0.119717
\(329\) 19.5384 1.07718
\(330\) 0 0
\(331\) 24.6874 1.35694 0.678472 0.734627i \(-0.262643\pi\)
0.678472 + 0.734627i \(0.262643\pi\)
\(332\) −10.7022 −0.587358
\(333\) 0 0
\(334\) −6.72561 −0.368009
\(335\) 11.0841 0.605588
\(336\) 0 0
\(337\) −21.0915 −1.14893 −0.574463 0.818531i \(-0.694789\pi\)
−0.574463 + 0.818531i \(0.694789\pi\)
\(338\) 9.74068 0.529823
\(339\) 0 0
\(340\) −5.85695 −0.317638
\(341\) 0 0
\(342\) 0 0
\(343\) −20.1608 −1.08858
\(344\) −22.9353 −1.23659
\(345\) 0 0
\(346\) −5.94704 −0.319715
\(347\) 18.8245 1.01055 0.505275 0.862958i \(-0.331391\pi\)
0.505275 + 0.862958i \(0.331391\pi\)
\(348\) 0 0
\(349\) 17.4884 0.936135 0.468067 0.883693i \(-0.344950\pi\)
0.468067 + 0.883693i \(0.344950\pi\)
\(350\) −1.68140 −0.0898748
\(351\) 0 0
\(352\) 17.5708 0.936529
\(353\) 8.38928 0.446517 0.223258 0.974759i \(-0.428331\pi\)
0.223258 + 0.974759i \(0.428331\pi\)
\(354\) 0 0
\(355\) 2.63719 0.139968
\(356\) 25.0931 1.32993
\(357\) 0 0
\(358\) −4.68742 −0.247738
\(359\) 27.6948 1.46168 0.730838 0.682551i \(-0.239130\pi\)
0.730838 + 0.682551i \(0.239130\pi\)
\(360\) 0 0
\(361\) 4.09146 0.215340
\(362\) −3.16215 −0.166199
\(363\) 0 0
\(364\) 1.92193 0.100737
\(365\) 14.7863 0.773948
\(366\) 0 0
\(367\) 9.00738 0.470181 0.235091 0.971973i \(-0.424461\pi\)
0.235091 + 0.971973i \(0.424461\pi\)
\(368\) −1.61072 −0.0839643
\(369\) 0 0
\(370\) 4.70218 0.244455
\(371\) −14.9883 −0.778153
\(372\) 0 0
\(373\) −25.0533 −1.29721 −0.648604 0.761126i \(-0.724647\pi\)
−0.648604 + 0.761126i \(0.724647\pi\)
\(374\) −9.68140 −0.500613
\(375\) 0 0
\(376\) 23.6181 1.21801
\(377\) −0.629813 −0.0324370
\(378\) 0 0
\(379\) −27.5725 −1.41631 −0.708153 0.706059i \(-0.750471\pi\)
−0.708153 + 0.706059i \(0.750471\pi\)
\(380\) 6.74037 0.345774
\(381\) 0 0
\(382\) −0.206360 −0.0105583
\(383\) 13.0724 0.667967 0.333983 0.942579i \(-0.391607\pi\)
0.333983 + 0.942579i \(0.391607\pi\)
\(384\) 0 0
\(385\) 6.52663 0.332628
\(386\) −8.27134 −0.421000
\(387\) 0 0
\(388\) −0.755134 −0.0383361
\(389\) 23.0797 1.17019 0.585095 0.810965i \(-0.301057\pi\)
0.585095 + 0.810965i \(0.301057\pi\)
\(390\) 0 0
\(391\) 8.70218 0.440088
\(392\) −5.96180 −0.301117
\(393\) 0 0
\(394\) 6.24791 0.314765
\(395\) 12.0650 0.607055
\(396\) 0 0
\(397\) −12.3129 −0.617966 −0.308983 0.951067i \(-0.599989\pi\)
−0.308983 + 0.951067i \(0.599989\pi\)
\(398\) 12.2154 0.612304
\(399\) 0 0
\(400\) 0.772866 0.0386433
\(401\) 36.4308 1.81927 0.909634 0.415410i \(-0.136362\pi\)
0.909634 + 0.415410i \(0.136362\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 19.3645 0.963419
\(405\) 0 0
\(406\) −1.68140 −0.0834466
\(407\) −18.2522 −0.904730
\(408\) 0 0
\(409\) −22.5842 −1.11672 −0.558359 0.829599i \(-0.688569\pi\)
−0.558359 + 0.829599i \(0.688569\pi\)
\(410\) −0.637193 −0.0314687
\(411\) 0 0
\(412\) 18.8556 0.928948
\(413\) −14.0415 −0.690939
\(414\) 0 0
\(415\) −7.62981 −0.374533
\(416\) 3.68878 0.180857
\(417\) 0 0
\(418\) 11.1417 0.544958
\(419\) 2.90854 0.142091 0.0710457 0.997473i \(-0.477366\pi\)
0.0710457 + 0.997473i \(0.477366\pi\)
\(420\) 0 0
\(421\) 27.1183 1.32166 0.660831 0.750534i \(-0.270204\pi\)
0.660831 + 0.750534i \(0.270204\pi\)
\(422\) 19.4928 0.948893
\(423\) 0 0
\(424\) −18.1179 −0.879885
\(425\) −4.17554 −0.202544
\(426\) 0 0
\(427\) −6.10318 −0.295354
\(428\) −12.9883 −0.627812
\(429\) 0 0
\(430\) −6.74037 −0.325050
\(431\) −19.5960 −0.943904 −0.471952 0.881624i \(-0.656450\pi\)
−0.471952 + 0.881624i \(0.656450\pi\)
\(432\) 0 0
\(433\) 24.1638 1.16124 0.580620 0.814175i \(-0.302810\pi\)
0.580620 + 0.814175i \(0.302810\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −12.0781 −0.578435
\(437\) −10.0148 −0.479071
\(438\) 0 0
\(439\) −30.4235 −1.45203 −0.726016 0.687678i \(-0.758630\pi\)
−0.726016 + 0.687678i \(0.758630\pi\)
\(440\) 7.88944 0.376114
\(441\) 0 0
\(442\) −2.03249 −0.0966758
\(443\) 32.3853 1.53867 0.769335 0.638846i \(-0.220588\pi\)
0.769335 + 0.638846i \(0.220588\pi\)
\(444\) 0 0
\(445\) 17.8894 0.848041
\(446\) 9.68140 0.458428
\(447\) 0 0
\(448\) 6.48508 0.306391
\(449\) 5.62243 0.265339 0.132670 0.991160i \(-0.457645\pi\)
0.132670 + 0.991160i \(0.457645\pi\)
\(450\) 0 0
\(451\) 2.47337 0.116466
\(452\) 15.8040 0.743357
\(453\) 0 0
\(454\) −5.76685 −0.270652
\(455\) 1.37019 0.0642353
\(456\) 0 0
\(457\) 3.33199 0.155864 0.0779320 0.996959i \(-0.475168\pi\)
0.0779320 + 0.996959i \(0.475168\pi\)
\(458\) −16.8436 −0.787048
\(459\) 0 0
\(460\) −2.92330 −0.136299
\(461\) −23.3437 −1.08722 −0.543612 0.839336i \(-0.682944\pi\)
−0.543612 + 0.839336i \(0.682944\pi\)
\(462\) 0 0
\(463\) 4.13735 0.192279 0.0961394 0.995368i \(-0.469351\pi\)
0.0961394 + 0.995368i \(0.469351\pi\)
\(464\) 0.772866 0.0358794
\(465\) 0 0
\(466\) −7.49278 −0.347096
\(467\) −35.7554 −1.65456 −0.827282 0.561786i \(-0.810114\pi\)
−0.827282 + 0.561786i \(0.810114\pi\)
\(468\) 0 0
\(469\) −24.1139 −1.11348
\(470\) 6.94103 0.320166
\(471\) 0 0
\(472\) −16.9735 −0.781270
\(473\) 26.1638 1.20301
\(474\) 0 0
\(475\) 4.80536 0.220485
\(476\) 12.7420 0.584031
\(477\) 0 0
\(478\) −8.27134 −0.378322
\(479\) 29.6107 1.35295 0.676474 0.736466i \(-0.263507\pi\)
0.676474 + 0.736466i \(0.263507\pi\)
\(480\) 0 0
\(481\) −3.83184 −0.174717
\(482\) −8.63149 −0.393154
\(483\) 0 0
\(484\) 2.80536 0.127516
\(485\) −0.538351 −0.0244453
\(486\) 0 0
\(487\) −4.33633 −0.196498 −0.0982489 0.995162i \(-0.531324\pi\)
−0.0982489 + 0.995162i \(0.531324\pi\)
\(488\) −7.37757 −0.333967
\(489\) 0 0
\(490\) −1.75209 −0.0791514
\(491\) −22.5193 −1.01628 −0.508140 0.861275i \(-0.669667\pi\)
−0.508140 + 0.861275i \(0.669667\pi\)
\(492\) 0 0
\(493\) −4.17554 −0.188057
\(494\) 2.33906 0.105239
\(495\) 0 0
\(496\) 0 0
\(497\) −5.73733 −0.257354
\(498\) 0 0
\(499\) −41.1638 −1.84275 −0.921373 0.388680i \(-0.872931\pi\)
−0.921373 + 0.388680i \(0.872931\pi\)
\(500\) 1.40268 0.0627297
\(501\) 0 0
\(502\) 11.0972 0.495291
\(503\) −31.5149 −1.40518 −0.702590 0.711595i \(-0.747973\pi\)
−0.702590 + 0.711595i \(0.747973\pi\)
\(504\) 0 0
\(505\) 13.8054 0.614330
\(506\) −4.83215 −0.214815
\(507\) 0 0
\(508\) −23.5457 −1.04467
\(509\) −22.6224 −1.00272 −0.501361 0.865238i \(-0.667167\pi\)
−0.501361 + 0.865238i \(0.667167\pi\)
\(510\) 0 0
\(511\) −32.1682 −1.42304
\(512\) −8.59162 −0.379699
\(513\) 0 0
\(514\) −5.49551 −0.242396
\(515\) 13.4426 0.592350
\(516\) 0 0
\(517\) −26.9427 −1.18494
\(518\) −10.2298 −0.449471
\(519\) 0 0
\(520\) 1.65629 0.0726332
\(521\) 2.06498 0.0904686 0.0452343 0.998976i \(-0.485597\pi\)
0.0452343 + 0.998976i \(0.485597\pi\)
\(522\) 0 0
\(523\) 24.5266 1.07247 0.536237 0.844067i \(-0.319845\pi\)
0.536237 + 0.844067i \(0.319845\pi\)
\(524\) 15.8040 0.690401
\(525\) 0 0
\(526\) 14.8556 0.647734
\(527\) 0 0
\(528\) 0 0
\(529\) −18.6566 −0.811157
\(530\) −5.32461 −0.231286
\(531\) 0 0
\(532\) −14.6640 −0.635764
\(533\) 0.519253 0.0224913
\(534\) 0 0
\(535\) −9.25963 −0.400329
\(536\) −29.1491 −1.25905
\(537\) 0 0
\(538\) −7.64321 −0.329522
\(539\) 6.80102 0.292941
\(540\) 0 0
\(541\) 0.389285 0.0167367 0.00836833 0.999965i \(-0.497336\pi\)
0.00836833 + 0.999965i \(0.497336\pi\)
\(542\) 10.6606 0.457913
\(543\) 0 0
\(544\) 24.4559 1.04854
\(545\) −8.61072 −0.368843
\(546\) 0 0
\(547\) 21.4352 0.916502 0.458251 0.888823i \(-0.348476\pi\)
0.458251 + 0.888823i \(0.348476\pi\)
\(548\) −26.4690 −1.13070
\(549\) 0 0
\(550\) 2.31860 0.0988653
\(551\) 4.80536 0.204715
\(552\) 0 0
\(553\) −26.2479 −1.11617
\(554\) 11.5668 0.491427
\(555\) 0 0
\(556\) −5.33769 −0.226369
\(557\) 19.7598 0.837249 0.418624 0.908159i \(-0.362512\pi\)
0.418624 + 0.908159i \(0.362512\pi\)
\(558\) 0 0
\(559\) 5.49278 0.232320
\(560\) −1.68140 −0.0710523
\(561\) 0 0
\(562\) 9.06467 0.382370
\(563\) −6.46165 −0.272326 −0.136163 0.990686i \(-0.543477\pi\)
−0.136163 + 0.990686i \(0.543477\pi\)
\(564\) 0 0
\(565\) 11.2670 0.474007
\(566\) 3.06194 0.128703
\(567\) 0 0
\(568\) −6.93533 −0.291000
\(569\) 14.9427 0.626431 0.313215 0.949682i \(-0.398594\pi\)
0.313215 + 0.949682i \(0.398594\pi\)
\(570\) 0 0
\(571\) 35.1521 1.47107 0.735535 0.677487i \(-0.236931\pi\)
0.735535 + 0.677487i \(0.236931\pi\)
\(572\) −2.65028 −0.110814
\(573\) 0 0
\(574\) 1.38624 0.0578606
\(575\) −2.08408 −0.0869122
\(576\) 0 0
\(577\) −0.740373 −0.0308222 −0.0154111 0.999881i \(-0.504906\pi\)
−0.0154111 + 0.999881i \(0.504906\pi\)
\(578\) −0.336329 −0.0139894
\(579\) 0 0
\(580\) 1.40268 0.0582431
\(581\) 16.5990 0.688642
\(582\) 0 0
\(583\) 20.6683 0.855994
\(584\) −38.8851 −1.60908
\(585\) 0 0
\(586\) −2.52496 −0.104305
\(587\) 18.5842 0.767054 0.383527 0.923530i \(-0.374709\pi\)
0.383527 + 0.923530i \(0.374709\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −4.98828 −0.205364
\(591\) 0 0
\(592\) 4.70218 0.193258
\(593\) −24.0268 −0.986662 −0.493331 0.869842i \(-0.664221\pi\)
−0.493331 + 0.869842i \(0.664221\pi\)
\(594\) 0 0
\(595\) 9.08408 0.372411
\(596\) 6.24791 0.255924
\(597\) 0 0
\(598\) −1.01445 −0.0414839
\(599\) −22.3585 −0.913542 −0.456771 0.889584i \(-0.650994\pi\)
−0.456771 + 0.889584i \(0.650994\pi\)
\(600\) 0 0
\(601\) −20.3511 −0.830138 −0.415069 0.909790i \(-0.636243\pi\)
−0.415069 + 0.909790i \(0.636243\pi\)
\(602\) 14.6640 0.597659
\(603\) 0 0
\(604\) −1.64891 −0.0670932
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 19.4928 0.791187 0.395594 0.918426i \(-0.370539\pi\)
0.395594 + 0.918426i \(0.370539\pi\)
\(608\) −28.1447 −1.14142
\(609\) 0 0
\(610\) −2.16816 −0.0877864
\(611\) −5.65629 −0.228829
\(612\) 0 0
\(613\) 39.1638 1.58181 0.790906 0.611938i \(-0.209610\pi\)
0.790906 + 0.611938i \(0.209610\pi\)
\(614\) −15.7936 −0.637379
\(615\) 0 0
\(616\) −17.1638 −0.691550
\(617\) −26.8321 −1.08022 −0.540111 0.841594i \(-0.681618\pi\)
−0.540111 + 0.841594i \(0.681618\pi\)
\(618\) 0 0
\(619\) −33.4278 −1.34358 −0.671788 0.740743i \(-0.734473\pi\)
−0.671788 + 0.740743i \(0.734473\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 8.50152 0.340880
\(623\) −38.9193 −1.55927
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −21.3839 −0.854672
\(627\) 0 0
\(628\) 1.12966 0.0450783
\(629\) −25.4044 −1.01294
\(630\) 0 0
\(631\) −32.3705 −1.28865 −0.644325 0.764752i \(-0.722861\pi\)
−0.644325 + 0.764752i \(0.722861\pi\)
\(632\) −31.7287 −1.26210
\(633\) 0 0
\(634\) −9.95275 −0.395274
\(635\) −16.7863 −0.666142
\(636\) 0 0
\(637\) 1.42779 0.0565711
\(638\) 2.31860 0.0917941
\(639\) 0 0
\(640\) −9.41006 −0.371965
\(641\) −33.0415 −1.30506 −0.652531 0.757762i \(-0.726293\pi\)
−0.652531 + 0.757762i \(0.726293\pi\)
\(642\) 0 0
\(643\) 27.4734 1.08344 0.541722 0.840558i \(-0.317773\pi\)
0.541722 + 0.840558i \(0.317773\pi\)
\(644\) 6.35976 0.250610
\(645\) 0 0
\(646\) 15.5075 0.610136
\(647\) 23.2023 0.912178 0.456089 0.889934i \(-0.349250\pi\)
0.456089 + 0.889934i \(0.349250\pi\)
\(648\) 0 0
\(649\) 19.3628 0.760057
\(650\) 0.486761 0.0190923
\(651\) 0 0
\(652\) 1.55776 0.0610066
\(653\) 28.9159 1.13157 0.565784 0.824554i \(-0.308574\pi\)
0.565784 + 0.824554i \(0.308574\pi\)
\(654\) 0 0
\(655\) 11.2670 0.440238
\(656\) −0.637193 −0.0248782
\(657\) 0 0
\(658\) −15.1005 −0.588680
\(659\) −36.1832 −1.40950 −0.704749 0.709456i \(-0.748941\pi\)
−0.704749 + 0.709456i \(0.748941\pi\)
\(660\) 0 0
\(661\) 45.0918 1.75387 0.876933 0.480612i \(-0.159585\pi\)
0.876933 + 0.480612i \(0.159585\pi\)
\(662\) −19.0801 −0.741567
\(663\) 0 0
\(664\) 20.0650 0.778672
\(665\) −10.4543 −0.405399
\(666\) 0 0
\(667\) −2.08408 −0.0806960
\(668\) −12.2064 −0.472278
\(669\) 0 0
\(670\) −8.56651 −0.330953
\(671\) 8.41607 0.324899
\(672\) 0 0
\(673\) −23.1491 −0.892331 −0.446165 0.894950i \(-0.647211\pi\)
−0.446165 + 0.894950i \(0.647211\pi\)
\(674\) 16.3009 0.627886
\(675\) 0 0
\(676\) 17.6784 0.679940
\(677\) −38.1521 −1.46630 −0.733152 0.680064i \(-0.761952\pi\)
−0.733152 + 0.680064i \(0.761952\pi\)
\(678\) 0 0
\(679\) 1.17121 0.0449468
\(680\) 10.9809 0.421098
\(681\) 0 0
\(682\) 0 0
\(683\) 0.228811 0.00875520 0.00437760 0.999990i \(-0.498607\pi\)
0.00437760 + 0.999990i \(0.498607\pi\)
\(684\) 0 0
\(685\) −18.8703 −0.720999
\(686\) 15.5816 0.594907
\(687\) 0 0
\(688\) −6.74037 −0.256974
\(689\) 4.33906 0.165305
\(690\) 0 0
\(691\) 16.2258 0.617257 0.308629 0.951183i \(-0.400130\pi\)
0.308629 + 0.951183i \(0.400130\pi\)
\(692\) −10.7933 −0.410301
\(693\) 0 0
\(694\) −14.5488 −0.552264
\(695\) −3.80536 −0.144345
\(696\) 0 0
\(697\) 3.44255 0.130396
\(698\) −13.5162 −0.511596
\(699\) 0 0
\(700\) −3.05159 −0.115339
\(701\) −9.69914 −0.366331 −0.183166 0.983082i \(-0.558634\pi\)
−0.183166 + 0.983082i \(0.558634\pi\)
\(702\) 0 0
\(703\) 29.2362 1.10266
\(704\) −8.94271 −0.337041
\(705\) 0 0
\(706\) −6.48379 −0.244021
\(707\) −30.0342 −1.12955
\(708\) 0 0
\(709\) 49.4884 1.85858 0.929289 0.369354i \(-0.120421\pi\)
0.929289 + 0.369354i \(0.120421\pi\)
\(710\) −2.03820 −0.0764921
\(711\) 0 0
\(712\) −47.0459 −1.76312
\(713\) 0 0
\(714\) 0 0
\(715\) −1.88944 −0.0706610
\(716\) −8.50722 −0.317930
\(717\) 0 0
\(718\) −21.4044 −0.798803
\(719\) −15.7139 −0.586029 −0.293015 0.956108i \(-0.594658\pi\)
−0.293015 + 0.956108i \(0.594658\pi\)
\(720\) 0 0
\(721\) −29.2449 −1.08914
\(722\) −3.16215 −0.117683
\(723\) 0 0
\(724\) −5.73901 −0.213289
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 6.51925 0.241786 0.120893 0.992666i \(-0.461424\pi\)
0.120893 + 0.992666i \(0.461424\pi\)
\(728\) −3.60334 −0.133548
\(729\) 0 0
\(730\) −11.4278 −0.422962
\(731\) 36.4161 1.34690
\(732\) 0 0
\(733\) −6.76716 −0.249951 −0.124975 0.992160i \(-0.539885\pi\)
−0.124975 + 0.992160i \(0.539885\pi\)
\(734\) −6.96149 −0.256953
\(735\) 0 0
\(736\) 12.2064 0.449932
\(737\) 33.2522 1.22486
\(738\) 0 0
\(739\) −41.1183 −1.51256 −0.756280 0.654248i \(-0.772985\pi\)
−0.756280 + 0.654248i \(0.772985\pi\)
\(740\) 8.53401 0.313717
\(741\) 0 0
\(742\) 11.5839 0.425259
\(743\) −36.5534 −1.34101 −0.670507 0.741903i \(-0.733924\pi\)
−0.670507 + 0.741903i \(0.733924\pi\)
\(744\) 0 0
\(745\) 4.45427 0.163192
\(746\) 19.3628 0.708923
\(747\) 0 0
\(748\) −17.5708 −0.642454
\(749\) 20.1447 0.736072
\(750\) 0 0
\(751\) −45.1980 −1.64930 −0.824649 0.565645i \(-0.808627\pi\)
−0.824649 + 0.565645i \(0.808627\pi\)
\(752\) 6.94103 0.253113
\(753\) 0 0
\(754\) 0.486761 0.0177268
\(755\) −1.17554 −0.0427824
\(756\) 0 0
\(757\) 53.4737 1.94353 0.971767 0.235943i \(-0.0758178\pi\)
0.971767 + 0.235943i \(0.0758178\pi\)
\(758\) 21.3099 0.774009
\(759\) 0 0
\(760\) −12.6372 −0.458399
\(761\) 49.4693 1.79326 0.896631 0.442778i \(-0.146007\pi\)
0.896631 + 0.442778i \(0.146007\pi\)
\(762\) 0 0
\(763\) 18.7330 0.678180
\(764\) −0.374525 −0.0135498
\(765\) 0 0
\(766\) −10.1032 −0.365043
\(767\) 4.06498 0.146778
\(768\) 0 0
\(769\) −14.2331 −0.513260 −0.256630 0.966510i \(-0.582612\pi\)
−0.256630 + 0.966510i \(0.582612\pi\)
\(770\) −5.04421 −0.181781
\(771\) 0 0
\(772\) −15.0117 −0.540284
\(773\) −21.7936 −0.783863 −0.391931 0.919994i \(-0.628193\pi\)
−0.391931 + 0.919994i \(0.628193\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.41576 0.0508229
\(777\) 0 0
\(778\) −17.8375 −0.639507
\(779\) −3.96180 −0.141946
\(780\) 0 0
\(781\) 7.91158 0.283099
\(782\) −6.72561 −0.240507
\(783\) 0 0
\(784\) −1.75209 −0.0625747
\(785\) 0.805358 0.0287444
\(786\) 0 0
\(787\) −14.1829 −0.505567 −0.252783 0.967523i \(-0.581346\pi\)
−0.252783 + 0.967523i \(0.581346\pi\)
\(788\) 11.3394 0.403948
\(789\) 0 0
\(790\) −9.32461 −0.331755
\(791\) −24.5119 −0.871542
\(792\) 0 0
\(793\) 1.76685 0.0627427
\(794\) 9.51621 0.337718
\(795\) 0 0
\(796\) 22.1698 0.785789
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) −37.5002 −1.32666
\(800\) −5.85695 −0.207074
\(801\) 0 0
\(802\) −28.1561 −0.994228
\(803\) 44.3588 1.56539
\(804\) 0 0
\(805\) 4.53401 0.159803
\(806\) 0 0
\(807\) 0 0
\(808\) −36.3055 −1.27722
\(809\) −30.6757 −1.07850 −0.539250 0.842146i \(-0.681292\pi\)
−0.539250 + 0.842146i \(0.681292\pi\)
\(810\) 0 0
\(811\) 36.6609 1.28734 0.643670 0.765303i \(-0.277411\pi\)
0.643670 + 0.765303i \(0.277411\pi\)
\(812\) −3.05159 −0.107090
\(813\) 0 0
\(814\) 14.1065 0.494434
\(815\) 1.11056 0.0389012
\(816\) 0 0
\(817\) −41.9088 −1.46620
\(818\) 17.4546 0.610285
\(819\) 0 0
\(820\) −1.15645 −0.0403849
\(821\) 36.1447 1.26146 0.630730 0.776002i \(-0.282756\pi\)
0.630730 + 0.776002i \(0.282756\pi\)
\(822\) 0 0
\(823\) −10.4395 −0.363898 −0.181949 0.983308i \(-0.558241\pi\)
−0.181949 + 0.983308i \(0.558241\pi\)
\(824\) −35.3514 −1.23152
\(825\) 0 0
\(826\) 10.8522 0.377597
\(827\) 34.1447 1.18733 0.593664 0.804713i \(-0.297681\pi\)
0.593664 + 0.804713i \(0.297681\pi\)
\(828\) 0 0
\(829\) 42.6874 1.48260 0.741298 0.671176i \(-0.234211\pi\)
0.741298 + 0.671176i \(0.234211\pi\)
\(830\) 5.89682 0.204682
\(831\) 0 0
\(832\) −1.87741 −0.0650875
\(833\) 9.46599 0.327977
\(834\) 0 0
\(835\) −8.70218 −0.301151
\(836\) 20.2211 0.699362
\(837\) 0 0
\(838\) −2.24791 −0.0776527
\(839\) −13.9926 −0.483079 −0.241539 0.970391i \(-0.577652\pi\)
−0.241539 + 0.970391i \(0.577652\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −20.9588 −0.722287
\(843\) 0 0
\(844\) 35.3776 1.21775
\(845\) 12.6033 0.433568
\(846\) 0 0
\(847\) −4.35109 −0.149505
\(848\) −5.32461 −0.182848
\(849\) 0 0
\(850\) 3.22713 0.110690
\(851\) −12.6797 −0.434655
\(852\) 0 0
\(853\) 34.6978 1.18803 0.594016 0.804453i \(-0.297542\pi\)
0.594016 + 0.804453i \(0.297542\pi\)
\(854\) 4.71694 0.161410
\(855\) 0 0
\(856\) 24.3511 0.832303
\(857\) −48.0077 −1.63991 −0.819956 0.572427i \(-0.806002\pi\)
−0.819956 + 0.572427i \(0.806002\pi\)
\(858\) 0 0
\(859\) 16.2627 0.554875 0.277438 0.960744i \(-0.410515\pi\)
0.277438 + 0.960744i \(0.410515\pi\)
\(860\) −12.2331 −0.417147
\(861\) 0 0
\(862\) 15.1450 0.515842
\(863\) −13.5605 −0.461604 −0.230802 0.973001i \(-0.574135\pi\)
−0.230802 + 0.973001i \(0.574135\pi\)
\(864\) 0 0
\(865\) −7.69480 −0.261631
\(866\) −18.6754 −0.634616
\(867\) 0 0
\(868\) 0 0
\(869\) 36.1950 1.22783
\(870\) 0 0
\(871\) 6.98090 0.236539
\(872\) 22.6446 0.766842
\(873\) 0 0
\(874\) 7.74006 0.261812
\(875\) −2.17554 −0.0735468
\(876\) 0 0
\(877\) −8.01476 −0.270639 −0.135320 0.990802i \(-0.543206\pi\)
−0.135320 + 0.990802i \(0.543206\pi\)
\(878\) 23.5132 0.793533
\(879\) 0 0
\(880\) 2.31860 0.0781599
\(881\) 44.1564 1.48767 0.743834 0.668364i \(-0.233005\pi\)
0.743834 + 0.668364i \(0.233005\pi\)
\(882\) 0 0
\(883\) −20.7022 −0.696684 −0.348342 0.937368i \(-0.613255\pi\)
−0.348342 + 0.937368i \(0.613255\pi\)
\(884\) −3.68878 −0.124067
\(885\) 0 0
\(886\) −25.0294 −0.840881
\(887\) −48.5387 −1.62977 −0.814884 0.579624i \(-0.803200\pi\)
−0.814884 + 0.579624i \(0.803200\pi\)
\(888\) 0 0
\(889\) 36.5193 1.22482
\(890\) −13.8261 −0.463453
\(891\) 0 0
\(892\) 17.5708 0.588315
\(893\) 43.1564 1.44418
\(894\) 0 0
\(895\) −6.06498 −0.202730
\(896\) 20.4720 0.683922
\(897\) 0 0
\(898\) −4.34538 −0.145007
\(899\) 0 0
\(900\) 0 0
\(901\) 28.7672 0.958373
\(902\) −1.91158 −0.0636487
\(903\) 0 0
\(904\) −29.6301 −0.985483
\(905\) −4.09146 −0.136005
\(906\) 0 0
\(907\) −26.5149 −0.880413 −0.440207 0.897896i \(-0.645095\pi\)
−0.440207 + 0.897896i \(0.645095\pi\)
\(908\) −10.4663 −0.347336
\(909\) 0 0
\(910\) −1.05897 −0.0351045
\(911\) −53.8469 −1.78403 −0.892014 0.452008i \(-0.850708\pi\)
−0.892014 + 0.452008i \(0.850708\pi\)
\(912\) 0 0
\(913\) −22.8894 −0.757530
\(914\) −2.57518 −0.0851794
\(915\) 0 0
\(916\) −30.5695 −1.01004
\(917\) −24.5119 −0.809453
\(918\) 0 0
\(919\) −11.4084 −0.376328 −0.188164 0.982138i \(-0.560254\pi\)
−0.188164 + 0.982138i \(0.560254\pi\)
\(920\) 5.48075 0.180695
\(921\) 0 0
\(922\) 18.0415 0.594167
\(923\) 1.66094 0.0546705
\(924\) 0 0
\(925\) 6.08408 0.200043
\(926\) −3.19761 −0.105080
\(927\) 0 0
\(928\) −5.85695 −0.192264
\(929\) −13.7554 −0.451301 −0.225651 0.974208i \(-0.572451\pi\)
−0.225651 + 0.974208i \(0.572451\pi\)
\(930\) 0 0
\(931\) −10.8938 −0.357029
\(932\) −13.5987 −0.445440
\(933\) 0 0
\(934\) 27.6342 0.904217
\(935\) −12.5266 −0.409665
\(936\) 0 0
\(937\) 1.29379 0.0422664 0.0211332 0.999777i \(-0.493273\pi\)
0.0211332 + 0.999777i \(0.493273\pi\)
\(938\) 18.6368 0.608514
\(939\) 0 0
\(940\) 12.5973 0.410879
\(941\) 49.8586 1.62534 0.812672 0.582721i \(-0.198012\pi\)
0.812672 + 0.582721i \(0.198012\pi\)
\(942\) 0 0
\(943\) 1.71823 0.0559534
\(944\) −4.98828 −0.162355
\(945\) 0 0
\(946\) −20.2211 −0.657445
\(947\) 21.1109 0.686011 0.343006 0.939333i \(-0.388555\pi\)
0.343006 + 0.939333i \(0.388555\pi\)
\(948\) 0 0
\(949\) 9.31258 0.302299
\(950\) −3.71390 −0.120495
\(951\) 0 0
\(952\) −23.8894 −0.774261
\(953\) 42.7672 1.38536 0.692682 0.721243i \(-0.256429\pi\)
0.692682 + 0.721243i \(0.256429\pi\)
\(954\) 0 0
\(955\) −0.267007 −0.00864013
\(956\) −15.0117 −0.485514
\(957\) 0 0
\(958\) −22.8851 −0.739384
\(959\) 41.0533 1.32568
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 2.96149 0.0954824
\(963\) 0 0
\(964\) −15.6653 −0.504547
\(965\) −10.7022 −0.344515
\(966\) 0 0
\(967\) −33.0342 −1.06231 −0.531154 0.847276i \(-0.678241\pi\)
−0.531154 + 0.847276i \(0.678241\pi\)
\(968\) −5.25963 −0.169051
\(969\) 0 0
\(970\) 0.416073 0.0133593
\(971\) 26.0841 0.837078 0.418539 0.908199i \(-0.362542\pi\)
0.418539 + 0.908199i \(0.362542\pi\)
\(972\) 0 0
\(973\) 8.27872 0.265404
\(974\) 3.35140 0.107386
\(975\) 0 0
\(976\) −2.16816 −0.0694012
\(977\) 43.6948 1.39792 0.698960 0.715161i \(-0.253646\pi\)
0.698960 + 0.715161i \(0.253646\pi\)
\(978\) 0 0
\(979\) 53.6683 1.71525
\(980\) −3.17988 −0.101578
\(981\) 0 0
\(982\) 17.4044 0.555395
\(983\) −2.55745 −0.0815700 −0.0407850 0.999168i \(-0.512986\pi\)
−0.0407850 + 0.999168i \(0.512986\pi\)
\(984\) 0 0
\(985\) 8.08408 0.257580
\(986\) 3.22713 0.102773
\(987\) 0 0
\(988\) 4.24518 0.135057
\(989\) 18.1759 0.577959
\(990\) 0 0
\(991\) 14.7287 0.467871 0.233936 0.972252i \(-0.424840\pi\)
0.233936 + 0.972252i \(0.424840\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 4.43419 0.140644
\(995\) 15.8054 0.501064
\(996\) 0 0
\(997\) 33.0992 1.04826 0.524130 0.851638i \(-0.324390\pi\)
0.524130 + 0.851638i \(0.324390\pi\)
\(998\) 31.8141 1.00706
\(999\) 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.q.1.2 3
3.2 odd 2 435.2.a.i.1.2 3
5.4 even 2 6525.2.a.bf.1.2 3
12.11 even 2 6960.2.a.cl.1.1 3
15.2 even 4 2175.2.c.m.349.4 6
15.8 even 4 2175.2.c.m.349.3 6
15.14 odd 2 2175.2.a.u.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.i.1.2 3 3.2 odd 2
1305.2.a.q.1.2 3 1.1 even 1 trivial
2175.2.a.u.1.2 3 15.14 odd 2
2175.2.c.m.349.3 6 15.8 even 4
2175.2.c.m.349.4 6 15.2 even 4
6525.2.a.bf.1.2 3 5.4 even 2
6960.2.a.cl.1.1 3 12.11 even 2