Properties

Label 3969.2.a.bi.1.6
Level $3969$
Weight $2$
Character 3969.1
Self dual yes
Analytic conductor $31.693$
Analytic rank $0$
Dimension $12$
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] = [3969,2,Mod(1,3969)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3969, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3969.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3969.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: \(31.6926245622\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 22 x^{10} + 92 x^{9} + 125 x^{8} - 620 x^{7} - 94 x^{6} + 1280 x^{5} - 234 x^{4} - 736 x^{3} + 96 x^{2} + 60 x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 441)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-1.48259\) of defining polynomial
Character \(\chi\) \(=\) 3969.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.0683740 q^{2} -1.99532 q^{4} +2.66379 q^{5} +0.273176 q^{8} +O(q^{10})\) \(q-0.0683740 q^{2} -1.99532 q^{4} +2.66379 q^{5} +0.273176 q^{8} -0.182134 q^{10} +1.59913 q^{11} +5.25380 q^{13} +3.97197 q^{16} +6.54721 q^{17} +1.90194 q^{19} -5.31513 q^{20} -0.109339 q^{22} +3.06837 q^{23} +2.09578 q^{25} -0.359223 q^{26} -6.38903 q^{29} +6.71923 q^{31} -0.817932 q^{32} -0.447659 q^{34} +4.22955 q^{37} -0.130043 q^{38} +0.727684 q^{40} -7.39296 q^{41} -11.2635 q^{43} -3.19078 q^{44} -0.209797 q^{46} -3.79918 q^{47} -0.143297 q^{50} -10.4830 q^{52} -8.89862 q^{53} +4.25974 q^{55} +0.436843 q^{58} +10.8928 q^{59} +2.71386 q^{61} -0.459420 q^{62} -7.88802 q^{64} +13.9950 q^{65} -3.32533 q^{67} -13.0638 q^{68} +12.3890 q^{71} +2.19863 q^{73} -0.289191 q^{74} -3.79498 q^{76} +0.813556 q^{79} +10.5805 q^{80} +0.505486 q^{82} -6.83684 q^{83} +17.4404 q^{85} +0.770132 q^{86} +0.436843 q^{88} -0.470572 q^{89} -6.12240 q^{92} +0.259765 q^{94} +5.06636 q^{95} -5.15246 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8} + 20 q^{11} + 12 q^{16} + 32 q^{23} + 12 q^{25} + 16 q^{29} + 48 q^{32} + 12 q^{37} + 56 q^{44} - 24 q^{46} - 4 q^{50} + 32 q^{53} + 48 q^{64} + 60 q^{65} + 12 q^{67} + 56 q^{71} + 68 q^{74} - 12 q^{79} - 12 q^{85} + 76 q^{86} + 16 q^{92} + 64 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0683740 −0.0483477 −0.0241739 0.999708i \(-0.507696\pi\)
−0.0241739 + 0.999708i \(0.507696\pi\)
\(3\) 0 0
\(4\) −1.99532 −0.997662
\(5\) 2.66379 1.19128 0.595642 0.803250i \(-0.296898\pi\)
0.595642 + 0.803250i \(0.296898\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0.273176 0.0965824
\(9\) 0 0
\(10\) −0.182134 −0.0575958
\(11\) 1.59913 0.482155 0.241077 0.970506i \(-0.422499\pi\)
0.241077 + 0.970506i \(0.422499\pi\)
\(12\) 0 0
\(13\) 5.25380 1.45714 0.728571 0.684970i \(-0.240185\pi\)
0.728571 + 0.684970i \(0.240185\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.97197 0.992993
\(17\) 6.54721 1.58793 0.793966 0.607963i \(-0.208013\pi\)
0.793966 + 0.607963i \(0.208013\pi\)
\(18\) 0 0
\(19\) 1.90194 0.436334 0.218167 0.975911i \(-0.429992\pi\)
0.218167 + 0.975911i \(0.429992\pi\)
\(20\) −5.31513 −1.18850
\(21\) 0 0
\(22\) −0.109339 −0.0233111
\(23\) 3.06837 0.639800 0.319900 0.947451i \(-0.396351\pi\)
0.319900 + 0.947451i \(0.396351\pi\)
\(24\) 0 0
\(25\) 2.09578 0.419157
\(26\) −0.359223 −0.0704495
\(27\) 0 0
\(28\) 0 0
\(29\) −6.38903 −1.18641 −0.593207 0.805050i \(-0.702138\pi\)
−0.593207 + 0.805050i \(0.702138\pi\)
\(30\) 0 0
\(31\) 6.71923 1.20681 0.603405 0.797435i \(-0.293810\pi\)
0.603405 + 0.797435i \(0.293810\pi\)
\(32\) −0.817932 −0.144591
\(33\) 0 0
\(34\) −0.447659 −0.0767728
\(35\) 0 0
\(36\) 0 0
\(37\) 4.22955 0.695333 0.347667 0.937618i \(-0.386974\pi\)
0.347667 + 0.937618i \(0.386974\pi\)
\(38\) −0.130043 −0.0210958
\(39\) 0 0
\(40\) 0.727684 0.115057
\(41\) −7.39296 −1.15459 −0.577293 0.816537i \(-0.695891\pi\)
−0.577293 + 0.816537i \(0.695891\pi\)
\(42\) 0 0
\(43\) −11.2635 −1.71767 −0.858836 0.512251i \(-0.828812\pi\)
−0.858836 + 0.512251i \(0.828812\pi\)
\(44\) −3.19078 −0.481028
\(45\) 0 0
\(46\) −0.209797 −0.0309329
\(47\) −3.79918 −0.554167 −0.277083 0.960846i \(-0.589368\pi\)
−0.277083 + 0.960846i \(0.589368\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −0.143297 −0.0202653
\(51\) 0 0
\(52\) −10.4830 −1.45374
\(53\) −8.89862 −1.22232 −0.611160 0.791507i \(-0.709297\pi\)
−0.611160 + 0.791507i \(0.709297\pi\)
\(54\) 0 0
\(55\) 4.25974 0.574383
\(56\) 0 0
\(57\) 0 0
\(58\) 0.436843 0.0573604
\(59\) 10.8928 1.41812 0.709060 0.705148i \(-0.249120\pi\)
0.709060 + 0.705148i \(0.249120\pi\)
\(60\) 0 0
\(61\) 2.71386 0.347475 0.173737 0.984792i \(-0.444416\pi\)
0.173737 + 0.984792i \(0.444416\pi\)
\(62\) −0.459420 −0.0583465
\(63\) 0 0
\(64\) −7.88802 −0.986002
\(65\) 13.9950 1.73587
\(66\) 0 0
\(67\) −3.32533 −0.406254 −0.203127 0.979152i \(-0.565110\pi\)
−0.203127 + 0.979152i \(0.565110\pi\)
\(68\) −13.0638 −1.58422
\(69\) 0 0
\(70\) 0 0
\(71\) 12.3890 1.47031 0.735154 0.677900i \(-0.237110\pi\)
0.735154 + 0.677900i \(0.237110\pi\)
\(72\) 0 0
\(73\) 2.19863 0.257331 0.128665 0.991688i \(-0.458931\pi\)
0.128665 + 0.991688i \(0.458931\pi\)
\(74\) −0.289191 −0.0336178
\(75\) 0 0
\(76\) −3.79498 −0.435314
\(77\) 0 0
\(78\) 0 0
\(79\) 0.813556 0.0915322 0.0457661 0.998952i \(-0.485427\pi\)
0.0457661 + 0.998952i \(0.485427\pi\)
\(80\) 10.5805 1.18294
\(81\) 0 0
\(82\) 0.505486 0.0558216
\(83\) −6.83684 −0.750441 −0.375220 0.926936i \(-0.622433\pi\)
−0.375220 + 0.926936i \(0.622433\pi\)
\(84\) 0 0
\(85\) 17.4404 1.89168
\(86\) 0.770132 0.0830455
\(87\) 0 0
\(88\) 0.436843 0.0465677
\(89\) −0.470572 −0.0498805 −0.0249403 0.999689i \(-0.507940\pi\)
−0.0249403 + 0.999689i \(0.507940\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6.12240 −0.638305
\(93\) 0 0
\(94\) 0.259765 0.0267927
\(95\) 5.06636 0.519798
\(96\) 0 0
\(97\) −5.15246 −0.523153 −0.261576 0.965183i \(-0.584242\pi\)
−0.261576 + 0.965183i \(0.584242\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −4.18177 −0.418177
\(101\) 1.84488 0.183572 0.0917862 0.995779i \(-0.470742\pi\)
0.0917862 + 0.995779i \(0.470742\pi\)
\(102\) 0 0
\(103\) 5.17802 0.510206 0.255103 0.966914i \(-0.417891\pi\)
0.255103 + 0.966914i \(0.417891\pi\)
\(104\) 1.43521 0.140734
\(105\) 0 0
\(106\) 0.608434 0.0590964
\(107\) 16.9489 1.63851 0.819256 0.573428i \(-0.194387\pi\)
0.819256 + 0.573428i \(0.194387\pi\)
\(108\) 0 0
\(109\) −8.49992 −0.814145 −0.407073 0.913396i \(-0.633450\pi\)
−0.407073 + 0.913396i \(0.633450\pi\)
\(110\) −0.291255 −0.0277701
\(111\) 0 0
\(112\) 0 0
\(113\) −3.90392 −0.367250 −0.183625 0.982996i \(-0.558783\pi\)
−0.183625 + 0.982996i \(0.558783\pi\)
\(114\) 0 0
\(115\) 8.17351 0.762183
\(116\) 12.7482 1.18364
\(117\) 0 0
\(118\) −0.744783 −0.0685628
\(119\) 0 0
\(120\) 0 0
\(121\) −8.44279 −0.767527
\(122\) −0.185558 −0.0167996
\(123\) 0 0
\(124\) −13.4070 −1.20399
\(125\) −7.73623 −0.691949
\(126\) 0 0
\(127\) 10.9533 0.971946 0.485973 0.873974i \(-0.338465\pi\)
0.485973 + 0.873974i \(0.338465\pi\)
\(128\) 2.17520 0.192262
\(129\) 0 0
\(130\) −0.956896 −0.0839253
\(131\) 4.45342 0.389097 0.194549 0.980893i \(-0.437676\pi\)
0.194549 + 0.980893i \(0.437676\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0.227366 0.0196414
\(135\) 0 0
\(136\) 1.78854 0.153366
\(137\) 19.5360 1.66907 0.834537 0.550952i \(-0.185735\pi\)
0.834537 + 0.550952i \(0.185735\pi\)
\(138\) 0 0
\(139\) 2.63079 0.223141 0.111570 0.993757i \(-0.464412\pi\)
0.111570 + 0.993757i \(0.464412\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −0.847087 −0.0710860
\(143\) 8.40149 0.702568
\(144\) 0 0
\(145\) −17.0190 −1.41335
\(146\) −0.150329 −0.0124413
\(147\) 0 0
\(148\) −8.43932 −0.693708
\(149\) 8.81281 0.721973 0.360987 0.932571i \(-0.382440\pi\)
0.360987 + 0.932571i \(0.382440\pi\)
\(150\) 0 0
\(151\) 4.66422 0.379569 0.189784 0.981826i \(-0.439221\pi\)
0.189784 + 0.981826i \(0.439221\pi\)
\(152\) 0.519564 0.0421422
\(153\) 0 0
\(154\) 0 0
\(155\) 17.8986 1.43765
\(156\) 0 0
\(157\) −4.07294 −0.325056 −0.162528 0.986704i \(-0.551965\pi\)
−0.162528 + 0.986704i \(0.551965\pi\)
\(158\) −0.0556261 −0.00442537
\(159\) 0 0
\(160\) −2.17880 −0.172249
\(161\) 0 0
\(162\) 0 0
\(163\) −12.1222 −0.949487 −0.474744 0.880124i \(-0.657459\pi\)
−0.474744 + 0.880124i \(0.657459\pi\)
\(164\) 14.7514 1.15189
\(165\) 0 0
\(166\) 0.467462 0.0362821
\(167\) 4.79902 0.371360 0.185680 0.982610i \(-0.440551\pi\)
0.185680 + 0.982610i \(0.440551\pi\)
\(168\) 0 0
\(169\) 14.6024 1.12326
\(170\) −1.19247 −0.0914582
\(171\) 0 0
\(172\) 22.4744 1.71366
\(173\) −5.03171 −0.382554 −0.191277 0.981536i \(-0.561263\pi\)
−0.191277 + 0.981536i \(0.561263\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 6.35169 0.478776
\(177\) 0 0
\(178\) 0.0321749 0.00241161
\(179\) 16.3979 1.22564 0.612819 0.790224i \(-0.290036\pi\)
0.612819 + 0.790224i \(0.290036\pi\)
\(180\) 0 0
\(181\) −14.4345 −1.07291 −0.536454 0.843930i \(-0.680237\pi\)
−0.536454 + 0.843930i \(0.680237\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0.838207 0.0617934
\(185\) 11.2666 0.828339
\(186\) 0 0
\(187\) 10.4698 0.765629
\(188\) 7.58059 0.552871
\(189\) 0 0
\(190\) −0.346407 −0.0251310
\(191\) −2.84131 −0.205590 −0.102795 0.994703i \(-0.532779\pi\)
−0.102795 + 0.994703i \(0.532779\pi\)
\(192\) 0 0
\(193\) 8.82886 0.635515 0.317758 0.948172i \(-0.397070\pi\)
0.317758 + 0.948172i \(0.397070\pi\)
\(194\) 0.352294 0.0252932
\(195\) 0 0
\(196\) 0 0
\(197\) −5.72354 −0.407785 −0.203893 0.978993i \(-0.565359\pi\)
−0.203893 + 0.978993i \(0.565359\pi\)
\(198\) 0 0
\(199\) −11.4150 −0.809191 −0.404596 0.914496i \(-0.632588\pi\)
−0.404596 + 0.914496i \(0.632588\pi\)
\(200\) 0.572518 0.0404832
\(201\) 0 0
\(202\) −0.126142 −0.00887530
\(203\) 0 0
\(204\) 0 0
\(205\) −19.6933 −1.37544
\(206\) −0.354042 −0.0246673
\(207\) 0 0
\(208\) 20.8679 1.44693
\(209\) 3.04144 0.210381
\(210\) 0 0
\(211\) −21.3837 −1.47212 −0.736059 0.676918i \(-0.763315\pi\)
−0.736059 + 0.676918i \(0.763315\pi\)
\(212\) 17.7556 1.21946
\(213\) 0 0
\(214\) −1.15886 −0.0792183
\(215\) −30.0037 −2.04623
\(216\) 0 0
\(217\) 0 0
\(218\) 0.581174 0.0393620
\(219\) 0 0
\(220\) −8.49956 −0.573041
\(221\) 34.3977 2.31384
\(222\) 0 0
\(223\) 7.16774 0.479987 0.239994 0.970774i \(-0.422855\pi\)
0.239994 + 0.970774i \(0.422855\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0.266926 0.0177557
\(227\) 13.7887 0.915187 0.457593 0.889162i \(-0.348712\pi\)
0.457593 + 0.889162i \(0.348712\pi\)
\(228\) 0 0
\(229\) −26.3943 −1.74418 −0.872092 0.489341i \(-0.837237\pi\)
−0.872092 + 0.489341i \(0.837237\pi\)
\(230\) −0.558855 −0.0368498
\(231\) 0 0
\(232\) −1.74533 −0.114587
\(233\) 12.6446 0.828375 0.414187 0.910192i \(-0.364066\pi\)
0.414187 + 0.910192i \(0.364066\pi\)
\(234\) 0 0
\(235\) −10.1202 −0.660170
\(236\) −21.7346 −1.41481
\(237\) 0 0
\(238\) 0 0
\(239\) 15.4328 0.998265 0.499133 0.866526i \(-0.333652\pi\)
0.499133 + 0.866526i \(0.333652\pi\)
\(240\) 0 0
\(241\) 1.17988 0.0760029 0.0380015 0.999278i \(-0.487901\pi\)
0.0380015 + 0.999278i \(0.487901\pi\)
\(242\) 0.577267 0.0371082
\(243\) 0 0
\(244\) −5.41504 −0.346662
\(245\) 0 0
\(246\) 0 0
\(247\) 9.99240 0.635801
\(248\) 1.83553 0.116557
\(249\) 0 0
\(250\) 0.528957 0.0334542
\(251\) −5.54970 −0.350294 −0.175147 0.984542i \(-0.556040\pi\)
−0.175147 + 0.984542i \(0.556040\pi\)
\(252\) 0 0
\(253\) 4.90672 0.308483
\(254\) −0.748919 −0.0469913
\(255\) 0 0
\(256\) 15.6273 0.976707
\(257\) −9.83076 −0.613226 −0.306613 0.951834i \(-0.599196\pi\)
−0.306613 + 0.951834i \(0.599196\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −27.9246 −1.73181
\(261\) 0 0
\(262\) −0.304498 −0.0188120
\(263\) −11.9322 −0.735774 −0.367887 0.929870i \(-0.619919\pi\)
−0.367887 + 0.929870i \(0.619919\pi\)
\(264\) 0 0
\(265\) −23.7041 −1.45613
\(266\) 0 0
\(267\) 0 0
\(268\) 6.63512 0.405304
\(269\) −29.9648 −1.82699 −0.913494 0.406853i \(-0.866626\pi\)
−0.913494 + 0.406853i \(0.866626\pi\)
\(270\) 0 0
\(271\) −7.09650 −0.431082 −0.215541 0.976495i \(-0.569151\pi\)
−0.215541 + 0.976495i \(0.569151\pi\)
\(272\) 26.0053 1.57680
\(273\) 0 0
\(274\) −1.33575 −0.0806959
\(275\) 3.35142 0.202098
\(276\) 0 0
\(277\) −9.82351 −0.590237 −0.295119 0.955461i \(-0.595359\pi\)
−0.295119 + 0.955461i \(0.595359\pi\)
\(278\) −0.179878 −0.0107883
\(279\) 0 0
\(280\) 0 0
\(281\) 23.8777 1.42443 0.712213 0.701963i \(-0.247693\pi\)
0.712213 + 0.701963i \(0.247693\pi\)
\(282\) 0 0
\(283\) −3.01595 −0.179280 −0.0896399 0.995974i \(-0.528572\pi\)
−0.0896399 + 0.995974i \(0.528572\pi\)
\(284\) −24.7201 −1.46687
\(285\) 0 0
\(286\) −0.574443 −0.0339675
\(287\) 0 0
\(288\) 0 0
\(289\) 25.8659 1.52153
\(290\) 1.16366 0.0683324
\(291\) 0 0
\(292\) −4.38699 −0.256729
\(293\) −17.0583 −0.996554 −0.498277 0.867018i \(-0.666034\pi\)
−0.498277 + 0.867018i \(0.666034\pi\)
\(294\) 0 0
\(295\) 29.0161 1.68938
\(296\) 1.15541 0.0671570
\(297\) 0 0
\(298\) −0.602567 −0.0349058
\(299\) 16.1206 0.932280
\(300\) 0 0
\(301\) 0 0
\(302\) −0.318911 −0.0183513
\(303\) 0 0
\(304\) 7.55444 0.433277
\(305\) 7.22917 0.413941
\(306\) 0 0
\(307\) 23.2178 1.32511 0.662554 0.749014i \(-0.269473\pi\)
0.662554 + 0.749014i \(0.269473\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.22380 −0.0695072
\(311\) −1.79093 −0.101555 −0.0507773 0.998710i \(-0.516170\pi\)
−0.0507773 + 0.998710i \(0.516170\pi\)
\(312\) 0 0
\(313\) 4.60917 0.260526 0.130263 0.991480i \(-0.458418\pi\)
0.130263 + 0.991480i \(0.458418\pi\)
\(314\) 0.278483 0.0157157
\(315\) 0 0
\(316\) −1.62331 −0.0913183
\(317\) 25.8841 1.45380 0.726898 0.686745i \(-0.240961\pi\)
0.726898 + 0.686745i \(0.240961\pi\)
\(318\) 0 0
\(319\) −10.2169 −0.572035
\(320\) −21.0120 −1.17461
\(321\) 0 0
\(322\) 0 0
\(323\) 12.4524 0.692869
\(324\) 0 0
\(325\) 11.0108 0.610771
\(326\) 0.828846 0.0459055
\(327\) 0 0
\(328\) −2.01958 −0.111513
\(329\) 0 0
\(330\) 0 0
\(331\) 0.161323 0.00886714 0.00443357 0.999990i \(-0.498589\pi\)
0.00443357 + 0.999990i \(0.498589\pi\)
\(332\) 13.6417 0.748687
\(333\) 0 0
\(334\) −0.328128 −0.0179544
\(335\) −8.85799 −0.483964
\(336\) 0 0
\(337\) −9.05351 −0.493176 −0.246588 0.969120i \(-0.579309\pi\)
−0.246588 + 0.969120i \(0.579309\pi\)
\(338\) −0.998425 −0.0543072
\(339\) 0 0
\(340\) −34.7993 −1.88725
\(341\) 10.7449 0.581869
\(342\) 0 0
\(343\) 0 0
\(344\) −3.07693 −0.165897
\(345\) 0 0
\(346\) 0.344038 0.0184956
\(347\) 5.81968 0.312417 0.156208 0.987724i \(-0.450073\pi\)
0.156208 + 0.987724i \(0.450073\pi\)
\(348\) 0 0
\(349\) 27.2619 1.45930 0.729648 0.683823i \(-0.239684\pi\)
0.729648 + 0.683823i \(0.239684\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.30798 −0.0697154
\(353\) 24.1896 1.28748 0.643741 0.765244i \(-0.277382\pi\)
0.643741 + 0.765244i \(0.277382\pi\)
\(354\) 0 0
\(355\) 33.0018 1.75155
\(356\) 0.938944 0.0497639
\(357\) 0 0
\(358\) −1.12119 −0.0592568
\(359\) 21.0376 1.11032 0.555161 0.831743i \(-0.312657\pi\)
0.555161 + 0.831743i \(0.312657\pi\)
\(360\) 0 0
\(361\) −15.3826 −0.809612
\(362\) 0.986944 0.0518726
\(363\) 0 0
\(364\) 0 0
\(365\) 5.85670 0.306554
\(366\) 0 0
\(367\) 35.0380 1.82897 0.914485 0.404620i \(-0.132596\pi\)
0.914485 + 0.404620i \(0.132596\pi\)
\(368\) 12.1875 0.635317
\(369\) 0 0
\(370\) −0.770345 −0.0400483
\(371\) 0 0
\(372\) 0 0
\(373\) 1.12862 0.0584377 0.0292189 0.999573i \(-0.490698\pi\)
0.0292189 + 0.999573i \(0.490698\pi\)
\(374\) −0.715863 −0.0370164
\(375\) 0 0
\(376\) −1.03784 −0.0535227
\(377\) −33.5667 −1.72877
\(378\) 0 0
\(379\) −21.9619 −1.12811 −0.564054 0.825738i \(-0.690759\pi\)
−0.564054 + 0.825738i \(0.690759\pi\)
\(380\) −10.1090 −0.518583
\(381\) 0 0
\(382\) 0.194272 0.00993981
\(383\) 23.0401 1.17729 0.588647 0.808390i \(-0.299661\pi\)
0.588647 + 0.808390i \(0.299661\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −0.603664 −0.0307257
\(387\) 0 0
\(388\) 10.2808 0.521930
\(389\) −15.7751 −0.799828 −0.399914 0.916553i \(-0.630960\pi\)
−0.399914 + 0.916553i \(0.630960\pi\)
\(390\) 0 0
\(391\) 20.0893 1.01596
\(392\) 0 0
\(393\) 0 0
\(394\) 0.391341 0.0197155
\(395\) 2.16714 0.109041
\(396\) 0 0
\(397\) 16.5055 0.828389 0.414195 0.910188i \(-0.364063\pi\)
0.414195 + 0.910188i \(0.364063\pi\)
\(398\) 0.780492 0.0391225
\(399\) 0 0
\(400\) 8.32439 0.416220
\(401\) −21.6600 −1.08165 −0.540823 0.841136i \(-0.681887\pi\)
−0.540823 + 0.841136i \(0.681887\pi\)
\(402\) 0 0
\(403\) 35.3015 1.75849
\(404\) −3.68113 −0.183143
\(405\) 0 0
\(406\) 0 0
\(407\) 6.76358 0.335258
\(408\) 0 0
\(409\) −30.5721 −1.51169 −0.755846 0.654750i \(-0.772774\pi\)
−0.755846 + 0.654750i \(0.772774\pi\)
\(410\) 1.34651 0.0664993
\(411\) 0 0
\(412\) −10.3318 −0.509013
\(413\) 0 0
\(414\) 0 0
\(415\) −18.2119 −0.893988
\(416\) −4.29725 −0.210690
\(417\) 0 0
\(418\) −0.207955 −0.0101714
\(419\) 21.6162 1.05602 0.528011 0.849237i \(-0.322938\pi\)
0.528011 + 0.849237i \(0.322938\pi\)
\(420\) 0 0
\(421\) −27.2434 −1.32776 −0.663881 0.747838i \(-0.731092\pi\)
−0.663881 + 0.747838i \(0.731092\pi\)
\(422\) 1.46209 0.0711735
\(423\) 0 0
\(424\) −2.43089 −0.118055
\(425\) 13.7215 0.665592
\(426\) 0 0
\(427\) 0 0
\(428\) −33.8186 −1.63468
\(429\) 0 0
\(430\) 2.05147 0.0989307
\(431\) −8.19687 −0.394829 −0.197415 0.980320i \(-0.563255\pi\)
−0.197415 + 0.980320i \(0.563255\pi\)
\(432\) 0 0
\(433\) 3.41468 0.164099 0.0820494 0.996628i \(-0.473853\pi\)
0.0820494 + 0.996628i \(0.473853\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 16.9601 0.812242
\(437\) 5.83585 0.279167
\(438\) 0 0
\(439\) 6.58831 0.314443 0.157221 0.987563i \(-0.449746\pi\)
0.157221 + 0.987563i \(0.449746\pi\)
\(440\) 1.16366 0.0554753
\(441\) 0 0
\(442\) −2.35191 −0.111869
\(443\) −28.6912 −1.36316 −0.681581 0.731743i \(-0.738707\pi\)
−0.681581 + 0.731743i \(0.738707\pi\)
\(444\) 0 0
\(445\) −1.25350 −0.0594218
\(446\) −0.490087 −0.0232063
\(447\) 0 0
\(448\) 0 0
\(449\) −0.457724 −0.0216013 −0.0108007 0.999942i \(-0.503438\pi\)
−0.0108007 + 0.999942i \(0.503438\pi\)
\(450\) 0 0
\(451\) −11.8223 −0.556689
\(452\) 7.78958 0.366391
\(453\) 0 0
\(454\) −0.942787 −0.0442472
\(455\) 0 0
\(456\) 0 0
\(457\) 20.2210 0.945900 0.472950 0.881089i \(-0.343189\pi\)
0.472950 + 0.881089i \(0.343189\pi\)
\(458\) 1.80468 0.0843273
\(459\) 0 0
\(460\) −16.3088 −0.760402
\(461\) −24.2071 −1.12744 −0.563719 0.825967i \(-0.690630\pi\)
−0.563719 + 0.825967i \(0.690630\pi\)
\(462\) 0 0
\(463\) −4.80483 −0.223299 −0.111650 0.993748i \(-0.535613\pi\)
−0.111650 + 0.993748i \(0.535613\pi\)
\(464\) −25.3770 −1.17810
\(465\) 0 0
\(466\) −0.864561 −0.0400500
\(467\) 27.2456 1.26078 0.630389 0.776279i \(-0.282895\pi\)
0.630389 + 0.776279i \(0.282895\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0.691959 0.0319177
\(471\) 0 0
\(472\) 2.97565 0.136965
\(473\) −18.0118 −0.828184
\(474\) 0 0
\(475\) 3.98605 0.182892
\(476\) 0 0
\(477\) 0 0
\(478\) −1.05520 −0.0482638
\(479\) −20.5255 −0.937834 −0.468917 0.883242i \(-0.655356\pi\)
−0.468917 + 0.883242i \(0.655356\pi\)
\(480\) 0 0
\(481\) 22.2212 1.01320
\(482\) −0.0806733 −0.00367457
\(483\) 0 0
\(484\) 16.8461 0.765733
\(485\) −13.7251 −0.623224
\(486\) 0 0
\(487\) 25.8449 1.17114 0.585571 0.810621i \(-0.300870\pi\)
0.585571 + 0.810621i \(0.300870\pi\)
\(488\) 0.741363 0.0335599
\(489\) 0 0
\(490\) 0 0
\(491\) −15.6155 −0.704718 −0.352359 0.935865i \(-0.614620\pi\)
−0.352359 + 0.935865i \(0.614620\pi\)
\(492\) 0 0
\(493\) −41.8303 −1.88394
\(494\) −0.683220 −0.0307395
\(495\) 0 0
\(496\) 26.6886 1.19835
\(497\) 0 0
\(498\) 0 0
\(499\) 21.2690 0.952133 0.476066 0.879409i \(-0.342062\pi\)
0.476066 + 0.879409i \(0.342062\pi\)
\(500\) 15.4363 0.690332
\(501\) 0 0
\(502\) 0.379455 0.0169359
\(503\) 16.3298 0.728110 0.364055 0.931377i \(-0.381392\pi\)
0.364055 + 0.931377i \(0.381392\pi\)
\(504\) 0 0
\(505\) 4.91437 0.218687
\(506\) −0.335492 −0.0149144
\(507\) 0 0
\(508\) −21.8553 −0.969674
\(509\) 13.4618 0.596683 0.298342 0.954459i \(-0.403567\pi\)
0.298342 + 0.954459i \(0.403567\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −5.41890 −0.239484
\(513\) 0 0
\(514\) 0.672168 0.0296481
\(515\) 13.7932 0.607800
\(516\) 0 0
\(517\) −6.07536 −0.267194
\(518\) 0 0
\(519\) 0 0
\(520\) 3.82311 0.167654
\(521\) 1.42619 0.0624826 0.0312413 0.999512i \(-0.490054\pi\)
0.0312413 + 0.999512i \(0.490054\pi\)
\(522\) 0 0
\(523\) −7.71060 −0.337161 −0.168581 0.985688i \(-0.553918\pi\)
−0.168581 + 0.985688i \(0.553918\pi\)
\(524\) −8.88603 −0.388188
\(525\) 0 0
\(526\) 0.815855 0.0355730
\(527\) 43.9922 1.91633
\(528\) 0 0
\(529\) −13.5851 −0.590656
\(530\) 1.62074 0.0704005
\(531\) 0 0
\(532\) 0 0
\(533\) −38.8411 −1.68240
\(534\) 0 0
\(535\) 45.1483 1.95193
\(536\) −0.908402 −0.0392370
\(537\) 0 0
\(538\) 2.04881 0.0883306
\(539\) 0 0
\(540\) 0 0
\(541\) 28.0456 1.20577 0.602886 0.797827i \(-0.294017\pi\)
0.602886 + 0.797827i \(0.294017\pi\)
\(542\) 0.485216 0.0208418
\(543\) 0 0
\(544\) −5.35517 −0.229601
\(545\) −22.6420 −0.969878
\(546\) 0 0
\(547\) −35.4610 −1.51620 −0.758101 0.652137i \(-0.773873\pi\)
−0.758101 + 0.652137i \(0.773873\pi\)
\(548\) −38.9807 −1.66517
\(549\) 0 0
\(550\) −0.229150 −0.00977099
\(551\) −12.1515 −0.517673
\(552\) 0 0
\(553\) 0 0
\(554\) 0.671672 0.0285366
\(555\) 0 0
\(556\) −5.24928 −0.222619
\(557\) −35.0419 −1.48477 −0.742386 0.669972i \(-0.766306\pi\)
−0.742386 + 0.669972i \(0.766306\pi\)
\(558\) 0 0
\(559\) −59.1763 −2.50289
\(560\) 0 0
\(561\) 0 0
\(562\) −1.63262 −0.0688677
\(563\) −16.0262 −0.675425 −0.337712 0.941249i \(-0.609653\pi\)
−0.337712 + 0.941249i \(0.609653\pi\)
\(564\) 0 0
\(565\) −10.3992 −0.437499
\(566\) 0.206213 0.00866776
\(567\) 0 0
\(568\) 3.38439 0.142006
\(569\) −0.371302 −0.0155658 −0.00778290 0.999970i \(-0.502477\pi\)
−0.00778290 + 0.999970i \(0.502477\pi\)
\(570\) 0 0
\(571\) 29.2304 1.22325 0.611626 0.791147i \(-0.290516\pi\)
0.611626 + 0.791147i \(0.290516\pi\)
\(572\) −16.7637 −0.700926
\(573\) 0 0
\(574\) 0 0
\(575\) 6.43065 0.268177
\(576\) 0 0
\(577\) −15.0570 −0.626833 −0.313417 0.949616i \(-0.601474\pi\)
−0.313417 + 0.949616i \(0.601474\pi\)
\(578\) −1.76856 −0.0735623
\(579\) 0 0
\(580\) 33.9585 1.41005
\(581\) 0 0
\(582\) 0 0
\(583\) −14.2300 −0.589347
\(584\) 0.600615 0.0248536
\(585\) 0 0
\(586\) 1.16634 0.0481811
\(587\) −1.67180 −0.0690026 −0.0345013 0.999405i \(-0.510984\pi\)
−0.0345013 + 0.999405i \(0.510984\pi\)
\(588\) 0 0
\(589\) 12.7795 0.526572
\(590\) −1.98395 −0.0816778
\(591\) 0 0
\(592\) 16.7996 0.690461
\(593\) 10.8174 0.444218 0.222109 0.975022i \(-0.428706\pi\)
0.222109 + 0.975022i \(0.428706\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −17.5844 −0.720286
\(597\) 0 0
\(598\) −1.10223 −0.0450736
\(599\) −16.6401 −0.679898 −0.339949 0.940444i \(-0.610410\pi\)
−0.339949 + 0.940444i \(0.610410\pi\)
\(600\) 0 0
\(601\) −25.8022 −1.05249 −0.526246 0.850332i \(-0.676401\pi\)
−0.526246 + 0.850332i \(0.676401\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −9.30663 −0.378681
\(605\) −22.4898 −0.914342
\(606\) 0 0
\(607\) −37.8049 −1.53445 −0.767227 0.641376i \(-0.778364\pi\)
−0.767227 + 0.641376i \(0.778364\pi\)
\(608\) −1.55566 −0.0630901
\(609\) 0 0
\(610\) −0.494287 −0.0200131
\(611\) −19.9601 −0.807499
\(612\) 0 0
\(613\) −12.9544 −0.523222 −0.261611 0.965173i \(-0.584254\pi\)
−0.261611 + 0.965173i \(0.584254\pi\)
\(614\) −1.58749 −0.0640659
\(615\) 0 0
\(616\) 0 0
\(617\) 32.4403 1.30600 0.652999 0.757359i \(-0.273511\pi\)
0.652999 + 0.757359i \(0.273511\pi\)
\(618\) 0 0
\(619\) −33.1974 −1.33431 −0.667157 0.744917i \(-0.732489\pi\)
−0.667157 + 0.744917i \(0.732489\pi\)
\(620\) −35.7136 −1.43429
\(621\) 0 0
\(622\) 0.122453 0.00490993
\(623\) 0 0
\(624\) 0 0
\(625\) −31.0866 −1.24346
\(626\) −0.315147 −0.0125958
\(627\) 0 0
\(628\) 8.12683 0.324296
\(629\) 27.6917 1.10414
\(630\) 0 0
\(631\) 32.2773 1.28494 0.642470 0.766311i \(-0.277910\pi\)
0.642470 + 0.766311i \(0.277910\pi\)
\(632\) 0.222244 0.00884040
\(633\) 0 0
\(634\) −1.76980 −0.0702877
\(635\) 29.1772 1.15786
\(636\) 0 0
\(637\) 0 0
\(638\) 0.698568 0.0276566
\(639\) 0 0
\(640\) 5.79428 0.229039
\(641\) −43.0814 −1.70161 −0.850806 0.525480i \(-0.823886\pi\)
−0.850806 + 0.525480i \(0.823886\pi\)
\(642\) 0 0
\(643\) −6.40176 −0.252461 −0.126230 0.992001i \(-0.540288\pi\)
−0.126230 + 0.992001i \(0.540288\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −0.851419 −0.0334986
\(647\) 3.88807 0.152856 0.0764278 0.997075i \(-0.475649\pi\)
0.0764278 + 0.997075i \(0.475649\pi\)
\(648\) 0 0
\(649\) 17.4189 0.683753
\(650\) −0.752854 −0.0295294
\(651\) 0 0
\(652\) 24.1878 0.947268
\(653\) −15.1035 −0.591044 −0.295522 0.955336i \(-0.595494\pi\)
−0.295522 + 0.955336i \(0.595494\pi\)
\(654\) 0 0
\(655\) 11.8630 0.463525
\(656\) −29.3646 −1.14650
\(657\) 0 0
\(658\) 0 0
\(659\) −14.2600 −0.555492 −0.277746 0.960655i \(-0.589587\pi\)
−0.277746 + 0.960655i \(0.589587\pi\)
\(660\) 0 0
\(661\) 19.4193 0.755323 0.377662 0.925944i \(-0.376728\pi\)
0.377662 + 0.925944i \(0.376728\pi\)
\(662\) −0.0110303 −0.000428706 0
\(663\) 0 0
\(664\) −1.86766 −0.0724794
\(665\) 0 0
\(666\) 0 0
\(667\) −19.6039 −0.759067
\(668\) −9.57561 −0.370492
\(669\) 0 0
\(670\) 0.605656 0.0233985
\(671\) 4.33981 0.167537
\(672\) 0 0
\(673\) 5.93126 0.228633 0.114317 0.993444i \(-0.463532\pi\)
0.114317 + 0.993444i \(0.463532\pi\)
\(674\) 0.619024 0.0238439
\(675\) 0 0
\(676\) −29.1366 −1.12064
\(677\) 36.9826 1.42136 0.710678 0.703518i \(-0.248388\pi\)
0.710678 + 0.703518i \(0.248388\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 4.76430 0.182703
\(681\) 0 0
\(682\) −0.734671 −0.0281320
\(683\) −13.1360 −0.502635 −0.251317 0.967905i \(-0.580864\pi\)
−0.251317 + 0.967905i \(0.580864\pi\)
\(684\) 0 0
\(685\) 52.0398 1.98834
\(686\) 0 0
\(687\) 0 0
\(688\) −44.7384 −1.70564
\(689\) −46.7516 −1.78109
\(690\) 0 0
\(691\) −14.7658 −0.561719 −0.280860 0.959749i \(-0.590620\pi\)
−0.280860 + 0.959749i \(0.590620\pi\)
\(692\) 10.0399 0.381659
\(693\) 0 0
\(694\) −0.397915 −0.0151046
\(695\) 7.00788 0.265824
\(696\) 0 0
\(697\) −48.4032 −1.83340
\(698\) −1.86401 −0.0705536
\(699\) 0 0
\(700\) 0 0
\(701\) 30.4627 1.15056 0.575281 0.817956i \(-0.304893\pi\)
0.575281 + 0.817956i \(0.304893\pi\)
\(702\) 0 0
\(703\) 8.04433 0.303398
\(704\) −12.6139 −0.475406
\(705\) 0 0
\(706\) −1.65394 −0.0622468
\(707\) 0 0
\(708\) 0 0
\(709\) 14.1030 0.529650 0.264825 0.964296i \(-0.414686\pi\)
0.264825 + 0.964296i \(0.414686\pi\)
\(710\) −2.25646 −0.0846836
\(711\) 0 0
\(712\) −0.128549 −0.00481758
\(713\) 20.6171 0.772117
\(714\) 0 0
\(715\) 22.3798 0.836958
\(716\) −32.7192 −1.22277
\(717\) 0 0
\(718\) −1.43842 −0.0536815
\(719\) 14.9958 0.559249 0.279624 0.960109i \(-0.409790\pi\)
0.279624 + 0.960109i \(0.409790\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.05177 0.0391429
\(723\) 0 0
\(724\) 28.8015 1.07040
\(725\) −13.3900 −0.497293
\(726\) 0 0
\(727\) −26.1055 −0.968199 −0.484099 0.875013i \(-0.660853\pi\)
−0.484099 + 0.875013i \(0.660853\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −0.400446 −0.0148212
\(731\) −73.7447 −2.72754
\(732\) 0 0
\(733\) −28.3821 −1.04832 −0.524159 0.851621i \(-0.675620\pi\)
−0.524159 + 0.851621i \(0.675620\pi\)
\(734\) −2.39569 −0.0884265
\(735\) 0 0
\(736\) −2.50972 −0.0925096
\(737\) −5.31763 −0.195877
\(738\) 0 0
\(739\) 46.5865 1.71371 0.856857 0.515555i \(-0.172414\pi\)
0.856857 + 0.515555i \(0.172414\pi\)
\(740\) −22.4806 −0.826403
\(741\) 0 0
\(742\) 0 0
\(743\) −0.339027 −0.0124377 −0.00621884 0.999981i \(-0.501980\pi\)
−0.00621884 + 0.999981i \(0.501980\pi\)
\(744\) 0 0
\(745\) 23.4755 0.860075
\(746\) −0.0771683 −0.00282533
\(747\) 0 0
\(748\) −20.8907 −0.763839
\(749\) 0 0
\(750\) 0 0
\(751\) −36.3662 −1.32702 −0.663510 0.748168i \(-0.730934\pi\)
−0.663510 + 0.748168i \(0.730934\pi\)
\(752\) −15.0902 −0.550284
\(753\) 0 0
\(754\) 2.29509 0.0835822
\(755\) 12.4245 0.452174
\(756\) 0 0
\(757\) −27.4703 −0.998424 −0.499212 0.866480i \(-0.666377\pi\)
−0.499212 + 0.866480i \(0.666377\pi\)
\(758\) 1.50162 0.0545415
\(759\) 0 0
\(760\) 1.38401 0.0502033
\(761\) 33.0357 1.19754 0.598771 0.800920i \(-0.295656\pi\)
0.598771 + 0.800920i \(0.295656\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 5.66934 0.205110
\(765\) 0 0
\(766\) −1.57534 −0.0569194
\(767\) 57.2285 2.06640
\(768\) 0 0
\(769\) −2.57751 −0.0929475 −0.0464738 0.998920i \(-0.514798\pi\)
−0.0464738 + 0.998920i \(0.514798\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −17.6164 −0.634030
\(773\) 6.72973 0.242051 0.121026 0.992649i \(-0.461382\pi\)
0.121026 + 0.992649i \(0.461382\pi\)
\(774\) 0 0
\(775\) 14.0820 0.505842
\(776\) −1.40753 −0.0505274
\(777\) 0 0
\(778\) 1.07860 0.0386698
\(779\) −14.0609 −0.503785
\(780\) 0 0
\(781\) 19.8116 0.708916
\(782\) −1.37358 −0.0491193
\(783\) 0 0
\(784\) 0 0
\(785\) −10.8495 −0.387234
\(786\) 0 0
\(787\) −28.6683 −1.02191 −0.510956 0.859607i \(-0.670709\pi\)
−0.510956 + 0.859607i \(0.670709\pi\)
\(788\) 11.4203 0.406832
\(789\) 0 0
\(790\) −0.148176 −0.00527187
\(791\) 0 0
\(792\) 0 0
\(793\) 14.2581 0.506320
\(794\) −1.12855 −0.0400507
\(795\) 0 0
\(796\) 22.7767 0.807300
\(797\) −22.9825 −0.814084 −0.407042 0.913410i \(-0.633440\pi\)
−0.407042 + 0.913410i \(0.633440\pi\)
\(798\) 0 0
\(799\) −24.8740 −0.879979
\(800\) −1.71421 −0.0606064
\(801\) 0 0
\(802\) 1.48098 0.0522951
\(803\) 3.51589 0.124073
\(804\) 0 0
\(805\) 0 0
\(806\) −2.41370 −0.0850191
\(807\) 0 0
\(808\) 0.503977 0.0177299
\(809\) 16.4779 0.579332 0.289666 0.957128i \(-0.406456\pi\)
0.289666 + 0.957128i \(0.406456\pi\)
\(810\) 0 0
\(811\) −40.4318 −1.41975 −0.709876 0.704326i \(-0.751249\pi\)
−0.709876 + 0.704326i \(0.751249\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −0.462453 −0.0162090
\(815\) −32.2911 −1.13111
\(816\) 0 0
\(817\) −21.4225 −0.749479
\(818\) 2.09033 0.0730868
\(819\) 0 0
\(820\) 39.2945 1.37222
\(821\) 28.1086 0.980998 0.490499 0.871442i \(-0.336814\pi\)
0.490499 + 0.871442i \(0.336814\pi\)
\(822\) 0 0
\(823\) 25.9058 0.903019 0.451510 0.892266i \(-0.350886\pi\)
0.451510 + 0.892266i \(0.350886\pi\)
\(824\) 1.41451 0.0492769
\(825\) 0 0
\(826\) 0 0
\(827\) 17.7998 0.618961 0.309480 0.950906i \(-0.399845\pi\)
0.309480 + 0.950906i \(0.399845\pi\)
\(828\) 0 0
\(829\) 15.7069 0.545523 0.272761 0.962082i \(-0.412063\pi\)
0.272761 + 0.962082i \(0.412063\pi\)
\(830\) 1.24522 0.0432223
\(831\) 0 0
\(832\) −41.4421 −1.43675
\(833\) 0 0
\(834\) 0 0
\(835\) 12.7836 0.442395
\(836\) −6.06866 −0.209889
\(837\) 0 0
\(838\) −1.47799 −0.0510563
\(839\) 7.39643 0.255353 0.127677 0.991816i \(-0.459248\pi\)
0.127677 + 0.991816i \(0.459248\pi\)
\(840\) 0 0
\(841\) 11.8197 0.407576
\(842\) 1.86274 0.0641942
\(843\) 0 0
\(844\) 42.6675 1.46868
\(845\) 38.8978 1.33812
\(846\) 0 0
\(847\) 0 0
\(848\) −35.3451 −1.21376
\(849\) 0 0
\(850\) −0.938196 −0.0321798
\(851\) 12.9778 0.444874
\(852\) 0 0
\(853\) −53.1262 −1.81901 −0.909503 0.415698i \(-0.863537\pi\)
−0.909503 + 0.415698i \(0.863537\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 4.63004 0.158251
\(857\) 3.81530 0.130328 0.0651640 0.997875i \(-0.479243\pi\)
0.0651640 + 0.997875i \(0.479243\pi\)
\(858\) 0 0
\(859\) 38.9768 1.32987 0.664936 0.746901i \(-0.268459\pi\)
0.664936 + 0.746901i \(0.268459\pi\)
\(860\) 59.8671 2.04145
\(861\) 0 0
\(862\) 0.560452 0.0190891
\(863\) −26.6736 −0.907978 −0.453989 0.891007i \(-0.650000\pi\)
−0.453989 + 0.891007i \(0.650000\pi\)
\(864\) 0 0
\(865\) −13.4034 −0.455730
\(866\) −0.233475 −0.00793380
\(867\) 0 0
\(868\) 0 0
\(869\) 1.30098 0.0441327
\(870\) 0 0
\(871\) −17.4706 −0.591970
\(872\) −2.32198 −0.0786321
\(873\) 0 0
\(874\) −0.399020 −0.0134971
\(875\) 0 0
\(876\) 0 0
\(877\) 24.0135 0.810879 0.405440 0.914122i \(-0.367118\pi\)
0.405440 + 0.914122i \(0.367118\pi\)
\(878\) −0.450469 −0.0152026
\(879\) 0 0
\(880\) 16.9196 0.570358
\(881\) 4.67326 0.157446 0.0787231 0.996897i \(-0.474916\pi\)
0.0787231 + 0.996897i \(0.474916\pi\)
\(882\) 0 0
\(883\) −35.6948 −1.20122 −0.600612 0.799541i \(-0.705076\pi\)
−0.600612 + 0.799541i \(0.705076\pi\)
\(884\) −68.6346 −2.30843
\(885\) 0 0
\(886\) 1.96173 0.0659058
\(887\) −29.1032 −0.977191 −0.488596 0.872510i \(-0.662491\pi\)
−0.488596 + 0.872510i \(0.662491\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0.0857071 0.00287291
\(891\) 0 0
\(892\) −14.3020 −0.478865
\(893\) −7.22579 −0.241802
\(894\) 0 0
\(895\) 43.6806 1.46008
\(896\) 0 0
\(897\) 0 0
\(898\) 0.0312964 0.00104437
\(899\) −42.9294 −1.43177
\(900\) 0 0
\(901\) −58.2611 −1.94096
\(902\) 0.808336 0.0269146
\(903\) 0 0
\(904\) −1.06646 −0.0354699
\(905\) −38.4505 −1.27814
\(906\) 0 0
\(907\) −41.2142 −1.36849 −0.684247 0.729250i \(-0.739869\pi\)
−0.684247 + 0.729250i \(0.739869\pi\)
\(908\) −27.5129 −0.913047
\(909\) 0 0
\(910\) 0 0
\(911\) 57.7239 1.91248 0.956239 0.292588i \(-0.0945163\pi\)
0.956239 + 0.292588i \(0.0945163\pi\)
\(912\) 0 0
\(913\) −10.9330 −0.361829
\(914\) −1.38259 −0.0457321
\(915\) 0 0
\(916\) 52.6652 1.74011
\(917\) 0 0
\(918\) 0 0
\(919\) −51.5598 −1.70080 −0.850400 0.526137i \(-0.823640\pi\)
−0.850400 + 0.526137i \(0.823640\pi\)
\(920\) 2.23281 0.0736135
\(921\) 0 0
\(922\) 1.65514 0.0545090
\(923\) 65.0895 2.14245
\(924\) 0 0
\(925\) 8.86422 0.291454
\(926\) 0.328525 0.0107960
\(927\) 0 0
\(928\) 5.22579 0.171545
\(929\) 50.2824 1.64971 0.824856 0.565343i \(-0.191256\pi\)
0.824856 + 0.565343i \(0.191256\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −25.2301 −0.826439
\(933\) 0 0
\(934\) −1.86289 −0.0609557
\(935\) 27.8894 0.912081
\(936\) 0 0
\(937\) 18.1400 0.592607 0.296303 0.955094i \(-0.404246\pi\)
0.296303 + 0.955094i \(0.404246\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 20.1931 0.658627
\(941\) 17.0332 0.555267 0.277633 0.960687i \(-0.410450\pi\)
0.277633 + 0.960687i \(0.410450\pi\)
\(942\) 0 0
\(943\) −22.6844 −0.738704
\(944\) 43.2658 1.40818
\(945\) 0 0
\(946\) 1.23154 0.0400408
\(947\) −31.5059 −1.02381 −0.511903 0.859044i \(-0.671059\pi\)
−0.511903 + 0.859044i \(0.671059\pi\)
\(948\) 0 0
\(949\) 11.5512 0.374967
\(950\) −0.272542 −0.00884243
\(951\) 0 0
\(952\) 0 0
\(953\) −16.0677 −0.520485 −0.260242 0.965543i \(-0.583803\pi\)
−0.260242 + 0.965543i \(0.583803\pi\)
\(954\) 0 0
\(955\) −7.56866 −0.244916
\(956\) −30.7935 −0.995932
\(957\) 0 0
\(958\) 1.40341 0.0453421
\(959\) 0 0
\(960\) 0 0
\(961\) 14.1480 0.456388
\(962\) −1.51935 −0.0489859
\(963\) 0 0
\(964\) −2.35425 −0.0758253
\(965\) 23.5182 0.757079
\(966\) 0 0
\(967\) −26.6098 −0.855713 −0.427857 0.903847i \(-0.640731\pi\)
−0.427857 + 0.903847i \(0.640731\pi\)
\(968\) −2.30637 −0.0741296
\(969\) 0 0
\(970\) 0.938438 0.0301314
\(971\) 56.5678 1.81535 0.907674 0.419676i \(-0.137856\pi\)
0.907674 + 0.419676i \(0.137856\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −1.76712 −0.0566221
\(975\) 0 0
\(976\) 10.7794 0.345040
\(977\) −53.7560 −1.71981 −0.859904 0.510456i \(-0.829477\pi\)
−0.859904 + 0.510456i \(0.829477\pi\)
\(978\) 0 0
\(979\) −0.752504 −0.0240501
\(980\) 0 0
\(981\) 0 0
\(982\) 1.06769 0.0340715
\(983\) −27.4102 −0.874249 −0.437125 0.899401i \(-0.644003\pi\)
−0.437125 + 0.899401i \(0.644003\pi\)
\(984\) 0 0
\(985\) −15.2463 −0.485788
\(986\) 2.86011 0.0910843
\(987\) 0 0
\(988\) −19.9381 −0.634315
\(989\) −34.5607 −1.09897
\(990\) 0 0
\(991\) −17.3374 −0.550740 −0.275370 0.961338i \(-0.588800\pi\)
−0.275370 + 0.961338i \(0.588800\pi\)
\(992\) −5.49587 −0.174494
\(993\) 0 0
\(994\) 0 0
\(995\) −30.4073 −0.963976
\(996\) 0 0
\(997\) −35.6638 −1.12948 −0.564742 0.825268i \(-0.691024\pi\)
−0.564742 + 0.825268i \(0.691024\pi\)
\(998\) −1.45425 −0.0460334
\(999\) 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 3969.2.a.bi.1.6 12
3.2 odd 2 3969.2.a.bh.1.7 12
7.6 odd 2 inner 3969.2.a.bi.1.5 12
9.2 odd 6 441.2.f.h.148.6 yes 24
9.4 even 3 1323.2.f.h.883.7 24
9.5 odd 6 441.2.f.h.295.6 yes 24
9.7 even 3 1323.2.f.h.442.7 24
21.20 even 2 3969.2.a.bh.1.8 12
63.2 odd 6 441.2.g.h.67.5 24
63.4 even 3 1323.2.g.h.667.8 24
63.5 even 6 441.2.h.h.214.8 24
63.11 odd 6 441.2.h.h.373.7 24
63.13 odd 6 1323.2.f.h.883.8 24
63.16 even 3 1323.2.g.h.361.8 24
63.20 even 6 441.2.f.h.148.5 24
63.23 odd 6 441.2.h.h.214.7 24
63.25 even 3 1323.2.h.h.226.6 24
63.31 odd 6 1323.2.g.h.667.7 24
63.32 odd 6 441.2.g.h.79.5 24
63.34 odd 6 1323.2.f.h.442.8 24
63.38 even 6 441.2.h.h.373.8 24
63.40 odd 6 1323.2.h.h.802.5 24
63.41 even 6 441.2.f.h.295.5 yes 24
63.47 even 6 441.2.g.h.67.6 24
63.52 odd 6 1323.2.h.h.226.5 24
63.58 even 3 1323.2.h.h.802.6 24
63.59 even 6 441.2.g.h.79.6 24
63.61 odd 6 1323.2.g.h.361.7 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.2.f.h.148.5 24 63.20 even 6
441.2.f.h.148.6 yes 24 9.2 odd 6
441.2.f.h.295.5 yes 24 63.41 even 6
441.2.f.h.295.6 yes 24 9.5 odd 6
441.2.g.h.67.5 24 63.2 odd 6
441.2.g.h.67.6 24 63.47 even 6
441.2.g.h.79.5 24 63.32 odd 6
441.2.g.h.79.6 24 63.59 even 6
441.2.h.h.214.7 24 63.23 odd 6
441.2.h.h.214.8 24 63.5 even 6
441.2.h.h.373.7 24 63.11 odd 6
441.2.h.h.373.8 24 63.38 even 6
1323.2.f.h.442.7 24 9.7 even 3
1323.2.f.h.442.8 24 63.34 odd 6
1323.2.f.h.883.7 24 9.4 even 3
1323.2.f.h.883.8 24 63.13 odd 6
1323.2.g.h.361.7 24 63.61 odd 6
1323.2.g.h.361.8 24 63.16 even 3
1323.2.g.h.667.7 24 63.31 odd 6
1323.2.g.h.667.8 24 63.4 even 3
1323.2.h.h.226.5 24 63.52 odd 6
1323.2.h.h.226.6 24 63.25 even 3
1323.2.h.h.802.5 24 63.40 odd 6
1323.2.h.h.802.6 24 63.58 even 3
3969.2.a.bh.1.7 12 3.2 odd 2
3969.2.a.bh.1.8 12 21.20 even 2
3969.2.a.bi.1.5 12 7.6 odd 2 inner
3969.2.a.bi.1.6 12 1.1 even 1 trivial