Properties

Label 7440.2.a.bp.1.2
Level $7440$
Weight $2$
Character 7440.1
Self dual yes
Analytic conductor $59.409$
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] = [7440,2,Mod(1,7440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7440, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("7440.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7440.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: \(59.4086991038\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 465)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 7440.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -0.688892 q^{7} +1.00000 q^{9} -2.00000 q^{11} +3.73975 q^{13} -1.00000 q^{15} +2.28100 q^{17} +0.688892 q^{21} -2.90321 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.02074 q^{29} -1.00000 q^{31} +2.00000 q^{33} -0.688892 q^{35} -8.79060 q^{37} -3.73975 q^{39} +2.75557 q^{41} -1.05086 q^{43} +1.00000 q^{45} -4.70964 q^{47} -6.52543 q^{49} -2.28100 q^{51} +2.14764 q^{53} -2.00000 q^{55} +6.44938 q^{59} -5.05086 q^{61} -0.688892 q^{63} +3.73975 q^{65} +5.44446 q^{67} +2.90321 q^{69} -8.96989 q^{71} -8.92396 q^{73} -1.00000 q^{75} +1.37778 q^{77} +8.38271 q^{79} +1.00000 q^{81} +10.5161 q^{83} +2.28100 q^{85} +4.02074 q^{87} +13.5002 q^{89} -2.57628 q^{91} +1.00000 q^{93} -12.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 6 q^{11} - 2 q^{13} - 3 q^{15} + 2 q^{21} - 2 q^{23} + 3 q^{25} - 3 q^{27} + 8 q^{29} - 3 q^{31} + 6 q^{33} - 2 q^{35} + 2 q^{39} + 8 q^{41} + 10 q^{43} + 3 q^{45}+ \cdots - 6 q^{99}+O(q^{100}) \) 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 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.688892 −0.260377 −0.130188 0.991489i \(-0.541558\pi\)
−0.130188 + 0.991489i \(0.541558\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 3.73975 1.03722 0.518610 0.855011i \(-0.326450\pi\)
0.518610 + 0.855011i \(0.326450\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 2.28100 0.553223 0.276611 0.960982i \(-0.410789\pi\)
0.276611 + 0.960982i \(0.410789\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0.688892 0.150329
\(22\) 0 0
\(23\) −2.90321 −0.605362 −0.302681 0.953092i \(-0.597882\pi\)
−0.302681 + 0.953092i \(0.597882\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.02074 −0.746633 −0.373317 0.927704i \(-0.621780\pi\)
−0.373317 + 0.927704i \(0.621780\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) −0.688892 −0.116444
\(36\) 0 0
\(37\) −8.79060 −1.44517 −0.722583 0.691284i \(-0.757045\pi\)
−0.722583 + 0.691284i \(0.757045\pi\)
\(38\) 0 0
\(39\) −3.73975 −0.598839
\(40\) 0 0
\(41\) 2.75557 0.430348 0.215174 0.976576i \(-0.430968\pi\)
0.215174 + 0.976576i \(0.430968\pi\)
\(42\) 0 0
\(43\) −1.05086 −0.160254 −0.0801270 0.996785i \(-0.525533\pi\)
−0.0801270 + 0.996785i \(0.525533\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −4.70964 −0.686971 −0.343485 0.939158i \(-0.611608\pi\)
−0.343485 + 0.939158i \(0.611608\pi\)
\(48\) 0 0
\(49\) −6.52543 −0.932204
\(50\) 0 0
\(51\) −2.28100 −0.319403
\(52\) 0 0
\(53\) 2.14764 0.295001 0.147501 0.989062i \(-0.452877\pi\)
0.147501 + 0.989062i \(0.452877\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.44938 0.839638 0.419819 0.907608i \(-0.362094\pi\)
0.419819 + 0.907608i \(0.362094\pi\)
\(60\) 0 0
\(61\) −5.05086 −0.646696 −0.323348 0.946280i \(-0.604808\pi\)
−0.323348 + 0.946280i \(0.604808\pi\)
\(62\) 0 0
\(63\) −0.688892 −0.0867923
\(64\) 0 0
\(65\) 3.73975 0.463859
\(66\) 0 0
\(67\) 5.44446 0.665147 0.332573 0.943077i \(-0.392083\pi\)
0.332573 + 0.943077i \(0.392083\pi\)
\(68\) 0 0
\(69\) 2.90321 0.349506
\(70\) 0 0
\(71\) −8.96989 −1.06453 −0.532265 0.846578i \(-0.678659\pi\)
−0.532265 + 0.846578i \(0.678659\pi\)
\(72\) 0 0
\(73\) −8.92396 −1.04447 −0.522235 0.852802i \(-0.674902\pi\)
−0.522235 + 0.852802i \(0.674902\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 1.37778 0.157013
\(78\) 0 0
\(79\) 8.38271 0.943128 0.471564 0.881832i \(-0.343690\pi\)
0.471564 + 0.881832i \(0.343690\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.5161 1.15429 0.577144 0.816643i \(-0.304167\pi\)
0.577144 + 0.816643i \(0.304167\pi\)
\(84\) 0 0
\(85\) 2.28100 0.247409
\(86\) 0 0
\(87\) 4.02074 0.431069
\(88\) 0 0
\(89\) 13.5002 1.43102 0.715511 0.698601i \(-0.246194\pi\)
0.715511 + 0.698601i \(0.246194\pi\)
\(90\) 0 0
\(91\) −2.57628 −0.270068
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 8.29529 0.825412 0.412706 0.910864i \(-0.364584\pi\)
0.412706 + 0.910864i \(0.364584\pi\)
\(102\) 0 0
\(103\) −2.98418 −0.294040 −0.147020 0.989134i \(-0.546968\pi\)
−0.147020 + 0.989134i \(0.546968\pi\)
\(104\) 0 0
\(105\) 0.688892 0.0672290
\(106\) 0 0
\(107\) −1.52543 −0.147469 −0.0737343 0.997278i \(-0.523492\pi\)
−0.0737343 + 0.997278i \(0.523492\pi\)
\(108\) 0 0
\(109\) 3.76049 0.360190 0.180095 0.983649i \(-0.442360\pi\)
0.180095 + 0.983649i \(0.442360\pi\)
\(110\) 0 0
\(111\) 8.79060 0.834367
\(112\) 0 0
\(113\) −9.18421 −0.863978 −0.431989 0.901879i \(-0.642188\pi\)
−0.431989 + 0.901879i \(0.642188\pi\)
\(114\) 0 0
\(115\) −2.90321 −0.270726
\(116\) 0 0
\(117\) 3.73975 0.345740
\(118\) 0 0
\(119\) −1.57136 −0.144046
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −2.75557 −0.248461
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 11.4795 1.01864 0.509320 0.860577i \(-0.329897\pi\)
0.509320 + 0.860577i \(0.329897\pi\)
\(128\) 0 0
\(129\) 1.05086 0.0925226
\(130\) 0 0
\(131\) −13.0716 −1.14207 −0.571035 0.820925i \(-0.693458\pi\)
−0.571035 + 0.820925i \(0.693458\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −11.1383 −0.951607 −0.475804 0.879552i \(-0.657843\pi\)
−0.475804 + 0.879552i \(0.657843\pi\)
\(138\) 0 0
\(139\) −10.2351 −0.868127 −0.434063 0.900882i \(-0.642921\pi\)
−0.434063 + 0.900882i \(0.642921\pi\)
\(140\) 0 0
\(141\) 4.70964 0.396623
\(142\) 0 0
\(143\) −7.47949 −0.625467
\(144\) 0 0
\(145\) −4.02074 −0.333905
\(146\) 0 0
\(147\) 6.52543 0.538208
\(148\) 0 0
\(149\) −1.51114 −0.123797 −0.0618986 0.998082i \(-0.519716\pi\)
−0.0618986 + 0.998082i \(0.519716\pi\)
\(150\) 0 0
\(151\) 1.76049 0.143267 0.0716334 0.997431i \(-0.477179\pi\)
0.0716334 + 0.997431i \(0.477179\pi\)
\(152\) 0 0
\(153\) 2.28100 0.184408
\(154\) 0 0
\(155\) −1.00000 −0.0803219
\(156\) 0 0
\(157\) −3.43801 −0.274383 −0.137191 0.990545i \(-0.543808\pi\)
−0.137191 + 0.990545i \(0.543808\pi\)
\(158\) 0 0
\(159\) −2.14764 −0.170319
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 0 0
\(163\) 10.5970 0.830023 0.415012 0.909816i \(-0.363778\pi\)
0.415012 + 0.909816i \(0.363778\pi\)
\(164\) 0 0
\(165\) 2.00000 0.155700
\(166\) 0 0
\(167\) 20.0415 1.55086 0.775428 0.631435i \(-0.217534\pi\)
0.775428 + 0.631435i \(0.217534\pi\)
\(168\) 0 0
\(169\) 0.985710 0.0758238
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −18.8988 −1.43685 −0.718423 0.695606i \(-0.755136\pi\)
−0.718423 + 0.695606i \(0.755136\pi\)
\(174\) 0 0
\(175\) −0.688892 −0.0520754
\(176\) 0 0
\(177\) −6.44938 −0.484765
\(178\) 0 0
\(179\) −16.2953 −1.21797 −0.608983 0.793183i \(-0.708422\pi\)
−0.608983 + 0.793183i \(0.708422\pi\)
\(180\) 0 0
\(181\) −1.70471 −0.126710 −0.0633552 0.997991i \(-0.520180\pi\)
−0.0633552 + 0.997991i \(0.520180\pi\)
\(182\) 0 0
\(183\) 5.05086 0.373370
\(184\) 0 0
\(185\) −8.79060 −0.646298
\(186\) 0 0
\(187\) −4.56199 −0.333606
\(188\) 0 0
\(189\) 0.688892 0.0501095
\(190\) 0 0
\(191\) 3.53188 0.255558 0.127779 0.991803i \(-0.459215\pi\)
0.127779 + 0.991803i \(0.459215\pi\)
\(192\) 0 0
\(193\) 17.4193 1.25387 0.626933 0.779073i \(-0.284310\pi\)
0.626933 + 0.779073i \(0.284310\pi\)
\(194\) 0 0
\(195\) −3.73975 −0.267809
\(196\) 0 0
\(197\) 10.0143 0.713489 0.356744 0.934202i \(-0.383887\pi\)
0.356744 + 0.934202i \(0.383887\pi\)
\(198\) 0 0
\(199\) −27.1798 −1.92672 −0.963361 0.268208i \(-0.913569\pi\)
−0.963361 + 0.268208i \(0.913569\pi\)
\(200\) 0 0
\(201\) −5.44446 −0.384023
\(202\) 0 0
\(203\) 2.76986 0.194406
\(204\) 0 0
\(205\) 2.75557 0.192457
\(206\) 0 0
\(207\) −2.90321 −0.201787
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 1.93978 0.133540 0.0667699 0.997768i \(-0.478731\pi\)
0.0667699 + 0.997768i \(0.478731\pi\)
\(212\) 0 0
\(213\) 8.96989 0.614607
\(214\) 0 0
\(215\) −1.05086 −0.0716677
\(216\) 0 0
\(217\) 0.688892 0.0467650
\(218\) 0 0
\(219\) 8.92396 0.603025
\(220\) 0 0
\(221\) 8.53035 0.573813
\(222\) 0 0
\(223\) −23.5526 −1.57720 −0.788600 0.614906i \(-0.789194\pi\)
−0.788600 + 0.614906i \(0.789194\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −19.8020 −1.31430 −0.657152 0.753758i \(-0.728239\pi\)
−0.657152 + 0.753758i \(0.728239\pi\)
\(228\) 0 0
\(229\) −24.4099 −1.61305 −0.806526 0.591199i \(-0.798655\pi\)
−0.806526 + 0.591199i \(0.798655\pi\)
\(230\) 0 0
\(231\) −1.37778 −0.0906516
\(232\) 0 0
\(233\) 3.13828 0.205595 0.102798 0.994702i \(-0.467221\pi\)
0.102798 + 0.994702i \(0.467221\pi\)
\(234\) 0 0
\(235\) −4.70964 −0.307223
\(236\) 0 0
\(237\) −8.38271 −0.544515
\(238\) 0 0
\(239\) −6.81579 −0.440877 −0.220438 0.975401i \(-0.570749\pi\)
−0.220438 + 0.975401i \(0.570749\pi\)
\(240\) 0 0
\(241\) 17.0923 1.10101 0.550507 0.834830i \(-0.314434\pi\)
0.550507 + 0.834830i \(0.314434\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −6.52543 −0.416894
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −10.5161 −0.666428
\(250\) 0 0
\(251\) −16.1748 −1.02095 −0.510473 0.859894i \(-0.670530\pi\)
−0.510473 + 0.859894i \(0.670530\pi\)
\(252\) 0 0
\(253\) 5.80642 0.365047
\(254\) 0 0
\(255\) −2.28100 −0.142842
\(256\) 0 0
\(257\) 19.0464 1.18808 0.594041 0.804435i \(-0.297532\pi\)
0.594041 + 0.804435i \(0.297532\pi\)
\(258\) 0 0
\(259\) 6.05578 0.376288
\(260\) 0 0
\(261\) −4.02074 −0.248878
\(262\) 0 0
\(263\) 6.23506 0.384470 0.192235 0.981349i \(-0.438426\pi\)
0.192235 + 0.981349i \(0.438426\pi\)
\(264\) 0 0
\(265\) 2.14764 0.131929
\(266\) 0 0
\(267\) −13.5002 −0.826201
\(268\) 0 0
\(269\) 20.8365 1.27043 0.635213 0.772337i \(-0.280912\pi\)
0.635213 + 0.772337i \(0.280912\pi\)
\(270\) 0 0
\(271\) −6.62222 −0.402271 −0.201135 0.979563i \(-0.564463\pi\)
−0.201135 + 0.979563i \(0.564463\pi\)
\(272\) 0 0
\(273\) 2.57628 0.155924
\(274\) 0 0
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) 4.79060 0.287839 0.143920 0.989589i \(-0.454029\pi\)
0.143920 + 0.989589i \(0.454029\pi\)
\(278\) 0 0
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −20.8256 −1.24235 −0.621177 0.783671i \(-0.713345\pi\)
−0.621177 + 0.783671i \(0.713345\pi\)
\(282\) 0 0
\(283\) 11.8731 0.705783 0.352891 0.935664i \(-0.385199\pi\)
0.352891 + 0.935664i \(0.385199\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.89829 −0.112053
\(288\) 0 0
\(289\) −11.7971 −0.693944
\(290\) 0 0
\(291\) 12.0000 0.703452
\(292\) 0 0
\(293\) −18.0143 −1.05241 −0.526203 0.850359i \(-0.676385\pi\)
−0.526203 + 0.850359i \(0.676385\pi\)
\(294\) 0 0
\(295\) 6.44938 0.375498
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 0 0
\(299\) −10.8573 −0.627893
\(300\) 0 0
\(301\) 0.723926 0.0417264
\(302\) 0 0
\(303\) −8.29529 −0.476552
\(304\) 0 0
\(305\) −5.05086 −0.289211
\(306\) 0 0
\(307\) −10.9240 −0.623463 −0.311732 0.950170i \(-0.600909\pi\)
−0.311732 + 0.950170i \(0.600909\pi\)
\(308\) 0 0
\(309\) 2.98418 0.169764
\(310\) 0 0
\(311\) −5.10324 −0.289378 −0.144689 0.989477i \(-0.546218\pi\)
−0.144689 + 0.989477i \(0.546218\pi\)
\(312\) 0 0
\(313\) −22.1082 −1.24963 −0.624814 0.780774i \(-0.714825\pi\)
−0.624814 + 0.780774i \(0.714825\pi\)
\(314\) 0 0
\(315\) −0.688892 −0.0388147
\(316\) 0 0
\(317\) −1.06515 −0.0598245 −0.0299123 0.999553i \(-0.509523\pi\)
−0.0299123 + 0.999553i \(0.509523\pi\)
\(318\) 0 0
\(319\) 8.04149 0.450237
\(320\) 0 0
\(321\) 1.52543 0.0851411
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 3.73975 0.207444
\(326\) 0 0
\(327\) −3.76049 −0.207956
\(328\) 0 0
\(329\) 3.24443 0.178871
\(330\) 0 0
\(331\) 24.9447 1.37108 0.685542 0.728033i \(-0.259565\pi\)
0.685542 + 0.728033i \(0.259565\pi\)
\(332\) 0 0
\(333\) −8.79060 −0.481722
\(334\) 0 0
\(335\) 5.44446 0.297463
\(336\) 0 0
\(337\) 0.453829 0.0247216 0.0123608 0.999924i \(-0.496065\pi\)
0.0123608 + 0.999924i \(0.496065\pi\)
\(338\) 0 0
\(339\) 9.18421 0.498818
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) 9.31756 0.503101
\(344\) 0 0
\(345\) 2.90321 0.156304
\(346\) 0 0
\(347\) 29.7560 1.59739 0.798694 0.601737i \(-0.205525\pi\)
0.798694 + 0.601737i \(0.205525\pi\)
\(348\) 0 0
\(349\) −9.65878 −0.517023 −0.258511 0.966008i \(-0.583232\pi\)
−0.258511 + 0.966008i \(0.583232\pi\)
\(350\) 0 0
\(351\) −3.73975 −0.199613
\(352\) 0 0
\(353\) −16.8617 −0.897459 −0.448730 0.893668i \(-0.648123\pi\)
−0.448730 + 0.893668i \(0.648123\pi\)
\(354\) 0 0
\(355\) −8.96989 −0.476072
\(356\) 0 0
\(357\) 1.57136 0.0831652
\(358\) 0 0
\(359\) −8.64296 −0.456158 −0.228079 0.973643i \(-0.573244\pi\)
−0.228079 + 0.973643i \(0.573244\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) −8.92396 −0.467101
\(366\) 0 0
\(367\) −6.95899 −0.363256 −0.181628 0.983367i \(-0.558137\pi\)
−0.181628 + 0.983367i \(0.558137\pi\)
\(368\) 0 0
\(369\) 2.75557 0.143449
\(370\) 0 0
\(371\) −1.47949 −0.0768115
\(372\) 0 0
\(373\) 21.0005 1.08736 0.543682 0.839291i \(-0.317030\pi\)
0.543682 + 0.839291i \(0.317030\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −15.0366 −0.774422
\(378\) 0 0
\(379\) −29.7146 −1.52633 −0.763167 0.646201i \(-0.776357\pi\)
−0.763167 + 0.646201i \(0.776357\pi\)
\(380\) 0 0
\(381\) −11.4795 −0.588112
\(382\) 0 0
\(383\) −4.05578 −0.207241 −0.103620 0.994617i \(-0.533043\pi\)
−0.103620 + 0.994617i \(0.533043\pi\)
\(384\) 0 0
\(385\) 1.37778 0.0702184
\(386\) 0 0
\(387\) −1.05086 −0.0534180
\(388\) 0 0
\(389\) −10.9985 −0.557644 −0.278822 0.960343i \(-0.589944\pi\)
−0.278822 + 0.960343i \(0.589944\pi\)
\(390\) 0 0
\(391\) −6.62222 −0.334900
\(392\) 0 0
\(393\) 13.0716 0.659375
\(394\) 0 0
\(395\) 8.38271 0.421780
\(396\) 0 0
\(397\) 0.193576 0.00971531 0.00485765 0.999988i \(-0.498454\pi\)
0.00485765 + 0.999988i \(0.498454\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −38.6242 −1.92880 −0.964401 0.264445i \(-0.914811\pi\)
−0.964401 + 0.264445i \(0.914811\pi\)
\(402\) 0 0
\(403\) −3.73975 −0.186290
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 17.5812 0.871468
\(408\) 0 0
\(409\) 2.94914 0.145826 0.0729129 0.997338i \(-0.476770\pi\)
0.0729129 + 0.997338i \(0.476770\pi\)
\(410\) 0 0
\(411\) 11.1383 0.549411
\(412\) 0 0
\(413\) −4.44293 −0.218622
\(414\) 0 0
\(415\) 10.5161 0.516213
\(416\) 0 0
\(417\) 10.2351 0.501213
\(418\) 0 0
\(419\) 26.0020 1.27028 0.635141 0.772397i \(-0.280942\pi\)
0.635141 + 0.772397i \(0.280942\pi\)
\(420\) 0 0
\(421\) −20.0143 −0.975437 −0.487718 0.873001i \(-0.662171\pi\)
−0.487718 + 0.873001i \(0.662171\pi\)
\(422\) 0 0
\(423\) −4.70964 −0.228990
\(424\) 0 0
\(425\) 2.28100 0.110645
\(426\) 0 0
\(427\) 3.47949 0.168385
\(428\) 0 0
\(429\) 7.47949 0.361113
\(430\) 0 0
\(431\) 2.77631 0.133730 0.0668651 0.997762i \(-0.478700\pi\)
0.0668651 + 0.997762i \(0.478700\pi\)
\(432\) 0 0
\(433\) −30.7590 −1.47818 −0.739091 0.673606i \(-0.764744\pi\)
−0.739091 + 0.673606i \(0.764744\pi\)
\(434\) 0 0
\(435\) 4.02074 0.192780
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −26.8385 −1.28093 −0.640467 0.767986i \(-0.721259\pi\)
−0.640467 + 0.767986i \(0.721259\pi\)
\(440\) 0 0
\(441\) −6.52543 −0.310735
\(442\) 0 0
\(443\) 11.7190 0.556787 0.278393 0.960467i \(-0.410198\pi\)
0.278393 + 0.960467i \(0.410198\pi\)
\(444\) 0 0
\(445\) 13.5002 0.639973
\(446\) 0 0
\(447\) 1.51114 0.0714743
\(448\) 0 0
\(449\) 33.8084 1.59552 0.797759 0.602976i \(-0.206019\pi\)
0.797759 + 0.602976i \(0.206019\pi\)
\(450\) 0 0
\(451\) −5.51114 −0.259509
\(452\) 0 0
\(453\) −1.76049 −0.0827151
\(454\) 0 0
\(455\) −2.57628 −0.120778
\(456\) 0 0
\(457\) 5.61930 0.262860 0.131430 0.991325i \(-0.458043\pi\)
0.131430 + 0.991325i \(0.458043\pi\)
\(458\) 0 0
\(459\) −2.28100 −0.106468
\(460\) 0 0
\(461\) −2.93825 −0.136848 −0.0684239 0.997656i \(-0.521797\pi\)
−0.0684239 + 0.997656i \(0.521797\pi\)
\(462\) 0 0
\(463\) 1.61285 0.0749554 0.0374777 0.999297i \(-0.488068\pi\)
0.0374777 + 0.999297i \(0.488068\pi\)
\(464\) 0 0
\(465\) 1.00000 0.0463739
\(466\) 0 0
\(467\) 11.5857 0.536120 0.268060 0.963402i \(-0.413617\pi\)
0.268060 + 0.963402i \(0.413617\pi\)
\(468\) 0 0
\(469\) −3.75065 −0.173189
\(470\) 0 0
\(471\) 3.43801 0.158415
\(472\) 0 0
\(473\) 2.10171 0.0966367
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.14764 0.0983338
\(478\) 0 0
\(479\) −26.0020 −1.18806 −0.594031 0.804442i \(-0.702464\pi\)
−0.594031 + 0.804442i \(0.702464\pi\)
\(480\) 0 0
\(481\) −32.8746 −1.49895
\(482\) 0 0
\(483\) −2.00000 −0.0910032
\(484\) 0 0
\(485\) −12.0000 −0.544892
\(486\) 0 0
\(487\) −40.8988 −1.85330 −0.926650 0.375925i \(-0.877325\pi\)
−0.926650 + 0.375925i \(0.877325\pi\)
\(488\) 0 0
\(489\) −10.5970 −0.479214
\(490\) 0 0
\(491\) −16.6953 −0.753450 −0.376725 0.926325i \(-0.622950\pi\)
−0.376725 + 0.926325i \(0.622950\pi\)
\(492\) 0 0
\(493\) −9.17130 −0.413055
\(494\) 0 0
\(495\) −2.00000 −0.0898933
\(496\) 0 0
\(497\) 6.17929 0.277179
\(498\) 0 0
\(499\) 26.3684 1.18041 0.590206 0.807253i \(-0.299046\pi\)
0.590206 + 0.807253i \(0.299046\pi\)
\(500\) 0 0
\(501\) −20.0415 −0.895388
\(502\) 0 0
\(503\) 9.47505 0.422472 0.211236 0.977435i \(-0.432251\pi\)
0.211236 + 0.977435i \(0.432251\pi\)
\(504\) 0 0
\(505\) 8.29529 0.369135
\(506\) 0 0
\(507\) −0.985710 −0.0437769
\(508\) 0 0
\(509\) 21.9003 0.970714 0.485357 0.874316i \(-0.338690\pi\)
0.485357 + 0.874316i \(0.338690\pi\)
\(510\) 0 0
\(511\) 6.14764 0.271956
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.98418 −0.131499
\(516\) 0 0
\(517\) 9.41927 0.414259
\(518\) 0 0
\(519\) 18.8988 0.829564
\(520\) 0 0
\(521\) −36.2034 −1.58610 −0.793050 0.609156i \(-0.791508\pi\)
−0.793050 + 0.609156i \(0.791508\pi\)
\(522\) 0 0
\(523\) 14.7971 0.647030 0.323515 0.946223i \(-0.395135\pi\)
0.323515 + 0.946223i \(0.395135\pi\)
\(524\) 0 0
\(525\) 0.688892 0.0300657
\(526\) 0 0
\(527\) −2.28100 −0.0993618
\(528\) 0 0
\(529\) −14.5714 −0.633537
\(530\) 0 0
\(531\) 6.44938 0.279879
\(532\) 0 0
\(533\) 10.3051 0.446365
\(534\) 0 0
\(535\) −1.52543 −0.0659500
\(536\) 0 0
\(537\) 16.2953 0.703194
\(538\) 0 0
\(539\) 13.0509 0.562140
\(540\) 0 0
\(541\) −41.9956 −1.80553 −0.902765 0.430134i \(-0.858466\pi\)
−0.902765 + 0.430134i \(0.858466\pi\)
\(542\) 0 0
\(543\) 1.70471 0.0731563
\(544\) 0 0
\(545\) 3.76049 0.161082
\(546\) 0 0
\(547\) 34.8004 1.48796 0.743980 0.668202i \(-0.232936\pi\)
0.743980 + 0.668202i \(0.232936\pi\)
\(548\) 0 0
\(549\) −5.05086 −0.215565
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −5.77478 −0.245569
\(554\) 0 0
\(555\) 8.79060 0.373140
\(556\) 0 0
\(557\) 26.3096 1.11477 0.557386 0.830253i \(-0.311804\pi\)
0.557386 + 0.830253i \(0.311804\pi\)
\(558\) 0 0
\(559\) −3.92993 −0.166218
\(560\) 0 0
\(561\) 4.56199 0.192607
\(562\) 0 0
\(563\) 31.8306 1.34150 0.670749 0.741684i \(-0.265973\pi\)
0.670749 + 0.741684i \(0.265973\pi\)
\(564\) 0 0
\(565\) −9.18421 −0.386383
\(566\) 0 0
\(567\) −0.688892 −0.0289308
\(568\) 0 0
\(569\) 23.6622 0.991970 0.495985 0.868331i \(-0.334807\pi\)
0.495985 + 0.868331i \(0.334807\pi\)
\(570\) 0 0
\(571\) −18.8889 −0.790477 −0.395238 0.918579i \(-0.629338\pi\)
−0.395238 + 0.918579i \(0.629338\pi\)
\(572\) 0 0
\(573\) −3.53188 −0.147546
\(574\) 0 0
\(575\) −2.90321 −0.121072
\(576\) 0 0
\(577\) 1.11108 0.0462548 0.0231274 0.999733i \(-0.492638\pi\)
0.0231274 + 0.999733i \(0.492638\pi\)
\(578\) 0 0
\(579\) −17.4193 −0.723920
\(580\) 0 0
\(581\) −7.24443 −0.300550
\(582\) 0 0
\(583\) −4.29529 −0.177893
\(584\) 0 0
\(585\) 3.73975 0.154620
\(586\) 0 0
\(587\) 17.3778 0.717258 0.358629 0.933480i \(-0.383244\pi\)
0.358629 + 0.933480i \(0.383244\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −10.0143 −0.411933
\(592\) 0 0
\(593\) −37.1753 −1.52661 −0.763304 0.646040i \(-0.776424\pi\)
−0.763304 + 0.646040i \(0.776424\pi\)
\(594\) 0 0
\(595\) −1.57136 −0.0644195
\(596\) 0 0
\(597\) 27.1798 1.11239
\(598\) 0 0
\(599\) −15.6207 −0.638244 −0.319122 0.947714i \(-0.603388\pi\)
−0.319122 + 0.947714i \(0.603388\pi\)
\(600\) 0 0
\(601\) −26.7239 −1.09009 −0.545046 0.838406i \(-0.683488\pi\)
−0.545046 + 0.838406i \(0.683488\pi\)
\(602\) 0 0
\(603\) 5.44446 0.221716
\(604\) 0 0
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) −14.3936 −0.584218 −0.292109 0.956385i \(-0.594357\pi\)
−0.292109 + 0.956385i \(0.594357\pi\)
\(608\) 0 0
\(609\) −2.76986 −0.112240
\(610\) 0 0
\(611\) −17.6128 −0.712540
\(612\) 0 0
\(613\) 6.09526 0.246185 0.123093 0.992395i \(-0.460719\pi\)
0.123093 + 0.992395i \(0.460719\pi\)
\(614\) 0 0
\(615\) −2.75557 −0.111115
\(616\) 0 0
\(617\) 1.34614 0.0541936 0.0270968 0.999633i \(-0.491374\pi\)
0.0270968 + 0.999633i \(0.491374\pi\)
\(618\) 0 0
\(619\) −5.71900 −0.229866 −0.114933 0.993373i \(-0.536665\pi\)
−0.114933 + 0.993373i \(0.536665\pi\)
\(620\) 0 0
\(621\) 2.90321 0.116502
\(622\) 0 0
\(623\) −9.30021 −0.372605
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −20.0513 −0.799499
\(630\) 0 0
\(631\) −37.3274 −1.48598 −0.742990 0.669302i \(-0.766593\pi\)
−0.742990 + 0.669302i \(0.766593\pi\)
\(632\) 0 0
\(633\) −1.93978 −0.0770992
\(634\) 0 0
\(635\) 11.4795 0.455550
\(636\) 0 0
\(637\) −24.4035 −0.966900
\(638\) 0 0
\(639\) −8.96989 −0.354843
\(640\) 0 0
\(641\) 36.1323 1.42714 0.713570 0.700584i \(-0.247077\pi\)
0.713570 + 0.700584i \(0.247077\pi\)
\(642\) 0 0
\(643\) 3.10123 0.122301 0.0611504 0.998129i \(-0.480523\pi\)
0.0611504 + 0.998129i \(0.480523\pi\)
\(644\) 0 0
\(645\) 1.05086 0.0413774
\(646\) 0 0
\(647\) −29.6227 −1.16459 −0.582294 0.812978i \(-0.697845\pi\)
−0.582294 + 0.812978i \(0.697845\pi\)
\(648\) 0 0
\(649\) −12.8988 −0.506321
\(650\) 0 0
\(651\) −0.688892 −0.0269998
\(652\) 0 0
\(653\) −25.2114 −0.986599 −0.493299 0.869860i \(-0.664209\pi\)
−0.493299 + 0.869860i \(0.664209\pi\)
\(654\) 0 0
\(655\) −13.0716 −0.510750
\(656\) 0 0
\(657\) −8.92396 −0.348157
\(658\) 0 0
\(659\) −40.6242 −1.58250 −0.791248 0.611496i \(-0.790568\pi\)
−0.791248 + 0.611496i \(0.790568\pi\)
\(660\) 0 0
\(661\) −38.7783 −1.50830 −0.754151 0.656701i \(-0.771951\pi\)
−0.754151 + 0.656701i \(0.771951\pi\)
\(662\) 0 0
\(663\) −8.53035 −0.331291
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 11.6731 0.451983
\(668\) 0 0
\(669\) 23.5526 0.910597
\(670\) 0 0
\(671\) 10.1017 0.389972
\(672\) 0 0
\(673\) 40.1813 1.54888 0.774438 0.632650i \(-0.218033\pi\)
0.774438 + 0.632650i \(0.218033\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 10.6811 0.410506 0.205253 0.978709i \(-0.434198\pi\)
0.205253 + 0.978709i \(0.434198\pi\)
\(678\) 0 0
\(679\) 8.26671 0.317247
\(680\) 0 0
\(681\) 19.8020 0.758813
\(682\) 0 0
\(683\) −31.6271 −1.21018 −0.605089 0.796158i \(-0.706863\pi\)
−0.605089 + 0.796158i \(0.706863\pi\)
\(684\) 0 0
\(685\) −11.1383 −0.425572
\(686\) 0 0
\(687\) 24.4099 0.931296
\(688\) 0 0
\(689\) 8.03164 0.305981
\(690\) 0 0
\(691\) −10.6508 −0.405175 −0.202588 0.979264i \(-0.564935\pi\)
−0.202588 + 0.979264i \(0.564935\pi\)
\(692\) 0 0
\(693\) 1.37778 0.0523377
\(694\) 0 0
\(695\) −10.2351 −0.388238
\(696\) 0 0
\(697\) 6.28544 0.238078
\(698\) 0 0
\(699\) −3.13828 −0.118700
\(700\) 0 0
\(701\) 39.3590 1.48657 0.743285 0.668974i \(-0.233266\pi\)
0.743285 + 0.668974i \(0.233266\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 4.70964 0.177375
\(706\) 0 0
\(707\) −5.71456 −0.214918
\(708\) 0 0
\(709\) 20.1847 0.758052 0.379026 0.925386i \(-0.376259\pi\)
0.379026 + 0.925386i \(0.376259\pi\)
\(710\) 0 0
\(711\) 8.38271 0.314376
\(712\) 0 0
\(713\) 2.90321 0.108726
\(714\) 0 0
\(715\) −7.47949 −0.279717
\(716\) 0 0
\(717\) 6.81579 0.254540
\(718\) 0 0
\(719\) −24.1017 −0.898842 −0.449421 0.893320i \(-0.648370\pi\)
−0.449421 + 0.893320i \(0.648370\pi\)
\(720\) 0 0
\(721\) 2.05578 0.0765611
\(722\) 0 0
\(723\) −17.0923 −0.635671
\(724\) 0 0
\(725\) −4.02074 −0.149327
\(726\) 0 0
\(727\) 14.1180 0.523608 0.261804 0.965121i \(-0.415683\pi\)
0.261804 + 0.965121i \(0.415683\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −2.39700 −0.0886561
\(732\) 0 0
\(733\) −23.1240 −0.854104 −0.427052 0.904227i \(-0.640448\pi\)
−0.427052 + 0.904227i \(0.640448\pi\)
\(734\) 0 0
\(735\) 6.52543 0.240694
\(736\) 0 0
\(737\) −10.8889 −0.401099
\(738\) 0 0
\(739\) 0.128907 0.00474193 0.00237097 0.999997i \(-0.499245\pi\)
0.00237097 + 0.999997i \(0.499245\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −28.0129 −1.02769 −0.513847 0.857882i \(-0.671780\pi\)
−0.513847 + 0.857882i \(0.671780\pi\)
\(744\) 0 0
\(745\) −1.51114 −0.0553638
\(746\) 0 0
\(747\) 10.5161 0.384763
\(748\) 0 0
\(749\) 1.05086 0.0383974
\(750\) 0 0
\(751\) 3.56247 0.129996 0.0649982 0.997885i \(-0.479296\pi\)
0.0649982 + 0.997885i \(0.479296\pi\)
\(752\) 0 0
\(753\) 16.1748 0.589444
\(754\) 0 0
\(755\) 1.76049 0.0640708
\(756\) 0 0
\(757\) 27.0573 0.983415 0.491707 0.870761i \(-0.336373\pi\)
0.491707 + 0.870761i \(0.336373\pi\)
\(758\) 0 0
\(759\) −5.80642 −0.210760
\(760\) 0 0
\(761\) 13.1447 0.476496 0.238248 0.971204i \(-0.423427\pi\)
0.238248 + 0.971204i \(0.423427\pi\)
\(762\) 0 0
\(763\) −2.59057 −0.0937850
\(764\) 0 0
\(765\) 2.28100 0.0824696
\(766\) 0 0
\(767\) 24.1191 0.870889
\(768\) 0 0
\(769\) 25.7418 0.928271 0.464136 0.885764i \(-0.346365\pi\)
0.464136 + 0.885764i \(0.346365\pi\)
\(770\) 0 0
\(771\) −19.0464 −0.685940
\(772\) 0 0
\(773\) 50.3007 1.80919 0.904595 0.426272i \(-0.140173\pi\)
0.904595 + 0.426272i \(0.140173\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) 0 0
\(777\) −6.05578 −0.217250
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 17.9398 0.641936
\(782\) 0 0
\(783\) 4.02074 0.143690
\(784\) 0 0
\(785\) −3.43801 −0.122708
\(786\) 0 0
\(787\) −8.70519 −0.310307 −0.155153 0.987890i \(-0.549587\pi\)
−0.155153 + 0.987890i \(0.549587\pi\)
\(788\) 0 0
\(789\) −6.23506 −0.221974
\(790\) 0 0
\(791\) 6.32693 0.224960
\(792\) 0 0
\(793\) −18.8889 −0.670765
\(794\) 0 0
\(795\) −2.14764 −0.0761691
\(796\) 0 0
\(797\) 39.0464 1.38309 0.691547 0.722331i \(-0.256929\pi\)
0.691547 + 0.722331i \(0.256929\pi\)
\(798\) 0 0
\(799\) −10.7427 −0.380048
\(800\) 0 0
\(801\) 13.5002 0.477007
\(802\) 0 0
\(803\) 17.8479 0.629839
\(804\) 0 0
\(805\) 2.00000 0.0704907
\(806\) 0 0
\(807\) −20.8365 −0.733481
\(808\) 0 0
\(809\) −24.0435 −0.845324 −0.422662 0.906287i \(-0.638904\pi\)
−0.422662 + 0.906287i \(0.638904\pi\)
\(810\) 0 0
\(811\) −45.5812 −1.60057 −0.800286 0.599618i \(-0.795319\pi\)
−0.800286 + 0.599618i \(0.795319\pi\)
\(812\) 0 0
\(813\) 6.62222 0.232251
\(814\) 0 0
\(815\) 10.5970 0.371198
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −2.57628 −0.0900226
\(820\) 0 0
\(821\) −32.2054 −1.12398 −0.561989 0.827145i \(-0.689964\pi\)
−0.561989 + 0.827145i \(0.689964\pi\)
\(822\) 0 0
\(823\) −33.1052 −1.15398 −0.576988 0.816752i \(-0.695772\pi\)
−0.576988 + 0.816752i \(0.695772\pi\)
\(824\) 0 0
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) 0.147643 0.00513406 0.00256703 0.999997i \(-0.499183\pi\)
0.00256703 + 0.999997i \(0.499183\pi\)
\(828\) 0 0
\(829\) −20.9175 −0.726495 −0.363247 0.931693i \(-0.618332\pi\)
−0.363247 + 0.931693i \(0.618332\pi\)
\(830\) 0 0
\(831\) −4.79060 −0.166184
\(832\) 0 0
\(833\) −14.8845 −0.515717
\(834\) 0 0
\(835\) 20.0415 0.693564
\(836\) 0 0
\(837\) 1.00000 0.0345651
\(838\) 0 0
\(839\) 42.9066 1.48130 0.740650 0.671891i \(-0.234518\pi\)
0.740650 + 0.671891i \(0.234518\pi\)
\(840\) 0 0
\(841\) −12.8336 −0.442539
\(842\) 0 0
\(843\) 20.8256 0.717273
\(844\) 0 0
\(845\) 0.985710 0.0339095
\(846\) 0 0
\(847\) 4.82225 0.165694
\(848\) 0 0
\(849\) −11.8731 −0.407484
\(850\) 0 0
\(851\) 25.5210 0.874848
\(852\) 0 0
\(853\) −49.2226 −1.68535 −0.842675 0.538422i \(-0.819021\pi\)
−0.842675 + 0.538422i \(0.819021\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 34.2449 1.16978 0.584892 0.811111i \(-0.301137\pi\)
0.584892 + 0.811111i \(0.301137\pi\)
\(858\) 0 0
\(859\) −14.5718 −0.497185 −0.248592 0.968608i \(-0.579968\pi\)
−0.248592 + 0.968608i \(0.579968\pi\)
\(860\) 0 0
\(861\) 1.89829 0.0646935
\(862\) 0 0
\(863\) −44.6164 −1.51876 −0.759380 0.650648i \(-0.774497\pi\)
−0.759380 + 0.650648i \(0.774497\pi\)
\(864\) 0 0
\(865\) −18.8988 −0.642577
\(866\) 0 0
\(867\) 11.7971 0.400649
\(868\) 0 0
\(869\) −16.7654 −0.568728
\(870\) 0 0
\(871\) 20.3609 0.689903
\(872\) 0 0
\(873\) −12.0000 −0.406138
\(874\) 0 0
\(875\) −0.688892 −0.0232888
\(876\) 0 0
\(877\) 8.05038 0.271842 0.135921 0.990720i \(-0.456601\pi\)
0.135921 + 0.990720i \(0.456601\pi\)
\(878\) 0 0
\(879\) 18.0143 0.607607
\(880\) 0 0
\(881\) −28.7130 −0.967366 −0.483683 0.875243i \(-0.660701\pi\)
−0.483683 + 0.875243i \(0.660701\pi\)
\(882\) 0 0
\(883\) 26.4197 0.889095 0.444548 0.895755i \(-0.353364\pi\)
0.444548 + 0.895755i \(0.353364\pi\)
\(884\) 0 0
\(885\) −6.44938 −0.216794
\(886\) 0 0
\(887\) 44.7926 1.50399 0.751994 0.659170i \(-0.229092\pi\)
0.751994 + 0.659170i \(0.229092\pi\)
\(888\) 0 0
\(889\) −7.90813 −0.265230
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −16.2953 −0.544691
\(896\) 0 0
\(897\) 10.8573 0.362514
\(898\) 0 0
\(899\) 4.02074 0.134099
\(900\) 0 0
\(901\) 4.89877 0.163202
\(902\) 0 0
\(903\) −0.723926 −0.0240907
\(904\) 0 0
\(905\) −1.70471 −0.0566666
\(906\) 0 0
\(907\) −7.21924 −0.239711 −0.119855 0.992791i \(-0.538243\pi\)
−0.119855 + 0.992791i \(0.538243\pi\)
\(908\) 0 0
\(909\) 8.29529 0.275137
\(910\) 0 0
\(911\) −21.1052 −0.699248 −0.349624 0.936890i \(-0.613691\pi\)
−0.349624 + 0.936890i \(0.613691\pi\)
\(912\) 0 0
\(913\) −21.0321 −0.696062
\(914\) 0 0
\(915\) 5.05086 0.166976
\(916\) 0 0
\(917\) 9.00492 0.297369
\(918\) 0 0
\(919\) −16.4415 −0.542357 −0.271178 0.962529i \(-0.587413\pi\)
−0.271178 + 0.962529i \(0.587413\pi\)
\(920\) 0 0
\(921\) 10.9240 0.359957
\(922\) 0 0
\(923\) −33.5451 −1.10415
\(924\) 0 0
\(925\) −8.79060 −0.289033
\(926\) 0 0
\(927\) −2.98418 −0.0980133
\(928\) 0 0
\(929\) 6.26470 0.205538 0.102769 0.994705i \(-0.467230\pi\)
0.102769 + 0.994705i \(0.467230\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 5.10324 0.167073
\(934\) 0 0
\(935\) −4.56199 −0.149193
\(936\) 0 0
\(937\) 37.1437 1.21343 0.606715 0.794919i \(-0.292487\pi\)
0.606715 + 0.794919i \(0.292487\pi\)
\(938\) 0 0
\(939\) 22.1082 0.721473
\(940\) 0 0
\(941\) 39.1318 1.27566 0.637830 0.770177i \(-0.279832\pi\)
0.637830 + 0.770177i \(0.279832\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) 0.688892 0.0224097
\(946\) 0 0
\(947\) 53.0968 1.72541 0.862707 0.505704i \(-0.168767\pi\)
0.862707 + 0.505704i \(0.168767\pi\)
\(948\) 0 0
\(949\) −33.3733 −1.08334
\(950\) 0 0
\(951\) 1.06515 0.0345397
\(952\) 0 0
\(953\) −6.57628 −0.213027 −0.106513 0.994311i \(-0.533969\pi\)
−0.106513 + 0.994311i \(0.533969\pi\)
\(954\) 0 0
\(955\) 3.53188 0.114289
\(956\) 0 0
\(957\) −8.04149 −0.259944
\(958\) 0 0
\(959\) 7.67307 0.247776
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) −1.52543 −0.0491562
\(964\) 0 0
\(965\) 17.4193 0.560746
\(966\) 0 0
\(967\) −54.2449 −1.74440 −0.872199 0.489151i \(-0.837307\pi\)
−0.872199 + 0.489151i \(0.837307\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 42.8879 1.37634 0.688169 0.725551i \(-0.258415\pi\)
0.688169 + 0.725551i \(0.258415\pi\)
\(972\) 0 0
\(973\) 7.05086 0.226040
\(974\) 0 0
\(975\) −3.73975 −0.119768
\(976\) 0 0
\(977\) 11.6316 0.372127 0.186064 0.982538i \(-0.440427\pi\)
0.186064 + 0.982538i \(0.440427\pi\)
\(978\) 0 0
\(979\) −27.0005 −0.862939
\(980\) 0 0
\(981\) 3.76049 0.120063
\(982\) 0 0
\(983\) 30.3595 0.968318 0.484159 0.874980i \(-0.339126\pi\)
0.484159 + 0.874980i \(0.339126\pi\)
\(984\) 0 0
\(985\) 10.0143 0.319082
\(986\) 0 0
\(987\) −3.24443 −0.103271
\(988\) 0 0
\(989\) 3.05086 0.0970115
\(990\) 0 0
\(991\) −42.5589 −1.35193 −0.675964 0.736934i \(-0.736273\pi\)
−0.675964 + 0.736934i \(0.736273\pi\)
\(992\) 0 0
\(993\) −24.9447 −0.791596
\(994\) 0 0
\(995\) −27.1798 −0.861656
\(996\) 0 0
\(997\) −33.5339 −1.06203 −0.531014 0.847363i \(-0.678189\pi\)
−0.531014 + 0.847363i \(0.678189\pi\)
\(998\) 0 0
\(999\) 8.79060 0.278122
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 7440.2.a.bp.1.2 3
4.3 odd 2 465.2.a.f.1.2 3
12.11 even 2 1395.2.a.i.1.2 3
20.3 even 4 2325.2.c.o.1024.3 6
20.7 even 4 2325.2.c.o.1024.4 6
20.19 odd 2 2325.2.a.q.1.2 3
60.59 even 2 6975.2.a.be.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.f.1.2 3 4.3 odd 2
1395.2.a.i.1.2 3 12.11 even 2
2325.2.a.q.1.2 3 20.19 odd 2
2325.2.c.o.1024.3 6 20.3 even 4
2325.2.c.o.1024.4 6 20.7 even 4
6975.2.a.be.1.2 3 60.59 even 2
7440.2.a.bp.1.2 3 1.1 even 1 trivial