Properties

Label 7728.2.a.bs.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
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] = [7728,2,Mod(1,7728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7728, 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("7728.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.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: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.11491\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.47283 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.47283 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.64207 q^{11} -0.169240 q^{13} +1.47283 q^{15} +2.58774 q^{17} -3.87189 q^{19} -1.00000 q^{21} +1.00000 q^{23} -2.83076 q^{25} -1.00000 q^{27} -0.357926 q^{29} +3.87189 q^{31} -1.64207 q^{33} -1.47283 q^{35} -4.35793 q^{37} +0.169240 q^{39} -9.87189 q^{41} +5.11491 q^{43} -1.47283 q^{45} -8.79811 q^{47} +1.00000 q^{49} -2.58774 q^{51} +13.5940 q^{53} -2.41850 q^{55} +3.87189 q^{57} +7.47283 q^{59} -3.11491 q^{61} +1.00000 q^{63} +0.249262 q^{65} +10.1623 q^{67} -1.00000 q^{69} -13.2361 q^{71} -12.3579 q^{73} +2.83076 q^{75} +1.64207 q^{77} -0.587741 q^{79} +1.00000 q^{81} +6.10170 q^{83} -3.81131 q^{85} +0.357926 q^{87} -14.5202 q^{89} -0.169240 q^{91} -3.87189 q^{93} +5.70265 q^{95} +4.81756 q^{97} +1.64207 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9} + 4 q^{11} - 5 q^{13} - q^{15} - 4 q^{17} + 2 q^{19} - 3 q^{21} + 3 q^{23} - 4 q^{25} - 3 q^{27} - 2 q^{29} - 2 q^{31} - 4 q^{33} + q^{35} - 14 q^{37} + 5 q^{39} - 16 q^{41} + 9 q^{43} + q^{45} - 10 q^{47} + 3 q^{49} + 4 q^{51} + q^{53} + 9 q^{55} - 2 q^{57} + 17 q^{59} - 3 q^{61} + 3 q^{63} - 20 q^{65} - 13 q^{67} - 3 q^{69} + q^{71} - 38 q^{73} + 4 q^{75} + 4 q^{77} + 10 q^{79} + 3 q^{81} - 8 q^{83} - 15 q^{85} + 2 q^{87} - q^{89} - 5 q^{91} + 2 q^{93} - q^{95} - 10 q^{97} + 4 q^{99}+O(q^{100}) \) 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
\(5\) −1.47283 −0.658671 −0.329336 0.944213i \(-0.606825\pi\)
−0.329336 + 0.944213i \(0.606825\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.64207 0.495104 0.247552 0.968875i \(-0.420374\pi\)
0.247552 + 0.968875i \(0.420374\pi\)
\(12\) 0 0
\(13\) −0.169240 −0.0469387 −0.0234693 0.999725i \(-0.507471\pi\)
−0.0234693 + 0.999725i \(0.507471\pi\)
\(14\) 0 0
\(15\) 1.47283 0.380284
\(16\) 0 0
\(17\) 2.58774 0.627619 0.313810 0.949486i \(-0.398395\pi\)
0.313810 + 0.949486i \(0.398395\pi\)
\(18\) 0 0
\(19\) −3.87189 −0.888272 −0.444136 0.895959i \(-0.646489\pi\)
−0.444136 + 0.895959i \(0.646489\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −2.83076 −0.566152
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −0.357926 −0.0664653 −0.0332326 0.999448i \(-0.510580\pi\)
−0.0332326 + 0.999448i \(0.510580\pi\)
\(30\) 0 0
\(31\) 3.87189 0.695412 0.347706 0.937604i \(-0.386961\pi\)
0.347706 + 0.937604i \(0.386961\pi\)
\(32\) 0 0
\(33\) −1.64207 −0.285848
\(34\) 0 0
\(35\) −1.47283 −0.248954
\(36\) 0 0
\(37\) −4.35793 −0.716439 −0.358219 0.933637i \(-0.616616\pi\)
−0.358219 + 0.933637i \(0.616616\pi\)
\(38\) 0 0
\(39\) 0.169240 0.0271000
\(40\) 0 0
\(41\) −9.87189 −1.54173 −0.770865 0.636999i \(-0.780176\pi\)
−0.770865 + 0.636999i \(0.780176\pi\)
\(42\) 0 0
\(43\) 5.11491 0.780016 0.390008 0.920811i \(-0.372472\pi\)
0.390008 + 0.920811i \(0.372472\pi\)
\(44\) 0 0
\(45\) −1.47283 −0.219557
\(46\) 0 0
\(47\) −8.79811 −1.28334 −0.641668 0.766982i \(-0.721757\pi\)
−0.641668 + 0.766982i \(0.721757\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.58774 −0.362356
\(52\) 0 0
\(53\) 13.5940 1.86728 0.933639 0.358216i \(-0.116615\pi\)
0.933639 + 0.358216i \(0.116615\pi\)
\(54\) 0 0
\(55\) −2.41850 −0.326111
\(56\) 0 0
\(57\) 3.87189 0.512844
\(58\) 0 0
\(59\) 7.47283 0.972880 0.486440 0.873714i \(-0.338295\pi\)
0.486440 + 0.873714i \(0.338295\pi\)
\(60\) 0 0
\(61\) −3.11491 −0.398823 −0.199412 0.979916i \(-0.563903\pi\)
−0.199412 + 0.979916i \(0.563903\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 0.249262 0.0309171
\(66\) 0 0
\(67\) 10.1623 1.24152 0.620760 0.784001i \(-0.286824\pi\)
0.620760 + 0.784001i \(0.286824\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −13.2361 −1.57083 −0.785416 0.618969i \(-0.787551\pi\)
−0.785416 + 0.618969i \(0.787551\pi\)
\(72\) 0 0
\(73\) −12.3579 −1.44639 −0.723193 0.690646i \(-0.757326\pi\)
−0.723193 + 0.690646i \(0.757326\pi\)
\(74\) 0 0
\(75\) 2.83076 0.326868
\(76\) 0 0
\(77\) 1.64207 0.187132
\(78\) 0 0
\(79\) −0.587741 −0.0661261 −0.0330630 0.999453i \(-0.510526\pi\)
−0.0330630 + 0.999453i \(0.510526\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.10170 0.669749 0.334875 0.942263i \(-0.391306\pi\)
0.334875 + 0.942263i \(0.391306\pi\)
\(84\) 0 0
\(85\) −3.81131 −0.413395
\(86\) 0 0
\(87\) 0.357926 0.0383737
\(88\) 0 0
\(89\) −14.5202 −1.53914 −0.769569 0.638563i \(-0.779529\pi\)
−0.769569 + 0.638563i \(0.779529\pi\)
\(90\) 0 0
\(91\) −0.169240 −0.0177411
\(92\) 0 0
\(93\) −3.87189 −0.401496
\(94\) 0 0
\(95\) 5.70265 0.585079
\(96\) 0 0
\(97\) 4.81756 0.489149 0.244574 0.969631i \(-0.421352\pi\)
0.244574 + 0.969631i \(0.421352\pi\)
\(98\) 0 0
\(99\) 1.64207 0.165035
\(100\) 0 0
\(101\) −5.09546 −0.507017 −0.253509 0.967333i \(-0.581585\pi\)
−0.253509 + 0.967333i \(0.581585\pi\)
\(102\) 0 0
\(103\) 4.45963 0.439420 0.219710 0.975565i \(-0.429489\pi\)
0.219710 + 0.975565i \(0.429489\pi\)
\(104\) 0 0
\(105\) 1.47283 0.143734
\(106\) 0 0
\(107\) 8.77643 0.848449 0.424225 0.905557i \(-0.360547\pi\)
0.424225 + 0.905557i \(0.360547\pi\)
\(108\) 0 0
\(109\) 7.32528 0.701634 0.350817 0.936444i \(-0.385904\pi\)
0.350817 + 0.936444i \(0.385904\pi\)
\(110\) 0 0
\(111\) 4.35793 0.413636
\(112\) 0 0
\(113\) −19.4464 −1.82937 −0.914683 0.404172i \(-0.867560\pi\)
−0.914683 + 0.404172i \(0.867560\pi\)
\(114\) 0 0
\(115\) −1.47283 −0.137342
\(116\) 0 0
\(117\) −0.169240 −0.0156462
\(118\) 0 0
\(119\) 2.58774 0.237218
\(120\) 0 0
\(121\) −8.30359 −0.754872
\(122\) 0 0
\(123\) 9.87189 0.890118
\(124\) 0 0
\(125\) 11.5334 1.03158
\(126\) 0 0
\(127\) 16.7764 1.48867 0.744334 0.667808i \(-0.232767\pi\)
0.744334 + 0.667808i \(0.232767\pi\)
\(128\) 0 0
\(129\) −5.11491 −0.450342
\(130\) 0 0
\(131\) 15.1949 1.32759 0.663794 0.747916i \(-0.268945\pi\)
0.663794 + 0.747916i \(0.268945\pi\)
\(132\) 0 0
\(133\) −3.87189 −0.335735
\(134\) 0 0
\(135\) 1.47283 0.126761
\(136\) 0 0
\(137\) 15.4247 1.31782 0.658912 0.752220i \(-0.271017\pi\)
0.658912 + 0.752220i \(0.271017\pi\)
\(138\) 0 0
\(139\) 4.18869 0.355280 0.177640 0.984096i \(-0.443154\pi\)
0.177640 + 0.984096i \(0.443154\pi\)
\(140\) 0 0
\(141\) 8.79811 0.740934
\(142\) 0 0
\(143\) −0.277904 −0.0232395
\(144\) 0 0
\(145\) 0.527166 0.0437788
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 7.51396 0.615568 0.307784 0.951456i \(-0.400413\pi\)
0.307784 + 0.951456i \(0.400413\pi\)
\(150\) 0 0
\(151\) −20.5808 −1.67484 −0.837420 0.546560i \(-0.815937\pi\)
−0.837420 + 0.546560i \(0.815937\pi\)
\(152\) 0 0
\(153\) 2.58774 0.209206
\(154\) 0 0
\(155\) −5.70265 −0.458048
\(156\) 0 0
\(157\) −18.5808 −1.48291 −0.741454 0.671004i \(-0.765863\pi\)
−0.741454 + 0.671004i \(0.765863\pi\)
\(158\) 0 0
\(159\) −13.5940 −1.07807
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 15.5940 1.22142 0.610708 0.791856i \(-0.290885\pi\)
0.610708 + 0.791856i \(0.290885\pi\)
\(164\) 0 0
\(165\) 2.41850 0.188280
\(166\) 0 0
\(167\) −5.98055 −0.462789 −0.231395 0.972860i \(-0.574329\pi\)
−0.231395 + 0.972860i \(0.574329\pi\)
\(168\) 0 0
\(169\) −12.9714 −0.997797
\(170\) 0 0
\(171\) −3.87189 −0.296091
\(172\) 0 0
\(173\) 14.3315 1.08961 0.544803 0.838564i \(-0.316605\pi\)
0.544803 + 0.838564i \(0.316605\pi\)
\(174\) 0 0
\(175\) −2.83076 −0.213985
\(176\) 0 0
\(177\) −7.47283 −0.561693
\(178\) 0 0
\(179\) −18.6219 −1.39187 −0.695933 0.718106i \(-0.745009\pi\)
−0.695933 + 0.718106i \(0.745009\pi\)
\(180\) 0 0
\(181\) −9.07378 −0.674449 −0.337224 0.941424i \(-0.609488\pi\)
−0.337224 + 0.941424i \(0.609488\pi\)
\(182\) 0 0
\(183\) 3.11491 0.230261
\(184\) 0 0
\(185\) 6.41850 0.471898
\(186\) 0 0
\(187\) 4.24926 0.310737
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 20.5808 1.48917 0.744587 0.667526i \(-0.232647\pi\)
0.744587 + 0.667526i \(0.232647\pi\)
\(192\) 0 0
\(193\) −17.6615 −1.27130 −0.635652 0.771976i \(-0.719269\pi\)
−0.635652 + 0.771976i \(0.719269\pi\)
\(194\) 0 0
\(195\) −0.249262 −0.0178500
\(196\) 0 0
\(197\) −22.5591 −1.60727 −0.803635 0.595123i \(-0.797103\pi\)
−0.803635 + 0.595123i \(0.797103\pi\)
\(198\) 0 0
\(199\) 4.64832 0.329510 0.164755 0.986334i \(-0.447317\pi\)
0.164755 + 0.986334i \(0.447317\pi\)
\(200\) 0 0
\(201\) −10.1623 −0.716792
\(202\) 0 0
\(203\) −0.357926 −0.0251215
\(204\) 0 0
\(205\) 14.5397 1.01549
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −6.35793 −0.439787
\(210\) 0 0
\(211\) −21.8844 −1.50658 −0.753291 0.657687i \(-0.771535\pi\)
−0.753291 + 0.657687i \(0.771535\pi\)
\(212\) 0 0
\(213\) 13.2361 0.906920
\(214\) 0 0
\(215\) −7.53341 −0.513774
\(216\) 0 0
\(217\) 3.87189 0.262841
\(218\) 0 0
\(219\) 12.3579 0.835071
\(220\) 0 0
\(221\) −0.437949 −0.0294596
\(222\) 0 0
\(223\) −8.56606 −0.573626 −0.286813 0.957987i \(-0.592596\pi\)
−0.286813 + 0.957987i \(0.592596\pi\)
\(224\) 0 0
\(225\) −2.83076 −0.188717
\(226\) 0 0
\(227\) −9.74154 −0.646569 −0.323284 0.946302i \(-0.604787\pi\)
−0.323284 + 0.946302i \(0.604787\pi\)
\(228\) 0 0
\(229\) −8.04113 −0.531373 −0.265686 0.964060i \(-0.585599\pi\)
−0.265686 + 0.964060i \(0.585599\pi\)
\(230\) 0 0
\(231\) −1.64207 −0.108041
\(232\) 0 0
\(233\) −27.5676 −1.80601 −0.903006 0.429628i \(-0.858645\pi\)
−0.903006 + 0.429628i \(0.858645\pi\)
\(234\) 0 0
\(235\) 12.9582 0.845297
\(236\) 0 0
\(237\) 0.587741 0.0381779
\(238\) 0 0
\(239\) 20.0147 1.29464 0.647322 0.762216i \(-0.275889\pi\)
0.647322 + 0.762216i \(0.275889\pi\)
\(240\) 0 0
\(241\) 5.99304 0.386046 0.193023 0.981194i \(-0.438171\pi\)
0.193023 + 0.981194i \(0.438171\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −1.47283 −0.0940959
\(246\) 0 0
\(247\) 0.655277 0.0416943
\(248\) 0 0
\(249\) −6.10170 −0.386680
\(250\) 0 0
\(251\) −14.2687 −0.900633 −0.450316 0.892869i \(-0.648689\pi\)
−0.450316 + 0.892869i \(0.648689\pi\)
\(252\) 0 0
\(253\) 1.64207 0.103236
\(254\) 0 0
\(255\) 3.81131 0.238674
\(256\) 0 0
\(257\) 31.0015 1.93382 0.966911 0.255115i \(-0.0821133\pi\)
0.966911 + 0.255115i \(0.0821133\pi\)
\(258\) 0 0
\(259\) −4.35793 −0.270788
\(260\) 0 0
\(261\) −0.357926 −0.0221551
\(262\) 0 0
\(263\) 7.65456 0.472000 0.236000 0.971753i \(-0.424163\pi\)
0.236000 + 0.971753i \(0.424163\pi\)
\(264\) 0 0
\(265\) −20.0217 −1.22992
\(266\) 0 0
\(267\) 14.5202 0.888622
\(268\) 0 0
\(269\) 5.72210 0.348882 0.174441 0.984668i \(-0.444188\pi\)
0.174441 + 0.984668i \(0.444188\pi\)
\(270\) 0 0
\(271\) −1.22134 −0.0741910 −0.0370955 0.999312i \(-0.511811\pi\)
−0.0370955 + 0.999312i \(0.511811\pi\)
\(272\) 0 0
\(273\) 0.169240 0.0102429
\(274\) 0 0
\(275\) −4.64832 −0.280304
\(276\) 0 0
\(277\) −14.8587 −0.892772 −0.446386 0.894841i \(-0.647289\pi\)
−0.446386 + 0.894841i \(0.647289\pi\)
\(278\) 0 0
\(279\) 3.87189 0.231804
\(280\) 0 0
\(281\) 18.7981 1.12140 0.560701 0.828019i \(-0.310532\pi\)
0.560701 + 0.828019i \(0.310532\pi\)
\(282\) 0 0
\(283\) −27.3378 −1.62506 −0.812531 0.582918i \(-0.801911\pi\)
−0.812531 + 0.582918i \(0.801911\pi\)
\(284\) 0 0
\(285\) −5.70265 −0.337796
\(286\) 0 0
\(287\) −9.87189 −0.582719
\(288\) 0 0
\(289\) −10.3036 −0.606094
\(290\) 0 0
\(291\) −4.81756 −0.282410
\(292\) 0 0
\(293\) 8.82452 0.515534 0.257767 0.966207i \(-0.417013\pi\)
0.257767 + 0.966207i \(0.417013\pi\)
\(294\) 0 0
\(295\) −11.0062 −0.640808
\(296\) 0 0
\(297\) −1.64207 −0.0952828
\(298\) 0 0
\(299\) −0.169240 −0.00978739
\(300\) 0 0
\(301\) 5.11491 0.294818
\(302\) 0 0
\(303\) 5.09546 0.292727
\(304\) 0 0
\(305\) 4.58774 0.262693
\(306\) 0 0
\(307\) −30.6825 −1.75114 −0.875571 0.483090i \(-0.839515\pi\)
−0.875571 + 0.483090i \(0.839515\pi\)
\(308\) 0 0
\(309\) −4.45963 −0.253699
\(310\) 0 0
\(311\) 23.2097 1.31610 0.658049 0.752975i \(-0.271382\pi\)
0.658049 + 0.752975i \(0.271382\pi\)
\(312\) 0 0
\(313\) −26.9582 −1.52376 −0.761882 0.647715i \(-0.775725\pi\)
−0.761882 + 0.647715i \(0.775725\pi\)
\(314\) 0 0
\(315\) −1.47283 −0.0829848
\(316\) 0 0
\(317\) 18.6483 1.04739 0.523697 0.851905i \(-0.324552\pi\)
0.523697 + 0.851905i \(0.324552\pi\)
\(318\) 0 0
\(319\) −0.587741 −0.0329072
\(320\) 0 0
\(321\) −8.77643 −0.489852
\(322\) 0 0
\(323\) −10.0194 −0.557497
\(324\) 0 0
\(325\) 0.479077 0.0265744
\(326\) 0 0
\(327\) −7.32528 −0.405089
\(328\) 0 0
\(329\) −8.79811 −0.485055
\(330\) 0 0
\(331\) −4.42074 −0.242986 −0.121493 0.992592i \(-0.538768\pi\)
−0.121493 + 0.992592i \(0.538768\pi\)
\(332\) 0 0
\(333\) −4.35793 −0.238813
\(334\) 0 0
\(335\) −14.9673 −0.817754
\(336\) 0 0
\(337\) −3.91302 −0.213156 −0.106578 0.994304i \(-0.533989\pi\)
−0.106578 + 0.994304i \(0.533989\pi\)
\(338\) 0 0
\(339\) 19.4464 1.05618
\(340\) 0 0
\(341\) 6.35793 0.344301
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 1.47283 0.0792947
\(346\) 0 0
\(347\) −15.8719 −0.852048 −0.426024 0.904712i \(-0.640086\pi\)
−0.426024 + 0.904712i \(0.640086\pi\)
\(348\) 0 0
\(349\) −12.0411 −0.644547 −0.322273 0.946647i \(-0.604447\pi\)
−0.322273 + 0.946647i \(0.604447\pi\)
\(350\) 0 0
\(351\) 0.169240 0.00903335
\(352\) 0 0
\(353\) 1.83299 0.0975605 0.0487802 0.998810i \(-0.484467\pi\)
0.0487802 + 0.998810i \(0.484467\pi\)
\(354\) 0 0
\(355\) 19.4945 1.03466
\(356\) 0 0
\(357\) −2.58774 −0.136958
\(358\) 0 0
\(359\) 18.4185 0.972091 0.486046 0.873933i \(-0.338439\pi\)
0.486046 + 0.873933i \(0.338439\pi\)
\(360\) 0 0
\(361\) −4.00848 −0.210973
\(362\) 0 0
\(363\) 8.30359 0.435826
\(364\) 0 0
\(365\) 18.2012 0.952693
\(366\) 0 0
\(367\) −23.0885 −1.20521 −0.602605 0.798040i \(-0.705871\pi\)
−0.602605 + 0.798040i \(0.705871\pi\)
\(368\) 0 0
\(369\) −9.87189 −0.513910
\(370\) 0 0
\(371\) 13.5940 0.705765
\(372\) 0 0
\(373\) −35.8455 −1.85601 −0.928004 0.372569i \(-0.878477\pi\)
−0.928004 + 0.372569i \(0.878477\pi\)
\(374\) 0 0
\(375\) −11.5334 −0.595583
\(376\) 0 0
\(377\) 0.0605754 0.00311979
\(378\) 0 0
\(379\) −10.1087 −0.519247 −0.259624 0.965710i \(-0.583598\pi\)
−0.259624 + 0.965710i \(0.583598\pi\)
\(380\) 0 0
\(381\) −16.7764 −0.859482
\(382\) 0 0
\(383\) −30.8565 −1.57669 −0.788345 0.615233i \(-0.789062\pi\)
−0.788345 + 0.615233i \(0.789062\pi\)
\(384\) 0 0
\(385\) −2.41850 −0.123258
\(386\) 0 0
\(387\) 5.11491 0.260005
\(388\) 0 0
\(389\) 2.16701 0.109872 0.0549358 0.998490i \(-0.482505\pi\)
0.0549358 + 0.998490i \(0.482505\pi\)
\(390\) 0 0
\(391\) 2.58774 0.130868
\(392\) 0 0
\(393\) −15.1949 −0.766483
\(394\) 0 0
\(395\) 0.865646 0.0435554
\(396\) 0 0
\(397\) −31.7563 −1.59380 −0.796901 0.604110i \(-0.793529\pi\)
−0.796901 + 0.604110i \(0.793529\pi\)
\(398\) 0 0
\(399\) 3.87189 0.193837
\(400\) 0 0
\(401\) −6.84396 −0.341771 −0.170886 0.985291i \(-0.554663\pi\)
−0.170886 + 0.985291i \(0.554663\pi\)
\(402\) 0 0
\(403\) −0.655277 −0.0326417
\(404\) 0 0
\(405\) −1.47283 −0.0731857
\(406\) 0 0
\(407\) −7.15604 −0.354712
\(408\) 0 0
\(409\) 3.50700 0.173410 0.0867050 0.996234i \(-0.472366\pi\)
0.0867050 + 0.996234i \(0.472366\pi\)
\(410\) 0 0
\(411\) −15.4247 −0.760847
\(412\) 0 0
\(413\) 7.47283 0.367714
\(414\) 0 0
\(415\) −8.98680 −0.441144
\(416\) 0 0
\(417\) −4.18869 −0.205121
\(418\) 0 0
\(419\) −37.3114 −1.82278 −0.911389 0.411545i \(-0.864989\pi\)
−0.911389 + 0.411545i \(0.864989\pi\)
\(420\) 0 0
\(421\) 3.03265 0.147802 0.0739012 0.997266i \(-0.476455\pi\)
0.0739012 + 0.997266i \(0.476455\pi\)
\(422\) 0 0
\(423\) −8.79811 −0.427779
\(424\) 0 0
\(425\) −7.32528 −0.355328
\(426\) 0 0
\(427\) −3.11491 −0.150741
\(428\) 0 0
\(429\) 0.277904 0.0134173
\(430\) 0 0
\(431\) −24.0995 −1.16083 −0.580415 0.814321i \(-0.697110\pi\)
−0.580415 + 0.814321i \(0.697110\pi\)
\(432\) 0 0
\(433\) 17.6087 0.846220 0.423110 0.906078i \(-0.360938\pi\)
0.423110 + 0.906078i \(0.360938\pi\)
\(434\) 0 0
\(435\) −0.527166 −0.0252757
\(436\) 0 0
\(437\) −3.87189 −0.185218
\(438\) 0 0
\(439\) −26.2617 −1.25340 −0.626702 0.779259i \(-0.715596\pi\)
−0.626702 + 0.779259i \(0.715596\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −9.13659 −0.434092 −0.217046 0.976161i \(-0.569642\pi\)
−0.217046 + 0.976161i \(0.569642\pi\)
\(444\) 0 0
\(445\) 21.3859 1.01379
\(446\) 0 0
\(447\) −7.51396 −0.355398
\(448\) 0 0
\(449\) −3.46035 −0.163304 −0.0816519 0.996661i \(-0.526020\pi\)
−0.0816519 + 0.996661i \(0.526020\pi\)
\(450\) 0 0
\(451\) −16.2104 −0.763316
\(452\) 0 0
\(453\) 20.5808 0.966969
\(454\) 0 0
\(455\) 0.249262 0.0116856
\(456\) 0 0
\(457\) 14.5132 0.678901 0.339450 0.940624i \(-0.389759\pi\)
0.339450 + 0.940624i \(0.389759\pi\)
\(458\) 0 0
\(459\) −2.58774 −0.120785
\(460\) 0 0
\(461\) −12.4185 −0.578387 −0.289194 0.957271i \(-0.593387\pi\)
−0.289194 + 0.957271i \(0.593387\pi\)
\(462\) 0 0
\(463\) −16.3315 −0.758990 −0.379495 0.925194i \(-0.623902\pi\)
−0.379495 + 0.925194i \(0.623902\pi\)
\(464\) 0 0
\(465\) 5.70265 0.264454
\(466\) 0 0
\(467\) 2.81756 0.130381 0.0651905 0.997873i \(-0.479234\pi\)
0.0651905 + 0.997873i \(0.479234\pi\)
\(468\) 0 0
\(469\) 10.1623 0.469250
\(470\) 0 0
\(471\) 18.5808 0.856157
\(472\) 0 0
\(473\) 8.39905 0.386189
\(474\) 0 0
\(475\) 10.9604 0.502897
\(476\) 0 0
\(477\) 13.5940 0.622426
\(478\) 0 0
\(479\) −35.3594 −1.61561 −0.807807 0.589447i \(-0.799346\pi\)
−0.807807 + 0.589447i \(0.799346\pi\)
\(480\) 0 0
\(481\) 0.737534 0.0336287
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −7.09546 −0.322188
\(486\) 0 0
\(487\) −4.80507 −0.217739 −0.108869 0.994056i \(-0.534723\pi\)
−0.108869 + 0.994056i \(0.534723\pi\)
\(488\) 0 0
\(489\) −15.5940 −0.705184
\(490\) 0 0
\(491\) 42.0202 1.89635 0.948174 0.317753i \(-0.102928\pi\)
0.948174 + 0.317753i \(0.102928\pi\)
\(492\) 0 0
\(493\) −0.926221 −0.0417149
\(494\) 0 0
\(495\) −2.41850 −0.108704
\(496\) 0 0
\(497\) −13.2361 −0.593718
\(498\) 0 0
\(499\) 29.1513 1.30499 0.652496 0.757792i \(-0.273722\pi\)
0.652496 + 0.757792i \(0.273722\pi\)
\(500\) 0 0
\(501\) 5.98055 0.267191
\(502\) 0 0
\(503\) 11.7151 0.522352 0.261176 0.965291i \(-0.415890\pi\)
0.261176 + 0.965291i \(0.415890\pi\)
\(504\) 0 0
\(505\) 7.50477 0.333958
\(506\) 0 0
\(507\) 12.9714 0.576078
\(508\) 0 0
\(509\) 12.9457 0.573807 0.286903 0.957960i \(-0.407374\pi\)
0.286903 + 0.957960i \(0.407374\pi\)
\(510\) 0 0
\(511\) −12.3579 −0.546682
\(512\) 0 0
\(513\) 3.87189 0.170948
\(514\) 0 0
\(515\) −6.56829 −0.289434
\(516\) 0 0
\(517\) −14.4471 −0.635385
\(518\) 0 0
\(519\) −14.3315 −0.629084
\(520\) 0 0
\(521\) 21.5404 0.943701 0.471850 0.881679i \(-0.343586\pi\)
0.471850 + 0.881679i \(0.343586\pi\)
\(522\) 0 0
\(523\) 6.82452 0.298415 0.149208 0.988806i \(-0.452328\pi\)
0.149208 + 0.988806i \(0.452328\pi\)
\(524\) 0 0
\(525\) 2.83076 0.123545
\(526\) 0 0
\(527\) 10.0194 0.436454
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 7.47283 0.324293
\(532\) 0 0
\(533\) 1.67072 0.0723667
\(534\) 0 0
\(535\) −12.9262 −0.558849
\(536\) 0 0
\(537\) 18.6219 0.803595
\(538\) 0 0
\(539\) 1.64207 0.0707291
\(540\) 0 0
\(541\) −16.9457 −0.728551 −0.364276 0.931291i \(-0.618683\pi\)
−0.364276 + 0.931291i \(0.618683\pi\)
\(542\) 0 0
\(543\) 9.07378 0.389393
\(544\) 0 0
\(545\) −10.7889 −0.462146
\(546\) 0 0
\(547\) 18.2012 0.778226 0.389113 0.921190i \(-0.372782\pi\)
0.389113 + 0.921190i \(0.372782\pi\)
\(548\) 0 0
\(549\) −3.11491 −0.132941
\(550\) 0 0
\(551\) 1.38585 0.0590392
\(552\) 0 0
\(553\) −0.587741 −0.0249933
\(554\) 0 0
\(555\) −6.41850 −0.272450
\(556\) 0 0
\(557\) 27.5529 1.16745 0.583726 0.811951i \(-0.301594\pi\)
0.583726 + 0.811951i \(0.301594\pi\)
\(558\) 0 0
\(559\) −0.865646 −0.0366129
\(560\) 0 0
\(561\) −4.24926 −0.179404
\(562\) 0 0
\(563\) −35.7610 −1.50715 −0.753573 0.657364i \(-0.771671\pi\)
−0.753573 + 0.657364i \(0.771671\pi\)
\(564\) 0 0
\(565\) 28.6414 1.20495
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −34.9582 −1.46552 −0.732761 0.680486i \(-0.761769\pi\)
−0.732761 + 0.680486i \(0.761769\pi\)
\(570\) 0 0
\(571\) 26.6964 1.11721 0.558605 0.829434i \(-0.311337\pi\)
0.558605 + 0.829434i \(0.311337\pi\)
\(572\) 0 0
\(573\) −20.5808 −0.859774
\(574\) 0 0
\(575\) −2.83076 −0.118051
\(576\) 0 0
\(577\) −45.7299 −1.90376 −0.951879 0.306473i \(-0.900851\pi\)
−0.951879 + 0.306473i \(0.900851\pi\)
\(578\) 0 0
\(579\) 17.6615 0.733988
\(580\) 0 0
\(581\) 6.10170 0.253141
\(582\) 0 0
\(583\) 22.3223 0.924496
\(584\) 0 0
\(585\) 0.249262 0.0103057
\(586\) 0 0
\(587\) −22.2081 −0.916628 −0.458314 0.888790i \(-0.651546\pi\)
−0.458314 + 0.888790i \(0.651546\pi\)
\(588\) 0 0
\(589\) −14.9915 −0.617715
\(590\) 0 0
\(591\) 22.5591 0.927957
\(592\) 0 0
\(593\) −21.3664 −0.877413 −0.438707 0.898630i \(-0.644563\pi\)
−0.438707 + 0.898630i \(0.644563\pi\)
\(594\) 0 0
\(595\) −3.81131 −0.156249
\(596\) 0 0
\(597\) −4.64832 −0.190243
\(598\) 0 0
\(599\) 9.31832 0.380736 0.190368 0.981713i \(-0.439032\pi\)
0.190368 + 0.981713i \(0.439032\pi\)
\(600\) 0 0
\(601\) −35.0257 −1.42873 −0.714364 0.699774i \(-0.753284\pi\)
−0.714364 + 0.699774i \(0.753284\pi\)
\(602\) 0 0
\(603\) 10.1623 0.413840
\(604\) 0 0
\(605\) 12.2298 0.497213
\(606\) 0 0
\(607\) 21.5356 0.874105 0.437052 0.899436i \(-0.356022\pi\)
0.437052 + 0.899436i \(0.356022\pi\)
\(608\) 0 0
\(609\) 0.357926 0.0145039
\(610\) 0 0
\(611\) 1.48899 0.0602381
\(612\) 0 0
\(613\) 42.8300 1.72989 0.864945 0.501867i \(-0.167353\pi\)
0.864945 + 0.501867i \(0.167353\pi\)
\(614\) 0 0
\(615\) −14.5397 −0.586295
\(616\) 0 0
\(617\) −36.2904 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(618\) 0 0
\(619\) −27.5481 −1.10725 −0.553626 0.832765i \(-0.686756\pi\)
−0.553626 + 0.832765i \(0.686756\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −14.5202 −0.581740
\(624\) 0 0
\(625\) −2.83299 −0.113320
\(626\) 0 0
\(627\) 6.35793 0.253911
\(628\) 0 0
\(629\) −11.2772 −0.449651
\(630\) 0 0
\(631\) 19.7368 0.785710 0.392855 0.919600i \(-0.371487\pi\)
0.392855 + 0.919600i \(0.371487\pi\)
\(632\) 0 0
\(633\) 21.8844 0.869826
\(634\) 0 0
\(635\) −24.7089 −0.980542
\(636\) 0 0
\(637\) −0.169240 −0.00670552
\(638\) 0 0
\(639\) −13.2361 −0.523610
\(640\) 0 0
\(641\) 28.3726 1.12065 0.560326 0.828272i \(-0.310676\pi\)
0.560326 + 0.828272i \(0.310676\pi\)
\(642\) 0 0
\(643\) −31.8432 −1.25578 −0.627888 0.778304i \(-0.716080\pi\)
−0.627888 + 0.778304i \(0.716080\pi\)
\(644\) 0 0
\(645\) 7.53341 0.296628
\(646\) 0 0
\(647\) −1.91998 −0.0754821 −0.0377411 0.999288i \(-0.512016\pi\)
−0.0377411 + 0.999288i \(0.512016\pi\)
\(648\) 0 0
\(649\) 12.2709 0.481677
\(650\) 0 0
\(651\) −3.87189 −0.151751
\(652\) 0 0
\(653\) 7.40753 0.289879 0.144940 0.989441i \(-0.453701\pi\)
0.144940 + 0.989441i \(0.453701\pi\)
\(654\) 0 0
\(655\) −22.3796 −0.874444
\(656\) 0 0
\(657\) −12.3579 −0.482129
\(658\) 0 0
\(659\) −7.27719 −0.283479 −0.141739 0.989904i \(-0.545270\pi\)
−0.141739 + 0.989904i \(0.545270\pi\)
\(660\) 0 0
\(661\) 9.27719 0.360841 0.180420 0.983590i \(-0.442254\pi\)
0.180420 + 0.983590i \(0.442254\pi\)
\(662\) 0 0
\(663\) 0.437949 0.0170085
\(664\) 0 0
\(665\) 5.70265 0.221139
\(666\) 0 0
\(667\) −0.357926 −0.0138590
\(668\) 0 0
\(669\) 8.56606 0.331183
\(670\) 0 0
\(671\) −5.11491 −0.197459
\(672\) 0 0
\(673\) 22.6266 0.872193 0.436096 0.899900i \(-0.356361\pi\)
0.436096 + 0.899900i \(0.356361\pi\)
\(674\) 0 0
\(675\) 2.83076 0.108956
\(676\) 0 0
\(677\) −24.4938 −0.941373 −0.470687 0.882300i \(-0.655994\pi\)
−0.470687 + 0.882300i \(0.655994\pi\)
\(678\) 0 0
\(679\) 4.81756 0.184881
\(680\) 0 0
\(681\) 9.74154 0.373297
\(682\) 0 0
\(683\) 22.4402 0.858650 0.429325 0.903150i \(-0.358752\pi\)
0.429325 + 0.903150i \(0.358752\pi\)
\(684\) 0 0
\(685\) −22.7181 −0.868013
\(686\) 0 0
\(687\) 8.04113 0.306788
\(688\) 0 0
\(689\) −2.30064 −0.0876475
\(690\) 0 0
\(691\) −11.4798 −0.436712 −0.218356 0.975869i \(-0.570069\pi\)
−0.218356 + 0.975869i \(0.570069\pi\)
\(692\) 0 0
\(693\) 1.64207 0.0623772
\(694\) 0 0
\(695\) −6.16924 −0.234013
\(696\) 0 0
\(697\) −25.5459 −0.967620
\(698\) 0 0
\(699\) 27.5676 1.04270
\(700\) 0 0
\(701\) 11.6204 0.438896 0.219448 0.975624i \(-0.429574\pi\)
0.219448 + 0.975624i \(0.429574\pi\)
\(702\) 0 0
\(703\) 16.8734 0.636393
\(704\) 0 0
\(705\) −12.9582 −0.488032
\(706\) 0 0
\(707\) −5.09546 −0.191635
\(708\) 0 0
\(709\) −45.5870 −1.71206 −0.856028 0.516929i \(-0.827075\pi\)
−0.856028 + 0.516929i \(0.827075\pi\)
\(710\) 0 0
\(711\) −0.587741 −0.0220420
\(712\) 0 0
\(713\) 3.87189 0.145003
\(714\) 0 0
\(715\) 0.409307 0.0153072
\(716\) 0 0
\(717\) −20.0147 −0.747463
\(718\) 0 0
\(719\) 31.5195 1.17548 0.587739 0.809050i \(-0.300018\pi\)
0.587739 + 0.809050i \(0.300018\pi\)
\(720\) 0 0
\(721\) 4.45963 0.166085
\(722\) 0 0
\(723\) −5.99304 −0.222884
\(724\) 0 0
\(725\) 1.01320 0.0376294
\(726\) 0 0
\(727\) −15.1560 −0.562106 −0.281053 0.959692i \(-0.590684\pi\)
−0.281053 + 0.959692i \(0.590684\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 13.2361 0.489553
\(732\) 0 0
\(733\) 45.1491 1.66762 0.833810 0.552052i \(-0.186155\pi\)
0.833810 + 0.552052i \(0.186155\pi\)
\(734\) 0 0
\(735\) 1.47283 0.0543263
\(736\) 0 0
\(737\) 16.6872 0.614681
\(738\) 0 0
\(739\) −16.5055 −0.607164 −0.303582 0.952805i \(-0.598183\pi\)
−0.303582 + 0.952805i \(0.598183\pi\)
\(740\) 0 0
\(741\) −0.655277 −0.0240722
\(742\) 0 0
\(743\) −53.5537 −1.96469 −0.982347 0.187070i \(-0.940101\pi\)
−0.982347 + 0.187070i \(0.940101\pi\)
\(744\) 0 0
\(745\) −11.0668 −0.405457
\(746\) 0 0
\(747\) 6.10170 0.223250
\(748\) 0 0
\(749\) 8.77643 0.320684
\(750\) 0 0
\(751\) −28.2570 −1.03111 −0.515557 0.856855i \(-0.672415\pi\)
−0.515557 + 0.856855i \(0.672415\pi\)
\(752\) 0 0
\(753\) 14.2687 0.519981
\(754\) 0 0
\(755\) 30.3121 1.10317
\(756\) 0 0
\(757\) 12.9821 0.471841 0.235921 0.971772i \(-0.424189\pi\)
0.235921 + 0.971772i \(0.424189\pi\)
\(758\) 0 0
\(759\) −1.64207 −0.0596035
\(760\) 0 0
\(761\) 7.47060 0.270809 0.135405 0.990790i \(-0.456767\pi\)
0.135405 + 0.990790i \(0.456767\pi\)
\(762\) 0 0
\(763\) 7.32528 0.265193
\(764\) 0 0
\(765\) −3.81131 −0.137798
\(766\) 0 0
\(767\) −1.26470 −0.0456657
\(768\) 0 0
\(769\) 4.22982 0.152531 0.0762655 0.997088i \(-0.475700\pi\)
0.0762655 + 0.997088i \(0.475700\pi\)
\(770\) 0 0
\(771\) −31.0015 −1.11649
\(772\) 0 0
\(773\) −12.8565 −0.462414 −0.231207 0.972905i \(-0.574268\pi\)
−0.231207 + 0.972905i \(0.574268\pi\)
\(774\) 0 0
\(775\) −10.9604 −0.393709
\(776\) 0 0
\(777\) 4.35793 0.156340
\(778\) 0 0
\(779\) 38.2229 1.36948
\(780\) 0 0
\(781\) −21.7346 −0.777725
\(782\) 0 0
\(783\) 0.357926 0.0127912
\(784\) 0 0
\(785\) 27.3664 0.976749
\(786\) 0 0
\(787\) 1.03016 0.0367212 0.0183606 0.999831i \(-0.494155\pi\)
0.0183606 + 0.999831i \(0.494155\pi\)
\(788\) 0 0
\(789\) −7.65456 −0.272510
\(790\) 0 0
\(791\) −19.4464 −0.691435
\(792\) 0 0
\(793\) 0.527166 0.0187202
\(794\) 0 0
\(795\) 20.0217 0.710096
\(796\) 0 0
\(797\) −6.61967 −0.234481 −0.117240 0.993104i \(-0.537405\pi\)
−0.117240 + 0.993104i \(0.537405\pi\)
\(798\) 0 0
\(799\) −22.7672 −0.805447
\(800\) 0 0
\(801\) −14.5202 −0.513046
\(802\) 0 0
\(803\) −20.2926 −0.716111
\(804\) 0 0
\(805\) −1.47283 −0.0519106
\(806\) 0 0
\(807\) −5.72210 −0.201427
\(808\) 0 0
\(809\) −21.2097 −0.745692 −0.372846 0.927893i \(-0.621618\pi\)
−0.372846 + 0.927893i \(0.621618\pi\)
\(810\) 0 0
\(811\) 34.0878 1.19698 0.598492 0.801129i \(-0.295767\pi\)
0.598492 + 0.801129i \(0.295767\pi\)
\(812\) 0 0
\(813\) 1.22134 0.0428342
\(814\) 0 0
\(815\) −22.9673 −0.804511
\(816\) 0 0
\(817\) −19.8044 −0.692867
\(818\) 0 0
\(819\) −0.169240 −0.00591371
\(820\) 0 0
\(821\) 37.3525 1.30361 0.651805 0.758386i \(-0.274012\pi\)
0.651805 + 0.758386i \(0.274012\pi\)
\(822\) 0 0
\(823\) 42.7959 1.49177 0.745885 0.666075i \(-0.232027\pi\)
0.745885 + 0.666075i \(0.232027\pi\)
\(824\) 0 0
\(825\) 4.64832 0.161834
\(826\) 0 0
\(827\) −30.9993 −1.07795 −0.538975 0.842322i \(-0.681188\pi\)
−0.538975 + 0.842322i \(0.681188\pi\)
\(828\) 0 0
\(829\) 56.0055 1.94515 0.972576 0.232585i \(-0.0747183\pi\)
0.972576 + 0.232585i \(0.0747183\pi\)
\(830\) 0 0
\(831\) 14.8587 0.515442
\(832\) 0 0
\(833\) 2.58774 0.0896599
\(834\) 0 0
\(835\) 8.80836 0.304826
\(836\) 0 0
\(837\) −3.87189 −0.133832
\(838\) 0 0
\(839\) −15.6443 −0.540102 −0.270051 0.962846i \(-0.587040\pi\)
−0.270051 + 0.962846i \(0.587040\pi\)
\(840\) 0 0
\(841\) −28.8719 −0.995582
\(842\) 0 0
\(843\) −18.7981 −0.647441
\(844\) 0 0
\(845\) 19.1047 0.657220
\(846\) 0 0
\(847\) −8.30359 −0.285315
\(848\) 0 0
\(849\) 27.3378 0.938230
\(850\) 0 0
\(851\) −4.35793 −0.149388
\(852\) 0 0
\(853\) 7.55286 0.258605 0.129302 0.991605i \(-0.458726\pi\)
0.129302 + 0.991605i \(0.458726\pi\)
\(854\) 0 0
\(855\) 5.70265 0.195026
\(856\) 0 0
\(857\) −36.6894 −1.25329 −0.626644 0.779306i \(-0.715572\pi\)
−0.626644 + 0.779306i \(0.715572\pi\)
\(858\) 0 0
\(859\) −19.7438 −0.673649 −0.336824 0.941567i \(-0.609353\pi\)
−0.336824 + 0.941567i \(0.609353\pi\)
\(860\) 0 0
\(861\) 9.87189 0.336433
\(862\) 0 0
\(863\) 20.5947 0.701052 0.350526 0.936553i \(-0.386003\pi\)
0.350526 + 0.936553i \(0.386003\pi\)
\(864\) 0 0
\(865\) −21.1079 −0.717692
\(866\) 0 0
\(867\) 10.3036 0.349928
\(868\) 0 0
\(869\) −0.965115 −0.0327393
\(870\) 0 0
\(871\) −1.71986 −0.0582753
\(872\) 0 0
\(873\) 4.81756 0.163050
\(874\) 0 0
\(875\) 11.5334 0.389900
\(876\) 0 0
\(877\) 10.8804 0.367404 0.183702 0.982982i \(-0.441192\pi\)
0.183702 + 0.982982i \(0.441192\pi\)
\(878\) 0 0
\(879\) −8.82452 −0.297644
\(880\) 0 0
\(881\) 52.7144 1.77599 0.887997 0.459849i \(-0.152097\pi\)
0.887997 + 0.459849i \(0.152097\pi\)
\(882\) 0 0
\(883\) 0.616384 0.0207430 0.0103715 0.999946i \(-0.496699\pi\)
0.0103715 + 0.999946i \(0.496699\pi\)
\(884\) 0 0
\(885\) 11.0062 0.369971
\(886\) 0 0
\(887\) 3.42698 0.115067 0.0575334 0.998344i \(-0.481676\pi\)
0.0575334 + 0.998344i \(0.481676\pi\)
\(888\) 0 0
\(889\) 16.7764 0.562663
\(890\) 0 0
\(891\) 1.64207 0.0550115
\(892\) 0 0
\(893\) 34.0653 1.13995
\(894\) 0 0
\(895\) 27.4270 0.916783
\(896\) 0 0
\(897\) 0.169240 0.00565075
\(898\) 0 0
\(899\) −1.38585 −0.0462207
\(900\) 0 0
\(901\) 35.1777 1.17194
\(902\) 0 0
\(903\) −5.11491 −0.170213
\(904\) 0 0
\(905\) 13.3642 0.444240
\(906\) 0 0
\(907\) −41.2291 −1.36899 −0.684495 0.729018i \(-0.739977\pi\)
−0.684495 + 0.729018i \(0.739977\pi\)
\(908\) 0 0
\(909\) −5.09546 −0.169006
\(910\) 0 0
\(911\) 13.7702 0.456227 0.228113 0.973635i \(-0.426744\pi\)
0.228113 + 0.973635i \(0.426744\pi\)
\(912\) 0 0
\(913\) 10.0194 0.331595
\(914\) 0 0
\(915\) −4.58774 −0.151666
\(916\) 0 0
\(917\) 15.1949 0.501781
\(918\) 0 0
\(919\) 26.1406 0.862299 0.431149 0.902281i \(-0.358108\pi\)
0.431149 + 0.902281i \(0.358108\pi\)
\(920\) 0 0
\(921\) 30.6825 1.01102
\(922\) 0 0
\(923\) 2.24007 0.0737327
\(924\) 0 0
\(925\) 12.3362 0.405613
\(926\) 0 0
\(927\) 4.45963 0.146473
\(928\) 0 0
\(929\) 19.4853 0.639293 0.319646 0.947537i \(-0.396436\pi\)
0.319646 + 0.947537i \(0.396436\pi\)
\(930\) 0 0
\(931\) −3.87189 −0.126896
\(932\) 0 0
\(933\) −23.2097 −0.759850
\(934\) 0 0
\(935\) −6.25846 −0.204673
\(936\) 0 0
\(937\) −13.1560 −0.429789 −0.214894 0.976637i \(-0.568941\pi\)
−0.214894 + 0.976637i \(0.568941\pi\)
\(938\) 0 0
\(939\) 26.9582 0.879746
\(940\) 0 0
\(941\) 2.93318 0.0956190 0.0478095 0.998856i \(-0.484776\pi\)
0.0478095 + 0.998856i \(0.484776\pi\)
\(942\) 0 0
\(943\) −9.87189 −0.321473
\(944\) 0 0
\(945\) 1.47283 0.0479113
\(946\) 0 0
\(947\) 25.7996 0.838375 0.419188 0.907900i \(-0.362315\pi\)
0.419188 + 0.907900i \(0.362315\pi\)
\(948\) 0 0
\(949\) 2.09145 0.0678914
\(950\) 0 0
\(951\) −18.6483 −0.604713
\(952\) 0 0
\(953\) 32.3338 1.04739 0.523697 0.851905i \(-0.324553\pi\)
0.523697 + 0.851905i \(0.324553\pi\)
\(954\) 0 0
\(955\) −30.3121 −0.980876
\(956\) 0 0
\(957\) 0.587741 0.0189990
\(958\) 0 0
\(959\) 15.4247 0.498091
\(960\) 0 0
\(961\) −16.0085 −0.516403
\(962\) 0 0
\(963\) 8.77643 0.282816
\(964\) 0 0
\(965\) 26.0125 0.837372
\(966\) 0 0
\(967\) −49.3400 −1.58667 −0.793334 0.608787i \(-0.791656\pi\)
−0.793334 + 0.608787i \(0.791656\pi\)
\(968\) 0 0
\(969\) 10.0194 0.321871
\(970\) 0 0
\(971\) −4.48131 −0.143812 −0.0719061 0.997411i \(-0.522908\pi\)
−0.0719061 + 0.997411i \(0.522908\pi\)
\(972\) 0 0
\(973\) 4.18869 0.134283
\(974\) 0 0
\(975\) −0.479077 −0.0153427
\(976\) 0 0
\(977\) −61.6304 −1.97173 −0.985865 0.167539i \(-0.946418\pi\)
−0.985865 + 0.167539i \(0.946418\pi\)
\(978\) 0 0
\(979\) −23.8432 −0.762033
\(980\) 0 0
\(981\) 7.32528 0.233878
\(982\) 0 0
\(983\) −10.1545 −0.323879 −0.161939 0.986801i \(-0.551775\pi\)
−0.161939 + 0.986801i \(0.551775\pi\)
\(984\) 0 0
\(985\) 33.2258 1.05866
\(986\) 0 0
\(987\) 8.79811 0.280047
\(988\) 0 0
\(989\) 5.11491 0.162645
\(990\) 0 0
\(991\) 13.1927 0.419080 0.209540 0.977800i \(-0.432803\pi\)
0.209540 + 0.977800i \(0.432803\pi\)
\(992\) 0 0
\(993\) 4.42074 0.140288
\(994\) 0 0
\(995\) −6.84620 −0.217039
\(996\) 0 0
\(997\) −47.2119 −1.49522 −0.747608 0.664141i \(-0.768798\pi\)
−0.747608 + 0.664141i \(0.768798\pi\)
\(998\) 0 0
\(999\) 4.35793 0.137879
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 7728.2.a.bs.1.1 3
4.3 odd 2 3864.2.a.q.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.q.1.1 3 4.3 odd 2
7728.2.a.bs.1.1 3 1.1 even 1 trivial