Properties

Label 7938.2.a.by.1.1
Level $7938$
Weight $2$
Character 7938.1
Self dual yes
Analytic conductor $63.385$
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] = [7938,2,Mod(1,7938)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7938, 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("7938.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7938.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: \(63.3852491245\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.69963\) of defining polynomial
Character \(\chi\) \(=\) 7938.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.69963 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.69963 q^{5} +1.00000 q^{8} -3.69963 q^{10} -1.47710 q^{11} -2.69963 q^{13} +1.00000 q^{16} +6.57598 q^{17} -0.888736 q^{19} -3.69963 q^{20} -1.47710 q^{22} +6.28799 q^{23} +8.68725 q^{25} -2.69963 q^{26} -2.51052 q^{29} -6.81089 q^{31} +1.00000 q^{32} +6.57598 q^{34} +2.77747 q^{37} -0.888736 q^{38} -3.69963 q^{40} -4.11126 q^{41} -0.0123797 q^{43} -1.47710 q^{44} +6.28799 q^{46} +6.98762 q^{47} +8.68725 q^{50} -2.69963 q^{52} +3.21015 q^{53} +5.46472 q^{55} -2.51052 q^{58} -6.90978 q^{59} +5.73305 q^{61} -6.81089 q^{62} +1.00000 q^{64} +9.98762 q^{65} -9.46472 q^{67} +6.57598 q^{68} -5.46472 q^{71} -12.0655 q^{73} +2.77747 q^{74} -0.888736 q^{76} +11.4523 q^{79} -3.69963 q^{80} -4.11126 q^{82} +4.47710 q^{83} -24.3287 q^{85} -0.0123797 q^{86} -1.47710 q^{88} -8.87636 q^{89} +6.28799 q^{92} +6.98762 q^{94} +3.28799 q^{95} -13.1767 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} - 5 q^{5} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{4} - 5 q^{5} + 3 q^{8} - 5 q^{10} + q^{11} - 2 q^{13} + 3 q^{16} - 4 q^{17} - 3 q^{19} - 5 q^{20} + q^{22} + 7 q^{23} + 2 q^{25} - 2 q^{26} + 5 q^{29} - 14 q^{31} + 3 q^{32} - 4 q^{34} + 9 q^{37} - 3 q^{38} - 5 q^{40} - 12 q^{41} - 18 q^{43} + q^{44} + 7 q^{46} + 3 q^{47} + 2 q^{50} - 2 q^{52} - 9 q^{53} - 7 q^{55} + 5 q^{58} + 4 q^{59} + 4 q^{61} - 14 q^{62} + 3 q^{64} + 12 q^{65} - 5 q^{67} - 4 q^{68} + 7 q^{71} - 25 q^{73} + 9 q^{74} - 3 q^{76} - 7 q^{79} - 5 q^{80} - 12 q^{82} + 8 q^{83} - 14 q^{85} - 18 q^{86} + q^{88} - 9 q^{89} + 7 q^{92} + 3 q^{94} - 2 q^{95} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −3.69963 −1.65452 −0.827262 0.561816i \(-0.810103\pi\)
−0.827262 + 0.561816i \(0.810103\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −3.69963 −1.16993
\(11\) −1.47710 −0.445362 −0.222681 0.974891i \(-0.571481\pi\)
−0.222681 + 0.974891i \(0.571481\pi\)
\(12\) 0 0
\(13\) −2.69963 −0.748742 −0.374371 0.927279i \(-0.622141\pi\)
−0.374371 + 0.927279i \(0.622141\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.57598 1.59491 0.797455 0.603378i \(-0.206179\pi\)
0.797455 + 0.603378i \(0.206179\pi\)
\(18\) 0 0
\(19\) −0.888736 −0.203890 −0.101945 0.994790i \(-0.532507\pi\)
−0.101945 + 0.994790i \(0.532507\pi\)
\(20\) −3.69963 −0.827262
\(21\) 0 0
\(22\) −1.47710 −0.314919
\(23\) 6.28799 1.31114 0.655568 0.755136i \(-0.272429\pi\)
0.655568 + 0.755136i \(0.272429\pi\)
\(24\) 0 0
\(25\) 8.68725 1.73745
\(26\) −2.69963 −0.529441
\(27\) 0 0
\(28\) 0 0
\(29\) −2.51052 −0.466192 −0.233096 0.972454i \(-0.574886\pi\)
−0.233096 + 0.972454i \(0.574886\pi\)
\(30\) 0 0
\(31\) −6.81089 −1.22327 −0.611636 0.791139i \(-0.709488\pi\)
−0.611636 + 0.791139i \(0.709488\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.57598 1.12777
\(35\) 0 0
\(36\) 0 0
\(37\) 2.77747 0.456614 0.228307 0.973589i \(-0.426681\pi\)
0.228307 + 0.973589i \(0.426681\pi\)
\(38\) −0.888736 −0.144172
\(39\) 0 0
\(40\) −3.69963 −0.584963
\(41\) −4.11126 −0.642072 −0.321036 0.947067i \(-0.604031\pi\)
−0.321036 + 0.947067i \(0.604031\pi\)
\(42\) 0 0
\(43\) −0.0123797 −0.00188789 −0.000943944 1.00000i \(-0.500300\pi\)
−0.000943944 1.00000i \(0.500300\pi\)
\(44\) −1.47710 −0.222681
\(45\) 0 0
\(46\) 6.28799 0.927114
\(47\) 6.98762 1.01925 0.509625 0.860397i \(-0.329784\pi\)
0.509625 + 0.860397i \(0.329784\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 8.68725 1.22856
\(51\) 0 0
\(52\) −2.69963 −0.374371
\(53\) 3.21015 0.440948 0.220474 0.975393i \(-0.429240\pi\)
0.220474 + 0.975393i \(0.429240\pi\)
\(54\) 0 0
\(55\) 5.46472 0.736863
\(56\) 0 0
\(57\) 0 0
\(58\) −2.51052 −0.329647
\(59\) −6.90978 −0.899576 −0.449788 0.893135i \(-0.648501\pi\)
−0.449788 + 0.893135i \(0.648501\pi\)
\(60\) 0 0
\(61\) 5.73305 0.734042 0.367021 0.930213i \(-0.380378\pi\)
0.367021 + 0.930213i \(0.380378\pi\)
\(62\) −6.81089 −0.864984
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 9.98762 1.23881
\(66\) 0 0
\(67\) −9.46472 −1.15630 −0.578150 0.815931i \(-0.696225\pi\)
−0.578150 + 0.815931i \(0.696225\pi\)
\(68\) 6.57598 0.797455
\(69\) 0 0
\(70\) 0 0
\(71\) −5.46472 −0.648543 −0.324271 0.945964i \(-0.605119\pi\)
−0.324271 + 0.945964i \(0.605119\pi\)
\(72\) 0 0
\(73\) −12.0655 −1.41216 −0.706078 0.708134i \(-0.749537\pi\)
−0.706078 + 0.708134i \(0.749537\pi\)
\(74\) 2.77747 0.322875
\(75\) 0 0
\(76\) −0.888736 −0.101945
\(77\) 0 0
\(78\) 0 0
\(79\) 11.4523 1.28849 0.644244 0.764820i \(-0.277172\pi\)
0.644244 + 0.764820i \(0.277172\pi\)
\(80\) −3.69963 −0.413631
\(81\) 0 0
\(82\) −4.11126 −0.454013
\(83\) 4.47710 0.491426 0.245713 0.969343i \(-0.420978\pi\)
0.245713 + 0.969343i \(0.420978\pi\)
\(84\) 0 0
\(85\) −24.3287 −2.63882
\(86\) −0.0123797 −0.00133494
\(87\) 0 0
\(88\) −1.47710 −0.157459
\(89\) −8.87636 −0.940892 −0.470446 0.882429i \(-0.655907\pi\)
−0.470446 + 0.882429i \(0.655907\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.28799 0.655568
\(93\) 0 0
\(94\) 6.98762 0.720718
\(95\) 3.28799 0.337341
\(96\) 0 0
\(97\) −13.1767 −1.33789 −0.668947 0.743310i \(-0.733255\pi\)
−0.668947 + 0.743310i \(0.733255\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 8.68725 0.868725
\(101\) −5.25457 −0.522849 −0.261425 0.965224i \(-0.584192\pi\)
−0.261425 + 0.965224i \(0.584192\pi\)
\(102\) 0 0
\(103\) −1.66621 −0.164176 −0.0820882 0.996625i \(-0.526159\pi\)
−0.0820882 + 0.996625i \(0.526159\pi\)
\(104\) −2.69963 −0.264720
\(105\) 0 0
\(106\) 3.21015 0.311797
\(107\) 10.7651 1.04070 0.520350 0.853953i \(-0.325801\pi\)
0.520350 + 0.853953i \(0.325801\pi\)
\(108\) 0 0
\(109\) 0.189108 0.0181132 0.00905662 0.999959i \(-0.497117\pi\)
0.00905662 + 0.999959i \(0.497117\pi\)
\(110\) 5.46472 0.521041
\(111\) 0 0
\(112\) 0 0
\(113\) 13.5636 1.27596 0.637978 0.770054i \(-0.279771\pi\)
0.637978 + 0.770054i \(0.279771\pi\)
\(114\) 0 0
\(115\) −23.2632 −2.16931
\(116\) −2.51052 −0.233096
\(117\) 0 0
\(118\) −6.90978 −0.636097
\(119\) 0 0
\(120\) 0 0
\(121\) −8.81818 −0.801652
\(122\) 5.73305 0.519046
\(123\) 0 0
\(124\) −6.81089 −0.611636
\(125\) −13.6414 −1.22013
\(126\) 0 0
\(127\) −2.85669 −0.253490 −0.126745 0.991935i \(-0.540453\pi\)
−0.126745 + 0.991935i \(0.540453\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 9.98762 0.875972
\(131\) −0.155687 −0.0136024 −0.00680122 0.999977i \(-0.502165\pi\)
−0.00680122 + 0.999977i \(0.502165\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −9.46472 −0.817627
\(135\) 0 0
\(136\) 6.57598 0.563886
\(137\) −3.41164 −0.291476 −0.145738 0.989323i \(-0.546556\pi\)
−0.145738 + 0.989323i \(0.546556\pi\)
\(138\) 0 0
\(139\) −13.5105 −1.14595 −0.572974 0.819574i \(-0.694210\pi\)
−0.572974 + 0.819574i \(0.694210\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −5.46472 −0.458589
\(143\) 3.98762 0.333462
\(144\) 0 0
\(145\) 9.28799 0.771326
\(146\) −12.0655 −0.998545
\(147\) 0 0
\(148\) 2.77747 0.228307
\(149\) 0.333792 0.0273453 0.0136727 0.999907i \(-0.495648\pi\)
0.0136727 + 0.999907i \(0.495648\pi\)
\(150\) 0 0
\(151\) −19.9098 −1.62023 −0.810117 0.586268i \(-0.800597\pi\)
−0.810117 + 0.586268i \(0.800597\pi\)
\(152\) −0.888736 −0.0720860
\(153\) 0 0
\(154\) 0 0
\(155\) 25.1978 2.02393
\(156\) 0 0
\(157\) 6.96286 0.555697 0.277848 0.960625i \(-0.410379\pi\)
0.277848 + 0.960625i \(0.410379\pi\)
\(158\) 11.4523 0.911099
\(159\) 0 0
\(160\) −3.69963 −0.292481
\(161\) 0 0
\(162\) 0 0
\(163\) −8.07413 −0.632414 −0.316207 0.948690i \(-0.602409\pi\)
−0.316207 + 0.948690i \(0.602409\pi\)
\(164\) −4.11126 −0.321036
\(165\) 0 0
\(166\) 4.47710 0.347490
\(167\) 19.4858 1.50785 0.753927 0.656959i \(-0.228157\pi\)
0.753927 + 0.656959i \(0.228157\pi\)
\(168\) 0 0
\(169\) −5.71201 −0.439385
\(170\) −24.3287 −1.86593
\(171\) 0 0
\(172\) −0.0123797 −0.000943944 0
\(173\) −22.5636 −1.71548 −0.857740 0.514085i \(-0.828132\pi\)
−0.857740 + 0.514085i \(0.828132\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.47710 −0.111341
\(177\) 0 0
\(178\) −8.87636 −0.665311
\(179\) −0.333792 −0.0249488 −0.0124744 0.999922i \(-0.503971\pi\)
−0.0124744 + 0.999922i \(0.503971\pi\)
\(180\) 0 0
\(181\) −23.2422 −1.72758 −0.863789 0.503853i \(-0.831915\pi\)
−0.863789 + 0.503853i \(0.831915\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 6.28799 0.463557
\(185\) −10.2756 −0.755478
\(186\) 0 0
\(187\) −9.71339 −0.710313
\(188\) 6.98762 0.509625
\(189\) 0 0
\(190\) 3.28799 0.238536
\(191\) −16.3214 −1.18098 −0.590488 0.807046i \(-0.701065\pi\)
−0.590488 + 0.807046i \(0.701065\pi\)
\(192\) 0 0
\(193\) −14.3214 −1.03088 −0.515439 0.856926i \(-0.672371\pi\)
−0.515439 + 0.856926i \(0.672371\pi\)
\(194\) −13.1767 −0.946034
\(195\) 0 0
\(196\) 0 0
\(197\) 2.42402 0.172704 0.0863520 0.996265i \(-0.472479\pi\)
0.0863520 + 0.996265i \(0.472479\pi\)
\(198\) 0 0
\(199\) −6.11126 −0.433216 −0.216608 0.976259i \(-0.569499\pi\)
−0.216608 + 0.976259i \(0.569499\pi\)
\(200\) 8.68725 0.614281
\(201\) 0 0
\(202\) −5.25457 −0.369710
\(203\) 0 0
\(204\) 0 0
\(205\) 15.2101 1.06232
\(206\) −1.66621 −0.116090
\(207\) 0 0
\(208\) −2.69963 −0.187186
\(209\) 1.31275 0.0908049
\(210\) 0 0
\(211\) −11.4451 −0.787910 −0.393955 0.919130i \(-0.628893\pi\)
−0.393955 + 0.919130i \(0.628893\pi\)
\(212\) 3.21015 0.220474
\(213\) 0 0
\(214\) 10.7651 0.735887
\(215\) 0.0458003 0.00312356
\(216\) 0 0
\(217\) 0 0
\(218\) 0.189108 0.0128080
\(219\) 0 0
\(220\) 5.46472 0.368431
\(221\) −17.7527 −1.19418
\(222\) 0 0
\(223\) −7.22253 −0.483656 −0.241828 0.970319i \(-0.577747\pi\)
−0.241828 + 0.970319i \(0.577747\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 13.5636 0.902238
\(227\) −13.6552 −0.906328 −0.453164 0.891427i \(-0.649705\pi\)
−0.453164 + 0.891427i \(0.649705\pi\)
\(228\) 0 0
\(229\) −17.3745 −1.14814 −0.574070 0.818807i \(-0.694636\pi\)
−0.574070 + 0.818807i \(0.694636\pi\)
\(230\) −23.2632 −1.53393
\(231\) 0 0
\(232\) −2.51052 −0.164824
\(233\) −15.2422 −0.998549 −0.499275 0.866444i \(-0.666400\pi\)
−0.499275 + 0.866444i \(0.666400\pi\)
\(234\) 0 0
\(235\) −25.8516 −1.68637
\(236\) −6.90978 −0.449788
\(237\) 0 0
\(238\) 0 0
\(239\) −18.9505 −1.22580 −0.612902 0.790159i \(-0.709998\pi\)
−0.612902 + 0.790159i \(0.709998\pi\)
\(240\) 0 0
\(241\) 24.5054 1.57853 0.789267 0.614051i \(-0.210461\pi\)
0.789267 + 0.614051i \(0.210461\pi\)
\(242\) −8.81818 −0.566854
\(243\) 0 0
\(244\) 5.73305 0.367021
\(245\) 0 0
\(246\) 0 0
\(247\) 2.39926 0.152661
\(248\) −6.81089 −0.432492
\(249\) 0 0
\(250\) −13.6414 −0.862761
\(251\) 12.1236 0.765238 0.382619 0.923906i \(-0.375022\pi\)
0.382619 + 0.923906i \(0.375022\pi\)
\(252\) 0 0
\(253\) −9.28799 −0.583931
\(254\) −2.85669 −0.179245
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.20877 0.512049 0.256025 0.966670i \(-0.417587\pi\)
0.256025 + 0.966670i \(0.417587\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 9.98762 0.619406
\(261\) 0 0
\(262\) −0.155687 −0.00961838
\(263\) −5.34617 −0.329659 −0.164830 0.986322i \(-0.552707\pi\)
−0.164830 + 0.986322i \(0.552707\pi\)
\(264\) 0 0
\(265\) −11.8764 −0.729559
\(266\) 0 0
\(267\) 0 0
\(268\) −9.46472 −0.578150
\(269\) 18.4844 1.12701 0.563506 0.826112i \(-0.309452\pi\)
0.563506 + 0.826112i \(0.309452\pi\)
\(270\) 0 0
\(271\) −7.35483 −0.446774 −0.223387 0.974730i \(-0.571711\pi\)
−0.223387 + 0.974730i \(0.571711\pi\)
\(272\) 6.57598 0.398728
\(273\) 0 0
\(274\) −3.41164 −0.206104
\(275\) −12.8319 −0.773795
\(276\) 0 0
\(277\) −9.09888 −0.546699 −0.273349 0.961915i \(-0.588132\pi\)
−0.273349 + 0.961915i \(0.588132\pi\)
\(278\) −13.5105 −0.810307
\(279\) 0 0
\(280\) 0 0
\(281\) −12.0087 −0.716377 −0.358188 0.933649i \(-0.616605\pi\)
−0.358188 + 0.933649i \(0.616605\pi\)
\(282\) 0 0
\(283\) −9.84294 −0.585102 −0.292551 0.956250i \(-0.594504\pi\)
−0.292551 + 0.956250i \(0.594504\pi\)
\(284\) −5.46472 −0.324271
\(285\) 0 0
\(286\) 3.98762 0.235793
\(287\) 0 0
\(288\) 0 0
\(289\) 26.2436 1.54374
\(290\) 9.28799 0.545410
\(291\) 0 0
\(292\) −12.0655 −0.706078
\(293\) 21.4203 1.25139 0.625694 0.780069i \(-0.284816\pi\)
0.625694 + 0.780069i \(0.284816\pi\)
\(294\) 0 0
\(295\) 25.5636 1.48837
\(296\) 2.77747 0.161437
\(297\) 0 0
\(298\) 0.333792 0.0193361
\(299\) −16.9752 −0.981704
\(300\) 0 0
\(301\) 0 0
\(302\) −19.9098 −1.14568
\(303\) 0 0
\(304\) −0.888736 −0.0509725
\(305\) −21.2101 −1.21449
\(306\) 0 0
\(307\) 5.68725 0.324588 0.162294 0.986742i \(-0.448111\pi\)
0.162294 + 0.986742i \(0.448111\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 25.1978 1.43114
\(311\) 11.7207 0.664618 0.332309 0.943171i \(-0.392172\pi\)
0.332309 + 0.943171i \(0.392172\pi\)
\(312\) 0 0
\(313\) 26.7738 1.51334 0.756671 0.653796i \(-0.226824\pi\)
0.756671 + 0.653796i \(0.226824\pi\)
\(314\) 6.96286 0.392937
\(315\) 0 0
\(316\) 11.4523 0.644244
\(317\) 1.90249 0.106855 0.0534273 0.998572i \(-0.482985\pi\)
0.0534273 + 0.998572i \(0.482985\pi\)
\(318\) 0 0
\(319\) 3.70829 0.207624
\(320\) −3.69963 −0.206816
\(321\) 0 0
\(322\) 0 0
\(323\) −5.84431 −0.325186
\(324\) 0 0
\(325\) −23.4523 −1.30090
\(326\) −8.07413 −0.447184
\(327\) 0 0
\(328\) −4.11126 −0.227007
\(329\) 0 0
\(330\) 0 0
\(331\) 5.56732 0.306008 0.153004 0.988226i \(-0.451105\pi\)
0.153004 + 0.988226i \(0.451105\pi\)
\(332\) 4.47710 0.245713
\(333\) 0 0
\(334\) 19.4858 1.06621
\(335\) 35.0159 1.91313
\(336\) 0 0
\(337\) 33.7738 1.83977 0.919887 0.392184i \(-0.128280\pi\)
0.919887 + 0.392184i \(0.128280\pi\)
\(338\) −5.71201 −0.310692
\(339\) 0 0
\(340\) −24.3287 −1.31941
\(341\) 10.0604 0.544799
\(342\) 0 0
\(343\) 0 0
\(344\) −0.0123797 −0.000667469 0
\(345\) 0 0
\(346\) −22.5636 −1.21303
\(347\) −30.4065 −1.63231 −0.816154 0.577834i \(-0.803898\pi\)
−0.816154 + 0.577834i \(0.803898\pi\)
\(348\) 0 0
\(349\) −12.5956 −0.674230 −0.337115 0.941464i \(-0.609451\pi\)
−0.337115 + 0.941464i \(0.609451\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.47710 −0.0787297
\(353\) 7.53156 0.400865 0.200432 0.979708i \(-0.435765\pi\)
0.200432 + 0.979708i \(0.435765\pi\)
\(354\) 0 0
\(355\) 20.2174 1.07303
\(356\) −8.87636 −0.470446
\(357\) 0 0
\(358\) −0.333792 −0.0176415
\(359\) 6.89602 0.363958 0.181979 0.983302i \(-0.441750\pi\)
0.181979 + 0.983302i \(0.441750\pi\)
\(360\) 0 0
\(361\) −18.2101 −0.958429
\(362\) −23.2422 −1.22158
\(363\) 0 0
\(364\) 0 0
\(365\) 44.6377 2.33645
\(366\) 0 0
\(367\) −23.1236 −1.20704 −0.603522 0.797346i \(-0.706237\pi\)
−0.603522 + 0.797346i \(0.706237\pi\)
\(368\) 6.28799 0.327784
\(369\) 0 0
\(370\) −10.2756 −0.534204
\(371\) 0 0
\(372\) 0 0
\(373\) 29.1643 1.51007 0.755036 0.655683i \(-0.227619\pi\)
0.755036 + 0.655683i \(0.227619\pi\)
\(374\) −9.71339 −0.502267
\(375\) 0 0
\(376\) 6.98762 0.360359
\(377\) 6.77747 0.349058
\(378\) 0 0
\(379\) −13.5622 −0.696645 −0.348322 0.937375i \(-0.613249\pi\)
−0.348322 + 0.937375i \(0.613249\pi\)
\(380\) 3.28799 0.168670
\(381\) 0 0
\(382\) −16.3214 −0.835076
\(383\) 2.83565 0.144895 0.0724475 0.997372i \(-0.476919\pi\)
0.0724475 + 0.997372i \(0.476919\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −14.3214 −0.728941
\(387\) 0 0
\(388\) −13.1767 −0.668947
\(389\) −18.6080 −0.943464 −0.471732 0.881742i \(-0.656371\pi\)
−0.471732 + 0.881742i \(0.656371\pi\)
\(390\) 0 0
\(391\) 41.3497 2.09115
\(392\) 0 0
\(393\) 0 0
\(394\) 2.42402 0.122120
\(395\) −42.3694 −2.13184
\(396\) 0 0
\(397\) −20.5760 −1.03268 −0.516340 0.856384i \(-0.672706\pi\)
−0.516340 + 0.856384i \(0.672706\pi\)
\(398\) −6.11126 −0.306330
\(399\) 0 0
\(400\) 8.68725 0.434362
\(401\) −6.75409 −0.337283 −0.168642 0.985677i \(-0.553938\pi\)
−0.168642 + 0.985677i \(0.553938\pi\)
\(402\) 0 0
\(403\) 18.3869 0.915916
\(404\) −5.25457 −0.261425
\(405\) 0 0
\(406\) 0 0
\(407\) −4.10260 −0.203358
\(408\) 0 0
\(409\) −15.3214 −0.757595 −0.378798 0.925480i \(-0.623662\pi\)
−0.378798 + 0.925480i \(0.623662\pi\)
\(410\) 15.2101 0.751176
\(411\) 0 0
\(412\) −1.66621 −0.0820882
\(413\) 0 0
\(414\) 0 0
\(415\) −16.5636 −0.813075
\(416\) −2.69963 −0.132360
\(417\) 0 0
\(418\) 1.31275 0.0642088
\(419\) −8.64283 −0.422230 −0.211115 0.977461i \(-0.567709\pi\)
−0.211115 + 0.977461i \(0.567709\pi\)
\(420\) 0 0
\(421\) −37.1272 −1.80947 −0.904735 0.425975i \(-0.859931\pi\)
−0.904735 + 0.425975i \(0.859931\pi\)
\(422\) −11.4451 −0.557137
\(423\) 0 0
\(424\) 3.21015 0.155899
\(425\) 57.1272 2.77108
\(426\) 0 0
\(427\) 0 0
\(428\) 10.7651 0.520350
\(429\) 0 0
\(430\) 0.0458003 0.00220869
\(431\) 9.42030 0.453760 0.226880 0.973923i \(-0.427148\pi\)
0.226880 + 0.973923i \(0.427148\pi\)
\(432\) 0 0
\(433\) 0.208771 0.0100329 0.00501645 0.999987i \(-0.498403\pi\)
0.00501645 + 0.999987i \(0.498403\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.189108 0.00905662
\(437\) −5.58836 −0.267328
\(438\) 0 0
\(439\) 9.96796 0.475745 0.237872 0.971296i \(-0.423550\pi\)
0.237872 + 0.971296i \(0.423550\pi\)
\(440\) 5.46472 0.260520
\(441\) 0 0
\(442\) −17.7527 −0.844410
\(443\) −15.6996 −0.745912 −0.372956 0.927849i \(-0.621656\pi\)
−0.372956 + 0.927849i \(0.621656\pi\)
\(444\) 0 0
\(445\) 32.8392 1.55673
\(446\) −7.22253 −0.341997
\(447\) 0 0
\(448\) 0 0
\(449\) 33.6253 1.58688 0.793439 0.608650i \(-0.208288\pi\)
0.793439 + 0.608650i \(0.208288\pi\)
\(450\) 0 0
\(451\) 6.07275 0.285955
\(452\) 13.5636 0.637978
\(453\) 0 0
\(454\) −13.6552 −0.640871
\(455\) 0 0
\(456\) 0 0
\(457\) 32.7083 1.53003 0.765015 0.644013i \(-0.222732\pi\)
0.765015 + 0.644013i \(0.222732\pi\)
\(458\) −17.3745 −0.811857
\(459\) 0 0
\(460\) −23.2632 −1.08465
\(461\) −4.14331 −0.192973 −0.0964865 0.995334i \(-0.530760\pi\)
−0.0964865 + 0.995334i \(0.530760\pi\)
\(462\) 0 0
\(463\) 16.6835 0.775349 0.387675 0.921796i \(-0.373278\pi\)
0.387675 + 0.921796i \(0.373278\pi\)
\(464\) −2.51052 −0.116548
\(465\) 0 0
\(466\) −15.2422 −0.706081
\(467\) 29.9171 1.38440 0.692198 0.721707i \(-0.256642\pi\)
0.692198 + 0.721707i \(0.256642\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −25.8516 −1.19245
\(471\) 0 0
\(472\) −6.90978 −0.318048
\(473\) 0.0182861 0.000840794 0
\(474\) 0 0
\(475\) −7.72067 −0.354249
\(476\) 0 0
\(477\) 0 0
\(478\) −18.9505 −0.866775
\(479\) 2.95930 0.135214 0.0676068 0.997712i \(-0.478464\pi\)
0.0676068 + 0.997712i \(0.478464\pi\)
\(480\) 0 0
\(481\) −7.49814 −0.341886
\(482\) 24.5054 1.11619
\(483\) 0 0
\(484\) −8.81818 −0.400826
\(485\) 48.7490 2.21358
\(486\) 0 0
\(487\) 28.0617 1.27160 0.635800 0.771854i \(-0.280671\pi\)
0.635800 + 0.771854i \(0.280671\pi\)
\(488\) 5.73305 0.259523
\(489\) 0 0
\(490\) 0 0
\(491\) −34.1469 −1.54103 −0.770513 0.637424i \(-0.780000\pi\)
−0.770513 + 0.637424i \(0.780000\pi\)
\(492\) 0 0
\(493\) −16.5091 −0.743534
\(494\) 2.39926 0.107948
\(495\) 0 0
\(496\) −6.81089 −0.305818
\(497\) 0 0
\(498\) 0 0
\(499\) −2.28071 −0.102099 −0.0510493 0.998696i \(-0.516257\pi\)
−0.0510493 + 0.998696i \(0.516257\pi\)
\(500\) −13.6414 −0.610064
\(501\) 0 0
\(502\) 12.1236 0.541105
\(503\) −13.9890 −0.623739 −0.311869 0.950125i \(-0.600955\pi\)
−0.311869 + 0.950125i \(0.600955\pi\)
\(504\) 0 0
\(505\) 19.4400 0.865067
\(506\) −9.28799 −0.412902
\(507\) 0 0
\(508\) −2.85669 −0.126745
\(509\) −25.6181 −1.13550 −0.567750 0.823201i \(-0.692186\pi\)
−0.567750 + 0.823201i \(0.692186\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 8.20877 0.362073
\(515\) 6.16435 0.271634
\(516\) 0 0
\(517\) −10.3214 −0.453935
\(518\) 0 0
\(519\) 0 0
\(520\) 9.98762 0.437986
\(521\) −41.8255 −1.83241 −0.916203 0.400714i \(-0.868762\pi\)
−0.916203 + 0.400714i \(0.868762\pi\)
\(522\) 0 0
\(523\) 15.7665 0.689420 0.344710 0.938709i \(-0.387977\pi\)
0.344710 + 0.938709i \(0.387977\pi\)
\(524\) −0.155687 −0.00680122
\(525\) 0 0
\(526\) −5.34617 −0.233104
\(527\) −44.7883 −1.95101
\(528\) 0 0
\(529\) 16.5388 0.719080
\(530\) −11.8764 −0.515876
\(531\) 0 0
\(532\) 0 0
\(533\) 11.0989 0.480746
\(534\) 0 0
\(535\) −39.8268 −1.72186
\(536\) −9.46472 −0.408814
\(537\) 0 0
\(538\) 18.4844 0.796918
\(539\) 0 0
\(540\) 0 0
\(541\) 42.1927 1.81400 0.907002 0.421126i \(-0.138365\pi\)
0.907002 + 0.421126i \(0.138365\pi\)
\(542\) −7.35483 −0.315917
\(543\) 0 0
\(544\) 6.57598 0.281943
\(545\) −0.699628 −0.0299688
\(546\) 0 0
\(547\) −40.6712 −1.73897 −0.869486 0.493957i \(-0.835550\pi\)
−0.869486 + 0.493957i \(0.835550\pi\)
\(548\) −3.41164 −0.145738
\(549\) 0 0
\(550\) −12.8319 −0.547155
\(551\) 2.23119 0.0950519
\(552\) 0 0
\(553\) 0 0
\(554\) −9.09888 −0.386575
\(555\) 0 0
\(556\) −13.5105 −0.572974
\(557\) −13.3759 −0.566754 −0.283377 0.959009i \(-0.591455\pi\)
−0.283377 + 0.959009i \(0.591455\pi\)
\(558\) 0 0
\(559\) 0.0334206 0.00141354
\(560\) 0 0
\(561\) 0 0
\(562\) −12.0087 −0.506555
\(563\) 32.7614 1.38073 0.690364 0.723463i \(-0.257451\pi\)
0.690364 + 0.723463i \(0.257451\pi\)
\(564\) 0 0
\(565\) −50.1803 −2.11110
\(566\) −9.84294 −0.413729
\(567\) 0 0
\(568\) −5.46472 −0.229295
\(569\) −16.7280 −0.701272 −0.350636 0.936512i \(-0.614035\pi\)
−0.350636 + 0.936512i \(0.614035\pi\)
\(570\) 0 0
\(571\) −27.4734 −1.14973 −0.574863 0.818250i \(-0.694945\pi\)
−0.574863 + 0.818250i \(0.694945\pi\)
\(572\) 3.98762 0.166731
\(573\) 0 0
\(574\) 0 0
\(575\) 54.6253 2.27803
\(576\) 0 0
\(577\) 2.83427 0.117992 0.0589962 0.998258i \(-0.481210\pi\)
0.0589962 + 0.998258i \(0.481210\pi\)
\(578\) 26.2436 1.09159
\(579\) 0 0
\(580\) 9.28799 0.385663
\(581\) 0 0
\(582\) 0 0
\(583\) −4.74171 −0.196382
\(584\) −12.0655 −0.499272
\(585\) 0 0
\(586\) 21.4203 0.884864
\(587\) −4.69591 −0.193821 −0.0969105 0.995293i \(-0.530896\pi\)
−0.0969105 + 0.995293i \(0.530896\pi\)
\(588\) 0 0
\(589\) 6.05308 0.249413
\(590\) 25.5636 1.05244
\(591\) 0 0
\(592\) 2.77747 0.114153
\(593\) 1.27205 0.0522367 0.0261184 0.999659i \(-0.491685\pi\)
0.0261184 + 0.999659i \(0.491685\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.333792 0.0136727
\(597\) 0 0
\(598\) −16.9752 −0.694169
\(599\) 43.8516 1.79173 0.895864 0.444329i \(-0.146558\pi\)
0.895864 + 0.444329i \(0.146558\pi\)
\(600\) 0 0
\(601\) −13.4327 −0.547930 −0.273965 0.961740i \(-0.588335\pi\)
−0.273965 + 0.961740i \(0.588335\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −19.9098 −0.810117
\(605\) 32.6240 1.32635
\(606\) 0 0
\(607\) 4.58465 0.186085 0.0930425 0.995662i \(-0.470341\pi\)
0.0930425 + 0.995662i \(0.470341\pi\)
\(608\) −0.888736 −0.0360430
\(609\) 0 0
\(610\) −21.2101 −0.858774
\(611\) −18.8640 −0.763155
\(612\) 0 0
\(613\) 22.1075 0.892915 0.446458 0.894805i \(-0.352685\pi\)
0.446458 + 0.894805i \(0.352685\pi\)
\(614\) 5.68725 0.229519
\(615\) 0 0
\(616\) 0 0
\(617\) −12.0087 −0.483450 −0.241725 0.970345i \(-0.577713\pi\)
−0.241725 + 0.970345i \(0.577713\pi\)
\(618\) 0 0
\(619\) 17.5636 0.705941 0.352970 0.935634i \(-0.385172\pi\)
0.352970 + 0.935634i \(0.385172\pi\)
\(620\) 25.1978 1.01197
\(621\) 0 0
\(622\) 11.7207 0.469956
\(623\) 0 0
\(624\) 0 0
\(625\) 7.03204 0.281282
\(626\) 26.7738 1.07009
\(627\) 0 0
\(628\) 6.96286 0.277848
\(629\) 18.2646 0.728258
\(630\) 0 0
\(631\) −44.9381 −1.78896 −0.894479 0.447110i \(-0.852453\pi\)
−0.894479 + 0.447110i \(0.852453\pi\)
\(632\) 11.4523 0.455550
\(633\) 0 0
\(634\) 1.90249 0.0755576
\(635\) 10.5687 0.419406
\(636\) 0 0
\(637\) 0 0
\(638\) 3.70829 0.146813
\(639\) 0 0
\(640\) −3.69963 −0.146241
\(641\) −28.9839 −1.14480 −0.572398 0.819976i \(-0.693987\pi\)
−0.572398 + 0.819976i \(0.693987\pi\)
\(642\) 0 0
\(643\) 12.0617 0.475669 0.237834 0.971306i \(-0.423562\pi\)
0.237834 + 0.971306i \(0.423562\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −5.84431 −0.229941
\(647\) 37.7651 1.48470 0.742349 0.670013i \(-0.233711\pi\)
0.742349 + 0.670013i \(0.233711\pi\)
\(648\) 0 0
\(649\) 10.2064 0.400637
\(650\) −23.4523 −0.919876
\(651\) 0 0
\(652\) −8.07413 −0.316207
\(653\) 37.4079 1.46388 0.731942 0.681366i \(-0.238614\pi\)
0.731942 + 0.681366i \(0.238614\pi\)
\(654\) 0 0
\(655\) 0.575984 0.0225056
\(656\) −4.11126 −0.160518
\(657\) 0 0
\(658\) 0 0
\(659\) −29.8713 −1.16362 −0.581810 0.813325i \(-0.697655\pi\)
−0.581810 + 0.813325i \(0.697655\pi\)
\(660\) 0 0
\(661\) −5.60803 −0.218127 −0.109063 0.994035i \(-0.534785\pi\)
−0.109063 + 0.994035i \(0.534785\pi\)
\(662\) 5.56732 0.216380
\(663\) 0 0
\(664\) 4.47710 0.173745
\(665\) 0 0
\(666\) 0 0
\(667\) −15.7861 −0.611242
\(668\) 19.4858 0.753927
\(669\) 0 0
\(670\) 35.0159 1.35278
\(671\) −8.46829 −0.326915
\(672\) 0 0
\(673\) 9.44506 0.364080 0.182040 0.983291i \(-0.441730\pi\)
0.182040 + 0.983291i \(0.441730\pi\)
\(674\) 33.7738 1.30092
\(675\) 0 0
\(676\) −5.71201 −0.219693
\(677\) −11.0617 −0.425137 −0.212569 0.977146i \(-0.568183\pi\)
−0.212569 + 0.977146i \(0.568183\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −24.3287 −0.932963
\(681\) 0 0
\(682\) 10.0604 0.385231
\(683\) 8.83922 0.338223 0.169112 0.985597i \(-0.445910\pi\)
0.169112 + 0.985597i \(0.445910\pi\)
\(684\) 0 0
\(685\) 12.6218 0.482254
\(686\) 0 0
\(687\) 0 0
\(688\) −0.0123797 −0.000471972 0
\(689\) −8.66621 −0.330156
\(690\) 0 0
\(691\) −25.0617 −0.953394 −0.476697 0.879068i \(-0.658166\pi\)
−0.476697 + 0.879068i \(0.658166\pi\)
\(692\) −22.5636 −0.857740
\(693\) 0 0
\(694\) −30.4065 −1.15422
\(695\) 49.9839 1.89600
\(696\) 0 0
\(697\) −27.0356 −1.02405
\(698\) −12.5956 −0.476752
\(699\) 0 0
\(700\) 0 0
\(701\) −43.4858 −1.64243 −0.821217 0.570616i \(-0.806705\pi\)
−0.821217 + 0.570616i \(0.806705\pi\)
\(702\) 0 0
\(703\) −2.46844 −0.0930989
\(704\) −1.47710 −0.0556703
\(705\) 0 0
\(706\) 7.53156 0.283454
\(707\) 0 0
\(708\) 0 0
\(709\) −22.7403 −0.854031 −0.427016 0.904244i \(-0.640435\pi\)
−0.427016 + 0.904244i \(0.640435\pi\)
\(710\) 20.2174 0.758747
\(711\) 0 0
\(712\) −8.87636 −0.332656
\(713\) −42.8268 −1.60388
\(714\) 0 0
\(715\) −14.7527 −0.551720
\(716\) −0.333792 −0.0124744
\(717\) 0 0
\(718\) 6.89602 0.257357
\(719\) 12.1236 0.452136 0.226068 0.974112i \(-0.427413\pi\)
0.226068 + 0.974112i \(0.427413\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −18.2101 −0.677712
\(723\) 0 0
\(724\) −23.2422 −0.863789
\(725\) −21.8095 −0.809985
\(726\) 0 0
\(727\) 46.1817 1.71278 0.856392 0.516327i \(-0.172701\pi\)
0.856392 + 0.516327i \(0.172701\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 44.6377 1.65212
\(731\) −0.0814088 −0.00301101
\(732\) 0 0
\(733\) 36.0297 1.33079 0.665394 0.746493i \(-0.268264\pi\)
0.665394 + 0.746493i \(0.268264\pi\)
\(734\) −23.1236 −0.853509
\(735\) 0 0
\(736\) 6.28799 0.231778
\(737\) 13.9803 0.514972
\(738\) 0 0
\(739\) −46.4239 −1.70773 −0.853865 0.520495i \(-0.825747\pi\)
−0.853865 + 0.520495i \(0.825747\pi\)
\(740\) −10.2756 −0.377739
\(741\) 0 0
\(742\) 0 0
\(743\) −1.19777 −0.0439419 −0.0219709 0.999759i \(-0.506994\pi\)
−0.0219709 + 0.999759i \(0.506994\pi\)
\(744\) 0 0
\(745\) −1.23491 −0.0452435
\(746\) 29.1643 1.06778
\(747\) 0 0
\(748\) −9.71339 −0.355157
\(749\) 0 0
\(750\) 0 0
\(751\) 48.1199 1.75592 0.877961 0.478733i \(-0.158904\pi\)
0.877961 + 0.478733i \(0.158904\pi\)
\(752\) 6.98762 0.254812
\(753\) 0 0
\(754\) 6.77747 0.246821
\(755\) 73.6588 2.68072
\(756\) 0 0
\(757\) 49.6006 1.80276 0.901382 0.433025i \(-0.142554\pi\)
0.901382 + 0.433025i \(0.142554\pi\)
\(758\) −13.5622 −0.492602
\(759\) 0 0
\(760\) 3.28799 0.119268
\(761\) 37.5402 1.36083 0.680416 0.732826i \(-0.261799\pi\)
0.680416 + 0.732826i \(0.261799\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −16.3214 −0.590488
\(765\) 0 0
\(766\) 2.83565 0.102456
\(767\) 18.6538 0.673551
\(768\) 0 0
\(769\) −26.9184 −0.970704 −0.485352 0.874319i \(-0.661308\pi\)
−0.485352 + 0.874319i \(0.661308\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −14.3214 −0.515439
\(773\) 50.2261 1.80651 0.903254 0.429107i \(-0.141172\pi\)
0.903254 + 0.429107i \(0.141172\pi\)
\(774\) 0 0
\(775\) −59.1679 −2.12537
\(776\) −13.1767 −0.473017
\(777\) 0 0
\(778\) −18.6080 −0.667130
\(779\) 3.65383 0.130912
\(780\) 0 0
\(781\) 8.07194 0.288837
\(782\) 41.3497 1.47866
\(783\) 0 0
\(784\) 0 0
\(785\) −25.7600 −0.919414
\(786\) 0 0
\(787\) 1.65892 0.0591342 0.0295671 0.999563i \(-0.490587\pi\)
0.0295671 + 0.999563i \(0.490587\pi\)
\(788\) 2.42402 0.0863520
\(789\) 0 0
\(790\) −42.3694 −1.50744
\(791\) 0 0
\(792\) 0 0
\(793\) −15.4771 −0.549608
\(794\) −20.5760 −0.730214
\(795\) 0 0
\(796\) −6.11126 −0.216608
\(797\) −30.7403 −1.08888 −0.544439 0.838800i \(-0.683258\pi\)
−0.544439 + 0.838800i \(0.683258\pi\)
\(798\) 0 0
\(799\) 45.9505 1.62561
\(800\) 8.68725 0.307141
\(801\) 0 0
\(802\) −6.75409 −0.238495
\(803\) 17.8219 0.628921
\(804\) 0 0
\(805\) 0 0
\(806\) 18.3869 0.647650
\(807\) 0 0
\(808\) −5.25457 −0.184855
\(809\) 2.88502 0.101432 0.0507159 0.998713i \(-0.483850\pi\)
0.0507159 + 0.998713i \(0.483850\pi\)
\(810\) 0 0
\(811\) −28.5461 −1.00239 −0.501195 0.865334i \(-0.667106\pi\)
−0.501195 + 0.865334i \(0.667106\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −4.10260 −0.143796
\(815\) 29.8713 1.04634
\(816\) 0 0
\(817\) 0.0110023 0.000384922 0
\(818\) −15.3214 −0.535701
\(819\) 0 0
\(820\) 15.2101 0.531161
\(821\) 7.96658 0.278036 0.139018 0.990290i \(-0.455605\pi\)
0.139018 + 0.990290i \(0.455605\pi\)
\(822\) 0 0
\(823\) 40.5461 1.41335 0.706675 0.707539i \(-0.250194\pi\)
0.706675 + 0.707539i \(0.250194\pi\)
\(824\) −1.66621 −0.0580451
\(825\) 0 0
\(826\) 0 0
\(827\) 1.22115 0.0424636 0.0212318 0.999775i \(-0.493241\pi\)
0.0212318 + 0.999775i \(0.493241\pi\)
\(828\) 0 0
\(829\) −14.1506 −0.491470 −0.245735 0.969337i \(-0.579029\pi\)
−0.245735 + 0.969337i \(0.579029\pi\)
\(830\) −16.5636 −0.574931
\(831\) 0 0
\(832\) −2.69963 −0.0935928
\(833\) 0 0
\(834\) 0 0
\(835\) −72.0901 −2.49478
\(836\) 1.31275 0.0454025
\(837\) 0 0
\(838\) −8.64283 −0.298561
\(839\) 2.39197 0.0825801 0.0412900 0.999147i \(-0.486853\pi\)
0.0412900 + 0.999147i \(0.486853\pi\)
\(840\) 0 0
\(841\) −22.6973 −0.782665
\(842\) −37.1272 −1.27949
\(843\) 0 0
\(844\) −11.4451 −0.393955
\(845\) 21.1323 0.726973
\(846\) 0 0
\(847\) 0 0
\(848\) 3.21015 0.110237
\(849\) 0 0
\(850\) 57.1272 1.95945
\(851\) 17.4647 0.598683
\(852\) 0 0
\(853\) −16.6800 −0.571111 −0.285556 0.958362i \(-0.592178\pi\)
−0.285556 + 0.958362i \(0.592178\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 10.7651 0.367943
\(857\) 13.8516 0.473162 0.236581 0.971612i \(-0.423973\pi\)
0.236581 + 0.971612i \(0.423973\pi\)
\(858\) 0 0
\(859\) −48.4944 −1.65461 −0.827304 0.561754i \(-0.810127\pi\)
−0.827304 + 0.561754i \(0.810127\pi\)
\(860\) 0.0458003 0.00156178
\(861\) 0 0
\(862\) 9.42030 0.320857
\(863\) −5.93082 −0.201887 −0.100944 0.994892i \(-0.532186\pi\)
−0.100944 + 0.994892i \(0.532186\pi\)
\(864\) 0 0
\(865\) 83.4769 2.83830
\(866\) 0.208771 0.00709433
\(867\) 0 0
\(868\) 0 0
\(869\) −16.9163 −0.573844
\(870\) 0 0
\(871\) 25.5512 0.865770
\(872\) 0.189108 0.00640399
\(873\) 0 0
\(874\) −5.58836 −0.189029
\(875\) 0 0
\(876\) 0 0
\(877\) 3.92944 0.132688 0.0663439 0.997797i \(-0.478867\pi\)
0.0663439 + 0.997797i \(0.478867\pi\)
\(878\) 9.96796 0.336402
\(879\) 0 0
\(880\) 5.46472 0.184216
\(881\) −37.6552 −1.26864 −0.634318 0.773072i \(-0.718719\pi\)
−0.634318 + 0.773072i \(0.718719\pi\)
\(882\) 0 0
\(883\) −53.2334 −1.79145 −0.895723 0.444613i \(-0.853341\pi\)
−0.895723 + 0.444613i \(0.853341\pi\)
\(884\) −17.7527 −0.597088
\(885\) 0 0
\(886\) −15.6996 −0.527439
\(887\) 36.9876 1.24192 0.620961 0.783841i \(-0.286742\pi\)
0.620961 + 0.783841i \(0.286742\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 32.8392 1.10077
\(891\) 0 0
\(892\) −7.22253 −0.241828
\(893\) −6.21015 −0.207815
\(894\) 0 0
\(895\) 1.23491 0.0412784
\(896\) 0 0
\(897\) 0 0
\(898\) 33.6253 1.12209
\(899\) 17.0989 0.570280
\(900\) 0 0
\(901\) 21.1099 0.703272
\(902\) 6.07275 0.202200
\(903\) 0 0
\(904\) 13.5636 0.451119
\(905\) 85.9875 2.85832
\(906\) 0 0
\(907\) −39.0159 −1.29550 −0.647752 0.761852i \(-0.724291\pi\)
−0.647752 + 0.761852i \(0.724291\pi\)
\(908\) −13.6552 −0.453164
\(909\) 0 0
\(910\) 0 0
\(911\) 25.6181 0.848764 0.424382 0.905483i \(-0.360491\pi\)
0.424382 + 0.905483i \(0.360491\pi\)
\(912\) 0 0
\(913\) −6.61312 −0.218862
\(914\) 32.7083 1.08189
\(915\) 0 0
\(916\) −17.3745 −0.574070
\(917\) 0 0
\(918\) 0 0
\(919\) −20.6735 −0.681956 −0.340978 0.940071i \(-0.610758\pi\)
−0.340978 + 0.940071i \(0.610758\pi\)
\(920\) −23.2632 −0.766966
\(921\) 0 0
\(922\) −4.14331 −0.136453
\(923\) 14.7527 0.485591
\(924\) 0 0
\(925\) 24.1286 0.793343
\(926\) 16.6835 0.548255
\(927\) 0 0
\(928\) −2.51052 −0.0824119
\(929\) 3.74033 0.122716 0.0613582 0.998116i \(-0.480457\pi\)
0.0613582 + 0.998116i \(0.480457\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −15.2422 −0.499275
\(933\) 0 0
\(934\) 29.9171 0.978916
\(935\) 35.9359 1.17523
\(936\) 0 0
\(937\) 27.1345 0.886445 0.443223 0.896412i \(-0.353835\pi\)
0.443223 + 0.896412i \(0.353835\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −25.8516 −0.843186
\(941\) −6.32870 −0.206310 −0.103155 0.994665i \(-0.532894\pi\)
−0.103155 + 0.994665i \(0.532894\pi\)
\(942\) 0 0
\(943\) −25.8516 −0.841844
\(944\) −6.90978 −0.224894
\(945\) 0 0
\(946\) 0.0182861 0.000594531 0
\(947\) 31.2792 1.01644 0.508218 0.861228i \(-0.330304\pi\)
0.508218 + 0.861228i \(0.330304\pi\)
\(948\) 0 0
\(949\) 32.5723 1.05734
\(950\) −7.72067 −0.250492
\(951\) 0 0
\(952\) 0 0
\(953\) 4.28937 0.138946 0.0694732 0.997584i \(-0.477868\pi\)
0.0694732 + 0.997584i \(0.477868\pi\)
\(954\) 0 0
\(955\) 60.3832 1.95395
\(956\) −18.9505 −0.612902
\(957\) 0 0
\(958\) 2.95930 0.0956105
\(959\) 0 0
\(960\) 0 0
\(961\) 15.3883 0.496395
\(962\) −7.49814 −0.241750
\(963\) 0 0
\(964\) 24.5054 0.789267
\(965\) 52.9839 1.70561
\(966\) 0 0
\(967\) 15.1840 0.488285 0.244142 0.969739i \(-0.421494\pi\)
0.244142 + 0.969739i \(0.421494\pi\)
\(968\) −8.81818 −0.283427
\(969\) 0 0
\(970\) 48.7490 1.56524
\(971\) 3.24729 0.104210 0.0521052 0.998642i \(-0.483407\pi\)
0.0521052 + 0.998642i \(0.483407\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 28.0617 0.899156
\(975\) 0 0
\(976\) 5.73305 0.183510
\(977\) 15.5439 0.497295 0.248647 0.968594i \(-0.420014\pi\)
0.248647 + 0.968594i \(0.420014\pi\)
\(978\) 0 0
\(979\) 13.1113 0.419038
\(980\) 0 0
\(981\) 0 0
\(982\) −34.1469 −1.08967
\(983\) −12.3832 −0.394961 −0.197481 0.980307i \(-0.563276\pi\)
−0.197481 + 0.980307i \(0.563276\pi\)
\(984\) 0 0
\(985\) −8.96796 −0.285743
\(986\) −16.5091 −0.525758
\(987\) 0 0
\(988\) 2.39926 0.0763305
\(989\) −0.0778435 −0.00247528
\(990\) 0 0
\(991\) 6.65521 0.211410 0.105705 0.994398i \(-0.466290\pi\)
0.105705 + 0.994398i \(0.466290\pi\)
\(992\) −6.81089 −0.216246
\(993\) 0 0
\(994\) 0 0
\(995\) 22.6094 0.716766
\(996\) 0 0
\(997\) −4.80208 −0.152083 −0.0760417 0.997105i \(-0.524228\pi\)
−0.0760417 + 0.997105i \(0.524228\pi\)
\(998\) −2.28071 −0.0721946
\(999\) 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 7938.2.a.by.1.1 3
3.2 odd 2 7938.2.a.bx.1.3 3
7.3 odd 6 1134.2.g.k.163.1 6
7.5 odd 6 1134.2.g.k.487.1 6
7.6 odd 2 7938.2.a.cb.1.3 3
9.2 odd 6 2646.2.f.n.1765.1 6
9.4 even 3 882.2.f.m.295.1 6
9.5 odd 6 2646.2.f.n.883.1 6
9.7 even 3 882.2.f.m.589.1 6
21.5 even 6 1134.2.g.n.487.3 6
21.17 even 6 1134.2.g.n.163.3 6
21.20 even 2 7938.2.a.bu.1.1 3
63.2 odd 6 2646.2.h.p.361.3 6
63.4 even 3 882.2.h.o.79.3 6
63.5 even 6 378.2.e.c.235.3 6
63.11 odd 6 2646.2.e.o.1549.1 6
63.13 odd 6 882.2.f.l.295.3 6
63.16 even 3 882.2.h.o.67.3 6
63.20 even 6 2646.2.f.o.1765.3 6
63.23 odd 6 2646.2.e.o.2125.1 6
63.25 even 3 882.2.e.p.373.3 6
63.31 odd 6 126.2.h.c.79.1 yes 6
63.32 odd 6 2646.2.h.p.667.3 6
63.34 odd 6 882.2.f.l.589.3 6
63.38 even 6 378.2.e.c.37.3 6
63.40 odd 6 126.2.e.d.25.1 6
63.41 even 6 2646.2.f.o.883.3 6
63.47 even 6 378.2.h.d.361.1 6
63.52 odd 6 126.2.e.d.121.1 yes 6
63.58 even 3 882.2.e.p.655.3 6
63.59 even 6 378.2.h.d.289.1 6
63.61 odd 6 126.2.h.c.67.1 yes 6
252.31 even 6 1008.2.t.g.961.3 6
252.47 odd 6 3024.2.t.g.1873.1 6
252.59 odd 6 3024.2.t.g.289.1 6
252.103 even 6 1008.2.q.h.529.3 6
252.115 even 6 1008.2.q.h.625.3 6
252.131 odd 6 3024.2.q.h.2881.3 6
252.187 even 6 1008.2.t.g.193.3 6
252.227 odd 6 3024.2.q.h.2305.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.e.d.25.1 6 63.40 odd 6
126.2.e.d.121.1 yes 6 63.52 odd 6
126.2.h.c.67.1 yes 6 63.61 odd 6
126.2.h.c.79.1 yes 6 63.31 odd 6
378.2.e.c.37.3 6 63.38 even 6
378.2.e.c.235.3 6 63.5 even 6
378.2.h.d.289.1 6 63.59 even 6
378.2.h.d.361.1 6 63.47 even 6
882.2.e.p.373.3 6 63.25 even 3
882.2.e.p.655.3 6 63.58 even 3
882.2.f.l.295.3 6 63.13 odd 6
882.2.f.l.589.3 6 63.34 odd 6
882.2.f.m.295.1 6 9.4 even 3
882.2.f.m.589.1 6 9.7 even 3
882.2.h.o.67.3 6 63.16 even 3
882.2.h.o.79.3 6 63.4 even 3
1008.2.q.h.529.3 6 252.103 even 6
1008.2.q.h.625.3 6 252.115 even 6
1008.2.t.g.193.3 6 252.187 even 6
1008.2.t.g.961.3 6 252.31 even 6
1134.2.g.k.163.1 6 7.3 odd 6
1134.2.g.k.487.1 6 7.5 odd 6
1134.2.g.n.163.3 6 21.17 even 6
1134.2.g.n.487.3 6 21.5 even 6
2646.2.e.o.1549.1 6 63.11 odd 6
2646.2.e.o.2125.1 6 63.23 odd 6
2646.2.f.n.883.1 6 9.5 odd 6
2646.2.f.n.1765.1 6 9.2 odd 6
2646.2.f.o.883.3 6 63.41 even 6
2646.2.f.o.1765.3 6 63.20 even 6
2646.2.h.p.361.3 6 63.2 odd 6
2646.2.h.p.667.3 6 63.32 odd 6
3024.2.q.h.2305.3 6 252.227 odd 6
3024.2.q.h.2881.3 6 252.131 odd 6
3024.2.t.g.289.1 6 252.59 odd 6
3024.2.t.g.1873.1 6 252.47 odd 6
7938.2.a.bu.1.1 3 21.20 even 2
7938.2.a.bx.1.3 3 3.2 odd 2
7938.2.a.by.1.1 3 1.1 even 1 trivial
7938.2.a.cb.1.3 3 7.6 odd 2