Properties

Label 8048.2.a.t.1.7
Level $8048$
Weight $2$
Character 8048.1
Self dual yes
Analytic conductor $64.264$
Analytic rank $0$
Dimension $21$
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] = [8048,2,Mod(1,8048)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8048, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8048.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8048.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: \(64.2636035467\)
Analytic rank: \(0\)
Dimension: \(21\)
Twist minimal: no (minimal twist has level 2012)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Character \(\chi\) \(=\) 8048.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.954017 q^{3} -2.55262 q^{5} -2.39192 q^{7} -2.08985 q^{9} +O(q^{10})\) \(q-0.954017 q^{3} -2.55262 q^{5} -2.39192 q^{7} -2.08985 q^{9} +1.06849 q^{11} -1.31853 q^{13} +2.43524 q^{15} +2.92040 q^{17} +4.34304 q^{19} +2.28193 q^{21} -8.73838 q^{23} +1.51585 q^{25} +4.85580 q^{27} -1.67364 q^{29} -1.33374 q^{31} -1.01936 q^{33} +6.10566 q^{35} -10.2523 q^{37} +1.25790 q^{39} -8.86010 q^{41} +0.916643 q^{43} +5.33459 q^{45} +5.35597 q^{47} -1.27871 q^{49} -2.78611 q^{51} -9.85191 q^{53} -2.72745 q^{55} -4.14334 q^{57} -7.51624 q^{59} -14.3465 q^{61} +4.99876 q^{63} +3.36570 q^{65} +6.54260 q^{67} +8.33656 q^{69} +2.20297 q^{71} +5.32927 q^{73} -1.44614 q^{75} -2.55575 q^{77} -4.78499 q^{79} +1.63704 q^{81} -8.01031 q^{83} -7.45466 q^{85} +1.59668 q^{87} -3.89360 q^{89} +3.15382 q^{91} +1.27241 q^{93} -11.0861 q^{95} +2.19229 q^{97} -2.23299 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) \) 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
\(3\) −0.954017 −0.550802 −0.275401 0.961329i \(-0.588811\pi\)
−0.275401 + 0.961329i \(0.588811\pi\)
\(4\) 0 0
\(5\) −2.55262 −1.14156 −0.570782 0.821101i \(-0.693360\pi\)
−0.570782 + 0.821101i \(0.693360\pi\)
\(6\) 0 0
\(7\) −2.39192 −0.904061 −0.452031 0.892002i \(-0.649300\pi\)
−0.452031 + 0.892002i \(0.649300\pi\)
\(8\) 0 0
\(9\) −2.08985 −0.696617
\(10\) 0 0
\(11\) 1.06849 0.322162 0.161081 0.986941i \(-0.448502\pi\)
0.161081 + 0.986941i \(0.448502\pi\)
\(12\) 0 0
\(13\) −1.31853 −0.365694 −0.182847 0.983141i \(-0.558531\pi\)
−0.182847 + 0.983141i \(0.558531\pi\)
\(14\) 0 0
\(15\) 2.43524 0.628776
\(16\) 0 0
\(17\) 2.92040 0.708301 0.354150 0.935188i \(-0.384770\pi\)
0.354150 + 0.935188i \(0.384770\pi\)
\(18\) 0 0
\(19\) 4.34304 0.996363 0.498181 0.867073i \(-0.334001\pi\)
0.498181 + 0.867073i \(0.334001\pi\)
\(20\) 0 0
\(21\) 2.28193 0.497959
\(22\) 0 0
\(23\) −8.73838 −1.82208 −0.911039 0.412321i \(-0.864718\pi\)
−0.911039 + 0.412321i \(0.864718\pi\)
\(24\) 0 0
\(25\) 1.51585 0.303169
\(26\) 0 0
\(27\) 4.85580 0.934500
\(28\) 0 0
\(29\) −1.67364 −0.310788 −0.155394 0.987853i \(-0.549665\pi\)
−0.155394 + 0.987853i \(0.549665\pi\)
\(30\) 0 0
\(31\) −1.33374 −0.239547 −0.119773 0.992801i \(-0.538217\pi\)
−0.119773 + 0.992801i \(0.538217\pi\)
\(32\) 0 0
\(33\) −1.01936 −0.177447
\(34\) 0 0
\(35\) 6.10566 1.03204
\(36\) 0 0
\(37\) −10.2523 −1.68547 −0.842737 0.538325i \(-0.819057\pi\)
−0.842737 + 0.538325i \(0.819057\pi\)
\(38\) 0 0
\(39\) 1.25790 0.201425
\(40\) 0 0
\(41\) −8.86010 −1.38371 −0.691857 0.722034i \(-0.743207\pi\)
−0.691857 + 0.722034i \(0.743207\pi\)
\(42\) 0 0
\(43\) 0.916643 0.139787 0.0698934 0.997554i \(-0.477734\pi\)
0.0698934 + 0.997554i \(0.477734\pi\)
\(44\) 0 0
\(45\) 5.33459 0.795234
\(46\) 0 0
\(47\) 5.35597 0.781248 0.390624 0.920550i \(-0.372259\pi\)
0.390624 + 0.920550i \(0.372259\pi\)
\(48\) 0 0
\(49\) −1.27871 −0.182673
\(50\) 0 0
\(51\) −2.78611 −0.390133
\(52\) 0 0
\(53\) −9.85191 −1.35326 −0.676632 0.736322i \(-0.736561\pi\)
−0.676632 + 0.736322i \(0.736561\pi\)
\(54\) 0 0
\(55\) −2.72745 −0.367769
\(56\) 0 0
\(57\) −4.14334 −0.548798
\(58\) 0 0
\(59\) −7.51624 −0.978531 −0.489265 0.872135i \(-0.662735\pi\)
−0.489265 + 0.872135i \(0.662735\pi\)
\(60\) 0 0
\(61\) −14.3465 −1.83688 −0.918440 0.395560i \(-0.870551\pi\)
−0.918440 + 0.395560i \(0.870551\pi\)
\(62\) 0 0
\(63\) 4.99876 0.629785
\(64\) 0 0
\(65\) 3.36570 0.417464
\(66\) 0 0
\(67\) 6.54260 0.799305 0.399653 0.916667i \(-0.369131\pi\)
0.399653 + 0.916667i \(0.369131\pi\)
\(68\) 0 0
\(69\) 8.33656 1.00360
\(70\) 0 0
\(71\) 2.20297 0.261445 0.130722 0.991419i \(-0.458270\pi\)
0.130722 + 0.991419i \(0.458270\pi\)
\(72\) 0 0
\(73\) 5.32927 0.623744 0.311872 0.950124i \(-0.399044\pi\)
0.311872 + 0.950124i \(0.399044\pi\)
\(74\) 0 0
\(75\) −1.44614 −0.166986
\(76\) 0 0
\(77\) −2.55575 −0.291254
\(78\) 0 0
\(79\) −4.78499 −0.538353 −0.269176 0.963091i \(-0.586751\pi\)
−0.269176 + 0.963091i \(0.586751\pi\)
\(80\) 0 0
\(81\) 1.63704 0.181893
\(82\) 0 0
\(83\) −8.01031 −0.879245 −0.439623 0.898183i \(-0.644888\pi\)
−0.439623 + 0.898183i \(0.644888\pi\)
\(84\) 0 0
\(85\) −7.45466 −0.808571
\(86\) 0 0
\(87\) 1.59668 0.171183
\(88\) 0 0
\(89\) −3.89360 −0.412720 −0.206360 0.978476i \(-0.566162\pi\)
−0.206360 + 0.978476i \(0.566162\pi\)
\(90\) 0 0
\(91\) 3.15382 0.330610
\(92\) 0 0
\(93\) 1.27241 0.131943
\(94\) 0 0
\(95\) −11.0861 −1.13741
\(96\) 0 0
\(97\) 2.19229 0.222594 0.111297 0.993787i \(-0.464500\pi\)
0.111297 + 0.993787i \(0.464500\pi\)
\(98\) 0 0
\(99\) −2.23299 −0.224424
\(100\) 0 0
\(101\) −5.11489 −0.508951 −0.254476 0.967079i \(-0.581903\pi\)
−0.254476 + 0.967079i \(0.581903\pi\)
\(102\) 0 0
\(103\) −9.32277 −0.918600 −0.459300 0.888281i \(-0.651900\pi\)
−0.459300 + 0.888281i \(0.651900\pi\)
\(104\) 0 0
\(105\) −5.82490 −0.568452
\(106\) 0 0
\(107\) −1.40644 −0.135966 −0.0679829 0.997686i \(-0.521656\pi\)
−0.0679829 + 0.997686i \(0.521656\pi\)
\(108\) 0 0
\(109\) −9.46174 −0.906270 −0.453135 0.891442i \(-0.649694\pi\)
−0.453135 + 0.891442i \(0.649694\pi\)
\(110\) 0 0
\(111\) 9.78090 0.928362
\(112\) 0 0
\(113\) −12.5187 −1.17766 −0.588831 0.808256i \(-0.700412\pi\)
−0.588831 + 0.808256i \(0.700412\pi\)
\(114\) 0 0
\(115\) 22.3057 2.08002
\(116\) 0 0
\(117\) 2.75553 0.254749
\(118\) 0 0
\(119\) −6.98536 −0.640347
\(120\) 0 0
\(121\) −9.85833 −0.896212
\(122\) 0 0
\(123\) 8.45268 0.762152
\(124\) 0 0
\(125\) 8.89370 0.795477
\(126\) 0 0
\(127\) −13.2558 −1.17626 −0.588130 0.808766i \(-0.700136\pi\)
−0.588130 + 0.808766i \(0.700136\pi\)
\(128\) 0 0
\(129\) −0.874493 −0.0769948
\(130\) 0 0
\(131\) 15.4209 1.34733 0.673665 0.739037i \(-0.264719\pi\)
0.673665 + 0.739037i \(0.264719\pi\)
\(132\) 0 0
\(133\) −10.3882 −0.900773
\(134\) 0 0
\(135\) −12.3950 −1.06679
\(136\) 0 0
\(137\) 7.40081 0.632294 0.316147 0.948710i \(-0.397611\pi\)
0.316147 + 0.948710i \(0.397611\pi\)
\(138\) 0 0
\(139\) 16.9459 1.43733 0.718665 0.695357i \(-0.244754\pi\)
0.718665 + 0.695357i \(0.244754\pi\)
\(140\) 0 0
\(141\) −5.10968 −0.430313
\(142\) 0 0
\(143\) −1.40884 −0.117813
\(144\) 0 0
\(145\) 4.27217 0.354785
\(146\) 0 0
\(147\) 1.21991 0.100617
\(148\) 0 0
\(149\) 3.17258 0.259908 0.129954 0.991520i \(-0.458517\pi\)
0.129954 + 0.991520i \(0.458517\pi\)
\(150\) 0 0
\(151\) 12.9261 1.05191 0.525955 0.850512i \(-0.323708\pi\)
0.525955 + 0.850512i \(0.323708\pi\)
\(152\) 0 0
\(153\) −6.10320 −0.493415
\(154\) 0 0
\(155\) 3.40452 0.273458
\(156\) 0 0
\(157\) 18.0154 1.43779 0.718893 0.695121i \(-0.244649\pi\)
0.718893 + 0.695121i \(0.244649\pi\)
\(158\) 0 0
\(159\) 9.39888 0.745380
\(160\) 0 0
\(161\) 20.9015 1.64727
\(162\) 0 0
\(163\) 1.30715 0.102384 0.0511921 0.998689i \(-0.483698\pi\)
0.0511921 + 0.998689i \(0.483698\pi\)
\(164\) 0 0
\(165\) 2.60203 0.202568
\(166\) 0 0
\(167\) −6.02686 −0.466372 −0.233186 0.972432i \(-0.574915\pi\)
−0.233186 + 0.972432i \(0.574915\pi\)
\(168\) 0 0
\(169\) −11.2615 −0.866268
\(170\) 0 0
\(171\) −9.07632 −0.694084
\(172\) 0 0
\(173\) 2.92019 0.222018 0.111009 0.993819i \(-0.464592\pi\)
0.111009 + 0.993819i \(0.464592\pi\)
\(174\) 0 0
\(175\) −3.62579 −0.274084
\(176\) 0 0
\(177\) 7.17061 0.538976
\(178\) 0 0
\(179\) −25.3006 −1.89106 −0.945529 0.325539i \(-0.894454\pi\)
−0.945529 + 0.325539i \(0.894454\pi\)
\(180\) 0 0
\(181\) −15.8590 −1.17879 −0.589397 0.807844i \(-0.700634\pi\)
−0.589397 + 0.807844i \(0.700634\pi\)
\(182\) 0 0
\(183\) 13.6868 1.01176
\(184\) 0 0
\(185\) 26.1703 1.92408
\(186\) 0 0
\(187\) 3.12042 0.228188
\(188\) 0 0
\(189\) −11.6147 −0.844845
\(190\) 0 0
\(191\) 18.9480 1.37103 0.685515 0.728058i \(-0.259577\pi\)
0.685515 + 0.728058i \(0.259577\pi\)
\(192\) 0 0
\(193\) 18.9777 1.36605 0.683023 0.730397i \(-0.260665\pi\)
0.683023 + 0.730397i \(0.260665\pi\)
\(194\) 0 0
\(195\) −3.21093 −0.229940
\(196\) 0 0
\(197\) 19.8295 1.41279 0.706395 0.707817i \(-0.250320\pi\)
0.706395 + 0.707817i \(0.250320\pi\)
\(198\) 0 0
\(199\) 2.42332 0.171784 0.0858922 0.996304i \(-0.472626\pi\)
0.0858922 + 0.996304i \(0.472626\pi\)
\(200\) 0 0
\(201\) −6.24174 −0.440259
\(202\) 0 0
\(203\) 4.00323 0.280971
\(204\) 0 0
\(205\) 22.6164 1.57960
\(206\) 0 0
\(207\) 18.2619 1.26929
\(208\) 0 0
\(209\) 4.64050 0.320990
\(210\) 0 0
\(211\) 23.5077 1.61834 0.809168 0.587577i \(-0.199918\pi\)
0.809168 + 0.587577i \(0.199918\pi\)
\(212\) 0 0
\(213\) −2.10167 −0.144004
\(214\) 0 0
\(215\) −2.33984 −0.159576
\(216\) 0 0
\(217\) 3.19020 0.216565
\(218\) 0 0
\(219\) −5.08421 −0.343559
\(220\) 0 0
\(221\) −3.85063 −0.259022
\(222\) 0 0
\(223\) 19.3050 1.29276 0.646378 0.763018i \(-0.276283\pi\)
0.646378 + 0.763018i \(0.276283\pi\)
\(224\) 0 0
\(225\) −3.16790 −0.211193
\(226\) 0 0
\(227\) −28.4805 −1.89032 −0.945158 0.326613i \(-0.894093\pi\)
−0.945158 + 0.326613i \(0.894093\pi\)
\(228\) 0 0
\(229\) −23.2164 −1.53418 −0.767091 0.641538i \(-0.778297\pi\)
−0.767091 + 0.641538i \(0.778297\pi\)
\(230\) 0 0
\(231\) 2.43822 0.160423
\(232\) 0 0
\(233\) 3.43202 0.224839 0.112420 0.993661i \(-0.464140\pi\)
0.112420 + 0.993661i \(0.464140\pi\)
\(234\) 0 0
\(235\) −13.6717 −0.891845
\(236\) 0 0
\(237\) 4.56496 0.296526
\(238\) 0 0
\(239\) −11.9941 −0.775834 −0.387917 0.921694i \(-0.626805\pi\)
−0.387917 + 0.921694i \(0.626805\pi\)
\(240\) 0 0
\(241\) 20.2705 1.30574 0.652869 0.757471i \(-0.273565\pi\)
0.652869 + 0.757471i \(0.273565\pi\)
\(242\) 0 0
\(243\) −16.1292 −1.03469
\(244\) 0 0
\(245\) 3.26406 0.208533
\(246\) 0 0
\(247\) −5.72643 −0.364364
\(248\) 0 0
\(249\) 7.64197 0.484290
\(250\) 0 0
\(251\) −18.1571 −1.14606 −0.573032 0.819533i \(-0.694233\pi\)
−0.573032 + 0.819533i \(0.694233\pi\)
\(252\) 0 0
\(253\) −9.33688 −0.587004
\(254\) 0 0
\(255\) 7.11187 0.445362
\(256\) 0 0
\(257\) 12.3951 0.773183 0.386592 0.922251i \(-0.373652\pi\)
0.386592 + 0.922251i \(0.373652\pi\)
\(258\) 0 0
\(259\) 24.5228 1.52377
\(260\) 0 0
\(261\) 3.49767 0.216500
\(262\) 0 0
\(263\) 7.61577 0.469609 0.234804 0.972043i \(-0.424555\pi\)
0.234804 + 0.972043i \(0.424555\pi\)
\(264\) 0 0
\(265\) 25.1481 1.54484
\(266\) 0 0
\(267\) 3.71455 0.227327
\(268\) 0 0
\(269\) 10.6787 0.651092 0.325546 0.945526i \(-0.394452\pi\)
0.325546 + 0.945526i \(0.394452\pi\)
\(270\) 0 0
\(271\) 24.7202 1.50165 0.750824 0.660502i \(-0.229657\pi\)
0.750824 + 0.660502i \(0.229657\pi\)
\(272\) 0 0
\(273\) −3.00880 −0.182101
\(274\) 0 0
\(275\) 1.61967 0.0976697
\(276\) 0 0
\(277\) −7.41110 −0.445290 −0.222645 0.974900i \(-0.571469\pi\)
−0.222645 + 0.974900i \(0.571469\pi\)
\(278\) 0 0
\(279\) 2.78732 0.166872
\(280\) 0 0
\(281\) −2.61269 −0.155860 −0.0779302 0.996959i \(-0.524831\pi\)
−0.0779302 + 0.996959i \(0.524831\pi\)
\(282\) 0 0
\(283\) −7.30214 −0.434067 −0.217034 0.976164i \(-0.569638\pi\)
−0.217034 + 0.976164i \(0.569638\pi\)
\(284\) 0 0
\(285\) 10.5763 0.626489
\(286\) 0 0
\(287\) 21.1927 1.25096
\(288\) 0 0
\(289\) −8.47127 −0.498310
\(290\) 0 0
\(291\) −2.09148 −0.122605
\(292\) 0 0
\(293\) 19.7368 1.15304 0.576518 0.817084i \(-0.304411\pi\)
0.576518 + 0.817084i \(0.304411\pi\)
\(294\) 0 0
\(295\) 19.1861 1.11706
\(296\) 0 0
\(297\) 5.18838 0.301060
\(298\) 0 0
\(299\) 11.5218 0.666323
\(300\) 0 0
\(301\) −2.19254 −0.126376
\(302\) 0 0
\(303\) 4.87969 0.280331
\(304\) 0 0
\(305\) 36.6211 2.09692
\(306\) 0 0
\(307\) −15.0292 −0.857762 −0.428881 0.903361i \(-0.641092\pi\)
−0.428881 + 0.903361i \(0.641092\pi\)
\(308\) 0 0
\(309\) 8.89408 0.505966
\(310\) 0 0
\(311\) 12.7086 0.720637 0.360318 0.932829i \(-0.382668\pi\)
0.360318 + 0.932829i \(0.382668\pi\)
\(312\) 0 0
\(313\) 15.0846 0.852634 0.426317 0.904574i \(-0.359811\pi\)
0.426317 + 0.904574i \(0.359811\pi\)
\(314\) 0 0
\(315\) −12.7599 −0.718940
\(316\) 0 0
\(317\) −5.60827 −0.314992 −0.157496 0.987520i \(-0.550342\pi\)
−0.157496 + 0.987520i \(0.550342\pi\)
\(318\) 0 0
\(319\) −1.78827 −0.100124
\(320\) 0 0
\(321\) 1.34177 0.0748902
\(322\) 0 0
\(323\) 12.6834 0.705725
\(324\) 0 0
\(325\) −1.99869 −0.110867
\(326\) 0 0
\(327\) 9.02665 0.499175
\(328\) 0 0
\(329\) −12.8111 −0.706296
\(330\) 0 0
\(331\) −13.4878 −0.741358 −0.370679 0.928761i \(-0.620875\pi\)
−0.370679 + 0.928761i \(0.620875\pi\)
\(332\) 0 0
\(333\) 21.4259 1.17413
\(334\) 0 0
\(335\) −16.7007 −0.912458
\(336\) 0 0
\(337\) 6.68255 0.364022 0.182011 0.983297i \(-0.441739\pi\)
0.182011 + 0.983297i \(0.441739\pi\)
\(338\) 0 0
\(339\) 11.9431 0.648658
\(340\) 0 0
\(341\) −1.42509 −0.0771729
\(342\) 0 0
\(343\) 19.8020 1.06921
\(344\) 0 0
\(345\) −21.2800 −1.14568
\(346\) 0 0
\(347\) 9.49596 0.509770 0.254885 0.966971i \(-0.417962\pi\)
0.254885 + 0.966971i \(0.417962\pi\)
\(348\) 0 0
\(349\) 11.0510 0.591547 0.295773 0.955258i \(-0.404423\pi\)
0.295773 + 0.955258i \(0.404423\pi\)
\(350\) 0 0
\(351\) −6.40252 −0.341741
\(352\) 0 0
\(353\) 7.33474 0.390389 0.195194 0.980765i \(-0.437466\pi\)
0.195194 + 0.980765i \(0.437466\pi\)
\(354\) 0 0
\(355\) −5.62334 −0.298456
\(356\) 0 0
\(357\) 6.66415 0.352704
\(358\) 0 0
\(359\) −19.3176 −1.01954 −0.509771 0.860310i \(-0.670270\pi\)
−0.509771 + 0.860310i \(0.670270\pi\)
\(360\) 0 0
\(361\) −0.137961 −0.00726108
\(362\) 0 0
\(363\) 9.40501 0.493635
\(364\) 0 0
\(365\) −13.6036 −0.712044
\(366\) 0 0
\(367\) 8.81863 0.460329 0.230164 0.973152i \(-0.426074\pi\)
0.230164 + 0.973152i \(0.426074\pi\)
\(368\) 0 0
\(369\) 18.5163 0.963920
\(370\) 0 0
\(371\) 23.5650 1.22343
\(372\) 0 0
\(373\) −34.4940 −1.78603 −0.893017 0.450024i \(-0.851416\pi\)
−0.893017 + 0.450024i \(0.851416\pi\)
\(374\) 0 0
\(375\) −8.48474 −0.438150
\(376\) 0 0
\(377\) 2.20675 0.113653
\(378\) 0 0
\(379\) −11.9305 −0.612829 −0.306414 0.951898i \(-0.599129\pi\)
−0.306414 + 0.951898i \(0.599129\pi\)
\(380\) 0 0
\(381\) 12.6462 0.647886
\(382\) 0 0
\(383\) 1.98209 0.101280 0.0506400 0.998717i \(-0.483874\pi\)
0.0506400 + 0.998717i \(0.483874\pi\)
\(384\) 0 0
\(385\) 6.52384 0.332486
\(386\) 0 0
\(387\) −1.91565 −0.0973779
\(388\) 0 0
\(389\) −25.5762 −1.29677 −0.648383 0.761314i \(-0.724555\pi\)
−0.648383 + 0.761314i \(0.724555\pi\)
\(390\) 0 0
\(391\) −25.5195 −1.29058
\(392\) 0 0
\(393\) −14.7118 −0.742112
\(394\) 0 0
\(395\) 12.2142 0.614565
\(396\) 0 0
\(397\) 23.5683 1.18286 0.591430 0.806356i \(-0.298564\pi\)
0.591430 + 0.806356i \(0.298564\pi\)
\(398\) 0 0
\(399\) 9.91054 0.496147
\(400\) 0 0
\(401\) 9.98546 0.498650 0.249325 0.968420i \(-0.419791\pi\)
0.249325 + 0.968420i \(0.419791\pi\)
\(402\) 0 0
\(403\) 1.75857 0.0876008
\(404\) 0 0
\(405\) −4.17874 −0.207643
\(406\) 0 0
\(407\) −10.9545 −0.542996
\(408\) 0 0
\(409\) 25.0108 1.23670 0.618352 0.785902i \(-0.287801\pi\)
0.618352 + 0.785902i \(0.287801\pi\)
\(410\) 0 0
\(411\) −7.06050 −0.348269
\(412\) 0 0
\(413\) 17.9782 0.884652
\(414\) 0 0
\(415\) 20.4472 1.00372
\(416\) 0 0
\(417\) −16.1666 −0.791683
\(418\) 0 0
\(419\) 20.5693 1.00488 0.502439 0.864613i \(-0.332436\pi\)
0.502439 + 0.864613i \(0.332436\pi\)
\(420\) 0 0
\(421\) 25.6881 1.25196 0.625980 0.779839i \(-0.284699\pi\)
0.625980 + 0.779839i \(0.284699\pi\)
\(422\) 0 0
\(423\) −11.1932 −0.544231
\(424\) 0 0
\(425\) 4.42688 0.214735
\(426\) 0 0
\(427\) 34.3157 1.66065
\(428\) 0 0
\(429\) 1.34405 0.0648915
\(430\) 0 0
\(431\) −2.59952 −0.125214 −0.0626072 0.998038i \(-0.519942\pi\)
−0.0626072 + 0.998038i \(0.519942\pi\)
\(432\) 0 0
\(433\) −27.6940 −1.33089 −0.665445 0.746447i \(-0.731758\pi\)
−0.665445 + 0.746447i \(0.731758\pi\)
\(434\) 0 0
\(435\) −4.07572 −0.195416
\(436\) 0 0
\(437\) −37.9512 −1.81545
\(438\) 0 0
\(439\) −5.33492 −0.254622 −0.127311 0.991863i \(-0.540635\pi\)
−0.127311 + 0.991863i \(0.540635\pi\)
\(440\) 0 0
\(441\) 2.67232 0.127253
\(442\) 0 0
\(443\) 0.872865 0.0414710 0.0207355 0.999785i \(-0.493399\pi\)
0.0207355 + 0.999785i \(0.493399\pi\)
\(444\) 0 0
\(445\) 9.93885 0.471147
\(446\) 0 0
\(447\) −3.02669 −0.143158
\(448\) 0 0
\(449\) −34.1930 −1.61367 −0.806833 0.590780i \(-0.798820\pi\)
−0.806833 + 0.590780i \(0.798820\pi\)
\(450\) 0 0
\(451\) −9.46693 −0.445781
\(452\) 0 0
\(453\) −12.3317 −0.579394
\(454\) 0 0
\(455\) −8.05049 −0.377413
\(456\) 0 0
\(457\) 10.7180 0.501367 0.250684 0.968069i \(-0.419345\pi\)
0.250684 + 0.968069i \(0.419345\pi\)
\(458\) 0 0
\(459\) 14.1809 0.661907
\(460\) 0 0
\(461\) −31.3775 −1.46140 −0.730698 0.682700i \(-0.760805\pi\)
−0.730698 + 0.682700i \(0.760805\pi\)
\(462\) 0 0
\(463\) 3.67678 0.170875 0.0854373 0.996344i \(-0.472771\pi\)
0.0854373 + 0.996344i \(0.472771\pi\)
\(464\) 0 0
\(465\) −3.24797 −0.150621
\(466\) 0 0
\(467\) −39.9647 −1.84935 −0.924673 0.380762i \(-0.875662\pi\)
−0.924673 + 0.380762i \(0.875662\pi\)
\(468\) 0 0
\(469\) −15.6494 −0.722621
\(470\) 0 0
\(471\) −17.1870 −0.791935
\(472\) 0 0
\(473\) 0.979425 0.0450340
\(474\) 0 0
\(475\) 6.58339 0.302067
\(476\) 0 0
\(477\) 20.5890 0.942707
\(478\) 0 0
\(479\) 7.51549 0.343392 0.171696 0.985150i \(-0.445075\pi\)
0.171696 + 0.985150i \(0.445075\pi\)
\(480\) 0 0
\(481\) 13.5180 0.616368
\(482\) 0 0
\(483\) −19.9404 −0.907319
\(484\) 0 0
\(485\) −5.59608 −0.254105
\(486\) 0 0
\(487\) 23.6836 1.07321 0.536603 0.843835i \(-0.319707\pi\)
0.536603 + 0.843835i \(0.319707\pi\)
\(488\) 0 0
\(489\) −1.24705 −0.0563934
\(490\) 0 0
\(491\) −15.2815 −0.689642 −0.344821 0.938668i \(-0.612060\pi\)
−0.344821 + 0.938668i \(0.612060\pi\)
\(492\) 0 0
\(493\) −4.88771 −0.220131
\(494\) 0 0
\(495\) 5.69996 0.256194
\(496\) 0 0
\(497\) −5.26934 −0.236362
\(498\) 0 0
\(499\) −36.0183 −1.61240 −0.806200 0.591643i \(-0.798479\pi\)
−0.806200 + 0.591643i \(0.798479\pi\)
\(500\) 0 0
\(501\) 5.74972 0.256879
\(502\) 0 0
\(503\) −1.00000 −0.0445878
\(504\) 0 0
\(505\) 13.0564 0.581000
\(506\) 0 0
\(507\) 10.7436 0.477142
\(508\) 0 0
\(509\) 16.9664 0.752024 0.376012 0.926615i \(-0.377295\pi\)
0.376012 + 0.926615i \(0.377295\pi\)
\(510\) 0 0
\(511\) −12.7472 −0.563903
\(512\) 0 0
\(513\) 21.0890 0.931101
\(514\) 0 0
\(515\) 23.7975 1.04864
\(516\) 0 0
\(517\) 5.72280 0.251689
\(518\) 0 0
\(519\) −2.78591 −0.122288
\(520\) 0 0
\(521\) −4.30395 −0.188559 −0.0942797 0.995546i \(-0.530055\pi\)
−0.0942797 + 0.995546i \(0.530055\pi\)
\(522\) 0 0
\(523\) −17.3360 −0.758050 −0.379025 0.925386i \(-0.623741\pi\)
−0.379025 + 0.925386i \(0.623741\pi\)
\(524\) 0 0
\(525\) 3.45906 0.150966
\(526\) 0 0
\(527\) −3.89505 −0.169671
\(528\) 0 0
\(529\) 53.3592 2.31997
\(530\) 0 0
\(531\) 15.7078 0.681662
\(532\) 0 0
\(533\) 11.6823 0.506017
\(534\) 0 0
\(535\) 3.59010 0.155214
\(536\) 0 0
\(537\) 24.1372 1.04160
\(538\) 0 0
\(539\) −1.36629 −0.0588504
\(540\) 0 0
\(541\) −13.3268 −0.572962 −0.286481 0.958086i \(-0.592486\pi\)
−0.286481 + 0.958086i \(0.592486\pi\)
\(542\) 0 0
\(543\) 15.1298 0.649281
\(544\) 0 0
\(545\) 24.1522 1.03457
\(546\) 0 0
\(547\) 15.7111 0.671759 0.335880 0.941905i \(-0.390966\pi\)
0.335880 + 0.941905i \(0.390966\pi\)
\(548\) 0 0
\(549\) 29.9821 1.27960
\(550\) 0 0
\(551\) −7.26871 −0.309658
\(552\) 0 0
\(553\) 11.4453 0.486704
\(554\) 0 0
\(555\) −24.9669 −1.05979
\(556\) 0 0
\(557\) 10.2458 0.434128 0.217064 0.976157i \(-0.430352\pi\)
0.217064 + 0.976157i \(0.430352\pi\)
\(558\) 0 0
\(559\) −1.20862 −0.0511192
\(560\) 0 0
\(561\) −2.97693 −0.125686
\(562\) 0 0
\(563\) 25.2671 1.06488 0.532440 0.846468i \(-0.321275\pi\)
0.532440 + 0.846468i \(0.321275\pi\)
\(564\) 0 0
\(565\) 31.9555 1.34438
\(566\) 0 0
\(567\) −3.91567 −0.164443
\(568\) 0 0
\(569\) 8.10550 0.339800 0.169900 0.985461i \(-0.445655\pi\)
0.169900 + 0.985461i \(0.445655\pi\)
\(570\) 0 0
\(571\) −0.372072 −0.0155707 −0.00778536 0.999970i \(-0.502478\pi\)
−0.00778536 + 0.999970i \(0.502478\pi\)
\(572\) 0 0
\(573\) −18.0767 −0.755166
\(574\) 0 0
\(575\) −13.2460 −0.552398
\(576\) 0 0
\(577\) 14.0765 0.586014 0.293007 0.956110i \(-0.405344\pi\)
0.293007 + 0.956110i \(0.405344\pi\)
\(578\) 0 0
\(579\) −18.1051 −0.752420
\(580\) 0 0
\(581\) 19.1600 0.794892
\(582\) 0 0
\(583\) −10.5267 −0.435970
\(584\) 0 0
\(585\) −7.03382 −0.290812
\(586\) 0 0
\(587\) −39.4904 −1.62994 −0.814972 0.579500i \(-0.803248\pi\)
−0.814972 + 0.579500i \(0.803248\pi\)
\(588\) 0 0
\(589\) −5.79249 −0.238675
\(590\) 0 0
\(591\) −18.9176 −0.778168
\(592\) 0 0
\(593\) 1.45894 0.0599113 0.0299557 0.999551i \(-0.490463\pi\)
0.0299557 + 0.999551i \(0.490463\pi\)
\(594\) 0 0
\(595\) 17.8310 0.730998
\(596\) 0 0
\(597\) −2.31188 −0.0946191
\(598\) 0 0
\(599\) 18.5660 0.758587 0.379293 0.925276i \(-0.376167\pi\)
0.379293 + 0.925276i \(0.376167\pi\)
\(600\) 0 0
\(601\) 7.95388 0.324446 0.162223 0.986754i \(-0.448134\pi\)
0.162223 + 0.986754i \(0.448134\pi\)
\(602\) 0 0
\(603\) −13.6731 −0.556810
\(604\) 0 0
\(605\) 25.1645 1.02308
\(606\) 0 0
\(607\) 23.2908 0.945344 0.472672 0.881238i \(-0.343290\pi\)
0.472672 + 0.881238i \(0.343290\pi\)
\(608\) 0 0
\(609\) −3.81914 −0.154760
\(610\) 0 0
\(611\) −7.06200 −0.285698
\(612\) 0 0
\(613\) −38.4507 −1.55301 −0.776504 0.630112i \(-0.783009\pi\)
−0.776504 + 0.630112i \(0.783009\pi\)
\(614\) 0 0
\(615\) −21.5764 −0.870046
\(616\) 0 0
\(617\) 34.9024 1.40512 0.702559 0.711625i \(-0.252041\pi\)
0.702559 + 0.711625i \(0.252041\pi\)
\(618\) 0 0
\(619\) −35.4749 −1.42586 −0.712928 0.701238i \(-0.752631\pi\)
−0.712928 + 0.701238i \(0.752631\pi\)
\(620\) 0 0
\(621\) −42.4318 −1.70273
\(622\) 0 0
\(623\) 9.31317 0.373124
\(624\) 0 0
\(625\) −30.2814 −1.21126
\(626\) 0 0
\(627\) −4.42712 −0.176802
\(628\) 0 0
\(629\) −29.9409 −1.19382
\(630\) 0 0
\(631\) −22.8627 −0.910150 −0.455075 0.890453i \(-0.650388\pi\)
−0.455075 + 0.890453i \(0.650388\pi\)
\(632\) 0 0
\(633\) −22.4267 −0.891382
\(634\) 0 0
\(635\) 33.8369 1.34278
\(636\) 0 0
\(637\) 1.68602 0.0668025
\(638\) 0 0
\(639\) −4.60389 −0.182127
\(640\) 0 0
\(641\) 35.3265 1.39531 0.697657 0.716432i \(-0.254226\pi\)
0.697657 + 0.716432i \(0.254226\pi\)
\(642\) 0 0
\(643\) 18.3940 0.725389 0.362695 0.931908i \(-0.381857\pi\)
0.362695 + 0.931908i \(0.381857\pi\)
\(644\) 0 0
\(645\) 2.23224 0.0878945
\(646\) 0 0
\(647\) 3.25083 0.127803 0.0639016 0.997956i \(-0.479646\pi\)
0.0639016 + 0.997956i \(0.479646\pi\)
\(648\) 0 0
\(649\) −8.03103 −0.315246
\(650\) 0 0
\(651\) −3.04350 −0.119284
\(652\) 0 0
\(653\) 0.0204648 0.000800848 0 0.000400424 1.00000i \(-0.499873\pi\)
0.000400424 1.00000i \(0.499873\pi\)
\(654\) 0 0
\(655\) −39.3636 −1.53806
\(656\) 0 0
\(657\) −11.1374 −0.434511
\(658\) 0 0
\(659\) 22.3422 0.870327 0.435164 0.900351i \(-0.356691\pi\)
0.435164 + 0.900351i \(0.356691\pi\)
\(660\) 0 0
\(661\) −28.1750 −1.09588 −0.547940 0.836518i \(-0.684588\pi\)
−0.547940 + 0.836518i \(0.684588\pi\)
\(662\) 0 0
\(663\) 3.67357 0.142670
\(664\) 0 0
\(665\) 26.5171 1.02829
\(666\) 0 0
\(667\) 14.6249 0.566280
\(668\) 0 0
\(669\) −18.4172 −0.712052
\(670\) 0 0
\(671\) −15.3291 −0.591773
\(672\) 0 0
\(673\) 9.84568 0.379523 0.189762 0.981830i \(-0.439229\pi\)
0.189762 + 0.981830i \(0.439229\pi\)
\(674\) 0 0
\(675\) 7.36066 0.283312
\(676\) 0 0
\(677\) 12.6973 0.487997 0.243999 0.969776i \(-0.421541\pi\)
0.243999 + 0.969776i \(0.421541\pi\)
\(678\) 0 0
\(679\) −5.24379 −0.201238
\(680\) 0 0
\(681\) 27.1709 1.04119
\(682\) 0 0
\(683\) 0.981954 0.0375734 0.0187867 0.999824i \(-0.494020\pi\)
0.0187867 + 0.999824i \(0.494020\pi\)
\(684\) 0 0
\(685\) −18.8914 −0.721805
\(686\) 0 0
\(687\) 22.1488 0.845030
\(688\) 0 0
\(689\) 12.9900 0.494881
\(690\) 0 0
\(691\) 7.45558 0.283624 0.141812 0.989894i \(-0.454707\pi\)
0.141812 + 0.989894i \(0.454707\pi\)
\(692\) 0 0
\(693\) 5.34113 0.202893
\(694\) 0 0
\(695\) −43.2563 −1.64080
\(696\) 0 0
\(697\) −25.8750 −0.980086
\(698\) 0 0
\(699\) −3.27420 −0.123842
\(700\) 0 0
\(701\) −46.7562 −1.76596 −0.882978 0.469414i \(-0.844465\pi\)
−0.882978 + 0.469414i \(0.844465\pi\)
\(702\) 0 0
\(703\) −44.5264 −1.67934
\(704\) 0 0
\(705\) 13.0431 0.491230
\(706\) 0 0
\(707\) 12.2344 0.460123
\(708\) 0 0
\(709\) 30.8689 1.15931 0.579654 0.814863i \(-0.303188\pi\)
0.579654 + 0.814863i \(0.303188\pi\)
\(710\) 0 0
\(711\) 9.99991 0.375026
\(712\) 0 0
\(713\) 11.6547 0.436472
\(714\) 0 0
\(715\) 3.59622 0.134491
\(716\) 0 0
\(717\) 11.4426 0.427331
\(718\) 0 0
\(719\) −9.08081 −0.338657 −0.169329 0.985560i \(-0.554160\pi\)
−0.169329 + 0.985560i \(0.554160\pi\)
\(720\) 0 0
\(721\) 22.2993 0.830471
\(722\) 0 0
\(723\) −19.3384 −0.719203
\(724\) 0 0
\(725\) −2.53699 −0.0942214
\(726\) 0 0
\(727\) 50.0727 1.85709 0.928546 0.371217i \(-0.121059\pi\)
0.928546 + 0.371217i \(0.121059\pi\)
\(728\) 0 0
\(729\) 10.4764 0.388014
\(730\) 0 0
\(731\) 2.67696 0.0990111
\(732\) 0 0
\(733\) 44.4826 1.64300 0.821502 0.570206i \(-0.193137\pi\)
0.821502 + 0.570206i \(0.193137\pi\)
\(734\) 0 0
\(735\) −3.11397 −0.114860
\(736\) 0 0
\(737\) 6.99070 0.257506
\(738\) 0 0
\(739\) −1.87749 −0.0690646 −0.0345323 0.999404i \(-0.510994\pi\)
−0.0345323 + 0.999404i \(0.510994\pi\)
\(740\) 0 0
\(741\) 5.46311 0.200692
\(742\) 0 0
\(743\) 15.6546 0.574313 0.287157 0.957884i \(-0.407290\pi\)
0.287157 + 0.957884i \(0.407290\pi\)
\(744\) 0 0
\(745\) −8.09838 −0.296702
\(746\) 0 0
\(747\) 16.7404 0.612498
\(748\) 0 0
\(749\) 3.36410 0.122921
\(750\) 0 0
\(751\) −10.8693 −0.396625 −0.198313 0.980139i \(-0.563546\pi\)
−0.198313 + 0.980139i \(0.563546\pi\)
\(752\) 0 0
\(753\) 17.3221 0.631254
\(754\) 0 0
\(755\) −32.9953 −1.20082
\(756\) 0 0
\(757\) 6.20265 0.225439 0.112720 0.993627i \(-0.464044\pi\)
0.112720 + 0.993627i \(0.464044\pi\)
\(758\) 0 0
\(759\) 8.90753 0.323323
\(760\) 0 0
\(761\) −44.6243 −1.61763 −0.808816 0.588062i \(-0.799891\pi\)
−0.808816 + 0.588062i \(0.799891\pi\)
\(762\) 0 0
\(763\) 22.6317 0.819324
\(764\) 0 0
\(765\) 15.5791 0.563265
\(766\) 0 0
\(767\) 9.91038 0.357843
\(768\) 0 0
\(769\) −28.7859 −1.03805 −0.519023 0.854761i \(-0.673704\pi\)
−0.519023 + 0.854761i \(0.673704\pi\)
\(770\) 0 0
\(771\) −11.8251 −0.425871
\(772\) 0 0
\(773\) 2.49010 0.0895628 0.0447814 0.998997i \(-0.485741\pi\)
0.0447814 + 0.998997i \(0.485741\pi\)
\(774\) 0 0
\(775\) −2.02174 −0.0726232
\(776\) 0 0
\(777\) −23.3951 −0.839296
\(778\) 0 0
\(779\) −38.4798 −1.37868
\(780\) 0 0
\(781\) 2.35386 0.0842276
\(782\) 0 0
\(783\) −8.12689 −0.290431
\(784\) 0 0
\(785\) −45.9864 −1.64133
\(786\) 0 0
\(787\) 10.0564 0.358472 0.179236 0.983806i \(-0.442637\pi\)
0.179236 + 0.983806i \(0.442637\pi\)
\(788\) 0 0
\(789\) −7.26557 −0.258661
\(790\) 0 0
\(791\) 29.9438 1.06468
\(792\) 0 0
\(793\) 18.9163 0.671737
\(794\) 0 0
\(795\) −23.9917 −0.850899
\(796\) 0 0
\(797\) −2.25957 −0.0800380 −0.0400190 0.999199i \(-0.512742\pi\)
−0.0400190 + 0.999199i \(0.512742\pi\)
\(798\) 0 0
\(799\) 15.6416 0.553359
\(800\) 0 0
\(801\) 8.13704 0.287508
\(802\) 0 0
\(803\) 5.69428 0.200947
\(804\) 0 0
\(805\) −53.3535 −1.88046
\(806\) 0 0
\(807\) −10.1877 −0.358622
\(808\) 0 0
\(809\) 56.3694 1.98184 0.990922 0.134441i \(-0.0429240\pi\)
0.990922 + 0.134441i \(0.0429240\pi\)
\(810\) 0 0
\(811\) 24.2213 0.850526 0.425263 0.905070i \(-0.360182\pi\)
0.425263 + 0.905070i \(0.360182\pi\)
\(812\) 0 0
\(813\) −23.5835 −0.827110
\(814\) 0 0
\(815\) −3.33666 −0.116878
\(816\) 0 0
\(817\) 3.98102 0.139278
\(818\) 0 0
\(819\) −6.59102 −0.230309
\(820\) 0 0
\(821\) −37.7502 −1.31749 −0.658745 0.752366i \(-0.728912\pi\)
−0.658745 + 0.752366i \(0.728912\pi\)
\(822\) 0 0
\(823\) −19.4724 −0.678764 −0.339382 0.940649i \(-0.610218\pi\)
−0.339382 + 0.940649i \(0.610218\pi\)
\(824\) 0 0
\(825\) −1.54519 −0.0537966
\(826\) 0 0
\(827\) 42.3298 1.47195 0.735975 0.677009i \(-0.236724\pi\)
0.735975 + 0.677009i \(0.236724\pi\)
\(828\) 0 0
\(829\) −10.0806 −0.350113 −0.175056 0.984558i \(-0.556011\pi\)
−0.175056 + 0.984558i \(0.556011\pi\)
\(830\) 0 0
\(831\) 7.07031 0.245267
\(832\) 0 0
\(833\) −3.73435 −0.129387
\(834\) 0 0
\(835\) 15.3843 0.532394
\(836\) 0 0
\(837\) −6.47638 −0.223856
\(838\) 0 0
\(839\) −2.69322 −0.0929803 −0.0464902 0.998919i \(-0.514804\pi\)
−0.0464902 + 0.998919i \(0.514804\pi\)
\(840\) 0 0
\(841\) −26.1989 −0.903411
\(842\) 0 0
\(843\) 2.49255 0.0858481
\(844\) 0 0
\(845\) 28.7462 0.988900
\(846\) 0 0
\(847\) 23.5803 0.810230
\(848\) 0 0
\(849\) 6.96636 0.239085
\(850\) 0 0
\(851\) 89.5888 3.07106
\(852\) 0 0
\(853\) −10.3274 −0.353602 −0.176801 0.984247i \(-0.556575\pi\)
−0.176801 + 0.984247i \(0.556575\pi\)
\(854\) 0 0
\(855\) 23.1684 0.792341
\(856\) 0 0
\(857\) 14.8792 0.508262 0.254131 0.967170i \(-0.418210\pi\)
0.254131 + 0.967170i \(0.418210\pi\)
\(858\) 0 0
\(859\) 43.4573 1.48274 0.741371 0.671095i \(-0.234176\pi\)
0.741371 + 0.671095i \(0.234176\pi\)
\(860\) 0 0
\(861\) −20.2181 −0.689033
\(862\) 0 0
\(863\) 40.3326 1.37294 0.686469 0.727159i \(-0.259160\pi\)
0.686469 + 0.727159i \(0.259160\pi\)
\(864\) 0 0
\(865\) −7.45413 −0.253448
\(866\) 0 0
\(867\) 8.08173 0.274470
\(868\) 0 0
\(869\) −5.11271 −0.173437
\(870\) 0 0
\(871\) −8.62661 −0.292301
\(872\) 0 0
\(873\) −4.58157 −0.155063
\(874\) 0 0
\(875\) −21.2730 −0.719160
\(876\) 0 0
\(877\) 19.4312 0.656145 0.328072 0.944653i \(-0.393601\pi\)
0.328072 + 0.944653i \(0.393601\pi\)
\(878\) 0 0
\(879\) −18.8292 −0.635094
\(880\) 0 0
\(881\) −57.6037 −1.94072 −0.970359 0.241669i \(-0.922305\pi\)
−0.970359 + 0.241669i \(0.922305\pi\)
\(882\) 0 0
\(883\) −25.9648 −0.873786 −0.436893 0.899513i \(-0.643921\pi\)
−0.436893 + 0.899513i \(0.643921\pi\)
\(884\) 0 0
\(885\) −18.3038 −0.615276
\(886\) 0 0
\(887\) 41.4058 1.39027 0.695135 0.718879i \(-0.255345\pi\)
0.695135 + 0.718879i \(0.255345\pi\)
\(888\) 0 0
\(889\) 31.7068 1.06341
\(890\) 0 0
\(891\) 1.74916 0.0585992
\(892\) 0 0
\(893\) 23.2612 0.778407
\(894\) 0 0
\(895\) 64.5828 2.15876
\(896\) 0 0
\(897\) −10.9920 −0.367012
\(898\) 0 0
\(899\) 2.23221 0.0744482
\(900\) 0 0
\(901\) −28.7715 −0.958518
\(902\) 0 0
\(903\) 2.09172 0.0696080
\(904\) 0 0
\(905\) 40.4820 1.34567
\(906\) 0 0
\(907\) 43.8161 1.45489 0.727444 0.686167i \(-0.240708\pi\)
0.727444 + 0.686167i \(0.240708\pi\)
\(908\) 0 0
\(909\) 10.6894 0.354544
\(910\) 0 0
\(911\) 11.9129 0.394692 0.197346 0.980334i \(-0.436768\pi\)
0.197346 + 0.980334i \(0.436768\pi\)
\(912\) 0 0
\(913\) −8.55894 −0.283260
\(914\) 0 0
\(915\) −34.9371 −1.15499
\(916\) 0 0
\(917\) −36.8856 −1.21807
\(918\) 0 0
\(919\) 25.8875 0.853949 0.426975 0.904264i \(-0.359579\pi\)
0.426975 + 0.904264i \(0.359579\pi\)
\(920\) 0 0
\(921\) 14.3381 0.472457
\(922\) 0 0
\(923\) −2.90468 −0.0956088
\(924\) 0 0
\(925\) −15.5410 −0.510984
\(926\) 0 0
\(927\) 19.4832 0.639913
\(928\) 0 0
\(929\) 30.0027 0.984357 0.492179 0.870494i \(-0.336201\pi\)
0.492179 + 0.870494i \(0.336201\pi\)
\(930\) 0 0
\(931\) −5.55350 −0.182009
\(932\) 0 0
\(933\) −12.1242 −0.396928
\(934\) 0 0
\(935\) −7.96523 −0.260491
\(936\) 0 0
\(937\) −49.8486 −1.62848 −0.814241 0.580527i \(-0.802847\pi\)
−0.814241 + 0.580527i \(0.802847\pi\)
\(938\) 0 0
\(939\) −14.3910 −0.469632
\(940\) 0 0
\(941\) −50.5492 −1.64786 −0.823929 0.566693i \(-0.808222\pi\)
−0.823929 + 0.566693i \(0.808222\pi\)
\(942\) 0 0
\(943\) 77.4229 2.52124
\(944\) 0 0
\(945\) 29.6479 0.964445
\(946\) 0 0
\(947\) −16.3095 −0.529987 −0.264994 0.964250i \(-0.585370\pi\)
−0.264994 + 0.964250i \(0.585370\pi\)
\(948\) 0 0
\(949\) −7.02680 −0.228100
\(950\) 0 0
\(951\) 5.35038 0.173498
\(952\) 0 0
\(953\) −38.5304 −1.24812 −0.624061 0.781375i \(-0.714518\pi\)
−0.624061 + 0.781375i \(0.714518\pi\)
\(954\) 0 0
\(955\) −48.3670 −1.56512
\(956\) 0 0
\(957\) 1.70604 0.0551485
\(958\) 0 0
\(959\) −17.7022 −0.571633
\(960\) 0 0
\(961\) −29.2211 −0.942617
\(962\) 0 0
\(963\) 2.93926 0.0947162
\(964\) 0 0
\(965\) −48.4428 −1.55943
\(966\) 0 0
\(967\) 30.1971 0.971074 0.485537 0.874216i \(-0.338624\pi\)
0.485537 + 0.874216i \(0.338624\pi\)
\(968\) 0 0
\(969\) −12.1002 −0.388714
\(970\) 0 0
\(971\) −51.4193 −1.65012 −0.825062 0.565042i \(-0.808860\pi\)
−0.825062 + 0.565042i \(0.808860\pi\)
\(972\) 0 0
\(973\) −40.5332 −1.29943
\(974\) 0 0
\(975\) 1.90678 0.0610659
\(976\) 0 0
\(977\) −48.8201 −1.56189 −0.780946 0.624598i \(-0.785263\pi\)
−0.780946 + 0.624598i \(0.785263\pi\)
\(978\) 0 0
\(979\) −4.16027 −0.132963
\(980\) 0 0
\(981\) 19.7736 0.631324
\(982\) 0 0
\(983\) 15.5739 0.496731 0.248365 0.968666i \(-0.420107\pi\)
0.248365 + 0.968666i \(0.420107\pi\)
\(984\) 0 0
\(985\) −50.6170 −1.61279
\(986\) 0 0
\(987\) 12.2220 0.389029
\(988\) 0 0
\(989\) −8.00997 −0.254702
\(990\) 0 0
\(991\) 26.9433 0.855883 0.427942 0.903806i \(-0.359239\pi\)
0.427942 + 0.903806i \(0.359239\pi\)
\(992\) 0 0
\(993\) 12.8676 0.408341
\(994\) 0 0
\(995\) −6.18579 −0.196103
\(996\) 0 0
\(997\) 40.2604 1.27506 0.637530 0.770426i \(-0.279956\pi\)
0.637530 + 0.770426i \(0.279956\pi\)
\(998\) 0 0
\(999\) −49.7833 −1.57508
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 8048.2.a.t.1.7 21
4.3 odd 2 2012.2.a.a.1.15 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2012.2.a.a.1.15 21 4.3 odd 2
8048.2.a.t.1.7 21 1.1 even 1 trivial