Properties

Label 4028.2.a.c.1.12
Level $4028$
Weight $2$
Character 4028.1
Self dual yes
Analytic conductor $32.164$
Analytic rank $1$
Dimension $19$
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] = [4028,2,Mod(1,4028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4028, 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("4028.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4028.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: \(32.1637419342\)
Analytic rank: \(1\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 5 x^{18} - 27 x^{17} + 161 x^{16} + 253 x^{15} - 2103 x^{14} - 683 x^{13} + 14442 x^{12} - 4144 x^{11} - 56325 x^{10} + 37245 x^{9} + 124233 x^{8} - 117486 x^{7} + \cdots - 4088 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(-0.440747\) of defining polynomial
Character \(\chi\) \(=\) 4028.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.440747 q^{3} +0.359379 q^{5} +0.921983 q^{7} -2.80574 q^{9} +O(q^{10})\) \(q+0.440747 q^{3} +0.359379 q^{5} +0.921983 q^{7} -2.80574 q^{9} +1.53021 q^{11} -2.01271 q^{13} +0.158396 q^{15} -1.43213 q^{17} +1.00000 q^{19} +0.406362 q^{21} +4.72144 q^{23} -4.87085 q^{25} -2.55887 q^{27} -7.88231 q^{29} +1.20119 q^{31} +0.674435 q^{33} +0.331342 q^{35} -8.80918 q^{37} -0.887096 q^{39} -4.83669 q^{41} +5.59135 q^{43} -1.00833 q^{45} +12.1547 q^{47} -6.14995 q^{49} -0.631208 q^{51} -1.00000 q^{53} +0.549925 q^{55} +0.440747 q^{57} -12.3656 q^{59} -0.555852 q^{61} -2.58685 q^{63} -0.723326 q^{65} +5.72210 q^{67} +2.08096 q^{69} -8.58675 q^{71} +11.9477 q^{73} -2.14681 q^{75} +1.41083 q^{77} -11.0735 q^{79} +7.28941 q^{81} -4.92651 q^{83} -0.514679 q^{85} -3.47411 q^{87} -10.1645 q^{89} -1.85568 q^{91} +0.529419 q^{93} +0.359379 q^{95} +0.927087 q^{97} -4.29337 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q - 5 q^{3} - 5 q^{5} - 10 q^{7} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q - 5 q^{3} - 5 q^{5} - 10 q^{7} + 22 q^{9} - 3 q^{11} - 23 q^{13} - 18 q^{15} - 7 q^{17} + 19 q^{19} - 4 q^{21} - 6 q^{23} + 18 q^{25} - 17 q^{27} - 4 q^{29} - 30 q^{31} - 10 q^{33} - q^{35} - 31 q^{37} + 5 q^{39} - 15 q^{41} - 29 q^{43} + 6 q^{45} - 18 q^{47} + 23 q^{49} - 5 q^{51} - 19 q^{53} - 19 q^{55} - 5 q^{57} + 8 q^{59} - 4 q^{61} - 64 q^{63} - 26 q^{65} - 62 q^{67} + 3 q^{69} - 17 q^{71} + q^{73} - 40 q^{75} - 14 q^{77} - 28 q^{79} + 11 q^{81} + 4 q^{83} - 31 q^{85} - 20 q^{87} + 33 q^{89} - 29 q^{91} - 59 q^{93} - 5 q^{95} + 5 q^{97} - 48 q^{99}+O(q^{100}) \) 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
\(3\) 0.440747 0.254466 0.127233 0.991873i \(-0.459390\pi\)
0.127233 + 0.991873i \(0.459390\pi\)
\(4\) 0 0
\(5\) 0.359379 0.160719 0.0803597 0.996766i \(-0.474393\pi\)
0.0803597 + 0.996766i \(0.474393\pi\)
\(6\) 0 0
\(7\) 0.921983 0.348477 0.174238 0.984704i \(-0.444254\pi\)
0.174238 + 0.984704i \(0.444254\pi\)
\(8\) 0 0
\(9\) −2.80574 −0.935247
\(10\) 0 0
\(11\) 1.53021 0.461375 0.230688 0.973028i \(-0.425902\pi\)
0.230688 + 0.973028i \(0.425902\pi\)
\(12\) 0 0
\(13\) −2.01271 −0.558225 −0.279112 0.960258i \(-0.590040\pi\)
−0.279112 + 0.960258i \(0.590040\pi\)
\(14\) 0 0
\(15\) 0.158396 0.0408975
\(16\) 0 0
\(17\) −1.43213 −0.347343 −0.173672 0.984804i \(-0.555563\pi\)
−0.173672 + 0.984804i \(0.555563\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.406362 0.0886754
\(22\) 0 0
\(23\) 4.72144 0.984488 0.492244 0.870457i \(-0.336177\pi\)
0.492244 + 0.870457i \(0.336177\pi\)
\(24\) 0 0
\(25\) −4.87085 −0.974169
\(26\) 0 0
\(27\) −2.55887 −0.492454
\(28\) 0 0
\(29\) −7.88231 −1.46371 −0.731854 0.681462i \(-0.761345\pi\)
−0.731854 + 0.681462i \(0.761345\pi\)
\(30\) 0 0
\(31\) 1.20119 0.215739 0.107870 0.994165i \(-0.465597\pi\)
0.107870 + 0.994165i \(0.465597\pi\)
\(32\) 0 0
\(33\) 0.674435 0.117404
\(34\) 0 0
\(35\) 0.331342 0.0560070
\(36\) 0 0
\(37\) −8.80918 −1.44822 −0.724110 0.689685i \(-0.757749\pi\)
−0.724110 + 0.689685i \(0.757749\pi\)
\(38\) 0 0
\(39\) −0.887096 −0.142049
\(40\) 0 0
\(41\) −4.83669 −0.755364 −0.377682 0.925935i \(-0.623279\pi\)
−0.377682 + 0.925935i \(0.623279\pi\)
\(42\) 0 0
\(43\) 5.59135 0.852673 0.426336 0.904565i \(-0.359804\pi\)
0.426336 + 0.904565i \(0.359804\pi\)
\(44\) 0 0
\(45\) −1.00833 −0.150312
\(46\) 0 0
\(47\) 12.1547 1.77294 0.886470 0.462786i \(-0.153150\pi\)
0.886470 + 0.462786i \(0.153150\pi\)
\(48\) 0 0
\(49\) −6.14995 −0.878564
\(50\) 0 0
\(51\) −0.631208 −0.0883869
\(52\) 0 0
\(53\) −1.00000 −0.137361
\(54\) 0 0
\(55\) 0.549925 0.0741519
\(56\) 0 0
\(57\) 0.440747 0.0583784
\(58\) 0 0
\(59\) −12.3656 −1.60987 −0.804934 0.593364i \(-0.797800\pi\)
−0.804934 + 0.593364i \(0.797800\pi\)
\(60\) 0 0
\(61\) −0.555852 −0.0711696 −0.0355848 0.999367i \(-0.511329\pi\)
−0.0355848 + 0.999367i \(0.511329\pi\)
\(62\) 0 0
\(63\) −2.58685 −0.325912
\(64\) 0 0
\(65\) −0.723326 −0.0897175
\(66\) 0 0
\(67\) 5.72210 0.699066 0.349533 0.936924i \(-0.386340\pi\)
0.349533 + 0.936924i \(0.386340\pi\)
\(68\) 0 0
\(69\) 2.08096 0.250518
\(70\) 0 0
\(71\) −8.58675 −1.01906 −0.509530 0.860453i \(-0.670181\pi\)
−0.509530 + 0.860453i \(0.670181\pi\)
\(72\) 0 0
\(73\) 11.9477 1.39837 0.699184 0.714941i \(-0.253547\pi\)
0.699184 + 0.714941i \(0.253547\pi\)
\(74\) 0 0
\(75\) −2.14681 −0.247893
\(76\) 0 0
\(77\) 1.41083 0.160779
\(78\) 0 0
\(79\) −11.0735 −1.24587 −0.622933 0.782275i \(-0.714059\pi\)
−0.622933 + 0.782275i \(0.714059\pi\)
\(80\) 0 0
\(81\) 7.28941 0.809935
\(82\) 0 0
\(83\) −4.92651 −0.540755 −0.270377 0.962754i \(-0.587148\pi\)
−0.270377 + 0.962754i \(0.587148\pi\)
\(84\) 0 0
\(85\) −0.514679 −0.0558247
\(86\) 0 0
\(87\) −3.47411 −0.372463
\(88\) 0 0
\(89\) −10.1645 −1.07744 −0.538718 0.842486i \(-0.681091\pi\)
−0.538718 + 0.842486i \(0.681091\pi\)
\(90\) 0 0
\(91\) −1.85568 −0.194528
\(92\) 0 0
\(93\) 0.529419 0.0548982
\(94\) 0 0
\(95\) 0.359379 0.0368715
\(96\) 0 0
\(97\) 0.927087 0.0941314 0.0470657 0.998892i \(-0.485013\pi\)
0.0470657 + 0.998892i \(0.485013\pi\)
\(98\) 0 0
\(99\) −4.29337 −0.431500
\(100\) 0 0
\(101\) 6.84479 0.681082 0.340541 0.940230i \(-0.389390\pi\)
0.340541 + 0.940230i \(0.389390\pi\)
\(102\) 0 0
\(103\) −5.55724 −0.547571 −0.273785 0.961791i \(-0.588276\pi\)
−0.273785 + 0.961791i \(0.588276\pi\)
\(104\) 0 0
\(105\) 0.146038 0.0142518
\(106\) 0 0
\(107\) −10.3781 −1.00329 −0.501644 0.865074i \(-0.667271\pi\)
−0.501644 + 0.865074i \(0.667271\pi\)
\(108\) 0 0
\(109\) 3.79003 0.363019 0.181510 0.983389i \(-0.441902\pi\)
0.181510 + 0.983389i \(0.441902\pi\)
\(110\) 0 0
\(111\) −3.88262 −0.368522
\(112\) 0 0
\(113\) −10.8505 −1.02073 −0.510363 0.859959i \(-0.670489\pi\)
−0.510363 + 0.859959i \(0.670489\pi\)
\(114\) 0 0
\(115\) 1.69679 0.158226
\(116\) 0 0
\(117\) 5.64714 0.522078
\(118\) 0 0
\(119\) −1.32040 −0.121041
\(120\) 0 0
\(121\) −8.65846 −0.787133
\(122\) 0 0
\(123\) −2.13176 −0.192214
\(124\) 0 0
\(125\) −3.54738 −0.317287
\(126\) 0 0
\(127\) −3.26100 −0.289367 −0.144684 0.989478i \(-0.546216\pi\)
−0.144684 + 0.989478i \(0.546216\pi\)
\(128\) 0 0
\(129\) 2.46437 0.216976
\(130\) 0 0
\(131\) −11.2967 −0.986995 −0.493497 0.869747i \(-0.664282\pi\)
−0.493497 + 0.869747i \(0.664282\pi\)
\(132\) 0 0
\(133\) 0.921983 0.0799461
\(134\) 0 0
\(135\) −0.919603 −0.0791469
\(136\) 0 0
\(137\) 13.5333 1.15623 0.578116 0.815955i \(-0.303788\pi\)
0.578116 + 0.815955i \(0.303788\pi\)
\(138\) 0 0
\(139\) −8.58140 −0.727865 −0.363932 0.931425i \(-0.618566\pi\)
−0.363932 + 0.931425i \(0.618566\pi\)
\(140\) 0 0
\(141\) 5.35714 0.451152
\(142\) 0 0
\(143\) −3.07986 −0.257551
\(144\) 0 0
\(145\) −2.83274 −0.235246
\(146\) 0 0
\(147\) −2.71057 −0.223564
\(148\) 0 0
\(149\) −12.3601 −1.01258 −0.506288 0.862364i \(-0.668983\pi\)
−0.506288 + 0.862364i \(0.668983\pi\)
\(150\) 0 0
\(151\) 7.64754 0.622348 0.311174 0.950353i \(-0.399278\pi\)
0.311174 + 0.950353i \(0.399278\pi\)
\(152\) 0 0
\(153\) 4.01819 0.324852
\(154\) 0 0
\(155\) 0.431681 0.0346735
\(156\) 0 0
\(157\) −18.2352 −1.45533 −0.727664 0.685934i \(-0.759394\pi\)
−0.727664 + 0.685934i \(0.759394\pi\)
\(158\) 0 0
\(159\) −0.440747 −0.0349535
\(160\) 0 0
\(161\) 4.35309 0.343071
\(162\) 0 0
\(163\) 14.1390 1.10745 0.553727 0.832698i \(-0.313205\pi\)
0.553727 + 0.832698i \(0.313205\pi\)
\(164\) 0 0
\(165\) 0.242378 0.0188691
\(166\) 0 0
\(167\) 10.3568 0.801430 0.400715 0.916203i \(-0.368762\pi\)
0.400715 + 0.916203i \(0.368762\pi\)
\(168\) 0 0
\(169\) −8.94901 −0.688385
\(170\) 0 0
\(171\) −2.80574 −0.214560
\(172\) 0 0
\(173\) −8.44245 −0.641867 −0.320934 0.947102i \(-0.603997\pi\)
−0.320934 + 0.947102i \(0.603997\pi\)
\(174\) 0 0
\(175\) −4.49084 −0.339475
\(176\) 0 0
\(177\) −5.45012 −0.409656
\(178\) 0 0
\(179\) 12.7966 0.956459 0.478230 0.878235i \(-0.341279\pi\)
0.478230 + 0.878235i \(0.341279\pi\)
\(180\) 0 0
\(181\) −2.19453 −0.163118 −0.0815592 0.996668i \(-0.525990\pi\)
−0.0815592 + 0.996668i \(0.525990\pi\)
\(182\) 0 0
\(183\) −0.244990 −0.0181102
\(184\) 0 0
\(185\) −3.16584 −0.232757
\(186\) 0 0
\(187\) −2.19146 −0.160255
\(188\) 0 0
\(189\) −2.35923 −0.171609
\(190\) 0 0
\(191\) 17.3246 1.25357 0.626783 0.779194i \(-0.284371\pi\)
0.626783 + 0.779194i \(0.284371\pi\)
\(192\) 0 0
\(193\) 20.4183 1.46974 0.734870 0.678208i \(-0.237243\pi\)
0.734870 + 0.678208i \(0.237243\pi\)
\(194\) 0 0
\(195\) −0.318804 −0.0228300
\(196\) 0 0
\(197\) 26.2928 1.87328 0.936641 0.350292i \(-0.113918\pi\)
0.936641 + 0.350292i \(0.113918\pi\)
\(198\) 0 0
\(199\) 1.20666 0.0855381 0.0427690 0.999085i \(-0.486382\pi\)
0.0427690 + 0.999085i \(0.486382\pi\)
\(200\) 0 0
\(201\) 2.52200 0.177888
\(202\) 0 0
\(203\) −7.26735 −0.510068
\(204\) 0 0
\(205\) −1.73821 −0.121402
\(206\) 0 0
\(207\) −13.2471 −0.920740
\(208\) 0 0
\(209\) 1.53021 0.105847
\(210\) 0 0
\(211\) −5.02684 −0.346062 −0.173031 0.984916i \(-0.555356\pi\)
−0.173031 + 0.984916i \(0.555356\pi\)
\(212\) 0 0
\(213\) −3.78459 −0.259316
\(214\) 0 0
\(215\) 2.00942 0.137041
\(216\) 0 0
\(217\) 1.10747 0.0751801
\(218\) 0 0
\(219\) 5.26590 0.355837
\(220\) 0 0
\(221\) 2.88246 0.193895
\(222\) 0 0
\(223\) −12.9145 −0.864816 −0.432408 0.901678i \(-0.642336\pi\)
−0.432408 + 0.901678i \(0.642336\pi\)
\(224\) 0 0
\(225\) 13.6663 0.911089
\(226\) 0 0
\(227\) −19.7954 −1.31386 −0.656932 0.753950i \(-0.728146\pi\)
−0.656932 + 0.753950i \(0.728146\pi\)
\(228\) 0 0
\(229\) −19.2426 −1.27159 −0.635793 0.771859i \(-0.719327\pi\)
−0.635793 + 0.771859i \(0.719327\pi\)
\(230\) 0 0
\(231\) 0.621818 0.0409126
\(232\) 0 0
\(233\) −23.0287 −1.50866 −0.754329 0.656497i \(-0.772038\pi\)
−0.754329 + 0.656497i \(0.772038\pi\)
\(234\) 0 0
\(235\) 4.36814 0.284946
\(236\) 0 0
\(237\) −4.88062 −0.317030
\(238\) 0 0
\(239\) 19.1442 1.23834 0.619168 0.785259i \(-0.287470\pi\)
0.619168 + 0.785259i \(0.287470\pi\)
\(240\) 0 0
\(241\) 1.20557 0.0776577 0.0388288 0.999246i \(-0.487637\pi\)
0.0388288 + 0.999246i \(0.487637\pi\)
\(242\) 0 0
\(243\) 10.8894 0.698554
\(244\) 0 0
\(245\) −2.21016 −0.141202
\(246\) 0 0
\(247\) −2.01271 −0.128066
\(248\) 0 0
\(249\) −2.17135 −0.137603
\(250\) 0 0
\(251\) 18.7902 1.18603 0.593015 0.805191i \(-0.297937\pi\)
0.593015 + 0.805191i \(0.297937\pi\)
\(252\) 0 0
\(253\) 7.22479 0.454218
\(254\) 0 0
\(255\) −0.226843 −0.0142055
\(256\) 0 0
\(257\) 0.178596 0.0111405 0.00557027 0.999984i \(-0.498227\pi\)
0.00557027 + 0.999984i \(0.498227\pi\)
\(258\) 0 0
\(259\) −8.12191 −0.504671
\(260\) 0 0
\(261\) 22.1157 1.36893
\(262\) 0 0
\(263\) −21.3040 −1.31366 −0.656831 0.754038i \(-0.728103\pi\)
−0.656831 + 0.754038i \(0.728103\pi\)
\(264\) 0 0
\(265\) −0.359379 −0.0220765
\(266\) 0 0
\(267\) −4.47998 −0.274170
\(268\) 0 0
\(269\) −3.90158 −0.237883 −0.118942 0.992901i \(-0.537950\pi\)
−0.118942 + 0.992901i \(0.537950\pi\)
\(270\) 0 0
\(271\) −23.3838 −1.42047 −0.710233 0.703967i \(-0.751410\pi\)
−0.710233 + 0.703967i \(0.751410\pi\)
\(272\) 0 0
\(273\) −0.817887 −0.0495008
\(274\) 0 0
\(275\) −7.45341 −0.449457
\(276\) 0 0
\(277\) −13.2607 −0.796756 −0.398378 0.917221i \(-0.630427\pi\)
−0.398378 + 0.917221i \(0.630427\pi\)
\(278\) 0 0
\(279\) −3.37022 −0.201770
\(280\) 0 0
\(281\) −14.2247 −0.848572 −0.424286 0.905528i \(-0.639475\pi\)
−0.424286 + 0.905528i \(0.639475\pi\)
\(282\) 0 0
\(283\) −21.6460 −1.28672 −0.643359 0.765564i \(-0.722460\pi\)
−0.643359 + 0.765564i \(0.722460\pi\)
\(284\) 0 0
\(285\) 0.158396 0.00938254
\(286\) 0 0
\(287\) −4.45934 −0.263227
\(288\) 0 0
\(289\) −14.9490 −0.879353
\(290\) 0 0
\(291\) 0.408611 0.0239532
\(292\) 0 0
\(293\) 2.95564 0.172670 0.0863352 0.996266i \(-0.472484\pi\)
0.0863352 + 0.996266i \(0.472484\pi\)
\(294\) 0 0
\(295\) −4.44395 −0.258737
\(296\) 0 0
\(297\) −3.91560 −0.227206
\(298\) 0 0
\(299\) −9.50288 −0.549566
\(300\) 0 0
\(301\) 5.15513 0.297137
\(302\) 0 0
\(303\) 3.01682 0.173312
\(304\) 0 0
\(305\) −0.199762 −0.0114383
\(306\) 0 0
\(307\) 18.1312 1.03480 0.517401 0.855743i \(-0.326900\pi\)
0.517401 + 0.855743i \(0.326900\pi\)
\(308\) 0 0
\(309\) −2.44934 −0.139338
\(310\) 0 0
\(311\) 12.2849 0.696611 0.348306 0.937381i \(-0.386757\pi\)
0.348306 + 0.937381i \(0.386757\pi\)
\(312\) 0 0
\(313\) 2.05996 0.116436 0.0582178 0.998304i \(-0.481458\pi\)
0.0582178 + 0.998304i \(0.481458\pi\)
\(314\) 0 0
\(315\) −0.929659 −0.0523803
\(316\) 0 0
\(317\) 25.2374 1.41747 0.708737 0.705473i \(-0.249265\pi\)
0.708737 + 0.705473i \(0.249265\pi\)
\(318\) 0 0
\(319\) −12.0616 −0.675318
\(320\) 0 0
\(321\) −4.57411 −0.255302
\(322\) 0 0
\(323\) −1.43213 −0.0796860
\(324\) 0 0
\(325\) 9.80359 0.543805
\(326\) 0 0
\(327\) 1.67045 0.0923759
\(328\) 0 0
\(329\) 11.2064 0.617828
\(330\) 0 0
\(331\) −3.76350 −0.206861 −0.103430 0.994637i \(-0.532982\pi\)
−0.103430 + 0.994637i \(0.532982\pi\)
\(332\) 0 0
\(333\) 24.7163 1.35444
\(334\) 0 0
\(335\) 2.05640 0.112353
\(336\) 0 0
\(337\) −3.78046 −0.205935 −0.102967 0.994685i \(-0.532834\pi\)
−0.102967 + 0.994685i \(0.532834\pi\)
\(338\) 0 0
\(339\) −4.78231 −0.259739
\(340\) 0 0
\(341\) 1.83806 0.0995367
\(342\) 0 0
\(343\) −12.1240 −0.654636
\(344\) 0 0
\(345\) 0.747855 0.0402632
\(346\) 0 0
\(347\) −13.3465 −0.716480 −0.358240 0.933630i \(-0.616623\pi\)
−0.358240 + 0.933630i \(0.616623\pi\)
\(348\) 0 0
\(349\) 28.6206 1.53203 0.766013 0.642825i \(-0.222238\pi\)
0.766013 + 0.642825i \(0.222238\pi\)
\(350\) 0 0
\(351\) 5.15025 0.274900
\(352\) 0 0
\(353\) −15.0178 −0.799316 −0.399658 0.916664i \(-0.630871\pi\)
−0.399658 + 0.916664i \(0.630871\pi\)
\(354\) 0 0
\(355\) −3.08590 −0.163783
\(356\) 0 0
\(357\) −0.581963 −0.0308008
\(358\) 0 0
\(359\) −17.3904 −0.917831 −0.458915 0.888480i \(-0.651762\pi\)
−0.458915 + 0.888480i \(0.651762\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −3.81620 −0.200298
\(364\) 0 0
\(365\) 4.29375 0.224745
\(366\) 0 0
\(367\) −24.2681 −1.26678 −0.633391 0.773832i \(-0.718338\pi\)
−0.633391 + 0.773832i \(0.718338\pi\)
\(368\) 0 0
\(369\) 13.5705 0.706452
\(370\) 0 0
\(371\) −0.921983 −0.0478670
\(372\) 0 0
\(373\) −1.60524 −0.0831162 −0.0415581 0.999136i \(-0.513232\pi\)
−0.0415581 + 0.999136i \(0.513232\pi\)
\(374\) 0 0
\(375\) −1.56350 −0.0807387
\(376\) 0 0
\(377\) 15.8648 0.817078
\(378\) 0 0
\(379\) 26.4231 1.35726 0.678632 0.734478i \(-0.262573\pi\)
0.678632 + 0.734478i \(0.262573\pi\)
\(380\) 0 0
\(381\) −1.43728 −0.0736340
\(382\) 0 0
\(383\) 0.550870 0.0281481 0.0140741 0.999901i \(-0.495520\pi\)
0.0140741 + 0.999901i \(0.495520\pi\)
\(384\) 0 0
\(385\) 0.507022 0.0258402
\(386\) 0 0
\(387\) −15.6879 −0.797460
\(388\) 0 0
\(389\) 0.0445884 0.00226072 0.00113036 0.999999i \(-0.499640\pi\)
0.00113036 + 0.999999i \(0.499640\pi\)
\(390\) 0 0
\(391\) −6.76172 −0.341955
\(392\) 0 0
\(393\) −4.97898 −0.251156
\(394\) 0 0
\(395\) −3.97959 −0.200235
\(396\) 0 0
\(397\) 9.31886 0.467700 0.233850 0.972273i \(-0.424868\pi\)
0.233850 + 0.972273i \(0.424868\pi\)
\(398\) 0 0
\(399\) 0.406362 0.0203435
\(400\) 0 0
\(401\) 30.6347 1.52982 0.764912 0.644135i \(-0.222782\pi\)
0.764912 + 0.644135i \(0.222782\pi\)
\(402\) 0 0
\(403\) −2.41764 −0.120431
\(404\) 0 0
\(405\) 2.61966 0.130172
\(406\) 0 0
\(407\) −13.4799 −0.668173
\(408\) 0 0
\(409\) 11.1673 0.552187 0.276094 0.961131i \(-0.410960\pi\)
0.276094 + 0.961131i \(0.410960\pi\)
\(410\) 0 0
\(411\) 5.96478 0.294221
\(412\) 0 0
\(413\) −11.4009 −0.561002
\(414\) 0 0
\(415\) −1.77049 −0.0869097
\(416\) 0 0
\(417\) −3.78223 −0.185217
\(418\) 0 0
\(419\) −5.61049 −0.274090 −0.137045 0.990565i \(-0.543761\pi\)
−0.137045 + 0.990565i \(0.543761\pi\)
\(420\) 0 0
\(421\) −30.9216 −1.50703 −0.753514 0.657432i \(-0.771643\pi\)
−0.753514 + 0.657432i \(0.771643\pi\)
\(422\) 0 0
\(423\) −34.1029 −1.65814
\(424\) 0 0
\(425\) 6.97569 0.338371
\(426\) 0 0
\(427\) −0.512486 −0.0248009
\(428\) 0 0
\(429\) −1.35744 −0.0655379
\(430\) 0 0
\(431\) 4.56872 0.220067 0.110034 0.993928i \(-0.464904\pi\)
0.110034 + 0.993928i \(0.464904\pi\)
\(432\) 0 0
\(433\) 6.13191 0.294681 0.147340 0.989086i \(-0.452929\pi\)
0.147340 + 0.989086i \(0.452929\pi\)
\(434\) 0 0
\(435\) −1.24852 −0.0598620
\(436\) 0 0
\(437\) 4.72144 0.225857
\(438\) 0 0
\(439\) −3.89651 −0.185970 −0.0929851 0.995668i \(-0.529641\pi\)
−0.0929851 + 0.995668i \(0.529641\pi\)
\(440\) 0 0
\(441\) 17.2552 0.821675
\(442\) 0 0
\(443\) 26.4701 1.25763 0.628817 0.777554i \(-0.283540\pi\)
0.628817 + 0.777554i \(0.283540\pi\)
\(444\) 0 0
\(445\) −3.65291 −0.173165
\(446\) 0 0
\(447\) −5.44767 −0.257666
\(448\) 0 0
\(449\) −30.8885 −1.45772 −0.728860 0.684663i \(-0.759949\pi\)
−0.728860 + 0.684663i \(0.759949\pi\)
\(450\) 0 0
\(451\) −7.40114 −0.348506
\(452\) 0 0
\(453\) 3.37063 0.158366
\(454\) 0 0
\(455\) −0.666894 −0.0312645
\(456\) 0 0
\(457\) −32.6517 −1.52738 −0.763692 0.645581i \(-0.776615\pi\)
−0.763692 + 0.645581i \(0.776615\pi\)
\(458\) 0 0
\(459\) 3.66463 0.171050
\(460\) 0 0
\(461\) −1.48479 −0.0691536 −0.0345768 0.999402i \(-0.511008\pi\)
−0.0345768 + 0.999402i \(0.511008\pi\)
\(462\) 0 0
\(463\) 31.5195 1.46483 0.732417 0.680856i \(-0.238392\pi\)
0.732417 + 0.680856i \(0.238392\pi\)
\(464\) 0 0
\(465\) 0.190262 0.00882321
\(466\) 0 0
\(467\) 15.4017 0.712704 0.356352 0.934352i \(-0.384020\pi\)
0.356352 + 0.934352i \(0.384020\pi\)
\(468\) 0 0
\(469\) 5.27568 0.243608
\(470\) 0 0
\(471\) −8.03712 −0.370331
\(472\) 0 0
\(473\) 8.55593 0.393402
\(474\) 0 0
\(475\) −4.87085 −0.223490
\(476\) 0 0
\(477\) 2.80574 0.128466
\(478\) 0 0
\(479\) 19.6443 0.897571 0.448786 0.893639i \(-0.351857\pi\)
0.448786 + 0.893639i \(0.351857\pi\)
\(480\) 0 0
\(481\) 17.7303 0.808432
\(482\) 0 0
\(483\) 1.91861 0.0872998
\(484\) 0 0
\(485\) 0.333176 0.0151287
\(486\) 0 0
\(487\) −14.4698 −0.655689 −0.327845 0.944732i \(-0.606322\pi\)
−0.327845 + 0.944732i \(0.606322\pi\)
\(488\) 0 0
\(489\) 6.23174 0.281809
\(490\) 0 0
\(491\) 0.920419 0.0415379 0.0207690 0.999784i \(-0.493389\pi\)
0.0207690 + 0.999784i \(0.493389\pi\)
\(492\) 0 0
\(493\) 11.2885 0.508409
\(494\) 0 0
\(495\) −1.54295 −0.0693504
\(496\) 0 0
\(497\) −7.91684 −0.355119
\(498\) 0 0
\(499\) 33.5889 1.50364 0.751822 0.659366i \(-0.229175\pi\)
0.751822 + 0.659366i \(0.229175\pi\)
\(500\) 0 0
\(501\) 4.56472 0.203936
\(502\) 0 0
\(503\) 30.3472 1.35312 0.676558 0.736390i \(-0.263471\pi\)
0.676558 + 0.736390i \(0.263471\pi\)
\(504\) 0 0
\(505\) 2.45988 0.109463
\(506\) 0 0
\(507\) −3.94425 −0.175170
\(508\) 0 0
\(509\) −29.0923 −1.28949 −0.644746 0.764396i \(-0.723037\pi\)
−0.644746 + 0.764396i \(0.723037\pi\)
\(510\) 0 0
\(511\) 11.0155 0.487299
\(512\) 0 0
\(513\) −2.55887 −0.112977
\(514\) 0 0
\(515\) −1.99716 −0.0880052
\(516\) 0 0
\(517\) 18.5992 0.817990
\(518\) 0 0
\(519\) −3.72099 −0.163333
\(520\) 0 0
\(521\) 7.33543 0.321371 0.160685 0.987006i \(-0.448630\pi\)
0.160685 + 0.987006i \(0.448630\pi\)
\(522\) 0 0
\(523\) −23.0612 −1.00840 −0.504198 0.863588i \(-0.668212\pi\)
−0.504198 + 0.863588i \(0.668212\pi\)
\(524\) 0 0
\(525\) −1.97932 −0.0863848
\(526\) 0 0
\(527\) −1.72026 −0.0749355
\(528\) 0 0
\(529\) −0.708008 −0.0307830
\(530\) 0 0
\(531\) 34.6948 1.50563
\(532\) 0 0
\(533\) 9.73484 0.421663
\(534\) 0 0
\(535\) −3.72967 −0.161248
\(536\) 0 0
\(537\) 5.64005 0.243386
\(538\) 0 0
\(539\) −9.41070 −0.405348
\(540\) 0 0
\(541\) −35.3976 −1.52186 −0.760932 0.648832i \(-0.775258\pi\)
−0.760932 + 0.648832i \(0.775258\pi\)
\(542\) 0 0
\(543\) −0.967235 −0.0415080
\(544\) 0 0
\(545\) 1.36206 0.0583442
\(546\) 0 0
\(547\) −15.7707 −0.674308 −0.337154 0.941449i \(-0.609464\pi\)
−0.337154 + 0.941449i \(0.609464\pi\)
\(548\) 0 0
\(549\) 1.55958 0.0665611
\(550\) 0 0
\(551\) −7.88231 −0.335798
\(552\) 0 0
\(553\) −10.2096 −0.434155
\(554\) 0 0
\(555\) −1.39533 −0.0592286
\(556\) 0 0
\(557\) −7.59077 −0.321631 −0.160816 0.986984i \(-0.551412\pi\)
−0.160816 + 0.986984i \(0.551412\pi\)
\(558\) 0 0
\(559\) −11.2538 −0.475983
\(560\) 0 0
\(561\) −0.965880 −0.0407795
\(562\) 0 0
\(563\) 21.7641 0.917246 0.458623 0.888631i \(-0.348343\pi\)
0.458623 + 0.888631i \(0.348343\pi\)
\(564\) 0 0
\(565\) −3.89943 −0.164050
\(566\) 0 0
\(567\) 6.72071 0.282243
\(568\) 0 0
\(569\) 42.6803 1.78925 0.894626 0.446816i \(-0.147442\pi\)
0.894626 + 0.446816i \(0.147442\pi\)
\(570\) 0 0
\(571\) −0.565043 −0.0236463 −0.0118231 0.999930i \(-0.503764\pi\)
−0.0118231 + 0.999930i \(0.503764\pi\)
\(572\) 0 0
\(573\) 7.63578 0.318989
\(574\) 0 0
\(575\) −22.9974 −0.959058
\(576\) 0 0
\(577\) −10.4967 −0.436982 −0.218491 0.975839i \(-0.570113\pi\)
−0.218491 + 0.975839i \(0.570113\pi\)
\(578\) 0 0
\(579\) 8.99930 0.373998
\(580\) 0 0
\(581\) −4.54216 −0.188440
\(582\) 0 0
\(583\) −1.53021 −0.0633747
\(584\) 0 0
\(585\) 2.02946 0.0839080
\(586\) 0 0
\(587\) −36.4173 −1.50310 −0.751551 0.659675i \(-0.770694\pi\)
−0.751551 + 0.659675i \(0.770694\pi\)
\(588\) 0 0
\(589\) 1.20119 0.0494940
\(590\) 0 0
\(591\) 11.5885 0.476686
\(592\) 0 0
\(593\) 23.7587 0.975652 0.487826 0.872941i \(-0.337790\pi\)
0.487826 + 0.872941i \(0.337790\pi\)
\(594\) 0 0
\(595\) −0.474525 −0.0194536
\(596\) 0 0
\(597\) 0.531834 0.0217665
\(598\) 0 0
\(599\) −41.8619 −1.71043 −0.855216 0.518272i \(-0.826575\pi\)
−0.855216 + 0.518272i \(0.826575\pi\)
\(600\) 0 0
\(601\) 38.0135 1.55060 0.775301 0.631591i \(-0.217598\pi\)
0.775301 + 0.631591i \(0.217598\pi\)
\(602\) 0 0
\(603\) −16.0547 −0.653799
\(604\) 0 0
\(605\) −3.11167 −0.126507
\(606\) 0 0
\(607\) 6.78146 0.275251 0.137625 0.990484i \(-0.456053\pi\)
0.137625 + 0.990484i \(0.456053\pi\)
\(608\) 0 0
\(609\) −3.20307 −0.129795
\(610\) 0 0
\(611\) −24.4638 −0.989699
\(612\) 0 0
\(613\) 2.09545 0.0846343 0.0423171 0.999104i \(-0.486526\pi\)
0.0423171 + 0.999104i \(0.486526\pi\)
\(614\) 0 0
\(615\) −0.766110 −0.0308925
\(616\) 0 0
\(617\) 11.8431 0.476784 0.238392 0.971169i \(-0.423380\pi\)
0.238392 + 0.971169i \(0.423380\pi\)
\(618\) 0 0
\(619\) −15.7568 −0.633319 −0.316660 0.948539i \(-0.602561\pi\)
−0.316660 + 0.948539i \(0.602561\pi\)
\(620\) 0 0
\(621\) −12.0815 −0.484815
\(622\) 0 0
\(623\) −9.37150 −0.375461
\(624\) 0 0
\(625\) 23.0794 0.923175
\(626\) 0 0
\(627\) 0.674435 0.0269343
\(628\) 0 0
\(629\) 12.6159 0.503029
\(630\) 0 0
\(631\) 6.53885 0.260307 0.130154 0.991494i \(-0.458453\pi\)
0.130154 + 0.991494i \(0.458453\pi\)
\(632\) 0 0
\(633\) −2.21557 −0.0880608
\(634\) 0 0
\(635\) −1.17194 −0.0465069
\(636\) 0 0
\(637\) 12.3780 0.490436
\(638\) 0 0
\(639\) 24.0922 0.953073
\(640\) 0 0
\(641\) 2.27237 0.0897533 0.0448766 0.998993i \(-0.485711\pi\)
0.0448766 + 0.998993i \(0.485711\pi\)
\(642\) 0 0
\(643\) −18.6275 −0.734597 −0.367298 0.930103i \(-0.619717\pi\)
−0.367298 + 0.930103i \(0.619717\pi\)
\(644\) 0 0
\(645\) 0.885645 0.0348722
\(646\) 0 0
\(647\) 45.6415 1.79435 0.897177 0.441671i \(-0.145614\pi\)
0.897177 + 0.441671i \(0.145614\pi\)
\(648\) 0 0
\(649\) −18.9220 −0.742753
\(650\) 0 0
\(651\) 0.488116 0.0191308
\(652\) 0 0
\(653\) 9.43679 0.369290 0.184645 0.982805i \(-0.440886\pi\)
0.184645 + 0.982805i \(0.440886\pi\)
\(654\) 0 0
\(655\) −4.05979 −0.158629
\(656\) 0 0
\(657\) −33.5221 −1.30782
\(658\) 0 0
\(659\) 32.9355 1.28298 0.641492 0.767130i \(-0.278316\pi\)
0.641492 + 0.767130i \(0.278316\pi\)
\(660\) 0 0
\(661\) 14.5298 0.565142 0.282571 0.959246i \(-0.408813\pi\)
0.282571 + 0.959246i \(0.408813\pi\)
\(662\) 0 0
\(663\) 1.27044 0.0493397
\(664\) 0 0
\(665\) 0.331342 0.0128489
\(666\) 0 0
\(667\) −37.2158 −1.44100
\(668\) 0 0
\(669\) −5.69201 −0.220066
\(670\) 0 0
\(671\) −0.850569 −0.0328359
\(672\) 0 0
\(673\) 9.80750 0.378051 0.189026 0.981972i \(-0.439467\pi\)
0.189026 + 0.981972i \(0.439467\pi\)
\(674\) 0 0
\(675\) 12.4638 0.479733
\(676\) 0 0
\(677\) 19.0133 0.730740 0.365370 0.930862i \(-0.380943\pi\)
0.365370 + 0.930862i \(0.380943\pi\)
\(678\) 0 0
\(679\) 0.854758 0.0328026
\(680\) 0 0
\(681\) −8.72475 −0.334333
\(682\) 0 0
\(683\) 21.7869 0.833653 0.416827 0.908986i \(-0.363142\pi\)
0.416827 + 0.908986i \(0.363142\pi\)
\(684\) 0 0
\(685\) 4.86360 0.185829
\(686\) 0 0
\(687\) −8.48113 −0.323575
\(688\) 0 0
\(689\) 2.01271 0.0766781
\(690\) 0 0
\(691\) 43.2260 1.64439 0.822197 0.569204i \(-0.192748\pi\)
0.822197 + 0.569204i \(0.192748\pi\)
\(692\) 0 0
\(693\) −3.95841 −0.150368
\(694\) 0 0
\(695\) −3.08398 −0.116982
\(696\) 0 0
\(697\) 6.92678 0.262370
\(698\) 0 0
\(699\) −10.1498 −0.383902
\(700\) 0 0
\(701\) 23.1213 0.873278 0.436639 0.899637i \(-0.356169\pi\)
0.436639 + 0.899637i \(0.356169\pi\)
\(702\) 0 0
\(703\) −8.80918 −0.332244
\(704\) 0 0
\(705\) 1.92524 0.0725089
\(706\) 0 0
\(707\) 6.31078 0.237341
\(708\) 0 0
\(709\) −11.2482 −0.422434 −0.211217 0.977439i \(-0.567743\pi\)
−0.211217 + 0.977439i \(0.567743\pi\)
\(710\) 0 0
\(711\) 31.0694 1.16519
\(712\) 0 0
\(713\) 5.67133 0.212393
\(714\) 0 0
\(715\) −1.10684 −0.0413934
\(716\) 0 0
\(717\) 8.43776 0.315114
\(718\) 0 0
\(719\) −8.44443 −0.314924 −0.157462 0.987525i \(-0.550331\pi\)
−0.157462 + 0.987525i \(0.550331\pi\)
\(720\) 0 0
\(721\) −5.12368 −0.190816
\(722\) 0 0
\(723\) 0.531352 0.0197612
\(724\) 0 0
\(725\) 38.3935 1.42590
\(726\) 0 0
\(727\) −3.64131 −0.135049 −0.0675244 0.997718i \(-0.521510\pi\)
−0.0675244 + 0.997718i \(0.521510\pi\)
\(728\) 0 0
\(729\) −17.0688 −0.632177
\(730\) 0 0
\(731\) −8.00755 −0.296170
\(732\) 0 0
\(733\) 17.6292 0.651150 0.325575 0.945516i \(-0.394442\pi\)
0.325575 + 0.945516i \(0.394442\pi\)
\(734\) 0 0
\(735\) −0.974124 −0.0359311
\(736\) 0 0
\(737\) 8.75600 0.322532
\(738\) 0 0
\(739\) 50.6494 1.86317 0.931585 0.363524i \(-0.118426\pi\)
0.931585 + 0.363524i \(0.118426\pi\)
\(740\) 0 0
\(741\) −0.887096 −0.0325883
\(742\) 0 0
\(743\) 35.3412 1.29654 0.648272 0.761409i \(-0.275492\pi\)
0.648272 + 0.761409i \(0.275492\pi\)
\(744\) 0 0
\(745\) −4.44195 −0.162741
\(746\) 0 0
\(747\) 13.8225 0.505739
\(748\) 0 0
\(749\) −9.56842 −0.349622
\(750\) 0 0
\(751\) −5.86347 −0.213961 −0.106981 0.994261i \(-0.534118\pi\)
−0.106981 + 0.994261i \(0.534118\pi\)
\(752\) 0 0
\(753\) 8.28175 0.301804
\(754\) 0 0
\(755\) 2.74837 0.100023
\(756\) 0 0
\(757\) −23.4695 −0.853013 −0.426507 0.904484i \(-0.640256\pi\)
−0.426507 + 0.904484i \(0.640256\pi\)
\(758\) 0 0
\(759\) 3.18431 0.115583
\(760\) 0 0
\(761\) 39.0176 1.41439 0.707193 0.707021i \(-0.249961\pi\)
0.707193 + 0.707021i \(0.249961\pi\)
\(762\) 0 0
\(763\) 3.49435 0.126504
\(764\) 0 0
\(765\) 1.44406 0.0522099
\(766\) 0 0
\(767\) 24.8884 0.898668
\(768\) 0 0
\(769\) 46.2642 1.66833 0.834165 0.551515i \(-0.185950\pi\)
0.834165 + 0.551515i \(0.185950\pi\)
\(770\) 0 0
\(771\) 0.0787159 0.00283488
\(772\) 0 0
\(773\) 26.0579 0.937239 0.468620 0.883400i \(-0.344752\pi\)
0.468620 + 0.883400i \(0.344752\pi\)
\(774\) 0 0
\(775\) −5.85079 −0.210167
\(776\) 0 0
\(777\) −3.57971 −0.128421
\(778\) 0 0
\(779\) −4.83669 −0.173292
\(780\) 0 0
\(781\) −13.1395 −0.470169
\(782\) 0 0
\(783\) 20.1698 0.720809
\(784\) 0 0
\(785\) −6.55336 −0.233899
\(786\) 0 0
\(787\) −38.0168 −1.35515 −0.677577 0.735452i \(-0.736970\pi\)
−0.677577 + 0.735452i \(0.736970\pi\)
\(788\) 0 0
\(789\) −9.38969 −0.334282
\(790\) 0 0
\(791\) −10.0039 −0.355699
\(792\) 0 0
\(793\) 1.11877 0.0397286
\(794\) 0 0
\(795\) −0.158396 −0.00561771
\(796\) 0 0
\(797\) −25.3162 −0.896748 −0.448374 0.893846i \(-0.647997\pi\)
−0.448374 + 0.893846i \(0.647997\pi\)
\(798\) 0 0
\(799\) −17.4071 −0.615818
\(800\) 0 0
\(801\) 28.5190 1.00767
\(802\) 0 0
\(803\) 18.2824 0.645173
\(804\) 0 0
\(805\) 1.56441 0.0551382
\(806\) 0 0
\(807\) −1.71961 −0.0605331
\(808\) 0 0
\(809\) 25.7411 0.905008 0.452504 0.891762i \(-0.350531\pi\)
0.452504 + 0.891762i \(0.350531\pi\)
\(810\) 0 0
\(811\) −37.0055 −1.29944 −0.649720 0.760174i \(-0.725114\pi\)
−0.649720 + 0.760174i \(0.725114\pi\)
\(812\) 0 0
\(813\) −10.3064 −0.361460
\(814\) 0 0
\(815\) 5.08128 0.177989
\(816\) 0 0
\(817\) 5.59135 0.195617
\(818\) 0 0
\(819\) 5.20656 0.181932
\(820\) 0 0
\(821\) −12.6894 −0.442862 −0.221431 0.975176i \(-0.571073\pi\)
−0.221431 + 0.975176i \(0.571073\pi\)
\(822\) 0 0
\(823\) −17.4609 −0.608647 −0.304324 0.952569i \(-0.598430\pi\)
−0.304324 + 0.952569i \(0.598430\pi\)
\(824\) 0 0
\(825\) −3.28507 −0.114371
\(826\) 0 0
\(827\) 9.17174 0.318933 0.159466 0.987203i \(-0.449023\pi\)
0.159466 + 0.987203i \(0.449023\pi\)
\(828\) 0 0
\(829\) 18.1335 0.629804 0.314902 0.949124i \(-0.398028\pi\)
0.314902 + 0.949124i \(0.398028\pi\)
\(830\) 0 0
\(831\) −5.84460 −0.202747
\(832\) 0 0
\(833\) 8.80754 0.305163
\(834\) 0 0
\(835\) 3.72201 0.128805
\(836\) 0 0
\(837\) −3.07367 −0.106242
\(838\) 0 0
\(839\) 21.0714 0.727464 0.363732 0.931504i \(-0.381502\pi\)
0.363732 + 0.931504i \(0.381502\pi\)
\(840\) 0 0
\(841\) 33.1308 1.14244
\(842\) 0 0
\(843\) −6.26948 −0.215933
\(844\) 0 0
\(845\) −3.21609 −0.110637
\(846\) 0 0
\(847\) −7.98295 −0.274298
\(848\) 0 0
\(849\) −9.54040 −0.327426
\(850\) 0 0
\(851\) −41.5920 −1.42576
\(852\) 0 0
\(853\) 6.07429 0.207980 0.103990 0.994578i \(-0.466839\pi\)
0.103990 + 0.994578i \(0.466839\pi\)
\(854\) 0 0
\(855\) −1.00833 −0.0344840
\(856\) 0 0
\(857\) 46.7400 1.59661 0.798304 0.602254i \(-0.205731\pi\)
0.798304 + 0.602254i \(0.205731\pi\)
\(858\) 0 0
\(859\) −38.7867 −1.32339 −0.661693 0.749775i \(-0.730162\pi\)
−0.661693 + 0.749775i \(0.730162\pi\)
\(860\) 0 0
\(861\) −1.96544 −0.0669822
\(862\) 0 0
\(863\) 30.4822 1.03763 0.518813 0.854887i \(-0.326374\pi\)
0.518813 + 0.854887i \(0.326374\pi\)
\(864\) 0 0
\(865\) −3.03404 −0.103161
\(866\) 0 0
\(867\) −6.58873 −0.223765
\(868\) 0 0
\(869\) −16.9448 −0.574811
\(870\) 0 0
\(871\) −11.5169 −0.390236
\(872\) 0 0
\(873\) −2.60117 −0.0880361
\(874\) 0 0
\(875\) −3.27062 −0.110567
\(876\) 0 0
\(877\) 9.57576 0.323350 0.161675 0.986844i \(-0.448310\pi\)
0.161675 + 0.986844i \(0.448310\pi\)
\(878\) 0 0
\(879\) 1.30269 0.0439387
\(880\) 0 0
\(881\) −3.84025 −0.129381 −0.0646907 0.997905i \(-0.520606\pi\)
−0.0646907 + 0.997905i \(0.520606\pi\)
\(882\) 0 0
\(883\) 7.42465 0.249859 0.124930 0.992166i \(-0.460129\pi\)
0.124930 + 0.992166i \(0.460129\pi\)
\(884\) 0 0
\(885\) −1.95866 −0.0658397
\(886\) 0 0
\(887\) 12.1102 0.406621 0.203310 0.979114i \(-0.434830\pi\)
0.203310 + 0.979114i \(0.434830\pi\)
\(888\) 0 0
\(889\) −3.00659 −0.100838
\(890\) 0 0
\(891\) 11.1543 0.373684
\(892\) 0 0
\(893\) 12.1547 0.406740
\(894\) 0 0
\(895\) 4.59882 0.153721
\(896\) 0 0
\(897\) −4.18837 −0.139846
\(898\) 0 0
\(899\) −9.46811 −0.315779
\(900\) 0 0
\(901\) 1.43213 0.0477112
\(902\) 0 0
\(903\) 2.27211 0.0756111
\(904\) 0 0
\(905\) −0.788670 −0.0262163
\(906\) 0 0
\(907\) 29.8920 0.992549 0.496275 0.868166i \(-0.334701\pi\)
0.496275 + 0.868166i \(0.334701\pi\)
\(908\) 0 0
\(909\) −19.2047 −0.636980
\(910\) 0 0
\(911\) −44.2395 −1.46572 −0.732860 0.680379i \(-0.761815\pi\)
−0.732860 + 0.680379i \(0.761815\pi\)
\(912\) 0 0
\(913\) −7.53858 −0.249491
\(914\) 0 0
\(915\) −0.0880445 −0.00291066
\(916\) 0 0
\(917\) −10.4153 −0.343945
\(918\) 0 0
\(919\) −36.0167 −1.18808 −0.594040 0.804436i \(-0.702468\pi\)
−0.594040 + 0.804436i \(0.702468\pi\)
\(920\) 0 0
\(921\) 7.99128 0.263322
\(922\) 0 0
\(923\) 17.2826 0.568864
\(924\) 0 0
\(925\) 42.9082 1.41081
\(926\) 0 0
\(927\) 15.5922 0.512114
\(928\) 0 0
\(929\) −53.6285 −1.75949 −0.879747 0.475442i \(-0.842288\pi\)
−0.879747 + 0.475442i \(0.842288\pi\)
\(930\) 0 0
\(931\) −6.14995 −0.201556
\(932\) 0 0
\(933\) 5.41453 0.177264
\(934\) 0 0
\(935\) −0.787565 −0.0257561
\(936\) 0 0
\(937\) −53.6935 −1.75409 −0.877044 0.480410i \(-0.840488\pi\)
−0.877044 + 0.480410i \(0.840488\pi\)
\(938\) 0 0
\(939\) 0.907920 0.0296289
\(940\) 0 0
\(941\) −14.2431 −0.464313 −0.232156 0.972678i \(-0.574578\pi\)
−0.232156 + 0.972678i \(0.574578\pi\)
\(942\) 0 0
\(943\) −22.8361 −0.743647
\(944\) 0 0
\(945\) −0.847859 −0.0275808
\(946\) 0 0
\(947\) 35.6890 1.15974 0.579869 0.814710i \(-0.303104\pi\)
0.579869 + 0.814710i \(0.303104\pi\)
\(948\) 0 0
\(949\) −24.0472 −0.780604
\(950\) 0 0
\(951\) 11.1233 0.360699
\(952\) 0 0
\(953\) −42.8183 −1.38702 −0.693511 0.720446i \(-0.743937\pi\)
−0.693511 + 0.720446i \(0.743937\pi\)
\(954\) 0 0
\(955\) 6.22611 0.201472
\(956\) 0 0
\(957\) −5.31611 −0.171845
\(958\) 0 0
\(959\) 12.4775 0.402920
\(960\) 0 0
\(961\) −29.5572 −0.953457
\(962\) 0 0
\(963\) 29.1182 0.938322
\(964\) 0 0
\(965\) 7.33790 0.236216
\(966\) 0 0
\(967\) 1.60778 0.0517027 0.0258514 0.999666i \(-0.491770\pi\)
0.0258514 + 0.999666i \(0.491770\pi\)
\(968\) 0 0
\(969\) −0.631208 −0.0202773
\(970\) 0 0
\(971\) −16.6691 −0.534938 −0.267469 0.963566i \(-0.586187\pi\)
−0.267469 + 0.963566i \(0.586187\pi\)
\(972\) 0 0
\(973\) −7.91190 −0.253644
\(974\) 0 0
\(975\) 4.32091 0.138380
\(976\) 0 0
\(977\) −52.8780 −1.69172 −0.845859 0.533407i \(-0.820911\pi\)
−0.845859 + 0.533407i \(0.820911\pi\)
\(978\) 0 0
\(979\) −15.5538 −0.497102
\(980\) 0 0
\(981\) −10.6339 −0.339513
\(982\) 0 0
\(983\) 5.20251 0.165934 0.0829672 0.996552i \(-0.473560\pi\)
0.0829672 + 0.996552i \(0.473560\pi\)
\(984\) 0 0
\(985\) 9.44907 0.301073
\(986\) 0 0
\(987\) 4.93919 0.157216
\(988\) 0 0
\(989\) 26.3992 0.839446
\(990\) 0 0
\(991\) 47.2436 1.50074 0.750372 0.661016i \(-0.229875\pi\)
0.750372 + 0.661016i \(0.229875\pi\)
\(992\) 0 0
\(993\) −1.65875 −0.0526389
\(994\) 0 0
\(995\) 0.433650 0.0137476
\(996\) 0 0
\(997\) 20.6048 0.652562 0.326281 0.945273i \(-0.394205\pi\)
0.326281 + 0.945273i \(0.394205\pi\)
\(998\) 0 0
\(999\) 22.5415 0.713182
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 4028.2.a.c.1.12 19
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4028.2.a.c.1.12 19 1.1 even 1 trivial