Properties

Label 8016.2.a.bf.1.2
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $12$
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] = [8016,2,Mod(1,8016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.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: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 31 x^{10} + 131 x^{9} + 309 x^{8} - 1453 x^{7} - 1072 x^{6} + 6350 x^{5} + \cdots + 3008 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4008)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.49351\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.49351 q^{5} -4.58171 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.49351 q^{5} -4.58171 q^{7} +1.00000 q^{9} -0.512480 q^{11} -0.325752 q^{13} +3.49351 q^{15} -2.49148 q^{17} -3.26410 q^{19} +4.58171 q^{21} +4.00623 q^{23} +7.20462 q^{25} -1.00000 q^{27} +1.72350 q^{29} -5.79709 q^{31} +0.512480 q^{33} +16.0063 q^{35} +7.03510 q^{37} +0.325752 q^{39} +1.69446 q^{41} +7.20067 q^{43} -3.49351 q^{45} -2.49012 q^{47} +13.9921 q^{49} +2.49148 q^{51} +2.26872 q^{53} +1.79036 q^{55} +3.26410 q^{57} +6.76876 q^{59} -1.57345 q^{61} -4.58171 q^{63} +1.13802 q^{65} -0.691665 q^{67} -4.00623 q^{69} -12.4421 q^{71} +3.41218 q^{73} -7.20462 q^{75} +2.34804 q^{77} +10.7644 q^{79} +1.00000 q^{81} +3.12519 q^{83} +8.70401 q^{85} -1.72350 q^{87} +15.5543 q^{89} +1.49250 q^{91} +5.79709 q^{93} +11.4032 q^{95} +8.68185 q^{97} -0.512480 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{3} + 4 q^{5} - 11 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{3} + 4 q^{5} - 11 q^{7} + 12 q^{9} - q^{11} + 8 q^{13} - 4 q^{15} + 3 q^{17} - 12 q^{19} + 11 q^{21} - 7 q^{23} + 18 q^{25} - 12 q^{27} + 5 q^{29} - 33 q^{31} + q^{33} - 15 q^{35} + 8 q^{37} - 8 q^{39} - 6 q^{41} - 16 q^{43} + 4 q^{45} - 18 q^{47} + 25 q^{49} - 3 q^{51} + 20 q^{53} - 39 q^{55} + 12 q^{57} - 4 q^{59} + 10 q^{61} - 11 q^{63} - 9 q^{67} + 7 q^{69} - 11 q^{71} + 22 q^{73} - 18 q^{75} + 24 q^{77} - 56 q^{79} + 12 q^{81} - 26 q^{83} + 15 q^{85} - 5 q^{87} - 15 q^{89} - 11 q^{91} + 33 q^{93} - 3 q^{95} + 8 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −3.49351 −1.56235 −0.781173 0.624315i \(-0.785378\pi\)
−0.781173 + 0.624315i \(0.785378\pi\)
\(6\) 0 0
\(7\) −4.58171 −1.73172 −0.865862 0.500283i \(-0.833229\pi\)
−0.865862 + 0.500283i \(0.833229\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.512480 −0.154519 −0.0772593 0.997011i \(-0.524617\pi\)
−0.0772593 + 0.997011i \(0.524617\pi\)
\(12\) 0 0
\(13\) −0.325752 −0.0903473 −0.0451737 0.998979i \(-0.514384\pi\)
−0.0451737 + 0.998979i \(0.514384\pi\)
\(14\) 0 0
\(15\) 3.49351 0.902021
\(16\) 0 0
\(17\) −2.49148 −0.604273 −0.302136 0.953265i \(-0.597700\pi\)
−0.302136 + 0.953265i \(0.597700\pi\)
\(18\) 0 0
\(19\) −3.26410 −0.748837 −0.374418 0.927260i \(-0.622158\pi\)
−0.374418 + 0.927260i \(0.622158\pi\)
\(20\) 0 0
\(21\) 4.58171 0.999812
\(22\) 0 0
\(23\) 4.00623 0.835356 0.417678 0.908595i \(-0.362844\pi\)
0.417678 + 0.908595i \(0.362844\pi\)
\(24\) 0 0
\(25\) 7.20462 1.44092
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.72350 0.320045 0.160023 0.987113i \(-0.448843\pi\)
0.160023 + 0.987113i \(0.448843\pi\)
\(30\) 0 0
\(31\) −5.79709 −1.04119 −0.520594 0.853804i \(-0.674289\pi\)
−0.520594 + 0.853804i \(0.674289\pi\)
\(32\) 0 0
\(33\) 0.512480 0.0892114
\(34\) 0 0
\(35\) 16.0063 2.70555
\(36\) 0 0
\(37\) 7.03510 1.15656 0.578282 0.815837i \(-0.303723\pi\)
0.578282 + 0.815837i \(0.303723\pi\)
\(38\) 0 0
\(39\) 0.325752 0.0521621
\(40\) 0 0
\(41\) 1.69446 0.264630 0.132315 0.991208i \(-0.457759\pi\)
0.132315 + 0.991208i \(0.457759\pi\)
\(42\) 0 0
\(43\) 7.20067 1.09809 0.549046 0.835792i \(-0.314991\pi\)
0.549046 + 0.835792i \(0.314991\pi\)
\(44\) 0 0
\(45\) −3.49351 −0.520782
\(46\) 0 0
\(47\) −2.49012 −0.363222 −0.181611 0.983370i \(-0.558131\pi\)
−0.181611 + 0.983370i \(0.558131\pi\)
\(48\) 0 0
\(49\) 13.9921 1.99887
\(50\) 0 0
\(51\) 2.49148 0.348877
\(52\) 0 0
\(53\) 2.26872 0.311633 0.155817 0.987786i \(-0.450199\pi\)
0.155817 + 0.987786i \(0.450199\pi\)
\(54\) 0 0
\(55\) 1.79036 0.241411
\(56\) 0 0
\(57\) 3.26410 0.432341
\(58\) 0 0
\(59\) 6.76876 0.881218 0.440609 0.897699i \(-0.354763\pi\)
0.440609 + 0.897699i \(0.354763\pi\)
\(60\) 0 0
\(61\) −1.57345 −0.201459 −0.100730 0.994914i \(-0.532118\pi\)
−0.100730 + 0.994914i \(0.532118\pi\)
\(62\) 0 0
\(63\) −4.58171 −0.577241
\(64\) 0 0
\(65\) 1.13802 0.141154
\(66\) 0 0
\(67\) −0.691665 −0.0845003 −0.0422501 0.999107i \(-0.513453\pi\)
−0.0422501 + 0.999107i \(0.513453\pi\)
\(68\) 0 0
\(69\) −4.00623 −0.482293
\(70\) 0 0
\(71\) −12.4421 −1.47661 −0.738304 0.674468i \(-0.764373\pi\)
−0.738304 + 0.674468i \(0.764373\pi\)
\(72\) 0 0
\(73\) 3.41218 0.399366 0.199683 0.979861i \(-0.436009\pi\)
0.199683 + 0.979861i \(0.436009\pi\)
\(74\) 0 0
\(75\) −7.20462 −0.831918
\(76\) 0 0
\(77\) 2.34804 0.267584
\(78\) 0 0
\(79\) 10.7644 1.21110 0.605548 0.795809i \(-0.292954\pi\)
0.605548 + 0.795809i \(0.292954\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.12519 0.343035 0.171517 0.985181i \(-0.445133\pi\)
0.171517 + 0.985181i \(0.445133\pi\)
\(84\) 0 0
\(85\) 8.70401 0.944083
\(86\) 0 0
\(87\) −1.72350 −0.184778
\(88\) 0 0
\(89\) 15.5543 1.64876 0.824378 0.566040i \(-0.191525\pi\)
0.824378 + 0.566040i \(0.191525\pi\)
\(90\) 0 0
\(91\) 1.49250 0.156457
\(92\) 0 0
\(93\) 5.79709 0.601130
\(94\) 0 0
\(95\) 11.4032 1.16994
\(96\) 0 0
\(97\) 8.68185 0.881508 0.440754 0.897628i \(-0.354711\pi\)
0.440754 + 0.897628i \(0.354711\pi\)
\(98\) 0 0
\(99\) −0.512480 −0.0515062
\(100\) 0 0
\(101\) −4.25125 −0.423015 −0.211508 0.977376i \(-0.567837\pi\)
−0.211508 + 0.977376i \(0.567837\pi\)
\(102\) 0 0
\(103\) 1.64310 0.161899 0.0809496 0.996718i \(-0.474205\pi\)
0.0809496 + 0.996718i \(0.474205\pi\)
\(104\) 0 0
\(105\) −16.0063 −1.56205
\(106\) 0 0
\(107\) −19.3290 −1.86861 −0.934305 0.356475i \(-0.883978\pi\)
−0.934305 + 0.356475i \(0.883978\pi\)
\(108\) 0 0
\(109\) 0.467960 0.0448224 0.0224112 0.999749i \(-0.492866\pi\)
0.0224112 + 0.999749i \(0.492866\pi\)
\(110\) 0 0
\(111\) −7.03510 −0.667743
\(112\) 0 0
\(113\) −10.0604 −0.946405 −0.473202 0.880954i \(-0.656902\pi\)
−0.473202 + 0.880954i \(0.656902\pi\)
\(114\) 0 0
\(115\) −13.9958 −1.30512
\(116\) 0 0
\(117\) −0.325752 −0.0301158
\(118\) 0 0
\(119\) 11.4152 1.04643
\(120\) 0 0
\(121\) −10.7374 −0.976124
\(122\) 0 0
\(123\) −1.69446 −0.152784
\(124\) 0 0
\(125\) −7.70186 −0.688875
\(126\) 0 0
\(127\) −7.02371 −0.623254 −0.311627 0.950205i \(-0.600874\pi\)
−0.311627 + 0.950205i \(0.600874\pi\)
\(128\) 0 0
\(129\) −7.20067 −0.633984
\(130\) 0 0
\(131\) 7.39191 0.645834 0.322917 0.946427i \(-0.395337\pi\)
0.322917 + 0.946427i \(0.395337\pi\)
\(132\) 0 0
\(133\) 14.9552 1.29678
\(134\) 0 0
\(135\) 3.49351 0.300674
\(136\) 0 0
\(137\) 8.76010 0.748426 0.374213 0.927343i \(-0.377913\pi\)
0.374213 + 0.927343i \(0.377913\pi\)
\(138\) 0 0
\(139\) −3.16611 −0.268546 −0.134273 0.990944i \(-0.542870\pi\)
−0.134273 + 0.990944i \(0.542870\pi\)
\(140\) 0 0
\(141\) 2.49012 0.209706
\(142\) 0 0
\(143\) 0.166941 0.0139603
\(144\) 0 0
\(145\) −6.02105 −0.500021
\(146\) 0 0
\(147\) −13.9921 −1.15405
\(148\) 0 0
\(149\) 10.8088 0.885488 0.442744 0.896648i \(-0.354005\pi\)
0.442744 + 0.896648i \(0.354005\pi\)
\(150\) 0 0
\(151\) −14.5677 −1.18550 −0.592750 0.805386i \(-0.701958\pi\)
−0.592750 + 0.805386i \(0.701958\pi\)
\(152\) 0 0
\(153\) −2.49148 −0.201424
\(154\) 0 0
\(155\) 20.2522 1.62670
\(156\) 0 0
\(157\) −3.35228 −0.267541 −0.133771 0.991012i \(-0.542709\pi\)
−0.133771 + 0.991012i \(0.542709\pi\)
\(158\) 0 0
\(159\) −2.26872 −0.179921
\(160\) 0 0
\(161\) −18.3554 −1.44661
\(162\) 0 0
\(163\) 4.58993 0.359511 0.179756 0.983711i \(-0.442469\pi\)
0.179756 + 0.983711i \(0.442469\pi\)
\(164\) 0 0
\(165\) −1.79036 −0.139379
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) −12.8939 −0.991837
\(170\) 0 0
\(171\) −3.26410 −0.249612
\(172\) 0 0
\(173\) 16.8765 1.28310 0.641549 0.767082i \(-0.278292\pi\)
0.641549 + 0.767082i \(0.278292\pi\)
\(174\) 0 0
\(175\) −33.0095 −2.49528
\(176\) 0 0
\(177\) −6.76876 −0.508771
\(178\) 0 0
\(179\) 24.6247 1.84053 0.920266 0.391292i \(-0.127972\pi\)
0.920266 + 0.391292i \(0.127972\pi\)
\(180\) 0 0
\(181\) 1.54891 0.115130 0.0575648 0.998342i \(-0.481666\pi\)
0.0575648 + 0.998342i \(0.481666\pi\)
\(182\) 0 0
\(183\) 1.57345 0.116312
\(184\) 0 0
\(185\) −24.5772 −1.80695
\(186\) 0 0
\(187\) 1.27683 0.0933714
\(188\) 0 0
\(189\) 4.58171 0.333271
\(190\) 0 0
\(191\) −1.85499 −0.134222 −0.0671111 0.997746i \(-0.521378\pi\)
−0.0671111 + 0.997746i \(0.521378\pi\)
\(192\) 0 0
\(193\) 19.6759 1.41630 0.708151 0.706061i \(-0.249530\pi\)
0.708151 + 0.706061i \(0.249530\pi\)
\(194\) 0 0
\(195\) −1.13802 −0.0814952
\(196\) 0 0
\(197\) 25.0223 1.78276 0.891382 0.453252i \(-0.149736\pi\)
0.891382 + 0.453252i \(0.149736\pi\)
\(198\) 0 0
\(199\) −23.2466 −1.64791 −0.823953 0.566659i \(-0.808236\pi\)
−0.823953 + 0.566659i \(0.808236\pi\)
\(200\) 0 0
\(201\) 0.691665 0.0487863
\(202\) 0 0
\(203\) −7.89657 −0.554230
\(204\) 0 0
\(205\) −5.91962 −0.413444
\(206\) 0 0
\(207\) 4.00623 0.278452
\(208\) 0 0
\(209\) 1.67279 0.115709
\(210\) 0 0
\(211\) 8.45455 0.582035 0.291018 0.956718i \(-0.406006\pi\)
0.291018 + 0.956718i \(0.406006\pi\)
\(212\) 0 0
\(213\) 12.4421 0.852520
\(214\) 0 0
\(215\) −25.1556 −1.71560
\(216\) 0 0
\(217\) 26.5606 1.80305
\(218\) 0 0
\(219\) −3.41218 −0.230574
\(220\) 0 0
\(221\) 0.811605 0.0545944
\(222\) 0 0
\(223\) −19.8078 −1.32643 −0.663215 0.748429i \(-0.730809\pi\)
−0.663215 + 0.748429i \(0.730809\pi\)
\(224\) 0 0
\(225\) 7.20462 0.480308
\(226\) 0 0
\(227\) 1.09680 0.0727970 0.0363985 0.999337i \(-0.488411\pi\)
0.0363985 + 0.999337i \(0.488411\pi\)
\(228\) 0 0
\(229\) −10.0699 −0.665441 −0.332720 0.943026i \(-0.607967\pi\)
−0.332720 + 0.943026i \(0.607967\pi\)
\(230\) 0 0
\(231\) −2.34804 −0.154490
\(232\) 0 0
\(233\) 5.01463 0.328519 0.164260 0.986417i \(-0.447477\pi\)
0.164260 + 0.986417i \(0.447477\pi\)
\(234\) 0 0
\(235\) 8.69927 0.567478
\(236\) 0 0
\(237\) −10.7644 −0.699226
\(238\) 0 0
\(239\) −16.8946 −1.09282 −0.546410 0.837518i \(-0.684006\pi\)
−0.546410 + 0.837518i \(0.684006\pi\)
\(240\) 0 0
\(241\) 14.0016 0.901924 0.450962 0.892543i \(-0.351081\pi\)
0.450962 + 0.892543i \(0.351081\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −48.8815 −3.12292
\(246\) 0 0
\(247\) 1.06329 0.0676554
\(248\) 0 0
\(249\) −3.12519 −0.198051
\(250\) 0 0
\(251\) −20.8610 −1.31673 −0.658367 0.752697i \(-0.728753\pi\)
−0.658367 + 0.752697i \(0.728753\pi\)
\(252\) 0 0
\(253\) −2.05311 −0.129078
\(254\) 0 0
\(255\) −8.70401 −0.545066
\(256\) 0 0
\(257\) 15.0215 0.937018 0.468509 0.883459i \(-0.344791\pi\)
0.468509 + 0.883459i \(0.344791\pi\)
\(258\) 0 0
\(259\) −32.2328 −2.00285
\(260\) 0 0
\(261\) 1.72350 0.106682
\(262\) 0 0
\(263\) −9.66555 −0.596003 −0.298002 0.954565i \(-0.596320\pi\)
−0.298002 + 0.954565i \(0.596320\pi\)
\(264\) 0 0
\(265\) −7.92581 −0.486879
\(266\) 0 0
\(267\) −15.5543 −0.951909
\(268\) 0 0
\(269\) 11.0300 0.672510 0.336255 0.941771i \(-0.390840\pi\)
0.336255 + 0.941771i \(0.390840\pi\)
\(270\) 0 0
\(271\) −27.3714 −1.66269 −0.831347 0.555754i \(-0.812430\pi\)
−0.831347 + 0.555754i \(0.812430\pi\)
\(272\) 0 0
\(273\) −1.49250 −0.0903303
\(274\) 0 0
\(275\) −3.69222 −0.222650
\(276\) 0 0
\(277\) −14.2246 −0.854675 −0.427338 0.904092i \(-0.640548\pi\)
−0.427338 + 0.904092i \(0.640548\pi\)
\(278\) 0 0
\(279\) −5.79709 −0.347063
\(280\) 0 0
\(281\) 7.17980 0.428311 0.214156 0.976800i \(-0.431300\pi\)
0.214156 + 0.976800i \(0.431300\pi\)
\(282\) 0 0
\(283\) 7.49336 0.445434 0.222717 0.974883i \(-0.428507\pi\)
0.222717 + 0.974883i \(0.428507\pi\)
\(284\) 0 0
\(285\) −11.4032 −0.675466
\(286\) 0 0
\(287\) −7.76354 −0.458267
\(288\) 0 0
\(289\) −10.7925 −0.634855
\(290\) 0 0
\(291\) −8.68185 −0.508939
\(292\) 0 0
\(293\) −22.4024 −1.30876 −0.654380 0.756166i \(-0.727070\pi\)
−0.654380 + 0.756166i \(0.727070\pi\)
\(294\) 0 0
\(295\) −23.6467 −1.37677
\(296\) 0 0
\(297\) 0.512480 0.0297371
\(298\) 0 0
\(299\) −1.30504 −0.0754722
\(300\) 0 0
\(301\) −32.9914 −1.90159
\(302\) 0 0
\(303\) 4.25125 0.244228
\(304\) 0 0
\(305\) 5.49685 0.314749
\(306\) 0 0
\(307\) −7.30538 −0.416940 −0.208470 0.978029i \(-0.566848\pi\)
−0.208470 + 0.978029i \(0.566848\pi\)
\(308\) 0 0
\(309\) −1.64310 −0.0934725
\(310\) 0 0
\(311\) 26.7279 1.51560 0.757800 0.652487i \(-0.226274\pi\)
0.757800 + 0.652487i \(0.226274\pi\)
\(312\) 0 0
\(313\) −19.0884 −1.07894 −0.539469 0.842005i \(-0.681375\pi\)
−0.539469 + 0.842005i \(0.681375\pi\)
\(314\) 0 0
\(315\) 16.0063 0.901851
\(316\) 0 0
\(317\) −0.844016 −0.0474047 −0.0237023 0.999719i \(-0.507545\pi\)
−0.0237023 + 0.999719i \(0.507545\pi\)
\(318\) 0 0
\(319\) −0.883258 −0.0494530
\(320\) 0 0
\(321\) 19.3290 1.07884
\(322\) 0 0
\(323\) 8.13245 0.452502
\(324\) 0 0
\(325\) −2.34692 −0.130184
\(326\) 0 0
\(327\) −0.467960 −0.0258782
\(328\) 0 0
\(329\) 11.4090 0.629000
\(330\) 0 0
\(331\) 0.805305 0.0442636 0.0221318 0.999755i \(-0.492955\pi\)
0.0221318 + 0.999755i \(0.492955\pi\)
\(332\) 0 0
\(333\) 7.03510 0.385521
\(334\) 0 0
\(335\) 2.41634 0.132019
\(336\) 0 0
\(337\) 1.49693 0.0815429 0.0407715 0.999168i \(-0.487018\pi\)
0.0407715 + 0.999168i \(0.487018\pi\)
\(338\) 0 0
\(339\) 10.0604 0.546407
\(340\) 0 0
\(341\) 2.97089 0.160883
\(342\) 0 0
\(343\) −32.0357 −1.72977
\(344\) 0 0
\(345\) 13.9958 0.753509
\(346\) 0 0
\(347\) 23.4606 1.25943 0.629715 0.776826i \(-0.283172\pi\)
0.629715 + 0.776826i \(0.283172\pi\)
\(348\) 0 0
\(349\) −19.0540 −1.01994 −0.509970 0.860192i \(-0.670343\pi\)
−0.509970 + 0.860192i \(0.670343\pi\)
\(350\) 0 0
\(351\) 0.325752 0.0173874
\(352\) 0 0
\(353\) −18.1020 −0.963471 −0.481735 0.876317i \(-0.659993\pi\)
−0.481735 + 0.876317i \(0.659993\pi\)
\(354\) 0 0
\(355\) 43.4667 2.30697
\(356\) 0 0
\(357\) −11.4152 −0.604159
\(358\) 0 0
\(359\) −8.02286 −0.423430 −0.211715 0.977331i \(-0.567905\pi\)
−0.211715 + 0.977331i \(0.567905\pi\)
\(360\) 0 0
\(361\) −8.34562 −0.439243
\(362\) 0 0
\(363\) 10.7374 0.563565
\(364\) 0 0
\(365\) −11.9205 −0.623948
\(366\) 0 0
\(367\) 6.45266 0.336826 0.168413 0.985716i \(-0.446136\pi\)
0.168413 + 0.985716i \(0.446136\pi\)
\(368\) 0 0
\(369\) 1.69446 0.0882101
\(370\) 0 0
\(371\) −10.3946 −0.539663
\(372\) 0 0
\(373\) 6.68132 0.345946 0.172973 0.984927i \(-0.444663\pi\)
0.172973 + 0.984927i \(0.444663\pi\)
\(374\) 0 0
\(375\) 7.70186 0.397722
\(376\) 0 0
\(377\) −0.561432 −0.0289152
\(378\) 0 0
\(379\) 32.6658 1.67793 0.838966 0.544184i \(-0.183161\pi\)
0.838966 + 0.544184i \(0.183161\pi\)
\(380\) 0 0
\(381\) 7.02371 0.359836
\(382\) 0 0
\(383\) −12.4445 −0.635884 −0.317942 0.948110i \(-0.602992\pi\)
−0.317942 + 0.948110i \(0.602992\pi\)
\(384\) 0 0
\(385\) −8.20289 −0.418058
\(386\) 0 0
\(387\) 7.20067 0.366031
\(388\) 0 0
\(389\) 1.21421 0.0615628 0.0307814 0.999526i \(-0.490200\pi\)
0.0307814 + 0.999526i \(0.490200\pi\)
\(390\) 0 0
\(391\) −9.98144 −0.504783
\(392\) 0 0
\(393\) −7.39191 −0.372873
\(394\) 0 0
\(395\) −37.6057 −1.89215
\(396\) 0 0
\(397\) 18.2956 0.918231 0.459116 0.888376i \(-0.348166\pi\)
0.459116 + 0.888376i \(0.348166\pi\)
\(398\) 0 0
\(399\) −14.9552 −0.748696
\(400\) 0 0
\(401\) −4.88106 −0.243748 −0.121874 0.992546i \(-0.538890\pi\)
−0.121874 + 0.992546i \(0.538890\pi\)
\(402\) 0 0
\(403\) 1.88841 0.0940686
\(404\) 0 0
\(405\) −3.49351 −0.173594
\(406\) 0 0
\(407\) −3.60535 −0.178711
\(408\) 0 0
\(409\) −11.2838 −0.557947 −0.278973 0.960299i \(-0.589994\pi\)
−0.278973 + 0.960299i \(0.589994\pi\)
\(410\) 0 0
\(411\) −8.76010 −0.432104
\(412\) 0 0
\(413\) −31.0125 −1.52603
\(414\) 0 0
\(415\) −10.9179 −0.535939
\(416\) 0 0
\(417\) 3.16611 0.155045
\(418\) 0 0
\(419\) −22.7280 −1.11034 −0.555168 0.831738i \(-0.687346\pi\)
−0.555168 + 0.831738i \(0.687346\pi\)
\(420\) 0 0
\(421\) −19.9399 −0.971810 −0.485905 0.874012i \(-0.661510\pi\)
−0.485905 + 0.874012i \(0.661510\pi\)
\(422\) 0 0
\(423\) −2.49012 −0.121074
\(424\) 0 0
\(425\) −17.9502 −0.870711
\(426\) 0 0
\(427\) 7.20907 0.348872
\(428\) 0 0
\(429\) −0.166941 −0.00806001
\(430\) 0 0
\(431\) 13.4845 0.649525 0.324763 0.945796i \(-0.394716\pi\)
0.324763 + 0.945796i \(0.394716\pi\)
\(432\) 0 0
\(433\) −30.6652 −1.47368 −0.736839 0.676068i \(-0.763682\pi\)
−0.736839 + 0.676068i \(0.763682\pi\)
\(434\) 0 0
\(435\) 6.02105 0.288687
\(436\) 0 0
\(437\) −13.0767 −0.625546
\(438\) 0 0
\(439\) 7.49335 0.357638 0.178819 0.983882i \(-0.442772\pi\)
0.178819 + 0.983882i \(0.442772\pi\)
\(440\) 0 0
\(441\) 13.9921 0.666290
\(442\) 0 0
\(443\) −6.21514 −0.295290 −0.147645 0.989040i \(-0.547169\pi\)
−0.147645 + 0.989040i \(0.547169\pi\)
\(444\) 0 0
\(445\) −54.3392 −2.57593
\(446\) 0 0
\(447\) −10.8088 −0.511237
\(448\) 0 0
\(449\) 18.3577 0.866354 0.433177 0.901309i \(-0.357393\pi\)
0.433177 + 0.901309i \(0.357393\pi\)
\(450\) 0 0
\(451\) −0.868378 −0.0408903
\(452\) 0 0
\(453\) 14.5677 0.684449
\(454\) 0 0
\(455\) −5.21407 −0.244439
\(456\) 0 0
\(457\) −29.7113 −1.38984 −0.694918 0.719089i \(-0.744559\pi\)
−0.694918 + 0.719089i \(0.744559\pi\)
\(458\) 0 0
\(459\) 2.49148 0.116292
\(460\) 0 0
\(461\) −10.4413 −0.486301 −0.243150 0.969989i \(-0.578181\pi\)
−0.243150 + 0.969989i \(0.578181\pi\)
\(462\) 0 0
\(463\) −12.4434 −0.578295 −0.289147 0.957285i \(-0.593372\pi\)
−0.289147 + 0.957285i \(0.593372\pi\)
\(464\) 0 0
\(465\) −20.2522 −0.939173
\(466\) 0 0
\(467\) 15.8612 0.733969 0.366985 0.930227i \(-0.380390\pi\)
0.366985 + 0.930227i \(0.380390\pi\)
\(468\) 0 0
\(469\) 3.16901 0.146331
\(470\) 0 0
\(471\) 3.35228 0.154465
\(472\) 0 0
\(473\) −3.69020 −0.169676
\(474\) 0 0
\(475\) −23.5166 −1.07902
\(476\) 0 0
\(477\) 2.26872 0.103878
\(478\) 0 0
\(479\) −17.6120 −0.804712 −0.402356 0.915483i \(-0.631809\pi\)
−0.402356 + 0.915483i \(0.631809\pi\)
\(480\) 0 0
\(481\) −2.29170 −0.104492
\(482\) 0 0
\(483\) 18.3554 0.835199
\(484\) 0 0
\(485\) −30.3301 −1.37722
\(486\) 0 0
\(487\) −12.9871 −0.588499 −0.294250 0.955729i \(-0.595070\pi\)
−0.294250 + 0.955729i \(0.595070\pi\)
\(488\) 0 0
\(489\) −4.58993 −0.207564
\(490\) 0 0
\(491\) 24.0244 1.08420 0.542102 0.840313i \(-0.317629\pi\)
0.542102 + 0.840313i \(0.317629\pi\)
\(492\) 0 0
\(493\) −4.29406 −0.193395
\(494\) 0 0
\(495\) 1.79036 0.0804705
\(496\) 0 0
\(497\) 57.0062 2.55708
\(498\) 0 0
\(499\) 36.3018 1.62509 0.812545 0.582899i \(-0.198082\pi\)
0.812545 + 0.582899i \(0.198082\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) 38.7818 1.72920 0.864598 0.502464i \(-0.167573\pi\)
0.864598 + 0.502464i \(0.167573\pi\)
\(504\) 0 0
\(505\) 14.8518 0.660896
\(506\) 0 0
\(507\) 12.8939 0.572638
\(508\) 0 0
\(509\) −36.7197 −1.62757 −0.813786 0.581165i \(-0.802597\pi\)
−0.813786 + 0.581165i \(0.802597\pi\)
\(510\) 0 0
\(511\) −15.6336 −0.691592
\(512\) 0 0
\(513\) 3.26410 0.144114
\(514\) 0 0
\(515\) −5.74018 −0.252942
\(516\) 0 0
\(517\) 1.27614 0.0561245
\(518\) 0 0
\(519\) −16.8765 −0.740797
\(520\) 0 0
\(521\) 9.85756 0.431867 0.215934 0.976408i \(-0.430720\pi\)
0.215934 + 0.976408i \(0.430720\pi\)
\(522\) 0 0
\(523\) −31.4813 −1.37658 −0.688290 0.725435i \(-0.741638\pi\)
−0.688290 + 0.725435i \(0.741638\pi\)
\(524\) 0 0
\(525\) 33.0095 1.44065
\(526\) 0 0
\(527\) 14.4433 0.629161
\(528\) 0 0
\(529\) −6.95013 −0.302180
\(530\) 0 0
\(531\) 6.76876 0.293739
\(532\) 0 0
\(533\) −0.551974 −0.0239087
\(534\) 0 0
\(535\) 67.5262 2.91941
\(536\) 0 0
\(537\) −24.6247 −1.06263
\(538\) 0 0
\(539\) −7.17067 −0.308863
\(540\) 0 0
\(541\) 11.6002 0.498729 0.249365 0.968410i \(-0.419778\pi\)
0.249365 + 0.968410i \(0.419778\pi\)
\(542\) 0 0
\(543\) −1.54891 −0.0664701
\(544\) 0 0
\(545\) −1.63482 −0.0700281
\(546\) 0 0
\(547\) −7.64269 −0.326778 −0.163389 0.986562i \(-0.552243\pi\)
−0.163389 + 0.986562i \(0.552243\pi\)
\(548\) 0 0
\(549\) −1.57345 −0.0671530
\(550\) 0 0
\(551\) −5.62567 −0.239662
\(552\) 0 0
\(553\) −49.3196 −2.09728
\(554\) 0 0
\(555\) 24.5772 1.04324
\(556\) 0 0
\(557\) 8.11952 0.344035 0.172018 0.985094i \(-0.444971\pi\)
0.172018 + 0.985094i \(0.444971\pi\)
\(558\) 0 0
\(559\) −2.34563 −0.0992097
\(560\) 0 0
\(561\) −1.27683 −0.0539080
\(562\) 0 0
\(563\) 20.6113 0.868663 0.434332 0.900753i \(-0.356985\pi\)
0.434332 + 0.900753i \(0.356985\pi\)
\(564\) 0 0
\(565\) 35.1462 1.47861
\(566\) 0 0
\(567\) −4.58171 −0.192414
\(568\) 0 0
\(569\) 12.2528 0.513664 0.256832 0.966456i \(-0.417321\pi\)
0.256832 + 0.966456i \(0.417321\pi\)
\(570\) 0 0
\(571\) 34.0736 1.42593 0.712967 0.701197i \(-0.247351\pi\)
0.712967 + 0.701197i \(0.247351\pi\)
\(572\) 0 0
\(573\) 1.85499 0.0774932
\(574\) 0 0
\(575\) 28.8633 1.20368
\(576\) 0 0
\(577\) 2.05077 0.0853745 0.0426873 0.999088i \(-0.486408\pi\)
0.0426873 + 0.999088i \(0.486408\pi\)
\(578\) 0 0
\(579\) −19.6759 −0.817702
\(580\) 0 0
\(581\) −14.3187 −0.594041
\(582\) 0 0
\(583\) −1.16268 −0.0481531
\(584\) 0 0
\(585\) 1.13802 0.0470513
\(586\) 0 0
\(587\) −27.2048 −1.12286 −0.561430 0.827524i \(-0.689749\pi\)
−0.561430 + 0.827524i \(0.689749\pi\)
\(588\) 0 0
\(589\) 18.9223 0.779680
\(590\) 0 0
\(591\) −25.0223 −1.02928
\(592\) 0 0
\(593\) −13.2090 −0.542429 −0.271215 0.962519i \(-0.587425\pi\)
−0.271215 + 0.962519i \(0.587425\pi\)
\(594\) 0 0
\(595\) −39.8793 −1.63489
\(596\) 0 0
\(597\) 23.2466 0.951419
\(598\) 0 0
\(599\) 33.8049 1.38123 0.690614 0.723223i \(-0.257340\pi\)
0.690614 + 0.723223i \(0.257340\pi\)
\(600\) 0 0
\(601\) 9.52887 0.388691 0.194345 0.980933i \(-0.437742\pi\)
0.194345 + 0.980933i \(0.437742\pi\)
\(602\) 0 0
\(603\) −0.691665 −0.0281668
\(604\) 0 0
\(605\) 37.5111 1.52504
\(606\) 0 0
\(607\) 17.2279 0.699258 0.349629 0.936888i \(-0.386308\pi\)
0.349629 + 0.936888i \(0.386308\pi\)
\(608\) 0 0
\(609\) 7.89657 0.319985
\(610\) 0 0
\(611\) 0.811162 0.0328161
\(612\) 0 0
\(613\) −44.0981 −1.78111 −0.890553 0.454880i \(-0.849682\pi\)
−0.890553 + 0.454880i \(0.849682\pi\)
\(614\) 0 0
\(615\) 5.91962 0.238702
\(616\) 0 0
\(617\) 31.7800 1.27941 0.639706 0.768619i \(-0.279056\pi\)
0.639706 + 0.768619i \(0.279056\pi\)
\(618\) 0 0
\(619\) −2.00109 −0.0804307 −0.0402153 0.999191i \(-0.512804\pi\)
−0.0402153 + 0.999191i \(0.512804\pi\)
\(620\) 0 0
\(621\) −4.00623 −0.160764
\(622\) 0 0
\(623\) −71.2655 −2.85519
\(624\) 0 0
\(625\) −9.11657 −0.364663
\(626\) 0 0
\(627\) −1.67279 −0.0668048
\(628\) 0 0
\(629\) −17.5278 −0.698880
\(630\) 0 0
\(631\) −42.1185 −1.67671 −0.838355 0.545125i \(-0.816482\pi\)
−0.838355 + 0.545125i \(0.816482\pi\)
\(632\) 0 0
\(633\) −8.45455 −0.336038
\(634\) 0 0
\(635\) 24.5374 0.973737
\(636\) 0 0
\(637\) −4.55795 −0.180593
\(638\) 0 0
\(639\) −12.4421 −0.492203
\(640\) 0 0
\(641\) 8.22836 0.325001 0.162500 0.986708i \(-0.448044\pi\)
0.162500 + 0.986708i \(0.448044\pi\)
\(642\) 0 0
\(643\) −7.04621 −0.277876 −0.138938 0.990301i \(-0.544369\pi\)
−0.138938 + 0.990301i \(0.544369\pi\)
\(644\) 0 0
\(645\) 25.1556 0.990502
\(646\) 0 0
\(647\) 45.5212 1.78962 0.894812 0.446444i \(-0.147310\pi\)
0.894812 + 0.446444i \(0.147310\pi\)
\(648\) 0 0
\(649\) −3.46886 −0.136165
\(650\) 0 0
\(651\) −26.5606 −1.04099
\(652\) 0 0
\(653\) 29.4865 1.15389 0.576947 0.816782i \(-0.304244\pi\)
0.576947 + 0.816782i \(0.304244\pi\)
\(654\) 0 0
\(655\) −25.8237 −1.00902
\(656\) 0 0
\(657\) 3.41218 0.133122
\(658\) 0 0
\(659\) 26.5112 1.03273 0.516364 0.856369i \(-0.327285\pi\)
0.516364 + 0.856369i \(0.327285\pi\)
\(660\) 0 0
\(661\) −16.4928 −0.641494 −0.320747 0.947165i \(-0.603934\pi\)
−0.320747 + 0.947165i \(0.603934\pi\)
\(662\) 0 0
\(663\) −0.811605 −0.0315201
\(664\) 0 0
\(665\) −52.2461 −2.02602
\(666\) 0 0
\(667\) 6.90472 0.267352
\(668\) 0 0
\(669\) 19.8078 0.765815
\(670\) 0 0
\(671\) 0.806360 0.0311292
\(672\) 0 0
\(673\) 44.9616 1.73314 0.866571 0.499054i \(-0.166319\pi\)
0.866571 + 0.499054i \(0.166319\pi\)
\(674\) 0 0
\(675\) −7.20462 −0.277306
\(676\) 0 0
\(677\) −27.9435 −1.07396 −0.536978 0.843596i \(-0.680434\pi\)
−0.536978 + 0.843596i \(0.680434\pi\)
\(678\) 0 0
\(679\) −39.7777 −1.52653
\(680\) 0 0
\(681\) −1.09680 −0.0420293
\(682\) 0 0
\(683\) 28.1762 1.07813 0.539067 0.842263i \(-0.318777\pi\)
0.539067 + 0.842263i \(0.318777\pi\)
\(684\) 0 0
\(685\) −30.6035 −1.16930
\(686\) 0 0
\(687\) 10.0699 0.384192
\(688\) 0 0
\(689\) −0.739041 −0.0281552
\(690\) 0 0
\(691\) 25.4670 0.968810 0.484405 0.874844i \(-0.339036\pi\)
0.484405 + 0.874844i \(0.339036\pi\)
\(692\) 0 0
\(693\) 2.34804 0.0891946
\(694\) 0 0
\(695\) 11.0608 0.419561
\(696\) 0 0
\(697\) −4.22172 −0.159909
\(698\) 0 0
\(699\) −5.01463 −0.189671
\(700\) 0 0
\(701\) −2.60309 −0.0983174 −0.0491587 0.998791i \(-0.515654\pi\)
−0.0491587 + 0.998791i \(0.515654\pi\)
\(702\) 0 0
\(703\) −22.9633 −0.866078
\(704\) 0 0
\(705\) −8.69927 −0.327633
\(706\) 0 0
\(707\) 19.4780 0.732546
\(708\) 0 0
\(709\) 26.4907 0.994878 0.497439 0.867499i \(-0.334274\pi\)
0.497439 + 0.867499i \(0.334274\pi\)
\(710\) 0 0
\(711\) 10.7644 0.403698
\(712\) 0 0
\(713\) −23.2245 −0.869763
\(714\) 0 0
\(715\) −0.583212 −0.0218109
\(716\) 0 0
\(717\) 16.8946 0.630939
\(718\) 0 0
\(719\) −53.1883 −1.98359 −0.991794 0.127850i \(-0.959192\pi\)
−0.991794 + 0.127850i \(0.959192\pi\)
\(720\) 0 0
\(721\) −7.52819 −0.280365
\(722\) 0 0
\(723\) −14.0016 −0.520726
\(724\) 0 0
\(725\) 12.4171 0.461161
\(726\) 0 0
\(727\) 40.2624 1.49325 0.746625 0.665245i \(-0.231673\pi\)
0.746625 + 0.665245i \(0.231673\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −17.9403 −0.663547
\(732\) 0 0
\(733\) 16.2660 0.600798 0.300399 0.953814i \(-0.402880\pi\)
0.300399 + 0.953814i \(0.402880\pi\)
\(734\) 0 0
\(735\) 48.8815 1.80302
\(736\) 0 0
\(737\) 0.354465 0.0130569
\(738\) 0 0
\(739\) 41.4651 1.52532 0.762660 0.646800i \(-0.223893\pi\)
0.762660 + 0.646800i \(0.223893\pi\)
\(740\) 0 0
\(741\) −1.06329 −0.0390609
\(742\) 0 0
\(743\) 36.6758 1.34551 0.672753 0.739867i \(-0.265112\pi\)
0.672753 + 0.739867i \(0.265112\pi\)
\(744\) 0 0
\(745\) −37.7605 −1.38344
\(746\) 0 0
\(747\) 3.12519 0.114345
\(748\) 0 0
\(749\) 88.5601 3.23592
\(750\) 0 0
\(751\) −34.0394 −1.24212 −0.621058 0.783765i \(-0.713297\pi\)
−0.621058 + 0.783765i \(0.713297\pi\)
\(752\) 0 0
\(753\) 20.8610 0.760217
\(754\) 0 0
\(755\) 50.8923 1.85216
\(756\) 0 0
\(757\) −32.5338 −1.18246 −0.591231 0.806503i \(-0.701358\pi\)
−0.591231 + 0.806503i \(0.701358\pi\)
\(758\) 0 0
\(759\) 2.05311 0.0745233
\(760\) 0 0
\(761\) −11.8709 −0.430319 −0.215159 0.976579i \(-0.569027\pi\)
−0.215159 + 0.976579i \(0.569027\pi\)
\(762\) 0 0
\(763\) −2.14406 −0.0776201
\(764\) 0 0
\(765\) 8.70401 0.314694
\(766\) 0 0
\(767\) −2.20494 −0.0796157
\(768\) 0 0
\(769\) 15.7830 0.569149 0.284574 0.958654i \(-0.408148\pi\)
0.284574 + 0.958654i \(0.408148\pi\)
\(770\) 0 0
\(771\) −15.0215 −0.540987
\(772\) 0 0
\(773\) 33.1698 1.19304 0.596518 0.802600i \(-0.296551\pi\)
0.596518 + 0.802600i \(0.296551\pi\)
\(774\) 0 0
\(775\) −41.7658 −1.50027
\(776\) 0 0
\(777\) 32.2328 1.15635
\(778\) 0 0
\(779\) −5.53090 −0.198165
\(780\) 0 0
\(781\) 6.37634 0.228164
\(782\) 0 0
\(783\) −1.72350 −0.0615927
\(784\) 0 0
\(785\) 11.7112 0.417992
\(786\) 0 0
\(787\) −8.27019 −0.294800 −0.147400 0.989077i \(-0.547091\pi\)
−0.147400 + 0.989077i \(0.547091\pi\)
\(788\) 0 0
\(789\) 9.66555 0.344103
\(790\) 0 0
\(791\) 46.0940 1.63891
\(792\) 0 0
\(793\) 0.512553 0.0182013
\(794\) 0 0
\(795\) 7.92581 0.281099
\(796\) 0 0
\(797\) 17.7455 0.628578 0.314289 0.949327i \(-0.398234\pi\)
0.314289 + 0.949327i \(0.398234\pi\)
\(798\) 0 0
\(799\) 6.20409 0.219485
\(800\) 0 0
\(801\) 15.5543 0.549585
\(802\) 0 0
\(803\) −1.74868 −0.0617095
\(804\) 0 0
\(805\) 64.1247 2.26010
\(806\) 0 0
\(807\) −11.0300 −0.388274
\(808\) 0 0
\(809\) −44.8874 −1.57816 −0.789079 0.614292i \(-0.789442\pi\)
−0.789079 + 0.614292i \(0.789442\pi\)
\(810\) 0 0
\(811\) −53.5750 −1.88127 −0.940636 0.339418i \(-0.889770\pi\)
−0.940636 + 0.339418i \(0.889770\pi\)
\(812\) 0 0
\(813\) 27.3714 0.959956
\(814\) 0 0
\(815\) −16.0350 −0.561681
\(816\) 0 0
\(817\) −23.5037 −0.822292
\(818\) 0 0
\(819\) 1.49250 0.0521522
\(820\) 0 0
\(821\) −28.2612 −0.986321 −0.493161 0.869938i \(-0.664158\pi\)
−0.493161 + 0.869938i \(0.664158\pi\)
\(822\) 0 0
\(823\) 30.5821 1.06603 0.533013 0.846107i \(-0.321060\pi\)
0.533013 + 0.846107i \(0.321060\pi\)
\(824\) 0 0
\(825\) 3.69222 0.128547
\(826\) 0 0
\(827\) −26.0995 −0.907567 −0.453784 0.891112i \(-0.649926\pi\)
−0.453784 + 0.891112i \(0.649926\pi\)
\(828\) 0 0
\(829\) 8.79496 0.305462 0.152731 0.988268i \(-0.451193\pi\)
0.152731 + 0.988268i \(0.451193\pi\)
\(830\) 0 0
\(831\) 14.2246 0.493447
\(832\) 0 0
\(833\) −34.8610 −1.20786
\(834\) 0 0
\(835\) 3.49351 0.120898
\(836\) 0 0
\(837\) 5.79709 0.200377
\(838\) 0 0
\(839\) −24.6667 −0.851591 −0.425795 0.904820i \(-0.640006\pi\)
−0.425795 + 0.904820i \(0.640006\pi\)
\(840\) 0 0
\(841\) −26.0296 −0.897571
\(842\) 0 0
\(843\) −7.17980 −0.247286
\(844\) 0 0
\(845\) 45.0449 1.54959
\(846\) 0 0
\(847\) 49.1955 1.69038
\(848\) 0 0
\(849\) −7.49336 −0.257172
\(850\) 0 0
\(851\) 28.1842 0.966143
\(852\) 0 0
\(853\) −10.7504 −0.368088 −0.184044 0.982918i \(-0.558919\pi\)
−0.184044 + 0.982918i \(0.558919\pi\)
\(854\) 0 0
\(855\) 11.4032 0.389981
\(856\) 0 0
\(857\) 8.67868 0.296458 0.148229 0.988953i \(-0.452643\pi\)
0.148229 + 0.988953i \(0.452643\pi\)
\(858\) 0 0
\(859\) −14.1922 −0.484232 −0.242116 0.970247i \(-0.577841\pi\)
−0.242116 + 0.970247i \(0.577841\pi\)
\(860\) 0 0
\(861\) 7.76354 0.264581
\(862\) 0 0
\(863\) 36.4638 1.24124 0.620622 0.784110i \(-0.286880\pi\)
0.620622 + 0.784110i \(0.286880\pi\)
\(864\) 0 0
\(865\) −58.9583 −2.00464
\(866\) 0 0
\(867\) 10.7925 0.366533
\(868\) 0 0
\(869\) −5.51657 −0.187137
\(870\) 0 0
\(871\) 0.225311 0.00763438
\(872\) 0 0
\(873\) 8.68185 0.293836
\(874\) 0 0
\(875\) 35.2877 1.19294
\(876\) 0 0
\(877\) 6.68456 0.225722 0.112861 0.993611i \(-0.463999\pi\)
0.112861 + 0.993611i \(0.463999\pi\)
\(878\) 0 0
\(879\) 22.4024 0.755613
\(880\) 0 0
\(881\) −1.97149 −0.0664211 −0.0332105 0.999448i \(-0.510573\pi\)
−0.0332105 + 0.999448i \(0.510573\pi\)
\(882\) 0 0
\(883\) 39.9784 1.34538 0.672690 0.739925i \(-0.265139\pi\)
0.672690 + 0.739925i \(0.265139\pi\)
\(884\) 0 0
\(885\) 23.6467 0.794877
\(886\) 0 0
\(887\) 31.6633 1.06315 0.531574 0.847012i \(-0.321601\pi\)
0.531574 + 0.847012i \(0.321601\pi\)
\(888\) 0 0
\(889\) 32.1806 1.07930
\(890\) 0 0
\(891\) −0.512480 −0.0171687
\(892\) 0 0
\(893\) 8.12802 0.271994
\(894\) 0 0
\(895\) −86.0265 −2.87555
\(896\) 0 0
\(897\) 1.30504 0.0435739
\(898\) 0 0
\(899\) −9.99126 −0.333227
\(900\) 0 0
\(901\) −5.65248 −0.188311
\(902\) 0 0
\(903\) 32.9914 1.09789
\(904\) 0 0
\(905\) −5.41113 −0.179872
\(906\) 0 0
\(907\) −3.86210 −0.128239 −0.0641195 0.997942i \(-0.520424\pi\)
−0.0641195 + 0.997942i \(0.520424\pi\)
\(908\) 0 0
\(909\) −4.25125 −0.141005
\(910\) 0 0
\(911\) −18.0805 −0.599035 −0.299518 0.954091i \(-0.596826\pi\)
−0.299518 + 0.954091i \(0.596826\pi\)
\(912\) 0 0
\(913\) −1.60160 −0.0530052
\(914\) 0 0
\(915\) −5.49685 −0.181720
\(916\) 0 0
\(917\) −33.8676 −1.11841
\(918\) 0 0
\(919\) −16.7072 −0.551118 −0.275559 0.961284i \(-0.588863\pi\)
−0.275559 + 0.961284i \(0.588863\pi\)
\(920\) 0 0
\(921\) 7.30538 0.240720
\(922\) 0 0
\(923\) 4.05305 0.133408
\(924\) 0 0
\(925\) 50.6852 1.66652
\(926\) 0 0
\(927\) 1.64310 0.0539664
\(928\) 0 0
\(929\) −40.5918 −1.33178 −0.665888 0.746052i \(-0.731947\pi\)
−0.665888 + 0.746052i \(0.731947\pi\)
\(930\) 0 0
\(931\) −45.6716 −1.49683
\(932\) 0 0
\(933\) −26.7279 −0.875032
\(934\) 0 0
\(935\) −4.46064 −0.145878
\(936\) 0 0
\(937\) 35.3223 1.15393 0.576965 0.816769i \(-0.304237\pi\)
0.576965 + 0.816769i \(0.304237\pi\)
\(938\) 0 0
\(939\) 19.0884 0.622925
\(940\) 0 0
\(941\) −20.9506 −0.682971 −0.341485 0.939887i \(-0.610930\pi\)
−0.341485 + 0.939887i \(0.610930\pi\)
\(942\) 0 0
\(943\) 6.78840 0.221061
\(944\) 0 0
\(945\) −16.0063 −0.520684
\(946\) 0 0
\(947\) 38.6755 1.25678 0.628392 0.777897i \(-0.283713\pi\)
0.628392 + 0.777897i \(0.283713\pi\)
\(948\) 0 0
\(949\) −1.11153 −0.0360817
\(950\) 0 0
\(951\) 0.844016 0.0273691
\(952\) 0 0
\(953\) −2.98349 −0.0966446 −0.0483223 0.998832i \(-0.515387\pi\)
−0.0483223 + 0.998832i \(0.515387\pi\)
\(954\) 0 0
\(955\) 6.48042 0.209701
\(956\) 0 0
\(957\) 0.883258 0.0285517
\(958\) 0 0
\(959\) −40.1363 −1.29607
\(960\) 0 0
\(961\) 2.60624 0.0840723
\(962\) 0 0
\(963\) −19.3290 −0.622870
\(964\) 0 0
\(965\) −68.7379 −2.21275
\(966\) 0 0
\(967\) −12.7366 −0.409580 −0.204790 0.978806i \(-0.565651\pi\)
−0.204790 + 0.978806i \(0.565651\pi\)
\(968\) 0 0
\(969\) −8.13245 −0.261252
\(970\) 0 0
\(971\) −26.6633 −0.855668 −0.427834 0.903857i \(-0.640723\pi\)
−0.427834 + 0.903857i \(0.640723\pi\)
\(972\) 0 0
\(973\) 14.5062 0.465047
\(974\) 0 0
\(975\) 2.34692 0.0751615
\(976\) 0 0
\(977\) −10.7975 −0.345443 −0.172721 0.984971i \(-0.555256\pi\)
−0.172721 + 0.984971i \(0.555256\pi\)
\(978\) 0 0
\(979\) −7.97129 −0.254763
\(980\) 0 0
\(981\) 0.467960 0.0149408
\(982\) 0 0
\(983\) 12.3650 0.394383 0.197192 0.980365i \(-0.436818\pi\)
0.197192 + 0.980365i \(0.436818\pi\)
\(984\) 0 0
\(985\) −87.4157 −2.78529
\(986\) 0 0
\(987\) −11.4090 −0.363153
\(988\) 0 0
\(989\) 28.8475 0.917299
\(990\) 0 0
\(991\) −20.4354 −0.649151 −0.324576 0.945860i \(-0.605221\pi\)
−0.324576 + 0.945860i \(0.605221\pi\)
\(992\) 0 0
\(993\) −0.805305 −0.0255556
\(994\) 0 0
\(995\) 81.2121 2.57460
\(996\) 0 0
\(997\) 33.1561 1.05006 0.525032 0.851083i \(-0.324053\pi\)
0.525032 + 0.851083i \(0.324053\pi\)
\(998\) 0 0
\(999\) −7.03510 −0.222581
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 8016.2.a.bf.1.2 12
4.3 odd 2 4008.2.a.l.1.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.l.1.2 12 4.3 odd 2
8016.2.a.bf.1.2 12 1.1 even 1 trivial