Properties

Label 8046.2.a.k.1.5
Level $8046$
Weight $2$
Character 8046.1
Self dual yes
Analytic conductor $64.248$
Analytic rank $0$
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] = [8046,2,Mod(1,8046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8046, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8046.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: \(64.2476334663\)
Analytic rank: \(0\)
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: \( x^{12} - 3 x^{11} - 31 x^{10} + 82 x^{9} + 334 x^{8} - 684 x^{7} - 1561 x^{6} + 1551 x^{5} + 3573 x^{4} + 345 x^{3} - 1607 x^{2} - 594 x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.815716\) of defining polynomial
Character \(\chi\) \(=\) 8046.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -0.815716 q^{5} -4.13057 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -0.815716 q^{5} -4.13057 q^{7} -1.00000 q^{8} +0.815716 q^{10} -5.38614 q^{11} +5.53585 q^{13} +4.13057 q^{14} +1.00000 q^{16} +1.97728 q^{17} -7.77630 q^{19} -0.815716 q^{20} +5.38614 q^{22} +4.49804 q^{23} -4.33461 q^{25} -5.53585 q^{26} -4.13057 q^{28} +4.21302 q^{29} -6.64708 q^{31} -1.00000 q^{32} -1.97728 q^{34} +3.36937 q^{35} -1.33356 q^{37} +7.77630 q^{38} +0.815716 q^{40} -4.44186 q^{41} -10.7772 q^{43} -5.38614 q^{44} -4.49804 q^{46} -3.59563 q^{47} +10.0616 q^{49} +4.33461 q^{50} +5.53585 q^{52} +6.44635 q^{53} +4.39356 q^{55} +4.13057 q^{56} -4.21302 q^{58} -5.75290 q^{59} -5.55640 q^{61} +6.64708 q^{62} +1.00000 q^{64} -4.51568 q^{65} +9.76096 q^{67} +1.97728 q^{68} -3.36937 q^{70} +1.83453 q^{71} -3.66077 q^{73} +1.33356 q^{74} -7.77630 q^{76} +22.2478 q^{77} -9.21104 q^{79} -0.815716 q^{80} +4.44186 q^{82} +1.93224 q^{83} -1.61290 q^{85} +10.7772 q^{86} +5.38614 q^{88} -14.0973 q^{89} -22.8662 q^{91} +4.49804 q^{92} +3.59563 q^{94} +6.34325 q^{95} -8.98324 q^{97} -10.0616 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{4} + 3 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{4} + 3 q^{5} - 6 q^{7} - 12 q^{8} - 3 q^{10} + 14 q^{11} - 3 q^{13} + 6 q^{14} + 12 q^{16} + 8 q^{17} - 4 q^{19} + 3 q^{20} - 14 q^{22} + 13 q^{23} + 11 q^{25} + 3 q^{26} - 6 q^{28} + 23 q^{29} - 14 q^{31} - 12 q^{32} - 8 q^{34} + 32 q^{35} - 19 q^{37} + 4 q^{38} - 3 q^{40} + 30 q^{41} - 15 q^{43} + 14 q^{44} - 13 q^{46} - q^{47} + 14 q^{49} - 11 q^{50} - 3 q^{52} + 16 q^{53} - 7 q^{55} + 6 q^{56} - 23 q^{58} + 26 q^{59} - 16 q^{61} + 14 q^{62} + 12 q^{64} + 8 q^{65} - 39 q^{67} + 8 q^{68} - 32 q^{70} + 15 q^{71} - 2 q^{73} + 19 q^{74} - 4 q^{76} + 34 q^{77} - 13 q^{79} + 3 q^{80} - 30 q^{82} + 6 q^{83} - 11 q^{85} + 15 q^{86} - 14 q^{88} + 18 q^{89} - 35 q^{91} + 13 q^{92} + q^{94} + 51 q^{95} + 19 q^{97} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −0.815716 −0.364799 −0.182400 0.983224i \(-0.558386\pi\)
−0.182400 + 0.983224i \(0.558386\pi\)
\(6\) 0 0
\(7\) −4.13057 −1.56121 −0.780604 0.625026i \(-0.785089\pi\)
−0.780604 + 0.625026i \(0.785089\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0.815716 0.257952
\(11\) −5.38614 −1.62398 −0.811991 0.583671i \(-0.801616\pi\)
−0.811991 + 0.583671i \(0.801616\pi\)
\(12\) 0 0
\(13\) 5.53585 1.53537 0.767684 0.640829i \(-0.221409\pi\)
0.767684 + 0.640829i \(0.221409\pi\)
\(14\) 4.13057 1.10394
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.97728 0.479561 0.239781 0.970827i \(-0.422924\pi\)
0.239781 + 0.970827i \(0.422924\pi\)
\(18\) 0 0
\(19\) −7.77630 −1.78401 −0.892003 0.452030i \(-0.850700\pi\)
−0.892003 + 0.452030i \(0.850700\pi\)
\(20\) −0.815716 −0.182400
\(21\) 0 0
\(22\) 5.38614 1.14833
\(23\) 4.49804 0.937906 0.468953 0.883223i \(-0.344631\pi\)
0.468953 + 0.883223i \(0.344631\pi\)
\(24\) 0 0
\(25\) −4.33461 −0.866922
\(26\) −5.53585 −1.08567
\(27\) 0 0
\(28\) −4.13057 −0.780604
\(29\) 4.21302 0.782339 0.391169 0.920319i \(-0.372071\pi\)
0.391169 + 0.920319i \(0.372071\pi\)
\(30\) 0 0
\(31\) −6.64708 −1.19385 −0.596925 0.802297i \(-0.703611\pi\)
−0.596925 + 0.802297i \(0.703611\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −1.97728 −0.339101
\(35\) 3.36937 0.569528
\(36\) 0 0
\(37\) −1.33356 −0.219236 −0.109618 0.993974i \(-0.534963\pi\)
−0.109618 + 0.993974i \(0.534963\pi\)
\(38\) 7.77630 1.26148
\(39\) 0 0
\(40\) 0.815716 0.128976
\(41\) −4.44186 −0.693702 −0.346851 0.937920i \(-0.612749\pi\)
−0.346851 + 0.937920i \(0.612749\pi\)
\(42\) 0 0
\(43\) −10.7772 −1.64350 −0.821752 0.569846i \(-0.807003\pi\)
−0.821752 + 0.569846i \(0.807003\pi\)
\(44\) −5.38614 −0.811991
\(45\) 0 0
\(46\) −4.49804 −0.663200
\(47\) −3.59563 −0.524477 −0.262238 0.965003i \(-0.584461\pi\)
−0.262238 + 0.965003i \(0.584461\pi\)
\(48\) 0 0
\(49\) 10.0616 1.43737
\(50\) 4.33461 0.613006
\(51\) 0 0
\(52\) 5.53585 0.767684
\(53\) 6.44635 0.885475 0.442737 0.896651i \(-0.354007\pi\)
0.442737 + 0.896651i \(0.354007\pi\)
\(54\) 0 0
\(55\) 4.39356 0.592427
\(56\) 4.13057 0.551971
\(57\) 0 0
\(58\) −4.21302 −0.553197
\(59\) −5.75290 −0.748964 −0.374482 0.927234i \(-0.622179\pi\)
−0.374482 + 0.927234i \(0.622179\pi\)
\(60\) 0 0
\(61\) −5.55640 −0.711424 −0.355712 0.934596i \(-0.615762\pi\)
−0.355712 + 0.934596i \(0.615762\pi\)
\(62\) 6.64708 0.844180
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.51568 −0.560101
\(66\) 0 0
\(67\) 9.76096 1.19249 0.596246 0.802802i \(-0.296658\pi\)
0.596246 + 0.802802i \(0.296658\pi\)
\(68\) 1.97728 0.239781
\(69\) 0 0
\(70\) −3.36937 −0.402717
\(71\) 1.83453 0.217719 0.108859 0.994057i \(-0.465280\pi\)
0.108859 + 0.994057i \(0.465280\pi\)
\(72\) 0 0
\(73\) −3.66077 −0.428461 −0.214231 0.976783i \(-0.568724\pi\)
−0.214231 + 0.976783i \(0.568724\pi\)
\(74\) 1.33356 0.155024
\(75\) 0 0
\(76\) −7.77630 −0.892003
\(77\) 22.2478 2.53537
\(78\) 0 0
\(79\) −9.21104 −1.03632 −0.518161 0.855283i \(-0.673383\pi\)
−0.518161 + 0.855283i \(0.673383\pi\)
\(80\) −0.815716 −0.0911998
\(81\) 0 0
\(82\) 4.44186 0.490521
\(83\) 1.93224 0.212091 0.106046 0.994361i \(-0.466181\pi\)
0.106046 + 0.994361i \(0.466181\pi\)
\(84\) 0 0
\(85\) −1.61290 −0.174944
\(86\) 10.7772 1.16213
\(87\) 0 0
\(88\) 5.38614 0.574164
\(89\) −14.0973 −1.49431 −0.747156 0.664649i \(-0.768581\pi\)
−0.747156 + 0.664649i \(0.768581\pi\)
\(90\) 0 0
\(91\) −22.8662 −2.39703
\(92\) 4.49804 0.468953
\(93\) 0 0
\(94\) 3.59563 0.370861
\(95\) 6.34325 0.650804
\(96\) 0 0
\(97\) −8.98324 −0.912110 −0.456055 0.889952i \(-0.650738\pi\)
−0.456055 + 0.889952i \(0.650738\pi\)
\(98\) −10.0616 −1.01638
\(99\) 0 0
\(100\) −4.33461 −0.433461
\(101\) −1.31930 −0.131275 −0.0656377 0.997844i \(-0.520908\pi\)
−0.0656377 + 0.997844i \(0.520908\pi\)
\(102\) 0 0
\(103\) −3.52663 −0.347489 −0.173745 0.984791i \(-0.555587\pi\)
−0.173745 + 0.984791i \(0.555587\pi\)
\(104\) −5.53585 −0.542835
\(105\) 0 0
\(106\) −6.44635 −0.626125
\(107\) 0.332526 0.0321465 0.0160732 0.999871i \(-0.494884\pi\)
0.0160732 + 0.999871i \(0.494884\pi\)
\(108\) 0 0
\(109\) −3.79744 −0.363729 −0.181864 0.983324i \(-0.558213\pi\)
−0.181864 + 0.983324i \(0.558213\pi\)
\(110\) −4.39356 −0.418909
\(111\) 0 0
\(112\) −4.13057 −0.390302
\(113\) −0.758572 −0.0713604 −0.0356802 0.999363i \(-0.511360\pi\)
−0.0356802 + 0.999363i \(0.511360\pi\)
\(114\) 0 0
\(115\) −3.66912 −0.342147
\(116\) 4.21302 0.391169
\(117\) 0 0
\(118\) 5.75290 0.529598
\(119\) −8.16730 −0.748695
\(120\) 0 0
\(121\) 18.0105 1.63731
\(122\) 5.55640 0.503053
\(123\) 0 0
\(124\) −6.64708 −0.596925
\(125\) 7.61439 0.681051
\(126\) 0 0
\(127\) 11.5200 1.02223 0.511117 0.859511i \(-0.329232\pi\)
0.511117 + 0.859511i \(0.329232\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 4.51568 0.396051
\(131\) −0.430517 −0.0376145 −0.0188072 0.999823i \(-0.505987\pi\)
−0.0188072 + 0.999823i \(0.505987\pi\)
\(132\) 0 0
\(133\) 32.1205 2.78520
\(134\) −9.76096 −0.843219
\(135\) 0 0
\(136\) −1.97728 −0.169551
\(137\) −18.9496 −1.61897 −0.809487 0.587138i \(-0.800255\pi\)
−0.809487 + 0.587138i \(0.800255\pi\)
\(138\) 0 0
\(139\) −10.9711 −0.930561 −0.465280 0.885163i \(-0.654046\pi\)
−0.465280 + 0.885163i \(0.654046\pi\)
\(140\) 3.36937 0.284764
\(141\) 0 0
\(142\) −1.83453 −0.153950
\(143\) −29.8168 −2.49341
\(144\) 0 0
\(145\) −3.43663 −0.285396
\(146\) 3.66077 0.302968
\(147\) 0 0
\(148\) −1.33356 −0.109618
\(149\) −1.00000 −0.0819232
\(150\) 0 0
\(151\) −6.47150 −0.526643 −0.263322 0.964708i \(-0.584818\pi\)
−0.263322 + 0.964708i \(0.584818\pi\)
\(152\) 7.77630 0.630741
\(153\) 0 0
\(154\) −22.2478 −1.79278
\(155\) 5.42213 0.435516
\(156\) 0 0
\(157\) 12.0774 0.963883 0.481941 0.876203i \(-0.339932\pi\)
0.481941 + 0.876203i \(0.339932\pi\)
\(158\) 9.21104 0.732791
\(159\) 0 0
\(160\) 0.815716 0.0644880
\(161\) −18.5795 −1.46427
\(162\) 0 0
\(163\) −5.69644 −0.446179 −0.223090 0.974798i \(-0.571614\pi\)
−0.223090 + 0.974798i \(0.571614\pi\)
\(164\) −4.44186 −0.346851
\(165\) 0 0
\(166\) −1.93224 −0.149971
\(167\) 17.9694 1.39051 0.695255 0.718763i \(-0.255291\pi\)
0.695255 + 0.718763i \(0.255291\pi\)
\(168\) 0 0
\(169\) 17.6456 1.35735
\(170\) 1.61290 0.123704
\(171\) 0 0
\(172\) −10.7772 −0.821752
\(173\) 20.4869 1.55759 0.778796 0.627278i \(-0.215831\pi\)
0.778796 + 0.627278i \(0.215831\pi\)
\(174\) 0 0
\(175\) 17.9044 1.35345
\(176\) −5.38614 −0.405995
\(177\) 0 0
\(178\) 14.0973 1.05664
\(179\) 14.2118 1.06224 0.531120 0.847296i \(-0.321771\pi\)
0.531120 + 0.847296i \(0.321771\pi\)
\(180\) 0 0
\(181\) 2.73904 0.203592 0.101796 0.994805i \(-0.467541\pi\)
0.101796 + 0.994805i \(0.467541\pi\)
\(182\) 22.8662 1.69496
\(183\) 0 0
\(184\) −4.49804 −0.331600
\(185\) 1.08781 0.0799772
\(186\) 0 0
\(187\) −10.6499 −0.778799
\(188\) −3.59563 −0.262238
\(189\) 0 0
\(190\) −6.34325 −0.460188
\(191\) −9.80574 −0.709518 −0.354759 0.934958i \(-0.615437\pi\)
−0.354759 + 0.934958i \(0.615437\pi\)
\(192\) 0 0
\(193\) −1.34693 −0.0969545 −0.0484772 0.998824i \(-0.515437\pi\)
−0.0484772 + 0.998824i \(0.515437\pi\)
\(194\) 8.98324 0.644959
\(195\) 0 0
\(196\) 10.0616 0.718686
\(197\) 22.1831 1.58048 0.790239 0.612798i \(-0.209956\pi\)
0.790239 + 0.612798i \(0.209956\pi\)
\(198\) 0 0
\(199\) −24.9250 −1.76689 −0.883444 0.468538i \(-0.844781\pi\)
−0.883444 + 0.468538i \(0.844781\pi\)
\(200\) 4.33461 0.306503
\(201\) 0 0
\(202\) 1.31930 0.0928257
\(203\) −17.4022 −1.22139
\(204\) 0 0
\(205\) 3.62329 0.253062
\(206\) 3.52663 0.245712
\(207\) 0 0
\(208\) 5.53585 0.383842
\(209\) 41.8842 2.89719
\(210\) 0 0
\(211\) 5.38099 0.370442 0.185221 0.982697i \(-0.440700\pi\)
0.185221 + 0.982697i \(0.440700\pi\)
\(212\) 6.44635 0.442737
\(213\) 0 0
\(214\) −0.332526 −0.0227310
\(215\) 8.79111 0.599549
\(216\) 0 0
\(217\) 27.4562 1.86385
\(218\) 3.79744 0.257195
\(219\) 0 0
\(220\) 4.39356 0.296213
\(221\) 10.9459 0.736303
\(222\) 0 0
\(223\) 12.5623 0.841237 0.420618 0.907238i \(-0.361813\pi\)
0.420618 + 0.907238i \(0.361813\pi\)
\(224\) 4.13057 0.275985
\(225\) 0 0
\(226\) 0.758572 0.0504594
\(227\) 3.13674 0.208193 0.104096 0.994567i \(-0.466805\pi\)
0.104096 + 0.994567i \(0.466805\pi\)
\(228\) 0 0
\(229\) 3.77927 0.249741 0.124870 0.992173i \(-0.460148\pi\)
0.124870 + 0.992173i \(0.460148\pi\)
\(230\) 3.66912 0.241935
\(231\) 0 0
\(232\) −4.21302 −0.276598
\(233\) −1.22631 −0.0803383 −0.0401691 0.999193i \(-0.512790\pi\)
−0.0401691 + 0.999193i \(0.512790\pi\)
\(234\) 0 0
\(235\) 2.93301 0.191329
\(236\) −5.75290 −0.374482
\(237\) 0 0
\(238\) 8.16730 0.529407
\(239\) 9.21295 0.595936 0.297968 0.954576i \(-0.403691\pi\)
0.297968 + 0.954576i \(0.403691\pi\)
\(240\) 0 0
\(241\) −20.6802 −1.33213 −0.666064 0.745895i \(-0.732022\pi\)
−0.666064 + 0.745895i \(0.732022\pi\)
\(242\) −18.0105 −1.15776
\(243\) 0 0
\(244\) −5.55640 −0.355712
\(245\) −8.20741 −0.524352
\(246\) 0 0
\(247\) −43.0484 −2.73910
\(248\) 6.64708 0.422090
\(249\) 0 0
\(250\) −7.61439 −0.481576
\(251\) −1.30517 −0.0823814 −0.0411907 0.999151i \(-0.513115\pi\)
−0.0411907 + 0.999151i \(0.513115\pi\)
\(252\) 0 0
\(253\) −24.2271 −1.52314
\(254\) −11.5200 −0.722829
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.03800 −0.376640 −0.188320 0.982108i \(-0.560304\pi\)
−0.188320 + 0.982108i \(0.560304\pi\)
\(258\) 0 0
\(259\) 5.50837 0.342274
\(260\) −4.51568 −0.280050
\(261\) 0 0
\(262\) 0.430517 0.0265974
\(263\) 31.5018 1.94249 0.971243 0.238091i \(-0.0765216\pi\)
0.971243 + 0.238091i \(0.0765216\pi\)
\(264\) 0 0
\(265\) −5.25839 −0.323020
\(266\) −32.1205 −1.96944
\(267\) 0 0
\(268\) 9.76096 0.596246
\(269\) 25.9304 1.58100 0.790501 0.612460i \(-0.209820\pi\)
0.790501 + 0.612460i \(0.209820\pi\)
\(270\) 0 0
\(271\) 2.66121 0.161657 0.0808285 0.996728i \(-0.474243\pi\)
0.0808285 + 0.996728i \(0.474243\pi\)
\(272\) 1.97728 0.119890
\(273\) 0 0
\(274\) 18.9496 1.14479
\(275\) 23.3468 1.40786
\(276\) 0 0
\(277\) 31.3539 1.88387 0.941936 0.335792i \(-0.109004\pi\)
0.941936 + 0.335792i \(0.109004\pi\)
\(278\) 10.9711 0.658006
\(279\) 0 0
\(280\) −3.36937 −0.201358
\(281\) 25.4005 1.51526 0.757632 0.652682i \(-0.226356\pi\)
0.757632 + 0.652682i \(0.226356\pi\)
\(282\) 0 0
\(283\) −21.6996 −1.28991 −0.644954 0.764221i \(-0.723124\pi\)
−0.644954 + 0.764221i \(0.723124\pi\)
\(284\) 1.83453 0.108859
\(285\) 0 0
\(286\) 29.8168 1.76311
\(287\) 18.3474 1.08301
\(288\) 0 0
\(289\) −13.0904 −0.770021
\(290\) 3.43663 0.201806
\(291\) 0 0
\(292\) −3.66077 −0.214231
\(293\) −31.8104 −1.85839 −0.929193 0.369594i \(-0.879497\pi\)
−0.929193 + 0.369594i \(0.879497\pi\)
\(294\) 0 0
\(295\) 4.69273 0.273221
\(296\) 1.33356 0.0775118
\(297\) 0 0
\(298\) 1.00000 0.0579284
\(299\) 24.9005 1.44003
\(300\) 0 0
\(301\) 44.5159 2.56585
\(302\) 6.47150 0.372393
\(303\) 0 0
\(304\) −7.77630 −0.446001
\(305\) 4.53244 0.259527
\(306\) 0 0
\(307\) −23.8509 −1.36124 −0.680622 0.732635i \(-0.738290\pi\)
−0.680622 + 0.732635i \(0.738290\pi\)
\(308\) 22.2478 1.26769
\(309\) 0 0
\(310\) −5.42213 −0.307956
\(311\) 26.7058 1.51435 0.757173 0.653215i \(-0.226580\pi\)
0.757173 + 0.653215i \(0.226580\pi\)
\(312\) 0 0
\(313\) 19.6816 1.11247 0.556235 0.831025i \(-0.312245\pi\)
0.556235 + 0.831025i \(0.312245\pi\)
\(314\) −12.0774 −0.681568
\(315\) 0 0
\(316\) −9.21104 −0.518161
\(317\) −9.46320 −0.531507 −0.265753 0.964041i \(-0.585621\pi\)
−0.265753 + 0.964041i \(0.585621\pi\)
\(318\) 0 0
\(319\) −22.6919 −1.27050
\(320\) −0.815716 −0.0455999
\(321\) 0 0
\(322\) 18.5795 1.03539
\(323\) −15.3759 −0.855540
\(324\) 0 0
\(325\) −23.9957 −1.33104
\(326\) 5.69644 0.315496
\(327\) 0 0
\(328\) 4.44186 0.245261
\(329\) 14.8520 0.818818
\(330\) 0 0
\(331\) 10.3797 0.570518 0.285259 0.958450i \(-0.407920\pi\)
0.285259 + 0.958450i \(0.407920\pi\)
\(332\) 1.93224 0.106046
\(333\) 0 0
\(334\) −17.9694 −0.983240
\(335\) −7.96217 −0.435020
\(336\) 0 0
\(337\) 32.3299 1.76112 0.880561 0.473934i \(-0.157166\pi\)
0.880561 + 0.473934i \(0.157166\pi\)
\(338\) −17.6456 −0.959795
\(339\) 0 0
\(340\) −1.61290 −0.0874718
\(341\) 35.8021 1.93879
\(342\) 0 0
\(343\) −12.6462 −0.682830
\(344\) 10.7772 0.581066
\(345\) 0 0
\(346\) −20.4869 −1.10138
\(347\) −21.3366 −1.14541 −0.572704 0.819763i \(-0.694105\pi\)
−0.572704 + 0.819763i \(0.694105\pi\)
\(348\) 0 0
\(349\) −5.35420 −0.286604 −0.143302 0.989679i \(-0.545772\pi\)
−0.143302 + 0.989679i \(0.545772\pi\)
\(350\) −17.9044 −0.957031
\(351\) 0 0
\(352\) 5.38614 0.287082
\(353\) 2.67919 0.142599 0.0712995 0.997455i \(-0.477285\pi\)
0.0712995 + 0.997455i \(0.477285\pi\)
\(354\) 0 0
\(355\) −1.49646 −0.0794236
\(356\) −14.0973 −0.747156
\(357\) 0 0
\(358\) −14.2118 −0.751118
\(359\) 35.7281 1.88566 0.942830 0.333275i \(-0.108154\pi\)
0.942830 + 0.333275i \(0.108154\pi\)
\(360\) 0 0
\(361\) 41.4708 2.18268
\(362\) −2.73904 −0.143961
\(363\) 0 0
\(364\) −22.8662 −1.19851
\(365\) 2.98615 0.156302
\(366\) 0 0
\(367\) 27.5484 1.43801 0.719007 0.695003i \(-0.244597\pi\)
0.719007 + 0.695003i \(0.244597\pi\)
\(368\) 4.49804 0.234477
\(369\) 0 0
\(370\) −1.08781 −0.0565524
\(371\) −26.6271 −1.38241
\(372\) 0 0
\(373\) −14.2867 −0.739740 −0.369870 0.929084i \(-0.620598\pi\)
−0.369870 + 0.929084i \(0.620598\pi\)
\(374\) 10.6499 0.550694
\(375\) 0 0
\(376\) 3.59563 0.185431
\(377\) 23.3227 1.20118
\(378\) 0 0
\(379\) 31.7017 1.62841 0.814203 0.580581i \(-0.197174\pi\)
0.814203 + 0.580581i \(0.197174\pi\)
\(380\) 6.34325 0.325402
\(381\) 0 0
\(382\) 9.80574 0.501705
\(383\) 1.71332 0.0875468 0.0437734 0.999041i \(-0.486062\pi\)
0.0437734 + 0.999041i \(0.486062\pi\)
\(384\) 0 0
\(385\) −18.1479 −0.924902
\(386\) 1.34693 0.0685572
\(387\) 0 0
\(388\) −8.98324 −0.456055
\(389\) −31.3579 −1.58991 −0.794953 0.606671i \(-0.792505\pi\)
−0.794953 + 0.606671i \(0.792505\pi\)
\(390\) 0 0
\(391\) 8.89389 0.449784
\(392\) −10.0616 −0.508188
\(393\) 0 0
\(394\) −22.1831 −1.11757
\(395\) 7.51359 0.378050
\(396\) 0 0
\(397\) −3.58737 −0.180045 −0.0900226 0.995940i \(-0.528694\pi\)
−0.0900226 + 0.995940i \(0.528694\pi\)
\(398\) 24.9250 1.24938
\(399\) 0 0
\(400\) −4.33461 −0.216730
\(401\) 11.6963 0.584087 0.292044 0.956405i \(-0.405665\pi\)
0.292044 + 0.956405i \(0.405665\pi\)
\(402\) 0 0
\(403\) −36.7972 −1.83300
\(404\) −1.31930 −0.0656377
\(405\) 0 0
\(406\) 17.4022 0.863656
\(407\) 7.18275 0.356036
\(408\) 0 0
\(409\) −22.7319 −1.12402 −0.562010 0.827131i \(-0.689972\pi\)
−0.562010 + 0.827131i \(0.689972\pi\)
\(410\) −3.62329 −0.178942
\(411\) 0 0
\(412\) −3.52663 −0.173745
\(413\) 23.7628 1.16929
\(414\) 0 0
\(415\) −1.57616 −0.0773707
\(416\) −5.53585 −0.271417
\(417\) 0 0
\(418\) −41.8842 −2.04862
\(419\) −12.6330 −0.617161 −0.308580 0.951198i \(-0.599854\pi\)
−0.308580 + 0.951198i \(0.599854\pi\)
\(420\) 0 0
\(421\) 0.516091 0.0251528 0.0125764 0.999921i \(-0.495997\pi\)
0.0125764 + 0.999921i \(0.495997\pi\)
\(422\) −5.38099 −0.261942
\(423\) 0 0
\(424\) −6.44635 −0.313063
\(425\) −8.57074 −0.415742
\(426\) 0 0
\(427\) 22.9511 1.11068
\(428\) 0.332526 0.0160732
\(429\) 0 0
\(430\) −8.79111 −0.423945
\(431\) 28.7458 1.38464 0.692319 0.721592i \(-0.256589\pi\)
0.692319 + 0.721592i \(0.256589\pi\)
\(432\) 0 0
\(433\) −18.6484 −0.896186 −0.448093 0.893987i \(-0.647897\pi\)
−0.448093 + 0.893987i \(0.647897\pi\)
\(434\) −27.4562 −1.31794
\(435\) 0 0
\(436\) −3.79744 −0.181864
\(437\) −34.9781 −1.67323
\(438\) 0 0
\(439\) 11.6890 0.557885 0.278942 0.960308i \(-0.410016\pi\)
0.278942 + 0.960308i \(0.410016\pi\)
\(440\) −4.39356 −0.209455
\(441\) 0 0
\(442\) −10.9459 −0.520645
\(443\) −19.7049 −0.936207 −0.468104 0.883674i \(-0.655063\pi\)
−0.468104 + 0.883674i \(0.655063\pi\)
\(444\) 0 0
\(445\) 11.4994 0.545124
\(446\) −12.5623 −0.594844
\(447\) 0 0
\(448\) −4.13057 −0.195151
\(449\) 3.01542 0.142306 0.0711532 0.997465i \(-0.477332\pi\)
0.0711532 + 0.997465i \(0.477332\pi\)
\(450\) 0 0
\(451\) 23.9245 1.12656
\(452\) −0.758572 −0.0356802
\(453\) 0 0
\(454\) −3.13674 −0.147214
\(455\) 18.6523 0.874434
\(456\) 0 0
\(457\) 13.4588 0.629576 0.314788 0.949162i \(-0.398067\pi\)
0.314788 + 0.949162i \(0.398067\pi\)
\(458\) −3.77927 −0.176593
\(459\) 0 0
\(460\) −3.66912 −0.171074
\(461\) 36.1944 1.68574 0.842870 0.538117i \(-0.180864\pi\)
0.842870 + 0.538117i \(0.180864\pi\)
\(462\) 0 0
\(463\) −29.1225 −1.35344 −0.676719 0.736241i \(-0.736599\pi\)
−0.676719 + 0.736241i \(0.736599\pi\)
\(464\) 4.21302 0.195585
\(465\) 0 0
\(466\) 1.22631 0.0568077
\(467\) −34.6912 −1.60532 −0.802658 0.596440i \(-0.796581\pi\)
−0.802658 + 0.596440i \(0.796581\pi\)
\(468\) 0 0
\(469\) −40.3183 −1.86173
\(470\) −2.93301 −0.135290
\(471\) 0 0
\(472\) 5.75290 0.264799
\(473\) 58.0473 2.66902
\(474\) 0 0
\(475\) 33.7072 1.54659
\(476\) −8.16730 −0.374348
\(477\) 0 0
\(478\) −9.21295 −0.421391
\(479\) −38.0159 −1.73699 −0.868494 0.495699i \(-0.834912\pi\)
−0.868494 + 0.495699i \(0.834912\pi\)
\(480\) 0 0
\(481\) −7.38240 −0.336608
\(482\) 20.6802 0.941957
\(483\) 0 0
\(484\) 18.0105 0.818657
\(485\) 7.32777 0.332737
\(486\) 0 0
\(487\) 16.5858 0.751575 0.375787 0.926706i \(-0.377372\pi\)
0.375787 + 0.926706i \(0.377372\pi\)
\(488\) 5.55640 0.251526
\(489\) 0 0
\(490\) 8.20741 0.370773
\(491\) 32.0610 1.44689 0.723446 0.690381i \(-0.242557\pi\)
0.723446 + 0.690381i \(0.242557\pi\)
\(492\) 0 0
\(493\) 8.33033 0.375179
\(494\) 43.0484 1.93684
\(495\) 0 0
\(496\) −6.64708 −0.298463
\(497\) −7.57766 −0.339905
\(498\) 0 0
\(499\) −35.4267 −1.58592 −0.792959 0.609275i \(-0.791460\pi\)
−0.792959 + 0.609275i \(0.791460\pi\)
\(500\) 7.61439 0.340526
\(501\) 0 0
\(502\) 1.30517 0.0582524
\(503\) −0.256073 −0.0114177 −0.00570886 0.999984i \(-0.501817\pi\)
−0.00570886 + 0.999984i \(0.501817\pi\)
\(504\) 0 0
\(505\) 1.07617 0.0478892
\(506\) 24.2271 1.07702
\(507\) 0 0
\(508\) 11.5200 0.511117
\(509\) 3.41418 0.151331 0.0756655 0.997133i \(-0.475892\pi\)
0.0756655 + 0.997133i \(0.475892\pi\)
\(510\) 0 0
\(511\) 15.1211 0.668917
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 6.03800 0.266325
\(515\) 2.87673 0.126764
\(516\) 0 0
\(517\) 19.3666 0.851740
\(518\) −5.50837 −0.242024
\(519\) 0 0
\(520\) 4.51568 0.198026
\(521\) −17.3318 −0.759321 −0.379661 0.925126i \(-0.623959\pi\)
−0.379661 + 0.925126i \(0.623959\pi\)
\(522\) 0 0
\(523\) 26.1909 1.14525 0.572623 0.819819i \(-0.305926\pi\)
0.572623 + 0.819819i \(0.305926\pi\)
\(524\) −0.430517 −0.0188072
\(525\) 0 0
\(526\) −31.5018 −1.37354
\(527\) −13.1432 −0.572525
\(528\) 0 0
\(529\) −2.76763 −0.120332
\(530\) 5.25839 0.228410
\(531\) 0 0
\(532\) 32.1205 1.39260
\(533\) −24.5895 −1.06509
\(534\) 0 0
\(535\) −0.271247 −0.0117270
\(536\) −9.76096 −0.421609
\(537\) 0 0
\(538\) −25.9304 −1.11794
\(539\) −54.1932 −2.33427
\(540\) 0 0
\(541\) 33.7760 1.45214 0.726071 0.687620i \(-0.241344\pi\)
0.726071 + 0.687620i \(0.241344\pi\)
\(542\) −2.66121 −0.114309
\(543\) 0 0
\(544\) −1.97728 −0.0847753
\(545\) 3.09763 0.132688
\(546\) 0 0
\(547\) 29.9610 1.28104 0.640521 0.767941i \(-0.278719\pi\)
0.640521 + 0.767941i \(0.278719\pi\)
\(548\) −18.9496 −0.809487
\(549\) 0 0
\(550\) −23.3468 −0.995510
\(551\) −32.7617 −1.39570
\(552\) 0 0
\(553\) 38.0468 1.61792
\(554\) −31.3539 −1.33210
\(555\) 0 0
\(556\) −10.9711 −0.465280
\(557\) −42.0778 −1.78289 −0.891446 0.453126i \(-0.850309\pi\)
−0.891446 + 0.453126i \(0.850309\pi\)
\(558\) 0 0
\(559\) −59.6608 −2.52338
\(560\) 3.36937 0.142382
\(561\) 0 0
\(562\) −25.4005 −1.07145
\(563\) 3.18789 0.134354 0.0671768 0.997741i \(-0.478601\pi\)
0.0671768 + 0.997741i \(0.478601\pi\)
\(564\) 0 0
\(565\) 0.618779 0.0260322
\(566\) 21.6996 0.912103
\(567\) 0 0
\(568\) −1.83453 −0.0769752
\(569\) −13.5411 −0.567674 −0.283837 0.958872i \(-0.591608\pi\)
−0.283837 + 0.958872i \(0.591608\pi\)
\(570\) 0 0
\(571\) −39.7747 −1.66452 −0.832259 0.554387i \(-0.812953\pi\)
−0.832259 + 0.554387i \(0.812953\pi\)
\(572\) −29.8168 −1.24670
\(573\) 0 0
\(574\) −18.3474 −0.765806
\(575\) −19.4972 −0.813091
\(576\) 0 0
\(577\) −33.9722 −1.41428 −0.707140 0.707073i \(-0.750015\pi\)
−0.707140 + 0.707073i \(0.750015\pi\)
\(578\) 13.0904 0.544487
\(579\) 0 0
\(580\) −3.43663 −0.142698
\(581\) −7.98127 −0.331119
\(582\) 0 0
\(583\) −34.7209 −1.43799
\(584\) 3.66077 0.151484
\(585\) 0 0
\(586\) 31.8104 1.31408
\(587\) 9.05854 0.373886 0.186943 0.982371i \(-0.440142\pi\)
0.186943 + 0.982371i \(0.440142\pi\)
\(588\) 0 0
\(589\) 51.6897 2.12984
\(590\) −4.69273 −0.193197
\(591\) 0 0
\(592\) −1.33356 −0.0548091
\(593\) −13.8029 −0.566818 −0.283409 0.958999i \(-0.591465\pi\)
−0.283409 + 0.958999i \(0.591465\pi\)
\(594\) 0 0
\(595\) 6.66219 0.273123
\(596\) −1.00000 −0.0409616
\(597\) 0 0
\(598\) −24.9005 −1.01826
\(599\) −28.9207 −1.18167 −0.590834 0.806793i \(-0.701201\pi\)
−0.590834 + 0.806793i \(0.701201\pi\)
\(600\) 0 0
\(601\) −25.3839 −1.03543 −0.517716 0.855553i \(-0.673217\pi\)
−0.517716 + 0.855553i \(0.673217\pi\)
\(602\) −44.5159 −1.81433
\(603\) 0 0
\(604\) −6.47150 −0.263322
\(605\) −14.6914 −0.597291
\(606\) 0 0
\(607\) 31.3343 1.27182 0.635909 0.771764i \(-0.280625\pi\)
0.635909 + 0.771764i \(0.280625\pi\)
\(608\) 7.77630 0.315371
\(609\) 0 0
\(610\) −4.53244 −0.183513
\(611\) −19.9049 −0.805265
\(612\) 0 0
\(613\) −0.743160 −0.0300160 −0.0150080 0.999887i \(-0.504777\pi\)
−0.0150080 + 0.999887i \(0.504777\pi\)
\(614\) 23.8509 0.962545
\(615\) 0 0
\(616\) −22.2478 −0.896390
\(617\) 32.4336 1.30573 0.652863 0.757476i \(-0.273568\pi\)
0.652863 + 0.757476i \(0.273568\pi\)
\(618\) 0 0
\(619\) 2.74612 0.110376 0.0551879 0.998476i \(-0.482424\pi\)
0.0551879 + 0.998476i \(0.482424\pi\)
\(620\) 5.42213 0.217758
\(621\) 0 0
\(622\) −26.7058 −1.07080
\(623\) 58.2299 2.33293
\(624\) 0 0
\(625\) 15.4619 0.618475
\(626\) −19.6816 −0.786636
\(627\) 0 0
\(628\) 12.0774 0.481941
\(629\) −2.63683 −0.105137
\(630\) 0 0
\(631\) 37.8303 1.50600 0.752999 0.658021i \(-0.228606\pi\)
0.752999 + 0.658021i \(0.228606\pi\)
\(632\) 9.21104 0.366395
\(633\) 0 0
\(634\) 9.46320 0.375832
\(635\) −9.39704 −0.372910
\(636\) 0 0
\(637\) 55.6995 2.20690
\(638\) 22.6919 0.898381
\(639\) 0 0
\(640\) 0.815716 0.0322440
\(641\) 20.1362 0.795331 0.397665 0.917531i \(-0.369821\pi\)
0.397665 + 0.917531i \(0.369821\pi\)
\(642\) 0 0
\(643\) −24.4318 −0.963496 −0.481748 0.876310i \(-0.659998\pi\)
−0.481748 + 0.876310i \(0.659998\pi\)
\(644\) −18.5795 −0.732134
\(645\) 0 0
\(646\) 15.3759 0.604958
\(647\) 8.54734 0.336031 0.168015 0.985784i \(-0.446264\pi\)
0.168015 + 0.985784i \(0.446264\pi\)
\(648\) 0 0
\(649\) 30.9859 1.21630
\(650\) 23.9957 0.941190
\(651\) 0 0
\(652\) −5.69644 −0.223090
\(653\) −2.59605 −0.101591 −0.0507957 0.998709i \(-0.516176\pi\)
−0.0507957 + 0.998709i \(0.516176\pi\)
\(654\) 0 0
\(655\) 0.351180 0.0137217
\(656\) −4.44186 −0.173425
\(657\) 0 0
\(658\) −14.8520 −0.578992
\(659\) −7.82570 −0.304846 −0.152423 0.988315i \(-0.548708\pi\)
−0.152423 + 0.988315i \(0.548708\pi\)
\(660\) 0 0
\(661\) 22.3756 0.870312 0.435156 0.900355i \(-0.356693\pi\)
0.435156 + 0.900355i \(0.356693\pi\)
\(662\) −10.3797 −0.403417
\(663\) 0 0
\(664\) −1.93224 −0.0749856
\(665\) −26.2012 −1.01604
\(666\) 0 0
\(667\) 18.9503 0.733760
\(668\) 17.9694 0.695255
\(669\) 0 0
\(670\) 7.96217 0.307605
\(671\) 29.9275 1.15534
\(672\) 0 0
\(673\) 23.4435 0.903681 0.451841 0.892099i \(-0.350768\pi\)
0.451841 + 0.892099i \(0.350768\pi\)
\(674\) −32.3299 −1.24530
\(675\) 0 0
\(676\) 17.6456 0.678677
\(677\) −20.3237 −0.781102 −0.390551 0.920581i \(-0.627715\pi\)
−0.390551 + 0.920581i \(0.627715\pi\)
\(678\) 0 0
\(679\) 37.1059 1.42399
\(680\) 1.61290 0.0618519
\(681\) 0 0
\(682\) −35.8021 −1.37093
\(683\) 42.7978 1.63761 0.818806 0.574070i \(-0.194636\pi\)
0.818806 + 0.574070i \(0.194636\pi\)
\(684\) 0 0
\(685\) 15.4575 0.590600
\(686\) 12.6462 0.482833
\(687\) 0 0
\(688\) −10.7772 −0.410876
\(689\) 35.6860 1.35953
\(690\) 0 0
\(691\) 35.9261 1.36669 0.683346 0.730094i \(-0.260524\pi\)
0.683346 + 0.730094i \(0.260524\pi\)
\(692\) 20.4869 0.778796
\(693\) 0 0
\(694\) 21.3366 0.809925
\(695\) 8.94934 0.339468
\(696\) 0 0
\(697\) −8.78281 −0.332673
\(698\) 5.35420 0.202659
\(699\) 0 0
\(700\) 17.9044 0.676723
\(701\) −41.3499 −1.56176 −0.780882 0.624679i \(-0.785230\pi\)
−0.780882 + 0.624679i \(0.785230\pi\)
\(702\) 0 0
\(703\) 10.3702 0.391119
\(704\) −5.38614 −0.202998
\(705\) 0 0
\(706\) −2.67919 −0.100833
\(707\) 5.44947 0.204948
\(708\) 0 0
\(709\) −19.8015 −0.743662 −0.371831 0.928300i \(-0.621270\pi\)
−0.371831 + 0.928300i \(0.621270\pi\)
\(710\) 1.49646 0.0561610
\(711\) 0 0
\(712\) 14.0973 0.528319
\(713\) −29.8988 −1.11972
\(714\) 0 0
\(715\) 24.3221 0.909593
\(716\) 14.2118 0.531120
\(717\) 0 0
\(718\) −35.7281 −1.33336
\(719\) −33.4253 −1.24655 −0.623276 0.782002i \(-0.714199\pi\)
−0.623276 + 0.782002i \(0.714199\pi\)
\(720\) 0 0
\(721\) 14.5670 0.542503
\(722\) −41.4708 −1.54338
\(723\) 0 0
\(724\) 2.73904 0.101796
\(725\) −18.2618 −0.678226
\(726\) 0 0
\(727\) 9.38165 0.347946 0.173973 0.984750i \(-0.444339\pi\)
0.173973 + 0.984750i \(0.444339\pi\)
\(728\) 22.8662 0.847478
\(729\) 0 0
\(730\) −2.98615 −0.110522
\(731\) −21.3095 −0.788161
\(732\) 0 0
\(733\) 27.4962 1.01560 0.507799 0.861476i \(-0.330459\pi\)
0.507799 + 0.861476i \(0.330459\pi\)
\(734\) −27.5484 −1.01683
\(735\) 0 0
\(736\) −4.49804 −0.165800
\(737\) −52.5739 −1.93658
\(738\) 0 0
\(739\) 16.6651 0.613035 0.306517 0.951865i \(-0.400836\pi\)
0.306517 + 0.951865i \(0.400836\pi\)
\(740\) 1.08781 0.0399886
\(741\) 0 0
\(742\) 26.6271 0.977512
\(743\) 8.50144 0.311888 0.155944 0.987766i \(-0.450158\pi\)
0.155944 + 0.987766i \(0.450158\pi\)
\(744\) 0 0
\(745\) 0.815716 0.0298855
\(746\) 14.2867 0.523075
\(747\) 0 0
\(748\) −10.6499 −0.389399
\(749\) −1.37352 −0.0501874
\(750\) 0 0
\(751\) 5.87802 0.214492 0.107246 0.994233i \(-0.465797\pi\)
0.107246 + 0.994233i \(0.465797\pi\)
\(752\) −3.59563 −0.131119
\(753\) 0 0
\(754\) −23.3227 −0.849361
\(755\) 5.27891 0.192119
\(756\) 0 0
\(757\) 21.8609 0.794548 0.397274 0.917700i \(-0.369956\pi\)
0.397274 + 0.917700i \(0.369956\pi\)
\(758\) −31.7017 −1.15146
\(759\) 0 0
\(760\) −6.34325 −0.230094
\(761\) −1.67436 −0.0606955 −0.0303477 0.999539i \(-0.509661\pi\)
−0.0303477 + 0.999539i \(0.509661\pi\)
\(762\) 0 0
\(763\) 15.6856 0.567856
\(764\) −9.80574 −0.354759
\(765\) 0 0
\(766\) −1.71332 −0.0619049
\(767\) −31.8472 −1.14994
\(768\) 0 0
\(769\) 0.908246 0.0327522 0.0163761 0.999866i \(-0.494787\pi\)
0.0163761 + 0.999866i \(0.494787\pi\)
\(770\) 18.1479 0.654004
\(771\) 0 0
\(772\) −1.34693 −0.0484772
\(773\) 24.9132 0.896066 0.448033 0.894017i \(-0.352125\pi\)
0.448033 + 0.894017i \(0.352125\pi\)
\(774\) 0 0
\(775\) 28.8125 1.03498
\(776\) 8.98324 0.322480
\(777\) 0 0
\(778\) 31.3579 1.12423
\(779\) 34.5412 1.23757
\(780\) 0 0
\(781\) −9.88104 −0.353571
\(782\) −8.89389 −0.318045
\(783\) 0 0
\(784\) 10.0616 0.359343
\(785\) −9.85174 −0.351624
\(786\) 0 0
\(787\) 10.2675 0.365996 0.182998 0.983113i \(-0.441420\pi\)
0.182998 + 0.983113i \(0.441420\pi\)
\(788\) 22.1831 0.790239
\(789\) 0 0
\(790\) −7.51359 −0.267321
\(791\) 3.13333 0.111409
\(792\) 0 0
\(793\) −30.7594 −1.09230
\(794\) 3.58737 0.127311
\(795\) 0 0
\(796\) −24.9250 −0.883444
\(797\) 9.46275 0.335188 0.167594 0.985856i \(-0.446400\pi\)
0.167594 + 0.985856i \(0.446400\pi\)
\(798\) 0 0
\(799\) −7.10958 −0.251519
\(800\) 4.33461 0.153252
\(801\) 0 0
\(802\) −11.6963 −0.413012
\(803\) 19.7174 0.695813
\(804\) 0 0
\(805\) 15.1556 0.534163
\(806\) 36.7972 1.29613
\(807\) 0 0
\(808\) 1.31930 0.0464129
\(809\) 45.0849 1.58510 0.792550 0.609807i \(-0.208753\pi\)
0.792550 + 0.609807i \(0.208753\pi\)
\(810\) 0 0
\(811\) 16.8789 0.592697 0.296349 0.955080i \(-0.404231\pi\)
0.296349 + 0.955080i \(0.404231\pi\)
\(812\) −17.4022 −0.610697
\(813\) 0 0
\(814\) −7.18275 −0.251755
\(815\) 4.64667 0.162766
\(816\) 0 0
\(817\) 83.8065 2.93202
\(818\) 22.7319 0.794802
\(819\) 0 0
\(820\) 3.62329 0.126531
\(821\) −28.2057 −0.984385 −0.492192 0.870486i \(-0.663804\pi\)
−0.492192 + 0.870486i \(0.663804\pi\)
\(822\) 0 0
\(823\) −43.4450 −1.51440 −0.757199 0.653185i \(-0.773433\pi\)
−0.757199 + 0.653185i \(0.773433\pi\)
\(824\) 3.52663 0.122856
\(825\) 0 0
\(826\) −23.7628 −0.826812
\(827\) 4.21075 0.146422 0.0732111 0.997316i \(-0.476675\pi\)
0.0732111 + 0.997316i \(0.476675\pi\)
\(828\) 0 0
\(829\) 47.6071 1.65346 0.826732 0.562596i \(-0.190197\pi\)
0.826732 + 0.562596i \(0.190197\pi\)
\(830\) 1.57616 0.0547094
\(831\) 0 0
\(832\) 5.53585 0.191921
\(833\) 19.8946 0.689308
\(834\) 0 0
\(835\) −14.6579 −0.507257
\(836\) 41.8842 1.44860
\(837\) 0 0
\(838\) 12.6330 0.436399
\(839\) 18.6965 0.645475 0.322738 0.946488i \(-0.395397\pi\)
0.322738 + 0.946488i \(0.395397\pi\)
\(840\) 0 0
\(841\) −11.2504 −0.387946
\(842\) −0.516091 −0.0177857
\(843\) 0 0
\(844\) 5.38099 0.185221
\(845\) −14.3938 −0.495162
\(846\) 0 0
\(847\) −74.3935 −2.55619
\(848\) 6.44635 0.221369
\(849\) 0 0
\(850\) 8.57074 0.293974
\(851\) −5.99842 −0.205623
\(852\) 0 0
\(853\) 12.3731 0.423647 0.211824 0.977308i \(-0.432060\pi\)
0.211824 + 0.977308i \(0.432060\pi\)
\(854\) −22.9511 −0.785370
\(855\) 0 0
\(856\) −0.332526 −0.0113655
\(857\) 32.1567 1.09845 0.549227 0.835673i \(-0.314922\pi\)
0.549227 + 0.835673i \(0.314922\pi\)
\(858\) 0 0
\(859\) −11.2702 −0.384534 −0.192267 0.981343i \(-0.561584\pi\)
−0.192267 + 0.981343i \(0.561584\pi\)
\(860\) 8.79111 0.299774
\(861\) 0 0
\(862\) −28.7458 −0.979087
\(863\) 3.08277 0.104939 0.0524693 0.998623i \(-0.483291\pi\)
0.0524693 + 0.998623i \(0.483291\pi\)
\(864\) 0 0
\(865\) −16.7115 −0.568208
\(866\) 18.6484 0.633699
\(867\) 0 0
\(868\) 27.4562 0.931925
\(869\) 49.6119 1.68297
\(870\) 0 0
\(871\) 54.0352 1.83091
\(872\) 3.79744 0.128597
\(873\) 0 0
\(874\) 34.9781 1.18315
\(875\) −31.4518 −1.06326
\(876\) 0 0
\(877\) −50.6336 −1.70977 −0.854887 0.518814i \(-0.826374\pi\)
−0.854887 + 0.518814i \(0.826374\pi\)
\(878\) −11.6890 −0.394484
\(879\) 0 0
\(880\) 4.39356 0.148107
\(881\) −51.6908 −1.74151 −0.870753 0.491720i \(-0.836368\pi\)
−0.870753 + 0.491720i \(0.836368\pi\)
\(882\) 0 0
\(883\) −34.7164 −1.16830 −0.584149 0.811646i \(-0.698572\pi\)
−0.584149 + 0.811646i \(0.698572\pi\)
\(884\) 10.9459 0.368152
\(885\) 0 0
\(886\) 19.7049 0.661999
\(887\) −14.0754 −0.472604 −0.236302 0.971680i \(-0.575935\pi\)
−0.236302 + 0.971680i \(0.575935\pi\)
\(888\) 0 0
\(889\) −47.5841 −1.59592
\(890\) −11.4994 −0.385461
\(891\) 0 0
\(892\) 12.5623 0.420618
\(893\) 27.9607 0.935669
\(894\) 0 0
\(895\) −11.5928 −0.387505
\(896\) 4.13057 0.137993
\(897\) 0 0
\(898\) −3.01542 −0.100626
\(899\) −28.0043 −0.933996
\(900\) 0 0
\(901\) 12.7463 0.424639
\(902\) −23.9245 −0.796597
\(903\) 0 0
\(904\) 0.758572 0.0252297
\(905\) −2.23428 −0.0742700
\(906\) 0 0
\(907\) 30.6751 1.01855 0.509275 0.860604i \(-0.329914\pi\)
0.509275 + 0.860604i \(0.329914\pi\)
\(908\) 3.13674 0.104096
\(909\) 0 0
\(910\) −18.6523 −0.618318
\(911\) 12.8968 0.427290 0.213645 0.976911i \(-0.431466\pi\)
0.213645 + 0.976911i \(0.431466\pi\)
\(912\) 0 0
\(913\) −10.4073 −0.344432
\(914\) −13.4588 −0.445178
\(915\) 0 0
\(916\) 3.77927 0.124870
\(917\) 1.77828 0.0587240
\(918\) 0 0
\(919\) 16.0395 0.529094 0.264547 0.964373i \(-0.414778\pi\)
0.264547 + 0.964373i \(0.414778\pi\)
\(920\) 3.66912 0.120967
\(921\) 0 0
\(922\) −36.1944 −1.19200
\(923\) 10.1557 0.334279
\(924\) 0 0
\(925\) 5.78047 0.190061
\(926\) 29.1225 0.957026
\(927\) 0 0
\(928\) −4.21302 −0.138299
\(929\) −31.8120 −1.04372 −0.521858 0.853032i \(-0.674761\pi\)
−0.521858 + 0.853032i \(0.674761\pi\)
\(930\) 0 0
\(931\) −78.2421 −2.56428
\(932\) −1.22631 −0.0401691
\(933\) 0 0
\(934\) 34.6912 1.13513
\(935\) 8.68730 0.284105
\(936\) 0 0
\(937\) −19.0768 −0.623213 −0.311606 0.950211i \(-0.600867\pi\)
−0.311606 + 0.950211i \(0.600867\pi\)
\(938\) 40.3183 1.31644
\(939\) 0 0
\(940\) 2.93301 0.0956643
\(941\) 29.3725 0.957517 0.478758 0.877947i \(-0.341087\pi\)
0.478758 + 0.877947i \(0.341087\pi\)
\(942\) 0 0
\(943\) −19.9797 −0.650627
\(944\) −5.75290 −0.187241
\(945\) 0 0
\(946\) −58.0473 −1.88728
\(947\) 13.3442 0.433629 0.216815 0.976213i \(-0.430433\pi\)
0.216815 + 0.976213i \(0.430433\pi\)
\(948\) 0 0
\(949\) −20.2655 −0.657846
\(950\) −33.7072 −1.09361
\(951\) 0 0
\(952\) 8.16730 0.264704
\(953\) −13.8031 −0.447127 −0.223563 0.974689i \(-0.571769\pi\)
−0.223563 + 0.974689i \(0.571769\pi\)
\(954\) 0 0
\(955\) 7.99869 0.258832
\(956\) 9.21295 0.297968
\(957\) 0 0
\(958\) 38.0159 1.22824
\(959\) 78.2726 2.52756
\(960\) 0 0
\(961\) 13.1837 0.425280
\(962\) 7.38240 0.238018
\(963\) 0 0
\(964\) −20.6802 −0.666064
\(965\) 1.09872 0.0353689
\(966\) 0 0
\(967\) 52.7669 1.69687 0.848434 0.529301i \(-0.177546\pi\)
0.848434 + 0.529301i \(0.177546\pi\)
\(968\) −18.0105 −0.578878
\(969\) 0 0
\(970\) −7.32777 −0.235281
\(971\) −7.77643 −0.249557 −0.124779 0.992185i \(-0.539822\pi\)
−0.124779 + 0.992185i \(0.539822\pi\)
\(972\) 0 0
\(973\) 45.3171 1.45280
\(974\) −16.5858 −0.531444
\(975\) 0 0
\(976\) −5.55640 −0.177856
\(977\) −3.56867 −0.114172 −0.0570859 0.998369i \(-0.518181\pi\)
−0.0570859 + 0.998369i \(0.518181\pi\)
\(978\) 0 0
\(979\) 75.9300 2.42673
\(980\) −8.20741 −0.262176
\(981\) 0 0
\(982\) −32.0610 −1.02311
\(983\) 37.1515 1.18495 0.592475 0.805589i \(-0.298151\pi\)
0.592475 + 0.805589i \(0.298151\pi\)
\(984\) 0 0
\(985\) −18.0951 −0.576557
\(986\) −8.33033 −0.265292
\(987\) 0 0
\(988\) −43.0484 −1.36955
\(989\) −48.4762 −1.54145
\(990\) 0 0
\(991\) 21.5462 0.684437 0.342218 0.939620i \(-0.388822\pi\)
0.342218 + 0.939620i \(0.388822\pi\)
\(992\) 6.64708 0.211045
\(993\) 0 0
\(994\) 7.57766 0.240349
\(995\) 20.3317 0.644559
\(996\) 0 0
\(997\) 7.42721 0.235222 0.117611 0.993060i \(-0.462476\pi\)
0.117611 + 0.993060i \(0.462476\pi\)
\(998\) 35.4267 1.12141
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8046.2.a.k.1.5 12
3.2 odd 2 8046.2.a.n.1.8 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8046.2.a.k.1.5 12 1.1 even 1 trivial
8046.2.a.n.1.8 yes 12 3.2 odd 2