Properties

Label 8046.2.a.o.1.4
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: \( x^{12} - 3 x^{11} - 29 x^{10} + 76 x^{9} + 320 x^{8} - 724 x^{7} - 1643 x^{6} + 3265 x^{5} + 3921 x^{4} - 6927 x^{3} - 3639 x^{2} + 5508 x + 423 \) 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 1.4
Root \(-1.76721\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.76721 q^{5} +2.53166 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.76721 q^{5} +2.53166 q^{7} +1.00000 q^{8} -1.76721 q^{10} -0.172733 q^{11} +1.33396 q^{13} +2.53166 q^{14} +1.00000 q^{16} +6.35073 q^{17} +0.244981 q^{19} -1.76721 q^{20} -0.172733 q^{22} +0.777947 q^{23} -1.87697 q^{25} +1.33396 q^{26} +2.53166 q^{28} -0.922435 q^{29} -2.24473 q^{31} +1.00000 q^{32} +6.35073 q^{34} -4.47398 q^{35} +4.59906 q^{37} +0.244981 q^{38} -1.76721 q^{40} +7.99362 q^{41} -9.41776 q^{43} -0.172733 q^{44} +0.777947 q^{46} +10.3563 q^{47} -0.590673 q^{49} -1.87697 q^{50} +1.33396 q^{52} -1.53474 q^{53} +0.305256 q^{55} +2.53166 q^{56} -0.922435 q^{58} +7.87122 q^{59} +1.82082 q^{61} -2.24473 q^{62} +1.00000 q^{64} -2.35738 q^{65} +10.3809 q^{67} +6.35073 q^{68} -4.47398 q^{70} -10.1375 q^{71} -14.6495 q^{73} +4.59906 q^{74} +0.244981 q^{76} -0.437303 q^{77} -2.04066 q^{79} -1.76721 q^{80} +7.99362 q^{82} +13.6488 q^{83} -11.2231 q^{85} -9.41776 q^{86} -0.172733 q^{88} -0.542300 q^{89} +3.37713 q^{91} +0.777947 q^{92} +10.3563 q^{94} -0.432933 q^{95} +13.9739 q^{97} -0.590673 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8} + 3 q^{10} + 10 q^{11} + 5 q^{13} + 6 q^{14} + 12 q^{16} + 8 q^{17} + 2 q^{19} + 3 q^{20} + 10 q^{22} + 9 q^{23} + 7 q^{25} + 5 q^{26} + 6 q^{28} + 19 q^{29} + 10 q^{31} + 12 q^{32} + 8 q^{34} + 20 q^{35} + 11 q^{37} + 2 q^{38} + 3 q^{40} + 8 q^{41} + 13 q^{43} + 10 q^{44} + 9 q^{46} + 11 q^{47} + 2 q^{49} + 7 q^{50} + 5 q^{52} + 24 q^{53} + 3 q^{55} + 6 q^{56} + 19 q^{58} + 10 q^{59} + 10 q^{62} + 12 q^{64} + 28 q^{65} + 21 q^{67} + 8 q^{68} + 20 q^{70} + 37 q^{71} - 2 q^{73} + 11 q^{74} + 2 q^{76} + 2 q^{77} + 7 q^{79} + 3 q^{80} + 8 q^{82} + 22 q^{83} + 15 q^{85} + 13 q^{86} + 10 q^{88} + 40 q^{89} + q^{91} + 9 q^{92} + 11 q^{94} + 11 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) \) 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
\(5\) −1.76721 −0.790320 −0.395160 0.918612i \(-0.629311\pi\)
−0.395160 + 0.918612i \(0.629311\pi\)
\(6\) 0 0
\(7\) 2.53166 0.956879 0.478440 0.878120i \(-0.341203\pi\)
0.478440 + 0.878120i \(0.341203\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.76721 −0.558841
\(11\) −0.172733 −0.0520811 −0.0260405 0.999661i \(-0.508290\pi\)
−0.0260405 + 0.999661i \(0.508290\pi\)
\(12\) 0 0
\(13\) 1.33396 0.369973 0.184987 0.982741i \(-0.440776\pi\)
0.184987 + 0.982741i \(0.440776\pi\)
\(14\) 2.53166 0.676616
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.35073 1.54028 0.770139 0.637876i \(-0.220187\pi\)
0.770139 + 0.637876i \(0.220187\pi\)
\(18\) 0 0
\(19\) 0.244981 0.0562025 0.0281012 0.999605i \(-0.491054\pi\)
0.0281012 + 0.999605i \(0.491054\pi\)
\(20\) −1.76721 −0.395160
\(21\) 0 0
\(22\) −0.172733 −0.0368269
\(23\) 0.777947 0.162213 0.0811065 0.996705i \(-0.474155\pi\)
0.0811065 + 0.996705i \(0.474155\pi\)
\(24\) 0 0
\(25\) −1.87697 −0.375394
\(26\) 1.33396 0.261611
\(27\) 0 0
\(28\) 2.53166 0.478440
\(29\) −0.922435 −0.171292 −0.0856460 0.996326i \(-0.527295\pi\)
−0.0856460 + 0.996326i \(0.527295\pi\)
\(30\) 0 0
\(31\) −2.24473 −0.403166 −0.201583 0.979471i \(-0.564609\pi\)
−0.201583 + 0.979471i \(0.564609\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.35073 1.08914
\(35\) −4.47398 −0.756241
\(36\) 0 0
\(37\) 4.59906 0.756081 0.378040 0.925789i \(-0.376598\pi\)
0.378040 + 0.925789i \(0.376598\pi\)
\(38\) 0.244981 0.0397411
\(39\) 0 0
\(40\) −1.76721 −0.279420
\(41\) 7.99362 1.24839 0.624197 0.781267i \(-0.285426\pi\)
0.624197 + 0.781267i \(0.285426\pi\)
\(42\) 0 0
\(43\) −9.41776 −1.43620 −0.718098 0.695943i \(-0.754987\pi\)
−0.718098 + 0.695943i \(0.754987\pi\)
\(44\) −0.172733 −0.0260405
\(45\) 0 0
\(46\) 0.777947 0.114702
\(47\) 10.3563 1.51062 0.755308 0.655370i \(-0.227487\pi\)
0.755308 + 0.655370i \(0.227487\pi\)
\(48\) 0 0
\(49\) −0.590673 −0.0843818
\(50\) −1.87697 −0.265444
\(51\) 0 0
\(52\) 1.33396 0.184987
\(53\) −1.53474 −0.210813 −0.105406 0.994429i \(-0.533614\pi\)
−0.105406 + 0.994429i \(0.533614\pi\)
\(54\) 0 0
\(55\) 0.305256 0.0411607
\(56\) 2.53166 0.338308
\(57\) 0 0
\(58\) −0.922435 −0.121122
\(59\) 7.87122 1.02475 0.512373 0.858763i \(-0.328767\pi\)
0.512373 + 0.858763i \(0.328767\pi\)
\(60\) 0 0
\(61\) 1.82082 0.233133 0.116566 0.993183i \(-0.462811\pi\)
0.116566 + 0.993183i \(0.462811\pi\)
\(62\) −2.24473 −0.285081
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.35738 −0.292397
\(66\) 0 0
\(67\) 10.3809 1.26822 0.634111 0.773242i \(-0.281366\pi\)
0.634111 + 0.773242i \(0.281366\pi\)
\(68\) 6.35073 0.770139
\(69\) 0 0
\(70\) −4.47398 −0.534743
\(71\) −10.1375 −1.20310 −0.601549 0.798836i \(-0.705450\pi\)
−0.601549 + 0.798836i \(0.705450\pi\)
\(72\) 0 0
\(73\) −14.6495 −1.71459 −0.857296 0.514824i \(-0.827857\pi\)
−0.857296 + 0.514824i \(0.827857\pi\)
\(74\) 4.59906 0.534630
\(75\) 0 0
\(76\) 0.244981 0.0281012
\(77\) −0.437303 −0.0498353
\(78\) 0 0
\(79\) −2.04066 −0.229592 −0.114796 0.993389i \(-0.536621\pi\)
−0.114796 + 0.993389i \(0.536621\pi\)
\(80\) −1.76721 −0.197580
\(81\) 0 0
\(82\) 7.99362 0.882747
\(83\) 13.6488 1.49815 0.749075 0.662485i \(-0.230498\pi\)
0.749075 + 0.662485i \(0.230498\pi\)
\(84\) 0 0
\(85\) −11.2231 −1.21731
\(86\) −9.41776 −1.01554
\(87\) 0 0
\(88\) −0.172733 −0.0184134
\(89\) −0.542300 −0.0574837 −0.0287419 0.999587i \(-0.509150\pi\)
−0.0287419 + 0.999587i \(0.509150\pi\)
\(90\) 0 0
\(91\) 3.37713 0.354020
\(92\) 0.777947 0.0811065
\(93\) 0 0
\(94\) 10.3563 1.06817
\(95\) −0.432933 −0.0444179
\(96\) 0 0
\(97\) 13.9739 1.41883 0.709416 0.704790i \(-0.248959\pi\)
0.709416 + 0.704790i \(0.248959\pi\)
\(98\) −0.590673 −0.0596669
\(99\) 0 0
\(100\) −1.87697 −0.187697
\(101\) 15.5965 1.55191 0.775956 0.630787i \(-0.217268\pi\)
0.775956 + 0.630787i \(0.217268\pi\)
\(102\) 0 0
\(103\) −9.33732 −0.920034 −0.460017 0.887910i \(-0.652157\pi\)
−0.460017 + 0.887910i \(0.652157\pi\)
\(104\) 1.33396 0.130805
\(105\) 0 0
\(106\) −1.53474 −0.149067
\(107\) 14.2996 1.38239 0.691196 0.722667i \(-0.257084\pi\)
0.691196 + 0.722667i \(0.257084\pi\)
\(108\) 0 0
\(109\) −9.28655 −0.889490 −0.444745 0.895657i \(-0.646706\pi\)
−0.444745 + 0.895657i \(0.646706\pi\)
\(110\) 0.305256 0.0291050
\(111\) 0 0
\(112\) 2.53166 0.239220
\(113\) −16.1535 −1.51959 −0.759796 0.650161i \(-0.774701\pi\)
−0.759796 + 0.650161i \(0.774701\pi\)
\(114\) 0 0
\(115\) −1.37479 −0.128200
\(116\) −0.922435 −0.0856460
\(117\) 0 0
\(118\) 7.87122 0.724605
\(119\) 16.0779 1.47386
\(120\) 0 0
\(121\) −10.9702 −0.997288
\(122\) 1.82082 0.164850
\(123\) 0 0
\(124\) −2.24473 −0.201583
\(125\) 12.1530 1.08700
\(126\) 0 0
\(127\) 11.5280 1.02294 0.511472 0.859300i \(-0.329101\pi\)
0.511472 + 0.859300i \(0.329101\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −2.35738 −0.206756
\(131\) −13.2407 −1.15685 −0.578424 0.815736i \(-0.696332\pi\)
−0.578424 + 0.815736i \(0.696332\pi\)
\(132\) 0 0
\(133\) 0.620210 0.0537790
\(134\) 10.3809 0.896769
\(135\) 0 0
\(136\) 6.35073 0.544571
\(137\) 10.2575 0.876353 0.438177 0.898889i \(-0.355624\pi\)
0.438177 + 0.898889i \(0.355624\pi\)
\(138\) 0 0
\(139\) −11.2051 −0.950403 −0.475201 0.879877i \(-0.657625\pi\)
−0.475201 + 0.879877i \(0.657625\pi\)
\(140\) −4.47398 −0.378121
\(141\) 0 0
\(142\) −10.1375 −0.850719
\(143\) −0.230419 −0.0192686
\(144\) 0 0
\(145\) 1.63014 0.135375
\(146\) −14.6495 −1.21240
\(147\) 0 0
\(148\) 4.59906 0.378040
\(149\) 1.00000 0.0819232
\(150\) 0 0
\(151\) −21.0529 −1.71326 −0.856632 0.515928i \(-0.827447\pi\)
−0.856632 + 0.515928i \(0.827447\pi\)
\(152\) 0.244981 0.0198706
\(153\) 0 0
\(154\) −0.437303 −0.0352389
\(155\) 3.96691 0.318630
\(156\) 0 0
\(157\) 1.29923 0.103690 0.0518448 0.998655i \(-0.483490\pi\)
0.0518448 + 0.998655i \(0.483490\pi\)
\(158\) −2.04066 −0.162346
\(159\) 0 0
\(160\) −1.76721 −0.139710
\(161\) 1.96950 0.155218
\(162\) 0 0
\(163\) 17.8886 1.40114 0.700572 0.713582i \(-0.252928\pi\)
0.700572 + 0.713582i \(0.252928\pi\)
\(164\) 7.99362 0.624197
\(165\) 0 0
\(166\) 13.6488 1.05935
\(167\) 23.5919 1.82560 0.912798 0.408412i \(-0.133917\pi\)
0.912798 + 0.408412i \(0.133917\pi\)
\(168\) 0 0
\(169\) −11.2206 −0.863120
\(170\) −11.2231 −0.860771
\(171\) 0 0
\(172\) −9.41776 −0.718098
\(173\) −5.42628 −0.412552 −0.206276 0.978494i \(-0.566135\pi\)
−0.206276 + 0.978494i \(0.566135\pi\)
\(174\) 0 0
\(175\) −4.75186 −0.359207
\(176\) −0.172733 −0.0130203
\(177\) 0 0
\(178\) −0.542300 −0.0406471
\(179\) 9.10535 0.680566 0.340283 0.940323i \(-0.389477\pi\)
0.340283 + 0.940323i \(0.389477\pi\)
\(180\) 0 0
\(181\) −4.26853 −0.317278 −0.158639 0.987337i \(-0.550711\pi\)
−0.158639 + 0.987337i \(0.550711\pi\)
\(182\) 3.37713 0.250330
\(183\) 0 0
\(184\) 0.777947 0.0573510
\(185\) −8.12750 −0.597546
\(186\) 0 0
\(187\) −1.09698 −0.0802194
\(188\) 10.3563 0.755308
\(189\) 0 0
\(190\) −0.432933 −0.0314082
\(191\) −19.1244 −1.38379 −0.691895 0.721998i \(-0.743224\pi\)
−0.691895 + 0.721998i \(0.743224\pi\)
\(192\) 0 0
\(193\) 18.6585 1.34307 0.671535 0.740973i \(-0.265635\pi\)
0.671535 + 0.740973i \(0.265635\pi\)
\(194\) 13.9739 1.00327
\(195\) 0 0
\(196\) −0.590673 −0.0421909
\(197\) 6.74520 0.480575 0.240288 0.970702i \(-0.422758\pi\)
0.240288 + 0.970702i \(0.422758\pi\)
\(198\) 0 0
\(199\) 16.9797 1.20366 0.601828 0.798625i \(-0.294439\pi\)
0.601828 + 0.798625i \(0.294439\pi\)
\(200\) −1.87697 −0.132722
\(201\) 0 0
\(202\) 15.5965 1.09737
\(203\) −2.33530 −0.163906
\(204\) 0 0
\(205\) −14.1264 −0.986630
\(206\) −9.33732 −0.650562
\(207\) 0 0
\(208\) 1.33396 0.0924933
\(209\) −0.0423164 −0.00292709
\(210\) 0 0
\(211\) −1.62775 −0.112059 −0.0560296 0.998429i \(-0.517844\pi\)
−0.0560296 + 0.998429i \(0.517844\pi\)
\(212\) −1.53474 −0.105406
\(213\) 0 0
\(214\) 14.2996 0.977499
\(215\) 16.6432 1.13505
\(216\) 0 0
\(217\) −5.68291 −0.385781
\(218\) −9.28655 −0.628965
\(219\) 0 0
\(220\) 0.305256 0.0205804
\(221\) 8.47161 0.569862
\(222\) 0 0
\(223\) 28.2695 1.89306 0.946531 0.322612i \(-0.104561\pi\)
0.946531 + 0.322612i \(0.104561\pi\)
\(224\) 2.53166 0.169154
\(225\) 0 0
\(226\) −16.1535 −1.07451
\(227\) −17.1689 −1.13954 −0.569769 0.821805i \(-0.692967\pi\)
−0.569769 + 0.821805i \(0.692967\pi\)
\(228\) 0 0
\(229\) 12.0079 0.793502 0.396751 0.917926i \(-0.370138\pi\)
0.396751 + 0.917926i \(0.370138\pi\)
\(230\) −1.37479 −0.0906513
\(231\) 0 0
\(232\) −0.922435 −0.0605608
\(233\) 20.2171 1.32447 0.662233 0.749298i \(-0.269609\pi\)
0.662233 + 0.749298i \(0.269609\pi\)
\(234\) 0 0
\(235\) −18.3017 −1.19387
\(236\) 7.87122 0.512373
\(237\) 0 0
\(238\) 16.0779 1.04218
\(239\) 5.11495 0.330859 0.165429 0.986222i \(-0.447099\pi\)
0.165429 + 0.986222i \(0.447099\pi\)
\(240\) 0 0
\(241\) 29.0965 1.87427 0.937136 0.348963i \(-0.113466\pi\)
0.937136 + 0.348963i \(0.113466\pi\)
\(242\) −10.9702 −0.705189
\(243\) 0 0
\(244\) 1.82082 0.116566
\(245\) 1.04384 0.0666886
\(246\) 0 0
\(247\) 0.326794 0.0207934
\(248\) −2.24473 −0.142541
\(249\) 0 0
\(250\) 12.1530 0.768626
\(251\) 21.0454 1.32837 0.664186 0.747567i \(-0.268778\pi\)
0.664186 + 0.747567i \(0.268778\pi\)
\(252\) 0 0
\(253\) −0.134377 −0.00844823
\(254\) 11.5280 0.723330
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −30.4128 −1.89710 −0.948549 0.316631i \(-0.897448\pi\)
−0.948549 + 0.316631i \(0.897448\pi\)
\(258\) 0 0
\(259\) 11.6433 0.723478
\(260\) −2.35738 −0.146199
\(261\) 0 0
\(262\) −13.2407 −0.818015
\(263\) 2.41012 0.148614 0.0743070 0.997235i \(-0.476326\pi\)
0.0743070 + 0.997235i \(0.476326\pi\)
\(264\) 0 0
\(265\) 2.71221 0.166610
\(266\) 0.620210 0.0380275
\(267\) 0 0
\(268\) 10.3809 0.634111
\(269\) 5.99638 0.365606 0.182803 0.983150i \(-0.441483\pi\)
0.182803 + 0.983150i \(0.441483\pi\)
\(270\) 0 0
\(271\) 21.7372 1.32044 0.660220 0.751072i \(-0.270463\pi\)
0.660220 + 0.751072i \(0.270463\pi\)
\(272\) 6.35073 0.385070
\(273\) 0 0
\(274\) 10.2575 0.619675
\(275\) 0.324215 0.0195509
\(276\) 0 0
\(277\) 6.50975 0.391133 0.195567 0.980690i \(-0.437345\pi\)
0.195567 + 0.980690i \(0.437345\pi\)
\(278\) −11.2051 −0.672036
\(279\) 0 0
\(280\) −4.47398 −0.267372
\(281\) −6.28645 −0.375018 −0.187509 0.982263i \(-0.560041\pi\)
−0.187509 + 0.982263i \(0.560041\pi\)
\(282\) 0 0
\(283\) −24.2940 −1.44413 −0.722064 0.691826i \(-0.756806\pi\)
−0.722064 + 0.691826i \(0.756806\pi\)
\(284\) −10.1375 −0.601549
\(285\) 0 0
\(286\) −0.230419 −0.0136250
\(287\) 20.2372 1.19456
\(288\) 0 0
\(289\) 23.3318 1.37246
\(290\) 1.63014 0.0957249
\(291\) 0 0
\(292\) −14.6495 −0.857296
\(293\) −10.7390 −0.627376 −0.313688 0.949526i \(-0.601565\pi\)
−0.313688 + 0.949526i \(0.601565\pi\)
\(294\) 0 0
\(295\) −13.9101 −0.809877
\(296\) 4.59906 0.267315
\(297\) 0 0
\(298\) 1.00000 0.0579284
\(299\) 1.03775 0.0600145
\(300\) 0 0
\(301\) −23.8426 −1.37427
\(302\) −21.0529 −1.21146
\(303\) 0 0
\(304\) 0.244981 0.0140506
\(305\) −3.21778 −0.184250
\(306\) 0 0
\(307\) 12.8387 0.732741 0.366370 0.930469i \(-0.380600\pi\)
0.366370 + 0.930469i \(0.380600\pi\)
\(308\) −0.437303 −0.0249177
\(309\) 0 0
\(310\) 3.96691 0.225305
\(311\) −10.6858 −0.605938 −0.302969 0.953000i \(-0.597978\pi\)
−0.302969 + 0.953000i \(0.597978\pi\)
\(312\) 0 0
\(313\) 15.2652 0.862842 0.431421 0.902151i \(-0.358012\pi\)
0.431421 + 0.902151i \(0.358012\pi\)
\(314\) 1.29923 0.0733197
\(315\) 0 0
\(316\) −2.04066 −0.114796
\(317\) 33.7500 1.89559 0.947794 0.318882i \(-0.103307\pi\)
0.947794 + 0.318882i \(0.103307\pi\)
\(318\) 0 0
\(319\) 0.159335 0.00892107
\(320\) −1.76721 −0.0987900
\(321\) 0 0
\(322\) 1.96950 0.109756
\(323\) 1.55581 0.0865675
\(324\) 0 0
\(325\) −2.50380 −0.138886
\(326\) 17.8886 0.990758
\(327\) 0 0
\(328\) 7.99362 0.441374
\(329\) 26.2186 1.44548
\(330\) 0 0
\(331\) −0.851427 −0.0467987 −0.0233993 0.999726i \(-0.507449\pi\)
−0.0233993 + 0.999726i \(0.507449\pi\)
\(332\) 13.6488 0.749075
\(333\) 0 0
\(334\) 23.5919 1.29089
\(335\) −18.3451 −1.00230
\(336\) 0 0
\(337\) 14.5751 0.793957 0.396979 0.917828i \(-0.370059\pi\)
0.396979 + 0.917828i \(0.370059\pi\)
\(338\) −11.2206 −0.610318
\(339\) 0 0
\(340\) −11.2231 −0.608657
\(341\) 0.387740 0.0209973
\(342\) 0 0
\(343\) −19.2170 −1.03762
\(344\) −9.41776 −0.507772
\(345\) 0 0
\(346\) −5.42628 −0.291719
\(347\) 13.5016 0.724806 0.362403 0.932021i \(-0.381956\pi\)
0.362403 + 0.932021i \(0.381956\pi\)
\(348\) 0 0
\(349\) 12.9484 0.693110 0.346555 0.938030i \(-0.387351\pi\)
0.346555 + 0.938030i \(0.387351\pi\)
\(350\) −4.75186 −0.253998
\(351\) 0 0
\(352\) −0.172733 −0.00920672
\(353\) −22.5635 −1.20093 −0.600466 0.799650i \(-0.705018\pi\)
−0.600466 + 0.799650i \(0.705018\pi\)
\(354\) 0 0
\(355\) 17.9151 0.950833
\(356\) −0.542300 −0.0287419
\(357\) 0 0
\(358\) 9.10535 0.481233
\(359\) 8.30844 0.438503 0.219251 0.975668i \(-0.429639\pi\)
0.219251 + 0.975668i \(0.429639\pi\)
\(360\) 0 0
\(361\) −18.9400 −0.996841
\(362\) −4.26853 −0.224349
\(363\) 0 0
\(364\) 3.37713 0.177010
\(365\) 25.8887 1.35508
\(366\) 0 0
\(367\) 16.2813 0.849876 0.424938 0.905223i \(-0.360296\pi\)
0.424938 + 0.905223i \(0.360296\pi\)
\(368\) 0.777947 0.0405533
\(369\) 0 0
\(370\) −8.12750 −0.422529
\(371\) −3.88545 −0.201722
\(372\) 0 0
\(373\) −18.5063 −0.958220 −0.479110 0.877755i \(-0.659040\pi\)
−0.479110 + 0.877755i \(0.659040\pi\)
\(374\) −1.09698 −0.0567237
\(375\) 0 0
\(376\) 10.3563 0.534083
\(377\) −1.23049 −0.0633734
\(378\) 0 0
\(379\) −25.2996 −1.29955 −0.649777 0.760125i \(-0.725138\pi\)
−0.649777 + 0.760125i \(0.725138\pi\)
\(380\) −0.432933 −0.0222090
\(381\) 0 0
\(382\) −19.1244 −0.978488
\(383\) −37.6264 −1.92262 −0.961310 0.275470i \(-0.911166\pi\)
−0.961310 + 0.275470i \(0.911166\pi\)
\(384\) 0 0
\(385\) 0.772806 0.0393859
\(386\) 18.6585 0.949694
\(387\) 0 0
\(388\) 13.9739 0.709416
\(389\) 26.1522 1.32597 0.662985 0.748633i \(-0.269289\pi\)
0.662985 + 0.748633i \(0.269289\pi\)
\(390\) 0 0
\(391\) 4.94053 0.249853
\(392\) −0.590673 −0.0298335
\(393\) 0 0
\(394\) 6.74520 0.339818
\(395\) 3.60627 0.181451
\(396\) 0 0
\(397\) −8.99278 −0.451335 −0.225667 0.974204i \(-0.572456\pi\)
−0.225667 + 0.974204i \(0.572456\pi\)
\(398\) 16.9797 0.851114
\(399\) 0 0
\(400\) −1.87697 −0.0938485
\(401\) 26.9760 1.34712 0.673558 0.739135i \(-0.264765\pi\)
0.673558 + 0.739135i \(0.264765\pi\)
\(402\) 0 0
\(403\) −2.99438 −0.149161
\(404\) 15.5965 0.775956
\(405\) 0 0
\(406\) −2.33530 −0.115899
\(407\) −0.794411 −0.0393775
\(408\) 0 0
\(409\) −0.843424 −0.0417047 −0.0208523 0.999783i \(-0.506638\pi\)
−0.0208523 + 0.999783i \(0.506638\pi\)
\(410\) −14.1264 −0.697653
\(411\) 0 0
\(412\) −9.33732 −0.460017
\(413\) 19.9273 0.980558
\(414\) 0 0
\(415\) −24.1203 −1.18402
\(416\) 1.33396 0.0654027
\(417\) 0 0
\(418\) −0.0423164 −0.00206976
\(419\) −32.4722 −1.58637 −0.793185 0.608980i \(-0.791579\pi\)
−0.793185 + 0.608980i \(0.791579\pi\)
\(420\) 0 0
\(421\) 31.3692 1.52884 0.764420 0.644719i \(-0.223026\pi\)
0.764420 + 0.644719i \(0.223026\pi\)
\(422\) −1.62775 −0.0792379
\(423\) 0 0
\(424\) −1.53474 −0.0745336
\(425\) −11.9201 −0.578211
\(426\) 0 0
\(427\) 4.60972 0.223080
\(428\) 14.2996 0.691196
\(429\) 0 0
\(430\) 16.6432 0.802604
\(431\) 32.0091 1.54182 0.770911 0.636942i \(-0.219801\pi\)
0.770911 + 0.636942i \(0.219801\pi\)
\(432\) 0 0
\(433\) −19.5149 −0.937825 −0.468912 0.883245i \(-0.655354\pi\)
−0.468912 + 0.883245i \(0.655354\pi\)
\(434\) −5.68291 −0.272788
\(435\) 0 0
\(436\) −9.28655 −0.444745
\(437\) 0.190582 0.00911678
\(438\) 0 0
\(439\) −38.2094 −1.82363 −0.911817 0.410597i \(-0.865320\pi\)
−0.911817 + 0.410597i \(0.865320\pi\)
\(440\) 0.305256 0.0145525
\(441\) 0 0
\(442\) 8.47161 0.402953
\(443\) 13.6766 0.649796 0.324898 0.945749i \(-0.394670\pi\)
0.324898 + 0.945749i \(0.394670\pi\)
\(444\) 0 0
\(445\) 0.958359 0.0454306
\(446\) 28.2695 1.33860
\(447\) 0 0
\(448\) 2.53166 0.119610
\(449\) −34.9553 −1.64964 −0.824821 0.565395i \(-0.808724\pi\)
−0.824821 + 0.565395i \(0.808724\pi\)
\(450\) 0 0
\(451\) −1.38076 −0.0650177
\(452\) −16.1535 −0.759796
\(453\) 0 0
\(454\) −17.1689 −0.805775
\(455\) −5.96810 −0.279789
\(456\) 0 0
\(457\) 19.5492 0.914471 0.457235 0.889346i \(-0.348840\pi\)
0.457235 + 0.889346i \(0.348840\pi\)
\(458\) 12.0079 0.561091
\(459\) 0 0
\(460\) −1.37479 −0.0641001
\(461\) 23.0442 1.07328 0.536638 0.843812i \(-0.319694\pi\)
0.536638 + 0.843812i \(0.319694\pi\)
\(462\) 0 0
\(463\) −2.30744 −0.107236 −0.0536180 0.998562i \(-0.517075\pi\)
−0.0536180 + 0.998562i \(0.517075\pi\)
\(464\) −0.922435 −0.0428230
\(465\) 0 0
\(466\) 20.2171 0.936539
\(467\) 12.1472 0.562106 0.281053 0.959692i \(-0.409316\pi\)
0.281053 + 0.959692i \(0.409316\pi\)
\(468\) 0 0
\(469\) 26.2808 1.21354
\(470\) −18.3017 −0.844193
\(471\) 0 0
\(472\) 7.87122 0.362302
\(473\) 1.62676 0.0747986
\(474\) 0 0
\(475\) −0.459822 −0.0210981
\(476\) 16.0779 0.736930
\(477\) 0 0
\(478\) 5.11495 0.233952
\(479\) −17.3346 −0.792037 −0.396018 0.918243i \(-0.629608\pi\)
−0.396018 + 0.918243i \(0.629608\pi\)
\(480\) 0 0
\(481\) 6.13495 0.279730
\(482\) 29.0965 1.32531
\(483\) 0 0
\(484\) −10.9702 −0.498644
\(485\) −24.6948 −1.12133
\(486\) 0 0
\(487\) −34.5694 −1.56649 −0.783244 0.621714i \(-0.786436\pi\)
−0.783244 + 0.621714i \(0.786436\pi\)
\(488\) 1.82082 0.0824249
\(489\) 0 0
\(490\) 1.04384 0.0471560
\(491\) −37.6299 −1.69821 −0.849106 0.528223i \(-0.822859\pi\)
−0.849106 + 0.528223i \(0.822859\pi\)
\(492\) 0 0
\(493\) −5.85814 −0.263837
\(494\) 0.326794 0.0147032
\(495\) 0 0
\(496\) −2.24473 −0.100791
\(497\) −25.6647 −1.15122
\(498\) 0 0
\(499\) 35.7585 1.60077 0.800386 0.599485i \(-0.204628\pi\)
0.800386 + 0.599485i \(0.204628\pi\)
\(500\) 12.1530 0.543501
\(501\) 0 0
\(502\) 21.0454 0.939301
\(503\) −23.8571 −1.06374 −0.531868 0.846827i \(-0.678510\pi\)
−0.531868 + 0.846827i \(0.678510\pi\)
\(504\) 0 0
\(505\) −27.5623 −1.22651
\(506\) −0.134377 −0.00597380
\(507\) 0 0
\(508\) 11.5280 0.511472
\(509\) −31.3319 −1.38876 −0.694381 0.719608i \(-0.744322\pi\)
−0.694381 + 0.719608i \(0.744322\pi\)
\(510\) 0 0
\(511\) −37.0876 −1.64066
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −30.4128 −1.34145
\(515\) 16.5010 0.727121
\(516\) 0 0
\(517\) −1.78887 −0.0786745
\(518\) 11.6433 0.511576
\(519\) 0 0
\(520\) −2.35738 −0.103378
\(521\) −16.5473 −0.724953 −0.362476 0.931993i \(-0.618069\pi\)
−0.362476 + 0.931993i \(0.618069\pi\)
\(522\) 0 0
\(523\) 41.5828 1.81829 0.909145 0.416479i \(-0.136736\pi\)
0.909145 + 0.416479i \(0.136736\pi\)
\(524\) −13.2407 −0.578424
\(525\) 0 0
\(526\) 2.41012 0.105086
\(527\) −14.2557 −0.620987
\(528\) 0 0
\(529\) −22.3948 −0.973687
\(530\) 2.71221 0.117811
\(531\) 0 0
\(532\) 0.620210 0.0268895
\(533\) 10.6631 0.461872
\(534\) 0 0
\(535\) −25.2704 −1.09253
\(536\) 10.3809 0.448385
\(537\) 0 0
\(538\) 5.99638 0.258522
\(539\) 0.102029 0.00439470
\(540\) 0 0
\(541\) 9.87819 0.424697 0.212348 0.977194i \(-0.431889\pi\)
0.212348 + 0.977194i \(0.431889\pi\)
\(542\) 21.7372 0.933692
\(543\) 0 0
\(544\) 6.35073 0.272285
\(545\) 16.4113 0.702982
\(546\) 0 0
\(547\) 26.4891 1.13259 0.566296 0.824202i \(-0.308376\pi\)
0.566296 + 0.824202i \(0.308376\pi\)
\(548\) 10.2575 0.438177
\(549\) 0 0
\(550\) 0.324215 0.0138246
\(551\) −0.225979 −0.00962703
\(552\) 0 0
\(553\) −5.16626 −0.219692
\(554\) 6.50975 0.276573
\(555\) 0 0
\(556\) −11.2051 −0.475201
\(557\) −6.61527 −0.280298 −0.140149 0.990130i \(-0.544758\pi\)
−0.140149 + 0.990130i \(0.544758\pi\)
\(558\) 0 0
\(559\) −12.5629 −0.531354
\(560\) −4.47398 −0.189060
\(561\) 0 0
\(562\) −6.28645 −0.265178
\(563\) −4.89783 −0.206419 −0.103209 0.994660i \(-0.532911\pi\)
−0.103209 + 0.994660i \(0.532911\pi\)
\(564\) 0 0
\(565\) 28.5466 1.20096
\(566\) −24.2940 −1.02115
\(567\) 0 0
\(568\) −10.1375 −0.425360
\(569\) −25.9447 −1.08766 −0.543829 0.839196i \(-0.683026\pi\)
−0.543829 + 0.839196i \(0.683026\pi\)
\(570\) 0 0
\(571\) 6.66808 0.279050 0.139525 0.990219i \(-0.455442\pi\)
0.139525 + 0.990219i \(0.455442\pi\)
\(572\) −0.230419 −0.00963431
\(573\) 0 0
\(574\) 20.2372 0.844683
\(575\) −1.46018 −0.0608938
\(576\) 0 0
\(577\) 1.48415 0.0617861 0.0308931 0.999523i \(-0.490165\pi\)
0.0308931 + 0.999523i \(0.490165\pi\)
\(578\) 23.3318 0.970475
\(579\) 0 0
\(580\) 1.63014 0.0676877
\(581\) 34.5542 1.43355
\(582\) 0 0
\(583\) 0.265101 0.0109794
\(584\) −14.6495 −0.606200
\(585\) 0 0
\(586\) −10.7390 −0.443622
\(587\) 3.09748 0.127847 0.0639233 0.997955i \(-0.479639\pi\)
0.0639233 + 0.997955i \(0.479639\pi\)
\(588\) 0 0
\(589\) −0.549916 −0.0226589
\(590\) −13.9101 −0.572670
\(591\) 0 0
\(592\) 4.59906 0.189020
\(593\) 30.2541 1.24239 0.621194 0.783657i \(-0.286648\pi\)
0.621194 + 0.783657i \(0.286648\pi\)
\(594\) 0 0
\(595\) −28.4131 −1.16482
\(596\) 1.00000 0.0409616
\(597\) 0 0
\(598\) 1.03775 0.0424367
\(599\) −23.2842 −0.951368 −0.475684 0.879616i \(-0.657799\pi\)
−0.475684 + 0.879616i \(0.657799\pi\)
\(600\) 0 0
\(601\) −2.07413 −0.0846057 −0.0423029 0.999105i \(-0.513469\pi\)
−0.0423029 + 0.999105i \(0.513469\pi\)
\(602\) −23.8426 −0.971752
\(603\) 0 0
\(604\) −21.0529 −0.856632
\(605\) 19.3866 0.788177
\(606\) 0 0
\(607\) −14.3319 −0.581715 −0.290858 0.956766i \(-0.593941\pi\)
−0.290858 + 0.956766i \(0.593941\pi\)
\(608\) 0.244981 0.00993529
\(609\) 0 0
\(610\) −3.21778 −0.130284
\(611\) 13.8148 0.558887
\(612\) 0 0
\(613\) −22.8495 −0.922882 −0.461441 0.887171i \(-0.652667\pi\)
−0.461441 + 0.887171i \(0.652667\pi\)
\(614\) 12.8387 0.518126
\(615\) 0 0
\(616\) −0.437303 −0.0176194
\(617\) −13.4688 −0.542234 −0.271117 0.962546i \(-0.587393\pi\)
−0.271117 + 0.962546i \(0.587393\pi\)
\(618\) 0 0
\(619\) 14.0564 0.564976 0.282488 0.959271i \(-0.408840\pi\)
0.282488 + 0.959271i \(0.408840\pi\)
\(620\) 3.96691 0.159315
\(621\) 0 0
\(622\) −10.6858 −0.428463
\(623\) −1.37292 −0.0550050
\(624\) 0 0
\(625\) −12.0921 −0.483685
\(626\) 15.2652 0.610122
\(627\) 0 0
\(628\) 1.29923 0.0518448
\(629\) 29.2074 1.16458
\(630\) 0 0
\(631\) −6.52689 −0.259831 −0.129916 0.991525i \(-0.541471\pi\)
−0.129916 + 0.991525i \(0.541471\pi\)
\(632\) −2.04066 −0.0811730
\(633\) 0 0
\(634\) 33.7500 1.34038
\(635\) −20.3724 −0.808453
\(636\) 0 0
\(637\) −0.787932 −0.0312190
\(638\) 0.159335 0.00630815
\(639\) 0 0
\(640\) −1.76721 −0.0698551
\(641\) 38.1961 1.50866 0.754328 0.656498i \(-0.227963\pi\)
0.754328 + 0.656498i \(0.227963\pi\)
\(642\) 0 0
\(643\) −10.0076 −0.394661 −0.197330 0.980337i \(-0.563227\pi\)
−0.197330 + 0.980337i \(0.563227\pi\)
\(644\) 1.96950 0.0776092
\(645\) 0 0
\(646\) 1.55581 0.0612124
\(647\) 7.87727 0.309688 0.154844 0.987939i \(-0.450513\pi\)
0.154844 + 0.987939i \(0.450513\pi\)
\(648\) 0 0
\(649\) −1.35962 −0.0533699
\(650\) −2.50380 −0.0982070
\(651\) 0 0
\(652\) 17.8886 0.700572
\(653\) 48.3734 1.89300 0.946498 0.322709i \(-0.104594\pi\)
0.946498 + 0.322709i \(0.104594\pi\)
\(654\) 0 0
\(655\) 23.3992 0.914281
\(656\) 7.99362 0.312098
\(657\) 0 0
\(658\) 26.2186 1.02211
\(659\) 26.3964 1.02826 0.514129 0.857713i \(-0.328115\pi\)
0.514129 + 0.857713i \(0.328115\pi\)
\(660\) 0 0
\(661\) −37.1213 −1.44385 −0.721925 0.691971i \(-0.756743\pi\)
−0.721925 + 0.691971i \(0.756743\pi\)
\(662\) −0.851427 −0.0330917
\(663\) 0 0
\(664\) 13.6488 0.529676
\(665\) −1.09604 −0.0425026
\(666\) 0 0
\(667\) −0.717605 −0.0277858
\(668\) 23.5919 0.912798
\(669\) 0 0
\(670\) −18.3451 −0.708735
\(671\) −0.314517 −0.0121418
\(672\) 0 0
\(673\) −21.8295 −0.841467 −0.420733 0.907184i \(-0.638227\pi\)
−0.420733 + 0.907184i \(0.638227\pi\)
\(674\) 14.5751 0.561413
\(675\) 0 0
\(676\) −11.2206 −0.431560
\(677\) 16.2095 0.622982 0.311491 0.950249i \(-0.399172\pi\)
0.311491 + 0.950249i \(0.399172\pi\)
\(678\) 0 0
\(679\) 35.3772 1.35765
\(680\) −11.2231 −0.430385
\(681\) 0 0
\(682\) 0.387740 0.0148473
\(683\) 28.1909 1.07869 0.539347 0.842084i \(-0.318671\pi\)
0.539347 + 0.842084i \(0.318671\pi\)
\(684\) 0 0
\(685\) −18.1271 −0.692600
\(686\) −19.2170 −0.733710
\(687\) 0 0
\(688\) −9.41776 −0.359049
\(689\) −2.04728 −0.0779951
\(690\) 0 0
\(691\) −16.1916 −0.615957 −0.307979 0.951393i \(-0.599653\pi\)
−0.307979 + 0.951393i \(0.599653\pi\)
\(692\) −5.42628 −0.206276
\(693\) 0 0
\(694\) 13.5016 0.512515
\(695\) 19.8017 0.751123
\(696\) 0 0
\(697\) 50.7653 1.92287
\(698\) 12.9484 0.490103
\(699\) 0 0
\(700\) −4.75186 −0.179603
\(701\) 11.9775 0.452382 0.226191 0.974083i \(-0.427373\pi\)
0.226191 + 0.974083i \(0.427373\pi\)
\(702\) 0 0
\(703\) 1.12668 0.0424936
\(704\) −0.172733 −0.00651014
\(705\) 0 0
\(706\) −22.5635 −0.849187
\(707\) 39.4852 1.48499
\(708\) 0 0
\(709\) −42.4090 −1.59270 −0.796351 0.604835i \(-0.793239\pi\)
−0.796351 + 0.604835i \(0.793239\pi\)
\(710\) 17.9151 0.672341
\(711\) 0 0
\(712\) −0.542300 −0.0203236
\(713\) −1.74628 −0.0653987
\(714\) 0 0
\(715\) 0.407199 0.0152284
\(716\) 9.10535 0.340283
\(717\) 0 0
\(718\) 8.30844 0.310068
\(719\) −36.5998 −1.36494 −0.682472 0.730912i \(-0.739095\pi\)
−0.682472 + 0.730912i \(0.739095\pi\)
\(720\) 0 0
\(721\) −23.6390 −0.880361
\(722\) −18.9400 −0.704873
\(723\) 0 0
\(724\) −4.26853 −0.158639
\(725\) 1.73138 0.0643020
\(726\) 0 0
\(727\) 4.46378 0.165552 0.0827762 0.996568i \(-0.473621\pi\)
0.0827762 + 0.996568i \(0.473621\pi\)
\(728\) 3.37713 0.125165
\(729\) 0 0
\(730\) 25.8887 0.958184
\(731\) −59.8097 −2.21214
\(732\) 0 0
\(733\) −33.4559 −1.23572 −0.617862 0.786287i \(-0.712001\pi\)
−0.617862 + 0.786287i \(0.712001\pi\)
\(734\) 16.2813 0.600953
\(735\) 0 0
\(736\) 0.777947 0.0286755
\(737\) −1.79312 −0.0660504
\(738\) 0 0
\(739\) −10.5841 −0.389344 −0.194672 0.980868i \(-0.562364\pi\)
−0.194672 + 0.980868i \(0.562364\pi\)
\(740\) −8.12750 −0.298773
\(741\) 0 0
\(742\) −3.88545 −0.142639
\(743\) −8.83031 −0.323953 −0.161976 0.986795i \(-0.551787\pi\)
−0.161976 + 0.986795i \(0.551787\pi\)
\(744\) 0 0
\(745\) −1.76721 −0.0647456
\(746\) −18.5063 −0.677564
\(747\) 0 0
\(748\) −1.09698 −0.0401097
\(749\) 36.2017 1.32278
\(750\) 0 0
\(751\) 11.7397 0.428389 0.214194 0.976791i \(-0.431287\pi\)
0.214194 + 0.976791i \(0.431287\pi\)
\(752\) 10.3563 0.377654
\(753\) 0 0
\(754\) −1.23049 −0.0448118
\(755\) 37.2050 1.35403
\(756\) 0 0
\(757\) −37.7776 −1.37305 −0.686525 0.727107i \(-0.740865\pi\)
−0.686525 + 0.727107i \(0.740865\pi\)
\(758\) −25.2996 −0.918924
\(759\) 0 0
\(760\) −0.432933 −0.0157041
\(761\) −9.75982 −0.353793 −0.176897 0.984229i \(-0.556606\pi\)
−0.176897 + 0.984229i \(0.556606\pi\)
\(762\) 0 0
\(763\) −23.5104 −0.851135
\(764\) −19.1244 −0.691895
\(765\) 0 0
\(766\) −37.6264 −1.35950
\(767\) 10.4999 0.379129
\(768\) 0 0
\(769\) −54.0317 −1.94843 −0.974217 0.225614i \(-0.927561\pi\)
−0.974217 + 0.225614i \(0.927561\pi\)
\(770\) 0.772806 0.0278500
\(771\) 0 0
\(772\) 18.6585 0.671535
\(773\) 42.5648 1.53095 0.765475 0.643465i \(-0.222504\pi\)
0.765475 + 0.643465i \(0.222504\pi\)
\(774\) 0 0
\(775\) 4.21329 0.151346
\(776\) 13.9739 0.501633
\(777\) 0 0
\(778\) 26.1522 0.937603
\(779\) 1.95828 0.0701628
\(780\) 0 0
\(781\) 1.75108 0.0626587
\(782\) 4.94053 0.176673
\(783\) 0 0
\(784\) −0.590673 −0.0210954
\(785\) −2.29601 −0.0819481
\(786\) 0 0
\(787\) 35.5826 1.26838 0.634191 0.773177i \(-0.281333\pi\)
0.634191 + 0.773177i \(0.281333\pi\)
\(788\) 6.74520 0.240288
\(789\) 0 0
\(790\) 3.60627 0.128305
\(791\) −40.8952 −1.45407
\(792\) 0 0
\(793\) 2.42890 0.0862529
\(794\) −8.99278 −0.319142
\(795\) 0 0
\(796\) 16.9797 0.601828
\(797\) −48.5637 −1.72021 −0.860107 0.510114i \(-0.829603\pi\)
−0.860107 + 0.510114i \(0.829603\pi\)
\(798\) 0 0
\(799\) 65.7698 2.32677
\(800\) −1.87697 −0.0663609
\(801\) 0 0
\(802\) 26.9760 0.952554
\(803\) 2.53045 0.0892978
\(804\) 0 0
\(805\) −3.48052 −0.122672
\(806\) −2.99438 −0.105472
\(807\) 0 0
\(808\) 15.5965 0.548684
\(809\) −37.5919 −1.32166 −0.660831 0.750535i \(-0.729796\pi\)
−0.660831 + 0.750535i \(0.729796\pi\)
\(810\) 0 0
\(811\) −11.1414 −0.391228 −0.195614 0.980681i \(-0.562670\pi\)
−0.195614 + 0.980681i \(0.562670\pi\)
\(812\) −2.33530 −0.0819529
\(813\) 0 0
\(814\) −0.794411 −0.0278441
\(815\) −31.6129 −1.10735
\(816\) 0 0
\(817\) −2.30717 −0.0807177
\(818\) −0.843424 −0.0294896
\(819\) 0 0
\(820\) −14.1264 −0.493315
\(821\) 1.38903 0.0484775 0.0242387 0.999706i \(-0.492284\pi\)
0.0242387 + 0.999706i \(0.492284\pi\)
\(822\) 0 0
\(823\) −24.3396 −0.848426 −0.424213 0.905563i \(-0.639449\pi\)
−0.424213 + 0.905563i \(0.639449\pi\)
\(824\) −9.33732 −0.325281
\(825\) 0 0
\(826\) 19.9273 0.693359
\(827\) 41.5712 1.44557 0.722786 0.691072i \(-0.242861\pi\)
0.722786 + 0.691072i \(0.242861\pi\)
\(828\) 0 0
\(829\) −34.7876 −1.20822 −0.604112 0.796900i \(-0.706472\pi\)
−0.604112 + 0.796900i \(0.706472\pi\)
\(830\) −24.1203 −0.837228
\(831\) 0 0
\(832\) 1.33396 0.0462467
\(833\) −3.75120 −0.129971
\(834\) 0 0
\(835\) −41.6918 −1.44280
\(836\) −0.0423164 −0.00146354
\(837\) 0 0
\(838\) −32.4722 −1.12173
\(839\) 22.5319 0.777887 0.388944 0.921262i \(-0.372840\pi\)
0.388944 + 0.921262i \(0.372840\pi\)
\(840\) 0 0
\(841\) −28.1491 −0.970659
\(842\) 31.3692 1.08105
\(843\) 0 0
\(844\) −1.62775 −0.0560296
\(845\) 19.8291 0.682141
\(846\) 0 0
\(847\) −27.7728 −0.954284
\(848\) −1.53474 −0.0527032
\(849\) 0 0
\(850\) −11.9201 −0.408857
\(851\) 3.57782 0.122646
\(852\) 0 0
\(853\) 30.8670 1.05687 0.528433 0.848975i \(-0.322780\pi\)
0.528433 + 0.848975i \(0.322780\pi\)
\(854\) 4.60972 0.157741
\(855\) 0 0
\(856\) 14.2996 0.488750
\(857\) −3.26408 −0.111499 −0.0557495 0.998445i \(-0.517755\pi\)
−0.0557495 + 0.998445i \(0.517755\pi\)
\(858\) 0 0
\(859\) 23.8097 0.812376 0.406188 0.913790i \(-0.366858\pi\)
0.406188 + 0.913790i \(0.366858\pi\)
\(860\) 16.6432 0.567527
\(861\) 0 0
\(862\) 32.0091 1.09023
\(863\) 24.1010 0.820408 0.410204 0.911994i \(-0.365457\pi\)
0.410204 + 0.911994i \(0.365457\pi\)
\(864\) 0 0
\(865\) 9.58937 0.326048
\(866\) −19.5149 −0.663142
\(867\) 0 0
\(868\) −5.68291 −0.192890
\(869\) 0.352490 0.0119574
\(870\) 0 0
\(871\) 13.8476 0.469209
\(872\) −9.28655 −0.314482
\(873\) 0 0
\(874\) 0.190582 0.00644653
\(875\) 30.7674 1.04013
\(876\) 0 0
\(877\) 9.84472 0.332433 0.166216 0.986089i \(-0.446845\pi\)
0.166216 + 0.986089i \(0.446845\pi\)
\(878\) −38.2094 −1.28950
\(879\) 0 0
\(880\) 0.305256 0.0102902
\(881\) 16.5879 0.558862 0.279431 0.960166i \(-0.409854\pi\)
0.279431 + 0.960166i \(0.409854\pi\)
\(882\) 0 0
\(883\) −47.2685 −1.59071 −0.795355 0.606143i \(-0.792716\pi\)
−0.795355 + 0.606143i \(0.792716\pi\)
\(884\) 8.47161 0.284931
\(885\) 0 0
\(886\) 13.6766 0.459475
\(887\) 0.359158 0.0120593 0.00602967 0.999982i \(-0.498081\pi\)
0.00602967 + 0.999982i \(0.498081\pi\)
\(888\) 0 0
\(889\) 29.1850 0.978833
\(890\) 0.958359 0.0321243
\(891\) 0 0
\(892\) 28.2695 0.946531
\(893\) 2.53708 0.0849003
\(894\) 0 0
\(895\) −16.0911 −0.537865
\(896\) 2.53166 0.0845770
\(897\) 0 0
\(898\) −34.9553 −1.16647
\(899\) 2.07062 0.0690590
\(900\) 0 0
\(901\) −9.74672 −0.324710
\(902\) −1.38076 −0.0459744
\(903\) 0 0
\(904\) −16.1535 −0.537257
\(905\) 7.54340 0.250751
\(906\) 0 0
\(907\) 15.7279 0.522236 0.261118 0.965307i \(-0.415909\pi\)
0.261118 + 0.965307i \(0.415909\pi\)
\(908\) −17.1689 −0.569769
\(909\) 0 0
\(910\) −5.96810 −0.197841
\(911\) −8.06624 −0.267246 −0.133623 0.991032i \(-0.542661\pi\)
−0.133623 + 0.991032i \(0.542661\pi\)
\(912\) 0 0
\(913\) −2.35760 −0.0780253
\(914\) 19.5492 0.646628
\(915\) 0 0
\(916\) 12.0079 0.396751
\(917\) −33.5211 −1.10696
\(918\) 0 0
\(919\) −13.4648 −0.444163 −0.222081 0.975028i \(-0.571285\pi\)
−0.222081 + 0.975028i \(0.571285\pi\)
\(920\) −1.37479 −0.0453256
\(921\) 0 0
\(922\) 23.0442 0.758921
\(923\) −13.5230 −0.445114
\(924\) 0 0
\(925\) −8.63230 −0.283828
\(926\) −2.30744 −0.0758273
\(927\) 0 0
\(928\) −0.922435 −0.0302804
\(929\) −19.3356 −0.634380 −0.317190 0.948362i \(-0.602739\pi\)
−0.317190 + 0.948362i \(0.602739\pi\)
\(930\) 0 0
\(931\) −0.144703 −0.00474246
\(932\) 20.2171 0.662233
\(933\) 0 0
\(934\) 12.1472 0.397469
\(935\) 1.93860 0.0633990
\(936\) 0 0
\(937\) −42.1949 −1.37845 −0.689224 0.724548i \(-0.742049\pi\)
−0.689224 + 0.724548i \(0.742049\pi\)
\(938\) 26.2808 0.858100
\(939\) 0 0
\(940\) −18.3017 −0.596935
\(941\) 18.3595 0.598502 0.299251 0.954174i \(-0.403263\pi\)
0.299251 + 0.954174i \(0.403263\pi\)
\(942\) 0 0
\(943\) 6.21861 0.202506
\(944\) 7.87122 0.256186
\(945\) 0 0
\(946\) 1.62676 0.0528906
\(947\) −15.5641 −0.505766 −0.252883 0.967497i \(-0.581379\pi\)
−0.252883 + 0.967497i \(0.581379\pi\)
\(948\) 0 0
\(949\) −19.5418 −0.634353
\(950\) −0.459822 −0.0149186
\(951\) 0 0
\(952\) 16.0779 0.521089
\(953\) −8.81570 −0.285569 −0.142784 0.989754i \(-0.545606\pi\)
−0.142784 + 0.989754i \(0.545606\pi\)
\(954\) 0 0
\(955\) 33.7968 1.09364
\(956\) 5.11495 0.165429
\(957\) 0 0
\(958\) −17.3346 −0.560055
\(959\) 25.9684 0.838564
\(960\) 0 0
\(961\) −25.9612 −0.837458
\(962\) 6.13495 0.197799
\(963\) 0 0
\(964\) 29.0965 0.937136
\(965\) −32.9735 −1.06146
\(966\) 0 0
\(967\) −7.39034 −0.237657 −0.118829 0.992915i \(-0.537914\pi\)
−0.118829 + 0.992915i \(0.537914\pi\)
\(968\) −10.9702 −0.352594
\(969\) 0 0
\(970\) −24.6948 −0.792901
\(971\) −43.6961 −1.40228 −0.701138 0.713026i \(-0.747324\pi\)
−0.701138 + 0.713026i \(0.747324\pi\)
\(972\) 0 0
\(973\) −28.3675 −0.909421
\(974\) −34.5694 −1.10767
\(975\) 0 0
\(976\) 1.82082 0.0582832
\(977\) 46.3239 1.48203 0.741016 0.671488i \(-0.234344\pi\)
0.741016 + 0.671488i \(0.234344\pi\)
\(978\) 0 0
\(979\) 0.0936734 0.00299382
\(980\) 1.04384 0.0333443
\(981\) 0 0
\(982\) −37.6299 −1.20082
\(983\) −53.7041 −1.71289 −0.856447 0.516235i \(-0.827333\pi\)
−0.856447 + 0.516235i \(0.827333\pi\)
\(984\) 0 0
\(985\) −11.9202 −0.379808
\(986\) −5.85814 −0.186561
\(987\) 0 0
\(988\) 0.326794 0.0103967
\(989\) −7.32652 −0.232970
\(990\) 0 0
\(991\) −32.7809 −1.04132 −0.520659 0.853765i \(-0.674314\pi\)
−0.520659 + 0.853765i \(0.674314\pi\)
\(992\) −2.24473 −0.0712703
\(993\) 0 0
\(994\) −25.6647 −0.814036
\(995\) −30.0066 −0.951274
\(996\) 0 0
\(997\) 20.9203 0.662554 0.331277 0.943534i \(-0.392521\pi\)
0.331277 + 0.943534i \(0.392521\pi\)
\(998\) 35.7585 1.13192
\(999\) 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.o.1.4 yes 12
3.2 odd 2 8046.2.a.j.1.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8046.2.a.j.1.9 12 3.2 odd 2
8046.2.a.o.1.4 yes 12 1.1 even 1 trivial