Properties

Label 6039.2.a.e.1.2
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
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] = [6039,2,Mod(1,6039)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6039.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.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: \(48.2216577807\)
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: \( x^{12} - x^{11} - 16 x^{10} + 13 x^{9} + 93 x^{8} - 59 x^{7} - 238 x^{6} + 108 x^{5} + 257 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2013)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.07880\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.07880 q^{2} +2.32140 q^{4} +2.51557 q^{5} +3.31485 q^{7} -0.668123 q^{8} +O(q^{10})\) \(q-2.07880 q^{2} +2.32140 q^{4} +2.51557 q^{5} +3.31485 q^{7} -0.668123 q^{8} -5.22935 q^{10} +1.00000 q^{11} -4.69636 q^{13} -6.89089 q^{14} -3.25391 q^{16} -2.84744 q^{17} -7.33856 q^{19} +5.83963 q^{20} -2.07880 q^{22} +2.08156 q^{23} +1.32807 q^{25} +9.76278 q^{26} +7.69508 q^{28} -0.764221 q^{29} +2.67183 q^{31} +8.10046 q^{32} +5.91926 q^{34} +8.33871 q^{35} +2.93137 q^{37} +15.2554 q^{38} -1.68071 q^{40} -3.67795 q^{41} -9.30987 q^{43} +2.32140 q^{44} -4.32715 q^{46} +0.466652 q^{47} +3.98820 q^{49} -2.76079 q^{50} -10.9021 q^{52} +7.82503 q^{53} +2.51557 q^{55} -2.21472 q^{56} +1.58866 q^{58} +7.75368 q^{59} -1.00000 q^{61} -5.55420 q^{62} -10.3314 q^{64} -11.8140 q^{65} -4.65473 q^{67} -6.61005 q^{68} -17.3345 q^{70} -7.30894 q^{71} -6.71256 q^{73} -6.09372 q^{74} -17.0357 q^{76} +3.31485 q^{77} -6.44921 q^{79} -8.18541 q^{80} +7.64571 q^{82} -2.42650 q^{83} -7.16293 q^{85} +19.3533 q^{86} -0.668123 q^{88} +2.54789 q^{89} -15.5677 q^{91} +4.83214 q^{92} -0.970075 q^{94} -18.4606 q^{95} -11.8241 q^{97} -8.29066 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8} - 8 q^{10} + 12 q^{11} - q^{13} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 20 q^{19} + 9 q^{20} - q^{22} + 9 q^{23} + 3 q^{25} + 18 q^{26} - 31 q^{28} - 18 q^{29} - 21 q^{31} - 18 q^{32} - 12 q^{34} + 4 q^{35} - 18 q^{37} + 2 q^{38} - 26 q^{40} - 15 q^{41} - 33 q^{43} + 9 q^{44} - 28 q^{46} + 20 q^{47} + 15 q^{49} + 2 q^{50} - 27 q^{52} + 3 q^{55} + 8 q^{56} - 11 q^{58} + 21 q^{59} - 12 q^{61} + 9 q^{62} - 12 q^{64} - 17 q^{65} - 34 q^{67} + 16 q^{68} - 36 q^{70} + 5 q^{71} - 2 q^{73} - 6 q^{74} - 27 q^{76} - 9 q^{77} - 31 q^{79} + 60 q^{80} - 12 q^{82} + 32 q^{83} - 40 q^{85} - 18 q^{86} - 6 q^{88} - 27 q^{89} - 45 q^{91} + 78 q^{92} - 13 q^{94} - 37 q^{95} - 19 q^{97} - 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.07880 −1.46993 −0.734966 0.678104i \(-0.762802\pi\)
−0.734966 + 0.678104i \(0.762802\pi\)
\(3\) 0 0
\(4\) 2.32140 1.16070
\(5\) 2.51557 1.12500 0.562498 0.826799i \(-0.309841\pi\)
0.562498 + 0.826799i \(0.309841\pi\)
\(6\) 0 0
\(7\) 3.31485 1.25289 0.626447 0.779464i \(-0.284509\pi\)
0.626447 + 0.779464i \(0.284509\pi\)
\(8\) −0.668123 −0.236217
\(9\) 0 0
\(10\) −5.22935 −1.65367
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −4.69636 −1.30254 −0.651268 0.758848i \(-0.725762\pi\)
−0.651268 + 0.758848i \(0.725762\pi\)
\(14\) −6.89089 −1.84167
\(15\) 0 0
\(16\) −3.25391 −0.813476
\(17\) −2.84744 −0.690606 −0.345303 0.938491i \(-0.612224\pi\)
−0.345303 + 0.938491i \(0.612224\pi\)
\(18\) 0 0
\(19\) −7.33856 −1.68358 −0.841790 0.539805i \(-0.818498\pi\)
−0.841790 + 0.539805i \(0.818498\pi\)
\(20\) 5.83963 1.30578
\(21\) 0 0
\(22\) −2.07880 −0.443201
\(23\) 2.08156 0.434036 0.217018 0.976168i \(-0.430367\pi\)
0.217018 + 0.976168i \(0.430367\pi\)
\(24\) 0 0
\(25\) 1.32807 0.265614
\(26\) 9.76278 1.91464
\(27\) 0 0
\(28\) 7.69508 1.45423
\(29\) −0.764221 −0.141912 −0.0709561 0.997479i \(-0.522605\pi\)
−0.0709561 + 0.997479i \(0.522605\pi\)
\(30\) 0 0
\(31\) 2.67183 0.479875 0.239938 0.970788i \(-0.422873\pi\)
0.239938 + 0.970788i \(0.422873\pi\)
\(32\) 8.10046 1.43197
\(33\) 0 0
\(34\) 5.91926 1.01514
\(35\) 8.33871 1.40950
\(36\) 0 0
\(37\) 2.93137 0.481914 0.240957 0.970536i \(-0.422539\pi\)
0.240957 + 0.970536i \(0.422539\pi\)
\(38\) 15.2554 2.47475
\(39\) 0 0
\(40\) −1.68071 −0.265743
\(41\) −3.67795 −0.574399 −0.287199 0.957871i \(-0.592724\pi\)
−0.287199 + 0.957871i \(0.592724\pi\)
\(42\) 0 0
\(43\) −9.30987 −1.41974 −0.709871 0.704332i \(-0.751247\pi\)
−0.709871 + 0.704332i \(0.751247\pi\)
\(44\) 2.32140 0.349964
\(45\) 0 0
\(46\) −4.32715 −0.638003
\(47\) 0.466652 0.0680682 0.0340341 0.999421i \(-0.489165\pi\)
0.0340341 + 0.999421i \(0.489165\pi\)
\(48\) 0 0
\(49\) 3.98820 0.569743
\(50\) −2.76079 −0.390435
\(51\) 0 0
\(52\) −10.9021 −1.51185
\(53\) 7.82503 1.07485 0.537425 0.843311i \(-0.319397\pi\)
0.537425 + 0.843311i \(0.319397\pi\)
\(54\) 0 0
\(55\) 2.51557 0.339199
\(56\) −2.21472 −0.295955
\(57\) 0 0
\(58\) 1.58866 0.208601
\(59\) 7.75368 1.00944 0.504722 0.863282i \(-0.331595\pi\)
0.504722 + 0.863282i \(0.331595\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) −5.55420 −0.705384
\(63\) 0 0
\(64\) −10.3314 −1.29142
\(65\) −11.8140 −1.46535
\(66\) 0 0
\(67\) −4.65473 −0.568666 −0.284333 0.958726i \(-0.591772\pi\)
−0.284333 + 0.958726i \(0.591772\pi\)
\(68\) −6.61005 −0.801586
\(69\) 0 0
\(70\) −17.3345 −2.07187
\(71\) −7.30894 −0.867412 −0.433706 0.901054i \(-0.642794\pi\)
−0.433706 + 0.901054i \(0.642794\pi\)
\(72\) 0 0
\(73\) −6.71256 −0.785645 −0.392823 0.919614i \(-0.628501\pi\)
−0.392823 + 0.919614i \(0.628501\pi\)
\(74\) −6.09372 −0.708381
\(75\) 0 0
\(76\) −17.0357 −1.95413
\(77\) 3.31485 0.377762
\(78\) 0 0
\(79\) −6.44921 −0.725593 −0.362797 0.931868i \(-0.618178\pi\)
−0.362797 + 0.931868i \(0.618178\pi\)
\(80\) −8.18541 −0.915157
\(81\) 0 0
\(82\) 7.64571 0.844327
\(83\) −2.42650 −0.266343 −0.133172 0.991093i \(-0.542516\pi\)
−0.133172 + 0.991093i \(0.542516\pi\)
\(84\) 0 0
\(85\) −7.16293 −0.776929
\(86\) 19.3533 2.08692
\(87\) 0 0
\(88\) −0.668123 −0.0712222
\(89\) 2.54789 0.270076 0.135038 0.990840i \(-0.456884\pi\)
0.135038 + 0.990840i \(0.456884\pi\)
\(90\) 0 0
\(91\) −15.5677 −1.63194
\(92\) 4.83214 0.503785
\(93\) 0 0
\(94\) −0.970075 −0.100056
\(95\) −18.4606 −1.89402
\(96\) 0 0
\(97\) −11.8241 −1.20055 −0.600277 0.799793i \(-0.704943\pi\)
−0.600277 + 0.799793i \(0.704943\pi\)
\(98\) −8.29066 −0.837483
\(99\) 0 0
\(100\) 3.08298 0.308298
\(101\) 12.8545 1.27907 0.639537 0.768760i \(-0.279126\pi\)
0.639537 + 0.768760i \(0.279126\pi\)
\(102\) 0 0
\(103\) −7.74643 −0.763279 −0.381639 0.924311i \(-0.624640\pi\)
−0.381639 + 0.924311i \(0.624640\pi\)
\(104\) 3.13774 0.307681
\(105\) 0 0
\(106\) −16.2667 −1.57996
\(107\) −15.8933 −1.53646 −0.768230 0.640174i \(-0.778862\pi\)
−0.768230 + 0.640174i \(0.778862\pi\)
\(108\) 0 0
\(109\) −3.51888 −0.337048 −0.168524 0.985698i \(-0.553900\pi\)
−0.168524 + 0.985698i \(0.553900\pi\)
\(110\) −5.22935 −0.498599
\(111\) 0 0
\(112\) −10.7862 −1.01920
\(113\) −8.00502 −0.753049 −0.376524 0.926407i \(-0.622881\pi\)
−0.376524 + 0.926407i \(0.622881\pi\)
\(114\) 0 0
\(115\) 5.23631 0.488288
\(116\) −1.77406 −0.164717
\(117\) 0 0
\(118\) −16.1183 −1.48381
\(119\) −9.43883 −0.865256
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.07880 0.188205
\(123\) 0 0
\(124\) 6.20239 0.556991
\(125\) −9.23698 −0.826180
\(126\) 0 0
\(127\) 5.63027 0.499605 0.249803 0.968297i \(-0.419634\pi\)
0.249803 + 0.968297i \(0.419634\pi\)
\(128\) 5.27597 0.466334
\(129\) 0 0
\(130\) 24.5589 2.15396
\(131\) −4.25978 −0.372179 −0.186089 0.982533i \(-0.559581\pi\)
−0.186089 + 0.982533i \(0.559581\pi\)
\(132\) 0 0
\(133\) −24.3262 −2.10935
\(134\) 9.67624 0.835900
\(135\) 0 0
\(136\) 1.90244 0.163133
\(137\) −0.938042 −0.0801423 −0.0400712 0.999197i \(-0.512758\pi\)
−0.0400712 + 0.999197i \(0.512758\pi\)
\(138\) 0 0
\(139\) 20.0152 1.69767 0.848834 0.528660i \(-0.177305\pi\)
0.848834 + 0.528660i \(0.177305\pi\)
\(140\) 19.3575 1.63601
\(141\) 0 0
\(142\) 15.1938 1.27504
\(143\) −4.69636 −0.392729
\(144\) 0 0
\(145\) −1.92245 −0.159651
\(146\) 13.9540 1.15484
\(147\) 0 0
\(148\) 6.80487 0.559357
\(149\) −13.2774 −1.08773 −0.543864 0.839173i \(-0.683039\pi\)
−0.543864 + 0.839173i \(0.683039\pi\)
\(150\) 0 0
\(151\) 11.8307 0.962771 0.481385 0.876509i \(-0.340134\pi\)
0.481385 + 0.876509i \(0.340134\pi\)
\(152\) 4.90306 0.397691
\(153\) 0 0
\(154\) −6.89089 −0.555284
\(155\) 6.72117 0.539858
\(156\) 0 0
\(157\) −12.3071 −0.982216 −0.491108 0.871099i \(-0.663408\pi\)
−0.491108 + 0.871099i \(0.663408\pi\)
\(158\) 13.4066 1.06657
\(159\) 0 0
\(160\) 20.3772 1.61096
\(161\) 6.90006 0.543801
\(162\) 0 0
\(163\) 20.8550 1.63349 0.816744 0.577001i \(-0.195777\pi\)
0.816744 + 0.577001i \(0.195777\pi\)
\(164\) −8.53798 −0.666704
\(165\) 0 0
\(166\) 5.04421 0.391506
\(167\) 3.99278 0.308970 0.154485 0.987995i \(-0.450628\pi\)
0.154485 + 0.987995i \(0.450628\pi\)
\(168\) 0 0
\(169\) 9.05577 0.696598
\(170\) 14.8903 1.14203
\(171\) 0 0
\(172\) −21.6119 −1.64789
\(173\) 25.5150 1.93987 0.969937 0.243358i \(-0.0782490\pi\)
0.969937 + 0.243358i \(0.0782490\pi\)
\(174\) 0 0
\(175\) 4.40235 0.332787
\(176\) −3.25391 −0.245272
\(177\) 0 0
\(178\) −5.29655 −0.396993
\(179\) −20.3940 −1.52432 −0.762158 0.647391i \(-0.775860\pi\)
−0.762158 + 0.647391i \(0.775860\pi\)
\(180\) 0 0
\(181\) −10.3544 −0.769633 −0.384816 0.922993i \(-0.625735\pi\)
−0.384816 + 0.922993i \(0.625735\pi\)
\(182\) 32.3621 2.39884
\(183\) 0 0
\(184\) −1.39074 −0.102527
\(185\) 7.37405 0.542151
\(186\) 0 0
\(187\) −2.84744 −0.208226
\(188\) 1.08329 0.0790067
\(189\) 0 0
\(190\) 38.3759 2.78408
\(191\) −2.57556 −0.186361 −0.0931805 0.995649i \(-0.529703\pi\)
−0.0931805 + 0.995649i \(0.529703\pi\)
\(192\) 0 0
\(193\) 21.0757 1.51706 0.758530 0.651638i \(-0.225918\pi\)
0.758530 + 0.651638i \(0.225918\pi\)
\(194\) 24.5799 1.76473
\(195\) 0 0
\(196\) 9.25820 0.661300
\(197\) −21.1509 −1.50694 −0.753469 0.657484i \(-0.771621\pi\)
−0.753469 + 0.657484i \(0.771621\pi\)
\(198\) 0 0
\(199\) −7.63056 −0.540916 −0.270458 0.962732i \(-0.587175\pi\)
−0.270458 + 0.962732i \(0.587175\pi\)
\(200\) −0.887315 −0.0627427
\(201\) 0 0
\(202\) −26.7220 −1.88015
\(203\) −2.53327 −0.177801
\(204\) 0 0
\(205\) −9.25212 −0.646196
\(206\) 16.1033 1.12197
\(207\) 0 0
\(208\) 15.2815 1.05958
\(209\) −7.33856 −0.507619
\(210\) 0 0
\(211\) −25.6471 −1.76562 −0.882811 0.469728i \(-0.844352\pi\)
−0.882811 + 0.469728i \(0.844352\pi\)
\(212\) 18.1650 1.24758
\(213\) 0 0
\(214\) 33.0389 2.25849
\(215\) −23.4196 −1.59720
\(216\) 0 0
\(217\) 8.85671 0.601233
\(218\) 7.31505 0.495437
\(219\) 0 0
\(220\) 5.83963 0.393708
\(221\) 13.3726 0.899539
\(222\) 0 0
\(223\) −21.6223 −1.44794 −0.723968 0.689834i \(-0.757684\pi\)
−0.723968 + 0.689834i \(0.757684\pi\)
\(224\) 26.8518 1.79411
\(225\) 0 0
\(226\) 16.6408 1.10693
\(227\) −7.41119 −0.491898 −0.245949 0.969283i \(-0.579100\pi\)
−0.245949 + 0.969283i \(0.579100\pi\)
\(228\) 0 0
\(229\) 12.2381 0.808718 0.404359 0.914600i \(-0.367495\pi\)
0.404359 + 0.914600i \(0.367495\pi\)
\(230\) −10.8852 −0.717750
\(231\) 0 0
\(232\) 0.510594 0.0335221
\(233\) −9.92929 −0.650489 −0.325245 0.945630i \(-0.605447\pi\)
−0.325245 + 0.945630i \(0.605447\pi\)
\(234\) 0 0
\(235\) 1.17389 0.0765764
\(236\) 17.9994 1.17166
\(237\) 0 0
\(238\) 19.6214 1.27187
\(239\) −11.4528 −0.740823 −0.370412 0.928868i \(-0.620783\pi\)
−0.370412 + 0.928868i \(0.620783\pi\)
\(240\) 0 0
\(241\) 1.38803 0.0894111 0.0447055 0.999000i \(-0.485765\pi\)
0.0447055 + 0.999000i \(0.485765\pi\)
\(242\) −2.07880 −0.133630
\(243\) 0 0
\(244\) −2.32140 −0.148612
\(245\) 10.0326 0.640958
\(246\) 0 0
\(247\) 34.4645 2.19292
\(248\) −1.78511 −0.113355
\(249\) 0 0
\(250\) 19.2018 1.21443
\(251\) 10.9919 0.693801 0.346900 0.937902i \(-0.387234\pi\)
0.346900 + 0.937902i \(0.387234\pi\)
\(252\) 0 0
\(253\) 2.08156 0.130867
\(254\) −11.7042 −0.734386
\(255\) 0 0
\(256\) 9.69512 0.605945
\(257\) −24.1795 −1.50827 −0.754137 0.656718i \(-0.771944\pi\)
−0.754137 + 0.656718i \(0.771944\pi\)
\(258\) 0 0
\(259\) 9.71703 0.603787
\(260\) −27.4250 −1.70083
\(261\) 0 0
\(262\) 8.85522 0.547077
\(263\) −0.0653591 −0.00403021 −0.00201511 0.999998i \(-0.500641\pi\)
−0.00201511 + 0.999998i \(0.500641\pi\)
\(264\) 0 0
\(265\) 19.6844 1.20920
\(266\) 50.5692 3.10060
\(267\) 0 0
\(268\) −10.8055 −0.660050
\(269\) −5.24648 −0.319884 −0.159942 0.987126i \(-0.551131\pi\)
−0.159942 + 0.987126i \(0.551131\pi\)
\(270\) 0 0
\(271\) −7.61564 −0.462617 −0.231308 0.972880i \(-0.574301\pi\)
−0.231308 + 0.972880i \(0.574301\pi\)
\(272\) 9.26531 0.561792
\(273\) 0 0
\(274\) 1.95000 0.117804
\(275\) 1.32807 0.0800857
\(276\) 0 0
\(277\) 16.4198 0.986570 0.493285 0.869868i \(-0.335796\pi\)
0.493285 + 0.869868i \(0.335796\pi\)
\(278\) −41.6075 −2.49545
\(279\) 0 0
\(280\) −5.57129 −0.332948
\(281\) 6.84038 0.408063 0.204032 0.978964i \(-0.434595\pi\)
0.204032 + 0.978964i \(0.434595\pi\)
\(282\) 0 0
\(283\) −6.59236 −0.391875 −0.195937 0.980616i \(-0.562775\pi\)
−0.195937 + 0.980616i \(0.562775\pi\)
\(284\) −16.9670 −1.00680
\(285\) 0 0
\(286\) 9.76278 0.577285
\(287\) −12.1918 −0.719661
\(288\) 0 0
\(289\) −8.89207 −0.523063
\(290\) 3.99638 0.234676
\(291\) 0 0
\(292\) −15.5825 −0.911898
\(293\) 0.825657 0.0482354 0.0241177 0.999709i \(-0.492322\pi\)
0.0241177 + 0.999709i \(0.492322\pi\)
\(294\) 0 0
\(295\) 19.5049 1.13562
\(296\) −1.95851 −0.113836
\(297\) 0 0
\(298\) 27.6010 1.59889
\(299\) −9.77576 −0.565347
\(300\) 0 0
\(301\) −30.8608 −1.77879
\(302\) −24.5937 −1.41521
\(303\) 0 0
\(304\) 23.8790 1.36955
\(305\) −2.51557 −0.144041
\(306\) 0 0
\(307\) −10.6787 −0.609463 −0.304732 0.952438i \(-0.598567\pi\)
−0.304732 + 0.952438i \(0.598567\pi\)
\(308\) 7.69508 0.438468
\(309\) 0 0
\(310\) −13.9720 −0.793554
\(311\) 3.76872 0.213704 0.106852 0.994275i \(-0.465923\pi\)
0.106852 + 0.994275i \(0.465923\pi\)
\(312\) 0 0
\(313\) 18.2136 1.02950 0.514748 0.857342i \(-0.327886\pi\)
0.514748 + 0.857342i \(0.327886\pi\)
\(314\) 25.5840 1.44379
\(315\) 0 0
\(316\) −14.9712 −0.842196
\(317\) −18.1257 −1.01804 −0.509019 0.860755i \(-0.669992\pi\)
−0.509019 + 0.860755i \(0.669992\pi\)
\(318\) 0 0
\(319\) −0.764221 −0.0427882
\(320\) −25.9893 −1.45285
\(321\) 0 0
\(322\) −14.3438 −0.799350
\(323\) 20.8961 1.16269
\(324\) 0 0
\(325\) −6.23710 −0.345972
\(326\) −43.3533 −2.40111
\(327\) 0 0
\(328\) 2.45732 0.135683
\(329\) 1.54688 0.0852822
\(330\) 0 0
\(331\) −23.3048 −1.28095 −0.640475 0.767979i \(-0.721262\pi\)
−0.640475 + 0.767979i \(0.721262\pi\)
\(332\) −5.63288 −0.309144
\(333\) 0 0
\(334\) −8.30017 −0.454165
\(335\) −11.7093 −0.639746
\(336\) 0 0
\(337\) −16.0798 −0.875924 −0.437962 0.898993i \(-0.644300\pi\)
−0.437962 + 0.898993i \(0.644300\pi\)
\(338\) −18.8251 −1.02395
\(339\) 0 0
\(340\) −16.6280 −0.901781
\(341\) 2.67183 0.144688
\(342\) 0 0
\(343\) −9.98366 −0.539067
\(344\) 6.22014 0.335368
\(345\) 0 0
\(346\) −53.0406 −2.85148
\(347\) −2.04396 −0.109725 −0.0548627 0.998494i \(-0.517472\pi\)
−0.0548627 + 0.998494i \(0.517472\pi\)
\(348\) 0 0
\(349\) −27.4212 −1.46782 −0.733912 0.679244i \(-0.762308\pi\)
−0.733912 + 0.679244i \(0.762308\pi\)
\(350\) −9.15160 −0.489173
\(351\) 0 0
\(352\) 8.10046 0.431756
\(353\) 26.7877 1.42576 0.712882 0.701284i \(-0.247390\pi\)
0.712882 + 0.701284i \(0.247390\pi\)
\(354\) 0 0
\(355\) −18.3861 −0.975835
\(356\) 5.91467 0.313477
\(357\) 0 0
\(358\) 42.3949 2.24064
\(359\) 20.1894 1.06556 0.532778 0.846255i \(-0.321148\pi\)
0.532778 + 0.846255i \(0.321148\pi\)
\(360\) 0 0
\(361\) 34.8544 1.83444
\(362\) 21.5246 1.13131
\(363\) 0 0
\(364\) −36.1388 −1.89419
\(365\) −16.8859 −0.883847
\(366\) 0 0
\(367\) −8.71927 −0.455142 −0.227571 0.973761i \(-0.573078\pi\)
−0.227571 + 0.973761i \(0.573078\pi\)
\(368\) −6.77321 −0.353078
\(369\) 0 0
\(370\) −15.3292 −0.796925
\(371\) 25.9388 1.34667
\(372\) 0 0
\(373\) 8.10112 0.419460 0.209730 0.977759i \(-0.432742\pi\)
0.209730 + 0.977759i \(0.432742\pi\)
\(374\) 5.91926 0.306077
\(375\) 0 0
\(376\) −0.311781 −0.0160789
\(377\) 3.58905 0.184846
\(378\) 0 0
\(379\) 16.1829 0.831259 0.415629 0.909534i \(-0.363561\pi\)
0.415629 + 0.909534i \(0.363561\pi\)
\(380\) −42.8545 −2.19839
\(381\) 0 0
\(382\) 5.35407 0.273938
\(383\) 11.5062 0.587940 0.293970 0.955815i \(-0.405023\pi\)
0.293970 + 0.955815i \(0.405023\pi\)
\(384\) 0 0
\(385\) 8.33871 0.424980
\(386\) −43.8120 −2.22997
\(387\) 0 0
\(388\) −27.4484 −1.39348
\(389\) 17.6156 0.893147 0.446573 0.894747i \(-0.352644\pi\)
0.446573 + 0.894747i \(0.352644\pi\)
\(390\) 0 0
\(391\) −5.92713 −0.299748
\(392\) −2.66461 −0.134583
\(393\) 0 0
\(394\) 43.9684 2.21509
\(395\) −16.2234 −0.816289
\(396\) 0 0
\(397\) 18.6556 0.936298 0.468149 0.883649i \(-0.344921\pi\)
0.468149 + 0.883649i \(0.344921\pi\)
\(398\) 15.8624 0.795110
\(399\) 0 0
\(400\) −4.32142 −0.216071
\(401\) 26.6759 1.33213 0.666066 0.745893i \(-0.267977\pi\)
0.666066 + 0.745893i \(0.267977\pi\)
\(402\) 0 0
\(403\) −12.5479 −0.625055
\(404\) 29.8405 1.48462
\(405\) 0 0
\(406\) 5.26616 0.261355
\(407\) 2.93137 0.145303
\(408\) 0 0
\(409\) −38.2979 −1.89371 −0.946856 0.321658i \(-0.895760\pi\)
−0.946856 + 0.321658i \(0.895760\pi\)
\(410\) 19.2333 0.949864
\(411\) 0 0
\(412\) −17.9826 −0.885937
\(413\) 25.7023 1.26473
\(414\) 0 0
\(415\) −6.10402 −0.299635
\(416\) −38.0426 −1.86519
\(417\) 0 0
\(418\) 15.2554 0.746165
\(419\) 4.67442 0.228360 0.114180 0.993460i \(-0.463576\pi\)
0.114180 + 0.993460i \(0.463576\pi\)
\(420\) 0 0
\(421\) 29.7448 1.44967 0.724836 0.688922i \(-0.241916\pi\)
0.724836 + 0.688922i \(0.241916\pi\)
\(422\) 53.3152 2.59534
\(423\) 0 0
\(424\) −5.22808 −0.253898
\(425\) −3.78161 −0.183435
\(426\) 0 0
\(427\) −3.31485 −0.160417
\(428\) −36.8946 −1.78337
\(429\) 0 0
\(430\) 48.6846 2.34778
\(431\) −14.2615 −0.686950 −0.343475 0.939162i \(-0.611604\pi\)
−0.343475 + 0.939162i \(0.611604\pi\)
\(432\) 0 0
\(433\) 40.0835 1.92629 0.963145 0.268983i \(-0.0866875\pi\)
0.963145 + 0.268983i \(0.0866875\pi\)
\(434\) −18.4113 −0.883771
\(435\) 0 0
\(436\) −8.16873 −0.391211
\(437\) −15.2757 −0.730734
\(438\) 0 0
\(439\) −41.7801 −1.99405 −0.997027 0.0770534i \(-0.975449\pi\)
−0.997027 + 0.0770534i \(0.975449\pi\)
\(440\) −1.68071 −0.0801246
\(441\) 0 0
\(442\) −27.7989 −1.32226
\(443\) 22.6161 1.07452 0.537262 0.843415i \(-0.319459\pi\)
0.537262 + 0.843415i \(0.319459\pi\)
\(444\) 0 0
\(445\) 6.40939 0.303834
\(446\) 44.9484 2.12837
\(447\) 0 0
\(448\) −34.2470 −1.61802
\(449\) −0.296481 −0.0139918 −0.00699590 0.999976i \(-0.502227\pi\)
−0.00699590 + 0.999976i \(0.502227\pi\)
\(450\) 0 0
\(451\) −3.67795 −0.173188
\(452\) −18.5828 −0.874063
\(453\) 0 0
\(454\) 15.4064 0.723056
\(455\) −39.1616 −1.83592
\(456\) 0 0
\(457\) −35.4014 −1.65601 −0.828004 0.560722i \(-0.810524\pi\)
−0.828004 + 0.560722i \(0.810524\pi\)
\(458\) −25.4406 −1.18876
\(459\) 0 0
\(460\) 12.1556 0.566756
\(461\) −24.8643 −1.15804 −0.579022 0.815312i \(-0.696566\pi\)
−0.579022 + 0.815312i \(0.696566\pi\)
\(462\) 0 0
\(463\) −23.6296 −1.09816 −0.549080 0.835770i \(-0.685022\pi\)
−0.549080 + 0.835770i \(0.685022\pi\)
\(464\) 2.48670 0.115442
\(465\) 0 0
\(466\) 20.6410 0.956175
\(467\) 24.7168 1.14376 0.571879 0.820338i \(-0.306215\pi\)
0.571879 + 0.820338i \(0.306215\pi\)
\(468\) 0 0
\(469\) −15.4297 −0.712478
\(470\) −2.44029 −0.112562
\(471\) 0 0
\(472\) −5.18042 −0.238448
\(473\) −9.30987 −0.428068
\(474\) 0 0
\(475\) −9.74613 −0.447183
\(476\) −21.9113 −1.00430
\(477\) 0 0
\(478\) 23.8081 1.08896
\(479\) 3.86168 0.176445 0.0882223 0.996101i \(-0.471881\pi\)
0.0882223 + 0.996101i \(0.471881\pi\)
\(480\) 0 0
\(481\) −13.7668 −0.627710
\(482\) −2.88544 −0.131428
\(483\) 0 0
\(484\) 2.32140 0.105518
\(485\) −29.7442 −1.35062
\(486\) 0 0
\(487\) 4.58072 0.207572 0.103786 0.994600i \(-0.466904\pi\)
0.103786 + 0.994600i \(0.466904\pi\)
\(488\) 0.668123 0.0302445
\(489\) 0 0
\(490\) −20.8557 −0.942164
\(491\) 33.5912 1.51595 0.757974 0.652284i \(-0.226189\pi\)
0.757974 + 0.652284i \(0.226189\pi\)
\(492\) 0 0
\(493\) 2.17608 0.0980055
\(494\) −71.6447 −3.22345
\(495\) 0 0
\(496\) −8.69389 −0.390367
\(497\) −24.2280 −1.08678
\(498\) 0 0
\(499\) 32.2036 1.44163 0.720816 0.693127i \(-0.243767\pi\)
0.720816 + 0.693127i \(0.243767\pi\)
\(500\) −21.4427 −0.958947
\(501\) 0 0
\(502\) −22.8499 −1.01984
\(503\) 39.4574 1.75932 0.879660 0.475603i \(-0.157770\pi\)
0.879660 + 0.475603i \(0.157770\pi\)
\(504\) 0 0
\(505\) 32.3364 1.43895
\(506\) −4.32715 −0.192365
\(507\) 0 0
\(508\) 13.0701 0.579892
\(509\) −21.0465 −0.932871 −0.466436 0.884555i \(-0.654462\pi\)
−0.466436 + 0.884555i \(0.654462\pi\)
\(510\) 0 0
\(511\) −22.2511 −0.984330
\(512\) −30.7061 −1.35703
\(513\) 0 0
\(514\) 50.2642 2.21706
\(515\) −19.4867 −0.858685
\(516\) 0 0
\(517\) 0.466652 0.0205233
\(518\) −20.1997 −0.887526
\(519\) 0 0
\(520\) 7.89320 0.346140
\(521\) −25.8419 −1.13216 −0.566078 0.824352i \(-0.691540\pi\)
−0.566078 + 0.824352i \(0.691540\pi\)
\(522\) 0 0
\(523\) 11.0550 0.483400 0.241700 0.970351i \(-0.422295\pi\)
0.241700 + 0.970351i \(0.422295\pi\)
\(524\) −9.88865 −0.431988
\(525\) 0 0
\(526\) 0.135868 0.00592414
\(527\) −7.60789 −0.331405
\(528\) 0 0
\(529\) −18.6671 −0.811613
\(530\) −40.9198 −1.77744
\(531\) 0 0
\(532\) −56.4708 −2.44832
\(533\) 17.2730 0.748175
\(534\) 0 0
\(535\) −39.9805 −1.72851
\(536\) 3.10993 0.134329
\(537\) 0 0
\(538\) 10.9064 0.470207
\(539\) 3.98820 0.171784
\(540\) 0 0
\(541\) −32.1200 −1.38095 −0.690473 0.723358i \(-0.742598\pi\)
−0.690473 + 0.723358i \(0.742598\pi\)
\(542\) 15.8314 0.680015
\(543\) 0 0
\(544\) −23.0656 −0.988929
\(545\) −8.85198 −0.379177
\(546\) 0 0
\(547\) −21.4050 −0.915211 −0.457605 0.889155i \(-0.651293\pi\)
−0.457605 + 0.889155i \(0.651293\pi\)
\(548\) −2.17757 −0.0930211
\(549\) 0 0
\(550\) −2.76079 −0.117721
\(551\) 5.60828 0.238921
\(552\) 0 0
\(553\) −21.3781 −0.909091
\(554\) −34.1334 −1.45019
\(555\) 0 0
\(556\) 46.4633 1.97048
\(557\) 14.8084 0.627453 0.313726 0.949513i \(-0.398423\pi\)
0.313726 + 0.949513i \(0.398423\pi\)
\(558\) 0 0
\(559\) 43.7225 1.84926
\(560\) −27.1334 −1.14659
\(561\) 0 0
\(562\) −14.2198 −0.599825
\(563\) −41.4376 −1.74639 −0.873193 0.487374i \(-0.837955\pi\)
−0.873193 + 0.487374i \(0.837955\pi\)
\(564\) 0 0
\(565\) −20.1371 −0.847176
\(566\) 13.7042 0.576029
\(567\) 0 0
\(568\) 4.88327 0.204898
\(569\) 24.3931 1.02261 0.511306 0.859399i \(-0.329162\pi\)
0.511306 + 0.859399i \(0.329162\pi\)
\(570\) 0 0
\(571\) −28.7749 −1.20419 −0.602097 0.798423i \(-0.705668\pi\)
−0.602097 + 0.798423i \(0.705668\pi\)
\(572\) −10.9021 −0.455840
\(573\) 0 0
\(574\) 25.3443 1.05785
\(575\) 2.76446 0.115286
\(576\) 0 0
\(577\) 5.15803 0.214732 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(578\) 18.4848 0.768867
\(579\) 0 0
\(580\) −4.46277 −0.185306
\(581\) −8.04348 −0.333700
\(582\) 0 0
\(583\) 7.82503 0.324080
\(584\) 4.48481 0.185583
\(585\) 0 0
\(586\) −1.71637 −0.0709028
\(587\) 36.0284 1.48705 0.743525 0.668708i \(-0.233152\pi\)
0.743525 + 0.668708i \(0.233152\pi\)
\(588\) 0 0
\(589\) −19.6074 −0.807909
\(590\) −40.5467 −1.66928
\(591\) 0 0
\(592\) −9.53839 −0.392026
\(593\) 18.1163 0.743946 0.371973 0.928243i \(-0.378681\pi\)
0.371973 + 0.928243i \(0.378681\pi\)
\(594\) 0 0
\(595\) −23.7440 −0.973409
\(596\) −30.8222 −1.26252
\(597\) 0 0
\(598\) 20.3218 0.831021
\(599\) 8.70761 0.355783 0.177892 0.984050i \(-0.443072\pi\)
0.177892 + 0.984050i \(0.443072\pi\)
\(600\) 0 0
\(601\) 23.7422 0.968464 0.484232 0.874940i \(-0.339099\pi\)
0.484232 + 0.874940i \(0.339099\pi\)
\(602\) 64.1533 2.61469
\(603\) 0 0
\(604\) 27.4638 1.11749
\(605\) 2.51557 0.102272
\(606\) 0 0
\(607\) 0.604695 0.0245438 0.0122719 0.999925i \(-0.496094\pi\)
0.0122719 + 0.999925i \(0.496094\pi\)
\(608\) −59.4457 −2.41084
\(609\) 0 0
\(610\) 5.22935 0.211730
\(611\) −2.19156 −0.0886612
\(612\) 0 0
\(613\) 19.6301 0.792853 0.396426 0.918067i \(-0.370250\pi\)
0.396426 + 0.918067i \(0.370250\pi\)
\(614\) 22.1988 0.895870
\(615\) 0 0
\(616\) −2.21472 −0.0892338
\(617\) 10.4332 0.420025 0.210012 0.977699i \(-0.432650\pi\)
0.210012 + 0.977699i \(0.432650\pi\)
\(618\) 0 0
\(619\) 21.6981 0.872121 0.436060 0.899917i \(-0.356373\pi\)
0.436060 + 0.899917i \(0.356373\pi\)
\(620\) 15.6025 0.626612
\(621\) 0 0
\(622\) −7.83440 −0.314131
\(623\) 8.44586 0.338376
\(624\) 0 0
\(625\) −29.8766 −1.19506
\(626\) −37.8625 −1.51329
\(627\) 0 0
\(628\) −28.5697 −1.14006
\(629\) −8.34690 −0.332813
\(630\) 0 0
\(631\) 2.07442 0.0825814 0.0412907 0.999147i \(-0.486853\pi\)
0.0412907 + 0.999147i \(0.486853\pi\)
\(632\) 4.30887 0.171398
\(633\) 0 0
\(634\) 37.6796 1.49645
\(635\) 14.1633 0.562054
\(636\) 0 0
\(637\) −18.7300 −0.742110
\(638\) 1.58866 0.0628957
\(639\) 0 0
\(640\) 13.2720 0.524624
\(641\) 13.9661 0.551628 0.275814 0.961211i \(-0.411053\pi\)
0.275814 + 0.961211i \(0.411053\pi\)
\(642\) 0 0
\(643\) −20.4874 −0.807943 −0.403971 0.914772i \(-0.632370\pi\)
−0.403971 + 0.914772i \(0.632370\pi\)
\(644\) 16.0178 0.631189
\(645\) 0 0
\(646\) −43.4388 −1.70908
\(647\) −11.4213 −0.449018 −0.224509 0.974472i \(-0.572078\pi\)
−0.224509 + 0.974472i \(0.572078\pi\)
\(648\) 0 0
\(649\) 7.75368 0.304359
\(650\) 12.9657 0.508555
\(651\) 0 0
\(652\) 48.4127 1.89599
\(653\) −17.7057 −0.692876 −0.346438 0.938073i \(-0.612609\pi\)
−0.346438 + 0.938073i \(0.612609\pi\)
\(654\) 0 0
\(655\) −10.7158 −0.418699
\(656\) 11.9677 0.467260
\(657\) 0 0
\(658\) −3.21565 −0.125359
\(659\) −17.5526 −0.683753 −0.341877 0.939745i \(-0.611063\pi\)
−0.341877 + 0.939745i \(0.611063\pi\)
\(660\) 0 0
\(661\) −37.1442 −1.44474 −0.722370 0.691506i \(-0.756947\pi\)
−0.722370 + 0.691506i \(0.756947\pi\)
\(662\) 48.4460 1.88291
\(663\) 0 0
\(664\) 1.62120 0.0629148
\(665\) −61.1941 −2.37301
\(666\) 0 0
\(667\) −1.59077 −0.0615950
\(668\) 9.26883 0.358622
\(669\) 0 0
\(670\) 24.3412 0.940384
\(671\) −1.00000 −0.0386046
\(672\) 0 0
\(673\) −24.8198 −0.956733 −0.478366 0.878160i \(-0.658771\pi\)
−0.478366 + 0.878160i \(0.658771\pi\)
\(674\) 33.4267 1.28755
\(675\) 0 0
\(676\) 21.0221 0.808541
\(677\) −20.8171 −0.800064 −0.400032 0.916501i \(-0.631001\pi\)
−0.400032 + 0.916501i \(0.631001\pi\)
\(678\) 0 0
\(679\) −39.1950 −1.50417
\(680\) 4.78572 0.183524
\(681\) 0 0
\(682\) −5.55420 −0.212681
\(683\) 21.9543 0.840058 0.420029 0.907511i \(-0.362020\pi\)
0.420029 + 0.907511i \(0.362020\pi\)
\(684\) 0 0
\(685\) −2.35971 −0.0901597
\(686\) 20.7540 0.792391
\(687\) 0 0
\(688\) 30.2934 1.15493
\(689\) −36.7491 −1.40003
\(690\) 0 0
\(691\) −0.279975 −0.0106508 −0.00532538 0.999986i \(-0.501695\pi\)
−0.00532538 + 0.999986i \(0.501695\pi\)
\(692\) 59.2306 2.25161
\(693\) 0 0
\(694\) 4.24897 0.161289
\(695\) 50.3496 1.90987
\(696\) 0 0
\(697\) 10.4727 0.396683
\(698\) 57.0032 2.15760
\(699\) 0 0
\(700\) 10.2196 0.386265
\(701\) −26.9968 −1.01965 −0.509827 0.860277i \(-0.670290\pi\)
−0.509827 + 0.860277i \(0.670290\pi\)
\(702\) 0 0
\(703\) −21.5120 −0.811341
\(704\) −10.3314 −0.389379
\(705\) 0 0
\(706\) −55.6861 −2.09578
\(707\) 42.6108 1.60254
\(708\) 0 0
\(709\) 17.4492 0.655319 0.327659 0.944796i \(-0.393740\pi\)
0.327659 + 0.944796i \(0.393740\pi\)
\(710\) 38.2210 1.43441
\(711\) 0 0
\(712\) −1.70230 −0.0637966
\(713\) 5.56159 0.208283
\(714\) 0 0
\(715\) −11.8140 −0.441818
\(716\) −47.3425 −1.76927
\(717\) 0 0
\(718\) −41.9697 −1.56629
\(719\) −24.9061 −0.928839 −0.464420 0.885615i \(-0.653737\pi\)
−0.464420 + 0.885615i \(0.653737\pi\)
\(720\) 0 0
\(721\) −25.6782 −0.956307
\(722\) −72.4553 −2.69651
\(723\) 0 0
\(724\) −24.0366 −0.893312
\(725\) −1.01494 −0.0376939
\(726\) 0 0
\(727\) −21.4670 −0.796166 −0.398083 0.917349i \(-0.630324\pi\)
−0.398083 + 0.917349i \(0.630324\pi\)
\(728\) 10.4011 0.385492
\(729\) 0 0
\(730\) 35.1023 1.29919
\(731\) 26.5093 0.980483
\(732\) 0 0
\(733\) −2.10577 −0.0777784 −0.0388892 0.999244i \(-0.512382\pi\)
−0.0388892 + 0.999244i \(0.512382\pi\)
\(734\) 18.1256 0.669028
\(735\) 0 0
\(736\) 16.8616 0.621527
\(737\) −4.65473 −0.171459
\(738\) 0 0
\(739\) 12.0592 0.443604 0.221802 0.975092i \(-0.428806\pi\)
0.221802 + 0.975092i \(0.428806\pi\)
\(740\) 17.1181 0.629274
\(741\) 0 0
\(742\) −53.9214 −1.97952
\(743\) −5.58320 −0.204828 −0.102414 0.994742i \(-0.532657\pi\)
−0.102414 + 0.994742i \(0.532657\pi\)
\(744\) 0 0
\(745\) −33.4002 −1.22369
\(746\) −16.8406 −0.616577
\(747\) 0 0
\(748\) −6.61005 −0.241687
\(749\) −52.6837 −1.92502
\(750\) 0 0
\(751\) 40.1470 1.46499 0.732493 0.680775i \(-0.238357\pi\)
0.732493 + 0.680775i \(0.238357\pi\)
\(752\) −1.51844 −0.0553719
\(753\) 0 0
\(754\) −7.46092 −0.271711
\(755\) 29.7610 1.08311
\(756\) 0 0
\(757\) 10.2101 0.371091 0.185546 0.982636i \(-0.440595\pi\)
0.185546 + 0.982636i \(0.440595\pi\)
\(758\) −33.6409 −1.22189
\(759\) 0 0
\(760\) 12.3340 0.447400
\(761\) −41.3818 −1.50009 −0.750045 0.661387i \(-0.769968\pi\)
−0.750045 + 0.661387i \(0.769968\pi\)
\(762\) 0 0
\(763\) −11.6646 −0.422285
\(764\) −5.97890 −0.216309
\(765\) 0 0
\(766\) −23.9191 −0.864232
\(767\) −36.4141 −1.31484
\(768\) 0 0
\(769\) −1.86294 −0.0671793 −0.0335896 0.999436i \(-0.510694\pi\)
−0.0335896 + 0.999436i \(0.510694\pi\)
\(770\) −17.3345 −0.624692
\(771\) 0 0
\(772\) 48.9250 1.76085
\(773\) 8.28658 0.298048 0.149024 0.988834i \(-0.452387\pi\)
0.149024 + 0.988834i \(0.452387\pi\)
\(774\) 0 0
\(775\) 3.54839 0.127462
\(776\) 7.89994 0.283591
\(777\) 0 0
\(778\) −36.6193 −1.31286
\(779\) 26.9908 0.967046
\(780\) 0 0
\(781\) −7.30894 −0.261535
\(782\) 12.3213 0.440609
\(783\) 0 0
\(784\) −12.9772 −0.463472
\(785\) −30.9594 −1.10499
\(786\) 0 0
\(787\) 1.03420 0.0368654 0.0184327 0.999830i \(-0.494132\pi\)
0.0184327 + 0.999830i \(0.494132\pi\)
\(788\) −49.0996 −1.74910
\(789\) 0 0
\(790\) 33.7252 1.19989
\(791\) −26.5354 −0.943490
\(792\) 0 0
\(793\) 4.69636 0.166773
\(794\) −38.7812 −1.37629
\(795\) 0 0
\(796\) −17.7136 −0.627841
\(797\) 22.6893 0.803696 0.401848 0.915706i \(-0.368368\pi\)
0.401848 + 0.915706i \(0.368368\pi\)
\(798\) 0 0
\(799\) −1.32877 −0.0470083
\(800\) 10.7580 0.380352
\(801\) 0 0
\(802\) −55.4538 −1.95814
\(803\) −6.71256 −0.236881
\(804\) 0 0
\(805\) 17.3575 0.611773
\(806\) 26.0845 0.918788
\(807\) 0 0
\(808\) −8.58841 −0.302139
\(809\) 19.9728 0.702206 0.351103 0.936337i \(-0.385807\pi\)
0.351103 + 0.936337i \(0.385807\pi\)
\(810\) 0 0
\(811\) 51.5452 1.81000 0.904999 0.425414i \(-0.139871\pi\)
0.904999 + 0.425414i \(0.139871\pi\)
\(812\) −5.88074 −0.206374
\(813\) 0 0
\(814\) −6.09372 −0.213585
\(815\) 52.4621 1.83767
\(816\) 0 0
\(817\) 68.3210 2.39025
\(818\) 79.6137 2.78363
\(819\) 0 0
\(820\) −21.4779 −0.750039
\(821\) −14.9141 −0.520506 −0.260253 0.965541i \(-0.583806\pi\)
−0.260253 + 0.965541i \(0.583806\pi\)
\(822\) 0 0
\(823\) −55.5932 −1.93786 −0.968929 0.247340i \(-0.920444\pi\)
−0.968929 + 0.247340i \(0.920444\pi\)
\(824\) 5.17557 0.180300
\(825\) 0 0
\(826\) −53.4298 −1.85906
\(827\) 9.76170 0.339448 0.169724 0.985492i \(-0.445712\pi\)
0.169724 + 0.985492i \(0.445712\pi\)
\(828\) 0 0
\(829\) −10.8861 −0.378090 −0.189045 0.981968i \(-0.560539\pi\)
−0.189045 + 0.981968i \(0.560539\pi\)
\(830\) 12.6890 0.440443
\(831\) 0 0
\(832\) 48.5199 1.68213
\(833\) −11.3562 −0.393468
\(834\) 0 0
\(835\) 10.0441 0.347590
\(836\) −17.0357 −0.589193
\(837\) 0 0
\(838\) −9.71717 −0.335674
\(839\) −30.5135 −1.05344 −0.526722 0.850038i \(-0.676579\pi\)
−0.526722 + 0.850038i \(0.676579\pi\)
\(840\) 0 0
\(841\) −28.4160 −0.979861
\(842\) −61.8334 −2.13092
\(843\) 0 0
\(844\) −59.5373 −2.04936
\(845\) 22.7804 0.783669
\(846\) 0 0
\(847\) 3.31485 0.113899
\(848\) −25.4619 −0.874365
\(849\) 0 0
\(850\) 7.86120 0.269637
\(851\) 6.10183 0.209168
\(852\) 0 0
\(853\) 46.5861 1.59508 0.797539 0.603267i \(-0.206135\pi\)
0.797539 + 0.603267i \(0.206135\pi\)
\(854\) 6.89089 0.235801
\(855\) 0 0
\(856\) 10.6187 0.362938
\(857\) 23.3679 0.798233 0.399116 0.916900i \(-0.369317\pi\)
0.399116 + 0.916900i \(0.369317\pi\)
\(858\) 0 0
\(859\) −29.1184 −0.993506 −0.496753 0.867892i \(-0.665475\pi\)
−0.496753 + 0.867892i \(0.665475\pi\)
\(860\) −54.3662 −1.85387
\(861\) 0 0
\(862\) 29.6467 1.00977
\(863\) −20.8386 −0.709353 −0.354676 0.934989i \(-0.615409\pi\)
−0.354676 + 0.934989i \(0.615409\pi\)
\(864\) 0 0
\(865\) 64.1848 2.18235
\(866\) −83.3255 −2.83151
\(867\) 0 0
\(868\) 20.5600 0.697851
\(869\) −6.44921 −0.218775
\(870\) 0 0
\(871\) 21.8603 0.740707
\(872\) 2.35105 0.0796165
\(873\) 0 0
\(874\) 31.7550 1.07413
\(875\) −30.6192 −1.03512
\(876\) 0 0
\(877\) 40.3955 1.36406 0.682030 0.731324i \(-0.261097\pi\)
0.682030 + 0.731324i \(0.261097\pi\)
\(878\) 86.8523 2.93112
\(879\) 0 0
\(880\) −8.18541 −0.275930
\(881\) −28.4606 −0.958863 −0.479432 0.877579i \(-0.659157\pi\)
−0.479432 + 0.877579i \(0.659157\pi\)
\(882\) 0 0
\(883\) 37.3212 1.25596 0.627980 0.778230i \(-0.283882\pi\)
0.627980 + 0.778230i \(0.283882\pi\)
\(884\) 31.0432 1.04409
\(885\) 0 0
\(886\) −47.0143 −1.57948
\(887\) 5.69722 0.191294 0.0956470 0.995415i \(-0.469508\pi\)
0.0956470 + 0.995415i \(0.469508\pi\)
\(888\) 0 0
\(889\) 18.6635 0.625953
\(890\) −13.3238 −0.446615
\(891\) 0 0
\(892\) −50.1940 −1.68062
\(893\) −3.42455 −0.114598
\(894\) 0 0
\(895\) −51.3024 −1.71485
\(896\) 17.4890 0.584267
\(897\) 0 0
\(898\) 0.616324 0.0205670
\(899\) −2.04187 −0.0681002
\(900\) 0 0
\(901\) −22.2813 −0.742299
\(902\) 7.64571 0.254574
\(903\) 0 0
\(904\) 5.34834 0.177883
\(905\) −26.0470 −0.865833
\(906\) 0 0
\(907\) −28.0847 −0.932536 −0.466268 0.884644i \(-0.654402\pi\)
−0.466268 + 0.884644i \(0.654402\pi\)
\(908\) −17.2043 −0.570946
\(909\) 0 0
\(910\) 81.4090 2.69868
\(911\) −5.56419 −0.184350 −0.0921749 0.995743i \(-0.529382\pi\)
−0.0921749 + 0.995743i \(0.529382\pi\)
\(912\) 0 0
\(913\) −2.42650 −0.0803055
\(914\) 73.5924 2.43422
\(915\) 0 0
\(916\) 28.4096 0.938679
\(917\) −14.1205 −0.466301
\(918\) 0 0
\(919\) 9.22080 0.304166 0.152083 0.988368i \(-0.451402\pi\)
0.152083 + 0.988368i \(0.451402\pi\)
\(920\) −3.49850 −0.115342
\(921\) 0 0
\(922\) 51.6878 1.70225
\(923\) 34.3254 1.12983
\(924\) 0 0
\(925\) 3.89307 0.128003
\(926\) 49.1211 1.61422
\(927\) 0 0
\(928\) −6.19054 −0.203214
\(929\) 18.9686 0.622339 0.311170 0.950354i \(-0.399279\pi\)
0.311170 + 0.950354i \(0.399279\pi\)
\(930\) 0 0
\(931\) −29.2676 −0.959208
\(932\) −23.0498 −0.755022
\(933\) 0 0
\(934\) −51.3812 −1.68124
\(935\) −7.16293 −0.234253
\(936\) 0 0
\(937\) 30.0732 0.982449 0.491224 0.871033i \(-0.336550\pi\)
0.491224 + 0.871033i \(0.336550\pi\)
\(938\) 32.0752 1.04729
\(939\) 0 0
\(940\) 2.72508 0.0888822
\(941\) 23.3429 0.760956 0.380478 0.924790i \(-0.375760\pi\)
0.380478 + 0.924790i \(0.375760\pi\)
\(942\) 0 0
\(943\) −7.65588 −0.249310
\(944\) −25.2298 −0.821159
\(945\) 0 0
\(946\) 19.3533 0.629231
\(947\) 16.1738 0.525579 0.262790 0.964853i \(-0.415358\pi\)
0.262790 + 0.964853i \(0.415358\pi\)
\(948\) 0 0
\(949\) 31.5246 1.02333
\(950\) 20.2602 0.657329
\(951\) 0 0
\(952\) 6.30630 0.204388
\(953\) 53.5723 1.73538 0.867689 0.497107i \(-0.165604\pi\)
0.867689 + 0.497107i \(0.165604\pi\)
\(954\) 0 0
\(955\) −6.47899 −0.209655
\(956\) −26.5866 −0.859873
\(957\) 0 0
\(958\) −8.02765 −0.259361
\(959\) −3.10946 −0.100410
\(960\) 0 0
\(961\) −23.8613 −0.769720
\(962\) 28.6183 0.922691
\(963\) 0 0
\(964\) 3.22218 0.103779
\(965\) 53.0172 1.70668
\(966\) 0 0
\(967\) −26.8918 −0.864781 −0.432390 0.901686i \(-0.642330\pi\)
−0.432390 + 0.901686i \(0.642330\pi\)
\(968\) −0.668123 −0.0214743
\(969\) 0 0
\(970\) 61.8323 1.98531
\(971\) −4.19888 −0.134749 −0.0673743 0.997728i \(-0.521462\pi\)
−0.0673743 + 0.997728i \(0.521462\pi\)
\(972\) 0 0
\(973\) 66.3473 2.12700
\(974\) −9.52238 −0.305117
\(975\) 0 0
\(976\) 3.25391 0.104155
\(977\) −4.88596 −0.156316 −0.0781578 0.996941i \(-0.524904\pi\)
−0.0781578 + 0.996941i \(0.524904\pi\)
\(978\) 0 0
\(979\) 2.54789 0.0814309
\(980\) 23.2896 0.743959
\(981\) 0 0
\(982\) −69.8293 −2.22834
\(983\) 41.8812 1.33580 0.667902 0.744250i \(-0.267193\pi\)
0.667902 + 0.744250i \(0.267193\pi\)
\(984\) 0 0
\(985\) −53.2064 −1.69530
\(986\) −4.52362 −0.144061
\(987\) 0 0
\(988\) 80.0058 2.54532
\(989\) −19.3791 −0.616219
\(990\) 0 0
\(991\) 20.1386 0.639724 0.319862 0.947464i \(-0.396363\pi\)
0.319862 + 0.947464i \(0.396363\pi\)
\(992\) 21.6431 0.687168
\(993\) 0 0
\(994\) 50.3651 1.59749
\(995\) −19.1952 −0.608528
\(996\) 0 0
\(997\) −2.25885 −0.0715385 −0.0357692 0.999360i \(-0.511388\pi\)
−0.0357692 + 0.999360i \(0.511388\pi\)
\(998\) −66.9448 −2.11910
\(999\) 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 6039.2.a.e.1.2 12
3.2 odd 2 2013.2.a.d.1.11 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.d.1.11 12 3.2 odd 2
6039.2.a.e.1.2 12 1.1 even 1 trivial