Properties

Label 3040.2.a.w.1.1
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
Analytic rank $0$
Dimension $5$
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] = [3040,2,Mod(1,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.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: \(24.2745222145\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.2363492.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 11x^{3} - 6x^{2} + 14x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.19018\) of defining polynomial
Character \(\chi\) \(=\) 3040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.78093 q^{3} +1.00000 q^{5} +0.646760 q^{7} +4.73359 q^{9} +O(q^{10})\) \(q-2.78093 q^{3} +1.00000 q^{5} +0.646760 q^{7} +4.73359 q^{9} +2.99272 q^{11} +6.15401 q^{13} -2.78093 q^{15} +2.72632 q^{17} +1.00000 q^{19} -1.79860 q^{21} +0.646760 q^{23} +1.00000 q^{25} -4.82100 q^{27} +3.81315 q^{29} -0.0145507 q^{31} -8.32257 q^{33} +0.646760 q^{35} +6.49997 q^{37} -17.1139 q^{39} +0.532818 q^{41} +2.45991 q^{43} +4.73359 q^{45} -11.8627 q^{47} -6.58170 q^{49} -7.58170 q^{51} -1.29862 q^{53} +2.99272 q^{55} -2.78093 q^{57} +13.0489 q^{59} -3.75343 q^{61} +3.06150 q^{63} +6.15401 q^{65} -13.2481 q^{67} -1.79860 q^{69} +1.46718 q^{71} -8.82100 q^{73} -2.78093 q^{75} +1.93557 q^{77} +9.17361 q^{79} -0.793890 q^{81} -11.3797 q^{83} +2.72632 q^{85} -10.6041 q^{87} +14.8408 q^{89} +3.98017 q^{91} +0.0404646 q^{93} +1.00000 q^{95} -6.34086 q^{97} +14.1663 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{3} + 5 q^{5} - 2 q^{7} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{3} + 5 q^{5} - 2 q^{7} + 19 q^{9} + 10 q^{11} + 2 q^{15} + 4 q^{17} + 5 q^{19} + 10 q^{21} - 2 q^{23} + 5 q^{25} + 8 q^{27} + 10 q^{29} - 10 q^{31} + 10 q^{33} - 2 q^{35} + 2 q^{37} - 10 q^{39} + 12 q^{41} - 2 q^{43} + 19 q^{45} - 22 q^{47} + 19 q^{49} + 14 q^{51} - 18 q^{53} + 10 q^{55} + 2 q^{57} + 4 q^{59} + 6 q^{61} - 10 q^{63} + 20 q^{67} + 10 q^{69} - 2 q^{71} - 12 q^{73} + 2 q^{75} - 4 q^{77} - 14 q^{79} + 13 q^{81} - 14 q^{83} + 4 q^{85} + 12 q^{87} + 22 q^{89} + 40 q^{91} + 8 q^{93} + 5 q^{95} - 10 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.78093 −1.60557 −0.802786 0.596267i \(-0.796650\pi\)
−0.802786 + 0.596267i \(0.796650\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.646760 0.244452 0.122226 0.992502i \(-0.460997\pi\)
0.122226 + 0.992502i \(0.460997\pi\)
\(8\) 0 0
\(9\) 4.73359 1.57786
\(10\) 0 0
\(11\) 2.99272 0.902340 0.451170 0.892438i \(-0.351007\pi\)
0.451170 + 0.892438i \(0.351007\pi\)
\(12\) 0 0
\(13\) 6.15401 1.70681 0.853407 0.521244i \(-0.174532\pi\)
0.853407 + 0.521244i \(0.174532\pi\)
\(14\) 0 0
\(15\) −2.78093 −0.718034
\(16\) 0 0
\(17\) 2.72632 0.661229 0.330614 0.943766i \(-0.392744\pi\)
0.330614 + 0.943766i \(0.392744\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.79860 −0.392486
\(22\) 0 0
\(23\) 0.646760 0.134859 0.0674294 0.997724i \(-0.478520\pi\)
0.0674294 + 0.997724i \(0.478520\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.82100 −0.927802
\(28\) 0 0
\(29\) 3.81315 0.708084 0.354042 0.935230i \(-0.384807\pi\)
0.354042 + 0.935230i \(0.384807\pi\)
\(30\) 0 0
\(31\) −0.0145507 −0.00261339 −0.00130669 0.999999i \(-0.500416\pi\)
−0.00130669 + 0.999999i \(0.500416\pi\)
\(32\) 0 0
\(33\) −8.32257 −1.44877
\(34\) 0 0
\(35\) 0.646760 0.109322
\(36\) 0 0
\(37\) 6.49997 1.06859 0.534295 0.845298i \(-0.320577\pi\)
0.534295 + 0.845298i \(0.320577\pi\)
\(38\) 0 0
\(39\) −17.1139 −2.74042
\(40\) 0 0
\(41\) 0.532818 0.0832122 0.0416061 0.999134i \(-0.486753\pi\)
0.0416061 + 0.999134i \(0.486753\pi\)
\(42\) 0 0
\(43\) 2.45991 0.375132 0.187566 0.982252i \(-0.439940\pi\)
0.187566 + 0.982252i \(0.439940\pi\)
\(44\) 0 0
\(45\) 4.73359 0.705642
\(46\) 0 0
\(47\) −11.8627 −1.73035 −0.865174 0.501473i \(-0.832792\pi\)
−0.865174 + 0.501473i \(0.832792\pi\)
\(48\) 0 0
\(49\) −6.58170 −0.940243
\(50\) 0 0
\(51\) −7.58170 −1.06165
\(52\) 0 0
\(53\) −1.29862 −0.178379 −0.0891897 0.996015i \(-0.528428\pi\)
−0.0891897 + 0.996015i \(0.528428\pi\)
\(54\) 0 0
\(55\) 2.99272 0.403539
\(56\) 0 0
\(57\) −2.78093 −0.368344
\(58\) 0 0
\(59\) 13.0489 1.69882 0.849410 0.527734i \(-0.176958\pi\)
0.849410 + 0.527734i \(0.176958\pi\)
\(60\) 0 0
\(61\) −3.75343 −0.480577 −0.240288 0.970702i \(-0.577242\pi\)
−0.240288 + 0.970702i \(0.577242\pi\)
\(62\) 0 0
\(63\) 3.06150 0.385712
\(64\) 0 0
\(65\) 6.15401 0.763311
\(66\) 0 0
\(67\) −13.2481 −1.61851 −0.809254 0.587459i \(-0.800128\pi\)
−0.809254 + 0.587459i \(0.800128\pi\)
\(68\) 0 0
\(69\) −1.79860 −0.216525
\(70\) 0 0
\(71\) 1.46718 0.174122 0.0870612 0.996203i \(-0.472252\pi\)
0.0870612 + 0.996203i \(0.472252\pi\)
\(72\) 0 0
\(73\) −8.82100 −1.03242 −0.516210 0.856462i \(-0.672658\pi\)
−0.516210 + 0.856462i \(0.672658\pi\)
\(74\) 0 0
\(75\) −2.78093 −0.321115
\(76\) 0 0
\(77\) 1.93557 0.220579
\(78\) 0 0
\(79\) 9.17361 1.03211 0.516056 0.856555i \(-0.327400\pi\)
0.516056 + 0.856555i \(0.327400\pi\)
\(80\) 0 0
\(81\) −0.793890 −0.0882100
\(82\) 0 0
\(83\) −11.3797 −1.24909 −0.624543 0.780990i \(-0.714715\pi\)
−0.624543 + 0.780990i \(0.714715\pi\)
\(84\) 0 0
\(85\) 2.72632 0.295710
\(86\) 0 0
\(87\) −10.6041 −1.13688
\(88\) 0 0
\(89\) 14.8408 1.57313 0.786563 0.617510i \(-0.211859\pi\)
0.786563 + 0.617510i \(0.211859\pi\)
\(90\) 0 0
\(91\) 3.98017 0.417235
\(92\) 0 0
\(93\) 0.0404646 0.00419599
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −6.34086 −0.643817 −0.321909 0.946771i \(-0.604324\pi\)
−0.321909 + 0.946771i \(0.604324\pi\)
\(98\) 0 0
\(99\) 14.1663 1.42377
\(100\) 0 0
\(101\) 18.9270 1.88331 0.941655 0.336579i \(-0.109270\pi\)
0.941655 + 0.336579i \(0.109270\pi\)
\(102\) 0 0
\(103\) 2.76110 0.272059 0.136030 0.990705i \(-0.456566\pi\)
0.136030 + 0.990705i \(0.456566\pi\)
\(104\) 0 0
\(105\) −1.79860 −0.175525
\(106\) 0 0
\(107\) −8.78093 −0.848885 −0.424442 0.905455i \(-0.639530\pi\)
−0.424442 + 0.905455i \(0.639530\pi\)
\(108\) 0 0
\(109\) 7.63948 0.731730 0.365865 0.930668i \(-0.380773\pi\)
0.365865 + 0.930668i \(0.380773\pi\)
\(110\) 0 0
\(111\) −18.0760 −1.71570
\(112\) 0 0
\(113\) −6.78287 −0.638079 −0.319039 0.947741i \(-0.603360\pi\)
−0.319039 + 0.947741i \(0.603360\pi\)
\(114\) 0 0
\(115\) 0.646760 0.0603107
\(116\) 0 0
\(117\) 29.1306 2.69312
\(118\) 0 0
\(119\) 1.76327 0.161639
\(120\) 0 0
\(121\) −2.04360 −0.185782
\(122\) 0 0
\(123\) −1.48173 −0.133603
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −9.68091 −0.859042 −0.429521 0.903057i \(-0.641318\pi\)
−0.429521 + 0.903057i \(0.641318\pi\)
\(128\) 0 0
\(129\) −6.84084 −0.602302
\(130\) 0 0
\(131\) 13.7897 1.20482 0.602408 0.798188i \(-0.294208\pi\)
0.602408 + 0.798188i \(0.294208\pi\)
\(132\) 0 0
\(133\) 0.646760 0.0560812
\(134\) 0 0
\(135\) −4.82100 −0.414926
\(136\) 0 0
\(137\) −3.25913 −0.278447 −0.139223 0.990261i \(-0.544461\pi\)
−0.139223 + 0.990261i \(0.544461\pi\)
\(138\) 0 0
\(139\) 9.19665 0.780049 0.390025 0.920804i \(-0.372466\pi\)
0.390025 + 0.920804i \(0.372466\pi\)
\(140\) 0 0
\(141\) 32.9893 2.77820
\(142\) 0 0
\(143\) 18.4173 1.54013
\(144\) 0 0
\(145\) 3.81315 0.316665
\(146\) 0 0
\(147\) 18.3033 1.50963
\(148\) 0 0
\(149\) −14.8772 −1.21878 −0.609392 0.792869i \(-0.708586\pi\)
−0.609392 + 0.792869i \(0.708586\pi\)
\(150\) 0 0
\(151\) 14.4817 1.17850 0.589251 0.807950i \(-0.299423\pi\)
0.589251 + 0.807950i \(0.299423\pi\)
\(152\) 0 0
\(153\) 12.9053 1.04333
\(154\) 0 0
\(155\) −0.0145507 −0.00116874
\(156\) 0 0
\(157\) 6.76070 0.539563 0.269781 0.962922i \(-0.413049\pi\)
0.269781 + 0.962922i \(0.413049\pi\)
\(158\) 0 0
\(159\) 3.61138 0.286401
\(160\) 0 0
\(161\) 0.418298 0.0329665
\(162\) 0 0
\(163\) 15.5691 1.21947 0.609733 0.792607i \(-0.291277\pi\)
0.609733 + 0.792607i \(0.291277\pi\)
\(164\) 0 0
\(165\) −8.32257 −0.647911
\(166\) 0 0
\(167\) −8.34808 −0.645994 −0.322997 0.946400i \(-0.604690\pi\)
−0.322997 + 0.946400i \(0.604690\pi\)
\(168\) 0 0
\(169\) 24.8718 1.91322
\(170\) 0 0
\(171\) 4.73359 0.361987
\(172\) 0 0
\(173\) 19.1200 1.45367 0.726833 0.686814i \(-0.240991\pi\)
0.726833 + 0.686814i \(0.240991\pi\)
\(174\) 0 0
\(175\) 0.646760 0.0488904
\(176\) 0 0
\(177\) −36.2881 −2.72758
\(178\) 0 0
\(179\) 6.64205 0.496450 0.248225 0.968702i \(-0.420153\pi\)
0.248225 + 0.968702i \(0.420153\pi\)
\(180\) 0 0
\(181\) −16.7897 −1.24797 −0.623984 0.781437i \(-0.714487\pi\)
−0.623984 + 0.781437i \(0.714487\pi\)
\(182\) 0 0
\(183\) 10.4380 0.771601
\(184\) 0 0
\(185\) 6.49997 0.477888
\(186\) 0 0
\(187\) 8.15911 0.596653
\(188\) 0 0
\(189\) −3.11803 −0.226803
\(190\) 0 0
\(191\) −5.54480 −0.401208 −0.200604 0.979672i \(-0.564290\pi\)
−0.200604 + 0.979672i \(0.564290\pi\)
\(192\) 0 0
\(193\) 17.1962 1.23781 0.618907 0.785465i \(-0.287576\pi\)
0.618907 + 0.785465i \(0.287576\pi\)
\(194\) 0 0
\(195\) −17.1139 −1.22555
\(196\) 0 0
\(197\) −18.8844 −1.34546 −0.672728 0.739890i \(-0.734878\pi\)
−0.672728 + 0.739890i \(0.734878\pi\)
\(198\) 0 0
\(199\) −15.5302 −1.10091 −0.550455 0.834865i \(-0.685546\pi\)
−0.550455 + 0.834865i \(0.685546\pi\)
\(200\) 0 0
\(201\) 36.8420 2.59863
\(202\) 0 0
\(203\) 2.46619 0.173093
\(204\) 0 0
\(205\) 0.532818 0.0372136
\(206\) 0 0
\(207\) 3.06150 0.212789
\(208\) 0 0
\(209\) 2.99272 0.207011
\(210\) 0 0
\(211\) 8.07485 0.555896 0.277948 0.960596i \(-0.410346\pi\)
0.277948 + 0.960596i \(0.410346\pi\)
\(212\) 0 0
\(213\) −4.08013 −0.279566
\(214\) 0 0
\(215\) 2.45991 0.167764
\(216\) 0 0
\(217\) −0.00941083 −0.000638849 0
\(218\) 0 0
\(219\) 24.5306 1.65763
\(220\) 0 0
\(221\) 16.7778 1.12859
\(222\) 0 0
\(223\) 16.6455 1.11467 0.557334 0.830289i \(-0.311824\pi\)
0.557334 + 0.830289i \(0.311824\pi\)
\(224\) 0 0
\(225\) 4.73359 0.315573
\(226\) 0 0
\(227\) −4.24812 −0.281957 −0.140979 0.990013i \(-0.545025\pi\)
−0.140979 + 0.990013i \(0.545025\pi\)
\(228\) 0 0
\(229\) −7.63478 −0.504520 −0.252260 0.967659i \(-0.581174\pi\)
−0.252260 + 0.967659i \(0.581174\pi\)
\(230\) 0 0
\(231\) −5.38270 −0.354156
\(232\) 0 0
\(233\) −16.5213 −1.08235 −0.541175 0.840910i \(-0.682020\pi\)
−0.541175 + 0.840910i \(0.682020\pi\)
\(234\) 0 0
\(235\) −11.8627 −0.773835
\(236\) 0 0
\(237\) −25.5112 −1.65713
\(238\) 0 0
\(239\) 14.0713 0.910196 0.455098 0.890441i \(-0.349604\pi\)
0.455098 + 0.890441i \(0.349604\pi\)
\(240\) 0 0
\(241\) −18.0290 −1.16135 −0.580675 0.814136i \(-0.697211\pi\)
−0.580675 + 0.814136i \(0.697211\pi\)
\(242\) 0 0
\(243\) 16.6708 1.06943
\(244\) 0 0
\(245\) −6.58170 −0.420490
\(246\) 0 0
\(247\) 6.15401 0.391570
\(248\) 0 0
\(249\) 31.6462 2.00550
\(250\) 0 0
\(251\) 15.7253 0.992574 0.496287 0.868159i \(-0.334696\pi\)
0.496287 + 0.868159i \(0.334696\pi\)
\(252\) 0 0
\(253\) 1.93557 0.121688
\(254\) 0 0
\(255\) −7.58170 −0.474785
\(256\) 0 0
\(257\) 21.4052 1.33522 0.667611 0.744510i \(-0.267317\pi\)
0.667611 + 0.744510i \(0.267317\pi\)
\(258\) 0 0
\(259\) 4.20392 0.261219
\(260\) 0 0
\(261\) 18.0499 1.11726
\(262\) 0 0
\(263\) 13.7023 0.844919 0.422460 0.906382i \(-0.361167\pi\)
0.422460 + 0.906382i \(0.361167\pi\)
\(264\) 0 0
\(265\) −1.29862 −0.0797737
\(266\) 0 0
\(267\) −41.2714 −2.52577
\(268\) 0 0
\(269\) 1.37365 0.0837532 0.0418766 0.999123i \(-0.486666\pi\)
0.0418766 + 0.999123i \(0.486666\pi\)
\(270\) 0 0
\(271\) −0.871561 −0.0529435 −0.0264718 0.999650i \(-0.508427\pi\)
−0.0264718 + 0.999650i \(0.508427\pi\)
\(272\) 0 0
\(273\) −11.0686 −0.669901
\(274\) 0 0
\(275\) 2.99272 0.180468
\(276\) 0 0
\(277\) −28.7500 −1.72742 −0.863711 0.503988i \(-0.831866\pi\)
−0.863711 + 0.503988i \(0.831866\pi\)
\(278\) 0 0
\(279\) −0.0688772 −0.00412357
\(280\) 0 0
\(281\) 0.343448 0.0204884 0.0102442 0.999948i \(-0.496739\pi\)
0.0102442 + 0.999948i \(0.496739\pi\)
\(282\) 0 0
\(283\) 6.95305 0.413316 0.206658 0.978413i \(-0.433741\pi\)
0.206658 + 0.978413i \(0.433741\pi\)
\(284\) 0 0
\(285\) −2.78093 −0.164728
\(286\) 0 0
\(287\) 0.344605 0.0203414
\(288\) 0 0
\(289\) −9.56720 −0.562777
\(290\) 0 0
\(291\) 17.6335 1.03369
\(292\) 0 0
\(293\) 4.02400 0.235085 0.117542 0.993068i \(-0.462498\pi\)
0.117542 + 0.993068i \(0.462498\pi\)
\(294\) 0 0
\(295\) 13.0489 0.759735
\(296\) 0 0
\(297\) −14.4279 −0.837193
\(298\) 0 0
\(299\) 3.98017 0.230179
\(300\) 0 0
\(301\) 1.59097 0.0917019
\(302\) 0 0
\(303\) −52.6348 −3.02379
\(304\) 0 0
\(305\) −3.75343 −0.214921
\(306\) 0 0
\(307\) 18.3178 1.04545 0.522727 0.852500i \(-0.324915\pi\)
0.522727 + 0.852500i \(0.324915\pi\)
\(308\) 0 0
\(309\) −7.67843 −0.436811
\(310\) 0 0
\(311\) −7.80592 −0.442633 −0.221317 0.975202i \(-0.571035\pi\)
−0.221317 + 0.975202i \(0.571035\pi\)
\(312\) 0 0
\(313\) 12.3173 0.696214 0.348107 0.937455i \(-0.386825\pi\)
0.348107 + 0.937455i \(0.386825\pi\)
\(314\) 0 0
\(315\) 3.06150 0.172496
\(316\) 0 0
\(317\) −7.35798 −0.413265 −0.206633 0.978419i \(-0.566251\pi\)
−0.206633 + 0.978419i \(0.566251\pi\)
\(318\) 0 0
\(319\) 11.4117 0.638932
\(320\) 0 0
\(321\) 24.4192 1.36295
\(322\) 0 0
\(323\) 2.72632 0.151696
\(324\) 0 0
\(325\) 6.15401 0.341363
\(326\) 0 0
\(327\) −21.2449 −1.17485
\(328\) 0 0
\(329\) −7.67229 −0.422987
\(330\) 0 0
\(331\) −12.9656 −0.712652 −0.356326 0.934362i \(-0.615971\pi\)
−0.356326 + 0.934362i \(0.615971\pi\)
\(332\) 0 0
\(333\) 30.7682 1.68609
\(334\) 0 0
\(335\) −13.2481 −0.723819
\(336\) 0 0
\(337\) −5.17113 −0.281689 −0.140845 0.990032i \(-0.544982\pi\)
−0.140845 + 0.990032i \(0.544982\pi\)
\(338\) 0 0
\(339\) 18.8627 1.02448
\(340\) 0 0
\(341\) −0.0435463 −0.00235817
\(342\) 0 0
\(343\) −8.78410 −0.474297
\(344\) 0 0
\(345\) −1.79860 −0.0968331
\(346\) 0 0
\(347\) −7.79389 −0.418398 −0.209199 0.977873i \(-0.567086\pi\)
−0.209199 + 0.977873i \(0.567086\pi\)
\(348\) 0 0
\(349\) −15.4066 −0.824698 −0.412349 0.911026i \(-0.635292\pi\)
−0.412349 + 0.911026i \(0.635292\pi\)
\(350\) 0 0
\(351\) −29.6685 −1.58359
\(352\) 0 0
\(353\) 13.2446 0.704938 0.352469 0.935823i \(-0.385342\pi\)
0.352469 + 0.935823i \(0.385342\pi\)
\(354\) 0 0
\(355\) 1.46718 0.0778699
\(356\) 0 0
\(357\) −4.90354 −0.259523
\(358\) 0 0
\(359\) 23.5166 1.24116 0.620579 0.784144i \(-0.286898\pi\)
0.620579 + 0.784144i \(0.286898\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 5.68311 0.298286
\(364\) 0 0
\(365\) −8.82100 −0.461712
\(366\) 0 0
\(367\) −18.3047 −0.955496 −0.477748 0.878497i \(-0.658547\pi\)
−0.477748 + 0.878497i \(0.658547\pi\)
\(368\) 0 0
\(369\) 2.52214 0.131298
\(370\) 0 0
\(371\) −0.839897 −0.0436053
\(372\) 0 0
\(373\) 21.9948 1.13885 0.569424 0.822044i \(-0.307166\pi\)
0.569424 + 0.822044i \(0.307166\pi\)
\(374\) 0 0
\(375\) −2.78093 −0.143607
\(376\) 0 0
\(377\) 23.4661 1.20857
\(378\) 0 0
\(379\) 1.98017 0.101714 0.0508571 0.998706i \(-0.483805\pi\)
0.0508571 + 0.998706i \(0.483805\pi\)
\(380\) 0 0
\(381\) 26.9220 1.37925
\(382\) 0 0
\(383\) 1.48647 0.0759553 0.0379777 0.999279i \(-0.487908\pi\)
0.0379777 + 0.999279i \(0.487908\pi\)
\(384\) 0 0
\(385\) 1.93557 0.0986460
\(386\) 0 0
\(387\) 11.6442 0.591907
\(388\) 0 0
\(389\) −34.3807 −1.74317 −0.871584 0.490245i \(-0.836907\pi\)
−0.871584 + 0.490245i \(0.836907\pi\)
\(390\) 0 0
\(391\) 1.76327 0.0891725
\(392\) 0 0
\(393\) −38.3484 −1.93442
\(394\) 0 0
\(395\) 9.17361 0.461574
\(396\) 0 0
\(397\) −1.63378 −0.0819971 −0.0409985 0.999159i \(-0.513054\pi\)
−0.0409985 + 0.999159i \(0.513054\pi\)
\(398\) 0 0
\(399\) −1.79860 −0.0900424
\(400\) 0 0
\(401\) 25.6962 1.28321 0.641604 0.767036i \(-0.278269\pi\)
0.641604 + 0.767036i \(0.278269\pi\)
\(402\) 0 0
\(403\) −0.0895453 −0.00446057
\(404\) 0 0
\(405\) −0.793890 −0.0394487
\(406\) 0 0
\(407\) 19.4526 0.964231
\(408\) 0 0
\(409\) −11.7713 −0.582054 −0.291027 0.956715i \(-0.593997\pi\)
−0.291027 + 0.956715i \(0.593997\pi\)
\(410\) 0 0
\(411\) 9.06343 0.447066
\(412\) 0 0
\(413\) 8.43949 0.415280
\(414\) 0 0
\(415\) −11.3797 −0.558608
\(416\) 0 0
\(417\) −25.5753 −1.25243
\(418\) 0 0
\(419\) 33.3881 1.63112 0.815558 0.578676i \(-0.196430\pi\)
0.815558 + 0.578676i \(0.196430\pi\)
\(420\) 0 0
\(421\) −17.5592 −0.855785 −0.427893 0.903830i \(-0.640744\pi\)
−0.427893 + 0.903830i \(0.640744\pi\)
\(422\) 0 0
\(423\) −56.1530 −2.73025
\(424\) 0 0
\(425\) 2.72632 0.132246
\(426\) 0 0
\(427\) −2.42756 −0.117478
\(428\) 0 0
\(429\) −51.2172 −2.47279
\(430\) 0 0
\(431\) −16.7253 −0.805628 −0.402814 0.915282i \(-0.631968\pi\)
−0.402814 + 0.915282i \(0.631968\pi\)
\(432\) 0 0
\(433\) −9.67485 −0.464943 −0.232472 0.972603i \(-0.574681\pi\)
−0.232472 + 0.972603i \(0.574681\pi\)
\(434\) 0 0
\(435\) −10.6041 −0.508428
\(436\) 0 0
\(437\) 0.646760 0.0309387
\(438\) 0 0
\(439\) −15.1882 −0.724891 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(440\) 0 0
\(441\) −31.1551 −1.48358
\(442\) 0 0
\(443\) 30.3690 1.44287 0.721437 0.692480i \(-0.243482\pi\)
0.721437 + 0.692480i \(0.243482\pi\)
\(444\) 0 0
\(445\) 14.8408 0.703523
\(446\) 0 0
\(447\) 41.3724 1.95685
\(448\) 0 0
\(449\) 25.4463 1.20089 0.600443 0.799668i \(-0.294991\pi\)
0.600443 + 0.799668i \(0.294991\pi\)
\(450\) 0 0
\(451\) 1.59458 0.0750858
\(452\) 0 0
\(453\) −40.2726 −1.89217
\(454\) 0 0
\(455\) 3.98017 0.186593
\(456\) 0 0
\(457\) 32.7592 1.53241 0.766205 0.642596i \(-0.222143\pi\)
0.766205 + 0.642596i \(0.222143\pi\)
\(458\) 0 0
\(459\) −13.1436 −0.613489
\(460\) 0 0
\(461\) −20.7606 −0.966917 −0.483459 0.875367i \(-0.660620\pi\)
−0.483459 + 0.875367i \(0.660620\pi\)
\(462\) 0 0
\(463\) −17.3550 −0.806554 −0.403277 0.915078i \(-0.632129\pi\)
−0.403277 + 0.915078i \(0.632129\pi\)
\(464\) 0 0
\(465\) 0.0404646 0.00187650
\(466\) 0 0
\(467\) −7.83356 −0.362494 −0.181247 0.983438i \(-0.558013\pi\)
−0.181247 + 0.983438i \(0.558013\pi\)
\(468\) 0 0
\(469\) −8.56831 −0.395648
\(470\) 0 0
\(471\) −18.8011 −0.866307
\(472\) 0 0
\(473\) 7.36182 0.338497
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) −6.14715 −0.281459
\(478\) 0 0
\(479\) 2.58604 0.118159 0.0590796 0.998253i \(-0.481183\pi\)
0.0590796 + 0.998253i \(0.481183\pi\)
\(480\) 0 0
\(481\) 40.0009 1.82388
\(482\) 0 0
\(483\) −1.16326 −0.0529301
\(484\) 0 0
\(485\) −6.34086 −0.287924
\(486\) 0 0
\(487\) −21.9744 −0.995754 −0.497877 0.867248i \(-0.665887\pi\)
−0.497877 + 0.867248i \(0.665887\pi\)
\(488\) 0 0
\(489\) −43.2966 −1.95794
\(490\) 0 0
\(491\) −19.8145 −0.894216 −0.447108 0.894480i \(-0.647546\pi\)
−0.447108 + 0.894480i \(0.647546\pi\)
\(492\) 0 0
\(493\) 10.3958 0.468205
\(494\) 0 0
\(495\) 14.1663 0.636729
\(496\) 0 0
\(497\) 0.948914 0.0425646
\(498\) 0 0
\(499\) 4.50356 0.201607 0.100803 0.994906i \(-0.467859\pi\)
0.100803 + 0.994906i \(0.467859\pi\)
\(500\) 0 0
\(501\) 23.2155 1.03719
\(502\) 0 0
\(503\) −8.22951 −0.366936 −0.183468 0.983026i \(-0.558732\pi\)
−0.183468 + 0.983026i \(0.558732\pi\)
\(504\) 0 0
\(505\) 18.9270 0.842242
\(506\) 0 0
\(507\) −69.1669 −3.07181
\(508\) 0 0
\(509\) −32.1685 −1.42584 −0.712922 0.701244i \(-0.752628\pi\)
−0.712922 + 0.701244i \(0.752628\pi\)
\(510\) 0 0
\(511\) −5.70507 −0.252377
\(512\) 0 0
\(513\) −4.82100 −0.212852
\(514\) 0 0
\(515\) 2.76110 0.121669
\(516\) 0 0
\(517\) −35.5017 −1.56136
\(518\) 0 0
\(519\) −53.1714 −2.33397
\(520\) 0 0
\(521\) 30.3308 1.32882 0.664408 0.747370i \(-0.268684\pi\)
0.664408 + 0.747370i \(0.268684\pi\)
\(522\) 0 0
\(523\) 20.5915 0.900402 0.450201 0.892927i \(-0.351352\pi\)
0.450201 + 0.892927i \(0.351352\pi\)
\(524\) 0 0
\(525\) −1.79860 −0.0784972
\(526\) 0 0
\(527\) −0.0396699 −0.00172805
\(528\) 0 0
\(529\) −22.5817 −0.981813
\(530\) 0 0
\(531\) 61.7681 2.68051
\(532\) 0 0
\(533\) 3.27897 0.142028
\(534\) 0 0
\(535\) −8.78093 −0.379633
\(536\) 0 0
\(537\) −18.4711 −0.797087
\(538\) 0 0
\(539\) −19.6972 −0.848419
\(540\) 0 0
\(541\) 33.5576 1.44275 0.721377 0.692542i \(-0.243509\pi\)
0.721377 + 0.692542i \(0.243509\pi\)
\(542\) 0 0
\(543\) 46.6910 2.00370
\(544\) 0 0
\(545\) 7.63948 0.327240
\(546\) 0 0
\(547\) 20.2577 0.866159 0.433079 0.901356i \(-0.357427\pi\)
0.433079 + 0.901356i \(0.357427\pi\)
\(548\) 0 0
\(549\) −17.7672 −0.758285
\(550\) 0 0
\(551\) 3.81315 0.162446
\(552\) 0 0
\(553\) 5.93312 0.252302
\(554\) 0 0
\(555\) −18.0760 −0.767283
\(556\) 0 0
\(557\) 0.531555 0.0225227 0.0112614 0.999937i \(-0.496415\pi\)
0.0112614 + 0.999937i \(0.496415\pi\)
\(558\) 0 0
\(559\) 15.1383 0.640281
\(560\) 0 0
\(561\) −22.6899 −0.957970
\(562\) 0 0
\(563\) −9.64160 −0.406345 −0.203173 0.979143i \(-0.565125\pi\)
−0.203173 + 0.979143i \(0.565125\pi\)
\(564\) 0 0
\(565\) −6.78287 −0.285358
\(566\) 0 0
\(567\) −0.513456 −0.0215631
\(568\) 0 0
\(569\) 2.57955 0.108140 0.0540702 0.998537i \(-0.482781\pi\)
0.0540702 + 0.998537i \(0.482781\pi\)
\(570\) 0 0
\(571\) −27.6284 −1.15621 −0.578107 0.815961i \(-0.696209\pi\)
−0.578107 + 0.815961i \(0.696209\pi\)
\(572\) 0 0
\(573\) 15.4197 0.644168
\(574\) 0 0
\(575\) 0.646760 0.0269717
\(576\) 0 0
\(577\) −47.3828 −1.97257 −0.986286 0.165044i \(-0.947223\pi\)
−0.986286 + 0.165044i \(0.947223\pi\)
\(578\) 0 0
\(579\) −47.8216 −1.98740
\(580\) 0 0
\(581\) −7.35994 −0.305342
\(582\) 0 0
\(583\) −3.88642 −0.160959
\(584\) 0 0
\(585\) 29.1306 1.20440
\(586\) 0 0
\(587\) −27.2206 −1.12351 −0.561757 0.827302i \(-0.689875\pi\)
−0.561757 + 0.827302i \(0.689875\pi\)
\(588\) 0 0
\(589\) −0.0145507 −0.000599553 0
\(590\) 0 0
\(591\) 52.5162 2.16023
\(592\) 0 0
\(593\) −11.1818 −0.459183 −0.229591 0.973287i \(-0.573739\pi\)
−0.229591 + 0.973287i \(0.573739\pi\)
\(594\) 0 0
\(595\) 1.76327 0.0722871
\(596\) 0 0
\(597\) 43.1886 1.76759
\(598\) 0 0
\(599\) −45.2965 −1.85076 −0.925382 0.379035i \(-0.876256\pi\)
−0.925382 + 0.379035i \(0.876256\pi\)
\(600\) 0 0
\(601\) −30.8585 −1.25874 −0.629372 0.777105i \(-0.716688\pi\)
−0.629372 + 0.777105i \(0.716688\pi\)
\(602\) 0 0
\(603\) −62.7109 −2.55379
\(604\) 0 0
\(605\) −2.04360 −0.0830841
\(606\) 0 0
\(607\) −26.9100 −1.09224 −0.546122 0.837706i \(-0.683896\pi\)
−0.546122 + 0.837706i \(0.683896\pi\)
\(608\) 0 0
\(609\) −6.85831 −0.277913
\(610\) 0 0
\(611\) −73.0029 −2.95338
\(612\) 0 0
\(613\) 7.22783 0.291929 0.145965 0.989290i \(-0.453371\pi\)
0.145965 + 0.989290i \(0.453371\pi\)
\(614\) 0 0
\(615\) −1.48173 −0.0597492
\(616\) 0 0
\(617\) 24.1382 0.971769 0.485884 0.874023i \(-0.338498\pi\)
0.485884 + 0.874023i \(0.338498\pi\)
\(618\) 0 0
\(619\) −44.8480 −1.80259 −0.901296 0.433204i \(-0.857383\pi\)
−0.901296 + 0.433204i \(0.857383\pi\)
\(620\) 0 0
\(621\) −3.11803 −0.125122
\(622\) 0 0
\(623\) 9.59846 0.384554
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −8.32257 −0.332371
\(628\) 0 0
\(629\) 17.7210 0.706582
\(630\) 0 0
\(631\) 21.2457 0.845779 0.422890 0.906181i \(-0.361016\pi\)
0.422890 + 0.906181i \(0.361016\pi\)
\(632\) 0 0
\(633\) −22.4556 −0.892531
\(634\) 0 0
\(635\) −9.68091 −0.384175
\(636\) 0 0
\(637\) −40.5038 −1.60482
\(638\) 0 0
\(639\) 6.94504 0.274741
\(640\) 0 0
\(641\) −26.5712 −1.04950 −0.524750 0.851256i \(-0.675841\pi\)
−0.524750 + 0.851256i \(0.675841\pi\)
\(642\) 0 0
\(643\) 47.8519 1.88709 0.943547 0.331239i \(-0.107467\pi\)
0.943547 + 0.331239i \(0.107467\pi\)
\(644\) 0 0
\(645\) −6.84084 −0.269358
\(646\) 0 0
\(647\) 37.4718 1.47317 0.736584 0.676346i \(-0.236438\pi\)
0.736584 + 0.676346i \(0.236438\pi\)
\(648\) 0 0
\(649\) 39.0517 1.53291
\(650\) 0 0
\(651\) 0.0261709 0.00102572
\(652\) 0 0
\(653\) −37.5750 −1.47043 −0.735213 0.677837i \(-0.762918\pi\)
−0.735213 + 0.677837i \(0.762918\pi\)
\(654\) 0 0
\(655\) 13.7897 0.538810
\(656\) 0 0
\(657\) −41.7550 −1.62902
\(658\) 0 0
\(659\) 3.03046 0.118050 0.0590249 0.998257i \(-0.481201\pi\)
0.0590249 + 0.998257i \(0.481201\pi\)
\(660\) 0 0
\(661\) −40.2386 −1.56510 −0.782550 0.622587i \(-0.786082\pi\)
−0.782550 + 0.622587i \(0.786082\pi\)
\(662\) 0 0
\(663\) −46.6579 −1.81204
\(664\) 0 0
\(665\) 0.646760 0.0250803
\(666\) 0 0
\(667\) 2.46619 0.0954912
\(668\) 0 0
\(669\) −46.2901 −1.78968
\(670\) 0 0
\(671\) −11.2330 −0.433644
\(672\) 0 0
\(673\) −18.0372 −0.695283 −0.347641 0.937628i \(-0.613017\pi\)
−0.347641 + 0.937628i \(0.613017\pi\)
\(674\) 0 0
\(675\) −4.82100 −0.185560
\(676\) 0 0
\(677\) −1.17918 −0.0453196 −0.0226598 0.999743i \(-0.507213\pi\)
−0.0226598 + 0.999743i \(0.507213\pi\)
\(678\) 0 0
\(679\) −4.10101 −0.157382
\(680\) 0 0
\(681\) 11.8137 0.452703
\(682\) 0 0
\(683\) 10.8165 0.413881 0.206940 0.978354i \(-0.433649\pi\)
0.206940 + 0.978354i \(0.433649\pi\)
\(684\) 0 0
\(685\) −3.25913 −0.124525
\(686\) 0 0
\(687\) 21.2318 0.810044
\(688\) 0 0
\(689\) −7.99173 −0.304461
\(690\) 0 0
\(691\) 21.3663 0.812813 0.406407 0.913692i \(-0.366782\pi\)
0.406407 + 0.913692i \(0.366782\pi\)
\(692\) 0 0
\(693\) 9.16221 0.348044
\(694\) 0 0
\(695\) 9.19665 0.348849
\(696\) 0 0
\(697\) 1.45263 0.0550223
\(698\) 0 0
\(699\) 45.9448 1.73779
\(700\) 0 0
\(701\) 39.6032 1.49579 0.747897 0.663815i \(-0.231064\pi\)
0.747897 + 0.663815i \(0.231064\pi\)
\(702\) 0 0
\(703\) 6.49997 0.245151
\(704\) 0 0
\(705\) 32.9893 1.24245
\(706\) 0 0
\(707\) 12.2412 0.460379
\(708\) 0 0
\(709\) 36.8989 1.38577 0.692884 0.721049i \(-0.256340\pi\)
0.692884 + 0.721049i \(0.256340\pi\)
\(710\) 0 0
\(711\) 43.4241 1.62853
\(712\) 0 0
\(713\) −0.00941083 −0.000352438 0
\(714\) 0 0
\(715\) 18.4173 0.688766
\(716\) 0 0
\(717\) −39.1313 −1.46139
\(718\) 0 0
\(719\) 27.1284 1.01172 0.505860 0.862616i \(-0.331175\pi\)
0.505860 + 0.862616i \(0.331175\pi\)
\(720\) 0 0
\(721\) 1.78577 0.0665055
\(722\) 0 0
\(723\) 50.1374 1.86463
\(724\) 0 0
\(725\) 3.81315 0.141617
\(726\) 0 0
\(727\) 45.2262 1.67735 0.838674 0.544634i \(-0.183332\pi\)
0.838674 + 0.544634i \(0.183332\pi\)
\(728\) 0 0
\(729\) −43.9786 −1.62884
\(730\) 0 0
\(731\) 6.70648 0.248048
\(732\) 0 0
\(733\) −25.7064 −0.949489 −0.474744 0.880124i \(-0.657459\pi\)
−0.474744 + 0.880124i \(0.657459\pi\)
\(734\) 0 0
\(735\) 18.3033 0.675126
\(736\) 0 0
\(737\) −39.6478 −1.46045
\(738\) 0 0
\(739\) −24.4766 −0.900387 −0.450193 0.892931i \(-0.648645\pi\)
−0.450193 + 0.892931i \(0.648645\pi\)
\(740\) 0 0
\(741\) −17.1139 −0.628694
\(742\) 0 0
\(743\) −42.8706 −1.57277 −0.786385 0.617737i \(-0.788050\pi\)
−0.786385 + 0.617737i \(0.788050\pi\)
\(744\) 0 0
\(745\) −14.8772 −0.545057
\(746\) 0 0
\(747\) −53.8669 −1.97089
\(748\) 0 0
\(749\) −5.67915 −0.207512
\(750\) 0 0
\(751\) −2.13320 −0.0778415 −0.0389207 0.999242i \(-0.512392\pi\)
−0.0389207 + 0.999242i \(0.512392\pi\)
\(752\) 0 0
\(753\) −43.7311 −1.59365
\(754\) 0 0
\(755\) 14.4817 0.527042
\(756\) 0 0
\(757\) −27.5713 −1.00210 −0.501048 0.865420i \(-0.667052\pi\)
−0.501048 + 0.865420i \(0.667052\pi\)
\(758\) 0 0
\(759\) −5.38270 −0.195380
\(760\) 0 0
\(761\) 28.1465 1.02031 0.510155 0.860083i \(-0.329588\pi\)
0.510155 + 0.860083i \(0.329588\pi\)
\(762\) 0 0
\(763\) 4.94091 0.178873
\(764\) 0 0
\(765\) 12.9053 0.466591
\(766\) 0 0
\(767\) 80.3029 2.89957
\(768\) 0 0
\(769\) 44.8313 1.61666 0.808329 0.588731i \(-0.200372\pi\)
0.808329 + 0.588731i \(0.200372\pi\)
\(770\) 0 0
\(771\) −59.5265 −2.14380
\(772\) 0 0
\(773\) −42.0321 −1.51179 −0.755895 0.654693i \(-0.772798\pi\)
−0.755895 + 0.654693i \(0.772798\pi\)
\(774\) 0 0
\(775\) −0.0145507 −0.000522678 0
\(776\) 0 0
\(777\) −11.6908 −0.419406
\(778\) 0 0
\(779\) 0.532818 0.0190902
\(780\) 0 0
\(781\) 4.39087 0.157118
\(782\) 0 0
\(783\) −18.3832 −0.656961
\(784\) 0 0
\(785\) 6.76070 0.241300
\(786\) 0 0
\(787\) 2.76957 0.0987245 0.0493623 0.998781i \(-0.484281\pi\)
0.0493623 + 0.998781i \(0.484281\pi\)
\(788\) 0 0
\(789\) −38.1051 −1.35658
\(790\) 0 0
\(791\) −4.38689 −0.155980
\(792\) 0 0
\(793\) −23.0986 −0.820256
\(794\) 0 0
\(795\) 3.61138 0.128083
\(796\) 0 0
\(797\) 39.1280 1.38599 0.692993 0.720944i \(-0.256292\pi\)
0.692993 + 0.720944i \(0.256292\pi\)
\(798\) 0 0
\(799\) −32.3414 −1.14416
\(800\) 0 0
\(801\) 70.2504 2.48218
\(802\) 0 0
\(803\) −26.3988 −0.931594
\(804\) 0 0
\(805\) 0.418298 0.0147431
\(806\) 0 0
\(807\) −3.82004 −0.134472
\(808\) 0 0
\(809\) 14.0375 0.493531 0.246766 0.969075i \(-0.420632\pi\)
0.246766 + 0.969075i \(0.420632\pi\)
\(810\) 0 0
\(811\) 23.1474 0.812817 0.406408 0.913692i \(-0.366781\pi\)
0.406408 + 0.913692i \(0.366781\pi\)
\(812\) 0 0
\(813\) 2.42375 0.0850047
\(814\) 0 0
\(815\) 15.5691 0.545361
\(816\) 0 0
\(817\) 2.45991 0.0860612
\(818\) 0 0
\(819\) 18.8405 0.658339
\(820\) 0 0
\(821\) 54.1127 1.88855 0.944273 0.329164i \(-0.106767\pi\)
0.944273 + 0.329164i \(0.106767\pi\)
\(822\) 0 0
\(823\) −46.1025 −1.60703 −0.803516 0.595284i \(-0.797040\pi\)
−0.803516 + 0.595284i \(0.797040\pi\)
\(824\) 0 0
\(825\) −8.32257 −0.289755
\(826\) 0 0
\(827\) −28.3825 −0.986955 −0.493478 0.869759i \(-0.664275\pi\)
−0.493478 + 0.869759i \(0.664275\pi\)
\(828\) 0 0
\(829\) 11.6104 0.403245 0.201623 0.979463i \(-0.435379\pi\)
0.201623 + 0.979463i \(0.435379\pi\)
\(830\) 0 0
\(831\) 79.9519 2.77350
\(832\) 0 0
\(833\) −17.9438 −0.621716
\(834\) 0 0
\(835\) −8.34808 −0.288897
\(836\) 0 0
\(837\) 0.0701491 0.00242471
\(838\) 0 0
\(839\) 26.1331 0.902215 0.451107 0.892470i \(-0.351029\pi\)
0.451107 + 0.892470i \(0.351029\pi\)
\(840\) 0 0
\(841\) −14.4599 −0.498618
\(842\) 0 0
\(843\) −0.955106 −0.0328956
\(844\) 0 0
\(845\) 24.8718 0.855617
\(846\) 0 0
\(847\) −1.32172 −0.0454148
\(848\) 0 0
\(849\) −19.3360 −0.663609
\(850\) 0 0
\(851\) 4.20392 0.144109
\(852\) 0 0
\(853\) −17.0511 −0.583818 −0.291909 0.956446i \(-0.594290\pi\)
−0.291909 + 0.956446i \(0.594290\pi\)
\(854\) 0 0
\(855\) 4.73359 0.161885
\(856\) 0 0
\(857\) 14.0556 0.480131 0.240065 0.970757i \(-0.422831\pi\)
0.240065 + 0.970757i \(0.422831\pi\)
\(858\) 0 0
\(859\) 40.8195 1.39274 0.696371 0.717682i \(-0.254797\pi\)
0.696371 + 0.717682i \(0.254797\pi\)
\(860\) 0 0
\(861\) −0.958325 −0.0326596
\(862\) 0 0
\(863\) 32.2655 1.09833 0.549165 0.835714i \(-0.314946\pi\)
0.549165 + 0.835714i \(0.314946\pi\)
\(864\) 0 0
\(865\) 19.1200 0.650099
\(866\) 0 0
\(867\) 26.6058 0.903579
\(868\) 0 0
\(869\) 27.4541 0.931316
\(870\) 0 0
\(871\) −81.5287 −2.76249
\(872\) 0 0
\(873\) −30.0150 −1.01586
\(874\) 0 0
\(875\) 0.646760 0.0218645
\(876\) 0 0
\(877\) −21.4974 −0.725916 −0.362958 0.931806i \(-0.618233\pi\)
−0.362958 + 0.931806i \(0.618233\pi\)
\(878\) 0 0
\(879\) −11.1905 −0.377445
\(880\) 0 0
\(881\) −40.5627 −1.36659 −0.683296 0.730141i \(-0.739454\pi\)
−0.683296 + 0.730141i \(0.739454\pi\)
\(882\) 0 0
\(883\) 9.41504 0.316841 0.158421 0.987372i \(-0.449360\pi\)
0.158421 + 0.987372i \(0.449360\pi\)
\(884\) 0 0
\(885\) −36.2881 −1.21981
\(886\) 0 0
\(887\) 24.2113 0.812935 0.406468 0.913665i \(-0.366760\pi\)
0.406468 + 0.913665i \(0.366760\pi\)
\(888\) 0 0
\(889\) −6.26122 −0.209995
\(890\) 0 0
\(891\) −2.37590 −0.0795955
\(892\) 0 0
\(893\) −11.8627 −0.396969
\(894\) 0 0
\(895\) 6.64205 0.222019
\(896\) 0 0
\(897\) −11.0686 −0.369569
\(898\) 0 0
\(899\) −0.0554841 −0.00185050
\(900\) 0 0
\(901\) −3.54045 −0.117950
\(902\) 0 0
\(903\) −4.42438 −0.147234
\(904\) 0 0
\(905\) −16.7897 −0.558108
\(906\) 0 0
\(907\) 15.8060 0.524830 0.262415 0.964955i \(-0.415481\pi\)
0.262415 + 0.964955i \(0.415481\pi\)
\(908\) 0 0
\(909\) 89.5928 2.97161
\(910\) 0 0
\(911\) 49.8333 1.65105 0.825525 0.564365i \(-0.190879\pi\)
0.825525 + 0.564365i \(0.190879\pi\)
\(912\) 0 0
\(913\) −34.0564 −1.12710
\(914\) 0 0
\(915\) 10.4380 0.345071
\(916\) 0 0
\(917\) 8.91865 0.294520
\(918\) 0 0
\(919\) 21.6976 0.715737 0.357869 0.933772i \(-0.383504\pi\)
0.357869 + 0.933772i \(0.383504\pi\)
\(920\) 0 0
\(921\) −50.9407 −1.67855
\(922\) 0 0
\(923\) 9.02905 0.297195
\(924\) 0 0
\(925\) 6.49997 0.213718
\(926\) 0 0
\(927\) 13.0699 0.429272
\(928\) 0 0
\(929\) 35.6115 1.16838 0.584188 0.811619i \(-0.301413\pi\)
0.584188 + 0.811619i \(0.301413\pi\)
\(930\) 0 0
\(931\) −6.58170 −0.215707
\(932\) 0 0
\(933\) 21.7078 0.710680
\(934\) 0 0
\(935\) 8.15911 0.266831
\(936\) 0 0
\(937\) 38.0134 1.24184 0.620922 0.783872i \(-0.286758\pi\)
0.620922 + 0.783872i \(0.286758\pi\)
\(938\) 0 0
\(939\) −34.2535 −1.11782
\(940\) 0 0
\(941\) 15.9582 0.520222 0.260111 0.965579i \(-0.416241\pi\)
0.260111 + 0.965579i \(0.416241\pi\)
\(942\) 0 0
\(943\) 0.344605 0.0112219
\(944\) 0 0
\(945\) −3.11803 −0.101429
\(946\) 0 0
\(947\) 60.5729 1.96836 0.984178 0.177185i \(-0.0566990\pi\)
0.984178 + 0.177185i \(0.0566990\pi\)
\(948\) 0 0
\(949\) −54.2845 −1.76215
\(950\) 0 0
\(951\) 20.4621 0.663528
\(952\) 0 0
\(953\) 29.3094 0.949423 0.474712 0.880141i \(-0.342552\pi\)
0.474712 + 0.880141i \(0.342552\pi\)
\(954\) 0 0
\(955\) −5.54480 −0.179425
\(956\) 0 0
\(957\) −31.7352 −1.02585
\(958\) 0 0
\(959\) −2.10788 −0.0680669
\(960\) 0 0
\(961\) −30.9998 −0.999993
\(962\) 0 0
\(963\) −41.5653 −1.33942
\(964\) 0 0
\(965\) 17.1962 0.553567
\(966\) 0 0
\(967\) 14.7396 0.473994 0.236997 0.971510i \(-0.423837\pi\)
0.236997 + 0.971510i \(0.423837\pi\)
\(968\) 0 0
\(969\) −7.58170 −0.243559
\(970\) 0 0
\(971\) 58.1559 1.86631 0.933155 0.359474i \(-0.117044\pi\)
0.933155 + 0.359474i \(0.117044\pi\)
\(972\) 0 0
\(973\) 5.94802 0.190685
\(974\) 0 0
\(975\) −17.1139 −0.548083
\(976\) 0 0
\(977\) 1.56869 0.0501869 0.0250935 0.999685i \(-0.492012\pi\)
0.0250935 + 0.999685i \(0.492012\pi\)
\(978\) 0 0
\(979\) 44.4145 1.41949
\(980\) 0 0
\(981\) 36.1622 1.15457
\(982\) 0 0
\(983\) −11.5312 −0.367788 −0.183894 0.982946i \(-0.558870\pi\)
−0.183894 + 0.982946i \(0.558870\pi\)
\(984\) 0 0
\(985\) −18.8844 −0.601706
\(986\) 0 0
\(987\) 21.3361 0.679137
\(988\) 0 0
\(989\) 1.59097 0.0505899
\(990\) 0 0
\(991\) −19.3870 −0.615850 −0.307925 0.951411i \(-0.599635\pi\)
−0.307925 + 0.951411i \(0.599635\pi\)
\(992\) 0 0
\(993\) 36.0564 1.14421
\(994\) 0 0
\(995\) −15.5302 −0.492342
\(996\) 0 0
\(997\) −36.5724 −1.15826 −0.579129 0.815236i \(-0.696607\pi\)
−0.579129 + 0.815236i \(0.696607\pi\)
\(998\) 0 0
\(999\) −31.3364 −0.991439
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 3040.2.a.w.1.1 yes 5
4.3 odd 2 3040.2.a.v.1.5 5
8.3 odd 2 6080.2.a.ck.1.1 5
8.5 even 2 6080.2.a.cj.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.v.1.5 5 4.3 odd 2
3040.2.a.w.1.1 yes 5 1.1 even 1 trivial
6080.2.a.cj.1.5 5 8.5 even 2
6080.2.a.ck.1.1 5 8.3 odd 2