Properties

Label 4028.2.a.c.1.4
Level $4028$
Weight $2$
Character 4028.1
Self dual yes
Analytic conductor $32.164$
Analytic rank $1$
Dimension $19$
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] = [4028,2,Mod(1,4028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4028, 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("4028.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4028.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: \(32.1637419342\)
Analytic rank: \(1\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 5 x^{18} - 27 x^{17} + 161 x^{16} + 253 x^{15} - 2103 x^{14} - 683 x^{13} + 14442 x^{12} - 4144 x^{11} - 56325 x^{10} + 37245 x^{9} + 124233 x^{8} - 117486 x^{7} + \cdots - 4088 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.14522\) of defining polynomial
Character \(\chi\) \(=\) 4028.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.14522 q^{3} +3.02869 q^{5} -3.82352 q^{7} +1.60196 q^{9} +O(q^{10})\) \(q-2.14522 q^{3} +3.02869 q^{5} -3.82352 q^{7} +1.60196 q^{9} -1.48254 q^{11} +3.10378 q^{13} -6.49720 q^{15} -2.84113 q^{17} +1.00000 q^{19} +8.20228 q^{21} +1.29632 q^{23} +4.17297 q^{25} +2.99910 q^{27} +3.11758 q^{29} -2.16500 q^{31} +3.18036 q^{33} -11.5803 q^{35} +8.28871 q^{37} -6.65829 q^{39} +1.09976 q^{41} -7.67575 q^{43} +4.85184 q^{45} -6.57658 q^{47} +7.61931 q^{49} +6.09485 q^{51} -1.00000 q^{53} -4.49015 q^{55} -2.14522 q^{57} +8.84950 q^{59} -15.3974 q^{61} -6.12513 q^{63} +9.40040 q^{65} +5.82077 q^{67} -2.78089 q^{69} +5.44343 q^{71} +10.1933 q^{73} -8.95193 q^{75} +5.66851 q^{77} -17.6867 q^{79} -11.2396 q^{81} +6.42418 q^{83} -8.60492 q^{85} -6.68788 q^{87} -16.1449 q^{89} -11.8674 q^{91} +4.64440 q^{93} +3.02869 q^{95} -3.90916 q^{97} -2.37496 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q - 5 q^{3} - 5 q^{5} - 10 q^{7} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q - 5 q^{3} - 5 q^{5} - 10 q^{7} + 22 q^{9} - 3 q^{11} - 23 q^{13} - 18 q^{15} - 7 q^{17} + 19 q^{19} - 4 q^{21} - 6 q^{23} + 18 q^{25} - 17 q^{27} - 4 q^{29} - 30 q^{31} - 10 q^{33} - q^{35} - 31 q^{37} + 5 q^{39} - 15 q^{41} - 29 q^{43} + 6 q^{45} - 18 q^{47} + 23 q^{49} - 5 q^{51} - 19 q^{53} - 19 q^{55} - 5 q^{57} + 8 q^{59} - 4 q^{61} - 64 q^{63} - 26 q^{65} - 62 q^{67} + 3 q^{69} - 17 q^{71} + q^{73} - 40 q^{75} - 14 q^{77} - 28 q^{79} + 11 q^{81} + 4 q^{83} - 31 q^{85} - 20 q^{87} + 33 q^{89} - 29 q^{91} - 59 q^{93} - 5 q^{95} + 5 q^{97} - 48 q^{99}+O(q^{100}) \) 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
\(3\) −2.14522 −1.23854 −0.619271 0.785177i \(-0.712572\pi\)
−0.619271 + 0.785177i \(0.712572\pi\)
\(4\) 0 0
\(5\) 3.02869 1.35447 0.677236 0.735766i \(-0.263178\pi\)
0.677236 + 0.735766i \(0.263178\pi\)
\(6\) 0 0
\(7\) −3.82352 −1.44515 −0.722577 0.691290i \(-0.757043\pi\)
−0.722577 + 0.691290i \(0.757043\pi\)
\(8\) 0 0
\(9\) 1.60196 0.533987
\(10\) 0 0
\(11\) −1.48254 −0.447002 −0.223501 0.974704i \(-0.571749\pi\)
−0.223501 + 0.974704i \(0.571749\pi\)
\(12\) 0 0
\(13\) 3.10378 0.860834 0.430417 0.902630i \(-0.358366\pi\)
0.430417 + 0.902630i \(0.358366\pi\)
\(14\) 0 0
\(15\) −6.49720 −1.67757
\(16\) 0 0
\(17\) −2.84113 −0.689076 −0.344538 0.938772i \(-0.611964\pi\)
−0.344538 + 0.938772i \(0.611964\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 8.20228 1.78989
\(22\) 0 0
\(23\) 1.29632 0.270301 0.135151 0.990825i \(-0.456848\pi\)
0.135151 + 0.990825i \(0.456848\pi\)
\(24\) 0 0
\(25\) 4.17297 0.834594
\(26\) 0 0
\(27\) 2.99910 0.577177
\(28\) 0 0
\(29\) 3.11758 0.578920 0.289460 0.957190i \(-0.406524\pi\)
0.289460 + 0.957190i \(0.406524\pi\)
\(30\) 0 0
\(31\) −2.16500 −0.388846 −0.194423 0.980918i \(-0.562283\pi\)
−0.194423 + 0.980918i \(0.562283\pi\)
\(32\) 0 0
\(33\) 3.18036 0.553630
\(34\) 0 0
\(35\) −11.5803 −1.95742
\(36\) 0 0
\(37\) 8.28871 1.36266 0.681328 0.731978i \(-0.261403\pi\)
0.681328 + 0.731978i \(0.261403\pi\)
\(38\) 0 0
\(39\) −6.65829 −1.06618
\(40\) 0 0
\(41\) 1.09976 0.171753 0.0858766 0.996306i \(-0.472631\pi\)
0.0858766 + 0.996306i \(0.472631\pi\)
\(42\) 0 0
\(43\) −7.67575 −1.17054 −0.585270 0.810838i \(-0.699012\pi\)
−0.585270 + 0.810838i \(0.699012\pi\)
\(44\) 0 0
\(45\) 4.85184 0.723270
\(46\) 0 0
\(47\) −6.57658 −0.959292 −0.479646 0.877462i \(-0.659235\pi\)
−0.479646 + 0.877462i \(0.659235\pi\)
\(48\) 0 0
\(49\) 7.61931 1.08847
\(50\) 0 0
\(51\) 6.09485 0.853450
\(52\) 0 0
\(53\) −1.00000 −0.137361
\(54\) 0 0
\(55\) −4.49015 −0.605451
\(56\) 0 0
\(57\) −2.14522 −0.284141
\(58\) 0 0
\(59\) 8.84950 1.15211 0.576054 0.817412i \(-0.304592\pi\)
0.576054 + 0.817412i \(0.304592\pi\)
\(60\) 0 0
\(61\) −15.3974 −1.97143 −0.985717 0.168413i \(-0.946136\pi\)
−0.985717 + 0.168413i \(0.946136\pi\)
\(62\) 0 0
\(63\) −6.12513 −0.771693
\(64\) 0 0
\(65\) 9.40040 1.16598
\(66\) 0 0
\(67\) 5.82077 0.711120 0.355560 0.934653i \(-0.384290\pi\)
0.355560 + 0.934653i \(0.384290\pi\)
\(68\) 0 0
\(69\) −2.78089 −0.334779
\(70\) 0 0
\(71\) 5.44343 0.646016 0.323008 0.946396i \(-0.395306\pi\)
0.323008 + 0.946396i \(0.395306\pi\)
\(72\) 0 0
\(73\) 10.1933 1.19303 0.596515 0.802602i \(-0.296552\pi\)
0.596515 + 0.802602i \(0.296552\pi\)
\(74\) 0 0
\(75\) −8.95193 −1.03368
\(76\) 0 0
\(77\) 5.66851 0.645987
\(78\) 0 0
\(79\) −17.6867 −1.98991 −0.994953 0.100344i \(-0.968006\pi\)
−0.994953 + 0.100344i \(0.968006\pi\)
\(80\) 0 0
\(81\) −11.2396 −1.24884
\(82\) 0 0
\(83\) 6.42418 0.705145 0.352573 0.935784i \(-0.385307\pi\)
0.352573 + 0.935784i \(0.385307\pi\)
\(84\) 0 0
\(85\) −8.60492 −0.933334
\(86\) 0 0
\(87\) −6.68788 −0.717016
\(88\) 0 0
\(89\) −16.1449 −1.71136 −0.855680 0.517505i \(-0.826861\pi\)
−0.855680 + 0.517505i \(0.826861\pi\)
\(90\) 0 0
\(91\) −11.8674 −1.24404
\(92\) 0 0
\(93\) 4.64440 0.481602
\(94\) 0 0
\(95\) 3.02869 0.310737
\(96\) 0 0
\(97\) −3.90916 −0.396915 −0.198457 0.980110i \(-0.563593\pi\)
−0.198457 + 0.980110i \(0.563593\pi\)
\(98\) 0 0
\(99\) −2.37496 −0.238693
\(100\) 0 0
\(101\) 14.5950 1.45226 0.726130 0.687558i \(-0.241317\pi\)
0.726130 + 0.687558i \(0.241317\pi\)
\(102\) 0 0
\(103\) −14.6436 −1.44287 −0.721436 0.692481i \(-0.756518\pi\)
−0.721436 + 0.692481i \(0.756518\pi\)
\(104\) 0 0
\(105\) 24.8422 2.42435
\(106\) 0 0
\(107\) −7.24789 −0.700680 −0.350340 0.936623i \(-0.613934\pi\)
−0.350340 + 0.936623i \(0.613934\pi\)
\(108\) 0 0
\(109\) −0.892124 −0.0854500 −0.0427250 0.999087i \(-0.513604\pi\)
−0.0427250 + 0.999087i \(0.513604\pi\)
\(110\) 0 0
\(111\) −17.7811 −1.68771
\(112\) 0 0
\(113\) −21.0126 −1.97670 −0.988348 0.152209i \(-0.951361\pi\)
−0.988348 + 0.152209i \(0.951361\pi\)
\(114\) 0 0
\(115\) 3.92615 0.366115
\(116\) 0 0
\(117\) 4.97214 0.459674
\(118\) 0 0
\(119\) 10.8631 0.995822
\(120\) 0 0
\(121\) −8.80209 −0.800190
\(122\) 0 0
\(123\) −2.35922 −0.212724
\(124\) 0 0
\(125\) −2.50482 −0.224038
\(126\) 0 0
\(127\) 16.4111 1.45625 0.728123 0.685446i \(-0.240393\pi\)
0.728123 + 0.685446i \(0.240393\pi\)
\(128\) 0 0
\(129\) 16.4661 1.44976
\(130\) 0 0
\(131\) 9.25479 0.808595 0.404297 0.914628i \(-0.367516\pi\)
0.404297 + 0.914628i \(0.367516\pi\)
\(132\) 0 0
\(133\) −3.82352 −0.331541
\(134\) 0 0
\(135\) 9.08335 0.781770
\(136\) 0 0
\(137\) −11.0887 −0.947372 −0.473686 0.880694i \(-0.657077\pi\)
−0.473686 + 0.880694i \(0.657077\pi\)
\(138\) 0 0
\(139\) 5.25379 0.445621 0.222810 0.974862i \(-0.428477\pi\)
0.222810 + 0.974862i \(0.428477\pi\)
\(140\) 0 0
\(141\) 14.1082 1.18812
\(142\) 0 0
\(143\) −4.60147 −0.384794
\(144\) 0 0
\(145\) 9.44218 0.784131
\(146\) 0 0
\(147\) −16.3451 −1.34812
\(148\) 0 0
\(149\) 12.8907 1.05605 0.528023 0.849230i \(-0.322934\pi\)
0.528023 + 0.849230i \(0.322934\pi\)
\(150\) 0 0
\(151\) 2.24419 0.182629 0.0913147 0.995822i \(-0.470893\pi\)
0.0913147 + 0.995822i \(0.470893\pi\)
\(152\) 0 0
\(153\) −4.55138 −0.367957
\(154\) 0 0
\(155\) −6.55712 −0.526681
\(156\) 0 0
\(157\) 6.21299 0.495850 0.247925 0.968779i \(-0.420251\pi\)
0.247925 + 0.968779i \(0.420251\pi\)
\(158\) 0 0
\(159\) 2.14522 0.170127
\(160\) 0 0
\(161\) −4.95650 −0.390627
\(162\) 0 0
\(163\) 1.06803 0.0836547 0.0418273 0.999125i \(-0.486682\pi\)
0.0418273 + 0.999125i \(0.486682\pi\)
\(164\) 0 0
\(165\) 9.63234 0.749877
\(166\) 0 0
\(167\) −20.0303 −1.54999 −0.774994 0.631969i \(-0.782247\pi\)
−0.774994 + 0.631969i \(0.782247\pi\)
\(168\) 0 0
\(169\) −3.36653 −0.258964
\(170\) 0 0
\(171\) 1.60196 0.122505
\(172\) 0 0
\(173\) −7.68062 −0.583946 −0.291973 0.956426i \(-0.594312\pi\)
−0.291973 + 0.956426i \(0.594312\pi\)
\(174\) 0 0
\(175\) −15.9554 −1.20612
\(176\) 0 0
\(177\) −18.9841 −1.42693
\(178\) 0 0
\(179\) −6.46057 −0.482885 −0.241443 0.970415i \(-0.577621\pi\)
−0.241443 + 0.970415i \(0.577621\pi\)
\(180\) 0 0
\(181\) 24.7458 1.83934 0.919670 0.392693i \(-0.128457\pi\)
0.919670 + 0.392693i \(0.128457\pi\)
\(182\) 0 0
\(183\) 33.0307 2.44170
\(184\) 0 0
\(185\) 25.1040 1.84568
\(186\) 0 0
\(187\) 4.21209 0.308018
\(188\) 0 0
\(189\) −11.4671 −0.834110
\(190\) 0 0
\(191\) 2.52081 0.182399 0.0911997 0.995833i \(-0.470930\pi\)
0.0911997 + 0.995833i \(0.470930\pi\)
\(192\) 0 0
\(193\) −0.691989 −0.0498104 −0.0249052 0.999690i \(-0.507928\pi\)
−0.0249052 + 0.999690i \(0.507928\pi\)
\(194\) 0 0
\(195\) −20.1659 −1.44411
\(196\) 0 0
\(197\) 0.955142 0.0680511 0.0340255 0.999421i \(-0.489167\pi\)
0.0340255 + 0.999421i \(0.489167\pi\)
\(198\) 0 0
\(199\) −5.83106 −0.413353 −0.206677 0.978409i \(-0.566265\pi\)
−0.206677 + 0.978409i \(0.566265\pi\)
\(200\) 0 0
\(201\) −12.4868 −0.880752
\(202\) 0 0
\(203\) −11.9201 −0.836629
\(204\) 0 0
\(205\) 3.33082 0.232635
\(206\) 0 0
\(207\) 2.07665 0.144337
\(208\) 0 0
\(209\) −1.48254 −0.102549
\(210\) 0 0
\(211\) 12.6821 0.873069 0.436535 0.899687i \(-0.356206\pi\)
0.436535 + 0.899687i \(0.356206\pi\)
\(212\) 0 0
\(213\) −11.6773 −0.800118
\(214\) 0 0
\(215\) −23.2475 −1.58546
\(216\) 0 0
\(217\) 8.27793 0.561942
\(218\) 0 0
\(219\) −21.8668 −1.47762
\(220\) 0 0
\(221\) −8.81826 −0.593180
\(222\) 0 0
\(223\) −22.9122 −1.53431 −0.767156 0.641460i \(-0.778329\pi\)
−0.767156 + 0.641460i \(0.778329\pi\)
\(224\) 0 0
\(225\) 6.68493 0.445662
\(226\) 0 0
\(227\) −25.1536 −1.66950 −0.834750 0.550629i \(-0.814388\pi\)
−0.834750 + 0.550629i \(0.814388\pi\)
\(228\) 0 0
\(229\) 16.1603 1.06790 0.533952 0.845515i \(-0.320706\pi\)
0.533952 + 0.845515i \(0.320706\pi\)
\(230\) 0 0
\(231\) −12.1602 −0.800082
\(232\) 0 0
\(233\) −11.8752 −0.777968 −0.388984 0.921244i \(-0.627174\pi\)
−0.388984 + 0.921244i \(0.627174\pi\)
\(234\) 0 0
\(235\) −19.9184 −1.29933
\(236\) 0 0
\(237\) 37.9418 2.46458
\(238\) 0 0
\(239\) −21.1080 −1.36536 −0.682680 0.730718i \(-0.739186\pi\)
−0.682680 + 0.730718i \(0.739186\pi\)
\(240\) 0 0
\(241\) 11.2722 0.726108 0.363054 0.931768i \(-0.381734\pi\)
0.363054 + 0.931768i \(0.381734\pi\)
\(242\) 0 0
\(243\) 15.1141 0.969570
\(244\) 0 0
\(245\) 23.0765 1.47431
\(246\) 0 0
\(247\) 3.10378 0.197489
\(248\) 0 0
\(249\) −13.7813 −0.873352
\(250\) 0 0
\(251\) −9.44066 −0.595889 −0.297945 0.954583i \(-0.596301\pi\)
−0.297945 + 0.954583i \(0.596301\pi\)
\(252\) 0 0
\(253\) −1.92184 −0.120825
\(254\) 0 0
\(255\) 18.4594 1.15597
\(256\) 0 0
\(257\) −23.8732 −1.48917 −0.744585 0.667527i \(-0.767353\pi\)
−0.744585 + 0.667527i \(0.767353\pi\)
\(258\) 0 0
\(259\) −31.6921 −1.96925
\(260\) 0 0
\(261\) 4.99424 0.309135
\(262\) 0 0
\(263\) −21.4779 −1.32438 −0.662191 0.749335i \(-0.730373\pi\)
−0.662191 + 0.749335i \(0.730373\pi\)
\(264\) 0 0
\(265\) −3.02869 −0.186051
\(266\) 0 0
\(267\) 34.6344 2.11959
\(268\) 0 0
\(269\) −30.6765 −1.87038 −0.935189 0.354149i \(-0.884771\pi\)
−0.935189 + 0.354149i \(0.884771\pi\)
\(270\) 0 0
\(271\) 6.36429 0.386603 0.193301 0.981139i \(-0.438080\pi\)
0.193301 + 0.981139i \(0.438080\pi\)
\(272\) 0 0
\(273\) 25.4581 1.54079
\(274\) 0 0
\(275\) −6.18658 −0.373065
\(276\) 0 0
\(277\) −26.7836 −1.60927 −0.804636 0.593768i \(-0.797640\pi\)
−0.804636 + 0.593768i \(0.797640\pi\)
\(278\) 0 0
\(279\) −3.46825 −0.207638
\(280\) 0 0
\(281\) −9.64873 −0.575595 −0.287797 0.957691i \(-0.592923\pi\)
−0.287797 + 0.957691i \(0.592923\pi\)
\(282\) 0 0
\(283\) −3.70423 −0.220194 −0.110097 0.993921i \(-0.535116\pi\)
−0.110097 + 0.993921i \(0.535116\pi\)
\(284\) 0 0
\(285\) −6.49720 −0.384861
\(286\) 0 0
\(287\) −4.20494 −0.248210
\(288\) 0 0
\(289\) −8.92796 −0.525174
\(290\) 0 0
\(291\) 8.38599 0.491595
\(292\) 0 0
\(293\) −21.4580 −1.25359 −0.626796 0.779183i \(-0.715634\pi\)
−0.626796 + 0.779183i \(0.715634\pi\)
\(294\) 0 0
\(295\) 26.8024 1.56050
\(296\) 0 0
\(297\) −4.44628 −0.257999
\(298\) 0 0
\(299\) 4.02349 0.232685
\(300\) 0 0
\(301\) 29.3484 1.69161
\(302\) 0 0
\(303\) −31.3095 −1.79868
\(304\) 0 0
\(305\) −46.6339 −2.67025
\(306\) 0 0
\(307\) −25.8194 −1.47359 −0.736796 0.676115i \(-0.763662\pi\)
−0.736796 + 0.676115i \(0.763662\pi\)
\(308\) 0 0
\(309\) 31.4136 1.78706
\(310\) 0 0
\(311\) −11.0750 −0.628007 −0.314003 0.949422i \(-0.601670\pi\)
−0.314003 + 0.949422i \(0.601670\pi\)
\(312\) 0 0
\(313\) −3.66099 −0.206931 −0.103466 0.994633i \(-0.532993\pi\)
−0.103466 + 0.994633i \(0.532993\pi\)
\(314\) 0 0
\(315\) −18.5511 −1.04524
\(316\) 0 0
\(317\) −3.71390 −0.208593 −0.104297 0.994546i \(-0.533259\pi\)
−0.104297 + 0.994546i \(0.533259\pi\)
\(318\) 0 0
\(319\) −4.62192 −0.258778
\(320\) 0 0
\(321\) 15.5483 0.867821
\(322\) 0 0
\(323\) −2.84113 −0.158085
\(324\) 0 0
\(325\) 12.9520 0.718447
\(326\) 0 0
\(327\) 1.91380 0.105833
\(328\) 0 0
\(329\) 25.1457 1.38633
\(330\) 0 0
\(331\) 7.05962 0.388032 0.194016 0.980998i \(-0.437849\pi\)
0.194016 + 0.980998i \(0.437849\pi\)
\(332\) 0 0
\(333\) 13.2782 0.727640
\(334\) 0 0
\(335\) 17.6293 0.963192
\(336\) 0 0
\(337\) −0.177661 −0.00967780 −0.00483890 0.999988i \(-0.501540\pi\)
−0.00483890 + 0.999988i \(0.501540\pi\)
\(338\) 0 0
\(339\) 45.0765 2.44822
\(340\) 0 0
\(341\) 3.20969 0.173815
\(342\) 0 0
\(343\) −2.36794 −0.127857
\(344\) 0 0
\(345\) −8.42245 −0.453449
\(346\) 0 0
\(347\) 13.9072 0.746576 0.373288 0.927715i \(-0.378230\pi\)
0.373288 + 0.927715i \(0.378230\pi\)
\(348\) 0 0
\(349\) 9.40474 0.503424 0.251712 0.967802i \(-0.419006\pi\)
0.251712 + 0.967802i \(0.419006\pi\)
\(350\) 0 0
\(351\) 9.30856 0.496854
\(352\) 0 0
\(353\) −28.5530 −1.51972 −0.759862 0.650084i \(-0.774734\pi\)
−0.759862 + 0.650084i \(0.774734\pi\)
\(354\) 0 0
\(355\) 16.4865 0.875010
\(356\) 0 0
\(357\) −23.3038 −1.23337
\(358\) 0 0
\(359\) 30.5193 1.61075 0.805373 0.592768i \(-0.201965\pi\)
0.805373 + 0.592768i \(0.201965\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 18.8824 0.991068
\(364\) 0 0
\(365\) 30.8722 1.61593
\(366\) 0 0
\(367\) −32.8374 −1.71410 −0.857049 0.515235i \(-0.827705\pi\)
−0.857049 + 0.515235i \(0.827705\pi\)
\(368\) 0 0
\(369\) 1.76177 0.0917139
\(370\) 0 0
\(371\) 3.82352 0.198507
\(372\) 0 0
\(373\) −26.3555 −1.36464 −0.682318 0.731055i \(-0.739028\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(374\) 0 0
\(375\) 5.37338 0.277480
\(376\) 0 0
\(377\) 9.67628 0.498354
\(378\) 0 0
\(379\) 0.543560 0.0279208 0.0139604 0.999903i \(-0.495556\pi\)
0.0139604 + 0.999903i \(0.495556\pi\)
\(380\) 0 0
\(381\) −35.2053 −1.80362
\(382\) 0 0
\(383\) 12.5427 0.640902 0.320451 0.947265i \(-0.396165\pi\)
0.320451 + 0.947265i \(0.396165\pi\)
\(384\) 0 0
\(385\) 17.1682 0.874971
\(386\) 0 0
\(387\) −12.2962 −0.625053
\(388\) 0 0
\(389\) 16.7321 0.848353 0.424177 0.905579i \(-0.360564\pi\)
0.424177 + 0.905579i \(0.360564\pi\)
\(390\) 0 0
\(391\) −3.68302 −0.186258
\(392\) 0 0
\(393\) −19.8535 −1.00148
\(394\) 0 0
\(395\) −53.5675 −2.69527
\(396\) 0 0
\(397\) 8.02006 0.402516 0.201258 0.979538i \(-0.435497\pi\)
0.201258 + 0.979538i \(0.435497\pi\)
\(398\) 0 0
\(399\) 8.20228 0.410628
\(400\) 0 0
\(401\) 24.2845 1.21271 0.606354 0.795195i \(-0.292631\pi\)
0.606354 + 0.795195i \(0.292631\pi\)
\(402\) 0 0
\(403\) −6.71969 −0.334732
\(404\) 0 0
\(405\) −34.0413 −1.69153
\(406\) 0 0
\(407\) −12.2883 −0.609110
\(408\) 0 0
\(409\) −16.8905 −0.835181 −0.417591 0.908635i \(-0.637125\pi\)
−0.417591 + 0.908635i \(0.637125\pi\)
\(410\) 0 0
\(411\) 23.7877 1.17336
\(412\) 0 0
\(413\) −33.8363 −1.66497
\(414\) 0 0
\(415\) 19.4569 0.955099
\(416\) 0 0
\(417\) −11.2705 −0.551920
\(418\) 0 0
\(419\) 33.0565 1.61492 0.807459 0.589924i \(-0.200842\pi\)
0.807459 + 0.589924i \(0.200842\pi\)
\(420\) 0 0
\(421\) −32.3865 −1.57842 −0.789211 0.614122i \(-0.789510\pi\)
−0.789211 + 0.614122i \(0.789510\pi\)
\(422\) 0 0
\(423\) −10.5354 −0.512249
\(424\) 0 0
\(425\) −11.8560 −0.575099
\(426\) 0 0
\(427\) 58.8722 2.84903
\(428\) 0 0
\(429\) 9.87116 0.476584
\(430\) 0 0
\(431\) 24.1246 1.16204 0.581021 0.813889i \(-0.302653\pi\)
0.581021 + 0.813889i \(0.302653\pi\)
\(432\) 0 0
\(433\) −25.6411 −1.23223 −0.616117 0.787655i \(-0.711295\pi\)
−0.616117 + 0.787655i \(0.711295\pi\)
\(434\) 0 0
\(435\) −20.2555 −0.971179
\(436\) 0 0
\(437\) 1.29632 0.0620114
\(438\) 0 0
\(439\) −15.2214 −0.726476 −0.363238 0.931696i \(-0.618329\pi\)
−0.363238 + 0.931696i \(0.618329\pi\)
\(440\) 0 0
\(441\) 12.2058 0.581230
\(442\) 0 0
\(443\) −20.2764 −0.963360 −0.481680 0.876347i \(-0.659973\pi\)
−0.481680 + 0.876347i \(0.659973\pi\)
\(444\) 0 0
\(445\) −48.8980 −2.31799
\(446\) 0 0
\(447\) −27.6533 −1.30796
\(448\) 0 0
\(449\) 13.3425 0.629671 0.314835 0.949146i \(-0.398051\pi\)
0.314835 + 0.949146i \(0.398051\pi\)
\(450\) 0 0
\(451\) −1.63043 −0.0767739
\(452\) 0 0
\(453\) −4.81427 −0.226194
\(454\) 0 0
\(455\) −35.9426 −1.68502
\(456\) 0 0
\(457\) 16.2859 0.761824 0.380912 0.924611i \(-0.375610\pi\)
0.380912 + 0.924611i \(0.375610\pi\)
\(458\) 0 0
\(459\) −8.52085 −0.397719
\(460\) 0 0
\(461\) 33.7784 1.57322 0.786608 0.617452i \(-0.211835\pi\)
0.786608 + 0.617452i \(0.211835\pi\)
\(462\) 0 0
\(463\) 15.2626 0.709313 0.354656 0.934997i \(-0.384598\pi\)
0.354656 + 0.934997i \(0.384598\pi\)
\(464\) 0 0
\(465\) 14.0665 0.652316
\(466\) 0 0
\(467\) 7.91781 0.366393 0.183196 0.983076i \(-0.441356\pi\)
0.183196 + 0.983076i \(0.441356\pi\)
\(468\) 0 0
\(469\) −22.2558 −1.02768
\(470\) 0 0
\(471\) −13.3282 −0.614131
\(472\) 0 0
\(473\) 11.3796 0.523233
\(474\) 0 0
\(475\) 4.17297 0.191469
\(476\) 0 0
\(477\) −1.60196 −0.0733487
\(478\) 0 0
\(479\) 7.31547 0.334252 0.167126 0.985936i \(-0.446551\pi\)
0.167126 + 0.985936i \(0.446551\pi\)
\(480\) 0 0
\(481\) 25.7264 1.17302
\(482\) 0 0
\(483\) 10.6328 0.483808
\(484\) 0 0
\(485\) −11.8396 −0.537610
\(486\) 0 0
\(487\) 11.1675 0.506048 0.253024 0.967460i \(-0.418575\pi\)
0.253024 + 0.967460i \(0.418575\pi\)
\(488\) 0 0
\(489\) −2.29116 −0.103610
\(490\) 0 0
\(491\) −24.7187 −1.11554 −0.557770 0.829995i \(-0.688343\pi\)
−0.557770 + 0.829995i \(0.688343\pi\)
\(492\) 0 0
\(493\) −8.85746 −0.398920
\(494\) 0 0
\(495\) −7.19303 −0.323303
\(496\) 0 0
\(497\) −20.8131 −0.933593
\(498\) 0 0
\(499\) 16.0080 0.716616 0.358308 0.933604i \(-0.383354\pi\)
0.358308 + 0.933604i \(0.383354\pi\)
\(500\) 0 0
\(501\) 42.9693 1.91973
\(502\) 0 0
\(503\) 21.1409 0.942627 0.471314 0.881966i \(-0.343780\pi\)
0.471314 + 0.881966i \(0.343780\pi\)
\(504\) 0 0
\(505\) 44.2038 1.96704
\(506\) 0 0
\(507\) 7.22195 0.320738
\(508\) 0 0
\(509\) 25.4724 1.12904 0.564522 0.825418i \(-0.309061\pi\)
0.564522 + 0.825418i \(0.309061\pi\)
\(510\) 0 0
\(511\) −38.9741 −1.72411
\(512\) 0 0
\(513\) 2.99910 0.132414
\(514\) 0 0
\(515\) −44.3508 −1.95433
\(516\) 0 0
\(517\) 9.75002 0.428805
\(518\) 0 0
\(519\) 16.4766 0.723242
\(520\) 0 0
\(521\) 0.752754 0.0329788 0.0164894 0.999864i \(-0.494751\pi\)
0.0164894 + 0.999864i \(0.494751\pi\)
\(522\) 0 0
\(523\) −16.5050 −0.721715 −0.360857 0.932621i \(-0.617516\pi\)
−0.360857 + 0.932621i \(0.617516\pi\)
\(524\) 0 0
\(525\) 34.2279 1.49383
\(526\) 0 0
\(527\) 6.15106 0.267944
\(528\) 0 0
\(529\) −21.3196 −0.926937
\(530\) 0 0
\(531\) 14.1765 0.615210
\(532\) 0 0
\(533\) 3.41341 0.147851
\(534\) 0 0
\(535\) −21.9516 −0.949051
\(536\) 0 0
\(537\) 13.8593 0.598074
\(538\) 0 0
\(539\) −11.2959 −0.486549
\(540\) 0 0
\(541\) −17.7793 −0.764391 −0.382195 0.924082i \(-0.624832\pi\)
−0.382195 + 0.924082i \(0.624832\pi\)
\(542\) 0 0
\(543\) −53.0851 −2.27810
\(544\) 0 0
\(545\) −2.70197 −0.115740
\(546\) 0 0
\(547\) −3.17104 −0.135584 −0.0677920 0.997699i \(-0.521595\pi\)
−0.0677920 + 0.997699i \(0.521595\pi\)
\(548\) 0 0
\(549\) −24.6660 −1.05272
\(550\) 0 0
\(551\) 3.11758 0.132813
\(552\) 0 0
\(553\) 67.6253 2.87572
\(554\) 0 0
\(555\) −53.8535 −2.28595
\(556\) 0 0
\(557\) 13.2902 0.563123 0.281561 0.959543i \(-0.409148\pi\)
0.281561 + 0.959543i \(0.409148\pi\)
\(558\) 0 0
\(559\) −23.8238 −1.00764
\(560\) 0 0
\(561\) −9.03584 −0.381493
\(562\) 0 0
\(563\) 26.1988 1.10415 0.552073 0.833795i \(-0.313837\pi\)
0.552073 + 0.833795i \(0.313837\pi\)
\(564\) 0 0
\(565\) −63.6406 −2.67738
\(566\) 0 0
\(567\) 42.9749 1.80477
\(568\) 0 0
\(569\) −33.5647 −1.40711 −0.703553 0.710642i \(-0.748404\pi\)
−0.703553 + 0.710642i \(0.748404\pi\)
\(570\) 0 0
\(571\) 0.546415 0.0228667 0.0114334 0.999935i \(-0.496361\pi\)
0.0114334 + 0.999935i \(0.496361\pi\)
\(572\) 0 0
\(573\) −5.40769 −0.225909
\(574\) 0 0
\(575\) 5.40950 0.225592
\(576\) 0 0
\(577\) −38.5698 −1.60568 −0.802840 0.596195i \(-0.796679\pi\)
−0.802840 + 0.596195i \(0.796679\pi\)
\(578\) 0 0
\(579\) 1.48447 0.0616923
\(580\) 0 0
\(581\) −24.5630 −1.01904
\(582\) 0 0
\(583\) 1.48254 0.0614004
\(584\) 0 0
\(585\) 15.0591 0.622616
\(586\) 0 0
\(587\) −11.4541 −0.472760 −0.236380 0.971661i \(-0.575961\pi\)
−0.236380 + 0.971661i \(0.575961\pi\)
\(588\) 0 0
\(589\) −2.16500 −0.0892073
\(590\) 0 0
\(591\) −2.04899 −0.0842841
\(592\) 0 0
\(593\) 21.6229 0.887947 0.443973 0.896040i \(-0.353568\pi\)
0.443973 + 0.896040i \(0.353568\pi\)
\(594\) 0 0
\(595\) 32.9011 1.34881
\(596\) 0 0
\(597\) 12.5089 0.511955
\(598\) 0 0
\(599\) −1.14548 −0.0468029 −0.0234014 0.999726i \(-0.507450\pi\)
−0.0234014 + 0.999726i \(0.507450\pi\)
\(600\) 0 0
\(601\) 41.9585 1.71152 0.855762 0.517369i \(-0.173089\pi\)
0.855762 + 0.517369i \(0.173089\pi\)
\(602\) 0 0
\(603\) 9.32464 0.379729
\(604\) 0 0
\(605\) −26.6588 −1.08383
\(606\) 0 0
\(607\) −22.5281 −0.914386 −0.457193 0.889367i \(-0.651145\pi\)
−0.457193 + 0.889367i \(0.651145\pi\)
\(608\) 0 0
\(609\) 25.5713 1.03620
\(610\) 0 0
\(611\) −20.4123 −0.825792
\(612\) 0 0
\(613\) 27.2153 1.09921 0.549607 0.835423i \(-0.314777\pi\)
0.549607 + 0.835423i \(0.314777\pi\)
\(614\) 0 0
\(615\) −7.14534 −0.288128
\(616\) 0 0
\(617\) 19.4037 0.781165 0.390582 0.920568i \(-0.372274\pi\)
0.390582 + 0.920568i \(0.372274\pi\)
\(618\) 0 0
\(619\) −37.1627 −1.49369 −0.746847 0.664996i \(-0.768433\pi\)
−0.746847 + 0.664996i \(0.768433\pi\)
\(620\) 0 0
\(621\) 3.88779 0.156012
\(622\) 0 0
\(623\) 61.7305 2.47318
\(624\) 0 0
\(625\) −28.4512 −1.13805
\(626\) 0 0
\(627\) 3.18036 0.127012
\(628\) 0 0
\(629\) −23.5493 −0.938974
\(630\) 0 0
\(631\) 40.1586 1.59869 0.799345 0.600873i \(-0.205180\pi\)
0.799345 + 0.600873i \(0.205180\pi\)
\(632\) 0 0
\(633\) −27.2058 −1.08133
\(634\) 0 0
\(635\) 49.7040 1.97244
\(636\) 0 0
\(637\) 23.6487 0.936995
\(638\) 0 0
\(639\) 8.72015 0.344964
\(640\) 0 0
\(641\) −26.4433 −1.04445 −0.522223 0.852809i \(-0.674897\pi\)
−0.522223 + 0.852809i \(0.674897\pi\)
\(642\) 0 0
\(643\) 10.9877 0.433311 0.216655 0.976248i \(-0.430485\pi\)
0.216655 + 0.976248i \(0.430485\pi\)
\(644\) 0 0
\(645\) 49.8709 1.96366
\(646\) 0 0
\(647\) 4.20933 0.165486 0.0827429 0.996571i \(-0.473632\pi\)
0.0827429 + 0.996571i \(0.473632\pi\)
\(648\) 0 0
\(649\) −13.1197 −0.514994
\(650\) 0 0
\(651\) −17.7580 −0.695989
\(652\) 0 0
\(653\) −22.1871 −0.868250 −0.434125 0.900853i \(-0.642942\pi\)
−0.434125 + 0.900853i \(0.642942\pi\)
\(654\) 0 0
\(655\) 28.0299 1.09522
\(656\) 0 0
\(657\) 16.3292 0.637062
\(658\) 0 0
\(659\) 25.8360 1.00643 0.503214 0.864162i \(-0.332151\pi\)
0.503214 + 0.864162i \(0.332151\pi\)
\(660\) 0 0
\(661\) −14.7374 −0.573217 −0.286608 0.958048i \(-0.592528\pi\)
−0.286608 + 0.958048i \(0.592528\pi\)
\(662\) 0 0
\(663\) 18.9171 0.734679
\(664\) 0 0
\(665\) −11.5803 −0.449063
\(666\) 0 0
\(667\) 4.04138 0.156483
\(668\) 0 0
\(669\) 49.1516 1.90031
\(670\) 0 0
\(671\) 22.8272 0.881234
\(672\) 0 0
\(673\) 51.0620 1.96830 0.984148 0.177349i \(-0.0567520\pi\)
0.984148 + 0.177349i \(0.0567520\pi\)
\(674\) 0 0
\(675\) 12.5152 0.481709
\(676\) 0 0
\(677\) 27.0726 1.04049 0.520243 0.854018i \(-0.325841\pi\)
0.520243 + 0.854018i \(0.325841\pi\)
\(678\) 0 0
\(679\) 14.9467 0.573603
\(680\) 0 0
\(681\) 53.9599 2.06775
\(682\) 0 0
\(683\) 31.4011 1.20153 0.600765 0.799426i \(-0.294863\pi\)
0.600765 + 0.799426i \(0.294863\pi\)
\(684\) 0 0
\(685\) −33.5843 −1.28319
\(686\) 0 0
\(687\) −34.6674 −1.32264
\(688\) 0 0
\(689\) −3.10378 −0.118245
\(690\) 0 0
\(691\) 2.59280 0.0986349 0.0493174 0.998783i \(-0.484295\pi\)
0.0493174 + 0.998783i \(0.484295\pi\)
\(692\) 0 0
\(693\) 9.08072 0.344948
\(694\) 0 0
\(695\) 15.9121 0.603581
\(696\) 0 0
\(697\) −3.12456 −0.118351
\(698\) 0 0
\(699\) 25.4748 0.963546
\(700\) 0 0
\(701\) 26.4060 0.997343 0.498671 0.866791i \(-0.333821\pi\)
0.498671 + 0.866791i \(0.333821\pi\)
\(702\) 0 0
\(703\) 8.28871 0.312615
\(704\) 0 0
\(705\) 42.7294 1.60928
\(706\) 0 0
\(707\) −55.8044 −2.09874
\(708\) 0 0
\(709\) −12.2019 −0.458253 −0.229126 0.973397i \(-0.573587\pi\)
−0.229126 + 0.973397i \(0.573587\pi\)
\(710\) 0 0
\(711\) −28.3333 −1.06258
\(712\) 0 0
\(713\) −2.80653 −0.105105
\(714\) 0 0
\(715\) −13.9364 −0.521193
\(716\) 0 0
\(717\) 45.2812 1.69106
\(718\) 0 0
\(719\) −21.1197 −0.787633 −0.393816 0.919189i \(-0.628845\pi\)
−0.393816 + 0.919189i \(0.628845\pi\)
\(720\) 0 0
\(721\) 55.9899 2.08517
\(722\) 0 0
\(723\) −24.1814 −0.899316
\(724\) 0 0
\(725\) 13.0096 0.483163
\(726\) 0 0
\(727\) 0.890804 0.0330381 0.0165191 0.999864i \(-0.494742\pi\)
0.0165191 + 0.999864i \(0.494742\pi\)
\(728\) 0 0
\(729\) 1.29578 0.0479917
\(730\) 0 0
\(731\) 21.8078 0.806591
\(732\) 0 0
\(733\) 3.01027 0.111187 0.0555934 0.998453i \(-0.482295\pi\)
0.0555934 + 0.998453i \(0.482295\pi\)
\(734\) 0 0
\(735\) −49.5042 −1.82599
\(736\) 0 0
\(737\) −8.62950 −0.317872
\(738\) 0 0
\(739\) −16.9331 −0.622892 −0.311446 0.950264i \(-0.600813\pi\)
−0.311446 + 0.950264i \(0.600813\pi\)
\(740\) 0 0
\(741\) −6.65829 −0.244598
\(742\) 0 0
\(743\) 25.3879 0.931392 0.465696 0.884945i \(-0.345804\pi\)
0.465696 + 0.884945i \(0.345804\pi\)
\(744\) 0 0
\(745\) 39.0419 1.43038
\(746\) 0 0
\(747\) 10.2913 0.376538
\(748\) 0 0
\(749\) 27.7124 1.01259
\(750\) 0 0
\(751\) 6.37394 0.232588 0.116294 0.993215i \(-0.462898\pi\)
0.116294 + 0.993215i \(0.462898\pi\)
\(752\) 0 0
\(753\) 20.2523 0.738034
\(754\) 0 0
\(755\) 6.79695 0.247366
\(756\) 0 0
\(757\) 20.4743 0.744153 0.372076 0.928202i \(-0.378646\pi\)
0.372076 + 0.928202i \(0.378646\pi\)
\(758\) 0 0
\(759\) 4.12277 0.149647
\(760\) 0 0
\(761\) −19.9087 −0.721691 −0.360846 0.932626i \(-0.617512\pi\)
−0.360846 + 0.932626i \(0.617512\pi\)
\(762\) 0 0
\(763\) 3.41106 0.123489
\(764\) 0 0
\(765\) −13.7847 −0.498388
\(766\) 0 0
\(767\) 27.4669 0.991773
\(768\) 0 0
\(769\) 32.8936 1.18617 0.593087 0.805138i \(-0.297909\pi\)
0.593087 + 0.805138i \(0.297909\pi\)
\(770\) 0 0
\(771\) 51.2133 1.84440
\(772\) 0 0
\(773\) −29.8542 −1.07378 −0.536890 0.843652i \(-0.680401\pi\)
−0.536890 + 0.843652i \(0.680401\pi\)
\(774\) 0 0
\(775\) −9.03449 −0.324528
\(776\) 0 0
\(777\) 67.9864 2.43900
\(778\) 0 0
\(779\) 1.09976 0.0394029
\(780\) 0 0
\(781\) −8.07008 −0.288770
\(782\) 0 0
\(783\) 9.34993 0.334139
\(784\) 0 0
\(785\) 18.8172 0.671615
\(786\) 0 0
\(787\) 34.5326 1.23096 0.615478 0.788154i \(-0.288963\pi\)
0.615478 + 0.788154i \(0.288963\pi\)
\(788\) 0 0
\(789\) 46.0747 1.64030
\(790\) 0 0
\(791\) 80.3420 2.85663
\(792\) 0 0
\(793\) −47.7901 −1.69708
\(794\) 0 0
\(795\) 6.49720 0.230432
\(796\) 0 0
\(797\) 3.89808 0.138077 0.0690385 0.997614i \(-0.478007\pi\)
0.0690385 + 0.997614i \(0.478007\pi\)
\(798\) 0 0
\(799\) 18.6849 0.661026
\(800\) 0 0
\(801\) −25.8635 −0.913843
\(802\) 0 0
\(803\) −15.1119 −0.533286
\(804\) 0 0
\(805\) −15.0117 −0.529093
\(806\) 0 0
\(807\) 65.8077 2.31654
\(808\) 0 0
\(809\) 29.2632 1.02884 0.514419 0.857539i \(-0.328008\pi\)
0.514419 + 0.857539i \(0.328008\pi\)
\(810\) 0 0
\(811\) −19.8879 −0.698357 −0.349179 0.937056i \(-0.613539\pi\)
−0.349179 + 0.937056i \(0.613539\pi\)
\(812\) 0 0
\(813\) −13.6528 −0.478824
\(814\) 0 0
\(815\) 3.23474 0.113308
\(816\) 0 0
\(817\) −7.67575 −0.268540
\(818\) 0 0
\(819\) −19.0111 −0.664300
\(820\) 0 0
\(821\) −38.5798 −1.34645 −0.673223 0.739440i \(-0.735091\pi\)
−0.673223 + 0.739440i \(0.735091\pi\)
\(822\) 0 0
\(823\) −6.46305 −0.225288 −0.112644 0.993635i \(-0.535932\pi\)
−0.112644 + 0.993635i \(0.535932\pi\)
\(824\) 0 0
\(825\) 13.2716 0.462057
\(826\) 0 0
\(827\) −16.6368 −0.578518 −0.289259 0.957251i \(-0.593409\pi\)
−0.289259 + 0.957251i \(0.593409\pi\)
\(828\) 0 0
\(829\) 2.49402 0.0866208 0.0433104 0.999062i \(-0.486210\pi\)
0.0433104 + 0.999062i \(0.486210\pi\)
\(830\) 0 0
\(831\) 57.4567 1.99315
\(832\) 0 0
\(833\) −21.6475 −0.750041
\(834\) 0 0
\(835\) −60.6655 −2.09942
\(836\) 0 0
\(837\) −6.49306 −0.224433
\(838\) 0 0
\(839\) 34.9880 1.20792 0.603959 0.797015i \(-0.293589\pi\)
0.603959 + 0.797015i \(0.293589\pi\)
\(840\) 0 0
\(841\) −19.2807 −0.664852
\(842\) 0 0
\(843\) 20.6986 0.712899
\(844\) 0 0
\(845\) −10.1962 −0.350760
\(846\) 0 0
\(847\) 33.6550 1.15640
\(848\) 0 0
\(849\) 7.94638 0.272719
\(850\) 0 0
\(851\) 10.7448 0.368328
\(852\) 0 0
\(853\) 28.6756 0.981834 0.490917 0.871206i \(-0.336662\pi\)
0.490917 + 0.871206i \(0.336662\pi\)
\(854\) 0 0
\(855\) 4.85184 0.165930
\(856\) 0 0
\(857\) −52.9259 −1.80792 −0.903958 0.427622i \(-0.859351\pi\)
−0.903958 + 0.427622i \(0.859351\pi\)
\(858\) 0 0
\(859\) −24.3655 −0.831340 −0.415670 0.909515i \(-0.636453\pi\)
−0.415670 + 0.909515i \(0.636453\pi\)
\(860\) 0 0
\(861\) 9.02052 0.307418
\(862\) 0 0
\(863\) 2.39312 0.0814627 0.0407313 0.999170i \(-0.487031\pi\)
0.0407313 + 0.999170i \(0.487031\pi\)
\(864\) 0 0
\(865\) −23.2622 −0.790939
\(866\) 0 0
\(867\) 19.1524 0.650450
\(868\) 0 0
\(869\) 26.2211 0.889491
\(870\) 0 0
\(871\) 18.0664 0.612157
\(872\) 0 0
\(873\) −6.26231 −0.211947
\(874\) 0 0
\(875\) 9.57721 0.323769
\(876\) 0 0
\(877\) 24.0858 0.813319 0.406659 0.913580i \(-0.366694\pi\)
0.406659 + 0.913580i \(0.366694\pi\)
\(878\) 0 0
\(879\) 46.0322 1.55263
\(880\) 0 0
\(881\) 15.6130 0.526015 0.263007 0.964794i \(-0.415286\pi\)
0.263007 + 0.964794i \(0.415286\pi\)
\(882\) 0 0
\(883\) −10.1886 −0.342875 −0.171438 0.985195i \(-0.554841\pi\)
−0.171438 + 0.985195i \(0.554841\pi\)
\(884\) 0 0
\(885\) −57.4970 −1.93274
\(886\) 0 0
\(887\) 7.54690 0.253400 0.126700 0.991941i \(-0.459561\pi\)
0.126700 + 0.991941i \(0.459561\pi\)
\(888\) 0 0
\(889\) −62.7480 −2.10450
\(890\) 0 0
\(891\) 16.6631 0.558236
\(892\) 0 0
\(893\) −6.57658 −0.220077
\(894\) 0 0
\(895\) −19.5671 −0.654055
\(896\) 0 0
\(897\) −8.63127 −0.288190
\(898\) 0 0
\(899\) −6.74956 −0.225110
\(900\) 0 0
\(901\) 2.84113 0.0946519
\(902\) 0 0
\(903\) −62.9586 −2.09513
\(904\) 0 0
\(905\) 74.9473 2.49133
\(906\) 0 0
\(907\) −2.08325 −0.0691733 −0.0345867 0.999402i \(-0.511011\pi\)
−0.0345867 + 0.999402i \(0.511011\pi\)
\(908\) 0 0
\(909\) 23.3806 0.775487
\(910\) 0 0
\(911\) −34.1346 −1.13093 −0.565464 0.824773i \(-0.691303\pi\)
−0.565464 + 0.824773i \(0.691303\pi\)
\(912\) 0 0
\(913\) −9.52408 −0.315201
\(914\) 0 0
\(915\) 100.040 3.30722
\(916\) 0 0
\(917\) −35.3859 −1.16854
\(918\) 0 0
\(919\) 28.2084 0.930509 0.465255 0.885177i \(-0.345963\pi\)
0.465255 + 0.885177i \(0.345963\pi\)
\(920\) 0 0
\(921\) 55.3883 1.82511
\(922\) 0 0
\(923\) 16.8952 0.556113
\(924\) 0 0
\(925\) 34.5886 1.13726
\(926\) 0 0
\(927\) −23.4584 −0.770474
\(928\) 0 0
\(929\) 38.6040 1.26656 0.633278 0.773925i \(-0.281709\pi\)
0.633278 + 0.773925i \(0.281709\pi\)
\(930\) 0 0
\(931\) 7.61931 0.249713
\(932\) 0 0
\(933\) 23.7583 0.777813
\(934\) 0 0
\(935\) 12.7571 0.417202
\(936\) 0 0
\(937\) −41.7681 −1.36450 −0.682252 0.731117i \(-0.738999\pi\)
−0.682252 + 0.731117i \(0.738999\pi\)
\(938\) 0 0
\(939\) 7.85361 0.256293
\(940\) 0 0
\(941\) −14.4863 −0.472241 −0.236121 0.971724i \(-0.575876\pi\)
−0.236121 + 0.971724i \(0.575876\pi\)
\(942\) 0 0
\(943\) 1.42564 0.0464251
\(944\) 0 0
\(945\) −34.7304 −1.12978
\(946\) 0 0
\(947\) −8.78786 −0.285567 −0.142784 0.989754i \(-0.545605\pi\)
−0.142784 + 0.989754i \(0.545605\pi\)
\(948\) 0 0
\(949\) 31.6376 1.02700
\(950\) 0 0
\(951\) 7.96712 0.258352
\(952\) 0 0
\(953\) −2.10693 −0.0682503 −0.0341251 0.999418i \(-0.510864\pi\)
−0.0341251 + 0.999418i \(0.510864\pi\)
\(954\) 0 0
\(955\) 7.63475 0.247055
\(956\) 0 0
\(957\) 9.91503 0.320508
\(958\) 0 0
\(959\) 42.3979 1.36910
\(960\) 0 0
\(961\) −26.3128 −0.848799
\(962\) 0 0
\(963\) −11.6108 −0.374154
\(964\) 0 0
\(965\) −2.09582 −0.0674668
\(966\) 0 0
\(967\) 24.5717 0.790173 0.395086 0.918644i \(-0.370715\pi\)
0.395086 + 0.918644i \(0.370715\pi\)
\(968\) 0 0
\(969\) 6.09485 0.195795
\(970\) 0 0
\(971\) −19.6548 −0.630754 −0.315377 0.948966i \(-0.602131\pi\)
−0.315377 + 0.948966i \(0.602131\pi\)
\(972\) 0 0
\(973\) −20.0880 −0.643991
\(974\) 0 0
\(975\) −27.7848 −0.889827
\(976\) 0 0
\(977\) 27.6754 0.885414 0.442707 0.896666i \(-0.354018\pi\)
0.442707 + 0.896666i \(0.354018\pi\)
\(978\) 0 0
\(979\) 23.9355 0.764981
\(980\) 0 0
\(981\) −1.42915 −0.0456292
\(982\) 0 0
\(983\) 9.66299 0.308202 0.154101 0.988055i \(-0.450752\pi\)
0.154101 + 0.988055i \(0.450752\pi\)
\(984\) 0 0
\(985\) 2.89283 0.0921733
\(986\) 0 0
\(987\) −53.9430 −1.71702
\(988\) 0 0
\(989\) −9.95022 −0.316398
\(990\) 0 0
\(991\) −13.5629 −0.430841 −0.215421 0.976521i \(-0.569112\pi\)
−0.215421 + 0.976521i \(0.569112\pi\)
\(992\) 0 0
\(993\) −15.1444 −0.480594
\(994\) 0 0
\(995\) −17.6605 −0.559875
\(996\) 0 0
\(997\) −41.0813 −1.30106 −0.650530 0.759481i \(-0.725453\pi\)
−0.650530 + 0.759481i \(0.725453\pi\)
\(998\) 0 0
\(999\) 24.8587 0.786494
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 4028.2.a.c.1.4 19
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4028.2.a.c.1.4 19 1.1 even 1 trivial