Properties

Label 6032.2.a.z.1.5
Level $6032$
Weight $2$
Character 6032.1
Self dual yes
Analytic conductor $48.166$
Analytic rank $1$
Dimension $10$
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] = [6032,2,Mod(1,6032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6032, 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("6032.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6032.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: \(48.1657624992\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3x^{9} - 17x^{8} + 47x^{7} + 104x^{6} - 235x^{5} - 283x^{4} + 364x^{3} + 330x^{2} + 12x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3016)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.131336\) of defining polynomial
Character \(\chi\) \(=\) 6032.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.131336 q^{3} -0.686438 q^{5} +3.43967 q^{7} -2.98275 q^{9} +O(q^{10})\) \(q-0.131336 q^{3} -0.686438 q^{5} +3.43967 q^{7} -2.98275 q^{9} +5.55935 q^{11} -1.00000 q^{13} +0.0901537 q^{15} -2.53126 q^{17} -1.08516 q^{19} -0.451750 q^{21} -3.63335 q^{23} -4.52880 q^{25} +0.785748 q^{27} -1.00000 q^{29} -5.76606 q^{31} -0.730140 q^{33} -2.36112 q^{35} +3.06448 q^{37} +0.131336 q^{39} +5.52019 q^{41} -10.6453 q^{43} +2.04747 q^{45} -1.05741 q^{47} +4.83130 q^{49} +0.332444 q^{51} -0.809282 q^{53} -3.81615 q^{55} +0.142520 q^{57} -8.17579 q^{59} -4.04037 q^{61} -10.2597 q^{63} +0.686438 q^{65} +8.59220 q^{67} +0.477188 q^{69} -14.5698 q^{71} -14.0357 q^{73} +0.594793 q^{75} +19.1223 q^{77} +7.02545 q^{79} +8.84506 q^{81} -8.52803 q^{83} +1.73755 q^{85} +0.131336 q^{87} -1.84349 q^{89} -3.43967 q^{91} +0.757289 q^{93} +0.744897 q^{95} +14.2896 q^{97} -16.5822 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9} - 14 q^{11} - 10 q^{13} - 7 q^{15} + 5 q^{17} - 11 q^{19} - 7 q^{23} + 10 q^{25} - 21 q^{27} - 10 q^{29} - 5 q^{31} + 5 q^{33} - 11 q^{35} + 8 q^{37} + 3 q^{39} + 14 q^{41} - 35 q^{43} + 7 q^{45} - 7 q^{49} - 20 q^{51} - 11 q^{53} - 8 q^{55} + 4 q^{57} - 23 q^{59} - 8 q^{61} - 43 q^{63} - 4 q^{65} - 27 q^{67} + 10 q^{69} - 3 q^{71} + 7 q^{73} - 23 q^{75} + 2 q^{77} - 9 q^{79} - 6 q^{81} - 48 q^{83} - 6 q^{85} + 3 q^{87} + 20 q^{89} - 3 q^{91} - 11 q^{93} - 11 q^{95} + q^{97} - 54 q^{99}+O(q^{100}) \) 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
\(3\) −0.131336 −0.0758266 −0.0379133 0.999281i \(-0.512071\pi\)
−0.0379133 + 0.999281i \(0.512071\pi\)
\(4\) 0 0
\(5\) −0.686438 −0.306984 −0.153492 0.988150i \(-0.549052\pi\)
−0.153492 + 0.988150i \(0.549052\pi\)
\(6\) 0 0
\(7\) 3.43967 1.30007 0.650036 0.759904i \(-0.274754\pi\)
0.650036 + 0.759904i \(0.274754\pi\)
\(8\) 0 0
\(9\) −2.98275 −0.994250
\(10\) 0 0
\(11\) 5.55935 1.67621 0.838104 0.545511i \(-0.183664\pi\)
0.838104 + 0.545511i \(0.183664\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0.0901537 0.0232776
\(16\) 0 0
\(17\) −2.53126 −0.613920 −0.306960 0.951722i \(-0.599312\pi\)
−0.306960 + 0.951722i \(0.599312\pi\)
\(18\) 0 0
\(19\) −1.08516 −0.248953 −0.124477 0.992223i \(-0.539725\pi\)
−0.124477 + 0.992223i \(0.539725\pi\)
\(20\) 0 0
\(21\) −0.451750 −0.0985800
\(22\) 0 0
\(23\) −3.63335 −0.757605 −0.378803 0.925478i \(-0.623664\pi\)
−0.378803 + 0.925478i \(0.623664\pi\)
\(24\) 0 0
\(25\) −4.52880 −0.905761
\(26\) 0 0
\(27\) 0.785748 0.151217
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −5.76606 −1.03562 −0.517808 0.855497i \(-0.673252\pi\)
−0.517808 + 0.855497i \(0.673252\pi\)
\(32\) 0 0
\(33\) −0.730140 −0.127101
\(34\) 0 0
\(35\) −2.36112 −0.399101
\(36\) 0 0
\(37\) 3.06448 0.503797 0.251899 0.967754i \(-0.418945\pi\)
0.251899 + 0.967754i \(0.418945\pi\)
\(38\) 0 0
\(39\) 0.131336 0.0210305
\(40\) 0 0
\(41\) 5.52019 0.862108 0.431054 0.902326i \(-0.358142\pi\)
0.431054 + 0.902326i \(0.358142\pi\)
\(42\) 0 0
\(43\) −10.6453 −1.62340 −0.811700 0.584075i \(-0.801457\pi\)
−0.811700 + 0.584075i \(0.801457\pi\)
\(44\) 0 0
\(45\) 2.04747 0.305219
\(46\) 0 0
\(47\) −1.05741 −0.154239 −0.0771194 0.997022i \(-0.524572\pi\)
−0.0771194 + 0.997022i \(0.524572\pi\)
\(48\) 0 0
\(49\) 4.83130 0.690185
\(50\) 0 0
\(51\) 0.332444 0.0465515
\(52\) 0 0
\(53\) −0.809282 −0.111163 −0.0555817 0.998454i \(-0.517701\pi\)
−0.0555817 + 0.998454i \(0.517701\pi\)
\(54\) 0 0
\(55\) −3.81615 −0.514569
\(56\) 0 0
\(57\) 0.142520 0.0188773
\(58\) 0 0
\(59\) −8.17579 −1.06440 −0.532198 0.846620i \(-0.678634\pi\)
−0.532198 + 0.846620i \(0.678634\pi\)
\(60\) 0 0
\(61\) −4.04037 −0.517316 −0.258658 0.965969i \(-0.583280\pi\)
−0.258658 + 0.965969i \(0.583280\pi\)
\(62\) 0 0
\(63\) −10.2597 −1.29260
\(64\) 0 0
\(65\) 0.686438 0.0851421
\(66\) 0 0
\(67\) 8.59220 1.04970 0.524852 0.851193i \(-0.324121\pi\)
0.524852 + 0.851193i \(0.324121\pi\)
\(68\) 0 0
\(69\) 0.477188 0.0574466
\(70\) 0 0
\(71\) −14.5698 −1.72912 −0.864561 0.502528i \(-0.832403\pi\)
−0.864561 + 0.502528i \(0.832403\pi\)
\(72\) 0 0
\(73\) −14.0357 −1.64275 −0.821377 0.570386i \(-0.806794\pi\)
−0.821377 + 0.570386i \(0.806794\pi\)
\(74\) 0 0
\(75\) 0.594793 0.0686808
\(76\) 0 0
\(77\) 19.1223 2.17919
\(78\) 0 0
\(79\) 7.02545 0.790425 0.395213 0.918590i \(-0.370671\pi\)
0.395213 + 0.918590i \(0.370671\pi\)
\(80\) 0 0
\(81\) 8.84506 0.982784
\(82\) 0 0
\(83\) −8.52803 −0.936073 −0.468037 0.883709i \(-0.655038\pi\)
−0.468037 + 0.883709i \(0.655038\pi\)
\(84\) 0 0
\(85\) 1.73755 0.188464
\(86\) 0 0
\(87\) 0.131336 0.0140807
\(88\) 0 0
\(89\) −1.84349 −0.195409 −0.0977047 0.995215i \(-0.531150\pi\)
−0.0977047 + 0.995215i \(0.531150\pi\)
\(90\) 0 0
\(91\) −3.43967 −0.360575
\(92\) 0 0
\(93\) 0.757289 0.0785272
\(94\) 0 0
\(95\) 0.744897 0.0764248
\(96\) 0 0
\(97\) 14.2896 1.45088 0.725442 0.688283i \(-0.241635\pi\)
0.725442 + 0.688283i \(0.241635\pi\)
\(98\) 0 0
\(99\) −16.5822 −1.66657
\(100\) 0 0
\(101\) −3.52214 −0.350466 −0.175233 0.984527i \(-0.556068\pi\)
−0.175233 + 0.984527i \(0.556068\pi\)
\(102\) 0 0
\(103\) 1.06761 0.105195 0.0525975 0.998616i \(-0.483250\pi\)
0.0525975 + 0.998616i \(0.483250\pi\)
\(104\) 0 0
\(105\) 0.310099 0.0302625
\(106\) 0 0
\(107\) −2.41480 −0.233447 −0.116724 0.993164i \(-0.537239\pi\)
−0.116724 + 0.993164i \(0.537239\pi\)
\(108\) 0 0
\(109\) −4.72653 −0.452719 −0.226360 0.974044i \(-0.572682\pi\)
−0.226360 + 0.974044i \(0.572682\pi\)
\(110\) 0 0
\(111\) −0.402475 −0.0382013
\(112\) 0 0
\(113\) −9.98385 −0.939202 −0.469601 0.882879i \(-0.655602\pi\)
−0.469601 + 0.882879i \(0.655602\pi\)
\(114\) 0 0
\(115\) 2.49407 0.232573
\(116\) 0 0
\(117\) 2.98275 0.275755
\(118\) 0 0
\(119\) −8.70668 −0.798140
\(120\) 0 0
\(121\) 19.9064 1.80967
\(122\) 0 0
\(123\) −0.724997 −0.0653708
\(124\) 0 0
\(125\) 6.54093 0.585039
\(126\) 0 0
\(127\) 10.0684 0.893422 0.446711 0.894678i \(-0.352595\pi\)
0.446711 + 0.894678i \(0.352595\pi\)
\(128\) 0 0
\(129\) 1.39811 0.123097
\(130\) 0 0
\(131\) 9.64133 0.842367 0.421184 0.906975i \(-0.361615\pi\)
0.421184 + 0.906975i \(0.361615\pi\)
\(132\) 0 0
\(133\) −3.73260 −0.323657
\(134\) 0 0
\(135\) −0.539367 −0.0464213
\(136\) 0 0
\(137\) 0.270066 0.0230733 0.0115366 0.999933i \(-0.496328\pi\)
0.0115366 + 0.999933i \(0.496328\pi\)
\(138\) 0 0
\(139\) 11.6477 0.987947 0.493974 0.869477i \(-0.335544\pi\)
0.493974 + 0.869477i \(0.335544\pi\)
\(140\) 0 0
\(141\) 0.138875 0.0116954
\(142\) 0 0
\(143\) −5.55935 −0.464896
\(144\) 0 0
\(145\) 0.686438 0.0570056
\(146\) 0 0
\(147\) −0.634521 −0.0523344
\(148\) 0 0
\(149\) −13.0255 −1.06709 −0.533543 0.845773i \(-0.679140\pi\)
−0.533543 + 0.845773i \(0.679140\pi\)
\(150\) 0 0
\(151\) 4.79500 0.390211 0.195106 0.980782i \(-0.437495\pi\)
0.195106 + 0.980782i \(0.437495\pi\)
\(152\) 0 0
\(153\) 7.55011 0.610390
\(154\) 0 0
\(155\) 3.95804 0.317918
\(156\) 0 0
\(157\) −8.66128 −0.691246 −0.345623 0.938374i \(-0.612332\pi\)
−0.345623 + 0.938374i \(0.612332\pi\)
\(158\) 0 0
\(159\) 0.106288 0.00842915
\(160\) 0 0
\(161\) −12.4975 −0.984941
\(162\) 0 0
\(163\) −20.0406 −1.56970 −0.784851 0.619685i \(-0.787260\pi\)
−0.784851 + 0.619685i \(0.787260\pi\)
\(164\) 0 0
\(165\) 0.501196 0.0390181
\(166\) 0 0
\(167\) 21.7096 1.67994 0.839970 0.542632i \(-0.182572\pi\)
0.839970 + 0.542632i \(0.182572\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 3.23677 0.247522
\(172\) 0 0
\(173\) 18.1395 1.37912 0.689562 0.724226i \(-0.257803\pi\)
0.689562 + 0.724226i \(0.257803\pi\)
\(174\) 0 0
\(175\) −15.5776 −1.17755
\(176\) 0 0
\(177\) 1.07377 0.0807096
\(178\) 0 0
\(179\) 23.8752 1.78452 0.892259 0.451525i \(-0.149120\pi\)
0.892259 + 0.451525i \(0.149120\pi\)
\(180\) 0 0
\(181\) 0.0264796 0.00196821 0.000984105 1.00000i \(-0.499687\pi\)
0.000984105 1.00000i \(0.499687\pi\)
\(182\) 0 0
\(183\) 0.530644 0.0392263
\(184\) 0 0
\(185\) −2.10357 −0.154658
\(186\) 0 0
\(187\) −14.0721 −1.02906
\(188\) 0 0
\(189\) 2.70271 0.196593
\(190\) 0 0
\(191\) −19.3928 −1.40322 −0.701608 0.712563i \(-0.747534\pi\)
−0.701608 + 0.712563i \(0.747534\pi\)
\(192\) 0 0
\(193\) −4.47916 −0.322417 −0.161209 0.986920i \(-0.551539\pi\)
−0.161209 + 0.986920i \(0.551539\pi\)
\(194\) 0 0
\(195\) −0.0901537 −0.00645604
\(196\) 0 0
\(197\) −15.4136 −1.09817 −0.549086 0.835766i \(-0.685024\pi\)
−0.549086 + 0.835766i \(0.685024\pi\)
\(198\) 0 0
\(199\) 2.79503 0.198134 0.0990672 0.995081i \(-0.468414\pi\)
0.0990672 + 0.995081i \(0.468414\pi\)
\(200\) 0 0
\(201\) −1.12846 −0.0795956
\(202\) 0 0
\(203\) −3.43967 −0.241417
\(204\) 0 0
\(205\) −3.78927 −0.264654
\(206\) 0 0
\(207\) 10.8374 0.753249
\(208\) 0 0
\(209\) −6.03280 −0.417297
\(210\) 0 0
\(211\) −11.3167 −0.779070 −0.389535 0.921012i \(-0.627364\pi\)
−0.389535 + 0.921012i \(0.627364\pi\)
\(212\) 0 0
\(213\) 1.91354 0.131113
\(214\) 0 0
\(215\) 7.30737 0.498358
\(216\) 0 0
\(217\) −19.8333 −1.34637
\(218\) 0 0
\(219\) 1.84339 0.124565
\(220\) 0 0
\(221\) 2.53126 0.170271
\(222\) 0 0
\(223\) −22.2875 −1.49248 −0.746241 0.665675i \(-0.768144\pi\)
−0.746241 + 0.665675i \(0.768144\pi\)
\(224\) 0 0
\(225\) 13.5083 0.900553
\(226\) 0 0
\(227\) −27.3103 −1.81265 −0.906325 0.422582i \(-0.861124\pi\)
−0.906325 + 0.422582i \(0.861124\pi\)
\(228\) 0 0
\(229\) −7.75270 −0.512313 −0.256156 0.966635i \(-0.582456\pi\)
−0.256156 + 0.966635i \(0.582456\pi\)
\(230\) 0 0
\(231\) −2.51144 −0.165241
\(232\) 0 0
\(233\) −13.9292 −0.912532 −0.456266 0.889843i \(-0.650813\pi\)
−0.456266 + 0.889843i \(0.650813\pi\)
\(234\) 0 0
\(235\) 0.725845 0.0473489
\(236\) 0 0
\(237\) −0.922692 −0.0599353
\(238\) 0 0
\(239\) −16.4918 −1.06677 −0.533384 0.845873i \(-0.679080\pi\)
−0.533384 + 0.845873i \(0.679080\pi\)
\(240\) 0 0
\(241\) 7.44009 0.479258 0.239629 0.970865i \(-0.422974\pi\)
0.239629 + 0.970865i \(0.422974\pi\)
\(242\) 0 0
\(243\) −3.51892 −0.225739
\(244\) 0 0
\(245\) −3.31638 −0.211876
\(246\) 0 0
\(247\) 1.08516 0.0690473
\(248\) 0 0
\(249\) 1.12003 0.0709793
\(250\) 0 0
\(251\) −24.2320 −1.52951 −0.764754 0.644322i \(-0.777140\pi\)
−0.764754 + 0.644322i \(0.777140\pi\)
\(252\) 0 0
\(253\) −20.1990 −1.26990
\(254\) 0 0
\(255\) −0.228202 −0.0142906
\(256\) 0 0
\(257\) 10.7838 0.672674 0.336337 0.941742i \(-0.390812\pi\)
0.336337 + 0.941742i \(0.390812\pi\)
\(258\) 0 0
\(259\) 10.5408 0.654973
\(260\) 0 0
\(261\) 2.98275 0.184628
\(262\) 0 0
\(263\) 28.8496 1.77894 0.889470 0.456994i \(-0.151074\pi\)
0.889470 + 0.456994i \(0.151074\pi\)
\(264\) 0 0
\(265\) 0.555522 0.0341254
\(266\) 0 0
\(267\) 0.242116 0.0148172
\(268\) 0 0
\(269\) −4.59148 −0.279948 −0.139974 0.990155i \(-0.544702\pi\)
−0.139974 + 0.990155i \(0.544702\pi\)
\(270\) 0 0
\(271\) −0.423401 −0.0257198 −0.0128599 0.999917i \(-0.504094\pi\)
−0.0128599 + 0.999917i \(0.504094\pi\)
\(272\) 0 0
\(273\) 0.451750 0.0273412
\(274\) 0 0
\(275\) −25.1772 −1.51824
\(276\) 0 0
\(277\) 8.11882 0.487812 0.243906 0.969799i \(-0.421571\pi\)
0.243906 + 0.969799i \(0.421571\pi\)
\(278\) 0 0
\(279\) 17.1987 1.02966
\(280\) 0 0
\(281\) 30.3456 1.81027 0.905134 0.425127i \(-0.139771\pi\)
0.905134 + 0.425127i \(0.139771\pi\)
\(282\) 0 0
\(283\) −16.3570 −0.972322 −0.486161 0.873869i \(-0.661603\pi\)
−0.486161 + 0.873869i \(0.661603\pi\)
\(284\) 0 0
\(285\) −0.0978314 −0.00579503
\(286\) 0 0
\(287\) 18.9876 1.12080
\(288\) 0 0
\(289\) −10.5927 −0.623102
\(290\) 0 0
\(291\) −1.87673 −0.110016
\(292\) 0 0
\(293\) 1.53200 0.0895002 0.0447501 0.998998i \(-0.485751\pi\)
0.0447501 + 0.998998i \(0.485751\pi\)
\(294\) 0 0
\(295\) 5.61217 0.326753
\(296\) 0 0
\(297\) 4.36825 0.253471
\(298\) 0 0
\(299\) 3.63335 0.210122
\(300\) 0 0
\(301\) −36.6164 −2.11054
\(302\) 0 0
\(303\) 0.462582 0.0265747
\(304\) 0 0
\(305\) 2.77346 0.158808
\(306\) 0 0
\(307\) −22.5274 −1.28571 −0.642853 0.765989i \(-0.722250\pi\)
−0.642853 + 0.765989i \(0.722250\pi\)
\(308\) 0 0
\(309\) −0.140215 −0.00797658
\(310\) 0 0
\(311\) 9.58495 0.543512 0.271756 0.962366i \(-0.412396\pi\)
0.271756 + 0.962366i \(0.412396\pi\)
\(312\) 0 0
\(313\) 10.2299 0.578230 0.289115 0.957294i \(-0.406639\pi\)
0.289115 + 0.957294i \(0.406639\pi\)
\(314\) 0 0
\(315\) 7.04262 0.396807
\(316\) 0 0
\(317\) 13.5846 0.762985 0.381492 0.924372i \(-0.375410\pi\)
0.381492 + 0.924372i \(0.375410\pi\)
\(318\) 0 0
\(319\) −5.55935 −0.311264
\(320\) 0 0
\(321\) 0.317149 0.0177015
\(322\) 0 0
\(323\) 2.74683 0.152837
\(324\) 0 0
\(325\) 4.52880 0.251213
\(326\) 0 0
\(327\) 0.620761 0.0343282
\(328\) 0 0
\(329\) −3.63713 −0.200521
\(330\) 0 0
\(331\) −4.15749 −0.228516 −0.114258 0.993451i \(-0.536449\pi\)
−0.114258 + 0.993451i \(0.536449\pi\)
\(332\) 0 0
\(333\) −9.14058 −0.500901
\(334\) 0 0
\(335\) −5.89801 −0.322243
\(336\) 0 0
\(337\) −23.6977 −1.29090 −0.645449 0.763803i \(-0.723330\pi\)
−0.645449 + 0.763803i \(0.723330\pi\)
\(338\) 0 0
\(339\) 1.31124 0.0712165
\(340\) 0 0
\(341\) −32.0556 −1.73591
\(342\) 0 0
\(343\) −7.45962 −0.402781
\(344\) 0 0
\(345\) −0.327560 −0.0176352
\(346\) 0 0
\(347\) −30.5937 −1.64236 −0.821179 0.570671i \(-0.806683\pi\)
−0.821179 + 0.570671i \(0.806683\pi\)
\(348\) 0 0
\(349\) 9.56661 0.512089 0.256044 0.966665i \(-0.417581\pi\)
0.256044 + 0.966665i \(0.417581\pi\)
\(350\) 0 0
\(351\) −0.785748 −0.0419401
\(352\) 0 0
\(353\) 25.2989 1.34653 0.673263 0.739403i \(-0.264892\pi\)
0.673263 + 0.739403i \(0.264892\pi\)
\(354\) 0 0
\(355\) 10.0013 0.530813
\(356\) 0 0
\(357\) 1.14350 0.0605203
\(358\) 0 0
\(359\) −8.24933 −0.435383 −0.217691 0.976018i \(-0.569853\pi\)
−0.217691 + 0.976018i \(0.569853\pi\)
\(360\) 0 0
\(361\) −17.8224 −0.938022
\(362\) 0 0
\(363\) −2.61441 −0.137221
\(364\) 0 0
\(365\) 9.63463 0.504300
\(366\) 0 0
\(367\) −31.5997 −1.64949 −0.824746 0.565504i \(-0.808682\pi\)
−0.824746 + 0.565504i \(0.808682\pi\)
\(368\) 0 0
\(369\) −16.4653 −0.857152
\(370\) 0 0
\(371\) −2.78366 −0.144520
\(372\) 0 0
\(373\) −21.2294 −1.09922 −0.549609 0.835422i \(-0.685223\pi\)
−0.549609 + 0.835422i \(0.685223\pi\)
\(374\) 0 0
\(375\) −0.859057 −0.0443615
\(376\) 0 0
\(377\) 1.00000 0.0515026
\(378\) 0 0
\(379\) −5.74408 −0.295053 −0.147527 0.989058i \(-0.547131\pi\)
−0.147527 + 0.989058i \(0.547131\pi\)
\(380\) 0 0
\(381\) −1.32233 −0.0677452
\(382\) 0 0
\(383\) 1.05900 0.0541122 0.0270561 0.999634i \(-0.491387\pi\)
0.0270561 + 0.999634i \(0.491387\pi\)
\(384\) 0 0
\(385\) −13.1263 −0.668977
\(386\) 0 0
\(387\) 31.7524 1.61407
\(388\) 0 0
\(389\) 15.7816 0.800161 0.400080 0.916480i \(-0.368982\pi\)
0.400080 + 0.916480i \(0.368982\pi\)
\(390\) 0 0
\(391\) 9.19693 0.465109
\(392\) 0 0
\(393\) −1.26625 −0.0638739
\(394\) 0 0
\(395\) −4.82254 −0.242648
\(396\) 0 0
\(397\) −31.3536 −1.57359 −0.786797 0.617212i \(-0.788262\pi\)
−0.786797 + 0.617212i \(0.788262\pi\)
\(398\) 0 0
\(399\) 0.490223 0.0245418
\(400\) 0 0
\(401\) 26.2074 1.30873 0.654367 0.756177i \(-0.272935\pi\)
0.654367 + 0.756177i \(0.272935\pi\)
\(402\) 0 0
\(403\) 5.76606 0.287228
\(404\) 0 0
\(405\) −6.07158 −0.301699
\(406\) 0 0
\(407\) 17.0365 0.844469
\(408\) 0 0
\(409\) 24.1192 1.19262 0.596309 0.802755i \(-0.296633\pi\)
0.596309 + 0.802755i \(0.296633\pi\)
\(410\) 0 0
\(411\) −0.0354693 −0.00174957
\(412\) 0 0
\(413\) −28.1220 −1.38379
\(414\) 0 0
\(415\) 5.85396 0.287360
\(416\) 0 0
\(417\) −1.52976 −0.0749127
\(418\) 0 0
\(419\) −29.8130 −1.45646 −0.728229 0.685334i \(-0.759656\pi\)
−0.728229 + 0.685334i \(0.759656\pi\)
\(420\) 0 0
\(421\) −13.4782 −0.656885 −0.328443 0.944524i \(-0.606524\pi\)
−0.328443 + 0.944524i \(0.606524\pi\)
\(422\) 0 0
\(423\) 3.15399 0.153352
\(424\) 0 0
\(425\) 11.4636 0.556065
\(426\) 0 0
\(427\) −13.8975 −0.672547
\(428\) 0 0
\(429\) 0.730140 0.0352515
\(430\) 0 0
\(431\) −3.98838 −0.192114 −0.0960568 0.995376i \(-0.530623\pi\)
−0.0960568 + 0.995376i \(0.530623\pi\)
\(432\) 0 0
\(433\) 28.7223 1.38030 0.690152 0.723664i \(-0.257544\pi\)
0.690152 + 0.723664i \(0.257544\pi\)
\(434\) 0 0
\(435\) −0.0901537 −0.00432254
\(436\) 0 0
\(437\) 3.94277 0.188608
\(438\) 0 0
\(439\) 26.9165 1.28465 0.642327 0.766431i \(-0.277969\pi\)
0.642327 + 0.766431i \(0.277969\pi\)
\(440\) 0 0
\(441\) −14.4106 −0.686217
\(442\) 0 0
\(443\) −10.3369 −0.491122 −0.245561 0.969381i \(-0.578972\pi\)
−0.245561 + 0.969381i \(0.578972\pi\)
\(444\) 0 0
\(445\) 1.26544 0.0599876
\(446\) 0 0
\(447\) 1.71071 0.0809136
\(448\) 0 0
\(449\) 1.37807 0.0650349 0.0325175 0.999471i \(-0.489648\pi\)
0.0325175 + 0.999471i \(0.489648\pi\)
\(450\) 0 0
\(451\) 30.6887 1.44507
\(452\) 0 0
\(453\) −0.629754 −0.0295884
\(454\) 0 0
\(455\) 2.36112 0.110691
\(456\) 0 0
\(457\) −10.4675 −0.489648 −0.244824 0.969568i \(-0.578730\pi\)
−0.244824 + 0.969568i \(0.578730\pi\)
\(458\) 0 0
\(459\) −1.98893 −0.0928353
\(460\) 0 0
\(461\) 10.3355 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(462\) 0 0
\(463\) 22.7588 1.05769 0.528846 0.848718i \(-0.322625\pi\)
0.528846 + 0.848718i \(0.322625\pi\)
\(464\) 0 0
\(465\) −0.519832 −0.0241066
\(466\) 0 0
\(467\) 11.5363 0.533837 0.266918 0.963719i \(-0.413995\pi\)
0.266918 + 0.963719i \(0.413995\pi\)
\(468\) 0 0
\(469\) 29.5543 1.36469
\(470\) 0 0
\(471\) 1.13753 0.0524148
\(472\) 0 0
\(473\) −59.1812 −2.72115
\(474\) 0 0
\(475\) 4.91449 0.225492
\(476\) 0 0
\(477\) 2.41389 0.110524
\(478\) 0 0
\(479\) −17.3214 −0.791433 −0.395716 0.918373i \(-0.629504\pi\)
−0.395716 + 0.918373i \(0.629504\pi\)
\(480\) 0 0
\(481\) −3.06448 −0.139728
\(482\) 0 0
\(483\) 1.64137 0.0746847
\(484\) 0 0
\(485\) −9.80889 −0.445399
\(486\) 0 0
\(487\) −23.1290 −1.04808 −0.524038 0.851695i \(-0.675575\pi\)
−0.524038 + 0.851695i \(0.675575\pi\)
\(488\) 0 0
\(489\) 2.63205 0.119025
\(490\) 0 0
\(491\) −15.3238 −0.691552 −0.345776 0.938317i \(-0.612384\pi\)
−0.345776 + 0.938317i \(0.612384\pi\)
\(492\) 0 0
\(493\) 2.53126 0.114002
\(494\) 0 0
\(495\) 11.3826 0.511611
\(496\) 0 0
\(497\) −50.1154 −2.24798
\(498\) 0 0
\(499\) 20.0357 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(500\) 0 0
\(501\) −2.85125 −0.127384
\(502\) 0 0
\(503\) 10.1268 0.451530 0.225765 0.974182i \(-0.427512\pi\)
0.225765 + 0.974182i \(0.427512\pi\)
\(504\) 0 0
\(505\) 2.41773 0.107588
\(506\) 0 0
\(507\) −0.131336 −0.00583282
\(508\) 0 0
\(509\) 41.7942 1.85250 0.926248 0.376915i \(-0.123015\pi\)
0.926248 + 0.376915i \(0.123015\pi\)
\(510\) 0 0
\(511\) −48.2781 −2.13570
\(512\) 0 0
\(513\) −0.852665 −0.0376461
\(514\) 0 0
\(515\) −0.732849 −0.0322932
\(516\) 0 0
\(517\) −5.87850 −0.258536
\(518\) 0 0
\(519\) −2.38237 −0.104574
\(520\) 0 0
\(521\) −12.1787 −0.533561 −0.266780 0.963757i \(-0.585960\pi\)
−0.266780 + 0.963757i \(0.585960\pi\)
\(522\) 0 0
\(523\) −11.3592 −0.496703 −0.248352 0.968670i \(-0.579889\pi\)
−0.248352 + 0.968670i \(0.579889\pi\)
\(524\) 0 0
\(525\) 2.04589 0.0892899
\(526\) 0 0
\(527\) 14.5954 0.635785
\(528\) 0 0
\(529\) −9.79879 −0.426035
\(530\) 0 0
\(531\) 24.3863 1.05828
\(532\) 0 0
\(533\) −5.52019 −0.239106
\(534\) 0 0
\(535\) 1.65761 0.0716646
\(536\) 0 0
\(537\) −3.13567 −0.135314
\(538\) 0 0
\(539\) 26.8589 1.15689
\(540\) 0 0
\(541\) −8.44789 −0.363203 −0.181602 0.983372i \(-0.558128\pi\)
−0.181602 + 0.983372i \(0.558128\pi\)
\(542\) 0 0
\(543\) −0.00347771 −0.000149243 0
\(544\) 0 0
\(545\) 3.24447 0.138978
\(546\) 0 0
\(547\) −22.4946 −0.961801 −0.480901 0.876775i \(-0.659690\pi\)
−0.480901 + 0.876775i \(0.659690\pi\)
\(548\) 0 0
\(549\) 12.0514 0.514341
\(550\) 0 0
\(551\) 1.08516 0.0462295
\(552\) 0 0
\(553\) 24.1652 1.02761
\(554\) 0 0
\(555\) 0.276274 0.0117272
\(556\) 0 0
\(557\) 42.5787 1.80412 0.902058 0.431615i \(-0.142056\pi\)
0.902058 + 0.431615i \(0.142056\pi\)
\(558\) 0 0
\(559\) 10.6453 0.450250
\(560\) 0 0
\(561\) 1.84817 0.0780299
\(562\) 0 0
\(563\) −4.63542 −0.195360 −0.0976799 0.995218i \(-0.531142\pi\)
−0.0976799 + 0.995218i \(0.531142\pi\)
\(564\) 0 0
\(565\) 6.85329 0.288320
\(566\) 0 0
\(567\) 30.4240 1.27769
\(568\) 0 0
\(569\) −29.7859 −1.24869 −0.624344 0.781149i \(-0.714634\pi\)
−0.624344 + 0.781149i \(0.714634\pi\)
\(570\) 0 0
\(571\) −46.6867 −1.95378 −0.976890 0.213744i \(-0.931434\pi\)
−0.976890 + 0.213744i \(0.931434\pi\)
\(572\) 0 0
\(573\) 2.54697 0.106401
\(574\) 0 0
\(575\) 16.4547 0.686209
\(576\) 0 0
\(577\) −11.2276 −0.467411 −0.233705 0.972307i \(-0.575085\pi\)
−0.233705 + 0.972307i \(0.575085\pi\)
\(578\) 0 0
\(579\) 0.588274 0.0244478
\(580\) 0 0
\(581\) −29.3336 −1.21696
\(582\) 0 0
\(583\) −4.49908 −0.186333
\(584\) 0 0
\(585\) −2.04747 −0.0846526
\(586\) 0 0
\(587\) 47.3001 1.95228 0.976142 0.217135i \(-0.0696712\pi\)
0.976142 + 0.217135i \(0.0696712\pi\)
\(588\) 0 0
\(589\) 6.25712 0.257820
\(590\) 0 0
\(591\) 2.02435 0.0832707
\(592\) 0 0
\(593\) 1.92154 0.0789084 0.0394542 0.999221i \(-0.487438\pi\)
0.0394542 + 0.999221i \(0.487438\pi\)
\(594\) 0 0
\(595\) 5.97659 0.245016
\(596\) 0 0
\(597\) −0.367087 −0.0150239
\(598\) 0 0
\(599\) 42.2462 1.72613 0.863067 0.505089i \(-0.168540\pi\)
0.863067 + 0.505089i \(0.168540\pi\)
\(600\) 0 0
\(601\) −29.7707 −1.21437 −0.607187 0.794559i \(-0.707702\pi\)
−0.607187 + 0.794559i \(0.707702\pi\)
\(602\) 0 0
\(603\) −25.6284 −1.04367
\(604\) 0 0
\(605\) −13.6645 −0.555540
\(606\) 0 0
\(607\) 10.9173 0.443119 0.221560 0.975147i \(-0.428885\pi\)
0.221560 + 0.975147i \(0.428885\pi\)
\(608\) 0 0
\(609\) 0.451750 0.0183059
\(610\) 0 0
\(611\) 1.05741 0.0427782
\(612\) 0 0
\(613\) −35.5720 −1.43674 −0.718370 0.695661i \(-0.755112\pi\)
−0.718370 + 0.695661i \(0.755112\pi\)
\(614\) 0 0
\(615\) 0.497665 0.0200678
\(616\) 0 0
\(617\) −36.8015 −1.48157 −0.740786 0.671741i \(-0.765547\pi\)
−0.740786 + 0.671741i \(0.765547\pi\)
\(618\) 0 0
\(619\) −16.4788 −0.662341 −0.331170 0.943571i \(-0.607444\pi\)
−0.331170 + 0.943571i \(0.607444\pi\)
\(620\) 0 0
\(621\) −2.85490 −0.114563
\(622\) 0 0
\(623\) −6.34098 −0.254046
\(624\) 0 0
\(625\) 18.1541 0.726163
\(626\) 0 0
\(627\) 0.792321 0.0316423
\(628\) 0 0
\(629\) −7.75699 −0.309291
\(630\) 0 0
\(631\) −11.4821 −0.457097 −0.228548 0.973533i \(-0.573398\pi\)
−0.228548 + 0.973533i \(0.573398\pi\)
\(632\) 0 0
\(633\) 1.48628 0.0590743
\(634\) 0 0
\(635\) −6.91130 −0.274267
\(636\) 0 0
\(637\) −4.83130 −0.191423
\(638\) 0 0
\(639\) 43.4582 1.71918
\(640\) 0 0
\(641\) 13.4324 0.530546 0.265273 0.964173i \(-0.414538\pi\)
0.265273 + 0.964173i \(0.414538\pi\)
\(642\) 0 0
\(643\) −2.41377 −0.0951899 −0.0475949 0.998867i \(-0.515156\pi\)
−0.0475949 + 0.998867i \(0.515156\pi\)
\(644\) 0 0
\(645\) −0.959717 −0.0377888
\(646\) 0 0
\(647\) 3.71669 0.146118 0.0730590 0.997328i \(-0.476724\pi\)
0.0730590 + 0.997328i \(0.476724\pi\)
\(648\) 0 0
\(649\) −45.4521 −1.78415
\(650\) 0 0
\(651\) 2.60482 0.102091
\(652\) 0 0
\(653\) 38.7029 1.51456 0.757280 0.653091i \(-0.226528\pi\)
0.757280 + 0.653091i \(0.226528\pi\)
\(654\) 0 0
\(655\) −6.61817 −0.258593
\(656\) 0 0
\(657\) 41.8650 1.63331
\(658\) 0 0
\(659\) −7.39422 −0.288038 −0.144019 0.989575i \(-0.546003\pi\)
−0.144019 + 0.989575i \(0.546003\pi\)
\(660\) 0 0
\(661\) −8.27897 −0.322015 −0.161007 0.986953i \(-0.551474\pi\)
−0.161007 + 0.986953i \(0.551474\pi\)
\(662\) 0 0
\(663\) −0.332444 −0.0129111
\(664\) 0 0
\(665\) 2.56220 0.0993577
\(666\) 0 0
\(667\) 3.63335 0.140684
\(668\) 0 0
\(669\) 2.92714 0.113170
\(670\) 0 0
\(671\) −22.4618 −0.867128
\(672\) 0 0
\(673\) 33.0124 1.27254 0.636268 0.771468i \(-0.280477\pi\)
0.636268 + 0.771468i \(0.280477\pi\)
\(674\) 0 0
\(675\) −3.55850 −0.136967
\(676\) 0 0
\(677\) 41.5826 1.59815 0.799075 0.601231i \(-0.205323\pi\)
0.799075 + 0.601231i \(0.205323\pi\)
\(678\) 0 0
\(679\) 49.1513 1.88625
\(680\) 0 0
\(681\) 3.58682 0.137447
\(682\) 0 0
\(683\) 9.49601 0.363355 0.181677 0.983358i \(-0.441847\pi\)
0.181677 + 0.983358i \(0.441847\pi\)
\(684\) 0 0
\(685\) −0.185383 −0.00708314
\(686\) 0 0
\(687\) 1.01820 0.0388469
\(688\) 0 0
\(689\) 0.809282 0.0308312
\(690\) 0 0
\(691\) −35.8653 −1.36438 −0.682189 0.731176i \(-0.738972\pi\)
−0.682189 + 0.731176i \(0.738972\pi\)
\(692\) 0 0
\(693\) −57.0371 −2.16666
\(694\) 0 0
\(695\) −7.99544 −0.303284
\(696\) 0 0
\(697\) −13.9730 −0.529266
\(698\) 0 0
\(699\) 1.82940 0.0691942
\(700\) 0 0
\(701\) 26.3423 0.994934 0.497467 0.867483i \(-0.334264\pi\)
0.497467 + 0.867483i \(0.334264\pi\)
\(702\) 0 0
\(703\) −3.32546 −0.125422
\(704\) 0 0
\(705\) −0.0953293 −0.00359031
\(706\) 0 0
\(707\) −12.1150 −0.455631
\(708\) 0 0
\(709\) 14.3348 0.538356 0.269178 0.963090i \(-0.413248\pi\)
0.269178 + 0.963090i \(0.413248\pi\)
\(710\) 0 0
\(711\) −20.9552 −0.785881
\(712\) 0 0
\(713\) 20.9501 0.784588
\(714\) 0 0
\(715\) 3.81615 0.142716
\(716\) 0 0
\(717\) 2.16596 0.0808894
\(718\) 0 0
\(719\) 10.5642 0.393979 0.196989 0.980406i \(-0.436884\pi\)
0.196989 + 0.980406i \(0.436884\pi\)
\(720\) 0 0
\(721\) 3.67223 0.136761
\(722\) 0 0
\(723\) −0.977148 −0.0363405
\(724\) 0 0
\(725\) 4.52880 0.168196
\(726\) 0 0
\(727\) 26.9021 0.997743 0.498872 0.866676i \(-0.333748\pi\)
0.498872 + 0.866676i \(0.333748\pi\)
\(728\) 0 0
\(729\) −26.0730 −0.965667
\(730\) 0 0
\(731\) 26.9461 0.996638
\(732\) 0 0
\(733\) 3.31499 0.122442 0.0612209 0.998124i \(-0.480501\pi\)
0.0612209 + 0.998124i \(0.480501\pi\)
\(734\) 0 0
\(735\) 0.435559 0.0160658
\(736\) 0 0
\(737\) 47.7671 1.75952
\(738\) 0 0
\(739\) −30.4617 −1.12055 −0.560277 0.828305i \(-0.689305\pi\)
−0.560277 + 0.828305i \(0.689305\pi\)
\(740\) 0 0
\(741\) −0.142520 −0.00523562
\(742\) 0 0
\(743\) 25.9403 0.951658 0.475829 0.879538i \(-0.342148\pi\)
0.475829 + 0.879538i \(0.342148\pi\)
\(744\) 0 0
\(745\) 8.94116 0.327579
\(746\) 0 0
\(747\) 25.4370 0.930691
\(748\) 0 0
\(749\) −8.30609 −0.303498
\(750\) 0 0
\(751\) 35.5404 1.29689 0.648443 0.761263i \(-0.275420\pi\)
0.648443 + 0.761263i \(0.275420\pi\)
\(752\) 0 0
\(753\) 3.18252 0.115977
\(754\) 0 0
\(755\) −3.29147 −0.119789
\(756\) 0 0
\(757\) −52.6251 −1.91269 −0.956346 0.292237i \(-0.905600\pi\)
−0.956346 + 0.292237i \(0.905600\pi\)
\(758\) 0 0
\(759\) 2.65285 0.0962925
\(760\) 0 0
\(761\) −33.8953 −1.22870 −0.614352 0.789032i \(-0.710583\pi\)
−0.614352 + 0.789032i \(0.710583\pi\)
\(762\) 0 0
\(763\) −16.2577 −0.588567
\(764\) 0 0
\(765\) −5.18268 −0.187380
\(766\) 0 0
\(767\) 8.17579 0.295211
\(768\) 0 0
\(769\) −9.69096 −0.349465 −0.174732 0.984616i \(-0.555906\pi\)
−0.174732 + 0.984616i \(0.555906\pi\)
\(770\) 0 0
\(771\) −1.41629 −0.0510066
\(772\) 0 0
\(773\) −29.7639 −1.07053 −0.535266 0.844684i \(-0.679789\pi\)
−0.535266 + 0.844684i \(0.679789\pi\)
\(774\) 0 0
\(775\) 26.1134 0.938020
\(776\) 0 0
\(777\) −1.38438 −0.0496644
\(778\) 0 0
\(779\) −5.99030 −0.214625
\(780\) 0 0
\(781\) −80.9988 −2.89837
\(782\) 0 0
\(783\) −0.785748 −0.0280803
\(784\) 0 0
\(785\) 5.94543 0.212202
\(786\) 0 0
\(787\) 43.5261 1.55154 0.775768 0.631018i \(-0.217363\pi\)
0.775768 + 0.631018i \(0.217363\pi\)
\(788\) 0 0
\(789\) −3.78897 −0.134891
\(790\) 0 0
\(791\) −34.3411 −1.22103
\(792\) 0 0
\(793\) 4.04037 0.143478
\(794\) 0 0
\(795\) −0.0729598 −0.00258762
\(796\) 0 0
\(797\) 41.0512 1.45411 0.727054 0.686580i \(-0.240889\pi\)
0.727054 + 0.686580i \(0.240889\pi\)
\(798\) 0 0
\(799\) 2.67657 0.0946903
\(800\) 0 0
\(801\) 5.49866 0.194286
\(802\) 0 0
\(803\) −78.0293 −2.75360
\(804\) 0 0
\(805\) 8.57875 0.302361
\(806\) 0 0
\(807\) 0.603025 0.0212275
\(808\) 0 0
\(809\) 45.5107 1.60007 0.800036 0.599952i \(-0.204814\pi\)
0.800036 + 0.599952i \(0.204814\pi\)
\(810\) 0 0
\(811\) 8.08353 0.283851 0.141926 0.989877i \(-0.454671\pi\)
0.141926 + 0.989877i \(0.454671\pi\)
\(812\) 0 0
\(813\) 0.0556076 0.00195024
\(814\) 0 0
\(815\) 13.7566 0.481874
\(816\) 0 0
\(817\) 11.5519 0.404151
\(818\) 0 0
\(819\) 10.2597 0.358502
\(820\) 0 0
\(821\) 34.1451 1.19167 0.595836 0.803106i \(-0.296821\pi\)
0.595836 + 0.803106i \(0.296821\pi\)
\(822\) 0 0
\(823\) 43.3908 1.51251 0.756255 0.654277i \(-0.227027\pi\)
0.756255 + 0.654277i \(0.227027\pi\)
\(824\) 0 0
\(825\) 3.30666 0.115123
\(826\) 0 0
\(827\) −3.17503 −0.110406 −0.0552032 0.998475i \(-0.517581\pi\)
−0.0552032 + 0.998475i \(0.517581\pi\)
\(828\) 0 0
\(829\) −37.6627 −1.30808 −0.654040 0.756460i \(-0.726927\pi\)
−0.654040 + 0.756460i \(0.726927\pi\)
\(830\) 0 0
\(831\) −1.06629 −0.0369892
\(832\) 0 0
\(833\) −12.2293 −0.423718
\(834\) 0 0
\(835\) −14.9023 −0.515715
\(836\) 0 0
\(837\) −4.53067 −0.156603
\(838\) 0 0
\(839\) −1.80602 −0.0623509 −0.0311755 0.999514i \(-0.509925\pi\)
−0.0311755 + 0.999514i \(0.509925\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −3.98546 −0.137266
\(844\) 0 0
\(845\) −0.686438 −0.0236142
\(846\) 0 0
\(847\) 68.4712 2.35270
\(848\) 0 0
\(849\) 2.14826 0.0737279
\(850\) 0 0
\(851\) −11.1343 −0.381679
\(852\) 0 0
\(853\) 2.85524 0.0977615 0.0488808 0.998805i \(-0.484435\pi\)
0.0488808 + 0.998805i \(0.484435\pi\)
\(854\) 0 0
\(855\) −2.22184 −0.0759854
\(856\) 0 0
\(857\) −56.8440 −1.94175 −0.970876 0.239581i \(-0.922990\pi\)
−0.970876 + 0.239581i \(0.922990\pi\)
\(858\) 0 0
\(859\) −4.86844 −0.166109 −0.0830545 0.996545i \(-0.526468\pi\)
−0.0830545 + 0.996545i \(0.526468\pi\)
\(860\) 0 0
\(861\) −2.49375 −0.0849867
\(862\) 0 0
\(863\) −0.313010 −0.0106550 −0.00532749 0.999986i \(-0.501696\pi\)
−0.00532749 + 0.999986i \(0.501696\pi\)
\(864\) 0 0
\(865\) −12.4517 −0.423370
\(866\) 0 0
\(867\) 1.39120 0.0472477
\(868\) 0 0
\(869\) 39.0570 1.32492
\(870\) 0 0
\(871\) −8.59220 −0.291136
\(872\) 0 0
\(873\) −42.6222 −1.44254
\(874\) 0 0
\(875\) 22.4986 0.760592
\(876\) 0 0
\(877\) −26.9944 −0.911534 −0.455767 0.890099i \(-0.650635\pi\)
−0.455767 + 0.890099i \(0.650635\pi\)
\(878\) 0 0
\(879\) −0.201206 −0.00678650
\(880\) 0 0
\(881\) 31.7567 1.06991 0.534956 0.844880i \(-0.320328\pi\)
0.534956 + 0.844880i \(0.320328\pi\)
\(882\) 0 0
\(883\) 33.1872 1.11684 0.558420 0.829558i \(-0.311408\pi\)
0.558420 + 0.829558i \(0.311408\pi\)
\(884\) 0 0
\(885\) −0.737078 −0.0247766
\(886\) 0 0
\(887\) 1.39174 0.0467302 0.0233651 0.999727i \(-0.492562\pi\)
0.0233651 + 0.999727i \(0.492562\pi\)
\(888\) 0 0
\(889\) 34.6318 1.16151
\(890\) 0 0
\(891\) 49.1728 1.64735
\(892\) 0 0
\(893\) 1.14746 0.0383983
\(894\) 0 0
\(895\) −16.3888 −0.547819
\(896\) 0 0
\(897\) −0.477188 −0.0159328
\(898\) 0 0
\(899\) 5.76606 0.192309
\(900\) 0 0
\(901\) 2.04850 0.0682455
\(902\) 0 0
\(903\) 4.80904 0.160035
\(904\) 0 0
\(905\) −0.0181766 −0.000604210 0
\(906\) 0 0
\(907\) 12.7666 0.423910 0.211955 0.977279i \(-0.432017\pi\)
0.211955 + 0.977279i \(0.432017\pi\)
\(908\) 0 0
\(909\) 10.5057 0.348451
\(910\) 0 0
\(911\) −10.3509 −0.342939 −0.171470 0.985189i \(-0.554852\pi\)
−0.171470 + 0.985189i \(0.554852\pi\)
\(912\) 0 0
\(913\) −47.4103 −1.56905
\(914\) 0 0
\(915\) −0.364254 −0.0120419
\(916\) 0 0
\(917\) 33.1630 1.09514
\(918\) 0 0
\(919\) −32.1403 −1.06021 −0.530106 0.847932i \(-0.677848\pi\)
−0.530106 + 0.847932i \(0.677848\pi\)
\(920\) 0 0
\(921\) 2.95865 0.0974908
\(922\) 0 0
\(923\) 14.5698 0.479572
\(924\) 0 0
\(925\) −13.8784 −0.456320
\(926\) 0 0
\(927\) −3.18442 −0.104590
\(928\) 0 0
\(929\) −13.8195 −0.453402 −0.226701 0.973964i \(-0.572794\pi\)
−0.226701 + 0.973964i \(0.572794\pi\)
\(930\) 0 0
\(931\) −5.24274 −0.171824
\(932\) 0 0
\(933\) −1.25884 −0.0412127
\(934\) 0 0
\(935\) 9.65965 0.315904
\(936\) 0 0
\(937\) 32.5613 1.06373 0.531865 0.846829i \(-0.321491\pi\)
0.531865 + 0.846829i \(0.321491\pi\)
\(938\) 0 0
\(939\) −1.34355 −0.0438452
\(940\) 0 0
\(941\) 15.5171 0.505844 0.252922 0.967487i \(-0.418608\pi\)
0.252922 + 0.967487i \(0.418608\pi\)
\(942\) 0 0
\(943\) −20.0568 −0.653138
\(944\) 0 0
\(945\) −1.85524 −0.0603510
\(946\) 0 0
\(947\) −8.09564 −0.263073 −0.131536 0.991311i \(-0.541991\pi\)
−0.131536 + 0.991311i \(0.541991\pi\)
\(948\) 0 0
\(949\) 14.0357 0.455618
\(950\) 0 0
\(951\) −1.78414 −0.0578546
\(952\) 0 0
\(953\) 38.7797 1.25620 0.628098 0.778134i \(-0.283834\pi\)
0.628098 + 0.778134i \(0.283834\pi\)
\(954\) 0 0
\(955\) 13.3120 0.430765
\(956\) 0 0
\(957\) 0.730140 0.0236021
\(958\) 0 0
\(959\) 0.928936 0.0299969
\(960\) 0 0
\(961\) 2.24748 0.0724994
\(962\) 0 0
\(963\) 7.20274 0.232105
\(964\) 0 0
\(965\) 3.07467 0.0989771
\(966\) 0 0
\(967\) 54.7054 1.75921 0.879603 0.475708i \(-0.157808\pi\)
0.879603 + 0.475708i \(0.157808\pi\)
\(968\) 0 0
\(969\) −0.360756 −0.0115892
\(970\) 0 0
\(971\) −9.37898 −0.300986 −0.150493 0.988611i \(-0.548086\pi\)
−0.150493 + 0.988611i \(0.548086\pi\)
\(972\) 0 0
\(973\) 40.0643 1.28440
\(974\) 0 0
\(975\) −0.594793 −0.0190486
\(976\) 0 0
\(977\) −38.4218 −1.22922 −0.614611 0.788830i \(-0.710687\pi\)
−0.614611 + 0.788830i \(0.710687\pi\)
\(978\) 0 0
\(979\) −10.2486 −0.327546
\(980\) 0 0
\(981\) 14.0980 0.450116
\(982\) 0 0
\(983\) −43.5714 −1.38971 −0.694856 0.719149i \(-0.744532\pi\)
−0.694856 + 0.719149i \(0.744532\pi\)
\(984\) 0 0
\(985\) 10.5805 0.337122
\(986\) 0 0
\(987\) 0.477685 0.0152049
\(988\) 0 0
\(989\) 38.6782 1.22990
\(990\) 0 0
\(991\) 5.06919 0.161028 0.0805141 0.996753i \(-0.474344\pi\)
0.0805141 + 0.996753i \(0.474344\pi\)
\(992\) 0 0
\(993\) 0.546026 0.0173276
\(994\) 0 0
\(995\) −1.91861 −0.0608241
\(996\) 0 0
\(997\) 23.3887 0.740727 0.370363 0.928887i \(-0.379233\pi\)
0.370363 + 0.928887i \(0.379233\pi\)
\(998\) 0 0
\(999\) 2.40791 0.0761829
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 6032.2.a.z.1.5 10
4.3 odd 2 3016.2.a.h.1.6 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3016.2.a.h.1.6 10 4.3 odd 2
6032.2.a.z.1.5 10 1.1 even 1 trivial