Properties

Label 1859.2.a.t.1.7
Level $1859$
Weight $2$
Character 1859.1
Self dual yes
Analytic conductor $14.844$
Analytic rank $0$
Dimension $21$
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] = [1859,2,Mod(1,1859)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1859, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1859.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1859.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: \(14.8441897358\)
Analytic rank: \(0\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Character \(\chi\) \(=\) 1859.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.50636 q^{2} -0.544988 q^{3} +0.269129 q^{4} +3.40659 q^{5} +0.820949 q^{6} -0.714897 q^{7} +2.60732 q^{8} -2.70299 q^{9} +O(q^{10})\) \(q-1.50636 q^{2} -0.544988 q^{3} +0.269129 q^{4} +3.40659 q^{5} +0.820949 q^{6} -0.714897 q^{7} +2.60732 q^{8} -2.70299 q^{9} -5.13156 q^{10} +1.00000 q^{11} -0.146672 q^{12} +1.07689 q^{14} -1.85655 q^{15} -4.46583 q^{16} +5.08876 q^{17} +4.07168 q^{18} -2.56104 q^{19} +0.916811 q^{20} +0.389610 q^{21} -1.50636 q^{22} +0.472206 q^{23} -1.42096 q^{24} +6.60485 q^{25} +3.10806 q^{27} -0.192399 q^{28} +7.13941 q^{29} +2.79664 q^{30} -7.93075 q^{31} +1.51252 q^{32} -0.544988 q^{33} -7.66551 q^{34} -2.43536 q^{35} -0.727452 q^{36} +1.68838 q^{37} +3.85786 q^{38} +8.88207 q^{40} +1.29725 q^{41} -0.586894 q^{42} +9.09702 q^{43} +0.269129 q^{44} -9.20797 q^{45} -0.711313 q^{46} +7.45791 q^{47} +2.43382 q^{48} -6.48892 q^{49} -9.94929 q^{50} -2.77331 q^{51} +7.78116 q^{53} -4.68186 q^{54} +3.40659 q^{55} -1.86397 q^{56} +1.39574 q^{57} -10.7545 q^{58} +11.1900 q^{59} -0.499651 q^{60} -14.3882 q^{61} +11.9466 q^{62} +1.93236 q^{63} +6.65326 q^{64} +0.820949 q^{66} -7.36614 q^{67} +1.36953 q^{68} -0.257346 q^{69} +3.66854 q^{70} -1.70293 q^{71} -7.04756 q^{72} -6.44348 q^{73} -2.54331 q^{74} -3.59956 q^{75} -0.689249 q^{76} -0.714897 q^{77} -5.05937 q^{79} -15.2132 q^{80} +6.41511 q^{81} -1.95414 q^{82} -13.8551 q^{83} +0.104855 q^{84} +17.3353 q^{85} -13.7034 q^{86} -3.89089 q^{87} +2.60732 q^{88} +8.30538 q^{89} +13.8705 q^{90} +0.127084 q^{92} +4.32216 q^{93} -11.2343 q^{94} -8.72441 q^{95} -0.824302 q^{96} +9.02088 q^{97} +9.77467 q^{98} -2.70299 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q + 6 q^{3} + 32 q^{4} + 7 q^{5} - 19 q^{6} + q^{7} - 3 q^{8} + 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 21 q + 6 q^{3} + 32 q^{4} + 7 q^{5} - 19 q^{6} + q^{7} - 3 q^{8} + 33 q^{9} + 18 q^{10} + 21 q^{11} + 23 q^{12} + 20 q^{14} + 16 q^{15} + 50 q^{16} + 16 q^{17} + 3 q^{18} - 11 q^{19} + 24 q^{20} - 5 q^{21} - 9 q^{23} - 54 q^{24} + 36 q^{25} - 11 q^{28} + 28 q^{29} + 21 q^{30} + 15 q^{31} - 61 q^{32} + 6 q^{33} - 6 q^{34} - 3 q^{35} + 45 q^{36} - 12 q^{37} + q^{38} + 55 q^{40} - 4 q^{41} - 34 q^{42} + 17 q^{43} + 32 q^{44} + 9 q^{45} + 11 q^{46} + 36 q^{47} + 24 q^{48} + 72 q^{49} - 9 q^{50} + 2 q^{51} + 19 q^{53} + q^{54} + 7 q^{55} + 44 q^{56} - 4 q^{57} - 33 q^{58} + 54 q^{59} + 64 q^{60} + 98 q^{61} - 29 q^{62} - 81 q^{63} + 63 q^{64} - 19 q^{66} + 25 q^{67} + 4 q^{68} + 89 q^{69} + 65 q^{70} + 37 q^{71} + 55 q^{72} + 8 q^{73} - 11 q^{74} + 24 q^{75} + 13 q^{76} + q^{77} + 24 q^{79} + 26 q^{80} + 81 q^{81} + 26 q^{82} - 34 q^{83} - 103 q^{84} - 11 q^{85} + 30 q^{86} + 32 q^{87} - 3 q^{88} + 6 q^{89} + 47 q^{90} - 80 q^{92} + 41 q^{93} + 40 q^{94} + 20 q^{95} - 98 q^{96} - 5 q^{98} + 33 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.50636 −1.06516 −0.532580 0.846380i \(-0.678777\pi\)
−0.532580 + 0.846380i \(0.678777\pi\)
\(3\) −0.544988 −0.314649 −0.157324 0.987547i \(-0.550287\pi\)
−0.157324 + 0.987547i \(0.550287\pi\)
\(4\) 0.269129 0.134564
\(5\) 3.40659 1.52347 0.761736 0.647887i \(-0.224347\pi\)
0.761736 + 0.647887i \(0.224347\pi\)
\(6\) 0.820949 0.335151
\(7\) −0.714897 −0.270206 −0.135103 0.990832i \(-0.543137\pi\)
−0.135103 + 0.990832i \(0.543137\pi\)
\(8\) 2.60732 0.921827
\(9\) −2.70299 −0.900996
\(10\) −5.13156 −1.62274
\(11\) 1.00000 0.301511
\(12\) −0.146672 −0.0423405
\(13\) 0 0
\(14\) 1.07689 0.287812
\(15\) −1.85655 −0.479359
\(16\) −4.46583 −1.11646
\(17\) 5.08876 1.23420 0.617102 0.786883i \(-0.288306\pi\)
0.617102 + 0.786883i \(0.288306\pi\)
\(18\) 4.07168 0.959704
\(19\) −2.56104 −0.587543 −0.293772 0.955876i \(-0.594910\pi\)
−0.293772 + 0.955876i \(0.594910\pi\)
\(20\) 0.916811 0.205005
\(21\) 0.389610 0.0850199
\(22\) −1.50636 −0.321158
\(23\) 0.472206 0.0984617 0.0492309 0.998787i \(-0.484323\pi\)
0.0492309 + 0.998787i \(0.484323\pi\)
\(24\) −1.42096 −0.290052
\(25\) 6.60485 1.32097
\(26\) 0 0
\(27\) 3.10806 0.598146
\(28\) −0.192399 −0.0363601
\(29\) 7.13941 1.32576 0.662878 0.748728i \(-0.269335\pi\)
0.662878 + 0.748728i \(0.269335\pi\)
\(30\) 2.79664 0.510594
\(31\) −7.93075 −1.42441 −0.712203 0.701974i \(-0.752302\pi\)
−0.712203 + 0.701974i \(0.752302\pi\)
\(32\) 1.51252 0.267377
\(33\) −0.544988 −0.0948702
\(34\) −7.66551 −1.31462
\(35\) −2.43536 −0.411651
\(36\) −0.727452 −0.121242
\(37\) 1.68838 0.277568 0.138784 0.990323i \(-0.455681\pi\)
0.138784 + 0.990323i \(0.455681\pi\)
\(38\) 3.85786 0.625827
\(39\) 0 0
\(40\) 8.88207 1.40438
\(41\) 1.29725 0.202597 0.101299 0.994856i \(-0.467700\pi\)
0.101299 + 0.994856i \(0.467700\pi\)
\(42\) −0.586894 −0.0905598
\(43\) 9.09702 1.38728 0.693642 0.720320i \(-0.256005\pi\)
0.693642 + 0.720320i \(0.256005\pi\)
\(44\) 0.269129 0.0405727
\(45\) −9.20797 −1.37264
\(46\) −0.711313 −0.104877
\(47\) 7.45791 1.08785 0.543924 0.839135i \(-0.316938\pi\)
0.543924 + 0.839135i \(0.316938\pi\)
\(48\) 2.43382 0.351292
\(49\) −6.48892 −0.926989
\(50\) −9.94929 −1.40704
\(51\) −2.77331 −0.388341
\(52\) 0 0
\(53\) 7.78116 1.06882 0.534412 0.845224i \(-0.320533\pi\)
0.534412 + 0.845224i \(0.320533\pi\)
\(54\) −4.68186 −0.637121
\(55\) 3.40659 0.459344
\(56\) −1.86397 −0.249083
\(57\) 1.39574 0.184870
\(58\) −10.7545 −1.41214
\(59\) 11.1900 1.45681 0.728406 0.685146i \(-0.240262\pi\)
0.728406 + 0.685146i \(0.240262\pi\)
\(60\) −0.499651 −0.0645046
\(61\) −14.3882 −1.84222 −0.921110 0.389304i \(-0.872716\pi\)
−0.921110 + 0.389304i \(0.872716\pi\)
\(62\) 11.9466 1.51722
\(63\) 1.93236 0.243454
\(64\) 6.65326 0.831657
\(65\) 0 0
\(66\) 0.820949 0.101052
\(67\) −7.36614 −0.899917 −0.449958 0.893050i \(-0.648561\pi\)
−0.449958 + 0.893050i \(0.648561\pi\)
\(68\) 1.36953 0.166080
\(69\) −0.257346 −0.0309809
\(70\) 3.66854 0.438474
\(71\) −1.70293 −0.202101 −0.101050 0.994881i \(-0.532220\pi\)
−0.101050 + 0.994881i \(0.532220\pi\)
\(72\) −7.04756 −0.830562
\(73\) −6.44348 −0.754152 −0.377076 0.926182i \(-0.623070\pi\)
−0.377076 + 0.926182i \(0.623070\pi\)
\(74\) −2.54331 −0.295654
\(75\) −3.59956 −0.415641
\(76\) −0.689249 −0.0790623
\(77\) −0.714897 −0.0814701
\(78\) 0 0
\(79\) −5.05937 −0.569223 −0.284612 0.958643i \(-0.591865\pi\)
−0.284612 + 0.958643i \(0.591865\pi\)
\(80\) −15.2132 −1.70089
\(81\) 6.41511 0.712790
\(82\) −1.95414 −0.215798
\(83\) −13.8551 −1.52079 −0.760396 0.649460i \(-0.774995\pi\)
−0.760396 + 0.649460i \(0.774995\pi\)
\(84\) 0.104855 0.0114407
\(85\) 17.3353 1.88028
\(86\) −13.7034 −1.47768
\(87\) −3.89089 −0.417147
\(88\) 2.60732 0.277941
\(89\) 8.30538 0.880369 0.440184 0.897907i \(-0.354913\pi\)
0.440184 + 0.897907i \(0.354913\pi\)
\(90\) 13.8705 1.46208
\(91\) 0 0
\(92\) 0.127084 0.0132494
\(93\) 4.32216 0.448187
\(94\) −11.2343 −1.15873
\(95\) −8.72441 −0.895106
\(96\) −0.824302 −0.0841300
\(97\) 9.02088 0.915932 0.457966 0.888970i \(-0.348578\pi\)
0.457966 + 0.888970i \(0.348578\pi\)
\(98\) 9.77467 0.987391
\(99\) −2.70299 −0.271661
\(100\) 1.77755 0.177755
\(101\) 11.9441 1.18848 0.594240 0.804288i \(-0.297453\pi\)
0.594240 + 0.804288i \(0.297453\pi\)
\(102\) 4.17761 0.413645
\(103\) 11.4370 1.12692 0.563460 0.826144i \(-0.309470\pi\)
0.563460 + 0.826144i \(0.309470\pi\)
\(104\) 0 0
\(105\) 1.32724 0.129526
\(106\) −11.7212 −1.13847
\(107\) −15.1805 −1.46755 −0.733776 0.679392i \(-0.762244\pi\)
−0.733776 + 0.679392i \(0.762244\pi\)
\(108\) 0.836468 0.0804891
\(109\) −6.09887 −0.584166 −0.292083 0.956393i \(-0.594348\pi\)
−0.292083 + 0.956393i \(0.594348\pi\)
\(110\) −5.13156 −0.489275
\(111\) −0.920145 −0.0873363
\(112\) 3.19261 0.301673
\(113\) 13.7747 1.29582 0.647909 0.761718i \(-0.275644\pi\)
0.647909 + 0.761718i \(0.275644\pi\)
\(114\) −2.10248 −0.196916
\(115\) 1.60861 0.150004
\(116\) 1.92142 0.178399
\(117\) 0 0
\(118\) −16.8562 −1.55174
\(119\) −3.63794 −0.333489
\(120\) −4.84062 −0.441886
\(121\) 1.00000 0.0909091
\(122\) 21.6738 1.96226
\(123\) −0.706988 −0.0637469
\(124\) −2.13439 −0.191674
\(125\) 5.46705 0.488988
\(126\) −2.91083 −0.259318
\(127\) 3.52528 0.312818 0.156409 0.987692i \(-0.450008\pi\)
0.156409 + 0.987692i \(0.450008\pi\)
\(128\) −13.0472 −1.15322
\(129\) −4.95777 −0.436507
\(130\) 0 0
\(131\) 9.18273 0.802299 0.401149 0.916013i \(-0.368611\pi\)
0.401149 + 0.916013i \(0.368611\pi\)
\(132\) −0.146672 −0.0127661
\(133\) 1.83088 0.158758
\(134\) 11.0961 0.958555
\(135\) 10.5879 0.911259
\(136\) 13.2680 1.13772
\(137\) 16.9513 1.44825 0.724123 0.689670i \(-0.242245\pi\)
0.724123 + 0.689670i \(0.242245\pi\)
\(138\) 0.387657 0.0329996
\(139\) 12.0540 1.02241 0.511205 0.859459i \(-0.329199\pi\)
0.511205 + 0.859459i \(0.329199\pi\)
\(140\) −0.655426 −0.0553936
\(141\) −4.06447 −0.342290
\(142\) 2.56523 0.215270
\(143\) 0 0
\(144\) 12.0711 1.00592
\(145\) 24.3210 2.01975
\(146\) 9.70621 0.803292
\(147\) 3.53638 0.291676
\(148\) 0.454391 0.0373507
\(149\) 14.1650 1.16044 0.580219 0.814460i \(-0.302967\pi\)
0.580219 + 0.814460i \(0.302967\pi\)
\(150\) 5.42224 0.442724
\(151\) −5.00340 −0.407171 −0.203585 0.979057i \(-0.565259\pi\)
−0.203585 + 0.979057i \(0.565259\pi\)
\(152\) −6.67745 −0.541613
\(153\) −13.7548 −1.11201
\(154\) 1.07689 0.0867787
\(155\) −27.0168 −2.17004
\(156\) 0 0
\(157\) 14.7018 1.17333 0.586665 0.809829i \(-0.300440\pi\)
0.586665 + 0.809829i \(0.300440\pi\)
\(158\) 7.62124 0.606313
\(159\) −4.24064 −0.336304
\(160\) 5.15252 0.407342
\(161\) −0.337579 −0.0266049
\(162\) −9.66348 −0.759235
\(163\) −11.1352 −0.872174 −0.436087 0.899904i \(-0.643636\pi\)
−0.436087 + 0.899904i \(0.643636\pi\)
\(164\) 0.349128 0.0272623
\(165\) −1.85655 −0.144532
\(166\) 20.8708 1.61988
\(167\) −1.28527 −0.0994573 −0.0497286 0.998763i \(-0.515836\pi\)
−0.0497286 + 0.998763i \(0.515836\pi\)
\(168\) 1.01584 0.0783737
\(169\) 0 0
\(170\) −26.1133 −2.00279
\(171\) 6.92246 0.529374
\(172\) 2.44827 0.186679
\(173\) 12.4553 0.946962 0.473481 0.880804i \(-0.342997\pi\)
0.473481 + 0.880804i \(0.342997\pi\)
\(174\) 5.86109 0.444328
\(175\) −4.72179 −0.356934
\(176\) −4.46583 −0.336624
\(177\) −6.09840 −0.458384
\(178\) −12.5109 −0.937733
\(179\) 8.97864 0.671095 0.335548 0.942023i \(-0.391079\pi\)
0.335548 + 0.942023i \(0.391079\pi\)
\(180\) −2.47813 −0.184709
\(181\) 24.2769 1.80449 0.902245 0.431224i \(-0.141918\pi\)
0.902245 + 0.431224i \(0.141918\pi\)
\(182\) 0 0
\(183\) 7.84139 0.579652
\(184\) 1.23119 0.0907647
\(185\) 5.75161 0.422867
\(186\) −6.51074 −0.477391
\(187\) 5.08876 0.372127
\(188\) 2.00714 0.146385
\(189\) −2.22194 −0.161623
\(190\) 13.1421 0.953430
\(191\) 20.5574 1.48748 0.743741 0.668468i \(-0.233050\pi\)
0.743741 + 0.668468i \(0.233050\pi\)
\(192\) −3.62594 −0.261680
\(193\) −5.62205 −0.404684 −0.202342 0.979315i \(-0.564855\pi\)
−0.202342 + 0.979315i \(0.564855\pi\)
\(194\) −13.5887 −0.975614
\(195\) 0 0
\(196\) −1.74635 −0.124740
\(197\) 21.1290 1.50538 0.752691 0.658374i \(-0.228756\pi\)
0.752691 + 0.658374i \(0.228756\pi\)
\(198\) 4.07168 0.289362
\(199\) 17.2175 1.22052 0.610259 0.792202i \(-0.291065\pi\)
0.610259 + 0.792202i \(0.291065\pi\)
\(200\) 17.2209 1.21770
\(201\) 4.01445 0.283158
\(202\) −17.9921 −1.26592
\(203\) −5.10395 −0.358227
\(204\) −0.746377 −0.0522569
\(205\) 4.41921 0.308651
\(206\) −17.2282 −1.20035
\(207\) −1.27637 −0.0887136
\(208\) 0 0
\(209\) −2.56104 −0.177151
\(210\) −1.99931 −0.137965
\(211\) −1.36491 −0.0939642 −0.0469821 0.998896i \(-0.514960\pi\)
−0.0469821 + 0.998896i \(0.514960\pi\)
\(212\) 2.09413 0.143826
\(213\) 0.928077 0.0635908
\(214\) 22.8673 1.56318
\(215\) 30.9898 2.11349
\(216\) 8.10370 0.551387
\(217\) 5.66967 0.384883
\(218\) 9.18712 0.622230
\(219\) 3.51161 0.237293
\(220\) 0.916811 0.0618114
\(221\) 0 0
\(222\) 1.38607 0.0930271
\(223\) 2.73109 0.182887 0.0914436 0.995810i \(-0.470852\pi\)
0.0914436 + 0.995810i \(0.470852\pi\)
\(224\) −1.08129 −0.0722469
\(225\) −17.8528 −1.19019
\(226\) −20.7497 −1.38025
\(227\) 11.5837 0.768834 0.384417 0.923160i \(-0.374402\pi\)
0.384417 + 0.923160i \(0.374402\pi\)
\(228\) 0.375632 0.0248769
\(229\) 8.05000 0.531959 0.265979 0.963979i \(-0.414305\pi\)
0.265979 + 0.963979i \(0.414305\pi\)
\(230\) −2.42315 −0.159778
\(231\) 0.389610 0.0256345
\(232\) 18.6147 1.22212
\(233\) −14.1199 −0.925023 −0.462511 0.886613i \(-0.653052\pi\)
−0.462511 + 0.886613i \(0.653052\pi\)
\(234\) 0 0
\(235\) 25.4060 1.65731
\(236\) 3.01154 0.196035
\(237\) 2.75729 0.179105
\(238\) 5.48006 0.355219
\(239\) −14.9340 −0.965999 −0.482999 0.875621i \(-0.660453\pi\)
−0.482999 + 0.875621i \(0.660453\pi\)
\(240\) 8.29103 0.535183
\(241\) −10.1399 −0.653170 −0.326585 0.945168i \(-0.605898\pi\)
−0.326585 + 0.945168i \(0.605898\pi\)
\(242\) −1.50636 −0.0968327
\(243\) −12.8203 −0.822425
\(244\) −3.87227 −0.247897
\(245\) −22.1051 −1.41224
\(246\) 1.06498 0.0679006
\(247\) 0 0
\(248\) −20.6780 −1.31305
\(249\) 7.55084 0.478515
\(250\) −8.23536 −0.520850
\(251\) −12.0199 −0.758692 −0.379346 0.925255i \(-0.623851\pi\)
−0.379346 + 0.925255i \(0.623851\pi\)
\(252\) 0.520053 0.0327603
\(253\) 0.472206 0.0296873
\(254\) −5.31035 −0.333201
\(255\) −9.44753 −0.591627
\(256\) 6.34738 0.396711
\(257\) 10.6786 0.666111 0.333055 0.942907i \(-0.391920\pi\)
0.333055 + 0.942907i \(0.391920\pi\)
\(258\) 7.46820 0.464950
\(259\) −1.20702 −0.0750004
\(260\) 0 0
\(261\) −19.2977 −1.19450
\(262\) −13.8325 −0.854576
\(263\) −26.7639 −1.65033 −0.825167 0.564889i \(-0.808919\pi\)
−0.825167 + 0.564889i \(0.808919\pi\)
\(264\) −1.42096 −0.0874539
\(265\) 26.5072 1.62832
\(266\) −2.75797 −0.169102
\(267\) −4.52633 −0.277007
\(268\) −1.98244 −0.121097
\(269\) −30.2655 −1.84532 −0.922659 0.385616i \(-0.873989\pi\)
−0.922659 + 0.385616i \(0.873989\pi\)
\(270\) −15.9492 −0.970636
\(271\) 2.68145 0.162887 0.0814434 0.996678i \(-0.474047\pi\)
0.0814434 + 0.996678i \(0.474047\pi\)
\(272\) −22.7255 −1.37794
\(273\) 0 0
\(274\) −25.5348 −1.54261
\(275\) 6.60485 0.398287
\(276\) −0.0692593 −0.00416892
\(277\) −12.7298 −0.764858 −0.382429 0.923985i \(-0.624912\pi\)
−0.382429 + 0.923985i \(0.624912\pi\)
\(278\) −18.1578 −1.08903
\(279\) 21.4367 1.28338
\(280\) −6.34977 −0.379471
\(281\) 20.0318 1.19500 0.597499 0.801869i \(-0.296161\pi\)
0.597499 + 0.801869i \(0.296161\pi\)
\(282\) 6.12256 0.364593
\(283\) 18.9846 1.12852 0.564259 0.825598i \(-0.309162\pi\)
0.564259 + 0.825598i \(0.309162\pi\)
\(284\) −0.458308 −0.0271956
\(285\) 4.75470 0.281644
\(286\) 0 0
\(287\) −0.927404 −0.0547429
\(288\) −4.08831 −0.240906
\(289\) 8.89544 0.523261
\(290\) −36.6363 −2.15136
\(291\) −4.91627 −0.288197
\(292\) −1.73412 −0.101482
\(293\) −24.9067 −1.45507 −0.727533 0.686073i \(-0.759333\pi\)
−0.727533 + 0.686073i \(0.759333\pi\)
\(294\) −5.32708 −0.310681
\(295\) 38.1197 2.21941
\(296\) 4.40214 0.255869
\(297\) 3.10806 0.180348
\(298\) −21.3376 −1.23605
\(299\) 0 0
\(300\) −0.968745 −0.0559305
\(301\) −6.50344 −0.374852
\(302\) 7.53693 0.433702
\(303\) −6.50937 −0.373954
\(304\) 11.4372 0.655966
\(305\) −49.0146 −2.80657
\(306\) 20.7198 1.18447
\(307\) −14.1232 −0.806051 −0.403026 0.915189i \(-0.632041\pi\)
−0.403026 + 0.915189i \(0.632041\pi\)
\(308\) −0.192399 −0.0109630
\(309\) −6.23302 −0.354584
\(310\) 40.6971 2.31144
\(311\) −23.0707 −1.30822 −0.654110 0.756400i \(-0.726956\pi\)
−0.654110 + 0.756400i \(0.726956\pi\)
\(312\) 0 0
\(313\) 1.92342 0.108718 0.0543589 0.998521i \(-0.482688\pi\)
0.0543589 + 0.998521i \(0.482688\pi\)
\(314\) −22.1462 −1.24978
\(315\) 6.58275 0.370896
\(316\) −1.36162 −0.0765971
\(317\) −17.0099 −0.955372 −0.477686 0.878531i \(-0.658524\pi\)
−0.477686 + 0.878531i \(0.658524\pi\)
\(318\) 6.38793 0.358218
\(319\) 7.13941 0.399730
\(320\) 22.6649 1.26701
\(321\) 8.27317 0.461763
\(322\) 0.508516 0.0283385
\(323\) −13.0325 −0.725148
\(324\) 1.72649 0.0959161
\(325\) 0 0
\(326\) 16.7736 0.929005
\(327\) 3.32381 0.183807
\(328\) 3.38236 0.186759
\(329\) −5.33164 −0.293943
\(330\) 2.79664 0.153950
\(331\) −27.1478 −1.49218 −0.746088 0.665847i \(-0.768070\pi\)
−0.746088 + 0.665847i \(0.768070\pi\)
\(332\) −3.72880 −0.204644
\(333\) −4.56367 −0.250087
\(334\) 1.93608 0.105938
\(335\) −25.0934 −1.37100
\(336\) −1.73993 −0.0949211
\(337\) 6.64985 0.362240 0.181120 0.983461i \(-0.442028\pi\)
0.181120 + 0.983461i \(0.442028\pi\)
\(338\) 0 0
\(339\) −7.50706 −0.407727
\(340\) 4.66543 0.253018
\(341\) −7.93075 −0.429474
\(342\) −10.4277 −0.563868
\(343\) 9.64319 0.520684
\(344\) 23.7189 1.27883
\(345\) −0.876673 −0.0471985
\(346\) −18.7623 −1.00867
\(347\) 28.0264 1.50454 0.752268 0.658857i \(-0.228960\pi\)
0.752268 + 0.658857i \(0.228960\pi\)
\(348\) −1.04715 −0.0561331
\(349\) 13.7845 0.737867 0.368933 0.929456i \(-0.379723\pi\)
0.368933 + 0.929456i \(0.379723\pi\)
\(350\) 7.11272 0.380191
\(351\) 0 0
\(352\) 1.51252 0.0806173
\(353\) 2.28197 0.121457 0.0607286 0.998154i \(-0.480658\pi\)
0.0607286 + 0.998154i \(0.480658\pi\)
\(354\) 9.18640 0.488252
\(355\) −5.80119 −0.307895
\(356\) 2.23522 0.118466
\(357\) 1.98263 0.104932
\(358\) −13.5251 −0.714823
\(359\) 25.1375 1.32671 0.663354 0.748306i \(-0.269132\pi\)
0.663354 + 0.748306i \(0.269132\pi\)
\(360\) −24.0081 −1.26534
\(361\) −12.4411 −0.654793
\(362\) −36.5699 −1.92207
\(363\) −0.544988 −0.0286044
\(364\) 0 0
\(365\) −21.9503 −1.14893
\(366\) −11.8120 −0.617422
\(367\) 33.0134 1.72328 0.861641 0.507518i \(-0.169437\pi\)
0.861641 + 0.507518i \(0.169437\pi\)
\(368\) −2.10879 −0.109928
\(369\) −3.50646 −0.182539
\(370\) −8.66401 −0.450420
\(371\) −5.56273 −0.288802
\(372\) 1.16322 0.0603100
\(373\) −2.61028 −0.135155 −0.0675775 0.997714i \(-0.521527\pi\)
−0.0675775 + 0.997714i \(0.521527\pi\)
\(374\) −7.66551 −0.396374
\(375\) −2.97948 −0.153859
\(376\) 19.4452 1.00281
\(377\) 0 0
\(378\) 3.34705 0.172154
\(379\) 26.7425 1.37367 0.686835 0.726813i \(-0.258999\pi\)
0.686835 + 0.726813i \(0.258999\pi\)
\(380\) −2.34799 −0.120449
\(381\) −1.92123 −0.0984278
\(382\) −30.9669 −1.58440
\(383\) −30.9425 −1.58109 −0.790544 0.612405i \(-0.790202\pi\)
−0.790544 + 0.612405i \(0.790202\pi\)
\(384\) 7.11059 0.362861
\(385\) −2.43536 −0.124118
\(386\) 8.46885 0.431053
\(387\) −24.5892 −1.24994
\(388\) 2.42778 0.123252
\(389\) 33.8926 1.71842 0.859212 0.511620i \(-0.170954\pi\)
0.859212 + 0.511620i \(0.170954\pi\)
\(390\) 0 0
\(391\) 2.40294 0.121522
\(392\) −16.9187 −0.854523
\(393\) −5.00448 −0.252442
\(394\) −31.8280 −1.60347
\(395\) −17.2352 −0.867196
\(396\) −0.727452 −0.0365558
\(397\) −19.3612 −0.971709 −0.485855 0.874040i \(-0.661492\pi\)
−0.485855 + 0.874040i \(0.661492\pi\)
\(398\) −25.9359 −1.30005
\(399\) −0.997808 −0.0499529
\(400\) −29.4961 −1.47481
\(401\) 25.4197 1.26940 0.634700 0.772758i \(-0.281124\pi\)
0.634700 + 0.772758i \(0.281124\pi\)
\(402\) −6.04722 −0.301608
\(403\) 0 0
\(404\) 3.21449 0.159927
\(405\) 21.8536 1.08592
\(406\) 7.68839 0.381569
\(407\) 1.68838 0.0836898
\(408\) −7.23091 −0.357983
\(409\) −6.74419 −0.333479 −0.166739 0.986001i \(-0.553324\pi\)
−0.166739 + 0.986001i \(0.553324\pi\)
\(410\) −6.65694 −0.328763
\(411\) −9.23825 −0.455689
\(412\) 3.07802 0.151643
\(413\) −7.99969 −0.393639
\(414\) 1.92267 0.0944941
\(415\) −47.1985 −2.31688
\(416\) 0 0
\(417\) −6.56930 −0.321700
\(418\) 3.85786 0.188694
\(419\) 13.9379 0.680911 0.340455 0.940261i \(-0.389419\pi\)
0.340455 + 0.940261i \(0.389419\pi\)
\(420\) 0.357199 0.0174295
\(421\) −28.7165 −1.39956 −0.699778 0.714361i \(-0.746718\pi\)
−0.699778 + 0.714361i \(0.746718\pi\)
\(422\) 2.05605 0.100087
\(423\) −20.1586 −0.980147
\(424\) 20.2880 0.985271
\(425\) 33.6105 1.63035
\(426\) −1.39802 −0.0677343
\(427\) 10.2861 0.497778
\(428\) −4.08550 −0.197480
\(429\) 0 0
\(430\) −46.6819 −2.25120
\(431\) 25.7254 1.23915 0.619574 0.784938i \(-0.287305\pi\)
0.619574 + 0.784938i \(0.287305\pi\)
\(432\) −13.8801 −0.667804
\(433\) 15.2106 0.730973 0.365486 0.930817i \(-0.380903\pi\)
0.365486 + 0.930817i \(0.380903\pi\)
\(434\) −8.54059 −0.409961
\(435\) −13.2547 −0.635513
\(436\) −1.64138 −0.0786079
\(437\) −1.20934 −0.0578505
\(438\) −5.28977 −0.252755
\(439\) 5.67704 0.270951 0.135475 0.990781i \(-0.456744\pi\)
0.135475 + 0.990781i \(0.456744\pi\)
\(440\) 8.88207 0.423436
\(441\) 17.5395 0.835213
\(442\) 0 0
\(443\) 6.03014 0.286501 0.143250 0.989686i \(-0.454245\pi\)
0.143250 + 0.989686i \(0.454245\pi\)
\(444\) −0.247637 −0.0117524
\(445\) 28.2930 1.34122
\(446\) −4.11401 −0.194804
\(447\) −7.71973 −0.365131
\(448\) −4.75640 −0.224719
\(449\) −4.47151 −0.211023 −0.105512 0.994418i \(-0.533648\pi\)
−0.105512 + 0.994418i \(0.533648\pi\)
\(450\) 26.8928 1.26774
\(451\) 1.29725 0.0610853
\(452\) 3.70717 0.174371
\(453\) 2.72679 0.128116
\(454\) −17.4492 −0.818931
\(455\) 0 0
\(456\) 3.63913 0.170418
\(457\) −7.92953 −0.370928 −0.185464 0.982651i \(-0.559379\pi\)
−0.185464 + 0.982651i \(0.559379\pi\)
\(458\) −12.1262 −0.566621
\(459\) 15.8162 0.738235
\(460\) 0.432923 0.0201852
\(461\) 1.29048 0.0601035 0.0300517 0.999548i \(-0.490433\pi\)
0.0300517 + 0.999548i \(0.490433\pi\)
\(462\) −0.586894 −0.0273048
\(463\) −23.9826 −1.11457 −0.557283 0.830322i \(-0.688156\pi\)
−0.557283 + 0.830322i \(0.688156\pi\)
\(464\) −31.8834 −1.48015
\(465\) 14.7238 0.682801
\(466\) 21.2696 0.985296
\(467\) −1.89329 −0.0876112 −0.0438056 0.999040i \(-0.513948\pi\)
−0.0438056 + 0.999040i \(0.513948\pi\)
\(468\) 0 0
\(469\) 5.26603 0.243163
\(470\) −38.2707 −1.76530
\(471\) −8.01230 −0.369187
\(472\) 29.1759 1.34293
\(473\) 9.09702 0.418282
\(474\) −4.15348 −0.190776
\(475\) −16.9153 −0.776126
\(476\) −0.979074 −0.0448758
\(477\) −21.0324 −0.963006
\(478\) 22.4960 1.02894
\(479\) −10.4681 −0.478300 −0.239150 0.970983i \(-0.576869\pi\)
−0.239150 + 0.970983i \(0.576869\pi\)
\(480\) −2.80806 −0.128170
\(481\) 0 0
\(482\) 15.2744 0.695730
\(483\) 0.183976 0.00837121
\(484\) 0.269129 0.0122331
\(485\) 30.7304 1.39540
\(486\) 19.3121 0.876013
\(487\) 30.9874 1.40417 0.702087 0.712091i \(-0.252252\pi\)
0.702087 + 0.712091i \(0.252252\pi\)
\(488\) −37.5146 −1.69821
\(489\) 6.06854 0.274429
\(490\) 33.2983 1.50426
\(491\) −38.2427 −1.72587 −0.862934 0.505317i \(-0.831376\pi\)
−0.862934 + 0.505317i \(0.831376\pi\)
\(492\) −0.190271 −0.00857806
\(493\) 36.3307 1.63625
\(494\) 0 0
\(495\) −9.20797 −0.413867
\(496\) 35.4174 1.59029
\(497\) 1.21742 0.0546088
\(498\) −11.3743 −0.509695
\(499\) −28.2177 −1.26320 −0.631600 0.775295i \(-0.717601\pi\)
−0.631600 + 0.775295i \(0.717601\pi\)
\(500\) 1.47134 0.0658003
\(501\) 0.700457 0.0312941
\(502\) 18.1064 0.808128
\(503\) −22.0249 −0.982040 −0.491020 0.871148i \(-0.663376\pi\)
−0.491020 + 0.871148i \(0.663376\pi\)
\(504\) 5.03828 0.224423
\(505\) 40.6886 1.81062
\(506\) −0.711313 −0.0316217
\(507\) 0 0
\(508\) 0.948754 0.0420941
\(509\) −12.0558 −0.534364 −0.267182 0.963646i \(-0.586093\pi\)
−0.267182 + 0.963646i \(0.586093\pi\)
\(510\) 14.2314 0.630177
\(511\) 4.60642 0.203776
\(512\) 16.5330 0.730664
\(513\) −7.95987 −0.351437
\(514\) −16.0858 −0.709514
\(515\) 38.9611 1.71683
\(516\) −1.33428 −0.0587383
\(517\) 7.45791 0.327998
\(518\) 1.81821 0.0798874
\(519\) −6.78801 −0.297961
\(520\) 0 0
\(521\) 21.1905 0.928374 0.464187 0.885737i \(-0.346347\pi\)
0.464187 + 0.885737i \(0.346347\pi\)
\(522\) 29.0694 1.27233
\(523\) −40.9438 −1.79035 −0.895173 0.445718i \(-0.852948\pi\)
−0.895173 + 0.445718i \(0.852948\pi\)
\(524\) 2.47134 0.107961
\(525\) 2.57332 0.112309
\(526\) 40.3162 1.75787
\(527\) −40.3577 −1.75801
\(528\) 2.43382 0.105918
\(529\) −22.7770 −0.990305
\(530\) −39.9295 −1.73442
\(531\) −30.2464 −1.31258
\(532\) 0.492743 0.0213631
\(533\) 0 0
\(534\) 6.81830 0.295056
\(535\) −51.7136 −2.23577
\(536\) −19.2059 −0.829567
\(537\) −4.89325 −0.211159
\(538\) 45.5908 1.96556
\(539\) −6.48892 −0.279498
\(540\) 2.84950 0.122623
\(541\) −3.10501 −0.133495 −0.0667475 0.997770i \(-0.521262\pi\)
−0.0667475 + 0.997770i \(0.521262\pi\)
\(542\) −4.03924 −0.173500
\(543\) −13.2306 −0.567781
\(544\) 7.69682 0.329999
\(545\) −20.7764 −0.889961
\(546\) 0 0
\(547\) −3.19998 −0.136821 −0.0684105 0.997657i \(-0.521793\pi\)
−0.0684105 + 0.997657i \(0.521793\pi\)
\(548\) 4.56208 0.194882
\(549\) 38.8911 1.65983
\(550\) −9.94929 −0.424239
\(551\) −18.2843 −0.778938
\(552\) −0.670984 −0.0285590
\(553\) 3.61693 0.153807
\(554\) 19.1757 0.814696
\(555\) −3.13456 −0.133055
\(556\) 3.24409 0.137580
\(557\) −18.5061 −0.784130 −0.392065 0.919938i \(-0.628239\pi\)
−0.392065 + 0.919938i \(0.628239\pi\)
\(558\) −32.2915 −1.36701
\(559\) 0 0
\(560\) 10.8759 0.459591
\(561\) −2.77331 −0.117089
\(562\) −30.1752 −1.27286
\(563\) −13.3004 −0.560547 −0.280273 0.959920i \(-0.590425\pi\)
−0.280273 + 0.959920i \(0.590425\pi\)
\(564\) −1.09386 −0.0460600
\(565\) 46.9248 1.97414
\(566\) −28.5977 −1.20205
\(567\) −4.58615 −0.192600
\(568\) −4.44009 −0.186302
\(569\) −6.15820 −0.258165 −0.129083 0.991634i \(-0.541203\pi\)
−0.129083 + 0.991634i \(0.541203\pi\)
\(570\) −7.16230 −0.299996
\(571\) −27.3358 −1.14397 −0.571983 0.820265i \(-0.693826\pi\)
−0.571983 + 0.820265i \(0.693826\pi\)
\(572\) 0 0
\(573\) −11.2035 −0.468034
\(574\) 1.39701 0.0583099
\(575\) 3.11885 0.130065
\(576\) −17.9837 −0.749320
\(577\) 42.8994 1.78592 0.892962 0.450132i \(-0.148623\pi\)
0.892962 + 0.450132i \(0.148623\pi\)
\(578\) −13.3998 −0.557357
\(579\) 3.06395 0.127333
\(580\) 6.54549 0.271787
\(581\) 9.90495 0.410927
\(582\) 7.40569 0.306976
\(583\) 7.78116 0.322263
\(584\) −16.8002 −0.695197
\(585\) 0 0
\(586\) 37.5185 1.54988
\(587\) −13.5755 −0.560322 −0.280161 0.959953i \(-0.590388\pi\)
−0.280161 + 0.959953i \(0.590388\pi\)
\(588\) 0.951742 0.0392492
\(589\) 20.3110 0.836899
\(590\) −57.4220 −2.36403
\(591\) −11.5151 −0.473667
\(592\) −7.54000 −0.309892
\(593\) −2.93581 −0.120559 −0.0602796 0.998182i \(-0.519199\pi\)
−0.0602796 + 0.998182i \(0.519199\pi\)
\(594\) −4.68186 −0.192099
\(595\) −12.3930 −0.508062
\(596\) 3.81220 0.156154
\(597\) −9.38335 −0.384035
\(598\) 0 0
\(599\) 9.71523 0.396953 0.198477 0.980106i \(-0.436401\pi\)
0.198477 + 0.980106i \(0.436401\pi\)
\(600\) −9.38521 −0.383149
\(601\) 0.492629 0.0200948 0.0100474 0.999950i \(-0.496802\pi\)
0.0100474 + 0.999950i \(0.496802\pi\)
\(602\) 9.79654 0.399277
\(603\) 19.9106 0.810821
\(604\) −1.34656 −0.0547907
\(605\) 3.40659 0.138498
\(606\) 9.80548 0.398320
\(607\) −17.0589 −0.692398 −0.346199 0.938161i \(-0.612528\pi\)
−0.346199 + 0.938161i \(0.612528\pi\)
\(608\) −3.87361 −0.157096
\(609\) 2.78159 0.112716
\(610\) 73.8338 2.98944
\(611\) 0 0
\(612\) −3.70182 −0.149637
\(613\) 6.67577 0.269632 0.134816 0.990871i \(-0.456956\pi\)
0.134816 + 0.990871i \(0.456956\pi\)
\(614\) 21.2746 0.858573
\(615\) −2.40842 −0.0971167
\(616\) −1.86397 −0.0751013
\(617\) −26.3268 −1.05988 −0.529938 0.848036i \(-0.677785\pi\)
−0.529938 + 0.848036i \(0.677785\pi\)
\(618\) 9.38918 0.377688
\(619\) −32.7137 −1.31487 −0.657437 0.753510i \(-0.728359\pi\)
−0.657437 + 0.753510i \(0.728359\pi\)
\(620\) −7.27100 −0.292010
\(621\) 1.46764 0.0588945
\(622\) 34.7528 1.39346
\(623\) −5.93749 −0.237881
\(624\) 0 0
\(625\) −14.4002 −0.576010
\(626\) −2.89736 −0.115802
\(627\) 1.39574 0.0557403
\(628\) 3.95667 0.157888
\(629\) 8.59175 0.342575
\(630\) −9.91601 −0.395063
\(631\) 12.0666 0.480364 0.240182 0.970728i \(-0.422793\pi\)
0.240182 + 0.970728i \(0.422793\pi\)
\(632\) −13.1914 −0.524725
\(633\) 0.743858 0.0295657
\(634\) 25.6231 1.01762
\(635\) 12.0092 0.476570
\(636\) −1.14128 −0.0452546
\(637\) 0 0
\(638\) −10.7545 −0.425776
\(639\) 4.60300 0.182092
\(640\) −44.4466 −1.75691
\(641\) −11.7037 −0.462267 −0.231134 0.972922i \(-0.574243\pi\)
−0.231134 + 0.972922i \(0.574243\pi\)
\(642\) −12.4624 −0.491851
\(643\) −16.5792 −0.653820 −0.326910 0.945055i \(-0.606007\pi\)
−0.326910 + 0.945055i \(0.606007\pi\)
\(644\) −0.0908521 −0.00358007
\(645\) −16.8891 −0.665007
\(646\) 19.6317 0.772399
\(647\) −24.6423 −0.968789 −0.484394 0.874850i \(-0.660960\pi\)
−0.484394 + 0.874850i \(0.660960\pi\)
\(648\) 16.7262 0.657069
\(649\) 11.1900 0.439245
\(650\) 0 0
\(651\) −3.08990 −0.121103
\(652\) −2.99680 −0.117364
\(653\) −19.6532 −0.769090 −0.384545 0.923106i \(-0.625642\pi\)
−0.384545 + 0.923106i \(0.625642\pi\)
\(654\) −5.00687 −0.195784
\(655\) 31.2818 1.22228
\(656\) −5.79331 −0.226191
\(657\) 17.4166 0.679488
\(658\) 8.03138 0.313096
\(659\) 26.2883 1.02405 0.512023 0.858971i \(-0.328896\pi\)
0.512023 + 0.858971i \(0.328896\pi\)
\(660\) −0.499651 −0.0194489
\(661\) −31.9156 −1.24137 −0.620687 0.784059i \(-0.713146\pi\)
−0.620687 + 0.784059i \(0.713146\pi\)
\(662\) 40.8944 1.58941
\(663\) 0 0
\(664\) −36.1246 −1.40191
\(665\) 6.23706 0.241863
\(666\) 6.87454 0.266383
\(667\) 3.37127 0.130536
\(668\) −0.345903 −0.0133834
\(669\) −1.48841 −0.0575452
\(670\) 37.7998 1.46033
\(671\) −14.3882 −0.555450
\(672\) 0.589292 0.0227324
\(673\) 5.89178 0.227111 0.113556 0.993532i \(-0.463776\pi\)
0.113556 + 0.993532i \(0.463776\pi\)
\(674\) −10.0171 −0.385844
\(675\) 20.5282 0.790133
\(676\) 0 0
\(677\) 47.2553 1.81617 0.908084 0.418787i \(-0.137545\pi\)
0.908084 + 0.418787i \(0.137545\pi\)
\(678\) 11.3084 0.434295
\(679\) −6.44901 −0.247490
\(680\) 45.1987 1.73329
\(681\) −6.31295 −0.241913
\(682\) 11.9466 0.457459
\(683\) 50.4460 1.93026 0.965132 0.261763i \(-0.0843039\pi\)
0.965132 + 0.261763i \(0.0843039\pi\)
\(684\) 1.86303 0.0712349
\(685\) 57.7461 2.20636
\(686\) −14.5261 −0.554611
\(687\) −4.38715 −0.167380
\(688\) −40.6257 −1.54884
\(689\) 0 0
\(690\) 1.32059 0.0502739
\(691\) −17.4666 −0.664462 −0.332231 0.943198i \(-0.607801\pi\)
−0.332231 + 0.943198i \(0.607801\pi\)
\(692\) 3.35209 0.127427
\(693\) 1.93236 0.0734043
\(694\) −42.2179 −1.60257
\(695\) 41.0631 1.55761
\(696\) −10.1448 −0.384538
\(697\) 6.60141 0.250046
\(698\) −20.7644 −0.785946
\(699\) 7.69515 0.291057
\(700\) −1.27077 −0.0480305
\(701\) −32.8524 −1.24082 −0.620409 0.784278i \(-0.713034\pi\)
−0.620409 + 0.784278i \(0.713034\pi\)
\(702\) 0 0
\(703\) −4.32401 −0.163083
\(704\) 6.65326 0.250754
\(705\) −13.8460 −0.521469
\(706\) −3.43748 −0.129371
\(707\) −8.53879 −0.321134
\(708\) −1.64125 −0.0616821
\(709\) −11.9069 −0.447171 −0.223586 0.974684i \(-0.571776\pi\)
−0.223586 + 0.974684i \(0.571776\pi\)
\(710\) 8.73869 0.327957
\(711\) 13.6754 0.512868
\(712\) 21.6548 0.811547
\(713\) −3.74495 −0.140249
\(714\) −2.98656 −0.111769
\(715\) 0 0
\(716\) 2.41641 0.0903055
\(717\) 8.13883 0.303950
\(718\) −37.8663 −1.41316
\(719\) 5.95330 0.222021 0.111010 0.993819i \(-0.464591\pi\)
0.111010 + 0.993819i \(0.464591\pi\)
\(720\) 41.1212 1.53250
\(721\) −8.17627 −0.304500
\(722\) 18.7408 0.697459
\(723\) 5.52614 0.205519
\(724\) 6.53362 0.242820
\(725\) 47.1547 1.75128
\(726\) 0.820949 0.0304683
\(727\) 4.11309 0.152546 0.0762730 0.997087i \(-0.475698\pi\)
0.0762730 + 0.997087i \(0.475698\pi\)
\(728\) 0 0
\(729\) −12.2584 −0.454015
\(730\) 33.0651 1.22379
\(731\) 46.2925 1.71219
\(732\) 2.11034 0.0780005
\(733\) −47.3957 −1.75060 −0.875301 0.483579i \(-0.839336\pi\)
−0.875301 + 0.483579i \(0.839336\pi\)
\(734\) −49.7301 −1.83557
\(735\) 12.0470 0.444360
\(736\) 0.714219 0.0263264
\(737\) −7.36614 −0.271335
\(738\) 5.28201 0.194433
\(739\) 45.5262 1.67471 0.837354 0.546661i \(-0.184102\pi\)
0.837354 + 0.546661i \(0.184102\pi\)
\(740\) 1.54792 0.0569028
\(741\) 0 0
\(742\) 8.37949 0.307621
\(743\) −12.3917 −0.454606 −0.227303 0.973824i \(-0.572991\pi\)
−0.227303 + 0.973824i \(0.572991\pi\)
\(744\) 11.2693 0.413151
\(745\) 48.2542 1.76790
\(746\) 3.93202 0.143962
\(747\) 37.4501 1.37023
\(748\) 1.36953 0.0500750
\(749\) 10.8525 0.396541
\(750\) 4.48817 0.163885
\(751\) 1.04500 0.0381326 0.0190663 0.999818i \(-0.493931\pi\)
0.0190663 + 0.999818i \(0.493931\pi\)
\(752\) −33.3057 −1.21453
\(753\) 6.55072 0.238722
\(754\) 0 0
\(755\) −17.0445 −0.620314
\(756\) −0.597989 −0.0217486
\(757\) 25.6560 0.932483 0.466241 0.884658i \(-0.345608\pi\)
0.466241 + 0.884658i \(0.345608\pi\)
\(758\) −40.2839 −1.46318
\(759\) −0.257346 −0.00934108
\(760\) −22.7473 −0.825133
\(761\) −0.152493 −0.00552788 −0.00276394 0.999996i \(-0.500880\pi\)
−0.00276394 + 0.999996i \(0.500880\pi\)
\(762\) 2.89408 0.104841
\(763\) 4.36007 0.157845
\(764\) 5.53259 0.200162
\(765\) −46.8571 −1.69412
\(766\) 46.6106 1.68411
\(767\) 0 0
\(768\) −3.45924 −0.124825
\(769\) −19.7792 −0.713257 −0.356629 0.934246i \(-0.616074\pi\)
−0.356629 + 0.934246i \(0.616074\pi\)
\(770\) 3.66854 0.132205
\(771\) −5.81969 −0.209591
\(772\) −1.51305 −0.0544560
\(773\) −20.9831 −0.754708 −0.377354 0.926069i \(-0.623166\pi\)
−0.377354 + 0.926069i \(0.623166\pi\)
\(774\) 37.0402 1.33138
\(775\) −52.3814 −1.88160
\(776\) 23.5203 0.844331
\(777\) 0.657810 0.0235988
\(778\) −51.0546 −1.83039
\(779\) −3.32232 −0.119035
\(780\) 0 0
\(781\) −1.70293 −0.0609357
\(782\) −3.61970 −0.129440
\(783\) 22.1897 0.792995
\(784\) 28.9784 1.03494
\(785\) 50.0830 1.78754
\(786\) 7.53855 0.268891
\(787\) −7.93451 −0.282835 −0.141417 0.989950i \(-0.545166\pi\)
−0.141417 + 0.989950i \(0.545166\pi\)
\(788\) 5.68643 0.202571
\(789\) 14.5860 0.519276
\(790\) 25.9624 0.923702
\(791\) −9.84752 −0.350137
\(792\) −7.04756 −0.250424
\(793\) 0 0
\(794\) 29.1649 1.03503
\(795\) −14.4461 −0.512350
\(796\) 4.63373 0.164238
\(797\) −26.1000 −0.924510 −0.462255 0.886747i \(-0.652960\pi\)
−0.462255 + 0.886747i \(0.652960\pi\)
\(798\) 1.50306 0.0532078
\(799\) 37.9515 1.34263
\(800\) 9.98993 0.353197
\(801\) −22.4493 −0.793209
\(802\) −38.2913 −1.35211
\(803\) −6.44348 −0.227385
\(804\) 1.08040 0.0381029
\(805\) −1.14999 −0.0405319
\(806\) 0 0
\(807\) 16.4943 0.580627
\(808\) 31.1420 1.09557
\(809\) 54.6289 1.92065 0.960325 0.278884i \(-0.0899644\pi\)
0.960325 + 0.278884i \(0.0899644\pi\)
\(810\) −32.9195 −1.15667
\(811\) 28.1892 0.989858 0.494929 0.868934i \(-0.335194\pi\)
0.494929 + 0.868934i \(0.335194\pi\)
\(812\) −1.37362 −0.0482045
\(813\) −1.46136 −0.0512521
\(814\) −2.54331 −0.0891430
\(815\) −37.9330 −1.32873
\(816\) 12.3851 0.433566
\(817\) −23.2979 −0.815089
\(818\) 10.1592 0.355208
\(819\) 0 0
\(820\) 1.18934 0.0415334
\(821\) 9.16089 0.319717 0.159859 0.987140i \(-0.448896\pi\)
0.159859 + 0.987140i \(0.448896\pi\)
\(822\) 13.9162 0.485382
\(823\) −30.9586 −1.07915 −0.539575 0.841938i \(-0.681415\pi\)
−0.539575 + 0.841938i \(0.681415\pi\)
\(824\) 29.8199 1.03882
\(825\) −3.59956 −0.125321
\(826\) 12.0504 0.419288
\(827\) −23.2966 −0.810102 −0.405051 0.914294i \(-0.632746\pi\)
−0.405051 + 0.914294i \(0.632746\pi\)
\(828\) −0.343507 −0.0119377
\(829\) 30.1486 1.04710 0.523552 0.851994i \(-0.324607\pi\)
0.523552 + 0.851994i \(0.324607\pi\)
\(830\) 71.0981 2.46785
\(831\) 6.93757 0.240662
\(832\) 0 0
\(833\) −33.0205 −1.14409
\(834\) 9.89575 0.342662
\(835\) −4.37839 −0.151520
\(836\) −0.689249 −0.0238382
\(837\) −24.6492 −0.852002
\(838\) −20.9955 −0.725278
\(839\) 42.8447 1.47916 0.739581 0.673067i \(-0.235024\pi\)
0.739581 + 0.673067i \(0.235024\pi\)
\(840\) 3.46054 0.119400
\(841\) 21.9712 0.757627
\(842\) 43.2574 1.49075
\(843\) −10.9171 −0.376005
\(844\) −0.367336 −0.0126442
\(845\) 0 0
\(846\) 30.3662 1.04401
\(847\) −0.714897 −0.0245642
\(848\) −34.7493 −1.19330
\(849\) −10.3464 −0.355087
\(850\) −50.6295 −1.73658
\(851\) 0.797262 0.0273298
\(852\) 0.249772 0.00855705
\(853\) −27.6138 −0.945478 −0.472739 0.881203i \(-0.656735\pi\)
−0.472739 + 0.881203i \(0.656735\pi\)
\(854\) −15.4946 −0.530213
\(855\) 23.5820 0.806487
\(856\) −39.5803 −1.35283
\(857\) 19.6565 0.671453 0.335726 0.941960i \(-0.391018\pi\)
0.335726 + 0.941960i \(0.391018\pi\)
\(858\) 0 0
\(859\) −42.1151 −1.43695 −0.718474 0.695553i \(-0.755159\pi\)
−0.718474 + 0.695553i \(0.755159\pi\)
\(860\) 8.34025 0.284400
\(861\) 0.505424 0.0172248
\(862\) −38.7518 −1.31989
\(863\) 37.7215 1.28405 0.642027 0.766682i \(-0.278094\pi\)
0.642027 + 0.766682i \(0.278094\pi\)
\(864\) 4.70099 0.159931
\(865\) 42.4302 1.44267
\(866\) −22.9126 −0.778602
\(867\) −4.84791 −0.164644
\(868\) 1.52587 0.0517915
\(869\) −5.05937 −0.171627
\(870\) 19.9663 0.676922
\(871\) 0 0
\(872\) −15.9017 −0.538500
\(873\) −24.3833 −0.825251
\(874\) 1.82170 0.0616200
\(875\) −3.90838 −0.132127
\(876\) 0.945076 0.0319312
\(877\) 7.22842 0.244086 0.122043 0.992525i \(-0.461055\pi\)
0.122043 + 0.992525i \(0.461055\pi\)
\(878\) −8.55169 −0.288605
\(879\) 13.5738 0.457834
\(880\) −15.2132 −0.512838
\(881\) 12.4762 0.420333 0.210167 0.977666i \(-0.432599\pi\)
0.210167 + 0.977666i \(0.432599\pi\)
\(882\) −26.4208 −0.889635
\(883\) 8.37451 0.281825 0.140912 0.990022i \(-0.454996\pi\)
0.140912 + 0.990022i \(0.454996\pi\)
\(884\) 0 0
\(885\) −20.7747 −0.698335
\(886\) −9.08358 −0.305169
\(887\) −36.3121 −1.21924 −0.609620 0.792694i \(-0.708678\pi\)
−0.609620 + 0.792694i \(0.708678\pi\)
\(888\) −2.39911 −0.0805090
\(889\) −2.52021 −0.0845252
\(890\) −42.6195 −1.42861
\(891\) 6.41511 0.214914
\(892\) 0.735014 0.0246101
\(893\) −19.1000 −0.639157
\(894\) 11.6287 0.388922
\(895\) 30.5865 1.02240
\(896\) 9.32744 0.311608
\(897\) 0 0
\(898\) 6.73571 0.224774
\(899\) −56.6209 −1.88841
\(900\) −4.80471 −0.160157
\(901\) 39.5964 1.31915
\(902\) −1.95414 −0.0650656
\(903\) 3.54429 0.117947
\(904\) 35.9151 1.19452
\(905\) 82.7015 2.74909
\(906\) −4.10754 −0.136464
\(907\) −30.5852 −1.01557 −0.507783 0.861485i \(-0.669535\pi\)
−0.507783 + 0.861485i \(0.669535\pi\)
\(908\) 3.11749 0.103458
\(909\) −32.2847 −1.07082
\(910\) 0 0
\(911\) 14.5722 0.482798 0.241399 0.970426i \(-0.422394\pi\)
0.241399 + 0.970426i \(0.422394\pi\)
\(912\) −6.23311 −0.206399
\(913\) −13.8551 −0.458536
\(914\) 11.9447 0.395097
\(915\) 26.7124 0.883084
\(916\) 2.16648 0.0715827
\(917\) −6.56471 −0.216786
\(918\) −23.8249 −0.786338
\(919\) −53.0844 −1.75109 −0.875547 0.483133i \(-0.839499\pi\)
−0.875547 + 0.483133i \(0.839499\pi\)
\(920\) 4.19416 0.138277
\(921\) 7.69695 0.253623
\(922\) −1.94393 −0.0640198
\(923\) 0 0
\(924\) 0.104855 0.00344949
\(925\) 11.1515 0.366658
\(926\) 36.1265 1.18719
\(927\) −30.9140 −1.01535
\(928\) 10.7985 0.354477
\(929\) 32.8684 1.07838 0.539189 0.842185i \(-0.318731\pi\)
0.539189 + 0.842185i \(0.318731\pi\)
\(930\) −22.1794 −0.727292
\(931\) 16.6184 0.544646
\(932\) −3.80006 −0.124475
\(933\) 12.5732 0.411630
\(934\) 2.85199 0.0933199
\(935\) 17.3353 0.566925
\(936\) 0 0
\(937\) −43.3227 −1.41529 −0.707646 0.706567i \(-0.750243\pi\)
−0.707646 + 0.706567i \(0.750243\pi\)
\(938\) −7.93255 −0.259007
\(939\) −1.04824 −0.0342079
\(940\) 6.83749 0.223014
\(941\) −39.7733 −1.29657 −0.648287 0.761396i \(-0.724514\pi\)
−0.648287 + 0.761396i \(0.724514\pi\)
\(942\) 12.0694 0.393243
\(943\) 0.612571 0.0199481
\(944\) −49.9725 −1.62647
\(945\) −7.56925 −0.246228
\(946\) −13.7034 −0.445537
\(947\) −22.1817 −0.720808 −0.360404 0.932796i \(-0.617361\pi\)
−0.360404 + 0.932796i \(0.617361\pi\)
\(948\) 0.742066 0.0241012
\(949\) 0 0
\(950\) 25.4805 0.826698
\(951\) 9.27020 0.300607
\(952\) −9.48527 −0.307419
\(953\) 47.1622 1.52773 0.763866 0.645375i \(-0.223299\pi\)
0.763866 + 0.645375i \(0.223299\pi\)
\(954\) 31.6824 1.02576
\(955\) 70.0306 2.26614
\(956\) −4.01916 −0.129989
\(957\) −3.89089 −0.125775
\(958\) 15.7688 0.509465
\(959\) −12.1184 −0.391325
\(960\) −12.3521 −0.398662
\(961\) 31.8968 1.02893
\(962\) 0 0
\(963\) 41.0326 1.32226
\(964\) −2.72895 −0.0878934
\(965\) −19.1520 −0.616525
\(966\) −0.277135 −0.00891667
\(967\) −8.94456 −0.287638 −0.143819 0.989604i \(-0.545938\pi\)
−0.143819 + 0.989604i \(0.545938\pi\)
\(968\) 2.60732 0.0838024
\(969\) 7.10256 0.228167
\(970\) −46.2912 −1.48632
\(971\) −24.8292 −0.796806 −0.398403 0.917210i \(-0.630436\pi\)
−0.398403 + 0.917210i \(0.630436\pi\)
\(972\) −3.45032 −0.110669
\(973\) −8.61740 −0.276261
\(974\) −46.6783 −1.49567
\(975\) 0 0
\(976\) 64.2552 2.05676
\(977\) −9.23253 −0.295375 −0.147687 0.989034i \(-0.547183\pi\)
−0.147687 + 0.989034i \(0.547183\pi\)
\(978\) −9.14142 −0.292310
\(979\) 8.30538 0.265441
\(980\) −5.94911 −0.190037
\(981\) 16.4852 0.526331
\(982\) 57.6073 1.83832
\(983\) −56.3251 −1.79649 −0.898246 0.439494i \(-0.855158\pi\)
−0.898246 + 0.439494i \(0.855158\pi\)
\(984\) −1.84334 −0.0587636
\(985\) 71.9779 2.29341
\(986\) −54.7272 −1.74287
\(987\) 2.90568 0.0924887
\(988\) 0 0
\(989\) 4.29567 0.136594
\(990\) 13.8705 0.440835
\(991\) −17.4706 −0.554972 −0.277486 0.960730i \(-0.589501\pi\)
−0.277486 + 0.960730i \(0.589501\pi\)
\(992\) −11.9954 −0.380854
\(993\) 14.7952 0.469511
\(994\) −1.83388 −0.0581671
\(995\) 58.6531 1.85943
\(996\) 2.03215 0.0643911
\(997\) −10.5546 −0.334268 −0.167134 0.985934i \(-0.553451\pi\)
−0.167134 + 0.985934i \(0.553451\pi\)
\(998\) 42.5061 1.34551
\(999\) 5.24758 0.166026
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 1859.2.a.t.1.7 yes 21
13.12 even 2 1859.2.a.s.1.15 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1859.2.a.s.1.15 21 13.12 even 2
1859.2.a.t.1.7 yes 21 1.1 even 1 trivial