Properties

Label 342.4.a.h.1.2
Level $342$
Weight $4$
Character 342.1
Self dual yes
Analytic conductor $20.179$
Analytic rank $0$
Dimension $2$
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] = [342,4,Mod(1,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 342.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: \(20.1786532220\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{73}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.77200\) of defining polynomial
Character \(\chi\) \(=\) 342.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} +17.3160 q^{5} -26.0880 q^{7} -8.00000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} +17.3160 q^{5} -26.0880 q^{7} -8.00000 q^{8} -34.6320 q^{10} +4.22800 q^{11} +64.0360 q^{13} +52.1760 q^{14} +16.0000 q^{16} +48.5440 q^{17} +19.0000 q^{19} +69.2640 q^{20} -8.45600 q^{22} -92.0360 q^{23} +174.844 q^{25} -128.072 q^{26} -104.352 q^{28} +88.2120 q^{29} -81.9681 q^{31} -32.0000 q^{32} -97.0880 q^{34} -451.740 q^{35} -23.6161 q^{37} -38.0000 q^{38} -138.528 q^{40} -17.7200 q^{41} +368.404 q^{43} +16.9120 q^{44} +184.072 q^{46} +497.812 q^{47} +337.584 q^{49} -349.688 q^{50} +256.144 q^{52} +536.876 q^{53} +73.2120 q^{55} +208.704 q^{56} -176.424 q^{58} +36.7000 q^{59} +630.692 q^{61} +163.936 q^{62} +64.0000 q^{64} +1108.85 q^{65} +282.556 q^{67} +194.176 q^{68} +903.480 q^{70} -595.552 q^{71} -597.048 q^{73} +47.2321 q^{74} +76.0000 q^{76} -110.300 q^{77} +427.224 q^{79} +277.056 q^{80} +35.4400 q^{82} -493.768 q^{83} +840.588 q^{85} -736.808 q^{86} -33.8240 q^{88} +921.136 q^{89} -1670.57 q^{91} -368.144 q^{92} -995.624 q^{94} +329.004 q^{95} +1082.74 q^{97} -675.168 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 8 q^{4} + 9 q^{5} - 18 q^{7} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 8 q^{4} + 9 q^{5} - 18 q^{7} - 16 q^{8} - 18 q^{10} + 17 q^{11} + 17 q^{13} + 36 q^{14} + 32 q^{16} + 80 q^{17} + 38 q^{19} + 36 q^{20} - 34 q^{22} - 73 q^{23} + 119 q^{25} - 34 q^{26} - 72 q^{28} - 3 q^{29} + 212 q^{31} - 64 q^{32} - 160 q^{34} - 519 q^{35} + 192 q^{37} - 76 q^{38} - 72 q^{40} + 50 q^{41} + 677 q^{43} + 68 q^{44} + 146 q^{46} + 389 q^{47} + 60 q^{49} - 238 q^{50} + 68 q^{52} + 1219 q^{53} - 33 q^{55} + 144 q^{56} + 6 q^{58} + 287 q^{59} + 313 q^{61} - 424 q^{62} + 128 q^{64} + 1500 q^{65} + 1223 q^{67} + 320 q^{68} + 1038 q^{70} - 200 q^{71} + 378 q^{73} - 384 q^{74} + 152 q^{76} - 7 q^{77} + 1350 q^{79} + 144 q^{80} - 100 q^{82} + 670 q^{83} + 579 q^{85} - 1354 q^{86} - 136 q^{88} + 236 q^{89} - 2051 q^{91} - 292 q^{92} - 778 q^{94} + 171 q^{95} + 1294 q^{97} - 120 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 17.3160 1.54879 0.774395 0.632702i \(-0.218054\pi\)
0.774395 + 0.632702i \(0.218054\pi\)
\(6\) 0 0
\(7\) −26.0880 −1.40862 −0.704310 0.709893i \(-0.748743\pi\)
−0.704310 + 0.709893i \(0.748743\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) −34.6320 −1.09516
\(11\) 4.22800 0.115890 0.0579450 0.998320i \(-0.481545\pi\)
0.0579450 + 0.998320i \(0.481545\pi\)
\(12\) 0 0
\(13\) 64.0360 1.36618 0.683092 0.730332i \(-0.260635\pi\)
0.683092 + 0.730332i \(0.260635\pi\)
\(14\) 52.1760 0.996045
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 48.5440 0.692568 0.346284 0.938130i \(-0.387443\pi\)
0.346284 + 0.938130i \(0.387443\pi\)
\(18\) 0 0
\(19\) 19.0000 0.229416
\(20\) 69.2640 0.774395
\(21\) 0 0
\(22\) −8.45600 −0.0819466
\(23\) −92.0360 −0.834384 −0.417192 0.908818i \(-0.636986\pi\)
−0.417192 + 0.908818i \(0.636986\pi\)
\(24\) 0 0
\(25\) 174.844 1.39875
\(26\) −128.072 −0.966038
\(27\) 0 0
\(28\) −104.352 −0.704310
\(29\) 88.2120 0.564847 0.282424 0.959290i \(-0.408862\pi\)
0.282424 + 0.959290i \(0.408862\pi\)
\(30\) 0 0
\(31\) −81.9681 −0.474900 −0.237450 0.971400i \(-0.576312\pi\)
−0.237450 + 0.971400i \(0.576312\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) −97.0880 −0.489719
\(35\) −451.740 −2.18166
\(36\) 0 0
\(37\) −23.6161 −0.104931 −0.0524656 0.998623i \(-0.516708\pi\)
−0.0524656 + 0.998623i \(0.516708\pi\)
\(38\) −38.0000 −0.162221
\(39\) 0 0
\(40\) −138.528 −0.547580
\(41\) −17.7200 −0.0674976 −0.0337488 0.999430i \(-0.510745\pi\)
−0.0337488 + 0.999430i \(0.510745\pi\)
\(42\) 0 0
\(43\) 368.404 1.30654 0.653268 0.757126i \(-0.273397\pi\)
0.653268 + 0.757126i \(0.273397\pi\)
\(44\) 16.9120 0.0579450
\(45\) 0 0
\(46\) 184.072 0.589999
\(47\) 497.812 1.54497 0.772483 0.635036i \(-0.219015\pi\)
0.772483 + 0.635036i \(0.219015\pi\)
\(48\) 0 0
\(49\) 337.584 0.984210
\(50\) −349.688 −0.989067
\(51\) 0 0
\(52\) 256.144 0.683092
\(53\) 536.876 1.39143 0.695713 0.718320i \(-0.255089\pi\)
0.695713 + 0.718320i \(0.255089\pi\)
\(54\) 0 0
\(55\) 73.2120 0.179489
\(56\) 208.704 0.498022
\(57\) 0 0
\(58\) −176.424 −0.399407
\(59\) 36.7000 0.0809818 0.0404909 0.999180i \(-0.487108\pi\)
0.0404909 + 0.999180i \(0.487108\pi\)
\(60\) 0 0
\(61\) 630.692 1.32380 0.661901 0.749592i \(-0.269750\pi\)
0.661901 + 0.749592i \(0.269750\pi\)
\(62\) 163.936 0.335805
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 1108.85 2.11593
\(66\) 0 0
\(67\) 282.556 0.515219 0.257610 0.966249i \(-0.417065\pi\)
0.257610 + 0.966249i \(0.417065\pi\)
\(68\) 194.176 0.346284
\(69\) 0 0
\(70\) 903.480 1.54266
\(71\) −595.552 −0.995480 −0.497740 0.867326i \(-0.665837\pi\)
−0.497740 + 0.867326i \(0.665837\pi\)
\(72\) 0 0
\(73\) −597.048 −0.957250 −0.478625 0.878020i \(-0.658865\pi\)
−0.478625 + 0.878020i \(0.658865\pi\)
\(74\) 47.2321 0.0741976
\(75\) 0 0
\(76\) 76.0000 0.114708
\(77\) −110.300 −0.163245
\(78\) 0 0
\(79\) 427.224 0.608436 0.304218 0.952602i \(-0.401605\pi\)
0.304218 + 0.952602i \(0.401605\pi\)
\(80\) 277.056 0.387198
\(81\) 0 0
\(82\) 35.4400 0.0477280
\(83\) −493.768 −0.652989 −0.326495 0.945199i \(-0.605868\pi\)
−0.326495 + 0.945199i \(0.605868\pi\)
\(84\) 0 0
\(85\) 840.588 1.07264
\(86\) −736.808 −0.923861
\(87\) 0 0
\(88\) −33.8240 −0.0409733
\(89\) 921.136 1.09708 0.548541 0.836124i \(-0.315184\pi\)
0.548541 + 0.836124i \(0.315184\pi\)
\(90\) 0 0
\(91\) −1670.57 −1.92443
\(92\) −368.144 −0.417192
\(93\) 0 0
\(94\) −995.624 −1.09246
\(95\) 329.004 0.355317
\(96\) 0 0
\(97\) 1082.74 1.13336 0.566680 0.823938i \(-0.308227\pi\)
0.566680 + 0.823938i \(0.308227\pi\)
\(98\) −675.168 −0.695942
\(99\) 0 0
\(100\) 699.376 0.699376
\(101\) 712.448 0.701893 0.350947 0.936395i \(-0.385860\pi\)
0.350947 + 0.936395i \(0.385860\pi\)
\(102\) 0 0
\(103\) −26.4797 −0.0253313 −0.0126656 0.999920i \(-0.504032\pi\)
−0.0126656 + 0.999920i \(0.504032\pi\)
\(104\) −512.288 −0.483019
\(105\) 0 0
\(106\) −1073.75 −0.983887
\(107\) 740.996 0.669484 0.334742 0.942310i \(-0.391351\pi\)
0.334742 + 0.942310i \(0.391351\pi\)
\(108\) 0 0
\(109\) −1983.08 −1.74261 −0.871304 0.490744i \(-0.836725\pi\)
−0.871304 + 0.490744i \(0.836725\pi\)
\(110\) −146.424 −0.126918
\(111\) 0 0
\(112\) −417.408 −0.352155
\(113\) 718.720 0.598332 0.299166 0.954201i \(-0.403292\pi\)
0.299166 + 0.954201i \(0.403292\pi\)
\(114\) 0 0
\(115\) −1593.70 −1.29229
\(116\) 352.848 0.282424
\(117\) 0 0
\(118\) −73.3999 −0.0572628
\(119\) −1266.42 −0.975565
\(120\) 0 0
\(121\) −1313.12 −0.986570
\(122\) −1261.38 −0.936069
\(123\) 0 0
\(124\) −327.872 −0.237450
\(125\) 863.100 0.617584
\(126\) 0 0
\(127\) 2610.72 1.82413 0.912063 0.410050i \(-0.134489\pi\)
0.912063 + 0.410050i \(0.134489\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) −2217.70 −1.49619
\(131\) 1216.69 0.811472 0.405736 0.913990i \(-0.367015\pi\)
0.405736 + 0.913990i \(0.367015\pi\)
\(132\) 0 0
\(133\) −495.672 −0.323160
\(134\) −565.112 −0.364315
\(135\) 0 0
\(136\) −388.352 −0.244860
\(137\) −1170.67 −0.730053 −0.365026 0.930997i \(-0.618940\pi\)
−0.365026 + 0.930997i \(0.618940\pi\)
\(138\) 0 0
\(139\) −271.083 −0.165417 −0.0827086 0.996574i \(-0.526357\pi\)
−0.0827086 + 0.996574i \(0.526357\pi\)
\(140\) −1806.96 −1.09083
\(141\) 0 0
\(142\) 1191.10 0.703910
\(143\) 270.744 0.158327
\(144\) 0 0
\(145\) 1527.48 0.874830
\(146\) 1194.10 0.676878
\(147\) 0 0
\(148\) −94.4642 −0.0524656
\(149\) −1841.19 −1.01232 −0.506161 0.862439i \(-0.668936\pi\)
−0.506161 + 0.862439i \(0.668936\pi\)
\(150\) 0 0
\(151\) 3322.32 1.79051 0.895254 0.445557i \(-0.146994\pi\)
0.895254 + 0.445557i \(0.146994\pi\)
\(152\) −152.000 −0.0811107
\(153\) 0 0
\(154\) 220.600 0.115432
\(155\) −1419.36 −0.735521
\(156\) 0 0
\(157\) 243.616 0.123839 0.0619194 0.998081i \(-0.480278\pi\)
0.0619194 + 0.998081i \(0.480278\pi\)
\(158\) −854.448 −0.430229
\(159\) 0 0
\(160\) −554.112 −0.273790
\(161\) 2401.04 1.17533
\(162\) 0 0
\(163\) −2598.11 −1.24847 −0.624233 0.781238i \(-0.714588\pi\)
−0.624233 + 0.781238i \(0.714588\pi\)
\(164\) −70.8801 −0.0337488
\(165\) 0 0
\(166\) 987.537 0.461733
\(167\) 491.064 0.227543 0.113772 0.993507i \(-0.463707\pi\)
0.113772 + 0.993507i \(0.463707\pi\)
\(168\) 0 0
\(169\) 1903.61 0.866460
\(170\) −1681.18 −0.758473
\(171\) 0 0
\(172\) 1473.62 0.653268
\(173\) −1648.56 −0.724496 −0.362248 0.932082i \(-0.617991\pi\)
−0.362248 + 0.932082i \(0.617991\pi\)
\(174\) 0 0
\(175\) −4561.33 −1.97031
\(176\) 67.6480 0.0289725
\(177\) 0 0
\(178\) −1842.27 −0.775754
\(179\) −2326.81 −0.971586 −0.485793 0.874074i \(-0.661469\pi\)
−0.485793 + 0.874074i \(0.661469\pi\)
\(180\) 0 0
\(181\) −4637.46 −1.90442 −0.952208 0.305449i \(-0.901193\pi\)
−0.952208 + 0.305449i \(0.901193\pi\)
\(182\) 3341.14 1.36078
\(183\) 0 0
\(184\) 736.288 0.294999
\(185\) −408.936 −0.162516
\(186\) 0 0
\(187\) 205.244 0.0802616
\(188\) 1991.25 0.772483
\(189\) 0 0
\(190\) −658.008 −0.251247
\(191\) −5260.38 −1.99281 −0.996407 0.0846903i \(-0.973010\pi\)
−0.996407 + 0.0846903i \(0.973010\pi\)
\(192\) 0 0
\(193\) 16.1833 0.00603575 0.00301787 0.999995i \(-0.499039\pi\)
0.00301787 + 0.999995i \(0.499039\pi\)
\(194\) −2165.49 −0.801407
\(195\) 0 0
\(196\) 1350.34 0.492105
\(197\) −3784.71 −1.36878 −0.684390 0.729116i \(-0.739931\pi\)
−0.684390 + 0.729116i \(0.739931\pi\)
\(198\) 0 0
\(199\) 73.2079 0.0260783 0.0130391 0.999915i \(-0.495849\pi\)
0.0130391 + 0.999915i \(0.495849\pi\)
\(200\) −1398.75 −0.494534
\(201\) 0 0
\(202\) −1424.90 −0.496313
\(203\) −2301.28 −0.795655
\(204\) 0 0
\(205\) −306.840 −0.104540
\(206\) 52.9594 0.0179119
\(207\) 0 0
\(208\) 1024.58 0.341546
\(209\) 80.3320 0.0265870
\(210\) 0 0
\(211\) −2945.44 −0.961006 −0.480503 0.876993i \(-0.659546\pi\)
−0.480503 + 0.876993i \(0.659546\pi\)
\(212\) 2147.50 0.695713
\(213\) 0 0
\(214\) −1481.99 −0.473397
\(215\) 6379.29 2.02355
\(216\) 0 0
\(217\) 2138.38 0.668954
\(218\) 3966.15 1.23221
\(219\) 0 0
\(220\) 292.848 0.0897446
\(221\) 3108.57 0.946175
\(222\) 0 0
\(223\) 3125.30 0.938499 0.469250 0.883066i \(-0.344524\pi\)
0.469250 + 0.883066i \(0.344524\pi\)
\(224\) 834.816 0.249011
\(225\) 0 0
\(226\) −1437.44 −0.423085
\(227\) 3577.80 1.04611 0.523055 0.852299i \(-0.324792\pi\)
0.523055 + 0.852299i \(0.324792\pi\)
\(228\) 0 0
\(229\) −4802.00 −1.38570 −0.692850 0.721082i \(-0.743645\pi\)
−0.692850 + 0.721082i \(0.743645\pi\)
\(230\) 3187.39 0.913785
\(231\) 0 0
\(232\) −705.696 −0.199704
\(233\) −5829.49 −1.63907 −0.819534 0.573031i \(-0.805768\pi\)
−0.819534 + 0.573031i \(0.805768\pi\)
\(234\) 0 0
\(235\) 8620.12 2.39283
\(236\) 146.800 0.0404909
\(237\) 0 0
\(238\) 2532.83 0.689828
\(239\) −1364.33 −0.369251 −0.184625 0.982809i \(-0.559107\pi\)
−0.184625 + 0.982809i \(0.559107\pi\)
\(240\) 0 0
\(241\) −2647.22 −0.707563 −0.353782 0.935328i \(-0.615104\pi\)
−0.353782 + 0.935328i \(0.615104\pi\)
\(242\) 2626.25 0.697610
\(243\) 0 0
\(244\) 2522.77 0.661901
\(245\) 5845.61 1.52434
\(246\) 0 0
\(247\) 1216.68 0.313424
\(248\) 655.745 0.167903
\(249\) 0 0
\(250\) −1726.20 −0.436698
\(251\) −1970.73 −0.495582 −0.247791 0.968814i \(-0.579705\pi\)
−0.247791 + 0.968814i \(0.579705\pi\)
\(252\) 0 0
\(253\) −389.128 −0.0966967
\(254\) −5221.44 −1.28985
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 7915.82 1.92131 0.960653 0.277752i \(-0.0895892\pi\)
0.960653 + 0.277752i \(0.0895892\pi\)
\(258\) 0 0
\(259\) 616.096 0.147808
\(260\) 4435.39 1.05797
\(261\) 0 0
\(262\) −2433.38 −0.573798
\(263\) −3287.96 −0.770892 −0.385446 0.922730i \(-0.625952\pi\)
−0.385446 + 0.922730i \(0.625952\pi\)
\(264\) 0 0
\(265\) 9296.55 2.15503
\(266\) 991.344 0.228508
\(267\) 0 0
\(268\) 1130.22 0.257610
\(269\) 4749.61 1.07654 0.538269 0.842773i \(-0.319078\pi\)
0.538269 + 0.842773i \(0.319078\pi\)
\(270\) 0 0
\(271\) 242.661 0.0543933 0.0271967 0.999630i \(-0.491342\pi\)
0.0271967 + 0.999630i \(0.491342\pi\)
\(272\) 776.704 0.173142
\(273\) 0 0
\(274\) 2341.34 0.516225
\(275\) 739.240 0.162101
\(276\) 0 0
\(277\) −4131.13 −0.896086 −0.448043 0.894012i \(-0.647879\pi\)
−0.448043 + 0.894012i \(0.647879\pi\)
\(278\) 542.167 0.116968
\(279\) 0 0
\(280\) 3613.92 0.771332
\(281\) −1007.19 −0.213822 −0.106911 0.994269i \(-0.534096\pi\)
−0.106911 + 0.994269i \(0.534096\pi\)
\(282\) 0 0
\(283\) 2333.63 0.490176 0.245088 0.969501i \(-0.421183\pi\)
0.245088 + 0.969501i \(0.421183\pi\)
\(284\) −2382.21 −0.497740
\(285\) 0 0
\(286\) −541.488 −0.111954
\(287\) 462.280 0.0950785
\(288\) 0 0
\(289\) −2556.48 −0.520350
\(290\) −3054.96 −0.618598
\(291\) 0 0
\(292\) −2388.19 −0.478625
\(293\) 1588.68 0.316763 0.158381 0.987378i \(-0.449372\pi\)
0.158381 + 0.987378i \(0.449372\pi\)
\(294\) 0 0
\(295\) 635.497 0.125424
\(296\) 188.928 0.0370988
\(297\) 0 0
\(298\) 3682.38 0.715820
\(299\) −5893.62 −1.13992
\(300\) 0 0
\(301\) −9610.93 −1.84041
\(302\) −6644.64 −1.26608
\(303\) 0 0
\(304\) 304.000 0.0573539
\(305\) 10921.1 2.05029
\(306\) 0 0
\(307\) 4057.46 0.754304 0.377152 0.926151i \(-0.376903\pi\)
0.377152 + 0.926151i \(0.376903\pi\)
\(308\) −441.200 −0.0816224
\(309\) 0 0
\(310\) 2838.72 0.520092
\(311\) −2871.92 −0.523638 −0.261819 0.965117i \(-0.584322\pi\)
−0.261819 + 0.965117i \(0.584322\pi\)
\(312\) 0 0
\(313\) 4322.67 0.780612 0.390306 0.920685i \(-0.372369\pi\)
0.390306 + 0.920685i \(0.372369\pi\)
\(314\) −487.232 −0.0875672
\(315\) 0 0
\(316\) 1708.90 0.304218
\(317\) −2513.56 −0.445349 −0.222674 0.974893i \(-0.571479\pi\)
−0.222674 + 0.974893i \(0.571479\pi\)
\(318\) 0 0
\(319\) 372.960 0.0654601
\(320\) 1108.22 0.193599
\(321\) 0 0
\(322\) −4802.07 −0.831084
\(323\) 922.336 0.158886
\(324\) 0 0
\(325\) 11196.3 1.91095
\(326\) 5196.22 0.882798
\(327\) 0 0
\(328\) 141.760 0.0238640
\(329\) −12986.9 −2.17627
\(330\) 0 0
\(331\) −4573.78 −0.759509 −0.379754 0.925087i \(-0.623992\pi\)
−0.379754 + 0.925087i \(0.623992\pi\)
\(332\) −1975.07 −0.326495
\(333\) 0 0
\(334\) −982.129 −0.160897
\(335\) 4892.74 0.797967
\(336\) 0 0
\(337\) 9001.71 1.45506 0.727529 0.686077i \(-0.240669\pi\)
0.727529 + 0.686077i \(0.240669\pi\)
\(338\) −3807.22 −0.612680
\(339\) 0 0
\(340\) 3362.35 0.536321
\(341\) −346.561 −0.0550361
\(342\) 0 0
\(343\) 141.289 0.0222417
\(344\) −2947.23 −0.461931
\(345\) 0 0
\(346\) 3297.12 0.512296
\(347\) −9358.68 −1.44784 −0.723920 0.689884i \(-0.757661\pi\)
−0.723920 + 0.689884i \(0.757661\pi\)
\(348\) 0 0
\(349\) 5787.76 0.887712 0.443856 0.896098i \(-0.353610\pi\)
0.443856 + 0.896098i \(0.353610\pi\)
\(350\) 9122.67 1.39322
\(351\) 0 0
\(352\) −135.296 −0.0204866
\(353\) −5784.59 −0.872188 −0.436094 0.899901i \(-0.643639\pi\)
−0.436094 + 0.899901i \(0.643639\pi\)
\(354\) 0 0
\(355\) −10312.6 −1.54179
\(356\) 3684.55 0.548541
\(357\) 0 0
\(358\) 4653.62 0.687015
\(359\) 10132.3 1.48959 0.744796 0.667292i \(-0.232547\pi\)
0.744796 + 0.667292i \(0.232547\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 9274.91 1.34663
\(363\) 0 0
\(364\) −6682.29 −0.962217
\(365\) −10338.5 −1.48258
\(366\) 0 0
\(367\) −6993.81 −0.994752 −0.497376 0.867535i \(-0.665703\pi\)
−0.497376 + 0.867535i \(0.665703\pi\)
\(368\) −1472.58 −0.208596
\(369\) 0 0
\(370\) 817.871 0.114917
\(371\) −14006.0 −1.95999
\(372\) 0 0
\(373\) 6523.15 0.905512 0.452756 0.891634i \(-0.350441\pi\)
0.452756 + 0.891634i \(0.350441\pi\)
\(374\) −410.488 −0.0567535
\(375\) 0 0
\(376\) −3982.50 −0.546228
\(377\) 5648.75 0.771685
\(378\) 0 0
\(379\) −9782.00 −1.32577 −0.662886 0.748720i \(-0.730669\pi\)
−0.662886 + 0.748720i \(0.730669\pi\)
\(380\) 1316.02 0.177658
\(381\) 0 0
\(382\) 10520.8 1.40913
\(383\) −9878.11 −1.31788 −0.658940 0.752196i \(-0.728995\pi\)
−0.658940 + 0.752196i \(0.728995\pi\)
\(384\) 0 0
\(385\) −1909.96 −0.252832
\(386\) −32.3666 −0.00426792
\(387\) 0 0
\(388\) 4330.98 0.566680
\(389\) 7891.25 1.02854 0.514270 0.857628i \(-0.328063\pi\)
0.514270 + 0.857628i \(0.328063\pi\)
\(390\) 0 0
\(391\) −4467.80 −0.577868
\(392\) −2700.67 −0.347971
\(393\) 0 0
\(394\) 7569.43 0.967874
\(395\) 7397.81 0.942340
\(396\) 0 0
\(397\) −2787.84 −0.352437 −0.176219 0.984351i \(-0.556387\pi\)
−0.176219 + 0.984351i \(0.556387\pi\)
\(398\) −146.416 −0.0184401
\(399\) 0 0
\(400\) 2797.50 0.349688
\(401\) −1264.42 −0.157461 −0.0787306 0.996896i \(-0.525087\pi\)
−0.0787306 + 0.996896i \(0.525087\pi\)
\(402\) 0 0
\(403\) −5248.91 −0.648801
\(404\) 2849.79 0.350947
\(405\) 0 0
\(406\) 4602.55 0.562613
\(407\) −99.8486 −0.0121605
\(408\) 0 0
\(409\) −8140.55 −0.984166 −0.492083 0.870548i \(-0.663764\pi\)
−0.492083 + 0.870548i \(0.663764\pi\)
\(410\) 613.680 0.0739207
\(411\) 0 0
\(412\) −105.919 −0.0126656
\(413\) −957.429 −0.114073
\(414\) 0 0
\(415\) −8550.10 −1.01134
\(416\) −2049.15 −0.241510
\(417\) 0 0
\(418\) −160.664 −0.0187998
\(419\) 9601.15 1.11944 0.559722 0.828680i \(-0.310908\pi\)
0.559722 + 0.828680i \(0.310908\pi\)
\(420\) 0 0
\(421\) 5702.48 0.660147 0.330074 0.943955i \(-0.392926\pi\)
0.330074 + 0.943955i \(0.392926\pi\)
\(422\) 5890.87 0.679534
\(423\) 0 0
\(424\) −4295.01 −0.491943
\(425\) 8487.63 0.968731
\(426\) 0 0
\(427\) −16453.5 −1.86473
\(428\) 2963.99 0.334742
\(429\) 0 0
\(430\) −12758.6 −1.43087
\(431\) 4025.72 0.449912 0.224956 0.974369i \(-0.427776\pi\)
0.224956 + 0.974369i \(0.427776\pi\)
\(432\) 0 0
\(433\) −1347.10 −0.149510 −0.0747548 0.997202i \(-0.523817\pi\)
−0.0747548 + 0.997202i \(0.523817\pi\)
\(434\) −4276.77 −0.473022
\(435\) 0 0
\(436\) −7932.31 −0.871304
\(437\) −1748.68 −0.191421
\(438\) 0 0
\(439\) 4109.36 0.446763 0.223381 0.974731i \(-0.428290\pi\)
0.223381 + 0.974731i \(0.428290\pi\)
\(440\) −585.696 −0.0634590
\(441\) 0 0
\(442\) −6217.13 −0.669047
\(443\) −6964.84 −0.746974 −0.373487 0.927635i \(-0.621838\pi\)
−0.373487 + 0.927635i \(0.621838\pi\)
\(444\) 0 0
\(445\) 15950.4 1.69915
\(446\) −6250.59 −0.663619
\(447\) 0 0
\(448\) −1669.63 −0.176078
\(449\) −3041.21 −0.319652 −0.159826 0.987145i \(-0.551093\pi\)
−0.159826 + 0.987145i \(0.551093\pi\)
\(450\) 0 0
\(451\) −74.9202 −0.00782229
\(452\) 2874.88 0.299166
\(453\) 0 0
\(454\) −7155.60 −0.739711
\(455\) −28927.6 −2.98055
\(456\) 0 0
\(457\) 11984.3 1.22670 0.613352 0.789810i \(-0.289821\pi\)
0.613352 + 0.789810i \(0.289821\pi\)
\(458\) 9604.01 0.979838
\(459\) 0 0
\(460\) −6374.79 −0.646143
\(461\) 12126.7 1.22515 0.612577 0.790411i \(-0.290133\pi\)
0.612577 + 0.790411i \(0.290133\pi\)
\(462\) 0 0
\(463\) −6399.19 −0.642323 −0.321162 0.947024i \(-0.604073\pi\)
−0.321162 + 0.947024i \(0.604073\pi\)
\(464\) 1411.39 0.141212
\(465\) 0 0
\(466\) 11659.0 1.15900
\(467\) −993.366 −0.0984315 −0.0492157 0.998788i \(-0.515672\pi\)
−0.0492157 + 0.998788i \(0.515672\pi\)
\(468\) 0 0
\(469\) −7371.32 −0.725748
\(470\) −17240.2 −1.69198
\(471\) 0 0
\(472\) −293.600 −0.0286314
\(473\) 1557.61 0.151414
\(474\) 0 0
\(475\) 3322.04 0.320896
\(476\) −5065.67 −0.487782
\(477\) 0 0
\(478\) 2728.65 0.261100
\(479\) −6639.36 −0.633320 −0.316660 0.948539i \(-0.602561\pi\)
−0.316660 + 0.948539i \(0.602561\pi\)
\(480\) 0 0
\(481\) −1512.28 −0.143355
\(482\) 5294.45 0.500323
\(483\) 0 0
\(484\) −5252.50 −0.493285
\(485\) 18748.8 1.75534
\(486\) 0 0
\(487\) −11088.8 −1.03179 −0.515894 0.856652i \(-0.672540\pi\)
−0.515894 + 0.856652i \(0.672540\pi\)
\(488\) −5045.54 −0.468034
\(489\) 0 0
\(490\) −11691.2 −1.07787
\(491\) 13215.2 1.21465 0.607324 0.794454i \(-0.292243\pi\)
0.607324 + 0.794454i \(0.292243\pi\)
\(492\) 0 0
\(493\) 4282.17 0.391195
\(494\) −2433.37 −0.221624
\(495\) 0 0
\(496\) −1311.49 −0.118725
\(497\) 15536.8 1.40225
\(498\) 0 0
\(499\) 410.640 0.0368393 0.0184196 0.999830i \(-0.494137\pi\)
0.0184196 + 0.999830i \(0.494137\pi\)
\(500\) 3452.40 0.308792
\(501\) 0 0
\(502\) 3941.45 0.350429
\(503\) 9407.88 0.833950 0.416975 0.908918i \(-0.363090\pi\)
0.416975 + 0.908918i \(0.363090\pi\)
\(504\) 0 0
\(505\) 12336.7 1.08709
\(506\) 778.256 0.0683749
\(507\) 0 0
\(508\) 10442.9 0.912063
\(509\) −10482.2 −0.912803 −0.456402 0.889774i \(-0.650862\pi\)
−0.456402 + 0.889774i \(0.650862\pi\)
\(510\) 0 0
\(511\) 15575.8 1.34840
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) −15831.6 −1.35857
\(515\) −458.523 −0.0392329
\(516\) 0 0
\(517\) 2104.75 0.179046
\(518\) −1232.19 −0.104516
\(519\) 0 0
\(520\) −8870.79 −0.748096
\(521\) 3181.02 0.267492 0.133746 0.991016i \(-0.457299\pi\)
0.133746 + 0.991016i \(0.457299\pi\)
\(522\) 0 0
\(523\) −4360.12 −0.364541 −0.182270 0.983248i \(-0.558345\pi\)
−0.182270 + 0.983248i \(0.558345\pi\)
\(524\) 4866.77 0.405736
\(525\) 0 0
\(526\) 6575.93 0.545103
\(527\) −3979.06 −0.328900
\(528\) 0 0
\(529\) −3696.37 −0.303803
\(530\) −18593.1 −1.52383
\(531\) 0 0
\(532\) −1982.69 −0.161580
\(533\) −1134.72 −0.0922142
\(534\) 0 0
\(535\) 12831.1 1.03689
\(536\) −2260.45 −0.182158
\(537\) 0 0
\(538\) −9499.22 −0.761227
\(539\) 1427.31 0.114060
\(540\) 0 0
\(541\) −23681.2 −1.88195 −0.940973 0.338481i \(-0.890087\pi\)
−0.940973 + 0.338481i \(0.890087\pi\)
\(542\) −485.322 −0.0384619
\(543\) 0 0
\(544\) −1553.41 −0.122430
\(545\) −34339.0 −2.69894
\(546\) 0 0
\(547\) −7373.25 −0.576339 −0.288169 0.957579i \(-0.593047\pi\)
−0.288169 + 0.957579i \(0.593047\pi\)
\(548\) −4682.69 −0.365026
\(549\) 0 0
\(550\) −1478.48 −0.114623
\(551\) 1676.03 0.129585
\(552\) 0 0
\(553\) −11145.4 −0.857055
\(554\) 8262.27 0.633628
\(555\) 0 0
\(556\) −1084.33 −0.0827086
\(557\) −4772.14 −0.363020 −0.181510 0.983389i \(-0.558098\pi\)
−0.181510 + 0.983389i \(0.558098\pi\)
\(558\) 0 0
\(559\) 23591.1 1.78497
\(560\) −7227.84 −0.545414
\(561\) 0 0
\(562\) 2014.38 0.151195
\(563\) −7276.49 −0.544702 −0.272351 0.962198i \(-0.587801\pi\)
−0.272351 + 0.962198i \(0.587801\pi\)
\(564\) 0 0
\(565\) 12445.4 0.926691
\(566\) −4667.26 −0.346607
\(567\) 0 0
\(568\) 4764.42 0.351955
\(569\) 10685.1 0.787245 0.393622 0.919272i \(-0.371222\pi\)
0.393622 + 0.919272i \(0.371222\pi\)
\(570\) 0 0
\(571\) 14856.1 1.08881 0.544404 0.838823i \(-0.316756\pi\)
0.544404 + 0.838823i \(0.316756\pi\)
\(572\) 1082.98 0.0791635
\(573\) 0 0
\(574\) −924.560 −0.0672306
\(575\) −16092.0 −1.16710
\(576\) 0 0
\(577\) 3212.67 0.231794 0.115897 0.993261i \(-0.463026\pi\)
0.115897 + 0.993261i \(0.463026\pi\)
\(578\) 5112.96 0.367943
\(579\) 0 0
\(580\) 6109.92 0.437415
\(581\) 12881.4 0.919814
\(582\) 0 0
\(583\) 2269.91 0.161252
\(584\) 4776.39 0.338439
\(585\) 0 0
\(586\) −3177.35 −0.223985
\(587\) 22321.1 1.56949 0.784745 0.619818i \(-0.212794\pi\)
0.784745 + 0.619818i \(0.212794\pi\)
\(588\) 0 0
\(589\) −1557.39 −0.108950
\(590\) −1270.99 −0.0886881
\(591\) 0 0
\(592\) −377.857 −0.0262328
\(593\) 8202.50 0.568021 0.284010 0.958821i \(-0.408335\pi\)
0.284010 + 0.958821i \(0.408335\pi\)
\(594\) 0 0
\(595\) −21929.3 −1.51095
\(596\) −7364.75 −0.506161
\(597\) 0 0
\(598\) 11787.2 0.806047
\(599\) −10583.3 −0.721906 −0.360953 0.932584i \(-0.617548\pi\)
−0.360953 + 0.932584i \(0.617548\pi\)
\(600\) 0 0
\(601\) −9051.94 −0.614370 −0.307185 0.951650i \(-0.599387\pi\)
−0.307185 + 0.951650i \(0.599387\pi\)
\(602\) 19221.9 1.30137
\(603\) 0 0
\(604\) 13289.3 0.895254
\(605\) −22738.1 −1.52799
\(606\) 0 0
\(607\) 8123.48 0.543199 0.271599 0.962410i \(-0.412447\pi\)
0.271599 + 0.962410i \(0.412447\pi\)
\(608\) −608.000 −0.0405554
\(609\) 0 0
\(610\) −21842.1 −1.44977
\(611\) 31877.9 2.11071
\(612\) 0 0
\(613\) 22384.7 1.47490 0.737448 0.675404i \(-0.236031\pi\)
0.737448 + 0.675404i \(0.236031\pi\)
\(614\) −8114.91 −0.533373
\(615\) 0 0
\(616\) 882.400 0.0577158
\(617\) −11349.1 −0.740517 −0.370259 0.928929i \(-0.620731\pi\)
−0.370259 + 0.928929i \(0.620731\pi\)
\(618\) 0 0
\(619\) −9106.25 −0.591294 −0.295647 0.955297i \(-0.595535\pi\)
−0.295647 + 0.955297i \(0.595535\pi\)
\(620\) −5677.44 −0.367760
\(621\) 0 0
\(622\) 5743.84 0.370268
\(623\) −24030.6 −1.54537
\(624\) 0 0
\(625\) −6910.06 −0.442244
\(626\) −8645.34 −0.551976
\(627\) 0 0
\(628\) 974.464 0.0619194
\(629\) −1146.42 −0.0726720
\(630\) 0 0
\(631\) −27784.2 −1.75289 −0.876444 0.481505i \(-0.840090\pi\)
−0.876444 + 0.481505i \(0.840090\pi\)
\(632\) −3417.79 −0.215115
\(633\) 0 0
\(634\) 5027.12 0.314909
\(635\) 45207.3 2.82519
\(636\) 0 0
\(637\) 21617.5 1.34461
\(638\) −745.921 −0.0462873
\(639\) 0 0
\(640\) −2216.45 −0.136895
\(641\) 16958.3 1.04495 0.522476 0.852654i \(-0.325008\pi\)
0.522476 + 0.852654i \(0.325008\pi\)
\(642\) 0 0
\(643\) 4754.37 0.291592 0.145796 0.989315i \(-0.453426\pi\)
0.145796 + 0.989315i \(0.453426\pi\)
\(644\) 9604.15 0.587665
\(645\) 0 0
\(646\) −1844.67 −0.112349
\(647\) 11254.0 0.683831 0.341916 0.939731i \(-0.388924\pi\)
0.341916 + 0.939731i \(0.388924\pi\)
\(648\) 0 0
\(649\) 155.167 0.00938498
\(650\) −22392.6 −1.35125
\(651\) 0 0
\(652\) −10392.4 −0.624233
\(653\) 15515.1 0.929793 0.464896 0.885365i \(-0.346092\pi\)
0.464896 + 0.885365i \(0.346092\pi\)
\(654\) 0 0
\(655\) 21068.2 1.25680
\(656\) −283.520 −0.0168744
\(657\) 0 0
\(658\) 25973.9 1.53885
\(659\) −17203.2 −1.01691 −0.508453 0.861090i \(-0.669783\pi\)
−0.508453 + 0.861090i \(0.669783\pi\)
\(660\) 0 0
\(661\) 2305.65 0.135672 0.0678361 0.997696i \(-0.478390\pi\)
0.0678361 + 0.997696i \(0.478390\pi\)
\(662\) 9147.55 0.537054
\(663\) 0 0
\(664\) 3950.15 0.230867
\(665\) −8583.06 −0.500507
\(666\) 0 0
\(667\) −8118.69 −0.471299
\(668\) 1964.26 0.113772
\(669\) 0 0
\(670\) −9785.48 −0.564248
\(671\) 2666.57 0.153415
\(672\) 0 0
\(673\) −14242.8 −0.815782 −0.407891 0.913031i \(-0.633736\pi\)
−0.407891 + 0.913031i \(0.633736\pi\)
\(674\) −18003.4 −1.02888
\(675\) 0 0
\(676\) 7614.45 0.433230
\(677\) 13480.0 0.765256 0.382628 0.923902i \(-0.375019\pi\)
0.382628 + 0.923902i \(0.375019\pi\)
\(678\) 0 0
\(679\) −28246.6 −1.59647
\(680\) −6724.71 −0.379236
\(681\) 0 0
\(682\) 693.122 0.0389164
\(683\) −27626.1 −1.54771 −0.773854 0.633365i \(-0.781673\pi\)
−0.773854 + 0.633365i \(0.781673\pi\)
\(684\) 0 0
\(685\) −20271.4 −1.13070
\(686\) −282.578 −0.0157272
\(687\) 0 0
\(688\) 5894.46 0.326634
\(689\) 34379.4 1.90094
\(690\) 0 0
\(691\) 17419.7 0.959009 0.479505 0.877539i \(-0.340816\pi\)
0.479505 + 0.877539i \(0.340816\pi\)
\(692\) −6594.24 −0.362248
\(693\) 0 0
\(694\) 18717.4 1.02378
\(695\) −4694.08 −0.256197
\(696\) 0 0
\(697\) −860.201 −0.0467467
\(698\) −11575.5 −0.627707
\(699\) 0 0
\(700\) −18245.3 −0.985155
\(701\) −5069.39 −0.273136 −0.136568 0.990631i \(-0.543607\pi\)
−0.136568 + 0.990631i \(0.543607\pi\)
\(702\) 0 0
\(703\) −448.705 −0.0240729
\(704\) 270.592 0.0144862
\(705\) 0 0
\(706\) 11569.2 0.616730
\(707\) −18586.3 −0.988701
\(708\) 0 0
\(709\) −16758.9 −0.887719 −0.443860 0.896096i \(-0.646391\pi\)
−0.443860 + 0.896096i \(0.646391\pi\)
\(710\) 20625.2 1.09021
\(711\) 0 0
\(712\) −7369.09 −0.387877
\(713\) 7544.02 0.396249
\(714\) 0 0
\(715\) 4688.21 0.245215
\(716\) −9307.24 −0.485793
\(717\) 0 0
\(718\) −20264.6 −1.05330
\(719\) 3885.84 0.201554 0.100777 0.994909i \(-0.467867\pi\)
0.100777 + 0.994909i \(0.467867\pi\)
\(720\) 0 0
\(721\) 690.803 0.0356822
\(722\) −722.000 −0.0372161
\(723\) 0 0
\(724\) −18549.8 −0.952208
\(725\) 15423.4 0.790081
\(726\) 0 0
\(727\) 6468.37 0.329984 0.164992 0.986295i \(-0.447240\pi\)
0.164992 + 0.986295i \(0.447240\pi\)
\(728\) 13364.6 0.680390
\(729\) 0 0
\(730\) 20677.0 1.04834
\(731\) 17883.8 0.904865
\(732\) 0 0
\(733\) 25245.5 1.27212 0.636059 0.771640i \(-0.280563\pi\)
0.636059 + 0.771640i \(0.280563\pi\)
\(734\) 13987.6 0.703396
\(735\) 0 0
\(736\) 2945.15 0.147500
\(737\) 1194.65 0.0597087
\(738\) 0 0
\(739\) 3229.28 0.160746 0.0803728 0.996765i \(-0.474389\pi\)
0.0803728 + 0.996765i \(0.474389\pi\)
\(740\) −1635.74 −0.0812582
\(741\) 0 0
\(742\) 28012.0 1.38592
\(743\) −18876.2 −0.932033 −0.466016 0.884776i \(-0.654311\pi\)
−0.466016 + 0.884776i \(0.654311\pi\)
\(744\) 0 0
\(745\) −31882.0 −1.56788
\(746\) −13046.3 −0.640294
\(747\) 0 0
\(748\) 820.976 0.0401308
\(749\) −19331.1 −0.943049
\(750\) 0 0
\(751\) 24895.8 1.20967 0.604833 0.796352i \(-0.293240\pi\)
0.604833 + 0.796352i \(0.293240\pi\)
\(752\) 7964.99 0.386241
\(753\) 0 0
\(754\) −11297.5 −0.545664
\(755\) 57529.3 2.77312
\(756\) 0 0
\(757\) −36203.2 −1.73821 −0.869107 0.494624i \(-0.835306\pi\)
−0.869107 + 0.494624i \(0.835306\pi\)
\(758\) 19564.0 0.937462
\(759\) 0 0
\(760\) −2632.03 −0.125624
\(761\) −11417.5 −0.543868 −0.271934 0.962316i \(-0.587663\pi\)
−0.271934 + 0.962316i \(0.587663\pi\)
\(762\) 0 0
\(763\) 51734.5 2.45467
\(764\) −21041.5 −0.996407
\(765\) 0 0
\(766\) 19756.2 0.931881
\(767\) 2350.12 0.110636
\(768\) 0 0
\(769\) 39414.5 1.84828 0.924138 0.382058i \(-0.124785\pi\)
0.924138 + 0.382058i \(0.124785\pi\)
\(770\) 3819.91 0.178779
\(771\) 0 0
\(772\) 64.7332 0.00301787
\(773\) 14268.5 0.663910 0.331955 0.943295i \(-0.392292\pi\)
0.331955 + 0.943295i \(0.392292\pi\)
\(774\) 0 0
\(775\) −14331.6 −0.664268
\(776\) −8661.95 −0.400704
\(777\) 0 0
\(778\) −15782.5 −0.727288
\(779\) −336.680 −0.0154850
\(780\) 0 0
\(781\) −2517.99 −0.115366
\(782\) 8935.59 0.408614
\(783\) 0 0
\(784\) 5401.35 0.246053
\(785\) 4218.46 0.191800
\(786\) 0 0
\(787\) −2922.28 −0.132361 −0.0661804 0.997808i \(-0.521081\pi\)
−0.0661804 + 0.997808i \(0.521081\pi\)
\(788\) −15138.9 −0.684390
\(789\) 0 0
\(790\) −14795.6 −0.666335
\(791\) −18750.0 −0.842823
\(792\) 0 0
\(793\) 40387.0 1.80856
\(794\) 5575.67 0.249211
\(795\) 0 0
\(796\) 292.832 0.0130391
\(797\) 7724.25 0.343296 0.171648 0.985158i \(-0.445091\pi\)
0.171648 + 0.985158i \(0.445091\pi\)
\(798\) 0 0
\(799\) 24165.8 1.06999
\(800\) −5595.01 −0.247267
\(801\) 0 0
\(802\) 2528.83 0.111342
\(803\) −2524.32 −0.110936
\(804\) 0 0
\(805\) 41576.4 1.82034
\(806\) 10497.8 0.458772
\(807\) 0 0
\(808\) −5699.58 −0.248157
\(809\) −42980.8 −1.86789 −0.933947 0.357412i \(-0.883659\pi\)
−0.933947 + 0.357412i \(0.883659\pi\)
\(810\) 0 0
\(811\) 28749.5 1.24480 0.622398 0.782701i \(-0.286158\pi\)
0.622398 + 0.782701i \(0.286158\pi\)
\(812\) −9205.11 −0.397827
\(813\) 0 0
\(814\) 199.697 0.00859875
\(815\) −44988.9 −1.93361
\(816\) 0 0
\(817\) 6999.68 0.299740
\(818\) 16281.1 0.695911
\(819\) 0 0
\(820\) −1227.36 −0.0522698
\(821\) 30274.8 1.28696 0.643482 0.765461i \(-0.277489\pi\)
0.643482 + 0.765461i \(0.277489\pi\)
\(822\) 0 0
\(823\) 17296.1 0.732568 0.366284 0.930503i \(-0.380630\pi\)
0.366284 + 0.930503i \(0.380630\pi\)
\(824\) 211.838 0.00895596
\(825\) 0 0
\(826\) 1914.86 0.0806615
\(827\) 2022.80 0.0850541 0.0425271 0.999095i \(-0.486459\pi\)
0.0425271 + 0.999095i \(0.486459\pi\)
\(828\) 0 0
\(829\) −43239.0 −1.81152 −0.905762 0.423786i \(-0.860701\pi\)
−0.905762 + 0.423786i \(0.860701\pi\)
\(830\) 17100.2 0.715128
\(831\) 0 0
\(832\) 4098.31 0.170773
\(833\) 16387.7 0.681632
\(834\) 0 0
\(835\) 8503.27 0.352417
\(836\) 321.328 0.0132935
\(837\) 0 0
\(838\) −19202.3 −0.791567
\(839\) −27435.9 −1.12895 −0.564477 0.825449i \(-0.690922\pi\)
−0.564477 + 0.825449i \(0.690922\pi\)
\(840\) 0 0
\(841\) −16607.6 −0.680948
\(842\) −11405.0 −0.466795
\(843\) 0 0
\(844\) −11781.7 −0.480503
\(845\) 32963.0 1.34196
\(846\) 0 0
\(847\) 34256.8 1.38970
\(848\) 8590.02 0.347857
\(849\) 0 0
\(850\) −16975.3 −0.684996
\(851\) 2173.53 0.0875530
\(852\) 0 0
\(853\) −20978.4 −0.842071 −0.421035 0.907044i \(-0.638333\pi\)
−0.421035 + 0.907044i \(0.638333\pi\)
\(854\) 32907.0 1.31857
\(855\) 0 0
\(856\) −5927.97 −0.236698
\(857\) 30822.4 1.22856 0.614279 0.789089i \(-0.289447\pi\)
0.614279 + 0.789089i \(0.289447\pi\)
\(858\) 0 0
\(859\) −39267.6 −1.55971 −0.779856 0.625959i \(-0.784708\pi\)
−0.779856 + 0.625959i \(0.784708\pi\)
\(860\) 25517.1 1.01178
\(861\) 0 0
\(862\) −8051.45 −0.318136
\(863\) 24131.3 0.951842 0.475921 0.879488i \(-0.342115\pi\)
0.475921 + 0.879488i \(0.342115\pi\)
\(864\) 0 0
\(865\) −28546.5 −1.12209
\(866\) 2694.21 0.105719
\(867\) 0 0
\(868\) 8553.54 0.334477
\(869\) 1806.30 0.0705116
\(870\) 0 0
\(871\) 18093.8 0.703885
\(872\) 15864.6 0.616105
\(873\) 0 0
\(874\) 3497.37 0.135355
\(875\) −22516.6 −0.869941
\(876\) 0 0
\(877\) 39380.6 1.51629 0.758147 0.652084i \(-0.226105\pi\)
0.758147 + 0.652084i \(0.226105\pi\)
\(878\) −8218.71 −0.315909
\(879\) 0 0
\(880\) 1171.39 0.0448723
\(881\) −30887.5 −1.18119 −0.590595 0.806968i \(-0.701107\pi\)
−0.590595 + 0.806968i \(0.701107\pi\)
\(882\) 0 0
\(883\) 28191.9 1.07444 0.537221 0.843441i \(-0.319474\pi\)
0.537221 + 0.843441i \(0.319474\pi\)
\(884\) 12434.3 0.473088
\(885\) 0 0
\(886\) 13929.7 0.528190
\(887\) −2760.58 −0.104500 −0.0522498 0.998634i \(-0.516639\pi\)
−0.0522498 + 0.998634i \(0.516639\pi\)
\(888\) 0 0
\(889\) −68108.5 −2.56950
\(890\) −31900.8 −1.20148
\(891\) 0 0
\(892\) 12501.2 0.469250
\(893\) 9458.43 0.354439
\(894\) 0 0
\(895\) −40291.0 −1.50478
\(896\) 3339.26 0.124506
\(897\) 0 0
\(898\) 6082.42 0.226028
\(899\) −7230.57 −0.268246
\(900\) 0 0
\(901\) 26062.1 0.963657
\(902\) 149.840 0.00553120
\(903\) 0 0
\(904\) −5749.76 −0.211542
\(905\) −80302.2 −2.94954
\(906\) 0 0
\(907\) −18969.1 −0.694443 −0.347222 0.937783i \(-0.612875\pi\)
−0.347222 + 0.937783i \(0.612875\pi\)
\(908\) 14311.2 0.523055
\(909\) 0 0
\(910\) 57855.3 2.10756
\(911\) 48732.9 1.77233 0.886164 0.463371i \(-0.153360\pi\)
0.886164 + 0.463371i \(0.153360\pi\)
\(912\) 0 0
\(913\) −2087.65 −0.0756749
\(914\) −23968.7 −0.867411
\(915\) 0 0
\(916\) −19208.0 −0.692850
\(917\) −31741.1 −1.14306
\(918\) 0 0
\(919\) −35850.4 −1.28683 −0.643414 0.765518i \(-0.722483\pi\)
−0.643414 + 0.765518i \(0.722483\pi\)
\(920\) 12749.6 0.456892
\(921\) 0 0
\(922\) −24253.4 −0.866315
\(923\) −38136.8 −1.36001
\(924\) 0 0
\(925\) −4129.13 −0.146773
\(926\) 12798.4 0.454191
\(927\) 0 0
\(928\) −2822.79 −0.0998518
\(929\) −22936.8 −0.810044 −0.405022 0.914307i \(-0.632736\pi\)
−0.405022 + 0.914307i \(0.632736\pi\)
\(930\) 0 0
\(931\) 6414.10 0.225793
\(932\) −23318.0 −0.819534
\(933\) 0 0
\(934\) 1986.73 0.0696016
\(935\) 3554.01 0.124308
\(936\) 0 0
\(937\) 47925.4 1.67092 0.835462 0.549548i \(-0.185200\pi\)
0.835462 + 0.549548i \(0.185200\pi\)
\(938\) 14742.6 0.513181
\(939\) 0 0
\(940\) 34480.5 1.19641
\(941\) 25842.2 0.895251 0.447626 0.894221i \(-0.352270\pi\)
0.447626 + 0.894221i \(0.352270\pi\)
\(942\) 0 0
\(943\) 1630.88 0.0563189
\(944\) 587.199 0.0202455
\(945\) 0 0
\(946\) −3115.22 −0.107066
\(947\) −36562.8 −1.25463 −0.627314 0.778766i \(-0.715846\pi\)
−0.627314 + 0.778766i \(0.715846\pi\)
\(948\) 0 0
\(949\) −38232.6 −1.30778
\(950\) −6644.07 −0.226908
\(951\) 0 0
\(952\) 10131.3 0.344914
\(953\) −29813.1 −1.01337 −0.506684 0.862132i \(-0.669129\pi\)
−0.506684 + 0.862132i \(0.669129\pi\)
\(954\) 0 0
\(955\) −91088.7 −3.08645
\(956\) −5457.31 −0.184625
\(957\) 0 0
\(958\) 13278.7 0.447825
\(959\) 30540.5 1.02837
\(960\) 0 0
\(961\) −23072.2 −0.774470
\(962\) 3024.56 0.101368
\(963\) 0 0
\(964\) −10588.9 −0.353782
\(965\) 280.230 0.00934811
\(966\) 0 0
\(967\) 30315.5 1.00815 0.504075 0.863660i \(-0.331833\pi\)
0.504075 + 0.863660i \(0.331833\pi\)
\(968\) 10505.0 0.348805
\(969\) 0 0
\(970\) −37497.6 −1.24121
\(971\) −26455.6 −0.874357 −0.437178 0.899375i \(-0.644022\pi\)
−0.437178 + 0.899375i \(0.644022\pi\)
\(972\) 0 0
\(973\) 7072.03 0.233010
\(974\) 22177.6 0.729584
\(975\) 0 0
\(976\) 10091.1 0.330950
\(977\) −30207.7 −0.989183 −0.494591 0.869126i \(-0.664682\pi\)
−0.494591 + 0.869126i \(0.664682\pi\)
\(978\) 0 0
\(979\) 3894.56 0.127141
\(980\) 23382.4 0.762168
\(981\) 0 0
\(982\) −26430.3 −0.858886
\(983\) 5878.48 0.190737 0.0953685 0.995442i \(-0.469597\pi\)
0.0953685 + 0.995442i \(0.469597\pi\)
\(984\) 0 0
\(985\) −65536.1 −2.11995
\(986\) −8564.33 −0.276616
\(987\) 0 0
\(988\) 4866.74 0.156712
\(989\) −33906.4 −1.09015
\(990\) 0 0
\(991\) −42532.3 −1.36335 −0.681676 0.731654i \(-0.738749\pi\)
−0.681676 + 0.731654i \(0.738749\pi\)
\(992\) 2622.98 0.0839513
\(993\) 0 0
\(994\) −31073.5 −0.991542
\(995\) 1267.67 0.0403898
\(996\) 0 0
\(997\) 6320.28 0.200767 0.100384 0.994949i \(-0.467993\pi\)
0.100384 + 0.994949i \(0.467993\pi\)
\(998\) −821.281 −0.0260493
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 342.4.a.h.1.2 2
3.2 odd 2 38.4.a.c.1.2 2
12.11 even 2 304.4.a.c.1.1 2
15.2 even 4 950.4.b.i.799.3 4
15.8 even 4 950.4.b.i.799.2 4
15.14 odd 2 950.4.a.e.1.1 2
21.20 even 2 1862.4.a.e.1.1 2
24.5 odd 2 1216.4.a.g.1.1 2
24.11 even 2 1216.4.a.p.1.2 2
57.56 even 2 722.4.a.f.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.4.a.c.1.2 2 3.2 odd 2
304.4.a.c.1.1 2 12.11 even 2
342.4.a.h.1.2 2 1.1 even 1 trivial
722.4.a.f.1.1 2 57.56 even 2
950.4.a.e.1.1 2 15.14 odd 2
950.4.b.i.799.2 4 15.8 even 4
950.4.b.i.799.3 4 15.2 even 4
1216.4.a.g.1.1 2 24.5 odd 2
1216.4.a.p.1.2 2 24.11 even 2
1862.4.a.e.1.1 2 21.20 even 2