Properties

Label 1617.4.a.be.1.11
Level $1617$
Weight $4$
Character 1617.1
Self dual yes
Analytic conductor $95.406$
Analytic rank $1$
Dimension $16$
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] = [1617,4,Mod(1,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1617.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1617.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: \(95.4060884793\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 92 x^{14} + 346 x^{13} + 3385 x^{12} - 11756 x^{11} - 63875 x^{10} + 199850 x^{9} + \cdots + 5479424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(2.12238\) of defining polynomial
Character \(\chi\) \(=\) 1617.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.12238 q^{2} -3.00000 q^{3} -3.49551 q^{4} -16.9491 q^{5} -6.36714 q^{6} -24.3978 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.12238 q^{2} -3.00000 q^{3} -3.49551 q^{4} -16.9491 q^{5} -6.36714 q^{6} -24.3978 q^{8} +9.00000 q^{9} -35.9724 q^{10} +11.0000 q^{11} +10.4865 q^{12} +28.2555 q^{13} +50.8473 q^{15} -23.8173 q^{16} +27.5145 q^{17} +19.1014 q^{18} -110.588 q^{19} +59.2457 q^{20} +23.3462 q^{22} +93.4721 q^{23} +73.1935 q^{24} +162.272 q^{25} +59.9688 q^{26} -27.0000 q^{27} +36.5480 q^{29} +107.917 q^{30} +210.528 q^{31} +144.633 q^{32} -33.0000 q^{33} +58.3963 q^{34} -31.4596 q^{36} +421.841 q^{37} -234.710 q^{38} -84.7665 q^{39} +413.521 q^{40} +14.6544 q^{41} -481.822 q^{43} -38.4506 q^{44} -152.542 q^{45} +198.383 q^{46} -287.451 q^{47} +71.4520 q^{48} +344.402 q^{50} -82.5436 q^{51} -98.7673 q^{52} +630.112 q^{53} -57.3042 q^{54} -186.440 q^{55} +331.765 q^{57} +77.5686 q^{58} -587.170 q^{59} -177.737 q^{60} -80.3881 q^{61} +446.820 q^{62} +497.505 q^{64} -478.905 q^{65} -70.0385 q^{66} +291.410 q^{67} -96.1773 q^{68} -280.416 q^{69} -114.985 q^{71} -219.580 q^{72} -54.3759 q^{73} +895.306 q^{74} -486.815 q^{75} +386.562 q^{76} -179.906 q^{78} +739.306 q^{79} +403.682 q^{80} +81.0000 q^{81} +31.1022 q^{82} -110.414 q^{83} -466.347 q^{85} -1022.61 q^{86} -109.644 q^{87} -268.376 q^{88} +268.479 q^{89} -323.752 q^{90} -326.733 q^{92} -631.584 q^{93} -610.080 q^{94} +1874.37 q^{95} -433.900 q^{96} -1526.63 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9} - 178 q^{10} + 176 q^{11} - 216 q^{12} - 104 q^{13} + 120 q^{15} + 220 q^{16} - 180 q^{17} + 36 q^{18} - 152 q^{19} - 298 q^{20} + 44 q^{22} + 4 q^{23} - 198 q^{24} + 588 q^{25} - 406 q^{26} - 432 q^{27} + 412 q^{29} + 534 q^{30} - 628 q^{31} + 592 q^{32} - 528 q^{33} + 88 q^{34} + 648 q^{36} + 148 q^{37} - 446 q^{38} + 312 q^{39} - 1376 q^{40} - 596 q^{41} - 260 q^{43} + 792 q^{44} - 360 q^{45} - 148 q^{46} - 2220 q^{47} - 660 q^{48} + 82 q^{50} + 540 q^{51} - 1046 q^{52} + 168 q^{53} - 108 q^{54} - 440 q^{55} + 456 q^{57} + 538 q^{58} + 48 q^{59} + 894 q^{60} - 1504 q^{61} - 1276 q^{62} + 630 q^{64} + 1224 q^{65} - 132 q^{66} + 116 q^{67} - 356 q^{68} - 12 q^{69} + 320 q^{71} + 594 q^{72} - 652 q^{73} + 1062 q^{74} - 1764 q^{75} + 594 q^{76} + 1218 q^{78} + 1136 q^{79} - 1970 q^{80} + 1296 q^{81} + 416 q^{82} - 3300 q^{83} - 1148 q^{85} + 1864 q^{86} - 1236 q^{87} + 726 q^{88} - 2416 q^{89} - 1602 q^{90} - 1064 q^{92} + 1884 q^{93} - 914 q^{94} + 1120 q^{95} - 1776 q^{96} - 3616 q^{97} + 1584 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.12238 0.750374 0.375187 0.926949i \(-0.377578\pi\)
0.375187 + 0.926949i \(0.377578\pi\)
\(3\) −3.00000 −0.577350
\(4\) −3.49551 −0.436939
\(5\) −16.9491 −1.51597 −0.757987 0.652270i \(-0.773817\pi\)
−0.757987 + 0.652270i \(0.773817\pi\)
\(6\) −6.36714 −0.433229
\(7\) 0 0
\(8\) −24.3978 −1.07824
\(9\) 9.00000 0.333333
\(10\) −35.9724 −1.13755
\(11\) 11.0000 0.301511
\(12\) 10.4865 0.252267
\(13\) 28.2555 0.602820 0.301410 0.953495i \(-0.402543\pi\)
0.301410 + 0.953495i \(0.402543\pi\)
\(14\) 0 0
\(15\) 50.8473 0.875247
\(16\) −23.8173 −0.372146
\(17\) 27.5145 0.392545 0.196272 0.980549i \(-0.437116\pi\)
0.196272 + 0.980549i \(0.437116\pi\)
\(18\) 19.1014 0.250125
\(19\) −110.588 −1.33530 −0.667650 0.744476i \(-0.732700\pi\)
−0.667650 + 0.744476i \(0.732700\pi\)
\(20\) 59.2457 0.662387
\(21\) 0 0
\(22\) 23.3462 0.226246
\(23\) 93.4721 0.847404 0.423702 0.905802i \(-0.360730\pi\)
0.423702 + 0.905802i \(0.360730\pi\)
\(24\) 73.1935 0.622523
\(25\) 162.272 1.29817
\(26\) 59.9688 0.452341
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 36.5480 0.234027 0.117014 0.993130i \(-0.462668\pi\)
0.117014 + 0.993130i \(0.462668\pi\)
\(30\) 107.917 0.656763
\(31\) 210.528 1.21974 0.609870 0.792502i \(-0.291222\pi\)
0.609870 + 0.792502i \(0.291222\pi\)
\(32\) 144.633 0.798993
\(33\) −33.0000 −0.174078
\(34\) 58.3963 0.294555
\(35\) 0 0
\(36\) −31.4596 −0.145646
\(37\) 421.841 1.87433 0.937165 0.348885i \(-0.113440\pi\)
0.937165 + 0.348885i \(0.113440\pi\)
\(38\) −234.710 −1.00197
\(39\) −84.7665 −0.348038
\(40\) 413.521 1.63459
\(41\) 14.6544 0.0558203 0.0279101 0.999610i \(-0.491115\pi\)
0.0279101 + 0.999610i \(0.491115\pi\)
\(42\) 0 0
\(43\) −481.822 −1.70877 −0.854385 0.519640i \(-0.826066\pi\)
−0.854385 + 0.519640i \(0.826066\pi\)
\(44\) −38.4506 −0.131742
\(45\) −152.542 −0.505324
\(46\) 198.383 0.635870
\(47\) −287.451 −0.892107 −0.446054 0.895006i \(-0.647171\pi\)
−0.446054 + 0.895006i \(0.647171\pi\)
\(48\) 71.4520 0.214859
\(49\) 0 0
\(50\) 344.402 0.974116
\(51\) −82.5436 −0.226636
\(52\) −98.7673 −0.263395
\(53\) 630.112 1.63307 0.816533 0.577299i \(-0.195893\pi\)
0.816533 + 0.577299i \(0.195893\pi\)
\(54\) −57.3042 −0.144410
\(55\) −186.440 −0.457083
\(56\) 0 0
\(57\) 331.765 0.770935
\(58\) 77.5686 0.175608
\(59\) −587.170 −1.29565 −0.647823 0.761791i \(-0.724320\pi\)
−0.647823 + 0.761791i \(0.724320\pi\)
\(60\) −177.737 −0.382429
\(61\) −80.3881 −0.168732 −0.0843659 0.996435i \(-0.526886\pi\)
−0.0843659 + 0.996435i \(0.526886\pi\)
\(62\) 446.820 0.915261
\(63\) 0 0
\(64\) 497.505 0.971689
\(65\) −478.905 −0.913859
\(66\) −70.0385 −0.130623
\(67\) 291.410 0.531364 0.265682 0.964061i \(-0.414403\pi\)
0.265682 + 0.964061i \(0.414403\pi\)
\(68\) −96.1773 −0.171518
\(69\) −280.416 −0.489249
\(70\) 0 0
\(71\) −114.985 −0.192201 −0.0961004 0.995372i \(-0.530637\pi\)
−0.0961004 + 0.995372i \(0.530637\pi\)
\(72\) −219.580 −0.359414
\(73\) −54.3759 −0.0871810 −0.0435905 0.999049i \(-0.513880\pi\)
−0.0435905 + 0.999049i \(0.513880\pi\)
\(74\) 895.306 1.40645
\(75\) −486.815 −0.749501
\(76\) 386.562 0.583444
\(77\) 0 0
\(78\) −179.906 −0.261159
\(79\) 739.306 1.05289 0.526446 0.850209i \(-0.323524\pi\)
0.526446 + 0.850209i \(0.323524\pi\)
\(80\) 403.682 0.564163
\(81\) 81.0000 0.111111
\(82\) 31.1022 0.0418861
\(83\) −110.414 −0.146019 −0.0730093 0.997331i \(-0.523260\pi\)
−0.0730093 + 0.997331i \(0.523260\pi\)
\(84\) 0 0
\(85\) −466.347 −0.595087
\(86\) −1022.61 −1.28222
\(87\) −109.644 −0.135116
\(88\) −268.376 −0.325102
\(89\) 268.479 0.319761 0.159881 0.987136i \(-0.448889\pi\)
0.159881 + 0.987136i \(0.448889\pi\)
\(90\) −323.752 −0.379182
\(91\) 0 0
\(92\) −326.733 −0.370263
\(93\) −631.584 −0.704217
\(94\) −610.080 −0.669414
\(95\) 1874.37 2.02428
\(96\) −433.900 −0.461299
\(97\) −1526.63 −1.59800 −0.798998 0.601334i \(-0.794636\pi\)
−0.798998 + 0.601334i \(0.794636\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
\(100\) −567.223 −0.567223
\(101\) 1514.02 1.49159 0.745795 0.666175i \(-0.232070\pi\)
0.745795 + 0.666175i \(0.232070\pi\)
\(102\) −175.189 −0.170062
\(103\) −1142.49 −1.09294 −0.546472 0.837477i \(-0.684030\pi\)
−0.546472 + 0.837477i \(0.684030\pi\)
\(104\) −689.372 −0.649986
\(105\) 0 0
\(106\) 1337.34 1.22541
\(107\) −1259.13 −1.13761 −0.568807 0.822471i \(-0.692595\pi\)
−0.568807 + 0.822471i \(0.692595\pi\)
\(108\) 94.3788 0.0840889
\(109\) −1026.36 −0.901906 −0.450953 0.892548i \(-0.648916\pi\)
−0.450953 + 0.892548i \(0.648916\pi\)
\(110\) −395.696 −0.342983
\(111\) −1265.52 −1.08215
\(112\) 0 0
\(113\) −934.910 −0.778309 −0.389155 0.921172i \(-0.627233\pi\)
−0.389155 + 0.921172i \(0.627233\pi\)
\(114\) 704.130 0.578490
\(115\) −1584.27 −1.28464
\(116\) −127.754 −0.102256
\(117\) 254.299 0.200940
\(118\) −1246.20 −0.972219
\(119\) 0 0
\(120\) −1240.56 −0.943728
\(121\) 121.000 0.0909091
\(122\) −170.614 −0.126612
\(123\) −43.9632 −0.0322279
\(124\) −735.902 −0.532951
\(125\) −631.723 −0.452024
\(126\) 0 0
\(127\) −631.889 −0.441504 −0.220752 0.975330i \(-0.570851\pi\)
−0.220752 + 0.975330i \(0.570851\pi\)
\(128\) −101.171 −0.0698623
\(129\) 1445.47 0.986559
\(130\) −1016.42 −0.685736
\(131\) −438.480 −0.292444 −0.146222 0.989252i \(-0.546711\pi\)
−0.146222 + 0.989252i \(0.546711\pi\)
\(132\) 115.352 0.0760613
\(133\) 0 0
\(134\) 618.482 0.398722
\(135\) 457.626 0.291749
\(136\) −671.295 −0.423258
\(137\) −18.2526 −0.0113826 −0.00569132 0.999984i \(-0.501812\pi\)
−0.00569132 + 0.999984i \(0.501812\pi\)
\(138\) −595.150 −0.367120
\(139\) −3132.69 −1.91159 −0.955797 0.294026i \(-0.905005\pi\)
−0.955797 + 0.294026i \(0.905005\pi\)
\(140\) 0 0
\(141\) 862.353 0.515058
\(142\) −244.042 −0.144222
\(143\) 310.810 0.181757
\(144\) −214.356 −0.124049
\(145\) −619.455 −0.354779
\(146\) −115.406 −0.0654184
\(147\) 0 0
\(148\) −1474.55 −0.818968
\(149\) 2132.14 1.17229 0.586146 0.810205i \(-0.300644\pi\)
0.586146 + 0.810205i \(0.300644\pi\)
\(150\) −1033.21 −0.562406
\(151\) −1602.55 −0.863669 −0.431835 0.901953i \(-0.642134\pi\)
−0.431835 + 0.901953i \(0.642134\pi\)
\(152\) 2698.11 1.43977
\(153\) 247.631 0.130848
\(154\) 0 0
\(155\) −3568.26 −1.84909
\(156\) 296.302 0.152071
\(157\) 3857.97 1.96114 0.980572 0.196160i \(-0.0628473\pi\)
0.980572 + 0.196160i \(0.0628473\pi\)
\(158\) 1569.09 0.790062
\(159\) −1890.33 −0.942851
\(160\) −2451.40 −1.21125
\(161\) 0 0
\(162\) 171.913 0.0833749
\(163\) 604.065 0.290270 0.145135 0.989412i \(-0.453638\pi\)
0.145135 + 0.989412i \(0.453638\pi\)
\(164\) −51.2246 −0.0243900
\(165\) 559.320 0.263897
\(166\) −234.341 −0.109569
\(167\) −344.448 −0.159606 −0.0798029 0.996811i \(-0.525429\pi\)
−0.0798029 + 0.996811i \(0.525429\pi\)
\(168\) 0 0
\(169\) −1398.63 −0.636608
\(170\) −989.764 −0.446538
\(171\) −995.294 −0.445100
\(172\) 1684.21 0.746628
\(173\) −469.487 −0.206326 −0.103163 0.994664i \(-0.532896\pi\)
−0.103163 + 0.994664i \(0.532896\pi\)
\(174\) −232.706 −0.101387
\(175\) 0 0
\(176\) −261.991 −0.112206
\(177\) 1761.51 0.748041
\(178\) 569.815 0.239941
\(179\) 1441.33 0.601845 0.300923 0.953649i \(-0.402705\pi\)
0.300923 + 0.953649i \(0.402705\pi\)
\(180\) 533.211 0.220796
\(181\) −1787.77 −0.734164 −0.367082 0.930188i \(-0.619643\pi\)
−0.367082 + 0.930188i \(0.619643\pi\)
\(182\) 0 0
\(183\) 241.164 0.0974173
\(184\) −2280.52 −0.913706
\(185\) −7149.82 −2.84144
\(186\) −1340.46 −0.528426
\(187\) 302.660 0.118357
\(188\) 1004.79 0.389796
\(189\) 0 0
\(190\) 3978.12 1.51897
\(191\) −5010.70 −1.89823 −0.949114 0.314934i \(-0.898018\pi\)
−0.949114 + 0.314934i \(0.898018\pi\)
\(192\) −1492.52 −0.561005
\(193\) 1792.51 0.668538 0.334269 0.942478i \(-0.391511\pi\)
0.334269 + 0.942478i \(0.391511\pi\)
\(194\) −3240.08 −1.19909
\(195\) 1436.71 0.527617
\(196\) 0 0
\(197\) 3011.66 1.08920 0.544599 0.838697i \(-0.316682\pi\)
0.544599 + 0.838697i \(0.316682\pi\)
\(198\) 210.115 0.0754154
\(199\) −793.917 −0.282810 −0.141405 0.989952i \(-0.545162\pi\)
−0.141405 + 0.989952i \(0.545162\pi\)
\(200\) −3959.08 −1.39975
\(201\) −874.230 −0.306783
\(202\) 3213.32 1.11925
\(203\) 0 0
\(204\) 288.532 0.0990259
\(205\) −248.379 −0.0846220
\(206\) −2424.80 −0.820117
\(207\) 841.249 0.282468
\(208\) −672.970 −0.224337
\(209\) −1216.47 −0.402608
\(210\) 0 0
\(211\) 4483.93 1.46297 0.731484 0.681858i \(-0.238828\pi\)
0.731484 + 0.681858i \(0.238828\pi\)
\(212\) −2202.56 −0.713550
\(213\) 344.956 0.110967
\(214\) −2672.35 −0.853637
\(215\) 8166.44 2.59045
\(216\) 658.741 0.207508
\(217\) 0 0
\(218\) −2178.33 −0.676767
\(219\) 163.128 0.0503340
\(220\) 651.703 0.199717
\(221\) 777.437 0.236634
\(222\) −2685.92 −0.812014
\(223\) −1262.74 −0.379191 −0.189596 0.981862i \(-0.560718\pi\)
−0.189596 + 0.981862i \(0.560718\pi\)
\(224\) 0 0
\(225\) 1460.45 0.432725
\(226\) −1984.23 −0.584023
\(227\) 5976.11 1.74735 0.873675 0.486511i \(-0.161730\pi\)
0.873675 + 0.486511i \(0.161730\pi\)
\(228\) −1159.69 −0.336851
\(229\) −437.923 −0.126370 −0.0631852 0.998002i \(-0.520126\pi\)
−0.0631852 + 0.998002i \(0.520126\pi\)
\(230\) −3362.42 −0.963961
\(231\) 0 0
\(232\) −891.691 −0.252338
\(233\) −5712.28 −1.60611 −0.803056 0.595904i \(-0.796794\pi\)
−0.803056 + 0.595904i \(0.796794\pi\)
\(234\) 539.719 0.150780
\(235\) 4872.03 1.35241
\(236\) 2052.46 0.566118
\(237\) −2217.92 −0.607887
\(238\) 0 0
\(239\) 1678.15 0.454186 0.227093 0.973873i \(-0.427078\pi\)
0.227093 + 0.973873i \(0.427078\pi\)
\(240\) −1211.05 −0.325720
\(241\) −5853.31 −1.56450 −0.782251 0.622963i \(-0.785929\pi\)
−0.782251 + 0.622963i \(0.785929\pi\)
\(242\) 256.808 0.0682158
\(243\) −243.000 −0.0641500
\(244\) 280.997 0.0737254
\(245\) 0 0
\(246\) −93.3065 −0.0241829
\(247\) −3124.72 −0.804945
\(248\) −5136.42 −1.31517
\(249\) 331.243 0.0843039
\(250\) −1340.75 −0.339187
\(251\) −919.616 −0.231258 −0.115629 0.993293i \(-0.536888\pi\)
−0.115629 + 0.993293i \(0.536888\pi\)
\(252\) 0 0
\(253\) 1028.19 0.255502
\(254\) −1341.11 −0.331293
\(255\) 1399.04 0.343574
\(256\) −4194.76 −1.02411
\(257\) −3382.61 −0.821017 −0.410509 0.911857i \(-0.634649\pi\)
−0.410509 + 0.911857i \(0.634649\pi\)
\(258\) 3067.82 0.740288
\(259\) 0 0
\(260\) 1674.02 0.399300
\(261\) 328.932 0.0780091
\(262\) −930.621 −0.219443
\(263\) 1782.55 0.417934 0.208967 0.977923i \(-0.432990\pi\)
0.208967 + 0.977923i \(0.432990\pi\)
\(264\) 805.128 0.187698
\(265\) −10679.8 −2.47568
\(266\) 0 0
\(267\) −805.438 −0.184614
\(268\) −1018.63 −0.232174
\(269\) −1932.41 −0.437997 −0.218998 0.975725i \(-0.570279\pi\)
−0.218998 + 0.975725i \(0.570279\pi\)
\(270\) 971.255 0.218921
\(271\) −553.223 −0.124007 −0.0620035 0.998076i \(-0.519749\pi\)
−0.0620035 + 0.998076i \(0.519749\pi\)
\(272\) −655.323 −0.146084
\(273\) 0 0
\(274\) −38.7389 −0.00854124
\(275\) 1784.99 0.391414
\(276\) 980.198 0.213772
\(277\) 3061.07 0.663977 0.331989 0.943283i \(-0.392280\pi\)
0.331989 + 0.943283i \(0.392280\pi\)
\(278\) −6648.76 −1.43441
\(279\) 1894.75 0.406580
\(280\) 0 0
\(281\) 6740.56 1.43099 0.715495 0.698618i \(-0.246201\pi\)
0.715495 + 0.698618i \(0.246201\pi\)
\(282\) 1830.24 0.386486
\(283\) −1021.58 −0.214582 −0.107291 0.994228i \(-0.534218\pi\)
−0.107291 + 0.994228i \(0.534218\pi\)
\(284\) 401.932 0.0839799
\(285\) −5623.11 −1.16872
\(286\) 659.657 0.136386
\(287\) 0 0
\(288\) 1301.70 0.266331
\(289\) −4155.95 −0.845909
\(290\) −1314.72 −0.266217
\(291\) 4579.88 0.922603
\(292\) 190.071 0.0380928
\(293\) 6893.14 1.37441 0.687204 0.726465i \(-0.258838\pi\)
0.687204 + 0.726465i \(0.258838\pi\)
\(294\) 0 0
\(295\) 9952.01 1.96416
\(296\) −10292.0 −2.02098
\(297\) −297.000 −0.0580259
\(298\) 4525.20 0.879658
\(299\) 2641.10 0.510832
\(300\) 1701.67 0.327486
\(301\) 0 0
\(302\) −3401.23 −0.648075
\(303\) −4542.06 −0.861170
\(304\) 2633.92 0.496926
\(305\) 1362.50 0.255793
\(306\) 525.566 0.0981851
\(307\) 1092.78 0.203154 0.101577 0.994828i \(-0.467611\pi\)
0.101577 + 0.994828i \(0.467611\pi\)
\(308\) 0 0
\(309\) 3427.48 0.631012
\(310\) −7573.19 −1.38751
\(311\) 1108.39 0.202093 0.101046 0.994882i \(-0.467781\pi\)
0.101046 + 0.994882i \(0.467781\pi\)
\(312\) 2068.12 0.375269
\(313\) −258.238 −0.0466342 −0.0233171 0.999728i \(-0.507423\pi\)
−0.0233171 + 0.999728i \(0.507423\pi\)
\(314\) 8188.07 1.47159
\(315\) 0 0
\(316\) −2584.25 −0.460049
\(317\) 5512.58 0.976710 0.488355 0.872645i \(-0.337597\pi\)
0.488355 + 0.872645i \(0.337597\pi\)
\(318\) −4012.01 −0.707491
\(319\) 402.028 0.0705618
\(320\) −8432.26 −1.47306
\(321\) 3777.39 0.656802
\(322\) 0 0
\(323\) −3042.78 −0.524164
\(324\) −283.136 −0.0485487
\(325\) 4585.07 0.782565
\(326\) 1282.05 0.217811
\(327\) 3079.09 0.520716
\(328\) −357.535 −0.0601877
\(329\) 0 0
\(330\) 1187.09 0.198021
\(331\) 9338.01 1.55064 0.775322 0.631566i \(-0.217587\pi\)
0.775322 + 0.631566i \(0.217587\pi\)
\(332\) 385.954 0.0638012
\(333\) 3796.57 0.624777
\(334\) −731.049 −0.119764
\(335\) −4939.13 −0.805534
\(336\) 0 0
\(337\) 519.619 0.0839925 0.0419962 0.999118i \(-0.486628\pi\)
0.0419962 + 0.999118i \(0.486628\pi\)
\(338\) −2968.42 −0.477694
\(339\) 2804.73 0.449357
\(340\) 1630.12 0.260016
\(341\) 2315.81 0.367765
\(342\) −2112.39 −0.333991
\(343\) 0 0
\(344\) 11755.4 1.84247
\(345\) 4752.80 0.741688
\(346\) −996.429 −0.154822
\(347\) −11391.8 −1.76238 −0.881191 0.472761i \(-0.843257\pi\)
−0.881191 + 0.472761i \(0.843257\pi\)
\(348\) 383.261 0.0590373
\(349\) 899.513 0.137965 0.0689825 0.997618i \(-0.478025\pi\)
0.0689825 + 0.997618i \(0.478025\pi\)
\(350\) 0 0
\(351\) −762.898 −0.116013
\(352\) 1590.96 0.240905
\(353\) 4777.73 0.720376 0.360188 0.932880i \(-0.382712\pi\)
0.360188 + 0.932880i \(0.382712\pi\)
\(354\) 3738.59 0.561311
\(355\) 1948.90 0.291371
\(356\) −938.472 −0.139716
\(357\) 0 0
\(358\) 3059.06 0.451609
\(359\) −853.360 −0.125456 −0.0627279 0.998031i \(-0.519980\pi\)
−0.0627279 + 0.998031i \(0.519980\pi\)
\(360\) 3721.69 0.544862
\(361\) 5370.76 0.783023
\(362\) −3794.32 −0.550898
\(363\) −363.000 −0.0524864
\(364\) 0 0
\(365\) 921.622 0.132164
\(366\) 511.842 0.0730994
\(367\) 8399.94 1.19475 0.597375 0.801962i \(-0.296210\pi\)
0.597375 + 0.801962i \(0.296210\pi\)
\(368\) −2226.26 −0.315358
\(369\) 131.890 0.0186068
\(370\) −15174.6 −2.13214
\(371\) 0 0
\(372\) 2207.71 0.307700
\(373\) −3386.21 −0.470058 −0.235029 0.971988i \(-0.575518\pi\)
−0.235029 + 0.971988i \(0.575518\pi\)
\(374\) 642.359 0.0888117
\(375\) 1895.17 0.260976
\(376\) 7013.18 0.961907
\(377\) 1032.68 0.141076
\(378\) 0 0
\(379\) −8867.11 −1.20178 −0.600888 0.799334i \(-0.705186\pi\)
−0.600888 + 0.799334i \(0.705186\pi\)
\(380\) −6551.88 −0.884485
\(381\) 1895.67 0.254903
\(382\) −10634.6 −1.42438
\(383\) 5452.97 0.727503 0.363752 0.931496i \(-0.381496\pi\)
0.363752 + 0.931496i \(0.381496\pi\)
\(384\) 303.514 0.0403350
\(385\) 0 0
\(386\) 3804.39 0.501653
\(387\) −4336.40 −0.569590
\(388\) 5336.34 0.698226
\(389\) 524.204 0.0683244 0.0341622 0.999416i \(-0.489124\pi\)
0.0341622 + 0.999416i \(0.489124\pi\)
\(390\) 3049.25 0.395910
\(391\) 2571.84 0.332644
\(392\) 0 0
\(393\) 1315.44 0.168843
\(394\) 6391.88 0.817305
\(395\) −12530.6 −1.59615
\(396\) −346.055 −0.0439140
\(397\) −10730.3 −1.35652 −0.678258 0.734824i \(-0.737265\pi\)
−0.678258 + 0.734824i \(0.737265\pi\)
\(398\) −1684.99 −0.212214
\(399\) 0 0
\(400\) −3864.88 −0.483110
\(401\) −10042.7 −1.25065 −0.625325 0.780365i \(-0.715034\pi\)
−0.625325 + 0.780365i \(0.715034\pi\)
\(402\) −1855.45 −0.230202
\(403\) 5948.57 0.735283
\(404\) −5292.27 −0.651734
\(405\) −1372.88 −0.168441
\(406\) 0 0
\(407\) 4640.25 0.565132
\(408\) 2013.88 0.244368
\(409\) 5230.60 0.632363 0.316182 0.948699i \(-0.397599\pi\)
0.316182 + 0.948699i \(0.397599\pi\)
\(410\) −527.154 −0.0634982
\(411\) 54.7577 0.00657177
\(412\) 3993.60 0.477550
\(413\) 0 0
\(414\) 1785.45 0.211957
\(415\) 1871.42 0.221360
\(416\) 4086.68 0.481649
\(417\) 9398.08 1.10366
\(418\) −2581.81 −0.302106
\(419\) −8177.82 −0.953491 −0.476746 0.879041i \(-0.658184\pi\)
−0.476746 + 0.879041i \(0.658184\pi\)
\(420\) 0 0
\(421\) −15485.0 −1.79262 −0.896312 0.443424i \(-0.853764\pi\)
−0.896312 + 0.443424i \(0.853764\pi\)
\(422\) 9516.60 1.09777
\(423\) −2587.06 −0.297369
\(424\) −15373.3 −1.76084
\(425\) 4464.83 0.509591
\(426\) 732.127 0.0832669
\(427\) 0 0
\(428\) 4401.30 0.497068
\(429\) −932.431 −0.104938
\(430\) 17332.3 1.94381
\(431\) −10033.9 −1.12139 −0.560694 0.828023i \(-0.689465\pi\)
−0.560694 + 0.828023i \(0.689465\pi\)
\(432\) 643.068 0.0716195
\(433\) −146.713 −0.0162831 −0.00814155 0.999967i \(-0.502592\pi\)
−0.00814155 + 0.999967i \(0.502592\pi\)
\(434\) 0 0
\(435\) 1858.36 0.204832
\(436\) 3587.66 0.394078
\(437\) −10336.9 −1.13154
\(438\) 346.218 0.0377693
\(439\) 9094.35 0.988724 0.494362 0.869256i \(-0.335402\pi\)
0.494362 + 0.869256i \(0.335402\pi\)
\(440\) 4548.73 0.492846
\(441\) 0 0
\(442\) 1650.01 0.177564
\(443\) −4491.35 −0.481694 −0.240847 0.970563i \(-0.577425\pi\)
−0.240847 + 0.970563i \(0.577425\pi\)
\(444\) 4423.65 0.472831
\(445\) −4550.48 −0.484749
\(446\) −2680.02 −0.284535
\(447\) −6396.41 −0.676823
\(448\) 0 0
\(449\) −14455.5 −1.51937 −0.759687 0.650289i \(-0.774648\pi\)
−0.759687 + 0.650289i \(0.774648\pi\)
\(450\) 3099.62 0.324705
\(451\) 161.198 0.0168304
\(452\) 3267.99 0.340073
\(453\) 4807.66 0.498640
\(454\) 12683.6 1.31117
\(455\) 0 0
\(456\) −8094.34 −0.831254
\(457\) −12296.9 −1.25869 −0.629347 0.777125i \(-0.716677\pi\)
−0.629347 + 0.777125i \(0.716677\pi\)
\(458\) −929.439 −0.0948250
\(459\) −742.893 −0.0755452
\(460\) 5537.82 0.561309
\(461\) 17486.9 1.76670 0.883349 0.468715i \(-0.155283\pi\)
0.883349 + 0.468715i \(0.155283\pi\)
\(462\) 0 0
\(463\) −5437.07 −0.545750 −0.272875 0.962050i \(-0.587974\pi\)
−0.272875 + 0.962050i \(0.587974\pi\)
\(464\) −870.475 −0.0870922
\(465\) 10704.8 1.06757
\(466\) −12123.6 −1.20518
\(467\) 6152.14 0.609609 0.304804 0.952415i \(-0.401409\pi\)
0.304804 + 0.952415i \(0.401409\pi\)
\(468\) −888.906 −0.0877985
\(469\) 0 0
\(470\) 10340.3 1.01481
\(471\) −11573.9 −1.13227
\(472\) 14325.7 1.39702
\(473\) −5300.04 −0.515214
\(474\) −4707.26 −0.456143
\(475\) −17945.3 −1.73345
\(476\) 0 0
\(477\) 5671.00 0.544355
\(478\) 3561.67 0.340809
\(479\) −13097.1 −1.24932 −0.624659 0.780897i \(-0.714762\pi\)
−0.624659 + 0.780897i \(0.714762\pi\)
\(480\) 7354.20 0.699317
\(481\) 11919.3 1.12988
\(482\) −12422.9 −1.17396
\(483\) 0 0
\(484\) −422.957 −0.0397217
\(485\) 25875.0 2.42252
\(486\) −515.738 −0.0481365
\(487\) −11122.7 −1.03494 −0.517472 0.855700i \(-0.673127\pi\)
−0.517472 + 0.855700i \(0.673127\pi\)
\(488\) 1961.29 0.181934
\(489\) −1812.19 −0.167587
\(490\) 0 0
\(491\) 5595.11 0.514265 0.257132 0.966376i \(-0.417222\pi\)
0.257132 + 0.966376i \(0.417222\pi\)
\(492\) 153.674 0.0140816
\(493\) 1005.60 0.0918661
\(494\) −6631.85 −0.604010
\(495\) −1677.96 −0.152361
\(496\) −5014.21 −0.453921
\(497\) 0 0
\(498\) 703.023 0.0632594
\(499\) −15905.5 −1.42691 −0.713453 0.700703i \(-0.752870\pi\)
−0.713453 + 0.700703i \(0.752870\pi\)
\(500\) 2208.19 0.197507
\(501\) 1033.34 0.0921485
\(502\) −1951.77 −0.173530
\(503\) 9783.89 0.867280 0.433640 0.901086i \(-0.357229\pi\)
0.433640 + 0.901086i \(0.357229\pi\)
\(504\) 0 0
\(505\) −25661.3 −2.26121
\(506\) 2182.22 0.191722
\(507\) 4195.88 0.367546
\(508\) 2208.77 0.192910
\(509\) −9275.87 −0.807752 −0.403876 0.914814i \(-0.632337\pi\)
−0.403876 + 0.914814i \(0.632337\pi\)
\(510\) 2969.29 0.257809
\(511\) 0 0
\(512\) −8093.51 −0.698605
\(513\) 2985.88 0.256978
\(514\) −7179.18 −0.616070
\(515\) 19364.2 1.65687
\(516\) −5052.64 −0.431066
\(517\) −3161.96 −0.268980
\(518\) 0 0
\(519\) 1408.46 0.119123
\(520\) 11684.2 0.985361
\(521\) −24.4500 −0.00205600 −0.00102800 0.999999i \(-0.500327\pi\)
−0.00102800 + 0.999999i \(0.500327\pi\)
\(522\) 698.118 0.0585360
\(523\) 10785.6 0.901765 0.450882 0.892583i \(-0.351109\pi\)
0.450882 + 0.892583i \(0.351109\pi\)
\(524\) 1532.71 0.127780
\(525\) 0 0
\(526\) 3783.24 0.313607
\(527\) 5792.58 0.478802
\(528\) 785.972 0.0647823
\(529\) −3429.97 −0.281907
\(530\) −22666.6 −1.85769
\(531\) −5284.53 −0.431882
\(532\) 0 0
\(533\) 414.067 0.0336496
\(534\) −1709.44 −0.138530
\(535\) 21341.1 1.72459
\(536\) −7109.77 −0.572939
\(537\) −4324.00 −0.347476
\(538\) −4101.31 −0.328661
\(539\) 0 0
\(540\) −1599.63 −0.127476
\(541\) −4418.05 −0.351103 −0.175551 0.984470i \(-0.556171\pi\)
−0.175551 + 0.984470i \(0.556171\pi\)
\(542\) −1174.15 −0.0930517
\(543\) 5363.30 0.423870
\(544\) 3979.52 0.313640
\(545\) 17395.9 1.36727
\(546\) 0 0
\(547\) 19944.6 1.55900 0.779499 0.626404i \(-0.215474\pi\)
0.779499 + 0.626404i \(0.215474\pi\)
\(548\) 63.8020 0.00497352
\(549\) −723.493 −0.0562439
\(550\) 3788.42 0.293707
\(551\) −4041.78 −0.312496
\(552\) 6841.55 0.527528
\(553\) 0 0
\(554\) 6496.74 0.498231
\(555\) 21449.5 1.64050
\(556\) 10950.4 0.835250
\(557\) 1189.41 0.0904791 0.0452396 0.998976i \(-0.485595\pi\)
0.0452396 + 0.998976i \(0.485595\pi\)
\(558\) 4021.38 0.305087
\(559\) −13614.1 −1.03008
\(560\) 0 0
\(561\) −907.980 −0.0683332
\(562\) 14306.0 1.07378
\(563\) −18622.6 −1.39405 −0.697024 0.717048i \(-0.745493\pi\)
−0.697024 + 0.717048i \(0.745493\pi\)
\(564\) −3014.36 −0.225049
\(565\) 15845.9 1.17990
\(566\) −2168.19 −0.161017
\(567\) 0 0
\(568\) 2805.39 0.207239
\(569\) −5657.38 −0.416819 −0.208409 0.978042i \(-0.566829\pi\)
−0.208409 + 0.978042i \(0.566829\pi\)
\(570\) −11934.4 −0.876975
\(571\) −3817.80 −0.279808 −0.139904 0.990165i \(-0.544679\pi\)
−0.139904 + 0.990165i \(0.544679\pi\)
\(572\) −1086.44 −0.0794167
\(573\) 15032.1 1.09594
\(574\) 0 0
\(575\) 15167.9 1.10008
\(576\) 4477.55 0.323896
\(577\) −19448.6 −1.40322 −0.701609 0.712562i \(-0.747535\pi\)
−0.701609 + 0.712562i \(0.747535\pi\)
\(578\) −8820.50 −0.634748
\(579\) −5377.53 −0.385980
\(580\) 2165.31 0.155017
\(581\) 0 0
\(582\) 9720.25 0.692298
\(583\) 6931.23 0.492388
\(584\) 1326.65 0.0940022
\(585\) −4310.14 −0.304620
\(586\) 14629.8 1.03132
\(587\) 10528.7 0.740314 0.370157 0.928969i \(-0.379304\pi\)
0.370157 + 0.928969i \(0.379304\pi\)
\(588\) 0 0
\(589\) −23281.9 −1.62872
\(590\) 21121.9 1.47386
\(591\) −9034.98 −0.628848
\(592\) −10047.1 −0.697524
\(593\) −3761.02 −0.260450 −0.130225 0.991484i \(-0.541570\pi\)
−0.130225 + 0.991484i \(0.541570\pi\)
\(594\) −630.346 −0.0435411
\(595\) 0 0
\(596\) −7452.91 −0.512220
\(597\) 2381.75 0.163281
\(598\) 5605.41 0.383315
\(599\) −10414.7 −0.710405 −0.355203 0.934789i \(-0.615588\pi\)
−0.355203 + 0.934789i \(0.615588\pi\)
\(600\) 11877.2 0.808143
\(601\) −18110.1 −1.22916 −0.614580 0.788854i \(-0.710675\pi\)
−0.614580 + 0.788854i \(0.710675\pi\)
\(602\) 0 0
\(603\) 2622.69 0.177121
\(604\) 5601.75 0.377370
\(605\) −2050.84 −0.137816
\(606\) −9639.97 −0.646200
\(607\) −17191.2 −1.14954 −0.574770 0.818315i \(-0.694909\pi\)
−0.574770 + 0.818315i \(0.694909\pi\)
\(608\) −15994.7 −1.06689
\(609\) 0 0
\(610\) 2891.75 0.191940
\(611\) −8122.07 −0.537780
\(612\) −865.596 −0.0571726
\(613\) −2876.07 −0.189500 −0.0947498 0.995501i \(-0.530205\pi\)
−0.0947498 + 0.995501i \(0.530205\pi\)
\(614\) 2319.30 0.152442
\(615\) 745.136 0.0488566
\(616\) 0 0
\(617\) 3937.58 0.256923 0.128461 0.991715i \(-0.458996\pi\)
0.128461 + 0.991715i \(0.458996\pi\)
\(618\) 7274.41 0.473495
\(619\) 4962.16 0.322207 0.161103 0.986938i \(-0.448495\pi\)
0.161103 + 0.986938i \(0.448495\pi\)
\(620\) 12472.9 0.807940
\(621\) −2523.75 −0.163083
\(622\) 2352.42 0.151645
\(623\) 0 0
\(624\) 2018.91 0.129521
\(625\) −9576.84 −0.612918
\(626\) −548.080 −0.0349931
\(627\) 3649.41 0.232446
\(628\) −13485.6 −0.856900
\(629\) 11606.8 0.735758
\(630\) 0 0
\(631\) 11653.3 0.735199 0.367600 0.929984i \(-0.380180\pi\)
0.367600 + 0.929984i \(0.380180\pi\)
\(632\) −18037.5 −1.13527
\(633\) −13451.8 −0.844646
\(634\) 11699.8 0.732898
\(635\) 10709.9 0.669309
\(636\) 6607.68 0.411968
\(637\) 0 0
\(638\) 853.255 0.0529478
\(639\) −1034.87 −0.0640669
\(640\) 1714.76 0.105909
\(641\) 16260.7 1.00196 0.500982 0.865458i \(-0.332972\pi\)
0.500982 + 0.865458i \(0.332972\pi\)
\(642\) 8017.06 0.492847
\(643\) −16247.1 −0.996459 −0.498230 0.867045i \(-0.666016\pi\)
−0.498230 + 0.867045i \(0.666016\pi\)
\(644\) 0 0
\(645\) −24499.3 −1.49560
\(646\) −6457.94 −0.393319
\(647\) −27096.7 −1.64650 −0.823248 0.567682i \(-0.807840\pi\)
−0.823248 + 0.567682i \(0.807840\pi\)
\(648\) −1976.22 −0.119805
\(649\) −6458.88 −0.390652
\(650\) 9731.25 0.587217
\(651\) 0 0
\(652\) −2111.51 −0.126830
\(653\) 24607.4 1.47467 0.737336 0.675526i \(-0.236083\pi\)
0.737336 + 0.675526i \(0.236083\pi\)
\(654\) 6534.99 0.390732
\(655\) 7431.85 0.443338
\(656\) −349.029 −0.0207733
\(657\) −489.383 −0.0290603
\(658\) 0 0
\(659\) −17794.2 −1.05184 −0.525921 0.850534i \(-0.676279\pi\)
−0.525921 + 0.850534i \(0.676279\pi\)
\(660\) −1955.11 −0.115307
\(661\) 15551.6 0.915109 0.457555 0.889181i \(-0.348725\pi\)
0.457555 + 0.889181i \(0.348725\pi\)
\(662\) 19818.8 1.16356
\(663\) −2332.31 −0.136621
\(664\) 2693.87 0.157443
\(665\) 0 0
\(666\) 8057.76 0.468816
\(667\) 3416.22 0.198315
\(668\) 1204.02 0.0697380
\(669\) 3788.23 0.218926
\(670\) −10482.7 −0.604451
\(671\) −884.269 −0.0508745
\(672\) 0 0
\(673\) −27136.0 −1.55426 −0.777129 0.629342i \(-0.783325\pi\)
−0.777129 + 0.629342i \(0.783325\pi\)
\(674\) 1102.83 0.0630258
\(675\) −4381.34 −0.249834
\(676\) 4888.92 0.278159
\(677\) 14990.2 0.850992 0.425496 0.904960i \(-0.360100\pi\)
0.425496 + 0.904960i \(0.360100\pi\)
\(678\) 5952.70 0.337186
\(679\) 0 0
\(680\) 11377.8 0.641647
\(681\) −17928.3 −1.00883
\(682\) 4915.02 0.275962
\(683\) 32077.5 1.79709 0.898544 0.438884i \(-0.144626\pi\)
0.898544 + 0.438884i \(0.144626\pi\)
\(684\) 3479.06 0.194481
\(685\) 309.364 0.0172558
\(686\) 0 0
\(687\) 1313.77 0.0729599
\(688\) 11475.7 0.635912
\(689\) 17804.1 0.984445
\(690\) 10087.2 0.556543
\(691\) −2469.29 −0.135943 −0.0679713 0.997687i \(-0.521653\pi\)
−0.0679713 + 0.997687i \(0.521653\pi\)
\(692\) 1641.10 0.0901519
\(693\) 0 0
\(694\) −24177.8 −1.32245
\(695\) 53096.3 2.89793
\(696\) 2675.07 0.145687
\(697\) 403.209 0.0219119
\(698\) 1909.11 0.103525
\(699\) 17136.8 0.927289
\(700\) 0 0
\(701\) 516.623 0.0278353 0.0139177 0.999903i \(-0.495570\pi\)
0.0139177 + 0.999903i \(0.495570\pi\)
\(702\) −1619.16 −0.0870530
\(703\) −46650.7 −2.50279
\(704\) 5472.56 0.292975
\(705\) −14616.1 −0.780815
\(706\) 10140.1 0.540552
\(707\) 0 0
\(708\) −6157.38 −0.326848
\(709\) 9574.56 0.507165 0.253583 0.967314i \(-0.418391\pi\)
0.253583 + 0.967314i \(0.418391\pi\)
\(710\) 4136.30 0.218637
\(711\) 6653.75 0.350964
\(712\) −6550.31 −0.344780
\(713\) 19678.5 1.03361
\(714\) 0 0
\(715\) −5267.95 −0.275539
\(716\) −5038.20 −0.262970
\(717\) −5034.45 −0.262224
\(718\) −1811.15 −0.0941388
\(719\) 32468.9 1.68412 0.842062 0.539381i \(-0.181342\pi\)
0.842062 + 0.539381i \(0.181342\pi\)
\(720\) 3633.14 0.188054
\(721\) 0 0
\(722\) 11398.8 0.587561
\(723\) 17559.9 0.903266
\(724\) 6249.16 0.320785
\(725\) 5930.70 0.303808
\(726\) −770.423 −0.0393844
\(727\) 27791.5 1.41779 0.708893 0.705316i \(-0.249195\pi\)
0.708893 + 0.705316i \(0.249195\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 1956.03 0.0991725
\(731\) −13257.1 −0.670769
\(732\) −842.992 −0.0425654
\(733\) −30827.5 −1.55340 −0.776698 0.629873i \(-0.783107\pi\)
−0.776698 + 0.629873i \(0.783107\pi\)
\(734\) 17827.9 0.896510
\(735\) 0 0
\(736\) 13519.2 0.677069
\(737\) 3205.51 0.160212
\(738\) 279.919 0.0139620
\(739\) −32412.6 −1.61342 −0.806709 0.590949i \(-0.798753\pi\)
−0.806709 + 0.590949i \(0.798753\pi\)
\(740\) 24992.3 1.24153
\(741\) 9374.17 0.464735
\(742\) 0 0
\(743\) −35098.6 −1.73303 −0.866517 0.499148i \(-0.833646\pi\)
−0.866517 + 0.499148i \(0.833646\pi\)
\(744\) 15409.3 0.759316
\(745\) −36137.8 −1.77716
\(746\) −7186.83 −0.352719
\(747\) −993.728 −0.0486729
\(748\) −1057.95 −0.0517146
\(749\) 0 0
\(750\) 4022.26 0.195830
\(751\) 6436.56 0.312748 0.156374 0.987698i \(-0.450020\pi\)
0.156374 + 0.987698i \(0.450020\pi\)
\(752\) 6846.32 0.331994
\(753\) 2758.85 0.133517
\(754\) 2191.74 0.105860
\(755\) 27161.8 1.30930
\(756\) 0 0
\(757\) −34769.8 −1.66939 −0.834697 0.550709i \(-0.814357\pi\)
−0.834697 + 0.550709i \(0.814357\pi\)
\(758\) −18819.4 −0.901781
\(759\) −3084.58 −0.147514
\(760\) −45730.6 −2.18266
\(761\) −28761.6 −1.37005 −0.685024 0.728521i \(-0.740208\pi\)
−0.685024 + 0.728521i \(0.740208\pi\)
\(762\) 4023.32 0.191272
\(763\) 0 0
\(764\) 17514.9 0.829409
\(765\) −4197.12 −0.198362
\(766\) 11573.3 0.545900
\(767\) −16590.8 −0.781041
\(768\) 12584.3 0.591272
\(769\) 26572.0 1.24605 0.623023 0.782203i \(-0.285904\pi\)
0.623023 + 0.782203i \(0.285904\pi\)
\(770\) 0 0
\(771\) 10147.8 0.474014
\(772\) −6265.74 −0.292110
\(773\) 19576.8 0.910903 0.455452 0.890260i \(-0.349478\pi\)
0.455452 + 0.890260i \(0.349478\pi\)
\(774\) −9203.47 −0.427406
\(775\) 34162.7 1.58343
\(776\) 37246.4 1.72303
\(777\) 0 0
\(778\) 1112.56 0.0512689
\(779\) −1620.60 −0.0745368
\(780\) −5022.05 −0.230536
\(781\) −1264.84 −0.0579507
\(782\) 5458.42 0.249607
\(783\) −986.795 −0.0450385
\(784\) 0 0
\(785\) −65389.1 −2.97304
\(786\) 2791.86 0.126695
\(787\) 18552.5 0.840313 0.420157 0.907452i \(-0.361975\pi\)
0.420157 + 0.907452i \(0.361975\pi\)
\(788\) −10527.3 −0.475912
\(789\) −5347.65 −0.241294
\(790\) −26594.6 −1.19771
\(791\) 0 0
\(792\) −2415.38 −0.108367
\(793\) −2271.40 −0.101715
\(794\) −22773.7 −1.01789
\(795\) 32039.5 1.42934
\(796\) 2775.14 0.123571
\(797\) 28888.3 1.28391 0.641955 0.766742i \(-0.278123\pi\)
0.641955 + 0.766742i \(0.278123\pi\)
\(798\) 0 0
\(799\) −7909.08 −0.350192
\(800\) 23469.9 1.03723
\(801\) 2416.31 0.106587
\(802\) −21314.5 −0.938455
\(803\) −598.135 −0.0262861
\(804\) 3055.88 0.134045
\(805\) 0 0
\(806\) 12625.1 0.551738
\(807\) 5797.23 0.252878
\(808\) −36938.8 −1.60830
\(809\) −26185.6 −1.13799 −0.568997 0.822339i \(-0.692669\pi\)
−0.568997 + 0.822339i \(0.692669\pi\)
\(810\) −2913.76 −0.126394
\(811\) −27785.6 −1.20306 −0.601532 0.798849i \(-0.705443\pi\)
−0.601532 + 0.798849i \(0.705443\pi\)
\(812\) 0 0
\(813\) 1659.67 0.0715955
\(814\) 9848.37 0.424060
\(815\) −10238.4 −0.440042
\(816\) 1965.97 0.0843415
\(817\) 53283.8 2.28172
\(818\) 11101.3 0.474509
\(819\) 0 0
\(820\) 868.210 0.0369746
\(821\) −30525.2 −1.29761 −0.648804 0.760956i \(-0.724730\pi\)
−0.648804 + 0.760956i \(0.724730\pi\)
\(822\) 116.217 0.00493129
\(823\) −15822.7 −0.670163 −0.335082 0.942189i \(-0.608764\pi\)
−0.335082 + 0.942189i \(0.608764\pi\)
\(824\) 27874.4 1.17846
\(825\) −5354.97 −0.225983
\(826\) 0 0
\(827\) −32901.3 −1.38342 −0.691710 0.722175i \(-0.743143\pi\)
−0.691710 + 0.722175i \(0.743143\pi\)
\(828\) −2940.59 −0.123421
\(829\) −15019.6 −0.629253 −0.314627 0.949215i \(-0.601879\pi\)
−0.314627 + 0.949215i \(0.601879\pi\)
\(830\) 3971.87 0.166103
\(831\) −9183.20 −0.383348
\(832\) 14057.2 0.585754
\(833\) 0 0
\(834\) 19946.3 0.828158
\(835\) 5838.08 0.241958
\(836\) 4252.18 0.175915
\(837\) −5684.25 −0.234739
\(838\) −17356.4 −0.715475
\(839\) −44553.9 −1.83334 −0.916669 0.399646i \(-0.869133\pi\)
−0.916669 + 0.399646i \(0.869133\pi\)
\(840\) 0 0
\(841\) −23053.2 −0.945231
\(842\) −32865.1 −1.34514
\(843\) −20221.7 −0.826182
\(844\) −15673.6 −0.639228
\(845\) 23705.5 0.965080
\(846\) −5490.72 −0.223138
\(847\) 0 0
\(848\) −15007.6 −0.607739
\(849\) 3064.75 0.123889
\(850\) 9476.07 0.382384
\(851\) 39430.4 1.58831
\(852\) −1205.80 −0.0484858
\(853\) 7099.01 0.284953 0.142477 0.989798i \(-0.454493\pi\)
0.142477 + 0.989798i \(0.454493\pi\)
\(854\) 0 0
\(855\) 16869.3 0.674759
\(856\) 30720.1 1.22662
\(857\) 34685.9 1.38255 0.691276 0.722591i \(-0.257049\pi\)
0.691276 + 0.722591i \(0.257049\pi\)
\(858\) −1978.97 −0.0787424
\(859\) 16370.1 0.650222 0.325111 0.945676i \(-0.394598\pi\)
0.325111 + 0.945676i \(0.394598\pi\)
\(860\) −28545.9 −1.13187
\(861\) 0 0
\(862\) −21295.8 −0.841460
\(863\) 29234.6 1.15314 0.576569 0.817049i \(-0.304391\pi\)
0.576569 + 0.817049i \(0.304391\pi\)
\(864\) −3905.10 −0.153766
\(865\) 7957.38 0.312785
\(866\) −311.381 −0.0122184
\(867\) 12467.8 0.488386
\(868\) 0 0
\(869\) 8132.37 0.317459
\(870\) 3944.15 0.153700
\(871\) 8233.93 0.320317
\(872\) 25041.0 0.972473
\(873\) −13739.7 −0.532665
\(874\) −21938.8 −0.849076
\(875\) 0 0
\(876\) −570.214 −0.0219929
\(877\) −23992.2 −0.923783 −0.461892 0.886936i \(-0.652829\pi\)
−0.461892 + 0.886936i \(0.652829\pi\)
\(878\) 19301.7 0.741913
\(879\) −20679.4 −0.793515
\(880\) 4440.50 0.170102
\(881\) −18970.8 −0.725473 −0.362737 0.931892i \(-0.618158\pi\)
−0.362737 + 0.931892i \(0.618158\pi\)
\(882\) 0 0
\(883\) −44844.7 −1.70911 −0.854556 0.519360i \(-0.826170\pi\)
−0.854556 + 0.519360i \(0.826170\pi\)
\(884\) −2717.54 −0.103394
\(885\) −29856.0 −1.13401
\(886\) −9532.34 −0.361451
\(887\) 31762.1 1.20233 0.601166 0.799125i \(-0.294703\pi\)
0.601166 + 0.799125i \(0.294703\pi\)
\(888\) 30876.0 1.16681
\(889\) 0 0
\(890\) −9657.84 −0.363743
\(891\) 891.000 0.0335013
\(892\) 4413.94 0.165683
\(893\) 31788.7 1.19123
\(894\) −13575.6 −0.507871
\(895\) −24429.3 −0.912381
\(896\) 0 0
\(897\) −7923.30 −0.294929
\(898\) −30680.1 −1.14010
\(899\) 7694.37 0.285452
\(900\) −5105.00 −0.189074
\(901\) 17337.2 0.641051
\(902\) 342.124 0.0126291
\(903\) 0 0
\(904\) 22809.8 0.839205
\(905\) 30301.1 1.11297
\(906\) 10203.7 0.374166
\(907\) −28138.3 −1.03012 −0.515059 0.857155i \(-0.672230\pi\)
−0.515059 + 0.857155i \(0.672230\pi\)
\(908\) −20889.5 −0.763485
\(909\) 13626.2 0.497197
\(910\) 0 0
\(911\) 5419.88 0.197111 0.0985557 0.995132i \(-0.468578\pi\)
0.0985557 + 0.995132i \(0.468578\pi\)
\(912\) −7901.75 −0.286900
\(913\) −1214.56 −0.0440263
\(914\) −26098.6 −0.944491
\(915\) −4087.51 −0.147682
\(916\) 1530.77 0.0552161
\(917\) 0 0
\(918\) −1576.70 −0.0566872
\(919\) 16582.6 0.595224 0.297612 0.954687i \(-0.403810\pi\)
0.297612 + 0.954687i \(0.403810\pi\)
\(920\) 38652.7 1.38515
\(921\) −3278.34 −0.117291
\(922\) 37113.9 1.32568
\(923\) −3248.97 −0.115862
\(924\) 0 0
\(925\) 68452.9 2.43321
\(926\) −11539.5 −0.409516
\(927\) −10282.4 −0.364315
\(928\) 5286.05 0.186986
\(929\) −1667.10 −0.0588760 −0.0294380 0.999567i \(-0.509372\pi\)
−0.0294380 + 0.999567i \(0.509372\pi\)
\(930\) 22719.6 0.801080
\(931\) 0 0
\(932\) 19967.3 0.701773
\(933\) −3325.16 −0.116678
\(934\) 13057.2 0.457435
\(935\) −5129.81 −0.179425
\(936\) −6204.35 −0.216662
\(937\) −42610.6 −1.48562 −0.742812 0.669501i \(-0.766508\pi\)
−0.742812 + 0.669501i \(0.766508\pi\)
\(938\) 0 0
\(939\) 774.715 0.0269243
\(940\) −17030.2 −0.590921
\(941\) 27501.4 0.952731 0.476366 0.879247i \(-0.341954\pi\)
0.476366 + 0.879247i \(0.341954\pi\)
\(942\) −24564.2 −0.849624
\(943\) 1369.78 0.0473023
\(944\) 13984.8 0.482169
\(945\) 0 0
\(946\) −11248.7 −0.386603
\(947\) 21993.8 0.754702 0.377351 0.926070i \(-0.376835\pi\)
0.377351 + 0.926070i \(0.376835\pi\)
\(948\) 7752.75 0.265609
\(949\) −1536.42 −0.0525545
\(950\) −38086.8 −1.30074
\(951\) −16537.7 −0.563904
\(952\) 0 0
\(953\) 1684.49 0.0572570 0.0286285 0.999590i \(-0.490886\pi\)
0.0286285 + 0.999590i \(0.490886\pi\)
\(954\) 12036.0 0.408470
\(955\) 84926.8 2.87766
\(956\) −5865.98 −0.198451
\(957\) −1206.08 −0.0407389
\(958\) −27797.1 −0.937456
\(959\) 0 0
\(960\) 25296.8 0.850469
\(961\) 14531.0 0.487764
\(962\) 25297.3 0.847836
\(963\) −11332.2 −0.379205
\(964\) 20460.3 0.683592
\(965\) −30381.4 −1.01349
\(966\) 0 0
\(967\) −16110.4 −0.535755 −0.267877 0.963453i \(-0.586322\pi\)
−0.267877 + 0.963453i \(0.586322\pi\)
\(968\) −2952.14 −0.0980220
\(969\) 9128.35 0.302626
\(970\) 54916.5 1.81780
\(971\) 13719.1 0.453416 0.226708 0.973963i \(-0.427204\pi\)
0.226708 + 0.973963i \(0.427204\pi\)
\(972\) 849.409 0.0280296
\(973\) 0 0
\(974\) −23606.6 −0.776596
\(975\) −13755.2 −0.451814
\(976\) 1914.63 0.0627928
\(977\) 22717.9 0.743919 0.371960 0.928249i \(-0.378686\pi\)
0.371960 + 0.928249i \(0.378686\pi\)
\(978\) −3846.16 −0.125753
\(979\) 2953.27 0.0964116
\(980\) 0 0
\(981\) −9237.27 −0.300635
\(982\) 11874.9 0.385891
\(983\) −22320.2 −0.724215 −0.362107 0.932136i \(-0.617943\pi\)
−0.362107 + 0.932136i \(0.617943\pi\)
\(984\) 1072.61 0.0347494
\(985\) −51044.9 −1.65119
\(986\) 2134.27 0.0689339
\(987\) 0 0
\(988\) 10922.5 0.351712
\(989\) −45036.9 −1.44802
\(990\) −3561.27 −0.114328
\(991\) −22364.4 −0.716882 −0.358441 0.933552i \(-0.616692\pi\)
−0.358441 + 0.933552i \(0.616692\pi\)
\(992\) 30449.3 0.974563
\(993\) −28014.0 −0.895265
\(994\) 0 0
\(995\) 13456.2 0.428733
\(996\) −1157.86 −0.0368356
\(997\) −4123.96 −0.131000 −0.0655000 0.997853i \(-0.520864\pi\)
−0.0655000 + 0.997853i \(0.520864\pi\)
\(998\) −33757.4 −1.07071
\(999\) −11389.7 −0.360715
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 1617.4.a.be.1.11 16
7.6 odd 2 1617.4.a.bf.1.11 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.4.a.be.1.11 16 1.1 even 1 trivial
1617.4.a.bf.1.11 yes 16 7.6 odd 2