Properties

Label 165.4.a.d.1.1
Level $165$
Weight $4$
Character 165.1
Self dual yes
Analytic conductor $9.735$
Analytic rank $1$
Dimension $3$
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] = [165,4,Mod(1,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, 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("165.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.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: \(9.73531515095\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.23612.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 20x + 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.26150\) of defining polynomial
Character \(\chi\) \(=\) 165.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.26150 q^{2} -3.00000 q^{3} +19.6833 q^{4} -5.00000 q^{5} +15.7845 q^{6} -10.3207 q^{7} -61.4719 q^{8} +9.00000 q^{9} +26.3075 q^{10} +11.0000 q^{11} -59.0500 q^{12} +63.9817 q^{13} +54.3024 q^{14} +15.0000 q^{15} +165.967 q^{16} +17.1461 q^{17} -47.3535 q^{18} +90.2104 q^{19} -98.4167 q^{20} +30.9621 q^{21} -57.8765 q^{22} -212.605 q^{23} +184.416 q^{24} +25.0000 q^{25} -336.639 q^{26} -27.0000 q^{27} -203.146 q^{28} +57.5461 q^{29} -78.9224 q^{30} -141.704 q^{31} -381.462 q^{32} -33.0000 q^{33} -90.2140 q^{34} +51.6035 q^{35} +177.150 q^{36} -257.963 q^{37} -474.642 q^{38} -191.945 q^{39} +307.359 q^{40} -225.914 q^{41} -162.907 q^{42} -347.445 q^{43} +216.517 q^{44} -45.0000 q^{45} +1118.62 q^{46} +404.364 q^{47} -497.902 q^{48} -236.483 q^{49} -131.537 q^{50} -51.4382 q^{51} +1259.37 q^{52} +259.568 q^{53} +142.060 q^{54} -55.0000 q^{55} +634.433 q^{56} -270.631 q^{57} -302.779 q^{58} -853.067 q^{59} +295.250 q^{60} -203.699 q^{61} +745.573 q^{62} -92.8864 q^{63} +679.320 q^{64} -319.908 q^{65} +173.629 q^{66} +266.890 q^{67} +337.492 q^{68} +637.814 q^{69} -271.512 q^{70} +92.4460 q^{71} -553.247 q^{72} -242.026 q^{73} +1357.27 q^{74} -75.0000 q^{75} +1775.64 q^{76} -113.528 q^{77} +1009.92 q^{78} -1021.60 q^{79} -829.837 q^{80} +81.0000 q^{81} +1188.65 q^{82} +706.415 q^{83} +609.438 q^{84} -85.7303 q^{85} +1828.08 q^{86} -172.638 q^{87} -676.191 q^{88} -440.218 q^{89} +236.767 q^{90} -660.336 q^{91} -4184.77 q^{92} +425.111 q^{93} -2127.56 q^{94} -451.052 q^{95} +1144.38 q^{96} -197.761 q^{97} +1244.25 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} + 12 q^{6} - 4 q^{7} - 48 q^{8} + 27 q^{9} + 20 q^{10} + 33 q^{11} - 66 q^{12} - 56 q^{14} + 45 q^{15} + 50 q^{16} - 218 q^{17} - 36 q^{18} + 146 q^{19} - 110 q^{20}+ \cdots + 297 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.26150 −1.86022 −0.930110 0.367281i \(-0.880289\pi\)
−0.930110 + 0.367281i \(0.880289\pi\)
\(3\) −3.00000 −0.577350
\(4\) 19.6833 2.46042
\(5\) −5.00000 −0.447214
\(6\) 15.7845 1.07400
\(7\) −10.3207 −0.557266 −0.278633 0.960398i \(-0.589881\pi\)
−0.278633 + 0.960398i \(0.589881\pi\)
\(8\) −61.4719 −2.71670
\(9\) 9.00000 0.333333
\(10\) 26.3075 0.831916
\(11\) 11.0000 0.301511
\(12\) −59.0500 −1.42052
\(13\) 63.9817 1.36502 0.682512 0.730874i \(-0.260887\pi\)
0.682512 + 0.730874i \(0.260887\pi\)
\(14\) 54.3024 1.03664
\(15\) 15.0000 0.258199
\(16\) 165.967 2.59324
\(17\) 17.1461 0.244620 0.122310 0.992492i \(-0.460970\pi\)
0.122310 + 0.992492i \(0.460970\pi\)
\(18\) −47.3535 −0.620073
\(19\) 90.2104 1.08925 0.544623 0.838681i \(-0.316673\pi\)
0.544623 + 0.838681i \(0.316673\pi\)
\(20\) −98.4167 −1.10033
\(21\) 30.9621 0.321738
\(22\) −57.8765 −0.560877
\(23\) −212.605 −1.92744 −0.963721 0.266913i \(-0.913996\pi\)
−0.963721 + 0.266913i \(0.913996\pi\)
\(24\) 184.416 1.56849
\(25\) 25.0000 0.200000
\(26\) −336.639 −2.53925
\(27\) −27.0000 −0.192450
\(28\) −203.146 −1.37111
\(29\) 57.5461 0.368484 0.184242 0.982881i \(-0.441017\pi\)
0.184242 + 0.982881i \(0.441017\pi\)
\(30\) −78.9224 −0.480307
\(31\) −141.704 −0.820991 −0.410496 0.911863i \(-0.634644\pi\)
−0.410496 + 0.911863i \(0.634644\pi\)
\(32\) −381.462 −2.10730
\(33\) −33.0000 −0.174078
\(34\) −90.2140 −0.455046
\(35\) 51.6035 0.249217
\(36\) 177.150 0.820139
\(37\) −257.963 −1.14619 −0.573093 0.819490i \(-0.694257\pi\)
−0.573093 + 0.819490i \(0.694257\pi\)
\(38\) −474.642 −2.02624
\(39\) −191.945 −0.788097
\(40\) 307.359 1.21494
\(41\) −225.914 −0.860533 −0.430266 0.902702i \(-0.641580\pi\)
−0.430266 + 0.902702i \(0.641580\pi\)
\(42\) −162.907 −0.598503
\(43\) −347.445 −1.23221 −0.616103 0.787666i \(-0.711289\pi\)
−0.616103 + 0.787666i \(0.711289\pi\)
\(44\) 216.517 0.741844
\(45\) −45.0000 −0.149071
\(46\) 1118.62 3.58547
\(47\) 404.364 1.25495 0.627473 0.778638i \(-0.284089\pi\)
0.627473 + 0.778638i \(0.284089\pi\)
\(48\) −497.902 −1.49721
\(49\) −236.483 −0.689455
\(50\) −131.537 −0.372044
\(51\) −51.4382 −0.141231
\(52\) 1259.37 3.35853
\(53\) 259.568 0.672726 0.336363 0.941732i \(-0.390803\pi\)
0.336363 + 0.941732i \(0.390803\pi\)
\(54\) 142.060 0.358000
\(55\) −55.0000 −0.134840
\(56\) 634.433 1.51392
\(57\) −270.631 −0.628877
\(58\) −302.779 −0.685462
\(59\) −853.067 −1.88237 −0.941185 0.337891i \(-0.890287\pi\)
−0.941185 + 0.337891i \(0.890287\pi\)
\(60\) 295.250 0.635277
\(61\) −203.699 −0.427558 −0.213779 0.976882i \(-0.568577\pi\)
−0.213779 + 0.976882i \(0.568577\pi\)
\(62\) 745.573 1.52722
\(63\) −92.8864 −0.185755
\(64\) 679.320 1.32680
\(65\) −319.908 −0.610458
\(66\) 173.629 0.323823
\(67\) 266.890 0.486653 0.243327 0.969944i \(-0.421761\pi\)
0.243327 + 0.969944i \(0.421761\pi\)
\(68\) 337.492 0.601866
\(69\) 637.814 1.11281
\(70\) −271.512 −0.463598
\(71\) 92.4460 0.154526 0.0772629 0.997011i \(-0.475382\pi\)
0.0772629 + 0.997011i \(0.475382\pi\)
\(72\) −553.247 −0.905566
\(73\) −242.026 −0.388040 −0.194020 0.980998i \(-0.562153\pi\)
−0.194020 + 0.980998i \(0.562153\pi\)
\(74\) 1357.27 2.13216
\(75\) −75.0000 −0.115470
\(76\) 1775.64 2.68000
\(77\) −113.528 −0.168022
\(78\) 1009.92 1.46603
\(79\) −1021.60 −1.45492 −0.727460 0.686150i \(-0.759299\pi\)
−0.727460 + 0.686150i \(0.759299\pi\)
\(80\) −829.837 −1.15973
\(81\) 81.0000 0.111111
\(82\) 1188.65 1.60078
\(83\) 706.415 0.934206 0.467103 0.884203i \(-0.345298\pi\)
0.467103 + 0.884203i \(0.345298\pi\)
\(84\) 609.438 0.791609
\(85\) −85.7303 −0.109397
\(86\) 1828.08 2.29217
\(87\) −172.638 −0.212745
\(88\) −676.191 −0.819116
\(89\) −440.218 −0.524304 −0.262152 0.965027i \(-0.584432\pi\)
−0.262152 + 0.965027i \(0.584432\pi\)
\(90\) 236.767 0.277305
\(91\) −660.336 −0.760682
\(92\) −4184.77 −4.74231
\(93\) 425.111 0.473999
\(94\) −2127.56 −2.33448
\(95\) −451.052 −0.487126
\(96\) 1144.38 1.21665
\(97\) −197.761 −0.207006 −0.103503 0.994629i \(-0.533005\pi\)
−0.103503 + 0.994629i \(0.533005\pi\)
\(98\) 1244.25 1.28254
\(99\) 99.0000 0.100504
\(100\) 492.084 0.492084
\(101\) 1400.62 1.37987 0.689937 0.723870i \(-0.257638\pi\)
0.689937 + 0.723870i \(0.257638\pi\)
\(102\) 270.642 0.262721
\(103\) −1345.70 −1.28734 −0.643669 0.765304i \(-0.722589\pi\)
−0.643669 + 0.765304i \(0.722589\pi\)
\(104\) −3933.07 −3.70836
\(105\) −154.811 −0.143885
\(106\) −1365.72 −1.25142
\(107\) −889.178 −0.803366 −0.401683 0.915779i \(-0.631575\pi\)
−0.401683 + 0.915779i \(0.631575\pi\)
\(108\) −531.450 −0.473508
\(109\) 1256.29 1.10395 0.551974 0.833861i \(-0.313875\pi\)
0.551974 + 0.833861i \(0.313875\pi\)
\(110\) 289.382 0.250832
\(111\) 773.890 0.661751
\(112\) −1712.90 −1.44512
\(113\) −2394.01 −1.99301 −0.996504 0.0835448i \(-0.973376\pi\)
−0.996504 + 0.0835448i \(0.973376\pi\)
\(114\) 1423.93 1.16985
\(115\) 1063.02 0.861978
\(116\) 1132.70 0.906626
\(117\) 575.835 0.455008
\(118\) 4488.41 3.50162
\(119\) −176.960 −0.136318
\(120\) −922.078 −0.701449
\(121\) 121.000 0.0909091
\(122\) 1071.76 0.795352
\(123\) 677.742 0.496829
\(124\) −2789.20 −2.01998
\(125\) −125.000 −0.0894427
\(126\) 488.721 0.345546
\(127\) 2065.57 1.44322 0.721612 0.692298i \(-0.243402\pi\)
0.721612 + 0.692298i \(0.243402\pi\)
\(128\) −522.548 −0.360837
\(129\) 1042.33 0.711414
\(130\) 1683.20 1.13559
\(131\) 785.526 0.523907 0.261953 0.965081i \(-0.415633\pi\)
0.261953 + 0.965081i \(0.415633\pi\)
\(132\) −649.550 −0.428304
\(133\) −931.035 −0.607000
\(134\) −1404.24 −0.905282
\(135\) 135.000 0.0860663
\(136\) −1054.00 −0.664558
\(137\) 1276.24 0.795885 0.397942 0.917410i \(-0.369724\pi\)
0.397942 + 0.917410i \(0.369724\pi\)
\(138\) −3355.86 −2.07007
\(139\) −2703.21 −1.64952 −0.824760 0.565482i \(-0.808690\pi\)
−0.824760 + 0.565482i \(0.808690\pi\)
\(140\) 1015.73 0.613178
\(141\) −1213.09 −0.724544
\(142\) −486.405 −0.287452
\(143\) 703.798 0.411570
\(144\) 1493.71 0.864413
\(145\) −287.731 −0.164791
\(146\) 1273.42 0.721840
\(147\) 709.449 0.398057
\(148\) −5077.58 −2.82010
\(149\) −2400.99 −1.32011 −0.660056 0.751217i \(-0.729467\pi\)
−0.660056 + 0.751217i \(0.729467\pi\)
\(150\) 394.612 0.214800
\(151\) −2517.30 −1.35665 −0.678326 0.734761i \(-0.737294\pi\)
−0.678326 + 0.734761i \(0.737294\pi\)
\(152\) −5545.40 −2.95916
\(153\) 154.315 0.0815399
\(154\) 597.326 0.312558
\(155\) 708.518 0.367158
\(156\) −3778.12 −1.93905
\(157\) 1391.42 0.707310 0.353655 0.935376i \(-0.384939\pi\)
0.353655 + 0.935376i \(0.384939\pi\)
\(158\) 5375.13 2.70647
\(159\) −778.705 −0.388398
\(160\) 1907.31 0.942412
\(161\) 2194.23 1.07410
\(162\) −426.181 −0.206691
\(163\) −2720.53 −1.30729 −0.653644 0.756802i \(-0.726761\pi\)
−0.653644 + 0.756802i \(0.726761\pi\)
\(164\) −4446.74 −2.11727
\(165\) 165.000 0.0778499
\(166\) −3716.80 −1.73783
\(167\) 2950.25 1.36705 0.683525 0.729927i \(-0.260446\pi\)
0.683525 + 0.729927i \(0.260446\pi\)
\(168\) −1903.30 −0.874064
\(169\) 1896.65 0.863292
\(170\) 451.070 0.203503
\(171\) 811.894 0.363082
\(172\) −6838.88 −3.03174
\(173\) 537.049 0.236018 0.118009 0.993013i \(-0.462349\pi\)
0.118009 + 0.993013i \(0.462349\pi\)
\(174\) 908.336 0.395752
\(175\) −258.018 −0.111453
\(176\) 1825.64 0.781891
\(177\) 2559.20 1.08679
\(178\) 2316.21 0.975320
\(179\) 2891.25 1.20728 0.603638 0.797259i \(-0.293717\pi\)
0.603638 + 0.797259i \(0.293717\pi\)
\(180\) −885.751 −0.366778
\(181\) 435.209 0.178723 0.0893615 0.995999i \(-0.471517\pi\)
0.0893615 + 0.995999i \(0.471517\pi\)
\(182\) 3474.36 1.41504
\(183\) 611.098 0.246851
\(184\) 13069.2 5.23628
\(185\) 1289.82 0.512590
\(186\) −2236.72 −0.881743
\(187\) 188.607 0.0737556
\(188\) 7959.23 3.08769
\(189\) 278.659 0.107246
\(190\) 2373.21 0.906161
\(191\) −3779.49 −1.43180 −0.715901 0.698202i \(-0.753984\pi\)
−0.715901 + 0.698202i \(0.753984\pi\)
\(192\) −2037.96 −0.766027
\(193\) 3751.91 1.39932 0.699660 0.714476i \(-0.253335\pi\)
0.699660 + 0.714476i \(0.253335\pi\)
\(194\) 1040.52 0.385076
\(195\) 959.725 0.352448
\(196\) −4654.78 −1.69635
\(197\) −3920.73 −1.41797 −0.708986 0.705223i \(-0.750847\pi\)
−0.708986 + 0.705223i \(0.750847\pi\)
\(198\) −520.888 −0.186959
\(199\) −597.084 −0.212694 −0.106347 0.994329i \(-0.533915\pi\)
−0.106347 + 0.994329i \(0.533915\pi\)
\(200\) −1536.80 −0.543340
\(201\) −800.669 −0.280969
\(202\) −7369.37 −2.56687
\(203\) −593.917 −0.205344
\(204\) −1012.48 −0.347488
\(205\) 1129.57 0.384842
\(206\) 7080.40 2.39473
\(207\) −1913.44 −0.642480
\(208\) 10618.9 3.53984
\(209\) 992.314 0.328420
\(210\) 814.536 0.267659
\(211\) −4384.55 −1.43054 −0.715272 0.698846i \(-0.753697\pi\)
−0.715272 + 0.698846i \(0.753697\pi\)
\(212\) 5109.17 1.65519
\(213\) −277.338 −0.0892155
\(214\) 4678.41 1.49444
\(215\) 1737.22 0.551059
\(216\) 1659.74 0.522829
\(217\) 1462.48 0.457510
\(218\) −6609.95 −2.05359
\(219\) 726.077 0.224035
\(220\) −1082.58 −0.331763
\(221\) 1097.03 0.333912
\(222\) −4071.82 −1.23100
\(223\) −2333.03 −0.700587 −0.350294 0.936640i \(-0.613918\pi\)
−0.350294 + 0.936640i \(0.613918\pi\)
\(224\) 3936.95 1.17433
\(225\) 225.000 0.0666667
\(226\) 12596.1 3.70743
\(227\) −2120.00 −0.619864 −0.309932 0.950759i \(-0.600306\pi\)
−0.309932 + 0.950759i \(0.600306\pi\)
\(228\) −5326.93 −1.54730
\(229\) 2347.12 0.677301 0.338651 0.940912i \(-0.390030\pi\)
0.338651 + 0.940912i \(0.390030\pi\)
\(230\) −5593.10 −1.60347
\(231\) 340.583 0.0970075
\(232\) −3537.47 −1.00106
\(233\) −375.499 −0.105578 −0.0527891 0.998606i \(-0.516811\pi\)
−0.0527891 + 0.998606i \(0.516811\pi\)
\(234\) −3029.75 −0.846415
\(235\) −2021.82 −0.561229
\(236\) −16791.2 −4.63142
\(237\) 3064.79 0.839998
\(238\) 931.072 0.253582
\(239\) −1428.15 −0.386524 −0.193262 0.981147i \(-0.561907\pi\)
−0.193262 + 0.981147i \(0.561907\pi\)
\(240\) 2489.51 0.669572
\(241\) 190.819 0.0510032 0.0255016 0.999675i \(-0.491882\pi\)
0.0255016 + 0.999675i \(0.491882\pi\)
\(242\) −636.641 −0.169111
\(243\) −243.000 −0.0641500
\(244\) −4009.49 −1.05197
\(245\) 1182.41 0.308334
\(246\) −3565.94 −0.924211
\(247\) 5771.81 1.48685
\(248\) 8710.79 2.23039
\(249\) −2119.24 −0.539364
\(250\) 657.687 0.166383
\(251\) −6294.80 −1.58297 −0.791483 0.611191i \(-0.790691\pi\)
−0.791483 + 0.611191i \(0.790691\pi\)
\(252\) −1828.31 −0.457036
\(253\) −2338.65 −0.581145
\(254\) −10868.0 −2.68471
\(255\) 257.191 0.0631605
\(256\) −2685.18 −0.655561
\(257\) 4459.44 1.08238 0.541191 0.840900i \(-0.317974\pi\)
0.541191 + 0.840900i \(0.317974\pi\)
\(258\) −5484.24 −1.32339
\(259\) 2662.36 0.638731
\(260\) −6296.87 −1.50198
\(261\) 517.915 0.122828
\(262\) −4133.04 −0.974581
\(263\) −4416.65 −1.03552 −0.517761 0.855525i \(-0.673234\pi\)
−0.517761 + 0.855525i \(0.673234\pi\)
\(264\) 2028.57 0.472917
\(265\) −1297.84 −0.300852
\(266\) 4898.64 1.12915
\(267\) 1320.65 0.302707
\(268\) 5253.28 1.19737
\(269\) 1914.86 0.434020 0.217010 0.976169i \(-0.430370\pi\)
0.217010 + 0.976169i \(0.430370\pi\)
\(270\) −710.302 −0.160102
\(271\) 6088.34 1.36472 0.682362 0.731014i \(-0.260953\pi\)
0.682362 + 0.731014i \(0.260953\pi\)
\(272\) 2845.69 0.634357
\(273\) 1981.01 0.439180
\(274\) −6714.91 −1.48052
\(275\) 275.000 0.0603023
\(276\) 12554.3 2.73798
\(277\) −832.321 −0.180539 −0.0902696 0.995917i \(-0.528773\pi\)
−0.0902696 + 0.995917i \(0.528773\pi\)
\(278\) 14222.9 3.06847
\(279\) −1275.33 −0.273664
\(280\) −3172.17 −0.677047
\(281\) −2545.32 −0.540360 −0.270180 0.962810i \(-0.587083\pi\)
−0.270180 + 0.962810i \(0.587083\pi\)
\(282\) 6382.67 1.34781
\(283\) −5911.71 −1.24175 −0.620874 0.783911i \(-0.713222\pi\)
−0.620874 + 0.783911i \(0.713222\pi\)
\(284\) 1819.65 0.380198
\(285\) 1353.16 0.281242
\(286\) −3703.03 −0.765611
\(287\) 2331.59 0.479545
\(288\) −3433.15 −0.702433
\(289\) −4619.01 −0.940161
\(290\) 1513.89 0.306548
\(291\) 593.282 0.119515
\(292\) −4763.87 −0.954742
\(293\) −6871.03 −1.37000 −0.685000 0.728543i \(-0.740198\pi\)
−0.685000 + 0.728543i \(0.740198\pi\)
\(294\) −3732.76 −0.740473
\(295\) 4265.34 0.841822
\(296\) 15857.5 3.11384
\(297\) −297.000 −0.0580259
\(298\) 12632.8 2.45570
\(299\) −13602.8 −2.63100
\(300\) −1476.25 −0.284105
\(301\) 3585.88 0.686666
\(302\) 13244.7 2.52367
\(303\) −4201.87 −0.796670
\(304\) 14972.0 2.82468
\(305\) 1018.50 0.191210
\(306\) −811.926 −0.151682
\(307\) 200.179 0.0372144 0.0186072 0.999827i \(-0.494077\pi\)
0.0186072 + 0.999827i \(0.494077\pi\)
\(308\) −2234.61 −0.413404
\(309\) 4037.10 0.743245
\(310\) −3727.87 −0.682995
\(311\) 5734.93 1.04565 0.522827 0.852439i \(-0.324878\pi\)
0.522827 + 0.852439i \(0.324878\pi\)
\(312\) 11799.2 2.14102
\(313\) −3077.36 −0.555727 −0.277864 0.960621i \(-0.589626\pi\)
−0.277864 + 0.960621i \(0.589626\pi\)
\(314\) −7320.97 −1.31575
\(315\) 464.432 0.0830723
\(316\) −20108.5 −3.57971
\(317\) 2142.38 0.379584 0.189792 0.981824i \(-0.439219\pi\)
0.189792 + 0.981824i \(0.439219\pi\)
\(318\) 4097.15 0.722506
\(319\) 633.007 0.111102
\(320\) −3396.60 −0.593362
\(321\) 2667.54 0.463823
\(322\) −11544.9 −1.99806
\(323\) 1546.75 0.266451
\(324\) 1594.35 0.273380
\(325\) 1599.54 0.273005
\(326\) 14314.0 2.43184
\(327\) −3768.86 −0.637365
\(328\) 13887.4 2.33781
\(329\) −4173.32 −0.699339
\(330\) −868.147 −0.144818
\(331\) −1618.23 −0.268719 −0.134359 0.990933i \(-0.542898\pi\)
−0.134359 + 0.990933i \(0.542898\pi\)
\(332\) 13904.6 2.29854
\(333\) −2321.67 −0.382062
\(334\) −15522.7 −2.54301
\(335\) −1334.45 −0.217638
\(336\) 5138.70 0.834343
\(337\) 2406.47 0.388988 0.194494 0.980904i \(-0.437694\pi\)
0.194494 + 0.980904i \(0.437694\pi\)
\(338\) −9979.23 −1.60591
\(339\) 7182.04 1.15066
\(340\) −1687.46 −0.269163
\(341\) −1558.74 −0.247538
\(342\) −4271.78 −0.675413
\(343\) 5980.67 0.941475
\(344\) 21358.1 3.34753
\(345\) −3189.07 −0.497663
\(346\) −2825.68 −0.439045
\(347\) −6612.89 −1.02305 −0.511525 0.859268i \(-0.670919\pi\)
−0.511525 + 0.859268i \(0.670919\pi\)
\(348\) −3398.10 −0.523441
\(349\) 349.871 0.0536623 0.0268311 0.999640i \(-0.491458\pi\)
0.0268311 + 0.999640i \(0.491458\pi\)
\(350\) 1357.56 0.207327
\(351\) −1727.50 −0.262699
\(352\) −4196.08 −0.635374
\(353\) 1723.29 0.259835 0.129917 0.991525i \(-0.458529\pi\)
0.129917 + 0.991525i \(0.458529\pi\)
\(354\) −13465.2 −2.02166
\(355\) −462.230 −0.0691060
\(356\) −8664.97 −1.29001
\(357\) 530.879 0.0787033
\(358\) −15212.3 −2.24580
\(359\) 5875.74 0.863816 0.431908 0.901918i \(-0.357841\pi\)
0.431908 + 0.901918i \(0.357841\pi\)
\(360\) 2766.24 0.404982
\(361\) 1278.92 0.186458
\(362\) −2289.85 −0.332464
\(363\) −363.000 −0.0524864
\(364\) −12997.6 −1.87159
\(365\) 1210.13 0.173537
\(366\) −3215.29 −0.459197
\(367\) −5368.28 −0.763548 −0.381774 0.924256i \(-0.624687\pi\)
−0.381774 + 0.924256i \(0.624687\pi\)
\(368\) −35285.5 −4.99832
\(369\) −2033.23 −0.286844
\(370\) −6786.37 −0.953531
\(371\) −2678.93 −0.374887
\(372\) 8367.60 1.16624
\(373\) 10393.9 1.44282 0.721412 0.692506i \(-0.243493\pi\)
0.721412 + 0.692506i \(0.243493\pi\)
\(374\) −992.354 −0.137202
\(375\) 375.000 0.0516398
\(376\) −24857.0 −3.40931
\(377\) 3681.90 0.502990
\(378\) −1466.16 −0.199501
\(379\) 10918.9 1.47986 0.739928 0.672686i \(-0.234860\pi\)
0.739928 + 0.672686i \(0.234860\pi\)
\(380\) −8878.21 −1.19853
\(381\) −6196.70 −0.833246
\(382\) 19885.8 2.66347
\(383\) −11663.6 −1.55609 −0.778044 0.628210i \(-0.783788\pi\)
−0.778044 + 0.628210i \(0.783788\pi\)
\(384\) 1567.64 0.208329
\(385\) 567.639 0.0751417
\(386\) −19740.7 −2.60304
\(387\) −3127.00 −0.410735
\(388\) −3892.59 −0.509321
\(389\) −5827.00 −0.759487 −0.379744 0.925092i \(-0.623988\pi\)
−0.379744 + 0.925092i \(0.623988\pi\)
\(390\) −5049.59 −0.655631
\(391\) −3645.34 −0.471490
\(392\) 14537.1 1.87304
\(393\) −2356.58 −0.302478
\(394\) 20628.9 2.63774
\(395\) 5107.99 0.650660
\(396\) 1948.65 0.247281
\(397\) 7366.99 0.931332 0.465666 0.884961i \(-0.345815\pi\)
0.465666 + 0.884961i \(0.345815\pi\)
\(398\) 3141.55 0.395658
\(399\) 2793.11 0.350452
\(400\) 4149.18 0.518648
\(401\) 14604.2 1.81870 0.909349 0.416035i \(-0.136581\pi\)
0.909349 + 0.416035i \(0.136581\pi\)
\(402\) 4212.72 0.522665
\(403\) −9066.43 −1.12067
\(404\) 27569.0 3.39507
\(405\) −405.000 −0.0496904
\(406\) 3124.89 0.381985
\(407\) −2837.60 −0.345588
\(408\) 3162.00 0.383683
\(409\) 12581.2 1.52103 0.760515 0.649320i \(-0.224946\pi\)
0.760515 + 0.649320i \(0.224946\pi\)
\(410\) −5943.23 −0.715891
\(411\) −3828.71 −0.459504
\(412\) −26487.9 −3.16739
\(413\) 8804.26 1.04898
\(414\) 10067.6 1.19516
\(415\) −3532.07 −0.417790
\(416\) −24406.6 −2.87651
\(417\) 8109.63 0.952351
\(418\) −5221.06 −0.610934
\(419\) −3776.01 −0.440263 −0.220131 0.975470i \(-0.570649\pi\)
−0.220131 + 0.975470i \(0.570649\pi\)
\(420\) −3047.19 −0.354018
\(421\) 12683.4 1.46830 0.734148 0.678989i \(-0.237582\pi\)
0.734148 + 0.678989i \(0.237582\pi\)
\(422\) 23069.3 2.66113
\(423\) 3639.27 0.418316
\(424\) −15956.2 −1.82759
\(425\) 428.652 0.0489239
\(426\) 1459.21 0.165960
\(427\) 2102.32 0.238263
\(428\) −17502.0 −1.97662
\(429\) −2111.39 −0.237620
\(430\) −9140.40 −1.02509
\(431\) 14152.6 1.58168 0.790841 0.612022i \(-0.209644\pi\)
0.790841 + 0.612022i \(0.209644\pi\)
\(432\) −4481.12 −0.499069
\(433\) −10950.2 −1.21532 −0.607661 0.794197i \(-0.707892\pi\)
−0.607661 + 0.794197i \(0.707892\pi\)
\(434\) −7694.84 −0.851070
\(435\) 863.192 0.0951423
\(436\) 24727.9 2.71618
\(437\) −19179.2 −2.09946
\(438\) −3820.25 −0.416755
\(439\) −11221.0 −1.21993 −0.609964 0.792429i \(-0.708816\pi\)
−0.609964 + 0.792429i \(0.708816\pi\)
\(440\) 3380.95 0.366320
\(441\) −2128.35 −0.229818
\(442\) −5772.04 −0.621149
\(443\) 9647.11 1.03465 0.517323 0.855790i \(-0.326929\pi\)
0.517323 + 0.855790i \(0.326929\pi\)
\(444\) 15232.7 1.62818
\(445\) 2201.09 0.234476
\(446\) 12275.2 1.30325
\(447\) 7202.96 0.762167
\(448\) −7011.07 −0.739379
\(449\) 6482.03 0.681305 0.340652 0.940189i \(-0.389352\pi\)
0.340652 + 0.940189i \(0.389352\pi\)
\(450\) −1183.84 −0.124015
\(451\) −2485.05 −0.259460
\(452\) −47122.2 −4.90363
\(453\) 7551.89 0.783264
\(454\) 11154.4 1.15308
\(455\) 3301.68 0.340187
\(456\) 16636.2 1.70847
\(457\) 11319.8 1.15868 0.579342 0.815085i \(-0.303310\pi\)
0.579342 + 0.815085i \(0.303310\pi\)
\(458\) −12349.4 −1.25993
\(459\) −462.944 −0.0470771
\(460\) 20923.9 2.12083
\(461\) −8406.73 −0.849329 −0.424664 0.905351i \(-0.639608\pi\)
−0.424664 + 0.905351i \(0.639608\pi\)
\(462\) −1791.98 −0.180455
\(463\) 9758.56 0.979523 0.489761 0.871857i \(-0.337084\pi\)
0.489761 + 0.871857i \(0.337084\pi\)
\(464\) 9550.78 0.955568
\(465\) −2125.55 −0.211979
\(466\) 1975.68 0.196399
\(467\) −16388.7 −1.62394 −0.811969 0.583701i \(-0.801604\pi\)
−0.811969 + 0.583701i \(0.801604\pi\)
\(468\) 11334.4 1.11951
\(469\) −2754.49 −0.271195
\(470\) 10637.8 1.04401
\(471\) −4174.27 −0.408365
\(472\) 52439.6 5.11384
\(473\) −3821.89 −0.371524
\(474\) −16125.4 −1.56258
\(475\) 2255.26 0.217849
\(476\) −3483.16 −0.335400
\(477\) 2336.11 0.224242
\(478\) 7514.20 0.719020
\(479\) 13829.0 1.31913 0.659567 0.751646i \(-0.270740\pi\)
0.659567 + 0.751646i \(0.270740\pi\)
\(480\) −5721.92 −0.544102
\(481\) −16504.9 −1.56457
\(482\) −1004.00 −0.0948771
\(483\) −6582.69 −0.620130
\(484\) 2381.68 0.223674
\(485\) 988.804 0.0925758
\(486\) 1278.54 0.119333
\(487\) 13264.4 1.23423 0.617113 0.786875i \(-0.288302\pi\)
0.617113 + 0.786875i \(0.288302\pi\)
\(488\) 12521.8 1.16155
\(489\) 8161.58 0.754763
\(490\) −6221.27 −0.573568
\(491\) −7468.22 −0.686428 −0.343214 0.939257i \(-0.611516\pi\)
−0.343214 + 0.939257i \(0.611516\pi\)
\(492\) 13340.2 1.22241
\(493\) 986.690 0.0901385
\(494\) −30368.4 −2.76586
\(495\) −495.000 −0.0449467
\(496\) −23518.2 −2.12903
\(497\) −954.109 −0.0861119
\(498\) 11150.4 1.00334
\(499\) −5276.64 −0.473377 −0.236688 0.971586i \(-0.576062\pi\)
−0.236688 + 0.971586i \(0.576062\pi\)
\(500\) −2460.42 −0.220067
\(501\) −8850.76 −0.789267
\(502\) 33120.1 2.94466
\(503\) −10956.8 −0.971253 −0.485626 0.874166i \(-0.661408\pi\)
−0.485626 + 0.874166i \(0.661408\pi\)
\(504\) 5709.90 0.504641
\(505\) −7003.12 −0.617098
\(506\) 12304.8 1.08106
\(507\) −5689.96 −0.498422
\(508\) 40657.3 3.55093
\(509\) 12734.4 1.10892 0.554462 0.832209i \(-0.312924\pi\)
0.554462 + 0.832209i \(0.312924\pi\)
\(510\) −1353.21 −0.117492
\(511\) 2497.87 0.216242
\(512\) 18308.4 1.58032
\(513\) −2435.68 −0.209626
\(514\) −23463.3 −2.01347
\(515\) 6728.51 0.575715
\(516\) 20516.6 1.75038
\(517\) 4448.00 0.378381
\(518\) −14008.0 −1.18818
\(519\) −1611.15 −0.136265
\(520\) 19665.4 1.65843
\(521\) −5650.70 −0.475166 −0.237583 0.971367i \(-0.576355\pi\)
−0.237583 + 0.971367i \(0.576355\pi\)
\(522\) −2725.01 −0.228487
\(523\) 14103.2 1.17914 0.589572 0.807716i \(-0.299297\pi\)
0.589572 + 0.807716i \(0.299297\pi\)
\(524\) 15461.8 1.28903
\(525\) 774.053 0.0643475
\(526\) 23238.2 1.92630
\(527\) −2429.66 −0.200830
\(528\) −5476.92 −0.451425
\(529\) 33033.8 2.71503
\(530\) 6828.59 0.559651
\(531\) −7677.60 −0.627457
\(532\) −18325.9 −1.49347
\(533\) −14454.4 −1.17465
\(534\) −6948.62 −0.563102
\(535\) 4445.89 0.359276
\(536\) −16406.2 −1.32209
\(537\) −8673.75 −0.697021
\(538\) −10075.0 −0.807372
\(539\) −2601.31 −0.207878
\(540\) 2657.25 0.211759
\(541\) −6391.90 −0.507965 −0.253983 0.967209i \(-0.581741\pi\)
−0.253983 + 0.967209i \(0.581741\pi\)
\(542\) −32033.8 −2.53869
\(543\) −1305.63 −0.103186
\(544\) −6540.57 −0.515486
\(545\) −6281.43 −0.493701
\(546\) −10423.1 −0.816971
\(547\) 20786.7 1.62482 0.812409 0.583088i \(-0.198156\pi\)
0.812409 + 0.583088i \(0.198156\pi\)
\(548\) 25120.6 1.95821
\(549\) −1833.29 −0.142519
\(550\) −1446.91 −0.112175
\(551\) 5191.26 0.401370
\(552\) −39207.6 −3.02317
\(553\) 10543.6 0.810777
\(554\) 4379.26 0.335843
\(555\) −3869.45 −0.295944
\(556\) −53208.2 −4.05851
\(557\) −15125.9 −1.15064 −0.575320 0.817928i \(-0.695123\pi\)
−0.575320 + 0.817928i \(0.695123\pi\)
\(558\) 6710.16 0.509075
\(559\) −22230.1 −1.68199
\(560\) 8564.50 0.646279
\(561\) −565.820 −0.0425828
\(562\) 13392.2 1.00519
\(563\) 8706.42 0.651744 0.325872 0.945414i \(-0.394342\pi\)
0.325872 + 0.945414i \(0.394342\pi\)
\(564\) −23877.7 −1.78268
\(565\) 11970.1 0.891300
\(566\) 31104.4 2.30992
\(567\) −835.977 −0.0619184
\(568\) −5682.83 −0.419800
\(569\) 7067.76 0.520731 0.260366 0.965510i \(-0.416157\pi\)
0.260366 + 0.965510i \(0.416157\pi\)
\(570\) −7119.63 −0.523172
\(571\) −3326.42 −0.243794 −0.121897 0.992543i \(-0.538898\pi\)
−0.121897 + 0.992543i \(0.538898\pi\)
\(572\) 13853.1 1.01264
\(573\) 11338.5 0.826651
\(574\) −12267.7 −0.892060
\(575\) −5315.12 −0.385488
\(576\) 6113.88 0.442266
\(577\) −3308.06 −0.238676 −0.119338 0.992854i \(-0.538077\pi\)
−0.119338 + 0.992854i \(0.538077\pi\)
\(578\) 24302.9 1.74891
\(579\) −11255.7 −0.807897
\(580\) −5663.50 −0.405455
\(581\) −7290.70 −0.520601
\(582\) −3121.55 −0.222324
\(583\) 2855.25 0.202834
\(584\) 14877.8 1.05419
\(585\) −2879.17 −0.203486
\(586\) 36151.9 2.54850
\(587\) −5694.88 −0.400431 −0.200215 0.979752i \(-0.564164\pi\)
−0.200215 + 0.979752i \(0.564164\pi\)
\(588\) 13964.3 0.979387
\(589\) −12783.1 −0.894262
\(590\) −22442.0 −1.56597
\(591\) 11762.2 0.818666
\(592\) −42813.5 −2.97234
\(593\) −3907.69 −0.270606 −0.135303 0.990804i \(-0.543201\pi\)
−0.135303 + 0.990804i \(0.543201\pi\)
\(594\) 1562.66 0.107941
\(595\) 884.798 0.0609633
\(596\) −47259.5 −3.24803
\(597\) 1791.25 0.122799
\(598\) 71571.1 4.89425
\(599\) 10727.2 0.731720 0.365860 0.930670i \(-0.380775\pi\)
0.365860 + 0.930670i \(0.380775\pi\)
\(600\) 4610.39 0.313697
\(601\) 3348.98 0.227301 0.113650 0.993521i \(-0.463746\pi\)
0.113650 + 0.993521i \(0.463746\pi\)
\(602\) −18867.1 −1.27735
\(603\) 2402.01 0.162218
\(604\) −49548.8 −3.33793
\(605\) −605.000 −0.0406558
\(606\) 22108.1 1.48198
\(607\) −21539.9 −1.44032 −0.720162 0.693806i \(-0.755932\pi\)
−0.720162 + 0.693806i \(0.755932\pi\)
\(608\) −34411.8 −2.29537
\(609\) 1781.75 0.118555
\(610\) −5358.82 −0.355692
\(611\) 25871.9 1.71303
\(612\) 3037.43 0.200622
\(613\) −9284.33 −0.611730 −0.305865 0.952075i \(-0.598946\pi\)
−0.305865 + 0.952075i \(0.598946\pi\)
\(614\) −1053.24 −0.0692270
\(615\) −3388.71 −0.222189
\(616\) 6978.77 0.456465
\(617\) −20711.3 −1.35139 −0.675695 0.737181i \(-0.736156\pi\)
−0.675695 + 0.737181i \(0.736156\pi\)
\(618\) −21241.2 −1.38260
\(619\) −13282.9 −0.862496 −0.431248 0.902233i \(-0.641927\pi\)
−0.431248 + 0.902233i \(0.641927\pi\)
\(620\) 13946.0 0.903363
\(621\) 5740.33 0.370936
\(622\) −30174.3 −1.94515
\(623\) 4543.36 0.292177
\(624\) −31856.6 −2.04373
\(625\) 625.000 0.0400000
\(626\) 16191.5 1.03378
\(627\) −2976.94 −0.189613
\(628\) 27387.9 1.74028
\(629\) −4423.06 −0.280380
\(630\) −2443.61 −0.154533
\(631\) −14789.9 −0.933086 −0.466543 0.884498i \(-0.654501\pi\)
−0.466543 + 0.884498i \(0.654501\pi\)
\(632\) 62799.5 3.95258
\(633\) 13153.6 0.825925
\(634\) −11272.1 −0.706109
\(635\) −10327.8 −0.645429
\(636\) −15327.5 −0.955622
\(637\) −15130.6 −0.941123
\(638\) −3330.57 −0.206675
\(639\) 832.014 0.0515086
\(640\) 2612.74 0.161371
\(641\) −5808.52 −0.357914 −0.178957 0.983857i \(-0.557272\pi\)
−0.178957 + 0.983857i \(0.557272\pi\)
\(642\) −14035.2 −0.862813
\(643\) −18891.2 −1.15862 −0.579311 0.815106i \(-0.696678\pi\)
−0.579311 + 0.815106i \(0.696678\pi\)
\(644\) 43189.8 2.64273
\(645\) −5211.67 −0.318154
\(646\) −8138.24 −0.495657
\(647\) −243.046 −0.0147684 −0.00738418 0.999973i \(-0.502350\pi\)
−0.00738418 + 0.999973i \(0.502350\pi\)
\(648\) −4979.22 −0.301855
\(649\) −9383.74 −0.567556
\(650\) −8415.98 −0.507849
\(651\) −4387.44 −0.264144
\(652\) −53549.0 −3.21648
\(653\) −15920.4 −0.954081 −0.477041 0.878881i \(-0.658291\pi\)
−0.477041 + 0.878881i \(0.658291\pi\)
\(654\) 19829.8 1.18564
\(655\) −3927.63 −0.234298
\(656\) −37494.4 −2.23157
\(657\) −2178.23 −0.129347
\(658\) 21957.9 1.30092
\(659\) −7476.38 −0.441940 −0.220970 0.975281i \(-0.570922\pi\)
−0.220970 + 0.975281i \(0.570922\pi\)
\(660\) 3247.75 0.191543
\(661\) −14920.1 −0.877948 −0.438974 0.898500i \(-0.644658\pi\)
−0.438974 + 0.898500i \(0.644658\pi\)
\(662\) 8514.30 0.499876
\(663\) −3291.10 −0.192784
\(664\) −43424.7 −2.53796
\(665\) 4655.18 0.271459
\(666\) 12215.5 0.710720
\(667\) −12234.6 −0.710232
\(668\) 58070.8 3.36351
\(669\) 6999.08 0.404484
\(670\) 7021.20 0.404854
\(671\) −2240.69 −0.128914
\(672\) −11810.9 −0.677997
\(673\) 563.692 0.0322864 0.0161432 0.999870i \(-0.494861\pi\)
0.0161432 + 0.999870i \(0.494861\pi\)
\(674\) −12661.6 −0.723602
\(675\) −675.000 −0.0384900
\(676\) 37332.5 2.12406
\(677\) 13280.2 0.753914 0.376957 0.926231i \(-0.376970\pi\)
0.376957 + 0.926231i \(0.376970\pi\)
\(678\) −37788.3 −2.14049
\(679\) 2041.03 0.115357
\(680\) 5270.01 0.297199
\(681\) 6359.99 0.357879
\(682\) 8201.30 0.460475
\(683\) 6856.80 0.384141 0.192070 0.981381i \(-0.438480\pi\)
0.192070 + 0.981381i \(0.438480\pi\)
\(684\) 15980.8 0.893334
\(685\) −6381.18 −0.355930
\(686\) −31467.3 −1.75135
\(687\) −7041.36 −0.391040
\(688\) −57664.5 −3.19541
\(689\) 16607.6 0.918287
\(690\) 16779.3 0.925763
\(691\) −28374.6 −1.56211 −0.781057 0.624460i \(-0.785319\pi\)
−0.781057 + 0.624460i \(0.785319\pi\)
\(692\) 10570.9 0.580702
\(693\) −1021.75 −0.0560073
\(694\) 34793.7 1.90310
\(695\) 13516.1 0.737688
\(696\) 10612.4 0.577963
\(697\) −3873.54 −0.210503
\(698\) −1840.84 −0.0998237
\(699\) 1126.50 0.0609556
\(700\) −5078.65 −0.274221
\(701\) 667.753 0.0359781 0.0179891 0.999838i \(-0.494274\pi\)
0.0179891 + 0.999838i \(0.494274\pi\)
\(702\) 9089.26 0.488678
\(703\) −23271.0 −1.24848
\(704\) 7472.52 0.400044
\(705\) 6065.45 0.324026
\(706\) −9067.10 −0.483349
\(707\) −14455.4 −0.768956
\(708\) 50373.6 2.67395
\(709\) 23667.8 1.25368 0.626842 0.779147i \(-0.284347\pi\)
0.626842 + 0.779147i \(0.284347\pi\)
\(710\) 2432.02 0.128552
\(711\) −9194.37 −0.484973
\(712\) 27061.0 1.42438
\(713\) 30126.9 1.58241
\(714\) −2793.22 −0.146405
\(715\) −3518.99 −0.184060
\(716\) 56909.5 2.97040
\(717\) 4284.45 0.223160
\(718\) −30915.2 −1.60689
\(719\) 11835.5 0.613896 0.306948 0.951726i \(-0.400692\pi\)
0.306948 + 0.951726i \(0.400692\pi\)
\(720\) −7468.53 −0.386577
\(721\) 13888.6 0.717390
\(722\) −6729.01 −0.346853
\(723\) −572.458 −0.0294467
\(724\) 8566.38 0.439733
\(725\) 1438.65 0.0736969
\(726\) 1909.92 0.0976362
\(727\) 15633.2 0.797530 0.398765 0.917053i \(-0.369439\pi\)
0.398765 + 0.917053i \(0.369439\pi\)
\(728\) 40592.1 2.06654
\(729\) 729.000 0.0370370
\(730\) −6367.08 −0.322817
\(731\) −5957.31 −0.301422
\(732\) 12028.5 0.607356
\(733\) −14870.7 −0.749335 −0.374668 0.927159i \(-0.622243\pi\)
−0.374668 + 0.927159i \(0.622243\pi\)
\(734\) 28245.2 1.42037
\(735\) −3547.24 −0.178016
\(736\) 81100.6 4.06169
\(737\) 2935.79 0.146731
\(738\) 10697.8 0.533593
\(739\) 20850.3 1.03788 0.518939 0.854812i \(-0.326327\pi\)
0.518939 + 0.854812i \(0.326327\pi\)
\(740\) 25387.9 1.26119
\(741\) −17315.4 −0.858432
\(742\) 14095.2 0.697372
\(743\) −29254.7 −1.44448 −0.722242 0.691641i \(-0.756888\pi\)
−0.722242 + 0.691641i \(0.756888\pi\)
\(744\) −26132.4 −1.28771
\(745\) 12004.9 0.590372
\(746\) −54687.2 −2.68397
\(747\) 6357.73 0.311402
\(748\) 3712.41 0.181470
\(749\) 9176.95 0.447688
\(750\) −1973.06 −0.0960613
\(751\) 10936.8 0.531411 0.265705 0.964054i \(-0.414395\pi\)
0.265705 + 0.964054i \(0.414395\pi\)
\(752\) 67111.2 3.25438
\(753\) 18884.4 0.913926
\(754\) −19372.3 −0.935672
\(755\) 12586.5 0.606714
\(756\) 5484.94 0.263870
\(757\) −8476.34 −0.406972 −0.203486 0.979078i \(-0.565227\pi\)
−0.203486 + 0.979078i \(0.565227\pi\)
\(758\) −57449.6 −2.75286
\(759\) 7015.96 0.335524
\(760\) 27727.0 1.32337
\(761\) 913.964 0.0435364 0.0217682 0.999763i \(-0.493070\pi\)
0.0217682 + 0.999763i \(0.493070\pi\)
\(762\) 32603.9 1.55002
\(763\) −12965.8 −0.615193
\(764\) −74393.0 −3.52283
\(765\) −771.573 −0.0364657
\(766\) 61367.9 2.89466
\(767\) −54580.6 −2.56948
\(768\) 8055.53 0.378488
\(769\) 32215.2 1.51067 0.755337 0.655337i \(-0.227473\pi\)
0.755337 + 0.655337i \(0.227473\pi\)
\(770\) −2986.63 −0.139780
\(771\) −13378.3 −0.624914
\(772\) 73850.2 3.44291
\(773\) 72.6900 0.00338225 0.00169112 0.999999i \(-0.499462\pi\)
0.00169112 + 0.999999i \(0.499462\pi\)
\(774\) 16452.7 0.764058
\(775\) −3542.59 −0.164198
\(776\) 12156.7 0.562373
\(777\) −7987.09 −0.368771
\(778\) 30658.7 1.41281
\(779\) −20379.8 −0.937332
\(780\) 18890.6 0.867169
\(781\) 1016.91 0.0465913
\(782\) 19179.9 0.877075
\(783\) −1553.75 −0.0709148
\(784\) −39248.5 −1.78792
\(785\) −6957.11 −0.316319
\(786\) 12399.1 0.562675
\(787\) 487.318 0.0220724 0.0110362 0.999939i \(-0.496487\pi\)
0.0110362 + 0.999939i \(0.496487\pi\)
\(788\) −77173.1 −3.48880
\(789\) 13249.9 0.597859
\(790\) −26875.7 −1.21037
\(791\) 24707.9 1.11064
\(792\) −6085.72 −0.273039
\(793\) −13033.0 −0.583627
\(794\) −38761.4 −1.73248
\(795\) 3893.52 0.173697
\(796\) −11752.6 −0.523317
\(797\) 31379.9 1.39465 0.697324 0.716756i \(-0.254374\pi\)
0.697324 + 0.716756i \(0.254374\pi\)
\(798\) −14695.9 −0.651917
\(799\) 6933.25 0.306985
\(800\) −9536.54 −0.421460
\(801\) −3961.96 −0.174768
\(802\) −76839.8 −3.38318
\(803\) −2662.28 −0.116999
\(804\) −15759.8 −0.691302
\(805\) −10971.2 −0.480351
\(806\) 47703.0 2.08470
\(807\) −5744.59 −0.250581
\(808\) −86099.0 −3.74870
\(809\) 1824.26 0.0792802 0.0396401 0.999214i \(-0.487379\pi\)
0.0396401 + 0.999214i \(0.487379\pi\)
\(810\) 2130.91 0.0924351
\(811\) −4364.52 −0.188976 −0.0944878 0.995526i \(-0.530121\pi\)
−0.0944878 + 0.995526i \(0.530121\pi\)
\(812\) −11690.3 −0.505232
\(813\) −18265.0 −0.787924
\(814\) 14930.0 0.642870
\(815\) 13602.6 0.584637
\(816\) −8537.06 −0.366246
\(817\) −31343.1 −1.34218
\(818\) −66196.1 −2.82945
\(819\) −5943.02 −0.253561
\(820\) 22233.7 0.946872
\(821\) 3306.16 0.140543 0.0702714 0.997528i \(-0.477613\pi\)
0.0702714 + 0.997528i \(0.477613\pi\)
\(822\) 20144.7 0.854779
\(823\) −19183.7 −0.812519 −0.406259 0.913758i \(-0.633167\pi\)
−0.406259 + 0.913758i \(0.633167\pi\)
\(824\) 82722.8 3.49731
\(825\) −825.000 −0.0348155
\(826\) −46323.6 −1.95134
\(827\) 20646.4 0.868131 0.434066 0.900881i \(-0.357079\pi\)
0.434066 + 0.900881i \(0.357079\pi\)
\(828\) −37663.0 −1.58077
\(829\) 5345.44 0.223950 0.111975 0.993711i \(-0.464282\pi\)
0.111975 + 0.993711i \(0.464282\pi\)
\(830\) 18584.0 0.777181
\(831\) 2496.96 0.104234
\(832\) 43464.0 1.81111
\(833\) −4054.75 −0.168654
\(834\) −42668.8 −1.77158
\(835\) −14751.3 −0.611363
\(836\) 19532.1 0.808051
\(837\) 3826.00 0.158000
\(838\) 19867.5 0.818985
\(839\) 29284.9 1.20504 0.602519 0.798105i \(-0.294164\pi\)
0.602519 + 0.798105i \(0.294164\pi\)
\(840\) 9516.50 0.390893
\(841\) −21077.4 −0.864219
\(842\) −66733.8 −2.73135
\(843\) 7635.97 0.311977
\(844\) −86302.6 −3.51974
\(845\) −9483.26 −0.386076
\(846\) −19148.0 −0.778159
\(847\) −1248.81 −0.0506605
\(848\) 43079.9 1.74454
\(849\) 17735.1 0.716923
\(850\) −2255.35 −0.0910092
\(851\) 54844.2 2.20921
\(852\) −5458.94 −0.219507
\(853\) −8070.62 −0.323954 −0.161977 0.986795i \(-0.551787\pi\)
−0.161977 + 0.986795i \(0.551787\pi\)
\(854\) −11061.4 −0.443222
\(855\) −4059.47 −0.162375
\(856\) 54659.5 2.18250
\(857\) −11344.2 −0.452169 −0.226085 0.974108i \(-0.572593\pi\)
−0.226085 + 0.974108i \(0.572593\pi\)
\(858\) 11109.1 0.442026
\(859\) 25470.6 1.01170 0.505848 0.862623i \(-0.331180\pi\)
0.505848 + 0.862623i \(0.331180\pi\)
\(860\) 34194.4 1.35584
\(861\) −6994.78 −0.276866
\(862\) −74463.6 −2.94228
\(863\) −14558.4 −0.574243 −0.287122 0.957894i \(-0.592698\pi\)
−0.287122 + 0.957894i \(0.592698\pi\)
\(864\) 10299.5 0.405550
\(865\) −2685.24 −0.105550
\(866\) 57614.6 2.26077
\(867\) 13857.0 0.542802
\(868\) 28786.5 1.12567
\(869\) −11237.6 −0.438675
\(870\) −4541.68 −0.176986
\(871\) 17076.0 0.664294
\(872\) −77226.3 −2.99910
\(873\) −1779.85 −0.0690019
\(874\) 100911. 3.90546
\(875\) 1290.09 0.0498434
\(876\) 14291.6 0.551220
\(877\) 185.528 0.00714350 0.00357175 0.999994i \(-0.498863\pi\)
0.00357175 + 0.999994i \(0.498863\pi\)
\(878\) 59039.1 2.26933
\(879\) 20613.1 0.790969
\(880\) −9128.21 −0.349672
\(881\) 18950.1 0.724681 0.362340 0.932046i \(-0.381978\pi\)
0.362340 + 0.932046i \(0.381978\pi\)
\(882\) 11198.3 0.427513
\(883\) 21258.3 0.810189 0.405095 0.914275i \(-0.367239\pi\)
0.405095 + 0.914275i \(0.367239\pi\)
\(884\) 21593.3 0.821563
\(885\) −12796.0 −0.486026
\(886\) −50758.2 −1.92467
\(887\) 35707.4 1.35168 0.675839 0.737049i \(-0.263781\pi\)
0.675839 + 0.737049i \(0.263781\pi\)
\(888\) −47572.5 −1.79778
\(889\) −21318.1 −0.804259
\(890\) −11581.0 −0.436177
\(891\) 891.000 0.0335013
\(892\) −45921.8 −1.72374
\(893\) 36477.8 1.36695
\(894\) −37898.4 −1.41780
\(895\) −14456.3 −0.539910
\(896\) 5393.06 0.201082
\(897\) 40808.4 1.51901
\(898\) −34105.2 −1.26738
\(899\) −8154.49 −0.302522
\(900\) 4428.75 0.164028
\(901\) 4450.58 0.164562
\(902\) 13075.1 0.482653
\(903\) −10757.6 −0.396447
\(904\) 147165. 5.41440
\(905\) −2176.05 −0.0799274
\(906\) −39734.2 −1.45704
\(907\) −6542.52 −0.239516 −0.119758 0.992803i \(-0.538212\pi\)
−0.119758 + 0.992803i \(0.538212\pi\)
\(908\) −41728.6 −1.52512
\(909\) 12605.6 0.459958
\(910\) −17371.8 −0.632823
\(911\) 31171.2 1.13364 0.566821 0.823841i \(-0.308173\pi\)
0.566821 + 0.823841i \(0.308173\pi\)
\(912\) −44915.9 −1.63083
\(913\) 7770.56 0.281674
\(914\) −59559.2 −2.15541
\(915\) −3055.49 −0.110395
\(916\) 46199.2 1.66644
\(917\) −8107.19 −0.291955
\(918\) 2435.78 0.0875737
\(919\) 12031.9 0.431877 0.215938 0.976407i \(-0.430719\pi\)
0.215938 + 0.976407i \(0.430719\pi\)
\(920\) −65346.1 −2.34174
\(921\) −600.537 −0.0214857
\(922\) 44232.0 1.57994
\(923\) 5914.85 0.210931
\(924\) 6703.82 0.238679
\(925\) −6449.08 −0.229237
\(926\) −51344.7 −1.82213
\(927\) −12111.3 −0.429113
\(928\) −21951.6 −0.776506
\(929\) −12546.2 −0.443085 −0.221542 0.975151i \(-0.571109\pi\)
−0.221542 + 0.975151i \(0.571109\pi\)
\(930\) 11183.6 0.394328
\(931\) −21333.2 −0.750986
\(932\) −7391.07 −0.259767
\(933\) −17204.8 −0.603708
\(934\) 86229.1 3.02088
\(935\) −943.034 −0.0329845
\(936\) −35397.7 −1.23612
\(937\) 17909.8 0.624427 0.312214 0.950012i \(-0.398930\pi\)
0.312214 + 0.950012i \(0.398930\pi\)
\(938\) 14492.7 0.504483
\(939\) 9232.08 0.320849
\(940\) −39796.2 −1.38086
\(941\) 829.893 0.0287500 0.0143750 0.999897i \(-0.495424\pi\)
0.0143750 + 0.999897i \(0.495424\pi\)
\(942\) 21962.9 0.759650
\(943\) 48030.4 1.65863
\(944\) −141581. −4.88144
\(945\) −1393.30 −0.0479618
\(946\) 20108.9 0.691116
\(947\) −17654.9 −0.605814 −0.302907 0.953020i \(-0.597957\pi\)
−0.302907 + 0.953020i \(0.597957\pi\)
\(948\) 60325.4 2.06675
\(949\) −15485.2 −0.529685
\(950\) −11866.0 −0.405248
\(951\) −6427.14 −0.219153
\(952\) 10878.0 0.370335
\(953\) 30736.6 1.04476 0.522380 0.852713i \(-0.325044\pi\)
0.522380 + 0.852713i \(0.325044\pi\)
\(954\) −12291.5 −0.417139
\(955\) 18897.4 0.640321
\(956\) −28110.8 −0.951012
\(957\) −1899.02 −0.0641449
\(958\) −72761.5 −2.45388
\(959\) −13171.7 −0.443519
\(960\) 10189.8 0.342578
\(961\) −9711.08 −0.325974
\(962\) 86840.6 2.91045
\(963\) −8002.61 −0.267789
\(964\) 3755.97 0.125489
\(965\) −18759.6 −0.625794
\(966\) 34634.8 1.15358
\(967\) 23645.2 0.786327 0.393163 0.919469i \(-0.371381\pi\)
0.393163 + 0.919469i \(0.371381\pi\)
\(968\) −7438.10 −0.246973
\(969\) −4640.26 −0.153836
\(970\) −5202.59 −0.172211
\(971\) 27402.0 0.905635 0.452818 0.891603i \(-0.350419\pi\)
0.452818 + 0.891603i \(0.350419\pi\)
\(972\) −4783.05 −0.157836
\(973\) 27899.1 0.919222
\(974\) −69790.6 −2.29593
\(975\) −4798.62 −0.157619
\(976\) −33807.5 −1.10876
\(977\) −49118.7 −1.60844 −0.804220 0.594332i \(-0.797416\pi\)
−0.804220 + 0.594332i \(0.797416\pi\)
\(978\) −42942.1 −1.40403
\(979\) −4842.40 −0.158084
\(980\) 23273.9 0.758630
\(981\) 11306.6 0.367983
\(982\) 39294.0 1.27691
\(983\) 52630.5 1.70768 0.853842 0.520533i \(-0.174267\pi\)
0.853842 + 0.520533i \(0.174267\pi\)
\(984\) −41662.1 −1.34973
\(985\) 19603.7 0.634136
\(986\) −5191.47 −0.167677
\(987\) 12520.0 0.403764
\(988\) 113609. 3.65827
\(989\) 73868.4 2.37500
\(990\) 2604.44 0.0836107
\(991\) 45472.7 1.45761 0.728803 0.684724i \(-0.240077\pi\)
0.728803 + 0.684724i \(0.240077\pi\)
\(992\) 54054.5 1.73007
\(993\) 4854.68 0.155145
\(994\) 5020.04 0.160187
\(995\) 2985.42 0.0951197
\(996\) −41713.8 −1.32706
\(997\) 55068.1 1.74927 0.874637 0.484779i \(-0.161100\pi\)
0.874637 + 0.484779i \(0.161100\pi\)
\(998\) 27763.0 0.880585
\(999\) 6965.01 0.220584
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 165.4.a.d.1.1 3
3.2 odd 2 495.4.a.l.1.3 3
5.2 odd 4 825.4.c.l.199.1 6
5.3 odd 4 825.4.c.l.199.6 6
5.4 even 2 825.4.a.s.1.3 3
11.10 odd 2 1815.4.a.s.1.3 3
15.14 odd 2 2475.4.a.s.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.d.1.1 3 1.1 even 1 trivial
495.4.a.l.1.3 3 3.2 odd 2
825.4.a.s.1.3 3 5.4 even 2
825.4.c.l.199.1 6 5.2 odd 4
825.4.c.l.199.6 6 5.3 odd 4
1815.4.a.s.1.3 3 11.10 odd 2
2475.4.a.s.1.1 3 15.14 odd 2