Properties

Label 693.4.a.l.1.1
Level $693$
Weight $4$
Character 693.1
Self dual yes
Analytic conductor $40.888$
Analytic rank $0$
Dimension $4$
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] = [693,4,Mod(1,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, 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("693.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 693.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: \(40.8883236340\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.522072.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 12x^{2} + 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.148103\) of defining polynomial
Character \(\chi\) \(=\) 693.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.60395 q^{2} +13.1964 q^{4} -1.84418 q^{5} -7.00000 q^{7} -23.9238 q^{8} +8.49053 q^{10} +11.0000 q^{11} +24.6401 q^{13} +32.2277 q^{14} +4.57310 q^{16} -17.8800 q^{17} +32.1459 q^{19} -24.3365 q^{20} -50.6435 q^{22} -14.1248 q^{23} -121.599 q^{25} -113.442 q^{26} -92.3745 q^{28} +41.5471 q^{29} +175.766 q^{31} +170.336 q^{32} +82.3187 q^{34} +12.9093 q^{35} +292.877 q^{37} -147.998 q^{38} +44.1199 q^{40} -154.296 q^{41} -277.144 q^{43} +145.160 q^{44} +65.0300 q^{46} +52.1450 q^{47} +49.0000 q^{49} +559.836 q^{50} +325.160 q^{52} -82.3907 q^{53} -20.2860 q^{55} +167.467 q^{56} -191.281 q^{58} -712.816 q^{59} -647.078 q^{61} -809.217 q^{62} -820.804 q^{64} -45.4408 q^{65} +260.867 q^{67} -235.951 q^{68} -59.4337 q^{70} -369.025 q^{71} +1145.77 q^{73} -1348.39 q^{74} +424.209 q^{76} -77.0000 q^{77} +488.885 q^{79} -8.43362 q^{80} +710.372 q^{82} -548.982 q^{83} +32.9740 q^{85} +1275.96 q^{86} -263.162 q^{88} -105.039 q^{89} -172.481 q^{91} -186.396 q^{92} -240.073 q^{94} -59.2829 q^{95} -1361.91 q^{97} -225.594 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 26 q^{4} - 10 q^{5} - 28 q^{7} + 18 q^{8} - 2 q^{10} + 44 q^{11} + 58 q^{13} - 14 q^{14} + 2 q^{16} - 4 q^{17} + 258 q^{19} - 182 q^{20} + 22 q^{22} - 8 q^{23} + 80 q^{25} + 482 q^{26} - 182 q^{28}+ \cdots + 98 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.60395 −1.62774 −0.813871 0.581045i \(-0.802644\pi\)
−0.813871 + 0.581045i \(0.802644\pi\)
\(3\) 0 0
\(4\) 13.1964 1.64955
\(5\) −1.84418 −0.164949 −0.0824744 0.996593i \(-0.526282\pi\)
−0.0824744 + 0.996593i \(0.526282\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) −23.9238 −1.05729
\(9\) 0 0
\(10\) 8.49053 0.268494
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) 24.6401 0.525687 0.262844 0.964838i \(-0.415340\pi\)
0.262844 + 0.964838i \(0.415340\pi\)
\(14\) 32.2277 0.615229
\(15\) 0 0
\(16\) 4.57310 0.0714546
\(17\) −17.8800 −0.255090 −0.127545 0.991833i \(-0.540710\pi\)
−0.127545 + 0.991833i \(0.540710\pi\)
\(18\) 0 0
\(19\) 32.1459 0.388146 0.194073 0.980987i \(-0.437830\pi\)
0.194073 + 0.980987i \(0.437830\pi\)
\(20\) −24.3365 −0.272090
\(21\) 0 0
\(22\) −50.6435 −0.490783
\(23\) −14.1248 −0.128054 −0.0640268 0.997948i \(-0.520394\pi\)
−0.0640268 + 0.997948i \(0.520394\pi\)
\(24\) 0 0
\(25\) −121.599 −0.972792
\(26\) −113.442 −0.855683
\(27\) 0 0
\(28\) −92.3745 −0.623470
\(29\) 41.5471 0.266038 0.133019 0.991113i \(-0.457533\pi\)
0.133019 + 0.991113i \(0.457533\pi\)
\(30\) 0 0
\(31\) 175.766 1.01834 0.509169 0.860667i \(-0.329953\pi\)
0.509169 + 0.860667i \(0.329953\pi\)
\(32\) 170.336 0.940983
\(33\) 0 0
\(34\) 82.3187 0.415222
\(35\) 12.9093 0.0623448
\(36\) 0 0
\(37\) 292.877 1.30131 0.650657 0.759372i \(-0.274494\pi\)
0.650657 + 0.759372i \(0.274494\pi\)
\(38\) −147.998 −0.631801
\(39\) 0 0
\(40\) 44.1199 0.174399
\(41\) −154.296 −0.587732 −0.293866 0.955847i \(-0.594942\pi\)
−0.293866 + 0.955847i \(0.594942\pi\)
\(42\) 0 0
\(43\) −277.144 −0.982887 −0.491443 0.870910i \(-0.663530\pi\)
−0.491443 + 0.870910i \(0.663530\pi\)
\(44\) 145.160 0.497357
\(45\) 0 0
\(46\) 65.0300 0.208438
\(47\) 52.1450 0.161832 0.0809162 0.996721i \(-0.474215\pi\)
0.0809162 + 0.996721i \(0.474215\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 559.836 1.58345
\(51\) 0 0
\(52\) 325.160 0.867145
\(53\) −82.3907 −0.213533 −0.106766 0.994284i \(-0.534050\pi\)
−0.106766 + 0.994284i \(0.534050\pi\)
\(54\) 0 0
\(55\) −20.2860 −0.0497339
\(56\) 167.467 0.399619
\(57\) 0 0
\(58\) −191.281 −0.433041
\(59\) −712.816 −1.57289 −0.786447 0.617658i \(-0.788081\pi\)
−0.786447 + 0.617658i \(0.788081\pi\)
\(60\) 0 0
\(61\) −647.078 −1.35819 −0.679097 0.734048i \(-0.737629\pi\)
−0.679097 + 0.734048i \(0.737629\pi\)
\(62\) −809.217 −1.65759
\(63\) 0 0
\(64\) −820.804 −1.60313
\(65\) −45.4408 −0.0867114
\(66\) 0 0
\(67\) 260.867 0.475672 0.237836 0.971305i \(-0.423562\pi\)
0.237836 + 0.971305i \(0.423562\pi\)
\(68\) −235.951 −0.420783
\(69\) 0 0
\(70\) −59.4337 −0.101481
\(71\) −369.025 −0.616833 −0.308417 0.951251i \(-0.599799\pi\)
−0.308417 + 0.951251i \(0.599799\pi\)
\(72\) 0 0
\(73\) 1145.77 1.83702 0.918509 0.395401i \(-0.129394\pi\)
0.918509 + 0.395401i \(0.129394\pi\)
\(74\) −1348.39 −2.11820
\(75\) 0 0
\(76\) 424.209 0.640264
\(77\) −77.0000 −0.113961
\(78\) 0 0
\(79\) 488.885 0.696251 0.348125 0.937448i \(-0.386818\pi\)
0.348125 + 0.937448i \(0.386818\pi\)
\(80\) −8.43362 −0.0117863
\(81\) 0 0
\(82\) 710.372 0.956676
\(83\) −548.982 −0.726008 −0.363004 0.931788i \(-0.618249\pi\)
−0.363004 + 0.931788i \(0.618249\pi\)
\(84\) 0 0
\(85\) 32.9740 0.0420769
\(86\) 1275.96 1.59989
\(87\) 0 0
\(88\) −263.162 −0.318786
\(89\) −105.039 −0.125102 −0.0625510 0.998042i \(-0.519924\pi\)
−0.0625510 + 0.998042i \(0.519924\pi\)
\(90\) 0 0
\(91\) −172.481 −0.198691
\(92\) −186.396 −0.211230
\(93\) 0 0
\(94\) −240.073 −0.263422
\(95\) −59.2829 −0.0640242
\(96\) 0 0
\(97\) −1361.91 −1.42557 −0.712787 0.701381i \(-0.752567\pi\)
−0.712787 + 0.701381i \(0.752567\pi\)
\(98\) −225.594 −0.232535
\(99\) 0 0
\(100\) −1604.66 −1.60466
\(101\) 1610.32 1.58647 0.793234 0.608917i \(-0.208396\pi\)
0.793234 + 0.608917i \(0.208396\pi\)
\(102\) 0 0
\(103\) −123.044 −0.117708 −0.0588540 0.998267i \(-0.518745\pi\)
−0.0588540 + 0.998267i \(0.518745\pi\)
\(104\) −589.485 −0.555805
\(105\) 0 0
\(106\) 379.323 0.347576
\(107\) 1740.90 1.57289 0.786446 0.617660i \(-0.211919\pi\)
0.786446 + 0.617660i \(0.211919\pi\)
\(108\) 0 0
\(109\) 248.938 0.218752 0.109376 0.994000i \(-0.465115\pi\)
0.109376 + 0.994000i \(0.465115\pi\)
\(110\) 93.3958 0.0809540
\(111\) 0 0
\(112\) −32.0117 −0.0270073
\(113\) 494.465 0.411641 0.205820 0.978590i \(-0.434014\pi\)
0.205820 + 0.978590i \(0.434014\pi\)
\(114\) 0 0
\(115\) 26.0488 0.0211223
\(116\) 548.271 0.438842
\(117\) 0 0
\(118\) 3281.77 2.56026
\(119\) 125.160 0.0964151
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 2979.12 2.21079
\(123\) 0 0
\(124\) 2319.47 1.67979
\(125\) 454.774 0.325410
\(126\) 0 0
\(127\) −979.104 −0.684106 −0.342053 0.939681i \(-0.611122\pi\)
−0.342053 + 0.939681i \(0.611122\pi\)
\(128\) 2416.25 1.66850
\(129\) 0 0
\(130\) 209.207 0.141144
\(131\) 1660.85 1.10770 0.553851 0.832616i \(-0.313158\pi\)
0.553851 + 0.832616i \(0.313158\pi\)
\(132\) 0 0
\(133\) −225.021 −0.146705
\(134\) −1201.02 −0.774271
\(135\) 0 0
\(136\) 427.758 0.269705
\(137\) 1618.17 1.00912 0.504559 0.863377i \(-0.331655\pi\)
0.504559 + 0.863377i \(0.331655\pi\)
\(138\) 0 0
\(139\) 695.736 0.424544 0.212272 0.977211i \(-0.431914\pi\)
0.212272 + 0.977211i \(0.431914\pi\)
\(140\) 170.356 0.102841
\(141\) 0 0
\(142\) 1698.97 1.00405
\(143\) 271.041 0.158501
\(144\) 0 0
\(145\) −76.6205 −0.0438826
\(146\) −5275.07 −2.99019
\(147\) 0 0
\(148\) 3864.90 2.14658
\(149\) 2081.84 1.14464 0.572318 0.820032i \(-0.306044\pi\)
0.572318 + 0.820032i \(0.306044\pi\)
\(150\) 0 0
\(151\) 2679.28 1.44395 0.721975 0.691919i \(-0.243234\pi\)
0.721975 + 0.691919i \(0.243234\pi\)
\(152\) −769.052 −0.410384
\(153\) 0 0
\(154\) 354.504 0.185498
\(155\) −324.144 −0.167973
\(156\) 0 0
\(157\) 2410.32 1.22525 0.612627 0.790372i \(-0.290113\pi\)
0.612627 + 0.790372i \(0.290113\pi\)
\(158\) −2250.80 −1.13332
\(159\) 0 0
\(160\) −314.131 −0.155214
\(161\) 98.8738 0.0483997
\(162\) 0 0
\(163\) 3629.08 1.74388 0.871938 0.489616i \(-0.162863\pi\)
0.871938 + 0.489616i \(0.162863\pi\)
\(164\) −2036.15 −0.969491
\(165\) 0 0
\(166\) 2527.49 1.18175
\(167\) 3826.04 1.77286 0.886432 0.462859i \(-0.153176\pi\)
0.886432 + 0.462859i \(0.153176\pi\)
\(168\) 0 0
\(169\) −1589.87 −0.723653
\(170\) −151.811 −0.0684903
\(171\) 0 0
\(172\) −3657.30 −1.62132
\(173\) 4310.67 1.89442 0.947209 0.320617i \(-0.103890\pi\)
0.947209 + 0.320617i \(0.103890\pi\)
\(174\) 0 0
\(175\) 851.193 0.367681
\(176\) 50.3040 0.0215444
\(177\) 0 0
\(178\) 483.593 0.203634
\(179\) −2491.12 −1.04019 −0.520097 0.854107i \(-0.674104\pi\)
−0.520097 + 0.854107i \(0.674104\pi\)
\(180\) 0 0
\(181\) −4315.49 −1.77220 −0.886098 0.463498i \(-0.846594\pi\)
−0.886098 + 0.463498i \(0.846594\pi\)
\(182\) 794.093 0.323418
\(183\) 0 0
\(184\) 337.920 0.135390
\(185\) −540.118 −0.214650
\(186\) 0 0
\(187\) −196.680 −0.0769127
\(188\) 688.124 0.266950
\(189\) 0 0
\(190\) 272.936 0.104215
\(191\) −2840.41 −1.07605 −0.538024 0.842930i \(-0.680829\pi\)
−0.538024 + 0.842930i \(0.680829\pi\)
\(192\) 0 0
\(193\) −1734.68 −0.646969 −0.323485 0.946233i \(-0.604854\pi\)
−0.323485 + 0.946233i \(0.604854\pi\)
\(194\) 6270.15 2.32047
\(195\) 0 0
\(196\) 646.622 0.235649
\(197\) 3098.42 1.12057 0.560287 0.828298i \(-0.310691\pi\)
0.560287 + 0.828298i \(0.310691\pi\)
\(198\) 0 0
\(199\) 4497.38 1.60206 0.801032 0.598622i \(-0.204285\pi\)
0.801032 + 0.598622i \(0.204285\pi\)
\(200\) 2909.11 1.02853
\(201\) 0 0
\(202\) −7413.85 −2.58236
\(203\) −290.830 −0.100553
\(204\) 0 0
\(205\) 284.550 0.0969456
\(206\) 566.491 0.191598
\(207\) 0 0
\(208\) 112.682 0.0375628
\(209\) 353.605 0.117030
\(210\) 0 0
\(211\) 1262.32 0.411857 0.205929 0.978567i \(-0.433979\pi\)
0.205929 + 0.978567i \(0.433979\pi\)
\(212\) −1087.26 −0.352232
\(213\) 0 0
\(214\) −8015.03 −2.56026
\(215\) 511.105 0.162126
\(216\) 0 0
\(217\) −1230.36 −0.384895
\(218\) −1146.10 −0.356071
\(219\) 0 0
\(220\) −267.702 −0.0820384
\(221\) −440.565 −0.134098
\(222\) 0 0
\(223\) 2931.38 0.880268 0.440134 0.897932i \(-0.354931\pi\)
0.440134 + 0.897932i \(0.354931\pi\)
\(224\) −1192.35 −0.355658
\(225\) 0 0
\(226\) −2276.49 −0.670045
\(227\) −4298.37 −1.25680 −0.628399 0.777891i \(-0.716289\pi\)
−0.628399 + 0.777891i \(0.716289\pi\)
\(228\) 0 0
\(229\) 698.500 0.201564 0.100782 0.994909i \(-0.467866\pi\)
0.100782 + 0.994909i \(0.467866\pi\)
\(230\) −119.927 −0.0343816
\(231\) 0 0
\(232\) −993.965 −0.281280
\(233\) −1887.78 −0.530783 −0.265391 0.964141i \(-0.585501\pi\)
−0.265391 + 0.964141i \(0.585501\pi\)
\(234\) 0 0
\(235\) −96.1649 −0.0266941
\(236\) −9406.57 −2.59456
\(237\) 0 0
\(238\) −576.231 −0.156939
\(239\) 6449.93 1.74565 0.872827 0.488030i \(-0.162284\pi\)
0.872827 + 0.488030i \(0.162284\pi\)
\(240\) 0 0
\(241\) 4636.98 1.23939 0.619697 0.784841i \(-0.287256\pi\)
0.619697 + 0.784841i \(0.287256\pi\)
\(242\) −557.078 −0.147977
\(243\) 0 0
\(244\) −8539.08 −2.24040
\(245\) −90.3650 −0.0235641
\(246\) 0 0
\(247\) 792.078 0.204043
\(248\) −4204.98 −1.07668
\(249\) 0 0
\(250\) −2093.76 −0.529683
\(251\) −2194.11 −0.551758 −0.275879 0.961192i \(-0.588969\pi\)
−0.275879 + 0.961192i \(0.588969\pi\)
\(252\) 0 0
\(253\) −155.373 −0.0386096
\(254\) 4507.75 1.11355
\(255\) 0 0
\(256\) −4557.87 −1.11276
\(257\) 2966.97 0.720133 0.360067 0.932927i \(-0.382754\pi\)
0.360067 + 0.932927i \(0.382754\pi\)
\(258\) 0 0
\(259\) −2050.14 −0.491850
\(260\) −599.654 −0.143034
\(261\) 0 0
\(262\) −7646.47 −1.80305
\(263\) −915.810 −0.214720 −0.107360 0.994220i \(-0.534240\pi\)
−0.107360 + 0.994220i \(0.534240\pi\)
\(264\) 0 0
\(265\) 151.944 0.0352220
\(266\) 1035.99 0.238799
\(267\) 0 0
\(268\) 3442.50 0.784642
\(269\) 164.462 0.0372768 0.0186384 0.999826i \(-0.494067\pi\)
0.0186384 + 0.999826i \(0.494067\pi\)
\(270\) 0 0
\(271\) 1502.60 0.336815 0.168407 0.985718i \(-0.446138\pi\)
0.168407 + 0.985718i \(0.446138\pi\)
\(272\) −81.7670 −0.0182274
\(273\) 0 0
\(274\) −7449.96 −1.64259
\(275\) −1337.59 −0.293308
\(276\) 0 0
\(277\) −7500.11 −1.62685 −0.813426 0.581669i \(-0.802400\pi\)
−0.813426 + 0.581669i \(0.802400\pi\)
\(278\) −3203.14 −0.691048
\(279\) 0 0
\(280\) −308.839 −0.0659167
\(281\) 2945.12 0.625235 0.312618 0.949879i \(-0.398794\pi\)
0.312618 + 0.949879i \(0.398794\pi\)
\(282\) 0 0
\(283\) 5215.29 1.09547 0.547733 0.836653i \(-0.315491\pi\)
0.547733 + 0.836653i \(0.315491\pi\)
\(284\) −4869.78 −1.01749
\(285\) 0 0
\(286\) −1247.86 −0.257998
\(287\) 1080.07 0.222142
\(288\) 0 0
\(289\) −4593.31 −0.934929
\(290\) 352.757 0.0714296
\(291\) 0 0
\(292\) 15120.0 3.03024
\(293\) −7407.99 −1.47706 −0.738531 0.674219i \(-0.764480\pi\)
−0.738531 + 0.674219i \(0.764480\pi\)
\(294\) 0 0
\(295\) 1314.56 0.259447
\(296\) −7006.72 −1.37587
\(297\) 0 0
\(298\) −9584.68 −1.86317
\(299\) −348.037 −0.0673161
\(300\) 0 0
\(301\) 1940.01 0.371496
\(302\) −12335.3 −2.35038
\(303\) 0 0
\(304\) 147.006 0.0277348
\(305\) 1193.33 0.224033
\(306\) 0 0
\(307\) −6850.01 −1.27345 −0.636727 0.771089i \(-0.719712\pi\)
−0.636727 + 0.771089i \(0.719712\pi\)
\(308\) −1016.12 −0.187983
\(309\) 0 0
\(310\) 1492.34 0.273417
\(311\) 5538.62 1.00986 0.504929 0.863161i \(-0.331519\pi\)
0.504929 + 0.863161i \(0.331519\pi\)
\(312\) 0 0
\(313\) 9361.13 1.69049 0.845243 0.534382i \(-0.179456\pi\)
0.845243 + 0.534382i \(0.179456\pi\)
\(314\) −11097.0 −1.99440
\(315\) 0 0
\(316\) 6451.50 1.14850
\(317\) −219.221 −0.0388413 −0.0194206 0.999811i \(-0.506182\pi\)
−0.0194206 + 0.999811i \(0.506182\pi\)
\(318\) 0 0
\(319\) 457.018 0.0802135
\(320\) 1513.71 0.264435
\(321\) 0 0
\(322\) −455.210 −0.0787822
\(323\) −574.769 −0.0990123
\(324\) 0 0
\(325\) −2996.21 −0.511384
\(326\) −16708.1 −2.83858
\(327\) 0 0
\(328\) 3691.35 0.621405
\(329\) −365.015 −0.0611669
\(330\) 0 0
\(331\) 3377.63 0.560880 0.280440 0.959872i \(-0.409520\pi\)
0.280440 + 0.959872i \(0.409520\pi\)
\(332\) −7244.57 −1.19758
\(333\) 0 0
\(334\) −17614.9 −2.88577
\(335\) −481.087 −0.0784615
\(336\) 0 0
\(337\) 7755.93 1.25369 0.626843 0.779145i \(-0.284347\pi\)
0.626843 + 0.779145i \(0.284347\pi\)
\(338\) 7319.66 1.17792
\(339\) 0 0
\(340\) 435.137 0.0694077
\(341\) 1933.42 0.307040
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) 6630.35 1.03920
\(345\) 0 0
\(346\) −19846.1 −3.08362
\(347\) 831.044 0.128567 0.0642835 0.997932i \(-0.479524\pi\)
0.0642835 + 0.997932i \(0.479524\pi\)
\(348\) 0 0
\(349\) 1840.94 0.282359 0.141180 0.989984i \(-0.454911\pi\)
0.141180 + 0.989984i \(0.454911\pi\)
\(350\) −3918.85 −0.598490
\(351\) 0 0
\(352\) 1873.70 0.283717
\(353\) −3409.11 −0.514019 −0.257009 0.966409i \(-0.582737\pi\)
−0.257009 + 0.966409i \(0.582737\pi\)
\(354\) 0 0
\(355\) 680.549 0.101746
\(356\) −1386.13 −0.206361
\(357\) 0 0
\(358\) 11469.0 1.69317
\(359\) −2199.75 −0.323394 −0.161697 0.986840i \(-0.551697\pi\)
−0.161697 + 0.986840i \(0.551697\pi\)
\(360\) 0 0
\(361\) −5825.64 −0.849343
\(362\) 19868.3 2.88468
\(363\) 0 0
\(364\) −2276.12 −0.327750
\(365\) −2113.01 −0.303014
\(366\) 0 0
\(367\) −855.008 −0.121611 −0.0608053 0.998150i \(-0.519367\pi\)
−0.0608053 + 0.998150i \(0.519367\pi\)
\(368\) −64.5942 −0.00915002
\(369\) 0 0
\(370\) 2486.68 0.349395
\(371\) 576.735 0.0807078
\(372\) 0 0
\(373\) 3193.40 0.443293 0.221646 0.975127i \(-0.428857\pi\)
0.221646 + 0.975127i \(0.428857\pi\)
\(374\) 905.505 0.125194
\(375\) 0 0
\(376\) −1247.51 −0.171104
\(377\) 1023.72 0.139853
\(378\) 0 0
\(379\) −5614.48 −0.760940 −0.380470 0.924793i \(-0.624238\pi\)
−0.380470 + 0.924793i \(0.624238\pi\)
\(380\) −782.319 −0.105611
\(381\) 0 0
\(382\) 13077.1 1.75153
\(383\) 1736.86 0.231721 0.115861 0.993265i \(-0.463037\pi\)
0.115861 + 0.993265i \(0.463037\pi\)
\(384\) 0 0
\(385\) 142.002 0.0187977
\(386\) 7986.38 1.05310
\(387\) 0 0
\(388\) −17972.2 −2.35155
\(389\) −8710.78 −1.13536 −0.567679 0.823250i \(-0.692159\pi\)
−0.567679 + 0.823250i \(0.692159\pi\)
\(390\) 0 0
\(391\) 252.552 0.0326652
\(392\) −1172.27 −0.151042
\(393\) 0 0
\(394\) −14265.0 −1.82401
\(395\) −901.593 −0.114846
\(396\) 0 0
\(397\) 11731.6 1.48311 0.741553 0.670894i \(-0.234089\pi\)
0.741553 + 0.670894i \(0.234089\pi\)
\(398\) −20705.7 −2.60775
\(399\) 0 0
\(400\) −556.084 −0.0695105
\(401\) 14408.8 1.79437 0.897183 0.441659i \(-0.145610\pi\)
0.897183 + 0.441659i \(0.145610\pi\)
\(402\) 0 0
\(403\) 4330.88 0.535327
\(404\) 21250.4 2.61695
\(405\) 0 0
\(406\) 1338.97 0.163674
\(407\) 3221.64 0.392361
\(408\) 0 0
\(409\) −2155.54 −0.260598 −0.130299 0.991475i \(-0.541594\pi\)
−0.130299 + 0.991475i \(0.541594\pi\)
\(410\) −1310.06 −0.157803
\(411\) 0 0
\(412\) −1623.74 −0.194165
\(413\) 4989.71 0.594498
\(414\) 0 0
\(415\) 1012.42 0.119754
\(416\) 4197.10 0.494663
\(417\) 0 0
\(418\) −1627.98 −0.190495
\(419\) −8443.41 −0.984458 −0.492229 0.870466i \(-0.663818\pi\)
−0.492229 + 0.870466i \(0.663818\pi\)
\(420\) 0 0
\(421\) −3070.28 −0.355430 −0.177715 0.984082i \(-0.556871\pi\)
−0.177715 + 0.984082i \(0.556871\pi\)
\(422\) −5811.67 −0.670398
\(423\) 0 0
\(424\) 1971.10 0.225767
\(425\) 2174.19 0.248150
\(426\) 0 0
\(427\) 4529.55 0.513349
\(428\) 22973.6 2.59456
\(429\) 0 0
\(430\) −2353.10 −0.263899
\(431\) 7004.40 0.782808 0.391404 0.920219i \(-0.371990\pi\)
0.391404 + 0.920219i \(0.371990\pi\)
\(432\) 0 0
\(433\) 7486.00 0.830841 0.415421 0.909629i \(-0.363634\pi\)
0.415421 + 0.909629i \(0.363634\pi\)
\(434\) 5664.52 0.626510
\(435\) 0 0
\(436\) 3285.07 0.360841
\(437\) −454.055 −0.0497034
\(438\) 0 0
\(439\) −13046.2 −1.41836 −0.709179 0.705029i \(-0.750934\pi\)
−0.709179 + 0.705029i \(0.750934\pi\)
\(440\) 485.319 0.0525833
\(441\) 0 0
\(442\) 2028.34 0.218277
\(443\) 11781.3 1.26354 0.631768 0.775157i \(-0.282329\pi\)
0.631768 + 0.775157i \(0.282329\pi\)
\(444\) 0 0
\(445\) 193.711 0.0206354
\(446\) −13495.9 −1.43285
\(447\) 0 0
\(448\) 5745.63 0.605927
\(449\) −7576.58 −0.796349 −0.398175 0.917310i \(-0.630356\pi\)
−0.398175 + 0.917310i \(0.630356\pi\)
\(450\) 0 0
\(451\) −1697.26 −0.177208
\(452\) 6525.14 0.679020
\(453\) 0 0
\(454\) 19789.5 2.04574
\(455\) 318.086 0.0327738
\(456\) 0 0
\(457\) 11793.0 1.20712 0.603560 0.797318i \(-0.293748\pi\)
0.603560 + 0.797318i \(0.293748\pi\)
\(458\) −3215.86 −0.328095
\(459\) 0 0
\(460\) 343.749 0.0348421
\(461\) −4228.32 −0.427185 −0.213592 0.976923i \(-0.568516\pi\)
−0.213592 + 0.976923i \(0.568516\pi\)
\(462\) 0 0
\(463\) 14448.8 1.45031 0.725154 0.688586i \(-0.241768\pi\)
0.725154 + 0.688586i \(0.241768\pi\)
\(464\) 189.999 0.0190096
\(465\) 0 0
\(466\) 8691.23 0.863977
\(467\) −16547.5 −1.63967 −0.819836 0.572599i \(-0.805935\pi\)
−0.819836 + 0.572599i \(0.805935\pi\)
\(468\) 0 0
\(469\) −1826.07 −0.179787
\(470\) 442.738 0.0434511
\(471\) 0 0
\(472\) 17053.3 1.66301
\(473\) −3048.59 −0.296351
\(474\) 0 0
\(475\) −3908.91 −0.377585
\(476\) 1651.66 0.159041
\(477\) 0 0
\(478\) −29695.2 −2.84148
\(479\) 2989.34 0.285149 0.142575 0.989784i \(-0.454462\pi\)
0.142575 + 0.989784i \(0.454462\pi\)
\(480\) 0 0
\(481\) 7216.51 0.684084
\(482\) −21348.4 −2.01741
\(483\) 0 0
\(484\) 1596.76 0.149959
\(485\) 2511.60 0.235147
\(486\) 0 0
\(487\) −5549.61 −0.516379 −0.258190 0.966094i \(-0.583126\pi\)
−0.258190 + 0.966094i \(0.583126\pi\)
\(488\) 15480.6 1.43601
\(489\) 0 0
\(490\) 416.036 0.0383563
\(491\) 7751.20 0.712438 0.356219 0.934403i \(-0.384066\pi\)
0.356219 + 0.934403i \(0.384066\pi\)
\(492\) 0 0
\(493\) −742.862 −0.0678638
\(494\) −3646.69 −0.332130
\(495\) 0 0
\(496\) 803.793 0.0727649
\(497\) 2583.17 0.233141
\(498\) 0 0
\(499\) −6841.83 −0.613792 −0.306896 0.951743i \(-0.599290\pi\)
−0.306896 + 0.951743i \(0.599290\pi\)
\(500\) 6001.36 0.536778
\(501\) 0 0
\(502\) 10101.6 0.898120
\(503\) 11518.5 1.02105 0.510523 0.859864i \(-0.329452\pi\)
0.510523 + 0.859864i \(0.329452\pi\)
\(504\) 0 0
\(505\) −2969.73 −0.261686
\(506\) 715.330 0.0628465
\(507\) 0 0
\(508\) −12920.6 −1.12846
\(509\) −8292.40 −0.722110 −0.361055 0.932545i \(-0.617583\pi\)
−0.361055 + 0.932545i \(0.617583\pi\)
\(510\) 0 0
\(511\) −8020.39 −0.694327
\(512\) 1654.21 0.142786
\(513\) 0 0
\(514\) −13659.8 −1.17219
\(515\) 226.917 0.0194158
\(516\) 0 0
\(517\) 573.595 0.0487943
\(518\) 9438.72 0.800606
\(519\) 0 0
\(520\) 1087.12 0.0916794
\(521\) −10891.6 −0.915868 −0.457934 0.888986i \(-0.651410\pi\)
−0.457934 + 0.888986i \(0.651410\pi\)
\(522\) 0 0
\(523\) 14662.0 1.22586 0.612928 0.790139i \(-0.289992\pi\)
0.612928 + 0.790139i \(0.289992\pi\)
\(524\) 21917.2 1.82721
\(525\) 0 0
\(526\) 4216.34 0.349508
\(527\) −3142.69 −0.259768
\(528\) 0 0
\(529\) −11967.5 −0.983602
\(530\) −699.541 −0.0573323
\(531\) 0 0
\(532\) −2969.46 −0.241997
\(533\) −3801.87 −0.308963
\(534\) 0 0
\(535\) −3210.54 −0.259446
\(536\) −6240.94 −0.502924
\(537\) 0 0
\(538\) −757.177 −0.0606770
\(539\) 539.000 0.0430730
\(540\) 0 0
\(541\) −19825.8 −1.57556 −0.787778 0.615960i \(-0.788768\pi\)
−0.787778 + 0.615960i \(0.788768\pi\)
\(542\) −6917.92 −0.548247
\(543\) 0 0
\(544\) −3045.61 −0.240036
\(545\) −459.087 −0.0360828
\(546\) 0 0
\(547\) 12706.1 0.993187 0.496593 0.867983i \(-0.334584\pi\)
0.496593 + 0.867983i \(0.334584\pi\)
\(548\) 21353.9 1.66459
\(549\) 0 0
\(550\) 6158.19 0.477430
\(551\) 1335.57 0.103262
\(552\) 0 0
\(553\) −3422.19 −0.263158
\(554\) 34530.1 2.64809
\(555\) 0 0
\(556\) 9181.19 0.700304
\(557\) 12599.1 0.958421 0.479211 0.877700i \(-0.340923\pi\)
0.479211 + 0.877700i \(0.340923\pi\)
\(558\) 0 0
\(559\) −6828.86 −0.516691
\(560\) 59.0354 0.00445482
\(561\) 0 0
\(562\) −13559.2 −1.01772
\(563\) −6004.47 −0.449482 −0.224741 0.974419i \(-0.572154\pi\)
−0.224741 + 0.974419i \(0.572154\pi\)
\(564\) 0 0
\(565\) −911.884 −0.0678996
\(566\) −24010.9 −1.78314
\(567\) 0 0
\(568\) 8828.47 0.652173
\(569\) 3145.89 0.231779 0.115890 0.993262i \(-0.463028\pi\)
0.115890 + 0.993262i \(0.463028\pi\)
\(570\) 0 0
\(571\) 23549.1 1.72592 0.862960 0.505273i \(-0.168608\pi\)
0.862960 + 0.505273i \(0.168608\pi\)
\(572\) 3576.76 0.261454
\(573\) 0 0
\(574\) −4972.60 −0.361590
\(575\) 1717.57 0.124569
\(576\) 0 0
\(577\) 327.335 0.0236172 0.0118086 0.999930i \(-0.496241\pi\)
0.0118086 + 0.999930i \(0.496241\pi\)
\(578\) 21147.4 1.52182
\(579\) 0 0
\(580\) −1011.11 −0.0723864
\(581\) 3842.88 0.274405
\(582\) 0 0
\(583\) −906.298 −0.0643825
\(584\) −27411.2 −1.94226
\(585\) 0 0
\(586\) 34106.0 2.40428
\(587\) 13270.8 0.933123 0.466562 0.884489i \(-0.345493\pi\)
0.466562 + 0.884489i \(0.345493\pi\)
\(588\) 0 0
\(589\) 5650.14 0.395263
\(590\) −6052.18 −0.422312
\(591\) 0 0
\(592\) 1339.35 0.0929849
\(593\) −5098.92 −0.353099 −0.176549 0.984292i \(-0.556494\pi\)
−0.176549 + 0.984292i \(0.556494\pi\)
\(594\) 0 0
\(595\) −230.818 −0.0159036
\(596\) 27472.7 1.88813
\(597\) 0 0
\(598\) 1602.35 0.109573
\(599\) −19358.2 −1.32046 −0.660230 0.751064i \(-0.729541\pi\)
−0.660230 + 0.751064i \(0.729541\pi\)
\(600\) 0 0
\(601\) 1238.87 0.0840841 0.0420420 0.999116i \(-0.486614\pi\)
0.0420420 + 0.999116i \(0.486614\pi\)
\(602\) −8931.71 −0.604700
\(603\) 0 0
\(604\) 35356.7 2.38186
\(605\) −223.146 −0.0149953
\(606\) 0 0
\(607\) −14175.6 −0.947888 −0.473944 0.880555i \(-0.657170\pi\)
−0.473944 + 0.880555i \(0.657170\pi\)
\(608\) 5475.60 0.365239
\(609\) 0 0
\(610\) −5494.04 −0.364667
\(611\) 1284.86 0.0850732
\(612\) 0 0
\(613\) 6906.67 0.455070 0.227535 0.973770i \(-0.426933\pi\)
0.227535 + 0.973770i \(0.426933\pi\)
\(614\) 31537.1 2.07286
\(615\) 0 0
\(616\) 1842.13 0.120490
\(617\) −12104.3 −0.789789 −0.394894 0.918727i \(-0.629219\pi\)
−0.394894 + 0.918727i \(0.629219\pi\)
\(618\) 0 0
\(619\) 14945.2 0.970437 0.485218 0.874393i \(-0.338740\pi\)
0.485218 + 0.874393i \(0.338740\pi\)
\(620\) −4277.52 −0.277080
\(621\) 0 0
\(622\) −25499.5 −1.64379
\(623\) 735.271 0.0472841
\(624\) 0 0
\(625\) 14361.2 0.919116
\(626\) −43098.2 −2.75168
\(627\) 0 0
\(628\) 31807.5 2.02111
\(629\) −5236.63 −0.331953
\(630\) 0 0
\(631\) −17711.9 −1.11743 −0.558717 0.829358i \(-0.688706\pi\)
−0.558717 + 0.829358i \(0.688706\pi\)
\(632\) −11696.0 −0.736141
\(633\) 0 0
\(634\) 1009.28 0.0632236
\(635\) 1805.65 0.112842
\(636\) 0 0
\(637\) 1207.36 0.0750982
\(638\) −2104.09 −0.130567
\(639\) 0 0
\(640\) −4456.01 −0.275218
\(641\) 17994.4 1.10879 0.554396 0.832253i \(-0.312949\pi\)
0.554396 + 0.832253i \(0.312949\pi\)
\(642\) 0 0
\(643\) 27947.8 1.71408 0.857039 0.515251i \(-0.172301\pi\)
0.857039 + 0.515251i \(0.172301\pi\)
\(644\) 1304.77 0.0798375
\(645\) 0 0
\(646\) 2646.21 0.161167
\(647\) 14336.2 0.871122 0.435561 0.900159i \(-0.356550\pi\)
0.435561 + 0.900159i \(0.356550\pi\)
\(648\) 0 0
\(649\) −7840.97 −0.474245
\(650\) 13794.4 0.832402
\(651\) 0 0
\(652\) 47890.7 2.87660
\(653\) −4315.79 −0.258637 −0.129318 0.991603i \(-0.541279\pi\)
−0.129318 + 0.991603i \(0.541279\pi\)
\(654\) 0 0
\(655\) −3062.91 −0.182714
\(656\) −705.611 −0.0419962
\(657\) 0 0
\(658\) 1680.51 0.0995640
\(659\) −4002.23 −0.236578 −0.118289 0.992979i \(-0.537741\pi\)
−0.118289 + 0.992979i \(0.537741\pi\)
\(660\) 0 0
\(661\) −12223.4 −0.719268 −0.359634 0.933094i \(-0.617098\pi\)
−0.359634 + 0.933094i \(0.617098\pi\)
\(662\) −15550.4 −0.912968
\(663\) 0 0
\(664\) 13133.7 0.767603
\(665\) 414.980 0.0241989
\(666\) 0 0
\(667\) −586.846 −0.0340671
\(668\) 50489.9 2.92442
\(669\) 0 0
\(670\) 2214.90 0.127715
\(671\) −7117.86 −0.409511
\(672\) 0 0
\(673\) −6121.53 −0.350621 −0.175310 0.984513i \(-0.556093\pi\)
−0.175310 + 0.984513i \(0.556093\pi\)
\(674\) −35707.9 −2.04068
\(675\) 0 0
\(676\) −20980.4 −1.19370
\(677\) −9626.46 −0.546492 −0.273246 0.961944i \(-0.588097\pi\)
−0.273246 + 0.961944i \(0.588097\pi\)
\(678\) 0 0
\(679\) 9533.34 0.538816
\(680\) −788.863 −0.0444875
\(681\) 0 0
\(682\) −8901.38 −0.499782
\(683\) −11554.4 −0.647315 −0.323658 0.946174i \(-0.604913\pi\)
−0.323658 + 0.946174i \(0.604913\pi\)
\(684\) 0 0
\(685\) −2984.19 −0.166453
\(686\) 1579.16 0.0878898
\(687\) 0 0
\(688\) −1267.41 −0.0702318
\(689\) −2030.12 −0.112251
\(690\) 0 0
\(691\) 2991.45 0.164689 0.0823444 0.996604i \(-0.473759\pi\)
0.0823444 + 0.996604i \(0.473759\pi\)
\(692\) 56885.2 3.12493
\(693\) 0 0
\(694\) −3826.08 −0.209274
\(695\) −1283.06 −0.0700279
\(696\) 0 0
\(697\) 2758.82 0.149925
\(698\) −8475.60 −0.459608
\(699\) 0 0
\(700\) 11232.7 0.606506
\(701\) −30772.8 −1.65802 −0.829010 0.559234i \(-0.811095\pi\)
−0.829010 + 0.559234i \(0.811095\pi\)
\(702\) 0 0
\(703\) 9414.77 0.505099
\(704\) −9028.84 −0.483363
\(705\) 0 0
\(706\) 15695.4 0.836690
\(707\) −11272.3 −0.599628
\(708\) 0 0
\(709\) −21698.3 −1.14936 −0.574681 0.818377i \(-0.694874\pi\)
−0.574681 + 0.818377i \(0.694874\pi\)
\(710\) −3133.21 −0.165616
\(711\) 0 0
\(712\) 2512.93 0.132269
\(713\) −2482.66 −0.130402
\(714\) 0 0
\(715\) −499.849 −0.0261445
\(716\) −32873.7 −1.71585
\(717\) 0 0
\(718\) 10127.5 0.526402
\(719\) 12364.2 0.641318 0.320659 0.947195i \(-0.396096\pi\)
0.320659 + 0.947195i \(0.396096\pi\)
\(720\) 0 0
\(721\) 861.311 0.0444895
\(722\) 26821.0 1.38251
\(723\) 0 0
\(724\) −56948.7 −2.92332
\(725\) −5052.09 −0.258800
\(726\) 0 0
\(727\) −16017.2 −0.817119 −0.408559 0.912732i \(-0.633969\pi\)
−0.408559 + 0.912732i \(0.633969\pi\)
\(728\) 4126.39 0.210075
\(729\) 0 0
\(730\) 9728.19 0.493228
\(731\) 4955.34 0.250725
\(732\) 0 0
\(733\) 10564.4 0.532340 0.266170 0.963926i \(-0.414242\pi\)
0.266170 + 0.963926i \(0.414242\pi\)
\(734\) 3936.42 0.197951
\(735\) 0 0
\(736\) −2405.97 −0.120496
\(737\) 2869.54 0.143420
\(738\) 0 0
\(739\) 1760.32 0.0876242 0.0438121 0.999040i \(-0.486050\pi\)
0.0438121 + 0.999040i \(0.486050\pi\)
\(740\) −7127.59 −0.354075
\(741\) 0 0
\(742\) −2655.26 −0.131371
\(743\) −11383.4 −0.562067 −0.281034 0.959698i \(-0.590677\pi\)
−0.281034 + 0.959698i \(0.590677\pi\)
\(744\) 0 0
\(745\) −3839.29 −0.188806
\(746\) −14702.3 −0.721566
\(747\) 0 0
\(748\) −2595.46 −0.126871
\(749\) −12186.3 −0.594497
\(750\) 0 0
\(751\) −15058.3 −0.731670 −0.365835 0.930680i \(-0.619217\pi\)
−0.365835 + 0.930680i \(0.619217\pi\)
\(752\) 238.464 0.0115637
\(753\) 0 0
\(754\) −4713.18 −0.227644
\(755\) −4941.08 −0.238178
\(756\) 0 0
\(757\) 38073.4 1.82801 0.914004 0.405706i \(-0.132974\pi\)
0.914004 + 0.405706i \(0.132974\pi\)
\(758\) 25848.8 1.23861
\(759\) 0 0
\(760\) 1418.27 0.0676923
\(761\) −15232.1 −0.725577 −0.362788 0.931872i \(-0.618175\pi\)
−0.362788 + 0.931872i \(0.618175\pi\)
\(762\) 0 0
\(763\) −1742.57 −0.0826803
\(764\) −37483.1 −1.77499
\(765\) 0 0
\(766\) −7996.40 −0.377182
\(767\) −17563.8 −0.826850
\(768\) 0 0
\(769\) 12013.1 0.563332 0.281666 0.959513i \(-0.409113\pi\)
0.281666 + 0.959513i \(0.409113\pi\)
\(770\) −653.771 −0.0305977
\(771\) 0 0
\(772\) −22891.5 −1.06720
\(773\) 14258.5 0.663443 0.331721 0.943377i \(-0.392371\pi\)
0.331721 + 0.943377i \(0.392371\pi\)
\(774\) 0 0
\(775\) −21372.9 −0.990630
\(776\) 32582.0 1.50725
\(777\) 0 0
\(778\) 40104.0 1.84807
\(779\) −4959.99 −0.228126
\(780\) 0 0
\(781\) −4059.27 −0.185982
\(782\) −1162.74 −0.0531706
\(783\) 0 0
\(784\) 224.082 0.0102078
\(785\) −4445.08 −0.202104
\(786\) 0 0
\(787\) −14377.9 −0.651227 −0.325613 0.945503i \(-0.605571\pi\)
−0.325613 + 0.945503i \(0.605571\pi\)
\(788\) 40887.9 1.84844
\(789\) 0 0
\(790\) 4150.89 0.186939
\(791\) −3461.26 −0.155585
\(792\) 0 0
\(793\) −15944.1 −0.713986
\(794\) −54011.8 −2.41412
\(795\) 0 0
\(796\) 59349.0 2.64268
\(797\) −38515.9 −1.71180 −0.855899 0.517144i \(-0.826995\pi\)
−0.855899 + 0.517144i \(0.826995\pi\)
\(798\) 0 0
\(799\) −932.352 −0.0412819
\(800\) −20712.7 −0.915381
\(801\) 0 0
\(802\) −66337.4 −2.92076
\(803\) 12603.5 0.553882
\(804\) 0 0
\(805\) −182.341 −0.00798347
\(806\) −19939.2 −0.871374
\(807\) 0 0
\(808\) −38525.1 −1.67736
\(809\) −11438.6 −0.497106 −0.248553 0.968618i \(-0.579955\pi\)
−0.248553 + 0.968618i \(0.579955\pi\)
\(810\) 0 0
\(811\) 15872.9 0.687265 0.343632 0.939104i \(-0.388343\pi\)
0.343632 + 0.939104i \(0.388343\pi\)
\(812\) −3837.89 −0.165867
\(813\) 0 0
\(814\) −14832.3 −0.638662
\(815\) −6692.70 −0.287650
\(816\) 0 0
\(817\) −8909.05 −0.381503
\(818\) 9924.00 0.424186
\(819\) 0 0
\(820\) 3755.03 0.159916
\(821\) 11988.8 0.509638 0.254819 0.966989i \(-0.417984\pi\)
0.254819 + 0.966989i \(0.417984\pi\)
\(822\) 0 0
\(823\) −222.224 −0.00941219 −0.00470610 0.999989i \(-0.501498\pi\)
−0.00470610 + 0.999989i \(0.501498\pi\)
\(824\) 2943.69 0.124452
\(825\) 0 0
\(826\) −22972.4 −0.967689
\(827\) −7524.36 −0.316382 −0.158191 0.987409i \(-0.550566\pi\)
−0.158191 + 0.987409i \(0.550566\pi\)
\(828\) 0 0
\(829\) −11807.1 −0.494663 −0.247332 0.968931i \(-0.579554\pi\)
−0.247332 + 0.968931i \(0.579554\pi\)
\(830\) −4661.15 −0.194929
\(831\) 0 0
\(832\) −20224.7 −0.842746
\(833\) −876.120 −0.0364415
\(834\) 0 0
\(835\) −7055.93 −0.292432
\(836\) 4666.30 0.193047
\(837\) 0 0
\(838\) 38873.1 1.60244
\(839\) 29112.9 1.19796 0.598980 0.800764i \(-0.295573\pi\)
0.598980 + 0.800764i \(0.295573\pi\)
\(840\) 0 0
\(841\) −22662.8 −0.929224
\(842\) 14135.4 0.578549
\(843\) 0 0
\(844\) 16658.1 0.679377
\(845\) 2932.00 0.119366
\(846\) 0 0
\(847\) −847.000 −0.0343604
\(848\) −376.781 −0.0152579
\(849\) 0 0
\(850\) −10009.9 −0.403924
\(851\) −4136.83 −0.166638
\(852\) 0 0
\(853\) −24892.1 −0.999169 −0.499584 0.866265i \(-0.666514\pi\)
−0.499584 + 0.866265i \(0.666514\pi\)
\(854\) −20853.8 −0.835601
\(855\) 0 0
\(856\) −41649.0 −1.66301
\(857\) 27299.2 1.08812 0.544062 0.839045i \(-0.316886\pi\)
0.544062 + 0.839045i \(0.316886\pi\)
\(858\) 0 0
\(859\) −6975.13 −0.277053 −0.138526 0.990359i \(-0.544237\pi\)
−0.138526 + 0.990359i \(0.544237\pi\)
\(860\) 6744.73 0.267434
\(861\) 0 0
\(862\) −32247.9 −1.27421
\(863\) 4481.56 0.176772 0.0883858 0.996086i \(-0.471829\pi\)
0.0883858 + 0.996086i \(0.471829\pi\)
\(864\) 0 0
\(865\) −7949.67 −0.312482
\(866\) −34465.2 −1.35240
\(867\) 0 0
\(868\) −16236.3 −0.634902
\(869\) 5377.73 0.209928
\(870\) 0 0
\(871\) 6427.80 0.250055
\(872\) −5955.54 −0.231284
\(873\) 0 0
\(874\) 2090.45 0.0809044
\(875\) −3183.42 −0.122993
\(876\) 0 0
\(877\) −7207.96 −0.277532 −0.138766 0.990325i \(-0.544314\pi\)
−0.138766 + 0.990325i \(0.544314\pi\)
\(878\) 60063.9 2.30872
\(879\) 0 0
\(880\) −92.7699 −0.00355372
\(881\) 38413.7 1.46900 0.734502 0.678607i \(-0.237416\pi\)
0.734502 + 0.678607i \(0.237416\pi\)
\(882\) 0 0
\(883\) −8705.04 −0.331764 −0.165882 0.986146i \(-0.553047\pi\)
−0.165882 + 0.986146i \(0.553047\pi\)
\(884\) −5813.86 −0.221200
\(885\) 0 0
\(886\) −54240.6 −2.05671
\(887\) 44100.0 1.66937 0.834687 0.550725i \(-0.185649\pi\)
0.834687 + 0.550725i \(0.185649\pi\)
\(888\) 0 0
\(889\) 6853.73 0.258568
\(890\) −891.834 −0.0335892
\(891\) 0 0
\(892\) 38683.6 1.45204
\(893\) 1676.25 0.0628146
\(894\) 0 0
\(895\) 4594.08 0.171579
\(896\) −16913.8 −0.630635
\(897\) 0 0
\(898\) 34882.2 1.29625
\(899\) 7302.55 0.270916
\(900\) 0 0
\(901\) 1473.15 0.0544702
\(902\) 7814.09 0.288449
\(903\) 0 0
\(904\) −11829.5 −0.435224
\(905\) 7958.54 0.292322
\(906\) 0 0
\(907\) 19921.2 0.729296 0.364648 0.931145i \(-0.381189\pi\)
0.364648 + 0.931145i \(0.381189\pi\)
\(908\) −56722.9 −2.07314
\(909\) 0 0
\(910\) −1464.45 −0.0533474
\(911\) 14446.7 0.525402 0.262701 0.964877i \(-0.415387\pi\)
0.262701 + 0.964877i \(0.415387\pi\)
\(912\) 0 0
\(913\) −6038.81 −0.218900
\(914\) −54294.4 −1.96488
\(915\) 0 0
\(916\) 9217.66 0.332489
\(917\) −11625.9 −0.418672
\(918\) 0 0
\(919\) 44381.1 1.59303 0.796516 0.604617i \(-0.206674\pi\)
0.796516 + 0.604617i \(0.206674\pi\)
\(920\) −623.186 −0.0223324
\(921\) 0 0
\(922\) 19467.0 0.695347
\(923\) −9092.80 −0.324261
\(924\) 0 0
\(925\) −35613.5 −1.26591
\(926\) −66521.6 −2.36073
\(927\) 0 0
\(928\) 7076.97 0.250337
\(929\) 3455.30 0.122029 0.0610145 0.998137i \(-0.480566\pi\)
0.0610145 + 0.998137i \(0.480566\pi\)
\(930\) 0 0
\(931\) 1575.15 0.0554494
\(932\) −24911.8 −0.875550
\(933\) 0 0
\(934\) 76183.8 2.66896
\(935\) 362.714 0.0126866
\(936\) 0 0
\(937\) −14962.8 −0.521678 −0.260839 0.965382i \(-0.583999\pi\)
−0.260839 + 0.965382i \(0.583999\pi\)
\(938\) 8407.14 0.292647
\(939\) 0 0
\(940\) −1269.03 −0.0440331
\(941\) 6116.95 0.211909 0.105955 0.994371i \(-0.466210\pi\)
0.105955 + 0.994371i \(0.466210\pi\)
\(942\) 0 0
\(943\) 2179.41 0.0752611
\(944\) −3259.77 −0.112390
\(945\) 0 0
\(946\) 14035.6 0.482384
\(947\) −53343.7 −1.83045 −0.915225 0.402943i \(-0.867987\pi\)
−0.915225 + 0.402943i \(0.867987\pi\)
\(948\) 0 0
\(949\) 28231.9 0.965696
\(950\) 17996.4 0.614611
\(951\) 0 0
\(952\) −2994.30 −0.101939
\(953\) −1979.62 −0.0672887 −0.0336443 0.999434i \(-0.510711\pi\)
−0.0336443 + 0.999434i \(0.510711\pi\)
\(954\) 0 0
\(955\) 5238.24 0.177493
\(956\) 85115.6 2.87954
\(957\) 0 0
\(958\) −13762.8 −0.464149
\(959\) −11327.2 −0.381411
\(960\) 0 0
\(961\) 1102.58 0.0370104
\(962\) −33224.4 −1.11351
\(963\) 0 0
\(964\) 61191.2 2.04444
\(965\) 3199.07 0.106717
\(966\) 0 0
\(967\) −38892.0 −1.29336 −0.646681 0.762761i \(-0.723843\pi\)
−0.646681 + 0.762761i \(0.723843\pi\)
\(968\) −2894.78 −0.0961175
\(969\) 0 0
\(970\) −11563.3 −0.382758
\(971\) 47826.1 1.58065 0.790325 0.612688i \(-0.209912\pi\)
0.790325 + 0.612688i \(0.209912\pi\)
\(972\) 0 0
\(973\) −4870.15 −0.160462
\(974\) 25550.1 0.840533
\(975\) 0 0
\(976\) −2959.15 −0.0970493
\(977\) −20840.5 −0.682442 −0.341221 0.939983i \(-0.610840\pi\)
−0.341221 + 0.939983i \(0.610840\pi\)
\(978\) 0 0
\(979\) −1155.43 −0.0377197
\(980\) −1192.49 −0.0388701
\(981\) 0 0
\(982\) −35686.2 −1.15966
\(983\) 54430.5 1.76609 0.883044 0.469290i \(-0.155490\pi\)
0.883044 + 0.469290i \(0.155490\pi\)
\(984\) 0 0
\(985\) −5714.05 −0.184837
\(986\) 3420.10 0.110465
\(987\) 0 0
\(988\) 10452.5 0.336579
\(989\) 3914.62 0.125862
\(990\) 0 0
\(991\) −35161.9 −1.12710 −0.563550 0.826082i \(-0.690565\pi\)
−0.563550 + 0.826082i \(0.690565\pi\)
\(992\) 29939.2 0.958238
\(993\) 0 0
\(994\) −11892.8 −0.379494
\(995\) −8293.99 −0.264258
\(996\) 0 0
\(997\) 26961.6 0.856451 0.428225 0.903672i \(-0.359139\pi\)
0.428225 + 0.903672i \(0.359139\pi\)
\(998\) 31499.4 0.999096
\(999\) 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 693.4.a.l.1.1 4
3.2 odd 2 77.4.a.d.1.4 4
12.11 even 2 1232.4.a.s.1.3 4
15.14 odd 2 1925.4.a.p.1.1 4
21.20 even 2 539.4.a.g.1.4 4
33.32 even 2 847.4.a.d.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.4.a.d.1.4 4 3.2 odd 2
539.4.a.g.1.4 4 21.20 even 2
693.4.a.l.1.1 4 1.1 even 1 trivial
847.4.a.d.1.1 4 33.32 even 2
1232.4.a.s.1.3 4 12.11 even 2
1925.4.a.p.1.1 4 15.14 odd 2