Properties

Label 189.4.a.e.1.1
Level $189$
Weight $4$
Character 189.1
Self dual yes
Analytic conductor $11.151$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [189,4,Mod(1,189)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(189, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 189.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: \(11.1513609911\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) 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 \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 189.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.73205 q^{2} +14.3923 q^{4} -9.92820 q^{5} +7.00000 q^{7} -30.2487 q^{8} +46.9808 q^{10} +3.71281 q^{11} -15.5692 q^{13} -33.1244 q^{14} +28.0000 q^{16} +33.4974 q^{17} +135.923 q^{19} -142.890 q^{20} -17.5692 q^{22} +87.7795 q^{23} -26.4308 q^{25} +73.6743 q^{26} +100.746 q^{28} -242.354 q^{29} -194.708 q^{31} +109.492 q^{32} -158.512 q^{34} -69.4974 q^{35} -239.708 q^{37} -643.195 q^{38} +300.315 q^{40} -470.338 q^{41} -448.215 q^{43} +53.4359 q^{44} -415.377 q^{46} -4.15906 q^{47} +49.0000 q^{49} +125.072 q^{50} -224.077 q^{52} +736.543 q^{53} -36.8616 q^{55} -211.741 q^{56} +1146.83 q^{58} -279.564 q^{59} -514.831 q^{61} +921.367 q^{62} -742.123 q^{64} +154.574 q^{65} -102.123 q^{67} +482.105 q^{68} +328.865 q^{70} +44.1539 q^{71} -901.615 q^{73} +1134.31 q^{74} +1956.25 q^{76} +25.9897 q^{77} +1054.31 q^{79} -277.990 q^{80} +2225.67 q^{82} -487.246 q^{83} -332.569 q^{85} +2120.98 q^{86} -112.308 q^{88} -963.682 q^{89} -108.985 q^{91} +1263.35 q^{92} +19.6809 q^{94} -1349.47 q^{95} +726.908 q^{97} -231.870 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{2} + 8 q^{4} - 6 q^{5} + 14 q^{7} - 12 q^{8} + 42 q^{10} - 48 q^{11} + 52 q^{13} - 42 q^{14} + 56 q^{16} - 30 q^{17} + 64 q^{19} - 168 q^{20} + 48 q^{22} - 60 q^{23} - 136 q^{25} - 12 q^{26}+ \cdots - 294 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.73205 −1.67303 −0.836516 0.547942i \(-0.815411\pi\)
−0.836516 + 0.547942i \(0.815411\pi\)
\(3\) 0 0
\(4\) 14.3923 1.79904
\(5\) −9.92820 −0.888005 −0.444003 0.896025i \(-0.646442\pi\)
−0.444003 + 0.896025i \(0.646442\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) −30.2487 −1.33682
\(9\) 0 0
\(10\) 46.9808 1.48566
\(11\) 3.71281 0.101769 0.0508843 0.998705i \(-0.483796\pi\)
0.0508843 + 0.998705i \(0.483796\pi\)
\(12\) 0 0
\(13\) −15.5692 −0.332163 −0.166082 0.986112i \(-0.553112\pi\)
−0.166082 + 0.986112i \(0.553112\pi\)
\(14\) −33.1244 −0.632347
\(15\) 0 0
\(16\) 28.0000 0.437500
\(17\) 33.4974 0.477901 0.238951 0.971032i \(-0.423197\pi\)
0.238951 + 0.971032i \(0.423197\pi\)
\(18\) 0 0
\(19\) 135.923 1.64120 0.820602 0.571500i \(-0.193638\pi\)
0.820602 + 0.571500i \(0.193638\pi\)
\(20\) −142.890 −1.59756
\(21\) 0 0
\(22\) −17.5692 −0.170262
\(23\) 87.7795 0.795795 0.397897 0.917430i \(-0.369740\pi\)
0.397897 + 0.917430i \(0.369740\pi\)
\(24\) 0 0
\(25\) −26.4308 −0.211446
\(26\) 73.6743 0.555720
\(27\) 0 0
\(28\) 100.746 0.679972
\(29\) −242.354 −1.55186 −0.775931 0.630818i \(-0.782719\pi\)
−0.775931 + 0.630818i \(0.782719\pi\)
\(30\) 0 0
\(31\) −194.708 −1.12808 −0.564041 0.825747i \(-0.690754\pi\)
−0.564041 + 0.825747i \(0.690754\pi\)
\(32\) 109.492 0.604865
\(33\) 0 0
\(34\) −158.512 −0.799544
\(35\) −69.4974 −0.335635
\(36\) 0 0
\(37\) −239.708 −1.06507 −0.532536 0.846407i \(-0.678761\pi\)
−0.532536 + 0.846407i \(0.678761\pi\)
\(38\) −643.195 −2.74579
\(39\) 0 0
\(40\) 300.315 1.18710
\(41\) −470.338 −1.79157 −0.895787 0.444484i \(-0.853387\pi\)
−0.895787 + 0.444484i \(0.853387\pi\)
\(42\) 0 0
\(43\) −448.215 −1.58959 −0.794793 0.606880i \(-0.792421\pi\)
−0.794793 + 0.606880i \(0.792421\pi\)
\(44\) 53.4359 0.183086
\(45\) 0 0
\(46\) −415.377 −1.33139
\(47\) −4.15906 −0.0129077 −0.00645384 0.999979i \(-0.502054\pi\)
−0.00645384 + 0.999979i \(0.502054\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 125.072 0.353756
\(51\) 0 0
\(52\) −224.077 −0.597575
\(53\) 736.543 1.90891 0.954453 0.298361i \(-0.0964400\pi\)
0.954453 + 0.298361i \(0.0964400\pi\)
\(54\) 0 0
\(55\) −36.8616 −0.0903711
\(56\) −211.741 −0.505269
\(57\) 0 0
\(58\) 1146.83 2.59631
\(59\) −279.564 −0.616884 −0.308442 0.951243i \(-0.599808\pi\)
−0.308442 + 0.951243i \(0.599808\pi\)
\(60\) 0 0
\(61\) −514.831 −1.08061 −0.540306 0.841469i \(-0.681691\pi\)
−0.540306 + 0.841469i \(0.681691\pi\)
\(62\) 921.367 1.88732
\(63\) 0 0
\(64\) −742.123 −1.44946
\(65\) 154.574 0.294963
\(66\) 0 0
\(67\) −102.123 −0.186214 −0.0931068 0.995656i \(-0.529680\pi\)
−0.0931068 + 0.995656i \(0.529680\pi\)
\(68\) 482.105 0.859762
\(69\) 0 0
\(70\) 328.865 0.561528
\(71\) 44.1539 0.0738043 0.0369021 0.999319i \(-0.488251\pi\)
0.0369021 + 0.999319i \(0.488251\pi\)
\(72\) 0 0
\(73\) −901.615 −1.44556 −0.722781 0.691077i \(-0.757137\pi\)
−0.722781 + 0.691077i \(0.757137\pi\)
\(74\) 1134.31 1.78190
\(75\) 0 0
\(76\) 1956.25 2.95259
\(77\) 25.9897 0.0384649
\(78\) 0 0
\(79\) 1054.31 1.50150 0.750752 0.660584i \(-0.229691\pi\)
0.750752 + 0.660584i \(0.229691\pi\)
\(80\) −277.990 −0.388502
\(81\) 0 0
\(82\) 2225.67 2.99736
\(83\) −487.246 −0.644364 −0.322182 0.946678i \(-0.604416\pi\)
−0.322182 + 0.946678i \(0.604416\pi\)
\(84\) 0 0
\(85\) −332.569 −0.424379
\(86\) 2120.98 2.65943
\(87\) 0 0
\(88\) −112.308 −0.136046
\(89\) −963.682 −1.14775 −0.573877 0.818942i \(-0.694561\pi\)
−0.573877 + 0.818942i \(0.694561\pi\)
\(90\) 0 0
\(91\) −108.985 −0.125546
\(92\) 1263.35 1.43167
\(93\) 0 0
\(94\) 19.6809 0.0215950
\(95\) −1349.47 −1.45740
\(96\) 0 0
\(97\) 726.908 0.760890 0.380445 0.924804i \(-0.375771\pi\)
0.380445 + 0.924804i \(0.375771\pi\)
\(98\) −231.870 −0.239005
\(99\) 0 0
\(100\) −380.400 −0.380400
\(101\) −268.872 −0.264889 −0.132444 0.991190i \(-0.542283\pi\)
−0.132444 + 0.991190i \(0.542283\pi\)
\(102\) 0 0
\(103\) 1108.35 1.06028 0.530142 0.847909i \(-0.322138\pi\)
0.530142 + 0.847909i \(0.322138\pi\)
\(104\) 470.949 0.444042
\(105\) 0 0
\(106\) −3485.36 −3.19366
\(107\) −84.4665 −0.0763148 −0.0381574 0.999272i \(-0.512149\pi\)
−0.0381574 + 0.999272i \(0.512149\pi\)
\(108\) 0 0
\(109\) −174.169 −0.153050 −0.0765248 0.997068i \(-0.524382\pi\)
−0.0765248 + 0.997068i \(0.524382\pi\)
\(110\) 174.431 0.151194
\(111\) 0 0
\(112\) 196.000 0.165359
\(113\) −1049.77 −0.873929 −0.436964 0.899479i \(-0.643946\pi\)
−0.436964 + 0.899479i \(0.643946\pi\)
\(114\) 0 0
\(115\) −871.492 −0.706670
\(116\) −3488.03 −2.79186
\(117\) 0 0
\(118\) 1322.91 1.03207
\(119\) 234.482 0.180630
\(120\) 0 0
\(121\) −1317.22 −0.989643
\(122\) 2436.20 1.80790
\(123\) 0 0
\(124\) −2802.29 −2.02946
\(125\) 1503.44 1.07577
\(126\) 0 0
\(127\) −293.969 −0.205398 −0.102699 0.994712i \(-0.532748\pi\)
−0.102699 + 0.994712i \(0.532748\pi\)
\(128\) 2635.83 1.82013
\(129\) 0 0
\(130\) −731.454 −0.493483
\(131\) −1110.76 −0.740820 −0.370410 0.928868i \(-0.620783\pi\)
−0.370410 + 0.928868i \(0.620783\pi\)
\(132\) 0 0
\(133\) 951.461 0.620317
\(134\) 483.251 0.311541
\(135\) 0 0
\(136\) −1013.25 −0.638866
\(137\) 238.113 0.148492 0.0742458 0.997240i \(-0.476345\pi\)
0.0742458 + 0.997240i \(0.476345\pi\)
\(138\) 0 0
\(139\) −189.108 −0.115395 −0.0576974 0.998334i \(-0.518376\pi\)
−0.0576974 + 0.998334i \(0.518376\pi\)
\(140\) −1000.23 −0.603819
\(141\) 0 0
\(142\) −208.939 −0.123477
\(143\) −57.8056 −0.0338038
\(144\) 0 0
\(145\) 2406.14 1.37806
\(146\) 4266.49 2.41847
\(147\) 0 0
\(148\) −3449.95 −1.91611
\(149\) −1107.59 −0.608975 −0.304487 0.952516i \(-0.598485\pi\)
−0.304487 + 0.952516i \(0.598485\pi\)
\(150\) 0 0
\(151\) 3512.55 1.89303 0.946515 0.322660i \(-0.104577\pi\)
0.946515 + 0.322660i \(0.104577\pi\)
\(152\) −4111.50 −2.19399
\(153\) 0 0
\(154\) −122.985 −0.0643531
\(155\) 1933.10 1.00174
\(156\) 0 0
\(157\) 524.169 0.266454 0.133227 0.991086i \(-0.457466\pi\)
0.133227 + 0.991086i \(0.457466\pi\)
\(158\) −4989.04 −2.51207
\(159\) 0 0
\(160\) −1087.06 −0.537123
\(161\) 614.456 0.300782
\(162\) 0 0
\(163\) 245.693 0.118062 0.0590311 0.998256i \(-0.481199\pi\)
0.0590311 + 0.998256i \(0.481199\pi\)
\(164\) −6769.25 −3.22311
\(165\) 0 0
\(166\) 2305.67 1.07804
\(167\) −586.785 −0.271897 −0.135948 0.990716i \(-0.543408\pi\)
−0.135948 + 0.990716i \(0.543408\pi\)
\(168\) 0 0
\(169\) −1954.60 −0.889667
\(170\) 1573.73 0.710000
\(171\) 0 0
\(172\) −6450.85 −2.85973
\(173\) −2957.80 −1.29987 −0.649936 0.759989i \(-0.725204\pi\)
−0.649936 + 0.759989i \(0.725204\pi\)
\(174\) 0 0
\(175\) −185.015 −0.0799192
\(176\) 103.959 0.0445238
\(177\) 0 0
\(178\) 4560.19 1.92023
\(179\) −2724.73 −1.13774 −0.568872 0.822426i \(-0.692620\pi\)
−0.568872 + 0.822426i \(0.692620\pi\)
\(180\) 0 0
\(181\) 446.138 0.183211 0.0916055 0.995795i \(-0.470800\pi\)
0.0916055 + 0.995795i \(0.470800\pi\)
\(182\) 515.720 0.210043
\(183\) 0 0
\(184\) −2655.22 −1.06383
\(185\) 2379.87 0.945791
\(186\) 0 0
\(187\) 124.370 0.0486354
\(188\) −59.8584 −0.0232214
\(189\) 0 0
\(190\) 6385.77 2.43828
\(191\) −3517.90 −1.33270 −0.666351 0.745638i \(-0.732145\pi\)
−0.666351 + 0.745638i \(0.732145\pi\)
\(192\) 0 0
\(193\) 4869.61 1.81618 0.908090 0.418776i \(-0.137541\pi\)
0.908090 + 0.418776i \(0.137541\pi\)
\(194\) −3439.76 −1.27299
\(195\) 0 0
\(196\) 705.223 0.257005
\(197\) 3427.46 1.23958 0.619788 0.784769i \(-0.287219\pi\)
0.619788 + 0.784769i \(0.287219\pi\)
\(198\) 0 0
\(199\) −4012.51 −1.42934 −0.714671 0.699461i \(-0.753424\pi\)
−0.714671 + 0.699461i \(0.753424\pi\)
\(200\) 799.497 0.282665
\(201\) 0 0
\(202\) 1272.32 0.443167
\(203\) −1696.48 −0.586548
\(204\) 0 0
\(205\) 4669.61 1.59093
\(206\) −5244.79 −1.77389
\(207\) 0 0
\(208\) −435.938 −0.145321
\(209\) 504.657 0.167023
\(210\) 0 0
\(211\) 4281.08 1.39678 0.698392 0.715715i \(-0.253899\pi\)
0.698392 + 0.715715i \(0.253899\pi\)
\(212\) 10600.6 3.43419
\(213\) 0 0
\(214\) 399.700 0.127677
\(215\) 4449.97 1.41156
\(216\) 0 0
\(217\) −1362.95 −0.426375
\(218\) 824.178 0.256057
\(219\) 0 0
\(220\) −530.523 −0.162581
\(221\) −521.529 −0.158741
\(222\) 0 0
\(223\) −764.153 −0.229469 −0.114734 0.993396i \(-0.536602\pi\)
−0.114734 + 0.993396i \(0.536602\pi\)
\(224\) 766.446 0.228617
\(225\) 0 0
\(226\) 4967.56 1.46211
\(227\) −3872.35 −1.13223 −0.566116 0.824325i \(-0.691555\pi\)
−0.566116 + 0.824325i \(0.691555\pi\)
\(228\) 0 0
\(229\) −3866.11 −1.11563 −0.557816 0.829965i \(-0.688360\pi\)
−0.557816 + 0.829965i \(0.688360\pi\)
\(230\) 4123.95 1.18228
\(231\) 0 0
\(232\) 7330.89 2.07455
\(233\) 5378.29 1.51220 0.756102 0.654454i \(-0.227102\pi\)
0.756102 + 0.654454i \(0.227102\pi\)
\(234\) 0 0
\(235\) 41.2920 0.0114621
\(236\) −4023.57 −1.10980
\(237\) 0 0
\(238\) −1109.58 −0.302199
\(239\) −4763.15 −1.28913 −0.644566 0.764548i \(-0.722962\pi\)
−0.644566 + 0.764548i \(0.722962\pi\)
\(240\) 0 0
\(241\) −318.030 −0.0850047 −0.0425023 0.999096i \(-0.513533\pi\)
−0.0425023 + 0.999096i \(0.513533\pi\)
\(242\) 6233.13 1.65571
\(243\) 0 0
\(244\) −7409.60 −1.94406
\(245\) −486.482 −0.126858
\(246\) 0 0
\(247\) −2116.22 −0.545148
\(248\) 5889.66 1.50804
\(249\) 0 0
\(250\) −7114.33 −1.79980
\(251\) −1577.00 −0.396571 −0.198285 0.980144i \(-0.563537\pi\)
−0.198285 + 0.980144i \(0.563537\pi\)
\(252\) 0 0
\(253\) 325.909 0.0809870
\(254\) 1391.08 0.343638
\(255\) 0 0
\(256\) −6535.88 −1.59567
\(257\) 6069.46 1.47316 0.736580 0.676350i \(-0.236439\pi\)
0.736580 + 0.676350i \(0.236439\pi\)
\(258\) 0 0
\(259\) −1677.95 −0.402560
\(260\) 2224.68 0.530650
\(261\) 0 0
\(262\) 5256.17 1.23942
\(263\) 3654.73 0.856883 0.428441 0.903570i \(-0.359063\pi\)
0.428441 + 0.903570i \(0.359063\pi\)
\(264\) 0 0
\(265\) −7312.55 −1.69512
\(266\) −4502.36 −1.03781
\(267\) 0 0
\(268\) −1469.78 −0.335005
\(269\) 435.652 0.0987441 0.0493721 0.998780i \(-0.484278\pi\)
0.0493721 + 0.998780i \(0.484278\pi\)
\(270\) 0 0
\(271\) 1230.98 0.275930 0.137965 0.990437i \(-0.455944\pi\)
0.137965 + 0.990437i \(0.455944\pi\)
\(272\) 937.928 0.209082
\(273\) 0 0
\(274\) −1126.76 −0.248431
\(275\) −98.1325 −0.0215186
\(276\) 0 0
\(277\) 3994.17 0.866377 0.433188 0.901303i \(-0.357388\pi\)
0.433188 + 0.901303i \(0.357388\pi\)
\(278\) 894.866 0.193059
\(279\) 0 0
\(280\) 2102.21 0.448682
\(281\) 3615.71 0.767598 0.383799 0.923417i \(-0.374616\pi\)
0.383799 + 0.923417i \(0.374616\pi\)
\(282\) 0 0
\(283\) −2353.42 −0.494332 −0.247166 0.968973i \(-0.579499\pi\)
−0.247166 + 0.968973i \(0.579499\pi\)
\(284\) 635.476 0.132777
\(285\) 0 0
\(286\) 273.539 0.0565549
\(287\) −3292.37 −0.677151
\(288\) 0 0
\(289\) −3790.92 −0.771611
\(290\) −11386.0 −2.30554
\(291\) 0 0
\(292\) −12976.3 −2.60062
\(293\) −3580.68 −0.713943 −0.356972 0.934115i \(-0.616191\pi\)
−0.356972 + 0.934115i \(0.616191\pi\)
\(294\) 0 0
\(295\) 2775.57 0.547796
\(296\) 7250.85 1.42381
\(297\) 0 0
\(298\) 5241.17 1.01883
\(299\) −1366.66 −0.264334
\(300\) 0 0
\(301\) −3137.51 −0.600807
\(302\) −16621.6 −3.16710
\(303\) 0 0
\(304\) 3805.85 0.718027
\(305\) 5111.34 0.959589
\(306\) 0 0
\(307\) −9077.92 −1.68764 −0.843818 0.536629i \(-0.819697\pi\)
−0.843818 + 0.536629i \(0.819697\pi\)
\(308\) 374.052 0.0691999
\(309\) 0 0
\(310\) −9147.51 −1.67595
\(311\) 8966.68 1.63490 0.817450 0.576000i \(-0.195387\pi\)
0.817450 + 0.576000i \(0.195387\pi\)
\(312\) 0 0
\(313\) 7191.74 1.29873 0.649363 0.760479i \(-0.275036\pi\)
0.649363 + 0.760479i \(0.275036\pi\)
\(314\) −2480.39 −0.445786
\(315\) 0 0
\(316\) 15173.9 2.70126
\(317\) 2288.95 0.405553 0.202776 0.979225i \(-0.435004\pi\)
0.202776 + 0.979225i \(0.435004\pi\)
\(318\) 0 0
\(319\) −899.814 −0.157931
\(320\) 7367.95 1.28713
\(321\) 0 0
\(322\) −2907.64 −0.503218
\(323\) 4553.07 0.784333
\(324\) 0 0
\(325\) 411.507 0.0702347
\(326\) −1162.63 −0.197522
\(327\) 0 0
\(328\) 14227.1 2.39501
\(329\) −29.1134 −0.00487864
\(330\) 0 0
\(331\) −6298.61 −1.04593 −0.522965 0.852354i \(-0.675174\pi\)
−0.522965 + 0.852354i \(0.675174\pi\)
\(332\) −7012.59 −1.15923
\(333\) 0 0
\(334\) 2776.70 0.454892
\(335\) 1013.90 0.165359
\(336\) 0 0
\(337\) 189.615 0.0306498 0.0153249 0.999883i \(-0.495122\pi\)
0.0153249 + 0.999883i \(0.495122\pi\)
\(338\) 9249.26 1.48844
\(339\) 0 0
\(340\) −4786.44 −0.763474
\(341\) −722.913 −0.114803
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 13557.9 2.12499
\(345\) 0 0
\(346\) 13996.5 2.17473
\(347\) −50.4728 −0.00780841 −0.00390421 0.999992i \(-0.501243\pi\)
−0.00390421 + 0.999992i \(0.501243\pi\)
\(348\) 0 0
\(349\) −1338.71 −0.205328 −0.102664 0.994716i \(-0.532737\pi\)
−0.102664 + 0.994716i \(0.532737\pi\)
\(350\) 875.503 0.133707
\(351\) 0 0
\(352\) 406.524 0.0615563
\(353\) −3192.99 −0.481433 −0.240717 0.970595i \(-0.577382\pi\)
−0.240717 + 0.970595i \(0.577382\pi\)
\(354\) 0 0
\(355\) −438.369 −0.0655386
\(356\) −13869.6 −2.06485
\(357\) 0 0
\(358\) 12893.6 1.90348
\(359\) −6661.22 −0.979292 −0.489646 0.871921i \(-0.662874\pi\)
−0.489646 + 0.871921i \(0.662874\pi\)
\(360\) 0 0
\(361\) 11616.1 1.69355
\(362\) −2111.15 −0.306518
\(363\) 0 0
\(364\) −1568.54 −0.225862
\(365\) 8951.42 1.28367
\(366\) 0 0
\(367\) 4592.77 0.653244 0.326622 0.945155i \(-0.394090\pi\)
0.326622 + 0.945155i \(0.394090\pi\)
\(368\) 2457.82 0.348160
\(369\) 0 0
\(370\) −11261.6 −1.58234
\(371\) 5155.80 0.721499
\(372\) 0 0
\(373\) 6982.57 0.969286 0.484643 0.874712i \(-0.338950\pi\)
0.484643 + 0.874712i \(0.338950\pi\)
\(374\) −588.524 −0.0813685
\(375\) 0 0
\(376\) 125.806 0.0172552
\(377\) 3773.26 0.515472
\(378\) 0 0
\(379\) −12060.4 −1.63457 −0.817286 0.576233i \(-0.804522\pi\)
−0.817286 + 0.576233i \(0.804522\pi\)
\(380\) −19422.0 −2.62192
\(381\) 0 0
\(382\) 16646.9 2.22965
\(383\) −9674.46 −1.29071 −0.645355 0.763883i \(-0.723291\pi\)
−0.645355 + 0.763883i \(0.723291\pi\)
\(384\) 0 0
\(385\) −258.031 −0.0341571
\(386\) −23043.3 −3.03853
\(387\) 0 0
\(388\) 10461.9 1.36887
\(389\) 11377.9 1.48299 0.741496 0.670957i \(-0.234117\pi\)
0.741496 + 0.670957i \(0.234117\pi\)
\(390\) 0 0
\(391\) 2940.39 0.380311
\(392\) −1482.19 −0.190974
\(393\) 0 0
\(394\) −16218.9 −2.07385
\(395\) −10467.4 −1.33334
\(396\) 0 0
\(397\) −15202.7 −1.92192 −0.960958 0.276693i \(-0.910762\pi\)
−0.960958 + 0.276693i \(0.910762\pi\)
\(398\) 18987.4 2.39134
\(399\) 0 0
\(400\) −740.062 −0.0925077
\(401\) 1396.02 0.173851 0.0869254 0.996215i \(-0.472296\pi\)
0.0869254 + 0.996215i \(0.472296\pi\)
\(402\) 0 0
\(403\) 3031.45 0.374707
\(404\) −3869.69 −0.476545
\(405\) 0 0
\(406\) 8027.81 0.981315
\(407\) −889.990 −0.108391
\(408\) 0 0
\(409\) −8673.54 −1.04860 −0.524302 0.851533i \(-0.675674\pi\)
−0.524302 + 0.851533i \(0.675674\pi\)
\(410\) −22096.9 −2.66167
\(411\) 0 0
\(412\) 15951.8 1.90749
\(413\) −1956.95 −0.233160
\(414\) 0 0
\(415\) 4837.48 0.572199
\(416\) −1704.71 −0.200914
\(417\) 0 0
\(418\) −2388.06 −0.279435
\(419\) −7313.92 −0.852765 −0.426382 0.904543i \(-0.640212\pi\)
−0.426382 + 0.904543i \(0.640212\pi\)
\(420\) 0 0
\(421\) −7695.26 −0.890841 −0.445420 0.895322i \(-0.646946\pi\)
−0.445420 + 0.895322i \(0.646946\pi\)
\(422\) −20258.3 −2.33687
\(423\) 0 0
\(424\) −22279.5 −2.55186
\(425\) −885.363 −0.101050
\(426\) 0 0
\(427\) −3603.81 −0.408433
\(428\) −1215.67 −0.137293
\(429\) 0 0
\(430\) −21057.5 −2.36159
\(431\) −6258.10 −0.699402 −0.349701 0.936861i \(-0.613717\pi\)
−0.349701 + 0.936861i \(0.613717\pi\)
\(432\) 0 0
\(433\) −11923.3 −1.32332 −0.661660 0.749804i \(-0.730148\pi\)
−0.661660 + 0.749804i \(0.730148\pi\)
\(434\) 6449.57 0.713339
\(435\) 0 0
\(436\) −2506.70 −0.275342
\(437\) 11931.3 1.30606
\(438\) 0 0
\(439\) 17115.6 1.86079 0.930393 0.366564i \(-0.119466\pi\)
0.930393 + 0.366564i \(0.119466\pi\)
\(440\) 1115.01 0.120810
\(441\) 0 0
\(442\) 2467.90 0.265579
\(443\) 8095.04 0.868187 0.434094 0.900868i \(-0.357069\pi\)
0.434094 + 0.900868i \(0.357069\pi\)
\(444\) 0 0
\(445\) 9567.63 1.01921
\(446\) 3616.01 0.383908
\(447\) 0 0
\(448\) −5194.86 −0.547844
\(449\) 6156.71 0.647111 0.323556 0.946209i \(-0.395122\pi\)
0.323556 + 0.946209i \(0.395122\pi\)
\(450\) 0 0
\(451\) −1746.28 −0.182326
\(452\) −15108.6 −1.57223
\(453\) 0 0
\(454\) 18324.2 1.89426
\(455\) 1082.02 0.111486
\(456\) 0 0
\(457\) 5657.23 0.579068 0.289534 0.957168i \(-0.406500\pi\)
0.289534 + 0.957168i \(0.406500\pi\)
\(458\) 18294.6 1.86649
\(459\) 0 0
\(460\) −12542.8 −1.27133
\(461\) 1414.61 0.142918 0.0714589 0.997444i \(-0.477235\pi\)
0.0714589 + 0.997444i \(0.477235\pi\)
\(462\) 0 0
\(463\) −5404.92 −0.542523 −0.271261 0.962506i \(-0.587441\pi\)
−0.271261 + 0.962506i \(0.587441\pi\)
\(464\) −6785.91 −0.678939
\(465\) 0 0
\(466\) −25450.3 −2.52996
\(467\) −16952.8 −1.67983 −0.839916 0.542717i \(-0.817396\pi\)
−0.839916 + 0.542717i \(0.817396\pi\)
\(468\) 0 0
\(469\) −714.861 −0.0703821
\(470\) −195.396 −0.0191765
\(471\) 0 0
\(472\) 8456.45 0.824661
\(473\) −1664.14 −0.161770
\(474\) 0 0
\(475\) −3592.55 −0.347027
\(476\) 3374.74 0.324960
\(477\) 0 0
\(478\) 22539.5 2.15676
\(479\) 18372.7 1.75255 0.876276 0.481810i \(-0.160021\pi\)
0.876276 + 0.481810i \(0.160021\pi\)
\(480\) 0 0
\(481\) 3732.06 0.353778
\(482\) 1504.94 0.142216
\(483\) 0 0
\(484\) −18957.8 −1.78041
\(485\) −7216.89 −0.675674
\(486\) 0 0
\(487\) −42.5560 −0.00395974 −0.00197987 0.999998i \(-0.500630\pi\)
−0.00197987 + 0.999998i \(0.500630\pi\)
\(488\) 15573.0 1.44458
\(489\) 0 0
\(490\) 2302.06 0.212237
\(491\) 6092.09 0.559943 0.279972 0.960008i \(-0.409675\pi\)
0.279972 + 0.960008i \(0.409675\pi\)
\(492\) 0 0
\(493\) −8118.23 −0.741636
\(494\) 10014.0 0.912051
\(495\) 0 0
\(496\) −5451.81 −0.493536
\(497\) 309.077 0.0278954
\(498\) 0 0
\(499\) 6128.64 0.549811 0.274906 0.961471i \(-0.411353\pi\)
0.274906 + 0.961471i \(0.411353\pi\)
\(500\) 21637.9 1.93535
\(501\) 0 0
\(502\) 7462.44 0.663476
\(503\) 3606.14 0.319661 0.159831 0.987144i \(-0.448905\pi\)
0.159831 + 0.987144i \(0.448905\pi\)
\(504\) 0 0
\(505\) 2669.41 0.235223
\(506\) −1542.22 −0.135494
\(507\) 0 0
\(508\) −4230.90 −0.369519
\(509\) 15207.2 1.32426 0.662128 0.749391i \(-0.269654\pi\)
0.662128 + 0.749391i \(0.269654\pi\)
\(510\) 0 0
\(511\) −6311.31 −0.546371
\(512\) 9841.49 0.849486
\(513\) 0 0
\(514\) −28721.0 −2.46465
\(515\) −11004.0 −0.941539
\(516\) 0 0
\(517\) −15.4418 −0.00131360
\(518\) 7940.16 0.673496
\(519\) 0 0
\(520\) −4675.68 −0.394311
\(521\) 5497.74 0.462304 0.231152 0.972918i \(-0.425750\pi\)
0.231152 + 0.972918i \(0.425750\pi\)
\(522\) 0 0
\(523\) 15158.7 1.26739 0.633693 0.773584i \(-0.281538\pi\)
0.633693 + 0.773584i \(0.281538\pi\)
\(524\) −15986.4 −1.33276
\(525\) 0 0
\(526\) −17294.4 −1.43359
\(527\) −6522.20 −0.539111
\(528\) 0 0
\(529\) −4461.77 −0.366711
\(530\) 34603.4 2.83599
\(531\) 0 0
\(532\) 13693.7 1.11597
\(533\) 7322.80 0.595095
\(534\) 0 0
\(535\) 838.601 0.0677680
\(536\) 3089.09 0.248933
\(537\) 0 0
\(538\) −2061.53 −0.165202
\(539\) 181.928 0.0145384
\(540\) 0 0
\(541\) 21418.8 1.70216 0.851080 0.525037i \(-0.175948\pi\)
0.851080 + 0.525037i \(0.175948\pi\)
\(542\) −5825.08 −0.461639
\(543\) 0 0
\(544\) 3667.71 0.289066
\(545\) 1729.19 0.135909
\(546\) 0 0
\(547\) 15536.1 1.21439 0.607197 0.794551i \(-0.292294\pi\)
0.607197 + 0.794551i \(0.292294\pi\)
\(548\) 3426.99 0.267142
\(549\) 0 0
\(550\) 464.368 0.0360013
\(551\) −32941.5 −2.54692
\(552\) 0 0
\(553\) 7380.15 0.567515
\(554\) −18900.6 −1.44948
\(555\) 0 0
\(556\) −2721.69 −0.207600
\(557\) −13975.1 −1.06309 −0.531547 0.847029i \(-0.678389\pi\)
−0.531547 + 0.847029i \(0.678389\pi\)
\(558\) 0 0
\(559\) 6978.36 0.528002
\(560\) −1945.93 −0.146840
\(561\) 0 0
\(562\) −17109.7 −1.28422
\(563\) −1117.89 −0.0836825 −0.0418412 0.999124i \(-0.513322\pi\)
−0.0418412 + 0.999124i \(0.513322\pi\)
\(564\) 0 0
\(565\) 10422.3 0.776054
\(566\) 11136.5 0.827034
\(567\) 0 0
\(568\) −1335.60 −0.0986628
\(569\) −14452.1 −1.06478 −0.532392 0.846498i \(-0.678707\pi\)
−0.532392 + 0.846498i \(0.678707\pi\)
\(570\) 0 0
\(571\) 17396.8 1.27501 0.637506 0.770445i \(-0.279966\pi\)
0.637506 + 0.770445i \(0.279966\pi\)
\(572\) −831.956 −0.0608144
\(573\) 0 0
\(574\) 15579.7 1.13290
\(575\) −2320.08 −0.168268
\(576\) 0 0
\(577\) 12251.2 0.883923 0.441961 0.897034i \(-0.354283\pi\)
0.441961 + 0.897034i \(0.354283\pi\)
\(578\) 17938.8 1.29093
\(579\) 0 0
\(580\) 34629.9 2.47918
\(581\) −3410.72 −0.243547
\(582\) 0 0
\(583\) 2734.65 0.194267
\(584\) 27272.7 1.93245
\(585\) 0 0
\(586\) 16943.9 1.19445
\(587\) −24620.6 −1.73118 −0.865589 0.500755i \(-0.833056\pi\)
−0.865589 + 0.500755i \(0.833056\pi\)
\(588\) 0 0
\(589\) −26465.3 −1.85141
\(590\) −13134.1 −0.916481
\(591\) 0 0
\(592\) −6711.81 −0.465969
\(593\) 3020.20 0.209148 0.104574 0.994517i \(-0.466652\pi\)
0.104574 + 0.994517i \(0.466652\pi\)
\(594\) 0 0
\(595\) −2327.98 −0.160400
\(596\) −15940.8 −1.09557
\(597\) 0 0
\(598\) 6467.09 0.442239
\(599\) 16830.3 1.14803 0.574013 0.818846i \(-0.305386\pi\)
0.574013 + 0.818846i \(0.305386\pi\)
\(600\) 0 0
\(601\) 8783.94 0.596180 0.298090 0.954538i \(-0.403650\pi\)
0.298090 + 0.954538i \(0.403650\pi\)
\(602\) 14846.8 1.00517
\(603\) 0 0
\(604\) 50553.7 3.40563
\(605\) 13077.6 0.878809
\(606\) 0 0
\(607\) 16803.3 1.12360 0.561799 0.827274i \(-0.310109\pi\)
0.561799 + 0.827274i \(0.310109\pi\)
\(608\) 14882.5 0.992707
\(609\) 0 0
\(610\) −24187.1 −1.60542
\(611\) 64.7533 0.00428746
\(612\) 0 0
\(613\) 1319.88 0.0869647 0.0434823 0.999054i \(-0.486155\pi\)
0.0434823 + 0.999054i \(0.486155\pi\)
\(614\) 42957.2 2.82347
\(615\) 0 0
\(616\) −786.155 −0.0514206
\(617\) 2586.39 0.168759 0.0843793 0.996434i \(-0.473109\pi\)
0.0843793 + 0.996434i \(0.473109\pi\)
\(618\) 0 0
\(619\) 5296.63 0.343925 0.171962 0.985104i \(-0.444989\pi\)
0.171962 + 0.985104i \(0.444989\pi\)
\(620\) 27821.7 1.80217
\(621\) 0 0
\(622\) −42430.8 −2.73524
\(623\) −6745.77 −0.433810
\(624\) 0 0
\(625\) −11622.6 −0.743844
\(626\) −34031.7 −2.17281
\(627\) 0 0
\(628\) 7544.00 0.479360
\(629\) −8029.59 −0.508999
\(630\) 0 0
\(631\) −15857.8 −1.00046 −0.500228 0.865894i \(-0.666751\pi\)
−0.500228 + 0.865894i \(0.666751\pi\)
\(632\) −31891.4 −2.00724
\(633\) 0 0
\(634\) −10831.4 −0.678503
\(635\) 2918.59 0.182395
\(636\) 0 0
\(637\) −762.892 −0.0474519
\(638\) 4257.97 0.264223
\(639\) 0 0
\(640\) −26169.0 −1.61628
\(641\) 30365.3 1.87107 0.935537 0.353228i \(-0.114916\pi\)
0.935537 + 0.353228i \(0.114916\pi\)
\(642\) 0 0
\(643\) 3746.71 0.229791 0.114896 0.993378i \(-0.463347\pi\)
0.114896 + 0.993378i \(0.463347\pi\)
\(644\) 8843.44 0.541119
\(645\) 0 0
\(646\) −21545.4 −1.31222
\(647\) −13032.7 −0.791915 −0.395957 0.918269i \(-0.629587\pi\)
−0.395957 + 0.918269i \(0.629587\pi\)
\(648\) 0 0
\(649\) −1037.97 −0.0627794
\(650\) −1947.27 −0.117505
\(651\) 0 0
\(652\) 3536.08 0.212398
\(653\) 3096.91 0.185592 0.0927960 0.995685i \(-0.470420\pi\)
0.0927960 + 0.995685i \(0.470420\pi\)
\(654\) 0 0
\(655\) 11027.8 0.657853
\(656\) −13169.5 −0.783813
\(657\) 0 0
\(658\) 137.766 0.00816213
\(659\) −11759.2 −0.695103 −0.347551 0.937661i \(-0.612987\pi\)
−0.347551 + 0.937661i \(0.612987\pi\)
\(660\) 0 0
\(661\) 2363.12 0.139054 0.0695270 0.997580i \(-0.477851\pi\)
0.0695270 + 0.997580i \(0.477851\pi\)
\(662\) 29805.4 1.74988
\(663\) 0 0
\(664\) 14738.6 0.861396
\(665\) −9446.30 −0.550845
\(666\) 0 0
\(667\) −21273.7 −1.23496
\(668\) −8445.19 −0.489153
\(669\) 0 0
\(670\) −4797.82 −0.276650
\(671\) −1911.47 −0.109972
\(672\) 0 0
\(673\) −19358.2 −1.10877 −0.554387 0.832259i \(-0.687047\pi\)
−0.554387 + 0.832259i \(0.687047\pi\)
\(674\) −897.267 −0.0512781
\(675\) 0 0
\(676\) −28131.2 −1.60055
\(677\) −7931.13 −0.450248 −0.225124 0.974330i \(-0.572279\pi\)
−0.225124 + 0.974330i \(0.572279\pi\)
\(678\) 0 0
\(679\) 5088.35 0.287589
\(680\) 10059.8 0.567317
\(681\) 0 0
\(682\) 3420.86 0.192070
\(683\) −7289.89 −0.408404 −0.204202 0.978929i \(-0.565460\pi\)
−0.204202 + 0.978929i \(0.565460\pi\)
\(684\) 0 0
\(685\) −2364.03 −0.131861
\(686\) −1623.09 −0.0903353
\(687\) 0 0
\(688\) −12550.0 −0.695444
\(689\) −11467.4 −0.634069
\(690\) 0 0
\(691\) −13338.6 −0.734335 −0.367167 0.930155i \(-0.619672\pi\)
−0.367167 + 0.930155i \(0.619672\pi\)
\(692\) −42569.6 −2.33852
\(693\) 0 0
\(694\) 238.840 0.0130637
\(695\) 1877.50 0.102471
\(696\) 0 0
\(697\) −15755.1 −0.856195
\(698\) 6334.84 0.343520
\(699\) 0 0
\(700\) −2662.80 −0.143778
\(701\) 12918.5 0.696042 0.348021 0.937487i \(-0.386854\pi\)
0.348021 + 0.937487i \(0.386854\pi\)
\(702\) 0 0
\(703\) −32581.8 −1.74800
\(704\) −2755.36 −0.147509
\(705\) 0 0
\(706\) 15109.4 0.805453
\(707\) −1882.10 −0.100118
\(708\) 0 0
\(709\) 2282.78 0.120919 0.0604596 0.998171i \(-0.480743\pi\)
0.0604596 + 0.998171i \(0.480743\pi\)
\(710\) 2074.38 0.109648
\(711\) 0 0
\(712\) 29150.1 1.53434
\(713\) −17091.3 −0.897721
\(714\) 0 0
\(715\) 573.906 0.0300180
\(716\) −39215.2 −2.04684
\(717\) 0 0
\(718\) 31521.2 1.63839
\(719\) 29360.8 1.52291 0.761456 0.648217i \(-0.224485\pi\)
0.761456 + 0.648217i \(0.224485\pi\)
\(720\) 0 0
\(721\) 7758.48 0.400750
\(722\) −54967.9 −2.83337
\(723\) 0 0
\(724\) 6420.96 0.329603
\(725\) 6405.60 0.328135
\(726\) 0 0
\(727\) −18139.1 −0.925368 −0.462684 0.886523i \(-0.653114\pi\)
−0.462684 + 0.886523i \(0.653114\pi\)
\(728\) 3296.64 0.167832
\(729\) 0 0
\(730\) −42358.6 −2.14762
\(731\) −15014.1 −0.759665
\(732\) 0 0
\(733\) −5292.54 −0.266691 −0.133345 0.991070i \(-0.542572\pi\)
−0.133345 + 0.991070i \(0.542572\pi\)
\(734\) −21733.2 −1.09290
\(735\) 0 0
\(736\) 9611.17 0.481348
\(737\) −379.163 −0.0189507
\(738\) 0 0
\(739\) −25955.7 −1.29201 −0.646007 0.763332i \(-0.723562\pi\)
−0.646007 + 0.763332i \(0.723562\pi\)
\(740\) 34251.8 1.70151
\(741\) 0 0
\(742\) −24397.5 −1.20709
\(743\) 25972.0 1.28240 0.641199 0.767375i \(-0.278437\pi\)
0.641199 + 0.767375i \(0.278437\pi\)
\(744\) 0 0
\(745\) 10996.4 0.540773
\(746\) −33041.9 −1.62165
\(747\) 0 0
\(748\) 1789.97 0.0874968
\(749\) −591.265 −0.0288443
\(750\) 0 0
\(751\) 6487.01 0.315199 0.157599 0.987503i \(-0.449625\pi\)
0.157599 + 0.987503i \(0.449625\pi\)
\(752\) −116.454 −0.00564711
\(753\) 0 0
\(754\) −17855.3 −0.862401
\(755\) −34873.3 −1.68102
\(756\) 0 0
\(757\) 22221.6 1.06692 0.533459 0.845826i \(-0.320892\pi\)
0.533459 + 0.845826i \(0.320892\pi\)
\(758\) 57070.6 2.73469
\(759\) 0 0
\(760\) 40819.8 1.94827
\(761\) −19820.0 −0.944120 −0.472060 0.881567i \(-0.656489\pi\)
−0.472060 + 0.881567i \(0.656489\pi\)
\(762\) 0 0
\(763\) −1219.19 −0.0578473
\(764\) −50630.6 −2.39758
\(765\) 0 0
\(766\) 45780.1 2.15940
\(767\) 4352.59 0.204906
\(768\) 0 0
\(769\) −3307.64 −0.155106 −0.0775531 0.996988i \(-0.524711\pi\)
−0.0775531 + 0.996988i \(0.524711\pi\)
\(770\) 1221.02 0.0571459
\(771\) 0 0
\(772\) 70085.0 3.26738
\(773\) −15964.0 −0.742801 −0.371401 0.928473i \(-0.621122\pi\)
−0.371401 + 0.928473i \(0.621122\pi\)
\(774\) 0 0
\(775\) 5146.28 0.238529
\(776\) −21988.0 −1.01717
\(777\) 0 0
\(778\) −53840.9 −2.48109
\(779\) −63929.8 −2.94034
\(780\) 0 0
\(781\) 163.935 0.00751096
\(782\) −13914.1 −0.636273
\(783\) 0 0
\(784\) 1372.00 0.0625000
\(785\) −5204.06 −0.236612
\(786\) 0 0
\(787\) −36394.0 −1.64842 −0.824210 0.566284i \(-0.808381\pi\)
−0.824210 + 0.566284i \(0.808381\pi\)
\(788\) 49329.1 2.23004
\(789\) 0 0
\(790\) 49532.2 2.23073
\(791\) −7348.38 −0.330314
\(792\) 0 0
\(793\) 8015.51 0.358940
\(794\) 71939.9 3.21543
\(795\) 0 0
\(796\) −57749.2 −2.57144
\(797\) 13384.0 0.594836 0.297418 0.954747i \(-0.403874\pi\)
0.297418 + 0.954747i \(0.403874\pi\)
\(798\) 0 0
\(799\) −139.318 −0.00616859
\(800\) −2893.97 −0.127896
\(801\) 0 0
\(802\) −6606.06 −0.290858
\(803\) −3347.53 −0.147113
\(804\) 0 0
\(805\) −6100.45 −0.267096
\(806\) −14345.0 −0.626898
\(807\) 0 0
\(808\) 8133.03 0.354108
\(809\) −13971.0 −0.607161 −0.303580 0.952806i \(-0.598182\pi\)
−0.303580 + 0.952806i \(0.598182\pi\)
\(810\) 0 0
\(811\) 18471.8 0.799793 0.399897 0.916560i \(-0.369046\pi\)
0.399897 + 0.916560i \(0.369046\pi\)
\(812\) −24416.2 −1.05522
\(813\) 0 0
\(814\) 4211.48 0.181342
\(815\) −2439.29 −0.104840
\(816\) 0 0
\(817\) −60922.8 −2.60884
\(818\) 41043.6 1.75435
\(819\) 0 0
\(820\) 67206.5 2.86214
\(821\) 28188.3 1.19827 0.599135 0.800648i \(-0.295511\pi\)
0.599135 + 0.800648i \(0.295511\pi\)
\(822\) 0 0
\(823\) −8071.14 −0.341849 −0.170925 0.985284i \(-0.554675\pi\)
−0.170925 + 0.985284i \(0.554675\pi\)
\(824\) −33526.3 −1.41741
\(825\) 0 0
\(826\) 9260.38 0.390085
\(827\) 41952.8 1.76402 0.882009 0.471233i \(-0.156191\pi\)
0.882009 + 0.471233i \(0.156191\pi\)
\(828\) 0 0
\(829\) 14443.2 0.605108 0.302554 0.953132i \(-0.402161\pi\)
0.302554 + 0.953132i \(0.402161\pi\)
\(830\) −22891.2 −0.957307
\(831\) 0 0
\(832\) 11554.3 0.481457
\(833\) 1641.37 0.0682716
\(834\) 0 0
\(835\) 5825.72 0.241446
\(836\) 7263.18 0.300481
\(837\) 0 0
\(838\) 34609.9 1.42670
\(839\) 26360.6 1.08471 0.542353 0.840151i \(-0.317534\pi\)
0.542353 + 0.840151i \(0.317534\pi\)
\(840\) 0 0
\(841\) 34346.4 1.40827
\(842\) 36414.4 1.49041
\(843\) 0 0
\(844\) 61614.5 2.51287
\(845\) 19405.7 0.790030
\(846\) 0 0
\(847\) −9220.51 −0.374050
\(848\) 20623.2 0.835146
\(849\) 0 0
\(850\) 4189.58 0.169061
\(851\) −21041.4 −0.847580
\(852\) 0 0
\(853\) −18939.8 −0.760241 −0.380120 0.924937i \(-0.624117\pi\)
−0.380120 + 0.924937i \(0.624117\pi\)
\(854\) 17053.4 0.683321
\(855\) 0 0
\(856\) 2555.00 0.102019
\(857\) 44583.6 1.77707 0.888533 0.458812i \(-0.151725\pi\)
0.888533 + 0.458812i \(0.151725\pi\)
\(858\) 0 0
\(859\) 7975.35 0.316782 0.158391 0.987376i \(-0.449369\pi\)
0.158391 + 0.987376i \(0.449369\pi\)
\(860\) 64045.4 2.53945
\(861\) 0 0
\(862\) 29613.7 1.17012
\(863\) −8258.42 −0.325747 −0.162874 0.986647i \(-0.552076\pi\)
−0.162874 + 0.986647i \(0.552076\pi\)
\(864\) 0 0
\(865\) 29365.7 1.15429
\(866\) 56421.7 2.21396
\(867\) 0 0
\(868\) −19616.0 −0.767064
\(869\) 3914.45 0.152806
\(870\) 0 0
\(871\) 1589.97 0.0618533
\(872\) 5268.40 0.204599
\(873\) 0 0
\(874\) −56459.3 −2.18508
\(875\) 10524.0 0.406603
\(876\) 0 0
\(877\) −43106.2 −1.65974 −0.829871 0.557956i \(-0.811586\pi\)
−0.829871 + 0.557956i \(0.811586\pi\)
\(878\) −80992.1 −3.11316
\(879\) 0 0
\(880\) −1032.12 −0.0395374
\(881\) −662.616 −0.0253395 −0.0126697 0.999920i \(-0.504033\pi\)
−0.0126697 + 0.999920i \(0.504033\pi\)
\(882\) 0 0
\(883\) 32497.0 1.23852 0.619258 0.785187i \(-0.287433\pi\)
0.619258 + 0.785187i \(0.287433\pi\)
\(884\) −7506.00 −0.285582
\(885\) 0 0
\(886\) −38306.1 −1.45251
\(887\) −3176.34 −0.120238 −0.0601190 0.998191i \(-0.519148\pi\)
−0.0601190 + 0.998191i \(0.519148\pi\)
\(888\) 0 0
\(889\) −2057.79 −0.0776332
\(890\) −45274.5 −1.70517
\(891\) 0 0
\(892\) −10997.9 −0.412823
\(893\) −565.312 −0.0211841
\(894\) 0 0
\(895\) 27051.7 1.01032
\(896\) 18450.8 0.687943
\(897\) 0 0
\(898\) −29133.8 −1.08264
\(899\) 47188.1 1.75063
\(900\) 0 0
\(901\) 24672.3 0.912268
\(902\) 8263.48 0.305037
\(903\) 0 0
\(904\) 31754.2 1.16828
\(905\) −4429.35 −0.162692
\(906\) 0 0
\(907\) −31726.2 −1.16147 −0.580733 0.814094i \(-0.697234\pi\)
−0.580733 + 0.814094i \(0.697234\pi\)
\(908\) −55732.0 −2.03693
\(909\) 0 0
\(910\) −5120.18 −0.186519
\(911\) 10882.3 0.395772 0.197886 0.980225i \(-0.436592\pi\)
0.197886 + 0.980225i \(0.436592\pi\)
\(912\) 0 0
\(913\) −1809.05 −0.0655760
\(914\) −26770.3 −0.968799
\(915\) 0 0
\(916\) −55642.2 −2.00706
\(917\) −7775.31 −0.280004
\(918\) 0 0
\(919\) −29610.9 −1.06286 −0.531432 0.847101i \(-0.678346\pi\)
−0.531432 + 0.847101i \(0.678346\pi\)
\(920\) 26361.5 0.944689
\(921\) 0 0
\(922\) −6694.02 −0.239106
\(923\) −687.442 −0.0245151
\(924\) 0 0
\(925\) 6335.66 0.225206
\(926\) 25576.4 0.907658
\(927\) 0 0
\(928\) −26535.9 −0.938667
\(929\) −52496.6 −1.85399 −0.926995 0.375074i \(-0.877617\pi\)
−0.926995 + 0.375074i \(0.877617\pi\)
\(930\) 0 0
\(931\) 6660.23 0.234458
\(932\) 77405.9 2.72051
\(933\) 0 0
\(934\) 80221.5 2.81041
\(935\) −1234.77 −0.0431885
\(936\) 0 0
\(937\) −1661.46 −0.0579269 −0.0289634 0.999580i \(-0.509221\pi\)
−0.0289634 + 0.999580i \(0.509221\pi\)
\(938\) 3382.76 0.117752
\(939\) 0 0
\(940\) 594.287 0.0206207
\(941\) −43066.9 −1.49197 −0.745983 0.665965i \(-0.768020\pi\)
−0.745983 + 0.665965i \(0.768020\pi\)
\(942\) 0 0
\(943\) −41286.0 −1.42572
\(944\) −7827.79 −0.269887
\(945\) 0 0
\(946\) 7874.79 0.270647
\(947\) 10024.6 0.343986 0.171993 0.985098i \(-0.444979\pi\)
0.171993 + 0.985098i \(0.444979\pi\)
\(948\) 0 0
\(949\) 14037.4 0.480163
\(950\) 17000.1 0.580587
\(951\) 0 0
\(952\) −7092.78 −0.241469
\(953\) 36134.8 1.22825 0.614125 0.789209i \(-0.289509\pi\)
0.614125 + 0.789209i \(0.289509\pi\)
\(954\) 0 0
\(955\) 34926.4 1.18345
\(956\) −68552.7 −2.31920
\(957\) 0 0
\(958\) −86940.7 −2.93208
\(959\) 1666.79 0.0561245
\(960\) 0 0
\(961\) 8120.07 0.272568
\(962\) −17660.3 −0.591883
\(963\) 0 0
\(964\) −4577.19 −0.152927
\(965\) −48346.5 −1.61278
\(966\) 0 0
\(967\) 22016.7 0.732171 0.366086 0.930581i \(-0.380698\pi\)
0.366086 + 0.930581i \(0.380698\pi\)
\(968\) 39844.1 1.32297
\(969\) 0 0
\(970\) 34150.7 1.13042
\(971\) −8087.76 −0.267300 −0.133650 0.991029i \(-0.542670\pi\)
−0.133650 + 0.991029i \(0.542670\pi\)
\(972\) 0 0
\(973\) −1323.75 −0.0436152
\(974\) 201.377 0.00662478
\(975\) 0 0
\(976\) −14415.3 −0.472768
\(977\) 19050.6 0.623830 0.311915 0.950110i \(-0.399030\pi\)
0.311915 + 0.950110i \(0.399030\pi\)
\(978\) 0 0
\(979\) −3577.97 −0.116805
\(980\) −7001.60 −0.228222
\(981\) 0 0
\(982\) −28828.1 −0.936803
\(983\) 35176.7 1.14136 0.570682 0.821171i \(-0.306679\pi\)
0.570682 + 0.821171i \(0.306679\pi\)
\(984\) 0 0
\(985\) −34028.5 −1.10075
\(986\) 38415.9 1.24078
\(987\) 0 0
\(988\) −30457.2 −0.980742
\(989\) −39344.1 −1.26498
\(990\) 0 0
\(991\) −26830.6 −0.860041 −0.430021 0.902819i \(-0.641494\pi\)
−0.430021 + 0.902819i \(0.641494\pi\)
\(992\) −21319.0 −0.682337
\(993\) 0 0
\(994\) −1462.57 −0.0466699
\(995\) 39837.0 1.26926
\(996\) 0 0
\(997\) 27316.3 0.867721 0.433860 0.900980i \(-0.357151\pi\)
0.433860 + 0.900980i \(0.357151\pi\)
\(998\) −29001.1 −0.919852
\(999\) 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 189.4.a.e.1.1 2
3.2 odd 2 189.4.a.i.1.2 yes 2
7.6 odd 2 1323.4.a.o.1.1 2
21.20 even 2 1323.4.a.x.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.a.e.1.1 2 1.1 even 1 trivial
189.4.a.i.1.2 yes 2 3.2 odd 2
1323.4.a.o.1.1 2 7.6 odd 2
1323.4.a.x.1.2 2 21.20 even 2