Properties

Label 363.4.a.q.1.1
Level $363$
Weight $4$
Character 363.1
Self dual yes
Analytic conductor $21.418$
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] = [363,4,Mod(1,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, 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("363.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.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: \(21.4176933321\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 28x^{2} - 2x + 72 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.63177\) of defining polynomial
Character \(\chi\) \(=\) 363.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.63177 q^{2} +3.00000 q^{3} +23.7168 q^{4} +5.58204 q^{5} -16.8953 q^{6} +16.3245 q^{7} -88.5134 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-5.63177 q^{2} +3.00000 q^{3} +23.7168 q^{4} +5.58204 q^{5} -16.8953 q^{6} +16.3245 q^{7} -88.5134 q^{8} +9.00000 q^{9} -31.4368 q^{10} +71.1504 q^{12} +36.6875 q^{13} -91.9358 q^{14} +16.7461 q^{15} +308.752 q^{16} +2.22540 q^{17} -50.6859 q^{18} +89.8470 q^{19} +132.388 q^{20} +48.9735 q^{21} +133.825 q^{23} -265.540 q^{24} -93.8408 q^{25} -206.615 q^{26} +27.0000 q^{27} +387.165 q^{28} +79.6014 q^{29} -94.3103 q^{30} -167.723 q^{31} -1030.71 q^{32} -12.5329 q^{34} +91.1241 q^{35} +213.451 q^{36} +38.2217 q^{37} -505.998 q^{38} +110.062 q^{39} -494.086 q^{40} -189.462 q^{41} -275.807 q^{42} +114.242 q^{43} +50.2384 q^{45} -753.672 q^{46} +162.032 q^{47} +926.257 q^{48} -76.5108 q^{49} +528.489 q^{50} +6.67620 q^{51} +870.110 q^{52} -211.325 q^{53} -152.058 q^{54} -1444.94 q^{56} +269.541 q^{57} -448.296 q^{58} -478.011 q^{59} +397.165 q^{60} +134.260 q^{61} +944.576 q^{62} +146.920 q^{63} +3334.73 q^{64} +204.791 q^{65} +809.611 q^{67} +52.7794 q^{68} +401.475 q^{69} -513.190 q^{70} +190.954 q^{71} -796.621 q^{72} +745.929 q^{73} -215.255 q^{74} -281.522 q^{75} +2130.89 q^{76} -619.846 q^{78} +56.5345 q^{79} +1723.47 q^{80} +81.0000 q^{81} +1067.01 q^{82} -595.935 q^{83} +1161.49 q^{84} +12.4223 q^{85} -643.382 q^{86} +238.804 q^{87} +1434.99 q^{89} -282.931 q^{90} +598.905 q^{91} +3173.90 q^{92} -503.168 q^{93} -912.527 q^{94} +501.530 q^{95} -3092.14 q^{96} -622.609 q^{97} +430.891 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 12 q^{3} + 25 q^{4} + 14 q^{5} - 3 q^{6} + 20 q^{7} - 75 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + 12 q^{3} + 25 q^{4} + 14 q^{5} - 3 q^{6} + 20 q^{7} - 75 q^{8} + 36 q^{9} + 3 q^{10} + 75 q^{12} + 32 q^{13} + 62 q^{14} + 42 q^{15} + 289 q^{16} + 92 q^{17} - 9 q^{18} - 34 q^{19} + 391 q^{20} + 60 q^{21} - 26 q^{23} - 225 q^{24} + 334 q^{25} - 181 q^{26} + 108 q^{27} + 692 q^{28} + 174 q^{29} + 9 q^{30} + 422 q^{31} - 1271 q^{32} - 477 q^{34} - 82 q^{35} + 225 q^{36} + 518 q^{37} - 798 q^{38} + 96 q^{39} + 423 q^{40} + 428 q^{41} + 186 q^{42} - 550 q^{43} + 126 q^{45} - 2004 q^{46} + 556 q^{47} + 867 q^{48} + 282 q^{49} + 2074 q^{50} + 276 q^{51} + 467 q^{52} + 882 q^{53} - 27 q^{54} - 2112 q^{56} - 102 q^{57} - 225 q^{58} - 158 q^{59} + 1173 q^{60} + 290 q^{61} + 1142 q^{62} + 180 q^{63} + 3097 q^{64} - 1636 q^{65} + 992 q^{67} - 2033 q^{68} - 78 q^{69} + 948 q^{70} - 42 q^{71} - 675 q^{72} + 1274 q^{73} + 1509 q^{74} + 1002 q^{75} + 632 q^{76} - 543 q^{78} + 362 q^{79} + 1423 q^{80} + 324 q^{81} + 2403 q^{82} + 1500 q^{83} + 2076 q^{84} - 2388 q^{85} - 1272 q^{86} + 522 q^{87} + 1428 q^{89} + 27 q^{90} + 1750 q^{91} + 896 q^{92} + 1266 q^{93} - 1728 q^{94} - 4452 q^{95} - 3813 q^{96} + 1052 q^{97} - 615 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.63177 −1.99113 −0.995565 0.0940734i \(-0.970011\pi\)
−0.995565 + 0.0940734i \(0.970011\pi\)
\(3\) 3.00000 0.577350
\(4\) 23.7168 2.96460
\(5\) 5.58204 0.499273 0.249637 0.968340i \(-0.419689\pi\)
0.249637 + 0.968340i \(0.419689\pi\)
\(6\) −16.8953 −1.14958
\(7\) 16.3245 0.881440 0.440720 0.897645i \(-0.354723\pi\)
0.440720 + 0.897645i \(0.354723\pi\)
\(8\) −88.5134 −3.91178
\(9\) 9.00000 0.333333
\(10\) −31.4368 −0.994118
\(11\) 0 0
\(12\) 71.1504 1.71161
\(13\) 36.6875 0.782713 0.391357 0.920239i \(-0.372006\pi\)
0.391357 + 0.920239i \(0.372006\pi\)
\(14\) −91.9358 −1.75506
\(15\) 16.7461 0.288256
\(16\) 308.752 4.82426
\(17\) 2.22540 0.0317493 0.0158747 0.999874i \(-0.494947\pi\)
0.0158747 + 0.999874i \(0.494947\pi\)
\(18\) −50.6859 −0.663710
\(19\) 89.8470 1.08486 0.542430 0.840101i \(-0.317505\pi\)
0.542430 + 0.840101i \(0.317505\pi\)
\(20\) 132.388 1.48015
\(21\) 48.9735 0.508900
\(22\) 0 0
\(23\) 133.825 1.21324 0.606619 0.794993i \(-0.292525\pi\)
0.606619 + 0.794993i \(0.292525\pi\)
\(24\) −265.540 −2.25847
\(25\) −93.8408 −0.750726
\(26\) −206.615 −1.55848
\(27\) 27.0000 0.192450
\(28\) 387.165 2.61312
\(29\) 79.6014 0.509711 0.254855 0.966979i \(-0.417972\pi\)
0.254855 + 0.966979i \(0.417972\pi\)
\(30\) −94.3103 −0.573954
\(31\) −167.723 −0.971739 −0.485869 0.874031i \(-0.661497\pi\)
−0.485869 + 0.874031i \(0.661497\pi\)
\(32\) −1030.71 −5.69395
\(33\) 0 0
\(34\) −12.5329 −0.0632171
\(35\) 91.1241 0.440079
\(36\) 213.451 0.988200
\(37\) 38.2217 0.169827 0.0849135 0.996388i \(-0.472939\pi\)
0.0849135 + 0.996388i \(0.472939\pi\)
\(38\) −505.998 −2.16010
\(39\) 110.062 0.451900
\(40\) −494.086 −1.95305
\(41\) −189.462 −0.721684 −0.360842 0.932627i \(-0.617511\pi\)
−0.360842 + 0.932627i \(0.617511\pi\)
\(42\) −275.807 −1.01329
\(43\) 114.242 0.405155 0.202578 0.979266i \(-0.435068\pi\)
0.202578 + 0.979266i \(0.435068\pi\)
\(44\) 0 0
\(45\) 50.2384 0.166424
\(46\) −753.672 −2.41571
\(47\) 162.032 0.502868 0.251434 0.967874i \(-0.419098\pi\)
0.251434 + 0.967874i \(0.419098\pi\)
\(48\) 926.257 2.78529
\(49\) −76.5108 −0.223064
\(50\) 528.489 1.49479
\(51\) 6.67620 0.0183305
\(52\) 870.110 2.32043
\(53\) −211.325 −0.547693 −0.273847 0.961773i \(-0.588296\pi\)
−0.273847 + 0.961773i \(0.588296\pi\)
\(54\) −152.058 −0.383193
\(55\) 0 0
\(56\) −1444.94 −3.44800
\(57\) 269.541 0.626344
\(58\) −448.296 −1.01490
\(59\) −478.011 −1.05477 −0.527387 0.849625i \(-0.676828\pi\)
−0.527387 + 0.849625i \(0.676828\pi\)
\(60\) 397.165 0.854563
\(61\) 134.260 0.281808 0.140904 0.990023i \(-0.454999\pi\)
0.140904 + 0.990023i \(0.454999\pi\)
\(62\) 944.576 1.93486
\(63\) 146.920 0.293813
\(64\) 3334.73 6.51314
\(65\) 204.791 0.390788
\(66\) 0 0
\(67\) 809.611 1.47626 0.738132 0.674657i \(-0.235708\pi\)
0.738132 + 0.674657i \(0.235708\pi\)
\(68\) 52.7794 0.0941241
\(69\) 401.475 0.700463
\(70\) −513.190 −0.876256
\(71\) 190.954 0.319184 0.159592 0.987183i \(-0.448982\pi\)
0.159592 + 0.987183i \(0.448982\pi\)
\(72\) −796.621 −1.30393
\(73\) 745.929 1.19595 0.597975 0.801514i \(-0.295972\pi\)
0.597975 + 0.801514i \(0.295972\pi\)
\(74\) −215.255 −0.338148
\(75\) −281.522 −0.433432
\(76\) 2130.89 3.21617
\(77\) 0 0
\(78\) −619.846 −0.899792
\(79\) 56.5345 0.0805143 0.0402572 0.999189i \(-0.487182\pi\)
0.0402572 + 0.999189i \(0.487182\pi\)
\(80\) 1723.47 2.40862
\(81\) 81.0000 0.111111
\(82\) 1067.01 1.43697
\(83\) −595.935 −0.788101 −0.394050 0.919089i \(-0.628926\pi\)
−0.394050 + 0.919089i \(0.628926\pi\)
\(84\) 1161.49 1.50868
\(85\) 12.4223 0.0158516
\(86\) −643.382 −0.806717
\(87\) 238.804 0.294282
\(88\) 0 0
\(89\) 1434.99 1.70908 0.854540 0.519386i \(-0.173839\pi\)
0.854540 + 0.519386i \(0.173839\pi\)
\(90\) −282.931 −0.331373
\(91\) 598.905 0.689915
\(92\) 3173.90 3.59677
\(93\) −503.168 −0.561034
\(94\) −912.527 −1.00128
\(95\) 501.530 0.541641
\(96\) −3092.14 −3.28740
\(97\) −622.609 −0.651715 −0.325857 0.945419i \(-0.605653\pi\)
−0.325857 + 0.945419i \(0.605653\pi\)
\(98\) 430.891 0.444149
\(99\) 0 0
\(100\) −2225.60 −2.22560
\(101\) −726.647 −0.715881 −0.357941 0.933744i \(-0.616521\pi\)
−0.357941 + 0.933744i \(0.616521\pi\)
\(102\) −37.5988 −0.0364984
\(103\) −1689.51 −1.61624 −0.808121 0.589017i \(-0.799515\pi\)
−0.808121 + 0.589017i \(0.799515\pi\)
\(104\) −3247.33 −3.06180
\(105\) 273.372 0.254080
\(106\) 1190.13 1.09053
\(107\) 1031.70 0.932133 0.466067 0.884750i \(-0.345671\pi\)
0.466067 + 0.884750i \(0.345671\pi\)
\(108\) 640.354 0.570538
\(109\) −1479.61 −1.30020 −0.650098 0.759851i \(-0.725272\pi\)
−0.650098 + 0.759851i \(0.725272\pi\)
\(110\) 0 0
\(111\) 114.665 0.0980497
\(112\) 5040.23 4.25229
\(113\) 1903.03 1.58427 0.792134 0.610347i \(-0.208970\pi\)
0.792134 + 0.610347i \(0.208970\pi\)
\(114\) −1517.99 −1.24713
\(115\) 747.018 0.605737
\(116\) 1887.89 1.51109
\(117\) 330.187 0.260904
\(118\) 2692.05 2.10019
\(119\) 36.3285 0.0279851
\(120\) −1482.26 −1.12759
\(121\) 0 0
\(122\) −756.123 −0.561116
\(123\) −568.387 −0.416665
\(124\) −3977.85 −2.88082
\(125\) −1221.58 −0.874091
\(126\) −827.422 −0.585021
\(127\) 2650.07 1.85162 0.925811 0.377987i \(-0.123384\pi\)
0.925811 + 0.377987i \(0.123384\pi\)
\(128\) −10534.7 −7.27456
\(129\) 342.725 0.233916
\(130\) −1153.34 −0.778110
\(131\) 2339.06 1.56003 0.780016 0.625759i \(-0.215211\pi\)
0.780016 + 0.625759i \(0.215211\pi\)
\(132\) 0 0
\(133\) 1466.71 0.956238
\(134\) −4559.54 −2.93943
\(135\) 150.715 0.0960852
\(136\) −196.978 −0.124196
\(137\) 1201.25 0.749125 0.374562 0.927202i \(-0.377793\pi\)
0.374562 + 0.927202i \(0.377793\pi\)
\(138\) −2261.02 −1.39471
\(139\) −1840.90 −1.12333 −0.561665 0.827365i \(-0.689839\pi\)
−0.561665 + 0.827365i \(0.689839\pi\)
\(140\) 2161.17 1.30466
\(141\) 486.096 0.290331
\(142\) −1075.41 −0.635537
\(143\) 0 0
\(144\) 2778.77 1.60809
\(145\) 444.339 0.254485
\(146\) −4200.90 −2.38129
\(147\) −229.532 −0.128786
\(148\) 906.496 0.503469
\(149\) 828.112 0.455313 0.227656 0.973742i \(-0.426894\pi\)
0.227656 + 0.973742i \(0.426894\pi\)
\(150\) 1585.47 0.863020
\(151\) −883.706 −0.476258 −0.238129 0.971233i \(-0.576534\pi\)
−0.238129 + 0.971233i \(0.576534\pi\)
\(152\) −7952.67 −4.24373
\(153\) 20.0286 0.0105831
\(154\) 0 0
\(155\) −936.236 −0.485163
\(156\) 2610.33 1.33970
\(157\) 1235.42 0.628010 0.314005 0.949421i \(-0.398329\pi\)
0.314005 + 0.949421i \(0.398329\pi\)
\(158\) −318.389 −0.160315
\(159\) −633.975 −0.316211
\(160\) −5753.50 −2.84284
\(161\) 2184.63 1.06940
\(162\) −456.173 −0.221237
\(163\) −1148.84 −0.552050 −0.276025 0.961150i \(-0.589017\pi\)
−0.276025 + 0.961150i \(0.589017\pi\)
\(164\) −4493.44 −2.13951
\(165\) 0 0
\(166\) 3356.17 1.56921
\(167\) 2316.46 1.07337 0.536686 0.843782i \(-0.319676\pi\)
0.536686 + 0.843782i \(0.319676\pi\)
\(168\) −4334.81 −1.99070
\(169\) −851.029 −0.387360
\(170\) −69.9594 −0.0315626
\(171\) 808.623 0.361620
\(172\) 2709.44 1.20112
\(173\) −1949.29 −0.856659 −0.428330 0.903623i \(-0.640898\pi\)
−0.428330 + 0.903623i \(0.640898\pi\)
\(174\) −1344.89 −0.585953
\(175\) −1531.90 −0.661720
\(176\) 0 0
\(177\) −1434.03 −0.608974
\(178\) −8081.50 −3.40300
\(179\) 1889.42 0.788947 0.394474 0.918907i \(-0.370927\pi\)
0.394474 + 0.918907i \(0.370927\pi\)
\(180\) 1191.49 0.493382
\(181\) 1180.63 0.484839 0.242419 0.970172i \(-0.422059\pi\)
0.242419 + 0.970172i \(0.422059\pi\)
\(182\) −3372.89 −1.37371
\(183\) 402.781 0.162702
\(184\) −11845.3 −4.74591
\(185\) 213.355 0.0847901
\(186\) 2833.73 1.11709
\(187\) 0 0
\(188\) 3842.88 1.49080
\(189\) 440.761 0.169633
\(190\) −2824.50 −1.07848
\(191\) 2773.75 1.05079 0.525396 0.850858i \(-0.323917\pi\)
0.525396 + 0.850858i \(0.323917\pi\)
\(192\) 10004.2 3.76036
\(193\) 2035.09 0.759010 0.379505 0.925190i \(-0.376094\pi\)
0.379505 + 0.925190i \(0.376094\pi\)
\(194\) 3506.39 1.29765
\(195\) 614.373 0.225622
\(196\) −1814.59 −0.661294
\(197\) −48.0363 −0.0173728 −0.00868640 0.999962i \(-0.502765\pi\)
−0.00868640 + 0.999962i \(0.502765\pi\)
\(198\) 0 0
\(199\) 3307.72 1.17828 0.589141 0.808030i \(-0.299466\pi\)
0.589141 + 0.808030i \(0.299466\pi\)
\(200\) 8306.17 2.93667
\(201\) 2428.83 0.852321
\(202\) 4092.30 1.42541
\(203\) 1299.45 0.449279
\(204\) 158.338 0.0543426
\(205\) −1057.59 −0.360318
\(206\) 9514.95 3.21815
\(207\) 1204.43 0.404413
\(208\) 11327.3 3.77601
\(209\) 0 0
\(210\) −1539.57 −0.505906
\(211\) −1334.95 −0.435554 −0.217777 0.975999i \(-0.569881\pi\)
−0.217777 + 0.975999i \(0.569881\pi\)
\(212\) −5011.96 −1.62369
\(213\) 572.862 0.184281
\(214\) −5810.30 −1.85600
\(215\) 637.701 0.202283
\(216\) −2389.86 −0.752822
\(217\) −2737.99 −0.856529
\(218\) 8332.84 2.58886
\(219\) 2237.79 0.690482
\(220\) 0 0
\(221\) 81.6443 0.0248506
\(222\) −645.766 −0.195230
\(223\) −3251.94 −0.976528 −0.488264 0.872696i \(-0.662370\pi\)
−0.488264 + 0.872696i \(0.662370\pi\)
\(224\) −16825.9 −5.01887
\(225\) −844.567 −0.250242
\(226\) −10717.4 −3.15449
\(227\) −1315.55 −0.384652 −0.192326 0.981331i \(-0.561603\pi\)
−0.192326 + 0.981331i \(0.561603\pi\)
\(228\) 6392.66 1.85686
\(229\) 2562.22 0.739372 0.369686 0.929157i \(-0.379465\pi\)
0.369686 + 0.929157i \(0.379465\pi\)
\(230\) −4207.03 −1.20610
\(231\) 0 0
\(232\) −7045.79 −1.99387
\(233\) 5033.07 1.41514 0.707570 0.706643i \(-0.249791\pi\)
0.707570 + 0.706643i \(0.249791\pi\)
\(234\) −1859.54 −0.519495
\(235\) 904.470 0.251069
\(236\) −11336.9 −3.12699
\(237\) 169.604 0.0464850
\(238\) −204.594 −0.0557221
\(239\) −4246.79 −1.14938 −0.574690 0.818371i \(-0.694877\pi\)
−0.574690 + 0.818371i \(0.694877\pi\)
\(240\) 5170.41 1.39062
\(241\) −5203.06 −1.39070 −0.695350 0.718671i \(-0.744751\pi\)
−0.695350 + 0.718671i \(0.744751\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 3184.22 0.835447
\(245\) −427.087 −0.111370
\(246\) 3201.03 0.829634
\(247\) 3296.26 0.849134
\(248\) 14845.7 3.80122
\(249\) −1787.80 −0.455010
\(250\) 6879.65 1.74043
\(251\) −232.010 −0.0583440 −0.0291720 0.999574i \(-0.509287\pi\)
−0.0291720 + 0.999574i \(0.509287\pi\)
\(252\) 3484.48 0.871039
\(253\) 0 0
\(254\) −14924.6 −3.68682
\(255\) 37.2669 0.00915192
\(256\) 32651.1 7.97146
\(257\) −1974.93 −0.479349 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(258\) −1930.15 −0.465758
\(259\) 623.949 0.149692
\(260\) 4856.99 1.15853
\(261\) 716.412 0.169904
\(262\) −13173.0 −3.10623
\(263\) 6288.51 1.47440 0.737198 0.675677i \(-0.236149\pi\)
0.737198 + 0.675677i \(0.236149\pi\)
\(264\) 0 0
\(265\) −1179.63 −0.273449
\(266\) −8260.16 −1.90400
\(267\) 4304.96 0.986738
\(268\) 19201.4 4.37653
\(269\) −6900.18 −1.56398 −0.781991 0.623290i \(-0.785796\pi\)
−0.781991 + 0.623290i \(0.785796\pi\)
\(270\) −848.793 −0.191318
\(271\) −5503.35 −1.23360 −0.616798 0.787121i \(-0.711571\pi\)
−0.616798 + 0.787121i \(0.711571\pi\)
\(272\) 687.098 0.153167
\(273\) 1796.71 0.398323
\(274\) −6765.18 −1.49160
\(275\) 0 0
\(276\) 9521.71 2.07659
\(277\) −181.153 −0.0392939 −0.0196470 0.999807i \(-0.506254\pi\)
−0.0196470 + 0.999807i \(0.506254\pi\)
\(278\) 10367.5 2.23669
\(279\) −1509.50 −0.323913
\(280\) −8065.70 −1.72149
\(281\) −4829.86 −1.02536 −0.512678 0.858581i \(-0.671347\pi\)
−0.512678 + 0.858581i \(0.671347\pi\)
\(282\) −2737.58 −0.578087
\(283\) −3157.70 −0.663272 −0.331636 0.943408i \(-0.607600\pi\)
−0.331636 + 0.943408i \(0.607600\pi\)
\(284\) 4528.82 0.946253
\(285\) 1504.59 0.312717
\(286\) 0 0
\(287\) −3092.88 −0.636121
\(288\) −9276.43 −1.89798
\(289\) −4908.05 −0.998992
\(290\) −2502.41 −0.506713
\(291\) −1867.83 −0.376268
\(292\) 17691.1 3.54552
\(293\) −5930.76 −1.18252 −0.591260 0.806481i \(-0.701369\pi\)
−0.591260 + 0.806481i \(0.701369\pi\)
\(294\) 1292.67 0.256429
\(295\) −2668.28 −0.526621
\(296\) −3383.13 −0.664326
\(297\) 0 0
\(298\) −4663.74 −0.906587
\(299\) 4909.71 0.949617
\(300\) −6676.81 −1.28495
\(301\) 1864.94 0.357120
\(302\) 4976.83 0.948293
\(303\) −2179.94 −0.413314
\(304\) 27740.5 5.23364
\(305\) 749.447 0.140699
\(306\) −112.796 −0.0210724
\(307\) −2344.18 −0.435797 −0.217898 0.975971i \(-0.569920\pi\)
−0.217898 + 0.975971i \(0.569920\pi\)
\(308\) 0 0
\(309\) −5068.54 −0.933137
\(310\) 5272.66 0.966023
\(311\) −2943.21 −0.536637 −0.268318 0.963330i \(-0.586468\pi\)
−0.268318 + 0.963330i \(0.586468\pi\)
\(312\) −9742.00 −1.76773
\(313\) 7534.35 1.36060 0.680298 0.732935i \(-0.261850\pi\)
0.680298 + 0.732935i \(0.261850\pi\)
\(314\) −6957.63 −1.25045
\(315\) 820.117 0.146693
\(316\) 1340.82 0.238693
\(317\) 7918.10 1.40292 0.701458 0.712710i \(-0.252533\pi\)
0.701458 + 0.712710i \(0.252533\pi\)
\(318\) 3570.40 0.629617
\(319\) 0 0
\(320\) 18614.6 3.25184
\(321\) 3095.10 0.538167
\(322\) −12303.3 −2.12931
\(323\) 199.946 0.0344436
\(324\) 1921.06 0.329400
\(325\) −3442.78 −0.587604
\(326\) 6470.00 1.09920
\(327\) −4438.84 −0.750668
\(328\) 16770.0 2.82307
\(329\) 2645.09 0.443248
\(330\) 0 0
\(331\) −4941.26 −0.820533 −0.410267 0.911966i \(-0.634564\pi\)
−0.410267 + 0.911966i \(0.634564\pi\)
\(332\) −14133.7 −2.33640
\(333\) 343.995 0.0566090
\(334\) −13045.8 −2.13722
\(335\) 4519.28 0.737059
\(336\) 15120.7 2.45506
\(337\) −10324.7 −1.66891 −0.834457 0.551073i \(-0.814218\pi\)
−0.834457 + 0.551073i \(0.814218\pi\)
\(338\) 4792.80 0.771284
\(339\) 5709.10 0.914678
\(340\) 294.617 0.0469937
\(341\) 0 0
\(342\) −4553.98 −0.720032
\(343\) −6848.30 −1.07806
\(344\) −10111.9 −1.58488
\(345\) 2241.05 0.349723
\(346\) 10978.0 1.70572
\(347\) −3806.42 −0.588875 −0.294437 0.955671i \(-0.595132\pi\)
−0.294437 + 0.955671i \(0.595132\pi\)
\(348\) 5663.67 0.872427
\(349\) 337.780 0.0518079 0.0259039 0.999664i \(-0.491754\pi\)
0.0259039 + 0.999664i \(0.491754\pi\)
\(350\) 8627.32 1.31757
\(351\) 990.562 0.150633
\(352\) 0 0
\(353\) 5488.35 0.827523 0.413761 0.910385i \(-0.364215\pi\)
0.413761 + 0.910385i \(0.364215\pi\)
\(354\) 8076.14 1.21255
\(355\) 1065.91 0.159360
\(356\) 34033.3 5.06674
\(357\) 108.986 0.0161572
\(358\) −10640.8 −1.57090
\(359\) −9157.77 −1.34632 −0.673159 0.739497i \(-0.735063\pi\)
−0.673159 + 0.739497i \(0.735063\pi\)
\(360\) −4446.77 −0.651015
\(361\) 1213.49 0.176920
\(362\) −6649.06 −0.965377
\(363\) 0 0
\(364\) 14204.1 2.04532
\(365\) 4163.81 0.597106
\(366\) −2268.37 −0.323960
\(367\) 2339.69 0.332782 0.166391 0.986060i \(-0.446789\pi\)
0.166391 + 0.986060i \(0.446789\pi\)
\(368\) 41318.8 5.85297
\(369\) −1705.16 −0.240561
\(370\) −1201.57 −0.168828
\(371\) −3449.78 −0.482759
\(372\) −11933.5 −1.66324
\(373\) −10249.9 −1.42284 −0.711421 0.702766i \(-0.751948\pi\)
−0.711421 + 0.702766i \(0.751948\pi\)
\(374\) 0 0
\(375\) −3664.74 −0.504657
\(376\) −14342.0 −1.96711
\(377\) 2920.37 0.398957
\(378\) −2482.27 −0.337762
\(379\) −1256.76 −0.170331 −0.0851654 0.996367i \(-0.527142\pi\)
−0.0851654 + 0.996367i \(0.527142\pi\)
\(380\) 11894.7 1.60575
\(381\) 7950.22 1.06903
\(382\) −15621.1 −2.09226
\(383\) −7948.33 −1.06042 −0.530210 0.847866i \(-0.677887\pi\)
−0.530210 + 0.847866i \(0.677887\pi\)
\(384\) −31604.1 −4.19997
\(385\) 0 0
\(386\) −11461.1 −1.51129
\(387\) 1028.17 0.135052
\(388\) −14766.3 −1.93207
\(389\) 952.983 0.124211 0.0621056 0.998070i \(-0.480218\pi\)
0.0621056 + 0.998070i \(0.480218\pi\)
\(390\) −3460.01 −0.449242
\(391\) 297.814 0.0385195
\(392\) 6772.23 0.872575
\(393\) 7017.17 0.900685
\(394\) 270.529 0.0345915
\(395\) 315.578 0.0401987
\(396\) 0 0
\(397\) 13876.1 1.75421 0.877105 0.480299i \(-0.159472\pi\)
0.877105 + 0.480299i \(0.159472\pi\)
\(398\) −18628.3 −2.34612
\(399\) 4400.12 0.552084
\(400\) −28973.6 −3.62170
\(401\) −13450.7 −1.67505 −0.837526 0.546397i \(-0.815999\pi\)
−0.837526 + 0.546397i \(0.815999\pi\)
\(402\) −13678.6 −1.69708
\(403\) −6153.32 −0.760593
\(404\) −17233.7 −2.12230
\(405\) 452.146 0.0554748
\(406\) −7318.21 −0.894574
\(407\) 0 0
\(408\) −590.933 −0.0717048
\(409\) 11241.4 1.35905 0.679525 0.733653i \(-0.262186\pi\)
0.679525 + 0.733653i \(0.262186\pi\)
\(410\) 5956.09 0.717440
\(411\) 3603.76 0.432507
\(412\) −40069.9 −4.79151
\(413\) −7803.29 −0.929721
\(414\) −6783.05 −0.805238
\(415\) −3326.54 −0.393478
\(416\) −37814.3 −4.45673
\(417\) −5522.69 −0.648554
\(418\) 0 0
\(419\) −13757.6 −1.60407 −0.802033 0.597280i \(-0.796248\pi\)
−0.802033 + 0.597280i \(0.796248\pi\)
\(420\) 6483.52 0.753246
\(421\) −7423.26 −0.859353 −0.429676 0.902983i \(-0.641372\pi\)
−0.429676 + 0.902983i \(0.641372\pi\)
\(422\) 7518.15 0.867246
\(423\) 1458.29 0.167623
\(424\) 18705.1 2.14245
\(425\) −208.833 −0.0238351
\(426\) −3226.23 −0.366928
\(427\) 2191.73 0.248397
\(428\) 24468.6 2.76340
\(429\) 0 0
\(430\) −3591.39 −0.402772
\(431\) −3919.09 −0.437995 −0.218997 0.975725i \(-0.570279\pi\)
−0.218997 + 0.975725i \(0.570279\pi\)
\(432\) 8336.32 0.928429
\(433\) 2196.59 0.243791 0.121896 0.992543i \(-0.461103\pi\)
0.121896 + 0.992543i \(0.461103\pi\)
\(434\) 15419.7 1.70546
\(435\) 1333.02 0.146927
\(436\) −35091.7 −3.85456
\(437\) 12023.8 1.31619
\(438\) −12602.7 −1.37484
\(439\) −7985.51 −0.868173 −0.434086 0.900871i \(-0.642929\pi\)
−0.434086 + 0.900871i \(0.642929\pi\)
\(440\) 0 0
\(441\) −688.597 −0.0743545
\(442\) −459.802 −0.0494809
\(443\) 2056.68 0.220577 0.110289 0.993900i \(-0.464822\pi\)
0.110289 + 0.993900i \(0.464822\pi\)
\(444\) 2719.49 0.290678
\(445\) 8010.15 0.853298
\(446\) 18314.2 1.94440
\(447\) 2484.34 0.262875
\(448\) 54437.7 5.74094
\(449\) −2956.70 −0.310769 −0.155384 0.987854i \(-0.549662\pi\)
−0.155384 + 0.987854i \(0.549662\pi\)
\(450\) 4756.40 0.498265
\(451\) 0 0
\(452\) 45133.9 4.69672
\(453\) −2651.12 −0.274968
\(454\) 7408.87 0.765893
\(455\) 3343.11 0.344456
\(456\) −23858.0 −2.45012
\(457\) 5264.75 0.538894 0.269447 0.963015i \(-0.413159\pi\)
0.269447 + 0.963015i \(0.413159\pi\)
\(458\) −14429.8 −1.47219
\(459\) 60.0858 0.00611016
\(460\) 17716.9 1.79577
\(461\) 6127.96 0.619105 0.309552 0.950882i \(-0.399821\pi\)
0.309552 + 0.950882i \(0.399821\pi\)
\(462\) 0 0
\(463\) −1362.20 −0.136732 −0.0683658 0.997660i \(-0.521778\pi\)
−0.0683658 + 0.997660i \(0.521778\pi\)
\(464\) 24577.1 2.45897
\(465\) −2808.71 −0.280109
\(466\) −28345.1 −2.81773
\(467\) −10269.1 −1.01756 −0.508778 0.860898i \(-0.669903\pi\)
−0.508778 + 0.860898i \(0.669903\pi\)
\(468\) 7830.99 0.773478
\(469\) 13216.5 1.30124
\(470\) −5093.76 −0.499910
\(471\) 3706.27 0.362582
\(472\) 42310.4 4.12604
\(473\) 0 0
\(474\) −955.168 −0.0925577
\(475\) −8431.32 −0.814432
\(476\) 861.597 0.0829648
\(477\) −1901.93 −0.182564
\(478\) 23916.9 2.28856
\(479\) 1487.98 0.141936 0.0709680 0.997479i \(-0.477391\pi\)
0.0709680 + 0.997479i \(0.477391\pi\)
\(480\) −17260.5 −1.64131
\(481\) 1402.26 0.132926
\(482\) 29302.4 2.76906
\(483\) 6553.88 0.617416
\(484\) 0 0
\(485\) −3475.43 −0.325384
\(486\) −1368.52 −0.127731
\(487\) −1712.51 −0.159346 −0.0796728 0.996821i \(-0.525388\pi\)
−0.0796728 + 0.996821i \(0.525388\pi\)
\(488\) −11883.8 −1.10237
\(489\) −3446.52 −0.318726
\(490\) 2405.25 0.221752
\(491\) 11924.4 1.09601 0.548005 0.836475i \(-0.315387\pi\)
0.548005 + 0.836475i \(0.315387\pi\)
\(492\) −13480.3 −1.23524
\(493\) 177.145 0.0161830
\(494\) −18563.8 −1.69074
\(495\) 0 0
\(496\) −51784.8 −4.68792
\(497\) 3117.23 0.281342
\(498\) 10068.5 0.905985
\(499\) 3853.81 0.345732 0.172866 0.984945i \(-0.444697\pi\)
0.172866 + 0.984945i \(0.444697\pi\)
\(500\) −28972.0 −2.59133
\(501\) 6949.38 0.619712
\(502\) 1306.63 0.116171
\(503\) 5342.09 0.473543 0.236771 0.971565i \(-0.423911\pi\)
0.236771 + 0.971565i \(0.423911\pi\)
\(504\) −13004.4 −1.14933
\(505\) −4056.17 −0.357420
\(506\) 0 0
\(507\) −2553.09 −0.223642
\(508\) 62851.3 5.48932
\(509\) −2310.50 −0.201201 −0.100600 0.994927i \(-0.532076\pi\)
−0.100600 + 0.994927i \(0.532076\pi\)
\(510\) −209.878 −0.0182227
\(511\) 12176.9 1.05416
\(512\) −99605.9 −8.59766
\(513\) 2425.87 0.208781
\(514\) 11122.3 0.954447
\(515\) −9430.95 −0.806946
\(516\) 8128.33 0.693469
\(517\) 0 0
\(518\) −3513.94 −0.298057
\(519\) −5847.88 −0.494592
\(520\) −18126.8 −1.52868
\(521\) −6312.10 −0.530783 −0.265391 0.964141i \(-0.585501\pi\)
−0.265391 + 0.964141i \(0.585501\pi\)
\(522\) −4034.67 −0.338300
\(523\) 21924.3 1.83305 0.916523 0.399982i \(-0.130984\pi\)
0.916523 + 0.399982i \(0.130984\pi\)
\(524\) 55474.9 4.62487
\(525\) −4595.71 −0.382044
\(526\) −35415.4 −2.93571
\(527\) −373.250 −0.0308521
\(528\) 0 0
\(529\) 5742.16 0.471946
\(530\) 6643.38 0.544472
\(531\) −4302.10 −0.351592
\(532\) 34785.6 2.83486
\(533\) −6950.90 −0.564872
\(534\) −24244.5 −1.96472
\(535\) 5759.00 0.465389
\(536\) −71661.4 −5.77481
\(537\) 5668.25 0.455499
\(538\) 38860.2 3.11409
\(539\) 0 0
\(540\) 3574.48 0.284854
\(541\) −12039.3 −0.956765 −0.478382 0.878152i \(-0.658777\pi\)
−0.478382 + 0.878152i \(0.658777\pi\)
\(542\) 30993.6 2.45625
\(543\) 3541.90 0.279922
\(544\) −2293.75 −0.180779
\(545\) −8259.27 −0.649153
\(546\) −10118.7 −0.793112
\(547\) −2517.28 −0.196766 −0.0983829 0.995149i \(-0.531367\pi\)
−0.0983829 + 0.995149i \(0.531367\pi\)
\(548\) 28489.9 2.22086
\(549\) 1208.34 0.0939359
\(550\) 0 0
\(551\) 7151.95 0.552964
\(552\) −35535.9 −2.74006
\(553\) 922.898 0.0709686
\(554\) 1020.21 0.0782394
\(555\) 640.065 0.0489536
\(556\) −43660.2 −3.33022
\(557\) −6507.93 −0.495062 −0.247531 0.968880i \(-0.579619\pi\)
−0.247531 + 0.968880i \(0.579619\pi\)
\(558\) 8501.18 0.644953
\(559\) 4191.23 0.317120
\(560\) 28134.8 2.12306
\(561\) 0 0
\(562\) 27200.6 2.04162
\(563\) −627.184 −0.0469497 −0.0234748 0.999724i \(-0.507473\pi\)
−0.0234748 + 0.999724i \(0.507473\pi\)
\(564\) 11528.6 0.860715
\(565\) 10622.8 0.790983
\(566\) 17783.4 1.32066
\(567\) 1322.28 0.0979378
\(568\) −16902.0 −1.24858
\(569\) −2336.62 −0.172155 −0.0860775 0.996288i \(-0.527433\pi\)
−0.0860775 + 0.996288i \(0.527433\pi\)
\(570\) −8473.51 −0.622660
\(571\) 12354.5 0.905462 0.452731 0.891647i \(-0.350450\pi\)
0.452731 + 0.891647i \(0.350450\pi\)
\(572\) 0 0
\(573\) 8321.24 0.606675
\(574\) 17418.4 1.26660
\(575\) −12558.3 −0.910809
\(576\) 30012.5 2.17105
\(577\) −2391.41 −0.172540 −0.0862699 0.996272i \(-0.527495\pi\)
−0.0862699 + 0.996272i \(0.527495\pi\)
\(578\) 27641.0 1.98912
\(579\) 6105.26 0.438214
\(580\) 10538.3 0.754446
\(581\) −9728.34 −0.694663
\(582\) 10519.2 0.749198
\(583\) 0 0
\(584\) −66024.7 −4.67829
\(585\) 1843.12 0.130263
\(586\) 33400.6 2.35455
\(587\) −24107.9 −1.69513 −0.847563 0.530694i \(-0.821931\pi\)
−0.847563 + 0.530694i \(0.821931\pi\)
\(588\) −5443.78 −0.381799
\(589\) −15069.4 −1.05420
\(590\) 15027.1 1.04857
\(591\) −144.109 −0.0100302
\(592\) 11801.0 0.819289
\(593\) −607.390 −0.0420616 −0.0210308 0.999779i \(-0.506695\pi\)
−0.0210308 + 0.999779i \(0.506695\pi\)
\(594\) 0 0
\(595\) 202.788 0.0139722
\(596\) 19640.2 1.34982
\(597\) 9923.17 0.680282
\(598\) −27650.3 −1.89081
\(599\) 10687.8 0.729035 0.364517 0.931197i \(-0.381234\pi\)
0.364517 + 0.931197i \(0.381234\pi\)
\(600\) 24918.5 1.69549
\(601\) 21602.2 1.46618 0.733088 0.680134i \(-0.238078\pi\)
0.733088 + 0.680134i \(0.238078\pi\)
\(602\) −10502.9 −0.711072
\(603\) 7286.49 0.492088
\(604\) −20958.7 −1.41192
\(605\) 0 0
\(606\) 12276.9 0.822963
\(607\) −19316.5 −1.29165 −0.645824 0.763486i \(-0.723486\pi\)
−0.645824 + 0.763486i \(0.723486\pi\)
\(608\) −92606.7 −6.17713
\(609\) 3898.36 0.259392
\(610\) −4220.71 −0.280150
\(611\) 5944.55 0.393602
\(612\) 475.014 0.0313747
\(613\) 25466.8 1.67797 0.838983 0.544158i \(-0.183151\pi\)
0.838983 + 0.544158i \(0.183151\pi\)
\(614\) 13201.9 0.867728
\(615\) −3172.76 −0.208030
\(616\) 0 0
\(617\) 19548.6 1.27552 0.637760 0.770235i \(-0.279861\pi\)
0.637760 + 0.770235i \(0.279861\pi\)
\(618\) 28544.9 1.85800
\(619\) −7285.93 −0.473096 −0.236548 0.971620i \(-0.576016\pi\)
−0.236548 + 0.971620i \(0.576016\pi\)
\(620\) −22204.5 −1.43831
\(621\) 3613.28 0.233488
\(622\) 16575.5 1.06851
\(623\) 23425.4 1.50645
\(624\) 33982.0 2.18008
\(625\) 4911.19 0.314316
\(626\) −42431.7 −2.70912
\(627\) 0 0
\(628\) 29300.3 1.86180
\(629\) 85.0585 0.00539190
\(630\) −4618.71 −0.292085
\(631\) −21199.2 −1.33744 −0.668722 0.743513i \(-0.733158\pi\)
−0.668722 + 0.743513i \(0.733158\pi\)
\(632\) −5004.06 −0.314954
\(633\) −4004.86 −0.251467
\(634\) −44592.9 −2.79339
\(635\) 14792.8 0.924465
\(636\) −15035.9 −0.937439
\(637\) −2806.99 −0.174595
\(638\) 0 0
\(639\) 1718.59 0.106395
\(640\) −58805.1 −3.63199
\(641\) 10801.4 0.665568 0.332784 0.943003i \(-0.392012\pi\)
0.332784 + 0.943003i \(0.392012\pi\)
\(642\) −17430.9 −1.07156
\(643\) −23096.4 −1.41654 −0.708269 0.705943i \(-0.750523\pi\)
−0.708269 + 0.705943i \(0.750523\pi\)
\(644\) 51812.4 3.17033
\(645\) 1913.10 0.116788
\(646\) −1126.05 −0.0685816
\(647\) −23521.2 −1.42923 −0.714616 0.699517i \(-0.753399\pi\)
−0.714616 + 0.699517i \(0.753399\pi\)
\(648\) −7169.59 −0.434642
\(649\) 0 0
\(650\) 19388.9 1.17000
\(651\) −8213.97 −0.494517
\(652\) −27246.8 −1.63661
\(653\) 2419.38 0.144989 0.0724945 0.997369i \(-0.476904\pi\)
0.0724945 + 0.997369i \(0.476904\pi\)
\(654\) 24998.5 1.49468
\(655\) 13056.7 0.778882
\(656\) −58497.0 −3.48159
\(657\) 6713.36 0.398650
\(658\) −14896.5 −0.882565
\(659\) 17627.3 1.04197 0.520987 0.853564i \(-0.325564\pi\)
0.520987 + 0.853564i \(0.325564\pi\)
\(660\) 0 0
\(661\) 750.997 0.0441912 0.0220956 0.999756i \(-0.492966\pi\)
0.0220956 + 0.999756i \(0.492966\pi\)
\(662\) 27828.1 1.63379
\(663\) 244.933 0.0143475
\(664\) 52748.2 3.08287
\(665\) 8187.23 0.477424
\(666\) −1937.30 −0.112716
\(667\) 10652.7 0.618400
\(668\) 54939.1 3.18212
\(669\) −9755.81 −0.563799
\(670\) −25451.5 −1.46758
\(671\) 0 0
\(672\) −50477.7 −2.89765
\(673\) −18959.8 −1.08595 −0.542977 0.839747i \(-0.682703\pi\)
−0.542977 + 0.839747i \(0.682703\pi\)
\(674\) 58146.5 3.32302
\(675\) −2533.70 −0.144477
\(676\) −20183.7 −1.14837
\(677\) −13858.9 −0.786767 −0.393383 0.919374i \(-0.628696\pi\)
−0.393383 + 0.919374i \(0.628696\pi\)
\(678\) −32152.3 −1.82124
\(679\) −10163.8 −0.574448
\(680\) −1099.54 −0.0620079
\(681\) −3946.65 −0.222079
\(682\) 0 0
\(683\) −15562.5 −0.871862 −0.435931 0.899980i \(-0.643581\pi\)
−0.435931 + 0.899980i \(0.643581\pi\)
\(684\) 19178.0 1.07206
\(685\) 6705.45 0.374018
\(686\) 38568.0 2.14655
\(687\) 7686.65 0.426876
\(688\) 35272.4 1.95457
\(689\) −7752.99 −0.428687
\(690\) −12621.1 −0.696343
\(691\) −18362.3 −1.01091 −0.505453 0.862854i \(-0.668675\pi\)
−0.505453 + 0.862854i \(0.668675\pi\)
\(692\) −46231.0 −2.53965
\(693\) 0 0
\(694\) 21436.9 1.17253
\(695\) −10276.0 −0.560848
\(696\) −21137.4 −1.15116
\(697\) −421.630 −0.0229130
\(698\) −1902.30 −0.103156
\(699\) 15099.2 0.817031
\(700\) −36331.9 −1.96174
\(701\) 7705.40 0.415163 0.207581 0.978218i \(-0.433441\pi\)
0.207581 + 0.978218i \(0.433441\pi\)
\(702\) −5578.61 −0.299931
\(703\) 3434.10 0.184238
\(704\) 0 0
\(705\) 2713.41 0.144955
\(706\) −30909.1 −1.64771
\(707\) −11862.1 −0.631007
\(708\) −34010.7 −1.80537
\(709\) −15368.2 −0.814053 −0.407027 0.913416i \(-0.633434\pi\)
−0.407027 + 0.913416i \(0.633434\pi\)
\(710\) −6002.98 −0.317307
\(711\) 508.811 0.0268381
\(712\) −127015. −6.68554
\(713\) −22445.5 −1.17895
\(714\) −613.782 −0.0321712
\(715\) 0 0
\(716\) 44810.9 2.33891
\(717\) −12740.4 −0.663595
\(718\) 51574.4 2.68070
\(719\) −9397.60 −0.487443 −0.243721 0.969845i \(-0.578368\pi\)
−0.243721 + 0.969845i \(0.578368\pi\)
\(720\) 15511.2 0.802874
\(721\) −27580.5 −1.42462
\(722\) −6834.11 −0.352270
\(723\) −15609.2 −0.802921
\(724\) 28000.9 1.43735
\(725\) −7469.86 −0.382653
\(726\) 0 0
\(727\) −28278.0 −1.44260 −0.721302 0.692620i \(-0.756456\pi\)
−0.721302 + 0.692620i \(0.756456\pi\)
\(728\) −53011.1 −2.69879
\(729\) 729.000 0.0370370
\(730\) −23449.6 −1.18892
\(731\) 254.233 0.0128634
\(732\) 9552.67 0.482346
\(733\) 33183.9 1.67214 0.836069 0.548625i \(-0.184848\pi\)
0.836069 + 0.548625i \(0.184848\pi\)
\(734\) −13176.6 −0.662613
\(735\) −1281.26 −0.0642993
\(736\) −137936. −6.90811
\(737\) 0 0
\(738\) 9603.08 0.478989
\(739\) 13324.8 0.663277 0.331638 0.943407i \(-0.392399\pi\)
0.331638 + 0.943407i \(0.392399\pi\)
\(740\) 5060.10 0.251369
\(741\) 9888.78 0.490248
\(742\) 19428.3 0.961236
\(743\) 1953.28 0.0964455 0.0482227 0.998837i \(-0.484644\pi\)
0.0482227 + 0.998837i \(0.484644\pi\)
\(744\) 44537.1 2.19464
\(745\) 4622.56 0.227326
\(746\) 57725.1 2.83306
\(747\) −5363.41 −0.262700
\(748\) 0 0
\(749\) 16842.0 0.821619
\(750\) 20638.9 1.00484
\(751\) 2848.62 0.138412 0.0692062 0.997602i \(-0.477953\pi\)
0.0692062 + 0.997602i \(0.477953\pi\)
\(752\) 50027.8 2.42596
\(753\) −696.031 −0.0336849
\(754\) −16446.9 −0.794376
\(755\) −4932.89 −0.237783
\(756\) 10453.5 0.502895
\(757\) −17909.1 −0.859864 −0.429932 0.902861i \(-0.641463\pi\)
−0.429932 + 0.902861i \(0.641463\pi\)
\(758\) 7077.78 0.339151
\(759\) 0 0
\(760\) −44392.1 −2.11878
\(761\) 15984.0 0.761392 0.380696 0.924700i \(-0.375684\pi\)
0.380696 + 0.924700i \(0.375684\pi\)
\(762\) −44773.8 −2.12859
\(763\) −24154.0 −1.14604
\(764\) 65784.4 3.11518
\(765\) 111.801 0.00528387
\(766\) 44763.2 2.11143
\(767\) −17537.0 −0.825586
\(768\) 97953.3 4.60232
\(769\) 12888.9 0.604402 0.302201 0.953244i \(-0.402279\pi\)
0.302201 + 0.953244i \(0.402279\pi\)
\(770\) 0 0
\(771\) −5924.79 −0.276752
\(772\) 48265.8 2.25016
\(773\) −20981.4 −0.976258 −0.488129 0.872772i \(-0.662320\pi\)
−0.488129 + 0.872772i \(0.662320\pi\)
\(774\) −5790.44 −0.268906
\(775\) 15739.2 0.729510
\(776\) 55109.2 2.54936
\(777\) 1871.85 0.0864249
\(778\) −5366.98 −0.247321
\(779\) −17022.6 −0.782926
\(780\) 14571.0 0.668878
\(781\) 0 0
\(782\) −1677.22 −0.0766973
\(783\) 2149.24 0.0980938
\(784\) −23622.9 −1.07612
\(785\) 6896.20 0.313549
\(786\) −39519.1 −1.79338
\(787\) 5493.86 0.248837 0.124419 0.992230i \(-0.460293\pi\)
0.124419 + 0.992230i \(0.460293\pi\)
\(788\) −1139.27 −0.0515034
\(789\) 18865.5 0.851242
\(790\) −1777.26 −0.0800408
\(791\) 31066.1 1.39644
\(792\) 0 0
\(793\) 4925.67 0.220575
\(794\) −78146.9 −3.49286
\(795\) −3538.88 −0.157876
\(796\) 78448.6 3.49314
\(797\) 15558.3 0.691474 0.345737 0.938331i \(-0.387629\pi\)
0.345737 + 0.938331i \(0.387629\pi\)
\(798\) −24780.5 −1.09927
\(799\) 360.586 0.0159657
\(800\) 96723.1 4.27460
\(801\) 12914.9 0.569693
\(802\) 75751.2 3.33525
\(803\) 0 0
\(804\) 57604.1 2.52679
\(805\) 12194.7 0.533921
\(806\) 34654.1 1.51444
\(807\) −20700.5 −0.902965
\(808\) 64318.0 2.80037
\(809\) −8185.89 −0.355748 −0.177874 0.984053i \(-0.556922\pi\)
−0.177874 + 0.984053i \(0.556922\pi\)
\(810\) −2546.38 −0.110458
\(811\) 10532.1 0.456021 0.228011 0.973659i \(-0.426778\pi\)
0.228011 + 0.973659i \(0.426778\pi\)
\(812\) 30818.9 1.33193
\(813\) −16510.1 −0.712218
\(814\) 0 0
\(815\) −6412.88 −0.275624
\(816\) 2061.29 0.0884310
\(817\) 10264.3 0.439536
\(818\) −63308.9 −2.70604
\(819\) 5390.14 0.229972
\(820\) −25082.6 −1.06820
\(821\) −20996.0 −0.892527 −0.446263 0.894902i \(-0.647246\pi\)
−0.446263 + 0.894902i \(0.647246\pi\)
\(822\) −20295.6 −0.861178
\(823\) 27292.2 1.15595 0.577974 0.816055i \(-0.303843\pi\)
0.577974 + 0.816055i \(0.303843\pi\)
\(824\) 149545. 6.32237
\(825\) 0 0
\(826\) 43946.3 1.85120
\(827\) −27817.4 −1.16966 −0.584829 0.811157i \(-0.698838\pi\)
−0.584829 + 0.811157i \(0.698838\pi\)
\(828\) 28565.1 1.19892
\(829\) 29670.5 1.24306 0.621531 0.783389i \(-0.286511\pi\)
0.621531 + 0.783389i \(0.286511\pi\)
\(830\) 18734.3 0.783465
\(831\) −543.459 −0.0226864
\(832\) 122343. 5.09792
\(833\) −170.267 −0.00708212
\(834\) 31102.5 1.29136
\(835\) 12930.6 0.535906
\(836\) 0 0
\(837\) −4528.51 −0.187011
\(838\) 77479.7 3.19390
\(839\) −47358.6 −1.94875 −0.974374 0.224932i \(-0.927784\pi\)
−0.974374 + 0.224932i \(0.927784\pi\)
\(840\) −24197.1 −0.993904
\(841\) −18052.6 −0.740195
\(842\) 41806.1 1.71108
\(843\) −14489.6 −0.591990
\(844\) −31660.8 −1.29125
\(845\) −4750.48 −0.193398
\(846\) −8212.74 −0.333759
\(847\) 0 0
\(848\) −65247.1 −2.64221
\(849\) −9473.10 −0.382940
\(850\) 1176.10 0.0474587
\(851\) 5115.02 0.206041
\(852\) 13586.5 0.546320
\(853\) 10456.2 0.419709 0.209855 0.977733i \(-0.432701\pi\)
0.209855 + 0.977733i \(0.432701\pi\)
\(854\) −12343.3 −0.494590
\(855\) 4513.77 0.180547
\(856\) −91319.3 −3.64630
\(857\) −33631.2 −1.34051 −0.670256 0.742130i \(-0.733816\pi\)
−0.670256 + 0.742130i \(0.733816\pi\)
\(858\) 0 0
\(859\) −41703.9 −1.65648 −0.828241 0.560372i \(-0.810658\pi\)
−0.828241 + 0.560372i \(0.810658\pi\)
\(860\) 15124.2 0.599689
\(861\) −9278.64 −0.367265
\(862\) 22071.4 0.872105
\(863\) −18699.3 −0.737579 −0.368790 0.929513i \(-0.620228\pi\)
−0.368790 + 0.929513i \(0.620228\pi\)
\(864\) −27829.3 −1.09580
\(865\) −10881.0 −0.427707
\(866\) −12370.7 −0.485420
\(867\) −14724.1 −0.576768
\(868\) −64936.4 −2.53927
\(869\) 0 0
\(870\) −7507.23 −0.292551
\(871\) 29702.6 1.15549
\(872\) 130966. 5.08607
\(873\) −5603.48 −0.217238
\(874\) −67715.2 −2.62071
\(875\) −19941.7 −0.770459
\(876\) 53073.2 2.04700
\(877\) 24095.9 0.927777 0.463889 0.885894i \(-0.346454\pi\)
0.463889 + 0.885894i \(0.346454\pi\)
\(878\) 44972.6 1.72864
\(879\) −17792.3 −0.682728
\(880\) 0 0
\(881\) 37466.9 1.43280 0.716398 0.697692i \(-0.245790\pi\)
0.716398 + 0.697692i \(0.245790\pi\)
\(882\) 3878.02 0.148050
\(883\) 28314.9 1.07913 0.539565 0.841944i \(-0.318589\pi\)
0.539565 + 0.841944i \(0.318589\pi\)
\(884\) 1936.34 0.0736722
\(885\) −8004.83 −0.304045
\(886\) −11582.7 −0.439198
\(887\) 3094.24 0.117130 0.0585651 0.998284i \(-0.481347\pi\)
0.0585651 + 0.998284i \(0.481347\pi\)
\(888\) −10149.4 −0.383549
\(889\) 43261.1 1.63209
\(890\) −45111.3 −1.69903
\(891\) 0 0
\(892\) −77125.6 −2.89502
\(893\) 14558.1 0.545541
\(894\) −13991.2 −0.523418
\(895\) 10546.8 0.393900
\(896\) −171973. −6.41209
\(897\) 14729.1 0.548262
\(898\) 16651.4 0.618781
\(899\) −13351.0 −0.495305
\(900\) −20030.4 −0.741868
\(901\) −470.283 −0.0173889
\(902\) 0 0
\(903\) 5594.81 0.206183
\(904\) −168444. −6.19730
\(905\) 6590.35 0.242067
\(906\) 14930.5 0.547497
\(907\) −3604.15 −0.131945 −0.0659723 0.997821i \(-0.521015\pi\)
−0.0659723 + 0.997821i \(0.521015\pi\)
\(908\) −31200.6 −1.14034
\(909\) −6539.82 −0.238627
\(910\) −18827.6 −0.685857
\(911\) 20320.4 0.739016 0.369508 0.929228i \(-0.379526\pi\)
0.369508 + 0.929228i \(0.379526\pi\)
\(912\) 83221.5 3.02164
\(913\) 0 0
\(914\) −29649.8 −1.07301
\(915\) 2248.34 0.0812326
\(916\) 60767.6 2.19194
\(917\) 38183.9 1.37507
\(918\) −338.389 −0.0121661
\(919\) −5615.52 −0.201566 −0.100783 0.994908i \(-0.532135\pi\)
−0.100783 + 0.994908i \(0.532135\pi\)
\(920\) −66121.1 −2.36951
\(921\) −7032.55 −0.251607
\(922\) −34511.2 −1.23272
\(923\) 7005.62 0.249830
\(924\) 0 0
\(925\) −3586.75 −0.127494
\(926\) 7671.58 0.272250
\(927\) −15205.6 −0.538747
\(928\) −82046.3 −2.90227
\(929\) −32968.4 −1.16433 −0.582163 0.813072i \(-0.697793\pi\)
−0.582163 + 0.813072i \(0.697793\pi\)
\(930\) 15818.0 0.557734
\(931\) −6874.27 −0.241993
\(932\) 119368. 4.19532
\(933\) −8829.63 −0.309827
\(934\) 57833.4 2.02609
\(935\) 0 0
\(936\) −29226.0 −1.02060
\(937\) 9150.95 0.319049 0.159524 0.987194i \(-0.449004\pi\)
0.159524 + 0.987194i \(0.449004\pi\)
\(938\) −74432.2 −2.59093
\(939\) 22603.0 0.785541
\(940\) 21451.1 0.744318
\(941\) 5093.14 0.176442 0.0882209 0.996101i \(-0.471882\pi\)
0.0882209 + 0.996101i \(0.471882\pi\)
\(942\) −20872.9 −0.721948
\(943\) −25354.8 −0.875575
\(944\) −147587. −5.08850
\(945\) 2460.35 0.0846933
\(946\) 0 0
\(947\) −3803.68 −0.130520 −0.0652602 0.997868i \(-0.520788\pi\)
−0.0652602 + 0.997868i \(0.520788\pi\)
\(948\) 4022.46 0.137809
\(949\) 27366.3 0.936087
\(950\) 47483.2 1.62164
\(951\) 23754.3 0.809974
\(952\) −3215.56 −0.109472
\(953\) 21263.6 0.722765 0.361383 0.932418i \(-0.382305\pi\)
0.361383 + 0.932418i \(0.382305\pi\)
\(954\) 10711.2 0.363510
\(955\) 15483.2 0.524632
\(956\) −100720. −3.40745
\(957\) 0 0
\(958\) −8379.94 −0.282613
\(959\) 19609.9 0.660308
\(960\) 55843.8 1.87745
\(961\) −1660.08 −0.0557241
\(962\) −7897.18 −0.264673
\(963\) 9285.30 0.310711
\(964\) −123400. −4.12287
\(965\) 11360.0 0.378953
\(966\) −36909.9 −1.22936
\(967\) 3365.02 0.111905 0.0559523 0.998433i \(-0.482181\pi\)
0.0559523 + 0.998433i \(0.482181\pi\)
\(968\) 0 0
\(969\) 599.837 0.0198860
\(970\) 19572.8 0.647882
\(971\) −18906.7 −0.624867 −0.312433 0.949940i \(-0.601144\pi\)
−0.312433 + 0.949940i \(0.601144\pi\)
\(972\) 5763.18 0.190179
\(973\) −30051.7 −0.990147
\(974\) 9644.46 0.317278
\(975\) −10328.3 −0.339253
\(976\) 41453.2 1.35951
\(977\) 7122.77 0.233242 0.116621 0.993176i \(-0.462794\pi\)
0.116621 + 0.993176i \(0.462794\pi\)
\(978\) 19410.0 0.634625
\(979\) 0 0
\(980\) −10129.1 −0.330167
\(981\) −13316.5 −0.433399
\(982\) −67155.5 −2.18230
\(983\) −10427.0 −0.338320 −0.169160 0.985589i \(-0.554105\pi\)
−0.169160 + 0.985589i \(0.554105\pi\)
\(984\) 50309.9 1.62990
\(985\) −268.141 −0.00867378
\(986\) −997.639 −0.0322224
\(987\) 7935.27 0.255909
\(988\) 78176.8 2.51734
\(989\) 15288.4 0.491549
\(990\) 0 0
\(991\) −12245.8 −0.392532 −0.196266 0.980551i \(-0.562882\pi\)
−0.196266 + 0.980551i \(0.562882\pi\)
\(992\) 172874. 5.53303
\(993\) −14823.8 −0.473735
\(994\) −17555.5 −0.560188
\(995\) 18463.9 0.588285
\(996\) −42401.0 −1.34892
\(997\) −46890.4 −1.48950 −0.744751 0.667342i \(-0.767432\pi\)
−0.744751 + 0.667342i \(0.767432\pi\)
\(998\) −21703.8 −0.688397
\(999\) 1031.98 0.0326832
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 363.4.a.q.1.1 4
3.2 odd 2 1089.4.a.bf.1.4 4
11.10 odd 2 363.4.a.s.1.4 yes 4
33.32 even 2 1089.4.a.ba.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.4.a.q.1.1 4 1.1 even 1 trivial
363.4.a.s.1.4 yes 4 11.10 odd 2
1089.4.a.ba.1.1 4 33.32 even 2
1089.4.a.bf.1.4 4 3.2 odd 2