Properties

Label 115.4.a.e.1.2
Level $115$
Weight $4$
Character 115.1
Self dual yes
Analytic conductor $6.785$
Analytic rank $0$
Dimension $5$
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] = [115,4,Mod(1,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, 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("115.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 115.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: \(6.78521965066\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.49214\) of defining polynomial
Character \(\chi\) \(=\) 115.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.49214 q^{2} +9.02447 q^{3} -5.77352 q^{4} -5.00000 q^{5} -13.4658 q^{6} +4.33445 q^{7} +20.5520 q^{8} +54.4411 q^{9} +O(q^{10})\) \(q-1.49214 q^{2} +9.02447 q^{3} -5.77352 q^{4} -5.00000 q^{5} -13.4658 q^{6} +4.33445 q^{7} +20.5520 q^{8} +54.4411 q^{9} +7.46070 q^{10} +48.1836 q^{11} -52.1029 q^{12} +53.0174 q^{13} -6.46761 q^{14} -45.1224 q^{15} +15.5216 q^{16} -61.0120 q^{17} -81.2338 q^{18} +12.0115 q^{19} +28.8676 q^{20} +39.1161 q^{21} -71.8968 q^{22} -23.0000 q^{23} +185.471 q^{24} +25.0000 q^{25} -79.1094 q^{26} +247.642 q^{27} -25.0250 q^{28} +198.213 q^{29} +67.3289 q^{30} -11.9661 q^{31} -187.577 q^{32} +434.832 q^{33} +91.0386 q^{34} -21.6723 q^{35} -314.317 q^{36} -230.018 q^{37} -17.9229 q^{38} +478.454 q^{39} -102.760 q^{40} -393.486 q^{41} -58.3668 q^{42} -167.496 q^{43} -278.189 q^{44} -272.206 q^{45} +34.3192 q^{46} +531.002 q^{47} +140.074 q^{48} -324.213 q^{49} -37.3035 q^{50} -550.602 q^{51} -306.097 q^{52} +373.570 q^{53} -369.516 q^{54} -240.918 q^{55} +89.0818 q^{56} +108.398 q^{57} -295.762 q^{58} -825.880 q^{59} +260.515 q^{60} -632.806 q^{61} +17.8551 q^{62} +235.972 q^{63} +155.718 q^{64} -265.087 q^{65} -648.830 q^{66} -38.7503 q^{67} +352.254 q^{68} -207.563 q^{69} +32.3381 q^{70} +1109.37 q^{71} +1118.88 q^{72} -446.171 q^{73} +343.219 q^{74} +225.612 q^{75} -69.3488 q^{76} +208.850 q^{77} -713.921 q^{78} +642.608 q^{79} -77.6081 q^{80} +764.926 q^{81} +587.137 q^{82} -292.575 q^{83} -225.838 q^{84} +305.060 q^{85} +249.927 q^{86} +1788.77 q^{87} +990.271 q^{88} -273.272 q^{89} +406.169 q^{90} +229.801 q^{91} +132.791 q^{92} -107.988 q^{93} -792.329 q^{94} -60.0577 q^{95} -1692.78 q^{96} -959.139 q^{97} +483.771 q^{98} +2623.17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9} - 30 q^{10} + 23 q^{11} + 47 q^{12} + 132 q^{13} + 93 q^{14} - 20 q^{15} + 282 q^{16} + 23 q^{17} - 15 q^{18} - 161 q^{19} - 110 q^{20} - 60 q^{21} + 193 q^{22} - 115 q^{23} + 105 q^{24} + 125 q^{25} - 257 q^{26} + 577 q^{27} + 17 q^{28} + 401 q^{29} - 95 q^{30} + 32 q^{31} + 670 q^{32} + 189 q^{33} - 663 q^{34} + 15 q^{35} - 659 q^{36} - 38 q^{37} - 875 q^{38} + 335 q^{39} - 690 q^{40} - 12 q^{41} - 798 q^{42} - 566 q^{43} + 47 q^{44} - 385 q^{45} - 138 q^{46} + 919 q^{47} - 773 q^{48} - 738 q^{49} + 150 q^{50} - 993 q^{51} - 305 q^{52} + 1156 q^{53} - 8 q^{54} - 115 q^{55} + 343 q^{56} + 114 q^{57} - 1042 q^{58} + 1324 q^{59} - 235 q^{60} - 1673 q^{61} + 565 q^{62} + 270 q^{63} + 2466 q^{64} - 660 q^{65} - 2781 q^{66} + 558 q^{67} - 2267 q^{68} - 92 q^{69} - 465 q^{70} - 108 q^{71} - 789 q^{72} + 1173 q^{73} + 1458 q^{74} + 100 q^{75} - 3477 q^{76} + 2608 q^{77} + 704 q^{78} + 656 q^{79} - 1410 q^{80} - 319 q^{81} + 3505 q^{82} - 82 q^{83} - 718 q^{84} - 115 q^{85} + 112 q^{86} + 2389 q^{87} + 2397 q^{88} + 570 q^{89} + 75 q^{90} - 1589 q^{91} - 506 q^{92} + 911 q^{93} - 948 q^{94} + 805 q^{95} - 5991 q^{96} + 633 q^{97} - 2555 q^{98} + 2021 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.49214 −0.527551 −0.263776 0.964584i \(-0.584968\pi\)
−0.263776 + 0.964584i \(0.584968\pi\)
\(3\) 9.02447 1.73676 0.868380 0.495899i \(-0.165161\pi\)
0.868380 + 0.495899i \(0.165161\pi\)
\(4\) −5.77352 −0.721689
\(5\) −5.00000 −0.447214
\(6\) −13.4658 −0.916231
\(7\) 4.33445 0.234038 0.117019 0.993130i \(-0.462666\pi\)
0.117019 + 0.993130i \(0.462666\pi\)
\(8\) 20.5520 0.908280
\(9\) 54.4411 2.01634
\(10\) 7.46070 0.235928
\(11\) 48.1836 1.32072 0.660360 0.750949i \(-0.270404\pi\)
0.660360 + 0.750949i \(0.270404\pi\)
\(12\) −52.1029 −1.25340
\(13\) 53.0174 1.13111 0.565553 0.824712i \(-0.308663\pi\)
0.565553 + 0.824712i \(0.308663\pi\)
\(14\) −6.46761 −0.123467
\(15\) −45.1224 −0.776703
\(16\) 15.5216 0.242525
\(17\) −61.0120 −0.870447 −0.435223 0.900323i \(-0.643331\pi\)
−0.435223 + 0.900323i \(0.643331\pi\)
\(18\) −81.2338 −1.06372
\(19\) 12.0115 0.145034 0.0725168 0.997367i \(-0.476897\pi\)
0.0725168 + 0.997367i \(0.476897\pi\)
\(20\) 28.8676 0.322749
\(21\) 39.1161 0.406469
\(22\) −71.8968 −0.696747
\(23\) −23.0000 −0.208514
\(24\) 185.471 1.57746
\(25\) 25.0000 0.200000
\(26\) −79.1094 −0.596716
\(27\) 247.642 1.76514
\(28\) −25.0250 −0.168903
\(29\) 198.213 1.26922 0.634609 0.772834i \(-0.281161\pi\)
0.634609 + 0.772834i \(0.281161\pi\)
\(30\) 67.3289 0.409751
\(31\) −11.9661 −0.0693284 −0.0346642 0.999399i \(-0.511036\pi\)
−0.0346642 + 0.999399i \(0.511036\pi\)
\(32\) −187.577 −1.03622
\(33\) 434.832 2.29377
\(34\) 91.0386 0.459205
\(35\) −21.6723 −0.104665
\(36\) −314.317 −1.45517
\(37\) −230.018 −1.02202 −0.511010 0.859575i \(-0.670728\pi\)
−0.511010 + 0.859575i \(0.670728\pi\)
\(38\) −17.9229 −0.0765126
\(39\) 478.454 1.96446
\(40\) −102.760 −0.406195
\(41\) −393.486 −1.49883 −0.749417 0.662098i \(-0.769666\pi\)
−0.749417 + 0.662098i \(0.769666\pi\)
\(42\) −58.3668 −0.214433
\(43\) −167.496 −0.594019 −0.297010 0.954874i \(-0.595989\pi\)
−0.297010 + 0.954874i \(0.595989\pi\)
\(44\) −278.189 −0.953149
\(45\) −272.206 −0.901734
\(46\) 34.3192 0.110002
\(47\) 531.002 1.64797 0.823984 0.566612i \(-0.191746\pi\)
0.823984 + 0.566612i \(0.191746\pi\)
\(48\) 140.074 0.421208
\(49\) −324.213 −0.945226
\(50\) −37.3035 −0.105510
\(51\) −550.602 −1.51176
\(52\) −306.097 −0.816307
\(53\) 373.570 0.968186 0.484093 0.875017i \(-0.339150\pi\)
0.484093 + 0.875017i \(0.339150\pi\)
\(54\) −369.516 −0.931200
\(55\) −240.918 −0.590644
\(56\) 89.0818 0.212572
\(57\) 108.398 0.251889
\(58\) −295.762 −0.669578
\(59\) −825.880 −1.82238 −0.911190 0.411987i \(-0.864835\pi\)
−0.911190 + 0.411987i \(0.864835\pi\)
\(60\) 260.515 0.560538
\(61\) −632.806 −1.32824 −0.664119 0.747627i \(-0.731193\pi\)
−0.664119 + 0.747627i \(0.731193\pi\)
\(62\) 17.8551 0.0365743
\(63\) 235.972 0.471900
\(64\) 155.718 0.304136
\(65\) −265.087 −0.505846
\(66\) −648.830 −1.21008
\(67\) −38.7503 −0.0706583 −0.0353292 0.999376i \(-0.511248\pi\)
−0.0353292 + 0.999376i \(0.511248\pi\)
\(68\) 352.254 0.628192
\(69\) −207.563 −0.362140
\(70\) 32.3381 0.0552163
\(71\) 1109.37 1.85434 0.927168 0.374645i \(-0.122236\pi\)
0.927168 + 0.374645i \(0.122236\pi\)
\(72\) 1118.88 1.83140
\(73\) −446.171 −0.715348 −0.357674 0.933847i \(-0.616430\pi\)
−0.357674 + 0.933847i \(0.616430\pi\)
\(74\) 343.219 0.539168
\(75\) 225.612 0.347352
\(76\) −69.3488 −0.104669
\(77\) 208.850 0.309099
\(78\) −713.921 −1.03635
\(79\) 642.608 0.915179 0.457589 0.889164i \(-0.348713\pi\)
0.457589 + 0.889164i \(0.348713\pi\)
\(80\) −77.6081 −0.108461
\(81\) 764.926 1.04928
\(82\) 587.137 0.790712
\(83\) −292.575 −0.386919 −0.193460 0.981108i \(-0.561971\pi\)
−0.193460 + 0.981108i \(0.561971\pi\)
\(84\) −225.838 −0.293344
\(85\) 305.060 0.389276
\(86\) 249.927 0.313376
\(87\) 1788.77 2.20433
\(88\) 990.271 1.19958
\(89\) −273.272 −0.325469 −0.162735 0.986670i \(-0.552031\pi\)
−0.162735 + 0.986670i \(0.552031\pi\)
\(90\) 406.169 0.475711
\(91\) 229.801 0.264722
\(92\) 132.791 0.150483
\(93\) −107.988 −0.120407
\(94\) −792.329 −0.869388
\(95\) −60.0577 −0.0648610
\(96\) −1692.78 −1.79967
\(97\) −959.139 −1.00398 −0.501989 0.864874i \(-0.667398\pi\)
−0.501989 + 0.864874i \(0.667398\pi\)
\(98\) 483.771 0.498655
\(99\) 2623.17 2.66302
\(100\) −144.338 −0.144338
\(101\) −339.881 −0.334845 −0.167423 0.985885i \(-0.553544\pi\)
−0.167423 + 0.985885i \(0.553544\pi\)
\(102\) 821.575 0.797530
\(103\) −316.048 −0.302341 −0.151170 0.988508i \(-0.548304\pi\)
−0.151170 + 0.988508i \(0.548304\pi\)
\(104\) 1089.61 1.02736
\(105\) −195.581 −0.181778
\(106\) −557.420 −0.510768
\(107\) 105.833 0.0956196 0.0478098 0.998856i \(-0.484776\pi\)
0.0478098 + 0.998856i \(0.484776\pi\)
\(108\) −1429.76 −1.27388
\(109\) −1903.06 −1.67229 −0.836146 0.548508i \(-0.815196\pi\)
−0.836146 + 0.548508i \(0.815196\pi\)
\(110\) 359.484 0.311595
\(111\) −2075.79 −1.77500
\(112\) 67.2777 0.0567602
\(113\) −94.0121 −0.0782647 −0.0391324 0.999234i \(-0.512459\pi\)
−0.0391324 + 0.999234i \(0.512459\pi\)
\(114\) −161.745 −0.132884
\(115\) 115.000 0.0932505
\(116\) −1144.39 −0.915981
\(117\) 2886.33 2.28069
\(118\) 1232.33 0.961399
\(119\) −264.454 −0.203718
\(120\) −927.356 −0.705464
\(121\) 990.663 0.744300
\(122\) 944.236 0.700714
\(123\) −3551.00 −2.60312
\(124\) 69.0866 0.0500336
\(125\) −125.000 −0.0894427
\(126\) −352.104 −0.248952
\(127\) 286.165 0.199945 0.0999727 0.994990i \(-0.468124\pi\)
0.0999727 + 0.994990i \(0.468124\pi\)
\(128\) 1268.26 0.875777
\(129\) −1511.56 −1.03167
\(130\) 395.547 0.266860
\(131\) 1226.25 0.817844 0.408922 0.912569i \(-0.365905\pi\)
0.408922 + 0.912569i \(0.365905\pi\)
\(132\) −2510.51 −1.65539
\(133\) 52.0635 0.0339434
\(134\) 57.8210 0.0372759
\(135\) −1238.21 −0.789393
\(136\) −1253.92 −0.790609
\(137\) 3159.26 1.97017 0.985086 0.172061i \(-0.0550426\pi\)
0.985086 + 0.172061i \(0.0550426\pi\)
\(138\) 309.713 0.191047
\(139\) 1095.59 0.668540 0.334270 0.942477i \(-0.391510\pi\)
0.334270 + 0.942477i \(0.391510\pi\)
\(140\) 125.125 0.0755357
\(141\) 4792.01 2.86213
\(142\) −1655.33 −0.978258
\(143\) 2554.57 1.49387
\(144\) 845.014 0.489013
\(145\) −991.067 −0.567611
\(146\) 665.750 0.377383
\(147\) −2925.85 −1.64163
\(148\) 1328.01 0.737581
\(149\) 2227.60 1.22478 0.612390 0.790556i \(-0.290208\pi\)
0.612390 + 0.790556i \(0.290208\pi\)
\(150\) −336.645 −0.183246
\(151\) −972.366 −0.524040 −0.262020 0.965062i \(-0.584389\pi\)
−0.262020 + 0.965062i \(0.584389\pi\)
\(152\) 246.862 0.131731
\(153\) −3321.56 −1.75511
\(154\) −311.633 −0.163066
\(155\) 59.8306 0.0310046
\(156\) −2762.36 −1.41773
\(157\) −2872.71 −1.46030 −0.730151 0.683285i \(-0.760551\pi\)
−0.730151 + 0.683285i \(0.760551\pi\)
\(158\) −958.862 −0.482804
\(159\) 3371.28 1.68151
\(160\) 937.883 0.463414
\(161\) −99.6924 −0.0488004
\(162\) −1141.38 −0.553550
\(163\) 1458.04 0.700631 0.350315 0.936632i \(-0.386074\pi\)
0.350315 + 0.936632i \(0.386074\pi\)
\(164\) 2271.80 1.08169
\(165\) −2174.16 −1.02581
\(166\) 436.563 0.204120
\(167\) −533.984 −0.247431 −0.123715 0.992318i \(-0.539481\pi\)
−0.123715 + 0.992318i \(0.539481\pi\)
\(168\) 803.916 0.369187
\(169\) 613.842 0.279400
\(170\) −455.193 −0.205363
\(171\) 653.922 0.292437
\(172\) 967.038 0.428697
\(173\) 2127.49 0.934973 0.467487 0.884000i \(-0.345160\pi\)
0.467487 + 0.884000i \(0.345160\pi\)
\(174\) −2669.10 −1.16290
\(175\) 108.361 0.0468077
\(176\) 747.888 0.320308
\(177\) −7453.13 −3.16504
\(178\) 407.760 0.171702
\(179\) −2827.30 −1.18057 −0.590286 0.807194i \(-0.700985\pi\)
−0.590286 + 0.807194i \(0.700985\pi\)
\(180\) 1571.58 0.650772
\(181\) 284.441 0.116809 0.0584043 0.998293i \(-0.481399\pi\)
0.0584043 + 0.998293i \(0.481399\pi\)
\(182\) −342.896 −0.139655
\(183\) −5710.75 −2.30683
\(184\) −472.697 −0.189389
\(185\) 1150.09 0.457061
\(186\) 161.133 0.0635208
\(187\) −2939.78 −1.14962
\(188\) −3065.75 −1.18932
\(189\) 1073.39 0.413110
\(190\) 89.6146 0.0342175
\(191\) −2683.22 −1.01650 −0.508248 0.861211i \(-0.669707\pi\)
−0.508248 + 0.861211i \(0.669707\pi\)
\(192\) 1405.27 0.528212
\(193\) 2173.31 0.810563 0.405281 0.914192i \(-0.367174\pi\)
0.405281 + 0.914192i \(0.367174\pi\)
\(194\) 1431.17 0.529650
\(195\) −2392.27 −0.878533
\(196\) 1871.85 0.682160
\(197\) 4564.42 1.65077 0.825385 0.564570i \(-0.190958\pi\)
0.825385 + 0.564570i \(0.190958\pi\)
\(198\) −3914.14 −1.40488
\(199\) −3842.37 −1.36873 −0.684367 0.729138i \(-0.739921\pi\)
−0.684367 + 0.729138i \(0.739921\pi\)
\(200\) 513.801 0.181656
\(201\) −349.701 −0.122717
\(202\) 507.150 0.176648
\(203\) 859.147 0.297046
\(204\) 3178.91 1.09102
\(205\) 1967.43 0.670299
\(206\) 471.588 0.159500
\(207\) −1252.15 −0.420436
\(208\) 822.915 0.274322
\(209\) 578.760 0.191549
\(210\) 291.834 0.0958974
\(211\) 679.119 0.221576 0.110788 0.993844i \(-0.464663\pi\)
0.110788 + 0.993844i \(0.464663\pi\)
\(212\) −2156.81 −0.698729
\(213\) 10011.5 3.22054
\(214\) −157.918 −0.0504442
\(215\) 837.478 0.265654
\(216\) 5089.54 1.60324
\(217\) −51.8666 −0.0162255
\(218\) 2839.63 0.882220
\(219\) −4026.46 −1.24239
\(220\) 1390.94 0.426261
\(221\) −3234.70 −0.984567
\(222\) 3097.37 0.936405
\(223\) 2187.32 0.656832 0.328416 0.944533i \(-0.393485\pi\)
0.328416 + 0.944533i \(0.393485\pi\)
\(224\) −813.042 −0.242516
\(225\) 1361.03 0.403268
\(226\) 140.279 0.0412887
\(227\) −5797.66 −1.69517 −0.847586 0.530658i \(-0.821945\pi\)
−0.847586 + 0.530658i \(0.821945\pi\)
\(228\) −625.837 −0.181785
\(229\) −6874.42 −1.98373 −0.991866 0.127286i \(-0.959374\pi\)
−0.991866 + 0.127286i \(0.959374\pi\)
\(230\) −171.596 −0.0491944
\(231\) 1884.76 0.536831
\(232\) 4073.69 1.15280
\(233\) −4274.38 −1.20182 −0.600910 0.799317i \(-0.705195\pi\)
−0.600910 + 0.799317i \(0.705195\pi\)
\(234\) −4306.80 −1.20318
\(235\) −2655.01 −0.736994
\(236\) 4768.23 1.31519
\(237\) 5799.20 1.58945
\(238\) 394.602 0.107472
\(239\) 2416.98 0.654148 0.327074 0.944999i \(-0.393937\pi\)
0.327074 + 0.944999i \(0.393937\pi\)
\(240\) −700.372 −0.188370
\(241\) 4246.06 1.13491 0.567453 0.823406i \(-0.307929\pi\)
0.567453 + 0.823406i \(0.307929\pi\)
\(242\) −1478.21 −0.392656
\(243\) 216.725 0.0572136
\(244\) 3653.52 0.958576
\(245\) 1621.06 0.422718
\(246\) 5298.60 1.37328
\(247\) 636.821 0.164048
\(248\) −245.928 −0.0629696
\(249\) −2640.34 −0.671986
\(250\) 186.518 0.0471856
\(251\) −4815.45 −1.21095 −0.605476 0.795864i \(-0.707017\pi\)
−0.605476 + 0.795864i \(0.707017\pi\)
\(252\) −1362.39 −0.340566
\(253\) −1108.22 −0.275389
\(254\) −426.999 −0.105481
\(255\) 2753.01 0.676079
\(256\) −3138.17 −0.766154
\(257\) 5309.07 1.28860 0.644301 0.764772i \(-0.277149\pi\)
0.644301 + 0.764772i \(0.277149\pi\)
\(258\) 2255.46 0.544259
\(259\) −997.002 −0.239192
\(260\) 1530.48 0.365064
\(261\) 10791.0 2.55917
\(262\) −1829.73 −0.431455
\(263\) −1045.05 −0.245022 −0.122511 0.992467i \(-0.539095\pi\)
−0.122511 + 0.992467i \(0.539095\pi\)
\(264\) 8936.68 2.08339
\(265\) −1867.85 −0.432986
\(266\) −77.6860 −0.0179069
\(267\) −2466.14 −0.565263
\(268\) 223.726 0.0509934
\(269\) −3226.10 −0.731221 −0.365611 0.930768i \(-0.619140\pi\)
−0.365611 + 0.930768i \(0.619140\pi\)
\(270\) 1847.58 0.416445
\(271\) 2535.37 0.568312 0.284156 0.958778i \(-0.408287\pi\)
0.284156 + 0.958778i \(0.408287\pi\)
\(272\) −947.005 −0.211105
\(273\) 2073.84 0.459759
\(274\) −4714.06 −1.03937
\(275\) 1204.59 0.264144
\(276\) 1198.37 0.261352
\(277\) −6453.28 −1.39978 −0.699891 0.714249i \(-0.746768\pi\)
−0.699891 + 0.714249i \(0.746768\pi\)
\(278\) −1634.78 −0.352689
\(279\) −651.449 −0.139789
\(280\) −445.409 −0.0950652
\(281\) −7534.53 −1.59955 −0.799773 0.600302i \(-0.795047\pi\)
−0.799773 + 0.600302i \(0.795047\pi\)
\(282\) −7150.35 −1.50992
\(283\) 4584.47 0.962964 0.481482 0.876456i \(-0.340099\pi\)
0.481482 + 0.876456i \(0.340099\pi\)
\(284\) −6404.96 −1.33826
\(285\) −541.989 −0.112648
\(286\) −3811.78 −0.788095
\(287\) −1705.55 −0.350785
\(288\) −10211.9 −2.08938
\(289\) −1190.53 −0.242323
\(290\) 1478.81 0.299444
\(291\) −8655.72 −1.74367
\(292\) 2575.98 0.516259
\(293\) 6890.03 1.37379 0.686894 0.726758i \(-0.258974\pi\)
0.686894 + 0.726758i \(0.258974\pi\)
\(294\) 4365.78 0.866045
\(295\) 4129.40 0.814993
\(296\) −4727.33 −0.928279
\(297\) 11932.3 2.33125
\(298\) −3323.89 −0.646134
\(299\) −1219.40 −0.235852
\(300\) −1302.57 −0.250680
\(301\) −726.001 −0.139023
\(302\) 1450.91 0.276458
\(303\) −3067.24 −0.581546
\(304\) 186.438 0.0351743
\(305\) 3164.03 0.594006
\(306\) 4956.24 0.925913
\(307\) −1461.21 −0.271646 −0.135823 0.990733i \(-0.543368\pi\)
−0.135823 + 0.990733i \(0.543368\pi\)
\(308\) −1205.80 −0.223074
\(309\) −2852.16 −0.525094
\(310\) −89.2757 −0.0163565
\(311\) −750.293 −0.136801 −0.0684007 0.997658i \(-0.521790\pi\)
−0.0684007 + 0.997658i \(0.521790\pi\)
\(312\) 9833.20 1.78428
\(313\) 8615.99 1.55592 0.777962 0.628311i \(-0.216254\pi\)
0.777962 + 0.628311i \(0.216254\pi\)
\(314\) 4286.49 0.770385
\(315\) −1179.86 −0.211040
\(316\) −3710.11 −0.660475
\(317\) 356.372 0.0631415 0.0315707 0.999502i \(-0.489949\pi\)
0.0315707 + 0.999502i \(0.489949\pi\)
\(318\) −5030.42 −0.887081
\(319\) 9550.64 1.67628
\(320\) −778.589 −0.136014
\(321\) 955.090 0.166068
\(322\) 148.755 0.0257447
\(323\) −732.849 −0.126244
\(324\) −4416.31 −0.757255
\(325\) 1325.43 0.226221
\(326\) −2175.61 −0.369619
\(327\) −17174.1 −2.90437
\(328\) −8086.93 −1.36136
\(329\) 2301.60 0.385688
\(330\) 3244.15 0.541166
\(331\) 7389.98 1.22716 0.613580 0.789633i \(-0.289729\pi\)
0.613580 + 0.789633i \(0.289729\pi\)
\(332\) 1689.19 0.279236
\(333\) −12522.4 −2.06074
\(334\) 796.779 0.130532
\(335\) 193.752 0.0315994
\(336\) 607.146 0.0985789
\(337\) 10614.8 1.71580 0.857898 0.513820i \(-0.171770\pi\)
0.857898 + 0.513820i \(0.171770\pi\)
\(338\) −915.939 −0.147398
\(339\) −848.410 −0.135927
\(340\) −1761.27 −0.280936
\(341\) −576.572 −0.0915633
\(342\) −975.744 −0.154275
\(343\) −2892.00 −0.455258
\(344\) −3442.37 −0.539536
\(345\) 1037.81 0.161954
\(346\) −3174.52 −0.493246
\(347\) −3472.76 −0.537256 −0.268628 0.963244i \(-0.586570\pi\)
−0.268628 + 0.963244i \(0.586570\pi\)
\(348\) −10327.5 −1.59084
\(349\) 4820.48 0.739353 0.369677 0.929161i \(-0.379469\pi\)
0.369677 + 0.929161i \(0.379469\pi\)
\(350\) −161.690 −0.0246935
\(351\) 13129.3 1.99656
\(352\) −9038.12 −1.36856
\(353\) 3056.54 0.460859 0.230430 0.973089i \(-0.425987\pi\)
0.230430 + 0.973089i \(0.425987\pi\)
\(354\) 11121.1 1.66972
\(355\) −5546.85 −0.829284
\(356\) 1577.74 0.234888
\(357\) −2386.56 −0.353809
\(358\) 4218.73 0.622812
\(359\) −10754.6 −1.58108 −0.790541 0.612409i \(-0.790201\pi\)
−0.790541 + 0.612409i \(0.790201\pi\)
\(360\) −5594.38 −0.819026
\(361\) −6714.72 −0.978965
\(362\) −424.426 −0.0616225
\(363\) 8940.21 1.29267
\(364\) −1326.76 −0.191047
\(365\) 2230.86 0.319913
\(366\) 8521.24 1.21697
\(367\) −10901.7 −1.55058 −0.775292 0.631602i \(-0.782398\pi\)
−0.775292 + 0.631602i \(0.782398\pi\)
\(368\) −356.997 −0.0505700
\(369\) −21421.8 −3.02216
\(370\) −1716.10 −0.241123
\(371\) 1619.22 0.226593
\(372\) 623.470 0.0868963
\(373\) 98.0242 0.0136072 0.00680362 0.999977i \(-0.497834\pi\)
0.00680362 + 0.999977i \(0.497834\pi\)
\(374\) 4386.57 0.606481
\(375\) −1128.06 −0.155341
\(376\) 10913.2 1.49682
\(377\) 10508.8 1.43562
\(378\) −1601.65 −0.217937
\(379\) −8819.20 −1.19528 −0.597641 0.801764i \(-0.703895\pi\)
−0.597641 + 0.801764i \(0.703895\pi\)
\(380\) 346.744 0.0468095
\(381\) 2582.49 0.347257
\(382\) 4003.74 0.536254
\(383\) 7622.61 1.01696 0.508482 0.861073i \(-0.330207\pi\)
0.508482 + 0.861073i \(0.330207\pi\)
\(384\) 11445.4 1.52101
\(385\) −1044.25 −0.138233
\(386\) −3242.89 −0.427614
\(387\) −9118.64 −1.19774
\(388\) 5537.60 0.724560
\(389\) −495.174 −0.0645406 −0.0322703 0.999479i \(-0.510274\pi\)
−0.0322703 + 0.999479i \(0.510274\pi\)
\(390\) 3569.60 0.463471
\(391\) 1403.28 0.181501
\(392\) −6663.22 −0.858530
\(393\) 11066.2 1.42040
\(394\) −6810.76 −0.870866
\(395\) −3213.04 −0.409280
\(396\) −15144.9 −1.92187
\(397\) −8510.26 −1.07586 −0.537931 0.842989i \(-0.680794\pi\)
−0.537931 + 0.842989i \(0.680794\pi\)
\(398\) 5733.35 0.722078
\(399\) 469.845 0.0589516
\(400\) 388.040 0.0485050
\(401\) −8022.24 −0.999031 −0.499516 0.866305i \(-0.666489\pi\)
−0.499516 + 0.866305i \(0.666489\pi\)
\(402\) 521.804 0.0647393
\(403\) −634.413 −0.0784177
\(404\) 1962.31 0.241654
\(405\) −3824.63 −0.469253
\(406\) −1281.97 −0.156707
\(407\) −11083.1 −1.34980
\(408\) −11316.0 −1.37310
\(409\) −2089.24 −0.252583 −0.126291 0.991993i \(-0.540307\pi\)
−0.126291 + 0.991993i \(0.540307\pi\)
\(410\) −2935.68 −0.353617
\(411\) 28510.6 3.42172
\(412\) 1824.71 0.218196
\(413\) −3579.74 −0.426507
\(414\) 1868.38 0.221801
\(415\) 1462.88 0.173036
\(416\) −9944.82 −1.17208
\(417\) 9887.15 1.16109
\(418\) −863.591 −0.101052
\(419\) 8022.73 0.935408 0.467704 0.883885i \(-0.345081\pi\)
0.467704 + 0.883885i \(0.345081\pi\)
\(420\) 1129.19 0.131188
\(421\) 5382.92 0.623153 0.311577 0.950221i \(-0.399143\pi\)
0.311577 + 0.950221i \(0.399143\pi\)
\(422\) −1013.34 −0.116893
\(423\) 28908.3 3.32286
\(424\) 7677.63 0.879383
\(425\) −1525.30 −0.174089
\(426\) −14938.5 −1.69900
\(427\) −2742.87 −0.310859
\(428\) −611.030 −0.0690076
\(429\) 23053.6 2.59450
\(430\) −1249.63 −0.140146
\(431\) 12908.1 1.44261 0.721303 0.692620i \(-0.243544\pi\)
0.721303 + 0.692620i \(0.243544\pi\)
\(432\) 3843.80 0.428090
\(433\) 4964.66 0.551008 0.275504 0.961300i \(-0.411155\pi\)
0.275504 + 0.961300i \(0.411155\pi\)
\(434\) 77.3923 0.00855979
\(435\) −8943.86 −0.985805
\(436\) 10987.3 1.20687
\(437\) −276.265 −0.0302416
\(438\) 6008.05 0.655424
\(439\) 6214.52 0.675633 0.337816 0.941212i \(-0.390312\pi\)
0.337816 + 0.941212i \(0.390312\pi\)
\(440\) −4951.36 −0.536470
\(441\) −17650.5 −1.90590
\(442\) 4826.63 0.519410
\(443\) −11783.3 −1.26375 −0.631875 0.775071i \(-0.717714\pi\)
−0.631875 + 0.775071i \(0.717714\pi\)
\(444\) 11984.6 1.28100
\(445\) 1366.36 0.145554
\(446\) −3263.78 −0.346513
\(447\) 20102.9 2.12715
\(448\) 674.952 0.0711796
\(449\) 13387.6 1.40713 0.703563 0.710633i \(-0.251591\pi\)
0.703563 + 0.710633i \(0.251591\pi\)
\(450\) −2030.85 −0.212744
\(451\) −18959.6 −1.97954
\(452\) 542.780 0.0564828
\(453\) −8775.09 −0.910132
\(454\) 8650.92 0.894291
\(455\) −1149.01 −0.118387
\(456\) 2227.80 0.228785
\(457\) 15194.2 1.55526 0.777632 0.628720i \(-0.216421\pi\)
0.777632 + 0.628720i \(0.216421\pi\)
\(458\) 10257.6 1.04652
\(459\) −15109.1 −1.53646
\(460\) −663.954 −0.0672979
\(461\) 6087.94 0.615063 0.307531 0.951538i \(-0.400497\pi\)
0.307531 + 0.951538i \(0.400497\pi\)
\(462\) −2812.32 −0.283206
\(463\) 14645.2 1.47002 0.735012 0.678055i \(-0.237177\pi\)
0.735012 + 0.678055i \(0.237177\pi\)
\(464\) 3076.59 0.307817
\(465\) 539.940 0.0538476
\(466\) 6377.98 0.634022
\(467\) 2233.21 0.221286 0.110643 0.993860i \(-0.464709\pi\)
0.110643 + 0.993860i \(0.464709\pi\)
\(468\) −16664.2 −1.64595
\(469\) −167.961 −0.0165368
\(470\) 3961.65 0.388802
\(471\) −25924.7 −2.53620
\(472\) −16973.5 −1.65523
\(473\) −8070.54 −0.784533
\(474\) −8653.23 −0.838515
\(475\) 300.289 0.0290067
\(476\) 1526.83 0.147021
\(477\) 20337.6 1.95219
\(478\) −3606.47 −0.345097
\(479\) −152.988 −0.0145934 −0.00729668 0.999973i \(-0.502323\pi\)
−0.00729668 + 0.999973i \(0.502323\pi\)
\(480\) 8463.90 0.804839
\(481\) −12194.9 −1.15601
\(482\) −6335.71 −0.598722
\(483\) −899.671 −0.0847546
\(484\) −5719.61 −0.537153
\(485\) 4795.69 0.448992
\(486\) −323.384 −0.0301831
\(487\) 8018.35 0.746090 0.373045 0.927813i \(-0.378314\pi\)
0.373045 + 0.927813i \(0.378314\pi\)
\(488\) −13005.5 −1.20641
\(489\) 13158.1 1.21683
\(490\) −2418.85 −0.223005
\(491\) 16353.2 1.50307 0.751536 0.659692i \(-0.229313\pi\)
0.751536 + 0.659692i \(0.229313\pi\)
\(492\) 20501.8 1.87864
\(493\) −12093.4 −1.10479
\(494\) −950.226 −0.0865439
\(495\) −13115.9 −1.19094
\(496\) −185.734 −0.0168139
\(497\) 4808.51 0.433986
\(498\) 3939.76 0.354507
\(499\) −14615.1 −1.31115 −0.655573 0.755132i \(-0.727573\pi\)
−0.655573 + 0.755132i \(0.727573\pi\)
\(500\) 721.689 0.0645499
\(501\) −4818.92 −0.429728
\(502\) 7185.34 0.638839
\(503\) −5505.02 −0.487985 −0.243993 0.969777i \(-0.578457\pi\)
−0.243993 + 0.969777i \(0.578457\pi\)
\(504\) 4849.71 0.428618
\(505\) 1699.40 0.149747
\(506\) 1653.63 0.145282
\(507\) 5539.60 0.485251
\(508\) −1652.18 −0.144299
\(509\) −7765.56 −0.676233 −0.338116 0.941104i \(-0.609790\pi\)
−0.338116 + 0.941104i \(0.609790\pi\)
\(510\) −4107.88 −0.356666
\(511\) −1933.91 −0.167419
\(512\) −5463.50 −0.471591
\(513\) 2974.56 0.256004
\(514\) −7921.88 −0.679804
\(515\) 1580.24 0.135211
\(516\) 8727.01 0.744545
\(517\) 25585.6 2.17650
\(518\) 1487.67 0.126186
\(519\) 19199.5 1.62382
\(520\) −5448.07 −0.459450
\(521\) −9808.85 −0.824824 −0.412412 0.910997i \(-0.635314\pi\)
−0.412412 + 0.910997i \(0.635314\pi\)
\(522\) −16101.6 −1.35009
\(523\) 5173.83 0.432573 0.216287 0.976330i \(-0.430605\pi\)
0.216287 + 0.976330i \(0.430605\pi\)
\(524\) −7079.75 −0.590229
\(525\) 977.904 0.0812937
\(526\) 1559.37 0.129262
\(527\) 730.078 0.0603467
\(528\) 6749.29 0.556298
\(529\) 529.000 0.0434783
\(530\) 2787.10 0.228422
\(531\) −44961.8 −3.67453
\(532\) −300.589 −0.0244966
\(533\) −20861.6 −1.69534
\(534\) 3679.82 0.298205
\(535\) −529.167 −0.0427624
\(536\) −796.398 −0.0641775
\(537\) −25514.9 −2.05037
\(538\) 4813.79 0.385757
\(539\) −15621.7 −1.24838
\(540\) 7148.82 0.569696
\(541\) 14529.6 1.15467 0.577336 0.816507i \(-0.304092\pi\)
0.577336 + 0.816507i \(0.304092\pi\)
\(542\) −3783.12 −0.299814
\(543\) 2566.93 0.202868
\(544\) 11444.4 0.901978
\(545\) 9515.28 0.747871
\(546\) −3094.45 −0.242547
\(547\) −8519.93 −0.665971 −0.332985 0.942932i \(-0.608056\pi\)
−0.332985 + 0.942932i \(0.608056\pi\)
\(548\) −18240.0 −1.42185
\(549\) −34450.7 −2.67818
\(550\) −1797.42 −0.139349
\(551\) 2380.85 0.184079
\(552\) −4265.84 −0.328924
\(553\) 2785.36 0.214187
\(554\) 9629.20 0.738457
\(555\) 10379.0 0.793805
\(556\) −6325.43 −0.482478
\(557\) 14706.3 1.11872 0.559360 0.828925i \(-0.311047\pi\)
0.559360 + 0.828925i \(0.311047\pi\)
\(558\) 972.054 0.0737461
\(559\) −8880.17 −0.671899
\(560\) −336.388 −0.0253839
\(561\) −26530.0 −1.99661
\(562\) 11242.6 0.843843
\(563\) 3719.27 0.278416 0.139208 0.990263i \(-0.455544\pi\)
0.139208 + 0.990263i \(0.455544\pi\)
\(564\) −27666.7 −2.06557
\(565\) 470.060 0.0350010
\(566\) −6840.68 −0.508013
\(567\) 3315.53 0.245572
\(568\) 22799.8 1.68426
\(569\) −16399.1 −1.20823 −0.604116 0.796896i \(-0.706474\pi\)
−0.604116 + 0.796896i \(0.706474\pi\)
\(570\) 808.724 0.0594276
\(571\) 2180.66 0.159821 0.0799105 0.996802i \(-0.474537\pi\)
0.0799105 + 0.996802i \(0.474537\pi\)
\(572\) −14748.9 −1.07811
\(573\) −24214.6 −1.76541
\(574\) 2544.91 0.185057
\(575\) −575.000 −0.0417029
\(576\) 8477.46 0.613242
\(577\) 10558.4 0.761790 0.380895 0.924618i \(-0.375616\pi\)
0.380895 + 0.924618i \(0.375616\pi\)
\(578\) 1776.44 0.127838
\(579\) 19613.0 1.40775
\(580\) 5721.94 0.409639
\(581\) −1268.15 −0.0905540
\(582\) 12915.6 0.919875
\(583\) 18000.0 1.27870
\(584\) −9169.72 −0.649736
\(585\) −14431.6 −1.01996
\(586\) −10280.9 −0.724743
\(587\) 21251.7 1.49430 0.747149 0.664656i \(-0.231422\pi\)
0.747149 + 0.664656i \(0.231422\pi\)
\(588\) 16892.4 1.18475
\(589\) −143.732 −0.0100549
\(590\) −6161.65 −0.429951
\(591\) 41191.5 2.86699
\(592\) −3570.25 −0.247865
\(593\) −9358.61 −0.648081 −0.324041 0.946043i \(-0.605041\pi\)
−0.324041 + 0.946043i \(0.605041\pi\)
\(594\) −17804.6 −1.22985
\(595\) 1322.27 0.0911054
\(596\) −12861.1 −0.883910
\(597\) −34675.3 −2.37716
\(598\) 1819.52 0.124424
\(599\) 2392.62 0.163205 0.0816025 0.996665i \(-0.473996\pi\)
0.0816025 + 0.996665i \(0.473996\pi\)
\(600\) 4636.78 0.315493
\(601\) −13565.8 −0.920730 −0.460365 0.887730i \(-0.652281\pi\)
−0.460365 + 0.887730i \(0.652281\pi\)
\(602\) 1083.30 0.0733420
\(603\) −2109.61 −0.142471
\(604\) 5613.97 0.378194
\(605\) −4953.31 −0.332861
\(606\) 4576.76 0.306796
\(607\) 10454.9 0.699097 0.349548 0.936918i \(-0.386335\pi\)
0.349548 + 0.936918i \(0.386335\pi\)
\(608\) −2253.08 −0.150287
\(609\) 7753.35 0.515897
\(610\) −4721.18 −0.313369
\(611\) 28152.3 1.86403
\(612\) 19177.1 1.26665
\(613\) 17858.8 1.17669 0.588344 0.808611i \(-0.299780\pi\)
0.588344 + 0.808611i \(0.299780\pi\)
\(614\) 2180.33 0.143307
\(615\) 17755.0 1.16415
\(616\) 4292.28 0.280748
\(617\) 13120.3 0.856082 0.428041 0.903759i \(-0.359204\pi\)
0.428041 + 0.903759i \(0.359204\pi\)
\(618\) 4255.83 0.277014
\(619\) −7081.28 −0.459807 −0.229904 0.973213i \(-0.573841\pi\)
−0.229904 + 0.973213i \(0.573841\pi\)
\(620\) −345.433 −0.0223757
\(621\) −5695.76 −0.368056
\(622\) 1119.54 0.0721697
\(623\) −1184.48 −0.0761723
\(624\) 7426.38 0.476431
\(625\) 625.000 0.0400000
\(626\) −12856.3 −0.820830
\(627\) 5223.00 0.332674
\(628\) 16585.7 1.05389
\(629\) 14033.9 0.889613
\(630\) 1760.52 0.111335
\(631\) −11411.8 −0.719961 −0.359981 0.932960i \(-0.617217\pi\)
−0.359981 + 0.932960i \(0.617217\pi\)
\(632\) 13206.9 0.831238
\(633\) 6128.69 0.384824
\(634\) −531.757 −0.0333104
\(635\) −1430.83 −0.0894183
\(636\) −19464.1 −1.21353
\(637\) −17188.9 −1.06915
\(638\) −14250.9 −0.884324
\(639\) 60395.3 3.73897
\(640\) −6341.30 −0.391659
\(641\) 2755.34 0.169781 0.0848903 0.996390i \(-0.472946\pi\)
0.0848903 + 0.996390i \(0.472946\pi\)
\(642\) −1425.13 −0.0876096
\(643\) −23204.2 −1.42315 −0.711574 0.702611i \(-0.752017\pi\)
−0.711574 + 0.702611i \(0.752017\pi\)
\(644\) 575.576 0.0352187
\(645\) 7557.80 0.461377
\(646\) 1093.51 0.0666002
\(647\) 30680.4 1.86425 0.932127 0.362133i \(-0.117951\pi\)
0.932127 + 0.362133i \(0.117951\pi\)
\(648\) 15720.8 0.953040
\(649\) −39793.9 −2.40685
\(650\) −1977.73 −0.119343
\(651\) −468.069 −0.0281798
\(652\) −8418.04 −0.505638
\(653\) −11329.3 −0.678942 −0.339471 0.940617i \(-0.610248\pi\)
−0.339471 + 0.940617i \(0.610248\pi\)
\(654\) 25626.1 1.53220
\(655\) −6131.23 −0.365751
\(656\) −6107.54 −0.363505
\(657\) −24290.1 −1.44238
\(658\) −3434.31 −0.203470
\(659\) 13794.3 0.815400 0.407700 0.913116i \(-0.366331\pi\)
0.407700 + 0.913116i \(0.366331\pi\)
\(660\) 12552.5 0.740314
\(661\) 28141.0 1.65591 0.827955 0.560795i \(-0.189504\pi\)
0.827955 + 0.560795i \(0.189504\pi\)
\(662\) −11026.9 −0.647390
\(663\) −29191.5 −1.70996
\(664\) −6013.01 −0.351431
\(665\) −260.317 −0.0151800
\(666\) 18685.2 1.08714
\(667\) −4558.91 −0.264650
\(668\) 3082.96 0.178568
\(669\) 19739.4 1.14076
\(670\) −289.105 −0.0166703
\(671\) −30490.9 −1.75423
\(672\) −7337.27 −0.421193
\(673\) 933.716 0.0534801 0.0267400 0.999642i \(-0.491487\pi\)
0.0267400 + 0.999642i \(0.491487\pi\)
\(674\) −15838.7 −0.905171
\(675\) 6191.04 0.353027
\(676\) −3544.03 −0.201640
\(677\) −10766.7 −0.611220 −0.305610 0.952157i \(-0.598860\pi\)
−0.305610 + 0.952157i \(0.598860\pi\)
\(678\) 1265.95 0.0717085
\(679\) −4157.34 −0.234969
\(680\) 6269.61 0.353571
\(681\) −52320.8 −2.94411
\(682\) 860.326 0.0483044
\(683\) −6057.52 −0.339363 −0.169681 0.985499i \(-0.554274\pi\)
−0.169681 + 0.985499i \(0.554274\pi\)
\(684\) −3775.43 −0.211048
\(685\) −15796.3 −0.881088
\(686\) 4315.27 0.240172
\(687\) −62038.1 −3.44527
\(688\) −2599.80 −0.144065
\(689\) 19805.7 1.09512
\(690\) −1548.57 −0.0854389
\(691\) 17012.4 0.936586 0.468293 0.883573i \(-0.344869\pi\)
0.468293 + 0.883573i \(0.344869\pi\)
\(692\) −12283.1 −0.674760
\(693\) 11370.0 0.623248
\(694\) 5181.85 0.283430
\(695\) −5477.97 −0.298980
\(696\) 36762.9 2.00215
\(697\) 24007.4 1.30466
\(698\) −7192.83 −0.390047
\(699\) −38574.0 −2.08727
\(700\) −625.626 −0.0337806
\(701\) 11722.7 0.631613 0.315806 0.948824i \(-0.397725\pi\)
0.315806 + 0.948824i \(0.397725\pi\)
\(702\) −19590.8 −1.05329
\(703\) −2762.87 −0.148227
\(704\) 7503.05 0.401679
\(705\) −23960.0 −1.27998
\(706\) −4560.79 −0.243127
\(707\) −1473.20 −0.0783667
\(708\) 43030.8 2.28417
\(709\) −6613.52 −0.350319 −0.175159 0.984540i \(-0.556044\pi\)
−0.175159 + 0.984540i \(0.556044\pi\)
\(710\) 8276.67 0.437490
\(711\) 34984.3 1.84531
\(712\) −5616.29 −0.295617
\(713\) 275.221 0.0144560
\(714\) 3561.08 0.186653
\(715\) −12772.8 −0.668080
\(716\) 16323.5 0.852006
\(717\) 21812.0 1.13610
\(718\) 16047.4 0.834102
\(719\) 4633.47 0.240333 0.120166 0.992754i \(-0.461657\pi\)
0.120166 + 0.992754i \(0.461657\pi\)
\(720\) −4225.07 −0.218693
\(721\) −1369.89 −0.0707594
\(722\) 10019.3 0.516455
\(723\) 38318.4 1.97106
\(724\) −1642.23 −0.0842995
\(725\) 4955.34 0.253844
\(726\) −13340.1 −0.681950
\(727\) −29659.7 −1.51309 −0.756547 0.653939i \(-0.773115\pi\)
−0.756547 + 0.653939i \(0.773115\pi\)
\(728\) 4722.88 0.240442
\(729\) −18697.2 −0.949914
\(730\) −3328.75 −0.168771
\(731\) 10219.2 0.517062
\(732\) 32971.1 1.66482
\(733\) 7552.02 0.380546 0.190273 0.981731i \(-0.439063\pi\)
0.190273 + 0.981731i \(0.439063\pi\)
\(734\) 16266.9 0.818013
\(735\) 14629.2 0.734160
\(736\) 4314.26 0.216068
\(737\) −1867.13 −0.0933198
\(738\) 31964.4 1.59434
\(739\) 27931.7 1.39037 0.695187 0.718829i \(-0.255322\pi\)
0.695187 + 0.718829i \(0.255322\pi\)
\(740\) −6640.06 −0.329856
\(741\) 5746.97 0.284913
\(742\) −2416.11 −0.119539
\(743\) −32808.3 −1.61995 −0.809974 0.586466i \(-0.800519\pi\)
−0.809974 + 0.586466i \(0.800519\pi\)
\(744\) −2219.37 −0.109363
\(745\) −11138.0 −0.547738
\(746\) −146.266 −0.00717852
\(747\) −15928.1 −0.780160
\(748\) 16972.9 0.829666
\(749\) 458.730 0.0223787
\(750\) 1683.22 0.0819502
\(751\) 1086.34 0.0527842 0.0263921 0.999652i \(-0.491598\pi\)
0.0263921 + 0.999652i \(0.491598\pi\)
\(752\) 8242.00 0.399674
\(753\) −43456.9 −2.10313
\(754\) −15680.5 −0.757363
\(755\) 4861.83 0.234358
\(756\) −6197.24 −0.298137
\(757\) −2008.83 −0.0964492 −0.0482246 0.998837i \(-0.515356\pi\)
−0.0482246 + 0.998837i \(0.515356\pi\)
\(758\) 13159.5 0.630573
\(759\) −10001.1 −0.478285
\(760\) −1234.31 −0.0589119
\(761\) 37676.6 1.79471 0.897356 0.441308i \(-0.145485\pi\)
0.897356 + 0.441308i \(0.145485\pi\)
\(762\) −3853.44 −0.183196
\(763\) −8248.71 −0.391380
\(764\) 15491.6 0.733594
\(765\) 16607.8 0.784911
\(766\) −11374.0 −0.536501
\(767\) −43786.0 −2.06130
\(768\) −28320.3 −1.33063
\(769\) −20834.3 −0.976988 −0.488494 0.872567i \(-0.662454\pi\)
−0.488494 + 0.872567i \(0.662454\pi\)
\(770\) 1558.17 0.0729252
\(771\) 47911.5 2.23799
\(772\) −12547.7 −0.584975
\(773\) 25149.2 1.17019 0.585094 0.810965i \(-0.301058\pi\)
0.585094 + 0.810965i \(0.301058\pi\)
\(774\) 13606.3 0.631871
\(775\) −299.153 −0.0138657
\(776\) −19712.2 −0.911892
\(777\) −8997.41 −0.415419
\(778\) 738.869 0.0340485
\(779\) −4726.37 −0.217381
\(780\) 13811.8 0.634028
\(781\) 53453.4 2.44906
\(782\) −2093.89 −0.0957509
\(783\) 49085.9 2.24034
\(784\) −5032.30 −0.229241
\(785\) 14363.6 0.653067
\(786\) −16512.4 −0.749333
\(787\) 30570.6 1.38465 0.692327 0.721584i \(-0.256585\pi\)
0.692327 + 0.721584i \(0.256585\pi\)
\(788\) −26352.8 −1.19134
\(789\) −9431.06 −0.425544
\(790\) 4794.31 0.215916
\(791\) −407.491 −0.0183169
\(792\) 53911.5 2.41876
\(793\) −33549.7 −1.50238
\(794\) 12698.5 0.567573
\(795\) −16856.4 −0.751993
\(796\) 22184.0 0.987801
\(797\) 1413.52 0.0628225 0.0314112 0.999507i \(-0.490000\pi\)
0.0314112 + 0.999507i \(0.490000\pi\)
\(798\) −701.075 −0.0311000
\(799\) −32397.5 −1.43447
\(800\) −4689.42 −0.207245
\(801\) −14877.2 −0.656256
\(802\) 11970.3 0.527040
\(803\) −21498.2 −0.944774
\(804\) 2019.01 0.0885633
\(805\) 498.462 0.0218242
\(806\) 946.633 0.0413694
\(807\) −29113.8 −1.26996
\(808\) −6985.24 −0.304133
\(809\) −13022.6 −0.565946 −0.282973 0.959128i \(-0.591321\pi\)
−0.282973 + 0.959128i \(0.591321\pi\)
\(810\) 5706.88 0.247555
\(811\) 6693.81 0.289829 0.144915 0.989444i \(-0.453709\pi\)
0.144915 + 0.989444i \(0.453709\pi\)
\(812\) −4960.30 −0.214375
\(813\) 22880.3 0.987021
\(814\) 16537.5 0.712089
\(815\) −7290.22 −0.313332
\(816\) −8546.22 −0.366639
\(817\) −2011.88 −0.0861527
\(818\) 3117.44 0.133250
\(819\) 12510.6 0.533769
\(820\) −11359.0 −0.483748
\(821\) 33631.6 1.42966 0.714830 0.699299i \(-0.246504\pi\)
0.714830 + 0.699299i \(0.246504\pi\)
\(822\) −42541.9 −1.80513
\(823\) 6220.84 0.263481 0.131740 0.991284i \(-0.457943\pi\)
0.131740 + 0.991284i \(0.457943\pi\)
\(824\) −6495.42 −0.274610
\(825\) 10870.8 0.458755
\(826\) 5341.47 0.225004
\(827\) −794.732 −0.0334166 −0.0167083 0.999860i \(-0.505319\pi\)
−0.0167083 + 0.999860i \(0.505319\pi\)
\(828\) 7229.28 0.303424
\(829\) −25334.6 −1.06141 −0.530703 0.847558i \(-0.678072\pi\)
−0.530703 + 0.847558i \(0.678072\pi\)
\(830\) −2182.82 −0.0912852
\(831\) −58237.4 −2.43109
\(832\) 8255.75 0.344010
\(833\) 19780.9 0.822769
\(834\) −14753.0 −0.612537
\(835\) 2669.92 0.110654
\(836\) −3341.48 −0.138239
\(837\) −2963.31 −0.122374
\(838\) −11971.0 −0.493476
\(839\) 5311.49 0.218561 0.109281 0.994011i \(-0.465145\pi\)
0.109281 + 0.994011i \(0.465145\pi\)
\(840\) −4019.58 −0.165106
\(841\) 14899.6 0.610913
\(842\) −8032.07 −0.328745
\(843\) −67995.2 −2.77803
\(844\) −3920.90 −0.159909
\(845\) −3069.21 −0.124952
\(846\) −43135.3 −1.75298
\(847\) 4293.98 0.174195
\(848\) 5798.41 0.234809
\(849\) 41372.5 1.67244
\(850\) 2275.96 0.0918411
\(851\) 5290.41 0.213106
\(852\) −57801.4 −2.32423
\(853\) 12483.5 0.501086 0.250543 0.968105i \(-0.419391\pi\)
0.250543 + 0.968105i \(0.419391\pi\)
\(854\) 4092.75 0.163994
\(855\) −3269.61 −0.130782
\(856\) 2175.09 0.0868493
\(857\) −27914.8 −1.11266 −0.556331 0.830961i \(-0.687791\pi\)
−0.556331 + 0.830961i \(0.687791\pi\)
\(858\) −34399.3 −1.36873
\(859\) −25257.2 −1.00322 −0.501610 0.865094i \(-0.667259\pi\)
−0.501610 + 0.865094i \(0.667259\pi\)
\(860\) −4835.19 −0.191719
\(861\) −15391.7 −0.609229
\(862\) −19260.8 −0.761049
\(863\) −23774.9 −0.937784 −0.468892 0.883255i \(-0.655347\pi\)
−0.468892 + 0.883255i \(0.655347\pi\)
\(864\) −46451.8 −1.82908
\(865\) −10637.5 −0.418133
\(866\) −7407.97 −0.290685
\(867\) −10743.9 −0.420856
\(868\) 299.453 0.0117098
\(869\) 30963.2 1.20869
\(870\) 13345.5 0.520063
\(871\) −2054.44 −0.0799220
\(872\) −39111.7 −1.51891
\(873\) −52216.6 −2.02436
\(874\) 412.227 0.0159540
\(875\) −541.806 −0.0209330
\(876\) 23246.8 0.896619
\(877\) 11398.5 0.438883 0.219441 0.975626i \(-0.429577\pi\)
0.219441 + 0.975626i \(0.429577\pi\)
\(878\) −9272.94 −0.356431
\(879\) 62178.9 2.38594
\(880\) −3739.44 −0.143246
\(881\) 28842.1 1.10297 0.551484 0.834186i \(-0.314062\pi\)
0.551484 + 0.834186i \(0.314062\pi\)
\(882\) 26337.0 1.00546
\(883\) −26840.4 −1.02293 −0.511467 0.859303i \(-0.670898\pi\)
−0.511467 + 0.859303i \(0.670898\pi\)
\(884\) 18675.6 0.710552
\(885\) 37265.7 1.41545
\(886\) 17582.3 0.666693
\(887\) 9384.94 0.355260 0.177630 0.984097i \(-0.443157\pi\)
0.177630 + 0.984097i \(0.443157\pi\)
\(888\) −42661.7 −1.61220
\(889\) 1240.37 0.0467949
\(890\) −2038.80 −0.0767874
\(891\) 36856.9 1.38581
\(892\) −12628.5 −0.474029
\(893\) 6378.15 0.239011
\(894\) −29996.4 −1.12218
\(895\) 14136.5 0.527968
\(896\) 5497.21 0.204965
\(897\) −11004.4 −0.409618
\(898\) −19976.2 −0.742332
\(899\) −2371.85 −0.0879928
\(900\) −7857.92 −0.291034
\(901\) −22792.3 −0.842754
\(902\) 28290.4 1.04431
\(903\) −6551.78 −0.241450
\(904\) −1932.14 −0.0710863
\(905\) −1422.21 −0.0522384
\(906\) 13093.7 0.480141
\(907\) 36100.4 1.32160 0.660802 0.750560i \(-0.270216\pi\)
0.660802 + 0.750560i \(0.270216\pi\)
\(908\) 33472.9 1.22339
\(909\) −18503.5 −0.675162
\(910\) 1714.48 0.0624554
\(911\) −21644.8 −0.787183 −0.393591 0.919285i \(-0.628768\pi\)
−0.393591 + 0.919285i \(0.628768\pi\)
\(912\) 1682.51 0.0610893
\(913\) −14097.3 −0.511012
\(914\) −22671.9 −0.820482
\(915\) 28553.7 1.03165
\(916\) 39689.6 1.43164
\(917\) 5315.10 0.191407
\(918\) 22544.9 0.810560
\(919\) −15016.8 −0.539018 −0.269509 0.962998i \(-0.586861\pi\)
−0.269509 + 0.962998i \(0.586861\pi\)
\(920\) 2363.48 0.0846975
\(921\) −13186.6 −0.471785
\(922\) −9084.07 −0.324477
\(923\) 58815.8 2.09745
\(924\) −10881.7 −0.387425
\(925\) −5750.45 −0.204404
\(926\) −21852.7 −0.775513
\(927\) −17206.0 −0.609621
\(928\) −37180.2 −1.31519
\(929\) 7097.82 0.250670 0.125335 0.992115i \(-0.460000\pi\)
0.125335 + 0.992115i \(0.460000\pi\)
\(930\) −805.667 −0.0284074
\(931\) −3894.29 −0.137089
\(932\) 24678.2 0.867341
\(933\) −6771.00 −0.237591
\(934\) −3332.26 −0.116740
\(935\) 14698.9 0.514124
\(936\) 59319.8 2.07151
\(937\) 8007.69 0.279189 0.139594 0.990209i \(-0.455420\pi\)
0.139594 + 0.990209i \(0.455420\pi\)
\(938\) 250.622 0.00872399
\(939\) 77754.7 2.70227
\(940\) 15328.7 0.531881
\(941\) 12382.9 0.428980 0.214490 0.976726i \(-0.431191\pi\)
0.214490 + 0.976726i \(0.431191\pi\)
\(942\) 38683.4 1.33797
\(943\) 9050.18 0.312528
\(944\) −12819.0 −0.441973
\(945\) −5366.95 −0.184748
\(946\) 12042.4 0.413881
\(947\) −14760.8 −0.506506 −0.253253 0.967400i \(-0.581501\pi\)
−0.253253 + 0.967400i \(0.581501\pi\)
\(948\) −33481.8 −1.14709
\(949\) −23654.8 −0.809134
\(950\) −448.073 −0.0153025
\(951\) 3216.07 0.109662
\(952\) −5435.06 −0.185033
\(953\) 37251.9 1.26622 0.633110 0.774062i \(-0.281778\pi\)
0.633110 + 0.774062i \(0.281778\pi\)
\(954\) −30346.6 −1.02988
\(955\) 13416.1 0.454591
\(956\) −13954.5 −0.472092
\(957\) 86189.5 2.91130
\(958\) 228.280 0.00769875
\(959\) 13693.7 0.461096
\(960\) −7026.36 −0.236224
\(961\) −29647.8 −0.995194
\(962\) 18196.6 0.609856
\(963\) 5761.69 0.192801
\(964\) −24514.7 −0.819050
\(965\) −10866.6 −0.362495
\(966\) 1342.44 0.0447124
\(967\) 13383.4 0.445069 0.222534 0.974925i \(-0.428567\pi\)
0.222534 + 0.974925i \(0.428567\pi\)
\(968\) 20360.1 0.676032
\(969\) −6613.57 −0.219256
\(970\) −7155.85 −0.236867
\(971\) −19630.5 −0.648788 −0.324394 0.945922i \(-0.605160\pi\)
−0.324394 + 0.945922i \(0.605160\pi\)
\(972\) −1251.26 −0.0412905
\(973\) 4748.80 0.156464
\(974\) −11964.5 −0.393601
\(975\) 11961.3 0.392892
\(976\) −9822.18 −0.322131
\(977\) 34825.7 1.14040 0.570200 0.821506i \(-0.306866\pi\)
0.570200 + 0.821506i \(0.306866\pi\)
\(978\) −19633.7 −0.641939
\(979\) −13167.2 −0.429854
\(980\) −9359.23 −0.305071
\(981\) −103605. −3.37190
\(982\) −24401.2 −0.792948
\(983\) 39998.8 1.29783 0.648913 0.760862i \(-0.275224\pi\)
0.648913 + 0.760862i \(0.275224\pi\)
\(984\) −72980.3 −2.36436
\(985\) −22822.1 −0.738247
\(986\) 18045.1 0.582832
\(987\) 20770.7 0.669848
\(988\) −3676.69 −0.118392
\(989\) 3852.40 0.123862
\(990\) 19570.7 0.628281
\(991\) −22462.6 −0.720029 −0.360015 0.932947i \(-0.617228\pi\)
−0.360015 + 0.932947i \(0.617228\pi\)
\(992\) 2244.57 0.0718398
\(993\) 66690.7 2.13128
\(994\) −7174.97 −0.228950
\(995\) 19211.8 0.612116
\(996\) 15244.0 0.484965
\(997\) 7603.55 0.241532 0.120766 0.992681i \(-0.461465\pi\)
0.120766 + 0.992681i \(0.461465\pi\)
\(998\) 21807.8 0.691697
\(999\) −56962.0 −1.80400
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 115.4.a.e.1.2 5
3.2 odd 2 1035.4.a.k.1.4 5
4.3 odd 2 1840.4.a.n.1.1 5
5.2 odd 4 575.4.b.i.24.4 10
5.3 odd 4 575.4.b.i.24.7 10
5.4 even 2 575.4.a.j.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.2 5 1.1 even 1 trivial
575.4.a.j.1.4 5 5.4 even 2
575.4.b.i.24.4 10 5.2 odd 4
575.4.b.i.24.7 10 5.3 odd 4
1035.4.a.k.1.4 5 3.2 odd 2
1840.4.a.n.1.1 5 4.3 odd 2