Properties

Label 435.4.a.g.1.3
Level $435$
Weight $4$
Character 435.1
Self dual yes
Analytic conductor $25.666$
Analytic rank $1$
Dimension $6$
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] = [435,4,Mod(1,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("435.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 435.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: \(25.6658308525\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 31x^{4} + 9x^{3} + 230x^{2} + 32x - 128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.691666\) of defining polynomial
Character \(\chi\) \(=\) 435.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.691666 q^{2} -3.00000 q^{3} -7.52160 q^{4} +5.00000 q^{5} +2.07500 q^{6} -14.5720 q^{7} +10.7358 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-0.691666 q^{2} -3.00000 q^{3} -7.52160 q^{4} +5.00000 q^{5} +2.07500 q^{6} -14.5720 q^{7} +10.7358 q^{8} +9.00000 q^{9} -3.45833 q^{10} +36.8326 q^{11} +22.5648 q^{12} -23.9199 q^{13} +10.0790 q^{14} -15.0000 q^{15} +52.7472 q^{16} +62.8914 q^{17} -6.22499 q^{18} -94.1599 q^{19} -37.6080 q^{20} +43.7161 q^{21} -25.4759 q^{22} +185.410 q^{23} -32.2073 q^{24} +25.0000 q^{25} +16.5446 q^{26} -27.0000 q^{27} +109.605 q^{28} -29.0000 q^{29} +10.3750 q^{30} -104.971 q^{31} -122.370 q^{32} -110.498 q^{33} -43.4999 q^{34} -72.8602 q^{35} -67.6944 q^{36} -259.799 q^{37} +65.1272 q^{38} +71.7597 q^{39} +53.6788 q^{40} +11.8069 q^{41} -30.2369 q^{42} -61.0501 q^{43} -277.040 q^{44} +45.0000 q^{45} -128.242 q^{46} +281.426 q^{47} -158.242 q^{48} -130.656 q^{49} -17.2916 q^{50} -188.674 q^{51} +179.916 q^{52} -170.846 q^{53} +18.6750 q^{54} +184.163 q^{55} -156.442 q^{56} +282.480 q^{57} +20.0583 q^{58} -636.987 q^{59} +112.824 q^{60} -379.603 q^{61} +72.6051 q^{62} -131.148 q^{63} -337.339 q^{64} -119.600 q^{65} +76.4276 q^{66} -623.333 q^{67} -473.044 q^{68} -556.231 q^{69} +50.3949 q^{70} +298.898 q^{71} +96.6218 q^{72} -524.124 q^{73} +179.694 q^{74} -75.0000 q^{75} +708.233 q^{76} -536.726 q^{77} -49.6337 q^{78} +563.264 q^{79} +263.736 q^{80} +81.0000 q^{81} -8.16645 q^{82} -885.342 q^{83} -328.815 q^{84} +314.457 q^{85} +42.2263 q^{86} +87.0000 q^{87} +395.426 q^{88} -447.652 q^{89} -31.1250 q^{90} +348.562 q^{91} -1394.58 q^{92} +314.914 q^{93} -194.653 q^{94} -470.799 q^{95} +367.109 q^{96} -558.895 q^{97} +90.3702 q^{98} +331.493 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - 18 q^{3} + 15 q^{4} + 30 q^{5} + 3 q^{6} + 23 q^{7} - 51 q^{8} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{2} - 18 q^{3} + 15 q^{4} + 30 q^{5} + 3 q^{6} + 23 q^{7} - 51 q^{8} + 54 q^{9} - 5 q^{10} - 111 q^{11} - 45 q^{12} - 83 q^{13} - 102 q^{14} - 90 q^{15} - 37 q^{16} - 35 q^{17} - 9 q^{18} - 76 q^{19} + 75 q^{20} - 69 q^{21} + 66 q^{22} + 166 q^{23} + 153 q^{24} + 150 q^{25} - 282 q^{26} - 162 q^{27} + 368 q^{28} - 174 q^{29} + 15 q^{30} - 164 q^{31} - 259 q^{32} + 333 q^{33} + 512 q^{34} + 115 q^{35} + 135 q^{36} - 538 q^{37} + 6 q^{38} + 249 q^{39} - 255 q^{40} - 1250 q^{41} + 306 q^{42} - 80 q^{43} - 1004 q^{44} + 270 q^{45} - 1190 q^{46} - 491 q^{47} + 111 q^{48} - 969 q^{49} - 25 q^{50} + 105 q^{51} - 336 q^{52} - 356 q^{53} + 27 q^{54} - 555 q^{55} - 1260 q^{56} + 228 q^{57} + 29 q^{58} - 646 q^{59} - 225 q^{60} - 1476 q^{61} - 4 q^{62} + 207 q^{63} - 845 q^{64} - 415 q^{65} - 198 q^{66} + 87 q^{67} - 558 q^{68} - 498 q^{69} - 510 q^{70} - 1312 q^{71} - 459 q^{72} - 1142 q^{73} - 2270 q^{74} - 450 q^{75} - 1594 q^{76} - 2561 q^{77} + 846 q^{78} - 142 q^{79} - 185 q^{80} + 486 q^{81} - 1544 q^{82} + 346 q^{83} - 1104 q^{84} - 175 q^{85} - 2056 q^{86} + 522 q^{87} + 1400 q^{88} - 2827 q^{89} - 45 q^{90} + 1379 q^{91} - 1434 q^{92} + 492 q^{93} + 978 q^{94} - 380 q^{95} + 777 q^{96} - 4 q^{97} - 1903 q^{98} - 999 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.691666 −0.244541 −0.122270 0.992497i \(-0.539018\pi\)
−0.122270 + 0.992497i \(0.539018\pi\)
\(3\) −3.00000 −0.577350
\(4\) −7.52160 −0.940200
\(5\) 5.00000 0.447214
\(6\) 2.07500 0.141186
\(7\) −14.5720 −0.786816 −0.393408 0.919364i \(-0.628704\pi\)
−0.393408 + 0.919364i \(0.628704\pi\)
\(8\) 10.7358 0.474458
\(9\) 9.00000 0.333333
\(10\) −3.45833 −0.109362
\(11\) 36.8326 1.00959 0.504793 0.863240i \(-0.331569\pi\)
0.504793 + 0.863240i \(0.331569\pi\)
\(12\) 22.5648 0.542825
\(13\) −23.9199 −0.510322 −0.255161 0.966899i \(-0.582128\pi\)
−0.255161 + 0.966899i \(0.582128\pi\)
\(14\) 10.0790 0.192409
\(15\) −15.0000 −0.258199
\(16\) 52.7472 0.824175
\(17\) 62.8914 0.897260 0.448630 0.893718i \(-0.351912\pi\)
0.448630 + 0.893718i \(0.351912\pi\)
\(18\) −6.22499 −0.0815136
\(19\) −94.1599 −1.13693 −0.568467 0.822706i \(-0.692463\pi\)
−0.568467 + 0.822706i \(0.692463\pi\)
\(20\) −37.6080 −0.420470
\(21\) 43.7161 0.454268
\(22\) −25.4759 −0.246885
\(23\) 185.410 1.68090 0.840450 0.541889i \(-0.182291\pi\)
0.840450 + 0.541889i \(0.182291\pi\)
\(24\) −32.2073 −0.273928
\(25\) 25.0000 0.200000
\(26\) 16.5446 0.124795
\(27\) −27.0000 −0.192450
\(28\) 109.605 0.739764
\(29\) −29.0000 −0.185695
\(30\) 10.3750 0.0631402
\(31\) −104.971 −0.608174 −0.304087 0.952644i \(-0.598351\pi\)
−0.304087 + 0.952644i \(0.598351\pi\)
\(32\) −122.370 −0.676003
\(33\) −110.498 −0.582885
\(34\) −43.4999 −0.219417
\(35\) −72.8602 −0.351875
\(36\) −67.6944 −0.313400
\(37\) −259.799 −1.15435 −0.577173 0.816622i \(-0.695844\pi\)
−0.577173 + 0.816622i \(0.695844\pi\)
\(38\) 65.1272 0.278027
\(39\) 71.7597 0.294635
\(40\) 53.6788 0.212184
\(41\) 11.8069 0.0449740 0.0224870 0.999747i \(-0.492842\pi\)
0.0224870 + 0.999747i \(0.492842\pi\)
\(42\) −30.2369 −0.111087
\(43\) −61.0501 −0.216513 −0.108256 0.994123i \(-0.534527\pi\)
−0.108256 + 0.994123i \(0.534527\pi\)
\(44\) −277.040 −0.949213
\(45\) 45.0000 0.149071
\(46\) −128.242 −0.411049
\(47\) 281.426 0.873409 0.436705 0.899605i \(-0.356145\pi\)
0.436705 + 0.899605i \(0.356145\pi\)
\(48\) −158.242 −0.475838
\(49\) −130.656 −0.380921
\(50\) −17.2916 −0.0489082
\(51\) −188.674 −0.518033
\(52\) 179.916 0.479805
\(53\) −170.846 −0.442782 −0.221391 0.975185i \(-0.571060\pi\)
−0.221391 + 0.975185i \(0.571060\pi\)
\(54\) 18.6750 0.0470619
\(55\) 184.163 0.451501
\(56\) −156.442 −0.373311
\(57\) 282.480 0.656409
\(58\) 20.0583 0.0454101
\(59\) −636.987 −1.40557 −0.702785 0.711402i \(-0.748060\pi\)
−0.702785 + 0.711402i \(0.748060\pi\)
\(60\) 112.824 0.242759
\(61\) −379.603 −0.796774 −0.398387 0.917218i \(-0.630430\pi\)
−0.398387 + 0.917218i \(0.630430\pi\)
\(62\) 72.6051 0.148723
\(63\) −131.148 −0.262272
\(64\) −337.339 −0.658865
\(65\) −119.600 −0.228223
\(66\) 76.4276 0.142539
\(67\) −623.333 −1.13660 −0.568300 0.822821i \(-0.692399\pi\)
−0.568300 + 0.822821i \(0.692399\pi\)
\(68\) −473.044 −0.843603
\(69\) −556.231 −0.970468
\(70\) 50.3949 0.0860477
\(71\) 298.898 0.499615 0.249808 0.968295i \(-0.419633\pi\)
0.249808 + 0.968295i \(0.419633\pi\)
\(72\) 96.6218 0.158153
\(73\) −524.124 −0.840329 −0.420165 0.907448i \(-0.638028\pi\)
−0.420165 + 0.907448i \(0.638028\pi\)
\(74\) 179.694 0.282284
\(75\) −75.0000 −0.115470
\(76\) 708.233 1.06895
\(77\) −536.726 −0.794358
\(78\) −49.6337 −0.0720502
\(79\) 563.264 0.802179 0.401090 0.916039i \(-0.368632\pi\)
0.401090 + 0.916039i \(0.368632\pi\)
\(80\) 263.736 0.368582
\(81\) 81.0000 0.111111
\(82\) −8.16645 −0.0109980
\(83\) −885.342 −1.17083 −0.585415 0.810734i \(-0.699068\pi\)
−0.585415 + 0.810734i \(0.699068\pi\)
\(84\) −328.815 −0.427103
\(85\) 314.457 0.401267
\(86\) 42.2263 0.0529462
\(87\) 87.0000 0.107211
\(88\) 395.426 0.479006
\(89\) −447.652 −0.533157 −0.266579 0.963813i \(-0.585893\pi\)
−0.266579 + 0.963813i \(0.585893\pi\)
\(90\) −31.1250 −0.0364540
\(91\) 348.562 0.401529
\(92\) −1394.58 −1.58038
\(93\) 314.914 0.351130
\(94\) −194.653 −0.213584
\(95\) −470.799 −0.508453
\(96\) 367.109 0.390290
\(97\) −558.895 −0.585022 −0.292511 0.956262i \(-0.594491\pi\)
−0.292511 + 0.956262i \(0.594491\pi\)
\(98\) 90.3702 0.0931507
\(99\) 331.493 0.336529
\(100\) −188.040 −0.188040
\(101\) −1138.78 −1.12191 −0.560956 0.827846i \(-0.689566\pi\)
−0.560956 + 0.827846i \(0.689566\pi\)
\(102\) 130.500 0.126680
\(103\) 1008.31 0.964583 0.482292 0.876011i \(-0.339805\pi\)
0.482292 + 0.876011i \(0.339805\pi\)
\(104\) −256.798 −0.242126
\(105\) 218.580 0.203155
\(106\) 118.168 0.108278
\(107\) 140.506 0.126946 0.0634730 0.997984i \(-0.479782\pi\)
0.0634730 + 0.997984i \(0.479782\pi\)
\(108\) 203.083 0.180942
\(109\) −850.530 −0.747394 −0.373697 0.927551i \(-0.621910\pi\)
−0.373697 + 0.927551i \(0.621910\pi\)
\(110\) −127.379 −0.110410
\(111\) 779.398 0.666461
\(112\) −768.634 −0.648474
\(113\) 1851.23 1.54115 0.770573 0.637352i \(-0.219970\pi\)
0.770573 + 0.637352i \(0.219970\pi\)
\(114\) −195.381 −0.160519
\(115\) 927.051 0.751721
\(116\) 218.126 0.174591
\(117\) −215.279 −0.170107
\(118\) 440.582 0.343719
\(119\) −916.456 −0.705978
\(120\) −161.036 −0.122505
\(121\) 25.6407 0.0192643
\(122\) 262.558 0.194844
\(123\) −35.4208 −0.0259657
\(124\) 789.552 0.571805
\(125\) 125.000 0.0894427
\(126\) 90.7108 0.0641362
\(127\) 903.124 0.631018 0.315509 0.948923i \(-0.397825\pi\)
0.315509 + 0.948923i \(0.397825\pi\)
\(128\) 1212.28 0.837122
\(129\) 183.150 0.125004
\(130\) 82.7229 0.0558098
\(131\) −685.958 −0.457499 −0.228750 0.973485i \(-0.573464\pi\)
−0.228750 + 0.973485i \(0.573464\pi\)
\(132\) 831.120 0.548028
\(133\) 1372.10 0.894558
\(134\) 431.138 0.277945
\(135\) −135.000 −0.0860663
\(136\) 675.187 0.425712
\(137\) −1152.95 −0.718999 −0.359500 0.933145i \(-0.617053\pi\)
−0.359500 + 0.933145i \(0.617053\pi\)
\(138\) 384.726 0.237319
\(139\) −438.465 −0.267555 −0.133777 0.991011i \(-0.542711\pi\)
−0.133777 + 0.991011i \(0.542711\pi\)
\(140\) 548.025 0.330833
\(141\) −844.279 −0.504263
\(142\) −206.738 −0.122176
\(143\) −881.032 −0.515214
\(144\) 474.725 0.274725
\(145\) −145.000 −0.0830455
\(146\) 362.518 0.205495
\(147\) 391.968 0.219925
\(148\) 1954.11 1.08532
\(149\) −405.570 −0.222991 −0.111495 0.993765i \(-0.535564\pi\)
−0.111495 + 0.993765i \(0.535564\pi\)
\(150\) 51.8749 0.0282371
\(151\) −78.9559 −0.0425519 −0.0212760 0.999774i \(-0.506773\pi\)
−0.0212760 + 0.999774i \(0.506773\pi\)
\(152\) −1010.88 −0.539428
\(153\) 566.023 0.299087
\(154\) 371.235 0.194253
\(155\) −524.856 −0.271984
\(156\) −539.748 −0.277015
\(157\) 314.869 0.160059 0.0800296 0.996792i \(-0.474499\pi\)
0.0800296 + 0.996792i \(0.474499\pi\)
\(158\) −389.590 −0.196166
\(159\) 512.537 0.255640
\(160\) −611.848 −0.302318
\(161\) −2701.80 −1.32256
\(162\) −56.0249 −0.0271712
\(163\) 816.812 0.392501 0.196251 0.980554i \(-0.437123\pi\)
0.196251 + 0.980554i \(0.437123\pi\)
\(164\) −88.8070 −0.0422845
\(165\) −552.489 −0.260674
\(166\) 612.361 0.286316
\(167\) −1364.36 −0.632200 −0.316100 0.948726i \(-0.602374\pi\)
−0.316100 + 0.948726i \(0.602374\pi\)
\(168\) 469.325 0.215531
\(169\) −1624.84 −0.739571
\(170\) −217.499 −0.0981261
\(171\) −847.439 −0.378978
\(172\) 459.194 0.203565
\(173\) 430.857 0.189350 0.0946748 0.995508i \(-0.469819\pi\)
0.0946748 + 0.995508i \(0.469819\pi\)
\(174\) −60.1749 −0.0262175
\(175\) −364.301 −0.157363
\(176\) 1942.82 0.832076
\(177\) 1910.96 0.811506
\(178\) 309.625 0.130379
\(179\) −3192.64 −1.33312 −0.666561 0.745451i \(-0.732234\pi\)
−0.666561 + 0.745451i \(0.732234\pi\)
\(180\) −338.472 −0.140157
\(181\) −621.160 −0.255085 −0.127543 0.991833i \(-0.540709\pi\)
−0.127543 + 0.991833i \(0.540709\pi\)
\(182\) −241.088 −0.0981903
\(183\) 1138.81 0.460017
\(184\) 1990.52 0.797517
\(185\) −1299.00 −0.516239
\(186\) −217.815 −0.0858655
\(187\) 2316.46 0.905861
\(188\) −2116.78 −0.821179
\(189\) 393.445 0.151423
\(190\) 325.636 0.124337
\(191\) 2943.37 1.11505 0.557526 0.830160i \(-0.311751\pi\)
0.557526 + 0.830160i \(0.311751\pi\)
\(192\) 1012.02 0.380396
\(193\) 3259.79 1.21578 0.607889 0.794022i \(-0.292017\pi\)
0.607889 + 0.794022i \(0.292017\pi\)
\(194\) 386.568 0.143062
\(195\) 358.799 0.131765
\(196\) 982.741 0.358142
\(197\) 2372.35 0.857983 0.428992 0.903308i \(-0.358869\pi\)
0.428992 + 0.903308i \(0.358869\pi\)
\(198\) −229.283 −0.0822950
\(199\) −1828.75 −0.651441 −0.325721 0.945466i \(-0.605607\pi\)
−0.325721 + 0.945466i \(0.605607\pi\)
\(200\) 268.394 0.0948916
\(201\) 1870.00 0.656216
\(202\) 787.657 0.274353
\(203\) 422.589 0.146108
\(204\) 1419.13 0.487055
\(205\) 59.0347 0.0201130
\(206\) −697.416 −0.235880
\(207\) 1668.69 0.560300
\(208\) −1261.71 −0.420595
\(209\) −3468.15 −1.14783
\(210\) −151.185 −0.0496797
\(211\) 4223.45 1.37798 0.688991 0.724770i \(-0.258054\pi\)
0.688991 + 0.724770i \(0.258054\pi\)
\(212\) 1285.03 0.416304
\(213\) −896.694 −0.288453
\(214\) −97.1832 −0.0310435
\(215\) −305.251 −0.0968275
\(216\) −289.865 −0.0913095
\(217\) 1529.65 0.478521
\(218\) 588.282 0.182768
\(219\) 1572.37 0.485164
\(220\) −1385.20 −0.424501
\(221\) −1504.36 −0.457891
\(222\) −539.083 −0.162977
\(223\) −327.670 −0.0983964 −0.0491982 0.998789i \(-0.515667\pi\)
−0.0491982 + 0.998789i \(0.515667\pi\)
\(224\) 1783.17 0.531889
\(225\) 225.000 0.0666667
\(226\) −1280.44 −0.376873
\(227\) 3662.12 1.07076 0.535382 0.844610i \(-0.320168\pi\)
0.535382 + 0.844610i \(0.320168\pi\)
\(228\) −2124.70 −0.617156
\(229\) −6562.57 −1.89374 −0.946871 0.321613i \(-0.895775\pi\)
−0.946871 + 0.321613i \(0.895775\pi\)
\(230\) −641.210 −0.183827
\(231\) 1610.18 0.458623
\(232\) −311.337 −0.0881046
\(233\) 2490.28 0.700189 0.350094 0.936714i \(-0.386150\pi\)
0.350094 + 0.936714i \(0.386150\pi\)
\(234\) 148.901 0.0415982
\(235\) 1407.13 0.390601
\(236\) 4791.16 1.32152
\(237\) −1689.79 −0.463138
\(238\) 633.881 0.172640
\(239\) −3584.40 −0.970106 −0.485053 0.874485i \(-0.661200\pi\)
−0.485053 + 0.874485i \(0.661200\pi\)
\(240\) −791.208 −0.212801
\(241\) −6543.94 −1.74910 −0.874548 0.484938i \(-0.838842\pi\)
−0.874548 + 0.484938i \(0.838842\pi\)
\(242\) −17.7348 −0.00471090
\(243\) −243.000 −0.0641500
\(244\) 2855.22 0.749126
\(245\) −653.279 −0.170353
\(246\) 24.4994 0.00634968
\(247\) 2252.29 0.580203
\(248\) −1126.95 −0.288553
\(249\) 2656.03 0.675979
\(250\) −86.4582 −0.0218724
\(251\) 528.100 0.132802 0.0664011 0.997793i \(-0.478848\pi\)
0.0664011 + 0.997793i \(0.478848\pi\)
\(252\) 986.445 0.246588
\(253\) 6829.14 1.69701
\(254\) −624.660 −0.154310
\(255\) −943.371 −0.231671
\(256\) 1860.22 0.454155
\(257\) 5020.78 1.21863 0.609314 0.792929i \(-0.291445\pi\)
0.609314 + 0.792929i \(0.291445\pi\)
\(258\) −126.679 −0.0305685
\(259\) 3785.81 0.908257
\(260\) 899.579 0.214575
\(261\) −261.000 −0.0618984
\(262\) 474.453 0.111877
\(263\) −422.212 −0.0989913 −0.0494957 0.998774i \(-0.515761\pi\)
−0.0494957 + 0.998774i \(0.515761\pi\)
\(264\) −1186.28 −0.276554
\(265\) −854.228 −0.198018
\(266\) −949.035 −0.218756
\(267\) 1342.96 0.307818
\(268\) 4688.46 1.06863
\(269\) −1777.19 −0.402815 −0.201407 0.979508i \(-0.564551\pi\)
−0.201407 + 0.979508i \(0.564551\pi\)
\(270\) 93.3749 0.0210467
\(271\) −1465.51 −0.328499 −0.164250 0.986419i \(-0.552520\pi\)
−0.164250 + 0.986419i \(0.552520\pi\)
\(272\) 3317.35 0.739499
\(273\) −1045.68 −0.231823
\(274\) 797.454 0.175825
\(275\) 920.815 0.201917
\(276\) 4183.74 0.912434
\(277\) −6081.50 −1.31914 −0.659570 0.751643i \(-0.729262\pi\)
−0.659570 + 0.751643i \(0.729262\pi\)
\(278\) 303.271 0.0654281
\(279\) −944.742 −0.202725
\(280\) −782.209 −0.166950
\(281\) −1194.50 −0.253587 −0.126794 0.991929i \(-0.540469\pi\)
−0.126794 + 0.991929i \(0.540469\pi\)
\(282\) 583.959 0.123313
\(283\) 3827.53 0.803968 0.401984 0.915647i \(-0.368321\pi\)
0.401984 + 0.915647i \(0.368321\pi\)
\(284\) −2248.19 −0.469738
\(285\) 1412.40 0.293555
\(286\) 609.380 0.125991
\(287\) −172.051 −0.0353862
\(288\) −1101.33 −0.225334
\(289\) −957.668 −0.194925
\(290\) 100.292 0.0203080
\(291\) 1676.68 0.337763
\(292\) 3942.25 0.790077
\(293\) −6821.00 −1.36002 −0.680012 0.733201i \(-0.738025\pi\)
−0.680012 + 0.733201i \(0.738025\pi\)
\(294\) −271.111 −0.0537806
\(295\) −3184.93 −0.628590
\(296\) −2789.14 −0.547688
\(297\) −994.480 −0.194295
\(298\) 280.519 0.0545303
\(299\) −4434.99 −0.857801
\(300\) 564.120 0.108565
\(301\) 889.624 0.170356
\(302\) 54.6111 0.0104057
\(303\) 3416.35 0.647736
\(304\) −4966.67 −0.937034
\(305\) −1898.02 −0.356328
\(306\) −391.499 −0.0731389
\(307\) −7350.74 −1.36654 −0.683272 0.730164i \(-0.739444\pi\)
−0.683272 + 0.730164i \(0.739444\pi\)
\(308\) 4037.04 0.746856
\(309\) −3024.94 −0.556902
\(310\) 363.025 0.0665111
\(311\) 3393.14 0.618673 0.309337 0.950953i \(-0.399893\pi\)
0.309337 + 0.950953i \(0.399893\pi\)
\(312\) 770.395 0.139792
\(313\) −10732.3 −1.93810 −0.969049 0.246867i \(-0.920599\pi\)
−0.969049 + 0.246867i \(0.920599\pi\)
\(314\) −217.784 −0.0391410
\(315\) −655.741 −0.117292
\(316\) −4236.65 −0.754209
\(317\) −8721.17 −1.54520 −0.772602 0.634890i \(-0.781045\pi\)
−0.772602 + 0.634890i \(0.781045\pi\)
\(318\) −354.504 −0.0625145
\(319\) −1068.15 −0.187475
\(320\) −1686.70 −0.294654
\(321\) −421.518 −0.0732923
\(322\) 1868.75 0.323420
\(323\) −5921.85 −1.02013
\(324\) −609.249 −0.104467
\(325\) −597.998 −0.102064
\(326\) −564.961 −0.0959825
\(327\) 2551.59 0.431508
\(328\) 126.756 0.0213383
\(329\) −4100.95 −0.687212
\(330\) 382.138 0.0637454
\(331\) −6093.02 −1.01179 −0.505895 0.862595i \(-0.668838\pi\)
−0.505895 + 0.862595i \(0.668838\pi\)
\(332\) 6659.18 1.10081
\(333\) −2338.20 −0.384782
\(334\) 943.682 0.154599
\(335\) −3116.66 −0.508303
\(336\) 2305.90 0.374397
\(337\) −6306.92 −1.01947 −0.509733 0.860333i \(-0.670256\pi\)
−0.509733 + 0.860333i \(0.670256\pi\)
\(338\) 1123.85 0.180855
\(339\) −5553.70 −0.889781
\(340\) −2365.22 −0.377271
\(341\) −3866.37 −0.614004
\(342\) 586.144 0.0926756
\(343\) 6902.13 1.08653
\(344\) −655.419 −0.102726
\(345\) −2781.15 −0.434007
\(346\) −298.009 −0.0463037
\(347\) 11395.7 1.76298 0.881491 0.472201i \(-0.156540\pi\)
0.881491 + 0.472201i \(0.156540\pi\)
\(348\) −654.379 −0.100800
\(349\) 8835.08 1.35510 0.677552 0.735475i \(-0.263041\pi\)
0.677552 + 0.735475i \(0.263041\pi\)
\(350\) 251.974 0.0384817
\(351\) 645.837 0.0982115
\(352\) −4507.19 −0.682483
\(353\) −3390.75 −0.511251 −0.255626 0.966776i \(-0.582281\pi\)
−0.255626 + 0.966776i \(0.582281\pi\)
\(354\) −1321.75 −0.198446
\(355\) 1494.49 0.223435
\(356\) 3367.06 0.501274
\(357\) 2749.37 0.407597
\(358\) 2208.24 0.326003
\(359\) −5160.27 −0.758632 −0.379316 0.925267i \(-0.623841\pi\)
−0.379316 + 0.925267i \(0.623841\pi\)
\(360\) 483.109 0.0707280
\(361\) 2007.08 0.292620
\(362\) 429.635 0.0623788
\(363\) −76.9222 −0.0111222
\(364\) −2621.74 −0.377518
\(365\) −2620.62 −0.375807
\(366\) −787.675 −0.112493
\(367\) 1110.17 0.157902 0.0789512 0.996878i \(-0.474843\pi\)
0.0789512 + 0.996878i \(0.474843\pi\)
\(368\) 9779.88 1.38536
\(369\) 106.262 0.0149913
\(370\) 898.472 0.126241
\(371\) 2489.57 0.348388
\(372\) −2368.66 −0.330132
\(373\) 5801.22 0.805297 0.402648 0.915355i \(-0.368090\pi\)
0.402648 + 0.915355i \(0.368090\pi\)
\(374\) −1602.21 −0.221520
\(375\) −375.000 −0.0516398
\(376\) 3021.32 0.414396
\(377\) 693.677 0.0947644
\(378\) −272.132 −0.0370290
\(379\) −12664.5 −1.71644 −0.858219 0.513284i \(-0.828429\pi\)
−0.858219 + 0.513284i \(0.828429\pi\)
\(380\) 3541.16 0.478047
\(381\) −2709.37 −0.364318
\(382\) −2035.83 −0.272675
\(383\) −13408.3 −1.78885 −0.894427 0.447215i \(-0.852416\pi\)
−0.894427 + 0.447215i \(0.852416\pi\)
\(384\) −3636.85 −0.483313
\(385\) −2683.63 −0.355248
\(386\) −2254.69 −0.297307
\(387\) −549.451 −0.0721710
\(388\) 4203.78 0.550038
\(389\) −4676.56 −0.609540 −0.304770 0.952426i \(-0.598580\pi\)
−0.304770 + 0.952426i \(0.598580\pi\)
\(390\) −248.169 −0.0322218
\(391\) 11660.7 1.50820
\(392\) −1402.69 −0.180731
\(393\) 2057.87 0.264137
\(394\) −1640.87 −0.209812
\(395\) 2816.32 0.358745
\(396\) −2493.36 −0.316404
\(397\) −5913.71 −0.747608 −0.373804 0.927508i \(-0.621947\pi\)
−0.373804 + 0.927508i \(0.621947\pi\)
\(398\) 1264.89 0.159304
\(399\) −4116.30 −0.516473
\(400\) 1318.68 0.164835
\(401\) 7756.80 0.965975 0.482987 0.875627i \(-0.339552\pi\)
0.482987 + 0.875627i \(0.339552\pi\)
\(402\) −1293.41 −0.160472
\(403\) 2510.90 0.310365
\(404\) 8565.46 1.05482
\(405\) 405.000 0.0496904
\(406\) −292.290 −0.0357294
\(407\) −9569.09 −1.16541
\(408\) −2025.56 −0.245785
\(409\) 7674.37 0.927808 0.463904 0.885886i \(-0.346448\pi\)
0.463904 + 0.885886i \(0.346448\pi\)
\(410\) −40.8323 −0.00491844
\(411\) 3458.84 0.415114
\(412\) −7584.13 −0.906901
\(413\) 9282.19 1.10592
\(414\) −1154.18 −0.137016
\(415\) −4426.71 −0.523611
\(416\) 2927.07 0.344979
\(417\) 1315.39 0.154473
\(418\) 2398.80 0.280692
\(419\) 11316.2 1.31940 0.659702 0.751527i \(-0.270682\pi\)
0.659702 + 0.751527i \(0.270682\pi\)
\(420\) −1644.07 −0.191006
\(421\) −15960.8 −1.84770 −0.923851 0.382753i \(-0.874976\pi\)
−0.923851 + 0.382753i \(0.874976\pi\)
\(422\) −2921.21 −0.336973
\(423\) 2532.84 0.291136
\(424\) −1834.16 −0.210081
\(425\) 1572.29 0.179452
\(426\) 620.213 0.0705385
\(427\) 5531.59 0.626914
\(428\) −1056.83 −0.119355
\(429\) 2643.10 0.297459
\(430\) 211.131 0.0236783
\(431\) 9181.83 1.02616 0.513078 0.858342i \(-0.328505\pi\)
0.513078 + 0.858342i \(0.328505\pi\)
\(432\) −1424.18 −0.158613
\(433\) 312.822 0.0347188 0.0173594 0.999849i \(-0.494474\pi\)
0.0173594 + 0.999849i \(0.494474\pi\)
\(434\) −1058.00 −0.117018
\(435\) 435.000 0.0479463
\(436\) 6397.34 0.702700
\(437\) −17458.2 −1.91107
\(438\) −1087.56 −0.118642
\(439\) −8018.14 −0.871720 −0.435860 0.900014i \(-0.643556\pi\)
−0.435860 + 0.900014i \(0.643556\pi\)
\(440\) 1977.13 0.214218
\(441\) −1175.90 −0.126974
\(442\) 1040.51 0.111973
\(443\) 553.480 0.0593603 0.0296802 0.999559i \(-0.490551\pi\)
0.0296802 + 0.999559i \(0.490551\pi\)
\(444\) −5862.32 −0.626607
\(445\) −2238.26 −0.238435
\(446\) 226.638 0.0240619
\(447\) 1216.71 0.128744
\(448\) 4915.71 0.518406
\(449\) −12055.2 −1.26708 −0.633540 0.773710i \(-0.718399\pi\)
−0.633540 + 0.773710i \(0.718399\pi\)
\(450\) −155.625 −0.0163027
\(451\) 434.880 0.0454051
\(452\) −13924.2 −1.44899
\(453\) 236.868 0.0245674
\(454\) −2532.96 −0.261845
\(455\) 1742.81 0.179569
\(456\) 3032.63 0.311439
\(457\) −268.261 −0.0274589 −0.0137295 0.999906i \(-0.504370\pi\)
−0.0137295 + 0.999906i \(0.504370\pi\)
\(458\) 4539.11 0.463097
\(459\) −1698.07 −0.172678
\(460\) −6972.91 −0.706768
\(461\) 3621.49 0.365877 0.182939 0.983124i \(-0.441439\pi\)
0.182939 + 0.983124i \(0.441439\pi\)
\(462\) −1113.70 −0.112152
\(463\) −7751.51 −0.778063 −0.389031 0.921225i \(-0.627190\pi\)
−0.389031 + 0.921225i \(0.627190\pi\)
\(464\) −1529.67 −0.153046
\(465\) 1574.57 0.157030
\(466\) −1722.44 −0.171225
\(467\) 8838.90 0.875836 0.437918 0.899015i \(-0.355716\pi\)
0.437918 + 0.899015i \(0.355716\pi\)
\(468\) 1619.24 0.159935
\(469\) 9083.22 0.894295
\(470\) −973.265 −0.0955178
\(471\) −944.607 −0.0924102
\(472\) −6838.54 −0.666884
\(473\) −2248.63 −0.218588
\(474\) 1168.77 0.113256
\(475\) −2354.00 −0.227387
\(476\) 6893.21 0.663760
\(477\) −1537.61 −0.147594
\(478\) 2479.20 0.237230
\(479\) −9904.02 −0.944731 −0.472366 0.881403i \(-0.656600\pi\)
−0.472366 + 0.881403i \(0.656600\pi\)
\(480\) 1835.54 0.174543
\(481\) 6214.38 0.589088
\(482\) 4526.22 0.427726
\(483\) 8105.41 0.763580
\(484\) −192.859 −0.0181123
\(485\) −2794.47 −0.261630
\(486\) 168.075 0.0156873
\(487\) 799.412 0.0743836 0.0371918 0.999308i \(-0.488159\pi\)
0.0371918 + 0.999308i \(0.488159\pi\)
\(488\) −4075.33 −0.378036
\(489\) −2450.44 −0.226611
\(490\) 451.851 0.0416583
\(491\) −10013.3 −0.920351 −0.460176 0.887828i \(-0.652214\pi\)
−0.460176 + 0.887828i \(0.652214\pi\)
\(492\) 266.421 0.0244130
\(493\) −1823.85 −0.166617
\(494\) −1557.84 −0.141883
\(495\) 1657.47 0.150500
\(496\) −5536.94 −0.501242
\(497\) −4355.55 −0.393105
\(498\) −1837.08 −0.165304
\(499\) 2556.35 0.229335 0.114667 0.993404i \(-0.463420\pi\)
0.114667 + 0.993404i \(0.463420\pi\)
\(500\) −940.200 −0.0840940
\(501\) 4093.08 0.365001
\(502\) −365.269 −0.0324756
\(503\) 11520.2 1.02120 0.510598 0.859820i \(-0.329424\pi\)
0.510598 + 0.859820i \(0.329424\pi\)
\(504\) −1407.98 −0.124437
\(505\) −5693.91 −0.501734
\(506\) −4723.48 −0.414989
\(507\) 4874.51 0.426992
\(508\) −6792.94 −0.593283
\(509\) 8055.10 0.701446 0.350723 0.936479i \(-0.385936\pi\)
0.350723 + 0.936479i \(0.385936\pi\)
\(510\) 652.498 0.0566531
\(511\) 7637.55 0.661184
\(512\) −10984.9 −0.948181
\(513\) 2542.32 0.218803
\(514\) −3472.70 −0.298004
\(515\) 5041.57 0.431375
\(516\) −1377.58 −0.117529
\(517\) 10365.7 0.881782
\(518\) −2618.51 −0.222106
\(519\) −1292.57 −0.109321
\(520\) −1283.99 −0.108282
\(521\) 22383.5 1.88222 0.941111 0.338097i \(-0.109783\pi\)
0.941111 + 0.338097i \(0.109783\pi\)
\(522\) 180.525 0.0151367
\(523\) 5260.45 0.439816 0.219908 0.975521i \(-0.429424\pi\)
0.219908 + 0.975521i \(0.429424\pi\)
\(524\) 5159.50 0.430141
\(525\) 1092.90 0.0908537
\(526\) 292.030 0.0242074
\(527\) −6601.79 −0.545690
\(528\) −5828.45 −0.480399
\(529\) 22210.0 1.82543
\(530\) 590.840 0.0484235
\(531\) −5732.88 −0.468523
\(532\) −10320.4 −0.841063
\(533\) −282.421 −0.0229512
\(534\) −928.876 −0.0752742
\(535\) 702.530 0.0567720
\(536\) −6691.95 −0.539269
\(537\) 9577.91 0.769678
\(538\) 1229.22 0.0985046
\(539\) −4812.40 −0.384573
\(540\) 1015.42 0.0809195
\(541\) −15052.8 −1.19625 −0.598123 0.801404i \(-0.704087\pi\)
−0.598123 + 0.801404i \(0.704087\pi\)
\(542\) 1013.64 0.0803315
\(543\) 1863.48 0.147274
\(544\) −7695.99 −0.606550
\(545\) −4252.65 −0.334245
\(546\) 723.264 0.0566902
\(547\) 13743.0 1.07424 0.537121 0.843505i \(-0.319512\pi\)
0.537121 + 0.843505i \(0.319512\pi\)
\(548\) 8672.00 0.676003
\(549\) −3416.43 −0.265591
\(550\) −636.896 −0.0493770
\(551\) 2730.64 0.211123
\(552\) −5971.56 −0.460446
\(553\) −8207.90 −0.631167
\(554\) 4206.36 0.322584
\(555\) 3896.99 0.298051
\(556\) 3297.96 0.251555
\(557\) 23296.7 1.77219 0.886097 0.463500i \(-0.153407\pi\)
0.886097 + 0.463500i \(0.153407\pi\)
\(558\) 653.446 0.0495745
\(559\) 1460.31 0.110491
\(560\) −3843.17 −0.290006
\(561\) −6949.37 −0.522999
\(562\) 826.196 0.0620124
\(563\) 6823.79 0.510814 0.255407 0.966834i \(-0.417791\pi\)
0.255407 + 0.966834i \(0.417791\pi\)
\(564\) 6350.33 0.474108
\(565\) 9256.17 0.689221
\(566\) −2647.37 −0.196603
\(567\) −1180.33 −0.0874240
\(568\) 3208.90 0.237046
\(569\) 17974.8 1.32433 0.662166 0.749358i \(-0.269638\pi\)
0.662166 + 0.749358i \(0.269638\pi\)
\(570\) −976.907 −0.0717862
\(571\) 22031.6 1.61470 0.807350 0.590072i \(-0.200901\pi\)
0.807350 + 0.590072i \(0.200901\pi\)
\(572\) 6626.77 0.484404
\(573\) −8830.11 −0.643775
\(574\) 119.002 0.00865338
\(575\) 4635.26 0.336180
\(576\) −3036.05 −0.219622
\(577\) 14520.1 1.04762 0.523811 0.851834i \(-0.324510\pi\)
0.523811 + 0.851834i \(0.324510\pi\)
\(578\) 662.386 0.0476672
\(579\) −9779.38 −0.701929
\(580\) 1090.63 0.0780793
\(581\) 12901.2 0.921227
\(582\) −1159.71 −0.0825968
\(583\) −6292.69 −0.447027
\(584\) −5626.87 −0.398701
\(585\) −1076.40 −0.0760743
\(586\) 4717.85 0.332581
\(587\) −998.184 −0.0701865 −0.0350932 0.999384i \(-0.511173\pi\)
−0.0350932 + 0.999384i \(0.511173\pi\)
\(588\) −2948.22 −0.206773
\(589\) 9884.08 0.691454
\(590\) 2202.91 0.153716
\(591\) −7117.04 −0.495357
\(592\) −13703.7 −0.951383
\(593\) 3431.22 0.237611 0.118806 0.992918i \(-0.462093\pi\)
0.118806 + 0.992918i \(0.462093\pi\)
\(594\) 687.848 0.0475130
\(595\) −4582.28 −0.315723
\(596\) 3050.54 0.209656
\(597\) 5486.26 0.376110
\(598\) 3067.53 0.209767
\(599\) −21059.5 −1.43651 −0.718255 0.695780i \(-0.755059\pi\)
−0.718255 + 0.695780i \(0.755059\pi\)
\(600\) −805.182 −0.0547857
\(601\) 8534.75 0.579267 0.289634 0.957138i \(-0.406467\pi\)
0.289634 + 0.957138i \(0.406467\pi\)
\(602\) −615.323 −0.0416589
\(603\) −5609.99 −0.378867
\(604\) 593.874 0.0400073
\(605\) 128.204 0.00861524
\(606\) −2362.97 −0.158398
\(607\) −17101.7 −1.14355 −0.571776 0.820409i \(-0.693746\pi\)
−0.571776 + 0.820409i \(0.693746\pi\)
\(608\) 11522.3 0.768571
\(609\) −1267.77 −0.0843555
\(610\) 1312.79 0.0871367
\(611\) −6731.69 −0.445720
\(612\) −4257.40 −0.281201
\(613\) −1048.59 −0.0690897 −0.0345449 0.999403i \(-0.510998\pi\)
−0.0345449 + 0.999403i \(0.510998\pi\)
\(614\) 5084.26 0.334176
\(615\) −177.104 −0.0116122
\(616\) −5762.16 −0.376890
\(617\) 2212.12 0.144338 0.0721689 0.997392i \(-0.477008\pi\)
0.0721689 + 0.997392i \(0.477008\pi\)
\(618\) 2092.25 0.136185
\(619\) 25602.7 1.66246 0.831228 0.555932i \(-0.187638\pi\)
0.831228 + 0.555932i \(0.187638\pi\)
\(620\) 3947.76 0.255719
\(621\) −5006.08 −0.323489
\(622\) −2346.92 −0.151291
\(623\) 6523.19 0.419496
\(624\) 3785.13 0.242831
\(625\) 625.000 0.0400000
\(626\) 7423.16 0.473944
\(627\) 10404.5 0.662702
\(628\) −2368.32 −0.150488
\(629\) −16339.2 −1.03575
\(630\) 453.554 0.0286826
\(631\) −25127.0 −1.58524 −0.792622 0.609713i \(-0.791285\pi\)
−0.792622 + 0.609713i \(0.791285\pi\)
\(632\) 6047.07 0.380600
\(633\) −12670.3 −0.795578
\(634\) 6032.14 0.377866
\(635\) 4515.62 0.282200
\(636\) −3855.10 −0.240353
\(637\) 3125.28 0.194392
\(638\) 738.800 0.0458454
\(639\) 2690.08 0.166538
\(640\) 6061.41 0.374372
\(641\) 2231.80 0.137521 0.0687605 0.997633i \(-0.478096\pi\)
0.0687605 + 0.997633i \(0.478096\pi\)
\(642\) 291.550 0.0179230
\(643\) −15105.0 −0.926415 −0.463207 0.886250i \(-0.653301\pi\)
−0.463207 + 0.886250i \(0.653301\pi\)
\(644\) 20321.9 1.24347
\(645\) 915.752 0.0559034
\(646\) 4095.94 0.249462
\(647\) 21013.9 1.27688 0.638441 0.769671i \(-0.279580\pi\)
0.638441 + 0.769671i \(0.279580\pi\)
\(648\) 869.596 0.0527176
\(649\) −23461.9 −1.41904
\(650\) 413.614 0.0249589
\(651\) −4588.94 −0.276274
\(652\) −6143.73 −0.369029
\(653\) 25780.5 1.54498 0.772489 0.635028i \(-0.219011\pi\)
0.772489 + 0.635028i \(0.219011\pi\)
\(654\) −1764.85 −0.105521
\(655\) −3429.79 −0.204600
\(656\) 622.783 0.0370665
\(657\) −4717.11 −0.280110
\(658\) 2836.49 0.168051
\(659\) 14065.8 0.831448 0.415724 0.909491i \(-0.363528\pi\)
0.415724 + 0.909491i \(0.363528\pi\)
\(660\) 4155.60 0.245086
\(661\) −13363.3 −0.786342 −0.393171 0.919465i \(-0.628622\pi\)
−0.393171 + 0.919465i \(0.628622\pi\)
\(662\) 4214.33 0.247424
\(663\) 4513.07 0.264364
\(664\) −9504.81 −0.555510
\(665\) 6860.50 0.400058
\(666\) 1617.25 0.0940948
\(667\) −5376.90 −0.312135
\(668\) 10262.2 0.594395
\(669\) 983.010 0.0568092
\(670\) 2155.69 0.124301
\(671\) −13981.8 −0.804412
\(672\) −5349.52 −0.307087
\(673\) −17727.5 −1.01537 −0.507686 0.861542i \(-0.669499\pi\)
−0.507686 + 0.861542i \(0.669499\pi\)
\(674\) 4362.28 0.249301
\(675\) −675.000 −0.0384900
\(676\) 12221.4 0.695345
\(677\) 15776.1 0.895605 0.447803 0.894132i \(-0.352207\pi\)
0.447803 + 0.894132i \(0.352207\pi\)
\(678\) 3841.31 0.217588
\(679\) 8144.23 0.460305
\(680\) 3375.94 0.190384
\(681\) −10986.4 −0.618206
\(682\) 2674.23 0.150149
\(683\) 14482.3 0.811350 0.405675 0.914018i \(-0.367037\pi\)
0.405675 + 0.914018i \(0.367037\pi\)
\(684\) 6374.09 0.356315
\(685\) −5764.73 −0.321546
\(686\) −4773.97 −0.265701
\(687\) 19687.7 1.09335
\(688\) −3220.22 −0.178445
\(689\) 4086.61 0.225961
\(690\) 1923.63 0.106132
\(691\) 19721.5 1.08573 0.542867 0.839819i \(-0.317339\pi\)
0.542867 + 0.839819i \(0.317339\pi\)
\(692\) −3240.74 −0.178026
\(693\) −4830.53 −0.264786
\(694\) −7882.04 −0.431121
\(695\) −2192.32 −0.119654
\(696\) 934.011 0.0508672
\(697\) 742.555 0.0403533
\(698\) −6110.92 −0.331378
\(699\) −7470.85 −0.404254
\(700\) 2740.12 0.147953
\(701\) 2004.45 0.107998 0.0539992 0.998541i \(-0.482803\pi\)
0.0539992 + 0.998541i \(0.482803\pi\)
\(702\) −446.704 −0.0240167
\(703\) 24462.7 1.31241
\(704\) −12425.1 −0.665181
\(705\) −4221.39 −0.225513
\(706\) 2345.27 0.125022
\(707\) 16594.4 0.882738
\(708\) −14373.5 −0.762978
\(709\) 10037.8 0.531705 0.265853 0.964014i \(-0.414347\pi\)
0.265853 + 0.964014i \(0.414347\pi\)
\(710\) −1033.69 −0.0546389
\(711\) 5069.38 0.267393
\(712\) −4805.88 −0.252961
\(713\) −19462.8 −1.02228
\(714\) −1901.64 −0.0996740
\(715\) −4405.16 −0.230411
\(716\) 24013.7 1.25340
\(717\) 10753.2 0.560091
\(718\) 3569.18 0.185516
\(719\) 10367.8 0.537768 0.268884 0.963173i \(-0.413345\pi\)
0.268884 + 0.963173i \(0.413345\pi\)
\(720\) 2373.63 0.122861
\(721\) −14693.2 −0.758949
\(722\) −1388.23 −0.0715576
\(723\) 19631.8 1.00984
\(724\) 4672.11 0.239831
\(725\) −725.000 −0.0371391
\(726\) 53.2044 0.00271984
\(727\) −32648.0 −1.66554 −0.832770 0.553619i \(-0.813246\pi\)
−0.832770 + 0.553619i \(0.813246\pi\)
\(728\) 3742.07 0.190509
\(729\) 729.000 0.0370370
\(730\) 1812.59 0.0919001
\(731\) −3839.53 −0.194268
\(732\) −8565.66 −0.432508
\(733\) 21036.1 1.06001 0.530004 0.847995i \(-0.322191\pi\)
0.530004 + 0.847995i \(0.322191\pi\)
\(734\) −767.863 −0.0386136
\(735\) 1959.84 0.0983534
\(736\) −22688.6 −1.13629
\(737\) −22959.0 −1.14750
\(738\) −73.4981 −0.00366599
\(739\) 24591.3 1.22410 0.612048 0.790820i \(-0.290346\pi\)
0.612048 + 0.790820i \(0.290346\pi\)
\(740\) 9770.54 0.485368
\(741\) −6756.88 −0.334980
\(742\) −1721.95 −0.0851950
\(743\) 3352.77 0.165547 0.0827734 0.996568i \(-0.473622\pi\)
0.0827734 + 0.996568i \(0.473622\pi\)
\(744\) 3380.84 0.166596
\(745\) −2027.85 −0.0997244
\(746\) −4012.50 −0.196928
\(747\) −7968.08 −0.390277
\(748\) −17423.4 −0.851690
\(749\) −2047.46 −0.0998831
\(750\) 259.375 0.0126280
\(751\) 10898.0 0.529525 0.264762 0.964314i \(-0.414706\pi\)
0.264762 + 0.964314i \(0.414706\pi\)
\(752\) 14844.5 0.719843
\(753\) −1584.30 −0.0766734
\(754\) −479.793 −0.0231738
\(755\) −394.779 −0.0190298
\(756\) −2959.33 −0.142368
\(757\) −34910.6 −1.67615 −0.838075 0.545555i \(-0.816319\pi\)
−0.838075 + 0.545555i \(0.816319\pi\)
\(758\) 8759.58 0.419739
\(759\) −20487.4 −0.979771
\(760\) −5054.39 −0.241239
\(761\) −17093.2 −0.814229 −0.407115 0.913377i \(-0.633465\pi\)
−0.407115 + 0.913377i \(0.633465\pi\)
\(762\) 1873.98 0.0890907
\(763\) 12393.9 0.588062
\(764\) −22138.8 −1.04837
\(765\) 2830.11 0.133756
\(766\) 9274.05 0.437448
\(767\) 15236.7 0.717293
\(768\) −5580.65 −0.262206
\(769\) 29371.4 1.37732 0.688660 0.725084i \(-0.258199\pi\)
0.688660 + 0.725084i \(0.258199\pi\)
\(770\) 1856.17 0.0868726
\(771\) −15062.3 −0.703575
\(772\) −24518.9 −1.14307
\(773\) −25646.0 −1.19330 −0.596651 0.802501i \(-0.703502\pi\)
−0.596651 + 0.802501i \(0.703502\pi\)
\(774\) 380.036 0.0176487
\(775\) −2624.28 −0.121635
\(776\) −6000.16 −0.277568
\(777\) −11357.4 −0.524382
\(778\) 3234.62 0.149057
\(779\) −1111.74 −0.0511325
\(780\) −2698.74 −0.123885
\(781\) 11009.2 0.504405
\(782\) −8065.32 −0.368817
\(783\) 783.000 0.0357371
\(784\) −6891.74 −0.313946
\(785\) 1574.35 0.0715806
\(786\) −1423.36 −0.0645923
\(787\) 18188.7 0.823834 0.411917 0.911221i \(-0.364859\pi\)
0.411917 + 0.911221i \(0.364859\pi\)
\(788\) −17843.8 −0.806676
\(789\) 1266.64 0.0571527
\(790\) −1947.95 −0.0877279
\(791\) −26976.2 −1.21260
\(792\) 3558.83 0.159669
\(793\) 9080.07 0.406611
\(794\) 4090.31 0.182821
\(795\) 2562.68 0.114326
\(796\) 13755.1 0.612485
\(797\) 12068.7 0.536379 0.268190 0.963366i \(-0.413575\pi\)
0.268190 + 0.963366i \(0.413575\pi\)
\(798\) 2847.11 0.126299
\(799\) 17699.3 0.783675
\(800\) −3059.24 −0.135201
\(801\) −4028.87 −0.177719
\(802\) −5365.11 −0.236220
\(803\) −19304.8 −0.848385
\(804\) −14065.4 −0.616974
\(805\) −13509.0 −0.591466
\(806\) −1736.71 −0.0758968
\(807\) 5331.57 0.232565
\(808\) −12225.7 −0.532300
\(809\) −18274.6 −0.794191 −0.397096 0.917777i \(-0.629982\pi\)
−0.397096 + 0.917777i \(0.629982\pi\)
\(810\) −280.125 −0.0121513
\(811\) −30620.5 −1.32581 −0.662903 0.748705i \(-0.730676\pi\)
−0.662903 + 0.748705i \(0.730676\pi\)
\(812\) −3178.54 −0.137371
\(813\) 4396.53 0.189659
\(814\) 6618.61 0.284990
\(815\) 4084.06 0.175532
\(816\) −9952.05 −0.426950
\(817\) 5748.47 0.246161
\(818\) −5308.10 −0.226887
\(819\) 3137.05 0.133843
\(820\) −444.035 −0.0189102
\(821\) −10076.0 −0.428326 −0.214163 0.976798i \(-0.568702\pi\)
−0.214163 + 0.976798i \(0.568702\pi\)
\(822\) −2392.36 −0.101512
\(823\) 37593.9 1.59227 0.796137 0.605116i \(-0.206873\pi\)
0.796137 + 0.605116i \(0.206873\pi\)
\(824\) 10825.0 0.457654
\(825\) −2762.45 −0.116577
\(826\) −6420.18 −0.270444
\(827\) 34830.2 1.46453 0.732264 0.681021i \(-0.238464\pi\)
0.732264 + 0.681021i \(0.238464\pi\)
\(828\) −12551.2 −0.526794
\(829\) −19925.3 −0.834784 −0.417392 0.908727i \(-0.637056\pi\)
−0.417392 + 0.908727i \(0.637056\pi\)
\(830\) 3061.80 0.128044
\(831\) 18244.5 0.761606
\(832\) 8069.12 0.336233
\(833\) −8217.14 −0.341785
\(834\) −909.814 −0.0377749
\(835\) −6821.81 −0.282729
\(836\) 26086.1 1.07919
\(837\) 2834.22 0.117043
\(838\) −7827.00 −0.322648
\(839\) 26090.0 1.07357 0.536786 0.843718i \(-0.319638\pi\)
0.536786 + 0.843718i \(0.319638\pi\)
\(840\) 2346.63 0.0963885
\(841\) 841.000 0.0344828
\(842\) 11039.5 0.451838
\(843\) 3583.51 0.146409
\(844\) −31767.1 −1.29558
\(845\) −8124.19 −0.330746
\(846\) −1751.88 −0.0711947
\(847\) −373.637 −0.0151574
\(848\) −9011.63 −0.364930
\(849\) −11482.6 −0.464171
\(850\) −1087.50 −0.0438833
\(851\) −48169.5 −1.94034
\(852\) 6744.57 0.271203
\(853\) −37626.4 −1.51032 −0.755160 0.655540i \(-0.772441\pi\)
−0.755160 + 0.655540i \(0.772441\pi\)
\(854\) −3826.01 −0.153306
\(855\) −4237.19 −0.169484
\(856\) 1508.44 0.0602306
\(857\) 25833.1 1.02969 0.514844 0.857284i \(-0.327850\pi\)
0.514844 + 0.857284i \(0.327850\pi\)
\(858\) −1828.14 −0.0727409
\(859\) 49111.2 1.95070 0.975351 0.220661i \(-0.0708215\pi\)
0.975351 + 0.220661i \(0.0708215\pi\)
\(860\) 2295.97 0.0910372
\(861\) 516.153 0.0204303
\(862\) −6350.76 −0.250937
\(863\) 14549.3 0.573886 0.286943 0.957948i \(-0.407361\pi\)
0.286943 + 0.957948i \(0.407361\pi\)
\(864\) 3303.98 0.130097
\(865\) 2154.29 0.0846797
\(866\) −216.368 −0.00849017
\(867\) 2873.00 0.112540
\(868\) −11505.4 −0.449905
\(869\) 20746.5 0.809869
\(870\) −300.875 −0.0117248
\(871\) 14910.1 0.580032
\(872\) −9131.08 −0.354607
\(873\) −5030.05 −0.195007
\(874\) 12075.2 0.467335
\(875\) −1821.50 −0.0703749
\(876\) −11826.7 −0.456151
\(877\) 501.781 0.0193203 0.00966017 0.999953i \(-0.496925\pi\)
0.00966017 + 0.999953i \(0.496925\pi\)
\(878\) 5545.88 0.213171
\(879\) 20463.0 0.785210
\(880\) 9714.09 0.372116
\(881\) 1569.47 0.0600189 0.0300095 0.999550i \(-0.490446\pi\)
0.0300095 + 0.999550i \(0.490446\pi\)
\(882\) 813.332 0.0310502
\(883\) −26159.0 −0.996967 −0.498483 0.866899i \(-0.666110\pi\)
−0.498483 + 0.866899i \(0.666110\pi\)
\(884\) 11315.2 0.430509
\(885\) 9554.80 0.362917
\(886\) −382.823 −0.0145160
\(887\) −29723.0 −1.12514 −0.562571 0.826749i \(-0.690188\pi\)
−0.562571 + 0.826749i \(0.690188\pi\)
\(888\) 8367.43 0.316208
\(889\) −13160.4 −0.496495
\(890\) 1548.13 0.0583071
\(891\) 2983.44 0.112176
\(892\) 2464.60 0.0925123
\(893\) −26499.1 −0.993009
\(894\) −841.557 −0.0314831
\(895\) −15963.2 −0.596190
\(896\) −17665.4 −0.658661
\(897\) 13305.0 0.495251
\(898\) 8338.16 0.309853
\(899\) 3044.17 0.112935
\(900\) −1692.36 −0.0626800
\(901\) −10744.7 −0.397290
\(902\) −300.792 −0.0111034
\(903\) −2668.87 −0.0983549
\(904\) 19874.4 0.731209
\(905\) −3105.80 −0.114078
\(906\) −163.833 −0.00600772
\(907\) −9772.21 −0.357752 −0.178876 0.983872i \(-0.557246\pi\)
−0.178876 + 0.983872i \(0.557246\pi\)
\(908\) −27545.0 −1.00673
\(909\) −10249.0 −0.373971
\(910\) −1205.44 −0.0439120
\(911\) −49823.7 −1.81200 −0.906001 0.423275i \(-0.860880\pi\)
−0.906001 + 0.423275i \(0.860880\pi\)
\(912\) 14900.0 0.540997
\(913\) −32609.4 −1.18205
\(914\) 185.547 0.00671483
\(915\) 5694.05 0.205726
\(916\) 49361.0 1.78050
\(917\) 9995.80 0.359968
\(918\) 1174.50 0.0422267
\(919\) 28235.9 1.01351 0.506756 0.862090i \(-0.330845\pi\)
0.506756 + 0.862090i \(0.330845\pi\)
\(920\) 9952.60 0.356660
\(921\) 22052.2 0.788974
\(922\) −2504.86 −0.0894720
\(923\) −7149.61 −0.254965
\(924\) −12111.1 −0.431197
\(925\) −6494.99 −0.230869
\(926\) 5361.45 0.190268
\(927\) 9074.82 0.321528
\(928\) 3548.72 0.125531
\(929\) −22846.1 −0.806842 −0.403421 0.915014i \(-0.632179\pi\)
−0.403421 + 0.915014i \(0.632179\pi\)
\(930\) −1089.08 −0.0384002
\(931\) 12302.5 0.433082
\(932\) −18730.9 −0.658317
\(933\) −10179.4 −0.357191
\(934\) −6113.56 −0.214178
\(935\) 11582.3 0.405113
\(936\) −2311.18 −0.0807088
\(937\) −45416.9 −1.58346 −0.791732 0.610868i \(-0.790820\pi\)
−0.791732 + 0.610868i \(0.790820\pi\)
\(938\) −6282.55 −0.218692
\(939\) 32196.9 1.11896
\(940\) −10583.9 −0.367243
\(941\) −39333.3 −1.36262 −0.681311 0.731994i \(-0.738590\pi\)
−0.681311 + 0.731994i \(0.738590\pi\)
\(942\) 653.352 0.0225981
\(943\) 2189.13 0.0755968
\(944\) −33599.3 −1.15844
\(945\) 1967.22 0.0677183
\(946\) 1555.30 0.0534538
\(947\) 31274.8 1.07317 0.536587 0.843845i \(-0.319713\pi\)
0.536587 + 0.843845i \(0.319713\pi\)
\(948\) 12709.9 0.435443
\(949\) 12537.0 0.428839
\(950\) 1628.18 0.0556054
\(951\) 26163.5 0.892124
\(952\) −9838.85 −0.334957
\(953\) −34174.1 −1.16160 −0.580802 0.814045i \(-0.697261\pi\)
−0.580802 + 0.814045i \(0.697261\pi\)
\(954\) 1063.51 0.0360928
\(955\) 14716.8 0.498666
\(956\) 26960.4 0.912093
\(957\) 3204.44 0.108239
\(958\) 6850.27 0.231025
\(959\) 16800.8 0.565720
\(960\) 5060.09 0.170118
\(961\) −18772.0 −0.630124
\(962\) −4298.27 −0.144056
\(963\) 1264.55 0.0423153
\(964\) 49220.9 1.64450
\(965\) 16299.0 0.543712
\(966\) −5606.24 −0.186726
\(967\) 37894.7 1.26020 0.630098 0.776515i \(-0.283015\pi\)
0.630098 + 0.776515i \(0.283015\pi\)
\(968\) 275.273 0.00914008
\(969\) 17765.5 0.588970
\(970\) 1932.84 0.0639792
\(971\) −39869.1 −1.31767 −0.658837 0.752286i \(-0.728951\pi\)
−0.658837 + 0.752286i \(0.728951\pi\)
\(972\) 1827.75 0.0603138
\(973\) 6389.33 0.210516
\(974\) −552.926 −0.0181898
\(975\) 1793.99 0.0589269
\(976\) −20023.0 −0.656681
\(977\) −34257.3 −1.12179 −0.560894 0.827887i \(-0.689543\pi\)
−0.560894 + 0.827887i \(0.689543\pi\)
\(978\) 1694.88 0.0554155
\(979\) −16488.2 −0.538268
\(980\) 4913.71 0.160166
\(981\) −7654.77 −0.249131
\(982\) 6925.83 0.225063
\(983\) 8665.65 0.281171 0.140586 0.990069i \(-0.455101\pi\)
0.140586 + 0.990069i \(0.455101\pi\)
\(984\) −380.269 −0.0123197
\(985\) 11861.7 0.383702
\(986\) 1261.50 0.0407446
\(987\) 12302.9 0.396762
\(988\) −16940.9 −0.545507
\(989\) −11319.3 −0.363937
\(990\) −1146.41 −0.0368034
\(991\) 22129.9 0.709362 0.354681 0.934987i \(-0.384589\pi\)
0.354681 + 0.934987i \(0.384589\pi\)
\(992\) 12845.3 0.411127
\(993\) 18279.0 0.584157
\(994\) 3012.59 0.0961302
\(995\) −9143.76 −0.291333
\(996\) −19977.6 −0.635555
\(997\) −14159.7 −0.449791 −0.224895 0.974383i \(-0.572204\pi\)
−0.224895 + 0.974383i \(0.572204\pi\)
\(998\) −1768.14 −0.0560817
\(999\) 7014.59 0.222154
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 435.4.a.g.1.3 6
3.2 odd 2 1305.4.a.i.1.4 6
5.4 even 2 2175.4.a.l.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.4.a.g.1.3 6 1.1 even 1 trivial
1305.4.a.i.1.4 6 3.2 odd 2
2175.4.a.l.1.4 6 5.4 even 2