Properties

Label 435.4.a.i.1.6
Level $435$
Weight $4$
Character 435.1
Self dual yes
Analytic conductor $25.666$
Analytic rank $0$
Dimension $7$
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: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 37x^{5} + 55x^{4} + 336x^{3} - 227x^{2} - 824x - 166 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-2.58378\) 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)
 
\(f(q)\) \(=\) \(q+2.58378 q^{2} -3.00000 q^{3} -1.32408 q^{4} +5.00000 q^{5} -7.75134 q^{6} -26.6398 q^{7} -24.0914 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.58378 q^{2} -3.00000 q^{3} -1.32408 q^{4} +5.00000 q^{5} -7.75134 q^{6} -26.6398 q^{7} -24.0914 q^{8} +9.00000 q^{9} +12.9189 q^{10} -8.03338 q^{11} +3.97225 q^{12} +49.9367 q^{13} -68.8314 q^{14} -15.0000 q^{15} -51.6541 q^{16} +116.617 q^{17} +23.2540 q^{18} +22.6617 q^{19} -6.62041 q^{20} +79.9195 q^{21} -20.7565 q^{22} +38.2616 q^{23} +72.2741 q^{24} +25.0000 q^{25} +129.025 q^{26} -27.0000 q^{27} +35.2733 q^{28} +29.0000 q^{29} -38.7567 q^{30} -8.82528 q^{31} +59.2681 q^{32} +24.1002 q^{33} +301.312 q^{34} -133.199 q^{35} -11.9167 q^{36} +301.022 q^{37} +58.5529 q^{38} -149.810 q^{39} -120.457 q^{40} +305.401 q^{41} +206.494 q^{42} +275.872 q^{43} +10.6369 q^{44} +45.0000 q^{45} +98.8596 q^{46} -479.282 q^{47} +154.962 q^{48} +366.680 q^{49} +64.5945 q^{50} -349.850 q^{51} -66.1204 q^{52} -496.683 q^{53} -69.7621 q^{54} -40.1669 q^{55} +641.790 q^{56} -67.9852 q^{57} +74.9296 q^{58} -563.505 q^{59} +19.8612 q^{60} +485.346 q^{61} -22.8026 q^{62} -239.758 q^{63} +566.369 q^{64} +249.684 q^{65} +62.2695 q^{66} +217.176 q^{67} -154.410 q^{68} -114.785 q^{69} -344.157 q^{70} +628.023 q^{71} -216.822 q^{72} -682.502 q^{73} +777.773 q^{74} -75.0000 q^{75} -30.0060 q^{76} +214.008 q^{77} -387.076 q^{78} +1159.26 q^{79} -258.271 q^{80} +81.0000 q^{81} +789.089 q^{82} +668.594 q^{83} -105.820 q^{84} +583.084 q^{85} +712.792 q^{86} -87.0000 q^{87} +193.535 q^{88} +798.300 q^{89} +116.270 q^{90} -1330.31 q^{91} -50.6616 q^{92} +26.4758 q^{93} -1238.36 q^{94} +113.309 q^{95} -177.804 q^{96} +867.116 q^{97} +947.421 q^{98} -72.3005 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 2 q^{2} - 21 q^{3} + 22 q^{4} + 35 q^{5} + 6 q^{6} - 50 q^{7} - 33 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 2 q^{2} - 21 q^{3} + 22 q^{4} + 35 q^{5} + 6 q^{6} - 50 q^{7} - 33 q^{8} + 63 q^{9} - 10 q^{10} + 76 q^{11} - 66 q^{12} + 30 q^{13} + 89 q^{14} - 105 q^{15} + 138 q^{16} - 140 q^{17} - 18 q^{18} + 90 q^{19} + 110 q^{20} + 150 q^{21} + 61 q^{22} + 34 q^{23} + 99 q^{24} + 175 q^{25} - 241 q^{26} - 189 q^{27} - 57 q^{28} + 203 q^{29} + 30 q^{30} + 524 q^{31} - 6 q^{32} - 228 q^{33} + 255 q^{34} - 250 q^{35} + 198 q^{36} - 28 q^{37} + 222 q^{38} - 90 q^{39} - 165 q^{40} + 1532 q^{41} - 267 q^{42} - 464 q^{43} + 1475 q^{44} + 315 q^{45} + 72 q^{46} - 360 q^{47} - 414 q^{48} + 569 q^{49} - 50 q^{50} + 420 q^{51} - 205 q^{52} + 282 q^{53} + 54 q^{54} + 380 q^{55} + 1102 q^{56} - 270 q^{57} - 58 q^{58} + 766 q^{59} - 330 q^{60} + 1200 q^{61} + 2856 q^{62} - 450 q^{63} + 701 q^{64} + 150 q^{65} - 183 q^{66} + 1546 q^{67} + 1801 q^{68} - 102 q^{69} + 445 q^{70} + 1802 q^{71} - 297 q^{72} - 220 q^{73} + 1594 q^{74} - 525 q^{75} + 1960 q^{76} + 3222 q^{77} + 723 q^{78} + 1298 q^{79} + 690 q^{80} + 567 q^{81} + 856 q^{82} + 1652 q^{83} + 171 q^{84} - 700 q^{85} + 7628 q^{86} - 609 q^{87} + 550 q^{88} + 2846 q^{89} - 90 q^{90} - 816 q^{91} + 472 q^{92} - 1572 q^{93} + 745 q^{94} + 450 q^{95} + 18 q^{96} + 1110 q^{97} - 761 q^{98} + 684 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.58378 0.913504 0.456752 0.889594i \(-0.349013\pi\)
0.456752 + 0.889594i \(0.349013\pi\)
\(3\) −3.00000 −0.577350
\(4\) −1.32408 −0.165510
\(5\) 5.00000 0.447214
\(6\) −7.75134 −0.527412
\(7\) −26.6398 −1.43842 −0.719208 0.694795i \(-0.755495\pi\)
−0.719208 + 0.694795i \(0.755495\pi\)
\(8\) −24.0914 −1.06470
\(9\) 9.00000 0.333333
\(10\) 12.9189 0.408531
\(11\) −8.03338 −0.220196 −0.110098 0.993921i \(-0.535116\pi\)
−0.110098 + 0.993921i \(0.535116\pi\)
\(12\) 3.97225 0.0955574
\(13\) 49.9367 1.06538 0.532691 0.846310i \(-0.321181\pi\)
0.532691 + 0.846310i \(0.321181\pi\)
\(14\) −68.8314 −1.31400
\(15\) −15.0000 −0.258199
\(16\) −51.6541 −0.807096
\(17\) 116.617 1.66375 0.831874 0.554964i \(-0.187268\pi\)
0.831874 + 0.554964i \(0.187268\pi\)
\(18\) 23.2540 0.304501
\(19\) 22.6617 0.273629 0.136815 0.990597i \(-0.456314\pi\)
0.136815 + 0.990597i \(0.456314\pi\)
\(20\) −6.62041 −0.0740185
\(21\) 79.9195 0.830470
\(22\) −20.7565 −0.201150
\(23\) 38.2616 0.346874 0.173437 0.984845i \(-0.444513\pi\)
0.173437 + 0.984845i \(0.444513\pi\)
\(24\) 72.2741 0.614704
\(25\) 25.0000 0.200000
\(26\) 129.025 0.973230
\(27\) −27.0000 −0.192450
\(28\) 35.2733 0.238073
\(29\) 29.0000 0.185695
\(30\) −38.7567 −0.235866
\(31\) −8.82528 −0.0511312 −0.0255656 0.999673i \(-0.508139\pi\)
−0.0255656 + 0.999673i \(0.508139\pi\)
\(32\) 59.2681 0.327413
\(33\) 24.1002 0.127130
\(34\) 301.312 1.51984
\(35\) −133.199 −0.643279
\(36\) −11.9167 −0.0551701
\(37\) 301.022 1.33750 0.668752 0.743486i \(-0.266829\pi\)
0.668752 + 0.743486i \(0.266829\pi\)
\(38\) 58.5529 0.249962
\(39\) −149.810 −0.615098
\(40\) −120.457 −0.476148
\(41\) 305.401 1.16331 0.581654 0.813436i \(-0.302406\pi\)
0.581654 + 0.813436i \(0.302406\pi\)
\(42\) 206.494 0.758637
\(43\) 275.872 0.978374 0.489187 0.872179i \(-0.337294\pi\)
0.489187 + 0.872179i \(0.337294\pi\)
\(44\) 10.6369 0.0364447
\(45\) 45.0000 0.149071
\(46\) 98.8596 0.316871
\(47\) −479.282 −1.48746 −0.743729 0.668481i \(-0.766945\pi\)
−0.743729 + 0.668481i \(0.766945\pi\)
\(48\) 154.962 0.465977
\(49\) 366.680 1.06904
\(50\) 64.5945 0.182701
\(51\) −349.850 −0.960566
\(52\) −66.1204 −0.176332
\(53\) −496.683 −1.28726 −0.643629 0.765338i \(-0.722572\pi\)
−0.643629 + 0.765338i \(0.722572\pi\)
\(54\) −69.7621 −0.175804
\(55\) −40.1669 −0.0984747
\(56\) 641.790 1.53148
\(57\) −67.9852 −0.157980
\(58\) 74.9296 0.169633
\(59\) −563.505 −1.24342 −0.621712 0.783246i \(-0.713563\pi\)
−0.621712 + 0.783246i \(0.713563\pi\)
\(60\) 19.8612 0.0427346
\(61\) 485.346 1.01872 0.509362 0.860552i \(-0.329881\pi\)
0.509362 + 0.860552i \(0.329881\pi\)
\(62\) −22.8026 −0.0467085
\(63\) −239.758 −0.479472
\(64\) 566.369 1.10619
\(65\) 249.684 0.476453
\(66\) 62.2695 0.116134
\(67\) 217.176 0.396004 0.198002 0.980202i \(-0.436555\pi\)
0.198002 + 0.980202i \(0.436555\pi\)
\(68\) −154.410 −0.275368
\(69\) −114.785 −0.200268
\(70\) −344.157 −0.587638
\(71\) 628.023 1.04975 0.524877 0.851178i \(-0.324111\pi\)
0.524877 + 0.851178i \(0.324111\pi\)
\(72\) −216.822 −0.354899
\(73\) −682.502 −1.09426 −0.547128 0.837049i \(-0.684279\pi\)
−0.547128 + 0.837049i \(0.684279\pi\)
\(74\) 777.773 1.22182
\(75\) −75.0000 −0.115470
\(76\) −30.0060 −0.0452885
\(77\) 214.008 0.316733
\(78\) −387.076 −0.561895
\(79\) 1159.26 1.65097 0.825484 0.564425i \(-0.190902\pi\)
0.825484 + 0.564425i \(0.190902\pi\)
\(80\) −258.271 −0.360944
\(81\) 81.0000 0.111111
\(82\) 789.089 1.06269
\(83\) 668.594 0.884189 0.442095 0.896968i \(-0.354236\pi\)
0.442095 + 0.896968i \(0.354236\pi\)
\(84\) −105.820 −0.137451
\(85\) 583.084 0.744051
\(86\) 712.792 0.893748
\(87\) −87.0000 −0.107211
\(88\) 193.535 0.234442
\(89\) 798.300 0.950782 0.475391 0.879775i \(-0.342307\pi\)
0.475391 + 0.879775i \(0.342307\pi\)
\(90\) 116.270 0.136177
\(91\) −1330.31 −1.53246
\(92\) −50.6616 −0.0574112
\(93\) 26.4758 0.0295206
\(94\) −1238.36 −1.35880
\(95\) 113.309 0.122371
\(96\) −177.804 −0.189032
\(97\) 867.116 0.907653 0.453826 0.891090i \(-0.350059\pi\)
0.453826 + 0.891090i \(0.350059\pi\)
\(98\) 947.421 0.976571
\(99\) −72.3005 −0.0733987
\(100\) −33.1021 −0.0331021
\(101\) 1270.57 1.25174 0.625872 0.779926i \(-0.284743\pi\)
0.625872 + 0.779926i \(0.284743\pi\)
\(102\) −903.936 −0.877481
\(103\) −884.596 −0.846232 −0.423116 0.906076i \(-0.639064\pi\)
−0.423116 + 0.906076i \(0.639064\pi\)
\(104\) −1203.04 −1.13431
\(105\) 399.597 0.371397
\(106\) −1283.32 −1.17591
\(107\) 925.937 0.836577 0.418288 0.908314i \(-0.362630\pi\)
0.418288 + 0.908314i \(0.362630\pi\)
\(108\) 35.7502 0.0318525
\(109\) 73.4989 0.0645864 0.0322932 0.999478i \(-0.489719\pi\)
0.0322932 + 0.999478i \(0.489719\pi\)
\(110\) −103.782 −0.0899570
\(111\) −903.065 −0.772208
\(112\) 1376.06 1.16094
\(113\) −2068.68 −1.72217 −0.861086 0.508459i \(-0.830215\pi\)
−0.861086 + 0.508459i \(0.830215\pi\)
\(114\) −175.659 −0.144315
\(115\) 191.308 0.155127
\(116\) −38.3984 −0.0307345
\(117\) 449.431 0.355127
\(118\) −1455.97 −1.13587
\(119\) −3106.65 −2.39316
\(120\) 361.371 0.274904
\(121\) −1266.46 −0.951514
\(122\) 1254.03 0.930609
\(123\) −916.204 −0.671637
\(124\) 11.6854 0.00846274
\(125\) 125.000 0.0894427
\(126\) −619.483 −0.437999
\(127\) −708.738 −0.495199 −0.247600 0.968862i \(-0.579642\pi\)
−0.247600 + 0.968862i \(0.579642\pi\)
\(128\) 989.228 0.683095
\(129\) −827.616 −0.564864
\(130\) 645.127 0.435242
\(131\) −2033.80 −1.35644 −0.678220 0.734859i \(-0.737248\pi\)
−0.678220 + 0.734859i \(0.737248\pi\)
\(132\) −31.9106 −0.0210414
\(133\) −603.705 −0.393593
\(134\) 561.135 0.361752
\(135\) −135.000 −0.0860663
\(136\) −2809.46 −1.77139
\(137\) 2635.96 1.64383 0.821917 0.569608i \(-0.192905\pi\)
0.821917 + 0.569608i \(0.192905\pi\)
\(138\) −296.579 −0.182945
\(139\) −1284.05 −0.783539 −0.391769 0.920063i \(-0.628137\pi\)
−0.391769 + 0.920063i \(0.628137\pi\)
\(140\) 176.367 0.106469
\(141\) 1437.85 0.858784
\(142\) 1622.67 0.958955
\(143\) −401.161 −0.234593
\(144\) −464.887 −0.269032
\(145\) 145.000 0.0830455
\(146\) −1763.43 −0.999608
\(147\) −1100.04 −0.617210
\(148\) −398.578 −0.221371
\(149\) 261.843 0.143966 0.0719832 0.997406i \(-0.477067\pi\)
0.0719832 + 0.997406i \(0.477067\pi\)
\(150\) −193.783 −0.105482
\(151\) 2760.20 1.48756 0.743781 0.668424i \(-0.233031\pi\)
0.743781 + 0.668424i \(0.233031\pi\)
\(152\) −545.952 −0.291333
\(153\) 1049.55 0.554583
\(154\) 552.949 0.289337
\(155\) −44.1264 −0.0228666
\(156\) 198.361 0.101805
\(157\) 1922.00 0.977019 0.488509 0.872559i \(-0.337541\pi\)
0.488509 + 0.872559i \(0.337541\pi\)
\(158\) 2995.26 1.50817
\(159\) 1490.05 0.743198
\(160\) 296.340 0.146424
\(161\) −1019.28 −0.498949
\(162\) 209.286 0.101500
\(163\) −2784.04 −1.33781 −0.668904 0.743349i \(-0.733236\pi\)
−0.668904 + 0.743349i \(0.733236\pi\)
\(164\) −404.376 −0.192540
\(165\) 120.501 0.0568544
\(166\) 1727.50 0.807711
\(167\) −1669.58 −0.773627 −0.386814 0.922158i \(-0.626424\pi\)
−0.386814 + 0.922158i \(0.626424\pi\)
\(168\) −1925.37 −0.884200
\(169\) 296.677 0.135037
\(170\) 1506.56 0.679694
\(171\) 203.956 0.0912098
\(172\) −365.277 −0.161931
\(173\) −3488.62 −1.53315 −0.766574 0.642156i \(-0.778040\pi\)
−0.766574 + 0.642156i \(0.778040\pi\)
\(174\) −224.789 −0.0979379
\(175\) −665.996 −0.287683
\(176\) 414.958 0.177719
\(177\) 1690.51 0.717892
\(178\) 2062.63 0.868543
\(179\) 4016.48 1.67713 0.838564 0.544803i \(-0.183396\pi\)
0.838564 + 0.544803i \(0.183396\pi\)
\(180\) −59.5837 −0.0246728
\(181\) −2275.81 −0.934583 −0.467291 0.884103i \(-0.654770\pi\)
−0.467291 + 0.884103i \(0.654770\pi\)
\(182\) −3437.22 −1.39991
\(183\) −1456.04 −0.588161
\(184\) −921.775 −0.369316
\(185\) 1505.11 0.598150
\(186\) 68.4077 0.0269672
\(187\) −936.827 −0.366351
\(188\) 634.610 0.246190
\(189\) 719.275 0.276823
\(190\) 292.765 0.111786
\(191\) 3204.35 1.21392 0.606961 0.794732i \(-0.292389\pi\)
0.606961 + 0.794732i \(0.292389\pi\)
\(192\) −1699.11 −0.638659
\(193\) −71.0310 −0.0264918 −0.0132459 0.999912i \(-0.504216\pi\)
−0.0132459 + 0.999912i \(0.504216\pi\)
\(194\) 2240.44 0.829144
\(195\) −749.051 −0.275080
\(196\) −485.515 −0.176937
\(197\) 1260.52 0.455881 0.227941 0.973675i \(-0.426801\pi\)
0.227941 + 0.973675i \(0.426801\pi\)
\(198\) −186.808 −0.0670500
\(199\) 2365.21 0.842539 0.421270 0.906935i \(-0.361585\pi\)
0.421270 + 0.906935i \(0.361585\pi\)
\(200\) −602.284 −0.212940
\(201\) −651.529 −0.228633
\(202\) 3282.87 1.14347
\(203\) −772.555 −0.267107
\(204\) 463.231 0.158984
\(205\) 1527.01 0.520247
\(206\) −2285.60 −0.773036
\(207\) 344.355 0.115625
\(208\) −2579.44 −0.859865
\(209\) −182.050 −0.0602521
\(210\) 1032.47 0.339273
\(211\) −1604.41 −0.523468 −0.261734 0.965140i \(-0.584294\pi\)
−0.261734 + 0.965140i \(0.584294\pi\)
\(212\) 657.649 0.213054
\(213\) −1884.07 −0.606076
\(214\) 2392.42 0.764216
\(215\) 1379.36 0.437542
\(216\) 650.467 0.204901
\(217\) 235.104 0.0735479
\(218\) 189.905 0.0589999
\(219\) 2047.50 0.631770
\(220\) 53.1843 0.0162986
\(221\) 5823.46 1.77253
\(222\) −2333.32 −0.705415
\(223\) −2789.26 −0.837591 −0.418795 0.908081i \(-0.637548\pi\)
−0.418795 + 0.908081i \(0.637548\pi\)
\(224\) −1578.89 −0.470956
\(225\) 225.000 0.0666667
\(226\) −5345.02 −1.57321
\(227\) 3091.67 0.903971 0.451985 0.892025i \(-0.350716\pi\)
0.451985 + 0.892025i \(0.350716\pi\)
\(228\) 90.0180 0.0261473
\(229\) −6479.99 −1.86991 −0.934955 0.354766i \(-0.884561\pi\)
−0.934955 + 0.354766i \(0.884561\pi\)
\(230\) 494.298 0.141709
\(231\) −642.024 −0.182866
\(232\) −698.650 −0.197710
\(233\) −3647.01 −1.02542 −0.512712 0.858561i \(-0.671359\pi\)
−0.512712 + 0.858561i \(0.671359\pi\)
\(234\) 1161.23 0.324410
\(235\) −2396.41 −0.665211
\(236\) 746.127 0.205800
\(237\) −3477.77 −0.953187
\(238\) −8026.90 −2.18616
\(239\) 3863.00 1.04551 0.522754 0.852483i \(-0.324905\pi\)
0.522754 + 0.852483i \(0.324905\pi\)
\(240\) 774.812 0.208391
\(241\) 6207.76 1.65924 0.829620 0.558329i \(-0.188557\pi\)
0.829620 + 0.558329i \(0.188557\pi\)
\(242\) −3272.27 −0.869212
\(243\) −243.000 −0.0641500
\(244\) −642.638 −0.168609
\(245\) 1833.40 0.478089
\(246\) −2367.27 −0.613543
\(247\) 1131.65 0.291520
\(248\) 212.613 0.0544393
\(249\) −2005.78 −0.510487
\(250\) 322.972 0.0817063
\(251\) 4942.68 1.24294 0.621472 0.783436i \(-0.286535\pi\)
0.621472 + 0.783436i \(0.286535\pi\)
\(252\) 317.460 0.0793575
\(253\) −307.370 −0.0763803
\(254\) −1831.22 −0.452367
\(255\) −1749.25 −0.429578
\(256\) −1975.00 −0.482179
\(257\) 3343.25 0.811465 0.405732 0.913992i \(-0.367016\pi\)
0.405732 + 0.913992i \(0.367016\pi\)
\(258\) −2138.38 −0.516006
\(259\) −8019.16 −1.92389
\(260\) −330.602 −0.0788579
\(261\) 261.000 0.0618984
\(262\) −5254.88 −1.23911
\(263\) 1053.61 0.247027 0.123514 0.992343i \(-0.460584\pi\)
0.123514 + 0.992343i \(0.460584\pi\)
\(264\) −580.606 −0.135355
\(265\) −2483.41 −0.575679
\(266\) −1559.84 −0.359549
\(267\) −2394.90 −0.548934
\(268\) −287.559 −0.0655428
\(269\) 2348.05 0.532205 0.266103 0.963945i \(-0.414264\pi\)
0.266103 + 0.963945i \(0.414264\pi\)
\(270\) −348.810 −0.0786219
\(271\) 4230.65 0.948316 0.474158 0.880440i \(-0.342753\pi\)
0.474158 + 0.880440i \(0.342753\pi\)
\(272\) −6023.74 −1.34280
\(273\) 3990.92 0.884767
\(274\) 6810.74 1.50165
\(275\) −200.835 −0.0440392
\(276\) 151.985 0.0331464
\(277\) 2146.72 0.465647 0.232823 0.972519i \(-0.425204\pi\)
0.232823 + 0.972519i \(0.425204\pi\)
\(278\) −3317.71 −0.715766
\(279\) −79.4275 −0.0170437
\(280\) 3208.95 0.684898
\(281\) −1445.80 −0.306937 −0.153469 0.988154i \(-0.549044\pi\)
−0.153469 + 0.988154i \(0.549044\pi\)
\(282\) 3715.08 0.784503
\(283\) 3775.85 0.793113 0.396557 0.918010i \(-0.370205\pi\)
0.396557 + 0.918010i \(0.370205\pi\)
\(284\) −831.554 −0.173745
\(285\) −339.926 −0.0706508
\(286\) −1036.51 −0.214301
\(287\) −8135.83 −1.67332
\(288\) 533.413 0.109138
\(289\) 8686.48 1.76806
\(290\) 374.648 0.0758624
\(291\) −2601.35 −0.524033
\(292\) 903.688 0.181111
\(293\) 6451.38 1.28633 0.643164 0.765729i \(-0.277622\pi\)
0.643164 + 0.765729i \(0.277622\pi\)
\(294\) −2842.26 −0.563824
\(295\) −2817.52 −0.556076
\(296\) −7252.02 −1.42404
\(297\) 216.901 0.0423767
\(298\) 676.544 0.131514
\(299\) 1910.66 0.369553
\(300\) 99.3062 0.0191115
\(301\) −7349.18 −1.40731
\(302\) 7131.74 1.35889
\(303\) −3811.70 −0.722695
\(304\) −1170.57 −0.220845
\(305\) 2426.73 0.455587
\(306\) 2711.81 0.506614
\(307\) 4197.07 0.780258 0.390129 0.920760i \(-0.372430\pi\)
0.390129 + 0.920760i \(0.372430\pi\)
\(308\) −283.364 −0.0524227
\(309\) 2653.79 0.488572
\(310\) −114.013 −0.0208887
\(311\) 6909.28 1.25977 0.629886 0.776687i \(-0.283102\pi\)
0.629886 + 0.776687i \(0.283102\pi\)
\(312\) 3609.13 0.654894
\(313\) 835.311 0.150845 0.0754226 0.997152i \(-0.475969\pi\)
0.0754226 + 0.997152i \(0.475969\pi\)
\(314\) 4966.01 0.892510
\(315\) −1198.79 −0.214426
\(316\) −1534.95 −0.273252
\(317\) −454.133 −0.0804626 −0.0402313 0.999190i \(-0.512809\pi\)
−0.0402313 + 0.999190i \(0.512809\pi\)
\(318\) 3849.96 0.678915
\(319\) −232.968 −0.0408894
\(320\) 2831.84 0.494703
\(321\) −2777.81 −0.482998
\(322\) −2633.60 −0.455792
\(323\) 2642.74 0.455250
\(324\) −107.251 −0.0183900
\(325\) 1248.42 0.213076
\(326\) −7193.34 −1.22209
\(327\) −220.497 −0.0372890
\(328\) −7357.53 −1.23857
\(329\) 12768.0 2.13958
\(330\) 311.347 0.0519367
\(331\) −6368.25 −1.05749 −0.528747 0.848779i \(-0.677338\pi\)
−0.528747 + 0.848779i \(0.677338\pi\)
\(332\) −885.274 −0.146343
\(333\) 2709.19 0.445835
\(334\) −4313.82 −0.706712
\(335\) 1085.88 0.177099
\(336\) −4128.17 −0.670269
\(337\) −8096.70 −1.30877 −0.654384 0.756162i \(-0.727072\pi\)
−0.654384 + 0.756162i \(0.727072\pi\)
\(338\) 766.547 0.123357
\(339\) 6206.05 0.994297
\(340\) −772.051 −0.123148
\(341\) 70.8968 0.0112589
\(342\) 526.976 0.0833205
\(343\) −630.841 −0.0993067
\(344\) −6646.13 −1.04167
\(345\) −573.924 −0.0895625
\(346\) −9013.82 −1.40054
\(347\) −2438.20 −0.377202 −0.188601 0.982054i \(-0.560395\pi\)
−0.188601 + 0.982054i \(0.560395\pi\)
\(348\) 115.195 0.0177446
\(349\) −777.284 −0.119218 −0.0596090 0.998222i \(-0.518985\pi\)
−0.0596090 + 0.998222i \(0.518985\pi\)
\(350\) −1720.79 −0.262800
\(351\) −1348.29 −0.205033
\(352\) −476.123 −0.0720950
\(353\) −9649.81 −1.45498 −0.727489 0.686119i \(-0.759313\pi\)
−0.727489 + 0.686119i \(0.759313\pi\)
\(354\) 4367.92 0.655797
\(355\) 3140.11 0.469465
\(356\) −1057.02 −0.157364
\(357\) 9319.95 1.38169
\(358\) 10377.7 1.53206
\(359\) 9657.56 1.41980 0.709898 0.704304i \(-0.248741\pi\)
0.709898 + 0.704304i \(0.248741\pi\)
\(360\) −1084.11 −0.158716
\(361\) −6345.45 −0.925127
\(362\) −5880.19 −0.853745
\(363\) 3799.39 0.549357
\(364\) 1761.43 0.253638
\(365\) −3412.51 −0.489367
\(366\) −3762.08 −0.537287
\(367\) −9173.93 −1.30484 −0.652419 0.757859i \(-0.726246\pi\)
−0.652419 + 0.757859i \(0.726246\pi\)
\(368\) −1976.37 −0.279961
\(369\) 2748.61 0.387770
\(370\) 3888.87 0.546412
\(371\) 13231.5 1.85161
\(372\) −35.0562 −0.00488597
\(373\) 2899.69 0.402521 0.201261 0.979538i \(-0.435496\pi\)
0.201261 + 0.979538i \(0.435496\pi\)
\(374\) −2420.56 −0.334663
\(375\) −375.000 −0.0516398
\(376\) 11546.6 1.58369
\(377\) 1448.17 0.197836
\(378\) 1858.45 0.252879
\(379\) −5819.59 −0.788739 −0.394370 0.918952i \(-0.629037\pi\)
−0.394370 + 0.918952i \(0.629037\pi\)
\(380\) −150.030 −0.0202536
\(381\) 2126.21 0.285903
\(382\) 8279.34 1.10892
\(383\) −5805.80 −0.774576 −0.387288 0.921959i \(-0.626588\pi\)
−0.387288 + 0.921959i \(0.626588\pi\)
\(384\) −2967.68 −0.394385
\(385\) 1070.04 0.141647
\(386\) −183.528 −0.0242004
\(387\) 2482.85 0.326125
\(388\) −1148.13 −0.150226
\(389\) 7690.82 1.00242 0.501208 0.865327i \(-0.332889\pi\)
0.501208 + 0.865327i \(0.332889\pi\)
\(390\) −1935.38 −0.251287
\(391\) 4461.95 0.577111
\(392\) −8833.83 −1.13820
\(393\) 6101.39 0.783141
\(394\) 3256.92 0.416449
\(395\) 5796.28 0.738336
\(396\) 95.7318 0.0121482
\(397\) −1638.24 −0.207105 −0.103553 0.994624i \(-0.533021\pi\)
−0.103553 + 0.994624i \(0.533021\pi\)
\(398\) 6111.18 0.769663
\(399\) 1811.11 0.227241
\(400\) −1291.35 −0.161419
\(401\) 3780.89 0.470845 0.235422 0.971893i \(-0.424353\pi\)
0.235422 + 0.971893i \(0.424353\pi\)
\(402\) −1683.41 −0.208857
\(403\) −440.705 −0.0544742
\(404\) −1682.34 −0.207177
\(405\) 405.000 0.0496904
\(406\) −1996.11 −0.244003
\(407\) −2418.22 −0.294513
\(408\) 8428.38 1.02271
\(409\) −330.980 −0.0400145 −0.0200073 0.999800i \(-0.506369\pi\)
−0.0200073 + 0.999800i \(0.506369\pi\)
\(410\) 3945.45 0.475248
\(411\) −7907.88 −0.949068
\(412\) 1171.28 0.140060
\(413\) 15011.7 1.78856
\(414\) 889.736 0.105624
\(415\) 3342.97 0.395422
\(416\) 2959.65 0.348820
\(417\) 3852.16 0.452376
\(418\) −470.378 −0.0550405
\(419\) 9913.43 1.15585 0.577927 0.816089i \(-0.303862\pi\)
0.577927 + 0.816089i \(0.303862\pi\)
\(420\) −529.100 −0.0614701
\(421\) 6286.56 0.727763 0.363881 0.931445i \(-0.381451\pi\)
0.363881 + 0.931445i \(0.381451\pi\)
\(422\) −4145.43 −0.478190
\(423\) −4313.54 −0.495819
\(424\) 11965.8 1.37054
\(425\) 2915.42 0.332750
\(426\) −4868.02 −0.553653
\(427\) −12929.5 −1.46535
\(428\) −1226.02 −0.138462
\(429\) 1203.48 0.135442
\(430\) 3563.96 0.399696
\(431\) 5798.56 0.648044 0.324022 0.946050i \(-0.394965\pi\)
0.324022 + 0.946050i \(0.394965\pi\)
\(432\) 1394.66 0.155326
\(433\) 11476.3 1.27371 0.636856 0.770983i \(-0.280235\pi\)
0.636856 + 0.770983i \(0.280235\pi\)
\(434\) 607.457 0.0671863
\(435\) −435.000 −0.0479463
\(436\) −97.3186 −0.0106897
\(437\) 867.075 0.0949149
\(438\) 5290.30 0.577124
\(439\) 14429.5 1.56875 0.784377 0.620285i \(-0.212983\pi\)
0.784377 + 0.620285i \(0.212983\pi\)
\(440\) 967.676 0.104846
\(441\) 3300.12 0.356346
\(442\) 15046.5 1.61921
\(443\) −11892.0 −1.27541 −0.637703 0.770283i \(-0.720115\pi\)
−0.637703 + 0.770283i \(0.720115\pi\)
\(444\) 1195.73 0.127808
\(445\) 3991.50 0.425203
\(446\) −7206.84 −0.765143
\(447\) −785.528 −0.0831190
\(448\) −15088.0 −1.59116
\(449\) −9881.53 −1.03862 −0.519308 0.854587i \(-0.673810\pi\)
−0.519308 + 0.854587i \(0.673810\pi\)
\(450\) 581.350 0.0609003
\(451\) −2453.40 −0.256156
\(452\) 2739.11 0.285037
\(453\) −8280.59 −0.858844
\(454\) 7988.20 0.825781
\(455\) −6651.53 −0.685337
\(456\) 1637.86 0.168201
\(457\) 11641.2 1.19158 0.595792 0.803139i \(-0.296838\pi\)
0.595792 + 0.803139i \(0.296838\pi\)
\(458\) −16742.9 −1.70817
\(459\) −3148.65 −0.320189
\(460\) −253.308 −0.0256751
\(461\) −8045.29 −0.812812 −0.406406 0.913693i \(-0.633218\pi\)
−0.406406 + 0.913693i \(0.633218\pi\)
\(462\) −1658.85 −0.167049
\(463\) −4340.54 −0.435684 −0.217842 0.975984i \(-0.569902\pi\)
−0.217842 + 0.975984i \(0.569902\pi\)
\(464\) −1497.97 −0.149874
\(465\) 132.379 0.0132020
\(466\) −9423.08 −0.936729
\(467\) −3461.65 −0.343011 −0.171506 0.985183i \(-0.554863\pi\)
−0.171506 + 0.985183i \(0.554863\pi\)
\(468\) −595.083 −0.0587772
\(469\) −5785.54 −0.569619
\(470\) −6191.80 −0.607673
\(471\) −5765.99 −0.564082
\(472\) 13575.6 1.32387
\(473\) −2216.18 −0.215434
\(474\) −8985.79 −0.870740
\(475\) 566.543 0.0547259
\(476\) 4113.46 0.396093
\(477\) −4470.14 −0.429086
\(478\) 9981.14 0.955076
\(479\) 6284.62 0.599481 0.299741 0.954021i \(-0.403100\pi\)
0.299741 + 0.954021i \(0.403100\pi\)
\(480\) −889.021 −0.0845377
\(481\) 15032.0 1.42495
\(482\) 16039.5 1.51572
\(483\) 3057.85 0.288068
\(484\) 1676.90 0.157485
\(485\) 4335.58 0.405915
\(486\) −627.858 −0.0586013
\(487\) −1809.63 −0.168382 −0.0841912 0.996450i \(-0.526831\pi\)
−0.0841912 + 0.996450i \(0.526831\pi\)
\(488\) −11692.7 −1.08463
\(489\) 8352.12 0.772384
\(490\) 4737.11 0.436736
\(491\) 9550.28 0.877796 0.438898 0.898537i \(-0.355369\pi\)
0.438898 + 0.898537i \(0.355369\pi\)
\(492\) 1213.13 0.111163
\(493\) 3381.89 0.308950
\(494\) 2923.94 0.266304
\(495\) −361.502 −0.0328249
\(496\) 455.862 0.0412678
\(497\) −16730.4 −1.50998
\(498\) −5182.50 −0.466332
\(499\) −13672.9 −1.22662 −0.613309 0.789843i \(-0.710162\pi\)
−0.613309 + 0.789843i \(0.710162\pi\)
\(500\) −165.510 −0.0148037
\(501\) 5008.73 0.446654
\(502\) 12770.8 1.13544
\(503\) 13406.0 1.18836 0.594181 0.804332i \(-0.297476\pi\)
0.594181 + 0.804332i \(0.297476\pi\)
\(504\) 5776.11 0.510493
\(505\) 6352.84 0.559797
\(506\) −794.177 −0.0697737
\(507\) −890.030 −0.0779638
\(508\) 938.428 0.0819606
\(509\) −13537.0 −1.17882 −0.589409 0.807835i \(-0.700639\pi\)
−0.589409 + 0.807835i \(0.700639\pi\)
\(510\) −4519.68 −0.392421
\(511\) 18181.7 1.57400
\(512\) −13016.8 −1.12357
\(513\) −611.867 −0.0526600
\(514\) 8638.23 0.741276
\(515\) −4422.98 −0.378446
\(516\) 1095.83 0.0934909
\(517\) 3850.26 0.327532
\(518\) −20719.8 −1.75748
\(519\) 10465.9 0.885163
\(520\) −6015.22 −0.507279
\(521\) −6612.84 −0.556073 −0.278036 0.960571i \(-0.589684\pi\)
−0.278036 + 0.960571i \(0.589684\pi\)
\(522\) 674.366 0.0565445
\(523\) −11689.8 −0.977363 −0.488682 0.872462i \(-0.662522\pi\)
−0.488682 + 0.872462i \(0.662522\pi\)
\(524\) 2692.91 0.224505
\(525\) 1997.99 0.166094
\(526\) 2722.29 0.225660
\(527\) −1029.18 −0.0850694
\(528\) −1244.87 −0.102606
\(529\) −10703.0 −0.879678
\(530\) −6416.59 −0.525885
\(531\) −5071.54 −0.414475
\(532\) 799.355 0.0651437
\(533\) 15250.7 1.23937
\(534\) −6187.89 −0.501454
\(535\) 4629.68 0.374128
\(536\) −5232.07 −0.421625
\(537\) −12049.4 −0.968290
\(538\) 6066.85 0.486172
\(539\) −2945.68 −0.235398
\(540\) 178.751 0.0142449
\(541\) −170.114 −0.0135190 −0.00675951 0.999977i \(-0.502152\pi\)
−0.00675951 + 0.999977i \(0.502152\pi\)
\(542\) 10931.1 0.866291
\(543\) 6827.42 0.539582
\(544\) 6911.65 0.544733
\(545\) 367.494 0.0288839
\(546\) 10311.7 0.808238
\(547\) −6453.08 −0.504413 −0.252206 0.967673i \(-0.581156\pi\)
−0.252206 + 0.967673i \(0.581156\pi\)
\(548\) −3490.23 −0.272071
\(549\) 4368.11 0.339575
\(550\) −518.912 −0.0402300
\(551\) 657.190 0.0508117
\(552\) 2765.33 0.213225
\(553\) −30882.4 −2.37478
\(554\) 5546.66 0.425370
\(555\) −4515.32 −0.345342
\(556\) 1700.19 0.129684
\(557\) −12800.7 −0.973759 −0.486879 0.873469i \(-0.661865\pi\)
−0.486879 + 0.873469i \(0.661865\pi\)
\(558\) −205.223 −0.0155695
\(559\) 13776.1 1.04234
\(560\) 6880.29 0.519188
\(561\) 2810.48 0.211513
\(562\) −3735.63 −0.280388
\(563\) 6454.48 0.483169 0.241584 0.970380i \(-0.422333\pi\)
0.241584 + 0.970380i \(0.422333\pi\)
\(564\) −1903.83 −0.142138
\(565\) −10343.4 −0.770179
\(566\) 9755.97 0.724512
\(567\) −2157.83 −0.159824
\(568\) −15129.9 −1.11767
\(569\) 2900.88 0.213728 0.106864 0.994274i \(-0.465919\pi\)
0.106864 + 0.994274i \(0.465919\pi\)
\(570\) −878.294 −0.0645398
\(571\) −13437.3 −0.984823 −0.492411 0.870363i \(-0.663884\pi\)
−0.492411 + 0.870363i \(0.663884\pi\)
\(572\) 531.170 0.0388275
\(573\) −9613.06 −0.700858
\(574\) −21021.2 −1.52859
\(575\) 956.541 0.0693748
\(576\) 5097.32 0.368730
\(577\) −5402.33 −0.389778 −0.194889 0.980825i \(-0.562435\pi\)
−0.194889 + 0.980825i \(0.562435\pi\)
\(578\) 22443.9 1.61513
\(579\) 213.093 0.0152951
\(580\) −191.992 −0.0137449
\(581\) −17811.2 −1.27183
\(582\) −6721.31 −0.478707
\(583\) 3990.04 0.283449
\(584\) 16442.4 1.16505
\(585\) 2247.15 0.158818
\(586\) 16669.0 1.17507
\(587\) −17702.2 −1.24471 −0.622357 0.782733i \(-0.713825\pi\)
−0.622357 + 0.782733i \(0.713825\pi\)
\(588\) 1456.55 0.102155
\(589\) −199.996 −0.0139910
\(590\) −7279.86 −0.507978
\(591\) −3781.57 −0.263203
\(592\) −15549.0 −1.07949
\(593\) −13235.5 −0.916558 −0.458279 0.888808i \(-0.651534\pi\)
−0.458279 + 0.888808i \(0.651534\pi\)
\(594\) 560.425 0.0387113
\(595\) −15533.3 −1.07025
\(596\) −346.701 −0.0238279
\(597\) −7095.63 −0.486440
\(598\) 4936.73 0.337588
\(599\) −22931.4 −1.56419 −0.782095 0.623159i \(-0.785849\pi\)
−0.782095 + 0.623159i \(0.785849\pi\)
\(600\) 1806.85 0.122941
\(601\) −11714.0 −0.795051 −0.397525 0.917591i \(-0.630131\pi\)
−0.397525 + 0.917591i \(0.630131\pi\)
\(602\) −18988.7 −1.28558
\(603\) 1954.59 0.132001
\(604\) −3654.73 −0.246207
\(605\) −6332.32 −0.425530
\(606\) −9848.60 −0.660185
\(607\) 4567.18 0.305397 0.152699 0.988273i \(-0.451204\pi\)
0.152699 + 0.988273i \(0.451204\pi\)
\(608\) 1343.12 0.0895898
\(609\) 2317.66 0.154214
\(610\) 6270.14 0.416181
\(611\) −23933.8 −1.58471
\(612\) −1389.69 −0.0917892
\(613\) −19389.4 −1.27754 −0.638768 0.769399i \(-0.720556\pi\)
−0.638768 + 0.769399i \(0.720556\pi\)
\(614\) 10844.3 0.712769
\(615\) −4581.02 −0.300365
\(616\) −5155.75 −0.337226
\(617\) −1918.52 −0.125181 −0.0625905 0.998039i \(-0.519936\pi\)
−0.0625905 + 0.998039i \(0.519936\pi\)
\(618\) 6856.81 0.446313
\(619\) 25281.1 1.64157 0.820785 0.571237i \(-0.193536\pi\)
0.820785 + 0.571237i \(0.193536\pi\)
\(620\) 58.4270 0.00378465
\(621\) −1033.06 −0.0667559
\(622\) 17852.1 1.15081
\(623\) −21266.6 −1.36762
\(624\) 7738.32 0.496443
\(625\) 625.000 0.0400000
\(626\) 2158.26 0.137798
\(627\) 546.151 0.0347866
\(628\) −2544.88 −0.161707
\(629\) 35104.2 2.22527
\(630\) −3097.41 −0.195879
\(631\) −21718.6 −1.37022 −0.685108 0.728442i \(-0.740245\pi\)
−0.685108 + 0.728442i \(0.740245\pi\)
\(632\) −27928.1 −1.75778
\(633\) 4813.22 0.302225
\(634\) −1173.38 −0.0735029
\(635\) −3543.69 −0.221460
\(636\) −1972.95 −0.123007
\(637\) 18310.8 1.13893
\(638\) −601.938 −0.0373526
\(639\) 5652.20 0.349918
\(640\) 4946.14 0.305489
\(641\) 9371.00 0.577429 0.288715 0.957415i \(-0.406772\pi\)
0.288715 + 0.957415i \(0.406772\pi\)
\(642\) −7177.25 −0.441220
\(643\) −7538.94 −0.462375 −0.231187 0.972909i \(-0.574261\pi\)
−0.231187 + 0.972909i \(0.574261\pi\)
\(644\) 1349.62 0.0825812
\(645\) −4138.08 −0.252615
\(646\) 6828.25 0.415873
\(647\) 15345.6 0.932455 0.466228 0.884665i \(-0.345613\pi\)
0.466228 + 0.884665i \(0.345613\pi\)
\(648\) −1951.40 −0.118300
\(649\) 4526.85 0.273797
\(650\) 3225.64 0.194646
\(651\) −705.312 −0.0424629
\(652\) 3686.30 0.221421
\(653\) −19648.0 −1.17747 −0.588735 0.808326i \(-0.700374\pi\)
−0.588735 + 0.808326i \(0.700374\pi\)
\(654\) −569.715 −0.0340636
\(655\) −10169.0 −0.606618
\(656\) −15775.2 −0.938902
\(657\) −6142.51 −0.364752
\(658\) 32989.7 1.95452
\(659\) −24207.8 −1.43096 −0.715481 0.698633i \(-0.753792\pi\)
−0.715481 + 0.698633i \(0.753792\pi\)
\(660\) −159.553 −0.00940999
\(661\) 3864.20 0.227383 0.113691 0.993516i \(-0.463733\pi\)
0.113691 + 0.993516i \(0.463733\pi\)
\(662\) −16454.2 −0.966026
\(663\) −17470.4 −1.02337
\(664\) −16107.3 −0.941395
\(665\) −3018.52 −0.176020
\(666\) 6999.96 0.407272
\(667\) 1109.59 0.0644129
\(668\) 2210.66 0.128043
\(669\) 8367.78 0.483583
\(670\) 2805.68 0.161780
\(671\) −3898.97 −0.224319
\(672\) 4736.67 0.271907
\(673\) 17574.4 1.00660 0.503300 0.864112i \(-0.332119\pi\)
0.503300 + 0.864112i \(0.332119\pi\)
\(674\) −20920.1 −1.19557
\(675\) −675.000 −0.0384900
\(676\) −392.825 −0.0223501
\(677\) −26672.9 −1.51421 −0.757106 0.653292i \(-0.773387\pi\)
−0.757106 + 0.653292i \(0.773387\pi\)
\(678\) 16035.1 0.908294
\(679\) −23099.8 −1.30558
\(680\) −14047.3 −0.792190
\(681\) −9275.01 −0.521908
\(682\) 183.182 0.0102850
\(683\) 2985.36 0.167250 0.0836249 0.996497i \(-0.473350\pi\)
0.0836249 + 0.996497i \(0.473350\pi\)
\(684\) −270.054 −0.0150962
\(685\) 13179.8 0.735145
\(686\) −1629.95 −0.0907171
\(687\) 19440.0 1.07959
\(688\) −14249.9 −0.789642
\(689\) −24802.7 −1.37142
\(690\) −1482.89 −0.0818157
\(691\) 15067.6 0.829518 0.414759 0.909931i \(-0.363866\pi\)
0.414759 + 0.909931i \(0.363866\pi\)
\(692\) 4619.22 0.253752
\(693\) 1926.07 0.105578
\(694\) −6299.76 −0.344576
\(695\) −6420.26 −0.350409
\(696\) 2095.95 0.114148
\(697\) 35614.9 1.93545
\(698\) −2008.33 −0.108906
\(699\) 10941.0 0.592029
\(700\) 881.833 0.0476145
\(701\) 32060.7 1.72741 0.863707 0.503994i \(-0.168137\pi\)
0.863707 + 0.503994i \(0.168137\pi\)
\(702\) −3483.69 −0.187298
\(703\) 6821.67 0.365980
\(704\) −4549.86 −0.243578
\(705\) 7189.24 0.384060
\(706\) −24933.0 −1.32913
\(707\) −33847.7 −1.80053
\(708\) −2238.38 −0.118818
\(709\) −6089.76 −0.322575 −0.161288 0.986907i \(-0.551565\pi\)
−0.161288 + 0.986907i \(0.551565\pi\)
\(710\) 8113.36 0.428858
\(711\) 10433.3 0.550323
\(712\) −19232.1 −1.01230
\(713\) −337.669 −0.0177361
\(714\) 24080.7 1.26218
\(715\) −2005.80 −0.104913
\(716\) −5318.15 −0.277582
\(717\) −11589.0 −0.603625
\(718\) 24953.0 1.29699
\(719\) 2735.79 0.141902 0.0709511 0.997480i \(-0.477397\pi\)
0.0709511 + 0.997480i \(0.477397\pi\)
\(720\) −2324.44 −0.120315
\(721\) 23565.5 1.21723
\(722\) −16395.2 −0.845107
\(723\) −18623.3 −0.957963
\(724\) 3013.36 0.154683
\(725\) 725.000 0.0371391
\(726\) 9816.80 0.501840
\(727\) −23072.1 −1.17702 −0.588511 0.808489i \(-0.700286\pi\)
−0.588511 + 0.808489i \(0.700286\pi\)
\(728\) 32048.9 1.63161
\(729\) 729.000 0.0370370
\(730\) −8817.17 −0.447038
\(731\) 32171.3 1.62777
\(732\) 1927.92 0.0973467
\(733\) 2341.92 0.118009 0.0590047 0.998258i \(-0.481207\pi\)
0.0590047 + 0.998258i \(0.481207\pi\)
\(734\) −23703.4 −1.19197
\(735\) −5500.21 −0.276025
\(736\) 2267.69 0.113571
\(737\) −1744.66 −0.0871986
\(738\) 7101.80 0.354229
\(739\) 30372.4 1.51186 0.755932 0.654651i \(-0.227184\pi\)
0.755932 + 0.654651i \(0.227184\pi\)
\(740\) −1992.89 −0.0990000
\(741\) −3394.96 −0.168309
\(742\) 34187.4 1.69145
\(743\) 16456.7 0.812566 0.406283 0.913747i \(-0.366825\pi\)
0.406283 + 0.913747i \(0.366825\pi\)
\(744\) −637.839 −0.0314305
\(745\) 1309.21 0.0643837
\(746\) 7492.17 0.367705
\(747\) 6017.35 0.294730
\(748\) 1240.44 0.0606349
\(749\) −24666.8 −1.20334
\(750\) −968.917 −0.0471731
\(751\) −28265.3 −1.37339 −0.686694 0.726947i \(-0.740939\pi\)
−0.686694 + 0.726947i \(0.740939\pi\)
\(752\) 24756.9 1.20052
\(753\) −14828.0 −0.717615
\(754\) 3741.74 0.180724
\(755\) 13801.0 0.665258
\(756\) −952.380 −0.0458171
\(757\) 30213.7 1.45064 0.725320 0.688412i \(-0.241692\pi\)
0.725320 + 0.688412i \(0.241692\pi\)
\(758\) −15036.5 −0.720517
\(759\) 922.111 0.0440982
\(760\) −2729.76 −0.130288
\(761\) 7521.30 0.358274 0.179137 0.983824i \(-0.442669\pi\)
0.179137 + 0.983824i \(0.442669\pi\)
\(762\) 5493.67 0.261174
\(763\) −1958.00 −0.0929020
\(764\) −4242.83 −0.200916
\(765\) 5247.76 0.248017
\(766\) −15000.9 −0.707578
\(767\) −28139.6 −1.32472
\(768\) 5925.01 0.278386
\(769\) −31256.5 −1.46572 −0.732859 0.680380i \(-0.761815\pi\)
−0.732859 + 0.680380i \(0.761815\pi\)
\(770\) 2764.75 0.129396
\(771\) −10029.8 −0.468499
\(772\) 94.0510 0.00438467
\(773\) 6657.91 0.309791 0.154895 0.987931i \(-0.450496\pi\)
0.154895 + 0.987931i \(0.450496\pi\)
\(774\) 6415.13 0.297916
\(775\) −220.632 −0.0102262
\(776\) −20890.0 −0.966376
\(777\) 24057.5 1.11076
\(778\) 19871.4 0.915711
\(779\) 6920.92 0.318315
\(780\) 991.805 0.0455286
\(781\) −5045.15 −0.231152
\(782\) 11528.7 0.527193
\(783\) −783.000 −0.0357371
\(784\) −18940.6 −0.862817
\(785\) 9609.98 0.436936
\(786\) 15764.6 0.715402
\(787\) 18918.1 0.856872 0.428436 0.903572i \(-0.359065\pi\)
0.428436 + 0.903572i \(0.359065\pi\)
\(788\) −1669.04 −0.0754531
\(789\) −3160.82 −0.142621
\(790\) 14976.3 0.674473
\(791\) 55109.4 2.47720
\(792\) 1741.82 0.0781475
\(793\) 24236.6 1.08533
\(794\) −4232.84 −0.189191
\(795\) 7450.24 0.332368
\(796\) −3131.73 −0.139449
\(797\) −16329.2 −0.725734 −0.362867 0.931841i \(-0.618202\pi\)
−0.362867 + 0.931841i \(0.618202\pi\)
\(798\) 4679.52 0.207585
\(799\) −55892.4 −2.47476
\(800\) 1481.70 0.0654826
\(801\) 7184.70 0.316927
\(802\) 9768.99 0.430119
\(803\) 5482.80 0.240951
\(804\) 862.678 0.0378412
\(805\) −5096.42 −0.223137
\(806\) −1138.69 −0.0497624
\(807\) −7044.16 −0.307269
\(808\) −30609.7 −1.33273
\(809\) −2755.55 −0.119753 −0.0598764 0.998206i \(-0.519071\pi\)
−0.0598764 + 0.998206i \(0.519071\pi\)
\(810\) 1046.43 0.0453924
\(811\) −8552.83 −0.370321 −0.185161 0.982708i \(-0.559281\pi\)
−0.185161 + 0.982708i \(0.559281\pi\)
\(812\) 1022.93 0.0442090
\(813\) −12692.0 −0.547511
\(814\) −6248.15 −0.269039
\(815\) −13920.2 −0.598286
\(816\) 18071.2 0.775269
\(817\) 6251.73 0.267712
\(818\) −855.181 −0.0365534
\(819\) −11972.8 −0.510820
\(820\) −2021.88 −0.0861063
\(821\) 20670.4 0.878686 0.439343 0.898319i \(-0.355211\pi\)
0.439343 + 0.898319i \(0.355211\pi\)
\(822\) −20432.2 −0.866977
\(823\) 474.045 0.0200780 0.0100390 0.999950i \(-0.496804\pi\)
0.0100390 + 0.999950i \(0.496804\pi\)
\(824\) 21311.1 0.900981
\(825\) 602.504 0.0254260
\(826\) 38786.8 1.63386
\(827\) 25547.5 1.07421 0.537106 0.843515i \(-0.319517\pi\)
0.537106 + 0.843515i \(0.319517\pi\)
\(828\) −455.954 −0.0191371
\(829\) −36219.9 −1.51745 −0.758726 0.651410i \(-0.774178\pi\)
−0.758726 + 0.651410i \(0.774178\pi\)
\(830\) 8637.50 0.361219
\(831\) −6440.17 −0.268841
\(832\) 28282.6 1.17851
\(833\) 42761.1 1.77861
\(834\) 9953.13 0.413248
\(835\) −8347.89 −0.345977
\(836\) 241.050 0.00997235
\(837\) 238.283 0.00984020
\(838\) 25614.1 1.05588
\(839\) −17875.0 −0.735535 −0.367767 0.929918i \(-0.619878\pi\)
−0.367767 + 0.929918i \(0.619878\pi\)
\(840\) −9626.85 −0.395426
\(841\) 841.000 0.0344828
\(842\) 16243.1 0.664814
\(843\) 4337.41 0.177210
\(844\) 2124.37 0.0866394
\(845\) 1483.38 0.0603905
\(846\) −11145.2 −0.452933
\(847\) 33738.4 1.36867
\(848\) 25655.7 1.03894
\(849\) −11327.6 −0.457904
\(850\) 7532.80 0.303968
\(851\) 11517.6 0.463945
\(852\) 2494.66 0.100312
\(853\) −1402.84 −0.0563097 −0.0281549 0.999604i \(-0.508963\pi\)
−0.0281549 + 0.999604i \(0.508963\pi\)
\(854\) −33407.1 −1.33860
\(855\) 1019.78 0.0407903
\(856\) −22307.1 −0.890702
\(857\) 32810.4 1.30780 0.653898 0.756582i \(-0.273132\pi\)
0.653898 + 0.756582i \(0.273132\pi\)
\(858\) 3109.53 0.123727
\(859\) −19007.6 −0.754982 −0.377491 0.926013i \(-0.623213\pi\)
−0.377491 + 0.926013i \(0.623213\pi\)
\(860\) −1826.39 −0.0724177
\(861\) 24407.5 0.966092
\(862\) 14982.2 0.591991
\(863\) −6248.92 −0.246484 −0.123242 0.992377i \(-0.539329\pi\)
−0.123242 + 0.992377i \(0.539329\pi\)
\(864\) −1600.24 −0.0630107
\(865\) −17443.1 −0.685645
\(866\) 29652.3 1.16354
\(867\) −26059.4 −1.02079
\(868\) −311.297 −0.0121729
\(869\) −9312.75 −0.363537
\(870\) −1123.94 −0.0437992
\(871\) 10845.1 0.421896
\(872\) −1770.69 −0.0687650
\(873\) 7804.05 0.302551
\(874\) 2240.33 0.0867051
\(875\) −3329.98 −0.128656
\(876\) −2711.07 −0.104564
\(877\) 18018.5 0.693777 0.346888 0.937906i \(-0.387238\pi\)
0.346888 + 0.937906i \(0.387238\pi\)
\(878\) 37282.7 1.43306
\(879\) −19354.2 −0.742661
\(880\) 2074.79 0.0794785
\(881\) 28136.0 1.07597 0.537983 0.842956i \(-0.319186\pi\)
0.537983 + 0.842956i \(0.319186\pi\)
\(882\) 8526.79 0.325524
\(883\) −41964.1 −1.59933 −0.799663 0.600450i \(-0.794988\pi\)
−0.799663 + 0.600450i \(0.794988\pi\)
\(884\) −7710.74 −0.293371
\(885\) 8452.57 0.321051
\(886\) −30726.2 −1.16509
\(887\) 10526.5 0.398474 0.199237 0.979951i \(-0.436154\pi\)
0.199237 + 0.979951i \(0.436154\pi\)
\(888\) 21756.1 0.822169
\(889\) 18880.7 0.712302
\(890\) 10313.2 0.388424
\(891\) −650.704 −0.0244662
\(892\) 3693.21 0.138630
\(893\) −10861.4 −0.407012
\(894\) −2029.63 −0.0759296
\(895\) 20082.4 0.750034
\(896\) −26352.8 −0.982575
\(897\) −5731.98 −0.213362
\(898\) −25531.7 −0.948780
\(899\) −255.933 −0.00949482
\(900\) −297.919 −0.0110340
\(901\) −57921.6 −2.14167
\(902\) −6339.06 −0.233999
\(903\) 22047.5 0.812510
\(904\) 49837.5 1.83359
\(905\) −11379.0 −0.417958
\(906\) −21395.2 −0.784557
\(907\) −41574.7 −1.52201 −0.761006 0.648744i \(-0.775294\pi\)
−0.761006 + 0.648744i \(0.775294\pi\)
\(908\) −4093.63 −0.149617
\(909\) 11435.1 0.417248
\(910\) −17186.1 −0.626058
\(911\) −978.851 −0.0355991 −0.0177996 0.999842i \(-0.505666\pi\)
−0.0177996 + 0.999842i \(0.505666\pi\)
\(912\) 3511.72 0.127505
\(913\) −5371.07 −0.194695
\(914\) 30078.4 1.08852
\(915\) −7280.19 −0.263034
\(916\) 8580.04 0.309490
\(917\) 54180.0 1.95112
\(918\) −8135.43 −0.292494
\(919\) −2296.62 −0.0824358 −0.0412179 0.999150i \(-0.513124\pi\)
−0.0412179 + 0.999150i \(0.513124\pi\)
\(920\) −4608.88 −0.165163
\(921\) −12591.2 −0.450482
\(922\) −20787.2 −0.742507
\(923\) 31361.4 1.11839
\(924\) 850.093 0.0302662
\(925\) 7525.54 0.267501
\(926\) −11215.0 −0.398000
\(927\) −7961.37 −0.282077
\(928\) 1718.77 0.0607991
\(929\) 29033.8 1.02537 0.512684 0.858578i \(-0.328651\pi\)
0.512684 + 0.858578i \(0.328651\pi\)
\(930\) 342.039 0.0120601
\(931\) 8309.61 0.292520
\(932\) 4828.95 0.169718
\(933\) −20727.8 −0.727330
\(934\) −8944.15 −0.313342
\(935\) −4684.14 −0.163837
\(936\) −10827.4 −0.378103
\(937\) 22945.7 0.800005 0.400002 0.916514i \(-0.369009\pi\)
0.400002 + 0.916514i \(0.369009\pi\)
\(938\) −14948.5 −0.520349
\(939\) −2505.93 −0.0870906
\(940\) 3173.05 0.110099
\(941\) 31712.3 1.09861 0.549304 0.835622i \(-0.314893\pi\)
0.549304 + 0.835622i \(0.314893\pi\)
\(942\) −14898.0 −0.515291
\(943\) 11685.1 0.403521
\(944\) 29107.3 1.00356
\(945\) 3596.38 0.123799
\(946\) −5726.13 −0.196800
\(947\) −7050.99 −0.241950 −0.120975 0.992656i \(-0.538602\pi\)
−0.120975 + 0.992656i \(0.538602\pi\)
\(948\) 4604.85 0.157762
\(949\) −34081.9 −1.16580
\(950\) 1463.82 0.0499923
\(951\) 1362.40 0.0464551
\(952\) 74843.5 2.54800
\(953\) −12178.0 −0.413938 −0.206969 0.978348i \(-0.566360\pi\)
−0.206969 + 0.978348i \(0.566360\pi\)
\(954\) −11549.9 −0.391972
\(955\) 16021.8 0.542882
\(956\) −5114.93 −0.173043
\(957\) 698.904 0.0236075
\(958\) 16238.1 0.547628
\(959\) −70221.5 −2.36452
\(960\) −8495.53 −0.285617
\(961\) −29713.1 −0.997386
\(962\) 38839.5 1.30170
\(963\) 8333.43 0.278859
\(964\) −8219.58 −0.274621
\(965\) −355.155 −0.0118475
\(966\) 7900.81 0.263151
\(967\) −13762.2 −0.457665 −0.228833 0.973466i \(-0.573491\pi\)
−0.228833 + 0.973466i \(0.573491\pi\)
\(968\) 30510.9 1.01308
\(969\) −7928.22 −0.262839
\(970\) 11202.2 0.370805
\(971\) −33646.0 −1.11200 −0.555999 0.831183i \(-0.687664\pi\)
−0.555999 + 0.831183i \(0.687664\pi\)
\(972\) 321.752 0.0106175
\(973\) 34206.9 1.12705
\(974\) −4675.69 −0.153818
\(975\) −3745.25 −0.123020
\(976\) −25070.1 −0.822208
\(977\) −42129.5 −1.37957 −0.689786 0.724013i \(-0.742296\pi\)
−0.689786 + 0.724013i \(0.742296\pi\)
\(978\) 21580.0 0.705576
\(979\) −6413.05 −0.209358
\(980\) −2427.58 −0.0791286
\(981\) 661.490 0.0215288
\(982\) 24675.8 0.801870
\(983\) −39740.6 −1.28945 −0.644724 0.764415i \(-0.723028\pi\)
−0.644724 + 0.764415i \(0.723028\pi\)
\(984\) 22072.6 0.715090
\(985\) 6302.62 0.203876
\(986\) 8738.05 0.282227
\(987\) −38304.0 −1.23529
\(988\) −1498.40 −0.0482495
\(989\) 10555.3 0.339372
\(990\) −934.042 −0.0299857
\(991\) 34300.8 1.09950 0.549748 0.835330i \(-0.314724\pi\)
0.549748 + 0.835330i \(0.314724\pi\)
\(992\) −523.057 −0.0167410
\(993\) 19104.8 0.610545
\(994\) −43227.7 −1.37938
\(995\) 11826.1 0.376795
\(996\) 2655.82 0.0844909
\(997\) −36545.8 −1.16090 −0.580449 0.814296i \(-0.697123\pi\)
−0.580449 + 0.814296i \(0.697123\pi\)
\(998\) −35327.7 −1.12052
\(999\) −8127.58 −0.257403
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.i.1.6 7
3.2 odd 2 1305.4.a.n.1.2 7
5.4 even 2 2175.4.a.n.1.2 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.4.a.i.1.6 7 1.1 even 1 trivial
1305.4.a.n.1.2 7 3.2 odd 2
2175.4.a.n.1.2 7 5.4 even 2