Properties

Label 354.4.a.h.1.1
Level $354$
Weight $4$
Character 354.1
Self dual yes
Analytic conductor $20.887$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [354,4,Mod(1,354)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(354, 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("354.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 354.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: \(20.8866761420\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 44x^{2} + 19x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.122980\) of defining polynomial
Character \(\chi\) \(=\) 354.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -14.5171 q^{5} -6.00000 q^{6} -18.9849 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -14.5171 q^{5} -6.00000 q^{6} -18.9849 q^{7} +8.00000 q^{8} +9.00000 q^{9} -29.0343 q^{10} +23.7792 q^{11} -12.0000 q^{12} -9.08972 q^{13} -37.9698 q^{14} +43.5514 q^{15} +16.0000 q^{16} +30.1895 q^{17} +18.0000 q^{18} +152.389 q^{19} -58.0685 q^{20} +56.9546 q^{21} +47.5585 q^{22} +7.74590 q^{23} -24.0000 q^{24} +85.7471 q^{25} -18.1794 q^{26} -27.0000 q^{27} -75.9395 q^{28} +86.3277 q^{29} +87.1028 q^{30} +143.628 q^{31} +32.0000 q^{32} -71.3377 q^{33} +60.3789 q^{34} +275.606 q^{35} +36.0000 q^{36} -15.9457 q^{37} +304.778 q^{38} +27.2691 q^{39} -116.137 q^{40} +326.380 q^{41} +113.909 q^{42} -216.856 q^{43} +95.1170 q^{44} -130.654 q^{45} +15.4918 q^{46} +87.4463 q^{47} -48.0000 q^{48} +17.4255 q^{49} +171.494 q^{50} -90.5684 q^{51} -36.3589 q^{52} +20.1580 q^{53} -54.0000 q^{54} -345.206 q^{55} -151.879 q^{56} -457.167 q^{57} +172.655 q^{58} +59.0000 q^{59} +174.206 q^{60} +298.414 q^{61} +287.256 q^{62} -170.864 q^{63} +64.0000 q^{64} +131.957 q^{65} -142.675 q^{66} -297.759 q^{67} +120.758 q^{68} -23.2377 q^{69} +551.212 q^{70} +313.600 q^{71} +72.0000 q^{72} +6.83173 q^{73} -31.8913 q^{74} -257.241 q^{75} +609.556 q^{76} -451.446 q^{77} +54.5383 q^{78} +75.3435 q^{79} -232.274 q^{80} +81.0000 q^{81} +652.760 q^{82} +448.906 q^{83} +227.819 q^{84} -438.264 q^{85} -433.712 q^{86} -258.983 q^{87} +190.234 q^{88} +1419.69 q^{89} -261.308 q^{90} +172.567 q^{91} +30.9836 q^{92} -430.884 q^{93} +174.893 q^{94} -2212.25 q^{95} -96.0000 q^{96} +622.392 q^{97} +34.8510 q^{98} +214.013 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} - 12 q^{3} + 16 q^{4} + 22 q^{5} - 24 q^{6} + 13 q^{7} + 32 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{2} - 12 q^{3} + 16 q^{4} + 22 q^{5} - 24 q^{6} + 13 q^{7} + 32 q^{8} + 36 q^{9} + 44 q^{10} + 24 q^{11} - 48 q^{12} + 20 q^{13} + 26 q^{14} - 66 q^{15} + 64 q^{16} + 91 q^{17} + 72 q^{18} + 141 q^{19} + 88 q^{20} - 39 q^{21} + 48 q^{22} + 13 q^{23} - 96 q^{24} + 278 q^{25} + 40 q^{26} - 108 q^{27} + 52 q^{28} + 295 q^{29} - 132 q^{30} + 311 q^{31} + 128 q^{32} - 72 q^{33} + 182 q^{34} + 551 q^{35} + 144 q^{36} + 609 q^{37} + 282 q^{38} - 60 q^{39} + 176 q^{40} + 677 q^{41} - 78 q^{42} + 170 q^{43} + 96 q^{44} + 198 q^{45} + 26 q^{46} + 17 q^{47} - 192 q^{48} + 651 q^{49} + 556 q^{50} - 273 q^{51} + 80 q^{52} + 166 q^{53} - 216 q^{54} + 108 q^{55} + 104 q^{56} - 423 q^{57} + 590 q^{58} + 236 q^{59} - 264 q^{60} + 651 q^{61} + 622 q^{62} + 117 q^{63} + 256 q^{64} + 700 q^{65} - 144 q^{66} - 894 q^{67} + 364 q^{68} - 39 q^{69} + 1102 q^{70} + 298 q^{71} + 288 q^{72} + 887 q^{73} + 1218 q^{74} - 834 q^{75} + 564 q^{76} - 79 q^{77} - 120 q^{78} - 784 q^{79} + 352 q^{80} + 324 q^{81} + 1354 q^{82} + 971 q^{83} - 156 q^{84} - 799 q^{85} + 340 q^{86} - 885 q^{87} + 192 q^{88} + 1321 q^{89} + 396 q^{90} - 2673 q^{91} + 52 q^{92} - 933 q^{93} + 34 q^{94} - 3133 q^{95} - 384 q^{96} - 1922 q^{97} + 1302 q^{98} + 216 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) −14.5171 −1.29845 −0.649226 0.760596i \(-0.724907\pi\)
−0.649226 + 0.760596i \(0.724907\pi\)
\(6\) −6.00000 −0.408248
\(7\) −18.9849 −1.02509 −0.512543 0.858661i \(-0.671297\pi\)
−0.512543 + 0.858661i \(0.671297\pi\)
\(8\) 8.00000 0.353553
\(9\) 9.00000 0.333333
\(10\) −29.0343 −0.918144
\(11\) 23.7792 0.651792 0.325896 0.945406i \(-0.394334\pi\)
0.325896 + 0.945406i \(0.394334\pi\)
\(12\) −12.0000 −0.288675
\(13\) −9.08972 −0.193926 −0.0969628 0.995288i \(-0.530913\pi\)
−0.0969628 + 0.995288i \(0.530913\pi\)
\(14\) −37.9698 −0.724846
\(15\) 43.5514 0.749661
\(16\) 16.0000 0.250000
\(17\) 30.1895 0.430707 0.215354 0.976536i \(-0.430910\pi\)
0.215354 + 0.976536i \(0.430910\pi\)
\(18\) 18.0000 0.235702
\(19\) 152.389 1.84002 0.920012 0.391891i \(-0.128179\pi\)
0.920012 + 0.391891i \(0.128179\pi\)
\(20\) −58.0685 −0.649226
\(21\) 56.9546 0.591834
\(22\) 47.5585 0.460887
\(23\) 7.74590 0.0702231 0.0351116 0.999383i \(-0.488821\pi\)
0.0351116 + 0.999383i \(0.488821\pi\)
\(24\) −24.0000 −0.204124
\(25\) 85.7471 0.685976
\(26\) −18.1794 −0.137126
\(27\) −27.0000 −0.192450
\(28\) −75.9395 −0.512543
\(29\) 86.3277 0.552781 0.276390 0.961045i \(-0.410862\pi\)
0.276390 + 0.961045i \(0.410862\pi\)
\(30\) 87.1028 0.530091
\(31\) 143.628 0.832141 0.416071 0.909332i \(-0.363407\pi\)
0.416071 + 0.909332i \(0.363407\pi\)
\(32\) 32.0000 0.176777
\(33\) −71.3377 −0.376312
\(34\) 60.3789 0.304556
\(35\) 275.606 1.33103
\(36\) 36.0000 0.166667
\(37\) −15.9457 −0.0708501 −0.0354250 0.999372i \(-0.511278\pi\)
−0.0354250 + 0.999372i \(0.511278\pi\)
\(38\) 304.778 1.30109
\(39\) 27.2691 0.111963
\(40\) −116.137 −0.459072
\(41\) 326.380 1.24322 0.621610 0.783327i \(-0.286479\pi\)
0.621610 + 0.783327i \(0.286479\pi\)
\(42\) 113.909 0.418490
\(43\) −216.856 −0.769075 −0.384537 0.923109i \(-0.625639\pi\)
−0.384537 + 0.923109i \(0.625639\pi\)
\(44\) 95.1170 0.325896
\(45\) −130.654 −0.432817
\(46\) 15.4918 0.0496552
\(47\) 87.4463 0.271391 0.135695 0.990751i \(-0.456673\pi\)
0.135695 + 0.990751i \(0.456673\pi\)
\(48\) −48.0000 −0.144338
\(49\) 17.4255 0.0508032
\(50\) 171.494 0.485059
\(51\) −90.5684 −0.248669
\(52\) −36.3589 −0.0969628
\(53\) 20.1580 0.0522437 0.0261219 0.999659i \(-0.491684\pi\)
0.0261219 + 0.999659i \(0.491684\pi\)
\(54\) −54.0000 −0.136083
\(55\) −345.206 −0.846320
\(56\) −151.879 −0.362423
\(57\) −457.167 −1.06234
\(58\) 172.655 0.390875
\(59\) 59.0000 0.130189
\(60\) 174.206 0.374831
\(61\) 298.414 0.626361 0.313181 0.949694i \(-0.398605\pi\)
0.313181 + 0.949694i \(0.398605\pi\)
\(62\) 287.256 0.588413
\(63\) −170.864 −0.341696
\(64\) 64.0000 0.125000
\(65\) 131.957 0.251803
\(66\) −142.675 −0.266093
\(67\) −297.759 −0.542941 −0.271470 0.962447i \(-0.587510\pi\)
−0.271470 + 0.962447i \(0.587510\pi\)
\(68\) 120.758 0.215354
\(69\) −23.2377 −0.0405433
\(70\) 551.212 0.941177
\(71\) 313.600 0.524190 0.262095 0.965042i \(-0.415587\pi\)
0.262095 + 0.965042i \(0.415587\pi\)
\(72\) 72.0000 0.117851
\(73\) 6.83173 0.0109533 0.00547667 0.999985i \(-0.498257\pi\)
0.00547667 + 0.999985i \(0.498257\pi\)
\(74\) −31.8913 −0.0500986
\(75\) −257.241 −0.396049
\(76\) 609.556 0.920012
\(77\) −451.446 −0.668144
\(78\) 54.5383 0.0791698
\(79\) 75.3435 0.107301 0.0536507 0.998560i \(-0.482914\pi\)
0.0536507 + 0.998560i \(0.482914\pi\)
\(80\) −232.274 −0.324613
\(81\) 81.0000 0.111111
\(82\) 652.760 0.879089
\(83\) 448.906 0.593661 0.296830 0.954930i \(-0.404070\pi\)
0.296830 + 0.954930i \(0.404070\pi\)
\(84\) 227.819 0.295917
\(85\) −438.264 −0.559252
\(86\) −433.712 −0.543818
\(87\) −258.983 −0.319148
\(88\) 190.234 0.230443
\(89\) 1419.69 1.69087 0.845434 0.534079i \(-0.179342\pi\)
0.845434 + 0.534079i \(0.179342\pi\)
\(90\) −261.308 −0.306048
\(91\) 172.567 0.198791
\(92\) 30.9836 0.0351116
\(93\) −430.884 −0.480437
\(94\) 174.893 0.191902
\(95\) −2212.25 −2.38918
\(96\) −96.0000 −0.102062
\(97\) 622.392 0.651488 0.325744 0.945458i \(-0.394385\pi\)
0.325744 + 0.945458i \(0.394385\pi\)
\(98\) 34.8510 0.0359233
\(99\) 214.013 0.217264
\(100\) 342.988 0.342988
\(101\) −416.885 −0.410709 −0.205354 0.978688i \(-0.565835\pi\)
−0.205354 + 0.978688i \(0.565835\pi\)
\(102\) −181.137 −0.175835
\(103\) 86.0527 0.0823206 0.0411603 0.999153i \(-0.486895\pi\)
0.0411603 + 0.999153i \(0.486895\pi\)
\(104\) −72.7177 −0.0685631
\(105\) −826.818 −0.768468
\(106\) 40.3160 0.0369419
\(107\) 348.798 0.315136 0.157568 0.987508i \(-0.449635\pi\)
0.157568 + 0.987508i \(0.449635\pi\)
\(108\) −108.000 −0.0962250
\(109\) 132.115 0.116095 0.0580474 0.998314i \(-0.481513\pi\)
0.0580474 + 0.998314i \(0.481513\pi\)
\(110\) −690.413 −0.598439
\(111\) 47.8370 0.0409053
\(112\) −303.758 −0.256272
\(113\) 353.219 0.294054 0.147027 0.989132i \(-0.453030\pi\)
0.147027 + 0.989132i \(0.453030\pi\)
\(114\) −914.334 −0.751187
\(115\) −112.448 −0.0911813
\(116\) 345.311 0.276390
\(117\) −81.8074 −0.0646419
\(118\) 118.000 0.0920575
\(119\) −573.143 −0.441512
\(120\) 348.411 0.265045
\(121\) −765.547 −0.575167
\(122\) 596.829 0.442904
\(123\) −979.140 −0.717773
\(124\) 574.512 0.416071
\(125\) 569.840 0.407744
\(126\) −341.728 −0.241615
\(127\) −1970.59 −1.37686 −0.688430 0.725303i \(-0.741700\pi\)
−0.688430 + 0.725303i \(0.741700\pi\)
\(128\) 128.000 0.0883883
\(129\) 650.568 0.444026
\(130\) 263.913 0.178052
\(131\) −1308.78 −0.872891 −0.436446 0.899731i \(-0.643763\pi\)
−0.436446 + 0.899731i \(0.643763\pi\)
\(132\) −285.351 −0.188156
\(133\) −2893.09 −1.88618
\(134\) −595.518 −0.383917
\(135\) 391.963 0.249887
\(136\) 241.516 0.152278
\(137\) 770.739 0.480647 0.240324 0.970693i \(-0.422746\pi\)
0.240324 + 0.970693i \(0.422746\pi\)
\(138\) −46.4754 −0.0286685
\(139\) 1567.36 0.956415 0.478207 0.878247i \(-0.341287\pi\)
0.478207 + 0.878247i \(0.341287\pi\)
\(140\) 1102.42 0.665513
\(141\) −262.339 −0.156687
\(142\) 627.200 0.370658
\(143\) −216.147 −0.126399
\(144\) 144.000 0.0833333
\(145\) −1253.23 −0.717759
\(146\) 13.6635 0.00774518
\(147\) −52.2765 −0.0293313
\(148\) −63.7827 −0.0354250
\(149\) −628.560 −0.345595 −0.172798 0.984957i \(-0.555281\pi\)
−0.172798 + 0.984957i \(0.555281\pi\)
\(150\) −514.482 −0.280049
\(151\) 1057.42 0.569881 0.284940 0.958545i \(-0.408026\pi\)
0.284940 + 0.958545i \(0.408026\pi\)
\(152\) 1219.11 0.650547
\(153\) 271.705 0.143569
\(154\) −902.892 −0.472449
\(155\) −2085.07 −1.08049
\(156\) 109.077 0.0559815
\(157\) −2184.80 −1.11061 −0.555305 0.831647i \(-0.687399\pi\)
−0.555305 + 0.831647i \(0.687399\pi\)
\(158\) 150.687 0.0758735
\(159\) −60.4741 −0.0301629
\(160\) −464.548 −0.229536
\(161\) −147.055 −0.0719848
\(162\) 162.000 0.0785674
\(163\) −847.798 −0.407391 −0.203695 0.979034i \(-0.565295\pi\)
−0.203695 + 0.979034i \(0.565295\pi\)
\(164\) 1305.52 0.621610
\(165\) 1035.62 0.488623
\(166\) 897.812 0.419782
\(167\) 1327.98 0.615343 0.307672 0.951493i \(-0.400450\pi\)
0.307672 + 0.951493i \(0.400450\pi\)
\(168\) 455.637 0.209245
\(169\) −2114.38 −0.962393
\(170\) −876.529 −0.395451
\(171\) 1371.50 0.613341
\(172\) −867.424 −0.384537
\(173\) 3125.82 1.37371 0.686854 0.726796i \(-0.258991\pi\)
0.686854 + 0.726796i \(0.258991\pi\)
\(174\) −517.966 −0.225672
\(175\) −1627.90 −0.703185
\(176\) 380.468 0.162948
\(177\) −177.000 −0.0751646
\(178\) 2839.39 1.19562
\(179\) 4682.86 1.95538 0.977692 0.210044i \(-0.0673607\pi\)
0.977692 + 0.210044i \(0.0673607\pi\)
\(180\) −522.617 −0.216409
\(181\) −132.858 −0.0545593 −0.0272796 0.999628i \(-0.508684\pi\)
−0.0272796 + 0.999628i \(0.508684\pi\)
\(182\) 345.134 0.140566
\(183\) −895.243 −0.361630
\(184\) 61.9672 0.0248276
\(185\) 231.485 0.0919954
\(186\) −861.769 −0.339720
\(187\) 717.883 0.280732
\(188\) 349.785 0.135695
\(189\) 512.592 0.197278
\(190\) −4424.50 −1.68941
\(191\) −3273.78 −1.24022 −0.620111 0.784514i \(-0.712913\pi\)
−0.620111 + 0.784514i \(0.712913\pi\)
\(192\) −192.000 −0.0721688
\(193\) 340.402 0.126957 0.0634784 0.997983i \(-0.479781\pi\)
0.0634784 + 0.997983i \(0.479781\pi\)
\(194\) 1244.78 0.460672
\(195\) −395.870 −0.145379
\(196\) 69.7020 0.0254016
\(197\) 4671.06 1.68934 0.844668 0.535291i \(-0.179798\pi\)
0.844668 + 0.535291i \(0.179798\pi\)
\(198\) 428.026 0.153629
\(199\) −1282.92 −0.457004 −0.228502 0.973543i \(-0.573383\pi\)
−0.228502 + 0.973543i \(0.573383\pi\)
\(200\) 685.976 0.242529
\(201\) 893.277 0.313467
\(202\) −833.770 −0.290415
\(203\) −1638.92 −0.566648
\(204\) −362.274 −0.124334
\(205\) −4738.10 −1.61426
\(206\) 172.105 0.0582095
\(207\) 69.7131 0.0234077
\(208\) −145.435 −0.0484814
\(209\) 3623.70 1.19931
\(210\) −1653.64 −0.543389
\(211\) 918.289 0.299610 0.149805 0.988716i \(-0.452135\pi\)
0.149805 + 0.988716i \(0.452135\pi\)
\(212\) 80.6321 0.0261219
\(213\) −940.800 −0.302641
\(214\) 697.595 0.222835
\(215\) 3148.13 0.998606
\(216\) −216.000 −0.0680414
\(217\) −2726.76 −0.853017
\(218\) 264.230 0.0820914
\(219\) −20.4952 −0.00632392
\(220\) −1380.83 −0.423160
\(221\) −274.414 −0.0835252
\(222\) 95.6740 0.0289244
\(223\) −4775.60 −1.43407 −0.717035 0.697037i \(-0.754501\pi\)
−0.717035 + 0.697037i \(0.754501\pi\)
\(224\) −607.516 −0.181211
\(225\) 771.723 0.228659
\(226\) 706.438 0.207927
\(227\) −3348.25 −0.978992 −0.489496 0.872006i \(-0.662819\pi\)
−0.489496 + 0.872006i \(0.662819\pi\)
\(228\) −1828.67 −0.531169
\(229\) 667.923 0.192741 0.0963703 0.995346i \(-0.469277\pi\)
0.0963703 + 0.995346i \(0.469277\pi\)
\(230\) −224.896 −0.0644749
\(231\) 1354.34 0.385753
\(232\) 690.621 0.195438
\(233\) 3641.88 1.02398 0.511990 0.858991i \(-0.328908\pi\)
0.511990 + 0.858991i \(0.328908\pi\)
\(234\) −163.615 −0.0457087
\(235\) −1269.47 −0.352387
\(236\) 236.000 0.0650945
\(237\) −226.031 −0.0619505
\(238\) −1146.29 −0.312196
\(239\) −4829.87 −1.30719 −0.653595 0.756844i \(-0.726740\pi\)
−0.653595 + 0.756844i \(0.726740\pi\)
\(240\) 696.822 0.187415
\(241\) 5698.29 1.52307 0.761533 0.648126i \(-0.224447\pi\)
0.761533 + 0.648126i \(0.224447\pi\)
\(242\) −1531.09 −0.406705
\(243\) −243.000 −0.0641500
\(244\) 1193.66 0.313181
\(245\) −252.968 −0.0659655
\(246\) −1958.28 −0.507542
\(247\) −1385.17 −0.356828
\(248\) 1149.02 0.294206
\(249\) −1346.72 −0.342750
\(250\) 1139.68 0.288319
\(251\) 2138.32 0.537727 0.268863 0.963178i \(-0.413352\pi\)
0.268863 + 0.963178i \(0.413352\pi\)
\(252\) −683.456 −0.170848
\(253\) 184.192 0.0457709
\(254\) −3941.17 −0.973587
\(255\) 1314.79 0.322885
\(256\) 256.000 0.0625000
\(257\) 4379.06 1.06287 0.531437 0.847098i \(-0.321652\pi\)
0.531437 + 0.847098i \(0.321652\pi\)
\(258\) 1301.14 0.313973
\(259\) 302.727 0.0726275
\(260\) 527.826 0.125902
\(261\) 776.949 0.184260
\(262\) −2617.56 −0.617227
\(263\) −6374.29 −1.49451 −0.747254 0.664539i \(-0.768628\pi\)
−0.747254 + 0.664539i \(0.768628\pi\)
\(264\) −570.702 −0.133047
\(265\) −292.637 −0.0678359
\(266\) −5786.18 −1.33373
\(267\) −4259.08 −0.976224
\(268\) −1191.04 −0.271470
\(269\) 8643.02 1.95901 0.979506 0.201417i \(-0.0645546\pi\)
0.979506 + 0.201417i \(0.0645546\pi\)
\(270\) 783.925 0.176697
\(271\) −7810.17 −1.75068 −0.875339 0.483510i \(-0.839362\pi\)
−0.875339 + 0.483510i \(0.839362\pi\)
\(272\) 483.032 0.107677
\(273\) −517.701 −0.114772
\(274\) 1541.48 0.339869
\(275\) 2039.00 0.447114
\(276\) −92.9508 −0.0202717
\(277\) −1071.20 −0.232354 −0.116177 0.993229i \(-0.537064\pi\)
−0.116177 + 0.993229i \(0.537064\pi\)
\(278\) 3134.72 0.676287
\(279\) 1292.65 0.277380
\(280\) 2204.85 0.470589
\(281\) 7587.19 1.61073 0.805363 0.592782i \(-0.201970\pi\)
0.805363 + 0.592782i \(0.201970\pi\)
\(282\) −524.678 −0.110795
\(283\) 8951.10 1.88017 0.940085 0.340941i \(-0.110746\pi\)
0.940085 + 0.340941i \(0.110746\pi\)
\(284\) 1254.40 0.262095
\(285\) 6636.76 1.37939
\(286\) −432.293 −0.0893777
\(287\) −6196.29 −1.27441
\(288\) 288.000 0.0589256
\(289\) −4001.60 −0.814491
\(290\) −2506.46 −0.507532
\(291\) −1867.18 −0.376137
\(292\) 27.3269 0.00547667
\(293\) −1422.92 −0.283713 −0.141856 0.989887i \(-0.545307\pi\)
−0.141856 + 0.989887i \(0.545307\pi\)
\(294\) −104.553 −0.0207403
\(295\) −856.511 −0.169044
\(296\) −127.565 −0.0250493
\(297\) −642.040 −0.125437
\(298\) −1257.12 −0.244373
\(299\) −70.4080 −0.0136181
\(300\) −1028.96 −0.198024
\(301\) 4116.98 0.788369
\(302\) 2114.85 0.402967
\(303\) 1250.65 0.237123
\(304\) 2438.23 0.460006
\(305\) −4332.12 −0.813300
\(306\) 543.410 0.101519
\(307\) 7310.99 1.35915 0.679577 0.733605i \(-0.262163\pi\)
0.679577 + 0.733605i \(0.262163\pi\)
\(308\) −1805.78 −0.334072
\(309\) −258.158 −0.0475278
\(310\) −4170.14 −0.764025
\(311\) 974.108 0.177610 0.0888048 0.996049i \(-0.471695\pi\)
0.0888048 + 0.996049i \(0.471695\pi\)
\(312\) 218.153 0.0395849
\(313\) 540.790 0.0976591 0.0488295 0.998807i \(-0.484451\pi\)
0.0488295 + 0.998807i \(0.484451\pi\)
\(314\) −4369.60 −0.785320
\(315\) 2480.45 0.443675
\(316\) 301.374 0.0536507
\(317\) 2666.13 0.472381 0.236190 0.971707i \(-0.424101\pi\)
0.236190 + 0.971707i \(0.424101\pi\)
\(318\) −120.948 −0.0213284
\(319\) 2052.81 0.360298
\(320\) −929.096 −0.162306
\(321\) −1046.39 −0.181944
\(322\) −294.110 −0.0509009
\(323\) 4600.55 0.792511
\(324\) 324.000 0.0555556
\(325\) −779.416 −0.133028
\(326\) −1695.60 −0.288069
\(327\) −396.345 −0.0670273
\(328\) 2611.04 0.439544
\(329\) −1660.16 −0.278199
\(330\) 2071.24 0.345509
\(331\) 1412.61 0.234574 0.117287 0.993098i \(-0.462580\pi\)
0.117287 + 0.993098i \(0.462580\pi\)
\(332\) 1795.62 0.296830
\(333\) −143.511 −0.0236167
\(334\) 2655.96 0.435114
\(335\) 4322.60 0.704983
\(336\) 911.274 0.147959
\(337\) −426.538 −0.0689466 −0.0344733 0.999406i \(-0.510975\pi\)
−0.0344733 + 0.999406i \(0.510975\pi\)
\(338\) −4228.75 −0.680514
\(339\) −1059.66 −0.169772
\(340\) −1753.06 −0.279626
\(341\) 3415.37 0.542383
\(342\) 2743.00 0.433698
\(343\) 6180.99 0.973009
\(344\) −1734.85 −0.271909
\(345\) 337.345 0.0526436
\(346\) 6251.63 0.971358
\(347\) 224.184 0.0346826 0.0173413 0.999850i \(-0.494480\pi\)
0.0173413 + 0.999850i \(0.494480\pi\)
\(348\) −1035.93 −0.159574
\(349\) 11037.8 1.69296 0.846479 0.532422i \(-0.178718\pi\)
0.846479 + 0.532422i \(0.178718\pi\)
\(350\) −3255.79 −0.497227
\(351\) 245.422 0.0373210
\(352\) 760.936 0.115222
\(353\) 9820.44 1.48071 0.740353 0.672219i \(-0.234658\pi\)
0.740353 + 0.672219i \(0.234658\pi\)
\(354\) −354.000 −0.0531494
\(355\) −4552.57 −0.680635
\(356\) 5678.78 0.845434
\(357\) 1719.43 0.254907
\(358\) 9365.73 1.38267
\(359\) 4391.53 0.645616 0.322808 0.946464i \(-0.395373\pi\)
0.322808 + 0.946464i \(0.395373\pi\)
\(360\) −1045.23 −0.153024
\(361\) 16363.4 2.38569
\(362\) −265.715 −0.0385792
\(363\) 2296.64 0.332073
\(364\) 690.269 0.0993953
\(365\) −99.1772 −0.0142224
\(366\) −1790.49 −0.255711
\(367\) −2173.35 −0.309123 −0.154561 0.987983i \(-0.549396\pi\)
−0.154561 + 0.987983i \(0.549396\pi\)
\(368\) 123.934 0.0175558
\(369\) 2937.42 0.414407
\(370\) 462.971 0.0650505
\(371\) −382.698 −0.0535544
\(372\) −1723.54 −0.240218
\(373\) 9445.30 1.31115 0.655575 0.755130i \(-0.272426\pi\)
0.655575 + 0.755130i \(0.272426\pi\)
\(374\) 1435.77 0.198507
\(375\) −1709.52 −0.235411
\(376\) 699.570 0.0959510
\(377\) −784.694 −0.107198
\(378\) 1025.18 0.139497
\(379\) −4028.86 −0.546039 −0.273019 0.962009i \(-0.588022\pi\)
−0.273019 + 0.962009i \(0.588022\pi\)
\(380\) −8849.01 −1.19459
\(381\) 5911.76 0.794930
\(382\) −6547.56 −0.876970
\(383\) 1787.00 0.238411 0.119206 0.992870i \(-0.461965\pi\)
0.119206 + 0.992870i \(0.461965\pi\)
\(384\) −384.000 −0.0510310
\(385\) 6553.70 0.867552
\(386\) 680.804 0.0897720
\(387\) −1951.70 −0.256358
\(388\) 2489.57 0.325744
\(389\) −5491.07 −0.715702 −0.357851 0.933779i \(-0.616490\pi\)
−0.357851 + 0.933779i \(0.616490\pi\)
\(390\) −791.740 −0.102798
\(391\) 233.845 0.0302456
\(392\) 139.404 0.0179617
\(393\) 3926.34 0.503964
\(394\) 9342.12 1.19454
\(395\) −1093.77 −0.139326
\(396\) 856.053 0.108632
\(397\) −11750.8 −1.48554 −0.742768 0.669549i \(-0.766487\pi\)
−0.742768 + 0.669549i \(0.766487\pi\)
\(398\) −2565.84 −0.323151
\(399\) 8679.26 1.08899
\(400\) 1371.95 0.171494
\(401\) −4123.69 −0.513535 −0.256767 0.966473i \(-0.582657\pi\)
−0.256767 + 0.966473i \(0.582657\pi\)
\(402\) 1786.55 0.221655
\(403\) −1305.54 −0.161374
\(404\) −1667.54 −0.205354
\(405\) −1175.89 −0.144272
\(406\) −3277.84 −0.400681
\(407\) −379.176 −0.0461795
\(408\) −724.547 −0.0879177
\(409\) −14142.8 −1.70982 −0.854910 0.518776i \(-0.826388\pi\)
−0.854910 + 0.518776i \(0.826388\pi\)
\(410\) −9476.20 −1.14145
\(411\) −2312.22 −0.277502
\(412\) 344.211 0.0411603
\(413\) −1120.11 −0.133455
\(414\) 139.426 0.0165517
\(415\) −6516.83 −0.770840
\(416\) −290.871 −0.0342815
\(417\) −4702.08 −0.552186
\(418\) 7247.40 0.848042
\(419\) −13489.9 −1.57285 −0.786425 0.617686i \(-0.788070\pi\)
−0.786425 + 0.617686i \(0.788070\pi\)
\(420\) −3307.27 −0.384234
\(421\) 11198.4 1.29638 0.648189 0.761479i \(-0.275527\pi\)
0.648189 + 0.761479i \(0.275527\pi\)
\(422\) 1836.58 0.211856
\(423\) 787.017 0.0904635
\(424\) 161.264 0.0184709
\(425\) 2588.66 0.295455
\(426\) −1881.60 −0.214000
\(427\) −5665.36 −0.642075
\(428\) 1395.19 0.157568
\(429\) 648.440 0.0729766
\(430\) 6296.25 0.706121
\(431\) −9819.96 −1.09747 −0.548737 0.835995i \(-0.684891\pi\)
−0.548737 + 0.835995i \(0.684891\pi\)
\(432\) −432.000 −0.0481125
\(433\) −6816.67 −0.756554 −0.378277 0.925692i \(-0.623483\pi\)
−0.378277 + 0.925692i \(0.623483\pi\)
\(434\) −5453.52 −0.603174
\(435\) 3759.69 0.414398
\(436\) 528.460 0.0580474
\(437\) 1180.39 0.129212
\(438\) −40.9904 −0.00447168
\(439\) −17813.6 −1.93666 −0.968331 0.249670i \(-0.919678\pi\)
−0.968331 + 0.249670i \(0.919678\pi\)
\(440\) −2761.65 −0.299219
\(441\) 156.830 0.0169344
\(442\) −548.827 −0.0590612
\(443\) −14194.9 −1.52239 −0.761194 0.648524i \(-0.775386\pi\)
−0.761194 + 0.648524i \(0.775386\pi\)
\(444\) 191.348 0.0204526
\(445\) −20609.9 −2.19551
\(446\) −9551.19 −1.01404
\(447\) 1885.68 0.199529
\(448\) −1215.03 −0.128136
\(449\) −1846.28 −0.194057 −0.0970284 0.995282i \(-0.530934\pi\)
−0.0970284 + 0.995282i \(0.530934\pi\)
\(450\) 1543.45 0.161686
\(451\) 7761.07 0.810321
\(452\) 1412.88 0.147027
\(453\) −3172.27 −0.329021
\(454\) −6696.50 −0.692252
\(455\) −2505.18 −0.258120
\(456\) −3657.34 −0.375593
\(457\) −10310.4 −1.05536 −0.527681 0.849443i \(-0.676938\pi\)
−0.527681 + 0.849443i \(0.676938\pi\)
\(458\) 1335.85 0.136288
\(459\) −815.116 −0.0828896
\(460\) −449.793 −0.0455907
\(461\) −118.971 −0.0120196 −0.00600982 0.999982i \(-0.501913\pi\)
−0.00600982 + 0.999982i \(0.501913\pi\)
\(462\) 2708.68 0.272768
\(463\) −16512.4 −1.65745 −0.828724 0.559658i \(-0.810933\pi\)
−0.828724 + 0.559658i \(0.810933\pi\)
\(464\) 1381.24 0.138195
\(465\) 6255.20 0.623824
\(466\) 7283.76 0.724064
\(467\) 9651.70 0.956375 0.478188 0.878258i \(-0.341294\pi\)
0.478188 + 0.878258i \(0.341294\pi\)
\(468\) −327.230 −0.0323209
\(469\) 5652.92 0.556562
\(470\) −2538.94 −0.249176
\(471\) 6554.39 0.641211
\(472\) 472.000 0.0460287
\(473\) −5156.67 −0.501277
\(474\) −452.061 −0.0438056
\(475\) 13066.9 1.26221
\(476\) −2292.57 −0.220756
\(477\) 181.422 0.0174146
\(478\) −9659.75 −0.924323
\(479\) −698.299 −0.0666098 −0.0333049 0.999445i \(-0.510603\pi\)
−0.0333049 + 0.999445i \(0.510603\pi\)
\(480\) 1393.64 0.132523
\(481\) 144.942 0.0137396
\(482\) 11396.6 1.07697
\(483\) 441.165 0.0415604
\(484\) −3062.19 −0.287584
\(485\) −9035.35 −0.845926
\(486\) −486.000 −0.0453609
\(487\) −495.933 −0.0461455 −0.0230728 0.999734i \(-0.507345\pi\)
−0.0230728 + 0.999734i \(0.507345\pi\)
\(488\) 2387.31 0.221452
\(489\) 2543.40 0.235207
\(490\) −505.937 −0.0466447
\(491\) 15858.8 1.45763 0.728815 0.684711i \(-0.240072\pi\)
0.728815 + 0.684711i \(0.240072\pi\)
\(492\) −3916.56 −0.358887
\(493\) 2606.19 0.238087
\(494\) −2770.35 −0.252315
\(495\) −3106.86 −0.282107
\(496\) 2298.05 0.208035
\(497\) −5953.66 −0.537340
\(498\) −2693.44 −0.242361
\(499\) −3175.90 −0.284915 −0.142458 0.989801i \(-0.545500\pi\)
−0.142458 + 0.989801i \(0.545500\pi\)
\(500\) 2279.36 0.203872
\(501\) −3983.95 −0.355269
\(502\) 4276.64 0.380230
\(503\) −2673.46 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(504\) −1366.91 −0.120808
\(505\) 6051.97 0.533286
\(506\) 368.383 0.0323649
\(507\) 6343.13 0.555638
\(508\) −7882.34 −0.688430
\(509\) −13398.4 −1.16675 −0.583374 0.812204i \(-0.698268\pi\)
−0.583374 + 0.812204i \(0.698268\pi\)
\(510\) 2629.59 0.228314
\(511\) −129.700 −0.0112281
\(512\) 512.000 0.0441942
\(513\) −4114.50 −0.354113
\(514\) 8758.13 0.751565
\(515\) −1249.24 −0.106889
\(516\) 2602.27 0.222013
\(517\) 2079.41 0.176890
\(518\) 605.453 0.0513554
\(519\) −9377.45 −0.793110
\(520\) 1055.65 0.0890258
\(521\) 5203.78 0.437585 0.218792 0.975771i \(-0.429788\pi\)
0.218792 + 0.975771i \(0.429788\pi\)
\(522\) 1553.90 0.130292
\(523\) 15363.7 1.28453 0.642264 0.766483i \(-0.277995\pi\)
0.642264 + 0.766483i \(0.277995\pi\)
\(524\) −5235.13 −0.436446
\(525\) 4883.69 0.405984
\(526\) −12748.6 −1.05678
\(527\) 4336.06 0.358409
\(528\) −1141.40 −0.0940781
\(529\) −12107.0 −0.995069
\(530\) −585.273 −0.0479672
\(531\) 531.000 0.0433963
\(532\) −11572.4 −0.943092
\(533\) −2966.70 −0.241092
\(534\) −8518.17 −0.690294
\(535\) −5063.54 −0.409189
\(536\) −2382.07 −0.191959
\(537\) −14048.6 −1.12894
\(538\) 17286.0 1.38523
\(539\) 414.366 0.0331131
\(540\) 1567.85 0.124944
\(541\) 4815.08 0.382655 0.191327 0.981526i \(-0.438721\pi\)
0.191327 + 0.981526i \(0.438721\pi\)
\(542\) −15620.3 −1.23792
\(543\) 398.573 0.0314998
\(544\) 966.063 0.0761390
\(545\) −1917.93 −0.150743
\(546\) −1035.40 −0.0811560
\(547\) 15456.4 1.20816 0.604082 0.796922i \(-0.293540\pi\)
0.604082 + 0.796922i \(0.293540\pi\)
\(548\) 3082.95 0.240324
\(549\) 2685.73 0.208787
\(550\) 4078.00 0.316157
\(551\) 13155.4 1.01713
\(552\) −185.902 −0.0143342
\(553\) −1430.39 −0.109993
\(554\) −2142.39 −0.164299
\(555\) −694.456 −0.0531135
\(556\) 6269.43 0.478207
\(557\) −15326.4 −1.16589 −0.582944 0.812512i \(-0.698099\pi\)
−0.582944 + 0.812512i \(0.698099\pi\)
\(558\) 2585.31 0.196138
\(559\) 1971.16 0.149143
\(560\) 4409.69 0.332756
\(561\) −2153.65 −0.162080
\(562\) 15174.4 1.13896
\(563\) −20801.2 −1.55713 −0.778565 0.627564i \(-0.784052\pi\)
−0.778565 + 0.627564i \(0.784052\pi\)
\(564\) −1049.36 −0.0783437
\(565\) −5127.73 −0.381815
\(566\) 17902.2 1.32948
\(567\) −1537.77 −0.113899
\(568\) 2508.80 0.185329
\(569\) −12363.3 −0.910889 −0.455445 0.890264i \(-0.650520\pi\)
−0.455445 + 0.890264i \(0.650520\pi\)
\(570\) 13273.5 0.975379
\(571\) 15498.3 1.13587 0.567935 0.823073i \(-0.307743\pi\)
0.567935 + 0.823073i \(0.307743\pi\)
\(572\) −864.586 −0.0631996
\(573\) 9821.34 0.716043
\(574\) −12392.6 −0.901143
\(575\) 664.188 0.0481714
\(576\) 576.000 0.0416667
\(577\) 11548.5 0.833223 0.416612 0.909085i \(-0.363217\pi\)
0.416612 + 0.909085i \(0.363217\pi\)
\(578\) −8003.19 −0.575932
\(579\) −1021.21 −0.0732986
\(580\) −5012.92 −0.358880
\(581\) −8522.43 −0.608554
\(582\) −3734.35 −0.265969
\(583\) 479.343 0.0340520
\(584\) 54.6539 0.00387259
\(585\) 1187.61 0.0839344
\(586\) −2845.84 −0.200615
\(587\) −17238.5 −1.21211 −0.606057 0.795421i \(-0.707250\pi\)
−0.606057 + 0.795421i \(0.707250\pi\)
\(588\) −209.106 −0.0146656
\(589\) 21887.4 1.53116
\(590\) −1713.02 −0.119532
\(591\) −14013.2 −0.975338
\(592\) −255.131 −0.0177125
\(593\) −13806.2 −0.956072 −0.478036 0.878340i \(-0.658651\pi\)
−0.478036 + 0.878340i \(0.658651\pi\)
\(594\) −1284.08 −0.0886977
\(595\) 8320.40 0.573282
\(596\) −2514.24 −0.172798
\(597\) 3848.76 0.263851
\(598\) −140.816 −0.00962943
\(599\) −28580.6 −1.94953 −0.974767 0.223223i \(-0.928342\pi\)
−0.974767 + 0.223223i \(0.928342\pi\)
\(600\) −2057.93 −0.140024
\(601\) −2557.49 −0.173581 −0.0867905 0.996227i \(-0.527661\pi\)
−0.0867905 + 0.996227i \(0.527661\pi\)
\(602\) 8233.97 0.557461
\(603\) −2679.83 −0.180980
\(604\) 4229.70 0.284940
\(605\) 11113.6 0.746827
\(606\) 2501.31 0.167671
\(607\) 5944.14 0.397471 0.198736 0.980053i \(-0.436316\pi\)
0.198736 + 0.980053i \(0.436316\pi\)
\(608\) 4876.45 0.325273
\(609\) 4916.76 0.327155
\(610\) −8664.24 −0.575090
\(611\) −794.862 −0.0526296
\(612\) 1086.82 0.0717845
\(613\) 9848.25 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(614\) 14622.0 0.961067
\(615\) 14214.3 0.931994
\(616\) −3611.57 −0.236224
\(617\) −13450.9 −0.877656 −0.438828 0.898571i \(-0.644606\pi\)
−0.438828 + 0.898571i \(0.644606\pi\)
\(618\) −516.316 −0.0336072
\(619\) −16407.2 −1.06537 −0.532683 0.846315i \(-0.678816\pi\)
−0.532683 + 0.846315i \(0.678816\pi\)
\(620\) −8340.27 −0.540247
\(621\) −209.139 −0.0135144
\(622\) 1948.22 0.125589
\(623\) −26952.7 −1.73329
\(624\) 436.306 0.0279908
\(625\) −18990.8 −1.21541
\(626\) 1081.58 0.0690554
\(627\) −10871.1 −0.692424
\(628\) −8739.19 −0.555305
\(629\) −481.391 −0.0305156
\(630\) 4960.91 0.313726
\(631\) −4817.91 −0.303959 −0.151979 0.988384i \(-0.548565\pi\)
−0.151979 + 0.988384i \(0.548565\pi\)
\(632\) 602.748 0.0379368
\(633\) −2754.87 −0.172980
\(634\) 5332.26 0.334024
\(635\) 28607.2 1.78779
\(636\) −241.896 −0.0150815
\(637\) −158.393 −0.00985205
\(638\) 4105.61 0.254769
\(639\) 2822.40 0.174730
\(640\) −1858.19 −0.114768
\(641\) −1980.91 −0.122061 −0.0610305 0.998136i \(-0.519439\pi\)
−0.0610305 + 0.998136i \(0.519439\pi\)
\(642\) −2092.79 −0.128654
\(643\) −14075.8 −0.863292 −0.431646 0.902043i \(-0.642067\pi\)
−0.431646 + 0.902043i \(0.642067\pi\)
\(644\) −588.220 −0.0359924
\(645\) −9444.38 −0.576546
\(646\) 9201.09 0.560390
\(647\) 13170.6 0.800296 0.400148 0.916451i \(-0.368959\pi\)
0.400148 + 0.916451i \(0.368959\pi\)
\(648\) 648.000 0.0392837
\(649\) 1402.98 0.0848561
\(650\) −1558.83 −0.0940653
\(651\) 8180.29 0.492490
\(652\) −3391.19 −0.203695
\(653\) 17815.8 1.06766 0.533832 0.845591i \(-0.320751\pi\)
0.533832 + 0.845591i \(0.320751\pi\)
\(654\) −792.690 −0.0473955
\(655\) 18999.8 1.13341
\(656\) 5222.08 0.310805
\(657\) 61.4856 0.00365111
\(658\) −3320.31 −0.196716
\(659\) −29695.3 −1.75533 −0.877666 0.479273i \(-0.840901\pi\)
−0.877666 + 0.479273i \(0.840901\pi\)
\(660\) 4142.48 0.244312
\(661\) 30236.1 1.77919 0.889596 0.456748i \(-0.150986\pi\)
0.889596 + 0.456748i \(0.150986\pi\)
\(662\) 2825.22 0.165869
\(663\) 823.241 0.0482233
\(664\) 3591.25 0.209891
\(665\) 41999.3 2.44912
\(666\) −287.022 −0.0166995
\(667\) 668.685 0.0388180
\(668\) 5311.93 0.307672
\(669\) 14326.8 0.827961
\(670\) 8645.21 0.498498
\(671\) 7096.07 0.408257
\(672\) 1822.55 0.104622
\(673\) −28150.8 −1.61238 −0.806191 0.591655i \(-0.798475\pi\)
−0.806191 + 0.591655i \(0.798475\pi\)
\(674\) −853.075 −0.0487526
\(675\) −2315.17 −0.132016
\(676\) −8457.51 −0.481196
\(677\) 8280.59 0.470087 0.235043 0.971985i \(-0.424477\pi\)
0.235043 + 0.971985i \(0.424477\pi\)
\(678\) −2119.32 −0.120047
\(679\) −11816.0 −0.667832
\(680\) −3506.12 −0.197726
\(681\) 10044.7 0.565221
\(682\) 6830.74 0.383523
\(683\) 23754.0 1.33078 0.665390 0.746496i \(-0.268265\pi\)
0.665390 + 0.746496i \(0.268265\pi\)
\(684\) 5486.01 0.306671
\(685\) −11188.9 −0.624097
\(686\) 12362.0 0.688021
\(687\) −2003.77 −0.111279
\(688\) −3469.69 −0.192269
\(689\) −183.231 −0.0101314
\(690\) 674.689 0.0372246
\(691\) 2606.85 0.143516 0.0717578 0.997422i \(-0.477139\pi\)
0.0717578 + 0.997422i \(0.477139\pi\)
\(692\) 12503.3 0.686854
\(693\) −4063.01 −0.222715
\(694\) 448.369 0.0245243
\(695\) −22753.6 −1.24186
\(696\) −2071.86 −0.112836
\(697\) 9853.24 0.535464
\(698\) 22075.7 1.19710
\(699\) −10925.6 −0.591195
\(700\) −6511.59 −0.351593
\(701\) −8619.02 −0.464388 −0.232194 0.972670i \(-0.574590\pi\)
−0.232194 + 0.972670i \(0.574590\pi\)
\(702\) 490.845 0.0263899
\(703\) −2429.95 −0.130366
\(704\) 1521.87 0.0814740
\(705\) 3808.41 0.203451
\(706\) 19640.9 1.04702
\(707\) 7914.51 0.421012
\(708\) −708.000 −0.0375823
\(709\) −22526.1 −1.19321 −0.596606 0.802534i \(-0.703484\pi\)
−0.596606 + 0.802534i \(0.703484\pi\)
\(710\) −9105.15 −0.481282
\(711\) 678.092 0.0357671
\(712\) 11357.6 0.597812
\(713\) 1112.53 0.0584355
\(714\) 3438.86 0.180247
\(715\) 3137.83 0.164123
\(716\) 18731.5 0.977692
\(717\) 14489.6 0.754707
\(718\) 8783.07 0.456520
\(719\) −14465.6 −0.750313 −0.375157 0.926961i \(-0.622411\pi\)
−0.375157 + 0.926961i \(0.622411\pi\)
\(720\) −2090.47 −0.108204
\(721\) −1633.70 −0.0843858
\(722\) 32726.9 1.68694
\(723\) −17094.9 −0.879343
\(724\) −531.430 −0.0272796
\(725\) 7402.34 0.379195
\(726\) 4593.28 0.234811
\(727\) −5354.94 −0.273183 −0.136591 0.990627i \(-0.543615\pi\)
−0.136591 + 0.990627i \(0.543615\pi\)
\(728\) 1380.54 0.0702831
\(729\) 729.000 0.0370370
\(730\) −198.354 −0.0100567
\(731\) −6546.77 −0.331246
\(732\) −3580.97 −0.180815
\(733\) −19575.6 −0.986413 −0.493206 0.869912i \(-0.664175\pi\)
−0.493206 + 0.869912i \(0.664175\pi\)
\(734\) −4346.70 −0.218583
\(735\) 758.905 0.0380852
\(736\) 247.869 0.0124138
\(737\) −7080.48 −0.353885
\(738\) 5874.84 0.293030
\(739\) −1620.91 −0.0806850 −0.0403425 0.999186i \(-0.512845\pi\)
−0.0403425 + 0.999186i \(0.512845\pi\)
\(740\) 925.941 0.0459977
\(741\) 4155.52 0.206015
\(742\) −765.395 −0.0378686
\(743\) −1746.91 −0.0862554 −0.0431277 0.999070i \(-0.513732\pi\)
−0.0431277 + 0.999070i \(0.513732\pi\)
\(744\) −3447.07 −0.169860
\(745\) 9124.89 0.448739
\(746\) 18890.6 0.927123
\(747\) 4040.16 0.197887
\(748\) 2871.53 0.140366
\(749\) −6621.88 −0.323042
\(750\) −3419.04 −0.166461
\(751\) 18511.7 0.899467 0.449734 0.893163i \(-0.351519\pi\)
0.449734 + 0.893163i \(0.351519\pi\)
\(752\) 1399.14 0.0678476
\(753\) −6414.96 −0.310457
\(754\) −1569.39 −0.0758007
\(755\) −15350.8 −0.739963
\(756\) 2050.37 0.0986390
\(757\) 24433.2 1.17311 0.586553 0.809911i \(-0.300485\pi\)
0.586553 + 0.809911i \(0.300485\pi\)
\(758\) −8057.72 −0.386108
\(759\) −552.575 −0.0264258
\(760\) −17698.0 −0.844703
\(761\) 22235.4 1.05918 0.529588 0.848255i \(-0.322347\pi\)
0.529588 + 0.848255i \(0.322347\pi\)
\(762\) 11823.5 0.562101
\(763\) −2508.19 −0.119007
\(764\) −13095.1 −0.620111
\(765\) −3944.38 −0.186417
\(766\) 3574.00 0.168582
\(767\) −536.293 −0.0252470
\(768\) −768.000 −0.0360844
\(769\) 18702.6 0.877027 0.438513 0.898725i \(-0.355505\pi\)
0.438513 + 0.898725i \(0.355505\pi\)
\(770\) 13107.4 0.613452
\(771\) −13137.2 −0.613650
\(772\) 1361.61 0.0634784
\(773\) −947.968 −0.0441087 −0.0220544 0.999757i \(-0.507021\pi\)
−0.0220544 + 0.999757i \(0.507021\pi\)
\(774\) −3903.41 −0.181273
\(775\) 12315.7 0.570829
\(776\) 4979.14 0.230336
\(777\) −908.180 −0.0419315
\(778\) −10982.1 −0.506078
\(779\) 49736.8 2.28755
\(780\) −1583.48 −0.0726893
\(781\) 7457.18 0.341663
\(782\) 467.689 0.0213869
\(783\) −2330.85 −0.106383
\(784\) 278.808 0.0127008
\(785\) 31717.0 1.44207
\(786\) 7852.69 0.356356
\(787\) −27044.5 −1.22495 −0.612473 0.790491i \(-0.709825\pi\)
−0.612473 + 0.790491i \(0.709825\pi\)
\(788\) 18684.2 0.844668
\(789\) 19122.9 0.862854
\(790\) −2187.54 −0.0985181
\(791\) −6705.82 −0.301431
\(792\) 1712.11 0.0768144
\(793\) −2712.50 −0.121468
\(794\) −23501.7 −1.05043
\(795\) 877.910 0.0391651
\(796\) −5131.68 −0.228502
\(797\) −1335.55 −0.0593569 −0.0296785 0.999559i \(-0.509448\pi\)
−0.0296785 + 0.999559i \(0.509448\pi\)
\(798\) 17358.5 0.770032
\(799\) 2639.96 0.116890
\(800\) 2743.91 0.121265
\(801\) 12777.3 0.563623
\(802\) −8247.38 −0.363124
\(803\) 162.454 0.00713930
\(804\) 3573.11 0.156734
\(805\) 2134.82 0.0934688
\(806\) −2611.08 −0.114108
\(807\) −25929.0 −1.13104
\(808\) −3335.08 −0.145208
\(809\) 10946.3 0.475714 0.237857 0.971300i \(-0.423555\pi\)
0.237857 + 0.971300i \(0.423555\pi\)
\(810\) −2351.78 −0.102016
\(811\) −19857.3 −0.859781 −0.429891 0.902881i \(-0.641448\pi\)
−0.429891 + 0.902881i \(0.641448\pi\)
\(812\) −6555.68 −0.283324
\(813\) 23430.5 1.01075
\(814\) −758.352 −0.0326538
\(815\) 12307.6 0.528977
\(816\) −1449.09 −0.0621672
\(817\) −33046.5 −1.41512
\(818\) −28285.6 −1.20903
\(819\) 1553.10 0.0662636
\(820\) −18952.4 −0.807130
\(821\) 16339.2 0.694569 0.347285 0.937760i \(-0.387104\pi\)
0.347285 + 0.937760i \(0.387104\pi\)
\(822\) −4624.43 −0.196223
\(823\) 26132.8 1.10684 0.553421 0.832902i \(-0.313322\pi\)
0.553421 + 0.832902i \(0.313322\pi\)
\(824\) 688.421 0.0291047
\(825\) −6117.00 −0.258141
\(826\) −2240.22 −0.0943669
\(827\) 40384.2 1.69806 0.849030 0.528345i \(-0.177187\pi\)
0.849030 + 0.528345i \(0.177187\pi\)
\(828\) 278.852 0.0117039
\(829\) 25800.6 1.08093 0.540465 0.841367i \(-0.318248\pi\)
0.540465 + 0.841367i \(0.318248\pi\)
\(830\) −13033.7 −0.545066
\(831\) 3213.59 0.134149
\(832\) −581.742 −0.0242407
\(833\) 526.067 0.0218813
\(834\) −9404.15 −0.390455
\(835\) −19278.5 −0.798994
\(836\) 14494.8 0.599656
\(837\) −3877.96 −0.160146
\(838\) −26979.8 −1.11217
\(839\) −6044.78 −0.248736 −0.124368 0.992236i \(-0.539690\pi\)
−0.124368 + 0.992236i \(0.539690\pi\)
\(840\) −6614.54 −0.271694
\(841\) −16936.5 −0.694433
\(842\) 22396.8 0.916678
\(843\) −22761.6 −0.929953
\(844\) 3673.16 0.149805
\(845\) 30694.7 1.24962
\(846\) 1574.03 0.0639674
\(847\) 14533.8 0.589596
\(848\) 322.528 0.0130609
\(849\) −26853.3 −1.08552
\(850\) 5177.32 0.208918
\(851\) −123.514 −0.00497531
\(852\) −3763.20 −0.151321
\(853\) 6900.19 0.276973 0.138487 0.990364i \(-0.455776\pi\)
0.138487 + 0.990364i \(0.455776\pi\)
\(854\) −11330.7 −0.454015
\(855\) −19910.3 −0.796394
\(856\) 2790.38 0.111417
\(857\) −17424.7 −0.694534 −0.347267 0.937766i \(-0.612890\pi\)
−0.347267 + 0.937766i \(0.612890\pi\)
\(858\) 1296.88 0.0516023
\(859\) −41478.4 −1.64753 −0.823764 0.566933i \(-0.808130\pi\)
−0.823764 + 0.566933i \(0.808130\pi\)
\(860\) 12592.5 0.499303
\(861\) 18588.9 0.735780
\(862\) −19639.9 −0.776031
\(863\) −14024.1 −0.553168 −0.276584 0.960990i \(-0.589202\pi\)
−0.276584 + 0.960990i \(0.589202\pi\)
\(864\) −864.000 −0.0340207
\(865\) −45377.9 −1.78369
\(866\) −13633.3 −0.534965
\(867\) 12004.8 0.470247
\(868\) −10907.0 −0.426508
\(869\) 1791.61 0.0699382
\(870\) 7519.38 0.293024
\(871\) 2706.54 0.105290
\(872\) 1056.92 0.0410457
\(873\) 5601.53 0.217163
\(874\) 2360.78 0.0913668
\(875\) −10818.3 −0.417973
\(876\) −81.9808 −0.00316196
\(877\) 7506.31 0.289019 0.144510 0.989503i \(-0.453840\pi\)
0.144510 + 0.989503i \(0.453840\pi\)
\(878\) −35627.1 −1.36943
\(879\) 4268.75 0.163802
\(880\) −5523.30 −0.211580
\(881\) 13812.2 0.528201 0.264101 0.964495i \(-0.414925\pi\)
0.264101 + 0.964495i \(0.414925\pi\)
\(882\) 313.659 0.0119744
\(883\) 11579.6 0.441318 0.220659 0.975351i \(-0.429179\pi\)
0.220659 + 0.975351i \(0.429179\pi\)
\(884\) −1097.65 −0.0417626
\(885\) 2569.53 0.0975976
\(886\) −28389.7 −1.07649
\(887\) 17903.3 0.677714 0.338857 0.940838i \(-0.389960\pi\)
0.338857 + 0.940838i \(0.389960\pi\)
\(888\) 382.696 0.0144622
\(889\) 37411.3 1.41140
\(890\) −41219.8 −1.55246
\(891\) 1926.12 0.0724213
\(892\) −19102.4 −0.717035
\(893\) 13325.9 0.499365
\(894\) 3771.36 0.141089
\(895\) −67981.7 −2.53897
\(896\) −2430.06 −0.0906057
\(897\) 211.224 0.00786239
\(898\) −3692.57 −0.137219
\(899\) 12399.1 0.459992
\(900\) 3086.89 0.114329
\(901\) 608.560 0.0225017
\(902\) 15522.1 0.572983
\(903\) −12350.9 −0.455165
\(904\) 2825.75 0.103964
\(905\) 1928.71 0.0708426
\(906\) −6344.55 −0.232653
\(907\) 48047.8 1.75899 0.879493 0.475912i \(-0.157882\pi\)
0.879493 + 0.475912i \(0.157882\pi\)
\(908\) −13393.0 −0.489496
\(909\) −3751.96 −0.136903
\(910\) −5010.36 −0.182518
\(911\) 32038.1 1.16517 0.582584 0.812770i \(-0.302042\pi\)
0.582584 + 0.812770i \(0.302042\pi\)
\(912\) −7314.68 −0.265585
\(913\) 10674.7 0.386943
\(914\) −20620.8 −0.746254
\(915\) 12996.4 0.469559
\(916\) 2671.69 0.0963703
\(917\) 24847.1 0.894790
\(918\) −1630.23 −0.0586118
\(919\) 41966.1 1.50635 0.753175 0.657821i \(-0.228522\pi\)
0.753175 + 0.657821i \(0.228522\pi\)
\(920\) −899.586 −0.0322375
\(921\) −21933.0 −0.784708
\(922\) −237.943 −0.00849916
\(923\) −2850.54 −0.101654
\(924\) 5417.35 0.192876
\(925\) −1367.29 −0.0486015
\(926\) −33024.9 −1.17199
\(927\) 774.474 0.0274402
\(928\) 2762.49 0.0977188
\(929\) −11790.5 −0.416399 −0.208200 0.978086i \(-0.566760\pi\)
−0.208200 + 0.978086i \(0.566760\pi\)
\(930\) 12510.4 0.441110
\(931\) 2655.46 0.0934792
\(932\) 14567.5 0.511990
\(933\) −2922.32 −0.102543
\(934\) 19303.4 0.676259
\(935\) −10421.6 −0.364516
\(936\) −654.460 −0.0228544
\(937\) −2254.98 −0.0786201 −0.0393101 0.999227i \(-0.512516\pi\)
−0.0393101 + 0.999227i \(0.512516\pi\)
\(938\) 11305.8 0.393549
\(939\) −1622.37 −0.0563835
\(940\) −5077.88 −0.176194
\(941\) 4345.03 0.150525 0.0752625 0.997164i \(-0.476021\pi\)
0.0752625 + 0.997164i \(0.476021\pi\)
\(942\) 13108.8 0.453405
\(943\) 2528.11 0.0873028
\(944\) 944.000 0.0325472
\(945\) −7441.36 −0.256156
\(946\) −10313.3 −0.354456
\(947\) −43486.4 −1.49221 −0.746103 0.665831i \(-0.768077\pi\)
−0.746103 + 0.665831i \(0.768077\pi\)
\(948\) −904.122 −0.0309752
\(949\) −62.0985 −0.00212413
\(950\) 26133.8 0.892519
\(951\) −7998.39 −0.272729
\(952\) −4585.15 −0.156098
\(953\) −11476.4 −0.390091 −0.195046 0.980794i \(-0.562485\pi\)
−0.195046 + 0.980794i \(0.562485\pi\)
\(954\) 362.844 0.0123140
\(955\) 47525.9 1.61037
\(956\) −19319.5 −0.653595
\(957\) −6158.42 −0.208018
\(958\) −1396.60 −0.0471002
\(959\) −14632.4 −0.492705
\(960\) 2787.29 0.0937077
\(961\) −9161.96 −0.307541
\(962\) 289.883 0.00971540
\(963\) 3139.18 0.105045
\(964\) 22793.2 0.761533
\(965\) −4941.66 −0.164847
\(966\) 882.330 0.0293877
\(967\) −30998.7 −1.03087 −0.515436 0.856928i \(-0.672370\pi\)
−0.515436 + 0.856928i \(0.672370\pi\)
\(968\) −6124.38 −0.203352
\(969\) −13801.6 −0.457557
\(970\) −18070.7 −0.598160
\(971\) −27050.2 −0.894009 −0.447004 0.894532i \(-0.647509\pi\)
−0.447004 + 0.894532i \(0.647509\pi\)
\(972\) −972.000 −0.0320750
\(973\) −29756.1 −0.980408
\(974\) −991.866 −0.0326298
\(975\) 2338.25 0.0768040
\(976\) 4774.63 0.156590
\(977\) 7821.75 0.256131 0.128066 0.991766i \(-0.459123\pi\)
0.128066 + 0.991766i \(0.459123\pi\)
\(978\) 5086.79 0.166317
\(979\) 33759.3 1.10210
\(980\) −1011.87 −0.0329828
\(981\) 1189.04 0.0386982
\(982\) 31717.5 1.03070
\(983\) 47680.9 1.54709 0.773543 0.633744i \(-0.218483\pi\)
0.773543 + 0.633744i \(0.218483\pi\)
\(984\) −7833.12 −0.253771
\(985\) −67810.3 −2.19352
\(986\) 5212.37 0.168353
\(987\) 4980.47 0.160618
\(988\) −5540.69 −0.178414
\(989\) −1679.74 −0.0540068
\(990\) −6213.72 −0.199480
\(991\) 11338.7 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(992\) 4596.10 0.147103
\(993\) −4237.83 −0.135431
\(994\) −11907.3 −0.379957
\(995\) 18624.3 0.593398
\(996\) −5386.87 −0.171375
\(997\) −20391.1 −0.647735 −0.323868 0.946102i \(-0.604983\pi\)
−0.323868 + 0.946102i \(0.604983\pi\)
\(998\) −6351.80 −0.201466
\(999\) 430.533 0.0136351
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 354.4.a.h.1.1 4
3.2 odd 2 1062.4.a.n.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
354.4.a.h.1.1 4 1.1 even 1 trivial
1062.4.a.n.1.4 4 3.2 odd 2