Properties

Label 65.4.a.d.1.1
Level $65$
Weight $4$
Character 65.1
Self dual yes
Analytic conductor $3.835$
Analytic rank $1$
Dimension $2$
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] = [65,4,Mod(1,65)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(65, 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("65.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 65.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: \(3.83512415037\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 65.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.267949 q^{2} +0.196152 q^{3} -7.92820 q^{4} -5.00000 q^{5} +0.0525589 q^{6} -11.0718 q^{7} -4.26795 q^{8} -26.9615 q^{9} +O(q^{10})\) \(q+0.267949 q^{2} +0.196152 q^{3} -7.92820 q^{4} -5.00000 q^{5} +0.0525589 q^{6} -11.0718 q^{7} -4.26795 q^{8} -26.9615 q^{9} -1.33975 q^{10} -10.1962 q^{11} -1.55514 q^{12} +13.0000 q^{13} -2.96668 q^{14} -0.980762 q^{15} +62.2820 q^{16} -48.5359 q^{17} -7.22432 q^{18} -36.4449 q^{19} +39.6410 q^{20} -2.17176 q^{21} -2.73205 q^{22} +11.8038 q^{23} -0.837169 q^{24} +25.0000 q^{25} +3.48334 q^{26} -10.5847 q^{27} +87.7795 q^{28} +103.033 q^{29} -0.262794 q^{30} +156.578 q^{31} +50.8320 q^{32} -2.00000 q^{33} -13.0052 q^{34} +55.3590 q^{35} +213.756 q^{36} -263.769 q^{37} -9.76537 q^{38} +2.54998 q^{39} +21.3397 q^{40} +185.951 q^{41} -0.581921 q^{42} -418.158 q^{43} +80.8372 q^{44} +134.808 q^{45} +3.16283 q^{46} -375.492 q^{47} +12.2168 q^{48} -220.415 q^{49} +6.69873 q^{50} -9.52043 q^{51} -103.067 q^{52} -189.397 q^{53} -2.83616 q^{54} +50.9808 q^{55} +47.2539 q^{56} -7.14875 q^{57} +27.6077 q^{58} -194.463 q^{59} +7.77568 q^{60} +21.8846 q^{61} +41.9550 q^{62} +298.513 q^{63} -484.636 q^{64} -65.0000 q^{65} -0.535898 q^{66} -658.715 q^{67} +384.802 q^{68} +2.31535 q^{69} +14.8334 q^{70} +357.755 q^{71} +115.070 q^{72} -341.205 q^{73} -70.6767 q^{74} +4.90381 q^{75} +288.942 q^{76} +112.890 q^{77} +0.683265 q^{78} -1005.58 q^{79} -311.410 q^{80} +725.885 q^{81} +49.8255 q^{82} +725.349 q^{83} +17.2182 q^{84} +242.679 q^{85} -112.045 q^{86} +20.2102 q^{87} +43.5167 q^{88} +1595.76 q^{89} +36.1216 q^{90} -143.933 q^{91} -93.5833 q^{92} +30.7132 q^{93} -100.613 q^{94} +182.224 q^{95} +9.97082 q^{96} +1037.94 q^{97} -59.0601 q^{98} +274.904 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 10 q^{3} - 2 q^{4} - 10 q^{5} - 38 q^{6} - 36 q^{7} - 12 q^{8} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 10 q^{3} - 2 q^{4} - 10 q^{5} - 38 q^{6} - 36 q^{7} - 12 q^{8} + 50 q^{9} - 20 q^{10} - 10 q^{11} - 62 q^{12} + 26 q^{13} - 96 q^{14} + 50 q^{15} - 14 q^{16} - 104 q^{17} + 280 q^{18} - 14 q^{19} + 10 q^{20} + 252 q^{21} - 2 q^{22} + 34 q^{23} + 78 q^{24} + 50 q^{25} + 52 q^{26} - 520 q^{27} - 60 q^{28} + 116 q^{29} + 190 q^{30} - 106 q^{31} - 172 q^{32} - 4 q^{33} - 220 q^{34} + 180 q^{35} + 670 q^{36} + 96 q^{37} + 74 q^{38} - 130 q^{39} + 60 q^{40} - 120 q^{41} + 948 q^{42} - 722 q^{43} + 82 q^{44} - 250 q^{45} + 86 q^{46} - 460 q^{47} + 790 q^{48} + 58 q^{49} + 100 q^{50} + 556 q^{51} - 26 q^{52} - 552 q^{53} - 1904 q^{54} + 50 q^{55} + 240 q^{56} - 236 q^{57} + 76 q^{58} + 342 q^{59} + 310 q^{60} - 268 q^{61} - 938 q^{62} - 1620 q^{63} - 706 q^{64} - 130 q^{65} - 8 q^{66} - 8 q^{67} + 56 q^{68} - 224 q^{69} + 480 q^{70} + 234 q^{71} - 480 q^{72} - 336 q^{73} + 1272 q^{74} - 250 q^{75} + 422 q^{76} + 108 q^{77} - 494 q^{78} - 868 q^{79} + 70 q^{80} + 3842 q^{81} - 1092 q^{82} + 1132 q^{83} + 1524 q^{84} + 520 q^{85} - 1246 q^{86} - 112 q^{87} + 42 q^{88} + 2180 q^{89} - 1400 q^{90} - 468 q^{91} + 38 q^{92} + 2708 q^{93} - 416 q^{94} + 70 q^{95} + 2282 q^{96} - 252 q^{97} + 980 q^{98} + 290 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.267949 0.0947343 0.0473672 0.998878i \(-0.484917\pi\)
0.0473672 + 0.998878i \(0.484917\pi\)
\(3\) 0.196152 0.0377496 0.0188748 0.999822i \(-0.493992\pi\)
0.0188748 + 0.999822i \(0.493992\pi\)
\(4\) −7.92820 −0.991025
\(5\) −5.00000 −0.447214
\(6\) 0.0525589 0.00357618
\(7\) −11.0718 −0.597821 −0.298910 0.954281i \(-0.596623\pi\)
−0.298910 + 0.954281i \(0.596623\pi\)
\(8\) −4.26795 −0.188618
\(9\) −26.9615 −0.998575
\(10\) −1.33975 −0.0423665
\(11\) −10.1962 −0.279478 −0.139739 0.990188i \(-0.544626\pi\)
−0.139739 + 0.990188i \(0.544626\pi\)
\(12\) −1.55514 −0.0374108
\(13\) 13.0000 0.277350
\(14\) −2.96668 −0.0566342
\(15\) −0.980762 −0.0168821
\(16\) 62.2820 0.973157
\(17\) −48.5359 −0.692452 −0.346226 0.938151i \(-0.612537\pi\)
−0.346226 + 0.938151i \(0.612537\pi\)
\(18\) −7.22432 −0.0945993
\(19\) −36.4449 −0.440054 −0.220027 0.975494i \(-0.570615\pi\)
−0.220027 + 0.975494i \(0.570615\pi\)
\(20\) 39.6410 0.443200
\(21\) −2.17176 −0.0225675
\(22\) −2.73205 −0.0264761
\(23\) 11.8038 0.107012 0.0535059 0.998568i \(-0.482960\pi\)
0.0535059 + 0.998568i \(0.482960\pi\)
\(24\) −0.837169 −0.00712026
\(25\) 25.0000 0.200000
\(26\) 3.48334 0.0262746
\(27\) −10.5847 −0.0754453
\(28\) 87.7795 0.592456
\(29\) 103.033 0.659752 0.329876 0.944024i \(-0.392993\pi\)
0.329876 + 0.944024i \(0.392993\pi\)
\(30\) −0.262794 −0.00159932
\(31\) 156.578 0.907170 0.453585 0.891213i \(-0.350145\pi\)
0.453585 + 0.891213i \(0.350145\pi\)
\(32\) 50.8320 0.280810
\(33\) −2.00000 −0.0105502
\(34\) −13.0052 −0.0655990
\(35\) 55.3590 0.267354
\(36\) 213.756 0.989613
\(37\) −263.769 −1.17198 −0.585992 0.810317i \(-0.699295\pi\)
−0.585992 + 0.810317i \(0.699295\pi\)
\(38\) −9.76537 −0.0416882
\(39\) 2.54998 0.0104698
\(40\) 21.3397 0.0843528
\(41\) 185.951 0.708310 0.354155 0.935187i \(-0.384769\pi\)
0.354155 + 0.935187i \(0.384769\pi\)
\(42\) −0.581921 −0.00213791
\(43\) −418.158 −1.48299 −0.741494 0.670960i \(-0.765882\pi\)
−0.741494 + 0.670960i \(0.765882\pi\)
\(44\) 80.8372 0.276970
\(45\) 134.808 0.446576
\(46\) 3.16283 0.0101377
\(47\) −375.492 −1.16534 −0.582672 0.812707i \(-0.697993\pi\)
−0.582672 + 0.812707i \(0.697993\pi\)
\(48\) 12.2168 0.0367362
\(49\) −220.415 −0.642610
\(50\) 6.69873 0.0189469
\(51\) −9.52043 −0.0261398
\(52\) −103.067 −0.274861
\(53\) −189.397 −0.490863 −0.245432 0.969414i \(-0.578930\pi\)
−0.245432 + 0.969414i \(0.578930\pi\)
\(54\) −2.83616 −0.00714726
\(55\) 50.9808 0.124986
\(56\) 47.2539 0.112760
\(57\) −7.14875 −0.0166118
\(58\) 27.6077 0.0625012
\(59\) −194.463 −0.429100 −0.214550 0.976713i \(-0.568828\pi\)
−0.214550 + 0.976713i \(0.568828\pi\)
\(60\) 7.77568 0.0167306
\(61\) 21.8846 0.0459350 0.0229675 0.999736i \(-0.492689\pi\)
0.0229675 + 0.999736i \(0.492689\pi\)
\(62\) 41.9550 0.0859401
\(63\) 298.513 0.596969
\(64\) −484.636 −0.946554
\(65\) −65.0000 −0.124035
\(66\) −0.535898 −0.000999463 0
\(67\) −658.715 −1.20112 −0.600559 0.799581i \(-0.705055\pi\)
−0.600559 + 0.799581i \(0.705055\pi\)
\(68\) 384.802 0.686238
\(69\) 2.31535 0.00403965
\(70\) 14.8334 0.0253276
\(71\) 357.755 0.597996 0.298998 0.954254i \(-0.403348\pi\)
0.298998 + 0.954254i \(0.403348\pi\)
\(72\) 115.070 0.188350
\(73\) −341.205 −0.547055 −0.273528 0.961864i \(-0.588190\pi\)
−0.273528 + 0.961864i \(0.588190\pi\)
\(74\) −70.6767 −0.111027
\(75\) 4.90381 0.00754991
\(76\) 288.942 0.436105
\(77\) 112.890 0.167078
\(78\) 0.683265 0.000991854 0
\(79\) −1005.58 −1.43210 −0.716052 0.698047i \(-0.754053\pi\)
−0.716052 + 0.698047i \(0.754053\pi\)
\(80\) −311.410 −0.435209
\(81\) 725.885 0.995727
\(82\) 49.8255 0.0671013
\(83\) 725.349 0.959245 0.479623 0.877475i \(-0.340774\pi\)
0.479623 + 0.877475i \(0.340774\pi\)
\(84\) 17.2182 0.0223649
\(85\) 242.679 0.309674
\(86\) −112.045 −0.140490
\(87\) 20.2102 0.0249053
\(88\) 43.5167 0.0527147
\(89\) 1595.76 1.90056 0.950281 0.311392i \(-0.100795\pi\)
0.950281 + 0.311392i \(0.100795\pi\)
\(90\) 36.1216 0.0423061
\(91\) −143.933 −0.165806
\(92\) −93.5833 −0.106051
\(93\) 30.7132 0.0342453
\(94\) −100.613 −0.110398
\(95\) 182.224 0.196798
\(96\) 9.97082 0.0106004
\(97\) 1037.94 1.08646 0.543230 0.839584i \(-0.317201\pi\)
0.543230 + 0.839584i \(0.317201\pi\)
\(98\) −59.0601 −0.0608773
\(99\) 274.904 0.279080
\(100\) −198.205 −0.198205
\(101\) −484.446 −0.477269 −0.238634 0.971109i \(-0.576700\pi\)
−0.238634 + 0.971109i \(0.576700\pi\)
\(102\) −2.55099 −0.00247633
\(103\) −947.040 −0.905967 −0.452984 0.891519i \(-0.649640\pi\)
−0.452984 + 0.891519i \(0.649640\pi\)
\(104\) −55.4833 −0.0523134
\(105\) 10.8588 0.0100925
\(106\) −50.7489 −0.0465016
\(107\) 1676.37 1.51458 0.757291 0.653077i \(-0.226522\pi\)
0.757291 + 0.653077i \(0.226522\pi\)
\(108\) 83.9175 0.0747682
\(109\) 1789.78 1.57275 0.786377 0.617747i \(-0.211954\pi\)
0.786377 + 0.617747i \(0.211954\pi\)
\(110\) 13.6603 0.0118405
\(111\) −51.7390 −0.0442418
\(112\) −689.574 −0.581773
\(113\) −565.069 −0.470418 −0.235209 0.971945i \(-0.575577\pi\)
−0.235209 + 0.971945i \(0.575577\pi\)
\(114\) −1.91550 −0.00157371
\(115\) −59.0192 −0.0478572
\(116\) −816.869 −0.653831
\(117\) −350.500 −0.276955
\(118\) −52.1061 −0.0406505
\(119\) 537.380 0.413962
\(120\) 4.18584 0.00318428
\(121\) −1227.04 −0.921892
\(122\) 5.86395 0.00435162
\(123\) 36.4748 0.0267384
\(124\) −1241.38 −0.899028
\(125\) −125.000 −0.0894427
\(126\) 79.9862 0.0565535
\(127\) −1396.08 −0.975448 −0.487724 0.872998i \(-0.662173\pi\)
−0.487724 + 0.872998i \(0.662173\pi\)
\(128\) −536.514 −0.370481
\(129\) −82.0226 −0.0559821
\(130\) −17.4167 −0.0117503
\(131\) −2673.75 −1.78326 −0.891629 0.452766i \(-0.850437\pi\)
−0.891629 + 0.452766i \(0.850437\pi\)
\(132\) 15.8564 0.0104555
\(133\) 403.510 0.263073
\(134\) −176.502 −0.113787
\(135\) 52.9234 0.0337402
\(136\) 207.149 0.130609
\(137\) 2502.70 1.56073 0.780366 0.625323i \(-0.215033\pi\)
0.780366 + 0.625323i \(0.215033\pi\)
\(138\) 0.620397 0.000382694 0
\(139\) 272.695 0.166400 0.0832002 0.996533i \(-0.473486\pi\)
0.0832002 + 0.996533i \(0.473486\pi\)
\(140\) −438.897 −0.264954
\(141\) −73.6537 −0.0439912
\(142\) 95.8602 0.0566508
\(143\) −132.550 −0.0775132
\(144\) −1679.22 −0.971770
\(145\) −515.167 −0.295050
\(146\) −91.4256 −0.0518249
\(147\) −43.2350 −0.0242582
\(148\) 2091.22 1.16147
\(149\) 1598.07 0.878652 0.439326 0.898328i \(-0.355217\pi\)
0.439326 + 0.898328i \(0.355217\pi\)
\(150\) 1.31397 0.000715236 0
\(151\) −2431.27 −1.31029 −0.655145 0.755503i \(-0.727392\pi\)
−0.655145 + 0.755503i \(0.727392\pi\)
\(152\) 155.545 0.0830023
\(153\) 1308.60 0.691465
\(154\) 30.2487 0.0158280
\(155\) −782.891 −0.405699
\(156\) −20.2168 −0.0103759
\(157\) 693.005 0.352279 0.176140 0.984365i \(-0.443639\pi\)
0.176140 + 0.984365i \(0.443639\pi\)
\(158\) −269.443 −0.135669
\(159\) −37.1508 −0.0185299
\(160\) −254.160 −0.125582
\(161\) −130.690 −0.0639739
\(162\) 194.500 0.0943295
\(163\) −431.962 −0.207570 −0.103785 0.994600i \(-0.533095\pi\)
−0.103785 + 0.994600i \(0.533095\pi\)
\(164\) −1474.26 −0.701953
\(165\) 10.0000 0.00471818
\(166\) 194.357 0.0908735
\(167\) −1054.76 −0.488740 −0.244370 0.969682i \(-0.578581\pi\)
−0.244370 + 0.969682i \(0.578581\pi\)
\(168\) 9.26896 0.00425664
\(169\) 169.000 0.0769231
\(170\) 65.0258 0.0293368
\(171\) 982.609 0.439427
\(172\) 3315.24 1.46968
\(173\) −2929.16 −1.28728 −0.643642 0.765327i \(-0.722577\pi\)
−0.643642 + 0.765327i \(0.722577\pi\)
\(174\) 5.41532 0.00235939
\(175\) −276.795 −0.119564
\(176\) −635.037 −0.271976
\(177\) −38.1443 −0.0161983
\(178\) 427.582 0.180049
\(179\) 3717.33 1.55221 0.776106 0.630602i \(-0.217192\pi\)
0.776106 + 0.630602i \(0.217192\pi\)
\(180\) −1068.78 −0.442568
\(181\) 1923.10 0.789739 0.394870 0.918737i \(-0.370790\pi\)
0.394870 + 0.918737i \(0.370790\pi\)
\(182\) −38.5668 −0.0157075
\(183\) 4.29271 0.00173402
\(184\) −50.3782 −0.0201844
\(185\) 1318.85 0.524127
\(186\) 8.22957 0.00324420
\(187\) 494.879 0.193525
\(188\) 2976.98 1.15489
\(189\) 117.191 0.0451028
\(190\) 48.8269 0.0186435
\(191\) −1941.49 −0.735505 −0.367752 0.929924i \(-0.619873\pi\)
−0.367752 + 0.929924i \(0.619873\pi\)
\(192\) −95.0625 −0.0357320
\(193\) 2334.49 0.870676 0.435338 0.900267i \(-0.356629\pi\)
0.435338 + 0.900267i \(0.356629\pi\)
\(194\) 278.115 0.102925
\(195\) −12.7499 −0.00468226
\(196\) 1747.50 0.636843
\(197\) 2160.54 0.781380 0.390690 0.920522i \(-0.372236\pi\)
0.390690 + 0.920522i \(0.372236\pi\)
\(198\) 73.6603 0.0264384
\(199\) −1985.74 −0.707365 −0.353683 0.935366i \(-0.615071\pi\)
−0.353683 + 0.935366i \(0.615071\pi\)
\(200\) −106.699 −0.0377237
\(201\) −129.209 −0.0453416
\(202\) −129.807 −0.0452138
\(203\) −1140.76 −0.394413
\(204\) 75.4799 0.0259052
\(205\) −929.756 −0.316766
\(206\) −253.759 −0.0858262
\(207\) −318.250 −0.106859
\(208\) 809.666 0.269905
\(209\) 371.597 0.122985
\(210\) 2.90961 0.000956104 0
\(211\) 1052.47 0.343389 0.171695 0.985150i \(-0.445076\pi\)
0.171695 + 0.985150i \(0.445076\pi\)
\(212\) 1501.58 0.486458
\(213\) 70.1745 0.0225741
\(214\) 449.181 0.143483
\(215\) 2090.79 0.663212
\(216\) 45.1749 0.0142304
\(217\) −1733.60 −0.542325
\(218\) 479.571 0.148994
\(219\) −66.9282 −0.0206511
\(220\) −404.186 −0.123865
\(221\) −630.967 −0.192052
\(222\) −13.8634 −0.00419122
\(223\) −3119.81 −0.936852 −0.468426 0.883503i \(-0.655179\pi\)
−0.468426 + 0.883503i \(0.655179\pi\)
\(224\) −562.802 −0.167874
\(225\) −674.038 −0.199715
\(226\) −151.410 −0.0445647
\(227\) −5135.43 −1.50154 −0.750772 0.660561i \(-0.770318\pi\)
−0.750772 + 0.660561i \(0.770318\pi\)
\(228\) 56.6767 0.0164628
\(229\) −1297.62 −0.374451 −0.187226 0.982317i \(-0.559950\pi\)
−0.187226 + 0.982317i \(0.559950\pi\)
\(230\) −15.8142 −0.00453372
\(231\) 22.1436 0.00630711
\(232\) −439.741 −0.124441
\(233\) −4251.77 −1.19546 −0.597732 0.801696i \(-0.703931\pi\)
−0.597732 + 0.801696i \(0.703931\pi\)
\(234\) −93.9161 −0.0262371
\(235\) 1877.46 0.521158
\(236\) 1541.74 0.425249
\(237\) −197.246 −0.0540613
\(238\) 143.990 0.0392164
\(239\) −229.286 −0.0620555 −0.0310278 0.999519i \(-0.509878\pi\)
−0.0310278 + 0.999519i \(0.509878\pi\)
\(240\) −61.0839 −0.0164289
\(241\) −2829.08 −0.756171 −0.378086 0.925771i \(-0.623418\pi\)
−0.378086 + 0.925771i \(0.623418\pi\)
\(242\) −328.784 −0.0873349
\(243\) 428.171 0.113034
\(244\) −173.505 −0.0455227
\(245\) 1102.08 0.287384
\(246\) 9.77339 0.00253304
\(247\) −473.783 −0.122049
\(248\) −668.268 −0.171109
\(249\) 142.279 0.0362111
\(250\) −33.4936 −0.00847330
\(251\) 3496.34 0.879230 0.439615 0.898186i \(-0.355115\pi\)
0.439615 + 0.898186i \(0.355115\pi\)
\(252\) −2366.67 −0.591611
\(253\) −120.354 −0.0299074
\(254\) −374.078 −0.0924084
\(255\) 47.6022 0.0116901
\(256\) 3733.33 0.911457
\(257\) −3688.28 −0.895208 −0.447604 0.894232i \(-0.647723\pi\)
−0.447604 + 0.894232i \(0.647723\pi\)
\(258\) −21.9779 −0.00530343
\(259\) 2920.40 0.700636
\(260\) 515.333 0.122922
\(261\) −2777.94 −0.658812
\(262\) −716.430 −0.168936
\(263\) −7176.39 −1.68257 −0.841284 0.540594i \(-0.818199\pi\)
−0.841284 + 0.540594i \(0.818199\pi\)
\(264\) 8.53590 0.00198996
\(265\) 946.987 0.219521
\(266\) 108.120 0.0249221
\(267\) 313.012 0.0717454
\(268\) 5222.43 1.19034
\(269\) 804.924 0.182443 0.0912214 0.995831i \(-0.470923\pi\)
0.0912214 + 0.995831i \(0.470923\pi\)
\(270\) 14.1808 0.00319635
\(271\) 2636.16 0.590906 0.295453 0.955357i \(-0.404529\pi\)
0.295453 + 0.955357i \(0.404529\pi\)
\(272\) −3022.91 −0.673864
\(273\) −28.2329 −0.00625909
\(274\) 670.597 0.147855
\(275\) −254.904 −0.0558956
\(276\) −18.3566 −0.00400339
\(277\) 1683.28 0.365122 0.182561 0.983195i \(-0.441561\pi\)
0.182561 + 0.983195i \(0.441561\pi\)
\(278\) 73.0683 0.0157638
\(279\) −4221.59 −0.905877
\(280\) −236.269 −0.0504278
\(281\) 2382.17 0.505724 0.252862 0.967502i \(-0.418628\pi\)
0.252862 + 0.967502i \(0.418628\pi\)
\(282\) −19.7355 −0.00416748
\(283\) −3634.50 −0.763423 −0.381711 0.924282i \(-0.624665\pi\)
−0.381711 + 0.924282i \(0.624665\pi\)
\(284\) −2836.35 −0.592629
\(285\) 35.7437 0.00742904
\(286\) −35.5167 −0.00734316
\(287\) −2058.81 −0.423442
\(288\) −1370.51 −0.280410
\(289\) −2557.27 −0.520510
\(290\) −138.038 −0.0279514
\(291\) 203.594 0.0410134
\(292\) 2705.14 0.542146
\(293\) 3491.94 0.696250 0.348125 0.937448i \(-0.386819\pi\)
0.348125 + 0.937448i \(0.386819\pi\)
\(294\) −11.5848 −0.00229809
\(295\) 972.314 0.191899
\(296\) 1125.75 0.221058
\(297\) 107.923 0.0210853
\(298\) 428.202 0.0832385
\(299\) 153.450 0.0296797
\(300\) −38.8784 −0.00748215
\(301\) 4629.76 0.886561
\(302\) −651.456 −0.124129
\(303\) −95.0252 −0.0180167
\(304\) −2269.86 −0.428241
\(305\) −109.423 −0.0205427
\(306\) 350.639 0.0655055
\(307\) 1710.16 0.317929 0.158965 0.987284i \(-0.449184\pi\)
0.158965 + 0.987284i \(0.449184\pi\)
\(308\) −895.013 −0.165578
\(309\) −185.764 −0.0341999
\(310\) −209.775 −0.0384336
\(311\) 8158.33 1.48751 0.743756 0.668451i \(-0.233042\pi\)
0.743756 + 0.668451i \(0.233042\pi\)
\(312\) −10.8832 −0.00197481
\(313\) 11014.7 1.98910 0.994552 0.104246i \(-0.0332428\pi\)
0.994552 + 0.104246i \(0.0332428\pi\)
\(314\) 185.690 0.0333729
\(315\) −1492.56 −0.266973
\(316\) 7972.42 1.41925
\(317\) −5890.38 −1.04365 −0.521825 0.853053i \(-0.674749\pi\)
−0.521825 + 0.853053i \(0.674749\pi\)
\(318\) −9.95452 −0.00175541
\(319\) −1050.54 −0.184386
\(320\) 2423.18 0.423312
\(321\) 328.823 0.0571748
\(322\) −35.0182 −0.00606053
\(323\) 1768.88 0.304716
\(324\) −5754.96 −0.986791
\(325\) 325.000 0.0554700
\(326\) −115.744 −0.0196640
\(327\) 351.071 0.0593708
\(328\) −793.630 −0.133600
\(329\) 4157.37 0.696667
\(330\) 2.67949 0.000446973 0
\(331\) 15.2227 0.00252784 0.00126392 0.999999i \(-0.499598\pi\)
0.00126392 + 0.999999i \(0.499598\pi\)
\(332\) −5750.71 −0.950636
\(333\) 7111.62 1.17031
\(334\) −282.622 −0.0463005
\(335\) 3293.58 0.537156
\(336\) −135.262 −0.0219617
\(337\) 10544.3 1.70441 0.852203 0.523212i \(-0.175266\pi\)
0.852203 + 0.523212i \(0.175266\pi\)
\(338\) 45.2834 0.00728726
\(339\) −110.840 −0.0177581
\(340\) −1924.01 −0.306895
\(341\) −1596.49 −0.253534
\(342\) 263.289 0.0416288
\(343\) 6238.02 0.981987
\(344\) 1784.68 0.279719
\(345\) −11.5768 −0.00180659
\(346\) −784.866 −0.121950
\(347\) −695.585 −0.107611 −0.0538054 0.998551i \(-0.517135\pi\)
−0.0538054 + 0.998551i \(0.517135\pi\)
\(348\) −160.231 −0.0246818
\(349\) −1430.88 −0.219465 −0.109732 0.993961i \(-0.534999\pi\)
−0.109732 + 0.993961i \(0.534999\pi\)
\(350\) −74.1670 −0.0113268
\(351\) −137.601 −0.0209248
\(352\) −518.291 −0.0784801
\(353\) −12657.8 −1.90851 −0.954256 0.298989i \(-0.903350\pi\)
−0.954256 + 0.298989i \(0.903350\pi\)
\(354\) −10.2207 −0.00153454
\(355\) −1788.78 −0.267432
\(356\) −12651.5 −1.88351
\(357\) 105.408 0.0156269
\(358\) 996.055 0.147048
\(359\) 5601.01 0.823426 0.411713 0.911313i \(-0.364931\pi\)
0.411713 + 0.911313i \(0.364931\pi\)
\(360\) −575.352 −0.0842325
\(361\) −5530.77 −0.806353
\(362\) 515.293 0.0748154
\(363\) −240.687 −0.0348010
\(364\) 1141.13 0.164318
\(365\) 1706.03 0.244651
\(366\) 1.15023 0.000164272 0
\(367\) −12630.3 −1.79644 −0.898221 0.439544i \(-0.855140\pi\)
−0.898221 + 0.439544i \(0.855140\pi\)
\(368\) 735.168 0.104139
\(369\) −5013.53 −0.707300
\(370\) 353.384 0.0496528
\(371\) 2096.97 0.293448
\(372\) −243.500 −0.0339379
\(373\) −4193.44 −0.582112 −0.291056 0.956706i \(-0.594007\pi\)
−0.291056 + 0.956706i \(0.594007\pi\)
\(374\) 132.603 0.0183335
\(375\) −24.5191 −0.00337642
\(376\) 1602.58 0.219805
\(377\) 1339.43 0.182982
\(378\) 31.4014 0.00427278
\(379\) 4003.82 0.542645 0.271323 0.962488i \(-0.412539\pi\)
0.271323 + 0.962488i \(0.412539\pi\)
\(380\) −1444.71 −0.195032
\(381\) −273.844 −0.0368227
\(382\) −520.221 −0.0696776
\(383\) −9524.94 −1.27076 −0.635381 0.772199i \(-0.719157\pi\)
−0.635381 + 0.772199i \(0.719157\pi\)
\(384\) −105.239 −0.0139855
\(385\) −564.449 −0.0747194
\(386\) 625.525 0.0824829
\(387\) 11274.2 1.48087
\(388\) −8228.98 −1.07671
\(389\) 9924.11 1.29350 0.646751 0.762701i \(-0.276127\pi\)
0.646751 + 0.762701i \(0.276127\pi\)
\(390\) −3.41633 −0.000443570 0
\(391\) −572.910 −0.0741006
\(392\) 940.721 0.121208
\(393\) −524.463 −0.0673172
\(394\) 578.914 0.0740236
\(395\) 5027.88 0.640456
\(396\) −2179.49 −0.276575
\(397\) −6473.93 −0.818431 −0.409216 0.912438i \(-0.634198\pi\)
−0.409216 + 0.912438i \(0.634198\pi\)
\(398\) −532.078 −0.0670118
\(399\) 79.1495 0.00993090
\(400\) 1557.05 0.194631
\(401\) −8383.05 −1.04396 −0.521982 0.852957i \(-0.674807\pi\)
−0.521982 + 0.852957i \(0.674807\pi\)
\(402\) −34.6213 −0.00429541
\(403\) 2035.52 0.251604
\(404\) 3840.79 0.472986
\(405\) −3629.42 −0.445303
\(406\) −305.667 −0.0373645
\(407\) 2689.43 0.327543
\(408\) 40.6327 0.00493044
\(409\) −11654.6 −1.40900 −0.704502 0.709702i \(-0.748830\pi\)
−0.704502 + 0.709702i \(0.748830\pi\)
\(410\) −249.127 −0.0300086
\(411\) 490.911 0.0589169
\(412\) 7508.32 0.897836
\(413\) 2153.05 0.256525
\(414\) −85.2748 −0.0101233
\(415\) −3626.74 −0.428988
\(416\) 660.816 0.0778826
\(417\) 53.4897 0.00628154
\(418\) 99.5692 0.0116509
\(419\) 2641.37 0.307970 0.153985 0.988073i \(-0.450789\pi\)
0.153985 + 0.988073i \(0.450789\pi\)
\(420\) −86.0908 −0.0100019
\(421\) −12311.3 −1.42521 −0.712607 0.701563i \(-0.752486\pi\)
−0.712607 + 0.701563i \(0.752486\pi\)
\(422\) 282.009 0.0325307
\(423\) 10123.8 1.16368
\(424\) 808.339 0.0925858
\(425\) −1213.40 −0.138490
\(426\) 18.8032 0.00213854
\(427\) −242.302 −0.0274609
\(428\) −13290.6 −1.50099
\(429\) −26.0000 −0.00292609
\(430\) 560.225 0.0628290
\(431\) 12155.0 1.35844 0.679219 0.733935i \(-0.262318\pi\)
0.679219 + 0.733935i \(0.262318\pi\)
\(432\) −659.236 −0.0734201
\(433\) −9899.29 −1.09868 −0.549341 0.835598i \(-0.685121\pi\)
−0.549341 + 0.835598i \(0.685121\pi\)
\(434\) −464.517 −0.0513768
\(435\) −101.051 −0.0111380
\(436\) −14189.8 −1.55864
\(437\) −430.190 −0.0470910
\(438\) −17.9334 −0.00195637
\(439\) −11922.9 −1.29624 −0.648121 0.761538i \(-0.724445\pi\)
−0.648121 + 0.761538i \(0.724445\pi\)
\(440\) −217.583 −0.0235747
\(441\) 5942.73 0.641695
\(442\) −169.067 −0.0181939
\(443\) 18171.1 1.94884 0.974420 0.224734i \(-0.0721514\pi\)
0.974420 + 0.224734i \(0.0721514\pi\)
\(444\) 410.197 0.0438448
\(445\) −7978.79 −0.849958
\(446\) −835.950 −0.0887520
\(447\) 313.466 0.0331687
\(448\) 5365.79 0.565870
\(449\) 13386.5 1.40701 0.703507 0.710689i \(-0.251617\pi\)
0.703507 + 0.710689i \(0.251617\pi\)
\(450\) −180.608 −0.0189199
\(451\) −1895.99 −0.197957
\(452\) 4479.98 0.466196
\(453\) −476.899 −0.0494629
\(454\) −1376.03 −0.142248
\(455\) 719.667 0.0741505
\(456\) 30.5105 0.00313330
\(457\) 15155.4 1.55129 0.775645 0.631170i \(-0.217425\pi\)
0.775645 + 0.631170i \(0.217425\pi\)
\(458\) −347.697 −0.0354734
\(459\) 513.737 0.0522423
\(460\) 467.917 0.0474277
\(461\) −10147.2 −1.02517 −0.512584 0.858637i \(-0.671312\pi\)
−0.512584 + 0.858637i \(0.671312\pi\)
\(462\) 5.93336 0.000597500 0
\(463\) 8417.13 0.844876 0.422438 0.906392i \(-0.361175\pi\)
0.422438 + 0.906392i \(0.361175\pi\)
\(464\) 6417.12 0.642042
\(465\) −153.566 −0.0153149
\(466\) −1139.26 −0.113251
\(467\) 12107.8 1.19974 0.599872 0.800096i \(-0.295218\pi\)
0.599872 + 0.800096i \(0.295218\pi\)
\(468\) 2778.83 0.274469
\(469\) 7293.16 0.718053
\(470\) 503.064 0.0493715
\(471\) 135.935 0.0132984
\(472\) 829.957 0.0809362
\(473\) 4263.60 0.414462
\(474\) −52.8520 −0.00512146
\(475\) −911.122 −0.0880108
\(476\) −4260.45 −0.410247
\(477\) 5106.44 0.490164
\(478\) −61.4369 −0.00587879
\(479\) −2068.14 −0.197277 −0.0986385 0.995123i \(-0.531449\pi\)
−0.0986385 + 0.995123i \(0.531449\pi\)
\(480\) −49.8541 −0.00474066
\(481\) −3429.00 −0.325050
\(482\) −758.051 −0.0716354
\(483\) −25.6351 −0.00241499
\(484\) 9728.21 0.913619
\(485\) −5189.69 −0.485880
\(486\) 114.728 0.0107082
\(487\) −90.2686 −0.00839930 −0.00419965 0.999991i \(-0.501337\pi\)
−0.00419965 + 0.999991i \(0.501337\pi\)
\(488\) −93.4022 −0.00866418
\(489\) −84.7303 −0.00783566
\(490\) 295.301 0.0272251
\(491\) 5422.85 0.498432 0.249216 0.968448i \(-0.419827\pi\)
0.249216 + 0.968448i \(0.419827\pi\)
\(492\) −289.179 −0.0264984
\(493\) −5000.81 −0.456847
\(494\) −126.950 −0.0115622
\(495\) −1374.52 −0.124808
\(496\) 9752.01 0.882818
\(497\) −3960.99 −0.357494
\(498\) 38.1235 0.00343043
\(499\) −10624.1 −0.953102 −0.476551 0.879147i \(-0.658113\pi\)
−0.476551 + 0.879147i \(0.658113\pi\)
\(500\) 991.025 0.0886400
\(501\) −206.893 −0.0184497
\(502\) 936.840 0.0832933
\(503\) 4451.06 0.394559 0.197279 0.980347i \(-0.436789\pi\)
0.197279 + 0.980347i \(0.436789\pi\)
\(504\) −1274.04 −0.112599
\(505\) 2422.23 0.213441
\(506\) −32.2487 −0.00283326
\(507\) 33.1498 0.00290381
\(508\) 11068.4 0.966694
\(509\) −8855.36 −0.771134 −0.385567 0.922680i \(-0.625994\pi\)
−0.385567 + 0.922680i \(0.625994\pi\)
\(510\) 12.7550 0.00110745
\(511\) 3777.75 0.327041
\(512\) 5292.45 0.456827
\(513\) 385.757 0.0332000
\(514\) −988.271 −0.0848069
\(515\) 4735.20 0.405161
\(516\) 650.292 0.0554797
\(517\) 3828.58 0.325688
\(518\) 782.518 0.0663743
\(519\) −574.562 −0.0485944
\(520\) 277.417 0.0233952
\(521\) −19004.5 −1.59808 −0.799042 0.601276i \(-0.794659\pi\)
−0.799042 + 0.601276i \(0.794659\pi\)
\(522\) −744.346 −0.0624121
\(523\) −7819.17 −0.653745 −0.326872 0.945068i \(-0.605995\pi\)
−0.326872 + 0.945068i \(0.605995\pi\)
\(524\) 21198.1 1.76725
\(525\) −54.2940 −0.00451349
\(526\) −1922.91 −0.159397
\(527\) −7599.66 −0.628172
\(528\) −124.564 −0.0102670
\(529\) −12027.7 −0.988548
\(530\) 253.744 0.0207961
\(531\) 5243.01 0.428488
\(532\) −3199.11 −0.260712
\(533\) 2417.37 0.196450
\(534\) 83.8713 0.00679675
\(535\) −8381.83 −0.677342
\(536\) 2811.36 0.226553
\(537\) 729.163 0.0585953
\(538\) 215.679 0.0172836
\(539\) 2247.39 0.179595
\(540\) −419.588 −0.0334374
\(541\) −15252.8 −1.21215 −0.606073 0.795409i \(-0.707256\pi\)
−0.606073 + 0.795409i \(0.707256\pi\)
\(542\) 706.358 0.0559791
\(543\) 377.221 0.0298123
\(544\) −2467.18 −0.194447
\(545\) −8948.92 −0.703357
\(546\) −7.56498 −0.000592951 0
\(547\) −9483.30 −0.741273 −0.370637 0.928778i \(-0.620860\pi\)
−0.370637 + 0.928778i \(0.620860\pi\)
\(548\) −19841.9 −1.54672
\(549\) −590.041 −0.0458695
\(550\) −68.3013 −0.00529523
\(551\) −3755.04 −0.290326
\(552\) −9.88181 −0.000761952 0
\(553\) 11133.5 0.856142
\(554\) 451.035 0.0345896
\(555\) 258.695 0.0197856
\(556\) −2161.98 −0.164907
\(557\) 16614.1 1.26384 0.631921 0.775033i \(-0.282267\pi\)
0.631921 + 0.775033i \(0.282267\pi\)
\(558\) −1131.17 −0.0858177
\(559\) −5436.05 −0.411307
\(560\) 3447.87 0.260177
\(561\) 97.0718 0.00730548
\(562\) 638.301 0.0479094
\(563\) −22459.9 −1.68130 −0.840652 0.541576i \(-0.817828\pi\)
−0.840652 + 0.541576i \(0.817828\pi\)
\(564\) 583.942 0.0435964
\(565\) 2825.35 0.210377
\(566\) −973.861 −0.0723223
\(567\) −8036.85 −0.595266
\(568\) −1526.88 −0.112793
\(569\) −2376.89 −0.175122 −0.0875609 0.996159i \(-0.527907\pi\)
−0.0875609 + 0.996159i \(0.527907\pi\)
\(570\) 9.57751 0.000703785 0
\(571\) 17585.3 1.28883 0.644414 0.764677i \(-0.277101\pi\)
0.644414 + 0.764677i \(0.277101\pi\)
\(572\) 1050.88 0.0768175
\(573\) −380.828 −0.0277650
\(574\) −551.658 −0.0401145
\(575\) 295.096 0.0214024
\(576\) 13066.5 0.945206
\(577\) −5989.96 −0.432176 −0.216088 0.976374i \(-0.569330\pi\)
−0.216088 + 0.976374i \(0.569330\pi\)
\(578\) −685.218 −0.0493102
\(579\) 457.916 0.0328676
\(580\) 4084.35 0.292402
\(581\) −8030.91 −0.573457
\(582\) 54.5529 0.00388538
\(583\) 1931.13 0.137185
\(584\) 1456.25 0.103185
\(585\) 1752.50 0.123858
\(586\) 935.662 0.0659588
\(587\) 11523.1 0.810240 0.405120 0.914264i \(-0.367230\pi\)
0.405120 + 0.914264i \(0.367230\pi\)
\(588\) 342.776 0.0240405
\(589\) −5706.47 −0.399204
\(590\) 260.531 0.0181795
\(591\) 423.795 0.0294968
\(592\) −16428.1 −1.14052
\(593\) 4672.21 0.323549 0.161775 0.986828i \(-0.448278\pi\)
0.161775 + 0.986828i \(0.448278\pi\)
\(594\) 28.9179 0.00199750
\(595\) −2686.90 −0.185130
\(596\) −12669.8 −0.870766
\(597\) −389.508 −0.0267027
\(598\) 41.1168 0.00281169
\(599\) −10245.3 −0.698850 −0.349425 0.936964i \(-0.613623\pi\)
−0.349425 + 0.936964i \(0.613623\pi\)
\(600\) −20.9292 −0.00142405
\(601\) 11755.8 0.797883 0.398942 0.916976i \(-0.369378\pi\)
0.398942 + 0.916976i \(0.369378\pi\)
\(602\) 1240.54 0.0839877
\(603\) 17760.0 1.19941
\(604\) 19275.6 1.29853
\(605\) 6135.19 0.412283
\(606\) −25.4619 −0.00170680
\(607\) −19364.3 −1.29484 −0.647422 0.762131i \(-0.724153\pi\)
−0.647422 + 0.762131i \(0.724153\pi\)
\(608\) −1852.57 −0.123571
\(609\) −223.764 −0.0148889
\(610\) −29.3198 −0.00194610
\(611\) −4881.40 −0.323208
\(612\) −10374.9 −0.685260
\(613\) −21110.0 −1.39091 −0.695454 0.718571i \(-0.744797\pi\)
−0.695454 + 0.718571i \(0.744797\pi\)
\(614\) 458.237 0.0301188
\(615\) −182.374 −0.0119578
\(616\) −481.808 −0.0315139
\(617\) −2427.03 −0.158361 −0.0791803 0.996860i \(-0.525230\pi\)
−0.0791803 + 0.996860i \(0.525230\pi\)
\(618\) −49.7754 −0.00323990
\(619\) −4293.14 −0.278765 −0.139383 0.990239i \(-0.544512\pi\)
−0.139383 + 0.990239i \(0.544512\pi\)
\(620\) 6206.92 0.402058
\(621\) −124.940 −0.00807354
\(622\) 2186.02 0.140919
\(623\) −17667.9 −1.13620
\(624\) 158.818 0.0101888
\(625\) 625.000 0.0400000
\(626\) 2951.39 0.188436
\(627\) 72.8897 0.00464264
\(628\) −5494.29 −0.349118
\(629\) 12802.3 0.811542
\(630\) −399.931 −0.0252915
\(631\) −3099.63 −0.195554 −0.0977769 0.995208i \(-0.531173\pi\)
−0.0977769 + 0.995208i \(0.531173\pi\)
\(632\) 4291.75 0.270121
\(633\) 206.445 0.0129628
\(634\) −1578.32 −0.0988695
\(635\) 6980.39 0.436234
\(636\) 294.539 0.0183636
\(637\) −2865.40 −0.178228
\(638\) −281.492 −0.0174677
\(639\) −9645.62 −0.597144
\(640\) 2682.57 0.165684
\(641\) 12932.1 0.796857 0.398429 0.917199i \(-0.369556\pi\)
0.398429 + 0.917199i \(0.369556\pi\)
\(642\) 88.1079 0.00541642
\(643\) 27965.0 1.71514 0.857568 0.514371i \(-0.171974\pi\)
0.857568 + 0.514371i \(0.171974\pi\)
\(644\) 1036.14 0.0633998
\(645\) 410.113 0.0250360
\(646\) 473.971 0.0288671
\(647\) −18467.6 −1.12216 −0.561080 0.827762i \(-0.689614\pi\)
−0.561080 + 0.827762i \(0.689614\pi\)
\(648\) −3098.04 −0.187813
\(649\) 1982.77 0.119924
\(650\) 87.0835 0.00525492
\(651\) −340.050 −0.0204725
\(652\) 3424.68 0.205707
\(653\) 4411.89 0.264396 0.132198 0.991223i \(-0.457797\pi\)
0.132198 + 0.991223i \(0.457797\pi\)
\(654\) 94.0691 0.00562445
\(655\) 13368.8 0.797498
\(656\) 11581.4 0.689296
\(657\) 9199.41 0.546276
\(658\) 1113.97 0.0659983
\(659\) 18059.6 1.06753 0.533765 0.845633i \(-0.320777\pi\)
0.533765 + 0.845633i \(0.320777\pi\)
\(660\) −79.2820 −0.00467583
\(661\) 18324.0 1.07825 0.539125 0.842226i \(-0.318755\pi\)
0.539125 + 0.842226i \(0.318755\pi\)
\(662\) 4.07890 0.000239473 0
\(663\) −123.766 −0.00724986
\(664\) −3095.75 −0.180931
\(665\) −2017.55 −0.117650
\(666\) 1905.55 0.110869
\(667\) 1216.19 0.0706013
\(668\) 8362.34 0.484354
\(669\) −611.958 −0.0353657
\(670\) 882.511 0.0508871
\(671\) −223.138 −0.0128378
\(672\) −110.395 −0.00633717
\(673\) 5321.92 0.304822 0.152411 0.988317i \(-0.451296\pi\)
0.152411 + 0.988317i \(0.451296\pi\)
\(674\) 2825.34 0.161466
\(675\) −264.617 −0.0150891
\(676\) −1339.87 −0.0762327
\(677\) −17817.6 −1.01150 −0.505751 0.862679i \(-0.668785\pi\)
−0.505751 + 0.862679i \(0.668785\pi\)
\(678\) −29.6994 −0.00168230
\(679\) −11491.8 −0.649509
\(680\) −1035.74 −0.0584102
\(681\) −1007.33 −0.0566826
\(682\) −427.779 −0.0240184
\(683\) 24498.3 1.37248 0.686238 0.727377i \(-0.259261\pi\)
0.686238 + 0.727377i \(0.259261\pi\)
\(684\) −7790.32 −0.435483
\(685\) −12513.5 −0.697981
\(686\) 1671.47 0.0930279
\(687\) −254.532 −0.0141354
\(688\) −26043.7 −1.44318
\(689\) −2462.17 −0.136141
\(690\) −3.10199 −0.000171146 0
\(691\) 36011.4 1.98254 0.991271 0.131839i \(-0.0420881\pi\)
0.991271 + 0.131839i \(0.0420881\pi\)
\(692\) 23223.0 1.27573
\(693\) −3043.68 −0.166840
\(694\) −186.381 −0.0101944
\(695\) −1363.47 −0.0744165
\(696\) −86.2563 −0.00469761
\(697\) −9025.31 −0.490471
\(698\) −383.403 −0.0207909
\(699\) −833.996 −0.0451282
\(700\) 2194.49 0.118491
\(701\) 19527.5 1.05213 0.526064 0.850445i \(-0.323667\pi\)
0.526064 + 0.850445i \(0.323667\pi\)
\(702\) −36.8700 −0.00198229
\(703\) 9613.03 0.515736
\(704\) 4941.42 0.264541
\(705\) 368.269 0.0196735
\(706\) −3391.64 −0.180802
\(707\) 5363.69 0.285321
\(708\) 302.416 0.0160530
\(709\) −10114.7 −0.535779 −0.267889 0.963450i \(-0.586326\pi\)
−0.267889 + 0.963450i \(0.586326\pi\)
\(710\) −479.301 −0.0253350
\(711\) 27111.9 1.43006
\(712\) −6810.62 −0.358481
\(713\) 1848.22 0.0970779
\(714\) 28.2441 0.00148040
\(715\) 662.750 0.0346650
\(716\) −29471.7 −1.53828
\(717\) −44.9750 −0.00234257
\(718\) 1500.79 0.0780068
\(719\) −22538.4 −1.16904 −0.584522 0.811378i \(-0.698718\pi\)
−0.584522 + 0.811378i \(0.698718\pi\)
\(720\) 8396.09 0.434589
\(721\) 10485.4 0.541606
\(722\) −1481.97 −0.0763893
\(723\) −554.932 −0.0285451
\(724\) −15246.7 −0.782652
\(725\) 2575.83 0.131950
\(726\) −64.4918 −0.00329685
\(727\) −16847.2 −0.859463 −0.429731 0.902957i \(-0.641392\pi\)
−0.429731 + 0.902957i \(0.641392\pi\)
\(728\) 614.300 0.0312740
\(729\) −19514.9 −0.991460
\(730\) 457.128 0.0231768
\(731\) 20295.7 1.02690
\(732\) −34.0335 −0.00171846
\(733\) 36464.0 1.83742 0.918709 0.394936i \(-0.129233\pi\)
0.918709 + 0.394936i \(0.129233\pi\)
\(734\) −3384.27 −0.170185
\(735\) 216.175 0.0108486
\(736\) 600.013 0.0300500
\(737\) 6716.36 0.335686
\(738\) −1343.37 −0.0670056
\(739\) 3230.48 0.160805 0.0804026 0.996762i \(-0.474379\pi\)
0.0804026 + 0.996762i \(0.474379\pi\)
\(740\) −10456.1 −0.519423
\(741\) −92.9337 −0.00460730
\(742\) 561.881 0.0277996
\(743\) −28359.4 −1.40028 −0.700138 0.714008i \(-0.746878\pi\)
−0.700138 + 0.714008i \(0.746878\pi\)
\(744\) −131.082 −0.00645929
\(745\) −7990.36 −0.392945
\(746\) −1123.63 −0.0551460
\(747\) −19556.5 −0.957878
\(748\) −3923.50 −0.191788
\(749\) −18560.4 −0.905449
\(750\) −6.56986 −0.000319863 0
\(751\) −5885.99 −0.285996 −0.142998 0.989723i \(-0.545674\pi\)
−0.142998 + 0.989723i \(0.545674\pi\)
\(752\) −23386.4 −1.13406
\(753\) 685.815 0.0331905
\(754\) 358.900 0.0173347
\(755\) 12156.3 0.585979
\(756\) −929.118 −0.0446980
\(757\) 29892.2 1.43521 0.717603 0.696452i \(-0.245239\pi\)
0.717603 + 0.696452i \(0.245239\pi\)
\(758\) 1072.82 0.0514072
\(759\) −23.6077 −0.00112899
\(760\) −777.724 −0.0371198
\(761\) −14562.1 −0.693661 −0.346830 0.937928i \(-0.612742\pi\)
−0.346830 + 0.937928i \(0.612742\pi\)
\(762\) −73.3763 −0.00348838
\(763\) −19816.1 −0.940225
\(764\) 15392.5 0.728904
\(765\) −6543.01 −0.309233
\(766\) −2552.20 −0.120385
\(767\) −2528.02 −0.119011
\(768\) 732.301 0.0344071
\(769\) 20090.7 0.942117 0.471058 0.882102i \(-0.343872\pi\)
0.471058 + 0.882102i \(0.343872\pi\)
\(770\) −151.244 −0.00707849
\(771\) −723.464 −0.0337937
\(772\) −18508.3 −0.862862
\(773\) −14955.4 −0.695871 −0.347936 0.937518i \(-0.613117\pi\)
−0.347936 + 0.937518i \(0.613117\pi\)
\(774\) 3020.90 0.140290
\(775\) 3914.45 0.181434
\(776\) −4429.87 −0.204927
\(777\) 572.843 0.0264487
\(778\) 2659.16 0.122539
\(779\) −6776.97 −0.311695
\(780\) 101.084 0.00464023
\(781\) −3647.73 −0.167127
\(782\) −153.511 −0.00701987
\(783\) −1090.58 −0.0497752
\(784\) −13727.9 −0.625361
\(785\) −3465.03 −0.157544
\(786\) −140.529 −0.00637725
\(787\) 14918.1 0.675695 0.337848 0.941201i \(-0.390301\pi\)
0.337848 + 0.941201i \(0.390301\pi\)
\(788\) −17129.2 −0.774368
\(789\) −1407.67 −0.0635162
\(790\) 1347.22 0.0606732
\(791\) 6256.33 0.281226
\(792\) −1173.28 −0.0526396
\(793\) 284.499 0.0127401
\(794\) −1734.68 −0.0775335
\(795\) 185.754 0.00828681
\(796\) 15743.4 0.701017
\(797\) −9626.31 −0.427831 −0.213916 0.976852i \(-0.568622\pi\)
−0.213916 + 0.976852i \(0.568622\pi\)
\(798\) 21.2080 0.000940798 0
\(799\) 18224.9 0.806945
\(800\) 1270.80 0.0561620
\(801\) −43024.1 −1.89785
\(802\) −2246.23 −0.0988992
\(803\) 3478.98 0.152890
\(804\) 1024.39 0.0449347
\(805\) 653.449 0.0286100
\(806\) 545.415 0.0238355
\(807\) 157.888 0.00688713
\(808\) 2067.59 0.0900218
\(809\) 28544.0 1.24048 0.620242 0.784410i \(-0.287034\pi\)
0.620242 + 0.784410i \(0.287034\pi\)
\(810\) −972.501 −0.0421855
\(811\) −11228.4 −0.486168 −0.243084 0.970005i \(-0.578159\pi\)
−0.243084 + 0.970005i \(0.578159\pi\)
\(812\) 9044.21 0.390874
\(813\) 517.090 0.0223064
\(814\) 720.631 0.0310296
\(815\) 2159.81 0.0928279
\(816\) −592.952 −0.0254381
\(817\) 15239.7 0.652594
\(818\) −3122.84 −0.133481
\(819\) 3880.66 0.165569
\(820\) 7371.30 0.313923
\(821\) 1636.05 0.0695474 0.0347737 0.999395i \(-0.488929\pi\)
0.0347737 + 0.999395i \(0.488929\pi\)
\(822\) 131.539 0.00558146
\(823\) −34916.5 −1.47887 −0.739437 0.673225i \(-0.764908\pi\)
−0.739437 + 0.673225i \(0.764908\pi\)
\(824\) 4041.92 0.170882
\(825\) −50.0000 −0.00211003
\(826\) 576.908 0.0243017
\(827\) −40106.8 −1.68640 −0.843198 0.537603i \(-0.819330\pi\)
−0.843198 + 0.537603i \(0.819330\pi\)
\(828\) 2523.15 0.105900
\(829\) 35438.8 1.48473 0.742364 0.669996i \(-0.233704\pi\)
0.742364 + 0.669996i \(0.233704\pi\)
\(830\) −971.783 −0.0406399
\(831\) 330.180 0.0137832
\(832\) −6300.27 −0.262527
\(833\) 10698.1 0.444977
\(834\) 14.3325 0.000595078 0
\(835\) 5273.79 0.218571
\(836\) −2946.10 −0.121882
\(837\) −1657.33 −0.0684417
\(838\) 707.752 0.0291753
\(839\) −32415.0 −1.33384 −0.666919 0.745130i \(-0.732387\pi\)
−0.666919 + 0.745130i \(0.732387\pi\)
\(840\) −46.3448 −0.00190363
\(841\) −13773.1 −0.564727
\(842\) −3298.80 −0.135017
\(843\) 467.269 0.0190908
\(844\) −8344.21 −0.340307
\(845\) −845.000 −0.0344010
\(846\) 2712.68 0.110241
\(847\) 13585.5 0.551126
\(848\) −11796.1 −0.477687
\(849\) −712.916 −0.0288189
\(850\) −325.129 −0.0131198
\(851\) −3113.49 −0.125416
\(852\) −556.358 −0.0223715
\(853\) 36881.7 1.48043 0.740214 0.672371i \(-0.234724\pi\)
0.740214 + 0.672371i \(0.234724\pi\)
\(854\) −64.9245 −0.00260149
\(855\) −4913.05 −0.196518
\(856\) −7154.64 −0.285678
\(857\) 30112.3 1.20025 0.600126 0.799906i \(-0.295117\pi\)
0.600126 + 0.799906i \(0.295117\pi\)
\(858\) −6.96668 −0.000277201 0
\(859\) 9640.14 0.382907 0.191454 0.981502i \(-0.438680\pi\)
0.191454 + 0.981502i \(0.438680\pi\)
\(860\) −16576.2 −0.657260
\(861\) −403.841 −0.0159848
\(862\) 3256.93 0.128691
\(863\) 22960.7 0.905668 0.452834 0.891595i \(-0.350413\pi\)
0.452834 + 0.891595i \(0.350413\pi\)
\(864\) −538.041 −0.0211858
\(865\) 14645.8 0.575691
\(866\) −2652.51 −0.104083
\(867\) −501.614 −0.0196490
\(868\) 13744.3 0.537458
\(869\) 10253.0 0.400241
\(870\) −27.0766 −0.00105515
\(871\) −8563.30 −0.333130
\(872\) −7638.71 −0.296651
\(873\) −27984.4 −1.08491
\(874\) −115.269 −0.00446113
\(875\) 1383.97 0.0534707
\(876\) 530.620 0.0204658
\(877\) 40801.9 1.57102 0.785508 0.618851i \(-0.212402\pi\)
0.785508 + 0.618851i \(0.212402\pi\)
\(878\) −3194.74 −0.122799
\(879\) 684.952 0.0262831
\(880\) 3175.19 0.121631
\(881\) −3875.57 −0.148208 −0.0741040 0.997251i \(-0.523610\pi\)
−0.0741040 + 0.997251i \(0.523610\pi\)
\(882\) 1592.35 0.0607905
\(883\) −18239.8 −0.695149 −0.347574 0.937652i \(-0.612995\pi\)
−0.347574 + 0.937652i \(0.612995\pi\)
\(884\) 5002.43 0.190328
\(885\) 190.722 0.00724411
\(886\) 4868.94 0.184622
\(887\) 2380.14 0.0900985 0.0450493 0.998985i \(-0.485656\pi\)
0.0450493 + 0.998985i \(0.485656\pi\)
\(888\) 220.819 0.00834483
\(889\) 15457.1 0.583143
\(890\) −2137.91 −0.0805202
\(891\) −7401.23 −0.278284
\(892\) 24734.5 0.928444
\(893\) 13684.8 0.512814
\(894\) 83.9928 0.00314222
\(895\) −18586.6 −0.694171
\(896\) 5940.17 0.221481
\(897\) 30.0996 0.00112040
\(898\) 3586.91 0.133292
\(899\) 16132.8 0.598507
\(900\) 5343.91 0.197923
\(901\) 9192.58 0.339899
\(902\) −508.028 −0.0187533
\(903\) 908.138 0.0334673
\(904\) 2411.69 0.0887295
\(905\) −9615.50 −0.353182
\(906\) −127.785 −0.00468583
\(907\) 2739.49 0.100290 0.0501452 0.998742i \(-0.484032\pi\)
0.0501452 + 0.998742i \(0.484032\pi\)
\(908\) 40714.8 1.48807
\(909\) 13061.4 0.476589
\(910\) 192.834 0.00702460
\(911\) −42446.3 −1.54370 −0.771850 0.635805i \(-0.780668\pi\)
−0.771850 + 0.635805i \(0.780668\pi\)
\(912\) −445.239 −0.0161659
\(913\) −7395.77 −0.268088
\(914\) 4060.87 0.146960
\(915\) −21.4636 −0.000775479 0
\(916\) 10287.8 0.371091
\(917\) 29603.3 1.06607
\(918\) 137.655 0.00494914
\(919\) 7246.81 0.260120 0.130060 0.991506i \(-0.458483\pi\)
0.130060 + 0.991506i \(0.458483\pi\)
\(920\) 251.891 0.00902674
\(921\) 335.453 0.0120017
\(922\) −2718.94 −0.0971187
\(923\) 4650.82 0.165854
\(924\) −175.559 −0.00625050
\(925\) −6594.23 −0.234397
\(926\) 2255.36 0.0800387
\(927\) 25533.6 0.904676
\(928\) 5237.39 0.185265
\(929\) −26804.8 −0.946648 −0.473324 0.880888i \(-0.656946\pi\)
−0.473324 + 0.880888i \(0.656946\pi\)
\(930\) −41.1479 −0.00145085
\(931\) 8033.01 0.282783
\(932\) 33708.9 1.18473
\(933\) 1600.28 0.0561529
\(934\) 3244.26 0.113657
\(935\) −2474.40 −0.0865470
\(936\) 1495.92 0.0522388
\(937\) 43808.7 1.52739 0.763696 0.645576i \(-0.223383\pi\)
0.763696 + 0.645576i \(0.223383\pi\)
\(938\) 1954.20 0.0680243
\(939\) 2160.57 0.0750878
\(940\) −14884.9 −0.516481
\(941\) −13038.6 −0.451697 −0.225848 0.974162i \(-0.572515\pi\)
−0.225848 + 0.974162i \(0.572515\pi\)
\(942\) 36.4236 0.00125981
\(943\) 2194.94 0.0757975
\(944\) −12111.5 −0.417581
\(945\) −585.957 −0.0201706
\(946\) 1142.43 0.0392638
\(947\) −45110.1 −1.54792 −0.773961 0.633233i \(-0.781727\pi\)
−0.773961 + 0.633233i \(0.781727\pi\)
\(948\) 1563.81 0.0535761
\(949\) −4435.67 −0.151726
\(950\) −244.134 −0.00833764
\(951\) −1155.41 −0.0393973
\(952\) −2293.51 −0.0780809
\(953\) 9455.70 0.321406 0.160703 0.987003i \(-0.448624\pi\)
0.160703 + 0.987003i \(0.448624\pi\)
\(954\) 1368.27 0.0464353
\(955\) 9707.46 0.328928
\(956\) 1817.82 0.0614986
\(957\) −206.067 −0.00696049
\(958\) −554.156 −0.0186889
\(959\) −27709.4 −0.933038
\(960\) 475.312 0.0159798
\(961\) −5274.28 −0.177043
\(962\) −918.797 −0.0307934
\(963\) −45197.4 −1.51242
\(964\) 22429.6 0.749385
\(965\) −11672.5 −0.389378
\(966\) −6.86891 −0.000228782 0
\(967\) −5616.97 −0.186794 −0.0933969 0.995629i \(-0.529773\pi\)
−0.0933969 + 0.995629i \(0.529773\pi\)
\(968\) 5236.94 0.173886
\(969\) 346.971 0.0115029
\(970\) −1390.57 −0.0460295
\(971\) 12009.7 0.396919 0.198459 0.980109i \(-0.436406\pi\)
0.198459 + 0.980109i \(0.436406\pi\)
\(972\) −3394.62 −0.112019
\(973\) −3019.22 −0.0994776
\(974\) −24.1874 −0.000795702 0
\(975\) 63.7495 0.00209397
\(976\) 1363.02 0.0447019
\(977\) −36835.7 −1.20622 −0.603111 0.797657i \(-0.706073\pi\)
−0.603111 + 0.797657i \(0.706073\pi\)
\(978\) −22.7034 −0.000742306 0
\(979\) −16270.6 −0.531165
\(980\) −8737.49 −0.284805
\(981\) −48255.3 −1.57051
\(982\) 1453.05 0.0472186
\(983\) 11311.1 0.367006 0.183503 0.983019i \(-0.441256\pi\)
0.183503 + 0.983019i \(0.441256\pi\)
\(984\) −155.673 −0.00504335
\(985\) −10802.7 −0.349444
\(986\) −1339.96 −0.0432791
\(987\) 815.479 0.0262989
\(988\) 3756.25 0.120954
\(989\) −4935.87 −0.158697
\(990\) −368.301 −0.0118236
\(991\) −45711.2 −1.46525 −0.732625 0.680632i \(-0.761705\pi\)
−0.732625 + 0.680632i \(0.761705\pi\)
\(992\) 7959.18 0.254742
\(993\) 2.98596 9.54246e−5 0
\(994\) −1061.34 −0.0338670
\(995\) 9928.72 0.316343
\(996\) −1128.02 −0.0358861
\(997\) −24127.3 −0.766418 −0.383209 0.923662i \(-0.625181\pi\)
−0.383209 + 0.923662i \(0.625181\pi\)
\(998\) −2846.71 −0.0902915
\(999\) 2791.91 0.0884206
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 65.4.a.d.1.1 2
3.2 odd 2 585.4.a.f.1.2 2
4.3 odd 2 1040.4.a.o.1.1 2
5.2 odd 4 325.4.b.c.274.3 4
5.3 odd 4 325.4.b.c.274.2 4
5.4 even 2 325.4.a.e.1.2 2
13.12 even 2 845.4.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.4.a.d.1.1 2 1.1 even 1 trivial
325.4.a.e.1.2 2 5.4 even 2
325.4.b.c.274.2 4 5.3 odd 4
325.4.b.c.274.3 4 5.2 odd 4
585.4.a.f.1.2 2 3.2 odd 2
845.4.a.c.1.2 2 13.12 even 2
1040.4.a.o.1.1 2 4.3 odd 2