Properties

Label 165.4.a.h.1.2
Level $165$
Weight $4$
Character 165.1
Self dual yes
Analytic conductor $9.735$
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] = [165,4,Mod(1,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, 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("165.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.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: \(9.73531515095\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.1540841.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 27x^{2} - 18x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.60719\) of defining polynomial
Character \(\chi\) \(=\) 165.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.607192 q^{2} +3.00000 q^{3} -7.63132 q^{4} +5.00000 q^{5} -1.82158 q^{6} +8.95080 q^{7} +9.49121 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-0.607192 q^{2} +3.00000 q^{3} -7.63132 q^{4} +5.00000 q^{5} -1.82158 q^{6} +8.95080 q^{7} +9.49121 q^{8} +9.00000 q^{9} -3.03596 q^{10} -11.0000 q^{11} -22.8940 q^{12} -0.460387 q^{13} -5.43485 q^{14} +15.0000 q^{15} +55.2876 q^{16} +128.395 q^{17} -5.46473 q^{18} -0.0245858 q^{19} -38.1566 q^{20} +26.8524 q^{21} +6.67911 q^{22} +171.528 q^{23} +28.4736 q^{24} +25.0000 q^{25} +0.279543 q^{26} +27.0000 q^{27} -68.3064 q^{28} -226.938 q^{29} -9.10788 q^{30} +195.637 q^{31} -109.500 q^{32} -33.0000 q^{33} -77.9604 q^{34} +44.7540 q^{35} -68.6819 q^{36} +338.584 q^{37} +0.0149283 q^{38} -1.38116 q^{39} +47.4560 q^{40} +136.972 q^{41} -16.3046 q^{42} +336.083 q^{43} +83.9445 q^{44} +45.0000 q^{45} -104.151 q^{46} -540.292 q^{47} +165.863 q^{48} -262.883 q^{49} -15.1798 q^{50} +385.185 q^{51} +3.51336 q^{52} -622.387 q^{53} -16.3942 q^{54} -55.0000 q^{55} +84.9539 q^{56} -0.0737574 q^{57} +137.795 q^{58} -9.86955 q^{59} -114.470 q^{60} +902.712 q^{61} -118.789 q^{62} +80.5572 q^{63} -375.813 q^{64} -2.30193 q^{65} +20.0373 q^{66} +146.979 q^{67} -979.823 q^{68} +514.585 q^{69} -27.1743 q^{70} -893.798 q^{71} +85.4209 q^{72} -1149.71 q^{73} -205.585 q^{74} +75.0000 q^{75} +0.187622 q^{76} -98.4588 q^{77} +0.838629 q^{78} -459.528 q^{79} +276.438 q^{80} +81.0000 q^{81} -83.1686 q^{82} -125.876 q^{83} -204.919 q^{84} +641.975 q^{85} -204.067 q^{86} -680.813 q^{87} -104.403 q^{88} +150.461 q^{89} -27.3236 q^{90} -4.12083 q^{91} -1308.99 q^{92} +586.912 q^{93} +328.061 q^{94} -0.122929 q^{95} -328.499 q^{96} -1264.58 q^{97} +159.621 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 12 q^{3} + 26 q^{4} + 20 q^{5} + 12 q^{6} + 34 q^{7} + 48 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 12 q^{3} + 26 q^{4} + 20 q^{5} + 12 q^{6} + 34 q^{7} + 48 q^{8} + 36 q^{9} + 20 q^{10} - 44 q^{11} + 78 q^{12} + 2 q^{13} - 52 q^{14} + 60 q^{15} + 66 q^{16} + 74 q^{17} + 36 q^{18} + 136 q^{19} + 130 q^{20} + 102 q^{21} - 44 q^{22} - 64 q^{23} + 144 q^{24} + 100 q^{25} - 320 q^{26} + 108 q^{27} - 20 q^{28} + 52 q^{29} + 60 q^{30} + 492 q^{31} + 208 q^{32} - 132 q^{33} + 244 q^{34} + 170 q^{35} + 234 q^{36} - 4 q^{37} - 404 q^{38} + 6 q^{39} + 240 q^{40} + 268 q^{41} - 156 q^{42} + 546 q^{43} - 286 q^{44} + 180 q^{45} + 368 q^{46} - 276 q^{47} + 198 q^{48} - 496 q^{49} + 100 q^{50} + 222 q^{51} - 1084 q^{52} - 184 q^{53} + 108 q^{54} - 220 q^{55} - 852 q^{56} + 408 q^{57} - 444 q^{58} - 1032 q^{59} + 390 q^{60} + 116 q^{61} - 1240 q^{62} + 306 q^{63} - 918 q^{64} + 10 q^{65} - 132 q^{66} - 552 q^{67} - 720 q^{68} - 192 q^{69} - 260 q^{70} - 920 q^{71} + 432 q^{72} + 926 q^{73} - 2856 q^{74} + 300 q^{75} + 1572 q^{76} - 374 q^{77} - 960 q^{78} + 1152 q^{79} + 330 q^{80} + 324 q^{81} - 1924 q^{82} - 134 q^{83} - 60 q^{84} + 370 q^{85} + 236 q^{86} + 156 q^{87} - 528 q^{88} - 1064 q^{89} + 180 q^{90} + 2780 q^{91} - 4896 q^{92} + 1476 q^{93} - 1432 q^{94} + 680 q^{95} + 624 q^{96} - 1648 q^{97} - 188 q^{98} - 396 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.607192 −0.214675 −0.107337 0.994223i \(-0.534232\pi\)
−0.107337 + 0.994223i \(0.534232\pi\)
\(3\) 3.00000 0.577350
\(4\) −7.63132 −0.953915
\(5\) 5.00000 0.447214
\(6\) −1.82158 −0.123943
\(7\) 8.95080 0.483298 0.241649 0.970364i \(-0.422312\pi\)
0.241649 + 0.970364i \(0.422312\pi\)
\(8\) 9.49121 0.419456
\(9\) 9.00000 0.333333
\(10\) −3.03596 −0.0960055
\(11\) −11.0000 −0.301511
\(12\) −22.8940 −0.550743
\(13\) −0.460387 −0.00982218 −0.00491109 0.999988i \(-0.501563\pi\)
−0.00491109 + 0.999988i \(0.501563\pi\)
\(14\) −5.43485 −0.103752
\(15\) 15.0000 0.258199
\(16\) 55.2876 0.863868
\(17\) 128.395 1.83179 0.915893 0.401423i \(-0.131484\pi\)
0.915893 + 0.401423i \(0.131484\pi\)
\(18\) −5.46473 −0.0715582
\(19\) −0.0245858 −0.000296862 0 −0.000148431 1.00000i \(-0.500047\pi\)
−0.000148431 1.00000i \(0.500047\pi\)
\(20\) −38.1566 −0.426604
\(21\) 26.8524 0.279032
\(22\) 6.67911 0.0647269
\(23\) 171.528 1.55505 0.777525 0.628852i \(-0.216475\pi\)
0.777525 + 0.628852i \(0.216475\pi\)
\(24\) 28.4736 0.242173
\(25\) 25.0000 0.200000
\(26\) 0.279543 0.00210857
\(27\) 27.0000 0.192450
\(28\) −68.3064 −0.461025
\(29\) −226.938 −1.45315 −0.726574 0.687088i \(-0.758889\pi\)
−0.726574 + 0.687088i \(0.758889\pi\)
\(30\) −9.10788 −0.0554288
\(31\) 195.637 1.13347 0.566734 0.823901i \(-0.308207\pi\)
0.566734 + 0.823901i \(0.308207\pi\)
\(32\) −109.500 −0.604907
\(33\) −33.0000 −0.174078
\(34\) −77.9604 −0.393238
\(35\) 44.7540 0.216137
\(36\) −68.6819 −0.317972
\(37\) 338.584 1.50440 0.752200 0.658935i \(-0.228993\pi\)
0.752200 + 0.658935i \(0.228993\pi\)
\(38\) 0.0149283 6.37287e−5 0
\(39\) −1.38116 −0.00567084
\(40\) 47.4560 0.187586
\(41\) 136.972 0.521744 0.260872 0.965373i \(-0.415990\pi\)
0.260872 + 0.965373i \(0.415990\pi\)
\(42\) −16.3046 −0.0599011
\(43\) 336.083 1.19191 0.595956 0.803017i \(-0.296773\pi\)
0.595956 + 0.803017i \(0.296773\pi\)
\(44\) 83.9445 0.287616
\(45\) 45.0000 0.149071
\(46\) −104.151 −0.333830
\(47\) −540.292 −1.67680 −0.838400 0.545055i \(-0.816509\pi\)
−0.838400 + 0.545055i \(0.816509\pi\)
\(48\) 165.863 0.498754
\(49\) −262.883 −0.766423
\(50\) −15.1798 −0.0429349
\(51\) 385.185 1.05758
\(52\) 3.51336 0.00936952
\(53\) −622.387 −1.61305 −0.806523 0.591203i \(-0.798653\pi\)
−0.806523 + 0.591203i \(0.798653\pi\)
\(54\) −16.3942 −0.0413142
\(55\) −55.0000 −0.134840
\(56\) 84.9539 0.202722
\(57\) −0.0737574 −0.000171393 0
\(58\) 137.795 0.311954
\(59\) −9.86955 −0.0217781 −0.0108890 0.999941i \(-0.503466\pi\)
−0.0108890 + 0.999941i \(0.503466\pi\)
\(60\) −114.470 −0.246300
\(61\) 902.712 1.89476 0.947380 0.320110i \(-0.103720\pi\)
0.947380 + 0.320110i \(0.103720\pi\)
\(62\) −118.789 −0.243327
\(63\) 80.5572 0.161099
\(64\) −375.813 −0.734010
\(65\) −2.30193 −0.00439261
\(66\) 20.0373 0.0373701
\(67\) 146.979 0.268005 0.134003 0.990981i \(-0.457217\pi\)
0.134003 + 0.990981i \(0.457217\pi\)
\(68\) −979.823 −1.74737
\(69\) 514.585 0.897809
\(70\) −27.1743 −0.0463992
\(71\) −893.798 −1.49400 −0.747002 0.664821i \(-0.768508\pi\)
−0.747002 + 0.664821i \(0.768508\pi\)
\(72\) 85.4209 0.139819
\(73\) −1149.71 −1.84333 −0.921664 0.387988i \(-0.873170\pi\)
−0.921664 + 0.387988i \(0.873170\pi\)
\(74\) −205.585 −0.322957
\(75\) 75.0000 0.115470
\(76\) 0.187622 0.000283181 0
\(77\) −98.4588 −0.145720
\(78\) 0.838629 0.00121739
\(79\) −459.528 −0.654443 −0.327221 0.944948i \(-0.606112\pi\)
−0.327221 + 0.944948i \(0.606112\pi\)
\(80\) 276.438 0.386334
\(81\) 81.0000 0.111111
\(82\) −83.1686 −0.112005
\(83\) −125.876 −0.166466 −0.0832331 0.996530i \(-0.526525\pi\)
−0.0832331 + 0.996530i \(0.526525\pi\)
\(84\) −204.919 −0.266173
\(85\) 641.975 0.819199
\(86\) −204.067 −0.255873
\(87\) −680.813 −0.838975
\(88\) −104.403 −0.126471
\(89\) 150.461 0.179201 0.0896003 0.995978i \(-0.471441\pi\)
0.0896003 + 0.995978i \(0.471441\pi\)
\(90\) −27.3236 −0.0320018
\(91\) −4.12083 −0.00474704
\(92\) −1308.99 −1.48339
\(93\) 586.912 0.654408
\(94\) 328.061 0.359967
\(95\) −0.122929 −0.000132761 0
\(96\) −328.499 −0.349243
\(97\) −1264.58 −1.32370 −0.661851 0.749635i \(-0.730229\pi\)
−0.661851 + 0.749635i \(0.730229\pi\)
\(98\) 159.621 0.164532
\(99\) −99.0000 −0.100504
\(100\) −190.783 −0.190783
\(101\) 690.792 0.680558 0.340279 0.940324i \(-0.389478\pi\)
0.340279 + 0.940324i \(0.389478\pi\)
\(102\) −233.881 −0.227036
\(103\) −11.7823 −0.0112713 −0.00563563 0.999984i \(-0.501794\pi\)
−0.00563563 + 0.999984i \(0.501794\pi\)
\(104\) −4.36963 −0.00411997
\(105\) 134.262 0.124787
\(106\) 377.908 0.346280
\(107\) −462.884 −0.418212 −0.209106 0.977893i \(-0.567055\pi\)
−0.209106 + 0.977893i \(0.567055\pi\)
\(108\) −206.046 −0.183581
\(109\) 330.525 0.290446 0.145223 0.989399i \(-0.453610\pi\)
0.145223 + 0.989399i \(0.453610\pi\)
\(110\) 33.3956 0.0289467
\(111\) 1015.75 0.868566
\(112\) 494.868 0.417506
\(113\) 1142.31 0.950965 0.475483 0.879725i \(-0.342273\pi\)
0.475483 + 0.879725i \(0.342273\pi\)
\(114\) 0.0447849 3.67938e−5 0
\(115\) 857.642 0.695440
\(116\) 1731.83 1.38618
\(117\) −4.14348 −0.00327406
\(118\) 5.99271 0.00467520
\(119\) 1149.24 0.885298
\(120\) 142.368 0.108303
\(121\) 121.000 0.0909091
\(122\) −548.119 −0.406757
\(123\) 410.917 0.301229
\(124\) −1492.97 −1.08123
\(125\) 125.000 0.0894427
\(126\) −48.9137 −0.0345839
\(127\) 629.589 0.439898 0.219949 0.975511i \(-0.429411\pi\)
0.219949 + 0.975511i \(0.429411\pi\)
\(128\) 1104.19 0.762480
\(129\) 1008.25 0.688151
\(130\) 1.39772 0.000942983 0
\(131\) 572.566 0.381873 0.190937 0.981602i \(-0.438848\pi\)
0.190937 + 0.981602i \(0.438848\pi\)
\(132\) 251.833 0.166055
\(133\) −0.220063 −0.000143473 0
\(134\) −89.2446 −0.0575340
\(135\) 135.000 0.0860663
\(136\) 1218.62 0.768354
\(137\) −948.680 −0.591615 −0.295807 0.955248i \(-0.595589\pi\)
−0.295807 + 0.955248i \(0.595589\pi\)
\(138\) −312.452 −0.192737
\(139\) 2488.86 1.51872 0.759362 0.650668i \(-0.225511\pi\)
0.759362 + 0.650668i \(0.225511\pi\)
\(140\) −341.532 −0.206177
\(141\) −1620.87 −0.968101
\(142\) 542.707 0.320725
\(143\) 5.06425 0.00296150
\(144\) 497.588 0.287956
\(145\) −1134.69 −0.649867
\(146\) 698.093 0.395716
\(147\) −788.650 −0.442495
\(148\) −2583.84 −1.43507
\(149\) −2186.43 −1.20214 −0.601072 0.799195i \(-0.705259\pi\)
−0.601072 + 0.799195i \(0.705259\pi\)
\(150\) −45.5394 −0.0247885
\(151\) 3668.21 1.97692 0.988459 0.151488i \(-0.0484064\pi\)
0.988459 + 0.151488i \(0.0484064\pi\)
\(152\) −0.233349 −0.000124520 0
\(153\) 1155.55 0.610595
\(154\) 59.7834 0.0312824
\(155\) 978.186 0.506902
\(156\) 10.5401 0.00540950
\(157\) 1418.28 0.720960 0.360480 0.932767i \(-0.382613\pi\)
0.360480 + 0.932767i \(0.382613\pi\)
\(158\) 279.022 0.140492
\(159\) −1867.16 −0.931293
\(160\) −547.499 −0.270523
\(161\) 1535.32 0.751552
\(162\) −49.1825 −0.0238527
\(163\) −601.009 −0.288802 −0.144401 0.989519i \(-0.546125\pi\)
−0.144401 + 0.989519i \(0.546125\pi\)
\(164\) −1045.28 −0.497699
\(165\) −165.000 −0.0778499
\(166\) 76.4309 0.0357361
\(167\) −2221.19 −1.02923 −0.514614 0.857422i \(-0.672065\pi\)
−0.514614 + 0.857422i \(0.672065\pi\)
\(168\) 254.862 0.117042
\(169\) −2196.79 −0.999904
\(170\) −389.802 −0.175861
\(171\) −0.221272 −9.89539e−5 0
\(172\) −2564.76 −1.13698
\(173\) −237.280 −0.104278 −0.0521390 0.998640i \(-0.516604\pi\)
−0.0521390 + 0.998640i \(0.516604\pi\)
\(174\) 413.384 0.180107
\(175\) 223.770 0.0966595
\(176\) −608.163 −0.260466
\(177\) −29.6086 −0.0125736
\(178\) −91.3588 −0.0384698
\(179\) 1065.77 0.445023 0.222512 0.974930i \(-0.428574\pi\)
0.222512 + 0.974930i \(0.428574\pi\)
\(180\) −343.409 −0.142201
\(181\) −1241.00 −0.509627 −0.254813 0.966990i \(-0.582014\pi\)
−0.254813 + 0.966990i \(0.582014\pi\)
\(182\) 2.50213 0.00101907
\(183\) 2708.14 1.09394
\(184\) 1628.01 0.652275
\(185\) 1692.92 0.672788
\(186\) −356.368 −0.140485
\(187\) −1412.34 −0.552304
\(188\) 4123.14 1.59952
\(189\) 241.672 0.0930107
\(190\) 0.0746415 2.85003e−5 0
\(191\) −1956.80 −0.741305 −0.370653 0.928772i \(-0.620866\pi\)
−0.370653 + 0.928772i \(0.620866\pi\)
\(192\) −1127.44 −0.423781
\(193\) 2778.02 1.03610 0.518048 0.855352i \(-0.326659\pi\)
0.518048 + 0.855352i \(0.326659\pi\)
\(194\) 767.846 0.284165
\(195\) −6.90580 −0.00253608
\(196\) 2006.15 0.731102
\(197\) 800.242 0.289416 0.144708 0.989474i \(-0.453776\pi\)
0.144708 + 0.989474i \(0.453776\pi\)
\(198\) 60.1120 0.0215756
\(199\) −2536.78 −0.903657 −0.451829 0.892105i \(-0.649228\pi\)
−0.451829 + 0.892105i \(0.649228\pi\)
\(200\) 237.280 0.0838912
\(201\) 440.938 0.154733
\(202\) −419.443 −0.146099
\(203\) −2031.27 −0.702303
\(204\) −2939.47 −1.00884
\(205\) 684.862 0.233331
\(206\) 7.15409 0.00241966
\(207\) 1543.76 0.518350
\(208\) −25.4537 −0.00848507
\(209\) 0.270444 8.95071e−5 0
\(210\) −81.5228 −0.0267886
\(211\) 1598.95 0.521688 0.260844 0.965381i \(-0.415999\pi\)
0.260844 + 0.965381i \(0.415999\pi\)
\(212\) 4749.63 1.53871
\(213\) −2681.40 −0.862564
\(214\) 281.060 0.0897797
\(215\) 1680.42 0.533039
\(216\) 256.263 0.0807244
\(217\) 1751.11 0.547802
\(218\) −200.692 −0.0623513
\(219\) −3449.12 −1.06425
\(220\) 419.722 0.128626
\(221\) −59.1113 −0.0179921
\(222\) −616.756 −0.186459
\(223\) −5074.55 −1.52384 −0.761922 0.647669i \(-0.775744\pi\)
−0.761922 + 0.647669i \(0.775744\pi\)
\(224\) −980.111 −0.292350
\(225\) 225.000 0.0666667
\(226\) −693.599 −0.204148
\(227\) 1048.19 0.306480 0.153240 0.988189i \(-0.451029\pi\)
0.153240 + 0.988189i \(0.451029\pi\)
\(228\) 0.562866 0.000163494 0
\(229\) −734.662 −0.211999 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(230\) −520.754 −0.149293
\(231\) −295.376 −0.0841313
\(232\) −2153.91 −0.609532
\(233\) −2012.78 −0.565929 −0.282965 0.959130i \(-0.591318\pi\)
−0.282965 + 0.959130i \(0.591318\pi\)
\(234\) 2.51589 0.000702858 0
\(235\) −2701.46 −0.749888
\(236\) 75.3176 0.0207744
\(237\) −1378.59 −0.377843
\(238\) −697.808 −0.190051
\(239\) 5566.41 1.50653 0.753267 0.657715i \(-0.228477\pi\)
0.753267 + 0.657715i \(0.228477\pi\)
\(240\) 829.313 0.223050
\(241\) −3361.25 −0.898412 −0.449206 0.893428i \(-0.648293\pi\)
−0.449206 + 0.893428i \(0.648293\pi\)
\(242\) −73.4702 −0.0195159
\(243\) 243.000 0.0641500
\(244\) −6888.88 −1.80744
\(245\) −1314.42 −0.342755
\(246\) −249.506 −0.0646663
\(247\) 0.0113190 2.91583e−6 0
\(248\) 1856.83 0.475440
\(249\) −377.628 −0.0961093
\(250\) −75.8990 −0.0192011
\(251\) −2930.46 −0.736929 −0.368464 0.929642i \(-0.620116\pi\)
−0.368464 + 0.929642i \(0.620116\pi\)
\(252\) −614.758 −0.153675
\(253\) −1886.81 −0.468865
\(254\) −382.281 −0.0944349
\(255\) 1925.92 0.472965
\(256\) 2336.05 0.570325
\(257\) −3412.89 −0.828367 −0.414184 0.910193i \(-0.635933\pi\)
−0.414184 + 0.910193i \(0.635933\pi\)
\(258\) −612.201 −0.147729
\(259\) 3030.59 0.727073
\(260\) 17.5668 0.00419018
\(261\) −2042.44 −0.484383
\(262\) −347.658 −0.0819785
\(263\) 3709.06 0.869622 0.434811 0.900522i \(-0.356815\pi\)
0.434811 + 0.900522i \(0.356815\pi\)
\(264\) −313.210 −0.0730179
\(265\) −3111.94 −0.721376
\(266\) 0.133620 3.07999e−5 0
\(267\) 451.384 0.103462
\(268\) −1121.64 −0.255654
\(269\) −5764.39 −1.30655 −0.653273 0.757122i \(-0.726605\pi\)
−0.653273 + 0.757122i \(0.726605\pi\)
\(270\) −81.9709 −0.0184763
\(271\) −1886.33 −0.422827 −0.211414 0.977397i \(-0.567807\pi\)
−0.211414 + 0.977397i \(0.567807\pi\)
\(272\) 7098.64 1.58242
\(273\) −12.3625 −0.00274070
\(274\) 576.031 0.127005
\(275\) −275.000 −0.0603023
\(276\) −3926.97 −0.856433
\(277\) −4095.99 −0.888462 −0.444231 0.895912i \(-0.646523\pi\)
−0.444231 + 0.895912i \(0.646523\pi\)
\(278\) −1511.22 −0.326032
\(279\) 1760.73 0.377822
\(280\) 424.770 0.0906601
\(281\) −8788.69 −1.86580 −0.932899 0.360138i \(-0.882729\pi\)
−0.932899 + 0.360138i \(0.882729\pi\)
\(282\) 984.182 0.207827
\(283\) −1767.75 −0.371314 −0.185657 0.982615i \(-0.559441\pi\)
−0.185657 + 0.982615i \(0.559441\pi\)
\(284\) 6820.86 1.42515
\(285\) −0.368787 −7.66493e−5 0
\(286\) −3.07497 −0.000635759 0
\(287\) 1226.01 0.252158
\(288\) −985.498 −0.201636
\(289\) 11572.3 2.35544
\(290\) 688.974 0.139510
\(291\) −3793.75 −0.764240
\(292\) 8773.78 1.75838
\(293\) 3079.85 0.614085 0.307043 0.951696i \(-0.400661\pi\)
0.307043 + 0.951696i \(0.400661\pi\)
\(294\) 478.862 0.0949924
\(295\) −49.3477 −0.00973944
\(296\) 3213.57 0.631030
\(297\) −297.000 −0.0580259
\(298\) 1327.58 0.258070
\(299\) −78.9695 −0.0152740
\(300\) −572.349 −0.110149
\(301\) 3008.21 0.576048
\(302\) −2227.31 −0.424394
\(303\) 2072.38 0.392921
\(304\) −1.35929 −0.000256449 0
\(305\) 4513.56 0.847363
\(306\) −701.643 −0.131079
\(307\) 2872.16 0.533950 0.266975 0.963703i \(-0.413976\pi\)
0.266975 + 0.963703i \(0.413976\pi\)
\(308\) 751.370 0.139004
\(309\) −35.3468 −0.00650747
\(310\) −593.947 −0.108819
\(311\) −317.978 −0.0579771 −0.0289885 0.999580i \(-0.509229\pi\)
−0.0289885 + 0.999580i \(0.509229\pi\)
\(312\) −13.1089 −0.00237867
\(313\) −8274.81 −1.49431 −0.747156 0.664648i \(-0.768581\pi\)
−0.747156 + 0.664648i \(0.768581\pi\)
\(314\) −861.166 −0.154772
\(315\) 402.786 0.0720458
\(316\) 3506.81 0.624283
\(317\) 1024.34 0.181492 0.0907459 0.995874i \(-0.471075\pi\)
0.0907459 + 0.995874i \(0.471075\pi\)
\(318\) 1133.73 0.199925
\(319\) 2496.32 0.438141
\(320\) −1879.07 −0.328259
\(321\) −1388.65 −0.241455
\(322\) −932.232 −0.161339
\(323\) −3.15669 −0.000543787 0
\(324\) −618.137 −0.105991
\(325\) −11.5097 −0.00196444
\(326\) 364.928 0.0619984
\(327\) 991.575 0.167689
\(328\) 1300.03 0.218849
\(329\) −4836.04 −0.810394
\(330\) 100.187 0.0167124
\(331\) 4575.83 0.759850 0.379925 0.925017i \(-0.375950\pi\)
0.379925 + 0.925017i \(0.375950\pi\)
\(332\) 960.600 0.158795
\(333\) 3047.25 0.501467
\(334\) 1348.69 0.220949
\(335\) 734.896 0.119856
\(336\) 1484.60 0.241047
\(337\) −917.888 −0.148370 −0.0741848 0.997245i \(-0.523635\pi\)
−0.0741848 + 0.997245i \(0.523635\pi\)
\(338\) 1333.87 0.214654
\(339\) 3426.92 0.549040
\(340\) −4899.11 −0.781446
\(341\) −2152.01 −0.341753
\(342\) 0.134355 2.12429e−5 0
\(343\) −5423.14 −0.853708
\(344\) 3189.84 0.499955
\(345\) 2572.93 0.401512
\(346\) 144.075 0.0223859
\(347\) 5402.26 0.835760 0.417880 0.908502i \(-0.362773\pi\)
0.417880 + 0.908502i \(0.362773\pi\)
\(348\) 5195.50 0.800311
\(349\) 5115.85 0.784656 0.392328 0.919825i \(-0.371670\pi\)
0.392328 + 0.919825i \(0.371670\pi\)
\(350\) −135.871 −0.0207504
\(351\) −12.4304 −0.00189028
\(352\) 1204.50 0.182386
\(353\) −3102.03 −0.467718 −0.233859 0.972270i \(-0.575135\pi\)
−0.233859 + 0.972270i \(0.575135\pi\)
\(354\) 17.9781 0.00269923
\(355\) −4468.99 −0.668139
\(356\) −1148.22 −0.170942
\(357\) 3447.71 0.511127
\(358\) −647.125 −0.0955352
\(359\) 3496.32 0.514008 0.257004 0.966410i \(-0.417265\pi\)
0.257004 + 0.966410i \(0.417265\pi\)
\(360\) 427.104 0.0625288
\(361\) −6859.00 −1.00000
\(362\) 753.522 0.109404
\(363\) 363.000 0.0524864
\(364\) 31.4474 0.00452827
\(365\) −5748.53 −0.824362
\(366\) −1644.36 −0.234841
\(367\) 7644.54 1.08731 0.543654 0.839309i \(-0.317040\pi\)
0.543654 + 0.839309i \(0.317040\pi\)
\(368\) 9483.39 1.34336
\(369\) 1232.75 0.173915
\(370\) −1027.93 −0.144431
\(371\) −5570.86 −0.779582
\(372\) −4478.91 −0.624249
\(373\) −5189.90 −0.720437 −0.360218 0.932868i \(-0.617298\pi\)
−0.360218 + 0.932868i \(0.617298\pi\)
\(374\) 857.564 0.118566
\(375\) 375.000 0.0516398
\(376\) −5128.02 −0.703344
\(377\) 104.479 0.0142731
\(378\) −146.741 −0.0199670
\(379\) 8573.91 1.16204 0.581019 0.813890i \(-0.302654\pi\)
0.581019 + 0.813890i \(0.302654\pi\)
\(380\) 0.938110 0.000126642 0
\(381\) 1888.77 0.253975
\(382\) 1188.15 0.159140
\(383\) −11740.2 −1.56631 −0.783157 0.621824i \(-0.786392\pi\)
−0.783157 + 0.621824i \(0.786392\pi\)
\(384\) 3312.57 0.440218
\(385\) −492.294 −0.0651679
\(386\) −1686.79 −0.222424
\(387\) 3024.75 0.397304
\(388\) 9650.45 1.26270
\(389\) −3763.68 −0.490556 −0.245278 0.969453i \(-0.578879\pi\)
−0.245278 + 0.969453i \(0.578879\pi\)
\(390\) 4.19315 0.000544431 0
\(391\) 22023.4 2.84852
\(392\) −2495.08 −0.321481
\(393\) 1717.70 0.220474
\(394\) −485.900 −0.0621302
\(395\) −2297.64 −0.292676
\(396\) 755.500 0.0958720
\(397\) 1151.35 0.145554 0.0727768 0.997348i \(-0.476814\pi\)
0.0727768 + 0.997348i \(0.476814\pi\)
\(398\) 1540.31 0.193992
\(399\) −0.660188 −8.28339e−5 0
\(400\) 1382.19 0.172774
\(401\) −10051.0 −1.25168 −0.625842 0.779950i \(-0.715245\pi\)
−0.625842 + 0.779950i \(0.715245\pi\)
\(402\) −267.734 −0.0332173
\(403\) −90.0688 −0.0111331
\(404\) −5271.66 −0.649195
\(405\) 405.000 0.0496904
\(406\) 1233.37 0.150767
\(407\) −3724.42 −0.453594
\(408\) 3655.87 0.443609
\(409\) −6263.39 −0.757225 −0.378612 0.925555i \(-0.623599\pi\)
−0.378612 + 0.925555i \(0.623599\pi\)
\(410\) −415.843 −0.0500903
\(411\) −2846.04 −0.341569
\(412\) 89.9142 0.0107518
\(413\) −88.3403 −0.0105253
\(414\) −937.356 −0.111277
\(415\) −629.380 −0.0744459
\(416\) 50.4123 0.00594150
\(417\) 7466.59 0.876836
\(418\) −0.164211 −1.92149e−5 0
\(419\) 8625.67 1.00571 0.502854 0.864372i \(-0.332283\pi\)
0.502854 + 0.864372i \(0.332283\pi\)
\(420\) −1024.60 −0.119036
\(421\) −9095.24 −1.05291 −0.526455 0.850203i \(-0.676479\pi\)
−0.526455 + 0.850203i \(0.676479\pi\)
\(422\) −970.868 −0.111993
\(423\) −4862.62 −0.558934
\(424\) −5907.21 −0.676602
\(425\) 3209.87 0.366357
\(426\) 1628.12 0.185171
\(427\) 8079.99 0.915733
\(428\) 3532.42 0.398939
\(429\) 15.1928 0.00170982
\(430\) −1020.34 −0.114430
\(431\) −4008.72 −0.448013 −0.224006 0.974588i \(-0.571914\pi\)
−0.224006 + 0.974588i \(0.571914\pi\)
\(432\) 1492.76 0.166251
\(433\) 2715.38 0.301369 0.150684 0.988582i \(-0.451852\pi\)
0.150684 + 0.988582i \(0.451852\pi\)
\(434\) −1063.26 −0.117599
\(435\) −3404.07 −0.375201
\(436\) −2522.34 −0.277060
\(437\) −4.21717 −0.000461635 0
\(438\) 2094.28 0.228467
\(439\) −11087.5 −1.20541 −0.602707 0.797962i \(-0.705911\pi\)
−0.602707 + 0.797962i \(0.705911\pi\)
\(440\) −522.017 −0.0565595
\(441\) −2365.95 −0.255474
\(442\) 35.8919 0.00386245
\(443\) 7132.98 0.765007 0.382503 0.923954i \(-0.375062\pi\)
0.382503 + 0.923954i \(0.375062\pi\)
\(444\) −7751.52 −0.828538
\(445\) 752.306 0.0801410
\(446\) 3081.23 0.327131
\(447\) −6559.29 −0.694058
\(448\) −3363.83 −0.354745
\(449\) 6291.28 0.661255 0.330628 0.943761i \(-0.392740\pi\)
0.330628 + 0.943761i \(0.392740\pi\)
\(450\) −136.618 −0.0143116
\(451\) −1506.70 −0.157312
\(452\) −8717.30 −0.907140
\(453\) 11004.6 1.14137
\(454\) −636.455 −0.0657936
\(455\) −20.6042 −0.00212294
\(456\) −0.700047 −7.18919e−5 0
\(457\) 7291.98 0.746400 0.373200 0.927751i \(-0.378261\pi\)
0.373200 + 0.927751i \(0.378261\pi\)
\(458\) 446.081 0.0455109
\(459\) 3466.66 0.352527
\(460\) −6544.94 −0.663390
\(461\) −16805.4 −1.69784 −0.848922 0.528519i \(-0.822748\pi\)
−0.848922 + 0.528519i \(0.822748\pi\)
\(462\) 179.350 0.0180609
\(463\) 7478.48 0.750658 0.375329 0.926892i \(-0.377530\pi\)
0.375329 + 0.926892i \(0.377530\pi\)
\(464\) −12546.8 −1.25533
\(465\) 2934.56 0.292660
\(466\) 1222.14 0.121491
\(467\) −7526.83 −0.745824 −0.372912 0.927867i \(-0.621641\pi\)
−0.372912 + 0.927867i \(0.621641\pi\)
\(468\) 31.6202 0.00312317
\(469\) 1315.58 0.129526
\(470\) 1640.30 0.160982
\(471\) 4254.83 0.416247
\(472\) −93.6739 −0.00913494
\(473\) −3696.92 −0.359375
\(474\) 837.066 0.0811133
\(475\) −0.614645 −5.93723e−5 0
\(476\) −8770.20 −0.844499
\(477\) −5601.48 −0.537682
\(478\) −3379.88 −0.323415
\(479\) 1258.80 0.120075 0.0600376 0.998196i \(-0.480878\pi\)
0.0600376 + 0.998196i \(0.480878\pi\)
\(480\) −1642.50 −0.156186
\(481\) −155.879 −0.0147765
\(482\) 2040.93 0.192866
\(483\) 4605.95 0.433909
\(484\) −923.389 −0.0867195
\(485\) −6322.92 −0.591978
\(486\) −147.548 −0.0137714
\(487\) 5127.97 0.477147 0.238574 0.971124i \(-0.423320\pi\)
0.238574 + 0.971124i \(0.423320\pi\)
\(488\) 8567.83 0.794769
\(489\) −1803.03 −0.166740
\(490\) 798.103 0.0735808
\(491\) 11932.3 1.09673 0.548366 0.836238i \(-0.315250\pi\)
0.548366 + 0.836238i \(0.315250\pi\)
\(492\) −3135.84 −0.287347
\(493\) −29137.7 −2.66185
\(494\) −0.00687279 −6.25955e−7 0
\(495\) −495.000 −0.0449467
\(496\) 10816.3 0.979166
\(497\) −8000.21 −0.722049
\(498\) 229.293 0.0206322
\(499\) 14716.3 1.32022 0.660111 0.751168i \(-0.270509\pi\)
0.660111 + 0.751168i \(0.270509\pi\)
\(500\) −953.915 −0.0853207
\(501\) −6663.57 −0.594225
\(502\) 1779.35 0.158200
\(503\) −5598.95 −0.496312 −0.248156 0.968720i \(-0.579825\pi\)
−0.248156 + 0.968720i \(0.579825\pi\)
\(504\) 764.585 0.0675741
\(505\) 3453.96 0.304355
\(506\) 1145.66 0.100654
\(507\) −6590.36 −0.577295
\(508\) −4804.60 −0.419625
\(509\) −16720.0 −1.45600 −0.727998 0.685579i \(-0.759549\pi\)
−0.727998 + 0.685579i \(0.759549\pi\)
\(510\) −1169.41 −0.101534
\(511\) −10290.8 −0.890877
\(512\) −10251.9 −0.884915
\(513\) −0.663817 −5.71310e−5 0
\(514\) 2072.28 0.177830
\(515\) −58.9113 −0.00504066
\(516\) −7694.27 −0.656437
\(517\) 5943.21 0.505574
\(518\) −1840.15 −0.156084
\(519\) −711.841 −0.0602050
\(520\) −21.8481 −0.00184251
\(521\) 3498.81 0.294214 0.147107 0.989121i \(-0.453004\pi\)
0.147107 + 0.989121i \(0.453004\pi\)
\(522\) 1240.15 0.103985
\(523\) −5681.71 −0.475036 −0.237518 0.971383i \(-0.576334\pi\)
−0.237518 + 0.971383i \(0.576334\pi\)
\(524\) −4369.44 −0.364274
\(525\) 671.310 0.0558064
\(526\) −2252.11 −0.186686
\(527\) 25118.8 2.07627
\(528\) −1824.49 −0.150380
\(529\) 17255.0 1.41818
\(530\) 1889.54 0.154861
\(531\) −88.8259 −0.00725935
\(532\) 1.67937 0.000136861 0
\(533\) −63.0603 −0.00512466
\(534\) −274.076 −0.0222106
\(535\) −2314.42 −0.187030
\(536\) 1395.01 0.112417
\(537\) 3197.30 0.256934
\(538\) 3500.09 0.280482
\(539\) 2891.72 0.231085
\(540\) −1030.23 −0.0820999
\(541\) 11641.7 0.925169 0.462585 0.886575i \(-0.346922\pi\)
0.462585 + 0.886575i \(0.346922\pi\)
\(542\) 1145.36 0.0907704
\(543\) −3722.99 −0.294233
\(544\) −14059.2 −1.10806
\(545\) 1652.63 0.129891
\(546\) 7.50640 0.000588360 0
\(547\) −16460.2 −1.28663 −0.643315 0.765602i \(-0.722441\pi\)
−0.643315 + 0.765602i \(0.722441\pi\)
\(548\) 7239.68 0.564350
\(549\) 8124.41 0.631587
\(550\) 166.978 0.0129454
\(551\) 5.57945 0.000431384 0
\(552\) 4884.04 0.376591
\(553\) −4113.15 −0.316291
\(554\) 2487.05 0.190730
\(555\) 5078.76 0.388434
\(556\) −18993.3 −1.44873
\(557\) −11769.7 −0.895331 −0.447666 0.894201i \(-0.647745\pi\)
−0.447666 + 0.894201i \(0.647745\pi\)
\(558\) −1069.10 −0.0811089
\(559\) −154.728 −0.0117072
\(560\) 2474.34 0.186714
\(561\) −4237.03 −0.318873
\(562\) 5336.42 0.400540
\(563\) 21123.9 1.58129 0.790644 0.612276i \(-0.209746\pi\)
0.790644 + 0.612276i \(0.209746\pi\)
\(564\) 12369.4 0.923486
\(565\) 5711.53 0.425285
\(566\) 1073.36 0.0797118
\(567\) 725.015 0.0536997
\(568\) −8483.23 −0.626670
\(569\) −21890.4 −1.61282 −0.806410 0.591357i \(-0.798592\pi\)
−0.806410 + 0.591357i \(0.798592\pi\)
\(570\) 0.223924 1.64547e−5 0
\(571\) 6630.99 0.485986 0.242993 0.970028i \(-0.421871\pi\)
0.242993 + 0.970028i \(0.421871\pi\)
\(572\) −38.6469 −0.00282502
\(573\) −5870.41 −0.427993
\(574\) −744.425 −0.0541319
\(575\) 4288.21 0.311010
\(576\) −3382.32 −0.244670
\(577\) 13361.3 0.964015 0.482007 0.876167i \(-0.339908\pi\)
0.482007 + 0.876167i \(0.339908\pi\)
\(578\) −7026.58 −0.505653
\(579\) 8334.07 0.598190
\(580\) 8659.17 0.619918
\(581\) −1126.69 −0.0804527
\(582\) 2303.54 0.164063
\(583\) 6846.26 0.486352
\(584\) −10912.1 −0.773196
\(585\) −20.7174 −0.00146420
\(586\) −1870.06 −0.131829
\(587\) −9451.34 −0.664563 −0.332282 0.943180i \(-0.607818\pi\)
−0.332282 + 0.943180i \(0.607818\pi\)
\(588\) 6018.44 0.422102
\(589\) −4.80990 −0.000336483 0
\(590\) 29.9635 0.00209081
\(591\) 2400.73 0.167094
\(592\) 18719.5 1.29960
\(593\) 712.344 0.0493296 0.0246648 0.999696i \(-0.492148\pi\)
0.0246648 + 0.999696i \(0.492148\pi\)
\(594\) 180.336 0.0124567
\(595\) 5746.19 0.395917
\(596\) 16685.3 1.14674
\(597\) −7610.35 −0.521727
\(598\) 47.9496 0.00327894
\(599\) 23951.9 1.63380 0.816901 0.576777i \(-0.195690\pi\)
0.816901 + 0.576777i \(0.195690\pi\)
\(600\) 711.841 0.0484346
\(601\) 18577.3 1.26087 0.630434 0.776243i \(-0.282877\pi\)
0.630434 + 0.776243i \(0.282877\pi\)
\(602\) −1826.56 −0.123663
\(603\) 1322.81 0.0893351
\(604\) −27993.3 −1.88581
\(605\) 605.000 0.0406558
\(606\) −1258.33 −0.0843501
\(607\) −6230.72 −0.416634 −0.208317 0.978061i \(-0.566799\pi\)
−0.208317 + 0.978061i \(0.566799\pi\)
\(608\) 2.69214 0.000179574 0
\(609\) −6093.82 −0.405475
\(610\) −2740.60 −0.181907
\(611\) 248.743 0.0164698
\(612\) −8818.40 −0.582456
\(613\) −17744.8 −1.16918 −0.584588 0.811330i \(-0.698744\pi\)
−0.584588 + 0.811330i \(0.698744\pi\)
\(614\) −1743.95 −0.114626
\(615\) 2054.59 0.134714
\(616\) −934.493 −0.0611230
\(617\) 15305.4 0.998656 0.499328 0.866413i \(-0.333580\pi\)
0.499328 + 0.866413i \(0.333580\pi\)
\(618\) 21.4623 0.00139699
\(619\) 17372.3 1.12803 0.564015 0.825764i \(-0.309256\pi\)
0.564015 + 0.825764i \(0.309256\pi\)
\(620\) −7464.85 −0.483541
\(621\) 4631.27 0.299270
\(622\) 193.074 0.0124462
\(623\) 1346.75 0.0866073
\(624\) −76.3610 −0.00489886
\(625\) 625.000 0.0400000
\(626\) 5024.40 0.320791
\(627\) 0.811331 5.16770e−5 0
\(628\) −10823.3 −0.687735
\(629\) 43472.4 2.75574
\(630\) −244.568 −0.0154664
\(631\) 23699.3 1.49517 0.747586 0.664165i \(-0.231213\pi\)
0.747586 + 0.664165i \(0.231213\pi\)
\(632\) −4361.48 −0.274510
\(633\) 4796.85 0.301197
\(634\) −621.973 −0.0389617
\(635\) 3147.95 0.196728
\(636\) 14248.9 0.888374
\(637\) 121.028 0.00752795
\(638\) −1515.74 −0.0940577
\(639\) −8044.19 −0.498002
\(640\) 5520.95 0.340992
\(641\) −15423.9 −0.950400 −0.475200 0.879878i \(-0.657624\pi\)
−0.475200 + 0.879878i \(0.657624\pi\)
\(642\) 843.179 0.0518343
\(643\) 13152.3 0.806652 0.403326 0.915056i \(-0.367854\pi\)
0.403326 + 0.915056i \(0.367854\pi\)
\(644\) −11716.5 −0.716917
\(645\) 5041.25 0.307750
\(646\) 1.91672 0.000116737 0
\(647\) −5601.87 −0.340390 −0.170195 0.985410i \(-0.554440\pi\)
−0.170195 + 0.985410i \(0.554440\pi\)
\(648\) 768.788 0.0466062
\(649\) 108.565 0.00656633
\(650\) 6.98858 0.000421715 0
\(651\) 5253.33 0.316274
\(652\) 4586.49 0.275492
\(653\) −27505.9 −1.64837 −0.824186 0.566319i \(-0.808367\pi\)
−0.824186 + 0.566319i \(0.808367\pi\)
\(654\) −602.076 −0.0359986
\(655\) 2862.83 0.170779
\(656\) 7572.87 0.450718
\(657\) −10347.4 −0.614443
\(658\) 2936.41 0.173971
\(659\) 16490.6 0.974785 0.487392 0.873183i \(-0.337948\pi\)
0.487392 + 0.873183i \(0.337948\pi\)
\(660\) 1259.17 0.0742622
\(661\) 12184.4 0.716973 0.358487 0.933535i \(-0.383293\pi\)
0.358487 + 0.933535i \(0.383293\pi\)
\(662\) −2778.41 −0.163121
\(663\) −177.334 −0.0103878
\(664\) −1194.72 −0.0698253
\(665\) −1.10031 −6.41629e−5 0
\(666\) −1850.27 −0.107652
\(667\) −38926.3 −2.25972
\(668\) 16950.6 0.981795
\(669\) −15223.7 −0.879791
\(670\) −446.223 −0.0257300
\(671\) −9929.83 −0.571292
\(672\) −2940.33 −0.168788
\(673\) −13598.5 −0.778876 −0.389438 0.921053i \(-0.627331\pi\)
−0.389438 + 0.921053i \(0.627331\pi\)
\(674\) 557.334 0.0318512
\(675\) 675.000 0.0384900
\(676\) 16764.4 0.953823
\(677\) 23318.1 1.32377 0.661883 0.749607i \(-0.269758\pi\)
0.661883 + 0.749607i \(0.269758\pi\)
\(678\) −2080.80 −0.117865
\(679\) −11319.0 −0.639742
\(680\) 6093.12 0.343618
\(681\) 3144.58 0.176947
\(682\) 1306.68 0.0733658
\(683\) 11493.5 0.643906 0.321953 0.946756i \(-0.395661\pi\)
0.321953 + 0.946756i \(0.395661\pi\)
\(684\) 1.68860 9.43935e−5 0
\(685\) −4743.40 −0.264578
\(686\) 3292.89 0.183270
\(687\) −2203.99 −0.122398
\(688\) 18581.2 1.02965
\(689\) 286.539 0.0158436
\(690\) −1562.26 −0.0861946
\(691\) −20253.1 −1.11500 −0.557501 0.830176i \(-0.688240\pi\)
−0.557501 + 0.830176i \(0.688240\pi\)
\(692\) 1810.76 0.0994724
\(693\) −886.129 −0.0485733
\(694\) −3280.21 −0.179416
\(695\) 12444.3 0.679194
\(696\) −6461.74 −0.351913
\(697\) 17586.6 0.955723
\(698\) −3106.30 −0.168446
\(699\) −6038.34 −0.326739
\(700\) −1707.66 −0.0922050
\(701\) 26895.3 1.44910 0.724551 0.689221i \(-0.242047\pi\)
0.724551 + 0.689221i \(0.242047\pi\)
\(702\) 7.54766 0.000405795 0
\(703\) −8.32435 −0.000446599 0
\(704\) 4133.94 0.221312
\(705\) −8104.37 −0.432948
\(706\) 1883.53 0.100407
\(707\) 6183.14 0.328912
\(708\) 225.953 0.0119941
\(709\) −18567.3 −0.983512 −0.491756 0.870733i \(-0.663645\pi\)
−0.491756 + 0.870733i \(0.663645\pi\)
\(710\) 2713.54 0.143433
\(711\) −4135.76 −0.218148
\(712\) 1428.06 0.0751668
\(713\) 33557.4 1.76260
\(714\) −2093.42 −0.109726
\(715\) 25.3213 0.00132442
\(716\) −8133.21 −0.424514
\(717\) 16699.2 0.869797
\(718\) −2122.94 −0.110344
\(719\) −19223.5 −0.997104 −0.498552 0.866860i \(-0.666135\pi\)
−0.498552 + 0.866860i \(0.666135\pi\)
\(720\) 2487.94 0.128778
\(721\) −105.461 −0.00544738
\(722\) 4164.73 0.214675
\(723\) −10083.8 −0.518698
\(724\) 9470.43 0.486140
\(725\) −5673.44 −0.290630
\(726\) −220.411 −0.0112675
\(727\) −33855.9 −1.72716 −0.863581 0.504210i \(-0.831784\pi\)
−0.863581 + 0.504210i \(0.831784\pi\)
\(728\) −39.1117 −0.00199117
\(729\) 729.000 0.0370370
\(730\) 3490.46 0.176970
\(731\) 43151.4 2.18333
\(732\) −20666.6 −1.04353
\(733\) 26532.1 1.33695 0.668477 0.743733i \(-0.266947\pi\)
0.668477 + 0.743733i \(0.266947\pi\)
\(734\) −4641.71 −0.233418
\(735\) −3943.25 −0.197890
\(736\) −18782.3 −0.940661
\(737\) −1616.77 −0.0808067
\(738\) −748.517 −0.0373351
\(739\) −15358.9 −0.764528 −0.382264 0.924053i \(-0.624856\pi\)
−0.382264 + 0.924053i \(0.624856\pi\)
\(740\) −12919.2 −0.641783
\(741\) 0.0339569 1.68345e−6 0
\(742\) 3382.58 0.167356
\(743\) −30995.4 −1.53043 −0.765216 0.643774i \(-0.777368\pi\)
−0.765216 + 0.643774i \(0.777368\pi\)
\(744\) 5570.50 0.274495
\(745\) −10932.2 −0.537615
\(746\) 3151.27 0.154660
\(747\) −1132.88 −0.0554887
\(748\) 10778.0 0.526851
\(749\) −4143.19 −0.202121
\(750\) −227.697 −0.0110858
\(751\) −32222.4 −1.56566 −0.782831 0.622234i \(-0.786225\pi\)
−0.782831 + 0.622234i \(0.786225\pi\)
\(752\) −29871.4 −1.44853
\(753\) −8791.38 −0.425466
\(754\) −63.4389 −0.00306407
\(755\) 18341.0 0.884105
\(756\) −1844.27 −0.0887243
\(757\) −36596.5 −1.75710 −0.878549 0.477651i \(-0.841488\pi\)
−0.878549 + 0.477651i \(0.841488\pi\)
\(758\) −5206.01 −0.249460
\(759\) −5660.44 −0.270700
\(760\) −1.16675 −5.56872e−5 0
\(761\) 24718.0 1.17743 0.588716 0.808340i \(-0.299634\pi\)
0.588716 + 0.808340i \(0.299634\pi\)
\(762\) −1146.84 −0.0545220
\(763\) 2958.46 0.140372
\(764\) 14933.0 0.707142
\(765\) 5777.77 0.273066
\(766\) 7128.58 0.336248
\(767\) 4.54381 0.000213908 0
\(768\) 7008.15 0.329277
\(769\) 36376.7 1.70582 0.852910 0.522058i \(-0.174836\pi\)
0.852910 + 0.522058i \(0.174836\pi\)
\(770\) 298.917 0.0139899
\(771\) −10238.7 −0.478258
\(772\) −21200.0 −0.988347
\(773\) 7525.40 0.350155 0.175078 0.984555i \(-0.443982\pi\)
0.175078 + 0.984555i \(0.443982\pi\)
\(774\) −1836.60 −0.0852911
\(775\) 4890.93 0.226693
\(776\) −12002.4 −0.555235
\(777\) 9091.78 0.419776
\(778\) 2285.28 0.105310
\(779\) −3.36758 −0.000154886 0
\(780\) 52.7004 0.00241920
\(781\) 9831.78 0.450459
\(782\) −13372.4 −0.611505
\(783\) −6127.32 −0.279658
\(784\) −14534.2 −0.662089
\(785\) 7091.38 0.322423
\(786\) −1042.97 −0.0473303
\(787\) −41044.5 −1.85906 −0.929530 0.368747i \(-0.879787\pi\)
−0.929530 + 0.368747i \(0.879787\pi\)
\(788\) −6106.90 −0.276078
\(789\) 11127.2 0.502077
\(790\) 1395.11 0.0628301
\(791\) 10224.5 0.459599
\(792\) −939.630 −0.0421569
\(793\) −415.597 −0.0186107
\(794\) −699.093 −0.0312467
\(795\) −9335.81 −0.416487
\(796\) 19359.0 0.862012
\(797\) −10836.9 −0.481635 −0.240817 0.970570i \(-0.577416\pi\)
−0.240817 + 0.970570i \(0.577416\pi\)
\(798\) 0.400861 1.77823e−5 0
\(799\) −69370.7 −3.07154
\(800\) −2737.50 −0.120981
\(801\) 1354.15 0.0597335
\(802\) 6102.91 0.268705
\(803\) 12646.8 0.555785
\(804\) −3364.93 −0.147602
\(805\) 7676.59 0.336104
\(806\) 54.6890 0.00239000
\(807\) −17293.2 −0.754335
\(808\) 6556.45 0.285464
\(809\) −16093.5 −0.699402 −0.349701 0.936861i \(-0.613717\pi\)
−0.349701 + 0.936861i \(0.613717\pi\)
\(810\) −245.913 −0.0106673
\(811\) 7289.26 0.315611 0.157805 0.987470i \(-0.449558\pi\)
0.157805 + 0.987470i \(0.449558\pi\)
\(812\) 15501.3 0.669937
\(813\) −5658.98 −0.244120
\(814\) 2261.44 0.0973751
\(815\) −3005.05 −0.129156
\(816\) 21295.9 0.913611
\(817\) −8.26288 −0.000353833 0
\(818\) 3803.08 0.162557
\(819\) −37.0875 −0.00158235
\(820\) −5226.40 −0.222578
\(821\) −33153.4 −1.40933 −0.704666 0.709539i \(-0.748903\pi\)
−0.704666 + 0.709539i \(0.748903\pi\)
\(822\) 1728.09 0.0733262
\(823\) −2781.57 −0.117812 −0.0589060 0.998264i \(-0.518761\pi\)
−0.0589060 + 0.998264i \(0.518761\pi\)
\(824\) −111.828 −0.00472780
\(825\) −825.000 −0.0348155
\(826\) 53.6395 0.00225951
\(827\) −4498.61 −0.189156 −0.0945780 0.995517i \(-0.530150\pi\)
−0.0945780 + 0.995517i \(0.530150\pi\)
\(828\) −11780.9 −0.494462
\(829\) 15630.3 0.654843 0.327421 0.944878i \(-0.393820\pi\)
0.327421 + 0.944878i \(0.393820\pi\)
\(830\) 382.155 0.0159817
\(831\) −12288.0 −0.512954
\(832\) 173.019 0.00720958
\(833\) −33752.9 −1.40392
\(834\) −4533.66 −0.188235
\(835\) −11106.0 −0.460284
\(836\) −2.06384 −8.53822e−5 0
\(837\) 5282.20 0.218136
\(838\) −5237.43 −0.215900
\(839\) −6182.34 −0.254396 −0.127198 0.991877i \(-0.540598\pi\)
−0.127198 + 0.991877i \(0.540598\pi\)
\(840\) 1274.31 0.0523426
\(841\) 27111.8 1.11164
\(842\) 5522.56 0.226033
\(843\) −26366.1 −1.07722
\(844\) −12202.1 −0.497646
\(845\) −10983.9 −0.447170
\(846\) 2952.55 0.119989
\(847\) 1083.05 0.0439362
\(848\) −34410.3 −1.39346
\(849\) −5303.26 −0.214378
\(850\) −1949.01 −0.0786476
\(851\) 58076.7 2.33942
\(852\) 20462.6 0.822813
\(853\) 31979.3 1.28365 0.641824 0.766852i \(-0.278178\pi\)
0.641824 + 0.766852i \(0.278178\pi\)
\(854\) −4906.11 −0.196585
\(855\) −1.10636 −4.42535e−5 0
\(856\) −4393.33 −0.175422
\(857\) −8439.63 −0.336397 −0.168199 0.985753i \(-0.553795\pi\)
−0.168199 + 0.985753i \(0.553795\pi\)
\(858\) −9.22492 −0.000367056 0
\(859\) 47640.1 1.89227 0.946134 0.323774i \(-0.104952\pi\)
0.946134 + 0.323774i \(0.104952\pi\)
\(860\) −12823.8 −0.508474
\(861\) 3678.04 0.145583
\(862\) 2434.06 0.0961770
\(863\) 27963.3 1.10299 0.551496 0.834177i \(-0.314057\pi\)
0.551496 + 0.834177i \(0.314057\pi\)
\(864\) −2956.50 −0.116414
\(865\) −1186.40 −0.0466346
\(866\) −1648.75 −0.0646962
\(867\) 34716.8 1.35991
\(868\) −13363.3 −0.522557
\(869\) 5054.81 0.197322
\(870\) 2066.92 0.0805462
\(871\) −67.6673 −0.00263240
\(872\) 3137.08 0.121829
\(873\) −11381.3 −0.441234
\(874\) 2.56063 9.91013e−5 0
\(875\) 1118.85 0.0432275
\(876\) 26321.3 1.01520
\(877\) −30584.0 −1.17759 −0.588796 0.808282i \(-0.700398\pi\)
−0.588796 + 0.808282i \(0.700398\pi\)
\(878\) 6732.23 0.258772
\(879\) 9239.56 0.354542
\(880\) −3040.82 −0.116484
\(881\) 14444.3 0.552372 0.276186 0.961104i \(-0.410929\pi\)
0.276186 + 0.961104i \(0.410929\pi\)
\(882\) 1436.58 0.0548439
\(883\) −19921.4 −0.759241 −0.379620 0.925142i \(-0.623945\pi\)
−0.379620 + 0.925142i \(0.623945\pi\)
\(884\) 451.097 0.0171630
\(885\) −148.043 −0.00562307
\(886\) −4331.09 −0.164228
\(887\) −13172.5 −0.498635 −0.249317 0.968422i \(-0.580206\pi\)
−0.249317 + 0.968422i \(0.580206\pi\)
\(888\) 9640.71 0.364325
\(889\) 5635.33 0.212602
\(890\) −456.794 −0.0172042
\(891\) −891.000 −0.0335013
\(892\) 38725.5 1.45362
\(893\) 13.2835 0.000497778 0
\(894\) 3982.75 0.148997
\(895\) 5328.83 0.199020
\(896\) 9883.38 0.368505
\(897\) −236.908 −0.00881844
\(898\) −3820.01 −0.141955
\(899\) −44397.5 −1.64710
\(900\) −1717.05 −0.0635943
\(901\) −79911.4 −2.95475
\(902\) 914.854 0.0337709
\(903\) 9024.64 0.332582
\(904\) 10841.9 0.398888
\(905\) −6204.98 −0.227912
\(906\) −6681.92 −0.245024
\(907\) 16309.7 0.597084 0.298542 0.954396i \(-0.403500\pi\)
0.298542 + 0.954396i \(0.403500\pi\)
\(908\) −7999.10 −0.292356
\(909\) 6217.13 0.226853
\(910\) 12.5107 0.000455741 0
\(911\) −14994.4 −0.545320 −0.272660 0.962110i \(-0.587903\pi\)
−0.272660 + 0.962110i \(0.587903\pi\)
\(912\) −4.07787 −0.000148061 0
\(913\) 1384.64 0.0501914
\(914\) −4427.63 −0.160233
\(915\) 13540.7 0.489225
\(916\) 5606.44 0.202229
\(917\) 5124.93 0.184558
\(918\) −2104.93 −0.0756787
\(919\) 40263.5 1.44524 0.722618 0.691248i \(-0.242939\pi\)
0.722618 + 0.691248i \(0.242939\pi\)
\(920\) 8140.06 0.291706
\(921\) 8616.48 0.308276
\(922\) 10204.1 0.364484
\(923\) 411.493 0.0146744
\(924\) 2254.11 0.0802541
\(925\) 8464.59 0.300880
\(926\) −4540.87 −0.161147
\(927\) −106.040 −0.00375709
\(928\) 24849.6 0.879019
\(929\) −9469.52 −0.334429 −0.167215 0.985921i \(-0.553477\pi\)
−0.167215 + 0.985921i \(0.553477\pi\)
\(930\) −1781.84 −0.0628267
\(931\) 6.46319 0.000227522 0
\(932\) 15360.2 0.539848
\(933\) −953.933 −0.0334731
\(934\) 4570.23 0.160110
\(935\) −7061.72 −0.246998
\(936\) −39.3266 −0.00137332
\(937\) 6356.51 0.221620 0.110810 0.993842i \(-0.464655\pi\)
0.110810 + 0.993842i \(0.464655\pi\)
\(938\) −798.810 −0.0278061
\(939\) −24824.4 −0.862742
\(940\) 20615.7 0.715329
\(941\) −5764.45 −0.199698 −0.0998489 0.995003i \(-0.531836\pi\)
−0.0998489 + 0.995003i \(0.531836\pi\)
\(942\) −2583.50 −0.0893577
\(943\) 23494.7 0.811338
\(944\) −545.663 −0.0188134
\(945\) 1208.36 0.0415956
\(946\) 2244.74 0.0771487
\(947\) −2284.74 −0.0783993 −0.0391997 0.999231i \(-0.512481\pi\)
−0.0391997 + 0.999231i \(0.512481\pi\)
\(948\) 10520.4 0.360430
\(949\) 529.310 0.0181055
\(950\) 0.373207 1.27457e−5 0
\(951\) 3073.03 0.104784
\(952\) 10907.7 0.371344
\(953\) −17209.0 −0.584947 −0.292474 0.956274i \(-0.594478\pi\)
−0.292474 + 0.956274i \(0.594478\pi\)
\(954\) 3401.18 0.115427
\(955\) −9784.02 −0.331522
\(956\) −42479.1 −1.43710
\(957\) 7488.95 0.252961
\(958\) −764.333 −0.0257771
\(959\) −8491.45 −0.285926
\(960\) −5637.20 −0.189521
\(961\) 8482.92 0.284748
\(962\) 94.6487 0.00317214
\(963\) −4165.96 −0.139404
\(964\) 25650.8 0.857008
\(965\) 13890.1 0.463356
\(966\) −2796.70 −0.0931493
\(967\) −7966.25 −0.264920 −0.132460 0.991188i \(-0.542288\pi\)
−0.132460 + 0.991188i \(0.542288\pi\)
\(968\) 1148.44 0.0381324
\(969\) −9.47008 −0.000313955 0
\(970\) 3839.23 0.127083
\(971\) −25547.5 −0.844343 −0.422172 0.906516i \(-0.638732\pi\)
−0.422172 + 0.906516i \(0.638732\pi\)
\(972\) −1854.41 −0.0611937
\(973\) 22277.3 0.733996
\(974\) −3113.66 −0.102431
\(975\) −34.5290 −0.00113417
\(976\) 49908.7 1.63682
\(977\) 12899.8 0.422417 0.211209 0.977441i \(-0.432260\pi\)
0.211209 + 0.977441i \(0.432260\pi\)
\(978\) 1094.78 0.0357948
\(979\) −1655.07 −0.0540310
\(980\) 10030.7 0.326959
\(981\) 2974.73 0.0968152
\(982\) −7245.18 −0.235441
\(983\) −7399.28 −0.240082 −0.120041 0.992769i \(-0.538303\pi\)
−0.120041 + 0.992769i \(0.538303\pi\)
\(984\) 3900.10 0.126352
\(985\) 4001.21 0.129431
\(986\) 17692.2 0.571433
\(987\) −14508.1 −0.467881
\(988\) −0.0863787 −2.78145e−6 0
\(989\) 57647.8 1.85348
\(990\) 300.560 0.00964891
\(991\) −38466.4 −1.23302 −0.616511 0.787346i \(-0.711454\pi\)
−0.616511 + 0.787346i \(0.711454\pi\)
\(992\) −21422.2 −0.685642
\(993\) 13727.5 0.438700
\(994\) 4857.66 0.155006
\(995\) −12683.9 −0.404128
\(996\) 2881.80 0.0916801
\(997\) −62092.5 −1.97241 −0.986203 0.165538i \(-0.947064\pi\)
−0.986203 + 0.165538i \(0.947064\pi\)
\(998\) −8935.60 −0.283418
\(999\) 9141.76 0.289522
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 165.4.a.h.1.2 4
3.2 odd 2 495.4.a.m.1.3 4
5.2 odd 4 825.4.c.p.199.4 8
5.3 odd 4 825.4.c.p.199.5 8
5.4 even 2 825.4.a.t.1.3 4
11.10 odd 2 1815.4.a.t.1.3 4
15.14 odd 2 2475.4.a.be.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.h.1.2 4 1.1 even 1 trivial
495.4.a.m.1.3 4 3.2 odd 2
825.4.a.t.1.3 4 5.4 even 2
825.4.c.p.199.4 8 5.2 odd 4
825.4.c.p.199.5 8 5.3 odd 4
1815.4.a.t.1.3 4 11.10 odd 2
2475.4.a.be.1.2 4 15.14 odd 2