Properties

Label 165.4.a.h.1.4
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 1.4
Root \(-4.17080\) 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)
 
\(f(q)\) \(=\) \(q+5.17080 q^{2} +3.00000 q^{3} +18.7372 q^{4} +5.00000 q^{5} +15.5124 q^{6} -11.1745 q^{7} +55.5199 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+5.17080 q^{2} +3.00000 q^{3} +18.7372 q^{4} +5.00000 q^{5} +15.5124 q^{6} -11.1745 q^{7} +55.5199 q^{8} +9.00000 q^{9} +25.8540 q^{10} -11.0000 q^{11} +56.2116 q^{12} -89.5310 q^{13} -57.7813 q^{14} +15.0000 q^{15} +137.185 q^{16} +58.3242 q^{17} +46.5372 q^{18} +24.5575 q^{19} +93.6859 q^{20} -33.5236 q^{21} -56.8788 q^{22} -111.696 q^{23} +166.560 q^{24} +25.0000 q^{25} -462.947 q^{26} +27.0000 q^{27} -209.379 q^{28} +109.954 q^{29} +77.5620 q^{30} +119.547 q^{31} +265.196 q^{32} -33.0000 q^{33} +301.583 q^{34} -55.8726 q^{35} +168.635 q^{36} -356.544 q^{37} +126.982 q^{38} -268.593 q^{39} +277.599 q^{40} +268.798 q^{41} -173.344 q^{42} +263.371 q^{43} -206.109 q^{44} +45.0000 q^{45} -577.557 q^{46} -206.732 q^{47} +411.554 q^{48} -218.130 q^{49} +129.270 q^{50} +174.973 q^{51} -1677.56 q^{52} +223.749 q^{53} +139.612 q^{54} -55.0000 q^{55} -620.408 q^{56} +73.6726 q^{57} +568.549 q^{58} +475.000 q^{59} +281.058 q^{60} -513.204 q^{61} +618.153 q^{62} -100.571 q^{63} +273.798 q^{64} -447.655 q^{65} -170.636 q^{66} -264.533 q^{67} +1092.83 q^{68} -335.087 q^{69} -288.906 q^{70} -1110.33 q^{71} +499.679 q^{72} +893.608 q^{73} -1843.62 q^{74} +75.0000 q^{75} +460.139 q^{76} +122.920 q^{77} -1388.84 q^{78} +1303.66 q^{79} +685.923 q^{80} +81.0000 q^{81} +1389.90 q^{82} +1049.37 q^{83} -628.138 q^{84} +291.621 q^{85} +1361.84 q^{86} +329.862 q^{87} -610.718 q^{88} -1417.91 q^{89} +232.686 q^{90} +1000.47 q^{91} -2092.86 q^{92} +358.640 q^{93} -1068.97 q^{94} +122.788 q^{95} +795.587 q^{96} +85.8091 q^{97} -1127.91 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 12 q^{3} + 26 q^{4} + 20 q^{5} + 12 q^{6} + 34 q^{7} + 48 q^{8} + 36 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.17080 1.82815 0.914077 0.405540i \(-0.132917\pi\)
0.914077 + 0.405540i \(0.132917\pi\)
\(3\) 3.00000 0.577350
\(4\) 18.7372 2.34215
\(5\) 5.00000 0.447214
\(6\) 15.5124 1.05549
\(7\) −11.1745 −0.603368 −0.301684 0.953408i \(-0.597549\pi\)
−0.301684 + 0.953408i \(0.597549\pi\)
\(8\) 55.5199 2.45365
\(9\) 9.00000 0.333333
\(10\) 25.8540 0.817575
\(11\) −11.0000 −0.301511
\(12\) 56.2116 1.35224
\(13\) −89.5310 −1.91011 −0.955056 0.296427i \(-0.904205\pi\)
−0.955056 + 0.296427i \(0.904205\pi\)
\(14\) −57.7813 −1.10305
\(15\) 15.0000 0.258199
\(16\) 137.185 2.14351
\(17\) 58.3242 0.832100 0.416050 0.909342i \(-0.363414\pi\)
0.416050 + 0.909342i \(0.363414\pi\)
\(18\) 46.5372 0.609385
\(19\) 24.5575 0.296520 0.148260 0.988948i \(-0.452633\pi\)
0.148260 + 0.988948i \(0.452633\pi\)
\(20\) 93.6859 1.04744
\(21\) −33.5236 −0.348355
\(22\) −56.8788 −0.551209
\(23\) −111.696 −1.01262 −0.506308 0.862353i \(-0.668990\pi\)
−0.506308 + 0.862353i \(0.668990\pi\)
\(24\) 166.560 1.41662
\(25\) 25.0000 0.200000
\(26\) −462.947 −3.49198
\(27\) 27.0000 0.192450
\(28\) −209.379 −1.41318
\(29\) 109.954 0.704066 0.352033 0.935988i \(-0.385490\pi\)
0.352033 + 0.935988i \(0.385490\pi\)
\(30\) 77.5620 0.472027
\(31\) 119.547 0.692621 0.346310 0.938120i \(-0.387434\pi\)
0.346310 + 0.938120i \(0.387434\pi\)
\(32\) 265.196 1.46501
\(33\) −33.0000 −0.174078
\(34\) 301.583 1.52121
\(35\) −55.8726 −0.269834
\(36\) 168.635 0.780716
\(37\) −356.544 −1.58420 −0.792100 0.610391i \(-0.791012\pi\)
−0.792100 + 0.610391i \(0.791012\pi\)
\(38\) 126.982 0.542085
\(39\) −268.593 −1.10280
\(40\) 277.599 1.09731
\(41\) 268.798 1.02388 0.511941 0.859021i \(-0.328927\pi\)
0.511941 + 0.859021i \(0.328927\pi\)
\(42\) −173.344 −0.636846
\(43\) 263.371 0.934038 0.467019 0.884247i \(-0.345328\pi\)
0.467019 + 0.884247i \(0.345328\pi\)
\(44\) −206.109 −0.706184
\(45\) 45.0000 0.149071
\(46\) −577.557 −1.85122
\(47\) −206.732 −0.641594 −0.320797 0.947148i \(-0.603951\pi\)
−0.320797 + 0.947148i \(0.603951\pi\)
\(48\) 411.554 1.23756
\(49\) −218.130 −0.635947
\(50\) 129.270 0.365631
\(51\) 174.973 0.480413
\(52\) −1677.56 −4.47376
\(53\) 223.749 0.579891 0.289946 0.957043i \(-0.406363\pi\)
0.289946 + 0.957043i \(0.406363\pi\)
\(54\) 139.612 0.351828
\(55\) −55.0000 −0.134840
\(56\) −620.408 −1.48046
\(57\) 73.6726 0.171196
\(58\) 568.549 1.28714
\(59\) 475.000 1.04813 0.524066 0.851678i \(-0.324415\pi\)
0.524066 + 0.851678i \(0.324415\pi\)
\(60\) 281.058 0.604740
\(61\) −513.204 −1.07720 −0.538598 0.842563i \(-0.681046\pi\)
−0.538598 + 0.842563i \(0.681046\pi\)
\(62\) 618.153 1.26622
\(63\) −100.571 −0.201123
\(64\) 273.798 0.534761
\(65\) −447.655 −0.854228
\(66\) −170.636 −0.318241
\(67\) −264.533 −0.482355 −0.241178 0.970481i \(-0.577534\pi\)
−0.241178 + 0.970481i \(0.577534\pi\)
\(68\) 1092.83 1.94890
\(69\) −335.087 −0.584634
\(70\) −288.906 −0.493299
\(71\) −1110.33 −1.85595 −0.927973 0.372648i \(-0.878450\pi\)
−0.927973 + 0.372648i \(0.878450\pi\)
\(72\) 499.679 0.817885
\(73\) 893.608 1.43273 0.716363 0.697728i \(-0.245806\pi\)
0.716363 + 0.697728i \(0.245806\pi\)
\(74\) −1843.62 −2.89616
\(75\) 75.0000 0.115470
\(76\) 460.139 0.694494
\(77\) 122.920 0.181922
\(78\) −1388.84 −2.01609
\(79\) 1303.66 1.85663 0.928314 0.371797i \(-0.121258\pi\)
0.928314 + 0.371797i \(0.121258\pi\)
\(80\) 685.923 0.958607
\(81\) 81.0000 0.111111
\(82\) 1389.90 1.87181
\(83\) 1049.37 1.38775 0.693873 0.720098i \(-0.255903\pi\)
0.693873 + 0.720098i \(0.255903\pi\)
\(84\) −628.138 −0.815898
\(85\) 291.621 0.372126
\(86\) 1361.84 1.70757
\(87\) 329.862 0.406493
\(88\) −610.718 −0.739805
\(89\) −1417.91 −1.68875 −0.844374 0.535754i \(-0.820028\pi\)
−0.844374 + 0.535754i \(0.820028\pi\)
\(90\) 232.686 0.272525
\(91\) 1000.47 1.15250
\(92\) −2092.86 −2.37170
\(93\) 358.640 0.399885
\(94\) −1068.97 −1.17293
\(95\) 122.788 0.132608
\(96\) 795.587 0.845826
\(97\) 85.8091 0.0898206 0.0449103 0.998991i \(-0.485700\pi\)
0.0449103 + 0.998991i \(0.485700\pi\)
\(98\) −1127.91 −1.16261
\(99\) −99.0000 −0.100504
\(100\) 468.430 0.468430
\(101\) 137.761 0.135721 0.0678603 0.997695i \(-0.478383\pi\)
0.0678603 + 0.997695i \(0.478383\pi\)
\(102\) 904.749 0.878269
\(103\) 419.397 0.401208 0.200604 0.979672i \(-0.435710\pi\)
0.200604 + 0.979672i \(0.435710\pi\)
\(104\) −4970.75 −4.68675
\(105\) −167.618 −0.155789
\(106\) 1156.96 1.06013
\(107\) −1190.45 −1.07557 −0.537783 0.843083i \(-0.680738\pi\)
−0.537783 + 0.843083i \(0.680738\pi\)
\(108\) 505.904 0.450747
\(109\) 1033.37 0.908066 0.454033 0.890985i \(-0.349985\pi\)
0.454033 + 0.890985i \(0.349985\pi\)
\(110\) −284.394 −0.246508
\(111\) −1069.63 −0.914638
\(112\) −1532.97 −1.29333
\(113\) −930.760 −0.774854 −0.387427 0.921900i \(-0.626636\pi\)
−0.387427 + 0.921900i \(0.626636\pi\)
\(114\) 380.946 0.312973
\(115\) −558.479 −0.452856
\(116\) 2060.23 1.64903
\(117\) −805.779 −0.636704
\(118\) 2456.13 1.91615
\(119\) −651.746 −0.502062
\(120\) 832.798 0.633531
\(121\) 121.000 0.0909091
\(122\) −2653.68 −1.96928
\(123\) 806.393 0.591138
\(124\) 2239.97 1.62222
\(125\) 125.000 0.0894427
\(126\) −520.031 −0.367683
\(127\) 94.4315 0.0659799 0.0329899 0.999456i \(-0.489497\pi\)
0.0329899 + 0.999456i \(0.489497\pi\)
\(128\) −705.814 −0.487389
\(129\) 790.112 0.539267
\(130\) −2314.74 −1.56166
\(131\) 1318.12 0.879122 0.439561 0.898213i \(-0.355134\pi\)
0.439561 + 0.898213i \(0.355134\pi\)
\(132\) −618.327 −0.407716
\(133\) −274.419 −0.178911
\(134\) −1367.85 −0.881820
\(135\) 135.000 0.0860663
\(136\) 3238.15 2.04169
\(137\) 2008.43 1.25249 0.626247 0.779624i \(-0.284590\pi\)
0.626247 + 0.779624i \(0.284590\pi\)
\(138\) −1732.67 −1.06880
\(139\) 2956.76 1.80424 0.902120 0.431486i \(-0.142011\pi\)
0.902120 + 0.431486i \(0.142011\pi\)
\(140\) −1046.90 −0.631992
\(141\) −620.196 −0.370425
\(142\) −5741.30 −3.39295
\(143\) 984.841 0.575920
\(144\) 1234.66 0.714504
\(145\) 549.769 0.314868
\(146\) 4620.67 2.61924
\(147\) −654.390 −0.367164
\(148\) −6680.62 −3.71043
\(149\) 878.768 0.483164 0.241582 0.970380i \(-0.422334\pi\)
0.241582 + 0.970380i \(0.422334\pi\)
\(150\) 387.810 0.211097
\(151\) −1679.35 −0.905058 −0.452529 0.891750i \(-0.649478\pi\)
−0.452529 + 0.891750i \(0.649478\pi\)
\(152\) 1363.43 0.727558
\(153\) 524.918 0.277367
\(154\) 635.594 0.332582
\(155\) 597.734 0.309749
\(156\) −5032.68 −2.58293
\(157\) 703.051 0.357386 0.178693 0.983905i \(-0.442813\pi\)
0.178693 + 0.983905i \(0.442813\pi\)
\(158\) 6740.99 3.39420
\(159\) 671.246 0.334800
\(160\) 1325.98 0.655174
\(161\) 1248.15 0.610980
\(162\) 418.835 0.203128
\(163\) 2934.66 1.41019 0.705093 0.709114i \(-0.250905\pi\)
0.705093 + 0.709114i \(0.250905\pi\)
\(164\) 5036.51 2.39808
\(165\) −165.000 −0.0778499
\(166\) 5426.06 2.53701
\(167\) −2146.05 −0.994407 −0.497204 0.867634i \(-0.665640\pi\)
−0.497204 + 0.867634i \(0.665640\pi\)
\(168\) −1861.22 −0.854742
\(169\) 5818.81 2.64852
\(170\) 1507.91 0.680305
\(171\) 221.018 0.0988401
\(172\) 4934.82 2.18766
\(173\) 321.356 0.141227 0.0706134 0.997504i \(-0.477504\pi\)
0.0706134 + 0.997504i \(0.477504\pi\)
\(174\) 1705.65 0.743131
\(175\) −279.363 −0.120674
\(176\) −1509.03 −0.646293
\(177\) 1425.00 0.605139
\(178\) −7331.75 −3.08729
\(179\) −2662.08 −1.11158 −0.555791 0.831322i \(-0.687585\pi\)
−0.555791 + 0.831322i \(0.687585\pi\)
\(180\) 843.173 0.349147
\(181\) 3306.02 1.35765 0.678825 0.734300i \(-0.262490\pi\)
0.678825 + 0.734300i \(0.262490\pi\)
\(182\) 5173.22 2.10695
\(183\) −1539.61 −0.621920
\(184\) −6201.33 −2.48461
\(185\) −1782.72 −0.708476
\(186\) 1854.46 0.731051
\(187\) −641.566 −0.250888
\(188\) −3873.57 −1.50271
\(189\) −301.712 −0.116118
\(190\) 634.910 0.242428
\(191\) 946.068 0.358404 0.179202 0.983812i \(-0.442648\pi\)
0.179202 + 0.983812i \(0.442648\pi\)
\(192\) 821.393 0.308744
\(193\) −3692.54 −1.37717 −0.688587 0.725154i \(-0.741769\pi\)
−0.688587 + 0.725154i \(0.741769\pi\)
\(194\) 443.702 0.164206
\(195\) −1342.97 −0.493189
\(196\) −4087.14 −1.48948
\(197\) −411.503 −0.148824 −0.0744122 0.997228i \(-0.523708\pi\)
−0.0744122 + 0.997228i \(0.523708\pi\)
\(198\) −511.909 −0.183736
\(199\) 1491.28 0.531226 0.265613 0.964080i \(-0.414426\pi\)
0.265613 + 0.964080i \(0.414426\pi\)
\(200\) 1388.00 0.490731
\(201\) −793.598 −0.278488
\(202\) 712.337 0.248118
\(203\) −1228.68 −0.424811
\(204\) 3278.49 1.12520
\(205\) 1343.99 0.457894
\(206\) 2168.62 0.733470
\(207\) −1005.26 −0.337539
\(208\) −12282.3 −4.09434
\(209\) −270.133 −0.0894042
\(210\) −866.719 −0.284806
\(211\) −899.947 −0.293625 −0.146813 0.989164i \(-0.546901\pi\)
−0.146813 + 0.989164i \(0.546901\pi\)
\(212\) 4192.42 1.35819
\(213\) −3330.99 −1.07153
\(214\) −6155.61 −1.96630
\(215\) 1316.85 0.417715
\(216\) 1499.04 0.472206
\(217\) −1335.88 −0.417905
\(218\) 5343.37 1.66009
\(219\) 2680.82 0.827184
\(220\) −1030.55 −0.315815
\(221\) −5221.83 −1.58940
\(222\) −5530.85 −1.67210
\(223\) 1935.29 0.581152 0.290576 0.956852i \(-0.406153\pi\)
0.290576 + 0.956852i \(0.406153\pi\)
\(224\) −2963.44 −0.883942
\(225\) 225.000 0.0666667
\(226\) −4812.77 −1.41655
\(227\) −2906.26 −0.849758 −0.424879 0.905250i \(-0.639683\pi\)
−0.424879 + 0.905250i \(0.639683\pi\)
\(228\) 1380.42 0.400966
\(229\) −5411.89 −1.56169 −0.780846 0.624724i \(-0.785212\pi\)
−0.780846 + 0.624724i \(0.785212\pi\)
\(230\) −2887.78 −0.827890
\(231\) 368.759 0.105033
\(232\) 6104.62 1.72753
\(233\) −4778.90 −1.34368 −0.671838 0.740699i \(-0.734495\pi\)
−0.671838 + 0.740699i \(0.734495\pi\)
\(234\) −4166.53 −1.16399
\(235\) −1033.66 −0.286930
\(236\) 8900.17 2.45488
\(237\) 3910.99 1.07192
\(238\) −3370.05 −0.917848
\(239\) 5883.88 1.59245 0.796227 0.604998i \(-0.206826\pi\)
0.796227 + 0.604998i \(0.206826\pi\)
\(240\) 2057.77 0.553452
\(241\) 1806.96 0.482974 0.241487 0.970404i \(-0.422365\pi\)
0.241487 + 0.970404i \(0.422365\pi\)
\(242\) 625.667 0.166196
\(243\) 243.000 0.0641500
\(244\) −9616.00 −2.52296
\(245\) −1090.65 −0.284404
\(246\) 4169.70 1.08069
\(247\) −2198.66 −0.566386
\(248\) 6637.22 1.69945
\(249\) 3148.10 0.801215
\(250\) 646.350 0.163515
\(251\) 1265.10 0.318137 0.159069 0.987268i \(-0.449151\pi\)
0.159069 + 0.987268i \(0.449151\pi\)
\(252\) −1884.41 −0.471059
\(253\) 1228.65 0.305315
\(254\) 488.287 0.120621
\(255\) 874.863 0.214847
\(256\) −5840.00 −1.42578
\(257\) −4366.53 −1.05983 −0.529916 0.848050i \(-0.677777\pi\)
−0.529916 + 0.848050i \(0.677777\pi\)
\(258\) 4085.51 0.985864
\(259\) 3984.21 0.955855
\(260\) −8387.80 −2.00073
\(261\) 989.585 0.234689
\(262\) 6815.76 1.60717
\(263\) −2950.96 −0.691878 −0.345939 0.938257i \(-0.612440\pi\)
−0.345939 + 0.938257i \(0.612440\pi\)
\(264\) −1832.16 −0.427126
\(265\) 1118.74 0.259335
\(266\) −1418.97 −0.327076
\(267\) −4253.74 −0.974999
\(268\) −4956.60 −1.12975
\(269\) −6048.65 −1.37098 −0.685488 0.728084i \(-0.740411\pi\)
−0.685488 + 0.728084i \(0.740411\pi\)
\(270\) 698.058 0.157342
\(271\) −2922.90 −0.655180 −0.327590 0.944820i \(-0.606236\pi\)
−0.327590 + 0.944820i \(0.606236\pi\)
\(272\) 8001.19 1.78362
\(273\) 3001.40 0.665396
\(274\) 10385.2 2.28975
\(275\) −275.000 −0.0603023
\(276\) −6278.59 −1.36930
\(277\) −5433.48 −1.17858 −0.589289 0.807922i \(-0.700592\pi\)
−0.589289 + 0.807922i \(0.700592\pi\)
\(278\) 15288.8 3.29843
\(279\) 1075.92 0.230874
\(280\) −3102.04 −0.662080
\(281\) −7981.02 −1.69433 −0.847167 0.531327i \(-0.821693\pi\)
−0.847167 + 0.531327i \(0.821693\pi\)
\(282\) −3206.91 −0.677194
\(283\) −2705.71 −0.568332 −0.284166 0.958775i \(-0.591717\pi\)
−0.284166 + 0.958775i \(0.591717\pi\)
\(284\) −20804.5 −4.34690
\(285\) 368.363 0.0765612
\(286\) 5092.42 1.05287
\(287\) −3003.69 −0.617777
\(288\) 2386.76 0.488338
\(289\) −1511.29 −0.307610
\(290\) 2842.75 0.575627
\(291\) 257.427 0.0518579
\(292\) 16743.7 3.35565
\(293\) 2207.71 0.440190 0.220095 0.975478i \(-0.429363\pi\)
0.220095 + 0.975478i \(0.429363\pi\)
\(294\) −3383.72 −0.671233
\(295\) 2375.00 0.468739
\(296\) −19795.2 −3.88708
\(297\) −297.000 −0.0580259
\(298\) 4543.94 0.883299
\(299\) 10000.2 1.93421
\(300\) 1405.29 0.270448
\(301\) −2943.04 −0.563569
\(302\) −8683.60 −1.65459
\(303\) 413.284 0.0783583
\(304\) 3368.92 0.635594
\(305\) −2566.02 −0.481737
\(306\) 2714.25 0.507069
\(307\) −3918.45 −0.728462 −0.364231 0.931309i \(-0.618668\pi\)
−0.364231 + 0.931309i \(0.618668\pi\)
\(308\) 2303.17 0.426089
\(309\) 1258.19 0.231638
\(310\) 3090.76 0.566270
\(311\) 8769.03 1.59886 0.799431 0.600758i \(-0.205134\pi\)
0.799431 + 0.600758i \(0.205134\pi\)
\(312\) −14912.3 −2.70590
\(313\) −5058.92 −0.913569 −0.456785 0.889577i \(-0.650999\pi\)
−0.456785 + 0.889577i \(0.650999\pi\)
\(314\) 3635.34 0.653357
\(315\) −502.854 −0.0899448
\(316\) 24427.0 4.34850
\(317\) −8329.52 −1.47581 −0.737906 0.674904i \(-0.764185\pi\)
−0.737906 + 0.674904i \(0.764185\pi\)
\(318\) 3470.88 0.612067
\(319\) −1209.49 −0.212284
\(320\) 1368.99 0.239152
\(321\) −3571.36 −0.620979
\(322\) 6453.92 1.11697
\(323\) 1432.30 0.246734
\(324\) 1517.71 0.260239
\(325\) −2238.28 −0.382022
\(326\) 15174.6 2.57804
\(327\) 3100.12 0.524272
\(328\) 14923.6 2.51225
\(329\) 2310.13 0.387118
\(330\) −853.182 −0.142322
\(331\) 2901.85 0.481874 0.240937 0.970541i \(-0.422545\pi\)
0.240937 + 0.970541i \(0.422545\pi\)
\(332\) 19662.2 3.25030
\(333\) −3208.89 −0.528067
\(334\) −11096.8 −1.81793
\(335\) −1322.66 −0.215716
\(336\) −4598.92 −0.746702
\(337\) 6000.75 0.969976 0.484988 0.874521i \(-0.338824\pi\)
0.484988 + 0.874521i \(0.338824\pi\)
\(338\) 30087.9 4.84191
\(339\) −2792.28 −0.447362
\(340\) 5464.16 0.871575
\(341\) −1315.02 −0.208833
\(342\) 1142.84 0.180695
\(343\) 6270.36 0.987078
\(344\) 14622.3 2.29181
\(345\) −1675.44 −0.261456
\(346\) 1661.67 0.258185
\(347\) −9059.04 −1.40148 −0.700742 0.713415i \(-0.747148\pi\)
−0.700742 + 0.713415i \(0.747148\pi\)
\(348\) 6180.68 0.952066
\(349\) −9362.35 −1.43597 −0.717987 0.696057i \(-0.754936\pi\)
−0.717987 + 0.696057i \(0.754936\pi\)
\(350\) −1444.53 −0.220610
\(351\) −2417.34 −0.367601
\(352\) −2917.15 −0.441718
\(353\) −274.260 −0.0413524 −0.0206762 0.999786i \(-0.506582\pi\)
−0.0206762 + 0.999786i \(0.506582\pi\)
\(354\) 7368.39 1.10629
\(355\) −5551.66 −0.830004
\(356\) −26567.7 −3.95530
\(357\) −1955.24 −0.289866
\(358\) −13765.1 −2.03214
\(359\) −1394.51 −0.205013 −0.102506 0.994732i \(-0.532686\pi\)
−0.102506 + 0.994732i \(0.532686\pi\)
\(360\) 2498.39 0.365769
\(361\) −6255.93 −0.912076
\(362\) 17094.8 2.48199
\(363\) 363.000 0.0524864
\(364\) 18745.9 2.69933
\(365\) 4468.04 0.640734
\(366\) −7961.03 −1.13697
\(367\) 11610.1 1.65134 0.825669 0.564155i \(-0.190798\pi\)
0.825669 + 0.564155i \(0.190798\pi\)
\(368\) −15322.9 −2.17055
\(369\) 2419.18 0.341294
\(370\) −9218.08 −1.29520
\(371\) −2500.29 −0.349888
\(372\) 6719.91 0.936590
\(373\) 5068.89 0.703639 0.351819 0.936068i \(-0.385563\pi\)
0.351819 + 0.936068i \(0.385563\pi\)
\(374\) −3317.41 −0.458661
\(375\) 375.000 0.0516398
\(376\) −11477.7 −1.57425
\(377\) −9844.28 −1.34484
\(378\) −1560.09 −0.212282
\(379\) 1623.62 0.220052 0.110026 0.993929i \(-0.464907\pi\)
0.110026 + 0.993929i \(0.464907\pi\)
\(380\) 2300.69 0.310587
\(381\) 283.295 0.0380935
\(382\) 4891.93 0.655217
\(383\) 5513.47 0.735575 0.367787 0.929910i \(-0.380115\pi\)
0.367787 + 0.929910i \(0.380115\pi\)
\(384\) −2117.44 −0.281394
\(385\) 614.599 0.0813581
\(386\) −19093.4 −2.51769
\(387\) 2370.34 0.311346
\(388\) 1607.82 0.210373
\(389\) 1591.80 0.207474 0.103737 0.994605i \(-0.466920\pi\)
0.103737 + 0.994605i \(0.466920\pi\)
\(390\) −6944.21 −0.901625
\(391\) −6514.57 −0.842598
\(392\) −12110.5 −1.56039
\(393\) 3954.37 0.507561
\(394\) −2127.80 −0.272074
\(395\) 6518.32 0.830310
\(396\) −1854.98 −0.235395
\(397\) −8032.19 −1.01543 −0.507713 0.861526i \(-0.669509\pi\)
−0.507713 + 0.861526i \(0.669509\pi\)
\(398\) 7711.11 0.971164
\(399\) −823.256 −0.103294
\(400\) 3429.62 0.428702
\(401\) 588.148 0.0732436 0.0366218 0.999329i \(-0.488340\pi\)
0.0366218 + 0.999329i \(0.488340\pi\)
\(402\) −4103.54 −0.509119
\(403\) −10703.2 −1.32298
\(404\) 2581.26 0.317878
\(405\) 405.000 0.0496904
\(406\) −6353.27 −0.776620
\(407\) 3921.98 0.477654
\(408\) 9714.46 1.17877
\(409\) 6945.63 0.839705 0.419853 0.907592i \(-0.362082\pi\)
0.419853 + 0.907592i \(0.362082\pi\)
\(410\) 6949.49 0.837100
\(411\) 6025.29 0.723128
\(412\) 7858.33 0.939689
\(413\) −5307.90 −0.632409
\(414\) −5198.01 −0.617073
\(415\) 5246.83 0.620618
\(416\) −23743.3 −2.79834
\(417\) 8870.28 1.04168
\(418\) −1396.80 −0.163445
\(419\) −3310.93 −0.386037 −0.193018 0.981195i \(-0.561828\pi\)
−0.193018 + 0.981195i \(0.561828\pi\)
\(420\) −3140.69 −0.364881
\(421\) −13906.6 −1.60989 −0.804946 0.593348i \(-0.797806\pi\)
−0.804946 + 0.593348i \(0.797806\pi\)
\(422\) −4653.45 −0.536792
\(423\) −1860.59 −0.213865
\(424\) 12422.5 1.42285
\(425\) 1458.11 0.166420
\(426\) −17223.9 −1.95892
\(427\) 5734.81 0.649946
\(428\) −22305.8 −2.51914
\(429\) 2954.52 0.332508
\(430\) 6809.18 0.763647
\(431\) 2713.07 0.303211 0.151606 0.988441i \(-0.451556\pi\)
0.151606 + 0.988441i \(0.451556\pi\)
\(432\) 3703.99 0.412519
\(433\) 668.058 0.0741451 0.0370725 0.999313i \(-0.488197\pi\)
0.0370725 + 0.999313i \(0.488197\pi\)
\(434\) −6907.57 −0.763995
\(435\) 1649.31 0.181789
\(436\) 19362.5 2.12683
\(437\) −2742.97 −0.300261
\(438\) 13862.0 1.51222
\(439\) 16512.6 1.79522 0.897611 0.440788i \(-0.145301\pi\)
0.897611 + 0.440788i \(0.145301\pi\)
\(440\) −3053.59 −0.330851
\(441\) −1963.17 −0.211982
\(442\) −27001.0 −2.90567
\(443\) −11305.6 −1.21252 −0.606258 0.795268i \(-0.707330\pi\)
−0.606258 + 0.795268i \(0.707330\pi\)
\(444\) −20041.9 −2.14222
\(445\) −7089.57 −0.755231
\(446\) 10007.0 1.06244
\(447\) 2636.31 0.278955
\(448\) −3059.56 −0.322657
\(449\) 2103.65 0.221108 0.110554 0.993870i \(-0.464738\pi\)
0.110554 + 0.993870i \(0.464738\pi\)
\(450\) 1163.43 0.121877
\(451\) −2956.77 −0.308712
\(452\) −17439.8 −1.81482
\(453\) −5038.06 −0.522536
\(454\) −15027.7 −1.55349
\(455\) 5002.34 0.515414
\(456\) 4090.29 0.420056
\(457\) −16013.6 −1.63914 −0.819570 0.572979i \(-0.805788\pi\)
−0.819570 + 0.572979i \(0.805788\pi\)
\(458\) −27983.8 −2.85501
\(459\) 1574.75 0.160138
\(460\) −10464.3 −1.06066
\(461\) 13332.9 1.34702 0.673509 0.739179i \(-0.264786\pi\)
0.673509 + 0.739179i \(0.264786\pi\)
\(462\) 1906.78 0.192016
\(463\) 2453.48 0.246270 0.123135 0.992390i \(-0.460705\pi\)
0.123135 + 0.992390i \(0.460705\pi\)
\(464\) 15084.0 1.50917
\(465\) 1793.20 0.178834
\(466\) −24710.8 −2.45645
\(467\) −771.767 −0.0764735 −0.0382367 0.999269i \(-0.512174\pi\)
−0.0382367 + 0.999269i \(0.512174\pi\)
\(468\) −15098.0 −1.49125
\(469\) 2956.03 0.291038
\(470\) −5344.85 −0.524552
\(471\) 2109.15 0.206337
\(472\) 26371.9 2.57175
\(473\) −2897.08 −0.281623
\(474\) 20223.0 1.95964
\(475\) 613.938 0.0593040
\(476\) −12211.9 −1.17590
\(477\) 2013.74 0.193297
\(478\) 30424.4 2.91125
\(479\) 2782.55 0.265423 0.132712 0.991155i \(-0.457632\pi\)
0.132712 + 0.991155i \(0.457632\pi\)
\(480\) 3977.94 0.378265
\(481\) 31921.7 3.02600
\(482\) 9343.44 0.882950
\(483\) 3744.44 0.352750
\(484\) 2267.20 0.212923
\(485\) 429.046 0.0401690
\(486\) 1256.50 0.117276
\(487\) −9716.69 −0.904118 −0.452059 0.891988i \(-0.649310\pi\)
−0.452059 + 0.891988i \(0.649310\pi\)
\(488\) −28493.0 −2.64307
\(489\) 8803.99 0.814172
\(490\) −5639.53 −0.519935
\(491\) 13582.7 1.24843 0.624213 0.781254i \(-0.285420\pi\)
0.624213 + 0.781254i \(0.285420\pi\)
\(492\) 15109.5 1.38453
\(493\) 6412.97 0.585853
\(494\) −11368.8 −1.03544
\(495\) −495.000 −0.0449467
\(496\) 16400.0 1.48464
\(497\) 12407.4 1.11982
\(498\) 16278.2 1.46474
\(499\) 12177.8 1.09249 0.546244 0.837626i \(-0.316057\pi\)
0.546244 + 0.837626i \(0.316057\pi\)
\(500\) 2342.15 0.209488
\(501\) −6438.14 −0.574121
\(502\) 6541.59 0.581604
\(503\) 5799.88 0.514123 0.257061 0.966395i \(-0.417246\pi\)
0.257061 + 0.966395i \(0.417246\pi\)
\(504\) −5583.67 −0.493485
\(505\) 688.807 0.0606961
\(506\) 6353.12 0.558164
\(507\) 17456.4 1.52913
\(508\) 1769.38 0.154535
\(509\) 19516.8 1.69954 0.849772 0.527151i \(-0.176740\pi\)
0.849772 + 0.527151i \(0.176740\pi\)
\(510\) 4523.74 0.392774
\(511\) −9985.65 −0.864460
\(512\) −24551.0 −2.11916
\(513\) 663.053 0.0570653
\(514\) −22578.5 −1.93754
\(515\) 2096.99 0.179426
\(516\) 14804.5 1.26304
\(517\) 2274.05 0.193448
\(518\) 20601.5 1.74745
\(519\) 964.068 0.0815374
\(520\) −24853.8 −2.09598
\(521\) 9489.76 0.797993 0.398996 0.916953i \(-0.369359\pi\)
0.398996 + 0.916953i \(0.369359\pi\)
\(522\) 5116.95 0.429047
\(523\) 5714.17 0.477750 0.238875 0.971050i \(-0.423221\pi\)
0.238875 + 0.971050i \(0.423221\pi\)
\(524\) 24697.9 2.05903
\(525\) −838.090 −0.0696709
\(526\) −15258.8 −1.26486
\(527\) 6972.47 0.576330
\(528\) −4527.09 −0.373137
\(529\) 308.942 0.0253918
\(530\) 5784.80 0.474105
\(531\) 4275.00 0.349377
\(532\) −5141.84 −0.419036
\(533\) −24065.7 −1.95573
\(534\) −21995.3 −1.78245
\(535\) −5952.27 −0.481008
\(536\) −14686.8 −1.18353
\(537\) −7986.24 −0.641772
\(538\) −31276.4 −2.50636
\(539\) 2399.43 0.191745
\(540\) 2529.52 0.201580
\(541\) −2530.63 −0.201110 −0.100555 0.994932i \(-0.532062\pi\)
−0.100555 + 0.994932i \(0.532062\pi\)
\(542\) −15113.8 −1.19777
\(543\) 9918.06 0.783839
\(544\) 15467.3 1.21904
\(545\) 5166.87 0.406100
\(546\) 15519.7 1.21645
\(547\) 18910.4 1.47816 0.739079 0.673619i \(-0.235261\pi\)
0.739079 + 0.673619i \(0.235261\pi\)
\(548\) 37632.3 2.93353
\(549\) −4618.83 −0.359066
\(550\) −1421.97 −0.110242
\(551\) 2700.19 0.208770
\(552\) −18604.0 −1.43449
\(553\) −14567.8 −1.12023
\(554\) −28095.4 −2.15462
\(555\) −5348.15 −0.409039
\(556\) 55401.4 4.22580
\(557\) 4480.34 0.340823 0.170411 0.985373i \(-0.445490\pi\)
0.170411 + 0.985373i \(0.445490\pi\)
\(558\) 5563.38 0.422073
\(559\) −23579.8 −1.78412
\(560\) −7664.87 −0.578393
\(561\) −1924.70 −0.144850
\(562\) −41268.3 −3.09750
\(563\) 1726.52 0.129244 0.0646218 0.997910i \(-0.479416\pi\)
0.0646218 + 0.997910i \(0.479416\pi\)
\(564\) −11620.7 −0.867590
\(565\) −4653.80 −0.346525
\(566\) −13990.7 −1.03900
\(567\) −905.137 −0.0670409
\(568\) −61645.5 −4.55385
\(569\) −10862.7 −0.800327 −0.400164 0.916444i \(-0.631047\pi\)
−0.400164 + 0.916444i \(0.631047\pi\)
\(570\) 1904.73 0.139966
\(571\) 14448.8 1.05895 0.529477 0.848324i \(-0.322388\pi\)
0.529477 + 0.848324i \(0.322388\pi\)
\(572\) 18453.2 1.34889
\(573\) 2838.20 0.206924
\(574\) −15531.5 −1.12939
\(575\) −2792.39 −0.202523
\(576\) 2464.18 0.178254
\(577\) −5335.95 −0.384988 −0.192494 0.981298i \(-0.561658\pi\)
−0.192494 + 0.981298i \(0.561658\pi\)
\(578\) −7814.56 −0.562358
\(579\) −11077.6 −0.795112
\(580\) 10301.1 0.737467
\(581\) −11726.2 −0.837321
\(582\) 1331.11 0.0948043
\(583\) −2461.23 −0.174844
\(584\) 49613.0 3.51541
\(585\) −4028.90 −0.284743
\(586\) 11415.6 0.804735
\(587\) 25633.8 1.80242 0.901208 0.433386i \(-0.142681\pi\)
0.901208 + 0.433386i \(0.142681\pi\)
\(588\) −12261.4 −0.859953
\(589\) 2935.77 0.205376
\(590\) 12280.7 0.856926
\(591\) −1234.51 −0.0859238
\(592\) −48912.3 −3.39575
\(593\) −16094.8 −1.11456 −0.557282 0.830324i \(-0.688156\pi\)
−0.557282 + 0.830324i \(0.688156\pi\)
\(594\) −1535.73 −0.106080
\(595\) −3258.73 −0.224529
\(596\) 16465.6 1.13164
\(597\) 4473.84 0.306704
\(598\) 51709.2 3.53603
\(599\) −17160.5 −1.17055 −0.585275 0.810835i \(-0.699014\pi\)
−0.585275 + 0.810835i \(0.699014\pi\)
\(600\) 4163.99 0.283324
\(601\) −14563.2 −0.988426 −0.494213 0.869341i \(-0.664544\pi\)
−0.494213 + 0.869341i \(0.664544\pi\)
\(602\) −15217.9 −1.03029
\(603\) −2380.79 −0.160785
\(604\) −31466.3 −2.11978
\(605\) 605.000 0.0406558
\(606\) 2137.01 0.143251
\(607\) −12834.7 −0.858230 −0.429115 0.903250i \(-0.641175\pi\)
−0.429115 + 0.903250i \(0.641175\pi\)
\(608\) 6512.55 0.434406
\(609\) −3686.05 −0.245265
\(610\) −13268.4 −0.880690
\(611\) 18508.9 1.22552
\(612\) 9835.48 0.649634
\(613\) −10814.5 −0.712548 −0.356274 0.934381i \(-0.615953\pi\)
−0.356274 + 0.934381i \(0.615953\pi\)
\(614\) −20261.5 −1.33174
\(615\) 4031.96 0.264365
\(616\) 6824.49 0.446374
\(617\) 5118.52 0.333977 0.166988 0.985959i \(-0.446596\pi\)
0.166988 + 0.985959i \(0.446596\pi\)
\(618\) 6505.86 0.423469
\(619\) 26144.1 1.69761 0.848804 0.528708i \(-0.177323\pi\)
0.848804 + 0.528708i \(0.177323\pi\)
\(620\) 11199.9 0.725479
\(621\) −3015.79 −0.194878
\(622\) 45342.9 2.92297
\(623\) 15844.5 1.01894
\(624\) −36846.9 −2.36387
\(625\) 625.000 0.0400000
\(626\) −26158.7 −1.67015
\(627\) −810.398 −0.0516175
\(628\) 13173.2 0.837051
\(629\) −20795.1 −1.31821
\(630\) −2600.16 −0.164433
\(631\) −15147.2 −0.955626 −0.477813 0.878462i \(-0.658570\pi\)
−0.477813 + 0.878462i \(0.658570\pi\)
\(632\) 72379.2 4.55552
\(633\) −2699.84 −0.169525
\(634\) −43070.3 −2.69801
\(635\) 472.158 0.0295071
\(636\) 12577.3 0.784152
\(637\) 19529.4 1.21473
\(638\) −6254.04 −0.388088
\(639\) −9992.98 −0.618648
\(640\) −3529.07 −0.217967
\(641\) 6959.04 0.428807 0.214404 0.976745i \(-0.431219\pi\)
0.214404 + 0.976745i \(0.431219\pi\)
\(642\) −18466.8 −1.13524
\(643\) 50.7662 0.00311357 0.00155678 0.999999i \(-0.499504\pi\)
0.00155678 + 0.999999i \(0.499504\pi\)
\(644\) 23386.8 1.43101
\(645\) 3950.56 0.241168
\(646\) 7406.13 0.451069
\(647\) −14853.2 −0.902537 −0.451268 0.892388i \(-0.649028\pi\)
−0.451268 + 0.892388i \(0.649028\pi\)
\(648\) 4497.11 0.272628
\(649\) −5225.00 −0.316023
\(650\) −11573.7 −0.698396
\(651\) −4007.64 −0.241278
\(652\) 54987.3 3.30287
\(653\) 18366.0 1.10064 0.550321 0.834953i \(-0.314505\pi\)
0.550321 + 0.834953i \(0.314505\pi\)
\(654\) 16030.1 0.958451
\(655\) 6590.62 0.393155
\(656\) 36874.9 2.19470
\(657\) 8042.47 0.477575
\(658\) 11945.2 0.707711
\(659\) −8660.13 −0.511913 −0.255957 0.966688i \(-0.582390\pi\)
−0.255957 + 0.966688i \(0.582390\pi\)
\(660\) −3091.64 −0.182336
\(661\) −11057.5 −0.650661 −0.325331 0.945600i \(-0.605476\pi\)
−0.325331 + 0.945600i \(0.605476\pi\)
\(662\) 15004.9 0.880940
\(663\) −15665.5 −0.917642
\(664\) 58260.6 3.40505
\(665\) −1372.09 −0.0800113
\(666\) −16592.5 −0.965387
\(667\) −12281.4 −0.712949
\(668\) −40210.9 −2.32905
\(669\) 5805.88 0.335528
\(670\) −6839.23 −0.394362
\(671\) 5645.24 0.324787
\(672\) −8890.32 −0.510344
\(673\) 1921.35 0.110048 0.0550241 0.998485i \(-0.482476\pi\)
0.0550241 + 0.998485i \(0.482476\pi\)
\(674\) 31028.7 1.77327
\(675\) 675.000 0.0384900
\(676\) 109028. 6.20324
\(677\) 25838.8 1.46686 0.733431 0.679764i \(-0.237918\pi\)
0.733431 + 0.679764i \(0.237918\pi\)
\(678\) −14438.3 −0.817847
\(679\) −958.877 −0.0541948
\(680\) 16190.8 0.913070
\(681\) −8718.78 −0.490608
\(682\) −6799.68 −0.381779
\(683\) −33038.7 −1.85094 −0.925469 0.378823i \(-0.876329\pi\)
−0.925469 + 0.378823i \(0.876329\pi\)
\(684\) 4141.25 0.231498
\(685\) 10042.2 0.560133
\(686\) 32422.8 1.80453
\(687\) −16235.7 −0.901643
\(688\) 36130.4 2.00212
\(689\) −20032.4 −1.10766
\(690\) −8663.35 −0.477983
\(691\) −2065.11 −0.113691 −0.0568454 0.998383i \(-0.518104\pi\)
−0.0568454 + 0.998383i \(0.518104\pi\)
\(692\) 6021.31 0.330774
\(693\) 1106.28 0.0606408
\(694\) −46842.5 −2.56213
\(695\) 14783.8 0.806880
\(696\) 18313.9 0.997393
\(697\) 15677.4 0.851972
\(698\) −48410.8 −2.62518
\(699\) −14336.7 −0.775771
\(700\) −5234.48 −0.282635
\(701\) −23504.1 −1.26639 −0.633194 0.773993i \(-0.718257\pi\)
−0.633194 + 0.773993i \(0.718257\pi\)
\(702\) −12499.6 −0.672031
\(703\) −8755.83 −0.469747
\(704\) −3011.77 −0.161236
\(705\) −3100.98 −0.165659
\(706\) −1418.14 −0.0755985
\(707\) −1539.42 −0.0818894
\(708\) 26700.5 1.41732
\(709\) −135.537 −0.00717943 −0.00358971 0.999994i \(-0.501143\pi\)
−0.00358971 + 0.999994i \(0.501143\pi\)
\(710\) −28706.5 −1.51738
\(711\) 11733.0 0.618876
\(712\) −78722.4 −4.14361
\(713\) −13352.9 −0.701359
\(714\) −10110.1 −0.529920
\(715\) 4924.21 0.257559
\(716\) −49879.9 −2.60349
\(717\) 17651.6 0.919404
\(718\) −7210.75 −0.374795
\(719\) 10752.3 0.557711 0.278856 0.960333i \(-0.410045\pi\)
0.278856 + 0.960333i \(0.410045\pi\)
\(720\) 6173.31 0.319536
\(721\) −4686.57 −0.242076
\(722\) −32348.2 −1.66742
\(723\) 5420.88 0.278845
\(724\) 61945.5 3.17982
\(725\) 2748.85 0.140813
\(726\) 1877.00 0.0959532
\(727\) −7521.06 −0.383687 −0.191844 0.981425i \(-0.561447\pi\)
−0.191844 + 0.981425i \(0.561447\pi\)
\(728\) 55545.8 2.82784
\(729\) 729.000 0.0370370
\(730\) 23103.4 1.17136
\(731\) 15360.9 0.777213
\(732\) −28848.0 −1.45663
\(733\) −5330.33 −0.268595 −0.134298 0.990941i \(-0.542878\pi\)
−0.134298 + 0.990941i \(0.542878\pi\)
\(734\) 60033.4 3.01890
\(735\) −3271.95 −0.164201
\(736\) −29621.2 −1.48350
\(737\) 2909.86 0.145436
\(738\) 12509.1 0.623938
\(739\) 29328.0 1.45988 0.729938 0.683513i \(-0.239549\pi\)
0.729938 + 0.683513i \(0.239549\pi\)
\(740\) −33403.1 −1.65936
\(741\) −6595.98 −0.327003
\(742\) −12928.5 −0.639649
\(743\) −1378.56 −0.0680679 −0.0340340 0.999421i \(-0.510835\pi\)
−0.0340340 + 0.999421i \(0.510835\pi\)
\(744\) 19911.7 0.981179
\(745\) 4393.84 0.216078
\(746\) 26210.2 1.28636
\(747\) 9444.29 0.462582
\(748\) −12021.1 −0.587616
\(749\) 13302.8 0.648962
\(750\) 1939.05 0.0944055
\(751\) −31066.4 −1.50950 −0.754748 0.656015i \(-0.772241\pi\)
−0.754748 + 0.656015i \(0.772241\pi\)
\(752\) −28360.4 −1.37526
\(753\) 3795.30 0.183677
\(754\) −50902.8 −2.45858
\(755\) −8396.76 −0.404754
\(756\) −5653.24 −0.271966
\(757\) 21971.1 1.05489 0.527445 0.849589i \(-0.323150\pi\)
0.527445 + 0.849589i \(0.323150\pi\)
\(758\) 8395.42 0.402290
\(759\) 3685.96 0.176274
\(760\) 6817.15 0.325374
\(761\) −35175.8 −1.67559 −0.837794 0.545987i \(-0.816155\pi\)
−0.837794 + 0.545987i \(0.816155\pi\)
\(762\) 1464.86 0.0696408
\(763\) −11547.5 −0.547898
\(764\) 17726.7 0.839435
\(765\) 2624.59 0.124042
\(766\) 28509.1 1.34474
\(767\) −42527.3 −2.00205
\(768\) −17520.0 −0.823176
\(769\) 32952.5 1.54525 0.772626 0.634862i \(-0.218943\pi\)
0.772626 + 0.634862i \(0.218943\pi\)
\(770\) 3177.97 0.148735
\(771\) −13099.6 −0.611894
\(772\) −69187.7 −3.22554
\(773\) −11158.7 −0.519213 −0.259607 0.965714i \(-0.583593\pi\)
−0.259607 + 0.965714i \(0.583593\pi\)
\(774\) 12256.5 0.569189
\(775\) 2988.67 0.138524
\(776\) 4764.11 0.220389
\(777\) 11952.6 0.551863
\(778\) 8230.86 0.379294
\(779\) 6601.00 0.303601
\(780\) −25163.4 −1.15512
\(781\) 12213.6 0.559589
\(782\) −33685.5 −1.54040
\(783\) 2968.75 0.135498
\(784\) −29924.1 −1.36316
\(785\) 3515.26 0.159828
\(786\) 20447.3 0.927901
\(787\) 17308.9 0.783985 0.391993 0.919968i \(-0.371786\pi\)
0.391993 + 0.919968i \(0.371786\pi\)
\(788\) −7710.41 −0.348569
\(789\) −8852.87 −0.399456
\(790\) 33704.9 1.51793
\(791\) 10400.8 0.467522
\(792\) −5496.47 −0.246602
\(793\) 45947.7 2.05757
\(794\) −41532.9 −1.85636
\(795\) 3356.23 0.149727
\(796\) 27942.4 1.24421
\(797\) −14802.4 −0.657876 −0.328938 0.944351i \(-0.606691\pi\)
−0.328938 + 0.944351i \(0.606691\pi\)
\(798\) −4256.90 −0.188838
\(799\) −12057.5 −0.533871
\(800\) 6629.90 0.293003
\(801\) −12761.2 −0.562916
\(802\) 3041.19 0.133901
\(803\) −9829.69 −0.431983
\(804\) −14869.8 −0.652260
\(805\) 6240.74 0.273239
\(806\) −55343.9 −2.41862
\(807\) −18146.0 −0.791534
\(808\) 7648.50 0.333011
\(809\) −8999.53 −0.391108 −0.195554 0.980693i \(-0.562651\pi\)
−0.195554 + 0.980693i \(0.562651\pi\)
\(810\) 2094.17 0.0908417
\(811\) −41368.5 −1.79117 −0.895587 0.444886i \(-0.853244\pi\)
−0.895587 + 0.444886i \(0.853244\pi\)
\(812\) −23022.1 −0.994970
\(813\) −8768.71 −0.378268
\(814\) 20279.8 0.873226
\(815\) 14673.3 0.630655
\(816\) 24003.6 1.02977
\(817\) 6467.73 0.276961
\(818\) 35914.5 1.53511
\(819\) 9004.21 0.384167
\(820\) 25182.6 1.07245
\(821\) 27772.2 1.18058 0.590289 0.807192i \(-0.299014\pi\)
0.590289 + 0.807192i \(0.299014\pi\)
\(822\) 31155.6 1.32199
\(823\) −9315.89 −0.394571 −0.197285 0.980346i \(-0.563213\pi\)
−0.197285 + 0.980346i \(0.563213\pi\)
\(824\) 23284.9 0.984426
\(825\) −825.000 −0.0348155
\(826\) −27446.1 −1.15614
\(827\) −20344.6 −0.855443 −0.427721 0.903911i \(-0.640684\pi\)
−0.427721 + 0.903911i \(0.640684\pi\)
\(828\) −18835.8 −0.790566
\(829\) 16916.7 0.708737 0.354368 0.935106i \(-0.384696\pi\)
0.354368 + 0.935106i \(0.384696\pi\)
\(830\) 27130.3 1.13459
\(831\) −16300.4 −0.680452
\(832\) −24513.4 −1.02145
\(833\) −12722.3 −0.529172
\(834\) 45866.5 1.90435
\(835\) −10730.2 −0.444712
\(836\) −5061.53 −0.209398
\(837\) 3227.76 0.133295
\(838\) −17120.2 −0.705735
\(839\) 23690.7 0.974844 0.487422 0.873167i \(-0.337937\pi\)
0.487422 + 0.873167i \(0.337937\pi\)
\(840\) −9306.12 −0.382252
\(841\) −12299.2 −0.504291
\(842\) −71908.1 −2.94313
\(843\) −23943.1 −0.978224
\(844\) −16862.5 −0.687714
\(845\) 29094.0 1.18446
\(846\) −9620.72 −0.390978
\(847\) −1352.12 −0.0548516
\(848\) 30694.9 1.24300
\(849\) −8117.14 −0.328127
\(850\) 7539.57 0.304241
\(851\) 39824.4 1.60419
\(852\) −62413.5 −2.50968
\(853\) −3804.67 −0.152719 −0.0763596 0.997080i \(-0.524330\pi\)
−0.0763596 + 0.997080i \(0.524330\pi\)
\(854\) 29653.6 1.18820
\(855\) 1105.09 0.0442026
\(856\) −66093.9 −2.63907
\(857\) −3125.89 −0.124596 −0.0622978 0.998058i \(-0.519843\pi\)
−0.0622978 + 0.998058i \(0.519843\pi\)
\(858\) 15277.3 0.607875
\(859\) −38044.2 −1.51112 −0.755559 0.655081i \(-0.772635\pi\)
−0.755559 + 0.655081i \(0.772635\pi\)
\(860\) 24674.1 0.978349
\(861\) −9011.06 −0.356674
\(862\) 14028.8 0.554317
\(863\) 728.739 0.0287446 0.0143723 0.999897i \(-0.495425\pi\)
0.0143723 + 0.999897i \(0.495425\pi\)
\(864\) 7160.29 0.281942
\(865\) 1606.78 0.0631586
\(866\) 3454.39 0.135549
\(867\) −4533.86 −0.177599
\(868\) −25030.6 −0.978796
\(869\) −14340.3 −0.559795
\(870\) 8528.24 0.332338
\(871\) 23683.9 0.921352
\(872\) 57372.7 2.22808
\(873\) 772.282 0.0299402
\(874\) −14183.4 −0.548924
\(875\) −1396.82 −0.0539669
\(876\) 50231.1 1.93739
\(877\) 18861.7 0.726241 0.363120 0.931742i \(-0.381711\pi\)
0.363120 + 0.931742i \(0.381711\pi\)
\(878\) 85383.3 3.28194
\(879\) 6623.12 0.254144
\(880\) −7545.16 −0.289031
\(881\) 24435.2 0.934442 0.467221 0.884141i \(-0.345255\pi\)
0.467221 + 0.884141i \(0.345255\pi\)
\(882\) −10151.2 −0.387537
\(883\) 3101.93 0.118220 0.0591101 0.998251i \(-0.481174\pi\)
0.0591101 + 0.998251i \(0.481174\pi\)
\(884\) −97842.4 −3.72262
\(885\) 7125.00 0.270626
\(886\) −58458.9 −2.21667
\(887\) 2129.33 0.0806043 0.0403021 0.999188i \(-0.487168\pi\)
0.0403021 + 0.999188i \(0.487168\pi\)
\(888\) −59385.7 −2.24421
\(889\) −1055.23 −0.0398101
\(890\) −36658.8 −1.38068
\(891\) −891.000 −0.0335013
\(892\) 36262.0 1.36114
\(893\) −5076.82 −0.190246
\(894\) 13631.8 0.509973
\(895\) −13310.4 −0.497115
\(896\) 7887.14 0.294075
\(897\) 30000.7 1.11672
\(898\) 10877.5 0.404219
\(899\) 13144.6 0.487651
\(900\) 4215.87 0.156143
\(901\) 13050.0 0.482527
\(902\) −15288.9 −0.564373
\(903\) −8829.13 −0.325376
\(904\) −51675.6 −1.90122
\(905\) 16530.1 0.607159
\(906\) −26050.8 −0.955276
\(907\) 33678.6 1.23294 0.616471 0.787378i \(-0.288562\pi\)
0.616471 + 0.787378i \(0.288562\pi\)
\(908\) −54455.1 −1.99026
\(909\) 1239.85 0.0452402
\(910\) 25866.1 0.942255
\(911\) −28917.6 −1.05168 −0.525841 0.850583i \(-0.676249\pi\)
−0.525841 + 0.850583i \(0.676249\pi\)
\(912\) 10106.7 0.366960
\(913\) −11543.0 −0.418421
\(914\) −82803.4 −2.99660
\(915\) −7698.06 −0.278131
\(916\) −101404. −3.65771
\(917\) −14729.4 −0.530434
\(918\) 8142.74 0.292756
\(919\) −8611.83 −0.309117 −0.154558 0.987984i \(-0.549395\pi\)
−0.154558 + 0.987984i \(0.549395\pi\)
\(920\) −31006.7 −1.11115
\(921\) −11755.4 −0.420578
\(922\) 68941.8 2.46256
\(923\) 99409.2 3.54506
\(924\) 6909.52 0.246003
\(925\) −8913.59 −0.316840
\(926\) 12686.5 0.450219
\(927\) 3774.58 0.133736
\(928\) 29159.3 1.03147
\(929\) −31195.3 −1.10171 −0.550853 0.834602i \(-0.685698\pi\)
−0.550853 + 0.834602i \(0.685698\pi\)
\(930\) 9272.29 0.326936
\(931\) −5356.73 −0.188571
\(932\) −89543.2 −3.14709
\(933\) 26307.1 0.923103
\(934\) −3990.65 −0.139805
\(935\) −3207.83 −0.112200
\(936\) −44736.8 −1.56225
\(937\) 24625.3 0.858561 0.429281 0.903171i \(-0.358767\pi\)
0.429281 + 0.903171i \(0.358767\pi\)
\(938\) 15285.0 0.532062
\(939\) −15176.8 −0.527449
\(940\) −19367.9 −0.672032
\(941\) 3605.28 0.124898 0.0624488 0.998048i \(-0.480109\pi\)
0.0624488 + 0.998048i \(0.480109\pi\)
\(942\) 10906.0 0.377216
\(943\) −30023.6 −1.03680
\(944\) 65162.7 2.24668
\(945\) −1508.56 −0.0519296
\(946\) −14980.2 −0.514850
\(947\) 11634.6 0.399231 0.199616 0.979874i \(-0.436031\pi\)
0.199616 + 0.979874i \(0.436031\pi\)
\(948\) 73281.0 2.51061
\(949\) −80005.7 −2.73666
\(950\) 3174.55 0.108417
\(951\) −24988.6 −0.852060
\(952\) −36184.8 −1.23189
\(953\) 27302.2 0.928021 0.464010 0.885830i \(-0.346410\pi\)
0.464010 + 0.885830i \(0.346410\pi\)
\(954\) 10412.6 0.353377
\(955\) 4730.34 0.160283
\(956\) 110247. 3.72976
\(957\) −3628.48 −0.122562
\(958\) 14388.0 0.485235
\(959\) −22443.3 −0.755715
\(960\) 4106.96 0.138075
\(961\) −15499.6 −0.520276
\(962\) 165061. 5.53199
\(963\) −10714.1 −0.358522
\(964\) 33857.4 1.13120
\(965\) −18462.7 −0.615891
\(966\) 19361.8 0.644881
\(967\) −19429.3 −0.646125 −0.323063 0.946378i \(-0.604712\pi\)
−0.323063 + 0.946378i \(0.604712\pi\)
\(968\) 6717.90 0.223059
\(969\) 4296.90 0.142452
\(970\) 2218.51 0.0734351
\(971\) 42852.0 1.41626 0.708129 0.706083i \(-0.249539\pi\)
0.708129 + 0.706083i \(0.249539\pi\)
\(972\) 4553.14 0.150249
\(973\) −33040.4 −1.08862
\(974\) −50243.1 −1.65287
\(975\) −6714.83 −0.220561
\(976\) −70403.7 −2.30898
\(977\) 6559.02 0.214782 0.107391 0.994217i \(-0.465750\pi\)
0.107391 + 0.994217i \(0.465750\pi\)
\(978\) 45523.7 1.48843
\(979\) 15597.1 0.509177
\(980\) −20435.7 −0.666117
\(981\) 9300.36 0.302689
\(982\) 70233.3 2.28232
\(983\) −15077.3 −0.489208 −0.244604 0.969623i \(-0.578658\pi\)
−0.244604 + 0.969623i \(0.578658\pi\)
\(984\) 44770.8 1.45045
\(985\) −2057.52 −0.0665563
\(986\) 33160.2 1.07103
\(987\) 6930.39 0.223502
\(988\) −41196.7 −1.32656
\(989\) −29417.4 −0.945822
\(990\) −2559.55 −0.0821694
\(991\) 45239.3 1.45013 0.725063 0.688683i \(-0.241811\pi\)
0.725063 + 0.688683i \(0.241811\pi\)
\(992\) 31703.3 1.01470
\(993\) 8705.56 0.278210
\(994\) 64156.4 2.04720
\(995\) 7456.40 0.237572
\(996\) 58986.5 1.87656
\(997\) 30975.1 0.983942 0.491971 0.870612i \(-0.336277\pi\)
0.491971 + 0.870612i \(0.336277\pi\)
\(998\) 62968.8 1.99724
\(999\) −9626.68 −0.304879
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.1.4 4
3.2 odd 2 495.4.a.m.1.1 4
5.2 odd 4 825.4.c.p.199.8 8
5.3 odd 4 825.4.c.p.199.1 8
5.4 even 2 825.4.a.t.1.1 4
11.10 odd 2 1815.4.a.t.1.1 4
15.14 odd 2 2475.4.a.be.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.h.1.4 4 1.1 even 1 trivial
495.4.a.m.1.1 4 3.2 odd 2
825.4.a.t.1.1 4 5.4 even 2
825.4.c.p.199.1 8 5.3 odd 4
825.4.c.p.199.8 8 5.2 odd 4
1815.4.a.t.1.1 4 11.10 odd 2
2475.4.a.be.1.4 4 15.14 odd 2