Properties

Label 322.4.a.f.1.4
Level $322$
Weight $4$
Character 322.1
Self dual yes
Analytic conductor $18.999$
Analytic rank $1$
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] = [322,4,Mod(1,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, 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("322.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 322.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: \(18.9986150218\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 42x^{2} + 26x + 46 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-6.22602\) of defining polynomial
Character \(\chi\) \(=\) 322.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +6.22602 q^{3} +4.00000 q^{4} -9.89045 q^{5} -12.4520 q^{6} +7.00000 q^{7} -8.00000 q^{8} +11.7633 q^{9} +19.7809 q^{10} -41.5157 q^{11} +24.9041 q^{12} +18.7390 q^{13} -14.0000 q^{14} -61.5782 q^{15} +16.0000 q^{16} -19.3383 q^{17} -23.5266 q^{18} -52.3251 q^{19} -39.5618 q^{20} +43.5821 q^{21} +83.0315 q^{22} -23.0000 q^{23} -49.8082 q^{24} -27.1789 q^{25} -37.4781 q^{26} -94.8639 q^{27} +28.0000 q^{28} -169.213 q^{29} +123.156 q^{30} -191.927 q^{31} -32.0000 q^{32} -258.478 q^{33} +38.6766 q^{34} -69.2332 q^{35} +47.0533 q^{36} +366.682 q^{37} +104.650 q^{38} +116.670 q^{39} +79.1236 q^{40} +81.0378 q^{41} -87.1643 q^{42} +202.640 q^{43} -166.063 q^{44} -116.345 q^{45} +46.0000 q^{46} +86.9907 q^{47} +99.6163 q^{48} +49.0000 q^{49} +54.3578 q^{50} -120.401 q^{51} +74.9561 q^{52} -708.249 q^{53} +189.728 q^{54} +410.609 q^{55} -56.0000 q^{56} -325.777 q^{57} +338.427 q^{58} -618.357 q^{59} -246.313 q^{60} -258.284 q^{61} +383.853 q^{62} +82.3432 q^{63} +64.0000 q^{64} -185.338 q^{65} +516.955 q^{66} -167.372 q^{67} -77.3531 q^{68} -143.198 q^{69} +138.466 q^{70} -281.024 q^{71} -94.1065 q^{72} -643.990 q^{73} -733.364 q^{74} -169.216 q^{75} -209.301 q^{76} -290.610 q^{77} -233.339 q^{78} -397.326 q^{79} -158.247 q^{80} -908.234 q^{81} -162.076 q^{82} +1439.93 q^{83} +174.329 q^{84} +191.264 q^{85} -405.280 q^{86} -1053.53 q^{87} +332.126 q^{88} -497.107 q^{89} +232.689 q^{90} +131.173 q^{91} -92.0000 q^{92} -1194.94 q^{93} -173.981 q^{94} +517.519 q^{95} -199.233 q^{96} +193.384 q^{97} -98.0000 q^{98} -488.363 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} - q^{3} + 16 q^{4} - 2 q^{5} + 2 q^{6} + 28 q^{7} - 32 q^{8} - 23 q^{9} + 4 q^{10} - 66 q^{11} - 4 q^{12} - 25 q^{13} - 56 q^{14} - 4 q^{15} + 64 q^{16} - 56 q^{17} + 46 q^{18} + 4 q^{19}+ \cdots - 108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 6.22602 1.19820 0.599099 0.800675i \(-0.295526\pi\)
0.599099 + 0.800675i \(0.295526\pi\)
\(4\) 4.00000 0.500000
\(5\) −9.89045 −0.884629 −0.442315 0.896860i \(-0.645843\pi\)
−0.442315 + 0.896860i \(0.645843\pi\)
\(6\) −12.4520 −0.847254
\(7\) 7.00000 0.377964
\(8\) −8.00000 −0.353553
\(9\) 11.7633 0.435678
\(10\) 19.7809 0.625527
\(11\) −41.5157 −1.13795 −0.568976 0.822354i \(-0.692660\pi\)
−0.568976 + 0.822354i \(0.692660\pi\)
\(12\) 24.9041 0.599099
\(13\) 18.7390 0.399790 0.199895 0.979817i \(-0.435940\pi\)
0.199895 + 0.979817i \(0.435940\pi\)
\(14\) −14.0000 −0.267261
\(15\) −61.5782 −1.05996
\(16\) 16.0000 0.250000
\(17\) −19.3383 −0.275895 −0.137948 0.990440i \(-0.544051\pi\)
−0.137948 + 0.990440i \(0.544051\pi\)
\(18\) −23.5266 −0.308071
\(19\) −52.3251 −0.631801 −0.315900 0.948792i \(-0.602307\pi\)
−0.315900 + 0.948792i \(0.602307\pi\)
\(20\) −39.5618 −0.442315
\(21\) 43.5821 0.452876
\(22\) 83.0315 0.804653
\(23\) −23.0000 −0.208514
\(24\) −49.8082 −0.423627
\(25\) −27.1789 −0.217431
\(26\) −37.4781 −0.282694
\(27\) −94.8639 −0.676169
\(28\) 28.0000 0.188982
\(29\) −169.213 −1.08352 −0.541761 0.840533i \(-0.682242\pi\)
−0.541761 + 0.840533i \(0.682242\pi\)
\(30\) 123.156 0.749506
\(31\) −191.927 −1.11197 −0.555984 0.831193i \(-0.687659\pi\)
−0.555984 + 0.831193i \(0.687659\pi\)
\(32\) −32.0000 −0.176777
\(33\) −258.478 −1.36349
\(34\) 38.6766 0.195088
\(35\) −69.2332 −0.334358
\(36\) 47.0533 0.217839
\(37\) 366.682 1.62925 0.814623 0.579990i \(-0.196944\pi\)
0.814623 + 0.579990i \(0.196944\pi\)
\(38\) 104.650 0.446750
\(39\) 116.670 0.479028
\(40\) 79.1236 0.312764
\(41\) 81.0378 0.308682 0.154341 0.988018i \(-0.450674\pi\)
0.154341 + 0.988018i \(0.450674\pi\)
\(42\) −87.1643 −0.320232
\(43\) 202.640 0.718658 0.359329 0.933211i \(-0.383006\pi\)
0.359329 + 0.933211i \(0.383006\pi\)
\(44\) −166.063 −0.568976
\(45\) −116.345 −0.385414
\(46\) 46.0000 0.147442
\(47\) 86.9907 0.269977 0.134988 0.990847i \(-0.456900\pi\)
0.134988 + 0.990847i \(0.456900\pi\)
\(48\) 99.6163 0.299549
\(49\) 49.0000 0.142857
\(50\) 54.3578 0.153747
\(51\) −120.401 −0.330577
\(52\) 74.9561 0.199895
\(53\) −708.249 −1.83558 −0.917788 0.397071i \(-0.870027\pi\)
−0.917788 + 0.397071i \(0.870027\pi\)
\(54\) 189.728 0.478124
\(55\) 410.609 1.00666
\(56\) −56.0000 −0.133631
\(57\) −325.777 −0.757022
\(58\) 338.427 0.766165
\(59\) −618.357 −1.36446 −0.682231 0.731137i \(-0.738990\pi\)
−0.682231 + 0.731137i \(0.738990\pi\)
\(60\) −246.313 −0.529980
\(61\) −258.284 −0.542129 −0.271064 0.962561i \(-0.587376\pi\)
−0.271064 + 0.962561i \(0.587376\pi\)
\(62\) 383.853 0.786281
\(63\) 82.3432 0.164671
\(64\) 64.0000 0.125000
\(65\) −185.338 −0.353666
\(66\) 516.955 0.964134
\(67\) −167.372 −0.305191 −0.152596 0.988289i \(-0.548763\pi\)
−0.152596 + 0.988289i \(0.548763\pi\)
\(68\) −77.3531 −0.137948
\(69\) −143.198 −0.249842
\(70\) 138.466 0.236427
\(71\) −281.024 −0.469739 −0.234869 0.972027i \(-0.575466\pi\)
−0.234869 + 0.972027i \(0.575466\pi\)
\(72\) −94.1065 −0.154036
\(73\) −643.990 −1.03251 −0.516256 0.856434i \(-0.672675\pi\)
−0.516256 + 0.856434i \(0.672675\pi\)
\(74\) −733.364 −1.15205
\(75\) −169.216 −0.260526
\(76\) −209.301 −0.315900
\(77\) −290.610 −0.430105
\(78\) −233.339 −0.338724
\(79\) −397.326 −0.565856 −0.282928 0.959141i \(-0.591306\pi\)
−0.282928 + 0.959141i \(0.591306\pi\)
\(80\) −158.247 −0.221157
\(81\) −908.234 −1.24586
\(82\) −162.076 −0.218271
\(83\) 1439.93 1.90425 0.952126 0.305705i \(-0.0988921\pi\)
0.952126 + 0.305705i \(0.0988921\pi\)
\(84\) 174.329 0.226438
\(85\) 191.264 0.244065
\(86\) −405.280 −0.508168
\(87\) −1053.53 −1.29827
\(88\) 332.126 0.402326
\(89\) −497.107 −0.592058 −0.296029 0.955179i \(-0.595663\pi\)
−0.296029 + 0.955179i \(0.595663\pi\)
\(90\) 232.689 0.272529
\(91\) 131.173 0.151106
\(92\) −92.0000 −0.104257
\(93\) −1194.94 −1.33236
\(94\) −173.981 −0.190902
\(95\) 517.519 0.558909
\(96\) −199.233 −0.211813
\(97\) 193.384 0.202425 0.101212 0.994865i \(-0.467728\pi\)
0.101212 + 0.994865i \(0.467728\pi\)
\(98\) −98.0000 −0.101015
\(99\) −488.363 −0.495781
\(100\) −108.716 −0.108716
\(101\) −1188.22 −1.17061 −0.585307 0.810812i \(-0.699026\pi\)
−0.585307 + 0.810812i \(0.699026\pi\)
\(102\) 240.801 0.233753
\(103\) 805.666 0.770725 0.385362 0.922765i \(-0.374076\pi\)
0.385362 + 0.922765i \(0.374076\pi\)
\(104\) −149.912 −0.141347
\(105\) −431.047 −0.400628
\(106\) 1416.50 1.29795
\(107\) 558.186 0.504317 0.252158 0.967686i \(-0.418860\pi\)
0.252158 + 0.967686i \(0.418860\pi\)
\(108\) −379.456 −0.338085
\(109\) 540.293 0.474777 0.237388 0.971415i \(-0.423709\pi\)
0.237388 + 0.971415i \(0.423709\pi\)
\(110\) −821.219 −0.711820
\(111\) 2282.97 1.95216
\(112\) 112.000 0.0944911
\(113\) 1060.50 0.882865 0.441433 0.897294i \(-0.354470\pi\)
0.441433 + 0.897294i \(0.354470\pi\)
\(114\) 651.555 0.535295
\(115\) 227.480 0.184458
\(116\) −676.853 −0.541761
\(117\) 220.433 0.174180
\(118\) 1236.71 0.964820
\(119\) −135.368 −0.104279
\(120\) 492.625 0.374753
\(121\) 392.556 0.294933
\(122\) 516.568 0.383343
\(123\) 504.543 0.369863
\(124\) −767.706 −0.555984
\(125\) 1505.12 1.07698
\(126\) −164.686 −0.116440
\(127\) 345.613 0.241482 0.120741 0.992684i \(-0.461473\pi\)
0.120741 + 0.992684i \(0.461473\pi\)
\(128\) −128.000 −0.0883883
\(129\) 1261.64 0.861094
\(130\) 370.675 0.250080
\(131\) 1303.42 0.869316 0.434658 0.900596i \(-0.356869\pi\)
0.434658 + 0.900596i \(0.356869\pi\)
\(132\) −1033.91 −0.681745
\(133\) −366.276 −0.238798
\(134\) 334.745 0.215803
\(135\) 938.247 0.598159
\(136\) 154.706 0.0975438
\(137\) 335.747 0.209378 0.104689 0.994505i \(-0.466615\pi\)
0.104689 + 0.994505i \(0.466615\pi\)
\(138\) 286.397 0.176665
\(139\) −1177.75 −0.718671 −0.359335 0.933209i \(-0.616997\pi\)
−0.359335 + 0.933209i \(0.616997\pi\)
\(140\) −276.933 −0.167179
\(141\) 541.606 0.323485
\(142\) 562.049 0.332156
\(143\) −777.964 −0.454942
\(144\) 188.213 0.108920
\(145\) 1673.60 0.958515
\(146\) 1287.98 0.730096
\(147\) 305.075 0.171171
\(148\) 1466.73 0.814623
\(149\) 597.570 0.328556 0.164278 0.986414i \(-0.447471\pi\)
0.164278 + 0.986414i \(0.447471\pi\)
\(150\) 338.433 0.184219
\(151\) 1732.94 0.933936 0.466968 0.884274i \(-0.345346\pi\)
0.466968 + 0.884274i \(0.345346\pi\)
\(152\) 418.601 0.223375
\(153\) −227.482 −0.120202
\(154\) 581.220 0.304130
\(155\) 1898.24 0.983680
\(156\) 466.678 0.239514
\(157\) 363.093 0.184573 0.0922867 0.995732i \(-0.470582\pi\)
0.0922867 + 0.995732i \(0.470582\pi\)
\(158\) 794.651 0.400121
\(159\) −4409.57 −2.19938
\(160\) 316.495 0.156382
\(161\) −161.000 −0.0788110
\(162\) 1816.47 0.880958
\(163\) 2426.28 1.16590 0.582948 0.812510i \(-0.301899\pi\)
0.582948 + 0.812510i \(0.301899\pi\)
\(164\) 324.151 0.154341
\(165\) 2556.46 1.20618
\(166\) −2879.86 −1.34651
\(167\) −3858.78 −1.78803 −0.894015 0.448037i \(-0.852123\pi\)
−0.894015 + 0.448037i \(0.852123\pi\)
\(168\) −348.657 −0.160116
\(169\) −1845.85 −0.840168
\(170\) −382.529 −0.172580
\(171\) −615.517 −0.275262
\(172\) 810.559 0.359329
\(173\) 565.736 0.248625 0.124313 0.992243i \(-0.460327\pi\)
0.124313 + 0.992243i \(0.460327\pi\)
\(174\) 2107.05 0.918018
\(175\) −190.252 −0.0821813
\(176\) −664.252 −0.284488
\(177\) −3849.90 −1.63490
\(178\) 994.213 0.418648
\(179\) 1320.44 0.551365 0.275682 0.961249i \(-0.411096\pi\)
0.275682 + 0.961249i \(0.411096\pi\)
\(180\) −465.378 −0.192707
\(181\) 3238.75 1.33002 0.665011 0.746833i \(-0.268427\pi\)
0.665011 + 0.746833i \(0.268427\pi\)
\(182\) −262.346 −0.106848
\(183\) −1608.08 −0.649578
\(184\) 184.000 0.0737210
\(185\) −3626.65 −1.44128
\(186\) 2389.88 0.942120
\(187\) 802.843 0.313956
\(188\) 347.963 0.134988
\(189\) −664.047 −0.255568
\(190\) −1035.04 −0.395208
\(191\) 2268.39 0.859344 0.429672 0.902985i \(-0.358629\pi\)
0.429672 + 0.902985i \(0.358629\pi\)
\(192\) 398.465 0.149775
\(193\) 4299.35 1.60349 0.801747 0.597664i \(-0.203904\pi\)
0.801747 + 0.597664i \(0.203904\pi\)
\(194\) −386.768 −0.143136
\(195\) −1153.91 −0.423762
\(196\) 196.000 0.0714286
\(197\) −214.502 −0.0775768 −0.0387884 0.999247i \(-0.512350\pi\)
−0.0387884 + 0.999247i \(0.512350\pi\)
\(198\) 976.725 0.350570
\(199\) −4983.74 −1.77532 −0.887658 0.460504i \(-0.847669\pi\)
−0.887658 + 0.460504i \(0.847669\pi\)
\(200\) 217.431 0.0768735
\(201\) −1042.06 −0.365679
\(202\) 2376.43 0.827749
\(203\) −1184.49 −0.409533
\(204\) −481.602 −0.165289
\(205\) −801.501 −0.273069
\(206\) −1611.33 −0.544985
\(207\) −270.556 −0.0908452
\(208\) 299.824 0.0999475
\(209\) 2172.32 0.718958
\(210\) 862.094 0.283286
\(211\) −1863.75 −0.608083 −0.304042 0.952659i \(-0.598336\pi\)
−0.304042 + 0.952659i \(0.598336\pi\)
\(212\) −2833.00 −0.917788
\(213\) −1749.66 −0.562840
\(214\) −1116.37 −0.356606
\(215\) −2004.20 −0.635746
\(216\) 758.911 0.239062
\(217\) −1343.49 −0.420285
\(218\) −1080.59 −0.335718
\(219\) −4009.49 −1.23715
\(220\) 1642.44 0.503332
\(221\) −362.381 −0.110300
\(222\) −4565.94 −1.38039
\(223\) 4056.06 1.21800 0.609000 0.793170i \(-0.291571\pi\)
0.609000 + 0.793170i \(0.291571\pi\)
\(224\) −224.000 −0.0668153
\(225\) −319.714 −0.0947301
\(226\) −2121.01 −0.624280
\(227\) 3197.93 0.935040 0.467520 0.883983i \(-0.345148\pi\)
0.467520 + 0.883983i \(0.345148\pi\)
\(228\) −1303.11 −0.378511
\(229\) 1986.22 0.573157 0.286579 0.958057i \(-0.407482\pi\)
0.286579 + 0.958057i \(0.407482\pi\)
\(230\) −454.961 −0.130431
\(231\) −1809.34 −0.515351
\(232\) 1353.71 0.383083
\(233\) 625.027 0.175738 0.0878688 0.996132i \(-0.471994\pi\)
0.0878688 + 0.996132i \(0.471994\pi\)
\(234\) −440.866 −0.123164
\(235\) −860.378 −0.238829
\(236\) −2473.43 −0.682231
\(237\) −2473.76 −0.678007
\(238\) 270.736 0.0737362
\(239\) −5688.03 −1.53945 −0.769724 0.638377i \(-0.779606\pi\)
−0.769724 + 0.638377i \(0.779606\pi\)
\(240\) −985.251 −0.264990
\(241\) 5167.63 1.38123 0.690614 0.723223i \(-0.257340\pi\)
0.690614 + 0.723223i \(0.257340\pi\)
\(242\) −785.111 −0.208549
\(243\) −3093.36 −0.816621
\(244\) −1033.14 −0.271064
\(245\) −484.632 −0.126376
\(246\) −1009.09 −0.261532
\(247\) −980.522 −0.252588
\(248\) 1535.41 0.393140
\(249\) 8965.04 2.28167
\(250\) −3010.24 −0.761536
\(251\) −493.136 −0.124010 −0.0620049 0.998076i \(-0.519749\pi\)
−0.0620049 + 0.998076i \(0.519749\pi\)
\(252\) 329.373 0.0823355
\(253\) 954.862 0.237279
\(254\) −691.227 −0.170754
\(255\) 1190.82 0.292438
\(256\) 256.000 0.0625000
\(257\) 3505.77 0.850910 0.425455 0.904980i \(-0.360114\pi\)
0.425455 + 0.904980i \(0.360114\pi\)
\(258\) −2523.28 −0.608886
\(259\) 2566.77 0.615797
\(260\) −741.350 −0.176833
\(261\) −1990.51 −0.472067
\(262\) −2606.84 −0.614699
\(263\) −1693.68 −0.397098 −0.198549 0.980091i \(-0.563623\pi\)
−0.198549 + 0.980091i \(0.563623\pi\)
\(264\) 2067.82 0.482067
\(265\) 7004.91 1.62380
\(266\) 732.552 0.168856
\(267\) −3094.99 −0.709403
\(268\) −669.490 −0.152596
\(269\) 1108.75 0.251308 0.125654 0.992074i \(-0.459897\pi\)
0.125654 + 0.992074i \(0.459897\pi\)
\(270\) −1876.49 −0.422962
\(271\) 784.351 0.175815 0.0879076 0.996129i \(-0.471982\pi\)
0.0879076 + 0.996129i \(0.471982\pi\)
\(272\) −309.412 −0.0689739
\(273\) 816.687 0.181055
\(274\) −671.493 −0.148052
\(275\) 1128.35 0.247426
\(276\) −572.794 −0.124921
\(277\) −7088.58 −1.53759 −0.768794 0.639497i \(-0.779143\pi\)
−0.768794 + 0.639497i \(0.779143\pi\)
\(278\) 2355.49 0.508177
\(279\) −2257.69 −0.484461
\(280\) 553.865 0.118214
\(281\) −7749.43 −1.64517 −0.822584 0.568644i \(-0.807468\pi\)
−0.822584 + 0.568644i \(0.807468\pi\)
\(282\) −1083.21 −0.228739
\(283\) −1837.38 −0.385939 −0.192969 0.981205i \(-0.561812\pi\)
−0.192969 + 0.981205i \(0.561812\pi\)
\(284\) −1124.10 −0.234869
\(285\) 3222.09 0.669684
\(286\) 1555.93 0.321692
\(287\) 567.265 0.116671
\(288\) −376.426 −0.0770178
\(289\) −4539.03 −0.923882
\(290\) −3347.19 −0.677772
\(291\) 1204.01 0.242545
\(292\) −2575.96 −0.516256
\(293\) −200.688 −0.0400147 −0.0200074 0.999800i \(-0.506369\pi\)
−0.0200074 + 0.999800i \(0.506369\pi\)
\(294\) −610.150 −0.121036
\(295\) 6115.83 1.20704
\(296\) −2933.45 −0.576026
\(297\) 3938.34 0.769447
\(298\) −1195.14 −0.232324
\(299\) −430.998 −0.0833620
\(300\) −676.865 −0.130263
\(301\) 1418.48 0.271627
\(302\) −3465.87 −0.660393
\(303\) −7397.86 −1.40263
\(304\) −837.202 −0.157950
\(305\) 2554.54 0.479583
\(306\) 454.965 0.0849954
\(307\) −2168.79 −0.403190 −0.201595 0.979469i \(-0.564612\pi\)
−0.201595 + 0.979469i \(0.564612\pi\)
\(308\) −1162.44 −0.215053
\(309\) 5016.09 0.923481
\(310\) −3796.48 −0.695567
\(311\) −9214.82 −1.68014 −0.840072 0.542475i \(-0.817487\pi\)
−0.840072 + 0.542475i \(0.817487\pi\)
\(312\) −933.356 −0.169362
\(313\) 855.244 0.154445 0.0772224 0.997014i \(-0.475395\pi\)
0.0772224 + 0.997014i \(0.475395\pi\)
\(314\) −726.187 −0.130513
\(315\) −814.412 −0.145673
\(316\) −1589.30 −0.282928
\(317\) −3621.69 −0.641686 −0.320843 0.947132i \(-0.603966\pi\)
−0.320843 + 0.947132i \(0.603966\pi\)
\(318\) 8819.15 1.55520
\(319\) 7025.01 1.23299
\(320\) −632.989 −0.110579
\(321\) 3475.28 0.604271
\(322\) 322.000 0.0557278
\(323\) 1011.88 0.174311
\(324\) −3632.94 −0.622931
\(325\) −509.306 −0.0869268
\(326\) −4852.56 −0.824413
\(327\) 3363.87 0.568877
\(328\) −648.302 −0.109136
\(329\) 608.935 0.102042
\(330\) −5112.92 −0.852901
\(331\) 2886.49 0.479323 0.239662 0.970856i \(-0.422964\pi\)
0.239662 + 0.970856i \(0.422964\pi\)
\(332\) 5759.72 0.952126
\(333\) 4313.39 0.709827
\(334\) 7717.55 1.26433
\(335\) 1655.39 0.269981
\(336\) 697.314 0.113219
\(337\) −2056.69 −0.332448 −0.166224 0.986088i \(-0.553158\pi\)
−0.166224 + 0.986088i \(0.553158\pi\)
\(338\) 3691.70 0.594088
\(339\) 6602.72 1.05785
\(340\) 765.058 0.122033
\(341\) 7967.97 1.26537
\(342\) 1231.03 0.194639
\(343\) 343.000 0.0539949
\(344\) −1621.12 −0.254084
\(345\) 1416.30 0.221017
\(346\) −1131.47 −0.175804
\(347\) 3641.32 0.563332 0.281666 0.959513i \(-0.409113\pi\)
0.281666 + 0.959513i \(0.409113\pi\)
\(348\) −4214.10 −0.649137
\(349\) 7487.45 1.14841 0.574204 0.818713i \(-0.305312\pi\)
0.574204 + 0.818713i \(0.305312\pi\)
\(350\) 380.505 0.0581109
\(351\) −1777.66 −0.270326
\(352\) 1328.50 0.201163
\(353\) 11902.3 1.79460 0.897302 0.441416i \(-0.145524\pi\)
0.897302 + 0.441416i \(0.145524\pi\)
\(354\) 7699.81 1.15605
\(355\) 2779.46 0.415545
\(356\) −1988.43 −0.296029
\(357\) −842.804 −0.124946
\(358\) −2640.88 −0.389874
\(359\) −4802.20 −0.705991 −0.352995 0.935625i \(-0.614837\pi\)
−0.352995 + 0.935625i \(0.614837\pi\)
\(360\) 930.756 0.136264
\(361\) −4121.08 −0.600828
\(362\) −6477.49 −0.940468
\(363\) 2444.06 0.353388
\(364\) 524.693 0.0755532
\(365\) 6369.35 0.913390
\(366\) 3216.16 0.459321
\(367\) −9127.37 −1.29822 −0.649108 0.760697i \(-0.724857\pi\)
−0.649108 + 0.760697i \(0.724857\pi\)
\(368\) −368.000 −0.0521286
\(369\) 953.273 0.134486
\(370\) 7253.30 1.01914
\(371\) −4957.74 −0.693782
\(372\) −4779.75 −0.666179
\(373\) −12900.8 −1.79083 −0.895413 0.445237i \(-0.853120\pi\)
−0.895413 + 0.445237i \(0.853120\pi\)
\(374\) −1605.69 −0.222000
\(375\) 9370.90 1.29043
\(376\) −695.926 −0.0954511
\(377\) −3170.89 −0.433181
\(378\) 1328.09 0.180714
\(379\) −1830.25 −0.248058 −0.124029 0.992279i \(-0.539582\pi\)
−0.124029 + 0.992279i \(0.539582\pi\)
\(380\) 2070.08 0.279455
\(381\) 2151.80 0.289343
\(382\) −4536.78 −0.607648
\(383\) −8010.59 −1.06873 −0.534363 0.845255i \(-0.679448\pi\)
−0.534363 + 0.845255i \(0.679448\pi\)
\(384\) −796.930 −0.105907
\(385\) 2874.27 0.380484
\(386\) −8598.70 −1.13384
\(387\) 2383.72 0.313104
\(388\) 773.537 0.101212
\(389\) 4424.06 0.576629 0.288315 0.957536i \(-0.406905\pi\)
0.288315 + 0.957536i \(0.406905\pi\)
\(390\) 2307.83 0.299645
\(391\) 444.780 0.0575282
\(392\) −392.000 −0.0505076
\(393\) 8115.12 1.04161
\(394\) 429.004 0.0548551
\(395\) 3929.73 0.500573
\(396\) −1953.45 −0.247890
\(397\) −6976.76 −0.881999 −0.440999 0.897507i \(-0.645376\pi\)
−0.440999 + 0.897507i \(0.645376\pi\)
\(398\) 9967.48 1.25534
\(399\) −2280.44 −0.286127
\(400\) −434.862 −0.0543578
\(401\) −9824.95 −1.22353 −0.611764 0.791040i \(-0.709540\pi\)
−0.611764 + 0.791040i \(0.709540\pi\)
\(402\) 2084.13 0.258574
\(403\) −3596.52 −0.444554
\(404\) −4752.87 −0.585307
\(405\) 8982.85 1.10213
\(406\) 2368.99 0.289583
\(407\) −15223.1 −1.85400
\(408\) 963.204 0.116877
\(409\) 8036.27 0.971559 0.485780 0.874081i \(-0.338536\pi\)
0.485780 + 0.874081i \(0.338536\pi\)
\(410\) 1603.00 0.193089
\(411\) 2090.36 0.250876
\(412\) 3222.66 0.385362
\(413\) −4328.50 −0.515718
\(414\) 541.112 0.0642373
\(415\) −14241.6 −1.68456
\(416\) −599.649 −0.0706736
\(417\) −7332.68 −0.861110
\(418\) −4344.63 −0.508380
\(419\) 4471.23 0.521322 0.260661 0.965430i \(-0.416060\pi\)
0.260661 + 0.965430i \(0.416060\pi\)
\(420\) −1724.19 −0.200314
\(421\) −8709.07 −1.00820 −0.504102 0.863644i \(-0.668177\pi\)
−0.504102 + 0.863644i \(0.668177\pi\)
\(422\) 3727.49 0.429980
\(423\) 1023.30 0.117623
\(424\) 5665.99 0.648974
\(425\) 525.593 0.0599883
\(426\) 3499.33 0.397988
\(427\) −1807.99 −0.204905
\(428\) 2232.75 0.252158
\(429\) −4843.62 −0.545110
\(430\) 4008.40 0.449540
\(431\) −11847.9 −1.32411 −0.662056 0.749455i \(-0.730316\pi\)
−0.662056 + 0.749455i \(0.730316\pi\)
\(432\) −1517.82 −0.169042
\(433\) 6306.04 0.699882 0.349941 0.936772i \(-0.386202\pi\)
0.349941 + 0.936772i \(0.386202\pi\)
\(434\) 2686.97 0.297186
\(435\) 10419.8 1.14849
\(436\) 2161.17 0.237388
\(437\) 1203.48 0.131740
\(438\) 8018.99 0.874799
\(439\) −26.8997 −0.00292449 −0.00146225 0.999999i \(-0.500465\pi\)
−0.00146225 + 0.999999i \(0.500465\pi\)
\(440\) −3284.88 −0.355910
\(441\) 576.402 0.0622398
\(442\) 724.761 0.0779941
\(443\) −12573.3 −1.34848 −0.674241 0.738512i \(-0.735529\pi\)
−0.674241 + 0.738512i \(0.735529\pi\)
\(444\) 9131.87 0.976080
\(445\) 4916.61 0.523752
\(446\) −8112.13 −0.861256
\(447\) 3720.48 0.393675
\(448\) 448.000 0.0472456
\(449\) 2972.66 0.312446 0.156223 0.987722i \(-0.450068\pi\)
0.156223 + 0.987722i \(0.450068\pi\)
\(450\) 639.428 0.0669843
\(451\) −3364.34 −0.351266
\(452\) 4242.01 0.441433
\(453\) 10789.3 1.11904
\(454\) −6395.86 −0.661173
\(455\) −1297.36 −0.133673
\(456\) 2606.22 0.267648
\(457\) 11841.7 1.21210 0.606050 0.795426i \(-0.292753\pi\)
0.606050 + 0.795426i \(0.292753\pi\)
\(458\) −3972.44 −0.405284
\(459\) 1834.50 0.186552
\(460\) 909.922 0.0922290
\(461\) −5145.33 −0.519831 −0.259915 0.965631i \(-0.583695\pi\)
−0.259915 + 0.965631i \(0.583695\pi\)
\(462\) 3618.69 0.364408
\(463\) −10743.6 −1.07840 −0.539200 0.842177i \(-0.681273\pi\)
−0.539200 + 0.842177i \(0.681273\pi\)
\(464\) −2707.41 −0.270880
\(465\) 11818.5 1.17864
\(466\) −1250.05 −0.124265
\(467\) −6047.05 −0.599195 −0.299597 0.954066i \(-0.596852\pi\)
−0.299597 + 0.954066i \(0.596852\pi\)
\(468\) 881.732 0.0870899
\(469\) −1171.61 −0.115351
\(470\) 1720.76 0.168878
\(471\) 2260.63 0.221155
\(472\) 4946.86 0.482410
\(473\) −8412.74 −0.817797
\(474\) 4947.51 0.479424
\(475\) 1422.14 0.137373
\(476\) −541.472 −0.0521393
\(477\) −8331.36 −0.799720
\(478\) 11376.1 1.08855
\(479\) −1039.14 −0.0991223 −0.0495612 0.998771i \(-0.515782\pi\)
−0.0495612 + 0.998771i \(0.515782\pi\)
\(480\) 1970.50 0.187376
\(481\) 6871.26 0.651357
\(482\) −10335.3 −0.976676
\(483\) −1002.39 −0.0944312
\(484\) 1570.22 0.147466
\(485\) −1912.66 −0.179071
\(486\) 6186.71 0.577438
\(487\) 4060.20 0.377793 0.188896 0.981997i \(-0.439509\pi\)
0.188896 + 0.981997i \(0.439509\pi\)
\(488\) 2066.27 0.191671
\(489\) 15106.1 1.39697
\(490\) 969.265 0.0893610
\(491\) 7755.82 0.712862 0.356431 0.934322i \(-0.383993\pi\)
0.356431 + 0.934322i \(0.383993\pi\)
\(492\) 2018.17 0.184931
\(493\) 3272.29 0.298939
\(494\) 1961.04 0.178606
\(495\) 4830.13 0.438582
\(496\) −3070.83 −0.277992
\(497\) −1967.17 −0.177545
\(498\) −17930.1 −1.61339
\(499\) 12482.6 1.11984 0.559919 0.828547i \(-0.310832\pi\)
0.559919 + 0.828547i \(0.310832\pi\)
\(500\) 6020.47 0.538488
\(501\) −24024.8 −2.14241
\(502\) 986.272 0.0876882
\(503\) 7263.00 0.643819 0.321910 0.946770i \(-0.395675\pi\)
0.321910 + 0.946770i \(0.395675\pi\)
\(504\) −658.746 −0.0582200
\(505\) 11752.0 1.03556
\(506\) −1909.72 −0.167782
\(507\) −11492.3 −1.00669
\(508\) 1382.45 0.120741
\(509\) −9298.85 −0.809753 −0.404876 0.914371i \(-0.632685\pi\)
−0.404876 + 0.914371i \(0.632685\pi\)
\(510\) −2381.63 −0.206785
\(511\) −4507.93 −0.390253
\(512\) −512.000 −0.0441942
\(513\) 4963.77 0.427204
\(514\) −7011.54 −0.601684
\(515\) −7968.40 −0.681805
\(516\) 5046.56 0.430547
\(517\) −3611.48 −0.307220
\(518\) −5133.55 −0.435434
\(519\) 3522.28 0.297902
\(520\) 1482.70 0.125040
\(521\) 17700.1 1.48840 0.744200 0.667957i \(-0.232831\pi\)
0.744200 + 0.667957i \(0.232831\pi\)
\(522\) 3981.02 0.333802
\(523\) −4985.36 −0.416815 −0.208408 0.978042i \(-0.566828\pi\)
−0.208408 + 0.978042i \(0.566828\pi\)
\(524\) 5213.68 0.434658
\(525\) −1184.51 −0.0984694
\(526\) 3387.36 0.280791
\(527\) 3711.53 0.306787
\(528\) −4135.64 −0.340873
\(529\) 529.000 0.0434783
\(530\) −14009.8 −1.14820
\(531\) −7273.93 −0.594467
\(532\) −1465.10 −0.119399
\(533\) 1518.57 0.123408
\(534\) 6189.99 0.501624
\(535\) −5520.72 −0.446133
\(536\) 1338.98 0.107901
\(537\) 8221.08 0.660644
\(538\) −2217.51 −0.177702
\(539\) −2034.27 −0.162564
\(540\) 3752.99 0.299079
\(541\) 15877.9 1.26182 0.630910 0.775856i \(-0.282682\pi\)
0.630910 + 0.775856i \(0.282682\pi\)
\(542\) −1568.70 −0.124320
\(543\) 20164.5 1.59363
\(544\) 618.825 0.0487719
\(545\) −5343.74 −0.420001
\(546\) −1633.37 −0.128026
\(547\) −8542.53 −0.667737 −0.333869 0.942620i \(-0.608354\pi\)
−0.333869 + 0.942620i \(0.608354\pi\)
\(548\) 1342.99 0.104689
\(549\) −3038.27 −0.236194
\(550\) −2256.70 −0.174957
\(551\) 8854.11 0.684569
\(552\) 1145.59 0.0883323
\(553\) −2781.28 −0.213873
\(554\) 14177.2 1.08724
\(555\) −22579.6 −1.72694
\(556\) −4710.99 −0.359335
\(557\) −14999.9 −1.14106 −0.570528 0.821278i \(-0.693261\pi\)
−0.570528 + 0.821278i \(0.693261\pi\)
\(558\) 4515.39 0.342565
\(559\) 3797.27 0.287312
\(560\) −1107.73 −0.0835896
\(561\) 4998.51 0.376181
\(562\) 15498.9 1.16331
\(563\) 19048.3 1.42592 0.712959 0.701206i \(-0.247355\pi\)
0.712959 + 0.701206i \(0.247355\pi\)
\(564\) 2166.42 0.161743
\(565\) −10488.9 −0.781008
\(566\) 3674.75 0.272900
\(567\) −6357.64 −0.470892
\(568\) 2248.20 0.166078
\(569\) −4524.55 −0.333355 −0.166678 0.986011i \(-0.553304\pi\)
−0.166678 + 0.986011i \(0.553304\pi\)
\(570\) −6444.17 −0.473538
\(571\) −21959.6 −1.60942 −0.804712 0.593666i \(-0.797680\pi\)
−0.804712 + 0.593666i \(0.797680\pi\)
\(572\) −3111.86 −0.227471
\(573\) 14123.0 1.02966
\(574\) −1134.53 −0.0824988
\(575\) 625.115 0.0453375
\(576\) 752.852 0.0544598
\(577\) 6815.71 0.491753 0.245877 0.969301i \(-0.420924\pi\)
0.245877 + 0.969301i \(0.420924\pi\)
\(578\) 9078.06 0.653283
\(579\) 26767.9 1.92130
\(580\) 6694.39 0.479257
\(581\) 10079.5 0.719740
\(582\) −2408.03 −0.171505
\(583\) 29403.5 2.08880
\(584\) 5151.92 0.365048
\(585\) −2180.18 −0.154085
\(586\) 401.376 0.0282947
\(587\) −10153.0 −0.713898 −0.356949 0.934124i \(-0.616183\pi\)
−0.356949 + 0.934124i \(0.616183\pi\)
\(588\) 1220.30 0.0855856
\(589\) 10042.6 0.702542
\(590\) −12231.7 −0.853508
\(591\) −1335.49 −0.0929523
\(592\) 5866.91 0.407312
\(593\) 5891.25 0.407968 0.203984 0.978974i \(-0.434611\pi\)
0.203984 + 0.978974i \(0.434611\pi\)
\(594\) −7876.69 −0.544081
\(595\) 1338.85 0.0922479
\(596\) 2390.28 0.164278
\(597\) −31028.8 −2.12718
\(598\) 861.995 0.0589458
\(599\) −26348.6 −1.79728 −0.898642 0.438683i \(-0.855445\pi\)
−0.898642 + 0.438683i \(0.855445\pi\)
\(600\) 1353.73 0.0921097
\(601\) −11213.6 −0.761088 −0.380544 0.924763i \(-0.624263\pi\)
−0.380544 + 0.924763i \(0.624263\pi\)
\(602\) −2836.96 −0.192069
\(603\) −1968.85 −0.132965
\(604\) 6931.74 0.466968
\(605\) −3882.55 −0.260906
\(606\) 14795.7 0.991807
\(607\) −17937.3 −1.19943 −0.599713 0.800215i \(-0.704719\pi\)
−0.599713 + 0.800215i \(0.704719\pi\)
\(608\) 1674.40 0.111688
\(609\) −7374.68 −0.490701
\(610\) −5109.09 −0.339116
\(611\) 1630.12 0.107934
\(612\) −909.929 −0.0601008
\(613\) 6735.03 0.443761 0.221880 0.975074i \(-0.428781\pi\)
0.221880 + 0.975074i \(0.428781\pi\)
\(614\) 4337.57 0.285098
\(615\) −4990.16 −0.327191
\(616\) 2324.88 0.152065
\(617\) 21561.2 1.40684 0.703421 0.710773i \(-0.251655\pi\)
0.703421 + 0.710773i \(0.251655\pi\)
\(618\) −10032.2 −0.652999
\(619\) −811.525 −0.0526946 −0.0263473 0.999653i \(-0.508388\pi\)
−0.0263473 + 0.999653i \(0.508388\pi\)
\(620\) 7592.96 0.491840
\(621\) 2181.87 0.140991
\(622\) 18429.6 1.18804
\(623\) −3479.75 −0.223777
\(624\) 1866.71 0.119757
\(625\) −11488.9 −0.735292
\(626\) −1710.49 −0.109209
\(627\) 13524.9 0.861454
\(628\) 1452.37 0.0922867
\(629\) −7091.00 −0.449502
\(630\) 1628.82 0.103006
\(631\) −6015.69 −0.379526 −0.189763 0.981830i \(-0.560772\pi\)
−0.189763 + 0.981830i \(0.560772\pi\)
\(632\) 3178.60 0.200060
\(633\) −11603.7 −0.728604
\(634\) 7243.39 0.453741
\(635\) −3418.27 −0.213622
\(636\) −17638.3 −1.09969
\(637\) 918.212 0.0571129
\(638\) −14050.0 −0.871859
\(639\) −3305.78 −0.204655
\(640\) 1265.98 0.0781909
\(641\) −28011.7 −1.72604 −0.863022 0.505166i \(-0.831431\pi\)
−0.863022 + 0.505166i \(0.831431\pi\)
\(642\) −6950.56 −0.427284
\(643\) −8896.79 −0.545653 −0.272827 0.962063i \(-0.587959\pi\)
−0.272827 + 0.962063i \(0.587959\pi\)
\(644\) −644.000 −0.0394055
\(645\) −12478.2 −0.761749
\(646\) −2023.76 −0.123256
\(647\) −15429.8 −0.937570 −0.468785 0.883312i \(-0.655308\pi\)
−0.468785 + 0.883312i \(0.655308\pi\)
\(648\) 7265.87 0.440479
\(649\) 25671.6 1.55269
\(650\) 1018.61 0.0614666
\(651\) −8364.57 −0.503584
\(652\) 9705.12 0.582948
\(653\) 11711.0 0.701820 0.350910 0.936409i \(-0.385872\pi\)
0.350910 + 0.936409i \(0.385872\pi\)
\(654\) −6727.75 −0.402257
\(655\) −12891.4 −0.769022
\(656\) 1296.60 0.0771706
\(657\) −7575.46 −0.449843
\(658\) −1217.87 −0.0721543
\(659\) 10814.5 0.639258 0.319629 0.947543i \(-0.396442\pi\)
0.319629 + 0.947543i \(0.396442\pi\)
\(660\) 10225.8 0.603092
\(661\) 4122.46 0.242580 0.121290 0.992617i \(-0.461297\pi\)
0.121290 + 0.992617i \(0.461297\pi\)
\(662\) −5772.99 −0.338933
\(663\) −2256.19 −0.132162
\(664\) −11519.4 −0.673255
\(665\) 3622.64 0.211248
\(666\) −8626.79 −0.501924
\(667\) 3891.91 0.225930
\(668\) −15435.1 −0.894015
\(669\) 25253.1 1.45941
\(670\) −3310.78 −0.190905
\(671\) 10722.8 0.616916
\(672\) −1394.63 −0.0800580
\(673\) −21192.2 −1.21382 −0.606909 0.794771i \(-0.707591\pi\)
−0.606909 + 0.794771i \(0.707591\pi\)
\(674\) 4113.38 0.235077
\(675\) 2578.30 0.147020
\(676\) −7383.40 −0.420084
\(677\) −29376.9 −1.66772 −0.833860 0.551976i \(-0.813874\pi\)
−0.833860 + 0.551976i \(0.813874\pi\)
\(678\) −13205.4 −0.748011
\(679\) 1353.69 0.0765093
\(680\) −1530.12 −0.0862901
\(681\) 19910.4 1.12036
\(682\) −15935.9 −0.894749
\(683\) −2923.77 −0.163800 −0.0818998 0.996641i \(-0.526099\pi\)
−0.0818998 + 0.996641i \(0.526099\pi\)
\(684\) −2462.07 −0.137631
\(685\) −3320.69 −0.185222
\(686\) −686.000 −0.0381802
\(687\) 12366.2 0.686756
\(688\) 3242.24 0.179664
\(689\) −13271.9 −0.733845
\(690\) −2832.60 −0.156283
\(691\) −19904.1 −1.09579 −0.547893 0.836548i \(-0.684570\pi\)
−0.547893 + 0.836548i \(0.684570\pi\)
\(692\) 2262.94 0.124313
\(693\) −3418.54 −0.187387
\(694\) −7282.64 −0.398336
\(695\) 11648.5 0.635757
\(696\) 8428.20 0.459009
\(697\) −1567.13 −0.0851641
\(698\) −14974.9 −0.812047
\(699\) 3891.43 0.210569
\(700\) −761.009 −0.0410906
\(701\) −15700.8 −0.845949 −0.422974 0.906142i \(-0.639014\pi\)
−0.422974 + 0.906142i \(0.639014\pi\)
\(702\) 3555.31 0.191149
\(703\) −19186.7 −1.02936
\(704\) −2657.01 −0.142244
\(705\) −5356.73 −0.286165
\(706\) −23804.6 −1.26898
\(707\) −8317.52 −0.442451
\(708\) −15399.6 −0.817448
\(709\) −11130.6 −0.589588 −0.294794 0.955561i \(-0.595251\pi\)
−0.294794 + 0.955561i \(0.595251\pi\)
\(710\) −5558.92 −0.293835
\(711\) −4673.87 −0.246531
\(712\) 3976.85 0.209324
\(713\) 4414.31 0.231862
\(714\) 1685.61 0.0883505
\(715\) 7694.42 0.402455
\(716\) 5281.76 0.275682
\(717\) −35413.8 −1.84456
\(718\) 9604.41 0.499211
\(719\) 26480.8 1.37353 0.686766 0.726879i \(-0.259030\pi\)
0.686766 + 0.726879i \(0.259030\pi\)
\(720\) −1861.51 −0.0963534
\(721\) 5639.66 0.291307
\(722\) 8242.16 0.424850
\(723\) 32173.7 1.65498
\(724\) 12955.0 0.665011
\(725\) 4599.03 0.235591
\(726\) −4888.12 −0.249883
\(727\) 15809.9 0.806542 0.403271 0.915081i \(-0.367873\pi\)
0.403271 + 0.915081i \(0.367873\pi\)
\(728\) −1049.39 −0.0534242
\(729\) 5263.02 0.267389
\(730\) −12738.7 −0.645864
\(731\) −3918.71 −0.198274
\(732\) −6432.32 −0.324789
\(733\) −16659.7 −0.839481 −0.419740 0.907644i \(-0.637879\pi\)
−0.419740 + 0.907644i \(0.637879\pi\)
\(734\) 18254.7 0.917977
\(735\) −3017.33 −0.151423
\(736\) 736.000 0.0368605
\(737\) 6948.59 0.347292
\(738\) −1906.55 −0.0950961
\(739\) −36041.7 −1.79407 −0.897033 0.441963i \(-0.854282\pi\)
−0.897033 + 0.441963i \(0.854282\pi\)
\(740\) −14506.6 −0.720640
\(741\) −6104.75 −0.302650
\(742\) 9915.49 0.490578
\(743\) −14639.2 −0.722828 −0.361414 0.932405i \(-0.617706\pi\)
−0.361414 + 0.932405i \(0.617706\pi\)
\(744\) 9559.51 0.471060
\(745\) −5910.24 −0.290650
\(746\) 25801.6 1.26631
\(747\) 16938.4 0.829642
\(748\) 3211.37 0.156978
\(749\) 3907.30 0.190614
\(750\) −18741.8 −0.912471
\(751\) −15263.2 −0.741629 −0.370814 0.928707i \(-0.620921\pi\)
−0.370814 + 0.928707i \(0.620921\pi\)
\(752\) 1391.85 0.0674941
\(753\) −3070.27 −0.148588
\(754\) 6341.79 0.306305
\(755\) −17139.5 −0.826187
\(756\) −2656.19 −0.127784
\(757\) −25812.5 −1.23933 −0.619663 0.784868i \(-0.712731\pi\)
−0.619663 + 0.784868i \(0.712731\pi\)
\(758\) 3660.51 0.175403
\(759\) 5944.99 0.284307
\(760\) −4140.16 −0.197604
\(761\) 8641.54 0.411637 0.205818 0.978590i \(-0.434014\pi\)
0.205818 + 0.978590i \(0.434014\pi\)
\(762\) −4303.59 −0.204597
\(763\) 3782.05 0.179449
\(764\) 9073.55 0.429672
\(765\) 2249.90 0.106334
\(766\) 16021.2 0.755703
\(767\) −11587.4 −0.545498
\(768\) 1593.86 0.0748874
\(769\) 7300.25 0.342333 0.171166 0.985242i \(-0.445246\pi\)
0.171166 + 0.985242i \(0.445246\pi\)
\(770\) −5748.53 −0.269042
\(771\) 21827.0 1.01956
\(772\) 17197.4 0.801747
\(773\) 16068.9 0.747684 0.373842 0.927492i \(-0.378040\pi\)
0.373842 + 0.927492i \(0.378040\pi\)
\(774\) −4767.43 −0.221398
\(775\) 5216.35 0.241777
\(776\) −1547.07 −0.0715679
\(777\) 15980.8 0.737847
\(778\) −8848.12 −0.407739
\(779\) −4240.31 −0.195026
\(780\) −4615.66 −0.211881
\(781\) 11666.9 0.534540
\(782\) −889.561 −0.0406786
\(783\) 16052.2 0.732644
\(784\) 784.000 0.0357143
\(785\) −3591.16 −0.163279
\(786\) −16230.2 −0.736531
\(787\) −754.116 −0.0341567 −0.0170783 0.999854i \(-0.505436\pi\)
−0.0170783 + 0.999854i \(0.505436\pi\)
\(788\) −858.007 −0.0387884
\(789\) −10544.9 −0.475802
\(790\) −7859.46 −0.353958
\(791\) 7423.52 0.333692
\(792\) 3906.90 0.175285
\(793\) −4839.99 −0.216738
\(794\) 13953.5 0.623667
\(795\) 43612.7 1.94564
\(796\) −19935.0 −0.887658
\(797\) −6159.16 −0.273737 −0.136869 0.990589i \(-0.543704\pi\)
−0.136869 + 0.990589i \(0.543704\pi\)
\(798\) 4560.88 0.202323
\(799\) −1682.25 −0.0744853
\(800\) 869.725 0.0384368
\(801\) −5847.62 −0.257947
\(802\) 19649.9 0.865165
\(803\) 26735.7 1.17495
\(804\) −4168.26 −0.182840
\(805\) 1592.36 0.0697185
\(806\) 7193.03 0.314347
\(807\) 6903.13 0.301117
\(808\) 9505.74 0.413875
\(809\) −1676.54 −0.0728602 −0.0364301 0.999336i \(-0.511599\pi\)
−0.0364301 + 0.999336i \(0.511599\pi\)
\(810\) −17965.7 −0.779321
\(811\) 16726.0 0.724205 0.362102 0.932138i \(-0.382059\pi\)
0.362102 + 0.932138i \(0.382059\pi\)
\(812\) −4737.97 −0.204766
\(813\) 4883.39 0.210662
\(814\) 30446.1 1.31098
\(815\) −23997.0 −1.03139
\(816\) −1926.41 −0.0826443
\(817\) −10603.2 −0.454048
\(818\) −16072.5 −0.686996
\(819\) 1543.03 0.0658338
\(820\) −3206.00 −0.136535
\(821\) −11588.4 −0.492617 −0.246309 0.969191i \(-0.579218\pi\)
−0.246309 + 0.969191i \(0.579218\pi\)
\(822\) −4180.73 −0.177396
\(823\) −42240.5 −1.78908 −0.894538 0.446991i \(-0.852496\pi\)
−0.894538 + 0.446991i \(0.852496\pi\)
\(824\) −6445.33 −0.272492
\(825\) 7025.14 0.296465
\(826\) 8657.00 0.364668
\(827\) 30885.9 1.29868 0.649341 0.760498i \(-0.275045\pi\)
0.649341 + 0.760498i \(0.275045\pi\)
\(828\) −1082.22 −0.0454226
\(829\) 11743.9 0.492017 0.246008 0.969268i \(-0.420881\pi\)
0.246008 + 0.969268i \(0.420881\pi\)
\(830\) 28483.1 1.19116
\(831\) −44133.7 −1.84233
\(832\) 1199.30 0.0499738
\(833\) −947.576 −0.0394136
\(834\) 14665.4 0.608897
\(835\) 38165.0 1.58174
\(836\) 8689.26 0.359479
\(837\) 18206.9 0.751879
\(838\) −8942.46 −0.368630
\(839\) 30404.4 1.25110 0.625552 0.780183i \(-0.284874\pi\)
0.625552 + 0.780183i \(0.284874\pi\)
\(840\) 3448.38 0.141643
\(841\) 4244.15 0.174019
\(842\) 17418.1 0.712908
\(843\) −48248.1 −1.97124
\(844\) −7454.98 −0.304042
\(845\) 18256.3 0.743237
\(846\) −2046.60 −0.0831720
\(847\) 2747.89 0.111474
\(848\) −11332.0 −0.458894
\(849\) −11439.5 −0.462431
\(850\) −1051.19 −0.0424181
\(851\) −8433.68 −0.339721
\(852\) −6998.65 −0.281420
\(853\) 1256.30 0.0504276 0.0252138 0.999682i \(-0.491973\pi\)
0.0252138 + 0.999682i \(0.491973\pi\)
\(854\) 3615.97 0.144890
\(855\) 6087.74 0.243505
\(856\) −4465.49 −0.178303
\(857\) −7524.50 −0.299921 −0.149960 0.988692i \(-0.547915\pi\)
−0.149960 + 0.988692i \(0.547915\pi\)
\(858\) 9687.24 0.385451
\(859\) 32543.0 1.29261 0.646306 0.763078i \(-0.276313\pi\)
0.646306 + 0.763078i \(0.276313\pi\)
\(860\) −8016.80 −0.317873
\(861\) 3531.80 0.139795
\(862\) 23695.8 0.936288
\(863\) 17286.5 0.681852 0.340926 0.940090i \(-0.389259\pi\)
0.340926 + 0.940090i \(0.389259\pi\)
\(864\) 3035.64 0.119531
\(865\) −5595.39 −0.219941
\(866\) −12612.1 −0.494891
\(867\) −28260.1 −1.10699
\(868\) −5373.94 −0.210142
\(869\) 16495.3 0.643916
\(870\) −20839.7 −0.812105
\(871\) −3136.40 −0.122012
\(872\) −4322.34 −0.167859
\(873\) 2274.84 0.0881920
\(874\) −2406.96 −0.0931539
\(875\) 10535.8 0.407058
\(876\) −16038.0 −0.618577
\(877\) −10416.4 −0.401068 −0.200534 0.979687i \(-0.564268\pi\)
−0.200534 + 0.979687i \(0.564268\pi\)
\(878\) 53.7994 0.00206793
\(879\) −1249.49 −0.0479456
\(880\) 6569.75 0.251666
\(881\) 48386.4 1.85037 0.925186 0.379514i \(-0.123909\pi\)
0.925186 + 0.379514i \(0.123909\pi\)
\(882\) −1152.80 −0.0440102
\(883\) 4310.85 0.164294 0.0821471 0.996620i \(-0.473822\pi\)
0.0821471 + 0.996620i \(0.473822\pi\)
\(884\) −1449.52 −0.0551501
\(885\) 38077.3 1.44628
\(886\) 25146.7 0.953521
\(887\) 4359.56 0.165028 0.0825139 0.996590i \(-0.473705\pi\)
0.0825139 + 0.996590i \(0.473705\pi\)
\(888\) −18263.7 −0.690193
\(889\) 2419.29 0.0912716
\(890\) −9833.22 −0.370349
\(891\) 37706.0 1.41773
\(892\) 16224.3 0.609000
\(893\) −4551.80 −0.170571
\(894\) −7440.96 −0.278370
\(895\) −13059.8 −0.487753
\(896\) −896.000 −0.0334077
\(897\) −2683.40 −0.0998842
\(898\) −5945.32 −0.220933
\(899\) 32476.5 1.20484
\(900\) −1278.86 −0.0473650
\(901\) 13696.3 0.506427
\(902\) 6728.69 0.248382
\(903\) 8831.48 0.325463
\(904\) −8484.03 −0.312140
\(905\) −32032.7 −1.17658
\(906\) −21578.6 −0.791281
\(907\) −50399.3 −1.84507 −0.922537 0.385908i \(-0.873888\pi\)
−0.922537 + 0.385908i \(0.873888\pi\)
\(908\) 12791.7 0.467520
\(909\) −13977.4 −0.510011
\(910\) 2594.73 0.0945212
\(911\) 46810.4 1.70241 0.851206 0.524832i \(-0.175872\pi\)
0.851206 + 0.524832i \(0.175872\pi\)
\(912\) −5212.44 −0.189256
\(913\) −59779.8 −2.16695
\(914\) −23683.3 −0.857085
\(915\) 15904.6 0.574635
\(916\) 7944.88 0.286579
\(917\) 9123.94 0.328570
\(918\) −3669.01 −0.131912
\(919\) 19768.2 0.709569 0.354785 0.934948i \(-0.384554\pi\)
0.354785 + 0.934948i \(0.384554\pi\)
\(920\) −1819.84 −0.0652157
\(921\) −13502.9 −0.483101
\(922\) 10290.7 0.367576
\(923\) −5266.12 −0.187797
\(924\) −7237.38 −0.257676
\(925\) −9966.01 −0.354249
\(926\) 21487.3 0.762545
\(927\) 9477.30 0.335788
\(928\) 5414.83 0.191541
\(929\) 30117.9 1.06366 0.531828 0.846852i \(-0.321505\pi\)
0.531828 + 0.846852i \(0.321505\pi\)
\(930\) −23637.0 −0.833427
\(931\) −2563.93 −0.0902572
\(932\) 2500.11 0.0878688
\(933\) −57371.7 −2.01314
\(934\) 12094.1 0.423695
\(935\) −7940.48 −0.277734
\(936\) −1763.46 −0.0615819
\(937\) −54880.6 −1.91342 −0.956708 0.291049i \(-0.905996\pi\)
−0.956708 + 0.291049i \(0.905996\pi\)
\(938\) 2343.21 0.0815657
\(939\) 5324.77 0.185056
\(940\) −3441.51 −0.119415
\(941\) 42264.7 1.46417 0.732087 0.681211i \(-0.238546\pi\)
0.732087 + 0.681211i \(0.238546\pi\)
\(942\) −4521.25 −0.156380
\(943\) −1863.87 −0.0643647
\(944\) −9893.72 −0.341115
\(945\) 6567.73 0.226083
\(946\) 16825.5 0.578270
\(947\) 16330.6 0.560373 0.280186 0.959946i \(-0.409604\pi\)
0.280186 + 0.959946i \(0.409604\pi\)
\(948\) −9895.03 −0.339004
\(949\) −12067.7 −0.412788
\(950\) −2844.28 −0.0971375
\(951\) −22548.7 −0.768867
\(952\) 1082.94 0.0368681
\(953\) −55484.1 −1.88595 −0.942973 0.332868i \(-0.891984\pi\)
−0.942973 + 0.332868i \(0.891984\pi\)
\(954\) 16662.7 0.565488
\(955\) −22435.4 −0.760201
\(956\) −22752.1 −0.769724
\(957\) 43737.9 1.47737
\(958\) 2078.28 0.0700901
\(959\) 2350.23 0.0791374
\(960\) −3941.00 −0.132495
\(961\) 7044.81 0.236474
\(962\) −13742.5 −0.460579
\(963\) 6566.12 0.219720
\(964\) 20670.5 0.690614
\(965\) −42522.6 −1.41850
\(966\) 2004.78 0.0667730
\(967\) 49602.8 1.64956 0.824778 0.565457i \(-0.191300\pi\)
0.824778 + 0.565457i \(0.191300\pi\)
\(968\) −3140.45 −0.104275
\(969\) 6299.97 0.208859
\(970\) 3825.31 0.126622
\(971\) 45053.6 1.48902 0.744510 0.667611i \(-0.232683\pi\)
0.744510 + 0.667611i \(0.232683\pi\)
\(972\) −12373.4 −0.408311
\(973\) −8244.23 −0.271632
\(974\) −8120.39 −0.267140
\(975\) −3170.95 −0.104156
\(976\) −4132.54 −0.135532
\(977\) −7977.63 −0.261235 −0.130618 0.991433i \(-0.541696\pi\)
−0.130618 + 0.991433i \(0.541696\pi\)
\(978\) −30212.1 −0.987810
\(979\) 20637.7 0.673733
\(980\) −1938.53 −0.0631878
\(981\) 6355.64 0.206850
\(982\) −15511.6 −0.504070
\(983\) 745.180 0.0241786 0.0120893 0.999927i \(-0.496152\pi\)
0.0120893 + 0.999927i \(0.496152\pi\)
\(984\) −4036.34 −0.130766
\(985\) 2121.52 0.0686267
\(986\) −6544.59 −0.211382
\(987\) 3791.24 0.122266
\(988\) −3922.09 −0.126294
\(989\) −4660.72 −0.149850
\(990\) −9660.26 −0.310124
\(991\) −30814.7 −0.987752 −0.493876 0.869532i \(-0.664420\pi\)
−0.493876 + 0.869532i \(0.664420\pi\)
\(992\) 6141.65 0.196570
\(993\) 17971.4 0.574324
\(994\) 3934.34 0.125543
\(995\) 49291.4 1.57050
\(996\) 35860.2 1.14084
\(997\) 37297.4 1.18477 0.592387 0.805654i \(-0.298186\pi\)
0.592387 + 0.805654i \(0.298186\pi\)
\(998\) −24965.3 −0.791845
\(999\) −34784.9 −1.10165
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 322.4.a.f.1.4 4
7.6 odd 2 2254.4.a.j.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.4.a.f.1.4 4 1.1 even 1 trivial
2254.4.a.j.1.1 4 7.6 odd 2